diff --git "a/process_2/tokenized_finally.jsonl" "b/process_2/tokenized_finally.jsonl"
new file mode 100644--- /dev/null
+++ "b/process_2/tokenized_finally.jsonl"
@@ -0,0 +1,9323 @@
+{"id": "3748.png", "formula": "\\begin{align*} U ( \\vec { a } , 1 ) \\left ( S , f _ \\alpha \\otimes \\rho ( E ^ \\alpha ) \\right ) : = & e ^ { - i [ \\hat { f } _ 0 \\cdot \\hat { H } ( \\vec { a } , \\rho ) + \\hat { f } _ 1 \\cdot \\hat { P } ( \\vec { a } , \\rho ) ] } \\left ( S + \\vec { a } , f _ \\alpha ( \\cdot - \\vec { a } ) \\otimes \\rho ( E ^ \\alpha ) \\right ) \\\\ = & e ^ { i [ a ^ 0 \\hat { H } ( \\rho ) - a ^ 1 \\hat { P } ( \\rho ) ] } \\left ( S + \\vec { a } , f _ \\alpha ( \\cdot - \\vec { a } ) \\otimes \\rho ( E ^ \\alpha ) \\right ) . \\end{align*}"}
+{"id": "6010.png", "formula": "\\begin{align*} \\pi _ { \\psi } [ ( x ^ { \\ast } , 0 ) + ( x , 0 ) + ( 0 , k ) ] g ( y ^ { \\ast } ) = \\psi ( k + \\langle x ^ { \\ast } + y ^ { \\ast } , x \\rangle ) g ( x ^ { \\ast } + y ^ { \\ast } ) , \\end{align*}"}
+{"id": "5139.png", "formula": "\\begin{align*} X ' : = \\{ g \\in L ^ 0 ( \\Omega ) : f g \\in L ^ 1 ( \\Omega ) \\} . \\end{align*}"}
+{"id": "4582.png", "formula": "\\begin{align*} & [ \\alpha _ { \\mathcal { L } } ( a ) , \\alpha _ { \\mathcal { L } } ( b ) , [ x , y , z ] _ { \\mathcal { L } } ] _ { \\mathcal { L } } \\\\ & = [ [ a , b , x ] _ { \\mathcal { L } } , \\alpha _ { \\mathcal { L } } ( y ) , \\alpha _ { \\mathcal { L } } ( z ) ] _ { \\mathcal { L } } + [ \\alpha _ { \\mathcal { L } } ( x ) , [ a , b , y ] _ { \\mathcal { L } } , \\alpha _ { \\mathcal { L } } ( z ) ] _ { \\mathcal { L } } + [ \\alpha _ { \\mathcal { L } } ( x ) , \\alpha _ { \\mathcal { L } } ( y ) , [ a , b , z ] _ { \\mathcal { L } } ] _ { \\mathcal { L } } , \\end{align*}"}
+{"id": "1357.png", "formula": "\\begin{align*} \\psi ( F _ { R } ) - \\psi ( F _ { L } ) & = \\int _ { F _ { L } } ^ { F _ { R } } \\eta ' ( y ) A ' ( t , y ) d y \\\\ & = - \\int _ { F _ { L } } ^ { F _ { R } } \\eta '' ( y ) ( A ( t , y ) - A ( t , F _ { L } ( t ) ) ) d y + \\left . \\eta ' ( \\cdot ) ( A ( t , \\cdot ) - A ( t , F _ { L } ) ) \\right | _ { F _ { L } } ^ { F _ { R } } \\\\ & = - \\int _ { F _ { L } } ^ { F _ { R } } \\eta '' ( y ) ( A ( t , y ) - A ( t , F _ { L } ) ) d y + \\eta ' ( F _ { R } ( t ) ) ( A ( t , F _ { R } ( t ) ) - A ( t , F _ { L } ( t ) ) ) , \\end{align*}"}
+{"id": "3587.png", "formula": "\\begin{gather*} a _ 0 ( 1 9 8 ) \\ = \\ 8 , a _ 1 ( 1 9 8 ) \\ = \\ 9 , a _ 2 ( 1 9 8 ) \\ = \\ 1 , \\\\ L ( 1 9 8 ) \\ = \\ 3 . \\end{gather*}"}
+{"id": "4138.png", "formula": "\\begin{align*} \\vec { v } _ { s , t , f , r } = A _ { 3 , r , 0 } A _ { 2 , f , t } \\overline { A _ { 1 , t , s } A _ { 3 , t , s } A _ { 2 , s , t } } \\end{align*}"}
+{"id": "5185.png", "formula": "\\begin{align*} P l ( k , n ) = \\sqcup _ { i = 0 } ^ { m } \\sigma ( \\widetilde { E } ( \\lambda ^ i ) ) . \\end{align*}"}
+{"id": "3038.png", "formula": "\\begin{align*} \\phi ( A _ 1 \\otimes \\cdots \\otimes A _ m ) = U ( \\varphi _ 1 ( A _ 1 ) \\otimes \\cdots \\otimes \\varphi _ m ( A _ m ) ) V \\end{align*}"}
+{"id": "6465.png", "formula": "\\begin{align*} V _ k ^ 2 ( t ) & = 2 \\int _ 0 ^ t \\int _ { 0 } ^ { s } R _ k ( s , \\varphi , \\varphi , R ( s ' , \\varphi , \\varphi , \\varphi ) ) \\dd s ' \\dd s + \\int _ 0 ^ t \\int _ { 0 } ^ { s } R _ k ( s , \\varphi , R ( s ' , \\varphi , \\varphi , \\varphi ) , \\varphi ) \\dd s ' \\dd s \\\\ & = : \\int _ 0 ^ t \\int _ 0 ^ s ( 2 A + B ) ( s , s ' ) \\dd s ' \\dd s . \\end{align*}"}
+{"id": "8544.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow 0 ^ { + } } \\mathcal { K } _ { \\nu , p } ( x ) & = - \\intop _ { 0 } ^ { \\infty } \\ , \\frac { y ^ { - \\nu - \\frac { 1 } { 2 } } ( y + 1 ) ^ { - \\nu - \\frac { 1 } { 2 } } } { e ^ { ( 2 y + 1 ) x } + 1 } d y = \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n } \\intop _ { 0 } ^ { \\infty } y ^ { - \\nu - \\frac { 1 } { 2 } } ( y + 1 ) ^ { - \\nu - \\frac { 1 } { 2 } } \\ , e ^ { - ( 2 y + 1 ) x n } d y \\\\ & = \\frac { \\Gamma \\left ( \\frac { 1 } { 2 } - \\nu \\right ) ( 2 x ) ^ { \\nu } } { \\sqrt { \\pi } } \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n } \\ , n ^ { \\nu } \\ , K _ { \\nu } ( x n ) . \\end{align*}"}
+{"id": "1642.png", "formula": "\\begin{align*} h _ i \\ , \\xi _ j = 0 . \\end{align*}"}
+{"id": "7535.png", "formula": "\\begin{align*} A ( s ) = ( 1 + s ) \\log ( 1 + s ) - s \\ , \\ , \\tilde { A } ( s ) = \\exp ( s ) - s - 1 . \\end{align*}"}
+{"id": "7258.png", "formula": "\\begin{align*} { \\tilde \\Delta } _ { n , k , 2 } ^ { ( 1 , 3 ) } ( i ) : = \\sum _ { j = i + 1 } ^ { k - 1 } \\big \\{ \\varphi ^ { ( 3 ) } _ { n - k } ( \\tilde S _ { k - j - 1 } ) ( Z _ { k - j } ( Z _ { k - i } Z _ k ) ^ { ( 0 ) } ) ^ { ( 0 ) } \\big \\} \\ , . \\end{align*}"}
+{"id": "2565.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = g ( u ) , \\ \\ u \\ge 0 , & \\ \\ B _ R , \\\\ u = 0 , \\ \\ & \\ \\ \\partial B _ R , \\end{cases} \\end{align*}"}
+{"id": "5658.png", "formula": "\\begin{align*} \\aligned & \\Bigl ( \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | \\eta _ \\epsilon | ^ { 2 ^ * _ \\mu } ) | \\eta _ \\epsilon | ^ { 2 ^ * _ { \\mu } } \\Bigr ) ^ { \\frac 1 { 2 ^ * _ \\mu } } \\leq C ( N , \\mu ) ^ { \\frac 1 { 2 ^ * _ { \\mu } } } | \\eta _ \\epsilon | ^ 2 _ { 2 ^ * } = C ( N , \\mu ) ^ { \\frac { N } { 2 } \\cdot \\frac { 1 } { 2 ^ * _ \\mu } } S ^ { \\frac { N } { 2 } } _ { H , L } + O ( \\epsilon ^ { N - 2 } ) . \\endaligned \\end{align*}"}
+{"id": "1990.png", "formula": "\\begin{align*} \\gamma _ 1 ( a ) : = \\sup \\{ \\gamma \\geq 0 : ~ \\exists u \\in C ^ 2 ( \\Omega ) , ~ ~ u < 0 ~ ~ \\textnormal { a n d } ~ ~ L _ a u \\geq - \\gamma u f \\} . \\end{align*}"}
+{"id": "7264.png", "formula": "\\begin{align*} \\Vert d _ { 0 , N } \\Vert _ 1 \\leq \\Vert d _ { 0 , N } \\Vert _ 2 \\leq \\limsup _ { n \\rightarrow \\infty } n ^ { - 1 / 2 } \\big \\Vert \\sum _ { i = 1 } ^ n d _ { i , N } \\big \\Vert _ 2 \\ , . \\end{align*}"}
+{"id": "7329.png", "formula": "\\begin{align*} U _ k = \\prod _ { p \\in { \\cal P } _ 1 } p ^ { u _ { k p } } , V _ k = \\prod _ { p \\in { \\cal P } _ 2 } p ^ { u _ { k p } } , W _ k = \\prod _ { p \\in { \\cal P } _ 3 } p ^ { u _ { k p } } . \\end{align*}"}
+{"id": "3750.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { \\sqrt { 2 \\pi } } \\right ) ^ 2 & \\int _ { S _ 0 } e ^ { - i [ q ^ 2 x ^ 2 + q ^ 3 x ^ 3 ] } F _ \\alpha \\left ( \\hat { H } ( \\rho ) , \\hat { P } ( \\rho ) , x ^ 2 , x ^ 3 \\right ) \\ d x ^ 2 d x ^ 3 \\otimes \\rho ( E ^ \\alpha ) \\\\ = & \\hat { F } _ \\alpha \\left ( \\hat { H } ( \\rho ) , \\hat { P } ( \\rho ) , q ^ 2 , q ^ 3 \\right ) \\otimes \\rho ( E ^ \\alpha ) . \\end{align*}"}
+{"id": "388.png", "formula": "\\begin{align*} \\partial \\Phi / \\partial U = \\Phi _ { U } = U ^ T P , \\Phi _ { U } \\lbrack P ^ { - 1 } ( ( A _ i U ) _ { x _ i } + A ^ T _ i U _ { x _ i } + C U ) \\rbrack = ( U ^ T A _ i U ) _ { x _ i } = ( \\Psi _ i ) _ { x _ i } . \\end{align*}"}
+{"id": "386.png", "formula": "\\begin{align*} \\Phi _ t + ( \\Psi _ i ) _ { x _ i } = 0 , \\end{align*}"}
+{"id": "4141.png", "formula": "\\begin{align*} \\left ( y ^ 2 + f y + r > 1 > y \\right ) \\left ( \\wedge 1 = \\frac { 1 } { | N ( 1 ) | } > \\frac { y } { | N ( y ) | } = y \\right ) \\wedge \\end{align*}"}
+{"id": "9035.png", "formula": "\\begin{align*} b _ { 2 , 3 } = b _ { 3 , 4 } = b _ { 4 , 2 } = 0 . \\end{align*}"}
+{"id": "2345.png", "formula": "\\begin{align*} I _ 2 \\leq & C ( t - s ) \\int _ { - R } ^ R \\int _ 0 ^ t \\left ( \\int _ { \\R } G _ { t - r } ( x - z ) d z \\right ) d r d x = C ( t - s ) R \\int _ 0 ^ t ( t - r ) d r \\leq C ( t - s ) R . \\end{align*}"}
+{"id": "8221.png", "formula": "\\begin{align*} P ^ \\omega ( ( C _ t \\cap D _ t ) ^ c \\mid S _ t ) & = 1 - P ^ \\omega ( C _ t \\cap D _ t \\mid S _ t ) \\\\ & \\leq 1 - P ^ \\omega ( D _ t \\mid C _ t , S _ t ) P ^ \\omega ( C _ t \\mid S _ t ) \\\\ & = 1 - ( 1 - P ^ \\omega ( D _ t ^ c \\mid C _ t , S _ t ) ) ( 1 - P ^ \\omega ( C _ t ^ c \\mid S _ t ) ) \\\\ & \\leq P ^ \\omega ( D _ t ^ c \\mid C _ t , S _ t ) + P ^ \\omega ( C _ t ^ c \\mid S _ t ) . \\end{align*}"}
+{"id": "8649.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { - 1 } M _ j ( \\tau _ N ) & = \\frac { \\nu q } { \\lambda } \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } y ^ { j - 1 } d y . \\end{align*}"}
+{"id": "3035.png", "formula": "\\begin{align*} \\phi \\left ( \\bigotimes \\limits _ { i = 1 } ^ { m - 1 } X _ i E _ { j _ i j _ i } X _ i ^ * \\otimes B \\right ) = W _ { X } \\left ( \\bigotimes \\limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \\otimes \\varphi _ { j _ 1 , \\ldots , j _ { m - 1 } , X } ( B ) \\right ) W _ { X } ^ * \\end{align*}"}
+{"id": "619.png", "formula": "\\begin{align*} \\omega ( a ^ * a ) = 1 \\end{align*}"}
+{"id": "4300.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial t } + \\nabla \\cdot ( u f ) - \\nu \\Delta f = 0 , \\end{align*}"}
+{"id": "783.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ n } u x _ l \\mathrm { d } A = O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } , \\ \\ l = 1 , 2 , \\cdots , n + 1 . \\end{align*}"}
+{"id": "3288.png", "formula": "\\begin{align*} ( t , w ) \\longmapsto \\left \\{ \\begin{array} { r l } \\widetilde { \\rho } _ w ( t , w ) = \\rho _ w ( t , t ^ s w ) ; & t \\ge 0 , w \\in \\C \\\\ \\frac { 1 } { 2 i } e ^ { i \\pi ( 1 - \\beta _ 1 ) } ; & t \\le 0 , w \\in \\C \\end{array} \\right . \\end{align*}"}
+{"id": "9300.png", "formula": "\\begin{align*} \\sum \\limits _ { j \\in I _ k } s _ j = 0 , k = 1 , \\ldots , m . \\end{align*}"}
+{"id": "6722.png", "formula": "\\begin{align*} \\varphi : = \\varphi _ W \\underline { \\varphi } : = \\varphi _ { \\underline { W } } ; \\end{align*}"}
+{"id": "3802.png", "formula": "\\begin{align*} \\Sigma ( \\delta ) = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\boldsymbol { K } _ 1 ( \\mu _ { 1 3 } , \\gamma _ { 1 3 } ) \\le \\delta _ 1 , \\boldsymbol { K } _ 2 ( \\mu _ { 2 3 } , \\gamma _ { 2 3 } ) \\le \\delta _ 2 \\right \\} , \\end{align*}"}
+{"id": "1972.png", "formula": "\\begin{align*} n ^ { - 1 } g \\left | \\sum _ { p = 1 } ^ n u ^ { p \\bar p } u _ { p \\bar p \\bar j } \\right | ^ 2 - g \\sum _ { p , q = 1 } ^ n u ^ { p \\bar p } u ^ { q \\bar q } | u _ { p \\bar q j } | ^ 2 + g \\sum _ { p = 1 } ^ n u ^ { p \\bar p } u _ { p \\bar p j \\bar j } = n ( 2 R e ( g _ { v j } v _ { \\bar j } ) + g _ v v _ { j \\bar j } + g _ { v v } | v _ j | ^ 2 + g _ { j \\bar j } ) . \\end{align*}"}
+{"id": "522.png", "formula": "\\begin{align*} \\mathcal { G } ( \\mathbb { R } ) = \\frac { C ^ { \\infty } } { C ^ { \\infty } } . \\end{align*}"}
+{"id": "5400.png", "formula": "\\begin{align*} L ( u , \\chi ) = \\prod _ { i = 1 } ^ { d ( \\chi ) } ( 1 - \\gamma _ i u ) \\end{align*}"}
+{"id": "4188.png", "formula": "\\begin{align*} \\norm { ( v _ 0 ^ { \\alpha , \\epsilon } , v _ 1 ^ { \\alpha , \\epsilon } ) } _ { \\dot { H } ^ 1 \\times L ^ 2 } ^ 2 = ( \\alpha / \\epsilon ) ^ { - 6 } \\alpha ^ 2 \\norm { ( v _ 0 , v _ 1 ) } _ { \\dot { H } ^ 1 \\times L ^ 2 } ^ 2 \\ , = 1 6 \\pi \\epsilon ^ 6 / 3 \\alpha ^ 4 . \\end{align*}"}
+{"id": "4325.png", "formula": "\\begin{gather*} u ^ { ( 2 , k , n ; L ) } ( ( x _ 1 , x _ 2 ) , t ) = u ^ { ( 1 , k , n ; L ) } ( ( x _ 2 , x _ 1 ) , t ) , \\\\ y _ t ^ { ( 2 , k , n ; L ) } ( ( x _ 1 , x _ 2 ) ) = y _ t ^ { ( 1 , k , n ; L ) } ( ( x _ 2 , x _ 1 ) ) . \\end{gather*}"}
+{"id": "4620.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } I ( k ) = + \\infty . \\end{align*}"}
+{"id": "328.png", "formula": "\\begin{align*} \\prod _ { \\substack { \\gcd ( j _ 1 , j _ 2 , \\ldots , j _ { m + 1 } , k ) = 1 \\\\ j _ 1 , j _ 2 , \\ldots , j _ { m + 1 } < k \\\\ j _ 1 , j _ 2 , \\ldots , j _ { m + 1 } \\geq 0 ; k \\geq 1 } } \\left ( \\frac { 1 } { 1 - y ^ { j _ 1 + j _ 2 + . . . + j _ { m + 1 } } z ^ k } \\right ) ^ { \\frac { 1 } { k } } \\end{align*}"}
+{"id": "1160.png", "formula": "\\begin{align*} T ( f ) = T ( f ) + ( n - 1 ) f T ( 1 ) + n B ( A ( f ) , A ( 1 ) ) , \\end{align*}"}
+{"id": "4522.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( V ^ { \\lambda } ) & = L ( V ^ { \\lambda } ) + N ( V ^ { \\lambda } ) + \\omega M ( V ^ { \\lambda } ) + \\mathbf { c } \\cdot \\mathbf { P } ( V ^ { \\lambda } ) \\\\ & = \\lambda ^ 2 L ( V ) + \\lambda ^ { \\frac { d } { 2 } + 1 } N ( V ) + \\omega M ( V ) + \\lambda \\mathbf { c } \\cdot \\mathbf { P } ( V ) , \\end{align*}"}
+{"id": "4370.png", "formula": "\\begin{align*} \\sup _ { | z | = 1 } \\big ( \\log | { \\rm d e t } ( z - U _ n ) | - \\log n + \\frac { 3 } { 4 } \\log \\log n \\big ) , \\end{align*}"}
+{"id": "3579.png", "formula": "\\begin{align*} 1 9 8 & \\ = \\ 2 \\cdot 3 ^ 2 \\cdot 1 1 , \\\\ 8 9 1 & \\ = \\ 3 ^ 4 \\cdot 1 1 , \\end{align*}"}
+{"id": "952.png", "formula": "\\begin{align*} B ( x _ 0 , r ) = \\{ x \\in { \\mathbb R } ^ n : | x - x _ 0 | < r \\} \\ , , { \\mathbb B } ^ n = B ( 0 , 1 ) \\ , , \\end{align*}"}
+{"id": "9342.png", "formula": "\\begin{align*} \\mathcal { Z } _ t = \\exp \\bigg \\{ \\int _ 0 ^ t e ^ { - \\frac { \\beta } { 2 } s } h ( x _ s ^ u , u _ s ) d \\xi _ s - \\frac { 1 } { 2 } \\int _ 0 ^ t e ^ { - \\beta s } | h ( x _ s ^ u , u _ s ) | ^ 2 d s \\bigg \\} \\end{align*}"}
+{"id": "4371.png", "formula": "\\begin{align*} \\alpha = 1 - \\frac { 3 } { 4 } \\frac { \\log n } { n } , \\end{align*}"}
+{"id": "3080.png", "formula": "\\begin{align*} \\frac { u _ 0 ( r ) } { u _ 0 ' ( r ) } \\cdot \\mathcal U ' ( r ) + \\mathcal U ( r ) = \\frac { u ' ( r ) u _ 0 ( r ) - u _ 0 ' ( r ) u ( r ) } { u _ 0 ( r ) u _ 0 ' ( r ) } + \\frac { u ( r ) } { u _ 0 ( r ) } = \\mathcal W ( r ) . \\end{align*}"}
+{"id": "3670.png", "formula": "\\begin{align*} \\mathcal { M } _ { q , \\alpha ^ 1 } f = \\mathcal { M } _ { q , \\alpha ^ 2 } f , f \\in H ^ { s _ M } ( \\Omega ^ c ) \\end{align*}"}
+{"id": "3454.png", "formula": "\\begin{align*} \\eta ^ { - 1 } S \\eta = \\iota . \\end{align*}"}
+{"id": "1502.png", "formula": "\\begin{align*} 0 < \\epsilon : = r _ * - \\underline { r } \\leq \\overline C _ 3 \\alpha ^ 2 . \\end{align*}"}
+{"id": "7275.png", "formula": "\\begin{align*} 0 = ( k x _ 1 ) ^ 2 x _ 1 ^ 4 + a x _ 1 y _ 1 + ( k x _ 1 ) y _ 1 ^ 3 = x _ 1 ( k ^ 2 x _ 1 ^ 5 + a y _ 1 + k y _ 1 ^ 3 ) . \\end{align*}"}
+{"id": "4632.png", "formula": "\\begin{align*} m ( x , 0 ) = m _ 0 ( x ) , \\ , c ( x , 0 ) = c _ 0 ( x ) , \\ , d ( x , 0 ) = d _ 0 ( x ) , \\end{align*}"}
+{"id": "2934.png", "formula": "\\begin{align*} C ^ n ( G , M ) & : = F ^ n ( G , M ) ^ G \\\\ d ^ n & : = d ^ n \\mid _ { C ^ n ( G , M ) } . \\end{align*}"}
+{"id": "1511.png", "formula": "\\begin{align*} | W _ t \\ ! - \\ ! W _ \\tau | ^ 2 = | W _ t | ^ 2 + | W _ \\tau | ^ 2 \\pm 2 | W _ t | | W _ \\tau | \\cos \\overline \\beta = \\varepsilon _ 1 ^ 2 \\sin ^ 2 \\overline \\beta + ( | W _ t | \\ ! \\pm \\ ! \\varepsilon _ 1 \\cos \\overline \\beta ) ^ 2 \\ge \\varepsilon _ 1 ^ 2 \\sin ^ 2 \\overline \\beta \\ge \\frac { 4 } { \\pi ^ 2 } \\varepsilon _ 1 ^ 2 \\overline \\beta ^ 2 \\end{align*}"}
+{"id": "4594.png", "formula": "\\begin{align*} c ( x ) \\triangleq \\left \\{ \\begin{array} { l r } 0 & x = 0 , \\\\ K + v x & x > 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1878.png", "formula": "\\begin{align*} \\mathrm { S p a n } \\{ \\tilde { \\lambda } \\} + \\mathrm { K e r } ( S ^ * ) = \\mathrm { S p a n } \\{ \\lambda ^ * \\} + \\mathrm { K e r } ( S ^ * ) \\end{align*}"}
+{"id": "1838.png", "formula": "\\begin{align*} \\delta _ t ( x + \\lambda y ) = \\delta _ { x , 0 } \\delta _ t ( \\lambda y ) + ( 1 - \\delta _ { x , 0 } ) \\delta _ t ( x + \\lambda y ) \\ . \\end{align*}"}
+{"id": "5332.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ 2 ^ 2 \\le \\left ( \\frac { e \\cdot 1 2 r } { \\theta } \\right ) ^ { d } \\gamma _ 1 \\cdot \\gamma \\leq \\left ( \\frac { 3 3 r } { d } \\right ) ^ d \\gamma _ 1 \\cdot \\gamma \\cdot \\log ^ { d } \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) , \\end{align*}"}
+{"id": "8280.png", "formula": "\\begin{align*} \\binom { m } { n } : = \\frac { m ( m - 1 ) \\cdots ( m - n + 1 ) } { n ! } . \\end{align*}"}
+{"id": "6929.png", "formula": "\\begin{align*} \\frac { 1 } { \\sum _ { j \\in \\N } f ( j ) ^ 2 } \\sum _ { v \\not \\in \\mathcal { L } } f ( v ) ^ 2 = o ( 1 ) \\textrm { a s } N \\to \\infty . \\end{align*}"}
+{"id": "1351.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ \\mathbb { R } \\mathrm { s g n } ( F _ { N } - \\alpha ) \\chi \\mathbf { S } [ F _ { N } ] ( t , x ) d x d t & = \\int _ 0 ^ T \\int _ \\mathbb { R } \\mathbf { 1 } _ { \\left \\{ F ( t , x ) \\neq \\alpha \\right \\} } \\mathrm { s g n } ( F _ { N } - \\alpha ) \\chi \\mathbf { S } [ F _ { N } ] ( t , x ) d x d t \\\\ & \\underset { N \\rightarrow \\infty } { \\rightarrow } \\int _ 0 ^ T \\int _ \\mathbb { R } \\mathrm { s g n } ( F - \\alpha ) \\chi \\mathbf { S } [ F ] ( t , x ) d x d t \\end{align*}"}
+{"id": "5586.png", "formula": "\\begin{align*} u _ { 1 } . ( u , v , x _ { 0 } ) & = ( u _ { 1 } u , v , x _ { 0 } ) , \\ u _ { 1 } \\in U , \\\\ s . ( u , v , x _ { 0 } ) & = \\left ( s u s ^ { - 1 } , s v s ^ { - 1 } , s . x _ { 0 } \\right ) = ( u , { \\rm I n t } ( s ) . v , s . x _ { 0 } ) . \\end{align*}"}
+{"id": "6898.png", "formula": "\\begin{align*} \\widehat \\psi _ r ( C _ r ) = & \\inf _ { x \\in [ 0 , 1 ] } \\left [ J _ r ( x , d _ r ( x ) ) + ( C _ r - d _ r ( x ) ) \\sup _ { \\beta \\in [ d _ r ( x ) , C _ r ] } \\theta ( x , \\beta ) \\right ] \\\\ \\leq & \\ , \\inf _ { x \\in [ 0 , 1 ] } ( C _ r - d _ r ( x ) ) \\sup _ { \\beta \\in [ d _ r ( x ) , C _ r ] } \\theta ( x , \\beta ) = 0 , \\end{align*}"}
+{"id": "3700.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) u ^ { ( 2 ) } ( x ) + F ^ { ( 1 ) } ( x ) u ^ { ( 2 ) } ( x ) + F ^ { ( 2 ) } ( x ) [ u ^ { ( 1 ) } ( x ) ] ^ 2 = 0 & \\Omega , \\\\ u ^ { ( 2 ) } = f ^ 2 & \\Omega ^ c . \\end{cases} \\end{align*}"}
+{"id": "7742.png", "formula": "\\begin{align*} y \\Bigl ( \\bigl \\{ T _ { 1 } ( Z ) , \\cdots , T _ { q - 1 } ( Z ) \\bigr \\} , \\ , T _ { q } ( Z ) \\Bigr ) = T _ { q } ( N ) , \\ \\forall q \\ge 1 . \\end{align*}"}
+{"id": "1181.png", "formula": "\\begin{align*} R ^ i ( f \\times f ) _ * ( W ) = \\R ^ i ( f \\times f ) _ * ( \\pi _ 1 ^ * ( V ) \\otimes \\pi _ 2 ^ * ( V ) ) \\cong \\bigoplus _ { t } \\R ^ t f _ * ( V ) \\otimes \\R ^ { i - t } f _ * ( V ) . \\end{align*}"}
+{"id": "6877.png", "formula": "\\begin{align*} u ( x ) = \\left ( \\frac { 1 } { N } \\sum _ { i \\in [ N ] \\setminus j } A _ N ( i , j ) - \\lambda \\right ) ^ { - 1 } \\frac { 1 } { N } \\sum _ { i \\in [ N ] \\setminus j } A _ N ( i , j ) \\overline { u } _ i . \\end{align*}"}
+{"id": "8352.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ m \\alpha _ j \\nabla u _ j \\Big \\| _ { p , q , \\mu } \\leq \\Big ( \\sum _ { j = 1 } ^ \\infty | a _ j | ^ q \\Big ) ^ \\frac 1 { q } \\end{align*}"}
+{"id": "8035.png", "formula": "\\begin{align*} t _ { j , j } & = \\sum _ { k } z _ { j , k } \\left ( \\sum _ { l } s _ { k , l } z _ { l , j } \\right ) \\\\ & = \\sum _ { k , l } z _ { j , k } s _ { k , l } z _ { l , j } \\\\ & = \\sum _ { k } z _ { j , k } s _ { k , k } z _ { k , j } + \\sum _ { k \\neq l } z _ { j , k } s _ { k , l } z _ { l , j } \\\\ & = 2 ^ { - p } { \\mathrm { T r \\ , } } S + \\sum _ { k < l } \\left ( s _ { k , l } z _ { j , k } z _ { l , j } + s _ { l , k } z _ { l , j } z _ { k , j } \\right ) \\\\ & = 2 ^ { - p } { \\mathrm { T r \\ , } } S . \\end{align*}"}
+{"id": "4515.png", "formula": "\\begin{align*} \\widetilde { \\mu } _ { \\omega , \\mathbf { c } } : = \\inf \\{ S _ { \\omega , \\mathbf { c } } ( \\Psi ) | \\ \\Psi \\in \\mathcal { H } ^ 1 \\backslash \\{ ( \\mathbf { 0 } , \\mathbf { 0 } , \\mathbf { 0 } ) \\} , \\ I _ { \\omega , \\mathbf { c } } ( \\Psi ) = 0 , \\ \\mathbf { c } \\cdot \\mathbf { P } ( \\Psi ) \\ge 0 \\} . \\end{align*}"}
+{"id": "8302.png", "formula": "\\begin{align*} \\mu ( \\Gamma ) = \\delta _ { x _ 0 } ( \\Gamma ) + \\int _ { \\textbf { X } } \\int _ { \\textbf { A } } p ( \\Gamma | y , a ) \\sigma ^ s ( d a | y ) \\mu ( d y ) , ~ \\Gamma \\in { \\cal B } ( \\textbf { X } ) . \\end{align*}"}
+{"id": "2370.png", "formula": "\\begin{align*} [ x ] = [ y ] \\Leftrightarrow \\prod _ { i \\neq j } x _ i y _ j = y _ i x _ j \\rlap { . } \\end{align*}"}
+{"id": "1271.png", "formula": "\\begin{align*} \\eta ( B ^ { \\prime } ) = \\eta ( B ) + 1 \\ge T + \\max _ { 1 \\le i \\le s } a _ i . \\end{align*}"}
+{"id": "2123.png", "formula": "\\begin{align*} r _ 2 ^ { q ^ m } ( u _ 1 x ^ 2 f _ 2 + u _ 2 x f _ 1 + u f _ 2 ) = x f _ 2 ( r _ 1 ^ { q ^ m } - r _ 1 r _ 2 ^ { q ^ m - 1 } ) . \\end{align*}"}
+{"id": "4057.png", "formula": "\\begin{align*} S _ { \\mathrm { m a i n } } = \\sum _ { d _ 1 , d _ 2 < D } \\frac { 1 } { \\sqrt { d _ 1 d _ 2 } } \\sum _ { I d _ 1 , J d _ 2 < D } \\alpha _ { I d _ 1 } \\bar { \\alpha } _ { J d _ 2 } & \\sum _ { I L = J M } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } . \\end{align*}"}
+{"id": "6786.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ; q ) _ \\infty ( x ^ \\alpha ; q ) _ \\infty ( q x ^ { - \\alpha } ; q ) _ \\infty } & = E _ 0 + E _ \\alpha + E _ { 2 \\alpha } + \\dotsm + q ( E _ { - \\alpha } + E _ { \\alpha } + 2 E _ { 2 \\alpha } + \\dotsm ) + q ^ 2 \\dotsm , \\end{align*}"}
+{"id": "6122.png", "formula": "\\begin{align*} A _ k = \\frac { 2 ^ { 2 s } } { b _ k } \\quad \\mbox { w i t h } b _ k = \\begin{cases} l & \\mbox { i f } A _ { 0 } A _ { 1 } \\cdots A _ { k - 1 } \\in \\left [ \\frac { 1 } { 2 } , 1 \\right ) , \\\\ l + 1 & \\mbox { i f } A _ { 0 } A _ { 1 } \\cdots A _ { k - 1 } \\in [ 1 , 2 ] . \\end{cases} \\end{align*}"}
+{"id": "5944.png", "formula": "\\begin{align*} h ^ { ( \\sigma , g ) } = h ^ { ( \\sigma , 1 ) ( 1 , g ) } = ( \\sum ^ m _ { k = 1 } ( a _ k - i b _ k ) f _ k , - t ) ^ g = ( [ \\sum ^ m _ { k = 1 } ( a _ k - i b _ k ) f _ k ] g , - t ) ; \\end{align*}"}
+{"id": "6823.png", "formula": "\\begin{align*} \\sum _ { s _ 1 \\geq \\dots \\geq s _ { r - 1 } \\geq 0 } \\frac { q ^ { s _ 1 ^ 2 + \\dots + s _ { r - 1 } ^ 2 + s _ { i } + \\dots + s _ { r - 1 } } } { ( q ) _ { s _ 1 - s _ 2 } \\dots ( q ) _ { s _ { r - 2 } - s _ { r - 1 } } ( q ) _ { s _ { r - 1 } } } = \\frac { ( q ^ { 2 r + 1 } , q ^ { i } , q ^ { 2 r - i + 1 } ; q ^ { 2 r + 1 } ) _ \\infty } { ( q ) _ \\infty } . \\end{align*}"}
+{"id": "1782.png", "formula": "\\begin{align*} h = h _ 1 * \\dots * h _ p \\hbox { a n d } \\mathcal { L } _ \\mathcal { A } ( h ) = \\mathcal { L } _ \\mathcal { A } ( h _ 1 ) * \\dots * \\mathcal { L } _ \\mathcal { A } ( h _ p ) . \\end{align*}"}
+{"id": "568.png", "formula": "\\begin{align*} ( R _ { 0 ( 1 , n ) } g ) ( x _ 1 , y _ 1 , x _ 2 , y _ 2 ) & = \\chi _ { \\R _ { + } } ( x _ 2 ) \\ , \\ , h _ 0 ( y _ { 1 } ) [ N ( y _ 2 ) ] ^ T g ( x _ 1 , x _ 2 ) \\\\ ( R _ { 0 ( n , 1 ) } g ) ( x _ 1 , y _ 1 , x _ 2 , y _ 2 ) & = \\chi _ { \\R _ { + } } ( x _ 2 ) \\ , \\ell _ { 0 } ( y _ { 2 } ) [ H ( y _ 1 ) ] ^ T g ( x _ 1 , x _ 2 ) , \\end{align*}"}
+{"id": "6732.png", "formula": "\\begin{align*} \\mathcal { P } ( X _ \\eta ) = \\mathcal { P } ( X _ \\varphi ) = \\{ N _ \\ast ( \\nu _ { \\eta * } \\vee \\nu _ \\eta \\vee \\kappa ) : \\kappa \\in \\mathcal { P } ( \\{ 0 , 1 \\} ^ { \\Z } ) \\} , \\end{align*}"}
+{"id": "403.png", "formula": "\\begin{align*} & ( \\sqrt { | \\Lambda ^ - | } W ^ - - R \\sqrt { \\Lambda ^ + } W ^ + - S G ) ^ T ( \\sqrt { | \\Lambda ^ - | } W ^ - - R \\sqrt { \\Lambda ^ + } W ^ + - S G ) + \\\\ & \\begin{bmatrix} \\sqrt { \\Lambda ^ + } W ^ + \\\\ G \\end{bmatrix} ^ T \\begin{bmatrix} I - R ^ T R & - R ^ T S \\\\ - S ^ T R & I - S ^ T S \\end{bmatrix} \\begin{bmatrix} \\sqrt { \\Lambda ^ + } W ^ + \\\\ G \\end{bmatrix} - G ^ T G \\end{align*}"}
+{"id": "6188.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\partial H } + 2 \\lambda f ' \\big ( | E \\Delta H | \\big ) \\left ( \\mathbf { 1 } _ { \\overline H } - \\mathbf 1 _ { \\overline E } \\right ) = \\mu \\partial H , \\end{align*}"}
+{"id": "2871.png", "formula": "\\begin{align*} \\Delta _ C = \\Delta - \\tau \\circ \\Delta . \\end{align*}"}
+{"id": "3598.png", "formula": "\\begin{align*} m + 1 & \\ = \\ L ( r ( p ) ) \\ = \\ L ( f q ^ \\beta ) \\ = \\ L ( f ) + L ( q ^ \\beta ) - [ f q ^ \\beta \\ < \\ 1 0 ^ { L ( f ) + L ( q ^ \\beta ) - 1 } ] \\\\ & \\ = \\ L ( f ) + L ( q ^ \\beta ) - [ r ( p ) \\ < \\ 1 0 ^ { L ( f ) + L ( q ^ \\beta ) - 1 } ] . \\end{align*}"}
+{"id": "8806.png", "formula": "\\begin{align*} & 2 u ^ 3 t + 6 u ^ 2 t ^ 2 + 9 u t ^ 3 - 2 u ^ 3 + 4 u ^ 2 t + 2 4 u t ^ 2 \\\\ & + 9 t ^ 3 - 1 0 u ^ 2 - 5 u t + 1 8 t ^ 2 - 2 8 u - 7 t - 2 0 = 0 \\end{align*}"}
+{"id": "8510.png", "formula": "\\begin{align*} \\zeta _ { p , p ^ { \\prime } } ( s , \\ , c ) = \\frac { \\pi } { \\sqrt { c } } \\ , \\frac { 1 } { s - 1 } + \\frac { \\pi } { \\sqrt { c } } \\left ( 2 C _ { p ^ { \\prime } } ^ { ( 1 ) } - \\log \\left ( 4 c \\right ) \\right ) + \\frac { \\pi ^ { 2 } } { 3 } \\ , \\frac { 1 + \\frac { 3 } { \\pi p } ( 1 + \\frac { 1 } { \\pi p } ) } { \\left ( 1 + \\frac { 1 } { \\pi p ^ { \\prime } } \\right ) \\left ( 1 + \\frac { 1 } { \\pi p } \\right ) ^ { 2 } } + H _ { p , p ^ { \\prime } } ( 1 , c ) + O \\left ( s - 1 \\right ) \\end{align*}"}
+{"id": "5712.png", "formula": "\\begin{align*} C _ { D ^ { k - 1 } ( f ) } ^ { + } ( p ^ { 2 m + 1 } ) = c C _ { F } ( p ^ { 2 m + 1 } ) + C _ { D ^ { k - 1 } ( h ) } ( p ^ { 2 m + 1 } ) . \\end{align*}"}
+{"id": "1954.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } L _ a \\phi _ 1 = - \\gamma _ 1 \\phi _ 1 f & \\textnormal { i n } & \\Omega , \\\\ \\phi _ 1 = 0 & \\textnormal { o n } & \\partial \\Omega \\\\ \\phi _ 1 < 0 & \\textnormal { i n } & \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "3386.png", "formula": "\\begin{align*} \\sum _ { u < n \\leq v } \\tau ( n ) & = v \\log v - u \\log u + ( 2 \\gamma - 1 ) ( v - u ) + O ( v ^ { 1 / 3 } ) \\\\ & = ( v - u ) \\log u + v \\log ( v / u ) + ( 2 \\gamma - 1 ) ( v - u ) + O ( v ^ { 1 / 3 } ) \\\\ & \\ll ( v - u ) \\log u + O ( v ^ { 1 / 3 } ) . \\end{align*}"}
+{"id": "4177.png", "formula": "\\begin{align*} \\big | \\overline { \\rho } ( \\alpha _ t ( f ) ) ( z ) - \\beta _ t ( \\overline { \\rho } ( f ) ) ( z ) \\big | & = \\big | \\alpha _ t ( f ) ( \\rho ( z ) ) - \\overline { \\rho } ( f ) ( \\eta _ { t ^ { - 1 } } ( z ) ) \\big | \\\\ & = \\big | f ( \\theta _ { t ^ { - 1 } } ( \\rho ( z ) ) ) - f ( \\rho ( \\eta _ { t ^ { - 1 } } ( z ) ) ) \\big | < \\varepsilon . \\end{align*}"}
+{"id": "1675.png", "formula": "\\begin{align*} ( \\widehat { \\phi _ 1 ^ { ( p ) } } ) ^ { ( p ) } \\circ \\phi _ 1 & = ( \\widehat { \\phi _ 1 ^ { ( p ) } } ) ^ { ( p ) } \\circ ( \\phi _ 1 ^ { ( p ) } ) ^ { ( p ) } = ( ( \\widehat { \\phi _ 1 ^ { ( p ) } } ) \\circ \\phi _ 1 ^ { ( p ) } ) ^ { ( p ) } \\\\ & = ( [ \\deg \\phi _ 1 ^ { ( p ) } ] _ { E _ 1 ^ { ( p ) } } ) ^ { ( p ) } = ( [ \\deg \\phi _ 1 ] _ { E _ 1 ^ { ( p ) } } ) ^ { ( p ) } \\\\ & = [ \\deg \\phi _ 1 ] _ { E _ 1 } , \\end{align*}"}
+{"id": "1422.png", "formula": "\\begin{align*} \\cosh ( b ) & = \\cosh ( a ) \\cosh ( r ) - \\sinh ( a ) \\sinh ( r ) \\cos ( \\theta ) \\\\ & = \\cosh ( a ) \\cosh ( r ) - \\sinh ( a ) \\cosh ( r ) \\tanh ( a ) \\\\ & = \\frac { \\cosh ( r ) } { \\cosh ( a ) } [ \\cosh ( a ) ^ 2 - \\sinh ^ 2 ( a ) ] = \\frac { \\cosh ( r ) } { \\cosh ( a ) } . \\end{align*}"}
+{"id": "4441.png", "formula": "\\begin{align*} f ( \\omega ) : = \\mu _ { \\omega , \\mathbf { c } } - S _ { \\omega , \\mathbf { c } } ( U _ 0 ) = \\mu _ { \\omega , \\sqrt { \\omega } \\ \\mathbf { c } _ 0 } - S _ { \\omega , \\sqrt { \\omega } \\ \\mathbf { c } _ 0 } ( U _ 0 ) . \\end{align*}"}
+{"id": "59.png", "formula": "\\begin{align*} \\mathrm { R e d } = \\prod _ { \\ell \\in \\mathcal { S } } \\mathrm { R e d } _ { \\mathcal { T } ( \\ell ) } , , \\Pi = \\prod _ { \\ell \\in \\mathcal { S } } \\Pi _ { \\mathcal { T } ( \\ell ) } , \\end{align*}"}
+{"id": "9195.png", "formula": "\\begin{align*} f = \\sum _ { k = 0 } ^ \\infty p _ k . \\ ; \\ ; \\ ; \\phantom { ( k \\geq 0 ) } \\end{align*}"}
+{"id": "1022.png", "formula": "\\begin{align*} & \\qquad \\quad \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 2 } { ( 1 ) _ { k } ^ 2 } = \\frac { \\binom { 2 k } { k } ^ 2 } { 1 6 ^ k } , \\\\ [ 1 m m ] & H _ k \\bigg ( - \\frac { 1 } { 2 } \\bigg ) = 2 H _ { 2 k } - H _ k , \\end{align*}"}
+{"id": "3234.png", "formula": "\\begin{align*} & d ( T _ 1 \\beta ) = T _ 1 ^ 3 + T _ 1 T _ 2 ^ 2 + T _ 1 ^ 2 T _ 2 , \\\\ & d ( T _ 2 \\beta ) = T _ 1 ^ 2 T _ 2 + T _ 2 ^ 3 + T _ 1 T _ 2 ^ 2 , \\\\ & d ( T _ 3 \\beta ) = ( a ^ 2 + a b + b ^ 2 ) T _ 3 ^ 3 \\mbox { a n d } \\\\ & d ( \\gamma ) = T _ 1 T _ 2 ^ 2 + T _ 1 ^ 2 T _ 2 + ( a ^ 2 b + a b ^ 2 ) T _ 3 ^ 3 \\end{align*}"}
+{"id": "749.png", "formula": "\\begin{align*} a _ 0 ^ 2 = O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 , \\ a _ 1 ^ 2 = O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 , \\end{align*}"}
+{"id": "4282.png", "formula": "\\begin{align*} \\tau _ 2 ( \\rho S _ t + \\rho u \\cdot S _ x ) + S = \\mu u _ x . \\end{align*}"}
+{"id": "1090.png", "formula": "\\begin{align*} x ( y z ) = \\lambda _ 1 ( x y ) z & + \\lambda _ 2 ( y x ) z + \\lambda _ 3 z ( x y ) + \\lambda _ 4 z ( y x ) \\\\ & + \\lambda _ 5 ( x z ) y + \\lambda _ 6 ( z x ) y + \\lambda _ 7 y ( x z ) + \\lambda _ 8 y ( z x ) \\intertext { a n d } ( y z ) x = \\mu _ 1 ( x y ) z & + \\mu _ 2 ( y x ) z + \\mu _ 3 z ( x y ) + \\mu _ 4 z ( y x ) \\\\ & + \\mu _ 5 ( x z ) y + \\mu _ 6 ( z x ) y + \\mu _ 7 y ( x z ) + \\mu _ 8 y ( z x ) \\end{align*}"}
+{"id": "3946.png", "formula": "\\begin{align*} \\Sigma _ { \\mathrm { D } } ( \\delta ) : = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } _ 1 \\times \\mathcal { S } _ 2 ) : \\boldsymbol { W } _ { p _ 1 } ( \\gamma _ 1 , \\mu _ 1 ) \\leq \\delta _ 1 ^ { 1 / p _ 1 } , \\ \\boldsymbol { W } _ { p _ 2 } ( \\gamma _ 2 , \\mu _ 2 ) \\leq \\delta _ 2 ^ { 1 / p _ 2 } \\right \\} . \\end{align*}"}
+{"id": "4706.png", "formula": "\\begin{align*} P _ { \\mathcal { H } _ \\chi ^ { \\rm s t i f f ( s o f t ) } } \\Pi _ \\chi ^ { \\rm s t i f f ( s o f t ) } = \\Pi _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\widehat { P } _ \\chi , \\bigl ( \\Pi _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\bigr ) ^ * P _ { \\mathcal { H } _ \\chi ^ { \\rm s t i f f ( s o f t ) } } = \\widehat { P } _ \\chi \\bigl ( \\Pi _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\bigr ) ^ * , \\end{align*}"}
+{"id": "4693.png", "formula": "\\begin{align*} \\Pi _ \\chi ^ { \\rm s t i f f } = \\Pi _ 0 ^ { \\rm s t i f f } + \\sum _ { n = 1 } ^ \\infty \\chi ^ n \\Pi _ { n } , \\mbox { i n $ H ^ 1 ( Y ; \\C ^ 3 ) $ , } \\end{align*}"}
+{"id": "5369.png", "formula": "\\begin{align*} v = \\frac 1 2 ( \\alpha ^ { - 1 } M _ 1 ^ H + \\beta ^ { - 1 } M _ 2 ^ H ) z , w = \\frac 1 2 ( \\alpha ^ { - 1 } M _ 1 ^ H - \\beta ^ { - 1 } M _ 2 ^ H ) z \\end{align*}"}
+{"id": "8107.png", "formula": "\\begin{align*} \\Lambda _ n : = X _ { \\bullet \\bullet \\cdots \\bullet } \\quad S _ n : = \\sum _ { | F | = n } X _ F \\end{align*}"}
+{"id": "7746.png", "formula": "\\begin{align*} Y _ { T } ( r ) & = N \\left ( y _ { T } ( Y _ { T } , r ) \\right ) \\\\ & = N \\left ( y _ { S } ( Y _ { T } , r ) \\right ) . \\end{align*}"}
+{"id": "3745.png", "formula": "\\begin{align*} { \\rm R e } \\ \\check { W } _ { \\vec { k } , \\vec { l } } ( \\vec { x } , \\vec { y } ) - { \\rm R e } \\ \\check { W } _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) & = 0 , \\\\ { \\rm I m } \\ \\check { W } _ { \\vec { k } , \\vec { l } } ( \\vec { x } , \\vec { y } ) + { \\rm I m } \\ \\check { W } _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) & = 0 , \\end{align*}"}
+{"id": "2383.png", "formula": "\\begin{align*} f _ N ( x ) = g ( T x ) x \\in A ^ k . \\end{align*}"}
+{"id": "4479.png", "formula": "\\begin{align*} \\| U \\| _ { \\mathcal { H } ^ s } ^ 2 : = \\| u _ 1 \\| _ { H ^ s ( \\R ^ d ) } ^ 2 + \\| u _ 2 \\| _ { H ^ s ( \\R ^ d ) } ^ 2 + \\| u _ 3 \\| _ { H ^ s ( \\R ^ d ) } ^ 2 . \\end{align*}"}
+{"id": "4794.png", "formula": "\\begin{align*} \\mathcal { K } _ t \\varphi = \\varphi ( \\Phi ( t ; x ) ) , \\forall x \\in \\mathbb { R } ^ d . \\end{align*}"}
+{"id": "8818.png", "formula": "\\begin{align*} \\varphi ( u ( t ) ) & = \\varphi ( u _ 0 ) + \\int _ 0 ^ t \\varphi ' ( u ( s ) ) b ( s ) \\ , d s + \\int _ 0 ^ t \\varphi ' ( u ( s ) ) G ( s ) \\ , d W ( s ) \\\\ & + \\frac 1 2 \\int _ 0 ^ t \\operatorname { T r } _ { G ( s ) } \\varphi '' ( u ( s ) ) \\ , d s . \\end{align*}"}
+{"id": "5130.png", "formula": "\\begin{align*} \\frac { V _ { i j } } { V } = e _ i e _ j + \\delta _ { i j } \\bigg ( \\frac { 1 } { 1 + \\lambda _ i ^ 2 } - 2 e _ i ^ 2 \\bigg ) \\geq \\frac { 1 - \\lambda _ i ^ 2 } { ( 1 + \\lambda _ i ^ 2 ) ^ 2 } \\geq C ( \\eta ) \\end{align*}"}
+{"id": "6457.png", "formula": "\\begin{align*} z ( s ) & = h ^ 2 \\beta _ h ( s ) + \\frac { h ^ 2 } { 2 } \\overline { \\beta _ h ( s ) } + \\frac { \\sigma ^ 2 } { 2 } \\overline { \\beta _ \\sigma ( s ) } , \\\\ \\zeta ( s ) & = i \\beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \\overline { \\beta _ h ( s ) } K _ 2 - i \\overline { \\beta _ \\sigma ( s ) } K , \\\\ \\widetilde \\gamma ( s ) & = - i s ( \\beta _ h ( s ) - 1 ) ( | K _ 1 | ^ 2 + | K _ 3 | ^ 2 ) + i s ( \\overline { \\beta _ h ( s ) } - 1 ) | K _ 2 | ^ 2 + i s ( \\overline { \\beta _ \\sigma ( s ) } - 1 ) | K | ^ 2 . \\end{align*}"}
+{"id": "1390.png", "formula": "\\begin{align*} b _ i = x _ i + \\sum _ { j > i } s _ { i j } x _ j = \\displaystyle { x _ i + \\sum _ { \\substack { j : f ( j ) < f ( i ) \\\\ g ( j ) \\ne g ( i ) } } s _ { i j } x _ j } . \\end{align*}"}
+{"id": "8194.png", "formula": "\\begin{align*} P ^ \\omega ( E _ t , S _ t ) \\leq P ^ \\omega ( E _ { 1 , t } ) \\prod _ { j = 2 } ^ { t / h ( t ) } P ^ \\omega \\left ( E _ { j , t } \\ : \\bigg \\vert \\ : S _ { j - 1 , t } \\ , , \\ : \\bigcap _ { k = 1 } ^ { j - 1 } ( E _ { k , t } , F _ { k , t } ) \\right ) + \\sum _ { j = 1 } ^ { t / h ( t ) } P ^ \\omega ( F _ { j , t } ^ c ) . \\end{align*}"}
+{"id": "1993.png", "formula": "\\begin{align*} \\mu _ 1 : = \\sup \\{ \\lambda \\geq 0 : ~ \\exists u \\in \\mathcal P _ 0 ( \\Omega ) , ~ \\textnormal { s o l u t i o n o f } ~ \\eqref { e q 3 1 9 } \\} , \\end{align*}"}
+{"id": "5270.png", "formula": "\\begin{align*} T _ { \\rho } ^ { ( i ) } = ( I \\otimes \\cdots \\otimes \\underbrace { T _ \\rho } _ i \\otimes \\cdots \\otimes I ) . \\end{align*}"}
+{"id": "2637.png", "formula": "\\begin{align*} T ^ { ( k ) } ( f _ 1 , f _ 2 ) = \\sum _ { | l | > \\Gamma } T _ l \\left ( \\Delta _ { \\mathfrak { r } _ 1 l + k _ 1 } ^ { ( 1 ) } f _ 1 , \\Delta _ { \\mathfrak { r } _ 2 l + k _ 2 } ^ { ( 2 ) } f _ 2 \\right ) . \\end{align*}"}
+{"id": "8953.png", "formula": "\\begin{align*} D _ { q } & \\equiv \\sum _ { n = 0 } ^ { \\lfloor q / s \\rfloor } C _ { q - n s , n } \\Big ( \\frac { \\Delta t } { \\Delta x ^ s } \\Big ) ^ { n } . \\end{align*}"}
+{"id": "2417.png", "formula": "\\begin{align*} N ( x ) = F ( x ) + O ( x ^ { \\theta } ) = A x + O ( x ^ { \\theta } + x ^ { 1 - \\theta } ) , \\end{align*}"}
+{"id": "2434.png", "formula": "\\begin{align*} y ( q x ) = \\Gamma ( x ) y ( x ) . \\end{align*}"}
+{"id": "8312.png", "formula": "\\begin{align*} \\sigma \\le q ^ { 4 n - 4 } \\sum _ { r _ 1 \\le \\o _ 1 , \\dots , r _ t \\le \\o _ t } \\ , \\prod _ { j = 1 } ^ t p _ j ^ { - r _ j ( n - 2 ) } = q ^ { 4 n - 4 } \\Theta ( n ) \\ , , \\end{align*}"}
+{"id": "3369.png", "formula": "\\begin{align*} P _ { m n } : = \\sum _ { i = 0 } ^ m \\sum _ { j = 0 } ^ { n } p _ { i j } \\to \\infty \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\max \\{ m , n \\} \\to \\infty . \\end{align*}"}
+{"id": "3178.png", "formula": "\\begin{align*} \\abs { \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } U ^ j ( \\mu ) ( x ) } \\leq \\frac { \\epsilon } { \\chi _ d } \\rho ( x ) x \\in ( e M e ) _ + n \\in \\{ 1 , \\ldots , N _ d \\} . \\end{align*}"}
+{"id": "4108.png", "formula": "\\begin{align*} \\left ( T _ \\lambda ^ { \\vec { v } } \\right ) _ { i , j } = \\sum _ { m = 1 } ^ { n - j } - x _ { j + m } \\alpha _ { n - i - m + 1 } \\end{align*}"}
+{"id": "629.png", "formula": "\\begin{align*} \\| x _ 1 - x _ { n + 1 } \\| & \\leq \\| x _ 1 - x _ n \\| + \\| x _ n - x _ { n + 1 } \\| < \\left ( 1 - \\frac { 1 } { 2 ^ n } \\right ) \\delta + \\frac { \\delta } { 2 ^ { n + 1 } } \\\\ & = \\left ( 1 - \\frac { 1 } { 2 ^ { n + 1 } } \\right ) \\delta < \\delta \\end{align*}"}
+{"id": "1962.png", "formula": "\\begin{align*} G ( z ) : = \\log \\beta ( z ) + \\frac { u ^ 2 } { 2 } + B \\rho \\end{align*}"}
+{"id": "2320.png", "formula": "\\begin{align*} | E ( H ' ) | \\ , \\leq \\ , C \\Delta ^ { 7 } \\log | V ( H ) | \\ , \\leq \\ , C \\Delta ^ { 7 } \\ , 2 \\epsilon R \\ , \\leq \\ , 2 C ( f ( 1 ) ) ^ { 7 } \\ , \\epsilon R \\ , = \\ , \\frac { 2 R } { 3 } \\ , \\leq \\ , R - 1 . \\end{align*}"}
+{"id": "7446.png", "formula": "\\begin{align*} \\langle u _ { p \\alpha + k } ^ { ( i ) } ( x _ 1 , \\ , \\cdot \\ , , t ) \\rangle _ { \\Upsilon _ i } : = \\int _ { \\Upsilon _ i } u _ { p \\alpha + k } ^ { ( i ) } \\big ( x _ i , \\bar { \\xi } _ i , t \\big ) \\ , d \\bar { \\xi } _ i = 0 . \\end{align*}"}
+{"id": "3875.png", "formula": "\\begin{align*} \\mathrm { R W } ( d ) & = \\sup _ { \\lambda \\ge 1 } \\left \\{ \\inf _ { \\pi \\in \\Pi ( \\mu _ { 1 , L + 1 } , \\dotsc , \\mu _ { L , L + 1 } ) } \\int _ { \\mathcal { V } } \\min _ { \\ell \\in [ L ] } \\{ y _ \\ell + \\phi _ { \\lambda , \\ell } ( x _ 1 , \\dotsc , x _ L ) \\} d \\pi ( s ) - \\langle \\lambda , \\delta \\rangle \\right \\} , \\end{align*}"}
+{"id": "2782.png", "formula": "\\begin{align*} | \\Im \\xi _ j | = | \\eta _ 2 | > 2 \\varpi , \\xi _ j \\cdot \\xi _ j = \\lambda , j = 1 , 2 , \\xi _ 1 + \\xi _ 2 = \\eta . \\end{align*}"}
+{"id": "1506.png", "formula": "\\begin{align*} & \\tau ^ s _ { r _ 0 } : = \\inf \\{ s ' \\geq s : \\ , R ^ z _ { s ' } \\leq r _ 0 \\} , \\overline \\tau _ { r _ 0 } : = \\inf \\{ s ' \\geq 0 : \\ , \\overline { R } ^ { | z | } _ { s ' } \\leq r _ 0 \\} , \\end{align*}"}
+{"id": "5231.png", "formula": "\\begin{align*} g ( s ) = 0 , g ( s + 1 ) = 0 , h ( s ) = 0 \\end{align*}"}
+{"id": "848.png", "formula": "\\begin{align*} L _ { c l } = 3 M _ { c } K _ { c } \\end{align*}"}
+{"id": "8437.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma ( z ) \\ , e ^ { 2 \\pi z } - 1 } = \\sum _ { m = 1 } ^ { \\infty } \\left ( \\frac { p - z } { p + z } \\right ) ^ { m } e ^ { - 2 \\pi m z } , \\ , \\ , \\ , \\ , \\ , ( z ) > 0 . \\end{align*}"}
+{"id": "3111.png", "formula": "\\begin{align*} S _ D ( n , k ) = \\frac { 1 } { 2 ^ k k ! } \\left [ \\sum _ { \\ell = 0 } ^ k D ( n , \\ell ) \\binom { n - \\ell } { k - \\ell } + n \\cdot 2 ^ { n - 1 } ( k - 1 ) ! S ( n - 1 , k - 1 ) \\right ] , \\end{align*}"}
+{"id": "8902.png", "formula": "\\begin{align*} \\mathsf { d } _ { c c } ( y ( \\{ X _ { n i } \\} _ { n = 1 , \\dots , N , i = 1 , \\dots , j } ) ) & = \\mathsf { d } _ { c c } \\Bigg ( \\prod _ { n = 1 } ^ N [ X _ { n 1 } , \\dots , X _ { n j } ] _ c \\Bigg ) \\\\ & \\leq \\sum _ { n = 1 } ^ N \\mathsf { d } _ { c c } ( [ X _ { n 1 } , \\dots , X _ { n j } ] _ c ) . \\end{align*}"}
+{"id": "1242.png", "formula": "\\begin{align*} \\hat { b } _ { n + 1 } - \\tfrac 1 2 = \\tfrac { 1 } { 2 } b _ { n } ^ { \\frac { 1 } { \\beta } } . \\end{align*}"}
+{"id": "7480.png", "formula": "\\begin{align*} \\tilde { \\alpha } _ i = ( a _ l , \\dots , a _ { j + 1 } , x , a _ j , \\dots , a _ 1 ) . \\end{align*}"}
+{"id": "6793.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = - x _ 1 ^ 2 + \\cdots + - x _ k ^ 2 + x _ { k + 1 } ^ 2 + \\cdots + x _ { 2 n } ^ 2 , \\end{align*}"}
+{"id": "449.png", "formula": "\\begin{align*} w ( x ) = \\frac { \\delta ( \\gamma - 1 ) } { 2 \\gamma } \\left ( 1 - | x | ^ 2 \\right ) , \\ \\ x \\in B _ 1 ( 0 ) , \\end{align*}"}
+{"id": "3436.png", "formula": "\\begin{align*} 0 = \\frac { 2 } { m } \\Delta \\left ( | X | ^ 2 \\right ) + \\frac { 2 } { m } \\nabla _ X \\left ( | X | ^ 2 \\right ) . \\end{align*}"}
+{"id": "8444.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\frac { 1 } { 2 } - \\mu - i \\infty } ^ { \\frac { 1 } { 2 } - \\mu + i \\infty } \\Gamma ( z ) \\ , \\Gamma ( s - z ) \\ , \\zeta _ { p } ( 2 z ) \\ , x ^ { 2 z } d z = \\sqrt { \\pi } \\ , x \\sum _ { m = 1 } ^ { \\infty } \\ , \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\mu - i \\infty } ^ { \\mu + i \\infty } \\ , \\Gamma \\left ( z \\right ) \\ , \\Gamma \\left ( s + z - \\frac { 1 } { 2 } \\right ) \\ , \\left ( 2 z , 2 \\pi p m \\right ) _ { m } \\ , ( \\pi x m ) ^ { - 2 z } \\ , d z , \\end{align*}"}
+{"id": "95.png", "formula": "\\begin{align*} \\# { \\rm R e s } ( P ) \\cap [ 1 - C _ 1 h , 1 + C _ 1 h ] + i [ - C _ 2 h , \\infty ) = \\mathcal { O } ( h ^ { - n } ) . \\end{align*}"}
+{"id": "6657.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\R ^ N } | v ( x ) | \\ , | \\left \\langle \\nabla \\zeta ( x ) , \\nabla \\gamma _ R ( x ) \\right \\rangle | \\ , d x & \\le \\frac { \\bar C } { R } \\ , \\int _ { B _ R \\setminus B _ { R / 2 } } | v ( x ) | \\ , | \\nabla \\zeta ( x ) | \\ , d x \\\\ & \\le \\frac { \\bar C C } { R } \\ , \\int _ { B _ R \\setminus B _ { R / 2 } } | v ( x ) | \\psi ( x ) \\ , d x . \\end{aligned} \\end{align*}"}
+{"id": "2338.png", "formula": "\\begin{align*} \\sup _ { \\substack { 0 \\leq r \\leq t \\\\ x ' \\in \\R } } \\int _ { \\R ^ 3 } \\big \\| \\Delta _ { h } ( r , z , t , x ) \\big \\| _ p \\big \\| \\Delta _ { h } ( r , z , t , x ' ) \\big \\| _ p | h | ^ { 2 H - 2 } d h d z d x = K _ 1 + K _ 2 + K _ 3 + K _ 4 , \\end{align*}"}
+{"id": "4925.png", "formula": "\\begin{align*} f ( k , t , \\mathbf { p } _ { n + 1 } ) = \\overline { F } ( k - 1 , t , \\mathbf { p } _ { n + 1 } ) - \\overline { F } ( k , t , \\mathbf { p } _ { n + 1 } ) . \\end{align*}"}
+{"id": "743.png", "formula": "\\begin{align*} { \\cal E } = d _ H ( z _ 1 , z _ 2 ) \\equiv \\max \\left ( \\max _ i ( \\min _ j ( | z _ 1 ( i ) - z _ 2 ( j ) | ) ) , \\max _ j ( \\min _ i ( | z _ 1 ( i ) - z _ 2 ( j ) | ) ) \\right ) \\ , , \\end{align*}"}
+{"id": "9004.png", "formula": "\\begin{align*} T ^ \\varphi _ { j i , i } = R ^ \\varphi _ { j i , i } = \\frac { 1 } { 2 } ( S ^ \\varphi ) _ j - \\alpha \\varphi ^ a _ { t t } \\varphi ^ a _ j = 0 . \\end{align*}"}
+{"id": "5756.png", "formula": "\\begin{align*} \\sup _ { Q } \\Big ( \\frac { 1 } { | Q | } \\int _ { Q } \\big ( v _ { \\vec { w } } ( x ) \\big ) ^ { q } d x \\Big ) ^ { 1 / q } \\prod _ { j = 1 } ^ { m } \\Big ( \\frac { 1 } { | Q | } \\int _ { Q } \\big ( w _ { j } ( x ) \\big ) ^ { - p ' _ { j } } d x \\Big ) ^ { 1 / p ' _ { j } } & < \\infty , \\end{align*}"}
+{"id": "6953.png", "formula": "\\begin{align*} b _ T ( \\ell _ j ) = b _ T ( ( \\ell _ j ) _ { \\sf p } ) + b _ T ( ( \\ell _ j ) _ { \\sf q } ) . \\end{align*}"}
+{"id": "4350.png", "formula": "\\begin{align*} \\widetilde { T } _ { i } : = \\partial f _ { i } \\cap \\left ( - \\sum _ { j \\neq i } \\partial f _ { j } \\right ) , i = 1 , \\ldots , m , \\end{align*}"}
+{"id": "3366.png", "formula": "\\begin{align*} \\pi _ { \\mu } ( f ) v = \\begin{pmatrix} f ( T ^ t ) & 0 \\\\ \\star & \\star \\end{pmatrix} \\begin{pmatrix} v _ 1 \\\\ v _ 2 \\end{pmatrix} = \\begin{pmatrix} v _ 1 \\\\ v _ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "7233.png", "formula": "\\begin{align*} \\sup _ { h \\in \\N } \\mathcal { G } ( u _ h , \\beta _ h , B _ r ) < + \\infty \\mbox { a n d } \\lim _ { h \\to + \\infty } \\Psi ( u _ h , \\beta _ h , B _ r ) = \\lim _ { h \\to + \\infty } \\mathcal { H } ^ 1 ( J _ { u _ h } ) = 0 . \\end{align*}"}
+{"id": "1491.png", "formula": "\\begin{align*} \\partial _ t \\partial _ x w ( t , x ) = \\frac 1 2 \\partial _ { x x } \\partial _ x w ( t , x ) + \\frac 1 x \\partial _ x \\partial _ x w ( t , x ) - \\frac { 1 } { x ^ 2 } \\partial _ x w ( t , x ) . \\end{align*}"}
+{"id": "3546.png", "formula": "\\begin{align*} F _ i ( x _ 1 , \\dots , x _ { n + 1 } ) = P _ i ( & x _ 1 , \\dots , x _ { n + 1 } , j ( i B ( x _ 1 ) ) , \\dots , j ( i B ( x _ { n + 1 } ) ) , \\\\ & i j ' ( i B ( x _ 1 ) ) , \\dots , i j ' ( i B ( x _ { n + 1 } ) ) , j '' ( i B ( x _ 1 ) ) , \\dots , j '' ( i B ( x _ { n + 1 } ) ) ) \\end{align*}"}
+{"id": "5439.png", "formula": "\\begin{align*} f _ { r _ 1 | \\Phi ( \\mathcal { A } ) > 0 } ( r ) = \\upsilon ( \\lambda , R _ S ) r e ^ { - \\lambda \\pi \\frac { R _ S } { R _ E } r ^ 2 } , R _ { m i n } \\leq r \\leq R _ { m a x } , \\end{align*}"}
+{"id": "4199.png", "formula": "\\begin{align*} W ^ * \\pi ( \\lambda ) g _ { q p t } & = W ^ * \\left ( \\pi ( \\lambda ) g + e ^ { \\frac { - 2 \\pi i t } { 3 } } e ^ { \\frac { 2 \\pi i k p } { M } } \\pi ( \\lambda ' ) g \\right ) \\\\ & = W ^ * \\pi ( \\lambda ) g + e ^ { \\frac { - 2 \\pi i t } { 3 } } e ^ { \\frac { 2 \\pi i k p } { M } } W ^ * \\pi ( \\lambda ' ) g . \\end{align*}"}
+{"id": "9099.png", "formula": "\\begin{align*} k + r = k ' + r ' . \\end{align*}"}
+{"id": "6508.png", "formula": "\\begin{align*} \\begin{cases} \\ , \\ , \\ , \\ , \\ , c _ { n , n } = 1 n \\geq 1 \\\\ \\sum _ { j = 1 } ^ n a _ { i , j } \\ , c _ { n , j } \\ , = 0 1 \\leq i \\leq n - 1 \\\\ \\sum _ { j = 1 } ^ n a _ { n , j } \\ , c _ { n , j } = \\frac { b _ n } { b _ { n - 1 } } \\ , \\ , n \\geq 1 . \\end{cases} \\end{align*}"}
+{"id": "692.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N - 1 } D \\varphi ( T ^ i x ) = \\sum _ { l = 0 } ^ { k } \\sum _ { 0 \\leq i < q ( l ) } S ( l ) D \\varphi ( ( T ^ { ( l ) } ) ^ i x ( l ) ) \\end{align*}"}
+{"id": "3984.png", "formula": "\\begin{align*} \\sup _ { x ^ \\prime \\in \\mathbb { R } ^ d } \\left [ \\varphi _ 1 ( x ^ \\prime , x _ 1 ) + \\varphi _ 2 ( x ^ \\prime , x _ 2 ) \\right ] = b + B ( x _ 1 - x _ 2 ) - ( x _ 1 - x _ 2 ) ^ \\top W ( x _ 1 - x _ 2 ) . \\end{align*}"}
+{"id": "6277.png", "formula": "\\begin{align*} g _ { n } ( x ) = \\left \\{ \\begin{tabular} { l } $ f _ { n } ( x ) $ \\ i f \\ $ x \\in E _ { n } $ \\\\ $ f _ { n + m } ( x ) $ i f $ x \\in E _ { n + m } \\backslash E _ { n + m - 1 } , \\ m = 1 , 2 , 3 , . . . . $ \\end{tabular} \\right . \\end{align*}"}
+{"id": "7643.png", "formula": "\\begin{align*} M ( c , 1 , y ) = ( 4 - c ^ 2 ) ^ 2 ( 3 6 - c ^ 2 ) \\leq 5 7 6 , \\ ; c \\in ( 0 , 2 ) , y \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "8266.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\frac { k ( k - 1 ) } { 2 } } \\frac { G ^ 2 ( k + 1 ) } { G ( 2 k + 1 ) } x ^ { \\frac { k ^ 2 - k - 1 } { 2 } } \\sum _ { i = 0 } ^ \\infty \\left ( \\sum _ { q = 0 } ^ { \\min ( i , \\lfloor \\frac { k + 1 } { 2 } \\rfloor ) } \\sum _ { m = 0 } ^ { k + 1 - 2 q } \\Big ( \\prod _ { s = 0 } ^ { m - 1 } ( i - q + \\frac { k ^ 2 } { 2 } - s ) \\Big ) b _ { q , m } ^ { ( k + 1 ) } a _ { i - q } \\right ) x ^ i = 0 . \\end{align*}"}
+{"id": "5012.png", "formula": "\\begin{align*} \\theta _ f : \\mathcal { X } \\ni x \\mapsto ( f _ j ( x ) ) _ { j = 1 } ^ n \\in \\ell ^ p ( [ n ] ) ; \\theta _ g : \\mathcal { X } \\ni x \\mapsto ( g _ k ( x ) ) _ { k = 1 } ^ n \\in \\ell ^ p ( [ n ] ) \\end{align*}"}
+{"id": "2232.png", "formula": "\\begin{align*} | D ( x , y ) | & < ( 1 - m ) \\left ( 1 - y ^ { - 1 } \\right ) \\frac { x } { \\log y } + M \\sqrt { x } \\log x + 1 \\\\ & \\le \\left ( ( 1 - m ) \\left ( 1 - y ^ { - 1 } \\right ) + M \\frac { \\log ^ 2 y } { \\sqrt { y } } + \\frac { \\log y } { y } \\right ) \\frac { x } { \\log y } , \\end{align*}"}
+{"id": "6211.png", "formula": "\\begin{align*} d ( c _ i ) = | a _ { i - 1 } b _ { i + 1 } - a _ { i + 1 } b _ { i - 1 } | , \\end{align*}"}
+{"id": "7001.png", "formula": "\\begin{align*} f _ { i j } = f _ { j k 0 } + f _ { j k 1 } Q _ k + \\ldots + f _ { j k s } Q _ k ^ s \\end{align*}"}
+{"id": "2206.png", "formula": "\\begin{align*} \\Phi ( x , y ) & = \\# \\{ n \\le x \\colon P ^ - ( n ) > y \\Omega ( n ) \\le 2 \\} \\\\ & = \\pi ( x ) - \\pi ( y ) + 1 + \\sum _ { y < p \\leq \\sqrt { x } } \\sum _ { p \\le q \\le x / p } 1 \\\\ & = \\pi ( x ) - \\pi ( y ) + 1 + \\sum _ { y < p \\leq \\sqrt { x } } ( \\pi ( x / p ) - \\pi ( p ) + 1 ) , \\end{align*}"}
+{"id": "8705.png", "formula": "\\begin{align*} h ( x ) - \\sum ^ { l } _ { s = 0 } \\frac { h ^ { ( s ) } ( 0 ) x ^ s } { s ! } = \\int ^ { x } _ { 0 } \\frac { h ^ { ( l + 1 ) } ( t ) ( x - t ) ^ l } { l ! } d t = r ( x ) . \\end{align*}"}
+{"id": "3082.png", "formula": "\\begin{align*} \\mathcal U _ \\infty = \\lim _ { r \\to \\infty } \\frac { u ( r ) } { u _ 0 ( r ) } = \\lim _ { r \\to \\infty } \\frac { u ' ( r ) } { u _ 0 ' ( r ) } = \\mathcal W _ \\infty \\end{align*}"}
+{"id": "2630.png", "formula": "\\begin{align*} Q ( t _ 1 , t _ 2 , t _ 3 ) = & \\left ( \\widetilde { P } _ 1 ' \\widetilde { P } _ 2 ' \\right ) ( t _ 2 ) \\left ( \\widetilde { P } _ 1 ' \\widetilde { P } _ 2 ' \\right ) ( t _ 3 ) \\left ( \\widetilde { P } _ 1 ' ( t _ 2 ) \\widetilde { P } _ 2 ' ( t _ 3 ) - \\widetilde { P } _ 1 ' ( t _ 3 ) \\widetilde { P } _ 2 ' ( t _ 2 ) \\right ) \\cdot \\\\ & \\left ( \\widetilde { P } _ 2 '' \\widetilde { P } _ 1 ' ( t _ 1 ) - \\widetilde { P } _ 1 '' \\widetilde { P } _ 2 ' ( t _ 1 ) \\right ) \\end{align*}"}
+{"id": "2073.png", "formula": "\\begin{align*} ( \\rho \\otimes \\varphi ) ^ { \\sigma ' , s ' } _ { \\sigma , s } ( g ) : = \\rho _ { g \\sigma } ^ { \\sigma ' } \\cdot \\varphi _ s ^ { s ' } ( g ) . \\end{align*}"}
+{"id": "7803.png", "formula": "\\begin{align*} T _ { 1 } = 2 n \\eta _ { ( 1 ) } , \\ , T _ { 2 } = 2 ( n - 1 ) ( \\eta _ { ( 2 ) } - \\eta _ { ( 1 ) } ) , \\ , \\cdots , \\ , T _ { n } = 2 ( \\eta _ { ( n ) } - \\eta _ { ( n - 1 ) } ) . \\end{align*}"}
+{"id": "7782.png", "formula": "\\begin{align*} x ^ { r + 1 } - z = 0 \\end{align*}"}
+{"id": "2305.png", "formula": "\\begin{align*} w _ b = \\sum _ { \\ell } c _ \\ell a _ \\ell + \\Psi _ b , \\end{align*}"}
+{"id": "7319.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\alpha _ i z _ { i p } = \\sum _ { i = 1 } ^ n \\beta _ i z _ { i p } \\leq \\sum _ { i = 1 } ^ n \\gamma _ i z _ { i p } , \\end{align*}"}
+{"id": "3570.png", "formula": "\\begin{align*} B ( T ) = - \\frac { \\pi ^ { 3 / 2 } \\sigma ^ { 3 } } { \\sqrt { 3 } } \\sqrt { \\frac { k T } { \\varepsilon } } \\exp \\left ( \\frac { \\varepsilon } { k T } \\right ) \\left ( 1 + \\frac { 3 5 } { 3 6 } \\frac { k T } { \\varepsilon } + \\frac { 5 0 0 5 } { 2 5 9 2 } \\left ( \\frac { k T } { \\varepsilon } \\right ) ^ { 2 } + \\mathcal { O } ( T ^ { 3 } ) \\right ) . \\end{align*}"}
+{"id": "588.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ g A _ k \\otimes X _ k \\leq I . \\end{align*}"}
+{"id": "6884.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ] } \\d y \\ , \\widehat { r } _ \\beta ( x , y ) = \\beta \\forall \\ , x \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "2034.png", "formula": "\\begin{align*} \\dfrac { \\partial u ^ { j \\bar k } } { \\partial u _ { p \\bar q } } = - u ^ { j \\bar q } u ^ { p \\bar k } . \\end{align*}"}
+{"id": "1029.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { k - 1 } \\frac { ( a ) _ k ( 1 - a ) _ k } { ( 1 ) _ { k } ( b ) _ { k } } \\sum _ { i = 1 } ^ k \\frac { 1 } { ( a - 1 + i ) ( - a + i ) } = \\frac { \\Gamma ( \\frac { b } { 2 } ) \\Gamma ( \\frac { 1 + b } { 2 } ) } { \\Gamma ( \\frac { a + b } { 2 } ) \\Gamma ( \\frac { 1 - a + b } { 2 } ) } \\\\ [ 1 m m ] & \\ : \\ : \\times \\frac { \\psi ( \\frac { 1 - a + b } { 2 } ) - \\psi ( \\frac { a + b } { 2 } ) } { 1 - 2 a } . \\end{align*}"}
+{"id": "7893.png", "formula": "\\begin{align*} \\alpha ( \\mathcal { K } ( n , k ) ) \\leq \\frac { 4 \\binom { n } { 4 } } { 1 - \\frac { 2 \\binom { n - k } { k } } { - 2 \\binom { n - k - 1 } { k - 1 } } } = 4 \\binom { n - 1 } { k - 1 } . \\end{align*}"}
+{"id": "3632.png", "formula": "\\begin{align*} a _ i \\ = \\ a _ { m - i } \\ = \\ 9 , . \\end{align*}"}
+{"id": "7632.png", "formula": "\\begin{align*} M ( c , 1 , 0 ) = M ( c , 1 , 1 ) = ( 4 - c ^ 2 ) ^ 2 ( 3 6 - 1 3 c ^ 2 ) \\leq 5 7 6 , \\ ; c \\in ( 0 , 2 ) . \\end{align*}"}
+{"id": "2642.png", "formula": "\\begin{align*} \\delta = \\exp \\left ( - \\exp \\left ( c \\varepsilon ^ { - 6 } \\right ) \\right ) \\end{align*}"}
+{"id": "8933.png", "formula": "\\begin{align*} \\mathsf { d } _ { c c } ( Z ) = \\mathsf { d } _ { c c } \\Bigg ( \\prod _ { j = 1 } ^ k y \\big ( \\bigl \\{ X _ { n i } ^ { ( j ) } \\bigr \\} \\big ) \\Bigg ) \\leq 2 ^ { k - 1 } \\mathsf { d } _ { \\mathrm { c o m } } ^ { ( k ) } \\big ( \\bigl \\{ X _ { n i } ^ { ( 1 ) } \\bigr \\} , \\dots , \\bigl \\{ X _ { n i } ^ { ( k ) } \\bigr \\} \\big ) \\leq 1 . \\end{align*}"}
+{"id": "8313.png", "formula": "\\begin{align*} \\sum _ { a , b } M ( a , b ) f ^ g ( b ) = \\sum _ { a , b } M ( a , b ) f ( g b ) = \\sum _ { a , b } M ( g a , g b ) f ( g b ) = \\mu f ( g a ) = \\mu f ^ g ( a ) \\ , , \\end{align*}"}
+{"id": "552.png", "formula": "\\begin{align*} S _ { \\varepsilon } ( t ) = \\left ( \\begin{array} { c c } a _ { \\varepsilon } ( t ) & 0 \\\\ 0 & 1 \\end{array} \\right ) , \\end{align*}"}
+{"id": "8160.png", "formula": "\\begin{align*} h _ 2 \\left ( \\frac { 1 - q t } { 1 - q } \\right ) = \\frac { ( 1 - q t ) ( 1 - q ^ 2 t ) } { ( 1 - q ) ( 1 - q ^ 2 ) } = \\frac { ( 1 + q x ) ( 1 + q + q ^ 2 x ) } { 1 + q } . \\end{align*}"}
+{"id": "2135.png", "formula": "\\begin{align*} | z _ { y _ { 1 } } ^ { j } | = 1 , \\ z _ { y _ { 1 } } ^ { j } \\cdot z _ { y _ { 2 } } ^ { j } = 0 \\mbox { a n d } | z _ { y _ { 2 } } ^ { j } | \\ge c , \\end{align*}"}
+{"id": "2053.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { \\nu _ k } { \\sigma ^ 2 _ k } = 0 ; \\end{align*}"}
+{"id": "8851.png", "formula": "\\begin{align*} w ( t ) = w _ 0 ( t ) + \\int _ 0 ^ t S ( t - s ) F ( s ) w ( s ) \\ , d s + \\int _ 0 ^ t S ( t - s ) \\Sigma ( s ) w ( s ) C \\ , d W ( s ) \\end{align*}"}
+{"id": "6728.png", "formula": "\\begin{align*} X _ \\eta = X _ \\varphi = \\overline { [ \\underline { \\varphi } , \\varphi ] } . \\end{align*}"}
+{"id": "3389.png", "formula": "\\begin{align*} G _ k = \\bigcap _ { l \\ , : \\ , \\tilde { X } _ { k - 1 } \\leq X _ { l - 1 } < \\tilde { X } _ k } \\bigcap _ { j = 1 } ^ J G _ { j , l } \\ , , & & I _ { j , l } = \\bigcap _ { r = 1 } ^ 3 I _ { j , l } ^ { ( r ) } \\ , , & & I _ { k } = \\bigcap _ { l \\ , : \\ , \\tilde { X } _ { k - 1 } \\leq X _ { l - 1 } < \\tilde { X } _ k } \\bigcap _ { j = 1 } ^ J I _ { j , l } \\ , . \\end{align*}"}
+{"id": "7411.png", "formula": "\\begin{align*} p > 2 : \\lim _ { N \\to \\infty } \\sum _ { i = 1 } ^ { M _ N } \\mathbb E \\Big [ \\big | X _ { N , M _ N } ^ { ( i ) } \\big | ^ p \\Big ] = 0 \\ , . \\end{align*}"}
+{"id": "2334.png", "formula": "\\begin{align*} f _ j ^ { ( n ) } ( \\pmb { t } _ { n - 1 } , \\pmb { x } _ { n - 1 } , r , z , t , x ) = & f _ n ( t _ 1 , \\ldots , t _ { j - 1 } , r , t _ { j } , \\dots , t _ { n - 1 } , x _ 1 , \\dots , x _ { j - 1 } , z , x _ { j } , \\ldots , x _ { n - 1 } , t , x ) \\\\ = & G _ { t - t _ { n - 1 } } ( x - x _ { n - 1 } ) \\times \\dots \\times G _ { t _ j - r } ( x _ j - z ) G _ { r - t _ { j - 1 } } ( z - x _ { j - 1 } ) \\\\ & \\times \\dots \\times G _ { t _ 2 - t _ 1 } ( x _ 2 - x _ 1 ) \\mathbf { 1 } _ { T _ { n - 1 } ^ { j } ( t , r ) } ( \\pmb { t } _ { n - 1 } ) , \\end{align*}"}
+{"id": "6931.png", "formula": "\\begin{align*} \\frac { 1 } { \\sum _ { j \\in \\N } f ( j ) ^ 2 } \\sum _ { v \\in L } f ( v ) ^ 2 = o ( 1 ) \\textrm { a s } N \\to \\infty . \\end{align*}"}
+{"id": "5243.png", "formula": "\\begin{align*} Q = L _ 1 \\circ L _ 2 \\circ L _ 3 . \\end{align*}"}
+{"id": "5745.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } p ' y ^ 2 = z \\cdots \\mbox { ( 6 $ - $ 1 ) } \\\\ x ^ 2 = w \\cdots \\mbox { ( 6 $ - $ 2 ) } \\\\ p ' w ^ 2 = p x + q z \\cdots \\mbox { ( 6 $ - $ 3 ) } \\\\ z ^ 2 = p y + q w \\cdots \\mbox { ( 6 $ - $ 4 ) } \\\\ p ' y w = a x + b z \\cdots \\mbox { ( 6 $ - $ 5 ) } \\\\ x z = a y + b w \\cdots \\mbox { ( 6 $ - $ 6 ) } \\\\ p ' y w = c x + d z \\cdots \\mbox { ( 6 $ - $ 7 ) } \\\\ x z = c y + d w \\cdots \\mbox { ( 6 $ - $ 8 ) } . \\end{array} \\right . \\end{align*}"}
+{"id": "982.png", "formula": "\\begin{align*} \\left ( { \\nabla } _ { X } C \\right ) \\xi = - \\omega A _ { \\xi } X - h ( X , B \\xi ) \\end{align*}"}
+{"id": "3357.png", "formula": "\\begin{align*} S _ i ^ * T T ^ * = S _ i ^ * T T ^ * \\left ( \\sum _ { j = 1 } ^ d S _ j S _ j ^ * \\right ) = T T ^ * S _ i ^ * . \\end{align*}"}
+{"id": "8910.png", "formula": "\\begin{align*} u _ j = \\sum _ { s _ 1 , \\dots , s _ j = 1 } ^ { d _ 1 } \\alpha _ { s _ 1 \\dots s _ j } X _ { s _ 1 } \\otimes \\cdots \\otimes X _ { s _ j } & = \\sum _ { s _ 1 , \\dots , s _ j = 1 } ^ { d _ 1 } \\big ( \\alpha _ { s _ 1 \\dots s _ j } ^ 1 X _ { s _ 1 } \\big ) \\otimes \\cdots \\otimes \\big ( \\alpha _ { s _ 1 \\dots s _ j } ^ j X _ { s _ j } \\big ) \\\\ & = : \\sum _ { n = 1 } ^ { d _ 1 ^ j } X _ { n 1 } \\otimes \\cdots \\otimes X _ { n j } , \\end{align*}"}
+{"id": "8374.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} { c c } 3 & 0 \\\\ 0 & 3 \\end{array} \\right ) , A = \\left ( \\begin{array} { c c c } 1 & 2 \\\\ 2 & 1 \\end{array} \\right ) , L = \\left ( \\begin{array} { c c c } 2 & - 2 \\\\ - 2 & 2 \\end{array} \\right ) . \\end{align*}"}
+{"id": "3950.png", "formula": "\\begin{align*} \\Sigma ( \\delta ) = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\boldsymbol { W } _ { p _ 1 } ( \\gamma _ { 1 } , \\mu _ 1 ) \\leq \\delta _ 1 ^ { 1 / p _ 1 } , \\ \\boldsymbol { W } _ { p _ 2 } ( \\gamma _ { 2 } , \\mu _ 2 ) \\leq \\delta _ 2 ^ { 1 / p _ 2 } \\right \\} . \\end{align*}"}
+{"id": "5496.png", "formula": "\\begin{align*} D _ I = \\left \\{ \\left ( \\begin{array} { c c } e ^ { - t _ 1 } \\mathrm { I } _ 2 & 0 \\\\ 0 & e ^ { - t _ 2 } \\mathrm { I } _ 3 \\\\ \\end{array} \\right ) , t _ 1 > t _ 2 \\right \\} , \\textrm { w h e r e } \\mathrm { I } _ k \\textrm { i s t h e $ k \\times k $ i d e n t i t y m a t r i x . } \\end{align*}"}
+{"id": "4257.png", "formula": "\\begin{align*} J \\left ( t , \\xi ; u \\right ) \\triangleq \\left . Y _ { s } ^ { t , \\xi ; u } \\right \\vert _ { s = t } \\end{align*}"}
+{"id": "7787.png", "formula": "\\begin{align*} \\alpha A \\circ \\beta B = \\alpha \\beta ( A \\circ B ) . \\end{align*}"}
+{"id": "180.png", "formula": "\\begin{align*} \\pi ^ 2 = 3 6 L i _ 2 \\left ( \\frac { 1 } { 2 } \\right ) - 3 6 L i _ 2 \\left ( \\frac { 1 } { 4 } \\right ) - 1 2 L i _ 2 \\left ( \\frac { 1 } { 8 } \\right ) + 6 L i _ 2 \\left ( \\frac { 1 } { 6 4 } \\right ) . \\end{align*}"}
+{"id": "172.png", "formula": "\\begin{align*} L i _ 2 ( - \\phi ^ { - 1 } ) = - \\frac { \\pi ^ 2 } { 1 5 } + \\frac { 1 } { 2 } ( \\log \\phi ) ^ 2 , \\end{align*}"}
+{"id": "4536.png", "formula": "\\begin{align*} | I _ { \\omega , \\mathbf { c } } ( U ( t ) ) - I _ { \\omega , \\mathbf { c } } ( \\Phi ) | & \\le 2 | L ( U ( t ) ) - L ( \\Phi ) | + \\frac { 5 } { 2 } | N ( U ( t ) ) - N ( \\Phi ) | + | \\mathbf { c } \\cdot ( \\mathbf { P } ( U ( t ) ) - \\mathbf { P } ( \\Phi ) | \\\\ & \\le C ( 1 + \\| U ( t ) \\| _ { \\mathcal { H } ^ 1 } ^ 2 + \\| \\Phi \\| _ { \\mathcal { H } ^ 1 } ^ 2 ) \\| U ( t ) - \\Phi \\| _ { \\mathcal { H } ^ 1 } \\\\ & \\le C ( 1 + M ^ 2 + \\mu _ { \\omega , \\mathbf { c } } ) \\| U ( t ) - \\Phi \\| _ { \\mathcal { H } ^ 1 } \\end{align*}"}
+{"id": "7138.png", "formula": "\\begin{align*} a _ \\varepsilon ( x _ n ) : = \\sup _ { x ^ \\prime \\in \\mathbb { R } ^ { n - 1 } } \\int _ { B _ { \\delta } ( \\hat { x } ^ \\prime ) \\times ( 0 , \\delta ) } \\rho _ \\varepsilon ( | A _ \\gamma ( \\hat { x } , \\hat { x } ) ( \\hat { x } - \\bar { y } ) | ) \\ ; y . \\end{align*}"}
+{"id": "800.png", "formula": "\\begin{align*} \\left ( { D } ^ { \\alpha } _ { N } U ^ { n - \\sigma } , U ^ { n , \\sigma } \\right ) \\ , \\le & \\ , \\| f ^ { n - \\sigma } \\| \\ , \\| U ^ { n , \\sigma } \\| . \\end{align*}"}
+{"id": "5952.png", "formula": "\\begin{align*} \\theta _ { X ^ { \\ast } , L } ( f ) ( \\epsilon , y ) & = \\int _ { X ^ { \\ast } / X ^ { \\ast } \\cap L } f ( [ \\dot { y } ^ { \\ast } , 0 ] + [ ( \\epsilon , y ) , 0 ] ) d \\dot { y } ^ { \\ast } \\\\ & = \\int _ { X ^ { \\ast } / X ^ { \\ast } \\cap L } \\psi ( \\epsilon \\tfrac { \\langle \\dot { y } ^ { \\ast } , y \\rangle } { 2 } ) f ( \\epsilon , y + \\epsilon \\dot { y } ^ { \\ast } ) d \\dot { y } ^ { \\ast } , \\end{align*}"}
+{"id": "4658.png", "formula": "\\begin{align*} i \\frac { \\partial \\psi } { \\partial t } + \\triangle \\psi - V ( x ) \\psi + h ( | \\psi | ^ { 2 } ) \\psi = 0 \\hbox { i n } \\mathbb { R } ^ N \\end{align*}"}
+{"id": "5967.png", "formula": "\\begin{align*} \\overline { \\overline { C } } _ { X ^ { \\ast } } ( k _ 1 , k _ 2 ) & = \\widetilde { C } _ { X ^ { \\ast } } ( k _ 1 , k _ 2 ) \\widetilde { s } ( k _ 1 ) \\widetilde { s } ( k _ 2 ) \\widetilde { s } ( k _ 1 k _ 2 ) ^ { - 1 } = 1 ; \\end{align*}"}
+{"id": "8658.png", "formula": "\\begin{align*} a - b = \\int _ 0 ^ 1 \\int _ 0 ^ x ( 1 - p y ) ^ { - 2 } ( 1 - p x ) ^ { - 2 } y ^ { j - 1 } x ^ { j - 1 } ( y - x ) ^ 2 d y d x > 0 \\end{align*}"}
+{"id": "7820.png", "formula": "\\begin{align*} \\textsf { E } \\exp ( C _ { 0 } \\lambda \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } ( \\max _ { j \\le n } \\eta _ { j } - \\textsf { E } \\max _ { j \\le n } \\eta _ { j } ) ) = \\prod _ { l = 1 } ^ { n } \\textsf { E } \\exp ( C _ { 0 } \\lambda \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } \\frac { T _ { l } - 2 } { 2 ( n - l + 1 ) } ) , \\end{align*}"}
+{"id": "4720.png", "formula": "\\begin{align*} \\abs { \\delta _ d } \\lesssim O \\left ( \\begin{cases} ( a ^ 3 \\rho _ 0 ) ^ { 6 / 1 5 } \\zeta ^ { - 3 / 5 } & d = 3 \\\\ ( a ^ 2 \\rho _ 0 ) ^ { 1 / 2 } \\zeta ^ { - 1 / 2 } & d = 2 \\\\ ( a \\rho _ 0 ) ^ { 1 / 2 } \\abs { \\log a \\rho _ 0 } ^ { 1 / 2 } & d = 1 \\end{cases} \\right ) + O \\left ( ( a ^ d \\rho _ 0 ) ^ { - 2 / d } \\left ( a ^ d \\rho _ 0 \\zeta ^ { d / 2 } \\abs { \\log a ^ d \\rho _ 0 } \\right ) ^ { n } \\right ) \\end{align*}"}
+{"id": "3850.png", "formula": "\\begin{align*} \\mathrm { R W } _ 0 ( d ) = \\sup _ { \\eta \\ge 1 } \\left \\{ \\mathbb { E } _ { \\mu } \\left [ \\max \\{ Y _ 2 + \\eta h _ 1 ( X ) , Y _ 1 + \\eta h _ 0 ( X ) \\} \\right ] - \\eta \\delta _ 0 \\right \\} , \\end{align*}"}
+{"id": "1434.png", "formula": "\\begin{align*} \\psi ^ 0 _ { \\mu , \\xi } ( x ) = \\mu \\frac { \\partial U _ { \\mu , \\xi } } { \\partial \\mu } , \\psi ^ h _ { \\mu , \\xi } ( x ) = \\mu \\frac { \\partial U _ { \\mu , \\xi } } { \\partial \\xi _ { h } } . \\end{align*}"}
+{"id": "2855.png", "formula": "\\begin{align*} \\widetilde { \\Delta } \\circ \\widetilde { \\varphi } ( x , m ) & = \\widetilde { \\Delta } ( \\varphi _ L ( x ) , \\varphi _ M ( m ) ) \\\\ & = \\Delta ( \\varphi _ L ( x ) ) + \\rho _ { \\beta } ( \\varphi _ M ( m ) ) - \\tau \\circ ( \\rho _ { \\beta } ( \\varphi _ M ( m ) ) ) . \\end{align*}"}
+{"id": "475.png", "formula": "\\begin{align*} C _ { \\gamma } ^ k [ 0 , 1 ] = \\big \\{ x \\in \\mathbb { R } ^ k : 0 < x _ 1 < \\dots < x _ k < 1 \\big \\} . \\end{align*}"}
+{"id": "2482.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { S } ( \\xi , - \\eta ) = \\min \\limits _ { ( u , v ) \\in \\mathcal { M } } \\mathcal { S } ( u , v ) . \\end{array} \\right . \\end{align*}"}
+{"id": "8805.png", "formula": "\\begin{align*} & 2 u ^ 3 t + 1 8 u ^ 2 t ^ 2 + 4 5 u t ^ 3 + 3 6 t ^ 4 + 4 u ^ 3 + 5 2 u ^ 2 t + 1 9 3 u t ^ 2 \\\\ & + 2 0 7 t ^ 3 + 3 2 u ^ 2 + 2 6 0 u t + 4 3 1 t ^ 2 + 1 0 8 u + 3 8 2 t + 1 2 0 \\\\ & = 4 u ^ 3 t + 2 4 u ^ 2 t ^ 2 + 5 4 u t ^ 3 + 3 6 t ^ 4 + 2 u ^ 3 + 5 6 u ^ 2 t + 2 1 7 u t ^ 2 \\\\ & + 2 1 6 t ^ 3 + 2 2 u ^ 2 + 2 5 5 u t + 4 4 9 t ^ 2 + 8 0 u + 3 7 5 t + 1 0 0 \\end{align*}"}
+{"id": "2614.png", "formula": "\\begin{align*} \\left \\| \\partial _ { x _ j } ^ k f _ j \\right \\| _ \\infty \\lesssim \\lambda ^ k \\| f _ j \\| _ \\infty , \\textrm { f o r $ j = 1 , 2 $ } . \\end{align*}"}
+{"id": "5775.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { m } f _ { j } ( y _ { j } ) & = \\prod _ { j = 1 } ^ { m } \\Big ( f _ { j } ^ { 0 } ( y _ { j } ) + f _ { j } ^ { \\infty } ( y _ { j } ) \\Big ) = \\sum _ { \\rho _ { 1 } , \\dots , \\rho _ { m } \\in \\{ 0 , \\infty \\} } f _ { 1 } ^ { \\rho _ { 1 } } ( y _ { 1 } ) \\cdots f _ { m } ^ { \\rho _ { m } } ( y _ { m } ) \\\\ & = \\prod _ { j = 1 } ^ { m } f _ { j } ^ { 0 } ( y _ { j } ) + \\sum _ { ( \\rho _ { 1 } , \\dots , \\rho _ { m } ) \\in \\rho } f _ { 1 } ^ { \\rho _ { 1 } } ( y _ { 1 } ) \\cdots f _ { m } ^ { \\rho _ { m } } ( y _ { m } ) , \\end{align*}"}
+{"id": "4824.png", "formula": "\\begin{align*} d _ { w } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } \\right ) & = \\int _ { - \\infty } ^ { \\infty } \\left | \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } \\leq x \\right ) - \\Phi ( x ) \\right | \\mbox { d } x \\\\ & \\leq \\frac { C } { n ^ { \\delta / 2 } } \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { 1 + | x | ^ { 1 + \\delta ^ { ' } } } \\mbox { d } x \\\\ & \\leq \\frac { C } { n ^ { \\delta / 2 } } , \\end{align*}"}
+{"id": "9288.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } w ^ { k } = w ^ { * } \\in W ^ { * } . \\end{align*}"}
+{"id": "1498.png", "formula": "\\begin{align*} & \\alpha \\leq \\overline { C } _ 1 \\max _ { q \\in S ^ 2 : \\ , q \\cdot v = 0 } \\ ; \\max _ { 0 \\le s \\le t } \\ ; ( q \\cdot W _ s ) \\leq \\overline { C } _ 1 \\varepsilon , \\quad \\widehat { \\alpha } \\leq \\overline { C } _ 2 \\varepsilon , \\epsilon \\leq \\overline { C } _ 3 \\ , \\alpha ^ 2 . \\end{align*}"}
+{"id": "2764.png", "formula": "\\begin{align*} \\varphi _ t = \\sum _ { N _ t } \\tau _ j ^ { - 1 } a _ j \\psi _ j , \\end{align*}"}
+{"id": "4905.png", "formula": "\\begin{align*} \\Delta _ \\varepsilon \\left [ f ( x ) \\right ] _ { x = \\alpha } \\overset { } { = } f ( \\alpha + \\varepsilon ) - f ( \\alpha ) . \\end{align*}"}
+{"id": "5957.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( g ) f ( [ \\epsilon , w ] ) & = \\widetilde { C } _ { X ^ { \\ast } } ( g , h _ { - \\epsilon } ) \\widetilde { \\beta } ( h _ { - 1 } g ^ { s ( - \\epsilon ) } ) ^ { - 1 } f ( [ - \\epsilon , w g ^ { s ( - \\epsilon ) } ] ) \\\\ & = e ^ { \\tfrac { \\pi i ( 1 - \\epsilon ) } { 4 } } ( - c , - \\epsilon ) _ { \\R } \\beta ( d , - \\epsilon c ) ^ { - 1 } f ( [ - \\epsilon , w g ^ { s ( - \\epsilon ) } ] ) . \\end{align*}"}
+{"id": "4949.png", "formula": "\\begin{align*} \\begin{aligned} \\Pr ( X _ t \\le k \\ , \\vert \\ , X _ 0 = r ) & = \\frac { n - k } { n ^ t } \\binom { n - r } { n - k } \\sum _ { j = 0 } ^ { k - r } ( - 1 ) ^ { k - r - j } \\binom { k - r } { j } \\frac { ( r + j ) ^ t } { n - r - j } \\\\ \\end{aligned} \\end{align*}"}
+{"id": "7021.png", "formula": "\\begin{align*} \\nu _ i \\left ( j f _ j Q _ i ' Q _ i ^ { j - 1 } \\right ) = v ( j ) + \\nu \\left ( f _ j Q _ i ^ j \\right ) + \\nu \\left ( Q ' _ i \\right ) - \\nu ( Q _ i ) \\geq \\nu _ i ( f ) + \\alpha _ i . \\end{align*}"}
+{"id": "6709.png", "formula": "\\begin{align*} \\partial _ j \\Phi & = \\lambda \\partial _ j \\Psi e ^ { \\lambda \\Psi } , \\partial _ { j , k } \\Phi = \\lambda \\partial _ { j , k } \\Psi e ^ { \\lambda \\Psi } + \\lambda ^ 2 ( \\partial _ j \\Psi ) ( \\partial _ k \\Psi ) e ^ { \\lambda \\Psi } , \\end{align*}"}
+{"id": "2925.png", "formula": "\\begin{align*} m ^ { 1 - r } \\cdot \\tilde { B } _ r ( x ) = \\sum _ { y m = x } \\tilde { B } _ r ( y ) . \\end{align*}"}
+{"id": "7883.png", "formula": "\\begin{align*} \\sigma \\left ( ( A ^ { * _ 1 } , \\underline { A } ^ { * _ 2 } ) \\right ) : = \\left ( \\sigma ( A ^ { * _ 1 } ) , \\sigma ( \\underline { A } ^ { * _ 2 } ) \\right ) . \\end{align*}"}
+{"id": "809.png", "formula": "\\begin{align*} y _ { 1 } & = y _ { 0 } + h f \\left ( x _ { 0 } , y _ { 0 } \\right ) + h \\int \\limits _ { x _ { 0 } } ^ { x _ { 0 } } K d t \\\\ & = y _ { 0 } + h f \\left ( x _ { 0 } , y _ { 0 } \\right ) . \\end{align*}"}
+{"id": "2886.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 8 , & ~ ~ t _ 2 \\geq 3 , \\\\ 6 , & ~ ~ t _ 2 = 2 , \\\\ 4 , & ~ ~ t _ 2 = 1 , \\\\ 2 - 2 t _ 2 , & ~ ~ t _ 2 \\leq 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "6011.png", "formula": "\\begin{align*} \\mathcal { F } _ { X ^ { \\ast } X } : & \\mathcal { H } ( X ) \\longrightarrow \\mathcal { H } ( X ^ { \\ast } ) ; \\\\ & f \\longmapsto \\mathcal { F } _ { X ^ { \\ast } X } ( f ) ( h ) = \\int _ { X ^ { \\ast } } f ( [ x ^ { \\ast } , 0 ] h ) d x ^ { \\ast } . \\end{align*}"}
+{"id": "4524.png", "formula": "\\begin{align*} C _ { \\omega , \\mathbf { c } } : = \\left \\{ U \\in \\mathcal { H } ^ 1 \\backslash \\{ ( \\mathbf { 0 } , \\mathbf { 0 } , \\mathbf { 0 } ) \\} \\ | \\ S _ { \\omega , \\mathbf { c } } ( U ) < \\widetilde { \\mu } _ { \\omega , \\mathbf { c } } , \\ I _ { \\omega , \\mathbf { c } } ( U ) < 0 , \\ \\mathbf { c } \\cdot \\mathbf { P } ( U ) \\ge 0 \\right \\} . \\end{align*}"}
+{"id": "4611.png", "formula": "\\begin{align*} \\mathbb { P } ( { E } _ { i , 2 } ) & = \\underset { \\substack { k _ 1 + \\cdots + k _ K \\\\ = m - i } } { \\sum } \\frac { ( m - i ) ! } { k _ 1 ! \\cdots k _ K ! } \\mathbb { P } _ K ^ { k _ 1 } \\mathbb { P } _ { e , 1 } ^ { k _ 1 } \\times \\cdots \\times \\mathbb { P } _ K ^ { k _ K } \\mathbb { P } _ { e , K } ^ { k _ K } , \\end{align*}"}
+{"id": "8494.png", "formula": "\\begin{align*} \\mathcal { J } _ { p } ( x , s ) : = \\intop _ { 0 } ^ { \\infty } \\ , \\frac { y ^ { s - \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( 2 \\pi x y ) } { \\sigma \\left ( y \\right ) e ^ { 2 \\pi y } - 1 } \\ , d y \\end{align*}"}
+{"id": "2579.png", "formula": "\\begin{align*} \\sigma ( c _ { i j k } ) = c _ { \\sigma i \\sigma j \\sigma k } \\end{align*}"}
+{"id": "1869.png", "formula": "\\begin{align*} \\begin{aligned} L ( K , u ^ * ) & = \\{ t v \\in \\mathcal { U } : \\ G ' ( u ^ * ) u ^ * + G ' ( u ^ * ) v \\in K , \\ \\forall \\ t > 0 \\} \\\\ & = \\{ t v \\in \\mathcal { U } : \\ G ' ( u ^ * ) u ^ * + G ' ( u ^ * ) v \\in \\mathcal { K } - G ( u ^ * ) + G ' ( u ^ * ) u ^ * , \\ \\forall \\ t > 0 \\} \\\\ & = \\{ t v \\in \\mathcal { U } : \\ G ( u ^ * ) + G ' ( u ^ * ) v \\in \\mathcal { K } , \\ \\forall \\ t > 0 \\} \\\\ & = L ( \\mathcal { K } , u ^ * ) . \\end{aligned} \\end{align*}"}
+{"id": "4149.png", "formula": "\\begin{align*} Q _ { 3 , \\vec { v } _ { s , t , f , 0 } ^ { ( 1 ) } } = \\left ( \\left ( \\begin{array} { r r r } - s & 0 & 1 \\\\ 1 & 0 & 0 \\\\ - t & 1 & 0 \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 0 & 1 & 0 \\\\ 0 & t & 1 \\\\ 1 & s & 0 \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) \\right ) . \\end{align*}"}
+{"id": "8161.png", "formula": "\\begin{align*} \\Gamma _ { B _ + ( T _ 1 \\cdots T _ k ) } ( A ) = B ( \\Gamma _ { T _ 1 } \\cdots \\Gamma _ { T _ k } ) , \\end{align*}"}
+{"id": "8135.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\varphi ^ + ( M _ I ) = P _ + ( \\varphi ^ - ( M _ { I ' } ) a ^ { i _ p } ) \\\\ \\varphi ^ - ( M _ I ) = - P _ - ( \\varphi ^ - ( M _ { I ' } ) a ^ { i _ p } ) . \\end{array} \\right . \\end{align*}"}
+{"id": "8233.png", "formula": "\\begin{align*} \\Gamma _ { s } ^ { N , \\phi } ( \\pi ^ N ) & = \\frac { 2 \\kappa } { N ^ 2 } \\sum _ { ( x , \\sigma ) \\in V } \\eta ^ N _ s ( x , \\sigma ) ( \\partial _ x \\phi ( \\tfrac { x } { N } , \\sigma ) ) ^ 2 + \\frac { 1 } { N ^ 2 } \\sum _ { ( x , \\sigma ) \\in V } \\sum _ { \\sigma ' \\in S } c ( \\sigma , \\sigma ' ) \\eta ^ N _ s ( x , \\sigma ) ( \\phi ( \\tfrac { x } { N } , \\sigma ' ) - \\phi ( \\tfrac { x } { N } , \\sigma ) ) ^ 2 \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ + R _ 4 ( \\phi , s , N , \\sigma ) , \\end{align*}"}
+{"id": "2912.png", "formula": "\\begin{align*} H _ \\delta = H \\cap \\left ( \\bigcap _ { \\substack { g \\in G \\\\ n _ g \\neq 0 } } g H g ^ { - 1 } \\right ) . \\end{align*}"}
+{"id": "7717.png", "formula": "\\begin{align*} & F _ 4 \\left ( \\frac { p - 1 } { 2 } , \\frac { p + q - 4 } { 2 } ; \\frac { p - 1 } { 2 } , \\frac { q - 1 } { 2 } ; - \\frac { | z ' | ^ 2 } { 4 } , - \\frac { | z '' | ^ 2 } { 4 } \\right ) \\\\ & = \\tau ( z ' , z '' ) ^ { - \\frac { p + q - 4 } { 2 } } { } _ 2 F _ 1 \\left ( \\frac { q - p } { 4 } , \\frac { p + q - 4 } { 4 } , \\frac { q - 1 } { 2 } ; \\frac { | z '' | ^ 2 } { \\tau ( z ' , z '' ) ^ { 2 } } \\right ) \\end{align*}"}
+{"id": "6342.png", "formula": "\\begin{align*} E _ t ( \\phi , \\omega , r ) : = G ( \\phi , \\omega , r ; t ) = \\big ( x ( \\phi , \\omega , r ; t ) , y ( \\phi , \\omega , r ; t ) , z ( \\phi , \\omega , r ; t ) \\big ) , \\end{align*}"}
+{"id": "7573.png", "formula": "\\begin{align*} F _ { f } ( z ) : = \\log \\dfrac { f ( z ) } { z } = 2 \\sum _ { n = 1 } ^ { \\infty } \\gamma _ { n } ( f ) z ^ n , \\ ; \\ ; z \\in \\mathbb { D } , \\ ; \\ ; \\log 1 : = 0 , \\end{align*}"}
+{"id": "865.png", "formula": "\\begin{align*} \\begin{pmatrix} \\Phi _ t & \\Phi _ { t + 1 } \\\\ \\Phi _ { t + 1 } & \\Phi _ { t + 2 } \\end{pmatrix} \\end{align*}"}
+{"id": "2209.png", "formula": "\\begin{align*} \\Phi ( x , y ) = \\Phi ( x , z ) + \\sum _ { y < p \\le z } \\Phi ( x / p , p ^ - ) , \\end{align*}"}
+{"id": "2802.png", "formula": "\\begin{align*} \\| w _ t \\| _ { H ^ 1 ( \\Gamma ) } = \\| \\mathcal { T } ^ \\ast v _ t \\| _ { H _ \\Gamma ^ 1 ( \\partial \\mathrm { M } ) } \\le t \\| v _ t \\| _ { L ^ 2 ( \\mathrm { M } _ 0 ) } . \\end{align*}"}
+{"id": "2645.png", "formula": "\\begin{align*} & x ^ { \\alpha } y ^ { \\beta } z ^ { \\gamma } + x ^ { \\beta } y ^ { \\gamma } z ^ { \\alpha } + x ^ { \\gamma } y ^ { \\alpha } z ^ { \\beta } - x ^ { \\beta } y ^ { \\alpha } z ^ { \\gamma } - x ^ { \\gamma } y ^ { \\beta } z ^ { \\alpha } - x ^ { \\alpha } y ^ { \\gamma } z ^ { \\beta } \\\\ = & ( x - y ) ( x - z ) ( y - z ) \\sum _ { i = 0 } ^ { \\beta - \\gamma - 1 } \\sum _ { j = 0 } ^ { \\alpha - \\beta - 1 } \\sum _ { k = 0 } ^ { \\alpha - \\beta - 1 - j + i } x ^ { \\gamma + i + j } y ^ { \\beta - 1 - i + k } z ^ { \\alpha - 2 - j - k } . \\end{align*}"}
+{"id": "5973.png", "formula": "\\begin{align*} \\overline { \\overline { C } } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) & = \\widetilde { C } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) \\widetilde { s } ( g _ 1 ) \\widetilde { s } ( g _ 2 ) \\widetilde { s } ( g _ 1 g _ 2 ) ^ { - 1 } . \\end{align*}"}
+{"id": "3478.png", "formula": "\\begin{align*} C _ { \\mathrm { U B F } } = \\Big ( \\frac { e } { \\theta } \\Big ) ^ { \\lceil \\frac { 2 \\eta } { b } \\rceil } = \\Big ( \\frac { 5 e ^ 2 \\Delta } { b ^ 2 } \\Big ) ^ { \\lceil \\frac { 2 \\eta } { b } \\rceil } , \\end{align*}"}
+{"id": "334.png", "formula": "\\begin{align*} s _ h ( 5 , 4 ) = \\frac { 8 9 0 } { 9 } \\zeta ( 9 ) + 6 6 \\zeta ( 4 ) \\zeta ( 5 ) - \\frac { 4 2 9 5 } { 2 4 } \\zeta ( 2 ) \\zeta ( 5 ) - 5 \\zeta ( 3 ) ^ 3 + \\frac { 2 6 5 } { 8 } \\zeta ( 2 ) \\zeta ( 7 ) . \\end{align*}"}
+{"id": "340.png", "formula": "\\begin{align*} \\| u _ n - u _ m \\| _ p ^ p = \\sum _ { x \\in V } \\mu ( x ) | u _ n ( x ) - u _ m ( x ) | ^ p < \\epsilon , \\quad \\forall n , m \\geq N . \\end{align*}"}
+{"id": "9178.png", "formula": "\\begin{align*} \\begin{array} { c c l } \\dot { q } ^ { 1 } & = & \\omega ^ { 1 } \\\\ \\dot { q } ^ { 2 } & = & \\omega ^ { 2 } \\\\ \\dot { q } ^ { 3 } & = & \\omega ^ { 3 } \\\\ \\dot { \\omega } ^ { 1 } & = & b _ { 1 } \\cos ( q ^ { 2 } ) \\sin ( q ^ { 3 } ) u ^ { 1 } \\\\ \\dot { \\omega } ^ { 2 } & = & a _ { 1 } \\sin ( q ^ { 2 } ) + a _ { 2 } \\cos ( q ^ { 2 } ) + b _ { 2 } \\cos ( q ^ { 3 } ) u ^ { 1 } \\\\ \\dot { \\omega } ^ { 3 } & = & a _ { 3 } \\cos ( q ^ { 2 } ) \\sin ( q ^ { 3 } ) + b _ { 3 } u ^ { 2 } \\end{array} \\end{align*}"}
+{"id": "128.png", "formula": "\\begin{align*} p _ k ( \\lambda , \\mu ) = \\sum \\limits _ { j = 0 } ^ { 2 n + 1 } ( - 1 ) ^ j \\lambda ^ { - j - 1 } \\mu ^ j \\end{align*}"}
+{"id": "5433.png", "formula": "\\begin{align*} s _ 1 ( U ) & = \\sum _ { j , k , l , m = 1 } ^ n R ( e _ j , \\bar e _ k , e _ l , \\bar e _ l ) R ( e _ k , \\bar e _ j , e _ m , \\bar e _ m ) = \\sum _ { j , k = 1 } ^ n | r ( e _ j , \\bar e _ k ) | ^ 2 = | r | ^ 2 \\\\ s _ 2 ( U ) & = \\sum _ { j , k , l , m = 1 } ^ n R ( e _ j , \\bar e _ k , e _ l , \\bar e _ m ) R ( e _ k , \\bar e _ j , e _ m , \\bar e _ l ) = | R | ^ 2 . \\end{align*}"}
+{"id": "6185.png", "formula": "\\begin{align*} A ( w , D _ \\tau w ) : D ^ 2 _ { \\tau } w + b ( w , D _ \\tau w ) \\cdot D _ \\tau w + \\mu w = ( n - 1 ) - \\mu - ( 1 + w ) R , \\end{align*}"}
+{"id": "262.png", "formula": "\\begin{align*} = \\left ( \\frac { 1 } { 1 - y z } \\right ) ^ { \\frac { y } { 1 - y } } \\exp \\left \\{ \\frac { y ( y + 1 ) } { ( 1 - y ) ^ 3 } ( L i _ 3 ( z ) - L i _ 3 ( y z ) ) - \\frac { 2 y } { ( 1 - y ) ^ 2 } L i _ 2 ( y z ) \\right \\} ; \\end{align*}"}
+{"id": "7788.png", "formula": "\\begin{align*} A \\circ B & = ( \\mathcal { R } A + i \\mathcal { I } A ) \\circ ( \\mathcal { R } B + i \\mathcal { I } B ) \\\\ & = ( \\mathcal { R } A \\circ \\mathcal { R } B - \\mathcal { I } A \\circ \\mathcal { I } B ) + i ( \\mathcal { R } A \\circ \\mathcal { I } B + \\mathcal { I } A \\circ \\mathcal { R } B ) . \\end{align*}"}
+{"id": "4653.png", "formula": "\\begin{align*} \\hat { m } ( x , t ) = \\begin{cases} \\bar { m } ( x , t ) & \\mbox { i f } \\ , x \\in \\Omega \\ , \\mbox { a n d } \\ , t \\in ( 0 , T ) , \\\\ 0 & \\mbox { i f } \\ , x \\in \\Omega \\ , \\mbox { a n d } \\ , t \\geq T , \\end{cases} \\end{align*}"}
+{"id": "611.png", "formula": "\\begin{align*} \\theta ( a ^ * a ) \\geq \\theta ( a ) ^ * \\theta ( a ) = \\sigma ( a ) ^ * \\sigma ( a ) = \\sigma ( a ^ * a ) . \\end{align*}"}
+{"id": "2771.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q _ 1 ) \\tilde { v } _ 2 = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } , \\tilde { v } _ 2 { _ { | \\partial M } } = v _ 2 { _ { | \\partial M } } , \\end{align*}"}
+{"id": "5484.png", "formula": "\\begin{align*} { \\rm E n t S p } \\left ( G , \\mu \\right ) : = \\left \\{ h _ { \\mu } ( X , \\nu ) : ( X , \\nu ) \\mbox { i s a n e r g o d i c } ( G , \\mu ) \\mbox { - s t a t i o n a r y s y s t e m } \\right \\} . \\end{align*}"}
+{"id": "8457.png", "formula": "\\begin{align*} x ^ { - 2 s } = x ^ { - 1 } \\left ( 1 - 2 \\log ( x ) \\ , \\left ( s - \\frac { 1 } { 2 } \\right ) + O \\left ( \\left ( s - \\frac { 1 } { 2 } \\right ) ^ { 2 } \\right ) \\right ) , \\end{align*}"}
+{"id": "6724.png", "formula": "\\begin{align*} \\underline { \\varphi } ( h ) = \\varphi ^ * ( \\Gamma _ { H , H ^ * } ( h ) ) . \\end{align*}"}
+{"id": "5343.png", "formula": "\\begin{align*} \\sqrt { q } ( r ' ) ^ { - 2 / q } = ( q \\cdot \\left ( \\frac { 1 } { ( r ' ) ^ 2 } \\right ) ^ { 2 / q } ) ^ { 1 / 2 } \\leq ( q \\cdot \\left ( \\frac { \\log ( 1 / \\gamma ) } { 2 } \\right ) ^ { 2 / q } ) ^ { 1 / 2 } = ( x _ 0 \\cdot a _ 0 ^ { 2 / x _ 0 } ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "8097.png", "formula": "\\begin{align*} \\mathbf { R } \\in L ^ { \\infty } ( \\mathcal { D } _ { a T } , \\mathbb { R } ^ { 3 \\times 3 } ) . \\end{align*}"}
+{"id": "8064.png", "formula": "\\begin{align*} X _ j = \\frac { \\partial } { \\partial x _ j } , \\enspace j = 1 , \\ldots , k , X _ { k + j } = | x | ^ { 2 \\alpha } \\frac { \\partial } { \\partial x _ { k + j } } , \\enspace j = 1 , \\ldots , N - k \\end{align*}"}
+{"id": "273.png", "formula": "\\begin{align*} = \\frac { \\left ( 1 - z \\right ) ^ { \\frac { 4 } { ( 1 - y ) ^ 5 } } } { \\left ( 1 - y z \\right ) ^ { \\frac { y ^ 5 - 4 y ^ 4 + 6 y ^ 3 + y } { ( 1 - y ) ^ 5 } } } \\end{align*}"}
+{"id": "1895.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\langle D _ u \\theta ( u ^ * ) , v \\rangle _ \\mathcal { U } + \\langle \\tilde { \\lambda } ^ * , S v \\rangle _ { * } = 0 \\forall \\ v \\in \\mathcal { U } ; \\\\ & - \\langle \\tilde { \\lambda } ^ * , \\zeta - S u ^ * \\rangle _ * \\geq 0 \\forall \\ \\zeta \\in \\overline { K \\cap R ( S ) } ^ { \\| \\cdot \\| _ * } , \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "700.png", "formula": "\\begin{align*} | c ( \\varphi ) | \\norm { S ( k ) ( \\bar { \\xi } - \\bar { \\xi } \\circ T ) } _ { \\sup } & = \\| S ( k ) \\chi \\| _ { \\sup } \\leq \\norm { S ( k ) \\varphi } _ { \\sup } + \\norm { S ( k ) ( \\widetilde { v } _ 0 \\circ T - \\widetilde { v } _ 0 ) } _ { \\sup } \\\\ & \\leq ( \\| v _ 0 \\| _ { \\sup } + o ( 1 ) ) \\norm { S ( k ) ( \\bar { \\xi } - \\bar { \\xi } \\circ T ) } _ { \\sup } . \\end{align*}"}
+{"id": "5437.png", "formula": "\\begin{align*} & \\quad \\Pr \\left ( > \\theta | \\Phi , \\Phi ( \\mathcal { A } ) > 0 \\right ) \\\\ & \\approx \\sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } \\prod \\limits _ { x _ { i } \\in \\Phi \\backslash \\{ x _ 1 \\} \\cap { \\mathcal { A } } } \\frac { 1 } { \\left ( 1 + \\frac { m \\eta \\theta r _ 1 ^ { \\alpha } } { M r _ i ^ { \\alpha } } \\right ) ^ M } . \\end{align*}"}
+{"id": "3701.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) u ^ { ( 2 ) } ( x ) + F ^ { ( 2 ) } ( x ) u ^ { ( 2 ) } ( x ) = 0 & \\Omega , \\\\ u ^ { ( 2 ) } = f ^ 2 & \\Omega ^ c : = \\mathbb { R } ^ n \\setminus \\overline { \\Omega } . \\end{cases} \\end{align*}"}
+{"id": "3871.png", "formula": "\\begin{align*} \\mathrm { R W } _ { C } ( d ) & = \\sum _ { \\ell = 1 } ^ L \\mathbb { E } _ { ( Y _ \\ell , X ) \\sim \\mu _ { \\ell , L + 1 } } \\left [ ( Y _ \\ell - \\delta _ 0 ) I ( D ( X ) = \\ell ) \\right ] \\\\ & = \\mathbb { E } _ { X } \\left [ \\sum _ { \\ell = 1 } ^ L \\left ( \\mathbb { E } [ Y _ \\ell \\mid X ] - \\delta _ 0 \\right ) I ( D ( X ) = \\ell ) \\right ] . \\end{align*}"}
+{"id": "7273.png", "formula": "\\begin{align*} y ^ m = P ( z ) , \\end{align*}"}
+{"id": "3648.png", "formula": "\\begin{align*} \\overline { \\mathrm { u m d i m } } _ { Y ^ \\mathbb { N } \\times I _ { \\varphi } ( G ) } ( F , D ) = \\overline { \\mathrm { u m d i m } } _ { I _ { \\varphi } ( G ) } ( G , d , \\mathbb { P } ) . \\end{align*}"}
+{"id": "2242.png", "formula": "\\begin{align*} \\mathcal { T } _ D ( f ) ( z ) = - \\frac { 1 } { 2 \\pi } \\iint _ D \\left ( \\frac { f ( \\zeta ) } { \\zeta } \\frac { \\zeta + z } { \\zeta - z } + \\frac { \\overline { f ( \\zeta ) } } { \\overline { \\zeta } } \\frac { 1 + z \\overline { \\zeta } } { 1 - z \\overline { \\zeta } } \\right ) \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "3221.png", "formula": "\\begin{align*} S _ n ' ( y _ 1 ' - S _ k ' ( y _ 1 ' ) ) & = S _ n ' \\Big [ \\frac { 1 } { k + 1 } \\sum _ { l = 0 } ^ { k } \\Big ( y _ 1 ' - \\sigma _ l ' ( y _ 1 ' ) \\Big ) \\Big ] \\\\ & = \\frac { 1 } { k + 1 } \\sum _ { l = 0 } ^ { k } \\Big [ S _ n ' \\Big ( y _ 1 ' - \\sigma _ l ' ( y _ 1 ' ) \\Big ) \\Big ] . \\end{align*}"}
+{"id": "7769.png", "formula": "\\begin{align*} M \\vec x ^ 2 & = c _ 1 \\odot M \\vec x ^ 1 , \\\\ \\vec x ^ 1 - \\vec z ^ { \\vec x ^ 1 } & = c _ 2 \\odot ( \\vec x ^ 2 - \\vec z ^ { \\vec x ^ 2 } ) . \\end{align*}"}
+{"id": "2251.png", "formula": "\\begin{align*} \\mathcal { T } _ D ^ k ( \\langle h _ k , P _ r ( \\theta - \\cdot ) + i Q _ r ( \\theta - \\cdot ) \\rangle ) = \\langle h _ k , ( P _ r ( \\theta - \\cdot ) + i Q _ r ( \\theta - \\cdot ) ) ( e ^ { i ( \\cdot ) } - r e ^ { i \\theta } + \\overline { e ^ { i ( \\cdot ) } - r e ^ { i \\theta } } ) ^ k \\rangle , \\end{align*}"}
+{"id": "8672.png", "formula": "\\begin{align*} ^ 2 ( P _ X , P _ Y ) = \\| \\mu _ { P _ { X } } - \\mu _ { P _ { Y } } \\| _ { \\mathcal { H } _ k } ^ 2 = E \\{ k ( X , X ' ) \\} + E \\{ k ( Y , Y ' ) \\} - 2 E \\{ k ( X , Y ) \\} , \\end{align*}"}
+{"id": "542.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ { t } U ( t , \\xi ) = i \\langle \\xi \\rangle A ( t ) U ( t , \\xi ) + i \\langle \\xi \\rangle ^ { - 1 } Q ( t ) U ( t , \\xi ) + F ( t , \\xi ) , \\quad ( t , \\xi ) \\in ( 0 , T ] \\times \\mathcal { I } _ { \\hbar } , \\\\ U ( 0 , \\xi ) = U _ { 0 } ( \\xi ) , \\quad \\xi \\in \\mathcal { I } _ { \\hbar } . \\end{array} \\right . \\end{align*}"}
+{"id": "3040.png", "formula": "\\begin{align*} s _ k ( A _ { s } ( \\mathrm { i , j } ) ) = 0 { \\rm f o r \\quad } s = 1 , \\ldots , m . \\end{align*}"}
+{"id": "1994.png", "formula": "\\begin{align*} \\mathcal { P } _ 0 ( \\Omega ) : = \\{ u \\in P S H ( \\Omega ) \\cap C ^ { 2 } ( \\Omega ) \\cap C ^ 0 ( \\bar \\Omega ) \\ , ; \\ , u _ { | _ { \\partial \\Omega } } = 0 \\} . \\end{align*}"}
+{"id": "5891.png", "formula": "\\begin{align*} L ^ 1 * M ^ { p , q } & = M ^ { p , q } , \\end{align*}"}
+{"id": "3498.png", "formula": "\\begin{align*} - X '' + R ( \\dot { \\gamma } , X ) \\dot { \\gamma } = 0 , \\end{align*}"}
+{"id": "2019.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } L v = - \\lambda _ 1 u G & \\textnormal { o n } & \\Omega \\\\ v = 0 & \\textnormal { i n } & \\partial \\Omega , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "9080.png", "formula": "\\begin{align*} \\omega ^ + = \\tilde { \\omega } ^ + + c _ 1 , \\ , \\ , \\omega ^ - = \\tilde { \\omega } ^ - + c _ 2 , \\ , \\ , \\omega ^ { ( 0 ) } = \\tilde { \\omega } ^ { ( 0 ) } + c _ 0 \\end{align*}"}
+{"id": "1883.png", "formula": "\\begin{align*} \\| F \\| _ { [ \\mathcal { X } ] ' } = \\| f \\| . \\end{align*}"}
+{"id": "3760.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty c _ { 1 , n } T _ 1 ^ n \\left ( \\{ x _ \\tau ^ 0 \\} _ { \\tau = 1 } ^ { r } \\right ) \\equiv & \\sum _ { n = 1 } ^ \\infty c _ { 1 , n } T _ 1 ^ n \\left ( \\{ x _ \\tau ^ 0 \\tilde { f } _ 0 ^ n \\} _ { \\tau = 1 } ^ { r } \\right ) , \\\\ \\sum _ { n = 1 } ^ \\infty c _ { 2 , n } T _ 2 ^ n \\left ( \\{ x _ \\theta ^ 0 \\} _ { \\theta = r + 1 } ^ { r + s } \\right ) \\equiv & \\sum _ { n = 1 } ^ \\infty c _ { 2 , n } T _ 2 ^ n \\left ( \\{ x _ \\theta ^ 0 \\tilde { f } _ 0 ^ n \\} _ { \\theta = r + 1 } ^ { r + s } \\right ) . \\end{align*}"}
+{"id": "7260.png", "formula": "\\begin{align*} \\Big \\Vert \\sup _ { k \\leq 2 ^ { L - m ( L ) } } \\Big | \\sum _ { \\ell = 1 } ^ k ( \\widetilde U _ { \\ell , L } - U _ { \\ell , L } ) \\Big | \\Big \\Vert _ 2 ^ 2 \\ll 2 ^ { L - m ( L ) } \\ , . \\end{align*}"}
+{"id": "290.png", "formula": "\\begin{align*} \\prod _ { \\substack { l , m , n \\geq 1 \\\\ l , m \\leq n ; \\ , \\gcd ( l , m , n ) = 1 } } \\left ( 1 - z ^ n \\right ) ^ { \\frac { l m ^ 2 } { n ^ 4 } } = \\sqrt [ 3 ] { \\left ( 1 - z \\right ) } \\ ; \\exp \\left \\{ - \\frac { 1 } { 1 2 } \\left ( L i _ 2 ( z ) + \\frac { z ( 7 - 5 z ) } { ( 1 - z ) ^ 2 } \\right ) \\right \\} . \\end{align*}"}
+{"id": "9191.png", "formula": "\\begin{align*} h & = \\psi ( g ^ { 1 } , \\ldots , g ^ { k } ) \\end{align*}"}
+{"id": "1038.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\frac { ( a ) _ k ( c ) _ k ( d ) _ k ( 1 - b ) _ { k } ( 1 + a - b - c ) _ k ( 1 + a - b - d ) _ { k } } { ( 1 ) _ { k } ( 1 + a - c ) _ { k } ( 1 + a - d ) _ { k } ( 2 + a - b - c - d ) _ { k } } \\\\ [ 1 m m ] & \\quad \\times \\frac { ( - 1 ) ^ k } { ( 1 + a - b ) _ { 2 k } } \\beta _ k ( a , b , c , d ) \\\\ [ 1 m m ] & \\ : = \\frac { \\Gamma ( 1 + a - b ) \\Gamma ( 1 + a - c ) \\Gamma ( 1 + a - d ) \\Gamma ( 2 + a - b - c - d ) } { \\Gamma ( 1 + a ) \\Gamma ( 1 + a - b - c ) \\Gamma ( 1 + a - b - d ) \\Gamma ( 1 + a - c - d ) } , \\end{align*}"}
+{"id": "7188.png", "formula": "\\begin{align*} \\abs { \\nabla \\left ( \\mathcal { P } _ a \\circ w \\right ) } = \\abs { \\nabla \\left ( \\mathcal { P } \\circ ( w ( x ) - a ) \\right ) } = \\abs { ( \\nabla \\mathcal { P } ) ( w ( x ) - a ) } \\abs { \\nabla w ( x ) } . \\end{align*}"}
+{"id": "3652.png", "formula": "\\begin{align*} \\Omega ( m ^ { 0 . 5 } \\cdot s ^ { \\omega } \\cdot m ^ 2 ) = \\Omega ( s ^ { \\omega } \\cdot m ^ { 2 . 5 } ) , \\end{align*}"}
+{"id": "5597.png", "formula": "\\begin{align*} \\mathrm { I } ( X , \\mathcal { F } ) = \\int _ { \\Omega } \\int _ { S } \\log \\frac { d P \\left ( X | \\mathcal { F } \\right ) } { d P ( X ) } d P \\left ( X | \\mathcal { F } \\right ) d \\mathbb { P } . \\end{align*}"}
+{"id": "1860.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\langle D _ u \\theta ( u ^ * ) , v \\rangle _ \\mathcal { U } + \\langle \\lambda ^ { * } , S v \\rangle = 0 \\forall \\ v \\in \\mathcal { U } ; \\\\ & - \\langle \\lambda ^ { * } , \\zeta - S u ^ * \\rangle \\geq 0 \\forall \\ \\zeta \\in \\overline { K \\cap R ( S ) } , \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "7561.png", "formula": "\\begin{align*} \\mathbb { V } ( \\overline { S } _ n ) = \\mathbb { E } ( \\overline { S } _ n ^ 2 ) - \\mathbb { E } ( \\overline { S } _ n ) ^ 2 . \\end{align*}"}
+{"id": "8824.png", "formula": "\\begin{align*} v ( t ) = v _ 0 ( t ) + \\int _ 0 ^ t S ( t - s ) f ' ( u ( s ) ) v ( s ) \\ , d s + \\int _ 0 ^ t S ( t - s ) \\sigma ' ( u ( s ) ) v ( s ) B \\ , d W ( s ) , \\end{align*}"}
+{"id": "3903.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { \\ell < 2 } K _ { \\ell } \\right ) \\cap K _ { 2 } = K _ 1 \\cap K _ 2 = \\{ 3 , 5 \\} \\in \\bigcup _ { \\ell < 2 } 2 ^ { K _ { \\ell } } = 2 ^ { K _ 1 } . \\end{align*}"}
+{"id": "3279.png", "formula": "\\begin{align*} f ( t ) = t ^ { s } \\kappa ( t ) , \\end{align*}"}
+{"id": "1269.png", "formula": "\\begin{align*} Z _ q ( G ) \\le T + \\max _ { 1 \\le i _ 1 < \\ldots < i _ q \\le s } \\sum _ { j = 1 } ^ q a _ { i _ j } . \\end{align*}"}
+{"id": "3789.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D M R } } ( \\delta ) : = \\sup _ { \\gamma \\in \\Sigma _ { \\mathrm { D M R } } ( \\delta ) } \\int _ { \\mathcal { X } } f \\ , d \\gamma , \\mbox { } \\delta \\ge 0 , \\end{align*}"}
+{"id": "4831.png", "formula": "\\begin{align*} c l V _ n \\cap c l V _ m = \\emptyset , m \\neq n , \\ ; \\ ; \\min _ { c l V _ n } | f _ n | \\geq 1 - 1 / n . \\end{align*}"}
+{"id": "2852.png", "formula": "\\begin{align*} \\rho _ { \\beta } \\circ \\varphi _ M ( m ) = ( \\varphi _ M ( m ) \\otimes \\alpha ) \\rho _ { \\beta } ( m ) + ( \\beta \\otimes \\varphi _ L ) \\rho _ { \\beta } ( m ) , m \\in M . \\end{align*}"}
+{"id": "8366.png", "formula": "\\begin{align*} \\frac { u _ 1 } { c ( v _ 1 ) } + \\cdots + \\frac { u _ n } { c ( v _ n ) } = 0 . \\end{align*}"}
+{"id": "6219.png", "formula": "\\begin{align*} g _ 2 ( x ) = \\frac { | x | ^ \\alpha } \\alpha - \\frac { | x | ^ \\beta } \\beta , \\qquad \\alpha > \\beta > 0 \\ , . \\end{align*}"}
+{"id": "2107.png", "formula": "\\begin{align*} F ( A ) = & \\left ( ( 2 m + 1 ) 2 ^ { k - 1 } - 1 \\right ) ^ 2 + ( d - m ) \\left ( ( 2 m + 1 ) 2 ^ { k - 1 } - 1 \\right ) - m d - d , \\\\ g ( A ) = & 2 ^ { k - 1 } ( 2 ^ k - 1 ) m ^ 2 + \\frac { 1 } { 2 } ( d - 1 ) ( 2 ^ k - 1 ) m + ( 2 ^ { 2 k - 1 } + k 2 ^ { k - 1 } - 2 ^ { k + 1 } ) m \\\\ & \\ \\ + 2 ^ { 2 k - 3 } + ( d + k ) 2 ^ { k - 2 } - 5 \\cdot 2 ^ { k - 2 } - d + 1 . \\end{align*}"}
+{"id": "5839.png", "formula": "\\begin{align*} \\theta _ j ^ y = X ^ { \\mathcal { X } _ j ^ y , \\mathcal { N } _ j ^ y } _ { T _ c ( X ^ { \\mathcal { X } _ j ^ y , \\mathcal { N } _ j ^ y } ) } \\end{align*}"}
+{"id": "7725.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\varepsilon \\in F ' / K \\\\ \\pi _ { K _ 1 } ( \\varepsilon ) = \\lambda } } \\sum _ { \\substack { \\mu \\in L ' / K \\\\ p _ K ( \\mu ) = \\varepsilon } } c ( m + q ( \\mu _ { K _ \\R ^ \\perp } ) , \\mu ) = \\sum _ { \\substack { \\gamma \\in L _ 1 ' \\\\ \\gamma | _ { K _ 1 } = \\lambda } } \\sum _ { \\substack { \\delta \\in M ' / L \\\\ \\pi ( \\delta ) = \\gamma } } c ( m + q ( \\gamma _ { ( K _ 1 ) _ \\R ^ \\perp } ) , \\delta ) \\end{align*}"}
+{"id": "8123.png", "formula": "\\begin{align*} f ( T ) = \\sum _ { F = F _ 1 F _ 2 } f ( F _ 1 ) B _ + ( f ( F _ 2 ) ) . \\end{align*}"}
+{"id": "1078.png", "formula": "\\begin{align*} \\Phi [ u ] ( t ) = e ^ { - t H ^ \\beta } u _ 0 + \\int _ 0 ^ t e ^ { - ( t - \\tau ) H ^ \\beta } \\left ( f ( u ( \\tau ) ) \\right ) \\ , d \\tau . \\end{align*}"}
+{"id": "5230.png", "formula": "\\begin{align*} \\begin{aligned} f ( \\rho ) & = \\rho ( \\rho + 1 ) \\cdots ( \\rho + 5 ) - p _ { 5 1 } \\rho ( \\rho + 1 ) \\cdots ( \\rho + 4 ) \\\\ & + p _ { 4 2 } \\rho ( \\rho + 1 ) \\cdots ( \\rho + 3 ) - p _ { 3 3 } \\rho ( \\rho + 1 ) ( \\rho + 2 ) \\\\ & + p _ { 2 2 } \\rho ( \\rho + 1 ) - p _ { 1 1 } \\rho + p _ 0 \\end{aligned} \\end{align*}"}
+{"id": "5357.png", "formula": "\\begin{align*} r '' : = \\sqrt { \\frac { d - 1 } { d } } r ' \\le r ' . \\end{align*}"}
+{"id": "3595.png", "formula": "\\begin{align*} q ( n + 1 ) \\ = \\ \\frac { 1 0 ^ n } { n + 2 } \\ = \\ \\frac { 1 0 ^ { n - 1 } } { n + 1 } \\cdot \\frac { 1 0 ( n + 1 ) } { n + 2 } \\ = \\ q ( n ) \\cdot \\frac { 1 0 ( n + 1 ) } { n + 2 } . \\end{align*}"}
+{"id": "9118.png", "formula": "\\begin{align*} \\delta ^ { - 1 } ( h ( \\dots , \\zeta _ { [ - 1 ] } , x , u , u _ { [ 1 ] } , u _ { [ 2 ] } , \\dots ) ) = h ( \\dots , \\zeta _ { [ - 2 ] } , \\psi _ { x } ( x , \\zeta _ { [ - 1 ] } ) , \\psi _ { u } ( x , \\zeta _ { [ - 1 ] } ) , u , u _ { [ 1 ] } , \\dots ) \\ , , \\end{align*}"}
+{"id": "2575.png", "formula": "\\begin{align*} 0 = \\int _ U \\nabla u \\cdot \\nabla ( w ^ t - w ^ 0 ) \\d x - \\int _ { U } f ( u ) ( w ^ t - w ^ 0 ) \\d x = : I _ 1 ( t ) - I _ 2 ( t ) , \\ \\ \\ t \\in [ 0 , 1 / \\kappa ) . \\end{align*}"}
+{"id": "1417.png", "formula": "\\begin{align*} V _ d ( r ) \\geq \\omega _ d \\int _ { a } ^ r \\sinh ( t ) ^ { d - 1 } d t & \\geq \\frac { \\omega _ d } { 2 \\cdot 2 ^ { d - 1 } } \\int _ { a } ^ r e ^ { t ( d - 1 ) } d t = \\frac { \\omega _ d } { 2 ( d - 1 ) 2 ^ { d - 1 } } [ e ^ { r ( d - 1 ) } - e ^ { a ( d - 1 ) } ] \\\\ & \\geq \\frac { \\omega _ d } { 4 ( d - 1 ) 2 ^ { d - 1 } } e ^ { r ( d - 1 ) } . \\end{align*}"}
+{"id": "8723.png", "formula": "\\begin{align*} & E \\{ ( \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { s } \\} + E \\{ ( \\| Y _ { 1 } - Y _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { s } \\} - 2 E \\{ ( \\| X _ { 1 } - Y _ { 1 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { s } \\} \\\\ & = \\sum ^ { s } _ { a = 0 } \\binom { s } { a } ( - p \\tau ) ^ { s - a } \\left ( E \\| X _ 1 - X _ 2 \\| ^ { 2 a } _ 2 + E \\| Y _ 1 - Y _ 2 \\| ^ { 2 a } _ 2 - 2 E \\| X _ 1 - Y _ 1 \\| ^ { 2 a } _ 2 \\right ) . \\end{align*}"}
+{"id": "7101.png", "formula": "\\begin{align*} & f \\in C ^ 0 ( [ - 1 , 1 ] ) \\cap C ^ 1 ( - 1 , 1 ) , \\\\ & \\lim \\limits _ { s \\rightarrow - 1 } f ^ \\prime ( s ) = - \\infty , \\ ; \\ ; \\ ; \\ ; \\lim \\limits _ { s \\rightarrow 1 } f ^ \\prime ( s ) = + \\infty , \\ ; \\ ; \\ ; \\ ; f ^ { \\prime \\prime } ( s ) \\geq \\alpha > 0 . \\end{align*}"}
+{"id": "5770.png", "formula": "\\begin{align*} | \\Omega ^ { * } | & \\le \\sum _ { j = 1 } ^ { m } | \\Omega _ { j } ^ { * } | \\le C \\sum _ { j = 1 } ^ { m } \\sum _ { k _ { j } } | Q _ { j , k _ { j } } | \\le C ( \\lambda \\gamma ) ^ { \\frac { - n } { m n - \\alpha } } . \\end{align*}"}
+{"id": "4251.png", "formula": "\\begin{align*} y ^ { i } \\left ( t \\right ) = \\xi ^ { i } + \\int _ { t } ^ { T } f ^ { i } \\left ( s , y _ { s } ^ { i } , z _ { s } ^ { i } \\right ) \\mathrm { d } s - \\int _ { t } ^ { T } z _ { s } ^ { i } \\mathrm { d } W _ { s } , \\end{align*}"}
+{"id": "7347.png", "formula": "\\begin{align*} \\left \\| \\frac { 1 } { m + 1 } \\sum _ { i = 0 } ^ m w _ { n + i } - w \\right \\| & < C \\left [ \\left \\| \\frac { 1 } { m + 1 } \\sum _ { i = 0 } ^ m x _ { n + i } - x \\right \\| + \\left \\| \\frac { 1 } { m + 1 } \\sum _ { i = 0 } ^ m y _ { n + i } - y \\right \\| \\right . \\\\ & + \\left . \\left \\| \\frac { 1 } { m + 1 } \\sum _ { i = 0 } ^ m z _ { n + i } - z \\right \\| \\right ] . \\end{align*}"}
+{"id": "5321.png", "formula": "\\begin{align*} \\mathbb { E } ^ 2 [ f ] \\left ( \\frac { C r ^ 2 \\log ( 1 / \\mathbb { E } [ f ] ) } { t } \\right ) ^ t = p ^ { 2 t } \\left ( C ' p ^ { - 2 } \\log ( 1 / p ) \\right ) ^ t = \\left ( C ' \\log ( 1 / p ) \\right ) ^ t , \\end{align*}"}
+{"id": "7989.png", "formula": "\\begin{align*} W = h \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\end{align*}"}
+{"id": "3267.png", "formula": "\\begin{align*} \\| f \\| _ { \\alpha } = \\max _ { z \\in G } | f ( z ) | + \\sup _ { z , w \\in G , z \\ne w } \\frac { | f ( z ) - f ( w ) | } { | z - w | ^ { \\alpha } } \\end{align*}"}
+{"id": "4002.png", "formula": "\\begin{align*} \\frac { \\partial \\mathrm { R } _ { \\mathrm { D } } ( \\lambda _ { \\mathrm { D } } ^ \\star , \\delta ) } { \\partial \\lambda _ \\ell } = 0 . \\end{align*}"}
+{"id": "2813.png", "formula": "\\begin{align*} S ( \\mathfrak M ( \\lambda , L 2 ) ) ( t ) = ( S ^ { 1 , 1 } _ k , \\dots , S ^ { 1 , q } _ k ) _ { k = 0 } ^ \\infty ; S _ k ^ { 1 , n } = ( L 2 ^ { \\ , k } ) _ { 0 , \\ , n - 1 } . \\end{align*}"}
+{"id": "9076.png", "formula": "\\begin{align*} \\lambda _ { m } = \\frac { \\langle g , M g \\rangle } { \\langle g , g \\rangle } = \\lambda _ { k } . \\end{align*}"}
+{"id": "3269.png", "formula": "\\begin{align*} \\overline { a } = r e ^ { h } , \\end{align*}"}
+{"id": "6964.png", "formula": "\\begin{align*} \\epsilon ( f ) : = \\max \\{ \\overline \\nu ( x - a ) \\mid a \\mbox { i s a r o o t o f } f \\} . \\end{align*}"}
+{"id": "2348.png", "formula": "\\begin{align*} b : = \\begin{cases} \\frac { b _ 1 } { b _ 2 } , & \\frac { d } { 4 } + \\frac { 1 } { 2 } \\leq \\alpha < \\frac { d } { 2 } + 1 , \\\\ 1 , & \\alpha \\geq \\frac { d } { 2 } + 1 . \\end{cases} . \\end{align*}"}
+{"id": "2352.png", "formula": "\\begin{align*} \\prod _ { x : X } f ( x ) ^ { k } ( m _ g ( x ) f ( x ) ^ { k _ h } - m _ h ( x ) f ( x ) ^ { k _ g } ) = 0 \\end{align*}"}
+{"id": "3062.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\frac { g _ j ( s ) } { s ^ { k _ j } } = 1 \\end{align*}"}
+{"id": "6614.png", "formula": "\\begin{align*} \\varphi _ { \\lambda + \\mu } ( [ x _ \\mu y ] , z ) = [ x _ \\mu \\varphi _ { \\lambda } ( y , z ) ] - ( - 1 ) ^ { | x | | y | } [ y _ { \\lambda } \\varphi _ { \\mu } ( x , z ) ] , \\end{align*}"}
+{"id": "3213.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\norm { S _ n ' ( y ' - \\sigma _ k ' ( y ' ) ) } = 0 . \\end{align*}"}
+{"id": "1405.png", "formula": "\\begin{align*} \\sharp \\{ p \\leq r \\leq q \\ , | \\ , D _ { \\lambda } ( r ) = \\times \\} \\geq \\sharp \\{ p \\leq r \\leq q \\ , | \\ , D _ { \\lambda } ( r ) = \\circ \\} \\end{align*}"}
+{"id": "8452.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { \\left ( \\left ( 2 n - 1 \\right ) ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } = \\frac { \\sqrt { \\pi } \\ , x ^ { 1 - 2 s } } { 4 \\Gamma ( s ) } \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) + \\frac { 2 ^ { \\frac { 1 } { 2 } - s } \\ , x ^ { \\frac { 1 } { 2 } - s } \\ , \\pi ^ { s } } { \\Gamma ( s ) } \\ , \\sum _ { m = 1 } ^ { \\infty } ( - 1 ) ^ { m } m ^ { s - \\frac { 1 } { 2 } } K _ { s - \\frac { 1 } { 2 } } ( \\pi x m ) . \\end{align*}"}
+{"id": "303.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } \\sum _ { n = 1 } ^ { \\infty } ( + x ^ { n + 1 } y ^ { n + 1 } - n x ^ { n + 1 } y ^ { n + 2 } + x ^ { n + 1 } y ^ { n + 2 } ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "2981.png", "formula": "\\begin{align*} \\theta ( ( F ( t _ i ) - F ( t _ { i - 1 } ) ) p _ i ) = \\pm \\theta ( ( F ( t _ 2 ) - F ( t _ { 1 } ) ) ( \\sigma _ i \\ast p _ i ) ) \\ . \\end{align*}"}
+{"id": "1302.png", "formula": "\\begin{align*} T _ i ( u ) = \\int _ 0 ^ u e ^ { 2 \\rho _ i ( v ) } d v \\xrightarrow [ u \\to \\infty ] { a . s . } T _ i ^ { \\infty } , \\end{align*}"}
+{"id": "6008.png", "formula": "\\begin{align*} \\mathcal { F } ' _ { X X ^ { \\ast } } : & L ^ 2 ( X ) \\longrightarrow L ^ 2 ( X ^ { \\ast } ) ; \\\\ & f \\longmapsto \\mathcal { F } ' _ { X X ^ { \\ast } } ( f ) ( x ^ { \\ast } ) = \\int _ { X } \\psi ( \\langle x , x ^ { \\ast } \\rangle ) f ( x ) d x , \\end{align*}"}
+{"id": "2576.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { M _ 1 ( t ) } | \\nabla u | ^ 2 \\d x - \\frac { 1 } { 2 } \\int _ { M _ 2 ( t ) } | \\nabla u | ^ 2 \\d x \\le \\frac { 1 } { 2 } \\int _ { M _ 1 ( t ) } ( | \\nabla u | ^ 2 - c ^ 2 ) \\d x - \\frac { 1 } { 2 } \\int _ { M _ 2 ( t ) } ( | \\nabla u | ^ 2 - c ^ 2 ) \\d x = o ( t ) . \\end{align*}"}
+{"id": "3872.png", "formula": "\\begin{align*} c _ { \\ell } ( s _ \\ell , s _ \\ell ' ) = | y _ \\ell - y _ \\ell ' | + \\| x _ { \\ell } - x _ \\ell ' \\| _ { 2 } , \\end{align*}"}
+{"id": "7036.png", "formula": "\\begin{align*} \\mathcal I _ 1 = \\left ( \\left . b _ { \\ell i } \\left ( X _ \\ell - \\sum \\limits _ { j = 1 } ^ { s _ { \\ell i } } b _ { \\ell i j } { \\textbf { X } } ^ { \\lambda _ j } \\right ) \\ \\right | \\ i , \\ell \\in I \\setminus \\{ i _ { } \\} , i < \\ell \\right ) . \\end{align*}"}
+{"id": "4339.png", "formula": "\\begin{gather*} \\theta : \\Omega \\times [ 0 , 3 \\cdot 2 ^ { - k } ] \\to \\mathbb { R } , \\\\ \\theta ( ( x _ 1 , x _ 2 ) , t ) = 2 \\left \\| f _ 0 \\right \\| _ { L ^ \\infty } C _ 0 ^ { - 1 } \\left ( \\mathrm { e r f } \\left ( \\frac { - x _ 1 } { \\sqrt { 4 \\nu t } } \\right ) + \\mathrm { e r f } \\left ( \\frac { x _ 1 - a } { \\sqrt { 4 \\nu t } } \\right ) \\right ) . \\end{gather*}"}
+{"id": "8252.png", "formula": "\\begin{align*} \\sum _ { h = 2 } ^ { j - 1 } ( - 1 ) ^ { h } S _ { h - 1 } ^ { ( 1 ) } H _ { k , Y _ { l - h , j - h } } = ( - 1 ) ^ { j } \\sum _ { h = 1 } ^ { k } ( - 1 ) ^ { h } ( \\alpha + k - 1 + l - h ) H _ { k , Y _ { l - 1 , h } } . \\end{align*}"}
+{"id": "6287.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\dot \\lambda _ t = \\vec h _ { \\bar u ( t ) } ^ \\nu ( \\lambda _ t ) & t \\in [ 0 , 1 ] , \\\\ \\displaystyle h _ { \\bar u ( t ) } ^ \\nu ( \\lambda _ t ) = \\max _ { v \\in \\R ^ k } h _ { v } ^ \\nu ( \\lambda _ t ) & t \\in [ 0 , 1 ] , \\\\ \\nu \\leq 0 . & \\end{cases} \\end{align*}"}
+{"id": "7911.png", "formula": "\\begin{align*} v ^ m : = \\prod _ { i \\in \\mathsf { I } } v ( i ) ^ { m ( i ) } . \\end{align*}"}
+{"id": "8758.png", "formula": "\\begin{align*} \\beta _ { i , k } : = \\begin{cases} 0 & \\mbox { i f } d _ { i , k } - x _ i \\leq 0 \\\\ c \\P ( \\omega _ k ) \\frac { x _ i - d _ { i , k } } { \\bar { y } } & \\mbox { i f } 0 \\leq d _ { i , k } - x _ i \\leq \\bar { y } \\\\ - c \\P ( \\omega _ k ) & \\mbox { i f } d _ { i , k } - x _ i \\geq \\bar { y } . \\end{cases} \\end{align*}"}
+{"id": "2796.png", "formula": "\\begin{align*} \\mathbf { n } _ \\lambda = \\lambda ^ { 7 / 2 } \\mathbf { e } _ \\lambda . \\end{align*}"}
+{"id": "7629.png", "formula": "\\begin{align*} \\begin{cases} h _ 1 ( c , x ) : = & x ^ 2 ( 4 - c ^ 2 ) ^ 2 ( 2 c ^ 2 + ( 3 6 - 1 3 c ^ 2 ) x ) + 2 c ^ 2 x ^ 2 ; \\\\ h _ 2 ( c , x ) : = & 8 c x ( 4 - c ^ 2 ) ^ 2 ( 1 + x ) ( 1 - x ^ 2 ) ; \\\\ h _ 3 ( c , x ) : = & 8 ( 4 - c ^ 2 ) ^ 2 ( 8 + x ^ 2 ) ( 1 - x ^ 2 ) ; \\\\ h _ 4 ( c , x ) : = & 7 2 x ( 4 - c ^ 2 ) ^ 2 ( 1 - x ^ 2 ) . \\end{cases} \\end{align*}"}
+{"id": "401.png", "formula": "\\begin{align*} W ^ T \\Lambda W = \\begin{bmatrix} \\sqrt { \\Lambda ^ + } W ^ + \\\\ G \\end{bmatrix} ^ T \\begin{bmatrix} I - R ^ T R & - R ^ T S \\\\ - S ^ T R & I - S ^ T S \\end{bmatrix} \\begin{bmatrix} \\sqrt { \\Lambda ^ + } W ^ + \\\\ G \\end{bmatrix} - G ^ T G . \\end{align*}"}
+{"id": "5687.png", "formula": "\\begin{align*} f ^ { - } ( z ) : = \\sum _ { n < 0 } C _ { h } ^ { - } ( n ) \\Gamma ( k - 1 , 4 \\pi \\left | n \\right | y ) q ^ { n } . \\end{align*}"}
+{"id": "3396.png", "formula": "\\begin{align*} \\Gamma _ k : = \\begin{cases} 1 , & k = 1 , \\\\ ( 1 - \\alpha _ k ) \\Gamma _ { k - 1 } , & k \\ge 2 . \\end{cases} \\end{align*}"}
+{"id": "6130.png", "formula": "\\begin{align*} p _ { h } = 2 \\left ( 1 + \\frac { s } { n } \\right ) ^ { h } , \\quad \\mbox { f o r } h = 0 , 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "9134.png", "formula": "\\begin{align*} \\begin{array} { c c c c l } v ^ { 1 } & = & \\varphi ^ { 1 } & = & x ^ { 1 } \\\\ v ^ { 2 } & = & \\varphi ^ { 2 } & = & x ^ { 2 } \\ , , \\end{array} \\end{align*}"}
+{"id": "6681.png", "formula": "\\begin{align*} T _ g ( u ) & : = \\int _ \\Omega \\langle \\nabla g , \\nabla u \\rangle \\ , d x - \\frac { C _ { N , s } } { 2 } \\iint _ { \\R ^ { 2 N } } \\frac { ( g ( x ) - g ( y ) ) ( \\hat { u } ( x ) - \\hat { u } ( y ) ) } { | x - y | ^ { N + 2 s } } \\ , d x \\ , d y \\\\ & \\qquad - \\int _ \\Omega V ( x ) g u \\ , d x - \\int _ \\Omega f u \\ , d x , \\end{align*}"}
+{"id": "5348.png", "formula": "\\begin{align*} q ( r ' ) ^ { - 2 / q } = \\sqrt { q } \\sqrt { q \\left ( \\frac { 1 } { ( r ' ) ^ 2 } \\right ) ^ { 2 / q } } \\le \\sqrt { 2 } \\sqrt { q } \\sqrt { \\log ( 1 / \\gamma ) } . \\end{align*}"}
+{"id": "7326.png", "formula": "\\begin{align*} & A = \\pm \\prod _ { p \\in { \\cal P } _ 3 } p ^ { a _ p - b _ p + \\sum _ { i = 1 } ^ n ( \\alpha _ i - \\beta _ i ) z _ { i p } ^ 0 } , \\\\ & B = \\pm \\prod _ { p \\in { \\cal P } _ 2 } p ^ { b _ p - c _ p + \\sum _ { i = 1 } ^ n ( \\beta _ i - \\gamma _ i ) z _ { i p } ^ 0 } , \\\\ & C = \\pm \\prod _ { p \\in { \\cal P } _ 1 } p ^ { c _ p - a _ p + \\sum _ { i = 1 } ^ n ( \\gamma _ i - \\alpha _ i ) z _ { i p } ^ 0 } . \\end{align*}"}
+{"id": "2907.png", "formula": "\\begin{align*} U L ( x ) & = \\begin{bmatrix} x & 0 \\\\ 0 & 1 \\end{bmatrix} & L R ( x ) & = \\begin{bmatrix} 1 & 0 \\\\ 0 & x \\end{bmatrix} & D ( x ) & = \\begin{bmatrix} x & 0 \\\\ 0 & x \\end{bmatrix} . \\end{align*}"}
+{"id": "4899.png", "formula": "\\begin{align*} R ^ \\times = \\mathfrak n \\times K ^ \\times . \\end{align*}"}
+{"id": "6922.png", "formula": "\\begin{align*} \\sum _ { m , v \\in \\mathcal { M } ' } \\sum _ { p \\in P } \\frac { r ( m ) r ( v ) } { p ^ { \\sigma } } \\Phi \\left ( T \\log \\frac { m p } { v } \\right ) \\ge \\sum _ { p \\in P } \\sum _ { \\substack { m , v \\in \\mathcal { M } \\\\ m p = v } } \\frac { f ( m ) f ( v ) } { p ^ { \\sigma } } = \\sum _ { v \\in \\mathcal { M } } f ( v ) ^ 2 \\sum _ { p | v } \\frac { 1 } { f ( p ) p ^ { \\sigma } } . \\end{align*}"}
+{"id": "2710.png", "formula": "\\begin{align*} T _ 6 = - \\frac { s } { 2 } \\iint _ Q \\sigma _ { t t } | u | ^ 2 d x d t \\geq - C s \\iint _ Q \\xi ^ { 3 / 2 } | u | ^ 2 d x d t . \\end{align*}"}
+{"id": "3924.png", "formula": "\\begin{align*} \\sup _ { \\pi \\in \\mathcal { G } _ { \\mathrm { D } , \\lambda } } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda d \\pi = \\sup _ { \\pi \\in \\Gamma \\left ( \\Pi ( \\mu _ 1 , \\mu _ 2 ) , \\varphi _ \\lambda \\right ) } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ { \\lambda } d \\pi . \\end{align*}"}
+{"id": "3640.png", "formula": "\\begin{align*} r ( p ) \\ = \\ r ( 4 \\underbrace { 9 \\cdots 9 } _ { m } ) \\ = \\ \\underbrace { 9 \\cdots 9 } _ { m } 4 \\ = \\ 2 \\cdot 4 \\underbrace { 9 \\cdots 9 } _ { m - 1 } 7 \\ = \\ 2 ( p - 2 ) . \\end{align*}"}
+{"id": "5388.png", "formula": "\\begin{align*} \\rho ' & = f ( S ) + d _ { G - S } ( T ) + \\gamma \\C T - q ' ( S ' , T ' ) \\\\ & = \\rho + q ( S , T ) + \\gamma \\C T - q ' ( S ' , T ' ) ~ \\ge ~ f ( T ) + \\gamma ( \\C T - 1 ) + q ( S , T ) - q ' ( S ' , T ' ) \\\\ & = f ' ( T ' ) + \\gamma ( \\C T - 1 ) - q ' ( S ' , T ' ) + q ( S , T ) , \\end{align*}"}
+{"id": "4548.png", "formula": "\\begin{align*} \\Phi ^ { \\lambda } ( x ) : = \\lambda ^ { \\frac { d } { 2 } } \\Phi ( \\lambda x ) . \\end{align*}"}
+{"id": "4625.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\sqrt { f ( \\chi ^ { - 1 } ( u - 1 ) ) } \\ , d u = \\int _ 0 ^ { \\infty } \\sqrt { f ( y ) } \\chi ' ( y ) \\ , d y \\\\ < \\int _ 0 ^ { \\infty } f ( y - 1 ) \\ , d y \\le V + 1 . \\end{align*}"}
+{"id": "2063.png", "formula": "\\begin{align*} \\check { \\psi } \\left ( x \\right ) = \\frac { 1 } { ( 2 \\pi \\lambda ) ^ d } \\sum _ { k \\in \\mathbb Z _ \\lambda ^ d } \\psi ( k ) e ^ { i k \\cdot x } . \\end{align*}"}
+{"id": "5828.png", "formula": "\\begin{align*} \\left \\{ \\hat { X } _ { n + 1 } ^ { y , \\Gamma } = \\hat { X } _ n ^ { y , \\Gamma } + 1 \\right \\} \\subseteq \\left \\{ X _ { n + 1 } ^ { y , \\Gamma } = X _ n ^ { y , \\Gamma } + e _ 2 \\right \\} . \\end{align*}"}
+{"id": "3444.png", "formula": "\\begin{align*} \\hat { g } = & \\ , \\frac { x ^ { n - 1 } d x ^ 2 } { 2 P ( x ) } + \\frac { 2 P ( x ) } { x ^ { n - 1 } } ( d \\phi + \\bar { \\sigma } ) ^ 2 + 2 x \\bar { g } , \\\\ \\hat { J } = & \\ , d \\left [ x ( d \\phi + \\bar \\sigma ) \\right ] . \\end{align*}"}
+{"id": "4456.png", "formula": "\\begin{align*} \\| \\Phi _ { \\omega , \\mathbf { c } } \\| _ { \\mathcal { H } ^ 1 } ^ 2 \\le C L _ { \\omega , \\mathbf { c } } ( \\Phi _ { \\omega , \\mathbf { c } } ) = 6 C \\mu _ { \\omega , \\mathbf { c } } . \\end{align*}"}
+{"id": "5513.png", "formula": "\\begin{align*} \\mathcal { I } : = \\left \\{ A \\in \\mathcal { B } ( W _ { \\Omega } ) : \\mathcal { S } ^ { - 1 } ( A ) = A \\right \\} . \\end{align*}"}
+{"id": "7305.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\alpha _ i z _ i = \\sum _ { i = 1 } ^ n \\beta _ i z _ i = \\sum _ { i = 1 } ^ n \\gamma _ i z _ i - 1 , z _ i \\geq 0 , i = 1 , \\dots , n , \\end{align*}"}
+{"id": "3001.png", "formula": "\\begin{align*} \\rho ^ f _ N ( t ) = \\Pi _ 0 ^ { i / N } \\exp f ( s ) \\ , d s , \\mbox { i f } \\frac i N \\leq t < \\frac { i + 1 } N . \\end{align*}"}
+{"id": "1622.png", "formula": "\\begin{align*} \\bar f ( X , \\sum a ^ i \\partial _ i ) = ( f X - \\sum a ^ i \\xi _ i , \\sum \\eta ^ j ( X ) \\partial _ j ) , \\bar Q ( X , \\sum a ^ i \\partial _ i ) = ( Q X , \\sum a ^ i \\partial _ i ) . \\end{align*}"}
+{"id": "6658.png", "formula": "\\begin{align*} G _ \\alpha ( r ) : = ( r ^ 2 + \\alpha ) ^ { \\frac p 2 } \\textrm { f o r a l l } \\ ; \\ ; r \\in \\R \\ , . \\end{align*}"}
+{"id": "4484.png", "formula": "\\begin{align*} Q ( U ( t ) ) = Q ( U _ 0 ) , \\ \\ E ( U ( t ) ) = E ( U _ 0 ) , \\ \\ \\mathbf { P } ( U ( t ) ) = \\mathbf { P } ( U _ 0 ) \\end{align*}"}
+{"id": "625.png", "formula": "\\begin{align*} \\limsup _ \\lambda \\| e _ \\lambda ^ * ( b - \\omega ( b ) I ) e _ \\lambda \\| & = \\limsup _ \\lambda \\| e _ \\lambda ^ * ( s ^ * + t ) e _ \\lambda \\| \\\\ & \\leq \\limsup _ \\lambda ( \\| e _ \\lambda s ^ * \\| + \\| t e _ \\lambda \\| ) \\\\ & = \\limsup _ \\lambda ( \\| s e _ \\lambda \\| + \\| t e _ \\lambda \\| ) \\\\ & = 0 . \\end{align*}"}
+{"id": "7721.png", "formula": "\\begin{align*} & v _ 0 \\left ( \\frac { z + z ' - q ( z ' ) z } { 2 } \\right ) = e _ 1 , v _ 0 ( ( K _ 1 ) _ \\R ) = \\langle e _ 2 , e _ 3 \\rangle , \\\\ & v _ 0 ( L _ 1 ( \\R ) ) = v _ 0 ( ( K _ 1 ) _ \\R ) \\oplus \\langle e _ 4 \\rangle , v _ 0 \\left ( \\frac { z - z ' + q ( z ' ) z } { 2 } \\right ) = e _ 5 , \\end{align*}"}
+{"id": "8593.png", "formula": "\\begin{align*} \\hat { S } _ { j , + } ( t ) : = \\sum _ { k = 1 } ^ { \\infty } \\hat { S } ^ k _ { j , + } ( t ) , \\hat { S } _ { j , - } ( t ) : = \\sum _ { k = 1 } ^ { \\infty } \\hat { S } ^ k _ { j , - } ( t ) . \\end{align*}"}
+{"id": "7636.png", "formula": "\\begin{align*} M ( 0 , x , 1 ) = 1 0 2 4 - 8 9 6 x ^ 2 + 5 7 6 x ^ 3 - 1 2 8 x ^ 4 \\leq 1 0 2 4 , \\ ; x \\in ( 0 , 1 ) \\end{align*}"}
+{"id": "683.png", "formula": "\\begin{align*} c & | \\log ( y - l _ \\alpha ) - \\log ( x - l _ \\alpha ) | = \\int _ x ^ { y } \\frac { c } { s - l _ \\alpha } d s \\leq \\Big | \\int _ x ^ { y } D ^ { n + 1 } \\varphi ( s ) d s \\Big | \\\\ & \\leq | D ^ { n } \\varphi ( x ) - D ^ { n } \\varphi ( y ) | \\leq \\| D ^ { n } \\varphi \\| _ { C ^ { \\tau } } | ( y - l _ \\alpha ) - ( x - l _ \\alpha ) | ^ { \\tau } . \\end{align*}"}
+{"id": "7521.png", "formula": "\\begin{align*} g _ n ( x ) = \\left \\{ \\begin{array} { c l } x & \\mbox { i f } x \\in I ^ n \\\\ x _ 0 & \\mbox { i f } x \\notin I ^ n \\end{array} , \\right . \\end{align*}"}
+{"id": "5237.png", "formula": "\\begin{gather*} A \\circ B = ( A \\circ f ) \\circ ( f ^ { - 1 } \\circ B ) , \\ f \\in \\mathbb { C } ( x ) , \\ f \\not = 0 , \\\\ \\partial ^ 2 = \\left ( \\partial + \\frac 1 { x - c } \\right ) \\circ \\left ( \\partial - \\frac 1 { x - c } \\right ) , \\ c \\in \\mathbb { C } . \\end{gather*}"}
+{"id": "495.png", "formula": "\\begin{align*} p ( \\bar { H } _ { ( m , n ) } | \\gamma _ { ( m , n ) } ) = \\mathcal { C N } ( \\bar { H } _ { ( m , n ) } ; 0 , \\gamma _ { ( m , n ) } ^ { - 1 } ) . \\end{align*}"}
+{"id": "6434.png", "formula": "\\begin{align*} \\theta ( a _ 1 \\cdots a _ k ) = \\theta ( a _ 1 ) \\cdots \\theta ( a _ k ) , \\end{align*}"}
+{"id": "8047.png", "formula": "\\begin{align*} | A ' | ^ 2 = V ' | B | ^ 2 V '^ * + W ' | B | ^ 2 W '^ * + X ' | B | ^ 2 X '^ * \\end{align*}"}
+{"id": "5292.png", "formula": "\\begin{align*} \\langle A h , g \\rangle _ { \\mu _ p } = 0 . \\end{align*}"}
+{"id": "4743.png", "formula": "\\begin{align*} A \\tilde { S } = J _ { 2 n } \\tilde { S } J _ { 2 n } ^ T A + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "5083.png", "formula": "\\begin{align*} A _ t ( x ) = x - \\gamma ( x , t ) - \\frac { 1 } { N } \\sigma ( t ) , \\end{align*}"}
+{"id": "6797.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = - x _ 1 ^ 2 + \\cdots + - x _ { k - 1 } ^ 2 + - x _ k ^ 4 + x _ { k + 1 } ^ 2 + \\cdots + x _ { 2 n } ^ 2 , \\end{align*}"}
+{"id": "232.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ 3 = \\frac { n ^ 2 } { 4 } + \\frac { n ^ 3 } { 3 } + \\frac { n ^ 4 } { 4 } , \\end{align*}"}
+{"id": "5216.png", "formula": "\\begin{align*} ( 1 - \\rho ) f _ 0 ( t ) + \\rho \\sum _ { r = 1 } ^ \\infty \\zeta _ { d , r } f _ r ( t ) \\end{align*}"}
+{"id": "5722.png", "formula": "\\begin{align*} 1 = \\langle D ^ { k - 1 } ( f ) , g \\rangle = \\langle c F , g \\rangle = c \\langle F , g \\rangle = c . \\end{align*}"}
+{"id": "7904.png", "formula": "\\begin{align*} n ! ! = \\begin{cases} n \\times ( n - 2 ) \\times \\ldots \\times 3 \\times 1 & \\mbox { i f $ n $ i s o d d } \\\\ n \\times ( n - 2 ) \\times \\ldots \\times 4 \\times 2 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
+{"id": "2806.png", "formula": "\\begin{align*} \\int _ \\mathrm { M } ( q _ 2 - q _ 1 ) v _ 1 v _ 2 d x = - \\int _ \\mathrm { M } [ \\Delta , \\psi ] v v _ 1 d x . \\end{align*}"}
+{"id": "5265.png", "formula": "\\begin{align*} D _ { S , x } T _ \\rho f = \\sum _ { T \\supseteq S } \\rho ^ { | S | } T _ \\rho D _ { S , x } [ f ^ { = T } ] = \\rho ^ { | S | } \\sum _ { T \\subseteq [ n ] } T _ \\rho D _ { S , x } [ f ^ { = T } ] = \\rho ^ { | S | } T _ \\rho D _ { S , x } f \\end{align*}"}
+{"id": "3018.png", "formula": "\\begin{align*} \\sum _ { i = \\ell + 1 } ^ { k } s _ i ^ p ( A ) = 0 \\hbox { a n d } \\displaystyle \\sum _ { i = k - \\ell + 1 } ^ { k } s _ k ^ p ( B ) = 0 , \\end{align*}"}
+{"id": "63.png", "formula": "\\begin{align*} \\Omega ( \\ell _ 1 ) \\left ( Z ( \\mathcal { G } ( \\ell _ 1 ) ) ( \\mathbb { Q } _ p ) \\mathcal { G } ( \\ell _ 1 ) ( \\mathbb { Q } ) \\right ) = \\mathbb { Q } _ p ^ \\times \\Omega ( \\ell _ 1 ) \\left ( \\mathcal { G } ( \\ell _ 1 ) ( \\mathbb { Q } ) \\right ) \\end{align*}"}
+{"id": "6857.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\| r _ N - r \\| _ \\infty = 0 \\end{align*}"}
+{"id": "5540.png", "formula": "\\begin{align*} S ( x , \\omega ) = S \\left ( x , \\zeta _ { + } ( x , \\omega ) , \\zeta _ { - } ( x , \\omega ) \\right ) \\subseteq L _ { x } \\backslash G \\end{align*}"}
+{"id": "5738.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } 0 = - q c ( b - d ) \\\\ 0 = - c ^ 2 \\\\ q ^ 2 = q b ^ 2 \\\\ q ( d - b ) = - c d . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "9333.png", "formula": "\\begin{align*} a ( X _ { 1 4 } ; T ) & = \\sum _ { d \\mid 6 } d ^ { 1 3 } \\tau ^ * ( 2 ( T ) / d ^ 2 ) \\\\ & = \\tau ^ * ( 2 ^ 2 \\cdot 3 ^ 2 \\cdot 5 ) + 2 ^ { 1 3 } \\tau ^ * ( 3 ^ 2 \\cdot 5 ) + 3 ^ { 1 3 } \\tau ^ * ( 2 ^ 2 \\cdot 5 ) + 6 ^ { 1 3 } \\tau ^ * ( 5 ) , \\end{align*}"}
+{"id": "5838.png", "formula": "\\begin{align*} \\mathcal { F } _ k = F _ { H _ { k + 1 } } \\cap \\bigcap _ { y \\in I _ { H _ { k + 1 } } } \\bigcap _ { j = 0 } ^ { l _ k - 1 } \\ ; D _ { H _ k } ^ { \\mathcal { X } _ j ^ y , \\mathcal { N } _ j ^ y } . \\end{align*}"}
+{"id": "9351.png", "formula": "\\begin{align*} \\mathbb { E } \\bigg ( \\int _ t ^ \\infty z _ n d W _ s - \\int _ t ^ \\infty z d W _ s \\bigg ) ^ 2 = \\mathbb { E } \\int _ t ^ \\infty | z _ n - z | ^ 2 d s \\leq \\mathbb { E } \\int _ t ^ \\infty e ^ { - \\beta s } | z _ n - z | ^ 2 d s \\rightarrow 0 , \\ \\ n \\rightarrow \\infty . \\end{align*}"}
+{"id": "8993.png", "formula": "\\begin{align*} w _ { i j , k } - w \\left ( R ^ \\varphi _ { i j , k } - \\frac { S ^ \\varphi _ k } { m - 1 } \\delta _ { i j } \\right ) - \\left ( R ^ \\varphi _ { i j } - \\frac { S ^ \\varphi } { m - 1 } \\delta _ { i j } \\right ) w _ k - R ^ \\varphi _ { i j , k } + \\frac { S ^ \\varphi _ k } { m } \\delta _ { i j } = 0 \\ , . \\end{align*}"}
+{"id": "3408.png", "formula": "\\begin{align*} E _ \\alpha = \\inf \\Big \\{ J ( u ) = \\frac 1 2 \\int _ { \\R ^ N } | \\nabla u | ^ 2 - \\int _ { \\R ^ N } F ( u ) \\ \\Big | \\ { u \\in \\mathcal M _ \\alpha } \\Big \\} , \\end{align*}"}
+{"id": "947.png", "formula": "\\begin{align*} Z _ 4 ( H ) = \\overline { \\{ ( p , q ) \\in X \\times X \\ , | \\ , \\exists \\ , C ' \\in | H | \\mbox { s m o o t h s . t . } p , q \\in C ' , \\ , 4 [ p - q ] = 0 \\in J C ' \\} } \\subseteq X \\times X \\end{align*}"}
+{"id": "2016.png", "formula": "\\begin{align*} \\lim _ { j \\to + \\infty } \\int _ \\Omega ( - w _ j ) ^ { n + 1 } d V _ g = \\int _ \\Omega ( - w ) ^ { n + 1 } d V _ g . \\end{align*}"}
+{"id": "1873.png", "formula": "\\begin{align*} T _ w ^ \\circ ( M , u ^ * ) = L ^ { \\circ } ( \\mathcal { K } , u ^ * ) . \\end{align*}"}
+{"id": "4748.png", "formula": "\\begin{align*} \\tilde { S } _ { \\gamma _ i \\gamma _ j } & = - J _ { 2 | \\alpha _ i | } \\tilde { S } _ { \\gamma _ i \\gamma _ j } J ^ T _ { 2 | \\alpha _ j | } + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "7361.png", "formula": "\\begin{align*} f ( x ) & = - a ( x - 1 ) + ( 1 + a ) \\sum _ { k = 1 } ^ \\infty \\frac { ( x - 1 ) ^ k \\log ^ k ( 1 + a ) } { k ! } , \\end{align*}"}
+{"id": "6288.png", "formula": "\\begin{align*} h _ { \\bar u ( t ) } ^ \\nu ( \\lambda _ t ) = \\max _ { v \\in \\R ^ k } h _ v ^ \\nu ( \\lambda _ t ) , \\qquad t \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "8522.png", "formula": "\\begin{align*} \\zeta _ { p } ( 2 s ) = \\frac { 1 } { 2 s - 1 } + C _ { p } ^ { ( 1 ) } + O \\left ( s - \\frac { 1 } { 2 } \\right ) \\end{align*}"}
+{"id": "8646.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\infty } e ^ { - \\lambda s } ( 1 - p _ 0 ( s ) ) d s = \\frac 1 { \\lambda } \\int _ 0 ^ { \\infty } \\frac { q } { 1 - p e ^ { - \\lambda s } } \\cdot \\lambda e ^ { - \\lambda s } d s . \\end{align*}"}
+{"id": "3686.png", "formula": "\\begin{align*} \\left \\langle \\mathcal { M } _ { q , \\alpha } f , g \\right \\rangle : = A _ q ( u _ f , g ) , \\end{align*}"}
+{"id": "6396.png", "formula": "\\begin{align*} ( s \\kappa ^ { p } + n \\sqrt { - m } ) = \\varepsilon \\eta ^ { p } , \\end{align*}"}
+{"id": "2010.png", "formula": "\\begin{align*} \\lim _ { j \\to + \\infty } \\int _ \\Omega ( - u _ j ) ^ { n + 1 } g d V = \\int _ \\Omega ( - u ) ^ { n + 1 } g d V . \\end{align*}"}
+{"id": "8139.png", "formula": "\\begin{align*} \\sum _ I ( - 1 ) ^ { \\ell ( I ) - 1 } S ^ I \\sum _ { w \\in S ( I ) } w & = \\sum _ I ( - 1 ) ^ { \\ell ( I ) - 1 } S ^ I \\sum _ { J \\ge I } \\sum _ { w \\in W ( J ) } w \\\\ & = \\sum _ J \\sum _ { w \\in W ( J ) } w \\sum _ { I \\le J } ( - 1 ) ^ { \\ell ( I ) - 1 } S ^ I \\end{align*}"}
+{"id": "1726.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\nu _ t ( x ) = - \\frac { \\delta V ^ { \\sigma } } { \\delta \\nu } ( \\nu _ t , \\mu _ t , x ) \\nu _ t ( x ) , \\\\ \\partial _ t \\mu _ t ( y ) = \\frac { \\delta V ^ { \\sigma } } { \\delta \\mu } ( \\nu _ t , \\mu _ t , y ) \\mu _ t ( y ) , \\end{cases} \\end{align*}"}
+{"id": "2763.png", "formula": "\\begin{align*} \\| T ^ \\ast \\varphi _ j \\| _ { H _ \\Gamma ^ { 3 / 2 } ( \\partial \\mathrm { M } ) } = \\tau _ j . \\end{align*}"}
+{"id": "6218.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ n \\{ h _ 1 g _ i , \\ldots , h _ { \\frac { d } { n } } g _ i \\} . \\end{align*}"}
+{"id": "8148.png", "formula": "\\begin{align*} \\left . \\frac { d } { d \\alpha } \\right | _ { \\alpha = 0 } \\chi _ F ( \\alpha ) = ( - 1 ) ^ { n - 1 } ( Y _ F , \\varphi _ n ) \\end{align*}"}
+{"id": "155.png", "formula": "\\begin{align*} L i _ 2 ( 1 - z ) + L i _ 2 ( 1 - \\frac { 1 } { z } ) = - \\frac { 1 } { 2 } L i _ 2 ( z ^ 2 ) , \\end{align*}"}
+{"id": "4578.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 } | w _ N ( x ) | ^ \\alpha \\d x = N ^ { 2 \\beta ( \\alpha - 1 ) } \\int | w ( x ) | ^ \\alpha \\d x , \\forall \\alpha > 0 . \\end{align*}"}
+{"id": "3163.png", "formula": "\\begin{align*} \\nu _ { y ' } ^ g ( x ) = \\langle y ' T _ { g } ( x ) \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } . \\end{align*}"}
+{"id": "7688.png", "formula": "\\begin{align*} H p ( x ) = \\sum _ { j = 0 } ^ { \\left [ \\frac { k } { 2 } \\right ] } \\frac { ( - 1 ) ^ j \\Gamma \\left ( \\frac { n } { 2 } + k - j - 1 \\right ) } { 4 ^ j j ! \\Gamma \\left ( \\frac { n } { 2 } + k - 1 \\right ) } \\| x \\| ^ { 2 j } \\Delta _ n ^ j p ( x ) . \\end{align*}"}
+{"id": "8876.png", "formula": "\\begin{align*} H _ 2 ( n , { \\textstyle \\frac { 1 } { 2 } } ) \\ge \\frac { \\sqrt { 2 \\pi } } { e ^ 2 } n ^ { 1 / 2 } \\frac { ( n - 1 ) ! } { 2 ^ { n + 1 } } ( 1 - o ( 1 ) ) = \\frac { \\sqrt { 2 \\pi } } { e ^ 2 } n ^ { 1 / 2 } E ( n , { \\textstyle \\frac { 1 } { 2 } } ) ( 1 - o ( 1 ) ) \\ ; . \\end{align*}"}
+{"id": "6529.png", "formula": "\\begin{align*} { } _ p F _ q ( a _ 1 , \\ldots , a _ p ; b _ 1 , \\ldots , b _ q ; x ) = \\sum _ { j = 0 } ^ \\infty \\frac { ( a _ 1 ) _ j \\cdots ( a _ p ) _ j } { ( b _ 1 ) _ j \\cdots ( b _ q ) _ j } \\frac { x ^ j } { j ! } , \\end{align*}"}
+{"id": "1777.png", "formula": "\\begin{align*} | K | _ { n a t } = \\int ^ { [ n ] \\in \\mathcal { A } } K _ n . | \\square _ S [ n ] | _ { n a t } \\end{align*}"}
+{"id": "8786.png", "formula": "\\begin{align*} & ( u ^ 2 + 4 t u + 4 t + 6 u + 6 ) c _ 1 \\\\ & = ( u ^ 2 + 2 t u + 2 t + 6 u + 4 ) b _ 1 + 2 u ( u + 1 ) c _ 2 + u ^ 2 b _ 3 + u ^ 2 c _ 3 \\end{align*}"}
+{"id": "4598.png", "formula": "\\begin{align*} \\begin{array} { l l l } V _ 1 ( x ) & = [ K + \\min _ { x \\leq y \\leq x + B } \\{ v ( y - x ) + L _ 1 ( y ) \\} - L _ 1 ( x ) ] ^ - ; \\end{array} \\end{align*}"}
+{"id": "2785.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q _ 1 ) \\tilde { u } _ 2 = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } , \\tilde { u } _ 2 { _ { | \\partial M } } = u _ 2 { _ { | \\partial M } } , \\end{align*}"}
+{"id": "9097.png", "formula": "\\begin{align*} \\mathfrak { S } ( f _ k ) = n - | E ( T ) | . \\end{align*}"}
+{"id": "4072.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( i ) \\lambda _ f ( j ) \\big | L ( f , n + 1 ) \\big | ^ 2 = 0 \\end{align*}"}
+{"id": "3014.png", "formula": "\\begin{align*} ( u _ i ^ { * } ( C + D ) u _ i ) ^ { \\gamma } + ( u _ i ^ { * } ( C - D ) u _ i ) ^ { \\gamma } \\geq 2 ( u _ i ^ { * } C u _ i ) ^ { \\gamma } \\hbox { f o r $ i = 1 , \\ldots , n . $ } \\end{align*}"}
+{"id": "9064.png", "formula": "\\begin{align*} \\overline { \\beta ( a , b , c ) } = \\sigma _ 1 \\sigma _ 2 ^ { a + 1 } \\sigma _ 1 \\sigma _ 2 ^ { b + 1 } \\sigma _ 1 \\sigma _ 2 ^ { c + 1 } . \\end{align*}"}
+{"id": "4419.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( V ) = \\frac { 1 } { 3 } K _ { \\omega , \\mathbf { c } } ( V ) + \\frac { 1 } { 6 } L _ { \\omega , \\mathbf { c } } ( V ) = \\mu _ { \\omega , \\mathbf { c } } \\end{align*}"}
+{"id": "8964.png", "formula": "\\begin{align*} e _ { i j } = D _ { 2 } \\Delta x ^ { 2 } \\big [ r _ x ^ { 2 ( i - j ) } - 1 \\big ] r _ x ^ { 2 ( j - 1 ) } + D _ { 3 } \\Delta x ^ { 3 } \\big [ r _ x ^ { 3 ( i - j ) } - 1 \\big ] r _ x ^ { 3 ( j - 1 ) } . \\end{align*}"}
+{"id": "9086.png", "formula": "\\begin{align*} a ^ + = n - z - \\mathfrak { S } ( f _ k ) , \\end{align*}"}
+{"id": "142.png", "formula": "\\begin{align*} u \\mapsto ( u _ j : = \\Pi _ { j , \\lambda } \\chi _ j u ) . \\end{align*}"}
+{"id": "5153.png", "formula": "\\begin{align*} \\| g \\| _ { X ( w ) ' } & = \\sup _ { \\| f \\| _ { X ( w ) } = 1 } \\int _ \\Omega \\ ! | f | w | g | w ^ { - 1 } \\ , \\mathrm { d } \\mu = \\sup _ { \\| h \\| _ { X } = 1 } \\int _ \\Omega \\ ! | h | | g | w ^ { - 1 } \\ , \\mathrm { d } \\mu \\\\ & = \\| g w ^ { - 1 } \\| _ { X ' } = \\| g \\| _ { X ' ( w ^ { - 1 } ) } . \\end{align*}"}
+{"id": "1443.png", "formula": "\\begin{align*} L _ { \\mu _ m , \\xi _ m } ( \\phi _ m ) = h _ m , \\| \\phi _ m \\| = 1 \\quad \\mbox { a n d } \\| h _ m \\| \\rightarrow 0 . \\end{align*}"}
+{"id": "5679.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { F } } _ { \\gamma } : = f ^ { + } - \\gamma E _ { g | V ( p ) } = \\sum _ { n \\gg - \\infty } \\left ( C _ { f } ^ { + } ( n ) - n ^ { 1 - k } C _ { g } ( n / p ) \\right ) q ^ { n } = \\sum _ { n \\gg - \\infty } n ^ { 1 - k } d _ { \\gamma } ( n ) q ^ { n } . \\end{align*}"}
+{"id": "4844.png", "formula": "\\begin{align*} \\bigcup _ { p = 1 } ^ { r + 1 } L ( p ) = \\{ i \\in [ n ] : b _ i \\in [ k ] \\} \\mbox { a n d } \\{ s ( 1 ) , s ( 2 ) , \\cdots , s ( r ) \\} = \\{ i \\in [ n ] : b _ i = v \\} . \\end{align*}"}
+{"id": "7744.png", "formula": "\\begin{align*} \\bigcap _ { j = 0 } ^ { q } \\Bigl ( T _ { j } ( \\eta _ { 1 } ) \\le a _ { j } \\Bigr ) \\cap \\Bigl ( T _ { q } ( \\eta _ { 1 } ) < y ^ { * } ( \\eta _ { 1 } , t ) \\le T _ { q + 1 } ( \\eta _ { 1 } ) \\Bigr ) \\end{align*}"}
+{"id": "2612.png", "formula": "\\begin{align*} P _ 2 ( t ) = b _ { \\sigma _ 2 } t ^ { \\sigma _ 2 } + b _ { \\sigma _ 2 + 1 } t ^ { \\sigma _ 2 + 1 } + \\cdots + b _ { d _ 2 } t ^ { d _ 2 } , \\end{align*}"}
+{"id": "2875.png", "formula": "\\begin{align*} \\textsf { A d } g ( X ) : = \\bar { g } X { \\bar { g } } ^ { - 1 } \\end{align*}"}
+{"id": "3195.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\sup _ { x \\in M _ + , x \\neq 0 } \\abs { ( B _ l ( \\nu _ k ) - \\bar { \\nu } _ k ) ( x ) } / \\rho ( x ) & = \\lim _ { l \\to \\infty } \\sup _ { x \\in M _ + , x \\neq 0 } \\abs { ( B _ l ( \\nu _ k ) - \\bar { \\nu } ) ( x ) } / \\rho ( x ) = 0 . \\end{align*}"}
+{"id": "3543.png", "formula": "\\begin{align*} F _ 1 = [ & - \\det \\Delta \\cdot ( \\Delta ^ { - 1 } ( \\partial F _ i / \\partial x _ 1 ) ) \\\\ & - \\det \\Delta \\cdot ( \\Delta ^ { - 1 } ( \\partial F _ i / \\partial x _ 0 ) ) ] . \\end{align*}"}
+{"id": "3156.png", "formula": "\\begin{align*} M _ a ( \\cdot ) : = \\begin{cases} \\frac { 1 } { a ^ d } \\sum _ { 0 \\leq i _ 1 < a } \\cdots \\sum _ { 0 \\leq i _ d < a } T _ 1 ^ { i _ 1 } \\cdots T _ d ^ { i _ d } ( \\cdot ) & G = \\Z _ + ^ d , ~ a \\in \\N , \\\\ \\\\ \\frac { 1 } { a ^ d } \\int _ { Q _ a } T _ t ( \\cdot ) d t & G = \\R _ + ^ d , ~ a \\in \\R _ + . \\end{cases} \\end{align*}"}
+{"id": "717.png", "formula": "\\begin{align*} D _ { D } [ \\partial _ { t } \\mu _ { S } ] ( e ) + A _ { B } ^ * [ \\partial _ { t } \\mu _ { B } ] ( e ) = G _ { D , 2 } ( e ) \\end{align*}"}
+{"id": "4781.png", "formula": "\\begin{align*} c l ( v ) : = \\frac { 1 } { \\sqrt { 2 } } ( v - i J v ) \\wedge ( \\cdot ) - \\frac { 1 } { \\sqrt { 2 } } ( v + i J v ) \\lrcorner ( \\cdot ) , \\end{align*}"}
+{"id": "802.png", "formula": "\\begin{align*} \\tau : = \\frac { 2 m _ 1 \\ , ( a _ { n , n } + b _ { n , n - 1 } ) } { ( 1 - \\sigma ) \\ , \\{ L C R _ 1 ( 1 - 2 \\sigma ) \\} ^ 2 } , \\end{align*}"}
+{"id": "7375.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta \\left ( \\frac { x _ i - x _ j + m } { \\ell } \\right ) & = \\int _ \\mathbb { R } \\chi \\left ( \\frac { x _ i + \\ell x _ 0 - x _ j + m } { \\ell } \\right ) \\chi ( x _ 0 ) \\ , d x _ 0 \\\\ & = \\frac { 1 } { \\ell } \\int _ \\mathbb { R } \\chi \\left ( \\frac { x _ i - x _ 0 ' + m } { \\ell } \\right ) \\chi \\left ( \\frac { x _ j - x _ 0 ' } { \\ell } \\right ) \\ , d x _ 0 ' \\end{aligned} \\end{align*}"}
+{"id": "8333.png", "formula": "\\begin{align*} u ^ \\bigstar _ \\mu ( x ) = \\int _ { C _ D | x | ^ D } ^ \\infty ( - u ^ * _ \\mu ) ' ( t ) \\dd t , \\end{align*}"}
+{"id": "8863.png", "formula": "\\begin{align*} f ^ { ( 1 ) } ( \\boldsymbol \\alpha ) \\triangleq & - \\log p ^ { ( 1 ) } ( \\mathbf R ; \\boldsymbol \\alpha ) - L \\log \\pi \\\\ = & \\log | \\boldsymbol \\Sigma ^ { ( 1 ) } _ { \\boldsymbol \\alpha } | + \\left ( \\boldsymbol \\Sigma _ { \\boldsymbol \\alpha } ^ { ( 1 ) - 1 } \\widehat { \\mathbf \\Sigma } _ { \\mathbf R } \\right ) . \\end{align*}"}
+{"id": "4873.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s u = V ( x ) u & , \\\\ u = 0 & , \\end{cases} u > 0 , \\end{align*}"}
+{"id": "5796.png", "formula": "\\begin{align*} \\frac { y ' } { y } = d A _ 1 ( z ) ^ { 1 / k } - \\frac { k - 1 } { 2 k } \\frac { A ' _ 1 ( z ) } { A ( z ) } + O ( r ^ { - 2 } ) , z \\in \\Omega , d ^ k = - 1 . \\end{align*}"}
+{"id": "4981.png", "formula": "\\begin{align*} \\hat { p } _ { k , n } ^ { ( t ) } = \\hat { p } _ { 1 , n } ^ { ( t ) } + ( n - 1 ) \\sum _ { j = 1 } ^ { k - 1 } \\frac { 1 } { j } \\hat { p } _ { j , n - 1 } ^ { ( t ) } , \\end{align*}"}
+{"id": "1806.png", "formula": "\\begin{align*} [ b _ k , D ^ * _ k ] = 0 \\forall k \\in \\Lambda ^ * \\ . \\end{align*}"}
+{"id": "6279.png", "formula": "\\begin{align*} \\left \\vert f \\left ( z \\right ) \\right \\vert \\geq \\left \\vert g \\left ( z \\right ) \\right \\vert - \\left \\vert g \\left ( z \\right ) - f \\left ( z \\right ) \\right \\vert > s - \\left ( s - n \\right ) = n \\end{align*}"}
+{"id": "5660.png", "formula": "\\begin{align*} m _ \\nu ( a , b ) = \\inf _ { \\mathcal { T } ( a , b ) } \\max _ { s > 0 } J _ \\nu ( s \\circ ( u , v ) ) . \\end{align*}"}
+{"id": "2687.png", "formula": "\\begin{align*} p - \\sum _ { i = 1 } ^ { n } \\Big ( m _ i ( p - x _ { i } ) \\ss _ i ^ { 3 } \\Big ) = 0 . \\end{align*}"}
+{"id": "4603.png", "formula": "\\begin{align*} \\bar { \\Delta } = \\underset { W \\rightarrow \\infty } { \\lim } \\frac { 1 } { W } \\int ^ { W } _ { 0 } \\Delta ( t ) d t , \\end{align*}"}
+{"id": "7747.png", "formula": "\\begin{align*} V ( Z _ \\ell ^ { ( k ) } ) = & \\{ v _ i ^ j \\colon i \\in [ \\ell ] , j \\in [ k - 1 ] \\} \\\\ E ( Z _ \\ell ^ { ( k ) } ) = & \\{ v _ i ^ 1 v _ i ^ 2 \\cdots v _ i ^ { k - 1 } v _ { i + 1 } ^ j \\colon i \\in [ \\ell ] , j \\in [ k - 1 ] \\} \\ , , \\end{align*}"}
+{"id": "2065.png", "formula": "\\begin{align*} Y = \\{ j \\in [ n ] \\ : \\ \\mu _ { C _ j } \\} \\end{align*}"}
+{"id": "8341.png", "formula": "\\begin{align*} ( \\nabla ( u _ { \\kappa } ) _ \\mu ^ \\bigstar ) _ \\mu ^ * ( t ) = \\kappa ( \\nabla u _ \\mu ^ \\bigstar ) _ \\mu ^ * ( \\kappa ^ D t ) . \\end{align*}"}
+{"id": "8254.png", "formula": "\\begin{align*} T _ 1 \\tau _ { k , ( l _ 1 , \\ldots , l _ s ) } ( x ) = \\tau _ { k , ( l _ 1 , \\ldots , l _ s , 1 ) } ( x ) + \\sum _ { i = 1 } ^ s \\tau _ { k , ( l _ 1 , \\ldots , l _ i + 1 , \\ldots , l _ s ) } ( x ) . \\end{align*}"}
+{"id": "1108.png", "formula": "\\begin{align*} \\Phi _ { k , i } ( \\alpha _ 1 , \\ldots , \\alpha _ k ) = 0 , \\forall i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "6053.png", "formula": "\\begin{align*} \\left | \\frac { A _ 1 } { A _ 2 } \\right | = \\left | \\frac { B _ 1 } { B _ 2 } \\right | = \\left | \\frac { C _ 1 } { C _ 2 } \\right | = 1 , \\end{align*}"}
+{"id": "3697.png", "formula": "\\begin{align*} \\left . u _ 1 \\right | _ W = f _ 1 = f _ 2 = \\left . u _ 2 \\right | _ W W . \\end{align*}"}
+{"id": "1836.png", "formula": "\\begin{align*} G _ k ( t - s ) = \\int _ { \\Lambda ^ * } \\chi ( r ) \\chi ^ \\perp ( r + k ) e ^ { - i ( t - s ) ( E _ r + E _ { r + k } ) } \\d r \\ . \\end{align*}"}
+{"id": "3073.png", "formula": "\\begin{align*} u _ 0 ( 0 ) = v _ 0 ( 0 ) = 0 , \\lim _ { r \\to 0 ^ + } \\frac { u _ 0 ( r ) } { u _ 0 ' ( r ) } < \\infty , \\lim _ { r \\to 0 ^ + } \\frac { v _ 0 ( r ) } { v _ 0 ' ( r ) } < \\infty . \\end{align*}"}
+{"id": "8942.png", "formula": "\\begin{align*} \\phi = \\phi _ { e } + \\sum _ { m = 1 } ^ { \\infty } C _ { q _ m } h ^ { q _ m } . \\end{align*}"}
+{"id": "8054.png", "formula": "\\begin{align*} E ^ 2 ( \\lambda ) & = \\left ( { \\mathrm { s o t } } \\lim _ { n \\to \\infty } f _ { \\lambda , n } ( A ) \\right ) ^ 2 \\\\ & = { \\mathrm { s o t } } \\lim _ { n \\to \\infty } f ^ 2 _ { \\lambda , n } ( A ) \\\\ & \\ge { \\mathrm { s o t } } \\lim _ { m \\to \\infty } f _ { \\lambda , m } ( A ) = E ( \\lambda ) \\end{align*}"}
+{"id": "6312.png", "formula": "\\begin{align*} [ 0 , 1 ) \\times V \\ni ( t , z ) \\mapsto d _ z \\phi _ t ( p , \\cdot ) = d _ z \\left ( \\exp _ z ( ( t - 1 ) d _ z \\mathfrak f _ p ) \\right ) \\end{align*}"}
+{"id": "7152.png", "formula": "\\begin{align*} \\partial _ t c = \\Delta c - f ^ \\prime ( c ) \\ ; \\ ; \\ ; \\Omega _ T \\end{align*}"}
+{"id": "7723.png", "formula": "\\begin{align*} C _ { L , K } ( m , \\delta ) = \\sum _ { \\substack { \\mu \\in L ' / K \\\\ p _ K ( \\mu ) = \\delta } } c ( m + q ( \\mu _ { K ^ \\perp } ) , \\mu + L ) \\end{align*}"}
+{"id": "5827.png", "formula": "\\begin{align*} g ( p , u ) = \\left \\{ \\begin{array} { l l } + e _ 1 & \\\\ - e _ 1 & \\\\ - e _ 2 & \\\\ + e _ 2 & \\end{array} \\right . \\end{align*}"}
+{"id": "7923.png", "formula": "\\begin{align*} { \\rm U } ( L ) = \\bigoplus _ { \\mathbf { m } \\in \\N _ 0 ^ { d } } { \\rm U } _ \\mathbf { m } . \\end{align*}"}
+{"id": "5870.png", "formula": "\\begin{align*} \\mathcal { M } _ { L , x _ 0 } = \\bigcap _ { E \\in \\sigma ^ { ( \\mathcal { I } , r e d ) } ( H _ { \\omega , \\Lambda _ { L } ( x _ 0 ) } ) } \\mathcal { M } _ { L , x _ 0 } ^ { ( E ) } . \\end{align*}"}
+{"id": "9285.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\| w ^ { k } - w ^ { * } \\| = \\sigma \\geq 0 \\end{align*}"}
+{"id": "2797.png", "formula": "\\begin{align*} \\varphi _ t = \\sum _ { N _ t } \\tau _ j ^ { - 1 } a _ j \\psi _ j . \\end{align*}"}
+{"id": "4822.png", "formula": "\\begin{align*} G _ { m , n } ( x ) = \\mathbb { P } \\left ( \\sum _ { i = m + 1 } ^ { n } \\frac { X _ { i } - \\mu } { \\sigma \\sqrt { n } } \\leq x \\right ) = G _ { n - m } \\left ( \\frac { x \\sqrt { n } } { \\sqrt { n - m } } \\right ) = G _ { n - m } \\Big ( x \\left ( 1 + R _ { n } \\right ) \\Big ) , \\end{align*}"}
+{"id": "5507.png", "formula": "\\begin{align*} h \\left ( G / P _ { \\mathsf { f } ( I ) } , \\bar { \\nu } _ { 0 } \\right ) = h \\left ( G \\times _ { P _ { \\mathsf { f } ( I ) } } X _ { 0 } , \\bar { \\nu } _ { 0 } \\times \\lambda \\right ) \\ge h ( X , \\nu ) . \\end{align*}"}
+{"id": "4480.png", "formula": "\\begin{align*} Q ( U ) & : = \\| u _ 1 \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 + \\frac { 1 } { 2 } \\| u _ 2 \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 + \\frac { 1 } { 2 } \\| u _ 3 \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 , \\\\ E ( U ) & : = L ( U ) + N ( U ) , \\end{align*}"}
+{"id": "3939.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } \\left \\{ \\langle \\lambda , \\delta \\rangle + \\mathcal { I } ^ { \\star } ( \\lambda ) \\right \\} = \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } \\left \\{ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\gamma \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\int _ { \\mathcal { V } } f _ \\lambda ( v ) \\ , d \\gamma ( v ) \\right \\} , \\end{align*}"}
+{"id": "6225.png", "formula": "\\begin{align*} \\psi _ { \\mu _ N } = \\frac 1 { N + 1 } \\ , \\sum _ { i = 1 } ^ { N + 1 } \\psi _ { \\delta _ { x _ i } } \\end{align*}"}
+{"id": "898.png", "formula": "\\begin{align*} S & \\ll \\widehat { Y } ^ { ( n + 1 ) / 2 + \\varepsilon } | d | \\left ( \\widehat { V } ^ n \\widehat { Z } ^ { - n / 2 } + \\widehat { Y } ^ { n / 3 } \\right ) \\sum _ { \\substack { | r | = \\widehat { Y } \\\\ d \\mid r } } | b _ 2 r _ 3 ' | ^ { 1 / 2 } | b _ 1 ' b _ 2 ' r _ 3 '' | \\\\ & = \\widehat { Y } ^ { ( n + 3 ) / 2 + \\varepsilon } | d | \\left ( \\widehat { V } ^ n \\widehat { Z } ^ { - n / 2 } + \\widehat { Y } ^ { n / 3 } \\right ) \\sum _ { \\substack { | r | = \\widehat { Y } \\\\ d \\mid r } } | b _ 1 | ^ { - 1 } | b _ 2 r ' _ 3 | ^ { - 1 / 2 } , \\end{align*}"}
+{"id": "5651.png", "formula": "\\begin{align*} A _ R ( a , b ) : = \\bigl \\{ ( u , v ) \\in \\mathcal { T } ( a , b ) : | \\nabla u | ^ 2 _ 2 + | \\nabla v | ^ 2 _ 2 < R ^ 2 \\bigr \\} . \\end{align*}"}
+{"id": "2727.png", "formula": "\\begin{align*} A ^ h : = \\begin{pmatrix} y ^ { \\alpha _ y } & 0 \\\\ 0 & ( x + h ) ^ { \\alpha _ x } \\end{pmatrix} , \\end{align*}"}
+{"id": "5646.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | u _ n | ^ { 2 ^ * _ \\mu } ) | u _ n | ^ { 2 ^ * _ \\mu } = \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | u | ^ { 2 ^ * _ \\mu } ) | u | ^ { 2 ^ * _ \\mu } + \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | \\tilde { u } _ n | ^ { 2 ^ * _ \\mu } ) | \\tilde { u } _ n | ^ { 2 ^ * _ \\mu } + o _ n ( 1 ) , \\end{align*}"}
+{"id": "1293.png", "formula": "\\begin{align*} \\tau = \\inf \\{ t \\ge 0 , \\ B ( t ) + \\theta - \\eta t = 0 \\} , \\end{align*}"}
+{"id": "8653.png", "formula": "\\begin{align*} & b = \\int _ 0 ^ { 1 } \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 2 } y ^ { j } ( 1 - p x ) ^ { - 1 } x ^ { j } d y d x , \\end{align*}"}
+{"id": "5247.png", "formula": "\\begin{align*} \\int \\omega ( x ) , \\omega ( x ) : = \\prod _ { i = 1 , 2 } t _ i ^ a ( t _ i - 1 ) ^ b ( t _ i - x ) ^ c \\cdot ( t _ 1 - t _ 2 ) ^ g \\ d t _ 1 \\wedge d t _ 2 \\end{align*}"}
+{"id": "6702.png", "formula": "\\begin{align*} p _ { \\Phi , f u l l } ( x , \\xi , \\tau ) & = | \\xi | ^ 2 - \\tau ^ 2 | \\nabla \\Phi ( x ) | ^ 2 + 2 i \\tau \\xi \\cdot \\nabla \\Phi ( x ) + \\tau \\Delta \\Phi ( x ) \\\\ p _ { \\Phi } ( x , \\xi , \\tau ) & = | \\xi | ^ 2 - \\tau ^ 2 | \\nabla \\Phi ( x ) | ^ 2 + 2 i \\tau \\xi \\cdot \\nabla \\Phi ( x ) \\end{align*}"}
+{"id": "2714.png", "formula": "\\begin{align*} \\begin{aligned} J _ 3 \\ge & - C s ^ 2 \\lambda ^ 2 \\iint _ { Q } \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 u ^ 2 d x d y d t - C s ^ 2 \\lambda ^ 2 \\int _ { 0 } ^ { T } \\int _ { \\omega _ 0 } \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 u ^ 2 d x d y d t , \\end{aligned} \\end{align*}"}
+{"id": "3335.png", "formula": "\\begin{align*} f ( z ) = \\frac { z - a } { 1 - \\overline { a } z } \\ , \\frac { z - \\overline { a } } { 1 - a z } \\ , \\widetilde { f } ( z ) , \\end{align*}"}
+{"id": "2245.png", "formula": "\\begin{align*} w ( z ) = i c - I + \\frac { 1 } { 2 \\pi } \\langle h _ b , P _ r ( \\theta - \\cdot ) \\rangle - \\frac { 1 } { 2 \\pi } \\iint _ { | \\zeta | < 1 } \\left ( \\frac { f ( \\zeta ) } { \\zeta } \\ , \\frac { \\zeta + z } { \\zeta - z } + \\frac { \\overline { f ( \\zeta ) } } { \\overline { \\zeta } } \\ , \\frac { 1 + z \\overline { \\zeta } } { 1 - z \\overline { \\zeta } } \\right ) \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "6959.png", "formula": "\\begin{align*} \\nu ( f _ 0 + f _ 1 g + \\ldots + f _ n g ^ n ) : = v ( f _ 0 ( \\eta ) ) . \\end{align*}"}
+{"id": "4132.png", "formula": "\\begin{align*} M _ B = \\begin{pmatrix} 2 0 & 4 5 & 1 6 \\\\ 1 6 & 3 6 & 1 3 \\\\ 1 3 & 2 9 & 1 0 \\end{pmatrix} = M _ 1 ^ 3 M _ 2 ^ { - 3 } . \\end{align*}"}
+{"id": "5486.png", "formula": "\\begin{align*} \\eta _ { \\omega } = \\lim _ { n \\to \\infty } \\omega _ { n } . \\eta \\mbox { e x i s t s } . \\end{align*}"}
+{"id": "6079.png", "formula": "\\begin{align*} \\lim _ { j \\to + \\infty } \\big \\| \\exp ( A ) ^ j \\exp ( B ) ^ { - \\lfloor \\varepsilon j \\rfloor } \\big \\| = \\infty , \\mbox { o r } \\lim _ { j \\to - \\infty } \\big \\| \\exp ( A ) ^ j \\exp ( B ) ^ { - \\lfloor \\varepsilon j \\rfloor } \\big \\| = \\infty . \\end{align*}"}
+{"id": "6131.png", "formula": "\\begin{align*} \\tilde { p } = \\begin{cases} p _ { h + 1 } & h < h _ { q } - 1 , \\\\ q & h = h _ { q } - 1 . \\end{cases} \\end{align*}"}
+{"id": "3359.png", "formula": "\\begin{align*} \\langle \\pi _ { \\zeta } ( B _ j B _ k ) \\xi , \\pi _ { \\zeta } ( B _ { \\ell } ) \\xi \\rangle = \\delta _ { j , \\ell } \\langle \\pi _ { \\zeta } ( B _ k ) \\xi , \\xi \\rangle = 0 . \\end{align*}"}
+{"id": "5594.png", "formula": "\\begin{align*} I & = \\left ( \\sum _ { A \\in \\mathcal { P } } \\left ( 1 - \\frac { \\alpha ' ( A ) } { \\alpha ( A ) } \\right ) \\alpha ( A ) \\log \\frac { \\alpha ( A ) } { \\beta ( A ) } \\right ) - H _ { \\alpha ' \\parallel \\alpha } \\left ( \\mathcal { P } \\right ) \\\\ & \\le \\sum _ { A \\in \\mathcal { P } } \\left ( 1 - \\frac { \\alpha ' ( A ) } { \\alpha ( A ) } \\right ) \\alpha ( A ) \\log \\frac { \\alpha ( A ) } { \\beta ( A ) } . \\end{align*}"}
+{"id": "7585.png", "formula": "\\begin{align*} Y ( A , B , C ) : = \\max \\{ | A + B z + C z ^ 2 | + 1 - | z | ^ 2 : z \\in \\overline { \\mathbb { D } } \\} . \\end{align*}"}
+{"id": "7613.png", "formula": "\\begin{align*} f ( z ) = z + \\sum _ { n = 2 } ^ { \\infty } a _ n z ^ n , \\ ; \\mbox { f o r } \\ ; z \\in \\mathbb { D } : = \\{ z : \\in \\mathbb { C } : | z | < 1 \\} \\end{align*}"}
+{"id": "8137.png", "formula": "\\begin{align*} W ( I ) = \\{ w | w _ { 1 : k } \\ge k \\ \\ w _ { 1 : k } < k \\ \\} , \\end{align*}"}
+{"id": "15.png", "formula": "\\begin{align*} X _ { \\mathbf { K } } ( \\mathbb { C } ) = \\mathbf { G } ( \\mathbb { Q } ) \\backslash \\left ( \\mathbf { X } \\times \\mathbf { G } ( \\mathbb { A } _ f ) / \\mathbf { K } \\right ) \\end{align*}"}
+{"id": "6859.png", "formula": "\\begin{align*} h ^ { G _ N } ( x , y ) = \\left \\{ \\begin{array} { l l } 1 , & , \\\\ 0 , & . \\end{array} \\right . \\end{align*}"}
+{"id": "6005.png", "formula": "\\begin{align*} \\theta _ { 1 / 2 } ( \\tfrac { a z + b } { c z + d } , \\epsilon ) = \\lambda ^ { \\pm } ( \\gamma , \\epsilon ) \\sqrt { \\det \\gamma ( c z + d ) } \\sum _ { n \\in \\Z } e ^ { i ( \\det \\gamma ) \\epsilon \\pi n ^ 2 z } , \\end{align*}"}
+{"id": "1032.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ 2 } { 3 2 ^ k ( 1 + k ) } \\bigg \\{ H _ { 2 k } ^ { ( 2 ) } - \\frac { 1 } { 4 } H _ { k } ^ { ( 2 ) } \\bigg \\} = \\Gamma \\bigg ( \\frac { 3 } { 4 } \\bigg ) ^ 2 \\frac { \\pi ^ 2 + 8 G - 1 6 } { 4 \\pi \\sqrt { \\pi } } . \\end{align*}"}
+{"id": "2862.png", "formula": "\\begin{align*} < \\beta \\circ \\rho _ { \\beta } ^ * ( f , \\gamma ) , m > & = < ( f , \\gamma ) , \\rho _ { \\beta } \\circ \\beta ( m ) > \\\\ & = < ( f , \\gamma ) , ( \\alpha \\otimes \\beta ) \\circ \\rho _ { \\beta } ( m ) > \\\\ & = < \\rho _ { \\beta } ^ * \\circ ( \\alpha \\otimes \\beta ) ( f , \\gamma ) , m > \\\\ & = < \\rho _ { \\beta } ^ * ( \\alpha ( f ) , \\beta ( \\gamma ) ) , m > \\end{align*}"}
+{"id": "4912.png", "formula": "\\begin{align*} T _ k = G _ { p _ 0 } + G _ { p _ 1 } + \\cdots + G _ { p _ { k - 1 } } , \\end{align*}"}
+{"id": "6246.png", "formula": "\\begin{align*} d ( J _ { \\mathrm { s e c t } ( \\delta / 2 ) } ( t ) ) = J _ { \\mathrm { s e c t } ( \\delta / 2 ) } ( t ^ { [ 1 ] } ) \\cdot \\omega ( t ^ { [ 2 ] } ) , \\end{align*}"}
+{"id": "6638.png", "formula": "\\begin{align*} C = Y _ 0 \\subset Y _ 1 \\subset \\dots \\subset Y _ k = Y \\end{align*}"}
+{"id": "5346.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ q \\leq ( 1 / \\rho ) ^ d \\gamma ' = \\gamma \\left ( 9 9 \\cdot 2 \\ , r \\ , r ' q \\ , ( r ' ) ^ { - 2 / q } \\sqrt { \\frac { \\log ( 1 / \\gamma ) } { d } } \\right ) ^ d = \\gamma \\left ( 9 9 \\cdot 2 \\cdot \\sqrt { 2 } \\ , r \\ , q \\ , ( r ' ) ^ { - 2 / q } \\right ) ^ d \\end{align*}"}
+{"id": "5361.png", "formula": "\\begin{align*} \\rho ^ { d } \\| f ^ { = d } \\| _ q = \\| T _ { \\rho } f ^ { = d } \\| _ q \\le \\gamma '^ { \\frac { q - 2 } { q } } \\| f ^ { = d } \\| _ 2 ^ { 2 / q } \\le \\gamma ' . \\end{align*}"}
+{"id": "7699.png", "formula": "\\begin{align*} v _ 1 ^ \\pm ( g _ z ^ { - 1 } ( \\lambda _ { L _ 1 } + c z ' ) ) & = v _ 1 ^ \\pm ( g _ z ^ { - 1 } ( \\lambda _ { L _ 1 } - c \\mu ( g _ z ) ) + a c z ^ \\ast ) \\\\ & = v _ 1 ^ { \\pm } ( g _ 1 ^ { - 1 } ( \\lambda _ { L _ 1 } - c \\mu ( g _ z ) _ { L _ 1 } ) ) . \\end{align*}"}
+{"id": "204.png", "formula": "\\begin{align*} = \\left ( 1 - x \\right ) ^ { \\frac { z ( 1 + z ) L i _ 2 ( y ) } { ( 1 - z ) ^ 3 } } , \\end{align*}"}
+{"id": "134.png", "formula": "\\begin{align*} \\chi _ K R ( \\lambda ) \\chi _ K = \\chi _ K R _ K ( \\lambda ) \\chi _ K : C ^ \\infty ( M ) \\to \\mathcal { D } ' ( M ) \\end{align*}"}
+{"id": "1770.png", "formula": "\\begin{align*} D _ \\Theta X _ T ^ { ( \\ast , n ) } \\le \\left ( D _ \\Theta X ^ { ( n ) } \\right ) _ T ^ \\ast : = \\sup _ { s \\in [ 0 , T ] } D _ \\Theta X _ s ^ { ( n ) } . \\end{align*}"}
+{"id": "1666.png", "formula": "\\begin{align*} \\theta = \\sum _ { j = 1 } ^ { \\ell } f _ j d \\log z _ j + \\sum _ { k = \\ell + 1 } ^ { n } f _ k d z _ k . \\end{align*}"}
+{"id": "856.png", "formula": "\\begin{align*} \\xi u + E _ { \\lambda } ( u _ { 0 } , v _ { 0 } ) \\leq - \\int _ { 0 } ^ { u } H _ { \\lambda } ( s ) \\ , d s + E _ { \\lambda } ( u _ { 0 } , v _ { 0 } ) = \\frac { 1 } { 2 } u _ { t } ^ 2 , \\end{align*}"}
+{"id": "4623.png", "formula": "\\begin{align*} c _ k ( D _ n ) = c _ k ( \\lambda E ( 1 , n ) ) , \\end{align*}"}
+{"id": "3846.png", "formula": "\\begin{align*} \\mathrm { R W } _ 0 ( d ) & : = \\inf _ { \\gamma \\in \\Sigma _ 0 ( \\delta ) } \\mathbb { E } [ Y _ 1 ( 1 - d ( X ) ) + Y _ 2 d ( X ) ] \\\\ \\mathrm { R W } ( d ) & : = \\inf _ { \\gamma \\in \\Sigma ( \\delta ) } \\mathbb { E } [ Y _ 1 ( 1 - d ( X ) ) + Y _ 2 d ( X ) ] , \\end{align*}"}
+{"id": "2707.png", "formula": "\\begin{align*} \\begin{aligned} T _ 4 = & s \\iint _ { Q } ( A \\nabla \\sigma ) _ i \\frac { \\partial A } { \\partial x _ i } \\nabla u \\cdot \\nabla u d x d t \\ge - C s \\lambda \\iint _ Q \\xi A \\nabla u \\cdot \\nabla u d x d t . \\end{aligned} \\end{align*}"}
+{"id": "2271.png", "formula": "\\begin{align*} T ( f ) ( z ) : = - \\frac { 1 } { \\pi } \\iint _ D \\frac { f ( \\zeta ) } { \\zeta - z } \\ , d \\xi \\ , d \\eta . \\end{align*}"}
+{"id": "3271.png", "formula": "\\begin{align*} \\gamma _ { \\xi } = \\overline { \\gamma _ { \\overline { \\xi } } } \\end{align*}"}
+{"id": "3264.png", "formula": "\\begin{align*} \\gamma = \\left \\{ w \\in \\C ; | w | = R \\left ( \\frac { w } { | w | } \\right ) \\right \\} \\end{align*}"}
+{"id": "3747.png", "formula": "\\begin{align*} \\langle z ^ { r } , & z ^ { r ' } \\rangle = \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } z ^ { r } \\cdot \\overline { z ^ { r ' } } e ^ { - | z ^ 2 | } \\ d x \\ d p , \\ z = x + \\sqrt { - 1 } p . \\end{align*}"}
+{"id": "2229.png", "formula": "\\begin{align*} h \\log y \\int _ { 1 } ^ { 2 } t ^ { - 1 } y ^ { h t - u } \\ , d t = \\frac { y ^ { 2 h - u } } { 2 } - y ^ { h - u } + \\int _ { 1 } ^ { 2 } t ^ { - 2 } y ^ { h t - u } \\ , d t \\end{align*}"}
+{"id": "8152.png", "formula": "\\begin{align*} \\Delta \\chi _ T ( t ) & = \\ < Y _ T , \\lambda _ 1 ^ t ( \\lambda _ 1 - 1 ) \\ > = \\ < \\Delta Y _ T , \\lambda _ 1 ^ t \\otimes ( \\lambda _ 1 - 1 ) \\ > \\\\ & = \\sum _ { ( T ) } \\ < Y _ { T ( 1 ) } \\otimes Y _ { T ( 2 ) } , \\lambda _ 1 ^ t \\otimes ( X _ \\bullet + X _ { \\bullet \\bullet } + \\cdots ) \\ > \\\\ & = \\ < Y _ { T _ 1 } \\cdots Y _ { T _ k } , \\lambda _ 1 ^ t \\ > \\quad \\\\ & = \\chi _ { T _ 1 } ( t ) \\cdots \\chi _ { T _ k } ( t ) . \\end{align*}"}
+{"id": "193.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c , d ) = 1 \\\\ a , b , c , d \\geq 1 } } \\left ( \\frac { 1 } { 1 - w ^ a x ^ b y ^ c z ^ d } \\right ) ^ { \\frac { 1 } { a ^ r b ^ s c ^ t d ^ u } } = \\exp \\left \\{ \\left ( \\sum _ { h = 1 } ^ { \\infty } \\frac { w ^ h } { h ^ r } \\right ) \\left ( \\sum _ { i = 1 } ^ { \\infty } \\frac { x ^ i } { i ^ s } \\right ) \\left ( \\sum _ { j = 1 } ^ { \\infty } \\frac { y ^ j } { j ^ t } \\right ) \\left ( \\sum _ { k = 1 } ^ { \\infty } \\frac { z ^ k } { k ^ u } \\right ) \\right \\} . \\end{align*}"}
+{"id": "9071.png", "formula": "\\begin{align*} f _ 1 ( x _ i ) = \\left \\{ \\begin{array} { l l } f ( x _ i ) , & \\hbox { $ i = 1 , 2 , \\ldots , n $ ; } \\\\ 0 , & \\hbox { $ i = n + 1 $ . } \\end{array} \\right . \\end{align*}"}
+{"id": "7449.png", "formula": "\\begin{align*} \\partial _ t { w } ^ { ( i ) } _ 0 ( x _ i , t ) \\ , + \\ , \\big ( v _ i ^ { ( i ) } ( x _ i ) \\ , w ^ { ( i ) } _ 0 ( x _ i , t ) \\big ) ^ \\prime = 0 , ( x _ i , t ) \\in I _ \\varepsilon ^ { ( i ) } \\times ( 0 , T ) , \\end{align*}"}
+{"id": "5109.png", "formula": "\\begin{align*} \\int _ { \\Omega } a ^ { i j , k l } ( D ^ { 2 } u ) u _ { i j } \\eta _ { k l } d x = 0 , \\forall \\eta \\in C _ { 0 } ^ { \\infty } ( \\Omega ) , \\end{align*}"}
+{"id": "8304.png", "formula": "\\begin{align*} A + B = \\{ a + b : a \\in A , \\ , b \\in B \\} , A \\cdot B = \\{ a b : a \\in A , \\ , b \\in B \\} \\ , . \\end{align*}"}
+{"id": "1046.png", "formula": "\\begin{align*} \\Delta ( f ) _ { \\mid p } \\coloneqq \\sum _ { i = 0 } ^ { \\infty } ( a ( i , \\ldots , i ) \\bmod p ) t ^ i \\in \\mathbb F _ p [ [ t ] ] \\ , . \\end{align*}"}
+{"id": "7843.png", "formula": "\\begin{align*} \\mathcal { O } _ i = \\left \\{ ( A , B ) \\in \\binom { [ n ] } { k } \\times \\binom { [ n ] } { k } : \\ | A \\cap B | = k - i \\right \\} . \\end{align*}"}
+{"id": "4102.png", "formula": "\\begin{align*} = \\left ( ( Q _ { 1 , \\vec { v } } ) _ 1 M ^ 3 _ { \\bullet , 1 } ( Q _ { 1 , \\vec { v } } ) _ 2 M ^ 3 _ { \\bullet , 1 } ( Q _ { 1 , \\vec { v } } ) _ 3 M ^ 3 _ { \\bullet , 1 } \\right ) \\end{align*}"}
+{"id": "5949.png", "formula": "\\begin{align*} \\overline { c } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) & = m _ { X ^ { \\ast } } ( g _ 1 g _ 2 ) ^ { - 1 } m _ { X ^ { \\ast } } ( g _ 1 ) m _ { X ^ { \\ast } } ( g _ 2 ) \\widetilde { c } _ { { X ^ { \\ast } } } ( g _ 1 , g _ 2 ) \\\\ & = [ \\widetilde { s } ( g _ 1 g _ 2 ) m _ { X ^ { \\ast } } ( g _ 1 g _ 2 ) ^ { - 1 } ] [ \\widetilde { s } ( g _ 1 ) m _ { X ^ { \\ast } } ( g _ 1 ) ^ { - 1 } ] ^ { - 1 } [ \\widetilde { s } ( g _ 2 ) m _ { X ^ { \\ast } } ( g _ 2 ) ^ { - 1 } ] ^ { - 1 } . \\end{align*}"}
+{"id": "3465.png", "formula": "\\begin{align*} f ( q ) V \\ : = \\ \\oplus _ { i \\in \\Z } V ^ { a _ i } \\{ i \\} , \\ \\ f ( q ) = \\sum _ i a _ i q ^ i , \\ a _ i \\in \\mathbb { N } , \\end{align*}"}
+{"id": "9104.png", "formula": "\\begin{align*} y _ { q s } ^ { k + \\ell + 1 } = \\frac { y _ { q p } ^ { k + \\ell } } { y _ { s p } ^ { k + \\ell } } . \\end{align*}"}
+{"id": "1610.png", "formula": "\\begin{align*} \\int _ { \\theta } v ^ { 2 } ( w ) & \\| ( \\chi _ { w } \\oplus \\xi _ { w } ) \\pi _ { F ( w ) \\oplus G ( w ) } S _ { \\chi \\oplus \\xi } ^ { - 1 } \\pi _ { S _ { \\chi \\oplus \\xi } ^ { - 1 } ( F ( w ) \\oplus G ( w ) ) } ( f , g ) \\| ^ { 2 } d \\mu ( w ) \\\\ & = \\int _ { \\Theta } v ^ { 2 } ( w ) \\| ( \\chi _ { w } \\oplus \\xi _ { w } ) \\pi _ { F ( w ) \\oplus G ( w ) } S _ { \\chi \\oplus \\xi } ^ { - 1 } ( f , g ) \\| ^ { 2 } d \\mu ( w ) \\\\ & \\leq B _ { 1 } \\| S _ { \\chi \\oplus \\xi } ^ { - 1 } \\| ^ { 2 } \\| ( f , g ) \\| ^ { 2 } . \\end{align*}"}
+{"id": "1781.png", "formula": "\\begin{align*} f : \\square [ \\underline { n } ] = ( \\square [ n _ 1 ] * \\dots * \\square [ n _ p ] ) * \\square [ n _ { p + 1 } ] \\longrightarrow \\square [ n ] \\end{align*}"}
+{"id": "2552.png", "formula": "\\begin{align*} \\lambda _ { j _ 1 } = \\lambda _ { j _ 1 ^ * } = \\lambda _ { j _ 2 } = \\lambda _ { j _ 2 ^ * } = \\lambda _ t \\end{align*}"}
+{"id": "7508.png", "formula": "\\begin{align*} \\tilde { \\Lambda } ( z ) = \\ln ' ( \\Lambda ( z ) ) \\pi ( z ) , \\end{align*}"}
+{"id": "9101.png", "formula": "\\begin{align*} \\mathfrak { S } ( f _ k ) + \\overline { \\mathfrak { S } } ( f _ k ) \\geq ( v _ l - z _ l - z _ r ) + 2 z _ r = v _ l - z _ l + z _ r . \\end{align*}"}
+{"id": "1870.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\langle D _ u f ( u ^ * ) , v \\rangle _ { \\mathcal { U } } + \\langle \\bar { \\lambda } , G ' ( u ^ * ) v \\rangle = 0 \\forall \\ v \\in \\mathcal { U } ; \\\\ & - \\langle \\bar { \\lambda } , \\zeta - G ' ( u ^ * ) u ^ * \\rangle \\geq 0 \\forall \\ \\zeta \\in K , \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "6080.png", "formula": "\\begin{align*} x _ { 1 } = \\exp ( X ) . \\end{align*}"}
+{"id": "4328.png", "formula": "\\begin{align*} f ( \\cdot , t ) = f _ 0 \\circ \\big ( y ^ { ( i , k , n ; L ) } _ t \\big ) ^ { - 1 } . \\end{align*}"}
+{"id": "125.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } + i B } \\hat { \\psi } ( \\lambda ) \\frac { d } { d \\lambda } \\log \\zeta _ 1 ( \\lambda ) d \\lambda = - 2 \\pi i \\left \\langle \\sum \\limits _ \\gamma \\frac { T _ \\gamma ^ \\# \\delta ( t - T _ \\gamma ) } { | \\det ( I - \\mathcal { P } _ \\gamma ) | } , \\psi \\right \\rangle \\end{align*}"}
+{"id": "662.png", "formula": "\\begin{align*} \\mathfrak { h } _ j ( \\varphi ) & = \\lim _ { k \\to \\infty } Q ( 0 , k ) ^ { - 1 } \\circ P _ { U ^ { ( k ) } _ j } \\circ \\mathcal { M } ^ { ( k ) } \\circ S ( k ) ( \\varphi ) \\\\ & = \\sum _ { l \\geq 0 } Q ( 0 , l ) ^ { - 1 } \\circ P _ { U ^ { ( l ) } _ j } \\circ \\big ( \\mathcal { M } ^ { ( l ) } \\circ S ( l ) - Z ( l ) \\circ \\mathcal { M } ^ { ( l - 1 ) } \\circ S ( l - 1 ) \\big ) ( \\varphi ) \\end{align*}"}
+{"id": "8909.png", "formula": "\\begin{align*} \\mathsf { d } _ { c c } ( Z ) & = \\mathsf { d } _ { c c } ( Z _ 1 \\cdot Z _ 2 ) = \\mathsf { d } _ { c c } ( Z _ 1 \\cdot Z _ 2 ) \\\\ & \\leq \\mathsf { d } _ { c c } ( Z _ 1 ) + \\mathsf { d } _ { c c } ( Z _ 2 ) \\leq \\| Z _ 1 \\| _ 1 + 2 \\mathsf { d } _ { \\mathrm { c o m } } ( \\{ X _ { n i } \\} ) \\leq \\frac { 1 } { 2 } + \\frac { 1 } { 2 } = 1 . \\end{align*}"}
+{"id": "4956.png", "formula": "\\begin{align*} f ( k , t , n ) = F ( k , t , n ) - F ( k - 1 , t , n ) . \\end{align*}"}
+{"id": "3060.png", "formula": "\\begin{align*} u _ 0 ( r ) = C _ \\lambda r ^ \\lambda = C _ \\lambda | x | ^ \\lambda , v _ 0 ( r ) = C _ \\mu r ^ \\mu = C _ \\mu | x | ^ \\mu \\end{align*}"}
+{"id": "1554.png", "formula": "\\begin{align*} n = 1 + \\ln \\sigma ^ { - \\frac { 1 } { \\ln 2 } } = 1 + \\log _ { \\frac { 1 } { \\eta ' } } \\sigma ^ { \\frac { \\ln \\eta ' } { \\ln 2 } } . \\end{align*}"}
+{"id": "210.png", "formula": "\\begin{align*} = e x p \\left \\{ \\frac { z ( 1 + 1 1 z + 1 1 z ^ 2 + z ^ 3 ) L i _ 2 ( x ) L i _ 3 ( y ) } { ( 1 - z ) ^ 5 } \\right \\} , \\end{align*}"}
+{"id": "7626.png", "formula": "\\begin{align*} H _ 3 ( 1 ) ( f ^ { - 1 } ) = \\frac { 1 } { 9 2 1 6 } ( g _ 1 ( c , \\delta ) + g _ 2 ( c , \\delta ) \\eta + g _ 3 ( c , \\delta ) \\eta ^ 2 ) + v ( c , \\delta , \\eta ) \\rho ) , \\end{align*}"}
+{"id": "3363.png", "formula": "\\begin{align*} f ( Z ) = ( I \\otimes x ^ * ) \\left ( I - \\sum _ { j = 1 } ^ d Z _ j \\otimes T _ { 0 , j } ^ * \\right ) ^ { - 1 } \\left ( \\sum _ { j = 1 } ^ d Z _ j \\otimes ( T _ j ^ * x ) \\right ) . \\end{align*}"}
+{"id": "2303.png", "formula": "\\begin{align*} w ( z ) = \\Phi _ { 0 } ( z ) + \\sum _ { k = 1 } ^ { n - 1 } \\left ( T ^ k ( \\Phi _ k ) ( z ) \\right ) + T ^ n ( f ) ( z ) , \\end{align*}"}
+{"id": "3182.png", "formula": "\\begin{align*} M _ 1 ( f ) = \\int _ { Q _ { 0 , \\ldots , 0 } } T _ t ( f ) d t = \\int _ { [ 0 , 1 ) ^ d } T _ t ( f ) d t . \\end{align*}"}
+{"id": "5669.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | u | ^ { p } ) | v | ^ { q } \\leq D _ 0 ( | \\nabla u | ^ 2 _ 2 + | \\nabla v | ^ 2 _ 2 ) ^ { \\frac { \\gamma _ p + \\gamma _ q } { 2 } } , \\ D _ 0 : = \\frac { 1 } { \\gamma _ p + \\gamma _ q } \\Bigl ( \\frac { \\gamma _ p + \\gamma _ q - 2 } { 2 ( \\gamma _ p + \\gamma _ q ) l ( a , b ) } \\Bigr ) ^ { \\frac { \\gamma _ p + \\gamma _ q - 2 } { 2 } } , \\end{align*}"}
+{"id": "2542.png", "formula": "\\begin{align*} \\lambda _ { j _ 1 } = \\lambda _ { j _ 1 ^ * } = \\dots = \\lambda _ { j _ r } = \\lambda _ { j _ r ^ * } \\end{align*}"}
+{"id": "6494.png", "formula": "\\begin{align*} \\det D _ n ( a , b ) = \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ { a - b } \\frac { ( a + b + i - j ) ( a + b + 2 i + j - 2 ) } { ( a + b + 2 i - j ) ( i + j - 1 ) } . \\end{align*}"}
+{"id": "6973.png", "formula": "\\begin{align*} I ' = \\left \\{ i \\in I ^ * \\ \\left | \\ Q _ i \\in \\textbf { Q } _ { n _ 0 } \\right . \\right \\} . \\end{align*}"}
+{"id": "341.png", "formula": "\\begin{align*} \\| T u \\| _ { ( L ^ p ( V ) ) ^ \\ast } = \\sup _ { \\| f \\| _ p = 1 } \\langle T u , f \\rangle \\leq \\| u \\| _ q . \\end{align*}"}
+{"id": "3077.png", "formula": "\\begin{align*} \\frac { B ( r ) } { B ' ( r ) } \\cdot \\left [ \\mathcal Y ( r ) ^ { p - 1 } \\right ] ' + \\mathcal Y ( r ) ^ { p - 1 } = Q _ 2 ( \\mathcal V ( r ) , v _ 0 ( r ) ) \\cdot Q _ 3 ( \\mathcal W ( r ) , u _ 0 ' ( r ) ) , \\end{align*}"}
+{"id": "5350.png", "formula": "\\begin{align*} \\rho ^ { d } \\| f ^ { = d } \\| _ q = \\| T _ { \\rho } f ^ { = d } \\| _ q \\le \\gamma ^ { \\frac { q - 2 } { 2 q } } \\| f ^ { = d } \\| _ 2 ^ { 2 / q } \\le \\sqrt { \\gamma } . \\end{align*}"}
+{"id": "4144.png", "formula": "\\begin{align*} Q _ { 1 , \\vec { v } _ { s , t , f , 0 } } = \\end{align*}"}
+{"id": "4167.png", "formula": "\\begin{align*} \\langle b \\delta _ t , \\lambda ( x ) b \\delta _ t \\rangle = \\big \\langle b \\delta _ t , \\sum _ { s \\in G } \\lambda _ s ( b _ s ) b \\delta _ t \\big \\rangle = \\sum _ { s \\in G } \\langle b \\delta _ t , b _ s b \\delta _ { s t } \\rangle = b ^ * b _ e b = b ^ * \\mathbb { E } ( x ) b . \\end{align*}"}
+{"id": "7007.png", "formula": "\\begin{align*} f ' = a _ 1 + 2 a _ 2 x + \\ldots + r a _ r x ^ { r - 1 } . \\end{align*}"}
+{"id": "2951.png", "formula": "\\begin{align*} \\mu ^ \\delta _ { K S } ( f ) ( \\tau ) = \\left ( \\frac { \\eta ^ \\delta ( \\tau ) } { \\eta ^ \\delta ( p \\tau ) } \\right ) ^ { 2 ( c ^ 2 - 1 ) } = \\left ( \\frac { \\Delta ^ \\delta ( \\tau ) } { \\Delta ^ \\delta ( p \\tau ) . } \\right ) ^ { ( c ^ 2 - 1 ) / 1 2 } \\end{align*}"}
+{"id": "6993.png", "formula": "\\begin{align*} \\mu ( f ) = \\gamma _ i + v \\left ( \\tilde { b } \\right ) . \\end{align*}"}
+{"id": "6752.png", "formula": "\\begin{align*} \\widetilde { R } ( h , x ) = \\begin{cases} ( R h , x ) & \\underline { \\varphi } ( h ) ( 0 ) = \\varphi ( h ) ( 0 ) , \\\\ ( R h , \\sigma x ) & 0 = \\underline { \\varphi } ( h ) ( 0 ) < \\varphi ( h ) ( 0 ) = 1 . \\end{cases} \\end{align*}"}
+{"id": "846.png", "formula": "\\begin{align*} \\mathcal { K } _ { c l r \\left ( i \\right ) } = \\mathcal { K } _ { c l r \\left ( i - 1 \\right ) } \\setminus k _ { n e w \\left ( i - 1 \\right ) } \\end{align*}"}
+{"id": "4068.png", "formula": "\\begin{align*} S _ { \\mathrm { o f f } } ^ { c < p ^ 2 } : = \\sum _ { d _ 1 , \\ , d _ 2 < D } \\ , \\ , \\ , \\ & \\sum _ { I d _ 1 , \\ , J d _ 2 < D } x _ { I d _ 1 } \\bar { x } _ { J d _ 2 } \\sqrt { I J } \\sum _ { L , M \\geq 1 } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } \\\\ & \\times \\left ( 2 \\pi i ^ { 2 k } \\sum _ { \\substack { p | c \\\\ c < p ^ 2 } } \\frac { \\mathcal { S } ( I L , J M , c ) } { c } \\ , J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { I L J M } } { c } \\right ) \\right ) \\end{align*}"}
+{"id": "1804.png", "formula": "\\begin{align*} \\lambda = 1 / N ^ { \\frac { 5 } { 3 } + \\alpha } 1 \\leq n \\leq N ^ { 1 / 3 } \\ , \\epsilon = 1 / N ^ \\beta \\end{align*}"}
+{"id": "3504.png", "formula": "\\begin{align*} - X '' + R X = \\lambda X , \\end{align*}"}
+{"id": "72.png", "formula": "\\begin{align*} \\mathbf { P G } ^ 1 ( \\mathbb { Q } _ p ) \\rightarrow \\prod _ { \\ell \\in \\mathcal { S } , \\sigma \\in \\mathcal { T } ( \\ell ) } \\mathbf { P G } ^ 1 [ \\ell , \\sigma ] \\backslash \\mathbf { P G } ^ 1 ( \\mathbb { Q } _ p ) = \\Gamma \\backslash \\prod _ { \\ell \\in \\mathcal { S } , \\sigma \\in \\mathcal { T } ( \\ell ) } \\mathbf { P G } ^ 1 ( \\mathbb { Q } _ p ) \\end{align*}"}
+{"id": "7734.png", "formula": "\\begin{align*} z \\bigl ( Y , \\ , t \\bigr ) = y ^ { * } ( N , \\ , t ) , \\ \\forall t , \\ \\pi \\end{align*}"}
+{"id": "3385.png", "formula": "\\begin{align*} m \\log ( m + 1 ) \\Delta _ { 1 0 } u _ { m n } = O ( 1 ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , n \\log ( n + 1 ) \\Delta _ { 0 1 } u _ { m n } = O ( 1 ) . \\end{align*}"}
+{"id": "1077.png", "formula": "\\begin{align*} | f ( u ) - f ( v ) | \\leq C | u - v | \\sum _ { k = 0 } ^ { \\infty } \\frac { \\lambda ^ k } { k ! } \\left ( | u | ^ { p k + m - 1 } + | v | ^ { p k + m - 1 } \\right ) . \\end{align*}"}
+{"id": "2476.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle G ( t ) = \\int _ { \\mathbb { R } ^ N } | x | ^ 2 \\left ( a _ 2 | \\phi | ^ 2 + a _ 1 | \\psi | ^ 2 \\right ) d x . \\end{array} \\right . \\end{align*}"}
+{"id": "5467.png", "formula": "\\begin{align*} & \\quad \\Pr \\left ( > \\theta | \\Phi , \\Phi ( \\mathcal { A } ) > 0 \\right ) \\\\ & \\approx \\sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } e ^ { - m \\eta \\theta r _ 1 ^ { \\alpha } / \\rho } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\prod \\limits _ { x _ { i } \\in \\Phi \\backslash \\{ x _ 1 \\} \\cap { \\mathcal { A } } } \\frac { 1 } { \\left ( 1 + \\frac { m \\eta \\theta r _ 1 ^ { \\alpha } } { M r _ i ^ { \\alpha } } \\right ) ^ M } . \\end{align*}"}
+{"id": "6496.png", "formula": "\\begin{align*} & \\det T _ { n , m } ( x ) \\\\ & = \\frac 1 { n ! } \\ , C T _ { \\vec { t } } \\left \\{ \\prod _ { i = 0 } ^ { n - 1 } \\left [ \\frac { ( 1 + t _ i ) ^ { x + m } ( t _ i - 1 ) } { t _ i ^ { x + n } } \\right ] \\cdot \\prod _ { i < j } ^ { 0 , n - 1 } ( t _ i - t _ j ) ( t _ j ^ { - 1 } - t _ i ^ { - 1 } ) ( 1 - t _ i t _ j ) \\right \\} \\\\ & = \\frac 1 { n ! } \\ , C T _ { \\vec { t } } \\left \\{ \\prod _ { i = 0 } ^ { n - 1 } \\left [ \\frac { ( 1 + t _ i ) ^ { x + m } ( t _ i - 1 ) } { t _ i ^ { x + 2 n - 1 } } \\right ] \\cdot \\prod _ { i < j } ^ { 0 , n - 1 } ( t _ i - t _ j ) ^ 2 ( 1 - t _ i t _ j ) \\right \\} . \\end{align*}"}
+{"id": "4663.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\epsilon ^ 2 \\Delta u + V ( x ) u = \\lambda u + u \\log u ^ 2 , \\ , \\ , \\ , \\ , \\hbox { i n } \\mathbb { R } ^ N , \\\\ & \\int _ { \\mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } \\epsilon ^ N , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7317.png", "formula": "\\begin{align*} a _ p + \\sum _ { i = 1 } ^ n \\alpha _ i z _ { i p } = c _ p + \\sum _ { i = 1 } ^ n \\gamma _ i z _ { i p } < b _ p + \\sum _ { i = 1 } ^ n \\beta _ i z _ { i p } , \\end{align*}"}
+{"id": "8387.png", "formula": "\\begin{align*} | a _ 0 ^ { - 1 } - a _ 1 ^ { - 1 } | \\geq \\bigl | a _ 0 ^ { - 1 } - ( a _ 0 + n ^ { - 1 } \\nu _ 1 / 2 ) ^ { - 1 } \\bigr | = | a _ 0 | ^ { - 1 } | a _ 0 + n ^ { - 1 } \\nu _ 1 / 2 | ^ { - 1 } \\frac { 1 } { 2 } n ^ { - 1 } \\nu _ 1 \\geq \\frac { 1 } { 2 } \\cdot \\frac { 1 } { 3 } \\cdot \\frac { 1 } { 2 } n ^ { - 1 } \\nu _ 1 = \\frac { 1 } { 1 2 } n ^ { - 1 } \\nu _ 1 . \\end{align*}"}
+{"id": "9250.png", "formula": "\\begin{align*} & d ^ { ( 0 ) } _ 0 = 1 , d ^ { ( 0 ) } _ k = 0 k \\ne 0 \\\\ & d ^ { ( n ) } _ k = 0 k < 0 \\quad k > 3 n / 2 \\\\ & 2 ( 3 n - 2 k ) d ^ { ( n ) } _ k = \\tfrac 1 2 d ^ { ( n - 1 ) } _ k + ( 1 - 2 \\sigma ) d ^ { ( n - 1 ) } _ { k - 1 } - 2 ( 3 n - 2 k ) ( 3 n - 2 k + 1 ) d ^ { ( n - 1 ) } _ { k - 2 } \\\\ & d ^ { ( n ) } _ { 3 n / 2 } = - \\sum _ { k = 0 } ^ { 3 n / 2 - 1 } ( - 1 ) ^ { 3 n / 2 - k } d ^ { ( n ) } _ k \\frac { ( 3 n - 2 k ) ! } { ( 3 n / 2 - k ) ! } , 3 n \\equiv 0 \\pmod 2 . \\end{align*}"}
+{"id": "7300.png", "formula": "\\begin{align*} x y - z t = 1 \\end{align*}"}
+{"id": "5387.png", "formula": "\\begin{align*} \\begin{bmatrix} x _ { 1 , k + 1 } \\\\ x _ { 2 , k + 1 } \\end{bmatrix} = \\begin{bmatrix} A _ 1 + B _ 1 D _ 2 ( I - D _ 1 D _ 2 ) ^ { - 1 } C _ 1 & B _ 1 C _ 2 + B _ 1 D _ 2 ( I - D _ 1 D _ 2 ) ^ { - 1 } D _ 1 C _ 2 \\\\ B _ 2 ( I - D _ 1 D _ 2 ) ^ { - 1 } C _ 1 & A _ 2 + B _ 2 ( I - D _ 1 D _ 2 ) ^ { - 1 } D _ 1 C _ 2 \\end{bmatrix} \\begin{bmatrix} x _ { 1 , k } \\\\ x _ { 2 , k } \\end{bmatrix} . \\end{align*}"}
+{"id": "3746.png", "formula": "\\begin{align*} W _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) & = \\overline { W _ { \\vec { k } , \\vec { l } } } ( \\vec { x } , \\vec { y } ) , \\tilde { W } _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) = \\overline { \\tilde { W } _ { \\vec { k } , \\vec { l } } } ( \\vec { x } , \\vec { y } ) , \\check { W } _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) = \\overline { \\check { W } _ { \\vec { k } , \\vec { l } } } ( \\vec { x } , \\vec { y } ) , \\end{align*}"}
+{"id": "5666.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | u _ n | ^ { 2 ^ * _ \\mu } ) | u _ n | ^ { 2 ^ * _ \\mu } = 1 , | \\nabla u _ n | ^ 2 _ 2 \\rightarrow S _ { H , L } , \\quad \\ n \\rightarrow \\infty . \\end{align*}"}
+{"id": "5728.png", "formula": "\\begin{align*} S ( p , q , a , b , c , d ) = \\begin{pmatrix} 0 & 1 \\\\ p & q \\\\ a & b \\\\ c & d \\end{pmatrix} . \\end{align*}"}
+{"id": "5697.png", "formula": "\\begin{align*} \\alpha _ { g } = \\displaystyle \\lim _ { m \\rightarrow \\infty } \\dfrac { C _ { D ^ { k - 1 } ( f ) } ( p ^ { 2 m + 1 } ) } { \\beta ^ { 2 m } } . \\end{align*}"}
+{"id": "5399.png", "formula": "\\begin{align*} \\chi _ { k , \\psi } = \\iota _ { \\xi _ { k , \\psi } } = \\iota _ { \\xi _ { k , \\psi } \\cdot \\chi _ { 0 } } . \\end{align*}"}
+{"id": "3417.png", "formula": "\\begin{align*} \\rho _ 1 = \\frac 1 2 \\min \\{ { C _ { t } ^ { - 1 } } , \\ell ^ { - 1 } \\alpha \\} \\end{align*}"}
+{"id": "2223.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial h } E _ 1 ( h ; y , u ) = - h ^ { - 1 } \\cdot \\frac { \\partial } { \\partial h } \\lambda ( y ^ u , y ^ h ) + h ^ { - 2 } \\lambda ( y ^ { u } , y ) - h ^ { - 2 } \\lambda ( y ^ { u - h } , y ^ h ) . \\end{align*}"}
+{"id": "6504.png", "formula": "\\begin{align*} & ( m + n ) ( 2 n + x - 1 ) ( 2 n + x + 1 ) ( 2 n + x ) ^ 2 ( m - n - x ) t _ { n + 1 } \\\\ & + n ( n + x ) ( m - 2 n - x - 1 ) ( m - 2 n - x ) ( m + 2 n + x - 1 ) ( m + 2 n + x ) t _ n = 0 . \\end{align*}"}
+{"id": "6019.png", "formula": "\\begin{align*} \\widetilde { c } _ { { X ^ { \\ast } } } ( g _ 1 , g _ 2 ) & = m _ { X ^ { \\ast } } ( g _ 1 g _ 2 ) m _ { X ^ { \\ast } } ( g _ 1 ) ^ { - 1 } m _ { X ^ { \\ast } } ( g _ 2 ) ^ { - 1 } \\overline { c } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) \\\\ & = [ \\overline { \\beta _ 1 } ( g _ 1 g _ 2 ) m _ { X ^ { \\ast } } ( g _ 1 g _ 2 ) ] [ \\overline { \\beta _ 1 } ( g _ 1 ) m _ { X ^ { \\ast } } ( g _ 1 ) ] ^ { - 1 } [ \\overline { \\beta _ 1 } ( g _ 2 ) m _ { X ^ { \\ast } } ( g _ 2 ) ] ^ { - 1 } . \\end{align*}"}
+{"id": "4287.png", "formula": "\\begin{align*} A ^ 0 ( U ) U _ t + A ^ 1 ( U ) U _ x + B ( U ) U = F ( U ) , \\end{align*}"}
+{"id": "5563.png", "formula": "\\begin{align*} \\phi _ { x , \\beta } ^ { t } \\left ( \\mho ( v ) \\right ) : = \\beta \\left ( \\left \\{ \\xi \\in \\partial F : L _ { x } \\xi _ { t } \\in { \\rm S h d } ( v ) \\right \\} \\right ) . \\end{align*}"}
+{"id": "6135.png", "formula": "\\begin{align*} \\gamma _ { \\tau } = \\frac { 2 n + 4 \\tau } { n + 2 s + 2 \\tau } . \\end{align*}"}
+{"id": "1238.png", "formula": "\\begin{align*} v _ { \\gamma , 1 } ( x ) = 2 ^ { \\alpha } x ^ { 1 + \\alpha } ( \\log 2 + \\log x ) \\forall x \\in ( 0 , \\tfrac 1 2 ] \\end{align*}"}
+{"id": "764.png", "formula": "\\begin{align*} \\sigma _ k ( A ) = \\dfrac { 1 } { k ! } \\delta _ { i _ 1 i _ 2 \\cdots i _ k } ^ { j _ 1 j _ 2 \\cdots j _ k } A ^ { i _ 1 } _ { j _ 1 } A ^ { i _ 2 } _ { j _ 2 } \\cdots A ^ { i _ k } _ { j _ k } . \\end{align*}"}
+{"id": "1747.png", "formula": "\\begin{align*} C _ s ^ { ( m ^ * ) } = \\max \\Big \\{ \\log _ 2 \\Big ( \\frac { 1 + \\Gamma _ { } ^ { ( m ^ * ) } } { 1 + \\Gamma _ { } ^ { ( m ^ * ) } } \\Big ) , 0 \\Big \\} , \\end{align*}"}
+{"id": "8984.png", "formula": "\\begin{align*} \\Lambda ( x ) = \\begin{cases} \\dfrac { S ^ \\varphi u ( x ) [ 1 - u ^ { \\frac { 4 } { m - 2 } } ( x ) ] } { c _ m [ 1 - u ( x ) ] } & \\ , u ( x ) \\neq 1 \\\\ [ 0 . 3 c m ] \\dfrac { S ^ \\varphi } { m - 1 } & \\ , u ( x ) = 1 . \\end{cases} \\end{align*}"}
+{"id": "7719.png", "formula": "\\begin{align*} z _ 1 = \\frac 1 2 ( e _ 2 + e _ { b - 1 } ) , z _ 1 ^ * = e _ 2 - e _ { b - 1 } , \\end{align*}"}
+{"id": "3446.png", "formula": "\\begin{align*} g = & \\ , ( d \\psi + \\beta x ( d \\phi + \\bar \\sigma ) ) ^ 2 + \\hat { g } , \\\\ X = & \\ , \\alpha \\frac { \\partial } { \\partial \\psi } , \\\\ \\beta = & \\ , \\sqrt { \\frac { 2 \\alpha ^ 2 } { m n } + \\frac { 2 \\lambda } { n } } . \\end{align*}"}
+{"id": "4941.png", "formula": "\\begin{align*} \\underset { t \\rightarrow \\infty } { \\lim } P ^ t = \\left [ O _ { ( n + 1 ) \\times n } \\ , \\Big \\vert \\ , \\mathbf { 1 } _ { n + 1 } \\right ] . \\end{align*}"}
+{"id": "5591.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\frac { d \\omega _ { n } \\lambda } { d \\lambda } \\left ( \\psi \\left ( x , { \\rm b n d } ( \\omega ) \\right ) \\right ) = h _ { \\mu } ( Z , \\lambda ) . \\end{align*}"}
+{"id": "8021.png", "formula": "\\begin{align*} \\frac { U A U ^ * + V A V ^ * } { 2 } = \\begin{pmatrix} X & 0 \\\\ 0 & T \\end{pmatrix} . \\end{align*}"}
+{"id": "8886.png", "formula": "\\begin{align*} \\operatorname { v o l } ( B _ { c c } ( R ) ) = \\operatorname { v o l } ( B _ { c c } ( 1 ) ) R ^ Q \\end{align*}"}
+{"id": "2830.png", "formula": "\\begin{align*} a _ i ( t ) = \\frac { b _ { i - p + 1 } ( t ) } { a _ { i - p + 1 } ( t ) \\cdots a _ { i - 1 } ( t ) } . \\end{align*}"}
+{"id": "4192.png", "formula": "\\begin{align*} \\abs { \\hat { w _ 1 } ( \\xi ) } \\leq \\int _ 0 ^ \\infty \\frac { e ^ { - 3 \\tau } - e ^ { - 6 \\tau } } { 2 } \\ , d \\tau = \\frac { 1 } { 1 2 } \\ , . \\end{align*}"}
+{"id": "3317.png", "formula": "\\begin{align*} d ( 0 + , f ( 0 + ) ) = - d ( 2 \\pi - , f ( 2 \\pi - ) ) \\end{align*}"}
+{"id": "6197.png", "formula": "\\begin{align*} V _ e ^ { ( i ) } : = \\{ v \\in V _ e \\mid m ( v ) = i \\} . \\end{align*}"}
+{"id": "9304.png", "formula": "\\begin{align*} S _ { k \\ell } [ a ] : = \\left ( \\begin{array} { c c c c } a _ 1 & a _ 2 & \\ldots & a _ { \\ell } \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ a _ 1 & a _ 2 & \\ldots & a _ { \\ell } \\\\ \\end{array} \\right ) . \\end{align*}"}
+{"id": "2861.png", "formula": "\\begin{align*} \\Delta ^ * ( f , g ) & = [ f , g ] , \\\\ \\varphi _ L ^ * : L ^ * \\rightarrow L ^ * \\ ; \\ & \\eta \\mapsto \\varphi _ L ^ * ( \\eta ) = \\eta \\circ \\varphi _ L , \\\\ \\alpha \\circ \\eta & = \\eta \\circ \\alpha . \\end{align*}"}
+{"id": "4807.png", "formula": "\\begin{align*} & ( a ) \\ - \\tau ^ { - 1 } \\langle c , A \\psi - \\phi \\rangle - \\tau ^ { - 1 } \\sum _ { i = 1 } ^ m \\langle c , B _ i \\psi \\rangle [ C \\chi ] _ i \\\\ & + \\sum _ { j = 1 } ^ J a _ j ( x ) \\sigma _ j ( x ) + \\sum _ { \\ell = 1 } ^ L b _ \\ell ( x ) \\rho _ \\ell ( x ) \\ \\mathrm { i s \\ S O S } , \\\\ & ( b ) \\ \\sigma _ j ( x ) \\ \\mathrm { i s \\ S O S } , \\end{align*}"}
+{"id": "2361.png", "formula": "\\begin{align*} e ( u _ j ) = e ( u _ j - u _ i + u _ i ) = e ( u ' _ j - u ' _ i + u _ i ) = u ' _ j - u ' _ i + e ( u _ i ) = u ' _ j - u ' _ i + u ' _ i = u ' _ j \\end{align*}"}
+{"id": "7633.png", "formula": "\\begin{align*} M ( c , 0 , 1 ) = 6 4 ( 4 - c ^ 2 ) ^ 2 \\leq 1 0 2 4 , \\ ; c \\in ( 0 , 2 ) , \\end{align*}"}
+{"id": "8744.png", "formula": "\\begin{align*} \\textup { M M D } _ { n , m } ^ { 2 } = \\Delta _ { 0 } + \\sum ^ { 2 ( l - 1 ) - 1 } _ { s = 1 } \\Delta _ { s } + \\widetilde { \\Delta } _ { 2 ( l - 1 ) } . \\end{align*}"}
+{"id": "6851.png", "formula": "\\begin{align*} \\alpha ' _ n & = \\frac { ( a q ^ { - N } ) _ { N + 1 } } { ( 1 - b _ 1 ) \\cdots ( 1 - b _ { N + 1 } ) } \\sum _ { j \\in \\Z } ( - 1 ) ^ j \\frac { q ^ { j n - j ( j + 1 ) / 2 } } { ( a q ^ { 2 n - N - j - 1 } ) _ { N + 2 } } ( 1 - a q ^ { 2 n - 1 - N } ) f _ { N + 1 , j , n } ( b _ 1 , \\ldots , b _ { N + 1 } ) \\alpha _ { n - j } \\\\ & = \\alpha _ n ^ { ( N + 1 ) } . \\end{align*}"}
+{"id": "2876.png", "formula": "\\begin{align*} \\chi ( X _ { - \\alpha } ) = 1 \\alpha \\in \\Pi _ 0 \\chi ( X _ \\beta ) = 0 \\beta \\in \\Phi _ 0 \\backslash \\Pi _ 0 \\end{align*}"}
+{"id": "4755.png", "formula": "\\begin{align*} \\tilde { S } = S Q + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "7047.png", "formula": "\\begin{align*} h _ i \\sum \\limits _ { j = 1 } ^ { s _ { { i _ { } i } } } b _ { i _ { } i j } { \\textbf { X } } ^ { \\lambda _ j } \\in \\mathcal I , \\end{align*}"}
+{"id": "9255.png", "formula": "\\begin{align*} S _ { \\alpha , \\beta } f ( x ) = \\sup _ { t > 3 x } t ^ { - \\beta } \\int _ { \\frac { t - x } 2 } ^ { t - x } ( t + x - z ) ^ { - \\alpha - 1 / 2 } ( t - x - z ) ^ { \\alpha + \\beta - 1 / 2 } \\abs { f ( z ) } \\ , d z , x > 0 . \\end{align*}"}
+{"id": "1780.png", "formula": "\\begin{align*} { \\delta _ { \\underline { f } } = f _ 1 * \\dots * f _ p : \\mathcal { A } [ \\underline { n } ] \\longrightarrow \\mathcal { A } [ \\underline { n } ] } . \\end{align*}"}
+{"id": "1085.png", "formula": "\\begin{align*} u _ 0 ( x ) : = \\begin{cases} \\alpha ( \\log \\frac { 1 } { | x | } ) ^ { \\frac 1 p } , & ~ ~ ~ | x | < 1 \\\\ 0 , & ~ ~ ~ ~ | x | \\geq 1 . \\end{cases} \\end{align*}"}
+{"id": "8130.png", "formula": "\\begin{align*} P _ + = - R , P _ - = I - P _ + , \\end{align*}"}
+{"id": "8363.png", "formula": "\\begin{align*} Q = S C _ E ^ { - 1 } S ^ t C _ V + T C _ E ^ { - 1 } T ^ t C _ V , A = S C _ E ^ { - 1 } T ^ t C _ V + T C _ E ^ { - 1 } S ^ t C _ V , L = ( S - T ) C _ E ^ { - 1 } ( S - T ) ^ t C _ V . \\end{align*}"}
+{"id": "6098.png", "formula": "\\begin{align*} \\int _ { s } ^ t ( \\delta h ) _ { s , \\tau } \\otimes \\mathrm { d } X _ { \\tau } : = \\lim _ { | \\pi | \\rightarrow 0 } \\int _ { \\pi } h _ { s , \\tau } \\circ \\mathrm { d } X _ { \\tau } = \\lim _ { | \\pi | \\rightarrow 0 } \\sum _ { 0 \\leq i < m } h _ { s , \\kappa _ i } \\otimes ( \\delta X ) _ { \\kappa _ i , \\kappa _ { i + 1 } } , \\end{align*}"}
+{"id": "9318.png", "formula": "\\begin{align*} \\begin{array} { l } f _ 1 = x _ 1 + \\log ( x _ 2 - x _ 3 ) , \\\\ f _ 2 = x _ 2 + \\log ( x _ 3 - x _ 1 ) , \\\\ f _ 3 = x _ 3 + \\log ( x _ 1 - x _ 2 ) \\\\ \\end{array} \\end{align*}"}
+{"id": "1935.png", "formula": "\\begin{align*} \\Theta _ { S , u } = a _ 0 + \\sum _ { T \\in \\mathcal { Q } _ m , T \\neq S } \\left ( \\prod _ { i \\in T } \\alpha _ i \\right ) f _ T ( u _ T ) \\in \\C . \\end{align*}"}
+{"id": "5593.png", "formula": "\\begin{align*} D \\left ( \\alpha \\parallel \\beta \\right ) = \\sup _ { n } H _ { \\alpha \\parallel \\beta } \\left ( \\mathcal { P } _ { n } \\right ) . \\end{align*}"}
+{"id": "7222.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 \\left ( \\Omega _ \\delta \\cap \\left ( \\overline { J _ u } \\setminus J _ u \\right ) \\right ) = 0 \\end{align*}"}
+{"id": "1609.png", "formula": "\\begin{align*} ( f , g ) & = S _ { \\chi \\oplus \\xi } S _ { \\chi \\oplus \\xi } ^ { - 1 } ( f , g ) \\\\ & = \\int _ { \\Theta } v ^ { 2 } ( w ) \\pi _ { F ( w ) \\oplus G ( w ) } ( \\chi _ { w } \\oplus \\xi _ { w } ) ^ { \\ast } ( \\chi _ { w } \\oplus \\xi _ { w } ) \\pi _ { F ( w ) \\oplus G ( w ) } S _ { \\chi \\oplus \\xi } ^ { - 1 } ( f , g ) d \\mu ( w ) \\end{align*}"}
+{"id": "3247.png", "formula": "\\begin{align*} \\langle f , g \\rangle & : = \\langle - \\ln { \\mathfrak R } \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } w ( Q ) \\psi _ Q ( \\cdot , x _ { Q } ) q _ { Q } h ( x _ { Q } ) , g \\rangle \\\\ & = - \\ln { \\mathfrak R } \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } w ( Q ) \\psi _ Q g ( x _ { Q } ) q _ { Q } h ( x _ { Q } ) , \\end{align*}"}
+{"id": "4329.png", "formula": "\\begin{align*} T _ { ( k , m , i , n ) } = \\sum _ { ( k ' , m ' , i ' , n ' ) < _ { \\mathrm { t i m e } } ( k , m , i , n ) } 3 \\cdot 2 ^ { - k ' } < 4 2 , \\end{align*}"}
+{"id": "8906.png", "formula": "\\begin{gather*} Z _ 2 = \\sum _ { n = 1 } ^ { d _ 1 ^ 2 } [ X _ { n 1 } , X _ { n 2 } ] , \\\\ \\nu = \\sqrt { \\sum _ { n , m = 1 } ^ { d _ 1 ^ 2 } \\langle X _ { n 1 } , X _ { m 1 } \\rangle _ 1 \\langle X _ { n 2 } , X _ { m 2 } \\rangle _ 1 } = \\sqrt { \\sum _ { n = 1 } ^ { d _ 1 ^ 2 } \\| X _ { n 1 } \\| _ 1 ^ 2 \\| X _ { n 2 } \\| _ 1 ^ 2 } , \\\\ \\| X _ { n 1 } \\| _ 1 = \\| X _ { n 2 } \\| _ 1 \\qquad n = 1 , \\dots , d _ 1 ^ 2 . \\end{gather*}"}
+{"id": "161.png", "formula": "\\begin{align*} L ( x ) + L ( y ) = L ( x y ) + L \\left ( \\frac { x ( 1 - y ) } { 1 - x y } \\right ) + L \\left ( \\frac { y ( 1 - x ) } { 1 - x y } \\right ) . \\end{align*}"}
+{"id": "406.png", "formula": "\\begin{align*} \\begin{bmatrix} W - \\epsilon \\Lambda ^ { - 1 } T ^ T \\tilde F \\\\ \\epsilon T ^ T \\tilde F \\end{bmatrix} ^ T \\begin{bmatrix} \\Lambda & 0 \\\\ 0 & - \\Lambda ^ { - 1 } \\end{bmatrix} \\begin{bmatrix} W - \\epsilon \\Lambda ^ { - 1 } T ^ T \\tilde F \\\\ \\epsilon T ^ T \\tilde F \\end{bmatrix} . \\end{align*}"}
+{"id": "6055.png", "formula": "\\begin{align*} H _ 1 ( z ) : = A _ 1 z ^ { n ' + m ' } + B _ 1 \\overline { z } ^ { m ' } + C _ 1 \\textrm { f o r a l l } z \\in \\mathbb { C } \\end{align*}"}
+{"id": "3506.png", "formula": "\\begin{align*} \\mathcal { J } _ { \\lambda } ^ t \\coloneqq \\{ X \\in \\mathcal { J } _ { \\lambda } \\mid X ( t ) = 0 \\} \\end{align*}"}
+{"id": "4113.png", "formula": "\\begin{align*} y < 1 \\iff 0 = f ( y ) < f ( 1 ) = 1 + \\alpha _ 2 + \\alpha _ 1 + \\alpha _ 0 . \\end{align*}"}
+{"id": "8285.png", "formula": "\\begin{align*} s _ { i , j } = \\sum _ { m = 0 } ^ { N + j - i } \\frac { ( 2 k - n + i - 1 ) ! } { ( 2 k - n + i - 1 + m ) ! } \\binom { i + k - n - 1 + m } { m } \\binom { i + m - 1 } { j - 1 } \\frac { N ! } { ( N + j - i - m ) ! } ( e ^ { \\i \\beta } - 1 ) ^ m , \\end{align*}"}
+{"id": "7144.png", "formula": "\\begin{align*} \\hat { I } _ { \\varepsilon , j } ^ 1 + \\Delta \\big ( \\varphi _ j c \\big ) = \\psi _ j \\mathcal { L } ^ { \\mathbb { R } ^ n _ { \\gamma _ j } } _ \\varepsilon \\big ( \\varphi _ j c \\big ) + \\Delta \\big ( \\varphi _ j c \\big ) = \\psi _ j \\Big ( \\mathcal { L } ^ { \\mathbb { R } ^ n _ { \\gamma _ j } } _ \\varepsilon \\big ( \\varphi _ j c \\big ) + \\Delta \\big ( \\varphi _ j c \\big ) \\Big ) . \\end{align*}"}
+{"id": "9046.png", "formula": "\\begin{align*} b _ { i , j } ' = b _ { g ( i ) , b ( j ) } . \\end{align*}"}
+{"id": "5886.png", "formula": "\\begin{align*} L _ { u ( t ) } \\ , S ^ k \\varrho ( t ) = ( \\nu _ { u _ 0 } + k ) S ^ k \\varrho ( t ) \\ , , \\end{align*}"}
+{"id": "166.png", "formula": "\\begin{align*} L i _ 2 ( - 1 ) = - \\frac { \\pi ^ 2 } { 1 2 } , \\end{align*}"}
+{"id": "7431.png", "formula": "\\begin{align*} \\displaystyle \\left ( \\Pi _ { i = 1 } ^ t \\sigma _ { ( 1 , 1 ) } \\right ) \\star \\sigma _ { \\lambda } = q \\sigma _ { ( \\lambda _ 2 + t ) } - \\left ( \\sum _ { | \\mu | + 2 n = | \\lambda | } a _ { \\mu } q \\sigma _ { ( \\mu _ 1 + t , \\mu _ 2 + t ) } \\right ) . \\end{align*}"}
+{"id": "3331.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial \\theta } = f \\frac { \\partial g } { \\partial \\theta } , \\end{align*}"}
+{"id": "1413.png", "formula": "\\begin{align*} S _ k & = \\{ x ^ * \\in S _ { k - 1 } : r _ { k - 1 } \\cos ( \\theta ) = \\langle \\log _ { y _ { k - 1 } } ( x ^ * ) , e _ k ^ { ( k - 1 ) } \\rangle = \\langle \\log _ { y _ { k - 1 } } ( x ^ * ) , e _ k \\rangle \\} \\\\ & = \\{ x ^ * \\in S _ { k - 1 } : r _ { k - 1 } \\cos ( \\theta ) = \\frac { r _ { k - 1 } } { \\sinh ( r _ { k - 1 } ) } \\langle x ^ * , e _ k \\rangle \\} . \\end{align*}"}
+{"id": "812.png", "formula": "\\begin{align*} K \\left ( y \\left ( t _ { j } \\right ) , t _ { j } \\right ) & = K \\left ( y _ { j } , x _ { j } \\right ) \\\\ K \\left ( y ^ { \\prime } \\left ( t _ { j } \\right ) , t _ { j } \\right ) & = K \\left ( y _ { j } ^ { \\prime } , x _ { j } \\right ) . \\end{align*}"}
+{"id": "7980.png", "formula": "\\begin{align*} ( \\mathrm { d i v } \\ , V _ k ) \\ , V _ k \\cdot \\nu - \\nabla V _ k \\ , V _ k \\cdot \\nu = ( \\mathrm { d i v } _ T V _ k ) \\ , V _ k \\cdot \\nu - \\nabla _ T V _ k \\ , ( V _ k ) _ T \\cdot \\nu \\quad \\partial \\Omega \\ , . \\end{align*}"}
+{"id": "1244.png", "formula": "\\begin{align*} ( T ^ { \\ell _ j - \\ell _ { j - 1 } } ) ' T ^ { \\ell _ { j - 1 } } x \\ge C \\frac { T ^ { p _ j } ( \\hat { b } _ { p _ j - 1 } ) - T ^ { p _ j } ( \\hat { b } _ { p _ j } ) } { \\hat { b } _ { p _ j - 1 } - \\hat { b } _ { p _ j } } = \\frac { C } { 2 } \\frac { 1 } { \\hat { b } _ { p _ j - 1 } - \\hat { b } _ { p _ j } } . \\end{align*}"}
+{"id": "24.png", "formula": "\\begin{align*} ( \\Lambda _ 0 , \\xi _ 0 , \\psi _ 0 ) \\leftrightarrow x _ 0 = ( A _ 0 , \\lambda _ 0 , \\psi _ 0 ) \\leftrightarrow [ z _ 0 , g _ 0 ] _ { \\mathbf { K } } \\in X _ { \\mathbf { K } } ( \\overline { \\mathbb { Q } } ) . \\end{align*}"}
+{"id": "3554.png", "formula": "\\begin{align*} I = \\int _ { 0 } ^ { \\infty } f ( x ) \\ , d x \\end{align*}"}
+{"id": "3440.png", "formula": "\\begin{align*} | X | ^ 2 \\equiv c ^ 2 = \\frac { m \\lambda } { 2 } + \\frac { 3 | m \\lambda | } { 2 } . \\end{align*}"}
+{"id": "9136.png", "formula": "\\begin{align*} \\mathrm { s p a n } \\{ \\mathrm { d } \\varphi _ { [ 0 , R ] } \\} = \\mathrm { s p a n } \\{ \\mathrm { d } \\varphi _ { [ 0 , A - 1 ] } , \\mathrm { d } \\varphi _ { [ A , R ] } \\} = \\mathrm { s p a n } \\{ \\mathrm { d } x , \\mathrm { d } \\varphi _ { c } , \\mathrm { d } \\varphi _ { [ A , R ] } \\} \\ , . \\end{align*}"}
+{"id": "2689.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { m = 1 \\\\ ( m . n ) = 1 } } ^ n ( m - 1 , n ) = \\phi ( n ) \\tau ( n ) . \\end{align*}"}
+{"id": "2540.png", "formula": "\\begin{align*} \\lambda _ i \\lambda _ j = \\lambda _ i \\lambda _ { j ^ * } = \\lambda _ j \\lambda _ { i ^ * } = \\lambda _ { j ^ * } \\lambda _ { i ^ * } = \\lambda _ i \\lambda _ { i ^ * } = \\lambda _ j \\lambda _ { j ^ * } , \\end{align*}"}
+{"id": "2031.png", "formula": "\\begin{align*} L H = u ^ { j \\bar k } H _ { j \\bar k } \\leq - \\alpha \\ , u ^ { j \\bar k } ( \\delta _ { j k } - \\alpha \\bar z _ j z _ k ) H . \\end{align*}"}
+{"id": "7218.png", "formula": "\\begin{align*} \\limsup _ { h \\to + \\infty } F _ h ( v _ h , \\gamma _ h , B _ \\rho ) \\leq \\limsup _ { h \\to + \\infty } F _ h \\left ( w _ h ^ { ( \\rho ) } , \\gamma _ h , B _ \\rho \\right ) \\end{align*}"}
+{"id": "4732.png", "formula": "\\begin{align*} T = \\begin{pmatrix} W & X \\\\ Y & Z \\end{pmatrix} , \\end{align*}"}
+{"id": "8523.png", "formula": "\\begin{align*} \\zeta _ { p , \\infty } ( s , c ) : = \\sum _ { m , n \\neq 0 } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p ( p + \\frac { 1 } { \\pi } ) + \\lambda _ { n } ^ { 2 } } \\cdot \\frac { 1 } { \\left ( \\lambda _ { m } ^ { 2 } + c \\ , n ^ { 2 } \\right ) ^ { s } } , \\ , \\ , \\ , \\ , ( s ) > 1 . \\end{align*}"}
+{"id": "8323.png", "formula": "\\begin{align*} \\varrho ^ t = e ^ { - t \\left ( \\frac { \\nu ^ 2 A } { 2 } + L \\right ) } \\varrho ^ 0 - \\int _ 0 ^ t e ^ { - ( t - \\tau ) \\left ( \\frac { \\nu ^ 2 A } { 2 } + L \\right ) } B ^ \\tau ( \\varrho ^ \\tau , \\varrho ^ \\tau ) d \\tau . \\end{align*}"}
+{"id": "8978.png", "formula": "\\begin{align*} \\kappa = \\frac { S ^ \\varphi } { m ( m - 1 ) } \\ , . \\end{align*}"}
+{"id": "1213.png", "formula": "\\begin{align*} & \\gamma ' ( t ) G ( t , \\sigma ( t ) ) + \\gamma ( t ) \\partial _ t G ( t , \\sigma ( t ) ) + \\gamma ( t ) \\sigma ' ( t ) \\partial _ s G ( t , \\sigma ( t ) ) \\\\ & = G ( t , \\sigma ( t ) ) \\left ( \\gamma ' ( t ) + \\varphi ( t ) \\gamma ( t ) \\right ) + \\gamma ( t ) \\partial _ s G ( t , \\sigma ( t ) ) \\left ( \\sigma ' ( t ) + q + \\varphi ( t ) \\sigma ( t ) \\right ) \\\\ & = 0 . \\end{align*}"}
+{"id": "2351.png", "formula": "\\begin{align*} g = \\frac { m _ g } { f ^ { k _ g } } \\quad h = \\frac { m _ h } { f ^ { k _ h } } \\rlap { . } \\end{align*}"}
+{"id": "4114.png", "formula": "\\begin{align*} y ^ 3 + \\alpha _ 2 y ^ 2 + \\alpha _ 1 y + \\alpha _ 0 = 0 \\end{align*}"}
+{"id": "4945.png", "formula": "\\begin{align*} \\frac { 1 } { n ^ { d } } \\sum \\limits _ { r \\le i _ 1 \\leqslant i _ 2 \\leqslant . . . \\leqslant i _ { d } \\le k } i _ 1 \\cdot i _ 2 \\cdots i _ d = \\frac { 1 } { n ^ { d } } h _ { d } ( r , r + 1 , . . . , k ) \\end{align*}"}
+{"id": "2151.png", "formula": "\\begin{align*} \\Theta ^ { \\{ \\partial W ^ 1 , \\partial W ^ 2 , \\dots , \\partial W ^ m \\} } ( x ) = \\sum _ { i = m + 1 } ^ n \\Theta ( x , \\partial W ^ i ) . \\end{align*}"}
+{"id": "4099.png", "formula": "\\begin{align*} \\mathcal { Q } _ { \\ell , \\vec { v } } = \\mathcal { Q } _ { \\ell , \\alpha \\vec { v } } \\end{align*}"}
+{"id": "1796.png", "formula": "\\begin{align*} f _ t = f _ 0 + \\lambda ^ 2 t \\ Q _ t [ f _ 0 ] + \\lambda ^ 2 t \\ B _ t [ f _ 0 ] \\end{align*}"}
+{"id": "5880.png", "formula": "\\begin{align*} c = - N \\left ( 1 + \\frac { 2 \\alpha } { \\beta } \\right ) \\ , . \\end{align*}"}
+{"id": "2480.png", "formula": "\\begin{align*} G ' ( 0 ) = - 2 I m \\int _ { \\mathbb { R } ^ N } \\left ( \\dfrac { a _ 2 } { m _ 1 } x \\phi _ 0 \\nabla \\overline { \\phi _ 0 } + \\dfrac { a _ 1 } { m _ 2 } x \\psi _ 0 \\nabla \\overline { \\psi _ 0 } \\right ) d x . \\end{align*}"}
+{"id": "5220.png", "formula": "\\begin{align*} ( g ^ { - 1 } ) ' ( z ) = \\frac { 1 } { g ' ( g ^ { - 1 } z ) } = \\frac { \\phi ( z ) } { f _ 0 ( g ^ { - 1 } z ) } , \\end{align*}"}
+{"id": "4253.png", "formula": "\\begin{align*} \\mathbb { G } _ { t , T } ^ { t , \\mu ; u } \\left [ \\Phi ^ { 0 } \\left ( \\rho _ { T } ^ { t , \\mu ; u } \\right ) \\right ] = \\mathbb { G } _ { t , t + \\delta } ^ { t , \\mu ; u } \\left [ Y _ { t + \\delta } ^ { t , \\mu ; u } \\right ] . \\end{align*}"}
+{"id": "3410.png", "formula": "\\begin{align*} t ^ 2 F ( u ) = t ^ 2 F ( \\sqrt { t ^ { - 2 } ( t u ) ^ 2 + ( 1 - t ^ { - 2 } ) 0 } ) < F ( t u ) + ( t ^ { 2 } - 1 ) F ( 0 ) = F ( t u ) , \\mbox { f o r } t > 1 , u \\neq 0 , \\end{align*}"}
+{"id": "2118.png", "formula": "\\begin{align*} \\pi _ 0 ^ { - 1 } \\left ( f ^ { - 1 } ( S _ 0 ) \\cap \\Gamma _ 0 \\right ) \\cap \\Gamma = \\left ( f \\circ \\pi _ 0 \\right ) ^ { - 1 } ( S _ 0 ) \\cap \\Gamma , \\end{align*}"}
+{"id": "4093.png", "formula": "\\begin{align*} \\vec { v } ^ T T _ \\lambda ^ { \\vec { v } } \\vec { e _ i } = \\lambda v _ i \\end{align*}"}
+{"id": "3838.png", "formula": "\\begin{align*} \\Sigma _ { \\mathrm { D } } ( \\delta ) = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { Y } _ 1 \\times \\mathcal { Y } _ 2 ) : \\boldsymbol { K } _ { Y _ 1 } ( \\mu _ { Y _ 1 } , \\gamma _ { 1 } ) \\le \\delta _ 1 , \\boldsymbol { K } _ { Y _ 1 } ( \\mu _ { Y _ 2 } , \\gamma _ { 2 } ) \\le \\delta _ 2 \\right \\} \\end{align*}"}
+{"id": "3024.png", "formula": "\\begin{align*} \\| T + x S \\| _ { ( p , k ) } ^ p + \\| T - x S \\| _ { ( p , k ) } ^ p = 2 \\| T \\| _ { ( p , k ) } ^ p + 2 \\| x S \\| _ { ( p , k ) } ^ p \\hbox { f o r a l l } 0 < x < 1 . \\end{align*}"}
+{"id": "5862.png", "formula": "\\begin{align*} L _ { \\hat { n } } = L _ 0 + 2 J \\sum _ { k = 1 } ^ { \\hat { n } } l _ k \\leq ( 1 + 2 J \\hat { n } ) L _ 0 . \\end{align*}"}
+{"id": "2688.png", "formula": "\\begin{align*} x _ i = \\sum _ { \\substack { 1 \\le j \\le n \\\\ j \\neq i } } ( m _ i \\ss _ { i j } ^ { 3 } ( x _ i - x _ j ) ) . \\end{align*}"}
+{"id": "6557.png", "formula": "\\begin{align*} D _ m = \\omega _ Q ^ { \\frac { 1 } { \\bar { p } _ 2 } - \\frac { 1 } { \\bar { p } _ 1 } } \\omega _ Q ^ m \\int _ 0 ^ \\infty . . . \\int _ 0 ^ \\infty \\frac { \\prod _ { i = 1 } ^ m | r | ^ { Q - \\frac { Q } { p _ i } - 1 } } { ( 1 + \\prod _ { i = 1 } ^ m | s | ) ^ m } d r _ 1 . . . d r _ m . \\end{align*}"}
+{"id": "9.png", "formula": "\\begin{align*} \\hat { f } ( a ^ s \\Lambda _ u ) = \\sum _ { ( \\bar { p } , \\bar { q } ) \\in \\Z ^ { m + n } } f \\left ( \\frac { N p _ 1 + v _ 1 + \\langle \\bar { u } _ 1 , N \\bar { q } + v '' \\rangle } { N } , \\ldots , \\frac { N p _ m + v _ m + \\langle \\bar { u } _ m , N \\bar { q } + v '' \\rangle } { N } , \\frac { N \\bar { q } + v '' } { N } \\right ) . \\end{align*}"}
+{"id": "419.png", "formula": "\\begin{align*} P U _ t + \\frac { 1 } { 2 } \\left [ ( A U ) _ x + A U _ x + ( B U ) _ y + B U _ y \\right ] = \\epsilon ( ( P U _ x ) _ x + ( P U _ y ) _ y ) \\end{align*}"}
+{"id": "1737.png", "formula": "\\begin{align*} \\log \\nu _ t ^ { ( n ) } ( x ) & = e ^ { - \\frac { \\sigma ^ 2 } { 2 } t } \\log \\nu _ 0 ( x ) \\\\ & - \\int _ 0 ^ t \\frac { \\sigma ^ 2 } { 2 } e ^ { - \\frac { \\sigma ^ 2 } { 2 } ( t - s ) } \\left ( \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta \\nu } ( \\nu _ s ^ { ( n - 1 ) } , \\mu _ s ^ { ( n - 1 ) } , x ) - \\log \\pi ( x ) - \\operatorname { D _ { K L } } ( \\nu _ s ^ { ( n - 1 ) } | \\pi ) \\right ) \\mathrm { d } s , \\\\ \\end{align*}"}
+{"id": "2056.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { \\nu _ k } { \\sigma ^ 2 _ k } = 0 . \\end{align*}"}
+{"id": "5479.png", "formula": "\\begin{align*} \\varphi \\circ f = f \\circ \\varphi ^ { \\otimes n } \\quad \\psi \\circ f = f \\circ \\psi ^ { \\otimes n } . \\end{align*}"}
+{"id": "105.png", "formula": "\\begin{align*} \\int \\chi ( x ) e ^ { - i x \\cdot \\xi / h } \\Tilde { R } _ h ( z ) ( \\psi ( x ) e ^ { i x \\cdot \\eta / h } ) d x = \\mathcal { O } ( h ^ \\infty ) \\langle \\xi , \\eta \\rangle ^ { - \\infty } , \\quad \\langle \\xi , \\eta \\rangle \\to \\infty . \\end{align*}"}
+{"id": "2991.png", "formula": "\\begin{align*} c _ { F , j } & = \\frac { 1 } { \\Delta } a _ { ( F ( t ) t ^ j , 0 , \\dots , 0 ) } = \\sum _ { i = 0 } ^ d \\mu _ i \\ , \\frac { 1 } { \\Delta } a _ { ( t ^ { i + j } , 0 , \\dots , 0 ) } = \\sum _ { i = 0 } ^ d \\mu _ i s _ { ( i + j - ( n - 1 ) , 0 , \\dots , 0 ) } \\\\ & = \\sum _ { i = 0 } ^ d \\mu _ i c _ { i + j - ( n - 1 ) } \\underset { ( * ) } { = } \\sum _ { i = 1 } ^ d \\mu _ i c _ { i + j - ( n - 1 ) } \\ , \\end{align*}"}
+{"id": "2159.png", "formula": "\\begin{align*} \\left ( u _ { i } , m _ { i } , k _ { i } \\right ) \\in S _ { 1 } ^ { 2 } \\left ( N _ { 2 } \\right ) \\times S _ { 2 } \\left ( N _ { 3 } \\right ) , i = 1 , 2 \\end{align*}"}
+{"id": "1500.png", "formula": "\\begin{align*} h = \\Delta + O ( \\Delta \\alpha ^ 2 ) , \\quad \\ ; \\ ; \\alpha \\leq \\overline C _ 1 \\max _ { q \\in S ^ 2 : \\ , q \\cdot v = 0 } \\ ; \\max _ { 0 \\le s \\le t } \\ ; ( q \\cdot W _ s ) . \\end{align*}"}
+{"id": "8041.png", "formula": "\\begin{align*} \\| X - \\tau I \\| _ { \\infty } - \\left \\| \\frac { | X - \\tau I | + | X ^ * - \\overline { \\tau } I | } { 2 } \\right \\| _ { \\infty } & = \\left \\| \\begin{pmatrix} | x | & 0 \\\\ 0 & | y | \\end{pmatrix} \\right \\| _ { \\infty } - \\frac { 1 } { 2 } \\left \\| \\begin{pmatrix} | x | + | y | & 0 \\\\ 0 & | x | + | y | \\end{pmatrix} \\right \\| _ { \\infty } \\\\ & = \\left | | x | - | y | \\right | \\\\ & = \\varepsilon ( X ) \\end{align*}"}
+{"id": "1107.png", "formula": "\\begin{align*} \\Phi _ { k , i } ( \\alpha _ 1 , \\ldots , \\alpha _ k ) = 0 , \\forall i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "1134.png", "formula": "\\begin{align*} B ( u , v ) = \\langle u , v \\rangle = \\sum _ { i = 1 } ^ { N } u _ { i } v _ { i } \\left ( u , v \\in \\mathbb { R } ^ { N } \\right ) , \\end{align*}"}
+{"id": "3740.png", "formula": "\\begin{align*} U ( \\vec { a } , \\Lambda ) & \\phi ^ { \\alpha , n } ( f ) U ( \\vec { a } , \\Lambda ) ^ { - 1 } \\left ( S , g _ \\beta \\otimes \\rho _ n ( E ^ \\beta ) \\right ) \\\\ & = \\phi ^ { \\alpha , n } [ f ( \\Lambda ^ { - 1 } ( \\cdot - \\vec { a } ) ) ] \\left ( S , g _ \\beta \\otimes \\rho _ n ( E ^ \\beta ) \\right ) . \\end{align*}"}
+{"id": "1693.png", "formula": "\\begin{align*} \\lambda = \\lambda ( a ) = a ^ { 2 } ( 1 - a ) ^ { - 2 } \\ , . \\end{align*}"}
+{"id": "7831.png", "formula": "\\begin{align*} I : = \\{ i : \\Vert B _ { i } \\Vert _ { 2 } > C \\log ^ { - \\frac { 5 } { 2 } } n \\} . \\end{align*}"}
+{"id": "7424.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } \\ , \\frac { \\theta _ N ^ 2 } { N ^ 4 } \\ , \\Xi ( 2 ) \\le \\frac { C } { M ^ 2 } \\ , , \\end{align*}"}
+{"id": "553.png", "formula": "\\begin{align*} S _ { \\varepsilon } A _ { \\varepsilon } - A _ { \\varepsilon } ^ { * } S _ { \\varepsilon } = 0 . \\end{align*}"}
+{"id": "4104.png", "formula": "\\begin{align*} \\left ( \\left ( Q _ { 1 , \\vec { v } } \\right ) _ i \\right ) _ { j , k } = \\begin{cases} 1 & i \\leq j , k = j - i + 1 \\\\ \\alpha _ { n - i + 1 + j - k } & 2 \\leq i \\leq j , k \\in \\{ j - i + 2 , \\dots , j \\} \\\\ - \\alpha _ { n - i + 1 + j - k } & j < i , j + 1 \\leq k \\leq n + j - i + 1 \\\\ 0 & \\end{cases} . \\end{align*}"}
+{"id": "724.png", "formula": "\\begin{align*} A _ { S } ^ * [ \\partial _ { t } \\mu _ { S } ] ( e ) = G _ { D , 1 } ( e ) \\ , , \\end{align*}"}
+{"id": "2051.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\nu _ k = 0 , \\lim _ { k \\to \\infty } \\sigma _ k = 0 , \\lim _ { k \\to 0 } \\frac { \\nu _ k } { \\sigma ^ 2 _ k } = 0 . \\end{align*}"}
+{"id": "885.png", "formula": "\\begin{align*} \\{ \\underline { a } \\colon \\underline { a } / \\varpi ^ k \\in L ( \\underline { c } ) \\} = \\{ a \\underline { c } ^ \\bot \\colon ( a , \\varpi ) = 1 , | a | < | \\varpi | ^ k \\} . \\end{align*}"}
+{"id": "1116.png", "formula": "\\begin{align*} ( f g ) x = \\lambda _ 1 ( f g ) x & + \\lambda _ 2 ( f x ) g + \\lambda _ 3 ( g f ) x + \\lambda _ 4 ( x f ) g \\\\ & + \\lambda _ 5 x ( f g ) + \\lambda _ 6 y ( f x ) + \\lambda _ 7 x ( g f ) + \\lambda _ 8 g ( x f ) \\\\ \\intertext { a n d } x ( f g ) = \\mu _ 1 ( f g ) x & + \\mu _ 2 ( f x ) g + \\mu _ 3 ( g f ) x + \\mu _ 4 ( x f ) g \\\\ & + \\mu _ 5 x ( f g ) + \\mu _ 6 g ( f x ) + \\mu _ 7 x ( g f ) + \\mu _ 8 g ( x f ) . \\end{align*}"}
+{"id": "1577.png", "formula": "\\begin{align*} \\sum _ { p \\in \\Z } \\mathrm { I } _ { \\mathrm { p } } ( 2 t ) = e ^ { 2 t } . \\end{align*}"}
+{"id": "4217.png", "formula": "\\begin{align*} K _ 1 & = K _ 1 '' + K _ 1 ' T ( \\phi _ R ) \\\\ & = T ( \\phi ) - T ( \\phi _ 0 \\cdots \\phi _ { R - 1 } ) T ( \\phi _ R ) \\\\ & + \\Big ( T ( \\phi _ 0 \\cdots \\phi _ { R - 1 } ) - T ( \\phi _ 0 ) \\cdots T ( \\phi _ { R - 1 } ) \\Big ) T ( \\phi _ R ) \\\\ & = T ( \\phi ) - T ( \\phi _ 0 ) \\cdots T ( \\phi _ R ) \\end{align*}"}
+{"id": "7245.png", "formula": "\\begin{align*} \\sup _ { k \\leq n } | S _ k - T _ k | = o ( n ^ { 1 / p } ( \\log n ) ^ { \\eta } ) \\end{align*}"}
+{"id": "7254.png", "formula": "\\begin{align*} x ^ 3 \\theta ( [ x ] ) \\leq 1 + 8 \\sum _ { k = 1 } ^ { [ x ] } k ^ 2 \\theta ( k ) \\leq 1 + 8 \\sum _ { k \\geq 1 } k ( k \\wedge x ) \\theta ( k ) \\ , . \\end{align*}"}
+{"id": "2277.png", "formula": "\\begin{align*} \\Phi _ b = \\sum _ { n } c _ n a _ n . \\end{align*}"}
+{"id": "2561.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\xi } _ x ( t ) = \\mathbf { v } ( \\xi _ x ( t ) ) , & \\\\ \\xi _ x ( 0 ) = x . \\ \\ & \\end{cases} \\end{align*}"}
+{"id": "7501.png", "formula": "\\begin{align*} \\nabla = \\partial _ z + u ( z ) \\check { \\alpha } + L ( z ) e \\end{align*}"}
+{"id": "548.png", "formula": "\\begin{align*} C _ { T } = c _ { 0 } ^ { - 1 } ( 1 + \\left \\| a \\right \\| _ { L ^ { \\infty } } ) e ^ { c _ { 0 } ^ { - 1 } \\left ( 1 + \\| a ^ { \\prime } \\| _ { L ^ { \\infty } } + \\| q \\| _ { L ^ { \\infty } } + 2 \\left \\| a \\right \\| _ { L ^ { \\infty } } \\right ) T } . \\end{align*}"}
+{"id": "6803.png", "formula": "\\begin{align*} ( k ^ + - 1 ) + ( 2 n - k ^ - - 1 ) - ( 2 n - 1 ) + ( 1 ) = k ^ + - k ^ - . \\end{align*}"}
+{"id": "9094.png", "formula": "\\begin{align*} x _ i - x _ j = 0 , \\ , \\ , \\{ i , j \\} \\in E ( H ) , \\ , Y _ { d + 1 } = 0 , \\ldots , Y _ { n - r } = 0 \\end{align*}"}
+{"id": "8357.png", "formula": "\\begin{align*} r _ { X / G } ( h G ) = r _ X ( h ) G , \\iota _ { X / G } ( h G ) = \\iota _ X ( h ) G . \\end{align*}"}
+{"id": "2410.png", "formula": "\\begin{align*} \\frac { 2 \\pi \\delta _ j } { 3 } \\le \\frac { t \\delta _ j } { 3 j } = \\frac { 2 \\pi K \\delta _ j } { 3 j } \\le \\beta ^ { - 1 } \\frac { 2 \\pi \\delta _ j } { 3 } = \\Big ( 1 + \\frac { \\alpha } { 4 \\pi } \\Big ) \\frac { 2 \\pi \\delta _ j } { 3 } \\le \\frac { 2 \\pi \\delta _ j } { 3 } + \\frac { \\alpha } { 2 } \\end{align*}"}
+{"id": "1004.png", "formula": "\\begin{align*} \\mathcal { D } _ x f ( x ) = \\frac { d } { d x } f ( x ) . \\end{align*}"}
+{"id": "5878.png", "formula": "\\begin{align*} \\nu _ { m - 2 } + 1 = \\nu _ { m } \\ , , S ^ * f _ { m } \\in E _ { \\nu _ { m - 2 } } \\end{align*}"}
+{"id": "9124.png", "formula": "\\begin{align*} \\begin{array} { c c c c l } y ^ { 1 } & = & \\varphi ^ { 1 } ( x ) & = & x ^ { 1 } \\\\ y ^ { 2 } & = & \\varphi ^ { 2 } ( x ) & = & x ^ { 2 } \\ , . \\end{array} \\end{align*}"}
+{"id": "1197.png", "formula": "\\begin{align*} \\mathcal { H } \\oplus \\mathcal { H } = \\operatorname { G r } ( T ) \\oplus W \\operatorname { G r } ( T ^ * ) \\end{align*}"}
+{"id": "7920.png", "formula": "\\begin{align*} \\uparrow ^ i X ^ \\mathbf { n } = X ^ { \\mathbf { n } + e _ i } \\end{align*}"}
+{"id": "2431.png", "formula": "\\begin{align*} & M _ { 0 , \\rho , \\rho ' } ( q x ) = m _ { 0 , \\rho , \\rho ' } ( x ) M _ { 0 , \\rho , \\rho ' } ( x ) \\\\ & M _ { \\infty , \\rho , \\rho ' } ( q x ) = m _ { \\infty , \\rho , \\rho ' } ^ { - 1 } ( x ) M _ { \\infty , \\rho , \\rho ' } ( x ) . \\end{align*}"}
+{"id": "8144.png", "formula": "\\begin{align*} S _ n ( ( 1 - q ) A ) = ( 1 - q ) \\sum _ { k = 0 } ^ n ( - q ) ^ k R _ { 1 ^ k , n - k } ( A ) , \\end{align*}"}
+{"id": "8508.png", "formula": "\\begin{align*} \\zeta _ { p ^ { \\prime } } ( 2 s - 1 ) = \\frac { 1 } { 2 ( s - 1 ) } + C _ { p ^ { \\prime } } ^ { ( 1 ) } + O \\left ( s - 1 \\right ) , \\end{align*}"}
+{"id": "2989.png", "formula": "\\begin{align*} J _ { F , \\Q } = \\left ( \\frac { q _ 0 } { \\Delta } , \\dots , \\frac { q _ { n - 2 } } { \\Delta } \\right ) \\ . \\end{align*}"}
+{"id": "8176.png", "formula": "\\begin{align*} R _ \\ell : = \\frac { R _ 0 } { 5 ^ { 1 / d } } ( \\log c \\ell ) ^ { 1 / d } , \\ : \\ : \\ : \\ell > 1 . \\end{align*}"}
+{"id": "8740.png", "formula": "\\begin{align*} E \\| X _ 1 - X _ 2 \\| ^ { 2 a } _ 2 + E \\| Y _ 1 - Y _ 2 \\| ^ { 2 a } _ 2 - 2 E \\| X _ 1 - Y _ 1 \\| ^ { 2 a } _ 2 = O ( p ^ { a - 2 r + 2 } ) , \\end{align*}"}
+{"id": "7196.png", "formula": "\\begin{gather*} \\forall h \\in \\N , \\mathcal { L } ^ 2 \\left ( \\left \\{ f _ h \\neq g _ h \\right \\} \\cap B _ 1 \\right ) \\lesssim \\left ( \\mathcal { H } ^ 1 \\left ( \\left \\{ f _ h \\neq g _ h \\right \\} \\cap B _ 1 \\right ) \\right ) ^ 2 , \\\\ \\left ( \\mathcal { H } ^ 1 \\left ( \\left \\{ f _ h \\neq g _ h \\right \\} \\cap B _ 1 \\right ) \\right ) ^ 2 = { \\rm o } ( \\gamma _ h ) \\mbox { a s } h \\to + \\infty . \\end{gather*}"}
+{"id": "1589.png", "formula": "\\begin{align*} \\mathrm { d e s c d } ( v ) & = \\# \\{ j \\in [ i + 1 , n ] : \\sigma ( i ) > \\sigma ( j ) \\} , \\\\ \\mathrm { d e p t h } ( v ) & = \\# \\{ j \\in [ i - 1 ] : \\sigma ( j ) > \\sigma ( i ) \\} . \\end{align*}"}
+{"id": "6716.png", "formula": "\\begin{align*} \\P ( \\omega _ i ^ w = \\omega _ i \\mid E _ 1 , \\dots , E _ { i - 1 } ) \\le \\ell / ( n - i \\ell ) . \\end{align*}"}
+{"id": "9275.png", "formula": "\\begin{align*} \\| T _ { \\eta } f \\| _ { L ^ p ( x ^ \\gamma d x ) } ^ p & \\lesssim \\int _ 0 ^ \\infty x ^ { - \\tau - 1 + \\gamma } \\int _ x ^ \\infty \\abs { f ( z ) } ^ p z ^ { \\tau } \\ , d z \\ , d x = \\int _ 0 ^ \\infty \\abs { f ( z ) } ^ p z ^ { \\tau } \\int _ 0 ^ z x ^ { - \\tau - 1 + \\gamma } \\ , d x \\ , d z \\\\ & \\simeq \\| f \\| _ { L ^ p ( x ^ \\gamma d x ) } ^ p , \\end{align*}"}
+{"id": "8728.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i } X _ { i } ^ { \\top } ( X _ { i } - \\bar { X } _ { - i } ) & = \\frac { 1 } { n } \\sum _ i \\| X _ i \\| _ { 2 } ^ { 2 } - \\frac { 1 } { n ( n - 1 ) } \\sum _ { i _ 1 \\neq i _ 2 } X _ { i _ 1 } ^ \\top X _ { i _ 2 } . \\end{align*}"}
+{"id": "6092.png", "formula": "\\begin{align*} \\Vert Z \\Vert _ { \\mathcal { D } _ { \\mathbf { X } , \\tilde { \\alpha } } ^ { \\gamma } ( I ) } : = \\Vert Z \\Vert _ { C ( I ; \\mathcal { B } _ { \\tilde { \\alpha } } ) } + \\Vert Z ^ { \\prime } \\Vert _ { \\mathcal { E } ^ { 0 , \\gamma } _ { \\tilde { \\alpha } - \\gamma ; I } } + \\Vert Z ^ { \\# } \\Vert _ { \\mathcal { E } ^ { \\gamma , 2 \\gamma } _ { \\tilde { \\alpha } ; I } } < \\infty \\end{align*}"}
+{"id": "5056.png", "formula": "\\begin{align*} \\psi ( x , t ) = \\phi ( x ) + \\tilde { v } ( x , t ) \\end{align*}"}
+{"id": "6674.png", "formula": "\\begin{align*} \\rho [ G _ \\alpha ( u ) ] _ t = \\rho G _ \\alpha ' ( u ) u _ t = - G _ \\alpha ' ( u ) [ - \\Delta u + ( - \\Delta ) ^ s u ] \\textrm { i n } \\ ; \\ , S _ T \\ , . \\end{align*}"}
+{"id": "1707.png", "formula": "\\begin{align*} C _ n ^ \\mu ( - z ) = ( - 1 ) ^ n C _ n ^ \\mu ( z ) . \\end{align*}"}
+{"id": "5082.png", "formula": "\\begin{align*} \\partial _ t E ( t ) = - 2 E ( t ) + E _ 1 ( t ) + E _ 2 ( t ) + E _ 3 ( t ) , \\end{align*}"}
+{"id": "242.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { m ^ 2 } { n ^ 3 } } = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { n } { 6 } + \\frac { n ^ 2 } { 2 } + \\frac { n ^ 3 } { 3 } \\right ) \\frac { z ^ n } { n ^ 3 } \\right \\} \\end{align*}"}
+{"id": "5111.png", "formula": "\\begin{align*} \\int _ { \\Omega ' } [ a ^ { i j , k l } ( D ^ { 2 } u ) u _ { i j } ] ^ { h _ { m } } \\eta _ { k l } d x = 0 . \\end{align*}"}
+{"id": "942.png", "formula": "\\begin{align*} Z ^ * : C H _ 0 ( X ) \\to C H _ 0 ( X ) , Z ^ * ( [ Y ] ) = [ { \\pi _ { 1 , X } } _ * ( Z \\cdot \\pi _ { 2 , X } ^ * ( Y ) ) ] \\end{align*}"}
+{"id": "6224.png", "formula": "\\begin{align*} \\mu _ N \\coloneqq \\frac { 1 } { N + 1 } \\sum _ { i = 1 } ^ { N + 1 } \\delta _ { x _ i } \\end{align*}"}
+{"id": "1948.png", "formula": "\\begin{align*} f ( z _ 1 , \\ldots , z _ { m } ) = a _ 0 + \\left ( \\sum _ { S \\in \\mathcal { Q } _ m } \\left ( \\prod _ { i \\in S } z _ i \\right ) f _ S ( z _ S ) \\right ) + f ^ * ( z _ 1 , \\ldots , z _ { m } ) , \\end{align*}"}
+{"id": "5018.png", "formula": "\\begin{align*} ( 1 - \\varepsilon - \\delta ) \\| x \\| \\leq \\| V \\theta _ g x \\| , \\forall x \\in \\mathcal { X } \\implies ( 1 - \\varepsilon - \\delta ) \\| y \\| = ( 1 - \\varepsilon - \\delta ) \\| \\theta _ g ^ { - 1 } y \\| \\leq \\| V y \\| , \\forall y \\in \\ell ^ p ( [ n ] ) , \\end{align*}"}
+{"id": "8917.png", "formula": "\\begin{align*} A _ l = \\sum _ { q \\geq 2 } \\sum _ { ( n _ 1 , \\dots , n _ q ) \\in \\mathbb { I } _ { d _ 1 ^ j } ^ q } \\sum _ { m _ 1 + \\cdots + m _ q = l } \\beta _ { ( n _ 1 , \\dots , n _ q ) } [ P _ { m _ 1 } ( U _ { n _ 1 } ) , \\dots , P _ { m _ q } ( U _ { n _ q } ) ] . \\end{align*}"}
+{"id": "2598.png", "formula": "\\begin{align*} \\nu _ { _ { \\Omega _ h } } ( 0 ) : = \\frac { T _ h \\nu _ { \\Omega } ( 0 ) } { | T _ h \\nu _ { \\Omega } ( 0 ) | } . \\end{align*}"}
+{"id": "7005.png", "formula": "\\begin{align*} \\deg \\left ( \\frac { \\textbf { Q } ^ { \\lambda _ j } } { Q _ i ^ { \\lambda _ j ( Q _ i ) } } \\right ) = \\deg \\left ( \\frac { \\tilde { \\textbf { Q } } ^ { \\lambda _ j } } { \\tilde Q _ i ^ { \\lambda _ j ( Q _ i ) } } \\right ) < \\deg ( Q _ i ) \\mbox { f o r e v e r y } j , 1 \\leq j \\leq r . \\end{align*}"}
+{"id": "8435.png", "formula": "\\begin{align*} \\zeta _ { p } ( \\sigma + i t ) = \\begin{cases} O \\left ( | t | ^ { 1 - \\sigma } \\log | t | \\right ) & \\frac { 1 } { 2 } \\leq \\sigma \\leq 1 , \\\\ O \\left ( | t | ^ { \\frac { 1 } { 2 } } \\log | t | \\right ) & 0 < \\sigma \\leq \\frac { 1 } { 2 } , \\\\ O \\left ( | t | ^ { \\frac { 1 } { 2 } - \\sigma } \\log | t | \\right ) & \\sigma \\leq 0 . \\end{cases} \\ , \\ , \\ , \\ , \\ , | t | \\rightarrow \\infty . \\end{align*}"}
+{"id": "9006.png", "formula": "\\begin{align*} ( 1 + w ) ( R ^ \\varphi _ { k j , i } - R ^ \\varphi _ { k i , j } ) = w _ t R _ { t k j i } + \\frac { S ^ \\varphi } { m - 1 } ( w _ i \\delta _ { j k } - w _ j \\delta _ { i k } ) - ( w _ i R ^ \\varphi _ { j k } - w _ j R ^ \\varphi _ { i k } ) \\ , . \\end{align*}"}
+{"id": "4853.png", "formula": "\\begin{align*} ( \\prod _ { i = 1 } ^ m a ( i ) \\ast f ( t ( i ) ) ) \\ast a ( m + 1 ) \\in B \\cap \\sigma ( W ) . \\end{align*}"}
+{"id": "5525.png", "formula": "\\begin{align*} W : = \\left \\{ \\left ( x , L _ { x } \\omega \\right ) : x \\in X , \\omega \\in G \\right \\} = \\bigsqcup _ { x \\in X } \\{ x \\} \\times L _ { x } \\backslash G . \\end{align*}"}
+{"id": "3514.png", "formula": "\\begin{align*} \\sigma _ K = \\{ ( q , p ) \\mid p ^ i = 0 , \\forall i \\in K , \\ q ^ j = 0 , \\forall j \\notin K \\} . \\end{align*}"}
+{"id": "2821.png", "formula": "\\begin{align*} S _ k ^ { 1 , n } ( t ) = Y _ n ( t ) ( S _ k ^ { 1 , n } ( 0 ) + \\int _ { 0 } ^ t Y _ n ( \\tau ) ^ { - 1 } S _ { k + q + 1 } ^ { 1 , n } ( \\tau ) d \\tau ) ; \\end{align*}"}
+{"id": "7395.png", "formula": "\\begin{align*} \\frac { \\Sigma _ N ^ 2 ( L , \\alpha ) - L } { L } = X _ N ( L , \\alpha ) - \\frac { L } { N } , \\end{align*}"}
+{"id": "5193.png", "formula": "\\begin{align*} p \\widetilde { \\mathbb { S } } _ { \\lambda ^ i } p \\widetilde { \\mathbb { S } } _ { \\lambda ^ j } = \\sum _ { \\lambda ^ \\ell \\succeq \\lambda ^ i , \\lambda ^ j } \\sum _ { P : \\delta P = \\Delta _ { \\sigma _ r \\lambda ^ i ~ \\sigma _ r \\lambda ^ j } ^ { \\sigma _ r \\lambda ^ \\ell } } \\sigma _ r w t ( P ) p \\widetilde { \\mathbb { S } } _ { \\lambda ^ \\ell } , \\end{align*}"}
+{"id": "8380.png", "formula": "\\begin{align*} P _ { n } ( \\epsilon , q ) \\leq \\sum \\limits _ { c = r } ^ { m } P \\left \\{ \\omega : \\sup _ { k \\geq n } { k \\choose r } ^ { - \\frac { 1 } { ( q + 1 ) } } \\left | \\left | C _ { m , k , c } ( \\omega ) \\right | \\right | _ { { \\cal { B } } \\otimes { \\cal { B } } } \\geq \\epsilon _ { c } \\right \\} , \\end{align*}"}
+{"id": "8926.png", "formula": "\\begin{align*} \\prod _ { l = 1 } ^ { j } y \\big ( \\bigl \\{ X _ { n i } ^ { ( l ) } \\bigr \\} \\big ) = \\sum _ { l = 1 } ^ { j } Z _ l + \\sum _ { l = j + 1 } ^ k B _ l ^ { ( j ) } . \\end{align*}"}
+{"id": "2347.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ d { k \\choose 2 } \\abs { A _ k } & = \\sum _ { k = 2 } ^ d \\sum _ { i = 1 } ^ { k - 1 } i \\abs { A _ k } = \\sum _ { i = 1 } ^ { d - 1 } i \\sum _ { k = i + 1 } ^ { d } \\abs { A _ k } = \\abs { A _ 2 } + \\biggl \\lvert \\bigcup _ { k \\geq 3 } A _ k \\biggr \\rvert + \\sum _ { i = 2 } ^ { d - 1 } i \\cdot \\biggl \\lvert \\bigcup _ { k \\geq i + 1 } A _ k \\biggr \\rvert . \\end{align*}"}
+{"id": "1568.png", "formula": "\\begin{align*} \\theta _ { s } : = \\frac { \\tau } { \\tau - s } M _ { \\tau } ( \\mathrm { P } _ 0 ) ^ { \\frac { s } { \\tau } } \\end{align*}"}
+{"id": "1265.png", "formula": "\\begin{align*} Z ( G ) = M ( G ) = n - 2 T + s _ 1 + 2 s _ 0 , \\end{align*}"}
+{"id": "7964.png", "formula": "\\begin{align*} i _ a = \\inf _ { t > 0 } \\frac { t \\ , a ' ( t ) } { a ( t ) } \\quad s _ a = \\sup _ { t > 0 } \\frac { t \\ , a ' ( t ) } { a ( t ) } , \\end{align*}"}
+{"id": "1476.png", "formula": "\\begin{align*} P \\psi ^ h _ { \\mu , \\xi } ( x ) = \\psi ^ h _ { \\mu , \\xi } ( x ) - \\alpha _ n \\mu ^ { \\frac { n } { 2 } } \\partial _ { \\xi _ h } H ( x , \\xi ) + O ( \\mu ^ { \\frac { n + 2 } { 2 } } ) , \\end{align*}"}
+{"id": "4139.png", "formula": "\\begin{align*} \\vec { v } _ { s , t , f , 0 } = A _ { 2 , f , t } \\overline { A _ { 1 , t , s } A _ { 3 , t , s } A _ { 2 , s , t } } . \\end{align*}"}
+{"id": "2535.png", "formula": "\\begin{align*} \\mu _ { a \\cup b } ( \\Pi ) = \\Pi - P _ a - P _ b + P _ a \\cup P _ b . \\end{align*}"}
+{"id": "556.png", "formula": "\\begin{align*} g _ { \\varepsilon } ( t , k ) : = \\left ( \\tilde { a } _ { \\varepsilon } - a _ { \\varepsilon } \\right ) ( t ) \\mathcal { H } _ { \\hbar , V } \\tilde { u } _ { \\varepsilon } ( t , k ) + \\left ( \\tilde { q } _ { \\varepsilon } - q _ { \\varepsilon } \\right ) ( t ) \\tilde { u } _ { \\varepsilon } ( t , k ) + ( f _ { \\varepsilon } - \\tilde { f } _ { \\varepsilon } ) ( t , k ) . \\end{align*}"}
+{"id": "3599.png", "formula": "\\begin{align*} L ( f ) & \\ = \\ m + 1 - L ( q ^ { \\beta } ) + [ r ( p ) \\ < \\ 1 0 ^ { L ( f ) + L ( q ^ { \\beta } ) - 1 } ] \\\\ & \\ \\le \\ m + 1 - \\ell + 1 \\ = \\ m + 2 - \\ell . \\end{align*}"}
+{"id": "486.png", "formula": "\\begin{align*} n \\mapsto k _ 1 , \\ ; a \\mapsto t ^ { k _ 1 } t _ 1 / t _ r , \\ ; b \\mapsto c ^ { 2 n - 2 } d ^ 2 t ^ { 2 - k _ 1 } / t _ 1 t _ r = t ^ { 1 - k _ 0 } t _ 2 / t _ r , \\end{align*}"}
+{"id": "7538.png", "formula": "\\begin{align*} p < 2 : \\abs { A _ 1 - A _ 2 } ^ p & \\lesssim _ { ( d , p ) } \\left ( \\abs { A _ 1 } ^ 2 + \\abs { A _ 2 } ^ 2 \\right ) ^ { \\frac { p - 2 } { 2 } } \\abs { A _ 1 - A _ 2 } ^ 2 \\\\ & + \\left ( \\left ( \\abs { A _ 1 } ^ 2 + \\abs { A _ 2 } ^ 2 \\right ) ^ { \\frac { p - 2 } { 2 } } \\abs { A _ 1 - A _ 2 } ^ 2 \\right ) ^ { { \\frac { p } { 2 } } } \\abs { A _ 1 } ^ { \\frac { p ( 2 - p ) } { 2 } } . \\end{align*}"}
+{"id": "6413.png", "formula": "\\begin{align*} \\pi _ { \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 } ( a \\otimes b ) & = \\pi _ { \\varphi _ 1 } ( a ) \\otimes \\pi _ { \\varphi _ 2 } ( b ) , a \\in M _ 1 , b \\in M _ 2 , \\\\ \\lambda ^ { \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 } ( t ) & = \\lambda ^ { \\varphi _ 1 } ( t ) \\otimes \\lambda ^ { \\varphi _ 2 } ( t ) , t \\in \\mathbb { R } . \\end{align*}"}
+{"id": "2825.png", "formula": "\\begin{align*} \\overset \\cdot b _ i = b _ i ( \\sum _ { j = 1 } ^ p b _ { i + j } - \\sum _ { j = 1 } ^ p b _ { i - j } ) , i \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "4344.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { \\alpha , p } = \\left ( \\sum \\limits _ { i = 1 } ^ { m } \\alpha _ { i } \\left \\Vert u _ { i } \\right \\Vert ^ { p } \\right ) ^ { 1 / p } . \\end{align*}"}
+{"id": "4695.png", "formula": "\\begin{align*} \\Lambda _ \\chi ^ { \\rm s t i f f } \\vect g : = \\vect h \\in H ^ { - 1 / 2 } ( \\Gamma ; \\C ^ 3 ) , \\mbox { w h e r e : } \\left \\{ \\begin{array} { c c } \\vect u = \\Pi _ \\chi ^ { \\rm s t i f f } \\vect g & \\\\ \\vect h = \\frac { \\partial \\vect u } { \\partial \\nu } | _ \\Gamma . & \\end{array} \\right . \\end{align*}"}
+{"id": "3400.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\ h ' ( r , x ) > h ( r , x ) 0 < r \\leq \\delta . \\\\ & \\ h ' ( r , x ) = \\big ( \\frac r \\delta \\big ) ^ 2 h ' ( \\delta , x ) r \\geq \\delta . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2919.png", "formula": "\\begin{align*} \\gamma = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} . \\end{align*}"}
+{"id": "5035.png", "formula": "\\begin{align*} g = \\widetilde g _ 0 \\cdot g _ 1 \\cdots g _ p , \\end{align*}"}
+{"id": "756.png", "formula": "\\begin{align*} D = \\sqrt { \\phi ^ 2 + \\rho ^ 2 | \\nabla u | ^ 2 } , \\end{align*}"}
+{"id": "1980.png", "formula": "\\begin{align*} 0 \\leq u _ { n \\bar n } ( 0 ) = u _ { x _ n x _ n } + u _ { y _ n y _ n } ( 0 ) \\leq C . \\end{align*}"}
+{"id": "8454.png", "formula": "\\begin{align*} \\frac { 1 } { \\left ( \\lambda _ { n } ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } = \\sum _ { m = 0 } ^ { \\infty } \\left ( \\begin{array} { c } - s \\\\ m \\end{array} \\right ) \\ , \\frac { x ^ { 2 m } } { \\lambda _ { n } ^ { 2 m + 2 s } } . \\end{align*}"}
+{"id": "3643.png", "formula": "\\begin{align*} \\overline { \\mathrm { d i m } } _ B Y + \\overline { \\mathrm { u m d i m } } _ Z \\left ( G , d , \\nu ^ \\mathbb { N } \\right ) = \\overline { \\mathrm { u m d i m } } _ { Y ^ { \\mathbb { N } } \\times Z } \\left ( F , D \\right ) . \\end{align*}"}
+{"id": "2178.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c } u _ { t t } ( x , t ) = \\Delta u ( x , t ) + f ( x , t ) , & ( x , t ) \\in \\Omega \\times ( 0 , T ] , \\\\ u = 0 , & ( x , t ) \\in \\partial \\Omega \\times ( 0 , T ] , \\\\ u ( x , 0 ) = \\psi _ 0 , ~ u _ t ( x , 0 ) = \\psi _ 1 , & x \\in \\Omega . \\end{array} \\right . \\ , \\end{align*}"}
+{"id": "1818.png", "formula": "\\begin{align*} [ \\N ^ \\ell , b _ { k } ( t ) b _ { - k } ( t ) ] = \\sum _ { n = 0 } ^ { \\ell - 1 } \\binom { \\ell } { n } 4 ^ { \\ell - n } ( \\N + 4 ) ^ { \\frac { n - 1 } { 2 } } b _ k ( t ) b _ { - k } ( t ) \\N ^ { \\frac { n + 1 } { 2 } } \\ . \\end{align*}"}
+{"id": "3510.png", "formula": "\\begin{align*} \\sum _ { t \\in ( a , b ) } \\dim ( \\sigma _ 0 ( t ) \\cap \\sigma ) - \\sum _ { \\lambda \\in ( \\lambda _ 0 , 0 ) } \\dim ( \\sigma _ \\lambda ( b ) \\cap \\sigma ) = 0 . \\end{align*}"}
+{"id": "604.png", "formula": "\\begin{align*} g ( 0 ) = 0 < g ( 1 ) = \\frac { 1 } { 2 } \\sqrt { C - 4 } < g ( s _ + ) . \\end{align*}"}
+{"id": "4420.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } G ( U _ n ) = \\limsup _ { n \\rightarrow \\infty } \\left \\{ ( 4 - 2 d ) \\omega Q ( U _ n ) + ( 3 - d ) \\mathbf { c } \\cdot \\mathbf { P } ( U _ n ) \\right \\} \\ge \\eta . \\end{align*}"}
+{"id": "8651.png", "formula": "\\begin{align*} & a : = \\left ( \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 2 } y ^ { j + 1 } d y \\right ) \\left ( \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } y ^ { j - 1 } d y \\right ) , \\\\ & b : = \\left ( \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 2 } y ^ { j } d y \\right ) \\left ( \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } y ^ { j } d y \\right ) . \\end{align*}"}
+{"id": "6318.png", "formula": "\\begin{align*} \\begin{cases} \\dot \\gamma ( t ) = \\displaystyle u _ 1 ( t ) X _ 1 ( \\gamma ( t ) ) + u _ 2 ( t ) X _ 2 ( \\gamma ( t ) ) , \\\\ \\displaystyle u ( t ) \\in B ^ { \\norm { \\cdot } } _ 1 ( 0 ) , \\quad \\gamma ( 0 ) = q _ 0 , \\quad \\quad \\gamma ( T ) = q _ 1 , \\\\ T \\to \\min . \\end{cases} \\end{align*}"}
+{"id": "1664.png", "formula": "\\begin{align*} \\nabla _ { { f } X } \\ , \\xi _ i - { f } \\ , \\nabla _ { X } \\ , \\xi _ i = 0 , X \\in \\mathfrak { X } _ M . \\end{align*}"}
+{"id": "207.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c ) = 1 \\\\ a , b , c \\geq 1 } } \\left ( \\frac { 1 } { 1 - x ^ a y ^ b z ^ c } \\right ) ^ { \\frac { c ^ 3 } { a ^ 2 b ^ 2 } } = \\exp \\left \\{ L i _ 2 ( x ) L i _ 2 ( y ) L i _ { - 3 } ( z ) \\right \\} \\end{align*}"}
+{"id": "1362.png", "formula": "\\begin{align*} v ( T ) : = \\sup \\{ | x ^ * ( T x ) | \\colon ( x , x ^ * ) \\in \\Pi ( X ) \\} . \\end{align*}"}
+{"id": "7895.png", "formula": "\\begin{align*} \\frac { 2 \\binom { n - k } { k } } { | H | } \\sum _ { h \\in H } \\chi ^ { \\left [ 2 , 1 ^ { n - 2 } \\right ] } ( x h ) = \\frac { 2 \\binom { n - k } { k } } { | H | } \\sum _ { h \\in H } \\chi ^ { [ 1 ^ n ] } ( x h ) \\chi ^ { [ n - 1 , 1 ] } ( x h ) = - 2 \\binom { n - k - 1 } { k - 1 } . \\end{align*}"}
+{"id": "9158.png", "formula": "\\begin{align*} v _ { [ 0 , R - \\kappa ] } = \\rho ( \\zeta _ { [ - q _ { 1 } , - 1 ] } , x , y _ { [ 0 , R ] } ^ { d } ) \\ , . \\end{align*}"}
+{"id": "6519.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\mathrm { e } ^ { \\beta t } t ^ \\nu K _ \\nu ( \\alpha t ) \\ , \\mathrm { d } t = \\frac { \\sqrt { \\pi } ( 2 \\alpha ) ^ \\nu } { ( \\alpha - \\beta ) ^ { 2 \\nu + 1 } } \\frac { \\Gamma ( 2 \\nu + 1 ) } { \\Gamma ( \\nu + 3 / 2 ) } { } _ 2 F _ 1 \\bigg ( 2 \\nu + 1 , \\nu + \\frac { 1 } { 2 } ; \\nu + \\frac { 3 } { 2 } ; - \\frac { \\alpha + \\beta } { \\alpha - \\beta } \\bigg ) , \\end{align*}"}
+{"id": "2519.png", "formula": "\\begin{align*} \\Delta ( 2 , 2 n ) = \\Big \\{ e _ 1 + e _ j \\ : : \\ : \\{ i , j \\} \\in \\binom { [ 2 n ] } { 2 } \\Big \\} \\subset H \\end{align*}"}
+{"id": "4143.png", "formula": "\\begin{align*} M _ 0 = \\begin{pmatrix} 1 & - f ^ { 2 } + f s - t + 1 & f + 1 \\\\ 1 & - 2 \\ , f + s + 1 & 1 \\\\ 1 & - f + s + 1 & - f + s + t + 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "1254.png", "formula": "\\begin{align*} \\int _ { b _ { \\ell + 1 } } ^ 1 E _ 1 ( x ) \\ , d x & = \\int _ { T ^ { - m } [ b _ { \\ell + 1 } , 1 ] } \\frac { | u ' ( x ) | } { ( T ^ m ) ' x } \\ , d x \\\\ & = \\sum _ { j = 0 } ^ \\infty \\int _ { T ^ { - m } [ b _ { \\ell + 1 } , 1 ] \\cap [ b _ { j + 1 } , b _ j ] } \\frac { | u ' ( x ) | } { ( T ^ m ) ' x } \\ , d x + \\int _ { T ^ { - m } [ b _ { \\ell + 1 } , 1 ] \\cap [ \\frac 1 2 , 1 ] } \\frac { | u ' ( x ) | } { ( T ^ m ) ' x } \\ , d x . \\end{align*}"}
+{"id": "2478.png", "formula": "\\begin{align*} G ( t ) = G ( 0 ) + G ' ( 0 ) t + \\int _ 0 ^ t ( t - s ) G '' ( s ) d x , 0 \\leq t < + \\infty . \\end{align*}"}
+{"id": "7954.png", "formula": "\\begin{align*} I _ H [ u ] = \\int _ \\Omega B ( H ( \\nabla u ) ) \\ , d x - \\int _ \\Omega f u \\ , d x \\end{align*}"}
+{"id": "7559.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\overline { S } _ n / \\mathbb { E } ( \\overline { S } _ n ) = 1 \\end{align*}"}
+{"id": "5999.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } ( p ) B ( [ \\epsilon , x ] ) = a ^ { \\tfrac { 3 } { 2 } } x e ^ { - ( \\det p ) \\epsilon \\pi x ^ 2 a ^ 2 } \\cdot e ^ { \\pi i \\epsilon ( \\det p ) x ^ 2 a b } = a ^ { \\tfrac { 3 } { 2 } } x e ^ { i \\epsilon \\pi x ^ 2 z _ p } . \\end{align*}"}
+{"id": "2844.png", "formula": "\\begin{align*} \\tilde C = \\{ a _ 0 a _ 1 + a _ 1 a _ 2 + a _ 2 a _ 3 + a _ 3 a _ 4 , \\ , - a _ 0 a _ 1 a _ 3 a _ 4 \\} = \\{ b _ 0 + b _ 1 + b _ 2 + b _ 3 , - b _ 0 b _ 3 \\} \\end{align*}"}
+{"id": "7795.png", "formula": "\\begin{align*} \\mathcal { R } \\Phi ( A \\circ B ) & = \\Phi ( \\mathcal { R } ( A \\circ B ) ) & & \\\\ & \\geq \\Phi ( \\mathcal { R } A \\circ \\mathcal { R } B ) & & \\\\ & = \\Phi ( \\mathcal { R } A ) \\circ \\Phi ( \\mathcal { R } B ) & & \\\\ & = \\mathcal { R } ( \\Phi ( A ) ) \\circ \\mathcal { R } ( \\Phi ( B ) ) . & & \\end{align*}"}
+{"id": "965.png", "formula": "\\begin{align*} = \\int \\limits _ { R _ * } ^ { R _ 0 } \\int \\limits _ { S ( z _ 1 , t ) } Q ( y ) \\cdot \\eta ^ { \\ , q } ( | y - z _ 1 | ) \\ , d \\mathcal { H } ^ { n - 1 } \\ , d t = \\frac { \\omega _ { n - 1 } } { I ^ { q - 1 } } < \\infty \\ , . \\end{align*}"}
+{"id": "1790.png", "formula": "\\begin{align*} [ ( \\varphi + \\psi ) ^ { \\otimes p } ] = \\sum ^ p _ { i = 0 } { p \\choose i } [ \\varphi ^ { \\otimes i } \\otimes \\psi ^ { \\otimes p - i } ] = [ \\varphi ^ { \\otimes p } ] + [ \\psi ^ { \\otimes p } ] \\ ; . \\end{align*}"}
+{"id": "2321.png", "formula": "\\begin{align*} O ( R \\log ^ \\ast n + R ( 2 \\Delta ) ^ { 4 R } + R q ) \\ , = \\ , O ( R \\log ^ \\ast n + R ( 2 \\Delta ) ^ { 4 R } + R ( 2 \\Delta ) ^ { 2 R } ) . \\end{align*}"}
+{"id": "5925.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( s ( - 1 ) ) f ( [ \\epsilon , y ] ) = f ( [ - \\epsilon , y ] ) , \\end{align*}"}
+{"id": "6604.png", "formula": "\\begin{align*} \\mathfrak { w } _ H : = ( H ^ + / ( H ^ + ) ^ 2 ) _ 1 \\ , ( = H _ 1 / ( H ^ + \\cap H _ 0 ) H _ 1 ) , \\end{align*}"}
+{"id": "4710.png", "formula": "\\begin{align*} \\mathcal { A } _ \\chi : = \\left ( \\left ( \\widehat { \\Pi } _ \\chi ^ { \\rm s t i f f } \\right ) ^ * \\right ) ^ { - 1 } \\widehat { \\Lambda } _ \\chi ^ { \\rm s t i f f } \\left ( \\widehat { \\Pi } _ \\chi ^ { \\rm s t i f f } \\right ) ^ { - 1 } , \\mathcal { B } _ \\chi ( z ) : = z I + \\left ( \\left ( \\widehat { \\Pi } _ \\chi ^ { \\rm s t i f f } \\right ) ^ * \\right ) ^ { - 1 } \\widehat { M } _ \\chi ^ { \\rm s o f t } ( z ) \\left ( \\widehat { \\Pi } _ \\chi ^ { \\rm s t i f f } \\right ) ^ { - 1 } , \\end{align*}"}
+{"id": "1768.png", "formula": "\\begin{align*} \\| F \\| _ { \\Theta ; 1 , p } : = \\| F \\| _ { L ^ p } + \\| D _ \\Theta F \\| _ { L ^ p } . \\end{align*}"}
+{"id": "6968.png", "formula": "\\begin{align*} \\Lambda - \\Delta = \\{ \\lambda - \\delta \\mid \\lambda \\in \\Lambda \\mbox { a n d } \\delta \\in \\Delta \\} . \\end{align*}"}
+{"id": "6795.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = - x _ 1 ^ 2 + \\cdots + - x _ k ^ 2 + x _ { k + 1 } ^ 2 + \\cdots + x _ { 2 n - 1 } ^ 2 + x _ { 2 n } ^ 3 , \\end{align*}"}
+{"id": "2668.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 } s _ \\alpha [ Y ] ( x ) = \\sigma [ Y ] ( x ) \\quad \\lim _ { \\alpha \\to 0 } t _ \\alpha [ Y ] ( x ) = \\sigma [ Y ] ( x ) \\end{align*}"}
+{"id": "1581.png", "formula": "\\begin{align*} \\frac { ( 2 n + | x | _ { 1 } ) ! } { \\prod _ { i = 1 } ^ { d } q _ { i } ! ( q _ { i } + | x _ { i } | ) ! } \\leq 2 ^ { 2 n + | x | _ { 1 } } \\frac { ( n + | x | _ { 1 } ) ! } { \\prod _ { i = 1 } ^ { d } ( q _ { i } + | x _ { i } | ) ! } C _ { d } \\frac { d ^ { n } } { n ^ { \\frac { d - 1 } { 2 } } } . \\end{align*}"}
+{"id": "7468.png", "formula": "\\begin{align*} \\| u _ \\varepsilon - \\mathfrak { U } _ { M } ^ { ( \\varepsilon ) } \\| _ { C ( \\overline { \\Omega } _ \\varepsilon \\times [ 0 , T ] ) } : = \\max _ { \\overline { \\Omega _ \\varepsilon } \\times [ 0 , T ] } | u _ \\varepsilon - \\mathfrak { U } _ { M } ^ { ( \\varepsilon ) } | \\le C _ M \\ , \\varepsilon ^ { \\gamma ( M + \\lfloor \\alpha \\rfloor - 1 ) } . \\end{align*}"}
+{"id": "943.png", "formula": "\\begin{align*} Z ^ 0 _ n ( L ) = \\{ ( x , y ) \\in X \\times X \\ , | \\ , \\exists \\ , C \\in | L | \\mbox { s m o o t h s . t . } x , y \\in C , \\ , n [ x - y ] = 0 \\in J C \\} , \\end{align*}"}
+{"id": "5441.png", "formula": "\\begin{align*} Q ( r _ 1 , \\theta ) \\approx & \\frac { \\pi ^ 2 \\lambda R _ S ( R _ { m a x } - r _ 1 ) } { N R _ E } \\sum _ { n = 1 } ^ N \\sqrt { 1 - \\phi _ n ^ 2 } c _ n \\\\ & \\times \\left ( 1 - \\prod _ { m = 1 } ^ { M } { \\left ( 1 + \\frac { m \\eta \\theta r _ 1 ^ { \\alpha } } { M c _ n ^ { \\alpha } } \\right ) ^ { - M b _ m } } \\right ) \\end{align*}"}
+{"id": "5868.png", "formula": "\\begin{align*} | \\Theta _ k ^ { n _ 1 } | \\leq L _ { n _ 1 } ^ d + \\sum _ { n = k } ^ { n _ 1 } K _ 2 ( 3 ( L _ { k - 1 } ) _ { n _ 1 } ) ^ d = L ^ { \\beta d } + ( n _ 1 - k + 1 ) K _ 2 3 ^ d L ^ { \\rho ^ { k - 1 } \\beta d } \\leq C L ^ { \\beta d } . \\end{align*}"}
+{"id": "7054.png", "formula": "\\begin{align*} \\mathcal J = ( d f _ j \\mid j \\in J ) . \\end{align*}"}
+{"id": "7049.png", "formula": "\\begin{align*} f _ { \\tilde { \\textbf { Q } } } = g ( x ) ^ r h ( x ) \\mbox { f o r s o m e } h \\in K [ x ] , r \\in \\N \\mbox { a n d } ( g , h ) = 1 . \\end{align*}"}
+{"id": "7040.png", "formula": "\\begin{align*} \\bar { \\mathcal I } = ( \\mathcal I _ 1 + ( g ( X _ 0 ) ) ) K [ \\textbf { X } ] . \\end{align*}"}
+{"id": "3422.png", "formula": "\\begin{align*} \\Gamma _ { \\varepsilon , K } ( u ) = \\frac { 1 } { 2 } \\int _ { \\R ^ N } ( | \\nabla u | ^ 2 + \\widetilde V ( \\varepsilon x ) u ^ 2 ) - \\int _ { \\mathbb R ^ N } F _ K ( u ) + \\Phi _ { \\varepsilon } ( u ) + \\Psi _ \\varepsilon ( u ) , u \\in Z ( \\frac { \\rho _ 1 } { 3 2 } , 3 \\delta _ 0 ) . \\end{align*}"}
+{"id": "8151.png", "formula": "\\begin{align*} \\Delta \\chi _ T ( t ) = \\chi _ { T _ 1 } ( t ) \\cdots \\chi _ { T _ k } ( t ) \\end{align*}"}
+{"id": "6782.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\lambda } ^ { \\vee } = \\mathbb { I } _ { \\lambda } , \\Delta _ { \\lambda } ^ { \\vee } = \\nabla _ { \\lambda } , ( \\overline { \\Delta } _ { \\lambda } ) ^ { \\vee } = \\overline { \\nabla } _ { \\lambda } . \\end{align*}"}
+{"id": "8841.png", "formula": "\\begin{align*} \\overline { v } _ \\lambda ( t ) = v _ 0 ( t ) + \\int _ 0 ^ t S ( t - s ) F _ \\lambda ( s ) \\overline { v } _ \\lambda ( s ) \\ , d s + \\int _ 0 ^ t S ( t - s ) \\sigma ' ( u ( s ) ) \\overline { v } _ \\lambda ( s ) B \\ , d W ( s ) , \\end{align*}"}
+{"id": "2400.png", "formula": "\\begin{align*} \\vec { \\delta } _ { i 0 j } = \\vec { u } ( A _ { 0 } , A _ { i } ) + \\vec { u } ( A _ { 0 } , A _ { j } ) \\end{align*}"}
+{"id": "2959.png", "formula": "\\begin{align*} \\mu _ k = \\left ( \\underbrace { \\frac { k } { n } , \\dots , \\frac { k } { n } } _ { n - k } , \\underbrace { \\frac { k - n } { n } , \\dots , \\frac { k - n } { n } } _ { k } \\right ) \\end{align*}"}
+{"id": "1414.png", "formula": "\\begin{align*} P _ { y _ { k + 1 } \\to y } { \\log _ { y _ { k + 1 } } ( x ^ * ) } = P _ { y _ k \\to y } P _ { y _ { k + 1 } \\to y _ k } { \\log _ { y _ { k + 1 } } ( x ^ * ) } . \\end{align*}"}
+{"id": "2367.png", "formula": "\\begin{align*} p _ 1 ( X _ 1 , \\dots , X _ n ) & = 0 \\rlap { , } \\\\ \\vdots \\quad \\quad \\ ; \\ ; \\\\ p _ m ( X _ 1 , \\dots , X _ n ) & = 0 \\rlap { , } \\end{align*}"}
+{"id": "6266.png", "formula": "\\begin{align*} L _ { \\Sigma } \\phi = \\lambda \\phi \\mbox { i n } \\partial \\Sigma \\mbox { a n d } \\nabla _ n \\phi = h ( \\nu , \\nu ) \\phi \\mbox { o n } \\partial \\Sigma . \\end{align*}"}
+{"id": "3820.png", "formula": "\\begin{align*} \\sup _ { \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) } \\left \\{ \\int _ { \\mathcal { S } } g \\ , d \\gamma - \\lambda _ 1 \\boldsymbol { K } _ 1 ( \\mu _ { 1 3 } , \\gamma _ { 1 3 } ) - \\lambda _ 2 \\boldsymbol { K } _ 2 ( \\mu _ { 2 3 } , \\gamma _ { 2 3 } ) : \\boldsymbol { K } _ \\ell ( \\mu _ { \\ell 3 } , \\gamma _ { \\ell 3 } ) < \\infty \\ell = 1 , 2 \\right \\} , \\end{align*}"}
+{"id": "2387.png", "formula": "\\begin{align*} \\cos \\alpha _ { 1 0 2 } = \\cos \\alpha _ { 3 0 4 } , \\end{align*}"}
+{"id": "8737.png", "formula": "\\begin{align*} \\| \\mathcal { T } _ { 1 , l } ^ { ( a _ 1 , a _ 2 ) } - \\mathcal { T } _ { 2 , l } ^ { ( a _ 1 , a _ 2 ) } \\| _ { \\rm F } ^ 2 & = \\sum _ { k _ 1 , k _ 2 } s _ { k _ 1 , k _ 2 } s _ { k _ 1 , k _ 2 } ^ { l - 2 a _ 1 - 1 } s _ { k _ 1 , k _ 1 } ^ { a _ 1 } s _ { k _ 2 , k _ 2 } ^ { a _ 1 } ( \\mu _ { k _ 1 , l } ^ { ( 1 ) } - \\mu _ { k _ 1 , l } ^ { ( 2 ) } ) ( \\mu _ { k _ 2 , l } ^ { ( 1 ) } - \\mu _ { k _ 2 , l } ^ { ( 2 ) } ) \\\\ & \\le C K ^ { l - 1 } \\sum _ { k _ 1 , k _ 2 } s _ { k _ 1 , k _ 2 } = O ( p ) . \\end{align*}"}
+{"id": "1317.png", "formula": "\\begin{align*} \\int \\nu _ V ^ { \\widetilde { W } ^ { ( u ) } , \\widetilde \\theta ^ { ( u ) } , \\widetilde \\eta ^ { ( u ) } } ( d \\widetilde { \\beta } ^ { ( u ) } ) = 1 . \\end{align*}"}
+{"id": "1884.png", "formula": "\\begin{align*} \\langle f , \\zeta \\rangle = F ( [ \\zeta ] ) \\forall \\ \\zeta \\in \\mathcal { X } . \\end{align*}"}
+{"id": "1991.png", "formula": "\\begin{align*} \\lambda _ 1 : = \\inf \\{ \\gamma _ 1 ( a ) : ~ ~ a \\in \\mathcal { A } ( \\Omega ) \\} . \\end{align*}"}
+{"id": "1272.png", "formula": "\\begin{align*} \\eta ( B ^ { \\prime } ) = \\eta ( B ) + 1 \\ge T + \\max _ { 1 \\le i \\le s - 1 } a _ i . \\end{align*}"}
+{"id": "3785.png", "formula": "\\begin{align*} \\sigma _ { j _ k + k } ( a ) = a - 1 , \\ \\sigma _ { j _ { k - 1 } + k - 1 } ( a - 1 ) = a - 2 , \\dots , \\sigma _ { j _ 1 + 1 } ( a - k + 1 ) = a - k . \\end{align*}"}
+{"id": "2879.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 2 t _ { 1 } t _ { 2 } - 2 t _ { 1 } - 2 t _ { 2 } + 2 , & ~ ~ t _ { 0 } \\geq 2 , \\\\ t _ { 1 } t _ { 2 } - 2 t _ { 1 } - 2 t _ { 2 } + 2 , & ~ ~ t _ { 0 } = 1 , \\\\ - 2 t _ { 1 } - 2 t _ { 2 } + 2 , & ~ ~ t _ 0 = 0 , \\\\ - t _ { 0 } t _ { 1 } t _ { 2 } , & ~ ~ t _ 0 \\leq - 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "3673.png", "formula": "\\begin{align*} \\begin{cases} \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) u = 0 & \\Omega , \\\\ \\mathcal { L } u = 0 & \\Omega ^ c , \\\\ u = 0 & W , \\end{cases} \\end{align*}"}
+{"id": "2136.png", "formula": "\\begin{align*} x _ { i } : = \\Upsilon _ { i } ( y ) = y _ { 3 } \\mathcal { N } _ { i } ^ { j } ( y _ { 1 } , y _ { 2 } ) + z _ { i } ^ { j } ( y _ { 1 } , y _ { 2 } ) \\mbox { f o r } i = 1 , 2 , 3 , \\end{align*}"}
+{"id": "6174.png", "formula": "\\begin{align*} \\alpha _ r ( u ( s , t ) ) u ( r , s t ) = u ( r , s ) u ( r s , t ) , r , s , t \\in G . \\end{align*}"}
+{"id": "4360.png", "formula": "\\begin{align*} \\sigma \\left ( b \\right ) : = \\left \\{ g _ { i } ( x ) \\leq b _ { i } , \\ ; i = 1 , \\ldots , m \\right \\} , \\end{align*}"}
+{"id": "7972.png", "formula": "\\begin{align*} \\mathrm { d i v } \\Big ( V \\ , \\mathrm { d i v } V \\Big ) = \\big ( \\mathrm { d i v } V \\big ) ^ 2 + V \\cdot \\nabla ( \\mathrm { d i v } V ) = \\big ( \\mathrm { d i v } V \\big ) ^ 2 + V ^ i \\ , \\partial _ i \\partial _ j V ^ j . \\end{align*}"}
+{"id": "1439.png", "formula": "\\begin{align*} \\Big \\langle V + \\phi - i ^ * [ f _ \\epsilon ( V + \\phi ) ] , P \\psi ^ h _ { \\mu _ { j _ \\epsilon } , \\xi _ { j _ \\epsilon } } \\Big \\rangle = 0 , \\quad \\mbox { f o r } \\ \\ h = 0 , \\cdots , n , \\end{align*}"}
+{"id": "7368.png", "formula": "\\begin{align*} \\langle S _ N ( \\ell ) \\rangle = \\int _ 0 ^ 1 S _ N ( \\ell ) \\ , d x _ 0 = \\sum _ { j = 1 } ^ N \\sum _ { n \\in \\mathbb { Z } } \\int _ 0 ^ 1 \\chi \\left ( \\frac { x _ j - x _ 0 + n } { \\ell } \\right ) \\ , d x _ 0 = L . \\end{align*}"}
+{"id": "9214.png", "formula": "\\begin{align*} & x \\oplus y = ( x + y ) ( 1 + \\eta _ 1 ) , & | \\eta _ 1 | \\le 2 ^ { - d } \\\\ & x \\ominus y = ( x - y ) ( 1 + \\eta _ 1 ) , & | \\eta _ 1 | \\le 2 ^ { - d } \\\\ & x \\otimes y = x y ( 1 + \\eta _ 1 ) , & | \\eta _ 1 | \\le 2 ^ { - d } , \\end{align*}"}
+{"id": "3956.png", "formula": "\\begin{align*} \\gamma _ 1 ^ \\star = \\left ( \\frac { \\eta } { \\eta _ 0 } \\right ) \\gamma _ 1 ^ { \\eta _ 0 , \\delta } + \\left ( \\frac { \\eta _ 0 - \\eta } { \\eta _ 0 } \\right ) \\mu _ 1 . \\end{align*}"}
+{"id": "7982.png", "formula": "\\begin{align*} \\mathcal { B } ^ H = \\nabla _ T \\big ( \\nabla _ \\xi H ( \\nu ) \\big ) = \\nabla _ \\xi ^ 2 H ( \\nu ) \\ , \\nabla _ T \\nu . \\end{align*}"}
+{"id": "2102.png", "formula": "\\begin{align*} R _ 1 & = \\Big \\{ ( x _ 1 , x _ 2 , . . . , x _ { k } ) \\mid 0 \\leq x _ k \\leq m - 1 ; \\ \\ \\ 0 \\leq x _ i \\leq 2 \\ \\ \\ \\ 1 \\leq i \\leq k - 1 , \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x _ i = 2 , \\ \\ \\ \\ x _ j = 0 \\ \\ \\ \\ j < i \\leq k - 1 \\Big \\} \\end{align*}"}
+{"id": "2131.png", "formula": "\\begin{align*} \\mathbb { K } : = \\mathbb { E } - \\mathbb { I } , \\end{align*}"}
+{"id": "8300.png", "formula": "\\begin{align*} P _ { n } ( \\lambda ) = & \\left ( \\lambda - \\frac { 1 } { 2 } \\right ) ^ n \\left [ \\left ( ( \\lambda - 3 ) ^ 2 - 4 \\right ) \\left ( \\lambda - \\frac { n + 4 } { 2 } \\right ) \\right ] - 2 \\left ( \\lambda - \\frac { 1 } { 2 } \\right ) ^ n ( \\lambda - 5 ) \\\\ & - \\frac { n } { 4 } \\left ( \\lambda - \\frac { 1 } { 2 } \\right ) ^ { n - 1 } \\left [ ( \\lambda - 3 ) ^ 2 - 4 \\right ] \\end{align*}"}
+{"id": "5379.png", "formula": "\\begin{align*} \\begin{bmatrix} x _ { 1 , k + 1 } \\\\ x _ { 2 , k + 1 } \\end{bmatrix} = \\hat A \\begin{bmatrix} x _ { 1 , k } \\\\ x _ { 2 , k } \\end{bmatrix} , \\| \\hat A z \\| _ { \\hat { \\mathcal { X } } } \\leq \\| z \\| _ { \\hat { \\mathcal { X } } } , \\quad \\| z \\| _ { \\hat { \\mathcal { X } } } : = \\left \\| \\begin{bmatrix} \\mathcal { X } _ 1 & 0 \\\\ 0 & \\mathcal { X } _ 2 \\end{bmatrix} z \\right \\| . \\end{align*}"}
+{"id": "8952.png", "formula": "\\begin{align*} \\phi = \\phi _ { e } + \\sum _ { m = 1 } ^ { \\infty } D _ { q _ m } \\Delta x ^ { q _ m } , \\end{align*}"}
+{"id": "621.png", "formula": "\\begin{align*} \\omega ( \\xi ) = \\lim _ \\lambda \\omega ( e _ \\lambda ) = 1 \\end{align*}"}
+{"id": "2917.png", "formula": "\\begin{align*} \\gamma = \\begin{bmatrix} a & b \\\\ 0 & d \\end{bmatrix} . \\end{align*}"}
+{"id": "8602.png", "formula": "\\begin{align*} \\varphi _ 1 ( p ) = \\begin{cases} \\dfrac 1 2 , & p = 0 , \\\\ - \\dfrac { p + q \\log ( q ) } { p \\log ( q ) } , & 0 < p < 1 , \\end{cases} \\end{align*}"}
+{"id": "7750.png", "formula": "\\begin{align*} B _ { c _ 0 } ( y ) \\cap \\bigcup _ { Q ' \\in \\mathcal { F } } Q ' = \\emptyset \\ , . \\end{align*}"}
+{"id": "6630.png", "formula": "\\begin{align*} \\nu ( \\{ n \\} ) : = \\phi ^ 2 ( n ) . \\end{align*}"}
+{"id": "8844.png", "formula": "\\begin{align*} J ^ F _ \\lambda : = \\frac { 1 } { 1 - \\lambda F } , \\end{align*}"}
+{"id": "6853.png", "formula": "\\begin{align*} \\lim _ { c _ 1 , c _ 2 \\to \\infty } F _ { \\lambda , k , r } ( c _ 1 , c _ 2 ; b _ 1 , \\dots , b _ \\lambda ; a ; q ) = \\lim _ { c _ 1 , c _ 2 \\to \\infty } G _ { \\lambda , k , r } ( c _ 1 , c _ 2 ; b _ 1 , \\dots , b _ \\lambda ; a ; q ) . \\end{align*}"}
+{"id": "206.png", "formula": "\\begin{align*} = \\left ( \\frac { 1 } { 1 - x } \\right ) ^ { \\frac { z ( 1 + 4 z + z ^ 2 ) L i _ 3 ( y ) } { ( 1 - z ) ^ 4 } } , \\end{align*}"}
+{"id": "7142.png", "formula": "\\begin{align*} z _ t & : = M _ t ( \\hat { x } , \\bar { y } ) ( \\hat { x } - \\bar { y } ) , \\\\ M _ t ( \\hat { x } , \\bar { y } ) & : = A _ \\gamma ( \\hat { x } , \\hat { x } ) + t \\big ( A _ \\gamma ( \\hat { x } , \\bar { y } ) - A _ \\gamma ( \\hat { x } , \\hat { x } ) \\big ) \\end{align*}"}
+{"id": "7851.png", "formula": "\\begin{align*} A ( X _ i / \\pi ) _ { 2 1 } = \\left | \\left \\{ ( x H , y H ) : \\ y H \\in \\mathcal { S } \\right \\} \\right | . \\end{align*}"}
+{"id": "7018.png", "formula": "\\begin{align*} \\gamma : = \\min \\{ \\alpha _ l \\mid Q _ l \\mbox { a p p e a r s i n t h e e x p a n s i o n o f s o m e } f _ j \\} , \\end{align*}"}
+{"id": "1216.png", "formula": "\\begin{align*} \\varphi ( t _ 0 ) = ( 1 - q ) G ( t _ 0 , 0 ) - 1 = ( 1 - q ) A ' ( t _ 0 ) G ( 0 , q A ( t _ 0 ) ) - 1 . \\end{align*}"}
+{"id": "9090.png", "formula": "\\begin{align*} r \\geq \\tilde { r } = e _ 0 - 2 z + c + | \\mathcal { F } | , \\end{align*}"}
+{"id": "5932.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( \\omega ) f ( [ \\epsilon , y ] ) = \\nu ( \\epsilon , g ) \\int _ { X ^ { \\ast } } \\psi ( \\langle \\epsilon y , y ^ { \\ast } \\rangle ) f ( [ \\epsilon , y ^ { \\ast } \\omega ^ { - 1 } ] ) d y ^ { \\ast } . \\end{align*}"}
+{"id": "1310.png", "formula": "\\begin{align*} \\Phi ( t ) = \\big ( \\log ( Z _ i ( t ) ) , ( T _ i ( u ) + t ) \\wedge T _ i ( v ) \\big ) _ { i \\in V } \\in \\R ^ V \\times \\R ^ V \\end{align*}"}
+{"id": "3333.png", "formula": "\\begin{align*} f ( \\xi ) = ( \\xi - 1 ) ^ { \\delta _ 1 } ( \\xi + 1 ) ^ { \\delta _ { - 1 } } \\kappa ( \\xi ) + f ( 1 ) \\frac { 1 + \\psi ( \\xi ) } { 2 } + f ( - 1 ) \\frac { 1 - \\psi ( \\xi ) } { 2 } . \\end{align*}"}
+{"id": "8615.png", "formula": "\\begin{align*} E \\left [ \\left ( S ^ k _ { j , + } ( t ) - \\bar { S } ^ k _ { j , + , \\Delta } ( t ) \\right ) ^ 2 \\right ] & = \\nu \\Delta \\sum _ { \\ell = 0 } ^ { \\lfloor t / \\Delta \\rfloor } p ^ k _ { j , + } ( t - \\ell \\Delta ) E [ Z _ 0 ( \\ell \\Delta ) ] \\\\ & + 2 \\sum _ { \\ell _ 1 < \\ell _ 2 } \\left ( I _ 1 ( \\ell _ 1 , \\ell _ 2 ) - I _ 2 ( \\ell _ 1 , \\ell _ 2 ) \\right ) + C _ 3 \\Delta e ^ { 3 \\lambda t } \\\\ & \\leq C _ 1 \\theta ^ k t e ^ { \\lambda t } + C _ 2 \\theta ^ k t ^ 2 e ^ { \\lambda t } + C _ 3 \\Delta e ^ { 3 \\lambda t } . \\end{align*}"}
+{"id": "5308.png", "formula": "\\begin{align*} \\Rightarrow \\ A \\geq \\sqrt { q \\cdot A } \\geq \\frac { 1 } { C ' \\ , r } \\frac { 1 } { \\frac { p } { 1 - p } } = \\frac { 1 } { C ' \\ , r } \\frac { 1 - p } { p } > \\frac { 1 } { 2 \\ , C ' \\ , r } \\frac { 1 } { p } \\end{align*}"}
+{"id": "2868.png", "formula": "\\begin{align*} ( \\alpha \\otimes \\Delta ) \\Delta ( l ) & = ( \\Delta \\otimes \\alpha ) \\Delta ( l ) \\\\ ( \\alpha \\otimes \\Delta ) ( l _ 1 \\otimes l _ 2 ) & = ( \\Delta \\otimes \\alpha ) ( l _ 1 \\otimes l _ 2 ) \\\\ \\alpha ( l _ 1 ) \\otimes l _ { 2 1 } \\otimes l _ { 2 2 } & = l _ { 1 1 } \\otimes l _ { 1 2 } \\otimes \\alpha ( l _ 2 ) \\end{align*}"}
+{"id": "5705.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ) = c F + D ^ { k - 1 } ( h ^ { \\sigma } ) + d ^ { \\sigma } g \\end{align*}"}
+{"id": "6138.png", "formula": "\\begin{align*} \\rho _ { i } = 2 ^ { j _ { 0 } } \\rho _ { z _ { i } } \\quad i . \\end{align*}"}
+{"id": "8140.png", "formula": "\\begin{align*} w = 4 0 2 0 1 2 0 0 0 1 0 \\end{align*}"}
+{"id": "1285.png", "formula": "\\begin{align*} D ( U \\otimes v _ 1 ) & = ( - \\sqrt { n } + 1 ) U \\otimes v _ 1 , \\\\ D ( U \\otimes v _ 2 ) & = ( - \\sqrt { n } + 1 ) U \\otimes v _ 2 , \\\\ D ( U \\otimes u _ i ) & = ( - \\sqrt { n } + 1 ) ( U \\otimes u _ i ) , \\mbox { f o r a n y } 2 \\leq i \\leq n \\\\ D ( V \\otimes v _ 1 ) & = - V \\otimes v _ 1 , \\\\ D ( V \\otimes v _ 2 ) & = - V \\otimes v _ 2 , \\\\ D ( V \\otimes u _ i ) & = - ( V \\otimes u _ i ) , \\mbox { f o r a n y } 2 \\leq i \\leq n . \\end{align*}"}
+{"id": "7311.png", "formula": "\\begin{align*} x _ i = \\left ( A \\prod _ { j = 1 } ^ n u _ j ^ { \\alpha ' _ j } \\right ) ^ { z _ i } \\left ( B \\prod _ { j = 1 } ^ n u _ j ^ { \\gamma ' _ j } \\right ) ^ { t _ i } w ^ { - z _ i - t _ i } \\cdot u _ i , i = 1 , \\dots , n , \\end{align*}"}
+{"id": "4674.png", "formula": "\\begin{align*} \\left [ \\mathcal { B } ( z ) \\right ] _ { i j } : = z \\delta _ { i j } + z ^ 2 \\sum _ { k = 1 } ^ { \\infty } \\frac { [ \\overline { \\varphi _ k } ] _ i [ \\overline { \\varphi _ k } ] _ j } { \\eta _ k - z } , i , j \\in { 1 , 2 , 3 } , \\end{align*}"}
+{"id": "1513.png", "formula": "\\begin{align*} & 0 = \\min _ { 0 \\le s \\le t } \\ , ( R ^ { \\overline { X } } _ s - \\widetilde { \\Lambda } _ { s } ) = \\overline { X } + \\min _ { 0 \\le s \\le t } \\bigg ( \\int _ 0 ^ s \\frac { 1 } { R ^ { \\overline { X } } _ r } \\ , \\mathrm { d } r + B _ s - \\widetilde { \\Lambda } _ { s } \\bigg ) \\ ; \\Longrightarrow \\ ; \\overline { X } \\le \\max _ { 0 \\le s \\le t } \\ , ( - B _ s + \\widetilde { \\Lambda } _ { s } ) , \\\\ & 0 = \\min _ { 0 \\le s \\le t } \\ , ( R ^ { \\underline { X } } _ s - \\Lambda _ { s } ) . \\end{align*}"}
+{"id": "4942.png", "formula": "\\begin{align*} f ( k , t , n ) = \\frac { ( n ) _ k } { n ^ t } \\frac { 1 } { k ! } \\Delta ^ { k } \\left [ x ^ t \\right ] _ { x = 0 } ^ { } . \\end{align*}"}
+{"id": "5656.png", "formula": "\\begin{align*} | \\eta _ \\epsilon | ^ 2 _ 2 = O ( \\epsilon ^ 2 | \\ln \\epsilon | ) \\ \\ N = 4 , | \\eta _ \\epsilon | ^ 2 _ 2 = O ( \\epsilon ) \\ \\ N = 3 , \\end{align*}"}
+{"id": "3938.png", "formula": "\\begin{align*} \\mathcal { I } ^ \\star ( \\lambda ) = \\sup _ { \\gamma \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\int _ { \\mathcal { V } } f _ \\lambda ( v ) \\ , d \\gamma ( v ) , \\end{align*}"}
+{"id": "1424.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\int _ 0 ^ t \\frac { \\tilde { H } _ { k 1 } ^ * ( u , t ) } { \\sqrt { n } \\ ! \\cdot \\ ! S ^ { ( 0 ) , * } ( \\boldsymbol { \\hat { \\beta } _ k } , u ) } M _ { k i } ^ * ( d u ) \\end{align*}"}
+{"id": "1098.png", "formula": "\\begin{align*} \\varphi \\colon B \\rightarrow [ X ] _ 2 , \\varphi ( b ) = b \\ast - \\end{align*}"}
+{"id": "4427.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( \\Psi ) \\le S _ { \\omega , \\mathbf { c } } ( \\Phi ) = \\mu _ { \\omega , \\mathbf { c } } . \\end{align*}"}
+{"id": "8224.png", "formula": "\\begin{align*} Y ^ N _ { t } = \\frac { 1 } { \\sqrt { N } } \\sum _ { x \\in \\mathbb { Z } } \\big ( \\eta _ t ( x , \\sigma ) - \\rho \\big ) \\delta _ { ( \\tfrac { x } { N } , \\sigma ) } . \\end{align*}"}
+{"id": "2391.png", "formula": "\\begin{align*} \\vec { u } ( A _ { 0 } , A _ { i } ) \\cdot \\vec { u } ( A _ { 0 } , A _ { j } ) = \\cos \\alpha _ { i 0 j } , \\end{align*}"}
+{"id": "4973.png", "formula": "\\begin{align*} [ \\widehat { P } ] _ { i , j } ^ { } = \\hat { p } _ { i , j } ^ { } = \\begin{dcases} \\frac { n - i } { n } & \\mbox { i f } \\ , j = i \\neq n ; \\\\ \\frac { n } { n } & \\mbox { i f } \\ , j = i = n ; \\\\ \\frac { i } { n } & \\mbox { i f } \\ , j = i + 1 ; \\\\ 0 & \\mbox { o t h e r w i s e } ; \\end{dcases} \\end{align*}"}
+{"id": "5632.png", "formula": "\\begin{align*} \\min \\{ p , q \\} > \\frac { 6 - \\mu } { 2 } , N = 3 , \\end{align*}"}
+{"id": "6664.png", "formula": "\\begin{align*} \\| u _ n \\| _ { C ^ { 2 , \\alpha } ( \\overline { \\mathcal { O } } } & \\leq C \\big ( \\| u _ n \\| _ { L ^ \\infty ( \\R ^ N ) } + \\| g \\| _ { C ^ \\gamma ( \\overline { O } _ { \\frac { 7 \\rho } { 8 } } \\times \\R } \\big ) \\big ( 1 + \\| g \\| _ { C ^ \\gamma ( \\overline { O } _ { \\frac { 7 \\rho } { 8 } } \\times \\R } \\big ) \\\\ & \\leq C ( | \\eta | + \\| c \\| _ { C ^ \\gamma ( \\R ^ N ) } ) ( 1 + \\| c \\| _ { C ^ \\gamma ( \\R ^ N ) } ) = : \\vartheta \\end{align*}"}
+{"id": "6137.png", "formula": "\\begin{align*} \\left \\{ Q _ { 2 ^ { j _ { 0 } } \\rho _ { z _ { i } } } ( z _ { i } ) \\right \\} _ { i \\in \\mathbb { N } } D _ { \\kappa \\lambda } \\subset \\bigcup \\limits _ { i = 1 } ^ { \\infty } Q _ { 5 \\times 2 ^ { j _ { 0 } } \\rho _ { z _ { i } } } ( z _ { i } ) . \\end{align*}"}
+{"id": "6106.png", "formula": "\\begin{align*} \\begin{cases} ( \\Phi ( \\xi ) - \\Phi ( \\xi ' ) ) ( \\xi - \\xi ' ) \\geq L ^ { - 1 } | \\xi - \\xi ' | ^ { 2 } , \\\\ | \\Phi ( \\xi ) - \\Phi ( \\xi ' ) | \\leq L | \\xi - \\xi ' | \\quad \\xi , \\xi ' \\in \\mathbb { R } . \\end{cases} \\end{align*}"}
+{"id": "415.png", "formula": "\\begin{align*} W ^ - = \\begin{bmatrix} U _ 1 ^ 2 + U _ n ^ 2 \\\\ U _ n U _ { \\tau } \\end{bmatrix} , \\Lambda ^ - = \\begin{bmatrix} \\frac { 1 } { 2 U _ n \\sqrt { U _ 1 } } & 0 \\\\ 0 & \\frac { 1 } { 2 U _ n \\sqrt { U _ 1 } } \\end{bmatrix} , W ^ + = U _ 1 ^ 2 , \\Lambda ^ + = - \\frac { 1 } { 2 U _ n \\sqrt { U _ 1 } } . \\end{align*}"}
+{"id": "6179.png", "formula": "\\begin{align*} \\| b \\| _ { L ^ 1 } = \\| \\Lambda _ 1 ( b ) \\| = \\max _ { x \\in X } \\sum _ { t \\in G } | b ( t ) ( x ) | , \\| b \\| _ { L ^ \\infty } = \\| \\Lambda _ { \\infty } ( b ) \\| = \\max _ { x \\in X } \\sum _ { t \\in G } | b ( t ) ( \\varphi _ { t } ( x ) ) | , \\end{align*}"}
+{"id": "8595.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } \\hat { S } _ { j , + } ( t ) = \\nu Y \\int _ 0 ^ \\infty e ^ { - \\lambda s } \\left ( \\sum _ { k = 1 } ^ \\infty p _ { j , + } ^ k ( s ) \\right ) d s . \\end{align*}"}
+{"id": "5362.png", "formula": "\\begin{align*} \\| D _ { S , x } [ f ^ { = d } ] \\| _ 2 = \\| ( D _ { S , x } [ f ] ) ^ { = d - | S | } \\| _ 2 \\le \\| D _ { S , x } [ f ] \\| _ 2 \\end{align*}"}
+{"id": "5768.png", "formula": "\\begin{align*} \\bigcup _ { q _ { 1 } , \\cdots , q _ { m } } \\prod _ { j = 1 } ^ { m } A _ { p _ { j } , q _ { j } } & \\subsetneq A _ { \\vec { P } , q } , \\end{align*}"}
+{"id": "7842.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ A \\subset [ n ] : \\ | A | = k \\mbox { a n d } S \\subset A \\} . \\end{align*}"}
+{"id": "1692.png", "formula": "\\begin{align*} s ( X ) = \\frac { 1 2 } { p - 1 } \\sum _ { j \\in J } w _ j ^ { - 1 } . \\end{align*}"}
+{"id": "864.png", "formula": "\\begin{align*} \\langle f , \\mu \\rangle = \\int _ X f ( x ) \\ , d \\mu ( x ) \\end{align*}"}
+{"id": "3320.png", "formula": "\\begin{align*} \\rho _ { \\theta } ( \\theta , f ( \\theta ) ) + 2 { \\rm R e } ( \\rho _ w ( \\theta , f ( \\theta ) ) f ( \\theta ) \\frac { \\partial g } { \\partial \\theta } ( \\theta ) ) = 0 \\end{align*}"}
+{"id": "845.png", "formula": "\\begin{align*} S _ { n \\left ( i \\right ) } = \\left ( S _ { n \\left ( i - 1 \\right ) } \\setminus k _ { e x \\left ( i - 1 \\right ) } \\right ) \\cup k _ { n e w \\left ( i - 1 \\right ) } \\end{align*}"}
+{"id": "5500.png", "formula": "\\begin{align*} Y : = \\xi \\left ( Q / P \\times _ { \\alpha } X _ { 0 } \\right ) , \\end{align*}"}
+{"id": "2124.png", "formula": "\\begin{align*} r _ 2 ^ { q ^ m } = x f _ 2 . \\end{align*}"}
+{"id": "7695.png", "formula": "\\begin{align*} v _ 1 ( x ) = v _ 0 ( x _ W ) , x \\in L _ \\R . \\end{align*}"}
+{"id": "3145.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } { \\phi } { H _ j } = \\int _ { \\R ^ N } { g } { Z _ j } . \\end{align*}"}
+{"id": "2570.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) , \\ \\ u > 0 , & \\ \\ D , \\\\ u = 0 , \\ \\ | \\nabla u | = c , & \\ \\ \\partial D , \\end{cases} \\end{align*}"}
+{"id": "4058.png", "formula": "\\begin{align*} S _ { \\mathrm { m a i n } } = \\sum _ { L = 1 } ^ { \\infty } L \\sum _ { U , \\ , V < \\frac { D } { L } } \\tau ( U ) \\tau ( V ) \\ , x _ { U L } \\bar { x } _ { V L } \\sum _ { T | L } \\frac { \\mu ( T ) } { T } \\sum _ { A = 1 } ^ { \\infty } \\frac { G _ k ( A ^ 2 T ^ 2 U V \\slash p ) } { A } , \\end{align*}"}
+{"id": "5000.png", "formula": "\\begin{align*} T _ { r : k } \\overset { } { = } G _ { \\frac { n - r } { n } } + G _ { \\frac { n - 1 } { n } } + . . . + G _ { \\frac { n - k - 1 } { n } } , \\end{align*}"}
+{"id": "3860.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\sup _ { \\gamma \\in \\Sigma _ { \\mathrm { D } } ( \\delta ) } \\int _ { \\mathcal { V } } g d \\gamma , \\end{align*}"}
+{"id": "7902.png", "formula": "\\begin{align*} \\mathcal { S } = \\left \\{ \\left ( A , \\underline { A } \\right ) ^ \\pm : \\ A \\in \\binom { [ n ] } { k } \\mbox { a n d } n \\in A \\right \\} . \\end{align*}"}
+{"id": "8512.png", "formula": "\\begin{align*} \\zeta _ { p , 0 } ( s , c ) : = \\sum _ { m , n \\neq 0 } \\frac { p ^ { 2 } + \\lambda _ { m } ^ { 2 } } { p ( p + \\frac { 1 } { \\pi } ) + \\lambda _ { m } ^ { 2 } } \\ , \\frac { 1 } { \\left ( \\lambda _ { m } ^ { 2 } + c \\ , \\left ( n - \\frac { 1 } { 2 } \\right ) ^ { 2 } \\right ) ^ { s } } , \\ , \\ , \\ , \\ , ( s ) > 1 . \\end{align*}"}
+{"id": "5739.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } 0 = a ^ 2 b \\\\ 0 = - q a ( b - d ) \\\\ 0 = a ^ 2 \\\\ q ^ 2 = a ^ 2 + q b ^ 2 + a b ^ 2 + a b d \\\\ q ( d - b ) = a b . \\end{array} \\right . \\end{align*}"}
+{"id": "8543.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow \\infty } \\mathcal { K } _ { \\nu , p } ( x ) & = \\intop _ { 0 } ^ { \\infty } \\ , \\frac { y ^ { - \\nu - \\frac { 1 } { 2 } } ( y + 1 ) ^ { - \\nu - \\frac { 1 } { 2 } } } { e ^ { ( 2 y + 1 ) x } - 1 } d y = \\sum _ { n = 1 } ^ { \\infty } \\intop _ { 0 } ^ { \\infty } y ^ { - \\nu - \\frac { 1 } { 2 } } ( y + 1 ) ^ { - \\nu - \\frac { 1 } { 2 } } \\ , e ^ { - ( 2 y + 1 ) x n } d y \\\\ & = \\frac { \\Gamma \\left ( \\frac { 1 } { 2 } - \\nu \\right ) ( 2 x ) ^ { \\nu } } { \\sqrt { \\pi } } \\sum _ { n = 1 } ^ { \\infty } n ^ { \\nu } \\ , K _ { \\nu } ( x n ) , \\end{align*}"}
+{"id": "395.png", "formula": "\\begin{align*} S ^ { - 1 } ( \\sqrt { | \\Lambda ^ - | } W ^ - - R \\sqrt { \\Lambda ^ + } W ^ + ) = G \\sqrt { | \\Lambda ^ - | } W ^ - = R \\sqrt { \\Lambda ^ + } W ^ + + S G \\end{align*}"}
+{"id": "4580.png", "formula": "\\begin{align*} & [ \\alpha ( a ) , \\alpha ( b ) , [ x , y , z ] ] = [ [ a , b , x ] , \\alpha ( y ) , \\alpha ( z ) ] + [ \\alpha ( x ) , [ a , b , y ] , \\alpha ( z ) ] + [ \\alpha ( x ) , \\alpha ( y ) , [ a , b , z ] ] , \\end{align*}"}
+{"id": "976.png", "formula": "\\begin{align*} g \\left ( h ( X , Y ) , H \\right ) = g ( X , Y ) g ( H , H ) \\end{align*}"}
+{"id": "1685.png", "formula": "\\begin{align*} \\rho = \\frac { - 1 } { p } \\alpha _ 1 \\alpha _ 2 = \\frac { 1 } { p } \\widehat { \\phi _ 1 } \\widehat { \\psi _ 1 } \\widehat { \\pi } \\pi \\phi _ 1 ^ { ( p ) } \\widehat { \\phi _ 2 ^ { ( p ) } } \\psi _ 2 \\phi _ 2 = \\widehat { \\phi _ 1 } \\widehat { \\psi _ 1 } \\phi _ 1 ^ { ( p ) } \\widehat { \\phi _ 2 ^ { ( p ) } } \\psi _ 2 \\phi _ 2 . \\end{align*}"}
+{"id": "850.png", "formula": "\\begin{align*} \\textup { P L } _ { m , k } = \\left \\{ \\begin{matrix} - \\textup { D } - 3 5 \\log _ { 1 0 } \\left ( d _ { m , k } \\right ) , \\textup { i f } d _ { m , k } > d _ { 1 } & \\\\ - \\textup { D } - 1 0 \\log _ { 1 0 } \\left ( d _ { 1 } ^ { 1 . 5 } d _ { m , k } ^ 2 \\right ) , \\textup { i f } d _ { 0 } < d _ { m , k } \\leq d _ { 1 } & \\\\ - \\textup { D } - 1 0 \\log _ { 1 0 } \\left ( d _ { 1 } ^ { 1 . 5 } d _ { 0 } ^ 2 \\right ) , \\textup { i f } d _ { m , k } \\leq d _ { 0 } & \\end{matrix} \\right . \\end{align*}"}
+{"id": "8177.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\mathbb { P } \\left ( A _ { n , k } ^ c \\right ) \\leq c ( n _ 0 ) + \\sum _ { n = n _ 0 } ^ \\infty \\left \\lceil e ^ { a ( n + 1 ) ^ { 3 / 2 } } \\right \\rceil e ^ { - n ^ { 1 4 / 9 } } < \\infty , \\end{align*}"}
+{"id": "723.png", "formula": "\\begin{align*} A _ { S } ^ * [ \\partial _ { t } \\mu _ { S } ] ( e ) = G _ { D , 1 } ( e ) \\ , , \\end{align*}"}
+{"id": "7835.png", "formula": "\\begin{align*} A x : = ( A ^ { 1 } , A ^ { 2 } , \\cdots , A ^ { N } ) ^ { * } x = ( A ^ { i _ { 1 } } , A ^ { i _ { 2 } } \\cdots , A ^ { i _ { | I | } } ) x \\end{align*}"}
+{"id": "647.png", "formula": "\\begin{align*} & \\max \\{ k \\geq 0 : \\exists _ { \\sigma \\in \\mathrm { S d } ( \\psi _ \\R ) \\cap M ' } \\mathfrak { o } ( \\sigma , k ) < r \\} + 1 = \\lceil m r + ( m - 2 ) \\rceil \\leq k _ r \\\\ & \\max \\{ k \\geq 0 : \\exists _ { \\sigma \\in \\mathrm { S d } ( \\psi _ \\R ) \\cap M ' } \\widehat { \\mathfrak { o } } ( \\sigma , k ) < r \\} + 1 = \\lceil r + ( m - 2 ) \\rceil \\leq k _ r . \\end{align*}"}
+{"id": "5570.png", "formula": "\\begin{align*} X = G / Q \\times _ { \\beta } \\left ( \\Omega _ { 0 } \\times _ { \\sigma } X _ { 0 } \\right ) \\mbox { e q u i p p e d w i t h m e a s u r e } \\eta _ { \\rho } ^ { \\mu } = \\nu _ { Q } \\times m _ { \\Omega _ { 0 } } \\times \\rho , \\end{align*}"}
+{"id": "1750.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\textrm { o u t } } & = 1 - \\mathcal { P } _ 1 - \\mathcal { P } _ 2 - \\mathcal { P } _ 3 , \\end{align*}"}
+{"id": "7812.png", "formula": "\\begin{align*} B A = B \\overline { A } + B \\tilde { A } , \\end{align*}"}
+{"id": "5866.png", "formula": "\\begin{align*} \\mathcal { N } _ { L , x _ 0 } = \\bigcap _ { n = 1 } ^ { n _ 1 } \\mathcal { N } _ { \\Lambda _ { L _ { n - 1 } , L _ n ( x _ 0 ) } } \\end{align*}"}
+{"id": "1387.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ n t _ i v _ i : t _ i \\in D ( X ) \\right \\} \\subseteq V \\end{align*}"}
+{"id": "8815.png", "formula": "\\begin{align*} \\bigl [ S ( t ) \\phi \\bigr ] ( x ) = \\int _ G K _ t ( x , y ) \\phi ( y ) \\ , d y \\end{align*}"}
+{"id": "5804.png", "formula": "\\begin{align*} | I m o ( 1 ) | = & \\left | - I m \\left [ \\omega \\left ( n \\left ( r _ n , \\frac { 1 } { f } \\right ) + \\frac { k - 1 } { 2 k } n \\left ( r _ n , \\frac { 1 } { A } \\right ) \\right ) \\right ] \\right | \\\\ & \\geq \\frac { k - 1 } { 2 k } | I m ( \\omega ) | \\\\ & \\geq \\frac { k - 1 } { 2 k } \\Delta , \\end{align*}"}
+{"id": "1497.png", "formula": "\\begin{align*} f ( \\Lambda _ { t _ 0 } - x ) = 1 - \\psi ( x ) , x \\in ( 0 , \\Lambda _ { t _ 0 } - \\Lambda _ T \\wedge \\widetilde { \\Lambda } _ T + \\varepsilon _ 2 ] \\end{align*}"}
+{"id": "5078.png", "formula": "\\begin{align*} \\mathring { \\phi } ( x , t ) = \\phi \\left ( x + \\gamma ( x , t ) + \\frac { 1 } { N } \\sigma ( t ) \\right ) , \\end{align*}"}
+{"id": "4267.png", "formula": "\\begin{align*} u ^ { \\ast } \\left ( s \\right ) = \\hat { u } \\left ( t , \\mathbb { P } _ { \\bar { X } _ { s } } ^ { W } \\right ) , t \\leq s < T \\end{align*}"}
+{"id": "4004.png", "formula": "\\begin{align*} \\rho _ \\ell ( y _ \\ell , y _ \\ell ^ \\prime ) = \\left ( y _ { \\ell } - y _ { \\ell } ^ { \\prime } \\right ) ^ { \\top } V _ { \\ell , Y Y } ^ { - 1 } \\left ( y _ { \\ell } - y _ { \\ell } ^ { \\prime } \\right ) . \\end{align*}"}
+{"id": "2839.png", "formula": "\\begin{align*} \\alpha _ { i , j } = \\sum _ { v = 0 } ^ { N + q } D _ v \\alpha _ { i - v - 1 , j } , D _ v \\in \\mathbb { C } , i \\ge N + q + 1 , \\end{align*}"}
+{"id": "1203.png", "formula": "\\begin{align*} \\norm { A _ n \\phi - A \\phi } ^ 2 = \\norm { \\Phi _ n \\phi - \\Phi \\phi } ^ 2 = \\int _ \\Omega \\abs { \\phi } ^ 2 \\abs { \\Phi _ n - \\Phi } ^ 2 \\ , d x \\leq \\norm { \\phi } _ \\infty ^ 2 \\int _ { \\operatorname { s u p p } \\phi } \\abs { \\Phi _ n - \\Phi } ^ 2 \\ , d x \\end{align*}"}
+{"id": "5004.png", "formula": "\\begin{align*} \\sum _ { d = t + 1 } ^ \\infty \\lambda ^ d = \\frac { \\lambda ^ { t + 1 } } { 1 - \\lambda } 0 \\le \\lambda < 1 . \\end{align*}"}
+{"id": "383.png", "formula": "\\begin{align*} \\oint \\limits _ { \\partial \\Omega } U ^ T ( n _ i A _ i ) \\\\ \\ U \\\\ \\ d s = \\oint \\limits _ { \\partial \\Omega } \\frac { 1 } { 2 } U ^ T ( ( n _ i A _ i ) + ( n _ i A _ i ) ^ T ) U \\\\ \\ d s \\geq 0 . \\end{align*}"}
+{"id": "8265.png", "formula": "\\begin{align*} \\tau _ { k , Y _ { k + 1 , k + 1 } } ( x ) = \\frac { 1 } { k + 1 } \\left ( \\sqrt { x } \\frac { d \\tau _ { k , Y _ { k , k } } ( x ) } { d x } - \\frac { k ^ 2 + k } { 2 \\sqrt { x } } \\tau _ { k , Y _ { k , k } } ( x ) \\right ) + \\frac { k } { \\sqrt { x } } \\tau _ { k , Y _ { k , k } } ( x ) - \\frac { 1 } { k + 1 } \\tau _ { k , Y _ { k - 1 , k - 1 } } ( x ) . \\end{align*}"}
+{"id": "4893.png", "formula": "\\begin{align*} G ( X , Y ) \\ge \\begin{cases} c | X - Y | ^ { 2 s - n } & { \\mbox { i f } } n > 2 s , \\\\ c | \\log | X - Y | | & { \\mbox { i f } } n = 2 s . \\end{cases} \\end{align*}"}
+{"id": "1192.png", "formula": "\\begin{align*} ( p ( u ) - p ( v ) ) ^ \\top S ( p ( u ) - p ( v ) ) = 0 u v \\in E ( G ) \\ell ^ \\top S \\ell = 0 . \\end{align*}"}
+{"id": "5302.png", "formula": "\\begin{align*} A \\chi ^ { p } _ { \\{ i \\} } ( x ) = \\frac { ( 1 - x ) - p } { \\sqrt { p ( 1 - p ) } } = - \\chi ^ { 1 - p } _ { \\{ i \\} } ( x ) , \\end{align*}"}
+{"id": "301.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } \\sum _ { n = 1 } ^ { \\infty } ( - 2 n ^ 2 x ^ { n + 2 } y ^ { n + 1 } + 2 n ^ 2 x ^ { n + 2 } y ^ { n + 2 } - n ^ 2 x y ^ { n + 1 } ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "1510.png", "formula": "\\begin{align*} & \\overline X : = \\sup \\big \\{ y \\in [ r _ 0 - \\varepsilon _ 2 , \\Lambda _ 0 + \\varepsilon _ 2 ] : \\ , \\max _ { [ 0 , t - s ] } \\ , ( x + B _ { \\cdot \\wedge \\tau ^ B _ { r _ 0 } } - \\Lambda _ { s + \\cdot } ) < 0 \\big \\} \\vee ( r _ 0 - \\varepsilon _ 2 ) , \\\\ & \\underline X : = \\inf \\big \\{ y \\in [ r _ 0 - \\varepsilon _ 2 , \\Lambda _ 0 + \\varepsilon _ 2 ] : \\ , \\max _ { [ 0 , t - s ] } \\ , ( x + B _ { \\cdot \\wedge \\tau ^ B _ { r _ 0 } } - \\widetilde \\Lambda _ { s + \\cdot } ) \\ge 0 \\big \\} . \\end{align*}"}
+{"id": "4626.png", "formula": "\\begin{align*} \\partial _ t m = D \\Delta m - \\chi \\nabla \\cdot \\left ( \\frac { m } { 1 + \\delta m } \\nabla c \\right ) + m \\left ( 1 - m \\right ) . \\end{align*}"}
+{"id": "672.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\log \\norm { S ( k ) h } _ { \\sup } } { k } = \\max \\big ( & \\{ \\lambda _ i - l \\lambda _ 1 : 0 \\leq l \\leq n , 1 \\leq | i | \\leq g , d ( h , h _ { i , l } ) \\neq 0 \\} \\\\ & \\cup \\{ - l \\lambda _ 1 : 0 \\leq l \\leq n , 1 \\leq s < \\gamma , d ( h , c _ { s , l } ) \\neq 0 \\} \\big ) . \\end{align*}"}
+{"id": "9055.png", "formula": "\\begin{align*} z _ 1 & = \\frac { e ^ s \\cos \\theta + i ( z \\cos \\theta - p _ \\theta \\sin \\theta ) } { \\sqrt { e ^ { 2 s } + z ^ 2 + p _ \\theta ^ 2 } } , \\\\ z _ 2 & = \\frac { e ^ s \\sin \\theta + i ( z \\sin \\theta + p _ \\theta \\cos \\theta ) } { \\sqrt { e ^ { 2 s } + z ^ 2 + p _ \\theta ^ 2 } } , \\\\ e ^ u & = e ^ { 2 s } + z ^ 2 + p _ \\theta ^ 2 . \\end{align*}"}
+{"id": "4913.png", "formula": "\\begin{align*} \\begin{aligned} \\Pr ( T _ k = t ) & = \\Pr ( \\{ X _ { t - 1 } = k - 1 \\} \\cap \\{ X _ t = k \\} ) \\\\ & = \\Pr ( X _ { t - 1 } = k - 1 ) \\cdot \\Pr ( X _ t = k \\ , \\vert \\ , X _ { t - 1 } = k - 1 ) \\\\ & = \\Pr ( X _ { t - 1 } = k - 1 ) \\cdot p _ { k - 1 } ^ { } , \\end{aligned} \\end{align*}"}
+{"id": "4477.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { H ^ { s } } : = \\left \\| \\langle \\nabla \\rangle ^ s f \\right \\| _ { L ^ 2 } = \\left \\Vert \\langle \\xi \\rangle ^ { s } \\hat { f } \\right \\Vert _ { L ^ { 2 } } , \\end{align*}"}
+{"id": "304.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } \\sum _ { n = 1 } ^ { \\infty } ( - n x ^ { n + 2 } y ^ { n + 1 } - n x ^ { n + 2 } y ^ { n + 2 } - n y ^ 2 x ^ { n + 1 } - y ^ 2 x ^ { n + 1 } ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "80.png", "formula": "\\begin{align*} { \\rm W F } ' ( B ) \\cap \\Delta ( T ^ * \\mathcal { M } ) = \\varnothing , \\end{align*}"}
+{"id": "8996.png", "formula": "\\begin{align*} \\begin{cases} w _ { j i } = ( 1 + w ) T ^ \\varphi _ { j i } - w \\frac { S ^ \\varphi } { m ( m - 1 ) } \\delta _ { j i } \\\\ \\varphi ^ a _ s w _ s = - ( 1 + w ) \\varphi ^ a _ { t t } \\ , . \\end{cases} \\end{align*}"}
+{"id": "2982.png", "formula": "\\begin{align*} \\det \\begin{pmatrix} F ( t _ 1 ) t _ 1 ^ { k _ 1 } & F ( t _ 2 ) t _ 2 ^ { k _ 1 } & \\dots & F ( t _ n ) t _ n ^ { k _ 1 } \\\\ t _ 1 ^ { k _ 2 } & t _ 2 ^ { k _ 2 } & \\dots & t _ n ^ { k _ 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ t _ 1 ^ { k _ n } & t _ 2 ^ { k _ n } & \\dots & t _ n ^ { k _ n } \\end{pmatrix} \\end{align*}"}
+{"id": "4568.png", "formula": "\\begin{align*} \\mathcal D ( A ) = \\bigl ( H ( 0 ) = ( 1 ^ 6 ) , \\ , H ( 1 ) = ( 0 , 2 , 2 , 2 , 2 , 0 ) , \\ , H ( 3 ) = ( 0 , 1 , 1 , 0 ) \\bigr ) \\end{align*}"}
+{"id": "4327.png", "formula": "\\begin{align*} \\begin{gathered} \\left | y _ t ( x ) - y _ s ( x ' ) \\right | \\le C ( | x - x ' | + | t - s | ) , \\\\ \\left | y _ t ^ { - 1 } ( x ) - y _ s ^ { - 1 } ( x ' ) \\right | \\le C ( | x - x ' | + | t - s | ) , \\end{gathered} \\end{align*}"}
+{"id": "2985.png", "formula": "\\begin{align*} \\theta ( ( F ( t _ 2 ) - F ( t _ 1 ) ) \\ , t _ 1 ^ { k _ 1 } \\cdots t _ n ^ { k _ n } ) = \\begin{cases} - 2 q _ { n - 2 } ( t _ 1 , \\dots , t _ n ) & k _ 2 = n - 2 \\ , \\\\ - q _ { k _ 2 } ( t _ 1 , \\dots , t _ n ) & k _ 2 \\neq n - 2 \\ . \\end{cases} \\end{align*}"}
+{"id": "1599.png", "formula": "\\begin{align*} f = S _ { \\chi } S _ { \\chi } ^ { - 1 } f & = \\int _ { \\Theta } v ^ { 2 } ( w ) \\pi _ { F ( w ) } \\chi _ { w } ^ { \\ast } \\chi _ { w } \\pi _ { F ( w ) } S _ { \\chi } ^ { - 1 } f d \\mu ( w ) \\\\ & = \\int _ { \\Theta } v ^ { 2 } ( w ) \\pi _ { F ( w ) } \\chi _ { w } ^ { \\ast } T _ { w } d \\mu ( w ) . \\end{align*}"}
+{"id": "3242.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } f ( x ) d \\omega ( x ) = 0 . \\end{align*}"}
+{"id": "3681.png", "formula": "\\begin{align*} A _ q ( u , v ) : = \\left \\langle \\alpha _ 1 , ( ( - \\Delta ) ^ { a _ 1 } u ) v \\right \\rangle + \\left \\langle \\alpha _ 2 , ( ( - \\Delta ) ^ { a _ 2 } u ) v \\right \\rangle + \\cdots + \\left \\langle \\alpha _ p , ( ( - \\Delta ) ^ { a _ p } u ) v \\right \\rangle + ( q u , v ) _ \\Omega . \\end{align*}"}
+{"id": "3307.png", "formula": "\\begin{align*} b ( \\xi ) = | \\xi - 1 | ^ { \\delta _ 1 } | \\xi + 1 | ^ { \\delta _ { - 1 } } \\widetilde { b } ( \\xi ) \\end{align*}"}
+{"id": "1067.png", "formula": "\\begin{align*} \\| u \\| _ { \\exp L ^ p } = \\inf \\left \\{ \\lambda > 0 : \\int _ { \\R ^ d } \\left ( e ^ { \\frac { | u ( x ) | ^ p } { \\lambda ^ p } } - 1 \\right ) d x \\leq 1 \\right \\} . \\end{align*}"}
+{"id": "2789.png", "formula": "\\begin{align*} \\| \\partial _ \\nu u \\| _ { L ^ 2 ( \\Sigma ) } & = \\| ( \\Lambda _ { q _ 1 , \\lambda } ^ 1 - \\Lambda _ { q _ 2 , \\lambda } ^ 1 ) ( u _ 2 { _ { | \\partial M } } ) \\| _ { L ^ 2 ( \\Sigma ) } \\\\ & \\le \\| \\Lambda _ { q _ 1 , \\lambda } ^ 1 - \\Lambda _ { q _ 2 , \\lambda } ^ 1 \\| \\| u _ 2 \\| _ { H ^ { 3 / 2 } ( \\partial \\mathrm { M } ) } \\end{align*}"}
+{"id": "3635.png", "formula": "\\begin{align*} ( \\{ 9 \\} ^ { I + 1 } , a _ { I + 1 } , \\ldots , a _ { m - I - 1 } , \\{ 9 \\} ^ { I } , 4 ) _ { 1 0 } \\ = \\ 2 ( 4 , \\{ 9 \\} ^ { I } , a _ { m - I - 1 } , \\ldots , a _ { I + 1 } , \\{ 9 \\} ^ { I } , 7 ) _ { 1 0 } . \\end{align*}"}
+{"id": "6202.png", "formula": "\\begin{align*} r \\colon V _ e \\to V _ e , r ( v ) = w v ) . \\end{align*}"}
+{"id": "1158.png", "formula": "\\begin{align*} T ( f ^ { n } ) = n f ^ { n - 1 } T ( f ) + n B ( A ( f ) , A ( f ^ { n - 1 } ) ) \\end{align*}"}
+{"id": "6946.png", "formula": "\\begin{align*} X ' = ( 1 / 4 + 2 a , - 1 / 4 + 2 b , 2 c ) , \\end{align*}"}
+{"id": "5094.png", "formula": "\\begin{align*} \\lim _ { t \\uparrow \\tau _ { \\max } } \\left \\| \\left ( \\gamma , \\gamma _ t , \\sigma , \\sigma _ t \\right ) \\right \\| _ { H ^ 4 _ N \\times H ^ 2 _ N \\times \\R \\times \\R } = \\infty . \\end{align*}"}
+{"id": "4009.png", "formula": "\\begin{align*} \\varphi _ { \\lambda , d } ( x _ 1 , x _ 2 ) = \\min _ { u \\in \\mathcal { X } : d ( u ) = d } \\sum _ { 1 \\leq \\ell \\leq 2 } \\lambda _ \\ell \\| x _ \\ell - u \\| _ 2 , \\end{align*}"}
+{"id": "67.png", "formula": "\\begin{align*} \\mathcal { T } = \\bigsqcup _ { i = 1 } ^ r \\mathcal { T } _ n \\end{align*}"}
+{"id": "9281.png", "formula": "\\begin{align*} F _ { I } = \\left \\{ x \\in \\mathbb { R } ^ { m n } \\ | \\ A _ { I } x = b _ { I } \\right \\} \\ \\mbox { a n d } \\ D _ { I } = \\left \\{ x \\in X \\ | \\ A _ { I } x = b _ { I } \\right \\} = X \\bigcap F _ { I } , \\end{align*}"}
+{"id": "7773.png", "formula": "\\begin{align*} x ^ 2 ( x ^ q y - x y ^ q ) + y ^ 2 ( y ^ q z - y z ^ q ) + ( z ^ 2 + k x ^ 2 ) ( z ^ q x - z x ^ q ) = 0 . \\end{align*}"}
+{"id": "4125.png", "formula": "\\begin{align*} ( T _ \\varepsilon ^ { \\vec { v } } ) _ { n , j - 1 } = - ( T _ \\varepsilon ^ { \\vec { v } } ) _ { 1 , j } \\alpha _ 0 , \\end{align*}"}
+{"id": "5786.png", "formula": "\\begin{align*} y ^ { ( k ) } + A _ 1 y = 0 \\end{align*}"}
+{"id": "7756.png", "formula": "\\begin{align*} a \\boxplus b = \\max \\{ a , b \\} , a \\odot b = a + b . \\end{align*}"}
+{"id": "122.png", "formula": "\\begin{align*} \\chi e ^ { - i t _ 0 h ^ { - 1 } P _ h ( z ) } R _ h ( z ) \\chi = & \\chi e ^ { - i t _ 0 h ^ { - 1 } \\tilde { P } _ h ( z ) } \\tilde { R } _ h ( z ) \\chi + \\chi ( R _ h ( z ) - \\tilde { R } _ h ( z ) ) \\chi \\\\ & - \\frac { i } { h } \\int _ 0 ^ { t _ 0 } \\chi ( e ^ { - i s h ^ { - 1 } P _ h ( z ) } - e ^ { - i s h ^ { - 1 } \\tilde { P } _ h ( z ) } ) \\chi d s . \\end{align*}"}
+{"id": "69.png", "formula": "\\begin{align*} \\mathbf { P } \\mathbf { G } ^ 1 ( \\mathbb { Q } _ p ) = \\mathbf { G } ^ 1 ( \\mathbb { Q } _ p ) / Z ( \\mathbf { G } ^ 1 ( \\mathbb { Q } _ p ) ) , \\end{align*}"}
+{"id": "2661.png", "formula": "\\begin{align*} & s _ \\alpha ' ( a ) = f ' ( a ) , s _ \\alpha ' ( b ) = f ' ( b ) , \\\\ & s _ \\alpha '' ( a ) = f '' ( a ) , s _ \\alpha '' ( b ) = f '' ( b ) . \\end{align*}"}
+{"id": "4467.png", "formula": "\\begin{align*} K _ { \\omega , c } ( \\lambda \\Phi _ { \\omega } ) & = 2 \\lambda ^ 2 L ( \\Phi _ { \\omega } ) + 3 \\lambda ^ 3 N ( \\Phi _ { \\omega } ) + 2 \\lambda ^ 2 \\omega Q ( \\Phi _ { \\omega } ) + 2 \\lambda ^ 2 c P ( \\Phi _ { \\omega } ) \\\\ & = \\lambda ^ 2 K _ { \\omega , 0 } ( \\Phi _ { \\omega } ) + \\lambda ^ 2 \\{ 3 ( \\lambda - 1 ) N ( \\Phi _ { \\omega } ) + 2 c P ( \\Phi _ { \\omega } ) \\} \\\\ & = 2 \\lambda ^ 2 \\{ - 3 ( \\lambda - 1 ) \\mu _ { \\omega , 0 } + c P ( \\Phi _ { \\omega } ) \\} . \\end{align*}"}
+{"id": "9069.png", "formula": "\\begin{align*} \\lambda _ k = \\min \\limits _ { P \\in P _ k } \\max \\limits _ { 0 \\neq g \\in P } \\frac { \\langle g , M g \\rangle } { \\langle g , g \\rangle } = \\max \\limits _ { P \\in P _ { k - 1 } ^ { \\perp } } \\min \\limits _ { 0 \\neq g \\in P \\in P _ k } \\frac { \\langle g , M g \\rangle } { \\langle g , g \\rangle } , \\end{align*}"}
+{"id": "5946.png", "formula": "\\begin{align*} \\phi ( h ) ^ { \\phi ( \\sigma , g ) } & = ( \\sum ^ m _ { k = 1 } ( a _ k e _ k + i b _ k e ^ { \\ast } _ k ) , t ) ^ { ( s ( - 1 ) , g ) } = ( \\sum ^ m _ { k = 1 } ( a _ k e _ k - i b _ k e ^ { \\ast } _ k ) , - t ) ^ { g } \\\\ & = ( [ \\sum ^ m _ { k = 1 } ( a _ k e _ k - i b _ k e ^ { \\ast } _ k ) ] g , - t ) = \\phi ( h ^ { ( \\sigma , g ) } ) . \\end{align*}"}
+{"id": "7121.png", "formula": "\\begin{align*} \\mathcal { R } _ \\varepsilon c ( x ) : = \\int _ { \\mathbb { R } ^ n _ - } J _ \\varepsilon ( | x - y | ) ( c ( x ) - \\tilde { c } ( y ) ) \\ : y \\end{align*}"}
+{"id": "6720.png", "formula": "\\begin{align*} \\mathcal { P } ( \\widetilde { X } _ { \\eta } ) = \\mathcal { P } ( \\widetilde { X } _ { \\eta ' } ) . \\end{align*}"}
+{"id": "8348.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ \\infty \\alpha _ j u _ j \\chi _ { B _ { R _ { j + 1 } } } \\Big \\| _ { p ^ * , q , \\mu } & \\leq \\sum _ { j = 1 } ^ \\infty | \\alpha _ j | \\| u _ j \\chi _ { B _ { R _ { j + 1 } } } \\| _ { p ^ * , q , \\mu } \\leq \\sum _ { j = 1 } ^ \\infty | \\alpha _ j | \\gamma _ j \\\\ & \\leq \\| \\{ \\alpha _ j \\} _ { j = 1 } ^ \\infty \\| _ { \\ell _ q } \\| \\{ \\gamma _ j \\} _ { j = 1 } ^ \\infty \\| _ { \\ell _ { q ' } } . \\end{align*}"}
+{"id": "7266.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } n ^ { - 1 / 2 } \\big \\Vert \\sum _ { i = 1 } ^ n d _ { i , N } \\big \\Vert _ 2 \\leq \\limsup _ { n \\rightarrow \\infty } n ^ { - 1 / 2 } \\big \\Vert \\sum _ { i = 1 } ^ n Y _ { i , N } \\big \\Vert _ 2 \\ , . \\end{align*}"}
+{"id": "5468.png", "formula": "\\begin{align*} M _ b ( \\theta ) = \\mathbb { E } _ { \\Phi } \\{ P _ s ^ b ( \\theta ) \\} \\end{align*}"}
+{"id": "5814.png", "formula": "\\begin{align*} { { \\boldsymbol { X } } _ { i ; k | k - 1 } } = { \\boldsymbol { f } } \\left ( { { { \\boldsymbol { \\xi } } _ { i ; k - 1 | k - 1 } } } \\right ) , \\left ( { i = 1 , 2 , \\cdots 2 n } \\right ) . \\end{align*}"}
+{"id": "5449.png", "formula": "\\begin{align*} & = \\frac { | h _ 1 | ^ 2 } { \\sum _ { x _ i \\in \\Phi \\backslash \\{ x _ 1 \\} \\cap \\mathcal { A } } | h _ i | ^ 2 } \\overset { \\Delta } { = } \\frac { | h _ 1 | ^ 2 } { I } , \\end{align*}"}
+{"id": "933.png", "formula": "\\begin{align*} L _ t = \\begin{cases} L _ { } & t = 1 , \\\\ 1 & L _ { } \\leq 2 \\land t > 1 , \\\\ 2 & 2 < L _ { } \\leq 1 0 \\land t > 1 , \\\\ 4 & 1 0 < L _ { } \\land t > 1 , \\\\ \\end{cases} \\end{align*}"}
+{"id": "788.png", "formula": "\\begin{align*} \\mathrm { V o l } ( \\Omega ) = \\mathrm { V o l } ( \\overline { B } _ { \\Omega } ) = \\mathrm { A r e a ( \\mathbb { S } ^ n ) } \\int _ { 0 } ^ R \\phi ^ n ( r ) \\mathrm { d } r . \\end{align*}"}
+{"id": "3654.png", "formula": "\\begin{align*} \\frac { v _ j } { \\lambda } \\sum _ { k \\in [ n ] , k \\neq j } ^ n ( t _ k t _ k ^ * ) = \\lambda v _ j . \\end{align*}"}
+{"id": "1001.png", "formula": "\\begin{align*} d = \\sum _ { E } \\delta ( \\mathbf { u } ( E ) ) , \\end{align*}"}
+{"id": "7945.png", "formula": "\\begin{align*} J _ H ( u ) = \\int _ \\Omega B ( H ( \\nabla u ) ) \\ , d x - \\int _ \\Omega f u \\ , d x \\end{align*}"}
+{"id": "1748.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\textrm { o u t } } = \\begin{cases} 1 - \\sum \\limits _ { m = 1 } ^ M \\mathcal { V } ^ { ( M , m ) } \\xi ^ m J _ { + } ^ { ( m ) } & \\textrm { i f ~ } \\mu _ { \\textrm { D } } ^ 2 \\Gamma _ 0 \\leq \\rho - 1 \\\\ 1 - \\sum \\limits _ { m = 1 } ^ M \\mathcal { V } ^ { ( M , m ) } \\xi ^ m \\big ( I _ + ^ { ( m ) } + I _ - ^ { ( m ) } \\big ) & \\textrm { i f ~ } \\mu _ { \\textrm { D } } ^ 2 \\Gamma _ 0 > \\rho - 1 \\end{cases} \\end{align*}"}
+{"id": "9072.png", "formula": "\\begin{align*} g _ { i } ( x ) = \\left \\{ \\begin{array} { l l } f _ k ( x ) , & \\hbox { i f $ x \\in \\Omega _ i $ ; } \\\\ 0 , & \\hbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
+{"id": "2127.png", "formula": "\\begin{align*} \\mathfrak { N } _ { f , a } ( L , L , H , H ) - \\mathfrak { N } _ { f , a } ( 1 , 1 , 1 , 1 ) \\geq & - \\sum _ { i = 1 } ^ t \\theta ( l _ i ) ( q ^ { m / 2 } + n ) - \\sum _ { i = 1 } ^ t \\theta ( l _ i ) ( n + 1 ) q ^ { m / 2 } \\\\ & - \\sum _ { i = 1 } ^ u \\Theta ( h _ i ) ( 2 q ^ { m / 2 } + n ) - \\sum _ { i = 1 } ^ u \\Theta ( h _ i ) ( n + 1 ) q ^ { m / 2 } \\\\ & - ( 2 \\epsilon _ 1 + 2 \\epsilon _ 2 ) \\mathfrak { N } _ { f , a } ( 1 , 1 , 1 , 1 ) . \\end{align*}"}
+{"id": "3304.png", "formula": "\\begin{align*} \\widetilde { B } ( \\xi ) = \\left \\{ \\begin{array} { r l } e ^ { - i \\frac { \\pi } { 4 } } B ( \\xi ) \\ , \\overline { \\xi } ^ { \\frac { 1 } { 2 } } = e ^ { i \\frac { 2 \\theta - \\pi } { 4 } } , & { \\rm i f \\ } 0 \\le \\theta \\le \\pi \\\\ e ^ { i \\frac { 3 \\pi } { 4 } } \\ , \\overline { \\xi } ^ { \\frac { 1 } { 2 } } = e ^ { i \\frac { 3 \\pi - 2 \\theta } { 4 } } , & { \\rm i f \\ } \\pi < \\theta < 2 \\pi . \\end{array} \\right . \\end{align*}"}
+{"id": "1676.png", "formula": "\\begin{align*} \\mu ^ 2 = \\pi \\psi \\pi \\psi = \\pi \\pi \\psi ^ { ( p ) } \\psi = \\pi _ E \\epsilon \\widehat { \\psi } \\psi = \\epsilon d \\pi _ E . \\end{align*}"}
+{"id": "620.png", "formula": "\\begin{align*} \\lim _ \\lambda \\| e ^ * _ \\lambda ( b - \\omega ( b ) I ) e _ \\lambda \\| = 0 , b \\in B . \\end{align*}"}
+{"id": "8453.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\left \\{ \\frac { 1 } { \\left ( \\lambda _ { n } ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } - \\sum _ { m = 0 } ^ { N - 1 } \\left ( \\begin{array} { c } - s \\\\ m \\end{array} \\right ) \\ , \\frac { x ^ { 2 m } } { \\lambda _ { n } ^ { 2 s + 2 m } } \\right \\} + \\sum _ { m = 0 } ^ { N - 1 } \\left ( \\begin{array} { c } - s \\\\ m \\end{array} \\right ) x ^ { 2 m } \\zeta _ { p } \\left ( 2 s + 2 m \\right ) . \\end{align*}"}
+{"id": "9108.png", "formula": "\\begin{align*} y _ { [ A ] } = ( y _ { [ a ^ { 1 } ] } ^ { 1 } , \\ldots , y _ { [ a ^ { m } ] } ^ { m } ) \\end{align*}"}
+{"id": "8614.png", "formula": "\\begin{align*} & I _ 1 ( \\ell _ 1 , \\ell _ 2 ) : = P ( W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 , W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) = 1 ) , \\\\ & I _ 2 ( \\ell _ 1 , \\ell _ 2 ) : = \\nu \\Delta p ^ k _ { j , + } ( t - \\Delta \\ell _ 2 ) E [ Z _ 0 ( \\Delta \\ell _ 2 ) ; W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 ] . \\end{align*}"}
+{"id": "7348.png", "formula": "\\begin{align*} \\sum _ { \\delta = 2 } ^ n \\varepsilon ^ \\delta + \\sum _ { \\delta = 1 } ^ n s \\delta \\varepsilon ^ \\delta & \\leq \\frac { \\varepsilon ^ 2 } { 1 - \\varepsilon } + \\frac { s \\varepsilon } { ( 1 - \\varepsilon ) ^ 2 } \\end{align*}"}
+{"id": "8746.png", "formula": "\\begin{align*} \\Delta _ 0 = - 2 f ^ { ( 1 ) } ( \\tau _ 1 ) \\frac { \\| \\mu _ 1 - \\mu _ 2 \\| _ 2 ^ 2 } { \\gamma } \\{ 1 + o ( 1 ) \\} , \\end{align*}"}
+{"id": "1119.png", "formula": "\\begin{align*} g ( f ( x y ) ) & = g ( ( f x ) y ) - g ( ( f y ) x ) , \\\\ ( f ( x y ) ) g & = ( ( f x ) y ) g - ( ( f y ) x ) g , \\\\ f ( ( g x ) y ) & = ( f ( g x ) ) y - ( f y ) ( g x ) , \\\\ f ( ( x g ) y ) & = ( f ( x g ) ) y - ( f y ) ( x g ) , \\\\ f ( x ( g y ) ) & = ( f x ) ( g y ) - ( f ( g y ) ) x , \\\\ f ( x ( y g ) ) & = ( f x ) ( y g ) - ( f ( y g ) ) x . \\end{align*}"}
+{"id": "6366.png", "formula": "\\begin{align*} \\rho _ t ( 0 , y ) = t ^ { - ( d + 2 \\beta ) / \\alpha } \\rho _ 1 ( 0 , t ^ { - 1 / \\alpha } y ) , t > 0 , \\ , y \\in \\Gamma , \\end{align*}"}
+{"id": "6401.png", "formula": "\\begin{align*} \\widetilde { \\varphi } : = \\widehat { \\varphi } \\circ T _ M , \\end{align*}"}
+{"id": "5677.png", "formula": "\\begin{align*} \\xi _ { 2 - k } : = 2 i y ^ { 2 - k } \\overline { ( \\dfrac { \\partial } { \\partial \\overline { z } } ) } : H _ { 2 - k } ( \\Gamma _ { 0 } ( N ) ) \\rightarrow S _ { k } ( \\Gamma _ { 0 } ( N ) ) . \\end{align*}"}
+{"id": "5867.png", "formula": "\\begin{align*} \\# \\sigma ^ { ( \\mathcal { I } , r e d ) } ( H _ { \\omega , \\Lambda _ { L } } ) & \\leq \\# \\{ \\{ E _ n \\} _ { n = 0 } ^ { n _ 1 } : E _ n \\in \\sigma ( H _ { \\omega , \\Lambda _ { L _ n } } ) ~ \\& ~ | E _ i - E _ j | \\leq 2 e ^ { - \\frac { \\hat { m } } { K } L _ { \\max { i , j } } } \\} \\\\ & : = \\# D _ 0 ^ { n _ 1 } \\end{align*}"}
+{"id": "7464.png", "formula": "\\begin{align*} \\mathcal { U } ^ { ( i ) } _ { p \\alpha + k - 1 } ( x , t ; \\varepsilon ) : = w _ { p \\alpha + k - 1 } ^ { ( i ) } ( x _ i ) + u _ { p \\alpha + k - 1 } ^ { ( i ) } \\big ( x _ i , \\tfrac { \\overline { x } _ i } { \\varepsilon } , t \\big ) , \\mathcal { P } ^ { ( i ) } _ { p \\alpha + k - 1 } ( x _ i , t ; \\varepsilon ) : = \\chi _ \\delta ^ { ( i ) } ( x _ i ) \\ , \\Pi _ { p \\alpha + k } ^ { ( i ) } \\big ( \\tfrac { \\ell _ i - x _ i } { \\varepsilon } , t \\big ) , \\end{align*}"}
+{"id": "5522.png", "formula": "\\begin{align*} \\frac { \\varphi _ { g ^ { - 1 } s } ( \\pi ( x ) ) } { \\varphi _ { g ^ { - 1 } } ( \\pi ( x ) ) } = \\frac { d g ^ { - 1 } s \\nu } { d g ^ { - 1 } \\nu } ( \\pi ( x ) ) = \\frac { d s \\nu } { d \\nu } ( g . \\pi ( x ) ) = \\frac { d s \\nu } { d \\nu } ( \\pi ( g . x ) ) . \\end{align*}"}
+{"id": "5873.png", "formula": "\\begin{align*} \\| u _ 0 \\| _ { L ^ 2 } = r \\ , . \\end{align*}"}
+{"id": "7976.png", "formula": "\\begin{align*} e ^ h \\mathrm { d i v } \\Big ( e ^ { - h } ( V \\cdot \\nabla ) V \\Big ) & = e ^ h \\partial _ j \\Big ( e ^ { - h } V ^ i \\ , \\partial _ i V ^ j \\Big ) = \\partial _ j V ^ i \\ , \\partial _ i V ^ j + V ^ i \\ , \\partial _ j \\partial _ i V ^ j - \\partial _ j h \\ , V ^ i \\ , \\partial _ i V ^ j \\\\ & = \\mathrm { t r } ( ( \\nabla V ) ^ 2 ) + V ^ i \\ , \\partial _ j \\partial _ i V ^ j - \\partial _ j h \\ , V ^ i \\ , \\partial _ i V ^ j . \\end{align*}"}
+{"id": "5047.png", "formula": "\\begin{align*} \\overline j _ f = \\left ( \\frac { 1 } { 4 } , \\frac { 1 } { 2 } , \\frac { 1 } { 1 2 } , \\frac { 1 } { 6 } \\right ) . \\end{align*}"}
+{"id": "8172.png", "formula": "\\begin{align*} \\underset { t \\rightarrow \\infty } { \\lim } ( \\log t ) ^ { 2 / d } \\left ( \\frac { \\log N _ t } { t } - \\beta \\right ) = - c ( d , \\nu ) \\ : \\ : \\widehat { P } ^ \\omega . \\end{align*}"}
+{"id": "4840.png", "formula": "\\begin{align*} g _ w ( y ) = \\prod _ { i = 1 } ^ n h _ { b _ i } ( n y + i ) . \\end{align*}"}
+{"id": "8659.png", "formula": "\\begin{align*} \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } ( 1 - y ) d y & = \\sum _ { k = 0 } ^ \\infty p ^ k \\int _ 0 ^ 1 y ^ k ( 1 - y ) d y \\\\ & = \\sum _ { k = 0 } ^ \\infty \\frac { p ^ k } { k + 1 } - \\sum _ { k = 0 } ^ \\infty \\frac { p ^ k } { k + 2 } . \\end{align*}"}
+{"id": "5050.png", "formula": "\\begin{align*} \\binom { t - k ( s ( t ) , t ) - \\left \\lfloor \\frac { t - k ( s ( t ) , t ) } { s ( t ) } \\right \\rfloor s ( t ) } { 2 } & = \\left ( 1 - \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 4 i } + o ( 1 ) \\right ) - ( i + o ( 1 ) ) \\left ( \\frac { 1 } { 2 i } + o ( 1 ) \\right ) \\right ) ^ 2 \\frac { t ^ 2 } { 2 } \\\\ & = \\left ( \\frac { 1 } { 1 6 i ^ 2 } + o ( 1 ) \\right ) \\binom { t } { 2 } . \\end{align*}"}
+{"id": "4140.png", "formula": "\\begin{align*} \\Delta = - 1 8 s t + 4 s ^ 3 + s ^ 2 t ^ 2 - 4 t ^ 3 - 2 7 = ( s ^ 2 - 4 t ) ( t ^ 2 + 4 s ) - 2 s t - 2 7 \\overset { t > \\frac { s ^ 2 } { 4 } } { < } 0 . \\end{align*}"}
+{"id": "5154.png", "formula": "\\begin{align*} w ( E ) : = \\int _ E \\ ! w \\ , \\mathrm { d } \\mu , \\end{align*}"}
+{"id": "8918.png", "formula": "\\begin{align*} \\| P _ m ( U _ n ) \\| _ m \\leq \\begin{cases} 0 , & m = 1 , \\dots , j - 1 , \\\\ \\| X _ { n 1 } \\| _ 1 ^ j , & m = j , \\\\ \\sum _ { ( i _ 1 , \\dots , i _ m ) \\in \\mathbb { I } _ j ^ m } | \\gamma _ { ( i _ 1 , \\dots , i _ m ) } | \\| X _ { n 1 } \\| _ 1 ^ m , & m = j + 1 , \\dots , k . \\end{cases} \\end{align*}"}
+{"id": "7306.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\alpha _ i t _ i = \\sum _ { i = 1 } ^ n \\beta _ i t _ i = \\sum _ { i = 1 } ^ n \\gamma _ i t _ i + 1 , t _ i \\geq 0 , i = 1 , \\dots , n , \\end{align*}"}
+{"id": "3388.png", "formula": "\\begin{align*} S _ { i , 0 } & = \\sum _ { \\substack { n \\leq x _ i \\\\ P ( n ) \\leq y _ 0 } } \\frac { f ( n ) } { \\sqrt { n } } , \\\\ S _ { i , j } & = \\sum _ { \\substack { y _ { j - 1 } < m \\leq x _ i \\\\ p | m \\Rightarrow p \\in ( y _ { j - 1 } , y _ j ] } } \\frac { f ( m ) } { \\sqrt { m } } \\sum _ { \\substack { n \\leq x _ i / m \\\\ P ( n ) \\leq y _ { j - 1 } } } \\frac { f ( n ) } { \\sqrt { n } } . \\end{align*}"}
+{"id": "530.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\hbar , V } u _ { \\xi } = \\lambda _ { \\xi } u _ { \\xi } , \\xi \\in \\mathcal { I } _ { \\hbar } . \\end{align*}"}
+{"id": "3825.png", "formula": "\\begin{align*} \\Theta _ { \\mathrm { D } } ( \\delta ) = \\left [ \\min _ { \\gamma \\in \\Sigma _ { \\mathrm { D } } ( \\delta ) } \\int _ { \\mathcal { S } _ 1 \\times \\mathcal { S } _ 2 } g \\ , d \\gamma , \\max _ { \\gamma \\in \\Sigma _ { \\mathrm { D } } ( \\delta ) } \\int _ { \\mathcal { S } _ 1 \\times \\mathcal { S } _ 2 } g \\ , d \\gamma \\right ] , \\end{align*}"}
+{"id": "1220.png", "formula": "\\begin{align*} q - I _ { \\boldsymbol { a } } ( z ) & = \\sum _ { n = 0 } ^ { \\infty } b _ n \\int _ { [ z , 1 / a _ { j _ 1 } ] } ( 1 - z ' a _ { j _ 1 } ) ^ { n + ( 1 - q ) \\nu ( j _ 1 ) / q } \\dd z ' \\\\ & = h ( z ) ( 1 - z a _ { j _ 1 } ) ^ { 1 + ( 1 - q ) \\nu ( j _ 1 ) / q } , \\end{align*}"}
+{"id": "3456.png", "formula": "\\begin{align*} S ^ { - 1 } r S = r ^ { - 1 } . \\end{align*}"}
+{"id": "8560.png", "formula": "\\begin{align*} M ( t ) : = \\sum _ { j = 1 } ^ \\infty S _ j ( t ) . \\end{align*}"}
+{"id": "7646.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x } M ( c , x , 0 ) = ( 4 - c ^ 2 ) ^ 2 ( 7 2 + 4 c ^ 2 x - ( 1 0 8 + 3 9 c ^ 2 ) x ^ 2 + 8 c ^ 2 x ^ 3 ) = 0 . \\end{align*}"}
+{"id": "4677.png", "formula": "\\begin{align*} \\Lambda \\vect g = \\Gamma _ 1 \\Pi \\vect g \\forall \\vect g \\in \\mathcal { D } ( \\Lambda ) , \\Pi ^ * \\vect f = \\Gamma _ 1 \\mathcal { A } _ 0 ^ { - 1 } \\vect f \\forall \\vect f \\in \\mathcal { H } . \\end{align*}"}
+{"id": "6759.png", "formula": "\\begin{align*} \\widehat { \\sigma x } _ { \\sigma z } = \\begin{cases} \\hat { x } _ z , & z ( 0 ) = 0 , \\\\ \\sigma \\hat { x } _ z , & z ( 0 ) = 1 , \\end{cases} \\end{align*}"}
+{"id": "2783.png", "formula": "\\begin{align*} u _ 1 u _ 2 = e ^ { - i \\eta \\cdot x } + \\varrho , \\varrho = e ^ { - i \\eta \\cdot x } ( w _ 1 + w _ 2 + w _ 1 w _ 2 ) , \\end{align*}"}
+{"id": "167.png", "formula": "\\begin{align*} L i _ 2 ( 0 ) = 0 , \\end{align*}"}
+{"id": "1915.png", "formula": "\\begin{align*} \\{ \\mathcal { D } _ { \\eta ^ { k } } ( \\hat { \\eta } , \\eta ^ { k } ; \\lambda ^ { k } ) \\} _ { k = 1 } ^ { + \\infty } \\ \\ \\mbox { i s \\ a \\ C a u c h y \\ s e q u e n c e } , \\end{align*}"}
+{"id": "1992.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u ) ^ n = ( 1 - \\lambda u ) ^ n f ^ n \\omega ^ n & \\textnormal { i n } & \\Omega , \\\\ u \\equiv 0 & \\textnormal { o n } & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "6663.png", "formula": "\\begin{align*} F ( a , b , c , - r ^ 2 ) = \\frac 1 { ( 1 + r ^ 2 ) ^ b } F \\left ( c - a , b , c , \\frac { r ^ 2 } { 1 + r ^ 2 } \\right ) \\textrm { f o r a l l } \\ , \\ ; r > 1 \\ , . \\end{align*}"}
+{"id": "613.png", "formula": "\\begin{align*} \\omega ( b ) = \\langle \\pi ( b ) \\xi , \\xi \\rangle , b \\in B . \\end{align*}"}
+{"id": "3959.png", "formula": "\\begin{align*} \\boldsymbol { K } _ 1 ( \\gamma _ 1 , \\gamma _ 1 ^ { \\eta _ 0 , \\delta } ) = \\mathbb { E } \\left [ c _ 1 ( S _ 1 , \\widetilde { S } _ 1 ) \\right ] \\leq \\eta _ 0 - \\eta . \\end{align*}"}
+{"id": "3225.png", "formula": "\\begin{align*} F _ { 0 } & = \\{ P \\in F ~ | ~ \\} \\\\ F _ { i j k } & = \\{ P \\in F ~ | ~ p _ { i } , p _ { j } , p _ { k } \\} \\\\ F _ { 1 2 3 4 } & = \\{ P \\in F ~ | ~ \\} \\end{align*}"}
+{"id": "689.png", "formula": "\\begin{align*} | \\varphi ( x ) | & \\leq \\frac { \\norm { \\varphi } _ { L ^ 1 } } { | I | } + p _ a ( \\varphi ) \\Big ( \\frac { 1 } { a \\min \\{ x - l _ \\alpha , r _ \\alpha - x \\} ^ { a } } + \\frac { 2 ^ { a + 2 } } { a ( 1 - a ) | I _ \\alpha | ^ { a } } \\Big ) 0 < a < 1 , \\\\ | \\varphi ( x ) | & \\leq \\frac { \\norm { \\varphi } _ { L ^ 1 } } { | I | } + p _ a ( \\varphi ) \\Big ( \\log \\frac { | I _ \\alpha | } { 2 \\min \\{ x - l _ \\alpha , r _ \\alpha - x \\} } + 2 \\Big ) a = 0 . \\end{align*}"}
+{"id": "3250.png", "formula": "\\begin{align*} & \\frac 1 { \\omega ( B ) \\ell ( B ) ^ \\beta } \\int _ B | g _ { 2 1 } ( y ) - ( g _ { 2 1 } ) _ B | d \\omega ( y ) + \\frac 1 { \\omega ( B ) \\ell ( B ) ^ \\beta } \\int _ B | g _ { \\sigma _ j 1 } ( y ) - ( g _ { \\sigma _ j 1 } ) _ B | d \\omega ( y ) \\\\ & \\le C \\| b \\| _ { \\Lambda ^ \\beta } \\big ( M _ t ( R _ { { \\rm D } , \\ell } f _ 1 ) ( x ) + M _ t ( R _ { { \\rm D } , \\ell } f _ { \\sigma _ j } ) ( x ) \\big ) \\qquad \\mbox { f o r } j = 1 , 2 , \\cdots , | G | - 1 . \\end{align*}"}
+{"id": "8582.png", "formula": "\\begin{align*} S _ j ( t ) = S _ { j , + } ( t ) - S _ { j , - } ( t ) . \\end{align*}"}
+{"id": "3532.png", "formula": "\\begin{align*} \\sum _ { t \\in ( a , b ) } \\dim ( \\sigma _ 0 ( t ) \\cap \\sigma ) = \\sum _ { \\lambda < 0 } \\dim ( \\sigma _ { \\lambda } ( b ) \\cap \\sigma ) . \\end{align*}"}
+{"id": "6446.png", "formula": "\\begin{align*} \\partial _ { a _ 1 } \\Psi & = ( 1 , 0 , 0 , - \\lambda ) , \\\\ \\partial _ { a _ 2 } \\Psi & = ( 0 , 1 , \\lambda , 0 ) , \\\\ \\partial _ { \\lambda } \\Psi & = ( 0 , 0 , a _ 1 , a _ 2 ) , \\end{align*}"}
+{"id": "4876.png", "formula": "\\begin{align*} \\frac { u } { d ^ s } ( x _ 0 ) = 0 . \\end{align*}"}
+{"id": "863.png", "formula": "\\begin{align*} \\alpha _ { \\tau , i } + \\alpha _ { \\tau , i + 2 } = 2 \\alpha _ { \\tau , i + 1 } \\end{align*}"}
+{"id": "2064.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ~ \\frac { \\delta _ { A , C } ( n ) } { \\delta _ { A , D } ( n ) } = \\frac { 1 } { 2 } \\left ( \\alpha + \\sqrt { \\alpha ^ 2 - 4 } \\right ) \\end{align*}"}
+{"id": "1397.png", "formula": "\\begin{align*} \\int _ M d d ^ c \\omega _ 0 ^ { n - k - 1 } \\wedge \\omega ^ k = \\int _ M \\omega _ 0 ^ { n - k - 1 } \\wedge d d ^ c \\omega ^ k = 0 , \\end{align*}"}
+{"id": "7328.png", "formula": "\\begin{align*} A \\prod _ { k = 1 } ^ { N _ 3 } W _ k ^ { g _ k } + B \\prod _ { k = 1 } ^ { N _ 2 } V _ k ^ { f _ k } + C \\prod _ { k = 1 } ^ { N _ 1 } U _ k ^ { e _ k } = 0 , \\end{align*}"}
+{"id": "338.png", "formula": "\\begin{align*} \\prod _ { \\substack { \\gcd ( j _ 1 , j _ 2 , j _ 3 , j _ 4 , j _ 5 , k ) = 1 \\\\ j _ 1 , j _ 2 , j _ 3 , j _ 4 , j _ 5 < k \\\\ j _ 1 , j _ 2 , j _ 3 , j _ 4 , j _ 5 \\geq 1 ; k \\geq 2 } } \\left ( \\frac { 1 } { 1 - y ^ { j _ 1 + j _ 2 + j _ 3 + j _ 4 + j _ 5 } z ^ k } \\right ) ^ { \\frac { 1 } { k } } , \\end{align*}"}
+{"id": "1762.png", "formula": "\\begin{align*} c ( z ) = \\frac { c _ \\alpha } { | z | ^ { 1 + \\alpha } } , \\textrm { ~ w h e r e ~ } c _ \\alpha = \\pi ^ { - 1 } \\Gamma ( \\alpha + 1 ) \\sin \\left ( \\frac { \\alpha \\pi } { 2 } \\right ) , \\end{align*}"}
+{"id": "1729.png", "formula": "\\begin{align*} \\int _ { \\mathcal { X } } \\Psi ( \\nu , \\mu ) \\left ( \\mathrm { d } x \\right ) = \\int _ { \\mathcal { X } } \\Psi ( \\nu , \\mu ) \\left ( x \\right ) \\mathrm { d } x = \\frac { 1 } { Z ( \\nu , \\mu ) } \\int _ { \\mathcal { X } } \\exp \\left ( - \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta \\nu } ( \\nu , \\mu , x ) - U ^ { \\pi } \\left ( x \\right ) \\right ) \\mathrm { d } x = 1 , \\end{align*}"}
+{"id": "614.png", "formula": "\\begin{align*} \\frac { 1 } { t - s } ( f ( t ) - f ( s ) ) & = \\frac { 1 } { t - s } \\psi \\left ( e ^ { s a * } e ^ { ( t - s ) a * } e ^ { ( t - s ) a } e ^ { s a } - e ^ { s a * } e ^ { s a } \\right ) \\\\ & = \\psi \\left ( e ^ { s a * } \\frac { 1 } { t - s } \\left ( e ^ { ( t - s ) a * } e ^ { ( t - s ) a } - I \\right ) e ^ { s a } \\right ) . \\end{align*}"}
+{"id": "4884.png", "formula": "\\begin{align*} L w ( x ) = L \\psi ( x ) - L u ^ - ( x ) \\le - \\frac { \\alpha _ r } { r ^ { 2 s } } + C _ { \\star } . \\end{align*}"}
+{"id": "598.png", "formula": "\\begin{align*} g ' ( s ) = \\rho \\left ( 2 t s \\cos ( \\gamma + \\beta ) + \\frac { 1 - 2 s ^ 2 } { \\sqrt { 1 - s ^ 2 } } \\right ) \\end{align*}"}
+{"id": "8103.png", "formula": "\\begin{align*} x _ \\tau = | { \\rm A u t } ( \\tau ) | \\sum _ { \\bar T = \\tau } X _ T \\end{align*}"}
+{"id": "3472.png", "formula": "\\begin{align*} \\mathcal P ^ \\tau : = \\{ ( u , s ) : u \\notin \\Lambda , s \\in \\Omega ^ \\tau _ u \\} \\end{align*}"}
+{"id": "200.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b ) = 1 \\\\ a , b \\geq 1 } } \\left ( \\frac { 1 } { 1 - y ^ a z ^ b } \\right ) ^ { \\frac { b ^ 2 } { a ^ 3 } } = \\exp \\left \\{ \\frac { z ( 1 + z ) L i _ 3 ( y ) } { ( 1 - z ) ^ 3 } \\right \\} . \\end{align*}"}
+{"id": "3633.png", "formula": "\\begin{align*} a _ { I + 1 } \\ = \\ a _ { m - I - 1 } \\ = \\ 9 . \\end{align*}"}
+{"id": "2283.png", "formula": "\\begin{align*} \\Phi : = w - \\widetilde { T } ( f ) \\end{align*}"}
+{"id": "6262.png", "formula": "\\begin{align*} \\Sigma _ t = \\psi ^ { - 1 } ( B \\cap \\{ x _ 3 = t \\} ) , \\end{align*}"}
+{"id": "4887.png", "formula": "\\begin{align*} { \\mathcal { C } } _ \\beta : = \\left \\{ x \\in \\Omega { \\mbox { s . t . } } \\frac { x - x _ 0 } { | x - x _ 0 | } \\cdot \\bar \\nu > c _ \\beta \\right \\} , \\qquad { \\mbox { w h e r e } } \\ , c _ \\beta : = \\cos \\left ( \\frac \\pi 2 - \\beta \\right ) > 0 . \\end{align*}"}
+{"id": "4485.png", "formula": "\\begin{align*} \\frac { d } { d t } P _ k ( U ) & = - \\sum _ { j = 1 } ^ 3 { \\rm R e } ( i \\partial _ t u _ j , \\partial _ k u _ j ) _ { L ^ 2 ( \\R ^ d ) } \\\\ & = { \\rm R e } ( \\nabla \\cdot u _ 3 , \\partial _ k u _ 1 \\cdot u _ 2 ) _ { L ^ 2 ( \\R ^ d ) } + { \\rm R e } ( \\nabla \\cdot \\overline { u _ 3 } , \\overline { u _ 1 } \\cdot \\partial _ k u _ 2 ) _ { L ^ 2 ( \\R ^ d ) } - { \\rm R e } ( \\nabla ( u _ 1 \\cdot \\overline { u _ 2 } ) , \\partial _ k u _ 3 ) _ { L ^ 2 ( \\R ^ d ) } \\\\ & = 0 \\end{align*}"}
+{"id": "868.png", "formula": "\\begin{align*} | \\underline { \\alpha } | \\coloneqq \\max _ { i = 1 , \\dots , n } | \\alpha _ i | \\dd \\underline { \\alpha } \\coloneqq \\dd \\alpha _ 1 \\dots \\dd \\alpha _ n \\end{align*}"}
+{"id": "897.png", "formula": "\\begin{align*} K _ r = \\sigma _ \\infty \\widehat { P } ^ { - 5 } , \\end{align*}"}
+{"id": "3811.png", "formula": "\\begin{align*} \\Sigma _ { \\Pi } ( \\delta _ 0 ) : = \\left \\{ \\gamma \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) : \\boldsymbol { K } _ c ( \\gamma , \\mu ) \\le \\delta _ 0 \\right \\} , \\end{align*}"}
+{"id": "6680.png", "formula": "\\begin{align*} \\mathcal { B } _ s ( u , u ) & = \\int _ \\Omega | \\nabla u | ^ 2 \\ , d x + \\frac { C _ { N , s } } { 2 } \\iint _ { \\R ^ { 2 N } } \\frac { | \\hat { u } ( x ) - \\hat { u } ( y ) | ^ 2 } { | x - y | ^ { N + 2 s } } \\ , d x \\ , d y + \\int _ \\Omega V ( x ) u ^ 2 \\ , d x \\\\ & \\geq \\int _ \\Omega | \\nabla u | ^ 2 \\ , d x = \\| u \\| _ { H _ 0 ^ 1 ( \\Omega ) } ^ 2 , \\end{align*}"}
+{"id": "517.png", "formula": "\\begin{align*} \\mathcal { H } _ { V } u ( x ) : = \\left ( - \\mathcal { L } + V \\right ) u ( x ) , x \\in \\mathbb { R } ^ { n } , \\end{align*}"}
+{"id": "7879.png", "formula": "\\begin{align*} \\left \\{ A ^ \\pm : \\ A \\in \\binom { [ n ] } { k } \\mbox { a n d } x \\in A \\right \\} , \\end{align*}"}
+{"id": "4224.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\det T _ n ( \\phi ) } { G [ \\phi _ 0 ] ^ n \\prod _ { k = 1 } ^ R \\det T _ n ( \\phi _ k ) } = E \\end{align*}"}
+{"id": "6217.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , T = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "1400.png", "formula": "\\begin{align*} I = \\int _ M J \\theta \\wedge d \\Omega ^ { n - 1 } = ( n - 1 ) \\int _ M J \\theta \\wedge \\theta \\wedge \\Omega ^ { n - 1 } = - 2 ( n - 1 ) \\int _ M \\sqrt { - 1 } \\theta ^ { ( 1 , 0 ) } \\wedge \\overline { \\theta ^ { ( 1 , 0 ) } } \\wedge \\Omega ^ { n - 1 } . \\end{align*}"}
+{"id": "4784.png", "formula": "\\begin{align*} \\int _ { \\rho } ^ { 2 \\rho } \\frac { \\exp \\left ( - \\frac { u ^ 2 } { 4 t ^ 2 } \\right ) } { \\sqrt { \\cosh u - \\cosh \\rho } } \\d u \\leq \\exp \\left ( - \\frac { \\rho ^ 2 } { 4 t ^ 2 } \\right ) \\int _ { \\rho } ^ { 2 \\rho } \\frac { \\d u } { \\sqrt { ( u - \\rho ) \\rho } } = 2 \\exp \\left ( - \\frac { \\rho ^ 2 } { 4 t ^ 2 } \\right ) . \\end{align*}"}
+{"id": "6789.png", "formula": "\\begin{align*} F = F ( Q _ { 2 \\rho } ) : = \\begin{cases*} \\mathbf { P } ^ f _ { p ' _ * } ( Q _ r ( x _ 0 , t _ 0 ) ) \\mbox { i f } p < 2 , \\\\ \\mathbf { M } ^ f _ p ( Q _ r ( x _ 0 , t _ 0 ) ) \\mbox { i f } p \\geq 2 . \\end{cases*} \\end{align*}"}
+{"id": "7270.png", "formula": "\\begin{align*} \\sum _ { i = m } ^ d a _ i x ^ i = 0 , \\ , \\ , a _ m , a _ { m + 1 } , \\dots , a _ d \\ , \\ , \\ , \\ , a _ m \\neq 0 \\ , \\ , \\ , \\ , a _ d \\neq 0 , \\end{align*}"}
+{"id": "2814.png", "formula": "\\begin{align*} a _ { i , k } = 0 k = i - q + 1 , \\dots , i + r - 1 \\end{align*}"}
+{"id": "8138.png", "formula": "\\begin{align*} S ( I ) = \\bigsqcup _ { J \\ge I } W ( J ) . \\end{align*}"}
+{"id": "6088.png", "formula": "\\begin{align*} \\begin{aligned} j _ 1 & \\colon X \\hookrightarrow X \\times X : x \\mapsto ( x , x _ 0 ) \\\\ j _ 2 & \\colon X \\hookrightarrow X \\times X : x \\mapsto ( x _ 0 , x ) \\end{aligned} \\end{align*}"}
+{"id": "7967.png", "formula": "\\begin{align*} \\mathrm { t r } \\big ( ( X Y ) ^ 2 \\big ) = \\mathrm { t r } ( X Y X Y ) \\geq \\l _ { { \\rm m i n } } \\ , \\mathrm { t r } ( Y X Y ) = \\l _ { { \\rm m i n } } \\ , \\mathrm { t r } ( X Y ^ 2 ) \\geq \\l _ { { \\rm m i n } } ^ 2 \\ , \\mathrm { t r } ( Y ^ 2 ) = \\l _ { { \\rm m i n } } ^ 2 \\ , | Y | ^ 2 , \\end{align*}"}
+{"id": "302.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } \\sum _ { n = 1 } ^ { \\infty } ( + 2 n ^ 2 x y ^ { n + 2 } - n ^ 2 x y ^ { n + 3 } + 3 n x ^ { n + 1 } y ^ { n + 1 } ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "569.png", "formula": "\\begin{align*} ( R _ { 0 ( 1 , n ) } ^ * f ) ( x _ { 1 } , x _ { 2 } ) & = \\int _ { \\R } \\int _ { \\R _ { + } } f ( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } ) h _ { 0 } ( y _ { 1 } ) N ( y _ 2 ) d y _ { 2 } d y _ { 1 } \\\\ ( R _ { 0 ( n , 1 ) } ^ * f ) ( x _ { 1 } , x _ { 2 } ) & = \\int _ { \\R } \\int _ { \\R _ { + } } f ( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } ) \\ell _ { 0 } ( y _ { 2 } ) H ( y _ 1 ) d y _ { 2 } d y _ { 1 } , \\end{align*}"}
+{"id": "5485.png", "formula": "\\begin{align*} { \\rm E n t S p } ( S , \\mu ) = \\bigcup _ { I \\subseteq \\{ 1 , \\ldots , d - 1 \\} } \\left [ h _ { \\mu } \\left ( G / P _ { I } , \\nu _ { I } \\right ) , h _ { \\mu } \\left ( G / P _ { \\mathsf { f } ( I ) } , \\nu _ { \\mathsf { f } ( I ) } \\right ) \\right ] , \\end{align*}"}
+{"id": "3346.png", "formula": "\\begin{align*} Z ( I \\otimes T ^ * T ) Z ^ * = T ^ * Y Y ^ * T = T ^ * T . \\end{align*}"}
+{"id": "1503.png", "formula": "\\begin{align*} & 2 \\underline { \\widehat { r } } < | x | + \\underline { r } = r _ * + \\widehat { r } _ * . \\end{align*}"}
+{"id": "8487.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { 1 } { \\left ( \\lambda _ { n } ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } = \\frac { \\sqrt { \\pi } \\ , 2 ^ { \\frac { 1 } { 2 } - s } } { \\Gamma ( s ) x ^ { s - \\frac { 1 } { 2 } } } \\intop _ { 0 } ^ { \\infty } y ^ { s - \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( x y ) \\ , \\sigma _ { p } ( y ) \\ , d y , \\ , \\ , \\ , \\ , ( s ) > \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "7726.png", "formula": "\\begin{align*} \\mathcal { N } _ { t } = \\sigma \\{ N ( s ) , \\ s \\le t \\} . \\end{align*}"}
+{"id": "446.png", "formula": "\\begin{align*} \\eta ( t , x ) = A ( t ) x , \\ \\ A \\in ^ + ( 3 , \\mathbb R ) , \\end{align*}"}
+{"id": "8465.png", "formula": "\\begin{align*} \\tilde { \\varphi } _ { p } ( x ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\lambda _ { n } ^ { s - \\frac { 1 } { 2 } } \\ , K _ { s - \\frac { 1 } { 2 } } ( 2 \\pi \\lambda _ { n } x ) , \\ , \\ , \\ , \\ , x > 0 . \\end{align*}"}
+{"id": "2216.png", "formula": "\\begin{align*} \\int _ { y } ^ { z } \\log \\left ( 1 - t ^ { - 1 } \\right ) \\frac { d t } { \\log t } = - \\log \\frac { \\log z } { \\log y } + \\int _ { y } ^ { z } \\left ( t ^ { - 1 } + \\log \\left ( 1 - t ^ { - 1 } \\right ) \\right ) \\frac { d t } { \\log t } . \\end{align*}"}
+{"id": "1625.png", "formula": "\\begin{align*} 0 = [ { f } , { f } ] ( X , \\xi _ i ) = { f } ^ 2 [ X , \\xi _ i ] - { f } [ { f } X , \\xi _ i ] = { f } \\ , ( \\pounds _ { \\xi _ i } { f } ) X . \\end{align*}"}
+{"id": "2169.png", "formula": "\\begin{align*} d = \\left [ \\frac { 3 } { 2 } + \\frac { \\left ( 1 + \\beta \\right ) \\left ( T / 2 - \\varepsilon \\right ) ^ { 2 } } { \\left ( \\varepsilon \\left ( T - \\varepsilon \\right ) \\right ) } \\right ] b ^ { 2 } . \\end{align*}"}
+{"id": "2449.png", "formula": "\\begin{align*} ( f * \\psi ) ( x ) = \\int _ { G _ { \\tau ( x ) } } f ( \\gamma ^ { - 1 } ) g ( \\gamma \\cdot x ) \\ d \\lambda _ { \\tau ( x ) } . \\end{align*}"}
+{"id": "3403.png", "formula": "\\begin{align*} i \\hbar \\frac { \\partial \\psi } { \\partial t } = - \\frac { \\hbar ^ 2 } { 2 m } \\Delta \\psi + V ( x ) \\psi - g ( | \\psi | ) \\psi = 0 . \\end{align*}"}
+{"id": "2677.png", "formula": "\\begin{align*} f ( \\sigma ( x ) ) = f ( x ) \\\\ f ( x _ 1 + x _ 2 ) \\leq F ( f ( x _ 1 ) , f ( x _ 2 ) ) \\\\ f ( x _ 1 x _ 2 ) \\leq F ( f ( x _ 1 ) , f ( x _ 2 ) ) \\end{align*}"}
+{"id": "2601.png", "formula": "\\begin{align*} 0 < a _ 1 < C _ \\epsilon h ^ { \\frac { 2 } { 3 } - \\epsilon } , \\ \\ C _ \\epsilon h ^ { \\frac { 1 } { 2 } + \\epsilon } < a _ i < C _ \\epsilon h ^ { \\frac { 1 } { 3 } - \\epsilon } \\ \\ i = 2 , \\cdots , n . \\end{align*}"}
+{"id": "5171.png", "formula": "\\begin{align*} \\Omega : = \\left \\{ \\begin{array} { c } \\forall k \\ge \\log n \\Gamma \\subseteq [ - n ^ 2 , n ^ 2 ] ^ d \\\\ k | \\{ e \\in \\Gamma : t _ e \\in B _ { \\delta _ 0 } \\} | \\ge k / 2 \\end{array} \\right \\} . \\end{align*}"}
+{"id": "2248.png", "formula": "\\begin{align*} h _ b = \\left ( \\frac { 1 } { 2 \\pi } \\langle h _ b , P _ r ( \\theta - \\cdot ) \\rangle \\right ) _ b \\end{align*}"}
+{"id": "5053.png", "formula": "\\begin{align*} \\| U ( a ) \\| _ 2 ^ 2 = \\sum _ { i = 1 } ^ n \\langle \\omega _ i , a \\rangle ^ 4 . \\end{align*}"}
+{"id": "5599.png", "formula": "\\begin{align*} \\mathrm { I } ( \\xi _ { 1 } ^ { x } , \\mathcal { T } _ { x } ) = \\inf _ { n } \\mathrm { I } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) = \\lim _ { n } \\mathrm { I } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) . \\end{align*}"}
+{"id": "7215.png", "formula": "\\begin{align*} \\lim _ { h \\to + \\infty } \\frac { \\lambda _ h ^ { ( s ) } } { t _ h } = 1 , \\lim _ { h \\to + \\infty } t _ h ^ { ( p ^ - ) ^ * } \\abs { 1 - \\frac { \\lambda _ h ^ { ( s ) } } { t _ h } } ^ { ( p ^ - ) ^ * } = d , \\end{align*}"}
+{"id": "2183.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\alpha } ^ { - { 1 } / { 2 } } \\mathcal { Y } \\mathcal { T } \\mathcal { P } _ { \\alpha } ^ { - { 1 } / { 2 } } = \\mathcal { Q } _ { \\alpha } - \\mathcal { E } _ \\alpha , \\end{align*}"}
+{"id": "3773.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 0 & \\dots & 0 & 0 & t + \\varepsilon \\\\ 1 & 0 & \\dots & 0 & 0 & 0 \\\\ 0 & 1 & \\dots & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & 1 & 0 & 0 \\\\ 0 & 0 & \\dots & 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "1608.png", "formula": "\\begin{align*} ( f , g ) & = S _ { \\chi \\oplus \\xi } ^ { - \\frac { 1 } { 2 } } S _ { \\chi \\oplus \\xi } S _ { \\chi \\oplus \\xi } ^ { - \\frac { 1 } { 2 } } ( f , g ) \\\\ & = \\int _ { \\Theta } v ^ { 2 } ( w ) S _ { \\chi \\oplus \\xi } ^ { - \\frac { 1 } { 2 } } \\pi _ { F ( w ) \\oplus G ( w ) } ( \\chi _ { w } \\oplus \\xi _ { w } ) ^ { \\ast } ( \\chi _ { w } \\oplus \\xi _ { w } ) \\pi _ { F ( w ) \\oplus G ( w ) } S _ { \\chi \\oplus \\xi } ^ { - \\frac { 1 } { 2 } } ( f , g ) d \\mu ( w ) . \\end{align*}"}
+{"id": "4281.png", "formula": "\\begin{align*} \\tau _ 1 ( \\theta ) ( \\rho q _ t + \\rho u \\cdot q _ x ) + q + \\kappa ( \\theta ) \\theta _ x = 0 , \\end{align*}"}
+{"id": "9156.png", "formula": "\\begin{align*} e _ { i , [ \\kappa _ { i } ^ { j _ { i } } ] } ^ { j _ { i } } + \\sum _ { \\beta = 0 } ^ { \\kappa _ { i } ^ { j _ { i } } - 1 } a _ { i } ^ { j _ { i } , \\beta } e _ { i , [ \\beta ] } ^ { j _ { i } } = 0 \\ , , j _ { i } = 1 , \\ldots , m _ { i } \\ , , i = 1 , \\ldots , s \\ , . \\end{align*}"}
+{"id": "365.png", "formula": "\\begin{align*} ( \\underline { X } , \\underline { A } ) ^ { \\emptyset } = ( \\underline { X } , \\underline { A } ) ^ { \\emptyset , c } = \\prod ^ m _ { j = 1 } A _ j \\qquad \\mbox { a n d } ( \\underline { Y } , \\underline { B } ) ^ { \\emptyset } = ( \\underline { Y } , \\underline { B } ) ^ { \\emptyset , c } = \\prod ^ m _ { j = 1 } B _ j , \\end{align*}"}
+{"id": "8104.png", "formula": "\\begin{align*} \\Delta ^ r Y _ F = \\sum _ { u \\in C ( F ) \\cap [ r ] ^ n } Y _ { F _ { ( 1 ) } } \\otimes Y _ { F _ { ( 2 ) } } \\cdots \\otimes Y _ { F _ { ( r ) } } \\end{align*}"}
+{"id": "3378.png", "formula": "\\begin{align*} \\frac { P _ \\mu } { P _ m } = \\frac { P _ { \\mu - 1 } } { P _ m } + \\frac { p _ \\mu } { P _ m } \\leq \\lambda + \\frac { p _ \\mu } { P _ \\mu } \\frac { P _ \\mu } { P _ m } = \\lambda ( 1 + o ( 1 ) ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\frac { P _ \\mu } { P _ m } \\geq \\left ( 1 + \\frac { \\delta } { 2 } \\right ) = \\frac { \\lambda + 1 } { 2 } \\end{align*}"}
+{"id": "4062.png", "formula": "\\begin{align*} S _ { \\mathrm { m a i n } } = \\sum _ { L = 1 } ^ { \\infty } L \\sum _ { U , \\ , V < \\frac { D } { L } } \\tau ( U ) \\tau ( V ) \\ , x _ { U L } \\bar { x } _ { V L } \\sum _ { T | L } \\frac { \\mu ( T ) } { T } \\sum _ { A = 1 } ^ { \\infty } \\frac { G _ k ( A ^ 2 T ^ 2 U V \\slash p ) } { A } , \\end{align*}"}
+{"id": "6155.png", "formula": "\\begin{align*} & \\forall \\ , x : n \\forall \\ , y : n \\forall \\ , z : m ( ( ( x + _ n y ) \\ast _ { n m } z ) = ( x \\ast _ { m n } z + _ { n + m } y \\ast _ { n m } z ) ) \\\\ & \\forall \\ , x : n \\forall \\ , y : m \\forall \\ , z : m ( ( x \\ast _ { m n } ( y + _ m z ) ) = ( x \\ast _ { n m } y + _ { n + m } x \\ast _ { n m } z ) ) \\\\ & \\forall \\ , x : n \\forall \\ , y : m ( h _ { n + m } ( x \\ast _ { n m } y ) = h _ n ( x ) \\ast _ { n m } h _ m ( y ) ) \\end{align*}"}
+{"id": "3816.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ \\lambda ( v , s ^ { \\prime } ) & : = f ( s ^ { \\prime } ) - \\lambda _ 1 c _ 1 ( s _ 1 , s _ 1 ^ { \\prime } ) - \\lambda _ 2 c _ 2 ( s _ 2 , s _ 2 ^ { \\prime } ) , \\end{aligned} \\end{align*}"}
+{"id": "323.png", "formula": "\\begin{align*} \\alpha _ h ( m , n ) = \\left ( 1 - \\frac { 2 } { 2 ^ { m + n } } \\right ) \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { k + 1 } H _ { k , m } } { k ^ n } , \\end{align*}"}
+{"id": "3810.png", "formula": "\\begin{align*} \\mathcal { I } _ \\Pi ( \\delta _ 0 ) : = \\sup _ { \\gamma \\in \\Sigma _ { \\Pi } ( \\delta ) } \\int _ { \\mathcal { V } } g \\ , d \\gamma , \\end{align*}"}
+{"id": "4198.png", "formula": "\\begin{align*} E = \\{ \\left ( ( k , \\ell ) , ( k ' , \\ell ' ) \\right ) \\colon k ' - k \\in Q , \\ell ' - \\ell \\in P \\} \\subset \\Lambda \\times \\Lambda . \\end{align*}"}
+{"id": "2965.png", "formula": "\\begin{align*} Y ^ { [ 2 ] } _ A & = \\{ ( w , z _ 1 , z _ 2 ) \\in Y ^ { [ 2 ] } \\ | \\ w \\in A \\} \\ . \\end{align*}"}
+{"id": "6938.png", "formula": "\\begin{align*} \\rho _ 0 = \\sum _ { i = 1 } ^ m \\frac { m - n + 1 - 2 i } { 2 } \\epsilon _ i + \\sum _ { j = 1 } ^ n \\frac { m + n + 1 - 2 j } { 2 } \\delta _ { j } , \\end{align*}"}
+{"id": "2152.png", "formula": "\\begin{align*} \\Theta ^ { \\{ \\partial W ^ 1 , \\partial W ^ 2 , \\dots , \\partial W ^ m \\} } ( x ) = \\Theta ^ { \\{ \\partial W ^ 1 , \\partial W ^ 2 , \\dots , \\partial W ^ { m + 1 } \\} } ( x ) + \\Theta ( x , \\partial W ^ { m + 1 } ) . \\end{align*}"}
+{"id": "4832.png", "formula": "\\begin{align*} \\| g _ n \\| = 1 \\ ; \\ ; \\max _ { \\partial _ A \\setminus V _ n } | g _ n | \\leq 1 / n . \\end{align*}"}
+{"id": "9345.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\mathbb { E } \\big [ | x _ T | ^ 2 e ^ { - \\beta T } \\big ] = 0 . \\end{align*}"}
+{"id": "3020.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } ( \\lambda _ i ( A ) + \\alpha b _ { i } ) ^ { \\gamma } \\leq \\sum _ { i = 1 } ^ { k } \\lambda _ i ^ { \\gamma } ( A ) + \\sum _ { i = 1 } ^ { k } \\lambda _ i ^ { \\gamma } ( \\alpha B ) \\hbox { f o r a l l $ 0 < \\alpha < 1 $ } . \\end{align*}"}
+{"id": "5743.png", "formula": "\\begin{align*} \\begin{pmatrix} f & 0 \\\\ 0 & p e \\end{pmatrix} \\ , \\ , ( p \\in \\mathcal H ) \\end{align*}"}
+{"id": "8958.png", "formula": "\\begin{align*} \\frac { \\partial \\phi } { \\partial t } + a \\frac { \\partial \\phi } { \\partial x } = \\nu \\frac { \\partial \\phi } { \\partial x ^ 2 } + 4 \\pi ^ 2 \\nu ^ 2 \\cos [ 2 \\pi ( x - a t ) ] \\end{align*}"}
+{"id": "4792.png", "formula": "\\begin{align*} \\mathcal { D } = \\{ x \\in \\mathbb { R } ^ d | \\ a _ j ( x ) \\geq 0 , b _ \\ell ( x ) = 0 , \\forall j , \\ell \\} . \\end{align*}"}
+{"id": "5858.png", "formula": "\\begin{align*} \\mathcal M _ { L , x _ 0 } ^ { ( E ) } : = \\bigcup _ { x \\in \\mathcal N _ l \\cap \\Lambda _ { 2 L _ + , L _ - } } \\left \\{ \\omega : \\Lambda _ l ( x ) ( \\omega , E _ 0 , m _ 0 , \\eta _ 0 ) - \\right \\} . \\end{align*}"}
+{"id": "1686.png", "formula": "\\begin{align*} Q ( x , y , z ) = p d d _ 2 ^ { 2 } x ^ 2 + p d d _ 1 ^ { 2 } y ^ 2 + z ^ 2 - t p x y , \\end{align*}"}
+{"id": "7462.png", "formula": "\\begin{align*} \\Pi _ { p \\alpha + k } ^ { ( i ) } ( \\eta _ 1 , t ) = \\mathcal { O } \\big ( e ^ { - \\theta _ i \\ , \\eta _ i } \\big ) \\eta _ i \\to + \\infty \\end{align*}"}
+{"id": "5844.png", "formula": "\\begin{align*} \\delta = \\frac { v _ + - v _ - } { 4 ( \\beta + 1 ) } . \\end{align*}"}
+{"id": "907.png", "formula": "\\begin{align*} \\mu = \\begin{cases} 1 / 2 & \\widehat { P } \\widehat { \\Theta } _ 1 \\ll \\widehat { \\Theta } _ 2 \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "2558.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\sigma } _ y ( t ) = \\nabla u ( \\sigma _ y ( t ) ) , & \\\\ \\sigma _ y ( 0 ) = y . \\ \\ & \\end{cases} \\end{align*}"}
+{"id": "909.png", "formula": "\\begin{align*} \\sum _ { \\substack { | r | } = \\widehat { Y } } | b _ 2 | ^ { 1 / 2 } | r _ 3 '' | ^ { ( 1 - \\delta ) / 2 } | r _ 3 | ^ { \\delta ( n - 2 ) / 2 } & \\ll \\widehat { Y } \\sum _ { | r _ 3 ' | \\leq \\widehat { Y } } | r _ 3 ' | ^ { \\delta ( n - 2 ) / 2 - 1 } \\sum _ { | r _ 3 '' b _ 2 | \\leq \\widehat { Y } | r _ 3 ' | ^ { - 1 } } | b _ 2 | ^ { - 1 / 2 } | r _ 3 '' | ^ { ( \\delta ( n - 3 ) - 1 ) / 2 } \\\\ & \\ll \\widehat { Y } ^ { 1 / 2 + \\delta ( n - 3 ) / 2 } \\sum _ { | r _ 3 ' | \\leq \\widehat { Y } } | r _ 3 ' | ^ { ( \\delta - 1 ) / 2 } \\end{align*}"}
+{"id": "1459.png", "formula": "\\begin{align*} \\Delta \\tilde { u } _ m ^ i ( y ) = ( \\mu _ m ^ i ) ^ { \\frac { n + 2 } { 2 } } \\bigg [ \\Big ( \\Delta u _ m ( x ) \\Big ) \\chi _ m ^ i ( x ) + 2 \\nabla u _ m ( x ) \\nabla \\chi _ m ^ i ( x ) + u _ m ( x ) \\Big ( \\Delta \\chi _ m ^ i ( x ) \\Big ) \\bigg ] . \\end{align*}"}
+{"id": "1639.png", "formula": "\\begin{align*} 2 \\ , g ( ( \\nabla _ { X } { f } ) Y , Z ) & = - g ( N ^ { \\ , ( 1 ) } ( Y , Z ) , { f } X ) + 2 \\ , g ( f X , f Y ) \\ , \\bar \\eta ( Z ) \\\\ & - 2 \\ , g ( f X , f Z ) \\ , \\bar \\eta ( Y ) + N ^ { \\ , ( 5 ) } ( X , Y , Z ) , \\end{align*}"}
+{"id": "8999.png", "formula": "\\begin{align*} \\div ( T ^ \\varphi ( \\nabla w , \\ , \\cdot \\ , ) ^ \\sharp ) & = ( T ^ \\varphi _ { i k } w _ k ) _ i \\\\ & = T ^ \\varphi _ { i k , i } w _ k + T ^ \\varphi _ { i k } w _ { k i } \\\\ & = \\left ( R ^ \\varphi _ { i k , i } - \\frac { S ^ \\varphi _ i } { m } \\delta _ { i k } \\right ) w _ k + T ^ \\varphi _ { i k } w _ { k i } \\ , . \\end{align*}"}
+{"id": "6291.png", "formula": "\\begin{align*} \\| \\hat \\lambda \\| _ * = \\| N ^ { - 1 } ( \\hat \\lambda ) \\| , \\end{align*}"}
+{"id": "7887.png", "formula": "\\begin{align*} X _ i = \\left \\{ 1 , 2 , \\ldots , k - i , x _ 1 , x _ 2 , \\ldots , x _ i \\right \\} , \\ \\mbox { f o r } 0 \\leq i \\leq k . \\end{align*}"}
+{"id": "2809.png", "formula": "\\begin{align*} M = \\begin{pmatrix} a _ { 0 0 } & \\dots & 1 _ { 0 \\ , r } & 0 & 0 & \\dots & \\dots & \\dots \\\\ a _ { 1 0 } & a _ { 1 1 } & \\dots & 1 & 0 & \\dots & \\dots & \\dots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\dots \\\\ a _ { q \\ , 0 } & a _ { q \\ , 1 } & a _ { q \\ , 2 } & \\dots & \\dots & 1 _ { q \\ , q + r } & 0 & \\dots \\\\ 0 & a _ { q + 1 \\ , 1 } & a _ { q + 1 \\ , 2 } & a _ { q + 1 \\ , 3 } & \\dots & \\dots & 1 & \\dots \\\\ 0 & 0 & \\ddots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{pmatrix} ; \\end{align*}"}
+{"id": "6466.png", "formula": "\\begin{align*} ( e ^ { i t \\Delta } g _ { K , \\epsilon } ) ( x ) = \\frac { \\beta _ \\epsilon ( t ) } { ( 2 \\pi ) ^ 2 } e ^ { - \\frac { \\epsilon ^ 2 \\beta _ \\epsilon ( t ) } { 2 } | x | ^ 2 + i \\beta _ { \\epsilon } ( t ) x \\cdot K - i t \\beta _ { \\epsilon } ( t ) | K | ^ 2 } \\end{align*}"}
+{"id": "644.png", "formula": "\\begin{align*} \\rho ( c ^ * c ) = \\rho ( c ) ^ * \\rho ( c ) \\end{align*}"}
+{"id": "271.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( + \\frac { y ^ 4 + 1 1 y ^ 3 + 1 1 y ^ 2 + z } { ( 1 - y ) ^ 5 } \\right ) \\frac { z ^ n } { n ^ 5 } \\right \\} \\end{align*}"}
+{"id": "6232.png", "formula": "\\begin{align*} \\mu ( Q _ 2 ) \\big ( \\psi _ 3 ( w ) & + \\psi _ 3 ( v ) \\big ) + \\mu ( Q _ 3 ) \\big ( \\psi _ 2 ( w ) + \\psi _ 2 ( v ) \\big ) \\\\ & = \\int _ { Q _ 2 } \\int _ { Q _ 3 } g ( p - v ) + g ( p - w ) + g ( q - v ) + g ( q - w ) \\ , d \\mu ( p ) \\ , d \\mu ( q ) \\\\ & \\geq - ( 3 + \\eta ) \\mu ( Q _ 2 ) \\mu ( Q _ 3 ) \\geq - ( 3 + \\eta ) \\bigg ( \\frac 1 3 + \\eta \\bigg ) ^ 2 \\ , . \\end{align*}"}
+{"id": "5226.png", "formula": "\\begin{align*} M ^ { - 1 } . x _ { i j } & = \\sum _ { \\ell = 1 } ^ n m _ { i \\ell } x _ { \\ell j } \\\\ M ^ { - 1 } . x _ { i _ 1 j _ 1 } \\dots x _ { i _ r j _ r } & = \\left ( \\sum _ { \\ell _ 1 = 1 } ^ n m _ { i _ 1 \\ell _ 1 } x _ { \\ell _ 1 j _ 1 } \\right ) \\cdots \\left ( \\sum _ { \\ell _ r = 1 } ^ n m _ { i _ r \\ell _ r } x _ { \\ell _ r j _ r } \\right ) \\\\ & = \\sum _ { \\ell _ 1 , \\dots , \\ell _ r = 1 } ^ n m _ { i _ 1 \\ell _ 1 } \\cdots m _ { i _ r \\ell _ r } ( x _ { \\ell _ 1 j _ 1 } \\dots x _ { \\ell _ r j _ r } ) \\end{align*}"}
+{"id": "6773.png", "formula": "\\begin{align*} E _ s : = \\Big \\{ n \\in \\mathbb N : 0 \\leq n < p ^ { k + \\ell } , \\ & n = \\lfloor a _ { k + \\ell - 1 } \\dots a _ 0 \\rfloor _ p \\\\ a _ { k - m - 1 } = p - 1 & a _ { k - m - 2 } \\neq 0 \\\\ a _ { k - m - i } = 0 \\forall \\ , 3 \\leq i \\leq \\ell & a _ { k - m - \\ell - 1 } < b _ { k - m - \\ell - 1 } \\\\ a _ j \\ge b _ j \\forall \\ , j \\in \\{ 0 , \\dots , & \\ , k - m - \\ell - 2 \\} \\cup \\{ k - m , \\dots , k \\} \\Big \\} . \\end{align*}"}
+{"id": "2122.png", "formula": "\\begin{align*} u _ 1 x + u _ 2 f ( x ) + u x ^ { - 1 } = \\R ( x ) ^ { q ^ m } - \\R ( x ) , \\end{align*}"}
+{"id": "7460.png", "formula": "\\begin{align*} N _ { p \\alpha + k } ( \\xi , t ) = w ^ { ( i ) } _ { p \\alpha + k } ( 0 , t ) + \\Psi ^ { ( i ) } _ { p \\alpha + k } ( \\xi _ i , t ) + \\mathcal { O } ( \\exp ( - \\beta _ 0 \\ , \\xi _ i ) ) \\mbox { a s } \\ \\ \\xi _ i \\to + \\infty , \\ \\ \\xi \\in \\Xi ^ { ( i ) } , \\end{align*}"}
+{"id": "7847.png", "formula": "\\begin{align*} k _ i : = | \\left \\{ y H \\in G / H : \\ ( H , y H ) \\in O _ i \\right \\} | . \\end{align*}"}
+{"id": "4182.png", "formula": "\\begin{align*} m ( t ; \\lambda ) = \\frac { 4 \\pi } { 3 } \\cdot \\begin{cases*} ( \\frac { 1 + t } { 1 + \\lambda } ) ^ 3 & i f $ 0 < \\lambda < \\frac { 2 t } { 1 - t } $ , \\\\ ( \\frac { 2 t } { \\lambda } ) ^ 3 & i f $ \\frac { 2 t } { 1 - t } \\leq \\lambda < 2 $ , \\\\ 0 & o t h e r w i s e . \\end{cases*} \\end{align*}"}
+{"id": "561.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ { t } ^ { 2 } u ( t , k ) + a ( t ) \\mathcal { H } _ { \\hbar , V } u ( t , k ) + q ( t ) u ( t , k ) = f ( t , k ) , ( t , k ) \\in ( 0 , T ] \\times \\hbar \\mathbb { Z } ^ { n } , \\\\ u ( 0 , k ) = u _ { 0 } ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\\\ \\partial _ { t } u ( 0 , k ) = u _ { 1 } ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\end{array} \\right . \\end{align*}"}
+{"id": "5943.png", "formula": "\\begin{align*} \\phi ^ { - 1 } ( h _ 1 ) \\phi ^ { - 1 } ( h _ 2 ) & = ( \\sum ^ m _ { k = 1 } ( a ^ 1 _ k + i b ^ 1 _ k ) f _ k , t _ 1 ) ( \\sum ^ m _ { k = 1 } ( a ^ 2 _ k + i b ^ 2 _ k ) f _ k , t _ 2 ) \\\\ & = ( \\sum ^ m _ { k = 1 } ( a ^ 1 _ k + i b ^ 1 _ k ) f _ k + \\sum ^ m _ { k = 1 } ( a ^ 2 _ k + i b ^ 2 _ k ) f _ k , t _ 1 + t _ 2 + \\tfrac { 1 } { 2 } \\sum ^ m _ { k = 1 } ( a ^ 1 _ k b ^ 2 _ k - b ^ 1 _ k a ^ 2 _ k ) ) . \\end{align*}"}
+{"id": "8581.png", "formula": "\\begin{align*} S _ { j , + } ( t ) : = \\sum _ { k = 1 } ^ \\infty S _ { j , + } ^ k ( t ) , S _ { j , - } ( t ) : = \\sum _ { k = 1 } ^ \\infty S _ { j , - } ^ k ( t ) . \\end{align*}"}
+{"id": "9248.png", "formula": "\\begin{align*} M _ n & = \\sum _ { j = 0 } ^ n ( n - j + 4 ) \\frac { 4 } { \\pi } 2 ^ { 2 j } \\frac { 1 } { ( 2 n - 2 j ) ! } \\left ( \\frac { \\pi } { 2 } \\right ) ^ { 2 n - 2 j } < \\\\ & < 2 ^ { 2 n } \\Bigl ( \\frac { 1 6 } { \\pi } \\cosh ( \\pi / 4 ) + \\frac 1 2 \\sinh ( \\pi / 4 ) \\Bigr ) < 7 . 2 \\cdot 2 ^ { 2 n } , \\\\ N _ n & = \\sum _ { j = 0 } ^ n \\frac { 4 } { \\pi } 2 ^ { 2 j } \\frac { 1 } { ( 2 n - 2 j ) ! } \\left ( \\frac { \\pi } { 2 } \\right ) ^ { 2 n - 2 j } < 2 ^ { 2 n } \\frac { 4 } { \\pi } \\cosh ( \\pi / 4 ) < 1 . 7 \\cdot 2 ^ { 2 n } . \\end{align*}"}
+{"id": "4469.png", "formula": "\\begin{align*} \\mu _ { \\omega , c } & \\le S _ { \\omega , c } ( \\lambda _ 0 \\Phi _ { \\omega } ) \\\\ & = \\lambda _ 0 ^ 2 L ( \\Phi _ { \\omega } ) + \\lambda _ 0 ^ 3 N ( \\Phi _ { \\omega } ) + \\lambda _ 0 ^ 2 \\omega Q ( \\Phi _ { \\omega } ) + \\lambda _ 0 ^ 2 c P ( \\Phi _ { \\omega } ) \\\\ & = \\lambda _ 0 ^ 2 S _ { \\omega , 0 } ( \\Phi _ { \\omega } ) + \\lambda _ 0 ^ 2 \\{ ( \\lambda _ 0 - 1 ) N ( \\Phi _ { \\omega } ) + c P ( \\Phi _ { \\omega } ) \\} \\\\ & = \\lambda _ 0 ^ 3 \\mu _ { \\omega , 0 } < 8 \\mu _ { \\omega , 0 } . \\end{align*}"}
+{"id": "4388.png", "formula": "\\begin{align*} \\sigma _ 0 : = \\min \\left \\{ \\frac { 2 } { \\alpha } , \\frac { 1 } { \\beta } , \\frac { 1 } { \\gamma } \\right \\} . \\end{align*}"}
+{"id": "8887.png", "formula": "\\begin{align*} \\delta _ t \\Bigg ( \\sum _ { j = 1 } ^ k X _ j \\Bigg ) = \\sum _ { j = 1 } ^ k t ^ j X _ j . \\end{align*}"}
+{"id": "8507.png", "formula": "\\begin{align*} \\frac { \\sqrt { \\pi } \\ , \\Gamma ( s - \\frac { 1 } { 2 } ) } { \\Gamma ( s ) } = \\pi - 2 \\pi \\log ( 2 ) \\ , ( s - 1 ) + O \\left ( s - 1 \\right ) ^ { 2 } \\end{align*}"}
+{"id": "9140.png", "formula": "\\begin{align*} \\begin{array} { r c l } x ^ { + } & = & f ( x , F _ { u } \\circ \\Psi ( x , z , v _ { [ 0 , R - A ] } ) ) \\\\ z ^ { + } & = & \\psi _ { c , [ 1 ] } ( x , z , v _ { [ 0 , R - A ] } ) \\end{array} \\end{align*}"}
+{"id": "9146.png", "formula": "\\begin{align*} y ^ { j } = \\varphi ^ { j } ( \\zeta _ { [ - q _ { 1 } ] } , \\dots , \\zeta _ { [ - 1 ] } , x , u ) \\ , , j = 1 , \\ldots , m \\end{align*}"}
+{"id": "5719.png", "formula": "\\begin{align*} \\langle D ^ { k - 1 } ( h ) , g \\rangle = \\sum _ { n < 0 } C _ { h } ( n ) C _ { g } ( n ) = ( \\xi _ { 2 - k } ( h ) , g ) = ( 0 , g ) = 0 . \\end{align*}"}
+{"id": "2733.png", "formula": "\\begin{align*} J _ 6 = - \\frac { s } { 2 } \\iint _ Q \\sigma _ { t t } | u | ^ 2 d x d y d t \\geq - C s \\iint _ Q \\xi ^ { 3 / 2 } | u | ^ 2 d x d y d t . \\end{align*}"}
+{"id": "7683.png", "formula": "\\begin{align*} z _ 0 = \\frac { e _ 1 + e _ b } { 2 } \\ \\hbox { a n d } \\ z _ 0 ^ \\ast = e _ 1 - e _ b , \\end{align*}"}
+{"id": "4719.png", "formula": "\\begin{align*} \\abs { \\delta _ d } \\leq \\begin{cases} C ( a ^ 3 \\rho _ 0 ) ^ { 1 / 3 9 } \\abs { \\log a ^ 3 \\rho _ 0 } ^ { 1 2 / 1 3 } & d = 3 , \\\\ C ( a ^ 2 \\rho _ 0 ) ^ { 1 / 5 } \\abs { \\log a ^ 2 \\rho _ 0 } ^ { 8 / 7 } & d = 2 , \\\\ C ( a \\rho _ 0 ) ^ { 1 / 7 } \\abs { \\log a \\rho _ 0 } ^ { 1 2 / 7 } & d = 1 . \\end{cases} \\end{align*}"}
+{"id": "2358.png", "formula": "\\begin{align*} \\left ( \\prod _ { x : X } M _ x \\right ) _ f = \\prod _ { x : X } \\left ( M _ x \\right ) _ { f ( x ) } = \\prod _ { x : X } \\left ( M _ x \\right ) ^ { D ( f ) } = \\left ( \\prod _ { x : X } M _ x \\right ) ^ { D ( f ) } \\rlap { . } \\end{align*}"}
+{"id": "5319.png", "formula": "\\begin{align*} f ( x ) = \\prod x _ i = \\prod \\left ( \\sqrt { p ( 1 - p ) } \\chi _ i + p \\right ) \\end{align*}"}
+{"id": "4249.png", "formula": "\\begin{align*} \\rho _ { s } ^ { t , \\mu ; u } = \\rho _ { s } ^ { \\tau , \\rho _ { \\tau } ^ { t , \\mu ; u } ; u ^ { \\tau } } , \\end{align*}"}
+{"id": "3597.png", "formula": "\\begin{align*} r ( p ) \\ = \\ f q ^ \\beta , \\end{align*}"}
+{"id": "5118.png", "formula": "\\begin{align*} \\int _ { B _ r } ( b ^ { i j , k l } ) _ r v _ { i j } v _ { k l } = \\int _ { B _ r } [ ( b ^ { i j , k l } ) _ r - b ^ { i j , k l } ( x ) ] f _ { i j } v _ { k l } . \\end{align*}"}
+{"id": "680.png", "formula": "\\begin{gather*} \\lim _ { k \\to \\infty } \\frac { 1 } { k } \\log \\| S ( k ) ( D ^ l \\mathfrak { r } _ { a , n } ( \\varphi ) ) \\| _ { \\sup } = - \\lambda _ 1 ( n - l - a ) 0 \\leq l < n \\\\ \\lim _ { k \\to \\infty } \\frac { 1 } { k } \\log \\big ( \\| S ( k ) ( D ^ l \\mathfrak { r } _ { a , n } ( \\varphi ) ) \\| _ { L ^ 1 ( I ^ { ( k ) } ) } / | I ^ { ( k ) } | \\big ) = - \\lambda _ 1 ( n - l - a ) 0 \\leq l \\leq n . \\end{gather*}"}
+{"id": "159.png", "formula": "\\begin{align*} L i _ 2 \\left ( \\frac { x } { 1 - x } . \\frac { y } { 1 - y } \\right ) = L i _ 2 \\left ( \\frac { x } { 1 - y } \\right ) + L i _ 2 \\left ( \\frac { y } { 1 - x } \\right ) + L i _ 2 \\left ( \\frac { - x } { 1 - x } \\right ) \\end{align*}"}
+{"id": "7074.png", "formula": "\\begin{align*} \\alpha = \\alpha ( \\{ \\beta _ j \\mid j \\in I ' \\} ) = \\alpha ( \\{ \\tilde { \\beta } _ j \\mid j \\in I ' \\} ) . \\end{align*}"}
+{"id": "575.png", "formula": "\\begin{align*} \\gamma ^ { c } ( x _ 1 , x _ 2 ) = \\int _ { \\R } \\int _ { \\R _ { + } } c \\left ( \\frac { - x _ { 1 } + y _ { 1 } } { 2 \\sqrt { x _ { 2 } } } , \\frac { y _ { 2 } } { 2 x _ { 2 } } \\right ) ( \\ell _ 0 ( y _ { 2 } ) ) ^ { 2 } H ( y _ { 1 } ) [ H ( y _ { 1 } ) ] ^ T d y _ { 2 } d y _ { 1 } . \\end{align*}"}
+{"id": "7854.png", "formula": "\\begin{align*} v = \\begin{bmatrix} 1 - \\frac { | \\mathcal { S } | } { | \\Omega | } \\\\ - \\frac { | \\mathcal { S } | } { | \\Omega | } \\end{bmatrix} . \\end{align*}"}
+{"id": "7733.png", "formula": "\\begin{align*} z ^ { * } \\bigl ( Y , \\ , t \\bigr ) = y ( N , \\ , t ) , \\ \\forall t , \\ \\pi \\end{align*}"}
+{"id": "6653.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s [ f ( x ) g ( x ) ] = f ( x ) ( - \\Delta ) ^ { s } g ( x ) + g ( x ) ( - \\Delta ) ^ s f ( x ) - \\mathcal B ( f , g ) ( x ) , \\end{align*}"}
+{"id": "8339.png", "formula": "\\begin{align*} \\| E \\| = \\sup _ { \\substack { u \\in \\mathcal F _ { r , R } \\\\ r \\in ( 0 , R ) } } \\frac { \\| u \\| _ { p ^ * , q , \\mu } } { \\| \\nabla u \\| _ { p , q , \\mu } } . \\end{align*}"}
+{"id": "4813.png", "formula": "\\begin{align*} \\mathcal { D } = \\{ ( x _ 1 , x _ 2 , x _ 3 ) \\in \\mathbb { R } ^ 3 | \\ \\eta ^ 2 - x _ 2 ^ 2 \\geq 0 , \\ 1 - x _ 1 ^ 2 - x _ 2 ^ 2 = 0 \\} \\end{align*}"}
+{"id": "4779.png", "formula": "\\begin{align*} \\left ( S _ { [ 1 ] } \\diamond \\cdots \\diamond S _ { [ j ] } \\right ) ^ T J _ { 2 | \\alpha _ i | } S _ { [ j + 1 ] } & = \\left ( S _ { [ 1 ] } \\diamond \\cdots \\diamond S _ { [ j ] } \\right ) ^ T J _ { 2 | \\alpha _ i | } W _ { [ j + 1 ] } R _ { [ j + 1 ] } ^ { - 1 } . \\end{align*}"}
+{"id": "1986.png", "formula": "\\begin{align*} \\sup _ { \\partial \\Omega } \\vert \\nabla u \\vert \\leq C _ 1 : = \\max \\{ \\sup _ { \\partial \\Omega } \\vert \\nabla \\underline u \\vert , \\Vert \\nabla w \\Vert _ { L ^ { \\infty } ( \\partial \\Omega ) } \\} \\cdot \\end{align*}"}
+{"id": "6758.png", "formula": "\\begin{align*} M _ H ( h , x ) & = ( \\Phi \\circ \\Psi ) ( h , x ) , \\\\ ( \\widetilde { R } \\circ \\Psi ) ( h , x ) & = ( \\Psi \\circ ( R \\times \\sigma ) ) ( h , x ) , \\\\ ( \\sigma \\circ \\Phi ) ( h , x ) & = ( \\Phi \\circ \\widetilde { R } ) ( h , x ) . \\end{align*}"}
+{"id": "7454.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 3 } \\int _ { \\Xi ^ { ( i ) } } \\big ( F _ { p \\alpha + k } - \\partial _ t { { N } } _ { p \\alpha + k - 1 } \\big ) \\ , d \\xi = \\int _ { \\Xi ^ { ( 0 ) } } \\partial _ t { { N } } _ { p \\alpha + k - 1 } \\ , d \\xi + \\delta _ { p \\alpha + k , \\alpha } \\int _ { \\Gamma _ 0 } \\varphi ^ { ( 0 ) } \\ , d \\sigma _ \\xi . \\end{align*}"}
+{"id": "1348.png", "formula": "\\begin{align*} \\frac { d } { d t } F _ { N } ( t , x ) & = \\frac { 1 } { N ^ { 2 } } \\stackrel [ l = 1 ] { ( i - 1 ) ^ { \\ast } ( j ) } { \\sum } \\stackrel [ r = 1 ] { N } { \\sum } m _ { l } ( t ) m _ { r } ( t ) S ( x _ { l } ( t ) - x _ { r } ( t ) ) \\\\ & = \\frac { 1 } { N ^ { 2 } } \\stackrel [ r = 1 ] { N } { \\sum } \\stackrel [ l = 1 ] { ( i - 1 ) ^ { \\ast } ( j ) } { \\sum } m _ { l } ( t ) m _ { r } ( t ) S ( x _ { l } ( t ) - x _ { r } ( t ) ) = \\mathbf { S } [ F _ { N } ] . \\end{align*}"}
+{"id": "3120.png", "formula": "\\begin{align*} S _ { r } [ n , k ] : = S _ { r } [ n - 1 , k - 1 ] + [ r k + 1 ] _ q \\ , S _ { r } [ n - 1 , k ] \\end{align*}"}
+{"id": "9224.png", "formula": "\\begin{align*} x _ 1 \\oplus x _ 2 \\oplus \\cdots \\oplus x _ n = x _ 1 ( 1 + \\eta _ n ) + x _ 2 ( 1 + \\eta _ { n - 1 } ) + \\cdots + x _ n ( 1 + \\eta _ 1 ) . \\end{align*}"}
+{"id": "4760.png", "formula": "\\begin{align*} z _ { 2 } & = x _ 2 + \\iota x _ 2 - \\langle w _ 1 , x _ 2 + \\iota y _ 2 \\rangle w _ 1 \\\\ & = y _ 2 + \\iota y _ 2 - \\langle x _ 1 + \\iota y _ 1 , x _ 2 + \\iota y _ 2 \\rangle w _ 1 + \\mathcal { O } ( \\| H \\| ) \\\\ & = x _ 2 + \\iota y _ 2 + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "527.png", "formula": "\\begin{align*} \\left \\| Q u \\right \\| _ { \\ell ^ { 2 } ( \\hbar \\mathbb { Z } ^ { n } ) } ^ { 2 } = \\sum _ { k \\in \\hbar \\mathbb { Z } ^ { n } } \\frac { 1 } { | V ( k ) + 1 | ^ { 2 } } | u ( k ) | ^ { 2 } \\leq \\sum _ { k \\in \\hbar \\mathbb { Z } ^ { n } } | u ( k ) | ^ { 2 } = \\left \\| u \\right \\| _ { \\ell ^ { 2 } ( \\hbar \\mathbb { Z } ^ { n } ) } ^ { 2 } . \\end{align*}"}
+{"id": "106.png", "formula": "\\begin{align*} \\| B _ { \\tilde { h } } W u \\| _ { H ^ N _ { h \\tilde { h } } } = \\mathcal { O } ( h ^ \\infty \\tilde { h } ^ \\infty ) \\| u \\| _ { H ^ { - N } _ { h \\tilde { h } } } . \\end{align*}"}
+{"id": "4940.png", "formula": "\\begin{align*} \\left [ U ^ { - 1 } \\right ] _ { { i , j } } = \\begin{cases} ( - 1 ) ^ { j - i } \\binom { n - i } { n - j } & \\textnormal { i f } \\ , i \\le j ; \\\\ 0 & \\textnormal { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "4042.png", "formula": "\\begin{align*} \\sum _ { p | c } \\frac { \\mathcal { S } ( i , j , c ) } { c } \\ , J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { i j } } { c } \\right ) = O \\left ( p ^ { - \\frac { 1 } { 2 } } \\ , ( i , j ) ^ { \\frac { 1 } { 2 } } \\ , ( i j ) ^ { \\frac { 1 } { 2 } } \\right ) . \\end{align*}"}
+{"id": "6100.png", "formula": "\\begin{align*} | h ( t ) - h ( s ) | ^ { 2 } \\leq C _ { 2 } ( \\gamma ^ \\prime ) ( t - s ) ^ { 2 \\gamma ^ \\prime - 1 } | h | _ { W ^ { \\gamma ^ \\prime , 2 } } ^ { 2 } = C _ { 2 } ( \\gamma ^ { \\prime } ) ( t - s ) ^ { 2 \\gamma ^ \\prime - 1 } \\int \\int _ { [ s , t ] ^ 2 } \\frac { | h ( u ) - h ( v ) | ^ 2 } { | u - v | ^ { 1 + 2 \\gamma ^ \\prime } } \\ , \\mathrm { d } u \\ , \\mathrm { d } v . \\end{align*}"}
+{"id": "9278.png", "formula": "\\begin{align*} \\texttt { L H S } \\eqref { i d : 9 } \\simeq ( b - a ) ^ { \\gamma } \\int _ { b - a } ^ { b - a + a / 4 } x ^ { - \\gamma } \\ , d x \\simeq ( b - a ) ^ { \\gamma } \\begin{cases} b ^ { - \\gamma + 1 } & \\gamma < 1 , \\\\ \\log \\big ( 1 + a / ( b - a ) \\big ) & \\gamma = 1 . \\end{cases} \\end{align*}"}
+{"id": "9231.png", "formula": "\\begin{align*} \\delta _ k = \\frac { \\varepsilon _ 4 a ^ k } { 4 } \\quad \\frac { T _ k } { a ^ k } \\cdot 1 . 0 6 \\cdot ( 2 k + 3 ) \\cdot 2 ^ { - d _ k } < \\frac { \\varepsilon _ 4 } { 2 } . \\end{align*}"}
+{"id": "5633.png", "formula": "\\begin{align*} S _ { H , L } : = \\inf _ { u \\in D ^ { 1 , 2 } ( \\mathbb { R } ^ N ) \\backslash \\{ 0 \\} } \\frac { | \\nabla u | ^ 2 _ 2 } { \\Bigl ( \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | u | ^ { 2 ^ * _ \\mu } ) | u | ^ { 2 ^ * _ \\mu } \\Bigr ) ^ { \\frac { 1 } { 2 ^ * _ \\mu } } } . \\end{align*}"}
+{"id": "3028.png", "formula": "\\begin{align*} \\| T + x S \\| _ { ( p , k ) } ^ p + \\| T - x S \\| _ { ( p , k ) } ^ p \\geq 2 \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T ^ * T + x ^ 2 S ^ * S ) \\end{align*}"}
+{"id": "431.png", "formula": "\\begin{align*} \\| \\vec U \\| _ { P \\otimes P _ { \\Omega } } ^ 2 \\leq \\| \\vec F \\| _ { P \\otimes P _ { \\Omega } } ^ 2 + 2 \\int _ 0 ^ t \\sum _ { j = 1 , N } \\lbrack \\vec G ^ T \\vec G ) \\rbrack _ j d s _ j \\ d t . \\end{align*}"}
+{"id": "8972.png", "formula": "\\begin{align*} \\Delta w + \\frac { S } { m - 1 } w = 0 , \\end{align*}"}
+{"id": "6190.png", "formula": "\\begin{align*} \\| u _ \\theta \\| _ { W ^ { 1 , \\ , \\infty } ( \\mathbb S ^ { n - 1 } ) } \\le \\theta ^ 2 , \\| \\Delta _ { \\mathbb S ^ { n - 1 } } u _ \\theta \\| _ { L ^ \\infty ( \\mathbb S ^ { n - 1 } ) } \\leq \\theta ^ 2 , \\| D ^ 2 _ { \\tau } u _ \\theta \\| _ { L ^ \\infty ( \\mathbb S ^ { n - 1 } ) } = \\theta ^ { - 1 } . \\end{align*}"}
+{"id": "3467.png", "formula": "\\begin{align*} t _ q ( \\Gamma ) \\ : = \\ \\sum ^ n _ { i = 0 } ( - 1 ) ^ i \\mathrm { g r a n k } ( \\tau ^ { - i } ( \\Gamma ) ) \\in \\Z [ q , q ^ { - 1 } ] , \\end{align*}"}
+{"id": "2903.png", "formula": "\\begin{align*} ( f \\sharp g ) _ { m + m ' } ( \\xi , \\mu ) = f _ m ( \\xi , \\mu ) g _ { m ' } ( \\xi , \\mu ) . \\end{align*}"}
+{"id": "1920.png", "formula": "\\begin{align*} \\theta ( \\bar { u } ) + I _ K ( \\bar { \\zeta } ) = \\lim \\limits _ { k \\rightarrow + \\infty } [ \\theta ( u ^ k ) + I _ K ( \\zeta ^ k ) ] . \\end{align*}"}
+{"id": "2506.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle R e \\int _ { \\mathbb { R } ^ N } \\psi \\overline { \\phi } ^ 2 d x < 0 A \\mu ^ 2 = \\dfrac { N } { 2 } B \\mu ^ \\frac { N } { 2 } . \\end{array} \\right . \\end{align*}"}
+{"id": "2836.png", "formula": "\\begin{align*} = ( b _ { k - p } \\cdots b _ { k - 2 p + 1 } ) ( b _ { k - 2 p } \\cdots b _ { k - 3 p + 1 } ) ^ 2 \\cdots ( b _ { k - ( h - 1 ) p } \\cdots b _ { k - h p + 1 } ) ^ { h - 1 } \\times , \\end{align*}"}
+{"id": "3338.png", "formula": "\\begin{align*} \\rho ( A ) = \\lim _ { n \\to \\infty } \\left \\| \\sum _ { | \\alpha | = n } A ^ { \\alpha } A ^ { \\alpha * } \\right \\| ^ { \\dfrac { 1 } { 2 n } } . \\end{align*}"}
+{"id": "2258.png", "formula": "\\begin{align*} f ( z ) = \\frac { 1 } { \\pi } \\langle f _ b , P ( x - \\cdot , y ) \\rangle , \\end{align*}"}
+{"id": "616.png", "formula": "\\begin{align*} f ( t ) = f ( 0 ) + f ' ( u ) t \\end{align*}"}
+{"id": "7180.png", "formula": "\\begin{align*} w ^ h _ 1 : = \\begin{cases} \\phi ^ i _ 1 ( u ) & \\mbox { o n } B _ h ^ i , \\ , \\ , i \\leq h , \\\\ u & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "6102.png", "formula": "\\begin{align*} N ( I , \\delta _ 1 , \\chi , \\mathbf { X } _ \\omega ) \\coloneqq \\inf \\lbrace n > 0 : \\ \\tau _ { n , \\omega } ^ { I } ( \\chi ) = b \\rbrace . \\end{align*}"}
+{"id": "2901.png", "formula": "\\begin{align*} f ^ { B , \\alpha } ( \\xi , \\mu ) : = \\partial _ x ^ \\alpha | _ { x = 0 } \\alpha _ { - x } \\big ( f ( \\xi + B x , \\mu ) \\big ) = \\sum _ { j = 0 } ^ { | \\alpha | } \\sum _ { \\substack { \\beta + \\gamma = \\alpha \\\\ | \\gamma | = j } } \\binom \\alpha \\beta ( i \\delta ) ^ \\beta \\partial _ { B , \\xi } ^ \\gamma f ( \\xi , \\mu ) . \\end{align*}"}
+{"id": "3671.png", "formula": "\\begin{align*} \\mathcal { M } _ { q , \\alpha ^ 1 } f = \\mathcal { M } _ { q , \\alpha ^ 2 } f \\end{align*}"}
+{"id": "4433.png", "formula": "\\begin{align*} \\mu _ { \\omega , \\mathbf { c } } = S _ { \\omega , \\mathbf { c } } ( \\Psi _ { \\omega } ) = \\omega ^ { 2 - \\frac { d } { 2 } } S _ { 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } } ( \\Psi ) = \\omega ^ { 2 - \\frac { d } { 2 } } \\mu _ { 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } } . \\end{align*}"}
+{"id": "891.png", "formula": "\\begin{align*} P _ 1 ( r ) \\coloneqq \\{ \\varpi ^ k \\parallel r \\colon ( \\varpi , \\Delta _ { F _ 2 } c _ 2 ) = 1 , \\varpi ^ k \\nmid d \\} \\quad P _ 2 ( r ) = \\{ \\varpi ^ k \\parallel r \\colon ( \\varpi , \\Delta _ { F _ 2 } ) = 1 , \\varpi ^ k \\mid d \\} . \\end{align*}"}
+{"id": "4191.png", "formula": "\\begin{align*} \\hat { w _ 0 } ( \\xi ) = \\int _ 0 ^ \\infty \\frac { e ^ { - 3 \\tau } } { 2 } \\cdot e ^ { - i \\xi \\tau } \\ , d \\tau = \\frac { 1 } { 6 + 2 i \\xi } \\ , . \\end{align*}"}
+{"id": "6569.png", "formula": "\\begin{align*} \\left \\| \\ , \\left | \\sum _ { i = 1 } ^ n B ^ * _ i A _ i \\right | ^ { \\alpha } \\ , \\right \\| ^ 2 \\le \\left \\| \\ , \\left | \\sum _ { i = 1 } ^ n B ^ * _ i B _ i \\right | ^ { \\alpha } \\ , \\right \\| \\left \\| \\ , \\left | \\sum _ { i = 1 } ^ n A ^ * _ i A _ i \\right | ^ { \\alpha } \\ , \\right \\| \\end{align*}"}
+{"id": "990.png", "formula": "\\begin{align*} \\alpha \\big ( ( \\ell _ E ) _ { E \\in \\mathbf { E } } , ( n _ E ) _ { E \\in \\mathbf { E } } \\big ) = { } & \\big ( ( n _ E ) _ { E \\in \\mathbf { E } } , ( 0 ) _ { 1 \\le i \\le r } , ( ( - 1 ) ^ { \\delta _ { v , v ' _ E } } \\ell _ E ) _ { v \\in E \\in \\mathbf { E } } \\big ) , \\\\ \\beta \\big ( ( n _ E ) _ { E \\in \\mathbf { E } } \\big ) = { } & \\big ( ( n _ E ) _ { v \\in E \\in \\mathbf { E } } \\big ) . \\end{align*}"}
+{"id": "1371.png", "formula": "\\begin{align*} g _ i v _ j = - v _ { - i + 2 j \\bmod 5 } , i , j \\in \\{ 1 , 2 , 3 , 4 , 5 \\} . \\end{align*}"}
+{"id": "2988.png", "formula": "\\begin{align*} E _ 1 ^ { p , q } = \\begin{cases} \\bigoplus _ { \\lvert I \\rvert = p + 1 } R _ F ( G _ I ) \\otimes \\Q & q , \\\\ 0 & q \\ . \\end{cases} \\end{align*}"}
+{"id": "1959.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c \\underline u ) ^ n \\geq \\psi ( z , \\underline u ) \\omega ^ n & \\textnormal { o n } & \\Omega , \\\\ \\underline u = 0 & \\textnormal { o n } & \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "7339.png", "formula": "\\begin{align*} e _ x v _ x + e _ y v _ x ^ 2 v _ y - e _ y v _ y u _ z ^ 2 v _ z ^ 2 = 0 . \\end{align*}"}
+{"id": "504.png", "formula": "\\begin{align*} L p _ { \\sigma ( j ) } ( y _ j ) = f ( y _ j ) , j = 1 , \\ldots , D . \\end{align*}"}
+{"id": "6619.png", "formula": "\\begin{align*} [ [ x _ \\mu [ { y _ 1 } _ \\eta y _ 2 ] ] _ { \\mu + \\gamma } \\varphi _ { \\lambda } ( w , v ) ] = [ ( \\varphi _ \\lambda ( x , [ { y _ 1 } _ \\eta y _ 2 ] ) ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] . \\end{align*}"}
+{"id": "7265.png", "formula": "\\begin{align*} n ^ { - 1 / 2 } \\big \\Vert \\sum _ { i = 1 } ^ n d _ { i , N } \\big \\Vert _ 2 \\leq n ^ { - 1 / 2 } \\Vert S _ n \\Vert _ 2 + 2 n ^ { - 1 / 2 } \\Vert g _ { 0 , N } \\Vert _ 2 + n ^ { - 1 / 2 } \\big \\Vert \\sum _ { i = 1 } ^ n Y _ { i , N } \\big \\Vert _ 2 \\ , . \\end{align*}"}
+{"id": "1196.png", "formula": "\\begin{align*} B ( \\phi _ 0 + \\phi _ + + U \\phi _ + ) = A \\phi _ 0 + i \\phi _ + - i U \\phi _ + \\end{align*}"}
+{"id": "3723.png", "formula": "\\begin{align*} \\sigma _ * ( \\varphi ) \\circ \\sigma _ { E _ \\bullet } = \\sigma _ { F _ \\bullet } \\circ \\varphi . \\end{align*}"}
+{"id": "7285.png", "formula": "\\begin{align*} x _ p = \\frac { m - l } { ( k , m - l ) } v _ p , y _ p = \\frac { k } { ( k , m - l ) } v _ p , \\end{align*}"}
+{"id": "3289.png", "formula": "\\begin{align*} { \\rm I m } ( e ^ { i \\pi ( 1 - \\beta _ 1 ) } \\kappa ' ( t ) ) = 0 { \\rm \\ f o r \\ } t < 0 \\end{align*}"}
+{"id": "6893.png", "formula": "\\begin{align*} \\sup _ { x \\in [ 0 , 1 ] } \\P ( N ^ { - 1 } d _ { i _ x } \\geq \\beta ) \\leq \\P \\left ( N ^ { - 1 } \\max _ { i \\in [ N ] } d _ i \\geq \\beta \\right ) \\leq \\sum _ { i \\in [ N ] } \\P ( N ^ { - 1 } d _ i \\geq \\beta ) \\leq N \\sup _ { x \\in [ 0 , 1 ] } \\P ( N ^ { - 1 } d _ { i _ x } \\geq \\beta ) . \\end{align*}"}
+{"id": "4606.png", "formula": "\\begin{align*} \\bar { \\rm P } _ { j , j + i } = & \\sum ^ { M - j } _ { m = i + 1 } { M - j \\choose m } \\mathbb { P } _ { \\rm T X } ^ m \\left ( 1 - \\mathbb { P } _ { \\rm T X } \\right ) ^ { M - j - m } \\\\ & \\times m \\cdots ( m - i + 1 ) \\mathbb { P } _ K ^ { m } \\gamma _ i \\gamma _ { m , i } \\\\ & + { M - j \\choose i } \\mathbb { P } _ { \\rm T X } ^ i \\left ( 1 - \\mathbb { P } _ { \\rm T X } \\right ) ^ { M - j - i } i ! \\mathbb { P } _ K ^ { i } \\gamma _ i \\end{align*}"}
+{"id": "9190.png", "formula": "\\begin{align*} u ^ { 1 } & = \\eta ^ { 1 } ( q ^ { 1 } , q ^ { 2 } , q ^ { 3 } , \\omega ^ { 1 } , \\omega ^ { 2 } , \\omega ^ { 3 } , y _ { 1 , [ 0 , 4 ] } ^ { 1 , d } , y _ { 2 , [ 0 , 4 ] } ^ { 1 , d } ) \\\\ u ^ { 2 } & = \\eta ^ { 2 } ( q ^ { 1 } , q ^ { 2 } , q ^ { 3 } , \\omega ^ { 1 } , \\omega ^ { 2 } , \\omega ^ { 3 } , y _ { 1 , [ 0 , 4 ] } ^ { 1 , d } , y _ { 2 , [ 0 , 4 ] } ^ { 1 , d } ) \\ , . \\end{align*}"}
+{"id": "1131.png", "formula": "\\begin{align*} \\sum _ { l = k } ^ { \\infty } \\left ( \\partial ^ 2 u _ l ( 0 ) - \\partial ^ 2 u _ { l + 1 } ( 0 ) \\right ) = \\partial ^ 2 u _ k ( 0 ) - \\partial ^ 2 u ( 0 ) . \\end{align*}"}
+{"id": "4262.png", "formula": "\\begin{align*} \\mathcal { V } \\left ( t , \\mu \\right ) = \\inf _ { u \\in \\mathcal { U } _ { t } ^ { 2 } } J \\left ( t , \\mu ; u \\right ) . \\end{align*}"}
+{"id": "2045.png", "formula": "\\begin{align*} a _ 0 + b _ 0 & \\leq a _ 1 + b _ 1 , \\\\ a _ 0 + b _ 0 & = a _ 1 + b _ 2 , \\\\ a _ 0 + b _ 0 & \\leq a _ 2 + b _ 1 , \\\\ a _ 0 + b _ 0 & \\leq a _ 2 + b _ 2 . \\end{align*}"}
+{"id": "2960.png", "formula": "\\begin{align*} \\Delta ^ \\ell = \\left \\{ ( t _ 1 , \\dots , t _ \\ell ) \\in \\R ^ { \\ell } \\ \\mid \\ \\sum _ { i = 1 } ^ \\ell t _ i \\leq 1 t _ j \\geq 0 \\ \\forall j \\in \\{ 1 , \\dots , \\ell \\} \\right \\} \\end{align*}"}
+{"id": "7052.png", "formula": "\\begin{align*} \\frac { d } { d x } ( f _ { \\tilde { \\textbf { Q } } } ) = \\frac { g ' } { b } \\cdot ( b h ) + g \\cdot \\frac { d } { d x } ( b h ) = \\frac { g ' } { b } \\cdot ( b h ) \\end{align*}"}
+{"id": "9261.png", "formula": "\\begin{align*} \\frac { 1 } { t ^ { \\eta } } \\int _ 0 ^ { t } z ^ { \\eta - 1 } | f ( z ) | \\ , d z & \\le \\frac { 1 } { x ^ { \\eta } } \\int _ 0 ^ { x } z ^ { \\eta - 1 } | f ( z ) | \\ , d z + \\frac { 1 } { t ^ { \\eta } } \\int _ x ^ t z ^ { \\eta - 1 } | f ( z ) | \\ , d z \\\\ & \\le \\frac { 1 } { x ^ { \\eta } } \\int _ 0 ^ { x } z ^ { \\eta - 1 } | f ( z ) | \\ , d z + \\int _ x ^ { \\infty } \\frac { | f ( z ) | } { z } \\ , d z = H _ { \\eta } | f | ( x ) + H _ 0 ^ { \\infty } | f | ( x ) . \\end{align*}"}
+{"id": "5626.png", "formula": "\\begin{align*} \\aligned \\left \\{ \\begin{array} { l l l } - \\Delta u + \\lambda _ 1 u = ( I _ \\mu \\ast | u | ^ { r _ 1 } ) | u | ^ { r _ 1 - 2 } u + \\nu p ( I _ \\mu \\ast | v | ^ q ) | u | ^ { p - 2 } u \\ & \\mathbb { R } ^ N , \\\\ - \\Delta v + \\lambda _ 2 v = ( I _ \\mu \\ast | v | ^ { r _ 2 } ) | v | ^ { r _ 2 - 2 } v + \\nu q ( I _ \\mu \\ast | u | ^ p ) | v | ^ { q - 2 } v \\ & \\mathbb { R } ^ N , \\\\ \\int _ { \\mathbb { R } ^ N } u ^ 2 = a ^ 2 , \\ \\int _ { \\mathbb { R } ^ N } v ^ 2 = b ^ 2 . \\end{array} \\right . \\endaligned \\end{align*}"}
+{"id": "1611.png", "formula": "\\begin{align*} f ^ 3 - f Q = 0 , Q \\ , \\xi = \\xi ( \\xi \\in \\ker f ) . \\end{align*}"}
+{"id": "5182.png", "formula": "\\begin{align*} w _ 1 - w _ { j } = b _ { r _ j } - b _ { s _ j } ; 2 \\leq j \\leq n . \\end{align*}"}
+{"id": "7903.png", "formula": "\\begin{align*} - \\frac { 2 \\binom { n - k } { k } \\binom { n - 1 } { k - 1 } } { 2 \\binom { n } { k } - 2 \\binom { n - 1 } { k - 1 } } . \\end{align*}"}
+{"id": "1382.png", "formula": "\\begin{align*} g \\colon \\Z / p \\Z \\to \\Z / p \\Z , g ( x ) = \\alpha x . \\end{align*}"}
+{"id": "2863.png", "formula": "\\begin{align*} ( 1 _ { L ^ { \\otimes ^ 3 } } - \\tau ^ { 1 2 } ) ( ( \\Delta _ 0 \\otimes \\alpha ) \\circ \\Delta _ 0 - ( \\alpha \\otimes \\Delta _ 0 ) \\circ \\Delta _ 0 ) = 0 \\end{align*}"}
+{"id": "8927.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { j + 1 } Z _ l + \\sum _ { l = j + 2 } ^ k B _ l ^ { ( j + 1 ) } & = \\prod _ { l = 1 } ^ { j + 1 } y \\big ( \\bigl \\{ X _ { n i } ^ { ( l ) } \\bigr \\} \\big ) = \\Bigg ( \\sum _ { l = 1 } ^ { j } Z _ l + \\sum _ { l = j + 1 } ^ k B _ l ^ { ( j ) } \\Bigg ) \\Bigg ( Z _ { j + 1 } + \\sum _ { l = j + 2 } ^ k A _ l \\Bigg ) . \\end{align*}"}
+{"id": "1118.png", "formula": "\\begin{align*} \\begin{aligned} ( f g ) x = ( f x ) g + f ( g x ) + \\alpha _ 1 \\big ( f ( x g ) + f ( g x ) \\big ) + \\alpha _ 2 \\big ( g ( x f ) + g ( f x ) \\big ) \\\\ x ( f g ) = ( x f ) g - ( x g ) f + \\beta _ 1 \\big ( f ( x g ) + f ( g x ) \\big ) + \\beta _ 2 \\big ( g ( x f ) + g ( f x ) \\big ) \\end{aligned} \\end{align*}"}
+{"id": "8762.png", "formula": "\\begin{align*} \\varphi ( y , d _ 0 ) : = \\begin{cases} \\max _ { p , v } & \\sum _ { i = 1 } ^ n ( 1 - c ) p _ i \\cdot \\min ( x _ i + y _ i , d _ i ) \\\\ s . t . \\quad & \\begin{cases} p _ i \\in [ p _ { i , \\min } , p _ { i , \\max } ] , \\forall i \\in I \\\\ v \\mbox { s o l v e s } F ( p ) . \\end{cases} \\end{cases} \\end{align*}"}
+{"id": "2086.png", "formula": "\\begin{align*} & F ( A ) = 2 a ^ 2 - \\left ( 3 - d + 2 \\left \\lfloor \\frac { a - 1 } { 3 } \\right \\rfloor \\right ) a - d . \\end{align*}"}
+{"id": "9181.png", "formula": "\\begin{align*} \\begin{array} { c c l } y _ { [ 4 ] } ^ { 1 } & = & v ^ { 1 } \\\\ y _ { [ 4 ] } ^ { 2 } & = & v ^ { 2 } \\ , . \\end{array} \\end{align*}"}
+{"id": "1173.png", "formula": "\\begin{align*} - \\frac { 2 } { ( f ^ { 2 } - 1 ) ^ { 2 } } T ( f ^ { 2 } ) + \\frac { 4 } { ( f ^ { 2 } - 1 ) ^ { 3 } } A ( f ^ { 2 } ) ^ { 2 } = - \\frac { 4 f } { ( f ^ 2 - 1 ) ^ { 2 } } T ( f ) + \\frac { 1 2 f ^ 2 + 4 } { ( f ^ { 2 } - 1 ) ^ 3 } A ( f ) ^ { 2 } . \\end{align*}"}
+{"id": "1023.png", "formula": "\\begin{align*} & \\qquad \\qquad \\frac { ( \\frac { 1 } { 3 } ) _ k ( \\frac { 2 } { 3 } ) _ k } { ( 1 ) _ { k } ^ 2 } = \\frac { \\binom { 2 k } { k } \\binom { 3 k } { k } } { 2 7 ^ k } , \\\\ [ 1 m m ] & H _ k \\bigg ( - \\frac { 1 } { 3 } \\bigg ) + H _ k \\bigg ( - \\frac { 2 } { 3 } \\bigg ) = 3 H _ { 3 k } - H _ k , \\end{align*}"}
+{"id": "3043.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ r a _ j ^ p + \\sum _ { j = r + 1 } ^ k b _ j ^ p \\leq k . \\end{align*}"}
+{"id": "7289.png", "formula": "\\begin{align*} b _ p + k x _ p + l y _ p = a _ p + n x _ p \\leq c _ p + m y _ p \\end{align*}"}
+{"id": "8392.png", "formula": "\\begin{align*} \\epsilon _ 1 ( n , A , \\rho ) : = \\frac { 1 } { 2 } ( 1 0 n ) ^ { - 3 } \\epsilon _ 0 ( n , A , \\rho ) , C _ 1 ( n , A , \\rho ) : = n \\epsilon _ 1 ( n , A , \\rho ) ^ { - 1 } . \\end{align*}"}
+{"id": "8569.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } S _ j ( t ) & = \\frac { \\nu q Y } { \\lambda } \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } ( 1 - y ) y ^ { j - 1 } d y \\\\ & = \\frac { \\nu q Y } { \\lambda } \\sum _ { k = 0 } ^ \\infty \\frac { p ^ k } { ( j + k ) ( j + k + 1 ) } , j \\geq 1 . \\end{align*}"}
+{"id": "258.png", "formula": "\\begin{align*} = \\left ( 1 - y z \\right ) ^ { \\frac { y } { 1 - y } } \\exp \\left \\{ \\frac { y } { ( 1 - y ) ^ 2 } \\left ( L i _ 2 ( z ) - L i _ 2 ( y z ) \\right ) \\right \\} ; \\end{align*}"}
+{"id": "3873.png", "formula": "\\begin{align*} \\mathrm { R W } ( d ) = \\sup _ { \\gamma \\in \\Sigma ( \\delta ) } \\mathbb { E } _ { \\gamma } \\left [ \\sum _ { \\ell = 1 } ^ L Y _ \\ell I ( d ( X ) = \\ell ) \\right ] . \\end{align*}"}
+{"id": "456.png", "formula": "\\begin{align*} \\chi _ t = - \\sqrt { \\chi _ 1 ^ 2 + 2 G \\left ( \\frac 1 { \\chi } - \\frac 1 { \\chi _ 0 } \\right ) } . \\end{align*}"}
+{"id": "1788.png", "formula": "\\begin{align*} \\xi _ { i _ 1 , \\ldots , i _ n } ( f _ { i _ n , \\ldots , i _ 1 , i _ 1 , \\ldots , i _ n } ) & = \\xi _ { i _ 1 , \\ldots , i _ n } ( \\psi ^ n _ { i _ n , \\ldots , i _ 1 , \\ldots , i _ n } ( f _ { i _ n , \\ldots , i _ 1 , \\ldots , i _ n } ) ) = \\\\ & = \\xi _ { i _ 2 , \\ldots , i _ n } ( \\delta ^ n _ { i _ n , \\ldots , i _ 1 , \\ldots , i _ n } ( f _ { i _ n , \\ldots , i _ 1 , \\ldots , i _ n } ) ) = \\xi _ { i _ 2 , \\ldots , i _ n } ( f _ { i _ n , \\ldots , i _ 2 , i _ 2 , \\ldots , i _ n } ) = \\\\ & = \\epsilon ( f ) . \\end{align*}"}
+{"id": "4814.png", "formula": "\\begin{align*} u _ * ( x _ 1 , x _ 2 , x _ 3 ) = 2 1 2 . 5 7 5 5 x _ 1 x _ 2 + 5 4 . 1 2 9 6 x _ 1 x _ 3 , \\end{align*}"}
+{"id": "357.png", "formula": "\\begin{align*} \\rho _ a ( x ) = \\underbrace { x \\cdot \\ldots \\cdot x } _ { a } . \\end{align*}"}
+{"id": "6343.png", "formula": "\\begin{align*} \\R \\ni s \\mapsto \\gamma ( s ) : = G ( \\phi , \\omega , r ; s ) . \\end{align*}"}
+{"id": "6414.png", "formula": "\\begin{align*} \\theta _ t = ( \\theta ^ { ( 1 ) } _ t \\bar { \\otimes } \\mathrm { i d } ) \\ ! \\upharpoonright _ { \\widetilde { M _ 1 \\bar { \\otimes } M _ 2 } } = ( \\mathrm { i d } \\bar { \\otimes } \\theta _ t ^ { ( 2 ) } ) \\ ! \\upharpoonright _ { \\widetilde { M _ 1 \\bar { \\otimes } M _ 2 } } , t \\in \\mathbb { R } . \\end{align*}"}
+{"id": "8124.png", "formula": "\\begin{align*} C _ F = \\sum _ { G \\le F } X _ G . \\end{align*}"}
+{"id": "2163.png", "formula": "\\begin{align*} \\widetilde { k } \\left ( x \\right ) = 2 \\left ( \\nabla u _ { 0 , 1 } \\left ( x \\right ) \\right ) ^ { - 2 } v \\left ( x , T / 2 \\right ) + F \\left ( x \\right ) , \\end{align*}"}
+{"id": "4637.png", "formula": "\\begin{align*} \\begin{cases} c _ t - \\Delta c + c = g , \\ \\ ( x , t ) \\in \\Omega \\times ( 0 , T ) , \\\\ \\displaystyle { \\frac { \\partial c } { \\partial \\nu } } = 0 , \\ \\ ( x , t ) \\in \\partial \\Omega \\times ( 0 , T ) , \\\\ c ( x , 0 ) = c _ 0 ( x ) , \\ \\ ( x , t ) \\in \\Omega . \\end{cases} \\end{align*}"}
+{"id": "2326.png", "formula": "\\begin{align*} \\int _ { T _ n ( t ) } \\prod _ { j = 1 } ^ n ( t _ { j + 1 } - t _ j ) ^ { \\alpha _ j } d \\pmb { t } _ n = \\frac { \\prod _ { j = 1 } ^ { n } \\Gamma ( \\alpha _ j + 1 ) } { \\Gamma ( | \\alpha | + n + 1 ) } t ^ { | \\alpha | + n } , \\end{align*}"}
+{"id": "4787.png", "formula": "\\begin{align*} R _ K ( X , \\varepsilon , t , a , b ) = \\mathcal { O } \\left ( \\frac { b + 1 } { \\sqrt { \\log g } } \\right ) . \\end{align*}"}
+{"id": "2224.png", "formula": "\\begin{align*} \\log y \\int _ { 1 } ^ { u / h } y ^ { h t - u } ( t \\omega ( t ) ) \\ , d t & = h ^ { - 1 } y ^ { h t - u } ( t \\omega ( t ) ) \\bigg | _ { 1 } ^ { u / h } - h ^ { - 1 } \\int _ { 1 } ^ { u / h } y ^ { h t - u } \\omega ( t - 1 ) \\ , d t \\\\ & = u h ^ { - 2 } \\omega ( u h ^ { - 1 } ) - h ^ { - 1 } y ^ { h - u } - h ^ { - 1 } y ^ h \\int _ { 1 } ^ { u / h - 1 } y ^ { h t - u } \\omega ( t ) \\ , d t \\\\ & = u h ^ { - 2 } \\omega ( u h ^ { - 1 } ) - h ^ { - 1 } y ^ { h - u } - \\left ( h ^ 2 e ^ \\gamma \\log y \\right ) ^ { - 1 } \\lambda ( y ^ { u - h } , y ^ h ) . \\end{align*}"}
+{"id": "3459.png", "formula": "\\begin{align*} m _ { 1 } t _ { 1 } ^ { m _ { 1 } } t _ { 2 } ^ { m _ { 2 } } \\mathbf { k } _ { 1 } + m _ { 2 } t _ { 1 } ^ { m _ { 1 } } t _ { 2 } ^ { m _ { 2 } } \\mathbf { k } _ { 2 } = 0 \\ m _ 1 , m _ 2 , n _ 1 , n _ 2 \\in \\Z . \\end{align*}"}
+{"id": "3057.png", "formula": "\\begin{align*} H ^ \\star = \\nu P ( \\Lambda ) P ^ T = \\nu ( P _ + P _ + ^ T - P _ - P _ - ^ T ) . \\end{align*}"}
+{"id": "5213.png", "formula": "\\begin{align*} { \\sqsupset _ \\psi ^ * * \\sqsupset _ \\phi ^ * } = { \\sqsupset _ \\psi * \\sqsupset _ \\phi } = { \\sqsupset _ { \\psi \\circ \\phi } ^ * } . \\end{align*}"}
+{"id": "6760.png", "formula": "\\begin{align*} \\widetilde { R } ( h , x ) = \\begin{cases} ( R h , x ) , & \\underline { \\varphi } ( h ) ( 0 ) = \\varphi ( h ) ( 0 ) , \\\\ ( R h , \\sigma x ) , & 0 = \\underline { \\varphi } ( h ) ( 0 ) < \\varphi ( h ) ( 0 ) = 1 . \\end{cases} \\end{align*}"}
+{"id": "7118.png", "formula": "\\begin{align*} \\Big | \\widehat { J _ \\varepsilon } ( 0 ) - \\widehat { J _ \\varepsilon } ( \\xi ) - | \\xi | ^ 2 \\Big | & = \\Big | \\int _ { \\mathbb { R } ^ n } J _ \\varepsilon ( | x | ) \\big ( f _ \\xi ( x ) - f _ \\xi ( 0 ) - \\nabla f _ \\xi ( 0 ) \\cdot x - \\frac { 1 } { 2 } x ^ T D ^ 2 f _ \\xi ( 0 ) x \\big ) \\ : x \\Big | \\\\ & \\leq C ( 1 + | \\xi | ^ 2 ) \\int _ { \\mathbb { R } ^ n } \\rho _ \\varepsilon ( | x | ) \\ : x \\leq C ( 1 + | \\xi | ^ 2 ) , \\end{align*}"}
+{"id": "7250.png", "formula": "\\begin{align*} \\alpha _ { { \\bf Y } , 4 } ( 0 ) = 1 \\alpha _ { { \\bf Y } , 4 } ( n ) = \\max _ { 1 \\leq l \\leq 4 } \\ \\sup _ { n \\leq i _ 1 \\leq \\cdots \\leq i _ l } \\alpha ( { \\mathcal F } _ 0 , ( Y _ { i _ 1 } , \\dots , Y _ { i _ l } ) ) \\ , . \\end{align*}"}
+{"id": "7088.png", "formula": "\\begin{align*} \\beta _ n = v ( p ) - p \\nu ( x - a _ n ) . \\end{align*}"}
+{"id": "9008.png", "formula": "\\begin{align*} \\Delta w = - \\frac { S ^ \\varphi } { m - 1 } w \\end{align*}"}
+{"id": "6456.png", "formula": "\\begin{align*} \\frac { 1 } { 4 z ( s ) } & = \\frac { 1 } { 2 \\sigma ^ 2 + \\mathcal { O } ( h ^ 2 + s \\sigma ^ 4 ) } \\\\ & = \\frac { 1 } { 2 \\sigma ^ 2 } \\cdot \\frac { 1 } { 1 + \\mathcal { O } ( h ^ 2 \\sigma ^ { - 2 } + s \\sigma ^ 2 ) } \\\\ & = \\frac { 1 } { 2 \\sigma ^ 2 } + \\mathcal { O } ( h ^ 2 \\sigma ^ { - 4 } + s ) \\end{align*}"}
+{"id": "4533.png", "formula": "\\begin{align*} M : = \\sup _ { t \\in [ 0 , T _ { \\max } ) } \\| U ( t ) \\| _ { \\mathcal { H } ^ 1 } < \\infty . \\end{align*}"}
+{"id": "704.png", "formula": "\\begin{align*} \\mathfrak { f } _ { \\bar t } ( \\xi _ { [ ( \\sigma , k , l ) ] } ) = 0 \\mathfrak { o } ( \\bar t ) < \\mathfrak { o } ( \\sigma , k ) . \\end{align*}"}
+{"id": "1546.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } M \\leq ( 1 - \\epsilon ) M = k - k _ \\epsilon \\leq | k _ \\epsilon | + | k | \\leq 2 M \\end{align*}"}
+{"id": "5689.png", "formula": "\\begin{align*} & M _ { k } ^ { \\# } ( \\Gamma _ { 0 } ( N ) ) : = \\{ f \\in M _ { k } ^ { ! } ( \\Gamma _ { 0 } ( N ) ) \\mid \\} , \\\\ & S _ { k } ^ { \\# , 0 } ( \\Gamma _ { 0 } ( N ) ) : = \\{ f \\in M _ { k } ^ { \\# } ( \\Gamma _ { 0 } ( N ) ) \\mid \\} . \\end{align*}"}
+{"id": "3713.png", "formula": "\\begin{align*} W = W ( \\beta ) = \\bigcup _ { s < t } [ \\beta ( s ) , \\beta ( t ) ] . \\end{align*}"}
+{"id": "3592.png", "formula": "\\begin{align*} v ( n ) & \\ = \\ v ( p ^ { \\alpha _ 1 } _ 1 \\cdots p ^ { \\alpha _ k } _ k ) \\ = \\ v ( p ^ { \\alpha _ 1 } _ 1 ) + \\cdots + v ( p ^ { \\alpha _ k } _ k ) \\\\ & \\ \\le \\ p ^ { \\alpha _ 1 } _ 1 + \\cdots + p ^ { \\alpha _ k } _ k \\ \\le \\ p ^ { \\alpha _ 1 } _ 1 \\cdots p ^ { \\alpha _ k } _ k \\ = \\ n . \\end{align*}"}
+{"id": "2057.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } \\sup _ { \\lambda \\in ( 1 , \\infty ) } \\sum _ j \\norm { P _ { \\geq N } g _ j ^ \\lambda } _ { L ^ 4 _ { \\lambda _ k } } ^ 2 = 0 . \\end{align*}"}
+{"id": "8749.png", "formula": "\\begin{align*} R ( Y ) : = \\bigcup _ { x \\in X } Y ( x ) \\subset \\bar { Y } . \\end{align*}"}
+{"id": "2241.png", "formula": "\\begin{align*} \\widetilde { T } ( f ) ( z ) = - \\frac { 1 } { \\pi } \\iint _ D \\left ( \\frac { f ( \\zeta ) } { \\zeta - z } + \\frac { z \\overline { f ( \\zeta ) } } { 1 - \\overline { \\zeta } z } \\right ) \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "6923.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } | R ( t ) | ^ { 2 } \\Phi \\left ( \\frac { t } { T } \\right ) d t = \\sqrt { 2 \\pi } T \\sum _ { m , n \\in \\mathcal { M } ' } r ( m ) r ( n ) \\Phi \\left ( T \\log { \\frac { m } { n } } \\right ) . \\end{align*}"}
+{"id": "4845.png", "formula": "\\begin{align*} g _ { w ( l ) } ( y ) = \\prod _ { i = 1 } ^ r c _ y ( i ) \\ast h _ l ( z _ y ( i ) ) ) \\ast c _ y ( r + 1 ) \\end{align*}"}
+{"id": "223.png", "formula": "\\begin{align*} \\omega ( s _ 1 , s _ { 2 } ) = \\sum _ { m _ 1 > 0 } \\frac { 1 } { { m _ 1 } ^ { s _ 1 } ( m _ 1 ) ^ { s _ { 2 } } } , \\end{align*}"}
+{"id": "5725.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } e ^ 2 = a _ 1 e + b _ 1 f \\\\ f ^ 2 = a _ 2 e + b _ 2 f \\\\ e f = a _ 3 e + b _ 3 f \\\\ f e = a _ 4 e + b _ 4 f \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "5166.png", "formula": "\\begin{align*} T ( u , v ) : = \\inf _ { p } T ( p ) , \\end{align*}"}
+{"id": "8309.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { q ^ n } | \\mu _ j | ^ 4 = \\sum _ { a , a ' } \\left | \\sum _ { b } M ( a , b ) M ( a ' , b ) \\right | ^ 2 : = \\sigma \\ , , \\end{align*}"}
+{"id": "4119.png", "formula": "\\begin{align*} M = { T ^ { \\vec { v } } _ \\varepsilon } ^ T , \\end{align*}"}
+{"id": "4668.png", "formula": "\\begin{align*} I _ \\mu ^ { \\prime } ( u ) = \\lambda _ a \\Psi ^ { \\prime } ( u ) , \\ , \\ , \\ , \\ , X ^ { \\prime } \\end{align*}"}
+{"id": "2615.png", "formula": "\\begin{align*} \\widetilde { T } _ l ( f _ 1 , f _ 2 ) = \\sum _ { \\mathbf { m } \\in ( \\mathbb { Z } ^ 2 ) ^ 2 } \\widetilde { T } _ l \\left ( f _ { 1 , m _ 1 } , f _ { 2 , m _ 2 } \\right ) \\textrm { w i t h $ \\mathbf { m } = ( m _ 1 , m _ 2 ) $ } . \\end{align*}"}
+{"id": "4617.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } e _ k ( \\mathrm { i n t } ( X ) ) = - \\infty . \\end{align*}"}
+{"id": "8078.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } \\left | \\left | u _ n \\right | ^ { p - 2 } u _ n - | u | ^ { p - 2 } u \\right | ^ { p ' } d x & \\leq C \\int _ { \\Omega } \\left | u _ n - u \\right | ^ { p ' } \\left ( \\left | u _ n \\right | + | u | \\right ) ^ { p ' ( p - 2 ) } d x \\\\ & \\leq C \\left \\| u _ n - u \\right \\| _ { p } ^ { p ' } \\left ( \\left \\| u _ n \\right \\| _ { p } + \\| u \\| _ { p } \\right ) ^ { p ' ( p - 2 ) } . \\end{aligned} \\end{align*}"}
+{"id": "1509.png", "formula": "\\begin{align*} \\phi ( x ; r ) = O \\big ( \\varepsilon ^ { - 2 } _ 2 ( r + \\varepsilon _ 2 - x ) ^ { 1 / 2 } ( x + \\varepsilon _ 2 - r ) ^ { 1 / 2 } \\big ) . \\end{align*}"}
+{"id": "7818.png", "formula": "\\begin{align*} \\sum _ { i \\le N } \\left \\langle \\tilde { g } _ { i j } B _ { i } , x \\right \\rangle ^ { 2 } \\le \\sum _ { i \\le N } ( \\max _ { j \\le n } \\tilde { g } _ { i j } ^ { 2 } ) \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } = : \\Xi ^ { 2 } . \\end{align*}"}
+{"id": "1941.png", "formula": "\\begin{align*} f _ { 1 , 0 } ^ * ( z _ 1 , \\ldots , z _ m ) : = \\delta _ { 1 , 0 } A _ 0 ( z _ 1 , \\ldots , z _ m ) = \\delta _ { 1 , 0 } z _ 1 \\cdots z _ m \\end{align*}"}
+{"id": "3693.png", "formula": "\\begin{align*} \\left . u _ 1 \\right | _ { \\Omega ^ c } = f _ 1 = f _ 2 = \\left . u _ 2 \\right | _ { \\Omega ^ c } \\Omega ^ c . \\end{align*}"}
+{"id": "7896.png", "formula": "\\begin{align*} \\underline { \\mathcal { I } } _ { n , k } ( A ) : = \\left \\{ g : [ n - k ] \\to [ n ] \\setminus \\operatorname { I m } ( A ) : \\ g \\mbox { i s i n j e c t i v e } \\right \\} . \\end{align*}"}
+{"id": "74.png", "formula": "\\begin{align*} \\zeta _ { R } ( \\lambda ) = \\prod _ { \\gamma ^ \\# } ( 1 - e ^ { - \\lambda T _ { \\gamma ^ \\# } } ) , \\Re \\lambda \\gg 1 . \\end{align*}"}
+{"id": "8612.png", "formula": "\\begin{align*} & E [ W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) ] = P ( W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) = 1 , W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 ) + O \\left ( e ^ { \\lambda \\Delta \\ell _ 1 } e ^ { 2 \\lambda \\Delta \\ell _ 2 } \\Delta ^ 3 \\right ) , \\\\ & E [ Z _ 0 ( \\ell _ 2 \\Delta ) W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) ] = E [ Z _ 0 ( \\ell _ 2 \\Delta ) ; W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 ] + O \\left ( e ^ { \\lambda \\Delta \\ell _ 1 } e ^ { 2 \\lambda \\Delta \\ell _ 2 } \\Delta ^ 2 \\right ) . \\end{align*}"}
+{"id": "6821.png", "formula": "\\begin{align*} \\beta _ n = \\sum _ { j = 0 } ^ n \\frac { \\alpha _ j } { ( q ) _ { n - j } ( a q ) _ { n + j } } \\ ; \\ ; \\ ; \\ ; \\forall \\ , n \\in \\mathbb { N } . \\end{align*}"}
+{"id": "8986.png", "formula": "\\begin{align*} \\begin{cases} \\Delta v + \\Lambda ( x ) v \\ge 0 & \\ , \\Omega \\\\ v \\equiv 0 & \\ , \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "7825.png", "formula": "\\begin{align*} \\textsf { P } \\{ \\Xi ^ { 2 } > C z \\log n \\sum _ { i \\le N } \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } \\} & = \\textsf { P } \\{ \\Xi ^ { 2 } > C \\log n \\sum _ { i \\le N } \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } \\} \\\\ & \\le \\exp ( - C _ { 1 } z \\log n \\sum _ { i \\le N } \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } ) . \\end{align*}"}
+{"id": "3891.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( \\operatorname { p r o j } _ { K _ N \\cap Q } \\circ { \\operatorname { p r o j } _ { K _ N } } ^ { - 1 } \\right ) \\# \\mu _ N & = \\left ( \\operatorname { p r o j } _ { K _ N \\cap Q } \\circ { \\operatorname { p r o j } _ { K _ \\ell } } ^ { - 1 } \\right ) \\# \\mu _ \\ell . \\end{aligned} \\end{align*}"}
+{"id": "6900.png", "formula": "\\begin{align*} J _ r ( x , \\beta ) = \\frac { 1 } { 2 v _ r ( x ) } \\ , ( \\beta - d _ r ( x ) ) ^ 2 + \\frac { 1 } { 6 } \\ , \\theta '' ( x , \\beta _ * ) \\ , ( \\beta - d _ r ( x ) ) ^ 3 , \\end{align*}"}
+{"id": "3437.png", "formula": "\\begin{align*} 0 = \\left \\langle X , \\pounds _ X g \\right \\rangle = \\nabla _ X X + \\frac 1 2 \\nabla \\left ( | X | ^ 2 \\right ) = \\nabla _ X X , \\end{align*}"}
+{"id": "1172.png", "formula": "\\begin{align*} T \\left ( \\frac { 1 } { f + 1 } \\right ) = - \\frac { 1 } { ( f + 1 ) ^ { 2 } } T ( f + 1 ) + \\frac { 2 } { ( f + 1 ) ^ { 3 } } A ( f + 1 ) ^ { 2 } . \\end{align*}"}
+{"id": "9052.png", "formula": "\\begin{align*} ( \\theta , p _ \\theta , z , s , t ) \\mapsto ( \\vartheta , r , p _ \\vartheta , p _ r , w ) = ( \\theta , e ^ s , e ^ s p _ \\theta , z , t + e ^ s z ) . \\end{align*}"}
+{"id": "3902.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { \\ell < 3 } K _ { \\ell } \\right ) \\cap S _ { 3 } = ( K _ 1 \\cup K _ 2 ) \\cap K _ 3 = \\{ 4 \\} \\in \\bigcup _ { \\ell < 3 } 2 ^ { K _ { \\ell } } = 2 ^ { K _ 1 } \\cup 2 ^ { K _ 2 } . \\end{align*}"}
+{"id": "6077.png", "formula": "\\begin{align*} \\mathcal { M } ^ { 0 } _ { ( N ) , A } f \\ ; = \\sup _ { \\substack { \\phi \\in \\mathcal { S } , \\| \\phi \\| _ { ( N ) } \\leq 1 } } M _ { \\phi , A } ^ { 0 } f , \\end{align*}"}
+{"id": "6659.png", "formula": "\\begin{align*} \\begin{aligned} & G _ \\alpha ' ( r ) = p r ( r ^ 2 + \\alpha ) ^ { \\frac p 2 - 1 } \\\\ & G _ \\alpha '' ( r ) = p ( r ^ 2 + \\alpha ) ^ { \\frac p 2 - 2 } [ \\alpha + r ^ 2 ( p - 1 ) ] \\geq 0 \\textrm { f o r a l l } \\ ; \\ ; r \\in \\R \\ , . \\end{aligned} \\end{align*}"}
+{"id": "8314.png", "formula": "\\begin{align*} | \\mu _ 2 | \\le \\left ( 3 m ^ { - 1 } q ^ { 4 n - 4 } \\Theta ( n ) \\right ) ^ { 1 / 4 } = q ^ { n - 1 } ( 3 \\Theta ( n ) m ^ { - 1 } ) ^ { 1 / 4 } \\end{align*}"}
+{"id": "2309.png", "formula": "\\begin{align*} w _ b = \\sum _ { j } c _ j a _ j + \\Psi _ b \\end{align*}"}
+{"id": "6453.png", "formula": "\\begin{align*} W _ { K _ 1 K _ 2 K _ 3 K } ( s ) = \\frac { \\pi } { z ( s ) } \\frac { 1 } { ( 2 \\pi ) ^ 8 } | \\beta _ h ( s ) | ^ 2 \\beta _ h ( s ) \\overline { \\beta _ { \\sigma } ( s ) } e ^ { \\gamma ( s ) + \\frac { \\zeta ( s ) \\cdot \\zeta ( s ) } { 4 z ( s ) } } \\end{align*}"}
+{"id": "5248.png", "formula": "\\begin{align*} D \\ ! F ( a , b , c , g ) = x ^ 2 ( x - 1 ) ^ 2 \\partial ^ 3 + D \\ ! F _ 1 \\partial ^ 2 + D \\ ! F _ 2 \\partial + D \\ ! F _ 3 , \\end{align*}"}
+{"id": "372.png", "formula": "\\begin{align*} \\eta ( \\sigma ) ^ * ( y _ { i _ 1 , j _ 1 } \\cup \\cdots \\cup y _ { i _ k , j _ k } ) = \\prod _ { \\ell = 1 } ^ { k } \\eta ( \\sigma ) ^ { \\ast } ( y _ { i _ { \\ell } , j _ { \\ell } } ) = \\prod ^ k _ { \\ell = 1 } \\left ( \\frac { c _ { i _ { \\ell } } ^ { \\sigma } } { c _ { i _ { \\ell } } ^ { \\{ i _ { \\ell } \\} } } x _ { i _ { \\ell } , j _ { \\ell } } \\right ) = \\left ( \\prod ^ k _ { \\ell = 1 } \\frac { c _ { i _ { \\ell } } ^ { \\sigma } } { c _ { i _ { \\ell } } ^ { \\{ i _ { \\ell } \\} } } \\right ) x _ { \\tau , \\mathfrak { u } } . \\end{align*}"}
+{"id": "3157.png", "formula": "\\begin{align*} S _ n ( x ) = \\frac { 1 } { n + 1 } \\sum _ { 0 } ^ { n } \\sigma _ k ( x ) , ~ n \\in \\N . \\end{align*}"}
+{"id": "4877.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s u = V ( x ) u & , \\\\ u = 0 & , \\end{cases} \\end{align*}"}
+{"id": "2167.png", "formula": "\\begin{align*} \\exp \\left [ - 2 \\lambda \\left ( \\alpha \\varepsilon \\left ( T - \\varepsilon \\right ) - b ^ { 2 } \\right ) \\right ] = \\exp \\left ( - 2 \\lambda \\beta b ^ { 2 } \\right ) . \\end{align*}"}
+{"id": "4818.png", "formula": "\\begin{align*} \\left | \\log \\kappa _ { n } \\right | = o \\left ( \\log n \\right ) , n \\rightarrow \\infty . \\end{align*}"}
+{"id": "2120.png", "formula": "\\begin{align*} X ( K ) \\cap \\Gamma = \\left ( X ( K ) \\cap \\tilde { \\Gamma } \\right ) \\cap \\Gamma , \\end{align*}"}
+{"id": "2731.png", "formula": "\\begin{align*} J _ { 5 1 } = & s \\lambda ^ 3 \\iint _ Q \\xi u A \\nabla u \\cdot \\nabla \\eta \\left ( A \\nabla \\eta \\cdot \\nabla \\eta \\right ) d x d y d t \\\\ \\geq & - C s ^ 2 \\lambda ^ 4 \\iint _ Q \\xi \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d y d t - C \\lambda ^ 2 \\iint _ Q \\xi | A \\nabla u \\cdot \\nabla \\eta | ^ 2 d x d y d t , \\end{align*}"}
+{"id": "8594.png", "formula": "\\begin{align*} e ^ { - \\lambda t } \\hat { S } _ { j , + } ^ k ( t ) & = \\nu e ^ { - \\lambda t } \\int _ 0 ^ t Y e ^ { \\lambda s } p _ { j , + } ^ k ( t - s ) d s \\\\ & = \\nu Y \\int _ 0 ^ t e ^ { - \\lambda s } p _ { j , + } ^ k ( s ) d s \\\\ & \\leq \\frac { \\nu Y } { \\lambda } \\theta ^ k . \\end{align*}"}
+{"id": "8153.png", "formula": "\\begin{align*} \\frac 1 { k ! } \\sum _ { \\ell ( I ) = k } \\varphi ^ I \\end{align*}"}
+{"id": "2041.png", "formula": "\\begin{align*} u _ { x _ n } ( 0 ) \\leq w _ { x _ n } ( 0 ) = : - \\delta _ 0 < 0 . \\end{align*}"}
+{"id": "8555.png", "formula": "\\begin{align*} p : = P ( \\Omega _ \\infty ^ c ) = P ( Z _ 0 ( t ) = 0 \\ ; \\} , \\end{align*}"}
+{"id": "3836.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\varpi \\in \\Pi ( \\mu _ { 1 3 } , \\mu _ { 2 3 } ) } \\int _ { \\mathcal { V } } f _ \\lambda \\ , d \\varpi \\right ] \\end{align*}"}
+{"id": "1378.png", "formula": "\\begin{align*} \\langle x _ 1 , x _ 2 , \\dots , x _ 7 : x _ i x _ j x _ i ^ { - 1 } = x _ { 3 i + 5 j \\bmod 7 } \\rangle . \\end{align*}"}
+{"id": "4186.png", "formula": "\\begin{align*} m ( t ; \\lambda ) = \\frac { 4 \\pi } { 3 } \\cdot \\begin{cases*} ( \\frac { t + 1 } { \\lambda + 1 } ) ^ 3 - ( \\frac { t - 1 } { 1 - \\lambda } ) ^ 3 & i f $ 0 < \\lambda < \\frac { 1 } { t } $ , \\\\ 0 & o t h e r w i s e . \\end{cases*} \\end{align*}"}
+{"id": "1361.png", "formula": "\\begin{align*} \\| A \\| = \\sup \\bigl \\{ \\| A ( x _ 1 , \\ldots , x _ k ) \\| \\colon ( x _ 1 , \\ldots , x _ k ) \\in S _ { X _ 1 } \\times \\cdots \\times S _ { X _ k } \\bigr \\} . \\end{align*}"}
+{"id": "9237.png", "formula": "\\begin{align*} \\texttt { F p [ m ] } = 0 , M \\le m \\le 3 L - 3 , \\end{align*}"}
+{"id": "7155.png", "formula": "\\begin{align*} c | _ { t = 0 } & = c _ { 0 } \\ ; \\ ; \\ ; \\Omega . \\end{align*}"}
+{"id": "6153.png", "formula": "\\begin{align*} & \\forall \\ , x : n \\forall \\ , y : n ( h _ n ( x + _ n y ) = h _ n ( x ) + _ { n + 1 } h _ n ( y ) ) \\\\ & \\forall \\ , x : n \\forall \\ , a : 0 ( h _ n ( a \\cdot _ n x ) = a \\cdot _ n h _ n ( x ) ) \\\\ & h _ n ( \\top _ n ) = \\top _ { n + 1 } \\end{align*}"}
+{"id": "5990.png", "formula": "\\begin{align*} \\overline { C } _ { X ^ { \\ast } } ( u _ - ( t _ 1 ) , u _ - ( t _ 2 ) ) & = \\overline { c } _ { X ^ { \\ast } } ( u _ - ( t _ 1 ) , u _ - ( t _ 2 ) ) \\\\ & = ( x ( u _ - ( t _ 1 ) ) , x ( u _ - ( t _ 2 ) ) _ \\R ( - x ( u _ - ( t _ 1 ) ) x ( u _ - ( t _ 2 ) ) , x ( u _ - ( t _ 1 + t _ 2 ) ) _ \\R ; \\end{align*}"}
+{"id": "7168.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 \\left ( J _ u \\cap B _ { 2 r } \\setminus J \\right ) = 0 , \\end{align*}"}
+{"id": "610.png", "formula": "\\begin{align*} \\phi ( b ) = \\langle \\theta ( b ) \\xi , \\xi \\rangle , b \\in B . \\end{align*}"}
+{"id": "2373.png", "formula": "\\begin{align*} ( R ) & \\xrightarrow { { \\sim } } \\Omega _ { c l } \\\\ I & \\mapsto ( I = ( 0 ) ) \\end{align*}"}
+{"id": "1772.png", "formula": "\\begin{align*} D _ \\Theta X _ T ^ \\ast \\le \\left ( D _ \\Theta X \\right ) _ T ^ \\ast : = \\sup _ { s \\in [ 0 , T ] } D _ \\Theta X _ s , \\end{align*}"}
+{"id": "202.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c ) = 1 \\\\ a , b , c \\geq 1 } } \\left ( \\frac { 1 } { 1 - x ^ a y ^ b z ^ c } \\right ) ^ { \\frac { c } { a b } } = \\exp \\left \\{ L i _ 1 ( x ) L i _ 1 ( y ) L i _ { - 1 } ( z ) \\right \\} = \\left ( 1 - x \\right ) ^ { \\frac { z \\log ( 1 - y ) } { ( 1 - z ) ^ 2 } } , \\end{align*}"}
+{"id": "2922.png", "formula": "\\begin{align*} v t ^ { - 1 } = \\bigcup _ { \\substack { w \\in V / L \\\\ w t = v } } w . \\end{align*}"}
+{"id": "8383.png", "formula": "\\begin{align*} H = \\begin{pmatrix} h | _ { L _ 1 } & & & \\\\ & h | _ { L _ 2 } & & \\\\ & & \\ddots & \\\\ & & & h | _ { L _ n } \\end{pmatrix} \\end{align*}"}
+{"id": "7796.png", "formula": "\\begin{align*} \\mathcal { R } ( A \\circ B ) ^ { - 1 } & \\leq \\mathcal { R } ^ { - 1 } ( A \\circ B ) & & \\\\ & \\leq ( \\mathcal { R } A \\circ \\mathcal { R } B ) ^ { - 1 } & & \\\\ & \\leq \\mathcal { R } ^ { - 1 } A \\circ \\mathcal { R } ^ { - 1 } B & & \\\\ & \\leq \\sec ^ 2 \\theta \\mathcal { R } A ^ { - 1 } \\circ \\sec ^ 2 \\theta \\mathcal { R } B ^ { - 1 } & & \\\\ & = \\sec ^ 4 \\theta ( \\mathcal { R } A ^ { - 1 } \\circ \\mathcal { R } B ^ { - 1 } ) . & & \\end{align*}"}
+{"id": "543.png", "formula": "\\begin{align*} S ( t ) = \\left ( \\begin{array} { c c } a ( t ) & 0 \\\\ 0 & 1 \\end{array} \\right ) , \\end{align*}"}
+{"id": "6235.png", "formula": "\\begin{align*} T _ { 1 } E _ 1 \\ , = \\ , E _ 1 \\ , , T _ { \\tau } E _ 1 \\ , = \\ , E _ 1 \\ , + \\ , 2 \\pi \\mathrm { i } \\ , . \\end{align*}"}
+{"id": "5214.png", "formula": "\\begin{align*} { \\sqsupset \\circ \\vartriangleright } \\ \\subseteq \\ { \\sqsupset * \\geq } \\ = \\ { \\sqsupset ^ * \\circ \\geq } \\ = \\ { \\sqsupset ^ * } . \\end{align*}"}
+{"id": "751.png", "formula": "\\begin{align*} \\phi ( r ) = \\left \\lbrace \\begin{array} { l l l } r , & r \\in [ 0 , + \\infty ) , & K = 0 , \\\\ \\sin r , & r \\in [ 0 , \\dfrac { \\pi } { 2 } ) , & K = 1 , \\\\ \\sinh r , & r \\in [ 0 , + \\infty ) , & K = - 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "3976.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\varpi \\in \\Pi \\left ( \\mu _ { Y _ 1 } , \\mu _ { Y _ 2 } \\right ) } \\int _ { \\mathbb { R } ^ 2 } ( f _ { \\mathcal { Y } } ) _ { \\lambda } ( y _ 1 , y _ 2 ) \\ , d \\varpi ( y _ 1 , y _ 2 ) \\right ] , \\end{align*}"}
+{"id": "4573.png", "formula": "\\begin{align*} \\begin{cases} \\i \\partial _ t \\varphi ( t , x ) = \\left ( - \\Delta _ x + b | \\varphi ( t , x ) | ^ 2 - \\mu ( t ) \\right ) \\varphi ( t , x ) , \\\\ \\varphi ( 0 , x ) = \\varphi _ 0 ( x ) \\ , , \\end{cases} \\end{align*}"}
+{"id": "4492.png", "formula": "\\begin{align*} K _ { \\omega , \\mathbf { c } } ( \\Psi ) = \\partial _ { \\lambda } S _ { \\omega , \\mathbf { c } } ( \\lambda \\Psi ) \\big | _ { \\lambda = 1 } = \\sum _ { j = 1 } ^ 3 \\langle D _ j S _ { \\omega , \\mathbf { c } } ( \\Psi ) , \\psi _ j \\rangle = 0 . \\end{align*}"}
+{"id": "8766.png", "formula": "\\begin{align*} e _ { i j } ( a , b ) = \\begin{cases} 1 \\quad & a = i b = j , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "5674.png", "formula": "\\begin{align*} e ^ { t _ \\nu } = ( 1 + o _ \\nu ( 1 ) ) \\Bigl ( \\frac { \\nu ( \\gamma _ p + \\gamma _ q ) \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | u _ 0 | ^ { p } ) | v _ 0 | ^ { q } } { | \\nabla u _ 0 | ^ 2 _ 2 + | \\nabla v _ 0 | ^ 2 _ 2 } \\Bigr ) ^ { \\frac { 1 } { 2 - \\gamma _ p - \\gamma _ q } } = ( 1 + o _ \\nu ( 1 ) ) \\nu ^ { \\frac { 1 } { 2 - \\gamma _ p - \\gamma _ q } } . \\end{align*}"}
+{"id": "4226.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ R \\det T _ n ( \\phi _ k ) & \\sim n ^ \\Omega E _ 4 \\end{align*}"}
+{"id": "1488.png", "formula": "\\begin{align*} J _ 3 = & \\int _ \\Omega \\Big ( f _ 0 ( V ) - f _ \\epsilon ( V ) \\Big ) \\psi ^ h _ { \\mu _ j , \\xi _ j } d x \\\\ = & \\epsilon \\int _ \\Omega V ^ p \\Big ( \\ln \\ln ( e + V ) \\Big ) \\psi ^ h _ { \\mu _ j , \\xi _ j } d x + O \\bigg ( \\epsilon ^ 2 \\Big ( \\ln \\Big | \\ln \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big | \\Big ) ^ 2 \\bigg ) . \\end{align*}"}
+{"id": "4718.png", "formula": "\\begin{align*} c _ d = \\begin{cases} 1 2 \\pi & d = 3 , \\\\ 4 \\pi & d = 2 , \\\\ 2 & d = 1 . \\end{cases} \\end{align*}"}
+{"id": "1505.png", "formula": "\\begin{align*} \\tau = \\inf \\{ s \\geq 0 : \\ , | W _ s | \\geq \\varepsilon _ { 2 } \\} , \\sigma ^ z = \\inf \\{ s \\geq 0 : \\ , | z + W _ s | \\geq \\widetilde { \\Lambda } _ { s } \\} , R _ s ^ z = | z + W _ s | . \\end{align*}"}
+{"id": "6683.png", "formula": "\\begin{align*} & \\partial _ t v _ i + a _ { k i } \\partial _ k q = 0 , i = 1 , 2 , 3 \\\\ & a _ { i k } \\partial _ i v _ k = 0 , \\end{align*}"}
+{"id": "3734.png", "formula": "\\begin{align*} [ F _ 1 ^ { S ^ \\flat } & \\cdot \\cdots F _ m ^ { S ^ \\flat } ] ( 0 , 0 , x ^ 2 , x ^ 3 ) ( \\hat { H } ( \\rho _ n ) , \\hat { P } ( \\rho _ n ) ) = \\prod _ { i = 1 } ^ m f _ i ( x ^ 2 , x ^ 3 ) . \\end{align*}"}
+{"id": "485.png", "formula": "\\begin{align*} t ^ { k _ r - k _ { r - 1 } + k _ n - 2 } t _ { 2 r - 1 } t _ { 2 r } t _ { 2 n + 1 } t _ { 2 n + 2 } t _ { 2 n + 3 } t _ { 2 n + 4 } = p q \\end{align*}"}
+{"id": "3050.png", "formula": "\\begin{align*} [ A x ] _ i = a _ { i i } x _ i + \\sum _ { j \\neq i } a _ { i j } x _ j = - \\frac { \\sum _ { j \\neq i } a _ { i j } x _ j } { x _ i } x _ i + \\sum _ { j \\neq i } a _ { i j } x _ j = 0 . \\end{align*}"}
+{"id": "6064.png", "formula": "\\begin{align*} \\mathcal { B } ( x _ { 0 } , r ) : = \\{ x \\in G \\colon \\| x _ { 0 } ^ { - 1 } x \\| < r \\} = x _ { 0 } \\mathcal { B } ( e , r ) . \\end{align*}"}
+{"id": "2977.png", "formula": "\\begin{align*} a _ { ( p , 1 , 0 , \\dots , 0 ) } = a _ { ( p , 0 , 0 , \\dots , 0 ) } \\cdot e _ 1 - a _ { ( q , 0 , 0 , \\dots , 0 ) } \\ . \\end{align*}"}
+{"id": "2014.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c w ) ^ n = ( - \\eta _ 1 w ) ^ n g \\omega ^ n & \\textnormal { o n } & \\Omega \\\\ w = 0 & \\textnormal { i n } & \\partial \\Omega \\\\ w < 0 . & & \\end{array} \\right . \\end{align*}"}
+{"id": "2369.png", "formula": "\\begin{align*} \\sum _ { z : C } \\exists _ { y : A } ( f ( y ) = x ) \\wedge g ( y ) = z \\end{align*}"}
+{"id": "1243.png", "formula": "\\begin{align*} m _ 1 ( \\tau \\ge n ) = 2 m ( [ \\tfrac 1 2 , \\hat { b } _ { n - 1 } ] ) \\le 2 ( \\hat { b } _ n - \\tfrac 1 2 ) \\le 2 ^ { \\frac { 1 } { \\beta } ( \\frac { 1 } { \\alpha ^ 2 } + \\frac { 2 } { \\alpha } ) } n ^ { - \\frac { 1 } { \\alpha \\beta } } \\le C ( B ) n ^ { - \\frac { 1 } { \\alpha _ u \\beta _ u } } . \\end{align*}"}
+{"id": "1531.png", "formula": "\\begin{align*} r _ { \\theta } ( t ) = \\begin{cases} 0 b = 0 , \\\\ b ^ { p \\theta } t ^ { p - 1 } + \\sum _ { m = 1 } ^ { p - 1 } \\binom { p - 1 } { m } \\left ( a ^ { p \\theta - m } b ^ m t ^ { p \\theta + p - 1 - m } + a ^ { - m } b ^ { p \\theta + m } t ^ { p - 1 - m } \\right ) . \\end{cases} \\end{align*}"}
+{"id": "6882.png", "formula": "\\begin{align*} d _ { \\overline { h } _ N } ( x ) = N \\int _ { [ \\frac { i - 1 } { N } , \\frac { i } { N } ) } \\d x \\ , h ( x ) \\leq \\| d _ h \\| _ { \\infty } . \\end{align*}"}
+{"id": "1453.png", "formula": "\\begin{align*} \\int _ \\Omega f _ \\epsilon ^ { ' } ( V _ m ) u _ m ^ 2 d x = o ( 1 ) . \\end{align*}"}
+{"id": "8500.png", "formula": "\\begin{align*} z ^ { - \\nu } J _ { \\nu } ( z ) = \\frac { 2 ^ { 1 - \\nu } } { \\sqrt { \\pi } \\Gamma ( \\nu + \\frac { 1 } { 2 } ) } \\ , \\intop _ { 0 } ^ { 1 } \\left ( 1 - u ^ { 2 } \\right ) ^ { \\nu - \\frac { 1 } { 2 } } \\ , \\cos ( z u ) \\ , \\ , d u , \\ , \\ , \\ , \\ , \\ , \\ , ( \\nu ) > - \\frac { 1 } { 2 } , \\ , \\ , \\ , z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "2653.png", "formula": "\\begin{align*} \\cosh ( \\alpha x _ { j - 1 } ) \\sinh ( \\alpha x _ { j } ) - \\cosh ( \\alpha x _ { j } ) \\sinh ( \\alpha x _ { j - 1 } ) = \\sinh ( \\alpha h _ j ) . \\end{align*}"}
+{"id": "5650.png", "formula": "\\begin{align*} \\bar { \\rho } = \\Bigl ( \\frac { 2 - \\gamma _ p - \\gamma _ q } { 2 2 ^ * _ \\mu - \\gamma _ p - \\gamma _ q } S ^ { 2 ^ * _ \\mu } _ { H , L } \\Bigr ) ^ { \\frac 1 { 2 2 ^ * _ \\mu - 2 } } , \\end{align*}"}
+{"id": "1367.png", "formula": "\\begin{align*} v _ 1 , \\ , v _ 2 = - g _ 3 v _ 1 , \\ , v _ 3 = - g _ 2 v _ 1 \\end{align*}"}
+{"id": "187.png", "formula": "\\begin{align*} L i _ 3 ( 0 ) = 0 , \\end{align*}"}
+{"id": "6966.png", "formula": "\\begin{align*} \\alpha _ i = \\nu ( Q _ i ' ) - \\nu ( Q _ i ) , \\ \\beta _ i = \\nu ( g ' ) - \\nu _ i ( g ) \\mbox { a n d } \\tilde { \\beta } _ i = \\nu _ i ( g ' ) - \\nu _ i ( g ) . \\end{align*}"}
+{"id": "5683.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ^ { + } ) = c F + D ^ { k - 1 } ( h ) + d g M _ { k } ^ { ! } ( \\Gamma _ { 0 } ( N ) ) . \\end{align*}"}
+{"id": "6087.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\vee \\otimes \\iota ^ * \\Big ( \\bigoplus _ { i \\in Q _ 0 } \\mathcal { U } _ i ^ \\vee \\boxtimes \\mathcal { U } _ i \\Big ) = \\bigoplus _ { i \\in Q _ 0 } ( ( \\iota ^ * \\circ p _ 1 ^ * ) ( \\mathcal { U } _ i ) \\otimes \\mathcal { L } ) ^ \\vee \\otimes ( \\iota ^ * \\circ p _ 2 ^ * ) ( \\mathcal { U } _ i ) \\end{align*}"}
+{"id": "7111.png", "formula": "\\begin{align*} ( J _ \\varepsilon * 1 ) ( x ) = \\mathcal { F } ( J _ \\varepsilon ) ( 0 ) \\end{align*}"}
+{"id": "2129.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\frac { d } { d t } x ( X , t ) = u ( x ( X , t ) , t ) , t > 0 , \\smallskip \\\\ x ( X , 0 ) = X , \\end{cases} \\end{align*}"}
+{"id": "268.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( - \\frac { n ^ 4 y ^ { n + 5 } + ( - 4 n ^ 4 - 4 n ^ 3 + 6 n ^ 2 - 4 n + 1 ) y ^ { n + 4 } } { ( 1 - y ) ^ 5 } \\right ) \\frac { z ^ n } { n ^ 5 } \\right \\} \\end{align*}"}
+{"id": "1843.png", "formula": "\\begin{align*} \\alpha ^ H ( h , k ) \\ = \\ \\frac { ( 2 / \\pi ) } { ( 2 \\pi ) ^ 3 } \\ , \\big | \\big \\{ x \\in \\Z ^ 2 : { p _ F ^ 2 - ( | h | + | k | ) ^ 2 } < | x | ^ 2 \\leq { p _ F ^ 2 - | h | ^ 2 } \\big \\} \\big | \\ . \\end{align*}"}
+{"id": "154.png", "formula": "\\begin{align*} L i _ 2 ( z ) + L i _ 2 ( - z ) = \\frac { 1 } { 2 } { L i _ 2 } ^ 2 ( z ) , \\end{align*}"}
+{"id": "719.png", "formula": "\\begin{align*} \\Big ( A _ { S } ^ * - C _ { D } A _ { B } ^ { * - 1 } D _ { D } \\Big ) [ \\partial _ { t } \\mu _ { S } ] = G _ { D , 1 } - C _ { D } A _ { B } ^ { * - 1 } G _ { D , 2 } \\ , . \\end{align*}"}
+{"id": "7530.png", "formula": "\\begin{align*} y _ m = \\sum _ { n = 1 } ^ { m } \\binom { m } { n } \\left \\{ \\sum _ { k = 1 } ^ n ( - 1 ) ^ { k } a _ { k , m - n } B _ { n , k } ( f _ { 1 , 0 } , \\ldots , f _ { n - k + 1 , 0 } ) \\right \\} . \\end{align*}"}
+{"id": "8220.png", "formula": "\\begin{align*} K = K ( \\omega ) = \\bigcup _ { x _ i \\in \\ : ( \\Pi ) } \\bar { B } ( x _ i , a ) \\end{align*}"}
+{"id": "4334.png", "formula": "\\begin{align*} \\begin{gathered} \\left \\| \\phi _ \\epsilon \\circ \\big ( \\tilde { y } _ { T _ m } ^ K \\big ) ^ { - 1 } - f _ 0 \\circ \\big ( \\tilde { y } _ { T _ m } ^ K \\big ) ^ { - 1 } \\right \\| _ { L ^ 1 ( \\mathbb { T } ^ 2 ) } \\le \\epsilon , \\\\ \\left \\| \\phi _ \\epsilon \\circ z _ m - f _ 0 \\circ z _ m \\right \\| _ { L ^ 1 ( \\mathbb { T } ^ 2 ) } \\le \\epsilon . \\end{gathered} \\end{align*}"}
+{"id": "2992.png", "formula": "\\begin{align*} \\bar { c } _ k ( t _ 1 , \\dots , t _ n ) = \\sum _ { 1 \\leq i _ 1 < \\dots < i _ k \\leq n } t _ { i _ 1 } t _ { i _ 2 } \\cdots t _ { i _ k } \\end{align*}"}
+{"id": "9082.png", "formula": "\\begin{align*} \\mathcal { F } : = \\{ x \\in V : f _ k ( x ) = 0 \\ , \\ , \\ , \\ , f _ k ( y ) = 0 \\ , \\ , \\ , \\ , y \\sim x \\} \\end{align*}"}
+{"id": "9143.png", "formula": "\\begin{align*} \\begin{array} { r c l c r c l } y ^ { 1 , + } & = & y _ { [ 1 ] } ^ { 1 } & \\cdots & y ^ { m , + } & = & y _ { [ 1 ] } ^ { m } \\\\ & \\vdots & & & & \\vdots \\\\ y _ { [ r ^ { 1 } - 1 ] } ^ { 1 , + } & = & y _ { [ r ^ { 1 } ] } ^ { 1 } & \\cdots & y _ { [ r ^ { m } - 1 ] } ^ { m , + } & = & y _ { [ r ^ { m } ] } ^ { m } \\ , . \\end{array} \\end{align*}"}
+{"id": "7286.png", "formula": "\\begin{align*} l ' = \\frac { l } { ( l , n - k ) } \\ , ; n ' = \\frac { n - k } { ( l , n - k ) } \\ , ; m ' = \\frac { m - l } { ( k , m - l ) } \\ , ; k ' = \\frac { k } { ( k , m - l ) } \\ , , \\end{align*}"}
+{"id": "230.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ 1 = \\frac { n } { 2 } + \\frac { n ^ 2 } { 2 } , \\end{align*}"}
+{"id": "6997.png", "formula": "\\begin{align*} \\epsilon ( \\Gamma \\mid \\Delta ) = | \\{ \\gamma \\in \\Gamma \\mid 0 \\leq \\gamma < \\Delta _ { > 0 } \\} | . \\end{align*}"}
+{"id": "4867.png", "formula": "\\begin{align*} \\mathcal { S } _ { D } \\left ( b \\right ) = \\left \\{ x \\in \\mathbb { R } ^ { n } : a _ { t } ^ { \\prime } x \\leq b _ { t } , \\ , t \\in T \\backslash D ; \\ ; a _ { t } ^ { \\prime } x = b _ { t } , \\ , t \\in D \\right \\} . \\end{align*}"}
+{"id": "376.png", "formula": "\\begin{align*} d ( \\tau , j ) _ i ^ { \\sigma } = \\begin{cases} p & j = i \\mbox { a n d } \\tau = \\sigma \\\\ 1 & \\mbox { e l s e } \\end{cases} \\end{align*}"}
+{"id": "4958.png", "formula": "\\begin{align*} f ( k , t , n ) = \\frac { n } { n - k } \\Big ( F ( k , t , n ) - F ( k , t + 1 , n ) \\Big ) . \\end{align*}"}
+{"id": "6524.png", "formula": "\\begin{align*} F _ X ( 0 ) & = \\frac { ( 1 - \\beta ^ 2 / \\alpha ^ 2 ) ^ { \\nu + 1 / 2 } } { 2 \\sqrt { \\pi } \\Gamma ( \\nu + 1 / 2 ) } \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { k ! } \\bigg ( \\frac { 2 \\beta } { \\alpha } \\bigg ) ^ k \\Gamma \\bigg ( \\frac { k + 1 } { 2 } \\bigg ) \\Gamma \\bigg ( \\nu + \\frac { k + 1 } { 2 } \\bigg ) \\\\ & = \\frac { ( 1 - \\beta ^ 2 / \\alpha ^ 2 ) ^ { \\nu + 1 / 2 } } { 2 \\sqrt { \\pi } \\Gamma ( \\nu + 1 / 2 ) } \\big ( S _ 1 + S _ 2 ) , \\end{align*}"}
+{"id": "60.png", "formula": "\\begin{align*} H = \\mathcal { G } [ \\ell ] \\bigcap \\mathcal { G } [ \\ell , \\sigma ] \\bigcap \\mathrm { K e r } ( \\mu ) , \\end{align*}"}
+{"id": "4992.png", "formula": "\\begin{align*} F ( k , t + 1 , \\mathbf { p } _ { n + 1 } ) = \\begin{dcases} p _ { k } ^ { } F ( k - 1 , t , \\mathbf { p } _ { n + 1 } ) + ( 1 - p _ { k } ^ { } ) F ( k , t , \\mathbf { p } _ { n + 1 } ) & \\textnormal { i f } \\ ; k < t ; \\\\ 1 & \\textnormal { i f } \\ ; k \\ge t ; \\end{dcases} \\end{align*}"}
+{"id": "178.png", "formula": "\\begin{align*} L i _ 2 \\left ( - \\frac { 1 } { 3 } \\right ) - \\frac { 1 } { 3 } L i _ 2 \\left ( \\frac { 1 } { 9 } \\right ) = - \\frac { \\pi ^ 2 } { 1 8 } + \\frac { 1 } { 6 } ( \\log 3 ) ^ 2 , \\end{align*}"}
+{"id": "6300.png", "formula": "\\begin{align*} E : D \\to M , E ( q , \\lambda ) = \\exp _ q ( \\lambda ) , \\end{align*}"}
+{"id": "1262.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 ) \\odot ( ( y _ 1 , y _ 2 ) \\oplus ( z _ 1 , z _ 2 ) ) = ( ( x _ 1 , x _ 2 ) \\odot ( y _ 1 , y _ 2 ) ) \\oplus ( ( x _ 1 , x _ 2 ) \\odot ( z _ 1 , z _ 2 ) ) \\end{align*}"}
+{"id": "1963.png", "formula": "\\begin{align*} \\log \\det ( u _ { i \\bar { j } } ) = \\log \\psi ( \\cdot , v ) = g , \\end{align*}"}
+{"id": "2180.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\alpha } ^ { - 1 } \\mathcal { Y T } { \\bf u } = \\mathcal { P } _ { \\alpha } ^ { - 1 } \\mathcal { Y } { \\bf f } . \\end{align*}"}
+{"id": "4151.png", "formula": "\\begin{align*} Q _ { 2 , \\vec { v } _ { s , t , f , 0 } ^ { ( 2 ) } } = \\left ( \\left ( \\begin{array} { r r r } t & 1 & 0 \\\\ s & 0 & 1 \\\\ 1 & 0 & 0 \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 0 & 0 & 1 \\\\ 1 & 0 & - t \\\\ 0 & 1 & - s \\end{array} \\right ) \\right ) , \\end{align*}"}
+{"id": "1999.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c v _ \\lambda ) ^ n = ( \\Vert u _ { \\lambda } \\Vert _ { C ^ 0 ( \\bar \\Omega ) } ^ { - 1 } - \\lambda v _ \\lambda ) ^ n f ^ n \\omega ^ n & \\textnormal { i n } & \\Omega \\\\ v _ \\lambda = 0 & \\textnormal { i n } & \\partial \\Omega \\\\ \\Vert v _ \\lambda \\Vert _ { C ^ 0 ( \\bar \\Omega ) } = 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "3238.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\omega ( Q ) \\lambda _ { Q } \\psi _ { Q } ( x , x _ { Q } ) \\end{align*}"}
+{"id": "6385.png", "formula": "\\begin{align*} \\phi & : C \\rightarrow E , \\\\ \\phi & ( x , y , r ) = ( 5 y / x , - 1 5 0 r / x ) , \\end{align*}"}
+{"id": "7819.png", "formula": "\\begin{align*} \\textsf { P } \\{ \\sum _ { i \\le N } ( \\max _ { j \\le n } \\tilde { g } _ { i j } ^ { 2 } ) \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } > t \\} \\le e ^ { - \\lambda t } \\prod _ { i = 1 } ^ { N } \\textsf { E } \\exp ( \\lambda ( \\max _ { j \\le n } \\tilde { g } _ { i j } ^ { 2 } ) \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } ) . \\end{align*}"}
+{"id": "5917.png", "formula": "\\begin{align*} \\nu ( \\alpha , g ) = ( \\det a _ 1 a _ 2 , y ) _ F \\gamma ( y , \\psi ^ { \\tfrac { 1 } { 2 } } ) ^ { - | S | } , \\end{align*}"}
+{"id": "5961.png", "formula": "\\begin{align*} ( \\tfrac { 1 } { \\sqrt { c ^ 2 + d ^ 2 } } ) \\tfrac { - c \\det g } { \\sqrt { c ^ 2 + d ^ 2 } } + ( \\tfrac { b d + a c } { \\sqrt { c ^ 2 + d ^ 2 } } ) \\tfrac { d } { \\sqrt { c ^ 2 + d ^ 2 } } & = \\tfrac { - c \\det g } { \\sqrt { c ^ 2 + d ^ 2 } } - \\tfrac { b d ^ 2 + a c d } { c ^ 2 + d ^ 2 } \\\\ & = \\tfrac { - c a d + b c ^ 2 + b d ^ 2 + a c d } { c ^ 2 + d ^ 2 } = b . \\end{align*}"}
+{"id": "3135.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\frac { ( x + k ) ^ { m + k } } { k ! } e ^ { - u ( x + k ) } u ^ k = \\frac { R _ m ( u , x ) } { ( 1 - u ) ^ { 2 m + 1 } } . \\end{align*}"}
+{"id": "7283.png", "formula": "\\begin{align*} k x _ p + l y _ p = m y _ p < n x _ p . \\end{align*}"}
+{"id": "5254.png", "formula": "\\begin{align*} \\| T _ { \\rho } f \\| _ q ^ q = \\| T _ { \\rho ' \\frac { 1 } { \\sqrt { q } } } f ' \\| _ q ^ q \\le \\sum _ { S \\subseteq [ n ] } \\left ( \\frac { \\beta } { \\sqrt { q } } \\right ) ^ { q | S | } \\mathbb { E } _ { x \\sim \\Omega ^ S } \\| D _ { S , x } [ f ' ] \\| _ 2 ^ q . \\end{align*}"}
+{"id": "5823.png", "formula": "\\begin{align*} { { { \\boldsymbol { \\hat x } } } _ { k ; t + 1 } } { } { \\boldsymbol { f } } \\left ( { { { { \\boldsymbol { \\hat x } } } _ { k ; t } } } \\right ) = { \\left ( { { \\boldsymbol { W } } _ k ^ T { { \\boldsymbol { \\Omega } } _ { k ; t } } { { \\boldsymbol { W } } _ k } } \\right ) ^ { - 1 } } { \\boldsymbol { W } } _ k ^ { } { { \\boldsymbol { \\Omega } } _ { k ; t } } { { \\boldsymbol { d } } _ k } , \\end{align*}"}
+{"id": "566.png", "formula": "\\begin{align*} E _ { - \\theta _ { j } v _ { j } } \\phi _ { \\hbar } ^ { ( 4 v _ { j } ) } ( \\xi ) = \\phi _ { \\hbar } ^ { ( 4 v _ { j } ) } ( \\xi + \\theta _ { j , \\xi } v _ { j } ) = \\hbar ^ { 4 } \\phi ^ { ( 4 v _ { j } ) } ( \\hbar \\xi + \\hbar \\theta _ { j , \\xi } v _ { j } ) = \\hbar ^ { 4 } E _ { - \\hbar \\theta _ { j } v _ { j } } \\phi ^ { ( 4 v _ { j } ) } ( \\hbar \\xi ) . \\end{align*}"}
+{"id": "1006.png", "formula": "\\begin{align*} \\psi ( x ) = \\frac { d } { d x } \\big \\{ \\log \\Gamma ( x ) \\big \\} , \\end{align*}"}
+{"id": "5902.png", "formula": "\\begin{align*} u & = u _ 0 + p _ \\mathrm { k } ( u ) + p _ \\mathrm { \\tilde k } ( u ) \\\\ u ' & = u _ 0 ' + p _ \\mathrm { k } ( u ' ) + p _ \\mathrm { \\tilde k } ( u ' ) \\ ; , \\end{align*}"}
+{"id": "6643.png", "formula": "\\begin{align*} \\alpha _ i x + \\beta _ i z = 0 , \\mbox { f o r } i = 1 , \\ldots , k , \\end{align*}"}
+{"id": "3072.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta _ p u _ 0 & = f _ 1 ( | x | ) \\cdot h _ 1 ( v _ 0 ) \\cdot | \\nabla u _ 0 | ^ { \\alpha } & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta _ p v _ 0 & = f _ 2 ( | x | ) \\cdot h _ 2 ( v _ 0 ) \\cdot h _ 3 ( | \\nabla u _ 0 | ) & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8489.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma ( x ) \\ , e ^ { 2 \\pi x } - 1 } = - \\frac { 1 } { 2 } + \\frac { 1 } { 2 \\pi x } \\cdot \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } + \\frac { 1 } { \\pi } \\ , \\intop _ { 0 } ^ { \\infty } \\sin ( x t ) \\ , \\sigma _ { p } ( t ) \\ , d t . \\end{align*}"}
+{"id": "3819.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\varpi \\in \\Pi ( \\mu _ { 1 3 } , \\mu _ { 2 3 } ) } \\int _ { \\mathcal { V } } f _ \\lambda \\ , d \\varpi \\right ] \\end{align*}"}
+{"id": "2554.png", "formula": "\\begin{align*} \\begin{cases} \\mathbf { v } \\cdot \\nabla \\mathbf { v } = - \\nabla P & \\ \\ , D , \\\\ \\nabla \\cdot \\mathbf { v } = 0 \\ , \\ \\ , \\ \\ \\ \\ \\ \\ \\ \\ \\ , & \\ \\ , D , \\end{cases} \\end{align*}"}
+{"id": "9079.png", "formula": "\\begin{align*} \\omega ^ + = a ^ + + m , \\ , \\ , \\omega ^ - = a ^ - + m , \\ , \\ , \\omega ^ { ( 0 ) } = n - \\omega ^ + - \\omega ^ - \\end{align*}"}
+{"id": "1701.png", "formula": "\\begin{align*} P _ n ^ { ( \\alpha , \\beta ) } ( \\cos \\theta ) : = \\frac { \\Gamma ( \\alpha + 1 + n ) } { \\Gamma ( n + 1 ) \\ , \\Gamma ( \\alpha + \\beta + 1 + n ) } \\sum _ { k = 0 } ^ n ( - 1 ) ^ k \\binom { n } { k } \\frac { \\Gamma ( \\alpha + \\beta + 1 + n + k ) } { \\Gamma ( \\alpha + 1 + k ) } \\sin ^ { 2 k } ( \\tfrac 1 2 \\theta ) , \\end{align*}"}
+{"id": "339.png", "formula": "\\begin{align*} \\| \\nabla ^ j u \\| _ \\infty \\leq C \\| u \\| _ { W ^ { m , \\mathcal { P } } ( V ) } , \\quad \\forall j = 0 , \\cdots , m . \\end{align*}"}
+{"id": "2376.png", "formula": "\\begin{align*} & \\left [ u + v ' \\right ] ( f _ 1 + g ' _ 1 , \\dots , f _ n + g ' _ n ) \\\\ = & u ( f _ 1 + g ' _ 1 , \\dots , f _ n + g ' _ n ) + v ' ( f _ 1 + g ' _ 1 , \\dots , f _ n + g ' _ n ) \\\\ = & u ( f _ 1 , \\dots , f _ n ) + [ u ( g _ 1 , \\dots , g _ n ) ] ' + [ v ( f _ 1 , \\dots , f _ n ) ] ' \\end{align*}"}
+{"id": "3718.png", "formula": "\\begin{align*} ( f ^ * h ) \\circ \\phi : M ^ { \\otimes r } \\overunderset { \\phi } { } { \\longrightarrow } f ^ * L \\overunderset { f ^ * h } { } { \\longrightarrow } f ^ * ( \\sigma ^ * L ) = ( \\sigma \\circ f ) ^ * L , \\end{align*}"}
+{"id": "7134.png", "formula": "\\begin{align*} | A _ \\gamma ( \\hat { x } , \\hat { x } ) ( \\hat { x } - \\bar { y } ) | ^ 2 = | A ^ \\prime ( \\hat { x } ) ( \\hat { x } ^ \\prime - y ^ \\prime ) | ^ 2 + | \\hat { x } _ n + y _ n | ^ 2 , \\end{align*}"}
+{"id": "3831.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\varpi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\int _ { \\mathcal { V } } g _ \\lambda \\ , d \\varpi \\right ] \\end{align*}"}
+{"id": "4045.png", "formula": "\\begin{align*} G _ k ( x ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\mathrm { R e } ( t ) = 3 \\slash 4 } \\frac { \\Gamma ( k + t ) ^ 2 } { ( 2 \\pi ) ^ { 2 t } x ^ t } \\frac { d t } { t } . \\end{align*}"}
+{"id": "6442.png", "formula": "\\begin{align*} i \\partial _ t u = - \\Delta u + \\varepsilon | u | ^ 2 u \\end{align*}"}
+{"id": "6995.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ s a _ i { \\tilde { \\textbf { Q } } } ^ { \\lambda _ i } \\end{align*}"}
+{"id": "8.png", "formula": "\\begin{align*} a = { d i a g } ( e ^ { w _ 1 } , \\ldots , e ^ { w _ m } , e ^ { - 1 } , \\ldots , e ^ { - 1 } ) . \\end{align*}"}
+{"id": "1495.png", "formula": "\\begin{align*} t _ 0 : = \\inf \\{ t \\in [ 0 , \\zeta \\wedge \\widetilde \\zeta ) : \\ , \\Lambda _ t \\neq \\widetilde { \\Lambda } _ t \\} . \\end{align*}"}
+{"id": "2568.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) , \\ \\ 0 < u < 1 , & \\ \\ D , \\\\ u = 1 , \\ \\ | \\nabla u | = c _ 1 > 0 , & \\ \\ , \\partial \\Omega _ 1 , \\\\ u = 0 , \\ \\ | \\nabla u | = 0 , & \\ \\ , \\partial \\Omega _ 2 . \\end{cases} \\end{align*}"}
+{"id": "7995.png", "formula": "\\begin{align*} \\mathcal T ^ { 1 , 1 } _ { \\rm l o c } ( \\Omega ) = \\left \\{ u \\ , \\hbox { i s m e a s u r a b l e i n $ \\Omega $ } : \\hbox { $ T _ { t } ( u ) \\in W ^ { 1 , 1 } _ { \\rm l o c } ( \\Omega ) $ f o r e v e r y $ t > 0 $ } \\right \\} . \\end{align*}"}
+{"id": "1277.png", "formula": "\\begin{align*} \\eta ( B ^ { \\prime } ) \\ge T + \\max _ { 1 \\le i _ 1 < \\ldots < i _ { q _ 0 } \\le s } \\sum _ { j = 1 } ^ { q _ 0 } a _ { i _ j } . \\end{align*}"}
+{"id": "7057.png", "formula": "\\begin{align*} \\nu \\left ( \\frac { a _ i h } { Q ' _ i } \\right ) = \\nu ( a _ i ) + \\nu ( h ) - \\nu ( Q _ i ' ) \\geq 0 . \\end{align*}"}
+{"id": "7665.png", "formula": "\\begin{align*} \\Delta _ g \\frac { | u | ^ p } { p } = ( p - 1 ) | u | ^ { p - 2 } | \\nabla _ g u | ^ 2 + | u | ^ { p - 2 } u \\Delta _ g u , \\end{align*}"}
+{"id": "3520.png", "formula": "\\begin{align*} S ( t ) = \\dfrac { \\sin t I + \\cos t S ( 0 ) } { \\cos t I - \\sin t S ( 0 ) } , \\end{align*}"}
+{"id": "2883.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 2 t _ { 1 } t _ { 2 } - 2 t _ { 1 } - 2 t _ { 2 } + 2 , & ~ ~ t _ { 0 } \\geq 2 , \\\\ t _ { 1 } t _ { 2 } - 2 t _ { 1 } - 2 t _ { 2 } + 2 , & ~ ~ t _ { 0 } = 1 , \\\\ - 2 t _ { 1 } - 2 t _ { 2 } + 2 , & ~ ~ t _ 0 = 0 , \\\\ - t _ { 0 } t _ { 1 } t _ { 2 } , & ~ ~ t _ 0 \\leq - 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "8403.png", "formula": "\\begin{align*} ( E _ 0 = M \\oplus K _ X ^ { n - 1 } \\oplus K _ X ^ { n - 2 } \\oplus \\cdots \\oplus K _ X ^ { 2 - n } \\oplus K _ X ^ { 1 - n } \\oplus M ^ { - 1 } , \\quad \\theta _ 0 = \\begin{pmatrix} 0 & & & & & \\\\ \\mu & 0 & & & & \\\\ & 1 & 0 & & & \\\\ & & \\ddots & \\ddots & \\\\ & & & 1 & 0 & \\\\ & & & & \\mu & 0 \\end{pmatrix} ) . \\end{align*}"}
+{"id": "7323.png", "formula": "\\begin{align*} z _ { i p } = z _ { i p } ^ 0 + \\sum _ { k = 1 } ^ { N _ 2 } u _ { k p } f _ { k i } , i = 1 , 2 , \\dots , n , z _ p ^ 0 = ( z _ { 1 p } ^ 0 , \\dots , z _ { n p } ^ 0 ) \\in S _ p ^ 2 , p \\in { \\cal P } _ 2 , \\end{align*}"}
+{"id": "3566.png", "formula": "\\begin{align*} J _ { T _ { * } } = - \\frac { 2 \\pi \\sigma ^ { 3 } } { n } T _ { * } ^ { - 3 / n } \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { k ! } \\Gamma \\left ( \\frac { k m - 3 } { n } \\right ) T _ { * } ^ { - ( n - m ) k / n } , \\end{align*}"}
+{"id": "1655.png", "formula": "\\begin{align*} g ( [ { f } , { f } ] ( X , Y ) , \\xi _ i ) = 2 \\ , g ( Q X , { f } Y ) . \\end{align*}"}
+{"id": "8488.png", "formula": "\\begin{align*} J _ { \\nu } ( x ) = \\frac { 2 ^ { \\nu + 1 } x ^ { - \\nu } } { \\sqrt { \\pi } \\Gamma \\left ( \\frac { 1 } { 2 } - \\nu \\right ) } \\ , \\intop _ { 1 } ^ { \\infty } \\frac { \\sin ( x t ) } { ( t ^ { 2 } - 1 ) ^ { \\nu + \\frac { 1 } { 2 } } } \\ , d t , \\ , \\ , \\ , \\ , \\ , | ( \\nu ) | < \\frac { 1 } { 2 } , \\ , x > 0 , \\end{align*}"}
+{"id": "2342.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\frac { \\sigma _ R ^ 2 ( t ) } { R } = \\lim _ { R \\to \\infty } 4 \\pi \\sum _ { n \\geq 2 } C _ H ^ n \\int _ { T _ n ( t ) } \\big ( \\ell _ R * g _ { \\pmb { t } _ n } ^ { ( n ) } \\big ) ( 0 ) d \\pmb { t } _ n = 4 \\pi \\sum _ { n \\geq 2 } C _ H ^ n \\int _ { T _ n ( t ) } g _ { \\pmb { t } _ n } ^ { ( n ) } ( 0 ) d \\pmb { t } _ n . \\end{align*}"}
+{"id": "3552.png", "formula": "\\begin{align*} B ( T _ { * } ) = - 2 \\pi \\int _ { 0 } ^ { \\infty } \\left [ \\exp \\left ( - \\frac { 1 } { T _ { * } } \\left [ \\left ( \\frac { \\sigma } { r } \\right ) ^ { n } - \\left ( \\frac { \\sigma } { r } \\right ) ^ { m } \\right ] \\right ) - 1 \\right ] \\ , r ^ { 2 } d r , \\end{align*}"}
+{"id": "5151.png", "formula": "\\begin{align*} \\| f \\| _ { X ( w ) } : = \\| f w \\| _ X . \\end{align*}"}
+{"id": "2835.png", "formula": "\\begin{align*} \\tilde \\Delta _ k = \\frac { \\Delta _ k } { S _ { k } ^ { k + 1 } ( t ) \\cdots S _ 0 ^ 1 ( t ) } = \\frac { a _ { k - 1 } a _ { k - 2 } ^ 2 \\cdots a _ 0 ^ { k } } { S _ { k } ^ { k + 1 } ( t ) \\cdots S _ 0 ^ 1 ( t ) } = 1 . \\end{align*}"}
+{"id": "2755.png", "formula": "\\begin{align*} u = \\Phi + R _ q ( \\lambda ) ( f + ( \\lambda - q ) \\Phi ) . \\end{align*}"}
+{"id": "7852.png", "formula": "\\begin{align*} | V | = \\left | \\bigcup _ { x H \\in \\Omega \\setminus \\mathcal { S } } \\{ ( x H , y H ) : \\ y H \\in \\mathcal { S } \\} \\right | = A ( X _ i / \\pi ) _ { 1 2 } | \\Omega \\setminus \\mathcal { S } | = A ( X _ i / \\pi ) _ { 2 1 } \\left ( | \\Omega | - | \\mathcal { S } | \\right ) . \\end{align*}"}
+{"id": "722.png", "formula": "\\begin{align*} \\Big ( A _ { S } - C _ { V } B _ { B } ^ { - 1 } D _ { V } \\Big ) [ \\partial _ { t } \\gamma _ { S } ] = G _ { V , 1 } - C _ { V } B _ { B } ^ { - 1 } G _ { V , 2 } \\ , . \\end{align*}"}
+{"id": "8894.png", "formula": "\\begin{align*} \\Bigg ( \\sum _ { i = 1 } ^ k X _ i \\Bigg ) \\Bigg ( \\sum _ { i = 1 } ^ k X _ { i + k } \\Bigg ) = \\sum _ { i = 1 } ^ { 2 k } X _ i + \\sum _ { p \\geq 2 } \\sum _ { ( i _ 1 , \\dots , i _ p ) \\in \\mathbb { I } _ { 2 k } ^ p } \\alpha _ { ( i _ 1 , \\dots , i _ p ) } [ X _ { i _ 1 } , \\dots , X _ { i _ p } ] . \\end{align*}"}
+{"id": "2956.png", "formula": "\\begin{align*} ( g , z _ 1 , z _ 2 ) \\cdot ( g , z _ 2 , z _ 3 ) = ( g , z _ 1 , z _ 3 ) \\ . \\end{align*}"}
+{"id": "6787.png", "formula": "\\begin{align*} \\rightarrow \\bigoplus _ { \\substack { \\sigma \\in W ^ { a f f } , \\\\ l ( \\sigma ) = r } } P _ { \\sigma ( \\rho ) - \\rho } \\rightarrow \\bigoplus _ { \\substack { \\sigma \\in W ^ { a f f } , \\\\ l ( \\sigma ) = r - 1 } } P _ { \\sigma ( \\rho ) - \\rho } \\rightarrow \\dots \\bigoplus _ { \\substack { \\sigma \\in W ^ { a f f } , \\\\ l ( \\sigma ) = 1 } } P _ { \\sigma ( \\rho ) - \\rho } \\rightarrow P _ 0 \\rightarrow V _ 0 . \\end{align*}"}
+{"id": "3362.png", "formula": "\\begin{align*} 1 \\geq | \\langle B \\eta , \\eta \\rangle | \\geq \\langle \\Re ( B ) \\eta , \\eta \\rangle = 1 . \\end{align*}"}
+{"id": "8033.png", "formula": "\\begin{align*} \\Omega : = W G W ^ * = [ \\Omega _ { s , t } ] \\end{align*}"}
+{"id": "8013.png", "formula": "\\begin{align*} g ( A ) & = g ( X ( A + B ) X ^ * ) \\\\ & \\leq U _ 0 X g ( A + B ) X ^ * U _ 0 ^ * \\\\ & = \\varepsilon U ^ * ( | g | ( A + B ) ) ^ { 1 / 2 } X ^ * X ( | g | ( A + B ) ) ^ { 1 / 2 } U , \\end{align*}"}
+{"id": "2928.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { n - 1 } ( 1 - q ^ \\alpha \\zeta ^ \\beta \\zeta ^ { j / n } ) = 1 - q ^ { n \\alpha } \\zeta ^ { n \\beta } . \\end{align*}"}
+{"id": "6355.png", "formula": "\\begin{align*} \\Psi _ t ( x ) = t ^ { - ( d + \\beta ) / \\alpha } \\Psi _ 1 \\big ( t ^ { - 1 / \\alpha } x \\big ) , \\end{align*}"}
+{"id": "5106.png", "formula": "\\begin{align*} D ^ { 2 } \\bar { w } = D ^ { 2 } w - \\left ( D ^ { 2 } w \\right ) _ { 1 } \\end{align*}"}
+{"id": "5975.png", "formula": "\\begin{align*} \\overline { \\overline { \\Pi } } _ { \\psi } [ u ( b ) ] f ( [ \\epsilon , x ] ) = e ^ { \\pi i \\epsilon x ^ 2 b } f ( [ \\epsilon , x ] ) ; \\end{align*}"}
+{"id": "4355.png", "formula": "\\begin{align*} \\frac { \\alpha _ { 1 } } { \\alpha _ { 2 } } = \\left ( \\frac { \\left \\Vert x - P _ { 2 } ( x ) \\right \\Vert } { \\left \\Vert x - P _ { 1 } ( x ) \\right \\Vert } \\right ) ^ { p - 1 } , \\end{align*}"}
+{"id": "9041.png", "formula": "\\begin{align*} b _ { i , j } ' = \\begin{cases} - b _ { i , j } & i = k j = k , \\\\ \\displaystyle b _ { i , j } + \\frac { | b _ { i , k } | b _ { k , j } + b _ { i , k } | b _ { k , j } | } { 2 } & . \\end{cases} \\end{align*}"}
+{"id": "5218.png", "formula": "\\begin{align*} ( 1 - \\rho ) \\phi ( z ) + \\rho \\phi ( z ) \\zeta _ \\infty ( z ) . \\end{align*}"}
+{"id": "8846.png", "formula": "\\begin{align*} d u _ \\lambda + A u _ \\lambda \\ , d t = f _ \\lambda ( u _ \\lambda ) \\ , d t + \\sigma ( u _ \\lambda ) B \\ , d W , u _ \\lambda ( 0 ) = u _ 0 \\in C ( \\overline { G } ) \\end{align*}"}
+{"id": "573.png", "formula": "\\begin{align*} \\gamma ^ { a } ( x _ { 1 } , x _ { 2 } ) = \\int _ { \\R } a \\left ( \\frac { - x _ { 1 } + y _ { 1 } } { 2 \\sqrt { x _ { 2 } } } \\right ) ( h _ 0 ( y _ { 1 } ) ) ^ { 2 } d y _ { 1 } I _ { n \\times n } . \\\\ \\end{align*}"}
+{"id": "4570.png", "formula": "\\begin{align*} \\begin{cases} & \\i \\partial _ t \\Psi _ N ( t ) = H _ N \\Psi _ N ( t ) , \\\\ & \\Psi _ N ( 0 ) = \\Psi _ { N , 0 } \\ , , \\end{cases} \\end{align*}"}
+{"id": "6455.png", "formula": "\\begin{align*} z ( s ) & = \\frac { \\sigma ^ 2 } { 2 } + \\mathcal { O } ( h ^ 2 + s \\sigma ^ 4 ) , \\\\ \\zeta ( s ) & = i ( K _ 1 + K _ 3 - K _ 2 - K ) + \\mathcal { O } ( s \\sigma ^ 2 ) \\end{align*}"}
+{"id": "8095.png", "formula": "\\begin{align*} p _ { } ( \\mathcal { D } _ a , C _ { } ) = 0 \\end{align*}"}
+{"id": "3765.png", "formula": "\\begin{align*} \\acute { \\rho } _ \\sigma ^ { a b } : = & \\frac { \\det \\acute { J } _ { a b } ^ \\sigma } { \\sqrt { \\det \\left [ \\acute { J } _ { a b } ^ { \\sigma , T } \\acute { J } _ { a b } ^ \\sigma + \\acute { J } _ { c d } ^ { \\sigma , T } \\acute { J } _ { c d } ^ \\sigma \\right ] } } , \\end{align*}"}
+{"id": "5629.png", "formula": "\\begin{align*} \\gamma _ p : = \\frac { N ( p - 2 ) + \\mu } { 2 } . \\end{align*}"}
+{"id": "718.png", "formula": "\\begin{align*} \\partial _ { t } \\mu _ { B } = A _ { B } ^ { * - 1 } \\Big [ G _ { D , 2 } - D _ { D } [ \\partial _ { t } \\mu _ { S } ] \\Big ] \\end{align*}"}
+{"id": "8931.png", "formula": "\\begin{align*} \\big \\| B _ l ^ { ( j + 1 ) } \\big \\| _ l \\leq \\sum _ { q = 2 } ^ k \\sum _ { ( p _ 1 , \\dots , p _ q ) \\in \\mathbb { I } _ { 2 k } ^ q , \\ , \\deg ( S _ { p _ 1 } ) + \\cdots + \\deg ( S _ { p _ q } ) = l } | \\alpha _ { p _ 1 , \\dots , p _ q } | 2 ^ { p _ 1 \\wedge \\cdots \\wedge p _ q } \\prod _ { i = 1 } ^ q \\tilde { Q } _ { p _ i } \\big ( \\nu _ 1 , \\dots , \\sqrt [ k ] { \\nu _ k } \\big ) , \\end{align*}"}
+{"id": "2676.png", "formula": "\\begin{align*} h ( \\sigma ( x ) ) = h ( x ) \\\\ h ( x _ 1 + \\dots + x _ r ) \\leq h ( x _ 1 ) + \\dots + h ( x _ r ) + \\log r \\\\ h ( x _ 1 \\dots x _ r ) \\leq h ( x _ 1 ) + \\dots + h ( x _ r ) . \\end{align*}"}
+{"id": "827.png", "formula": "\\begin{align*} \\begin{gathered} \\sum _ { M = 0 } ^ \\infty M ^ { \\# S - 1 - k } e ^ { - M } \\sum _ { N = 0 } ^ { \\lfloor w \\rfloor + M } N ^ k \\leq \\sum _ { M = 0 } ^ \\infty M ^ { \\# S - 1 - k } e ^ { - M } ( w + M ) ^ { k + 1 } \\\\ \\lesssim \\sum _ { M = 0 } ^ \\infty M ^ { \\# S } e ^ { - M } \\end{gathered} \\end{align*}"}
+{"id": "7724.png", "formula": "\\begin{align*} & \\{ \\mu \\in M ' / K \\mid \\pi _ { K _ 1 } ( p _ K ( \\mu ) ) = \\lambda / n \\} \\\\ & \\simeq \\{ ( \\gamma , \\delta ) \\in L ' _ 1 \\times M ' / L \\mid n | \\gamma , \\ \\gamma | _ { K _ 1 } = \\lambda , \\ \\pi ( \\delta ) = \\gamma / n \\} , \\end{align*}"}
+{"id": "2013.png", "formula": "\\begin{align*} \\eta _ 1 ^ n = \\frac { E ( w ) } { I _ g ( w ) } \\cdot \\end{align*}"}
+{"id": "5122.png", "formula": "\\begin{align*} \\int _ { \\Omega ' } \\beta ^ { i j , k l } u ^ { h _ m } _ { i j } \\eta _ { k l } d x = 0 \\forall \\eta \\in C _ 0 ^ \\infty ( \\Omega ' ) \\end{align*}"}
+{"id": "7651.png", "formula": "\\begin{align*} \\max M ( c , x , 1 ) = 1 0 2 4 \\end{align*}"}
+{"id": "1014.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ 2 } { x ^ k } ( 2 H _ { 2 k } - H _ k ) = \\frac { 1 } { 2 } \\bigg ( \\log \\frac { x } { x - 1 6 } \\bigg ) \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ 2 } { x ^ k } , \\end{align*}"}
+{"id": "6426.png", "formula": "\\begin{align*} \\widetilde { D } _ { \\alpha , z } ( \\psi | | \\varphi ) : = \\frac { 1 } { \\alpha - 1 } \\log \\frac { \\widetilde { Q } _ { \\alpha , z } ( \\psi | | \\varphi ) } { \\psi ( 1 ) } \\end{align*}"}
+{"id": "5516.png", "formula": "\\begin{align*} P _ { \\mu , x } \\left ( L _ { x } g , A \\right ) = \\int _ { G } { \\bf 1 } _ { A } \\left ( L _ { x } g s \\right ) \\frac { \\varphi _ { g s } ( \\pi ( x ) ) } { \\varphi _ { g } ( \\pi ( x ) ) } d \\mu ( s ) , \\end{align*}"}
+{"id": "5442.png", "formula": "\\begin{align*} M _ b ( \\theta ) = \\mathbb { E } _ { \\Phi | \\Phi ( \\mathcal { A } ) > 0 } \\{ P _ s ^ b ( \\theta ) \\} \\Pr ( \\Phi ( \\mathcal { A } ) > 0 ) . \\end{align*}"}
+{"id": "8580.png", "formula": "\\begin{align*} & S _ { j , + } ^ k ( t ) : = \\sum _ { i \\in { \\cal M } _ t } 1 _ { \\{ I _ { j , + } ^ i ( t ) \\geq k \\} } , S _ { j , - } ^ k ( t ) : = \\sum _ { i \\in { \\cal M } _ t } 1 _ { \\{ I _ { j , - } ^ i ( t ) \\geq k \\} } . \\end{align*}"}
+{"id": "7627.png", "formula": "\\begin{align*} \\begin{cases} g _ 1 ( c , \\delta ) : = \\quad \\delta ^ 2 ( 4 - c ^ 2 ) ^ 2 ( 2 c ^ 2 - ( 3 6 - 1 3 c ^ 2 ) \\delta ) + 2 c ^ 2 \\delta ^ 2 ; \\\\ g _ 2 ( c , \\delta ) : = - 8 c \\delta ( 4 - c ^ 2 ) ^ 2 ( 1 + \\delta ) ( 1 - | \\delta | ^ 2 ) ; \\\\ g _ 3 ( c , \\delta ) : = - 8 ( 4 - c ^ 2 ) ^ 2 ( 8 + | \\delta | ^ 2 ) ( 1 - | \\delta | ^ 2 ) ; \\\\ v ( c , \\delta , \\eta ) : = \\quad 7 2 \\delta ( 4 - c ^ 2 ) ^ 2 ( 1 - | \\delta | ^ 2 ) ( 1 - | \\eta | ^ 2 ) . \\end{cases} \\end{align*}"}
+{"id": "785.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ n } u \\mathrm { d } A = - \\dfrac { n } { 2 } \\dfrac { \\phi ' ( \\rho ) } { \\phi ( \\rho ) } \\rho \\int _ { \\mathbb { S } ^ n } u ^ 2 \\mathrm { d } A + O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 , \\end{align*}"}
+{"id": "38.png", "formula": "\\begin{align*} \\psi ( X ) = \\mathbf { K } \\psi ( X ) ^ { \\square } \\end{align*}"}
+{"id": "1887.png", "formula": "\\begin{align*} \\mathrm { K e r } ( \\lambda ^ * ) \\cap \\overline { R ( S ) } = Y \\subset \\mathrm { K e r } ( \\tilde { \\lambda } ) \\cap \\overline { R ( S ) } . \\end{align*}"}
+{"id": "8088.png", "formula": "\\begin{align*} < \\mathcal { R } A , B > = ( A , B ) _ { H _ 0 ^ 1 ( \\mathcal { D } _ a ; K ) } , \\forall A , B \\in H _ 0 ^ 1 ( \\mathcal { D } _ a ; K ) . \\end{align*}"}
+{"id": "1730.png", "formula": "\\begin{align*} \\nu _ t ( x ) = e ^ { - \\alpha t } \\nu _ 0 ( x ) + \\int _ 0 ^ t \\alpha e ^ { - \\alpha ( t - s ) } \\Psi ( \\nu _ s , \\mu _ s ) ( x ) \\mathrm { d } s , \\end{align*}"}
+{"id": "3908.png", "formula": "\\begin{align*} \\widehat { \\mathcal { I } } _ { \\mathrm { D } } ( \\delta _ 1 , \\delta _ 2 ) = \\sup _ { \\gamma \\in \\widehat { \\Sigma } _ { \\mathrm { D } } ( \\delta _ 1 , \\delta _ 2 ) } \\int _ { \\mathcal { V } } g ( s _ 1 , s _ 2 ) \\ , d \\gamma ( s _ 1 , s _ 2 ) . \\end{align*}"}
+{"id": "6184.png", "formula": "\\begin{align*} \\frac { n - 1 } { \\sqrt { 1 + | D _ \\tau \\xi | ^ 2 } } - { \\rm d i v } _ \\tau \\bigg ( \\frac { D _ \\tau \\xi } { \\sqrt { 1 + | D _ \\tau \\xi | ^ 2 } } \\bigg ) - \\mu e ^ { \\xi } - R e ^ { \\xi } = 0 . \\end{align*}"}
+{"id": "7900.png", "formula": "\\begin{align*} \\sigma \\left ( ( S , \\underline { S } ) ^ * \\right ) : = \\begin{cases} ( \\sigma ( S ) , \\sigma ( \\underline { S } ) ) ^ { \\mathsf { s g n } ( \\sigma ( \\hat { S } ) , \\sigma ( \\underline { \\hat { S } } ) ) } & \\mbox { i f } * = + , \\\\ ( \\sigma ( S ) , \\sigma ( \\underline { S } ) ) ^ { - \\mathsf { s g n } ( \\sigma ( \\hat { S } ) , \\sigma ( \\underline { \\hat { S } } ) ) } & \\mbox { i f } * = - . \\end{cases} \\end{align*}"}
+{"id": "3036.png", "formula": "\\begin{align*} \\phi \\left ( \\bigotimes \\limits _ { i = 1 } ^ { m - 1 } X _ i E _ { j _ i j _ i } X _ i ^ * \\otimes B \\right ) = \\left ( Z _ { X } \\left ( \\bigotimes \\limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \\right ) Z _ { X } ^ * \\right ) \\otimes \\varphi _ { j _ 1 , \\ldots , j _ { m - 1 } , X } ( B ) \\end{align*}"}
+{"id": "379.png", "formula": "\\begin{align*} d ( \\tau , j ) ^ { \\delta } _ { j } = \\prod _ { \\tau \\subseteq \\sigma } ( c ( \\sigma , j ) ^ { \\delta } _ { j } ) ^ { ( - 1 ) ^ { | \\sigma \\setminus \\tau | } } = \\prod _ { \\tau \\subseteq \\sigma \\subseteq \\delta } ( c ( \\sigma , j ) ^ { \\delta } _ { j } ) ^ { ( - 1 ) ^ { | \\sigma \\setminus \\tau | } } . \\end{align*}"}
+{"id": "7512.png", "formula": "\\begin{align*} Z ^ H | _ { W _ i } = \\begin{pmatrix} \\zeta _ i & 0 \\\\ 0 & - \\zeta _ i - \\sum _ { j \\neq i } { a _ { j i } } \\zeta _ j \\end{pmatrix} . \\end{align*}"}
+{"id": "278.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - y ^ m z ^ n } \\right ) ^ { \\frac { m ^ 3 } { n ^ 4 } } = \\left ( 1 - y z \\right ) ^ { \\frac { y } { 1 - y } } \\end{align*}"}
+{"id": "6346.png", "formula": "\\begin{align*} D _ \\epsilon : = \\{ ( s , t ) \\in [ 0 , 1 ] ^ 2 \\ , : \\ , | t - s | < \\epsilon \\} , \\end{align*}"}
+{"id": "5132.png", "formula": "\\begin{align*} \\Delta _ g & = \\frac { 1 } { \\sqrt { g } } \\partial _ i ( \\sqrt { g } g ^ { i j } \\partial _ j ) \\\\ & = g ^ { i j } \\partial _ { i j } + \\frac { 1 } { \\sqrt { g } } \\partial _ i ( \\sqrt { g } g ^ { i j } ) \\partial _ j \\\\ & = g ^ { i j } \\partial _ { i j } - g ^ { j p } \\Theta _ q u _ { p q } \\partial _ j . \\end{align*}"}
+{"id": "8748.png", "formula": "\\begin{align*} \\frac { \\Delta _ 0 } { \\surd { ( \\Delta _ { 1 } ) } } = \\frac { \\| \\mu _ 1 - \\mu _ 2 \\| _ 2 ^ 2 } { \\surd { \\frac { 2 } { n ( n - 1 ) } ( \\Sigma _ { 1 } ^ { 2 } ) + \\frac { 2 } { m ( m - 1 ) } ( \\Sigma _ { 2 } ^ { 2 } ) + \\frac { 4 } { n m } ( \\Sigma _ { 1 } \\Sigma _ { 2 } ) } } \\{ 1 + o ( 1 ) \\} , \\end{align*}"}
+{"id": "698.png", "formula": "\\begin{align*} S ( k ) \\varphi = S ( k ) ( \\widetilde { v } _ 0 \\circ T - \\widetilde { v } _ 0 ) + Q ( k ) \\chi . \\end{align*}"}
+{"id": "3297.png", "formula": "\\begin{align*} \\widetilde { \\rho } ( - i ^ - , w ) = - \\widetilde { \\rho } ( - i ^ + , w ) \\ \\ { \\rm a n d } \\ \\ \\widetilde { \\rho } _ w ( - i ^ - , w ) = - \\widetilde { \\rho } _ w ( - i ^ + , w ) , \\end{align*}"}
+{"id": "1499.png", "formula": "\\begin{align*} - \\Sigma _ 1 = \\Sigma _ 1 , \\quad - \\Sigma _ 2 = \\Sigma _ 3 , \\quad \\Sigma _ 1 \\cup \\Sigma _ 2 \\cup \\Sigma _ 3 = S ^ 2 , \\end{align*}"}
+{"id": "4096.png", "formula": "\\begin{align*} \\left ( Q _ { \\ell , \\vec { v } } \\right ) _ i M _ { \\bullet , \\ell } = \\left ( \\mathcal { Q } _ { \\ell , \\vec { v } } \\left ( M _ { \\bullet , \\ell } \\right ) \\right ) _ { \\bullet , i } . \\end{align*}"}
+{"id": "455.png", "formula": "\\begin{align*} E ( t ) : = \\frac 1 2 \\chi _ t ^ 2 - \\frac { G ( r ) } { \\chi } = E ( 0 ) \\end{align*}"}
+{"id": "3906.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { \\ell < m } K _ { \\ell } \\right ) \\cap K _ { m } = \\bigcup _ { \\ell < m } ( K _ { \\ell } \\cap K _ { m } ) = ( K _ { 1 } \\cap K _ { m } ) \\in 2 ^ { K _ { 1 } } \\subset \\bigcup _ { \\ell < m } 2 ^ { K _ { \\ell } } . \\end{align*}"}
+{"id": "7120.png", "formula": "\\begin{align*} \\big \\| \\mathcal { R } _ \\varepsilon c \\big \\| _ { L ^ 2 ( \\Omega ^ \\prime ) } ^ 2 & \\leq K \\int _ { \\Omega ^ \\prime } \\left | \\int _ { \\Omega ^ c } J _ \\varepsilon ( | x - y | ) c ( x ) \\ : y \\right | ^ 2 \\ : x + K \\int _ { \\Omega ^ \\prime } \\left | \\int _ { \\Omega ^ c } J _ \\varepsilon ( | x - y | ) \\tilde { c } ( y ) \\ : y \\right | ^ 2 \\ : x \\\\ & = : K \\big ( I _ \\varepsilon ^ 1 + I _ \\varepsilon ^ 2 \\big ) . \\end{align*}"}
+{"id": "5146.png", "formula": "\\begin{align*} \\| f \\| _ { X } = \\sup _ { \\| g \\| _ { X ' } = 1 } \\| f g \\| _ { L ^ 1 ( \\Omega ) } . \\end{align*}"}
+{"id": "2075.png", "formula": "\\begin{align*} F ( A ) & = F ( a , B ) = \\max _ { r \\in \\lbrace 0 , 1 , . . . , a - 1 \\rbrace } N _ r - a , \\\\ g ( A ) & = g ( a , B ) = \\frac { 1 } { a } \\sum _ { r = 1 } ^ { a - 1 } N _ r - \\frac { a - 1 } { 2 } . \\end{align*}"}
+{"id": "2851.png", "formula": "\\begin{align*} \\psi \\circ \\varphi _ { L _ 1 } = \\varphi _ { L _ 2 } \\circ \\psi . \\end{align*}"}
+{"id": "3468.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda _ n } g _ n ' ( 0 , 0 ) = \\rho - | \\alpha | \\frac { d } { d \\lambda _ n } h _ n ' ( 0 ) . \\end{align*}"}
+{"id": "7690.png", "formula": "\\begin{align*} \\vartheta _ L ( f ) ( g ) = \\sum _ { h \\in A } \\vartheta _ L ( f ; h ) ( g ) h ^ \\vee \\end{align*}"}
+{"id": "4145.png", "formula": "\\begin{align*} Q _ { 2 , \\vec { v } _ { s , t , f , 0 } } = \\end{align*}"}
+{"id": "973.png", "formula": "\\begin{align*} \\overline { \\nabla } _ { X } Y = \\nabla _ { X } Y + h ( X , Y ) \\end{align*}"}
+{"id": "8499.png", "formula": "\\begin{align*} \\frac { 1 } { \\left ( x ^ { 2 } + t ^ { 2 } \\right ) ^ { s } } = \\frac { \\sqrt { \\pi } \\ , 2 ^ { \\frac { 1 } { 2 } - s } } { \\Gamma ( s ) t ^ { s - \\frac { 1 } { 2 } } } \\intop _ { 0 } ^ { \\infty } y ^ { s - \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( t y ) \\ , e ^ { - x y } \\ , d y , \\end{align*}"}
+{"id": "5073.png", "formula": "\\begin{align*} v ( x , t ) : = \\psi \\left ( x - \\gamma ( x , t ) - \\frac { 1 } { N } \\sigma ( t ) , t \\right ) - \\phi ( x ) , \\end{align*}"}
+{"id": "1460.png", "formula": "\\begin{align*} - \\Delta \\tilde { u } ^ i = f _ 0 ^ { ' } ( U _ { 1 , \\sigma _ i } ) \\tilde { u } ^ i \\quad \\mbox { i n } \\ \\R ^ n , \\end{align*}"}
+{"id": "8686.png", "formula": "\\begin{align*} _ { n , m } ^ { 2 } = \\Delta _ { 0 } + \\sum ^ { l - 1 } _ { s = 1 } \\Delta _ { s } + \\widetilde { \\Delta } _ { l } , \\end{align*}"}
+{"id": "2492.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle 2 \\left ( \\dfrac { a _ 2 } { 2 m _ 1 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\xi | ^ 2 d x + \\dfrac { a _ 1 } { 4 m _ 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\eta | ^ 2 d x \\right ) - \\dfrac { N } { 2 } a _ 1 a _ 2 \\int _ { \\mathbb { R } ^ N } \\eta \\xi ^ 2 d x = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "3181.png", "formula": "\\begin{align*} M _ n ( f ) : = \\frac { 1 } { n ^ d } \\sum _ { ( j _ 1 , \\ldots , j _ d ) \\in \\Z ^ d _ + \\cap Q _ n } \\int _ { Q _ { j _ 1 , \\ldots , j _ d } } T _ t ( f ) d t , \\end{align*}"}
+{"id": "8934.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { k - 1 } y \\big ( \\bigl \\{ t X _ { n i } ^ { ( j ) } \\bigr \\} \\big ) = \\sum _ { j = 1 } ^ { k - 1 } t ^ j Z _ j + B _ k ^ { ( k - 1 ) } . \\end{align*}"}
+{"id": "7986.png", "formula": "\\begin{align*} \\mathrm { d i v } _ T \\Big ( \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\big ) \\Big ) \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\cdot \\nu & - \\nabla _ T \\big [ \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\big ] \\ , \\big [ \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\big ] _ T \\cdot \\nu \\\\ & = H ^ 2 ( \\nabla v ) \\ , H ( \\nu ) \\ , \\mathrm { t r } \\ , \\mathcal { B } ^ H \\ , . \\end{align*}"}
+{"id": "7448.png", "formula": "\\begin{align*} \\partial _ t { w } ^ { ( i ) } _ { \\alpha - 1 } ( x _ i , t ) \\ , + \\ , \\big ( v _ i ^ { ( i ) } ( x _ i ) \\ , w ^ { ( i ) } _ { \\alpha - 1 } ( x _ i , t ) \\big ) ^ \\prime = - \\widehat { \\varphi } ^ { ( i ) } ( x _ i , t ) , ( x _ i , t ) \\in I _ \\varepsilon ^ { ( i ) } \\times ( 0 , T ) , \\end{align*}"}
+{"id": "1823.png", "formula": "\\begin{align*} V _ F ( t ) = \\int _ { \\Lambda ^ { * 4 } } \\bigg [ \\int _ { \\Lambda ^ { * } } \\hat V ( k ) d _ t ( - k , p _ 1 , q _ 1 ) \\ , d _ t ( k , p _ 2 , q _ 2 ) \\d k \\bigg ] a _ { p _ 1 } ^ * a _ { q _ 1 } a _ { p _ 2 } ^ * a _ { q _ 2 } \\d p _ 1 \\d p _ 2 \\d q _ 1 \\d q _ 2 \\ . \\end{align*}"}
+{"id": "2967.png", "formula": "\\begin{align*} l _ z ( x ) - l _ w ( x ) = \\begin{cases} - 1 & z < x < w \\ , \\\\ 1 & w < x < z \\ , \\\\ 0 & \\ . \\end{cases} \\end{align*}"}
+{"id": "4381.png", "formula": "\\begin{align*} \\mathcal { Q } ( s ) = \\sum _ { e ^ { \\ell } \\leq \\log p \\leq e ^ { n _ { \\mathcal { L } } } } \\left ( \\frac { a ( p ) } { p ^ s } + \\frac { b ( p ) } { p ^ { 2 s } } \\right ) , \\end{align*}"}
+{"id": "3311.png", "formula": "\\begin{align*} f _ 0 ( \\xi ) = ( \\xi - 1 ) ^ { \\delta _ 1 } ( \\xi + 1 ) ^ { \\delta _ { - 1 } } \\kappa _ 0 ( \\xi ) + w _ 1 \\frac { 1 + \\psi ( \\xi ) } { 2 } + w _ { - 1 } \\frac { 1 - \\psi ( \\xi ) } { 2 } , \\end{align*}"}
+{"id": "4986.png", "formula": "\\begin{align*} [ P ] _ { i , j } ^ { } = \\frac { 1 } { 2 } \\times \\begin{dcases} 1 & j = i \\neq n ; \\\\ 2 & j = i = n ; \\\\ 1 & j = i + 1 ; \\\\ 0 & . \\end{dcases} \\end{align*}"}
+{"id": "5281.png", "formula": "\\begin{align*} | 1 + y | ^ q = 1 + q y + \\binom { q } { 2 } y ^ 2 + \\binom { q } { 3 } | 1 + y ' | ^ { q - 3 } ( 1 + y ' ) y ^ 3 , \\end{align*}"}
+{"id": "1595.png", "formula": "\\begin{align*} t _ n \\le 2 ^ r \\ , t _ { n - 1 } , \\mbox { f o r } \\ ; 2 \\le n \\le N . \\\\ \\end{align*}"}
+{"id": "8268.png", "formula": "\\begin{align*} b _ { q , l - 2 q } ^ { ( l ) } = \\frac { 1 } { l } b _ { q , l - 2 q - 1 } ^ { ( l - 1 ) } + \\frac { l - k - 2 } { l } b _ { q - 1 , l - 2 q } ^ { ( l - 2 ) } , \\end{align*}"}
+{"id": "4120.png", "formula": "\\begin{align*} M \\vec { v } = \\varepsilon \\vec { v } , \\end{align*}"}
+{"id": "1099.png", "formula": "\\begin{align*} b \\ast ( b ' \\ast x ) = 0 , \\forall b , b ' \\in B , \\ ; \\forall x \\in X , \\end{align*}"}
+{"id": "8242.png", "formula": "\\begin{align*} G ( 1 + z ) = ( 2 \\pi ) ^ { z / 2 } \\exp \\left ( - \\frac { z + z ^ 2 ( 1 + \\gamma ) } { 2 } \\right ) \\ , \\prod _ { k = 1 } ^ \\infty \\left \\{ \\left ( 1 + \\frac { z } { k } \\right ) ^ k \\exp \\left ( \\frac { z ^ 2 } { 2 k } - z \\right ) \\right \\} . \\end{align*}"}
+{"id": "8889.png", "formula": "\\begin{gather*} \\phi _ 2 ( X _ 1 \\otimes X _ 2 ) = [ X _ 1 , X _ 2 ] , \\\\ \\phi _ i ( X _ 1 \\otimes \\cdots \\otimes X _ i ) = [ X _ 1 , \\phi _ { i - 1 } ( X _ 2 \\otimes \\cdots \\otimes X _ i ) ] , i \\geq 3 . \\end{gather*}"}
+{"id": "1314.png", "formula": "\\begin{align*} H _ \\beta = 2 \\beta K _ { 1 / 2 \\beta } = 2 \\beta \\widetilde { K } ^ { ( u ) } K ^ { ( u ) } = 2 \\beta \\widetilde { T } ( u ) \\widetilde { H } ^ { ( u ) } K ^ { ( u ) } , \\end{align*}"}
+{"id": "5288.png", "formula": "\\begin{align*} 1 + q \\rho d X + \\binom { q } { 2 } \\rho ^ 2 d ^ 2 X ^ 2 + \\frac { q ^ 3 } { 6 ( \\alpha q - 1 ) } e ^ { \\frac { ( q - 3 ) } { \\alpha q - 1 } } \\rho ^ 2 d ^ 2 | X | ^ 2 . \\end{align*}"}
+{"id": "6509.png", "formula": "\\begin{align*} \\det \\widetilde { T } _ { c , a } ( b ) = \\prod _ { i = 1 } ^ c \\prod _ { j = 1 } ^ a \\frac { ( b + i - j ) ( b + 2 i + j - 1 ) } { ( b + 2 i - j - 1 ) ( i + j - 1 ) } . \\end{align*}"}
+{"id": "5778.png", "formula": "\\begin{align*} | K _ { \\alpha } ( z , \\vec { y } ) - K _ { \\alpha } ( x , \\vec { y } ) | & \\le \\dfrac { A } { \\Big ( \\sum \\limits _ { j = 1 } ^ { m } | z - y _ { j } | \\Big ) ^ { m n - \\alpha } } \\omega \\bigg ( \\frac { | z - x | } { \\sum \\limits _ { j = 1 } ^ { m } | z - y _ { j } | } \\bigg ) \\le \\dfrac { C \\omega ( 2 ^ { - k } ) } { | 2 ^ { k } \\sqrt { n } Q | ^ { m - \\alpha / n } } . \\end{align*}"}
+{"id": "8900.png", "formula": "\\begin{align*} Y ( \\{ X _ { n i } \\} ) = \\sum _ { n = 1 } ^ N [ X _ { n 1 } , \\dots , X _ { n j } ] . \\end{align*}"}
+{"id": "2660.png", "formula": "\\begin{align*} & s _ \\alpha ' ( a ) = y ' _ 0 , s _ \\alpha ' ( b ) = y ' _ N , \\\\ & s _ \\alpha '' ( a ) = s _ \\alpha '' ( b ) = 0 , \\\\ & s _ \\alpha '' ( a ) = y '' _ 0 , s _ \\alpha '' ( b ) = y '' _ N . \\end{align*}"}
+{"id": "656.png", "formula": "\\begin{align*} K ^ { a , i , \\tau } _ { k } ( T ) & \\leq C _ \\tau e ^ { ( \\lambda _ { 1 } a + 5 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k ) } , \\\\ C ^ { a , i , \\tau } _ { k } ( T ) & \\leq C _ \\tau e ^ { ( \\max \\{ \\lambda _ { i } , \\lambda _ 1 a \\} + 3 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k ) } . \\end{align*}"}
+{"id": "9115.png", "formula": "\\begin{align*} \\begin{array} { c c c } x ^ { + } & = & f ( x , u ) \\\\ \\zeta & = & g ( x , u ) \\end{array} \\end{align*}"}
+{"id": "7654.png", "formula": "\\begin{align*} \\top & = ( a \\lor \\neg a ) ^ k \\to ( a \\to b ) = ( a ^ k \\lor ( \\neg a ) ^ k ) \\to ( a \\to b ) \\\\ & = ( a ^ k \\to ( a \\to b ) ) \\land ( ( \\neg a ) ^ k \\to ( a \\to b ) ) = a ^ { k + 1 } \\to b , \\end{align*}"}
+{"id": "2304.png", "formula": "\\begin{align*} w _ b = \\Phi _ { 0 , b } + \\Psi _ { b } , \\end{align*}"}
+{"id": "2770.png", "formula": "\\begin{align*} u _ 1 u _ 2 = ( u _ 1 - v _ 1 ) u _ 2 + ( v _ 1 - u _ 1 ) ( u _ 2 - v _ 2 ) + u _ 1 ( u _ 2 - v _ 2 ) + v _ 1 v _ 2 , \\end{align*}"}
+{"id": "5067.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\| { S } _ { N } ( t ) v \\right \\| _ { L _ N ^ 2 } \\leq C ( 1 + t ) ^ { - \\frac { 3 } { 4 } } \\| v \\| _ { L _ N ^ 1 \\cap L _ N ^ 2 } , & & v \\in L _ N ^ 2 , \\end{aligned} \\end{align*}"}
+{"id": "2395.png", "formula": "\\begin{align*} \\vec { u } ( A _ { 0 } , A _ { 4 } ) = - \\vec { u } ( A _ { 0 } , A _ { 1 } ) + \\vec { u } ( A _ { 0 } , A _ { 2 } ) + \\vec { u } ( A _ { 0 } , A _ { 3 } ) ) . \\end{align*}"}
+{"id": "1030.png", "formula": "\\begin{align*} \\lim _ { a \\to \\frac { 1 } { 2 } } \\frac { \\psi ( \\frac { 1 - a + b } { 2 } ) - \\psi ( \\frac { a + b } { 2 } ) } { 1 - 2 a } = \\frac { \\psi \\ , ' ( \\frac { 1 + 2 b } { 4 } ) } { 2 } . \\end{align*}"}
+{"id": "7175.png", "formula": "\\begin{align*} \\nu : = \\frac { x - y } { \\abs { x - y } } \\end{align*}"}
+{"id": "1102.png", "formula": "\\begin{align*} \\varphi ( b ) = \\varphi ( b ' ) = 1 _ { X } \\end{align*}"}
+{"id": "5402.png", "formula": "\\begin{align*} \\sum _ { i \\ge 1 } i \\widetilde { A } _ i u ^ { i } = F ( u ) G ( u ) , \\end{align*}"}
+{"id": "7117.png", "formula": "\\begin{align*} C _ n = \\int _ { \\mathbb { S } ^ { n - 1 } } | e _ 1 \\cdot \\sigma | ^ 2 \\ : \\mathcal { H } ^ { n - 1 } ( \\sigma ) = \\int _ { \\mathbb { S } ^ { n - 1 } } | e _ j \\cdot \\sigma | ^ 2 \\ : \\mathcal { H } ^ { n - 1 } ( \\sigma ) = \\frac { 1 } { n } \\int _ { \\mathbb { S } ^ { n - 1 } } | \\sigma | ^ 2 \\ : \\mathcal { H } ^ { n - 1 } ( \\sigma ) = \\frac { \\omega _ n } { n } . \\end{align*}"}
+{"id": "5163.png", "formula": "\\begin{align*} \\begin{aligned} D ( p _ { \\mu _ 1 , \\sigma _ 1 , h } | | p _ { \\mu _ 2 , \\sigma _ 2 , h } ) & = D ( h ( x ) f _ { \\mu _ 1 , \\sigma _ 1 } ( y ) | | h ( x ) f _ { \\mu _ 2 , \\sigma _ 2 } ( y ) ) \\\\ & \\le c ( \\left \\| { \\mu _ 1 - \\mu _ 2 } \\right \\| ^ 2 _ { 2 , h } + \\left \\| { \\sigma _ 1 - \\sigma _ 2 } \\right \\| ^ 2 _ { 2 , h } ) \\\\ & \\le c C ( \\left \\| { \\mu _ 1 - \\mu _ 2 } \\right \\| ^ 2 _ { 2 , h _ 0 } + \\left \\| { \\sigma _ 1 - \\sigma _ 2 } \\right \\| ^ 2 _ { 2 , h _ 0 } ) , \\end{aligned} \\end{align*}"}
+{"id": "3293.png", "formula": "\\begin{align*} f ( \\xi ) = ( \\xi - 1 ) ^ { \\delta _ 1 } ( \\xi + 1 ) ^ { \\delta _ { - 1 } } \\kappa ( \\xi ) + w _ 1 \\frac { 1 + \\psi ( \\xi ) } { 2 } + w _ { - 1 } \\frac { 1 - \\psi ( \\xi ) } { 2 } , \\end{align*}"}
+{"id": "5896.png", "formula": "\\begin{align*} \\abs { F _ \\mu ( x ) } ^ { 1 / p } = \\sup \\{ \\abs { f ( x ) - f ( 0 ) } \\ , : \\ , f \\in W _ { 1 , q } ( \\mu ) \\} . \\end{align*}"}
+{"id": "8903.png", "formula": "\\begin{align*} [ X _ { n 1 } , X _ { n 2 } , X _ { n 3 } ] _ c & = [ X _ { n 1 } , [ X _ { n 2 } , X _ { n 3 } ] _ c ] _ c \\\\ & = X _ { n 1 } \\cdot [ X _ { n 2 } , X _ { n 3 } ] _ c \\cdot X _ { n 1 } ^ { - 1 } \\cdot [ X _ { n 2 , X _ { n 3 } } ] _ c ^ { - 1 } \\\\ & = X _ { n 1 } \\cdot \\bigl ( X _ { n 2 } \\cdot X _ { n 3 } \\cdot X _ { n 2 } ^ { - 1 } \\cdot X _ { n 3 } ^ { - 1 } \\bigr ) \\cdot X _ { n 1 } ^ { - 1 } \\cdot \\bigl ( X _ { n 3 } \\cdot X _ { n 2 } \\cdot X _ { n 3 } ^ { - 1 } \\cdot X _ { n 2 } ^ { - 1 } \\bigr ) . \\end{align*}"}
+{"id": "3000.png", "formula": "\\begin{align*} r ( t ) = \\Pi _ 0 ^ t \\exp f ( s ) \\ , d s . \\end{align*}"}
+{"id": "5175.png", "formula": "\\begin{align*} c r ^ d ( 3 / 8 ) ^ { k } \\mu \\le \\sum _ { \\| h \\| \\le r } \\sum _ { e \\in E ( \\mathbb Z ^ d ) } q _ e p _ { e - h } = \\sum _ { e \\in E ( \\mathbb Z ^ d ) } q _ e \\cdot \\mathbb E [ | \\gamma \\cap \\Lambda _ e | ] \\le C r \\sum _ { e \\in E ( \\mathbb Z ^ d ) } q _ e , \\end{align*}"}
+{"id": "6090.png", "formula": "\\begin{align*} \\vert S _ { t } x \\vert _ { \\beta + { \\sigma } _ 1 } & \\lesssim t ^ { - { \\sigma } _ 1 } \\vert x \\vert _ { { \\beta } } , \\\\ \\vert ( I - S _ { t } ) x \\vert _ { { \\beta } } & \\lesssim t ^ { { \\sigma } _ 1 } \\vert x \\vert _ { \\beta + { \\sigma } _ 1 } . \\end{align*}"}
+{"id": "7664.png", "formula": "\\begin{align*} | \\xi | ^ { p } \\geq | \\eta | ^ { p } + p | \\eta | ^ { p - 2 } ( \\xi - \\eta ) \\eta = p | \\eta | ^ { p - 2 } \\xi \\eta + ( 1 - p ) | \\eta | ^ { p } , \\qquad \\forall \\xi , \\eta , \\end{align*}"}
+{"id": "7165.png", "formula": "\\begin{align*} \\norm { u } _ { W ^ { 1 , p ( \\cdot ) } ( \\Omega ) } : = \\norm { u } _ { L ^ { p ( \\cdot ) } ( \\Omega ) } + \\norm { \\nabla u } _ { L ^ { p ( \\cdot ) } ( \\Omega ) } , \\end{align*}"}
+{"id": "9249.png", "formula": "\\begin{align*} M _ n & = \\sum _ { j = 0 } ^ n ( n - j + 4 ) \\frac { 4 } { \\pi } 2 ^ { 2 j } \\frac { 1 } { ( n - j ) ! } \\left ( \\frac { \\pi } { 2 } \\right ) ^ { n - j } < \\Bigl ( \\frac { 1 6 } { \\pi } + \\frac 1 2 \\Bigr ) e ^ { \\pi / 8 } 2 ^ { 2 n } < 8 . 3 \\cdot 2 ^ { 2 n } , \\\\ N _ n & = \\sum _ { j = 0 } ^ n \\frac { 4 } { \\pi } 2 ^ { 2 j } \\frac { 1 } { ( n - j ) ! } \\left ( \\frac { \\pi } { 2 } \\right ) ^ { n - j } < \\frac { 4 } { \\pi } e ^ { \\pi / 8 } 2 ^ { 2 n } < 1 . 9 \\cdot 2 ^ { 2 n } . \\end{align*}"}
+{"id": "2793.png", "formula": "\\begin{align*} \\mathfrak { I } : H ^ { - 1 / 2 } _ \\Gamma ( \\partial \\mathrm { M } ) \\rightarrow H ^ { 1 / 2 } _ \\Gamma ( \\partial \\mathrm { M } ) : \\langle u , v \\rangle = ( \\mathfrak { I } u | v ) _ { H ^ { 1 / 2 } _ \\Gamma ( \\mathrm { M } ) } , \\end{align*}"}
+{"id": "6696.png", "formula": "\\begin{align*} S a = - a S \\nabla \\eta a - r a - a . \\end{align*}"}
+{"id": "1951.png", "formula": "\\begin{align*} ( \\det b ) ^ \\frac { 1 } { n } = \\frac { 1 } { n } \\inf \\{ t r ( a \\cdot b ) : ~ a \\in \\mathcal { H } _ n \\} , \\end{align*}"}
+{"id": "5879.png", "formula": "\\begin{align*} { L } _ { u } \\ , S ^ * f _ m = ( \\nu _ m - 1 ) S ^ * f _ m \\ , . \\end{align*}"}
+{"id": "104.png", "formula": "\\begin{align*} \\Omega _ + = \\{ ( e ^ { t H _ p } ( y , \\eta ) , y , \\eta ) \\ , : \\ , p ( y , \\eta ) = 0 , y \\in \\mathcal { U } , \\varphi ^ t ( y ) \\in \\mathcal { U } , t \\geq 0 \\} . \\end{align*}"}
+{"id": "2078.png", "formula": "\\begin{align*} o p t _ B ( M ) : = \\min \\left \\{ \\sum _ { i = 1 } ^ k x _ i \\mid \\sum _ { i = 1 } ^ k b _ i x _ i = M , \\ \\ M , x _ i \\in \\mathbb { N } , 1 \\leq i \\leq k \\right \\} . \\end{align*}"}
+{"id": "7438.png", "formula": "\\begin{align*} q _ i ( 0 ) = \\frac { d q _ i } { d t } ( 0 ) = 0 , i \\in \\{ 1 , 2 , 3 \\} , \\varphi ^ { ( i ) } \\big | _ { t = 0 } = 0 , i \\in \\{ 0 , 1 , 2 , 3 \\} . \\end{align*}"}
+{"id": "1381.png", "formula": "\\begin{align*} c \\colon V _ p \\otimes V _ p \\to V _ p \\otimes V _ p , c ( v _ i \\otimes v _ j ) = \\lambda v _ { 2 i - j \\bmod p } \\otimes v _ i , \\end{align*}"}
+{"id": "2363.png", "formula": "\\begin{align*} \\forall r \\geq l > 0 \\forall i _ 0 , \\dots , i _ { r - l } . H ^ l ( U _ { i _ 0 , \\dots , i _ { r - l } } , \\mathcal F ) = 0 \\rlap { . } \\end{align*}"}
+{"id": "5545.png", "formula": "\\begin{align*} D \\left ( \\beta _ { x , 1 } \\parallel \\beta _ { x , 2 } \\right ) = \\sup _ { n } H _ { \\beta _ { x , 1 } \\parallel \\beta _ { x , 2 } } ( \\mathcal { P } _ { x , n } ) , \\quad \\textrm { w h e r e } H _ { \\beta _ { x , 1 } \\parallel \\beta _ { x , 2 } } ( \\mathcal { P } _ { x , n } ) = \\sum _ { A \\in \\mathcal { P } _ { x , n } } \\beta _ { x , 1 } ( A ) \\log \\frac { \\beta _ { x , 1 } ( A ) } { \\beta _ { x , 2 } ( A ) } . \\end{align*}"}
+{"id": "6495.png", "formula": "\\begin{align*} \\det B _ { n , m } ( x ) : & = \\det \\left [ \\binom { x + m + 2 i } { i - j + m } - \\binom { x + m + 2 i } { i - j + m - 1 } \\right ] _ { i , j = 0 } ^ { n - 1 } \\\\ & = \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ m \\frac { ( x + i - j ) ( x + 2 i + j - 2 ) } { ( x + 2 i - j ) ( i + j - 1 ) } . \\end{align*}"}
+{"id": "3636.png", "formula": "\\begin{align*} ( \\{ 9 \\} ^ { I + 1 } , a _ { I + 1 } , \\ldots , 5 , \\{ 9 \\} ^ { I } , 4 ) _ { 1 0 } \\ = \\ 2 ( 4 , \\{ 9 \\} ^ { I } , 5 , \\ldots , a _ { I + 1 } , \\{ 9 \\} ^ { I } , 7 ) _ { 1 0 } \\end{align*}"}
+{"id": "2564.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta w = f ( w ) , \\ \\ 0 < w < 1 & \\ \\ \\Omega _ { a , b } , \\\\ w ( x ) \\equiv 1 , \\ \\ & \\ \\ | x | \\le a , \\\\ w \\in H ^ 1 _ 0 ( B _ b ) . \\end{cases} \\end{align*}"}
+{"id": "6868.png", "formula": "\\begin{align*} C _ r ^ 0 = \\int _ { [ 0 , 1 ] ^ 2 } \\log \\tfrac { 1 } { 1 - r } . \\end{align*}"}
+{"id": "8928.png", "formula": "\\begin{align*} S _ p = \\begin{cases} Z _ p , & p = 1 , \\dots , j , \\\\ B _ p ^ { ( j ) } , & p = j + 1 , \\dots , k , \\\\ 0 , & p = k + 1 , \\dots , k + j , \\\\ Z _ { j + 1 } , & p = k + j + 1 , \\\\ A _ { p - k } , & p = k + j + 2 , \\dots , 2 k . \\end{cases} \\end{align*}"}
+{"id": "7388.png", "formula": "\\begin{align*} \\Delta \\left ( \\frac { m } { a } \\right ) = \\begin{cases} 1 - \\tfrac { | m | } { a } , & | m | \\leq a \\\\ 0 , & | m | > a . \\end{cases} \\end{align*}"}
+{"id": "7190.png", "formula": "\\begin{align*} w _ s \\vert _ { \\partial \\Omega } = v _ s \\vert _ { \\partial \\Omega } = v \\vert _ { \\partial \\Omega } . \\end{align*}"}
+{"id": "3122.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\frac { q ^ { r \\binom { k + 1 } { 2 } + ( 1 - r ) k } [ r ] _ q ^ k \\ , [ k ] _ { q ^ r } ! S _ { r } [ n , k ] \\ , t ^ k } { \\prod _ { i = 0 } ^ k ( 1 - t q ^ { r i } ) } = \\sum _ { m = 0 } ^ { \\infty } [ r m + 1 ] _ q ^ { n } \\ , t ^ { m } . \\end{align*}"}
+{"id": "1671.png", "formula": "\\begin{align*} q _ { i j } ^ * T _ { \\varrho _ { i j } } = q _ { i j } ^ * T _ { \\eta _ { i j } } \\leq T _ { \\psi ^ * \\delta _ { i j } } . \\end{align*}"}
+{"id": "2784.png", "formula": "\\begin{align*} \\hat { q } ( \\eta ) = \\int _ { \\mathrm { M } _ 0 } q u _ 1 u _ 2 d x - \\int _ { \\mathrm { M } _ 0 } q \\varrho d x , \\end{align*}"}
+{"id": "3383.png", "formula": "\\begin{align*} \\lim _ { \\kappa \\to 1 ^ + } \\limsup _ { m , n \\to \\infty } \\max _ { n \\leq j \\leq n ^ \\kappa } | u _ { m j } - u _ { m n } | = 0 , \\end{align*}"}
+{"id": "3253.png", "formula": "\\begin{align*} & \\Big \\| \\Big \\{ \\sum _ { k \\in \\mathbb Z } { \\mathfrak R } ^ { - k \\beta } | D _ k ( a ) | \\Big \\} \\Big \\| _ { L ^ { p ' } _ { \\omega } } \\\\ & \\le \\Big \\| \\Big \\{ \\sum _ { k = k _ 0 + 1 } ^ \\infty { \\mathfrak R } ^ { - k \\beta } | D _ k ( a ) | \\Big \\} \\Big \\| _ { L ^ { p ' } _ { \\omega } } + \\Big \\| \\Big \\{ \\sum _ { k = - \\infty } ^ { k _ 0 } { \\mathfrak R } ^ { - k \\beta } | D _ k ( a ) | \\Big \\} \\Big \\| _ { L ^ { p ' } _ { \\omega } } \\\\ & = : I _ 1 + I _ 2 . \\end{align*}"}
+{"id": "6985.png", "formula": "\\begin{align*} \\mu ( f ) = \\min \\{ \\mu ( a q ) , \\mu ( b ) \\} . \\end{align*}"}
+{"id": "3429.png", "formula": "\\begin{align*} | d X | ^ 2 = 4 c ^ 2 \\left ( \\lambda + \\frac { c ^ 2 } { m } \\right ) . \\end{align*}"}
+{"id": "1886.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\bar { \\Lambda } ( [ \\bar { \\zeta } _ 0 ] ) > 0 , \\\\ & \\langle \\lambda ^ * , \\bar { \\zeta } _ 0 \\rangle > 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7304.png", "formula": "\\begin{align*} a \\prod _ { i = 1 } ^ n x _ i ^ { \\alpha _ i } + b \\prod _ { i = 1 } ^ n x _ i ^ { \\beta _ i } = c \\prod _ { i = 1 } ^ n x _ i ^ { \\gamma _ i } , \\end{align*}"}
+{"id": "6816.png", "formula": "\\begin{align*} ( W ^ { \\natural } ; \\mathcal { L } : = \\mathcal { L } ' \\cup \\mathcal { L } '' ) , \\end{align*}"}
+{"id": "8703.png", "formula": "\\begin{align*} \\bigg | h ( x ) - \\sum ^ { l } _ { s = 0 } \\frac { h ^ { ( s ) } ( 0 ) x ^ s } { s ! } \\bigg | \\leq C | x | ^ { l + 1 } \\end{align*}"}
+{"id": "6393.png", "formula": "\\begin{align*} a w _ { 2 } ^ { p } - b w _ { 1 } ^ { 2 p } = c \\end{align*}"}
+{"id": "6970.png", "formula": "\\begin{align*} \\min \\ , \\alpha = \\alpha _ \\ell = \\tilde { \\beta } _ i . \\end{align*}"}
+{"id": "2888.png", "formula": "\\begin{align*} r _ 2 = ( \\frac { - t _ 0 } { 1 } \\ominus \\frac { - 1 } { 0 } ) \\bullet \\frac { 1 } { 0 } - ( \\frac { 1 } { 0 } \\ominus \\frac { 1 } { 1 } ) \\bullet \\frac { 1 } { 0 } - ( \\frac { 1 } { 1 } \\ominus \\frac { 1 } { 2 } ) \\bullet \\frac { 1 } { 0 } - \\cdots - ( \\frac { 1 } { - t _ { 2 } - 1 } \\ominus \\frac { 1 } { - t _ { 2 } } ) \\bullet \\frac { 1 } { 0 } = t _ { 2 } - 1 . \\end{align*}"}
+{"id": "9327.png", "formula": "\\begin{align*} & X _ { 1 0 } : = \\frac { 1 7 } { 1 6 1 2 8 0 } \\left ( E _ { 4 , \\mathbb { H } } E _ { 6 , \\mathbb { H } } - E _ { 1 0 , \\mathbb { H } } \\right ) , \\\\ & X _ { 1 2 } : = \\frac { 2 1 4 2 1 } { 2 0 3 2 1 2 8 0 0 } \\left ( \\frac { 4 4 1 } { 6 9 1 } E _ { 4 , \\mathbb { H } } ^ 3 + \\frac { 2 5 0 } { 6 9 1 } E _ { 6 , \\mathbb { H } } ^ 2 - E _ { 1 2 , \\mathbb { H } } \\right ) . \\end{align*}"}
+{"id": "7099.png", "formula": "\\begin{align*} c | _ { t = 0 } = c _ { 0 } \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Omega \\end{align*}"}
+{"id": "499.png", "formula": "\\begin{align*} \\hat { s } _ { n } = \\left \\{ \\begin{array} { r l } 1 , & - ( 1 + a ) \\sum _ { i _ { r } = ( n - 1 ) M + 1 } ^ { n M } ( \\log ( \\frac { \\hat { x } _ { i _ { r } } ^ { 2 } + \\hat { x } _ { i _ { r } + M N } ^ { 2 } } { b } + 1 ) \\\\ & + \\frac { \\hat { x } _ { i _ { r } } ^ { 2 } + \\hat { x } _ { i _ { r } + M N } ^ { 2 } } { \\epsilon } ) > \\varpi _ { t h } , \\\\ \\\\ 0 , & \\end{array} \\right . \\end{align*}"}
+{"id": "7097.png", "formula": "\\begin{align*} \\partial _ t c & = m \\Delta \\mu \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Omega _ T , \\\\ \\mu & = \\mathcal { L } _ \\varepsilon c + f ^ \\prime ( c ) \\ ; \\ ; \\ ; \\ ; \\Omega _ T , \\end{align*}"}
+{"id": "3048.png", "formula": "\\begin{align*} \\| ( x , y ) \\| _ { X _ { 1 } \\times X _ { 2 } } ^ { 2 } : = \\| x \\| _ { X _ { 1 } } ^ { 2 } + \\| y \\| _ { X _ { 2 } } ^ { 2 } \\ , . \\end{align*}"}
+{"id": "373.png", "formula": "\\begin{align*} y _ { i _ 1 , j _ 1 } \\cdots y _ { i _ k , j _ k } = \\left ( \\prod ^ k _ { \\ell = 1 } \\frac { c _ { i _ { \\ell } } ^ { \\tau } } { c _ { i _ { \\ell } } ^ { \\{ i _ { \\ell } \\} } } \\right ) y _ { \\tau , \\mathfrak { u } } . \\end{align*}"}
+{"id": "6652.png", "formula": "\\begin{align*} & \\bullet \\ , \\ , L ^ p _ f ( \\R ^ N ) = \\Big \\{ u : \\R ^ N \\to \\R : \\ , \\int _ { \\R ^ N } | u | ^ p f \\ , d x < \\infty \\Big \\} \\quad ; \\\\ [ 0 . 1 c m ] & \\bullet \\ , \\ , L ^ \\infty _ f ( \\R ^ N ) = \\big \\{ u : \\R ^ N \\to \\R : \\ , \\mathrm { e s s \\ , s u p } _ { \\R ^ N } ( | u | f ) < \\infty \\} . \\end{align*}"}
+{"id": "867.png", "formula": "\\begin{align*} N ( P ) = c \\widehat { P } ^ { n - 5 } + O \\left ( \\widehat { P } ^ { n - 5 - \\delta } \\right ) , \\end{align*}"}
+{"id": "1847.png", "formula": "\\begin{align*} - 3 F ( 2 \\pi / 3 ) + \\sum _ { i = 1 } ^ { 3 } F ( \\theta _ { i } ) \\leq - \\frac { 3 } { 2 \\pi ^ { 2 } } \\Big ( \\lambda _ { 1 , n } ^ { r , s } + \\frac { 1 } { 2 5 } \\lambda _ { 5 , n } ^ { r , s } \\Big ) \\leq - \\frac { 1 8 } { 7 } \\frac { 3 } { 2 \\pi ^ { 2 } } \\Big ( \\lambda _ { 1 , n } ^ { r , s } + \\frac { 1 } { 2 5 } \\lambda _ { 5 , n } ^ { r , s } \\Big ) \\sum _ { i = 1 } ^ { 3 } \\Big ( \\frac { \\theta _ { i } } { 2 \\pi } - 1 / 3 \\Big ) ^ { 2 } . \\end{align*}"}
+{"id": "5116.png", "formula": "\\begin{align*} \\int _ { B _ { 1 } } b ^ { i j , k l } f _ { i j } \\tau ^ { 4 } f _ { k l } d x = - \\int _ { B _ { 1 } } b ^ { i j , k l } f _ { i j } \\left ( ( \\tau ^ { 4 } ) _ { k l } f + ( \\tau ^ { 4 } ) _ { l } f _ { k } + ( \\tau ^ { 4 } ) _ { k } f _ { l } \\right ) d x . \\end{align*}"}
+{"id": "8417.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { - 1 } ( n ) \\ , e ^ { - 2 n \\alpha } - \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { - 1 } ( n ) \\ , e ^ { - 2 n \\beta } = \\frac { \\beta - \\alpha } { 1 2 } + \\frac { 1 } { 4 } \\log \\left ( \\frac { \\alpha } { \\beta } \\right ) . \\end{align*}"}
+{"id": "1177.png", "formula": "\\begin{align*} \\psi ^ 2 ( \\R ^ \\bullet f _ * ( V ) ) = \\R ^ \\bullet f _ * ( \\Theta ^ 2 ( \\Omega _ f ) ^ { - 1 } \\otimes \\psi ^ 2 ( V ) ) \\end{align*}"}
+{"id": "955.png", "formula": "\\begin{align*} \\| u \\mid W ^ 1 _ p ( D ) \\| = \\| u \\mid L _ p ( D ) \\| + \\| \\nabla u \\mid L _ p ( D ) \\| , \\end{align*}"}
+{"id": "5613.png", "formula": "\\begin{align*} P _ { \\mu , x } ^ { 1 } ( L _ { x } , \\xi _ { 1 } ^ { x } ) = \\sum _ { \\{ s : \\xi _ { 1 } ^ { x } = L _ { x } s \\} } \\mu ( s ) \\frac { d s \\nu } { d \\nu } ( \\pi ( x ) ) . \\end{align*}"}
+{"id": "5427.png", "formula": "\\begin{align*} y ' ( t ) = - 5 y ( t ) , y ( 0 ) = 1 , \\end{align*}"}
+{"id": "6833.png", "formula": "\\begin{align*} \\sum _ { s _ 1 \\geq \\dots \\geq s _ { r } } \\frac { a ^ { s _ 1 + \\dots + s _ r } q ^ { s _ 1 ^ 2 + \\dots + s _ { r } ^ 2 - s _ 1 - \\dots - s _ { i } } } { ( q ) _ { s _ 1 - s _ 2 } \\dots ( q ) _ { s _ { r - 1 } - s _ r } } \\beta _ { s _ r } = \\frac { 1 } { ( a q ) _ \\infty } \\sum _ { j \\in \\mathbb { Z } } a ^ { r j } q ^ { r j ^ 2 - i j } \\frac { 1 - a ^ { i + 1 } q ^ { 2 j ( i + 1 ) } } { 1 - a q ^ { 2 j } } \\alpha _ j , \\end{align*}"}
+{"id": "5081.png", "formula": "\\begin{align*} B [ \\phi ] = \\left ( \\begin{array} { c c } 3 \\phi _ { r } ^ 2 + \\phi _ { i } ^ 2 & 2 \\phi _ { r } \\phi _ { i } \\\\ 2 \\phi _ { r } \\phi _ { i } & \\phi _ { r } ^ 2 + 3 \\phi _ { i } ^ 2 \\end{array} \\right ) , \\end{align*}"}
+{"id": "433.png", "formula": "\\begin{align*} \\dfrac { 1 } { 2 } \\dfrac { d } { d t } \\| \\vec U \\| _ { P \\otimes P _ { \\Omega } } ^ 2 + \\vec U ^ T ( I _ n \\otimes E ^ T P _ { \\partial \\Omega } N _ { i } E ) { \\bf A _ i } \\vec U + ( \\vec U , { \\vec L _ D } ) = 0 . \\end{align*}"}
+{"id": "7038.png", "formula": "\\begin{align*} \\mathcal I _ 2 = \\left ( \\left . h _ i \\sum \\limits _ { j = 1 } ^ { s _ { i _ { } } } b _ { i _ { } i j } { \\textbf { X } } ^ { \\lambda _ j } \\ \\right | \\ i \\in I \\setminus \\{ i _ { } \\} \\right ) . \\end{align*}"}
+{"id": "7307.png", "formula": "\\begin{align*} x _ i = \\left ( a \\prod _ { j = 1 } ^ n u _ j ^ { \\alpha _ j } + b \\prod _ { j = 1 } ^ n u _ j ^ { \\beta _ j } \\right ) ^ { z _ i } \\left ( c \\prod _ { j = 1 } ^ n u _ j ^ { \\gamma _ j } \\right ) ^ { t _ i } w ^ { - z _ i - t _ i } \\cdot u _ i , i = 1 , \\dots , n , \\end{align*}"}
+{"id": "633.png", "formula": "\\begin{align*} 1 = \\| x \\| ^ 2 = \\phi ( x ^ * x ) = \\psi ( x ^ * x ) \\end{align*}"}
+{"id": "5951.png", "formula": "\\begin{align*} \\theta _ { L , X ^ { \\ast } } ( f ' ) ( \\epsilon , w ) = \\sum _ { l \\in L / L \\cap X ^ { \\ast } } f ' ( \\epsilon , w + l ) \\psi ( \\epsilon \\tfrac { \\langle l , w \\rangle } { 2 } + \\tfrac { \\langle x _ { l } , x ^ { \\ast } _ l \\rangle } { 2 } ) , \\end{align*}"}
+{"id": "882.png", "formula": "\\begin{align*} \\norm { r ' \\alpha _ 2 z } = \\norm { \\theta ' ( z _ 1 t + z _ 2 ) + c z _ 2 / t } . \\end{align*}"}
+{"id": "3008.png", "formula": "\\begin{align*} \\| \\phi ( A \\otimes B ) \\| _ { ( p , k ) } = \\| A \\otimes B \\| _ { ( p , k ) } \\hbox { f o r a l l $ A \\in M _ m $ a n d $ B \\in M _ n $ , } \\end{align*}"}
+{"id": "5307.png", "formula": "\\begin{align*} 1 \\leq e \\sum _ { d = 1 } ^ m \\left ( \\frac { p } { 1 - p } \\ , C \\ , r \\ , \\sqrt { q \\cdot A } \\right ) ^ d . \\end{align*}"}
+{"id": "1630.png", "formula": "\\begin{align*} \\nabla _ { \\xi _ i } \\ , \\xi _ j + \\nabla _ { \\xi _ j } \\ , \\xi _ i = 0 , \\end{align*}"}
+{"id": "8060.png", "formula": "\\begin{align*} | A + A ' | & \\geq | A ^ * + A ' | = | A ^ * + \\{ a _ 1 , \\dots , a _ c \\} | + \\sum _ { i = m + 1 } ^ d | A ^ * + \\{ b _ i \\} | \\\\ & \\geq ( 1 - 1 0 0 ^ { - d ^ 2 } ) | A ^ { * } + A ^ { * } | + ( d - m ) | A ^ * | \\\\ & \\geq ( 1 - 1 0 0 ^ { - d ^ 2 } ) ( m + 1 + 1 / 1 0 ) | A ^ * | + ( d - m ) | A ^ * | - m ( m + 1 ) / 2 \\\\ & \\geq ( d + 1 + 1 / 1 5 ) | A ^ * | - m ( m + 1 ) / 2 \\\\ & \\geq ( d + 1 + 2 / 3 1 ) | A | - d ( d + 1 ) , \\end{align*}"}
+{"id": "2534.png", "formula": "\\begin{align*} \\dim { \\mathcal G } _ { M , { \\mathcal H } } = \\dim \\mathcal G _ M + \\dim T _ M ' - \\dim T _ { M } \\end{align*}"}
+{"id": "1944.png", "formula": "\\begin{align*} f _ 1 ^ * ( z _ 1 , \\ldots , z _ m ) : = f _ { 1 , 1 } ^ * ( z _ 1 , \\ldots , z _ m ) , \\end{align*}"}
+{"id": "7162.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } : = \\inf \\{ \\lambda > 0 : \\varrho _ { p ( \\cdot ) } ( u / \\lambda ) \\leq 1 \\} . \\end{align*}"}
+{"id": "4393.png", "formula": "\\begin{align*} \\delta _ 1 = \\sqrt { \\frac { \\alpha } { 2 \\omega } } , \\ \\ \\delta _ 2 = \\sqrt { \\frac { \\beta } { \\omega } } , \\ \\ \\delta _ 3 = \\sqrt { \\frac { \\gamma } { \\omega } } . \\end{align*}"}
+{"id": "3298.png", "formula": "\\begin{align*} 2 { \\rm R e } \\left ( \\widetilde { \\rho } _ w ( \\theta , \\kappa ( \\theta ) ) \\frac { \\partial \\kappa } { \\partial \\theta } \\right ) = - \\widetilde { \\rho } _ { \\theta } ( \\theta , \\kappa ( \\theta ) ) , \\end{align*}"}
+{"id": "6628.png", "formula": "\\begin{align*} \\mathcal { B } _ p [ \\mathcal { H } ; X ] : = & \\left [ \\mathcal { B } _ 1 [ \\mathcal { H } ; X ] , \\mathcal { B } _ \\infty [ \\mathcal { H } ; X ] \\right ] _ { 1 / p } \\mbox { ( i n t h e s e n s e o f c o m p l e x i n t e r p o l a t i o n ) } \\\\ \\mathcal { B } _ { p , r } [ \\mathcal { H } ; X ] : = & \\left [ \\mathcal { B } _ 1 [ \\mathcal { H } ; X ] , \\mathcal { B } _ \\infty [ \\mathcal { H } ; X ] \\right ] _ { 1 / p , r } \\mbox { ( i n t h e s e n s e o f r e a l i n t e r p o l a t i o n ) } . \\end{align*}"}
+{"id": "3013.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\lambda _ { i } ^ { \\gamma } ( C + D ) = \\sum _ { i = 1 } ^ k \\lambda _ { i } \\big ( ( C + D ) ^ { \\gamma } \\big ) \\geq \\mathrm { t r } ( \\hat { U } ^ * ( C + D ) ^ { \\gamma } \\hat { U } ) \\end{align*}"}
+{"id": "6527.png", "formula": "\\begin{align*} S _ 2 = - \\frac { 2 \\beta } { \\alpha } \\Gamma ( \\nu + 1 ) { } _ 2 F _ 1 \\bigg ( 1 , \\nu + 1 ; \\frac { 3 } { 2 } ; \\frac { \\beta ^ 2 } { \\alpha ^ 2 } \\bigg ) . \\end{align*}"}
+{"id": "2372.png", "formula": "\\begin{align*} g & = \\alpha \\cdot { ( X - a _ 1 ) } ^ { e _ 1 } \\dots { ( X - a _ k ) } ^ { e _ k } { ( X - b _ 1 ) } ^ { e ' _ 1 } \\dots { ( X - b _ l ) } ^ { e ' _ l } \\rlap { , } \\\\ h & = { ( X - c _ 1 ) } ^ { e '' _ 1 } \\dots { ( X - c _ m ) } ^ { e '' _ m } \\rlap { . } \\end{align*}"}
+{"id": "6935.png", "formula": "\\begin{align*} \\sum _ { p \\in P } f ( p ) ^ 2 = ( 1 + o ( 1 ) ) \\frac { \\log { N } } { | \\log { ( 2 \\sigma - 1 ) } | } ( e ^ 2 - e ) . \\end{align*}"}
+{"id": "3287.png", "formula": "\\begin{align*} ( t , w ) \\longmapsto \\left \\{ \\begin{array} { r l } \\widetilde { \\rho } ( t , w ) = \\frac { 1 } { t ^ s } \\rho ( t , t ^ s w ) ; & t \\ge 0 , w \\in \\C \\\\ { \\rm I m } ( e ^ { i \\pi ( 1 - \\beta _ 1 ) } w ) ; & t \\le 0 , w \\in \\C \\end{array} \\right . \\end{align*}"}
+{"id": "1290.png", "formula": "\\begin{align*} \\sigma ( A ) = \\left \\{ n ( \\ell - 1 ) ^ { ( 1 ) } , - n ^ { ( \\ell - 1 ) } , 0 ^ { ( \\ell ( n - 1 ) ) } \\right \\} , \\end{align*}"}
+{"id": "979.png", "formula": "\\begin{align*} F X = \\phi X + \\omega X \\end{align*}"}
+{"id": "6309.png", "formula": "\\begin{align*} \\phi _ t ( p , z ) = \\exp _ z ( ( t - 1 ) d _ z \\mathfrak f _ p ) , \\forall \\ , z \\in \\mathcal O _ q . \\end{align*}"}
+{"id": "8279.png", "formula": "\\begin{align*} d _ { i j } = \\sum _ { m = 0 } ^ { 4 M } \\frac { ( 2 k - n + i - 1 ) ! } { ( 2 k - n + i - 1 + m ) ! } \\binom { 2 k - n - 1 + m } { m } \\binom { i + m - 1 } { j - 1 } ( \\i N \\beta ) ^ m , \\end{align*}"}
+{"id": "3881.png", "formula": "\\begin{align*} g ( s ) \\ge \\sum _ { \\ell = 1 } ^ { L } a _ \\ell ( s _ \\ell ) , \\forall s = ( s _ 1 , \\ldots , s _ L ) \\in \\prod _ { \\ell \\in [ L ] } \\mathcal { S } _ \\ell . \\end{align*}"}
+{"id": "8049.png", "formula": "\\begin{align*} E _ j = P _ 1 + P _ { 2 } + \\cdots P _ { j } , j = 1 , 2 \\cdots \\end{align*}"}
+{"id": "7489.png", "formula": "\\begin{align*} \\Phi ( \\sigma ) = \\tilde { \\delta } ^ { \\prime } \\mu _ m ^ { \\prime } \\tilde { \\alpha } _ m ^ { \\prime } \\mu _ { m - 1 } ^ { \\prime } \\tilde { \\alpha } _ { m - 1 } ^ { \\prime } \\cdots \\mu _ 1 ^ { \\prime } \\tilde { \\alpha } _ 1 ^ { \\prime } \\mu _ 0 ^ { \\prime } \\end{align*}"}
+{"id": "7027.png", "formula": "\\begin{align*} S _ i \\left ( f ' \\right ) = \\{ l - 1 \\mid l \\in S _ i ( f ) \\setminus \\{ 0 \\} \\mbox { a n d } p \\nmid l \\} . \\end{align*}"}
+{"id": "1480.png", "formula": "\\begin{align*} & \\Big | f _ \\epsilon ( V ) - \\sum _ { i = 1 } ^ k ( - 1 ) ^ i f _ 0 ( P U _ { \\mu _ i , \\xi _ i } ) \\Big | _ { \\frac { 2 n } { n + 2 } } \\\\ = & \\Big | f _ \\epsilon ( V ) - f _ 0 ( V ) \\Big | _ { \\frac { 2 n } { n + 2 } } + \\Big | f _ 0 ( V ) - \\sum _ { i = 1 } ^ k ( - 1 ) ^ i f _ 0 ( P U _ { \\mu _ i , \\xi _ i } ) \\Big | _ { \\frac { 2 n } { n + 2 } } . \\end{align*}"}
+{"id": "2559.png", "formula": "\\begin{align*} \\begin{cases} \\left ( \\sigma _ y ( t ) , \\partial D \\right ) \\to 0 \\ \\ , \\ \\ , u ( \\sigma _ y ( t ) ) \\to 0 \\ \\ , \\ \\ , t \\to t _ y ^ - , \\ \\ & \\\\ | \\sigma _ y ( t ) | \\to 0 \\ \\ , \\ \\ , u ( \\sigma _ y ( t ) ) \\to u ( 0 ) \\ \\ , \\ \\ , t \\to t _ y ^ + . & \\end{cases} \\end{align*}"}
+{"id": "9293.png", "formula": "\\begin{align*} p ^ { k } _ { i } = y ^ { k - 1 } _ { i } + \\lambda _ { k } ( x _ { s } ^ { k - 1 } - x _ { t } ^ { k - 1 } ) \\forall i = ( s , t ) , \\ i \\in \\mathcal { N } ^ { - } _ { ( k ) } ( s ) . \\end{align*}"}
+{"id": "3481.png", "formula": "\\begin{align*} H ( \\sigma ) = - \\sum _ { \\{ v , u \\} \\in E } K ( \\sigma _ v , \\sigma _ u ) - \\sum _ { \\{ v , u \\} \\in E : u \\in V , v \\in ( \\partial V ) _ { \\tau } } K ( \\sigma _ v , \\tau _ v ) - \\sum _ { v \\in V } U ( \\sigma _ v ) , \\end{align*}"}
+{"id": "593.png", "formula": "\\begin{align*} \\psi ( b _ k d b _ j ^ * ) = \\langle \\pi ( d ) \\pi ( b _ j ) ^ * \\xi , \\pi ( b _ k ) ^ * \\xi \\rangle = \\psi ( b _ { k } ) \\psi ( d ) \\psi ( b _ j ) ^ * = \\omega ( b _ { k } ) \\psi ( d ) \\omega ( b _ j ) ^ * \\end{align*}"}
+{"id": "6107.png", "formula": "\\begin{align*} \\mathcal { L } ^ { \\Phi } _ { A } u = ( - \\Delta ) ^ { \\frac { s } { 2 } } f + g \\end{align*}"}
+{"id": "6349.png", "formula": "\\begin{align*} f ( \\alpha _ 0 ) = f ( \\beta _ 0 ) = 0 \\qquad f > 0 \\ , \\ \\ \\ , ( \\alpha _ 0 , \\beta _ 0 ) . \\end{align*}"}
+{"id": "7066.png", "formula": "\\begin{align*} \\beta _ j - \\tilde { \\beta } _ j = \\nu ( g ' ) - \\nu _ j ( g ' ) < \\epsilon \\mbox { f o r e v e r y } j \\in I ' \\mbox { w i t h } j \\geq j _ 0 . \\end{align*}"}
+{"id": "2447.png", "formula": "\\begin{align*} X ^ M ( m ) = \\tfrac { d } { d r } ( e ^ { r X } \\tau ( m ) ) \\cdot m \\Big | _ { r = 0 } & & m \\in M . \\end{align*}"}
+{"id": "2873.png", "formula": "\\begin{align*} ( \\alpha \\otimes \\Delta _ c ) \\circ \\Delta _ c ( l ) & = \\alpha ( l _ 1 ) \\otimes T ( l _ { 2 1 } ) \\otimes T ( l _ { 2 2 } ) - \\alpha ( l _ 1 ) \\otimes T ( l _ { 2 2 } ) \\otimes T ( l _ { 2 1 } ) \\\\ & - T ( \\alpha ( l _ 2 ) ) \\otimes l _ { 1 1 } \\otimes T ( l _ { 1 2 } ) + T ( \\alpha ( l _ 2 ) ) \\otimes T ( l _ { 1 2 } ) \\otimes l _ { 1 1 } \\end{align*}"}
+{"id": "3762.png", "formula": "\\begin{align*} D _ { g _ 0 ^ n } & H ^ n \\left ( x ^ 0 , \\vec { a } , t \\right ) \\\\ & = i m _ n \\left [ c _ r k _ { 1 2 } ^ n \\left ( \\{ x ^ 0 \\} _ { \\tau = 1 } ^ { r + s } , \\vec { a } \\right ) \\cos ( t c _ r m _ n ) \\mp c _ s k _ { 2 1 } ^ n \\left ( \\{ x ^ 0 \\} _ { \\tau = 1 } ^ { r + s } , \\vec { a } \\right ) \\cos ( t c _ s m _ n ) \\right ] . \\end{align*}"}
+{"id": "5924.png", "formula": "\\begin{align*} \\Pi _ { \\psi } [ ( x , 0 ) + ( x ^ { \\ast } , 0 ) + ( 0 , k ) ] f ( [ \\epsilon , y ] ) = \\psi ( \\epsilon k + \\epsilon \\langle x + y , x ^ { \\ast } \\rangle ) f ( [ \\epsilon , x + y ] ) , \\end{align*}"}
+{"id": "2569.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) , \\ \\ u \\in ( 0 , 1 ) , & \\ \\ D , \\\\ u = 1 , \\ \\ | \\nabla u | = c _ 1 , & \\ \\ \\partial \\Omega _ 1 , \\\\ u = 0 , \\ \\ | \\nabla u | = c _ 2 , & \\ \\ \\partial \\Omega _ 2 , \\\\ \\end{cases} \\end{align*}"}
+{"id": "7013.png", "formula": "\\begin{align*} \\alpha _ { i ' } = \\min _ { \\ell \\in I _ 0 ( f , i ) } \\alpha _ \\ell . \\end{align*}"}
+{"id": "8747.png", "formula": "\\begin{align*} ( \\Delta _ { 1 } ) & = \\frac { 8 } { \\gamma ^ 2 } \\{ f ^ { ( 1 ) } ( \\tau _ 1 ) \\} ^ { 2 } \\bigg \\{ \\frac { 1 } { n ( n - 1 ) } ( \\Sigma _ { 1 } ^ { 2 } ) + \\frac { 1 } { m ( m - 1 ) } ( \\Sigma _ { 2 } ^ { 2 } ) + \\frac { 2 } { n m } ( \\Sigma _ { 1 } \\Sigma _ { 2 } ) \\bigg \\} \\{ 1 + o ( 1 ) \\} . \\end{align*}"}
+{"id": "8491.png", "formula": "\\begin{align*} \\intop _ { 1 } ^ { \\infty } \\frac { 1 } { ( t ^ { 2 } - 1 ) ^ { s } } \\ , \\frac { d t } { t } = \\frac { \\pi } { 2 \\sin ( \\pi s ) } , \\ , \\ , \\ , \\ , \\ , \\ , 0 < ( s ) < 1 , \\end{align*}"}
+{"id": "7558.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { \\mathbb { E } ( \\overline { S } _ n ) - \\mathbb { E } ( \\overline { S } _ { n - n ' } ) } { \\mathbb { E } ( \\overline { S } _ n ) } = 0 \\ \\ \\ \\ n ' \\ll n \\log ^ { - B } n , \\end{align*}"}
+{"id": "3826.png", "formula": "\\begin{align*} \\Theta ( \\delta ) = \\left [ \\min _ { \\gamma \\in \\Sigma ( \\delta ) } \\int _ { \\mathcal { S } } f \\ , d \\gamma , \\max _ { \\gamma \\in \\Sigma ( \\delta ) } \\int _ { \\mathcal { S } } f \\ , d \\gamma \\right ] , \\end{align*}"}
+{"id": "2401.png", "formula": "\\begin{align*} \\vec { \\delta } _ { 1 0 2 } = ( 1 + \\cos \\alpha _ { 1 0 2 } , \\sin \\alpha _ { 1 0 2 } , 0 ) \\end{align*}"}
+{"id": "813.png", "formula": "\\begin{align*} Y _ { i } ^ { 3 } - Y _ { i } ^ { 5 } & = K _ { 3 } h ^ { 3 } + \\ldots - \\left ( K _ { 5 } h ^ { 5 } + \\ldots \\right ) \\\\ & \\approx K _ { 3 } ^ { i } h ^ { 3 } \\end{align*}"}
+{"id": "5956.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( g ) f ( [ \\epsilon , w ] ) = e ^ { \\tfrac { \\pi i ( 1 - \\epsilon ) } { 4 } } ( c , \\epsilon ) _ { \\R } \\beta ( d , \\epsilon c ) ^ { - 1 } _ { \\R } f ( [ \\epsilon , w g ^ { s ( \\epsilon ) } ] ) . \\end{align*}"}
+{"id": "9112.png", "formula": "\\begin{align*} x ^ { i , + } = f ^ { i } ( x , u ) \\ , , i = 1 , \\dots , n \\end{align*}"}
+{"id": "6285.png", "formula": "\\begin{align*} \\dot \\gamma ( t ) = \\sum _ { i = 1 } ^ k \\bar u _ i ( t ) X _ i ( \\gamma ( t ) ) , \\qquad \\norm { \\dot \\gamma ( t ) } _ { \\gamma ( t ) } = \\norm { \\bar u ( t ) } , \\qquad t \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "6192.png", "formula": "\\begin{align*} | E \\Delta F | & \\le | { \\rm c o v } ( E ) \\setminus E | + | { \\rm c o v } ( E ) \\setminus F | = 2 | { \\rm c o v } ( E ) \\setminus E | \\\\ & \\le 2 ( \\sqrt \\pi ) ^ { - 1 } | { \\rm c o v } ( E ) | ^ { \\frac 1 2 } ( P ( E ) - P ( F ) ) \\leq 2 \\sqrt { 2 } ( P ( E ) - P ( F ) ) . \\end{align*}"}
+{"id": "1800.png", "formula": "\\begin{align*} \\nu ( a _ p ^ * a _ q ) = \\delta ( p - q ) \\Big ( \\delta ( p - h _ 1 ) + \\ldots + \\delta ( p - h _ n ) + \\delta ( p - p _ 1 ) + \\ldots + \\delta ( p - p _ n ) \\Big ) \\ . \\end{align*}"}
+{"id": "5673.png", "formula": "\\begin{align*} \\aligned m _ \\nu ( a , b ) = J _ \\nu ( u _ \\nu , v _ \\nu ) \\geq \\bar { f } \\Bigl ( \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | u _ \\nu | ^ { p } ) | v _ \\nu | ^ { q } \\Bigr ) . \\endaligned \\end{align*}"}
+{"id": "4407.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } S _ { \\omega , \\mathbf { c } } ( U _ n ) = \\mu _ { \\omega , \\mathbf { c } } , \\ \\ \\lim _ { n \\rightarrow \\infty } K _ { \\omega , \\mathbf { c } } ( U _ n ) = 0 . \\end{align*}"}
+{"id": "1764.png", "formula": "\\begin{align*} \\| \\mathbf { v } \\| _ { \\mathbb { V } _ p } : = \\| \\nabla _ z \\mathbf { v } \\| _ { \\mathbb { L } _ p ^ 1 } + \\| \\mathbf { v } \\varrho \\| _ { \\mathbb { L } _ p ^ 1 } < \\infty , \\end{align*}"}
+{"id": "7331.png", "formula": "\\begin{align*} m = \\frac { s } { v _ i ^ d } \\frac { ( q ^ * ) ^ d } { q } ( U ' ) ^ d . \\end{align*}"}
+{"id": "7226.png", "formula": "\\begin{align*} \\Omega _ u : = \\bigcup _ { \\delta > 0 } \\Omega _ u ^ { ( \\delta ) } \\end{align*}"}
+{"id": "7892.png", "formula": "\\begin{align*} 2 ( - 1 ) ^ j \\binom { n - k - j } { k - j } , \\end{align*}"}
+{"id": "7678.png", "formula": "\\begin{align*} f ( a , b ) : = a ( p - 1 ) ( 2 ( b + 1 ) n - ( 2 + b ) ^ 2 p ) + a p ( 2 b ( p - 1 ) + 4 p - n - 2 ) \\gamma - a p ^ 2 \\gamma ^ 2 - ( p - 1 ) a ^ { \\frac { p } { p - 1 } } \\ge C . \\end{align*}"}
+{"id": "6957.png", "formula": "\\begin{align*} b _ { \\varepsilon ' , \\hat { T } } ( ( \\hat { \\ell } ) _ { \\sf p } ) + b _ { \\varepsilon '' , \\hat { T } } ( ( \\hat { \\ell } ) _ { \\sf p } ) = b _ { \\varepsilon ' , \\hat { T } } ( ( \\hat { \\ell } ) _ { \\sf q } ) + b _ { \\varepsilon '' , \\hat { T } } ( ( \\hat { \\ell } ) _ { \\sf q } ) \\end{align*}"}
+{"id": "5805.png", "formula": "\\begin{align*} Y ( z ) = \\frac { y ' } { y } = c A ( z ) ^ { 1 / k } - \\frac { k - 1 } { 2 k } \\frac { A ' ( z ) } { A ( z ) } + O ( r ^ { - 2 } ) , c ^ k = - 1 , \\end{align*}"}
+{"id": "3128.png", "formula": "\\begin{align*} \\frac { t A _ n ( t , q ) } { \\prod _ { i = 0 } ^ n ( 1 - t q ^ { i } ) } = \\sum _ { k = 0 } ^ n \\frac { q ^ { \\binom { k } { 2 } } [ k ] _ { q } ! S [ n , k ] t ^ k } { \\prod _ { i = 0 } ^ k ( 1 - t q ^ { i } ) } . \\end{align*}"}
+{"id": "1159.png", "formula": "\\begin{align*} T ( f ^ { n } ) = n f ^ { n - 1 } T ( f ) + n B ( A ( f ) , A ( f ^ { n - 1 } ) ) \\end{align*}"}
+{"id": "1294.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( u ) = \\log ( \\theta ) + B ( u ) - \\frac { 1 } { 2 } u - \\int _ 0 ^ u \\eta e ^ { \\rho ( v ) } d v \\end{aligned} \\end{align*}"}
+{"id": "6460.png", "formula": "\\begin{align*} \\widetilde \\gamma ( s ) + \\frac { \\zeta ( s ) \\cdot \\zeta ( s ) } { 4 z ( s ) } & = - 2 h ^ 2 s ^ 2 ( | K _ 1 | ^ 2 + | K _ 2 | ^ 2 + | K _ 3 | ^ 2 ) + 4 s ^ 2 h ^ 2 ( K _ 1 + K _ 2 + K _ 3 ) \\cdot K - 6 s ^ 2 h ^ 2 | K | ^ 2 \\\\ & \\quad + \\mathcal { O } ( s ^ 2 \\frac { h ^ 4 } { \\sigma ^ 2 } + s ^ 3 \\sigma ^ 4 ) \\\\ & = - 2 h ^ 2 s ^ 2 | K _ 1 - K | ^ 2 - 2 h ^ 2 s ^ 2 | K _ 2 - K | ^ 2 - 2 h ^ 2 s ^ 2 | K _ 3 - K | ^ 2 + \\mathcal { O } ( s ^ 2 \\frac { h ^ 4 } { \\sigma ^ 2 } + s ^ 3 \\sigma ^ 4 ) \\end{align*}"}
+{"id": "1856.png", "formula": "\\begin{align*} \\mu _ { 0 i _ 1 } = \\min \\{ \\mu _ { 0 j } \\ | \\ 0 < j \\leq n , \\ b _ { n - j } ( x ) \\ne 0 \\} . \\end{align*}"}
+{"id": "9002.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta ( 1 + w ) ^ 2 = - \\frac { 1 } { m - 1 } S ^ \\varphi w ( 1 + w ) + | \\nabla w | ^ 2 \\ , , \\end{align*}"}
+{"id": "126.png", "formula": "\\begin{align*} \\langle u _ k , \\psi \\rangle = \\int _ { \\tilde { \\gamma } _ k ^ 3 } \\hat { \\psi } ( \\lambda ) \\frac { d } { d \\lambda } \\log \\zeta _ 1 ( \\lambda ) d \\lambda , \\end{align*}"}
+{"id": "3621.png", "formula": "\\begin{align*} a _ 0 \\ \\equiv \\ b _ 0 + 2 \\ \\equiv \\ \\begin{cases} 3 , & , \\\\ 8 , & . \\end{cases} \\end{align*}"}
+{"id": "5010.png", "formula": "\\begin{align*} \\Pr ( X _ t \\le k ) = ( 1 - \\frac { 1 } { n } ) ^ t \\sum _ { j = 0 } ^ k \\Pr ( \\widetilde { X } _ t = k \\ , \\vert \\ , \\widetilde { X } _ 0 = j ) , \\end{align*}"}
+{"id": "3443.png", "formula": "\\begin{align*} \\hat \\omega = & \\ , \\sqrt { \\frac { \\alpha ^ 2 } { 2 m n } + \\frac { \\lambda } { 2 n } } \\hat { J } , \\\\ \\hat { R } _ { i j } = & \\ , \\left ( \\frac { \\alpha ^ 2 } { m n } + \\left ( 1 + \\frac { 1 } { n } \\right ) \\lambda \\right ) \\hat { g } _ { i j } \\end{align*}"}
+{"id": "422.png", "formula": "\\begin{align*} ( T ^ T \\tilde F ) ^ T \\Lambda ^ { - 1 } ( T ^ T \\tilde F ) = \\begin{bmatrix} \\tilde F _ n \\\\ \\tilde F _ { \\tau } \\\\ - \\tilde F _ n / u _ n \\end{bmatrix} ^ T \\begin{bmatrix} - 1 / u _ n & 0 & 0 \\\\ 0 & - 1 / u _ n & 0 \\\\ 0 & 0 & u _ n \\end{bmatrix} \\begin{bmatrix} \\tilde F _ n \\\\ \\tilde F _ { \\tau } \\\\ - \\tilde F _ n / u _ n \\end{bmatrix} = - \\tilde F _ { \\tau } ^ 2 / u _ n \\end{align*}"}
+{"id": "3980.png", "formula": "\\begin{align*} \\phi _ { \\ell } ( z _ \\ell ' , x ' ; x _ \\ell ) & = a _ \\ell z _ \\ell ' - \\lambda _ \\ell \\ \\begin{bmatrix} z _ \\ell ' \\\\ x ' - x _ \\ell \\end{bmatrix} ^ { \\top } Q _ { \\ell } \\begin{bmatrix} z _ \\ell ' \\\\ x ' - x _ \\ell \\end{bmatrix} . \\end{align*}"}
+{"id": "5681.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ^ { + } ) = - \\eta ( 3 z ) ^ { 8 } ( \\dfrac { \\eta ( z ) ^ { 3 } } { \\eta ( 9 z ) ^ { 3 } } + 3 ) ^ { 2 } = \\sum C ( n ) q ^ { n } \\end{align*}"}
+{"id": "8630.png", "formula": "\\begin{align*} Z _ 0 ( \\tau _ N ) & = Y e ^ { \\lambda \\tau _ N } + \\omega _ 0 \\left ( \\tau _ N \\right ) \\\\ & = N e ^ { \\lambda ( \\tau _ N - t _ N ) } + \\omega _ 0 \\left ( \\tau _ N \\right ) , \\end{align*}"}
+{"id": "5587.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ { N } s ^ { n } . \\left ( f \\circ \\xi \\right ) \\left ( u , v , x _ { 0 } \\right ) = \\mathbb { E } _ { \\lambda } \\left ( \\tilde { f } \\left ( u , \\cdot \\right ) | \\mathcal { F } ^ { s } \\right ) ( x _ { 0 } ) , \\end{align*}"}
+{"id": "3809.png", "formula": "\\begin{align*} c _ 2 ( ( y _ 2 , x ) , ( y _ 2 , x ' ) ) = \\| x - x ' \\| _ { p } + \\kappa _ 2 \\| y _ 2 - y _ 2 ' \\| _ { p ' } \\end{align*}"}
+{"id": "5893.png", "formula": "\\begin{align*} \\psi \\equiv \\lim _ { n \\to \\infty } \\psi _ n = \\sum _ { k = 1 } ^ \\infty \\frac { ( 2 r ) ^ { k - 1 } } { ( 2 r + 1 ) ^ k } \\phi _ k , \\end{align*}"}
+{"id": "7308.png", "formula": "\\begin{align*} a \\prod _ { i = 1 } ^ n x _ i ^ { \\alpha _ i } = b \\prod _ { i = 1 } ^ n x _ i ^ { \\gamma _ i } , \\end{align*}"}
+{"id": "1132.png", "formula": "\\begin{align*} T ( f \\cdot g ) = f T ( g ) + T ( f ) g + 2 B ( A ( f ) , A ( g ) ) \\left ( f , g \\in P \\right ) , \\end{align*}"}
+{"id": "4016.png", "formula": "\\begin{align*} g ( \\alpha ) = \\frac { a \\alpha + b } { c \\alpha + d } , \\ , \\ \\ , \\ g = \\small { \\left ( \\begin{array} { c c c } a & b \\\\ c & d \\end{array} \\right ) } . \\end{align*}"}
+{"id": "5604.png", "formula": "\\begin{align*} h ( Z , \\lambda ) = \\int _ { G ^ { \\mathbb { N } } } \\int _ { X } \\log \\frac { d \\omega _ { 1 } . \\lambda } { d \\lambda } \\left ( \\psi \\left ( x , \\omega \\right ) \\right ) d \\eta _ { \\omega } ( x ) d \\mathbb { P } _ { \\mu } ( \\omega ) . \\end{align*}"}
+{"id": "5765.png", "formula": "\\begin{align*} \\dfrac { 1 } { q ( x ) } = \\dfrac { 1 } { q _ { 1 } ( x ) } + \\cdots + \\dfrac { 1 } { q _ { m } ( x ) } \\mbox { f o r a . e . } \\ x \\in \\mathbb { R } ^ { n } . \\end{align*}"}
+{"id": "9196.png", "formula": "\\begin{align*} W ( G _ { n , r , s } ) = \\binom { n } { 2 } + ( r - 1 ) ^ 2 n - r ( r - 1 ) ^ 2 \\end{align*}"}
+{"id": "4215.png", "formula": "\\begin{align*} K _ 1 & = T ( \\phi ) - T ( \\phi _ 0 ) T ( \\phi _ 1 ) \\cdots T ( \\phi _ R ) \\\\ K _ 2 & = T ( \\tilde { \\phi } ) - T ( \\tilde { \\phi } _ 0 ) T ( \\tilde { \\phi } _ 1 ) \\cdots T ( \\tilde { \\phi } _ R ) \\end{align*}"}
+{"id": "6369.png", "formula": "\\begin{align*} 0 = G ^ \\prime ( \\widetilde { y } , \\widetilde { z } ) \\widetilde { g } ( \\widetilde { y } , \\widetilde { z } ) = F ^ \\prime ( \\widetilde { y } ) \\widetilde { g } _ Y ( \\widetilde { y } , \\widetilde { z } ) - \\widetilde { g } _ Z ( \\widetilde { y } , \\widetilde { z } ) , \\end{align*}"}
+{"id": "5181.png", "formula": "\\begin{align*} b _ i = w _ { \\lambda ^ i } : = a + \\sum _ { j = 1 } ^ { k } w _ { \\lambda _ j ^ i } , \\end{align*}"}
+{"id": "2408.png", "formula": "\\begin{align*} \\frac { \\vec { \\delta } _ { 1 0 2 } } { | \\vec { \\delta } _ { 1 0 2 } | } \\cdot \\frac { \\vec { \\delta } _ { 3 0 4 } } { | \\vec { \\delta } _ { 3 0 4 } | } = \\frac { 1 } { \\sqrt { 2 ( 1 + \\cos \\alpha _ { 1 0 2 } ) } } \\frac { 1 } { \\sqrt { 2 ( 1 + \\cos \\alpha _ { 3 0 4 } ) } } ( - 2 ( 1 + \\cos \\alpha _ { 1 0 2 } ) ) . \\end{align*}"}
+{"id": "2746.png", "formula": "\\begin{align*} \\beta _ K = \\prod _ { j = 1 } ^ n \\frac { k _ j ^ 2 } { \\mu _ j ^ 2 } , K = ( k _ 1 , \\ldots , k _ n ) \\in \\mathbb { N } ^ n . \\end{align*}"}
+{"id": "3180.png", "formula": "\\begin{align*} M _ a ( f ) : = \\frac { 1 } { a ^ d } \\int _ { Q _ a } T _ t ( f ) d t . \\end{align*}"}
+{"id": "2894.png", "formula": "\\begin{align*} \\partial _ x ^ \\alpha \\big | _ { x = 0 } f ( \\xi + B x ) = \\partial _ { B , \\xi } ^ \\alpha f ( \\xi ) . \\end{align*}"}
+{"id": "6581.png", "formula": "\\begin{align*} \\boldsymbol { \\Phi } = { \\rm d i a g } ( e ^ { j \\phi _ { 1 , n _ t } } , \\cdots , e ^ { j \\phi _ { l , n _ t } } , \\cdots , e ^ { j \\phi _ { L , n _ t } } ) , \\end{align*}"}
+{"id": "1386.png", "formula": "\\begin{align*} X ^ { \\star } = \\{ f \\in H ^ * : f ( X ) \\subseteq R \\} . \\end{align*}"}
+{"id": "6690.png", "formula": "\\begin{align*} \\div v = ( \\delta _ { j i } - a _ { j i } ) \\partial _ j v _ i . \\end{align*}"}
+{"id": "3275.png", "formula": "\\begin{align*} \\rho ( t , w ) = \\rho ( 0 , 0 ) + \\rho _ t ( 0 , 0 ) t + 2 { \\rm R e } ( \\rho _ w ( 0 , 0 ) w ) + \\frac { 1 } { 2 } \\rho _ { t t } ( 0 , 0 ) t ^ 2 + \\end{align*}"}
+{"id": "397.png", "formula": "\\begin{align*} I - S ^ T S - ( R ^ T S ) ^ T \\left ( \\sum ^ { \\infty } _ { k = 0 } ( R ^ T R ) ^ k \\right ) ( R ^ T S ) \\geq 0 . \\end{align*}"}
+{"id": "6564.png", "formula": "\\begin{align*} \\int _ { \\mathbb H ^ n } \\frac { | y | _ h ^ { - Q / p } } { \\max ( 1 , | y | _ h ^ Q ) } d y = \\frac { \\omega _ Q Q } { ( Q - Q / p ) Q / p } , \\end{align*}"}
+{"id": "3649.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { \\mathrm { m d i m } } _ M ( Y ^ \\mathbb { N } \\times X , F , D ) = \\overline { \\mathrm { m d i m } } _ { I _ \\psi ( F ) } ( F , D ) & \\le \\overline { \\mathrm { m d i m } } _ { Y ^ \\mathbb { N } \\times I _ \\psi ( F ) } ( F , D ) \\\\ & \\le \\overline { \\mathrm { u m d i m } } _ { Y ^ \\mathbb { N } \\times I _ \\psi ( F ) } ( F , D ) . \\end{aligned} \\end{align*}"}
+{"id": "5935.png", "formula": "\\begin{align*} \\Pi _ { \\psi } [ - 1 , 0 ] f ( \\epsilon , x + x ^ { \\ast } ) = f ( - \\epsilon , x - x ^ { \\ast } ) , \\end{align*}"}
+{"id": "1388.png", "formula": "\\begin{align*} x v = \\lambda v , \\end{align*}"}
+{"id": "7606.png", "formula": "\\begin{align*} H _ { 2 , 1 } ( F _ { f } / 2 ) & \\leq \\frac { 1 } { 2 3 0 4 } \\left ( | - 3 \\tau ^ 4 _ { 1 } + 1 2 ( 1 - \\tau ^ 2 _ { 1 } ) \\tau ^ 2 _ { 1 } \\tau _ { 2 } - 8 ( 1 - \\tau ^ 2 _ { 1 } ) ( 2 + \\tau ^ 2 _ { 1 } ) \\tau ^ 2 _ { 2 } | \\right . \\\\ & \\left . + 2 4 \\tau _ { 1 } ( 1 - \\tau ^ 2 _ { 1 } ) ( 1 - | \\tau ^ 2 _ { 2 } | ) \\right ) \\\\ & = \\frac { 1 } { 9 6 } \\tau _ { 1 } ( 1 - \\tau ^ 2 _ { 1 } ) \\left ( | A + B \\tau _ { 2 } + C \\tau ^ 2 _ { 2 } | + 1 - | \\tau _ { 2 } | ^ 2 \\right ) , \\end{align*}"}
+{"id": "6411.png", "formula": "\\begin{align*} [ D \\widetilde { \\psi } : D \\tau _ M ] _ t = [ D \\psi : D \\varphi ] _ t \\lambda ^ { \\varphi } ( t ) \\end{align*}"}
+{"id": "3638.png", "formula": "\\begin{align*} ( \\{ 9 \\} ^ { I + 1 } , a _ { I + 1 } , \\ldots , \\{ 9 \\} ^ { I + 1 } , 4 ) _ { 1 0 } \\ = \\ 2 ( 4 , \\{ 9 \\} ^ { I + 1 } , \\ldots , a _ { I + 1 } , \\{ 9 \\} ^ { I } , 7 ) _ { 1 0 } \\end{align*}"}
+{"id": "6750.png", "formula": "\\begin{align*} \\mathcal { P } ( X _ \\eta ) = \\mathcal { P } ( X _ { \\eta ' } ) = \\mathcal { P } ( X _ { \\eta ^ * } ) . \\end{align*}"}
+{"id": "1862.png", "formula": "\\begin{align*} M = \\{ u \\in \\mathcal { U } : \\ G ( u ) \\in \\mathcal { K } \\} . \\end{align*}"}
+{"id": "3352.png", "formula": "\\begin{align*} I - \\mathfrak { z } _ 1 T _ { 1 , 0 } ^ * - \\mathfrak { z } _ 2 T _ { 2 , 0 } ^ * = \\begin{pmatrix} 1 & \\frac { 1 } { 2 } ( \\mathfrak { z } _ 1 + \\mathfrak { z } _ 2 ) & \\frac { 1 } { 2 } ( \\mathfrak { z } _ 1 + \\mathfrak { z } _ 2 ) \\\\ 0 & 1 + \\frac { 1 } { 2 } \\mathfrak { z } _ 2 & - \\frac { 1 } { 2 } \\mathfrak { z } _ 2 \\\\ 0 & - \\frac { 1 } { 2 } \\mathfrak { z } _ 1 & 1 + \\frac { 1 } { 2 } \\mathfrak { z } _ 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "7032.png", "formula": "\\begin{align*} f _ { \\textbf { Q } } : = f \\left ( \\textbf { Q } \\right ) = \\sum _ { j = 1 } ^ s b _ j \\textbf { Q } ^ { \\lambda _ j } \\mbox { a n d } f _ { \\tilde { \\textbf { Q } } } : = f \\left ( \\tilde { \\textbf { Q } } \\right ) = \\sum _ { j = 1 } ^ s b _ j \\tilde { \\textbf { Q } } ^ { \\lambda _ j } . \\end{align*}"}
+{"id": "7841.png", "formula": "\\begin{align*} \\mathbf { 1 } _ H ^ G = \\sum _ { i = 1 } ^ t m _ i \\phi _ i , \\end{align*}"}
+{"id": "4630.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t m = \\nabla \\cdot \\left ( D ( m ) \\nabla m - \\chi f ( m ) \\nabla c \\right ) + M ( m ) , \\\\ \\partial _ t c = \\Delta c - c + d + m , \\\\ \\partial _ t d = h ( m ) \\left ( 1 - d \\right ) , \\end{cases} x \\in \\Omega , \\ , t > 0 , \\end{align*}"}
+{"id": "7580.png", "formula": "\\begin{align*} & c _ 1 = 2 \\tau _ 1 , \\\\ & c _ 2 = 2 \\tau ^ 2 _ 1 + 2 ( 1 - \\tau ^ 2 _ 1 ) \\tau _ 2 \\end{align*}"}
+{"id": "5662.png", "formula": "\\begin{align*} s ' ( \\nu ) = - \\frac { \\frac { \\partial F } { \\partial \\mu } } { \\frac { \\partial F } { \\partial s } } \\Bigl | _ { \\mu = \\nu } = - \\frac { ( \\gamma _ p + \\gamma _ q ) B } { \\nu ( \\gamma _ p + \\gamma _ q ) ( \\gamma _ p + \\gamma _ q - 2 ) B + ( 2 2 ^ * _ \\mu - 2 ) C } , \\end{align*}"}
+{"id": "9020.png", "formula": "\\begin{align*} C ( p , p ' , t , \\tau , \\rho ) = \\frac { \\bar C ( p , p ' , \\rho ) } { \\tau - t } . \\end{align*}"}
+{"id": "4087.png", "formula": "\\begin{align*} r = r _ 1 + r _ 2 - 1 , \\end{align*}"}
+{"id": "6276.png", "formula": "\\begin{align*} \\mathcal { G } ^ M _ { r } ( K , \\gamma ) \\cap B ^ + _ { \\rho ( r ) } ( q ) = C N ^ + _ { r } ( \\xi ) \\cap B ^ + _ { \\rho ( r ) } ( q ) , \\end{align*}"}
+{"id": "7608.png", "formula": "\\begin{align*} | B | - 2 ( 1 - | C | ) & = \\frac { \\tau _ { 1 } } { 2 } - 2 \\left ( 1 - \\frac { ( 2 + \\tau ^ 2 _ { 1 } ) } { 3 \\tau _ { 1 } } \\right ) \\\\ & = \\frac { 7 \\tau ^ 2 _ { 1 } - 1 2 \\tau _ { 1 } + 8 } { 6 \\tau _ { 1 } } > 0 , \\end{align*}"}
+{"id": "1105.png", "formula": "\\begin{align*} b \\ast ( b ' \\ast x ) = 0 , \\forall b , b ' \\in B , \\ ; \\forall x \\in X . \\end{align*}"}
+{"id": "5572.png", "formula": "\\begin{align*} \\Psi & : Z \\to M : = G / Q \\times _ { \\beta } \\left ( \\Omega _ { 0 } \\times _ { \\sigma } M _ { F } \\right ) \\\\ & \\left ( \\left ( y , r , H \\right ) , A \\right ) \\mapsto \\left ( y , r , \\left ( H , \\bar { \\psi } _ { H } \\left ( r ^ { - 1 } \\bar { p } _ { k } \\left ( \\tau ( y ) ^ { - 1 } A \\right ) \\right ) \\right ) \\right ) , \\end{align*}"}
+{"id": "7355.png", "formula": "\\begin{align*} \\Pr [ \\hat { M } = M ] \\leq \\int _ { \\mathbb { R } ^ { n \\times n } } \\max _ { m } \\frac { s ! } { ( n + s ) ! } f ( g - \\mu m ) d g \\end{align*}"}
+{"id": "7392.png", "formula": "\\begin{align*} \\langle X _ N ( L , \\alpha ) ^ 2 \\rangle = \\int _ 0 ^ 1 \\big | X _ N ( L , \\alpha ) \\big | ^ 2 \\ , d \\alpha \\ll _ \\varepsilon L N ^ { - 3 + \\varepsilon } E _ N ( \\mathcal { A } ) \\end{align*}"}
+{"id": "1212.png", "formula": "\\begin{align*} \\gamma ( t ) G ( t , \\sigma ( t ) ) = \\gamma ( 0 ) G ( 0 , \\sigma ( 0 ) ) 0 \\leq t \\leq t _ 0 , \\end{align*}"}
+{"id": "6761.png", "formula": "\\begin{align*} h ( H \\times \\{ 0 , 1 \\} ^ \\Z , \\widetilde { R } ) = m _ H ( W \\setminus \\overline { W } ) = \\overline { d } - d ^ * = h ( X _ \\eta ) \\end{align*}"}
+{"id": "5489.png", "formula": "\\begin{align*} \\eta = \\int _ { G / H } \\boldsymbol { \\beta } _ { \\eta } ( w ) d \\nu _ { H } ( w ) = \\int _ { G / H } g . \\lambda d \\nu _ { H } ( g H ) . \\end{align*}"}
+{"id": "7129.png", "formula": "\\begin{align*} \\mathcal { R } _ \\varepsilon \\tilde { c } ( x ) : = \\int _ { ( \\mathbb { R } ^ n _ \\gamma ) ^ c } J _ \\varepsilon ( | x - y | ) \\big ( c ( x ) - \\tilde { c } ( y ) \\big ) \\ ; y \\end{align*}"}
+{"id": "7767.png", "formula": "\\begin{align*} z ^ { \\vec x } _ j : = \\begin{cases} ( M \\vec x ) _ { i } - \\max _ k ( M \\vec x ) _ k & j \\in J _ i , \\\\ 0 & j \\notin \\cup _ { i \\in [ n ] } J _ i . \\end{cases} \\end{align*}"}
+{"id": "549.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ { t } U _ { \\varepsilon } ( t , \\xi ) = i \\langle \\xi \\rangle A _ { \\varepsilon } ( t ) U _ { \\varepsilon } ( t , \\xi ) + i \\langle \\xi \\rangle ^ { - 1 } Q _ { \\varepsilon } ( t ) U _ { \\varepsilon } ( t , \\xi ) + F _ { \\varepsilon } ( t , \\xi ) , \\xi \\in \\mathcal { I } _ { \\hbar } , \\\\ U _ { \\varepsilon } ( 0 , \\xi ) = U _ { 0 } ( \\xi ) , \\xi \\in \\mathcal { I } _ { \\hbar } , \\end{array} \\right . \\end{align*}"}
+{"id": "4508.png", "formula": "\\begin{align*} - \\partial _ { x } ^ 2 \\phi + \\left ( \\omega - \\frac { c ^ 2 } { 4 } \\right ) \\phi + \\frac { c } { 2 } \\phi ^ 3 - \\frac { 3 } { 1 6 } \\phi ^ 5 = 0 . \\end{align*}"}
+{"id": "8129.png", "formula": "\\begin{align*} D ( I ) : = \\{ i _ 1 , i _ 1 + i _ 2 , \\dots , i _ 1 + \\dots + i _ { r - 1 } \\} = \\{ 1 \\leq i \\leq n - 1 \\ ; \\ \\varepsilon _ i = - \\} . \\end{align*}"}
+{"id": "4588.png", "formula": "\\begin{align*} & \\alpha ( f ( u ) ) = f ( \\beta ( u ) ) , \\\\ & f ( \\rho ( x , f ( u ) ) v ) = [ x , f ( u ) , f ( v ) ] , \\end{align*}"}
+{"id": "6733.png", "formula": "\\begin{align*} h ( X _ \\eta ) = 0 \\iff \\mathcal { P } ( X _ \\eta ) = \\{ \\nu _ { \\eta } \\} \\iff X _ \\eta \\end{align*}"}
+{"id": "5070.png", "formula": "\\begin{align*} \\lim _ { t \\uparrow T _ { \\max } } \\left \\| \\psi ( t ) \\right \\| _ { H ^ 2 _ N } = \\infty . \\end{align*}"}
+{"id": "7372.png", "formula": "\\begin{align*} R _ N ^ 2 ( L , \\alpha , \\Delta ) = L + o ( 1 ) \\end{align*}"}
+{"id": "1603.png", "formula": "\\begin{align*} \\langle S _ { \\xi \\chi } f , g \\rangle & = \\langle \\int _ { \\Theta } v ( w ) s ( w ) \\pi _ { G ( w ) } \\xi _ { w } ^ { \\ast } \\chi _ { w } \\pi _ { F ( w ) } f d \\mu ( w ) , g \\rangle \\\\ & = \\int _ { \\Theta } v ( w ) s ( w ) \\langle \\chi _ { w } \\pi _ { F ( w ) } f , \\xi _ { w } \\pi _ { G ( w ) } g \\rangle d \\mu ( w ) . \\end{align*}"}
+{"id": "554.png", "formula": "\\begin{align*} E _ { \\varepsilon } ( t , \\xi ) : = ( S _ { \\varepsilon } ( t ) U _ { \\varepsilon } ( t , \\xi ) , U _ { \\varepsilon } ( t , \\xi ) ) , \\end{align*}"}
+{"id": "9274.png", "formula": "\\begin{align*} \\| T _ { \\eta } f \\| _ { L ^ p ( x ^ \\gamma d x ) } ^ p & \\lesssim \\int _ 0 ^ \\infty x ^ { - 1 - \\tau + \\gamma } \\int _ x ^ \\infty \\abs { f ( z ) } ^ p z ^ { \\tau } \\ , d z \\ , d x = \\int _ 0 ^ \\infty \\abs { f ( z ) } ^ p z ^ { \\tau } \\int _ 0 ^ z x ^ { - \\tau - 1 + \\gamma } \\ , d x \\ , d z \\\\ & \\simeq \\| f \\| _ { L ^ p ( x ^ \\gamma d x ) } ^ p , \\end{align*}"}
+{"id": "8663.png", "formula": "\\begin{align*} u v ' + v u ' = u ' v + v ' u = - ( u , v ) \\in \\mathbb { F } \\subseteq \\mathcal { C } \\mathrm { \\ f o r \\ e v e r y \\ } u \\mathrm { \\ a n d \\ } v \\mathrm { \\ i n \\ } V . \\end{align*}"}
+{"id": "8325.png", "formula": "\\begin{align*} y _ { B 1 } = \\sqrt { 1 - \\alpha } h _ { A B } x + \\sqrt { \\alpha } h _ { C B } z _ d + n _ { B 1 } , \\end{align*}"}
+{"id": "7198.png", "formula": "\\begin{align*} a _ h = \\mathcal { L } ^ 2 \\left ( \\left \\{ f _ h \\neq g _ h \\right \\} \\cap B _ 1 \\right ) \\end{align*}"}
+{"id": "5903.png", "formula": "\\begin{align*} L u _ n + C _ n u _ n = f = L u _ f + K f \\ , , \\end{align*}"}
+{"id": "6251.png", "formula": "\\begin{align*} & \\mathrm { i n t } ( F _ 0 ( \\mathcal G ) ) = \\mathrm { i n t } ( O _ S [ g _ 1 ] ) = \\mathrm { i n t } ( \\partial ( O _ S [ g _ 1 ] ) + \\sum _ { s \\in S } T _ { \\tilde s } ( \\mathrm { S p a n } _ { \\mathbb C } \\{ g _ n \\ , | \\ , n \\geq 0 \\} ) ) \\\\ & = \\mathrm { i n t } ( \\partial ( O _ S [ g _ 1 ] ) ) + \\sum _ { s \\in S , n \\geq 0 } \\mathbb C \\cdot \\mathrm { i n t } ( T _ { \\tilde s } ( g _ n ) ) \\subset F _ 0 ( \\mathcal G ) + \\sum _ { s \\in S , n \\geq 0 } F _ 1 ( \\mathcal G ) = F _ 1 ( \\mathcal G ) \\end{align*}"}
+{"id": "6228.png", "formula": "\\begin{align*} \\partial ^ 2 _ v \\psi _ { \\mu _ N } = \\frac 1 { N + 1 } \\bigg ( - 1 + \\sum _ { i = 2 } ^ { N + 1 } \\partial ^ 2 _ v \\psi _ { \\delta _ { x _ i } } ( x _ 1 ) \\bigg ) \\geq \\frac { - 1 + ( \\alpha - 2 ) K _ N } { N + 1 } \\ , . \\end{align*}"}
+{"id": "5329.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ q = \\rho ^ { - d } \\| \\mathrm { T } _ { \\rho } f ^ { = d } \\| _ q \\le \\rho ^ { - d } \\| f ^ { = d } \\| _ 2 ^ { 2 / q } \\gamma ^ { 1 - 2 / q } \\le \\rho ^ { - d } \\gamma . \\end{align*}"}
+{"id": "8654.png", "formula": "\\begin{align*} a - b & = \\int _ 0 ^ 1 \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 2 } ( 1 - p x ) ^ { - 1 } y ^ j x ^ { j - 1 } ( y - x ) d y d x \\\\ & = \\int _ 0 ^ 1 \\int _ 0 ^ x ( 1 - p y ) ^ { - 2 } ( 1 - p x ) ^ { - 1 } y ^ j x ^ { j - 1 } ( y - x ) d y d x \\\\ & \\quad + \\int _ 0 ^ 1 \\int _ x ^ 1 ( 1 - p y ) ^ { - 2 } ( 1 - p x ) ^ { - 1 } y ^ j x ^ { j - 1 } ( y - x ) d y d x . \\end{align*}"}
+{"id": "4250.png", "formula": "\\begin{align*} Y _ { t } = \\xi + \\int _ { t } ^ { T } f \\left ( s , Y _ { s } , Z _ { s } \\right ) \\mathrm { d } s - \\int _ { t } ^ { T } Z _ { s } \\mathrm { d } W _ { s } \\end{align*}"}
+{"id": "2393.png", "formula": "\\begin{align*} \\vec { u } ( A _ { 0 } , A _ { 2 } ) + \\vec { u } ( A _ { 0 } , A _ { 3 } ) = - ( \\vec { u } ( A _ { 0 } , A _ { 1 } ) + \\vec { u } ( A _ { 0 } , A _ { 4 } ) ) \\end{align*}"}
+{"id": "7119.png", "formula": "\\begin{align*} \\mathcal { R } _ \\varepsilon c ( x ) : = \\int _ { \\Omega ^ c } J _ \\varepsilon ( | x - y | ) \\big ( c ( x ) - \\tilde { c } ( y ) \\big ) \\ : y \\end{align*}"}
+{"id": "5850.png", "formula": "\\begin{align*} \\theta ( I ) = \\int _ I Q ( E ) d \\rho ( E ) . \\end{align*}"}
+{"id": "815.png", "formula": "\\begin{align*} y \\left ( x \\right ) & = \\widetilde { y } \\left ( \\frac { x - 2 } { 3 } \\right ) = 9 \\left ( \\frac { x - 2 } { 3 } \\right ) ^ { 2 } + 1 2 \\left ( \\frac { x - 2 } { 3 } \\right ) + 4 \\\\ & = x ^ { 2 } \\end{align*}"}
+{"id": "4332.png", "formula": "\\begin{align*} \\tilde { y } _ { T _ { ( k , m , i , n ) } } ^ { \\mathcal { K } ' } \\circ \\big ( \\tilde { y } _ { t } ^ { \\mathcal { K } ' } \\big ) ^ { - 1 } = \\tilde { y } _ { T _ { ( k , m , i , n ) } } ^ { \\mathcal { K } } \\circ \\big ( \\tilde { y } _ t ^ { \\mathcal { K } } \\big ) ^ { - 1 } . \\end{align*}"}
+{"id": "4338.png", "formula": "\\begin{gather*} \\mathrm { e r f } ( x ) = \\int _ { - \\infty } ^ x e ^ { - y ^ 2 } \\ ; d y , \\\\ C _ 0 = \\mathrm { e r f } \\left ( 0 \\right ) , \\\\ a = 1 - 2 ^ { - \\lfloor k / 2 \\rfloor } . \\end{gather*}"}
+{"id": "8433.png", "formula": "\\begin{align*} \\left \\{ \\intop _ { \\mu - i T } ^ { \\mu + i T } + \\intop _ { \\mu + i T } ^ { \\frac { 1 } { 2 } - \\mu + i T } + \\intop _ { \\frac { 1 } { 2 } - \\mu + i T } ^ { \\frac { 1 } { 2 } - \\mu - i T } + \\intop _ { \\frac { 1 } { 2 } - \\mu - i T } ^ { \\mu - i T } \\right \\} \\ , \\frac { \\zeta _ { p ^ { \\prime } } ( 1 - s ) \\ , \\zeta _ { p } ( 2 s ) } { 2 \\cos \\left ( \\frac { \\pi s } { 2 } \\right ) } \\ , \\left ( \\frac { x } { 2 \\pi } \\right ) ^ { - s } \\ , d s = 2 \\pi i \\ , R _ { p , p ^ { \\prime } } ( x ) , \\end{align*}"}
+{"id": "1516.png", "formula": "\\begin{align*} & \\upsilon _ 0 = \\upsilon \\wedge \\frac { ( ( \\Lambda _ 0 ) ^ 3 - \\widehat { r } _ 0 ^ 3 ) ^ 2 } { 3 6 C _ \\varepsilon ^ 2 \\Lambda _ 0 ^ 4 } . \\end{align*}"}
+{"id": "2749.png", "formula": "\\begin{align*} u = \\sum _ { k \\ge 1 } \\frac { ( f | \\phi _ q ^ k ) } { \\lambda _ q ^ k - \\lambda } \\phi _ q ^ k , \\end{align*}"}
+{"id": "5354.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ q = \\rho ^ { - d } \\| \\mathrm { T } _ { \\rho } f ^ { = d } \\| _ q \\le \\rho ^ { - d } \\| f ^ { = d } \\| _ 2 ^ { 2 / q } \\gamma ^ { 1 - 2 / q } \\le \\rho ^ { - d } \\gamma . \\end{align*}"}
+{"id": "6230.png", "formula": "\\begin{align*} \\psi _ i ( a ) = \\int _ { Q _ i } g ( b - a ) \\ , d \\mu ( b ) \\forall \\ , i \\in \\{ 1 , \\ , 2 , \\ , 3 \\} \\ , , & & \\psi _ \\infty ( a ) = \\int _ { \\R ^ 2 \\setminus ( Q _ 1 \\cup Q _ 2 \\cup Q _ 3 ) } g ( b - a ) \\ , d \\mu ( b ) \\ , . \\end{align*}"}
+{"id": "4408.png", "formula": "\\begin{align*} L _ { \\omega , \\mathbf { c } } ( U _ n ) = 6 S _ { \\omega , \\mathbf { c } } ( U _ n ) - 2 K _ { \\omega , \\mathbf { c } } ( U _ n ) \\ \\rightarrow \\ 6 \\mu _ { \\omega , \\mathbf { c } } . \\end{align*}"}
+{"id": "7342.png", "formula": "\\begin{align*} x ^ n + y ^ m = z ^ k \\end{align*}"}
+{"id": "439.png", "formula": "\\begin{align*} \\vec U ^ T ( I _ 2 \\otimes P _ x ) \\vec U _ t + \\vec U ^ T ( I _ 2 \\otimes Q _ x ) { \\bf A } \\vec U ) + \\vec U ^ T ( I _ 2 \\otimes P _ x ) { \\bf A } ^ T { \\bf D _ x } ( \\vec U ) = 0 . \\end{align*}"}
+{"id": "2247.png", "formula": "\\begin{align*} h ( r e ^ { i \\theta } ) = \\frac { 1 } { 2 \\pi } \\langle h _ b , P _ r ( \\theta - \\cdot ) \\rangle . \\end{align*}"}
+{"id": "4447.png", "formula": "\\begin{align*} K _ { \\omega , \\mathbf { c } } ( V ) = 2 S _ { \\omega , \\mathbf { c } } ( V ) + N ( V ) < 0 . \\end{align*}"}
+{"id": "4526.png", "formula": "\\begin{align*} f ' ( \\lambda ) = 2 \\lambda L ( U ) + \\left ( \\frac { d } { 2 } + 1 \\right ) \\lambda ^ { \\frac { d } { 2 } } N ( U ) + \\mathbf { c } \\cdot \\mathbf { P } ( U ) . \\end{align*}"}
+{"id": "5285.png", "formula": "\\begin{align*} \\| 1 + d Z \\| _ { q } ^ { q } \\ge 1 + \\binom { \\lfloor q \\rfloor } { 2 } d ^ 2 \\ge 1 + \\frac { q ^ 2 } { 6 } d ^ 2 . \\end{align*}"}
+{"id": "6511.png", "formula": "\\begin{align*} C T _ { \\vec { t } } \\ , \\prod _ { i = 0 } ^ { c - 1 } ( 1 + t _ i ) ^ a \\left ( 1 + t _ i ^ { - 1 } \\right ) ^ b \\prod _ { i \\neq j } ^ { 0 , c - 1 } \\left ( 1 - t _ j t _ i ^ { - 1 } \\right ) ^ m = \\prod _ { \\ell = 0 } ^ { c - 1 } \\frac { ( a + b + \\ell m ) ! \\ , ( ( \\ell + 1 ) m ) ! } { ( a + \\ell m ) ! \\ , ( b + \\ell m ) ! \\ , m ! } . \\end{align*}"}
+{"id": "559.png", "formula": "\\begin{align*} \\| w _ { \\varepsilon } ( t , \\cdot ) \\| ^ { 2 } _ { \\mathrm { H } _ { \\mathcal { H } _ { \\hbar , V } } ^ { 1 + s } } + \\| \\partial _ { t } w _ { \\varepsilon } ( t , \\cdot ) \\| ^ { 2 } _ { \\mathrm { H } _ { \\mathcal { H } _ { \\hbar , V } } ^ { s } } \\lesssim \\varepsilon ^ { - 2 L _ { 1 } - L _ { 2 } - 1 } \\varepsilon ^ { 2 L _ { 1 } + L _ { 2 } + 1 + q } = \\varepsilon ^ { q } , q \\in \\mathbb { N } _ { 0 } , \\end{align*}"}
+{"id": "3030.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T ^ * T + x ^ 2 S ^ * S ) \\leq \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T ^ * T ) + \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( x ^ 2 S ^ * S ) \\end{align*}"}
+{"id": "3351.png", "formula": "\\begin{align*} T _ { 1 , 0 } ^ * = T _ 1 ^ * ( I - e _ 1 e _ 1 ^ * ) T _ { 2 , 0 } ^ * = T _ 2 ^ * ( I - e _ 1 e _ 1 ^ * ) . \\end{align*}"}
+{"id": "658.png", "formula": "\\begin{align*} p _ { a } ( D ^ { n } \\varphi ) : = \\max _ { \\alpha \\in \\mathcal { A } } \\sup _ { x \\in ( l _ \\alpha , r _ \\alpha ) } \\max \\{ | D ^ { n + 1 } \\varphi ( x ) ( x - l _ \\alpha ) ^ { 1 + a } | , | D ^ { n + 1 } \\varphi ( x ) ( r _ \\alpha - x ) ^ { 1 + a } | \\} \\end{align*}"}
+{"id": "4443.png", "formula": "\\begin{align*} \\mu _ { \\omega , \\sqrt { \\omega } \\ \\mathbf { c } _ 0 } = \\omega \\mu _ { 1 , \\mathbf { c } _ 0 } = \\omega S _ { 1 , \\mathbf { c } _ 0 } ( \\Phi ) = \\omega ( Q ( \\Phi ) - E ( \\Phi ) ) . \\end{align*}"}
+{"id": "7919.png", "formula": "\\begin{align*} \\Psi \\left [ \\sigma \\curvearrowright ^ { \\mathbf { n } } \\tau \\right ] = \\tfrac { 1 } { \\mathbf { n } ! } \\Psi \\left [ \\sigma \\right ] D ^ { ( \\mathbf { n } ) } \\Psi \\left [ \\tau \\right ] . \\end{align*}"}
+{"id": "8438.png", "formula": "\\begin{align*} F _ { p } ( a ) : = \\left \\{ \\frac { e ^ { i \\frac { \\pi } { 4 } } } { \\sigma \\left ( \\sqrt { a } \\ , e ^ { i \\frac { \\pi } { 4 } } \\right ) \\ , e ^ { 2 \\pi \\sqrt { a } \\ , e ^ { i \\frac { \\pi } { 4 } } } - 1 } \\right \\} = \\frac { G _ { p } ( a ) } { 2 \\sqrt { 2 } } , \\end{align*}"}
+{"id": "8065.png", "formula": "\\begin{align*} X _ 1 = \\frac { \\partial } { \\partial x _ 1 } , \\enspace \\ldots , \\enspace X _ { N - 1 } = \\frac { \\partial } { \\partial x _ { N - 1 } } , \\enspace X _ { N } = x _ { 1 } ^ \\beta \\frac { \\partial } { \\partial x _ { N } } \\end{align*}"}
+{"id": "2563.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) , \\ \\ 0 \\le u \\le 1 & \\ \\ B _ R ( 0 ) \\backslash Q _ r ( 0 ) , \\\\ u \\equiv 1 , \\ \\ & \\ \\ Q _ r ( 0 ) . \\end{cases} \\end{align*}"}
+{"id": "3192.png", "formula": "\\begin{align*} \\langle B _ l ' ( y ' ) x \\Omega _ { \\rho } , \\Omega _ { \\rho } \\rangle _ { \\rho } = \\langle y ' B _ l ( x ) \\Omega _ { \\rho } , \\Omega _ { \\rho } \\rangle _ { \\rho } , x \\in M , y ' \\in M ' . \\end{align*}"}
+{"id": "5865.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\Lambda _ { L , L ' } } = \\bigcap _ { E \\in \\sigma ( H _ { \\omega , \\Lambda _ { L ' } } ) } \\mathcal { N } _ { \\Lambda _ { L , L ' } } ^ { ( E ) } \\in \\mathcal { F } _ { \\Lambda _ L } \\end{align*}"}
+{"id": "1572.png", "formula": "\\begin{align*} & \\sum _ { x \\in \\Z ^ { d } } \\sum _ { w \\in \\mathcal { W } _ 2 ^ { S A W } ( 0 , x ) } \\prod _ { j = 0 } ^ { n - 1 } D ( w _ { j } , w _ { j + 1 } ) = \\sum _ { x \\neq 0 } \\sum _ { x _ { 1 } \\neq 0 , x } D ( 0 , x _ { 1 } ) D ( x _ { 1 } , x ) \\\\ & = \\sum _ { x _ { 1 } \\neq 0 } D ( 0 , x _ { 1 } ) \\sum _ { x \\neq x _ { 1 } , 0 } D ( x _ { 1 } , x ) \\leq \\sum _ { x _ { 1 } \\neq 0 } D ( 0 , x _ { 1 } ) \\sum _ { x \\neq x _ { 1 } } D ( x _ { 1 } , x ) = \\left ( \\sum _ { z \\neq 0 } D ( 0 , z ) \\right ) ^ { 2 } . \\end{align*}"}
+{"id": "3337.png", "formula": "\\begin{align*} \\widetilde { \\gamma } _ { \\xi } = \\frac { 1 - a \\xi } { \\xi - a } \\ , \\gamma _ { \\xi } . \\end{align*}"}
+{"id": "615.png", "formula": "\\begin{align*} a ^ * a - a a ^ * = 2 i ( b c - c b ) \\end{align*}"}
+{"id": "1902.png", "formula": "\\begin{align*} y = \\mathcal { L } u , \\end{align*}"}
+{"id": "4805.png", "formula": "\\begin{align*} \\tilde { \\mathcal { L } } ( V ) = \\tau ^ { - 1 } \\langle c , A \\psi - \\phi \\rangle + \\tau ^ { - 1 } \\sum _ { i = 1 } ^ m \\langle c , B _ i \\psi \\rangle u _ i < 0 . \\end{align*}"}
+{"id": "8133.png", "formula": "\\begin{align*} \\varphi ^ + ( F _ I ) = \\tilde F _ I ( - X ) = ( - 1 ) ^ { | I | } \\tilde F _ { \\bar I ^ \\sim } , \\varphi ^ - ( F _ I ) = V \\varphi _ + ( \\tilde F _ I ) . \\end{align*}"}
+{"id": "4664.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta u + V ( \\epsilon x ) u = \\lambda u + u \\log u ^ 2 , \\hbox { i n } \\mathbb { R } ^ N , \\\\ & \\int _ { \\mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "412.png", "formula": "\\begin{align*} S ^ { - 1 } \\left ( \\begin{bmatrix} \\sqrt { | u _ n | } & 0 \\\\ 0 & \\sqrt { | u _ n | } \\end{bmatrix} \\begin{bmatrix} u _ n + p / u _ n \\\\ u _ { \\tau } \\end{bmatrix} - \\begin{bmatrix} R _ 1 \\\\ R _ 2 \\end{bmatrix} p / \\sqrt { | u _ n | } \\right ) = G \\end{align*}"}
+{"id": "3116.png", "formula": "\\begin{align*} t ^ n = \\sum _ { k = 0 } ^ { n } S _ B [ n , k ] ( t ) _ { k , q } ^ B , \\end{align*}"}
+{"id": "6020.png", "formula": "\\begin{align*} \\chi ( g n _ 2 ^ - ) & = ( \\tfrac { c ' + d } { d } ) e ^ { \\tfrac { - \\pi i } { 1 2 } [ - \\tfrac { c + 2 d } { d } + \\tfrac { b } { d } + 1 2 s ( c + 2 d , d ) ] } \\\\ & = ( \\tfrac { c ' } { d } ) e ^ { \\tfrac { - \\pi i } { 1 2 } [ - \\tfrac { c } { d } + \\tfrac { b } { d } + 1 2 s ( c , d ) ] } e ^ { \\tfrac { \\pi i } { 6 } } \\\\ & = \\chi ( g ) \\chi ( n _ 2 ^ - ) . \\end{align*}"}
+{"id": "528.png", "formula": "\\begin{align*} Q _ { m } u ( k ) : = \\left \\{ \\begin{array} { c c } Q u ( k ) , & | k | \\leq m , \\\\ 0 , & \\mathrm { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
+{"id": "2179.png", "formula": "\\begin{align*} \\mathcal { Y } \\mathcal { T } \\mathbf { u } = \\mathcal { Y } \\mathbf { f } , \\end{align*}"}
+{"id": "2407.png", "formula": "\\begin{align*} \\vec { \\delta } _ { 1 0 2 } \\cdot \\vec { \\delta } _ { 1 0 3 } = 1 + \\cos \\alpha _ { 1 0 2 } + \\cos \\alpha _ { 1 0 3 } + \\cos \\alpha _ { 2 0 3 } . \\end{align*}"}
+{"id": "4638.png", "formula": "\\begin{align*} 2 \\nabla f \\cdot \\nabla \\Delta f = \\Delta | \\nabla f | ^ 2 - 2 | D ^ 2 f | ^ 2 , \\end{align*}"}
+{"id": "8874.png", "formula": "\\begin{align*} \\frac { p \\binom { n } { r } - i + 1 } { q \\binom { n } { r } - i + 1 } & \\ge \\frac { p \\binom { n } { r } - n + 1 } { q \\binom { n } { r } - n + 1 } \\\\ & \\ge \\frac { p n / 2 - 1 } { q n / 2 - 1 } \\\\ & = \\frac { p } { q } \\left ( 1 - \\frac { q / p - 1 } { q n / 2 - 1 } \\right ) \\\\ & > \\frac { p } { q } \\left ( 1 - \\frac { 2 / p } { n } \\right ) \\ ; . \\end{align*}"}
+{"id": "370.png", "formula": "\\begin{align*} \\eta ( \\sigma ) ^ * ( y _ { \\tau , \\mathfrak { u } } ) = \\left ( \\prod ^ k _ { \\ell = 1 } \\frac { c _ { i _ { \\ell } } ^ { \\sigma } } { c _ { i _ { \\ell } } ^ { \\tau } } \\right ) x _ { \\tau , \\mathfrak { u } } \\in H ^ * ( ( \\underline { X } , \\underline { \\ast } ) ^ { \\sigma } ; R ) . \\end{align*}"}
+{"id": "1766.png", "formula": "\\begin{align*} \\mathbf { v } ( t , z ) = 0 , \\forall z \\in U . \\end{align*}"}
+{"id": "7504.png", "formula": "\\begin{align*} W ( q _ - , q _ + ) ( z ) + 2 \\zeta q _ - ( z ) q _ + ( z ) = \\Lambda ( z ) , \\end{align*}"}
+{"id": "4154.png", "formula": "\\begin{align*} \\vec { v } _ { s , t , f , r } = A _ { 3 , r , 0 } A _ { 2 , f , t } \\overline { A _ { 1 , t , s } A _ { 3 , t , s } A _ { 2 , s , t } } \\end{align*}"}
+{"id": "4514.png", "formula": "\\begin{align*} L ( \\Phi ^ { \\lambda } ) = \\lambda ^ 2 L ( \\Phi ) , \\ \\ N ( \\Phi ^ { \\lambda } ) = \\lambda ^ { \\frac { d } { 2 } + 1 } N ( \\Phi ) , \\ \\ M ( \\Phi ^ { \\lambda } ) = M ( \\Phi ) , \\ \\ \\mathbf { P } ( \\Phi ^ { \\lambda } ) = \\lambda \\mathbf { P } ( \\Phi ) , \\end{align*}"}
+{"id": "7252.png", "formula": "\\begin{align*} B _ n ( t ) = \\frac { 1 } { \\sqrt { n } } \\Big ( \\sum _ { k = 1 } ^ { [ n t ] } X _ k + ( n t - [ n t ] ) X _ { [ n t ] } \\Big ) \\ , . \\end{align*}"}
+{"id": "8529.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow 0 ^ { + } , p ^ { \\prime } \\rightarrow \\infty } \\tilde { \\zeta } _ { p , p ^ { \\prime } } ( s , c ) = \\sum _ { m \\neq 0 , \\ , n \\in \\mathbb { Z } } \\frac { ( - 1 ) ^ { m } } { \\left ( m ^ { 2 } + c \\ , n ^ { 2 } \\right ) ^ { s } } , \\end{align*}"}
+{"id": "6043.png", "formula": "\\begin{align*} g ( z ) : = | A | z ^ { n + m } + | B | \\overline { z } ^ m - | C | , z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "8713.png", "formula": "\\begin{align*} h ( \\widetilde { X } _ { i _ 1 , i _ 2 } ) = \\sum ^ { l } _ { s = 0 } \\frac { h ^ { ( s ) } ( 0 ) \\widetilde { X } _ { i _ 1 , i _ 2 } ^ s } { s ! } + c _ { l + 1 , \\tau _ 1 } ( \\widetilde { X } _ { i _ 1 , i _ 2 } ) \\widetilde { X } _ { i _ 1 , i _ 2 } ^ { l + 1 } , \\end{align*}"}
+{"id": "1358.png", "formula": "\\begin{align*} \\begin{aligned} \\psi ( F _ { L } ) - \\psi ( F _ { R } ) & = \\int _ { F _ { L } } ^ { F _ { R } } \\eta '' ( y ) ( A ( t , y ) - A ( t , F _ { L } ) ) d y - \\eta ' ( F _ { R } ) ( A ( t , F _ { R } ) - A ( t , F _ { L } ) ) \\\\ & \\geq \\dot { s } ( t ) \\int _ { F _ { L } } ^ { F _ { R } } \\eta '' ( y ) ( y - F _ { L } ) d y - \\eta ' ( F _ { R } ) ( A ( t , F _ { R } ) - A ( t , F _ { L } ) . \\end{aligned} \\end{align*}"}
+{"id": "1518.png", "formula": "\\begin{align*} I ( A , B ) : = H ( A ) - H ( A | B ) = H ( B ) - H ( B | A ) = H ( A ) + H ( B ) - H ( A , B ) . \\end{align*}"}
+{"id": "7205.png", "formula": "\\begin{align*} \\abs { v _ h } = t _ h \\mbox { a . e . i n } B _ 1 , \\end{align*}"}
+{"id": "5798.png", "formula": "\\begin{align*} A _ 1 ^ { \\frac { 1 } { k } } ( z ) & = ( A + h ) ^ { \\frac { 1 } { k } } \\\\ & = A ^ { \\frac { 1 } { k } } \\left ( 1 + \\frac { h } { A } \\right ) ^ { \\frac { 1 } { k } } \\\\ & = A ^ { \\frac { 1 } { k } } \\left ( 1 + O \\left ( { \\frac { | h | } { | A | } } \\right ) \\right ) , \\end{align*}"}
+{"id": "7685.png", "formula": "\\begin{align*} \\Bigl \\langle \\sum _ { \\gamma \\in D _ L } a _ { \\gamma } \\mathfrak { e } _ \\gamma , \\sum _ { \\delta \\in D _ L } b _ \\delta \\mathfrak { e } _ \\delta \\Bigr \\rangle _ L = \\sum _ { \\gamma \\in D _ L } a _ \\gamma \\overline { b _ \\gamma } . \\end{align*}"}
+{"id": "3787.png", "formula": "\\begin{align*} \\boldsymbol { K } _ c ( \\mu , \\nu ) = \\inf _ { \\pi \\in \\Pi ( \\mu , \\nu ) } \\int _ { \\mathcal { X } \\times \\mathcal { X } } c \\ , d \\pi . \\end{align*}"}
+{"id": "3296.png", "formula": "\\begin{align*} \\widetilde { \\rho } _ j ( \\xi , w ) = { \\rm I m } \\left ( e ^ { i \\pi ( 1 - \\beta _ j ) } \\frac { ( \\xi - 1 ) ^ { \\delta _ 1 } ( \\xi + 1 ) ^ { \\delta _ { - 1 } } } { T _ j ( \\xi ) ^ { \\delta _ j } } w \\right ) . \\end{align*}"}
+{"id": "8624.png", "formula": "\\begin{align*} \\frac { P ( Z _ 0 ( \\Delta \\ell _ 1 ) = n ) } { P ( W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 , Z _ 0 ( \\Delta \\ell _ 1 ) = n ) } & = \\frac 1 { P ( W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 | Z _ 0 ( \\Delta \\ell _ 1 ) = n ) } \\\\ & = \\frac 1 { n \\nu \\Delta p ^ k _ { j , + } ( t - \\Delta \\ell _ 1 ) } , \\end{align*}"}
+{"id": "9038.png", "formula": "\\begin{align*} R _ { \\theta _ 0 } ( z _ 1 , z _ 2 , u ) & = ( z _ 1 \\cos \\theta _ 0 - z _ 2 \\sin \\theta _ 0 , z _ 1 \\sin \\theta _ 0 + z _ 2 \\cos \\theta _ 0 , u ) ; \\\\ \\eta ( z _ 1 , z _ 2 , u ) & = ( \\bar z _ 1 , \\bar z _ 2 , u ) . \\end{align*}"}
+{"id": "5998.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } ( p ) A ( [ \\epsilon , x ] ) = a ^ { \\tfrac { 1 } { 2 } } e ^ { - ( \\det p ) \\epsilon \\pi x ^ 2 a ^ 2 } \\cdot e ^ { \\pi i \\epsilon ( \\det p ) x ^ 2 a b } = a ^ { \\tfrac { 1 } { 2 } } e ^ { i \\epsilon \\pi x ^ 2 z _ p } . \\end{align*}"}
+{"id": "4142.png", "formula": "\\begin{align*} \\left ( \\lfloor y ^ 2 + f y + r \\rfloor = r \\right ) \\wedge \\left ( \\lfloor y \\rfloor \\right ) = 0 . \\end{align*}"}
+{"id": "7220.png", "formula": "\\begin{align*} \\limsup _ { h \\to + \\infty } { ( _ h ) } = c \\left ( \\alpha ( \\rho ) - \\alpha ( \\rho ' ) \\right ) , \\end{align*}"}
+{"id": "5115.png", "formula": "\\begin{align*} \\int _ { B _ { 1 } } { b } ^ { i j , k l } f _ { i j } [ \\tau ^ { 4 } f ] _ { k l } d x = 0 , \\end{align*}"}
+{"id": "2160.png", "formula": "\\begin{align*} \\delta _ { 0 } = \\delta _ { 0 } \\left ( N , \\varepsilon , \\Omega , T , c , \\rho \\right ) \\in \\left ( 0 , 1 \\right ) , \\end{align*}"}
+{"id": "2485.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle Q ( \\xi _ n , - \\eta _ n ) = Q \\left ( ( u ^ * _ n ) _ { \\beta _ n } , - ( v ^ * _ n ) _ { \\beta _ n } \\right ) = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "8409.png", "formula": "\\begin{align*} ( E _ 1 = L K _ X ^ { \\frac { 1 } { 2 } } \\oplus L K _ X ^ { - \\frac { 1 } { 2 } } \\oplus L ^ { - 1 } K _ X ^ { \\frac { 1 } { 2 } } \\oplus L ^ { - 1 } K _ X ^ { - \\frac { 1 } { 2 } } , \\theta _ 1 = \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} ) , \\end{align*}"}
+{"id": "757.png", "formula": "\\begin{align*} g _ { i j } = \\rho ^ 2 u _ i u _ j + \\phi ^ 2 s _ { i j } . \\end{align*}"}
+{"id": "1722.png", "formula": "\\begin{align*} \\Psi ( \\nu , \\mu ) ( x ) = \\frac { 1 } { Z ( \\nu , \\mu ) } \\exp \\left ( { - \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta \\nu } ( \\nu , \\mu , x ) - U ^ { \\pi } ( x ) } \\right ) , \\end{align*}"}
+{"id": "7760.png", "formula": "\\begin{align*} M _ { r , c ( i ) } \\in \\begin{cases} - \\infty & \\emph { i f } \\ , \\ , r \\neq i , \\\\ \\R & \\emph { i f } \\ , \\ , r = i . \\end{cases} \\end{align*}"}
+{"id": "2787.png", "formula": "\\begin{align*} 0 = \\int _ M \\psi ( q _ 2 - q _ 1 ) u _ 2 u _ 1 d x + \\int _ M [ \\Delta , \\psi ] u u _ 1 d x . \\end{align*}"}
+{"id": "9100.png", "formula": "\\begin{align*} \\mathfrak { S } ( f _ k ) = \\mathfrak { S } ( f _ k | _ { \\tilde { G } } ) \\geq \\tilde { k } + \\tilde { r } - 1 - \\ell ' + \\ell _ + , \\end{align*}"}
+{"id": "1888.png", "formula": "\\begin{align*} \\min \\limits _ { u \\in H _ 0 ^ 1 ( \\Omega ) } \\int _ \\Omega \\nabla u \\cdot \\nabla u \\ d x \\mbox { s . t . } \\ \\ \\| u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 = 1 . \\end{align*}"}
+{"id": "808.png", "formula": "\\begin{align*} ( a ) K & = K \\left ( x , t \\right ) \\\\ ( b ) K & = K \\left ( y \\left ( t \\right ) , t \\right ) \\\\ ( c ) K & = K \\left ( y ^ { \\prime } \\left ( t \\right ) , t \\right ) \\\\ ( d ) K & = K _ { 1 } \\left ( x \\right ) K _ { 2 } \\left ( y \\left ( t \\right ) , t \\right ) . \\end{align*}"}
+{"id": "3404.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) + \\lambda u , \\\\ \\int _ { \\R ^ N } u ^ 2 = \\alpha . \\end{cases} \\end{align*}"}
+{"id": "2457.png", "formula": "\\begin{align*} ( \\rho \\cdot \\varphi ) ( x ) = \\rho ( \\tau ( x ) ) \\varphi ( x ) & & \\rho \\in C ^ \\infty ( B ) , \\varphi \\in C _ c ^ \\infty ( M ) , x \\in M . \\end{align*}"}
+{"id": "7635.png", "formula": "\\begin{align*} x _ 1 = \\sqrt { \\frac { 2 } { 3 } } \\approx 0 . 8 1 6 4 9 7 . \\end{align*}"}
+{"id": "7154.png", "formula": "\\begin{align*} \\partial _ t c = - \\mathcal { L } _ \\varepsilon c - f ^ \\prime ( c ) \\ ; \\ ; \\ ; \\Omega _ T \\end{align*}"}
+{"id": "902.png", "formula": "\\begin{align*} S ' & \\ll \\widehat { C } _ 1 \\widehat { C } _ 2 | d | \\widehat { V } ^ { n ( 1 - \\kappa ) } \\widehat { Y } ^ { n / 2 + 1 + \\varepsilon } \\sum _ { \\substack { | r | } = \\widehat { Y } } | b _ 2 | ^ { 1 / 2 } | r _ 3 '' | ^ { ( 1 - \\kappa ) / 2 } | r _ 3 | ^ { \\kappa ( n - 1 ) / 2 } \\\\ & \\ll \\widehat { C } _ 1 \\widehat { C } _ 2 | d | \\widehat { V } ^ { n ( 1 - \\kappa ) } \\widehat { Y } ^ { n / 2 + 2 + \\varepsilon } \\sum _ { | b _ 2 r _ 3 ' r _ 3 '' | \\leq \\widehat { Y } } | b _ 2 | ^ { - 1 / 2 } | r _ 3 ' | ^ { ( n - 1 ) / ( 2 n - 4 ) - 1 } . \\end{align*}"}
+{"id": "6911.png", "formula": "\\begin{align*} \\mathcal { M ' } : = \\{ m _ j : j \\in \\mathcal { J } , m _ j = \\min \\{ [ ( 1 + T ^ { - 1 } ) ^ { j } , ( 1 + T ^ { - 1 } ) ^ { j + 1 } ) \\cap \\mathcal { M } \\} . \\end{align*}"}
+{"id": "405.png", "formula": "\\begin{align*} U ^ T ( n _ i A _ i ) U - \\epsilon U ^ T ( n _ i D _ i ) = U ^ T \\tilde A U - \\epsilon U ^ T \\tilde F - \\epsilon \\tilde F ^ T U = \\begin{bmatrix} U \\\\ \\epsilon \\tilde F \\end{bmatrix} ^ T \\begin{bmatrix} \\tilde A & - I \\\\ - I & 0 \\end{bmatrix} \\begin{bmatrix} U \\\\ \\epsilon \\tilde F \\end{bmatrix} \\end{align*}"}
+{"id": "2593.png", "formula": "\\begin{align*} | q _ i | \\leq 2 \\epsilon \\ \\ \\ i = 1 , \\cdots , n - 1 . \\end{align*}"}
+{"id": "744.png", "formula": "\\begin{align*} \\eta _ { 0 } ( x ) = A \\ , \\cos ( k x ) \\ A = \\frac { 1 } { 2 } \\ \\ k = 1 \\ , . \\end{align*}"}
+{"id": "876.png", "formula": "\\begin{align*} - c _ 2 F _ 1 ( x _ 1 , \\dots , x _ n ) = g ( 0 , x _ 1 , \\dots , x _ n ) h ( 0 , x _ 1 , \\dots , x _ n ) . \\end{align*}"}
+{"id": "3774.png", "formula": "\\begin{align*} F _ \\varepsilon ^ n = ( t + \\varepsilon ) { \\rm I d } . \\end{align*}"}
+{"id": "675.png", "formula": "\\begin{align*} \\mathfrak { h } _ { 0 } & \\Big ( \\varphi - \\sum _ { 1 \\leq i < i _ a } d ^ + _ { i , 0 } ( D ^ 2 \\varphi ) h _ { i , 2 } - \\sum _ { 1 \\leq i \\leq g } d ^ + _ { i , 1 } ( D \\varphi ) h _ { i , 1 } - \\sum _ { 1 \\leq s < \\gamma } d ^ 0 _ { s , 1 } ( D \\varphi ) c _ { s , 1 } \\Big ) \\\\ & = \\sum _ { 1 \\leq i \\leq g } d ^ + _ { i , 2 } ( \\varphi ) h _ i + \\sum _ { 1 \\leq s < \\gamma } d ^ 0 _ { s , 2 } ( \\varphi ) c _ s + \\sum _ { 1 \\leq j \\leq g } d ^ - _ { - j , 2 } ( \\varphi ) h _ { - j } . \\end{align*}"}
+{"id": "255.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( 1 - z ^ n \\right ) ^ { \\frac { m ^ 4 } { n ^ 5 } } = \\sqrt [ 3 ] { 1 - z } \\ ; \\exp \\left \\{ \\frac { z ( 2 z - 7 ) } { 1 0 ( 1 - z ) ^ 2 } + \\frac { 1 } { 3 0 } L i _ 3 ( z ) \\right \\} . \\end{align*}"}
+{"id": "3123.png", "formula": "\\begin{align*} \\frac { A _ n ^ { r } ( t , q ) } { \\prod _ { i = 0 } ^ n ( 1 - t q ^ { r i } ) } = \\sum _ { k = 0 } ^ { \\infty } [ r k + 1 ] _ q ^ { n } \\ , t ^ { k } . \\end{align*}"}
+{"id": "5275.png", "formula": "\\begin{align*} \\left | 1 + \\rho d X \\right | ^ { q } \\le 1 + q \\rho d X + \\binom { q } { 2 } \\rho ^ 2 d ^ 2 X ^ 2 + \\binom { q } { 3 } \\rho ^ 3 | d X | ^ 3 ( 1 + | \\rho d X | ) ^ { q - 3 } . \\end{align*}"}
+{"id": "2403.png", "formula": "\\begin{align*} \\vec { \\delta } _ { 1 0 4 } = ( 1 + \\cos a _ { 4 , 1 0 2 } \\cos \\omega _ { 4 , 1 0 2 } , \\cos a _ { 4 , 1 0 2 } \\sin \\omega _ { 4 , 1 0 2 } , \\sin a _ { 4 , 1 0 2 } ) \\end{align*}"}
+{"id": "6587.png", "formula": "\\begin{align*} y = \\sqrt { P _ s } \\sum \\nolimits _ { l = 1 } ^ L h _ l e ^ { j \\phi _ { l , n _ t } } g _ { l , n _ t } + n _ 0 , \\end{align*}"}
+{"id": "353.png", "formula": "\\begin{align*} ( \\underline { X } , \\underline { A } ) ^ { \\sigma } = \\{ ( x _ 1 , \\ldots , x _ m ) \\in \\prod ^ m _ { i = 1 } X _ i | x _ i \\in A _ i i \\notin \\sigma \\} . \\end{align*}"}
+{"id": "1810.png", "formula": "\\begin{align*} [ b _ k ( s ) , a _ p ^ * ( t ) ] & \\ = \\ \\chi ( p - k ) \\ , e ^ { i ( t - s ) E _ p } a _ { p - k } ( s ) \\ , \\\\ [ b _ k ( s ) , a _ h ^ * ( t ) ] & \\ = \\ - \\chi ^ \\perp ( h + k ) \\ , e ^ { i ( t - s ) E _ h } a _ { h + k } ( s ) \\ . \\end{align*}"}
+{"id": "138.png", "formula": "\\begin{align*} | \\rho ( x ) | \\leq 2 \\epsilon , \\rho ( x ) \\neq - \\epsilon , X \\rho ( x ) = 0 \\Longrightarrow X ^ 2 \\rho ( x ) < 0 . \\end{align*}"}
+{"id": "3839.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\sup _ { \\gamma \\in \\Sigma _ { \\mathrm { D } } ( \\delta ) } \\int _ { \\mathcal { Y } _ 1 \\times \\mathcal { Y } _ 2 } f ( y _ 1 , y _ 2 ) \\ , d \\gamma ( y _ 1 , y _ 2 ) \\mathcal { I } ( \\delta ) = \\sup _ { \\gamma \\in \\Sigma ( \\delta ) } \\int _ { \\mathcal { S } } f ( y _ 1 , y _ 2 ) \\ , d \\gamma ( y _ 1 , y _ 2 , x ) . \\end{align*}"}
+{"id": "3582.png", "formula": "\\begin{align*} 5 \\cdot 1 0 ^ m - 3 \\ = \\ 4 \\underbrace { 9 \\cdots 9 } _ { m - 1 } 7 \\end{align*}"}
+{"id": "8981.png", "formula": "\\begin{align*} c _ m \\Delta u - S ^ \\varphi u + \\tilde S ^ { \\tilde \\varphi } u ^ { \\frac { m + 2 } { m - 2 } } = 0 \\ , \\Omega , \\end{align*}"}
+{"id": "3251.png", "formula": "\\begin{align*} \\frac 1 { \\omega ( B ) } \\| ( b - b _ B ) f _ 1 \\| ^ t _ { L ^ t _ \\omega } & = \\frac 1 { \\omega ( B ) } \\int _ { 5 B } | b ( y ) - b _ B | ^ t | f | ^ t d \\omega ( y ) \\\\ & \\le { \\Big ( \\frac 1 { \\omega ( B ) } \\int _ { 5 B } | b ( y ) - b _ B | ^ { t s ' } d \\omega ( y ) \\Big ) ^ { \\frac 1 { s ' } } } \\Big ( \\frac 1 { \\omega ( B ) } \\int _ { 5 B } | f | ^ { t s } d \\omega ( y ) \\Big ) ^ { \\frac 1 s } . \\end{align*}"}
+{"id": "4403.png", "formula": "\\begin{align*} L _ { \\omega , \\mathbf { c } } ( \\Psi ) = - 3 N ( \\Psi ) \\lesssim \\| \\nabla \\cdot \\psi _ 3 \\| _ { L ^ 2 } \\| \\psi _ 1 \\| _ { L ^ 4 } \\| \\psi _ 2 \\| _ { L ^ 4 } \\lesssim \\| \\Psi \\| _ { \\mathcal { H } ^ 1 } ^ 3 . \\end{align*}"}
+{"id": "2973.png", "formula": "\\begin{align*} \\bigoplus _ { j = 1 } ^ n V _ j ^ { \\oplus q _ j ( \\xi _ I - \\sigma \\cdot \\xi _ I ) } \\to \\bigoplus _ { j = 1 } ^ n V _ j ^ { \\oplus q _ j ( \\rho \\cdot \\xi _ I - \\rho \\cdot \\sigma \\cdot \\xi _ I ) } \\end{align*}"}
+{"id": "1139.png", "formula": "\\begin{align*} A ^ { \\ast } ( k x ) = k ^ { n } A ^ { \\ast } ( x ) \\left ( x \\in G \\right ) . \\end{align*}"}
+{"id": "2307.png", "formula": "\\begin{align*} \\widetilde { \\Psi } ( z ) : = \\widetilde { T } ( \\Phi _ 1 + \\widetilde { T } ( \\Phi _ 2 + \\widetilde { T } ( \\cdots \\Phi _ { n - 1 } + \\widetilde { T } ( f ) ) \\cdots ) ) , \\end{align*}"}
+{"id": "3199.png", "formula": "\\begin{align*} \\tau ( T ^ * _ { g } ( x ) ) = \\tau ( x ) x \\in M g \\in G . \\end{align*}"}
+{"id": "2289.png", "formula": "\\begin{align*} P _ r ( \\theta ) = \\frac { 1 - r ^ 2 } { 1 - 2 r \\cos ( \\theta ) + r ^ 2 } \\end{align*}"}
+{"id": "5251.png", "formula": "\\begin{align*} \\{ E _ { T ^ c } L _ T \\} _ { T \\subseteq [ n ] } = \\bigotimes _ { i \\in T } L _ i \\otimes \\bigotimes _ { i \\in T ^ c } E _ i . \\end{align*}"}
+{"id": "7977.png", "formula": "\\begin{align*} e ^ h \\mathrm { d i v } \\Big ( e ^ { - h } V \\ , \\mathcal { L } _ h V \\Big ) & = \\big ( \\mathcal { L } _ h V \\big ) ^ 2 + V \\cdot \\nabla \\mathcal { L } _ h V = \\\\ & = \\big ( \\mathcal { L } _ h V \\big ) ^ 2 + V ^ i \\ , \\partial _ i \\partial _ j V ^ j - \\partial _ j h \\ , V ^ i \\ , \\partial _ i V ^ j - \\partial _ { i j } h \\ , V ^ i \\ , V ^ j . \\end{align*}"}
+{"id": "5487.png", "formula": "\\begin{align*} \\eta = \\int _ { B } \\boldsymbol { \\beta } _ { \\eta } ( b ) d \\nu _ { B } ( b ) , \\end{align*}"}
+{"id": "6040.png", "formula": "\\begin{align*} \\left | \\frac { A _ 1 } { A _ 2 } \\right | = \\left | \\frac { B _ 1 } { B _ 2 } \\right | = \\left | \\frac { C _ 1 } { C _ 2 } \\right | \\end{align*}"}
+{"id": "5641.png", "formula": "\\begin{align*} \\aligned \\left \\{ \\begin{array} { l l l } - \\Delta u + \\mu _ 1 u = ( I _ \\mu \\ast | u | ^ { 2 ^ * _ \\mu } ) | u | ^ { 2 ^ * _ \\mu - 2 } u + p ( I _ \\mu \\ast | v | ^ q ) | u | ^ { p - 2 } u \\ & \\mathbb { R } ^ N , \\\\ - \\Delta v + \\mu _ 2 v = ( I _ \\mu \\ast | v | ^ { 2 ^ * _ \\mu } ) | v | ^ { 2 ^ * _ \\mu - 2 } v + q ( I _ \\mu \\ast | u | ^ p ) | v | ^ { q - 2 } v \\ & \\mathbb { R } ^ N , \\end{array} \\right . \\endaligned \\end{align*}"}
+{"id": "428.png", "formula": "\\begin{align*} ( \\vec U , D _ { x _ i } \\vec V ) = \\vec U ^ T P _ { \\Omega } ( D _ { x _ i } \\vec V ) = - ( D _ { x _ i } \\vec U , \\vec V ) + ( E \\vec U ) ^ T P _ { \\partial \\Omega } N _ { i } ( E \\vec V ) . \\end{align*}"}
+{"id": "5550.png", "formula": "\\begin{align*} \\Delta _ { x , g } ( n ) : = 2 \\left ( \\frac { C _ { x , g } } { \\left ( 1 - \\varepsilon _ { x } ( e , n ) \\right ) \\left ( 1 - \\varepsilon _ { x } ( g , n ) \\right ) } \\right ) ^ { 1 / 2 } \\varepsilon _ { x } ( g , n ) + \\varepsilon _ { x } ( e , n ) . \\end{align*}"}
+{"id": "8514.png", "formula": "\\begin{align*} \\zeta _ { 0 , 0 } ( s , c ) = \\frac { \\pi } { \\sqrt { c } } \\ , \\frac { 1 } { s - 1 } + \\frac { \\pi } { \\sqrt { c } } \\left ( 2 \\gamma - \\log \\left ( \\frac { c } { 4 } \\right ) - 8 \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { 2 n - 1 } \\cdot \\frac { 1 } { e ^ { ( 2 n - 1 ) \\pi \\sqrt { c } } + 1 } \\right ) + O ( s - 1 ) . \\end{align*}"}
+{"id": "3513.png", "formula": "\\begin{align*} J _ K ( q ^ i , p ^ i ) = \\begin{cases} ( - p ^ i , q ^ i ) , & \\mbox { i f } i \\in K ; \\\\ ( q ^ i , p ^ i ) , & \\mbox { i f } i \\notin K , \\end{cases} \\end{align*}"}
+{"id": "9107.png", "formula": "\\begin{align*} \\tau ^ 3 _ { e , F } = ( \\tau ^ 1 _ { e , F } ) ^ { - 1 } \\circ \\tau ^ 2 _ { e , F } . \\end{align*}"}
+{"id": "5036.png", "formula": "\\begin{align*} | G _ f ^ { g r } | = | G _ { f _ 0 } ^ d | \\cdot | G _ { f _ 1 } ^ { g r } | \\cdots | G _ { f _ p } ^ { g r } | . \\end{align*}"}
+{"id": "3805.png", "formula": "\\begin{align*} \\Sigma _ 0 ( \\delta _ 0 ) : = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\ , \\boldsymbol { K } _ c ( \\mu , \\gamma ) \\le \\delta _ 0 \\right \\} , \\end{align*}"}
+{"id": "2222.png", "formula": "\\begin{align*} \\psi ( x , y ) = \\psi ( y ^ u , y ^ h ) ( 1 - H _ y ( h ) ) + \\int _ { 1 } ^ { h } \\psi ( y ^ { u - v } , ( y ^ v ) ^ - ) \\ , d H _ y ( v ) . \\end{align*}"}
+{"id": "5474.png", "formula": "\\begin{align*} G ( \\theta ) = & \\mathbb { E } _ { \\Phi | \\Phi ( \\mathcal { A } ) > 0 } \\left \\{ \\left ( \\sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } e ^ { - m \\eta \\theta r _ 1 ^ { \\alpha } / \\rho } \\right . \\right . \\\\ & \\left . \\left . \\prod \\limits _ { x _ { i } \\in \\Phi \\backslash \\{ x _ 1 \\} \\cap { \\mathcal { A } } } \\frac { 1 } { \\left ( 1 + \\frac { m \\eta \\theta r _ 1 ^ { \\alpha } } { M r _ i ^ { \\alpha } } \\right ) ^ M } \\right ) ^ b \\right \\} , \\end{align*}"}
+{"id": "4764.png", "formula": "\\begin{align*} \\hat { S } = S Q , \\end{align*}"}
+{"id": "9128.png", "formula": "\\begin{align*} v _ { [ 1 ] } ^ { 1 } = \\delta ^ { 2 } ( \\varphi ^ { 1 } ) = x ^ { 1 } + u ^ { 1 } + u _ { [ 1 ] } ^ { 1 } \\ , . \\end{align*}"}
+{"id": "988.png", "formula": "\\begin{align*} f _ { i ( v _ 1 ) } ( p _ { E , v _ 1 } ) = f _ { i ( v _ 2 ) } ( p _ { E , v _ 2 } ) , \\end{align*}"}
+{"id": "5661.png", "formula": "\\begin{align*} a ' ( \\nu ) = - \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | u _ \\nu | ^ p ) | v _ \\nu | ^ q . \\end{align*}"}
+{"id": "4970.png", "formula": "\\begin{align*} F ( k , t ; p , n ) = ( n - k ) \\binom { n } { k } \\sum _ { j = 0 } ^ k ( - 1 ) ^ { k - j } \\binom { k } { j } \\dfrac { q _ { j } ^ t } { ( n - j ) } \\end{align*}"}
+{"id": "1034.png", "formula": "\\begin{align*} & \\psi \\ , ' \\bigg ( \\frac { 5 } { 4 } \\bigg ) = \\psi \\ , ' \\bigg ( \\frac { 1 } { 4 } \\bigg ) - 1 6 = \\pi ^ 2 + 8 G - 1 6 . \\end{align*}"}
+{"id": "2329.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { n } | \\eta _ j - \\eta _ { j - 1 } | ^ { 1 - 2 H } \\leq \\sum _ { \\pmb { \\alpha } _ n \\in D _ n } \\prod _ { j = 1 } ^ n | \\eta _ j | ^ { \\alpha _ j } , \\end{align*}"}
+{"id": "4982.png", "formula": "\\begin{align*} \\hat { p } _ { k , n } ^ { ( t ) } = \\hat { p } _ { k - 1 , n } ^ { ( t ) } + \\frac { n - 1 } { k - 1 } \\hat { p } _ { k - 1 , n - 1 } ^ { ( t ) } , \\end{align*}"}
+{"id": "6475.png", "formula": "\\begin{align*} \\mathcal F \\left ( e ^ { - z | \\cdot | ^ 2 } \\right ) = \\frac { \\pi } { z } e ^ { - \\frac { | \\xi | ^ 2 } { 4 z } } . \\end{align*}"}
+{"id": "4537.png", "formula": "\\begin{align*} \\eta = \\frac { 2 ( \\mu _ { \\omega , \\mathbf { c } } - S _ { \\omega , \\mathbf { c } } ( \\Phi ^ { \\lambda _ 0 } _ 0 ) ) } { C ( 1 + M ^ 2 + \\mu _ { \\omega , \\mathbf { c } } ) } . \\end{align*}"}
+{"id": "8393.png", "formula": "\\begin{align*} \\nu _ 2 : = \\frac { 1 } { 2 } \\epsilon _ 1 ( n , A , \\rho ) ^ { - 1 } \\epsilon < \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "2066.png", "formula": "\\begin{align*} \\epsilon _ j = \\begin{cases} 1 & j \\notin Y \\\\ - 1 & j \\in Y \\end{cases} \\end{align*}"}
+{"id": "5595.png", "formula": "\\begin{align*} \\frac { 1 - M ^ { - 1 } } { \\log M } \\sum _ { A \\in \\mathcal { P } _ { + } } \\beta ( A ) \\frac { \\alpha ( A ) } { \\beta ( A ) } \\log \\frac { \\alpha ( A ) } { \\beta ( A ) } \\le \\sum _ { A \\in \\mathcal { P } _ { + } } \\beta ( A ) \\left ( \\frac { \\alpha ( A ) } { \\beta ( A ) } - 1 \\right ) = d _ { { \\rm T V } } ( \\alpha , \\beta ) . \\end{align*}"}
+{"id": "1202.png", "formula": "\\begin{align*} \\operatorname { G r } ( B ) = \\operatorname { G r } ( A _ n ) \\oplus \\mathbb { C } ( \\phi _ + ^ n - \\phi _ - ^ n , i \\phi _ + ^ n + i \\phi _ - ^ n ) = : \\operatorname { G r } ( A _ n ) \\oplus \\mathbb { C } v _ n , \\end{align*}"}
+{"id": "585.png", "formula": "\\begin{align*} e ^ { 4 ( x _ 2 - t _ 2 ) r ^ 2 + 4 ( x _ 1 \\sqrt { x _ 2 } - t _ 1 \\sqrt { t _ 2 } ) r + x _ 1 ^ 2 - t _ 1 ^ 2 } = C \\forall r \\in \\R . \\end{align*}"}
+{"id": "1848.png", "formula": "\\begin{align*} \\begin{aligned} & - 3 F ( 2 \\pi / 3 ) + \\sum _ { i = 1 } ^ { 3 } F ( \\theta _ { i } ) \\leq - 3 F ( 2 \\pi / 3 ) + F ( \\pi / 3 ) + \\max ( F ( \\pi / 3 ) , 2 F ( \\pi / 6 ) ) \\\\ & \\quad \\leq \\max \\Big ( - F ( 2 \\pi / 3 ) - 2 [ F ( 2 \\pi / 3 ) - F ( \\pi / 3 ) ] \\ , , \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - F ( 2 \\pi / 3 ) + F ( \\pi / 3 ) - 2 [ F ( 2 \\pi / 3 ) - F ( \\pi / 6 ) ] \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "4680.png", "formula": "\\begin{align*} S ( z ) = \\Pi + z ( \\mathcal { A } _ 0 - z I ) ^ { - 1 } \\Pi . \\end{align*}"}
+{"id": "1956.png", "formula": "\\begin{align*} D _ \\xi ( \\det ) ( x ) = D ( \\det ) ( x ) \\cdot \\xi = t r ( \\tilde { x } ^ \\tau \\xi ) , \\end{align*}"}
+{"id": "581.png", "formula": "\\begin{align*} \\phi ^ { + } ( t ) : = \\int ^ { \\infty } _ { t } H ( s ) [ H ( s ) ] ^ T d s . \\end{align*}"}
+{"id": "6165.png", "formula": "\\begin{align*} l _ R ( - c d ) l _ R ( a d ) & = l _ R ( - c d ) l _ R ( a ) + l _ R ( - c d ) l _ R ( d ) = \\\\ & = l _ R ( - c d ) l _ R ( a ) + l _ R ( c ) l _ R ( d ) = l _ R ( a ) l _ R ( - c d ) + l _ R ( a ) l _ R ( b ) \\\\ & = l _ R ( a ) l _ R ( - b c d ) = l _ R ( a ) l _ R ( - a ) = 0 . \\end{align*}"}
+{"id": "3862.png", "formula": "\\begin{align*} v _ { A , \\ell } & = \\begin{cases} v _ { \\ell } & \\ell \\in A , \\\\ 0 & \\ell \\notin A . \\end{cases} \\end{align*}"}
+{"id": "1544.png", "formula": "\\begin{align*} \\boldsymbol Y _ n = | A _ n | / | Q _ n | \\end{align*}"}
+{"id": "3027.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T T ^ * + x ^ 2 S S ^ * ) \\leq \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T T ^ * ) + \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( x ^ 2 S S ^ * ) \\end{align*}"}
+{"id": "3431.png", "formula": "\\begin{align*} \\Delta \\left ( | X | ^ 2 \\right ) = 2 | \\nabla X | ^ 2 + 2 \\left \\langle X , \\Delta X \\right \\rangle . \\end{align*}"}
+{"id": "3646.png", "formula": "\\begin{align*} \\begin{aligned} M _ w ^ B ( Z , \\lambda , N , \\frac { \\varepsilon } { 4 } , G , d ) & \\ge \\inf _ { \\mathcal { G } _ w ( \\mathcal { U } ) } \\left \\{ \\sum _ { ( w _ { \\mathbf { U } } , \\mathbf { U } ) \\in \\mathcal { G } _ w ( \\mathcal { U } ) } e ^ { - \\lambda \\mathfrak { l } ( \\mathbf { U } ) } \\right \\} \\\\ & \\ge M _ w ( Z , \\lambda , N , \\varepsilon , G , d ) . \\end{aligned} \\end{align*}"}
+{"id": "2724.png", "formula": "\\begin{align*} T _ { 5 1 } = & s \\lambda ^ 3 \\iint _ Q \\xi u A \\nabla u \\cdot \\nabla \\eta \\left ( A \\nabla \\eta \\cdot \\nabla \\eta \\right ) d x d t \\\\ \\geq & - C s ^ 2 \\lambda ^ 4 \\iint _ Q \\xi \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d t - C \\lambda ^ 2 \\iint _ Q \\xi | A \\nabla u \\cdot \\nabla \\eta | ^ 2 d x d t , \\end{align*}"}
+{"id": "5644.png", "formula": "\\begin{align*} J ' _ \\nu ( u , v ) + \\lambda _ 1 ( u , 0 ) + \\lambda _ 2 ( 0 , v ) = 0 , \\ u \\geq 0 , \\ v \\geq 0 , \\end{align*}"}
+{"id": "9205.png", "formula": "\\begin{align*} W ( D ) & \\ge W ( H ) + \\sum _ { i = n _ 0 + 1 } ^ { n } \\left ( 2 i - 1 + ( r - 1 ) ^ 2 \\right ) \\\\ & > 2 \\binom { n _ 0 } { 2 } + 2 \\sum _ { i ' = n _ 0 } ^ { n - 1 } i ' + ( n - n _ 0 ) + ( n - n _ 0 ) ( r - 1 ) ^ 2 \\\\ & \\ge 2 \\binom { n } { 2 } + a n _ 0 + ( n - n _ 0 ) a \\\\ & = 2 \\binom { n } { 2 } + a n \\end{align*}"}
+{"id": "8873.png", "formula": "\\begin{align*} H ( G ) = F _ 1 ( G ) & \\le \\frac { 1 } { 2 } P e r ( A _ G ) \\\\ & \\le ( 1 + o ( 1 ) ) \\frac { 1 } { e } ( \\sqrt { 2 \\pi } ) ^ { \\frac { 1 } { p } - 1 } p ^ { \\frac { 1 } { 2 p } } n ^ { \\frac { 1 } { 2 } + \\frac { 1 } { 2 p } } p ^ { n } \\frac { ( n - 1 ) ! } { 2 } \\\\ & = ( 1 + o ( 1 ) ) c _ p n ^ { \\frac { 1 } { 2 } + \\frac { 1 } { 2 p } } E ( n , p ) \\end{align*}"}
+{"id": "460.png", "formula": "\\begin{align*} \\phi = \\phi _ { } + \\tau ^ m \\frac H r , \\end{align*}"}
+{"id": "8426.png", "formula": "\\begin{align*} \\Gamma ( s ) \\ , ( s , \\lambda ) _ { k } = \\Gamma ( s ) ( - 1 ) ^ { k } + \\Gamma ( s ) \\ , k ^ { s } e ^ { \\lambda } \\ , \\sum _ { \\ell = 1 } ^ { k } \\left ( \\begin{array} { c } k \\\\ \\ell \\end{array} \\right ) \\left ( \\frac { 2 \\lambda } { k } \\right ) ^ { \\ell } ( - 1 ) ^ { k - \\ell } \\ , \\intop _ { k } ^ { \\infty } t ^ { - s } e ^ { - \\frac { \\lambda } { k } t } \\ , \\frac { ( t - k ) ^ { \\ell - 1 } } { ( \\ell - 1 ) ! } \\ , d t , \\end{align*}"}
+{"id": "2426.png", "formula": "\\begin{align*} y ( q x ) = m ( x ) y ( x ) + r ( x ) , \\end{align*}"}
+{"id": "6994.png", "formula": "\\begin{align*} \\Gamma = \\bigcup _ { i = 1 } ^ \\epsilon \\left ( \\gamma _ i + v K \\right ) . \\end{align*}"}
+{"id": "3591.png", "formula": "\\begin{align*} v ( p ^ \\alpha ) \\ = \\ p + \\alpha \\ \\le \\ p \\alpha \\ = \\ \\underbrace { p + \\cdots + p } _ { \\alpha } \\ \\le \\ p ^ \\alpha . \\end{align*}"}
+{"id": "3146.png", "formula": "\\begin{align*} \\phi = \\sum _ { j = 0 } ^ N c _ j Z _ j , \\end{align*}"}
+{"id": "2602.png", "formula": "\\begin{align*} | k _ i | \\leq C \\frac { a _ 1 ^ { \\frac { 1 } { 2 } } } { a _ 1 } = C a _ 1 ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "7696.png", "formula": "\\begin{align*} L = L _ 1 \\oplus \\Z \\zeta \\oplus \\Z z \\end{align*}"}
+{"id": "5582.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } \\left [ \\tilde { f } ( g , \\cdot ) | \\mathcal { F } ^ { s } \\right ] = \\mathbb { E } _ { \\lambda } \\left [ \\tilde { f } ( e , \\cdot ) | \\mathcal { F } ^ { s } \\right ] \\mbox { } \\lambda \\mbox { - a . e . } , \\end{align*}"}
+{"id": "3963.png", "formula": "\\begin{align*} \\widehat { S } \\equiv ( \\widehat { Y } _ 1 , \\widehat { Y } _ 2 , \\widehat { X } ) = \\varepsilon ( Y _ 1 , Y _ 2 , X ) + ( 1 - \\varepsilon ) ( \\widetilde { Y } _ 1 , \\widetilde { Y } _ 2 , \\widetilde { X } ) , \\end{align*}"}
+{"id": "1504.png", "formula": "\\begin{align*} \\max _ { q \\in S ^ { 2 } : \\ , q \\cdot v ' = 0 } ( q \\cdot x ) = \\max _ { q \\in S ^ { 2 } : \\ , q \\cdot ( O v ) = 0 } ( q \\cdot x ) = \\max _ { q \\in S ^ { 2 } : \\ , ( O ^ \\intercal q ) \\cdot v = 0 } ( ( O ^ \\intercal q ) \\cdot ( O ^ \\intercal x ) ) = \\max _ { q \\in S ^ { 2 } : \\ , q \\cdot v = 0 } ( q \\cdot ( O ^ \\intercal x ) ) . \\end{align*}"}
+{"id": "8006.png", "formula": "\\begin{align*} A \\mapsto \\langle h , A h \\rangle = \\sum _ { i = 1 } ^ n w _ i \\lambda _ i ( A ) \\end{align*}"}
+{"id": "6046.png", "formula": "\\begin{align*} \\frac { | A _ 1 | } { | A _ 2 | } = \\frac { | B _ 1 | } { | B _ 2 | } = \\frac { | C _ 1 | } { | C _ 2 | } = : r \\end{align*}"}
+{"id": "5434.png", "formula": "\\begin{align*} & = \\frac { | h _ 1 | ^ 2 } { \\sum _ { x _ i \\in \\Phi \\backslash \\{ x _ 1 \\} \\cap \\mathcal { A } } | h _ i | ^ 2 } \\overset { \\Delta } { = } \\frac { | h _ 1 | ^ 2 } { I } , \\end{align*}"}
+{"id": "7610.png", "formula": "\\begin{align*} | B | - 2 ( 1 - | C | ) & = \\frac { \\tau _ { 1 } } { 2 } - 2 \\left ( 1 - \\frac { ( 2 + \\tau ^ 2 _ { 1 } ) } { 3 \\tau _ { 1 } } \\right ) \\\\ & = \\frac { 7 \\tau ^ 2 _ { 1 } - 1 2 \\tau _ { 1 } + 8 } { 6 \\tau _ { 1 } } < 0 , \\end{align*}"}
+{"id": "7167.png", "formula": "\\begin{align*} \\pi _ 0 ( \\mathcal { M } ) = \\dots = \\pi _ j ( \\mathcal { M } ) = 0 . \\end{align*}"}
+{"id": "5877.png", "formula": "\\begin{align*} \\nu _ { m - 1 } + 1 = \\nu _ { m + 1 } \\ , , S f _ { m - 1 } \\in E _ { \\nu _ { m + 1 } } \\end{align*}"}
+{"id": "147.png", "formula": "\\begin{align*} \\Pi _ \\lambda = \\chi \\Pi _ \\lambda \\chi + \\sum \\limits _ j \\chi _ j \\Pi _ \\lambda \\chi _ j + \\sum \\limits _ { j \\neq j ' } \\chi _ { j } \\Pi _ \\lambda \\chi _ { j ' } + \\sum \\limits _ { j } ( \\chi \\Pi _ \\lambda \\chi _ j + \\chi _ j \\Pi _ \\lambda \\chi ) \\end{align*}"}
+{"id": "3813.png", "formula": "\\begin{align*} g _ { \\lambda _ 1 , 1 } ( s _ 1 , s _ 2 ) & = \\sup _ { s _ 1 ' \\in \\mathcal { S } _ 1 } \\{ g ( s _ 1 ' , s _ 2 ) - \\lambda _ 1 c _ 1 ( s _ 1 , s _ 1 ' ) \\} \\\\ g _ { \\lambda _ 2 , 2 } ( s _ 1 , s _ 2 ) & = \\sup _ { s _ 2 ' \\in \\mathcal { S } _ 2 } \\{ g ( s _ 1 , s _ 2 ' ) - \\lambda _ 2 c _ 2 ( s _ 2 , s _ 2 ' ) \\} . \\end{align*}"}
+{"id": "6956.png", "formula": "\\begin{align*} b _ { \\delta ' , \\hat { T } } ( \\hat { \\ell } ) = b _ { \\delta ' , \\hat { T } } ( ( \\hat { \\ell } ) _ { \\sf p } ) + b _ { \\delta ' , \\hat { T } } ( ( \\hat { \\ell } ) _ { \\sf q } ) \\end{align*}"}
+{"id": "7956.png", "formula": "\\begin{align*} \\Psi _ { \\mathcal { B } ^ H } ( r ) \\coloneqq \\begin{cases} \\sup \\limits _ { x \\in \\partial \\Omega } | | \\mathcal { B } ^ H | | _ { L ^ { n - 1 , \\infty } ( \\partial \\Omega \\cap B _ r ( x ) ) } \\quad n \\geq 3 , \\\\ \\\\ \\sup \\limits _ { x \\in \\partial \\Omega } | | \\mathcal { B } ^ H | | _ { L ^ { 1 , \\infty } \\log L ( \\partial \\Omega \\cap B _ r ( x ) ) } \\quad n = 2 . \\end{cases} \\end{align*}"}
+{"id": "4661.png", "formula": "\\begin{align*} V _ { \\infty } : = \\lim _ { \\vert x \\vert \\rightarrow \\infty } \\ , V ( x ) > \\inf _ { x \\in \\mathbb { R } ^ N } \\ , V ( x ) = V _ { 0 } > - 1 . \\end{align*}"}
+{"id": "7114.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } J _ \\varepsilon ( | x | ) x _ l x _ m \\ : x = 0 \\end{align*}"}
+{"id": "9213.png", "formula": "\\begin{align*} & \\texttt { m p . p r e c = d } \\\\ & \\texttt { x 1 = r o u n d ( x , d ) ; y 1 = r o u n d ( y , d ) } \\\\ & \\texttt { z = x 1 + y 1 } . \\end{align*}"}
+{"id": "7913.png", "formula": "\\begin{align*} \\tilde { p } : = p + q . \\end{align*}"}
+{"id": "1636.png", "formula": "\\begin{align*} 3 \\ , d \\Phi ( X , \\xi _ i , \\xi _ j ) = g ( [ \\xi _ i , \\xi _ j ] , f X ) , 2 \\ , d \\eta ^ k ( \\xi _ j , \\xi _ i ) = g ( [ \\xi _ i , \\xi _ j ] , \\xi _ k ) \\end{align*}"}
+{"id": "6827.png", "formula": "\\begin{align*} \\alpha ' _ n = ( 1 - a ) \\left ( \\frac { q ^ n \\alpha _ n } { 1 - a q ^ { 2 n } } - \\frac { q ^ { n - 1 } \\alpha _ { n - 1 } } { 1 - a q ^ { 2 n - 2 } } \\right ) , \\beta ' _ n = q ^ n \\beta _ n , \\end{align*}"}
+{"id": "4923.png", "formula": "\\begin{align*} F ( \\underline { k } , t , \\mathbf { p } _ { n + 1 } ) = \\left ( \\prod _ { j = 0 } ^ { k } p _ j ^ { } \\right ) \\sum _ { j = 0 } ^ { k } \\frac { q _ j ^ { t - k } } { 1 - q _ j } \\prod _ { \\substack { i = 0 \\\\ i \\neq j } } ^ { k } \\frac { 1 } { q _ j ^ { } - q _ i ^ { } } \\ , . \\end{align*}"}
+{"id": "9152.png", "formula": "\\begin{align*} y _ { i , [ 0 , \\kappa _ { i } - 1 ] } ^ { j _ { i } } = \\varphi _ { i , [ 0 , \\kappa _ { i } - 1 ] } ^ { j _ { i } } ( \\zeta _ { [ - q _ { 1 } , - 1 ] } , x , v _ { 1 } , v _ { 1 , [ 1 ] } , \\ldots , v _ { i - 1 } , v _ { i - 1 , [ 1 ] } , \\ldots ) \\ , , j _ { i } = 1 , \\ldots , m _ { i } \\ , , i = 1 , \\ldots , s \\end{align*}"}
+{"id": "4003.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) = \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\mathrm { R } _ { \\mathrm { D } } ( \\lambda , \\delta ) = \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) . \\end{align*}"}
+{"id": "4264.png", "formula": "\\begin{align*} x _ { t } = x _ { 0 } + \\int _ { 0 } ^ { t } b _ { r } \\mathrm { d } r + \\int _ { 0 } ^ { t } \\sigma _ { r } \\mathrm { d } W _ { r } , 0 \\leq t \\leq T , \\end{align*}"}
+{"id": "8093.png", "formula": "\\begin{align*} _ { \\alpha } : = \\{ \\mathbf { W } \\in M ( \\mathbb { R } ^ { 3 \\times 3 } ) : \\mathbf { W } \\geq \\alpha ^ 3 , \\ ; ( \\mathbf { W } ) \\geq 2 \\alpha ^ 2 , \\ ; \\mathbf { W } \\geq 3 \\alpha \\} \\end{align*}"}
+{"id": "3795.png", "formula": "\\begin{align*} \\mathcal { I } ( 0 ) = \\int _ { \\mathcal { X } } \\left [ \\sup _ { \\gamma ( \\cdot | x ) \\in \\Pi ( \\mu _ { 1 | 3 } , \\mu _ { 2 | 3 } ) } \\int _ { \\mathcal { Y } _ 1 \\times \\mathcal { Y } _ 2 } f ( y _ 1 , y _ 2 , x ) \\ , d \\gamma ( y _ 1 , y _ 2 | x ) \\right ] d \\gamma _ X ( x ) , \\end{align*}"}
+{"id": "2911.png", "formula": "\\begin{align*} - ^ \\delta : M & \\to M \\\\ m & \\mapsto m ^ \\delta : = \\delta \\ast m . \\end{align*}"}
+{"id": "5320.png", "formula": "\\begin{align*} \\Rightarrow \\| f ^ { = t } \\| _ 2 ^ 2 = ( p ( 1 - p ) ) ^ t \\approx p ^ t . \\end{align*}"}
+{"id": "5335.png", "formula": "\\begin{align*} r ' _ d = \\frac { \\sqrt { d } } { \\log ^ { 1 / 2 } ( \\gamma _ 2 / \\gamma _ 1 ) } \\mbox { a n d } \\gamma ' _ d = \\left ( \\frac { 3 3 r } { \\sqrt { d } } \\right ) ^ d \\gamma _ 1 \\log ^ { d / 2 } \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) . \\end{align*}"}
+{"id": "3299.png", "formula": "\\begin{align*} \\widetilde { \\rho } _ w ( - i ^ - , \\kappa ( - i ) ) = - \\widetilde { \\rho } _ w ( - i ^ + , \\kappa ( - i ) ) \\end{align*}"}
+{"id": "7382.png", "formula": "\\begin{align*} R _ N ^ 2 ( L , \\alpha , \\Delta ) = \\sum _ { k \\in \\mathbb { Z } } b _ { k , N } ( L ) e ( k \\alpha ) . \\end{align*}"}
+{"id": "2816.png", "formula": "\\begin{align*} \\overset \\cdot { \\ ; \\ ; R ^ 1 _ { i , 0 } } = a _ 0 \\dots a _ r R _ { i , r + 1 } ^ 1 ; i = 0 , \\dots , r - 1 . \\end{align*}"}
+{"id": "170.png", "formula": "\\begin{align*} L i _ 2 ( 2 ) = - \\frac { \\pi ^ 2 } { 4 } - i \\pi \\log 2 , \\end{align*}"}
+{"id": "9150.png", "formula": "\\begin{align*} \\delta ^ { k _ { s } ^ { j } - 1 } ( \\varphi _ { r e s t _ { s - 1 } } ^ { j } ) & = \\varphi _ { r e s t _ { s - 1 } , [ k _ { s } ^ { j } - 1 ] } ^ { j } ( x , v _ { 1 } , v _ { 1 , [ 1 ] } , \\ldots , v _ { s - 1 } , v _ { s - 1 , [ 1 ] } , \\ldots ) \\\\ \\delta ^ { k _ { s } ^ { j } } ( \\varphi _ { r e s t _ { s - 1 } } ^ { j } ) & = \\varphi _ { r e s t _ { s - 1 } , [ k _ { s } ^ { j } ] } ^ { j } ( x , v _ { 1 } , v _ { 1 , [ 1 ] } , \\ldots , v _ { s - 1 } , v _ { s - 1 , [ 1 ] } , \\ldots , u _ { r e s t _ { s - 1 } } ) \\end{align*}"}
+{"id": "252.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { m ^ 3 } { n ^ 4 } } = \\sqrt [ 3 ] { \\frac { 1 } { 1 - z } } \\ ; \\exp \\left \\{ \\frac { 1 } { 4 } L i _ 2 ( z ) + \\frac { 1 } { 4 } \\frac { z } { 1 - z } \\right \\} , \\end{align*}"}
+{"id": "6282.png", "formula": "\\begin{align*} \\tau _ { K , N } ^ { ( t ) } ( \\theta ) : = t ^ { \\frac { 1 } { N } } \\left [ \\sigma _ { K , N - 1 } ^ { ( t ) } ( \\theta ) \\right ] ^ { 1 - \\frac { 1 } { N } } , \\end{align*}"}
+{"id": "6032.png", "formula": "\\begin{align*} \\int f \\otimes \\bigl ( \\otimes _ { i = 1 } ^ n f _ i \\bigr ) \\otimes g \\ , d \\rho _ n = \\int f \\cdot \\bigl ( \\prod _ { i = 1 } ^ n V f _ i \\bigr ) \\cdot J g \\ , d \\mu = \\int f \\cdot \\bigl ( \\prod _ { i = 1 } ^ n V f _ i \\bigr ) \\otimes g \\ , d \\eta . \\end{align*}"}
+{"id": "9084.png", "formula": "\\begin{align*} \\mathfrak { S } ( f _ k ) = k + r - 1 , \\ , \\ , \\ , \\ , \\ , \\overline { \\mathfrak { S } } ( f _ k ) = n - k + 1 . \\end{align*}"}
+{"id": "1342.png", "formula": "\\begin{align*} \\partial _ { t } \\mu ( t , x ) - \\partial _ { x } \\left ( \\mu ( t , x ) \\left ( 2 \\int ^ x _ { - \\infty } \\mu ( t , y ) d y - 1 \\right ) \\right ) = \\mu ( t , x ) S \\star \\mu ( t , x ) , \\mu ( 0 , x ) = \\mu ^ { i n } . \\end{align*}"}
+{"id": "871.png", "formula": "\\begin{align*} \\left | \\frac { a _ i } { r } - \\frac { a ' _ i } { r ' } \\right | = \\left | \\frac { a _ { 1 , i } } { r } - \\frac { a _ { 2 , i } } { r ' } + \\frac { d _ i - d _ i ' } { d } \\right | \\end{align*}"}
+{"id": "8421.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow \\infty } \\zeta _ { p } ( s ) = \\zeta ( s ) , \\ , \\ , \\ , \\ , \\ , \\lim _ { p \\rightarrow 0 ^ { + } } \\zeta _ { p } ( s ) = ( 2 ^ { s } - 1 ) \\ , \\zeta ( s ) . \\end{align*}"}
+{"id": "2184.png", "formula": "\\begin{align*} \\Lambda _ { \\alpha , i } = { \\rm d i a g } ( g _ { \\lambda _ i } ( \\alpha \\theta _ n ^ { k } ) ) _ { k = 0 } ^ { n - 1 } \\textrm { a n d } g _ { \\lambda _ i } ( z ) : = \\lambda _ i - 2 z + \\lambda _ i z ^ 2 . \\end{align*}"}
+{"id": "5195.png", "formula": "\\begin{align*} \\sigma _ r w t ( p _ s ) . ( p \\widetilde { \\mathbb { S } } _ { \\lambda ^ i } | _ { z = 0 } ) = { \\bf { b } } w t ( p _ s ) . ( p \\widetilde { \\mathbb { S } } _ { \\lambda ^ i } | _ { z = 0 } ) + \\frac { { \\bf { b } } ( p _ s ) } { b _ { 0 } } ( p \\widetilde { \\mathbb { S } } _ { \\lambda ^ 1 } | _ { z = 0 } ) ( p \\widetilde { \\mathbb { S } } _ { \\lambda ^ i } | _ { z = 0 } ) . \\end{align*}"}
+{"id": "9183.png", "formula": "\\begin{align*} \\varphi ^ { 2 } & = q ^ { 1 } \\\\ \\delta ( \\varphi ^ { 2 } ) & = q ^ { 1 } + T \\omega ^ { 1 } \\\\ \\delta ^ { 2 } ( \\varphi ^ { 2 } ) & = q ^ { 1 } + 2 T \\omega ^ { 1 } + T ^ { 2 } b _ { 1 } \\cos ( q ^ { 2 } ) \\sin ( q ^ { 3 } ) u ^ { 1 } \\ , , \\end{align*}"}
+{"id": "7759.png", "formula": "\\begin{align*} W _ p ^ - ( \\mu , \\nu ) = \\inf _ { \\beta \\in \\Phi ^ - ( \\nu , m ) } W _ p ( \\mu , \\beta ) , \\\\ W _ p ^ + ( \\mu , \\nu ) = \\inf _ { \\alpha \\in \\Phi ^ + ( \\mu , n ) } W _ p ( \\alpha , \\nu ) . \\end{align*}"}
+{"id": "3358.png", "formula": "\\begin{align*} B ^ * B = \\sum _ { i , j = 1 } ^ d \\zeta _ i \\overline { \\zeta _ j } S _ i ^ * S _ j = 1 . \\end{align*}"}
+{"id": "741.png", "formula": "\\begin{align*} c ( m ) : = \\sqrt { g h _ { 0 } \\Big ( 1 + \\frac { \\eta _ 1 ( m ) } { h _ { 0 } } \\Big ) \\Big ( 1 + \\frac { \\eta _ 2 ( m ) } { h _ { 0 } } \\Big ) \\Big ( 1 + \\frac { \\eta _ 3 ( m ) } { h _ { 0 } } \\Big ) } \\end{align*}"}
+{"id": "844.png", "formula": "\\begin{align*} k _ { n e w \\left ( 1 \\right ) } = \\underset { k \\in \\mathcal { K } _ { c l r \\left ( 1 \\right ) } } { { \\arg \\max } } ~ \\textbf { g } _ { k } ^ { H } \\textbf { g } _ { k } \\end{align*}"}
+{"id": "73.png", "formula": "\\begin{align*} H = \\prod _ { \\ell \\in \\mathcal { S } } \\prod _ { i = 1 } ^ { r ( \\ell ) } \\Delta ^ { \\mathcal { T } ( \\ell ) _ i } ( \\mathbf { P G } ^ 1 ( \\mathbb { Q } _ p ) ) . \\end{align*}"}
+{"id": "7698.png", "formula": "\\begin{align*} \\mu ( g _ z ) = - z ' + a g _ z z ^ \\ast = - z ' + z ^ \\ast + u - q ( u ) z . \\end{align*}"}
+{"id": "6910.png", "formula": "\\begin{align*} \\mathcal { J } : = \\left \\{ j \\in \\Z : [ ( 1 + T ^ { - 1 } ) ^ { j } , ( 1 + T ^ { - 1 } ) ^ { j + 1 } ) \\cap \\mathcal { M } \\not = \\emptyset \\right \\} \\end{align*}"}
+{"id": "7506.png", "formula": "\\begin{align*} \\tilde { \\Lambda } ( z ) = p _ { + } ( z ) p _ { - } ( z ) . \\end{align*}"}
+{"id": "2706.png", "formula": "\\begin{align*} \\begin{aligned} T _ 3 \\ge & - C s ^ 2 \\lambda ^ 2 \\iint _ { Q } \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 u ^ 2 d x d t - C s ^ 2 \\lambda ^ 2 \\int _ { 0 } ^ { T } \\int _ { \\omega } \\xi ^ 3 u ^ 2 d x d t , \\end{aligned} \\end{align*}"}
+{"id": "1406.png", "formula": "\\begin{align*} M _ { i + 1 } - ( k _ { i + 1 } - \\overline { \\lambda } _ { i + 1 } - 1 ) - 1 & = \\sharp \\{ j | - \\overline { \\lambda } ^ { i + 1 } _ { - j } \\leq \\overline { \\lambda } _ { i + 1 } \\} - 1 \\leq \\sharp \\{ j | - \\overline { \\lambda } ^ { i } _ { - j } \\leq \\overline { \\lambda } _ { i + 1 } \\} \\\\ & < \\sharp \\{ j | - \\overline { \\lambda } ^ { i } _ { - j } \\leq \\overline { \\lambda } _ { i } \\} = M _ { i } - ( k _ { i } - \\overline { \\lambda } _ { i } - 1 ) . \\end{align*}"}
+{"id": "6000.png", "formula": "\\begin{align*} \\theta _ { L , X ^ { \\ast } } ( A ) ( [ \\epsilon , 0 ] ) = \\sum _ { l \\in L \\cap X } A ( \\epsilon , l ) = \\sum _ { n \\in \\Z } A ( \\epsilon , n ) . \\end{align*}"}
+{"id": "685.png", "formula": "\\begin{align*} ( x _ - , x _ + ) = \\bigcup _ { 1 \\leq q \\leq q ( k ) } J ^ { ( k ) } ( q ) \\cup \\bigcup _ { l > k } \\bigcup _ { \\epsilon = \\pm } \\bigcup _ { 1 \\leq q \\leq q _ \\epsilon ( l ) } J _ \\epsilon ^ { ( l ) } ( q ) . \\end{align*}"}
+{"id": "4146.png", "formula": "\\begin{align*} Q _ { 1 , \\vec { v } _ { s , t , f , 0 } } = \\end{align*}"}
+{"id": "1682.png", "formula": "\\begin{align*} \\pi \\psi ( G ) = \\pi ( \\pi ( G ) ) = G , \\end{align*}"}
+{"id": "1428.png", "formula": "\\begin{align*} \\mu _ i = \\Big ( \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big ) ^ { \\frac { 2 i - 1 } { n - 2 } } d _ i , \\quad \\mbox { w i t h } d _ i > 0 , i = 1 , \\cdots , k . \\end{align*}"}
+{"id": "5892.png", "formula": "\\begin{align*} \\tau - \\psi _ m = \\sum _ { k = 1 } ^ m \\frac { ( 2 r ) ^ { k - 1 } } { ( 2 r + 1 ) ^ k } ( T ( \\phi ) \\phi _ k - \\phi _ k ) . \\end{align*}"}
+{"id": "6560.png", "formula": "\\begin{align*} G = \\omega _ Q ^ { 1 / \\bar { p } _ 2 - 1 / \\bar { p } _ 1 } \\int _ { \\mathbb H ^ n } \\frac { | y | _ h ^ { - Q / p } } { \\max ( 1 , | y | _ h ^ Q ) } d y . \\end{align*}"}
+{"id": "1988.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u _ { j + 1 } ) ^ n = \\psi ( \\cdot , u _ j ) \\ , \\omega ^ n & \\textnormal { o n } & \\Omega \\\\ u _ { j + 1 } = 0 & \\textnormal { o n } & \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "4590.png", "formula": "\\begin{align*} l _ T ( u , v , x ) & = [ T u , T v , x ] , \\\\ m _ T ( u , x , v ) & = [ T u , x , T v ] - T \\rho ( T u , x ) v , \\\\ r _ T ( x , u , v ) & = [ x , T u , T v ] - T \\rho ( x , T u ) v , \\end{align*}"}
+{"id": "7502.png", "formula": "\\begin{align*} U ( z ) \\nabla U ^ { - 1 } ( z ) = \\partial _ z + Z . \\end{align*}"}
+{"id": "3663.png", "formula": "\\begin{align*} \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) u : = \\sum _ { i = 1 } ^ { M ' } \\tilde { \\alpha } _ i ( - \\Delta ) ^ { \\tilde { s } _ i } u \\tilde { s } = ( \\tilde { s } _ 1 , \\dots , \\tilde { s } _ { M ' } ) , 0 < \\tilde { s } _ 1 < \\cdots < \\tilde { s } _ { M ' } \\leq s _ M , \\tilde { \\alpha } _ i \\in L ^ \\infty ( \\mathbb { R } ^ n ) \\end{align*}"}
+{"id": "7844.png", "formula": "\\begin{align*} \\langle \\phi , \\psi \\rangle = \\frac { 1 } { | G | } \\sum _ { g \\in G } \\phi ( g ) \\overline { \\psi ( g ) } . \\end{align*}"}
+{"id": "7207.png", "formula": "\\begin{align*} \\forall ( y , \\xi ) \\in B _ 1 \\times \\R ^ k , f _ 0 ( y , \\xi ) \\equiv f _ 0 ( \\xi ) : = \\abs { \\xi } ^ { p _ 0 } . \\end{align*}"}
+{"id": "7334.png", "formula": "\\begin{align*} - 1 = \\sum _ { k = 1 } ^ { N _ 1 } u ' _ k e _ k \\end{align*}"}
+{"id": "3140.png", "formula": "\\begin{align*} p _ m ( u , x ) : = \\sum _ { k = 0 } ^ { m } \\sum _ { i = 0 } ^ { m - k } p ( m , k , i ) u ^ k x ^ i \\end{align*}"}
+{"id": "2143.png", "formula": "\\begin{align*} u _ { + } = 0 , \\mathbb { F } _ { + } = \\mathbb { I } . \\end{align*}"}
+{"id": "3576.png", "formula": "\\begin{align*} A ( n ) \\ = \\ \\sum _ { i = 1 } ^ { k } p _ i \\cdot \\alpha _ i . \\end{align*}"}
+{"id": "8570.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } S _ j ( t ) = \\frac { \\nu Y } { \\lambda } \\frac 1 { j ( j + 1 ) } . \\end{align*}"}
+{"id": "6365.png", "formula": "\\begin{align*} \\rho _ t ( 0 , y ) : = \\lim _ { \\Gamma \\ni x \\to 0 } \\rho _ t ( x , y ) \\in ( 0 , \\infty ) , t > 0 , \\ , y \\in \\Gamma , \\end{align*}"}
+{"id": "1142.png", "formula": "\\begin{align*} p = \\sum _ { k = 0 } ^ { n } A ^ { \\ast } _ { k } , \\end{align*}"}
+{"id": "4985.png", "formula": "\\begin{align*} f ( k , n ) = \\dfrac { \\binom { n } { k } } { 2 ^ n } , 0 \\le k \\le n . \\end{align*}"}
+{"id": "199.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b ) = 1 \\\\ a , b \\geq 1 } } \\left ( \\frac { 1 } { 1 - y ^ a z ^ b } \\right ) ^ { \\frac { b } { a ^ 2 } } = \\exp \\left \\{ \\frac { z L i _ 2 ( y ) } { ( 1 - z ) ^ 2 } \\right \\} \\end{align*}"}
+{"id": "6525.png", "formula": "\\begin{align*} S _ 1 & = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { ( 2 k ) ! } \\bigg ( \\frac { 2 \\beta } { \\alpha } \\bigg ) ^ { 2 k } \\Gamma \\bigg ( k + \\frac { 1 } { 2 } \\bigg ) \\Gamma \\bigg ( \\nu + k + \\frac { 1 } { 2 } \\bigg ) , \\\\ S _ 2 & = - \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { ( 2 k + 1 ) ! } \\bigg ( \\frac { 2 \\beta } { \\alpha } \\bigg ) ^ { 2 k + 1 } k ! \\Gamma ( \\nu + k + 1 ) . \\end{align*}"}
+{"id": "779.png", "formula": "\\begin{align*} 0 = \\int _ { \\Omega } \\left . \\nabla ^ { \\mathbb { N } ^ { n + 1 } ( K ) } _ p \\left ( d _ K ^ 2 ( y , p ) \\right ) \\right | _ { p = O } \\mathrm { d } \\mu _ K ( y ) = \\int _ { \\Omega } \\left . 2 d _ K ( y , O ) \\nabla ^ { \\mathbb { N } ^ { n + 1 } ( K ) } _ p [ d _ K ( y , p ) ] \\right | _ { p = O } \\mathrm { d } \\mu _ K ( y ) . \\end{align*}"}
+{"id": "1767.png", "formula": "\\begin{align*} F ( \\omega ) = f ( W ( h _ 1 ) , \\ldots , W ( h _ { m _ 1 } ) , N ( g _ 1 ) , \\ldots , N ( g _ { m _ 2 } ) ) , \\end{align*}"}
+{"id": "4077.png", "formula": "\\begin{align*} \\langle z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace , f \\rangle = 0 \\end{align*}"}
+{"id": "1198.png", "formula": "\\begin{align*} \\norm { ( \\phi _ n , B _ n \\phi _ n ) - ( \\phi , B \\phi ) } ^ 2 & = \\norm { \\phi _ n - \\phi } ^ 2 + \\norm { B _ n \\phi _ n - B \\phi } ^ 2 = \\norm { \\phi _ n - \\phi } ^ 2 + \\norm { i \\phi - i \\phi _ n } ^ 2 \\\\ & = 2 \\norm { ( B _ n + i ) ^ { - 1 } \\psi - ( B + i ) ^ { - 1 } \\psi } ^ 2 \\longrightarrow 0 \\end{align*}"}
+{"id": "5079.png", "formula": "\\begin{align*} \\widetilde { \\mathcal R } ( \\gamma , \\gamma _ t , \\sigma _ t ) & = - \\beta J \\left ( \\phi '' \\left ( \\cdot + \\gamma ( \\cdot , t ) + \\frac { 1 } { N } \\sigma ( t ) \\right ) \\left ( 2 \\gamma _ x + \\gamma _ x ^ 2 \\right ) + \\phi ' \\left ( \\cdot + \\gamma ( \\cdot , t ) + \\frac { 1 } { N } \\sigma ( t ) \\right ) \\gamma _ { x x } \\right ) \\\\ & + \\ , \\phi ' \\left ( \\cdot + \\gamma ( \\cdot , t ) + \\frac { 1 } { N } \\sigma ( t ) \\right ) \\left ( \\gamma _ t + \\frac { 1 } { N } \\sigma _ t \\right ) . \\end{align*}"}
+{"id": "5304.png", "formula": "\\begin{align*} \\langle T _ { ( 1 - p ) \\rightarrow p } A h , \\ , g \\rangle = \\sum _ { d = 0 } ^ m \\left ( - \\frac { p } { 1 - p } \\right ) ^ d \\langle h ^ { = d } , g \\rangle \\geq \\mu _ { p } ( h ) \\mu _ { p } ( g ) - \\sum _ { d = 1 } ^ m \\left ( \\frac { p } { 1 - p } \\right ) ^ d | \\langle h ^ { = d } , g \\rangle | . \\end{align*}"}
+{"id": "2453.png", "formula": "\\begin{align*} f * ( \\psi * g ) = ( f * \\psi ) * g = \\mu _ ! ( f \\ltimes \\psi \\rtimes g ) & & f \\in C _ c ^ \\infty ( G _ 1 ) , \\psi \\in C _ c ^ \\infty ( M ) , g \\in C _ c ^ \\infty ( G _ 2 ) . \\end{align*}"}
+{"id": "1204.png", "formula": "\\begin{align*} e ^ t M _ \\ell ( \\boldsymbol { a } , t ) = a _ \\ell + \\int _ 0 ^ t e ^ u M _ \\ell ( \\boldsymbol { a } , u ) \\left ( q M _ \\ell ( \\boldsymbol { a } , u ) + \\phi ( u ) + 1 \\right ) . \\end{align*}"}
+{"id": "8431.png", "formula": "\\begin{align*} \\zeta _ { p ^ { \\prime } } ( 1 - s ) = 2 \\cos \\left ( \\frac { \\pi s } { 2 } \\right ) \\intop _ { 0 } ^ { \\infty } \\frac { x ^ { s - 1 } } { \\sigma ^ { \\prime } ( x ) \\ , e ^ { 2 \\pi x } - 1 } \\ , d x . \\end{align*}"}
+{"id": "2850.png", "formula": "\\begin{align*} R ( l _ 1 ) \\otimes R ( l _ 2 ) = R ( l ) _ 1 \\otimes R ( R ( l ) _ 2 ) + R ( R ( l ) _ 1 ) \\otimes R ( l ) _ 2 + \\lambda R ( l ) _ 1 \\otimes R ( l ) _ 2 . \\end{align*}"}
+{"id": "4356.png", "formula": "\\begin{align*} x = \\frac { \\alpha _ { 1 } ^ { \\frac { 1 } { p - 1 } } } { \\alpha _ { 1 } ^ { \\frac { 1 } { p - 1 } } + \\alpha _ { 2 } ^ { \\frac { 1 } { p - 1 } } } P _ { 1 } \\left ( x \\right ) + \\frac { \\alpha _ { 2 } ^ { \\frac { 1 } { p - 1 } } } { \\alpha _ { 1 } ^ { \\frac { 1 } { p - 1 } } + \\alpha _ { 2 } ^ { \\frac { 1 } { p - 1 } } } P _ { 2 } \\left ( x \\right ) \\end{align*}"}
+{"id": "1522.png", "formula": "\\begin{align*} x = & \\phantom { . } e _ { 1 , 2 } ( r _ { 1 , 2 } ) e _ { 2 , 3 } ( r _ { 2 , 3 } ) \\cdot \\ldots \\cdot e _ { n - 1 , n } ( r _ { n - 1 , n } ) \\cdot \\\\ & \\cdot e _ { 1 , 3 } ( r _ { 1 , 3 } ) \\cdot \\ldots \\cdot e _ { n - 2 , n } ( r _ { n - 2 , n } ) \\cdot \\ldots \\ldots \\cdot e _ { 1 , n } ( r _ { 1 , n } ) . \\end{align*}"}
+{"id": "2282.png", "formula": "\\begin{align*} w _ b = \\Phi _ b + ( \\widetilde { T } ( f ) ) _ b . \\end{align*}"}
+{"id": "5141.png", "formula": "\\begin{align*} \\iota ( g ) ( f ) : = \\int _ \\Omega \\ ! f g \\ , \\mathrm { d } \\mu , f \\in X , \\end{align*}"}
+{"id": "8375.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} { c c } 3 & 0 \\\\ 0 & 3 \\end{array} \\right ) , A = \\left ( \\begin{array} { c c c } 2 & 3 \\\\ 1 & 0 \\end{array} \\right ) , L = \\left ( \\begin{array} { c c c } 1 & - 3 \\\\ - 1 & 3 \\end{array} \\right ) . \\end{align*}"}
+{"id": "3837.png", "formula": "\\begin{align*} \\Theta _ { \\mathrm { D } } ( \\delta ) : = \\left \\{ \\int _ { \\mathcal { Y } _ 1 \\times \\mathcal { Y } _ 2 } f ( y _ 1 , y _ 2 ) \\ , d \\gamma ( y _ 1 , y _ 2 ) : \\gamma \\in \\Sigma _ { \\mathrm { D } } ( \\delta ) \\right \\} , \\end{align*}"}
+{"id": "434.png", "formula": "\\begin{align*} \\vec U ^ T ( I _ n \\otimes E ^ T P _ { \\partial \\Omega } N _ { x _ i } E ) { \\bf A _ i } \\vec U = \\sum _ { j = 1 , N } \\lbrack ( E \\vec U ) ^ T ( N _ { i } { \\bf A _ i } ) ( E \\vec U ) \\rbrack _ j d s _ j = \\sum _ { j = 1 , N } \\lbrack \\vec W ^ T \\Lambda \\vec W \\rbrack _ j d s _ j . \\end{align*}"}
+{"id": "7461.png", "formula": "\\begin{gather*} \\Pi _ { \\alpha - 1 } ^ { ( i ) } ( \\eta _ i , t ) = \\Phi _ { \\alpha - 1 } ( t ) \\ , e ^ { - v _ i ^ { ( i ) } ( \\ell _ i ) \\ , \\eta _ i } , \\Pi _ 0 ^ { ( i ) } ( \\eta _ i , t ) = \\Phi _ 0 ( t ) \\ , e ^ { - v _ i ^ { ( i ) } ( \\ell _ i ) \\ , \\eta _ i } , \\\\ \\Pi _ { \\alpha } ^ { ( i ) } ( \\eta _ 1 , t ) = \\left ( \\Phi _ { \\alpha } ( t ) - \\frac { \\partial _ t { \\Phi } _ { \\alpha - 1 } ( t ) } { v _ i ^ { ( i ) } ( \\ell _ 3 ) } \\ , \\eta _ 1 \\right ) e ^ { - v _ i ^ { ( i ) } ( \\ell _ i ) \\ , \\eta _ i } \\end{gather*}"}
+{"id": "8448.png", "formula": "\\begin{align*} K _ { \\nu } ( x ) = \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\mu - i \\infty } ^ { \\mu + i \\infty } 2 ^ { s - 2 } \\Gamma \\left ( \\frac { s - \\nu } { 2 } \\right ) \\Gamma \\left ( \\frac { s + \\nu } { 2 } \\right ) \\ , x ^ { - s } d s , \\ , \\ , \\ , \\ , \\ , x > 0 , \\ , \\ , \\ , \\mu : = ( s ) > \\max \\{ 0 , ( \\nu ) \\} . \\end{align*}"}
+{"id": "2880.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 8 , & ~ ~ t _ { 0 } \\geq 6 , \\\\ 7 , & ~ ~ t _ { 0 } = 5 , \\\\ 6 , & ~ ~ t _ 0 = 4 , \\\\ 4 - t _ 0 , & ~ ~ t _ 0 \\leq 3 . \\end{array} \\right . \\end{align*}"}
+{"id": "8665.png", "formula": "\\begin{align*} \\operatorname { O } _ { V ^ { \\perp } } ( V , q ) : = \\big \\{ \\varphi \\in \\operatorname { O } ( V , q ) \\big | \\varphi | _ { V ^ { \\perp } } = \\operatorname { I d } _ { V ^ { \\perp } } \\big \\} = \\{ \\varphi \\in \\operatorname { O } ( V , q ) | \\varphi ( v ) = v \\ \\forall v \\in V ^ { \\perp } \\} . \\end{align*}"}
+{"id": "3696.png", "formula": "\\begin{align*} ( q _ 1 ( x ) - q _ 2 ( x ) ) u _ 1 ( x ) = 0 \\quad E , \\end{align*}"}
+{"id": "3780.png", "formula": "\\begin{align*} \\varphi ( p ( \\mathcal J ) ) _ { k + 1 + i } = \\varphi ( p ( \\mathcal J ) ) _ { k + i } \\oplus z ^ { n + i + 2 - x } , \\end{align*}"}
+{"id": "6310.png", "formula": "\\begin{align*} \\Omega _ t ^ p : = \\{ \\phi _ t ( p , q ) : q \\in \\Omega \\} . \\end{align*}"}
+{"id": "7374.png", "formula": "\\begin{align*} \\begin{aligned} \\langle S _ N ( \\ell ) ^ 2 \\rangle & = \\sum _ { i , j = 1 } ^ N \\sum _ { m , n \\in \\mathbb { Z } } \\int _ 0 ^ 1 \\chi \\left ( \\frac { x _ i - x _ 0 + m } { \\ell } \\right ) \\chi \\left ( \\frac { x _ j - x _ 0 + n } { \\ell } \\right ) \\ , d x _ 0 \\\\ & = \\sum _ { i , j = 1 } ^ N \\sum _ { m \\in \\mathbb { Z } } \\int _ \\mathbb { R } \\chi \\left ( \\frac { x _ i - x _ 0 + m } { \\ell } \\right ) \\chi \\left ( \\frac { x _ j - x _ 0 } { \\ell } \\right ) \\ , d x _ 0 . \\end{aligned} \\end{align*}"}
+{"id": "4195.png", "formula": "\\begin{align*} \\beta ( M , C ) = \\min _ { \\substack { P \\subset \\mathbb { Z } _ M \\\\ P \\ne \\emptyset } } \\left \\lbrace \\vert P \\vert \\colon \\Vert P \\Vert _ u \\leq \\frac { C - 3 } { C - 1 } \\mathbf { P } ( P ) \\right \\rbrace . \\end{align*}"}
+{"id": "2828.png", "formula": "\\begin{align*} a _ { i } : = \\frac { b _ { i } } { a _ { i + 1 } \\cdots a _ { i + p - 1 } } , i \\le I - 1 ; \\end{align*}"}
+{"id": "7722.png", "formula": "\\begin{align*} \\vartheta _ L ( f ; h _ \\kappa ^ 3 ) ( g _ K ) = \\bigl ( \\Theta _ { ( L , K ) } ( f ; \\tau ) , \\Theta _ K ( \\tau , g _ K ; h _ \\kappa ^ 3 ) \\bigr ) _ \\tau \\end{align*}"}
+{"id": "3511.png", "formula": "\\begin{align*} ( q , p ) = ( q ^ 1 , \\ldots , q ^ n , p ^ 1 , \\ldots , p ^ n ) = q + i p . \\end{align*}"}
+{"id": "6636.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } x _ 0 y _ 0 + x _ 1 y _ 1 + x _ 2 y _ 2 = 0 , \\\\ \\\\ \\displaystyle \\sum _ { \\substack { 0 \\le i , j \\le 2 } } a _ { i , j } x _ i y _ j = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1191.png", "formula": "\\begin{align*} ( p ( u ) - p ( v ) ) ^ \\top ( A ^ \\top A - I _ d ) ( p ( u ) - p ( v ) ) = 0 . \\end{align*}"}
+{"id": "5720.png", "formula": "\\begin{align*} \\langle D ^ { k - 1 } ( f ) , g \\rangle = 1 . \\end{align*}"}
+{"id": "6066.png", "formula": "\\begin{align*} \\delta _ { t } ^ I \\delta _ { r } ^ { \\Lambda } x = \\delta _ { r } ^ { \\Lambda } \\delta _ { t } ^ I x , r , t > 0 . \\end{align*}"}
+{"id": "8183.png", "formula": "\\begin{align*} n _ t : = \\left \\lceil \\frac { k t } { \\rho ( t ) / ( 2 \\sqrt { d } ) } \\right \\rceil ^ d \\end{align*}"}
+{"id": "4148.png", "formula": "\\begin{align*} Q _ { 2 , \\vec { v } _ { s , t , f , 0 } ^ { ( 1 ) } } = \\left ( \\left ( \\begin{array} { r r r } s ^ { 2 } - t & 1 & - s \\\\ - s & 0 & 1 \\\\ s t + 1 & 0 & - t \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , \\left ( \\begin{array} { r r r } - s & 0 & 1 \\\\ 1 & 0 & 0 \\\\ - t & 1 & 0 \\end{array} \\right ) \\right ) , \\end{align*}"}
+{"id": "2590.png", "formula": "\\begin{align*} ( D v ) _ { \\# } ( g + f \\chi _ { _ { \\Omega \\backslash \\overline { U } } } ) ) = f . \\end{align*}"}
+{"id": "5501.png", "formula": "\\begin{align*} \\xi _ { \\ast } \\left ( p _ { \\ast } \\left ( m _ { K \\cap Q } \\right ) \\times \\lambda \\right ) = \\int _ { K \\cap Q } \\vartheta ( k P ) . \\lambda d m _ { K \\cap Q } ( k ) = \\int _ { K \\cap Q } k . \\lambda d m _ { K \\cap Q } ( k ) = m _ { K \\cap Q } \\ast \\lambda = \\eta . \\end{align*}"}
+{"id": "1957.png", "formula": "\\begin{align*} ( \\det a ) ^ \\frac { 1 } { n } & \\geq \\int _ { 0 } ^ 1 \\Big [ \\det \\Big ( [ \\det x ( t ) ] ^ { \\frac { 1 } { n } - 1 } \\tilde { x } ( t ) \\Big ) \\Big ] ^ \\frac { 1 } { n } d t \\\\ & = \\int _ { 0 } ^ 1 [ \\det x ( t ) ] ^ { \\frac { 1 } { n } - 1 } [ \\det \\tilde { x } ( t ) ] ^ \\frac { 1 } { n } d t . \\end{align*}"}
+{"id": "442.png", "formula": "\\begin{align*} T _ { \\mu \\nu } = ( \\rho + p ) u _ { \\mu } u _ \\nu + p g _ { \\mu \\nu } , \\end{align*}"}
+{"id": "2607.png", "formula": "\\begin{align*} A ^ * _ { h _ 0 } \\mathcal { C } _ { K , r } = A ^ * _ { h _ 0 } T _ { h _ 0 } ^ { - 1 } T _ { h _ 0 } \\mathcal { C } _ { K , r } \\rightarrow \\{ x \\in \\mathbb { R } ^ n : x \\cdot e _ 0 \\geq 0 \\} \\end{align*}"}
+{"id": "2993.png", "formula": "\\begin{align*} c _ { F , j } = ( - 1 ) ^ { n - j } \\frac { \\partial V } { \\partial \\bar { c } _ { n - ( j + 1 ) } } \\end{align*}"}
+{"id": "5316.png", "formula": "\\begin{align*} \\| D _ { i , x } [ f ^ { = 1 } ] \\| _ 2 = | \\hat { f } ( \\{ i \\} ) | \\ , | \\chi ^ p _ i ( x _ i ) | \\leq \\frac { \\sqrt { 2 } \\sigma } { \\sqrt { m p } } \\end{align*}"}
+{"id": "8617.png", "formula": "\\begin{align*} E [ Z _ 0 ( s ) ^ 2 ] = c _ 1 e ^ { 2 \\lambda s } - c _ 2 e ^ { \\lambda s } . \\end{align*}"}
+{"id": "8817.png", "formula": "\\begin{align*} u ( t ) = u _ 0 + \\int _ 0 ^ t b ( s ) \\ , d s + \\int _ 0 ^ t G ( s ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "1637.png", "formula": "\\begin{align*} [ \\xi _ i , \\xi _ j ] & = 0 , 1 \\le i , j \\le p . \\end{align*}"}
+{"id": "42.png", "formula": "\\begin{align*} \\mathcal { G } _ { \\overline { \\psi } _ 0 } ^ { \\mathbf { K } } : = \\mathcal { G } ( \\mathbb { Q } ) \\bigcap \\mathbf { K } _ { { \\overline { \\psi } _ 0 } } ^ p \\end{align*}"}
+{"id": "8027.png", "formula": "\\begin{align*} \\left \\| \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} \\right \\| \\le \\| A \\| + \\| B \\| . \\end{align*}"}
+{"id": "4190.png", "formula": "\\begin{align*} \\tfrac { d } { d \\tau } \\bigl [ e ^ \\tau F ( t _ 0 , x _ 0 , e ^ \\tau ) - e ^ \\tau \\tilde F ( t _ 0 , x _ 0 , e ^ \\tau ) \\bigr ] = 0 , \\end{align*}"}
+{"id": "6606.png", "formula": "\\begin{align*} f * g ( d ) = ( - 1 ) ^ { | f | | d _ { ( 2 ) } | } f ( d _ { ( 1 ) } ) g ( d _ { ( 2 ) } ) , \\end{align*}"}
+{"id": "1366.png", "formula": "\\begin{align*} \\langle x _ 1 , x _ 2 , x _ 3 : x _ 1 x _ 2 = x _ 3 x _ 1 , \\ , x _ 1 x _ 3 = x _ 2 x _ 1 , \\ , x _ 2 x _ 3 = x _ 1 x _ 2 \\rangle . \\end{align*}"}
+{"id": "567.png", "formula": "\\begin{align*} \\psi \\left ( \\begin{pmatrix} b _ 1 & b _ 2 \\\\ b _ 3 & b _ 4 \\end{pmatrix} \\right ) = \\begin{pmatrix} b _ 1 ^ 2 & b _ 1 b _ 2 & b _ 2 ^ 2 \\\\ 2 b _ 1 b _ 3 & b _ 1 b _ 4 + b _ 2 b _ 3 & 2 b _ 2 b _ 4 \\\\ b _ 3 ^ 2 & b _ 3 b _ 4 & b _ 4 ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "5022.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\overline { p } _ { - t } ( n ) q ^ n = \\frac { f _ 2 ^ t } { f _ 1 ^ { 2 t } } = \\left ( \\frac { f _ 2 } { f _ 1 ^ { 2 } } \\right ) ^ t . \\end{align*}"}
+{"id": "776.png", "formula": "\\begin{align*} \\phi ' ( \\rho ( 1 + u ) ) = \\phi ' ( \\rho ) - K \\phi ( \\rho ) \\rho u - \\dfrac { 1 } { 2 } K \\phi ' ( \\rho ) \\rho ^ 2 u ^ 2 + o ( u ^ 2 ) , \\end{align*}"}
+{"id": "173.png", "formula": "\\begin{align*} L i _ 2 ( \\phi ^ { - 2 } ) = \\frac { \\pi ^ 2 } { 1 5 } - ( \\log \\phi ) ^ 2 , \\end{align*}"}
+{"id": "1062.png", "formula": "\\begin{align*} f \\circ _ { } g \\coloneqq \\sum _ { i = 0 } ^ { \\infty } \\left ( \\sum _ { k = 0 } ^ { i } \\binom { i } { k } a ( k ) b ( i - k ) \\right ) t ^ { i } \\in k [ [ t ] ] \\ , , \\end{align*}"}
+{"id": "8067.png", "formula": "\\begin{align*} \\| u \\| : = \\left ( \\int _ { \\Omega } \\left | X u \\right | ^ { p } d x \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
+{"id": "5457.png", "formula": "\\begin{align*} M _ b ( \\theta ) = \\mathbb { E } _ { \\Phi | \\Phi ( \\mathcal { A } ) > 0 } \\{ P _ s ^ b ( \\theta ) \\} \\Pr ( \\Phi ( \\mathcal { A } ) > 0 ) . \\end{align*}"}
+{"id": "1741.png", "formula": "\\begin{align*} V ^ { \\sigma } ( \\hat { \\nu } ^ * , \\mu ^ * ) - V ^ { \\sigma } ( \\nu ^ * , \\mu ^ * ) & \\geq \\int _ { \\mathcal { X } } \\left ( \\frac { \\delta F } { \\delta \\nu } ( \\nu ^ * , \\mu ^ * , x ) + \\frac { \\sigma ^ 2 } { 2 } \\log \\left ( \\frac { \\nu ^ * ( x ) } { \\pi ( x ) } \\right ) \\right ) ( \\hat { \\nu } ^ * - \\nu ^ * ) ( \\mathrm { d } x ) \\\\ & + \\frac { \\sigma ^ 2 } { 2 } \\operatorname { D _ { K L } } ( \\hat { \\nu } ^ * | \\nu ^ * ) = \\frac { \\sigma ^ 2 } { 2 } \\operatorname { D _ { K L } } ( \\hat { \\nu } ^ * | \\nu ^ * ) , \\end{align*}"}
+{"id": "5718.png", "formula": "\\begin{align*} \\langle \\sum _ { n \\gg - \\infty } a _ { n } q ^ { n } , \\sum _ { n > 0 } b _ { n } q ^ { n } \\rangle : = \\displaystyle \\sum _ { n < 0 } \\dfrac { a _ { n } b _ { - n } } { n ^ { k - 1 } } . \\end{align*}"}
+{"id": "7555.png", "formula": "\\begin{align*} c _ 3 = \\sum _ r \\frac { \\mu ( r ) } { r ^ 2 } f _ 1 ( r ) f _ 2 ( r ) \\end{align*}"}
+{"id": "2748.png", "formula": "\\begin{align*} \\| R _ q ( \\lambda ) f \\| _ { H ^ j ( \\mathrm { M } ) } \\le c _ 1 \\lambda ^ { j / 2 } \\mathbf { e } _ \\lambda \\| f \\| _ { L ^ 2 ( \\mathrm { M } ) } , j = 1 , 2 . \\end{align*}"}
+{"id": "1779.png", "formula": "\\begin{align*} \\mathcal { A } [ \\underline { n } ] = \\mathcal { A } [ n _ 1 ] * \\dots * \\mathcal { A } [ n _ p ] \\end{align*}"}
+{"id": "7321.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\beta _ i z _ { i p } = \\sum _ { i = 1 } ^ n \\gamma _ i z _ { i p } < \\sum _ { i = 1 } ^ n \\alpha _ i z _ { i p } . \\end{align*}"}
+{"id": "7804.png", "formula": "\\begin{align*} \\eta _ { ( n ) } = \\sum _ { i = 1 } ^ { n } \\frac { T _ { i } } { 2 ( n - i + 1 ) } , \\end{align*}"}
+{"id": "3469.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda _ n } g _ n ' ( 0 , 0 ) = \\rho - | \\alpha | \\overline { P \\left ( \\frac { d \\mu _ { \\lambda , n } } { d \\lambda _ n } \\right ) ( 1 ) } . \\end{align*}"}
+{"id": "3538.png", "formula": "\\begin{align*} \\wp ' ( z + w ) = \\frac { \\wp ( w ) \\wp ' ( z ) - \\wp ' ( w ) \\wp ( z ) - \\wp ( z + w ) ( \\wp ' ( z ) - \\wp ' ( w ) ) } { \\wp ( z ) - \\wp ( w ) } . \\end{align*}"}
+{"id": "5527.png", "formula": "\\begin{align*} \\mathrm { I } \\left ( \\xi _ { 1 } , \\mathcal { T } | X , \\eta \\right ) & : = \\int _ { X } \\mathrm { I } \\left ( \\xi _ { 1 } ^ { x } , \\mathcal { T } _ { x } \\right ) d \\eta ( x ) . \\end{align*}"}
+{"id": "6943.png", "formula": "\\begin{align*} x _ { i } & = E _ i + k - 1 \\\\ x _ { m - j _ { 2 k } } & = E _ { m - j _ { 2 k } } + k - 1 + \\frac 1 2 ( m - j _ { 2 k } - i ) + \\frac 1 2 \\\\ x _ { m + k } & = F _ k - j _ { 2 k } - ( m - j _ { 2 k } - i ) = F _ k - m + i . \\end{align*}"}
+{"id": "9070.png", "formula": "\\begin{align*} f _ 1 & \\approx ( 1 . 7 6 , 1 . 6 2 , 2 . 8 4 , 1 . 6 2 , 1 , 1 ) ^ T , \\\\ f _ 2 & = ( 0 , - 1 , 0 , 1 , 0 , 0 ) ^ T , \\\\ f _ 3 & = ( 0 , 0 , 0 , 0 , - 1 , 1 ) ^ T , \\\\ f _ 4 & \\approx ( - 2 . 4 4 , - 0 . 6 2 , 1 . 5 1 , - 0 . 6 2 , 1 , 1 ) ^ T , \\\\ f _ 5 & \\approx ( 0 . 8 2 , - 0 . 6 2 , - 0 . 5 1 , - 0 . 6 2 , 1 , 1 ) ^ T , \\\\ f _ 6 & \\approx ( - 1 . 1 4 , 1 . 6 2 , - 1 . 8 4 , 1 . 6 2 , 1 , 1 ) ^ T . \\end{align*}"}
+{"id": "4534.png", "formula": "\\begin{align*} I _ { \\omega , \\mathbf { c } } ( U ( t ) ) \\le 2 ( S _ { \\omega , \\mathbf { c } } ( U ( t ) ) - S _ { \\omega , \\mathbf { c } } ( \\Phi _ 0 ) ) = 2 ( S _ { \\omega , \\mathbf { c } } ( U ( t ) ) - \\mu _ { \\omega , \\mathbf { c } } ) \\end{align*}"}
+{"id": "7009.png", "formula": "\\begin{align*} \\textbf { Q } ^ { \\lambda _ j } = Q _ { i _ 1 } ^ { \\lambda _ j ( Q _ { i _ 1 } ) } \\cdots Q _ { i _ r } ^ { \\lambda _ j ( Q _ { i _ r } ) } . \\end{align*}"}
+{"id": "935.png", "formula": "\\begin{align*} \\mathbb { E } ( X ) = \\sum _ { \\substack { 0 \\leq i < j \\leq h \\\\ j - i \\leq t } } \\mathbb { P } ( s _ i = s _ j ) + \\sum _ { \\substack { 0 \\leq i \\leq h < j \\leq k \\\\ j - i \\leq t } } \\mathbb { P } ( s _ i = s _ j ) + \\sum _ { \\substack { h < i < j \\leq k \\\\ j - i \\leq t } } \\mathbb { P } ( s _ i = s _ j ) . \\end{align*}"}
+{"id": "8235.png", "formula": "\\begin{align*} \\left ( \\int _ s ^ t \\pi _ { r } ^ N ( A \\phi _ j ) d r \\right ) ^ 2 = \\frac { 1 } { N ^ 2 } \\left [ \\int _ s ^ t \\sum _ { ( x , \\sigma ) \\in V } \\eta _ r ^ N ( x , \\sigma ) \\cdot ( A \\phi _ j ) ( \\tfrac { x } { N } , \\sigma ) d r \\right ] ^ 2 . \\end{align*}"}
+{"id": "2328.png", "formula": "\\begin{align*} \\| u ( s , 0 ) - u ( s , h ) \\| _ p \\leq \\sum _ { n \\geq 1 } ( p - 1 ) ^ { n / 2 } \\| I _ n ( { f } _ { n , h } ( \\cdot , 0 , s ) ) \\| _ 2 = \\sum _ { n \\geq 1 } ( p - 1 ) ^ { n / 2 } [ J _ { n , h } ( s ) ] ^ { 1 / 2 } \\end{align*}"}
+{"id": "368.png", "formula": "\\begin{align*} \\eta ( \\sigma ) ^ * ( y _ { \\tau , \\mathfrak { u } } ) = \\left ( \\prod ^ k _ { \\ell = 1 } \\frac { c _ { i _ { \\ell } } ^ { \\sigma } } { c _ { i _ { \\ell } } ^ { \\tau } } \\right ) x _ { \\tau , \\mathfrak { u } } . \\end{align*}"}
+{"id": "9187.png", "formula": "\\begin{align*} \\begin{array} { c c l } y _ { 1 , [ 2 ] } ^ { 1 } & = & v _ { 1 } ^ { 1 } \\\\ y _ { 2 , [ 4 ] } ^ { 1 } & = & v _ { 2 } ^ { 1 } \\ , . \\end{array} \\end{align*}"}
+{"id": "4826.png", "formula": "\\begin{align*} I _ { 3 } = \\mathbb { P } \\left ( \\sum _ { i = 1 } ^ { n } \\eta _ { n , i } \\geq \\frac { x } { 2 } \\right ) \\ \\ \\ \\ \\ \\ I _ { 4 } = \\mathbb { P } \\left ( \\frac { \\log W _ { n } } { \\sigma \\sqrt { n } } \\geq \\frac { x } { 2 } \\right ) . \\end{align*}"}
+{"id": "8935.png", "formula": "\\begin{align*} \\partial _ t v + v \\cdot \\nabla v - \\Delta v = - \\nabla q , { \\rm d i v } \\ , v = 0 \\end{align*}"}
+{"id": "6754.png", "formula": "\\begin{align*} \\hat { x } _ z ( 0 ) = x ( \\min \\{ k \\geq 0 : z ( k ) = 1 \\} ) . \\end{align*}"}
+{"id": "795.png", "formula": "\\begin{align*} l ( U ^ { n , \\sigma } ) - d \\ , = \\ , 0 , \\end{align*}"}
+{"id": "7239.png", "formula": "\\begin{align*} \\mathcal { B } _ a : = \\left \\{ y + a : y \\in \\mathcal { B } \\right \\} \\mbox { a n d } \\mathcal { X } _ a : = \\left \\{ y + a : y \\in \\mathcal { X } \\right \\} \\end{align*}"}
+{"id": "5773.png", "formula": "\\begin{align*} \\mathbb { R } ^ { n } \\setminus \\Omega ^ { * } & \\subset \\mathbb { R } ^ { n } \\setminus \\big ( Q _ { 1 , k _ { 1 } } ^ { * } \\bigcup Q _ { 2 , k _ { 2 } } ^ { * } \\big ) \\subset \\bigcup _ { h = 1 } ^ { \\infty } \\bigcup _ { i = 1 } ^ { \\infty } \\big ( \\mathcal { Q } _ { 1 , k _ { 1 } } ^ { i } \\bigcap \\mathcal { Q } _ { 2 , k _ { 2 } } ^ { h } \\big ) . \\end{align*}"}
+{"id": "3332.png", "formula": "\\begin{align*} \\frac { \\partial g } { \\partial \\theta } = \\left ( e ^ { - i ( d + i H d ) } \\right ) \\left ( e ^ { i ( d + i H d ) } \\frac { \\partial g } { \\partial \\theta } \\right ) . \\end{align*}"}
+{"id": "4837.png", "formula": "\\begin{align*} ( \\prod \\limits _ { i = 1 } ^ m a ( i ) \\ast f ( t ( i ) ) ) \\ast a ( m + 1 ) \\in A \\cap \\sigma ( L ) , \\end{align*}"}
+{"id": "2503.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle G ( t ) = \\int _ { \\mathbb { R } ^ N } | x | ^ 2 \\left ( a _ 2 | \\phi | ^ 2 + a _ 1 | \\psi | ^ 2 \\right ) d x , \\end{array} \\right . \\end{align*}"}
+{"id": "7169.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 ( J \\cap \\partial B _ R ) = 0 . \\end{align*}"}
+{"id": "9042.png", "formula": "\\begin{align*} y _ i ' = \\begin{cases} \\displaystyle { y } _ { i } { y } _ { k } ^ { \\max \\{ b _ { i , k } , 0 \\} } ( 1 + { y } _ { k } ) ^ { - b _ { i , k } } & i \\neq k , \\\\ { y } _ { k } ^ { - 1 } & . \\end{cases} \\end{align*}"}
+{"id": "1003.png", "formula": "\\begin{align*} ( x ) _ 0 = 1 \\quad ( x ) _ m = x ( x + 1 ) \\cdots ( x + m - 1 ) m \\in \\mathbb { Z } ^ { + } . \\end{align*}"}
+{"id": "6029.png", "formula": "\\begin{align*} \\int \\otimes _ { i = 1 } ^ n f _ i \\otimes g \\ , d \\tilde \\rho = \\int \\bigl ( \\prod _ { i = 1 } ^ n V f _ i \\bigr ) \\ , J ^ { \\ast } g \\ , d \\mu , \\end{align*}"}
+{"id": "2269.png", "formula": "\\begin{align*} \\lim _ { r \\nearrow 1 } \\int _ 0 ^ { 2 \\pi } | T ( f ) ( e ^ { i \\theta } ) - T ( f ) ( r e ^ { i \\theta } ) | ^ \\gamma \\ , d \\theta = 0 , \\end{align*}"}
+{"id": "2048.png", "formula": "\\begin{align*} \\mathcal { W A } _ \\lambda [ u ] = \\frac { 1 } { 2 } \\norm { u } _ { L ^ 2 _ \\lambda } ^ 2 ; \\end{align*}"}
+{"id": "215.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c , d , e ) = 1 \\\\ a , b , c , d , e \\geq 1 } } \\left ( \\frac { 1 } { 1 - v ^ a w ^ b x ^ c y ^ d z ^ e } \\right ) ^ { \\frac { 1 } { a ^ q b ^ r c ^ s d ^ t e ^ u } } \\end{align*}"}
+{"id": "8112.png", "formula": "\\begin{align*} X _ T \\prec X _ F = X _ { F T } , \\end{align*}"}
+{"id": "6822.png", "formula": "\\begin{align*} \\alpha _ n = ( - 1 ) ^ n q ^ { \\left ( { n \\atop 2 } \\right ) } \\frac { 1 - a q ^ { 2 n } } { 1 - a } \\frac { ( a ) _ n } { ( q ) _ n } , \\beta _ n = \\delta _ { n , 0 } . \\end{align*}"}
+{"id": "1392.png", "formula": "\\begin{align*} x _ i = b _ i + \\sum _ { j > i } t _ { i j } b _ j \\end{align*}"}
+{"id": "6551.png", "formula": "\\begin{align*} C = \\int _ { \\mathbb { R } ^ n } K \\left ( x , \\hat { e } _ 1 \\right ) | x | ^ { - n / p ^ { \\prime } } d x \\end{align*}"}
+{"id": "8009.png", "formula": "\\begin{align*} \\lambda _ k [ f ( A _ { \\mathcal { S ' } } ) ] & \\le \\min _ { h \\in { \\mathcal { F } } ; \\ \\Vert h \\Vert = 1 } \\langle h , f ( A ) h \\rangle \\\\ & \\le \\lambda _ k [ f ( A ) _ { \\mathcal { S ' } } ] . \\end{align*}"}
+{"id": "2643.png", "formula": "\\begin{align*} I : = \\int _ { [ 0 , 1 ] ^ 3 } \\ ! \\ ! f ( x , y ) f \\left ( x + P _ 1 ( t ) , y \\right ) f \\left ( x , y + P _ 2 ( t ) \\right ) \\ , \\mathrm { d } t \\mathrm { d } x \\mathrm { d } y > \\delta \\end{align*}"}
+{"id": "4896.png", "formula": "\\begin{align*} H ^ { n - 1 } ( \\partial B _ 1 ) = \\int _ { \\partial B _ 1 } | e - \\omega | ^ { 2 s - n } \\ , d H ^ { n - 1 } _ \\omega . \\end{align*}"}
+{"id": "1635.png", "formula": "\\begin{align*} ( \\pounds _ { \\xi _ i } \\ , d \\eta ^ j ) ( X , Y ) = ( \\pounds _ { \\xi _ i } \\ , g ) ( X , { f } Y ) + g ( X , ( \\pounds _ { \\xi _ i } { f } ) Y ) . \\end{align*}"}
+{"id": "970.png", "formula": "\\begin{align*} M _ q ( \\Gamma ^ { \\ , * } _ m ) \\geqslant C _ 0 \\cdot m ^ { \\frac { q - n } { n } } ( D ) \\cdot m ^ { - \\frac { q - n } { n } } ( D ) = C _ 0 . \\end{align*}"}
+{"id": "8695.png", "formula": "\\begin{align*} \\Delta _ { 0 } = 0 \\mu _ { 1 } = \\mu _ { 2 } \\textup { t r } ( \\Sigma _ 1 ) = \\textup { t r } ( \\Sigma _ 2 ) . \\end{align*}"}
+{"id": "1275.png", "formula": "\\begin{align*} \\eta ( B ) \\ge T _ 0 - 1 + \\max _ { 1 \\le i _ 1 < \\ldots < i _ { q _ 0 } \\le s - 1 } \\sum _ { j = 1 } ^ { q _ 0 } a _ { i _ j } , \\end{align*}"}
+{"id": "4435.png", "formula": "\\begin{align*} K _ { \\omega , \\mathbf { c } } ( \\Phi ) = 2 L ( \\Phi ) + 3 N ( \\Phi ) + 2 \\omega Q ( \\Phi ) + 2 \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) = 0 . \\end{align*}"}
+{"id": "4557.png", "formula": "\\begin{align*} P _ { \\ell , A } = ( 5 , 3 , 1 ) \\ell = a x + b y a b \\not = 0 , P _ { x , A } = P _ { y , A } = ( 3 , 3 , 3 ) . \\end{align*}"}
+{"id": "646.png", "formula": "\\begin{align*} \\mathfrak { d } ^ k _ { \\sigma , j } ( f ) = \\sum _ { 0 \\leq n \\leq \\frac { k - j } { m _ \\sigma } } \\frac { \\binom { k } { j + n m _ \\sigma } \\binom { \\frac { ( m _ \\sigma - 1 ) - j } { m _ \\sigma } - 1 } { n } } { \\binom { \\frac { ( k - j ) - ( m _ \\sigma - 1 ) } { m _ \\sigma } } { n } } \\frac { \\partial ^ k ( f \\cdot V ) } { \\partial z ^ { j + n m _ \\sigma } \\partial \\overline { z } ^ { k - j - n m _ \\sigma } } ( 0 , 0 ) . \\end{align*}"}
+{"id": "7669.png", "formula": "\\begin{align*} \\nabla _ g u _ \\delta & = ( \\phi ' ( \\rho ) - s \\phi ( \\rho ) ) e ^ { - s \\rho } \\nabla _ g \\rho , \\\\ \\Delta _ g u _ \\delta & = \\left [ - 2 s \\phi ' ( \\rho ) + s ^ 2 \\phi ( \\rho ) + ( \\phi ' ( \\rho ) - s \\phi ( \\rho ) ) p s \\coth ( \\kappa \\rho ) \\right ] e ^ { - s \\rho } \\\\ & = \\left [ s ( p \\coth ( \\kappa \\rho ) - 2 ) \\phi ' ( \\rho ) + s ^ 2 ( 1 - p \\coth ( \\kappa \\rho ) ) \\phi ( \\rho ) \\right ] e ^ { - s \\rho } . \\end{align*}"}
+{"id": "2648.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } A ^ m = x \\cdot y ^ T = \\frac { 1 } { n } J = \\frac { 1 } { n } P _ { \\sigma _ 1 } + \\cdots + \\frac { 1 } { n } P _ { \\sigma _ n } , \\end{align*}"}
+{"id": "117.png", "formula": "\\begin{align*} \\chi e ^ { - i t h ^ { - 1 } \\tilde { P } _ h ( 0 ) } q _ 1 e ^ { - ( t _ 0 - t ) X } \\tilde { R } _ h ( z ) \\chi = i h ^ { - 1 } \\int _ 0 ^ \\infty \\chi e ^ { - i t h ^ { - 1 } \\tilde { P } _ h ( 0 ) } q _ 1 e ^ { - ( t _ 0 - t ) X } e ^ { - i s h ^ { - 1 } \\tilde { P } _ h ( z ) } \\chi d s . \\end{align*}"}
+{"id": "3515.png", "formula": "\\begin{align*} J _ K \\begin{bmatrix} I \\\\ S \\end{bmatrix} = J _ K \\begin{bmatrix} I _ { k \\times k } & 0 \\\\ 0 & I _ { ( n - k ) \\times ( n - k ) } \\\\ S _ 1 & S _ 2 \\\\ S _ 3 & S _ 4 \\end{bmatrix} = \\begin{bmatrix} - S _ 1 & - S _ 2 \\\\ 0 & I \\\\ I & 0 \\\\ S _ 3 & S _ 4 \\\\ \\end{bmatrix} . \\end{align*}"}
+{"id": "129.png", "formula": "\\begin{align*} ( \\lambda + \\mu ) ^ { - 1 } - p _ k ( \\lambda , \\mu ) = \\frac { \\mu ^ { 2 n + 2 } } { \\lambda ^ { 2 n + 2 } ( \\lambda + \\mu ) } \\end{align*}"}
+{"id": "537.png", "formula": "\\begin{align*} \\sum _ { k \\in \\hbar \\mathbb { Z } ^ { n } } | f ( k ) | ^ { 2 } = \\left ( \\sum _ { \\xi \\in \\mathcal { I } _ { \\hbar } } \\widehat { f } ( \\xi ) u _ { \\xi } , \\sum _ { \\eta \\in \\mathcal { I } _ { \\hbar } } \\widehat { f } ( \\eta ) u _ { \\eta } \\right ) = \\sum \\limits _ { \\xi , \\eta \\in \\mathcal { I } _ { \\hbar } } \\widehat { f } ( \\xi ) \\overline { \\widehat { f } ( \\eta ) } \\left ( u _ { \\xi } , u _ { \\eta } \\right ) = \\sum _ { \\xi \\in \\mathcal { I } _ { \\hbar } } | \\widehat { f } ( \\xi ) | ^ { 2 } . \\end{align*}"}
+{"id": "3502.png", "formula": "\\begin{gather*} 0 = \\int _ a ^ t \\langle L _ { \\lambda } [ X ] , X \\rangle d s = - \\int _ a ^ t \\langle X '' , X \\rangle d s + \\int _ a ^ t \\langle ( R - \\lambda I ) X , X \\rangle d s \\\\ = \\int _ a ^ t \\| X ' \\| ^ 2 d s - \\langle X ' , X \\rangle | _ a ^ t + \\int _ a ^ t \\langle ( R - \\lambda I ) X , X \\rangle d s \\\\ = \\int _ a ^ t \\| X ' \\| ^ 2 + \\langle ( R - \\lambda I ) X , X \\rangle d s . \\end{gather*}"}
+{"id": "4307.png", "formula": "\\begin{gather*} x ^ { ( 3 ) } - x ^ { ( 2 ) } = - \\left ( x ^ { ( 1 ) } - x ^ { ( 0 ) } \\right ) , \\\\ x ^ { ( 4 ) } - x ^ { ( 3 ) } = - \\left ( x ^ { ( 2 ) } - x ^ { ( 1 ) } \\right ) , \\end{gather*}"}
+{"id": "7484.png", "formula": "\\begin{align*} \\alpha _ i ^ { \\prime } = a _ 1 \\cdots a _ p a _ { p + 1 } \\cdots a _ q x a _ { q + 1 } \\cdots a _ l \\quad \\quad \\tilde { \\alpha } _ i ^ { \\prime } = ( a _ l , \\dots , a _ { q + 1 } , x , a _ q , \\dots , a _ { p + 1 } , a _ p , \\dots , a _ 1 ) ; \\end{align*}"}
+{"id": "961.png", "formula": "\\begin{align*} M _ q ( \\Gamma ) \\leqslant \\frac { 1 } { \\varepsilon _ 0 ^ q } \\int \\limits _ { { \\mathbb R } ^ n } Q ( z ) \\ , d m ( z ) = \\frac { \\Vert Q \\Vert _ 1 } { { | f ( x ) - f ( y ) | } ^ { q } } \\ , . \\end{align*}"}
+{"id": "4959.png", "formula": "\\begin{align*} \\frac { n } { n - k } \\Big ( F ( k , t , n ) - F ( k , t + 1 , n ) \\Big ) = F ( k , t , n ) - F ( k - 1 , t , n ) , \\end{align*}"}
+{"id": "6124.png", "formula": "\\begin{align*} \\mathcal { A } = \\left \\{ \\mathcal { K } \\times I \\in \\overline { \\mathcal { A } } \\ , : \\mathcal { K } \\times I \\not \\subset \\mathcal { Q } _ { 5 \\rho _ { { i } } } \\left ( z _ { i } \\right ) i \\right \\} . \\end{align*}"}
+{"id": "7135.png", "formula": "\\begin{align*} \\int _ { B _ { \\delta } ( \\hat { x } ^ \\prime ) \\times ( 0 , \\delta ) } J _ \\varepsilon ( | A _ \\gamma ( \\hat { x } , \\hat { x } ) ( \\hat { x } - \\bar { y } ) | ) \\nabla _ { x ^ \\prime } u ( \\hat { x } ) \\cdot ( \\hat { x } ^ \\prime - \\bar { y } ^ \\prime ) \\big | \\det F _ \\gamma ( \\hat { x } ) \\big | \\ ; y = 0 . \\end{align*}"}
+{"id": "7455.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ 3 \\mathrm { v } _ i \\ , h _ i ^ 2 \\ , w _ { p \\alpha + k } ^ { ( i ) } ( 0 , t ) = \\boldsymbol { d _ { p \\alpha + k } } ( t ) , \\end{align*}"}
+{"id": "1096.png", "formula": "\\begin{align*} d ( x y ) = - d ( x ) y - d ( y ) x , \\forall x , y \\in X . \\end{align*}"}
+{"id": "7105.png", "formula": "\\begin{align*} \\lim \\limits _ { \\varepsilon \\searrow 0 } \\mathcal { E } _ \\varepsilon ( c ) = \\frac { 1 } { 2 } \\int _ { \\Omega } \\big | \\nabla c ( x ) \\big | ^ 2 \\ : x \\end{align*}"}
+{"id": "1379.png", "formula": "\\begin{align*} v _ 1 , \\ , v _ 2 = - g _ 6 v _ 1 , \\ , v _ 3 = - g _ 4 v _ 1 , \\ , v _ 4 = - g _ 2 v _ 1 , \\ , v _ 5 = - g _ 7 v _ 1 , \\ , v _ 6 = - g _ 5 v _ 1 , \\ , v _ 7 = - g _ 3 v _ 1 , \\end{align*}"}
+{"id": "7143.png", "formula": "\\begin{align*} \\overline { \\Omega } \\subseteq \\bigcup _ { j = 0 } ^ N U _ j \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Omega \\cap U _ j = \\mathbb { R } ^ n _ { \\gamma _ j } \\cap U _ j \\ ; \\ ; \\ ; j = 0 , \\ldots , N . \\end{align*}"}
+{"id": "2130.png", "formula": "\\begin{align*} \\mathbb { F } ( x ( X , t ) , t ) = \\widetilde { \\mathbb { F } } ( X , t ) . \\end{align*}"}
+{"id": "2083.png", "formula": "\\begin{align*} w ( r _ 1 ) & = \\sum _ { i = 1 } ^ k 2 ^ i x _ i ^ { \\prime } = \\sum _ { i = 1 } ^ { k - 1 } 2 ^ i x _ i ^ { \\prime } + 2 ^ k x _ k ^ { \\prime } \\leq 2 ^ k + 2 ^ k x _ k ^ { \\prime } = 2 ^ k ( 1 + x _ k ^ { \\prime } ) \\leq 2 ^ k x _ k \\leq w ( r _ 2 ) . \\end{align*}"}
+{"id": "4085.png", "formula": "\\begin{align*} t _ \\delta ( x ) = \\delta x \\end{align*}"}
+{"id": "577.png", "formula": "\\begin{align*} \\gamma ^ { a } ( x _ { 1 } , x _ { 2 } ) = \\int _ { \\R } a \\left ( \\frac { - x _ { 1 } + y _ { 1 } } { 2 \\sqrt { x _ { 2 } } } \\right ) H ( y _ { 1 } ) [ H ( y _ { 1 } ) ] ^ T d y _ { 1 } . \\\\ \\end{align*}"}
+{"id": "3314.png", "formula": "\\begin{align*} \\rho _ w ( \\xi , w ) = \\rho _ { \\overline { w } } ( \\overline { \\xi } , \\overline { w } ) = \\overline { \\rho _ w ( \\overline { \\xi } , \\overline { w } ) } { \\rm \\ f o r \\ } \\xi \\ne \\pm 1 \\end{align*}"}
+{"id": "2028.png", "formula": "\\begin{align*} \\Vert g \\Vert _ { C ^ { 1 , \\bar 1 } ( \\bar \\Omega \\times I ) } : = \\max \\left \\{ \\Vert g \\Vert , \\Vert \\nabla g \\Vert , \\Vert \\Delta g \\Vert , \\Vert g _ t \\Vert , \\Vert g _ { t t } \\Vert \\right \\} \\leq A , \\end{align*}"}
+{"id": "3334.png", "formula": "\\begin{align*} { \\rm R e } \\left ( r e ^ { i ( v + i H v ) } \\frac { \\partial \\kappa } { \\partial \\theta } \\right ) = - e ^ { - u } e ^ { - ( H v ) } \\widetilde { \\rho } _ { \\theta } ( \\theta , \\kappa ) , \\end{align*}"}
+{"id": "6325.png", "formula": "\\begin{align*} \\frac { \\partial x } { \\partial \\omega } = O ( t ^ 2 ) \\qquad \\qquad \\frac { \\partial y } { \\partial \\omega } = O ( t ^ 2 ) , \\end{align*}"}
+{"id": "7048.png", "formula": "\\begin{align*} \\nu _ i \\left ( g ^ { ( i ) } ( x ) \\right ) = 0 \\end{align*}"}
+{"id": "9123.png", "formula": "\\begin{align*} \\begin{array} { c c l } x ^ { 1 , + } & = & x ^ { 1 } + u ^ { 1 } \\\\ x ^ { 2 , + } & = & \\frac { x ^ { 3 } } { u ^ { 1 } + 1 } \\\\ x ^ { 3 , + } & = & u ^ { 2 } \\end{array} \\end{align*}"}
+{"id": "2609.png", "formula": "\\begin{align*} \\det \\ , D ^ 2 v _ 0 = c _ 0 \\chi _ { _ { \\Omega ^ * _ 0 } } \\quad \\R ^ n \\end{align*}"}
+{"id": "3849.png", "formula": "\\begin{align*} c ( s , s ' ) = | y _ 1 - y _ 1 ' | + | y _ 2 - y _ 2 ' | + \\| x ' - x \\| _ 2 , \\end{align*}"}
+{"id": "3895.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { I } ( \\lambda \\delta + ( 1 - \\lambda ) \\delta ^ \\prime ) & = \\sup _ { \\nu \\in \\Sigma \\left ( \\lambda \\delta + ( 1 - \\lambda ) \\delta ^ \\prime \\right ) } \\int _ { \\mathcal { S } } f ( s ) d \\nu ( s ) \\\\ & \\geq \\int _ { \\mathcal { S } } f d \\gamma ^ { \\prime \\prime } = \\lambda \\int _ { \\mathcal { S } } f d \\gamma + ( 1 - \\lambda ) \\int _ { \\mathcal { S } } f d \\gamma ^ { \\prime } . \\end{aligned} \\end{align*}"}
+{"id": "8189.png", "formula": "\\begin{align*} \\mathcal { R } ( I ) = \\bigcup _ { s \\in I } ( Z _ s ) . \\end{align*}"}
+{"id": "4385.png", "formula": "\\begin{align*} \\begin{aligned} U _ j & = y + \\alpha j - 1 0 \\log ( j \\wedge ( n - j ) ) , \\\\ L _ j & = - 1 0 + ( \\alpha + \\frac { y } { n _ \\mathcal L } ) j - ( j \\wedge ( n - j ) ) ^ { 3 / 4 } . \\end{aligned} \\end{align*}"}
+{"id": "2586.png", "formula": "\\begin{align*} B _ \\vee ( 1 ) = & \\ ; 1 2 + 2 1 , \\\\ B _ \\vee ( 1 2 + 2 1 ) = & \\ ; \\prescript { \\uparrow } { } { 1 } \\ , \\vert \\ , \\prescript { \\uparrow } { } { 2 } ^ { \\uparrow } + \\prescript { \\uparrow } { } { 2 } ^ { \\uparrow } \\ , \\vert \\ , 1 ^ { \\uparrow } \\\\ = & \\ ; 1 2 3 + 1 3 2 + 2 1 2 + 2 1 2 + 2 3 1 + 3 2 1 . \\end{align*}"}
+{"id": "3534.png", "formula": "\\begin{align*} ( \\wp ' ( z ) ) ^ 2 = 4 \\wp ^ 3 ( z ) - g _ 2 \\wp ( z ) - g _ 3 . \\end{align*}"}
+{"id": "390.png", "formula": "\\begin{align*} B _ i - A _ i ^ T = 0 , C + C ^ T = 0 , U _ { x _ i } ^ T D _ i \\geq 0 , i = 1 , 2 , . . , k . \\end{align*}"}
+{"id": "6848.png", "formula": "\\begin{align*} \\left [ { N + 1 \\atop j } \\right ] = q ^ j \\left [ { N \\atop j } \\right ] + \\left [ { N \\atop j - 1 } \\right ] , \\end{align*}"}
+{"id": "1297.png", "formula": "\\begin{align*} e ^ { \\rho ( u ) } = X ( T ( u ) ) . \\end{align*}"}
+{"id": "5002.png", "formula": "\\begin{align*} \\Pr ( T _ { r : k { + } 1 } = t ) = \\frac { n - k } { n } \\Pr ( X _ { t - 1 } = k \\ , \\vert \\ , X _ 0 = r ) . \\end{align*}"}
+{"id": "767.png", "formula": "\\begin{align*} A ^ j _ s [ T _ m ] ^ i _ j ( A ) = \\delta ^ i _ s \\sigma _ { m + 1 } ( A ) - [ T _ { m + 1 } ] ^ i _ s ( A ) . \\end{align*}"}
+{"id": "8591.png", "formula": "\\begin{align*} E \\big | \\bar { S } ^ k _ { j , + } ( t ) - \\hat { S } ^ k _ { j , + } ( t ) \\big | = O ( \\theta ^ k e ^ { \\lambda t / 2 } ) . \\end{align*}"}
+{"id": "1441.png", "formula": "\\begin{align*} \\bar { G } _ h ( s _ 1 , \\xi ) = & \\frac { \\alpha _ n } { 2 } a _ 2 \\partial _ { \\xi _ h } \\varphi ( \\xi ) s _ 1 ^ { n - 2 } , \\quad \\mbox { f o r } h = 1 , \\cdots , n , \\\\ \\bar { G } _ 0 ( \\bar { s } , \\bar { \\sigma } , \\xi ) = & \\alpha _ n a _ 1 s _ 1 ^ { n - 2 } \\varphi ( \\xi ) + a _ 3 \\sum \\limits _ { i = 2 } ^ { k } s _ i ^ { \\frac { n - 2 } { 2 } } g ( \\sigma _ i ) - a _ 4 \\sum \\limits _ { i = 1 } ^ k \\frac { 2 } { 2 i - 1 } | \\ln s _ i | . \\end{align*}"}
+{"id": "7148.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\partial _ t u ( - \\Delta _ N ) ^ { - 1 } u \\ : x = \\frac { } { t } \\int _ { \\Omega } \\frac { 1 } { 2 } \\big | ( - \\Delta _ N ) ^ { - 1 / 2 } u \\big | ^ 2 \\ : x = \\frac { } { t } \\frac { 1 } { 2 } \\| u \\| ^ 2 _ { H ^ { - 1 } ( \\Omega ) } . \\end{align*}"}
+{"id": "1416.png", "formula": "\\begin{align*} \\cosh ( s _ { k + 1 } ) & = \\cosh ( s _ k ) / \\cosh ( \\eta _ k \\| g _ k \\| ) = \\cosh ( s _ k ) \\sqrt { 1 - \\tanh ( \\eta _ k \\| g _ k \\| ) ^ 2 } \\\\ & = \\cosh ( s _ k ) \\sqrt { 1 - \\frac { 1 } { \\zeta _ { s _ k } ^ 2 } \\cdot \\frac { ( f ( x _ k ) - f ^ * ) ^ 2 } { \\| g _ k \\| ^ 2 } } , \\end{align*}"}
+{"id": "2776.png", "formula": "\\begin{align*} \\mathfrak { C } = \\{ \\mathfrak { c } \\in C ^ \\infty ( \\mathrm { M } , \\mathbb { R } ) ; \\ ; \\mathfrak { c } > 0 \\} . \\end{align*}"}
+{"id": "2659.png", "formula": "\\begin{align*} \\| f - s _ \\alpha \\| _ { L ^ \\infty [ a , b ] } = \\| \\cosh ( \\alpha \\cdot ) \\left ( g - t _ \\alpha \\right ) \\| _ { L ^ { \\infty } [ a , b ] } \\leq C \\overline { h } ^ 2 \\end{align*}"}
+{"id": "3051.png", "formula": "\\begin{align*} A ( \\alpha ) = \\left [ \\begin{matrix} B & 0 \\\\ 0 & C \\end{matrix} \\right ] . \\end{align*}"}
+{"id": "7482.png", "formula": "\\begin{align*} \\sigma = \\alpha _ 1 ^ { \\prime } \\alpha _ 2 ^ { \\prime } \\cdots \\alpha _ m ^ { \\prime } \\beta ^ { \\prime } \\end{align*}"}
+{"id": "2077.png", "formula": "\\begin{align*} N _ { d r } = \\min \\left \\{ O _ { B } ^ { H } ( m a + r ) \\cdot a + ( m a + r ) d \\mid m \\in \\mathbb { N } \\right \\} . \\end{align*}"}
+{"id": "5573.png", "formula": "\\begin{align*} \\alpha _ { \\rho } ^ { x } = \\left ( \\bar { \\psi } _ { H } \\right ) _ { \\ast } \\left ( r ^ { - 1 } ( \\bar { p } _ { k } ) _ { \\ast } \\left ( \\tau ( y ) ^ { - 1 } . \\nu _ { P } ^ { y } \\right ) \\right ) \\mbox { w h e r e } x = ( y , r , H ) . \\end{align*}"}
+{"id": "1914.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { + \\infty } \\| r ^ { k } \\| ^ 2 < \\infty \\end{align*}"}
+{"id": "9001.png", "formula": "\\begin{align*} C ^ \\varphi _ { i j k , k } R ^ \\varphi _ { i j } = ( C ^ \\varphi _ { i j k } R ^ \\varphi _ { i j } ) _ k - \\frac { 1 } { 2 } | C ^ \\varphi | ^ 2 \\ , . \\end{align*}"}
+{"id": "8831.png", "formula": "\\begin{align*} d u _ \\lambda + A u _ \\lambda \\ , d t = f _ \\lambda ( u _ \\lambda ) \\ , d t + \\sigma ( u _ \\lambda ) B \\ , d W , u _ \\lambda ( 0 ) = u _ 0 . \\end{align*}"}
+{"id": "8466.png", "formula": "\\begin{align*} I _ { m , p } ^ { \\star } ( s , x ) : = \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\sigma - i \\infty } ^ { \\sigma + i \\infty } \\Gamma ( z ) \\ , \\Gamma ( s - z ) \\ , ( 2 z , 2 \\pi p m ) _ { m } \\ , \\left ( \\frac { x } { m } \\right ) ^ { 2 z } \\ , d z = I _ { m , p } ^ { \\star ( 1 ) } ( s , x ) + I _ { m , p } ^ { \\star ( 2 ) } ( s , x ) , \\end{align*}"}
+{"id": "500.png", "formula": "\\begin{align*} \\ell = \\ell _ \\gamma : Z = \\N _ \\star ^ + \\ni \\phi \\mapsto \\phi ^ \\gamma \\in ( L _ { 1 / \\gamma } ( \\N ) ) ^ + \\end{align*}"}
+{"id": "7414.png", "formula": "\\begin{align*} q _ { n } ^ f ( x ) : = \\sum _ { z \\in \\Z ^ 2 } q _ { n } ( x - z ) \\ , f ( z ) \\ , , \\ f : \\Z ^ 2 \\to \\R \\ , , \\end{align*}"}
+{"id": "572.png", "formula": "\\begin{align*} \\gamma ^ { b } ( x _ { 1 } , x _ { 2 } ) = \\gamma ^ { b } ( x _ { 2 } ) = \\int _ { \\R _ { + } } b \\left ( \\frac { y _ { 2 } } { 2 x _ { 2 } } \\right ) N ( y _ 2 ) [ N ( y _ 2 ) ] ^ T d y _ { 2 } . \\end{align*}"}
+{"id": "4037.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( i ) \\lambda _ f ( j ) \\big | L ( f , n + 1 ) \\big | ^ 2 . \\end{align*}"}
+{"id": "5846.png", "formula": "\\begin{align*} \\pi _ 1 ( X ^ { y , \\Gamma } _ n ) & \\leqslant \\pi _ 1 ( Z _ { j _ 0 - 1 } ) + \\frac { \\delta H } { 5 } \\\\ & < \\pi _ 1 ( w _ { j _ 0 } ) + \\frac { 4 \\delta H } { 5 } + \\frac { \\delta H } { 5 } & \\mbox { u s i n g \\eqref { e : l o c a l i z a t i o n _ x _ 2 } } \\\\ & = \\pi _ 1 ( w _ { j _ 0 } ) + \\delta H . \\end{align*}"}
+{"id": "6324.png", "formula": "\\begin{align*} \\det \\big ( M _ 2 ( \\phi , \\omega , r ; t ) \\big ) , \\ , \\det \\big ( M _ 3 ( \\phi , \\omega , r ; t ) \\big ) = O ( t ^ 3 ) , t \\to 0 . \\end{align*}"}
+{"id": "3106.png", "formula": "\\begin{align*} S _ B ( n , k ) & = \\sum _ { j = k } ^ { n } 2 ^ { j - k } \\binom { n } { j } S ( j , k ) ; \\\\ S _ B ( n , k ) & = S _ D ( n , k ) + n \\cdot 2 ^ { n - k - 1 } S ( n - 1 , k ) . \\end{align*}"}
+{"id": "4091.png", "formula": "\\begin{align*} A ^ { ( 0 ) } \\cdots A ^ { ( i - 2 ) } A ^ { ( i - 1 ) } \\vec { v } ^ { ( i ) } = \\vec { v } ^ { ( 0 ) } \\end{align*}"}
+{"id": "1065.png", "formula": "\\begin{align*} Q ( t ) = \\Delta \\left ( { \\frac { \\left ( t _ 2 ^ 2 - 1 \\right ) \\left ( t _ 3 ^ 2 - 1 \\right ) } { 1 - t _ 1 \\left ( t _ 2 ^ 2 t _ 3 ^ 2 + t _ 2 t _ 3 ^ 2 + t _ 2 ^ 2 + t _ 3 ^ 2 + t _ 2 + 1 \\right ) } } \\right ) \\ , , \\end{align*}"}
+{"id": "1708.png", "formula": "\\begin{align*} ( 1 - z ^ 2 ) \\frac { \\dd ^ 2 w } { \\dd z ^ 2 } + \\left ( \\beta - \\alpha - z ( \\alpha + \\beta + 2 ) \\right ) \\frac { \\dd w } { \\dd z } + \\gamma ( \\alpha + \\beta + \\gamma + 1 ) w = 0 , \\end{align*}"}
+{"id": "1569.png", "formula": "\\begin{align*} p ( x , y ) = \\frac { D ( x , y ) } { \\sum _ { z \\neq x } D ( x , z ) } . \\end{align*}"}
+{"id": "3239.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) = \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\omega ( Q ) \\psi _ { j } ( x , x _ { Q } ) q _ { j } h ( x _ { Q } ) , \\end{aligned} \\end{align*}"}
+{"id": "993.png", "formula": "\\begin{align*} n _ { v _ 1 } + \\ell _ { v _ 1 , E } \\mathbf { u } _ { v _ 1 } ( E ) = n _ { v _ 2 } + \\ell _ { v _ 2 , E } \\mathbf { u } _ { v _ 2 } ( E ) . \\end{align*}"}
+{"id": "4759.png", "formula": "\\begin{align*} w _ 1 = z _ 1 + \\mathcal { O } ( \\| H \\| ) = x _ 1 + \\iota y _ 1 + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "2530.png", "formula": "\\begin{align*} \\mu _ T ( \\overline { T _ { } x } ) = \\{ \\phi _ { i , j } = \\varepsilon _ i + \\varepsilon _ j \\ : : \\ : \\{ i , j \\} \\in { \\mathcal B } ( x ) \\} . \\end{align*}"}
+{"id": "315.png", "formula": "\\begin{align*} \\zeta ( s , t ) + \\zeta ( t , s ) = \\zeta ( s ) \\zeta ( t ) - \\zeta ( s + t ) \\Re ( s ) > 1 , \\ ; \\Re ( t ) > 1 , \\end{align*}"}
+{"id": "8726.png", "formula": "\\begin{align*} ( \\widetilde { \\Delta } _ { 2 } ) & = O ( N ^ { - 1 } p ^ { - 2 } ) = o ( ( \\Delta _ { 1 } ) ) , \\\\ E ( \\widetilde { \\Delta } _ { 2 } ) - T _ { 1 } & = O ( p ^ { - \\frac { 3 } { 2 } } ) = o ( \\surd { ( \\Delta _ { 1 } ) } ) , \\end{align*}"}
+{"id": "4615.png", "formula": "\\begin{align*} \\mathrm { d i m } _ M ( \\partial X ) = 4 - \\liminf _ { d \\to 0 } \\frac { \\ln V _ d ( U ) } { \\ln ( d ) } . \\end{align*}"}
+{"id": "1833.png", "formula": "\\begin{align*} M ^ \\delta = M ^ \\delta _ 0 + M ^ \\delta _ 1 \\end{align*}"}
+{"id": "7092.png", "formula": "\\begin{align*} \\partial _ t c & = m \\Delta \\mu \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Omega _ T , \\\\ \\mu & = - \\Delta c + f ^ \\prime ( c ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Omega _ T , \\end{align*}"}
+{"id": "7043.png", "formula": "\\begin{align*} n _ + = \\min \\left \\{ n ' \\in \\N \\ \\left | \\ n ' > n , \\textbf { Q } _ { n ' } \\ne \\emptyset \\right . \\right \\} . \\end{align*}"}
+{"id": "472.png", "formula": "\\begin{align*} \\langle L \\rangle = \\sum _ { C \\subset \\# L } \\langle L , C \\rangle \\in \\Bbb Z [ A , A ^ { - 1 } ] \\end{align*}"}
+{"id": "5498.png", "formula": "\\begin{align*} \\varphi \\circ \\xi \\left ( g , x _ { 0 } \\right ) = g Q \\mbox { f o r a l l } g \\in G . \\end{align*}"}
+{"id": "2900.png", "formula": "\\begin{align*} f ^ \\star \\sharp f ( \\xi ) + h ^ \\star \\sharp h ( \\xi ) & = f ( \\xi ) ^ * f ( \\xi ) + h ( \\xi ) ^ * h ( \\xi ) + b ( \\xi ) \\\\ & = f ( \\xi ) ^ * f ( \\xi ) + 1 + g ( \\xi ) + b ( \\xi ) \\\\ & = f ( \\xi ) ^ * f ( \\xi ) + 1 + C - f ( \\xi ) ^ * f ( \\xi ) + b ( \\xi ) \\\\ & = 1 + C + b ( \\xi ) . \\end{align*}"}
+{"id": "5054.png", "formula": "\\begin{align*} f _ \\Theta ( a ) \\coloneqq \\sum _ { i = 1 } ^ n y _ i [ \\Theta ^ { - 1 } U ( a ) ] _ i = \\sum _ { i = 1 } ^ n [ \\Theta ^ { - 1 } y ] _ i U ( a ) _ i , \\end{align*}"}
+{"id": "4.png", "formula": "\\begin{align*} \\| \\Tilde { \\theta } \\| _ { \\C ^ k } \\ll \\rho ^ { - ( 2 k + 2 N ) } , N = n ^ 2 + m ^ 2 + n m - 1 \\end{align*}"}
+{"id": "514.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\hbar } u ( k ) : = \\sum \\limits _ { j = 1 } ^ { n } \\left ( u \\left ( k + \\hbar v _ { j } \\right ) + u \\left ( k - \\hbar v _ { j } \\right ) \\right ) - 2 n u ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\end{align*}"}
+{"id": "8231.png", "formula": "\\begin{align*} \\frac { \\dd ^ 2 Z _ { t } ^ { ( \\varepsilon ) } } { \\dd t ^ 2 } = \\lambda ^ 2 \\partial _ { x x } Z _ { t } ^ { ( \\varepsilon ) } + 2 \\gamma \\frac { \\dd Z _ { t } ^ { ( \\varepsilon ) } } { \\dd t } + \\varepsilon 2 \\lambda \\sqrt { \\gamma \\rho } \\partial _ x \\frac { \\dd \\tilde { W } _ { t } } { \\dd t } . \\end{align*}"}
+{"id": "4543.png", "formula": "\\begin{align*} \\| \\partial _ k \\varphi _ 3 \\| _ { H ^ m } & = \\| \\partial _ k ( - \\gamma \\Delta + \\omega + i \\mathbf { c } \\cdot \\nabla ) ^ { - 1 } \\nabla ( \\varphi _ 1 \\cdot \\overline { \\varphi _ 2 } ) \\| _ { H ^ m } \\lesssim \\| \\varphi _ 1 \\cdot \\varphi _ 2 \\| _ { H ^ m } \\lesssim \\| \\varphi _ 1 \\| _ { H ^ m } \\| \\varphi _ 2 \\| _ { H ^ m } < \\infty \\end{align*}"}
+{"id": "3577.png", "formula": "\\begin{align*} F ( n ) & \\ = \\ \\sum _ { i = 1 } ^ { k } \\left ( p _ i + \\alpha _ i \\right ) , \\\\ G ( n ) & \\ = \\ \\prod _ { i = 1 } ^ k p _ i \\cdot \\alpha _ i , \\end{align*}"}
+{"id": "6531.png", "formula": "\\begin{align*} t _ { \\mu , \\nu } ( x ) & = \\frac { x ^ { \\mu + 1 } } { ( \\mu - \\nu + 1 ) ( \\mu + \\nu + 1 ) } { } _ 1 F _ 2 \\bigg ( 1 ; \\frac { \\mu - \\nu + 3 } { 2 } , \\frac { \\mu + \\nu + 3 } { 2 } ; \\frac { x ^ 2 } { 4 } \\bigg ) \\\\ & = 2 ^ { \\mu - 1 } \\Gamma \\bigg ( \\frac { \\mu - \\nu + 1 } { 2 } \\bigg ) \\Gamma \\bigg ( \\frac { \\mu + \\nu + 1 } { 2 } \\bigg ) \\sum _ { k = 0 } ^ \\infty \\frac { ( \\frac { 1 } { 2 } x ) ^ { \\mu + 2 k + 1 } } { \\Gamma \\big ( k + \\frac { \\mu - \\nu + 3 } { 2 } \\big ) \\Gamma \\big ( k + \\frac { \\mu + \\nu + 3 } { 2 } \\big ) } . \\end{align*}"}
+{"id": "8248.png", "formula": "\\begin{align*} \\widetilde { M } _ k ^ { ( 1 ) } = \\Big ( ( \\alpha + j + h - 1 + t _ { k - j } ) a _ { \\alpha + i + j + h - 1 + t _ { k - j } } \\Big ) _ { i , j = 0 , \\ldots , k - 1 } , \\end{align*}"}
+{"id": "2777.png", "formula": "\\begin{align*} \\Delta _ { \\mathfrak { c } g } u = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } , u _ { | \\partial \\mathrm { M } } = \\varphi . \\end{align*}"}
+{"id": "107.png", "formula": "\\begin{align*} \\int \\chi ( x ) e ^ { - i x \\cdot \\xi _ 0 / h \\Tilde { h } } ( - i h X - z - i ( \\tilde { h } ^ { - 1 } Q _ { \\infty , \\tilde { h } } + \\tilde { h } ^ { - 1 } q _ { 1 } + W ) ) ^ { - 1 } ( \\psi ( x ) e ^ { i x \\cdot r \\eta _ 0 / h \\Tilde { h } } ) d x = \\mathcal { O } ( h ^ \\infty \\Tilde { h } ^ \\infty ) . \\end{align*}"}
+{"id": "5049.png", "formula": "\\begin{align*} f ( k , s , t ) = \\binom { k } { 2 } + k ( t - k ) + \\left \\lfloor \\frac { t - k } { s } \\right \\rfloor \\binom { s } { 2 } + \\binom { t - k - \\left \\lfloor \\frac { t - k } { s } \\right \\rfloor s } { 2 } \\end{align*}"}
+{"id": "4859.png", "formula": "\\begin{align*} \\| m ( x , y ) - m ( y , z ) \\| & = \\| \\frac { 1 } { 2 } ( x + z ) - \\frac { 1 } { 2 } ( y + z ) \\| \\\\ & = \\frac { 1 } { 2 } \\| x - y \\| . \\end{align*}"}
+{"id": "8526.png", "formula": "\\begin{align*} \\zeta _ { p , \\infty } \\left ( \\frac { 1 } { 2 } , c \\right ) = 2 C _ { p } ^ { ( 1 ) } + \\log \\left ( \\frac { c } { 4 } \\right ) - 2 \\log ( 2 \\pi ) + \\theta \\ , \\frac { 2 ^ { 5 / 2 } e ^ { - \\pi \\sqrt { c } } } { c ^ { 1 / 4 } } > 0 \\end{align*}"}
+{"id": "5364.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ q \\le \\sqrt { \\gamma } ( 1 2 r q ) ^ d \\le \\gamma ( 1 2 e ^ 2 r q ) ^ { d } \\end{align*}"}
+{"id": "5072.png", "formula": "\\begin{align*} \\breve { v } ( x , t ) : = \\psi ( x , t ) - \\phi \\left ( x + \\sigma ( t ) \\right ) , \\end{align*}"}
+{"id": "103.png", "formula": "\\begin{align*} & \\ ; e ^ { - i t h ^ { - 1 } ( - i h X - i ( q _ 1 + W ) ) } - e ^ { - i t h ^ { - 1 } ( - i h X - i W ) } \\\\ = & \\ ; - h ^ { - 1 } \\int _ 0 ^ t e ^ { - i ( t - s ) h ^ { - 1 } ( - i h X - i W ) } q _ 1 e ^ { - i s h ^ { - 1 } ( - i h X - i ( q _ 1 + W ) ) } d s . \\end{align*}"}
+{"id": "1346.png", "formula": "\\begin{align*} \\partial _ { t } F _ { N } + \\partial _ { x } ( A _ { N } ( t , F _ { N } ) ) = \\mathbf { S } [ F _ { N } ] ( t , x ) . \\end{align*}"}
+{"id": "6443.png", "formula": "\\begin{align*} i \\partial _ t u = - \\Delta u + | u | ^ 2 u \\end{align*}"}
+{"id": "7548.png", "formula": "\\begin{align*} \\left \\lvert \\sum _ { l \\equiv v ( \\bmod a ) } \\binom n l \\alpha ^ l ( 1 - \\alpha ) ^ { n - l } - \\frac 1 a \\right \\rvert = O \\left ( n ^ { - \\frac 1 2 } \\right ) . \\end{align*}"}
+{"id": "4259.png", "formula": "\\begin{align*} V \\left ( t , \\xi \\right ) = \\inf _ { u \\in \\Lambda _ { t , \\xi } } J \\left ( t , \\xi ; u \\right ) , \\end{align*}"}
+{"id": "3129.png", "formula": "\\begin{align*} n _ \\alpha : = \\frac { n } { \\gcd ( n , Q ( \\alpha ^ \\vee ) ) } . \\end{align*}"}
+{"id": "7828.png", "formula": "\\begin{align*} \\max _ { i , j } \\vert a _ { i j } \\vert = ( \\frac { M n } { \\log ( 2 N ) } ) ^ { \\frac { 1 } { 2 + \\varepsilon / 4 } } . \\end{align*}"}
+{"id": "7241.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } ( n \\log n ) ^ { - 1 / p } \\big | S _ n - \\sum _ { k = 1 } ^ n g _ k \\big | > 0 \\mbox { a . s . } \\end{align*}"}
+{"id": "9239.png", "formula": "\\begin{align*} H _ m : = \\sum _ { k = m } ^ { J - 1 } 2 ^ k | c _ k | . \\end{align*}"}
+{"id": "6595.png", "formula": "\\begin{align*} \\alpha _ { l , { \\hat { n } _ t } } e ^ { - j \\omega } = \\alpha _ { l , { \\hat { n } _ t } } \\cos \\omega - j \\alpha _ { l , { \\hat { n } _ t } } \\sin \\omega . \\end{align*}"}
+{"id": "4049.png", "formula": "\\begin{align*} \\sum _ { d _ 1 , d _ 2 < D } \\frac { 1 } { \\sqrt { d _ 1 d _ 2 } } \\sum _ { I d _ 1 , J d _ 2 < D } \\alpha _ { I d _ 1 } \\bar { \\alpha } _ { J d _ 2 } \\sum _ { L , M \\geq 1 } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( I L ) \\lambda _ f ( J M ) = 0 . \\end{align*}"}
+{"id": "85.png", "formula": "\\begin{align*} \\Omega = \\{ \\langle \\xi \\rangle ^ { - 1 } p = 0 \\} \\setminus { \\rm C o n } _ p ( { \\rm e l l } _ h ( B ) ; { \\rm e l l } _ h ( B _ 1 ) ) \\end{align*}"}
+{"id": "2117.png", "formula": "\\begin{align*} X ( K ) \\cap \\Gamma = \\pi _ 0 ^ { - 1 } \\left ( X _ 0 ( K ) \\cap \\Gamma _ 0 \\right ) \\cap \\Gamma = \\left ( \\pi _ 0 | _ { \\Gamma } \\right ) ^ { - 1 } \\left ( X _ 0 ( K ) \\cap \\Gamma _ 0 \\right ) . \\end{align*}"}
+{"id": "3883.png", "formula": "\\begin{align*} f _ { \\lambda , [ L ] } ( s ) \\ge \\sum _ { \\ell = 1 } ^ { L } a _ { \\lambda , \\ell } ( s _ \\ell ) , \\forall s = ( s _ 1 , \\ldots , s _ L ) \\in \\prod _ { \\ell \\in [ L ] } \\mathcal { S } _ \\ell . \\end{align*}"}
+{"id": "4671.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta u = \\lambda u + u \\log u ^ 2 , \\hbox { i n } \\ , \\ , \\mathbb { R } ^ N , \\\\ & \\int _ { \\mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4836.png", "formula": "\\begin{align*} \\widehat { f _ j } = ( f _ j ( 1 ) , f _ j ( 2 ) , \\cdots , f _ j ( r ) , s ^ 0 _ j , s ^ 1 _ j , \\cdots ) \\mbox { i n } S , \\end{align*}"}
+{"id": "1567.png", "formula": "\\begin{align*} M = G _ m ^ \\Delta q ^ { n - m \\Delta } > \\left ( q ^ m ( 1 - 2 ( 0 . 9 9 ) ^ m ) \\right ) ^ \\Delta q ^ { n - m \\Delta } = q ^ { n } \\left ( 1 - 2 ( 0 . 9 9 ) ^ m \\right ) ^ \\Delta . \\end{align*}"}
+{"id": "4097.png", "formula": "\\begin{align*} M _ { i , \\ell } = ( ( \\lambda v _ i ) _ { \\vec { v } } ) _ \\ell \\end{align*}"}
+{"id": "8781.png", "formula": "\\begin{align*} & ( 4 u ^ 2 + 2 t u + 2 t + 8 u + 5 ) b _ 3 \\\\ & = ( 4 t u + 4 t + 4 u + 3 ) c _ 3 + 2 ( u + 1 ) b _ 2 + ( 1 + u ) ( b _ 2 + c _ 2 ) - u ( b _ 3 + c _ 3 ) \\end{align*}"}
+{"id": "3364.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ d T _ j ^ t \\otimes T _ j ^ * \\right ) \\mathrm { v e c } ( X ) = \\mathrm { v e c } \\left ( \\sum _ { j = 1 } ^ d T _ j ^ * X T _ j \\right ) . \\end{align*}"}
+{"id": "4667.png", "formula": "\\begin{align*} I _ \\mu ( u ) = & \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ N } \\left ( | \\nabla u | ^ 2 + ( \\mu + 1 ) u ^ 2 \\right ) d x + \\int _ { \\mathbb { R } ^ N } F _ 1 ( u ) d x - \\int _ { \\mathbb { R } ^ N } F _ 2 ( u ) d x \\\\ \\geq & \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ 2 d x + \\int _ { \\mathbb { R } ^ N } F _ 1 ( u ) d x - C C _ 1 a ^ { \\left ( 1 - \\beta _ p \\right ) p } \\left ( \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ 2 d x \\right ) ^ { \\frac { p \\beta _ p } { 2 } } . \\end{align*}"}
+{"id": "3528.png", "formula": "\\begin{align*} E _ 1 = \\begin{bmatrix} I & 0 \\\\ 0 & 0 \\end{bmatrix} \\end{align*}"}
+{"id": "5947.png", "formula": "\\begin{align*} \\overline { C } _ { X ^ { \\ast } } ( p , h ) & = \\nu _ 2 ( \\det h , g _ p ) \\overline { c } _ { X ^ { \\ast } } ( ( g _ p ) ^ { \\det h } , g _ h ) \\\\ & = ( \\det h , a _ 1 ) _ { \\R } ( x ( g _ p ^ { \\det h } ) , x ( g _ h ) ) _ \\R ( - x ( g _ p ^ { \\det h } ) x ( g _ h ) , x ( ( g _ p ) ^ { \\det h } g _ h ) ) _ \\R \\\\ & = 1 ; \\end{align*}"}
+{"id": "6051.png", "formula": "\\begin{align*} \\widetilde { g } _ { u , v } ( z ) : = u | A | z ^ { n ' + m ' } + v | B | \\overline { z } ^ { m ' } + | C | , z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "482.png", "formula": "\\begin{align*} t ^ { k _ 1 + k _ n - 2 } t _ 1 t _ 2 t _ { 2 n + 1 } t _ { 2 n + 2 } t _ { 2 n + 3 } t _ { 2 n + 4 } = p q \\end{align*}"}
+{"id": "2846.png", "formula": "\\begin{align*} W _ { 1 } = \\left \\langle \\left [ \\theta _ { 1 } \\right ] , \\left [ \\theta _ { 2 } \\right ] , \\dots , \\left [ \\theta _ { s } \\right ] \\right \\rangle , W _ { 2 } = \\left \\langle \\left [ \\vartheta _ { 1 } \\right ] , \\left [ \\vartheta _ { 2 } \\right ] , \\dots , \\left [ \\vartheta _ { s } \\right ] \\right \\rangle \\in G _ { s } \\left ( { \\rm H ^ { 2 } } \\left ( { \\bf A } , \\mathbb C \\right ) \\right ) , \\end{align*}"}
+{"id": "319.png", "formula": "\\begin{align*} \\zeta ( 4 , 1 ) = 2 \\zeta ( 5 ) - \\frac { 1 } { 2 } \\zeta ( 2 ) ^ 2 , \\end{align*}"}
+{"id": "2654.png", "formula": "\\begin{align*} s _ j ( x ) = \\dfrac { \\sinh ( \\alpha ( x _ { j } - x ) ) } { \\sinh ( \\alpha h _ j ) } y _ { j - 1 } + \\dfrac { \\sinh ( \\alpha ( x - x _ { j - 1 } ) ) } { \\sinh ( \\alpha h _ j ) } y _ j . \\end{align*}"}
+{"id": "4081.png", "formula": "\\begin{align*} B _ j \\rho ^ j & = { \\rm m a x } _ { j ' \\leq j } \\{ b _ { j ' } \\rho ^ j \\} \\\\ & = { \\rm m a x } \\{ { \\rm m a x } _ { j ' < j _ { \\varepsilon , 0 } } \\{ b _ { j ' } \\rho ^ { j ' } \\rho ^ { j - j ' } \\} , { \\rm m a x } _ { j _ { \\varepsilon , 0 } \\leq j ' \\leq j } \\{ b _ { j ' } \\rho ^ { j ' } \\rho ^ { j - j ' } \\} \\} < \\varepsilon \\end{align*}"}
+{"id": "5032.png", "formula": "\\begin{align*} ( 0 , \\dots , 0 , s _ p , s _ { p + 1 } , \\dots , s _ { N } ) , \\{ 1 , \\dots , p - 1 \\} = I _ g , \\end{align*}"}
+{"id": "309.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } ( ( x - 1 ) x ( y - 1 ) ^ 2 y \\log ( 1 - x y z ) ) \\right \\} \\end{align*}"}
+{"id": "794.png", "formula": "\\begin{align*} \\left ( { D } ^ { \\alpha } _ { N } U ^ { n - \\sigma } , v _ h \\right ) \\ , + \\ , a \\big ( l ( U ^ { n , \\sigma } ) \\big ) \\ , ( \\nabla U ^ { n , \\sigma } , \\nabla v _ h ) = & \\ , ( f ^ { n - \\sigma } , v _ h ) , \\\\ U ^ 0 = & \\ , u ^ 0 _ h , \\end{align*}"}
+{"id": "4172.png", "formula": "\\begin{align*} b ^ * b + ( c + \\alpha 1 _ { \\tilde { B _ e } } ) ^ * ( c + \\alpha 1 _ { \\tilde { B _ e } } ) = \\| b ^ * b \\| 1 _ { \\tilde { B _ e } } = \\| b \\| ^ 2 1 _ { \\tilde { B _ e } } . \\end{align*}"}
+{"id": "1224.png", "formula": "\\begin{align*} L = \\{ z \\in K \\ , ; \\ , \\lim _ { m \\to \\infty } d _ { K , q , m } ^ { S } ( f ) e ^ { m q ( z ) } = 0 \\} \\neq \\varnothing \\end{align*}"}
+{"id": "1393.png", "formula": "\\begin{align*} s _ { i j } + t _ { i j } + \\sum _ { i < k < j } s _ { i k } t _ { k j } = 0 \\end{align*}"}
+{"id": "5876.png", "formula": "\\begin{align*} L _ u S f _ { n - 1 } = & \\ , S L _ u f _ { n - 1 } + S f _ { n - 1 } \\\\ = & \\ , ( \\nu _ { n - 1 } + 1 ) \\ , S f _ { n - 1 } \\\\ = & \\ , \\nu _ { n } \\ , S f _ { n - 1 } \\ , , \\end{align*}"}
+{"id": "221.png", "formula": "\\begin{align*} L i _ n ( z ) = \\int _ { 0 } ^ { z } \\frac { L i _ { n - 1 } ( t ) } { t } d t . \\end{align*}"}
+{"id": "8603.png", "formula": "\\begin{align*} \\widehat { p } = \\widehat { p } ( x ) : = \\varphi _ 1 ^ { - 1 } ( x ) . \\end{align*}"}
+{"id": "2032.png", "formula": "\\begin{align*} L ( H ) = u ^ { j \\bar k } H _ { j \\bar k } \\leq - \\alpha ( 1 - \\alpha \\ ; \\textnormal { { d i a m } } ^ 2 ( \\Omega ) ) \\sum _ { j = 1 } ^ { n } u ^ { j \\bar j } H . \\end{align*}"}
+{"id": "2931.png", "formula": "\\begin{align*} H _ \\delta = \\left ( \\bigcap _ { g \\in G ; n _ g \\neq 0 } g H g ^ { - 1 } \\right ) \\cap H . \\end{align*}"}
+{"id": "6767.png", "formula": "\\begin{align*} [ 1 ] ^ { s + 1 } ( n + 1 ) = [ 1 ] ^ s ( n ) + [ 1 ] ^ { s + 1 } ( n ) = \\binom { n } { s } + \\binom { n } { s + 1 } = \\binom { n + 1 } { s + 1 } . \\end{align*}"}
+{"id": "996.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p a _ i f _ i + a _ { p + 1 } s + g \\end{align*}"}
+{"id": "805.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } p ^ { ( k ) } _ { k - j } \\ , j ^ { r ( l _ N - \\alpha ) } \\ , \\le & \\ , \\frac { 1 1 \\ , \\Gamma ( 1 + l _ n - \\alpha ) } { 4 \\ , \\Gamma { ( 1 + l _ N ) } } \\ ; T ^ { \\alpha } \\ , N ^ { r ( l _ N - \\alpha ) } . \\end{align*}"}
+{"id": "5464.png", "formula": "\\begin{align*} & = \\frac { | h _ 1 | ^ 2 } { \\sum _ { x _ i \\in \\Phi \\backslash \\{ x _ 1 \\} \\cap \\mathcal { A } } | h _ i | ^ 2 + 1 / \\rho } \\overset { \\Delta } { = } \\frac { | h _ 1 | ^ 2 } { I + 1 / \\rho } , \\end{align*}"}
+{"id": "4634.png", "formula": "\\begin{align*} \\gamma > \\max \\left \\{ 2 - \\frac { 2 } { n } , 1 \\right \\} \\ , \\mbox { f o r } \\ , n = 1 , 2 , 3 . \\end{align*}"}
+{"id": "9301.png", "formula": "\\begin{align*} x _ j + \\varphi \\left ( \\sum _ { k = 1 } ^ n a _ { j k } x _ k \\right ) , j = 1 , \\ldots , n \\end{align*}"}
+{"id": "9179.png", "formula": "\\begin{align*} y = ( q ^ { 2 } , q ^ { 1 } ) \\ , . \\end{align*}"}
+{"id": "8519.png", "formula": "\\begin{align*} C _ { p } ^ { ( 2 ) } : = \\lim _ { n \\rightarrow \\infty } \\left \\{ \\sum _ { k = 1 } ^ { n - 1 } \\frac { ( 1 , 2 \\pi p k ) _ { k } } { k } - \\frac { \\log ( n ) } { 1 + \\frac { 1 } { \\pi p } } \\right \\} \\end{align*}"}
+{"id": "8400.png", "formula": "\\begin{align*} F _ 1 ( W ) = M , F _ { 2 i + 1 } ( W ) = M \\oplus K _ X ^ { n - 2 } \\oplus \\cdots \\oplus K _ X ^ { n - 2 j } \\ , \\ , \\ , ( i = 1 , \\ldots , n - 1 ) , F _ { 2 n + 1 } ( W ) = W , \\end{align*}"}
+{"id": "5048.png", "formula": "\\begin{align*} \\sum _ { v \\in X } \\deg _ G ( v ) & = \\sum _ { v \\in X } | N ( v ) \\cap X | + \\sum _ { v \\in X } | N ( v ) \\setminus X | \\\\ & = \\left ( \\frac { 1 } { 2 } \\sum _ { v \\in X } | N ( v ) \\cap X | + \\sum _ { v \\in X } | N ( v ) \\setminus X | \\right ) + \\frac { 1 } { 2 } \\sum _ { v \\in X } | N ( v ) \\cap X | \\\\ & = M + \\frac { 1 } { 2 } \\sum _ { v \\in X } | N ( v ) \\cap X | \\\\ & > \\frac { t | X | } { 2 } + \\frac { | X | \\delta ( G [ X ] ) } { 2 } \\\\ & \\geq \\frac { 3 t | X | } { 4 } . \\end{align*}"}
+{"id": "4088.png", "formula": "\\begin{align*} u = \\zeta u _ 1 ^ { m _ 1 } \\dots u _ r ^ { m _ r } , \\end{align*}"}
+{"id": "6839.png", "formula": "\\begin{align*} \\sum _ { s _ 1 \\geq \\dots \\geq s _ { r - 1 } \\geq 0 } \\frac { q ^ { s _ 1 ^ 2 + \\dots + s _ { r - 1 } ^ 2 - s _ 1 - \\dots - s _ i } } { ( q ) _ { s _ 1 - s _ 2 } \\dots ( q ) _ { s _ { r - 2 } - s _ { r - 1 } } ( q ^ 2 ; q ^ 2 ) _ { s _ { r - 1 } } } = \\sum _ { k = 0 } ^ { i } \\frac { ( q ^ { 2 r } , q ^ { r - i + 2 k } , q ^ { r + i - 2 k } ; q ^ { 2 r } ) _ \\infty } { ( q ) _ \\infty } , \\end{align*}"}
+{"id": "2656.png", "formula": "\\begin{align*} s _ \\alpha \\big | _ { [ x _ { j - 1 } , x _ j ] } ( x ) & = y _ { j - 1 } + \\dfrac { y _ { j } - y _ { j - 1 } } { h _ j } ( x - x _ { j - 1 } ) + O ( \\alpha ^ 2 ) \\\\ t _ \\alpha \\big | _ { [ x _ { j - 1 } , x _ j ] } ( x ) & = y _ { j - 1 } + \\dfrac { y _ { j } - y _ { j - 1 } } { h _ j } ( x - x _ { j - 1 } ) + O ( \\alpha ^ 2 ) . \\end{align*}"}
+{"id": "8299.png", "formula": "\\begin{align*} \\displaystyle \\lambda _ n ( x _ i - x _ j ) \\leq & \\ ; \\ ; ( d _ i + d _ j ) ( x _ i - x _ j ) - C _ S ( i , j ) ( x _ i - x _ j ) - \\frac { 1 } { 2 } C _ { \\frac { 1 } { 2 } } ( i , j ) ( x _ i - x _ j ) \\\\ & - \\frac { 1 } { 4 } C _ { \\frac { 1 } { 4 } } ( i , j ) ( x _ i - x _ j ) - \\left ( \\sum _ { l = 2 } ^ { r - 1 } l C _ { l } ( i , j ) \\right ) ( x _ i - x _ j ) . \\end{align*}"}
+{"id": "6227.png", "formula": "\\begin{align*} \\partial ^ 2 _ v \\psi _ { \\mu _ N } = \\frac 1 { N + 1 } \\sum _ { i = 2 } ^ { N + 1 } \\partial ^ 2 _ v \\psi _ { \\delta _ { x _ i } } ( x _ 1 ) \\geq \\frac { ( \\alpha - \\beta ) K _ N } { N + 1 } \\geq \\frac { \\alpha - \\beta } { 2 ( N + 1 ) } \\ , , \\end{align*}"}
+{"id": "7785.png", "formula": "\\begin{align*} A \\circ B = V ^ * ( A \\otimes B ) V . \\end{align*}"}
+{"id": "7811.png", "formula": "\\begin{align*} \\overline { a } _ { i j } : = a _ { i j } \\mathbb { I } _ { ( \\vert a _ { i j } \\vert \\le a ) } , \\tilde { a } _ { i j } : = a _ { i j } \\mathbb { I } _ { ( \\vert a _ { i j } \\vert > a ) } . \\end{align*}"}
+{"id": "2681.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\Big ( q _ i ( p - x _ { i } ) \\prod _ { \\substack { 1 \\le j \\le n \\\\ j \\neq i } } ( \\sum _ { 1 \\le k \\le d } ( p _ k - x _ { j k } ) ^ 2 ) ^ { \\frac { m + 2 } { 2 } } \\Big ) = 0 . \\end{align*}"}
+{"id": "1050.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\sum _ { { \\bf j } \\in \\mathcal S _ i } c _ { i , { \\bf j } , \\ell } a _ { i , { \\bf k } - { \\bf j } } = 0 \\end{align*}"}
+{"id": "6237.png", "formula": "\\begin{align*} e _ r \\ , : = \\ , \\sum _ { \\lambda \\in \\Lambda \\smallsetminus \\{ 0 \\} } \\frac { 1 } { \\lambda ^ r } \\ , \\in \\mathbb C \\ , , \\end{align*}"}
+{"id": "3387.png", "formula": "\\begin{align*} y _ 0 = \\exp \\biggl ( \\frac { c e ^ l } { 6 l } \\biggr ) , \\ ; \\ ; y _ j = y _ { j - 1 } ^ { e ^ { \\alpha } } , , \\end{align*}"}
+{"id": "5508.png", "formula": "\\begin{align*} \\int _ { \\tilde { X } } f d \\tilde { \\nu } & = \\int _ { G / \\Gamma } \\int _ { X } f _ { z } d \\phi _ { \\tau ( z ) } d m _ { G / \\Gamma } ( z ) \\\\ & = \\int _ { G / \\Gamma } \\int _ { G / P } \\int _ { X } f _ { z } d \\boldsymbol { \\beta } _ { \\nu } ( w ) d \\nu _ { P } ( \\tau ( z ) . w ) d m _ { G / \\Gamma } ( z ) . \\end{align*}"}
+{"id": "5125.png", "formula": "\\begin{align*} \\Theta = \\sum _ { i = 1 } ^ n \\arctan \\lambda _ i \\end{align*}"}
+{"id": "731.png", "formula": "\\begin{align*} \\cal A _ { D , N } : = A ^ * _ { S , N } - C _ { D , N } A ^ { * - 1 } _ { B , N } D _ { D , N } \\ , , \\end{align*}"}
+{"id": "2432.png", "formula": "\\begin{align*} y ( q x ) = ( 1 - x ^ 3 ) y ( x ) , \\end{align*}"}
+{"id": "1575.png", "formula": "\\begin{align*} \\sum _ { \\abs { u } \\leq \\frac { \\abs { x } } { 3 } < \\abs { v } } = \\sum _ { \\abs { u } \\leq \\frac { \\abs { x } } { 3 } , \\ : \\ : \\frac { \\abs { x } } { 2 } < \\abs { v } } + \\sum _ { \\abs { u } \\leq \\frac { \\abs { x } } { 3 } < \\abs { v } \\leq \\frac { \\abs { x } } { 2 } } . \\end{align*}"}
+{"id": "7597.png", "formula": "\\begin{align*} | B | - 2 ( 1 - | C | ) & = \\frac { \\tau _ { 1 } } { 4 } - 2 \\left ( 1 - \\frac { ( 3 + \\tau ^ 2 _ { 1 } ) } { 4 \\tau _ { 1 } } \\right ) \\\\ & = \\frac { 3 \\tau ^ 2 _ { 1 } - 8 \\tau _ { 1 } + 6 } { 4 \\tau _ { 1 } } < 0 \\end{align*}"}
+{"id": "7748.png", "formula": "\\begin{align*} b _ 1 b _ 2 = e _ 1 \\triangleright e _ 2 \\triangleright \\cdots \\triangleright e _ { \\ell - 2 } \\ , . \\end{align*}"}
+{"id": "1057.png", "formula": "\\begin{align*} \\mathcal E _ 1 \\coloneqq \\left \\{ ( a _ 1 , \\ldots , a _ n , b ) \\in \\mathbb N ^ { n + 1 } : b \\leq d , \\forall i , a _ i \\leq h _ i , \\exists i , a _ i = 0 \\right \\} \\end{align*}"}
+{"id": "6918.png", "formula": "\\begin{align*} K ( u ) & = \\left ( \\frac { \\sin { ( u / 2 ) } } { u } \\right ) ^ 2 ( 1 + \\cos { ( \\theta + u \\log { ( e \\log { N } \\log _ { 2 } { N } ) } ) } ) \\\\ & + \\left ( \\frac { \\sin { ( u / 2 ) } } { u } \\right ) ^ 2 ( 1 + \\cos { ( \\theta + u \\log { ( e ^ { 3 / 2 } \\log { N } \\log _ { 2 } { N } ) } ) } ) \\\\ & + \\left ( \\frac { \\sin { ( u / 2 ) } } { u } \\right ) ^ 2 ( 1 + \\cos { ( \\theta + u \\log { ( e ^ { 2 } \\log { N } \\log _ { 2 } { N } ) } ) } ) . \\end{align*}"}
+{"id": "701.png", "formula": "\\begin{align*} \\varphi = V _ n ( \\varphi ) \\circ T - V _ n ( \\varphi ) + \\Gamma _ n ( \\varphi ) . \\end{align*}"}
+{"id": "7993.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\cdot ( \\partial _ \\nu w ) \\nu = H ^ 2 ( \\nabla w ) \\ , . \\end{align*}"}
+{"id": "244.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { m ^ 3 } { n ^ 4 } } = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { n ^ 2 } { 4 } + \\frac { n ^ 3 } { 3 } + \\frac { n ^ 4 } { 4 } \\right ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "8178.png", "formula": "\\begin{align*} \\Omega _ s = \\{ \\omega : \\exists n _ 1 = n _ 1 ( \\omega ) \\ : \\ : \\forall n \\geq n _ 1 , \\ : \\ : \\ : \\ : C _ { 1 , n } , \\ldots , C _ { f ( n + 1 ) , n } \\ : \\ : \\} . \\end{align*}"}
+{"id": "6573.png", "formula": "\\begin{align*} \\begin{bmatrix} | Z ^ * | & Z \\\\ Z ^ * & | Z | \\end{bmatrix} \\end{align*}"}
+{"id": "1150.png", "formula": "\\begin{align*} \\mathcal { A } ( f , f ) = T ( f ^ { 2 } ) - 2 f T ( f ) - 2 B ( A ( f ) , A ( f ) ) \\left ( f \\in P \\right ) \\end{align*}"}
+{"id": "2497.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle \\frac { 1 } { 2 } \\Lambda \\left ( \\dfrac { a _ 2 } { m _ 1 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\xi | ^ 2 d x + \\dfrac { a _ 1 } { 2 m _ 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\eta | ^ 2 d x \\right ) = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "1321.png", "formula": "\\begin{align*} \\P [ A _ u ] = \\int _ { A _ u } D ( u ) d \\widehat { \\P } , \\end{align*}"}
+{"id": "4277.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } 0 & = & \\varphi ^ { \\prime } \\left ( t \\right ) + \\left [ A + \\bar { A } + \\beta \\left ( C + \\bar { C } \\right ) \\right ] ^ { \\top } \\varphi \\left ( t \\right ) \\\\ & & - \\Pi _ { 5 } ^ { \\top } \\left ( t \\right ) \\Pi _ { 2 } ^ { - 1 } \\left ( t \\right ) \\Pi _ { 4 } ^ { \\top } \\left ( t \\right ) , \\\\ \\varphi \\left ( T \\right ) & = & 0 \\end{array} \\right . \\end{align*}"}
+{"id": "8204.png", "formula": "\\begin{align*} p _ s = \\exp \\left [ - \\frac { \\lambda _ d s } { ( \\hat { r } ( s ) ) ^ 2 } ( 1 + o ( 1 ) ) \\right ] . \\end{align*}"}
+{"id": "3263.png", "formula": "\\begin{align*} \\frac 1 { \\omega ( B ) } \\| ( b - b _ B ) f _ 1 \\| ^ s _ { L ^ s _ \\omega } & = \\frac 1 { \\omega ( B ) } \\int _ { 5 B } | b ( y ) - b _ B | ^ s | f | ^ s d \\omega ( y ) \\\\ & \\le { \\Big ( \\frac 1 { \\omega ( B ) } \\int _ { 5 B } | b ( y ) - b _ B | ^ { s s _ 1 ' } d \\omega ( y ) \\Big ) ^ { \\frac 1 { s _ 1 ' } } } \\Big ( \\frac 1 { \\omega ( B ) } \\int _ { 5 B } | f | ^ { s s _ 1 } d \\omega ( y ) \\Big ) ^ { \\frac 1 { s _ 1 } } \\\\ & \\lesssim \\ell ( B ) ^ { \\beta s } \\| b \\| ^ s _ { \\Lambda ^ \\beta } M ^ s _ { s s _ 1 } ( f ) ( x ) \\end{align*}"}
+{"id": "7137.png", "formula": "\\begin{align*} \\partial _ { x _ n } u ( \\hat { x } ) = \\partial _ { x _ n } u ( \\hat { x } ) - \\partial _ { x _ n } u ( \\hat { x } ^ \\prime , 0 ) = \\Big ( \\int _ { 0 } ^ 1 \\partial _ { x _ n } ^ 2 u ( \\hat { x } ^ \\prime , t x _ n ) \\ : t \\Big ) x _ n . \\end{align*}"}
+{"id": "2154.png", "formula": "\\begin{align*} \\left . \\begin{array} { c } Y \\left ( x , y \\right ) = \\delta \\left ( x _ { 1 } - y _ { 1 } \\right ) \\overline { Y } \\left ( \\overline { x } , \\overline { y } \\right ) , \\\\ \\overline { Y } \\in L _ { \\infty } \\left ( \\Omega _ { 1 } \\times \\Omega _ { 1 } \\right ) , \\left \\Vert \\overline { Y } \\right \\Vert _ { L _ { \\infty } \\left ( \\Omega _ { 1 } \\times \\Omega _ { 1 } \\right ) } \\leq N _ { 1 } . \\end{array} \\right . \\end{align*}"}
+{"id": "5389.png", "formula": "\\begin{align*} \\rho ' & = f ' ( S ' ) + d _ { G ' - S ' } ( T ' ) - q ' ( S ' , T ' ) \\\\ & \\ge f ( S ) + d _ { G - S } ( T ) + 2 \\gamma \\C T + ( \\gamma - 1 ) \\C R ~ = ~ \\rho + q ( S , T ) + 2 \\gamma \\C T + ( \\gamma - 1 ) \\C R \\\\ & \\ge f ( T ) + \\gamma ( 2 \\C T - 1 ) + q ( S , T ) ~ \\ge ~ f ' ( T ' ) + \\gamma ( 2 \\C T - 2 ) + q ( S , T ) . \\end{align*}"}
+{"id": "3474.png", "formula": "\\begin{align*} T _ { m i x } ( P _ { \\phi } ) = \\Big ( \\frac { e ^ 2 \\Delta } { b } \\Big ) ^ { 9 + 4 \\lceil \\frac { 2 \\eta } { b } \\rceil } \\cdot O ( \\log n ) . \\end{align*}"}
+{"id": "4931.png", "formula": "\\begin{align*} p _ { i , j } ^ { ( t + 1 ) } = \\begin{dcases} 0 ^ { t + 1 } & \\mbox { i f } i = j = 0 ; \\\\ \\frac { j } { n } p _ { i , j } ^ { ( t ) } \\ , + \\ , \\frac { n - j + 1 } { n } p _ { i , j - 1 } ^ { ( t ) } & \\mbox { i f } \\ ; 0 < i \\le j \\le n j < t ; \\\\ 0 & \\mbox { o t h e r w i s e } ; \\\\ \\end{dcases} \\end{align*}"}
+{"id": "8020.png", "formula": "\\begin{align*} A \\simeq \\begin{pmatrix} X & \\ast \\\\ \\ast & \\ast \\end{pmatrix} . \\end{align*}"}
+{"id": "4352.png", "formula": "\\begin{align*} \\nabla d _ { S _ { i } } \\left ( x \\right ) = \\left ( x - P _ { i } ( x ) \\right ) / \\left \\Vert x - P _ { i } ( x ) \\right \\Vert . \\end{align*}"}
+{"id": "2173.png", "formula": "\\begin{align*} \\delta \\in \\left ( 0 , \\delta _ { 0 } \\right ) , \\delta _ { 0 } = \\delta _ { 0 } \\left ( N , \\varepsilon , \\Omega , T , c , \\rho \\right ) = \\exp \\left ( - \\frac { \\lambda _ { 1 } d } { \\rho } \\right ) \\in \\left ( 0 , 1 \\right ) . \\end{align*}"}
+{"id": "2396.png", "formula": "\\begin{align*} \\vec { u } ( A _ { 0 } , A _ { 1 } ) = ( 1 , 0 , 0 ) , \\end{align*}"}
+{"id": "4768.png", "formula": "\\begin{align*} \\tilde { S } _ { \\gamma _ i \\gamma _ i } = M _ { [ 1 ] } \\diamond \\cdots \\diamond M _ { [ | \\alpha _ i | ] } . \\end{align*}"}
+{"id": "2125.png", "formula": "\\begin{align*} u _ 1 x ^ 2 f _ 2 + u _ 2 x f _ 1 + u f _ 2 = r _ 1 ^ { q ^ m } - r _ 1 r _ 2 ^ { q ^ m - 1 } . \\end{align*}"}
+{"id": "9259.png", "formula": "\\begin{align*} \\texttt { L H S } \\eqref { T 7 . 1 } \\le \\int _ { t - l } ^ { t } ( t - z ) ^ { \\alpha + \\beta - 1 / 2 } \\ , d z = \\int _ 0 ^ { l } z ^ { \\alpha + \\beta - 1 / 2 } \\ , d z \\simeq l ^ { \\alpha + \\beta + 1 / 2 } = \\texttt { R H S } \\eqref { T 7 . 1 } . \\end{align*}"}
+{"id": "3142.png", "formula": "\\begin{align*} p _ m ( u , x + 1 ) - p _ m ( u , x ) = \\biggl [ m + ( 1 + u ) \\frac { \\partial } { \\partial { u } } \\biggr ] p _ { m - 1 } ( u , x ) . \\end{align*}"}
+{"id": "114.png", "formula": "\\begin{align*} \\| A \\chi e ^ { - i t _ 0 h ^ { - 1 } \\tilde { P } _ h ( z ) } \\tilde { R } _ h ( z ) \\chi B \\| _ { L ^ 2 \\to L ^ 2 } = \\mathcal { O } ( h ^ { - 2 } ) . \\end{align*}"}
+{"id": "8803.png", "formula": "\\begin{align*} & ( 4 u ^ 3 + 3 6 u ^ 2 t + 9 0 u t ^ 2 + 7 2 t ^ 3 + 3 2 u ^ 2 + 2 0 6 u t + 2 7 0 t ^ 2 + 1 0 8 u + 3 2 2 t + 1 2 0 ) c _ 3 \\\\ & = ( 4 u ^ 3 + 2 4 u ^ 2 t + 5 4 u t ^ 2 + 3 6 t ^ 3 + 4 4 u ^ 2 + 1 9 0 u t + 1 9 8 t ^ 2 + 1 6 0 u + 3 5 0 t + 2 0 0 ) b _ 3 \\end{align*}"}
+{"id": "3630.png", "formula": "\\begin{align*} p & \\ = \\ ( 4 , a _ { m - 1 } , \\ldots , a _ 1 , 9 ) _ { 1 0 } , \\\\ r ( p ) & \\ = \\ ( 9 , a _ 1 , \\ldots , a _ { m - 1 } , 4 ) _ { 1 0 } , \\\\ p - 2 & \\ = \\ ( 4 , a _ { m - 1 } , \\ldots , a _ 1 , 7 ) _ { 1 0 } . \\end{align*}"}
+{"id": "1356.png", "formula": "\\begin{align*} \\frac { A ( t , F _ { L } ^ { i , j } ( t ) ) - A ( t , F _ { R } ^ { i , j } ( t ) ) } { F _ { L } ^ { i , j } ( t ) - F _ { R } ^ { i , j } ( t ) } = \\dot { s } _ { i } ^ { j } ( t ) \\end{align*}"}
+{"id": "432.png", "formula": "\\begin{align*} \\vec U ^ T ( P \\otimes P _ { \\Omega } ) \\vec U _ t + ( \\vec U , { \\bf D _ { x _ i } } { \\bf A _ i } \\vec U ) + ( { \\bf A _ i } \\vec U , { \\bf D _ { x _ i } } \\vec U ) + ( \\vec U , { \\vec L _ D ) } = 0 , \\end{align*}"}
+{"id": "5493.png", "formula": "\\begin{align*} \\mathfrak { n } ^ { I } = \\oplus _ { \\alpha > 0 , \\alpha \\notin [ I ] } \\mathfrak { g } _ { \\alpha } , \\ \\mathfrak { n } ^ { - I } = \\oplus _ { \\alpha < 0 , \\alpha \\notin [ I ] } \\mathfrak { g } _ { \\alpha } . \\end{align*}"}
+{"id": "6322.png", "formula": "\\begin{align*} \\norm { Q _ \\psi + s v } _ * = 1 + C s ^ 2 + o ( s ^ 2 ) , s \\to 0 . \\end{align*}"}
+{"id": "5311.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\hat f ( \\{ i \\} ) ^ 2 \\geq m \\frac { p } { 1 - p } \\alpha ^ 2 . \\end{align*}"}
+{"id": "4164.png", "formula": "\\begin{align*} \\pi ( x ) = \\sum _ { t \\in G } \\pi _ t ( x ( t ) ) . \\end{align*}"}
+{"id": "3005.png", "formula": "\\begin{align*} \\phi ( _ g F ) = \\phi ( F ) \\end{align*}"}
+{"id": "5440.png", "formula": "\\begin{align*} & G ( \\theta ) \\approx \\sum _ { b _ 1 , \\cdots , b _ M } { b \\choose b _ 1 , \\cdots , b _ M } \\prod _ { m = 1 } ^ { M } \\left ( C _ { M } ^ { m } ( - 1 ) ^ { m + 1 } \\right ) ^ { b _ m } \\\\ & \\frac { ( R _ { m a x } \\ ! \\ ! - \\ ! \\ ! R _ { m i n } ) \\pi } { 2 N } \\ ! \\sum _ { k = 1 } ^ { K } \\ ! \\sqrt { 1 \\ ! \\ ! - \\ ! \\psi _ k ^ 2 } \\exp \\left ( - Q ( d _ k , \\theta ) \\right ) f _ { r _ 1 | \\Phi ( \\mathcal { A } ) > 0 } ( d _ k ) , \\end{align*}"}
+{"id": "7064.png", "formula": "\\begin{align*} \\tilde \\beta _ i = \\min \\ , \\tilde \\beta = \\min \\ , \\alpha . \\end{align*}"}
+{"id": "687.png", "formula": "\\begin{align*} \\mathrm { o s c } ( v , I ^ { ( k ) } ) = O ( e ^ { ( - \\lambda _ 1 ( 1 - a ) + \\tau ) k } ) c _ 1 ( \\varphi ) . \\end{align*}"}
+{"id": "1691.png", "formula": "\\begin{align*} \\Lambda _ { \\alpha _ 1 \\alpha _ 2 } : = \\mathbb Z + \\mathbb Z \\alpha _ 1 + \\mathbb Z \\alpha _ 2 + \\mathbb Z \\alpha _ 1 \\alpha _ 2 . \\end{align*}"}
+{"id": "3483.png", "formula": "\\begin{align*} P _ { \\phi } : = \\prod _ { i = 1 } ^ { n } P _ i \\prod _ { i = 0 } ^ { n - 1 } P _ { n - i } . \\end{align*}"}
+{"id": "2723.png", "formula": "\\begin{align*} \\begin{aligned} T _ 4 = & s \\iint _ { Q } ( A \\nabla \\sigma ) _ i \\frac { \\partial A } { \\partial x _ i } \\nabla u \\cdot \\nabla u d x d t \\ge - C s \\lambda \\iint _ Q \\xi A \\nabla u \\cdot \\nabla u d x d t . \\end{aligned} \\end{align*}"}
+{"id": "6072.png", "formula": "\\begin{align*} H ^ p _ A = \\{ f \\in \\mathcal { S } ' \\colon M ^ 0 _ { \\phi , A } f \\in L ^ p \\} , \\end{align*}"}
+{"id": "2537.png", "formula": "\\begin{align*} \\overline { _ N } = \\overline { G _ M \\cap { \\mathcal H } } . \\end{align*}"}
+{"id": "7984.png", "formula": "\\begin{align*} \\Psi _ \\Omega ^ H ( r ) = \\begin{cases} \\sup \\limits _ { x \\in \\partial \\Omega } | | \\mathcal B ^ H | | _ { L ^ { n - 1 , \\infty } ( \\partial \\Omega \\cap B _ r ( x ) ) } \\quad n \\geq 3 , \\\\ \\\\ \\sup \\limits _ { x \\in \\partial \\Omega } | | \\mathcal B ^ H | | _ { L ^ { 1 , \\infty } \\log L ( \\partial \\Omega \\cap B _ r ( x ) ) } \\quad n = 2 \\end{cases} \\end{align*}"}
+{"id": "5824.png", "formula": "\\begin{align*} { { \\boldsymbol { \\hat x } } _ { k ; t + 1 } } = { { \\boldsymbol { \\hat x } } _ { k | k - 1 } } + { { \\boldsymbol { K } } _ { k ; t } } \\left ( { { { \\boldsymbol { y } } _ k } - { { { \\boldsymbol { \\hat y } } } _ { k | k - 1 } } } \\right ) \\end{align*}"}
+{"id": "4994.png", "formula": "\\begin{align*} \\begin{aligned} F ( k , t + 1 , \\mathbf { p } _ { n + 1 } ) = F ( k , t , \\mathbf { p } _ { n + 1 } ) - p _ { k } ^ { } f ( k , t , \\mathbf { p } _ { n + 1 } ) , \\end{aligned} \\end{align*}"}
+{"id": "7320.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\alpha _ i z _ { i p } = \\sum _ { i = 1 } ^ n \\gamma _ i z _ { i p } < \\sum _ { i = 1 } ^ n \\beta _ i z _ { i p } , \\end{align*}"}
+{"id": "8691.png", "formula": "\\begin{align*} \\mathcal { T } _ { 1 , a } ^ { ( a _ 1 , a _ 2 ) } = ( \\mu _ { i _ 1 , \\ldots , i _ { a - a _ 1 - a _ 2 } } ^ { ( 1 ) } ) _ { 1 \\leq i _ 1 , \\ldots , i _ { a - a _ 1 - a _ 2 } \\leq p } , \\quad \\mathcal { T } _ { 2 , a } ^ { ( a _ 1 , a _ 2 ) } = ( \\mu _ { i _ 1 , \\ldots , i _ { a - a _ 1 - a _ 2 } } ^ { ( 2 ) } ) _ { 1 \\leq i _ 1 , \\ldots , i _ { a - a _ 1 - a _ 2 } \\leq p } , \\end{align*}"}
+{"id": "2325.png", "formula": "\\begin{align*} v ^ { ( r , z ) } ( t , x ) = 2 G _ { t - r } ( x - z ) v ^ { ( r , z ) } ( t , x ) ; \\end{align*}"}
+{"id": "18.png", "formula": "\\begin{align*} \\mathbf { G } ( R ) : = \\{ g \\in \\mathrm { G L } _ { 2 n } ( R ) \\mid g ^ { \\mathrm { t } } J _ n g = \\nu ( g ) J _ n \\ , \\nu ( g ) \\in R ^ \\times \\} . \\end{align*}"}
+{"id": "2892.png", "formula": "\\begin{align*} f \\sharp g ( \\xi ) \\sim \\sum _ \\alpha \\frac { ( - i ) ^ { | \\alpha | } } { \\alpha ! } ( \\partial _ \\xi ^ \\alpha f ) ( \\xi ) \\partial _ x ^ \\alpha | _ { x = 0 } \\Big ( \\alpha _ { - x } \\big ( g ( \\xi + B x ) \\big ) \\Big ) . \\end{align*}"}
+{"id": "3223.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { x \\in M _ + , x \\neq 0 } \\abs { ( S _ n ( \\nu _ k ) - \\bar { \\nu } _ k ) ( x ) } / \\rho ( x ) & = \\lim _ { n \\to \\infty } \\sup _ { x \\in M _ + , x \\neq 0 } \\abs { ( S _ n ( \\nu _ k ) - \\bar { \\nu } ) ( x ) } / \\rho ( x ) = 0 . \\end{align*}"}
+{"id": "8524.png", "formula": "\\begin{align*} 8 \\sum _ { n = 1 } ^ { \\infty } d ( n ) \\ , K _ { 0 } \\left ( \\pi n \\ , \\sqrt { c } \\right ) < \\frac { 2 ^ { 5 / 2 } \\ , e ^ { - \\pi \\sqrt { c } } } { c ^ { 1 / 4 } } . \\end{align*}"}
+{"id": "8250.png", "formula": "\\begin{align*} W _ { h , q } ( i ) = \\begin{cases} h & i = q ; \\\\ 1 & k - j + h \\leq i \\leq k - 2 i \\neq q \\\\ l - j + 1 & i = k - 1 ; \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "1154.png", "formula": "\\begin{align*} \\mathcal { A } ( f , f ) = \\mathcal { Q } ( f ) \\left ( f \\in P \\right ) \\end{align*}"}
+{"id": "9045.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & - 1 \\\\ - 3 & 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "6330.png", "formula": "\\begin{align*} f ' _ \\psi ( 0 ) = 0 f '' _ \\psi ( 0 ) = C ' _ \\circ ( \\psi ) > 0 . \\end{align*}"}
+{"id": "218.png", "formula": "\\begin{align*} 2 ( \\log 2 ) ^ 3 - 7 \\zeta ( 3 ) = - 2 4 L i _ 3 \\left ( \\frac { 1 } { 2 } \\right ) + 1 8 L i _ 3 \\left ( \\frac { 1 } { 4 } \\right ) + 4 L i _ 3 \\left ( \\frac { 1 } { 8 } \\right ) - L i _ 3 \\left ( \\frac { 1 } { 6 4 } \\right ) , \\end{align*}"}
+{"id": "1598.png", "formula": "\\begin{align*} u ^ n - U ^ n = u ^ n - R _ h u ^ n + R _ h u ^ n - U ^ n = \\rho ^ n + \\theta ^ n , \\end{align*}"}
+{"id": "3284.png", "formula": "\\begin{align*} \\widetilde { \\rho } ( t , \\kappa ( t ) ) = 0 { \\rm \\ f o r \\ } t \\ge 0 , \\end{align*}"}
+{"id": "2042.png", "formula": "\\begin{align*} \\begin{array} { r l l l l } \\sigma ^ \\star D _ 1 & = & - D _ 1 & & \\\\ \\sigma ^ \\star D _ 2 & = & - D _ 2 & & \\\\ \\sigma ^ \\star D _ 3 & = & D _ 1 + D _ 3 + \\div ( h _ 3 ) & & h _ 3 = \\frac { y } { x ^ 3 + a ^ { 5 8 } x ^ 2 + a ^ 2 x + a ^ { 5 4 } } \\\\ \\sigma ^ \\star D _ 4 & = & D _ 2 + D _ 4 + \\div ( h _ 4 ) & & h _ 4 = \\frac { y } { x ^ 3 + a ^ { 8 0 } x ^ 2 + a ^ { 1 0 3 } x + a ^ { 1 1 4 } } \\\\ \\sigma ^ \\star D _ 5 & = & D _ 5 \\end{array} \\end{align*}"}
+{"id": "9184.png", "formula": "\\begin{align*} v _ { 1 } ^ { 1 } & = \\varphi _ { 1 , [ 2 ] } ^ { 1 } = q ^ { 2 } + 2 T \\omega ^ { 2 } + T ^ { 2 } \\left ( a _ { 1 } \\sin ( q ^ { 2 } ) + a _ { 2 } \\cos ( q ^ { 2 } ) + b _ { 2 } \\cos ( q ^ { 3 } ) u ^ { 1 } \\right ) \\\\ v _ { 1 , [ 1 ] } ^ { 1 } & = \\varphi _ { 1 , [ 3 ] } ^ { 1 } ( q ^ { 2 } , q ^ { 3 } , \\omega ^ { 2 } , \\omega ^ { 3 } , u ^ { 1 } , u _ { [ 1 ] } ^ { 1 } ) \\ , , \\\\ & \\ : \\ : \\vdots \\end{align*}"}
+{"id": "7968.png", "formula": "\\begin{align*} | X Y | ^ 2 = \\mathrm { t r } \\big ( X Y ( X Y ) ^ t \\big ) = \\mathrm { t r } ( X Y Y X ) = \\mathrm { t r } ( X ^ 2 Y ^ 2 ) \\leq \\l _ { { \\rm m a x } } ^ 2 \\mathrm { t r } ( Y ^ 2 ) = \\l _ { { \\rm m a x } } ^ 2 | Y | ^ 2 , \\end{align*}"}
+{"id": "4500.png", "formula": "\\begin{align*} G ( \\Psi ) : = ( 4 - 2 d ) \\omega Q ( \\Psi ) + ( 3 - d ) \\mathbf { c } \\cdot \\mathbf { P } ( \\Psi ) . \\end{align*}"}
+{"id": "8029.png", "formula": "\\begin{align*} \\lambda _ { 1 + j } \\left ( \\oplus ^ m A \\right ) = \\lambda _ { \\langle ( 1 + j ) / m \\rangle } ( A ) \\end{align*}"}
+{"id": "5921.png", "formula": "\\begin{align*} \\overline { c } _ { X ^ { \\ast } } ( g , g ' ) = \\overline { c } _ { X ^ { \\ast } } ( g ^ { \\alpha } , g ^ { ' \\alpha } ) \\nu _ 2 ( \\alpha , g ) \\nu _ 2 ( \\alpha , g ' ) \\nu _ 2 ( \\alpha , g g ' ) ^ { - 1 } . \\end{align*}"}
+{"id": "6611.png", "formula": "\\begin{align*} [ \\partial x _ \\lambda y ] & = - \\lambda [ x _ \\lambda y ] , [ x _ \\lambda \\partial y ] = ( \\partial + \\lambda ) [ x _ \\lambda y ] ( c o n f o r m a l \\ s e s q u i l i n e a r i t y ) , \\\\ { } [ x _ \\lambda y ] & = - ( - 1 ) ^ { | x | | y | } [ y _ { - \\lambda - \\partial } x ] ( s k e w s y m m e t r y ) , \\\\ { } [ x _ \\lambda [ y _ \\mu z ] ] & = [ [ x _ \\lambda y ] _ { \\lambda + \\mu } z ] + ( - 1 ) ^ { | x | | y | } [ y _ \\mu [ x _ \\lambda z ] ] ( J a c o b i \\ i d e n t i t y ) , \\end{align*}"}
+{"id": "2462.png", "formula": "\\begin{align*} \\varphi & = g _ 1 * \\Phi _ 1 ( \\varphi ) + \\ldots + g _ N * \\Phi ( \\varphi ) \\\\ & = g _ i * ( \\rho \\cdot \\Psi _ 1 ( \\varphi ) ) + \\ldots + g _ N * ( \\rho \\cdot \\Psi _ N ( \\varphi ) ) \\\\ & = f _ 1 * \\Psi _ 1 ( \\varphi ) + \\ldots + f _ N * \\Psi ( \\varphi ) \\end{align*}"}
+{"id": "2186.png", "formula": "\\begin{align*} | g _ { \\lambda _ i } ( z ) ^ { - \\frac { 1 } { 2 } } - \\lambda _ i ^ { - \\frac { 1 } { 2 } } | \\leq \\sqrt { 2 } \\lambda _ i ^ { - \\frac { 3 } { 2 } } | \\lambda _ i z ^ 2 - 2 z | , i = 1 , 2 , . . . , m . \\end{align*}"}
+{"id": "796.png", "formula": "\\begin{align*} F _ i ( U ^ n , \\ , d ) \\ , = \\ , 0 , \\mbox { f o r } \\ ; 1 \\le i \\le M + 1 , \\end{align*}"}
+{"id": "1094.png", "formula": "\\begin{align*} x ( y z ) = ( x y ) z \\end{align*}"}
+{"id": "7551.png", "formula": "\\begin{align*} W _ i = \\begin{cases} 1 , & \\ ~ \\alpha , \\\\ 0 , & \\ ~ 1 - \\alpha . \\end{cases} \\end{align*}"}
+{"id": "5191.png", "formula": "\\begin{align*} H _ { T ^ { n + 1 } } ^ { * } ( P l ( k , n ) ; \\mathbb { Q } ) = \\Big \\{ \\alpha \\in \\bigoplus _ { i = 0 } ^ m \\mathbb { Q } [ y _ 1 , \\dots , y _ n , z ] / ( Y _ { \\lambda ^ i } + z ) ~ \\big { | } ~ & ( \\alpha _ { i } - \\alpha _ { j } ) \\in ( Y _ { \\lambda ^ i } + z , Y _ { \\lambda ^ j } + z ) \\\\ & | \\lambda ^ i \\cap \\lambda ^ j | = k - 1 \\Big \\} . \\end{align*}"}
+{"id": "2725.png", "formula": "\\begin{align*} T _ 5 \\geq & - C s ^ 2 \\lambda ^ 4 \\iint _ Q \\xi \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d t - C \\lambda ^ 2 \\iint _ { Q } \\xi \\left | A \\nabla u \\cdot \\nabla \\eta \\right | ^ 2 d x d t \\\\ & - C s ^ 2 \\lambda ^ 3 \\int _ 0 ^ T \\int _ { \\omega } \\xi | u | ^ 2 d x d t - C \\lambda \\iint _ Q \\xi A \\nabla u \\cdot \\nabla u d x d t . \\end{align*}"}
+{"id": "4541.png", "formula": "\\begin{align*} \\frac { 1 } { q } : = \\frac { 1 } { p } - \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "8589.png", "formula": "\\begin{align*} \\bar { S } _ { j , + , \\Delta } ^ k ( t ) : = \\nu \\Delta \\sum _ { \\ell = 0 } ^ { \\lfloor t / \\Delta \\rfloor } Z _ 0 ( \\ell \\Delta ) p _ { j , + } ^ k ( t - \\ell \\Delta ) . \\end{align*}"}
+{"id": "1819.png", "formula": "\\begin{align*} | \\ < \\Psi , [ \\N ^ \\ell , V _ B ( t ) ] \\Psi \\ > | & \\leq \\sum _ { n = 0 } ^ { \\ell - 1 } \\binom { \\ell } { n } 4 ^ { \\ell - n } \\int _ { \\Lambda ^ * } | \\hat V ( k ) | \\| ( \\N + 4 ) ^ { \\frac { n + 1 } { 2 } } \\Psi \\| \\ \\| b _ k ( t ) b _ { - k } ( t ) \\N ^ { \\frac { n - 1 } { 2 } } \\Psi \\| \\d k \\ . \\end{align*}"}
+{"id": "8373.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} { c c c } 3 & 0 & 0 \\\\ 0 & 3 & 0 \\\\ 0 & 0 & 3 \\end{array} \\right ) , A = \\left ( \\begin{array} { c c c } 1 & 2 & 2 \\\\ 1 & 0 & 1 \\\\ 1 & 1 & 0 \\end{array} \\right ) , L = \\left ( \\begin{array} { c c c } 2 & - 2 & - 2 \\\\ - 1 & 3 & - 1 \\\\ - 1 & - 1 & 3 \\end{array} \\right ) . \\end{align*}"}
+{"id": "1283.png", "formula": "\\begin{align*} \\sigma ( A ) & = \\left \\{ \\sqrt { n } ^ { ( 1 ) } , - \\sqrt { n } ^ { ( 1 ) } , 0 ^ { ( n - 1 ) } \\right \\} , \\\\ \\sigma ( B ) & = \\left \\{ \\sqrt { n } + 1 ^ { ( 1 ) } , - \\sqrt { n } + 1 ^ { ( 1 ) } , 1 ^ { ( n - 1 ) } , \\sqrt { n } - 1 ^ { ( 1 ) } , - \\sqrt { n } - 1 ^ { ( 1 ) } , - 1 ^ { ( n - 1 ) } \\right \\} . \\end{align*}"}
+{"id": "4449.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\omega , \\mathbf { c } } : = \\left \\{ \\left . \\left ( ( \\sqrt { \\omega } - \\tau ) ^ 2 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } ( \\sqrt { \\omega } - \\tau ) \\right ) \\ \\right | \\ \\tau < \\sqrt { \\omega } \\ \\right \\} \\end{align*}"}
+{"id": "5668.png", "formula": "\\begin{align*} \\ K ( u ^ * , v ^ * ) \\leq K ( u , v ) , \\ L ( u ^ * , v ^ * ) \\leq L ( u , v ) = 0 . \\end{align*}"}
+{"id": "7351.png", "formula": "\\begin{align*} \\log \\log \\Pr [ \\hat { M } = M ] ^ { - 1 } \\geq ( 1 - \\beta ) \\log n + \\log \\log n + \\omega ( 1 ) , \\end{align*}"}
+{"id": "4084.png", "formula": "\\begin{align*} \\Q ( \\alpha ) = \\{ a _ 0 + a _ 1 \\alpha + \\dots + a _ { n - 1 } \\alpha ^ { n - 1 } | a _ i \\in \\Q \\} . \\end{align*}"}
+{"id": "8810.png", "formula": "\\begin{align*} & \\frac { 2 ( 3 + t + 2 u ) ( 1 + 2 t ) u } { 3 ( 1 + t ) ( \\displaystyle \\sum _ { k = 0 } ^ { n - 1 } u ^ k ) ^ 2 } \\\\ & = \\frac { 2 ( 2 + t ) ( 1 + 2 t ) u } { 3 ( 1 + t ) ( \\displaystyle \\sum _ { k = 0 } ^ { n - 1 } u ^ k ) ^ 2 } + \\frac { 3 ( 1 + t ) ( 1 + u ) u } { 3 ( 1 + t ) ( \\displaystyle \\sum _ { k = 0 } ^ { n - 1 } u ^ k ) ^ 2 } + p _ { 1 , 3 } \\end{align*}"}
+{"id": "1897.png", "formula": "\\begin{align*} \\tilde { \\mathcal { X } } = S ( \\mathcal { U } ) - \\mathcal { K } _ { K , \\zeta ^ * } . \\end{align*}"}
+{"id": "409.png", "formula": "\\begin{align*} U ^ T ( n _ 1 A + n _ 2 B ) U = U ^ T _ n \\tilde A U _ n = W ^ T \\Lambda W = ( W ^ + ) ^ T \\Lambda ^ + W ^ + + ( W ^ - ) ^ T \\Lambda ^ - W ^ - \\end{align*}"}
+{"id": "470.png", "formula": "\\begin{align*} A _ N \\begin{pmatrix} D _ { N } \\\\ W _ N \\end{pmatrix} = \\begin{pmatrix} S ^ 1 _ { N } \\\\ S ^ 2 _ N \\end{pmatrix} , \\ \\ N \\ge 2 , \\end{align*}"}
+{"id": "2088.png", "formula": "\\begin{align*} F ( A ) & = ( m ( 2 ^ n - 1 ) + d - 2 + m - 1 + 2 ) \\cdot m ( 2 ^ n - 1 ) - d \\\\ & = m ^ 2 \\cdot 2 ^ { 2 n } - ( m ^ 2 + m - m d ) 2 ^ n - d m - d + m . \\end{align*}"}
+{"id": "8044.png", "formula": "\\begin{align*} \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} = U \\begin{bmatrix} A & 0 \\\\ 0 & 0 \\end{bmatrix} U ^ * + V \\begin{bmatrix} 0 & 0 \\\\ 0 & B \\end{bmatrix} V ^ * , \\end{align*}"}
+{"id": "6854.png", "formula": "\\begin{align*} \\mathcal { W } = \\bigl \\{ h \\colon \\ , [ 0 , 1 ] ^ 2 \\to [ 0 , 1 ] \\colon \\ , h ( x , y ) = h ( y , x ) \\ , \\ , \\forall \\ , x , y \\in [ 0 , 1 ] \\bigr \\} \\end{align*}"}
+{"id": "249.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( 1 - z ^ n \\right ) ^ { \\frac { m ^ 1 } { n ^ 2 } } = \\sqrt { 1 - z } \\ ; \\exp \\left \\{ \\frac { - 1 } { 2 } \\frac { z } { 1 - z } \\right \\} . \\end{align*}"}
+{"id": "663.png", "formula": "\\begin{align*} \\mathfrak { h } ^ * _ j ( \\varphi ) = \\lim _ { l \\to \\infty } Q ( 0 , l ) ^ { - 1 } \\circ P _ { U ^ { ( l ) } _ { - j } } \\circ \\mathcal { M } ^ { ( l ) } \\circ S ( l ) ( \\varphi ) \\in U _ { - j } \\end{align*}"}
+{"id": "8911.png", "formula": "\\begin{gather*} Z _ j = \\sum _ { n = 1 } ^ { d _ 1 ^ j } [ X _ { n 1 } , \\dots , X _ { n j } ] , \\\\ \\nu _ j = \\sqrt { \\sum _ { n , m = 1 } ^ { d _ 1 ^ j } \\langle X _ { n 1 } , X _ { m 1 } \\rangle _ 1 \\cdots \\langle X _ { n j } , Y _ { n j } \\rangle _ 1 } = \\sqrt { \\sum _ { n = 1 } ^ { d _ 1 ^ j } \\| X _ { n 1 } \\| _ 1 ^ 2 \\cdots \\| X _ { n j } \\| _ 1 ^ 2 } , \\\\ \\| X _ { n 1 } \\| _ 1 = \\cdots = \\| X _ { n j } \\| _ 1 n = 1 , \\dots , d _ 1 ^ j . \\end{gather*}"}
+{"id": "4409.png", "formula": "\\begin{align*} N ( U _ n ) = \\left | { \\rm R e } ( u _ { 3 n } , \\nabla ( u _ { 1 n } \\cdot \\overline { u _ { 2 n } } ) _ { L ^ 2 } \\right | \\le \\| \\nabla \\cdot u _ { 3 n } \\| _ { L ^ 2 } \\| u _ { 1 n } \\| _ { L ^ 4 } \\| u _ { 2 n } \\| _ { L ^ 4 } \\le C \\| U _ n \\| _ { L ^ 4 } ^ 2 . \\end{align*}"}
+{"id": "6960.png", "formula": "\\begin{align*} \\nu _ q ( f ) = \\min \\left \\{ \\nu \\left ( f _ i q ^ i \\right ) \\right \\} , \\end{align*}"}
+{"id": "6826.png", "formula": "\\begin{align*} \\alpha ' _ n = ( 1 - a ) \\left ( \\frac { \\alpha _ n } { 1 - a q ^ { 2 n } } - \\frac { a q ^ { 2 n - 2 } \\alpha _ { n - 1 } } { 1 - a q ^ { 2 n - 2 } } \\right ) , \\beta ' _ n = \\beta _ n , \\end{align*}"}
+{"id": "8023.png", "formula": "\\begin{align*} Q _ { { \\mathcal { H } } _ { n s } } \\simeq \\begin{pmatrix} I & 0 \\\\ R & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "2872.png", "formula": "\\begin{align*} \\tau \\circ ( 1 \\otimes T ) \\Delta ( l ) = \\tau \\circ ( l _ 1 \\otimes T ( l _ 2 ) ) = T ( l _ 2 ) \\otimes l _ 1 = ( T \\otimes 1 ) ( l _ 2 \\otimes l _ 1 ) = ( T \\otimes 1 ) \\tau \\circ \\Delta ( l ) . \\end{align*}"}
+{"id": "4279.png", "formula": "\\begin{align*} Y _ { t } ^ { t , \\xi ; u } = \\sum _ { i = 1 } ^ { N } I _ { A _ { i } } Y _ { t } ^ { i } . \\end{align*}"}
+{"id": "430.png", "formula": "\\begin{align*} & \\vec L _ D = ( I _ n \\otimes P _ { \\Omega } ) ^ { - 1 } ) ( D C ) \\vec L _ C & & \\ \\ \\ D C = ( I _ n \\otimes E ^ T ) ( P _ { e r m } ) ^ T ( P _ { \\partial \\Omega } \\otimes I _ n ) , & \\\\ & \\vec L _ C = ( ( L _ C ) _ 1 , ( L _ C ) _ 2 , . . . , ( L _ C ) _ N ) ^ T & & ( L _ C ) _ j = ( 2 ( J ^ - T ^ { - 1 } ) ^ T \\Sigma ( \\sqrt { | \\Lambda ^ - | } \\vec W ^ - - R \\sqrt { \\Lambda ^ + } \\vec W ^ + - S \\vec G ) ) _ j . & \\end{align*}"}
+{"id": "5635.png", "formula": "\\begin{align*} \\tilde { U } _ { \\epsilon , \\xi } ( x ) = S ^ { \\frac { ( N - \\mu ) ( 2 - N ) } { 4 ( N - \\mu + 2 ) } } \\bigl ( C ( N , \\mu ) \\bigr ) ^ { \\frac { 2 - \\mu } { 2 ( N - \\mu + 2 ) } } U _ { \\epsilon , \\xi } ( x ) , \\end{align*}"}
+{"id": "8644.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 ( 1 - y ) y ^ { j - 1 } d y = \\frac 1 { j ( j + 1 ) } . \\end{align*}"}
+{"id": "6427.png", "formula": "\\begin{align*} \\widetilde { Q } _ { \\alpha , z } ( \\psi _ 1 \\bar { \\otimes } \\psi _ 2 | | \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 ) = \\widetilde { Q } _ { \\alpha , z } ( \\psi _ 1 | | \\varphi _ 1 ) \\widetilde { Q } _ { \\alpha , z } ( \\psi _ 2 | | \\varphi _ 2 ) \\end{align*}"}
+{"id": "2857.png", "formula": "\\begin{align*} ( \\widetilde { \\alpha } \\otimes \\widetilde { \\varphi } ) \\circ \\widetilde { \\Delta } ( x , m ) = ( \\alpha \\otimes \\varphi _ L ) \\Delta ( x ) + ( \\beta \\otimes \\varphi _ L ) ( \\rho _ { \\beta } ( m ) ) - ( \\alpha _ M \\otimes \\varphi _ L ) ( \\tau \\circ ( \\rho _ { \\beta } ( m ) ) ) . \\end{align*}"}
+{"id": "6565.png", "formula": "\\begin{align*} \\omega _ Q ^ { 1 / \\bar { p } _ 2 - 1 / \\bar { p } _ 1 } \\int _ { \\mathbb H ^ n } \\frac { | y | _ h ^ { - Q / p } } { \\max ( 1 , | y | _ h ^ Q ) } d y = \\omega _ Q ^ { 1 / \\bar { p } _ 2 - 1 / \\bar { p } _ 1 } \\frac { \\omega _ Q Q } { ( Q - Q / p ) Q / p } , \\end{align*}"}
+{"id": "3081.png", "formula": "\\begin{align*} 0 > \\mathcal U ' ( r ) \\cdot \\frac { u _ 0 ( r ) } { u _ 0 ' ( r ) } = \\frac { u ' ( r ) u _ 0 ( r ) - u _ 0 ' ( r ) u ( r ) } { u _ 0 ( r ) u _ 0 ' ( r ) } = \\mathcal W ( r ) - \\mathcal U ( r ) , \\end{align*}"}
+{"id": "7019.png", "formula": "\\begin{align*} f ' = f _ 0 ' + f _ 1 Q _ i ' + f _ 1 ' Q _ i + \\ldots + f _ r ' Q ^ r + r f _ r Q _ i ' Q _ i ^ { r - 1 } . \\end{align*}"}
+{"id": "932.png", "formula": "\\begin{align*} V _ A ^ { } ( \\mathbf { h } _ t , a _ t ) & = ( 1 - \\eta ) V _ A ( \\mathbf { h } _ t , a _ t ) + \\eta \\big ( r _ t + \\gamma _ { _ { } } \\max _ { a _ { t + 1 } \\in \\mathcal { A } } V _ B ( \\mathbf { h } _ { t + 1 } , a _ { t + 1 } ) \\big ) , \\end{align*}"}
+{"id": "5424.png", "formula": "\\begin{align*} Y _ { n + 1 } = ( A \\otimes I ) Y _ n + h \\Phi ( t _ n , Y _ n , h ) \\quad ( n \\in \\mathbb { N } ) , \\end{align*}"}
+{"id": "1165.png", "formula": "\\begin{align*} T ( f \\cdot g ) = T ( f ) \\cdot g + f \\cdot T ( g ) + 2 A ( f ) \\cdot A ( g ) \\end{align*}"}
+{"id": "9147.png", "formula": "\\begin{align*} \\delta ^ { k _ { 1 } ^ { j } - 1 } ( \\varphi ^ { j } ) & = \\varphi _ { [ k _ { 1 } ^ { j } - 1 ] } ^ { j } ( \\zeta _ { [ - q _ { 1 } , - 1 ] } , x ) \\\\ \\delta ^ { k _ { 1 } ^ { j } } ( \\varphi ^ { j } ) & = \\varphi _ { [ k _ { 1 } ^ { j } ] } ^ { j } ( \\zeta _ { [ - q _ { 1 } , - 1 ] } , x , u ) \\end{align*}"}
+{"id": "2411.png", "formula": "\\begin{align*} \\frac { 2 \\pi \\delta } { m } \\le \\frac { t \\delta } { M + ( j - 1 ) m } \\le \\frac { 2 \\pi \\delta } { \\beta m } = \\frac { 2 \\pi \\delta } { m } + \\frac { 2 \\pi \\delta } { 8 m ^ { 2 } } , \\end{align*}"}
+{"id": "4987.png", "formula": "\\begin{align*} \\overline { S } ( k , n ) \\overset { } { = } \\sum _ { j = k + 1 } ^ n \\binom { n } { j } = 2 ^ n - \\sum _ { j = 0 } ^ { k } \\binom { n } { j } \\end{align*}"}
+{"id": "6655.png", "formula": "\\begin{align*} \\gamma _ R ( x ) : = \\gamma \\left ( \\frac { | x | } { R } \\right ) \\textrm { f o r a l l } \\ ; \\ ; x \\in \\R ^ N \\ , . \\end{align*}"}
+{"id": "8683.png", "formula": "\\begin{align*} ( \\Delta _ { 1 , 1 } ) = \\frac { 8 } { p ^ { 2 } } \\bigg [ \\frac { 1 } { n ( n - 1 ) } \\{ f ^ { ( 1 ) } ( \\tau _ { 1 } ) \\} ^ { 2 } ( \\Sigma _ { 1 } ^ { 2 } ) + \\frac { 1 } { m ( m - 1 ) } \\{ f ^ { ( 1 ) } ( \\tau _ { 2 } ) \\} ^ { 2 } ( \\Sigma _ { 2 } ^ { 2 } ) + \\frac { 2 } { n m } \\{ f ^ { ( 1 ) } ( \\tau _ { 3 } ) \\} ^ { 2 } ( \\Sigma _ { 1 } \\Sigma _ { 2 } ) \\bigg ] \\end{align*}"}
+{"id": "6259.png", "formula": "\\begin{align*} \\Pi : = \\big \\{ \\psi _ t ( \\Sigma _ t ) \\ , | \\ , \\{ \\psi _ t \\} _ { t \\in I ^ n } \\ , \\textrm { i s a n i s o t o p y w i t h } \\ , \\psi _ t = \\mathrm { i d } \\ , \\mathrm { f o r } \\ , t \\in \\partial I ^ n \\big \\} . \\end{align*}"}
+{"id": "8709.png", "formula": "\\begin{align*} E ( U ^ \\top A U ) = \\sum _ { i , j } a _ { i j } E ( u _ i u _ j ) = ( A ) , \\end{align*}"}
+{"id": "7539.png", "formula": "\\begin{align*} - \\Delta _ p v = 0 \\ , \\ , B _ r ( x _ 0 ) . \\end{align*}"}
+{"id": "3665.png", "formula": "\\begin{align*} \\mathcal { M } _ { q _ 1 , \\alpha } f = \\mathcal { M } _ { q _ 2 , \\alpha } f , f \\in H ^ { s _ M } ( \\Omega ^ c ) \\end{align*}"}
+{"id": "2718.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } z - \\sum _ { i , j = 1 } ^ n \\partial _ { x _ i } ( A _ { i j } \\partial _ { x _ j } z ) + b z = \\chi _ { \\omega } g , & \\mbox { i n } \\ Q , \\\\ z = 0 \\ \\mbox { o r } \\ A \\nabla z \\cdot \\nu = 0 , & \\mbox { o n } \\ \\Sigma , \\\\ z ( 0 ) = z _ { 0 } , & \\mbox { i n } \\ \\Omega . \\end{cases} \\end{align*}"}
+{"id": "4228.png", "formula": "\\begin{align*} 0 & = T ( \\tilde { a } ) ^ { - 1 } H ( \\tilde { a } ) + H ( \\tilde { a } ^ { - 1 } ) T ( a ^ { - 1 } ) ^ { - 1 } \\end{align*}"}
+{"id": "7935.png", "formula": "\\begin{align*} \\beta ( \\xi , 0 ) + \\sum _ \\mathbf { n } \\beta ( \\mathbf { n } ) = 1 , \\end{align*}"}
+{"id": "8019.png", "formula": "\\begin{align*} { \\mathrm { T r \\ , } } & \\left \\{ g ( X ^ * A X + Y ^ * B Y ) + g ( Y ^ * A Y + X ^ * B X ) \\right \\} \\\\ & \\le { \\mathrm { T r \\ , } } \\left \\{ X ^ * g ( A ) X + Y ^ * g ( B ) Y ) + Y ^ * g ( A ) Y + X ^ * g ( B ) X \\right \\} \\\\ & = { \\mathrm { T r \\ , } } \\left \\{ ( g ( A ) + g ( B ) ) ( X X ^ * + Y Y ^ * ) \\right \\} = { \\mathrm { T r \\ , } } \\left \\{ g ( A ) + g ( B ) \\right \\} \\end{align*}"}
+{"id": "2276.png", "formula": "\\begin{align*} w _ b = \\Phi _ b + T ( f ) _ b , \\end{align*}"}
+{"id": "2756.png", "formula": "\\begin{align*} T : H _ \\Gamma ^ { 3 / 2 } ( \\partial \\mathrm { M } ) \\rightarrow H : \\varphi \\mapsto T \\varphi : = u _ { | \\mathrm { M } _ 0 } , \\end{align*}"}
+{"id": "916.png", "formula": "\\begin{align*} u h _ u + v h _ v & = - ( \\nu _ { 1 2 3 } + \\mu / 2 + 1 / 4 ) h , \\\\ u g _ u + v g _ v & = - ( \\nu _ { 1 2 3 } + \\mu / 2 + 1 / 4 ) g , \\end{align*}"}
+{"id": "1690.png", "formula": "\\begin{align*} Q ( x , y , z ) = p d d _ 2 ^ { 2 } x ^ 2 + p d d _ 1 ^ { 2 } y ^ 2 + z ^ 2 - t p x y , \\end{align*}"}
+{"id": "2441.png", "formula": "\\begin{align*} \\chi _ + : U _ + \\to \\R & & \\chi _ + ( x , y ) & = x \\\\ \\chi _ - : U _ - \\to \\R & & \\chi _ - ( x , y ) & = \\begin{cases} x & x \\leq 0 \\\\ \\theta _ + ( x ) & x > 0 . \\end{cases} \\end{align*}"}
+{"id": "838.png", "formula": "\\begin{align*} \\textbf { R } = \\rho _ { f } \\hat { \\textbf { G } } ^ { T } \\textbf { P } \\textbf { P } ^ { H } \\hat { \\textbf { G } } ^ { \\ast } \\left ( \\rho _ { f } \\tilde { \\textbf { G } } ^ { T } \\textbf { P } \\textbf { P } ^ { H } \\tilde { \\textbf { G } } ^ { \\ast } + \\sigma _ { w } ^ { 2 } \\textbf { I } _ K \\right ) ^ { - 1 } \\end{align*}"}
+{"id": "5337.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ 2 ^ 2 \\le \\left ( \\frac { 3 3 r ' _ d } { d } \\right ) ^ d \\gamma _ 1 \\| f ^ { = d } \\| _ 2 \\log ^ d \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) . \\end{align*}"}
+{"id": "1448.png", "formula": "\\begin{align*} u _ m = \\phi _ m - h _ m - \\omega _ m , \\ \\| u _ m \\| \\rightarrow 1 . \\end{align*}"}
+{"id": "139.png", "formula": "\\begin{align*} E _ { + } ^ * : = \\lim \\limits _ { t \\to - \\infty } ( d \\varphi ^ t ( x ) ) ^ { T } \\tau _ { x _ 0 \\to \\varphi ^ t ( x ) } E _ u ^ * ( x _ 0 ) \\end{align*}"}
+{"id": "6936.png", "formula": "\\begin{align*} \\frac { \\sum _ { v \\in \\mathcal { L } } f ( v ) ^ 2 } { \\sum _ { n \\in \\mathcal { M } } f ( n ) ^ 2 } = 1 - o ( 1 ) , \\textrm { a s } N \\to \\infty . \\end{align*}"}
+{"id": "6879.png", "formula": "\\begin{align*} \\| d _ { h ^ { G _ N } } - d _ r \\| _ \\infty \\leq \\| d _ { h ^ { G _ N } } - d _ { r _ N } \\| _ \\infty + \\| d _ { r _ N } - d _ r \\| _ \\infty . \\end{align*}"}
+{"id": "203.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c ) = 1 \\\\ a , b , c \\geq 1 } } \\left ( \\frac { 1 } { 1 - x ^ a y ^ b z ^ c } \\right ) ^ { \\frac { c ^ 2 } { a b ^ 2 } } = \\exp \\left \\{ L i _ 1 ( x ) L i _ 2 ( y ) L i _ { - 2 } ( z ) \\right \\} \\end{align*}"}
+{"id": "2418.png", "formula": "\\begin{align*} X = \\prod _ { j = 1 } ^ { \\infty } [ a _ { j } , b _ { j } ] = \\bigl \\{ \\mathcal { N } = ( n _ { j } ) _ { j \\ge 1 } : a _ { j } \\le n _ { j } \\le b _ { j } \\bigr \\} . \\end{align*}"}
+{"id": "179.png", "formula": "\\begin{align*} L i _ 2 \\left ( - \\frac { 1 } { 8 } \\right ) + L i _ 2 \\left ( \\frac { 1 } { 9 } \\right ) = - \\frac { 1 } { 2 } \\left ( \\log \\left ( \\frac { 9 } { 8 } \\right ) \\right ) ^ 2 . \\end{align*}"}
+{"id": "7684.png", "formula": "\\begin{align*} \\varGamma _ L = \\{ g \\in \\mathrm { S O } ^ + ( L ) \\mid g \\lambda - \\lambda \\in L \\hbox { f o r a l l } \\lambda \\in L ' \\} . \\end{align*}"}
+{"id": "6313.png", "formula": "\\begin{align*} \\det \\big ( d _ z \\phi _ t ( p , \\cdot ) \\big ) = \\sum _ { i = 0 } ^ N a _ i ( z ) t ^ i + t ^ { N + 1 } R _ N ( t , z ) , \\forall \\ , z \\in V , \\end{align*}"}
+{"id": "7650.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x } M ( c , x , 1 ) & = ( 4 - c ^ 2 ) ^ 2 [ c ^ 2 x ( 4 - 3 9 x + 8 x ^ 2 ) - 4 x ( 2 8 - 2 7 x + 8 x ^ 2 ) \\\\ & \\quad - 8 c ( - 1 - 2 x + 3 x ^ 2 + 4 x ^ 3 ) ] = 0 . \\end{align*}"}
+{"id": "1525.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\mathrm { d e g } ( h ) } h _ i t ^ i & = \\sum _ { i = 0 } ^ { d _ 0 } h _ i t ^ i = \\sum _ { i = 0 } ^ { p d + p - 1 } h _ i t ^ i \\ , h _ i = 0 i > \\mathrm { d e g } ( h ) \\\\ & = \\sum _ { \\theta = 0 } ^ { d } \\left ( h _ { p \\theta } t ^ { p \\theta } + h _ { p \\theta + 1 } t ^ { p \\theta + 1 } + \\cdots + h _ { p \\theta + p - 1 } t ^ { p \\theta + p - 1 } \\right ) \\\\ & = \\sum _ { \\ell = 0 } ^ { p - 1 } \\sum _ { \\theta = 0 } ^ { d } h _ { p \\theta + \\ell } t ^ { p \\theta + \\ell } . \\end{align*}"}
+{"id": "267.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - y ^ m z ^ n } \\right ) ^ { \\frac { m ^ 4 } { n ^ 5 } } \\end{align*}"}
+{"id": "7221.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 } \\frac { 1 } { \\rho } F ( u , B _ { \\rho } ( x ) ) = 0 , \\end{align*}"}
+{"id": "2139.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\frac { d } { d t } x _ \\pm ( X _ \\pm , t ) = u _ \\pm ( x _ \\pm ( X , t ) , t ) , t > 0 , \\\\ x _ \\pm ( X _ \\pm , 0 ) = X , \\end{cases} \\end{align*}"}
+{"id": "945.png", "formula": "\\begin{align*} Z _ 3 ( H ) = \\overline { \\{ ( p , q ) \\in X \\times X \\ , | \\ , \\exists \\ , C \\in | H | \\mbox { s m o o t h s . t . } p , q \\in C , \\ , 3 [ p - q ] = 0 \\in J C \\} } \\subseteq X \\times X \\end{align*}"}
+{"id": "2524.png", "formula": "\\begin{align*} s = \\sum _ { i = 1 } ^ n x _ { i , i ^ * } , \\end{align*}"}
+{"id": "745.png", "formula": "\\begin{align*} \\hat F = \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\tanh \\left ( \\frac { 2 \\ , | k | \\ , \\pi / N _ S - \\xi _ 0 } { d } \\right ) \\ , . \\end{align*}"}
+{"id": "7704.png", "formula": "\\begin{align*} L _ \\R = \\Bigl ( \\bigoplus _ { j = 0 } ^ r \\langle z _ j , z _ j ' \\rangle \\Bigr ) \\oplus ( L _ { r + 1 } ) _ \\R \\end{align*}"}
+{"id": "2947.png", "formula": "\\begin{align*} \\Delta ^ \\delta ( \\tau ) = \\prod _ { D \\mid N } \\Delta ( D \\tau ) ^ { n _ D } . \\end{align*}"}
+{"id": "3189.png", "formula": "\\begin{align*} M _ a ( \\cdot ) : = \\begin{cases} \\frac { 1 } { a ^ d } \\sum _ { 0 \\leq i _ 1 < a } \\cdots \\sum _ { 0 \\leq i _ d < a } T _ 1 ^ { i _ 1 } \\cdots T _ d ^ { i _ d } ( \\cdot ) & G = \\Z _ + ^ d , ~ a \\in \\N , \\\\ \\\\ \\frac { 1 } { a ^ d } \\int _ { Q _ a } T _ t ( \\cdot ) d t & G = \\R _ + ^ d , ~ a \\in \\R _ + . \\end{cases} \\end{align*}"}
+{"id": "8455.png", "formula": "\\begin{align*} C _ { p } ^ { ( 1 ) } : = \\lim _ { n \\rightarrow \\infty } \\left \\{ \\sum _ { j = 1 } ^ { n - 1 } \\frac { p ^ { 2 } + \\lambda _ { j } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { j } ^ { 2 } } \\cdot \\frac { 1 } { \\lambda _ { j } } - \\log \\left ( \\lambda _ { n } \\right ) \\right \\} . \\end{align*}"}
+{"id": "5630.png", "formula": "\\begin{align*} \\aligned P _ \\nu ( u , v ) = & | \\nabla u | ^ 2 _ 2 + | \\nabla v | ^ 2 _ 2 - \\int _ { \\mathbb { R } ^ N } \\bigl [ ( I _ \\mu * | u | ^ { 2 ^ * _ \\mu } ) | u | ^ { 2 ^ * _ \\mu } + ( I _ \\mu * | v | ^ { 2 ^ * _ \\mu } ) | v | ^ { 2 ^ * _ \\mu } \\bigr ] \\\\ & - \\nu ( \\gamma _ p + \\gamma _ q ) \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | u | ^ { p } ) | v | ^ { q } = 0 . \\endaligned \\end{align*}"}
+{"id": "2675.png", "formula": "\\begin{align*} s _ { \\alpha } ( x ) & = y _ j + y ' _ j ( x - x _ j ) \\\\ & \\quad + \\left ( \\dfrac { 3 ( y _ { j + 1 } - y _ j ) - h _ { j + 1 } ( 2 y ' _ j + y ' _ { j + 1 } ) } { h _ { j + 1 } ^ 2 } + O ( \\alpha ^ 2 ) \\right ) ( x - x _ j ) ^ 2 \\\\ & \\qquad + \\left ( \\dfrac { h _ { j + 1 } ( y ' _ j + y ' _ { j + 1 } ) - 2 ( y _ { j + 1 } - y _ j ) } { h _ { j + 1 } ^ 3 } + O ( \\alpha ^ 2 ) \\right ) ( x - x _ j ) ^ 3 + O ( \\alpha ^ 2 ) \\\\ & = \\sigma ( x ) + O ( \\alpha ^ 2 h _ { \\max } ^ 2 ) . \\end{align*}"}
+{"id": "5749.png", "formula": "\\begin{align*} \\left | \\begin{matrix} x & y \\\\ z & w \\end{matrix} \\right | = x w - y z = x ^ 3 - \\frac { a } { p ' x } \\ , p ' \\left ( \\frac { a } { p ' x } \\right ) ^ 2 = x ^ 3 - \\frac { a ^ 3 } { p '^ 2 x ^ 3 } = x ^ 3 - x ^ 3 = 0 , \\end{align*}"}
+{"id": "7937.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) u = u \\xi + c _ { e _ { ( \\xi , 0 ) } + 2 e _ { ( \\xi , e _ 0 ) } } u . \\end{align*}"}
+{"id": "3944.png", "formula": "\\begin{align*} \\sup _ { \\pi \\in \\mathcal { G } _ { \\lambda } } \\int _ { \\mathcal { V } \\times \\mathcal { S } } \\phi _ \\lambda \\ , d \\pi = \\sup _ { \\pi \\in \\Gamma \\left ( \\Pi ( \\mu _ 1 , \\mu _ 2 ) , \\phi _ \\lambda \\right ) } \\int _ { \\mathcal { V } \\times \\mathcal { S } } \\phi _ { \\lambda } \\ , d \\pi \\end{align*}"}
+{"id": "6239.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } g _ n \\ , \\alpha ^ { n - 1 } \\ , : = \\ , \\frac { 1 } { \\alpha } \\ , \\exp \\Big ( - \\sum _ { r \\geq 1 } \\frac { ( - \\alpha ) ^ r } { r } \\ , ( E _ r - e _ r ) \\Big ) \\ , . \\end{align*}"}
+{"id": "1740.png", "formula": "\\begin{align*} & \\left | \\frac { \\delta F } { \\delta \\mu } ( \\nu _ t , \\mu _ t , y ) - \\frac { \\sigma ^ 2 } { 2 } \\log \\left ( \\frac { \\mu _ t ( y ) } { \\rho ( y ) } \\right ) + \\frac { \\sigma ^ 2 } { 2 } \\operatorname { D _ { K L } } ( \\mu _ t | \\rho ) \\right | \\\\ & \\leq 3 C _ { \\mu } + \\frac { \\sigma ^ 2 } { 2 } \\left ( \\max \\{ | \\log r _ { 1 , \\mu } | , \\log R _ { 1 , \\mu } \\} + 2 \\log R _ { \\mu } \\right ) = : C _ { V , \\mu } . \\end{align*}"}
+{"id": "9169.png", "formula": "\\begin{align*} \\varphi ^ { 2 } & = x ^ { 1 } \\sin \\left ( \\tfrac { \\zeta _ { [ - 1 ] } ^ { 1 } + x ^ { 3 } } { 2 } \\right ) - x ^ { 2 } \\cos \\left ( \\tfrac { \\zeta _ { [ - 1 ] } ^ { 1 } + x ^ { 3 } } { 2 } \\right ) \\\\ \\delta ( \\varphi ^ { 2 } ) & = x ^ { 1 } \\sin ( \\bar { u } ^ { 2 } ) - x ^ { 2 } \\cos ( \\bar { u } ^ { 2 } ) \\ , , \\end{align*}"}
+{"id": "6610.png", "formula": "\\begin{align*} d a = \\rho ( a ) ( d ) \\end{align*}"}
+{"id": "9010.png", "formula": "\\begin{align*} R _ { i j k t , i } = R ^ \\varphi _ { j t , k } - R ^ \\varphi _ { j k , t } + \\alpha ( \\varphi ^ a _ { j k } \\varphi ^ a _ t - \\varphi ^ a _ { j t } \\varphi ^ a _ k ) \\ , . \\end{align*}"}
+{"id": "2162.png", "formula": "\\begin{align*} v \\left ( x , t \\right ) = \\widetilde { u } _ { t } \\left ( x , t \\right ) , q \\left ( x , t \\right ) = \\widetilde { m } _ { t } \\left ( x , t \\right ) . \\end{align*}"}
+{"id": "8014.png", "formula": "\\begin{align*} \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} = U \\begin{bmatrix} A & 0 \\\\ 0 & 0 \\end{bmatrix} U ^ * + V \\begin{bmatrix} 0 & 0 \\\\ 0 & B \\end{bmatrix} V ^ * \\end{align*}"}
+{"id": "4296.png", "formula": "\\begin{align*} E _ t + ( u E + u p - u S + q ) _ x = 0 , \\end{align*}"}
+{"id": "9159.png", "formula": "\\begin{align*} u & = \\eta ( \\zeta _ { [ - q _ { 1 } , - 1 ] } , x , y _ { [ 0 , R ] } ^ { d } ) \\ , . \\end{align*}"}
+{"id": "8596.png", "formula": "\\begin{align*} E \\big | S _ { j , + } ( t ) - \\hat { S } _ { j , + } ( t ) \\big | & \\leq \\sum _ { k = 1 } ^ \\infty E \\big | S ^ k _ { j , + } ( t ) - \\hat { S } ^ k _ { j , + } ( t ) \\big | = O \\big ( t e ^ { \\lambda t / 2 } \\big ) , \\end{align*}"}
+{"id": "9047.png", "formula": "\\begin{align*} b _ { I , J } ^ G = \\sum _ { i \\in I } b _ { i , j } \\end{align*}"}
+{"id": "8682.png", "formula": "\\begin{align*} ( \\widetilde { \\Delta } _ { 2 } ) = O ( N ^ { - 2 } p ^ { - 2 } ) , \\end{align*}"}
+{"id": "3855.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\varpi \\in \\Pi ( \\mu _ { 1 3 } , \\mu _ { 2 3 } ) } \\int _ { \\mathcal { V } } f _ { \\theta , \\lambda } \\ , d \\varpi \\right ] , \\end{align*}"}
+{"id": "64.png", "formula": "\\begin{align*} \\Omega _ 1 ( \\mathcal { G } _ 1 ( \\mathbb { Q } ) ) = \\Omega _ 2 ( \\mathcal { G } _ 2 ( \\mathbb { Q } ) ) . \\end{align*}"}
+{"id": "2481.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow T ^ * } G ( t ) = \\lim \\limits _ { t \\rightarrow T ^ * } \\int _ { \\mathbb { R } ^ N } | x | ^ 2 \\left ( a _ 2 | \\phi | ^ 2 + a _ 1 | \\psi | ^ 2 \\right ) d x = 0 . \\end{align*}"}
+{"id": "3025.png", "formula": "\\begin{align*} \\| T + x S \\| _ { ( p , k ) } ^ p + \\| T - x S \\| _ { ( p , k ) } ^ p \\geq 2 \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T ^ * T + x ^ 2 S ^ * S ) \\end{align*}"}
+{"id": "6880.png", "formula": "\\begin{align*} \\P _ { N , r _ N } ( U _ \\varepsilon ^ x ) = \\P _ { N , h _ N } ( U _ \\varepsilon ^ x ) \\ , \\frac { 1 } { \\P _ { N , h _ N } ( U _ \\varepsilon ^ x ) } \\int _ { U _ \\varepsilon ^ x } \\exp \\left ( - \\log \\frac { \\d \\P _ { N , h _ N } } { \\d \\P _ { N , r _ N } } \\right ) \\ , \\d \\P _ { N , h _ N } . \\end{align*}"}
+{"id": "8413.png", "formula": "\\begin{align*} a ^ { s } \\Gamma ( s ) \\ , \\zeta _ { Q } ( s ) & = 2 \\Gamma ( s ) \\zeta ( 2 s ) + 2 k ^ { 1 - 2 s } \\pi ^ { 1 / 2 } \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) \\zeta ( 2 s - 1 ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , + \\ , 8 k ^ { 1 / 2 - s } \\pi ^ { s } \\sum _ { n = 1 } ^ { \\infty } n ^ { s - 1 / 2 } \\sigma _ { 1 - 2 s } ( n ) \\ , \\cos \\left ( n \\pi b / a \\right ) K _ { s - 1 / 2 } \\left ( 2 \\pi k \\ , n \\right ) . \\end{align*}"}
+{"id": "1384.png", "formula": "\\begin{align*} g _ i \\triangleright g _ j & = g _ { ( 1 - \\alpha ) i + \\alpha j } = g _ { j + ( 1 - \\alpha ) ( i - j ) } , \\\\ g _ i ^ { - 1 } \\triangleright g _ j & = g _ { j + ( 1 - \\alpha ^ { - 1 } ) ( i - j ) } \\end{align*}"}
+{"id": "3921.png", "formula": "\\begin{align*} \\pi _ n ( \\cdot ) = \\frac { \\pi \\left ( \\cdot \\cap ( A _ { 1 n } \\cap A _ { 2 n } ) \\right ) } { \\pi ( A _ { 1 n } \\cap A _ { 2 n } ) } , \\end{align*}"}
+{"id": "9154.png", "formula": "\\begin{align*} u = F _ { u } \\circ \\phi ( x , v _ { [ 0 , R - \\kappa ] } ) \\ , , \\end{align*}"}
+{"id": "4089.png", "formula": "\\begin{align*} \\mathcal { I } = \\{ I _ 1 , I _ 2 , \\dots \\} \\end{align*}"}
+{"id": "1326.png", "formula": "\\begin{align*} \\frac { 1 } { T _ i ^ { \\infty } } + \\frac { 1 } { \\theta _ i ^ 2 T ^ * _ i ( u ) } = \\frac { 1 } { T _ i ( u ) } . \\end{align*}"}
+{"id": "9078.png", "formula": "\\begin{align*} \\mathfrak { S } ( f _ k ) + \\overline { \\mathfrak { S } } ( f _ k ) = n + 1 . \\end{align*}"}
+{"id": "8086.png", "formula": "\\begin{align*} \\gamma _ { n } : = \\inf _ { \\mathcal { A } \\in \\mathbb { F } _ n } \\sup _ { u \\in \\mathcal { A } } G ( u ) . \\end{align*}"}
+{"id": "4555.png", "formula": "\\begin{align*} P _ { \\ell , M } = \\Delta _ \\ell ^ { \\ , \\vee } . \\end{align*}"}
+{"id": "7140.png", "formula": "\\begin{align*} \\partial _ { x _ n } u ( \\hat { x } ) _ { \\big | \\{ x _ n = 0 \\} } = \\partial _ { x _ n } \\big ( c \\circ F _ \\gamma \\big ) ( \\hat { x } ) _ { \\big | \\{ x _ n = 0 \\} } = \\nabla c ( x ^ \\prime , \\gamma ( x ^ \\prime ) ) \\cdot ( - \\mathbf { n } ( x ^ \\prime , \\gamma ( x ^ \\prime ) ) ) = 0 , \\end{align*}"}
+{"id": "7694.png", "formula": "\\begin{align*} L _ \\R = W _ \\R \\oplus \\langle z , z ^ \\ast \\rangle , W _ \\R = \\langle z ^ + \\rangle ^ \\perp \\cap \\langle z ^ - \\rangle ^ \\perp . \\end{align*}"}
+{"id": "5553.png", "formula": "\\begin{align*} \\int _ { S _ { y } } D \\left ( \\alpha _ { x , g } \\parallel \\alpha _ { x , e } \\right ) d \\eta ^ { y } ( x ) = \\sup _ { n } \\left \\{ \\int _ { S _ { y } } H _ { q _ { x , g } ^ { n } \\parallel q _ { x , e } ^ { n } } \\left ( \\mathcal { P } _ { x , n } \\right ) d \\eta ^ { y } - \\delta _ { n } \\right \\} . \\end{align*}"}
+{"id": "7668.png", "formula": "\\begin{align*} | \\nabla _ g \\rho | = 1 \\quad \\mbox { a n d } \\quad \\Delta _ g \\rho = ( n - 1 ) \\kappa \\coth ( \\kappa \\rho ) = p s \\coth ( \\kappa \\rho ) . \\end{align*}"}
+{"id": "6965.png", "formula": "\\begin{align*} I _ \\gamma : = \\{ y \\in L \\ | \\ v ( y ) \\ge a a \\in \\gamma \\} . \\end{align*}"}
+{"id": "3022.png", "formula": "\\begin{align*} \\phi ( X E _ { i i } X ^ * \\otimes B ) = W _ X ( E _ { i i } \\otimes \\varphi _ { i , X } ( B ) ) { W } _ X ^ * \\hbox { f o r a l l \\quad } B \\in M _ n , \\end{align*}"}
+{"id": "5297.png", "formula": "\\begin{align*} \\mu _ { p } ( h ) \\mu _ { p } ( g ) \\leq e \\mu _ { p } ( h ) \\mu _ { p } ( g ) \\sum _ { d = 1 } ^ n \\left ( \\frac { p } { 1 - p } \\ , 4 0 0 \\ , r \\ , \\sqrt { q \\cdot A } \\right ) ^ d . \\end{align*}"}
+{"id": "5960.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\sqrt { c ^ 2 + d ^ 2 } } \\tfrac { d \\det g } { \\sqrt { c ^ 2 + d ^ 2 } } + ( \\tfrac { b d + a c } { \\sqrt { c ^ 2 + d ^ 2 } } ) \\tfrac { c } { \\sqrt { c ^ 2 + d ^ 2 } } & = \\tfrac { d \\det g } { \\sqrt { c ^ 2 + d ^ 2 } } + \\tfrac { c b d + a c ^ 2 } { c ^ 2 + d ^ 2 } \\\\ & = \\tfrac { d a d - d b c + c b d + a c ^ 2 } { c ^ 2 + d ^ 2 } = a ; \\end{align*}"}
+{"id": "8479.png", "formula": "\\begin{align*} \\eta _ { p } ( s ) = \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } \\ , \\frac { 1 } { s - 1 } + C _ { p } ^ { ( 2 ) } - \\frac { 2 e ^ { 2 \\pi p } Q _ { 2 \\pi p } ( 0 ) } { 1 + \\frac { 1 } { \\pi p } } + O \\left ( s - 1 \\right ) , \\end{align*}"}
+{"id": "3637.png", "formula": "\\begin{align*} ( \\{ 9 \\} ^ { I + 1 } , a _ { I + 1 } , \\ldots , 7 , \\{ 9 \\} ^ { I } , 4 ) _ { 1 0 } \\ = \\ 2 ( 4 , \\{ 9 \\} ^ { I } , 7 , \\ldots , a _ { I + 1 } , \\{ 9 \\} ^ { I } , 7 ) _ { 1 0 } \\end{align*}"}
+{"id": "2138.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 3 } a _ { 1 i } a _ { 3 i } = \\sum _ { i = 1 } ^ { 3 } a _ { 2 i } a _ { 3 i } = 0 , \\sum _ { i = 1 } ^ { 3 } a _ { 3 i } ^ 2 = | \\mathcal { N } | ^ 2 = 1 \\end{align*}"}
+{"id": "6450.png", "formula": "\\begin{align*} W _ { K _ 1 K _ 2 K _ 3 K } ( s ) & = \\frac { 1 } { ( 2 \\pi ) ^ 8 } | \\beta _ h ( s ) | ^ 2 \\beta _ h ( s ) \\overline { \\beta _ { \\sigma } ( s ) } \\int _ { \\mathbb { R } ^ 2 } F _ { K _ 1 K _ 2 K _ 3 K } ( s , x ) \\dd x \\end{align*}"}
+{"id": "275.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( 1 - y ^ m z ^ n \\right ) ^ { \\frac { m ^ 1 } { n ^ 2 } } = \\left ( \\frac { 1 } { 1 - y z } \\right ) ^ { \\frac { y } { 1 - y } } \\exp \\left \\{ - \\frac { y } { ( 1 - y ) ^ 2 } \\left ( L i _ 2 ( z ) - L i _ 2 ( y z ) \\right ) \\right \\} . \\end{align*}"}
+{"id": "6418.png", "formula": "\\begin{align*} \\big [ D \\widetilde { \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 } : D \\tau _ { M _ 1 \\bar { \\otimes } M _ 2 } \\big ] _ t = \\lambda ^ { \\varphi _ 1 } ( t ) \\otimes \\lambda ^ { \\varphi _ 2 } ( t ) , t \\in \\mathbb { R } \\end{align*}"}
+{"id": "860.png", "formula": "\\begin{align*} & M ( \\lambda ) : = \\left \\{ \\left ( \\phi _ { 2 } ( \\lambda ) - \\phi _ { 1 } ( \\lambda ) , V \\right ) \\mid V \\geq 0 \\right \\} ; \\\\ & M _ { \\Phi } ( \\lambda ) : = \\left \\{ \\left ( \\phi _ { 2 } ( \\lambda ) - \\phi _ { 1 } ( \\lambda ) , \\Phi \\left ( \\phi _ { 2 } ( \\lambda ) - \\phi _ { 1 } ( \\lambda ) ; \\lambda , \\alpha \\right ) \\right ) \\mid \\alpha \\geq 0 \\right \\} . \\end{align*}"}
+{"id": "5431.png", "formula": "\\begin{align*} s _ 1 = \\sum _ { j , k = 1 } ^ n R ( e _ j , \\bar e _ j , e _ k , \\bar e _ k ) , s _ 2 = \\sum _ { j , k = 1 } ^ n R ( e _ j , e _ k , \\bar e _ k , \\bar e _ j ) . \\end{align*}"}
+{"id": "3926.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\left [ \\langle \\lambda , \\delta \\rangle + \\int _ { \\mathcal { V } } g _ \\lambda \\ , d \\pi \\right ] = \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\int _ { \\mathcal { V } } g _ \\lambda \\ , d \\pi \\right ] . \\end{align*}"}
+{"id": "7618.png", "formula": "\\begin{align*} f ^ { - 1 } ( w ) = w + A _ 2 w ^ 2 + A _ 3 w ^ 3 + A _ 4 w ^ 4 + \\cdots . \\end{align*}"}
+{"id": "7672.png", "formula": "\\begin{align*} u _ \\delta = \\phi ( \\rho ) e ^ { - s \\rho } \\end{align*}"}
+{"id": "5240.png", "formula": "\\begin{align*} \\begin{array} { l l l l } x \\partial & = \\theta , & \\partial x & = \\theta + 1 , \\\\ x ^ 2 \\partial ^ 2 & = \\theta ( \\theta - 1 ) , & \\partial ^ 2 x ^ 2 & = ( \\theta + 1 ) ( \\theta + 2 ) , \\\\ x ^ 3 \\partial ^ 3 & = \\theta ( \\theta - 1 ) ( \\theta - 2 ) , \\quad & \\partial ^ 3 x ^ 3 & = ( \\theta + 1 ) ( \\theta + 2 ) ( \\theta + 3 ) , \\end{array} \\end{align*}"}
+{"id": "3095.png", "formula": "\\begin{align*} k ! S ( n , k ) = \\sum _ { \\ell = 1 } ^ k A _ { n , \\ell - 1 } \\binom { n - \\ell } { k - \\ell } \\end{align*}"}
+{"id": "7059.png", "formula": "\\begin{align*} \\mathcal J = \\mathcal J _ 1 + \\mathcal J _ 2 , \\end{align*}"}
+{"id": "1147.png", "formula": "\\begin{align*} T ( f ^ { 2 } ) = 2 f T ( f ) + 2 B ( A ( f ) , A ( f ) ) \\end{align*}"}
+{"id": "4359.png", "formula": "\\begin{align*} \\nabla d _ { S _ { i } } ^ { p } \\left ( x \\right ) = p \\left \\Vert x - P _ { i } ( x ) \\right \\Vert ^ { p - 2 } \\left ( x - P _ { i } \\left ( x \\right ) \\right ) \\end{align*}"}
+{"id": "3721.png", "formula": "\\begin{align*} \\sigma ^ * u : = \\left ( f _ { \\sigma } : \\sigma ^ { - 1 } U \\to X , \\ , \\sigma _ U ^ { - 1 } M , \\ , \\phi _ { \\sigma } , \\ , \\sigma _ U ^ * ( t ) \\right ) \\end{align*}"}
+{"id": "5703.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ) = c ^ { \\sigma } F + D ^ { k - 1 } ( h ^ { \\sigma } ) + d ^ { \\sigma } g \\end{align*}"}
+{"id": "7046.png", "formula": "\\begin{align*} h _ i \\sum \\limits _ { j = 1 } ^ { s _ { { i _ { } i } } } b _ { i _ { } i j } { \\textbf { X } } ^ { \\lambda _ j } = g _ i ^ { ( i ) } . \\end{align*}"}
+{"id": "4373.png", "formula": "\\begin{align*} \\begin{aligned} G _ 0 & = [ - \\tfrac 1 2 , \\tfrac 1 2 ] \\cap e ^ { - ( n _ \\mathcal L - n _ 0 ) } \\mathbb Z , \\\\ G _ { \\ell } & = \\{ h \\in G _ 0 : S _ { k } ( h ) \\in [ L _ k , U _ k ] k \\leq \\ell \\} . \\end{aligned} \\end{align*}"}
+{"id": "9212.png", "formula": "\\begin{align*} \\circ ( x , d ) = x ( 1 + \\eta _ 1 ) | \\eta _ 1 | \\le 2 ^ { - d } . \\end{align*}"}
+{"id": "7915.png", "formula": "\\begin{align*} z _ \\mathbf { n } = \\tfrac { 1 } { \\mathbf { n } ! } \\partial ^ \\mathbf { n } p ( x ) . \\end{align*}"}
+{"id": "7116.png", "formula": "\\begin{align*} \\frac { 1 } { 2 n } \\int _ { \\mathbb { R } ^ n } \\rho _ \\varepsilon ( | x | ) \\ : x = \\frac { 1 } { 2 n } \\ : \\omega _ n \\int _ { 0 } ^ \\infty \\rho ( r ) r ^ { n - 1 } \\ : r = \\frac { \\omega _ n } { n } \\frac { 1 } { C _ n } = 1 , \\end{align*}"}
+{"id": "3657.png", "formula": "\\begin{align*} \\mp _ s = \\begin{bmatrix} \\mp _ { s , 1 } & \\mp _ { s , 2 } & \\cdots & \\mp _ { s , 2 ^ { t - s } } \\end{bmatrix} . \\end{align*}"}
+{"id": "2980.png", "formula": "\\begin{align*} q _ i ( t _ 1 , \\dots , t _ n ) = \\det \\begin{pmatrix} F ( t _ 1 ) t _ 1 ^ i & F ( t _ 2 ) t _ 2 ^ i & \\dots & F ( t _ n ) t _ n ^ i \\\\ t _ 1 ^ { n - 2 } & t _ 2 ^ { n - 2 } & \\dots & t _ n ^ { n - 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ t _ 1 & t _ 2 & \\dots & t _ n \\\\ 1 & 1 & \\dots & 1 \\end{pmatrix} \\end{align*}"}
+{"id": "854.png", "formula": "\\begin{align*} E _ { \\lambda } ( u ( t ) , u _ { t } ( t ) ) + \\alpha \\int _ { 0 } ^ { t } f ( u _ { t } ( r ) ) u _ { t } ( r ) \\ , d r = E _ { \\lambda } ( u _ { 0 } , v _ { 0 } ) . \\end{align*}"}
+{"id": "2757.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q ) u = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } , u _ { | \\partial \\mathrm { M } } = \\overline { \\varphi } . \\end{align*}"}
+{"id": "6765.png", "formula": "\\begin{align*} f _ I : = \\Delta ^ { \\bar { k } t } f f _ N : = f - f _ I . \\end{align*}"}
+{"id": "7774.png", "formula": "\\begin{align*} x ^ 2 ( x ^ q y - x y ^ q ) + y ^ 2 ( y ^ q z - y z ^ q ) + ( z ^ 2 + k x y ) ( z ^ q x - z x ^ q ) = 0 . \\end{align*}"}
+{"id": "414.png", "formula": "\\begin{align*} U _ t + ( A U ) _ x + A ^ T U _ x + ( B U ) _ y + B ^ T U _ y + C U = 0 , \\end{align*}"}
+{"id": "8730.png", "formula": "\\begin{align*} \\bigg ( \\frac { 1 } { n ( n - 1 ) } \\sum _ { i _ 1 \\neq i _ 2 } X _ { i _ 1 } ^ \\top X _ { i _ 2 } \\bigg ) = \\frac { 2 } { n ( n - 1 ) } ( \\Sigma _ { 1 } ^ { 2 } ) + \\frac { 4 } { n } \\mu _ { 1 } ^ { \\top } \\Sigma _ { 1 } \\mu _ { 1 } = O ( p N ^ { - 1 } ) . \\end{align*}"}
+{"id": "8946.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 1 & 1 \\\\ 1 & r ^ { p _ 1 } & r ^ { p _ 2 } \\\\ 1 & r ^ { 2 p _ 1 } & r ^ { 2 p _ 2 } \\end{bmatrix} \\begin{bmatrix} \\Tilde { \\phi } _ { e } \\\\ C _ { p _ 1 } h ^ { p _ 1 } \\\\ C _ { p _ 2 } h ^ { p _ 2 } \\end{bmatrix} & = \\begin{bmatrix} \\phi _ 1 \\\\ \\phi _ 2 \\\\ \\phi _ 3 \\end{bmatrix} . \\end{align*}"}
+{"id": "4730.png", "formula": "\\begin{align*} ( A '' , I '' ) & = ( A ' , I ' ) \\otimes _ { ( A , I ) } ( A ' , I ' ) \\\\ ( B ' , J ' ) & = ( A ' , I ' ) \\otimes _ { ( A , I ) } ( B , J ) \\\\ ( B '' , J '' ) & = ( A '' , I '' ) \\otimes _ { ( A , I ) } ( B , J ) . \\end{align*}"}
+{"id": "4207.png", "formula": "\\begin{align*} L _ k = \\frac { 1 } { 2 \\pi i } \\log \\left ( \\phi ( \\tau _ k + 0 ) ^ { - 1 } \\phi ( \\tau _ k - 0 ) \\right ) \\end{align*}"}
+{"id": "2739.png", "formula": "\\begin{align*} R _ q ( \\mu ) = ( A _ q - \\mu ) ^ { - 1 } , \\mu \\in \\rho ( A _ q ) . \\end{align*}"}
+{"id": "2220.png", "formula": "\\begin{align*} \\eta _ 1 ( y ) = \\sup _ { t \\ge y } \\frac { \\log t } { t R ( t ) } + C _ 3 ( y _ 0 ) + 2 ( 1 + C _ 3 ( y _ 0 ) R ( y ) ) . \\end{align*}"}
+{"id": "6977.png", "formula": "\\begin{align*} B ( F ) = \\{ b \\in \\{ 1 , \\ldots , D - 1 \\} \\ | \\ \\nu ( a _ { Q b } Q ^ b ) \\in \\delta ^ L \\mbox { f o r e v e r y } Q \\mbox { w i t h l a r g e e n o u g h v a l u e } \\} . \\end{align*}"}
+{"id": "1103.png", "formula": "\\begin{align*} \\varphi ( b ) ( \\varphi ( b ' ) ( x ) ) = 1 _ X ( 1 _ X ( x ) ) = x \\neq 0 . \\end{align*}"}
+{"id": "6684.png", "formula": "\\begin{align*} \\eta _ t ( x , t ) & = v ( x , t ) \\\\ \\eta ( x , 0 ) & = x \\end{align*}"}
+{"id": "7568.png", "formula": "\\begin{align*} \\Xi _ 2 ( x ) = \\sum _ { x ' \\le n \\le x } \\beta _ n \\Omega ( n ) \\end{align*}"}
+{"id": "4769.png", "formula": "\\begin{align*} R _ { [ 1 ] } & = \\begin{pmatrix} 1 & 0 \\\\ 0 & u _ 1 ^ T J _ { 2 | \\alpha _ i | } v _ 1 \\end{pmatrix} = I _ 2 + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "4347.png", "formula": "\\begin{align*} N _ { S } \\left ( x \\right ) : = \\left \\{ x ^ { \\ast } \\in X ^ { \\ast } \\mid \\left \\langle s - x , x ^ { \\ast } \\right \\rangle \\leq 0 , s \\in S \\right \\} , \\end{align*}"}
+{"id": "4128.png", "formula": "\\begin{align*} M = \\pm \\left ( \\left ( { T _ { \\varepsilon _ 1 } ^ { \\vec { v } } } ^ T \\right ) ^ { m _ 1 } \\left ( { T _ { \\varepsilon _ 2 } ^ { \\vec { v } } } ^ T \\right ) ^ { m _ 2 } \\right ) \\end{align*}"}
+{"id": "7128.png", "formula": "\\begin{align*} \\Big | \\int _ { B _ { \\delta } ( x ^ \\prime ) \\times ( 0 , \\delta ) } J _ \\varepsilon ( | x - \\hat { y } | ) \\partial _ { x _ n } c ( x ) ( x _ n + y _ n ) y \\Big | & \\leq \\int _ 0 ^ 1 K | \\partial _ { x _ n } ^ 2 c ( x ^ \\prime , t x _ n ) | t \\\\ & = K \\frac { 1 } { x _ n } \\int _ 0 ^ { x _ n } | \\partial _ { x _ n } ^ 2 c ( x ^ \\prime , z _ n ) | z _ n , \\end{align*}"}
+{"id": "2929.png", "formula": "\\begin{align*} \\Theta ( u , \\tau ) = q ^ { 1 / 1 2 } ( t ^ { 1 / 2 } - t ^ { - 1 / 2 } ) \\prod _ { n > 0 } ( 1 - q ^ n t ) ( 1 - q ^ n t ^ { - 1 } ) \\end{align*}"}
+{"id": "4686.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { R } _ { \\beta _ 0 , \\beta _ 1 } ( z ) : = & ( \\mathcal { A } _ 0 - z I ) ^ { - 1 } - \\left ( I - z \\mathcal { A } _ 0 ^ { - 1 } \\right ) ^ { - 1 } \\Pi ( \\overline { \\beta _ 0 + \\beta _ 1 M ( z ) } ) ^ { - 1 } \\beta _ 1 \\bigl ( \\Pi ^ * \\left ( I - z \\mathcal { A } _ 0 ^ { - 1 } \\right ) ^ { - 1 } \\bigr ) \\\\ [ 0 . 3 e m ] = & ( \\mathcal { A } _ 0 - z I ) ^ { - 1 } - S ( z ) ( \\overline { \\beta _ 0 + \\beta _ 1 M ( z ) } ) ^ { - 1 } \\beta _ 1 S ^ * ( \\overline { z } ) . \\end{aligned} \\end{align*}"}
+{"id": "5840.png", "formula": "\\begin{align*} T ( x _ 0 ) = \\left \\{ \\begin{array} { l l } \\sup \\ , \\{ t \\leqslant \\tau ^ { x _ 0 } _ { a + H } , \\ , \\pi _ 2 ( X ^ { x _ 0 } _ t ) = \\pi _ 2 ( x _ 0 ' ) \\} & \\\\ \\infty & \\end{array} \\right . \\\\ T ( x _ 0 ' ) = \\left \\{ \\begin{array} { l l } \\sup \\ , \\{ t \\leqslant \\tau ^ { x _ 0 ' } _ { H } , \\ , \\pi _ 2 ( X ^ { x _ 0 ' } _ t ) = \\pi _ 2 ( x _ 0 ) \\} & \\\\ \\infty & \\end{array} \\right . \\end{align*}"}
+{"id": "6909.png", "formula": "\\begin{align*} \\mathcal { M } = \\left \\{ m \\in \\N : \\begin{matrix} m \\textrm { i s s q u a r e - f r e e , a l l p r i m e f a c t o r s o f } m \\textrm { a r e i n } P \\\\ \\textrm { a n d } m \\textrm { h a s a t m o s t } \\frac { a \\log { N } } { | \\log ( 2 \\sigma - 1 ) | } \\textrm { p r i m e f a c t o r s } \\end{matrix} \\right \\} . \\end{align*}"}
+{"id": "1091.png", "formula": "\\begin{align*} l ' ( \\varphi ( b ) , x ) = l ( b , x ) , r ' ( x , \\varphi ( b ) ) = r ( x , b ) , \\end{align*}"}
+{"id": "919.png", "formula": "\\begin{align*} P ^ { ( \\alpha , \\beta ) } _ k ( x ) = \\frac { ( \\alpha + 1 ) _ k } { k ! } { } _ 2 F _ 1 \\left ( \\begin{matrix} - k , \\ , k + \\alpha + \\beta + 1 \\\\ \\alpha + 1 \\end{matrix} \\ , ; \\ , \\frac { 1 - x } { 2 } \\right ) . \\end{align*}"}
+{"id": "8281.png", "formula": "\\begin{align*} V _ A ( \\beta ) ^ { n _ 1 } V _ A ( 2 \\beta ) ^ { n _ 2 } V _ A ( 0 ) ^ { 2 k - n _ 1 - n _ 2 } = e ^ { - \\i N \\frac { \\beta } { 2 } n _ 1 } e ^ { - \\i N \\beta n _ 2 } \\mathcal { Z } _ A ( 0 ) ^ k \\overline { \\mathcal { Z } _ A ( 0 ) } ^ { k - n _ 1 - n _ 2 } \\overline { \\mathcal { Z } _ A ( \\beta ) } ^ { n _ 1 } \\overline { \\mathcal { Z } _ A ( 2 \\beta ) } ^ { n _ 2 } . \\end{align*}"}
+{"id": "6171.png", "formula": "\\begin{align*} \\rho _ F ( a ) \\rho _ F ( a c d ) + \\rho _ F ( c ) \\rho _ F ( d ) & = \\rho _ F ( a ) \\rho _ F ( a c d ) + \\rho _ F ( c ) \\rho _ F ( d ) + \\rho _ F ( a c ) \\rho _ F ( a d ) \\\\ & = \\rho _ F ( a ) \\rho _ F ( a c ) + \\rho _ F ( a ) \\rho _ F ( d ) + \\rho _ F ( c ) \\rho _ F ( d ) + \\rho _ F ( a c ) \\rho _ F ( a d ) \\\\ & = [ \\rho _ F ( a ) \\rho _ F ( a c ) + \\rho _ F ( a c ) \\rho _ F ( a d ) ] + [ \\rho _ F ( a ) \\rho _ F ( d ) + \\rho _ F ( c ) \\rho _ F ( d ) ] \\\\ & = \\rho _ F ( d ) \\rho _ F ( a c ) + \\rho _ F ( d ) \\rho _ F ( a c ) = 0 , \\end{align*}"}
+{"id": "7261.png", "formula": "\\begin{align*} D _ { L , 2 } = O ( 2 ^ { L / p } L ^ { 1 / 2 - 1 / p } ) \\end{align*}"}
+{"id": "5217.png", "formula": "\\begin{align*} \\phi ( z ) \\left ( 1 - \\rho + \\rho \\zeta _ { k } ( g ^ { - 1 } z ) \\right ) , \\end{align*}"}
+{"id": "4694.png", "formula": "\\begin{align*} \\left \\lVert \\widetilde { \\Pi } _ { \\chi , k } \\right \\rVert _ { H ^ { 1 / 2 } ( \\Gamma ; \\C ^ 3 ) \\to H ^ 1 ( Y _ { \\rm s t i f f } ; \\C ^ 3 ) } \\leq C | \\chi | ^ k , k = 1 , \\dots , n , \\left \\lVert \\widetilde { \\Pi } _ { \\chi , n } ^ { \\rm e r r o r } \\right \\rVert _ { H ^ { 1 / 2 } ( \\Gamma ; \\C ^ 3 ) \\to H ^ 1 ( Y _ { \\rm s t i f f } ; \\C ^ 3 ) } \\leq C | \\chi | ^ { n + 1 } , \\end{align*}"}
+{"id": "7192.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 \\left ( \\Omega \\cap \\left ( \\overline { J _ u } \\setminus J _ u \\right ) \\right ) = 0 . \\end{align*}"}
+{"id": "1859.png", "formula": "\\begin{align*} t \\deg \\phi ( x ) \\leq \\deg h ( x ) = \\deg h _ t ( x ) + t \\deg \\phi ( x ) \\leq ( t + 1 ) \\deg \\phi ( x ) - 1 . \\end{align*}"}
+{"id": "3283.png", "formula": "\\begin{align*} { \\rm I m } ( e ^ { i \\pi ( 1 - \\beta _ 1 ) } \\kappa ( t ) ) = 0 { \\rm \\ f o r \\ } t \\le 0 \\end{align*}"}
+{"id": "3010.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\lambda _ i ( A ) = \\max \\limits _ { U ^ * U = I _ k } \\mathrm { t r } ( U ^ * A U ) \\hbox { a n d } \\sum _ { i = 1 } ^ k \\lambda _ { n - i + 1 } ( A ) = \\min \\limits _ { U ^ * U = I _ k } \\mathrm { t r } ( U ^ * A U ) , \\end{align*}"}
+{"id": "1852.png", "formula": "\\begin{align*} - 3 F ( 2 \\pi / 3 ) + \\sum _ { i = 1 } ^ { 3 } F ( \\theta _ { i } ) \\leq - \\frac { 1 } { \\pi ^ { 2 } } ( . 7 2 8 ) \\lambda _ { 1 , n } ^ { r , s } . \\leq - \\frac { 1 } { \\pi ^ { 2 } } ( . 7 2 8 ) \\frac { 4 8 } { 7 } \\lambda _ { 1 , n } ^ { r , s } \\sum _ { i = 1 } ^ { 3 } \\Big ( \\frac { \\theta _ { i } } { 2 \\pi } - 1 / 3 \\Big ) ^ { 2 } . \\end{align*}"}
+{"id": "5851.png", "formula": "\\begin{align*} i _ - \\chi _ { I } ( H _ \\omega ) i _ + = \\int _ { I } P _ \\omega ( E ) d \\mu _ \\omega ( E ) \\end{align*}"}
+{"id": "4222.png", "formula": "\\begin{align*} A _ n & = P _ n + P _ n L _ 1 P _ n + W _ n L _ 2 W _ n + C _ n \\end{align*}"}
+{"id": "6386.png", "formula": "\\begin{align*} E : Y ^ 2 = X ^ 3 - 1 1 2 5 . \\end{align*}"}
+{"id": "6479.png", "formula": "\\begin{align*} [ P : Q ] & : = \\mathrm { I n d } ( Q P | _ { R ( P ) } : R ( P ) \\to R ( Q ) ) \\\\ & = \\dim ( N ( Q ) \\cap R ( P ) ) - \\dim ( R ( Q ) \\cap N ( P ) ) \\end{align*}"}
+{"id": "2680.png", "formula": "\\begin{align*} f _ 1 ( x _ 1 , \\dots , x _ m ) = \\dots = f _ p ( x _ 1 , \\dots , x _ m ) = 0 . \\end{align*}"}
+{"id": "3373.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 1 ^ + } \\limsup _ { m , n \\to \\infty } \\max _ { P _ m \\leq P _ i \\leq \\lambda P _ { m } } \\left | u _ { i n } - u _ { m n } \\right | = 0 , \\end{align*}"}
+{"id": "3753.png", "formula": "\\begin{align*} \\left \\langle A _ r ^ n P _ 0 B _ s ^ n 1 , 1 \\right \\rangle & = \\sum _ { Q \\in \\Gamma } c _ Q \\int _ Q \\{ h _ { \\theta } \\} _ { \\theta = 1 } ^ { r + s } , \\end{align*}"}
+{"id": "1727.png", "formula": "\\begin{align*} \\frac { \\delta ^ 2 F } { \\delta \\nu \\delta \\mu } ( \\nu , \\mu , y , x ) = \\frac { \\delta ^ 2 F } { \\delta \\mu \\delta \\nu } ( \\nu , \\mu , x , y ) , \\end{align*}"}
+{"id": "8870.png", "formula": "\\begin{align*} P ^ \\top & = [ e _ { i _ 1 } , \\ldots , e _ { i _ k } , e _ { i _ { k + 1 } } , \\ldots , e _ { i _ \\ell } ] \\intertext { a n c o n s i d e r } P @ @ A _ { I ( u ) } P ^ \\top & = \\begin{bmatrix} A _ { I ( v ) } & A _ { I ( v ) I ( w ) } \\\\ A _ { I ( w ) I ( v ) } & A _ { I ( w ) } \\\\ \\end{bmatrix} . \\end{align*}"}
+{"id": "3943.png", "formula": "\\begin{align*} B _ { \\ell n } = \\left \\{ ( ( s _ 1 , s _ 2 ) , ( y _ 1 , y _ 2 , x ) ) \\in \\mathcal { V } \\times \\mathcal { S } : c _ \\ell ( s _ \\ell , ( y _ \\ell , x ) ) < n \\right \\} , \\end{align*}"}
+{"id": "7691.png", "formula": "\\begin{align*} L _ \\Q = ( L _ 1 ) _ \\Q \\oplus \\langle z , z ' \\rangle _ \\Q , \\end{align*}"}
+{"id": "999.png", "formula": "\\begin{align*} \\mu N _ { \\beta } = N _ { \\tilde \\beta } \\prod _ { i = 1 } ^ n { \\mu \\tilde w _ i \\over w _ i } . \\end{align*}"}
+{"id": "2150.png", "formula": "\\begin{align*} d ( \\uparrow _ x ^ y , \\eta ) = d ( \\uparrow _ x ^ y , \\Sigma _ x ( M \\cap \\partial W _ 0 ) ) \\le \\frac { \\beta } { 2 } = \\frac { \\pi } { 2 } - \\frac 1 2 \\Theta ^ M ( x , \\partial W _ 0 ) . \\end{align*}"}
+{"id": "8422.png", "formula": "\\begin{align*} \\eta _ { p } ( s ) : = \\sum _ { k = 1 } ^ { \\infty } \\frac { ( s , 2 \\pi p k ) _ { k } } { k ^ { s } } , \\ , \\ , \\ , \\ , \\ , ( s ) > 1 , \\end{align*}"}
+{"id": "1647.png", "formula": "\\begin{align*} [ { f } , { f } ] ( X , \\xi _ i ) = { f } ^ 2 [ X , \\xi _ i ] - { f } [ { f } X , \\xi _ i ] = f N ^ { \\ , ( 3 ) } _ i ( X ) . \\end{align*}"}
+{"id": "760.png", "formula": "\\begin{align*} N = \\dfrac { - \\rho s ^ { i j } u _ i \\frac { \\partial } { \\partial \\theta _ j } + \\phi ^ 2 \\frac { \\partial } { \\partial r } } { \\phi D } . \\end{align*}"}
+{"id": "3518.png", "formula": "\\begin{align*} \\begin{bmatrix} I \\\\ S ( t ) \\end{bmatrix} , \\begin{bmatrix} \\cos t I & - \\sin t I \\\\ \\sin t I & \\cos t I \\end{bmatrix} \\begin{bmatrix} I \\\\ S ( 0 ) \\end{bmatrix} \\end{align*}"}
+{"id": "1074.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\left \\| e ^ { - t H ^ { \\beta } } g - g \\right \\| _ { L ^ q } = 0 \\end{align*}"}
+{"id": "6378.png", "formula": "\\begin{align*} ( x - 4 r ) ^ 3 + ( x - 3 r ) ^ 3 + ( x - 2 r ) ^ 3 + ( x - r ) ^ 3 + x ^ 3 + ( x + r ) ^ 3 + ( x + 2 r ) ^ 3 + ( x + 3 r ) ^ 3 + ( x + 4 r ) ^ 3 = y ^ p \\end{align*}"}
+{"id": "8045.png", "formula": "\\begin{align*} { \\mathbf { R } } ^ * _ l { \\mathbf { R } } _ l = \\left ( A ^ * _ i A _ j \\right ) _ { \\alpha _ { l } \\le i , j < \\alpha _ { l + 1 } } \\end{align*}"}
+{"id": "4778.png", "formula": "\\begin{align*} S _ { [ 1 ] } \\diamond \\cdots \\diamond S _ { [ j + 1 ] } & = \\left ( S _ { [ 1 ] } \\diamond \\cdots \\diamond S _ { [ j ] } \\right ) \\diamond S _ { [ j + 1 ] } . \\end{align*}"}
+{"id": "8461.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\left \\{ \\frac { 1 } { \\sqrt { \\left ( 2 n - 1 \\right ) ^ { 2 } + x ^ { 2 } } } - \\frac { 1 } { 2 n - 1 } \\right \\} + \\frac { \\gamma } { 2 } + \\frac { \\log \\left ( x \\right ) } { 2 } = \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n } \\ , K _ { 0 } ( \\pi n x ) . \\end{align*}"}
+{"id": "2280.png", "formula": "\\begin{align*} \\widetilde { T } ( f ) ( z ) = - \\frac { 1 } { \\pi } \\iint _ D \\left ( \\frac { f ( \\zeta ) } { \\zeta - z } + \\frac { z \\overline { f ( \\zeta ) } } { 1 - \\overline { \\zeta } z } \\right ) \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "9309.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } a _ { 1 1 } ^ 2 + a _ { 2 1 } a _ { 2 2 } = 0 , \\\\ a _ { 1 1 } a _ { 1 2 } + a _ { 2 2 } ^ 2 = 0 , \\\\ a _ { 1 2 } a _ { 2 2 } ( a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } ) = 0 , \\\\ a _ { 1 1 } a _ { 2 1 } ( a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } ) = 0 , \\\\ ( a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } ) ( a _ { 1 1 } a _ { 2 2 } + a _ { 1 2 } a _ { 2 1 } ) = 0 . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "8227.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\varrho } _ t ( x ) = \\frac { \\kappa } { 2 } \\partial _ { x x } \\varrho _ t ( x ) - \\lambda \\partial _ x \\Delta _ t ( x ) , \\\\ \\\\ \\dot { \\Delta } _ t ( x ) = \\frac { \\kappa } { 2 } \\partial _ { x x } \\Delta _ t ( x ) - \\lambda \\partial _ x \\varrho _ t ( x ) - 2 \\gamma \\Delta _ t ( x ) . \\end{cases} \\end{align*}"}
+{"id": "1241.png", "formula": "\\begin{align*} X _ { \\gamma , 2 } ( x ) = ( \\log 2 + \\log ( \\tfrac 1 2 x ^ { \\frac { 1 } { \\beta } } ) ) x = \\tfrac { 1 } { \\beta } \\log ( x ) x \\forall x \\in ( 0 , 1 ] . \\end{align*}"}
+{"id": "8187.png", "formula": "\\begin{align*} \\underset { t \\to \\infty } { \\lim } \\ : P ^ \\omega \\left ( \\mathcal { R } ( t ) \\cap \\widehat { \\Phi } _ t ^ \\omega = \\emptyset \\ : \\big \\vert \\ : S _ t \\right ) = 0 . \\end{align*}"}
+{"id": "6651.png", "formula": "\\begin{align*} \\exists \\ , \\ , ( - \\Delta ) ^ s u ( x ) = \\mathfrak F ^ { - 1 } \\big ( | \\xi | ^ { 2 s } \\mathfrak F u \\big ) ( x ) . \\end{align*}"}
+{"id": "5585.png", "formula": "\\begin{align*} \\textrm { a n d } s = \\left ( \\begin{array} { c c c } e ^ { - t _ 1 } & 0 & 0 \\\\ 0 & e ^ { - t _ 1 } & 0 \\\\ 0 & 0 & e ^ { - t _ 2 } \\\\ \\end{array} \\right ) , \\textrm { f o r } t _ 1 < t _ 2 . \\end{align*}"}
+{"id": "6160.png", "formula": "\\begin{align*} x \\ast _ { n , n } x & = ( y + z ) \\ast _ { n , n } ( y + z ) = y \\ast _ { n , n } y + y \\ast _ { n , n } z + z \\ast _ { n , n } y + z \\ast _ { n , n } z \\\\ & = y \\ast _ { n , n } y + z \\ast _ { n , n } z = \\top _ n \\ast _ { n , n } y + \\top _ n \\ast _ { n , n } z \\\\ & = \\top _ n \\ast _ { n , n } ( y + z ) = \\top _ n \\ast _ { n , n } x \\end{align*}"}
+{"id": "4153.png", "formula": "\\begin{align*} \\vec { v } _ { s , t , f , 0 } = A _ { 2 , f , t } \\overline { A _ { 1 , t , s } A _ { 3 , t , s } A _ { 2 , s , t } } \\end{align*}"}
+{"id": "708.png", "formula": "\\begin{align*} { \\rm p v } \\int \\cot \\Big ( \\frac { z ( e ) - z ( e ' ) } { L / \\pi } \\Big ) f ( e ' ) \\d e ' = \\int \\cot \\Big ( \\frac { z ( e ) - z ( e ' ) } { L / \\pi } \\Big ) \\frac { f ( e ' ) z _ { e } ( e ) - f ( e ) z _ { e } ( e ' ) } { z _ { e } ( e ) } \\d e ' \\ , , \\end{align*}"}
+{"id": "8652.png", "formula": "\\begin{align*} & a = \\int _ 0 ^ { 1 } \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 2 } y ^ { j + 1 } ( 1 - p x ) ^ { - 1 } x ^ { j - 1 } d y d x \\\\ \\end{align*}"}
+{"id": "6583.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { \\hat h } ^ { L o S } ( \\varphi ) = & [ 1 , e ^ { j \\frac { 2 \\pi d } { \\lambda } \\sin \\varphi } , \\cdots , e ^ { j \\frac { 2 \\pi d } { \\lambda } l \\sin \\varphi } , \\\\ & \\cdots , e ^ { j \\frac { 2 \\pi d } { \\lambda } ( L - 1 ) \\sin \\varphi } ] ^ T , \\end{aligned} \\end{align*}"}
+{"id": "8848.png", "formula": "\\begin{align*} d v _ \\lambda ^ h + A v _ \\lambda ^ h = \\bigl ( \\sigma ( u _ \\lambda ) Q h + f ' _ \\lambda ( u _ \\lambda ) v _ \\lambda ^ h \\bigr ) \\ , d t + \\sigma ' ( u _ \\lambda ) v _ \\lambda ^ h B \\ , d W , v _ \\lambda ^ h ( 0 ) = 0 . \\end{align*}"}
+{"id": "5298.png", "formula": "\\begin{align*} \\log ( 1 / \\mu _ p ( g ) ) = q > c / p , \\end{align*}"}
+{"id": "1844.png", "formula": "\\begin{align*} C _ { d } ^ { ( \\frac { n } { 2 } - 1 ) } ( 1 ) = \\frac { ( n - 2 ) _ { d } } { d ! } . \\end{align*}"}
+{"id": "8826.png", "formula": "\\begin{align*} \\operatorname { s u p p } \\chi _ \\lambda = [ - n - 1 , n + 1 ] , \\bigl . \\chi _ n \\bigr \\vert _ { [ - n , n ] } = 1 . \\end{align*}"}
+{"id": "3405.png", "formula": "\\begin{align*} E _ \\alpha = \\inf \\Big \\{ J ( u ) = \\frac 1 2 \\int _ { \\R ^ N } | \\nabla u | ^ 2 - \\int _ { \\R ^ N } F ( u ) \\ \\Big | \\ { u \\in \\mathcal M _ \\alpha } \\Big \\} , \\end{align*}"}
+{"id": "4256.png", "formula": "\\begin{align*} Y _ { t } ^ { t , \\xi ; u } = \\sum _ { i = 1 } ^ { N } I _ { A _ { i } } Y _ { t } ^ { i } . \\end{align*}"}
+{"id": "5834.png", "formula": "\\begin{align*} \\begin{array} { c } V _ { H , w } ^ { y , \\Gamma } = \\displaystyle \\frac { 1 } { H } \\ ; \\left ( \\pi _ 1 ( X ^ { y , \\Gamma } _ { \\tau _ { H , w } } ) - \\pi _ 1 ( y ) \\right ) . \\end{array} \\end{align*}"}
+{"id": "2412.png", "formula": "\\begin{align*} \\delta = \\delta _ { j } , \\delta = \\delta _ { j } \\pm 1 , \\dotsc , \\delta = \\delta _ { j } \\pm \\lfloor k / 2 \\rfloor \\mod m , \\delta \\in \\Delta _ { j } . \\end{align*}"}
+{"id": "254.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { m ^ 4 } { n ^ 5 } } = \\sqrt [ 3 ] { \\frac { 1 } { 1 - z } } \\ ; \\exp \\left \\{ \\frac { z ( 7 - 2 z ) } { 1 0 ( 1 - z ) ^ 2 } - \\frac { 1 } { 3 0 } L i _ 3 ( z ) \\right \\} , \\end{align*}"}
+{"id": "5452.png", "formula": "\\begin{align*} & \\quad \\Pr \\left ( > \\theta | \\Phi , \\Phi ( \\mathcal { A } ) > 0 \\right ) \\\\ & \\approx \\sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } \\prod \\limits _ { x _ { i } \\in \\Phi \\backslash \\{ x _ 1 \\} \\cap { \\mathcal { A } } } \\frac { 1 } { \\left ( 1 + \\frac { m \\eta \\theta r _ 1 ^ { \\alpha } } { M r _ i ^ { \\alpha } } \\right ) ^ M } . \\end{align*}"}
+{"id": "592.png", "formula": "\\begin{align*} \\psi ( b _ { k } d ) & = \\langle \\pi ( d ) \\xi , \\pi ( b _ k ) ^ * \\xi \\rangle = \\psi ( b _ { k } ) \\psi ( d ) = \\omega ( b _ k ) \\psi ( d ) \\end{align*}"}
+{"id": "4697.png", "formula": "\\begin{align*} \\left \\langle \\Lambda _ \\chi ^ { \\rm s t i f f } \\vect g , \\vect v \\right \\rangle _ { H ^ { - 1 / 2 } ( \\Gamma ; \\C ^ 3 ) , H ^ { 1 / 2 } ( \\Gamma ; \\C ^ 3 ) } = \\lambda _ \\chi ^ { \\rm s t i f f } ( \\vect g , \\vect v ) , \\forall \\vect g , \\vect h \\in H ^ { 1 / 2 } ( \\Gamma ; \\C ^ 3 ) . \\end{align*}"}
+{"id": "3965.png", "formula": "\\begin{align*} \\widehat { \\gamma } = \\left ( \\eta / \\eta _ 0 \\right ) \\gamma ^ { \\eta _ 0 , \\delta } + \\left ( 1 - \\eta / \\eta _ 0 \\right ) \\widetilde { \\gamma } , \\ \\widetilde { \\gamma } = ( \\widetilde { Y } _ 1 , \\widetilde { Y } _ 2 , \\widetilde { X } ) . \\end{align*}"}
+{"id": "26.png", "formula": "\\begin{align*} K ^ \\circ = \\{ x \\in K ^ \\times \\mid c ( x ) x \\in \\mathbb { Q } ^ \\times \\} . \\end{align*}"}
+{"id": "3090.png", "formula": "\\begin{align*} ( \\lambda C _ \\lambda ) ^ { p - 1 } \\cdot ( \\lambda ( p - 1 ) + n - p ) & = c _ 1 C _ \\mu ^ { k _ 1 } \\cdot ( \\lambda C _ \\lambda ) ^ \\alpha , \\\\ ( \\mu C _ \\mu ) ^ { p - 1 } \\cdot ( \\mu ( p - 1 ) + n - p ) & = c _ 2 C _ \\mu ^ { k _ 2 } \\cdot ( \\lambda C _ \\lambda ) ^ { k _ 3 } . \\end{align*}"}
+{"id": "5562.png", "formula": "\\begin{align*} \\mathcal { P } _ { x , n } = \\left \\{ \\mho ( v ) \\right \\} _ { v \\in S _ { x } ( n ) } , \\end{align*}"}
+{"id": "266.png", "formula": "\\begin{align*} = \\left ( 1 - y z \\right ) ^ { \\frac { y } { 1 - y } } \\exp \\left \\{ \\frac { y ( y ^ 2 + 4 y + 1 ) } { ( 1 - y ) ^ 4 } ( L i _ 4 ( z ) - L i _ 4 ( y z ) ) + \\frac { 3 y ( 1 + y ) } { ( 1 - y ) ^ 3 } L i _ 3 ( y z ) + \\frac { 3 y } { ( 1 - y ) ^ 2 } L i _ 2 ( y z ) \\right \\} ; \\end{align*}"}
+{"id": "4821.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\left | D _ { m , n } \\right | > \\alpha _ { n } \\right ) \\leq \\displaystyle \\frac { \\mathbb { E } \\left | D _ { m , n } \\right | } { \\alpha _ { n } } = & \\frac { \\sqrt { n } \\ , \\mathbb { E } \\left | \\log W _ { n } - \\log W _ { m } \\right | } { \\sigma \\sqrt { n } } \\leq \\displaystyle C \\gamma ^ { m } \\\\ \\leq & \\ , C \\frac { 1 } { n ^ { \\delta / 2 } } \\frac { 1 } { 1 + | x | ^ { 2 + \\delta } } . \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{align*}"}
+{"id": "6144.png", "formula": "\\begin{align*} b _ j : = ( x - y ) ^ 2 \\cdot z ^ j \\cdot w ^ { n - 1 - j } . \\end{align*}"}
+{"id": "7104.png", "formula": "\\begin{align*} \\mathcal { E } _ \\varepsilon ( c ) : = \\frac { 1 } { 4 } \\int _ { \\Omega } \\int _ { \\Omega } J _ \\varepsilon ( x - y ) \\big | c ( x ) - c ( y ) \\big | ^ 2 \\ : y x \\end{align*}"}
+{"id": "1721.png", "formula": "\\begin{align*} \\mu ^ * ( y ) = \\frac { 1 } { Z ' ( \\nu ^ * , \\mu ^ * ) } \\exp { \\left ( \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta \\mu } ( \\nu ^ * , \\mu ^ * , y ) - U ^ { \\rho } ( y ) \\right ) } , \\end{align*}"}
+{"id": "3580.png", "formula": "\\begin{align*} 2 + ( 3 + 2 ) + 1 1 \\ = \\ ( 3 + 4 ) + 1 1 . \\end{align*}"}
+{"id": "3464.png", "formula": "\\begin{align*} [ n ] : = \\frac { q ^ n - q ^ { - n } } { q - q ^ { - 1 } } = q ^ { n - 1 } + q ^ { n - 3 } + \\dots + q ^ { 1 - n } . \\end{align*}"}
+{"id": "2603.png", "formula": "\\begin{align*} \\mathcal { C } _ { K , r } : = \\{ x _ n \\geq K | x ' | \\} \\cap B _ r ( 0 ) \\subset U . \\end{align*}"}
+{"id": "4038.png", "formula": "\\begin{align*} \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( r ) \\lambda _ f ( s ) = \\delta _ { r s } + 2 \\pi i ^ { 2 k } \\sum _ { p | c } \\frac { \\mathcal { S } ( r , s , c ) } { c } \\ , J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { r s } } { c } \\right ) , \\end{align*}"}
+{"id": "7617.png", "formula": "\\begin{align*} H _ { 3 } ( 1 ) ( f ) : = \\begin{vmatrix} a _ { 1 } & a _ { 2 } & a _ { 3 } \\\\ a _ { 2 } & a _ { 3 } & a _ { 4 } \\\\ a _ { 3 } & a _ { 4 } & a _ { 5 } \\end{vmatrix} = 2 a _ 2 a _ 3 a _ 4 - a _ 3 ^ 2 - a _ 4 ^ 2 + a _ 3 a _ 5 - a _ 2 ^ 2 a _ 5 \\end{align*}"}
+{"id": "4517.png", "formula": "\\begin{align*} I _ { \\omega , \\mathbf { c } } ( V ) = 0 , \\ \\ \\mathbf { c } \\cdot \\mathbf { P } ( V ) \\ge 0 . \\end{align*}"}
+{"id": "4825.png", "formula": "\\begin{align*} I _ { 2 } = \\mathbb { P } \\left ( W _ { n } \\geq \\exp \\left \\{ x ^ { 2 } \\right \\} \\right ) \\leq \\exp \\left \\{ - x ^ { 2 } \\right \\} \\mathbb { E } W _ { n } = \\exp \\left \\{ - x ^ { 2 } \\right \\} . \\end{align*}"}
+{"id": "2322.png", "formula": "\\begin{align*} R _ H ( x , y ) = H ( 2 H - 1 ) \\int _ { 0 } ^ { x } \\int _ { 0 } ^ { y } | u - v | ^ { 2 H - 2 } d u d v , \\end{align*}"}
+{"id": "653.png", "formula": "\\begin{align*} & \\{ 0 \\} = U _ 1 \\subset U _ 2 \\subset \\ldots \\subset U _ { g } \\subset U _ { g + 1 } \\subset U _ { - g } \\subset \\ldots \\subset U _ { - 1 } \\subset U _ 0 = \\Gamma \\\\ & U _ { g + 1 } \\subset H ( \\pi ) E _ j ( \\pi , \\lambda ) \\oplus U _ j = \\Gamma - g \\leq j \\leq g + 1 . \\end{align*}"}
+{"id": "7164.png", "formula": "\\begin{align*} W ^ { 1 , p ( \\cdot ) } \\left ( \\Omega , \\R ^ k \\right ) : = \\left \\{ u \\in \\left ( W ^ { 1 , 1 } \\cap L ^ { p ( \\cdot ) } \\right ) \\left ( \\Omega , \\R ^ k \\right ) : \\nabla u \\in L ^ { p ( \\cdot ) } \\left ( \\Omega , \\R ^ { k \\times n } \\right ) \\right \\} . \\end{align*}"}
+{"id": "2436.png", "formula": "\\begin{align*} \\delta y ( x ) = 0 , \\delta = m _ 0 ( x ) + m _ 1 ( x ) \\sigma _ q + . . . + m _ n ( x ) \\sigma _ q ^ n , \\end{align*}"}
+{"id": "7474.png", "formula": "\\begin{align*} \\pi = \\alpha _ 1 \\cdots \\alpha _ k \\beta \\qquad \\qquad \\phi ( \\pi , x ) = \\alpha _ 1 ^ { \\prime } \\cdots \\alpha _ m ^ { \\prime } \\beta ^ { \\prime } \\end{align*}"}
+{"id": "5887.png", "formula": "\\begin{align*} u ( t , x ) : = u _ 0 ( x - c t ) \\ , , c \\in \\R \\ , . \\end{align*}"}
+{"id": "5396.png", "formula": "\\begin{align*} S ( n , \\alpha \\cdot \\chi _ { 0 } ) = S ( n , \\iota _ \\alpha ) \\end{align*}"}
+{"id": "1135.png", "formula": "\\begin{align*} T ( f ^ { 2 } ) = 2 f T ( f ) + 2 B ( A ( f ) , A ( f ) ) \\end{align*}"}
+{"id": "3188.png", "formula": "\\begin{align*} M _ a ( \\mu ) ( x ) \\leq \\frac { \\chi _ d } { ( [ a ] + 1 ) _ d } \\sum _ { j = 0 } ^ { ( [ a ] + 1 ) _ d - 1 } U ^ j ( M _ 1 ( \\mu ) ) ( x ) \\leq \\epsilon \\rho ( x ) x \\in ( e M e ) _ + . \\end{align*}"}
+{"id": "3433.png", "formula": "\\begin{align*} S _ { i j } : = & \\ , \\pounds _ X g _ { i j } = \\nabla _ i X _ j + \\nabla _ j X _ i , \\\\ A _ { i j } : = & \\ , ( d X ) _ { i j } = \\nabla _ i X _ j - \\nabla _ j X _ i . \\end{align*}"}
+{"id": "1183.png", "formula": "\\begin{align*} & M ^ { \\rm O P T } _ { n m } : = b _ { n m } \\times \\underset { \\eqref { e q : O T S _ N P _ F l o w _ S } - \\eqref { e q : O T S _ N P _ s l a c k } \\ , \\cap \\ , \\mathcal { X } _ { n m } } { \\max } ( \\theta _ n - \\theta _ m ) \\\\ & M ^ { \\rm O P T } _ { m n } : = b _ { n m } \\times \\underset { \\eqref { e q : O T S _ N P _ F l o w _ S } - \\eqref { e q : O T S _ N P _ s l a c k } \\ , \\cap \\ , \\mathcal { X } _ { n m } } { \\max } ( \\theta _ m - \\theta _ n ) \\end{align*}"}
+{"id": "5552.png", "formula": "\\begin{align*} \\int _ { S _ { y } } D \\left ( \\alpha _ { x , g } \\parallel \\alpha _ { x , e } \\right ) d \\eta ^ { y } ( x ) = \\lim _ { n } \\int _ { S _ { y } } H _ { q _ { x , g } ^ { n } \\parallel q _ { x , e } ^ { n } } \\left ( \\mathcal { P } _ { x , n } \\right ) d \\eta ^ { y } ( x ) . \\end{align*}"}
+{"id": "2829.png", "formula": "\\begin{align*} a _ { i } ( t ) = a _ { i } ( 0 ) e ^ { \\displaystyle \\int _ { 0 } ^ { t } b _ { i + 1 } ( \\tau ) d \\tau } ; \\ ; a _ i ( 0 ) \\in \\mathbb { C } , \\ , a _ i ( 0 ) \\ne 0 ; \\ ; i = 0 , \\dots , p - 2 ; \\end{align*}"}
+{"id": "2672.png", "formula": "\\begin{align*} \\| \\delta '' \\| _ { L ^ { \\infty } [ a , b ] } = O \\left ( ( \\alpha \\overline { h } ) ^ 2 \\right ) \\max \\left \\{ \\| \\sigma ''' \\| _ { L ^ { \\infty } ( [ a , b ] \\setminus X ) } , \\| \\sigma '' \\| _ { L ^ { \\infty } ( X ) } \\right \\} ; \\alpha \\to 0 . \\end{align*}"}
+{"id": "6129.png", "formula": "\\begin{align*} N D ^ { d } _ { \\lambda } ( \\mathcal { H } \\times J ) = \\left \\{ \\mathcal { K } \\times I \\in N D _ { \\lambda } ^ { d } \\ , : P ^ { d } \\mathcal { K } \\times I \\subset \\mathcal { H } \\times J \\right \\} \\quad ( d = 1 , 2 ) \\end{align*}"}
+{"id": "6027.png", "formula": "\\begin{align*} d f _ { i j } + \\sum _ { k } f _ { k j } \\circ f _ { i k } = 0 \\end{align*}"}
+{"id": "3185.png", "formula": "\\begin{align*} U = \\sum _ { u \\in Z _ + ^ d } a ( u ) S _ 1 ^ { u _ 1 } \\cdots S _ d ^ { u _ d } \\end{align*}"}
+{"id": "7208.png", "formula": "\\begin{align*} \\widetilde { f } _ h ( \\xi ) : = \\gamma _ h \\sigma _ h f \\left ( x _ j , ( \\gamma _ h \\sigma _ h ) ^ { - 1 / p _ h ^ 0 } \\xi \\right ) = \\abs { \\xi } ^ { p ^ 0 _ h } ( \\xi \\in \\R ^ k ) , \\end{align*}"}
+{"id": "177.png", "formula": "\\begin{align*} L i _ 2 \\left ( \\frac { 1 } { 4 } \\right ) + \\frac { 1 } { 3 } L i _ 2 \\left ( \\frac { 1 } { 9 } \\right ) = \\frac { \\pi ^ 2 } { 1 8 } + 2 \\log 2 \\log 3 - 2 ( \\log 2 ) ^ 2 - \\frac { 2 } { 3 } ( \\log 3 ) ^ 2 , \\end{align*}"}
+{"id": "468.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } W ( x ) = 1 , \\ \\ \\lim _ { x \\to 0 ^ + } W ( x ) = \\frac 1 3 . \\end{align*}"}
+{"id": "1794.png", "formula": "\\begin{align*} \\omega ( t ; x , y ) = \\sum _ { i = 1 } ^ N e _ { k _ i } ( x - y ) \\end{align*}"}
+{"id": "2475.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle \\dfrac { d } { d t } G ( t ) = 4 I m \\int _ { \\mathbb { R } ^ N } \\left ( \\dfrac { a _ 2 } { 2 m _ 1 } x \\overline { \\phi } \\nabla \\phi + \\dfrac { a _ 1 } { 2 m _ 2 } x \\overline { \\psi } \\nabla \\psi \\right ) d x , \\end{array} \\right . \\end{align*}"}
+{"id": "1472.png", "formula": "\\begin{align*} \\int _ \\Omega \\Delta ( V + \\phi ) P \\psi ^ h _ { \\mu _ j , \\xi _ j } d x + \\int _ \\Omega f _ \\epsilon ( V + \\phi ) P \\psi ^ h _ { \\mu _ j , \\xi _ j } d x . = 0 , \\end{align*}"}
+{"id": "2684.png", "formula": "\\begin{align*} f ' _ m ( p ) g ( p ) - f ( p ) g ' _ m ( p ) = 0 , \\end{align*}"}
+{"id": "7335.png", "formula": "\\begin{align*} x ^ 3 - y ^ 2 z - z = 0 \\end{align*}"}
+{"id": "6691.png", "formula": "\\begin{align*} \\partial _ 3 v _ 3 = ( \\delta _ { j i } - a _ { j i } ) \\partial _ j v _ i - \\partial _ 1 v _ 1 - \\partial _ 2 v _ 2 , \\end{align*}"}
+{"id": "7324.png", "formula": "\\begin{align*} z _ { i p } = z _ { i p } ^ 0 + \\sum _ { k = 1 } ^ { N _ 3 } u _ { k p } g _ { k i } , i = 1 , 2 , \\dots , n , z _ p ^ 0 = ( z _ { 1 p } ^ 0 , \\dots , z _ { n p } ^ 0 ) \\in S _ p ^ 3 , p \\in { \\cal P } _ 3 , \\end{align*}"}
+{"id": "5113.png", "formula": "\\begin{align*} \\int _ { B _ { 2 s } ( x _ 0 ) } b ^ { i j , k l } v _ { i j } \\eta _ { k l } d x = \\int _ { B _ { 2 s } ( x _ 0 ) } b ^ { i j , k l } w _ { i j } \\eta _ { k l } d x . \\end{align*}"}
+{"id": "8782.png", "formula": "\\begin{align*} & ( 4 u ^ 2 + 2 t u + 2 t + 9 u + 5 ) b _ 3 \\\\ & = ( 4 t u + 4 t + 3 u + 3 ) c _ 3 + 3 ( u + 1 ) b _ 2 + ( u + 1 ) c _ 2 \\end{align*}"}
+{"id": "4875.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s u = V ( x ) u & , \\\\ u = 0 & . \\end{cases} \\end{align*}"}
+{"id": "4170.png", "formula": "\\begin{align*} f \\big ( \\rho ( \\eta _ t ( z ) ) \\big ) & = \\varphi ( f ) ( \\eta _ t ( z ) ) = \\beta _ { t ^ { - 1 } } ( \\varphi ( f ) ) ( z ) = v _ { t ^ { - 1 } } \\varphi ( f ) v _ { t ^ { - 1 } } ^ * = \\varphi ( \\alpha _ { t ^ { - 1 } } ( f ) ) ( z ) \\\\ & = \\alpha _ { t ^ { - 1 } } ( f ) ( \\rho ( z ) ) = f \\big ( \\theta _ t ( \\rho ( z ) ) \\big ) . \\end{align*}"}
+{"id": "1557.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\geq 1 } f _ n n ^ { \\frac { \\lambda - 1 } { 2 } } e ( n z ) , \\end{align*}"}
+{"id": "699.png", "formula": "\\begin{align*} | S ( k ) ( \\widetilde { v } _ 0 \\circ T - \\widetilde v _ 0 ) ( x ) | = | \\widetilde { v } _ 0 ( T ^ { Q _ \\alpha ( k ) } x ) - \\widetilde v _ 0 ( x ) | \\leq \\| v _ 0 \\| _ { \\sup } | x - T ^ { Q _ \\alpha ( k ) } x | . \\end{align*}"}
+{"id": "1894.png", "formula": "\\begin{align*} \\tilde { \\lambda } ( \\zeta ) = \\lim \\limits _ { k \\rightarrow + \\infty } \\langle \\lambda ^ { k } , \\zeta \\rangle \\forall \\ \\zeta \\in \\mathrm { R } ( S ) . \\end{align*}"}
+{"id": "185.png", "formula": "\\begin{align*} L i _ 3 ( z ) + L i _ 3 ( 1 - z ) + L i _ 3 ( 1 - z ^ { - 1 } ) = \\zeta ( 3 ) + \\frac { 1 } { 6 } ( \\log z ) ^ 3 + \\frac { \\pi ^ 2 } { 6 } \\log z - \\frac { 1 } { 2 } ( \\log z ) ^ 2 \\log ( 1 - z ) . \\end{align*}"}
+{"id": "1229.png", "formula": "\\begin{align*} a = \\varliminf _ { m \\to \\infty } \\big ( d ( m S , \\N ^ n \\setminus m S ) \\big ) ^ { 1 / { m } } \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "3061.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac { u ( r ) } { u _ 0 ( r ) } = \\lim _ { r \\to \\infty } \\frac { v ( r ) } { v _ 0 ( r ) } = 1 . \\end{align*}"}
+{"id": "1801.png", "formula": "\\begin{align*} \\sigma = \\sigma _ { H H } + \\sigma _ { P P } + \\sigma _ { H P } + \\sigma _ { P H } \\end{align*}"}
+{"id": "6115.png", "formula": "\\begin{align*} u _ { t } + \\mathcal { L } _ { A } ^ { \\Phi } u = ( - \\Delta ) ^ { \\frac { s } { 2 } } f + g \\quad Q _ { 2 } . \\end{align*}"}
+{"id": "3882.png", "formula": "\\begin{align*} \\mathcal { J } _ { \\mathrm { D } } ( \\delta ) & = \\inf _ { \\substack { \\lambda \\in \\mathbb { R } _ { + } ^ L , \\ \\psi _ \\ell > - \\infty \\\\ ( \\psi _ \\ell ) _ { \\ell \\in [ L ] } \\in \\prod _ { \\ell \\in [ L ] } L ^ 1 ( \\mu _ \\ell ) } } \\Bigg \\{ \\langle \\lambda , \\delta \\rangle + \\sum _ { \\ell = 1 } ^ L \\int \\psi _ \\ell d \\mu _ \\ell : \\sum _ { \\ell = 1 } ^ L \\psi _ \\ell ( s _ \\ell ) \\ge g _ { \\lambda , [ L ] } ( s ) , \\forall s \\Bigg \\} . \\end{align*}"}
+{"id": "6553.png", "formula": "\\begin{align*} E _ m = \\omega _ Q ^ { \\frac { 1 } { \\bar { p } _ 2 } - \\frac { 1 } { \\bar { p } _ 1 } } \\int _ { \\mathbb H ^ n } K ( e , y ) | y | _ h ^ { - Q / p } d y \\| f _ j \\| _ { L _ { | x | _ h } ^ p L _ \\theta ^ { \\bar { p } _ 2 } ( \\mathbb H ^ n ) } . \\end{align*}"}
+{"id": "5717.png", "formula": "\\begin{align*} ( \\xi _ { 2 - k } ( f ) , g ) = \\sum _ { s : } \\sum _ { n < 0 } C _ { f , s } ^ { + } ( n ) C _ { g , s } ( n ) \\end{align*}"}
+{"id": "6947.png", "formula": "\\begin{align*} \\langle \\cdot , \\cdot \\rangle : W ^ { \\otimes d } \\times ( W ^ * ) ^ { \\otimes d } \\to \\C \\ , \\ \\langle v _ 1 \\otimes \\cdots \\otimes v _ d , v _ 1 ^ * \\otimes \\cdots \\otimes v _ d ^ * \\rangle : = \\prod _ { i = 1 } ^ d \\langle v _ { d - i + 1 } , v _ i ^ * \\rangle , \\end{align*}"}
+{"id": "3623.png", "formula": "\\begin{align*} ( b _ m \\cdots 1 ) _ { 1 0 } \\ = \\ ( 2 \\cdots 3 ) _ { 1 0 } - 2 , \\end{align*}"}
+{"id": "5762.png", "formula": "\\begin{align*} \\| T _ { \\alpha } ( \\vec { f } ) \\| _ { L ^ { q ( \\cdot ) } ( \\mathbb { R } ^ { n } ) } & \\le C \\prod _ { j = 1 } ^ { m } \\| f _ { j } \\| _ { L ^ { p _ { j } ( \\cdot ) } ( \\mathbb { R } ^ { n } ) } . \\end{align*}"}
+{"id": "8102.png", "formula": "\\begin{align*} C _ T = \\sum _ { T ' \\le T } X _ { T ' } \\end{align*}"}
+{"id": "5202.png", "formula": "\\begin{align*} d = \\bigcap \\{ e \\in C \\cup D : x _ d \\in e \\} \\qquad d \\neq \\{ x _ d \\} \\ \\Rightarrow \\ d \\notin C . \\end{align*}"}
+{"id": "2421.png", "formula": "\\begin{align*} \\sum _ { n _ { j } \\le x } n _ { j } ^ { - \\sigma } = A \\frac { x ^ { 1 - \\sigma } - 1 } { 1 - \\sigma } + O \\Bigl ( \\frac { \\sigma } { \\sigma - 1 / 2 } \\Bigr ) , \\end{align*}"}
+{"id": "5029.png", "formula": "\\begin{align*} J : = \\langle j _ f \\rangle \\subseteq G _ f ^ d . \\end{align*}"}
+{"id": "333.png", "formula": "\\begin{align*} s _ h ( 4 , 3 ) = - \\frac { 1 0 9 } { 8 } \\zeta ( 7 ) + \\frac { 3 7 } { 2 } \\zeta ( 3 ) \\zeta ( 4 ) - 5 \\zeta ( 2 ) \\zeta ( 5 ) , \\end{align*}"}
+{"id": "5060.png", "formula": "\\begin{align*} v ( x , t ) = \\psi \\left ( x - \\gamma _ { \\mathrm { n l } } ( x , t ) , t \\right ) - \\phi ( x ) , \\end{align*}"}
+{"id": "9114.png", "formula": "\\begin{align*} \\begin{bmatrix} \\partial _ { x } f & \\partial _ { u } f \\\\ \\partial _ { x } g & \\partial _ { u } g \\end{bmatrix} \\end{align*}"}
+{"id": "2523.png", "formula": "\\begin{align*} B \\overline { x } = ( B ' [ x ] ) \\cap _ { 2 n } / P \\end{align*}"}
+{"id": "6280.png", "formula": "\\begin{align*} | f ( \\alpha , x ) - f _ \\xi ( \\alpha , x ) | = & | h ( x ) - f _ \\xi ( \\alpha , x ) | \\\\ = & | h ( x ) - f _ \\xi ( \\gamma , x ) + f _ \\xi ( \\gamma , x ) - f _ \\xi ( \\alpha , x ) | \\\\ \\leq & | h ( x ) - h _ \\xi ( x ) | + | f _ \\xi ( \\gamma , x ) - f _ \\xi ( \\alpha , x ) | \\\\ < & \\frac { \\epsilon } { 2 } + \\frac { \\epsilon } { 2 } = \\epsilon . \\end{align*}"}
+{"id": "97.png", "formula": "\\begin{align*} \\tilde { R } _ h ( z ) f = \\frac { i } { h } \\int _ 0 ^ \\infty e ^ { - i t h ^ { - 1 } \\tilde { P } _ h ( z ) } f d t = \\frac { i } { h } \\int _ 0 ^ \\infty e ^ { - i t h ^ { - 1 } ( - i h X - z - i ( q _ 1 + W ) ) } f d t \\end{align*}"}
+{"id": "9137.png", "formula": "\\begin{align*} \\varphi _ { [ 0 , R ] } = \\Psi ( x , \\varphi _ { c } , \\varphi _ { [ A , R ] } ) \\end{align*}"}
+{"id": "5854.png", "formula": "\\begin{align*} \\mathbb { E } \\left \\{ \\sup \\limits _ { t } \\left \\Vert \\langle X - x \\rangle ^ { b d } e ^ { - i t H _ \\omega } P _ \\omega ( I ) \\delta _ x \\right \\Vert ^ s \\right \\} & \\leq C \\sum \\limits _ k 2 ^ { \\frac { s b d } { 2 } + ( k + 1 ) s b d - k ( p d - s \\nu ) } \\\\ & \\leq C \\sum \\limits _ k \\left ( 2 ^ { s b d - p d + s \\nu } \\right ) ^ k < \\infty . \\end{align*}"}
+{"id": "5571.png", "formula": "\\begin{align*} M = G / Q \\times _ { \\beta } \\left ( \\Omega _ { 0 } \\times _ { \\sigma } M _ { F } \\right ) . \\end{align*}"}
+{"id": "9319.png", "formula": "\\begin{align*} a _ 1 b _ 1 ^ d + a _ 2 b _ 2 ^ d + \\ldots a _ { n - 1 } b _ { n - 1 } ^ d = 0 . \\end{align*}"}
+{"id": "3996.png", "formula": "\\begin{align*} \\underset { x ^ \\prime \\in \\mathbb { R } ^ d } { \\arg \\max } \\left [ \\sum _ { 1 \\leq \\ell \\leq 2 } \\varphi _ \\ell ( x ^ { \\prime } , \\lambda _ { \\mathrm { D } , \\ell } ^ { \\star } ) \\right ] = \\underset { x ^ \\prime \\in \\mathbb { R } ^ d } { \\arg \\max } \\ \\varphi _ \\ell ( x ^ { \\prime } , \\lambda _ { \\mathrm { D } , \\ell } ^ { \\star } ) . \\end{align*}"}
+{"id": "7587.png", "formula": "\\begin{align*} \\frac { z f ^ { \\prime } ( z ) } { f ( z ) } = w ( z ) + \\sqrt { 1 + w ^ 2 ( z ) } , \\end{align*}"}
+{"id": "7514.png", "formula": "\\begin{align*} \\partial _ z q ^ i ( z ) + \\langle Z , \\alpha _ i \\rangle q ^ i ( z ) = \\Big [ L _ i ( z ) { p ^ i ( z ) } \\Big ] ^ { 2 } \\prod _ { j \\neq i , j \\in I _ w \\cup I _ g } p ^ j ( z ) ^ { 2 a _ { j i } } \\prod _ { j \\neq i , j \\in I _ b } p ^ j ( z ) ^ { a _ { j i } } . \\end{align*}"}
+{"id": "3354.png", "formula": "\\begin{align*} U T _ 1 U = \\begin{pmatrix} - 1 & 0 & 0 \\\\ 0 & 0 & - \\frac { 1 } { \\sqrt { 2 } } \\\\ 0 & 0 & 0 \\end{pmatrix} U T _ 2 U = \\begin{pmatrix} 0 & 0 & 0 \\\\ - \\frac { 1 } { 2 \\sqrt { 2 } } & - \\frac { 1 } { 2 } & - \\frac { 1 } { 2 \\sqrt { 2 } } \\\\ \\frac { 1 } { 2 } & - \\frac { 1 } { \\sqrt { 2 } } & \\frac { 1 } { 2 } \\end{pmatrix} . \\end{align*}"}
+{"id": "4657.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\epsilon ^ 2 \\Delta u + V ( x ) u = \\lambda u + u \\log u ^ 2 , \\hbox { i n } \\mathbb { R } ^ N , \\\\ & \\int _ { \\mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } \\epsilon ^ N , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7242.png", "formula": "\\begin{align*} \\Big \\Vert \\sup _ { k \\leq n } | S _ k - T _ k | \\Big \\Vert _ 2 = O ( n ^ { 1 / p } ( \\log n ) ^ { ( 1 / 2 - 1 / p ) } ) \\ , . \\end{align*}"}
+{"id": "2544.png", "formula": "\\begin{align*} \\lambda _ { j _ 2 } = \\lambda _ { j _ 2 ^ * } = \\lambda _ { j _ 3 } = \\lambda _ { j _ 3 ^ * } = \\lambda _ { j _ 1 } = \\lambda _ { j _ 1 ^ * } , \\end{align*}"}
+{"id": "2663.png", "formula": "\\begin{align*} E _ { j - 1 } = \\dfrac { \\alpha h _ { j } \\cosh ( \\alpha h _ { j } ) - \\sinh ( \\alpha h _ { j } ) } { 2 \\alpha ^ 2 h _ { j } ( \\tanh ( \\alpha x _ j ) - \\tanh ( \\alpha x _ { j - 1 } ) - \\alpha h _ j \\tanh ( \\alpha x _ { j - 1 } ) \\tanh ( \\alpha x _ j ) ) } , \\end{align*}"}
+{"id": "8979.png", "formula": "\\begin{align*} \\ , c > 0 \\ , , & \\liminf _ { r \\to \\infty } \\frac { 1 } { r } \\log \\left ( \\int _ { B _ r \\cap \\Omega } w ^ 2 \\psi _ + ^ 2 \\right ) > 0 \\ , ; \\\\ \\ , c = 0 \\ , , & \\liminf _ { r \\to \\infty } \\frac { 1 } { r ^ 2 } \\int _ { B _ r \\cap \\Omega } w ^ 2 \\psi _ + ^ 2 > 0 \\end{align*}"}
+{"id": "3442.png", "formula": "\\begin{align*} \\hat { R } _ { i j } = & \\ , 2 \\hat \\omega _ { i k } \\hat \\omega _ { j } ^ { \\ ; k } + \\lambda \\hat { g } _ { i j } , \\\\ \\hat { \\nabla } ^ i \\hat \\omega _ { i j } = & \\ , 0 , \\\\ \\hat \\omega _ { i j } \\hat \\omega ^ { i j } = & \\ , \\frac { \\alpha ^ 2 } { m } + \\lambda . \\end{align*}"}
+{"id": "8038.png", "formula": "\\begin{align*} \\left \\| \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} \\right \\| _ { \\infty } & = \\left \\| \\begin{bmatrix} F & 0 \\\\ 0 & F \\end{bmatrix} \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} \\begin{bmatrix} F & 0 \\\\ 0 & F \\end{bmatrix} \\right \\| _ { \\infty } \\\\ & = \\left \\| \\begin{bmatrix} F A F & F X F \\\\ F X ^ * F & F B F \\end{bmatrix} \\right \\| _ { \\infty } . \\end{align*}"}
+{"id": "9221.png", "formula": "\\begin{align*} \\texttt { a = A + T u r i n g ( e p s ) } \\end{align*}"}
+{"id": "7214.png", "formula": "\\begin{align*} \\avg { z _ h ^ { ( s ) } } _ { B _ s } = t _ h \\avg { \\widetilde { z } _ h ^ { ( s ) } } _ { B _ s } , \\mbox { s o t h a t } \\lambda _ h ^ { ( s ) } : = \\abs { \\avg { { z } _ h ^ { ( s ) } } _ { B _ s } } = t _ h \\widetilde \\lambda _ h ^ { ( s ) } . \\end{align*}"}
+{"id": "2918.png", "formula": "\\begin{align*} \\gamma = \\begin{bmatrix} a / \\alpha & 0 \\\\ 0 & d \\end{bmatrix} \\begin{bmatrix} 1 & \\alpha b / a \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} \\alpha & 0 \\\\ 0 & 1 \\end{bmatrix} , \\end{align*}"}
+{"id": "9061.png", "formula": "\\begin{align*} \\tilde \\beta _ 0 ( a , b , c ) = ( \\sigma _ 2 \\sigma _ { 1 , 3 } \\sigma _ 2 ) \\sigma _ 2 ^ { a - 1 } \\sigma _ 1 ^ { b - 1 } \\sigma _ 3 ^ { c - 1 } \\quad \\tilde \\beta ( a , b , c ) = ( \\sigma _ 2 \\sigma _ { 1 , 3 } \\sigma _ 2 ) ^ 3 \\sigma _ 2 ^ { a - 1 } \\sigma _ 1 ^ { b + 1 } \\sigma _ 3 ^ { c + 1 } , \\end{align*}"}
+{"id": "2215.png", "formula": "\\begin{align*} \\Phi ( x , y ) & = \\pi ( x ) - \\pi ( y ) + 1 + \\sum _ { y < p \\leq \\sqrt { x } } ( \\pi ( x / p ) - \\pi ( p ) + 1 ) \\\\ & = \\pi ( x ) - M ( x , y ) + \\sum _ { y < p \\le x ^ { 1 / 2 } } \\pi ( x / p ) , \\end{align*}"}
+{"id": "6763.png", "formula": "\\begin{align*} \\underline { d } ( x , Y ) : = \\inf _ { y \\in Y } \\underline { d } ( x , y ) \\underline { d } ^ H ( X , Y ) : = \\max \\{ \\sup _ { x \\in X } \\underline { d } ( x , Y ) , \\sup _ { y \\in Y } \\underline { d } ( y , X ) \\} \\end{align*}"}
+{"id": "7562.png", "formula": "\\begin{align*} \\mathbb { E } ( \\overline { S } _ n ) ^ 2 = c _ n ^ 2 + O ( \\log ^ { - A } n ) . \\end{align*}"}
+{"id": "5784.png", "formula": "\\begin{align*} A _ 1 = A + h , \\end{align*}"}
+{"id": "6394.png", "formula": "\\begin{align*} D \\lambda ^ { p } - E \\mu ^ { 2 p } = F \\nu , \\end{align*}"}
+{"id": "5448.png", "formula": "\\begin{align*} \\kappa = \\frac { M _ 1 M _ 2 - M _ 1 ^ 2 } { M _ 1 ^ 2 - M _ 2 } , \\beta = \\frac { ( 1 - M _ 1 ) ( M _ 2 - M _ 1 ) } { M _ 1 ^ 2 - M _ 2 } , \\end{align*}"}
+{"id": "6902.png", "formula": "\\begin{align*} \\nu ( \\sigma ) = \\begin{cases} ( 1 - \\sigma ) ^ { - 1 } + O ( | \\log { ( 1 - \\sigma ) } | ) , & \\sigma \\to 1 ^ { - } \\\\ ( 1 / \\sqrt { 2 } + o ( 1 ) ) \\sqrt { | \\log { ( 2 \\sigma - 1 ) } | } , & \\sigma \\to 1 / 2 ^ { + } \\end{cases} \\end{align*}"}
+{"id": "8428.png", "formula": "\\begin{align*} \\sigma ( t ) : = \\frac { p + t } { p - t } , \\ , \\ , \\ , \\sigma ^ { \\prime } ( t ) : = \\frac { p ^ { \\prime } + t } { p ^ { \\prime } - t } . \\end{align*}"}
+{"id": "6397.png", "formula": "\\begin{align*} w _ { 2 } ^ { 1 7 } - 2 ^ { 2 8 } \\cdot 3 ^ { 3 0 } \\cdot w _ { 1 } ^ { 3 4 } = 5 \\cdot 3 9 0 6 2 5 ^ 2 . \\end{align*}"}
+{"id": "6820.png", "formula": "\\begin{align*} \\left ( \\tilde { W } _ 1 ^ { \\sharp } ; \\ , \\tilde { \\mathcal { L } } \\cup \\mathcal { L } ' , \\ , \\mathcal { L } '' \\cup L \\right ) = \\left ( \\tilde { W } _ 1 ^ { \\sharp } ; \\ , \\tilde { \\mathcal { L } } \\cup \\mathcal { L } \\cup L \\right ) \\end{align*}"}
+{"id": "7871.png", "formula": "\\begin{align*} \\sigma ( \\hat { A } ) = \\sigma ( ( 1 , 2 , \\ldots , k ) ) = \\sigma _ { | \\{ 1 , 2 \\ldots , k \\} } ( ( 1 , 2 , \\ldots , k ) ) = ( \\sigma ( 1 ) , \\sigma ( 2 ) , \\ldots , \\sigma ( k ) ) . \\end{align*}"}
+{"id": "8483.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n } \\ , \\intop _ { 0 } ^ { \\infty } e ^ { - 2 \\pi n y } \\ , | J _ { 1 } ( 2 \\pi x y ) | \\ , d y \\leq \\frac { D } { \\sqrt { 2 \\pi x } } \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n } \\ , \\intop _ { 0 } ^ { \\infty } \\frac { e ^ { - 2 \\pi n y } } { \\sqrt { y } } \\ , d y \\leq \\frac { D ^ { \\prime } } { \\sqrt { x } } \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { 3 / 2 } } , \\end{align*}"}
+{"id": "1536.png", "formula": "\\begin{align*} X - a Y & = h _ m \\\\ - a X + Y & = h _ { \\lambda _ m } \\end{align*}"}
+{"id": "6498.png", "formula": "\\begin{align*} & C T _ { \\vec { t } } \\left \\{ \\prod _ { i = 0 } ^ { n - 1 } \\left [ \\frac { ( 1 + t _ i ) ^ { x + m } ( t _ i - 1 ) } { t _ i ^ { x + 2 n - 1 } } \\right ] \\prod _ { i < j } ^ { 0 , n - 1 } ( t _ i - t _ j ) ^ 2 ( 1 - t _ i t _ j ) \\right \\} \\\\ & = n ! \\cdot \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ m \\frac { ( x + i - j ) ( x + 2 i + j - 2 ) } { ( x + 2 i - j ) ( i + j - 1 ) } . \\end{align*}"}
+{"id": "8919.png", "formula": "\\begin{align*} \\mathsf { d } _ { \\mathrm { c o m } } ( \\{ X _ { n i } \\} ) = j \\sum _ { n = 1 } ^ { d _ 1 ^ j } \\| X _ { n 1 } \\| _ 1 . \\end{align*}"}
+{"id": "6006.png", "formula": "\\begin{align*} \\theta _ { 3 / 2 } ( \\tfrac { a z + b } { c z + d } , \\epsilon ) = \\lambda ^ { \\pm } ( \\gamma , \\epsilon ) ( \\sqrt { \\det \\gamma ( c z + d ) } ) ^ 3 \\sum _ { n \\in \\Z } n e ^ { i ( \\det \\gamma ) \\epsilon \\pi n ^ 2 z } , \\end{align*}"}
+{"id": "8007.png", "formula": "\\begin{align*} { \\mathrm { T r \\ , } } f \\left ( \\sum _ { i = 1 } ^ m Z ^ * _ i A _ i Z _ i \\right ) \\le { \\mathrm { T r \\ , } } \\sum _ { i = 1 } ^ m Z ^ * _ i f ( A ) _ i Z _ i . \\end{align*}"}
+{"id": "829.png", "formula": "\\begin{gather*} \\sum _ { ( v _ \\nu ) \\in \\mathcal { U } _ \\gamma } \\prod _ { \\nu \\in S } \\min ( 1 , e ^ { v _ \\nu } ) \\\\ \\lesssim \\sum _ { M = 0 } ^ \\infty \\sum _ { k = 1 } ^ { \\# S - 1 } \\begin{pmatrix} \\# S - 1 \\\\ k \\end{pmatrix} e ^ { - M } ( M ^ { k + 1 } + M ^ { \\# S } ) \\\\ \\lesssim \\sum _ { M = 0 } ^ \\infty e ^ { - M } M ^ { \\# S } < \\infty \\end{gather*}"}
+{"id": "3206.png", "formula": "\\begin{align*} \\bar { \\mu } = \\norm { \\cdot } _ 1 - \\lim _ { n \\to \\infty } S _ n ( \\mu ) , \\end{align*}"}
+{"id": "2092.png", "formula": "\\begin{align*} P F ( A ) & = \\{ N _ { d r } - m ( 2 ^ n - 1 ) \\mid r = m ( 2 ^ n - 1 ) - \\{ 1 , 2 , . . . , n - 1 \\} \\} \\\\ & = \\left \\{ m ^ 2 2 ^ { 2 n } - 2 ^ n ( m ^ 2 - m d + m ) + m - m d - d - \\{ 0 , d , 2 d , . . . , ( n - 2 ) d \\} \\right \\} \\\\ & = \\{ F ( A ) , F ( A ) - d , . . . , F ( A ) - ( n - 2 ) d \\} . \\end{align*}"}
+{"id": "4495.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( \\Psi _ { \\omega } ) = \\omega ^ { 2 - \\frac { d } { 2 } } S _ { 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } } ( \\Psi ) . \\end{align*}"}
+{"id": "5733.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } \\mathcal { E C S } _ { 0 0 1 } = \\{ S ( p , q , a , b , c , d ) \\in \\mathcal { E C } : p = a = 0 , c \\ne 0 \\} \\\\ \\mathcal { E C S } _ { 0 1 0 } = \\{ S ( p , q , a , b , c , d ) \\in \\mathcal { E C } : p = c = 0 , a \\ne 0 \\} \\\\ \\mathcal { E C S } _ { 1 0 0 } = \\{ S ( p , q , a , b , c , d ) \\in \\mathcal { E C } : a = c = 0 , p \\ne 0 \\} . \\end{array} \\right . \\end{align*}"}
+{"id": "929.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { h } _ t ( i ) = - \\sum _ { \\zeta \\in \\mathbb { F } _ q } & \\left ( s _ i = \\zeta \\mid \\mathbf { y } _ t , \\dots , \\mathbf { y } _ 0 \\right ) \\log _ 2 \\left ( s _ i = \\zeta \\mid \\mathbf { y } _ t , \\dots , \\mathbf { y } _ 0 \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "5222.png", "formula": "\\begin{align*} \\varphi _ { k , \\mathrm { e s t } } = \\frac { 1 } { \\mu } \\mathrm { u n w r a p } \\left ( \\arg \\left ( \\sum _ { k ' = k - K } ^ { k + K } z _ { k ' } ^ \\mu \\right ) \\right ) , \\end{align*}"}
+{"id": "1210.png", "formula": "\\begin{align*} F : [ 0 , 1 / ( q \\| \\boldsymbol { a } \\| _ { \\infty } ) ) \\to \\R _ + , F ( s ) = ( 1 - q ) \\sum _ { j } \\frac { { a } _ j } { 1 - q s { a } _ j } \\nu ( j ) . \\end{align*}"}
+{"id": "1840.png", "formula": "\\begin{align*} L _ t [ f ] ( h ) & = 2 \\pi \\int _ { \\Z ^ d } | \\hat V ( k ) | ^ 2 \\Big ( \\rho ^ { H } _ t ( h - k , k ) f ( h - k ) \\widetilde f ( h ) - \\rho ^ { H } _ t ( h , k ) f ( h ) \\widetilde f ( h + k ) \\Big ) \\d k . \\end{align*}"}
+{"id": "6924.png", "formula": "\\begin{align*} \\sum _ { m , n \\in \\mathcal { M } ' } r ( m ) r ( n ) \\Phi \\left ( T \\log { \\frac { m } { n } } \\right ) \\le \\sum _ { m \\in \\mathcal { M } ' } r ( m ) ^ 2 + \\sum _ { \\substack { m , n \\in \\mathcal { M } ' \\\\ m \\not = n } } r ( m ) r ( n ) \\Phi \\left ( T \\log { \\frac { m } { n } } \\right ) . \\end{align*}"}
+{"id": "133.png", "formula": "\\begin{align*} X _ 0 \\rho ( x ) = 0 \\Rightarrow X _ 0 ^ 2 \\rho ( x ) < 0 \\end{align*}"}
+{"id": "6894.png", "formula": "\\begin{align*} & \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\P \\left ( N ^ { - 1 } \\max _ { i \\in [ N ] } d _ i \\geq \\beta \\right ) = \\lim _ { N \\to \\infty } \\sup _ { x \\in [ 0 , 1 ] } \\frac { 1 } { N } \\log \\P \\left ( N ^ { - 1 } d _ { i _ x } \\geq \\beta \\right ) \\\\ & = \\sup _ { x \\in [ 0 , 1 ] } \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\P \\left ( N ^ { - 1 } d _ { i _ x } \\geq \\beta \\right ) = - \\inf _ { x \\in [ 0 , 1 ] } J _ r ( x , \\beta ) = - \\widehat { \\psi } _ r ( \\beta ) . \\end{align*}"}
+{"id": "5001.png", "formula": "\\begin{align*} \\Pr ( T _ { r : k { + } 1 } > t ) = \\Pr ( X _ t \\le k \\ , \\vert \\ , X _ 0 = r ) . \\end{align*}"}
+{"id": "1996.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c \\underline u ) ^ n = ( 1 + \\mu _ 1 M ) ^ n f ^ n \\omega ^ n & \\textnormal { i n } & \\Omega , \\\\ \\underline u = 0 & \\textnormal { o n } & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "3711.png", "formula": "\\begin{align*} \\| x \\| = \\max \\{ | x _ 1 | , \\ldots , | x _ k | \\} . \\end{align*}"}
+{"id": "4134.png", "formula": "\\begin{align*} \\mathcal { I } _ { A J P A } = \\{ I _ { 1 , j , k } , I _ { 2 , j , k } , I _ { 3 , j , k } \\colon j , k \\in \\N _ 3 \\} \\end{align*}"}
+{"id": "746.png", "formula": "\\begin{align*} \\pi \\cot ( \\pi x ) = \\lim _ { N \\to \\infty } \\sum _ { k = - N } ^ N \\frac 1 { x + n } = \\frac 1 x + \\lim _ { N \\to \\infty } \\sum _ { k = 1 } ^ N \\frac { 2 x } { x ^ 2 - n ^ 2 } \\end{align*}"}
+{"id": "3585.png", "formula": "\\begin{align*} v ( p ^ \\alpha ) \\ : = \\ p + \\iota ( \\alpha ) \\end{align*}"}
+{"id": "4014.png", "formula": "\\begin{align*} \\mathbb { M } _ 2 = ( F \\slash R ) \\slash ( F \\slash R ) _ { \\mathrm { t o r } } , \\end{align*}"}
+{"id": "4242.png", "formula": "\\begin{align*} \\Theta _ m & = ( 2 n + 1 ) X _ 2 + ( 2 n + 1 ) 2 n X _ 1 + 2 n Y _ 2 + ( 2 n + 1 ) Y _ 1 \\\\ \\Theta _ b & = X _ 2 + 2 n X _ 1 + Y _ 2 \\\\ \\Psi _ r & = ( 2 n - 1 ) X _ 2 + ( 2 ( n - 1 ) ( 2 n + 1 ) + 4 ) X _ 1 + ( 2 n + 1 ) Y _ 2 + ( 2 n - 1 ) Y _ 1 \\\\ \\Psi _ g & = X _ 2 + ( 2 n - 1 ) X _ 1 + Y _ 2 + 2 Y _ 1 \\end{align*}"}
+{"id": "7705.png", "formula": "\\begin{align*} g _ 0 = [ u _ 1 , \\ldots , u _ { r + 1 } , a _ 0 , \\ldots , a _ r , g _ { r + 1 } ] \\in P _ { \\le r } . \\end{align*}"}
+{"id": "5158.png", "formula": "\\begin{align*} L ^ p ( \\Omega ) ( w ) ' = L ^ { p ' } ( \\Omega ) ( w ^ { - 1 } ) , \\end{align*}"}
+{"id": "2335.png", "formula": "\\begin{align*} f _ j ^ { ( n ) } ( \\pmb { t } _ { n - 1 } , \\pmb { x } _ { n - 1 } , r , z , t , x ) = & f _ { j } ( \\pmb { t } _ { j - 1 } , \\pmb { x } _ { j - 1 } , r , z ) g _ { n - j } ( \\pmb { t } _ { j : n - 1 } , \\pmb { x } _ { j ; n - 1 } , r , z , t , x ) , \\end{align*}"}
+{"id": "5143.png", "formula": "\\begin{align*} \\int _ \\Omega \\ ! | f g | \\ , \\mathrm { d } x = | \\iota ( g ) ( \\widetilde { f } ) | \\leq \\| \\iota ( g ) \\| _ { X ^ \\ast } \\| \\widetilde { f } \\| _ X \\leq \\| \\iota ( g ) \\| _ { X ^ \\ast } \\| f \\| _ X . \\end{align*}"}
+{"id": "2741.png", "formula": "\\begin{align*} H _ \\Gamma ^ s ( \\partial \\mathrm { M } ) = \\{ \\varphi \\in H ^ s ( \\partial \\mathrm { M } ) ; \\ ; \\mathrm { s u p p } ( \\varphi ) \\subset \\Gamma \\} . \\end{align*}"}
+{"id": "3542.png", "formula": "\\begin{align*} F _ 0 = [ & - \\det \\Delta \\cdot ( \\Delta ^ { - 1 } ( \\partial F _ i / \\partial x _ 0 ) ) \\\\ & - \\det \\Delta \\cdot ( \\Delta ^ { - 1 } ( \\partial F _ i / \\partial x _ 1 ) ) ] \\end{align*}"}
+{"id": "6743.png", "formula": "\\begin{align*} ( \\underline { \\varphi } \\otimes \\varphi ) _ \\ast ( m _ H ) ( \\underline { A } \\times A ) = m _ H ( \\underline { \\varphi } ^ { - 1 } ( \\underline { A } ) \\cap \\varphi ^ { - 1 } ( A ) ) = m _ H ( \\underline { C } \\cap C ) . \\end{align*}"}
+{"id": "8571.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { - 1 } S _ j ( \\tau _ N ) & = \\frac { \\nu q } { \\lambda } \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } ( 1 - y ) y ^ { j - 1 } d y \\\\ & = \\frac { \\nu q } { \\lambda } \\sum _ { k = 0 } ^ \\infty \\frac { p ^ k } { ( j + k ) ( j + k + 1 ) } , j \\geq 1 . \\end{align*}"}
+{"id": "4389.png", "formula": "\\begin{align*} \\theta _ { \\epsilon } ( x ) : = \\exp \\left ( \\frac { p \\sqrt { 1 + | x | ^ 2 } } { 1 + \\epsilon \\sqrt { 1 + | x | ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "7827.png", "formula": "\\begin{align*} ( \\textsf { E } \\Vert B A ^ { \\prime } \\Vert ^ { p } ) ^ { 1 / p } = ( \\textsf { E } _ { \\tilde { g } } \\textsf { E } _ { g } \\Vert B A ^ { \\prime } \\Vert ^ { p } ) ^ { 1 / p } \\lesssim \\sqrt { n } p . \\end{align*}"}
+{"id": "4565.png", "formula": "\\begin{align*} \\eta ( p , \\nu ) = \\eta ( p , j + 1 - \\nu - p ) . \\end{align*}"}
+{"id": "1529.png", "formula": "\\begin{align*} ( a t + b ) ^ { p \\theta + \\ell } & = ( a t + b ) ^ { p \\theta } ( a t + b ) ^ \\ell \\\\ & = \\begin{cases} a ^ { p \\theta + \\ell } t ^ { p \\theta + \\ell } & b = 0 , \\\\ ( a ^ { p \\theta } t ^ { p \\theta } + b ^ { p \\theta } ) \\left ( \\sum _ { m = 0 } ^ \\ell \\binom { \\ell } { m } a ^ { \\ell - m } b ^ m t ^ { \\ell - m } \\right ) & . \\end{cases} \\end{align*}"}
+{"id": "3277.png", "formula": "\\begin{align*} { \\rm R e } ( i e ^ { - i \\pi \\beta _ 1 } w ) = - { \\rm I m } ( e ^ { - i \\pi \\beta _ 1 } w ) = { \\rm I m } ( e ^ { i \\pi ( 1 - \\beta _ 1 ) } w ) \\end{align*}"}
+{"id": "7287.png", "formula": "\\begin{align*} b _ p + k x _ p + l y _ p > a _ p + n x _ p = c _ p + m y _ p . \\end{align*}"}
+{"id": "9295.png", "formula": "\\begin{align*} y ^ { k } _ { i } = y ^ { k - 1 } _ { i } + \\lambda _ { k } ( x _ { s } ^ { k } - x _ { t } ^ { k } ) \\forall i = ( s , t ) , \\ i \\in \\mathcal { N } ^ { - } _ { ( k ) } ( s ) . \\end{align*}"}
+{"id": "6829.png", "formula": "\\begin{align*} \\beta ' _ n = \\frac { ( b q ) _ n } { ( b ) _ n } \\beta _ n . \\end{align*}"}
+{"id": "7202.png", "formula": "\\begin{align*} p ^ 0 _ h : = p _ h ( 0 ) = p ( x _ h ) , \\end{align*}"}
+{"id": "7778.png", "formula": "\\begin{align*} r x ^ { r - 1 } \\cdot ( x ^ q y - x y ^ q ) + x ^ r \\cdot ( - y ^ q ) + k r x ^ { r - 1 } \\cdot ( z ^ q x - z x ^ q ) + ( z ^ r + k x ^ r ) \\cdot z ^ q = 0 \\end{align*}"}
+{"id": "2060.png", "formula": "\\begin{align*} P _ { N / 8 \\leq \\cdot \\leq 8 N } P _ N u = P _ N u . \\end{align*}"}
+{"id": "9153.png", "formula": "\\begin{align*} u = F _ { u } \\circ \\phi ( \\zeta _ { [ - q _ { 1 } , - 1 ] } , x , v _ { [ 0 , R - \\kappa ] } ) \\ , , \\end{align*}"}
+{"id": "7822.png", "formula": "\\begin{align*} & e ^ { - \\lambda t } \\prod _ { i = 1 } ^ { N } \\textsf { E } \\exp ( C _ { 0 } \\lambda \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } ( \\max _ { j \\le n } \\eta _ { j } - \\textsf { E } \\max _ { j \\le n } \\eta _ { j } ) ) \\le \\exp ( - \\lambda t + C _ { 1 } ^ { 2 } \\lambda ^ { 2 } \\sum _ { i \\le N } \\Vert B _ { i } \\Vert _ { 2 } ^ { 4 } ) . \\end{align*}"}
+{"id": "6568.png", "formula": "\\begin{align*} X \\# Y = X ^ { 1 / 2 } W Y ^ { 1 / 2 } \\end{align*}"}
+{"id": "5535.png", "formula": "\\begin{align*} p \\mapsto H ( \\xi _ { n } | X , \\eta _ { p } ) = \\int _ { X } H ( \\xi _ { n } ^ { x } ) d \\eta _ { p } ( x ) = \\int _ { Y } \\int _ { \\pi ^ { - 1 } ( y ) } H ( \\theta _ { L _ { x } \\ast } \\mathbb { P } _ { \\mu , n } ^ { y } ) d \\eta _ { p } ^ { y } ( x ) d \\nu ( y ) \\end{align*}"}
+{"id": "8568.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { S _ j ( t ) } { M _ j ( t ) } = \\lim _ { N \\to \\infty } \\frac { S _ j ( \\tau _ N ) } { M _ j ( \\tau _ N ) } = \\frac { \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s } { \\int _ 0 ^ \\infty e ^ { - \\lambda s } \\big ( \\sum _ { k = j } ^ \\infty p _ k ( s ) \\big ) d s } , j \\geq 1 , \\end{align*}"}
+{"id": "9062.png", "formula": "\\begin{align*} \\tilde \\beta _ 0 ( a , b , b ) = ( \\sigma _ 2 \\sigma _ { 1 , 3 } \\sigma _ 2 ) \\sigma _ 2 ^ { a - 1 } \\sigma _ { 1 , 3 } ^ { b - 1 } \\quad \\tilde \\beta ( a , b , b ) = ( \\sigma _ 2 \\sigma _ { 1 , 3 } \\sigma _ 2 ) ^ 3 \\sigma _ 2 ^ { a - 1 } \\sigma _ { 1 , 3 } ^ { b - 1 } , \\end{align*}"}
+{"id": "7490.png", "formula": "\\begin{align*} \\tilde { \\delta } = ( d _ 1 , d _ p , \\dots , d _ 2 ) \\quad \\quad \\tilde { \\delta } ^ { \\prime } = ( x , \\pi _ { l - 1 } , \\dots , \\pi _ { j + 1 } , \\pi _ j , d _ p , \\dots , d _ 2 , d _ 1 , a _ q , \\dots , a _ 2 , a _ 1 ) . \\end{align*}"}
+{"id": "43.png", "formula": "\\begin{align*} \\mathcal { G } _ { \\overline { \\psi } _ 0 } ^ { 1 , \\mathbf { K } } : = \\mathcal { G } ^ 1 ( \\mathbb { Q } ) \\bigcap \\mathbf { K } _ { { \\overline { \\psi } _ 0 } } ^ p \\end{align*}"}
+{"id": "5688.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ^ { - } ) = 0 , \\xi _ { 2 - k } ( f ^ { + } ) = 0 . \\end{align*}"}
+{"id": "3801.png", "formula": "\\begin{align*} \\Theta ( \\delta ) : = \\left \\{ \\int _ { \\mathcal { S } } f ( y _ 1 , y _ 2 ) \\ , d \\gamma ( y _ 1 , y _ 2 , x ) : \\gamma \\in \\Sigma ( \\delta ) \\right \\} , \\end{align*}"}
+{"id": "2526.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { C } } ( { \\rm S p } \\Sigma _ { \\{ i , j \\} } ) = \\begin{cases} p ( i ) + p ( j ) - 3 , & p ( i ) + p ( j ) < 2 n + 1 \\\\ p ( i ) + p ( j ) - 4 , & p ( i ) + p ( j ) > 2 n + 1 , \\end{cases} \\end{align*}"}
+{"id": "7498.png", "formula": "\\begin{align*} | \\Phi ( \\Pi ) | = | \\Pi | = | \\Pi ^ { \\prime } | \\qquad \\qquad | \\Phi ( \\hat { \\Pi } ) | = | \\hat { \\Pi } | = | \\hat { \\Pi } ^ { \\prime } | , \\end{align*}"}
+{"id": "2501.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle a _ 2 \\int _ { \\mathbb { R } ^ N } | \\phi | ^ 2 d x + a _ 1 \\int _ { \\mathbb { R } ^ N } | \\psi | ^ 2 d x = a _ 2 \\int _ { \\mathbb { R } ^ N } | \\phi _ 0 | ^ 2 d x + a _ 1 \\int _ { \\mathbb { R } ^ N } | \\psi _ 0 | ^ 2 d x , \\end{array} \\right . \\end{align*}"}
+{"id": "908.png", "formula": "\\begin{align*} E _ { 1 , b } ( Y , \\Theta _ 1 , \\Theta _ 2 ) & \\ll \\widehat { P } ^ n \\widehat { Y } ^ { 5 / 2 - n / 2 + \\varepsilon } \\widehat { \\Theta } _ 1 \\widehat { \\Theta } _ 2 \\widehat { Z } ^ { 1 - n / 2 - \\mu } \\left ( \\widehat { V } ^ n + \\widehat { Y } ^ { n / 3 } \\right ) \\\\ & = \\widehat { Y } ^ { 5 / 2 + n / 2 + \\varepsilon } \\widehat { \\Theta } _ 1 \\widehat { \\Theta } _ 2 \\widehat { Z } ^ { 1 + n / 2 - \\mu } + \\widehat { P } ^ { n - 5 } \\widehat { Y } ^ { 5 / 2 - n / 6 + \\varepsilon } \\widehat { Z } ^ { 3 - n / 2 - \\mu } . \\end{align*}"}
+{"id": "6847.png", "formula": "\\begin{align*} \\alpha ^ { ( N ) } _ n = ( 1 - a q ^ { 2 n - N } ) ( a q ^ { 1 - N } ) _ N \\sum _ { j \\in \\mathbb { Z } } ( - 1 ) ^ j \\frac { q ^ { N ( n - j ) + j ( j - 1 ) / 2 } } { ( a q ^ { 2 n - N - j } ) _ { N + 1 } } \\left [ { N \\atop j } \\right ] \\alpha _ { n - j } , \\end{align*}"}
+{"id": "6407.png", "formula": "\\begin{align*} f _ t ( \\lambda ) : = \\begin{cases} \\lambda ^ { i t } = e ^ { i t \\log \\lambda } & ( \\lambda > 0 ) , \\\\ 0 & ( \\lambda = 0 ) . \\end{cases} \\end{align*}"}
+{"id": "4535.png", "formula": "\\begin{align*} I _ { \\omega , \\mathbf { c } } ( U ( t ) ) \\le 2 ( S _ { \\omega , \\mathbf { c } } ( \\Phi _ 0 ^ { \\lambda _ 0 } ) - S _ { \\omega , \\mathbf { c } } ( \\Phi _ 0 ) ) = - 2 ( \\mu _ { \\omega , \\mathbf { c } } - S _ { \\omega , \\mathbf { c } } ( \\Phi _ 0 ^ { \\lambda _ 0 } ) ) < 0 . \\end{align*}"}
+{"id": "5425.png", "formula": "\\begin{align*} \\mathfrak { M } _ J = \\left ( \\mathfrak { M } _ { J _ 1 } , \\mathfrak { M } _ { J _ 2 } , \\ldots , \\mathfrak { M } _ { J _ n } \\right ) , \\end{align*}"}
+{"id": "9145.png", "formula": "\\begin{align*} \\begin{array} { r c l } x ^ { + } & = & F _ { x } ( y _ { [ 1 , R ] } ) \\circ \\Psi ( x , z , v _ { [ 0 , R - A ] } ) \\\\ z ^ { + } & = & y _ { c , [ 1 ] } \\circ \\Psi ( x , z , v _ { [ 0 , R - A ] } ) \\\\ v _ { [ 0 , R - A - 1 ] } ^ { + } & = & y _ { [ A + 1 , R ] } \\circ \\Psi ( x , z , v _ { [ 0 , R - A ] } ) \\ , . \\end{array} \\end{align*}"}
+{"id": "1268.png", "formula": "\\begin{align*} M _ q ( G ) = Z _ q ( G ) = T + \\max _ { 1 \\le i _ 1 < \\ldots < i _ q \\le s } \\sum _ { j = 1 } ^ q a _ { i _ j } . \\end{align*}"}
+{"id": "6615.png", "formula": "\\begin{align*} Z _ L ( S ) = \\{ x \\in L : [ x _ \\lambda y ] = 0 , ~ ~ ~ ~ \\forall y \\in S \\} . \\end{align*}"}
+{"id": "9056.png", "formula": "\\begin{align*} ( \\R ^ 3 , d z ' + x ' d y ' - y ' d x ' ) & \\to ( J ^ 1 \\R , d z - y d x ) ; \\\\ ( x ' , y ' , z ' ) & \\mapsto ( x , y , z ) = ( x ' , 2 y ' , z ' + x ' y ' ) , \\end{align*}"}
+{"id": "5539.png", "formula": "\\begin{align*} \\tilde { T } ( x , \\omega ) = \\left ( \\omega _ { 1 } ^ { - 1 } . x , \\left ( \\omega _ { 1 } ^ { - 1 } \\omega _ { n + 1 } \\right ) _ { n \\in \\mathbb { Z } } \\right ) . \\end{align*}"}
+{"id": "1336.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { T _ i ^ { \\infty } } + \\frac { 1 } { \\theta _ i ^ 2 T _ i ^ * ( u ) } , \\frac { Z _ i ^ * ( u ) ^ 2 } { T _ i ^ * ( u ) } + \\frac { T _ i ^ { \\infty } Z _ i ^ * ( u ) ^ 2 } { \\theta _ i ^ 2 T _ i ^ * ( u ) ^ 2 } \\right ) _ { i \\in V } & \\overset { l a w } = \\left ( 2 \\delta _ i + \\frac { 1 } { \\theta _ i ^ 2 T _ i ^ * ( u ) } , \\frac { z _ i ^ 2 } { \\theta _ i ^ 2 T _ i ^ * ( u ) } + \\frac { z _ i ^ 2 } { 2 \\delta _ i ( \\theta _ i ^ 2 T _ i ^ * ( u ) ) ^ 2 } \\right ) _ { i \\in V } \\end{align*}"}
+{"id": "4051.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\sum _ { l , \\ , m \\geq 1 } G _ k \\left ( \\frac { l m } { p } \\right ) \\frac { 1 } { \\sqrt { l m } } \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( i ) \\lambda _ f ( j ) \\lambda _ f ( l ) \\lambda _ f ( m ) . \\end{align*}"}
+{"id": "2740.png", "formula": "\\begin{align*} \\mathbf { e } _ \\lambda = \\max \\left ( \\frac { 1 } { \\mathrm { d i s t } ( \\lambda , \\sigma _ \\lambda ) } , 1 \\right ) , \\end{align*}"}
+{"id": "298.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } \\sum _ { n = 1 } ^ { \\infty } ( n ^ 3 x ^ { n + 1 } y ^ { n + 1 } - 2 n ^ 3 x ^ { n + 1 } y ^ { n + 2 } + n ^ 3 x ^ { n + 1 } y ^ { n + 3 } ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "6305.png", "formula": "\\begin{align*} \\sqrt { 2 H ( \\Phi ( p ) ) } = \\ell ( \\gamma ) = \\delta ( p ) \\in ( 0 , \\epsilon ) , \\end{align*}"}
+{"id": "8184.png", "formula": "\\begin{align*} \\mathcal { R } ( t ) = \\bigcup _ { 0 \\leq s \\leq t } ( Z _ s ) . \\end{align*}"}
+{"id": "3651.png", "formula": "\\begin{align*} \\frac { a } { a + b } \\ : x + \\frac { b } { a + b } \\ : y = z \\end{align*}"}
+{"id": "811.png", "formula": "\\begin{align*} y _ { i + 1 } & = y _ { i } + h f \\left ( x _ { i } , y _ { i } \\right ) + h \\int \\limits _ { x _ { 0 } } ^ { x _ { i } } K d t \\\\ & = y _ { i } + h f \\left ( x _ { i } , y _ { i } \\right ) + \\frac { h ^ { 2 } } { 2 } \\left ( \\sum \\limits _ { j = 0 } ^ { j = i } 2 K _ { j } - \\left ( K _ { 0 } + K _ { i } \\right ) \\right ) . \\end{align*}"}
+{"id": "4772.png", "formula": "\\begin{align*} W _ { [ j + 1 ] } = M _ { [ j + 1 ] } - \\left ( S _ { [ 1 ] } \\diamond \\cdots \\diamond S _ { [ j ] } \\right ) J ^ T _ { 2 j } \\left ( S _ { [ 1 ] } \\diamond \\cdots \\diamond S _ { [ j ] } \\right ) ^ T J _ { 2 | \\alpha _ i | } M _ { [ j + 1 ] } . \\end{align*}"}
+{"id": "5863.png", "formula": "\\begin{align*} \\mathcal { Q } _ { x _ 0 , L _ { \\hat { n } } } = \\bigcap _ { k = 0 } ^ { \\hat { n } } Y _ { k } ( \\omega _ { k - 1 } ) \\end{align*}"}
+{"id": "9202.png", "formula": "\\begin{align*} \\sum _ { u \\in V ^ * , u \\not = v } \\left ( d ( v , u ) - 1 \\right ) + \\sum _ { 1 \\le i \\le r ' } \\left ( d ( u _ i , v ) - 1 \\right ) \\ge ( r - 1 ) ^ 2 . \\end{align*}"}
+{"id": "4736.png", "formula": "\\begin{align*} T \\oplus ^ { \\operatorname { s } } T ' & = \\begin{pmatrix} W \\oplus W ' & X \\oplus X ' \\\\ Y \\oplus Y ' & Z \\oplus Z ' \\end{pmatrix} . \\end{align*}"}
+{"id": "8721.png", "formula": "\\begin{align*} ( \\widetilde { \\Delta } _ { 2 } ) & = p ^ { - 4 } \\times ( P _ { 1 } + P _ { 2 } + P _ { 3 } ) = \\frac { 2 } { p ^ { 4 } } \\times \\bigg \\{ \\frac { 1 } { n ( n - 1 ) } + \\frac { 1 } { m ( m - 1 ) } + \\frac { 2 } { n m } \\bigg \\} ( Q _ { 2 } - 2 Q _ { 1 } ) \\\\ & \\le \\frac { 6 } { p ^ { 4 } } \\times \\left \\{ \\frac { 1 } { n ( n - 1 ) } + \\frac { 1 } { m ( m - 1 ) } + \\frac { 2 } { n m } \\right \\} Q _ { 2 } , \\end{align*}"}
+{"id": "160.png", "formula": "\\begin{align*} L ( x ) = \\frac { 6 } { \\pi ^ 2 } \\left ( L i _ 2 ( x ) + \\frac { 1 } { 2 } \\log ( x ) \\log ( 1 - x ) \\right ) = \\frac { 6 } { \\pi ^ 2 } \\left ( \\sum _ { k = 1 } ^ { \\infty } \\frac { x ^ k } { k ^ 2 } + \\frac { 1 } { 2 } \\log ( x ) \\log ( 1 - x ) \\right ) . \\end{align*}"}
+{"id": "2976.png", "formula": "\\begin{align*} d _ k ( y ) = x _ F \\wedge y \\ . \\end{align*}"}
+{"id": "162.png", "formula": "\\begin{align*} t ^ 3 + 2 t ^ 2 - t - 1 = 0 , \\end{align*}"}
+{"id": "4728.png", "formula": "\\begin{align*} \\begin{aligned} \\abs { \\xi _ { = 1 } } & \\leq C I _ g \\rho _ 0 ^ { 3 + 2 / d } | x _ 1 - x _ 2 | ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "5236.png", "formula": "\\begin{align*} s a R _ 2 : \\left ( \\begin{array} { c c c } x = 0 : & 0 & 0 \\\\ x = 1 : & 0 & 0 \\\\ x = \\infty : & 1 / 2 & 1 / 2 \\end{array} \\right ) . \\end{align*}"}
+{"id": "3847.png", "formula": "\\begin{align*} \\Sigma _ 0 ( \\delta _ 0 ) & = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\boldsymbol { K } ( \\mu , \\gamma ) \\le \\delta _ 0 \\right \\} \\\\ \\Sigma ( \\delta ) & = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\boldsymbol { K } _ \\ell ( \\mu _ { \\ell , 3 } , \\gamma _ { \\ell , 3 } ) \\le \\delta _ \\ell , \\ \\forall \\ell = 1 , 2 \\right \\} . \\end{align*}"}
+{"id": "5325.png", "formula": "\\begin{align*} \\alpha ^ 2 \\log ( 1 / \\alpha ) = p ^ 2 \\log ( 1 / p ) < W ^ { = 1 } [ f ] . \\end{align*}"}
+{"id": "7159.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 \\left ( \\Omega \\cap \\left ( \\overline { J _ u } \\setminus J _ u \\right ) \\right ) = 0 . \\end{align*}"}
+{"id": "5353.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ 2 ^ 2 = \\langle f , f ^ { = d } \\rangle \\le \\| f ^ { = d } \\| _ { q } \\| f \\| _ { q ' } . \\end{align*}"}
+{"id": "4101.png", "formula": "\\begin{align*} \\left ( ( Q _ { 1 , \\vec { v } } ) _ 1 M _ { \\bullet , 1 } ( Q _ { 1 , \\vec { v } } ) _ 2 M _ { \\bullet , 1 } ( Q _ { 1 , \\vec { v } } ) _ 3 M _ { \\bullet , 1 } \\right ) = \\begin{pmatrix} 1 + 0 + 0 & 0 + 0 + 2 & 0 + 2 + 0 \\\\ 0 + 1 + 0 & 1 + 0 + 0 & 0 + 0 + 2 \\\\ 0 + 0 + 1 & 0 + 1 + 0 & 1 + 0 + 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "4439.png", "formula": "\\begin{align*} \\| U ( t ) \\| _ { \\mathcal { H } ^ 1 } ^ 2 \\le \\frac { 6 } { C } S _ { \\omega , \\mathbf { c } } ( U ( t ) ) = \\frac { 6 } { C } S _ { \\omega , \\mathbf { c } } ( U _ 0 ) \\end{align*}"}
+{"id": "2962.png", "formula": "\\begin{align*} A _ i & = \\left \\{ ( t _ 1 , \\dots , t _ \\ell ) \\in \\Delta ^ \\ell \\ | \\ t _ i \\geq 1 - \\delta _ \\ell \\right \\} 1 \\leq i \\leq \\ell \\ , \\\\ A _ 0 & = \\left \\{ ( t _ 1 , \\dots , t _ \\ell ) \\in \\Delta ^ \\ell \\ | \\ \\sum _ { j = 1 } ^ n t _ j \\leq \\delta _ \\ell \\right \\} \\ . \\end{align*}"}
+{"id": "591.png", "formula": "\\begin{align*} \\psi ( b _ k b _ k ^ * ) = | \\omega ( b _ k ) | ^ 2 = | \\psi ( b _ k ) | ^ 2 , 1 \\leq k \\leq n . \\end{align*}"}
+{"id": "1662.png", "formula": "\\begin{align*} 0 & = 2 \\ , g ( [ { f } , { f } ] ( X , Y ) , Z ) \\\\ & = N ^ { \\ , ( 5 ) } ( X , Y , { f } Z ) + N ^ { \\ , ( 5 ) } ( { f } X , Y , Z ) - N ^ { \\ , ( 5 ) } ( Y , X , { f } Z ) - N ^ { \\ , ( 5 ) } ( { f } Y , X , Z ) . \\end{align*}"}
+{"id": "9182.png", "formula": "\\begin{align*} \\varphi ^ { 1 } & = q ^ { 2 } \\\\ \\delta ( \\varphi ^ { 1 } ) & = q ^ { 2 } + T \\omega ^ { 2 } \\\\ \\delta ^ { 2 } ( \\varphi ^ { 1 } ) & = q ^ { 2 } + 2 T \\omega ^ { 2 } + T ^ { 2 } \\left ( a _ { 1 } \\sin ( q ^ { 2 } ) + a _ { 2 } \\cos ( q ^ { 2 } ) + b _ { 2 } \\cos ( q ^ { 3 } ) u ^ { 1 } \\right ) \\end{align*}"}
+{"id": "6770.png", "formula": "\\begin{align*} f ^ { s + 1 } = \\Sigma ( f ^ { s } ) = & \\Sigma ( \\Delta ^ { \\overline { - s } } f - \\sum _ { j = 0 } ^ { s - 1 } [ e _ { \\overline { j - s } } ] ^ { j } ) \\\\ = & \\Delta ^ { \\overline { - s - 1 } } f - [ e _ { \\overline { - s - 1 } } ] - \\sum _ { j = 1 } ^ { s } [ e _ { \\overline { j - 1 - s } } ] ^ { j } \\\\ = & \\Delta ^ { \\overline { - ( s + 1 ) } } f - \\sum _ { j = 0 } ^ { s } [ e _ { \\overline { j - ( s + 1 ) } } ] ^ { j } . \\end{align*}"}
+{"id": "1814.png", "formula": "\\begin{align*} [ \\beta _ k , \\mathcal D _ \\ell ] = \\int _ { \\Lambda ^ { * } } \\ 1 _ { \\S ( 1 ) } ( p - \\ell ) a _ { p + k - \\ell } a _ p \\d p + \\int _ { \\Lambda ^ { * } } \\ 1 _ { \\S ( 1 ) } ( p ) a _ { p + k - \\ell } a _ p \\d p \\ . \\end{align*}"}
+{"id": "4380.png", "formula": "\\begin{align*} \\rho _ j = \\begin{cases} \\mathfrak { s } _ j ^ 2 - \\varepsilon _ j & j \\leq \\log | h - h ' | ^ { - 1 } , \\\\ \\varepsilon _ j & j > \\log | h - h ' | ^ { - 1 } . \\end{cases} \\end{align*}"}
+{"id": "8511.png", "formula": "\\begin{align*} H _ { p , p ^ { \\prime } } ( 1 , c ) = \\frac { 4 \\pi } { \\sqrt { c } } \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { \\prime 2 } + \\lambda _ { n } ^ { \\prime 2 } } { p ^ { \\prime } \\left ( p ^ { \\prime } + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { \\prime 2 } } \\cdot \\frac { \\lambda _ { n } ^ { \\prime - 1 } } { \\sigma \\left ( \\sqrt { c } \\lambda _ { n } ^ { \\prime } \\right ) e ^ { 2 \\pi \\sqrt { c } \\lambda _ { n } ^ { \\prime } } - 1 } \\end{align*}"}
+{"id": "8531.png", "formula": "\\begin{align*} _ { s = 1 } \\tilde { \\zeta } _ { p , p ^ { \\prime } } ( s , c ) = \\frac { \\pi } { \\sqrt { c } \\left ( 1 + \\frac { 1 } { \\pi p } \\right ) } . \\end{align*}"}
+{"id": "602.png", "formula": "\\begin{align*} s _ + = \\sqrt { \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\sqrt { \\frac { C - 4 } { C } } } \\end{align*}"}
+{"id": "4083.png", "formula": "\\begin{align*} K = \\Q ( \\alpha _ 1 , \\dots , \\alpha _ n ) : = \\bigcap \\{ T | T \\mathbb { C } , \\alpha _ 1 , \\dots , \\alpha _ n \\in T \\} . \\end{align*}"}
+{"id": "6062.png", "formula": "\\begin{align*} H ^ p _ A = \\{ f \\in \\mathcal { S } ' \\colon M _ { \\phi , A } ^ 0 f \\in L ^ p \\} \\end{align*}"}
+{"id": "7430.png", "formula": "\\begin{align*} \\left ( \\Pi _ { i = 1 } ^ t \\sigma _ { ( 1 , 1 ) } \\right ) \\star \\sigma _ \\mu = \\left ( \\Pi _ { i = 1 } ^ t \\tau _ { ( 1 , 1 ) } \\right ) \\star \\tau _ \\mu = \\tau _ { ( \\mu _ 1 + t , \\mu _ 2 + t ) } = \\sigma _ { ( \\mu _ 1 + t , \\mu _ 2 + t ) } . \\end{align*}"}
+{"id": "5160.png", "formula": "\\begin{align*} Y _ i = \\mu ( X _ i ) + \\sigma ( X _ i ) \\epsilon _ i , i = 1 , 2 , \\cdots , n , \\end{align*}"}
+{"id": "1496.png", "formula": "\\begin{align*} \\Lambda ^ 3 _ t - \\widetilde { \\Lambda } ^ 3 _ t = \\max _ { t _ 0 \\le s \\le t } \\ , | \\Lambda ^ 3 _ s - \\widetilde { \\Lambda } ^ 3 _ s | > 0 . \\end{align*}"}
+{"id": "6203.png", "formula": "\\begin{align*} V _ e = \\{ r ^ i ( v ) : 0 \\le i < | V _ e | \\} . \\end{align*}"}
+{"id": "3620.png", "formula": "\\begin{align*} ( a _ 0 \\cdots a _ m ) _ { 1 0 } \\ = \\ f ( b _ m \\cdots b _ 0 ) _ { 1 0 } . \\end{align*}"}
+{"id": "1365.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 & \\subseteq \\{ y ^ * ( A x ) \\colon x \\in S _ X , \\ y ^ * \\in S _ { Y ^ * } , \\ y ^ * ( T x ) = \\| T \\| \\} \\\\ & = \\{ y ^ * ( A x ) \\colon x \\in M _ T , \\ y ^ * \\in S _ { Y ^ * } , \\ y ^ * ( T x ) = \\| T \\| \\} = \\mathcal { A } _ 2 . \\end{align*}"}
+{"id": "1095.png", "formula": "\\begin{align*} x ( y z ) = \\alpha ( x y ) z + \\beta ( x z ) y , \\end{align*}"}
+{"id": "3569.png", "formula": "\\begin{align*} \\frac { 1 } { T _ { * } } = \\frac { 2 7 \\varepsilon } { 4 k T } . \\end{align*}"}
+{"id": "6264.png", "formula": "\\begin{align*} L _ { \\Sigma } = \\Delta + | A | ^ 2 + \\mathrm { R i c } ( \\nu , \\nu ) . \\end{align*}"}
+{"id": "3845.png", "formula": "\\begin{align*} c _ { \\ell } ( s _ \\ell , s _ \\ell ' ) = ( s _ \\ell - s _ \\ell ' ) ^ { \\top } V _ \\ell ^ { - 1 } ( s _ \\ell - s _ \\ell ' ) . \\end{align*}"}
+{"id": "7545.png", "formula": "\\begin{align*} \\gamma = \\frac { 1 } { 8 } \\sigma = \\sigma ( d , p ) \\in \\left ( 0 , \\frac { 1 } { 2 } \\right ) , \\end{align*}"}
+{"id": "4312.png", "formula": "\\begin{align*} f ^ K ( \\cdot , t ) = f _ 0 \\circ \\big ( \\tilde { y } _ t ^ K \\big ) ^ { - 1 } , \\end{align*}"}
+{"id": "141.png", "formula": "\\begin{align*} \\chi _ j R ( \\lambda ) \\chi _ j = \\chi _ j R _ j ( \\lambda ) \\chi _ j \\end{align*}"}
+{"id": "2728.png", "formula": "\\begin{align*} \\left \\| P _ 1 u \\right \\| ^ 2 + \\left \\| P _ 2 u \\right \\| ^ 2 + 2 \\left ( P _ 1 u , P _ 2 u \\right ) = \\left \\| e ^ { - s \\sigma } f \\right \\| ^ 2 . \\end{align*}"}
+{"id": "7230.png", "formula": "\\begin{align*} m _ j : = m ^ { ( s _ { h _ j } ) } _ { \\rho _ { h _ j } } \\to m \\mbox { a s } j \\to + \\infty . \\end{align*}"}
+{"id": "5964.png", "formula": "\\begin{align*} \\overline { \\overline { C } } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) = \\overline { C } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) \\overline { s } ( g _ 1 ) \\overline { s } ( g _ 2 ) \\overline { s } ( g _ 1 g _ 2 ) ^ { - 1 } . \\end{align*}"}
+{"id": "6022.png", "formula": "\\begin{align*} \\pi _ { \\psi } ( g ) f = \\theta _ { L , X ^ { \\ast } } [ \\pi _ { \\psi } ( g ) ( f ' ) ] = \\theta _ { L , X ^ { \\ast } } [ \\pi _ { \\psi } ( g ) \\theta _ { X ^ { \\ast } , L } ( f ) ] . \\end{align*}"}
+{"id": "8547.png", "formula": "\\begin{align*} \\eta _ { p } ( s ) = - \\frac { 1 } { 2 } + \\left ( \\frac { 1 } { 2 } C _ { p } ^ { ( 1 ) } - \\frac { \\gamma } { 2 } - \\frac { \\log ( 2 \\pi ) } { 2 } \\right ) \\ , s + O \\left ( s ^ { 2 } \\right ) . \\end{align*}"}
+{"id": "4227.png", "formula": "\\begin{align*} \\Omega & = - \\sum _ { k = 1 } ^ R \\sum _ { j = 1 } ^ N ( \\beta ^ { ( j ) } _ k ) ^ 2 \\\\ E _ 4 & = \\prod _ { k = 1 } ^ R \\prod _ { j = 1 } ^ N G ( 1 + \\beta ^ { ( j ) } _ k ) G ( 1 - \\beta ^ { ( j ) } _ k ) . \\end{align*}"}
+{"id": "2984.png", "formula": "\\begin{align*} \\theta ( ( F ( t _ 2 ) - F ( t _ 1 ) ) \\ , t _ 1 ^ { k _ 1 } \\cdots t _ n ^ { k _ n } ) = - q _ { k _ 1 } ( t _ 1 , \\dots , t _ n ) \\ , \\end{align*}"}
+{"id": "3519.png", "formula": "\\begin{align*} \\begin{bmatrix} I \\\\ S ( t ) \\end{bmatrix} = \\begin{bmatrix} \\cos t I & - \\sin t I \\\\ \\sin t I & \\cos t I \\end{bmatrix} \\begin{bmatrix} I \\\\ S ( 0 ) \\end{bmatrix} G ( t ) . \\end{align*}"}
+{"id": "4575.png", "formula": "\\begin{align*} U _ N ( t ) = \\bigoplus _ { k = 0 } ^ N q ( t ) ^ { \\otimes k } \\left ( \\frac { a ( u _ N ( t ) ) ^ { N - k } } { \\sqrt { ( N - k ) ! } } \\right ) \\end{align*}"}
+{"id": "728.png", "formula": "\\begin{align*} A _ { B , N } ^ { * - 1 } = 2 ( { \\rm I } _ { N } - R _ { B , N } ) ^ { - 1 } = 2 \\sum _ { k = 0 } ^ { + \\infty } R ^ k _ { B , N } \\ , . \\end{align*}"}
+{"id": "5938.png", "formula": "\\begin{align*} \\widetilde { C } _ { X ^ { \\ast } } ( I _ { 1 } \\otimes g _ 2 , s _ 1 ( - 1 ) \\otimes I _ 2 ) & = \\widetilde { C } _ { X ^ { \\ast } } ( [ 1 , I _ { 1 } \\otimes g _ 2 ] , [ - 1 , 1 ] ) \\\\ & = \\nu ( - 1 , I _ { 1 } \\otimes g _ 2 ) \\widetilde { c } _ { X ^ { \\ast } } ( [ I _ { 1 } \\otimes g _ 2 ] ^ { s ( - 1 ) } , 1 ) \\\\ & = \\nu ( - 1 , I _ { 1 } \\otimes g _ 2 ) = ( ( \\det g _ 2 ) ^ { n _ 1 } , - 1 ) _ { \\R } . \\end{align*}"}
+{"id": "3168.png", "formula": "\\begin{align*} \\mathcal { L } : = \\Big \\{ ( \\underbar { x } , \\underbar { y } ) : = ( x _ 1 , \\ldots , x _ N , y _ 1 , \\ldots , y _ N ) \\in R : x _ i , y _ i \\geq 0 \\ \\sum _ { i = 1 } ^ { N } ( x _ i + y _ i ) \\leq 1 \\Big \\} . \\end{align*}"}
+{"id": "637.png", "formula": "\\begin{align*} P _ V a | _ V = \\omega ( a ) I , a \\in M \\end{align*}"}
+{"id": "842.png", "formula": "\\begin{align*} \\textbf { D } _ { n } = \\begin{bmatrix} \\sqrt { p _ { 1 } } & 0 & \\cdots & 0 \\\\ 0 & \\sqrt { p _ { 2 } } & \\cdots & 0 \\\\ \\vdots & \\vdots & \\cdots & \\vdots \\\\ 0 & 0 & \\cdots & \\sqrt { p _ { n } } \\end{bmatrix} \\end{align*}"}
+{"id": "6904.png", "formula": "\\begin{align*} & \\frac { 2 } { \\pi } \\int _ { - ( \\log { t } ) ^ { 2 } } ^ { ( \\log { t } ) ^ { 2 } } \\log \\zeta ( \\sigma + i ( t + u ) ) \\left ( \\frac { \\sin { \\psi u } } { u } \\right ) ^ 2 e ^ { i H u } d u \\\\ & = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\Lambda ( n ) \\max ( 0 , \\psi - | H - \\log { n } | ) } { n ^ { \\sigma + i t } \\log { n } } + O \\left ( \\frac { e ^ { | H | + 2 \\psi } } { ( \\log { t } ) ^ { 2 } } \\right ) . \\end{align*}"}
+{"id": "9125.png", "formula": "\\begin{align*} \\begin{array} { c c l } x ^ { 1 } & = & y ^ { 1 } \\\\ x ^ { 2 } & = & y ^ { 2 } \\\\ x ^ { 3 } & = & y _ { [ 1 ] } ^ { 2 } \\left ( 1 - y ^ { 1 } + y _ { [ 1 ] } ^ { 1 } \\right ) \\\\ u ^ { 1 } & = & y _ { [ 1 ] } ^ { 1 } - y ^ { 1 } \\\\ u ^ { 2 } & = & y _ { [ 2 ] } ^ { 2 } \\left ( 1 - y _ { [ 1 ] } ^ { 1 } + y _ { [ 2 ] } ^ { 1 } \\right ) \\ , , \\end{array} \\end{align*}"}
+{"id": "8976.png", "formula": "\\begin{align*} f = - \\log ( 1 + w ) \\end{align*}"}
+{"id": "8142.png", "formula": "\\begin{align*} \\bar \\Psi _ n = \\sum _ { | T | = n } X _ T . \\end{align*}"}
+{"id": "8890.png", "formula": "\\begin{align*} [ X _ 1 , \\dots , X _ i ] : = \\phi _ i ( X _ 1 \\otimes \\cdots \\otimes X _ i ) . \\end{align*}"}
+{"id": "2974.png", "formula": "\\begin{align*} & F \\ ! \\left ( \\sum _ { j = 1 } ^ n q _ j ( \\xi _ I - \\rho \\cdot \\xi _ I ) t _ j \\right ) \\cdot ( \\rho \\ast \\kappa _ I ( x ) ) \\\\ = \\ & \\psi ( c _ I ( \\rho ) ) \\cdot ( \\rho \\ast \\kappa _ I ( x ) ) = \\varphi _ I ( \\rho ) \\cdot \\kappa _ I ( x ) \\ , \\end{align*}"}
+{"id": "7377.png", "formula": "\\begin{align*} \\Delta \\left ( \\frac { m } { \\ell } \\right ) = \\max \\left \\{ 1 - \\left | \\frac { m } { \\ell } \\right | , 0 \\right \\} = \\begin{cases} 1 , & m = 0 \\\\ 0 , & m \\neq 0 . \\end{cases} \\end{align*}"}
+{"id": "1612.png", "formula": "\\begin{align*} f ^ 2 = Q - \\sum \\nolimits _ { i } \\eta ^ i \\otimes \\xi _ i , \\eta ^ i ( \\xi _ j ) = \\delta ^ i _ j \\ , . \\end{align*}"}
+{"id": "8072.png", "formula": "\\begin{align*} \\begin{aligned} X \\cdot \\left ( \\left | X u \\right | ^ { p - 2 } X u \\right ) & = - \\lambda | u | ^ { p - 2 } u & & \\Omega , \\\\ u & = 0 & & \\partial \\Omega , \\end{aligned} \\end{align*}"}
+{"id": "4078.png", "formula": "\\begin{align*} \\langle z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace , f \\rangle = \\langle T ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) , T f \\rangle = 0 . \\end{align*}"}
+{"id": "5600.png", "formula": "\\begin{align*} \\mathrm { I } ( \\xi _ { 1 } ^ { x } , \\mathcal { T } _ { x } ) & = \\int _ { G ^ { \\mathbb { N } } } \\log \\frac { d \\left ( \\omega _ { 1 } \\lambda \\right ) _ { x } } { d \\lambda _ { x } } ( \\psi _ { x } ( \\omega ) ) d \\mathbb { P } _ { \\mu } ^ { \\pi ( x ) } ( \\omega ) \\\\ & = \\int _ { G } \\int _ { Z _ { x } } \\log \\frac { d ( g \\lambda ) _ { x } } { d \\lambda _ { x } } ( z ) d ( g \\lambda ) _ { x } ( z ) \\varphi _ { g } ( \\pi ( x ) ) d \\mu ( g ) . \\end{align*}"}
+{"id": "5384.png", "formula": "\\begin{align*} \\begin{bmatrix} u _ { 1 , k } ^ 1 \\\\ u _ { 2 , k } ^ 1 \\end{bmatrix} = - \\begin{bmatrix} D _ 1 ^ { 1 1 } & - I \\\\ - I & D _ 2 ^ { 1 1 } \\end{bmatrix} ^ { - 1 } \\begin{bmatrix} C _ 1 ^ 1 & 0 \\\\ 0 & C _ 2 ^ 1 \\end{bmatrix} \\begin{bmatrix} x _ { 1 , k } \\\\ x _ { 2 , k } \\end{bmatrix} . \\end{align*}"}
+{"id": "9280.png", "formula": "\\begin{align*} D = \\left \\{ x \\in X \\ | \\ A x = b \\right \\} , \\end{align*}"}
+{"id": "6304.png", "formula": "\\begin{align*} \\phi : M \\to \\R ; \\phi ( z ) : = \\frac 1 2 \\delta ^ 2 ( z ) . \\end{align*}"}
+{"id": "124.png", "formula": "\\begin{align*} w ( t , x ) : = e ^ { - h ^ { - 1 } \\int _ 0 ^ t q _ 1 ( \\varphi ^ { - s } ( x ) ) d s } f ( \\varphi ^ { - t } ( x ) ) = e ^ { - i t h ^ { - 1 } ( - i h X - i ( Q _ \\infty + q _ 1 ) ) } f ( x ) . \\end{align*}"}
+{"id": "6278.png", "formula": "\\begin{align*} A _ { n } = \\left \\{ f \\in C _ { s } \\left ( X \\right ) : \\underset { x \\in X } { \\sup } \\left \\vert f \\left ( x \\right ) \\right \\vert \\leq n \\right \\} \\end{align*}"}
+{"id": "6815.png", "formula": "\\begin{align*} ( ( W ^ { \\flat } , \\lambda ^ { \\flat } , \\phi ^ { \\flat } ) ; \\ , \\mathcal { L } = ( L _ 1 , \\dots , L _ m ) ) , \\end{align*}"}
+{"id": "8895.png", "formula": "\\begin{align*} X _ 1 \\cdots X _ N = \\sum _ { n = 1 } ^ N X _ n + \\sum _ { p \\geq 2 } \\sum _ { ( n _ 1 , \\dots , n _ p ) \\in \\mathbb { I } _ N ^ p } \\beta _ { ( n _ 1 , \\dots , n _ p ) } [ X _ { n _ 1 } , \\dots , X _ { n _ p } ] . \\end{align*}"}
+{"id": "8113.png", "formula": "\\begin{align*} x _ \\tau = | { \\rm A u t } ( \\tau ) | \\sum _ { \\bar T = \\tau } X _ T \\end{align*}"}
+{"id": "2286.png", "formula": "\\begin{align*} w _ b = \\sum _ { n } c _ n a _ n + T ( f ) _ b \\end{align*}"}
+{"id": "9034.png", "formula": "\\begin{align*} b _ { 2 , 3 } = b _ { 3 , 4 } = b _ { 4 , 2 } \\neq 0 \\end{align*}"}
+{"id": "1045.png", "formula": "\\begin{align*} \\mathcal Z ( f ) \\coloneqq \\{ n \\in \\mathbb N : a ( n ) = 0 \\} \\end{align*}"}
+{"id": "8966.png", "formula": "\\begin{align*} \\phi = \\phi _ { e } + \\sum _ { m = 1 } ^ { \\infty } C _ { q _ m } h ^ { q _ m } , \\end{align*}"}
+{"id": "8179.png", "formula": "\\begin{align*} R _ \\ell + b \\leq R _ { n + 1 } + b \\leq R _ n + 1 + b = R _ n + k . \\end{align*}"}
+{"id": "8847.png", "formula": "\\begin{align*} y _ \\lambda ( t ) = v _ { 0 , \\lambda } ( t ) + \\int _ 0 ^ t S ( t - s ) \\sigma ' ( u _ \\lambda ( s ) ) y _ \\lambda ( s ) B \\ , d W ( s ) . \\end{align*}"}
+{"id": "7262.png", "formula": "\\begin{align*} D _ { L , 2 } = O ( \\lambda _ L ) = O ( 2 ^ { L / p } L ^ { 1 / 2 + \\varepsilon / p } ) \\end{align*}"}
+{"id": "3783.png", "formula": "\\begin{align*} \\sigma _ j ( a ) = \\begin{cases} a - 1 , & a = j \\mod n , \\\\ a + n - 1 , & \\end{cases} . \\end{align*}"}
+{"id": "6914.png", "formula": "\\begin{align*} R ( t ) : = \\sum _ { m \\in \\mathcal { M } ' } \\frac { r ( m ) } { m ^ { i t } } . \\end{align*}"}
+{"id": "9273.png", "formula": "\\begin{align*} \\| T _ { \\eta } f \\| _ { L ^ p ( x ^ \\gamma d x ) } ^ p & \\lesssim \\int _ 0 ^ \\infty x ^ { - \\tau - 1 + \\gamma } \\int _ x ^ \\infty \\abs { f ( z ) } ^ p z ^ { \\tau } \\ , d z \\ , d x = \\int _ 0 ^ \\infty \\abs { f ( z ) } ^ p z ^ { \\tau } \\int _ 0 ^ z x ^ { - \\tau - 1 + \\gamma } \\ , d x \\ , d z \\\\ & \\simeq \\| f \\| _ { L ^ p ( x ^ \\gamma d x ) } ^ p , \\end{align*}"}
+{"id": "4136.png", "formula": "\\begin{align*} T _ { 2 3 } = \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 1 \\end{array} \\right ) , T _ { 3 1 } = \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 1 \\end{array} \\right ) , T _ { 3 2 } = \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 1 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "989.png", "formula": "\\begin{align*} \\pi \\big ( ( \\ell _ E ) _ { E \\in \\mathbf { E } } , ( n _ E ) _ { E \\in \\mathbf { E } } \\big ) = \\big ( \\ell _ E \\mathbf { u } ( E , v _ E ) - n _ E ) _ { E \\in \\mathbf { E } } , \\end{align*}"}
+{"id": "3313.png", "formula": "\\begin{align*} \\rho ( \\xi , w ) = \\rho ( \\overline { \\xi } , \\overline { w } ) { \\rm \\ f o r \\ } \\xi \\ne \\pm 1 . \\end{align*}"}
+{"id": "2231.png", "formula": "\\begin{align*} | \\eta ( x , y ) | \\le e ^ { \\gamma } \\left ( \\eta _ 1 ( y _ 0 ) + \\sum _ { k = 2 } ^ { \\infty } \\xi _ { k } ( y _ 0 ) \\right ) R ( y ) \\end{align*}"}
+{"id": "2249.png", "formula": "\\begin{align*} P _ r ( \\theta ) = \\frac { 1 - r ^ 2 } { 1 - 2 r \\cos ( \\theta ) + r ^ 2 } , \\end{align*}"}
+{"id": "8801.png", "formula": "\\begin{align*} & 2 b _ 2 = ( 3 u + 2 t + 5 ) b _ 3 - ( u + 4 t + 3 ) c _ 3 \\end{align*}"}
+{"id": "5080.png", "formula": "\\begin{align*} M [ \\phi ] = 2 \\left ( \\begin{array} { c c } - 2 \\phi _ r \\phi _ i & \\phi _ r ^ 2 - \\phi _ i ^ 2 \\\\ \\phi _ r ^ 2 - \\phi _ i ^ 2 & 2 \\phi _ r \\phi _ i \\end{array} \\right ) , \\end{align*}"}
+{"id": "7524.png", "formula": "\\begin{align*} f ( x , y ) = \\sum _ { m , n \\geq 0 } f _ { m , n } \\frac { x ^ m y ^ n } { m ! n ! } \\end{align*}"}
+{"id": "1517.png", "formula": "\\begin{align*} 0 \\le \\sqrt { t } \\ , \\frac { w _ n ( t , \\Lambda _ t ^ n \\pm y ) } { y } \\le \\frac { \\varepsilon } { 2 } \\Big ( 1 + \\sup _ { 0 < t \\le \\upsilon _ 0 } \\ ; | \\sqrt { t } \\ , ( \\Lambda _ t ^ n ) ' | ^ 2 \\Big ) \\le \\frac \\varepsilon 2 ( 1 + C _ \\varepsilon ^ 2 ) \\le C _ \\varepsilon , \\ ; y \\in ( 0 , y _ 0 ] , \\ ; t \\in ( 0 , \\upsilon \\wedge \\upsilon _ 0 = \\upsilon _ 0 ) . \\end{align*}"}
+{"id": "2711.png", "formula": "\\begin{align*} A ^ h : = \\begin{pmatrix} y ^ { \\alpha _ y } & 0 \\\\ 0 & ( x + h ) ^ { \\alpha _ x } \\end{pmatrix} , \\end{align*}"}
+{"id": "2103.png", "formula": "\\begin{align*} R _ 2 & = \\Big \\{ ( x _ 1 , x _ 2 , . . . , x _ { k } ) \\mid x _ k = m ; \\ x _ { k - 1 } = 0 ; \\ \\ 0 \\leq x _ i \\leq 2 \\ \\ \\ 1 \\leq i \\leq k - 2 , \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x _ i = 2 , \\ \\ \\ \\ x _ j = 0 \\ \\ \\ \\ j < i \\leq k - 2 \\Big \\} . \\end{align*}"}
+{"id": "108.png", "formula": "\\begin{align*} \\int \\chi ( x ) e ^ { - i x \\cdot \\xi _ 0 / h \\Tilde { h } } \\tilde { R } _ h ( z ) ( \\psi ( x ) e ^ { i x \\cdot r \\eta _ 0 / h \\Tilde { h } } ) d x = \\mathcal { O } ( h ^ \\infty \\Tilde { h } ^ \\infty ) . \\end{align*}"}
+{"id": "7420.png", "formula": "\\begin{align*} \\overline { q } ^ { f } _ L ( x ) : = \\max _ { 1 \\le n \\le L } q _ n ^ { f } ( x ) \\ , . \\end{align*}"}
+{"id": "8635.png", "formula": "\\begin{align*} E \\big | M _ + ( t ) - \\hat { M } _ + ( t ) \\big | = O ( t e ^ { \\lambda t / 2 } ) , \\end{align*}"}
+{"id": "2585.png", "formula": "\\begin{align*} K ( y , z ) = \\sum _ { k , \\ell \\in \\mathbb N _ 0 } c _ { k \\ell } \\ , y ^ k z ^ \\ell , \\end{align*}"}
+{"id": "2226.png", "formula": "\\begin{align*} E _ 1 ( h ; y , u ) = e ^ { \\gamma } \\log y \\int _ { 1 } ^ { h } t ^ { - 1 } y ^ { t - u } \\ , d t < e ^ { \\gamma } y ^ { h - u } . \\end{align*}"}
+{"id": "5885.png", "formula": "\\begin{align*} u ( z ) = \\frac { P ( z ) } { Q ( z ) } \\ , , P \\in \\C _ { \\leq N } [ z ] \\ , , \\end{align*}"}
+{"id": "6616.png", "formula": "\\begin{align*} Z _ L ( L ) = Z ( L ) = \\{ x \\in L : [ x _ \\lambda y ] = 0 , ~ ~ \\forall y \\in L \\} . \\end{align*}"}
+{"id": "4470.png", "formula": "\\begin{align*} \\mu _ { \\omega , c } = \\frac { 1 } { 6 } L _ { \\omega , c } ( \\Phi ) \\ge C _ { \\omega } ( 4 \\omega - \\sigma c ^ 2 ) \\| \\Phi \\| _ { \\mathcal { H } ^ 1 } ^ 2 \\end{align*}"}
+{"id": "5426.png", "formula": "\\begin{align*} \\mathfrak { M } _ { J \\otimes I } = \\left ( \\mathfrak { M } _ { J _ 1 \\otimes I } , \\ldots , \\mathfrak { M } _ { J _ n \\otimes I } \\right ) . \\end{align*}"}
+{"id": "5583.png", "formula": "\\begin{align*} \\int _ { X _ { 0 } } \\mathbb { E } _ { \\lambda } \\left [ \\tilde { f } ( g , \\cdot ) | \\mathcal { F } ^ { s } \\right ] d \\lambda = \\int _ { X _ { 0 } } \\tilde { f } \\left ( g , x _ { 0 } \\right ) d \\lambda ( x _ { 0 } ) = \\int _ { X } f \\left ( g . x _ { 0 } \\right ) d \\lambda ( x _ { 0 } ) = \\int _ { X } f \\left ( x \\right ) d g . \\lambda ( x ) . \\end{align*}"}
+{"id": "7962.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } \\nabla ^ 2 _ \\xi H ^ 2 ( \\xi ) = H ( \\xi ) \\ , \\nabla ^ 2 _ \\xi H ( \\xi ) + \\nabla _ \\xi H ( \\xi ) \\otimes \\nabla _ \\xi H ( \\xi ) \\end{align*}"}
+{"id": "4631.png", "formula": "\\begin{align*} \\frac { \\partial m } { \\partial \\mathbf { n } } \\bigg | _ { \\partial \\Omega } = \\frac { \\partial c } { \\partial \\mathbf { n } } \\bigg | _ { \\partial \\Omega } = \\frac { \\partial d } { \\partial \\mathbf { n } } \\bigg | _ { \\partial \\Omega } = 0 , \\end{align*}"}
+{"id": "7838.png", "formula": "\\begin{align*} t \\ge t _ { 0 } = e ^ { - \\frac { \\tilde { c } \\sqrt { N } } { d } } \\end{align*}"}
+{"id": "4727.png", "formula": "\\begin{align*} & \\sum _ { \\pi : \\pi \\sim ( 1 3 ) ( 2 4 ) \\textnormal { o r } ( 1 3 2 4 ) } \\Gamma _ { \\pi ^ { - 1 } , G } \\\\ & \\ , = \\frac { 2 } { L ^ { 2 d } } \\sum _ { k _ 1 , \\ldots , k _ 4 } \\hat \\gamma ^ { ( 1 ) } ( k _ 1 ) \\cdots \\hat \\gamma ^ { ( 1 ) } ( k _ 4 ) \\chi _ { ( k _ 1 + k _ 2 - k _ 3 - k _ 4 = 0 ) } \\left [ \\hat g ( k _ 2 - k _ 4 ) \\hat g ( k _ 4 - k _ 2 ) - \\hat g ( k _ 2 - k _ 4 ) \\hat g ( k _ 4 - k _ 1 ) \\right ] . \\end{align*}"}
+{"id": "316.png", "formula": "\\begin{align*} \\zeta ( s , 1 ) = \\frac { 1 } { 2 } s \\ , \\zeta ( s + 1 ) - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ { s - 2 } \\zeta ( k + 1 ) \\zeta ( s - k ) , 1 < s \\in \\mathbb { Z } , \\end{align*}"}
+{"id": "6952.png", "formula": "\\begin{align*} D ^ \\lambda w _ j ^ * = \\sum _ { i = 1 } ^ { d _ \\lambda } w _ i ^ * \\partial _ { w _ i } w _ j ^ * = d ! \\sum _ { i = 1 } ^ { d _ \\lambda } w _ i ^ * \\langle w _ i , w _ j ^ * \\rangle = d ! w _ j ^ * . \\end{align*}"}
+{"id": "168.png", "formula": "\\begin{align*} L i _ 2 ( \\frac { 1 } { 2 } ) = \\frac { \\pi ^ 2 } { 1 2 } - \\frac { 1 } { 2 } ( \\log 2 ) ^ 2 , \\end{align*}"}
+{"id": "7353.png", "formula": "\\begin{align*} \\log \\Pr [ d ( \\hat { M } ' , M ) \\leq n ^ { 1 - \\beta } ] ^ { - 1 } & \\geq \\log \\Pr [ \\hat { M } ' = M ] ^ { - 1 } - n ^ { 1 - \\beta } ( 2 + \\max ( 1 , \\alpha + \\beta ) \\log n ) \\\\ & \\geq \\omega ( n ^ { 1 - \\beta } \\log n ) - O ( n ^ { 1 - \\beta } \\log n ) \\\\ & \\geq \\omega ( n ^ { 1 - \\beta } \\log n ) \\geq \\omega ( \\log n ) . \\end{align*}"}
+{"id": "2314.png", "formula": "\\begin{align*} \\gamma = \\begin{cases} \\| r _ { \\ell - 1 } ^ { ( N ) } \\| _ \\infty , & i = 1 , \\\\ \\| r _ { \\ell } ^ { ( i - 1 ) } \\| _ \\infty , & i > 1 , \\\\ \\end{cases} w ^ * _ { \\mathrm { a d d } } = \\begin{cases} 5 , & i = 1 , \\\\ 4 , & i > 1 , \\\\ \\end{cases} \\end{align*}"}
+{"id": "1442.png", "formula": "\\begin{align*} L _ { \\mu , \\xi } ( \\phi ) = \\phi - \\Pi ^ { \\bot } _ { \\mu , \\xi } \\Big ( i ^ * [ f _ \\epsilon ^ { ' } ( V ) \\phi ] \\Big ) , \\end{align*}"}
+{"id": "7108.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n - 1 } } \\int _ { \\mathbb { R } ^ { n - 1 } } a ( x - y ) \\ : b ( ( 1 - t ) y + t x ) \\ : \\textup { d } x \\textup { d } y & = \\left ( \\int _ { \\mathbb { R } ^ { n - 1 } } a ( x ) \\ : x \\right ) \\left ( \\int _ { \\mathbb { R } ^ { n - 1 } } b ( y ) \\ : y \\right ) \\\\ & = \\| a \\| _ { L ^ 1 ( \\mathbb { R } ^ { n - 1 } ) } \\| b \\| _ { L ^ 1 ( \\mathbb { R } ^ { n - 1 } ) } \\end{align*}"}
+{"id": "875.png", "formula": "\\begin{align*} - c _ 2 F _ 1 + x _ 0 c _ 1 F _ 2 = g h \\end{align*}"}
+{"id": "7729.png", "formula": "\\begin{align*} y ^ { * } ( N , \\ , T _ { k } ( Y ) ) = T _ { k } ( N ) . \\end{align*}"}
+{"id": "2767.png", "formula": "\\begin{align*} \\| v _ t \\| _ { L ^ 2 ( \\mathrm { M } _ 0 ) } ^ 2 = ( v _ t | v _ t ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } = ( v _ t + T g _ t | v _ t ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } = ( u | v _ t ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } . \\end{align*}"}
+{"id": "3241.png", "formula": "\\begin{align*} \\begin{aligned} | f ( x ) - f ( x ' ) | & \\le C \\Big ( \\frac { \\| x - x ' \\| } { r } \\Big ) ^ \\beta \\Big \\{ \\frac { 1 } { V ( x , x _ 0 , r + d ( x , x _ 0 ) ) } \\Big ( \\frac { r } { r + { \\| x - x _ 0 \\| } } \\Big ) ^ \\gamma \\\\ & \\qquad + \\frac { 1 } { V ( x ' , x _ 0 , r + d ( x ' , x _ 0 ) ) } \\Big ( \\frac { r } { r + { \\| x ' - x _ 0 \\| } } \\Big ) ^ \\gamma \\Big \\} ; \\end{aligned} \\end{align*}"}
+{"id": "6156.png", "formula": "\\begin{align*} & \\forall \\ , x : n \\forall \\ , y : m \\forall \\ , z : p ( ( x \\ast _ { n , m } y ) \\ast _ { ( m + n ) , p } z = x \\ast _ { n , ( m + p ) } ( y \\ast _ { m , p } z ) ) \\\\ & \\forall \\ , x : n \\forall \\ , y : m ( x \\ast _ { n , m } y = y \\ast _ { m , n } x ) \\end{align*}"}
+{"id": "3381.png", "formula": "\\begin{align*} \\frac { P _ { m } } { p _ m } \\Delta _ { 1 0 } u _ { m n } = O ( 1 ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\frac { Q _ { n } } { q _ n } \\Delta _ { 0 1 } u _ { m n } = O ( 1 ) , \\end{align*}"}
+{"id": "354.png", "formula": "\\begin{align*} ( \\underline { X } , \\underline { A } ) ^ { K } = \\bigcup _ { \\sigma \\subseteq K } ( \\underline { X } , \\underline { A } ) ^ { \\sigma } \\subseteq \\prod ^ m _ { i = 1 } X _ i . \\end{align*}"}
+{"id": "2702.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } z - \\sum _ { i , j = 1 } ^ n \\partial _ { x _ i } ( A _ { i j } \\partial _ { x _ j } z ) + b z = \\chi _ { \\omega } g , & \\mbox { i n } \\ Q , \\\\ z = 0 \\ \\mbox { o r } \\ A \\nabla z \\cdot \\nu = 0 , & \\mbox { o n } \\ \\Sigma , \\\\ z ( 0 ) = z _ { 0 } , & \\mbox { i n } \\ \\Omega . \\end{cases} \\end{align*}"}
+{"id": "9335.png", "formula": "\\begin{align*} & a ( G _ { k , \\mathbb { H } } ; T ) = \\sum _ { d \\mid \\varepsilon ( T ) } d ^ { k - 1 } b _ k ^ * ( 2 { \\rm d e t } ( T ) / d ^ 2 ) \\\\ & b _ k ^ * ( \\ell ) = \\sigma _ { k - 3 } ( \\ell ) - 2 ^ { k - 2 } \\sigma _ { k - 3 } ( \\ell / 4 ) . \\end{align*}"}
+{"id": "1607.png", "formula": "\\begin{align*} f & = \\int _ { \\Theta } T _ { w } ( f ) d \\mu ( w ) \\\\ & = \\int _ { \\Theta } v ( w ) s ( w ) K \\pi _ { G ( w ) } \\xi ^ { \\ast } \\chi _ { w } \\pi _ { F ( w ) } f d \\mu ( w ) \\\\ & = K \\int _ { \\Theta } v ( w ) s ( w ) \\pi _ { G ( w ) } \\xi _ { w } ^ { \\ast } \\chi _ { w } \\pi _ { F ( w ) } f d \\mu ( w ) \\\\ & = K S _ { \\xi \\chi } f . \\end{align*}"}
+{"id": "2265.png", "formula": "\\begin{align*} \\langle f _ b , \\varphi \\rangle : = \\lim _ { r \\nearrow 1 } \\int _ 0 ^ { 2 \\pi } f ( r e ^ { i \\theta } ) \\ , \\varphi ( \\theta ) \\ , d \\theta \\end{align*}"}
+{"id": "3495.png", "formula": "\\begin{align*} - Y '' + R ( \\dot \\gamma , X ) \\dot \\gamma = \\lambda X , Y ( a ) = Y ( b ) = 0 , \\end{align*}"}
+{"id": "8119.png", "formula": "\\begin{align*} \\varphi ^ + ( Y _ T ) & = P _ + ( \\varphi ^ + ( Y _ F ) a ) , \\\\ \\varphi ^ - ( Y _ T ) & = - P _ - ( \\varphi ^ + ( Y _ F ) a ) . \\end{align*}"}
+{"id": "5928.png", "formula": "\\begin{align*} \\Pi _ { \\psi } [ \\begin{pmatrix} 1 & b \\\\ 0 & 1 \\end{pmatrix} ] f ( [ \\epsilon , y ] ) = \\psi ( \\tfrac { 1 } { 2 } \\langle y , \\epsilon y b \\rangle ) f ( [ \\epsilon , y ] ) . \\end{align*}"}
+{"id": "5734.png", "formula": "\\begin{align*} \\mathcal { E C S } _ { { \\rm { I I } } } = \\mathcal { E C S } _ { 0 1 1 } \\cup \\mathcal { E C S } _ { 1 0 1 } \\cup \\mathcal { E C S } _ { 1 1 0 } , \\end{align*}"}
+{"id": "6360.png", "formula": "\\begin{align*} \\rho _ t ( x , y ) = \\frac { p _ t ^ { \\Gamma } ( x , y ) } { M _ { \\Gamma } ( x ) M _ { \\Gamma } ( y ) } , x , y \\in \\Gamma , \\ ; t > 0 . \\end{align*}"}
+{"id": "7675.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to \\infty } L ^ { ( k ) } ( \\delta ) = \\begin{cases} p s , & \\mbox { i f } k = 0 , \\\\ 0 , & \\mbox { i f } k \\ge 1 . \\end{cases} \\end{align*}"}
+{"id": "7809.png", "formula": "\\begin{align*} \\textsf { E } f ( x _ { 1 } \\varepsilon _ { 1 } , x _ { 2 } \\varepsilon _ { 2 } , \\cdots , x _ { n } \\varepsilon _ { n } ) = \\textsf { E } f ( - x _ { 1 } \\varepsilon _ { 1 } , x _ { 2 } \\varepsilon _ { 2 } , \\cdots , x _ { n } \\varepsilon _ { n } ) \\end{align*}"}
+{"id": "1217.png", "formula": "\\begin{align*} H ' = \\frac { A '' \\circ A ^ { - 1 } } { H } = H F - 1 . \\end{align*}"}
+{"id": "7732.png", "formula": "\\begin{align*} y ^ { * } ( N ^ { a } ) ( t ) & = N ^ { a } \\left ( y ^ { * } ( N , t ) \\right ) \\\\ & = N \\bigl ( y ^ { * } ( N , t ) \\wedge a \\bigr ) \\end{align*}"}
+{"id": "1548.png", "formula": "\\begin{align*} \\big | A _ { k _ \\epsilon , \\varrho } ( t ) \\big | & = \\big | A _ { k _ \\epsilon , ( 1 - \\sigma ) \\varrho } ( t ) \\cup ( A _ { k _ \\epsilon , \\varrho } ( t ) \\setminus A _ { k _ \\epsilon , ( 1 - \\sigma ) \\varrho } ( t ) ) \\big | \\\\ & \\leq \\big | A _ { k _ \\epsilon , ( 1 - \\sigma ) \\varrho } ( t ) \\big | + | K _ \\varrho \\setminus K _ { ( 1 - \\sigma ) \\varrho } | \\\\ & \\leq \\big | A _ { k _ \\epsilon , ( 1 - \\sigma ) \\varrho } ( t ) \\big | + N \\sigma | K _ \\varrho | . \\end{align*}"}
+{"id": "8867.png", "formula": "\\begin{align*} f ^ { ( 2 ) } ( \\boldsymbol \\beta ) \\triangleq & - \\log p ^ { ( 2 ) } ( \\mathbf R ; \\boldsymbol \\beta ) - L \\log \\pi \\\\ = & \\log | \\boldsymbol \\Sigma ^ { ( 2 ) } _ { \\boldsymbol \\beta } | + \\left ( \\boldsymbol \\Sigma _ { \\boldsymbol \\beta } ^ { ( 2 ) - 1 } \\widehat { \\mathbf \\Sigma } _ { \\mathbf R } \\right ) . \\end{align*}"}
+{"id": "4474.png", "formula": "\\begin{align*} \\eta _ { 0 } = \\eta _ 0 ( \\omega ) : = \\frac { B _ { \\omega } } { 2 } , \\ \\ c _ 0 = c _ 0 ( \\omega ) : = \\min \\left \\{ c _ 1 ( \\omega ) , c _ 2 ( \\omega ) , \\frac { B _ { \\omega } } { 6 A _ { \\omega } ^ 2 } \\right \\} > 0 . \\end{align*}"}
+{"id": "4316.png", "formula": "\\begin{align*} \\tilde { y } _ t ^ { K + 2 } = \\tilde { y } _ t ^ K . \\end{align*}"}
+{"id": "3684.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p \\norm { u } _ { H ^ { \\alpha _ i } ( \\mathbb { R } ^ n ) } \\leq C \\left ( \\norm { F } _ { ( H ^ \\alpha ( \\Omega ) ) ^ * } + \\norm { f } _ { ( H ^ \\alpha ( \\Omega ^ c ) ) ^ * } \\right ) , \\end{align*}"}
+{"id": "4619.png", "formula": "\\begin{align*} \\sum _ i a _ i d ( k ) _ i = c _ k ( M ) ; \\end{align*}"}
+{"id": "5619.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } - \\frac { 1 } { n } \\log P _ { \\mu , { x } } ^ { n } \\left ( L _ { x } , L _ { x } \\omega _ { n } \\right ) = \\lim _ { n \\to \\infty } \\frac { 1 } { n } H ( \\xi _ { n } | X ) = h _ { \\mu } ( Z , \\lambda ) - h _ { \\mu } ( X , \\eta ) , \\end{align*}"}
+{"id": "2020.png", "formula": "\\begin{align*} ( d d ^ c u ) ^ n = e ^ { B u + \\gamma ( v ) } f \\omega ^ n . \\end{align*}"}
+{"id": "4552.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } S _ { \\omega , \\mathbf { c } } ( \\Phi ^ { \\lambda } ) | _ { \\lambda = 1 } & = \\frac { d } { d \\lambda } \\left . \\left ( L ( \\Phi ^ { \\lambda } ) + N ( \\Phi ^ { \\lambda } ) + \\omega Q ( \\Phi ^ { \\lambda } ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ^ { \\lambda } ) \\right ) \\right | _ { \\lambda = 1 } \\\\ & = 2 L ( \\Phi ) + \\left ( \\frac { d } { 2 } + 1 \\right ) N ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) \\end{align*}"}
+{"id": "6889.png", "formula": "\\begin{align*} ( ( D F ) ( g , \\mu , u ) ) ( f , \\nu , w ) = F ( f , \\nu , u ) + F ( g , \\mu , w ) . \\end{align*}"}
+{"id": "4621.png", "formula": "\\begin{align*} \\Delta f > \\frac { 1 } { n } ( n ^ { 1 / ( p + 1 ) + \\epsilon } - n ^ { 1 / ( p + 1 ) } ) = n ^ { \\frac { - p } { p + 1 } } ( n ^ { \\epsilon } - 1 ) . \\end{align*}"}
+{"id": "5505.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\varrho , s } : C ( Y ) & \\to \\tilde { L } ^ { \\infty } \\left ( Y \\right ) , \\\\ f & \\mapsto \\mathcal { E } _ { \\varrho , s } f , \\mbox { w h e r e } \\left ( \\mathcal { E } _ { \\varrho , s } f \\right ) ( \\bar { u } , \\bar { v } , \\cdot ) = \\mathbb { E } _ { \\lambda } \\left [ \\tilde { f } \\left ( \\bar { u } , \\cdot \\right ) | \\mathcal { F } ^ { s } \\right ] , \\end{align*}"}
+{"id": "2990.png", "formula": "\\begin{align*} F ( t ) = \\sum _ { i = 0 } ^ d \\mu _ i t ^ i \\ . \\end{align*}"}
+{"id": "3270.png", "formula": "\\begin{align*} { \\rm I m } ( f _ { \\ast } ( \\xi ) ) = 0 { \\rm \\ \\ f o r \\ \\ } \\xi \\in L \\end{align*}"}
+{"id": "5280.png", "formula": "\\begin{align*} \\| 1 + d Z \\| _ { q } ^ { q } \\ge 1 + \\binom { \\lfloor q \\rfloor } { 2 } d ^ 2 \\ge 1 + \\frac { q ^ 2 } { 9 } d ^ 2 , \\end{align*}"}
+{"id": "7187.png", "formula": "\\begin{align*} N : = \\left \\{ ( x , y ) \\in \\Omega \\times \\R ^ k : w ( x ) - y \\in \\mathcal { X } \\right \\} \\end{align*}"}
+{"id": "464.png", "formula": "\\begin{align*} \\mathcal L _ \\xi g = 2 g . \\end{align*}"}
+{"id": "7247.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } ( n \\log n ) ^ { - 1 / p } \\big | S _ n - \\sum _ { k = 1 } ^ n g _ k \\big | > 0 \\mbox { a . s . } \\end{align*}"}
+{"id": "4616.png", "formula": "\\begin{align*} c _ k ( X _ \\Omega ) = c _ k ( \\sqcup _ i B ( a _ i ) ) . \\end{align*}"}
+{"id": "326.png", "formula": "\\begin{align*} s _ h ( 5 , 4 ) = \\frac { 8 9 0 } { 9 } \\zeta ( 9 ) + 6 6 \\zeta ( 4 ) \\zeta ( 5 ) - \\frac { 4 2 9 5 } { 2 4 } \\zeta ( 2 ) \\zeta ( 5 ) - 5 \\zeta ( 3 ) ^ 3 + \\frac { 2 6 5 } { 8 } \\zeta ( 2 ) \\zeta ( 7 ) , \\end{align*}"}
+{"id": "2044.png", "formula": "\\begin{align*} \\varphi _ { v _ 0 } ( e _ 1 , e _ 2 , f _ 1 , f _ 2 ) & = ( a _ 0 , b _ 0 ) = ( 0 , 0 ) , \\\\ \\varphi _ { v _ 1 } ( e _ 1 , e _ 2 , f _ 1 , f _ 2 ) & = ( a _ 1 , b _ 1 ) = ( - e _ 1 , f _ 1 + f _ 2 ) , \\\\ \\varphi _ { v _ 2 } ( e _ 1 , e _ 2 , f _ 1 , f _ 2 ) & = ( a _ 2 , b _ 2 ) = ( e _ 2 , f _ 1 ) . \\end{align*}"}
+{"id": "7671.png", "formula": "\\begin{align*} \\max _ { b , C } f ( b , C ) = f \\left ( \\frac { ( n - 1 ) \\kappa } { 2 } , \\frac { ( n - 1 ) ^ 2 \\kappa ^ 2 } { 4 } \\right ) = \\frac { ( n - 1 ) ^ 2 \\kappa ^ 2 } { 4 } , \\end{align*}"}
+{"id": "3668.png", "formula": "\\begin{align*} \\mathcal { M } _ { F _ 1 , \\alpha } f = \\mathcal { M } _ { F _ 2 , \\alpha } f , f \\in H ^ { s _ M } ( \\Omega ^ c ) \\end{align*}"}
+{"id": "10.png", "formula": "\\begin{align*} \\{ ( \\bar { x } , \\bar { y } ) \\in \\R ^ { m + n } : 1 - \\epsilon \\leq \\| N \\bar { y } \\| \\leq e + \\epsilon , | N x _ i | < \\vartheta _ { i } ( \\epsilon ) \\| N \\bar { y } \\| ^ { - w _ i } i = 1 , \\ldots , m \\} . \\end{align*}"}
+{"id": "4337.png", "formula": "\\begin{align*} \\begin{gathered} \\left \\| f ^ { n + 1 , \\nu } - f ^ { n , \\nu } \\right \\| _ { L ^ \\infty ( [ T _ { \\mathcal { K } _ { n + 1 } } , T _ { \\mathcal { K } _ { n + 1 } } + 3 \\cdot 2 ^ { - k } ] ; L ^ 1 ( \\mathbb { T } ^ 2 ) ) } \\\\ \\le 2 \\left \\| f _ 0 \\right \\| _ { L ^ \\infty } 2 ^ { - \\lfloor k / 2 \\rfloor } + C \\left \\| f _ 0 \\right \\| _ { L ^ \\infty } \\sqrt { \\nu 2 ^ { - k } } , \\end{gathered} \\end{align*}"}
+{"id": "4563.png", "formula": "\\begin{align*} p _ 1 \\ge \\cdots \\ge p _ s \\qquad { \\nu _ k \\le \\nu _ { k + 1 } } { p _ k = p _ { k + 1 } } . \\end{align*}"}
+{"id": "7227.png", "formula": "\\begin{align*} \\Omega \\cap \\overline { S _ u } = \\Omega \\cap \\overline { J _ u } = \\Omega \\setminus \\Omega _ u . \\end{align*}"}
+{"id": "3625.png", "formula": "\\begin{align*} a _ 0 \\ = \\ \\begin{cases} 4 , & , \\\\ 9 , & . \\end{cases} \\end{align*}"}
+{"id": "8504.png", "formula": "\\begin{align*} _ { s = 1 } \\zeta _ { p , p ^ \\prime } ( s , c ) = \\frac { \\pi } { \\sqrt { c } } . \\end{align*}"}
+{"id": "4040.png", "formula": "\\begin{align*} S _ { \\mathrm { m a i n } } : = \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\ , \\ \\delta _ { i j } \\ , \\big | L ( f , n + 1 ) \\big | ^ 2 \\geq \\sum _ { i = 1 } ^ D | \\alpha _ i | ^ 2 \\ , c , \\end{align*}"}
+{"id": "7338.png", "formula": "\\begin{align*} x + x ^ 2 y - y z ^ 2 = 0 . \\end{align*}"}
+{"id": "4414.png", "formula": "\\begin{align*} K _ { \\omega , \\mathbf { c } } ( V _ n ) = K _ { \\omega , \\mathbf { c } } ( U _ n ) \\rightarrow 0 , \\end{align*}"}
+{"id": "1112.png", "formula": "\\begin{align*} \\langle ( f \\ast - , - \\ast f ) , ( g \\ast - , - \\ast g ) \\rangle = ( f \\ast ( g \\ast - ) , ( - \\ast f ) \\ast g ) \\end{align*}"}
+{"id": "893.png", "formula": "\\begin{align*} s _ 1 = \\prod _ { \\substack { \\varpi ^ k \\parallel s \\\\ \\varpi ^ { 1 + v _ \\varpi } \\nmid a _ 1 } } \\varpi ^ k \\quad s _ 2 = \\prod _ { \\substack { \\varpi ^ k \\parallel s \\\\ \\varpi ^ { 1 + v _ \\varpi } \\mid a _ 1 } } \\varpi ^ k , \\end{align*}"}
+{"id": "2142.png", "formula": "\\begin{align*} F _ { c o n s e r v a t i v e } ^ - = F _ { d i s s i p a t i v e } ^ - , \\end{align*}"}
+{"id": "682.png", "formula": "\\begin{align*} \\frac { c } { 2 a ( 1 - a ) } & | ( y - l _ \\alpha ) ^ { 1 - a } - ( x - l _ \\alpha ) ^ { 1 - a } | = \\int _ x ^ { y } \\frac { c } { 2 a ( s - l _ \\alpha ) ^ a } d s \\leq \\Big | \\int _ x ^ { y } D ^ { n } \\varphi ( s ) d s \\Big | \\\\ & \\leq | D ^ { n - 1 } \\varphi ( x ) - D ^ { n - 1 } \\varphi ( y ) | \\leq \\| D ^ { n - 1 } \\varphi \\| _ { C ^ { 1 - a + \\tau } } | ( y - l _ \\alpha ) - ( x - l _ \\alpha ) | ^ { 1 - a + \\tau } . \\end{align*}"}
+{"id": "7991.png", "formula": "\\begin{align*} \\mathrm { d i v } _ T \\Big ( & \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\big ) \\Big ) \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\cdot \\nu \\\\ & - \\nabla _ T \\big [ \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\big ] \\ , \\big [ \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\big ] _ T \\cdot \\nu = H ^ 2 ( \\nabla w ) \\ , H ( \\nu ) \\ , \\mathrm { t r } \\ , \\mathcal { B } ^ H \\ , . \\end{align*}"}
+{"id": "6417.png", "formula": "\\begin{align*} \\pi _ \\sigma ( a _ 1 \\otimes a _ 2 ) \\lambda ^ \\sigma ( t , t ) = \\pi _ { \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 } ( a _ 1 \\otimes a _ 2 ) \\lambda ^ { \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 } ( t ) , a \\in M _ 1 , b \\in M _ 2 , ( t , t ) \\in H \\end{align*}"}
+{"id": "7569.png", "formula": "\\begin{align*} \\Xi _ 3 ( x ) = & \\sum _ { x ' \\le n \\le x } \\beta _ n \\sum _ { r \\mid n } \\mu ( r ) \\Omega ( r , n ) \\end{align*}"}
+{"id": "7821.png", "formula": "\\begin{align*} \\prod _ { l = 1 } ^ { n } \\textsf { E } \\exp ( \\frac { C ^ { 2 } \\Vert B _ { i } \\Vert _ { 2 } ^ { 4 } \\lambda ^ { 2 } } { l ^ { 2 } } ) \\le \\exp ( C _ { 1 } ^ { 2 } \\Vert B _ { i } \\Vert _ { 2 } ^ { 4 } \\lambda ^ { 2 } ) , \\end{align*}"}
+{"id": "8092.png", "formula": "\\begin{align*} p _ { } ( \\mathcal { D } _ a , C ) = p _ { } ( \\mathcal { D } _ a , C ) + p _ { } ( \\mathcal { D } _ a , C ) \\forall C \\in \\mathcal { C } . \\end{align*}"}
+{"id": "8920.png", "formula": "\\begin{align*} j d _ 1 ^ { \\frac { 2 j - 1 } { 2 } } \\sqrt [ 2 j ] { \\sum _ { n = 1 } ^ { d _ 1 ^ j } \\| X _ { n 1 } \\| _ 1 ^ { 2 j } } = j d _ 1 ^ { \\frac { 2 j - 1 } { 2 } } \\nu _ j ^ { \\frac { 1 } { j } } . \\end{align*}"}
+{"id": "1267.png", "formula": "\\begin{align*} ( 0 ^ { ( k _ 1 ) } , 1 ^ { ( t _ 1 ) } , \\ldots , 0 ^ { ( k _ s ) } , 1 ^ { ( t _ s ) } ) : = ( \\underbrace { 0 , \\ldots , 0 } _ { k _ 1 } , \\underbrace { 1 , \\ldots , 1 } _ { t _ 1 } , \\ldots , \\underbrace { 0 , \\ldots , 0 } _ { k _ s } , \\underbrace { 1 , \\ldots , 1 } _ { t _ s } ) , \\end{align*}"}
+{"id": "3531.png", "formula": "\\begin{align*} \\sum _ { t \\in ( a ' , b ' ) } \\dim ( \\sigma _ 0 ( t ) \\cap \\sigma ) - \\sum _ { \\lambda \\in ( \\lambda _ 0 , \\lambda ' ) } \\dim ( \\sigma _ { \\lambda } ( b ) \\cap \\sigma ) = 0 . \\end{align*}"}
+{"id": "4791.png", "formula": "\\begin{align*} p ( x ) = \\sum _ { i = 1 } ^ k q _ i ( x ) ^ 2 . \\end{align*}"}
+{"id": "4520.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } K _ { \\omega , \\mathbf { c } } ( V ^ { \\lambda } ) = - \\infty \\end{align*}"}
+{"id": "5207.png", "formula": "\\begin{align*} p \\vartriangleleft q \\qquad \\Leftrightarrow \\qquad \\exists C \\in \\mathsf { C } \\mathbb { P } \\ ( C p = \\{ q \\} ) . \\end{align*}"}
+{"id": "1149.png", "formula": "\\begin{align*} \\mathcal { A } ( f , g ) = T ( f \\cdot g ) - f T ( g ) - T ( f ) g - 2 B ( A ( f ) , A ( g ) ) \\left ( f , g \\in P \\right ) . \\end{align*}"}
+{"id": "3045.png", "formula": "\\begin{align*} \\phi ( E _ { j _ 1 j _ 1 } \\otimes \\cdots \\otimes E _ { j _ { m } j _ { m } } ) = E _ { j _ 1 j _ 1 } \\otimes \\cdots \\otimes E _ { j _ m j _ m } . \\end{align*}"}
+{"id": "6071.png", "formula": "\\begin{align*} M ^ { 0 } _ { \\phi , A } f ( x ) = \\sup _ { t > 0 } | f \\ast \\phi ^ A _ { t } ( x ) | , x \\in G . \\end{align*}"}
+{"id": "4454.png", "formula": "\\begin{align*} \\frac { h '' ( 0 ) } { 2 } - Q ( \\Phi ) = \\frac { 1 } { 2 \\omega } \\left \\{ ( 4 - 2 d ) \\omega Q ( \\Phi ) + ( 3 - d ) \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) ) \\right \\} \\ge \\frac { \\eta } { 2 \\omega } \\ ( > 0 ) \\end{align*}"}
+{"id": "1333.png", "formula": "\\begin{align*} \\theta _ i ^ 2 \\frac { Z _ i ^ * ( u ) } { e _ i ^ * ( u ) } + \\frac { T _ i ^ { \\infty } } { e _ i ^ * ( u ) ^ 2 } & = \\theta _ i ^ 2 \\frac { Z _ i ( u ) } { \\theta _ i } \\left ( \\frac { \\theta _ i } { e ^ { \\rho _ i ( u ) } } - \\frac { \\theta _ i Z _ i ( u ) } { T _ i ^ { \\infty } } \\right ) + T _ i ^ { \\infty } \\left ( \\frac { \\theta _ i } { e ^ { \\rho _ i ( u ) } } - \\frac { \\theta _ i Z _ i ( u ) } { T _ i ^ { \\infty } } \\right ) ^ 2 \\\\ & = \\theta _ i ^ 2 \\left ( \\frac { T _ i ^ { \\infty } } { e ^ { 2 \\rho _ i ( u ) } } - \\frac { Z _ i ( u ) } { e ^ { \\rho _ i ( u ) } } \\right ) . \\end{align*}"}
+{"id": "3935.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta _ 1 , 0 ) = \\widehat { \\mathcal { I } } ( \\delta _ 1 , \\delta _ 2 ) = \\inf _ { \\lambda _ 1 \\in \\mathbb { R } _ { + } } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\varpi \\in \\Pi \\left ( \\mu _ { 1 3 } , \\mu _ { 2 3 } \\right ) } \\int _ { \\mathcal { V } } f _ { \\lambda } ( s _ 1 , s _ 2 ) \\ , d \\varpi ( s _ 1 , s _ 2 ) \\right ] , \\end{align*}"}
+{"id": "5832.png", "formula": "\\begin{align*} H ' = \\left \\lceil H ^ { 1 / 2 } \\right \\rceil . \\end{align*}"}
+{"id": "7604.png", "formula": "\\begin{align*} | H _ { 2 , 1 } ( F _ { f } / 2 ) | = \\frac { 1 } { 7 6 8 } . \\end{align*}"}
+{"id": "2502.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle R e \\mathcal { S } \\left ( \\bar { \\phi } ( t ) , \\psi ( t ) \\right ) = R e \\mathcal { S } ( \\bar { \\phi } _ 0 , \\psi _ 0 ) . \\end{array} \\right . \\end{align*}"}
+{"id": "3564.png", "formula": "\\begin{align*} J _ { T _ { * } } ^ { ( 2 ) } = - \\frac { 2 \\pi \\sigma ^ { 3 } } { n T _ { * } ^ { 3 / n } } \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { k ! } \\Gamma \\left ( \\frac { k m - 3 } { n } \\right ) \\left ( \\frac { 1 } { T _ { * } } \\right ) ^ { ( n - m ) k / n } . \\end{align*}"}
+{"id": "7772.png", "formula": "\\begin{align*} W _ { \\mathrm { t r } , p } ^ + ( \\mu , \\nu ) & = \\inf _ { \\phi \\in \\mathcal { M } _ { \\mathrm { t r } } } \\{ W _ { \\mathrm { t r } , p } ( \\alpha , \\nu ) : \\mu = \\phi ( \\alpha ) \\} \\geq \\inf _ { \\phi \\in \\mathcal { M } _ { \\mathrm { t r } } } \\{ W _ { \\mathrm { t r } , p } ( \\mu , \\phi ( \\nu ) ) \\} = W _ { \\mathrm { t r } , p } ^ - ( \\mu , \\nu ) . \\end{align*}"}
+{"id": "9203.png", "formula": "\\begin{align*} \\sum _ { 1 \\le i \\le r ' } \\left ( d ( v , u _ i ) - 1 \\right ) + \\sum _ { r ' - r + 1 \\le i \\le r } \\left ( d ( u _ i , v ) - 1 \\right ) & \\ge \\sum _ { 1 \\le i \\le r ' } ( i - 1 ) + \\sum _ { r ' - r + 1 \\le i \\le r } ( r - ( i - 1 ) - 1 ) \\\\ & = \\frac { r ' ( r ' - 1 ) } { 2 } + \\frac { ( 2 r - r ' ) ( 2 r - r ' - 1 ) } { 2 } \\\\ & = \\frac { ( r '^ 2 + ( 2 r - r ' ) ^ 2 ) } { 2 } - r \\\\ & \\ge r ^ 2 - r > ( r - 1 ) ^ 2 . \\end{align*}"}
+{"id": "954.png", "formula": "\\begin{align*} | f ( x ) - f ( y ) | \\leqslant C _ n \\cdot \\frac { ( \\Vert Q \\Vert _ 1 ) ^ { \\frac { 1 } { q } } } { \\log ^ { \\frac { 1 } { n } } \\left ( 1 + \\frac { r _ 0 } { 2 | x - y | } \\right ) } , \\ , \\ , r _ 0 = d ( K , \\partial { \\mathbb B } ^ n ) , \\end{align*}"}
+{"id": "4457.png", "formula": "\\begin{align*} h ( \\mp \\tau _ 0 ) = h ( 0 ) \\mp \\tau _ 0 h ' ( 0 ) + \\frac { \\tau _ 0 ^ 2 } { 2 } h '' ( \\theta ) . \\end{align*}"}
+{"id": "7525.png", "formula": "\\begin{align*} D ^ k ( y ) = y _ k + \\sum _ { n \\geq 1 } y _ { k + n } \\frac { x ^ n } { n ! } . \\end{align*}"}
+{"id": "9066.png", "formula": "\\begin{align*} C G _ { 2 k } ( z ) & = \\prod _ { n = 0 } ^ \\infty ( 1 + z q ^ { n + 1 } ) ^ { 2 k } ( 1 + z ^ { - 1 } q ^ n ) ^ { 2 k } \\\\ & \\equiv \\prod _ { n = 0 } ^ \\infty ( 1 + z ^ 2 q ^ { 2 n + 2 } ) ^ { k } ( 1 + z ^ { - 2 } q ^ { 2 n } ) ^ { k } \\pmod { 2 } . \\end{align*}"}
+{"id": "2472.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } E ( \\phi ( t , x ) , \\psi ( t , x ) ) = E ( \\phi _ 0 ( x ) , \\psi _ 0 ( x ) ) , \\end{array} \\right . \\end{align*}"}
+{"id": "545.png", "formula": "\\begin{align*} E ( t , \\xi ) : = ( S ( t ) U ( t , \\xi ) , U ( t , \\xi ) ) , \\end{align*}"}
+{"id": "9023.png", "formula": "\\begin{align*} p ^ n \\to p \\ \\Rightarrow \\ p ^ n = p \\quad n \\geq \\bar n . \\end{align*}"}
+{"id": "7858.png", "formula": "\\begin{align*} \\mbox { $ \\sigma ( w ) = \\tau ( w ) \\Leftrightarrow $ $ v _ { \\sigma ( \\mathcal { S } ) } = v _ { \\tau ( S ) } \\Leftrightarrow \\tau ^ { - 1 } \\sigma ( \\mathcal { S } ) = \\mathcal { S } $ . } \\end{align*}"}
+{"id": "6500.png", "formula": "\\begin{align*} c _ { n , j } : = c _ { n , j } ( m , x ) : = ( - 1 ) ^ { n + j } \\ , \\frac { \\det A _ n ^ { ( n , j ) } } { \\det A _ { n - 1 } } . \\end{align*}"}
+{"id": "3056.png", "formula": "\\begin{align*} G \\cdot H ^ \\star / 2 = \\frac { \\nu } { 2 } \\sum _ { i = 1 } ^ n | \\lambda _ i | . \\end{align*}"}
+{"id": "9165.png", "formula": "\\begin{align*} y = ( \\zeta _ { [ - 1 ] } ^ { 1 } , x ^ { 1 } \\sin \\left ( \\tfrac { \\zeta _ { [ - 1 ] } ^ { 1 } + x ^ { 3 } } { 2 } \\right ) - x ^ { 2 } \\cos \\left ( \\tfrac { \\zeta _ { [ - 1 ] } ^ { 1 } + x ^ { 3 } } { 2 } \\right ) ) \\ , . \\end{align*}"}
+{"id": "4592.png", "formula": "\\begin{align*} T _ t \\circ \\beta = & \\alpha \\circ T _ t , \\\\ [ T _ t u , T _ t v , T _ t w ] = & T _ t \\rho ( T _ t u , T _ t v ) w , \\end{align*}"}
+{"id": "7400.png", "formula": "\\begin{align*} R _ N ^ 2 ( L , \\alpha , \\Delta ) \\leq ( 1 + C / m ) ( L + o ( 1 ) ) = L + o ( 1 ) . \\end{align*}"}
+{"id": "4397.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ 3 \\left ( \\| \\theta _ { \\epsilon } ^ { \\frac { 1 } { 2 } } | \\nabla \\varphi _ j | \\| _ { L ^ 2 } ^ 2 + \\| \\theta _ { \\epsilon } ^ { \\frac { 1 } { 2 } } | \\varphi _ j | \\| _ { L ^ 2 } ^ 2 \\right ) \\le 2 ( 3 + p ) C \\theta _ { \\epsilon } ( R ) \\prod _ { j = 1 } ^ 3 \\| \\varphi _ j \\| _ { H ^ 1 } . \\end{align*}"}
+{"id": "3315.png", "formula": "\\begin{align*} \\rho _ w ( \\xi , w ) w = \\rho _ { \\overline { w } } ( \\overline { \\xi } , \\overline { w } ) w = \\overline { \\rho _ w ( \\overline { \\xi } , \\overline { w } ) \\overline { w } } . \\end{align*}"}
+{"id": "8829.png", "formula": "\\begin{align*} \\operatorname { s u p p } \\chi _ \\lambda = [ - 1 / \\lambda - 1 , 1 / \\lambda + 1 ] , \\bigl . \\chi _ \\lambda \\bigr \\vert _ { [ - 1 / \\lambda , 1 / \\lambda ] } = 1 , \\end{align*}"}
+{"id": "7702.png", "formula": "\\begin{align*} P _ { \\le r } = \\bigcap _ { 0 \\leq j \\leq r } P _ { U _ j } \\end{align*}"}
+{"id": "7837.png", "formula": "\\begin{align*} _ { J } ( K _ { 1 } , K _ { 2 } ) : = \\{ y \\in \\mathbb { S } ^ { n - 1 } \\cap \\mathbb { R } ^ { J } : \\frac { K _ { 1 } } { \\sqrt { d } } \\le \\vert y _ { k } \\vert \\le \\frac { K _ { 2 } } { \\sqrt { d } } , \\forall k \\in J \\} , \\end{align*}"}
+{"id": "227.png", "formula": "\\begin{align*} = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\prod _ { j = 1 } ^ { M } L i _ { s _ j } \\left ( e ^ { i \\theta } \\right ) \\prod _ { k = 1 } ^ { N } L i _ { t _ k } \\left ( e ^ { - i \\theta } \\right ) d \\theta . \\end{align*}"}
+{"id": "5390.png", "formula": "\\begin{align*} \\rho ' & = f ' ( S ' ) + d _ { G ' - S ' } ( T ' ) - q ' ( S ' , T ' ) = f ( S ) + \\gamma \\C R - q ' ( S ' , T ' ) \\ge f ( S ) + ( \\gamma - 1 ) \\C R . \\end{align*}"}
+{"id": "3132.png", "formula": "\\begin{align*} \\frac { e ^ { x u } } { 1 - u } & = \\sum _ { n = 0 } ^ { \\infty } ( x + n ) ^ n \\frac { z ^ n } { n ! } , \\\\ \\frac { e ^ { x u } - 1 } { x } & = \\sum _ { n = 1 } ^ { \\infty } ( x + n ) ^ { n - 1 } \\frac { z ^ n } { n ! } . \\end{align*}"}
+{"id": "3215.png", "formula": "\\begin{align*} S _ n ' ( y ' - \\sigma _ { k + 1 } ' ( y ' ) ) & = S _ n ' \\Big [ y ' - \\frac { 1 } { w } \\Big ( \\sigma ' _ 1 \\circ \\sigma ' _ k ( y ' ) - ( 1 - w ) \\sigma _ { k - 1 } ' ( y ' ) \\Big ) \\Big ] \\\\ & = S _ n ' \\Big ( y ' - \\sigma _ { k - 1 } ' ( y ' ) \\Big ) - \\frac { 1 } { w } S _ n ' \\Big ( \\sigma ' _ 1 \\circ \\sigma ' _ k ( y ' ) - \\sigma _ { k - 1 } ' ( y ' ) \\Big ) . \\end{align*}"}
+{"id": "3776.png", "formula": "\\begin{align*} \\big ( \\mathcal { T } _ q \\big ) _ { k , l } : = \\delta _ { k , l + q } = \\begin{cases} 1 & { \\rm i f } \\ k = l + q , \\\\ 0 & { \\rm o t h e r w i s e } \\end{cases} \\mathrm { f o r } \\ k , l \\in [ m ] . \\end{align*}"}
+{"id": "7006.png", "formula": "\\begin{align*} f = a _ 0 + a _ 1 x + \\ldots + a _ r x ^ r , \\end{align*}"}
+{"id": "7794.png", "formula": "\\begin{align*} \\Phi ( A \\circ B ) = \\Phi ( A ) \\circ \\Phi ( B ) . \\end{align*}"}
+{"id": "2076.png", "formula": "\\begin{align*} \\{ N _ 0 , N _ 1 , N _ 2 , . . . , N _ { a - 1 } \\} = \\{ N _ { d \\cdot 0 } , N _ { d \\cdot 1 } , N _ { d \\cdot 2 } , . . . , N _ { d \\cdot ( a - 1 ) } \\} . \\end{align*}"}
+{"id": "5131.png", "formula": "\\begin{align*} \\int _ { B _ 1 } \\Theta \\Delta _ g \\eta d \\mu _ g = 0 . \\end{align*}"}
+{"id": "369.png", "formula": "\\begin{align*} ( c ^ { \\wedge \\sigma / \\tau } ) ^ * ( x ) = \\left ( \\prod _ { i \\in \\tau } \\frac { c _ i ^ { \\sigma } } { c _ i ^ { \\tau } } \\right ) x \\end{align*}"}
+{"id": "4950.png", "formula": "\\begin{align*} P ^ t \\Sigma = U \\Lambda ^ t U ^ { - 1 } \\Sigma = U \\Lambda ^ t V ^ { - 1 } , \\end{align*}"}
+{"id": "2833.png", "formula": "\\begin{align*} \\Delta _ k = a _ { k - 1 } a _ { k - 2 } ^ 2 \\cdots a _ 0 ^ { k } , k \\ge 1 . \\end{align*}"}
+{"id": "3897.png", "formula": "\\begin{align*} \\varphi ^ \\star ( \\lambda ) = \\sup _ { x \\in \\mathbb { R } ^ n _ + } \\left \\{ \\varphi ( x ) - \\langle \\lambda , x \\rangle \\right \\} . \\end{align*}"}
+{"id": "4885.png", "formula": "\\begin{align*} \\begin{dcases} L ( u - w ) \\ge 0 & B _ r ( x _ r ) \\setminus \\overline { B _ { r / 2 } ( x _ r ) } , \\\\ u - w \\ge 0 & \\R ^ n \\setminus \\big ( B _ r ( x _ r ) \\setminus \\overline { B _ { r / 2 } ( x _ r ) } \\big ) , \\end{dcases} \\end{align*}"}
+{"id": "4163.png", "formula": "\\begin{align*} ( a \\delta _ s ) ( b \\delta _ t ) & = a b \\delta _ { s t } \\\\ ( a \\delta _ s ) ^ * & = a ^ * \\delta _ { s ^ { - 1 } } . \\end{align*}"}
+{"id": "3999.png", "formula": "\\begin{align*} \\delta _ 1 ^ { 1 / 2 } V _ { 1 , Y Y } ^ { - 1 / 2 } V _ { 1 , X Y } + \\delta _ 2 ^ { 1 / 2 } V _ { 2 , Y Y } ^ { - 1 / 2 } V _ { 2 , X Y } = 0 . \\end{align*}"}
+{"id": "8240.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } G _ k ( x , t ) = T _ 2 G _ k ( x , t ) = G _ { k , Y _ { 2 , 1 } } - G _ { k , Y _ { 2 , 2 } } . \\end{align*}"}
+{"id": "5968.png", "formula": "\\begin{align*} \\overline { \\overline { C } } _ { X ^ { \\ast } } ( g , k ) = \\overline { \\overline { C } } _ { X ^ { \\ast } } ( p _ g k _ g , k ) & = \\overline { \\overline { C } } _ { X ^ { \\ast } } ( p _ g , k _ g ) ^ { - 1 } \\overline { \\overline { C } } _ { X ^ { \\ast } } ( p _ g , k _ g k ) \\overline { \\overline { C } } _ { X ^ { \\ast } } ( k _ g , k ) = 1 . \\end{align*}"}
+{"id": "4583.png", "formula": "\\begin{align*} & T \\circ \\beta = \\alpha \\circ T , \\\\ & [ T u , T v , T w ] = T ( \\rho ( T u , T v ) w ) , \\end{align*}"}
+{"id": "6180.png", "formula": "\\begin{align*} \\| \\xi _ i \\| ^ p & = \\sum _ { t \\in G } \\| v _ { t ^ { - 1 } } \\pi ( g _ i ( t ) ) \\xi \\| ^ p = \\sum _ { t \\in G } \\int | \\pi _ 0 ( g _ i ( t ) ) \\xi | ^ p \\ , d \\mu = \\int \\pi _ 0 \\left ( \\sum _ { t \\in G } g _ i ( t ) ^ p \\right ) | \\xi | ^ p \\ , d \\mu \\\\ & \\longrightarrow \\int | \\xi | ^ p \\ , d \\mu = \\| \\xi \\| ^ p . \\end{align*}"}
+{"id": "4509.png", "formula": "\\begin{align*} \\phi _ { \\omega , c } ( x ) : = \\left [ \\frac { \\sqrt { \\omega } } { 4 \\omega - c ^ 2 } \\left \\{ \\cosh ( \\sqrt { ( 4 \\omega - c ^ 2 ) x } ) - \\frac { c } { \\sqrt { 4 \\omega } } \\right \\} \\right ] ^ { - 1 / 2 } \\end{align*}"}
+{"id": "6314.png", "formula": "\\begin{align*} m : = \\min \\{ i \\leq N \\ , : \\ , \\exists \\ , z \\in V a _ i ( z ) \\neq 0 \\} . \\end{align*}"}
+{"id": "1985.png", "formula": "\\begin{align*} \\sup _ { \\bar \\Omega } \\vert u \\vert \\leq C _ 0 : = \\sup _ { \\bar \\Omega } \\vert \\underline u \\vert , \\end{align*}"}
+{"id": "8861.png", "formula": "\\begin{align*} \\mathbf H _ { n , m } \\triangleq \\begin{bmatrix} h _ { n , m , 1 } & h _ { n , m , L } & \\cdots & h _ { n , m , 2 } \\\\ h _ { n , m , 2 } & h _ { n , m , 1 } & \\cdots & h _ { n , m , 3 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ h _ { n , m , L } & h _ { n , m , L - 1 } & \\cdots & h _ { n , m , 1 } \\\\ \\end{bmatrix} \\in \\mathbb { C } ^ { L \\times L } , \\end{align*}"}
+{"id": "3409.png", "formula": "\\begin{align*} F _ 1 ( t ) = \\begin{cases} - F ( t ) , & | t | \\in [ 0 , t _ 0 ) , \\\\ - F ( t _ 0 ) , & | t | \\in [ t _ 0 , + \\infty ) , \\end{cases} F _ 2 ( t ) = \\begin{cases} 0 , & | t | \\in [ 0 , t _ 0 ) , \\\\ F ( t ) - F ( t _ 0 ) , & | t | \\in [ t _ 0 , + \\infty ) . \\end{cases} \\end{align*}"}
+{"id": "7630.png", "formula": "\\begin{align*} \\frac { \\partial M ( c , x , y ) } { \\partial y } = 0 \\ ; \\mbox { i m p l i e s t h a t } \\ ; 8 ( 4 - c ^ 2 ) ^ 2 ( 1 - x ^ 2 ) [ c x ( 1 + x ) + 2 ( 1 - x ) ( 8 - x ) y ] = 0 \\end{align*}"}
+{"id": "94.png", "formula": "\\begin{align*} n ( 1 ) \\leq \\frac { 1 } { \\log 2 } \\int _ 1 ^ 2 \\frac { n ( r ) } { r } d r = \\mathcal { O } ( h ^ { - n } ) . \\end{align*}"}
+{"id": "7401.png", "formula": "\\begin{align*} R _ N ^ 2 ( L , \\alpha , \\Delta ) \\geq \\frac { N _ { m - 1 } } { N } R _ { N _ { m - 1 } } ^ 2 ( L N _ { m - 1 } / N , \\alpha , \\Delta ) = L - o ( 1 ) . \\end{align*}"}
+{"id": "5443.png", "formula": "\\begin{align*} \\Pr ( \\Phi ( \\mathcal { A } ) > 0 ) = 1 - \\exp \\left ( - 2 \\pi \\lambda R _ { m i n } R _ S \\right ) . \\end{align*}"}
+{"id": "3376.png", "formula": "\\begin{align*} m \\log ( m + 1 ) \\Delta _ { 1 0 } u _ { m n } = O _ L ( 1 ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , n \\log ( n + 1 ) \\Delta _ { 0 1 } u _ { m n } = O _ L ( 1 ) . \\end{align*}"}
+{"id": "5024.png", "formula": "\\begin{align*} f & = x _ 1 ^ { a _ 1 } & , \\\\ f & = x _ 1 ^ { a _ 1 } + x _ 1 x _ 2 ^ { a _ 2 } + \\dots + x _ { N - 1 } x _ N ^ { a _ N } & , \\\\ f & = x _ 1 ^ { a _ 1 } x _ 2 + x _ 2 ^ { a _ 2 } x _ 3 + \\dots + x _ { N - 1 } ^ { a _ { N - 1 } } x _ N + x _ N ^ { a _ N } x _ 1 & . \\end{align*}"}
+{"id": "2581.png", "formula": "\\begin{align*} \\sigma ( m _ { i j } ) = m _ { \\sigma i \\sigma j } . \\end{align*}"}
+{"id": "6164.png", "formula": "\\begin{align*} l _ R ( a ) l _ R ( - d ) - l _ R ( b ) l _ R ( - c ) & = l _ R ( c ) l _ R ( a d ) - l _ R ( a d ) l _ R ( d ) = \\\\ & = l _ R ( c ) l _ R ( a d ) - l _ R ( d ) l _ R ( a d ) = l _ R ( - c d ) l _ R ( a d ) . \\end{align*}"}
+{"id": "5300.png", "formula": "\\begin{align*} 0 = \\sum _ { d = 0 } ^ m \\left ( - \\frac { p } { 1 - p } \\right ) ^ d \\langle h ^ { = d } , g \\rangle \\geq \\mu _ { p } ( h ) \\mu _ { p } ( g ) - \\sum _ { d = 1 } ^ m \\left ( \\frac { p } { 1 - p } \\right ) ^ d | \\langle h ^ { = d } , g \\rangle | . \\end{align*}"}
+{"id": "4423.png", "formula": "\\begin{align*} D _ j S _ { \\omega , \\mathbf { c } } ( \\Psi ) = \\eta D _ j K _ { \\omega , \\mathbf { c } } ( \\Psi ) , \\ \\ j = 1 , 2 , 3 . \\end{align*}"}
+{"id": "490.png", "formula": "\\begin{align*} x ' : = \\big ( x _ 1 , \\dots , x _ { k _ 1 } , c ^ { n - 1 } d t ^ { k _ 1 - k _ 2 + 1 } / t _ 3 , c ^ { n - 1 } d t ^ { k _ 1 - k _ 2 + 2 } / t _ 3 , \\dots , c ^ { n - 1 } d / t _ 3 \\big ) . \\end{align*}"}
+{"id": "2743.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { H ^ s ( \\Sigma ) } = \\inf \\{ \\| h \\| _ { H ^ s ( \\partial \\mathrm { M } ) } ; \\ ; h _ { | \\Sigma } = \\varphi \\} . \\end{align*}"}
+{"id": "5797.png", "formula": "\\begin{align*} & \\limsup _ { r \\to \\infty } \\frac { \\log \\log M ( r , h ) } { \\log r } = \\rho ( h ) , \\\\ \\implies & | h ( z ) | \\leq e ^ { r ^ { \\rho ( h ) + o ( 1 ) } } , \\end{align*}"}
+{"id": "3967.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { E } \\left [ c _ 1 \\left ( ( \\widehat { Y } _ 1 , \\widehat { X } ) , ( Y _ 1 , X ) \\right ) \\right ] \\leq \\left ( 1 - \\frac { \\eta } { \\eta _ 0 } \\right ) \\mathbb { E } \\left [ c _ 1 \\left ( ( \\widetilde { Y } _ 1 , \\widetilde { X } ) , ( Y _ 1 , X ) \\right ) \\right ] \\\\ \\end{aligned} \\leq ( \\eta - \\eta _ 0 ) \\end{align*}"}
+{"id": "5108.png", "formula": "\\begin{align*} \\int _ { B _ { 1 } } \\left \\vert D ^ { 2 } \\bar { w } \\right \\vert ^ { p _ 0 } = \\int _ { B _ { 1 } } \\left \\vert D ^ { 2 } w - \\left ( D ^ { 2 } w \\right ) _ { 1 } \\right \\vert ^ { p _ 0 } . \\end{align*}"}
+{"id": "1656.png", "formula": "\\begin{align*} g ( N ^ { \\ , ( 1 ) } ( X , Y ) , \\xi _ i ) = 2 \\ , g ( \\widetilde { Q } X , { f } Y ) . \\end{align*}"}
+{"id": "2293.png", "formula": "\\begin{align*} H = \\langle h , P _ r ( \\theta - \\cdot ) + i Q _ r ( \\theta - \\cdot ) \\rangle \\in H ^ p ( D ) . \\end{align*}"}
+{"id": "7292.png", "formula": "\\begin{align*} x _ p = x ^ 0 _ p + m ' v _ p , y _ p = y ^ 0 _ p + k ' v _ p v _ p \\geq 0 , \\end{align*}"}
+{"id": "8419.png", "formula": "\\begin{align*} \\zeta _ { p } ( s ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { 1 } { \\lambda _ { n } ^ { s } } , \\ , \\ , \\ , \\ , \\ , ( s ) > 1 , \\end{align*}"}
+{"id": "2405.png", "formula": "\\begin{align*} \\vec { \\delta } _ { 2 0 4 } = ( \\cos \\alpha _ { 1 0 2 } + \\cos a _ { 4 , 1 0 2 } \\cos \\omega _ { 4 , 1 0 2 } , \\sin \\alpha _ { 1 0 2 } + \\cos a _ { 4 , 1 0 2 } \\sin \\omega _ { 4 , 1 0 2 } , \\sin a _ { 4 , 1 0 2 } ) \\end{align*}"}
+{"id": "1036.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\frac { ( c ) _ k ( d ) _ k ( e ) _ k ( 1 + a - b - c ) _ k ( 1 + a - b - d ) _ { k } ( 1 + a - b - e ) _ { k } } { ( 1 + a - c ) _ { k } ( 1 + a - d ) _ { k } ( 1 + a - e ) _ { k } ( 1 + 2 a - b - c - d - e ) _ { k } } \\\\ [ 1 m m ] & \\quad \\times \\frac { ( - 1 ) ^ k } { ( 1 + a - b ) _ { 2 k } } \\alpha _ k ( a , b , c , d , e ) \\\\ [ 1 m m ] & \\ : = \\sum _ { k = 0 } ^ { \\infty } ( a + 2 k ) \\frac { ( b ) _ k ( c ) _ k ( d ) _ k ( e ) _ k } { ( 1 + a - b ) _ { k } ( 1 + a - c ) _ { k } ( 1 + a - d ) _ { k } ( 1 + a - e ) _ { k } } , \\end{align*}"}
+{"id": "7560.png", "formula": "\\begin{align*} \\sum _ { l \\equiv v ( \\bmod a ) } 1 _ { I } ( l ) = 0 ~ ~ 1 . \\end{align*}"}
+{"id": "6245.png", "formula": "\\begin{align*} d ( J _ { \\mathrm { s e c t } ( \\delta / 2 ) } ( [ \\ , t \\ , | \\left ( \\begin{smallmatrix} n \\\\ a \\end{smallmatrix} \\right ) ] ) = J _ { \\mathrm { s e c t } ( \\delta / 2 ) } ( t ) \\cdot \\omega _ { n , a } \\end{align*}"}
+{"id": "7325.png", "formula": "\\begin{align*} \\pm \\prod _ { p \\in { \\cal P } _ 3 } p ^ { a _ p - b _ p + \\sum _ { i = 1 } ^ n ( \\alpha _ i - \\beta _ i ) z _ { i p } } \\pm \\prod _ { p \\in { \\cal P } _ 2 } p ^ { b _ p - c _ p + \\sum _ { i = 1 } ^ n ( \\beta _ i - \\gamma _ i ) z _ { i p } } \\pm \\prod _ { p \\in { \\cal P } _ 1 } p ^ { c _ p - a _ p + \\sum _ { i = 1 } ^ n ( \\gamma _ i - \\alpha _ i ) z _ { i p } } = 0 . \\end{align*}"}
+{"id": "4943.png", "formula": "\\begin{align*} \\begin{aligned} p ^ { ( t ) } _ { r , k } = \\frac { 1 } { n ^ t } \\ , \\binom { n - r } { n - k } \\sum _ { j = 0 } ^ { k - r } ( - 1 ) ^ { k - r - j } \\binom { k - r } { j } ( j + r ) ^ t . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "5825.png", "formula": "\\begin{align*} { { \\boldsymbol { K } } _ { k ; t } } = { \\left ( { { \\boldsymbol { W } } _ k ^ T { { \\boldsymbol { \\Omega } } _ { k ; t } } { { \\boldsymbol { W } } _ k } } \\right ) ^ { - 1 } } \\left ( { { \\boldsymbol { \\bar P } } _ { k | k - 1 ; t } ^ { y x } + { \\boldsymbol { H } } _ k ^ T { \\boldsymbol { \\bar P } } _ { k | k - 1 ; t } ^ y } \\right ) . \\end{align*}"}
+{"id": "7924.png", "formula": "\\begin{align*} \\iota _ \\mathbf { n } ( U _ 1 U _ 2 ) = \\rho ( U _ 1 ) \\iota _ \\mathbf { n } ( U _ 2 ) + \\sum _ \\mathbf { m } \\tbinom { \\mathbf { n } + \\mathbf { m } } { \\mathbf { m } } \\iota _ { \\mathbf { n } + \\mathbf { m } } ( U _ 1 ) \\epsilon _ \\mathbf { m } ( U _ 2 ) , \\end{align*}"}
+{"id": "3694.png", "formula": "\\begin{align*} \\begin{cases} \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) \\tilde { u } = 0 & \\Omega , \\\\ \\tilde { u } = 0 & \\Omega ^ c , \\end{cases} \\end{align*}"}
+{"id": "2197.png", "formula": "\\begin{align*} \\mathcal { T } = B _ 1 ^ { ( 0 ) } \\otimes L + B _ 2 ^ { ( 0 ) } \\otimes ( - 2 I _ { m } ) , \\end{align*}"}
+{"id": "5112.png", "formula": "\\begin{align*} \\int _ { \\Omega ' } b ^ { i j , k l } f _ { i j } \\eta _ { k l } d x = 0 \\forall \\eta \\in C _ 0 ^ \\infty ( \\Omega ' ) , \\end{align*}"}
+{"id": "4455.png", "formula": "\\begin{align*} \\omega _ { \\pm } = \\omega _ { \\pm } ( \\tau _ 0 ) : = ( \\sqrt { \\omega } \\pm \\tau _ 0 ) ^ 2 , \\ \\ \\mathbf { c } _ { \\pm } = \\mathbf { c } _ { \\pm } ( \\tau _ 0 ) : = \\frac { \\mathbf { c } } { \\sqrt { \\omega } } ( \\sqrt { \\omega } \\pm \\tau _ 0 ) , \\end{align*}"}
+{"id": "1964.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ n u ^ { p \\bar p } u _ { p \\bar p j } = g _ j , \\end{align*}"}
+{"id": "4532.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } S _ { \\omega , \\mathbf { c } } ( \\Phi _ 0 ^ { \\lambda } ) = \\frac { 5 } { 2 } N ( \\Phi _ 0 ) \\lambda ^ { \\frac { 3 } { 2 } } + 2 \\lambda L ( \\Phi _ 0 ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi _ 0 ) = \\frac { I _ { \\omega , \\mathbf { c } } ( \\Phi _ 0 ^ { \\lambda } ) } { \\lambda } . \\end{align*}"}
+{"id": "4766.png", "formula": "\\begin{align*} \\tilde { S } = M Q + \\mathcal { O } ( \\| H \\| ) , \\end{align*}"}
+{"id": "5843.png", "formula": "\\begin{align*} \\left ( X _ { T + T _ { n _ 1 } ^ { o u t } - T ' } , \\ldots , X _ { T + T _ { n _ 1 } ^ { o u t } - T _ { n _ 1 } ^ { i n } } \\right ) = \\left ( X ' _ { T _ { n _ 1 } ^ { i n } } , X ' _ { T _ { n _ 1 } ^ { i n } + 1 } , \\ldots , X ' _ { T ' } \\right ) . \\end{align*}"}
+{"id": "8711.png", "formula": "\\begin{align*} | E ( U ^ { \\top } A V ) ^ { L } \\} | = E | E \\{ ( U ^ { \\top } A V ) ^ { L } \\mid U \\} | \\leq C \\times E ( \\| A ^ { \\top } U \\| _ { 2 } ^ { L } ) , \\end{align*}"}
+{"id": "4713.png", "formula": "\\begin{align*} \\left \\langle \\mathcal { A } x , x \\right \\rangle = \\left \\langle \\Re \\mathcal { A } x , x \\right \\rangle + i \\left \\langle \\Im \\mathcal { A } x , x \\right \\rangle \\end{align*}"}
+{"id": "5347.png", "formula": "\\begin{align*} ( r ' ) ^ { - 2 / q } = \\left ( \\frac { 1 } { ( r ' ) ^ 2 } \\right ) ^ { 1 / q } \\leq \\sqrt { ( q / 2 ) ^ { 2 / q } } < \\sqrt { 2 } . \\end{align*}"}
+{"id": "2098.png", "formula": "\\begin{align*} N _ { d r } & = \\left ( \\sum _ { i = 1 } ^ { n + 1 } 2 ^ i x _ i \\right ) a + \\sum _ { i = 1 } ^ { n + 1 } ( 2 ^ i - 1 ) x _ i d \\\\ & = \\left ( \\sum _ { i = 1 } ^ { n + 1 } 2 ^ i x _ i \\right ) ( a + d ) - \\sum _ { i = 1 } ^ { n + 1 } x _ i d . \\end{align*}"}
+{"id": "4974.png", "formula": "\\begin{align*} n ^ t \\ , \\overline { \\widehat { F } } ( k , t , n ) = k ! h _ { t - k } ( n , n { - } 1 , n { - } 2 , . . . , n { - } k ) = ( n + 1 ) ^ { t } \\ , \\hat { f } ( k + 1 , t , n + 1 ) , \\end{align*}"}
+{"id": "2330.png", "formula": "\\begin{align*} v ^ { ( r , z ) } ( t , x ) = G _ { t - r } ( x - z ) + \\sum _ { n \\geq 1 } I _ n ( g _ n ( \\cdot , r , z , t , x ) ) , \\end{align*}"}
+{"id": "9242.png", "formula": "\\begin{align*} c _ { 2 n } = ( - 1 ) ^ { n + 1 } \\frac { i } { \\sqrt { 2 } } \\sum _ { k = 0 } ^ n ( - 1 ) ^ k V ( k ) W ( 2 n - 2 k ) + \\frac { e ^ { 3 \\pi i / 8 } } { 2 } \\sum _ { k = 0 } ^ n i ^ { n - k } V ( k ) W ( n - k ) \\end{align*}"}
+{"id": "212.png", "formula": "\\begin{align*} = e x p \\left \\{ \\frac { z ( 1 + 2 6 z + 6 6 z ^ 2 + 2 6 z ^ 3 + z ^ 4 ) L i _ 3 ( x ) L i _ 3 ( y ) } { ( 1 - z ) ^ 6 } \\right \\} . \\end{align*}"}
+{"id": "7824.png", "formula": "\\begin{align*} \\textsf { E } ( \\Xi ^ { 2 } ) ^ { p } = & \\int _ { 0 } ^ { \\infty } \\textsf { P } \\{ ( \\Xi ^ { 2 } ) ^ { p } > t \\} \\ , d t = \\int _ { 0 } ^ { \\infty } \\textsf { P } \\{ \\Xi ^ { 2 } > u \\} p u ^ { p - 1 } \\ , d u \\\\ = & ( C \\log n \\sum _ { i \\le N } \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } ) ^ { p } \\int _ { 0 } ^ { \\infty } \\textsf { P } \\{ \\Xi ^ { 2 } > C z \\log n \\sum _ { i \\le N } \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } \\} p z ^ { p - 1 } \\ , d z . \\end{align*}"}
+{"id": "8970.png", "formula": "\\begin{align*} \\Tilde { \\phi } _ { e } & = \\phi _ 1 - C _ { \\Tilde { q } } h ^ { \\Tilde { q } } , \\\\ \\Tilde { q } & = \\ln \\Big ( \\frac { \\phi _ 3 - \\phi _ 2 } { \\phi _ 2 - \\phi _ 1 } \\Big ) / \\ln r , \\\\ C _ { \\Tilde { q } } h ^ { \\Tilde { q } } & = \\frac { \\phi _ 2 - \\phi _ 1 } { r ^ { \\Tilde { q } } - 1 } . \\end{align*}"}
+{"id": "5180.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ j t ^ { b _ { r _ j } + b _ { s _ j } } z _ { r _ j } z _ { s _ j } = 0 . \\end{align*}"}
+{"id": "3087.png", "formula": "\\begin{align*} \\Delta _ p ( C _ \\lambda r ^ \\lambda ) = ( \\lambda C _ \\lambda ) ^ { p - 1 } \\ , ( \\lambda ( p - 1 ) + n - p ) \\cdot r ^ { ( p - 1 ) \\lambda - p } . \\end{align*}"}
+{"id": "1320.png", "formula": "\\begin{align*} \\langle e ^ { \\rho ( u ) } , \\partial _ u ( \\widetilde { W } ^ { ( u ) } ) e ^ { \\rho ( u ) } \\rangle & = \\langle e ^ { \\rho ( u ) } , W ( H ^ { ( u ) } ) ^ { - 1 } T ( u ) ^ { - 1 } e ^ { 2 \\rho ( u ) } T ( u ) ^ { - 1 } ( H ^ { ( u ) } ) ^ { - 1 } W e ^ { \\rho ( u ) } \\rangle \\\\ & = \\langle e ^ { \\rho ( u ) } , \\widetilde { W } ^ { ( u ) } e ^ { 2 \\rho ( u ) } \\widetilde { W } ^ { ( u ) } e ^ { \\rho ( u ) } \\rangle = \\sum _ { i \\in V } ( \\widetilde { W } ^ { ( u ) } e ^ { \\rho ( u ) } ) _ i ^ 2 e ^ { 2 \\rho _ i ( u ) } , \\end{align*}"}
+{"id": "2147.png", "formula": "\\begin{align*} | \\varphi \\circ d _ q ( x ) - ( d _ q ( x ) - r ) | & = \\frac \\epsilon { b } ( d _ q ( x ) - r ) ^ 2 \\le \\frac \\epsilon { b } d ^ 2 _ p ( x ) \\le 2 \\epsilon \\cdot d _ p ( x ) . \\end{align*}"}
+{"id": "2424.png", "formula": "\\begin{align*} \\psi ( x ) = x + O ( x ^ { 1 / 2 } ) , N ( x ) = x + O \\bigl ( x ^ { 1 / 2 } \\exp ( c ( \\log x ) ^ { 2 / 3 } ) \\bigr ) , c > 0 \\end{align*}"}
+{"id": "4815.png", "formula": "\\begin{align*} u _ * ( \\theta , \\dot \\theta ) = 2 1 2 . 5 7 5 5 \\cos ( \\theta ) \\sin ( \\theta ) + 5 4 . 1 2 9 6 \\cos ( \\theta ) \\dot \\theta . \\end{align*}"}
+{"id": "7736.png", "formula": "\\begin{align*} Z ( t ) = N \\Bigl ( y ( Z , t ) \\Bigr ) \\end{align*}"}
+{"id": "7670.png", "formula": "\\begin{align*} f ( b , C ) : = 2 \\sqrt { C b ( n - 1 ) \\kappa - C b ^ 2 } - C . \\end{align*}"}
+{"id": "2819.png", "formula": "\\begin{align*} \\overset \\cdot { \\ ; \\ ; R ^ 2 _ { 0 , 0 } } = R _ { q + 1 , 0 } ^ 2 ; \\ ; \\overset \\cdot { \\ ; \\ ; R ^ 2 _ { 0 , j } } = R _ { q + 1 , j } ^ 2 - ( \\sum _ { l = 1 } ^ j b _ l ) R _ { 0 , j } ^ 2 ; j = 1 , \\dots , q - 1 . \\end{align*}"}
+{"id": "346.png", "formula": "\\begin{align*} \\sigma _ 1 ( 1 ) = 1 , \\ \\ \\sigma _ 1 ( 2 ) = 3 , \\ \\ \\sigma _ 1 ( 3 ) = 2 , \\ \\ \\sigma _ 1 ( 4 ) = 4 , \\end{align*}"}
+{"id": "8344.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ \\infty \\alpha _ j u _ j \\Big \\| _ { p ^ * , q , \\mu } \\geq \\Bigg ( \\frac { \\lambda } { 1 + \\varepsilon _ 1 } - \\varepsilon _ 2 \\Bigg ) \\Big ( \\sum _ { j = 1 } ^ \\infty | \\alpha _ j | ^ q \\Big ) ^ { \\frac 1 { q } } . \\end{align*}"}
+{"id": "3608.png", "formula": "\\begin{align*} 1 0 ^ m - 6 \\ \\le \\ v ( f ) \\ = \\ v ( 2 ^ \\eta ) + v ( 3 ^ \\delta ) + v ( 5 ^ \\gamma ) \\ \\le \\ 1 0 + \\eta + \\delta + \\gamma . \\end{align*}"}
+{"id": "6693.png", "formula": "\\begin{align*} D ( f g ) = D f \\tau g + f D g . \\end{align*}"}
+{"id": "519.png", "formula": "\\begin{align*} \\mathrm { H } _ { \\mathcal { H } _ { V } } ^ { s } : = \\left \\{ f \\in \\mathcal { D } _ { \\mathcal { H } _ { V } } ^ { \\prime } \\left ( \\mathbb { R } ^ { n } \\right ) : \\left ( I + \\mathcal { H } _ { V } \\right ) ^ { s / 2 } f \\in L ^ { 2 } \\left ( \\mathbb { R } ^ { n } \\right ) \\right \\} , s \\in \\mathbb { R } , \\end{align*}"}
+{"id": "4610.png", "formula": "\\begin{align*} \\mathbb { P } ( \\bar { E } _ { i , 1 } ) = \\mathbb { P } _ K ^ { k _ 1 } \\left ( 1 - \\mathbb { P } _ { e , 1 } \\right ) ^ { k _ 1 } \\cdots \\mathbb { P } _ K ^ { k _ K } \\left ( 1 - \\mathbb { P } _ { e , K } \\right ) ^ { k _ K } , \\end{align*}"}
+{"id": "2580.png", "formula": "\\begin{align*} c _ { i j k } c _ { i k \\ell } = c _ { i j \\ell } c _ { j k \\ell } \\end{align*}"}
+{"id": "3302.png", "formula": "\\begin{align*} f ( \\xi ) = ( \\xi - 1 ) ^ { \\delta _ 1 } ( \\xi + 1 ) ^ { \\delta _ { - 1 } } \\kappa ( \\xi ) + w _ 1 \\frac { 1 + \\psi ( \\xi ) } { 2 } + w _ { - 1 } \\frac { 1 - \\psi ( \\xi ) } { 2 } , \\end{align*}"}
+{"id": "8693.png", "formula": "\\begin{align*} T _ { l - 1 } = \\frac { f ^ { ( l ) } ( \\tau ) } { p ^ { l } } \\sum _ { \\substack { 0 \\leq a _ 1 + a _ 2 \\leq l \\\\ a _ 1 = a _ 2 } } ( - 2 ) ^ { l - a _ 1 - a _ 2 } \\frac { 1 } { a _ 1 ! a _ 2 ! ( l - a _ 1 - a _ 2 ) ! } \\| \\mathcal { T } _ { 1 , l } ^ { ( a _ 1 , a _ 2 ) } - \\mathcal { T } _ { 2 , l } ^ { ( a _ 1 , a _ 2 ) } \\| _ { \\rm F } ^ 2 . \\end{align*}"}
+{"id": "4453.png", "formula": "\\begin{align*} F ( \\tau ) = \\frac { h '' ( \\tau ) } { 2 } - \\sup _ { \\Phi \\in \\mathcal { M } _ { \\omega , \\mathbf { c } } ^ * ( \\eta ) } Q ( \\Phi ) . \\end{align*}"}
+{"id": "9322.png", "formula": "\\begin{align*} J _ n ( u ) = J ( u ) , \\end{align*}"}
+{"id": "1248.png", "formula": "\\begin{align*} \\partial _ 2 g _ { \\beta , 2 } ' ( x ) = - \\frac { X ' _ { \\beta , 2 } ( x ) } { f _ { \\beta , 2 } ' ( g _ { \\beta , 2 } ( x ) ) } + X _ { \\beta , 2 } \\frac { f _ { \\beta , 2 } '' ( g _ { \\beta , 2 } ( x ) ) } { ( f _ { \\beta , 2 } ' ( g _ { \\beta , 2 } ( x ) ) ) ^ 3 } . \\end{align*}"}
+{"id": "1082.png", "formula": "\\begin{align*} & t ^ { \\sigma } \\| \\Phi ( u ) ( t ) - \\Phi ( v ) ( t ) \\| _ { L ^ a } \\\\ & \\leq C \\sum _ { k = 0 } ^ { \\infty } \\frac { \\lambda ^ k } { k ! } t ^ { \\sigma } \\int _ 0 ^ t ( t - \\tau ) ^ { - \\frac d { 2 \\beta } ( \\frac 1 r - \\frac 1 a ) } \\| ( u - v ) \\| _ { L ^ a } \\\\ & \\left [ \\| u \\| _ { L ^ { a } } ^ { ( p k + m - 1 ) \\theta } \\ , \\| u \\| _ { L ^ { \\rho } } ^ { ( p k + m - 1 ) ( 1 - \\theta ) } + \\| v \\| _ { L ^ { a } } ^ { ( p k + m - 1 ) \\theta } \\ , \\| v \\| _ { L ^ { \\rho } } ^ { ( p k + m - 1 ) ( 1 - \\theta ) } \\right ] \\ , d \\tau . \\end{align*}"}
+{"id": "8521.png", "formula": "\\begin{align*} \\Gamma \\left ( s \\right ) = \\sqrt { \\pi } - \\sqrt { \\pi } \\left ( 2 \\log ( 2 ) + \\gamma \\right ) \\left ( s - \\frac { 1 } { 2 } \\right ) + O \\left ( s - \\frac { 1 } { 2 } \\right ) ^ { 2 } . \\end{align*}"}
+{"id": "5751.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } k \\omega ( 2 ^ { - k } ) & \\approx \\int _ { 0 } ^ { 1 } \\omega ( t ) ( 1 + \\log t ^ { - 1 } ) \\frac { \\mathrm { d } t } { t } < \\infty . \\end{align*}"}
+{"id": "3167.png", "formula": "\\begin{align*} \\overline { \\psi } _ { y _ 1 , y _ 2 } = \\norm { \\cdot } _ 1 - \\lim _ { l \\to \\infty } B _ l ( \\psi _ { y _ 1 , y _ 2 } ) . \\end{align*}"}
+{"id": "5404.png", "formula": "\\begin{align*} B _ { L , k , 1 } & \\ll T \\sum _ { p \\mid k } \\nu _ p ( k ) \\log p = T \\log k . \\end{align*}"}
+{"id": "191.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b ) = 1 \\\\ a , b \\geq 1 } } \\left ( \\frac { 1 } { 1 - y ^ a z ^ b } \\right ) ^ { \\frac { 1 } { a ^ s b ^ t } } = \\exp \\left \\{ \\left ( \\sum _ { j = 1 } ^ { \\infty } \\frac { y ^ j } { j ^ s } \\right ) \\left ( \\sum _ { k = 1 } ^ { \\infty } \\frac { z ^ k } { k ^ t } \\right ) \\right \\} . \\end{align*}"}
+{"id": "7151.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\mathcal { L } ^ { \\Omega } _ \\varepsilon u u \\ : x = \\frac { 1 } { 4 } \\int _ { \\Omega } \\int _ { \\Omega } J _ \\varepsilon ( x - y ) \\big | u ( x ) - u ( y ) \\big | ^ 2 \\ : x \\ : y = \\mathcal { E } _ \\varepsilon ( u ) \\end{align*}"}
+{"id": "2070.png", "formula": "\\begin{align*} f ( t ) = \\alpha - \\frac { t } { \\alpha t - 1 } \\end{align*}"}
+{"id": "1395.png", "formula": "\\begin{align*} d d ^ c \\omega _ 0 ^ { n - k - 1 } & = d J d \\omega _ 0 ^ { n - k - 1 } = ( n - k - 1 ) d J ( \\theta \\wedge \\omega _ 0 ^ { n - k - 1 } ) \\\\ & = ( - 1 ) ^ { n - k - 1 } ( n - k - 1 ) d ( J \\theta \\wedge ( d J \\theta ) ^ { n - k - 1 } ) \\\\ & = - ( n - k - 1 ) ( - d J \\theta ) ^ { n - k } \\leq 0 . \\end{align*}"}
+{"id": "1645.png", "formula": "\\begin{align*} 2 \\ , g ( ( h _ i - h _ i ^ * ) X , Y ) = N ^ { \\ , ( 5 ) } ( \\xi _ i , X , Y ) - N ^ { \\ , ( 2 ) } _ i ( X , Y ) . \\end{align*}"}
+{"id": "766.png", "formula": "\\begin{align*} [ T _ k ] _ i ^ j ( A ) = [ T _ k ] _ i ^ j ( \\underbrace { A , A , \\cdots , A } _ k ) . \\end{align*}"}
+{"id": "1450.png", "formula": "\\begin{align*} u _ m = i ^ * \\Big [ f _ \\epsilon ^ { ' } ( V _ m ) u _ m + f _ \\epsilon ^ { ' } ( V _ m ) ( h _ m + \\omega _ m ) \\Big ] . \\end{align*}"}
+{"id": "5123.png", "formula": "\\begin{align*} \\beta ^ { i j , k l } ( D ^ 2 u ( x ) ) = \\int _ { 0 } ^ { 1 } \\frac { \\partial F ^ { i j } } { \\partial u _ { k l } } ( D ^ 2 u ( x ) + t [ D ^ 2 u ( x + h _ m ) - D ^ 2 u ( x ) ] ) d t . \\end{align*}"}
+{"id": "2204.png", "formula": "\\begin{align*} \\Phi ( x , y ) = \\frac { x } { \\log y } \\left ( \\omega ( u ) + O \\left ( \\frac { 1 } { \\log y } \\right ) \\right ) \\end{align*}"}
+{"id": "6979.png", "formula": "\\begin{align*} \\Gamma _ v = v K \\mbox { a n d } \\Gamma _ \\mu = \\mu ( K [ x ] ) . \\end{align*}"}
+{"id": "1010.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ 2 } { 3 2 ^ k } = \\frac { \\Gamma ( 1 / 4 ) ^ 2 } { 2 \\pi \\sqrt { \\pi } } , \\end{align*}"}
+{"id": "4507.png", "formula": "\\begin{align*} u _ { \\omega , c } ( t , x ) : = \\phi _ { \\omega , c } ( x - c t ) \\exp \\left \\{ i \\omega t + i \\frac { c } { 2 } ( x - c t ) - \\frac { 3 i } { 4 } \\int _ { - \\infty } ^ { x - c t } | \\phi _ { \\omega , c } ( \\eta ) | ^ 2 d \\eta \\right \\} . \\end{align*}"}
+{"id": "6810.png", "formula": "\\begin{align*} \\beta : = - \\textstyle \\sum _ { i = 1 } ^ { n - 1 } y _ i \\ , d x _ i - y ' \\ , d x ' + \\phi ( x ' ) \\ , d y ' . \\end{align*}"}
+{"id": "6723.png", "formula": "\\begin{align*} \\Gamma _ { H , H ^ { * } } ( \\underline { W } ) = W ^ { * } \\Gamma _ { H , H ^ { * } } ( H \\setminus \\underline { W } ) = H ^ { * } \\setminus W ^ { * } . \\end{align*}"}
+{"id": "6449.png", "formula": "\\begin{align*} \\hat f ( k ) & = \\int _ { \\R ^ 2 } e ^ { - z | x | ^ 2 + \\xi \\cdot x } e ^ { - i k \\cdot x } \\dd x = \\frac { \\pi } { z } e ^ { \\frac { 1 } { 4 z } ( \\xi - i k ) ( \\xi - i k ) } = \\frac { \\pi } { z } e ^ { \\frac { 1 } { 4 z } ( - | k | ^ 2 - 2 i \\xi \\cdot k + \\xi \\cdot \\xi ) } . \\end{align*}"}
+{"id": "8859.png", "formula": "\\begin{align*} \\mathbf s _ n = \\mathbf F ^ H \\tilde { \\mathbf s } _ n \\in \\mathbb { C } ^ { L } , n \\in \\mathcal { N } . \\end{align*}"}
+{"id": "2176.png", "formula": "\\begin{align*} \\frac { \\rho \\beta b ^ { 2 } } { d } = \\frac { \\rho \\beta } { 3 / 2 + \\left ( 1 + \\beta \\right ) \\left ( T / 2 - \\varepsilon \\right ) ^ { 2 } / \\left ( \\varepsilon \\left ( T - \\varepsilon \\right ) \\right ) } \\geq 1 - \\rho . \\end{align*}"}
+{"id": "4510.png", "formula": "\\begin{align*} \\| \\phi _ { \\omega , c } \\| _ { L ^ 2 ( \\R ) } ^ 2 = 8 \\tan ^ { - 1 } \\sqrt { \\frac { \\sqrt { 4 \\omega } + c } { \\sqrt { 4 \\omega } - c } } < 4 \\pi . \\end{align*}"}
+{"id": "6530.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( a , b ; c ; x ) = \\frac { \\sin ^ { - 1 } ( \\sqrt { x } ) } { \\sqrt { x ( 1 - x ) } } \\end{align*}"}
+{"id": "7737.png", "formula": "\\begin{align*} \\dot z ^ { * } \\bigl ( Y , y ^ { * } ( N , t ) \\bigr ) \\times \\ , \\dot y ^ { * } ( N , t ) = 1 , \\ , \\pi - \\end{align*}"}
+{"id": "3821.png", "formula": "\\begin{align*} \\inf _ { \\gamma \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\mathbb { E } _ { \\gamma } [ g ( S _ 1 , S _ 2 ) ] & = \\sup _ { x \\in \\mathbb { R } } \\max \\left \\{ \\mu _ 1 ( x ) + \\mu _ 2 ( z - x ) - 1 , 0 \\right \\} \\ \\\\ \\sup _ { \\gamma \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\mathbb { E } _ { \\gamma } [ g ( S _ 1 , S _ 2 ) ] & = 1 + \\inf _ { x \\in \\mathbb { R } } \\min \\left \\{ \\mu _ 1 ( x ) + \\mu _ 2 ( z - x ) - 1 , 0 \\right \\} . \\end{align*}"}
+{"id": "8360.png", "formula": "\\begin{align*} Q = S S ^ t + T T ^ t , A = S T ^ t + T S ^ t , L = Q - A = ( S - T ) ( S - T ) ^ t . \\end{align*}"}
+{"id": "5894.png", "formula": "\\begin{align*} c _ v & = \\sup _ { x \\in Q } ( v ( x ) , 1 ) , \\ Q = [ 0 , 1 ] ^ d \\end{align*}"}
+{"id": "8719.png", "formula": "\\begin{align*} & \\quad \\{ c _ { 2 , \\tau } ( \\widetilde { X } _ { 1 , 2 } ) ( \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { 2 } \\} \\\\ & = \\{ c _ { 2 , \\tau } ( \\widetilde { Y } _ { 1 , 2 } ) ( \\| Y _ { 1 } - Y _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { 2 } \\} \\\\ & = \\{ c _ { 2 , \\tau } ( \\widetilde { Z } _ { 1 , 1 } ) ( \\| X _ { 1 } - Y _ { 1 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { 2 } \\} = Q _ { 2 } . \\end{align*}"}
+{"id": "6940.png", "formula": "\\begin{align*} \\overline { \\mathrm { o r b } ( x ) } \\supset \\{ u \\in \\C ^ { m | n } \\mid u _ i - u _ { i _ 0 } = 2 u _ i + u _ { m + j } = \\pm \\frac 1 2 , u _ s = x _ s \\forall s \\notin \\{ i , i _ 0 , m + j \\} \\} , \\end{align*}"}
+{"id": "1683.png", "formula": "\\begin{align*} \\rho ( E _ 3 [ \\deg \\tau ] ) = \\tau \\pi \\psi \\widehat { \\tau } ( E _ 3 [ \\deg \\tau ] ) = \\tau \\pi \\psi ( \\ker \\tau ) = \\tau \\pi \\psi ( G ) = \\tau ( G ) = 0 . \\end{align*}"}
+{"id": "2662.png", "formula": "\\begin{align*} b _ 0 t '' _ 0 + c _ 0 t '' _ { 1 } & = d _ 0 , \\\\ a _ j t _ { j - 1 } '' + b _ j t '' _ j + c _ j t '' _ { j + 1 } & = d _ j ; 1 \\leq j \\leq N - 1 , \\\\ a _ { N } t '' _ { N - 1 } + b _ { N } t '' _ { N } & = d _ N . \\end{align*}"}
+{"id": "6834.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { Z } } ( - 1 ) ^ j z ^ j q ^ { j ( j - 1 ) / 2 } = ( q , z , q / z ; q ) _ \\infty , \\end{align*}"}
+{"id": "3118.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } S _ B [ n , k ] ( t ) _ { k , q } ^ D = \\sum _ { k = 0 } ^ { n } S _ D [ n , k ] ( t ) _ { k , q } ^ D + \\sum _ { k = 0 } ^ { n - 1 } n \\cdot [ 2 ] _ q ^ { n - k - 1 } q ^ { n - k - 1 } S [ n - 1 , k ] _ { q ^ 2 } ( t ) _ { k , q } ^ D . \\end{align*}"}
+{"id": "7113.png", "formula": "\\begin{align*} \\big | I _ \\varepsilon ^ 1 \\big | = \\left | \\int _ { \\mathbb { R } ^ n } J _ \\varepsilon ( | x | ) \\big ( f _ \\xi ( x ) - f _ \\xi ( 0 ) - \\nabla f _ \\xi ( 0 ) \\cdot x - \\frac { 1 } { 2 } x ^ T D ^ 2 f _ \\xi ( 0 ) x \\big ) \\ : x \\right | . \\end{align*}"}
+{"id": "3971.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\leq \\mathcal { I } ( \\eta _ 0 , \\delta ) - \\mathcal { I } ( \\eta , 0 ) & = \\mathcal { I } ( \\eta _ 0 , \\delta ) - \\mathcal { I } ( \\eta , \\delta ) + \\mathcal { I } ( \\eta , \\delta ) - \\mathcal { I } ( \\eta , 0 ) \\\\ & \\leq \\Psi _ 1 \\left ( \\eta _ 0 - \\eta , M ( 1 - \\eta / \\eta _ 0 ) \\right ) + \\Psi _ 2 \\left ( M \\delta , \\delta \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "6648.png", "formula": "\\begin{align*} \\psi _ \\beta ( x ) : = ( 1 + | x | ^ 2 ) ^ { - \\frac { \\beta } 2 } \\ ; \\ , \\ ; ( x \\in \\R ^ N ) \\ , , \\end{align*}"}
+{"id": "4855.png", "formula": "\\begin{align*} \\mu : = \\frac 1 2 \\delta _ x + \\frac 1 2 \\delta _ y . \\end{align*}"}
+{"id": "2695.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { k = 1 } } ^ { n ^ s } \\psi ( k ; n ^ s ) ( k , n ^ s ) _ s & = \\sum \\limits _ { \\substack { k = 1 } } ^ { n ^ s } \\psi ( k ; n ^ s ) \\sum \\limits _ { d ^ s | ( k , n ^ s ) _ s } \\Phi _ s ( d ^ s ) \\\\ & = \\sum \\limits _ { \\substack { k = 1 } } ^ { n ^ s } \\psi ( k ; n ^ s ) \\sum \\limits _ { \\substack { d ^ s \\mid k \\\\ d ^ s \\mid n ^ s } } \\Phi _ s ( d ^ s ) . \\end{align*}"}
+{"id": "4189.png", "formula": "\\begin{align*} H * w = \\tilde { H } * w , \\end{align*}"}
+{"id": "7876.png", "formula": "\\begin{align*} \\mathcal { S } = \\left \\{ A ^ \\pm : \\ A \\in \\binom { [ n ] } { k } \\mbox { a n d } n \\in A \\right \\} \\mbox { a n d } \\Omega \\setminus \\mathcal { S } . \\end{align*}"}
+{"id": "3604.png", "formula": "\\begin{align*} 1 0 ^ m - 6 \\ \\le \\ v ( f ) \\ = \\ v ( 2 ^ \\gamma ) \\ \\le \\ 2 + \\gamma . \\end{align*}"}
+{"id": "3421.png", "formula": "\\begin{align*} 0 < \\varepsilon < \\varepsilon _ L : = ( \\frac \\delta { 4 L } ) ^ 4 , \\end{align*}"}
+{"id": "2319.png", "formula": "\\begin{align*} \\| v \\| _ A = \\langle A v , v \\rangle ^ \\frac { 1 } { 2 } = \\gamma \\langle e _ p , A ^ { - 1 } e _ p \\rangle ^ \\frac { 1 } { 2 } = \\gamma ^ \\frac { 1 } { 2 } = \\langle e _ p , A ^ { - 1 } e _ p \\rangle ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "8077.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } \\left | \\left | u _ n \\right | ^ { p - 2 } u _ n - | u | ^ { p - 2 } u \\right | ^ { p ' } d x & \\leq C \\int _ { \\Omega } \\left | u _ n - u \\right | ^ { p ' ( p - 1 ) } d x \\\\ & = C \\left \\| u _ n - u \\right \\| _ { p } ^ { p } . \\end{aligned} \\end{align*}"}
+{"id": "3962.png", "formula": "\\begin{align*} \\boldsymbol { K } _ 1 \\left ( \\gamma _ { 1 , 3 } ^ { \\eta _ 1 , \\delta } , \\mu _ 1 \\right ) = \\mathbb { E } \\left [ c _ 1 ( S _ 1 , \\widetilde { S } _ 1 ) \\right ] \\leq \\eta _ 0 , S _ 1 = ( Y _ 1 , X ) \\widetilde { S } _ 1 = ( \\widetilde { Y } _ 1 , \\widetilde { X } ) . \\end{align*}"}
+{"id": "1606.png", "formula": "\\begin{align*} f & = K S _ { \\xi \\chi } f \\\\ & = K \\int _ { \\Theta } v ^ { 2 } ( w ) s ^ { 2 } ( w ) \\pi _ { G ( w ) } \\xi _ { w } ^ { \\ast } \\chi _ { w } \\pi _ { F ( w ) } f d \\mu ( w ) \\\\ & = \\int _ { \\Theta } v ^ { 2 } ( w ) s ^ { 2 } ( w ) K \\pi _ { G ( w ) } \\xi _ { w } ^ { \\ast } \\chi _ { w } \\pi _ { F ( w ) } f d \\mu ( w ) \\\\ & = \\int _ { \\Theta } T _ { w } f d \\mu ( w ) . \\end{align*}"}
+{"id": "1734.png", "formula": "\\begin{align*} \\frac { \\delta F } { \\delta \\mu } ( \\nu _ t , \\mu _ t , y ) - \\frac { \\sigma ^ 2 } { 2 } \\log \\frac { \\Phi ( \\nu _ t , \\mu _ t ) ( y ) } { \\rho ( y ) } = \\Tilde { C _ t } , \\end{align*}"}
+{"id": "4761.png", "formula": "\\begin{align*} u _ j + \\iota v _ j = x _ j + \\iota y _ j + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "2049.png", "formula": "\\begin{align*} P _ { \\leq N } u = u _ { \\leq N } , P _ { \\geq N } u = u _ { \\geq N } . \\end{align*}"}
+{"id": "3983.png", "formula": "\\begin{align*} ( f _ { \\mathcal { S } } ) _ \\lambda ( s _ 1 , s _ 2 ) = \\sup _ { x ^ \\prime \\in \\mathbb { R } ^ d } \\left [ \\varphi _ 1 ( x ^ \\prime - x _ 1 , \\lambda _ 1 ) + \\varphi _ 2 ( x ^ \\prime - x _ 2 , \\lambda _ 2 ) \\right ] . \\end{align*}"}
+{"id": "3055.png", "formula": "\\begin{align*} H ^ \\star = \\nu P ( \\Lambda ) P ^ T . \\end{align*}"}
+{"id": "2514.png", "formula": "\\begin{align*} \\| d _ c \\| \\le \\kappa _ c ^ \\frac { 1 } { 2 } \\| d _ c \\| _ { A _ c } = \\kappa _ c ^ \\frac { 1 } { 2 } \\| B _ c A _ c ^ { - 1 } r _ c \\| _ { A _ c } \\le 2 \\kappa _ c ^ \\frac { 1 } { 2 } \\| A _ c ^ { - 1 } P ^ t r _ \\mu \\| _ { A _ c } \\le 2 \\kappa _ c ^ \\frac { 1 } { 2 } \\| A ^ { - 1 } r \\| _ A . \\end{align*}"}
+{"id": "5609.png", "formula": "\\begin{align*} I ( \\xi _ { 1 } , \\xi _ { n } | X ) : = \\int _ { X } I ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) d \\eta ( x ) \\quad \\textrm { a n d } H ( \\xi _ { n } | X ) : = \\int _ { X } H ( \\xi _ { n } ^ { x } ) d \\eta ( x ) . \\end{align*}"}
+{"id": "768.png", "formula": "\\begin{align*} \\dfrac { 1 } { ( k - 1 ) ! } \\delta _ { i _ 1 i _ 2 \\cdots i _ k } ^ { j _ 1 j _ 2 \\cdots j _ k } ( w v _ 1 ^ t ) ^ { i _ 1 } _ { j _ 1 } ( w v _ 2 ^ t ) ^ { i _ 2 } _ { j _ 2 } ( A _ 1 ) ^ { i _ 3 } _ { j _ 3 } \\cdots ( A _ { k - 2 } ) ^ { i _ k } _ { j _ k } = 0 . \\end{align*}"}
+{"id": "8701.png", "formula": "\\begin{align*} \\frac { T _ 1 } { \\surd { ( \\Delta _ { 1 } ) } } = \\frac { f ^ { ( 2 ) } ( \\tau ) } { | f ^ { ( 1 ) } ( \\tau ) | } \\frac { \\gamma ^ { - 1 } \\| \\Sigma _ 1 - \\Sigma _ 2 \\| _ { \\rm F } ^ 2 } { \\surd { \\{ \\frac { 2 } { n ( n - 1 ) } ( \\Sigma _ { 1 } ^ { 2 } ) + \\frac { 2 } { m ( m - 1 ) } ( \\Sigma _ { 2 } ^ { 2 } ) + \\frac { 4 } { n m } ( \\Sigma _ { 1 } \\Sigma _ { 2 } ) \\} } } \\{ 1 + o ( 1 ) \\} . \\end{align*}"}
+{"id": "5242.png", "formula": "\\begin{align*} \\begin{aligned} Q _ 1 ^ { ( n + 1 ) } \\circ \\partial & = \\partial \\circ Q _ 1 ^ { ( n ) } \\circ f ^ { ( n ) } , \\\\ Q _ 2 ^ { ( n + 1 ) } \\circ \\partial & = \\partial \\circ \\frac 1 { f ^ { ( n ) } } \\circ Q _ 2 ^ { ( n ) } , \\\\ Q _ 1 ^ { ( n + 1 ) } \\circ Q _ 2 ^ { ( n + 1 ) } \\circ \\partial & = \\partial \\circ Q _ 1 ^ { ( n ) } \\circ Q _ 2 ^ { ( n ) } . \\end{aligned} \\end{align*}"}
+{"id": "4489.png", "formula": "\\begin{align*} \\langle D _ { j } S _ { \\omega , \\mathbf { c } } ( \\Phi ) , \\psi \\rangle : = \\lim _ { \\epsilon \\rightarrow 0 } \\frac { S _ { \\omega , \\mathbf { c } } ( \\Phi + \\epsilon E _ j ( \\psi ) ) - S _ { \\omega , \\mathbf { c } } ( \\Phi ) } { \\epsilon } , \\ \\ \\Phi \\in \\mathcal { H } ^ 1 , \\ \\psi \\in ( H ^ 1 ( \\R ^ d ) ) ^ { d } , \\ j = 1 , 2 , 3 , \\end{align*}"}
+{"id": "347.png", "formula": "\\begin{align*} \\sigma _ 1 ( 1 ) = 1 , \\ \\ \\sigma _ 1 ( 2 ) = 3 , \\ \\ \\sigma _ 1 ( 3 ) = 4 , \\ \\ \\sigma _ 1 ( 4 ) = 2 , \\end{align*}"}
+{"id": "1706.png", "formula": "\\begin{align*} ( z ^ 2 - 1 ) ^ \\alpha : = ( z + 1 ) ^ \\alpha ( z - 1 ) ^ \\alpha , \\end{align*}"}
+{"id": "2768.png", "formula": "\\begin{align*} \\| v _ t \\| _ { L ^ 2 ( \\mathrm { M } _ 0 ) } ^ 2 = ( u | ( \\Delta + \\lambda - q ) w _ t ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } . \\end{align*}"}
+{"id": "8031.png", "formula": "\\begin{align*} Q _ j = \\left \\{ \\otimes ^ { j - 1 } \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} \\right \\} \\otimes \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\otimes \\left \\{ \\otimes ^ { \\beta - j } \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} \\right \\} . \\end{align*}"}
+{"id": "1231.png", "formula": "\\begin{align*} S = \\{ s \\in \\R ^ 2 \\ , ; \\ , 0 \\leq s _ 2 \\leq f ( s _ 1 ) , 0 \\leq s _ 1 \\leq 1 \\} . \\end{align*}"}
+{"id": "3347.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ d X _ j ^ t \\otimes Y _ j \\right ) ( \\varphi \\otimes \\xi ) = \\sum _ { j = 1 } ^ d ( \\varphi \\circ X _ j ) \\otimes Y _ j \\xi . \\end{align*}"}
+{"id": "2905.png", "formula": "\\begin{align*} ( P - \\lambda ) ^ { - 1 } = Q ( \\lambda ) + Q ( \\lambda ) \\sum _ { j \\geq 1 } R ( \\lambda ) ^ j . \\end{align*}"}
+{"id": "2545.png", "formula": "\\begin{align*} \\lambda _ { j _ 1 } = \\lambda _ { j _ 1 ^ { - 1 } } = \\dots = \\lambda _ { j _ r } = \\lambda _ { j _ r ^ { - 1 } } = \\pm 1 . \\end{align*}"}
+{"id": "4476.png", "formula": "\\begin{align*} \\| U _ { \\lambda } ( 0 ) \\| _ { \\dot { H } ^ s } = \\lambda ^ { \\frac { d } { 2 } - 1 - s } \\| U _ 0 \\| _ { \\dot { H } ^ s } , \\end{align*}"}
+{"id": "8807.png", "formula": "\\begin{align*} & ( t - 1 ) ( 2 u ^ 3 + 6 u ^ 2 t + 9 u t ^ 2 + 1 0 u ^ 2 + 3 3 u t + 9 t ^ 2 + 2 8 u + 2 7 t + 2 0 ) = 0 \\end{align*}"}
+{"id": "5305.png", "formula": "\\begin{align*} | \\langle h ^ { = d } , g \\rangle | \\leq \\| h ^ { = d } \\| _ q \\ , \\| g \\| _ { q ' } \\leq \\| h ^ { = d } \\| _ q \\cdot \\mu _ { p } ( g ) / \\mu _ { p } ( g ) ^ { 1 / q } \\end{align*}"}
+{"id": "7201.png", "formula": "\\begin{align*} & \\widetilde { u } _ h : B _ 1 \\to \\S ^ { k - 1 } , \\widetilde { u } _ h ( y ) : = u _ h ( x _ h + \\sigma _ h y ) , \\\\ & p _ h : B _ 1 \\to ( 1 , + \\infty ) , p _ h ( y ) : = p ( x _ h + \\sigma _ h y ) , \\end{align*}"}
+{"id": "833.png", "formula": "\\begin{align*} \\tilde \\phi _ 0 ( u _ n ^ N ) = u _ n \\tilde \\phi ( u ^ G _ g ) = u _ g \\otimes \\lambda ^ { G / N } _ { q ( g ) } , n \\in N , \\ g \\in G , \\end{align*}"}
+{"id": "5102.png", "formula": "\\begin{align*} \\begin{cases} u = g & \\\\ D u = f & , \\end{cases} \\end{align*}"}
+{"id": "6377.png", "formula": "\\begin{align*} f _ { \\lvert \\tau \\rvert } ^ { ( m ) } ( 0 ) = \\sum _ { \\pi \\in S _ m } \\mathrm { d } y _ { \\pi ( 1 ) } \\otimes \\cdots \\otimes \\mathrm { d } y _ { \\pi ( m ) } , \\end{align*}"}
+{"id": "1644.png", "formula": "\\begin{align*} g ( ( \\pounds _ { \\xi _ i } { f } ) Y , X ) & = N ^ { \\ , ( 5 ) } ( \\xi _ i , Y , X ) + g ( { f } \\nabla _ { Y } \\ , \\xi _ i - \\nabla _ { { f } Y } \\ , \\xi _ i , \\ X ) . \\end{align*}"}
+{"id": "3758.png", "formula": "\\begin{align*} \\tilde { f } _ 0 ^ n & : = \\frac { 1 } { \\sqrt { \\hat { H } ( \\rho _ n ) ^ 2 + \\hat { P } ( \\rho _ n ) ^ 2 } } [ \\hat { H } ( \\rho _ n ) e _ 0 - \\hat { P } ( \\rho _ n ) e _ 1 ] , \\\\ \\tilde { f } _ 1 ^ n & : = \\frac { 1 } { \\sqrt { \\hat { H } ( \\rho _ n ) ^ 2 + \\hat { P } ( \\rho _ n ) ^ 2 } } [ \\hat { P } ( \\rho _ n ) e _ 0 + \\hat { H } ( \\rho _ n ) e _ 1 ] . \\end{align*}"}
+{"id": "8988.png", "formula": "\\begin{align*} \\Delta w _ \\delta = \\Delta w = \\mu ( S ^ \\varphi - m \\lambda ( x ) ) ( 1 + w ) \\ , . \\end{align*}"}
+{"id": "3605.png", "formula": "\\begin{align*} r ( p ) \\ = \\ 2 ^ { \\gamma } 3 ^ { \\beta } \\ \\geq \\ 2 ^ { 1 0 ^ m - 8 } \\cdot 3 \\ \\geq \\ 1 0 ^ { m + 1 } \\end{align*}"}
+{"id": "565.png", "formula": "\\begin{align*} \\phi ( \\xi + \\mathbf { h } ) = \\sum _ { | \\alpha | \\leq 3 } \\frac { \\partial ^ { \\alpha } \\phi ( \\xi ) } { \\alpha ! } \\mathbf { h } ^ { \\alpha } + \\sum _ { | \\alpha | = 4 } \\frac { \\partial ^ { \\alpha } \\phi ( \\xi + \\theta _ { \\xi } \\mathbf { h } ) } { \\alpha ! } \\mathbf { h } ^ { \\alpha } , \\end{align*}"}
+{"id": "9117.png", "formula": "\\begin{align*} \\delta ( h ( \\dots , \\zeta _ { [ - 2 ] } , \\zeta _ { [ - 1 ] } , x , u , u _ { [ 1 ] } , \\dots ) ) = h ( \\dots , \\zeta _ { [ - 1 ] } , g ( x , u ) , f ( x , u ) , u _ { [ 1 ] } , u _ { [ 2 ] } , \\dots ) \\ , . \\end{align*}"}
+{"id": "5881.png", "formula": "\\begin{align*} \\| u _ 0 \\| _ { L ^ 2 } = r \\ , . \\end{align*}"}
+{"id": "1402.png", "formula": "\\begin{align*} \\mathrm { V o l } _ { \\omega _ 0 } ( F _ b ) = \\int _ { F _ b } \\Omega _ 0 ^ m = \\int _ { { F _ { \\tilde { \\pi } ^ { - 1 } ( \\gamma ( b ) ) } } } \\gamma ^ * ( \\Omega _ 0 ) ^ m = \\int _ { { F _ { \\tilde { \\pi } ^ { - 1 } ( \\gamma ( b ) ) } } } { { c _ \\gamma ^ m } } \\cdot \\Omega _ 0 ^ m = { { c _ \\gamma ^ m } } \\int _ { F _ b } \\Omega _ 0 ^ m , \\end{align*}"}
+{"id": "1559.png", "formula": "\\begin{align*} f _ { \\l } ( y ) = \\frac { \\chi _ { ( 0 , 1 ) } ( y ) } { \\sqrt { 1 - y ^ { 2 } } } \\cos ( \\l \\arccos ( y ) ) . \\end{align*}"}
+{"id": "2145.png", "formula": "\\begin{align*} \\int _ { M _ i } K _ { g _ i } \\operatorname d v o l _ { g _ i } = 4 \\pi = \\int _ Z K _ x \\operatorname d s . \\end{align*}"}
+{"id": "6254.png", "formula": "\\begin{align*} & d \\ , G _ r ( \\xi , \\sigma _ 1 , \\ldots , \\sigma _ r , \\alpha _ 1 , \\ldots , \\alpha _ r , \\beta _ 1 , \\ldots , \\beta _ r ) \\ , = \\ , \\\\ & \\textbf { e } ( ( \\xi - \\sigma _ r ) \\beta _ r ) \\ , F ( \\sigma _ r - \\xi , \\alpha _ r - \\tau \\beta _ r ) \\ , G _ { r - 1 } ( \\xi , \\sigma _ 1 , \\ldots , \\sigma _ { r - 1 } , \\alpha _ 1 , \\ldots , \\alpha _ { r - 1 } , \\beta _ 1 , \\ldots , \\beta _ { r - 1 } ) \\ , d \\xi , \\end{align*}"}
+{"id": "9233.png", "formula": "\\begin{align*} 2 ^ { d _ { k , j } } = \\frac { 4 } { \\varepsilon _ 4 a ^ k } 2 ^ { - j } \\pi ^ { j - 2 k } D ^ { ( k ) } _ j F _ { 3 k - 2 j } \\le \\frac { 4 A ( \\sigma ) } { \\varepsilon _ 4 } \\frac { \\Gamma ( k + 1 / 2 ) ^ { 1 / 2 } } { ( B _ 1 a \\sqrt { \\pi } ) ^ k } , 0 \\le j \\le 3 k / 2 . \\end{align*}"}
+{"id": "9058.png", "formula": "\\begin{align*} S ( \\beta ) = \\beta _ 1 \\sigma _ j \\beta _ 2 \\end{align*}"}
+{"id": "459.png", "formula": "\\begin{align*} \\mathcal G : = \\{ ( t , r ) \\ , \\big | \\ , 1 - g ( r ) t > 0 \\} . \\end{align*}"}
+{"id": "7297.png", "formula": "\\begin{align*} a x _ i ^ n v ^ { n m ' - k m ' - l k ' } + b x _ i ^ k y _ i ^ l + c y _ i ^ m u ^ { m n ' - k l ' - l n ' } = 0 , \\end{align*}"}
+{"id": "3667.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) u + F ( x , u ) = 0 & \\Omega , \\\\ u = f & \\Omega ^ c . \\end{cases} \\end{align*}"}
+{"id": "5926.png", "formula": "\\begin{align*} \\begin{aligned} ( [ 1 , g ] , 1 ) ( [ \\epsilon , 1 ] , 1 ) & = ( [ \\epsilon , g ^ { s ( \\epsilon ) } ] , \\widetilde { C } _ { X ^ { \\ast } } ( [ 1 , g ] , [ \\epsilon , 1 ] ) ) \\\\ & = ( [ \\epsilon , g ^ { s ( \\epsilon ) } ] , \\nu ( \\epsilon , g ) \\widetilde { c } _ { X ^ { \\ast } } ( g ^ { s ( \\epsilon ) } , 1 ) ) \\\\ & = ( [ \\epsilon , g ^ { s ( \\epsilon ) } ] , \\nu ( \\epsilon , g ) ) \\\\ & = ( [ \\epsilon , 1 ] , 1 ) ( [ 1 , g ^ { s ( \\epsilon ) } ] , \\nu ( \\epsilon , g ) ) . \\end{aligned} \\end{align*}"}
+{"id": "6561.png", "formula": "\\begin{align*} \\| H ( f ) \\| _ { L _ { | x | _ h } ^ p L _ { \\theta } ^ { \\bar { p } _ 1 } ( \\mathbb H ^ n ) \\rightarrow L _ { | x | _ h } ^ p L _ { \\theta } ^ { \\bar { p } _ 2 } ( \\mathbb H ^ n ) } = G . \\end{align*}"}
+{"id": "1461.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\nabla \\psi _ { 1 , \\sigma _ i } ^ h \\nabla \\tilde { u } ^ i d x = 0 , h = 0 , 1 , \\cdots , n . \\end{align*}"}
+{"id": "5918.png", "formula": "\\begin{align*} \\beta _ y ^ { - 1 } \\chi _ y ( \\tfrac { 1 } { s } + \\tfrac { 1 } { t } ) = ( \\tfrac { 1 } { s } + \\tfrac { 1 } { t } , y ) _ F \\gamma ( y , \\psi ^ { \\tfrac { 1 } { 2 } } ) . \\end{align*}"}
+{"id": "5192.png", "formula": "\\begin{align*} p \\widetilde { \\mathbb { S } } _ { \\lambda ^ i } : = \\pi ^ { * } ( \\widetilde { \\mathbb { S } } _ { \\lambda ^ i } ) . \\end{align*}"}
+{"id": "5940.png", "formula": "\\begin{align*} h _ 1 h _ 2 = ( w _ 1 + w _ 2 , t _ 1 + t _ 2 + \\tfrac { \\langle w _ 1 , w _ 2 \\rangle } { 2 } ) = ( w _ 1 + w _ 2 , t _ 1 + t _ 2 + \\tfrac { 1 } { 2 } \\sum _ { k = 1 } ^ m ( a _ k ^ 1 b _ k ^ 2 - a _ k ^ 2 b _ k ^ 1 ) ) ; \\end{align*}"}
+{"id": "5023.png", "formula": "\\begin{align*} j _ f \\cdot ( x _ 1 , \\dots , x _ N ) : = ( e ^ { 2 \\pi \\sqrt { - 1 } d _ 1 / d _ 0 } x _ 1 , \\dots , e ^ { 2 \\pi \\sqrt { - 1 } d _ N / d _ 0 } x _ N ) . \\end{align*}"}
+{"id": "5138.png", "formula": "\\begin{align*} \\| f \\| _ X : = \\int _ 0 ^ 1 \\ ! | f | \\ , \\mathrm { d } x , \\end{align*}"}
+{"id": "5006.png", "formula": "\\begin{align*} F ( k , t , n ) = \\frac { ( n ) _ { k + 1 } } { n ^ t } \\frac { 1 } { k ! } \\Delta ^ k \\left [ \\frac { x ^ t } { n - x } \\right ] _ { x = 0 } ^ { } , \\end{align*}"}
+{"id": "8329.png", "formula": "\\begin{align*} y _ { D 1 } ^ { } = \\left \\{ \\begin{array} { c c c c c c c c c c } \\sqrt { 1 - \\alpha } h _ { A D } + \\sqrt { \\alpha } h _ { C D } z _ { d } + n _ { D 1 } , & \\mbox { i f } x = 1 ; \\\\ \\sqrt { \\alpha } h _ { C D } z _ d + n _ { D 1 } , & \\mbox { i f } x = 0 ; \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "8121.png", "formula": "\\begin{align*} \\varphi ^ + ( Y _ T ) = \\sum _ { F \\ge T } a _ F z ^ { - r ( F ) } \\end{align*}"}
+{"id": "4856.png", "formula": "\\begin{align*} t _ 0 = \\frac { ( 1 - \\lambda ) ^ { p - 1 } } { \\lambda ^ { p - 1 } + ( 1 - \\lambda ) ^ { p - 1 } } . \\end{align*}"}
+{"id": "4243.png", "formula": "\\begin{align*} D _ \\Psi = \\begin{pmatrix} 3 & 2 \\\\ - 2 & - 1 \\end{pmatrix} D _ \\Theta = \\begin{pmatrix} 1 & 0 \\\\ - 1 & 1 \\end{pmatrix} \\end{align*}"}
+{"id": "6298.png", "formula": "\\begin{align*} \\lambda _ t : = ( P _ { t , 1 } ^ u ) ^ * \\lambda _ 1 \\in T ^ * _ { \\gamma ( t ) } M , \\qquad \\forall \\ , t \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "7029.png", "formula": "\\begin{align*} Q _ j = Q _ i - ( Q _ i - Q _ j ) . \\end{align*}"}
+{"id": "5370.png", "formula": "\\begin{align*} & ~ ~ ~ ~ ( \\beta M _ 1 ^ H - \\alpha M _ 2 ^ H ) ^ H ( \\beta M _ 1 ^ H - \\alpha M _ 2 ^ H ) - ( \\beta M _ 1 ^ H + \\alpha M _ 2 ^ H ) ^ H ( \\beta M _ 1 ^ H + \\alpha M _ 2 ^ H ) \\\\ & = - \\overline { \\beta } \\alpha M _ 1 M _ 2 ^ H - \\overline { \\alpha } \\beta M _ 2 M _ 1 ^ H - \\overline { \\beta } \\alpha M _ 1 M _ 2 ^ H - \\overline { \\alpha } \\beta M _ 2 M _ 1 ^ H \\\\ & = - 2 \\alpha \\overline { \\beta } ( M _ 1 M _ 2 ^ H + M _ 2 M _ 1 ^ H ) \\\\ & \\leq 0 . \\end{align*}"}
+{"id": "7793.png", "formula": "\\begin{align*} \\Phi ( \\mathcal { R } A ) = \\mathcal { R } ( \\Phi ( A ) ) , \\end{align*}"}
+{"id": "2199.png", "formula": "\\begin{align*} { \\mathcal { C } } _ { \\alpha } = ( I _ n \\otimes U ) \\Pi \\big ( \\Lambda \\otimes B _ 1 ^ { ( \\alpha ) } + ( - 2 I _ { m } ) \\otimes B _ 2 ^ { ( \\alpha ) } \\big ) \\Pi ^ T ( I _ n \\otimes U ) ^ T . \\end{align*}"}
+{"id": "9270.png", "formula": "\\begin{align*} \\int _ a ^ { b } ( A + x ) ^ { \\xi } x ^ { \\lambda } \\ , d x \\simeq \\begin{cases} ( b - a ) b ^ { \\lambda } ( b + A ) ^ { \\xi } & \\xi + \\lambda > - 1 , \\\\ b ^ { \\lambda } ( b + A ) ^ { - \\lambda } \\log \\Big ( 1 + \\frac { b - a } { a + A } \\Big ) & \\xi + \\lambda = - 1 , \\\\ ( b - a ) b ^ { \\lambda } ( a + A ) ^ { \\xi + \\lambda + 1 } ( b + A ) ^ { - \\lambda - 1 } & \\xi + \\lambda < - 1 , \\end{cases} \\end{align*}"}
+{"id": "6133.png", "formula": "\\begin{align*} \\tau = \\frac { \\sigma q } { q - 2 } < \\left ( 1 - \\frac { 2 } { q } \\right ) \\min \\{ s , 1 - s \\} . \\end{align*}"}
+{"id": "5363.png", "formula": "\\begin{align*} \\rho ^ { d } \\| f ^ { = d } \\| _ q = \\| T _ { \\rho } f ^ { = d } \\| _ q \\le \\gamma ^ { \\frac { q - 2 } { 2 q } } \\| f ^ { = d } \\| _ 2 ^ { 2 / q } \\le \\sqrt { \\gamma } . \\end{align*}"}
+{"id": "7369.png", "formula": "\\begin{align*} \\Sigma _ N ^ 2 ( L ) : = \\langle ( S _ N ( \\ell ) - L ) ^ 2 \\rangle = \\langle S _ N ( \\ell ) ^ 2 \\rangle - L ^ 2 . \\end{align*}"}
+{"id": "6142.png", "formula": "\\begin{align*} d ( x , y ) : = \\max _ { 0 \\le j \\le n - 1 } ( x - y ) ^ 2 \\cdot y ^ j \\cdot x ^ { n - 1 - j } . \\end{align*}"}
+{"id": "2766.png", "formula": "\\begin{align*} v _ t = \\sum _ { j \\in N _ t ^ 0 } a _ j \\varphi _ j . \\end{align*}"}
+{"id": "5385.png", "formula": "\\begin{align*} \\| \\hat { \\mathbf { A } } \\hat { \\mathbf { x } } _ k + \\hat { \\mathbf { B } } \\hat { \\mathbf { u } } _ k \\| _ { \\hat { \\mathcal { X } } } ^ 2 - \\| \\hat { \\mathbf { x } } _ k \\| _ { \\hat { \\mathcal { X } } } ^ 2 = \\| \\hat { \\mathbf { x } } _ { k + 1 } \\| _ { \\hat { \\mathcal { X } } } ^ 2 - \\| \\hat { \\mathbf { x } } _ k \\| _ { \\hat { \\mathcal { X } } } ^ 2 \\leq \\| \\hat { \\mathbf { u } } _ k \\| ^ 2 - \\| \\hat { \\mathbf { y } } _ k \\| ^ 2 = \\| \\hat { \\mathbf { u } } _ k \\| ^ 2 - \\| \\hat { \\mathbf { C } } \\hat { \\mathbf { x } } _ k + \\hat { \\mathbf { D } } \\hat { \\mathbf { u } } _ k \\| ^ 2 , \\end{align*}"}
+{"id": "3113.png", "formula": "\\begin{align*} t ^ n = \\sum _ { k = 0 } ^ { n } S _ B ( n , k ) ( t ) _ k ^ B , \\end{align*}"}
+{"id": "4636.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\nu } = 0 \\ , \\mbox { o n } \\ , \\partial \\Omega \\times ( 0 , + \\infty ) . \\end{align*}"}
+{"id": "1080.png", "formula": "\\begin{align*} t ^ { \\sigma } \\| \\Phi ( u ) ( t ) - \\Phi ( v ) ( t ) \\| _ { L ^ a } \\leq C \\rho ( u , v ) \\sum _ { k = 0 } ^ { \\infty } ( C \\lambda ) ^ k M ^ { p k + m - 1 } . \\end{align*}"}
+{"id": "1771.png", "formula": "\\begin{align*} X _ T ^ \\ast : = \\sup _ { s \\in [ 0 , T ] } X _ s . \\end{align*}"}
+{"id": "2532.png", "formula": "\\begin{align*} t _ i \\ , t _ j = \\lambda \\quad \\quad \\{ i , j \\} \\in M . \\end{align*}"}
+{"id": "2450.png", "formula": "\\begin{align*} & ( \\gamma , m ) \\mapsto m \\\\ & ( \\gamma , m ) \\mapsto \\gamma \\cdot m \\\\ & ( \\gamma ' , \\gamma \\cdot m ) ( \\gamma , m ) = ( \\gamma ' \\gamma , m ) \\\\ & ( \\gamma , m ) \\mapsto ( \\gamma ^ { - 1 } , \\gamma \\cdot m ) \\end{align*}"}
+{"id": "8198.png", "formula": "\\begin{align*} P ^ \\omega \\left ( E _ { j , t } \\mid A _ { j - 1 , t } \\right ) \\leq 1 - e ^ { - 2 h ( t ) \\log h ( t ) } , j = 2 , \\ldots , t / h ( t ) . \\end{align*}"}
+{"id": "8118.png", "formula": "\\begin{align*} \\varphi ^ + ( Y _ T ) + \\varphi ^ + ( Y _ F ) \\alpha ( Y _ \\bullet ) = \\varphi ^ + ( Y _ T ) - \\varphi ^ + ( Y _ F ) a = \\varphi ^ - ( Y _ T ) \\end{align*}"}
+{"id": "8622.png", "formula": "\\begin{align*} & P ( W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) = 1 , A | Z _ 0 ( \\Delta \\ell _ 2 ) = m , Z _ 0 ( \\Delta \\ell _ 1 ) = n , X _ { \\ell _ 1 , \\Delta } = 1 , B _ { \\ell _ 1 \\Delta } ( \\ell _ 2 \\Delta ) = i ) \\\\ & = i \\nu \\Delta p ^ k _ { j , + } ( t - \\Delta \\ell _ 2 ) . \\end{align*}"}
+{"id": "7595.png", "formula": "\\begin{align*} | B | - 2 ( 1 - | C | ) & = \\frac { \\tau _ { 1 } } { 4 } - 2 \\left ( 1 - \\frac { ( 3 + \\tau ^ 2 _ { 1 } ) } { 4 \\tau _ { 1 } } \\right ) \\\\ & = \\frac { 3 \\tau ^ 2 _ { 1 } - 8 \\tau _ { 1 } + 6 } { 4 \\tau _ { 1 } } > 0 \\end{align*}"}
+{"id": "5331.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ 2 ^ 2 \\leq \\rho ^ { - d } \\gamma \\cdot \\gamma _ 1 \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) ^ \\theta = ( e / \\rho ) ^ { d } \\gamma _ 1 \\cdot \\gamma . \\end{align*}"}
+{"id": "6623.png", "formula": "\\begin{align*} 0 = [ ( - \\varphi _ { \\lambda } ( x , [ { y _ 1 } _ \\eta y _ 2 ] ) + \\varphi _ \\lambda ( [ x _ \\mu y _ 1 ] , y _ 2 ) + ( - 1 ) ^ { x y _ 1 } \\varphi _ \\lambda ( y _ 1 , [ x _ \\mu y _ 2 ] ) ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] . \\end{align*}"}
+{"id": "8261.png", "formula": "\\begin{align*} \\tilde { c } ^ { ( 2 ) } _ { i , j } = \\begin{cases} c ^ { ( 2 ) } _ { i , j } & j \\leq l _ 0 - 2 ; \\\\ ( - 1 ) ^ { i + j } / l _ 0 & l _ 0 - 1 \\leq j \\leq l _ 2 . \\end{cases} \\end{align*}"}
+{"id": "1657.png", "formula": "\\begin{align*} g ( \\nabla _ { X } \\ , \\xi _ i , { f } Z ) = - \\eta ^ i ( X ) \\ , \\bar \\eta ( Z ) + g ( X , Q Z ) + \\frac 1 2 \\ , N ^ { \\ , ( 5 ) } ( X , \\xi _ i , Z ) . \\end{align*}"}
+{"id": "8416.png", "formula": "\\begin{align*} \\zeta _ { Q } \\left ( s \\right ) : = \\zeta _ { 2 } ( s , c ) = \\frac { \\pi } { \\sqrt { c } } \\ , \\frac { 1 } { s - 1 } + \\frac { \\pi } { \\sqrt { c } } \\ , \\left ( 2 \\gamma - \\log \\left ( 4 c \\right ) - 4 \\log \\left ( \\left | \\eta \\left ( i \\sqrt { c } \\right ) \\right | \\right ) \\right ) + O ( s - 1 ) . \\end{align*}"}
+{"id": "6656.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\R ^ N } | v ( x ) | \\zeta ( x ) | \\Delta \\gamma _ R | \\ , d x & \\le \\frac { \\bar C } { R ^ 2 } \\int _ { B _ R \\setminus B _ { R / 2 } } | v ( x ) | \\zeta ( x ) \\ , d x \\\\ & \\le \\frac { \\bar C C } { R ^ 2 } \\int _ { B _ R \\setminus B _ { R / 2 } } | v ( x ) | \\psi ( x ) \\ , d x , \\end{aligned} \\end{align*}"}
+{"id": "5764.png", "formula": "\\begin{align*} \\dfrac { 1 } { r ( x ) } = \\dfrac { 1 } { p ( x ) } + \\dfrac { 1 } { q ( x ) } \\mbox { f o r a . e . } \\ x \\in \\mathbb { R } ^ { n } . \\end{align*}"}
+{"id": "1713.png", "formula": "\\begin{align*} { \\mathcal P } _ { k , l } ^ { ( \\alpha , \\beta ) } ( w , \\phi ) : = w ^ { k - l } P _ l ^ { ( \\alpha - \\beta - 1 , \\beta + k - l ) } ( 2 w ^ 2 \\ ! - \\ ! 1 ) C _ { k - l } ^ \\beta ( \\cos \\phi ) , \\end{align*}"}
+{"id": "5966.png", "formula": "\\begin{align*} \\overline { \\overline { C } } _ { X ^ { \\ast } } ( p _ g , k _ g ) & = \\widetilde { C } _ { X ^ { \\ast } } ( p _ g , k _ g ) \\widetilde { s } ( p _ g ) \\widetilde { s } ( k _ g ) \\widetilde { s } ( g ) ^ { - 1 } \\\\ & = \\widetilde { s } ( p _ g ) \\widetilde { s } ( k _ g ) \\widetilde { s } ( g ) ^ { - 1 } = 1 ; \\end{align*}"}
+{"id": "7940.png", "formula": "\\begin{align*} | \\beta | & = \\sum _ { k _ 0 \\in \\N _ 0 } \\big ( ( - \\tfrac { 3 } { 2 } - ) + 2 k _ 0 \\big ) \\beta ( \\xi , k _ 0 e _ 0 ) + \\sum _ { k _ 0 \\in \\N _ 0 } 2 k _ 0 \\beta ( 0 , k _ 0 e _ 0 ) \\\\ & + \\sum _ { k _ 0 \\in \\N _ 0 } ( 2 k _ 0 + 1 ) \\beta ( 0 , k _ 0 e _ 0 + e _ { ( 0 , 1 ) } ) + \\sum _ { k _ 0 \\in \\N _ 0 } ( 2 k _ 0 + 2 ) \\beta ( 0 , k _ 0 e _ 0 + 2 e _ { ( 0 , 1 ) } ) \\\\ & + \\sum _ { \\mathbf { n } \\in \\N _ 0 ^ d } ( | \\mathbf { n } | - 2 ) \\beta ( \\mathbf { n } ) . \\end{align*}"}
+{"id": "924.png", "formula": "\\begin{align*} \\phi ( z ) = P _ k ^ { ( - N - 2 \\nu _ 1 , - N - 2 \\nu _ 2 ) } ( 1 + 2 z ) , \\end{align*}"}
+{"id": "5671.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | t _ \\nu \\star { u _ 0 } | ^ { p } ) | t _ \\nu \\star { v _ 0 } | ^ { q } \\leq \\mathbb { A } , \\ \\ \\mathbb { A } = \\Bigl ( \\frac { 2 2 ^ * _ \\mu - \\gamma _ p - \\gamma _ q } { 2 2 ^ * _ \\mu - 2 } \\nu ( \\gamma _ p + \\gamma _ q ) D ^ { \\frac { 2 } { \\gamma _ p + \\gamma _ q } } _ 0 \\Bigr ) ^ { \\frac { \\gamma _ p + \\gamma _ q } { 2 - \\gamma _ p - \\gamma _ q } } . \\end{align*}"}
+{"id": "5447.png", "formula": "\\begin{align*} \\mathbb { E } \\{ Z \\} = \\frac { \\kappa } { \\kappa + \\beta } , \\mathbb { E } \\{ Z ^ 2 \\} = \\frac { \\kappa ( \\kappa + 1 ) } { \\kappa + \\beta ( \\kappa + \\beta + 1 ) } . \\end{align*}"}
+{"id": "6830.png", "formula": "\\begin{align*} \\alpha ^ { ( N ) } _ n = \\frac { ( 1 - a q ^ { 2 n - N } ) ( a q ^ { 1 - N } ) _ N } { ( 1 - b _ 1 ) \\cdots ( 1 - b _ N ) } \\sum _ { j \\in \\Z } ( - 1 ) ^ j \\frac { q ^ { j n - j ( j + 1 ) / 2 } f _ { N , j , n } ( b _ 1 , \\ldots , b _ N ) } { ( a q ^ { 2 n - N - j } ) _ { N + 1 } } \\alpha _ { n - j } , \\end{align*}"}
+{"id": "2773.png", "formula": "\\begin{align*} 0 = \\int _ M \\chi ( q _ 2 - q _ 1 ) v _ 2 v _ 1 d x + \\int _ M [ \\Delta , \\psi ] v v _ 1 d x . \\end{align*}"}
+{"id": "6150.png", "formula": "\\begin{align*} \\rho ( a b ) \\rho ( - a b ) & = \\rho ( a ) \\rho ( ( - a ) b ) + \\rho ( b ) \\rho ( ( - b ) a ) \\\\ & = \\rho ( a ) \\rho ( - a ) + \\rho ( a ) \\rho ( b ) + \\rho ( b ) \\rho ( - b ) + \\rho ( b ) \\rho ( a ) \\\\ & = \\rho ( a ) \\rho ( b ) + \\rho ( b ) \\rho ( a ) . \\end{align*}"}
+{"id": "4230.png", "formula": "\\begin{align*} \\phi _ 0 = e ^ { \\eta _ 0 } e ^ { \\eta _ 1 } \\cdots e ^ { \\eta _ S } , \\end{align*}"}
+{"id": "6892.png", "formula": "\\begin{align*} J _ r ( x , \\beta ) = \\frac { 1 } { 2 v _ r ( x ) } ( \\beta - d _ r ( x ) ) ^ 2 + \\frac { 1 } { 6 } \\theta '' ( x , \\beta _ * ) ( \\beta - d _ r ( x ) ) ^ 3 \\end{align*}"}
+{"id": "4196.png", "formula": "\\begin{align*} \\langle x , \\pi ( k , \\ell ) g \\rangle \\overline { \\langle x , \\pi ( k + q , \\ell + p ) g \\rangle } = \\\\ \\frac { e ^ { 2 \\pi i k p / M } } { 3 } \\sum _ { t = 0 } ^ { 2 } e ^ { 2 \\pi i t / 3 } \\vert \\langle x , \\pi ( k , \\ell ) g _ { q p t } \\rangle \\vert ^ 2 \\end{align*}"}
+{"id": "9317.png", "formula": "\\begin{align*} C ( 0 , 1 , - 1 ) = \\left ( \\begin{array} { r r r } 0 & 1 & - 1 \\\\ - 1 & 0 & 1 \\\\ 1 & - 1 & 0 \\\\ \\end{array} \\right ) \\end{align*}"}
+{"id": "5995.png", "formula": "\\begin{align*} & d \\overline { \\Pi } _ { \\psi } ( e ^ + ) A ( [ \\epsilon , x ] ) - d \\overline { \\Pi } _ { \\psi } ( e ^ - ) A ( [ \\epsilon , x ] ) \\\\ & = \\pi i \\epsilon x ^ 2 A ( [ \\epsilon , x ] ) - \\frac { \\epsilon i } { 4 \\pi } \\frac { d ^ 2 } { d x ^ 2 } A ( [ \\epsilon , x ] ) \\\\ & = \\pi i \\epsilon x ^ 2 A ( [ \\epsilon , x ] ) - \\pi i \\epsilon x ^ 2 A ( [ \\epsilon , x ] ) - \\frac { \\epsilon i } { 4 \\pi } [ - 2 \\epsilon \\pi e ^ { - \\epsilon \\pi x ^ 2 } ] \\\\ & = \\tfrac { i } { 2 } A ( [ \\epsilon , x ] ) ; \\end{align*}"}
+{"id": "7255.png", "formula": "\\begin{align*} \\varphi ( t ) = \\left \\{ \\begin{array} { l l } 0 & \\\\ \\frac { 1 } { 2 4 } ( t - \\frac { x } { 2 } ) ^ 4 & \\\\ \\frac { x ^ 4 } { 2 4 \\times 2 ^ 4 } + \\frac { x } { 1 2 } ( t - x ) ^ 3 + \\frac { x ^ 2 } { 1 6 } ( t - x ) ^ 2 + \\frac { x ^ 3 } { 4 8 } ( t - x ) & \\ , . \\end{array} \\right . \\end{align*}"}
+{"id": "2651.png", "formula": "\\begin{align*} t _ \\alpha ( x ) = p ( x ) + q ( x ) \\tanh ( \\alpha x ) , \\end{align*}"}
+{"id": "834.png", "formula": "\\begin{align*} \\nabla h ( x ) = - \\frac { 1 } { m \\omega _ m } \\begin{cases} x & \\ | x | < 1 \\\\ \\frac { x } { | x | ^ m } & \\ | x | \\ge 1 \\end{cases} \\end{align*}"}
+{"id": "9262.png", "formula": "\\begin{align*} \\mathcal { V } f ( x ) = x ^ { \\lambda } \\mathcal { U } \\big ( ( \\cdot ) ^ { - \\lambda } f \\big ) ( x ) , x > 0 . \\end{align*}"}
+{"id": "5268.png", "formula": "\\begin{align*} ( 1 + \\rho | d x | ) ^ { q - 2 } \\leq ( 1 + \\omega ) ^ { q - 2 } \\leq e ^ { \\omega ( q - 2 ) } = \\rho ^ { - 1 / 2 } . \\end{align*}"}
+{"id": "6954.png", "formula": "\\begin{align*} b _ { 1 , T } ( ( \\ell _ j ) _ { \\sf p } ) + b _ { 2 , T } ( ( \\ell _ j ) _ { \\sf p } ) = b _ { 1 , T } ( ( \\ell _ j ) _ { \\sf q } ) + b _ { 2 , T } ( ( \\ell _ j ) _ { \\sf q } ) . \\end{align*}"}
+{"id": "5920.png", "formula": "\\begin{align*} \\widetilde { C } _ { X ^ { \\ast } } ( \\begin{pmatrix} y _ 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\begin{pmatrix} y _ 2 & 0 \\\\ 0 & 1 \\end{pmatrix} ) & = \\widetilde { C } _ { X ^ { \\ast } } ( [ y _ 1 , g _ 1 ] , [ y _ 2 , g _ 2 ] ) \\\\ & = \\nu ( y _ 2 , g _ 1 ) \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 ^ { y _ 2 } , g _ 2 ) \\\\ & = ( y _ 1 ^ { m } , y _ 2 ) _ F = ( y _ 1 , y _ 2 ) _ F . \\end{align*}"}
+{"id": "710.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { L _ { S } } \\mu _ { S , 0 } ( e ) \\d e = 0 \\ , . \\end{align*}"}
+{"id": "807.png", "formula": "\\begin{align*} y ^ { \\left ( n \\right ) } \\left ( x \\right ) = f \\left ( x , y \\right ) + \\int \\limits _ { x _ { 0 } } ^ { x } K d t , x > x _ { 0 } \\end{align*}"}
+{"id": "466.png", "formula": "\\begin{align*} x : = \\tilde r ( y ) , \\end{align*}"}
+{"id": "4333.png", "formula": "\\begin{align*} f ^ { \\mathcal { K } } ( \\cdot , t ) = f ^ { \\mathcal { K } } ( \\cdot , T _ { m - 1 } ) \\circ \\tilde { y } _ { T _ { m - 1 } } ^ { \\mathcal { K } } \\circ \\big ( \\tilde { y } _ t ^ { \\mathcal { K } } \\big ) ^ { - 1 } . \\end{align*}"}
+{"id": "6392.png", "formula": "\\begin{align*} C _ { 1 } X ^ 2 + C _ { 2 } = w _ { 2 } ^ { p } , \\end{align*}"}
+{"id": "7520.png", "formula": "\\begin{align*} w = a + \\zeta \\xi \\end{align*}"}
+{"id": "8492.png", "formula": "\\begin{align*} \\intop _ { 1 } ^ { \\infty } \\frac { d t } { ( t ^ { 2 } - 1 ) ^ { s } } = \\frac { \\Gamma \\left ( 1 - s \\right ) \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) } { 2 \\sqrt { \\pi } } , \\ , \\ , \\ , \\frac { 1 } { 2 } < ( s ) < 1 , \\end{align*}"}
+{"id": "6625.png", "formula": "\\begin{align*} \\varphi _ { \\lambda } ( x \\otimes a , y \\otimes b ) = \\sum _ { t \\in T } \\varphi _ { \\lambda t } ( x \\otimes a , y \\otimes b ) \\otimes b _ t \\end{align*}"}
+{"id": "5483.png", "formula": "\\begin{align*} h _ { \\mu } ( X , \\eta ) = - \\int _ { G } \\int _ { X } \\log \\frac { d g ^ { - 1 } \\eta } { d \\eta } ( x ) d \\eta ( x ) d \\mu ( g ) . \\end{align*}"}
+{"id": "5093.png", "formula": "\\begin{align*} & \\left \\| V ( \\gamma _ 1 , \\sigma _ 1 , t ) - V ( \\gamma _ 2 , \\sigma _ 2 , t ) \\right \\| _ { L ^ 2 _ N } \\leq \\| \\psi ( t ) \\| _ { W ^ { 1 , \\infty } } \\left ( \\| \\gamma _ 1 - \\gamma _ 2 \\| _ { L ^ 2 _ N } + \\frac { 1 } { \\sqrt { N } } | \\sigma _ 1 - \\sigma _ 2 | \\right ) \\end{align*}"}
+{"id": "5340.png", "formula": "\\begin{align*} \\rho ^ { d } \\| f ^ { = d } \\| _ q = \\| T _ { \\rho } f ^ { = d } \\| _ q \\le \\gamma '^ { \\frac { q - 2 } { q } } \\| f ^ { = d } \\| _ 2 ^ { 2 / q } \\le \\gamma ' . \\end{align*}"}
+{"id": "7082.png", "formula": "\\begin{align*} \\nu ( g ' ( a _ i ) ) - \\nu _ j ( g ) = \\nu ( g ' ) - \\nu _ j ( g ) = \\beta _ j < \\alpha _ i = - \\nu ( x - a _ i ) . \\end{align*}"}
+{"id": "3413.png", "formula": "\\begin{align*} \\widetilde V ( x ) = \\begin{cases} V ( x ) , \\ & | x | < M _ 0 ; \\\\ \\max \\{ V ( x ) , | x | ^ 2 \\} , \\ & | x | \\geq M _ 0 . \\end{cases} \\end{align*}"}
+{"id": "1799.png", "formula": "\\begin{align*} \\nu \\Big ( \\prod _ { i = 1 } ^ { k } a _ { p _ i } ^ * \\prod _ { j = 1 } ^ { k ' } a _ { q _ j } \\Big ) \\ , = \\ , \\delta _ { k , k ' } ( - 1 ) ^ \\frac { k ( k - 1 ) } { 2 } \\det \\big [ \\nu ( a _ { p _ i } ^ * a _ { q _ j } ) \\big ] _ { 1 \\leq i , j \\leq k } \\ . \\end{align*}"}
+{"id": "3807.png", "formula": "\\begin{align*} \\Sigma _ { \\mathrm { R D J } } ( \\delta ) : = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\ \\boldsymbol { K } _ 1 ( \\widehat { \\mu } _ { 1 3 } , \\gamma _ { 1 3 } ) \\leq \\delta _ 1 , \\ \\boldsymbol { K } _ 2 ( \\widehat { \\mu } _ { 2 3 } , \\gamma _ { 2 3 } ) \\leq \\delta _ 2 \\right \\} , \\end{align*}"}
+{"id": "4152.png", "formula": "\\begin{align*} Q _ { 3 , \\vec { v } _ { s , t , f , 0 } ^ { ( 2 ) } } = \\left ( \\left ( \\begin{array} { r r r } t ^ { 2 } + s & t & 1 \\\\ s t + 1 & s & 0 \\\\ t & 1 & 0 \\end{array} \\right ) , \\left ( \\begin{array} { r r r } t & 1 & 0 \\\\ s & 0 & 1 \\\\ 1 & 0 & 0 \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) \\right ) . \\end{align*}"}
+{"id": "5557.png", "formula": "\\begin{align*} C _ { n } = \\left \\{ \\omega : \\log \\left | \\omega _ { n } ^ { - 1 } \\omega _ { n + 1 } \\right | \\le n \\epsilon , H _ { 0 } \\omega _ { n } \\in A _ { n } , x _ { 0 } \\in \\left [ H _ { 0 } \\omega _ { n } ; \\omega _ { n } ^ { - 1 } \\omega _ { n + 1 } \\right ] \\right \\} . \\end{align*}"}
+{"id": "8814.png", "formula": "\\begin{align*} d u + A u \\ , d t = f ( u ) \\ , d t + \\sigma ( u ) B \\ , d W ( t ) , u ( 0 ) = u _ 0 , \\end{align*}"}
+{"id": "3321.png", "formula": "\\begin{align*} \\rho _ { \\theta } ( \\theta , f ( \\theta ) ) + 2 { \\rm R e } ( e ^ { c ( \\theta , f ( \\theta ) ) + i d ( \\theta , f ( \\theta ) ) } \\frac { \\partial g } { \\partial \\theta } ( \\theta ) ) = 0 . \\end{align*}"}
+{"id": "5101.png", "formula": "\\begin{align*} \\mathcal { A } [ g ] = \\{ u \\in W ^ { 2 , 2 } ( B _ 1 ) : u = g \\partial B _ 1 D u = D g \\partial B _ 1 \\} \\end{align*}"}
+{"id": "8084.png", "formula": "\\begin{align*} \\nu _ n = \\inf _ { M \\in \\mathcal { F } _ n } \\sup _ { u \\in M } G ( u ) . \\end{align*}"}
+{"id": "4273.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\mathrm { d } X _ { s } & = & u _ { s } \\mathrm { d } W _ { s } , \\\\ - \\mathrm { d } Y _ { s } & = & \\Big [ \\left \\langle Q _ { s } X _ { s } , X _ { s } \\right \\rangle + \\left \\langle R _ { s } u _ { s } , u _ { s } \\right \\rangle + \\kappa Z _ { s } \\Big ] \\mathrm { d } s - Z _ { s } \\mathrm { d } W ( s ) , \\\\ X _ { 0 } & = & x , Y _ { T } = \\left \\langle G X _ { T } , X _ { T } \\right \\rangle , \\end{array} \\right . \\end{align*}"}
+{"id": "6607.png", "formula": "\\begin{align*} k : = L ^ D \\ , ( = K ^ D ) . \\end{align*}"}
+{"id": "8237.png", "formula": "\\begin{align*} Z _ { A } ( s ) : = e ^ { - \\pi \\i N / 2 } e ^ { \\i \\sum _ { n = 1 } ^ { N } \\theta _ { n } / 2 } s ^ { - N / 2 } \\Lambda _ { A } ( s ) , \\end{align*}"}
+{"id": "5453.png", "formula": "\\begin{align*} M _ b ( \\theta ) = \\mathbb { E } _ { \\Phi } \\{ P _ s ^ b ( \\theta ) \\} \\end{align*}"}
+{"id": "228.png", "formula": "\\begin{align*} \\omega ( s _ 1 , . . . , s _ { M + 1 } ) = \\omega ( s _ 1 , . . . , s _ M \\ , \\mid \\ , s _ { M + 1 } ) . \\end{align*}"}
+{"id": "461.png", "formula": "\\begin{align*} \\mathcal A _ N ( \\omega _ 0 , \\omega _ 1 , \\rho _ 1 ) = \\begin{pmatrix} - 2 N + 2 - 2 N \\frac { \\omega _ 1 } { \\omega _ 0 } & - \\frac { 2 \\rho _ 1 } { \\omega _ 0 } - 2 \\\\ - 2 & - 2 N - 4 + \\frac 2 { \\omega _ 0 } - ( 2 N + 2 ) \\frac { \\omega _ 1 } { \\omega _ 0 } \\end{pmatrix} . \\end{align*}"}
+{"id": "3222.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\norm { S _ n ( \\nu _ k ) - \\overline { \\nu } } = 0 . \\end{align*}"}
+{"id": "8332.png", "formula": "\\begin{align*} \\| f \\| _ { p , q , \\mu } = \\Big ( p \\int _ 0 ^ \\infty t ^ { q - 1 } f _ { * \\mu } ( t ) ^ \\frac { q } { p } \\dd t \\Big ) ^ { \\frac 1 { q } } \\end{align*}"}
+{"id": "3330.png", "formula": "\\begin{align*} \\left | | \\beta _ j | - \\frac { 1 } { 2 } \\right | < \\frac { \\beta _ 0 } { 2 } , \\ \\ \\ j = \\pm 1 . \\end{align*}"}
+{"id": "8406.png", "formula": "\\begin{align*} ( E = N \\oplus N ^ { - 1 } K _ X \\oplus N ^ { - 1 } \\oplus N K _ X ^ { - 1 } , \\quad \\theta = \\begin{pmatrix} 0 & 0 & \\nu & q _ 2 \\\\ 0 & 0 & q _ 2 & \\mu \\\\ 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\end{pmatrix} ) . \\end{align*}"}
+{"id": "7083.png", "formula": "\\begin{align*} a _ n = \\sum _ { i = 0 } ^ n a ^ { \\frac { 1 } { p ^ i } } \\end{align*}"}
+{"id": "6069.png", "formula": "\\begin{align*} \\delta _ { 1 / r } ^ { A } \\delta _ { r } ^ { B } \\{ x \\in G \\colon \\| x \\| = 1 \\} \\subseteq \\mathcal { B } ^ { A } ( e , D ) \\subseteq \\mathcal { B } ( e , R ) , \\end{align*}"}
+{"id": "3936.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta _ 1 , 0 ) = \\inf _ { \\lambda _ 1 \\in \\mathbb { R } _ { + } } \\left [ \\lambda _ 1 \\delta _ 1 + \\sup _ { \\varpi \\in \\Pi \\left ( \\mu _ { 1 3 } , \\mu _ { 2 3 } \\right ) } \\int _ { \\mathcal { V } } f _ { \\lambda , 1 } ( v ) \\ , d \\varpi ( v ) \\right ] . \\end{align*}"}
+{"id": "3614.png", "formula": "\\begin{align*} q \\ = \\ p - 4 \\ \\geq \\ 1 0 ^ m + 3 - 4 \\ = \\ 1 0 ^ m - 1 . \\end{align*}"}
+{"id": "3197.png", "formula": "\\begin{align*} \\lim _ { a \\to \\infty } \\sup _ { x \\in e M _ + e , x \\neq 0 } \\frac { \\abs { ( M _ a ( \\mu ) - \\bar { \\mu } ) ( x ) } } { \\tau ( x ) } = 0 . \\end{align*}"}
+{"id": "2780.png", "formula": "\\begin{align*} | \\eta _ 2 | ^ 2 = | \\eta | ^ 2 / 4 + | \\eta _ 1 | ^ 2 - \\lambda . \\end{align*}"}
+{"id": "8645.png", "formula": "\\begin{align*} 1 - p _ 0 ( t ) = \\frac { ( 1 - p ) e ^ { \\lambda t } } { e ^ { \\lambda t } - p } = \\frac { q e ^ { \\lambda t } } { e ^ { \\lambda t } - p } . \\end{align*}"}
+{"id": "634.png", "formula": "\\begin{align*} \\langle a v , v \\rangle = \\phi ( a ) , a \\in M . \\end{align*}"}
+{"id": "2113.png", "formula": "\\begin{align*} X ( K ) \\cap \\Gamma = \\left \\{ p ^ n Q \\colon n \\ge 0 \\right \\} . \\end{align*}"}
+{"id": "540.png", "formula": "\\begin{align*} U ( t , \\xi ) : = \\left ( \\begin{array} { c } i \\langle \\xi \\rangle \\widehat { u } ( t , \\xi ) \\\\ \\partial _ { t } \\widehat { u } ( t , \\xi ) \\end{array} \\right ) , U _ { 0 } ( \\xi ) : = \\left ( \\begin{array} { c } i \\langle \\xi \\rangle \\widehat { u } _ { 0 } ( \\xi ) \\\\ \\widehat { u } _ { 1 } ( \\xi ) \\end{array} \\right ) , \\end{align*}"}
+{"id": "4717.png", "formula": "\\begin{align*} \\Pi _ \\chi ^ { \\rm s t i f f } \\widehat { P } _ \\chi \\vect e _ j = \\Pi _ 0 ^ { \\rm s t i f f } \\widehat { P } _ 0 \\vect e _ j + \\mathcal { O } ( | \\chi | ) = \\Pi _ 0 ^ { \\rm s t i f f } \\vect e _ j + \\mathcal { O } ( | \\chi | ) = \\vect e _ j + \\mathcal { O } ( | \\chi | ) , j = 1 , 2 , 3 , \\end{align*}"}
+{"id": "870.png", "formula": "\\begin{align*} \\left | \\frac { a _ i } { r } - \\frac { a _ i ' } { r ' } \\right | \\geq \\frac { | c _ i ^ \\bot | } { | r r ' | } i = 1 , 2 \\quad \\max _ { i = 1 , 2 } \\left \\{ | c _ i | \\left | \\frac { a _ i } { r } - \\frac { a _ i ' } { r ' } \\right | \\right \\} \\geq 1 . \\end{align*}"}
+{"id": "4019.png", "formula": "\\begin{align*} g ( P \\otimes \\lbrace \\alpha , \\beta \\rbrace ) = ( g P ) \\otimes \\lbrace g ( \\alpha ) , g ( \\beta ) \\rbrace . \\end{align*}"}
+{"id": "4231.png", "formula": "\\begin{align*} \\phi _ 0 = e ^ { \\eta _ 0 } b . \\end{align*}"}
+{"id": "6729.png", "formula": "\\begin{align*} [ w , x ] : = \\{ \\sigma ^ m y \\in \\{ 0 , 1 \\} ^ \\Z : w \\leq y \\leq x , \\ m \\in \\Z \\} . \\end{align*}"}
+{"id": "6520.png", "formula": "\\begin{align*} F _ X ( x ) = \\frac { 1 } { 2 } + \\frac { \\alpha ( x - \\mu ) } { 2 } \\bigg [ K _ { \\nu } ( \\alpha | x - \\mu | ) \\mathbf { L } _ { \\nu - 1 } ( \\alpha | x - \\mu | ) + \\mathbf { L } _ { \\nu } ( \\alpha | x - \\mu | ) K _ { \\nu - 1 } ( \\alpha | x - \\mu | ) \\bigg ] , \\end{align*}"}
+{"id": "7628.png", "formula": "\\begin{align*} M ( c , x , y ) : = ( h _ 1 ( c , x ) + h _ 2 ( c , x ) y + h _ 3 ( c , x ) y ^ 2 + h _ 4 ( c , x ) ( 1 - y ^ 2 ) ) \\end{align*}"}
+{"id": "2486.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle \\dfrac { a _ 2 } { m _ 1 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\xi _ n | ^ 2 d x + \\dfrac { a _ 1 } { 2 m _ 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\eta _ n | ^ 2 d x - \\dfrac { N } { 2 } a _ 1 a _ 2 \\int _ { \\mathbb { R } ^ N } \\eta _ n \\xi _ n ^ 2 d x = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1335.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { T _ i ( u ) } , \\int _ 0 ^ { + \\infty } e ^ { 2 \\rho _ i ( u + v ) - 2 \\rho _ i ( u ) } d v \\right ) _ { i \\in V } = \\left ( \\frac { 1 } { T _ i ^ { \\infty } } + \\frac { 1 } { \\theta _ i ^ 2 T _ i ^ * ( u ) } , \\frac { Z _ i ^ * ( u ) } { e _ i ^ * ( u ) } + \\frac { T _ i ^ { \\infty } } { \\theta _ i ^ 2 e _ i ^ * ( u ) ^ 2 } \\right ) _ { i \\in V } . \\end{align*}"}
+{"id": "5199.png", "formula": "\\begin{align*} \\mathsf { B } \\mathbb { P } & = \\{ B \\in \\mathsf { F } \\mathbb { P } : \\mathbb { P } = B ^ \\lessgtr \\} . \\\\ \\mathsf { C } \\mathbb { P } & = \\{ C \\in \\mathsf { P } \\mathbb { P } : \\exists B \\in \\mathsf { B } \\mathbb { P } \\ ( B \\leq C ) \\} . \\end{align*}"}
+{"id": "5623.png", "formula": "\\begin{align*} \\aligned \\left \\{ \\begin{array} { l l l } - i \\partial _ t \\Psi _ 1 = \\Delta \\Psi _ 1 + ( K ( x ) \\ast | \\Psi _ 1 | ^ { r _ 1 } ) | \\Psi _ 1 | ^ { r _ 1 - 2 } \\Psi _ 1 + \\nu p ( K ( x ) \\ast | \\Psi _ 2 | ^ { q } ) | \\Psi _ 1 | ^ { p - 2 } \\Psi _ 1 , \\\\ - i \\partial _ t \\Psi _ 2 = \\Delta \\Psi _ 2 + ( K ( x ) \\ast | \\Psi _ 2 | ^ { r _ 2 } ) | \\Psi _ 2 | ^ { r _ 2 - 2 } \\Psi _ 2 + \\nu q ( K ( x ) \\ast | \\Psi _ 1 | ^ { p } ) | \\Psi _ 2 | ^ { q - 2 } \\Psi _ 2 , \\end{array} \\right . \\endaligned \\end{align*}"}
+{"id": "9048.png", "formula": "\\begin{align*} b _ { I _ 1 , I _ 2 } ^ G & = \\sum _ { i \\in I _ 1 } b _ { i , 2 } = b _ { 1 , 2 } = - 1 , \\\\ b _ { I _ 2 , I _ 1 } ^ G & = \\sum _ { i \\in I _ 2 } b _ { i , 1 } = b _ { 2 , 1 } + b _ { 3 , 1 } + b _ { 4 , 1 } = 3 . \\end{align*}"}
+{"id": "1784.png", "formula": "\\begin{align*} f = \\sum _ { n = 0 } ^ \\infty \\sum _ { ( i _ 1 , \\ldots , i _ n ) \\in \\{ 1 , 2 \\} ^ n } f _ { i _ 1 , \\ldots , i _ n } \\end{align*}"}
+{"id": "16.png", "formula": "\\begin{align*} Z ( { \\mathcal { G } ( \\mathbb { Q } _ p ) } ) \\mathcal { G } ( \\mathbb { Q } ) \\cap \\mathbf { T } _ { K ^ \\circ } ( \\mathbb { Q } _ p ) = Z ( { \\mathcal { G } ( \\mathbb { Q } _ p ) } ) \\mathbf { T } _ { K ^ \\circ } ( \\mathbb { Q } ) , \\end{align*}"}
+{"id": "5269.png", "formula": "\\begin{align*} \\| 1 + d Z \\| _ { q } ^ { q } \\ge \\| 1 + d Z \\| _ { \\lfloor q \\rfloor } ^ { \\lfloor q \\rfloor } \\geq 1 + \\binom { \\lfloor q \\rfloor } { 2 } d ^ 2 \\ge 1 + \\frac { 1 } { 9 } q ^ 2 d ^ 2 , \\end{align*}"}
+{"id": "5415.png", "formula": "\\begin{align*} \\deg _ A ( f _ 1 ^ { n _ 1 } \\cdots f _ k ^ { n _ k } ) = ( ( f _ 1 ) ^ { n _ 1 } _ * \\cdots ( f _ k ) ^ { n _ k } _ * A ) \\cdot A . \\end{align*}"}
+{"id": "2986.png", "formula": "\\begin{align*} & \\theta ( ( F ( t _ 2 ) - F ( t _ 1 ) ) \\ , t _ 1 ^ { k _ 1 } \\cdots t _ n ^ { k _ n } ) = - a _ { ( p , 1 , 0 , \\dots , 0 ) } ( t _ 1 , \\dots , t _ n ) \\\\ = \\ & ( a _ { ( p , 0 , \\dots , 0 ) } \\cdot e _ 1 ) ( t _ 1 , \\dots , t _ n ) - a _ { ( q , 0 , \\dots , 0 ) } ( t _ 1 , \\dots , t _ n ) \\\\ = \\ & q _ { k _ 2 } ( t _ 1 , \\dots , t _ n ) \\cdot e _ 1 ( t _ 1 , \\dots , t _ n ) - q _ { k _ 2 + 1 } ( t _ 1 , \\dots , t _ n ) \\ . \\end{align*}"}
+{"id": "8398.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\triangle u = \\frac { 1 } { \\max _ { i = 1 , \\cdots , n - 1 } r _ i } \\big ( \\prod _ { i = 1 } ^ { n - 1 } | \\gamma _ i | ^ { i ( n - i ) } \\big ) ^ { \\frac { 1 2 } { n ( n ^ 2 - 1 ) } } \\cdot e ^ { u } . \\end{align*}"}
+{"id": "6558.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\frac { 1 } { ( 1 + t ) ^ a t ^ { \\beta } } d t = \\int _ 0 ^ 1 ( 1 - t ) ^ { - \\beta } t ^ { a + \\beta - 2 } d t = B ( 1 - \\beta , a + \\beta - 1 ) . \\\\ \\end{align*}"}
+{"id": "7051.png", "formula": "\\begin{align*} f _ { \\tilde { \\textbf { Q } } } = \\frac { g } { b } \\cdot ( b h ) . \\end{align*}"}
+{"id": "4830.png", "formula": "\\begin{align*} \\left | \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } \\leq x \\right ) - \\Phi ( x ) \\right | & = \\left | \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } > x \\right ) - ( 1 - \\Phi ( x ) ) \\right | \\\\ & \\leq \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } > x \\right ) + 1 - \\Phi ( x ) . \\end{align*}"}
+{"id": "5682.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ^ { + } ) = c F \\widehat { S _ { k } } ^ { \\# , 0 } ( \\Gamma _ { 0 } ( N ) ) \\end{align*}"}
+{"id": "3879.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ { L } \\phi _ \\ell ( x _ \\ell ) \\geq f ( x ) , \\forall x = ( x _ 1 , \\ldots , x _ L ) \\in \\mathcal { X } . \\end{align*}"}
+{"id": "7095.png", "formula": "\\begin{align*} \\mathcal { E } ^ { C H } ( c ) : = \\int _ { \\Omega } \\frac { 1 } { 2 } | \\nabla c | ^ 2 + f ( c ) \\ : x . \\end{align*}"}
+{"id": "7885.png", "formula": "\\begin{align*} \\mathcal { O } _ i ^ + : = \\left \\{ \\left ( ( A ^ + , \\underline { A } ^ + ) , ( B ^ + , \\underline { B } ^ + ) \\right ) , \\left ( ( A ^ + , \\underline { A } ^ + ) , ( B ^ - , \\underline { B } ^ - ) \\right ) : A , B \\in \\binom { [ n ] } { k } \\mbox { a n d } | A \\cap B | = k - i \\right \\} , \\end{align*}"}
+{"id": "7421.png", "formula": "\\begin{align*} t = \\tfrac { 1 } { \\sqrt { N } } N \\ge L \\ , . \\end{align*}"}
+{"id": "4689.png", "formula": "\\begin{align*} \\ker \\bigl ( \\Pi _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\bigr ) = \\left \\{ 0 \\right \\} , \\mathcal { D } \\bigl ( \\mathcal { A } _ { 0 , \\chi } ^ { \\rm s t i f f ( s o f t ) } \\bigr ) \\cap \\mathcal { R } \\bigl ( \\Pi _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\bigr ) = \\{ 0 \\} . \\end{align*}"}
+{"id": "3835.png", "formula": "\\begin{align*} \\mathcal { I } ( 0 ) = \\sup _ { \\gamma \\in \\mathcal { F } ( \\mu _ { 1 3 } , \\mu _ { 2 3 } ) } \\int _ \\mathcal { S } f \\ , d \\gamma . \\end{align*}"}
+{"id": "3537.png", "formula": "\\begin{align*} \\begin{vmatrix} \\wp ( z ) & \\wp ' ( z ) & 1 \\\\ \\wp ( w ) & \\wp ' ( w ) & 1 \\\\ \\wp ( z + w ) & - \\wp ' ( z + w ) & 1 \\end{vmatrix} = 0 , \\end{align*}"}
+{"id": "987.png", "formula": "\\begin{align*} \\chi _ g \\cdot ( \\chi _ { f _ i } \\circ \\bar g ^ { \\flat } ) = \\chi _ { f _ i ' } \\cdot ( \\chi _ { g _ i } \\circ \\theta _ i ' ) \\end{align*}"}
+{"id": "2137.png", "formula": "\\begin{align*} J = \\Upsilon _ { y _ { 1 } } \\times \\Upsilon _ { y _ { 2 } } \\cdot \\mathcal { N } = | z _ { y _ { 2 } } | + ( \\alpha _ { 1 } | z _ { y _ { 2 } } | + \\beta _ { 2 } ) y _ { 3 } + ( \\alpha _ { 1 } \\beta _ { 2 } - \\beta _ { 1 } \\alpha _ { 2 } ) y _ { 3 } ^ 2 . \\end{align*}"}
+{"id": "6054.png", "formula": "\\begin{align*} P _ { * , j } & = \\frac { ( n + m ) ( \\beta _ j - \\gamma _ j - \\pi ) + m ( \\alpha _ j - \\gamma _ j - \\pi ) } { 2 \\pi } \\textrm { a n d } \\\\ \\omega _ { * , j } & = \\frac { ( n + m ) w _ 1 - m w _ 2 } { 2 \\pi } , \\end{align*}"}
+{"id": "3854.png", "formula": "\\begin{align*} \\varphi _ { \\lambda , 0 } ( x _ 1 , x _ 2 ) & = \\min _ { x ' : d ( x ' ) = 0 } \\bigg ( \\lambda _ 1 \\| x _ 1 - x ' \\| _ 2 + \\lambda _ 2 \\| x _ 2 - x ' \\| _ 2 \\bigg ) , \\\\ \\varphi _ { \\lambda , 1 } ( x _ 1 , x _ 2 ) & = \\min _ { x ' : d ( x ' ) = 1 } \\bigg ( \\lambda _ 1 \\| x _ 1 - x ' \\| _ 2 + \\lambda _ 2 \\| x _ 2 - x ' \\| _ 2 \\bigg ) ; \\end{align*}"}
+{"id": "2262.png", "formula": "\\begin{align*} f ( x + i y ) & = \\lim _ { k \\to \\infty } \\frac { 1 } { \\pi } \\langle F _ k ( \\cdot , 0 ) , P ( x - \\cdot , y ) \\rangle = \\lim _ { k \\to \\infty } \\frac { 1 } { \\pi } \\int _ { - \\infty } ^ \\infty F _ k ( t , 0 ) P ( x - t , y ) \\ , d t \\\\ & = \\lim _ { k \\to \\infty } \\frac { 1 } { \\pi } \\int _ { - \\infty } ^ \\infty f ( t + i y _ k ) P ( x - t , y ) \\ , d t = \\frac { 1 } { \\pi } \\langle f _ b , P ( x - \\cdot , y ) \\rangle . \\end{align*}"}
+{"id": "7599.png", "formula": "\\begin{align*} h ( z ) = \\int _ { 0 } ^ { z } \\frac { h _ { 0 } ( x ) } { x } d x = z + \\frac { \\sqrt { 6 9 } } { 1 2 \\sqrt { 1 7 } } z ^ 3 + \\frac { 1 } { 2 0 } \\left ( \\frac { 6 9 } { 1 3 6 } + \\frac { \\sqrt { 6 9 } } { 4 \\sqrt { 1 7 } } \\right ) z ^ 5 + \\cdots , \\end{align*}"}
+{"id": "9162.png", "formula": "\\begin{align*} \\begin{array} { c c l } \\bar { u } ^ { 1 } & = & 2 u ^ { 1 } \\tfrac { \\sin \\left ( u ^ { 2 } \\tfrac { T } { 2 } \\right ) } { u ^ { 2 } } \\\\ \\bar { u } ^ { 2 } & = & x ^ { 3 } + u ^ { 2 } \\tfrac { T } { 2 } \\ , , \\end{array} \\end{align*}"}
+{"id": "7080.png", "formula": "\\begin{align*} \\beta _ i = \\nu ( g ' ) - \\nu _ i ( g ) \\mbox { f o r e v e r y } i \\in I ^ * \\mbox { w i t h } i \\geq i _ 0 . \\end{align*}"}
+{"id": "1863.png", "formula": "\\begin{align*} T ( M , \\bar { u } ) = \\{ v \\in \\mathcal { U } : \\ \\exists \\{ u _ n \\} \\subset M , \\ \\{ t _ n \\} \\subset \\mathbb { R } ^ + , \\ u _ n \\rightarrow \\bar { u } , \\ t _ n \\rightarrow 0 ^ + , \\frac { 1 } { t _ n } ( u _ n - \\bar { u } ) \\rightarrow v \\} , \\end{align*}"}
+{"id": "8478.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , K _ { 0 } ( 2 \\pi \\lambda _ { n } x ) = \\frac { 1 } { 4 x } + \\frac { C _ { p } ^ { ( 2 ) } } { 2 } - \\frac { e ^ { 2 \\pi p } Q _ { 2 \\pi p } ( 0 ) } { 1 + \\frac { 1 } { \\pi p } } + \\frac { \\log \\left ( \\frac { x } { 2 } \\right ) } { 2 \\left ( 1 + \\frac { 1 } { \\pi p } \\right ) } + \\pi \\ , \\intop _ { 0 } ^ { \\infty } \\ , \\frac { J _ { 0 } ( 2 \\pi x y ) - 1 } { \\sigma \\left ( y \\right ) e ^ { 2 \\pi y } - 1 } \\ , d y . \\end{align*}"}
+{"id": "127.png", "formula": "\\begin{align*} \\hat { u } _ k ( \\mu ) = \\int _ { \\tilde { \\gamma } ^ 3 _ k } \\frac { 1 } { i ( \\lambda + \\mu ) } \\frac { d } { d \\lambda } \\log \\zeta _ 1 ( \\lambda ) d \\lambda \\end{align*}"}
+{"id": "9168.png", "formula": "\\begin{align*} \\varphi ^ { 1 } & = \\zeta _ { [ - 1 ] } ^ { 1 } \\\\ \\delta ( \\varphi ^ { 1 } ) & = x ^ { 3 } \\\\ \\delta ^ { 2 } ( \\varphi ^ { 1 } ) & = 2 \\bar { u } ^ { 2 } - x ^ { 3 } \\end{align*}"}
+{"id": "3617.png", "formula": "\\begin{align*} 1 \\ = \\ L ( f ) \\ \\le \\ m + 1 - \\beta m . \\end{align*}"}
+{"id": "6951.png", "formula": "\\begin{align*} \\partial _ v ( ( v ^ * ) ^ { \\otimes d } ) = d \\langle v , v ^ * \\rangle ( v ^ * ) ^ { \\otimes ( d - 1 ) } , \\end{align*}"}
+{"id": "3424.png", "formula": "\\begin{align*} \\begin{aligned} I + I I : = & \\int _ { \\R ^ N } \\left ( \\nabla u _ n \\nabla v _ n + \\widetilde V _ \\varepsilon u _ n v _ n - \\bar f ( u _ n ) v _ n - \\lambda _ n u _ n v _ n \\right ) \\\\ & + 4 \\Phi _ { \\varepsilon _ n } ( u _ n ) ^ { \\frac 1 2 } \\xi _ 1 ( \\Upsilon ( u _ n ) ) \\int _ { \\R ^ { N } } \\chi _ { u _ n } u _ n v _ n \\mathrm d x = o _ n ( 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "759.png", "formula": "\\begin{align*} g ^ { i j } = \\dfrac { s ^ { i j } } { \\phi ^ 2 } - \\dfrac { 1 } { \\phi ^ 2 } \\cdot \\dfrac { \\rho ^ 2 u _ k u _ l s ^ { i k } s ^ { j l } } { D ^ 2 } , \\end{align*}"}
+{"id": "560.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ { t } ^ { 2 } \\tilde { u } ( t , k ) + a ( t ) \\mathcal { H } _ { \\hbar , V } \\tilde { u } ( t , k ) + q ( t ) \\tilde { u } ( t , k ) = f ( t , k ) , \\quad ( t , k ) \\in ( 0 , T ] \\times \\hbar \\mathbb { Z } ^ { n } , \\\\ \\tilde { u } ( 0 , k ) = u _ { 0 } ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\\\ \\partial _ { t } \\tilde { u } ( 0 , k ) = u _ { 1 } ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\end{array} \\right . \\end{align*}"}
+{"id": "3285.png", "formula": "\\begin{align*} \\widetilde { \\rho } ( t , w ) = A t ^ { \\beta _ 1 } + { \\rm I m } ( e ^ { i \\pi ( 1 - \\beta _ 1 ) } w ) + B t ^ { 1 + \\beta _ 1 } + C t ^ { 1 - \\beta _ 1 } | w | ^ 2 + \\end{align*}"}
+{"id": "2399.png", "formula": "\\begin{align*} \\vec { u } ( A _ { 0 } , A _ { 4 } ) = ( \\cos a _ { 4 , 1 0 2 } \\cos \\omega _ { 4 , 1 0 2 } , \\cos a _ { 4 , 1 0 2 } \\sin \\omega _ { 4 , 1 0 2 } , \\sin a _ { 4 , 1 0 2 } ) . \\end{align*}"}
+{"id": "1025.png", "formula": "\\begin{align*} & \\qquad \\qquad \\qquad \\frac { ( \\frac { 1 } { 6 } ) _ k ( \\frac { 5 } { 6 } ) _ k } { ( 1 ) _ { k } ^ 2 } = \\frac { \\binom { 3 k } { k } \\binom { 6 k } { 3 k } } { 4 3 2 ^ k } , \\\\ [ 1 m m ] & H _ k ( - \\tfrac { 1 } { 6 } ) + H _ k ( - \\tfrac { 5 } { 6 } ) = 6 H _ { 6 k } - 3 H _ { 3 k } - 2 H _ { 2 k } + H _ k , \\end{align*}"}
+{"id": "8145.png", "formula": "\\begin{align*} \\varphi : = \\log \\sigma _ 1 = \\left . \\frac { d } { d \\alpha } \\right | _ { \\alpha = 0 } \\exp { \\alpha \\varphi } = \\left . \\frac { d } { d \\alpha } \\right | _ { \\alpha = 0 } \\sigma _ 1 ( \\alpha A ) , \\end{align*}"}
+{"id": "1360.png", "formula": "\\begin{align*} & \\lim \\phi _ { \\sigma ( n ) } ( u ) = \\phi _ 0 ( u ) , & & \\lim \\psi _ { \\sigma ( n ) } ( u ) = \\psi _ 0 ( u ) , & & \\lim \\xi _ { \\sigma ( n ) } ( u ) = \\xi _ 0 ( u ) , \\\\ & \\lim \\phi _ { \\sigma ( n ) } ( z ) = \\phi _ 0 ( z ) , & & \\lim \\psi _ { \\sigma ( n ) } ( z ) = \\psi _ 0 ( z ) , & & \\lim \\xi _ { \\sigma ( n ) } ( z ) = \\xi _ 0 ( z ) . \\end{align*}"}
+{"id": "6017.png", "formula": "\\begin{align*} \\theta _ { L , X ^ { \\ast } } ( f ' ) ( y e _ 1 + z e _ 1 ^ { \\ast } ) & = \\sum _ { t \\in \\Z } f ' ( y e _ 1 + z e _ 1 ^ { \\ast } + t e _ 1 ) \\psi ( \\tfrac { \\langle t e _ 1 , y e _ 1 + z e _ 1 ^ { \\ast } \\rangle } { 2 } ) \\\\ & = \\sum _ { t \\in \\Z } f ' ( y e _ 1 + z e _ 1 ^ { \\ast } + t e _ 1 ) \\psi ( \\tfrac { t z } { 2 } ) . \\end{align*}"}
+{"id": "595.png", "formula": "\\begin{align*} g = \\begin{bmatrix} t e ^ { i \\gamma } & 1 \\\\ 0 & 0 \\end{bmatrix} \\end{align*}"}
+{"id": "3951.png", "formula": "\\begin{align*} \\Gamma ( \\mathcal { G } _ 1 , \\mathcal { G } _ 2 ) : = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\gamma _ { 1 3 } \\in \\mathcal { G } _ 1 , \\gamma _ { 2 3 } \\in \\mathcal { G } _ 2 \\right \\} , \\end{align*}"}
+{"id": "7658.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\lambda _ { } ( B _ R ) = \\frac { ( n - 1 ) ^ 4 } { 1 6 } , \\end{align*}"}
+{"id": "7181.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 ( J _ { w _ 1 ^ h } \\setminus J _ u ) = 0 \\ , . \\end{align*}"}
+{"id": "6193.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\cdot z = \\begin{cases} \\frac { a z + b } { c z + d } & a d - b c = 1 , \\\\ \\frac { a \\overline { z } + b } { c \\overline { z } + d } & a d - b c = - 1 . \\end{cases} \\end{align*}"}
+{"id": "8091.png", "formula": "\\begin{align*} \\mathbf { A } ^ { 2 n + 1 } = ( - 1 ) ^ n \\mathbf { A } , \\mathbf { A } ^ { 2 n + 2 } = ( - 1 ) ^ { n + 1 } \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "8878.png", "formula": "\\begin{align*} H ( T ( n , k ) ) = ( 1 - o ( 1 ) ) c _ k n ^ { 1 / 2 } E ( n , { \\textstyle \\frac { k - 1 } { k } } ) \\end{align*}"}
+{"id": "601.png", "formula": "\\begin{align*} s ^ 2 = \\frac { 1 } { 2 } \\pm \\frac { 1 } { 2 } \\sqrt { \\frac { C - 4 } { C } } . \\end{align*}"}
+{"id": "6249.png", "formula": "\\begin{align*} \\mathrm { S p a n } _ { \\mathbb C } \\{ g _ n \\ , | \\ , n \\geq 0 \\} + \\partial ( O [ g _ 1 ] ) = O [ g _ 1 ] ; \\end{align*}"}
+{"id": "2317.png", "formula": "\\begin{align*} \\begin{aligned} \\| A y - r \\| _ \\infty / \\| r \\| _ \\infty \\gets & \\| - c _ 2 A E r + ( c _ 1 - c _ 2 ) A r - r \\| _ \\infty / \\| r \\| _ \\infty \\\\ & \\leq \\| - c _ 2 A E + ( c _ 1 - c _ 2 ) A - I \\| _ \\infty \\\\ & \\leq 2 c _ 2 + 2 | c _ 1 - c _ 2 | + 1 \\\\ & = 2 c _ 1 + 1 , \\end{aligned} \\end{align*}"}
+{"id": "5395.png", "formula": "\\begin{align*} S = \\{ \\lceil 1 2 \\log _ q n \\rceil \\le d \\le n : \\gcd ( k , q ^ d - 1 ) < q ^ { d / 3 } \\} \\end{align*}"}
+{"id": "6199.png", "formula": "\\begin{align*} | V _ e | = \\sum _ { v \\in V _ i } | \\Phi ^ { - 1 } ( v ) | . \\end{align*}"}
+{"id": "1261.png", "formula": "\\begin{align*} \\mu _ { X _ 2 } ( x _ 2 , y _ 2 ) > 0 \\wedge \\mu _ { X _ 2 } ( y _ 2 , x _ 2 ) > 0 \\Rightarrow x _ 2 = y _ 2 \\end{align*}"}
+{"id": "5436.png", "formula": "\\begin{align*} P _ s ( \\theta ) & \\overset { \\Delta } { = } \\left ( > \\theta , \\Phi ( \\mathcal { A } ) > 0 | \\Phi \\right ) \\\\ & = \\left ( > \\theta | \\Phi , \\Phi ( \\mathcal { A } ) > 0 \\right ) \\mathbf { 1 } ( \\Phi ( \\mathcal { A } ) > 0 | \\Phi ) , \\end{align*}"}
+{"id": "1344.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } _ { i } ^ { N } ( t ) = \\frac { 1 } { N } \\underset { 1 \\leq j \\leq N : j \\neq i } { \\sum } m _ { j } ^ { N } ( t ) \\mathrm { s g n } ( x _ { j } ^ { N } ( t ) - x _ { i } ^ { N } ( t ) ) , & x _ { i } ( 0 ) = x _ { i } ^ { 0 , N } , \\\\ \\dot { m } _ { i } ^ { N } ( t ) = \\psi _ { i } ^ { N } ( \\mathbf { x } _ { N } ( t ) , \\mathbf { m } _ { N } ( t ) ) , & m _ { i } ( 0 ) = m _ { i } ^ { 0 , N } . \\end{cases} \\end{align*}"}
+{"id": "8362.png", "formula": "\\begin{align*} ( C _ V ) _ { u v } = c ( u ) \\delta _ { u v } , ( C _ E ) _ { e f } = c ( e ) \\delta _ { e f } \\end{align*}"}
+{"id": "4644.png", "formula": "\\begin{align*} P = \\frac { 4 n q } { ( p + \\gamma - 1 ) ( 2 q + n - 2 - \\theta ( q - 1 ) ) } , R = 2 , Q = \\frac { 2 } { p + \\gamma - 1 } = S , \\end{align*}"}
+{"id": "7755.png", "formula": "\\begin{align*} \\mu ( \\partial Q ) = 0 \\mbox { f o r e v e r y d y a d i c c u b e $ Q $ . } \\end{align*}"}
+{"id": "9120.png", "formula": "\\begin{align*} \\begin{array} { c c l } y _ { [ 1 ] } & = & \\delta ( \\varphi ( \\zeta _ { [ - q _ { 1 } ] } , \\dots , \\zeta _ { [ - 1 ] } , x , u , \\dots , u _ { [ q _ { 2 } ] } ) ) \\\\ y _ { [ 2 ] } & = & \\delta ^ { 2 } ( \\varphi ( \\zeta _ { [ - q _ { 1 } ] } , \\dots , \\zeta _ { [ - 1 ] } , x , u , \\dots , u _ { [ q _ { 2 } ] } ) ) \\\\ & \\vdots \\end{array} \\end{align*}"}
+{"id": "7936.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) u = & u \\xi + ( c _ { e _ { ( \\xi , 0 ) } + e _ { ( \\xi , e _ 0 ) } } + c _ { e _ { ( \\xi , 0 ) } + 2 e _ { ( \\xi , e _ 0 ) } } + c _ { e _ { ( \\xi , 0 ) } + 3 e _ { ( \\xi , e _ 0 ) } } ) u \\\\ & + c _ { 2 e _ { ( \\xi , e _ 0 ) } + e _ { e _ 1 } } \\partial _ x u . \\end{align*}"}
+{"id": "8018.png", "formula": "\\begin{align*} H ^ * \\begin{bmatrix} A & 0 \\\\ 0 & B \\end{bmatrix} H = \\begin{bmatrix} X ^ * A X + Y ^ * B Y & \\star \\\\ \\star & Y ^ * A Y + X ^ * B X \\end{bmatrix} \\end{align*}"}
+{"id": "5075.png", "formula": "\\begin{align*} \\mathcal { Q } ( v , \\gamma ) = ( 1 - \\gamma _ x ) \\mathcal { J } \\left [ \\left ( \\begin{array} { c c } 3 v _ r ^ 2 + v _ i ^ 2 & 2 v _ r v _ i \\\\ 2 v _ r v _ i & v _ r ^ 2 + 3 v _ i ^ 2 \\end{array} \\right ) \\phi + | v | ^ 2 v \\right ] , \\end{align*}"}
+{"id": "6362.png", "formula": "\\begin{align*} \\rho _ { s t } ( t ^ { 1 / \\alpha } x , t ^ { 1 / \\alpha } y ) = t ^ { - ( d + 2 \\beta ) / \\alpha } \\rho _ s ( x , y ) , x , y \\in \\Gamma , \\ ; s , t > 0 . \\end{align*}"}
+{"id": "6586.png", "formula": "\\begin{align*} y = \\sqrt { P _ s } \\mathbf { h } \\boldsymbol { \\Phi } \\mathbf { g } _ { n _ t } x + n _ 0 , \\end{align*}"}
+{"id": "7749.png", "formula": "\\begin{align*} \\mathcal L = \\{ ( x _ 1 , x _ 2 ) \\in V ^ 2 \\colon x _ 1 , x _ 2 \\in N ( e _ { \\ell - 2 } ) \\setminus R x _ 1 x _ 2 \\in E ( L _ v ) \\} \\ , . \\end{align*}"}
+{"id": "7853.png", "formula": "\\begin{align*} | V | = \\left | \\bigcup _ { y H \\in \\mathcal { S } } \\left \\{ ( x H , y H ) : \\ x H \\in \\Omega \\setminus \\mathcal { S } \\right \\} \\right | = | \\mathcal { S } | A ( X _ i / \\pi ) _ { 1 2 } = | \\mathcal { S } | ( k _ i - a _ i ) . \\end{align*}"}
+{"id": "9256.png", "formula": "\\begin{align*} E & = \\big \\{ ( t , x , z ) \\in \\mathbb { R } ^ 3 _ + : | t - x | < z < t + x \\big \\} , \\\\ F & = \\big \\{ ( t , x , z ) \\in \\mathbb { R } ^ 3 _ + : z < t - x \\big \\} . \\end{align*}"}
+{"id": "3412.png", "formula": "\\begin{align*} \\mu _ 0 = \\min _ { x \\in O ^ { 3 \\delta _ 0 } } V ( x ) . \\end{align*}"}
+{"id": "2366.png", "formula": "\\begin{align*} \\check { H } ^ 2 ( \\{ U \\} , \\mathcal F ) = H ^ 2 ( X , \\mathcal F ) \\rlap { . } \\end{align*}"}
+{"id": "3041.png", "formula": "\\begin{align*} A _ { s } ( \\mathrm { i , j } ) \\perp B _ s ( \\mathrm { i , j } ) \\quad \\hbox { a n d } s _ k ( A _ s ( \\mathrm { i , j } ) ) = 0 \\end{align*}"}
+{"id": "8674.png", "formula": "\\begin{align*} \\frac { n m } { n + m } _ { n , m } ^ { 2 } \\rightarrow \\sum _ { l = 1 } ^ { \\infty } \\lambda _ { l } ( z _ { l } ^ { 2 } - 1 ) n , m \\rightarrow \\infty , \\end{align*}"}
+{"id": "263.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - y ^ m z ^ n } \\right ) ^ { \\frac { m ^ 3 } { n ^ 4 } } = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { n ^ 3 y ^ { n + 4 } + ( 3 n ^ 3 + 6 n ^ 2 - 4 ) y ^ { n + 2 } } { ( 1 - y ) ^ 4 } \\right ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "7295.png", "formula": "\\begin{align*} n l ' - k l ' - l n ' = ( n - k ) \\frac { l } { ( l , n - k ) } - l \\frac { n - k } { ( l , n - k ) } = 0 , \\end{align*}"}
+{"id": "8936.png", "formula": "\\begin{align*} f ( R ) + M ( R ) \\leq g ( R ) : = c _ * \\ln ^ \\alpha \\ln ^ \\frac 1 2 ( 1 / R ) \\end{align*}"}
+{"id": "3778.png", "formula": "\\begin{align*} U _ { k + i + 1 } = U _ { k + i } \\oplus \\begin{cases} z ^ { i + 1 } , & n \\notin J _ { i + 1 } , \\\\ z ^ { n + i + 2 - x } , & n \\in J _ { i + 1 } , J _ { i + 2 } = s _ 1 ( J _ { i + 1 } \\setminus \\{ n \\} ) \\cup \\{ x \\} . \\end{cases} \\end{align*}"}
+{"id": "6499.png", "formula": "\\begin{align*} a _ { i , j } : = a _ { i , j } ( m , x ) : = \\binom { m + x } { m - i + j } - \\binom { m + x } { m - i - j + 1 } . \\end{align*}"}
+{"id": "6063.png", "formula": "\\begin{align*} \\mathcal { B } ^ { A } ( x _ 0 , r ) = x _ 0 \\mathcal { B } ^ A ( e , r ) , \\mathcal { B } ^ { A } ( e , r ) = \\delta _ { r } ^ { A } \\mathcal { B } ^ { A } ( e , 1 ) . \\end{align*}"}
+{"id": "828.png", "formula": "\\begin{align*} \\begin{gathered} \\sum _ { M = 0 } ^ \\infty \\sum _ { N = \\lfloor w \\rfloor + M } ^ \\infty N ^ k e ^ { w - N } \\leq \\sum _ { M = 0 } ^ \\infty ( M + 1 ) ( w + M ) ^ k e ^ { - M } \\\\ \\lesssim \\sum _ { M = 0 } ^ \\infty M ^ { k + 1 } e ^ { - M } \\end{gathered} \\end{align*}"}
+{"id": "5021.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ n a _ j \\tau _ j \\right \\| = \\sum _ { j = 1 } ^ n | a _ j | ^ p . \\end{align*}"}
+{"id": "4059.png", "formula": "\\begin{align*} \\sum _ { A = 1 } ^ { \\infty } \\frac { G _ k ( A ^ 2 \\slash X ) } { A } = \\frac { \\left ( ( k - 1 ) ! \\right ) ^ 2 } { 2 } \\log X + c _ 0 + O \\left ( \\frac { 1 } { X } \\right ) , \\end{align*}"}
+{"id": "7091.png", "formula": "\\begin{align*} - \\gamma = \\inf _ { n \\in \\N } \\{ - \\nu ( x - a _ n ) \\} = \\inf _ { n \\in \\N } \\alpha _ n = \\inf _ { n \\in \\N } \\beta _ n = v ( p ) - p \\gamma . \\end{align*}"}
+{"id": "225.png", "formula": "\\begin{align*} \\omega ( s _ 1 , s _ 2 , s _ { 3 } , s _ { 4 } ) = \\sum _ { m _ 1 , \\ , m _ 2 , \\ , m _ 3 > 0 } \\frac { 1 } { { m _ 1 } ^ { s _ 1 } { m _ 2 } ^ { s _ 2 } { m _ 3 } ^ { s _ 3 } ( m _ 1 + m _ 2 + m _ 3 ) ^ { s _ { 4 } } } , \\end{align*}"}
+{"id": "4530.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( U ^ { \\lambda _ 0 } ) \\ge \\widetilde { \\mu } _ { \\omega , \\mathbf { c } } = S _ { \\omega , \\mathbf { c } } ( \\Phi ) \\end{align*}"}
+{"id": "6216.png", "formula": "\\begin{align*} g t g ^ { - 1 } ( g h _ i g ^ { - 1 } ) \\cdot K = ( g h _ j g ^ { - 1 } ) \\cdot K . \\end{align*}"}
+{"id": "913.png", "formula": "\\begin{align*} A ^ { ( S ) } _ { 0 } = \\sum _ { i \\in S } A ^ { ( i ) } _ { 0 } , A ^ { ( S ) } _ { \\pm } = \\sum _ { i \\in S } A ^ { ( i ) } _ { \\pm } , P ^ { ( S ) } = \\prod _ { i \\in S } P ^ { ( i ) } , \\end{align*}"}
+{"id": "9349.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d y _ { ( n , t ) } & = \\bigg [ G \\big ( y _ { ( n , t ) } , z _ { ( n , t ) } , \\tilde { z } _ { ( n , t ) } , \\gamma _ { ( n , t , e ) } \\big ) + \\varphi ( t ) \\bigg ] d t - z _ { ( n , t ) } d W _ t - \\tilde { z } _ { ( n , t ) } d \\xi _ t \\\\ & \\quad - \\int _ { \\mathcal { E } } \\gamma _ { ( n , t , e ) } \\tilde { N } ( d e , d t ) , \\ t \\in [ 0 , n ] , \\\\ y _ { ( n , n ) } & = 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "9229.png", "formula": "\\begin{align*} \\texttt { w p t e r m [ k ] } = \\max \\Bigl \\{ \\texttt { m a g } ( 4 0 ( L + 2 ) ) , \\texttt { m a g } \\Bigl ( 6 8 \\frac { ( L + 2 ) ^ 2 \\ , A } { \\varepsilon _ 3 } \\frac { \\Gamma ( k + \\frac 1 2 ) ^ { 1 / 2 } } { ( B _ 1 a \\sqrt { \\pi } ) ^ k } \\Bigr ) \\Bigr \\} . \\end{align*}"}
+{"id": "6860.png", "formula": "\\begin{align*} J _ r ( \\widetilde h ) = \\inf _ { \\phi \\in \\mathcal { M } } I _ r ( h ^ \\phi ) , \\end{align*}"}
+{"id": "7929.png", "formula": "\\begin{align*} \\beta ( \\xi , 0 ) + \\beta ( 0 , 0 ) - \\beta ( 0 , 2 ) - 2 \\beta ( 0 , 3 ) + \\sum _ { \\mathbf { n } \\in \\N _ 0 ^ 4 } \\beta ( \\mathbf { n } ) = 1 , \\end{align*}"}
+{"id": "4005.png", "formula": "\\begin{align*} \\rho _ \\ell ( y _ \\ell , y _ \\ell ^ \\prime ) = \\min _ { ( x _ \\ell , x ^ \\prime _ \\ell ) \\in \\mathcal { X } _ \\ell \\times \\mathcal { X } _ \\ell } \\left ( s _ { \\ell } - s _ { \\ell } ^ { \\prime } \\right ) ^ { \\top } V _ { \\ell } ^ { - 1 } \\left ( s _ { \\ell } - s _ { \\ell } ^ { \\prime } \\right ) \\leq c _ \\ell ( s _ \\ell , s _ \\ell ^ \\prime ) , \\forall s _ \\ell , s _ \\ell ^ \\prime \\in \\mathcal { S } _ \\ell . \\end{align*}"}
+{"id": "2484.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle ( u _ { n } ) _ { \\lambda } = \\lambda ^ { \\frac { N } { 2 } } u _ { n } ( \\lambda x ) , ( v _ { n } ) _ { \\lambda } = \\lambda ^ { \\frac { N } { 2 } } v _ { n } ( \\lambda x ) . \\end{array} \\right . \\end{align*}"}
+{"id": "6738.png", "formula": "\\begin{align*} X _ z \\subseteq \\overline { [ \\eta ^ { * } , \\eta ' ] } = X _ { \\varphi ' } , \\end{align*}"}
+{"id": "3348.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ d ( \\varphi \\circ X _ j ) \\otimes Y _ j \\xi \\right ) v = \\sum _ { j = 1 } ^ d \\varphi ( X _ j v ) Y _ j \\xi = \\left ( \\sum _ { j = 1 } ^ d Y _ j ( \\varphi \\otimes \\xi ) X _ j \\right ) v . \\end{align*}"}
+{"id": "1971.png", "formula": "\\begin{align*} n ^ { - 2 } g u ^ { p \\bar p } u ^ { q \\bar q } u _ { p \\bar p \\bar j } u _ { q \\bar q j } - n ^ { - 1 } g u ^ { p \\bar p } u ^ { q \\bar q } | u _ { p \\bar q j } | ^ 2 + n ^ { - 1 } g u ^ { p \\bar p } u _ { p \\bar p j \\bar j } = 2 R e ( g _ { v j } v _ { \\bar j } ) + g _ v v _ { j \\bar j } + g _ { j \\bar j } , \\end{align*}"}
+{"id": "8869.png", "formula": "\\begin{align*} I ( v ) & = \\{ j \\in I ( u ) : a _ { i j } > f ( u ) \\} \\cup \\{ i \\} \\quad \\\\ I ( w ) & = \\{ j \\in I ( u ) \\setminus \\{ i \\} : a _ { i j } = f ( u ) \\} . \\end{align*}"}
+{"id": "6897.png", "formula": "\\begin{align*} \\widehat \\psi _ r ( \\beta - \\varepsilon ) = & \\inf _ { x \\in [ 0 , 1 ] } J _ r ( x , \\beta - \\varepsilon ) \\\\ \\geq \\ , & \\inf _ { x \\in [ 0 , 1 ] } [ J _ r ( x , \\beta ) - \\theta ( x , \\beta ) \\varepsilon ] \\\\ \\geq \\ , & \\widehat \\psi _ r ( \\beta ) - \\sup _ { x \\in [ 0 , 1 ] } \\theta ( x , \\beta ) \\varepsilon \\rightarrow \\widehat \\psi _ r ( \\beta ) , \\varepsilon \\downarrow 0 . \\end{align*}"}
+{"id": "823.png", "formula": "\\begin{gather*} \\# \\mathcal { Z } _ { M , 0 } = \\sum _ { k = 1 } ^ n \\tfrac { ( \\# S - 1 ) ! } { ( \\# S - 1 - k ) ! } p _ k ( M ) \\leq C \\sum _ { k = 1 } ^ n \\tfrac { ( \\# S - 1 ) ! } { ( \\# S - 1 - k ) ! } M ^ { k - 1 } \\left ( \\tfrac { e } { k } \\right ) ^ { 2 k } \\leq C M ^ { n - 1 } \\end{gather*}"}
+{"id": "2287.png", "formula": "\\begin{align*} | | \\sum _ { n } c _ n a _ n | | _ { a t } : = \\left ( \\inf \\sum _ { n } | c _ n | ^ p \\right ) ^ { 1 / p } , \\end{align*}"}
+{"id": "3757.png", "formula": "\\begin{align*} \\tilde { G } ( a ) = \\left \\{ \\begin{array} { l l } a + \\bar { k } , & \\hbox { $ 1 \\leq a \\leq \\bar { k } $ ; } \\\\ a - \\bar { l } + \\bar { k } , & \\hbox { $ r + 1 \\leq a \\leq r + \\bar { l } - \\bar { k } $ ; } \\\\ a + \\underline { k } + \\bar { k } - \\bar { l } \\equiv a + \\underline { l } , & \\hbox { $ r + \\bar { l } - \\bar { k } + 1 \\leq a \\leq r + \\underline { k } $ . } \\end{array} \\right . \\end{align*}"}
+{"id": "8608.png", "formula": "\\begin{align*} E [ W ^ k _ { \\ell \\Delta , t } ( j ) | { \\cal F } _ { ( \\ell + 1 ) \\Delta } ] & = E \\left [ E \\left [ W ^ k _ { \\ell \\Delta , t } ( j ) | X _ { \\ell , \\Delta } , { \\cal F } _ { ( \\ell + 1 ) \\Delta } \\right ] | { \\cal F } _ { ( \\ell + 1 ) \\Delta } \\right ] \\\\ & = p ^ k _ { j , + } ( t - \\ell \\Delta ) E \\left [ X _ { \\ell , \\Delta } | { \\cal F } _ { ( \\ell + 1 ) \\Delta } \\right ] \\\\ & = \\Delta \\nu p ^ k _ { j , + } ( t - \\ell \\Delta ) Z _ 0 ( \\ell \\Delta ) , \\end{align*}"}
+{"id": "1867.png", "formula": "\\begin{align*} \\overline { L ( \\mathcal { K } , \\bar { u } ) } = T ( M , \\bar { u } ) = T _ w ( M , \\bar { u } ) . \\end{align*}"}
+{"id": "1156.png", "formula": "\\begin{align*} T ( f \\cdot g ) = f T ( g ) + T ( f ) g + 2 B ( A ( f ) , A ( g ) ) \\end{align*}"}
+{"id": "939.png", "formula": "\\begin{align*} \\begin{cases} y _ { k - 1 1 } + y _ { k - 1 0 } + y _ { k - 9 } ' + y _ { k - 8 } ' = 0 ; \\\\ y _ { k - 1 0 } + y _ { k - 9 } ' + y _ { k - 8 } '' = 0 ; \\\\ y _ { k - 1 1 } + y _ { k - 1 0 } + y _ { k - 9 } '' + y _ { k - 8 } '' = 0 ; \\\\ y _ { k - 1 0 } + y _ { k - 9 } '' + y _ { k - 8 } ' = 0 . \\end{cases} \\end{align*}"}
+{"id": "2961.png", "formula": "\\begin{align*} \\varphi _ I ( \\sigma ) = \\left ( \\xi _ I - \\sigma \\cdot \\xi _ I , \\ \\sigma \\right ) \\ . \\end{align*}"}
+{"id": "8495.png", "formula": "\\begin{align*} \\mathcal { J } _ { p } ( x , s ) = \\intop _ { 0 } ^ { \\infty } \\ , \\frac { y ^ { s - \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( 2 \\pi x y ) } { \\sigma \\left ( y \\right ) e ^ { 2 \\pi y } - 1 } \\ , d y = \\frac { 2 \\ , \\pi ^ { - s } x ^ { \\frac { 1 } { 2 } - s } } { \\Gamma \\left ( 1 - s \\right ) } \\ , \\intop _ { 1 } ^ { \\infty } \\frac { 1 } { \\left ( t ^ { 2 } - 1 \\right ) ^ { s } } \\ , \\intop _ { 0 } ^ { \\infty } \\frac { \\sin ( 2 \\pi x y t ) } { \\sigma \\left ( y \\right ) e ^ { 2 \\pi y } - 1 } \\ , d y \\ , d t , \\end{align*}"}
+{"id": "1975.png", "formula": "\\begin{align*} 0 \\geq ( b - b ^ 2 r ^ 2 ) \\sum _ { p = 1 } ^ n u ^ { p \\bar p } - \\frac { A _ 1 ( 1 + \\Vert \\nabla v \\Vert ^ 2 + \\Delta v ) } { g \\Delta u } , \\end{align*}"}
+{"id": "1452.png", "formula": "\\begin{align*} \\int _ \\Omega f _ \\epsilon ^ { ' } ( V _ m ) ( h _ m + \\omega _ m ) u _ m d x \\leq & | f _ \\epsilon ^ { ' } ( V _ m ) | _ { \\frac { n } { 2 } } | h _ m + \\omega _ m ) | _ { \\frac { 2 n } { n - 2 } } | u _ m | _ { \\frac { 2 n } { n - 2 } } \\\\ \\leq & \\| h _ m + \\omega _ m \\| \\| u _ m \\| = o ( 1 ) . \\end{align*}"}
+{"id": "936.png", "formula": "\\begin{align*} \\mathbb { E } ( X ) \\leq 0 + \\frac { t ^ 2 } { \\ell } + \\sum _ { \\substack { h < i < j \\leq k \\\\ j - i \\leq t } } \\mathbb { P } ( s _ i = s _ j ) . \\end{align*}"}
+{"id": "2427.png", "formula": "\\begin{align*} f _ a ( q x ) = \\left ( 1 - \\dfrac { x } { a } \\right ) f _ a ( x ) , \\end{align*}"}
+{"id": "2546.png", "formula": "\\begin{align*} \\lambda _ { j _ 1 } \\lambda _ { j _ 1 ^ * } = \\dots = \\lambda _ { j _ s } \\lambda _ { j _ s ^ * } = \\lambda _ { j _ 1 } \\lambda _ { j _ 2 } = , \\end{align*}"}
+{"id": "4952.png", "formula": "\\begin{align*} \\begin{aligned} \\left [ P ^ t \\Sigma \\right ] _ { r , k } ^ { } & = \\left [ U \\Lambda ^ t U ^ { - 1 } \\Sigma \\right ] _ { r , k } ^ { } = \\left [ U \\Lambda ^ t ( U ^ { - 1 } \\Sigma ) \\right ] _ { r , k } ^ { } \\\\ & = [ U ] _ { r , \\mathbf { : } } ^ { } \\ , \\Lambda ^ t \\ , [ V ^ { - 1 } ] _ { \\mathbf { : } , k } ^ { } \\\\ & = \\frac { 1 } { n ^ t } \\sum _ { j = r } ^ k ( - 1 ) ^ { k - j } \\binom { n - j - 1 } { n - k - 1 } \\binom { n - r } { n - j } j ^ t . \\end{aligned} \\end{align*}"}
+{"id": "1614.png", "formula": "\\begin{align*} [ { f } , { f } ] ( X , Y ) & = { f } ^ 2 [ X , Y ] + [ { f } X , { f } Y ] - { f } [ { f } X , Y ] - { f } [ X , { f } Y ] , \\ X , Y \\in \\mathfrak { X } _ M , \\\\ d \\eta ^ i ( X , Y ) & = \\frac 1 2 \\ , \\{ X ( \\eta ^ i ( Y ) ) - Y ( \\eta ^ i ( X ) ) - \\eta ^ i ( [ X , Y ] ) \\} , X , Y \\in \\mathfrak { X } _ M . \\end{align*}"}
+{"id": "5611.png", "formula": "\\begin{align*} \\mathrm { I } \\left ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } \\right ) = \\sum _ { \\xi _ { 1 } ^ { x } } \\sum _ { \\xi _ { n } ^ { x } } \\log \\left ( \\frac { P _ { \\mu , x } ^ { n - 1 } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) } { P _ { \\mu , x } ^ { n } ( L _ { x } , \\xi _ { n } ^ { x } ) } \\right ) P _ { \\mu , x } ^ { 1 } ( L _ { x } , \\xi _ { 1 } ^ { x } ) P _ { \\mu , x } ^ { n - 1 } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) \\end{align*}"}
+{"id": "4033.png", "formula": "\\begin{align*} \\langle T _ i ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) , f \\rangle = \\langle ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) , \\ , T _ i f \\rangle \\ , \\ i = 1 , \\ldots , D . \\end{align*}"}
+{"id": "6932.png", "formula": "\\begin{align*} \\frac { 1 } { \\sum _ { j \\in \\N } f ( j ) ^ 2 } \\sum _ { v \\in L } f ( v ) ^ 2 = \\frac { 1 } { \\prod _ { p \\in P } ( 1 + f ( p ) ^ 2 ) } \\sum _ { v \\in L ' } f ( v ) ^ 2 \\end{align*}"}
+{"id": "9230.png", "formula": "\\begin{align*} | \\sqrt { 1 + \\eta _ 1 } - 1 | & = \\Bigl | \\eta _ 1 \\sum _ { n = 1 } ^ \\infty \\binom { 1 / 2 } { n } \\eta _ 1 ^ { ( n - 1 ) } \\Bigr | \\le 0 . 5 8 5 7 8 7 \\times 2 ^ { - d ' } \\\\ \\Bigl | \\frac { 1 } { \\sqrt { 1 + \\eta _ 1 } } - 1 \\Bigr | & = \\Bigl | \\eta _ 1 \\sum _ { n = 1 } ^ \\infty \\binom { - 1 / 2 } { n } \\eta _ 1 ^ { ( n - 1 ) } \\Bigr | \\le 0 . 8 2 8 4 2 8 \\times 2 ^ { - d ' } \\end{align*}"}
+{"id": "7236.png", "formula": "\\begin{align*} & \\zeta = 1 \\mbox { o n } B _ { \\sigma '' } , \\norm { \\nabla \\zeta } _ { L ^ \\infty ( B _ { \\sigma ' } , \\R ^ { n \\times K } ) } \\leq 2 ( \\sigma ' - \\sigma '' ) ^ { - 1 } , \\\\ & \\varphi = 1 \\mbox { o n } B _ { \\sigma } , \\norm { \\nabla \\varphi } _ { L ^ \\infty ( B _ { \\overline { \\sigma } } , \\R ^ { n \\times K } ) } \\leq 2 ( \\overline { \\sigma } - \\sigma ' ) ^ { - 1 } . \\end{align*}"}
+{"id": "4927.png", "formula": "\\begin{align*} \\overline { F } ( k , t + 1 , \\mathbf { p } _ { n + 1 } ) = p _ { k } ^ { } \\overline { F } ( k - 1 , t , \\mathbf { p } _ { n + 1 } ) + ( 1 - p _ { k } ^ { } ) \\overline { F } ( k , t , \\mathbf { p } _ { n + 1 } ) , \\end{align*}"}
+{"id": "7985.png", "formula": "\\begin{align*} \\mathrm { d i v } _ T \\Big ( h \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\Big ) \\ , h \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\cdot \\nu & - h ^ 2 \\ , \\nabla _ T \\Big [ h \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\Big ] \\ , \\Big [ \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\Big ] _ T \\cdot \\nu \\\\ & = h \\ , H ( \\nu ) \\ , H ^ 2 ( \\nabla v ) \\ , \\mathrm { t r } \\ , \\mathcal { B } ^ H \\end{align*}"}
+{"id": "967.png", "formula": "\\begin{align*} \\overline { B ( \\overline { x _ 1 } , \\delta _ 0 ) } \\cap | \\gamma _ 2 | = \\varnothing = \\overline { U _ 2 } \\cap | \\gamma _ 1 | \\ , , \\overline { B ( \\overline { x _ 1 } , \\delta _ 0 ) } \\cap \\overline { U _ 2 } = \\varnothing \\ , . \\end{align*}"}
+{"id": "8631.png", "formula": "\\begin{align*} M ( t ) = M _ + ( t ) - M _ - ( t ) , \\end{align*}"}
+{"id": "969.png", "formula": "\\begin{align*} M _ q ( \\Gamma ^ { \\ , * } _ m ) \\leqslant \\sum \\limits _ { i = 0 } ^ { N _ 0 } M _ q ( \\Gamma _ { i m } ) \\leqslant \\sum \\limits _ { i = 1 } ^ { N _ 0 } \\frac { \\omega _ { n - 1 } } { I _ i ^ { q - 1 } } + \\frac { \\omega _ { n - 1 } } { I _ 0 ^ { p - 1 } } : = C _ 0 \\ , , m = 1 , 2 , \\ldots \\ , . \\end{align*}"}
+{"id": "7761.png", "formula": "\\begin{align*} x _ { a ^ * } - x _ { b ^ * } = M _ { i ^ * b ^ * } - M _ { i ^ * a ^ * } . \\end{align*}"}
+{"id": "3395.png", "formula": "\\begin{align*} & \\alpha _ k : = \\frac { 2 } { k + 1 } , \\ \\beta _ k : = \\frac { 1 } { 2 L _ { E f } } , \\ \\lambda _ k : = \\frac { k \\beta _ k } { 2 } , m _ k : = \\left \\lceil \\frac { L _ { E f } k } { \\tilde { D } ^ 2 } \\right \\rceil , \\ a n d \\\\ & \\Pr ( R = k ) : = \\frac { \\Gamma _ k ^ { - 1 } \\beta _ k ( 1 - L _ { E f } \\beta _ k ) } { \\sum _ { { \\tau } = 1 } ^ N \\Gamma _ { \\tau } ^ { - 1 } \\beta _ { \\tau } ( 1 - L _ { E f } \\beta _ { \\tau } ) } , \\mathrm { f o r } \\ k = 1 , 2 , \\dots , N , \\end{align*}"}
+{"id": "5663.png", "formula": "\\begin{align*} \\nu _ 0 = \\frac { 1 } { 2 } C ^ { - 1 } _ { N , p , q } ( a ^ 2 + b ^ 2 ) ^ { - \\frac { p + q - 2 } { 2 } } . \\end{align*}"}
+{"id": "2250.png", "formula": "\\begin{align*} Q _ r ( \\theta ) = \\frac { 2 r \\sin ( \\theta ) } { 1 - 2 r \\cos ( \\theta ) + r ^ 2 } , \\end{align*}"}
+{"id": "6439.png", "formula": "\\begin{align*} { } ^ c \\theta _ m : = { } ^ c \\Phi ( \\theta _ m ) . \\end{align*}"}
+{"id": "8415.png", "formula": "\\begin{align*} \\eta ( \\tau ) = e ^ { \\frac { \\pi i \\tau } { 1 2 } } \\ , \\prod _ { m = 1 } ^ { \\infty } \\left ( 1 - e ^ { 2 \\pi i m \\tau } \\right ) , \\ , \\ , \\ , \\ , \\ , ( \\tau ) > 0 . \\end{align*}"}
+{"id": "4444.png", "formula": "\\begin{align*} f ( \\omega ) & = \\omega ( Q ( \\Phi ) - E ( \\Phi ) ) - ( E ( U _ 0 ) + \\omega Q ( U _ 0 ) + \\sqrt { \\omega } \\ \\mathbf { c } _ 0 \\cdot \\mathbf { P } ( U _ 0 ) ) \\\\ & = \\omega ( Q ( \\Phi ) - E ( \\Phi ) - Q ( U _ 0 ) ) - \\sqrt { \\omega } \\ \\mathbf { c } _ 0 \\cdot \\mathbf { P } ( U _ 0 ) - E ( U _ 0 ) . \\end{align*}"}
+{"id": "1667.png", "formula": "\\begin{align*} f ^ * ( t . [ \\varrho ] ) = t . [ f ^ * \\varrho ] . \\end{align*}"}
+{"id": "6340.png", "formula": "\\begin{align*} \\left | \\frac { \\partial z } { \\partial \\omega } \\det ( M _ 1 ) \\right | ( \\phi , \\omega , r ; t ) = C ( 1 + O ( \\varepsilon ) ) | t | ^ 5 , \\forall \\ , t \\in [ - \\rho , \\rho ] \\end{align*}"}
+{"id": "1340.png", "formula": "\\begin{align*} \\partial _ { t } \\mu ( t , x ) - \\mathrm { d i v } ( \\mu \\nabla V \\star \\mu ) ( t , x ) = h [ \\mu ] ( t , x ) , \\mu ( 0 , \\cdot ) = \\mu ^ { i n } . \\end{align*}"}
+{"id": "8319.png", "formula": "\\begin{align*} = h ^ { - 2 } \\sum _ { a , a ' , h , h ' ~ : ~ ( a + h ) ( a ' + h ' ) = 1 } A ( a ) A ( a ' ) H ( h ) H ( h ' ) \\chi ( a + h ) + \\mathcal { E } = \\sigma + \\mathcal { E } \\ , , \\end{align*}"}
+{"id": "7008.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ r \\tilde b _ j \\tilde { \\textbf { Q } } ^ { \\lambda _ j } \\end{align*}"}
+{"id": "1512.png", "formula": "\\begin{align*} \\mathrm { d } R ^ x _ s = \\frac { 1 } { R ^ x _ s } \\ , \\mathrm { d } s + \\mathrm { d } B _ s , R ^ x _ 0 = x . \\end{align*}"}
+{"id": "9199.png", "formula": "\\begin{align*} W ( D _ { 2 r , r , 1 } ) & = 2 \\sum _ { i < j } \\left ( d ( v _ i , v _ j ) + d ( v _ j , v _ i ) \\right ) + 2 \\sum _ { 1 \\le i , j \\le r } d ( v _ i , w _ j ) \\\\ & = 2 \\sum _ { 1 \\le i < j \\le r } \\left ( ( j - i ) + 1 \\right ) + 2 r \\sum _ { 1 \\le j \\le r } j \\\\ & = 2 \\sum _ { 2 \\le j \\le r } \\left ( \\binom { j + 1 } { 2 } - 1 \\right ) + r ^ 2 ( r + 1 ) \\\\ & = 2 \\binom { r + 2 } { 3 } - 2 r + r ^ 2 ( r + 1 ) \\\\ \\end{align*}"}
+{"id": "4194.png", "formula": "\\begin{align*} \\abs { \\hat { w _ 1 } ( \\xi ) } \\leq \\frac { 1 } { \\abs { \\xi } } \\int _ 0 ^ \\infty \\abs { B ( \\tau ) - A ( \\tau ) } \\ , d \\tau \\leq \\frac { 1 } { \\abs { \\xi } } \\int _ 0 ^ \\infty A ( \\tau ) - \\frac { 1 } { 3 } B ( \\tau ) \\ , d \\tau = \\frac { 1 } { 3 \\abs { \\xi } } \\ , . \\end{align*}"}
+{"id": "1696.png", "formula": "\\begin{align*} \\kappa _ \\ell = \\ell \\chi + K + \\Theta ( e ^ { - c \\ell } ) \\ , , \\end{align*}"}
+{"id": "9296.png", "formula": "\\begin{align*} \\frac { F _ m + E _ m } { F _ 1 + E _ 1 } = \\frac { F _ m } { F _ 1 } + \\frac { E _ m - \\frac { F _ m } { F _ 1 } E _ 1 } { F _ 1 + E _ 1 } \\end{align*}"}
+{"id": "3892.png", "formula": "\\begin{align*} \\left ( \\mathord { \\operatorname { p r o j } } _ { K _ N \\cap Q } \\circ { \\mathrm { p r o j } _ { Q } } ^ { - 1 } \\right ) \\# \\gamma = \\left ( \\mathord { \\operatorname { p r o j } } _ { K _ N \\cap Q } \\circ { \\mathrm { p r o j } _ { K _ \\ell } } ^ { - 1 } \\right ) \\# \\mu _ \\ell . \\end{align*}"}
+{"id": "8136.png", "formula": "\\begin{align*} w _ { 1 : k } : = \\sum _ { i = 1 } ^ k w _ i \\end{align*}"}
+{"id": "798.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } p ^ { ( k ) } _ { k - j } \\ , \\le \\ , \\frac { 1 1 \\ , t _ k ^ { \\alpha } } { 4 \\ , \\Gamma { ( 1 + \\alpha ) } } \\ , \\le \\ , \\frac { 1 1 \\ , T ^ { \\alpha } } { 4 \\ , \\Gamma { ( 1 + \\alpha ) } } . \\end{align*}"}
+{"id": "8327.png", "formula": "\\begin{align*} y _ { B 2 } = h _ { C B } z + n _ { B 2 } , \\end{align*}"}
+{"id": "3159.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\norm { e ( X _ n - X ) e } = 0 . \\end{align*}"}
+{"id": "8940.png", "formula": "\\begin{align*} | \\sigma ( r , x _ 3 , t ) | = r | v _ \\theta ( r , x _ 3 , t ) | \\leq \\frac { c C _ 1 } { l n ^ 3 ( e / r ) } \\end{align*}"}
+{"id": "1278.png", "formula": "\\begin{align*} M _ q ( G ) = Z _ q ( G ) = T + \\max _ { 1 \\le i _ 1 < \\ldots < i _ q \\le s } \\sum _ { j = 1 } ^ q a _ { i _ j } . \\end{align*}"}
+{"id": "1903.png", "formula": "\\begin{align*} \\min \\limits _ { u \\in \\mathcal { U } , \\zeta \\in \\mathcal { X } } \\theta ( u ) + I _ K ( \\zeta ) \\ \\ \\mbox { s . t . } \\ \\ S u = \\zeta , \\end{align*}"}
+{"id": "4471.png", "formula": "\\begin{align*} c _ { 1 } = c _ 1 ( \\omega ) : = \\min \\left \\{ \\frac { \\mu _ { \\omega , 0 } } { 2 \\| \\Phi _ { \\omega } \\| _ { \\mathcal { H } ^ 1 } ^ 2 } , \\sqrt { \\frac { 2 \\omega } { \\sigma } } \\right \\} . \\end{align*}"}
+{"id": "8085.png", "formula": "\\begin{align*} \\mathbb { F } _ n : = \\{ \\mathcal { A } \\subset \\mathcal { F } : \\enspace \\left . h : S ^ { n - 1 } \\rightarrow \\mathcal { A } \\right \\} , \\end{align*}"}
+{"id": "9306.png", "formula": "\\begin{align*} \\begin{array} { c } a _ { 1 1 } = 0 , \\\\ a _ { 2 2 } + a _ { 3 3 } + a _ { 4 4 } = 0 , \\\\ { [ } A ] _ { 1 2 } + { [ } A ] _ { 1 3 } + { [ } A ] _ { 1 4 } = 0 . \\end{array} \\end{align*}"}
+{"id": "5290.png", "formula": "\\begin{align*} \\alpha = \\mu _ p ( f ) , \\sigma ^ 2 = \\sum _ { i = 1 } ^ n \\hat f ( \\{ i \\} ) ^ 2 , \\delta = \\max _ i | \\hat f ( \\{ i \\} ) | , m = m ( f ) = | \\{ i : \\hat f ( \\{ i \\} ) ^ 2 \\geq \\frac { \\delta ^ 2 } { 2 } \\} | , \\end{align*}"}
+{"id": "2174.png", "formula": "\\begin{align*} \\exp \\left ( - 2 \\lambda \\left ( \\delta \\right ) \\beta b ^ { 2 } \\right ) = \\delta ^ { \\left ( 2 \\rho \\beta b ^ { 2 } \\right ) / d } . \\end{align*}"}
+{"id": "6600.png", "formula": "\\begin{align*} \\begin{aligned} \\bar P _ b & = \\int _ 0 ^ \\infty Q \\left ( \\sqrt { \\frac { \\rho \\zeta ^ 2 x } { 2 ( 1 + \\rho ( 1 - \\zeta ^ 2 ) \\sigma _ e ^ 2 L ) } } \\right ) f ( x ) d x , \\end{aligned} \\end{align*}"}
+{"id": "7728.png", "formula": "\\begin{align*} y \\bigl ( \\omega , y ^ { * } ( \\omega , t ) \\bigr ) = y ^ { * } \\bigl ( \\omega , y ( \\omega , t ) \\bigr ) = t \\tau _ { y ^ { * } } ( X ) ( t ) = X \\Bigl ( y ^ { * } ( t ) \\Bigr ) , t \\ge 0 , \\ \\mu \\end{align*}"}
+{"id": "8872.png", "formula": "\\begin{align*} P e r ( A _ G ) = \\sum _ { k = 0 } ^ { \\infty } 2 ^ k F _ k ( G ) \\ ; . \\end{align*}"}
+{"id": "4488.png", "formula": "\\begin{align*} N ( U ) = - 2 S _ { \\omega , \\mathbf { c } } ( U ) + K _ { \\omega , \\mathbf { c } } ( U ) \\end{align*}"}
+{"id": "2890.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 8 , & ~ ~ t _ 2 \\geq 3 , \\\\ 6 , & ~ ~ t _ 2 = 2 , \\\\ 4 , & ~ ~ t _ 2 = 1 , \\\\ 2 - 2 t _ { 2 } , & ~ ~ t _ 2 \\leq 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "8670.png", "formula": "\\begin{align*} H _ 0 : P _ X = P _ Y H _ a : P _ X \\neq P _ Y . \\end{align*}"}
+{"id": "4073.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( i ) \\lambda _ f ( j ) \\big | L ( f , n + 1 ) \\big | ^ 2 = \\large { S _ { \\mathrm { o f f } } + S _ { \\mathrm { m a i n } } } . \\end{align*}"}
+{"id": "7810.png", "formula": "\\begin{align*} \\textsf { E } f ( \\delta _ { 1 } h _ { 1 } , \\cdots , \\delta _ { n } h _ { n } ) = \\textsf { E } \\textsf { E } _ { \\delta } f ( \\delta _ { 1 } h _ { 1 } , \\cdots , \\delta _ { n } h _ { n } ) \\ge \\textsf { E } f ( h _ { 1 } / c , \\cdots , h _ { n } / c ) . \\end{align*}"}
+{"id": "6908.png", "formula": "\\begin{align*} \\mathcal { M } : = \\textrm { s u p p } ( f ) \\setminus M \\end{align*}"}
+{"id": "2536.png", "formula": "\\begin{align*} \\sigma _ { P _ a | i } ( \\Pi ) = \\Pi - P _ a + ( P _ { a } - \\{ i \\} ) + \\{ i \\} \\end{align*}"}
+{"id": "3997.png", "formula": "\\begin{align*} \\underset { x ^ \\prime \\in \\mathbb { R } ^ d } { \\arg \\max } \\left [ \\sum _ { 1 \\leq \\ell \\leq 2 } \\varphi _ \\ell ( x ^ { \\prime } , \\lambda _ \\ell ) \\right ] = \\left ( \\sum _ { 1 \\leq \\ell \\leq 2 } \\lambda _ { \\ell } V _ { \\ell , X X } ^ { - 1 } \\right ) ^ { - 1 } \\left ( \\sum _ { 1 \\leq \\ell \\leq 2 } a _ { \\ell } V _ { \\ell , X X } ^ { - 1 } V _ { \\ell , X Y } \\right ) , \\end{align*}"}
+{"id": "6429.png", "formula": "\\begin{align*} | x | \\bar { \\otimes } | y | = \\int _ { [ 0 , \\infty ) ^ 2 } \\lambda _ 1 \\lambda _ 2 \\ , e _ { | x | , | y | } ( d \\lambda _ 1 , d \\lambda _ 2 ) \\end{align*}"}
+{"id": "8367.png", "formula": "\\begin{align*} \\frac { c _ v } { c ( v _ 1 ) } u ' _ 1 + \\cdots + \\frac { c _ v } { c ( v _ n ) } u ' _ n = 0 . \\end{align*}"}
+{"id": "2473.png", "formula": "\\begin{align*} \\displaystyle \\Sigma : = \\left \\{ u : ~ ~ u \\in H ^ 1 ( \\mathbb { R } ^ N ) , ~ ~ x u \\in L ^ 2 ( \\mathbb { R } ^ N ) \\right \\} . \\end{align*}"}
+{"id": "4000.png", "formula": "\\begin{align*} \\mathrm { R } _ { \\mathrm { D } } ( \\lambda , \\delta ) = \\langle \\lambda , \\delta \\rangle + \\sum _ { 1 \\leq \\ell \\leq 2 } \\frac { V _ \\ell / V _ { \\ell , X X } } { 4 \\lambda _ \\ell } + \\frac { 1 } { 4 } V _ o ^ { \\top } { \\underbrace { \\left ( \\lambda _ 1 V _ { 1 , X X } ^ { - 1 } + \\lambda _ 2 V _ { 2 , X X } ^ { - 1 } \\right ) } _ { = \\Lambda _ \\lambda } } ^ { - 1 } V _ o \\end{align*}"}
+{"id": "5342.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ q \\leq \\gamma ' \\rho ^ { - d } = \\gamma \\left ( 9 9 \\cdot 2 \\ , r \\ , r ' q \\ , ( r ' ) ^ { - 2 / q } \\sqrt { \\frac { \\log ( 1 / \\gamma ) } { d } } \\right ) ^ d = \\gamma \\left ( 9 9 \\cdot 2 \\cdot \\sqrt { 2 } \\ , r \\ , q \\ , ( r ' ) ^ { - 2 / q } \\right ) ^ d . \\end{align*}"}
+{"id": "3920.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ^ \\star ( \\lambda ) = \\sup _ { \\pi \\in \\mathcal { G } _ { \\mathrm { D } , \\lambda } } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda d \\pi . \\end{align*}"}
+{"id": "8836.png", "formula": "\\begin{align*} C _ 1 = C _ 1 ( q , B ) & : = \\Bigl ( \\frac 1 2 ( q - 2 ) + \\frac 1 2 q ( q - 1 ) L _ \\sigma ^ 2 \\norm [ \\big ] { B } ^ 2 _ { \\gamma ( U ; X ) } \\Bigr ) ^ { 1 / q } , \\\\ C _ 2 = C _ 2 ( q , B ) & : = ( q - 1 ) ^ { 1 / q } \\bigl ( L _ \\sigma { \\norm { B } } _ { \\gamma ( U ; E _ \\eta ) } \\bigr ) ^ { 2 / q } . \\end{align*}"}
+{"id": "4795.png", "formula": "\\begin{align*} \\mathcal { L } \\varphi = \\lim _ { t \\to 0 ^ + } \\frac { \\mathcal { K } _ t \\varphi - \\varphi } { t } . \\end{align*}"}
+{"id": "4404.png", "formula": "\\begin{align*} K _ { \\omega , \\mathbf { c } } ( \\lambda U ) = L _ { \\omega , \\mathbf { c } } ( \\lambda U ) + 3 N ( \\lambda U ) = \\lambda ^ 2 L _ { \\omega , \\mathbf { c } } ( U ) + 3 \\lambda ^ 3 N ( U ) . \\end{align*}"}
+{"id": "5257.png", "formula": "\\begin{align*} \\beta = ( 3 \\rho ' ) \\cdot \\frac { 1 } { 3 } \\left ( 1 + \\frac { 2 ( q - 2 ) } { \\log ( 1 / \\rho ' ) } \\right ) \\le \\frac { 3 \\rho ' q } { \\min \\{ q , \\log ( 1 / \\rho ' ) \\} } \\le 3 \\rho ' q . \\end{align*}"}
+{"id": "3790.png", "formula": "\\begin{align*} \\Sigma _ { \\mathrm { D M R } } ( \\delta ) : = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { X } ) : \\boldsymbol { K } _ c ( \\mu , \\gamma ) \\le \\delta \\right \\} . \\end{align*}"}
+{"id": "1601.png", "formula": "\\begin{align*} \\int _ { \\Theta } v ^ { 2 } ( w ) \\| ( \\chi _ { w } + \\xi _ { w } ) \\pi _ { F ( w ) } ( L + G ) ^ { \\ast } \\pi _ { ( L + G ) F ( w ) } f \\| ^ { 2 } d \\mu ( w ) & \\geq \\int _ { \\Theta } v ^ { 2 } ( w ) \\| \\chi _ { w } \\pi _ { F ( w ) } ( L + G ) ^ { \\ast } f \\| ^ { 2 } d \\mu ( w ) \\\\ & \\geq A _ { 1 } \\| K ^ { \\ast } ( L + G ) ^ { \\ast } f \\| ^ { 2 } \\\\ & = A _ { 1 } \\| ( L + G ) ^ { \\ast } K ^ { \\ast } f \\| ^ { 2 } \\\\ & \\geq A _ { 1 } \\| ( L + G ) ^ { - 1 } \\| ^ { - 2 } \\| K ^ { \\ast } f \\| ^ { 2 } . \\end{align*}"}
+{"id": "2100.png", "formula": "\\begin{align*} \\max \\nolimits _ { \\preceq _ { \\langle A \\rangle } } A p e ( A , a ) = \\Big \\{ ( 2 ^ n + 2 ^ { n + 1 } ) ( a + d ) - \\{ 2 d , 3 d , . . . , ( n + 1 ) d \\} \\Big \\} \\bigcup \\{ 2 ^ { n + 1 } ( a + d ) - ( n + 1 ) d \\} , \\end{align*}"}
+{"id": "2626.png", "formula": "\\begin{align*} \\mathcal { E } = \\{ ( \\mathbf { z } , t ) \\in K ' : | \\alpha ( \\mathbf { z } ) - H ( t ) \\beta ( \\mathbf { z } + \\mathbf { p } ( t ) ) | \\leq \\varepsilon \\} , \\end{align*}"}
+{"id": "4789.png", "formula": "\\begin{align*} \\dot { x } = f ( x ) + \\sum _ { i = 1 } ^ m g _ i ( x ) u _ i , x \\in \\mathbb { R } ^ d , \\ u _ i \\in \\mathbb { R } , \\end{align*}"}
+{"id": "5291.png", "formula": "\\begin{align*} \\forall x , y \\in \\mathcal { A } \\ \\exists i \\in [ n ] : \\ x _ i = y _ i . \\end{align*}"}
+{"id": "1143.png", "formula": "\\begin{align*} p _ { n } = A _ { n } ^ { \\ast } , \\end{align*}"}
+{"id": "6162.png", "formula": "\\begin{align*} l _ R ( a ) l _ R ( - d ) & = l _ R ( a ) l _ R ( - a b c ) = l _ R ( a ) l _ R ( b c ) = l _ R ( a ) l _ R ( b ) + l _ R ( a ) l _ R ( c ) \\\\ & = l _ R ( c ) l _ R ( d ) + l _ R ( a ) l _ R ( c ) = l _ R ( c ) l _ R ( d ) + l _ R ( c ) l _ R ( a ) = l _ R ( c ) l _ R ( a d ) . \\end{align*}"}
+{"id": "6459.png", "formula": "\\begin{align*} \\zeta ( s ) & = i \\beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \\overline { \\beta _ h ( s ) } K _ 2 - i \\overline { \\beta _ \\sigma ( s ) } K \\\\ & = 2 s h ^ 2 ( K _ 1 + K _ 3 + K _ 2 ) + 2 s \\sigma ^ 2 K + \\mathcal { O } ( s ^ 2 \\sigma ^ 4 ) . \\end{align*}"}
+{"id": "8462.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { 1 } { x ^ { 2 } + \\lambda _ { n } ^ { 2 } } = \\frac { \\pi } { 2 x } - \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } \\cdot \\frac { 1 } { 2 x ^ { 2 } } + \\frac { \\pi } { x } \\ , \\frac { 1 } { \\sigma ( x ) \\ , e ^ { 2 \\pi x } - 1 } , \\end{align*}"}
+{"id": "7692.png", "formula": "\\begin{align*} & n _ z ( u ) z = z , n _ z ( u ) x = - ( u , x ) z + x , n _ z ( u ) z ' = - q ( u ) z + u + z ' , \\\\ & m _ z ( a , g _ 1 ) z = a z , m _ z ( a , g _ 1 ) x = g _ 1 x , m _ z ( a , g _ 1 ) z ' = a ^ { - 1 } z ' + ( a - a ^ { - 1 } ) q ( z ' ) z \\end{align*}"}
+{"id": "1055.png", "formula": "\\begin{align*} N \\coloneqq ( d + 1 ) \\cdot \\min \\left \\{ \\prod _ { i = 1 } ^ n ( h _ i + 1 ) - \\prod _ { i = 1 } ^ n h _ i , \\binom { n + h } { n } - \\binom { h } { n } \\right \\} \\ , . \\end{align*}"}
+{"id": "231.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ 2 = \\frac { n } { 6 } + \\frac { n ^ 2 } { 2 } + \\frac { n ^ 3 } { 3 } , \\end{align*}"}
+{"id": "2822.png", "formula": "\\begin{align*} S _ k ^ { m , 1 } ( t ) = \\frac { \\displaystyle \\sum _ { l = 0 } ^ { \\infty } \\frac { S _ { k + ( r + 1 ) l } ^ { m , 1 } ( 0 ) t ^ l } { l ! } } { \\displaystyle \\sum _ { l = 0 } ^ { \\infty } \\frac { S _ { ( r + 1 ) l } ^ { 1 , 1 } ( 0 ) t ^ l } { l ! } } ; S _ k ^ { 1 , n } ( t ) = \\frac { \\displaystyle \\sum _ { l = 0 } ^ { \\infty } \\frac { S _ { k + ( q + 1 ) l } ^ { 1 , n } ( 0 ) t ^ l } { l ! } } { \\displaystyle \\sum _ { l = 0 } ^ { \\infty } \\frac { S _ { n - 1 + ( q + 1 ) l } ^ { 1 , n } ( 0 ) t ^ l } { l ! } } ; k \\in \\mathbb { Z } _ + . \\end{align*}"}
+{"id": "3165.png", "formula": "\\begin{align*} \\bar { \\mu } = \\norm { \\cdot } _ 1 - \\lim _ { l \\to \\infty } B _ l ( \\mu ) , \\end{align*}"}
+{"id": "4118.png", "formula": "\\begin{align*} M \\vec { v } = \\lambda \\vec { v } , \\end{align*}"}
+{"id": "5391.png", "formula": "\\begin{align*} C _ { f _ * \\mathcal F } = \\frac { \\lambda _ \\Delta ^ + ( f , x , \\delta ) C _ { \\mathcal F } + t ( E _ x , \\Delta , \\delta ) t ( f _ * E _ x , \\Delta ' , \\delta ' ) \\| D f \\| _ { C ^ { \\theta } } } { \\lambda _ \\Delta ^ - ( f , x , \\delta ) \\lambda _ { \\mathcal F } ( f , x , \\delta ) ^ { \\theta } } , \\end{align*}"}
+{"id": "9007.png", "formula": "\\begin{align*} ( 1 + w ) C ^ \\varphi _ { k j i } = w _ t R _ { i j k t } + \\frac { S ^ \\varphi } { m - 1 } ( w _ i \\delta _ { j k } - w _ j \\delta _ { i k } ) - ( w _ i R ^ \\varphi _ { j k } - w _ j R ^ \\varphi _ { i k } ) \\ , . \\end{align*}"}
+{"id": "3970.png", "formula": "\\begin{align*} M = N \\boldsymbol { W } _ { p _ 2 } \\left ( \\mu _ 2 , \\delta _ { s _ 2 } \\right ) ^ { p _ 2 } + N \\left [ \\delta ^ { 1 / p _ 2 } + \\boldsymbol { W } _ { p _ 2 } \\left ( \\mu _ 2 , \\delta _ { s _ 2 } \\right ) \\right ] ^ { p _ 2 } < \\infty . \\end{align*}"}
+{"id": "881.png", "formula": "\\begin{align*} \\int _ { | \\theta | < \\widehat { Z } ^ { - 1 } } \\psi ( x \\theta ) \\dd \\theta = \\begin{cases} \\widehat { Z } ^ { - 1 } & | x | < \\widehat { Z } \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "8111.png", "formula": "\\begin{align*} M _ I ( - X ) = ( - 1 ) ^ { \\ell ( I ) } \\sum _ { J \\le I } M _ J ( X ) \\quad F _ I ( - X ) = ( - 1 ) ^ { | I | } F _ { \\bar I ^ \\sim } ( X ) . \\end{align*}"}
+{"id": "6982.png", "formula": "\\begin{align*} v ( a ^ { - 1 } ) = \\mu ( q ) . \\end{align*}"}
+{"id": "8625.png", "formula": "\\begin{align*} & I _ 1 ( \\ell _ 1 , \\ell _ 2 ) - I _ 2 ( \\ell _ 1 , \\ell _ 2 ) \\\\ & \\leq \\Delta \\nu p ^ k _ { j , + } ( t - \\Delta \\ell _ 2 ) \\\\ & \\quad \\ ; \\cdot \\sum _ { n = 1 } ^ \\infty \\sum _ { m = 1 } ^ \\infty \\sum _ { i = 1 } ^ m i P ( Z _ 0 ( \\Delta \\ell _ 2 ) = m , Z _ 0 ( \\Delta \\ell _ 1 ) = n , X _ { \\ell _ 1 , \\Delta } = 1 , B _ { \\ell _ 1 \\Delta } ( \\Delta \\ell _ 2 ) = i ) . \\end{align*}"}
+{"id": "2860.png", "formula": "\\begin{align*} [ f , g ] ( l ) = ( f \\otimes g ) \\Delta ( l ) = f ( l _ 1 ) g ( l _ 2 ) ; l , l _ 1 , l _ 2 \\in L . \\end{align*}"}
+{"id": "1445.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ k \\sum \\limits _ { l = 0 } ^ n c ^ { i l } _ m \\langle P \\psi ^ l _ { \\mu _ m ^ i , \\xi _ m ^ i } , P \\psi ^ h _ { \\mu _ m ^ j , \\xi _ m ^ j } \\rangle = \\int _ \\Omega f _ \\epsilon ^ { ' } ( V _ m ) \\phi _ m P \\psi ^ h _ { \\mu _ m ^ j , \\xi _ m ^ j } d x . \\end{align*}"}
+{"id": "4900.png", "formula": "\\begin{align*} h _ { d } ( X _ 0 , X _ 1 , . . . , X _ k ) \\overset { } { = } \\sum \\limits _ { \\substack { c _ 0 ^ { } + c _ 1 ^ { } + \\cdots + c _ k ^ { } = d \\\\ c _ j ^ { } \\geqslant 0 } } X _ { 0 } ^ { c _ 0 ^ { } } \\cdot X _ { 1 } ^ { c _ 1 ^ { } } \\cdots X _ { k } ^ { c _ k ^ { } } \\ , , \\end{align*}"}
+{"id": "8496.png", "formula": "\\begin{align*} \\intop _ { 0 } ^ { \\infty } \\frac { \\sin ( 2 \\pi x y t ) } { \\sigma \\left ( y \\right ) e ^ { 2 \\pi y } - 1 } \\ , d y = \\frac { \\sigma _ { p } ( 2 \\pi x t ) } { 2 } + \\frac { 1 } { 4 } \\ , \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } - \\frac { 1 } { 4 \\pi x t } , \\end{align*}"}
+{"id": "5183.png", "formula": "\\begin{align*} R ( \\lambda ^ i ) = \\{ \\lambda ^ j : \\lambda ^ j = ( \\lambda ^ i _ s , r ) \\lambda ^ i ( \\lambda ^ i _ s , r ) \\in \\rm { r e v } ( \\lambda ^ i ) \\} . \\end{align*}"}
+{"id": "5950.png", "formula": "\\begin{align*} \\overline { C } _ { X ^ { \\ast } } '' ( g , s ( - 1 ) ) & = \\overline { C } _ { X ^ { \\ast } } ( g , s ( - 1 ) ) \\overline { \\beta } ( g ) \\overline { \\beta } ( g ^ { s ( - 1 ) } ) ^ { - 1 } . \\end{align*}"}
+{"id": "2500.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } c _ 1 | \\xi | \\leq \\left | g ^ * _ 1 ( \\xi , \\eta ) \\right | = \\left | g _ 1 ( \\xi , - \\eta ) \\right | \\leq d _ 1 | \\xi | , c _ 2 | \\eta | \\leq \\left | g ^ * _ 2 ( \\xi , \\eta ) \\right | = \\left | - g _ 2 ( \\xi , - \\eta ) \\right | \\leq d _ 2 | \\eta | . \\end{array} \\right . \\end{align*}"}
+{"id": "3934.png", "formula": "\\begin{align*} \\widehat { \\mathcal { I } } ( \\delta _ 1 , \\delta _ 2 ) = \\sup _ { \\gamma \\in \\widehat { \\Sigma } ( \\delta _ 1 , \\delta _ 2 ) } \\int _ { \\mathcal { V } } f ( v ) \\ , d \\gamma ( v ) . \\end{align*}"}
+{"id": "6471.png", "formula": "\\begin{align*} a ( s , s ' ) & : = \\frac { z ( s ) } { 1 + 4 i s z ( s ) } + \\frac { \\widetilde z ( s ' ) } { 1 - 4 i s ' \\widetilde z ( s ' ) } , \\\\ b ( s , s ' ) & : = \\frac { \\zeta ( s ) } { 1 + 4 i s z ( s ) } + \\frac { \\widetilde \\zeta ( s ' ) } { 1 - 4 i s ' \\widetilde z ( s ' ) } , \\\\ c ( s , s ' ) & : = \\frac { i s } { 1 + 4 i s z ( s ) } \\zeta ( s ) \\cdot \\zeta ( s ) + \\frac { i s ' } { 1 - 4 i s ' \\widetilde z ( s ' ) } \\widetilde \\zeta ( s ' ) \\cdot \\widetilde \\zeta ( s ' ) + \\gamma ( s ) + \\widetilde \\gamma ( s ' ) . \\end{align*}"}
+{"id": "2002.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } \\Delta w \\leq \\Delta v _ { \\lambda _ j } & \\textnormal { i n } & \\Omega \\\\ w = v _ { \\lambda _ j } = 0 & \\textnormal { o n } & \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "8990.png", "formula": "\\begin{align*} \\Lambda ( x ) = \\frac { S ^ \\varphi } { m - 1 } = \\frac { \\mu } { 1 - e ^ { \\mu f } } ( m \\lambda ( x ) - S ^ \\varphi ) \\ , \\Omega . \\end{align*}"}
+{"id": "8318.png", "formula": "\\begin{align*} 2 ^ { - 2 } \\left | \\sum _ { a , b } c _ A ( a ) c _ B ( b ) \\sum _ { g \\in G ~ : ~ g a = b } \\chi ( \\gamma a + \\d ) \\right | \\end{align*}"}
+{"id": "2995.png", "formula": "\\begin{align*} V ( t _ 1 , \\dots , t _ n ) = \\sum _ { i = 1 } ^ d \\mu _ i \\sum _ { k = 1 } ^ n \\frac { t _ k ^ { i } } { i } = \\sum _ { i = 0 } ^ d \\mu _ i V _ { i } \\ . \\end{align*}"}
+{"id": "4392.png", "formula": "\\begin{align*} | I _ { 1 , j } | \\le \\sum _ { k = 1 } ^ d \\int _ { \\R ^ d } | \\nabla \\varphi _ j ^ { ( k ) } ( x ) | p \\theta _ { \\epsilon } ( x ) | \\varphi _ j ^ { ( k ) } ( x ) | d x \\le p \\left ( \\frac { \\delta _ j } { 2 } \\| \\theta _ { \\epsilon } ^ { \\frac { 1 } { 2 } } | \\nabla \\varphi _ j | \\| _ { L ^ 2 } ^ 2 + \\frac { 1 } { 2 \\delta _ j } \\| \\theta _ { \\epsilon } ^ { \\frac { 1 } { 2 } } | \\varphi _ j | \\| _ { L ^ 2 } ^ 2 \\right ) \\end{align*}"}
+{"id": "5584.png", "formula": "\\begin{align*} H = P = \\left ( \\begin{array} { c c c } \\ast & \\ast & \\ast \\\\ 0 & \\ast & \\ast \\\\ 0 & 0 & \\ast \\\\ \\end{array} \\right ) , U = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ \\ast & 1 & 0 \\\\ 0 & 0 & 1 \\\\ \\end{array} \\right ) , V = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ \\ast & \\ast & 1 \\\\ \\end{array} \\right ) , W = \\left ( \\begin{array} { c c c } 1 & 0 & \\ast \\\\ 0 & 1 & \\ast \\\\ 0 & 0 & 1 \\\\ \\end{array} \\right ) , \\end{align*}"}
+{"id": "8779.png", "formula": "\\begin{align*} c \\beta _ i - c \\beta _ j + \\alpha _ { i j } + \\gamma _ i = 0 \\implies \\gamma _ i = c \\beta _ j - c \\beta _ i - \\alpha _ { i j } \\leq 2 p _ { \\max } , \\end{align*}"}
+{"id": "8016.png", "formula": "\\begin{align*} R \\begin{bmatrix} A & 0 \\\\ 0 & B \\end{bmatrix} R = \\begin{bmatrix} ( 1 - x ) A + x B & \\star \\\\ \\star & x A + ( 1 - x ) B \\end{bmatrix} \\end{align*}"}
+{"id": "7350.png", "formula": "\\begin{align*} \\sum _ { \\delta = 0 } ^ \\infty \\delta \\varepsilon ^ \\delta & = \\frac { \\varepsilon } { ( 1 - \\varepsilon ) ^ 2 } . \\end{align*}"}
+{"id": "1965.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j \\leq n } u _ { j p } u _ { \\bar j } + u _ p u _ { p \\bar p } = - \\beta ( u u _ p + B \\rho _ p ) , \\end{align*}"}
+{"id": "6819.png", "formula": "\\begin{align*} W _ 0 ^ { \\sharp } & : = ( W _ 0 ^ { \\flat } \\times \\{ \\theta = 0 \\} ) \\cup N ( \\mathring { \\Lambda } _ 0 ) , \\\\ W _ 1 ^ { \\sharp } & : = ( W _ 1 ^ { \\flat } \\times \\{ \\theta = 0 \\} ) \\cup N ( \\mathring { \\Lambda } _ 0 ) . \\end{align*}"}
+{"id": "5737.png", "formula": "\\begin{align*} \\mathcal { E C S } _ { { \\rm { ( r a n k \\ , 2 ) } } } = \\mathcal { E C S } _ { { \\rm { I } } } \\sqcup \\mathcal { E C S } _ { { \\rm { I I } } } \\sqcup \\mathcal { E C S } _ { { \\rm { I I I } } } . \\end{align*}"}
+{"id": "9348.png", "formula": "\\begin{align*} \\begin{aligned} - d y _ t & = \\big [ g ( x _ t , y _ t , z _ t , \\tilde { z } _ t , \\gamma _ { ( t , e ) } ) - g ( x _ t , 0 , 0 , 0 , 0 ) + g ( x _ t , 0 , 0 , 0 , 0 ) \\big ] d t \\\\ & - z _ t d W _ t - \\tilde { z } _ t d \\xi _ t - \\int _ { \\mathcal { E } } \\gamma _ { ( t , e ) } \\tilde { N } ( d e , d t ) \\\\ & : = \\big [ G ( y _ t , z _ t , \\tilde { z } _ t , \\gamma _ { ( t , e ) } ) + \\varphi ( t ) \\big ] d t - z _ t d W _ t - \\tilde { z } _ t d \\xi _ t - \\int _ { \\mathcal { E } } \\gamma _ { ( t , e ) } \\tilde { N } ( d e , d t ) . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "2567.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) , \\ \\ u > 0 , & \\ \\ \\Omega _ 2 , \\\\ u = 0 , \\ \\ | \\nabla u | = c _ 2 , & \\ \\ , \\partial \\Omega _ 2 . \\end{cases} \\end{align*}"}
+{"id": "188.png", "formula": "\\begin{align*} L i _ 3 \\left ( \\frac { 1 } { 2 } \\right ) = \\frac { 7 } { 8 } \\zeta ( 3 ) - \\frac { 1 } { 1 2 } \\pi ^ 2 \\log 2 + \\frac { 1 } { 6 } ( \\log 2 ) ^ 3 , \\end{align*}"}
+{"id": "4607.png", "formula": "\\begin{align*} \\bar { \\rm P } _ { j , j + i } = & \\sum ^ { M - j } _ { m = i + 1 } { M - j \\choose m } \\mathbb { P } _ { \\rm T X } ^ m \\left ( 1 - \\mathbb { P } _ { \\rm T X } \\right ) ^ { M - j - m } \\bar { \\rm P } _ { m , i } , \\end{align*}"}
+{"id": "6382.png", "formula": "\\begin{align*} X = ( A ^ { 2 } - 5 B ^ { 2 } ) ^ 2 , Y = ( A ^ { 2 } - 5 B ^ { 2 } ) ( ( A ^ { 2 } + 5 B ^ { 2 } ) ^ 2 + 2 0 A ^ 2 B ^ 2 ) , \\end{align*}"}
+{"id": "3411.png", "formula": "\\begin{align*} - \\tau F _ 1 ( t ) - ( 1 + \\tau ^ { - 1 } ) F _ 1 ( s ) \\leq F _ 1 ( | t | - | s | ) - F _ 1 ( t ) = F _ 1 ( t + s ) - F _ 1 ( t ) \\leq F _ 1 ( s ) . \\end{align*}"}
+{"id": "3922.png", "formula": "\\begin{align*} \\int _ { \\mathcal { V } \\times \\mathcal { V } } g \\ , d \\pi = \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathcal { V } \\times \\mathcal { V } } g \\ , d \\pi _ n \\leq \\sup _ { \\gamma \\in \\mathcal { P } _ { \\mathrm { D } } } \\int _ { \\mathcal { V } } g \\ , d \\gamma . \\end{align*}"}
+{"id": "8671.png", "formula": "\\begin{align*} \\mathcal { M } ( P _ X , P _ Y ) = \\sup _ { f \\in \\mathcal { F } } \\left | E \\{ f ( X ) \\} - E \\{ f ( Y ) \\} \\right | , \\end{align*}"}
+{"id": "6402.png", "formula": "\\begin{align*} T _ M ( a ) : = \\int _ \\mathbb { R } \\theta ^ M _ t ( a ) \\ , d t , a \\in \\widetilde { M } _ + . \\end{align*}"}
+{"id": "8028.png", "formula": "\\begin{align*} \\| A \\| _ p = \\{ { \\mathrm { T r \\ , } } ( A ^ * A ) ^ { p / 2 } \\} ^ { 1 / p } , \\end{align*}"}
+{"id": "5642.png", "formula": "\\begin{align*} 0 < \\nu < \\frac 1 2 ( a ^ 2 + b ^ 2 ) ^ { - \\frac { p + q - 2 } { 2 } } C ^ { - 1 } _ { N , p , q } , \\quad \\ \\ p + q = 4 + \\frac { 4 - 2 \\mu } { N } , \\end{align*}"}
+{"id": "7567.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\le x \\\\ n \\equiv v ( \\bmod d ) } } \\beta _ n = \\frac 1 d \\sum _ { n \\le x } \\beta _ n + O ( 1 ) . \\end{align*}"}
+{"id": "425.png", "formula": "\\begin{align*} ( P \\otimes I _ M ) \\vec U _ t + { \\bf D _ { x _ i } } { \\bf A _ i } \\vec U + { \\bf A _ i ^ T } { \\bf D _ { x _ i } } \\vec U + { \\bf C } \\vec U + { \\vec L _ D } = 0 , \\vec U ( 0 ) = \\vec F \\end{align*}"}
+{"id": "7149.png", "formula": "\\begin{align*} \\int _ { \\Omega } w u \\ : x = \\int _ { \\Omega } \\mathcal { L } ^ { \\Omega } _ \\varepsilon c _ \\varepsilon u \\ : x + \\int _ { \\Omega } \\Delta c u \\ : x + \\int _ { \\Omega } \\big ( f ^ \\prime ( c _ \\varepsilon ) - f ^ \\prime ( c ) \\big ) u \\ : x . \\end{align*}"}
+{"id": "34.png", "formula": "\\begin{align*} ( - ) ^ \\ast \\colon \\mathcal { B } \\rightarrow \\mathcal { B } , b = ( b _ { i , j } ) _ { i , j = 1 } ^ n \\mapsto b ^ \\ast = ( b _ { i , j } ^ \\ast ) ^ { \\mathrm { t } } . \\end{align*}"}
+{"id": "6594.png", "formula": "\\begin{align*} f _ { \\omega } ( x ) = \\left \\{ \\begin{aligned} & \\frac { 1 } { 2 \\pi } ( 1 + \\frac { x } { 2 \\pi } ) , \\ \\ \\ x \\in [ - 2 \\pi , 0 ) , \\\\ & \\frac { 1 } { 2 \\pi } ( 1 - \\frac { x } { 2 \\pi } ) , \\ \\ \\ x \\in [ 0 , 2 \\pi ) . \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "276.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - y ^ m z ^ n } \\right ) ^ { \\frac { m ^ 2 } { n ^ 3 } } = \\left ( \\frac { 1 } { 1 - y z } \\right ) ^ { \\frac { y } { 1 - y } } \\end{align*}"}
+{"id": "6903.png", "formula": "\\begin{align*} c _ { \\beta } ( \\sigma ) = ( \\sqrt { 2 } + o ( 1 ) ) ( 1 - \\beta ) ^ { 1 - \\sigma } \\sqrt { | \\log { ( \\sigma - 1 / 2 ) } | } , \\textrm { a s } \\sigma \\to 1 / 2 ^ { + } . \\end{align*}"}
+{"id": "6469.png", "formula": "\\begin{align*} e ^ { i s \\Delta } ( \\overline { e ^ { i s \\Delta } g _ { K , \\sigma } } e ^ { i s \\Delta } g _ { K _ 1 , h } \\overline { e ^ { i s \\Delta } g _ { K _ 2 , h } } ) & = \\frac { | \\beta _ h ( s ) | ^ 2 \\overline { \\beta _ \\sigma ( s ) } } { ( 2 \\pi ) ^ 6 \\big ( 1 + 4 i s z ( s ) \\big ) } \\\\ & \\quad \\times e ^ { - \\frac { z ( s ) } { 1 + 4 i s z ( s ) } | x | ^ 2 + \\frac { 1 } { 1 + 4 i s z ( s ) } x \\cdot \\zeta ( s ) + \\frac { i s } { 1 + 4 i s z ( s ) } \\zeta ( s ) \\cdot \\zeta ( s ) } e ^ { \\gamma ( s ) } \\end{align*}"}
+{"id": "6463.png", "formula": "\\begin{align*} \\frac { \\zeta ' ( s ) \\zeta ( s ) } { 2 z ( s ) } = \\big ( i \\partial _ s \\beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \\overline { \\partial _ s \\beta _ h ( s ) } K _ 2 - i \\overline { \\partial _ s \\beta _ \\sigma ( s ) } K \\big ) \\frac { \\zeta ( s ) } { 2 z ( s ) } \\end{align*}"}
+{"id": "4596.png", "formula": "\\begin{align*} C _ n ( x ) = \\left \\{ \\begin{array} { l r r } K - v x + G _ n ( x + B ) & x \\leq s _ m - B & \\mbox { ( s a t u r a t e d ) } \\\\ K + \\min _ { x \\leq y \\leq x + B } \\{ G _ n ( y ) - v x \\} & s _ m - B < x \\leq s _ m & \\mbox { ( u n s a t u r a t e d o r s a t u r a t e d ) } \\\\ - v x + G _ n ( x ) & x > s _ m & \\mbox { ( n o o r d e r ) } , \\end{array} \\right . \\end{align*}"}
+{"id": "8132.png", "formula": "\\begin{align*} \\varphi ^ + ( M _ I ) = \\widetilde { M _ I ( - X ) } , \\varphi ^ - ( M _ I ) = V \\widetilde { M _ I ( - X ) } , \\end{align*}"}
+{"id": "3844.png", "formula": "\\begin{align*} c _ { Y _ \\ell } ( y _ \\ell , y _ \\ell ' ) & = \\min _ { x _ \\ell , x _ \\ell ' \\in \\mathcal { X } _ \\ell ' } c _ { \\ell } ( s _ \\ell , s _ \\ell ' ) = ( y _ \\ell - y _ \\ell ' ) ^ { \\top } V _ { \\ell , Y Y } ^ { - 1 } ( y _ \\ell - y _ \\ell ' ) , \\end{align*}"}
+{"id": "2218.png", "formula": "\\begin{align*} C _ 3 ( y _ 0 ) & = \\sup _ { t \\ge y _ 0 } \\frac { \\exp ( C _ 1 ( y _ 0 ) R ( t ) ) - 1 } { R ( t ) } = \\frac { \\exp ( C _ 1 ( y _ 0 ) R ( y _ 0 ) ) - 1 } { R ( y _ 0 ) } , \\\\ C _ 4 ( y _ 0 ) & = \\sup _ { t \\ge y _ 0 } \\frac { 1 - \\exp ( - C _ 2 ( y _ 0 ) R ( t ) ) } { R ( t ) } = C _ 2 ( y _ 0 ) . \\end{align*}"}
+{"id": "8370.png", "formula": "\\begin{align*} \\det ( I _ n - A + ( Q - I _ n ) ) = \\det L = 0 , \\end{align*}"}
+{"id": "2551.png", "formula": "\\begin{align*} \\lambda _ t \\lambda _ { j _ 1 } = \\lambda _ t \\lambda _ { j _ 1 ^ * } = \\lambda _ t \\lambda _ { j _ 2 } = \\lambda _ t \\lambda _ { j _ 2 ^ * } = \\lambda _ { j _ 1 } \\lambda _ { j _ 2 } = \\lambda _ { j _ 1 } \\lambda _ { j _ 2 ^ * } = \\lambda _ { j _ 1 ^ * } \\lambda _ { j _ 2 } = \\lambda _ { j _ 1 ^ * } \\lambda _ { j _ 2 ^ * } \\end{align*}"}
+{"id": "4902.png", "formula": "\\begin{align*} h _ d ( X _ 0 , . . . , X _ j , . . . , X _ k ) = X _ j h _ { d - 1 } ( X _ 0 , . . . , X _ j , . . . , X _ k ) + h _ { d } ( X _ 0 , . . . , X _ { j - 1 } , X _ { j + 1 } , . . . , X _ { k } ) , \\end{align*}"}
+{"id": "305.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } \\sum _ { n = 1 } ^ { \\infty } ( + n y ^ 2 x ^ { n + 2 } - n y x ^ { n + 1 } - y x ^ { n + 1 } + n y x ^ { n + 2 } - 2 n x y ^ { n + 1 } ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "3301.png", "formula": "\\begin{align*} { \\rm R e } \\left ( \\xi ^ { - N } e ^ { i ( v + i H v ) } \\frac { \\partial \\kappa } { \\partial \\theta } \\right ) = - e ^ { - u } e ^ { - ( H v ) } \\widetilde { \\rho } _ { \\theta } ( \\theta , \\kappa ) . \\end{align*}"}
+{"id": "5761.png", "formula": "\\begin{align*} \\| T _ { \\alpha } ( \\vec { f } ) \\| _ { L ^ { q , \\infty } ( \\mathbb { R } ^ { n } ) } & \\le C \\prod _ { j = 1 } ^ { m } \\| f _ { j } \\| _ { L ^ { p _ { j } } ( \\mathbb { R } ^ { n } ) } . \\end{align*}"}
+{"id": "4745.png", "formula": "\\begin{align*} \\mu _ i \\tilde { S } _ { \\gamma _ i \\gamma _ j } & = \\mu _ j J _ { 2 | \\alpha _ i | } \\tilde { S } _ { \\gamma _ i \\gamma _ j } J ^ T _ { 2 | \\alpha _ j | } + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "4863.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( b \\right ) & : = \\left \\{ x \\in \\mathbb { R } ^ { n } : a _ { t } ^ { \\prime } x \\leq b _ { t } t \\in T \\right \\} , \\\\ \\mathcal { F } ^ { o p } \\left ( c , b \\right ) & : = \\arg \\min \\left \\{ c ^ { \\prime } x : x \\in \\mathcal { F } \\left ( b \\right ) \\right \\} . \\end{align*}"}
+{"id": "2678.png", "formula": "\\begin{align*} a _ { 1 6 } & = 1 4 2 6 6 4 1 0 7 3 0 5 \\\\ a _ { 1 7 } & = 1 8 3 6 6 5 2 1 7 3 3 6 3 . \\end{align*}"}
+{"id": "4013.png", "formula": "\\begin{align*} \\lbrace \\alpha , \\beta \\rbrace + \\lbrace \\beta , \\gamma \\rbrace + \\lbrace \\gamma , \\alpha \\rbrace = 0 \\end{align*}"}
+{"id": "7771.png", "formula": "\\begin{align*} W _ { \\mathrm { t r } , p } ^ - ( \\mu , \\nu ) = \\inf _ { \\beta \\in \\Phi _ { \\mathrm { t r } } ^ - ( \\nu , m ) } W _ { \\mathrm { t r } , p } ( \\mu , \\beta ) , \\\\ W _ { \\mathrm { t r } , p } ^ + ( \\mu , \\nu ) = \\inf _ { \\alpha \\in \\Phi _ { \\mathrm { t r } } ^ + ( \\mu , n ) } W _ { \\mathrm { t r } , p } ( \\alpha , \\nu ) . \\end{align*}"}
+{"id": "2908.png", "formula": "\\begin{align*} ( [ U ] \\mid \\gamma ) ( v ) = [ U ] ( v \\gamma ^ { - 1 } ) = [ U \\gamma ] ( v ) . \\end{align*}"}
+{"id": "3864.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta _ { A } ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ L } \\left [ \\langle \\lambda , \\delta _ { A } \\rangle + \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\dotsc , \\mu _ L ) } \\int _ { \\mathcal { V } } g _ { \\lambda , A } \\ , d \\pi \\right ] . \\end{align*}"}
+{"id": "960.png", "formula": "\\begin{align*} = | f ( y ) - f ( x ) | + | f ( x ) - z ^ 1 | = | z ^ 1 - f ( y ) | = \\varepsilon ^ { ( 2 ) } \\ , , \\end{align*}"}
+{"id": "9313.png", "formula": "\\begin{align*} \\begin{array} { c c } 1 + d a _ { 1 1 } ( a _ { 1 1 } x _ 1 + a _ { 1 2 } x _ 2 ) ^ { d - 1 } + d a _ { 2 2 } ( a _ { 2 1 } x _ 1 + a _ { 2 2 } x _ 2 ) ^ { d - 1 } + \\\\ d ^ 2 ( a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } ) ( a _ { 1 1 } x _ 1 + a _ { 1 2 } x _ 2 ) ^ { 2 d - 2 } ( a _ { 2 1 } x _ 1 + a _ { 2 2 } x _ 2 ) ^ { 2 d - 2 } . \\end{array} \\end{align*}"}
+{"id": "7284.png", "formula": "\\begin{align*} x _ p = \\frac { l } { ( l , n - k ) } u _ p , y _ p = \\frac { n - k } { ( l , n - k ) } u _ p \\end{align*}"}
+{"id": "5752.png", "formula": "\\begin{align*} | K _ { \\alpha } ( x , \\vec { y } ) - K _ { \\alpha } ( x ' , \\vec { y } ) | & \\le \\dfrac { A } { \\Big ( \\sum \\limits _ { j = 1 } ^ { m } | x - y _ { j } | \\Big ) ^ { m n - \\alpha } } \\omega \\Big ( \\frac { | x - x ' | } { | x - y _ { 1 } | + \\cdots + | x - y _ { m } | } \\Big ) \\end{align*}"}
+{"id": "3259.png", "formula": "\\begin{align*} | K _ \\alpha ( x , y ) - K _ \\alpha ( x , y ' ) | & \\leqslant c \\int _ 0 ^ \\infty | h _ t ( x , y ) - h _ t ( x , y ' ) | \\frac { d t } { t ^ { 1 - \\alpha / 2 } } \\\\ & \\leqslant c \\bigg ( \\int _ 0 ^ { d ( x , y ) ^ 2 } + \\int _ { d ( x , y ) ^ 2 } ^ \\infty \\bigg ) | h _ t ( x , y ) - h _ t ( x , y ' ) | \\frac { d t } { t ^ { 1 - \\alpha / 2 } } \\\\ & = : I \\ ! I _ 1 + I \\ ! I _ 2 . \\end{align*}"}
+{"id": "9029.png", "formula": "\\begin{align*} V ' _ { p _ 2 , p _ 1 } ( \\varphi _ { p _ 3 , p _ 2 , p _ 1 } ( t ) ) = \\frac { \\bar C ( p _ 3 , p _ 2 , \\rho ) } { ( \\varphi _ { p _ 3 , p _ 2 , p _ 1 } ( t ) - t ) ^ 2 } . \\end{align*}"}
+{"id": "8532.png", "formula": "\\begin{align*} \\tilde { G } _ { p , p ^ { \\prime } } ( s , c ) : = \\frac { 2 } { 1 + \\frac { 1 } { \\pi p ^ { \\prime } } } \\ , \\eta _ { p } ( 2 s ) + \\frac { 2 \\sqrt { \\pi } \\ , c ^ { \\frac { 1 } { 2 } - s } } { \\Gamma ( s ) } \\cdot \\frac { \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) } { 1 + \\frac { 1 } { \\pi p } } \\ , \\zeta _ { p ^ { \\prime } } ( 2 s - 1 ) . \\end{align*}"}
+{"id": "8540.png", "formula": "\\begin{align*} \\mathcal { K } _ { \\nu , p } ( x ) = \\intop _ { 0 } ^ { \\infty } \\ , \\frac { y ^ { - \\nu - \\frac { 1 } { 2 } } ( y + 1 ) ^ { - \\nu - \\frac { 1 } { 2 } } } { \\sigma \\left ( \\frac { x } { 2 \\pi } \\ , ( 2 y + 1 ) \\ , \\right ) e ^ { ( 2 y + 1 ) x } - 1 } d y , \\ , \\ , \\ , \\ , \\ , x > 0 , \\ , \\ , ( \\nu ) < \\frac { 1 } { 2 } , \\end{align*}"}
+{"id": "7946.png", "formula": "\\begin{align*} B ( t ) = \\int _ 0 ^ t b ( s ) \\ , d s \\end{align*}"}
+{"id": "2522.png", "formula": "\\begin{align*} ( k , 2 n ) = _ { 2 n } / P . \\end{align*}"}
+{"id": "5295.png", "formula": "\\begin{align*} | \\langle h ^ { = d } , g \\rangle | \\leq \\| h ^ { = d } \\| _ q \\ , \\| g \\| _ { q ' } = \\| h ^ { = d } \\| _ q \\cdot \\mu _ { p } ( g ) / \\mu _ { p } ( g ) ^ { 1 / q } \\end{align*}"}
+{"id": "2111.png", "formula": "\\begin{align*} \\left \\{ \\alpha _ 0 + \\sum _ { i = 1 } ^ r F ^ { k n _ i } ( \\alpha _ i ) \\colon n _ i \\in \\mathbb { N } \\right \\} , \\end{align*}"}
+{"id": "1372.png", "formula": "\\begin{align*} \\langle x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 : x _ i x _ j x _ i ^ { - 1 } = x _ { 3 ( i + j ) \\bmod 5 } \\rangle . \\end{align*}"}
+{"id": "3101.png", "formula": "\\begin{align*} B _ { n , k } ( q ) = q ^ { 2 n k - n ^ 2 } B _ { n , n - k } ( q ) \\end{align*}"}
+{"id": "4018.png", "formula": "\\begin{align*} ( g P ) ( X , Y ) = P ( d X - b Y , - c X + a Y ) . \\end{align*}"}
+{"id": "9277.png", "formula": "\\begin{align*} \\widetilde { S } _ { \\alpha , \\beta } f ( x ) & = x ^ { - \\beta } S _ { \\alpha , \\beta } \\big ( ( \\cdot ) ^ { \\beta } f \\big ) ( x ) \\\\ & \\simeq x ^ { - \\beta } \\sup _ { t > 2 x } \\int _ { t / 2 } ^ { t } ( t + x - z ) ^ { - \\alpha - 1 / 2 } ( t - z ) ^ { \\alpha + \\beta - 1 / 2 } \\abs { f ( z ) } \\ , d z , x > 0 . \\end{align*}"}
+{"id": "2936.png", "formula": "\\begin{align*} \\tilde { \\mu } _ \\delta \\{ r \\to s \\} ( p U ) = \\tilde { \\mu } _ \\delta \\{ r \\to s \\} ( U ) . \\end{align*}"}
+{"id": "8391.png", "formula": "\\begin{align*} F _ { U _ 2 , U _ 1 } ( g ) = [ s , g ] = s _ { | U _ 2 } \\circ g - g \\circ s _ { | U _ 1 } . \\end{align*}"}
+{"id": "335.png", "formula": "\\begin{align*} s _ h ( m + 1 , m ) = \\sum _ { k = 1 } ^ { \\infty } \\left ( \\frac { 1 } { 1 } + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { k } \\right ) ^ { m + 1 } \\frac { 1 } { ( k + 1 ) ^ m } . \\end{align*}"}
+{"id": "1489.png", "formula": "\\begin{align*} ( 1 - c ) ^ { k + 1 } \\leq ( 1 - c ) ^ { \\frac { \\log ( y _ 0 / y ) } { \\log ( L ) } } = ( y / y _ 0 ) ^ \\varrho . \\end{align*}"}
+{"id": "5322.png", "formula": "\\begin{align*} \\| f ^ { = 1 } \\| _ 2 ^ 2 \\approx n p ^ { 2 s - 1 } , \\end{align*}"}
+{"id": "1423.png", "formula": "\\begin{align*} \\| P _ { T \\to 0 } ^ c \\nabla \\hat { f } ( p ) - \\nabla \\hat { f } ( p ) \\| = \\| \\int _ { 0 } ^ T P _ { t \\to 0 } ^ c \\nabla ^ 2 \\hat { f } ( c ( t ) ) c ' ( t ) d t \\| \\leq L T . \\end{align*}"}
+{"id": "5515.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } ^ { y } ( g , A ) = \\int _ { G } { \\bf 1 } _ { A } \\left ( g s \\right ) \\frac { \\varphi _ { g s } ( y ) } { \\varphi _ { g } ( y ) } d \\mu ( s ) . \\end{align*}"}
+{"id": "1507.png", "formula": "\\begin{align*} \\chi ( x ) : = C ^ { - 1 } _ 1 \\int _ { r _ 0 } ^ { \\Lambda _ { 0 } } \\phi ( x ; r ) \\ , f ( r ) \\ , r ^ 2 \\ , \\mathrm { d } r = C ^ { - 1 } _ 1 \\int _ { ( x - \\varepsilon _ 2 ) \\vee r _ 0 } ^ { ( x + \\varepsilon _ { 2 } ) \\wedge \\Lambda _ 0 } \\phi ( x ; r ) \\ , f ( r ) \\ , r ^ 2 \\ , \\mathrm { d } r , x \\in [ r _ 0 - \\varepsilon _ 2 , \\Lambda _ 0 + \\varepsilon _ 2 ] . \\end{align*}"}
+{"id": "4054.png", "formula": "\\begin{align*} 0 = \\sum _ { d _ 1 , d _ 2 < D } \\frac { 1 } { \\sqrt { d _ 1 d _ 2 } } \\sum _ { I d _ 1 , J d _ 2 < D } \\alpha _ { I d _ 1 } \\bar { \\alpha } _ { J d _ 2 } \\sum _ { L , M \\geq 1 } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( I L ) \\lambda _ f ( J M ) . \\end{align*}"}
+{"id": "8190.png", "formula": "\\begin{align*} E _ t \\cap S _ t = \\bigcap _ { j = 1 } ^ { t / h ( t ) } ( E _ { j , t } \\cap S _ { j , t } ) . \\end{align*}"}
+{"id": "7336.png", "formula": "\\begin{align*} x ^ 3 - y ^ 2 z - y = 0 , \\end{align*}"}
+{"id": "7171.png", "formula": "\\begin{align*} \\alpha : = c _ 1 / ( 8 c _ 2 ) \\end{align*}"}
+{"id": "1751.png", "formula": "\\begin{align*} \\mathcal { I } _ { + } ^ { ( m , \\mathbf { k } ) } & = \\int _ { \\frac { \\mu ^ 2 _ { } \\Gamma _ 0 } { \\rho } } ^ { \\infty } \\exp \\Big ( - \\frac { \\Big ( { \\sqrt { \\frac { \\rho x } { \\Gamma _ 0 } } - \\mu _ { \\textrm { D } } } \\Big ) ^ 2 } { 2 ( \\sigma ^ { ( m , \\mathbf { k } ) } _ { } ) ^ 2 } \\Big ) f _ { \\Gamma _ { } ^ { ( m ) } } ( x ) d x . \\end{align*}"}
+{"id": "4168.png", "formula": "\\begin{align*} \\pi ( a ) & = 0 \\Rightarrow \\pi ( a ^ * a ) = 0 \\Rightarrow g \\pi ( a ^ * a ) = 0 \\Rightarrow f \\phi ( a ^ * a ) = 0 \\\\ & \\Rightarrow f ( \\phi ( a ) ^ * \\phi ( a ) ) = 0 \\Rightarrow \\phi ( a ) = 0 \\end{align*}"}
+{"id": "2685.png", "formula": "\\begin{align*} w ^ T M _ { h } = 0 , \\end{align*}"}
+{"id": "1679.png", "formula": "\\begin{align*} P ' = ( P ' - [ a ] Q ) + [ a ] Q \\in \\ker \\phi _ 1 + \\langle Q \\rangle = \\langle P \\rangle + \\langle Q \\rangle , \\end{align*}"}
+{"id": "1426.png", "formula": "\\begin{align*} U ( x ) = \\alpha _ n \\frac { 1 } { ( 1 + | x | ^ 2 ) ^ { \\frac { n - 2 } { 2 } } } , U _ { \\mu , \\xi } ( x ) = \\frac { \\alpha _ n \\mu ^ { \\frac { n - 2 } { 2 } } } { ( \\mu ^ 2 + | x - \\xi | ^ 2 ) ^ { \\frac { n - 2 } { 2 } } } , \\mbox { w i t h } \\ \\alpha _ n = ( n ( n - 2 ) ) ^ { \\frac { n - 2 } { 4 } } , \\end{align*}"}
+{"id": "5375.png", "formula": "\\begin{align*} \\| x _ { k + 1 } \\| ^ 2 + \\| f _ { k , R } \\| ^ 2 + \\| y _ k \\| ^ 2 = \\| x _ k \\| ^ 2 + \\| e _ { k , R } \\| ^ 2 + \\| u _ k \\| ^ 2 , k \\geq 0 . \\end{align*}"}
+{"id": "883.png", "formula": "\\begin{align*} \\{ \\underline { a } \\colon \\underline { a } / \\varpi ^ k \\in L ( \\varpi ^ m \\underline { c } ) \\} = & \\{ a \\underline { c } ^ \\bot + \\varpi ^ { k - m } \\underline { d } \\colon | a | < | \\varpi | ^ { k - m } , ( a , \\varpi ) = 1 , | \\underline { d } | < | \\varpi | ^ m \\} \\setminus \\\\ & \\{ a \\underline { c } ^ \\bot + \\varpi ^ { k - m + 1 } \\underline { d } \\colon | a | < | \\varpi | ^ { k - m + 1 } , | \\underline { d } | < | \\varpi | ^ { m - 1 } , ( a , \\varpi ) = 1 \\} \\end{align*}"}
+{"id": "4321.png", "formula": "\\begin{align*} ( \\rho * f _ 0 ) ( x ) = ( \\rho * f _ 0 ) ( x + 2 \\tau _ k e _ { i _ k } ) . \\end{align*}"}
+{"id": "4460.png", "formula": "\\begin{align*} N ( U _ n ( t _ n ) ) \\ \\rightarrow \\ - 2 h ( 0 ) = - 2 \\mu _ { \\omega , \\mathbf { c } } . \\end{align*}"}
+{"id": "1257.png", "formula": "\\begin{align*} \\mu _ p ( ( x _ 1 , x _ 2 ) , ( x _ 1 , x _ 2 ) ) = 1 \\end{align*}"}
+{"id": "9131.png", "formula": "\\begin{align*} \\psi ( \\ldots , \\zeta ( k - 2 ) , \\zeta ( k - 1 ) , x ( k ) , v ( k ) , v ( k + 1 ) , \\ldots ) = 0 \\ , . \\end{align*}"}
+{"id": "1787.png", "formula": "\\begin{align*} \\theta ^ \\ell _ { i _ 1 , \\ldots , i _ { \\ell - 1 } , 1 , i _ { \\ell + 1 } , \\ldots , i _ n } ( f _ { i _ 1 , \\ldots , i _ { \\ell - 1 } , 1 , i _ { \\ell + 1 } , \\ldots , i _ n } ) = \\theta ^ \\ell _ { i _ 1 , \\ldots , i _ { \\ell - 1 } , 2 , i _ { \\ell + 1 } , \\ldots , i _ n } ( f _ { i _ 1 , \\ldots , i _ { \\ell - 1 } , 2 , i _ { \\ell + 1 } , \\ldots , i _ n } ) \\end{align*}"}
+{"id": "6726.png", "formula": "\\begin{align*} [ \\underline { \\varphi } , \\varphi ] : = \\{ x \\in \\{ 0 , 1 \\} ^ \\Z : \\underline { \\varphi } ( h ) \\leq x \\leq \\varphi ( h ) h \\in H \\} . \\end{align*}"}
+{"id": "6151.png", "formula": "\\begin{align*} f _ { n + 1 } \\circ h _ n ( a _ n ) & = f _ { n + 1 } \\circ ( \\ast _ { 1 n } ( \\top _ 1 , a _ n ) ) = f _ 1 ( \\top _ 1 ) \\ast _ { 1 n } f _ n ( a _ n ) \\\\ & = \\top _ 1 \\ast _ { 1 n } f _ n ( a _ n ) = l _ n ( f _ n ( a _ n ) ) = l _ n \\circ f _ n ( a _ n ) . \\end{align*}"}
+{"id": "8688.png", "formula": "\\begin{align*} ( \\Delta _ s ) = o ( N ^ { - 2 } p ^ { - 1 } ) s = 2 , \\ldots , l - 1 . \\end{align*}"}
+{"id": "1669.png", "formula": "\\begin{align*} M ( \\tau ( x ) ) & = M ( \\sum _ { j _ 1 , \\ldots , j _ \\ell = 1 , 2 } \\tau _ { j _ 1 } ( x _ { i _ 1 } ) \\cdots \\tau _ { j _ \\ell } ( x _ { i _ \\ell } ) ) \\\\ & \\geq \\min _ { { j _ 1 , \\ldots , j _ \\ell = 1 , 2 } } \\{ M ( \\tau _ { j _ 1 } ( x _ { i _ 1 } ) \\cdots \\tau _ { j _ \\ell } ( x _ { i _ \\ell } ) ) \\} . \\end{align*}"}
+{"id": "9192.png", "formula": "\\begin{align*} z ^ { i } = g ^ { i } \\ , , i = 1 , \\ldots , l \\end{align*}"}
+{"id": "1854.png", "formula": "\\begin{align*} \\lambda _ { 1 , 2 } ^ { r , s } = \\frac { I _ { 1 } ( \\rho r s / ( 1 - \\rho ^ { 2 } ) ) } { I _ { 0 } ( \\rho r s / ( 1 - \\rho ^ { 2 } ) ) } \\geq \\frac { \\rho r s / ( 1 - \\rho ^ { 2 } ) } { 1 / 2 + \\sqrt { 9 / 4 + [ \\rho r s / ( 1 - \\rho ^ { 2 } ) ] ^ { 2 } } } \\geq . 9 3 ( 1 - e ^ { - \\frac { \\rho r s / ( 1 - \\rho ^ { 2 } ) } { 2 } } ) . \\end{align*}"}
+{"id": "590.png", "formula": "\\begin{align*} 1 & = \\sum _ { k = 1 } ^ n | \\omega ( b _ k ) | ^ 2 \\leq \\sum _ { k = 1 } ^ n \\psi ( b _ k b _ k ^ * ) = \\psi \\left ( \\sum _ { k = 1 } ^ n b _ k b _ k ^ * \\right ) \\leq 1 \\end{align*}"}
+{"id": "239.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , \\gcd ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - y ^ m z ^ n } \\right ) ^ { \\frac { 1 } { m ^ a n ^ b } } = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\sum _ { m = 1 } ^ { n } \\frac { y ^ m } { m ^ a } \\right ) \\frac { z ^ n } { n ^ b } \\right \\} . \\end{align*}"}
+{"id": "6213.png", "formula": "\\begin{align*} A = \\begin{pmatrix} - a _ i & a _ { i + 1 } \\\\ - b _ i & b _ { i + 1 } \\end{pmatrix} \\end{align*}"}
+{"id": "6755.png", "formula": "\\begin{align*} H _ \\infty : = \\{ h \\in H : R ^ n h \\in W \\setminus \\underline { W } \\} \\end{align*}"}
+{"id": "4229.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\det T _ n ( \\phi ) } { G [ \\phi _ 0 ] ^ n \\prod _ { k = 1 } ^ R \\det T _ n ( \\phi _ k ) } = E \\end{align*}"}
+{"id": "7210.png", "formula": "\\begin{align*} \\forall s , t \\in [ 1 / 2 , 1 ) , \\widetilde { z } _ h ^ { ( s ) } = \\widetilde { z } _ h ^ { ( t ) } = \\widetilde { u } _ h \\mbox { a . e . o n } B _ 1 \\setminus \\left ( \\cup _ { \\mathcal { F } ^ { ( s ) } _ h } B \\bigcup \\cup _ { \\mathcal { F } ^ { ( t ) } _ h } B \\right ) . \\end{align*}"}
+{"id": "4799.png", "formula": "\\begin{align*} \\Phi = \\begin{bmatrix} \\phi ( y _ 1 ) & \\phi ( y _ 2 ) & \\cdots & \\phi ( y _ n ) \\end{bmatrix} , \\end{align*}"}
+{"id": "6646.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + \\rho \\ , c u = 0 \\quad \\ , \\ , \\ , \\R ^ N \\ , . \\end{align*}"}
+{"id": "4472.png", "formula": "\\begin{align*} \\mu _ { \\omega , c } = - \\frac { 1 } { 2 } N ( \\Phi ) \\le C \\| \\Phi \\| _ { \\mathcal { H } ^ 1 } ^ 3 , \\end{align*}"}
+{"id": "676.png", "formula": "\\begin{align*} \\mathfrak { h } _ { 0 } & \\Big ( \\varphi - \\sum _ { 1 \\leq i \\leq g } d ^ + _ { i , l - 2 } ( D ^ 2 \\varphi ) h _ { i , 2 } - \\sum _ { 1 \\leq i \\leq g } d ^ + _ { i , l - 1 } ( D \\varphi ) h _ { i , 1 } - \\sum _ { 1 \\leq s < \\gamma } d ^ 0 _ { s , l - 1 } ( D \\varphi ) c _ { s , 1 } \\Big ) \\\\ & = \\sum _ { 1 \\leq i \\leq g } d ^ + _ { i , l } ( \\varphi ) h _ i + \\sum _ { 1 \\leq s < \\gamma } d ^ 0 _ { s , l } ( \\varphi ) c _ s + \\sum _ { 1 \\leq j \\leq g } d ^ - _ { - j , l } ( \\varphi ) h _ { - j } . \\end{align*}"}
+{"id": "2641.png", "formula": "\\begin{align*} T _ l ' ( f _ 1 , f _ 2 ) ( x , y ) = \\int _ \\mathbb { R } \\ ! f _ 1 \\left ( x + \\widetilde { P } _ 1 ( t ) , y \\right ) f _ 2 \\left ( x , y + \\widetilde { P } _ 2 ( t ) \\right ) \\psi ( t ) t ^ { - 1 } \\ , \\mathrm { d } t . \\end{align*}"}
+{"id": "3071.png", "formula": "\\begin{align*} \\left [ u ' ( r ) ^ { p - 1 - \\alpha } \\right ] ' & \\geq \\frac { \\delta } { n - 1 } \\ , f _ 1 ( r ) g _ 1 ( v ( r ) ) - \\frac { \\delta ^ 2 } { ( n - 1 ) ( \\delta + 1 ) } \\ , f _ 1 ( r ) g _ 1 ( v ( r ) ) \\\\ & = \\frac { p - 1 - \\alpha } { n ( p - 1 - \\alpha ) + \\alpha } \\ , f _ 1 ( r ) g _ 1 ( v ( r ) ) \\end{align*}"}
+{"id": "6271.png", "formula": "\\begin{align*} \\eta _ { t } ( x ) = \\begin{cases*} 0 & i f $ r ( x ) \\leq t ^ 2 $ \\\\ ( \\log t ^ 2 - \\log r ) / \\log ( t ) & $ t \\leq r ( x ) \\leq t ^ 2 $ \\\\ 1 & i f $ r ( x ) \\geq t $ . \\end{cases*} \\end{align*}"}
+{"id": "2775.png", "formula": "\\begin{align*} \\mathbf { m } _ \\lambda = \\max \\left ( \\lambda ^ 2 , \\mathbf { b } _ \\lambda \\right ) \\lambda ^ 5 \\mathbf { e } _ \\lambda ^ 2 . \\end{align*}"}
+{"id": "9207.png", "formula": "\\begin{align*} \\lvert A ( D ) \\rvert \\le n _ 0 ( n _ 0 - 1 ) + \\sum _ { i = n _ 0 + 1 } ^ n 2 ( i - 1 ) - ( 2 r - 2 ) & = n ^ 2 - ( 2 r - 1 ) n + ( 2 r - 2 ) n _ 0 \\\\ & < \\lvert A ( D _ { n , r , 1 } ) \\rvert . \\end{align*}"}
+{"id": "7592.png", "formula": "\\begin{align*} | H _ { 2 , 1 } ( F _ { f } / 2 ) | = \\frac { 1 } { 1 6 } | \\tau _ { 2 } | ^ 2 \\leq \\frac { 1 } { 1 6 } . \\end{align*}"}
+{"id": "5942.png", "formula": "\\begin{align*} \\phi ^ { - 1 } ( h _ 1 h _ 2 ) = ( \\sum ^ m _ { k = 1 } ( a ^ 1 _ k + i b ^ 1 _ k ) f _ k + \\sum ^ m _ { k = 1 } ( a ^ 2 _ k + i b ^ 2 _ k ) f _ k , t _ 1 + t _ 2 + \\tfrac { 1 } { 2 } \\sum _ { k = 1 } ^ m ( a _ k ^ 1 b _ k ^ 2 - a _ k ^ 2 b _ k ^ 1 ) ) ; \\end{align*}"}
+{"id": "4122.png", "formula": "\\begin{align*} \\left ( \\frac { M ^ { ( s ) } _ { i , j } } { M ^ { ( s ) } _ { k , j } } \\right ) _ { s = P } ^ { + \\infty } \\end{align*}"}
+{"id": "8334.png", "formula": "\\begin{align*} \\phi ( t ) = D C _ D ^ \\frac 1 { D } t ^ { \\frac { D - 1 } { D } } ( - u _ \\mu ^ * ) ' ( t ) , \\ t \\in ( 0 , \\infty ) . \\end{align*}"}
+{"id": "515.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ { t } ^ { 2 } u ( t , k ) + a ( t ) \\mathcal { H } _ { \\hbar , V } u ( t , k ) + q ( t ) u ( t , k ) = f ( t , k ) , \\quad ( t , k ) \\in ( 0 , T ] \\times \\hbar \\mathbb { Z } ^ { n } , \\\\ u ( 0 , k ) = u _ { 0 } ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\\\ \\partial _ { t } u ( 0 , k ) = u _ { 1 } ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\end{array} \\right . \\end{align*}"}
+{"id": "1100.png", "formula": "\\begin{align*} ( b , x ) \\ast _ \\varphi ( b ' , x ' ) = ( b b ' , x x ' + b \\ast x ' + b ' \\ast x ) , \\forall ( b , x ) , ( b ' , x ' ) \\in B \\oplus X . \\end{align*}"}
+{"id": "2761.png", "formula": "\\begin{align*} T ^ \\ast T \\psi _ j = \\mu _ j \\psi _ j . \\end{align*}"}
+{"id": "1437.png", "formula": "\\begin{align*} \\Pi _ { \\mu , \\xi } \\Big ( V + \\phi - i ^ * [ f _ \\epsilon ( V + \\phi ) ] \\Big ) = 0 . \\end{align*}"}
+{"id": "5772.png", "formula": "\\begin{align*} E _ { 2 } & = \\{ x \\in \\mathbb { R } ^ { n } \\setminus \\Omega ^ { * } : | T _ { \\alpha } ( b _ { 1 } , g _ { 2 } ) ( x ) | > \\lambda / 4 \\} , \\\\ E _ { 3 } & = \\{ x \\in \\mathbb { R } ^ { n } \\setminus \\Omega ^ { * } : | T _ { \\alpha } ( g _ { 1 } , b _ { 2 } ) ( x ) | > \\lambda / 4 \\} , \\\\ E _ { 4 } & = \\{ x \\in \\mathbb { R } ^ { n } \\setminus \\Omega ^ { * } : | T _ { \\alpha } ( b _ { 1 } , b _ { 2 } ) ( x ) | > \\lambda / 4 \\} . \\end{align*}"}
+{"id": "8891.png", "formula": "\\begin{align*} \\langle u _ 1 \\otimes \\cdots \\otimes u _ i , v _ 1 \\otimes \\cdots \\otimes v _ i \\rangle _ { \\otimes i } = \\prod _ { j = 1 } ^ i \\langle u _ j , v _ j \\rangle _ 1 . \\end{align*}"}
+{"id": "4232.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\det T _ n ( \\phi ) } { G [ \\phi _ 0 ] ^ n \\prod _ { k = 1 } ^ R \\det T _ n ( \\phi _ k ) } = E \\end{align*}"}
+{"id": "2792.png", "formula": "\\begin{align*} \\mathcal { S } _ { q , \\lambda } ^ 1 = \\{ u \\in H ^ 2 ( \\mathrm { M } ) ; \\ ; ( \\Delta + \\lambda - q ) u = 0 , \\ ; ( \\partial _ \\nu - i a ) u \\in H ^ { 1 / 2 } _ \\Gamma ( \\partial \\mathrm { M } ) \\} . \\end{align*}"}
+{"id": "3243.png", "formula": "\\begin{align*} \\begin{aligned} | f ( x ) - f ( x ' ) | & \\leqslant C \\Big ( \\frac { \\| x - x ' \\| } { r } \\Big ) ^ \\beta \\Big \\{ \\frac { 1 } { V ( x , x _ 0 , r + d ( x , x _ 0 ) ) } \\Big ( \\frac { r } { r + { d ( x , x _ 0 ) } } \\Big ) ^ \\gamma \\\\ & \\qquad + \\frac { 1 } { V ( x ' , x _ 0 , r + d ( x ' , x _ 0 ) ) } \\Big ( \\frac { r } { r + { d ( x ' , x _ 0 ) } } \\Big ) ^ \\gamma \\Big \\} ; \\end{aligned} \\end{align*}"}
+{"id": "4396.png", "formula": "\\begin{align*} \\lim _ { | x | \\rightarrow \\infty } \\varphi _ 3 ( x ) = 0 . \\end{align*}"}
+{"id": "4050.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( i ) \\lambda _ f ( j ) \\big | L ( f , k ) \\big | ^ 2 . \\end{align*}"}
+{"id": "716.png", "formula": "\\begin{align*} A _ { S } ^ * [ \\partial _ { t } \\mu _ { S } ] ( e ) + C _ { D } [ \\partial _ { t } \\mu _ { B } ] ( e ) = G _ { D , 1 } ( e ) \\ , , \\end{align*}"}
+{"id": "3512.png", "formula": "\\begin{gather*} \\omega ( ( q , S q ) , ( r , S r ) ) = - \\langle q , S r \\rangle + \\langle S q , r \\rangle = \\langle q , ( S ^ { \\mathsf { T } } - S ) r \\rangle = 0 , \\end{gather*}"}
+{"id": "861.png", "formula": "\\begin{align*} \\chi _ { \\tau , i } ( z ) = z ^ { a _ { \\tau , i } } \\overline { z } ^ { b _ { \\tau , i } } \\end{align*}"}
+{"id": "8784.png", "formula": "\\begin{align*} & 2 ( u + 1 ) ( u + 2 t + 3 ) c _ 1 \\\\ & = 2 ( u + 1 ) ( 2 + t ) b _ 1 + 2 ( u + 1 ) a _ 3 + 2 u ( u + 1 ) a _ 2 + 2 u ( u + 1 ) c _ 2 \\end{align*}"}
+{"id": "253.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( 1 - z ^ n \\right ) ^ { \\frac { m ^ 3 } { n ^ 4 } } = \\sqrt [ 3 ] { 1 - z } \\ ; \\exp \\left \\{ \\frac { - 1 } { 4 } L i _ 2 ( z ) - \\frac { 1 } { 4 } \\frac { z } { 1 - z } \\right \\} . \\end{align*}"}
+{"id": "7496.png", "formula": "\\begin{align*} \\sigma = \\varphi _ { S } ( \\pi ) = ( \\varphi _ { x _ { k } } \\circ \\cdots \\circ \\varphi _ { x _ { 2 } } \\circ \\varphi _ { x _ { 1 } } ) ( \\pi ) . \\end{align*}"}
+{"id": "4445.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( V ) = \\frac { 1 } { 2 } L _ { \\omega , \\mathbf { c } } ( V ) + N ( V ) > \\mu _ { \\omega , \\mathbf { c } } , \\end{align*}"}
+{"id": "4263.png", "formula": "\\begin{align*} J \\left ( t , \\mu ; u ^ { t , \\omega ^ { 0 } } \\right ) = \\left . J \\left ( t , \\mu ; u \\right ) \\right \\vert _ { t , \\mu , u ^ { t , \\omega ^ { 0 } } } , \\mathbb { P } ^ { 0 } \\end{align*}"}
+{"id": "8074.png", "formula": "\\begin{align*} \\lambda _ { 1 } = \\inf _ { v \\in S _ { 0 } ^ { 1 , p } ( \\Omega ) \\setminus \\{ 0 \\} } \\left \\{ \\frac { \\int _ { \\Omega } | X v | ^ { p } d x } { \\int _ { \\Omega } | v | ^ { p } d x } \\right \\} = \\frac { \\int _ { \\Omega } | X u _ { 1 } | ^ { p } d x } { \\int _ { \\Omega } | u _ { 1 } | ^ { p } d x } , \\end{align*}"}
+{"id": "911.png", "formula": "\\begin{align*} E _ 2 & \\ll \\widehat { P } ^ { n - 5 } \\widehat { Y } ^ { 4 - n / 2 - 2 / n + \\varepsilon } \\widehat { Z } ^ { 3 - n / 2 } \\widehat { V } ^ { n - 2 } \\\\ & = \\widehat { P } ^ { - 3 } \\widehat { Z } ^ { 1 + n / 2 } \\widehat { Y } ^ { 2 + n / 2 - 2 / n + \\varepsilon } \\\\ & \\ll \\widehat { P } ^ { 5 n / 6 - 3 + 5 / 3 } \\widehat { Y } ^ { 1 - 2 / n + \\varepsilon } \\\\ & \\ll \\widehat { P } ^ { 5 n / 6 - 3 + 1 0 / 3 + \\varepsilon } , \\end{align*}"}
+{"id": "4034.png", "formula": "\\begin{align*} \\langle T _ i ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) , f \\rangle = \\lambda _ f ( i ) \\ , \\langle ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) , \\ , f \\rangle \\ , \\ i = 1 , \\ldots , D . \\end{align*}"}
+{"id": "4662.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta u = \\lambda u + g ( u ) , \\hbox { i n } \\mathbb { R } ^ N , \\\\ & \\int _ { \\mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4318.png", "formula": "\\begin{align*} E & = \\left ( \\bigcup _ { ( k , m ) \\in \\mathcal { D } } [ t _ { k , m } , t _ { k , m } + 2 ^ { - 2 k } ] \\right ) \\cup \\left \\{ t _ { k , m } + 2 ^ { 2 - 2 k } : ( k , m ) \\in \\mathcal { D } \\right \\} \\\\ & \\subset [ 0 , 1 ] , \\end{align*}"}
+{"id": "2837.png", "formula": "\\begin{align*} = a _ { p + h _ 1 - 1 } ^ { h } a _ { p + h 1 - 2 } ^ { 2 h } \\cdots a _ { h _ 1 + 1 } ^ { ( p - 1 ) h } ( a _ { h _ 1 } \\cdots a _ 0 ) ^ { p h } . \\end{align*}"}
+{"id": "5533.png", "formula": "\\begin{align*} \\mathrm { I } ( \\xi _ { 1 } , \\xi _ { n } | X , \\eta _ { p } ) = \\int _ { X } \\mathrm { I } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) d \\eta _ { p } ( x ) = \\int _ { Y } \\int _ { S _ { y } } \\mathrm { I } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) d \\eta _ { p } ^ { y } ( x ) d \\nu ( y ) \\end{align*}"}
+{"id": "5643.png", "formula": "\\begin{align*} \\lambda _ { 1 , n } = - a ^ { - 2 } \\langle J ' _ \\nu ( u _ n , v _ n ) , ( u _ n , 0 ) \\rangle + o _ n ( 1 ) , \\ \\lambda _ { 2 , n } = - b ^ { - 2 } \\langle J ' _ \\nu ( u _ n , v _ n ) , ( 0 , v _ n ) \\rangle + o _ n ( 1 ) . \\end{align*}"}
+{"id": "6545.png", "formula": "\\begin{align*} g _ F ( t ) : = - \\frac { m _ 0 } { 2 } \\ , t ^ 2 + i B _ F t + \\sum _ { \\gamma \\not = 0 } m _ \\gamma \\frac { e ^ { - i \\gamma t } - 1 } { \\gamma ^ 2 } , B _ F : = i \\frac { \\xi _ F ^ \\prime } { \\xi _ F } \\left ( \\frac { 1 } { 2 } \\right ) \\end{align*}"}
+{"id": "5517.png", "formula": "\\begin{align*} g \\mapsto \\varphi _ { g } ( \\pi ( x ) ) = \\frac { d g \\nu } { d \\nu } ( \\pi ( x ) ) \\end{align*}"}
+{"id": "6984.png", "formula": "\\begin{align*} \\mu ( f ) = \\min \\limits _ { i \\in \\{ 1 , \\dots , s \\} } v ( a _ i ) . \\end{align*}"}
+{"id": "6368.png", "formula": "\\begin{align*} \\widetilde { g } \\bigl ( \\widetilde { y } , F ( \\widetilde { y } ) \\bigr ) = \\bigl ( \\widetilde { f } ( \\widetilde { y } ) , F ^ \\prime ( \\widetilde { y } ) \\widetilde { f } ( \\widetilde { y } ) \\bigr ) , \\end{align*}"}
+{"id": "8752.png", "formula": "\\begin{align*} f ( y , \\xi ( \\omega ) ) : = y ^ 2 \\delta _ 0 ( \\xi ( \\omega ) ) + ( 1 - y ) ^ 2 \\delta _ 1 ( \\xi ( \\omega ) ) . \\end{align*}"}
+{"id": "7593.png", "formula": "\\begin{align*} | H _ { 2 , 1 } ( F _ { f } / 2 ) | & = \\frac { 1 } { 1 9 2 } \\left ( | - 3 \\tau ^ 4 _ { 1 } + 4 ( 1 - \\tau ^ 2 _ { 1 } ) \\tau ^ 2 _ { 1 } \\tau _ { 2 } - 4 ( 1 - \\tau ^ 2 _ { 1 } ) ( 3 + \\tau ^ 2 _ { 1 } ) \\tau ^ 2 _ { 2 } | \\right . \\\\ & \\left . + 1 6 \\tau _ { 1 } ( 1 - \\tau ^ 2 _ { 1 } ) ( 1 - | \\tau ^ 2 _ { 2 } | ) \\right ) \\\\ & = \\frac { 1 } { 1 2 } \\tau _ { 1 } ( 1 - \\tau ^ 2 _ { 1 } ) \\left ( | A + B \\tau _ { 2 } + C \\tau ^ 2 _ { 2 } | + 1 - | \\tau _ { 2 } | ^ 2 \\right ) , \\end{align*}"}
+{"id": "7418.png", "formula": "\\begin{align*} \\mathbb E [ \\xi _ { \\beta } ^ J ] : = \\prod _ { i : \\ , | J ^ i | \\geq 2 } \\mathbb E [ \\xi _ { \\beta } ^ { | J ^ i | } ] \\ , . \\end{align*}"}
+{"id": "2238.png", "formula": "\\begin{align*} \\varphi ( q _ { n * } ( \\xi , \\ldots , \\xi , a , b ) ) = q _ { n * } ( \\varphi ( \\xi ) , \\ldots , \\varphi ( \\xi ) , \\varphi ( a ) , \\varphi ( b ) ) \\end{align*}"}
+{"id": "3491.png", "formula": "\\begin{align*} C _ { \\mathrm { U B F } } = \\left ( \\frac { e } { \\theta } \\right ) ^ { \\lceil \\frac { 2 \\eta } { b } \\rceil } = \\left ( \\frac { 5 e ^ 2 \\Delta ^ 2 } { b ^ 2 } \\right ) ^ { \\lceil \\frac { 2 \\eta } { b } \\rceil } . \\end{align*}"}
+{"id": "5512.png", "formula": "\\begin{align*} \\mathcal { S } \\left ( x , \\left ( L _ { x } \\omega _ { 1 } , L _ { x } \\omega _ { 2 } , \\ldots \\right ) \\right ) = \\left ( x , \\left ( L _ { x } \\omega _ { 2 } , L _ { x } \\omega _ { 3 } , \\ldots \\right ) \\right ) , \\end{align*}"}
+{"id": "5208.png", "formula": "\\begin{align*} { \\wedge } = { \\wedge } \\circ { \\vartriangleleft } . \\end{align*}"}
+{"id": "8684.png", "formula": "\\begin{align*} \\widehat { ( \\Delta _ { 1 } ) } = \\frac { 8 } { p ^ { 2 } } \\bigg [ \\frac { 1 } { n ( n - 1 ) } \\{ f ^ { ( 1 ) } ( \\widehat { \\tau _ { 1 } } ) \\} ^ { 2 } \\widehat { ( \\Sigma _ { 1 } ^ { 2 } ) } + \\frac { 1 } { m ( m - 1 ) } \\{ f ^ { ( 1 ) } ( \\widehat { \\tau _ { 2 } } ) \\} ^ { 2 } \\widehat { ( \\Sigma _ { 2 } ^ { 2 } ) } + \\frac { 2 } { n m } \\{ f ^ { ( 1 ) } ( \\widehat { \\tau _ { 3 } } ) \\} ^ { 2 } \\widehat { ( \\Sigma _ { 1 } \\Sigma _ { 2 } ) } \\bigg ] , \\end{align*}"}
+{"id": "1291.png", "formula": "\\begin{align*} Z _ q ( G _ { n , \\ell } ) & = \\left \\{ \\begin{aligned} & n ( \\ell - 1 ) & \\mbox { i f } q = 0 , \\\\ & n \\ell - 2 & \\mbox { i f } q \\geq 1 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3544.png", "formula": "\\begin{align*} W = \\{ & z \\in \\R ^ { 3 n + 3 } : P _ 2 ( z ) = 0 , \\dots , P _ { 2 n + 1 } ( z ) = 0 ( \\partial P _ i / \\partial y _ j ) ( z ) \\ne 0 \\\\ & i = 2 , \\dots , 2 n + 1 , j = 2 , \\dots , 3 n + 2 \\} . \\end{align*}"}
+{"id": "1469.png", "formula": "\\begin{align*} H _ 1 \\leq C \\Big ( \\Big | | \\phi | ^ p \\Big | _ { \\frac { 2 n } { n + 2 } } + | \\phi ^ 2 | _ { \\frac { 2 n } { n + 2 } } \\Big ) = C \\bigg ( | \\phi | ^ p _ { \\frac { 2 n } { n - 2 } } + \\Big ( \\int \\limits _ \\Omega | \\phi | ^ { \\frac { 2 n } { n - 2 } } d x \\Big ) ^ { \\frac { n + 2 } { 2 n } } \\bigg ) = 2 C | \\phi | _ { p + 1 } ^ p \\leq 2 C \\| \\phi \\| ^ 2 . \\end{align*}"}
+{"id": "579.png", "formula": "\\begin{align*} \\lim _ { ( x _ { 1 } , x _ { 2 } ) \\rightarrow ( x _ { 0 } , 0 ) } \\gamma ^ { a } ( x _ { 1 } , x _ { 2 } ) = \\ & a _ { - } \\int _ { - \\infty } ^ { x _ { 0 } } H ( y _ 1 ) [ H ( y _ 1 ) ] ^ T d y _ { 1 } \\\\ & + a _ { + } \\int _ { x _ { 0 } } ^ { + \\infty } H ( y _ 1 ) [ H ( y _ 1 ) ] ^ T d y _ { 1 } . \\end{align*}"}
+{"id": "6589.png", "formula": "\\begin{align*} [ { \\hat n _ t } ] = \\arg \\min \\limits _ { n _ t \\in \\{ 1 , \\cdots , N _ t \\} } \\left | y - \\sqrt { P _ s } \\zeta \\sum \\nolimits _ { l = 1 } ^ L \\alpha _ { l , { n _ t } } \\hat \\beta _ l \\right | ^ 2 . \\end{align*}"}
+{"id": "7697.png", "formula": "\\begin{align*} \\pi ( \\lambda ) = \\lambda _ { L _ 1 } - \\frac { ( \\lambda , z ) } { N } \\zeta _ { L _ 1 } , \\lambda \\in M ' \\end{align*}"}
+{"id": "896.png", "formula": "\\begin{align*} \\# \\{ | \\underline { a } _ 2 | < | e _ 2 k ' | \\colon \\underline { a } _ 2 / r _ 2 ' \\in L ( d _ 2 \\underline { c } ) \\} \\leq \\left ( \\frac { | e _ 2 k ' | } { | r _ 2 ' d _ 2 ^ { - 1 } | } \\right ) ^ { 2 } | r _ 2 ' d _ 2 ^ { - 1 } | = | e _ 2 k ' | ^ 2 | r _ 2 ' | ^ { - 1 } | d _ 2 | . \\end{align*}"}
+{"id": "2541.png", "formula": "\\begin{align*} B ( N ) = \\left \\{ \\{ j _ \\ell , j _ k \\} , \\{ j _ \\ell , j _ k ^ * \\} , \\{ j _ k , j _ \\ell ^ * \\} , \\{ j _ k ^ * , j _ \\ell ^ * \\} \\ , | \\ , 1 \\le j < k \\le r \\right \\} \\end{align*}"}
+{"id": "8463.png", "formula": "\\begin{align*} K _ { \\frac { 1 } { 2 } } ( x ) = \\sqrt { \\frac { \\pi } { 2 x } } \\ , e ^ { - x } , \\ , \\ , \\ , \\ , \\ , x > 0 , \\end{align*}"}
+{"id": "8396.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\triangle ( w _ 1 + w _ 2 + \\cdots + w _ { n - 1 } ) = | \\gamma _ 1 | ^ 2 e ^ { 2 w _ 1 - w _ 2 } + \\sum _ { i = 2 } ^ { n - 2 } | \\gamma _ i | ^ 2 e ^ { 2 w _ i - w _ { i - 1 } - w _ { i + 1 } } + | \\gamma _ { n - 1 } | ^ 2 e ^ { 2 w _ { n - 1 } - w _ { n - 2 } } \\end{align*}"}
+{"id": "8858.png", "formula": "\\begin{align*} ( \\theta _ r w ) _ 2 - ( \\phi _ r w ) _ 2 = [ ( \\theta _ r ) _ 2 - ( \\phi _ r ) _ 2 ] w _ 0 + [ ( \\theta _ r ) _ 1 - ( \\phi _ r ) _ ] w _ 1 = [ ( \\theta _ { r - 1 } ) _ 1 + ( \\phi _ { r - 1 } ) _ 1 ] w _ 0 + [ ( \\theta _ { r - 1 } ) _ 0 + ( \\phi _ { r - 1 } ) _ 0 ] w _ 1 \\end{align*}"}
+{"id": "6208.png", "formula": "\\begin{align*} | a d - b c | = d ( v , w ) - 1 . \\end{align*}"}
+{"id": "2998.png", "formula": "\\begin{align*} \\partial ^ { l o g } f ( t ) = f ^ { \\prime } ( t ) \\cdot f ( t ) ^ { - 1 } , \\end{align*}"}
+{"id": "7673.png", "formula": "\\begin{align*} \\nabla _ g u _ \\delta = ( \\phi ' ( \\rho ) - s \\phi ( \\rho ) ) e ^ { - s \\rho } \\nabla _ g \\rho , \\end{align*}"}
+{"id": "8167.png", "formula": "\\begin{align*} K = K ( \\omega ) : = \\bigcup _ { x _ i \\in \\ , ( \\Pi ) } \\bar { B } ( x _ i , a ) , \\end{align*}"}
+{"id": "5518.png", "formula": "\\begin{align*} \\left ( g . \\overline { \\mathbb { P } } _ { \\mu } \\right ) _ { x } = \\overline { \\mathbb { P } } _ { \\mu , x , g } , \\end{align*}"}
+{"id": "8283.png", "formula": "\\begin{align*} ( \\ref { i n t e g r a l } ) = M _ N ( 2 k ) \\det ( s _ { i , j } ) _ { i , j = 1 , \\ldots , n } , \\end{align*}"}
+{"id": "4566.png", "formula": "\\begin{align*} \\mathcal { S } ^ \\vee ( A ) = \\{ ( p _ i , n _ i + ( p _ 1 - 1 ) ) ^ { \\eta ( p _ i , n _ i ) } \\} . \\end{align*}"}
+{"id": "8620.png", "formula": "\\begin{align*} & E [ W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) ; W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) > 1 ] + E [ W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) ; W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) > 1 ] \\\\ & = O \\left ( e ^ { \\lambda \\Delta \\ell _ 1 } e ^ { 2 \\lambda \\Delta \\ell _ 2 } \\Delta ^ 3 \\right ) , \\end{align*}"}
+{"id": "1201.png", "formula": "\\begin{align*} ( B + i ) ( \\phi _ + + U \\phi _ + ) = 2 i \\phi _ + \\qquad \\qquad ( B _ n + i ) ( \\phi _ + ^ n + U _ n \\phi _ + ^ n ) = 2 i \\phi _ + ^ n \\end{align*}"}
+{"id": "4421.png", "formula": "\\begin{align*} | G ( V _ n ) - G ( V ) | & \\lesssim \\| V _ n - V \\| _ { L ^ 2 } ^ 2 + \\| V _ n - V \\| _ { L ^ 2 } \\| \\nabla ( V _ n - V ) \\| _ { L ^ 2 } + \\| V \\| _ { L ^ 2 } \\| \\nabla ( V _ n - V ) \\| _ { L ^ 2 } + \\| V _ n - V \\| _ { L ^ 2 } \\| \\nabla V \\| _ { L ^ 2 } \\\\ & \\lesssim \\| V _ n - V \\| _ { \\mathcal { H } ^ 1 } ^ 2 + \\| V _ n - V \\| _ { \\mathcal { H } ^ 1 } \\ \\rightarrow \\ 0 \\end{align*}"}
+{"id": "3824.png", "formula": "\\begin{align*} \\Theta _ { \\mathrm { D } } ( \\delta ) & : = \\left \\{ \\int _ { \\mathcal { S } _ 1 \\times \\mathcal { S } _ 2 } g \\ , d \\gamma : \\gamma \\in \\Sigma _ { \\mathrm { D } } ( \\delta ) \\right \\} \\Theta ( \\delta ) : = \\left \\{ \\int _ { \\mathcal { S } } f \\ , d \\gamma : \\gamma \\in \\Sigma ( \\delta ) \\right \\} . \\end{align*}"}
+{"id": "9276.png", "formula": "\\begin{align*} \\widetilde { T } _ { \\eta } f ( x ) = x ^ { \\eta - 1 } T _ { \\eta } \\big ( ( \\cdot ) ^ { 1 - \\eta } f \\big ) ( x ) = x ^ { \\eta - 1 } \\sup _ { t > 2 x } \\int _ { t / 2 } ^ t \\frac { \\abs { f ( z ) } } { ( t - z + x ) ^ { \\eta } } \\ , d z , x > 0 . \\end{align*}"}
+{"id": "2958.png", "formula": "\\begin{align*} \\Lambda = \\{ ( \\lambda _ 1 , \\dots , \\lambda _ n ) \\in \\Z ^ n \\ | \\ \\lambda _ 1 + \\dots + \\lambda _ n = 0 \\} \\subset \\Z ^ n \\ . \\end{align*}"}
+{"id": "3486.png", "formula": "\\begin{align*} P _ \\phi \\preceq _ { \\mu } \\frac { 1 } { k } \\sum _ { j = 1 } ^ k K _ j = P _ { \\mathcal { B } } . \\end{align*}"}
+{"id": "7290.png", "formula": "\\begin{align*} b _ p + k x _ p + l y _ p = c _ p + m y _ p < a _ p + n x _ p . \\end{align*}"}
+{"id": "4774.png", "formula": "\\begin{align*} R _ { [ j + 1 ] } & = \\begin{pmatrix} 1 & 0 \\\\ 0 & w _ { j + 1 } ^ T J _ { 2 | \\alpha _ i | } z _ { j + 1 } \\end{pmatrix} . \\end{align*}"}
+{"id": "233.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ 4 = \\frac { - n } { 3 0 } + \\frac { n ^ 3 } { 3 } + \\frac { n ^ 4 } { 2 } + \\frac { n ^ 5 } { 5 } , \\end{align*}"}
+{"id": "6988.png", "formula": "\\begin{align*} 0 \\leq \\mu ( f ) = \\min \\left \\{ \\mu \\left ( f _ i q ^ i \\right ) \\right \\} \\leq \\mu ( f _ i ) , \\mbox { f o r e v e r y } i , 1 \\leq i \\leq r , \\end{align*}"}
+{"id": "7014.png", "formula": "\\begin{align*} \\alpha = \\alpha ( \\alpha _ i \\mid i \\in I \\} ) \\mbox { a n d } \\beta = \\alpha ( \\{ \\beta _ i \\mid i \\in I \\} ) \\end{align*}"}
+{"id": "2311.png", "formula": "\\begin{align*} \\| z \\| _ A = \\mathcal { O } ( \\kappa ^ { \\frac { 1 } { 2 } } n ) , \\end{align*}"}
+{"id": "3488.png", "formula": "\\begin{align*} r \\ge \\frac { b ^ { 5 + 4 \\kappa } } { k \\Delta ^ { 2 + 4 \\kappa } \\cdot e ^ { 8 \\kappa } } , \\end{align*}"}
+{"id": "778.png", "formula": "\\begin{align*} Y _ 0 = \\dfrac { 1 } { \\sqrt { \\mathrm { A r e a } ( \\mathbb { S } ^ n ) } } , \\ \\ Y _ 1 = \\dfrac { \\sqrt { n + 1 } } { \\sqrt { \\mathrm { A r e a } ( \\mathbb { S } ^ n ) } } x \\cdot v , \\end{align*}"}
+{"id": "6593.png", "formula": "\\begin{align*} \\eta - \\hat \\eta = \\sum _ { l = 1 } ^ L \\hat \\beta _ l \\left ( \\alpha _ { l , { { n } _ t } } - \\alpha _ { l , { \\hat { n } _ t } } e ^ { - j \\omega } \\right ) . \\end{align*}"}
+{"id": "1370.png", "formula": "\\begin{align*} v _ 1 , \\ , v _ 2 = - g _ 5 v _ 1 , \\ , v _ 3 = - g _ 4 v _ 1 , \\ , v _ 4 = - g _ 3 v _ 1 , \\ , v _ 5 = - g _ 2 v _ 1 \\end{align*}"}
+{"id": "1824.png", "formula": "\\begin{align*} V _ F ( t ) & = \\int _ { \\Lambda ^ { * 4 } } \\phi _ t ( p _ 1 , p _ 2 , q _ 2 , q _ 1 ) a _ { p _ 1 } ^ * a _ { p _ 2 } ^ * a _ { q _ 2 } a _ { q _ 1 } \\d p _ 1 \\d p _ 2 \\d q _ 1 \\d q _ 2 \\\\ & + \\int _ { \\Lambda ^ { * 2 } } \\bigg [ \\ \\int _ { \\Lambda ^ { * 2 } } \\phi _ t ( p _ 1 , p _ 2 , q _ 2 , q _ 1 ) \\delta ( q _ 1 - p _ 2 ) \\d p _ 2 \\d q _ 1 \\bigg ] a _ { p _ 1 } ^ * a _ { q _ 2 } \\d p _ 1 \\d q _ 2 \\ . \\end{align*}"}
+{"id": "2355.png", "formula": "\\begin{align*} \\prod _ { x : U _ { i j } } \\frac { m _ i ( x ) } { f ( x ) ^ k } = \\varphi ( x ) = \\frac { m _ j ( x ) } { f ( x ) ^ k } \\end{align*}"}
+{"id": "3562.png", "formula": "\\begin{align*} B ( T _ { * } ) = J _ { T _ { * } } - \\lim \\limits _ { \\Lambda \\rightarrow \\infty } J _ { \\Lambda } . \\end{align*}"}
+{"id": "1908.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & D _ u \\theta ( u ^ { k + 1 } ) + S ^ * \\lambda ^ { k + 1 } = 0 ; \\\\ & I _ K ( \\zeta ) - I _ K ( \\zeta ^ { k + 1 } ) - \\langle \\lambda ^ { k + 1 } , \\zeta - \\zeta ^ { k + 1 } \\rangle \\geq 0 \\forall \\ \\zeta \\in \\mathcal { X } ; \\\\ & \\beta r ^ { k + 1 } = \\lambda ^ { k + 1 } - \\lambda ^ k , \\end{aligned} \\right . \\end{align*}"}
+{"id": "320.png", "formula": "\\begin{align*} \\zeta ( 5 , 1 ) = \\frac { 5 } { 2 } \\zeta ( 6 ) - \\zeta ( 2 ) \\zeta ( 3 ) . \\end{align*}"}
+{"id": "5042.png", "formula": "\\begin{align*} \\rho _ { h , g } = \\prod _ { \\substack { k = 1 , \\dots , N \\\\ \\lambda ' _ k \\neq 1 } } \\lambda _ k \\end{align*}"}
+{"id": "5621.png", "formula": "\\begin{align*} \\sum _ { m = n } ^ { \\infty } \\overline { \\mathbb { P } } _ { \\mu } \\left ( L _ { x } \\omega _ { m } \\in A _ { n } \\left ( x , \\zeta \\right ) , ( x , L _ { x } \\omega ) \\in V | \\left ( x , \\zeta \\right ) \\in U \\right ) \\le c e ^ { - \\frac { 1 } { 2 } \\epsilon n } . \\end{align*}"}
+{"id": "4462.png", "formula": "\\begin{align*} K _ { \\omega , c } ( U _ n ( t _ n ) ) = N ( U _ n ( t _ n ) ) + 2 S _ { \\omega , \\mathbf { c } } ( U _ n ( t _ n ) ) \\ \\rightarrow \\ 0 . \\end{align*}"}
+{"id": "3029.png", "formula": "\\begin{align*} \\| T + x S \\| _ { ( p , k ) } ^ p + \\| T - x S \\| _ { ( p , k ) } ^ p & \\leq 2 \\| T \\| _ { ( p , k ) } ^ p + 2 \\| x S \\| _ { ( p , k ) } ^ p \\\\ & = 2 \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T ^ * T ) + 2 \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( x ^ 2 S ^ * S ) . \\end{align*}"}
+{"id": "1719.png", "formula": "\\begin{align*} h _ 1 = V ^ n h _ { 1 , n } + E _ n h _ { 2 , n } h _ 2 = X ^ n h _ { 2 , n } \\end{align*}"}
+{"id": "6421.png", "formula": "\\begin{align*} \\Vert x _ 1 \\bar { \\otimes } x _ 2 \\Vert _ { p , \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 , \\eta } = \\Vert x _ 1 \\Vert _ { p , \\varphi _ 1 , \\eta } \\Vert x _ 2 \\Vert _ { p , \\varphi _ 2 , \\eta } \\end{align*}"}
+{"id": "9220.png", "formula": "\\begin{align*} a \\otimes b = a b ( 1 + \\eta _ 3 ) . \\end{align*}"}
+{"id": "6320.png", "formula": "\\begin{align*} \\rho ( t ) = \\big \\| \\hat \\lambda _ t \\big \\| _ * , \\dot \\psi ( t ) = \\frac { h _ 1 ( t ) \\dot h _ 2 ( t ) - \\dot h _ 1 ( t ) h _ 2 ( t ) } { \\rho ^ 2 ( t ) } . \\end{align*}"}
+{"id": "4754.png", "formula": "\\begin{align*} Q = Q _ { [ 1 ] } \\oplus ^ { \\operatorname { s } } \\cdots \\oplus ^ { \\operatorname { s } } Q _ { [ r ] } , \\end{align*}"}
+{"id": "2378.png", "formula": "\\begin{align*} & a + b : = m ( a , 0 , b ) , - a : = m ( 0 , a , 0 ) , \\\\ & r a : = r ( a , 0 ) \\end{align*}"}
+{"id": "6870.png", "formula": "\\begin{align*} S _ r ( \\beta ) = \\{ x \\in [ 0 , 1 ] \\colon \\ , d _ r ( x ) \\geq \\beta \\} \\end{align*}"}
+{"id": "3688.png", "formula": "\\begin{align*} B ( ( - \\Delta ) ^ b ) \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) u ( x ) = \\sum _ { j = 1 } ^ { J } c _ j ( - \\Delta ) ^ { n _ j } u ( x ) + c _ r ( - \\Delta ) ^ r u ( x ) = 0 \\quad x \\in \\mathbb { R } ^ n . \\end{align*}"}
+{"id": "839.png", "formula": "\\begin{align*} R _ { c l } = \\sum _ { c = 1 } ^ { C } R _ { c } \\end{align*}"}
+{"id": "7130.png", "formula": "\\begin{align*} F _ \\gamma ( \\hat { x } ) - F _ \\gamma ( \\bar { y } ) = \\int _ { 0 } ^ 1 F _ \\gamma ( \\bar { y } + t ( \\hat { x } - \\bar { y } ) ) ( \\hat { x } - \\bar { y } ) \\ : t = : A _ \\gamma ( \\hat { x } , \\bar { y } ) ( \\hat { x } - \\bar { y } ) . \\end{align*}"}
+{"id": "7957.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } \\nabla ^ 2 _ \\xi H ^ 2 ( \\xi ) \\ , \\xi \\cdot \\xi = H ^ 2 ( \\xi ) \\end{align*}"}
+{"id": "3796.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) : = \\sup _ { \\gamma \\in \\Sigma ( \\delta ) } \\int _ \\mathcal { S } f \\ , d \\gamma , \\ \\delta \\in \\mathbb { R } _ { + } ^ 2 , \\end{align*}"}
+{"id": "7789.png", "formula": "\\begin{align*} \\mathcal { R } ( A \\circ B ) = \\mathcal { R } A \\circ \\mathcal { R } B - \\mathcal { I } A \\circ \\mathcal { I } B . \\end{align*}"}
+{"id": "2467.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } Q ( u , v ) = & \\displaystyle \\dfrac { a _ 2 } { m _ 1 } \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ 2 d x + \\dfrac { a _ 1 } { 2 m _ 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla v | ^ 2 d x + \\dfrac { N } { 2 } a _ 1 a _ 2 \\int _ { \\mathbb { R } ^ N } v u ^ 2 d x , \\end{array} \\right . \\end{align*}"}
+{"id": "3742.png", "formula": "\\begin{align*} { \\rm R e } \\ W _ { \\vec { k } , \\vec { l } } ( \\vec { x } , \\vec { y } ) - { \\rm R e } \\ W _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) = 0 , \\end{align*}"}
+{"id": "9244.png", "formula": "\\begin{align*} P _ 1 ( n ) : & = ( - 1 ) ^ { n + 1 } i \\sum _ { k = 0 } ^ n ( - 1 ) ^ k V _ k W _ { 2 n - 2 k } \\\\ P _ 2 ( n ) : & = \\sum _ { k = 0 } ^ n i ^ { n - k } V _ k W _ { n - k } . \\end{align*}"}
+{"id": "6432.png", "formula": "\\begin{align*} f ( | x | \\bar { \\otimes } | y | ) = \\int _ { [ 0 , \\infty ) ^ 2 } f ( \\lambda _ 1 ) f ( \\lambda _ 2 ) \\ , e _ { | x | , | y | } ( d \\lambda _ 1 , d \\lambda _ 2 ) = f ( | x | ) \\bar { \\otimes } f ( | y | ) . \\end{align*}"}
+{"id": "4704.png", "formula": "\\begin{align*} M _ { \\chi , \\varepsilon } ( z ) = \\Lambda _ { \\chi , \\varepsilon } + \\mathcal { B } _ { \\chi , \\varepsilon } ( z ) = \\varepsilon ^ { - 2 } \\Lambda _ { \\chi } ^ { \\rm s t i f f } + \\Lambda _ { \\chi } ^ { \\rm s o f t } + \\mathcal { B } _ { \\chi , \\varepsilon } ( z ) , \\end{align*}"}
+{"id": "5068.png", "formula": "\\begin{align*} \\widetilde s _ { p , N } ( t ) = \\chi ( t ) s _ { p , N } ( t ) , \\widetilde { S } _ N ( t ) = ( 1 - \\chi ( t ) ) \\left ( \\mathcal { P } _ { 0 , N } + \\phi ' \\widetilde s _ { p , N } ( t ) \\right ) + { S } _ { N } ( t ) . \\end{align*}"}
+{"id": "3641.png", "formula": "\\begin{align*} v ( r ( p ) ) \\ = \\ v ( 2 ( p - 2 ) ) \\ = \\ 2 + ( p - 2 ) \\ = \\ p . \\end{align*}"}
+{"id": "9116.png", "formula": "\\begin{align*} ( x , u ) = \\psi ( x ^ { + } , \\zeta ) \\ , , \\end{align*}"}
+{"id": "3151.png", "formula": "\\begin{align*} \\begin{aligned} \\ 1 + \\ 2 + \\ 4 + & \\ 3 + \\ 5 \\leq C _ { \\rm { d } 1 } \\\\ & ( \\ 6 + \\ 8 + \\ 7 + \\ 9 ) . \\end{aligned} \\end{align*}"}
+{"id": "6802.png", "formula": "\\begin{gather*} \\dim W ^ { \\mathrm { u } } ( p ^ - ) = 2 n - k ^ - , \\dim W ^ { \\mathrm { s } } ( p ^ + ) = k ^ + . \\end{gather*}"}
+{"id": "2826.png", "formula": "\\begin{align*} a _ { I + p - 1 } = \\frac { b _ I } { a _ I \\cdots a _ { I + p - 2 } } ; \\end{align*}"}
+{"id": "5133.png", "formula": "\\begin{align*} \\int _ { B _ 1 } \\Theta \\bigg ( g ^ { i j } \\partial _ { i j } \\eta - g ^ { j p } \\Theta _ q u _ { p q } \\partial _ j \\eta \\bigg ) d \\mu _ g = 0 . \\end{align*}"}
+{"id": "917.png", "formula": "\\begin{align*} h ( u , v ) = u ^ N \\phi ( v / u ) , \\end{align*}"}
+{"id": "2438.png", "formula": "\\begin{align*} d ' ( a _ 0 \\otimes \\ldots \\otimes a _ n ) & = \\sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ i a _ 0 \\otimes \\ldots \\otimes a _ i a _ { i + 1 } \\otimes \\ldots \\otimes a _ n & & n \\geq 1 . \\end{align*}"}
+{"id": "8307.png", "formula": "\\begin{align*} M = O ( \\log q / \\log \\log q ) \\end{align*}"}
+{"id": "1020.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\frac { ( a ) _ k ( b ) _ k } { ( 1 ) _ { k } ( a + b ) _ { k } } x ^ k H _ k ( a - 1 ) \\\\ [ 1 m m ] & \\ : \\ : = - \\{ \\log ( 1 - x ) \\} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( a ) _ k ( b ) _ k } { ( 1 ) _ { k } ( a + b ) _ { k } } x ^ k - \\sum _ { k = 0 } ^ { \\infty } \\frac { ( a ) _ k ( b ) _ k } { ( 1 ) _ { k } ( a + b ) _ { k } } x ^ k H _ k ( b - 1 ) . \\end{align*}"}
+{"id": "4820.png", "formula": "\\begin{align*} \\frac { \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } \\geq x \\right ) } { 1 - \\Phi ( x ) } = 1 + o ( 1 ) \\quad \\frac { \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } \\leq - x \\right ) } { \\Phi ( - x ) } = 1 + o ( 1 ) \\end{align*}"}
+{"id": "1654.png", "formula": "\\begin{align*} g ( \\nabla _ X \\ , \\xi _ i + { f } X ^ \\top , \\ , { f } \\ , Z ) = 0 . \\end{align*}"}
+{"id": "1052.png", "formula": "\\begin{align*} N ' \\coloneqq ( d + 1 ) \\cdot \\min \\left \\{ \\prod _ { i = 1 } ^ n ( h _ i + 1 ) , \\binom { n + h } { n } \\right \\} \\ , . \\end{align*}"}
+{"id": "6478.png", "formula": "\\begin{align*} A ^ \\dagger - B ^ \\dagger = - A ^ \\dagger ( A - B ) B ^ \\dagger + A ^ \\dagger ( A ^ * ) ^ \\dagger ( A ^ * - B ^ * ) ( I - B B ^ \\dagger ) + ( I - A ^ \\dagger A ) ( A ^ * - B ^ * ) ( B ^ * ) ^ \\dagger B . \\end{align*}"}
+{"id": "8937.png", "formula": "\\begin{align*} \\partial _ t \\sigma + \\Big ( v + 2 \\frac { x ' } { | x ' | ^ 2 } \\Big ) \\cdot \\nabla \\sigma - \\Delta \\sigma = 0 , \\end{align*}"}
+{"id": "5262.png", "formula": "\\begin{align*} D _ { S , x } [ f ^ { = T } ] = ( L _ S E _ { T ^ c } L _ T [ f ] ) _ { S \\rightarrow x } , \\end{align*}"}
+{"id": "7315.png", "formula": "\\begin{align*} a \\prod _ { i = 1 } ^ n x _ i ^ { \\alpha _ i } + b \\prod _ { i = 1 } ^ n x _ i ^ { \\beta _ i } + c \\prod _ { i = 1 } ^ n x _ i ^ { \\gamma _ i } = 0 , \\end{align*}"}
+{"id": "2157.png", "formula": "\\begin{align*} \\left . \\begin{array} { c } \\max _ { \\overline { Q } _ { T } } \\varphi _ { \\lambda } \\left ( x _ { 1 } , t \\right ) = \\varphi _ { \\lambda } \\left ( b , T / 2 \\right ) = e ^ { 2 \\lambda b ^ { 2 } } , \\\\ \\min _ { \\overline { Q } _ { T , \\varepsilon } } \\varphi _ { \\lambda } \\left ( x _ { 1 } , t \\right ) = \\exp \\left [ 2 \\lambda \\left ( a ^ { 2 } - \\alpha \\left ( T / 2 - \\varepsilon \\right ) ^ { 2 } \\right ) \\right ] . \\end{array} \\right . \\end{align*}"}
+{"id": "880.png", "formula": "\\begin{align*} M ( P ) = c \\widehat { P } ^ { n - 5 } + O \\left ( \\widehat { P } ^ { n - 5 - \\kappa ' } \\right ) \\quad E _ 1 ( P ) \\ll \\widehat { P } ^ { n - 5 - \\kappa '' } , \\end{align*}"}
+{"id": "3361.png", "formula": "\\begin{align*} 1 = \\mu ( T ) = \\langle \\sigma ( T ) \\eta , \\eta \\rangle = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\langle \\Re ( \\sigma ( B ) ) \\eta , \\eta \\rangle . \\end{align*}"}
+{"id": "8475.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\sigma - i \\infty } ^ { \\sigma + i \\infty } \\Gamma ( z ) \\ , \\Gamma ( s - z ) \\ , \\left ( \\frac { t ^ { 2 } } { x ^ { 2 } } \\right ) ^ { - z } d z = \\frac { \\Gamma ( s ) \\ , x ^ { 2 s } } { \\left ( x ^ { 2 } + t ^ { 2 } \\right ) ^ { s } } - \\frac { \\Gamma ( s ) \\ , x ^ { 2 s } } { t ^ { 2 s } } , \\ , \\ , \\ , \\ , \\ , 0 < ( s ) < \\frac { 1 } { 2 } , \\end{align*}"}
+{"id": "8474.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\intop _ { 0 } ^ { \\infty } \\ , y ^ { ( s ) - \\frac { 1 } { 2 } } | J _ { s - \\frac { 1 } { 2 } } ( 2 \\pi x y ) | \\ , e ^ { - 2 \\pi n y } \\ , d y \\leq C _ { s } \\ , \\sum _ { n = 1 } ^ { \\infty } \\intop _ { 0 } ^ { \\infty } \\ , y ^ { ( s ) - 1 } \\ , e ^ { - 2 \\pi n y } \\ , d y \\leq D _ { s } \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { ( s ) } } < \\infty , \\end{align*}"}
+{"id": "7713.png", "formula": "\\begin{align*} \\tau ( u ) ^ 2 = \\tau ( u ^ + , u ^ - ) ^ 2 = & ~ \\left ( 1 - \\frac { \\Vert u ^ + \\Vert ^ 2 - \\Vert u ^ - \\Vert ^ 2 } { 4 } \\right ) ^ 2 + \\Vert u ^ + \\Vert ^ 2 = \\left ( 1 - \\frac { Q ( u ) } { 2 } \\right ) ^ 2 + \\Vert u ^ + \\Vert ^ 2 \\\\ = & ~ \\left ( 1 + \\frac { \\Vert u ^ + \\Vert ^ 2 - \\Vert u ^ - \\Vert ^ 2 } { 4 } \\right ) ^ 2 + \\Vert u ^ - \\Vert ^ 2 = \\left ( 1 + \\frac { Q ( u ) } { 2 } \\right ) ^ 2 + \\Vert u ^ - \\Vert ^ 2 . \\end{align*}"}
+{"id": "7349.png", "formula": "\\begin{align*} \\textrm { b o u n d e d b y } n ^ { 1 - \\beta } \\textrm { i f } s = 0 \\textrm { a n d } x & \\geq 1 + \\beta \\\\ \\textrm { b o u n d e d b y } n ^ { 1 - \\beta } ( 1 + o ( 1 ) ) \\textrm { i f } s \\geq 1 \\textrm { a n d } x & \\geq 2 ( \\alpha + \\beta ) \\end{align*}"}
+{"id": "6374.png", "formula": "\\begin{align*} \\dot { \\widetilde { y } } = \\widetilde { f } ( \\widetilde { y } ) , \\dot { \\widetilde { \\xi } } = \\widetilde { f } ^ \\prime ( \\widetilde { y } ) \\widetilde { \\xi } , \\dot { \\widetilde { \\eta } } = \\widetilde { f } ^ \\prime ( \\widetilde { y } ) \\widetilde { \\eta } . \\end{align*}"}
+{"id": "4313.png", "formula": "\\begin{align*} \\tilde { y } _ t ^ K \\circ \\big ( \\tilde { y } _ { t _ { k , m } } ^ K \\big ) ^ { - 1 } = \\begin{cases} y _ { t - t _ { k , m } } ^ { ( i _ k ; L _ k ) } & t \\in [ t _ { k , m } , t _ { k , m } + \\tau _ k ] , \\\\ y _ { \\tau _ k } ^ { ( i _ k ; L _ k ) } & t \\in [ t _ { k , m } + \\tau _ k , t _ { \\mathrm { s u c } _ K ( k , m ) } ] . \\end{cases} \\end{align*}"}
+{"id": "6778.png", "formula": "\\begin{align*} { \\mathbb T } _ \\lambda : = \\nabla _ \\lambda \\otimes _ { \\mathcal { A } _ \\lambda } ( \\Delta _ \\lambda ^ \\vee ) ^ o . \\end{align*}"}
+{"id": "934.png", "formula": "\\begin{align*} h _ { ( 0 , \\dots , ( k - 1 ) _ 2 ) , \\ell } ( z , \\dots , z _ { \\ell } ) : = \\frac { q _ { ( 0 , \\dots , ( k - 1 ) _ 2 ) } ( x _ 1 , \\dots , x _ h , z _ { 1 } , \\dots , z _ { \\ell } ) } { q _ { ( 0 , \\dots , ( h - 1 ) _ 2 ) } ( x _ 1 , \\cdots , x _ h ) \\prod _ { 1 \\leq i \\leq 1 < j \\leq \\ell , \\ a _ j = a _ i } ( z _ j - x _ i ) } . \\end{align*}"}
+{"id": "1545.png", "formula": "\\begin{align*} A _ { k , \\rho } ( t ) = [ u ( \\cdot , t ) < k ] \\cap K _ { \\rho } . \\end{align*}"}
+{"id": "3375.png", "formula": "\\begin{align*} \\frac { P _ { m } } { p _ m } \\Delta _ { 1 0 } u _ { m n } = O _ L ( 1 ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\frac { Q _ { n } } { q _ n } \\Delta _ { 0 1 } u _ { m n } = O _ L ( 1 ) , \\end{align*}"}
+{"id": "4968.png", "formula": "\\begin{align*} \\left [ P \\right ] _ { i , j } ^ { } = \\begin{dcases} q _ i = \\frac { n - ( n - j ) p } { n } & \\mbox { i f } \\ , j = i ; \\\\ p _ i = \\frac { ( n - j ) p } { n } & \\mbox { i f } \\ , j = i + 1 ; \\\\ 0 & \\mbox { o t h e r w i s e } ; \\end{dcases} \\end{align*}"}
+{"id": "247.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { - 1 } { 3 0 } L i _ 3 ( z ) + \\frac { 1 } { 3 } \\log \\left ( \\frac { 1 } { 1 - z } \\right ) + \\frac { 1 } { 2 } \\frac { z } { ( 1 - z ) ^ 2 } + \\frac { 1 } { 5 } \\frac { z } { 1 - z } \\right \\} . \\end{align*}"}
+{"id": "6370.png", "formula": "\\begin{align*} \\dot { y } = f ( y ) , \\dot { \\eta } = f ^ \\prime ( y ) \\eta , \\end{align*}"}
+{"id": "2234.png", "formula": "\\begin{align*} \\varphi ( q _ { n * } ( \\xi , \\ldots , \\xi , a , b ) ) = q _ { n * } ( \\varphi ( \\xi ) , \\ldots , \\varphi ( \\xi ) , \\varphi ( a ) , \\varphi ( b ) ) \\end{align*}"}
+{"id": "7783.png", "formula": "\\begin{align*} 1 - z ^ r x - k x ^ { r + 1 } = 0 . \\end{align*}"}
+{"id": "775.png", "formula": "\\begin{align*} \\Phi ( \\rho ( 1 + u ) ) = \\Phi ( \\rho ) + \\phi ( \\rho ) \\rho u + \\dfrac { 1 } { 2 } \\phi ' ( \\rho ) \\rho ^ 2 u ^ 2 + o ( u ^ 2 ) , \\end{align*}"}
+{"id": "4839.png", "formula": "\\begin{align*} ( \\prod _ { i = 1 } ^ m a ( i ) \\ast f ( t ( i ) ) ) \\ast a ( m + 1 ) \\in A \\cap \\sigma ( L ) . \\end{align*}"}
+{"id": "7228.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } \\left ( J \\cap \\partial B _ R ^ n \\right ) = 0 \\end{align*}"}
+{"id": "7519.png", "formula": "\\begin{align*} \\tilde { \\sigma } _ v ( w ) - \\sigma _ v ( w ) = \\gamma ^ { ( m - 3 ) } ( w ) \\end{align*}"}
+{"id": "873.png", "formula": "\\begin{align*} \\frac { 1 } { | r _ 1 r _ 2 | } \\leq \\left | \\frac { \\underline { a } _ 1 } { r _ 1 } - \\frac { \\underline { a } _ 2 } { r _ 2 } \\right | = \\left | \\left ( \\frac { \\underline { a } _ 1 } { r _ 1 } - \\underline { \\alpha } \\right ) + \\left ( \\underline { \\alpha } - \\frac { \\underline { a } _ 2 } { r _ 2 } \\right ) \\right | < \\max \\left \\{ \\frac { 1 } { | r _ 1 | \\widehat { R } _ 2 } , \\frac { 1 } { | r _ 2 | \\widehat { R } _ 2 } \\right \\} , \\end{align*}"}
+{"id": "5541.png", "formula": "\\begin{align*} \\left \\{ \\tilde { T } ^ { n } ( x , \\omega ) \\in A \\right \\} & = \\left \\{ L _ { \\omega _ { n } ^ { - 1 } . x } \\in S \\left ( \\omega _ { n } ^ { - 1 } . x , \\left ( \\omega _ { n } ^ { - 1 } \\omega _ { m + n } \\right ) _ { m \\in \\mathbb { Z } } \\right ) \\right \\} \\\\ & = \\left \\{ L _ { x } \\omega _ { n } \\in S \\left ( x , \\omega \\right ) \\right \\} \\ \\ \\mbox { b y c o m p a t i b i l i t y ( i ) } . \\end{align*}"}
+{"id": "1151.png", "formula": "\\begin{align*} T ( f \\cdot g ) = f T ( g ) + T ( f ) g + 2 B ( A ( f ) , A ( g ) ) \\end{align*}"}
+{"id": "2374.png", "formula": "\\begin{align*} D ( g _ { i 1 } , \\dots , g _ { i b _ i } ) = U \\cap D ( f _ i ) \\rlap { . } \\end{align*}"}
+{"id": "7541.png", "formula": "\\begin{align*} M : = \\omega _ { d } \\left ( \\int _ { B _ { 4 r } } \\abs { \\partial \\phi } ^ d \\right ) ^ { \\frac { 1 } { d } } . \\end{align*}"}
+{"id": "4920.png", "formula": "\\begin{align*} \\Pr ( \\widetilde { X } _ t = k + 1 ) + \\sum _ { j = 0 } ^ k \\Pr ( \\widetilde { X } _ t = j ) = 1 . \\end{align*}"}
+{"id": "8257.png", "formula": "\\begin{align*} G _ { k , Y _ { 1 , 1 } } = \\sqrt { x } \\frac { \\partial G _ { k } } { \\partial x } - \\frac { 1 } { 2 \\sqrt { x } } k ^ 2 G _ { k } - \\frac { t } { \\sqrt { x } } F _ { 1 } , \\end{align*}"}
+{"id": "7199.png", "formula": "\\begin{align*} a _ h = \\mathcal { L } ^ 2 \\left ( \\left \\{ f _ h \\neq g _ h \\right \\} \\cap B _ 1 \\right ) \\lesssim \\left ( \\mathcal { H } ^ 1 \\left ( \\left \\{ f _ h \\neq g _ h \\right \\} \\cap B _ 1 \\right ) \\right ) ^ 2 = { \\rm o } ( \\gamma _ h ) \\mbox { a s } h \\to + \\infty \\end{align*}"}
+{"id": "4823.png", "formula": "\\begin{align*} \\delta ^ { ' } = 3 + \\delta - \\frac { 3 + 2 \\delta } { \\tau } . \\end{align*}"}
+{"id": "3047.png", "formula": "\\begin{align*} \\phi \\left ( \\bigotimes \\limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \\otimes B \\right ) = \\bigotimes \\limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \\otimes \\varphi _ { j _ 1 , \\ldots , j _ { m - 1 } , \\mathrm { I } } ( B ) \\hbox { f o r a l l \\quad } B \\in M _ { n _ m } . \\end{align*}"}
+{"id": "3964.png", "formula": "\\begin{align*} \\begin{aligned} \\widehat { \\gamma } ( A ) & = \\mathbb { P } ( \\widehat { S } \\in A ) = \\mathbb { E } \\left [ \\mathbb { P } ( \\widehat { S } \\in A | \\varepsilon ) \\right ] \\\\ & = \\left ( \\eta / \\eta _ 0 \\right ) \\mathbb { P } ( S \\in A ) + \\left ( 1 - \\eta / \\eta _ 0 \\right ) \\mathbb { P } ( \\widetilde { S } \\in A ) . \\end{aligned} \\end{align*}"}
+{"id": "5859.png", "formula": "\\begin{align*} \\Vert \\psi _ E \\Vert _ { \\ell ^ 2 ( \\Lambda _ { 2 L , L } ) } & \\leq l ^ { d - 1 } e ^ { - m '' l } \\sup _ { x \\in \\Lambda _ { 2 L _ + , L _ - } ( x _ 0 ) } | \\psi _ E ( x ) | \\\\ & \\leq ( \\tfrac { L } { 1 0 0 } ) ^ { d - 1 } e ^ { - \\frac { m '' } { 1 0 0 } L } ( 2 L _ + ) ^ \\nu \\Vert T _ { x _ 0 } ^ { - 1 } \\psi _ E \\Vert _ { \\ell ^ 2 } \\\\ & \\leq e ^ { - \\frac { m } { 1 0 0 } L } \\Vert T _ { x _ 0 } ^ { - 1 } \\psi _ E \\Vert _ { \\ell ^ 2 } \\end{align*}"}
+{"id": "3252.png", "formula": "\\begin{align*} & \\| x - z \\| \\ge \\max ( \\| x - \\sigma ( x ) \\| / 2 , { \\mathfrak R } ^ k ) \\quad \\mbox { f o r } z \\in \\sigma ( \\Gamma ) , \\\\ & 5 r \\le \\| \\sigma ( x ) - z \\| = d ( x , z ) \\le \\| x - z \\| \\quad \\mbox { f o r } z \\in \\sigma ( \\Gamma ) , z \\notin { \\mathcal O } ( 5 B ) . \\end{align*}"}
+{"id": "3130.png", "formula": "\\begin{align*} \\psi _ u ( n ) = \\psi ( \\kappa ( u , \\log ( n ) ) ) . \\end{align*}"}
+{"id": "3868.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta _ { A } ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ L } \\left [ \\langle \\lambda , \\delta _ { A } \\rangle + \\sup _ { \\pi \\in \\Pi ( \\mu _ { 1 , L + 1 } , \\dotsc , \\mu _ { L , L + 1 } ) } \\int _ { \\mathcal { V } } f _ { \\lambda , A } \\ , d \\pi \\right ] . \\end{align*}"}
+{"id": "8641.png", "formula": "\\begin{align*} \\begin{array} { l l } & x = ( 1 - y ) / ( 1 - p y ) , \\\\ & d x = - \\big ( q / ( 1 - p y ) ^ 2 \\big ) d y , \\\\ & 1 - p x = q / ( 1 - p y ) . \\end{array} \\end{align*}"}
+{"id": "5601.png", "formula": "\\begin{align*} \\mathrm { I } ( \\xi _ { 1 } ^ { x } , \\mathcal { T } _ { x } ) = \\int _ { ( L _ { x } \\backslash G ) ^ { \\mathbb { N } } } \\log \\frac { d \\overline { \\mathbb { P } } _ { \\mu , x , \\mathrm { \\xi _ { 1 } ^ { x } } } | _ { \\mathcal { T } _ { x } } } { d \\overline { \\mathbb { P } } _ { \\mu , x , e } | _ { \\mathcal { T } _ { x } } } d \\overline { \\mathbb { P } } _ { \\mu , x , e } , \\end{align*}"}
+{"id": "8962.png", "formula": "\\begin{align*} C _ { q _ m } ( x _ o ) = \\Gamma _ a C _ { q _ m } ( x _ a ) + \\Gamma _ b C _ { q _ m } ( x _ b ) + \\mathcal { O } ( h ^ { 2 } ) \\end{align*}"}
+{"id": "4268.png", "formula": "\\begin{align*} Y _ { t } ^ { u } = \\Phi ^ { 0 } \\left ( \\mathbb { P } _ { X _ { T } ^ { u } } ^ { W } \\right ) + \\int _ { t } ^ { T } f ^ { 0 } \\left ( \\mathbb { P } _ { X _ { s } ^ { u } } ^ { W } , Y _ { s } ^ { u } , Z _ { s } ^ { u } , \\mathbb { P } _ { \\left ( Y _ { s } ^ { u } , Z _ { s } ^ { u } \\right ) } ^ { W } , u _ { s } \\right ) \\mathrm { d } s - \\int _ { t } ^ { T } Z _ { s } ^ { u } \\mathrm { d } W _ { s } . \\end{align*}"}
+{"id": "968.png", "formula": "\\begin{align*} \\Gamma ^ { \\ , * } _ m = \\bigcup \\limits _ { i = 0 } ^ { N _ 0 } \\Gamma _ { m i } \\ , , \\end{align*}"}
+{"id": "4490.png", "formula": "\\begin{align*} E _ 1 ( \\psi ) = ( \\psi , \\mathbf { 0 } , \\mathbf { 0 } ) , \\ E _ 2 ( \\psi ) = ( \\mathbf { 0 } , \\psi , \\mathbf { 0 } ) , \\ E _ 3 ( \\psi ) = ( \\mathbf { 0 } , \\mathbf { 0 } , \\psi ) \\in \\mathcal { H } ^ 1 , \\ \\ \\mathbf { 0 } = ( 0 , \\cdots , 0 ) \\in ( H ^ 1 ( \\R ^ d ) ) ^ d . \\end{align*}"}
+{"id": "4643.png", "formula": "\\begin{align*} \\alpha = \\frac { n q } { 2 q + n - 2 - \\theta ( q - 1 ) } \\quad \\mbox { a n d } \\quad \\beta = \\frac { n q } { ( q - 1 ) ( n - 2 + \\theta ) } . \\end{align*}"}
+{"id": "6585.png", "formula": "\\begin{align*} \\begin{aligned} & E ( \\alpha _ { n _ t , l } ) = { \\sqrt { \\pi } } / { 2 } , \\ \\ \\ V a r ( \\alpha _ { n _ t , l } ) = ( 4 - \\pi ) / { 4 } . \\end{aligned} \\end{align*}"}
+{"id": "4833.png", "formula": "\\begin{align*} h _ n = \\sum \\limits _ { i = 0 } ^ { 2 n } \\bigg { ( } 1 - \\frac { 1 } { \\sqrt { n } } \\bigg { ) } ^ { | i - n | } ( T ^ { \\prime \\prime } ) ^ i g _ n \\end{align*}"}
+{"id": "501.png", "formula": "\\begin{align*} x _ j & = a + h j , j = 0 , \\ldots , N , h = ( b - a ) / N , \\\\ G _ i & = [ x _ { i - 1 } , x _ { i + 1 } ] , i = 1 , \\ldots , N - 1 , \\\\ P _ i & = \\Pi _ 2 , . \\end{align*}"}
+{"id": "1011.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ 2 } { 3 2 ^ k } \\bigg \\{ H _ { 2 k } ^ { ( 2 ) } - \\frac { 1 } { 4 } H _ { k } ^ { ( 2 ) } \\bigg \\} = \\Gamma \\bigg ( \\frac { 1 } { 4 } \\bigg ) ^ 2 \\frac { \\pi ^ 2 - 8 G } { 3 2 \\pi \\sqrt { \\pi } } . \\end{align*}"}
+{"id": "14.png", "formula": "\\begin{align*} \\sum _ { p \\in \\Z } \\int _ { [ 0 , N ] ^ n } h ( N p + v _ { m + 1 } + \\langle \\bar { u } , N \\bar { q } + v '' \\rangle ) \\ , d \\bar { u } & = \\int _ { \\R ^ n / N \\Z ^ n } \\Tilde { \\psi } ( S x ) \\ , d \\bar { u } \\\\ & = \\int _ { \\R ^ n / N \\Z ^ n } \\Tilde { \\psi } ( x ) \\ , d \\bar { u } = N ^ { n - 1 } \\int _ { \\R } h ( u _ 1 ) \\ , d u _ 1 , \\end{align*}"}
+{"id": "5735.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } \\mathcal { E C S } _ { 0 1 1 } = \\{ S ( p , q , a , b , c , d ) \\in \\mathcal { E C } : p = 0 , a , c \\ne 0 \\} \\\\ \\mathcal { E C S } _ { 1 0 1 } = \\{ S ( p , q , a , b , c , d ) \\in \\mathcal { E C } : a = 0 , p , c \\ne 0 \\} \\\\ \\mathcal { E C S } _ { 1 1 0 } = \\{ S ( p , q , a , b , c , d ) \\in \\mathcal { E C } : c = 0 , p , a \\ne 0 \\} . \\end{array} \\right . \\end{align*}"}
+{"id": "4995.png", "formula": "\\begin{align*} \\overline { F } ( k , t + 1 , \\mathbf { p } _ { n + 1 } ) = p _ { k } ^ { } \\overline { F } ( k - 1 , t , \\mathbf { p } _ { n + 1 } ) + ( 1 - p _ { k } ^ { } ) \\overline { F } ( k , t , \\mathbf { p } _ { n + 1 } ) , \\end{align*}"}
+{"id": "6024.png", "formula": "\\begin{align*} \\pi _ { \\psi } ( g ) f ( w ) = \\epsilon _ g f ( w g ) . \\end{align*}"}
+{"id": "8290.png", "formula": "\\begin{align*} f ( z + 1 ) = f ( z ) \\ \\ f ( z + \\eta ) = - e ^ { - 2 \\pi i n z } f ( z ) . \\end{align*}"}
+{"id": "6645.png", "formula": "\\begin{align*} - \\Delta u - g ( b , \\ , \\nabla u ) + c \\ , u = 0 \\ , \\ , \\ , M , \\end{align*}"}
+{"id": "3751.png", "formula": "\\begin{align*} A _ r ^ n = & \\psi ^ { \\alpha _ 1 , n } ( f _ 1 ) \\cdots \\psi ^ { \\alpha _ r , n } ( f _ r ) , B _ s ^ n = \\psi ^ { \\beta _ 1 , n } ( g _ 1 ) \\cdots \\psi ^ { \\beta _ s , n } ( g _ s ) . \\end{align*}"}
+{"id": "703.png", "formula": "\\begin{align*} \\widehat { \\xi } _ { [ ( \\sigma , k , l ) ] } ( s ) & = \\frac { ( r _ \\alpha - s ) ^ { \\frac { k - ( m _ \\sigma - 2 ) } { m _ \\sigma } } } { m _ \\sigma ^ 2 k ! } k \\neq m _ \\sigma - 2 \\ \\operatorname { m o d } m _ \\sigma \\\\ \\widehat { \\xi } _ { [ ( \\sigma , k , l ) ] } ( s ) & = - \\frac { ( r _ \\alpha - s ) ^ { \\frac { k - ( m _ \\sigma - 2 ) } { m _ \\sigma } } \\log ( r _ \\alpha - s ) } { m _ \\sigma ^ 2 k ! } k = m _ \\sigma - 2 \\ \\operatorname { m o d } m _ \\sigma . \\end{align*}"}
+{"id": "3890.png", "formula": "\\begin{align*} \\mathrm { p r o j } _ { Q } \\# \\pi = \\left ( \\mathord { \\operatorname { p r o j } } _ { Q } \\circ { \\operatorname { p r o j } _ { K _ j } } ^ { - 1 } \\right ) \\# \\mu _ j , \\forall j \\in [ N ] . \\end{align*}"}
+{"id": "8655.png", "formula": "\\begin{align*} & \\int _ 0 ^ 1 \\int _ x ^ 1 ( 1 - p y ) ^ { - 2 } ( 1 - p x ) ^ { - 1 } y ^ j x ^ { j - 1 } ( y - x ) d y d x \\\\ & = \\int _ 0 ^ 1 \\int _ 0 ^ y ( 1 - p y ) ^ { - 2 } ( 1 - p x ) ^ { - 1 } y ^ j x ^ { j - 1 } ( y - x ) d x d y \\\\ & = - \\int _ 0 ^ 1 \\int _ 0 ^ x ( 1 - p x ) ^ { - 2 } ( 1 - p y ) ^ { - 1 } x ^ j y ^ { j - 1 } ( y - x ) d y d x \\end{align*}"}
+{"id": "5941.png", "formula": "\\begin{align*} \\phi ^ { - 1 } ( h _ l ) = ( \\sum ^ m _ { k = 1 } ( a ^ l _ k + i b ^ l _ k ) f _ k , t _ l ) ; \\end{align*}"}
+{"id": "3631.png", "formula": "\\begin{align*} ( 9 , a _ 1 , \\ldots , a _ { m - 1 } , 4 ) _ { 1 0 } \\ = \\ 2 ( 4 , a _ { m - 1 } , \\ldots , a _ 1 , 7 ) _ { 1 0 } . \\end{align*}"}
+{"id": "6233.png", "formula": "\\begin{align*} k _ 2 = \\min \\big \\{ | \\theta _ { a , z } \\cdot \\nu | ^ 2 , \\ , a \\in S , \\ , z \\in B _ 2 \\big \\} \\ , , & & k _ 3 = \\min \\big \\{ | \\theta _ { a , w } \\cdot \\nu | ^ 2 , \\ , a \\in S , \\ , w \\in B _ 3 \\big \\} \\ , , \\end{align*}"}
+{"id": "9283.png", "formula": "\\begin{align*} \\bigcap \\limits ^ { \\infty } _ { k = 1 } W ^ { * } _ { ( k ) } \\neq \\varnothing , \\end{align*}"}
+{"id": "7356.png", "formula": "\\begin{align*} & \\frac { s ! } { ( n + s ) ! } \\sum _ k \\binom { n } { k } \\frac { ( n + s ) ! } { ( k + s ) ! } ( 1 - Q ( ( 1 - \\gamma ) \\mu ) ) ^ { n - k } ( 1 - Q ( \\gamma \\mu ) ) ^ { k ( k + s ) } e ^ { k \\theta } \\\\ & = \\sum _ k \\binom { n } { k } \\frac { s ! } { ( k + s ) ! } ( 1 - Q ( ( 1 - \\gamma ) \\mu ) ) ^ { n - k } ( 1 - Q ( \\gamma \\mu ) ) ^ { k ( k + s ) } e ^ { k \\theta } \\end{align*}"}
+{"id": "6361.png", "formula": "\\begin{align*} \\rho _ t ( x , y ) = \\frac { t ^ { - d / \\alpha } p _ 1 ^ { \\Gamma } ( t ^ { - 1 / \\alpha } x , t ^ { - 1 / \\alpha } y ) } { t ^ { 2 \\beta / \\alpha } M _ { \\Gamma } ( t ^ { - 1 / \\alpha } x ) M _ { \\Gamma } ( t ^ { - 1 / \\alpha } y ) } = t ^ { - ( d + 2 \\beta ) / \\alpha } \\rho _ 1 ( t ^ { - 1 / \\alpha } x , t ^ { - 1 / \\alpha } y ) . \\end{align*}"}
+{"id": "2679.png", "formula": "\\begin{align*} a _ { 4 3 } & = 1 6 8 5 2 1 6 6 9 0 6 \\\\ a _ { 4 4 } & = 2 7 0 9 4 7 0 5 9 1 6 0 . \\end{align*}"}
+{"id": "8837.png", "formula": "\\begin{align*} C _ 3 = C _ 3 ( q , B ) & : = ( 2 q - 1 ) ^ { 1 / q } L _ \\sigma ^ { 1 / q } \\norm { B } ^ { 1 / q } _ { \\gamma ( U ; X ) } , \\\\ C _ 4 = C _ 4 ( q , B ) & : = L _ \\sigma ^ { 1 / q } \\norm { B } ^ { 1 / q } _ { \\gamma ( U ; X ) } \\end{align*}"}
+{"id": "6085.png", "formula": "\\begin{align*} \\| a \\| _ { H ^ { 1 } _ { A } } \\approx \\int \\mathcal { M } _ { ( N ) , A } ^ { 0 } a \\ , \\dd \\mu & = \\int \\lim _ { i \\to \\infty } \\mathcal { M } _ { ( N ) , A } ^ { 0 } a _ { j _ { i } } \\ , \\dd \\mu \\\\ & \\leq \\liminf _ { i \\to \\infty } \\int \\mathcal { M } _ { ( N ) , A } ^ { 0 } a _ { j _ { i } } \\ , \\dd \\mu \\leq C \\liminf _ { i \\to \\infty } \\| a _ { j _ { i } } \\| _ { H ^ { 1 } _ { A } } \\leq C ' , \\end{align*}"}
+{"id": "2079.png", "formula": "\\begin{align*} & ( x _ 1 ^ { \\prime } , x _ 2 ^ { \\prime } , . . . , x _ k ^ { \\prime } ) \\preceq ( x _ 1 , x _ 2 , . . . , x _ k ) \\\\ \\Longleftrightarrow & x _ i ^ { \\prime } = x _ i \\ \\ \\ \\ i > 0 \\ \\ \\\\ & x _ j ^ { \\prime } < x _ j , x _ i ^ { \\prime } = x _ i \\ \\ \\ \\ j > 0 \\ \\ \\ \\ i > j . \\end{align*}"}
+{"id": "4160.png", "formula": "\\begin{align*} \\langle \\lambda _ s ( b ) \\xi , \\eta \\rangle & = \\sum _ { t \\in G } \\lambda _ s ( b ) \\xi ( t ) ^ * \\eta ( t ) = \\sum _ { t \\in G } ( b \\xi ( s ^ { - 1 } t ) ) ^ * \\eta ( t ) = \\sum _ { t \\in G } \\xi ( s ^ { - 1 } t ) ^ * b ^ * \\eta ( t ) \\\\ & \\stackrel { r = s ^ { - 1 } t } { = } \\sum _ { r \\in G } \\xi ( r ) ^ * b ^ * \\eta ( s r ) = \\sum _ { r \\in G } \\xi ( r ) ^ * \\lambda _ { s ^ { - 1 } } ( b ^ * ) \\eta ( r ) = \\langle \\xi , \\lambda _ { s ^ { - 1 } } ( b ^ * ) \\eta \\rangle . \\end{align*}"}
+{"id": "6799.png", "formula": "\\begin{align*} ( 2 n - k ^ - - 1 + 2 ) + ( k ^ + - 1 + 2 ) - ( 2 n - 1 + 2 ) = k ^ + - k ^ - + 1 . \\end{align*}"}
+{"id": "7492.png", "formula": "\\begin{align*} d _ 2 \\cdots d _ l x \\pi _ j \\cdots \\pi _ n = \\alpha _ { k + 1 } ^ { \\prime } \\cdots \\alpha _ m ^ { \\prime } \\underset { \\beta ^ { \\prime } } { \\underbrace { d _ p d _ { p + 1 } \\cdots d _ l x \\pi _ j \\cdots \\pi _ n } } \\end{align*}"}
+{"id": "6549.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } t \\ , e ^ { i z t } \\ , d t = - \\frac { 1 } { z ^ 2 } , \\int _ { 0 } ^ { \\infty } \\frac { t ^ 2 } { 2 } \\ , e ^ { i z t } \\ , d t = - \\frac { i } { z ^ 3 } , \\Im ( z ) > 0 , \\end{align*}"}
+{"id": "5581.png", "formula": "\\begin{align*} h _ { \\mu } \\left ( G / Q , \\nu _ { Q } \\right ) = \\sum _ { \\ell = 1 } ^ { k } \\sum _ { r _ { \\ell - 1 } < i \\le r _ { \\ell } , j > r _ { \\ell } } \\lambda _ { i } - \\lambda _ { j } = h _ { \\mu } \\left ( G / P , \\nu _ { P } \\right ) - \\sum _ { \\ell = 1 } ^ { k } \\sum _ { r _ { \\ell - 1 } \\le i < j \\le r _ { \\ell } } \\lambda _ { i } - \\lambda _ { j } . \\end{align*}"}
+{"id": "3656.png", "formula": "\\begin{align*} \\mp _ s = \\begin{bmatrix} \\mp _ { s , 1 } & \\mp _ { s , 2 } & \\cdots & \\mp _ { s , 2 ^ { t - s } } \\end{bmatrix} . \\end{align*}"}
+{"id": "5723.png", "formula": "\\begin{align*} \\begin{pmatrix} f & 0 \\\\ 0 & p e \\end{pmatrix} \\ , \\ , ( p \\in \\mathcal H ) \\end{align*}"}
+{"id": "2953.png", "formula": "\\begin{align*} \\chi _ F = \\frac { 1 } { t - t ^ { - 1 } } \\ , \\det \\begin{pmatrix} F ( t ) & F ( t ^ { - 1 } ) \\\\ 1 & 1 \\end{pmatrix} \\ . \\end{align*}"}
+{"id": "6700.png", "formula": "\\begin{align*} e ^ { \\tau \\Phi } D _ j ^ { \\alpha _ j } e ^ { - \\tau \\Phi } & = ( e ^ { \\tau \\Phi } D _ j e ^ { - \\tau \\Phi } ) ( e ^ { \\tau \\Phi } D _ j e ^ { - \\tau \\Phi } ) \\cdots ( e ^ { \\tau \\Phi } D _ j e ^ { - \\tau \\Phi } ) ( \\alpha _ j ) . \\end{align*}"}
+{"id": "3176.png", "formula": "\\begin{align*} U = \\overline { \\xi } ( T _ 1 ) \\circ \\overline { \\xi } ( T _ 2 ) . \\end{align*}"}
+{"id": "6380.png", "formula": "\\begin{align*} \\psi _ { 3 } ( X ) = 3 X ^ { 4 } + 6 \\cdot 2 0 r ^ { 2 } X ^ { 2 } - 2 0 ^ { 2 } r ^ { 4 } ; \\end{align*}"}
+{"id": "6644.png", "formula": "\\begin{align*} \\it { L } u - c u = f \\quad \\quad \\ , \\ , \\ , \\Omega \\ , ; \\end{align*}"}
+{"id": "555.png", "formula": "\\begin{align*} a _ { \\varepsilon } ( t ) = \\left ( a * \\psi _ { \\omega ( \\varepsilon ) } \\right ) ( t ) = \\langle a , \\tau _ { t } \\tilde { \\psi } _ { \\omega ( \\varepsilon ) } \\rangle \\geq \\tilde { a } _ { 0 } > 0 , \\end{align*}"}
+{"id": "443.png", "formula": "\\begin{align*} g _ { \\mu \\nu } u ^ \\mu u ^ \\nu = - 1 . \\end{align*}"}
+{"id": "1697.png", "formula": "\\begin{align*} \\bar \\kappa ^ { \\pm } _ { a - i } = 2 \\kappa ^ { \\pm } _ { a } - \\kappa ^ { \\pm } _ { a + i } , i \\geq 2 , \\mathcal { K } ^ { \\pm } = \\big \\{ \\kappa ^ { \\pm } _ { i } \\big \\} _ { i \\geq a } \\bigcup \\big \\{ \\bar \\kappa ^ { \\pm } _ { a - i } \\big \\} _ { i \\geq 1 } . \\end{align*}"}
+{"id": "1967.png", "formula": "\\begin{align*} 2 n K _ 1 \\beta ^ { - 1 / 2 } | \\nabla v | \\psi ^ { - 1 \\slash n } ( \\cdot , v ) \\leq 2 n K _ 1 e ^ { C _ 0 ^ 2 + 1 } \\psi ^ { - 1 \\slash n } ( \\cdot , v ) \\leq ( B - 2 K _ 1 - 1 ) \\sum _ { p = 1 } ^ n u ^ { p \\bar p } . \\end{align*}"}
+{"id": "4628.png", "formula": "\\begin{align*} \\partial _ t d = r m \\frac { m } { 1 + \\delta m } \\left ( 1 - d \\right ) . \\end{align*}"}
+{"id": "378.png", "formula": "\\begin{align*} d ( \\tau , j ) ^ { \\delta } _ { i } = \\prod _ { \\tau \\subseteq \\sigma } ( c ( \\sigma , j ) ^ { \\delta } _ { i } ) ^ { ( - 1 ) ^ { | \\sigma \\setminus \\tau | } } \\end{align*}"}
+{"id": "498.png", "formula": "\\begin{align*} p ( X _ { ( m , n ) } | s _ { n } = 0 ) = \\mathcal { C N } ( X _ { ( m , n ) } ; 0 , \\epsilon ) , \\end{align*}"}
+{"id": "5215.png", "formula": "\\begin{align*} { \\sqsupset ^ { * * } } = { \\sqsupset ^ * } = { \\sqsupset * \\vartriangleright } . \\end{align*}"}
+{"id": "2735.png", "formula": "\\begin{align*} H _ \\Delta ( \\mathrm { M } ) = \\{ u \\in H ^ 1 ( \\mathrm { M } ) ; \\ ; \\Delta u \\in L ^ 2 ( \\mathrm { M } ) \\} , \\end{align*}"}
+{"id": "7037.png", "formula": "\\begin{align*} v ( h _ i ) = - \\nu _ i ( g ) . \\end{align*}"}
+{"id": "444.png", "formula": "\\begin{align*} \\eta ( t , x ) = \\chi ( t , r ) x , \\end{align*}"}
+{"id": "8017.png", "formula": "\\begin{align*} \\begin{bmatrix} A & 0 \\\\ 0 & B \\end{bmatrix} = U _ 1 \\begin{bmatrix} ( 1 - x ) A + x B & 0 \\\\ 0 & 0 \\end{bmatrix} U _ 1 ^ * + V _ 1 \\begin{bmatrix} 0 & 0 \\\\ 0 & x A + ( 1 - x ) B \\end{bmatrix} V _ 1 ^ * \\end{align*}"}
+{"id": "900.png", "formula": "\\begin{align*} \\sum _ { \\substack { | r | = \\widehat { Y } \\\\ d \\mid r } } | b _ 2 r _ 3 | ^ { 1 / 2 } \\ll \\widehat { Y } ^ { 1 + \\varepsilon } . \\end{align*}"}
+{"id": "1746.png", "formula": "\\begin{align*} F _ { \\Gamma _ { } ^ { ( m ) } } ( x ) & = 1 - \\xi Q \\Bigg ( \\frac { \\sqrt { \\frac { x } { \\Gamma _ 0 } } - \\mu _ { \\textrm { D } } } { \\sigma _ { } } \\Bigg ) ~ ~ \\textrm { f o r ~ } x \\ge 0 , \\end{align*}"}
+{"id": "6035.png", "formula": "\\begin{align*} H ( z ) : = A z ^ { n + m } + B \\overline { z } ^ m + C \\textrm { f o r a l l } z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "4012.png", "formula": "\\begin{align*} \\mathcal { I } ^ \\star ( \\lambda ) = \\sup _ { \\gamma \\in \\bar { \\mathcal { P } } } I _ { \\lambda } [ \\gamma ] = \\sup _ { \\pi \\in \\Gamma ( \\Pi , \\phi _ \\lambda ) } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda d \\pi = \\sup _ { \\pi \\in \\Pi } \\int _ { \\mathcal { V } } f _ \\lambda d \\pi . \\end{align*}"}
+{"id": "4881.png", "formula": "\\begin{align*} \\begin{dcases} L \\varphi \\le - 1 & B _ 1 \\setminus \\overline { B _ { 1 / 2 } } , \\\\ \\varphi \\ge ( 1 - | x | ^ 2 ) _ + ^ s & \\\\ \\varphi \\le C & B _ { 1 / 2 } , \\\\ \\varphi = 0 & \\R ^ n \\setminus B _ 1 . \\end{dcases} \\end{align*}"}
+{"id": "1206.png", "formula": "\\begin{align*} \\partial _ t G ( t , s ) = ( q + s \\varphi ( t ) ) \\partial _ s G ( t , s ) + \\varphi ( t ) G ( t , s ) , \\end{align*}"}
+{"id": "5065.png", "formula": "\\begin{align*} \\| f \\| _ { H ^ m _ N } ^ 2 = \\| f \\| _ { L ^ 2 _ N } ^ 2 + \\sum _ { k = 1 } ^ m \\big \\| f ^ { ( k ) } \\big \\| _ { L ^ 2 _ N } ^ 2 , \\end{align*}"}
+{"id": "5074.png", "formula": "\\begin{align*} \\mathring { v } ( x , t ) : = \\psi \\left ( x , t \\right ) - \\phi \\left ( x + \\gamma ( x , t ) + \\frac { 1 } { N } \\sigma ( t ) \\right ) , \\end{align*}"}
+{"id": "1961.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u ) ^ n = \\psi ( \\cdot , v ) \\omega ^ n & \\textnormal { i n } & \\Omega , \\\\ u = 0 & \\textnormal { i n } & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "8549.png", "formula": "\\begin{align*} \\sqrt { \\alpha } \\left ( \\sum _ { m , n = 1 } ^ { \\infty } ( - 1 ) ^ { m } \\ , K _ { 0 } \\left ( \\left ( 2 n - 1 \\right ) m \\ , \\alpha \\right ) - \\frac { \\log ( 2 ) } { 4 } \\right ) = \\sqrt { \\beta } \\left ( \\sum _ { m , n = 1 } ^ { \\infty } ( - 1 ) ^ { m } K _ { 0 } \\left ( \\left ( 2 n - 1 \\right ) m \\ , \\beta \\right ) - \\frac { \\log ( 2 ) } { 4 } \\right ) . \\end{align*}"}
+{"id": "1292.png", "formula": "\\begin{align*} \\binom { n - 1 } { 2 } - 1 \\leq Z _ 1 ( G _ n ) \\leq \\binom { n - 1 } { 2 } . \\end{align*}"}
+{"id": "6001.png", "formula": "\\begin{align*} \\theta _ { 1 / 2 } ( p _ g , \\epsilon ) & = \\theta _ { L , X ^ { \\ast } } \\Big ( J ( p _ g , i ) \\overline { \\Pi } _ { \\psi } ( p _ g ) A \\Big ) [ \\epsilon , 0 ] \\\\ & = \\sum _ { n \\in \\Z } J ( p _ g , i ) \\overline { \\Pi } _ { \\psi } ( p _ g ) A ( \\epsilon , n ) \\\\ & = \\sum _ { n \\in \\Z } e ^ { i \\epsilon \\pi n ^ 2 z _ g } . \\end{align*}"}
+{"id": "7150.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\big ( f ^ \\prime ( c _ \\varepsilon ) - f ^ \\prime ( c ) \\big ) u \\ : x \\geq - \\alpha ( c _ \\varepsilon - c ) u = - \\alpha | u | ^ 2 . \\end{align*}"}
+{"id": "491.png", "formula": "\\begin{align*} t ^ { k _ 1 } ( d t _ 3 t _ 4 t _ 5 / t ) ( d t _ 6 ) = t ^ { 2 k _ 1 - k _ 0 - 2 } t _ 1 t _ 2 t _ 3 t _ 4 t _ 5 t _ 6 = p q , \\end{align*}"}
+{"id": "7158.png", "formula": "\\begin{align*} \\lambda \\cdot ( [ x : y : z ] , [ u : v : w ] ) = ( [ \\lambda x : y : z ] , [ \\lambda ^ { - 1 } u : v : w ] ) . \\end{align*}"}
+{"id": "1871.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\langle D _ u f ( u ^ * ) , v \\rangle _ { \\mathcal { U } } + \\langle \\bar { \\lambda } , G ' ( u ^ * ) v \\rangle = 0 \\forall \\ v \\in \\mathcal { U } ; \\\\ & - \\langle \\bar { \\lambda } , \\zeta - G ( u ^ * ) \\rangle \\geq 0 \\forall \\ \\zeta \\in \\mathcal { K } . \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "2425.png", "formula": "\\begin{align*} \\zeta ( s ) = \\sum _ { n _ j \\leq x } n _ j ^ { - s } - A \\frac { x ^ { 1 - s } } { 1 - s } + O \\Big ( \\Big ( 1 + \\frac { \\abs { s } } { \\sigma - \\theta } \\Big ) x ^ { \\theta - \\sigma } \\Big ) , x \\geq 1 , \\sigma = \\Re ( s ) > \\theta . \\end{align*}"}
+{"id": "8208.png", "formula": "\\begin{align*} A _ { m ( t ) } = \\left \\{ \\exists \\ , z _ 0 = z _ 0 ( \\omega ) \\in [ - k m ( t ) , k m ( t ) ] ^ d \\ : \\ : \\ : \\ : Z _ { m ( t ) } \\left ( B ( z _ 0 , c [ \\log \\log m ( t ) ] ^ { 1 / d } ) \\right ) \\geq I ( t ) \\right \\} \\end{align*}"}
+{"id": "8785.png", "formula": "\\begin{align*} & 2 ( u + 1 ) ( u + 2 t + 3 ) c _ 1 \\\\ & = 2 ( u + 1 ) ( 2 + t ) b _ 1 + u ( u + 1 ) ( b _ 1 + c _ 1 ) + u ( b _ 1 + c _ 1 + u b _ 3 + u c _ 3 ) + 2 u ( u + 1 ) c _ 2 \\end{align*}"}
+{"id": "2261.png", "formula": "\\begin{align*} F _ k ( x , y ) & = \\frac { 1 } { \\pi } \\int _ { - \\infty } ^ \\infty F _ k ( t , 0 ) \\frac { y } { ( x - t ) ^ 2 + y ^ 2 } \\ , d t \\\\ f ( x + i ( y + y _ k ) ) & = \\frac { 1 } { \\pi } \\langle F _ k ( \\cdot , 0 ) , P ( x - \\cdot , y ) \\rangle . \\end{align*}"}
+{"id": "9264.png", "formula": "\\begin{align*} \\log N \\simeq \\int _ { 1 / 2 } ^ N x ^ { - 1 } \\ , d x \\lesssim \\int _ { N } ^ { N + 1 } \\ , d x = 1 , N > 1 , \\end{align*}"}
+{"id": "2207.png", "formula": "\\begin{align*} \\int _ { 2 ^ - } ^ { u } \\frac { G ( v ) } { ( v - 1 ) ^ 2 } \\ , d v & \\ge \\int _ { 2 } ^ { u } \\frac { 1 } { ( v - 1 ) ^ 2 } \\left ( \\log \\frac { v } { 2 } - \\frac { c _ 1 } { \\log ^ 2 y } \\right ) \\ , d v \\\\ & = - \\frac { 1 } { u - 1 } \\log \\frac { u } { 2 } + \\int _ { 2 } ^ { u } \\frac { 1 } { v ( v - 1 ) } \\ , d v - \\frac { c _ 1 } { \\log ^ 2 y } \\left ( 1 - \\frac { 1 } { u - 1 } \\right ) \\\\ & = - \\frac { u } { u - 1 } \\log \\frac { u } { 2 } + \\log ( u - 1 ) - \\frac { c _ 1 } { \\log ^ 2 y } \\left ( 1 - \\frac { 1 } { u - 1 } \\right ) . \\end{align*}"}
+{"id": "4258.png", "formula": "\\begin{align*} Y _ { t + \\delta } ^ { t , \\mu ; u } = Y _ { t + \\delta } ^ { t + \\delta , \\rho _ { t + \\delta } ^ { t , \\mu ; u } ; u } = J \\left ( t + \\delta , \\rho _ { t + \\delta } ^ { t , \\mu ; u } ; u \\right ) . \\end{align*}"}
+{"id": "6912.png", "formula": "\\begin{align*} \\mathcal { L } : = \\mathcal { M } \\setminus L . \\end{align*}"}
+{"id": "7041.png", "formula": "\\begin{align*} \\mathcal I _ 1 K [ \\textbf { X } ] = \\ker \\ , ( \\bf e ) \\end{align*}"}
+{"id": "243.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { 1 } { 6 } L i _ 2 ( z ) + \\frac { 1 } { 2 } \\log \\left ( \\frac { 1 } { 1 - z } \\right ) + \\frac { 1 } { 3 } \\frac { z } { 1 - z } \\right \\} , \\end{align*}"}
+{"id": "477.png", "formula": "\\begin{align*} \\mathcal { K } _ c ( x ; y ; t ; p , q ) = \\mathcal { K } _ c ( y ; x ; t ; p , q ) = \\mathcal { K } _ c ( x ; y ; t ; q , p ) = \\mathcal { K } _ { - c } ( - x ; y ; t ; p , q ) . \\end{align*}"}
+{"id": "121.png", "formula": "\\begin{align*} \\zeta _ 1 ( \\lambda ) = \\exp \\left ( - \\sum \\limits _ \\gamma \\frac { T _ \\gamma ^ \\# e ^ { i \\lambda T _ \\gamma } } { T _ \\gamma | \\det ( I - \\mathcal { P } _ \\gamma ) | } \\right ) , \\quad \\Im \\lambda \\gg 1 \\end{align*}"}
+{"id": "4491.png", "formula": "\\begin{align*} D _ { 1 } S _ { \\omega , \\mathbf { c } } ( \\Phi ) = D _ { 2 } S _ { \\omega , \\mathbf { c } } ( \\Phi ) = D _ { 3 } S _ { \\omega , \\mathbf { c } } ( \\Phi ) = 0 . \\end{align*}"}
+{"id": "4861.png", "formula": "\\begin{align*} W _ \\vartheta ( \\delta _ a , \\mu ) = \\inf S . \\end{align*}"}
+{"id": "538.png", "formula": "\\begin{align*} \\left ( \\mathcal { F } _ { \\mathcal { H } _ { \\hbar , V } } \\mathcal { H } _ { \\hbar , V } f \\right ) ( \\xi ) = ( \\mathcal { H } _ { \\hbar , V } f , u _ { \\xi } ) = ( f , \\mathcal { H } _ { \\hbar , V } u _ { \\xi } ) = ( f , \\lambda _ { \\xi } u _ { \\xi } ) = \\lambda _ { \\xi } \\widehat { f } ( \\xi ) , \\xi \\in \\mathcal { I } _ { \\hbar } , \\end{align*}"}
+{"id": "1594.png", "formula": "\\begin{align*} \\tau _ n \\le C \\ , r \\ , T \\ , N ^ { - r } n ^ { r - 1 } \\ , \\le \\ , C \\ , r \\ , T ^ { \\frac { 1 } { r } } \\ , N ^ { - 1 } \\ , t _ n ^ { 1 - \\frac { 1 } { r } } , \\mbox { f o r } \\ ; 1 \\le n \\le N , \\\\ \\end{align*}"}
+{"id": "6597.png", "formula": "\\begin{align*} & E ( \\alpha _ { l , { { n } _ t } } - \\alpha _ { l , { \\hat { n } _ t } } e ^ { - j \\omega } ) = \\frac { \\sqrt { \\pi } } { 2 } , \\\\ & V a r ( \\alpha _ { l , { { n } _ t } } - \\alpha _ { l , { \\hat { n } _ t } } e ^ { - j \\omega } ) = \\frac { 8 - \\pi } { 4 } . \\end{align*}"}
+{"id": "4851.png", "formula": "\\begin{align*} ( \\prod _ { i = 1 } ^ u d ( i ) \\ast h _ l ( q ( i ) ) ) \\ast d ( u + 1 ) = ( \\prod _ { i = 1 } ^ m a ( i ) \\ast g _ { w ( l ) } ( t ( m ) ) ) \\ast a ( m + 1 ) \\in A _ 1 \\cap \\sigma ( L ) \\subseteq A _ 1 \\cap \\sigma ( L _ 1 ) . \\end{align*}"}
+{"id": "6835.png", "formula": "\\begin{align*} \\alpha _ n = ( - 1 ) ^ n q ^ { \\left ( { n \\atop 2 } \\right ) } \\mbox { a n d } \\quad \\beta _ n = ( q ) _ m ( - 1 ) ^ n q ^ { \\left ( { n \\atop 2 } \\right ) } \\left [ { m + n \\atop m + 2 n } \\right ] . \\end{align*}"}
+{"id": "1786.png", "formula": "\\begin{align*} \\delta ^ \\ell _ { i _ 1 , \\ldots , i _ n } ( f _ { i _ 1 , \\ldots , i _ n } ) = f _ { i _ 1 , \\ldots , i _ { \\ell - 1 } , i _ { \\ell + 1 } , \\ldots , i _ n } \\end{align*}"}
+{"id": "7934.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) u = ( \\lambda _ 0 + \\lambda _ { e _ 0 } u ) \\xi . \\end{align*}"}
+{"id": "8698.png", "formula": "\\begin{align*} \\Delta _ 0 + T _ 1 = \\omega ( \\max \\{ p ^ { - 1 / 2 } N ^ { - 1 } , p ^ { - 1 } N ^ { - 1 / 2 } , p ^ { - 3 / 2 } \\} ) , \\end{align*}"}
+{"id": "1419.png", "formula": "\\begin{align*} T _ { y , s } \\dot { s } = P _ { y , s } \\dot { s } _ { | | } + \\frac { \\sinh ( \\| s \\| ) } { \\| s \\| } P _ { y , s } \\dot { s } _ { \\perp } , T _ { y , s } ^ { - 1 } \\ddot { s } = P _ { y , s } ^ { - 1 } \\ddot { s } _ { | | } + \\frac { \\| s \\| } { \\sinh ( \\| s \\| ) } P _ { y , s } ^ { - 1 } \\ddot { s } _ { \\perp } . \\end{align*}"}
+{"id": "7583.png", "formula": "\\begin{align*} p ( z ) = \\frac { 1 + ( \\overline { \\tau _ 1 } \\tau _ 2 + \\tau _ 1 ) z + \\tau _ 2 z ^ 2 } { 1 + ( \\overline { \\tau _ 1 } \\tau _ 2 - \\tau _ 1 ) z - \\tau _ 2 z ^ 2 } , \\ ; \\ ; z \\in \\mathbb { D } . \\end{align*}"}
+{"id": "325.png", "formula": "\\begin{align*} s _ h ( 4 , 3 ) = - \\frac { 1 0 9 } { 8 } \\zeta ( 7 ) + \\frac { 3 7 } { 2 } \\zeta ( 3 ) \\zeta ( 4 ) - 5 \\zeta ( 2 ) \\zeta ( 5 ) , \\end{align*}"}
+{"id": "7249.png", "formula": "\\begin{align*} \\alpha _ { \\infty , 4 } ( n ) = \\sup _ { i _ 4 > i _ 3 > i _ 2 > i _ 1 \\geq n } \\alpha ( { \\mathcal F } _ { 0 } , \\sigma ( Y _ { i _ 1 } , Y _ { i _ 2 } , Y _ { i _ 3 } , Y _ { i _ 4 } ) ) \\ , , \\end{align*}"}
+{"id": "7745.png", "formula": "\\begin{align*} B & = \\bigcap _ { j = 0 } ^ { q } \\Bigl ( y ^ { * } \\bigl ( \\eta _ { 1 } , T _ { j } ( \\eta _ { 2 } ) \\bigr ) \\le a _ { j } \\Bigr ) \\cap \\Bigl ( T _ { q } ( \\eta _ { 2 } ) < t \\le T _ { q + 1 } ( \\eta _ { 2 } ) \\Bigr ) \\\\ & = \\bigcap _ { j = 0 } ^ { q } \\Bigl ( T _ { j } ( \\eta _ { 2 } ) \\le y ( \\eta _ { 1 } , a _ { j } ) \\Bigr ) \\cap \\Bigl ( T _ { q } ( \\eta _ { 2 } ) < t \\le T _ { q + 1 } ( \\eta _ { 2 } ) \\Bigr ) \\end{align*}"}
+{"id": "3887.png", "formula": "\\begin{align*} \\mu _ 1 = \\mathcal { N } \\left ( 0 , \\left [ \\begin{array} { c c } 2 & - 1 \\\\ - 1 & 4 \\end{array} \\right ] \\right ) , \\mu _ 2 = \\mathcal { N } \\left ( 0 , \\left [ \\begin{array} { c c } 4 & - 2 \\\\ - 2 & 4 \\end{array} \\right ] \\right ) , \\mu _ 3 = \\mathcal { N } \\left ( 0 , \\left [ \\begin{array} { c c } 2 & - 2 \\\\ - 2 & 4 \\end{array} \\right ] \\right ) . \\end{align*}"}
+{"id": "5856.png", "formula": "\\begin{align*} G _ { \\omega , E , \\Lambda _ L ( x ) } - G _ { \\omega , E _ 0 , \\Lambda _ L ( x ) } = ( E _ 0 - E ) G _ { \\omega , E , \\Lambda _ L ( x ) } G _ { \\omega , E _ 0 , \\Lambda _ L ( x ) } . \\end{align*}"}
+{"id": "9241.png", "formula": "\\begin{align*} m = 1 & \\texttt { c [ k ] } = ( k + 1 ) \\texttt { c [ k + 1 ] } ( 1 + \\eta _ 2 ) \\\\ m = 2 & \\texttt { c [ k ] } = ( k + 1 ) ( k + 2 ) \\texttt { c [ k + 2 ] } ( 1 + \\eta _ 3 ) \\\\ \\dots \\\\ m & \\texttt { c [ k ] } = ( k + 1 ) ( k + 2 ) \\cdots ( k + m ) \\texttt { c [ k + m ] } ( 1 + \\eta _ { m + 1 } ) \\end{align*}"}
+{"id": "8155.png", "formula": "\\begin{align*} [ \\alpha ^ k ] \\Gamma _ F ( \\alpha ) = ( - 1 ) ^ { | F | } [ \\alpha ^ k ] \\chi _ F ( \\alpha ) . \\end{align*}"}
+{"id": "3857.png", "formula": "\\begin{align*} \\widehat { \\mathcal { I } } ( \\delta ) & : = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\varpi \\in \\Pi ( \\widehat { \\mu } _ { 1 3 } , \\widehat { \\mu } _ { 2 3 } ) } \\int _ { \\mathcal { V } } f _ { \\theta , \\lambda } \\ , d \\varpi \\right ] . \\end{align*}"}
+{"id": "8577.png", "formula": "\\begin{align*} E [ S _ j ( \\tau _ N ) ] = \\frac { \\nu N } { \\lambda } \\cdot \\frac 1 { j ( j + 1 ) } , j = 2 , \\ldots , N - 1 . \\end{align*}"}
+{"id": "4331.png", "formula": "\\begin{align*} f ^ \\mathcal { K } ( \\cdot , t ) = f _ 0 \\circ \\big ( \\tilde { y } _ t ^ { \\mathcal { K } } \\big ) ^ { - 1 } , \\end{align*}"}
+{"id": "2235.png", "formula": "\\begin{align*} \\varphi ( q _ { n * } ( \\xi , \\ldots , \\xi , a , b ) ) = q _ { n * } ( \\varphi ( \\xi ) , \\ldots , \\varphi ( \\xi ) , \\varphi ( a ) , \\varphi ( b ) ) \\end{align*}"}
+{"id": "1660.png", "formula": "\\begin{align*} g ( \\nabla _ { X } \\ , \\xi _ i , \\ , Q Z ) = - \\frac 1 2 \\ , N ^ { \\ , ( 5 ) } ( X , \\xi _ i , { f } Z ) . \\end{align*}"}
+{"id": "182.png", "formula": "\\begin{align*} \\textmd { w h e r e } \\ ; \\alpha _ 1 = ( x ^ { 1 0 } + x ^ 9 - x ^ 7 - x ^ 6 - x ^ 5 - x ^ 4 - x ^ 3 + x + 1 ) _ 2 \\approx 1 . 1 7 6 2 8 0 8 1 8 2 5 9 9 1 7 5 0 6 5 4 4 \\end{align*}"}
+{"id": "1088.png", "formula": "\\begin{align*} b \\ast x = s ( b ) i ( x ) , x \\ast b = i ( x ) s ( b ) , \\forall b \\in B , \\ ; \\forall x \\in X \\end{align*}"}
+{"id": "8490.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { 1 } { \\left ( \\lambda _ { n } ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } = \\frac { 2 \\sin ( \\pi s ) } { x ^ { 2 s - 1 } } \\ , \\intop _ { 1 } ^ { \\infty } \\left \\{ \\frac { 1 } { \\sigma ( x t ) e ^ { 2 \\pi x t } - 1 } + \\frac { 1 } { 2 } - \\frac { 1 } { 2 \\pi x t } \\cdot \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } \\right \\} \\ , \\frac { d t } { ( t ^ { 2 } - 1 ) ^ { s } } . \\end{align*}"}
+{"id": "754.png", "formula": "\\begin{align*} \\langle \\frac { \\partial } { \\partial \\theta _ i } , \\frac { \\partial } { \\partial r } \\rangle = 0 , \\ \\ \\langle \\frac { \\partial } { \\partial r } , \\frac { \\partial } { \\partial r } \\rangle = 1 , \\ \\ \\langle \\frac { \\partial } { \\partial \\theta _ i } , \\frac { \\partial } { \\partial \\theta _ j } \\rangle = \\phi ^ 2 ( r ) s _ { i j } . \\end{align*}"}
+{"id": "4935.png", "formula": "\\begin{align*} P U = U \\Lambda , \\end{align*}"}
+{"id": "1353.png", "formula": "\\begin{align*} I I _ { \\delta } = 0 . \\end{align*}"}
+{"id": "7537.png", "formula": "\\begin{align*} \\left | \\abs { A _ 1 } ^ { \\frac { p - 2 } { 2 } } A _ 1 - \\abs { A _ 2 } ^ { \\frac { p - 2 } { 2 } } A _ 2 \\right | ^ 2 & \\lesssim _ { ( d , p ) } \\left ( \\abs { A _ 1 } ^ 2 + \\abs { A _ 2 } ^ 2 \\right ) ^ { \\frac { p - 2 } { 2 } } \\abs { A _ 1 - A _ 2 } ^ 2 \\\\ & \\lesssim _ { ( d , p ) } \\left < \\abs { A _ 1 } ^ { p - 2 } A _ 1 - \\abs { A _ 2 } ^ { p - 2 } A _ 2 , A _ 1 - A _ 2 \\right > , \\end{align*}"}
+{"id": "5413.png", "formula": "\\begin{align*} ( f _ i ) ^ n _ * A = A + n R _ i + \\frac { n ( n - 1 ) t _ { i i } } { 2 } D _ 0 , \\end{align*}"}
+{"id": "2290.png", "formula": "\\begin{align*} Q _ r ( \\theta ) = \\frac { 2 r \\sin ( \\theta ) } { 1 - 2 r \\cos ( \\theta ) + r ^ 2 } \\end{align*}"}
+{"id": "6591.png", "formula": "\\begin{align*} \\begin{aligned} P _ b = & \\Pr ( - 2 \\Re \\{ y \\sqrt { P _ s } \\zeta \\eta \\} + | \\sqrt { P _ s } \\zeta \\eta | ^ 2 \\\\ & > - 2 \\Re \\{ y \\sqrt { P _ s } \\zeta \\hat \\eta \\} + | \\sqrt { P _ s } \\zeta \\hat \\eta | ^ 2 ) \\\\ = & \\Pr \\left ( 2 \\Re \\{ y \\sqrt { P _ s } \\zeta ( \\hat \\eta - \\eta ) \\} \\right . \\\\ & + \\left . | \\sqrt { P _ s } \\zeta \\eta | ^ 2 - | \\sqrt { P _ s } \\zeta \\hat \\eta | ^ 2 > 0 \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "3158.png", "formula": "\\begin{align*} \\tau ( X ) : = \\int _ { 0 } ^ { \\infty } \\lambda d \\tau ( e _ { \\lambda } ) , \\end{align*}"}
+{"id": "6891.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\binom { N } { 2 } } \\log \\P \\left ( N ^ { - 1 } \\max _ { i \\in [ N ] } d _ i \\leq \\beta \\right ) \\geq \\frac { 1 } { \\binom { N } { 2 } } \\sum _ { i = 1 } ^ N \\log \\P ( N ^ { - 1 } d _ i \\leq \\beta ) \\\\ & = \\frac { 2 N } { N - 1 } \\frac { 1 } { N } \\sum _ { i \\in [ N ] } \\frac { 1 } { N } \\log \\P ( N ^ { - 1 } d _ i \\leq \\beta ) = \\frac { 2 N } { N - 1 } \\int _ { [ 0 , 1 ] } \\d x \\ , \\frac { 1 } { N } \\log \\P ( N ^ { - 1 } d _ { i _ x } \\leq \\beta ) . \\end{aligned} \\end{align*}"}
+{"id": "2323.png", "formula": "\\begin{align*} \\begin{dcases} \\dfrac { \\partial ^ 2 u } { \\partial t ^ 2 } ( t , x ) = \\dfrac { \\partial ^ 2 u } { \\partial x ^ 2 } ( t , x ) + u ( t , x ) \\dot { W } ( t , x ) , & ( t , x ) \\in \\R _ + \\times \\R , \\\\ u ( 0 , x ) = 1 , \\ \\dfrac { \\partial u } { \\partial t } ( 0 , x ) = 0 , & \\forall x \\in \\R . \\end{dcases} \\end{align*}"}
+{"id": "1585.png", "formula": "\\begin{align*} f ^ { ( 2 ) } _ n ( 1 2 3 ) = \\sum _ { d \\ge 1 } f _ { n , d } ^ { ( 2 ) } ( 1 2 3 ) . \\end{align*}"}
+{"id": "8751.png", "formula": "\\begin{align*} \\delta _ i ( \\xi ) = \\begin{cases} 1 & \\mbox { i f } \\xi = i \\\\ 0 & \\quad \\mbox { o t h e r w i s e , } \\end{cases} \\qquad \\mbox { f o r } i = 0 , 1 . \\end{align*}"}
+{"id": "8297.png", "formula": "\\begin{align*} x ^ t L _ { e - e ' } x = & ( l - 1 ) x ^ 2 _ 1 - ( l - 1 ) x _ 1 x _ l + ( l - 2 ) x ^ 2 _ 2 - ( l - 2 ) x _ 2 x _ l - \\cdots \\\\ & + 2 x ^ 2 _ { l - 2 } - 2 x _ { l - 2 } x _ l + \\mu _ { ( l - 1 ) 1 } x ^ 2 _ { l - 1 } - \\mu _ { ( l - 1 ) 1 } x _ { l - 1 } x _ r - ( l - 1 ) x _ 1 x _ l \\\\ & - ( l - 2 ) x _ 2 x _ l - \\cdots - 2 x _ { l - 2 } x _ l - \\mu _ { ( l - 1 ) 1 } x _ { l - 1 } x _ l + d _ l x ^ 2 _ l \\\\ & = ( l - 1 ) ( x _ 1 - x _ r ) ^ 2 + \\cdots + 2 ( x _ { l - 1 } - x _ l ) ^ 2 + \\mu _ { ( l - 1 ) 1 } ( x _ { l - 1 } - x _ l ) ^ 2 \\end{align*}"}
+{"id": "4071.png", "formula": "\\begin{align*} J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { I L J M } } { c } \\right ) = \\sum _ { \\ell = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ \\ell } { \\ell ! \\ , \\ \\Gamma ( \\ell + 2 k ) } \\left ( \\frac { 2 \\pi \\sqrt { I J L M } } { c } \\right ) ^ { 2 k + 2 \\ell - 1 } . \\end{align*}"}
+{"id": "1909.png", "formula": "\\begin{align*} \\lambda ^ k \\in \\partial I _ K ( \\zeta ^ k ) \\forall \\ k = 2 , 3 , \\dots . \\end{align*}"}
+{"id": "375.png", "formula": "\\begin{align*} \\mathfrak { c } ( \\tau ) _ { \\sigma } = \\begin{cases} p & \\tau \\subseteq \\sigma \\\\ 1 & \\mbox { e l s e } . \\end{cases} \\end{align*}"}
+{"id": "4724.png", "formula": "\\begin{align*} \\frac { 1 } { L ^ d } \\sum _ { k \\in \\frac { 2 \\pi } { L } \\Z ^ d } \\frac { | k | ^ p \\hat \\gamma ( k ) ^ n } { ( 1 + \\hat \\gamma ( k ) ) ^ m } = \\frac { 1 } { L ^ d } \\sum _ { k \\in \\frac { 2 \\pi } { L } \\Z ^ d } \\frac { | k | ^ p e ^ { - n \\beta | k | ^ 2 } } { \\left ( 1 + e ^ { - \\beta | k | ^ 2 } \\right ) ^ m } \\leq C \\rho _ 0 ^ { 1 + p / d } \\end{align*}"}
+{"id": "3969.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\rho _ 2 ( \\widetilde { Y } _ 2 , y _ 2 ) \\right ] - 1 \\leq \\mathbb { E } \\left [ \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( \\widetilde { S } _ 2 , s _ 2 ) ^ { p _ 2 } \\right ] = \\boldsymbol { W } _ { p _ 2 } \\left ( \\mu _ 2 , \\delta _ { s _ 2 } \\right ) ^ { p _ 2 } . \\end{align*}"}
+{"id": "7093.png", "formula": "\\begin{align*} c | _ { t = 0 } = c _ 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Omega \\end{align*}"}
+{"id": "5577.png", "formula": "\\begin{align*} ( y , r ) \\times { \\rm T r e e } _ { F , H _ { 0 } } & \\to \\mathbb { R } \\\\ x = ( y , r , H ) & \\mapsto H _ { \\alpha _ { x , g } \\parallel \\alpha _ { x , e } } \\left ( \\mathcal { P } _ { x , n } \\right ) \\end{align*}"}
+{"id": "5418.png", "formula": "\\begin{align*} r _ n ^ { [ 1 ] } ( h ) : = \\frac { 2 ^ p \\cdot y _ { 2 n } \\left ( \\frac { h } { 2 } \\right ) - y _ n ( h ) } { 2 ^ p - 1 } \\end{align*}"}
+{"id": "3647.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } \\psi \\left ( F ^ j ( \\omega , x ) \\right ) = \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } \\varphi \\left ( f _ { \\overline { \\omega | _ { [ 1 , j ] } } } ( x ) \\right ) . \\end{align*}"}
+{"id": "9013.png", "formula": "\\begin{align*} ( \\ast ) = w _ j C ^ \\varphi _ { k j i } R ^ \\varphi _ { i k } + \\frac { 1 } { 2 } \\langle \\nabla w , \\nabla | T ^ \\varphi | ^ 2 \\rangle + \\frac { 1 } { 2 } ( 1 + w ) | C ^ \\varphi | ^ 2 + w _ { t i } R ^ \\varphi _ { i k , t k } . \\end{align*}"}
+{"id": "2364.png", "formula": "\\begin{align*} \\check { H } ^ 1 ( \\{ U \\} , \\mathcal F ) = H ^ 1 ( X , \\mathcal F ) \\rlap { . } \\end{align*}"}
+{"id": "1399.png", "formula": "\\begin{align*} I : = \\int _ M J \\theta \\wedge d \\Omega ^ { n - 1 } , \\end{align*}"}
+{"id": "7477.png", "formula": "\\begin{align*} \\Phi ( \\pi ) = ( 9 , 6 , 7 ) ( 8 , 1 ) ( 4 , 3 , 2 ) ( 5 ) ( 1 0 ) ( 1 1 ) . \\end{align*}"}
+{"id": "1127.png", "formula": "\\begin{align*} J E _ I ( z , 0 ) = I d . \\end{align*}"}
+{"id": "6939.png", "formula": "\\begin{align*} d _ { i j } = \\begin{cases} - \\frac 1 2 & j = 2 i - 1 \\mbox { o r } 2 i \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "5529.png", "formula": "\\begin{align*} h _ { \\mu } ( Z , \\lambda ) = h _ { \\mu } ( Y , \\nu ) + \\mathrm { I } \\left ( \\xi _ { 1 } , \\mathcal { T } | X , \\eta \\right ) . \\end{align*}"}
+{"id": "1933.png", "formula": "\\begin{align*} X = \\bigcup _ { S \\in \\mathcal { P } _ { m } } X _ { S } , \\end{align*}"}
+{"id": "5608.png", "formula": "\\begin{align*} { \\rm I } ( \\xi _ { 1 } , \\xi _ { n } ) = H \\left ( \\mu ^ { ( n ) } \\right ) - H \\left ( \\mu ^ { ( n - 1 ) } \\right ) \\end{align*}"}
+{"id": "3842.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) = \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) & = \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + \\sqrt { \\delta _ 1 } + \\sqrt { \\delta _ 2 } . \\end{align*}"}
+{"id": "5015.png", "formula": "\\begin{align*} \\theta _ f : \\mathcal { X } \\ni x \\mapsto ( f _ j ( x ) ) _ { j = 1 } ^ n \\in \\ell ^ p ( [ n ] ) , \\theta _ \\tau : \\ell ^ p ( [ n ] ) \\ni ( a _ j ) _ { j = 1 } ^ n \\mapsto \\sum _ { j = 1 } ^ { n } a _ j \\tau _ j \\in \\mathcal { X } . \\end{align*}"}
+{"id": "5077.png", "formula": "\\begin{align*} \\lim _ { t \\uparrow \\tau _ { \\max } } \\left \\| \\left ( \\sigma ( t ) , \\partial _ t \\sigma ( t ) , \\gamma ( t ) , \\partial _ t \\gamma ( t ) \\right ) \\right \\| _ { \\R \\times \\R \\times H ^ 4 _ N \\times H ^ 2 _ N } = \\infty . \\end{align*}"}
+{"id": "1907.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & D _ u \\theta ( u ^ { k + 1 } ) + S ^ * [ \\lambda ^ k + \\beta ( S u ^ { k + 1 } - \\zeta ^ { k + 1 } ) ] = 0 ; \\\\ & I _ K ( \\zeta ) - I _ K ( \\zeta ^ { k + 1 } ) - \\langle \\lambda ^ k + \\beta ( S u ^ { k + 1 } - \\zeta ^ { k + 1 } ) , \\zeta - \\zeta ^ { k + 1 } \\rangle \\geq 0 \\forall \\ \\zeta \\in \\mathcal { X } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "9257.png", "formula": "\\begin{align*} \\texttt { L H S } \\eqref { R 4 . 1 } \\le \\int _ { B - l _ 2 } ^ B ( B - z ) ^ { \\alpha + \\beta - 1 / 2 } \\ , d z \\simeq l _ 2 ^ { \\alpha + \\beta + 1 / 2 } = \\texttt { R H S } \\eqref { R 4 . 1 } . \\end{align*}"}
+{"id": "7776.png", "formula": "\\begin{align*} & ( b + a ^ 3 - k ) x ^ { q + 2 } z + ( 2 a ^ 2 b ) x ^ { q + 1 } z ^ 2 + ( b ^ 2 a - 1 ) x ^ q z ^ 3 + \\\\ & ( - b - a ^ 3 + k ) x ^ 3 z ^ q + ( - 2 a b ) x ^ 2 z ^ { q + 1 } + ( - a b ^ 2 + 1 ) x z ^ { q + 2 } = 0 \\end{align*}"}
+{"id": "186.png", "formula": "\\begin{align*} L i _ 3 ( - 1 ) = - \\frac { 3 } { 4 } \\zeta ( 3 ) , \\end{align*}"}
+{"id": "5948.png", "formula": "\\begin{align*} \\overline { C } _ { X ^ { \\ast } } ( h , p ) & = \\nu _ 2 ( \\det p , g _ h ) \\overline { c } _ { X ^ { \\ast } } ( ( g _ h ) ^ { \\det p } , g _ p ) = \\nu _ 2 ( \\det p , g _ h ) = \\left \\{ \\begin{array} { c l } ( \\det p , a ) _ { \\R } & \\textrm { i f } c = 0 , \\\\ 1 & \\textrm { i f } c \\neq 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "3940.png", "formula": "\\begin{align*} \\sup _ { \\gamma \\in \\bar { \\mathcal { P } } } I _ { \\lambda } [ \\gamma ] = \\sup _ { \\gamma \\in \\bar { \\mathcal { P } } } \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ { 1 3 } , \\mu _ { 2 3 } , \\gamma \\right ) } \\int _ { \\mathcal { V } \\times \\mathcal { S } } \\phi _ \\lambda ( v , s ^ \\prime ) \\ , d \\pi ( v , s ^ \\prime ) . \\end{align*}"}
+{"id": "6123.png", "formula": "\\begin{align*} \\kappa = \\frac { \\tau } { 2 ^ { \\frac { 2 0 n } { s } } \\times \\left ( 2 0 \\sqrt { n } \\right ) ^ { 1 0 n } ( c _ { n } + 1 ) } , \\end{align*}"}
+{"id": "9320.png", "formula": "\\begin{align*} \\begin{cases} | b _ { x } ( t , x , v ) | + | b _ { v } ( t , x , v ) | + | \\sigma _ { x } ( t , x , v ) | + | \\sigma _ { v } ( t , x , v ) | \\leq L , \\\\ | \\ell _ x ( t , x , v ) | + | \\ell _ v ( t , x , v ) | \\leq L ( 1 + | x | + | v | ) , \\\\ | g _ x ( x ) | + | f _ x ( t , x ) | \\leq L ( 1 + | x | ) , \\\\ | b ( t , 0 , 0 ) | + | \\sigma ( t , 0 , 0 ) | + | \\ell ( t , 0 , 0 ) | + | g ( 0 ) | + | f ( t , 0 ) | \\leq L , \\end{cases} \\end{align*}"}
+{"id": "5819.png", "formula": "\\begin{align*} { { \\boldsymbol { y } } _ k } = { { \\boldsymbol { \\hat y } } _ { k | k - 1 } } + { { \\boldsymbol { H } } _ k } { { \\boldsymbol { \\varepsilon } } _ { k | k - 1 } } + { { \\boldsymbol { r } } _ k } + { { \\boldsymbol { v } } _ k } , \\end{align*}"}
+{"id": "1000.png", "formula": "\\begin{align*} x ^ { w _ i / \\tilde w _ i } - t ^ { w _ i } \\varphi ( t ) = 0 . \\end{align*}"}
+{"id": "6323.png", "formula": "\\begin{align*} J ( \\phi , \\omega , r ; t ) = \\left | \\frac { \\partial z } { \\partial \\omega } ( \\phi , \\omega , r ; t ) \\det ( M _ 1 ) - \\frac { \\partial z } { \\partial \\phi } ( \\phi , \\omega , r ; t ) \\det ( M _ 2 ) + \\frac { \\partial z } { \\partial r } ( \\phi , \\omega , r ; t ) \\det ( M _ 3 ) \\right | \\end{align*}"}
+{"id": "4747.png", "formula": "\\begin{align*} \\tilde { S } _ { \\gamma _ i \\gamma _ j } & = J _ { 2 | \\alpha _ i | } \\tilde { S } _ { \\gamma _ j \\gamma _ j } J ^ T _ { 2 | \\alpha _ j | } + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "7602.png", "formula": "\\begin{align*} H _ { 2 , 1 } ( F _ { f } / 2 ) & = \\dfrac { 1 } { 4 8 } \\left ( a ^ 4 _ 2 - 1 2 a ^ 2 _ 3 + 1 2 a _ 2 a _ 4 \\right ) \\\\ & = \\dfrac { 1 } { 3 6 8 6 4 } \\left ( - 7 c ^ 4 _ { 1 } - 8 c ^ 2 _ { 1 } c _ 2 - 6 4 c ^ 2 _ { 2 } + 9 6 c _ 1 c _ 3 \\right ) . \\end{align*}"}
+{"id": "5045.png", "formula": "\\begin{align*} q _ \\bullet ( \\xi _ u ) + q _ \\bullet ( \\xi _ v ) = q _ \\bullet ( \\xi _ { u v } ) \\end{align*}"}
+{"id": "1280.png", "formula": "\\begin{align*} Z ( G ) = M ( G ) = n - 2 T + s _ 1 + 2 s _ 0 = & ~ n - 2 T + ( s - p ) + 2 \\ , \\sum _ { j = 1 } ^ s ( t _ j - 1 ) \\\\ = & ~ n - 2 T + s - p + 2 T - 2 s = n - s - p . \\end{align*}"}
+{"id": "6885.png", "formula": "\\begin{align*} J _ r ^ { ( k ) } ( x , \\beta ) = \\theta ^ { ( k - 1 ) } ( x , \\beta ) . \\end{align*}"}
+{"id": "5014.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ n a _ j \\tau _ j \\right \\| = \\left ( \\sum _ { j = 1 } ^ n | a _ j | ^ p \\right ) ^ \\frac { 1 } { p } . \\end{align*}"}
+{"id": "2698.png", "formula": "\\begin{align*} ( a _ 1 ' ) ^ * I C _ { M _ s } = I C _ { U _ v } = ( a _ 1 ' ) ^ * \\iota _ s ^ * I C _ M [ \\dim { S } - \\dim { M } ] . \\end{align*}"}
+{"id": "877.png", "formula": "\\begin{align*} N ( P ) = M ( P ) + E _ 1 ( P ) + E _ 2 ( P ) , \\end{align*}"}
+{"id": "2975.png", "formula": "\\begin{align*} \\partial _ k ( c _ { i _ 1 , \\dots , i _ k } ) = \\sum _ { j = 1 } ^ k ( - 1 ) ^ { j - 1 } ( s _ j - 1 ) \\ , c _ { i _ 1 , \\dots , \\check { i } _ j , \\dots , i _ k } \\end{align*}"}
+{"id": "1600.png", "formula": "\\begin{align*} \\int _ { \\Theta } v ^ { 2 } ( w ) \\| T _ { w } ( f ) \\| ^ { 2 } d \\mu ( w ) & = \\int _ { \\Theta } v ^ { 2 } ( w ) \\| \\chi _ { w } \\pi _ { F ( w ) } S _ { \\chi } ^ { - 1 } f \\| ^ { 2 } d \\mu ( w ) \\\\ & \\leq D \\| S _ { \\chi } ^ { - 1 } f \\| ^ { 2 } \\\\ & \\leq D \\| S _ { \\chi } ^ { - 1 } \\| ^ { 2 } \\| f \\| ^ { 2 } \\\\ & \\leq \\frac { D } { C ^ { 2 } } \\| f \\| ^ { 2 } . \\end{align*}"}
+{"id": "6906.png", "formula": "\\begin{align*} M : = \\left \\{ m \\in \\N : m \\textrm { h a s a t l e a s t } \\frac { a \\log { N } } { | \\log ( 2 \\sigma - 1 ) | } \\textrm { p r i m e f a c t o r s i n } P \\right \\} \\end{align*}"}
+{"id": "8458.png", "formula": "\\begin{align*} \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) = \\frac { 1 } { s - \\frac { 1 } { 2 } } - \\gamma + O \\left ( s - \\frac { 1 } { 2 } \\right ) \\end{align*}"}
+{"id": "6112.png", "formula": "\\begin{align*} G ( x , y , t ) = g ( x , t ) . \\end{align*}"}
+{"id": "4770.png", "formula": "\\begin{align*} M _ { [ 1 ] } = W _ { [ 1 ] } = S _ { [ 1 ] } + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "7376.png", "formula": "\\begin{align*} \\langle S _ N ( \\ell ) ^ 2 \\rangle = \\ell \\sum _ { i , j = 1 } ^ N \\sum _ { m \\in \\mathbb { Z } } \\Delta \\left ( \\frac { x _ i - x _ j + m } { \\ell } \\right ) . \\end{align*}"}
+{"id": "940.png", "formula": "\\begin{align*} Z _ * : C H _ i ( X ) \\to C H _ i ( X ) , Z _ * [ D ] = p _ { 2 , X * } ( Z \\cdot p _ { 1 , X } ^ * ( [ D ] ) ) \\end{align*}"}
+{"id": "8175.png", "formula": "\\begin{align*} \\underset { t \\rightarrow \\infty } { \\lim } \\ , \\frac { 1 } { r ( t ) } \\log P \\left ( n _ t < \\gamma _ t p _ t e ^ { \\beta t } \\right ) = - \\kappa . \\end{align*}"}
+{"id": "7259.png", "formula": "\\begin{align*} \\Big \\Vert \\sup _ { k \\leq 2 ^ { L - m ( L ) } } \\Big | \\sum _ { \\ell = 1 } ^ k ( \\widetilde U _ { \\ell , L } - V _ { \\ell , L } ) \\Big | \\Big \\Vert ^ 2 _ 2 \\leq 1 6 \\sum _ { \\ell = 1 } ^ { 2 ^ { L - m ( L ) } } \\Vert U _ { \\ell , L } - V _ { \\ell , L } \\Vert ^ 2 _ 2 \\ , . \\end{align*}"}
+{"id": "8217.png", "formula": "\\begin{align*} P ^ \\omega \\left ( | \\widehat { Z } _ s | < e ^ { - \\sqrt { \\beta / 2 } \\ , \\widehat { R } ( s ) } p _ s e ^ { \\beta s } \\ : \\big | \\ : C _ t \\right ) & \\leq P _ { y _ 0 } \\left ( \\big | Z _ s ^ { \\widehat { B } _ s } \\big | < e ^ { - \\sqrt { \\beta / 2 } \\ , \\widehat { R } ( s ) } p _ s e ^ { \\beta s } \\right ) \\\\ & = \\exp \\left [ - \\sqrt { \\beta / 2 } \\ , \\widehat { R } ( s ) ( 1 + o ( 1 ) ) \\right ] . \\end{align*}"}
+{"id": "2165.png", "formula": "\\begin{align*} \\lambda _ { 1 } = \\lambda _ { 1 } \\left ( N , \\varepsilon , \\Omega , T , c \\right ) \\geq \\lambda _ { 0 } \\geq 1 \\end{align*}"}
+{"id": "6030.png", "formula": "\\begin{align*} \\int \\otimes _ { i = 1 } ^ { n + 1 } f _ i \\otimes g \\ , d \\tilde \\rho _ { n + 1 } = \\int V f _ { n + 1 } \\otimes \\bigl ( \\otimes _ { i = 1 } ^ n f _ i \\bigr ) \\otimes g \\ , d \\rho _ n . \\end{align*}"}
+{"id": "2843.png", "formula": "\\begin{align*} S _ 3 ^ 1 = ( a _ 1 + a _ 3 ) S _ 1 ^ 2 - ( a _ 0 a _ 3 ) S _ 0 ^ 1 . \\end{align*}"}
+{"id": "1471.png", "formula": "\\begin{align*} K _ 1 = & \\Big | f _ \\epsilon ( V + \\phi _ 2 ) - f _ \\epsilon ( V + \\phi _ 1 ) - f ^ { ' } _ \\epsilon ( V ) ( \\phi _ 2 - \\phi _ 1 ) \\Big | _ { \\frac { 2 n } { n + 2 } } \\\\ = & \\Big | \\Big ( f _ \\epsilon ^ { ' } ( V + \\phi _ \\varrho ) - f ^ { ' } _ \\epsilon ( V ) \\Big ) ( \\phi _ 2 - \\phi _ 1 ) \\Big | _ { \\frac { 2 n } { n + 2 } } . \\end{align*}"}
+{"id": "8759.png", "formula": "\\begin{align*} \\begin{aligned} & - M z _ { i j } \\leq \\lambda _ { i j } \\leq M z _ { i j } \\\\ & - M ( 1 - z _ { i j } ) \\leq v _ { i j } \\leq M ( 1 - z _ { i j } ) \\end{aligned} \\end{align*}"}
+{"id": "6253.png", "formula": "\\begin{align*} d H _ r ( \\xi , & \\sigma _ 1 , \\ldots , \\sigma _ r , \\alpha _ 1 , \\ldots , \\alpha _ r , \\beta _ 1 , \\ldots , \\beta _ r ) \\ , = \\\\ & \\textbf { e } ( ( \\xi - \\sigma _ r ) \\beta _ r ) \\hat { F } ( \\sigma _ r - \\xi , \\alpha _ r ) H _ { r - 1 } ( \\xi , \\sigma _ 1 , \\ldots , \\sigma _ { r - 1 } , \\alpha _ 1 , \\ldots , \\alpha _ { r - 1 } , \\beta _ 1 - \\beta _ 2 , \\ldots , \\beta _ { r - 1 } - \\beta _ { r } ) \\ , . \\end{align*}"}
+{"id": "8225.png", "formula": "\\begin{align*} \\Gamma _ { s } ^ { N , \\phi } ( Y ^ N ) : = L _ N \\big ( Y _ { s } ^ N ( \\phi ) ^ 2 \\big ) - 2 Y _ { s } ^ N ( \\phi ) L _ N Y _ { s } ^ N ( \\phi ) . \\end{align*}"}
+{"id": "8423.png", "formula": "\\begin{align*} \\left ( s , \\nu k \\right ) _ { k } : = \\frac { 1 } { \\Gamma ( s ) } \\ , \\intop _ { 0 } ^ { \\infty } x ^ { s - 1 } e ^ { - x } \\left ( \\frac { k \\nu - x } { k \\nu + x } \\right ) ^ { k } \\ , d x . \\end{align*}"}
+{"id": "6512.png", "formula": "\\begin{align*} C T _ { \\vec { t } } \\ , \\prod _ { i = 0 } ^ { c - 1 } ( 1 + t _ i ) ^ { i + a } \\left ( 1 + \\frac 1 { t _ i } \\right ) ^ b \\cdot \\prod _ { i \\neq j } ^ { 0 , c - 1 } \\left ( 1 - \\frac { t _ j } { t _ i } \\right ) = c ! \\cdot \\prod _ { i = 0 } ^ { a - 1 } \\prod _ { j = 0 } ^ { b - 1 } \\prod _ { k = 0 } ^ { c - 1 } \\frac { i + j + k + 2 } { i + j + k + 1 } . \\end{align*}"}
+{"id": "1755.png", "formula": "\\begin{align*} I _ { + } ^ { ( m ) } & = \\sum _ { \\mathbf { k } \\in \\mathcal { S } _ m } \\binom { m } { \\mathbf { k } } \\frac { w _ 1 ^ { k _ 1 } w _ 2 ^ { k _ 2 } w _ 3 ^ { k _ 3 } } { 2 ^ { k _ 1 + k _ 2 + k _ 3 - 1 } } \\mathcal { I } _ { + } ^ { ( m , \\mathbf { k } ) } , \\end{align*}"}
+{"id": "2557.png", "formula": "\\begin{align*} \\Delta u + f ( u ) = 0 \\ \\ \\ \\ \\ D . \\end{align*}"}
+{"id": "4783.png", "formula": "\\begin{align*} C ( X , t ) = o \\left ( \\frac { b } { \\sqrt { g } } \\right ) . \\end{align*}"}
+{"id": "7100.png", "formula": "\\begin{align*} \\partial _ { \\mathbf { n } } \\mu = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\partial \\Omega \\times ( 0 , T ) . \\end{align*}"}
+{"id": "4947.png", "formula": "\\begin{align*} \\left [ \\Sigma ^ { - 1 } \\right ] _ { i , j } ^ { } = \\begin{cases} \\phantom { - } 1 & j = i ; \\\\ - 1 & j = i + 1 ; \\\\ \\phantom { - } 0 & . \\end{cases} \\end{align*}"}
+{"id": "4276.png", "formula": "\\begin{align*} u ^ { \\ast } \\left ( t , x , \\mu \\right ) = - \\frac { 1 } { 2 } \\Pi _ { 1 } ^ { - 1 } \\left ( t \\right ) \\Pi _ { 3 } ^ { \\top } \\left ( t \\right ) \\left ( x - \\bar { \\mu } \\right ) - \\Pi _ { 2 } ^ { - 1 } \\left ( t \\right ) \\Pi _ { 4 } \\left ( t \\right ) \\bar { \\mu } - \\frac { 1 } { 2 } \\Pi _ { 2 } ^ { - 1 } \\left ( t \\right ) \\Pi _ { 5 } \\left ( t \\right ) . \\end{align*}"}
+{"id": "673.png", "formula": "\\begin{align*} \\limsup _ { k \\rightarrow \\infty } \\frac { \\log \\ ! \\norm { S ( k ) \\varphi } _ { \\sup } } { k } \\ ! \\leq \\ ! - l \\lambda _ 1 + \\lambda ( D ^ l \\varphi ) \\lambda ( D ^ l \\varphi ) \\ ! = \\ ! \\lim _ { k \\rightarrow \\infty } \\ ! \\frac { \\log \\ ! \\norm { Q ( k ) D ^ l \\varphi } } { k } . \\end{align*}"}
+{"id": "5726.png", "formula": "\\begin{align*} A ' = \\widetilde { X ^ { - 1 } } A X . \\end{align*}"}
+{"id": "1576.png", "formula": "\\begin{align*} \\mathrm { I } _ { \\mathrm { p } } ( 2 t ) = \\sum _ { \\mathrm { q } \\geq 0 , p + q \\geq 0 } \\frac { 1 } { \\mathrm { q } ! ( \\mathrm { p } + \\mathrm { q } ) ! } t ^ { 2 \\mathrm { q } + \\mathrm { p } } \\end{align*}"}
+{"id": "7204.png", "formula": "\\begin{align*} v _ h : B _ 1 \\to \\R ^ k , v _ h : = t _ h \\widetilde { u } _ h , \\end{align*}"}
+{"id": "2610.png", "formula": "\\begin{align*} \\Delta _ h v : = v ( h e _ n ) + v ( - h e _ n ) - 2 v ( 0 ) \\geq 2 h _ 0 . \\end{align*}"}
+{"id": "3306.png", "formula": "\\begin{align*} { \\rm I m } ( f ( \\xi ) ) = 0 { \\rm \\ f o r \\ } \\xi \\in L \\end{align*}"}
+{"id": "9247.png", "formula": "\\begin{align*} \\texttt { P } = \\sum _ { k = 0 } ^ n ( V _ k + \\alpha _ k ) ( W _ { n - k } + \\beta _ { n - k } ) ( 1 + \\eta _ 3 ) ( 1 + \\eta _ { n - k + 1 } ) \\end{align*}"}
+{"id": "6737.png", "formula": "\\begin{align*} X _ \\eta \\subseteq \\overline { [ \\eta ^ * , \\eta ] } \\subseteq \\overline { [ \\underline { \\varphi } , \\varphi ] } = X _ \\varphi . \\end{align*}"}
+{"id": "4811.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 , x _ 3 ) = ( \\cos ( \\theta ) , \\sin ( \\theta ) , \\dot \\theta ) . \\end{align*}"}
+{"id": "2291.png", "formula": "\\begin{align*} h = \\sum _ { n } c _ n a _ n . \\end{align*}"}
+{"id": "143.png", "formula": "\\begin{align*} \\chi _ j \\Pi _ \\lambda \\chi _ j u = \\chi _ j \\Pi _ { j , \\lambda } \\chi _ j u = \\chi _ j u _ j = 0 . \\end{align*}"}
+{"id": "436.png", "formula": "\\begin{align*} \\dfrac { 1 } { 2 } \\dfrac { d } { d t } \\| \\vec U \\| _ { P \\otimes P _ { \\Omega } } ^ 2 + \\sum _ { j = 1 , N } \\lbrack \\vec W ^ T \\Lambda \\vec W + 2 ( \\vec W ^ - ) ^ T \\Sigma ( \\sqrt { | \\Lambda ^ - | } \\vec W ^ - - R \\sqrt { \\Lambda ^ + } \\vec W ^ + - S \\vec G ) \\rbrack _ j d s _ j = 0 . \\end{align*}"}
+{"id": "2556.png", "formula": "\\begin{align*} \\mathbf { v } = 0 \\ \\ \\ \\ \\ , \\partial D . \\end{align*}"}
+{"id": "1627.png", "formula": "\\begin{align*} 0 & = g ( [ { f } , { f } ] ( { f } X , Y ) - 2 \\sum \\nolimits _ { j } d \\eta ^ j ( { f } X , Y ) \\ , \\xi _ j , \\ \\xi _ i ) \\\\ & = g ( [ { f } ^ 2 X , { f } Y ] , \\xi _ i ) - ( { f } X ) ( \\eta ^ i ( Y ) ) + \\eta ^ i ( [ { f } X , Y ] ) , 1 \\le i \\le p . \\end{align*}"}
+{"id": "8106.png", "formula": "\\begin{align*} \\Delta \\check \\S ^ { 3 1 2 4 } = 1 \\otimes \\check \\S ^ { 3 1 2 4 } + \\check \\S ^ { 1 } \\otimes \\check \\S ^ { 1 2 3 } + \\check \\S ^ { 1 } \\otimes \\check \\S ^ { 2 1 3 } + \\check \\S ^ { 1 2 } \\otimes \\check \\S ^ { 1 2 } + \\check \\S ^ { 2 1 } \\otimes \\check \\S ^ { 1 2 } + \\check \\S ^ { 3 1 2 } \\otimes \\check \\S ^ { 1 } + \\check \\S ^ { 3 1 2 4 } \\otimes 1 , \\end{align*}"}
+{"id": "6686.png", "formula": "\\begin{align*} v _ 3 = 0 \\Gamma _ 0 , \\end{align*}"}
+{"id": "951.png", "formula": "\\begin{align*} M _ q ( \\Gamma ) = \\inf \\limits _ { \\rho \\in \\ , { \\rm a d m } \\ , \\Gamma } \\int \\limits _ D \\rho ^ { \\ , q } ( x ) \\ , d m ( x ) \\ , . \\end{align*}"}
+{"id": "5589.png", "formula": "\\begin{align*} \\mathcal { B } _ { W } ( X _ { 0 } ) : = \\{ A \\in \\mathcal { B } ( X _ { 0 } ) : g . A = A \\mbox { f o r a l l } g \\in W \\} . \\end{align*}"}
+{"id": "4882.png", "formula": "\\begin{align*} \\alpha _ r : = \\frac 1 C \\ , \\inf _ { B _ { r / 2 } ( x _ r ) } u \\qquad { \\mbox { a n d } } \\psi ( x ) : = \\alpha _ r \\ , \\varphi \\left ( \\frac { x - x _ r } { r } \\right ) . \\end{align*}"}
+{"id": "120.png", "formula": "\\begin{align*} \\Omega _ 0 = \\Omega \\cap \\{ - 1 \\leq \\Re \\lambda \\leq 1 \\} , \\Omega _ { \\pm k } = \\Omega \\cap \\{ k \\leq \\pm \\Re \\lambda \\leq k + 1 \\} , k > 0 . \\end{align*}"}
+{"id": "3720.png", "formula": "\\begin{align*} \\phi _ { \\sigma } : \\sigma _ U ^ * M ^ { \\otimes r } \\stackrel { \\sigma ^ * _ U \\phi } { \\longrightarrow } f _ \\sigma ^ * \\sigma ^ * L \\stackrel { f _ { \\sigma } ^ * ( \\widehat { \\sigma _ L } ) } { \\longrightarrow } f _ { \\sigma } ^ * L \\end{align*}"}
+{"id": "4562.png", "formula": "\\begin{align*} \\mathcal S _ { \\ell , A } = \\sum _ { p } \\sum _ \\nu ( p , \\nu ) ^ { \\eta ( p , \\nu ) } , \\end{align*}"}
+{"id": "462.png", "formula": "\\begin{align*} \\bar y _ \\ast & : = \\inf _ { y _ \\ast \\in Y } y _ \\ast , \\end{align*}"}
+{"id": "1565.png", "formula": "\\begin{align*} f \\left ( \\frac { a } { c } - \\frac { t } { i c } \\right ) = \\sum _ { n \\ge 1 } f _ { n } n ^ { \\frac { \\l - 1 } { 2 } } e ^ { - 2 \\pi n \\frac { t } { c } } e \\left ( n \\frac { a } { c } \\right ) . \\end{align*}"}
+{"id": "8247.png", "formula": "\\begin{align*} \\begin{pmatrix} H _ { k , \\{ X ; Y _ { l , 1 } \\} } \\\\ \\vdots \\\\ \\vdots \\\\ H _ { k , \\{ X ; Y _ { l , l } \\} } \\\\ \\end{pmatrix} = B ^ { ( l ) } \\begin{pmatrix} \\vdots \\\\ \\sum _ { h = 1 } ^ { j - 1 } ( - 1 ) ^ h T _ { h } H _ { k , \\{ X ; Y _ { l - h , j - h } \\} } \\\\ \\vdots \\\\ T _ { l } H _ { k , X } \\\\ \\end{pmatrix} _ { j = 2 , . . , l } . \\end{align*}"}
+{"id": "294.png", "formula": "\\begin{align*} B ( n , y ) = \\frac { - n ^ 2 y ^ { n + 3 } + ( 2 n ^ 2 + 2 n - 1 ) y ^ { n + 2 } - ( n ^ 2 + 2 n + 1 ) y ^ { n + 1 } + y ^ 2 + y } { ( 1 - y ) ^ 3 } , \\end{align*}"}
+{"id": "981.png", "formula": "\\begin{align*} \\left ( { \\nabla } _ { X } \\omega \\right ) Y + h ( X , \\phi Y ) = C h ( X , Y ) \\end{align*}"}
+{"id": "4094.png", "formula": "\\begin{align*} \\vec { e _ i } ^ T ( T _ \\lambda ^ { \\vec { v } } ) ^ T \\vec { v } = \\lambda v _ i . \\end{align*}"}
+{"id": "7707.png", "formula": "\\begin{align*} \\Lambda _ { j + 1 } ^ + = \\{ \\lambda \\in L _ { j + 1 } ' \\mid q ( \\lambda ) > 0 \\} \\end{align*}"}
+{"id": "3607.png", "formula": "\\begin{align*} r ( p ) \\ = \\ 2 ^ { \\delta } 3 ^ { \\gamma } 5 ^ { \\beta } \\ \\geq \\ 2 ^ { \\delta + \\gamma } \\cdot 5 \\ \\geq \\ 2 ^ { 1 0 ^ m - 1 1 } \\cdot 5 \\ \\geq \\ 1 0 ^ { m + 1 } \\end{align*}"}
+{"id": "3955.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta _ 1 , \\delta _ 2 ) - \\mathcal { I } _ { \\mathrm { D } } ( 0 , 0 ) = \\sup _ { \\widetilde { \\gamma } \\in \\Sigma _ { \\mathrm { D } } ( \\delta ) } \\int _ { \\mathcal { V } } g \\ , d \\widetilde { \\gamma } - \\mathcal { I } _ { \\mathrm { D } } ( 0 , 0 ) \\leq \\Psi ( \\delta _ 1 + \\epsilon , \\delta _ 2 + \\epsilon ) . \\end{align*}"}
+{"id": "5062.png", "formula": "\\begin{align*} \\mathring { v } ( x , t ) = \\psi \\left ( x , t \\right ) - \\phi \\left ( x + \\gamma _ { \\mathrm { n l } } ( x , t ) \\right ) . \\end{align*}"}
+{"id": "6158.png", "formula": "\\begin{align*} \\{ x \\in R _ n : x & = y _ l \\ast _ { l 1 } ( \\top _ 1 + _ 1 a _ 1 ) \\ast _ { 1 1 } a _ 1 \\ast _ { 1 r } z _ r , \\\\ & \\mbox { w i t h } a _ 1 \\in R _ 1 , \\ , y _ l \\in R _ l , \\ , z _ r \\in R _ r , \\ , l + r = n - 2 \\} . \\end{align*}"}
+{"id": "3560.png", "formula": "\\begin{align*} J _ { T _ { * } } = - 2 \\pi \\int _ { 0 } ^ { \\infty } \\left [ \\exp \\left ( - \\frac { 1 } { T _ { * } } \\left [ \\left ( \\frac { \\sigma } { r } \\right ) ^ { n } - \\left ( \\frac { \\sigma } { r } \\right ) ^ { m } \\right ] \\right ) \\right ] \\ , r ^ { 2 } d r , \\end{align*}"}
+{"id": "8040.png", "formula": "\\begin{align*} f ( t ) = t h ^ 2 ( t ^ { 1 / 2 } ) , g ( t ) = t h ^ { - 2 } ( t ^ { 1 / 2 } ) , \\end{align*}"}
+{"id": "6599.png", "formula": "\\begin{align*} \\mu = \\frac { \\sqrt { \\pi } L E ( \\hat \\beta _ l ) } { 2 } , \\ \\ \\sigma ^ 2 = \\frac { L [ 8 - \\pi E ^ 2 ( \\hat \\beta _ l ) ] } { 4 } . \\end{align*}"}
+{"id": "3523.png", "formula": "\\begin{align*} \\begin{cases} \\Phi ' ( t ) = A ( t ) \\Phi ( t ) , \\\\ \\Phi ( t _ 0 ) = I _ { 2 n \\times 2 n } ; \\end{cases} \\end{align*}"}
+{"id": "318.png", "formula": "\\begin{align*} \\zeta ( 3 , 1 ) = \\frac { 3 } { 2 } \\zeta ( 4 ) - \\frac { 1 } { 2 } \\zeta ( 2 ) , \\end{align*}"}
+{"id": "1463.png", "formula": "\\begin{align*} \\mu _ m ^ i \\int _ { \\mathcal { A } _ m ^ i } f _ 0 ^ { ' } \\Big ( U _ { \\mu ^ i _ m , \\xi ^ i _ m } ( x ) \\Big ) \\psi _ { \\mu ^ i _ m , \\xi ^ i _ m } ^ h ( x ) u _ m ( x ) d x = o ( 1 ) . \\end{align*}"}
+{"id": "5495.png", "formula": "\\begin{align*} R _ { I } & : = \\left \\{ \\exp ( s ) : s \\in \\mathfrak { a } , \\alpha _ { 1 } ( s ) \\le 0 , \\alpha _ { 2 } ( s ) < 0 \\mbox { f o r a l l } \\alpha _ { 1 } \\in \\Delta \\mbox { a n d } \\alpha _ { 2 } \\in \\Delta \\setminus I \\right \\} , \\\\ D _ { I } & : = R _ { I } \\cap A _ { I } = \\left \\{ \\exp ( s ) : s \\in \\mathfrak { a } _ { I } , \\alpha ( s ) < 0 \\mbox { f o r a l l } \\alpha \\in \\Delta \\setminus I \\right \\} . \\end{align*}"}
+{"id": "163.png", "formula": "\\begin{align*} L ( \\alpha ) - L ( \\alpha ^ 2 ) = \\frac { 1 } { 7 } , \\end{align*}"}
+{"id": "4932.png", "formula": "\\begin{align*} \\lambda _ j = \\frac { j } { n } , 0 \\le j \\le n . \\end{align*}"}
+{"id": "7177.png", "formula": "\\begin{align*} r _ x : = \\max \\left \\{ \\frac { \\lambda _ x } { 2 ^ k } : k \\in \\N , \\ , \\ , \\mathcal { H } ^ 1 \\left ( J \\cap B _ { \\lambda _ x / 2 ^ k } ( x ) \\right ) \\geq \\eta \\frac { \\lambda _ x } { 2 ^ k } \\right \\} . \\end{align*}"}
+{"id": "6566.png", "formula": "\\begin{align*} 2 \\lambda _ j ^ { \\downarrow } ( | X | ) \\le \\lambda _ j ^ { \\downarrow } \\left ( \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} \\right ) \\end{align*}"}
+{"id": "1334.png", "formula": "\\begin{align*} \\theta _ i ^ 2 \\frac { Z _ i ^ * ( u ) } { e _ i ^ * ( u ) } + \\frac { T _ i ^ { \\infty } } { e _ i ^ * ( u ) ^ 2 } & = \\theta _ i ^ 2 \\frac { T _ i ^ { \\infty } - T _ i ( u ) } { e ^ { 2 \\rho _ i ( u ) } } \\\\ & = \\theta _ i ^ 2 \\int _ 0 ^ { + \\infty } e ^ { 2 \\rho _ i ( u + v ) - 2 \\rho _ i ( u ) } d v . \\end{align*}"}
+{"id": "3961.png", "formula": "\\begin{align*} ( Y _ 1 , Y _ 2 , X ) = \\gamma ^ { \\eta _ 0 , \\delta } , ( \\widetilde { Y } _ 1 , \\widetilde { X } ) = \\mu _ 1 , ( \\tilde { Y } _ 2 , \\widetilde { X } ) = \\mu _ 2 , \\end{align*}"}
+{"id": "1467.png", "formula": "\\begin{align*} H _ 1 = \\Big | f _ \\epsilon ( V + \\phi ) - f _ \\epsilon ( V ) - f ^ { ' } _ \\epsilon ( V ) \\phi ) \\Big | _ { \\frac { 2 n } { n + 2 } } = \\Big | f _ \\epsilon ( V + t \\phi ) - f ^ { ' } _ \\epsilon ( V ) \\phi ) \\Big | _ { \\frac { 2 n } { n + 2 } } . \\end{align*}"}
+{"id": "382.png", "formula": "\\begin{align*} B _ i - A _ i ^ T = 0 , C + C ^ T = 0 , i = 1 , 2 , . . , k \\end{align*}"}
+{"id": "6436.png", "formula": "\\begin{align*} \\imath ( a _ { - l } \\ldots a _ { r } ) : = a _ 0 . \\end{align*}"}
+{"id": "4234.png", "formula": "\\begin{align*} \\phi ( e ^ { i \\theta } ) = \\lambda I - \\frac { 1 } { \\Lambda ( \\theta ) } M ( \\theta ) , \\end{align*}"}
+{"id": "4400.png", "formula": "\\begin{align*} & \\alpha - 2 A _ 1 > 0 , \\ \\ \\beta - 2 A _ 2 > 0 , \\ \\ \\gamma - 2 A _ 3 > 0 , \\\\ & 2 \\omega - \\frac { | \\mathbf { c } | ^ 2 } { 8 A _ 1 } > 0 , \\ \\ \\omega - \\frac { | \\mathbf { c } | ^ 2 } { 8 A _ 2 } > 0 , \\ \\ \\omega - \\frac { | \\mathbf { c } | ^ 2 } { 8 A _ 3 } > 0 . \\end{align*}"}
+{"id": "8634.png", "formula": "\\begin{align*} \\bar { M } _ { + , \\Delta } ( t ) : = \\nu \\Delta \\sum _ { \\ell = 0 } ^ { \\lfloor t / \\Delta \\rfloor } Z _ 0 ( \\ell \\Delta ) . \\end{align*}"}
+{"id": "3728.png", "formula": "\\begin{gather*} c a p \\times c u p = ( a _ { 1 } b _ { 1 } + \\dots + a _ { n } b _ { n } ) , c u p \\times c a p ( a _ { 1 } b _ { 1 } , \\dots , a _ { n } b _ { n } ) . \\end{gather*}"}
+{"id": "5378.png", "formula": "\\begin{align*} y _ { 1 , k } ^ 1 = u _ { 2 , k } ^ 1 , u _ { 1 , k } ^ 1 = y _ { 2 , k } ^ 1 \\end{align*}"}
+{"id": "3507.png", "formula": "\\begin{align*} X ( t ) = x ^ i ( t ) E _ i ( t ) \\longmapsto ( x ^ 1 ( t ) , \\ldots , x ^ n ( t ) ) , \\end{align*}"}
+{"id": "3977.png", "formula": "\\begin{align*} ( f _ { \\mathcal { Y } } ) _ { \\lambda } ( y _ 1 , y _ 2 ) = y _ 2 - y _ 1 + \\frac { V _ { 1 , Y Y } } { 4 \\lambda _ 1 } + \\frac { V _ { 2 , Y Y } } { 4 \\lambda _ 2 } . \\end{align*}"}
+{"id": "8718.png", "formula": "\\begin{align*} E ( | \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau _ { 1 } | ^ { 3 } ) & \\le \\surd { E \\{ ( \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau _ { 1 } ) ^ { 2 } \\} } \\times \\surd { E \\{ ( \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau _ { 1 } ) ^ { 4 } \\} } \\\\ & = \\surd { O ( p ) } \\times \\surd { O ( p ^ { 2 } ) } = O ( p ^ { \\frac { 3 } { 2 } } ) . \\end{align*}"}
+{"id": "726.png", "formula": "\\begin{align*} 0 = e _ { B , 1 } < e _ { B , 2 } < \\dots < e _ { B , N } < L _ { B } \\end{align*}"}
+{"id": "1084.png", "formula": "\\begin{align*} t ^ { \\sigma } \\| \\Phi ( u ) ( t ) - \\Phi ( v ) ( t ) \\| _ { L ^ a } \\leq C M ^ { m - 1 } \\ , \\sup _ { \\tau > 0 } ( \\tau ^ { \\sigma } \\| ( u - v ) \\| _ { L ^ a } ) = C M ^ { m - 1 } \\ , \\rho ( u , v ) . \\end{align*}"}
+{"id": "2164.png", "formula": "\\begin{align*} w \\left ( x , t \\right ) = v _ { t } \\left ( x , t \\right ) , r \\left ( x , t \\right ) = q _ { t } \\left ( x , t \\right ) . \\end{align*}"}
+{"id": "9315.png", "formula": "\\begin{align*} c _ { 1 } ^ { d } v _ 1 + c _ { 2 } ^ { d } v _ 2 + \\ldots + c _ { n } ^ { d } v _ n = 0 . \\end{align*}"}
+{"id": "1301.png", "formula": "\\begin{align*} X _ i ( T _ i ( u ) ) = e ^ { \\rho _ i ( u ) } \\end{align*}"}
+{"id": "6332.png", "formula": "\\begin{align*} F _ { n , m } : = \\{ s \\in \\R \\ , : \\ , \\phi + \\omega s \\in E _ { n , m } \\} . \\end{align*}"}
+{"id": "790.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ n } u \\mathrm { d } A = - \\int _ { \\mathbb { S } ^ n } \\dfrac { n } { 2 } u ^ 2 \\mathrm { d } A + O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 , \\end{align*}"}
+{"id": "197.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b ) = 1 \\\\ a , b \\geq 1 } } \\left ( 1 - \\frac { z ^ b } { 2 ^ a } \\right ) ^ { \\frac { b ^ { 2 } } { a ^ 3 } } = \\exp \\left \\{ \\left ( \\frac { 7 } { 8 } \\zeta ( 3 ) - \\frac { 1 } { 1 2 } \\pi ^ 2 \\log 2 + \\frac { 1 } { 6 } ( \\log 2 ) ^ 3 \\right ) \\left ( \\frac { z ( 1 + z ) } { ( 1 - z ) ^ 3 } \\right ) \\right \\} , \\end{align*}"}
+{"id": "2933.png", "formula": "\\begin{align*} ( \\gamma \\ast f ) ( \\mathbf { g } ) = \\gamma \\ast ( f ( \\gamma ^ { - 1 } \\mathbf { g } ) ) . \\end{align*}"}
+{"id": "2697.png", "formula": "\\begin{align*} ( a _ 1 ' ) ^ * \\iota _ s ^ * \\mathcal { I C } _ M = \\iota _ v ^ * a _ 1 ^ * \\mathcal { I C } _ M = \\iota _ v ^ * \\mathcal { I C } _ U = \\mathcal { I C } _ { U _ v } . \\end{align*}"}
+{"id": "1465.png", "formula": "\\begin{align*} & \\int _ { \\Omega \\setminus B ( \\xi _ m , \\rho ) } \\bigg | f _ 0 ^ { ' } \\Big ( U _ { \\mu ^ i _ m , \\xi ^ i _ m } ( x ) \\Big ) \\psi _ { \\mu ^ i _ m , \\xi ^ i _ m } ^ h ( x ) u _ m ( x ) \\bigg | d x \\\\ \\leq & C | \\psi _ { \\mu ^ i _ m , \\xi ^ i _ m } ^ h | _ { \\frac { 2 n } { n - 2 } } | u _ m | _ { \\frac { 2 n } { n - 2 } } \\Big ( \\int _ { \\Omega \\setminus B ( \\xi _ m , \\rho ) } U _ { \\mu ^ i _ m , \\xi ^ i _ m } ^ { \\frac { 2 n } { n - 2 } } d x \\Big ) ^ { \\frac { 2 } { n } } = O ( \\mu ^ i _ m ) . \\end{align*}"}
+{"id": "3236.png", "formula": "\\begin{align*} \\| f \\| _ { \\Lambda ^ \\beta } : = & \\sup _ { x \\ne y } \\frac { | f ( x ) - f ( y ) | } { \\| x - y \\| ^ \\beta } < \\infty . \\end{align*}"}
+{"id": "3744.png", "formula": "\\begin{align*} { \\rm R e } \\ \\tilde { W } _ { \\vec { k } , \\vec { l } } ( \\vec { x } , \\vec { y } ) - { \\rm R e } \\ \\tilde { W } _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) & = 0 , \\\\ { \\rm I m } \\ \\tilde { W } _ { \\vec { k } , \\vec { l } } ( \\vec { x } , \\vec { y } ) + { \\rm I m } \\ \\tilde { W } _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) & = 0 , \\end{align*}"}
+{"id": "8181.png", "formula": "\\begin{align*} \\rho ( t ) = ( \\log t ) ^ { 2 / 3 } , t > 1 . \\end{align*}"}
+{"id": "4957.png", "formula": "\\begin{align*} F ( k , t + 1 , n ) = F ( k , t , n ) - \\frac { n - k } { n } f ( k , t , n ) , \\end{align*}"}
+{"id": "7404.png", "formula": "\\begin{align*} \\sigma _ { \\beta _ N } ^ 2 = \\frac { \\hat { \\beta } ^ 2 } { R _ N } \\sim \\frac { \\hat { \\beta } ^ 2 \\ , \\pi } { \\log N } \\ , , \\hat \\beta \\in ( 0 , \\infty ) \\ , . \\end{align*}"}
+{"id": "7910.png", "formula": "\\begin{align*} | \\mathcal { Q } _ k | = ( 2 k + 1 ) ! ! . \\end{align*}"}
+{"id": "1209.png", "formula": "\\begin{align*} \\partial _ t G ( t , s ) & = \\sum _ { k = 1 } ^ \\infty k s ^ { k - 1 } \\left ( q R _ { k + 1 } ( t ) + \\varphi ( t ) R _ k ( t ) \\right ) \\\\ & = \\partial _ s \\left ( \\sum _ { k = 1 } ^ \\infty s ^ k \\left ( q R _ { k + 1 } ( t ) + \\varphi ( t ) R _ k ( t ) \\right ) \\right ) \\\\ & = \\partial _ s \\left ( q G ( t , s ) + \\varphi ( t ) s G ( t , s ) \\right ) . \\end{align*}"}
+{"id": "4366.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\left [ A x - \\overline { b } \\right ] _ { + } = h , \\\\ A ^ { \\prime } h = 0 _ { n } . \\end{array} \\right . \\end{align*}"}
+{"id": "4569.png", "formula": "\\begin{align*} H ( 0 ) = ( 1 , 2 , 2 , 2 , 1 ) , \\ , H ( 1 ) = ( 0 , 1 , 1 , 0 ) . \\end{align*}"}
+{"id": "8228.png", "formula": "\\begin{align*} \\ddot { \\varrho } _ t ( x ) = \\frac { \\kappa } { 2 } \\partial _ { x x } \\dot { \\varrho } _ t ( x ) - \\lambda \\partial _ x \\dot { \\Delta } _ t ( x ) = \\frac { \\kappa } { 2 } \\partial _ { x x } \\dot { \\varrho } _ t ( x ) - \\lambda \\partial _ x \\left ( \\frac { \\kappa } { 2 } \\partial _ { x x } \\Delta _ t ( x ) - \\lambda \\partial _ x \\varrho _ t ( x ) - 2 \\gamma \\Delta _ t ( x ) \\right ) . \\end{align*}"}
+{"id": "1725.png", "formula": "\\begin{align*} \\begin{cases} \\mathrm { d } \\nu _ t ( x ) = \\alpha \\left ( \\Psi ( \\nu _ t , \\mu _ t ) ( x ) - \\nu _ t ( x ) \\right ) \\mathrm { d } t , \\\\ \\mathrm { d } \\mu _ t ( y ) = \\alpha \\left ( \\Phi ( \\nu _ t , \\mu _ t ) ( y ) - \\mu _ t ( y ) \\right ) \\mathrm { d } t , t \\geq 0 , \\end{cases} \\end{align*}"}
+{"id": "1246.png", "formula": "\\begin{align*} \\psi ( x ) = F ( x ) + \\lambda x ^ { \\frac { 1 } { \\beta } - \\alpha - 1 } . \\end{align*}"}
+{"id": "5091.png", "formula": "\\begin{align*} \\Phi _ \\xi = \\phi ' + \\mathcal { O } ( | \\xi | ) . \\end{align*}"}
+{"id": "9132.png", "formula": "\\begin{align*} \\psi ( \\ldots , \\zeta _ { [ - 2 ] } , \\zeta _ { [ - 1 ] } , x , \\delta ^ { A } ( \\varphi ) , \\delta ^ { A + 1 } ( \\varphi ) , \\ldots ) = 0 \\ , . \\end{align*}"}
+{"id": "3245.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot { \\mathcal F } ^ { \\alpha , q } _ { p , { \\rm D } } } & = \\bigg \\{ \\sum _ { k \\in \\Bbb Z } \\Big ( { \\mathfrak R } ^ { k \\alpha } \\Big \\| q _ { k } \\Big ( \\sum _ { j \\in \\Bbb Z } \\sum _ { Q \\in Q ^ j } \\omega ( Q ) { \\widetilde D } _ j ( x , x _ { Q } ) D _ j ( f ) ( x _ { Q } ) \\Big ) \\Big \\| _ p \\Big ) ^ q \\bigg \\} ^ { 1 / q } . \\end{align*}"}
+{"id": "4765.png", "formula": "\\begin{align*} \\| \\tilde { S } - S \\| = \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "8809.png", "formula": "\\begin{align*} & ( 2 + 2 u ) q _ { i , j } = q _ { i + 1 , j } + q _ { i , j + 1 } + u q _ { i - 1 , j } + u q _ { i , j - 1 } \\end{align*}"}
+{"id": "7185.png", "formula": "\\begin{align*} w ^ h _ l : = \\begin{cases} \\phi ^ i ( w _ { l - 1 } ) & \\mbox { o n } B ^ i _ l , \\ , i \\leq h , \\\\ w _ { l - 1 } & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
+{"id": "8192.png", "formula": "\\begin{align*} F _ { j , t } = \\{ M _ { j h ( t ) } \\leq k j h ( t ) \\} , \\ : \\ : F _ t = F _ { 1 , t } \\cap \\ldots \\cap F _ { t / h ( t ) , t } . \\end{align*}"}
+{"id": "237.png", "formula": "\\begin{align*} = \\frac { n ^ 3 z ^ { n + 4 } + ( 3 n ^ 3 + 6 n ^ 2 - 4 ) z ^ { n + 2 } - ( 3 n ^ 3 + 3 n ^ 2 - 3 n + 1 ) z ^ { n + 3 } - ( n + 1 ) ^ 3 z ^ { n + 1 } + z ^ 3 + 4 z ^ 2 + z } { ( 1 - z ) ^ 4 } , \\end{align*}"}
+{"id": "3137.png", "formula": "\\begin{align*} R _ 0 ( u , x ) & = 1 , \\\\ R _ 1 ( u , x ) & = x + ( 1 - x ) u , \\\\ R _ 2 ( u , x ) & = x ^ 2 + ( 1 + 3 x - 2 x ^ 2 ) u + ( 2 - 3 x + x ^ 2 ) u ^ 2 . \\end{align*}"}
+{"id": "4473.png", "formula": "\\begin{align*} L ( \\Phi ) + \\omega Q ( \\Phi ) \\ge C ' ( 1 + \\omega ) \\| \\Phi \\| _ { \\mathcal { H } ^ 1 } ^ 2 \\ge \\frac { 4 C ' C _ { \\omega } ^ 2 \\omega ^ 2 ( 1 + \\omega ) } { C ^ 2 } = : \\widetilde { B } _ { \\omega } , \\end{align*}"}
+{"id": "1240.png", "formula": "\\begin{align*} X _ { \\gamma , 1 } ( x ) & = 2 ^ { \\alpha } g _ { \\alpha , 1 } ( x ) ^ { 1 + \\alpha } ( \\log 2 + \\log g _ { \\alpha , 1 } ( x ) ) \\forall x \\in ( 0 , 1 ] \\end{align*}"}
+{"id": "5034.png", "formula": "\\begin{align*} & f _ 0 = x _ 1 ^ { a _ 1 } f _ 0 = x _ 1 ^ { a _ 1 } x _ 2 + \\dots + x _ K ^ { a _ K } x _ 1 , \\\\ & f _ 1 = x _ 1 x _ { K + 1 } ^ { b _ 1 } + x _ { K + 1 } x _ { K + 2 } ^ { b _ 2 } + \\dots + x _ { K + L - 1 } x _ { K + L } ^ { b _ L } \\end{align*}"}
+{"id": "3233.png", "formula": "\\begin{align*} d ( \\beta ) & = \\sum _ { i = 1 } ^ n ( a _ i ^ 2 + a _ i b _ i + b _ i ^ 2 ) T _ i ^ 2 , \\\\ d ( \\gamma ) & = \\sum _ { i = 1 } ^ n ( a _ i ^ 2 b _ i + a _ i b _ i ^ 2 ) T _ i ^ 3 , \\end{align*}"}
+{"id": "3931.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( 0 ) = \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } \\int g _ \\lambda \\ , d \\pi = \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\int g _ \\lambda \\ , d \\pi = \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\int g \\ , d \\pi . \\end{align*}"}
+{"id": "9197.png", "formula": "\\begin{align*} W ( D _ { n , r , s } ) = 2 \\binom { n } { 2 } + ( r - 1 ) ^ 2 n - 4 \\binom { r } { 3 } . \\end{align*}"}
+{"id": "3426.png", "formula": "\\begin{align*} \\begin{aligned} - \\Delta w _ t + \\left [ \\frac { f _ 1 ( u _ t ) - f _ 1 ( u ) } { u _ t - u } - \\lambda \\right ] w _ t = \\frac { f _ 2 ( u _ t ) } { u _ t } u _ t - \\frac { f _ 2 ( u ) } { u } u \\geq \\frac { f _ 2 ( u ) } { u } w _ t \\geq 0 , \\end{aligned} \\end{align*}"}
+{"id": "2994.png", "formula": "\\begin{align*} \\frac { \\partial V _ m } { \\partial \\bar { c } _ j } = ( - 1 ) ^ { j - 1 } c _ { m - j } \\ . \\end{align*}"}
+{"id": "2867.png", "formula": "\\begin{align*} ( R \\otimes 1 _ L ) ( \\alpha \\otimes \\varphi _ L ) \\Delta ( l ) & = ( R \\otimes 1 _ L ) ( \\alpha \\otimes \\varphi _ L ) ( l _ 1 \\otimes l _ 2 ) \\\\ & = ( R \\otimes 1 _ L ) ( \\alpha ( l _ 1 ) \\otimes \\varphi _ L ( l _ 2 ) ) \\\\ & = R ( \\alpha ( l _ 1 ) ) \\otimes \\varphi _ L ( l _ 2 ) \\\\ & \\overset { \\eqref { e q 3 . 3 } } { = } \\alpha ( R ( l _ 1 ) ) \\otimes \\varphi _ L ( l _ 2 ) \\\\ & = ( \\alpha \\otimes \\varphi _ L ) ( R \\otimes 1 _ L ) \\Delta ( l ) . \\end{align*}"}
+{"id": "1906.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( u ^ { k + 1 } , \\zeta ^ { k + 1 } ) & = \\arg \\min \\limits _ { ( u , \\zeta ) \\in \\mathcal { U } \\times \\mathcal { X } } L _ { \\beta } ( u , \\zeta ; \\lambda ^ k ) , \\\\ \\lambda ^ { k + 1 } & = \\lambda ^ k + \\beta ( S u ^ { k + 1 } - \\zeta ^ { k + 1 } ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7077.png", "formula": "\\begin{align*} \\nu ( g ' ) - \\nu _ { j _ 0 } ( g ' ) = \\beta _ { j _ 0 } - \\tilde { \\beta } _ { j _ 0 } < \\epsilon . \\end{align*}"}
+{"id": "5229.png", "formula": "\\begin{align*} T = p _ 6 ( x ) \\partial ^ 6 + \\cdots + p _ 1 ( x ) \\partial + p _ 0 , \\end{align*}"}
+{"id": "5256.png", "formula": "\\begin{align*} \\| T _ { \\rho } f \\| _ q ^ q \\le \\sum _ { S \\subseteq [ n ] } \\gamma ^ { q - 2 } \\mathbb { E } _ { x \\sim \\Omega ^ S } \\| D _ { S , x } T _ { \\frac { 1 } { \\sqrt { 2 } } } f \\| _ 2 ^ 2 = \\gamma ^ { q - 2 } \\sum _ { S \\subseteq [ n ] } \\| L _ S T _ { 1 / \\sqrt { 2 } } f \\| _ 2 ^ 2 = \\gamma ^ { q - 2 } \\| f \\| _ 2 ^ 2 , \\end{align*}"}
+{"id": "4926.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { F } ( k , t + 1 , \\mathbf { p } _ { n + 1 } ) = \\overline { F } ( k , t , \\mathbf { p } _ { n + 1 } ) + p _ { k } ^ { } f ( k , t , \\mathbf { p } _ { n + 1 } ) , \\end{aligned} \\end{align*}"}
+{"id": "4106.png", "formula": "\\begin{align*} \\left ( T _ \\lambda ^ { \\vec { v } } \\right ) _ { 1 , j } = x _ j , \\left ( T _ \\lambda ^ { \\vec { v } } \\right ) _ { 2 , j } = x _ { j - 1 } + x _ j \\alpha _ { n - 1 } \\end{align*}"}
+{"id": "8607.png", "formula": "\\begin{align*} S _ { j , + } ^ k ( t ) = \\sum _ { \\ell = 0 } ^ { \\lfloor t / \\Delta \\rfloor } W ^ k _ { \\ell \\Delta , t } ( j ) . \\end{align*}"}
+{"id": "9044.png", "formula": "\\begin{align*} c _ { i , j } = \\begin{cases} 2 & i = j , \\\\ - | b _ { i , j } | & . \\end{cases} \\end{align*}"}
+{"id": "5549.png", "formula": "\\begin{align*} \\max _ { A \\in \\mathcal { P } _ { x , n } } \\left | 1 - \\frac { \\alpha _ { x , g } ( A ) } { q _ { x , g } ^ { n } ( A ) } \\right | \\quad \\textrm { a n d } \\quad \\lim _ { n \\to \\infty } \\varepsilon _ { x } ( g , n ) = 0 . \\end{align*}"}
+{"id": "750.png", "formula": "\\begin{align*} \\alpha ( \\Omega ) = \\inf \\left \\lbrace \\mathrm { V o l } \\left ( \\Omega \\Delta \\overline { B } _ { \\rho } ( x ) \\right ) : x \\in \\mathbb { N } ^ { n + 1 } ( K ) , \\mathrm { V o l } ( \\Omega ) = \\mathrm { V o l } ( \\overline { B } _ { \\rho } ( x ) ) \\right \\rbrace , \\end{align*}"}
+{"id": "1017.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 3 k } { k } \\binom { 6 k } { 3 k } } { x ^ k } ( 6 H _ { 6 k } - 3 H _ { 3 k } - 2 H _ { 2 k } + H _ k ) = \\bigg ( \\log \\frac { x } { x - 4 3 2 } \\bigg ) \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 3 k } { k } \\binom { 6 k } { 3 k } } { x ^ k } , \\end{align*}"}
+{"id": "1190.png", "formula": "\\begin{align*} E = \\{ \\{ i + \\gamma , j + ( \\gamma + z ( e ) ) \\} : ( i , j ; z ( e ) ) \\in \\hat { E } ( \\hat { G } ) , \\gamma \\in \\mathbb { Z } \\} . \\end{align*}"}
+{"id": "4983.png", "formula": "\\begin{align*} \\widehat { T } _ k = G _ { \\frac { 1 } { n } } + G _ { \\frac { 2 } { n } } + \\cdots + G _ { \\frac { k - 1 } { n } } , \\end{align*}"}
+{"id": "8950.png", "formula": "\\begin{align*} \\phi = \\phi _ { e } + \\sum _ { m = q _ x } ^ { \\infty } C _ { m , 0 } \\Delta x ^ { m } + \\sum _ { m = q _ t } ^ { \\infty } C _ { 0 , m } \\Delta t ^ { m } + \\sum _ { m = q _ x + q _ t } ^ { \\infty } \\sum _ { n = q _ t } ^ { m - q _ x } C _ { m - n , n } \\Delta x ^ { m - n } \\Delta t ^ { n } . \\end{align*}"}
+{"id": "3974.png", "formula": "\\begin{align*} c _ { Y _ { \\ell } } \\left ( y _ { \\ell } , y _ { \\ell } ^ { \\prime } \\right ) = \\inf _ { x _ \\ell , x _ \\ell ^ \\prime \\in \\mathcal { X } } c _ \\ell \\left ( ( y _ \\ell , x _ \\ell ) , ( y ^ \\prime _ \\ell , x ^ \\prime _ \\ell ) \\right ) \\leq c _ \\ell \\left ( ( y _ \\ell , x _ \\ell ) , ( y ^ \\prime _ \\ell , x ^ \\prime _ \\ell ) \\right ) . \\end{align*}"}
+{"id": "1571.png", "formula": "\\begin{align*} \\mathrm { c } ^ { D } _ n ( x ) = \\mathrm { c } ^ { D , \\mathrm { S A W } } _ n ( x ) & : = \\sum _ { w \\in \\mathcal { W } _ n ^ { S A W } ( 0 , x ) } \\prod _ { j = 0 } ^ { n - 1 } D ( w _ { j } , w _ { j + 1 } ) , \\end{align*}"}
+{"id": "6867.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\binom { N } { 2 } ^ { - 1 } \\log \\mathbb { P } ^ * _ N ( \\lambda _ N / N \\leq \\beta ) = - \\psi _ r ( \\beta ) , \\beta \\in [ 0 , C _ r ] , \\end{align*}"}
+{"id": "1141.png", "formula": "\\begin{align*} \\Delta ^ { n } _ { y } A ^ { \\ast } ( x ) = n ! A ^ { \\ast } ( y ) . \\end{align*}"}
+{"id": "4202.png", "formula": "\\begin{align*} \\phi ( t ) = \\phi _ 0 ( t ) \\prod _ { k = 1 } ^ R u _ { \\beta _ k , \\tau _ k } ( t ) \\end{align*}"}
+{"id": "3200.png", "formula": "\\begin{align*} \\sigma _ n : = \\frac { 1 } { \\abs { W _ n } } \\sum _ { a \\in W _ n } \\phi ( a ) , ~ n \\in \\N , \\end{align*}"}
+{"id": "62.png", "formula": "\\begin{align*} a y d ^ \\ast - b y ^ \\ast c ^ \\ast = ( a y b ^ \\ast - b y ^ \\ast a ^ \\ast s ^ \\ast ( t ^ \\ast ) ^ { - 1 } ) t ^ \\ast , a y b ^ \\ast - b y ^ \\ast a ^ \\ast \\in B . \\end{align*}"}
+{"id": "3901.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { \\ell < 2 } K _ { \\ell } \\right ) \\cap K _ { 2 } = K _ 1 \\cap K _ 2 = \\{ 3 \\} \\in \\bigcup _ { \\ell < 2 } 2 ^ { K _ { \\ell } } = 2 ^ { K _ 1 } . \\end{align*}"}
+{"id": "4501.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\omega , c } ^ * ( \\eta ) = \\begin{cases} \\{ \\Psi \\in \\mathcal { M } _ { \\omega , c } \\ | \\ \\omega Q ( \\Psi ) + c P ( \\Psi ) \\ge \\eta \\} , & \\ d = 1 , \\\\ \\{ \\Psi \\in \\mathcal { M } _ { \\omega , \\mathbf { c } } \\ | \\ \\mathbf { c } \\cdot \\mathbf { P } ( \\Psi ) \\ge \\eta \\} , & \\ d = 2 , \\\\ \\emptyset , & \\ d = 3 . \\end{cases} \\end{align*}"}
+{"id": "8286.png", "formula": "\\begin{align*} \\frac { ( 2 k - n + i - 1 ) ! } { ( 2 k - n + i - 1 + m ) ! } = \\frac { 1 } { ( 2 k - n + i - 1 + m ) ( 2 k - n + i - 1 + m - 1 ) \\cdots ( 2 k - n + i ) } . \\end{align*}"}
+{"id": "5044.png", "formula": "\\begin{align*} ( g ^ k ) ^ * ( \\xi _ { g ^ l } ) = \\exp \\left ( - 2 \\pi i \\cdot \\frac { k } { r } \\right ) \\xi _ { g ^ l } . \\end{align*}"}
+{"id": "4857.png", "formula": "\\begin{align*} W _ p ^ p ( \\lambda \\mu + ( 1 - \\lambda ) \\nu , \\delta _ b ) = \\lambda W _ p ^ p ( \\mu , \\delta _ b ) + ( 1 - \\lambda ) W _ p ^ p ( \\nu , \\delta _ b ) \\geq \\lambda W _ p ^ p ( \\mu , \\delta _ a ) + ( 1 - \\lambda ) W _ p ^ p ( \\nu , \\delta _ a ) , \\end{align*}"}
+{"id": "4361.png", "formula": "\\begin{align*} d _ { H } \\left ( x \\right ) = \\dfrac { [ \\left \\langle a , x \\right \\rangle - b ] _ { + } } { \\left \\Vert a \\right \\Vert _ { \\ast } } . \\end{align*}"}
+{"id": "8450.png", "formula": "\\begin{align*} \\left ( \\frac { z } { 2 } \\right ) ^ { \\nu } \\ , K _ { \\nu } ( z ) = \\frac { \\sqrt { \\pi } } { \\Gamma \\left ( \\frac { 1 } { 2 } - \\nu \\right ) } \\ , \\intop _ { 1 } ^ { \\infty } e ^ { - z t } \\left ( t ^ { 2 } - 1 \\right ) ^ { - \\nu - \\frac { 1 } { 2 } } d t , \\ , \\ , \\ , ( \\nu ) < \\frac { 1 } { 2 } , \\ , \\ , \\ , | \\arg ( z ) | < \\frac { \\pi } { 2 } , \\end{align*}"}
+{"id": "5365.png", "formula": "\\begin{align*} E x _ { k + 1 } & = A x _ k + B u _ k , y _ k = C x _ k + D u _ k , k \\geq 0 , x _ 0 = x ^ 0 , \\end{align*}"}
+{"id": "5392.png", "formula": "\\begin{align*} F ( \\lambda ) = \\sum _ { i = 1 } ^ { k } a _ i \\lambda ^ i = 0 . \\end{align*}"}
+{"id": "1809.png", "formula": "\\begin{align*} [ a _ r ^ * a _ r , a _ p ^ * a _ q ] = \\big ( \\delta ( r - q ) - \\delta ( r - p ) \\big ) a _ p ^ * a _ q \\end{align*}"}
+{"id": "7859.png", "formula": "\\begin{align*} U _ { k } : = \\left \\{ \\{ B _ 1 , B _ 2 \\} : \\ B _ 1 \\cup B _ 2 = [ 2 k ] , \\ B _ 1 \\cap B _ 2 = \\varnothing , \\ | B _ 1 | = | B _ 2 | = k \\right \\} . \\end{align*}"}
+{"id": "3325.png", "formula": "\\begin{align*} H ( \\widetilde { d } ) = H ( { \\rm R e } ( q ) ) + H ( \\widetilde { e } ) \\end{align*}"}
+{"id": "1789.png", "formula": "\\begin{align*} \\pi _ 0 ( G _ 1 \\ast _ H G _ 2 ) = \\pi _ 0 ( \\pi _ 0 ( G _ 1 ) \\ast _ { \\pi _ 0 ( H ) } \\pi _ 0 ( G _ 2 ) ) \\ ; . \\end{align*}"}
+{"id": "5414.png", "formula": "\\begin{align*} ( f _ 1 ) ^ { n _ 1 } _ * \\cdots ( f _ k ) ^ { n _ k } _ * A = A + \\sum _ i n _ i R _ i + \\left ( \\sum _ i \\frac { n _ i ( n _ i - 1 ) } { 2 } t _ { i i } + \\sum _ { i < j } n _ i n _ j t _ { i j } \\right ) D _ 0 . \\end{align*}"}
+{"id": "8476.png", "formula": "\\begin{align*} \\frac { \\Gamma ( s ) \\ , x ^ { 2 s } } { \\left ( x ^ { 2 } + t ^ { 2 } \\right ) ^ { s } } - \\frac { \\Gamma ( s ) \\ , x ^ { 2 s } } { t ^ { 2 s } } = \\sqrt { \\pi } 2 ^ { \\frac { 1 } { 2 } - s } x ^ { s + \\frac { 1 } { 2 } } \\ , \\intop _ { 0 } ^ { \\infty } y ^ { s - \\frac { 1 } { 2 } } \\left \\{ J _ { s - \\frac { 1 } { 2 } } ( x y ) - \\frac { 2 ^ { \\frac { 1 } { 2 } - s } x ^ { s - \\frac { 1 } { 2 } } } { \\Gamma ( s + \\frac { 1 } { 2 } ) } y ^ { s - \\frac { 1 } { 2 } } \\right \\} \\ , e ^ { - t y } d y . \\end{align*}"}
+{"id": "1532.png", "formula": "\\begin{align*} h ( t ) - \\alpha ( g ( t ) ) & = \\sum _ { \\theta = 1 } ^ d ( h _ { p \\theta + p - 1 } - g _ { p \\theta + p - 1 } a ^ { p \\theta } ) t ^ { p \\theta + p - 1 } - \\sum _ { \\theta = 1 } ^ d r _ { \\theta } ( t ) + f _ 0 ( t ) \\\\ & = \\sum _ { \\theta = 1 } ^ d ( h _ { p \\theta + p - 1 } - g _ { p \\theta + p - 1 } a ^ { p \\theta } ) t ^ { p \\theta + p - 1 } + f _ { h , g } ( t ) , \\end{align*}"}
+{"id": "2298.png", "formula": "\\begin{align*} w _ b = \\sum _ { n } c _ n a _ n . \\end{align*}"}
+{"id": "3769.png", "formula": "\\begin{align*} v \\cdot u = & - \\cosh ( \\phi - \\theta ) = \\left ( \\begin{array} { c } \\cosh ( \\phi - \\theta ) \\\\ \\sinh ( \\phi - \\theta ) \\\\ \\end{array} \\right ) \\cdot \\left ( \\begin{array} { c } 1 \\\\ 0 \\\\ \\end{array} \\right ) , \\end{align*}"}
+{"id": "5204.png", "formula": "\\begin{align*} C _ 1 & = \\{ [ 0 , \\tfrac { 3 } { 4 } ) , ( \\tfrac { 1 } { 4 } , 1 ] \\} , \\\\ C _ 2 & = \\{ [ 0 , \\tfrac { 2 } { 3 } ) , ( \\tfrac { 1 } { 3 } , 1 ] \\} , \\\\ C _ 3 & = \\{ [ 0 , \\tfrac { 1 } { 2 } ) , ( \\tfrac { 1 } { 4 } , \\tfrac { 2 } { 3 } ) , ( \\tfrac { 1 } { 3 } , \\tfrac { 3 } { 4 } ) , ( \\tfrac { 1 } { 2 } , 1 ] \\} . \\end{align*}"}
+{"id": "2435.png", "formula": "\\begin{align*} y ( q x ) = \\sin \\left ( \\dfrac { 2 } { x } \\right ) y ( x ) . \\end{align*}"}
+{"id": "5715.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { m \\rightarrow \\infty } \\dfrac { C _ { F } ( p ^ { 2 m + 1 } ) } { \\beta ^ { 2 m } } \\not = 0 \\end{align*}"}
+{"id": "3800.png", "formula": "\\begin{align*} F _ { Y _ 2 | X } ( y | x ) = \\mathbb { P } ( Y _ 2 \\leq y | X = x ) = \\mathbb { P } ( Y \\leq y | X = x , D = 1 ) . \\end{align*}"}
+{"id": "4602.png", "formula": "\\begin{align*} \\log \\left ( 1 + \\frac { P _ k } { 1 + P _ { k + 1 } } \\right ) = R , 1 \\leq k \\leq K - 1 , \\end{align*}"}
+{"id": "5051.png", "formula": "\\begin{align*} & \\left ( \\left ( \\frac { 1 } { 4 } - \\frac { 1 } { 4 i } + \\frac { 1 } { 1 6 i ^ 2 } + o ( 1 ) \\right ) + \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 8 i ^ 2 } + o ( 1 ) \\right ) + \\left ( \\frac { 1 } { 4 i } + o ( 1 ) \\right ) + \\left ( \\frac { 1 } { 1 6 i ^ 2 } + o ( 1 ) \\right ) \\right ) \\binom { t } { 2 } \\\\ & = \\left ( \\frac { 3 } { 4 } + o ( 1 ) \\right ) \\binom { t } { 2 } , \\end{align*}"}
+{"id": "5110.png", "formula": "\\begin{align*} b ^ { i j , k l } ( D ^ 2 u ( x ) ) = \\int _ { 0 } ^ 1 \\frac { \\partial } { \\partial { u _ { i j } } } \\bigg [ a ^ { p q , k l } ( D ^ 2 u ( x ) + t [ D ^ 2 u ( x + h _ m ) - D ^ 2 u ( x ) ] ) u _ { p q } ( x ) \\bigg ] d t \\end{align*}"}
+{"id": "4589.png", "formula": "\\begin{align*} l ( \\alpha ( x ) , \\alpha ( y ) , \\beta ( u ) ) & = \\beta ( l ( x , y , u ) ) , \\\\ m ( \\alpha ( x ) , \\beta ( u ) , \\alpha ( z ) ) & = \\beta ( m ( x , u , z ) ) , \\\\ r ( \\beta ( u ) , \\alpha ( y ) , \\alpha ( z ) ) & = \\beta ( r ( u , y , z ) ) \\end{align*}"}
+{"id": "3524.png", "formula": "\\begin{align*} L ( t ) = \\Phi ( t ) L ( t _ 0 ) G ( t ) . \\end{align*}"}
+{"id": "8206.png", "formula": "\\begin{align*} P ^ \\omega \\left ( A _ t ^ c \\mid S \\right ) & = \\frac { P ^ \\omega ( A _ t ^ c \\cap S ) } { P ^ \\omega ( S ) } \\leq c ( \\omega ) P ^ \\omega ( A _ t ^ c \\cap S ) \\\\ & \\leq c ( \\omega ) P ^ \\omega ( A _ t ^ c \\cap S _ { \\alpha t } ) \\to 0 , \\ : \\ : \\ : t \\to \\infty . \\end{align*}"}
+{"id": "2133.png", "formula": "\\begin{align*} \\frac { 1 } { p _ 0 } = \\frac { 1 } { p _ 1 } + \\frac { 1 } { p _ 2 } = \\frac { 1 } { p _ 3 } + \\frac { 1 } { p _ 4 } . \\end{align*}"}
+{"id": "551.png", "formula": "\\begin{align*} A _ { \\varepsilon } ( t ) : = \\left ( \\begin{array} { c c } 0 & 1 \\\\ a _ { \\varepsilon } ( t ) & 0 \\end{array} \\right ) , Q _ { \\varepsilon } ( t ) : = \\left ( \\begin{array} { c c } 0 & 0 \\\\ q _ { \\varepsilon } ( t ) - a _ { \\varepsilon } ( t ) & 0 \\end{array} \\right ) , F _ { \\varepsilon } ( t , \\xi ) : = \\left ( \\begin{array} { c } 0 \\\\ \\widehat { f } _ { \\varepsilon } ( t , \\xi ) \\end{array} \\right ) . \\end{align*}"}
+{"id": "3342.png", "formula": "\\begin{align*} F _ a = \\left \\{ \\mu \\in K ( B ) \\mid \\mu ( a ^ * a ) = 1 \\right \\} . \\end{align*}"}
+{"id": "7662.png", "formula": "\\begin{align*} { \\bf M } _ { - \\kappa ^ 2 } ^ n = \\begin{cases} \\mathbb { R } ^ n \\mbox { - - t h e E u c l i d e a n s p a c e } , & \\mbox { i f } \\kappa = 0 , \\\\ \\mathbb H ^ n _ { - \\kappa ^ 2 } \\mbox { - - t h e H y p e r b o l i c s p a c e } , & \\mbox { i f } \\kappa > 0 . \\end{cases} \\end{align*}"}
+{"id": "6034.png", "formula": "\\begin{align*} T ( z ) : = A z ^ { n + m } + B z ^ m + C \\textrm { f o r a l l } z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "148.png", "formula": "\\begin{align*} \\tilde { \\varphi } ( m , e ) = ( \\varphi ( m ) , g ( m ) e ) : E \\to E . \\end{align*}"}
+{"id": "1002.png", "formula": "\\begin{align*} \\hbox { $ \\sum _ { i = 1 } ^ \\ell u _ i = 0 $ i n $ \\Lambda _ x / \\Lambda _ { \\tilde \\rho _ { j _ k } } $ } \\end{align*}"}
+{"id": "83.png", "formula": "\\begin{align*} Y _ 1 Y _ 2 = 1 + \\mathcal { O } ( h ^ \\infty ) L , L \\subset { \\rm e l l } _ h ( Z ) \\end{align*}"}
+{"id": "8317.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { q ^ 2 } \\mu _ j ^ 4 = \\sum _ { a , a ' } \\left | \\sum _ { b } M ( a , b ) M ( a ' , b ) \\right | ^ 2 \\le 1 6 q ^ 4 + 2 q ^ 2 ( 2 q ) ^ 2 = 2 4 q ^ 4 \\ , . \\end{align*}"}
+{"id": "4330.png", "formula": "\\begin{align*} T _ m = \\sum _ { \\substack { m ' \\le m \\\\ ( k ' , m ' , i ' , n ' ) \\in \\mathcal { N } } } 3 \\cdot 2 ^ { - k ' } < 4 2 . \\end{align*}"}
+{"id": "1958.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u ) ^ n = \\psi ( \\cdot , u ) \\omega ^ n & \\textnormal { i n } & \\Omega , \\\\ u = 0 & \\textnormal { i n } & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "8361.png", "formula": "\\begin{align*} L _ { u v } = Q _ { u v } - A _ { u v } , Q _ { u v } = \\delta _ { u v } \\sum _ { h \\in T _ v X } \\frac { c ( v ) } { c ( h ) } , A _ { u v } = \\sum _ { h \\in T _ v X : \\ , r _ X ( \\iota _ X ( h ) ) = u } \\frac { c ( v ) } { c ( h ) } . \\end{align*}"}
+{"id": "2531.png", "formula": "\\begin{align*} \\Delta ( 2 , 2 n ) = \\{ \\varepsilon _ { i , j } \\ : : \\ : \\{ i , j \\} \\in J _ n ^ 2 \\} \\subset \\mathbb { R } ^ n . \\end{align*}"}
+{"id": "8001.png", "formula": "\\begin{align*} T _ i ( P _ i ) = 0 \\ , , T _ i ( \\nu ^ i + P _ i ) = e _ n \\ , , G ( P _ i ) \\coloneqq \\nu ^ i \\in \\mathbb { S } ^ { n - 1 } \\ , , \\end{align*}"}
+{"id": "307.png", "formula": "\\begin{align*} \\prod _ { \\substack { l , m , n \\geq 1 \\\\ l , m \\leq n ; \\ , \\gcd ( l , m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - x ^ l y ^ m z ^ n } \\right ) ^ { \\frac { l m ^ 2 } { n ^ 4 } } \\end{align*}"}
+{"id": "345.png", "formula": "\\begin{align*} \\int _ { V } | u _ { k } - u | ^ { p } d \\mu = \\int _ { \\{ x : \\rho ( x ) \\geq k + 1 \\} } | u | ^ p d \\mu = o _ k ( 1 ) . \\end{align*}"}
+{"id": "7648.png", "formula": "\\begin{align*} M ( c , x , 1 ) = ( 4 - c ^ 2 ) ^ 2 [ x ^ 2 ( 2 c ^ 2 x ^ 2 + ( 3 6 - 1 3 c ^ 2 ) x + 2 c ^ 2 ) + 8 c x ( 1 + x ) ( 1 - x ^ 2 ) + 8 ( 8 + x ^ 2 ) ( 1 - x ^ 2 ) ] . \\end{align*}"}
+{"id": "804.png", "formula": "\\begin{align*} \\| ^ { c } _ { 0 } { D } ^ { \\alpha } _ { t _ { n - \\sigma } } R _ h u - \\ , ^ { c } _ { 0 } { D } ^ { \\alpha } _ { t _ { n - \\sigma } } u \\| \\ ; \\le \\ ; C \\ , h ^ 2 \\ , \\| \\Delta \\ , ^ { c } _ { 0 } { D } ^ { \\alpha } _ { t _ { n - \\sigma } } u \\| \\ ; \\le \\ ; C \\ , h ^ 2 . \\end{align*}"}
+{"id": "6337.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial s _ 1 } F ( s _ 1 , s _ 2 ) & = \\frac 1 2 f ' ( s _ 1 ) ( s _ 2 - s _ 1 ) + \\frac 1 2 \\big ( f ( s _ 1 ) - f ( s _ 2 ) \\big ) \\\\ \\frac { \\partial } { \\partial s _ 2 } F ( s _ 1 , s _ 2 ) & = \\frac 1 2 f ' ( s _ 2 ) ( s _ 2 - s _ 1 ) + \\frac 1 2 \\big ( f ( s _ 1 ) - f ( s _ 2 ) \\big ) . \\end{align*}"}
+{"id": "3463.png", "formula": "\\begin{align*} \\deg ( F ) = - \\sum _ { 1 \\le i < j \\le 4 } \\chi ( F _ { i j } ( c ) ) . \\end{align*}"}
+{"id": "8125.png", "formula": "\\begin{align*} \\varphi ^ - ( x ) & = - P _ - \\big ( \\varphi ( x ) + \\sum _ { ( x ) } \\varphi ^ - ( x ' ) \\varphi ( x '' ) \\big ) \\\\ \\varphi ^ + ( x ) & = P _ + \\big ( \\varphi ( x ) + \\sum _ { ( x ) } \\varphi ^ - ( x ' ) \\varphi ( x '' ) \\big ) . \\end{align*}"}
+{"id": "7634.png", "formula": "\\begin{align*} M ( 0 , x , 0 ) = - 5 7 6 x ^ 3 + 1 1 5 2 x \\leq M ( 0 , x _ 1 , 0 ) \\leq 2 5 6 \\sqrt { 6 } \\approx 6 2 7 . 0 6 9 , \\ ; x \\in ( 0 , 1 ) \\end{align*}"}
+{"id": "5963.png", "formula": "\\begin{align*} \\widetilde { \\widetilde { C } } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) = \\widetilde { C } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) \\widetilde { s } ( g _ 1 ) \\widetilde { s } ( g _ 2 ) \\widetilde { s } ( g _ 1 g _ 2 ) ^ { - 1 } , \\end{align*}"}
+{"id": "7442.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { + \\infty } \\varepsilon ^ { k - 1 } \\ , \\mathfrak { B } _ { k - 1 } \\ + \\ \\sum \\limits _ { k = 0 } ^ { + \\infty } \\varepsilon ^ { \\alpha + k - 1 } \\ , \\mathfrak { B } _ { \\alpha + k - 1 } . \\end{align*}"}
+{"id": "3917.png", "formula": "\\begin{align*} \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda d \\widetilde { \\pi } \\geq \\int _ { \\mathcal { V } } g \\ , d \\gamma - \\lambda _ 1 \\boldsymbol { K } _ { 1 } ( \\mu _ 1 , \\gamma _ 1 ) - \\lambda _ 2 \\boldsymbol { K } _ { 2 } ( \\mu _ 2 , \\gamma _ 2 ) - \\epsilon = I _ { \\mathrm { D } , \\lambda } [ \\gamma ] - \\epsilon . \\end{align*}"}
+{"id": "4797.png", "formula": "\\begin{align*} \\phi = \\begin{bmatrix} \\phi _ 1 & \\cdots & \\phi _ p \\end{bmatrix} ^ T , \\psi = \\begin{bmatrix} \\psi _ 1 & \\cdots & \\psi _ q \\end{bmatrix} ^ T . \\end{align*}"}
+{"id": "6552.png", "formula": "\\begin{align*} H f ( x ) = \\int _ { \\mathbb { H } ^ { n } } K ( x , y ) f ( y ) d y . \\end{align*}"}
+{"id": "1195.png", "formula": "\\begin{align*} \\left \\langle \\begin{pmatrix} \\hat { P } & \\ell \\end{pmatrix} ^ \\top S \\begin{pmatrix} \\hat { P } & \\ell \\end{pmatrix} , F _ e \\right \\rangle = 0 \\end{align*}"}
+{"id": "473.png", "formula": "\\begin{align*} \\Delta ( x ) : = \\prod _ { 1 \\leq i < j \\leq k } ( x _ i - x _ j ) \\quad \\Delta ( x ; y ) : = \\prod _ { i = 1 } ^ k \\prod _ { j = 1 } ^ { \\ell } ( x _ i - y _ j ) \\end{align*}"}
+{"id": "6408.png", "formula": "\\begin{align*} [ D \\psi : D \\varphi ] _ t & = s ( \\psi ) [ D \\chi : D \\varphi ] _ t , \\\\ [ D \\widetilde { \\psi } : D \\widetilde { \\varphi } ] _ t & = s ( \\psi ) [ D \\widetilde { \\chi } : D \\widetilde { \\varphi } ] _ t , \\\\ [ D \\widetilde { \\psi } : D \\tau _ M ] _ t & = s ( \\psi ) [ D \\widetilde { \\chi } : D \\tau _ M ] _ t \\end{align*}"}
+{"id": "4291.png", "formula": "\\begin{align*} \\frac { \\rho } { \\theta } ( C _ v \\theta + a ( \\theta ) q ^ 2 ) _ t + \\frac { \\rho u } { \\theta } ( C _ v \\theta + a ( \\theta ) q ^ 2 ) _ x + R \\rho u _ x + \\frac { q _ x } { \\theta } = \\frac { 1 } { \\mu \\theta } S ^ 2 . \\end{align*}"}
+{"id": "3430.png", "formula": "\\begin{align*} \\nabla _ j \\left ( | X | ^ 2 \\right ) = 2 X ^ i \\nabla _ j X _ i , \\end{align*}"}
+{"id": "3021.png", "formula": "\\begin{align*} f ( \\alpha ) = f ( 0 ) + f ^ { ' } ( 0 ) \\alpha + o ( \\alpha ) = \\left [ \\sum _ { i = 1 } ^ { k } \\lambda _ i ^ { \\gamma - 1 } ( A ) b _ { i } \\gamma \\right ] \\alpha + o ( \\alpha ) , \\end{align*}"}
+{"id": "7156.png", "formula": "\\begin{align*} \\partial _ t u = - \\mathcal { L } ^ \\Omega _ \\varepsilon c _ \\varepsilon - f ^ \\prime ( c _ \\varepsilon ) - \\Delta c + f ^ \\prime ( c ) . \\end{align*}"}
+{"id": "8997.png", "formula": "\\begin{align*} \\div ( T ^ \\varphi ( \\nabla w , \\ , \\cdot \\ , ) ^ \\sharp ) = \\alpha ( 1 + w ) | \\tau ( \\varphi ) | ^ 2 + ( 1 + w ) | T ^ \\varphi | ^ 2 \\ , . \\end{align*}"}
+{"id": "3445.png", "formula": "\\begin{align*} P ( x ) = x ^ n - \\left ( \\frac { \\alpha ^ 2 } { m ( n + 1 ) } + \\lambda \\right ) \\frac { x ^ { n + 1 } } { n } + c , \\end{align*}"}
+{"id": "9188.png", "formula": "\\begin{align*} v _ { 1 } ^ { 1 } & = y _ { 1 , [ 2 ] } ^ { 1 , d } - a _ { 1 } ^ { 1 , 1 } ( y _ { 1 , [ 1 ] } ^ { 1 } - y _ { 1 , [ 1 ] } ^ { 1 , d } ) - a _ { 1 } ^ { 1 , 0 } ( y _ { 1 } ^ { 1 } - y _ { 1 } ^ { 1 , d } ) \\\\ v _ { 2 } ^ { 1 } & = y _ { 2 , [ 4 ] } ^ { 1 , d } - \\sum _ { \\beta = 0 } ^ { 3 } a _ { 2 } ^ { 1 , \\beta } ( y _ { 2 , [ \\beta ] } ^ { 1 } - y _ { 2 , [ \\beta ] } ^ { 1 , d } ) \\end{align*}"}
+{"id": "6622.png", "formula": "\\begin{align*} [ [ x _ \\mu [ { y _ 1 } _ \\eta y _ 2 ] ] _ { \\mu + \\gamma } \\varphi _ { \\lambda } ( w , v ) ] = [ \\varphi _ \\lambda ( [ x _ \\mu y _ 1 ] , y _ 2 ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] + ( - 1 ) ^ { x y _ 1 } [ \\varphi _ \\lambda ( y _ 1 , [ x _ \\mu y _ 2 ] ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] , \\end{align*}"}
+{"id": "6229.png", "formula": "\\begin{align*} A _ x = \\big \\{ y : \\ , 1 - \\xi < | y - x | < 1 + \\xi \\big \\} \\ \\end{align*}"}
+{"id": "5266.png", "formula": "\\begin{align*} \\| 1 + \\rho d X \\| _ { q } ^ { q } \\le \\left ( \\rho \\left ( 1 + \\frac { 1 } { \\omega } \\right ) \\right ) ^ q \\| d X \\| _ { q } ^ { q } + \\| \\left ( 1 + \\rho d X \\right ) 1 _ { \\rho | d X | < \\omega } \\| _ { q } ^ { q } = \\beta ^ q \\| d X \\| _ { q } ^ { q } + \\| \\left ( 1 + \\rho d X \\right ) 1 _ { \\rho | d X | < \\omega } \\| _ { q } ^ { q } . \\end{align*}"}
+{"id": "359.png", "formula": "\\begin{align*} c ^ { \\sigma / \\tau } = c ^ { \\sigma / \\tau } ( K ) : ( \\underline { X } , \\underline { A } ) ^ K \\longrightarrow ( \\underline { X } , \\underline { A } ) ^ K \\end{align*}"}
+{"id": "4702.png", "formula": "\\begin{align*} S _ { \\chi } ^ { \\rm s t i f f } ( \\varepsilon ^ 2 z ) = \\Bigl ( I + \\varepsilon ^ 2 z \\bigl ( \\mathcal { A } _ { 0 , \\chi } ^ { \\rm s t i f f } - \\varepsilon ^ 2 z I \\bigr ) ^ { - 1 } \\Bigr ) \\Pi ^ { \\rm s t i f f } _ \\chi = \\Pi ^ { \\rm s t i f f } _ \\chi + \\mathcal { O } ( \\varepsilon ^ 2 ) , \\end{align*}"}
+{"id": "110.png", "formula": "\\begin{align*} T ( z ) = \\mathcal { O } ( h ^ { - 2 n - 2 } ) . \\end{align*}"}
+{"id": "7556.png", "formula": "\\begin{align*} \\frac { \\binom n l \\alpha ^ l ( 1 - \\alpha ) ^ { n - l } } { \\binom n { l _ 0 } \\alpha ^ { l _ 0 } ( 1 - \\alpha ) ^ { n - l _ 0 } } = \\frac { l _ 0 ( l _ 0 - 1 ) \\cdots ( l _ 0 - h + 1 ) } { ( n - l _ 0 + 1 ) ( n - l _ 0 + 2 ) \\cdots ( n - l _ 0 + h ) } \\alpha ^ { - h } ( 1 - \\alpha ) ^ { h } . \\end{align*}"}
+{"id": "3973.png", "formula": "\\begin{align*} \\left ( f _ { \\mathcal { S } } \\right ) _ \\lambda ( s _ 1 , s _ 2 ) = \\sup _ { ( y _ 1 ^ \\prime , y _ 2 ^ \\prime , x ^ \\prime ) \\in \\mathcal { S } } \\left \\{ f _ 1 ( y _ 1 ^ \\prime ) + f _ 2 ( y _ 2 ^ \\prime ) - \\sum _ { 1 \\leq \\ell \\leq 2 } \\lambda _ \\ell c _ \\ell \\left ( ( y _ \\ell , x _ \\ell ) , ( y ^ \\prime _ \\ell , x ^ \\prime ) \\right ) \\right \\} . \\end{align*}"}
+{"id": "830.png", "formula": "\\begin{gather*} \\lambda ( g _ { - s } E _ { T , c } \\cap c ^ { - 1 } g _ { - t } E _ { T , c } ) = \\lambda \\left ( g _ { \\tau } \\left ( g _ { - s } E _ { T , c } \\cap c ^ { - 1 } g _ { - t } E _ { T , c } \\right ) \\right ) \\\\ = \\lambda ( g _ { \\tau - s } E _ { T , c } \\cap c ^ { - 1 } g _ { \\tau - t } E _ { T , c } ) \\end{gather*}"}
+{"id": "8853.png", "formula": "\\begin{align*} ( x - a ) ^ N \\prod _ { k = 1 } ^ { { T \\over 2 } } ( f ( x ) - v _ { 2 k - 1 } ^ 2 ( x ) ) \\sim \\prod _ { k = 0 } ^ { { T \\over 2 } - 1 } ( f ( x ) - v _ { 2 k } ^ 2 ( x ) ) \\end{align*}"}
+{"id": "4245.png", "formula": "\\begin{align*} \\Omega ( X _ 2 , r ) & = 2 ( n + 1 ) - 3 = 2 n - 1 \\\\ \\Omega ( Y _ 1 , r ) & = n - 1 \\\\ \\Omega ( Y _ 2 , r ) & = 2 n n - ( 2 n - 1 ) = 2 n ( n - 1 ) + 1 \\\\ \\end{align*}"}
+{"id": "3730.png", "formula": "\\begin{gather*} t _ { 1 2 3 } = - 1 , t _ { 1 3 2 } = 1 , t _ { 2 1 3 } = - 1 , t _ { 2 3 1 } = 1 , t _ { 3 2 1 } = - 1 , t _ { 3 1 2 } = 1 , \\end{gather*}"}
+{"id": "6111.png", "formula": "\\begin{align*} \\gamma = \\frac { 2 ( n + 2 s ) } { n + 4 s } . \\end{align*}"}
+{"id": "1653.png", "formula": "\\begin{align*} g ( Q \\ , \\nabla _ { X } \\ , \\xi _ i , Z ) = g ( ( { f } Q - h _ i { f } ) Z , X ) = g ( { f } ( h ^ * _ i - Q ) X , Z ) . \\end{align*}"}
+{"id": "4320.png", "formula": "\\begin{gather*} f ( \\cdot , 1 ) = f ( \\cdot , 1 - 2 ^ { 2 - 2 k } ) , \\\\ f ( \\cdot , 1 ) = f ( \\cdot , 1 - 2 ^ { 2 - 2 k } ) \\circ \\big ( y _ { 2 \\tau _ k } ^ { ( i _ k ; L _ k ) } \\big ) ^ { - 1 } . \\end{gather*}"}
+{"id": "1415.png", "formula": "\\begin{align*} \\cos ( \\theta _ k ) & = \\frac { f ( x _ k ) - f ^ * } { s _ k \\| g _ k \\| } = \\tanh ( \\eta _ k \\| g _ k \\| ) / \\tanh ( s _ k ) , \\sin ( \\theta _ k ) = \\sinh ( s _ { k + 1 } ) / \\sinh ( s _ k ) , \\\\ \\cosh ( s _ { k + 1 } ) & = \\cosh ( s _ k ) \\sqrt { 1 - \\frac { 1 } { \\zeta _ { s _ k } ^ 2 } \\cdot \\frac { ( f ( x _ k ) - f ^ * ) ^ 2 } { \\| g _ k \\| ^ 2 } } , \\end{align*}"}
+{"id": "4828.png", "formula": "\\begin{align*} \\exp \\left \\{ - \\frac { x ^ { 2 } } { 2 } + C \\frac { x ^ { 3 } } { \\sqrt { n } } \\right \\} = \\exp \\left \\{ - \\frac { x ^ { 2 } } { 2 } \\left ( 1 - C \\frac { x } { \\sqrt { n } } \\right ) \\right \\} \\leq \\exp \\left \\{ - \\frac { x ^ { 2 } } { 2 ( 1 + C x / \\sqrt { n } ) } \\right \\} . \\end{align*}"}
+{"id": "669.png", "formula": "\\begin{gather*} \\mathfrak { h } _ { 0 } ( \\varphi ) = 0 , \\mathfrak { h } _ { - g , 2 } ( D \\varphi ) = 0 , \\mathfrak { h } _ { 2 } ( D ^ 2 \\varphi ) = 0 \\\\ \\| S ( k ) D \\varphi \\| _ { \\sup } = O ( e ^ { - \\rho r ( 0 , k ) } ) c ( D \\varphi ) \\rho > 0 , \\end{gather*}"}
+{"id": "7293.png", "formula": "\\begin{align*} u = \\prod _ { p \\in { \\cal P } _ 1 } p ^ { u _ p } \\cdot \\prod _ { p \\in { \\cal P } _ 4 } p ^ { u _ p } , v = \\prod _ { p \\in { \\cal P } _ 2 } p ^ { v _ p } \\cdot \\prod _ { p \\in { \\cal P } _ 5 } p ^ { v _ p } . \\end{align*}"}
+{"id": "3390.png", "formula": "\\begin{align*} \\mathcal { C } _ { i , j } = \\mathbf { 1 } _ { G _ { j , l } \\cap I _ { j , l } } \\int _ { y _ { j - 1 } } ^ { x _ i } \\Biggl | \\sum _ { \\substack { n \\leq x _ i / t \\\\ P ( n ) \\leq y _ { j - 1 } } } \\frac { f ( n ) } { \\sqrt { n } } \\Biggr | ^ 2 \\sum _ { \\substack { t / ( 1 + 1 / V ) \\leq m \\leq t \\\\ p | m \\Rightarrow p \\in ( y _ { j - 1 } , y _ j ] } } \\frac { V \\tau _ q ( m ) } { m ^ 2 } d t \\ , . \\end{align*}"}
+{"id": "7517.png", "formula": "\\begin{align*} \\begin{gathered} \\langle \\alpha _ i , Z \\rangle + \\partial _ z \\log \\Big [ F _ i ( z ) ( z - w ^ i _ \\ell ) ^ 2 \\Big ] \\Bigg | _ { z = w ^ i _ \\ell } = 0 , \\\\ i = 1 , \\dots , r ; \\ell = 1 , \\dots , \\deg ( q ^ i _ + ( z ) ) . \\end{gathered} \\end{align*}"}
+{"id": "730.png", "formula": "\\begin{align*} \\| A _ { B , N } ^ { * - 1 } - U _ { n } \\| = 2 \\| \\sum _ { k = n + 1 } ^ { + \\infty } R _ { B , N } ^ k \\| \\leq 2 \\| R _ { B , N } ^ { n + 1 } \\| \\frac 1 { 1 - \\| R _ { B , N } \\| } = \\frac { \\| U _ { n + 1 } - U _ { n } \\| } { 1 - \\| R _ { B , N } \\| } \\ , . \\end{align*}"}
+{"id": "4684.png", "formula": "\\begin{align*} \\left ( \\beta _ 0 \\Gamma _ 0 + \\beta _ 1 \\Gamma _ 1 \\right ) \\vect u = 0 , \\end{align*}"}
+{"id": "5086.png", "formula": "\\begin{align*} \\left | A _ t ^ { - 1 } ( x ) - \\left ( x + \\gamma ( x , t ) + \\frac { 1 } { N } \\sigma ( t ) \\right ) \\right | & = \\left | \\gamma ( A _ t ^ { - 1 } ( x ) , t ) - \\gamma ( x , t ) \\right | \\\\ & \\leq \\| \\gamma _ x ( t ) \\| _ { L ^ \\infty } \\left | A _ t ^ { - 1 } ( x ) - x \\right | \\\\ & \\leq \\| \\gamma _ x ( t ) \\| _ { L ^ \\infty } \\left ( \\left | \\gamma \\left ( A _ t ^ { - 1 } ( x ) , t \\right ) \\right | + \\frac { 1 } { N } | \\sigma ( t ) | \\right ) , \\end{align*}"}
+{"id": "3987.png", "formula": "\\begin{align*} \\sup _ { \\varpi \\in \\Pi ( \\mu _ { 1 3 } , \\mu _ { 2 3 } ) } \\int ( f _ { \\mathcal { S } } ) _ { \\lambda , \\ell } \\ , d \\varpi = \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + \\frac { V _ { \\ell } / V _ { \\ell , X X } } { 4 \\lambda _ \\ell } , \\ell \\in [ 2 ] . \\end{align*}"}
+{"id": "2667.png", "formula": "\\begin{align*} I I : & b _ 0 = b _ N = 1 , a _ 0 = c _ 0 = 0 , d _ 0 = d _ N = 0 \\\\ I I I : & b _ 0 = b _ N = 1 , a _ 0 = c _ 0 = 0 , d _ 0 = y '' _ 0 , d _ N = y '' _ N . \\end{align*}"}
+{"id": "2751.png", "formula": "\\begin{align*} \\int _ { \\mathrm { M } } | \\nabla u | ^ 2 d \\mu = \\int _ { \\mathrm { M } } ( \\lambda - q ) u ^ 2 d \\mu + \\int _ { \\mathrm { M } } f u d \\mu . \\end{align*}"}
+{"id": "321.png", "formula": "\\begin{align*} s _ h ( m , n ) = \\sum _ { k = 1 } ^ { \\infty } \\left ( \\gamma + \\Psi ( k + 1 ) \\right ) ^ m \\frac { ( - 1 ) ^ { k + 1 } } { ( k + 1 ) ^ n } , \\end{align*}"}
+{"id": "8162.png", "formula": "\\begin{align*} \\Gamma _ T ( X _ { q , t } ) = \\omega ( \\Gamma _ T ) \\left ( \\frac { 1 - q t | } { | 1 - q } \\right ) . \\end{align*}"}
+{"id": "7754.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\frac { h ( r ) } { r ^ k } = 0 \\ , , \\end{align*}"}
+{"id": "4584.png", "formula": "\\begin{align*} & ( \\alpha \\oplus \\beta ) ( T u + u ) = \\alpha ( T u ) + \\beta ( u ) , \\\\ & [ T u + u , T v + v , T w + w ] _ { \\rho } = [ T u , T v , T w ] + \\rho ( T u , T v ) w , \\end{align*}"}
+{"id": "872.png", "formula": "\\begin{align*} C = \\begin{pmatrix} \\underline { a } / r - \\underline { b } / r _ 1 \\\\ \\underline { a } / r - \\underline { a } ' / r ' \\end{pmatrix} . \\end{align*}"}
+{"id": "3399.png", "formula": "\\begin{align*} V ' ( \\rho ) = - S ( \\rho ) \\end{align*}"}
+{"id": "3779.png", "formula": "\\begin{align*} \\varphi ( p ( \\mathcal J ) ) _ { k + 1 + i } = \\varphi ( p ( \\mathcal J ) ) _ { k + i } \\oplus z ^ { i + 1 } , \\end{align*}"}
+{"id": "424.png", "formula": "\\begin{align*} P U _ t + ( A _ i U ) _ { x _ i } + A ^ T _ i U _ { x _ i } + C U + L _ C = 0 , t \\geq 0 , \\vec x = ( x _ 1 , x _ 2 , . . , x _ k ) \\in \\Omega . \\end{align*}"}
+{"id": "1171.png", "formula": "\\begin{align*} T \\left ( \\frac { 1 } { f - 1 } \\right ) = - \\frac { 1 } { ( f - 1 ) ^ { 2 } } T ( f - 1 ) + \\frac { 2 } { ( f - 1 ) ^ { 3 } } A ( f - 1 ) ^ { 2 } \\end{align*}"}
+{"id": "4478.png", "formula": "\\begin{align*} U : = ( u _ 1 , u _ 2 , u _ 3 ) , \\ \\ \\ \\ \\ U _ 0 : = ( u _ { 1 , 0 } , u _ { 2 , 0 } , u _ { 3 , 0 } ) . \\end{align*}"}
+{"id": "3558.png", "formula": "\\begin{align*} \\phi _ { n } = \\frac { ( - 1 ) ^ { n } } { n ! } = \\frac { ( - 1 ) ^ { n } } { \\Gamma ( n + 1 ) } . \\end{align*}"}
+{"id": "3799.png", "formula": "\\begin{align*} F _ { Y _ 1 | X } ( y | x ) = \\mathbb { P } ( Y _ 1 \\leq y | X = x ) = \\mathbb { P } ( Y \\leq y | X = x , D = 0 ) \\end{align*}"}
+{"id": "2479.png", "formula": "\\begin{align*} G ( 0 ) = \\int _ { \\mathbb { R } ^ N } | x | ^ 2 \\left ( a _ 2 | \\phi _ 0 | ^ 2 + a _ 1 | \\psi _ 0 | ^ 2 \\right ) d x \\geq 0 , \\end{align*}"}
+{"id": "3712.png", "formula": "\\begin{align*} \\tau _ M ( W _ M ^ + ( h ) ) = W _ M ^ + ( h ) \\mbox { a n d } \\tau _ M ( W ( \\gamma ) ) = W ( \\gamma ) . \\end{align*}"}
+{"id": "6682.png", "formula": "\\begin{align*} \\iint _ { \\R ^ { 2 N } } \\frac { | u ( x ) - u ( y ) | ^ 2 } { | x - y | ^ { N + 2 s } } \\ , d x \\ , d y & = \\iint _ { \\R ^ { 2 N } } \\frac { | v ( x ) - v ( y ) | ^ 2 } { | x - y | ^ { N + 2 s } } \\ , d x \\ , d y \\leq \\gamma \\ , \\| v \\| _ { H ^ 1 _ 0 ( \\Omega ) } = \\gamma \\ , \\| u - \\alpha \\| _ { H ^ 1 _ 0 ( \\Omega ) } , \\end{align*}"}
+{"id": "1705.png", "formula": "\\begin{align*} \\binom { z } { n } = \\frac { ( - 1 ) ^ n ( - z ) _ n } { n ! } . \\end{align*}"}
+{"id": "471.png", "formula": "\\begin{align*} f ' + a [ x ; \\R , W ] f = b [ x ; \\R , W ] , \\end{align*}"}
+{"id": "1752.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\textrm { o u t } } & = 1 - \\mathcal { P } _ 1 - \\mathcal { P } _ 3 = e ^ { - \\frac { \\mu ^ 2 _ { } \\Gamma _ 0 } { \\rho \\lambda _ { } } } + \\sum ^ M _ { m = 1 } \\mathcal { V } ^ { ( M , m ) } I _ + ^ { ( m ) } . \\end{align*}"}
+{"id": "5228.png", "formula": "\\begin{align*} \\Phi M & = M ^ { - 1 } \\Phi . \\\\ \\intertext { M u l t i p l y i n g b y $ \\Phi ^ { - 1 } $ o n t h e l e f t a n d r i g h t o f b o t h s i d e s y i e l d s a n a n a l o g o u s s t a t e m e n t f o r $ \\Phi ^ { - 1 } $ : } M \\Phi ^ { - 1 } & = \\Phi ^ { - 1 } M ^ { - 1 } . \\end{align*}"}
+{"id": "209.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c ) = 1 \\\\ a , b , c \\geq 1 } } \\left ( \\frac { 1 } { 1 - x ^ a y ^ b z ^ c } \\right ) ^ { \\frac { c ^ 4 } { a ^ 2 b ^ 3 } } = \\exp \\left \\{ L i _ 2 ( x ) L i _ 3 ( y ) L i _ { - 4 } ( z ) \\right \\} \\end{align*}"}
+{"id": "7237.png", "formula": "\\begin{align*} \\overline { u } _ h : = \\zeta ( v + m _ h ) + ( 1 - \\zeta ) \\left ( \\varphi z _ h + ( 1 - \\varphi ) u _ h \\right ) \\end{align*}"}
+{"id": "2618.png", "formula": "\\begin{align*} \\mathcal { D } _ s ^ { ( j ) } f _ { j , m _ j } ( x , y ) = \\sum _ { k \\in \\mathbb { Z } ^ 2 } a _ { j , m _ j , k , s } e ^ { \\pi i \\lambda ^ \\gamma k \\cdot ( x , y ) } \\textrm { f o r $ ( x , y ) \\in Q _ { m _ j } $ } , \\end{align*}"}
+{"id": "2652.png", "formula": "\\begin{align*} \\begin{bmatrix} \\cosh ( \\alpha x _ { j - 1 } ) & \\sinh ( \\alpha x _ { j - 1 } ) \\\\ \\cosh ( \\alpha x _ j ) & \\sinh ( \\alpha x _ j ) \\end{bmatrix} \\begin{bmatrix} a _ j \\\\ b _ j \\end{bmatrix} = \\begin{bmatrix} y _ { j - 1 } \\\\ y _ { j } \\end{bmatrix} . \\end{align*}"}
+{"id": "4910.png", "formula": "\\begin{align*} F ( k , t , \\mathbf { p } _ { n + 1 } ) \\overset { } { = } \\Pr ( X _ t \\le k \\ , \\vert \\ , X _ 0 = 0 , \\mathbf { p } _ { n + 1 } ) , k \\in S , \\end{align*}"}
+{"id": "7309.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n x _ i ^ { e _ i } = r , \\end{align*}"}
+{"id": "2236.png", "formula": "\\begin{align*} \\varphi ( q _ { n * } ( \\xi , \\ldots , \\xi , a , b ) ) = q _ { n * } ( \\varphi ( \\xi ) , \\ldots , \\varphi ( \\xi ) , \\varphi ( a ) , \\varphi ( b ) ) \\end{align*}"}
+{"id": "4874.png", "formula": "\\begin{align*} \\begin{dcases} L u = f ( x , u ) & , \\\\ u = 0 & , \\end{dcases} \\end{align*}"}
+{"id": "6570.png", "formula": "\\begin{align*} \\begin{pmatrix} A & X ^ * \\\\ X & B \\end{pmatrix} \\ge 0 \\Longrightarrow | X | \\le A \\# ( V ^ * B V ) \\le \\frac { A + V ^ * B V } { 2 } , \\end{align*}"}
+{"id": "5097.png", "formula": "\\begin{align*} \\check { A } ( t ) = \\begin{cases} A ( t ) , & t \\in [ 0 , \\tau _ { \\max } - \\frac { \\delta } { 2 } ] , \\\\ \\widetilde A ( t ) , & t \\in [ \\tau _ { \\max } - \\frac { \\delta } { 2 } , \\min \\{ \\tau _ { \\max } + \\frac { \\delta } { 2 } , T _ { \\max } \\} ] , \\end{cases} \\end{align*}"}
+{"id": "6283.png", "formula": "\\begin{align*} \\sigma _ { K , N } ^ { ( t ) } ( \\theta ) : = \\begin{cases} \\displaystyle \\frac { \\sin ( t \\theta \\sqrt { K / N } ) } { \\sin ( \\theta \\sqrt { K / N } ) } & \\textrm { i f } \\ N \\pi ^ { 2 } > K \\theta ^ { 2 } > 0 , \\crcr t & \\textrm { i f } \\ K = 0 , \\crcr \\displaystyle \\frac { \\sinh ( t \\theta \\sqrt { - K / N } ) } { \\sinh ( \\theta \\sqrt { - K / N } ) } & \\textrm { i f } \\ K < 0 . \\end{cases} \\end{align*}"}
+{"id": "5534.png", "formula": "\\begin{align*} \\mathrm { I } ( \\xi _ { 1 } , \\mathcal { T } | X , \\eta _ { p } ) = \\inf _ { n } \\int _ { X } \\mathrm { I } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) d \\eta _ { p } ( x ) = \\inf _ { n } \\mathrm { I } ( \\xi _ { 1 } , \\xi _ { n } | X , \\eta _ { p } ) . \\end{align*}"}
+{"id": "7170.png", "formula": "\\begin{align*} x ' _ { h , j } : = R _ h \\left ( \\cos \\frac { 2 \\pi j } { 2 ^ { h } } , \\sin \\frac { 2 \\pi j } { 2 ^ { h } } \\right ) , \\mbox { f o r } j \\in \\left \\{ 1 , \\dots , 2 ^ { h } \\right \\} . \\end{align*}"}
+{"id": "1808.png", "formula": "\\begin{align*} \\nu ( \\N ) \\equiv \\int _ { \\Lambda ^ { * } } \\nu ( a _ p ^ * a _ p ) \\d p = \\int _ { \\Lambda ^ { * } } f _ 0 ( p ) \\d p < \\infty \\ , \\end{align*}"}
+{"id": "9012.png", "formula": "\\begin{align*} w _ t \\varphi ^ a _ { t i } = - w _ { t i } \\varphi ^ a _ t , \\end{align*}"}
+{"id": "2624.png", "formula": "\\begin{align*} \\widetilde { T } _ \\sharp & = \\widetilde { T } _ l ( f _ { 1 , \\sharp } , f _ { 2 , \\sharp } ) , \\\\ \\widetilde { T } _ \\flat & = \\widetilde { T } _ l ( f _ { 1 , \\flat } , f _ 2 ) + \\widetilde { T } _ l ( f _ { 1 , \\sharp } , f _ { 2 , \\flat } ) , \\\\ \\widetilde { T } _ { } & = \\widetilde { T } _ l ( f _ { 1 , \\sharp } , f _ { 2 , } ) + \\widetilde { T } _ l ( f _ { 1 , } , f _ 2 ) . \\end{align*}"}
+{"id": "7003.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ r b _ j \\underline { a } ^ { \\lambda _ j } \\tilde { \\textbf { Q } } ^ { \\lambda _ j } = \\sum _ { j = 1 } ^ r \\tilde b _ j \\tilde { \\textbf { Q } } ^ { \\lambda _ j } \\end{align*}"}
+{"id": "8790.png", "formula": "\\begin{align*} & ( u ^ 3 + 8 t u ^ 2 + 1 2 t ^ 2 u + 7 u ^ 2 + 3 8 t u + 1 2 t ^ 2 + 2 2 u + 3 0 t + 1 6 ) c _ 2 \\\\ & = ( u ^ 3 + 4 t u ^ 2 + 6 t ^ 2 u + 1 1 u ^ 2 + 2 8 t u + 6 t ^ 2 + 3 4 u + 2 4 t + 2 4 ) b _ 2 \\\\ & + ( u ^ 3 + 4 t u ^ 2 + 4 t u + 7 u ^ 2 + 6 u ) c _ 3 - ( u ^ 3 + 2 t u ^ 2 + 2 t u + 5 u ^ 2 + 4 u ) b _ 3 \\end{align*}"}
+{"id": "3829.png", "formula": "\\begin{align*} \\boldsymbol { d } _ { \\mathcal { V } } ( ( s _ 1 , s _ 2 ) , ( s _ 1 ^ \\prime , s _ 2 ^ \\prime ) ) = \\boldsymbol { d } _ { \\mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \\prime ) \\vee \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \\prime ) . \\end{align*}"}
+{"id": "819.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ N P _ i ( x _ 1 , \\ldots , x _ m ) G _ i ( x _ 1 , \\ldots , x _ m ) = 0 \\end{align*}"}
+{"id": "7711.png", "formula": "\\begin{align*} \\lambda \\in ( U _ r ) _ \\R ^ \\perp \\cap L _ { j + 1 } ' = ( ( U _ r ) _ \\R \\oplus ( L _ { r + 1 } ) _ \\R ) \\cap L _ { j + 1 } ' . \\end{align*}"}
+{"id": "1805.png", "formula": "\\begin{align*} D ^ * _ k = D _ { - k } \\forall k \\in \\Lambda ^ * \\ . \\end{align*}"}
+{"id": "2945.png", "formula": "\\begin{align*} \\gamma = \\begin{bmatrix} \\alpha & x \\\\ \\beta & y \\end{bmatrix} . \\end{align*}"}
+{"id": "8451.png", "formula": "\\begin{align*} \\left ( \\frac { z } { 2 } \\right ) ^ { \\nu } K _ { \\nu } ( z ) = \\frac { 2 ^ { - 2 \\nu } \\ , \\sqrt { \\pi } \\ , e ^ { - z } } { \\Gamma \\left ( \\frac { 1 } { 2 } - \\nu \\right ) } \\ , \\intop _ { 0 } ^ { \\infty } e ^ { - 2 z t } t ^ { - \\nu - \\frac { 1 } { 2 } } ( t + 1 ) ^ { - \\nu - \\frac { 1 } { 2 } } d t , \\ , \\ , \\ , \\ , ( \\nu ) < \\frac { 1 } { 2 } , \\ , \\ , \\ , \\ , | \\arg ( z ) | < \\frac { \\pi } { 2 } . \\end{align*}"}
+{"id": "8390.png", "formula": "\\begin{align*} [ s , f ] = \\sum _ { U _ 1 , U _ 2 = V ^ { ( \\nu _ 1 ) } , W , \\kappa ( W ) } [ s , f _ { U _ 1 , U _ 2 } ] . \\end{align*}"}
+{"id": "7961.png", "formula": "\\begin{align*} \\nabla _ \\xi ^ 2 H ( \\xi ) \\ , \\xi = 0 \\end{align*}"}
+{"id": "4171.png", "formula": "\\begin{align*} \\| \\varphi _ s ( a ) \\varphi _ t ( b ) - \\varphi _ { s t } ( a b ) \\| & = \\| \\varphi ( \\lambda _ s ( a ) ) \\varphi ( \\lambda _ t ( b ) ) - \\varphi ( \\lambda _ { s t } ( a b ) ) \\| \\\\ & = \\| \\varphi ( \\lambda _ s ( a ) ) \\varphi ( \\lambda _ t ( b ) ) - \\varphi ( \\lambda _ { s } ( a ) \\lambda _ t ( b ) ) \\| < \\varepsilon . \\end{align*}"}
+{"id": "5822.png", "formula": "\\begin{align*} { { \\boldsymbol { x } } _ k } { } { \\left ( { { \\boldsymbol { W } } _ k ^ T { { \\boldsymbol { \\Omega } } _ k } { { \\boldsymbol { W } } _ k } } \\right ) ^ { - 1 } } { \\boldsymbol { W } } _ k ^ { } { { \\boldsymbol { \\Omega } } _ k } { { \\boldsymbol { d } } _ k } , \\end{align*}"}
+{"id": "8382.png", "formula": "\\begin{align*} h ^ { \\lor } ( u ^ { \\lor } , v ^ { \\lor } ) = h \\bigl ( \\Psi _ h ^ { - 1 } ( v ^ { \\lor } ) , \\Psi _ h ^ { - 1 } ( u ^ { \\lor } ) \\bigr ) . \\end{align*}"}
+{"id": "6541.png", "formula": "\\begin{align*} \\widehat { \\mu } ( t ) = \\exp \\left [ - \\frac { 1 } { 2 } a t ^ 2 + i b _ 0 t + \\int _ { - \\infty } ^ { \\infty } ( e ^ { i t \\lambda } - 1 ) \\ , \\nu ( d \\lambda ) \\right ] \\end{align*}"}
+{"id": "7263.png", "formula": "\\begin{align*} X _ i = d _ { i , N } + g _ { i - 1 , N } - g _ { i , N } + Y _ { i , N } \\ , . \\end{align*}"}
+{"id": "3310.png", "formula": "\\begin{align*} f ( \\xi ) = ( \\xi - 1 ) ^ { \\delta _ 1 } ( \\xi + 1 ) ^ { \\delta _ { - 1 } } \\kappa ( \\xi ) , \\end{align*}"}
+{"id": "8429.png", "formula": "\\begin{align*} \\sigma _ { p } ( z ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , e ^ { - \\lambda _ { n } z } , \\ , \\ , \\ , \\ , \\ , \\ , ( z ) > 0 . \\end{align*}"}
+{"id": "3818.png", "formula": "\\begin{align*} f _ { \\lambda _ 1 , 1 } ( s _ 1 , s _ 2 ) & = \\sup _ { y _ 1 ' \\in \\mathcal { Y } _ 1 } \\{ f ( y _ 1 ' , y _ 2 , x _ 2 ) - \\lambda _ 1 c _ 1 ( ( y _ 1 , x _ 1 ) , ( y _ 1 ' , x _ 2 ) ) \\} \\\\ f _ { \\lambda _ 2 , 2 } ( s _ 1 , s _ 2 ) & = \\sup _ { y _ 2 ' \\in \\mathcal { Y } _ 2 } \\{ f ( y _ 1 , y _ 2 ' , x _ 1 ) - \\lambda _ 2 c _ 2 ( ( y _ 2 , x _ 2 ) , ( y _ 2 ' , x _ 1 ) \\} , \\end{align*}"}
+{"id": "5596.png", "formula": "\\begin{align*} I I = \\sum _ { A \\in \\mathcal { P } } \\alpha ' ( A ) \\log \\frac { \\beta ' ( A ) } { \\beta ( A ) } \\le \\sum _ { A \\in \\mathcal { P } } \\alpha ' ( A ) \\log \\left ( \\max _ { A \\in \\mathcal { P } } \\frac { \\beta ' ( A ) } { \\beta ( A ) } \\right ) = \\log \\left ( \\max _ { A \\in \\mathcal { P } } \\frac { \\beta ' ( A ) } { \\beta ( A ) } \\right ) . \\end{align*}"}
+{"id": "1369.png", "formula": "\\begin{align*} \\langle x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 : x _ i x _ j x _ i ^ { - 1 } = x _ { - i + 2 j \\bmod 5 } \\rangle . \\end{align*}"}
+{"id": "7251.png", "formula": "\\begin{align*} \\left ( \\int \\sup _ { k \\leq n } \\left | \\sum _ { i = 1 } ^ k ( f \\circ T _ { \\gamma } ^ { i } ( x ) - \\nu _ { \\gamma } ( f ) ) - \\sum _ { i = 1 } ^ k Z _ i ( x , y ) \\right | ^ 2 \\ ! \\ ! \\ ! \\nu _ { \\gamma } ( d x ) d y \\ ! \\right ) ^ { \\frac 1 2 } \\ ! \\ ! \\ ! = o ( n ^ { \\max ( \\gamma , 1 / 4 ) } ( \\log n ) ^ { \\eta } ) \\ , , \\end{align*}"}
+{"id": "4937.png", "formula": "\\begin{align*} U \\Lambda = \\frac { 1 } { n } \\left [ \\mathbf { u } _ 0 , \\mathbf { u } _ 1 , 2 \\mathbf { u } _ 2 , \\ , . . . , \\ , k \\mathbf { u } _ k , \\ , . . . . , \\ , ( n { - } 1 ) \\mathbf { u } _ { n - 1 } , n \\mathbf { u } _ n \\right ] . \\end{align*}"}
+{"id": "8255.png", "formula": "\\begin{align*} T _ { 1 } G _ { k , Y } = \\sqrt { x } \\frac { d G _ { k , Y } } { d x } - \\frac { k ^ 2 + m } { 2 \\sqrt { x } } G _ { k , Y } - \\frac { t } { \\sqrt { x } } T _ 2 G _ { k , Y } . \\end{align*}"}
+{"id": "1646.png", "formula": "\\begin{align*} g ( ( \\nabla _ { X } { f } ) \\xi _ i , Z ) & = - \\frac 1 2 \\ , g ( N ^ { \\ , ( 1 ) } ( \\xi _ i , Z ) , { f } X ) - g ( { f } X , { f } Z ) + \\frac 1 2 \\ , N ^ { \\ , ( 5 ) } ( X , \\xi _ i , Z ) . \\end{align*}"}
+{"id": "6559.png", "formula": "\\begin{align*} & I _ m ( a , \\beta _ 1 , . . . , \\beta _ m ) \\\\ & = \\int _ 0 ^ \\infty . . . \\int _ 0 ^ \\infty \\frac { t _ 1 ^ { - \\beta _ 1 } . . . t _ { m - 1 } ^ { - \\beta _ { m - 1 } } } { ( 1 + t _ 1 + . . . + t _ { m - 1 } ) ^ { a - 1 + { \\beta _ m } } } d t _ 1 . . . d t _ { m - 1 } \\int _ 0 ^ \\infty \\frac { 1 } { ( 1 + q _ m ) ^ a q _ m ^ { \\beta _ m } } d q _ m \\\\ & = B ( 1 - \\beta _ m , a + \\beta _ m - 1 ) I _ m ( a - 1 + \\beta _ m , \\beta _ 1 , . . . , \\beta _ m ) \\\\ & = \\frac { \\prod _ { i = 1 } ^ m \\Gamma ( 1 - \\beta _ i ) \\Gamma ( a - m + \\sum _ { i = 1 } ^ m \\beta _ i ) } { \\Gamma ( a ) } \\end{align*}"}
+{"id": "8053.png", "formula": "\\begin{align*} \\| Z _ { n + p } h - Z _ n h \\| ^ 2 & = \\langle h , ( Z _ n - Z _ { n + p } ) ^ 2 h \\rangle \\\\ & = \\langle h , ( Z _ n - Z _ { n + p } ) ^ { 1 / 2 } ( Z _ n - Z _ { n + p } ) ( Z _ n - Z _ { n + p } ) ^ { 1 / 2 } h \\rangle \\\\ & \\le \\| Z _ n - Z _ { n + p } \\| \\langle h , ( Z _ n - Z _ { n + p } ) h \\rangle . \\end{align*}"}
+{"id": "2461.png", "formula": "\\begin{align*} \\varphi = & \\pi _ 1 ( f _ 0 ) \\varphi + \\pi _ 1 ( f _ 1 ) P ( X ^ M _ 1 ) \\varphi \\\\ = & \\pi _ 1 ( f _ 0 ) \\pi _ 2 ( f _ 0 ) \\varphi + \\pi _ 1 ( f _ 0 ) \\pi _ 2 ( f _ 1 ) P ( X ^ M _ 2 ) \\varphi \\\\ & + \\pi _ 1 ( f _ 1 ) \\pi _ 2 ( f _ 0 ) P ( X _ M ^ 1 ) \\varphi + \\pi _ 1 ( f _ 1 ) \\pi _ 2 ( f _ 1 ) P ( X ^ M _ 2 ) P ( X ^ M _ 1 ) \\varphi \\\\ \\vdots & \\\\ = & \\sum _ { i _ 1 , \\ldots , i _ k \\in \\{ 0 , 1 \\} } \\pi _ 1 ( f _ { i _ 1 } ) \\cdots \\pi _ k ( f _ { i _ k } ) P ( X ^ M _ k ) ^ { i _ k } \\cdots P ( X ^ M _ 1 ) ^ { i _ 1 } \\varphi \\end{align*}"}
+{"id": "8239.png", "formula": "\\begin{align*} f _ l ( x ) = \\frac { \\partial ^ l } { \\partial t ^ l } G _ k ( x , t ) \\Big | _ { t = 0 } , \\end{align*}"}
+{"id": "4785.png", "formula": "\\begin{align*} R _ K ( X , \\varepsilon , t , a , b ) = R ^ - _ K ( X , \\varepsilon , t , a , b , L ) + R ^ + _ K ( X , \\varepsilon , t , a , b , L ) . \\end{align*}"}
+{"id": "2805.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q _ 1 ) w = ( q _ 2 - q _ 1 ) v _ 2 + [ \\Delta , \\psi ] v \\ ; \\mathrm { i n } \\ ; \\mathrm { M } . \\end{align*}"}
+{"id": "713.png", "formula": "\\begin{align*} \\mu _ { S , 0 } ( e ) = \\int _ { 0 } ^ e \\tilde \\gamma _ { S } ( e ' ) \\d e ' - \\frac 1 { L _ { S } } \\int _ { 0 } ^ { L _ { S } } \\int _ { 0 } ^ e \\tilde \\gamma _ { S } ( e ' ) \\d e ' \\d e \\ , . \\end{align*}"}
+{"id": "4060.png", "formula": "\\begin{align*} \\sum _ { A = 1 } ^ { \\infty } \\frac { G _ k ( A ^ 2 \\slash X ) } { A } & = \\frac { 1 } { 2 \\pi i } \\sum _ { A = 1 } ^ { \\infty } \\int _ { \\mathrm { R e } ( t ) = \\frac { 3 } { 4 } } \\frac { \\Gamma ( k + t ) ^ 2 } { ( 2 \\pi ) ^ { 2 t } } \\frac { X ^ t } { A ^ { 2 t + 1 } } \\frac { d t } { t } \\\\ & = \\frac { 1 } { 2 \\pi i } \\int _ { \\mathrm { R e } ( t ) = \\frac { 3 } { 4 } } \\frac { \\Gamma ( k + t ) ^ 2 } { ( 2 \\pi ) ^ { 2 t } } X ^ t \\ , \\zeta ( 2 t + 1 ) \\frac { d t } { t } . \\end{align*}"}
+{"id": "857.png", "formula": "\\begin{align*} u _ { t t } ( 0 ) = - H _ { \\lambda } ( u _ { 0 } ) \\geq \\xi > 0 . \\end{align*}"}
+{"id": "5652.png", "formula": "\\begin{align*} m _ \\nu ( a , b ) = \\inf _ { \\mathcal { T } ( a , b ) } \\max _ { t \\in \\mathbb { R } } J _ \\nu ( t \\star ( u , v ) ) , m _ { r , \\nu } ( a , b ) = \\inf _ { \\mathcal { T } _ r ( a , b ) } \\max _ { t \\in \\mathbb { R } } J _ \\nu ( t \\star ( u , v ) ) . \\end{align*}"}
+{"id": "7440.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { + \\infty } \\sum \\limits _ { p = 0 } ^ { 1 } \\varepsilon ^ { p \\alpha + k - 1 } \\ , N _ { p \\alpha + k - 1 } \\left ( \\frac { x } { \\varepsilon } , t \\right ) \\end{align*}"}
+{"id": "4559.png", "formula": "\\begin{align*} h \\circ F = h ( \\partial / \\partial X _ 1 , \\ldots , \\partial / \\partial X _ r ) \\circ F , \\end{align*}"}
+{"id": "5993.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } ( [ u ( - t ) , 1 ] ) f ( [ - 1 , x ] ) = e ^ { \\pi i x ^ 2 t } f ( [ - 1 , x ] ) \\end{align*}"}
+{"id": "7693.png", "formula": "\\begin{align*} z ^ \\ast = \\frac { z ^ + - z ^ - } { 2 \\Vert z ^ + \\Vert ^ 2 } \\in L _ \\R , \\end{align*}"}
+{"id": "3434.png", "formula": "\\begin{align*} | \\nabla X | ^ 2 = & \\ , \\frac { 1 } { 4 } \\left ( | S | ^ 2 + | A | ^ 2 \\right ) , \\\\ \\nabla ^ i X ^ j \\nabla _ j X _ i = & \\ , \\frac { 1 } { 4 } \\left ( | S | ^ 2 - | A | ^ 2 \\right ) , \\\\ \\left \\langle S , A \\right \\rangle = & \\ 0 . \\end{align*}"}
+{"id": "330.png", "formula": "\\begin{align*} \\prod _ { \\substack { \\gcd ( j _ 1 , j _ 2 , j _ 3 , j _ 4 , k ) = 1 \\\\ j _ 1 , j _ 2 , j _ 3 , j _ 4 < k \\\\ j _ 1 , j _ 2 , j _ 3 , j _ 4 \\geq 1 ; k \\geq 2 } } \\left ( \\frac { 1 } { 1 - y ^ { j _ 1 + j _ 2 + j _ 3 + j _ 4 } z ^ k } \\right ) ^ { \\frac { 1 } { k } } = \\exp \\left [ \\sum _ { k = 1 } ^ { \\infty } \\left ( \\frac { y } { 1 } + \\frac { y ^ 2 } { 2 } + \\cdots + \\frac { y ^ k } { k } \\right ) ^ { 4 } \\frac { z ^ { k + 1 } } { ( k + 1 ) ^ 3 } \\right ] . \\end{align*}"}
+{"id": "2217.png", "formula": "\\begin{align*} C _ 1 ( y _ 0 ) & = 2 \\log \\left ( 1 - y _ 0 ^ { - 1 } \\right ) ^ { - 1 } \\frac { y _ 0 } { \\log y _ 0 } + \\frac { C _ 0 ( y _ 0 ) y _ 0 } { y _ 0 - 1 } , \\\\ C _ 2 ( y _ 0 ) & = C _ 1 ( y _ 0 ) + \\sup _ { t \\ge y _ 0 } \\frac { 1 } { 2 ( t - 1 ) R ( t ) } . \\end{align*}"}
+{"id": "2213.png", "formula": "\\begin{align*} \\frac { d } { d v } ( v \\omega ( v - 1 ) ) & = \\omega ( v - 2 ) + \\omega ' ( v - 1 ) \\ge \\frac { 1 } { 2 } - \\rho ( v - 1 ) \\ge \\frac { 1 } { 2 } - \\rho ( 2 ) = \\log 2 - \\frac { 1 } { 2 } , \\\\ \\frac { d } { d v } ( v \\omega ( v - 1 ) ) & \\le 1 + \\rho ( v - 1 ) \\le 1 + \\rho ( 2 ) = 2 - \\log 2 , \\end{align*}"}
+{"id": "6547.png", "formula": "\\begin{align*} \\xi _ { F ^ \\ast } ( s ) = \\overline { \\xi _ F ( \\bar { s } ) } . \\end{align*}"}
+{"id": "7550.png", "formula": "\\begin{align*} P _ i = P _ { i - 1 } + W _ i , \\ \\ \\ \\ \\ i = 1 , 2 , \\dots , \\end{align*}"}
+{"id": "1793.png", "formula": "\\begin{align*} \\psi _ { \\mathrm { S l a t e r } } ( x _ 1 , \\ldots , x _ N ) \\equiv ( 1 / \\sqrt { N ! } ) \\det [ e _ { k _ i } ( x _ j ) ] _ { i , j = 1 } ^ N \\end{align*}"}
+{"id": "2983.png", "formula": "\\begin{align*} \\det \\begin{pmatrix} t _ 1 ^ { k _ 1 } & t _ 2 ^ { k _ 1 } & \\dots & t _ n ^ { k _ 1 } \\\\ F ( t _ 1 ) t _ 1 ^ { k _ 2 } & F ( t _ 2 ) t _ 2 ^ { k _ 2 } & \\dots & F ( t _ n ) t _ n ^ { k _ 2 } \\\\ t _ 1 ^ { k _ 3 } & t _ 2 ^ { k _ 3 } & \\dots & t _ n ^ { k _ 3 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ t _ 1 ^ { k _ n } & t _ 2 ^ { k _ n } & \\dots & t _ n ^ { k _ n } \\end{pmatrix} \\end{align*}"}
+{"id": "5084.png", "formula": "\\begin{align*} A _ t ( N T ) - A _ t ( 0 ) = N T , \\end{align*}"}
+{"id": "7011.png", "formula": "\\begin{align*} I _ 0 ( f , i ) = \\{ \\ell \\in I \\ | \\ \\lambda _ j ( Q _ \\ell ) \\neq 0 \\mbox { f o r s o m e } j , \\ 0 \\leq j \\leq r \\} . \\end{align*}"}
+{"id": "5178.png", "formula": "\\begin{align*} \\lambda \\preceq \\mu \\lambda _ i \\leq \\mu _ i i = 1 , 2 , \\dots , k . \\end{align*}"}
+{"id": "7661.png", "formula": "\\begin{align*} | \\nabla _ g d _ { x _ 0 } | = 1 . \\end{align*}"}
+{"id": "6742.png", "formula": "\\begin{align*} \\underline { C } _ K : = \\bigcap _ { s \\in \\underline { S } } \\Gamma _ { H , H ^ * } ^ { - 1 } ( R ^ * ) ^ { - s } ( W _ { K } ^ * ) ^ c . \\end{align*}"}
+{"id": "1072.png", "formula": "\\begin{align*} \\| e ^ { - t H ^ { \\beta } } g \\| _ { \\exp L ^ p } & = \\inf \\left \\{ \\lambda > 0 : \\int _ { \\R ^ d } \\left ( \\exp \\left | \\frac { e ^ { - t H ^ { \\beta } } g } { \\lambda } \\right | ^ p - 1 \\right ) \\ , d x \\leq 1 \\right \\} \\\\ & \\leq \\inf \\left \\{ \\lambda > 0 : \\int _ { \\R ^ d } \\left ( \\exp \\left | \\frac { { C } g } { \\lambda } \\right | ^ p - 1 \\right ) \\ , d x \\leq 1 \\right \\} = { C } \\| g \\| _ { \\exp L ^ { p } } . \\end{align*}"}
+{"id": "2452.png", "formula": "\\begin{align*} & ( m , \\gamma ) \\mapsto m \\cdot \\gamma \\\\ & ( m , \\gamma ) \\mapsto m \\\\ & ( m , \\gamma ) ( m \\cdot \\gamma , \\gamma ' ) = ( m , \\gamma \\gamma ' ) \\\\ & ( m , \\gamma ) \\mapsto ( m \\cdot \\gamma , \\gamma ^ { - 1 } ) \\end{align*}"}
+{"id": "977.png", "formula": "\\begin{align*} F ^ { 2 } = I ( F \\not = \\overset { - } { + } I ) \\end{align*}"}
+{"id": "3349.png", "formula": "\\begin{align*} \\psi ( X ) = \\sum _ { j = 1 } ^ d \\pi ( S _ j ) X \\overline { T _ j } = ( I \\otimes S ^ * ) \\left ( \\sum _ { j = 1 } ^ d \\pi ( S _ j ) X Z _ j ^ * \\right ) ( I \\otimes S ^ { - 1 * } ) . \\end{align*}"}
+{"id": "5826.png", "formula": "\\begin{align*} { { \\boldsymbol { P } } _ { k | k } } = \\left ( { { \\boldsymbol { I } } - { { \\boldsymbol { K } } _ k } { { \\boldsymbol { H } } _ k } } \\right ) { { \\boldsymbol { P } } _ { k | k - 1 } } { \\left ( { { \\boldsymbol { I } } - { { \\boldsymbol { K } } _ k } { { \\boldsymbol { H } } _ k } } \\right ) ^ { } } + { { \\boldsymbol { K } } _ k } { { \\boldsymbol { R } } _ k } { \\boldsymbol { K } } _ k ^ T . \\end{align*}"}
+{"id": "2267.png", "formula": "\\begin{align*} w ( z ) = \\Phi ( z ) - \\frac { 1 } { \\pi } \\iint _ D \\frac { f ( \\zeta ) } { \\zeta - z } \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "6627.png", "formula": "\\begin{align*} \\varPsi _ { \\lambda } ( u \\otimes a ) = \\sum _ { t \\in T } \\varPsi _ { \\lambda t } ( u \\otimes a ) \\otimes b _ t \\end{align*}"}
+{"id": "6352.png", "formula": "\\begin{align*} g ( t ) : = \\frac { d ( s _ 1 ) } { d ( s _ 2 ) } \\left [ f \\left ( \\alpha ( s _ 2 ) + \\frac { d ( s _ 2 ) } { d ( s _ 1 ) } ( t - \\alpha ( s _ 1 ) ) \\right ) - s _ 2 \\right ] , \\qquad \\forall \\ , t \\in \\R . \\end{align*}"}
+{"id": "2331.png", "formula": "\\begin{align*} g _ n ( \\pmb { t } _ n , \\pmb { x } _ n , r , z , t , x ) = G _ { t - t _ { n } } ( x - x _ { n } ) \\times \\dots \\times G _ { t _ 1 - r } ( x _ 1 - z ) . \\end{align*}"}
+{"id": "6148.png", "formula": "\\begin{align*} a + b & = \\begin{cases} H _ p \\mbox { i f } a = b , \\ , a , b \\ne 0 \\\\ \\{ a , b \\} \\mbox { i f } a \\ne b , \\ , a , b \\ne 0 \\\\ \\{ a \\} \\mbox { i f } b = 0 \\\\ \\{ b \\} \\mbox { i f } a = 0 \\end{cases} \\\\ a \\cdot b & = k \\mbox { w h e r e } 0 \\le k < p \\mbox { a n d } k \\equiv a b \\mbox { m o d p } . \\end{align*}"}
+{"id": "5520.png", "formula": "\\begin{align*} P _ { \\mu , x } ^ { n } \\left ( L _ { x } g ^ { - 1 } , A \\right ) = P _ { \\mu , g . x } ^ { n } \\left ( L _ { g . x } , g . A \\right ) . \\end{align*}"}
+{"id": "5223.png", "formula": "\\begin{align*} z ' _ { k , l } = \\sigma \\left ( s _ l ( | z _ k | - \\theta _ { 0 , l } ) \\right ) \\sigma \\left ( s _ l ( \\theta _ { 1 , l } - | z _ k | ) \\right ) \\frac { z _ k } { | z _ k | } , \\end{align*}"}
+{"id": "2636.png", "formula": "\\begin{align*} T _ l ( f _ 1 , f _ 2 ) ( x , y ) = \\int _ { \\mathbb { R } } \\ ! f _ 1 \\left ( x + P _ 1 ( t ) , y \\right ) f _ 2 \\left ( x , y + P _ 2 ( t ) \\right ) \\psi \\left ( 2 ^ l t \\right ) t ^ { - 1 } \\ , \\mathrm { d } t . \\end{align*}"}
+{"id": "423.png", "formula": "\\begin{align*} \\tilde W ^ T \\tilde \\Lambda \\tilde W = \\begin{bmatrix} W - \\epsilon \\Lambda ^ { - 1 } T ^ T \\tilde F \\\\ \\ , \\ - \\epsilon \\tilde F _ { \\tau } \\end{bmatrix} ^ T \\begin{bmatrix} \\Lambda & 0 \\\\ 0 & - 1 / u _ n \\end{bmatrix} \\begin{bmatrix} W - \\epsilon \\Lambda ^ { - 1 } T ^ T \\tilde F \\\\ \\ , \\ \\ - \\epsilon \\tilde F _ { \\tau } \\end{bmatrix} . \\end{align*}"}
+{"id": "8753.png", "formula": "\\begin{align*} \\theta _ + ( x , y , \\xi ( \\omega ) ) & = \\frac { 1 } { 2 } \\big ( x ^ 2 \\delta _ 0 ( \\xi ( \\omega ) ) + ( 1 - x ) ^ 2 \\delta _ 1 ( \\xi ( \\omega ) ) \\big ) + f ( x , y , \\xi ( \\omega ) ) \\\\ \\theta _ { - } ( x , y , \\xi ( \\omega ) ) & = \\frac { 1 } { 2 } \\big ( x ^ 2 \\delta _ 0 ( \\xi ( \\omega ) ) - ( 1 - x ) ^ 2 \\delta _ 1 ( \\xi ( \\omega ) ) \\big ) - f ( x , y , \\xi ( \\omega ) ) \\end{align*}"}
+{"id": "2422.png", "formula": "\\begin{align*} \\max _ { \\abs { s - z } = r } \\abs { \\log \\Bigl ( \\zeta ( s ) \\frac { s - 1 } { s } \\Bigr ) } \\le \\frac { 2 r } { R - r } \\max _ { \\abs { s - z } = R } \\log \\abs { \\zeta ( s ) \\frac { s - 1 } { s } } + \\frac { R + r } { R - r } \\abs { \\log \\Bigl ( \\zeta ( z ) \\frac { z - 1 } { z } \\Bigr ) } . \\end{align*}"}
+{"id": "8484.png", "formula": "\\begin{align*} \\sigma _ { p } ( z ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , e ^ { - \\lambda _ { n } z } , \\ , \\ , \\ , \\ , \\ , \\ , ( z ) > 0 , \\end{align*}"}
+{"id": "6303.png", "formula": "\\begin{align*} F : \\R ^ n \\to \\R ; F ( x ) : = f \\circ \\varphi ^ { - 1 } ( x ) - C \\frac { | x | ^ 2 } { 2 } , \\end{align*}"}
+{"id": "84.png", "formula": "\\begin{align*} { \\rm W F } _ h ( A ) \\subset { \\rm e l l } _ h ( B _ 1 ) , L \\subset { \\rm e l l } _ h ( A ) , L \\cap { \\rm W F } _ h ( B ) = \\varnothing \\\\ q \\geq 0 { \\rm W F } _ h ( B _ 1 ) , { \\rm I m } \\ , P \\lesssim - h H ^ s _ h L . \\end{align*}"}
+{"id": "8199.png", "formula": "\\begin{align*} \\underset { t \\to \\infty } { \\lim } \\ : P ^ \\omega ( E _ t \\cap S _ t ) = 0 . \\end{align*}"}
+{"id": "4431.png", "formula": "\\begin{align*} \\Theta ^ { \\omega } ( x ) : = \\omega ^ { - \\frac { 1 } { 2 } } \\Theta ( \\omega ^ { - \\frac { 1 } { 2 } } x ) . \\end{align*}"}
+{"id": "1019.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\frac { ( a ) _ k ( b ) _ k } { ( 1 ) _ { k } ( c ) _ { k } } x ^ k H _ k ( a - 1 ) \\\\ [ 1 m m ] & \\ : \\ : = - ( 1 - x ) ^ { c - a - b } \\{ \\log ( 1 - x ) \\} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( c - a ) _ k ( c - b ) _ k } { ( 1 ) _ { k } ( c ) _ { k } } x ^ k \\\\ [ 1 m m ] & \\ : \\ : \\quad - ( 1 - x ) ^ { c - a - b } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( c - a ) _ k ( c - b ) _ k } { ( 1 ) _ { k } ( c ) _ { k } } x ^ k H _ k ( c - a - 1 ) . \\end{align*}"}
+{"id": "217.png", "formula": "\\begin{align*} \\frac { 3 5 } { 2 } \\zeta ( 3 ) - \\pi ^ 2 \\log 2 = 3 6 L i _ 3 \\left ( \\frac { 1 } { 2 } \\right ) - 1 8 L i _ 3 \\left ( \\frac { 1 } { 4 } \\right ) - 4 L i _ 3 \\left ( \\frac { 1 } { 8 } \\right ) + L i _ 3 \\left ( \\frac { 1 } { 6 4 } \\right ) , \\end{align*}"}
+{"id": "2573.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\sup \\left \\{ d ( x ) : x \\in M _ 1 ( t ) \\cup M _ 2 ( t ) \\right \\} = 0 . \\end{align*}"}
+{"id": "2474.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle G ( t ) = \\int _ { \\mathbb { R } ^ N } | x | ^ 2 \\left ( a _ 2 | \\phi | ^ 2 + a _ 1 | \\psi | ^ 2 \\right ) d x , \\end{array} \\right . \\end{align*}"}
+{"id": "9133.png", "formula": "\\begin{align*} \\begin{array} { r c c } \\mathrm { d } x ^ { 1 } \\\\ \\mathrm { d } x ^ { 2 } \\\\ \\mathrm { d } x ^ { 3 } \\\\ \\mathrm { d } \\delta ( \\varphi ^ { 1 } ) & = & \\mathrm { d } x ^ { 1 } + \\mathrm { d } u ^ { 1 } \\end{array} \\end{align*}"}
+{"id": "2202.png", "formula": "\\begin{align*} \\Phi ( x , y ) = \\sum _ { d \\mid P ( y ) } \\mu ( d ) \\left \\lfloor \\frac { x } { d } \\right \\rfloor , \\end{align*}"}
+{"id": "4629.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t m = \\nabla \\cdot \\left ( D ( m ) \\nabla m \\right ) - \\chi \\nabla \\cdot \\left ( f ( m ) \\nabla c \\right ) + M ( m ) , \\\\ \\tau \\partial _ t c = \\alpha \\Delta c + \\lambda d - c + \\beta m , \\\\ \\partial _ t d = r h ( m ) \\left ( 1 - d \\right ) , \\end{cases} \\end{align*}"}
+{"id": "2533.png", "formula": "\\begin{align*} \\sum _ { \\{ j , j ^ * \\} \\in M } \\lambda _ j \\ , \\lambda _ { j ^ * } a _ { j , j ^ * } = 0 . \\end{align*}"}
+{"id": "1263.png", "formula": "\\begin{align*} ( ( x _ 1 , x _ 2 ) \\odot ( y _ 1 , y _ 2 ) ) \\oplus ( ( z _ 1 , z _ 2 ) \\odot ( y _ 1 , y _ 2 ) ) = ( ( ( x _ 1 , x _ 2 ) \\odot ( y _ 1 , y _ 2 ) ) \\oplus ( z _ 1 , z _ 2 ) ) \\odot ( y _ 1 , y _ 2 ) \\end{align*}"}
+{"id": "4383.png", "formula": "\\begin{align*} \\mathfrak S ( h ) = \\{ \\mathcal G _ k ( h ) \\in [ L _ k - 1 , U _ k + 1 ] , \\forall n _ 0 < k \\leq n _ \\mathcal L \\} , h \\in [ - 1 , 1 ] . \\end{align*}"}
+{"id": "7594.png", "formula": "\\begin{align*} A : = \\frac { - 3 \\tau ^ 3 _ { 1 } } { 1 6 ( 1 - \\tau ^ 2 _ { 1 } ) } , \\ ; \\ ; B : = \\frac { \\tau _ { 1 } } { 4 } \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; C : = \\frac { - ( 3 + \\tau ^ 2 _ { 1 } ) } { 4 \\tau _ { 1 } } . \\end{align*}"}
+{"id": "1256.png", "formula": "\\begin{align*} & \\frac { 2 ^ \\beta ( y - \\tfrac 1 2 ) ^ { \\beta } } { b _ 1 } \\biggl [ \\frac { \\beta - 1 } { \\beta 2 ^ \\beta ( y - \\tfrac 1 2 ) ^ \\beta } + \\frac { b _ 1 } { y \\beta 2 ^ { \\beta } ( y - \\tfrac 1 2 ) ^ { \\beta - 1 } } \\biggr ] \\\\ & = \\frac { 1 } { b _ 1 } ( 1 - \\frac { 1 } { \\beta } ) + \\frac { 1 } { \\beta } - \\frac { 1 } { 2 y \\beta } \\le 1 , \\end{align*}"}
+{"id": "7901.png", "formula": "\\begin{align*} S = \\left \\{ ( X _ i , \\underline { X _ i } ) ^ + : \\ 0 \\leq i \\leq k \\right \\} \\cup \\left \\{ ( X _ i , \\underline { X _ i } ) ^ - : \\ 0 \\leq i \\leq k \\right \\} \\end{align*}"}
+{"id": "7800.png", "formula": "\\begin{align*} W = B A = B _ { I } A _ { I } + B _ { I ^ { c } } A _ { I ^ { c } } . \\end{align*}"}
+{"id": "2027.png", "formula": "\\begin{align*} B : = 4 n \\sup _ { \\bar \\Omega \\times I } \\vert \\nabla \\psi ^ { 1 / n } \\vert + 1 , \\end{align*}"}
+{"id": "1129.png", "formula": "\\begin{align*} | \\Gamma ( x , y ) | \\le c \\frac { 1 } { d ( x , y ) ^ { q - 2 } } , | X _ j \\Gamma ( x , y ) | \\le c \\frac { 1 } { d ( x , y ) ^ { q - 2 + d e g ( X _ j ) } } , \\\\ | X _ i X _ j \\Gamma ( x , y ) | \\le c \\frac { 1 } { d ( x , y ) ^ { q - 2 + d e g ( X _ j ) + d e g ( X _ i ) } } \\end{align*}"}
+{"id": "8379.png", "formula": "\\begin{align*} J _ { r } ( g _ { r } ) = \\int \\limits _ { { \\cal { X } } } \\ldots \\int \\limits _ { { \\cal { X } } } g _ { r } ( x _ 1 , \\ldots , x _ r ) W ( d x _ 1 ) \\ldots W ( d x _ r ) , \\end{align*}"}
+{"id": "2069.png", "formula": "\\begin{align*} \\frac { x ' } { y ' } = \\frac { \\alpha ( \\alpha x - y ) - x } { \\alpha x - y } = \\alpha - \\frac { x } { \\alpha x - y } = \\alpha - \\frac { x / y } { \\alpha ( x / y ) - 1 } . \\end{align*}"}
+{"id": "4130.png", "formula": "\\begin{align*} M = \\pm ( M _ 1 ^ { m _ 1 } M _ 2 ^ { m _ 2 } ) \\end{align*}"}
+{"id": "284.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { n ^ 2 } { 1 2 } + \\frac { n ^ 3 } { 3 } + \\frac { 5 n ^ 4 } { 1 2 } + \\frac { n ^ 5 } { 6 } \\right ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "3885.png", "formula": "\\begin{align*} \\mathcal { F } ( \\mathcal { S } ; \\mu _ 1 , \\mu _ 2 ) = \\left \\{ \\pi \\in \\mathcal { P } ( \\mathcal { X } _ 1 \\times \\mathcal { X } _ 2 ) : \\pi \\circ \\operatorname { p r o j } _ { \\{ j \\} } ^ { - 1 } = \\mu _ j , \\ \\forall j = 1 , 2 \\right \\} . \\end{align*}"}
+{"id": "1910.png", "formula": "\\begin{align*} \\beta S ^ * r ^ { k + 1 } = S ^ * ( \\lambda ^ { k + 1 } - \\lambda ^ k ) = S ^ * \\lambda ^ { k + 1 } - S ^ * \\lambda ^ k = D _ u \\theta ( u ^ { k } ) - D _ u \\theta ( u ^ { k + 1 } ) \\end{align*}"}
+{"id": "2105.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { a - 1 } \\left ( \\sum _ { i = 1 } ^ { k - 1 } x _ i \\right ) _ r = & m \\left ( \\sum _ { i = 0 } ^ { k - 1 } i \\binom { k - 1 } { i } + \\sum _ { j = 1 } ^ { k - 1 } \\left ( \\sum _ { i = 0 } ^ { k - j - 1 } ( i + 2 ) \\binom { k - j - 1 } { i } \\right ) \\right ) \\\\ & \\ \\ + \\left ( \\sum _ { i = 0 } ^ { k - 2 } i \\binom { k - 2 } { i } + \\sum _ { j = 1 } ^ { k - 2 } \\left ( \\sum _ { i = 0 } ^ { k - j - 2 } ( i + 2 ) \\binom { k - j - 2 } { i } \\right ) \\right ) \\\\ = & m ( 2 ^ { k - 1 } k - 1 ) + 2 ^ { k - 2 } ( k - 1 ) - 1 . \\end{align*}"}
+{"id": "7481.png", "formula": "\\begin{align*} \\tilde { \\delta } = ( d _ 1 , d _ l , d _ { l - 1 } , \\dots , d _ 3 , d _ 2 ) . \\end{align*}"}
+{"id": "52.png", "formula": "\\begin{align*} X = Y _ \\mathbb { Q } Y _ f ^ { - 1 } , \\overline { \\lambda } _ g = Y _ \\mathbb { Q } ^ \\vee \\overline { \\lambda } _ 0 Y _ \\mathbb { Q } , \\overline { \\lambda } _ g = Y _ q ^ \\vee \\overline { \\lambda } _ 0 Y _ q , \\ , \\forall q . \\end{align*}"}
+{"id": "3775.png", "formula": "\\begin{align*} \\bigcup _ { \\substack { \\omega \\ge 1 \\\\ 1 \\le k \\le n - 1 } } { \\rm G r } _ { k \\omega } ( \\overline T _ { n , \\omega } ) = \\bigsqcup _ { i = 0 } ^ { n - 1 } \\widehat { S L } _ n / P _ i . \\end{align*}"}
+{"id": "4544.png", "formula": "\\begin{align*} \\| \\partial _ k \\varphi _ 1 \\| _ { H ^ m } & = \\| \\partial _ k ( - \\alpha \\Delta + 2 \\omega + i \\mathbf { c } \\cdot \\nabla ) ^ { - 1 } \\{ ( \\nabla \\cdot \\varphi _ 3 ) \\varphi _ 2 \\} \\| _ { H ^ m } \\lesssim \\| ( \\nabla \\cdot \\varphi _ 3 ) \\varphi _ 2 \\| _ { H ^ m } \\lesssim \\| \\varphi _ 3 \\| _ { H ^ { m + 1 } } \\| \\varphi _ 2 \\| _ { H ^ m } < \\infty \\end{align*}"}
+{"id": "6454.png", "formula": "\\begin{align*} \\langle V ^ 1 \\rangle _ { K , \\sigma } & = \\frac { \\sigma ^ 2 } { ( 2 \\pi ) ^ 5 } \\sum _ { K _ 1 , K _ 2 , K _ 3 } \\eta _ { K _ 1 } \\overline { \\eta _ { K _ 2 } } \\eta _ { K _ 3 } \\int _ 0 ^ t \\frac { \\pi } { z ( s ) } | \\beta _ h ( s ) | ^ 2 \\beta _ h ( s ) \\overline { \\beta _ { \\sigma } ( s ) } e ^ { \\gamma ( s ) + \\frac { \\zeta ( s ) \\cdot \\zeta ( s ) } { 4 z ( s ) } } \\dd s . \\end{align*}"}
+{"id": "6962.png", "formula": "\\begin{align*} \\deg ( q ) \\leq \\deg ( f ) \\mbox { a n d } \\nu ( f ) = \\nu _ q ( f ) . \\end{align*}"}
+{"id": "6546.png", "formula": "\\begin{align*} \\nu _ F ( d \\lambda ) = \\sum _ { \\gamma \\not = 0 } \\frac { m _ \\gamma } { \\gamma ^ 2 } \\ , \\delta _ { - \\gamma } ( d \\lambda ) , \\end{align*}"}
+{"id": "1526.png", "formula": "\\begin{align*} h ( t ) - \\alpha ( g ( t ) ) = \\sum _ { \\theta = 1 } ^ { d } ( h _ { p \\theta + p - 1 } - g _ { p \\theta + p - 1 } a ^ { p \\theta } ) t ^ { p \\theta + p - 1 } + f _ { h , g } ( t ) , \\end{align*}"}
+{"id": "4895.png", "formula": "\\begin{align*} \\int _ { \\partial B _ 1 } \\frac { ( \\rho ^ 2 - \\rho _ 0 ^ 2 ) ^ s } { | \\rho e - \\rho _ 0 \\omega | ^ { n } } \\ , d H ^ { n - 1 } _ \\omega \\ , G ( \\rho _ 0 e , 0 ) = \\rho ^ { 2 s - n } \\int _ { \\partial B _ 1 } G ( \\rho _ 0 e , \\rho _ 0 \\omega ) \\ , d H ^ { n - 1 } _ \\omega . \\end{align*}"}
+{"id": "7397.png", "formula": "\\begin{align*} \\frac { N _ m } { N } = 1 + O ( 1 / m ) , \\end{align*}"}
+{"id": "1982.png", "formula": "\\begin{align*} u _ { \\alpha \\bar \\beta } ( 0 ) = - u _ { x _ n } ( 0 ) \\rho _ { \\alpha \\bar \\beta } ( 0 ) , ~ ~ \\alpha , \\beta \\leq n - 1 . \\end{align*}"}
+{"id": "1432.png", "formula": "\\begin{align*} - \\Delta \\psi = f _ 0 ^ { ' } ( U _ { \\mu , \\xi } ) \\psi \\quad \\mbox { i n } \\ \\mathbb { R } ^ n , \\end{align*}"}
+{"id": "7358.png", "formula": "\\begin{align*} 2 \\ell x & = \\theta - \\log ( y / 3 ) \\\\ & = \\frac { ( 2 \\gamma - 1 ) \\mu ^ 2 } { 2 } + \\log ( 3 q ( ( 1 - \\gamma ) \\mu ) ) + \\frac { ( 1 - \\gamma ) ^ 2 \\mu ^ 2 } { 2 } \\\\ & = \\frac { \\gamma ^ 2 \\mu ^ 2 } { 2 } + \\log ( 3 q ( ( 1 - \\gamma ) \\mu ) ) . \\end{align*}"}
+{"id": "4652.png", "formula": "\\begin{align*} \\hat { D } _ \\epsilon ( z ) = \\begin{cases} D _ \\epsilon ( 0 ) & z < 0 , \\\\ D _ \\epsilon ( z ) & 0 \\leq z \\leq K + 1 , \\\\ D _ \\epsilon ( K + 1 ) & z > K + 1 , \\end{cases} \\end{align*}"}
+{"id": "3044.png", "formula": "\\begin{align*} \\phi \\left ( \\bigotimes \\limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \\otimes X _ m E _ { j _ { m } j _ { m } } X _ m ^ * \\right ) = U _ { X _ m } ( E _ { j _ 1 j _ 1 } \\otimes \\cdots \\otimes E _ { j _ m j _ m } ) V _ { X _ m } ^ * \\end{align*}"}
+{"id": "6238.png", "formula": "\\begin{align*} \\forall r \\geq 0 , \\wp ^ { ( r ) } = ( - 1 ) ^ r ( r + 1 ) ! E _ { r + 2 } - e _ 2 \\delta _ { r , 0 } . \\end{align*}"}
+{"id": "5900.png", "formula": "\\begin{align*} Y _ k = x ^ * ( X _ k ) \\textrm { a n d } T _ k = \\sum \\limits _ { i = 1 } ^ k Y _ i . \\end{align*}"}
+{"id": "7581.png", "formula": "\\begin{align*} c _ 3 = 2 \\tau ^ 3 _ 1 + 4 ( 1 - \\tau ^ 2 _ 1 ) \\tau _ 1 \\tau _ 2 - 2 ( 1 - \\tau ^ 2 _ 1 ) \\tau _ 1 \\tau ^ 2 _ 2 + 2 ( 1 - \\tau ^ 2 _ 1 ) ( 1 - | \\tau _ 2 | ^ 2 ) \\tau _ 3 \\end{align*}"}
+{"id": "8080.png", "formula": "\\begin{align*} \\mu _ n = \\sup _ { M \\in \\mathcal { G } _ n } \\inf _ { u \\in M } F ( u ) . \\end{align*}"}
+{"id": "7303.png", "formula": "\\begin{align*} P ( x _ 1 ^ k x _ 2 , x _ 3 , \\dots , x _ n ) = 0 , \\end{align*}"}
+{"id": "6438.png", "formula": "\\begin{align*} p ( \\theta ) : = \\mbox { l c m } \\{ p ( a , b ) : a , b \\mbox { a r e a p e r i o d i c p a i r } \\} , \\end{align*}"}
+{"id": "6865.png", "formula": "\\begin{align*} \\sigma _ { \\mathrm { e s s } } ( S ) = \\sigma ( S ) \\setminus \\sigma _ d ( S ) . \\end{align*}"}
+{"id": "5634.png", "formula": "\\begin{align*} U _ { \\epsilon , \\xi } ( x ) = [ N ( N - 2 ) ] ^ { \\frac { N - 2 } { 4 } } \\epsilon ^ { \\frac { N - 2 } { 2 } } ( \\epsilon ^ 2 + | x - \\xi | ^ 2 ) ^ { - \\frac { N - 2 } { 2 } } , \\ \\ x , \\xi \\in \\mathbb { R } ^ N , \\ \\epsilon > 0 . \\end{align*}"}
+{"id": "810.png", "formula": "\\begin{align*} y _ { 3 } & = y _ { 2 } + h f \\left ( x _ { 2 } , y _ { 2 } \\right ) + h \\int \\limits _ { x _ { 0 } } ^ { x _ { 3 } } K d t \\\\ & = y _ { 2 } + h f \\left ( x _ { 2 } , y _ { 2 } \\right ) + \\frac { h ^ { 2 } } { 2 } \\left ( K _ { 0 } + K _ { 1 } \\right ) + \\frac { h ^ { 2 } } { 2 } \\left ( K _ { 1 } + K _ { 2 } \\right ) \\\\ & = y _ { 2 } + h f \\left ( x _ { 2 } , y _ { 2 } \\right ) + \\frac { h ^ { 2 } } { 2 } \\left ( K _ { 0 } + 2 K _ { 1 } + K _ { 2 } \\right ) \\end{align*}"}
+{"id": "1661.png", "formula": "\\begin{align*} 2 \\ , g ( ( \\nabla _ { X } { f } ) Y , Z ) = - g ( [ { f } , { f } ] ( Y , Z ) , { f } X ) + N ^ { \\ , ( 5 ) } ( X , Y , Z ) . \\end{align*}"}
+{"id": "1551.png", "formula": "\\begin{align*} k _ j = \\frac { \\epsilon M } { 2 ^ j } j \\geq 1 \\end{align*}"}
+{"id": "5700.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ) = D ^ { k - 1 } ( h ) + d g . \\end{align*}"}
+{"id": "8774.png", "formula": "\\begin{align*} \\mathcal { L } ( v ) = \\sum _ { i = 1 } ^ n c p _ i \\max \\{ - x _ i - y _ i , - d _ i \\} + \\sum _ { i \\neq j } \\alpha _ { i j } v _ { i j } - \\langle \\lambda , v \\rangle + \\sum _ { i \\in I } \\gamma _ i ( \\sum _ { j \\neq i } v _ { i j } - x _ { 0 i } ) . \\end{align*}"}
+{"id": "9238.png", "formula": "\\begin{align*} \\sum _ { k = m } ^ { J - 1 } 2 ^ k \\delta _ k < \\frac { 1 } { 3 } \\frac { \\varepsilon _ m } { m ! } , 0 \\le m < M . \\end{align*}"}
+{"id": "4744.png", "formula": "\\begin{align*} \\tilde { S } = \\begin{pmatrix} \\tilde { W } & \\tilde { X } \\\\ \\tilde { Y } & \\tilde { Z } \\end{pmatrix} , \\end{align*}"}
+{"id": "6447.png", "formula": "\\begin{align*} V _ k ^ 1 ( t ) = & \\int _ 0 ^ t R _ k ( s , \\varphi , \\varphi , \\varphi ) \\dd s , \\\\ V _ k ^ 2 ( t ) = & 2 \\int _ 0 ^ t \\int _ { 0 } ^ { s } R _ k ( s , \\varphi , \\varphi , R ( s ' , \\varphi , \\varphi , \\varphi ) ) \\dd s ' \\dd s + \\int _ 0 ^ t \\int _ { 0 } ^ { s } R _ k ( s , \\varphi , R ( s ' , \\varphi , \\varphi , \\varphi ) , \\varphi ) \\dd s ' \\dd s . \\end{align*}"}
+{"id": "6470.png", "formula": "\\begin{align*} W _ { K _ 1 K _ 2 K _ 3 K _ 4 K _ 5 K } ( s , s ' ) & : = \\frac { | \\beta _ h ( s ) | ^ 2 \\overline { \\beta _ \\sigma ( s ) } | \\beta _ h ( s ' ) | ^ 2 \\beta _ h ( s ' ) } { ( 2 \\pi ) ^ { 1 2 } \\big ( 1 + 4 i s z ( s ) \\big ) \\big ( 1 - 4 i s ' \\widetilde z ( s ' ) \\big ) } e ^ { c ( s , s ' ) + \\frac { b ( s , s ' ) \\cdot b ( s , s ' ) } { 4 a ( s , s ' ) } } \\end{align*}"}
+{"id": "3919.png", "formula": "\\begin{align*} \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda \\ , d \\pi = \\int _ { \\mathcal { V } } g \\ , d \\gamma - \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\left ( \\lambda _ 1 c _ 1 + \\lambda _ 2 c _ 2 \\right ) \\ , d \\pi < \\infty , \\end{align*}"}
+{"id": "7288.png", "formula": "\\begin{align*} x _ p = x ^ 0 _ p + \\frac { m } { \\gcd ( n , m ) } u _ p , y _ p = y ^ 0 _ p + \\frac { n } { \\gcd ( n , m ) } u _ p u _ p \\geq 0 . \\end{align*}"}
+{"id": "7860.png", "formula": "\\begin{align*} \\mathcal { O } _ i = \\left \\{ \\left ( \\mathcal { B } , \\mathcal { B } ^ \\prime \\right ) \\in U _ { k } \\times U _ k : \\ \\mbox { t h e r e e x i s t s $ B \\in \\mathcal { B } , \\ B ^ \\prime \\in \\mathcal { B } ^ \\prime $ s u c h t h a t $ | B \\cap B ^ \\prime | = k - i $ } \\right \\} , \\end{align*}"}
+{"id": "393.png", "formula": "\\begin{align*} P U _ t + ( A _ i ( V ) U ) _ { x _ i } + A ^ T _ i ( V ) U _ { x _ i } + C ( V ) U + L _ C ( U ) = 0 , t \\geq 0 , \\vec x = ( x _ 1 , x _ 2 , . . , x _ k ) \\in \\Omega . \\end{align*}"}
+{"id": "5953.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( g ) f = \\theta _ { L , X ^ { \\ast } } [ \\Pi _ { \\psi } ( g ) ( f ' ) ] = \\theta _ { L , X ^ { \\ast } } [ \\Pi _ { \\psi } ( g ) \\theta _ { X ^ { \\ast } , L } ( f ) ] . \\end{align*}"}
+{"id": "7978.png", "formula": "\\begin{align*} V ( x ) = 0 \\end{align*}"}
+{"id": "3258.png", "formula": "\\begin{align*} | K _ \\alpha ( x , y ) | & \\lesssim \\int _ 0 ^ \\infty \\frac 1 { V ( x , y , \\sqrt t ) } \\frac t { \\| x - y \\| ^ 2 } e ^ { - c d ( x , y ) ^ 2 / t } \\frac { d t } { t ^ { 1 - \\alpha / 2 } } \\\\ & \\leqslant \\frac 1 { \\| x - y \\| ^ 2 } \\bigg ( \\int _ 0 ^ { d ( x , y ) ^ 2 } + \\int _ { d ( x , y ) ^ 2 } ^ \\infty \\bigg ) \\frac 1 { V ( x , y , \\sqrt t ) } e ^ { - c d ( x , y ) ^ 2 / t } t ^ \\alpha d t \\\\ & = : I _ 1 + I _ 2 . \\end{align*}"}
+{"id": "7589.png", "formula": "\\begin{align*} H _ { 2 , 1 } ( F _ { f } / 2 ) & = \\frac { 1 } { 4 8 } \\left ( a ^ 4 _ 2 - 1 2 a ^ 2 _ 3 + 1 2 a _ 2 a _ 4 \\right ) \\\\ & = \\frac { 1 } { 3 0 7 2 } \\left ( - 3 c ^ 4 _ { 1 } - 8 c ^ 2 _ { 1 } c _ 2 - 4 8 c ^ 2 _ { 2 } + 6 4 c _ 1 c _ 3 \\right ) . \\end{align*}"}
+{"id": "7157.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\partial _ t u u \\ ; x = \\frac { } { t } \\frac { 1 } { 2 } \\| u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 . \\end{align*}"}
+{"id": "529.png", "formula": "\\begin{align*} \\left ( \\mathcal { H } _ { \\hbar , V } + I \\right ) ^ { - 1 } = \\left ( \\mathcal { H } _ { \\hbar , V } + I \\right ) ^ { - 1 } \\left ( - \\hbar ^ { - 2 } \\mathcal { L } _ { \\hbar } \\right ) \\left ( V + I \\right ) ^ { - 1 } + \\left ( V + I \\right ) ^ { - 1 } . \\end{align*}"}
+{"id": "6481.png", "formula": "\\begin{align*} F ^ * ( \\sigma , V ) : = ( \\sigma \\circ F , V ) . \\end{align*}"}
+{"id": "6694.png", "formula": "\\begin{align*} R = \\partial ^ \\alpha ( a \\nabla \\eta ) - \\partial ^ \\alpha a \\nabla \\eta - a \\partial ^ \\alpha \\nabla \\eta . \\end{align*}"}
+{"id": "8834.png", "formula": "\\begin{align*} v _ \\lambda ( t ) = v _ { 0 , \\lambda } ( t ) + \\int _ 0 ^ t S ( t - s ) f ' _ \\lambda ( u _ \\lambda ( s ) ) v _ \\lambda ( s ) \\ , d s + \\int _ 0 ^ t S ( t - s ) \\sigma ' ( u _ \\lambda ( s ) ) v _ \\lambda ( s ) B \\ , d W ( s ) . \\end{align*}"}
+{"id": "3458.png", "formula": "\\begin{align*} a ^ { \\sigma } ( z ) = \\sum _ { r \\in \\Z } a ^ { ( r ) } \\left ( \\frac { r } { T } \\right ) z ^ { - \\frac { r } { T } - 1 } , \\end{align*}"}
+{"id": "3942.png", "formula": "\\begin{align*} B _ \\ell = \\{ ( ( s _ 1 , s _ 2 ) , ( y _ 1 , y _ 2 , x ) ) \\in \\mathcal { V } \\times \\mathcal { S } : c _ \\ell ( s _ \\ell , ( y _ \\ell , x ) ) < \\infty \\} , \\end{align*}"}
+{"id": "3099.png", "formula": "\\begin{align*} S _ { n , k } ( q ) = ( 1 + q ) ^ { k } q ^ { k ^ 2 } S _ B [ n , k ] . \\end{align*}"}
+{"id": "4365.png", "formula": "\\begin{align*} \\sigma \\left ( \\overline { b } \\right ) = \\left \\{ \\left \\langle a _ { i } , x \\right \\rangle \\leq \\overline { b } _ { i } , i = 1 , . . . , m \\right \\} \\end{align*}"}
+{"id": "8835.png", "formula": "\\begin{align*} d \\widetilde { v } _ \\lambda + A \\widetilde { v } _ \\lambda \\ , d t = f ' _ \\lambda ( u _ \\lambda ) \\bigl ( \\widetilde { v } _ \\lambda + v _ { 0 , \\lambda } \\bigr ) \\ , d t + \\sigma ' ( u _ \\lambda ) \\bigl ( \\widetilde { v } _ \\lambda + v _ { 0 , \\lambda } \\bigr ) B \\ , d W \\end{align*}"}
+{"id": "488.png", "formula": "\\begin{align*} t / b d ^ 2 \\mapsto c ^ { n - 1 } t ^ { k _ 0 - 2 k _ 1 + k _ 2 + 1 } t _ 3 / t _ 1 t _ 2 = c ^ { n - 1 } t / t _ 4 , \\end{align*}"}
+{"id": "2210.png", "formula": "\\begin{align*} \\Phi ( x , y ) = \\Phi \\left ( x , x ^ { 1 / 3 } \\right ) + \\sum _ { y < p \\le x ^ { 1 / 3 } } \\Phi ( x / p , p ^ - ) . \\end{align*}"}
+{"id": "6114.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } v _ t + \\mathcal { L } ^ { \\Phi } _ { A _ { 2 } ( t ) } v & = 0 \\quad \\mbox { i n } Q _ 2 , \\\\ v & = w \\quad \\mbox { i n } ( \\mathbb { R } ^ n \\setminus B _ 2 ) \\times \\Lambda _ 2 \\cup B _ 2 \\times \\{ - 2 ^ { 2 s } \\} . \\end{array} \\right . \\end{align*}"}
+{"id": "3794.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { M } } ( \\mu _ { 1 3 } , \\mu _ { 2 3 } ) : = \\sup _ { \\gamma \\in \\mathcal { F } ( \\mu _ { 1 3 } , \\mu _ { 2 3 } ) } \\int _ { \\mathcal { S } } f \\ , d \\gamma . \\end{align*}"}
+{"id": "2018.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u ) ^ n = ( - \\lambda _ 1 u ) ^ n f ^ n \\omega ^ n & \\textnormal { o n } & \\Omega \\\\ u = 0 & \\textnormal { i n } & \\partial \\Omega . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "9353.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d \\hat { y } ^ n _ t & = g \\big ( x _ t , \\hat { y } ^ n _ t , \\hat { z } ^ n _ t , \\hat { \\tilde { z } } ^ n _ t , \\hat { \\gamma } ^ n _ { ( t , e ) } \\big ) d t - \\hat { z } ^ n _ t d W _ t - \\hat { \\tilde { z } } ^ n _ t d \\xi _ t - \\int _ { \\mathcal { E } } \\hat { \\gamma } ^ n _ { ( t , e ) } \\tilde { N } ( d e , d t ) , \\ t \\in [ 0 , n ] , \\\\ \\hat { y } _ n ^ n & = \\zeta _ n : = \\mathbb { E } [ \\zeta | \\mathcal { F } _ n ] , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1307.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { K } \\to \\R ^ V \\times \\R ^ V \\\\ ( \\rho , t ) \\mapsto \\bigg ( - \\frac { 1 } { 2 } - e ^ { \\rho _ i } \\Big ( W K _ t ^ { - 1 } ( e ^ \\rho + t \\eta ) + \\eta \\Big ) _ i \\ , , \\ , e ^ { 2 \\rho _ i } \\bigg ) _ { i \\in V } \\end{cases} \\end{align*}"}
+{"id": "6578.png", "formula": "\\begin{align*} \\Psi ( X ) = J ^ * ( \\oplus ^ 6 X ) J \\end{align*}"}
+{"id": "5724.png", "formula": "\\begin{align*} \\widetilde { X } = \\begin{pmatrix} a ^ 2 & b ^ 2 & a b & a b \\\\ c ^ 2 & d ^ 2 & c d & c d \\\\ a c & b d & a d & b c \\\\ a c & b d & b c & a d \\end{pmatrix} , \\end{align*}"}
+{"id": "3986.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta _ 1 , 0 ) = \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + V _ { 1 , Y Y } ^ { 1 / 2 } \\delta _ 1 ^ 2 \\mathcal { I } _ { \\mathrm { D } } ( 0 , \\delta _ 2 ) = \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + V _ { 2 , Y Y } ^ { 1 / 2 } \\delta _ 2 ^ 2 . \\end{align*}"}
+{"id": "8427.png", "formula": "\\begin{align*} \\eta _ { p } ( s ) : = \\sum _ { k = 1 } ^ { \\infty } \\frac { ( s , 2 \\pi p k ) _ { k } } { k ^ { s } } = \\left ( 2 ^ { 1 - s } - 1 \\right ) \\zeta ( s ) + \\sum _ { k = 1 } ^ { \\infty } e ^ { 2 \\pi p k } \\sum _ { \\ell = 1 } ^ { k } \\left ( \\begin{array} { c } k \\\\ \\ell \\end{array} \\right ) \\left ( 4 \\pi p \\right ) ^ { \\ell } ( - 1 ) ^ { k - \\ell } \\ , \\intop _ { k } ^ { \\infty } t ^ { - s } e ^ { - 2 \\pi p t } \\ , \\frac { ( t - k ) ^ { \\ell - 1 } } { ( \\ell - 1 ) ! } \\ , d t , \\end{align*}"}
+{"id": "2055.png", "formula": "\\begin{align*} N _ k & = 1 1 \\leq k \\leq k _ 2 \\\\ N _ k & = 2 k _ 2 < k \\leq k _ 3 \\\\ N _ k & = 3 k _ 3 < k \\leq k _ 4 \\\\ \\ldots & \\ldots . \\end{align*}"}
+{"id": "8477.png", "formula": "\\begin{align*} \\mathcal { J } _ { N } ( x , s ) : = \\intop _ { 0 } ^ { \\infty } \\ , \\frac { y ^ { s - \\frac { 1 } { 2 } } \\left \\{ J _ { s - \\frac { 1 } { 2 } } ( 2 \\pi x y ) - \\sum _ { k = 0 } ^ { N } \\frac { \\left ( - 1 \\right ) ^ { k } \\left ( \\pi x y \\right ) ^ { s + 2 k - \\frac { 1 } { 2 } } } { k ! \\Gamma \\left ( s + k + \\frac { 1 } { 2 } \\right ) } \\right \\} } { \\sigma \\left ( y \\right ) e ^ { 2 \\pi y } - 1 } \\ , d y \\end{align*}"}
+{"id": "1309.png", "formula": "\\begin{align*} Z = Z ^ { ( u , v ) } : t \\mapsto \\Big ( X _ i \\big ( ( T _ i ( u ) + t ) \\wedge T _ i ( v ) \\big ) \\Big ) _ { i \\in V } . \\end{align*}"}
+{"id": "6395.png", "formula": "\\begin{align*} \\mathfrak { G } = \\{ \\mathcal { P } \\subset \\mathcal { O } : \\mathcal { P } \\mathcal { P } \\mid r \\mathcal { P } \\mid 2 n \\sqrt { { - m } } \\} . \\end{align*}"}
+{"id": "8039.png", "formula": "\\begin{align*} \\lambda _ { 1 + 2 j } \\left ( \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} \\right ) & \\le \\inf _ { \\dim { \\mathcal { S } } = n - j } \\lambda _ { 1 } \\left ( A _ { \\mathcal { S } } + B _ { \\mathcal { S } } \\right ) + \\delta _ 2 ( X ) \\\\ & = \\lambda _ { 1 + j } ( A + B ) + \\delta _ 2 ( X ) \\end{align*}"}
+{"id": "4547.png", "formula": "\\begin{align*} 2 L ( \\Phi ) + \\left ( \\frac { d } { 2 } + 1 \\right ) N ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) = 0 . \\end{align*}"}
+{"id": "7624.png", "formula": "\\begin{align*} & A _ 2 = - \\frac { c _ 1 } { 2 } , \\\\ & A _ 3 = \\frac { 3 } { 8 } c _ 1 ^ 2 - \\frac { 1 } { 4 } c _ 2 , \\\\ & A _ 4 = - \\frac { 1 } { 3 } c _ 1 ^ 3 + \\frac { 1 } { 2 } c _ 1 c _ 2 - \\frac { 1 } { 6 } c _ 3 , \\\\ & A _ 5 = \\frac { 1 2 5 } { 3 8 4 } c _ 1 ^ 4 - \\frac { 2 5 } { 3 2 } c _ 1 ^ 2 c _ 2 + \\frac { 5 } { 3 2 } c _ 2 ^ 2 + \\frac { 5 } { 1 2 } c _ 1 c _ 3 - \\frac { 1 } { 8 } c _ 4 . \\end{align*}"}
+{"id": "8385.png", "formula": "\\begin{align*} P ^ { - 1 } = Q ^ { - 1 } + \\sum _ { m \\geq 1 } ( - 1 ) ^ m ( Q ^ { - 1 } R ) ^ m Q ^ { - 1 } . \\end{align*}"}
+{"id": "820.png", "formula": "\\begin{align*} P _ R ( x _ 1 , \\ldots , x _ m , n ) g _ { n + R } + \\cdots + P _ 0 ( x _ 1 , \\ldots , x _ m , n ) g _ n = 0 \\end{align*}"}
+{"id": "3322.png", "formula": "\\begin{align*} 2 { \\rm R e } ( e ^ { i ( d ( \\theta , f ( \\theta ) ) + i H d ( \\theta , f ( \\theta ) ) ) } \\frac { \\partial g } { \\partial \\theta } ( \\theta ) ) = - \\rho _ { \\theta } ( \\theta , f ( \\theta ) ) e ^ { - c ( \\theta , f ( \\theta ) ) - H d ( \\theta , f ( \\theta ) ) } . \\end{align*}"}
+{"id": "8267.png", "formula": "\\begin{align*} D _ { l , 0 } ( n ) = \\frac { n ( 2 k - 1 + n ) } { 2 } \\prod _ { s = 3 } ^ l \\frac { ( 2 k - s + 1 ) ( s - 1 ) + n } { s } . \\end{align*}"}
+{"id": "1331.png", "formula": "\\begin{align*} \\left ( Z _ i ( u ) , \\frac { 1 } { T _ i ( v ) } \\right ) _ { u \\geq 0 , v \\geq 0 , i \\in V } = \\left ( \\theta _ i Z ^ * _ i ( u ) , \\frac { 1 } { T _ i ^ { \\infty } } + \\frac { 1 } { \\theta _ i ^ 2 T _ i ^ * ( v ) } \\right ) _ { u \\geq 0 , v \\geq 0 , i \\in V } . \\end{align*}"}
+{"id": "946.png", "formula": "\\begin{align*} T ^ 0 = \\{ ( p , q , r ) \\in X \\times X \\times X \\setminus \\Delta _ T \\ , | \\ , X \\cdot H ^ p _ { q , r } = 4 p + q + r \\in Z _ 0 ( X ) \\} \\end{align*}"}
+{"id": "8340.png", "formula": "\\begin{align*} ( u _ { \\kappa } ) _ \\mu ^ * ( t ) = u _ \\mu ^ * ( \\kappa ^ D t ) \\end{align*}"}
+{"id": "4372.png", "formula": "\\begin{align*} U _ k & = \\frac { y } { 1 0 } + \\alpha ( k - n _ 0 ) - 1 0 \\mathcal { S } _ { \\mathcal { L } } ( x \\mapsto \\log ( x ) ) ( k ) , \\\\ L _ k & = - 1 0 y + \\alpha ( k - n _ 0 ) - \\mathcal { S } _ { \\mathcal { L } } ( x \\mapsto x ^ { 3 / 4 } ) ( k ) . \\end{align*}"}
+{"id": "6713.png", "formula": "\\begin{align*} | g ( z ) | & \\leq C e ^ { c | z | } , z \\in Q _ 1 , \\\\ | g ( z ) | & \\leq 1 , z \\in \\partial Q _ 1 = \\R _ + \\cup i \\R _ + . \\end{align*}"}
+{"id": "6699.png", "formula": "\\begin{align*} ( \\partial _ t ^ 2 - \\Delta ) u = 0 \\R \\times \\R ^ d , \\end{align*}"}
+{"id": "6177.png", "formula": "\\begin{align*} v _ { s } v _ { t } & = \\lim _ { i } \\lim _ { j } \\psi ( \\mu _ j \\delta _ { s } \\mu _ i \\delta _ { t } ) = \\lim _ { i } \\lim _ { j } \\psi ( \\alpha _ { s } ( \\mu _ i ) u ( s , t ) \\mu _ j \\delta _ { s t } ) \\\\ & = \\lim _ { i } \\pi ( \\alpha _ { s } ( \\mu _ i ) u ( s , t ) ) v _ { s t } = \\overline { \\pi } ( u ( s , t ) ) v _ { s t } . \\end{align*}"}
+{"id": "2294.png", "formula": "\\begin{align*} w = H + \\widetilde { T } ( f ) . \\end{align*}"}
+{"id": "4538.png", "formula": "\\begin{align*} \\frac { 1 } { p _ 0 } = \\frac { 1 } { 2 } + \\frac { 1 } { q _ 0 } . \\end{align*}"}
+{"id": "4993.png", "formula": "\\begin{align*} f ( k , t , \\mathbf { p } _ { n + 1 } ) = F ( k , t , \\mathbf { p } _ { n + 1 } ) - F ( k - 1 , t , \\mathbf { p } _ { n + 1 } ) . \\end{align*}"}
+{"id": "6886.png", "formula": "\\begin{align*} J _ r ' ( x , \\beta ) = \\theta ( x , \\beta ) . \\end{align*}"}
+{"id": "3268.png", "formula": "\\begin{align*} \\| f \\| _ { W _ p ^ k } = \\left ( \\sum _ { j = 0 } ^ { k } \\| D ^ j f \\| _ p \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
+{"id": "6293.png", "formula": "\\begin{align*} h ^ { - 1 } _ { \\bar u ( t ) } ( \\lambda _ t ) = \\max _ { u \\in \\R ^ k } h ^ { - 1 } _ u ( \\lambda _ t ) = H ( \\lambda _ t ) , \\qquad \\forall \\ , t \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "3470.png", "formula": "\\begin{align*} \\mu ( \\sigma ) = \\frac { e ^ { - H ( \\sigma ) } } { Z } , \\end{align*}"}
+{"id": "6639.png", "formula": "\\begin{align*} p _ a ( Y ) = - \\sum _ { j = 1 } ^ { k } { ( \\alpha + 1 + d _ i ) } . \\end{align*}"}
+{"id": "4314.png", "formula": "\\begin{align*} t _ { \\mathrm { s u c } _ K ( k , m ) } = \\begin{cases} t _ { k , m } + 2 ^ { - 2 k } & k < K , \\\\ t _ { k , m } + 2 ^ { 1 - 2 k } & k = K . \\\\ \\end{cases} \\end{align*}"}
+{"id": "8505.png", "formula": "\\begin{align*} \\zeta _ { p , p ^ { \\prime } } ( s , c ) = \\frac { 2 } { 1 + \\frac { 1 } { \\pi p ^ { \\prime } } } \\ , \\zeta _ { p } ( 2 s ) + \\frac { 2 \\sqrt { \\pi } \\ , c ^ { \\frac { 1 } { 2 } - s } \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) } { \\Gamma ( s ) } \\zeta _ { p ^ { \\prime } } ( 2 s - 1 ) + H _ { p , p ^ { \\prime } } ( s , c ) , \\ , \\ , \\ , \\ , \\ , ( s ) > 1 , \\end{align*}"}
+{"id": "1822.png", "formula": "\\begin{align*} D _ k ( t ) = \\int _ { ( \\Lambda ^ * ) ^ 2 } d _ t ( k , p , q ) a _ { p } ^ * a _ { q } \\d p \\d q \\end{align*}"}
+{"id": "1058.png", "formula": "\\begin{align*} \\mathcal E _ 2 \\coloneqq \\left \\{ ( a _ 1 , \\ldots , a _ n , b ) \\in \\mathbb N ^ { n + 1 } : b \\leq d , \\sum _ { i = 1 } ^ n a _ i \\leq h , \\exists i , a _ i = 0 \\right \\} \\ , . \\end{align*}"}
+{"id": "6717.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\frac { 1 } { \\ell _ i } \\sum _ { \\ell \\leq \\ell _ i } \\delta _ { \\sigma ^ \\ell \\eta } = \\nu _ \\eta , \\end{align*}"}
+{"id": "7981.png", "formula": "\\begin{align*} \\int _ \\Omega \\big ( \\mathrm { d i v } \\ , V \\big ) ^ 2 \\ , d x = \\int _ \\Omega \\mathrm { t r } \\big ( ( \\nabla \\ , V ) ^ 2 \\big ) d x + \\int _ { \\partial \\Omega } \\Big \\{ ( \\mathrm { d i v } _ T V ) \\ , V \\cdot \\nu - \\nabla _ T V \\ , V _ T \\cdot \\nu \\Big \\} d \\mathcal { H } ^ { n - 1 } . \\end{align*}"}
+{"id": "5041.png", "formula": "\\begin{align*} h ^ \\ast \\left ( \\xi _ { h g h ^ { - 1 } } \\right ) = \\rho _ { h , g } \\xi _ { g } . \\end{align*}"}
+{"id": "5558.png", "formula": "\\begin{align*} { \\rm T r e e } _ { F } ^ { H _ { 0 } } = \\left \\{ H \\in { \\rm T r e e } _ { F } : H < H _ { 0 } ^ { \\gamma } \\mbox { f o r s o m e } \\gamma \\in F \\right \\} , \\end{align*}"}
+{"id": "5263.png", "formula": "\\begin{align*} D _ { S , x } [ f ^ { = T } ] = ( L _ S E _ { T ^ c } L _ T [ f ] ) _ { S \\rightarrow x } = ( E _ { T ^ c } L _ T [ f ] ) _ { S \\rightarrow x } \\end{align*}"}
+{"id": "3815.png", "formula": "\\begin{align*} \\sup _ { \\gamma \\in \\mathcal { P } ( \\mathcal { V } ) } \\left \\{ \\int _ { \\mathcal { V } } g d \\gamma - \\lambda _ 1 \\boldsymbol { K } _ 1 ( \\mu _ 1 , \\gamma _ 1 ) - \\lambda _ 2 \\boldsymbol { K } _ 2 ( \\mu _ 2 , \\gamma _ 2 ) : \\boldsymbol { K } _ \\ell ( \\mu _ \\ell , \\gamma _ \\ell ) < \\infty \\ell = 1 , 2 \\right \\} \\end{align*}"}
+{"id": "6121.png", "formula": "\\begin{align*} \\rho _ { i } = 2 ^ { j _ { 0 } } \\rho _ { z _ { i } } \\quad i . \\end{align*}"}
+{"id": "4299.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial t } + \\nabla \\cdot ( u f ) = 0 , \\end{align*}"}
+{"id": "5164.png", "formula": "\\begin{align*} \\begin{aligned} & \\int \\dot { \\xi } ^ 2 _ { ( 0 , 1 ) _ \\sigma } d y \\\\ & = \\int ( - \\frac { 1 } { 2 } \\sqrt { g ( y ) } + y \\dot { \\xi } _ { ( 0 , 1 ) _ \\mu } ) ^ 2 d y \\\\ & = \\int [ \\frac { 1 } { 4 } g ( y ) + y ^ 2 \\dot { \\xi } ^ 2 _ { ( 0 , 1 ) _ \\mu } - y \\sqrt { g ( y ) } \\dot { \\xi } _ { ( 0 , 1 ) _ \\mu } ] d y . \\end{aligned} \\end{align*}"}
+{"id": "7068.png", "formula": "\\begin{align*} \\epsilon : = \\beta _ j - \\tilde { \\beta } _ j = \\nu ( g ' ) - \\nu _ j ( g ' ) \\in \\Delta \\mbox { f o r e v e r y } j \\in I ' , j \\geq j _ 0 . \\end{align*}"}
+{"id": "5872.png", "formula": "\\begin{align*} i \\partial _ t u + \\partial _ x ^ 2 u \\ , \\pm \\ , \\frac { 2 } { i } \\ , \\partial _ x \\Pi ( | u | ^ 2 ) u = 0 \\ , , x \\in \\mathbb { T } \\ , , \\end{align*}"}
+{"id": "4645.png", "formula": "\\begin{align*} \\bar { a } _ 1 = \\frac { \\frac { p + \\gamma - 1 } { 2 } - \\frac { ( p + \\gamma - 1 ) ( 2 q + n - 2 - \\theta ( q - 1 ) ) } { 4 n q } } { \\frac { p + \\gamma - 1 } { 2 } + \\frac { 1 } { n } - \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "7126.png", "formula": "\\begin{align*} \\partial _ { x _ n } c ( x ^ \\prime , x _ n ) = \\partial _ { x _ n } c ( x ^ \\prime , x _ n ) - \\partial _ { x _ n } c ( x ^ \\prime , 0 ) = \\int _ 0 ^ 1 \\partial _ { x _ n } ^ 2 c ( x ^ \\prime , t x _ n ) x _ n \\ : t . \\end{align*}"}
+{"id": "2543.png", "formula": "\\begin{align*} \\lambda _ { j _ 1 } = \\lambda _ { j _ 1 ^ { - 1 } } = \\dots = \\lambda _ { j _ r } = \\lambda _ { j _ r ^ { - 1 } } = \\pm 1 \\end{align*}"}
+{"id": "2520.png", "formula": "\\begin{align*} P _ { k _ i } = \\begin{cases} \\{ 1 , \\ldots , k _ 1 \\} , & i = 1 , \\\\ \\{ k _ 1 + \\cdots + k _ { i - 1 } + 1 , \\ldots , k _ 1 + \\cdots + k _ { i } \\} , & i > 1 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "1537.png", "formula": "\\begin{align*} t ^ { \\ell + y } f ( t ) - a t ^ { 2 k + \\ell + x } f ( t ^ { - 1 } ) & = \\sum _ { m \\in \\Z } \\left ( f _ m t ^ { m + \\ell + y } - a f _ m t ^ { 2 k + \\ell + x - m } \\right ) \\\\ & = \\sum _ { m \\in \\Z } ( f _ { m - \\ell - y } - a f _ { 2 k + \\ell + x - m } ) t ^ m \\\\ & = \\sum _ { m \\in \\Z } h _ m t ^ m = h ( t ) , \\end{align*}"}
+{"id": "5188.png", "formula": "\\begin{align*} ( t _ 1 , t _ 2 , \\dots , t _ n ) \\sum _ { i = 0 } ^ m [ a _ { i } e _ { \\lambda ^ i } ] = \\sum _ { i = 0 } ^ m [ t _ { \\lambda ^ i } a _ { i } e _ { \\lambda ^ i } ] \\end{align*}"}
+{"id": "484.png", "formula": "\\begin{align*} d ^ 2 : = c ^ { 2 - 2 n } t ^ { k _ 1 - k _ 0 - 1 } t _ 1 t _ 2 \\end{align*}"}
+{"id": "6978.png", "formula": "\\begin{align*} L = K ( \\eta ) = K [ x ] / ( g ) \\end{align*}"}
+{"id": "4916.png", "formula": "\\begin{align*} f ( \\underline { k } , t , \\mathbf { p } _ { n + 1 } ) = h _ { t - k } ( q _ 0 ^ { } , q _ 1 ^ { } , . . . , q _ k ^ { } ) \\prod _ { i = 0 } ^ { k - 1 } p _ i ^ { } , \\end{align*}"}
+{"id": "7917.png", "formula": "\\begin{align*} | \\mathbf { n } | _ { \\mathfrak { s } } : = \\sum _ { i = 1 } ^ d \\mathfrak { s } _ i \\mathbf { n } ( i ) ; \\end{align*}"}
+{"id": "6505.png", "formula": "\\begin{align*} t _ n = \\frac { \\Gamma ( n ) \\ , \\Gamma ( n + x ) \\ , \\Gamma ( 2 n + x - m ) \\ , \\Gamma ( 2 n + x + m - 1 ) } { \\Gamma ( n + m ) \\ , \\Gamma ( n + x - m ) \\ , \\Gamma ( 2 n + x ) \\ , \\Gamma ( 2 n + x - 1 ) } . \\end{align*}"}
+{"id": "45.png", "formula": "\\begin{align*} x _ 0 = ( A _ 0 , \\lambda _ 0 , \\psi _ 0 ) \\leftrightarrow ( \\Lambda _ 0 , \\xi _ 0 , \\psi _ { 0 } ) \\end{align*}"}
+{"id": "4707.png", "formula": "\\begin{align*} \\mathcal { D } \\bigl ( \\mathcal { A } _ { \\chi , \\varepsilon } ^ { \\rm a p p } \\bigr ) : = \\bigl \\{ ( \\vect u , \\widehat { \\vect u } ) \\in \\mathcal { H } ^ { \\rm s o f t } \\oplus \\widehat { \\mathcal { H } } _ \\chi ^ { \\rm s t i f f } , \\vect u \\in \\mathcal { D } ( \\widehat { \\mathcal { A } } _ \\chi ^ { \\rm s o f t } ) , \\widehat { \\vect u } = \\widehat { \\Pi } _ \\chi ^ { \\rm s t i f f } \\widehat { \\Gamma } _ { 0 , \\chi } ^ { \\rm s o f t } \\vect u \\bigr \\} , \\end{align*}"}
+{"id": "8369.png", "formula": "\\begin{align*} \\zeta ( u , X ) ^ { - 1 } = ( 1 - u ^ 2 ) ^ { m - n } \\det ( I _ n - A u + ( Q - I _ n ) u ^ 2 ) . \\end{align*}"}
+{"id": "4384.png", "formula": "\\begin{align*} \\overline { \\mathcal G } _ { j } = \\frac { { \\mathcal G } _ { j } + \\mathcal G ' _ { j } } { 2 } , \\mathcal G ^ \\perp _ { j } = \\frac { { \\mathcal G } _ { j } - \\mathcal G ' _ { j } } { 2 } , n _ 0 < j \\leq n _ \\mathcal L . \\end{align*}"}
+{"id": "2700.png", "formula": "\\begin{align*} ( R ^ i \\overline { f } _ * ( \\mathcal { I C } _ { M } ) ) [ d i m ( S ) ] = { } ^ p \\mathcal { H } ^ { d i m ( S ) + i } ( R \\overline { f } _ * ( \\mathcal { I C } _ { M } ) ) . \\end{align*}"}
+{"id": "3246.png", "formula": "\\begin{align*} & \\Big \\| \\sum _ { | j | > \\ell } \\sum _ { Q \\in Q ^ j } w ( Q ) \\psi _ { j } ( x , x _ { Q } ) q _ { j } h ( x _ { Q } ) \\Big \\| _ { \\dot { \\mathcal F } ^ { \\alpha , q } _ { p , { \\rm D } } } \\\\ & = C \\Big \\| \\Big \\{ \\sum _ { k \\in \\mathbb Z } \\sum _ { Q \\in Q ^ k } \\Big ( { \\mathfrak R } ^ { k \\alpha } \\Big | q _ k \\Big ( \\sum _ { | j | > \\ell } \\sum _ { Q \\in Q ^ j } w ( Q ) \\psi _ { j } ( \\cdot , x _ { Q } ) q _ { j } h ( x _ { Q } ) \\Big ) ( x _ Q ) \\Big | \\Big ) ^ q \\chi _ Q \\Big \\} ^ { 1 / q } \\Big \\| _ { L ^ p _ \\omega } , \\end{align*}"}
+{"id": "4046.png", "formula": "\\begin{align*} & \\left ( \\frac { \\sqrt { p } } { 2 \\pi } \\right ) ^ { k + t } \\Gamma ( k + t ) \\ , L ( f , k + t ) = i ^ { - 2 k } \\epsilon _ f \\left ( \\frac { \\sqrt { p } } { 2 \\pi } \\right ) ^ { k - t } \\Gamma ( k - t ) \\ , L ( f , k - t ) \\\\ \\implies & \\left ( \\frac { p } { 4 \\pi ^ 2 } \\right ) ^ t \\Gamma ( k + t ) ^ 2 L ( f , k + t ) ^ 2 = \\left ( \\frac { p } { 4 \\pi ^ 2 } \\right ) ^ { - t } \\Gamma ( k - t ) ^ 2 L ( f , k - t ) ^ . \\end{align*}"}
+{"id": "1580.png", "formula": "\\begin{align*} \\frac { ( 2 n + | x | _ { 1 } ) ! } { n ! \\ , ( n + | x | _ { 1 } ) ! } \\leq 2 ^ { 2 n + | x | _ { 1 } } \\mbox { a n d } \\frac { n ! } { \\prod _ { i = 1 } ^ { d } q _ { j } ! } \\leq \\frac { n ! } { \\left ( \\lfloor { \\frac { n } { d } \\rfloor } \\right ) ! ^ { d } } \\leq C _ { d } \\frac { d ^ { n } } { n ^ { \\frac { d - 1 } { 2 } } } , \\end{align*}"}
+{"id": "6273.png", "formula": "\\begin{align*} \\Lambda ^ S _ { h , t } = \\{ \\exp _ x ( h \\phi _ t ( x ) \\nu ( x ) ) \\in M \\ ; | \\ ; x \\in \\Sigma _ 0 \\setminus B _ t \\} . \\end{align*}"}
+{"id": "8249.png", "formula": "\\begin{align*} \\sum _ { h = 2 } ^ { j - 1 } ( - 1 ) ^ { h } S _ { h - 1 } ^ { ( 1 ) } H _ { k , Y _ { l - h , j - h } } = \\sum _ { h = 2 } ^ { j - 1 } ( - 1 ) ^ { h } ( \\alpha + k - 1 + l - j + h ) H _ { k , Y _ { l - 1 , j - h } } + H _ { k , Y _ { l - 1 , j - 1 } } \\sum _ { h = 2 } ^ { j - 1 } ( \\alpha + k - j + h ) . \\end{align*}"}
+{"id": "6222.png", "formula": "\\begin{align*} H '' = \\bigg \\{ x \\notin H ' : \\ , \\frac { x \\cdot z ^ + } { | z ^ + | } \\leq \\frac 1 2 \\ , R ^ + \\bigg \\} \\ , , & & H ''' = \\bigg \\{ x \\notin H ' : \\ , \\frac { x \\cdot z ^ + } { | z ^ + | } > \\frac 1 2 \\ , R ^ + \\bigg \\} \\ , , \\end{align*}"}
+{"id": "465.png", "formula": "\\begin{align*} s = \\frac { \\tau } { a } , \\ \\ y = \\frac { R } { a } , \\ \\ a > 0 . \\end{align*}"}
+{"id": "787.png", "formula": "\\begin{align*} a _ 0 ^ 2 = O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 + O ( \\varepsilon ) \\| \\nabla u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 . \\end{align*}"}
+{"id": "1454.png", "formula": "\\begin{align*} \\mathcal { A } _ i = B ( \\xi , \\sqrt { \\mu _ i \\mu _ { i - 1 } } ) \\setminus B ( \\xi , \\sqrt { \\mu _ i \\mu _ { i + 1 } } ) , i = 1 , \\cdots , k , \\end{align*}"}
+{"id": "3603.png", "formula": "\\begin{align*} 1 0 ^ m - 6 \\ = \\ 1 0 ^ m + 3 - 9 \\ \\le \\ v ( f ) \\ = \\ p - v ( q ^ \\beta ) \\ \\le \\ 1 0 ^ { m + 1 } - 3 - 3 \\ = \\ 1 0 ^ { m + 1 } - 6 . \\end{align*}"}
+{"id": "7087.png", "formula": "\\begin{align*} \\nu _ n ( g ) = p \\nu ( x - a _ n ) \\end{align*}"}
+{"id": "818.png", "formula": "\\begin{align*} D ^ n _ { x _ 1 } G = \\left ( \\sum _ { j = 0 } ^ n { n \\choose j } \\alpha _ 1 ^ { n - j } D _ { x _ 1 } ^ j D _ { x _ m } ^ { n - j } F \\right ) \\Biggr \\rvert _ { \\substack ( x _ 1 , x _ 2 , \\ldots , \\alpha _ 1 x _ 1 + \\cdots + \\alpha _ { m - 1 } x _ { m - 1 } ) } . \\end{align*}"}
+{"id": "9172.png", "formula": "\\begin{align*} \\bar { u } = F _ { \\bar { u } } \\circ \\phi ( \\zeta _ { [ - 1 ] } ^ { 1 } , x ^ { 1 } , x ^ { 2 } , x ^ { 3 } , v _ { 1 } ^ { 1 } , v _ { 1 , [ 1 ] } ^ { 1 } , v _ { 2 } ^ { 1 } ) \\end{align*}"}
+{"id": "7291.png", "formula": "\\begin{align*} x _ p = x ^ 0 _ p + l ' u _ p , y _ p = y ^ 0 _ p + n ' u _ p u _ p \\geq 0 , \\end{align*}"}
+{"id": "2946.png", "formula": "\\begin{align*} \\mu _ { D D , \\delta } \\{ \\infty \\to \\tfrac { \\alpha } { \\beta } \\} ( f ) = \\sum _ { D \\mid N } n _ D \\cdot C ( \\alpha , \\beta / D , a , b D ) . \\end{align*}"}
+{"id": "2346.png", "formula": "\\begin{align*} \\mathrm { V a r } \\big ( \\big \\langle D F _ R ( t _ i ) , - D L ^ { - 1 } F _ R ( t _ j ) \\big \\rangle _ { \\mathcal { H } } \\big ) \\leq C R , \\mbox { f o r a n y $ i , j = 1 , \\ldots , m $ } . \\end{align*}"}
+{"id": "7045.png", "formula": "\\begin{align*} h _ i \\sum \\limits _ { j = 1 } ^ { s _ { { i _ { } i } } } b _ { i _ { } i j } { \\textbf { X } } ^ { \\lambda _ j } \\in \\mathcal I _ 2 . \\end{align*}"}
+{"id": "1553.png", "formula": "\\begin{align*} 2 ^ n \\sigma = 2 \\end{align*}"}
+{"id": "5098.png", "formula": "\\begin{align*} \\int _ { \\Omega } F ^ { i j } ( D ^ { 2 } u ) \\eta _ { i j } d x = 0 , \\end{align*}"}
+{"id": "7451.png", "formula": "\\begin{align*} \\partial _ t { w } _ { p \\alpha + k - 1 } ^ { ( i ) } ( x _ i , t ) + \\Big ( v ^ { ( i ) } _ i ( x _ i ) \\ , w ^ { ( i ) } _ { p \\alpha + k - 1 } ( x _ i , t ) \\Big ) ^ \\prime = \\Big ( w ^ { ( i ) } _ { p \\alpha + k - 2 } ( x _ i , t ) \\Big ) ^ { \\prime \\prime } , ( x _ i , t ) \\in I _ \\varepsilon ^ { ( i ) } \\times ( 0 , T ) . \\end{align*}"}
+{"id": "55.png", "formula": "\\begin{align*} \\mathbf { K } \\overline { \\psi } _ { g } ^ { \\square } Y _ f Z _ f ^ { - 1 } = \\mathbf { K } \\psi ( X ) ^ { \\square } Y _ f = \\mathbf { K } \\overline { \\psi } _ { g } ^ { \\square } Y _ f \\end{align*}"}
+{"id": "3341.png", "formula": "\\begin{align*} 1 = \\| \\varphi ( a ) \\| ^ 2 \\leq \\| \\varphi ( a ^ * a ) \\| \\leq 1 \\end{align*}"}
+{"id": "5405.png", "formula": "\\begin{align*} S = \\{ m \\le d \\le n : \\gcd ( k , q ^ d - 1 ) < q ^ { d / 3 } \\} . \\end{align*}"}
+{"id": "6437.png", "formula": "\\begin{align*} F _ \\beta ( v . u ) = \\lim _ n \\beta \\theta ^ n ( v _ { - 1 } . u _ 0 ) & = \\lim _ n \\eta ^ n ( \\beta ( v _ { - 1 } ) . \\beta ( u _ 0 ) ) \\\\ & = \\eta ( \\lim _ n \\eta ^ n ( \\beta ( v _ { - 1 } ) . \\beta ( u _ 0 ) ) ) = \\eta ( F _ \\beta ( v . u ) ) . \\end{align*}"}
+{"id": "1953.png", "formula": "\\begin{align*} ( \\det u _ { j \\bar k } ) ^ \\frac { 1 } { n } = \\inf \\Big \\{ L _ a u \\ , ; \\ , ~ a \\in \\mathcal { A } ( \\Omega ) \\Big \\} . \\end{align*}"}
+{"id": "7927.png", "formula": "\\begin{align*} \\Pi _ x = \\Gamma _ { x y } ^ * \\Pi _ y , \\ , \\ , \\ , \\Gamma _ { x y } ^ * = \\Gamma _ { x z } ^ * \\Gamma _ { z y } ^ * \\end{align*}"}
+{"id": "6270.png", "formula": "\\begin{align*} N ( s , t , \\mu ) : = \\bigcup _ { r \\in [ s , t ] } \\partial R ( \\Sigma _ r ' , \\mu ) . \\end{align*}"}
+{"id": "8277.png", "formula": "\\begin{align*} g _ \\beta ( x , t _ 1 , t _ 2 , \\ldots , t _ { d } ) = \\sum _ { n _ 1 = 0 } ^ \\infty \\cdots \\sum _ { n _ d = 0 } ^ \\infty \\frac { t _ 1 ^ { n _ 1 } \\cdots t _ d ^ { n _ d } } { n _ 1 ! \\cdots n _ d ! } I _ { \\beta + \\sum _ { i = 1 } ^ { d } ( i + 1 ) n _ i } ( 2 \\sqrt { x } ) , \\end{align*}"}
+{"id": "3439.png", "formula": "\\begin{align*} X ^ k C _ { k i j } = - \\frac { 1 } { m ( n - 2 ) } | X | ^ 2 \\left ( \\nabla _ i X _ j - \\nabla _ j X _ i \\right ) = - \\frac { 1 } { m ( n - 2 ) } | X | ^ 2 ( d X ) _ { i j } . \\end{align*}"}
+{"id": "20.png", "formula": "\\begin{align*} \\mathbf { X } \\simeq \\left \\{ z = z _ 1 + i z _ 2 \\in \\mathrm { S y m } _ { n } ( \\mathbb { C } ) \\mid z _ 1 , z _ 2 \\in \\mathrm { S y m } _ { n } ( \\mathbb { R } ) , \\ , z _ 2 \\right \\} . \\end{align*}"}
+{"id": "2099.png", "formula": "\\begin{align*} & ( 2 ^ n + 2 ^ { n + 1 } ) ( a + d ) - ( t + 1 ) d - ( ( 2 ^ n + 2 ^ n ) ( a + d ) - t d ) \\\\ = & 2 ^ n ( a + d ) - d = 2 ^ n a + ( 2 ^ n - 1 ) d \\in \\langle A \\rangle , \\end{align*}"}
+{"id": "2889.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 4 - 4 t _ 1 , & ~ ~ t _ 0 \\geq 4 , \\\\ 4 - 3 t _ 1 , & ~ ~ t _ 0 = 3 , \\\\ 4 - 2 t _ 1 , & ~ ~ t _ 0 = 2 , \\\\ 4 - 4 t _ 1 + t _ 0 t _ 1 , & ~ ~ t _ 0 \\leq 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "435.png", "formula": "\\begin{align*} ( \\vec U , { \\vec L _ D } ) = \\sum _ { j = 1 , N } \\lbrack 2 ( \\vec W ^ - ) ^ T \\Sigma ( \\sqrt { | \\Lambda ^ - | } \\vec W ^ - - R \\sqrt { \\Lambda ^ + } \\vec W ^ + - S \\vec G ) \\rbrack _ j d s _ j . \\end{align*}"}
+{"id": "8619.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } S _ { j , + } ( t ) = \\hat { S } \\end{align*}"}
+{"id": "2195.png", "formula": "\\begin{align*} \\begin{bmatrix} L _ { \\bar { a } } & & & & \\\\ - 2 I _ { m } & L _ { \\bar { a } } & & & \\\\ L _ { \\bar { a } } & - 2 I _ { m } & L _ { \\bar { a } } & & \\\\ & \\ddots & \\ddots & \\ddots & \\\\ & & L _ { \\bar { a } } & - 2 I _ { m } & L _ { \\bar { a } } \\end{bmatrix} , \\end{align*}"}
+{"id": "1391.png", "formula": "\\begin{align*} x _ { i _ 0 } + \\sum _ { \\substack { j : f ( j ) < f ( i _ 0 ) \\\\ g ( j ) = g ( i _ 0 ) } } s _ { { i _ 0 } j } x _ j . \\end{align*}"}
+{"id": "1974.png", "formula": "\\begin{align*} g \\sum _ { p = 1 } ^ n u ^ { p \\bar p } u _ { p \\bar p j \\bar j } \\geq - A _ 1 ( 1 + \\Vert \\nabla v \\Vert ^ 2 + \\Delta v ) , \\end{align*}"}
+{"id": "7125.png", "formula": "\\begin{align*} \\int _ { B _ { \\delta } ( x ^ \\prime ) } J _ \\varepsilon ( | x - \\hat { y } | ) \\nabla _ { x ^ \\prime } c ( x ) \\cdot ( x ^ \\prime - y ^ \\prime ) y ^ \\prime = 0 , \\end{align*}"}
+{"id": "8623.png", "formula": "\\begin{align*} & P ( Z _ 0 ( \\Delta \\ell _ 2 ) = m | W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 , Z _ 0 ( \\Delta \\ell _ 1 ) = n ) \\\\ & \\cdot P ( B _ { \\ell _ 1 \\Delta } ( \\ell _ 2 \\Delta ) = i | Z _ 0 ( \\Delta \\ell _ 2 ) = m , Z _ 0 ( \\Delta \\ell _ 1 ) = n , W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 ) \\\\ & = \\frac { P ( B _ { \\ell _ 1 \\Delta } ( \\ell _ 2 \\Delta ) = i , Z _ 0 ( \\Delta \\ell _ 2 ) = m , Z _ 0 ( \\Delta \\ell _ 1 ) = n , W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 ) } { P ( W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 , Z _ 0 ( \\Delta \\ell _ 1 ) = n ) } \\end{align*}"}
+{"id": "1322.png", "formula": "\\begin{align*} \\lim _ { u \\to + \\infty } T _ i ( u ) = \\tau _ i , \\end{align*}"}
+{"id": "582.png", "formula": "\\begin{align*} \\langle \\phi ^ { + } ( t ) v , v \\rangle = \\int _ { t } ^ { \\infty } | \\langle H ( s ) , v \\rangle | ^ 2 d s , \\end{align*}"}
+{"id": "6983.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ s a _ i \\textbf { Q } ^ { \\lambda _ i } \\end{align*}"}
+{"id": "5694.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ) = D ^ { k - 1 } ( f ^ { + } ) = \\sum _ { n \\gg - \\infty } ( \\dfrac { n } { h } ) ^ { k - 1 } C _ { s } ( n ) q _ { h } ^ { n } . \\end{align*}"}
+{"id": "9324.png", "formula": "\\begin{align*} \\widetilde { Y } _ { t } = U _ { t } ^ { - 1 } \\left ( U _ 0 \\widetilde { Y } _ 0 + \\int _ { 0 } ^ { t } U _ { s } d \\widetilde { \\mu } _ { s } \\right ) \\end{align*}"}
+{"id": "3672.png", "formula": "\\begin{align*} \\begin{cases} \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) u = 0 & \\Omega , \\\\ u = 0 & \\Omega ^ c , \\end{cases} \\end{align*}"}
+{"id": "158.png", "formula": "\\begin{align*} L i _ 2 \\left ( \\frac { x } { 1 - x } . \\frac { y } { 1 - y } \\right ) = L i _ 2 \\left ( \\frac { x } { 1 - y } \\right ) + L i _ 2 \\left ( \\frac { y } { 1 - x } \\right ) - L i _ 2 ( x ) - L i _ 2 ( y ) - \\log ( 1 - x ) \\log ( 1 - y ) . \\end{align*}"}
+{"id": "6480.png", "formula": "\\begin{align*} ( \\rho ( g ) ( \\varphi ) ) ( x ) = \\varphi ( x g ) . \\end{align*}"}
+{"id": "1540.png", "formula": "\\begin{align*} l = \\frac { m } { 2 ^ { j } } j \\geq 1 Q = K _ { \\rho } \\times J \\end{align*}"}
+{"id": "7020.png", "formula": "\\begin{align*} \\nu _ i \\left ( f ' _ j Q _ i ^ j \\right ) = \\nu _ i \\left ( f _ j Q _ i ^ j \\frac { f ' _ j } { f _ j } \\right ) > \\nu _ i ( f ) + \\alpha _ i . \\end{align*}"}
+{"id": "7459.png", "formula": "\\begin{align*} N _ \\alpha ( \\xi , t ) = w ^ { ( i ) } _ { \\alpha } ( 0 , t ) + \\xi _ i \\ , \\partial _ { x _ i } w ^ { ( i ) } _ { \\alpha - 1 } ( 0 , t ) + \\mathcal { O } ( \\exp ( - \\beta _ 0 \\xi _ i ) ) \\mbox { a s } \\ \\ \\xi _ i \\to + \\infty , \\ \\ \\xi \\in \\Xi ^ { ( i ) } , i \\in \\{ 1 , 2 , 3 \\} . \\end{align*}"}
+{"id": "6364.png", "formula": "\\begin{align*} \\ell _ t ( x , y ) : = \\rho _ { 1 - e ^ { - t } } ( e ^ { - t / \\alpha } x , y ) , x , y \\in \\Gamma , \\ ; t > 0 , \\end{align*}"}
+{"id": "3173.png", "formula": "\\begin{align*} \\frac { 1 } { n ^ d } \\sum _ { 0 \\leq i _ 1 < n } \\cdots \\sum _ { 0 \\leq i _ d < n } T _ 1 ^ { i _ 1 } \\cdots T _ d ^ { i _ d } f \\leq \\frac { \\chi _ d } { n _ d } \\sum _ { j = 0 } ^ { n _ d - 1 } U ^ j f \\end{align*}"}
+{"id": "971.png", "formula": "\\begin{align*} { \\rm c a p } _ { q , Q } \\ , ( E , F , D ) \\quad = \\quad \\inf \\limits _ { u \\ , \\in \\ , W _ 0 ( E , F ) } \\quad \\int \\limits _ D \\ , Q ( x ) \\cdot | \\nabla u | ^ q \\ , \\ , d m ( x ) \\ , . \\end{align*}"}
+{"id": "950.png", "formula": "\\begin{align*} Z ^ * : C H _ i ( X ) \\to C H _ i ( X ) , Z ^ * ( [ Y ] ) = [ { \\pi _ { 1 , X } } _ * ( Z \\cdot \\pi _ { 2 , X } ^ * ( Y ) ) ] \\end{align*}"}
+{"id": "7467.png", "formula": "\\begin{align*} - \\varepsilon \\ , \\partial _ { \\boldsymbol { \\nu } _ \\varepsilon } u _ \\varepsilon + u _ \\varepsilon \\ , \\overrightarrow { V _ \\varepsilon } \\boldsymbol { \\cdot } \\boldsymbol { \\nu } _ \\varepsilon = 0 \\end{align*}"}
+{"id": "8828.png", "formula": "\\begin{align*} f _ \\lambda ( x ) : = f ( 0 ) + \\int _ 0 ^ x f ' ( y ) \\chi _ \\lambda ( y ) \\ , d y , \\end{align*}"}
+{"id": "1940.png", "formula": "\\begin{align*} A _ n ( z _ 1 , \\ldots , z _ { m } ) : = \\prod _ { j = 1 } ^ { n } ( \\mu _ { n , 1 } ^ { ( j ) } z _ 1 + \\cdots + \\mu _ { n , m } ^ { ( j ) } z _ m - \\lambda _ n ^ { ( j ) } ) , \\end{align*}"}
+{"id": "3529.png", "formula": "\\begin{align*} E _ 2 = - E _ 1 = \\begin{bmatrix} - I & 0 \\\\ 0 & 0 \\end{bmatrix} \\end{align*}"}
+{"id": "536.png", "formula": "\\begin{align*} f ( k ) = \\sum _ { \\xi \\in \\mathcal { I } _ { \\hbar } } \\widehat { f } ( \\xi ) u _ { \\xi } ( k ) , f \\in \\mathrm { H } ^ { \\infty } _ { \\mathcal { H } _ { \\hbar , V } } . \\end{align*}"}
+{"id": "3612.png", "formula": "\\begin{align*} \\beta \\ \\le \\ \\frac { m } { m - 1 } \\ = \\ 1 + \\frac { 1 } { m - 1 } , \\end{align*}"}
+{"id": "2521.png", "formula": "\\begin{align*} ( k , 2 n ) = _ { 2 n } / _ { _ { 2 n } } ( W ) , \\end{align*}"}
+{"id": "6223.png", "formula": "\\begin{align*} \\int _ V \\hat f - f _ j = \\int _ { U ^ c } f _ j - \\hat f = \\eta \\ , . \\end{align*}"}
+{"id": "4639.png", "formula": "\\begin{align*} \\nabla | \\nabla f | ^ { 2 q - 2 } = ( q - 1 ) | \\nabla f | ^ { 2 q - 4 } \\nabla | \\nabla f | ^ 2 , \\end{align*}"}
+{"id": "8108.png", "formula": "\\begin{align*} \\left ( \\sum _ { n \\ge 0 } ( - 1 ) ^ n X _ { \\bullet \\bullet \\cdots \\bullet } \\right ) ^ { - 1 } = \\sum _ { n \\ge 0 } \\sum _ { | F | = n } X _ F . \\end{align*}"}
+{"id": "1245.png", "formula": "\\begin{align*} ( T ^ m ) ' x & \\ge C _ 1 ( m - \\ell _ k + 1 ) ^ { 1 + \\frac { 1 } { \\alpha \\beta } } ( \\ell _ 1 + 1 ) ^ { 1 + \\frac { 1 } { \\alpha } } \\times \\prod _ { j = 2 } ^ k \\max \\{ \\lambda , C _ 2 ( \\ell _ { j } - \\ell _ { j - 1 } ) ^ { 1 + \\frac { 1 } { \\alpha } } \\} . \\end{align*}"}
+{"id": "5282.png", "formula": "\\begin{align*} \\binom { q } { 3 } | 1 + y ' | ^ { q - 3 } ( 1 + y ' ) y ^ 3 \\le \\binom { q } { 3 } | y | ^ 3 ( 1 + | y | ) ^ { q - 3 } . \\end{align*}"}
+{"id": "8965.png", "formula": "\\begin{align*} e _ { i j } = \\sum _ { m = 1 } ^ { k } D _ { p _ m } h ^ { p _ m } \\big [ r ^ { ( i - j ) p _ m } - 1 \\big ] r ^ { ( j - 1 ) p _ m } \\end{align*}"}
+{"id": "5103.png", "formula": "\\begin{align*} \\int _ { B _ r } c _ { 0 } ^ { i j , k l } w _ { i j } \\eta _ { k l } d x = 0 \\forall \\eta \\in C _ { 0 } ^ { \\infty } ( B _ { r } ( 0 ) ) , \\end{align*}"}
+{"id": "5260.png", "formula": "\\begin{align*} \\| ( f _ { n , d , j } ) _ { S \\rightarrow 1 } \\| _ 2 ^ 2 = ( | S | ( 1 - p ) ) ^ 2 + ( \\frac { n } { d } - | S | ) ( p ( 1 - p ) ) \\le | S | ^ 2 + 1 , \\end{align*}"}
+{"id": "37.png", "formula": "\\begin{align*} X = Y _ \\mathbb { Q } Y _ f ^ { - 1 } Y _ \\mathbb { Q } \\in \\mathcal { R } ^ 1 ( \\mathbb { Q } ) Y _ f = ( Y _ q ) _ q \\in \\mathcal { R } ^ 1 ( \\widehat { \\mathbb { Z } } ) . \\end{align*}"}
+{"id": "4705.png", "formula": "\\begin{align*} \\bigl ( \\widehat { S } _ \\chi ^ { \\rm s t i f f ( s o f t ) } ( z ) \\bigr ) ^ * = \\widehat { P } _ \\chi \\bigl ( S _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\bigr ) ^ * , \\bigl ( \\widehat { \\Pi } _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\bigr ) ^ * = \\widehat { P } _ \\chi \\bigl ( \\Pi _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\bigr ) ^ * . \\end{align*}"}
+{"id": "8324.png", "formula": "\\begin{align*} R _ { n m } ^ { 1 2 } ( z + u - v + a h ) = \\prod _ { i = 1 , \\dots , n } ^ { \\longrightarrow } \\prod _ { j = n + 1 , \\ldots , n + m } ^ { \\longleftarrow } R _ { i j } ( z + u _ i - v _ { j - n } + a h ) . \\end{align*}"}
+{"id": "4540.png", "formula": "\\begin{align*} \\alpha | \\xi | ^ 2 + 2 \\omega - \\mathbf { c } \\cdot \\xi = \\alpha \\left | \\xi - \\frac { \\mathbf { c } } { 2 \\alpha } \\right | ^ 2 + 2 \\left ( \\omega - \\frac { | \\mathbf { c } | ^ 2 } { 8 \\alpha } \\right ) \\sim 1 + | \\xi | ^ 2 \\end{align*}"}
+{"id": "3732.png", "formula": "\\begin{align*} U & ( \\vec { a } , \\Lambda ) \\left ( S , f _ \\alpha \\otimes \\rho ( E ^ \\alpha ) \\right ) \\\\ & : = \\left ( \\Lambda S + \\vec { a } , e ^ { - i [ \\vec { a } \\cdot ( \\hat { H } ( \\rho _ n ) \\hat { f } _ 0 + \\hat { P } ( \\rho _ n ) \\hat { f } _ 1 ) ] } f _ { \\alpha } ( \\Lambda ^ { - 1 } ( \\cdot - \\vec { a } ) ) \\otimes \\rho ( E ^ \\alpha ) \\right ) . \\end{align*}"}
+{"id": "151.png", "formula": "\\begin{align*} | \\det ( I - d _ x \\varphi ^ { - t } ( x _ 0 ) ) | = ( - 1 ) ^ s \\det ( I - d _ x \\varphi ^ { - t } ( x _ 0 ) ) . \\end{align*}"}
+{"id": "2589.png", "formula": "\\begin{align*} \\partial _ { n t } \\Omega : = \\{ x \\in \\partial \\Omega \\cap \\overline { \\Omega \\cap \\partial U } : \\langle D u ( x ) - x , z \\rangle \\leq 0 \\ \\ \\forall z \\in \\Omega \\} . \\end{align*}"}
+{"id": "7552.png", "formula": "\\begin{align*} c _ x = \\prod _ { p \\le \\exp ( \\log ^ \\beta x ) } ( 1 - p ^ { - 1 } ) . \\end{align*}"}
+{"id": "6734.png", "formula": "\\begin{align*} \\overline { [ \\underline { \\varphi ^ { \\prime } } , \\varphi ^ { \\prime } ] } = X _ { \\varphi ^ { \\prime } } = X _ { \\eta ' } \\subseteq X _ \\eta \\subseteq X _ \\varphi = \\overline { [ \\underline { \\varphi } , \\varphi ] } . \\end{align*}"}
+{"id": "3752.png", "formula": "\\begin{align*} \\psi ^ { \\alpha _ \\theta , n } ( h ) & = \\phi ^ { \\alpha _ \\theta , n } ( h ) ^ \\ast , \\ 1 \\leq \\theta \\leq r , \\\\ \\psi ^ { \\beta _ { \\theta - r } , n } ( h ) & = \\phi ^ { \\beta _ { \\theta - r } , n } ( h ) ^ \\ast , \\ r + 1 \\leq \\theta \\leq r + s , \\end{align*}"}
+{"id": "666.png", "formula": "\\begin{align*} s _ k & = O ( e ^ { ( \\max \\{ \\lambda _ { i } , \\lambda _ 1 a \\} - \\lambda _ 1 + 6 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k + 1 ) } ) \\\\ V _ { k } ^ { j , \\tau } ( T , \\bar { s } ) & = O ( e ^ { ( \\max \\{ \\lambda _ { i } , \\lambda _ 1 a \\} - \\lambda _ 1 + 8 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k ) } ) \\\\ W _ { k } ^ { j , \\tau } ( T , \\bar { s } ) & = O ( e ^ { ( \\max \\{ \\lambda _ { i } - \\lambda _ 1 , \\lambda _ 1 a - \\lambda _ 1 , - \\lambda _ j \\} + 9 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k ) } ) . \\end{align*}"}
+{"id": "4605.png", "formula": "\\begin{align*} { \\rm P } _ { j , j } = 1 - \\sum ^ { \\min \\{ K , M - j \\} } _ { i = 1 } \\bar { \\rm P } _ { j , j + i } , \\end{align*}"}
+{"id": "2356.png", "formula": "\\begin{align*} M ( U ) _ f = \\left ( \\prod _ { x : U } ( M _ x ) _ { f ( x ) } \\right ) \\to \\left ( \\prod _ { x : U } ( M _ x ) ^ { D ( f ( x ) ) } \\right ) = \\left ( \\prod _ { x : D ( f ) } M _ x \\right ) = M ( D ( f ) ) \\end{align*}"}
+{"id": "2726.png", "formula": "\\begin{align*} T _ 6 = - \\frac { s } { 2 } \\iint _ Q \\sigma _ { t t } | u | ^ 2 d x d t \\geq - C s \\iint _ Q \\xi ^ { 3 / 2 } | u | ^ 2 d x d t . \\end{align*}"}
+{"id": "1778.png", "formula": "\\begin{align*} | K | _ { q } = \\int ^ { [ n ] \\in \\mathcal { A } } K _ n . ( \\{ 0 < 1 \\} ^ n ) ^ { c o f } \\end{align*}"}
+{"id": "7680.png", "formula": "\\begin{align*} \\Lambda _ { , 1 } ( k , p ) = \\prod _ { s = 1 } ^ k \\left ( \\frac { n } { p } - 2 s \\right ) ^ p \\left ( \\frac { n ( p - 1 ) } { p } + 2 s - 2 \\right ) ^ p . \\end{align*}"}
+{"id": "4816.png", "formula": "\\begin{align*} p _ 0 \\left ( \\xi _ 0 \\right ) = 0 \\end{align*}"}
+{"id": "2869.png", "formula": "\\begin{align*} \\Delta \\circ T = ( T \\otimes T ) \\Delta . \\end{align*}"}
+{"id": "5165.png", "formula": "\\begin{align*} T ( p ) : = \\sum _ { e \\in p } t _ e , \\end{align*}"}
+{"id": "3736.png", "formula": "\\begin{align*} T ( f ) = & \\int _ { I ^ 2 } f ^ { \\hat { S } ^ \\flat } ( \\sigma ( \\hat { t } ) ) h ( \\hat { t } ) \\ d \\hat { t } . \\end{align*}"}
+{"id": "119.png", "formula": "\\begin{align*} \\| \\tilde { A } \\tilde { R } _ h ( z ) \\tilde { B } \\| _ { L ^ 2 \\to L ^ 2 } = \\mathcal { O } ( h ^ { - 1 } ) , \\end{align*}"}
+{"id": "1969.png", "formula": "\\begin{align*} n ^ { - 2 } g u ^ { k \\bar l } u _ { k \\bar l \\bar j } u ^ { p \\bar q } u _ { p \\bar q j } + n ^ { - 1 } g ( u ^ { p \\bar q } ) _ { \\bar j } u _ { p \\bar q j } + n ^ { - 1 } g u ^ { p \\bar q } u _ { p \\bar q j \\bar j } = 2 R e ( g _ { v j } v _ { \\bar j } ) + g _ v v _ { j \\bar j } + g _ { v v } | v _ j | ^ 2 + g _ { j \\bar j } . \\end{align*}"}
+{"id": "2313.png", "formula": "\\begin{align*} \\begin{aligned} u '''' & = f , \\\\ u = u ' & = 0 , \\end{aligned} \\end{align*}"}
+{"id": "4274.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\mathrm { d } \\Psi _ { s } & = & \\kappa \\Psi _ { s } \\mathrm { d } W _ { s } , \\\\ \\Psi _ { 0 } & = & 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "4591.png", "formula": "\\begin{align*} ( \\delta _ T f ) ( u , v , w ) = & - f ( [ u , v , w ] _ T ) + l _ T ( u , v , f ( w ) ) + m _ T ( u , f ( v ) , w ) + r _ T ( f ( u ) , v , w ) \\\\ = & - f ( \\rho ( T u , T v ) w ) + [ T u , T v , f ( w ) ] + [ T u , f ( v ) , T w ] - T \\rho ( T u , f ( v ) ) w \\\\ & + [ f ( u ) , T v , T w ] - T \\rho ( f ( u ) , T v ) w \\end{align*}"}
+{"id": "5511.png", "formula": "\\begin{align*} W _ { \\Omega } : = \\left \\{ \\left ( x , \\left ( L _ { x } \\omega _ { 1 } , L _ { x } \\omega _ { 2 } , \\dots \\right ) \\right ) : x \\in X , ( \\omega _ { 1 } , \\omega _ { 2 } \\dots ) \\in G ^ { \\mathbb { N } } \\right \\} = \\bigsqcup _ { x \\in X } \\{ x \\} \\times ( L _ { x } \\backslash G ) ^ { \\mathbb { N } } \\end{align*}"}
+{"id": "1573.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Z ^ { d } } \\sum _ { w \\in \\mathcal { W } _ n ^ { S A W } ( 0 , x ) } \\prod _ { j = 0 } ^ { n - 1 } D ( w _ { j } , w _ { j + 1 } ) \\leq \\left ( \\sum _ { z \\neq 0 } D ( 0 , z ) \\right ) ^ { n } , \\end{align*}"}
+{"id": "8024.png", "formula": "\\begin{align*} Q \\simeq P \\oplus \\begin{pmatrix} I & 0 \\\\ R & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "8186.png", "formula": "\\begin{align*} \\Phi _ t ^ \\omega = \\{ x \\in \\mathbb { R } ^ d : B ( x , r ( t ) ) \\subseteq K ^ c ( \\omega ) \\} , \\quad \\ : \\ : \\widehat { \\Phi } _ t ^ \\omega = \\Phi _ t ^ \\omega \\cap [ - k t , k t ] ^ d . \\end{align*}"}
+{"id": "4465.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } _ { \\omega , c _ * } : = \\bigcup _ { | c | \\le c _ * } \\mathcal { M } _ { \\omega , c } , \\end{align*}"}
+{"id": "9272.png", "formula": "\\begin{align*} T _ \\eta f ( x ) \\gtrsim \\begin{cases} ( b - a ) b ^ { - 1 } & \\eta \\le 0 , \\\\ ( b - a ) b ^ { \\eta - 1 } ( b - a + x ) ^ { - \\eta } & \\eta \\in ( 0 , 1 ) , \\\\ \\log \\Big ( 1 + \\frac { b - a } { x } \\Big ) & \\eta = 1 , \\\\ ( b - a ) b ^ { \\eta - 1 } ( b - a + x ) ^ { - 1 } x ^ { - \\eta + 1 } & \\eta > 1 , \\end{cases} \\end{align*}"}
+{"id": "8590.png", "formula": "\\begin{align*} \\lim _ { \\Delta \\to 0 } E \\big | \\bar { S } ^ k _ { j , + , \\Delta } ( t ) - \\bar { S } ^ k _ { j , + } ( t ) \\big | = 0 , t \\geq 0 . \\end{align*}"}
+{"id": "3705.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) _ 1 \\tilde { u } + q \\tilde { u } = \\left ( \\alpha _ m ^ 2 - \\alpha _ m ^ 1 \\right ) ( - \\Delta ) ^ { s _ m } u _ 2 & \\Omega , \\\\ \\mathcal { L } \\tilde { u } = 0 & \\Omega ^ c , \\\\ \\tilde { u } = 0 & \\Omega ^ c . \\end{cases} \\end{align*}"}
+{"id": "8525.png", "formula": "\\begin{align*} \\zeta _ { p , p ^ { \\prime } } \\left ( \\frac { 1 } { 2 } , c \\right ) = \\frac { 2 C _ { p } ^ { ( 1 ) } + \\log \\left ( \\frac { c } { 4 } \\right ) - 4 e ^ { 2 \\pi p ^ { \\prime } } Q _ { 2 \\pi p ^ { \\prime } } ( 0 ) } { 1 + \\frac { 1 } { \\pi p ^ { \\prime } } } + 2 C _ { p ^ { \\prime } } ^ { ( 2 ) } - 2 \\log ( 2 \\pi ) - 2 \\gamma + \\frac { 2 ^ { 5 / 2 } \\theta e ^ { - \\pi \\sqrt { c } } } { c ^ { 1 / 4 } } . \\end{align*}"}
+{"id": "7209.png", "formula": "\\begin{align*} \\lim _ { h \\to + \\infty } \\left ( \\frac { 1 } { \\gamma _ h \\sigma _ h } \\right ) ^ { \\frac { p _ h ( y ) } { p _ h ^ 0 } - 1 } = 1 . \\end{align*}"}
+{"id": "3784.png", "formula": "\\begin{gather*} \\sigma _ { j _ { m + 1 } + m + 1 } \\dots \\sigma _ { j _ k + k } ( j _ m + k ) = j _ m + m , \\\\ \\sigma _ { j _ { m } + m } ( j _ m + m ) = j _ m + m + n - 1 , \\\\ \\sigma _ { j _ 1 + 1 } \\dots \\sigma _ { j _ { m - 1 } + m - 1 } ( j _ m + m + n - 1 ) = j _ m + n . \\end{gather*}"}
+{"id": "2616.png", "formula": "\\begin{align*} \\lambda ^ { - 2 \\gamma } \\ ! & \\int _ { \\mathbb { R } ^ 4 } \\ ! f _ { 1 , m _ 1 } \\left ( x + \\widetilde { P } _ 1 ( t + s ) , y \\right ) \\overline { f _ { 1 , m _ 1 } \\left ( x + \\widetilde { P } _ 1 ( t ) , y \\right ) } \\\\ & f _ { 2 , m _ 2 } \\left ( x , y + \\widetilde { P } _ 2 ( t + s ) \\right ) \\overline { f _ { 2 , m _ 2 } \\left ( x , y + \\widetilde { P } _ 2 ( t ) \\right ) } \\widetilde { \\zeta } _ \\mathbf { m } ( x , y , t , s ) \\ , \\mathrm { d } x \\mathrm { d } y \\mathrm { d } t \\mathrm { d } s , \\end{align*}"}
+{"id": "1989.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u ) ^ n = \\psi _ \\varepsilon ( \\cdot , u ) \\ , \\omega ^ n & \\textnormal { i n } & \\Omega \\\\ u = \\phi & \\textnormal { o n } & \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "1161.png", "formula": "\\begin{align*} ( n - 1 ) T ( 1 ) f + n B ( A ( f ) , A ( 1 ) ) = 0 \\end{align*}"}
+{"id": "1117.png", "formula": "\\begin{align*} \\begin{aligned} f ( x y ) & = ( f x ) y - ( f y ) x \\\\ ( x y ) f & = ( x f ) y + x ( y f ) \\\\ x ( f y ) & = x ( y f ) \\end{aligned} \\end{align*}"}
+{"id": "2762.png", "formula": "\\begin{align*} ( \\varphi _ j | \\varphi _ k ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } & = \\mu _ j ^ { - 1 / 2 } \\mu _ k ^ { - 1 / 2 } ( T \\psi _ j | T \\psi _ k ) _ { L ^ 2 ( \\mathrm { M } ) } \\\\ & = \\mu _ j ^ { - 1 / 2 } \\mu _ k ^ { - 1 / 2 } ( T ^ \\ast T \\psi _ j | \\psi _ k ) _ { H _ \\Gamma ^ { 3 / 2 } ( \\partial \\mathrm { M } ) } = \\delta _ { j k } . \\end{align*}"}
+{"id": "2402.png", "formula": "\\begin{align*} \\vec { \\delta } _ { 1 0 3 } = ( 1 + \\cos a _ { 3 , 1 0 2 } \\cos \\omega _ { 3 , 1 0 2 } , \\cos a _ { 3 , 1 0 2 } \\sin \\omega _ { 3 , 1 0 2 } , \\sin a _ { 3 , 1 0 2 } ) \\end{align*}"}
+{"id": "3559.png", "formula": "\\begin{align*} \\frac { 1 } { | \\det ( A ) | } C ( n _ { 1 } ^ { * } , \\cdots , n _ { k } ^ { * } ) \\prod _ { j = 1 } ^ { k } \\Gamma ( - n _ { j } ^ { * } ) \\end{align*}"}
+{"id": "8209.png", "formula": "\\begin{align*} R ( t ) + 1 = R _ { \\ell ( t ) } + 1 \\asymp R _ 0 [ \\log ( \\ell ( t ) ) ] ^ { 1 / d } \\asymp R _ 0 [ \\log t ] ^ { 1 / d } , \\end{align*}"}
+{"id": "733.png", "formula": "\\begin{align*} \\tilde A ^ { - 1 } = { \\rm I } _ { N _ { S } } - \\frac 1 { 1 + \\sum a _ { k } } ( a _ { j } ) _ { i , j } . \\end{align*}"}
+{"id": "1076.png", "formula": "\\begin{align*} u ( t ) = e ^ { - t H ^ { \\beta } } u _ 0 + \\int _ 0 ^ t e ^ { - ( t - s ) H ^ { \\beta } } \\ , f ( u ( s ) ) \\ , d s \\end{align*}"}
+{"id": "2664.png", "formula": "\\begin{align*} a _ { j } & = \\dfrac { \\cosh ( \\alpha x _ { j - 1 } ) } { \\cosh ( \\alpha x _ j ) } E _ { j - 1 } = \\dfrac { h _ j } { 6 } + O \\left ( ( \\alpha \\overline { h } ) ^ 2 \\right ) ; \\alpha \\to 0 , \\\\ c _ { j } & = \\dfrac { \\cosh ( \\alpha x _ { j + 1 } ) } { \\cosh ( \\alpha x _ j ) } E _ { j } = \\dfrac { h _ { j + 1 } } { 6 } + O \\left ( ( \\alpha \\overline { h } ) ^ 2 \\right ) ; \\alpha \\to 0 , \\\\ d _ { j } & = \\dfrac { y _ { j + 1 } - y _ j } { h _ { j + 1 } } - \\dfrac { y _ { j } - y _ { j - 1 } } { h _ j } . \\end{align*}"}
+{"id": "6544.png", "formula": "\\begin{align*} \\aligned \\xi _ F ( s ) & = s ^ { m _ F } ( s - 1 ) ^ { m _ F } Q ^ s \\prod _ { j = 1 } ^ { r } \\Gamma ( \\lambda _ { j } s + \\mu _ { j } ) \\ , F ( s ) , \\endaligned \\end{align*}"}
+{"id": "4079.png", "formula": "\\begin{align*} \\varphi : \\mathbb { T } \\to \\bigoplus _ { n = 0 } ^ { 2 k - 2 } \\prod _ { [ f ] \\in \\mathcal { E } _ n } \\mathbb { T } \\slash \\mathrm { A n n } _ { [ f ] _ n } . \\end{align*}"}
+{"id": "1850.png", "formula": "\\begin{align*} - 3 F ( 2 \\pi / 3 ) + \\sum _ { i = 1 } ^ { 3 } F ( \\theta _ { i } ) \\leq - \\frac { 1 } { \\pi ^ { 2 } } \\frac { 1 } { 2 } \\lambda _ { 1 , n } ^ { r , s } . \\leq - \\frac { 9 } { 4 \\pi ^ { 2 } } \\lambda _ { 1 , n } ^ { r , s } \\sum _ { i = 1 } ^ { 3 } \\Big ( \\frac { \\theta _ { i } } { 2 \\pi } - 1 / 3 \\Big ) ^ { 2 } . \\end{align*}"}
+{"id": "8358.png", "formula": "\\begin{align*} L _ X = r _ X \\circ ( I d - \\iota _ X ) \\circ \\tau _ X , L _ X ( v ) = \\sum _ { h \\in T _ v X } ( v - r _ X ( \\iota _ X ( h ) ) ) . \\end{align*}"}
+{"id": "4446.png", "formula": "\\begin{align*} N ( U ) = \\frac { 1 } { 3 } ( K _ { \\omega , \\mathbf { c } } ( U ) - L _ { \\omega , \\mathbf { c } } ( U ) ) < - 2 \\mu _ { \\omega , \\mathbf { c } } . \\end{align*}"}
+{"id": "2161.png", "formula": "\\begin{align*} \\left \\Vert \\partial _ { t } ^ { s } \\widetilde { u } \\right \\Vert _ { H ^ { 2 , 1 } \\left ( Q _ { \\varepsilon , T } \\right ) } , \\left \\Vert \\partial _ { t } ^ { s } \\widetilde { m } \\right \\Vert _ { H ^ { 2 , 1 } \\left ( Q _ { \\varepsilon , T } \\right ) } , \\left \\Vert \\widetilde { k } \\right \\Vert _ { L _ { 2 } \\left ( \\Omega \\right ) } \\leq C \\delta ^ { 1 - \\rho } , \\forall \\delta \\in \\left ( 0 , \\delta _ { 0 } \\right ) , s = 0 , 1 , 2 . \\end{align*}"}
+{"id": "5438.png", "formula": "\\begin{align*} M _ b ( \\theta ) = \\mathbb { E } _ { \\Phi } \\{ P _ s ^ b ( \\theta ) \\} \\end{align*}"}
+{"id": "5480.png", "formula": "\\begin{align*} \\begin{array} { l l } ( \\delta _ { B i H } ^ n f ) ( y ; a _ 1 , \\cdots , a _ { n + 1 } ) & = \\varphi ^ { n - 1 } ( a _ 1 ) \\bullet _ 0 ^ y f ( d _ o ( y ) ; a _ 2 \\cdots , a _ { n + 1 } ) \\\\ & + \\sum ^ n _ { i = 1 } ( - 1 ) ^ i f ( d _ i ( y ) ; \\alpha ( a _ 1 ) , \\cdots , a _ i \\bullet _ i ^ y a _ { i + 1 } , \\psi ( a _ { i + 2 } ) , \\cdots , \\psi ( a _ { n + 1 } ) ) \\\\ & + ( - 1 ) ^ { n + 1 } f ( d _ { n + 1 } ( y ) ; a _ 1 , \\cdots , a _ n ) \\bullet _ { n + 1 } ^ y \\psi ^ { n - 1 } ( a _ { n + 1 } ) , \\end{array} \\end{align*}"}
+{"id": "4800.png", "formula": "\\begin{align*} \\Psi = \\begin{bmatrix} \\psi ( x _ 1 ) & \\psi ( x _ 2 ) & \\cdots & \\psi ( x _ n ) \\\\ \\psi ( x _ 1 ) u _ { 1 , 1 } & \\psi ( x _ 2 ) u _ { 1 , 2 } & \\cdots & \\psi ( x _ n ) u _ { 1 , n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\psi ( x _ 1 ) u _ { m , 1 } & \\psi ( x _ 2 ) u _ { m , 2 } & \\cdots & \\psi ( x _ n ) u _ { m , n } \\end{bmatrix} , \\end{align*}"}
+{"id": "7971.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\nabla V \\ , V ) = \\partial _ j \\Big ( V ^ i \\ , \\partial _ i V ^ j \\Big ) = \\partial _ j V ^ i \\ , \\partial _ i V ^ j + V ^ i \\ , \\partial _ j \\partial _ i V ^ j = \\mathrm { t r } ( ( \\nabla V ) ^ 2 ) + V ^ i \\ , \\partial _ j \\partial _ i V ^ j \\ , , \\end{align*}"}
+{"id": "5194.png", "formula": "\\begin{align*} { \\bf { b } } ( p _ s ) : = b _ { e } - b _ { f } . \\end{align*}"}
+{"id": "774.png", "formula": "\\begin{align*} \\begin{aligned} B ^ m = & \\left . ( - 1 ) ^ m \\rho ^ { m + 2 } { n - m \\choose k - m } \\dfrac { k + n - 2 m } { n - m } \\left [ \\frac { \\phi '^ { k - m } } { D ^ { k + 1 } } \\phi ^ { n - m - 1 } \\right ] \\right | _ { u = | \\nabla u | ^ 2 = 0 } \\\\ = & ( - 1 ) ^ m \\rho ^ { m + 2 } { n - m \\choose k - m } \\dfrac { k + n - 2 m } { n - m } \\phi '^ { k - m } ( \\rho ) \\phi ^ { n - m - 1 } ( \\rho ) . \\end{aligned} \\end{align*}"}
+{"id": "260.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { - n ^ 2 y ^ { n + 3 } + ( 2 n ^ 2 + 2 n - 1 ) y ^ { n + 2 } - ( n ^ 2 + 2 n + 1 ) y ^ { n + 1 } + y ^ 2 + y } { ( 1 - y ) ^ 3 } \\right ) \\frac { z ^ n } { n ^ 3 } \\right \\} \\end{align*}"}
+{"id": "1214.png", "formula": "\\begin{align*} \\gamma ' + \\varphi \\gamma = 0 \\sigma ' + \\varphi \\sigma + q = 0 . \\end{align*}"}
+{"id": "1146.png", "formula": "\\begin{align*} ( A f ) ( x ) = - \\frac { \\sqrt { 2 } } { 2 } \\Delta _ { h } f ( x ) = \\frac { \\sqrt { 2 } } { 2 } \\left ( f ( x ) - f ( x + h ) \\right ) \\end{align*}"}
+{"id": "1592.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { m } \\ , \\mu _ s \\ , ^ { c } _ { 0 } { D } ^ { \\alpha _ s } _ { t } u ( x , t ) - \\Delta u ( x , t ) & = f ( x , t ) & \\mbox { i n } & \\Omega \\times ( 0 , T ] , \\\\ u ( x , t ) & = 0 & \\mbox { o n } & \\partial \\Omega \\times ( 0 , T ] , \\\\ u ( x , 0 ) & = u _ 0 ( x ) & \\mbox { i n } & \\Omega , \\end{align*}"}
+{"id": "3174.png", "formula": "\\begin{align*} \\overline { \\xi } ( x ) = \\sum _ { \\lambda = 0 } ^ { \\infty } \\alpha _ { \\lambda + 1 } ^ { ( 1 ) } x ^ \\lambda . \\end{align*}"}
+{"id": "3133.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( x + k ) ^ { m + k } } { k ! } e ^ { - u ( x + k ) } u ^ k = \\sum _ { k = 1 } ^ { m + 1 } \\frac { \\psi _ k ( m , x ) } { ( 1 - u ) ^ { m + k } } . \\end{align*}"}
+{"id": "2090.png", "formula": "\\begin{align*} & F ( S ( n ) ) = 2 ^ { 2 n } - 2 ^ n - 1 , \\\\ & g ( S ( n ) ) = 2 ^ { n - 1 } ( 2 ^ n + n - 3 ) . \\end{align*}"}
+{"id": "941.png", "formula": "\\begin{align*} Z \\cdot p _ { 1 , X } ^ * ( D ) = \\delta _ { X \\times X } ^ { ! } ( Z \\times p _ { , X 1 } ^ * ( D ) ) \\in C H _ * ( | Z | \\cap p _ { 1 , X } ^ { - 1 } ( | D | ) ) \\end{align*}"}
+{"id": "8229.png", "formula": "\\begin{align*} \\left < \\phi , Z ^ N _ t \\right > = \\left < \\phi , Y _ { t , 1 } ^ N \\right > + \\left < \\phi , Y _ { t , - 1 } ^ N \\right > . \\end{align*}"}
+{"id": "6489.png", "formula": "\\begin{align*} \\operatorname { W e i g h t } ( \\pi ) : & = \\operatorname { s i g n } ( \\pi ) \\ , a _ { 1 , \\pi _ 1 } \\cdots a _ { n , \\pi _ n } \\ ; , \\end{align*}"}
+{"id": "7212.png", "formula": "\\begin{align*} z _ h ^ { ( s ) } : = t _ h \\widetilde { z } _ h ^ { ( s ) } . \\end{align*}"}
+{"id": "8584.png", "formula": "\\begin{align*} & p _ { j , + } ^ k ( t ) : = P ( \\tau _ j ^ + ( k ) \\leq t ) , p _ { j , - } ^ k ( t ) : = P ( \\tau _ j ^ - ( k ) \\leq t ) , \\end{align*}"}
+{"id": "5754.png", "formula": "\\begin{align*} 1 / p = 1 / p _ { _ { 1 } } + 1 / p _ { _ { 2 } } + \\cdots + 1 / p _ { _ { m } } , & 1 / q = 1 / p - \\alpha / n , \\end{align*}"}
+{"id": "5592.png", "formula": "\\begin{align*} H _ { \\alpha \\parallel \\beta } \\left ( \\mathcal { P } \\right ) : = \\sum _ { A \\in \\mathcal { P } } \\alpha ( A ) \\log \\frac { \\alpha ( A ) } { \\beta ( A ) } . \\end{align*}"}
+{"id": "1590.png", "formula": "\\begin{align*} \\mathrm { f p } ( T ) & : = \\{ v \\in V ( T ) : \\mathrm { d e s c d } ( v ) = \\mathrm { d e p t h } ( v ) \\} \\\\ \\mathrm { a d t } ( T ) & : = \\{ ( v , w ) \\in V ( T ) : \\mathrm { d e s c d } ( w ) = \\mathrm { d e p t h } ( v ) , w v \\} . \\end{align*}"}
+{"id": "7990.png", "formula": "\\begin{align*} & \\mathrm { d i v } _ T \\Big ( h \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\Big ) \\ , h \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\cdot \\nu \\\\ & - h ^ 2 \\ , \\nabla _ T \\Big [ h \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\Big ] \\ , \\Big [ \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\Big ] _ T \\cdot \\nu = h \\ , H ( \\nu ) \\ , H ^ 2 ( \\nabla w ) \\ , \\mathrm { t r } \\ , \\mathcal { B } ^ H \\end{align*}"}
+{"id": "6018.png", "formula": "\\begin{align*} \\overline { c } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) & = m _ { X ^ { \\ast } } ( g _ 1 g _ 2 ) ^ { - 1 } m _ { X ^ { \\ast } } ( g _ 1 ) m _ { X ^ { \\ast } } ( g _ 2 ) \\widetilde { c } _ { { X ^ { \\ast } } } ( g _ 1 , g _ 2 ) \\\\ & = [ \\widetilde { \\beta } ( g _ 1 g _ 2 ) m _ { X ^ { \\ast } } ( g _ 1 g _ 2 ) ^ { - 1 } ] [ \\widetilde { \\beta } ( g _ 1 ) m _ { X ^ { \\ast } } ( g _ 1 ) ^ { - 1 } ] ^ { - 1 } [ \\widetilde { \\beta } ( g _ 2 ) m _ { X ^ { \\ast } } ( g _ 2 ) ^ { - 1 } ] ^ { - 1 } . \\end{align*}"}
+{"id": "1715.png", "formula": "\\begin{align*} \\lim _ { \\mu \\to 0 } \\frac { n + \\mu } { \\mu } C _ n ^ \\mu ( x ) = \\epsilon _ n T _ n ( x ) , \\end{align*}"}
+{"id": "5987.png", "formula": "\\begin{align*} d \\overline { \\Pi } _ { \\psi } ( e ^ + ) f ( [ \\epsilon , x ] ) & = \\frac { d } { d t } \\Big \\{ \\Pi _ { \\psi } ( [ e ^ { t e ^ + } , 1 ] ) f ( [ \\epsilon , x ] ) \\Big \\} \\Big | _ { t = 0 } \\\\ & = \\frac { d } { d t } \\Big \\{ e ^ { \\pi i \\epsilon x ^ 2 t } f ( [ \\epsilon , x ] ) \\Big \\} \\Big | _ { t = 0 } \\\\ & = \\pi i \\epsilon x ^ 2 f ( [ \\epsilon , x ] ) . \\end{align*}"}
+{"id": "6766.png", "formula": "\\begin{align*} 0 & = f ( 0 ) \\\\ 0 & = \\Delta f ( 0 ) = f ( 1 ) \\\\ & \\ \\ , \\vdots \\\\ 0 & = \\Delta ^ { \\eta - 1 } f ( 0 ) = \\Delta ^ { \\eta - 2 } f ( 1 ) = \\cdots f ( \\eta - 1 ) \\\\ 0 & = f ( 0 ) = \\Delta ^ { \\eta } f ( 0 ) = \\Delta ^ { \\eta - 1 } f ( 1 ) = \\cdots = f ( \\eta ) \\\\ & \\ \\ , \\vdots \\end{align*}"}
+{"id": "2631.png", "formula": "\\begin{align*} \\triangle _ 1 = \\varepsilon ^ { \\frac { 1 } { 8 ( d _ 1 + d _ 2 ) - 1 7 } } \\textrm { a n d } \\triangle _ 2 = \\varepsilon ^ { \\frac { 4 ( d _ 1 + d _ 2 ) - 1 0 } { 8 ( d _ 1 + d _ 2 ) - 1 7 } } . \\end{align*}"}
+{"id": "6678.png", "formula": "\\begin{align*} & 0 \\geq - \\int _ { \\Omega } \\langle \\nabla u , \\nabla u ^ - \\rangle \\ , d x = \\int _ { \\Omega } | \\nabla u ^ - | ^ 2 \\ , d x \\geq 0 . \\end{align*}"}
+{"id": "8338.png", "formula": "\\begin{align*} 0 \\leq | \\nabla v _ n | \\leq | \\nabla u _ \\mu ^ \\bigstar | & \\lim _ { n \\to \\infty } | \\nabla v _ n - \\nabla u _ \\mu ^ \\bigstar | ( x ) = 0 . \\end{align*}"}
+{"id": "8311.png", "formula": "\\begin{align*} | \\mathcal { A } ( r _ 1 , \\dots , r _ n ) | \\le \\frac { q ^ { 2 n } } { \\prod _ { j = 1 } ^ t p ^ { r _ j n } _ j } \\ , . \\end{align*}"}
+{"id": "3639.png", "formula": "\\begin{align*} ( \\{ 9 \\} ^ { I + 1 } , 4 , \\ldots , \\{ 9 \\} ^ { I + 1 } , 4 ) _ { 1 0 } \\ = \\ 2 ( 4 , \\{ 9 \\} ^ { I + 1 } , \\ldots , 4 , \\{ 9 \\} ^ { I } , 7 ) _ { 1 0 } . \\end{align*}"}
+{"id": "6210.png", "formula": "\\begin{align*} \\begin{cases} | \\alpha Y - \\beta X | & = d ( v _ i , v _ { i - 1 } ) - 1 , \\\\ \\gamma Y - \\delta X & = - 1 . \\end{cases} \\end{align*}"}
+{"id": "1562.png", "formula": "\\begin{align*} h ( v ) = \\begin{cases} 0 & v \\in [ - 1 , 0 ) , \\\\ 2 ( b + c v ^ { 2 } ) & v \\in [ 0 , 1 ] . \\end{cases} \\end{align*}"}
+{"id": "7899.png", "formula": "\\begin{align*} \\Omega _ { n , k } : = \\left \\{ ( S , \\underline { S } ) ^ \\pm : \\ S \\in \\binom { [ n ] } { k } \\right \\} . \\end{align*}"}
+{"id": "3011.png", "formula": "\\begin{align*} ( U ( A + B ) V ) ^ { * } ( U C V ) = 0 \\hbox { a n d } ( U ( A + B ) V ) ( U C V ) ^ * = 0 , \\end{align*}"}
+{"id": "5758.png", "formula": "\\begin{align*} \\| T _ { \\alpha } ( \\vec { f } ) \\| _ { L ^ { q } ( v _ { \\vec { w } } ^ { q } ) } & \\le C \\prod _ { j = 1 } ^ { m } \\| f _ { j } \\| _ { L ^ { p _ { j } } ( w _ { j } ^ { p _ { j } } ) } . \\end{align*}"}
+{"id": "3675.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta v + q v = 0 & \\Omega , \\\\ v = f & \\partial \\Omega \\end{cases} \\end{align*}"}
+{"id": "4798.png", "formula": "\\begin{align*} \\mathrm { d e g } ( \\nabla V ) + \\mathrm { d e g } ( F ) = \\mathrm { d e g } ( V ) - 1 + \\mathrm { d e g } ( F ) \\geq \\mathrm { d e g } ( V ) , \\end{align*}"}
+{"id": "417.png", "formula": "\\begin{align*} P \\Phi _ t + ( A \\Phi ) _ x + A ^ T \\Phi _ x + ( B \\Phi ) _ y + B ^ T \\Phi _ y = 0 , \\end{align*}"}
+{"id": "7441.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { + \\infty } \\sum \\limits _ { p = 0 } ^ { 1 } \\varepsilon ^ { p \\alpha + k - 1 } \\ , \\Pi ^ { ( i ) } _ { p \\alpha + k - 1 } \\Big ( \\frac { \\ell _ i - x _ i } { \\varepsilon } , \\dfrac { \\overline { x } _ i } { \\varepsilon } , t \\Big ) \\end{align*}"}
+{"id": "5009.png", "formula": "\\begin{align*} U _ { n } ^ { - 1 } \\Sigma = \\left [ \\begin{array} { c | c } U _ { n - 1 } ^ { - 1 } & \\mathbf { 0 } \\\\ \\hline \\mathbf { 0 ' } & 1 \\end{array} \\right ] , \\Sigma ^ { - 1 } U _ { n } = \\left [ \\begin{array} { c | c } U _ { n - 1 } & \\mathbf { 0 } \\\\ \\hline \\mathbf { 0 ' } & 1 \\end{array} \\right ] . \\end{align*}"}
+{"id": "4880.png", "formula": "\\begin{align*} \\begin{dcases} L v = k & \\\\ v = 0 & , \\end{dcases} \\end{align*}"}
+{"id": "8772.png", "formula": "\\begin{align*} \\begin{array} { c } \\sum _ { k = 1 } ^ m \\left ( \\beta _ { i , k } - \\beta _ { j , k } \\right ) + \\alpha _ { i j } + \\gamma _ i = 0 \\\\ \\implies \\\\ \\gamma _ i = \\sum _ { k = 1 } ^ m \\left ( \\beta _ { j , k } - \\beta _ { i , k } \\right ) - \\alpha _ { i j } \\leq 2 m p _ { \\max } , \\end{array} \\end{align*}"}
+{"id": "6045.png", "formula": "\\begin{align*} g ( z ) : = | A | z ^ { n + m } - | B | \\overline { z } ^ m + | C | , z \\in \\mathbb { C } . \\end{align*}"}
+{"id": "479.png", "formula": "\\begin{align*} t _ 4 = t v _ 1 \\quad c ^ 2 t ^ { k - 1 } t _ 1 t _ 2 t _ 3 t _ 4 = p q . \\end{align*}"}
+{"id": "2944.png", "formula": "\\begin{align*} \\theta ( a , b ) ( \\tau ) = \\Theta ( a \\tau + b , \\tau ) \\end{align*}"}
+{"id": "4418.png", "formula": "\\begin{align*} \\| V _ n - V \\| _ { \\mathcal { H } ^ 1 } ^ 2 & \\lesssim L _ { \\omega , \\mathbf { c } } ( V _ n - V ) \\\\ & = L _ { \\omega , \\mathbf { c } } ( V _ n ) - L _ { \\omega , \\mathbf { c } } ( V ) - ( L _ { \\omega , \\mathbf { c } } ( V _ n ) - L _ { \\omega , \\mathbf { c } } ( V _ n - V ) - L _ { \\omega , \\mathbf { c } } ( V ) ) \\\\ & \\rightarrow 0 . \\end{align*}"}
+{"id": "1739.png", "formula": "\\begin{align*} & \\left | \\frac { \\delta F } { \\delta \\nu } ( \\nu _ t , \\mu _ t , x ) + \\frac { \\sigma ^ 2 } { 2 } \\log \\left ( \\frac { \\nu _ t ( x ) } { \\pi ( x ) } \\right ) - \\frac { \\sigma ^ 2 } { 2 } \\operatorname { D _ { K L } } ( \\nu _ t | \\pi ) \\right | \\\\ & \\leq 3 C _ { \\nu } + \\frac { \\sigma ^ 2 } { 2 } \\left ( \\max \\{ | \\log r _ { 1 , \\nu } | , \\log R _ { 1 , \\nu } \\} + 2 \\log R _ { \\nu } \\right ) = : C _ { V , \\nu } , \\end{align*}"}
+{"id": "8575.png", "formula": "\\begin{align*} \\frac { \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s } { \\int _ 0 ^ \\infty e ^ { - \\lambda s } ( 1 - p _ 0 ( s ) ) d s } & = \\begin{cases} \\dfrac 1 { j ( j + 1 ) } , & p = 0 , \\\\ \\displaystyle - \\frac { p } { \\log ( q ) } \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } ( 1 - y ) y ^ { j - 1 } d y , & 0 < p < 1 , \\end{cases} \\end{align*}"}
+{"id": "6284.png", "formula": "\\begin{align*} \\dot \\gamma ( t ) = \\sum _ { i = 1 } ^ k u _ i ( t ) X _ i ( \\gamma ( t ) ) , \\qquad t \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "149.png", "formula": "\\begin{align*} \\det ( I - \\mathcal { P } _ \\gamma ^ n | _ { E _ s } ) > 0 , ( - 1 ) ^ s \\det ( I - \\mathcal { P } _ \\gamma ^ n | _ { E _ s } ) = \\det ( I - \\mathcal { P } _ \\gamma ^ { - n } | _ { E _ s } ) \\det ( \\mathcal { P } _ \\gamma ^ n | _ { E _ s } ) > 0 . \\end{align*}"}
+{"id": "609.png", "formula": "\\begin{align*} \\theta ( b ) = P _ H \\pi ( b ) | _ H , b \\in B \\end{align*}"}
+{"id": "3260.png", "formula": "\\begin{align*} | \\triangle _ { \\rm D } ^ { - \\alpha / 2 } f ( x ) | & = \\bigg | \\int _ { \\mathbb R ^ N } K _ \\alpha ( x , y ) f ( y ) d \\omega ( y ) \\bigg | \\\\ & \\leq \\int _ { d ( x , y ) \\leq s } \\frac { d ( x , y ) ^ { 2 + \\alpha } } { \\| x - y \\| ^ 2 } \\frac 1 { \\omega ( B ( x , d ( x , y ) ) ) } | f ( y ) | d \\omega ( y ) \\\\ & \\quad + \\int _ { d ( x , y ) > s } \\frac { d ( x , y ) ^ { 2 + \\alpha } } { \\| x - y \\| ^ 2 } \\frac 1 { \\omega ( B ( x , d ( x , y ) ) ) } | f ( y ) | d \\omega ( y ) \\\\ & = : J _ 1 + J _ 2 . \\end{align*}"}
+{"id": "6937.png", "formula": "\\begin{align*} u \\sim \\begin{cases} u - e _ i + e _ { m + j } & \\\\ u + e _ i - e _ { m + j } & \\end{cases} \\end{align*}"}
+{"id": "5777.png", "formula": "\\begin{align*} c & = \\sum \\limits _ { ( \\rho _ { 1 } , \\dots , \\rho _ { m } ) \\in \\rho } T _ { \\alpha } ( f _ { 1 } ^ { \\rho _ { 1 } } , \\dots , f _ { m } ^ { \\rho _ { m } } ) ( x ) , \\end{align*}"}
+{"id": "8516.png", "formula": "\\begin{align*} \\zeta _ { 0 , \\infty } ( s , c ) = \\frac { \\pi } { \\sqrt { c } } \\ , \\frac { 1 } { s - 1 } + \\pi ^ { 2 } + \\frac { \\pi } { \\sqrt { c } } \\left ( 2 \\gamma - \\log \\left ( 4 c \\right ) - 4 \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n } \\cdot \\frac { 1 } { e ^ { 2 \\pi \\sqrt { c } n } + 1 } \\right ) + O ( s - 1 ) \\end{align*}"}
+{"id": "1407.png", "formula": "\\begin{align*} \\sharp \\{ p \\leq r \\leq q \\ , | \\ , D _ { \\lambda } ( r ) = \\times \\} \\geq \\sharp \\{ p \\leq r \\leq q \\ , | \\ , D _ { \\lambda } ( r ) = \\circ \\} \\end{align*}"}
+{"id": "9332.png", "formula": "\\begin{align*} & a ( X _ { 1 4 } ; T ) = \\sum _ { d \\mid \\varepsilon ( T ) } d ^ { 1 3 } \\tau ^ * ( 2 { \\rm d e t } ( T ) / d ^ 2 ) \\\\ & \\tau ^ * ( \\ell ) = \\tau ( \\ell ) - 2 ^ { 1 2 } \\tau ( \\ell / 4 ) , \\end{align*}"}
+{"id": "1520.png", "formula": "\\begin{align*} H ( B , C , D | A ) & = H ( A , B , C , D | A ) \\\\ & \\leq H ( X | A ) \\\\ & = H ( X ) + H ( A | X ) - H ( A ) \\\\ & \\leq H ( X ) + H ( A , B ) - H ( A B ) - H ( A ) . \\end{align*}"}
+{"id": "4847.png", "formula": "\\begin{align*} \\begin{array} { r l } g _ { w ( l ) } ( y ) = & \\prod \\limits _ { i = 1 } ^ n h _ { d _ i } ( n y + i ) \\\\ = & ( \\prod \\limits _ { p = 1 } ^ r ( \\prod \\limits _ { i \\in L ( p ) } h _ { b _ i } ( n y + i ) ) \\ast h _ l ( n y + s ( p ) ) ) \\ast \\prod \\limits _ { i \\in { L ( r + 1 ) } } h _ { b _ i } ( n y + i ) \\\\ = & ( \\prod \\limits _ { p = 1 } ^ r c _ y ( p ) \\ast h _ l ( z _ y ( p ) ) ) \\ast c _ y ( r + 1 ) \\end{array} \\end{align*}"}
+{"id": "8372.png", "formula": "\\begin{align*} 4 D _ a = 4 D _ b = 4 D _ c = D _ a + D _ b + D _ c = 0 . \\end{align*}"}
+{"id": "1120.png", "formula": "\\begin{align*} ( f g ) ( x y ) = ( ( f g ) x ) y - ( ( f g ) y ) x , \\end{align*}"}
+{"id": "7299.png", "formula": "\\begin{align*} x ^ n + x ^ k y ^ l + y ^ m = 0 , \\end{align*}"}
+{"id": "6128.png", "formula": "\\begin{align*} \\begin{aligned} J \\subset \\Lambda _ { 2 ^ { \\frac { 2 } { s } } \\times 1 0 \\sqrt { n } \\mathfrak { l } \\left ( \\mathcal { H } \\right ) } ( t _ { 0 } ) . \\end{aligned} \\end{align*}"}
+{"id": "7316.png", "formula": "\\begin{align*} a _ p + \\sum _ { i = 1 } ^ n \\alpha _ i z _ { i p } = b _ p + \\sum _ { i = 1 } ^ n \\beta _ i z _ { i p } \\leq c _ p + \\sum _ { i = 1 } ^ n \\gamma _ i z _ { i p } , \\end{align*}"}
+{"id": "2818.png", "formula": "\\begin{align*} S _ k ^ { m , 1 } ( t ) = X ( t ) ( S _ k ^ { m , 1 } ( 0 ) + \\int _ { 0 } ^ t X ( \\tau ) ^ { - 1 } S _ { k + r + 1 } ^ { m , 1 } ( \\tau ) d \\tau ) ; \\end{align*}"}
+{"id": "1462.png", "formula": "\\begin{align*} f _ \\epsilon ^ { ' } \\Big ( V _ m ( \\mu _ m ^ i y + \\xi _ m ) \\Big ) = & f _ \\epsilon ^ { ' } \\Big ( P U _ { \\mu _ { i n } , \\xi _ { i n } } ( \\mu _ m ^ i y + \\xi _ m ) + \\sum \\limits _ { i = 1 , j \\neq i } ^ k P U _ { \\mu _ { j m } , \\xi _ { j m } } ( \\mu _ m ^ i y + \\xi _ m ) \\Big ) \\\\ = & f _ \\epsilon ^ { ' } \\Big ( ( \\mu _ m ^ { i } ) ^ { - \\frac { n - 2 } { 2 } } U ( y - \\sigma _ { i n } ) + U _ { \\mu _ { j m } , \\xi _ { j m } } ( \\mu _ m ^ i y + \\xi _ m ) + o ( 1 ) \\Big ) , \\end{align*}"}
+{"id": "5344.png", "formula": "\\begin{align*} \\sqrt { q } ( r ' ) ^ { - 2 / q } \\leq \\max ( 2 \\frac { \\log ( 1 / \\gamma ) } { 2 } , \\log ( 1 / \\gamma ) \\left ( \\frac { \\log ( 1 / \\gamma ) } { 2 } \\right ) ^ { 2 / \\log ( 1 / \\gamma ) } ) ^ { 1 / 2 } = \\sqrt { \\log ( 1 / \\gamma ) } \\sqrt { \\max ( 1 , 2 ) } = \\sqrt { 2 } \\sqrt { \\log ( 1 / \\gamma ) } , \\end{align*}"}
+{"id": "7584.png", "formula": "\\begin{align*} p ( z ) = \\frac { 1 + ( \\overline { \\tau _ 2 } \\tau _ 3 + \\overline { \\tau _ 1 } \\tau _ 2 + \\tau _ 1 ) z + ( \\overline { \\tau _ 1 } \\tau _ 3 + \\tau _ 1 \\overline { \\tau _ 2 } \\tau _ 3 + \\tau _ 2 ) z ^ 2 + \\tau _ 3 z ^ 3 } { 1 + ( \\overline { \\tau _ 2 } \\tau _ 3 + \\overline { \\tau _ 1 } \\tau _ 2 - \\tau _ 1 ) z + ( \\overline { \\tau _ 1 } \\tau _ 3 - \\tau _ 1 \\overline { \\tau _ 2 } \\tau _ 3 - \\tau _ 2 ) z ^ 2 - \\tau _ 3 z ^ 3 } , \\ ; \\ ; z \\in \\mathbb { D } . \\end{align*}"}
+{"id": "220.png", "formula": "\\begin{align*} L i _ n ( z ) = \\sum _ { r = 1 } ^ { \\infty } \\frac { z ^ r } { r ^ n } , \\ ; \\ ; | z | \\leq 1 . \\end{align*}"}
+{"id": "1383.png", "formula": "\\begin{align*} g _ x g _ y ^ { - 1 } & = g _ { x + z } g _ { y + z } ^ { - 1 } , \\\\ g _ 0 g _ k ^ { - 1 } & = \\gamma ^ k , \\\\ g _ x \\gamma & = \\gamma ^ \\alpha g _ x \\end{align*}"}
+{"id": "3907.png", "formula": "\\begin{align*} \\widehat { \\Sigma } _ { \\mathrm { D } } ( \\delta _ 1 , \\delta _ 2 ) = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } _ 1 \\times \\mathcal { S } _ 2 ) : \\boldsymbol { K } _ 1 ( \\gamma _ 1 , \\mu _ 1 ) \\leq \\delta _ 1 , \\widehat { \\boldsymbol { K } } _ 2 ( \\gamma _ 2 , \\mu _ 2 ) \\leq \\delta _ 2 \\right \\} . \\end{align*}"}
+{"id": "7360.png", "formula": "\\begin{align*} f ( x ) & = - a x + \\sum _ { k = 1 } ^ \\infty \\frac { x ^ k \\log ^ k ( 1 + a ) } { k ! } , \\end{align*}"}
+{"id": "270.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( - \\frac { ( - 4 n ^ 4 - 1 2 n ^ 3 - 6 n ^ 2 + 1 2 n + 1 1 ) y ^ { n + 2 } + ( n + 1 ) ^ 4 y ^ { n + 1 } } { ( 1 - y ) ^ 5 } \\right ) \\frac { z ^ n } { n ^ 5 } \\right \\} \\end{align*}"}
+{"id": "7344.png", "formula": "\\begin{align*} u ^ 3 v ^ 2 = w ^ 3 + 1 \\end{align*}"}
+{"id": "9065.png", "formula": "\\begin{align*} 2 4 r + 1 & \\equiv 3 6 k ^ 2 - 1 2 k + 1 \\pmod { p } \\\\ & = ( 6 k - 1 ) ^ 2 . \\end{align*}"}
+{"id": "3916.png", "formula": "\\begin{align*} \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda \\ , d \\widetilde { \\pi } = \\int _ { \\mathcal { V } } g \\ , d \\gamma - \\int _ { \\mathcal { S } _ { 1 } \\times \\mathcal { S } _ { 1 } } \\lambda _ 1 c _ { 1 } \\ , d \\nu _ { 1 } - \\int _ { \\mathcal { S } _ { 2 } \\times \\mathcal { S } _ { 2 } } \\lambda _ 2 c _ { 2 } \\ , d \\nu _ { 2 } < \\infty , \\end{align*}"}
+{"id": "2455.png", "formula": "\\begin{align*} \\pi ( f ) \\varphi = \\alpha _ ! ( f \\otimes \\varphi ) & & f \\in C _ c ^ \\infty ( \\R ) , \\varphi \\in C _ c ^ \\infty ( M ) \\end{align*}"}
+{"id": "7436.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 3 } h _ i ^ 2 ( 0 ) \\ , \\mathrm { v } _ i = 0 \\end{align*}"}
+{"id": "4741.png", "formula": "\\begin{align*} \\tilde { S } ^ T ( A + H ) \\tilde { S } = A + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "6746.png", "formula": "\\begin{align*} \\varphi _ K \\circ R = \\sigma \\circ \\varphi _ K \\underline { \\varphi } _ K \\circ R = \\sigma \\circ \\underline { \\varphi } _ K . \\end{align*}"}
+{"id": "4367.png", "formula": "\\begin{align*} 0 _ { n } = \\sum \\limits _ { i = 1 } ^ { m } \\left \\Vert a _ { i } \\right \\Vert ^ { 2 } \\nabla d _ { S _ { i } } ^ { 2 } \\left ( x ^ { 0 } \\right ) = 2 \\sum \\limits _ { i = 1 } ^ { m } \\left \\Vert a _ { i } \\right \\Vert d _ { S _ { i } } \\left ( x ^ { 0 } \\right ) a _ { i } = 2 \\sum \\limits _ { i = 1 } ^ { m } \\left [ a _ { i } ^ { \\prime } x ^ { 0 } - \\overline { b } _ { i } \\right ] _ { + } a _ { i } ; \\end{align*}"}
+{"id": "1111.png", "formula": "\\begin{align*} x ( y z ) & = ( x y ) z - ( x z ) y , \\\\ ( y z ) x & = ( y x ) z + y ( z x ) , \\end{align*}"}
+{"id": "914.png", "formula": "\\begin{align*} A ^ { ( 1 2 3 ) } _ { - } F = 0 , \\end{align*}"}
+{"id": "3184.png", "formula": "\\begin{align*} M _ a ( f ) \\leq \\frac { 1 } { n ^ d } \\sum _ { ( j _ 1 , \\ldots , j _ d ) \\in \\Z ^ d _ + \\cap Q _ { n + 1 } } S _ 1 ^ { j _ 1 } \\cdots S _ d ^ { j _ d } M _ 1 ( f ) = \\frac { 1 } { n ^ d } \\sum _ { 0 \\leq j _ 1 < n + 1 } \\cdots \\sum _ { 0 \\leq j _ d < n + 1 } S _ 1 ^ { j _ 1 } \\cdots S _ d ^ { j _ d } M _ 1 ( f ) \\end{align*}"}
+{"id": "6440.png", "formula": "\\begin{align*} & \\alpha = \\{ 1 , 3 , 4 , 6 , 8 , 9 , A , C , D , E , N , O , P , Q , R , Z , a , b , c , d , j , k , l , m , n \\} , \\\\ & \\beta = \\{ 5 , B , G , H , I , J , K , L , M , T , U , V , W , X , Y , e , f , g , h , i , o \\} , \\mbox { a n d } \\\\ & \\gamma = \\{ 0 , 2 , 7 , F , S \\} . \\end{align*}"}
+{"id": "7546.png", "formula": "\\begin{align*} \\sum _ { j = j _ 0 } ^ { j _ 1 } \\mathfrak { E } _ p ( \\partial u , B _ { j + 1 } ) \\leq \\frac { 1 } { 3 2 } \\left ( \\frac { \\sigma } { 2 } \\right ) ^ { \\frac { d } { p } } \\lambda . \\end{align*}"}
+{"id": "8430.png", "formula": "\\begin{align*} \\tan ( \\pi y ) = - \\frac { y } { p } , \\ , \\ , \\ , \\ , \\tan ( \\pi y ) = - \\frac { y } { p ^ { \\prime } } . \\end{align*}"}
+{"id": "8034.png", "formula": "\\begin{align*} \\Omega _ { s , t } = - \\Omega _ { t , s } . \\end{align*}"}
+{"id": "7969.png", "formula": "\\begin{align*} \\big ( \\mathrm { d i v } \\ , V \\big ) ^ 2 = \\mathrm { t r } \\big ( ( \\nabla V ) ^ 2 \\big ) + \\mathrm { d i v } \\Big ( V \\ , \\mathrm { d i v } V - \\nabla V \\ , V \\Big ) \\ , . \\end{align*}"}
+{"id": "2525.png", "formula": "\\begin{align*} H _ i ( ( 2 , 2 n ) , \\mathbb { Z } ) = \\begin{cases} H _ i ( G ( 2 , 2 n ) , \\mathbb { Z } ) , & i \\leq 4 n - 5 , \\\\ H _ { i + 2 } ( G ( 2 , 2 n ) , \\mathbb { Z } ) , & i \\geq 4 n - 4 . \\end{cases} \\end{align*}"}
+{"id": "9017.png", "formula": "\\begin{align*} & \\big ( 2 , t _ 0 , t _ 1 , t _ 2 \\leq T , ( 0 , 1 , 0 ) , ( 1 , 1 , 0 ) , ( 1 , 1 , 1 ) \\big ) , \\\\ & \\big ( 2 , t _ 0 , t _ 1 , t _ 2 = T , ( 0 , 0 , 0 ) , ( 1 , 0 , 0 ) , ( 1 , 0 , 1 ) \\big ) , \\end{align*}"}
+{"id": "8856.png", "formula": "\\begin{align*} N _ k = N _ { k - 2 } + \\sum _ { j = 2 } ^ { \\lfloor \\frac { k } { 2 } \\rfloor - 1 } N _ { k - 2 j } + N _ { k - 3 } + \\sum _ { j = 2 } ^ { \\lfloor \\frac { k } { 2 } \\rfloor - 1 } N _ { k - 1 - 2 j } + 2 \\end{align*}"}
+{"id": "2168.png", "formula": "\\begin{align*} \\exp \\left [ 2 \\lambda \\left ( 3 b ^ { 2 } / 2 + \\alpha \\left ( T / 2 - \\varepsilon \\right ) ^ { 2 } \\right ) \\right ] = \\exp \\left ( 2 \\lambda d \\right ) , \\end{align*}"}
+{"id": "2308.png", "formula": "\\begin{align*} w _ b = \\Phi _ { 0 , b } + \\widetilde { \\Psi } _ b . \\end{align*}"}
+{"id": "6642.png", "formula": "\\begin{align*} Y = Y ^ { ( 1 ) } _ { ( 1 , 1 ) } = C _ 1 \\cup C _ 2 \\end{align*}"}
+{"id": "7022.png", "formula": "\\begin{align*} f _ j Q _ i ' = a _ j + b _ j Q _ i \\end{align*}"}
+{"id": "6757.png", "formula": "\\begin{align*} m _ H ( W \\setminus \\underline { W } ) & = m _ H ( W ) - m _ H ( \\underline { W } ) \\\\ & = m _ H ( W ) - m _ H ( \\Gamma _ { H , H ^ { * } } ^ { - 1 } ( W ^ * ) ) \\\\ & = m _ H ( W ) - ( \\Gamma _ { H , H ^ { * } } ) _ { \\ast } ( m _ H ) ( W ^ * ) \\\\ & = m _ H ( W ) - m _ { H ^ * } ( W ^ * ) \\end{align*}"}
+{"id": "2849.png", "formula": "\\begin{gather*} \\sum _ { \\{ S \\colon a \\cup b \\to \\{ U , V , U V \\} \\ , | \\ , S ( b ) = \\{ U \\} \\} } \\Gamma _ S - \\sum _ { \\{ S ' \\colon a \\cup b \\to \\{ U , V , U V \\} \\ , | \\ , S ' ( a ) = \\{ U \\} \\} } \\Gamma _ { S ' } . \\end{gather*}"}
+{"id": "2350.png", "formula": "\\begin{align*} M ( X ) _ f = \\prod _ { x : X } ( M _ x ) _ { f ( x ) } \\end{align*}"}
+{"id": "9226.png", "formula": "\\begin{align*} \\texttt { r s b o u n d } : = c \\sqrt { \\pi } / b a . \\end{align*}"}
+{"id": "1977.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ n u ^ { p \\bar p } \\leq A _ 1 g ^ { - 1 } . \\end{align*}"}
+{"id": "1699.png", "formula": "\\begin{align*} \\sigma _ { j + 1 } = \\inf _ { r > 0 } \\Big \\{ \\tilde Z _ { \\sigma _ { j } + 2 r } \\in \\{ \\sqrt { m } - 1 , \\sqrt { m } , \\eta ( m ) - 1 , \\eta ( m ) \\} \\Big \\} \\end{align*}"}
+{"id": "2466.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } S ( u , v ) = & \\displaystyle \\dfrac { a _ 2 } { 2 m _ 1 } \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ 2 d x + \\dfrac { a _ 1 } { 4 m _ 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla v | ^ 2 d x \\\\ \\\\ & \\displaystyle + a _ 2 \\omega _ 1 \\int _ { \\mathbb { R } ^ N } | u | ^ 2 d x + \\frac { a _ 1 } { 2 } \\omega _ 2 \\int _ { \\mathbb { R } ^ N } | v | ^ 2 d x + a _ 1 a _ 2 \\int _ { \\mathbb { R } ^ N } v u ^ 2 d x . \\end{array} \\right . \\end{align*}"}
+{"id": "8987.png", "formula": "\\begin{align*} \\zeta _ \\delta = \\frac { v } { w _ \\delta } \\ , \\Omega \\ , . \\end{align*}"}
+{"id": "9352.png", "formula": "\\begin{align*} \\begin{aligned} & \\leq C \\mathbb { E } \\int _ 0 ^ \\infty \\big ( | y _ n - y | ^ 2 + | z _ n - z | ^ 2 + | \\tilde { z } _ n - \\tilde { z } | ^ 2 + | | \\gamma _ n - \\gamma | | ^ 2 \\big ) e ^ { - \\beta s } d s \\rightarrow 0 , \\ \\ n \\rightarrow \\infty , \\end{aligned} \\end{align*}"}
+{"id": "3078.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 ^ + } \\frac { A ( r ) ^ 2 } { A ' ( r ) } \\cdot \\left [ \\mathcal W ( r ) ^ { p - 1 - \\alpha } \\right ] ' & = \\lim _ { r \\to 0 ^ + } A ( r ) \\cdot \\left [ Q _ 1 ( \\mathcal V ( r ) , v _ 0 ( r ) ) - \\mathcal W ( r ) ^ { p - 1 - \\alpha } \\right ] \\\\ & = \\lim _ { r \\to 0 ^ + } \\left [ r ^ \\delta u _ 0 ' ( r ) ^ { p - 1 - \\alpha } Q _ 1 ( \\mathcal V ( r ) , v _ 0 ( r ) ) - r ^ \\delta u ' ( r ) ^ { p - 1 - \\alpha } \\right ] \\end{align*}"}
+{"id": "8742.png", "formula": "\\begin{align*} \\| \\mathcal { T } _ { 1 , a } ^ { ( a _ 1 , a _ 2 ) } - \\mathcal { T } _ { 2 , a } ^ { ( a _ 1 , a _ 2 ) } \\| _ { \\rm F } ^ 2 = O ( p ^ { a - a _ 1 - a _ 2 + 2 ( a _ 1 - c _ 2 + 1 ) } ) = O ( p ^ { a + a _ 1 - a _ 2 - 2 c _ 2 + 2 } ) = O ( p ^ { a + a _ 1 - a _ 2 - 2 r + 2 } ) , \\end{align*}"}
+{"id": "7510.png", "formula": "\\begin{align*} u ^ i ( z ) = \\zeta _ i + \\ln ' ( p ^ i ( z ) ) . \\end{align*}"}
+{"id": "4416.png", "formula": "\\begin{align*} L _ { \\omega , \\mathbf { c } } ( V _ n ) = L _ { \\omega , \\mathbf { c } } ( U _ n ) \\rightarrow 6 \\mu _ { \\omega , \\mathbf { c } } \\end{align*}"}
+{"id": "4075.png", "formula": "\\begin{align*} \\displaystyle \\bigoplus _ { n = 0 } ^ { 2 k - 2 } \\bigcap _ { [ f ] \\in \\mathcal { E } _ n } \\mathrm { A n n } _ { [ f ] _ n } = \\displaystyle \\bigoplus _ { n = 0 } ^ { 2 k - 2 } \\mathrm { A n n } ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) . \\end{align*}"}
+{"id": "3244.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } f ( x ) d \\omega ( x ) = 0 . \\end{align*}"}
+{"id": "2703.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } z - ( y ^ { \\alpha _ y } \\partial _ { x x } z + x ^ { \\alpha _ x } \\partial _ { y y } z ) = \\chi _ { \\omega } g , & \\mbox { i n } \\ Q , \\\\ z ( x , y , t ) = 0 \\ \\mbox { o r } \\ A \\nabla z \\cdot \\nu = 0 , & \\mbox { o n } \\ \\Sigma , \\\\ z ( x , y , 0 ) = z _ { 0 } ( x , y ) , & \\mbox { i n } \\ \\Omega , \\end{cases} \\end{align*}"}
+{"id": "4738.png", "formula": "\\begin{align*} \\alpha _ i & \\coloneqq \\{ j : d _ j ( A ) = \\mu _ i , j = 1 , \\ldots , n \\} , \\\\ \\beta _ i & \\coloneqq \\{ j + n : j \\in \\alpha _ i \\} , \\\\ \\gamma _ i & \\coloneqq \\alpha _ i \\cup \\beta _ i . \\end{align*}"}
+{"id": "1188.png", "formula": "\\begin{align*} \\delta ^ L _ { n m } = 1 0 0 \\frac { F _ { n m } + F _ { m n } } { 2 F ^ 0 _ { n m } } \\end{align*}"}
+{"id": "4244.png", "formula": "\\begin{align*} \\Omega ( X _ 1 , r ) = ( n + 1 ) - 1 = n , \\end{align*}"}
+{"id": "9060.png", "formula": "\\begin{align*} S ( \\beta ) & = \\Delta _ { N + 1 } ( \\beta _ 0 \\sigma _ N ) \\Delta _ { N + 1 } = ( \\sigma _ 1 \\dots \\sigma _ N ) ( \\Delta _ N \\beta _ 0 \\Delta _ N ) ( \\sigma _ N \\dots \\sigma _ 1 ) \\sigma _ 1 \\mathrel { \\dot { = } } \\beta ( \\sigma _ N \\dots \\sigma _ 2 \\sigma _ 1 ^ 3 \\sigma _ 2 \\dots \\sigma _ N ) , \\end{align*}"}
+{"id": "11.png", "formula": "\\begin{align*} \\Theta _ { \\infty } ( s ) & : = \\frac { 1 } { \\zeta _ N ( m + n ) } \\left ( \\sum _ { \\substack { k _ 1 \\geq 1 \\\\ g c d ( k _ 1 , q ) = 1 } } \\sum _ { \\substack { k _ 2 \\in \\Z \\setminus \\{ 0 \\} \\\\ k _ 2 = k _ 1 ( m o d \\ { q } ) } } \\int _ { \\R ^ { m + n } } \\left ( ( \\chi \\circ a ^ s ) ( k _ 1 x ) \\chi ( k _ 2 x ) \\right ) \\ , d x \\right ) . \\end{align*}"}
+{"id": "3866.png", "formula": "\\begin{align*} \\Sigma ( \\delta ) = \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\boldsymbol { K } _ \\ell ( \\mu _ { \\ell , L + 1 } , \\gamma _ { \\ell , L + 1 } ) \\le \\delta _ \\ell \\ell \\in [ L ] \\} , \\end{align*}"}
+{"id": "3991.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\mathbb { E } \\left [ Y _ 2 \\right ] - \\mathbb { E } \\left [ Y _ 1 \\right ] + \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\mathrm { R } _ { \\mathrm { D } } ( \\lambda , \\delta ) , \\end{align*}"}
+{"id": "3423.png", "formula": "\\begin{gather*} \\| u _ n \\| _ { \\varepsilon _ n } , \\ \\int _ { \\R ^ N } f _ 1 ( u _ n ) u _ n , \\ \\int _ { \\R ^ N } F _ 1 ( u _ n ) , \\ \\int _ { \\R ^ N } \\bar f _ 2 ( u _ n ) u _ n \\ \\int _ { \\R ^ N } \\bar F _ 2 ( u _ n ) , \\ | \\lambda _ n | \\leq C _ 0 , \\\\ \\int _ { \\R ^ N \\setminus \\cup _ { j = 1 } ^ \\ell B ( \\Upsilon _ j ( u _ n ) , \\frac { 1 } { 1 0 } \\xi _ 1 ( \\Upsilon ( u _ n ) ) ) } \\left ( | \\nabla u _ n | ^ 2 + u _ n ^ 2 \\right ) \\mathrm d x \\leq C e ^ { - c \\xi _ 1 ( \\Upsilon ( u _ n ) ) } + o _ n ( 1 ) . \\end{gather*}"}
+{"id": "8566.png", "formula": "\\begin{align*} & \\lim _ { N \\to \\infty } N ^ { - 1 } M ( \\tau _ N ) = \\nu \\int _ 0 ^ \\infty e ^ { - \\lambda s } ( 1 - p _ 0 ( s ) ) d s , \\end{align*}"}
+{"id": "7248.png", "formula": "\\begin{align*} \\alpha ( \\mathcal { U } , \\mathcal { V } ) = \\sup \\{ | { \\ \\mathbb { P } } ( U \\cap V ) - { \\mathbb { P } } ( U ) { \\mathbb { P } } ( V ) | : U \\in \\mathcal { U } , V \\in \\mathcal { V } \\} \\ , . \\end{align*}"}
+{"id": "931.png", "formula": "\\begin{align*} \\mathcal { F } ( \\mathbf { h } _ t ) = \\mathcal { G } ( \\tilde { \\mathcal { F } } ( \\mathbf { h } _ t ) , \\mathbf { h } _ t ) , \\end{align*}"}
+{"id": "7720.png", "formula": "\\begin{align*} K = L \\cap ( K _ 1 \\oplus \\langle z , z ' \\rangle ) , \\end{align*}"}
+{"id": "576.png", "formula": "\\begin{align*} \\gamma ^ { b } ( x _ { 1 } , x _ { 2 } ) = \\gamma ^ { b } ( x _ { 2 } ) = \\int _ { \\R _ { + } } b \\left ( \\frac { y _ { 2 } } { 2 x _ { 2 } } \\right ) ( \\ell _ 0 ( y _ 2 ) ) ^ 2 d y _ { 2 } I _ { n \\times n } . \\end{align*}"}
+{"id": "7948.png", "formula": "\\begin{align*} B ( t ) = t ^ p , \\end{align*}"}
+{"id": "1250.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { I } { \\delta } = - \\frac { v _ { \\beta , 2 } ' ( g _ { \\beta , 2 } ( x ) ) } { ( f _ { \\beta , 2 } ' ( g _ { \\beta , 2 } ( x ) ) ) ^ 2 } = - \\frac { X ' _ { \\beta , 2 } ( x ) } { f _ { \\beta , 2 } ' ( g _ { \\beta , 2 } ( x ) ) } . \\end{align*}"}
+{"id": "5514.png", "formula": "\\begin{align*} \\eta _ { \\omega } = \\eta ^ { \\boldsymbol { \\beta } _ { Y } ( \\omega ) } \\mbox { f o r } \\mathbb { P } _ { \\mu } \\mbox { - a . e . } \\omega . \\end{align*}"}
+{"id": "8122.png", "formula": "\\begin{align*} f ( F ) = \\sum _ { G \\ge F } G . \\end{align*}"}
+{"id": "102.png", "formula": "\\begin{align*} \\partial _ t w = - i h ^ { - 1 } ( - i h X - i W ) w , w | _ { t = 0 } = f . \\end{align*}"}
+{"id": "5303.png", "formula": "\\begin{align*} T _ { ( 1 - p ) \\rightarrow p } A \\chi ^ { p } _ S = \\left ( - \\frac { p } { 1 - p } \\right ) ^ { | S | } \\chi ^ { p } _ S . \\end{align*}"}
+{"id": "6430.png", "formula": "\\begin{align*} e _ { | x | \\bar { \\otimes } | y | } ( \\Lambda ) : = \\int _ { [ 0 , \\infty ) ^ 2 } \\mathbf { 1 } _ \\Lambda ( \\lambda _ 1 \\lambda _ 2 ) \\ , e _ { | x | , | y | } ( d \\lambda _ 1 , d \\lambda _ 2 ) , \\Lambda \\subset [ 0 , \\infty ) . \\end{align*}"}
+{"id": "6209.png", "formula": "\\begin{align*} \\begin{cases} | - b + 0 \\cdot a | & = d ( v _ 1 , v _ 2 ) - 1 , \\\\ 0 \\cdot b - a & = - 1 , \\end{cases} \\end{align*}"}
+{"id": "1893.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} x ^ { k + 1 } - 1 & = \\frac { 1 } { 1 + 5 \\beta } ( x ^ { k } - 1 ) ; \\\\ \\lambda ^ { k + 1 } + \\frac { 1 } { 5 } \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} & = J \\left [ \\lambda ^ k + \\frac { 1 } { 5 } \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} \\right ] ; \\\\ \\lambda ^ { k + 1 } _ 1 + 2 \\lambda ^ { k + 1 } _ 2 + 1 & = \\frac { 1 } { 1 + 5 \\beta } ( \\lambda ^ { k } + 2 \\lambda ^ { k } + 1 ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4201.png", "formula": "\\begin{align*} e _ 1 \\cdot [ X : Y : Z ] & = [ \\zeta _ 5 ^ 2 X : \\zeta _ 5 ^ 4 Y : Z ] \\\\ e _ 2 \\cdot [ X : Y : Z ] & = [ \\zeta _ 5 X : \\zeta _ 5 ^ 3 Y : Z ] . \\end{align*}"}
+{"id": "2527.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { C } } ( { \\rm S p } \\Sigma _ { \\{ i , j \\} } ) = \\begin{cases} \\dim _ { \\mathbb { C } } ( \\Sigma _ { \\{ p ( i ) , p ( j ) \\} } ) , & p ( i ) + p ( j ) < 2 n + 1 \\\\ \\dim _ { \\mathbb { C } } ( \\Sigma _ { \\{ p ( i ) , p ( j ) \\} } ) - 1 , & p ( i ) + p ( j ) > 2 n + 1 \\end{cases} \\end{align*}"}
+{"id": "7277.png", "formula": "\\begin{align*} a x ^ n + b x ^ k y ^ l + c y ^ m = 0 \\end{align*}"}
+{"id": "4375.png", "formula": "\\begin{align*} \\begin{aligned} \\ 1 ( x \\in [ 0 , \\Delta ^ { - 1 } ] ) & \\leq ( 1 + C e ^ { - \\Delta ^ { A - 1 } } ) | \\mathcal D ^ + _ { \\Delta , A } ( x ) | ^ 2 \\\\ | \\mathcal D ^ + _ { \\Delta , A } ( x ) | ^ 2 & \\leq \\mathbf { 1 } ( x \\in [ - \\Delta ^ { - A / 2 } , \\Delta ^ { - 1 } + \\Delta ^ { - A / 2 } ] ) + C e ^ { - \\Delta ^ { A - 1 } } , \\end{aligned} \\end{align*}"}
+{"id": "8743.png", "formula": "\\begin{align*} \\| \\mathcal { T } _ { 1 , a } ^ { ( a _ 1 , a _ 2 ) } - \\mathcal { T } _ { 2 , a } ^ { ( a _ 1 , a _ 2 ) } \\| _ { \\rm F } ^ 2 = O ( p ^ { a - a _ 1 - a _ 2 + 2 ( a _ 1 - c _ 2 + 1 ) } ) = O ( p ^ { a + a _ 1 - a _ 2 - 2 c _ 2 + 2 } ) = O ( p ^ { a + a _ 1 - a _ 2 - 2 r + 2 } ) . \\end{align*}"}
+{"id": "6084.png", "formula": "\\begin{align*} \\| \\exp ( Q _ { j } ) Z _ { j } \\| = \\| \\exp ( Q _ { j } ) \\| = : \\tau _ j . \\end{align*}"}
+{"id": "2799.png", "formula": "\\begin{align*} v _ t = \\sum _ { j \\in N _ t ^ 0 } a _ j \\varphi _ j . \\end{align*}"}
+{"id": "7500.png", "formula": "\\begin{align*} u _ i ( z ) = \\zeta _ i + \\ln ' \\big [ p _ i ( z ) \\big ] , 1 , \\dots , r . \\end{align*}"}
+{"id": "994.png", "formula": "\\begin{align*} \\omega _ r = \\mathrm { C o n v } \\{ F ^ 1 _ r , \\ldots , F ^ p _ r , \\sigma _ r \\} + G \\end{align*}"}
+{"id": "7781.png", "formula": "\\begin{align*} - x ^ r + z ^ { r + 1 } + k x ^ r z = 0 \\end{align*}"}
+{"id": "7522.png", "formula": "\\begin{align*} g _ n ( x ) = \\left \\{ \\begin{array} { c l } x & \\mbox { i f } x \\in I ^ n \\\\ x _ 0 & \\mbox { i f } x \\notin I ^ n \\end{array} \\right . . \\end{align*}"}
+{"id": "5481.png", "formula": "\\begin{align*} \\begin{array} { l l } ( f \\circ _ i g ) = y ( f ; \\overbrace { i d , \\cdots , i d , \\underbrace { g } _ { i - } i d \\cdots , i d } ^ { m - } ) , { f o r } f \\in \\Theta ( m ) \\end{array} \\end{align*}"}
+{"id": "8888.png", "formula": "\\begin{align*} \\operatorname { v o l } ( B _ { c c } ( R ) ) = \\operatorname { v o l } ( B _ { c c } ( 1 ) ) R ^ Q , \\end{align*}"}
+{"id": "2600.png", "formula": "\\begin{align*} e = c _ 1 e _ 1 + c _ 2 e _ 2 + \\cdots + c _ { n - 1 } e _ { n - 1 } . \\end{align*}"}
+{"id": "6226.png", "formula": "\\begin{align*} \\partial ^ 2 _ v \\psi _ { \\delta _ { x _ 1 } } ( x _ 1 ) = g '' ( 0 ) \\ , . \\end{align*}"}
+{"id": "3220.png", "formula": "\\begin{align*} \\langle S _ n ' ( y ' ) x \\Omega _ { \\rho } , \\Omega _ { \\rho } \\rangle _ { \\rho } = \\langle y ' S _ n ( x ) \\Omega _ { \\rho } , \\Omega _ { \\rho } \\rangle _ { \\rho } , x \\in M , y ' \\in M ' . \\end{align*}"}
+{"id": "3861.png", "formula": "\\begin{align*} \\Sigma _ { \\mathrm { D } } ( \\delta ) = \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { V } ) : \\boldsymbol { K } _ \\ell ( \\mu _ \\ell , \\gamma _ \\ell ) \\le \\delta _ \\ell , \\ \\forall \\ell \\in [ L ] \\} \\end{align*}"}
+{"id": "3109.png", "formula": "\\begin{align*} S _ D [ n , k ] : = \\frac { 1 } { q ^ { k ^ 2 } [ 2 ] _ q ^ { k } } \\sum _ { \\pi \\in D _ { \\subseteq } ( [ n ] , k ) } q ^ { m ( \\pi ) } . \\end{align*}"}
+{"id": "7557.png", "formula": "\\begin{align*} \\left \\lvert \\sum _ { l \\equiv v ( \\bmod a ) } \\binom n l \\alpha ^ l ( 1 - \\alpha ) ^ { n - l } - \\frac 1 a \\right \\rvert = O \\left ( n ^ { - \\frac 1 2 } \\right ) . \\end{align*}"}
+{"id": "8219.png", "formula": "\\begin{align*} p _ s = \\exp \\left [ - \\frac { c ( d , \\nu ) t } { ( \\log t ) ^ { 2 / d } } ( 1 + o ( 1 ) ) \\right ] . \\end{align*}"}
+{"id": "2674.png", "formula": "\\begin{align*} \\| D ^ { i } ( s _ { \\alpha } [ f ] - f ) \\| _ { L ^ { \\infty } [ a , b ] } = \\| D ^ { i } ( \\cosh ( \\alpha \\cdot ) ( t _ { \\alpha } [ g ] - g ) ) \\| _ { L ^ { \\infty } [ a , b ] } \\end{align*}"}
+{"id": "7780.png", "formula": "\\begin{align*} y ^ r \\cdot y ^ q + r z ^ { r - 1 } \\cdot ( z ^ q x - z x ^ q ) + ( z ^ r + k x ^ r ) \\cdot ( - x ^ q ) = 0 . \\end{align*}"}
+{"id": "1670.png", "formula": "\\begin{align*} \\ker \\sigma \\cap \\Im [ \\pi _ 1 ( F ) \\to \\pi _ 1 ( X ) ] = H \\cap \\Im [ \\pi _ 1 ( F ) \\to \\pi _ 1 ( X ) ] . \\end{align*}"}
+{"id": "7389.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { n \\ne 0 } \\widehat { \\Delta } ( a n ) ^ { 2 } & \\leq \\frac { 1 } { a } \\sum _ { | m | \\leq a } \\left ( 1 - \\frac { | m | } { a } \\right ) - 1 = \\frac { 1 } { a } \\left ( 2 \\lfloor a \\rfloor + 1 - \\frac { \\lfloor a \\rfloor ( \\lfloor a \\rfloor + 1 ) } { a } \\right ) - 1 \\\\ & = \\frac { 1 } { a } \\left ( \\left ( 1 + \\lfloor a \\rfloor - a \\right ) + \\frac { \\lfloor a \\rfloor } { a } \\left ( a - \\lfloor a \\rfloor - 1 \\right ) \\right ) < \\frac { 1 } { a } , \\end{aligned} \\end{align*}"}
+{"id": "5422.png", "formula": "\\begin{align*} A : = \\left ( \\begin{array} { c c c c c } - \\alpha _ { k - 1 } / \\alpha _ k & - \\alpha _ { k - 2 } / \\alpha _ k & \\ldots & - \\alpha _ { 1 } / \\alpha _ k & - \\alpha _ { 0 } / \\alpha _ k \\\\ 1 & 0 & \\ldots & 0 & 0 \\\\ 0 & 1 & \\ldots & 0 & 0 \\\\ 0 & 0 & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\ldots & 1 & 0 \\\\ \\end{array} \\right ) \\end{align*}"}
+{"id": "9119.png", "formula": "\\begin{align*} y ^ { j } = \\varphi ^ { j } ( \\zeta _ { [ - q _ { 1 } ] } , \\dots , \\zeta _ { [ - 1 ] } , x , u , \\dots , u _ { [ q _ { 2 } ] } ) \\ , , j = 1 , \\ldots , m \\end{align*}"}
+{"id": "1328.png", "formula": "\\begin{align*} ( \\theta _ i Z _ i ^ * ( u ) ) _ { u \\geq 0 } = ( Z _ i ( u ) ) _ { u \\geq 0 } . \\end{align*}"}
+{"id": "612.png", "formula": "\\begin{align*} \\phi ( a _ \\lambda ^ * a _ \\lambda ) \\geq \\omega ( a _ \\lambda ^ * a _ \\lambda ) = 1 \\end{align*}"}
+{"id": "6764.png", "formula": "\\begin{align*} \\Sigma _ c f ( n ) : = \\begin{cases} c n = 0 \\\\ f ( n - 1 ) + \\Sigma _ c f ( n - 1 ) n \\ge 1 . \\end{cases} \\end{align*}"}
+{"id": "2046.png", "formula": "\\begin{align*} \\varphi _ { v _ 0 } ( e _ 1 , e _ 2 , f _ 2 ) & = ( a _ 0 , b _ 0 ) = ( 0 , 0 ) , \\\\ \\varphi _ { v _ 1 } ( e _ 1 , e _ 2 , f _ 2 ) & = ( a _ 1 , b _ 1 ) = ( - e _ 1 , e _ 1 + f _ 2 ) , \\\\ \\varphi _ { v _ 2 } ( e _ 1 , e _ 2 , f _ 2 ) & = ( a _ 2 , b _ 2 ) = ( e _ 2 , e _ 1 ) . \\end{align*}"}
+{"id": "5977.png", "formula": "\\begin{align*} \\overline { \\overline { \\Pi } } _ { \\psi } [ h _ { - 1 } , t ] f ( [ \\epsilon , x ] ) = t f ( [ - \\epsilon , x ] ) . \\end{align*}"}
+{"id": "972.png", "formula": "\\begin{align*} { \\rm c a p } _ { q , Q } \\ , ( f ( E ) , f ( F ) , f ( D ) ) = M _ { q , Q } \\ , ( f ( E ) , f ( F ) , f ( D ) ) \\ , , \\end{align*}"}
+{"id": "9235.png", "formula": "\\begin{align*} \\gamma _ { k , \\ell } = \\frac { 1 } { 4 } \\cdot \\frac { \\delta _ k } { 8 \\cdot 2 ^ { - \\ell } \\pi ^ { \\ell - 2 k } F _ m } = \\frac { \\sqrt { 2 \\pi } } { 1 2 8 } \\Bigl ( \\frac { \\pi a ^ 2 } { 8 } \\Bigr ) ^ { k / 2 } \\frac { 2 ^ { 2 \\ell } \\varepsilon _ 4 } { \\Gamma \\bigl ( \\frac { 3 k - 2 \\ell + 1 } { 2 } \\bigr ) } \\end{align*}"}
+{"id": "5760.png", "formula": "\\begin{align*} \\| T _ { \\alpha } ( \\vec { f } ) \\| _ { L ^ { q } ( \\mathbb { R } ^ { n } ) } & \\le C \\prod _ { j = 1 } ^ { m } \\| f _ { j } \\| _ { L ^ { p _ { j } } ( \\mathbb { R } ^ { n } ) } . \\end{align*}"}
+{"id": "959.png", "formula": "\\begin{align*} | f ( x ) - f ( y ) | : = \\varepsilon _ 0 \\ , . \\end{align*}"}
+{"id": "9289.png", "formula": "\\begin{align*} \\min \\to \\left \\{ \\sum \\limits ^ { m } _ { i = 1 } f _ { i } ( v ) \\ \\vrule \\ \\bigcap \\limits ^ { m } _ { i = 1 } X _ { i } \\right \\} , \\end{align*}"}
+{"id": "6178.png", "formula": "\\begin{align*} v _ s v _ t \\xi ( x , r ) & = u ( s , s ^ { - 1 } r ) ( x ) u ( t , ( s t ) ^ { - 1 } r ) ( \\varphi _ { s ^ { - 1 } } ( x ) ) \\xi ( \\varphi _ { ( s t ) ^ { - 1 } } ( x ) , ( s t ) ^ { - 1 } r ) \\\\ & \\stackrel { \\eqref { e q : c o c y c l e _ i d e n t i t y } } { = } u ( s , t ) ( x ) u ( s t , ( s t ) ^ { - 1 } r ) ( \\varphi _ { ( s t ) ^ { - 1 } } ( x ) ) \\xi ( \\varphi _ { ( s t ) ^ { - 1 } } ( x ) , ( s t ) ^ { - 1 } r ) = \\overline { \\pi } ( u ( s , t ) ) v _ { s t } \\xi ( x , r ) . \\end{align*}"}
+{"id": "1943.png", "formula": "\\begin{align*} f _ { 1 , 1 } ^ * ( z _ 1 , \\ldots , z _ m ) : = f _ { 1 , 0 } ^ * ( z _ 1 , \\ldots , z _ m ) + \\delta _ { 1 , 1 } z _ 1 \\cdots z _ m A _ 1 ^ { ( 1 ) } ( z _ 1 , \\ldots , z _ m ) \\end{align*}"}
+{"id": "6967.png", "formula": "\\begin{align*} \\alpha = \\alpha \\left ( \\left \\{ \\left . \\alpha _ i \\ \\right | \\ i \\in I \\right \\} \\right ) \\mbox { a n d } \\beta = \\alpha \\left ( \\left \\{ \\left . \\beta _ i \\ \\right | \\ i \\in I \\right \\} \\right ) . \\end{align*}"}
+{"id": "4656.png", "formula": "\\begin{align*} F _ { i + \\frac { 1 } { 2 } } ^ { m } = \\max ( \\vartheta _ { } ^ { i + 1 } , 0 ) \\Big [ \\widetilde { m } _ { i } + \\frac { \\Delta x } { 2 } ( m _ { x } ) _ { i } \\Big ] + \\min ( \\vartheta _ { m } ^ { i + 1 } , 0 ) \\Big [ \\widetilde { m } _ { i } - \\frac { \\Delta x } { 2 } ( m _ { x } ) ^ { i } \\Big ] , \\end{align*}"}
+{"id": "643.png", "formula": "\\begin{align*} b _ k = u _ k | _ F , 1 \\leq k \\leq n . \\end{align*}"}
+{"id": "492.png", "formula": "\\begin{align*} t ^ { 2 k - 2 } t _ 1 t _ 2 t _ { 2 n + 1 } t _ { 2 n + 2 } t _ { 2 n + 3 } t _ { 2 n + 4 } & = t ^ { k - 2 } t _ 3 t _ 4 t _ { 2 n + 1 } t _ { 2 n + 2 } t _ { 2 n + 3 } t _ { 2 n + 4 } \\\\ & = \\cdots = t ^ { k - 2 } t _ { 2 n - 1 } t _ { 2 n } t _ { 2 n + 1 } t _ { 2 n + 2 } t _ { 2 n + 3 } t _ { 2 n + 4 } = p q . \\end{align*}"}
+{"id": "1552.png", "formula": "\\begin{align*} X = \\frac { x } { \\rho } Y = \\frac { Y } { \\rho } T = \\frac { t } { \\delta \\rho ^ { s p } } w = \\frac { ( u - M ) _ { - } } { M } \\end{align*}"}
+{"id": "5598.png", "formula": "\\begin{align*} \\int _ { Y } { \\rm I } \\left ( \\mathbb { P } _ { \\mu , 1 } ^ { y } , \\mathbb { P } _ { \\mu , n } ^ { y } \\right ) d \\nu ( y ) = { \\rm I } \\left ( \\mathbb { P } _ { \\mu , 1 } , \\mathbb { P } _ { \\mu , n } \\right ) - h _ { \\mu } ( Y , \\nu ) . \\end{align*}"}
+{"id": "5628.png", "formula": "\\begin{align*} 2 _ { \\mu , * } : = \\frac { 2 N - \\mu } { N } , 2 ^ * _ \\mu : = \\frac { 2 N - \\mu } { N - 2 } . \\end{align*}"}
+{"id": "6899.png", "formula": "\\begin{align*} \\widehat \\psi _ r ( 1 ) = \\inf _ { x \\in [ 0 , 1 ] } J _ r ( x , 1 ) = \\inf _ { x \\in [ 0 , 1 ] } \\int _ { [ 0 , 1 ] } \\d y \\ , \\log \\frac { 1 } { r ( x , y ) } . \\end{align*}"}
+{"id": "2344.png", "formula": "\\begin{align*} \\int _ { - R } ^ R \\big ( G _ { t - r } ( x - z ) - G _ { s - r } ( x - z ) \\big ) d x = \\frac { 1 } { 2 } \\int _ { - R } ^ { R } 1 _ { \\{ s - r < | x - z | < t - r \\} } d x \\in [ 0 , t - s ] . \\end{align*}"}
+{"id": "5477.png", "formula": "\\begin{align*} \\mathbb { E } \\{ Z \\} = \\frac { \\kappa } { \\kappa + \\beta } , \\mathbb { E } \\{ Z ^ 2 \\} = \\frac { \\kappa ( \\kappa + 1 ) } { \\kappa + \\beta ( \\kappa + \\beta + 1 ) } . \\end{align*}"}
+{"id": "1923.png", "formula": "\\begin{align*} \\langle D _ u \\theta ( u ^ * ) , v \\rangle _ \\mathcal { U } + \\lim \\limits _ { k \\rightarrow + \\infty } \\langle \\lambda ^ { k } , S v \\rangle = 0 \\forall \\ v \\in \\mathcal { U } , \\end{align*}"}
+{"id": "7183.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 \\left ( J _ { w ^ h _ 1 } \\cap \\cup _ { i \\in \\N } B ^ i _ 1 \\right ) = \\mathcal { H } ^ 1 \\left ( J _ u \\cap \\cup _ { i \\geq h + 1 } B _ 1 ^ i \\right ) , \\end{align*}"}
+{"id": "5287.png", "formula": "\\begin{align*} \\left | 1 + \\rho d X \\right | ^ { q } \\le 1 + q \\rho d X + \\binom { q } { 2 } \\rho ^ 2 d ^ 2 X ^ 2 + \\binom { q } { 3 } \\rho ^ 3 | d X | ^ 3 ( 1 + | \\rho d X | ) ^ { q - 3 } . \\end{align*}"}
+{"id": "3527.png", "formula": "\\begin{align*} \\begin{bmatrix} - S _ 1 ' & - S _ 2 ' \\\\ 0 & 0 \\\\ S _ 3 ' & S _ 4 ' \\\\ 0 & 0 \\end{bmatrix} = \\begin{bmatrix} I - S _ 2 G _ 3 ' & - S _ 2 G _ 4 ' \\\\ S _ 3 + G _ 3 ' & S _ 4 + G _ 4 ' \\\\ G _ 1 ' & - R _ 1 S _ 2 + R _ 2 + G _ 2 ' \\\\ S _ 3 G _ 1 ' + S _ 4 G _ 3 ' & - R _ 3 S _ 2 + R _ 4 + S _ 3 G _ 2 ' + S _ 4 G _ 4 ' \\end{bmatrix} , \\end{align*}"}
+{"id": "2384.png", "formula": "\\begin{align*} C ^ { ( 1 ) } = D ^ { ( 1 ) } . \\end{align*}"}
+{"id": "3583.png", "formula": "\\begin{align*} [ P ] \\ : = \\ \\begin{cases} 1 , & , \\\\ 0 , & . \\end{cases} \\end{align*}"}
+{"id": "1398.png", "formula": "\\begin{align*} \\int _ M ( d J \\theta ) ^ { n - p } \\wedge \\omega = \\int _ M ( J \\theta \\wedge ( d J \\theta ) ^ { n - p - 1 } ) \\wedge d \\omega = 0 , \\end{align*}"}
+{"id": "6856.png", "formula": "\\begin{align*} \\delta _ { \\square } ( h _ 1 , h _ 2 ) = \\inf _ { \\phi \\in \\mathcal { M } } d _ { \\square } ( h _ 1 , h _ 2 ^ { \\phi } ) , \\end{align*}"}
+{"id": "7786.png", "formula": "\\begin{align*} ( A \\otimes B ) ( C \\otimes D ) = A C \\otimes B D , \\end{align*}"}
+{"id": "7757.png", "formula": "\\begin{align*} \\mathbf { x } \\sim \\mathbf { y } \\Leftrightarrow \\mathbf { x } = a \\odot \\mathbf { y } ; \\end{align*}"}
+{"id": "3550.png", "formula": "\\begin{align*} u ( r ) = \\varepsilon A \\left [ \\left ( \\frac { \\sigma } { r } \\right ) ^ { n } - \\left ( \\frac { \\sigma } { r } \\right ) ^ { m } \\right ] , \\end{align*}"}
+{"id": "4458.png", "formula": "\\begin{align*} S _ { \\omega _ { \\pm } , \\mathbf { c } _ { \\pm } } ( U _ 0 ) = h ( \\mp \\tau _ 0 ) - \\tau _ 0 ^ 2 \\left ( \\frac { h '' ( \\theta ) } { 2 } - Q ( \\Phi _ { \\omega , \\mathbf { c } } ) \\right ) + O ( \\delta ) \\le h ( \\mp \\tau _ 0 ) - \\tau _ 0 ^ 2 F ( \\theta ) + O ( \\delta ) . \\end{align*}"}
+{"id": "7318.png", "formula": "\\begin{align*} b _ p + \\sum _ { i = 1 } ^ n \\beta _ i z _ { i p } = c _ p + \\sum _ { i = 1 } ^ n \\gamma _ i z _ { i p } < a _ p + \\sum _ { i = 1 } ^ n \\alpha _ i z _ { i p } . \\end{align*}"}
+{"id": "6025.png", "formula": "\\begin{align*} \\pi _ { \\psi } ( g _ 1 ) \\pi _ { \\psi } ( g _ 2 ) f ( w ) & = \\epsilon _ { g _ 1 } \\epsilon _ { g _ 2 } f ( w g _ 1 g _ 2 ) \\\\ & = \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) \\pi _ { \\psi } ( g _ 1 g _ 2 ) f ( w ) \\\\ & = \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) \\epsilon _ { g _ 1 g _ 2 } f ( w g _ 1 g _ 2 ) . \\end{align*}"}
+{"id": "2259.png", "formula": "\\begin{align*} P ( x , y ) = \\frac { y } { x ^ 2 + y ^ 2 } \\end{align*}"}
+{"id": "1440.png", "formula": "\\begin{align*} & G _ 0 ( \\bar { d } , \\bar { \\sigma } , \\xi ) = \\alpha _ n a _ 1 d _ 1 ^ { n - 2 } \\varphi ( \\xi ) + a _ 3 \\sum \\limits _ { i = 1 } ^ { k - 1 } \\Big ( \\frac { d _ { i + 1 } } { d _ i } \\Big ) ^ { \\frac { n - 2 } { 2 } } g ( \\sigma _ i ) - a _ 4 \\sum \\limits _ { i = 1 } ^ k \\frac { 2 } { 2 i - 1 } | \\ln d _ i | , \\\\ & G _ h ( d _ 1 , \\xi ) = \\frac { \\alpha _ n } { 2 } a _ 2 \\partial _ { \\xi _ h } \\varphi ( \\xi ) d _ 1 ^ { n - 2 } , \\quad \\mbox { f o r } \\ \\ h = 1 , \\cdots , n . \\end{align*}"}
+{"id": "4984.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\binom { n } { k } = 2 ^ n \\end{align*}"}
+{"id": "4362.png", "formula": "\\begin{align*} d _ { p } \\left ( \\overline { b } , \\Theta _ { c } \\right ) ^ { p } \\leq \\inf _ { x \\in C } \\sum \\limits _ { i = 1 } ^ { m } \\left ( u _ { i } \\right ) ^ { p } d _ { S _ { i } } ^ { p } \\left ( x \\right ) = \\inf _ { x \\in \\mathbb { R } ^ { n } } \\sum \\limits _ { i = 1 } ^ { m } \\left ( u _ { i } \\right ) ^ { p } d _ { S _ { i } } ^ { p } \\left ( x \\right ) + I _ { C } \\left ( x \\right ) , \\end{align*}"}
+{"id": "4269.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\mathrm { d } \\zeta ( s ) & = & - g ( \\kappa ( s ) ) \\mathrm { d } s + \\kappa ( s ) \\mathrm { d } W ( s ) , \\\\ \\zeta ( T ) & = & \\chi , 0 \\leq s \\leq T , \\end{array} \\right . \\end{align*}"}
+{"id": "4368.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } - \\dbinom { 0 _ { n } } { 2 h ^ { 0 } } = \\dbinom { A ^ { \\prime } } { - I _ { m } } \\lambda + \\dbinom { 0 _ { n \\times m } } { - I _ { m } } \\mu , \\\\ \\left ( A x ^ { 0 } - \\overline { b } - h ^ { 0 } \\right ) ^ { \\prime } \\lambda = 0 , - \\left ( h ^ { 0 } \\right ) ^ { \\prime } \\mu = 0 , \\\\ A x ^ { 0 } - \\overline { b } - h ^ { 0 } \\leq 0 _ { m } , ~ h ^ { 0 } \\geq 0 _ { m } . \\end{array} \\right . \\end{align*}"}
+{"id": "3256.png", "formula": "\\begin{align*} | b ( x ) - b ( y ) | & = | b ( x ) - m _ b ( \\tilde B ) + m _ b ( \\widetilde B ) - b ( y ) | \\\\ & = | b ( x ) - m _ b ( \\widetilde B ) | + | m _ b ( \\widetilde B ) - b ( y ) | \\ge | b ( x ) - m _ b ( \\widetilde B ) | . \\end{align*}"}
+{"id": "5419.png", "formula": "\\begin{align*} r _ n ^ { [ 2 ] } ( h ) : = \\frac { 2 ^ { 2 p + 1 } \\cdot y _ { 4 n } \\left ( \\frac { h } { 4 } \\right ) - 3 \\cdot 2 ^ p \\cdot y _ { 2 n } \\left ( \\frac { h } { 2 } \\right ) + y _ n ( h ) } { \\left ( 2 ^ p - 1 \\right ) \\left ( 2 ^ { p + 1 } - 1 \\right ) } \\end{align*}"}
+{"id": "2333.png", "formula": "\\begin{align*} D _ { r , z } u ( t , x ) = \\sum _ { n \\geq 1 } n I _ { n - 1 } ( \\bar { f } _ n ( \\cdot , r , z , t , x ) ) = \\sum _ { n \\geq 1 } \\sum _ { j = 1 } ^ { n } I _ { n - 1 } ( f _ j ^ { ( n ) } ( \\cdot , r , z , t , x ) ) , \\end{align*}"}
+{"id": "8769.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ m ( p _ i \\beta _ { i , k } - p _ j \\beta _ { j , k } ) + \\alpha _ { i j } - \\lambda _ { i j } + \\gamma _ { i } = 0 , \\forall i , j \\in \\{ 1 , \\ldots , n \\} , \\ i \\neq j \\end{align*}"}
+{"id": "7381.png", "formula": "\\begin{align*} \\Sigma _ N ^ 2 ( L , \\alpha ) = L - L ^ 2 + L R _ N ^ 2 ( L , \\alpha , \\Delta ) . \\end{align*}"}
+{"id": "4842.png", "formula": "\\begin{align*} ( \\exists f \\in F _ i ) , ( \\prod \\limits _ { i = 1 } ^ m a ( i ) \\ast f ( t ( i ) ) \\ast a ( m + 1 ) \\notin A _ i \\cap \\sigma ( L _ i ) . \\end{align*}"}
+{"id": "3161.png", "formula": "\\begin{align*} \\tau ( \\beta _ g ( x ) Y ) = \\tau ( x \\hat { \\beta } _ g ( Y ) ) x \\in M Y \\in L ^ 1 ( M , \\tau ) . \\end{align*}"}
+{"id": "2647.png", "formula": "\\begin{align*} C o n ( X \\cdot Y ) = C o n ( X ) C o n ( Y ) , \\end{align*}"}
+{"id": "4971.png", "formula": "\\begin{align*} \\Pr ( X _ t = k \\ , \\vert \\ , X _ 0 = r ) = p _ { r , k } ^ { ( t ) } = \\binom { n - r } { n - k } \\sum _ { j = 0 } ^ { k - r } ( - 1 ) ^ { k - r - j } \\binom { k - r } { j } q _ { j + r } ^ t , \\end{align*}"}
+{"id": "8771.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ m ( p _ i \\beta _ { i , k } - p _ j \\beta _ { j , k } ) + \\alpha _ { i j } - \\lambda _ { i j } + \\gamma _ { i } = 0 , \\forall j \\neq i . \\end{align*}"}
+{"id": "6698.png", "formula": "\\begin{align*} \\psi S A = - a S \\nabla ( \\psi E ) a + a ( S \\nabla ( \\psi E ) - \\psi S \\nabla E ) a - \\psi A S \\nabla \\tilde \\eta a - \\psi \\tilde a S \\nabla \\tilde \\eta A - \\psi R a - \\psi \\tilde r A - \\psi A . \\end{align*}"}
+{"id": "1475.png", "formula": "\\begin{align*} P \\psi ^ 0 _ { \\mu , \\xi } ( x ) = \\psi ^ 0 _ { \\mu , \\xi } ( x ) - \\frac { n - 2 } { 2 } \\alpha _ n \\mu ^ { \\frac { n - 2 } { 2 } } H ( x , \\xi ) + O ( \\mu ^ { \\frac { n + 4 } { 2 } } ) , \\end{align*}"}
+{"id": "5831.png", "formula": "\\begin{align*} \\left \\vert \\pi _ 1 ( X _ n ) \\left ( \\frac { 1 } { \\pi _ 2 ( X _ n ) } - \\frac { 1 } { H _ n } \\right ) \\right \\vert = \\left \\vert \\frac { \\pi _ 1 ( X _ n ) } { \\pi _ 2 ( X _ n ) } \\right \\vert \\frac { H _ n - \\pi _ 2 ( X _ n ) } { H _ n } \\leqslant \\frac { n } { H _ n / 2 } \\frac { H _ n ^ { 1 / 2 } } { H _ n } = 2 n H _ n ^ { - 3 / 2 } . \\end{align*}"}
+{"id": "1306.png", "formula": "\\begin{align*} \\frac { X ( t ) } { \\tau - t } = \\frac { \\theta } { \\tau } e ^ { \\widehat { B } ( U ( t ) ) + \\frac { 1 } { 2 } U ( t ) } . \\end{align*}"}
+{"id": "219.png", "formula": "\\begin{align*} 1 0 ( \\log 2 ) ^ 3 - 2 \\pi ^ 2 \\log 2 = - 4 8 L i _ 3 \\left ( \\frac { 1 } { 2 } \\right ) + 5 4 L i _ 3 \\left ( \\frac { 1 } { 4 } \\right ) + 1 2 L i _ 3 \\left ( \\frac { 1 } { 8 } \\right ) - 3 L i _ 3 \\left ( \\frac { 1 } { 6 4 } \\right ) . \\end{align*}"}
+{"id": "8610.png", "formula": "\\begin{align*} & E \\left [ Z _ 0 ( \\Delta \\ell _ 1 ) \\left ( \\nu \\Delta Z _ 0 ( \\Delta \\ell _ 2 ) p ^ k _ { j , + } ( t - \\Delta \\ell _ 2 ) - W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) \\right ) \\right ] \\\\ & = E \\left [ E \\left [ Z _ 0 ( \\Delta \\ell _ 1 ) \\big ( \\nu \\Delta Z _ 0 ( \\Delta \\ell _ 2 ) p ^ k _ { j , + } ( t - \\Delta \\ell _ 2 ) - E \\left [ W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) | { \\cal F } _ { ( \\ell _ 2 + 1 ) \\Delta } \\right ] \\big ) | { \\cal F } _ { ( \\ell _ 1 + 1 ) \\Delta } \\right ] \\right ] . \\end{align*}"}
+{"id": "7206.png", "formula": "\\begin{align*} \\forall ( y , \\xi ) \\in B _ 1 \\times \\R ^ k , f _ h ( y , \\xi ) : = ( \\gamma _ h \\sigma _ h ) ^ { 1 - \\frac { p _ h ( y ) } { p _ h ^ 0 } } \\abs { \\xi } ^ { p _ h ( y ) } , \\end{align*}"}
+{"id": "8008.png", "formula": "\\begin{align*} \\lambda _ k [ f ( A _ { \\mathcal { S ' } } ) ] & = \\min _ { h \\in { \\mathcal { F } } ; \\ \\Vert h \\Vert = 1 } \\langle h , f ( A _ { \\mathcal { F } } ) h \\rangle \\\\ & = \\min \\{ f ( \\lambda _ 1 ( A _ { \\mathcal { F } } ) ) \\ , ; \\ , f ( \\lambda _ k ( A _ { \\mathcal { F } } ) ) \\} \\\\ & = \\min _ { h \\in { \\mathcal { F } } ; \\ \\Vert h \\Vert = 1 } f ( \\langle h , A _ { \\mathcal { F } } h \\rangle ) \\\\ & = \\min _ { h \\in { \\mathcal { F } } ; \\ \\Vert h \\Vert = 1 } f ( \\langle h , A h \\rangle ) \\end{align*}"}
+{"id": "1354.png", "formula": "\\begin{align*} \\underset { \\delta \\rightarrow 0 } { \\lim } \\left | I I I _ { \\delta } \\right | \\leq 2 \\underset { \\overline { t } \\in [ 0 , T ] } { \\sup } \\left | A _ { - } ( \\overline { t } , \\overline { t } , \\cdot ) \\right | _ { \\mathrm { L i p } } ( \\tau - \\sigma ) = \\underset { \\overline { t } \\in [ 0 , T ] } { \\sup } \\left | A ( \\overline { t } , \\cdot ) - \\widetilde { A } ( \\overline { t } , \\cdot ) \\right | _ { \\mathrm { L i p } } ( \\tau - \\sigma ) . \\end{align*}"}
+{"id": "7637.png", "formula": "\\begin{align*} \\\\ M ( 0 , 0 , y ) = 1 0 2 4 y ^ 2 \\leq 1 0 2 4 , y \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "3428.png", "formula": "\\begin{align*} x & = \\varepsilon ^ { - 1 } ( R _ { n } R _ { n + 1 } ) \\log ^ { 2 N - 3 } \\left ( \\left ( \\frac { R _ { n } R _ { n + 1 } } { \\varepsilon } \\right ) ^ { \\frac { R _ { n } R _ { n + 1 } } { 2 } } \\frac { N - 1 } { \\eta } \\right ) , \\\\ y & = \\log ^ 4 \\left ( \\left ( \\frac { R _ { n } R _ { n + 1 } } { \\varepsilon } \\right ) ^ { \\frac { 1 } { 2 } } \\log ^ { N - 1 } \\left ( \\left ( \\frac { R _ { n } R _ { n + 1 } } { \\varepsilon } \\right ) ^ { \\frac { R _ { n } R _ { n + 1 } } { 2 } } \\frac { N - 1 } { \\eta } \\right ) \\right ) \\log \\prod _ { j \\ne n } I _ j , \\end{align*}"}
+{"id": "3655.png", "formula": "\\begin{align*} \\lambda ^ 2 = \\sum _ { k \\in [ n ] , k \\neq j } ^ n ( t _ k t _ k ^ * ) = \\sum _ { k \\in [ n ] , k \\neq j } ^ n \\norm { t _ k } ^ 2 . \\end{align*}"}
+{"id": "4200.png", "formula": "\\begin{align*} e _ 1 \\cdot [ X : Y : Z ] & = [ \\zeta _ 5 X : Y : Z ] \\\\ e _ 2 \\cdot [ X : Y : Z ] & = [ X : \\zeta _ 5 Y : Z ] . \\end{align*}"}
+{"id": "3127.png", "formula": "\\begin{align*} q ^ { \\binom { k + 1 } { 2 } } [ k ] _ { q } ! \\ , S [ n + 1 , k + 1 ] = \\sum _ { \\ell = 0 } ^ { k } q ^ { k ( k - \\ell ) } A _ { n , \\ell } ( q ) { n - \\ell \\brack k - \\ell } _ { q } , \\end{align*}"}
+{"id": "2172.png", "formula": "\\begin{align*} \\lambda \\left ( \\delta \\right ) \\geq \\lambda _ { 1 } = \\lambda _ { 1 } \\left ( N , \\varepsilon , \\Omega , T , c \\right ) \\geq 1 , \\end{align*}"}
+{"id": "2423.png", "formula": "\\begin{align*} \\psi _ { C } ( x ) = x - \\log x - 1 + \\sum _ { k = 1 } ^ { \\infty } R _ { k } ( x ) , \\end{align*}"}
+{"id": "6550.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } g _ F ( t ) \\ , e ^ { i z t } \\ , d t = \\frac { 1 } { z ^ 2 } \\frac { \\xi _ F ' } { \\xi _ F } \\left ( \\frac { 1 } { 2 } - i z \\right ) , \\Im ( z ) > 1 / 2 \\end{align*}"}
+{"id": "3166.png", "formula": "\\begin{align*} u _ g ( x \\Omega _ { \\rho } ) = T _ g ( x ) \\Omega _ { \\rho } , x \\in M _ s , g \\in G . \\end{align*}"}
+{"id": "8301.png", "formula": "\\begin{align*} \\mathsf { M } _ { x _ 0 } ^ { \\sigma ^ s } ( \\Gamma _ X \\times \\Gamma _ A ) = \\int _ { \\Gamma _ X } \\sigma ^ s ( \\Gamma _ A | x ) \\mathsf { M } _ { x _ 0 } ^ { \\sigma ^ s } ( d x \\times \\textbf { A } ) , ~ \\Gamma _ X \\in { \\cal B } ( \\textbf { X } ) , ~ \\Gamma _ A \\in { \\cal B } ( \\textbf { A } ) \\end{align*}"}
+{"id": "3143.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ m \\sum _ { i = 0 } ^ { m - k } p ( m , k , i ) u ^ k ( ( x + 1 ) ^ i - x ^ i ) \\\\ = \\sum _ { k = 0 } ^ { m - 1 } \\sum _ { i = 0 } ^ { m - k - 1 } \\left ( m p ( m - 1 , k , i ) + ( k + 1 ) p ( m - 1 , k + 1 , i ) + k p ( m - 1 , k , i ) \\right ) u ^ k x ^ i . \\end{gather*}"}
+{"id": "3978.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + V _ { 1 , Y Y } ^ { 1 / 2 } \\ ; \\delta _ 1 ^ { 1 / 2 } + V _ { 2 , Y Y } ^ { 1 / 2 } \\ ; \\delta _ 2 ^ { 1 / 2 } \\end{align*}"}
+{"id": "7278.png", "formula": "\\begin{align*} x y - z t = 0 \\end{align*}"}
+{"id": "734.png", "formula": "\\begin{align*} \\tilde A \\Big ( { \\rm I } _ { N _ { S } } - \\frac 1 { 1 + \\sum a _ { k } } ( a _ { j } ) _ { i , j } \\Big ) & = { \\rm I } _ { N _ { S } } + ( a _ { j } ) _ { i , j } - \\frac 1 { 1 + \\sum a _ { k } } ( a _ { j } ) _ { i , j } - \\frac 1 { 1 + \\sum a _ { k } } ( a _ { j } ) _ { i , j } ^ 2 \\\\ & = { \\rm I } _ { N _ { S } } + \\Bigg ( \\frac { a _ { j } ( 1 + \\sum a _ { k } ) } { 1 + \\sum a _ { k } } - \\frac { a _ { j } } { 1 + \\sum a _ { k } } - \\frac { a _ { j } \\sum a _ { k } } { 1 + \\sum a _ { k } } \\Bigg ) _ { i , j } = { \\rm I } _ { N _ { S } } . \\end{align*}"}
+{"id": "1438.png", "formula": "\\begin{align*} V + \\phi - i ^ * [ f _ \\epsilon ( V + \\phi ) ] = \\sum _ { i = 1 } ^ k \\sum _ { l = 0 } ^ n c _ { i l } P \\psi ^ l _ { \\mu _ i , \\xi _ i } , \\end{align*}"}
+{"id": "4801.png", "formula": "\\begin{align*} K = \\begin{bmatrix} A & B _ 1 & \\cdots & B _ m \\end{bmatrix} , \\end{align*}"}
+{"id": "3164.png", "formula": "\\begin{align*} \\nu _ { y ' } ^ g ( x ) & = \\langle y ' T _ { g } ( x ) \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } \\\\ & \\leq \\norm { y ' } _ { \\infty } \\langle T _ { g } ( x ) \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } \\\\ & = \\norm { y ' } _ { \\infty } \\rho ( T _ { g } ( x ) ) \\\\ & = \\norm { y ' } _ { \\infty } \\rho ( x ) \\\\ & = \\norm { y ' } _ { \\infty } \\langle x \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } . \\end{align*}"}
+{"id": "7815.png", "formula": "\\begin{align*} W _ { j } ^ { \\prime } = \\sum _ { i \\le N } g _ { i j } \\tilde { g } _ { i j } B _ { i } , \\end{align*}"}
+{"id": "4292.png", "formula": "\\begin{align*} R \\rho u _ x = - R \\rho ( ( \\ln \\rho ) _ t + u ( \\ln \\rho ) _ x ) \\end{align*}"}
+{"id": "1597.png", "formula": "\\begin{align*} \\| { w - R _ h w } \\| \\ , + h \\ , \\| { \\nabla ( w - R _ h w ) } \\| \\ , \\le & \\ , C h ^ 2 \\ , \\| { \\Delta w } \\| , \\forall w \\in H ^ 2 \\cap H ^ 1 _ 0 . \\\\ \\end{align*}"}
+{"id": "7565.png", "formula": "\\begin{align*} a = g _ 1 g _ 2 \\cdots g _ s \\end{align*}"}
+{"id": "3536.png", "formula": "\\begin{align*} \\wp ( z + w ) = \\frac { 1 } { 4 } \\left ( \\frac { \\wp ' ( z ) - \\wp ' ( w ) } { \\wp ( z ) - \\wp ( w ) } \\right ) ^ 2 - \\wp ( z ) - \\wp ( w ) . \\end{align*}"}
+{"id": "6626.png", "formula": "\\begin{align*} \\bar { \\varphi } _ { \\lambda t } ( [ x _ \\mu y ] , z ) = [ x _ \\mu { \\bar { \\varphi } _ { ( \\lambda - \\mu ) t } ( y , z ) } ] - ( - 1 ) ^ { | y | | x | } [ y _ { \\lambda - \\mu } { \\bar { \\varphi } _ { \\mu t } ( x , z ) } ] . \\end{align*}"}
+{"id": "1349.png", "formula": "\\begin{align*} \\partial _ { t } F + \\partial _ { x } ( A ( t , F ) ) = \\mathbf { S } [ F ] ( t , x ) \\end{align*}"}
+{"id": "3186.png", "formula": "\\begin{align*} M _ a ( f ) \\leq \\frac { \\chi _ d } { ( [ a ] + 1 ) _ d } \\sum _ { j = 0 } ^ { ( [ a ] + 1 ) _ d - 1 } U ^ j M _ 1 ( f ) . \\end{align*}"}
+{"id": "6506.png", "formula": "\\begin{align*} t _ n = \\frac { \\det A _ n } { \\det A _ { n - 1 } } , \\end{align*}"}
+{"id": "5815.png", "formula": "\\begin{align*} { { \\boldsymbol { \\gamma } } _ { i ; k } } = { \\boldsymbol { h } } \\left ( { { { \\boldsymbol { \\xi } } _ { i ; k | k - 1 } } } \\right ) , \\left ( { i = 1 , 2 , \\cdots , 2 n } \\right ) . \\end{align*}"}
+{"id": "8467.png", "formula": "\\begin{align*} I _ { m , p } ^ { \\star ( 1 ) } ( s , x ) = \\frac { ( - 1 ) ^ { m } } { 2 \\pi i } \\intop _ { \\sigma - i \\infty } ^ { \\sigma + i \\infty } \\Gamma ( z ) \\ , \\Gamma ( s - z ) \\ , \\left ( \\frac { x } { m } \\right ) ^ { 2 z } d z = \\frac { ( - 1 ) ^ { m } \\Gamma ( s ) x ^ { 2 s } } { \\left ( x ^ { 2 } + m ^ { 2 } \\right ) ^ { s } } . \\end{align*}"}
+{"id": "1717.png", "formula": "\\begin{align*} L : = \\lim _ { n \\to \\infty } T ^ n = \\begin{bmatrix} I & 0 \\\\ 0 & A \\end{bmatrix} , \\end{align*}"}
+{"id": "2126.png", "formula": "\\begin{align*} \\mathfrak { N } _ { f , a } \\geq & \\theta ( d ) ^ 2 \\Theta ( g ) ^ 2 \\Bigg [ \\lambda \\left ( \\frac { 2 \\sum _ { i = 1 } ^ { r } \\theta ( p _ i ) + 2 \\sum _ { i = 1 } ^ { s } \\Theta ( g _ i ) + 1 } { \\lambda } \\right ) \\left ( - ( 2 n + 1 ) W ( d ) ^ 2 W ( g ) ^ 2 q ^ { m / 2 } \\right ) \\\\ + & \\left \\{ q ^ { m - 1 } - ( n + 1 ) - ( 2 n + 1 ) q ^ { m / 2 } \\right \\} \\Bigg ] \\end{align*}"}
+{"id": "8727.png", "formula": "\\begin{align*} ( \\widetilde { \\Delta } _ { 2 } ) & = O ( N ^ { - 1 } p ^ { - 2 } ) = o ( ( \\Delta _ { 1 } ) ) , \\\\ E ( \\widetilde { \\Delta } _ { 2 } ) - T _ { 1 } & = O ( p ^ { - \\frac { 3 } { 2 } } ) = o ( \\surd { ( \\Delta _ { 1 } ) } ) . \\end{align*}"}
+{"id": "3355.png", "formula": "\\begin{align*} T = \\psi _ { S , Z } ( T ) = S ( I \\otimes T ) Z ^ * . \\end{align*}"}
+{"id": "1792.png", "formula": "\\begin{align*} H \\equiv \\frac { \\hbar ^ 2 } { 2 } \\sum _ { i = 1 } ^ N ( - \\Delta _ { x _ i } ) + \\lambda \\sum _ { i < j } ^ N V ( x _ i - x _ j ) \\ . \\end{align*}"}
+{"id": "4398.png", "formula": "\\begin{align*} I _ { j , k } : = \\frac { 1 } { A _ j } \\left \\| A _ j \\partial _ k u _ j - \\frac { c _ k } { 4 } i u _ j \\right \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 \\ \\ ( j = 1 , 2 , 3 , \\ k = 1 , \\cdots , d ) . \\end{align*}"}
+{"id": "2571.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) , \\ \\ 0 < u < 1 , & \\ \\ D , \\\\ u = 1 , \\ \\ | \\nabla u | = c , & \\ \\ , \\partial \\Omega _ 1 , \\\\ u = 0 , \\ \\ | \\nabla u | = 0 , & \\ \\ , \\partial \\Omega _ 2 . \\end{cases} \\end{align*}"}
+{"id": "8717.png", "formula": "\\begin{align*} c _ { 2 , \\tau _ 2 } ( \\widetilde { Y } _ { j _ 1 , j _ 2 } ) & = \\frac { 1 } { 2 } f ^ { ( 2 ) } ( \\tau _ 2 ) + c _ { 3 , \\tau _ 2 } ( \\widetilde { Y } _ { j _ 1 , j _ 2 } ) \\widetilde { Y } _ { j _ 1 , j _ 2 } , \\\\ c _ { 2 , \\tau _ 3 } ( \\widetilde { Z } _ { i , j } ) & = \\frac { 1 } { 2 } f ^ { ( 2 ) } ( \\tau _ 3 ) + c _ { 3 , \\tau _ 3 } ( \\widetilde { Z } _ { i , j } ) \\widetilde { Z } _ { i , j } . \\end{align*}"}
+{"id": "737.png", "formula": "\\begin{align*} \\partial _ { t } \\mu _ { S } = \\cal A _ { D , N } ^ { - 1 } { \\rm R H S } = 2 ( { \\rm I } _ { N _ { B } } - \\cal R ) ^ { - 1 } \\tilde { \\cal A } _ { - 1 } { \\rm R H S } = 2 \\sum _ { k = 0 } ^ { + \\infty } \\cal R ^ k \\tilde { \\cal A } _ { - 1 } { \\rm R H S } . \\end{align*}"}
+{"id": "6075.png", "formula": "\\begin{align*} P = \\sum _ { I \\in \\mathbb { N } _ 0 ^ n } c _ I \\eta _ A ^ I , \\eta _ A ^ I : = \\eta _ { 1 , A } ^ { i _ 1 } \\cdots \\eta _ { n , A } ^ { i _ n } , \\end{align*}"}
+{"id": "420.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| U \\| ^ 2 _ P + \\epsilon ( \\| U _ x \\| ^ 2 _ P + \\| U _ y \\| ^ 2 _ P ) + \\frac { 1 } { 2 } \\oint \\limits _ { \\partial \\Omega } U ^ T ( n _ 1 A + n _ 2 B ) U - 2 \\epsilon U ^ T ( n _ 1 P U _ x + n _ 2 P U _ y ) \\\\ \\ d s = 0 . \\end{align*}"}
+{"id": "2295.png", "formula": "\\begin{align*} \\widetilde { T } ( f ) ( z ) & = - \\iint _ D \\left ( \\frac { f ( \\zeta ) } { \\zeta - z } + \\frac { z \\overline { f ( \\zeta ) } } { 1 - z \\overline { \\zeta } } \\right ) d \\xi \\ , d \\eta \\\\ & = - \\iint _ D \\left ( \\frac { f ( \\zeta ) } { \\zeta - z } - \\overline { \\left ( \\frac { f ( \\zeta ) } { \\zeta - z } \\right ) } \\right ) d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "1732.png", "formula": "\\begin{align*} \\nu _ t \\left ( x \\right ) \\log \\frac { \\nu _ t \\left ( x \\right ) } { e ^ { - U ^ { \\pi } \\left ( x \\right ) } } = \\frac { \\nu _ t \\left ( x \\right ) } { e ^ { - U ^ { \\pi } \\left ( x \\right ) } } e ^ { - U ^ { \\pi } \\left ( x \\right ) } \\log \\frac { \\nu _ t \\left ( x \\right ) } { e ^ { - U ^ { \\pi } \\left ( x \\right ) } } \\geq - \\frac 1 e e ^ { - U ^ { \\pi } \\left ( x \\right ) } = : - f _ 2 \\left ( x \\right ) , \\end{align*}"}
+{"id": "2440.png", "formula": "\\begin{align*} M & = \\left ( ( - \\infty , 0 ] \\times \\{ - 1 , 1 \\} \\right ) \\cup \\left ( ( 0 , \\infty ) \\times \\{ 0 \\} \\right ) \\\\ U _ + & = \\left ( ( - \\infty , 0 ] \\times \\{ 1 \\} \\right ) \\cup \\left ( ( 0 , \\infty ) \\times \\{ 0 \\} \\right ) \\\\ U _ - & = \\left ( ( - \\infty , 0 ] \\times \\{ - 1 \\} \\right ) \\cup \\left ( ( 0 , \\infty ) \\times \\{ 0 \\} \\right ) \\end{align*}"}
+{"id": "2753.png", "formula": "\\begin{align*} H _ { 0 , \\Sigma } ^ 1 ( \\mathrm { M } ) = \\{ u \\in H ^ 1 ( \\mathrm { M } ) ; \\ ; u _ { | \\Sigma } = 0 \\} . \\end{align*}"}
+{"id": "6482.png", "formula": "\\begin{align*} D ^ { 1 / 3 } \\psi = X \\cdot X ( \\psi ) + Y \\cdot Y ( \\psi ) + Z \\cdot T ( \\psi ) + T \\cdot Z ( \\psi ) + { \\textstyle \\frac 1 2 } X Y Z \\cdot \\psi . \\end{align*}"}
+{"id": "664.png", "formula": "\\begin{align*} x _ { k } = \\sum _ { j = 0 } ^ { k } Q ( j , k ) \\Delta x _ j . \\end{align*}"}
+{"id": "7166.png", "formula": "\\begin{align*} W ^ { 1 , p ( \\cdot ) } \\left ( \\Omega , \\mathcal { M } \\right ) : = \\left \\{ W ^ { 1 , p ( \\cdot ) } \\left ( \\Omega , \\R ^ k \\right ) : u ( x ) \\in \\mathcal { M } \\ , \\ , \\mbox { f o r a . e . } x \\in \\Omega \\right \\} . \\end{align*}"}
+{"id": "9294.png", "formula": "\\begin{align*} \\displaystyle x _ { s } ^ { k } = \\arg \\min _ { x _ { s } \\in X _ { s } } \\left \\{ f _ { s } ( x _ { s } ) + \\langle v _ { s } ^ { k } , x _ { s } \\rangle + 0 . 5 \\lambda _ { k } ^ { - 1 } \\| x _ { s } - x _ { s } ^ { k - 1 } \\| ^ { 2 } \\right \\} \\end{align*}"}
+{"id": "2513.png", "formula": "\\begin{align*} \\| A _ c ^ { - 1 } P ^ t r _ \\mu \\| _ { A _ c } = \\langle P A _ c ^ { - 1 } P ^ t r _ \\mu , r _ \\mu \\rangle ^ \\frac { 1 } { 2 } \\le \\langle A ^ { - 1 } r _ \\mu , r _ \\mu \\rangle ^ \\frac { 1 } { 2 } = \\| A ^ { - 1 } r _ \\mu \\| _ A \\le \\| A ^ { - 1 } r \\| _ A . \\end{align*}"}
+{"id": "3460.png", "formula": "\\begin{align*} h _ i ( m ) . v = ( \\sum _ { j = 1 } ^ { k _ i } p _ { i , j } b _ { i , j } ^ m ) v \\end{align*}"}
+{"id": "686.png", "formula": "\\begin{align*} \\sup \\{ | v ( x ) - v ( y ) | : x , y \\in I \\} \\leq 2 \\sum _ { l = 0 } ^ \\infty \\norm { Z ( l + 1 ) } \\norm { S ( l ) \\varphi } _ { \\sup } . \\end{align*}"}
+{"id": "3096.png", "formula": "\\begin{align*} q ^ { \\binom { k } { 2 } } [ k ] _ q ! S [ n , k ] = \\sum _ { \\ell = 1 } ^ k q ^ { k ( k - \\ell ) } A _ { n , \\ell - 1 } ( q ) { n - \\ell \\brack k - \\ell } _ { q } \\end{align*}"}
+{"id": "2604.png", "formula": "\\begin{align*} a _ i \\geq C _ \\epsilon h _ 0 ^ { \\frac { 1 } { 2 } + \\epsilon } , \\ \\ i = 1 , \\cdots , n - 1 . \\end{align*}"}
+{"id": "5349.png", "formula": "\\begin{align*} \\| D _ { S , x } [ f ^ { = d } ] \\| _ 2 = \\| ( D _ { S , x } [ f ] ) ^ { = d - | S | } \\| _ 2 \\le \\| D _ { S , x } [ f ] \\| _ 2 , \\end{align*}"}
+{"id": "4554.png", "formula": "\\begin{align*} M = \\langle S _ 1 \\rangle \\oplus \\cdots \\oplus \\langle S _ s \\rangle . \\end{align*}"}
+{"id": "7056.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ s b _ l \\frac { d } { d x } ( \\tilde Q _ l ) \\in I _ \\alpha . \\end{align*}"}
+{"id": "1457.png", "formula": "\\begin{align*} \\mbox { i f } \\ h \\in L ^ 1 _ { r a d } ( \\R ^ n ) , \\ , i \\neq l , \\int _ { \\frac { \\mathcal { A } _ m ^ l - \\xi _ m } { \\mu _ m ^ i } } h ( | x | ) d x = O \\bigg ( \\Big ( \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big ) ^ { \\frac { n } { n - 2 } } \\bigg ) . \\end{align*}"}
+{"id": "7407.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) \\ , = \\ , \\frac { 1 } { 2 } \\Delta _ x u ( t , x ) \\ , + \\ , \\beta \\ , \\dot { W } ( t , x ) \\ , u ( t , x ) \\ , , \\end{align*}"}
+{"id": "6116.png", "formula": "\\begin{align*} \\tilde { q } = \\frac { 1 } { 2 } \\left ( q + \\frac { 2 s } { s - \\tau } \\right ) \\end{align*}"}
+{"id": "7280.png", "formula": "\\begin{align*} x = x _ 0 + \\frac { m } { ( n , m ) } u , y = y _ 0 + \\frac { n } { ( n , m ) } u , u \\geq 0 , \\end{align*}"}
+{"id": "2987.png", "formula": "\\begin{align*} \\theta ( ( F ( t _ 2 ) - F ( t _ 1 ) ) \\ , t _ 1 ^ { k _ 1 } \\cdots t _ n ^ { k _ n } ) & = \\left ( - a _ { ( p , 1 , 0 , \\dots , 0 ) } + a _ { ( q , 0 , \\dots , 0 ) } \\right ) ( t _ 1 , \\dots , t _ n ) \\\\ & = q _ { n - 2 } ( t _ 1 , \\dots , t _ n ) \\cdot e _ 1 ( t _ 1 , \\dots , t _ n ) \\end{align*}"}
+{"id": "8515.png", "formula": "\\begin{align*} \\zeta _ { p , \\infty } ( s , c ) = \\sum _ { m , n \\neq 0 } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p ( p + \\frac { 1 } { \\pi } ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { 1 } { \\left ( \\lambda _ { m } ^ { 2 } + c n ^ { 2 } \\right ) ^ { s } } , \\ , \\ , \\ , \\ , \\ , ( s ) > 1 . \\end{align*}"}
+{"id": "5985.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } [ h _ { - 1 } , t ] f ( [ \\epsilon , x ] ) = t f ( [ - \\epsilon , x ] ) . \\end{align*}"}
+{"id": "9222.png", "formula": "\\begin{align*} 3 L + 4 < \\frac { 8 } { 2 5 } a ^ 2 = \\frac { 4 t } { 2 5 \\pi } \\quad 3 L + 2 + \\sigma \\ge 0 . \\end{align*}"}
+{"id": "7997.png", "formula": "\\begin{align*} w ( y ' , y _ n ) = \\begin{cases} v \\circ \\psi ^ { - 1 } ( y ' , y _ n ) & \\\\ v \\circ \\psi ^ { - 1 } ( y ' , - y _ n ) & \\end{cases} \\end{align*}"}
+{"id": "5922.png", "formula": "\\begin{align*} \\overline { C } _ { X ^ { \\ast } } ( [ y _ 1 , g _ 1 ] , [ y _ 2 , g _ 2 ] ) = m _ { X ^ { \\ast } } ( g _ 1 ^ { y _ 2 } g _ 2 ) ^ { - 1 } m _ { X ^ { \\ast } } ( g _ 1 ) m _ { X ^ { \\ast } } ( g _ 2 ) \\widetilde { C } _ { X ^ { \\ast } } ( [ y _ 1 , g _ 1 ] , [ y _ 2 , g _ 2 ] ) . \\end{align*}"}
+{"id": "4751.png", "formula": "\\begin{align*} \\tilde { S } _ { \\gamma _ i \\gamma _ i } ^ T J _ { 2 | \\alpha _ i | } \\tilde { S } _ { \\gamma _ i \\gamma _ i } J _ { 2 | \\alpha _ i | } ^ T = I _ { 2 | \\alpha _ i | } + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "2299.png", "formula": "\\begin{align*} w ( z ) = \\Phi _ 0 ( z ) + \\Psi ( z ) , \\end{align*}"}
+{"id": "8051.png", "formula": "\\begin{align*} f ( A ) = { \\mathrm { s o t } } \\lim _ { n \\to \\infty } f ( A _ n ) \\end{align*}"}
+{"id": "4185.png", "formula": "\\begin{align*} u _ \\mathrm { l i n } ( t , r ) = \\begin{cases*} \\frac { 1 + r - t } { r } & i f $ t - 1 \\leq r < t $ , \\\\ \\frac { 1 - r + t } { r } & i f $ t \\leq r < t + 1 $ , \\\\ 0 & o t h e r w i s e . \\end{cases*} \\end{align*}"}
+{"id": "8326.png", "formula": "\\begin{align*} y _ { C 1 } = \\sqrt { 1 - \\alpha } h _ { A C } x + h _ { C C } + n _ { C 1 } , \\end{align*}"}
+{"id": "2560.png", "formula": "\\begin{align*} f ( \\tau ) = - \\Delta u ( \\sigma _ y ( g ^ { - 1 } ( \\tau ) ) ) \\ \\ \\ \\ \\ \\tau \\in ( 0 , u ( 0 ) ) . \\end{align*}"}
+{"id": "4579.png", "formula": "\\begin{align*} \\delta = \\begin{cases} \\frac { 3 - 2 \\beta } { 4 } & \\beta \\ge \\frac 1 2 \\\\ \\frac 1 2 & \\beta \\le \\frac 1 2 \\end{cases} \\end{align*}"}
+{"id": "7026.png", "formula": "\\begin{align*} \\nu _ i \\left ( f ' \\right ) = \\nu _ i ( f ) + \\alpha _ i . \\end{align*}"}
+{"id": "2717.png", "formula": "\\begin{align*} J _ 6 = - \\frac { s } { 2 } \\iint _ Q \\sigma _ { t t } | u | ^ 2 d x d y d t \\geq - C s \\iint _ Q \\xi ^ { 3 / 2 } | u | ^ 2 d x d y d t . \\end{align*}"}
+{"id": "2504.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle \\dfrac { d ^ 2 } { d t ^ 2 } G ( t ) = \\dfrac { 2 } { m _ 1 } R e Q ( \\bar { \\phi } , \\psi ) , \\end{array} \\right . \\end{align*}"}
+{"id": "1515.png", "formula": "\\begin{align*} \\underline { f } ( x ) : = \\begin{cases} 1 - \\eta ^ { - 1 } \\psi ( x ) , & x \\in [ 0 , \\varepsilon _ 2 ] , \\\\ 0 , & x \\in ( \\varepsilon _ 2 , \\infty ) \\end{cases} \\end{align*}"}
+{"id": "8745.png", "formula": "\\begin{align*} 0 & = \\frac { 1 } { 2 } \\{ f ( \\tau _ 1 ) + f ( \\tau _ 2 ) \\} - f ( \\tau _ 3 ) \\geq f \\left ( \\frac { \\tau _ 1 + \\tau _ 2 } { 2 } \\right ) - f ( \\tau _ 3 ) \\geq 0 , \\end{align*}"}
+{"id": "3194.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\norm { B _ l ( \\nu _ k ) - \\overline { \\nu } } = 0 . \\end{align*}"}
+{"id": "4604.png", "formula": "\\begin{align*} { \\rm P } _ { j , j + i } = & \\sum ^ { M - j } _ { m = i + 1 } { M - j \\choose m } \\mathbb { P } _ { \\rm T X } ^ m \\left ( 1 - \\mathbb { P } _ { \\rm T X } \\right ) ^ { M - j - m } \\\\ & \\times \\frac { M - j - i } { M - j } m \\cdots ( m - i + 1 ) \\mathbb { P } _ K ^ { m } \\gamma _ i \\gamma _ { m , i } \\\\ & + { M - j \\choose i } \\mathbb { P } _ { \\rm T X } ^ i \\left ( 1 - \\mathbb { P } _ { \\rm T X } \\right ) ^ { M - j - i } \\frac { M - j - i } { M - j } i ! \\mathbb { P } _ K ^ { i } \\gamma _ i , \\end{align*}"}
+{"id": "7889.png", "formula": "\\begin{align*} \\sigma _ { | X _ 1 } = \\sigma _ { | X _ 0 \\setminus \\{ k \\} } \\mbox { i s a n e v e n p e r m u t a t i o n } . \\end{align*}"}
+{"id": "5707.png", "formula": "\\begin{align*} C _ { h } ( - 1 ) = C _ { f } ^ { + } ( - 1 ) + 1 \\end{align*}"}
+{"id": "1820.png", "formula": "\\begin{align*} \\mathcal { O } ( k ) : = \\int _ { \\Lambda ^ { * } } \\ 1 _ \\S ( p ) g ( p ) a _ { p + k } ^ * a _ p \\d p \\end{align*}"}
+{"id": "6.png", "formula": "\\begin{align*} D ( t _ 1 , \\ldots , t _ r ) : = \\min \\{ t _ i , | t _ i - t _ j | : 1 \\leq i \\neq j \\leq r \\} . \\end{align*}"}
+{"id": "7094.png", "formula": "\\begin{align*} \\partial _ { \\mathbf { n } } c = 0 , \\ ; \\ ; \\partial _ { \\mathbf { n } } \\mu = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\partial \\Omega \\times ( 0 , T ) . \\end{align*}"}
+{"id": "6523.png", "formula": "\\begin{align*} F _ X ( x ) & = F _ X ( 0 ) + M \\mathrm { s g n } ( x ) \\int _ 0 ^ x \\mathrm { e } ^ { \\beta t } | t | ^ \\nu K _ { \\nu } ( \\alpha | t | ) \\ , \\mathrm { d } t \\\\ & = F _ X ( 0 ) + M \\mathrm { s g n } ( x ) \\sum _ { k = 0 } ^ \\infty \\frac { \\beta ^ k } { k ! } \\int _ 0 ^ x ( - 1 ) ^ k | t | ^ { \\nu + k } K _ \\nu ( \\alpha | t | ) \\ , \\mathrm { d } t \\\\ & = F _ X ( 0 ) + M \\sum _ { k = 0 } ^ \\infty \\frac { \\beta ^ k } { k ! } ( \\mathrm { s g n } ( x ) ) ^ { k + 1 } \\int _ 0 ^ { | x | } t ^ { \\nu + k } K _ \\nu ( \\alpha t ) \\ , \\mathrm { d } t . \\end{align*}"}
+{"id": "6575.png", "formula": "\\begin{align*} H = \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "5157.png", "formula": "\\begin{align*} \\| f w \\| _ { L ^ p ( \\Omega ) } = \\Big ( \\int _ \\Omega \\ ! | f | ^ p w ^ p \\ , \\mathrm { d } \\mu \\Big ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
+{"id": "1704.png", "formula": "\\begin{align*} ( a ) _ n = \\frac { \\Gamma ( a + n ) } { \\Gamma ( a ) } , \\end{align*}"}
+{"id": "9298.png", "formula": "\\begin{align*} \\widehat { | \\cdot | \\eta } ( \\xi ) = \\int _ { - 1 } ^ 1 | x | e ^ { 2 \\pi i x \\xi } d x = \\int _ { - 1 } ^ { 0 } ( - x ) e ^ { 2 \\pi i x \\xi } d x + \\int _ { 0 } ^ { 1 } x e ^ { 2 \\pi i x \\xi } d x = 2 \\frac { \\sin ( 2 \\pi \\xi ) } { 2 \\pi \\xi } - \\frac { \\sin ^ 2 ( \\pi \\xi ) } { ( \\pi \\xi ) ^ 2 } . \\end{align*}"}
+{"id": "9265.png", "formula": "\\begin{align*} \\int _ { \\big \\{ x \\ , : \\ , x ^ { \\eta - 1 } \\| f \\| _ { L ^ 1 ( x ^ { - \\eta } d x ) } > \\lambda \\big \\} } x ^ { - \\eta } \\ , d x = \\int _ { \\big \\{ x \\ , : \\ , x < \\big ( \\lambda ^ { - 1 } \\| f \\| _ { L ^ 1 ( x ^ { - \\eta } d x ) } \\big ) ^ { 1 / ( 1 - \\eta ) } \\big \\} } x ^ { - \\eta } \\ , d x \\simeq \\frac { \\| f \\| _ { L ^ 1 ( x ^ { - \\eta } d x ) } } { \\lambda } , \\end{align*}"}
+{"id": "5211.png", "formula": "\\begin{align*} \\phi _ \\sqsupset ( S ) = S ^ { \\sqsubset \\vartriangleleft } . \\end{align*}"}
+{"id": "7294.png", "formula": "\\begin{align*} a x _ i ^ n u ^ { n l ' - k l ' - l n ' } v ^ { n m ' - k m ' - l k ' } + b x _ i ^ k y _ i ^ l + c y _ i ^ m u ^ { m n ' - k l ' - l n ' } v ^ { m k ' - k m ' - l k ' } = 0 . \\end{align*}"}
+{"id": "3340.png", "formula": "\\begin{align*} \\Psi \\Phi v = A ^ { - 1 } B D ^ { - 1 } C v = A ^ { - 1 } A v = v . \\end{align*}"}
+{"id": "3210.png", "formula": "\\begin{align*} S _ n ( \\mu ) = \\frac { 1 } { n + 1 } \\sum _ { k = 0 } ^ { n } \\sigma _ k ( \\mu ) . \\end{align*}"}
+{"id": "5848.png", "formula": "\\begin{align*} ( H _ { \\omega } \\phi ) ( n ) : = \\sum _ { | m - n | = 1 } ( \\phi ( m ) - \\phi ( n ) ) + V _ { \\omega } ( n ) \\phi ( n ) . \\end{align*}"}
+{"id": "9314.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c c c } c _ 1 v _ 1 & c _ 1 v _ 2 & \\ldots & c _ 1 v _ n \\\\ c _ 2 v _ 1 & c _ 2 v _ 2 & \\ldots & c _ 2 v _ n \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ c _ n v _ 1 & c _ n v _ 2 & \\ldots & c _ n v _ n \\\\ \\end{array} \\right ) = c \\otimes v , \\end{align*}"}
+{"id": "816.png", "formula": "\\begin{align*} H ( \\alpha _ 1 , \\ldots , \\alpha _ m ) : = H ( [ \\alpha _ 1 : \\cdots : \\alpha _ m : 1 ] ) . \\end{align*}"}
+{"id": "7839.png", "formula": "\\begin{align*} ( A z , H _ { J ^ { c } } ) = \\Vert \\tilde { W } z \\Vert _ { 2 } . \\end{align*}"}
+{"id": "5491.png", "formula": "\\begin{align*} \\int _ { C \\times X } f d \\lambda = \\int _ { C } \\int _ { X } f ( c , x ) d \\beta _ { c } ( x ) d \\nu _ { C } ( c ) . \\end{align*}"}
+{"id": "4740.png", "formula": "\\begin{align*} \\tilde { D } = D + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "2670.png", "formula": "\\begin{align*} \\delta \\big | _ { [ x _ { j - 1 } , x _ { j } ] } ( x ) = \\int _ { x _ j } ^ x \\delta ' ( u ) { \\rm d } u \\delta ' \\big | _ { [ x _ { j - 1 } , x _ { j } ] } ( x ) = \\int _ { \\xi _ j } ^ x \\delta '' ( u ) { \\rm d } u . \\end{align*}"}
+{"id": "7109.png", "formula": "\\begin{align*} ( 1 - \\Delta ) w _ j & = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ : \\mathbb { R } ^ n _ + , \\\\ \\mathbf { n } \\cdot \\nabla w _ j & = \\mathbf { n } \\cdot \\nabla \\tilde { c } _ j \\ ; \\ ; \\ ; \\partial \\mathbb { R } ^ n _ + \\end{align*}"}
+{"id": "123.png", "formula": "\\begin{align*} & R _ h ^ 0 ( z ) - \\tilde { R } _ h ( z ) = - i \\tilde { R } _ h ( z ) W R _ h ^ 0 ( z ) \\\\ & = i h ^ 2 \\left ( \\int _ 0 ^ \\infty e ^ { - i s h ^ { - 1 } ( - i h X - i ( Q _ \\infty + q _ 1 + W ) - z ) } d s \\right ) W \\left ( \\int _ 0 ^ \\infty e ^ { - i t h ^ { - 1 } ( - i h X - i ( Q _ \\infty + q _ 1 ) - z ) } d t \\right ) . \\end{align*}"}
+{"id": "4978.png", "formula": "\\begin{align*} \\hat { p } _ { k , n } ^ { ( t ) } = \\hat { p } _ { k - 1 , n } ^ { ( t ) } + \\frac { n - 1 } { k - 1 } \\hat { p } _ { k - 1 , n - 1 } ^ { ( t ) } \\end{align*}"}
+{"id": "7412.png", "formula": "\\begin{align*} q _ n ( x ) : = \\P ( S _ n = x \\ , | \\ , S _ 0 = 0 ) \\ , . \\end{align*}"}
+{"id": "7174.png", "formula": "\\begin{align*} \\phi ( u ) ( x - y ) = \\phi ( u ) ( x ) - \\phi ( u ) ( y ) = \\left ( \\nabla \\phi ( u ) \\right ) ( x - y ) = u ( x ) - u ( y ) . \\end{align*}"}
+{"id": "5523.png", "formula": "\\begin{align*} P _ { \\mu , x } ^ { n } \\left ( L _ { x } g ^ { - 1 } , A \\right ) = \\int _ { G } { \\bf 1 } _ { g . A } \\left ( g . x , L _ { g . x } s \\right ) \\frac { d s \\nu } { d \\nu } ( \\pi ( g . x ) ) d \\mu ^ { ( n ) } ( s ) = P _ { \\mu , g . x } ^ { n } \\left ( L _ { g . x } , g . A \\right ) . \\end{align*}"}
+{"id": "7679.png", "formula": "\\begin{align*} \\max _ { a , b } f ( a , b ) = f \\left ( \\left ( \\frac { n } { p } - 2 + \\gamma \\right ) ^ { p - 1 } \\left ( \\frac { n ( p - 1 ) } { p } - \\gamma \\right ) ^ { p - 1 } , \\frac { n } { p } - 2 + \\gamma \\right ) = \\left ( \\frac { n } { p } - 2 + \\gamma \\right ) ^ p \\left ( \\frac { n ( p - 1 ) } { p } - \\gamma \\right ) ^ p , \\end{align*}"}
+{"id": "3957.png", "formula": "\\begin{align*} \\boldsymbol { K } _ 1 ( \\gamma _ 1 ^ { \\eta _ 0 , \\delta } , \\gamma _ 1 ^ \\star ) = \\int _ { \\mathcal { S } _ 1 \\times \\mathcal { S } _ 1 } c _ 1 \\ , d \\nu . \\end{align*}"}
+{"id": "4891.png", "formula": "\\begin{align*} \\Psi \\left ( - \\frac 1 2 , x ' , 0 \\right ) = \\lim _ { m } U _ m \\left ( y _ m ^ x , 0 \\right ) = 0 | x ' | < \\frac 1 4 , \\end{align*}"}
+{"id": "6007.png", "formula": "\\begin{align*} \\mathcal { F } : & L ^ 2 ( \\R ) \\longrightarrow L ^ 2 ( \\R ) ; \\\\ & f \\longmapsto \\mathcal { F } ( f ) ( x ) = \\int _ { \\R } \\psi ( - x t ) f ( t ) d t \\end{align*}"}
+{"id": "8055.png", "formula": "\\begin{align*} A = \\{ 0 , e _ 1 , \\dots , e _ { d - 1 } \\} \\times \\{ 1 , 2 , \\dots , N \\} , \\end{align*}"}
+{"id": "398.png", "formula": "\\begin{align*} \\Sigma = \\sqrt { | \\Lambda ^ - | } . \\end{align*}"}
+{"id": "5445.png", "formula": "\\begin{align*} \\bar { F } _ { P _ s } ( \\theta , x ) \\overset { \\Delta } { = } \\Pr \\left ( P _ s ( \\theta ) > x \\right ) , x \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "8898.png", "formula": "\\begin{align*} [ x _ 1 , \\dots , x _ n ] _ c : = \\psi _ n ( x _ 1 , \\dots , x _ n ) . \\end{align*}"}
+{"id": "7865.png", "formula": "\\begin{align*} S & = \\left \\{ A \\in \\binom { [ n ] } { k } : \\ n \\in A \\right \\} . \\end{align*}"}
+{"id": "4452.png", "formula": "\\begin{align*} F ( \\tau ) : = \\inf _ { \\Phi \\in \\mathcal { M } _ { \\omega , \\mathbf { c } } ^ * ( \\eta ) } \\left ( \\frac { h '' ( \\tau ) } { 2 } - Q ( \\Phi ) \\right ) . \\end{align*}"}
+{"id": "7042.png", "formula": "\\begin{align*} ( g ( x ) ) K [ x ] = \\ker \\ , ( _ \\eta ) . \\end{align*}"}
+{"id": "3659.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) u + q u : = \\sum _ { i = 1 } ^ M \\alpha _ i ( - \\Delta ) ^ { s _ i } u + q u = 0 & \\Omega , \\\\ u = f & \\Omega ^ c : = \\mathbb { R } ^ n \\backslash \\overline { \\Omega } \\end{cases} \\end{align*}"}
+{"id": "7476.png", "formula": "\\begin{align*} \\pi = \\alpha _ 1 \\alpha _ 2 \\cdots \\alpha _ k \\beta . \\end{align*}"}
+{"id": "8542.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow 0 ^ { + } } \\mathcal { K } _ { \\nu , p } ( x ) = \\frac { \\Gamma \\left ( \\frac { 1 } { 2 } - \\nu \\right ) ( 2 x ) ^ { \\nu } } { \\sqrt { \\pi } } \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n } \\ , n ^ { \\nu } \\ , K _ { \\nu } ( x n ) . \\end{align*}"}
+{"id": "343.png", "formula": "\\begin{align*} \\int _ V u h d \\mu = 0 , \\quad \\forall u \\in L ^ q ( V ) . \\end{align*}"}
+{"id": "4919.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ n \\Pr ( X _ t = j ) = \\sum _ { j = 0 } ^ k \\Pr ( X _ t = j ) + \\sum _ { j = k + 1 } ^ n \\Pr ( X _ t = j ) = 1 , \\end{align*}"}
+{"id": "4187.png", "formula": "\\begin{align*} \\norm { ( u _ 0 ^ { \\alpha , \\epsilon } , u _ 1 ^ { \\alpha , \\epsilon } ) } _ { \\dot { H } ^ 1 \\times L ^ 2 } ^ 2 = \\epsilon ^ 2 \\norm { ( u _ 0 , u _ 1 ) } _ { \\dot { H } ^ 1 \\times L ^ 2 } ^ 2 = 1 6 \\pi \\epsilon ^ 2 . \\end{align*}"}
+{"id": "5451.png", "formula": "\\begin{align*} P _ s ( \\theta ) & \\overset { \\Delta } { = } \\left ( > \\theta , \\Phi ( \\mathcal { A } ) > 0 | \\Phi \\right ) \\\\ & = \\left ( > \\theta | \\Phi , \\Phi ( \\mathcal { A } ) > 0 \\right ) \\mathbf { 1 } ( \\Phi ( \\mathcal { A } ) > 0 | \\Phi ) , \\end{align*}"}
+{"id": "7996.png", "formula": "\\begin{align*} \\nabla \\big ( T _ { t } ( u ) \\big ) = \\chi _ { \\{ | u | < t \\} } Z _ u \\hbox { a . e . i n $ \\Omega $ } \\end{align*}"}
+{"id": "2926.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { n - 1 } ( 1 - x e ^ { 2 \\pi i j / n } ) = 1 - x ^ n . \\end{align*}"}
+{"id": "1182.png", "formula": "\\begin{align*} \\Theta ^ 2 ( \\Omega _ { H / S } | _ Y ) ^ { - 1 } \\otimes \\Theta ^ 2 ( N _ { Y / H } ) = { 1 \\over 2 } \\sum _ j E _ j b _ j \\end{align*}"}
+{"id": "1663.png", "formula": "\\begin{align*} [ \\xi _ i , \\xi _ j ] ^ \\bot = 0 , 1 \\le i , j \\le p . \\end{align*}"}
+{"id": "1945.png", "formula": "\\begin{align*} f ^ * _ { n , 0 } ( z _ 1 , \\ldots , z _ m ) : = f ^ * _ { n - 1 } ( z _ 1 , \\ldots , z _ m ) + \\delta _ { n , 0 } z _ 1 ^ n z _ 2 \\cdots z _ m A _ { n - 1 } ( z _ 1 , \\ldots , z _ m ) \\end{align*}"}
+{"id": "7085.png", "formula": "\\begin{align*} p \\cdot { \\rm d i s t } ( \\eta , K ) = { \\rm d i s t } ( \\eta , K ) . \\end{align*}"}
+{"id": "1586.png", "formula": "\\begin{align*} A _ { 2 3 1 } ( x ) = 1 + x + x ^ 2 - \\frac { 1 } { 1 + T _ { 2 3 1 } ( x ) } . \\end{align*}"}
+{"id": "3954.png", "formula": "\\begin{align*} \\lim _ { \\Delta \\rightarrow 0 } \\boldsymbol { W } _ { p } \\left ( \\pi ^ \\prime , \\frac { \\nu ( t _ 0 ) - \\Delta \\pi ^ \\prime } { 1 - \\Delta } \\right ) = \\boldsymbol { W } _ { p } \\left ( \\pi ^ \\prime , \\nu ( t _ 0 ) \\right ) < \\infty . \\end{align*}"}
+{"id": "1325.png", "formula": "\\begin{align*} \\theta _ i ^ 2 T _ i ^ * ( u ) = \\frac { T _ i ^ { \\infty } T _ i ( u ) } { T _ i ^ { \\infty } - T _ i ( u ) } . \\end{align*}"}
+{"id": "1167.png", "formula": "\\begin{align*} T ( 1 ) = - T ( 1 ) + 2 A ( 1 ) ^ { 2 } \\end{align*}"}
+{"id": "6841.png", "formula": "\\begin{align*} \\sum _ { s _ 1 \\geq \\dots \\geq s _ { r } \\geq - \\left \\lfloor m / 2 \\right \\rfloor } \\frac { q ^ { s _ 1 ^ 2 + \\dots + s _ { r } ^ 2 + m ( s _ 1 + \\dots + s _ { r - 1 } ) - s _ 1 - \\dots - s _ { i } } ( - q ) _ { m + 2 s _ r - 1 } } { ( q ) _ { s _ 1 - s _ 2 } \\dots ( q ) _ { s _ { r - 2 } - s _ { r - 1 } } ( q ^ 2 ; q ^ 2 ) _ { s _ { r - 1 } - s _ { r } } } ( - 1 ) ^ { s _ r } \\left [ { m + s _ r \\atop m + 2 s _ r } \\right ] _ { q ^ 2 } \\\\ = a _ m , \\end{align*}"}
+{"id": "6373.png", "formula": "\\begin{align*} \\dot { y } = f ( y ) , \\dot { \\xi } = f ^ \\prime ( y ) \\xi , \\dot { \\eta } = f ^ \\prime ( y ) \\eta . \\end{align*}"}
+{"id": "3985.png", "formula": "\\begin{align*} \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ { 1 3 } , \\mu _ { 2 3 } \\right ) } \\int _ { \\mathbb { R } ^ { 2 d + 2 } } ( f _ { \\mathcal { S } } ) _ \\lambda \\ , d \\pi = \\frac { 1 } { 4 \\lambda _ 1 } Q _ { 1 , Y Y } ^ { - 1 } + \\frac { 1 } { 4 \\lambda _ 2 } Q _ { 2 , Y Y } ^ { - 1 } + b \\end{align*}"}
+{"id": "7200.png", "formula": "\\begin{align*} p _ 0 : = p ( x _ 0 ) . \\end{align*}"}
+{"id": "1138.png", "formula": "\\begin{align*} A ( x _ { 1 } , \\ldots , x _ { i - 1 } , k x _ { i } , x _ { i + 1 } , \\ldots , x _ n ) = k A ( x _ { 1 } , \\ldots , x _ { i - 1 } , x _ { i } , x _ { i + 1 } , \\ldots , x _ { n } ) \\left ( x _ { 1 } , \\ldots , x _ { n } \\in G \\right ) \\end{align*}"}
+{"id": "374.png", "formula": "\\begin{align*} c ( \\tau , j ) _ i ^ { \\sigma } = \\begin{cases} p & j = i \\mbox { a n d } \\tau \\subseteq \\sigma \\\\ 1 & \\mbox { e l s e } \\end{cases} \\end{align*}"}
+{"id": "4007.png", "formula": "\\begin{align*} \\sup _ { y _ 1 ' \\in \\mathcal { Y } _ 1 } \\{ - y _ 1 ' ( 1 - d ( x ' ) ) - \\lambda _ 1 | y _ 1 - y _ 1 ' | \\} = \\begin{cases} \\infty & 0 \\le \\lambda _ 1 < 1 \\\\ - y _ 1 ( 1 - d ( x ' ) ) & \\lambda _ 1 \\ge 1 \\end{cases} . \\end{align*}"}
+{"id": "2166.png", "formula": "\\begin{align*} \\alpha = \\frac { \\left ( 1 + \\beta \\right ) b ^ { 2 } } { \\varepsilon \\left ( T - \\varepsilon \\right ) } . \\end{align*}"}
+{"id": "400.png", "formula": "\\begin{align*} W ^ T \\Lambda W = ( \\sqrt { \\Lambda ^ + } W ^ + ) ^ T ( \\sqrt { \\Lambda ^ + } W ^ + ) - ( R ( \\sqrt { \\Lambda ^ + } W ^ + ) + S G ) ^ T ( R ( \\sqrt { \\Lambda ^ + } W ^ + ) + S G ) . \\end{align*}"}
+{"id": "7453.png", "formula": "\\begin{align*} N _ { p \\alpha + k } ( \\xi , t ) = \\sum \\limits _ { i = 1 } ^ 3 \\big ( w ^ { ( i ) } _ { p \\alpha + k } ( 0 , t ) + \\Psi ^ { ( i ) } _ { p \\alpha + k } ( \\xi _ i , t ) \\big ) \\ , \\chi _ { \\ell _ 0 } ( \\xi _ i ) + \\widetilde { N } _ { p \\alpha + k } ( \\xi , t ) , \\end{align*}"}
+{"id": "7884.png", "formula": "\\begin{align*} \\sigma ( A ^ + ) = A ^ + \\mbox { a n d } \\sigma ( \\underline { A } ^ + ) = \\underline { A } ^ + . \\end{align*}"}
+{"id": "5740.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } p q = p b ^ 2 \\\\ 0 = ( b - d ) p ( b + d ) \\\\ p ( d - b ) = 0 \\\\ q ^ 2 + p d = q b ^ 2 \\\\ q ( d - b ) = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "4165.png", "formula": "\\begin{align*} \\pi _ t ( b ) = \\pi ( b \\delta _ t ) , b \\in B _ t \\end{align*}"}
+{"id": "6747.png", "formula": "\\begin{align*} \\lim _ { K \\to \\infty } \\overline { d } _ { ( \\ell _ { i } ) } ( \\{ n \\in \\N : ( \\underline { \\eta } _ { K } ( n ) , \\eta _ { K } ( n ) ) \\neq ( \\eta ^ * ( n ) , \\eta ( n ) ) \\} ) = 0 . \\end{align*}"}
+{"id": "3496.png", "formula": "\\begin{align*} \\mathrm { i n d } ( H _ b ) = \\sum _ { \\lambda < 0 } \\mathrm { n u l } ( H _ b - \\lambda I ) = \\sum _ { \\lambda \\in ( \\lambda _ 0 , 0 ) } \\mathrm { n u l } ( H _ b - \\lambda I ) = \\sum _ { t \\in ( a , b ) } \\mathrm { n u l } ( H _ t ) , \\end{align*}"}
+{"id": "7768.png", "formula": "\\begin{align*} z ^ { c \\odot \\vec x } _ j & = \\begin{cases} ( M ( c \\odot \\vec x ) _ { i } - \\max _ k ( M ( c \\odot \\vec x ) _ k ) = M ( \\vec x ) _ { i } - \\max _ k ( M \\vec x ) _ k & j \\in J _ i \\\\ 0 \\qquad & j \\notin \\cup _ i J _ i , \\end{cases} \\end{align*}"}
+{"id": "8503.png", "formula": "\\begin{align*} \\zeta _ { p , p ^ { \\prime } } ( s , c ) : = \\sum _ { m , n \\neq 0 } \\frac { \\left ( p ^ { 2 } + \\lambda _ { n } ^ { 2 } \\right ) \\cdot \\left ( p ^ { \\prime 2 } + \\lambda _ { n } ^ { \\prime 2 } \\right ) } { \\left ( p ( p + \\frac { 1 } { \\pi } ) + \\lambda _ { n } ^ { 2 } \\right ) \\cdot \\left ( p ^ { \\prime } ( p ^ { \\prime } + \\frac { 1 } { \\pi } ) + \\lambda _ { n } ^ { \\prime 2 } \\right ) } \\ , \\frac { 1 } { \\left ( \\lambda _ { m } ^ { 2 } + c \\lambda _ { n } ^ { \\prime 2 } \\right ) ^ { s } } , \\ , \\ , \\ , \\ , ( s ) > 1 , \\ , \\ , c > 0 , \\end{align*}"}
+{"id": "344.png", "formula": "\\begin{align*} \\| \\mathbf { f } _ n - \\mathbf { f } _ m \\| _ p ^ p = \\sum _ { x \\in V } \\mu ( x ) | \\mathbf { f } _ n ( x ) - \\mathbf { f } _ m ( x ) | ^ p < \\epsilon , \\quad \\forall n , m \\geq N . \\end{align*}"}
+{"id": "2877.png", "formula": "\\begin{align*} Z _ k = \\begin{cases} X _ { - ( \\epsilon _ i - \\epsilon _ { i + 1 } ) } ^ { p - 1 } X _ { \\epsilon _ { i + 1 } + \\epsilon _ j } & \\beta _ k = ( \\epsilon _ i + \\epsilon _ j ) \\in \\Phi _ { 1 } \\cr X _ { - ( \\epsilon _ i - \\epsilon _ { i + 1 } ) } X ^ { p - 1 } _ { \\epsilon _ { i + 1 } - \\epsilon _ j } & \\beta _ k = \\epsilon _ i - \\epsilon _ j \\in \\Phi _ 0 ^ + \\backslash \\Pi _ 0 \\end{cases} i < j . \\end{align*}"}
+{"id": "2033.png", "formula": "\\begin{align*} \\left ( \\det ( u _ { j \\bar k } ) \\right ) ^ { 1 / n } = \\psi ^ { 1 \\slash n } ( z , u ( z ) ) , \\end{align*}"}
+{"id": "654.png", "formula": "\\begin{align*} \\| P _ { E ^ { ( k ) } _ j } \\| \\leq C \\norm { Q ( k ) } ^ \\tau , & \\| P _ { U ^ { ( k ) } _ j } \\| \\leq C \\norm { Q ( k ) } ^ \\tau . \\end{align*}"}
+{"id": "9279.png", "formula": "\\begin{align*} \\widetilde { S } _ { \\alpha , \\beta } ^ { \\ , \\log } f ( x ) & = x ^ { - \\beta } S _ { \\alpha , \\beta } ^ { \\log } \\big ( ( \\cdot ) ^ { \\beta } f \\big ) ( x ) \\\\ & \\simeq x ^ { - \\beta } \\sup _ { t > 2 x } \\int _ { t / 2 } ^ { t } ( t + x - z ) ^ { - \\alpha - 1 / 2 } \\log \\Big ( 2 + \\frac { x } { t - z } \\Big ) \\abs { f ( z ) } \\ , d z , x > 0 . \\end{align*}"}
+{"id": "3975.png", "formula": "\\begin{align*} \\begin{aligned} ( f _ { \\mathcal { S } } ) _ \\lambda ( s _ 1 , s _ 2 ) & \\leq \\sup _ { ( y _ 1 ^ \\prime , y _ 2 ^ \\prime , x ^ \\prime ) \\in \\mathcal { S } } \\left \\{ f _ 1 ( y _ 1 ^ \\prime ) + f _ 2 ( y _ 2 ^ \\prime ) - \\sum _ { 1 \\leq \\ell \\leq 2 } \\lambda _ \\ell c _ { Y _ \\ell } \\left ( y _ \\ell , y _ \\ell ^ \\prime \\right ) \\right \\} \\\\ & = ( f _ 1 ) _ { \\lambda _ 1 } ( y _ 1 ) + ( f _ 2 ) _ { \\lambda _ 2 } ( y _ 2 ) . \\end{aligned} \\end{align*}"}
+{"id": "1633.png", "formula": "\\begin{align*} ( N ^ { \\ , ( 1 ) } ( X , Y ) ) ^ \\bot = 2 \\ , g ( X , f \\widetilde { Q } Y ) \\ , \\bar \\xi \\ , ; \\end{align*}"}
+{"id": "2827.png", "formula": "\\begin{align*} a _ { i } : = \\frac { b _ { i - p + 1 } } { a _ { i - p + 1 } \\cdots a _ { i - 1 } } , i \\ge I + p , \\end{align*}"}
+{"id": "2935.png", "formula": "\\begin{align*} \\tilde { \\mu } _ \\delta \\{ \\gamma r \\to \\gamma s \\} ( \\gamma U ) = \\tilde { \\mu } _ \\delta \\{ r \\to s \\} ( U ) . \\end{align*}"}
+{"id": "5338.png", "formula": "\\begin{align*} \\left ( \\frac { 3 3 r } { \\sqrt { d } } \\right ) ^ d \\gamma _ 1 \\log ^ { d / 2 } \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) = \\gamma ' _ d < \\| f ^ { = d } \\| _ 2 \\leq \\left ( \\frac { 3 3 \\sqrt { d } } { d \\log ^ { 1 / 2 } \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) } \\right ) ^ { d } \\gamma _ 1 \\log ^ { d } \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) , \\end{align*}"}
+{"id": "6143.png", "formula": "\\begin{align*} d ( x , ( 1 - t ) x + t y ) = t ^ 2 \\max _ { 0 \\le j \\le n - 1 } ( x - y ) ^ 2 \\cdot ( ( 1 - t ) x + t y ) ^ j \\cdot x ^ { n - 1 - j } \\lesssim t ^ 2 d ( x , y ) \\end{align*}"}
+{"id": "3725.png", "formula": "\\begin{align*} \\sigma _ Y ^ * P & \\simeq \\{ ( v , y ) \\in P \\times Y \\ , \\mid \\ , h ( v ) = \\sigma _ Y ( y ) \\} \\\\ & = \\{ ( v , y ) \\ , \\mid \\ , y = ( \\sigma _ Y \\circ h ) ( v ) \\} \\\\ & = \\{ ( v , \\left ( \\sigma _ Y \\circ h ) ( v ) \\right ) \\ , \\mid \\ , v \\in P \\} \\end{align*}"}
+{"id": "150.png", "formula": "\\begin{align*} \\det ( I - \\mathcal { P } _ \\gamma ^ n | _ { E _ s } ) > 0 , ( - 1 ) ^ { s + n } \\det ( I - \\mathcal { P } _ \\gamma ^ n | _ { E _ s } ) = \\det ( I - \\mathcal { P } _ \\gamma ^ { - n } | _ { E _ s } ) ( - 1 ) ^ n \\det ( \\mathcal { P } _ \\gamma ^ n | _ { E _ s } ) > 0 . \\end{align*}"}
+{"id": "4266.png", "formula": "\\begin{align*} F \\left ( \\xi , r , p , P \\right ) = - \\mathcal { H } \\left ( \\xi , r , p , P \\right ) . \\end{align*}"}
+{"id": "2022.png", "formula": "\\begin{align*} 0 \\geq \\sum _ { p = 1 } ^ n u ^ { p \\bar { p } } G _ { p \\bar { p } } = \\sum _ { p = 1 } ^ n u ^ { p \\bar { p } } \\left ( \\frac { \\beta _ { p \\bar { p } } } { \\beta } - \\frac { | \\beta _ p | ^ 2 } { \\beta ^ 2 } + u u _ { p \\bar p } + \\vert u _ p \\vert ^ 2 + B \\right ) . \\end{align*}"}
+{"id": "1738.png", "formula": "\\begin{align*} \\log \\mu _ t ^ { ( n ) } ( y ) & = e ^ { - \\frac { \\sigma ^ 2 } { 2 } t } \\log \\mu _ 0 ( y ) \\\\ & + \\int _ 0 ^ t \\frac { \\sigma ^ 2 } { 2 } e ^ { - \\frac { \\sigma ^ 2 } { 2 } ( t - s ) } \\left ( \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta \\mu } ( \\nu _ s ^ { ( n - 1 ) } , \\mu _ s ^ { ( n - 1 ) } , y ) + \\log \\rho ( y ) + \\operatorname { D _ { K L } } ( \\mu _ s ^ { ( n - 1 ) } | \\rho ) \\right ) \\mathrm { d } s . \\end{align*}"}
+{"id": "3160.png", "formula": "\\begin{align*} \\tau ( \\alpha ^ * _ g ( x ) Y ) = \\tau ( x \\alpha _ g ( Y ) ) x \\in M Y \\in L ^ 1 ( M , \\tau ) . \\end{align*}"}
+{"id": "8289.png", "formula": "\\begin{align*} S ( a , b , c ) = \\langle x _ 1 , x _ 2 , x _ 3 \\rangle / ( a x _ 1 ^ 2 + b x _ 2 x _ 3 + c x _ 3 x _ 2 , a x _ 2 ^ 2 + b x _ 3 x _ 1 + c x _ 1 x _ 3 , a x _ 3 ^ 2 + b x _ 1 x _ 2 + c x _ 2 x _ 1 ) \\end{align*}"}
+{"id": "7625.png", "formula": "\\begin{align*} H _ 3 ( 1 ) ( f ^ { - 1 } ) = \\frac { 1 } { 9 2 1 6 } ( 1 7 c _ 1 ^ 6 - 1 0 2 c _ 1 ^ 4 c _ 2 + & 3 2 c _ 1 ^ 3 c _ 3 + 1 8 0 c _ 1 ^ 2 c _ 2 ^ 2 - 1 4 4 c _ 1 ^ 2 c _ 4 + 1 9 2 c _ 1 c _ 2 c _ 3 - 2 1 6 c _ 2 ^ 3 \\quad \\quad \\quad & \\\\ + 2 8 8 c _ 2 c _ 4 - 2 5 6 c _ 3 ^ 2 ) \\end{align*}"}
+{"id": "1650.png", "formula": "\\begin{align*} 2 \\ , g ( ( \\nabla _ { \\xi _ i } { f } ) Y , \\xi _ j ) \\overset { \\eqref { 3 . 1 A A } } = N ^ { \\ , ( 5 ) } ( \\xi _ i , Y , \\xi _ j ) \\overset { \\eqref { K K } } = 0 . \\end{align*}"}
+{"id": "2823.png", "formula": "\\begin{align*} a _ i = \\frac { \\Delta _ { i + 1 } \\Delta _ { i - 1 } } { \\Delta _ i ^ 2 } , b _ i = \\frac { \\Delta _ { i + p } \\Delta _ { i - 1 } } { \\Delta _ { i + p - 1 } \\Delta _ i } , i \\ge 0 , \\end{align*}"}
+{"id": "1148.png", "formula": "\\begin{align*} T ( f ^ { 2 } ) = 2 f T ( f ) + 2 B ( A ( f ) , A ( f ) ) \\end{align*}"}
+{"id": "4933.png", "formula": "\\begin{align*} \\left [ U \\right ] _ { { i , k } } = \\begin{cases} \\binom { n - i } { n - k } & \\textnormal { i f } \\ , i \\le k ; \\\\ 0 & \\textnormal { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "8842.png", "formula": "\\begin{align*} \\overline { v } _ \\lambda ( t ) - \\overline { v } _ \\mu ( t ) & = \\int _ 0 ^ t S ( t - s ) \\bigl ( F _ \\lambda ( s ) \\overline { v } _ \\lambda ( s ) - F _ \\mu ( s ) \\overline { v } _ \\mu ( s ) \\bigr ) \\ , d s \\\\ & + \\int _ 0 ^ t S ( t - s ) \\sigma ' ( u ( s ) ) \\bigl ( \\overline { v } _ \\lambda ( s ) - \\overline { v } _ \\mu ( s ) \\bigr ) B \\ , d W ( s ) , \\end{align*}"}
+{"id": "2800.png", "formula": "\\begin{align*} \\| v _ t \\| _ { L ^ 2 ( \\mathrm { M } _ 0 ) } ^ 2 = ( v _ t | v _ t ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } = ( v _ t + T g _ t | v _ t ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } = ( u | v _ t ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } . \\end{align*}"}
+{"id": "279.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( 1 - y ^ m z ^ n \\right ) ^ { \\frac { m ^ 3 } { n ^ 4 } } = \\left ( \\frac { 1 } { 1 - y z } \\right ) ^ { \\frac { y } { 1 - y } } \\end{align*}"}
+{"id": "6328.png", "formula": "\\begin{align*} g ( t ) = C ( 1 + O ( \\varepsilon ) ) f ( t ) , \\forall \\ , t \\in [ - \\rho , \\rho ] . \\end{align*}"}
+{"id": "1431.png", "formula": "\\begin{align*} \\bar { d } = ( d _ 1 , \\cdots , d _ k ) \\in \\R ^ k _ + , \\ \\bar { \\sigma } = ( \\sigma _ 1 , \\cdots , \\sigma _ k ) \\in ( \\R _ + ^ { n } ) ^ k . \\end{align*}"}
+{"id": "1401.png", "formula": "\\begin{align*} \\left ( \\Omega ^ { n - 1 } \\right ) ^ { ( n - 1 , n - 1 ) } = \\sum _ { k = 0 } ^ { \\lfloor \\frac { n - 1 } { 2 } \\rfloor } \\binom { n - 1 } { k } ( \\Omega ^ { ( 2 , 0 ) } \\wedge \\Omega ^ { ( 0 , 2 ) } ) ^ k \\wedge ( \\Omega ^ { ( 1 , 1 ) } ) ^ { n - 2 k - 1 } . \\end{align*}"}
+{"id": "7922.png", "formula": "\\begin{align*} { \\rm U } ( L ) = \\bigoplus _ { \\nu } { \\rm U } _ \\mathbf { \\nu } , \\end{align*}"}
+{"id": "3371.png", "formula": "\\begin{align*} P _ { m } : = \\sum _ { i = 0 } ^ { m } p _ { i } \\to \\infty \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , Q _ { n } : = \\sum _ { j = 0 } ^ { n } q _ { j } \\to \\infty \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , m , n \\to \\infty . \\end{align*}"}
+{"id": "8497.png", "formula": "\\begin{align*} \\intop _ { 0 } ^ { \\infty } \\frac { x ^ { \\nu + 1 } J _ { \\nu } ( b x ) } { \\left ( x ^ { 2 } + a ^ { 2 } \\right ) ^ { \\mu + 1 } } \\ , d x = \\frac { a ^ { \\nu - \\mu } b ^ { \\mu } } { 2 ^ { \\mu } \\Gamma ( \\mu + 1 ) } K _ { \\nu - \\mu } ( a b ) , \\ , \\ , \\ , \\ , \\ , a , b > 0 , \\ , \\ , \\ , - 1 < ( \\nu ) < 2 \\ , ( \\mu ) + \\frac { 3 } { 2 } . \\end{align*}"}
+{"id": "6041.png", "formula": "\\begin{align*} z _ 1 : = \\frac { - 1 - \\sqrt { 1 3 } } { 2 \\sqrt { 1 2 } } \\approx - 0 . 6 6 4 \\textrm { a n d } z _ 2 : = \\frac { - 1 + \\sqrt { 1 3 } } { 2 \\sqrt { 1 2 } } \\approx 0 . 3 7 6 . \\end{align*}"}
+{"id": "2627.png", "formula": "\\begin{align*} \\begin{cases} ( \\bar { \\mathbf { z } } , t _ 1 ) \\in \\mathcal { E } , \\\\ ( \\bar { \\mathbf { z } } + \\mathbf { p } ( t _ 1 ) - \\mathbf { p } ( t _ 2 ) , t _ 2 ) \\in \\mathcal { E } , \\\\ ( \\bar { \\mathbf { z } } + \\mathbf { p } ( t _ 1 ) - \\mathbf { p } ( t _ 2 ) , t _ 3 ) \\in \\mathcal { E } , \\end{cases} \\end{align*}"}
+{"id": "9110.png", "formula": "\\begin{align*} y = ( \\underbrace { y ^ { 1 } , \\ldots , y ^ { m _ { 1 } } } _ { y _ { 1 } } , \\underbrace { y ^ { m _ { 1 } + 1 } , \\ldots , y ^ { m } } _ { y _ { 2 } } ) \\ , . \\end{align*}"}
+{"id": "3615.png", "formula": "\\begin{align*} q \\ = \\ p - v ( f ) \\ \\geq \\ 1 0 ^ m + 3 - 3 4 \\ = \\ 1 0 ^ m - 3 1 , \\end{align*}"}
+{"id": "5055.png", "formula": "\\begin{align*} f _ \\Theta ( a ) = \\underbrace { \\sum _ { i \\in S ( \\eta ) } [ \\Theta ^ { - 1 } y ] _ i U ( a ) _ i } _ { \\eqqcolon f _ 1 ( \\eta , a ) } + \\underbrace { \\sum _ { i \\notin S ( \\eta ) } [ \\Theta ^ { - 1 } y ] _ i U ( a ) _ i } _ { \\eqqcolon f _ 2 ( \\eta , a ) } . \\end{align*}"}
+{"id": "5184.png", "formula": "\\begin{align*} \\{ p t \\} = X ^ 0 \\subset X ^ 1 \\subset X ^ 2 \\subset \\dots \\subset X ^ m . \\end{align*}"}
+{"id": "4553.png", "formula": "\\begin{align*} { \\mathcal { B } = \\{ \\ell ^ i z _ k \\mid 1 \\leq k \\leq s , \\ , 0 \\leq i \\leq p _ k - 1 \\} } , \\end{align*}"}
+{"id": "894.png", "formula": "\\begin{align*} d _ 3 ' = e _ 3 ' f _ 3 ' , \\quad f ' _ 3 = \\prod _ { \\nu _ \\varpi ( d _ 3 ' ) = \\nu _ { \\varpi } ( e _ 3 ) + 1 } \\varpi , \\end{align*}"}
+{"id": "4739.png", "formula": "\\begin{align*} \\tilde { S } ^ T ( A + H ) \\tilde { S } = \\tilde { D } \\oplus \\tilde { D } , \\end{align*}"}
+{"id": "557.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } W _ { \\varepsilon } ^ { \\prime } ( t , \\xi ) = i \\langle \\xi \\rangle A _ { \\varepsilon } ( t ) W _ { \\varepsilon } ( t , \\xi ) + i \\langle \\xi \\rangle ^ { - 1 } Q _ { \\varepsilon } ( t ) W _ { \\varepsilon } ( t , \\xi ) + G _ { \\varepsilon } ( t , \\xi ) , \\quad \\xi \\in \\mathcal { I } _ { \\hbar } , \\\\ W _ { \\varepsilon } ( 0 , \\xi ) = 0 , \\xi \\in \\mathcal { I } _ { \\hbar } , \\end{array} \\right . \\end{align*}"}
+{"id": "712.png", "formula": "\\begin{align*} \\Phi _ { S } ( e ) = ( \\phi _ { F } + \\phi _ { A } ) ( z _ { S } ( e ) ) \\ , , \\end{align*}"}
+{"id": "8100.png", "formula": "\\begin{align*} \\mathbf { A } _ { 3 k + i + j - 1 } : = \\xi _ k \\left ( \\mathbf { e } _ i \\otimes \\mathbf { e } _ j - \\mathbf { e } _ j \\otimes \\mathbf { e } _ i \\right ) . \\end{align*}"}
+{"id": "1973.png", "formula": "\\begin{align*} n ^ { - 1 } \\left | \\sum _ { p = 1 } ^ n u ^ { p \\bar p } u _ { p \\bar p \\bar j } \\right | ^ 2 - \\sum _ { p , q = 1 } ^ n u ^ { p \\bar p } u ^ { q \\bar q } | u _ { p \\bar q j } | ^ 2 \\leq n ^ { - 1 } \\left | \\sum _ { p = 1 } ^ n u ^ { p \\bar p } u _ { p \\bar p \\bar j } \\right | ^ 2 - \\sum _ { p = 1 } ^ n | u ^ { p \\bar p } u _ { p \\bar p j } | ^ 2 \\leq 0 . \\end{align*}"}
+{"id": "7405.png", "formula": "\\begin{align*} \\sigma _ { \\beta _ N } ^ 2 = \\frac { 1 } { R _ N } \\bigg ( 1 - \\frac { \\theta _ N } { \\log N } \\bigg ) 1 \\ll \\theta _ N \\ll \\log N \\ , . \\end{align*}"}
+{"id": "7330.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { N _ 1 } U _ k ^ { e _ k } = \\frac { m } { C } \\end{align*}"}
+{"id": "7586.png", "formula": "\\begin{align*} g ( z ) = z \\exp \\left ( \\int _ { 0 } ^ { z } \\frac { x ^ 2 + \\sqrt { 1 + x ^ 4 } - 1 } { x } d x \\right ) = z + \\frac { z ^ 3 } { 2 } + \\frac { z ^ 5 } { 4 } + \\cdots . \\end{align*}"}
+{"id": "1327.png", "formula": "\\begin{align*} \\frac { 1 } { \\theta _ i ^ 2 } \\frac { e _ i ^ * ( u ) ^ 2 } { T _ i ^ * ( u ) ^ 2 } = \\frac { e ^ { 2 \\rho _ i ( u ) } } { T _ i ( u ) ^ 2 } . \\end{align*}"}
+{"id": "8096.png", "formula": "\\begin{gather*} \\boldsymbol { \\Phi } ( \\mathbf { a } , t ) = \\mathbf { a } , \\ , \\mathbf { U } ( \\mathbf { a } , t ) = \\mathbf { 0 } , \\ , \\mathbf { R } ( \\mathbf { a } , t ) \\mathbf { = I } , \\ , \\mathbf { W } ( \\mathbf { a } , t ) \\mathbf { = I , \\ } , \\ , \\mathbf { Z } ( \\mathbf { a } , t ) = \\mathbf { 0 . } \\end{gather*}"}
+{"id": "7090.png", "formula": "\\begin{align*} \\inf _ { n \\in \\N } \\alpha _ n = \\inf _ { n \\in \\N } \\beta _ n . \\end{align*}"}
+{"id": "1681.png", "formula": "\\begin{align*} \\psi ( G ) = \\pi ( G ) . \\end{align*}"}
+{"id": "2712.png", "formula": "\\begin{align*} \\left \\| P _ 1 u \\right \\| ^ 2 + \\left \\| P _ 2 u \\right \\| ^ 2 + 2 \\left ( P _ 1 u , P _ 2 u \\right ) = \\left \\| e ^ { - s \\sigma } f \\right \\| ^ 2 . \\end{align*}"}
+{"id": "2272.png", "formula": "\\begin{align*} w _ b = \\sum _ { n } c _ n a _ n + T ( f ) _ b . \\end{align*}"}
+{"id": "3669.png", "formula": "\\begin{align*} \\mathcal { M } _ { F _ 1 , \\alpha } f = \\mathcal { M } _ { F _ 2 , \\alpha } f \\end{align*}"}
+{"id": "2650.png", "formula": "\\begin{align*} S _ { \\alpha } ^ k ( X ) : = \\{ s \\in C ^ { 2 k - 2 } [ a , b ] : ( D ^ 2 - \\alpha ^ 2 ) ^ k s = 0 [ a , b ] \\setminus X \\} . \\end{align*}"}
+{"id": "5471.png", "formula": "\\begin{align*} Q ( r _ 1 , \\theta ) \\approx & \\frac { \\pi ^ 2 \\lambda R _ S ( R _ { m a x } - r _ 1 ) } { N R _ E } \\sum _ { n = 1 } ^ N \\sqrt { 1 - \\phi _ n ^ 2 } c _ n \\\\ & \\times \\left ( 1 - \\prod _ { m = 1 } ^ { M } { \\left ( 1 + \\frac { m \\eta \\theta r _ 1 ^ { \\alpha } } { M c _ n ^ { \\alpha } } \\right ) ^ { - M b _ m } } \\right ) \\end{align*}"}
+{"id": "39.png", "formula": "\\begin{align*} \\lambda ( X ' ) = ( \\varphi _ { A ( X ) } ^ { - 1 } \\varphi _ { A ( X ' ) } ) ^ \\vee \\lambda ( X ) ( \\varphi _ { A ( X ) } ^ { - 1 } \\varphi _ { A ( X ' ) } ) , \\psi ^ { \\square } ( X ' ) = \\psi ( X ) ^ { \\square } ( \\varphi _ { A ( X ) } ^ { - 1 } \\varphi _ { A ( X ' ) } ) . \\end{align*}"}
+{"id": "2101.png", "formula": "\\begin{align*} F ( A ) & = \\left ( m ( 2 ^ k - 1 ) + 2 ^ { k - 1 } - 1 + d - 2 + m + 2 \\right ) \\cdot ( m ( 2 ^ k - 1 ) + 2 ^ { k - 1 } - 1 ) - d \\\\ & = \\left ( ( 2 m + 1 ) 2 ^ { k - 1 } - 1 \\right ) ^ 2 + ( d - m ) \\left ( ( 2 m + 1 ) 2 ^ { k - 1 } - 1 \\right ) - m d - d . \\end{align*}"}
+{"id": "2572.png", "formula": "\\begin{align*} \\lim _ { s \\to 1 } \\sup \\left \\{ d ( x ) : s \\le u ( x ) < 1 \\right \\} = 0 . \\end{align*}"}
+{"id": "8816.png", "formula": "\\begin{align*} S \\ast g : = \\int _ 0 ^ \\cdot S ( \\cdot - s ) g ( s ) \\ , d s S \\diamond G : = \\int _ 0 ^ \\cdot S ( \\cdot - s ) G ( s ) \\ , d W ( s ) . \\end{align*}"}
+{"id": "31.png", "formula": "\\begin{align*} R : = \\mathrm { E n d } _ { \\kappa } ( E ) = \\mathrm { E n d } _ { \\overline { \\mathbb { F } } _ \\ell } ( E ) . \\end{align*}"}
+{"id": "8439.png", "formula": "\\begin{align*} - \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\cdot \\frac { 1 } { e ^ { \\lambda _ { n } ^ { 2 } x } + 1 } = \\frac { 1 } { 4 \\left ( 1 + \\frac { 1 } { \\pi p } \\right ) } + \\frac { 1 } { 2 } \\sqrt { \\frac { \\pi } { x } } \\left ( \\sqrt { 2 } - 1 \\right ) \\zeta \\left ( \\frac { 1 } { 2 } \\right ) + \\sqrt { \\frac { \\pi } { 2 x } } \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { G _ { p } \\left ( \\frac { \\pi ( 2 n - 1 ) } { x } \\right ) } { \\sqrt { 2 n - 1 } } , \\end{align*}"}
+{"id": "1339.png", "formula": "\\begin{align*} \\partial _ { t } \\mu ( t , x ) - \\mathrm { d i v } ( \\mu \\nabla V \\star \\mu ) ( t , x ) = 0 , \\mu ( 0 , \\cdot ) = \\mu ^ { i n } \\end{align*}"}
+{"id": "4294.png", "formula": "\\begin{align*} \\theta - \\ln \\theta - 1 = \\frac { 1 } { 2 \\xi ^ 2 } ( \\theta - 1 ) ^ 2 , \\\\ \\rho \\ln \\rho - \\rho + 1 = \\frac { 1 } { 2 \\eta } ( \\rho - 1 ) ^ 2 , \\end{align*}"}
+{"id": "1813.png", "formula": "\\begin{align*} D _ k ( t ) = \\int _ { \\Lambda ^ { * } } f ^ { ( 1 ) } ( t , k , p ) a _ { p - k } ^ * a _ p \\d p + \\int _ { \\Lambda ^ { * } } f ^ { ( 2 ) } ( t , k , h ) a _ { h + k } ^ * a _ h \\d h \\end{align*}"}
+{"id": "6536.png", "formula": "\\begin{align*} \\int _ 0 ^ x t ^ \\mu K _ \\nu ( a t ) \\ , \\mathrm { d } t & = \\frac { 2 ^ { \\mu - 1 } } { a ^ \\mu } \\Gamma \\bigg ( \\frac { \\mu - \\nu + 1 } { 2 } \\bigg ) \\Gamma \\bigg ( \\frac { \\mu + \\nu + 1 } { 2 } \\bigg ) G _ { \\mu , \\nu } ( a x ) , \\\\ \\int _ x ^ \\infty t ^ \\mu K _ \\nu ( a t ) \\ , \\mathrm { d } t & = \\frac { 2 ^ { \\mu - 1 } } { a ^ \\mu } \\Gamma \\bigg ( \\frac { \\mu - \\nu + 1 } { 2 } \\bigg ) \\Gamma \\bigg ( \\frac { \\mu + \\nu + 1 } { 2 } \\bigg ) \\tilde { G } _ { \\mu , \\nu } ( a x ) , \\end{align*}"}
+{"id": "5203.png", "formula": "\\begin{align*} L _ F & = \\{ x \\in X : x ^ \\in \\subseteq F \\} = X \\setminus \\bigcup ( C \\setminus F ) . \\\\ X _ F & = \\{ x \\in X : x ^ \\in = F \\} = L _ F \\setminus \\bigcup _ { G \\subsetneqq F } L _ G . \\end{align*}"}
+{"id": "9201.png", "formula": "\\begin{align*} \\sum _ { u \\in V , u \\not = v } \\left ( d ( v , u ) - 1 \\right ) + \\sum _ { 1 \\le i \\le r ' } \\left ( d ( u _ i , v ) - 1 \\right ) \\ge ( r - 1 ) ^ 2 \\end{align*}"}
+{"id": "8330.png", "formula": "\\begin{align*} y _ { D 2 } ^ { } = \\left \\{ \\begin{array} { c c c c c c c c c c } h _ { C D } z + n _ { D 2 } , & \\mbox { i f } \\hat { x } = 1 ; \\\\ \\sqrt { 2 - \\alpha } h _ { C D } e ^ { \\iota \\frac { \\pi } { M } } z + n _ { D 2 } , & \\mbox { i f } \\hat { x } = 0 ; \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "4437.png", "formula": "\\begin{align*} L ( \\Phi ) = \\frac { d + 2 } { 4 - d } \\omega Q ( \\Phi ) + \\frac { d - 1 } { 4 - d } \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) , \\ \\ \\ \\ \\ \\ N ( \\Phi ) = - \\frac { 2 } { 4 - d } \\left ( 2 \\omega Q ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) \\right ) . \\end{align*}"}
+{"id": "5755.png", "formula": "\\begin{align*} \\sup _ { Q } \\Big ( \\frac { 1 } { | Q | } \\int _ { Q } u _ { \\vec { w } } ( x ) d x \\Big ) ^ { 1 / p } \\prod _ { j = 1 } ^ { m } \\Big ( \\frac { 1 } { | Q | } \\int _ { Q } \\big ( w _ { j } ( x ) \\big ) ^ { 1 - p ' _ { j } } d x \\Big ) ^ { 1 / p ' _ { j } } & < \\infty , \\end{align*}"}
+{"id": "4897.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s u = V ( x ) u & , \\\\ u = 0 & , \\end{cases} u > 0 \\ ; { } . \\end{align*}"}
+{"id": "6668.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s V ( x ) = \\vartheta | x | ^ { - \\beta - 2 s } \\quad , \\end{align*}"}
+{"id": "7941.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) \\Pi _ { e _ { ( \\xi , 0 ) } + e _ { ( \\xi , e _ 0 ) } } = \\Pi _ { e _ { ( \\xi , 0 ) } } \\xi + c _ { e _ { ( \\xi , 0 ) } + e _ { ( \\xi , e _ 0 ) } } , \\end{align*}"}
+{"id": "2574.png", "formula": "\\begin{align*} - \\Delta u = f ( u ) \\ \\ \\ \\ \\ , D . \\end{align*}"}
+{"id": "4252.png", "formula": "\\begin{align*} Y _ { t } ^ { i } = \\zeta ^ { i } + \\int _ { t } ^ { T } g ^ { i } \\left ( Y _ { s } ^ { i } , Z _ { s } ^ { i } , \\mathbb { P } _ { \\left ( Y _ { s } ^ { i } , Z _ { s } ^ { i } \\right ) } \\right ) \\mathrm { d } s - \\int _ { t } ^ { T } Z _ { s } ^ { i } W _ { s } \\mathrm { d } s , i = 1 , 2 , \\end{align*}"}
+{"id": "3947.png", "formula": "\\begin{align*} \\Sigma _ { \\mathrm { D } } ( \\delta ) = \\Pi \\left ( \\mathcal { B } _ 1 , \\mathcal { B } _ 2 \\right ) , \\mathcal { B } _ 1 = B _ { p _ 1 } ( \\mu _ 1 , \\delta _ 1 ^ { 1 / p _ 1 } ) \\mathcal { B } _ 2 = B _ { p _ 2 } ( \\mu _ 2 , \\delta _ 2 ^ { 1 / p _ 2 } ) . \\end{align*}"}
+{"id": "8893.png", "formula": "\\begin{align*} X _ 1 \\cdot X _ 2 = X _ 1 + X _ 2 + \\frac { 1 } { 2 } [ X _ 1 , X _ 2 ] + \\sum _ { p \\geq 3 } \\sum _ { ( i _ 1 , \\dots , i _ p ) \\in \\{ 1 , 2 \\} ^ p } c _ { ( i _ 1 , \\dots , i _ p ) } [ X _ { i _ 1 } , \\dots , X _ { i _ p } ] . \\end{align*}"}
+{"id": "2640.png", "formula": "\\begin{align*} H _ 2 ( t , x , y ) = 2 ^ { \\sigma _ 2 l + k _ 2 } \\int _ { n _ 2 2 ^ { - ( \\sigma _ 2 l + k _ 2 ) } + P _ 2 ( 2 ^ { - l } t ) } ^ { ( n _ 2 + 1 ) 2 ^ { - ( \\sigma _ 2 l + k _ 2 ) } + P _ 2 ( 2 ^ { - l } t ) } | f _ 2 | \\left ( x , y + v \\right ) \\ , \\mathrm { d } v . \\end{align*}"}
+{"id": "1896.png", "formula": "\\begin{align*} \\mathcal { X } = S ( \\mathcal { U } ) - \\mathcal { K } _ { K , \\zeta ^ * } , \\end{align*}"}
+{"id": "3384.png", "formula": "\\begin{align*} \\lim _ { \\lambda , \\kappa \\to 1 ^ + } \\limsup _ { m , n \\to \\infty } \\max _ { \\substack { m \\leq i \\leq m ^ \\lambda \\\\ n \\leq j \\leq n ^ \\kappa } } | u _ { i j } - u _ { i n } | = 0 , \\end{align*}"}
+{"id": "806.png", "formula": "\\begin{align*} \\| \\theta ^ { n } \\| & \\le C \\ , \\big ( h ^ 2 + N ^ { - \\min \\{ 2 , \\ , r \\alpha \\} } \\big ) , \\\\ \\end{align*}"}
+{"id": "8376.png", "formula": "\\begin{align*} L _ { 0 , u v } = \\begin{cases} | \\{ \\mbox { n o n - l o o p e d g e s a t } u \\} | + 4 | \\{ \\mbox { o d d l o o p s a t } u \\} | + 2 | \\{ \\mbox { o d d l e g s a t } u \\} | + | \\mbox { n u l l l e g s a t } u \\} | , & u = v , \\\\ | \\{ \\mbox { o d d e d g e s b e t w e e n } u \\mbox { a n d } v \\} | - | \\{ \\mbox { e v e n e d g e s b e t w e e n } u \\mbox { a n d } v \\} | , & u \\neq v . \\end{cases} \\end{align*}"}
+{"id": "408.png", "formula": "\\begin{align*} P = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} , A = \\begin{bmatrix} u & 0 & 1 \\\\ 0 & u & 0 \\\\ 1 & 0 & 0 \\end{bmatrix} , B = \\begin{bmatrix} v & 0 & 0 \\\\ 0 & v & 1 \\\\ 0 & 1 & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "3054.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} M _ 1 & 0 & N _ 1 ^ T \\\\ 0 & M _ 2 & N _ 2 ^ T \\\\ N _ 1 & N _ 2 & M _ 3 \\end{matrix} \\right ] \\end{align*}"}
+{"id": "1468.png", "formula": "\\begin{align*} H _ 1 \\leq & C \\Big ( | \\phi | ^ p _ { \\frac { 2 n } { n + 2 } } + | U _ { \\mu _ i , \\xi _ i } ^ { p - 2 } \\phi ^ 2 | _ { \\frac { 2 n } { n + 2 } } \\Big ) \\leq C \\Big ( | \\phi | ^ { p - 2 } _ { \\frac { 2 n } { n - 2 } } + | U _ { \\mu _ i , \\xi _ i } | _ { p - 2 } ^ { p - 2 } \\Big ) | \\phi | ^ 2 _ { \\frac { 2 n } { n - 2 } } = C \\Big ( | \\phi | ^ { p - 2 } + 1 \\Big ) \\| \\phi \\| ^ 2 . \\end{align*}"}
+{"id": "1479.png", "formula": "\\begin{align*} & \\int _ { \\Omega \\setminus B ( \\xi , \\rho ) } \\Big | V ^ { p - 1 } - \\sum \\limits _ { i = 1 } ^ k ( - 1 ) ^ i ( P U _ { \\mu _ i , \\xi _ i } ) ^ { p - 1 } \\Big | ^ { \\frac { n } { 2 } } d x \\\\ \\leq & \\sum \\limits _ { i = 1 } ^ k \\int _ { \\Omega \\setminus B ( \\xi , \\rho ) } U _ { \\mu _ i , \\xi _ i } ^ { ( p - 1 ) \\frac { n } { 2 } } d x \\leq C \\sum \\limits _ { i = 1 } ^ k \\mu _ i ^ n = O \\bigg ( \\Big ( \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big ) ^ { \\frac { n } { n - 2 } } \\bigg ) . \\end{align*}"}
+{"id": "8710.png", "formula": "\\begin{align*} E \\{ ( U ^ \\top A U ) ^ 2 \\} & = \\sum _ { i _ 1 , j _ 1 } \\sum _ { i _ 2 , j _ 2 } a _ { i _ 1 j _ 1 } a _ { i _ 2 j _ 2 } E ( u _ { i _ 1 } u _ { i _ 2 } u _ { j _ 1 } u _ { j _ 2 } ) \\\\ & = \\sum _ i E ( u _ i ^ 4 ) a _ { i i } ^ 2 + 2 \\sum _ { i \\neq j } a _ { i j } ^ 2 + \\sum _ { i \\neq j } a _ { i i } a _ { j j } \\\\ & = \\sum _ i \\delta _ i a _ { i i } ^ 2 + 2 ( A ^ 2 ) + \\{ ( A ) \\} ^ 2 . \\end{align*}"}
+{"id": "6712.png", "formula": "\\begin{align*} q ( x , \\xi _ x ) : = \\sum _ { i , j = 1 } ^ d g ^ { i j } ( x ) \\xi _ i \\xi _ j \\geq C | \\xi _ x | ^ 2 , \\textnormal { f o r a l l } ( x , \\xi _ x ) \\in \\Omega _ x \\times \\R ^ d . \\end{align*}"}
+{"id": "1178.png", "formula": "\\begin{align*} ( \\phi \\cdot \\psi ) ( y ) = \\phi ( y ) \\cdot \\psi ( y ) . \\end{align*}"}
+{"id": "8048.png", "formula": "\\begin{align*} A = \\sum _ { j = 1 } ^ k \\lambda _ j ^ { \\uparrow } ( A ) P _ j \\end{align*}"}
+{"id": "156.png", "formula": "\\begin{align*} L i _ 2 ( z ) + L i _ 2 ( 1 - z ) = \\frac { \\pi ^ 2 } { 6 } - \\log ( z ) \\log ( 1 - z ) , \\end{align*}"}
+{"id": "112.png", "formula": "\\begin{align*} e ^ { - t _ 0 X } - e ^ { - i t _ 0 h ^ { - 1 } \\tilde { P } _ h ( 0 ) } = h ^ { - 1 } \\int _ 0 ^ { t _ 0 } e ^ { - ( t _ 0 - t ) X } ( Q _ \\infty + q _ 1 + W ) e ^ { - i t h ^ { - 1 } \\tilde { P } _ h ( 0 ) } d t . \\end{align*}"}
+{"id": "8750.png", "formula": "\\begin{align*} S T O ( z _ 1 ) : = \\left \\{ \\begin{array} { l } \\displaystyle \\min _ { x , y } \\ , \\ , \\mathbb { E } \\left [ \\theta \\left ( x , y , z _ 1 , \\zeta ( \\omega ) \\right ) \\right ] \\\\ \\left \\{ \\begin{array} { l l } x \\in X & \\\\ y \\in D ( x ) . & \\end{array} \\right . \\end{array} \\right . \\end{align*}"}
+{"id": "5793.png", "formula": "\\begin{align*} \\inf _ { | z | = r \\in J _ 1 } \\log | A ( z ) | \\geq r ^ \\gamma , \\end{align*}"}
+{"id": "4571.png", "formula": "\\begin{align*} \\Psi _ N ( t , x _ 1 , . . . , x _ N ) = \\Psi _ N ( t , x _ { \\sigma ( 1 ) } , . . . , x _ { \\sigma ( N ) } ) \\forall \\sigma \\in S _ N , \\ ; \\forall x _ i \\in \\R ^ 2 \\ , , \\end{align*}"}
+{"id": "8694.png", "formula": "\\begin{align*} \\textup { M M D } ^ 2 ( P _ X , P _ Y ) = O ( p ^ { - 2 r + 2 } ) . \\end{align*}"}
+{"id": "5809.png", "formula": "\\begin{align*} h = k M Y ^ { k - 1 } + R _ { k - 2 } ( Y , M ) . \\end{align*}"}
+{"id": "991.png", "formula": "\\begin{align*} \\gamma \\big ( ( n _ E ) _ { E \\in \\mathbf { E } } , ( s _ i ) _ { 1 \\le i \\le r } , ( \\ell _ { E , v } ) _ { v \\in E \\in \\mathbf { E } } \\big ) = { } & \\big ( ( s _ i ) _ { 1 \\le i \\le r } , ( \\ell _ { E , v _ E } + \\ell _ { E , v ' _ E } ) _ { E \\in \\mathbf { E } } \\big ) , \\\\ \\delta \\big ( ( n _ { E , v } ) _ { v \\in E \\in \\mathbf { E } } \\big ) = { } & \\big ( n _ { E , v _ E } - n _ { E , v ' _ E } \\big ) _ { E \\in \\mathbf { E } } . \\end{align*}"}
+{"id": "6989.png", "formula": "\\begin{align*} f _ i = \\sum _ { j = 1 } ^ { s _ i } a _ { i , j } \\textbf { Q } ^ { \\lambda _ { i , j } } \\end{align*}"}
+{"id": "5696.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { F } } _ { \\gamma } : = f ^ { + } - \\gamma E _ { g | V ( p ) } . \\end{align*}"}
+{"id": "8328.png", "formula": "\\begin{align*} y _ { B 2 } = \\sqrt { 2 - \\alpha } h _ { C B } z e ^ { \\frac { \\iota \\pi } { M } } + n _ { B 2 } . \\end{align*}"}
+{"id": "4742.png", "formula": "\\begin{align*} \\| \\tilde { S } \\| ^ 2 & = \\| ( A + H ) ^ { - 1 / 2 } ( A + H ) ^ { 1 / 2 } \\tilde { S } \\| ^ 2 \\\\ & \\leq \\| ( A + H ) ^ { - 1 / 2 } \\| ^ 2 \\| ( A + H ) ^ { 1 / 2 } \\tilde { S } \\| ^ 2 \\\\ & = \\| ( A + H ) ^ { - 1 } \\| \\| \\tilde { S } ^ T ( A + H ) \\tilde { S } \\| \\\\ & = 2 \\| ( A + H ) ^ { - 1 } \\| d _ { 1 } ( A + H ) \\\\ & \\leq 2 \\| ( A + H ) ^ { - 1 } \\| \\| A + H \\| = 2 \\kappa ( A + H ) , \\end{align*}"}
+{"id": "3382.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 1 ^ + } \\limsup _ { m , n \\to \\infty } \\max _ { m \\leq i \\leq m ^ \\lambda } | u _ { i n } - u _ { m n } | = 0 , \\end{align*}"}
+{"id": "910.png", "formula": "\\begin{align*} E _ 2 ' & \\ll \\widehat { P } ^ { n - 5 } \\widehat { Y } ^ { 3 + - n / 2 } \\widehat { Z } ^ { 3 - n / 2 } \\widehat { V } ^ { n ( 1 - 1 / n - 1 ) ) } \\\\ & = \\widehat { P } ^ { n / 3 - 5 } \\widehat { Y } ^ { 2 + n / 3 } \\widehat { Z } ^ { 3 + n / 6 } \\\\ & \\ll \\widehat { P } ^ { 1 1 n / 1 8 } \\widehat { Y } ^ { n / 6 - 1 } \\\\ & \\ll \\widehat { P } ^ { 8 n / 9 - 5 / 3 } \\end{align*}"}
+{"id": "4270.png", "formula": "\\begin{align*} g ( \\kappa ) = 0 \\Longleftrightarrow \\kappa = 0 . \\end{align*}"}
+{"id": "6905.png", "formula": "\\begin{align*} & \\frac { 2 } { \\pi } \\int _ { - ( \\log { t } ) ^ { 2 } } ^ { ( \\log { t } ) ^ { 2 } } \\log \\zeta ( \\sigma + i ( t + u ) ) \\left ( \\frac { \\sin { ( u / 2 ) } } { u } \\right ) ^ 2 ( 1 + \\cos ( \\theta + u \\log { x } ) ) d u \\\\ & = \\frac { 1 } { 2 } e ^ { i \\theta } \\sum _ { e ^ { - 1 / 2 } x \\le n \\le e ^ { 1 / 2 } x } \\frac { \\Lambda ( n ) } { n ^ { \\sigma + i t } \\log { n } } \\left ( \\frac { 1 } { 2 } - \\left | \\log \\frac { n } { x } \\right | \\right ) + O \\left ( \\frac { x } { ( \\log { t } ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "2024.png", "formula": "\\begin{align*} \\log \\det ( u _ { i \\bar { j } } ) = \\log \\psi ( z , u ) = : f ( z , u ) \\end{align*}"}
+{"id": "5196.png", "formula": "\\begin{align*} { \\bf { b } } \\widetilde { \\mathbb { S } } _ { \\lambda ^ i } { \\bf { b } } \\widetilde { \\mathbb { S } } _ { \\lambda ^ j } & = \\sum _ { \\lambda ^ q \\succeq \\lambda ^ i , \\lambda ^ j } \\sum _ { P : \\delta P = \\Delta _ { \\sigma _ r \\lambda ^ i ~ \\sigma _ r \\lambda ^ j } ^ { \\sigma _ r \\lambda ^ q } } \\sum _ { s = 0 } ^ { | P | } a ^ { ( P ) } _ s ( { \\bf { b } } \\widetilde { \\mathbb { S } } _ { \\lambda ^ 1 } ) ^ s { \\bf { b } } \\widetilde { \\mathbb { S } } _ { \\lambda ^ q } \\\\ \\end{align*}"}
+{"id": "4841.png", "formula": "\\begin{align*} ( \\prod \\limits _ { i = 1 } ^ m a ( i ) \\ast g _ w ( t ( i ) ) \\ast a ( m + 1 ) \\in A \\cap \\sigma ( L ) \\mbox { f o r a l l } g _ w \\in G . \\end{align*}"}
+{"id": "3344.png", "formula": "\\begin{align*} 1 = \\| T v \\| ^ 2 = \\langle \\psi _ { Y , Z } ( T ) v , T v \\rangle = \\langle ( I \\otimes T ) Z ^ * v , Y ^ * T v \\rangle . \\end{align*}"}
+{"id": "3193.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\norm { B _ l ' ( y _ 1 ' - B _ k ' ( y _ 1 ' ) ) } \\leq \\frac { 1 } { m ( I _ k ) } \\int _ { I _ k } \\lim _ { l \\to \\infty } \\frac { m ( I _ l \\Delta h I _ l ) } { m ( I _ l ) } d m ( h ) = 0 . \\end{align*}"}
+{"id": "4354.png", "formula": "\\begin{align*} \\alpha _ { 1 } \\left \\Vert x - P _ { 1 } ( x ) \\right \\Vert ^ { p - 2 } \\left ( x - P _ { 1 } ( x ) \\right ) + \\alpha _ { 2 } \\left \\Vert x - P _ { 2 } ( x ) \\right \\Vert ^ { p - 2 } \\left ( x - P _ { 2 } ( x ) \\right ) = 0 , \\end{align*}"}
+{"id": "7061.png", "formula": "\\begin{align*} \\mathcal J _ 2 = \\left ( \\left . h _ i d \\left ( \\sum \\limits _ { j = 1 } ^ { s _ \\ell } b _ { i _ { } i j } { \\textbf { X } } ^ { \\lambda _ j } \\right ) \\ \\right | \\ i \\in I \\setminus \\{ i _ { } \\} \\right ) . \\end{align*}"}
+{"id": "2084.png", "formula": "\\begin{align*} w ( r _ 1 ) & = \\sum _ { i = 1 } ^ k 2 ^ i x _ i ^ { \\prime } = \\sum _ { i = 1 } ^ { u } 2 ^ i x _ i ^ { \\prime } + \\sum _ { j = u + 1 } ^ { k } 2 ^ j x _ j ^ { \\prime } \\\\ & \\leq 2 ^ u ( 1 + x _ u ^ { \\prime } ) + \\sum _ { j = u + 1 } ^ { k } 2 ^ j x _ j ^ { \\prime } = 2 ^ u ( 1 + x _ u ^ { \\prime } ) + \\sum _ { j = u + 1 } ^ { k } 2 ^ j x _ j \\\\ & \\leq 2 ^ u x _ u + \\sum _ { j = u + 1 } ^ { k } 2 ^ j x _ j \\leq w ( r _ 2 ) . \\end{align*}"}
+{"id": "1889.png", "formula": "\\begin{align*} \\int _ \\Omega \\nabla u ^ * \\cdot \\nabla v \\ d x - \\lambda ^ * \\int _ { \\Omega } u ^ * v \\ d x \\forall \\ v \\in H _ 0 ^ 1 ( \\Omega ) \\mbox { a n d } \\| u ^ * \\| _ { L ^ 2 ( \\Omega ) } = 1 . \\end{align*}"}
+{"id": "5916.png", "formula": "\\begin{align*} \\tfrac { \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 ^ h , g _ 2 ^ h ) } { \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) } = [ \\widetilde { c } _ { X ^ { \\ast } } ( h ^ { - 1 } , g _ 1 g _ 2 h ) \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 g _ 2 , h ) ] [ \\widetilde { c } _ { X ^ { \\ast } } ( h ^ { - 1 } , g _ 1 h ) \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 , h ) ] ^ { - 1 } [ \\widetilde { c } _ { X ^ { \\ast } } ( h ^ { - 1 } , g _ 2 h ) \\widetilde { c } _ { X ^ { \\ast } } ( g _ 2 , h ) ] ^ { - 1 } . \\end{align*}"}
+{"id": "6082.png", "formula": "\\begin{align*} \\tilde a _ 0 ( x ) = \\begin{cases} f ( x ) & \\mbox { i f } x \\in \\mathcal { B } ( e , \\theta ) , \\\\ 1 & \\mbox { i f } x \\in \\mathcal { B } ( x _ { 1 } , \\epsilon ) , \\\\ 0 & \\mbox { i f } x \\notin \\mathcal { B } ( e , \\theta ) \\cup \\mathcal { B } ( x _ { 1 } , \\epsilon ) . \\end{cases} \\end{align*}"}
+{"id": "5339.png", "formula": "\\begin{align*} r ' = \\frac { \\sqrt { d } } { \\log ^ { 1 / 2 } ( 1 / \\sqrt { \\gamma } ) } \\mbox { a n d } \\gamma ' = \\left ( \\frac { 3 3 \\cdot ( 2 r ) } { \\sqrt { d } } \\right ) ^ d \\gamma \\log ^ { d / 2 } \\left ( \\frac { 1 } { \\sqrt { \\gamma } } \\right ) . \\end{align*}"}
+{"id": "4209.png", "formula": "\\begin{align*} \\phi ( t ) & = \\phi _ 0 ( t ) \\phi _ 1 ( t ) \\cdots \\phi _ R ( t ) \\end{align*}"}
+{"id": "6783.png", "formula": "\\begin{align*} ( q ; q ) _ \\infty ^ { \\mathrm { r k } ( \\mathfrak { s } ) } \\prod _ { \\alpha \\in \\Delta _ + } ( 1 - x ^ \\alpha ) \\prod _ { \\alpha \\in \\Delta } ( q x ^ \\alpha ; q ) _ \\infty & = \\sum _ { k = 0 } ^ \\infty a _ k ( x ) q ^ k . \\end{align*}"}
+{"id": "1534.png", "formula": "\\begin{align*} X - a Y & = \\ell \\\\ - a X + Y & = m \\end{align*}"}
+{"id": "4080.png", "formula": "\\begin{align*} \\lambda = { \\rm m a x } _ { j ' < j _ { \\varepsilon , 0 } } b _ { j ' } \\rho ^ { j ' } . \\end{align*}"}
+{"id": "9155.png", "formula": "\\begin{align*} v _ { i } ^ { j _ { i } } = y _ { i , [ \\kappa _ { i } ^ { j _ { i } } ] } ^ { j _ { i } , d } - \\sum _ { \\beta = 0 } ^ { \\kappa _ { i } ^ { j _ { i } } - 1 } a _ { i } ^ { j _ { i } , \\beta } ( y _ { i , [ \\beta ] } ^ { j _ { i } } - y _ { i , [ \\beta ] } ^ { j _ { i } , d } ) \\ , , j _ { i } = 1 , \\ldots , m _ { i } \\ , , i = 1 , \\ldots , s \\end{align*}"}
+{"id": "1579.png", "formula": "\\begin{align*} \\frac { ( 2 n + | x | _ { 1 } ) ! } { \\prod _ { i = 1 } ^ { d } q _ { i } ! ( q _ { i } + | x _ { i } | ) ! } = \\frac { ( 2 n + | x | _ { 1 } ) ! } { n ! \\ , ( n + | x | _ { 1 } ) ! } \\ \\frac { ( n + | x | _ { 1 } ) ! } { \\prod _ { i = 1 } ^ { d } ( q _ { i } + | x _ { i } | ) ! } \\ \\frac { n ! } { \\prod _ { i = 1 } ^ { d } q _ { i } ! } . \\end{align*}"}
+{"id": "4301.png", "formula": "\\begin{gather*} \\left \\| f ( \\cdot , t ) \\right \\| _ { L ^ p ( \\mathbb { T } ^ d ) } \\le \\left \\| f _ 0 \\right \\| _ { L ^ p } , \\\\ \\int _ { \\mathbb { T } ^ d } f ( \\cdot , t ) \\ ; d x = \\int _ { \\mathbb { T } ^ d } f _ 0 \\ ; d x . \\end{gather*}"}
+{"id": "9347.png", "formula": "\\begin{align*} \\begin{aligned} - d \\hat { y } _ t & = \\big [ g ^ 1 ( x , y _ 1 , z _ 1 , \\tilde { z } _ 1 , \\gamma _ { ( 1 , t , e ) } ) - g ^ 2 ( x , y _ 2 , z _ 2 , \\tilde { z } _ 2 , \\gamma _ { ( 2 , t , e ) } ) \\big ] d t \\\\ & \\quad - \\hat { z } _ t d W _ t - \\hat { \\tilde { z } } _ t d \\xi _ t - \\int _ { \\mathcal { E } } \\hat { \\gamma } _ { t , e } \\tilde { N } ( d e , d t ) , \\ t \\in [ 0 , \\infty ] . \\end{aligned} \\end{align*}"}
+{"id": "729.png", "formula": "\\begin{align*} U _ { n + 1 } = 2 \\sum _ { k = 0 } ^ { n + 1 } R ^ k _ { B , N } = R _ { B , N } \\Big ( 2 \\sum _ { k = 0 } ^ { n } R _ { B , N } ^ k \\Big ) + 2 { \\rm I } _ { N } = R _ { B , N } U _ { n } + 2 { \\rm I } _ { N } , U _ { 0 } = 2 { \\rm I } _ { N } \\ , . \\end{align*}"}
+{"id": "1221.png", "formula": "\\begin{align*} q - I _ { \\boldsymbol { a } } ( z ) = h ( z ) ( 1 - z a _ { j _ 1 } ) ^ { \\beta } , z \\in \\mathbb { D } _ 1 \\cap B ( 1 / a _ { j _ 1 } , r _ 1 ) . \\end{align*}"}
+{"id": "6317.png", "formula": "\\begin{align*} X _ 1 = \\partial _ x - \\frac { y } 2 \\partial _ z , X _ 2 = \\partial _ y + \\frac { x } 2 \\partial _ z , \\end{align*}"}
+{"id": "9051.png", "formula": "\\begin{align*} \\{ ( x , y , z ) \\in \\R ^ 3 \\mid ( x ^ 2 - z ^ 2 ) ( y - z ) = 0 \\} . \\end{align*}"}
+{"id": "6294.png", "formula": "\\begin{align*} \\dot \\lambda _ t = \\vec H ( \\lambda _ t ) . \\end{align*}"}
+{"id": "3319.png", "formula": "\\begin{align*} \\rho _ { \\theta } ( \\theta , f ( \\theta ) ) + 2 { \\rm R e } ( \\rho _ w ( \\theta , f ( \\theta ) ) \\frac { \\partial f } { \\partial \\theta } ( \\theta ) ) = 0 . \\end{align*}"}
+{"id": "1703.png", "formula": "\\begin{align*} P _ n ( \\cos \\theta _ 1 \\cos \\theta _ 2 \\pm \\sin \\theta _ 1 \\sin \\theta _ 2 \\cos \\phi ) = \\sum _ { k = - n } ^ n ( \\pm 1 ) ^ k \\frac { ( n - k ) ! } { ( n + k ) ! } { \\sf P } _ n ^ k ( \\cos \\theta _ 1 ) { \\sf P } _ n ^ k ( \\cos \\theta _ 2 ) \\cos ( k \\phi ) , \\end{align*}"}
+{"id": "661.png", "formula": "\\begin{align*} & p _ a ( S ( k , l ) \\varphi ) \\leq C p _ a ( \\varphi ) 0 < a < 1 , \\\\ & p _ a ( S ( k , l ) \\varphi ) \\leq C ( 1 + \\log \\| Q ( k , l ) \\| ) p _ a ( \\varphi ) a = 0 . \\end{align*}"}
+{"id": "6669.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { L } V ( x ) & = C \\big [ \\beta ( \\beta + 2 - N ) | x | ^ { - \\beta - 2 } - \\vartheta | x | ^ { - \\beta - 2 s } \\big ] \\\\ & \\leq - C \\vartheta | x | ^ { - \\beta - 2 s } \\leq - C \\vartheta | x | ^ { - \\alpha } \\quad , \\end{aligned} \\end{align*}"}
+{"id": "3124.png", "formula": "\\begin{align*} q ^ { r \\binom { k + 1 } { 2 } + ( 1 - r ) k } [ r ] _ q ^ k \\ , [ k ] _ { q ^ r } ! S _ { r } [ n , k ] = \\sum _ { \\ell = 0 } ^ { k } q ^ { r k ( k - \\ell ) } A _ { n , \\ell } ^ { r } ( q ) { n - \\ell \\brack k - \\ell } _ { q ^ r } \\end{align*}"}
+{"id": "2488.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { S } \\big ( \\xi , - \\eta \\big ) = \\min \\limits _ { ( u , v ) \\in \\mathcal { M } } \\mathcal { S } ( u , v ) . \\end{array} \\right . \\end{align*}"}
+{"id": "3294.png", "formula": "\\begin{align*} T _ 1 ( \\xi ) = \\frac { ( \\xi - 1 ) } { i ( \\xi + 1 ) } \\ \\ \\ { \\rm a n d } \\ \\ \\ T _ { - 1 } ( \\xi ) = \\frac { 1 } { T _ 1 ( \\xi ) } = \\frac { i ( \\xi + 1 ) } { ( \\xi - 1 ) } . \\end{align*}"}
+{"id": "5714.png", "formula": "\\begin{align*} \\alpha _ { g } = \\displaystyle \\lim _ { m \\rightarrow \\infty } \\dfrac { C _ { D ^ { k - 1 } ( f ) } ( p ^ { 2 m + 1 } ) } { \\beta ^ { 2 m } } = c \\displaystyle \\lim _ { m \\rightarrow \\infty } \\dfrac { C _ { F } ( p ^ { 2 m + 1 } ) } { \\beta ^ { 2 m } } . \\end{align*}"}
+{"id": "3249.png", "formula": "\\begin{align*} f = f _ 1 + f _ 2 + \\sum _ { j = 1 } ^ { | G | - 1 } f _ { \\sigma _ j } , \\quad \\mbox { w h e r e } f _ 1 = f \\chi _ { 5 B } , \\ f _ 2 = f \\chi _ { ( { \\mathcal O } ( 5 B ) ) ^ c } , \\ f _ { \\sigma _ j } = f \\chi _ { U _ j } . \\end{align*}"}
+{"id": "884.png", "formula": "\\begin{align*} \\{ \\underline { a } \\colon \\underline { a } / \\varpi ^ k \\in L ( \\varpi ^ k \\underline { c } ) \\} = & \\{ \\underline { d } \\colon ( \\underline { d } , \\varpi ) = 1 , | \\underline { d } | < | \\varpi | ^ k \\} \\setminus \\\\ & \\{ a \\underline { c } ^ \\bot + \\varpi \\underline { d } \\colon ( a , \\varpi ) = 1 , | a | < | \\varpi | , | \\underline { d } | < | \\varpi | ^ { k - 1 } \\} . \\end{align*}"}
+{"id": "7327.png", "formula": "\\begin{align*} g _ k = \\sum _ { i = 1 } ^ n ( \\alpha _ i - \\beta _ i ) g _ { k i } , f _ k = \\sum _ { i = 1 } ^ n ( \\beta _ i - \\gamma _ i ) f _ { k i } , e _ k = \\sum _ { i = 1 } ^ n ( \\gamma _ i - \\alpha _ i ) e _ { k i } , \\end{align*}"}
+{"id": "5837.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\mathcal { X } _ j ^ y = X ^ y _ { \\tau _ { j H _ k } } ; \\\\ \\mathcal { N } _ j ^ y = N ^ y _ { \\tau _ { j H _ k } } . \\end{array} \\right . \\end{align*}"}
+{"id": "4378.png", "formula": "\\begin{align*} \\mathcal Y _ k ( h ) = \\mathcal S _ { k } ( h ) - \\mathcal S _ { k - 1 } ( h ) , k \\leq n _ { \\mathcal L } . \\end{align*}"}
+{"id": "9266.png", "formula": "\\begin{align*} \\int _ { \\big \\{ x \\ , : \\ , x ^ { \\eta - 1 } \\| f \\| _ { L ^ 1 ( x ^ { - \\eta } d x ) } > \\lambda \\big \\} } x ^ { - \\eta } \\ , d x = \\int _ { \\big \\{ x \\ , : \\ , x > \\big ( \\lambda \\| f \\| _ { L ^ 1 ( x ^ { - \\eta } d x ) } ^ { - 1 } \\big ) ^ { 1 / ( \\eta - 1 ) } \\big \\} } x ^ { - \\eta } \\ , d x \\simeq \\frac { \\| f \\| _ { L ^ 1 ( x ^ { - \\eta } d x ) } } { \\lambda } . \\end{align*}"}
+{"id": "5117.png", "formula": "\\begin{align*} \\int _ { B _ { r } } ( b ^ { i j , k l } ) _ { r } w _ { i j } \\eta _ { k l } d x & = 0 , \\forall \\eta \\in C _ { 0 } ^ { \\infty } ( B _ { r } ) \\\\ w & = f \\partial B _ { r } \\\\ D w & = D f \\partial B _ { r } . \\end{align*}"}
+{"id": "3401.png", "formula": "\\begin{align*} T _ k = \\sup \\big \\{ t \\geq 0 : E _ t ( u _ k ) \\subset B ( k ) \\big \\} . \\end{align*}"}
+{"id": "5701.png", "formula": "\\begin{align*} C _ { f } ^ { + } ( n ) = C _ { h } ( n ) . \\end{align*}"}
+{"id": "9210.png", "formula": "\\begin{align*} n _ 1 + 2 \\sum _ { i = 1 } ^ { r - 1 } n _ i n _ { i + 1 } & = r + 2 \\sum _ { i = 1 } ^ { r - 1 } x _ i + x _ { r } + \\sum _ { i = 1 } ^ { r - 1 } x _ i x _ { i + 1 } \\\\ & \\le r + 2 ( n - r - 1 ) + x _ o x _ e \\\\ & \\le r + 2 ( n - r - 1 ) + \\left \\lfloor \\frac { ( n - r - 1 ) ^ 2 } 4 \\right \\rfloor \\end{align*}"}
+{"id": "5274.png", "formula": "\\begin{align*} \\binom { q } { 3 } | 1 + y ' | ^ { q - 3 } ( 1 + y ' ) y ^ 3 \\le \\binom { q } { 3 } | y | ^ 3 ( 1 + | y | ) ^ { q - 3 } . \\end{align*}"}
+{"id": "4516.png", "formula": "\\begin{align*} I _ { \\omega , \\mathbf { c } } ( \\Phi ) = 0 , \\ \\ \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) \\ge 0 . \\end{align*}"}
+{"id": "4237.png", "formula": "\\begin{align*} \\Theta _ r & = a X _ 1 + b X _ 2 + c Y _ 1 + d Y _ 2 \\end{align*}"}
+{"id": "792.png", "formula": "\\begin{align*} ^ { c } _ { 0 } { D } ^ { \\alpha } _ { t } u ( x , t ) - a ( l ( u ) ) \\ : \\Delta u ( x , t ) & = f ( x , t ) & \\mbox { i n } & \\Omega \\times ( 0 , T ] , \\\\ u ( x , t ) & = 0 & \\mbox { o n } & \\partial \\Omega \\times ( 0 , T ] , \\\\ u ( x , 0 ) & = u _ 0 ( x ) & \\mbox { i n } & \\Omega , \\end{align*}"}
+{"id": "8951.png", "formula": "\\begin{align*} \\phi & = \\phi _ { e } \\\\ & + C _ { 2 , 0 } \\Delta x ^ { 2 } + C _ { 3 , 0 } \\Delta x ^ { 3 } + \\dots \\\\ & + C _ { 0 , 2 } \\Delta t ^ { 2 } + C _ { 0 , 3 } \\Delta t ^ { 3 } + \\dots \\\\ & + C _ { 2 , 2 } \\Delta t ^ { 2 } \\Delta x ^ { 2 } + C _ { 2 , 3 } \\Delta x ^ { 2 } \\Delta t ^ { 3 } + \\dots . \\end{align*}"}
+{"id": "1208.png", "formula": "\\begin{align*} R _ k ' ( t ) & = k \\sum _ j \\nu ( j ) M _ j ( \\boldsymbol { a } , t ) ^ k \\frac { M ' _ j ( \\boldsymbol { a } , t ) } { M _ j ( \\boldsymbol { a } , t ) } \\\\ & = k \\left ( q R _ { k + 1 } ( t ) + \\varphi ( t ) R _ k ( t ) \\right ) . \\end{align*}"}
+{"id": "4868.png", "formula": "\\begin{align*} T _ { b ^ { \\mu } } \\left ( x ^ { \\mu } \\right ) = T _ { \\overline { b } } \\left ( \\overline { x } \\right ) \\cap T _ { b } \\left ( x \\right ) , \\mu \\in ] 0 , 1 [ . \\end{align*}"}
+{"id": "8754.png", "formula": "\\begin{align*} x _ { j } ( v ) = x _ { 0 , j } + \\sum _ { i \\neq j } v _ { i j } - \\sum _ { k \\neq j } v _ { j k } , \\quad \\forall j \\in I . \\end{align*}"}
+{"id": "2233.png", "formula": "\\begin{align*} \\varphi ( q _ { n * } ( x _ 1 , x _ 2 , . . . , x _ n ) ) = q _ { n * } ( \\varphi ( x _ 1 ) , \\varphi ( x _ 2 ) , . . . , \\varphi ( x _ i ) , . . . , \\varphi ( x _ n ) ) , \\end{align*}"}
+{"id": "40.png", "formula": "\\begin{align*} \\lambda ' = \\varphi _ { A ' } ^ \\vee Y _ { \\mathbb { Q } } ^ \\vee \\overline { \\lambda } _ 0 Y _ { \\mathbb { Q } } \\varphi _ { A ' } ; \\end{align*}"}
+{"id": "7302.png", "formula": "\\begin{align*} y z t = x ^ 2 + 1 . \\end{align*}"}
+{"id": "7797.png", "formula": "\\begin{align*} \\Vert \\xi \\Vert _ { \\psi _ { 2 } } = \\inf \\{ K > 0 : \\textsf { E } \\exp ( \\frac { \\xi ^ { 2 } } { K ^ { 2 } } ) \\le 2 \\} . \\end{align*}"}
+{"id": "7419.png", "formula": "\\begin{align*} U _ { m , \\beta } ( x ) : = \\sum _ { k = 1 } ^ \\infty ( \\sigma _ \\beta ^ 2 ) ^ { k } \\ ! \\sum _ { \\substack { 0 = : n _ 0 < n _ 1 < \\dots < n _ k : = m \\\\ x _ 0 : = 0 , \\ ; x _ 1 , \\ldots , x _ { k - 1 } \\in \\Z ^ 2 , \\ ; x _ k : = x } } \\ \\prod _ { i = 1 } ^ k q _ { n _ i - n _ { i - 1 } } ( x _ i - x _ { i - 1 } ) ^ 2 \\ , . \\end{align*}"}
+{"id": "274.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - y ^ m z ^ n } \\right ) ^ { \\frac { m ^ 1 } { n ^ 2 } } = \\left ( 1 - y z \\right ) ^ { \\frac { y } { 1 - y } } \\exp \\left \\{ \\frac { y } { ( 1 - y ) ^ 2 } \\left ( L i _ 2 ( z ) - L i _ 2 ( y z ) \\right ) \\right \\} , \\end{align*}"}
+{"id": "589.png", "formula": "\\begin{align*} \\psi ( a b ^ * ) = \\langle \\pi ( a b ^ * ) \\xi , \\xi \\rangle , a , b \\in A . \\end{align*}"}
+{"id": "3162.png", "formula": "\\begin{align*} \\langle T _ g ' ( y ' ) x \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } = \\langle y ' T _ { g } ( x ) \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } x \\in M , y ' \\in M ' , g \\in G . \\end{align*}"}
+{"id": "88.png", "formula": "\\begin{align*} P _ 0 = \\frac { h } { i } X - i ( Q _ \\infty + q _ 1 ) \\in \\Psi ^ 1 _ h , \\Tilde { P } _ h ( z ) = P _ 0 - i M h W - z \\in \\Psi ^ 1 _ h . \\end{align*}"}
+{"id": "6533.png", "formula": "\\begin{align*} K _ { m + 1 / 2 } ( x ) = \\sqrt { \\frac { \\pi } { 2 x } } \\sum _ { j = 0 } ^ m \\frac { ( m + j ) ! } { ( m - j ) ! j ! } ( 2 x ) ^ { - j } \\mathrm { e } ^ { - x } . \\end{align*}"}
+{"id": "6486.png", "formula": "\\begin{align*} D ^ t = i \\sqrt { \\pi c } \\left ( \\begin{array} { c c } 0 & \\mp 2 \\\\ 3 t - 1 & 0 \\end{array} \\right ) . \\end{align*}"}
+{"id": "8197.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { t / h ( t ) } P ^ \\omega ( F _ { j , t } ^ c ) \\leq t \\exp [ - h ( t ) ( k ^ 2 / 2 - \\beta ) ( 1 + o ( 1 ) ) ] . \\end{align*}"}
+{"id": "1930.png", "formula": "\\begin{align*} \\psi ( z _ 1 , \\ldots , z _ m ) : = \\frac { f ( z _ 1 , \\ldots , z _ m ) + \\overline { f ( \\overline { z _ 1 } , \\ldots , \\overline { z _ m } ) } } { 2 } . \\end{align*}"}
+{"id": "4737.png", "formula": "\\begin{align*} J _ { 2 ( k + \\ell ) } & = \\left ( J _ { 2 k } ^ T \\diamond O _ { 2 k , 2 \\ell } \\right ) ^ T \\diamond \\left ( O _ { 2 \\ell , 2 k } \\diamond J _ { 2 \\ell } ^ T \\right ) ^ T , \\end{align*}"}
+{"id": "8539.png", "formula": "\\begin{align*} \\left ( \\frac { \\pi } { \\sqrt { c } } \\right ) ^ { - s } \\Gamma ( s ) \\ , \\eta _ { p } ( 2 s ) = c ^ { 1 / 4 } \\left [ \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } \\cdot \\frac { 1 } { 2 s - 1 } + C _ { p } ^ { ( 2 ) } - \\frac { 2 e ^ { 2 \\pi p } } { 1 + \\frac { 1 } { \\pi p } } Q _ { 2 \\pi p } ( 0 ) - \\frac { 1 } { 2 } \\left ( \\gamma + \\log \\left ( \\frac { 4 \\pi } { \\sqrt { c } } \\right ) \\right ) + O \\left ( s - \\frac { 1 } { 2 } \\right ) \\right ] . \\end{align*}"}
+{"id": "2001.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } \\Delta w = \\frac { \\mu _ 2 } { 3 } f \\theta _ B & \\textnormal { i n } & \\Omega \\\\ w = 0 & \\textnormal { o n } & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "145.png", "formula": "\\begin{align*} \\tilde { \\chi } _ j u = \\tilde { \\chi } _ j \\Pi _ \\lambda u = \\tilde { \\chi } _ j \\Pi _ \\lambda ( 1 - \\chi _ j ) u + \\tilde { \\chi } _ j \\Pi _ \\lambda \\chi _ j u = 0 , \\end{align*}"}
+{"id": "7224.png", "formula": "\\begin{align*} \\mu \\left ( \\left ( \\Omega _ \\delta \\cap \\left ( \\overline { J _ u } \\setminus J _ u \\right ) \\setminus \\Sigma _ \\delta \\right ) \\right ) = \\mathcal { H } ^ 1 ( \\underbrace { J _ u \\cap ( \\Omega _ \\delta \\cap \\left ( \\overline { J _ u } \\setminus J _ u \\right ) \\setminus \\Sigma _ \\delta } _ { = \\varnothing } ) ) = 0 , \\end{align*}"}
+{"id": "6140.png", "formula": "\\begin{align*} \\d ( x , y ) : = F ( x ) - F ( y ) - \\langle F ' ( y ) , x - y \\rangle . \\end{align*}"}
+{"id": "4593.png", "formula": "\\begin{align*} \\mathfrak { I } _ 2 u - \\mathfrak { I } _ 1 u = & T \\beta ^ { - 1 } ( \\rho ( a , b ) u ) - \\alpha ^ { - 1 } ( [ a , b , T u ] ) \\\\ = & T \\rho ( a , b ) \\beta ^ { - 1 } ( u ) - [ a , b , T \\beta ^ { - 1 } ( u ) ] \\\\ = & \\wp ( a , b ) u \\in \\mathcal { B } ^ 1 _ T ( V , L ) , \\end{align*}"}
+{"id": "3777.png", "formula": "\\begin{align*} E _ { i + 1 } \\mathcal { T } _ { q _ i } ( v _ l ^ { ( i ) } ) = \\mathcal { T } _ { q _ { i + 1 } } E _ i ( v _ l ^ { ( i ) } ) \\mathrm { f o r } \\ \\mathrm { a l l } \\ i \\in \\mathbb { Z } _ r , \\ , l \\in [ \\omega n ] . \\end{align*}"}
+{"id": "2054.png", "formula": "\\begin{align*} D ( k ) = \\frac { 1 } { \\abs { k } ^ { 2 \\alpha } } + \\abs { k } ^ { 2 \\gamma } , \\end{align*}"}
+{"id": "8667.png", "formula": "\\begin{align*} \\widetilde { \\Gamma } ( M , q ) : = \\{ \\alpha \\in \\mathcal { C } ^ { \\times } | \\alpha ( \\mathbb { F } \\oplus V ) \\alpha '^ { - 1 } = \\mathbb { F } \\oplus V \\} . \\end{align*}"}
+{"id": "3848.png", "formula": "\\begin{align*} c _ { \\ell } ( s _ \\ell , s _ \\ell ' ) = c _ { Y _ \\ell } ( y _ \\ell , y _ \\ell ' ) + b ( x , x ' ) , \\end{align*}"}
+{"id": "6679.png", "formula": "\\begin{align*} \\mathcal { B } _ s ( u , v ) = \\int _ { \\Omega } \\langle \\nabla u , \\nabla v \\rangle \\ , d x + \\frac { C _ { N , s } } { 2 } \\iint _ { \\R ^ { 2 N } } \\frac { ( \\hat { u } ( x ) - \\hat { u } ( y ) ) ( \\hat { u } ( x ) - \\hat { u } ( y ) } { | x - y | ^ { N + 2 s } } \\ , d x \\ , d y + \\int _ \\Omega V ( x ) u v \\ , d x , \\end{align*}"}
+{"id": "7112.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } J _ \\varepsilon ( | x | ) x _ i \\ : x = 0 \\end{align*}"}
+{"id": "6762.png", "formula": "\\begin{align*} \\underline { d } ( x , y ) : = \\liminf _ { n \\to \\infty } \\frac { 1 } { n } | \\{ 1 \\leq i \\leq n : x ( i ) \\neq y ( i ) \\} | \\end{align*}"}
+{"id": "2190.png", "formula": "\\begin{align*} | g _ { \\lambda _ i } ( z ) ^ { - \\frac { 1 } { 2 } } - \\lambda _ i ^ { - \\frac { 1 } { 2 } } | & = \\frac { | \\lambda _ i z ^ 2 - 2 z | } { | \\lambda _ i g _ { \\lambda _ i } ( z ) | ^ { \\frac { 1 } { 2 } } | g _ { \\lambda _ i } ( z ) ^ { \\frac { 1 } { 2 } } + \\lambda _ i ^ { \\frac { 1 } { 2 } } | } \\\\ & \\leq \\frac { | \\lambda _ i z ^ 2 - 2 z | } { | \\lambda _ i \\lambda _ i / 2 | ^ { \\frac { 1 } { 2 } } \\lambda _ i ^ { \\frac { 1 } { 2 } } } = \\sqrt { 2 } \\lambda _ i ^ { - \\frac { 3 } { 2 } } | \\lambda _ i z ^ 2 - 2 z | , \\end{align*}"}
+{"id": "8985.png", "formula": "\\begin{align*} y ( t ) = \\frac { t ( t ^ { \\frac { 4 } { m - 2 } } - 1 ) } { c _ m ( t - 1 ) } \\end{align*}"}
+{"id": "8345.png", "formula": "\\begin{align*} \\| \\{ \\gamma _ j \\} _ { j = 1 } ^ \\infty \\| _ { \\ell _ { q ' } } = \\Big ( \\sum _ { j = 1 } ^ \\infty | \\gamma _ j | ^ { q ' } \\Big ) ^ \\frac 1 { q ' } \\leq \\varepsilon _ 2 . \\end{align*}"}
+{"id": "677.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\log \\norm { S ( k ) \\mathfrak { r } _ { a , n } ( \\varphi ) } _ { \\sup } } { k } = \\lim _ { k \\rightarrow \\infty } \\frac { \\log \\frac { \\norm { S ( k ) \\mathfrak { r } _ { a , n } ( \\varphi ) } _ { L ^ 1 ( I ^ { ( k ) } ) } } { | I ^ { ( k ) } | } } { k } = ( a - n ) \\lambda _ 1 . \\end{align*}"}
+{"id": "3015.png", "formula": "\\begin{align*} \\| A + B \\| _ { ( p , k ) } ^ p + \\| A - B \\| _ { ( p , k ) } ^ p \\geq 2 \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( A ^ * A + B ^ * B ) \\end{align*}"}
+{"id": "3990.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 d + 2 } } \\left ( f _ { \\mathcal { S } } \\right ) _ \\lambda d \\pi = \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + \\mathrm { R } ( \\lambda , \\delta ) , \\end{align*}"}
+{"id": "7235.png", "formula": "\\begin{align*} \\lim _ { h \\to + \\infty } \\mathcal { G } ( u _ h , \\beta _ h , B _ s ) = : \\Lambda ( s ) \\end{align*}"}
+{"id": "9292.png", "formula": "\\begin{align*} F _ { I } = \\left \\{ x \\in \\mathbb { R } ^ { m n } \\ | \\ x _ { s } - x _ { t } = \\mathbf { 0 } , \\ i = ( s , t ) \\in I \\right \\} , \\end{align*}"}
+{"id": "8696.png", "formula": "\\begin{align*} T _ 1 = 2 p ^ { - 2 } f ^ { ( 2 ) } ( \\tau ) \\| \\Sigma _ 1 - \\Sigma _ 2 \\| _ { \\rm F } ^ 2 . \\end{align*}"}
+{"id": "8141.png", "formula": "\\begin{align*} U ( I ) = \\{ u | u _ { 1 : k } \\ge 0 \\ \\ u _ { 1 : k } < 0 \\ \\} . \\end{align*}"}
+{"id": "4123.png", "formula": "\\begin{align*} { T _ { \\varepsilon } ^ { \\vec { v } } } = { T _ { \\widehat { \\varepsilon } } ^ { \\widehat { \\vec { v } } } } . \\end{align*}"}
+{"id": "4780.png", "formula": "\\begin{align*} \\tilde { S } _ { \\gamma _ i \\gamma _ i } = M _ { [ 1 ] } \\diamond \\cdots \\diamond M _ { [ | \\alpha _ i | ] } = N _ { [ i ] } + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "6242.png", "formula": "\\begin{align*} \\mathrm { S h } ( V ) = \\oplus _ { k \\geq 0 } [ \\underbrace { \\left ( \\begin{smallmatrix} 1 \\\\ 0 \\end{smallmatrix} \\right ) | \\ldots | \\left ( \\begin{smallmatrix} 1 \\\\ 0 \\end{smallmatrix} \\right ) } _ { k } | \\mathrm { S h } ^ * ( V ) ] . \\end{align*}"}
+{"id": "7193.png", "formula": "\\begin{align*} \\lim _ { h \\to + \\infty } t _ h \\abs { 1 - \\frac { \\lambda _ h ^ { ( 1 / 2 ) } } { t _ h } } = d , \\end{align*}"}
+{"id": "8407.png", "formula": "\\begin{align*} ( E _ 0 = N \\oplus N K _ X ^ { - 1 } \\oplus N ^ { - 1 } K _ X \\oplus N ^ { - 1 } , \\theta _ 0 = \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & \\mu & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} ) . \\end{align*}"}
+{"id": "7862.png", "formula": "\\begin{align*} A & = \\{ 1 , 2 , \\ldots , k - i - 2 , k + 1 , k + 2 , \\ldots , k + i , 2 k - 1 , 2 k \\} \\\\ B & = \\{ 1 , 2 , \\ldots , k - i - 2 , k + i + 1 , k + i + 2 , \\ldots , k + 2 i , 2 k - 1 , 2 k \\} . \\end{align*}"}
+{"id": "5284.png", "formula": "\\begin{align*} 1 + q \\rho d X + \\binom { q } { 2 } \\rho ^ 2 d ^ 2 X ^ 2 + e \\frac { q ( q - 2 ) } { 6 } \\rho ^ 3 | d X | ^ 2 , \\end{align*}"}
+{"id": "5155.png", "formula": "\\begin{align*} X ^ \\dag = X ' ( w ) . \\end{align*}"}
+{"id": "7582.png", "formula": "\\begin{align*} p ( z ) = \\frac { 1 + \\tau _ 1 z } { 1 - \\tau _ 1 z } , \\ ; \\ ; z \\in \\mathbb { D } . \\end{align*}"}
+{"id": "1279.png", "formula": "\\begin{align*} Z ( G ) = T + \\sum _ { j = 1 } ^ s a _ { i } = T + \\sum _ { j = 1 } ^ p ( k _ { i _ j } - 2 ) = & T + \\sum _ { j = 1 } ^ p k _ { i _ j } - 2 p \\\\ = & T + ( \\# \\ , o f \\ , z e r o s ) - ( s - p ) - 2 p \\\\ = & T + n - T - p - s = n - p - s , \\end{align*}"}
+{"id": "5356.png", "formula": "\\begin{align*} \\gamma ' = \\left ( \\frac { 1 0 0 r } { \\sqrt { d } } \\right ) ^ d \\gamma _ 1 \\log ^ { d / 2 } \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) . \\end{align*}"}
+{"id": "2750.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ 2 ( \\mathrm { M } ) } ^ 2 = \\sum _ { k \\ge 1 } \\frac { | ( f | \\phi _ q ^ k ) | ^ 2 } { | \\lambda _ q ^ k - \\lambda | ^ 2 } . \\end{align*}"}
+{"id": "895.png", "formula": "\\begin{align*} | \\varpi | ^ { 2 m + k } N _ 3 ( \\varpi ^ k ) \\ll | \\varpi | ^ { m + n k } + | \\varpi | ^ { m + n k } \\sum _ { l = 0 } ^ { k - 1 } | \\varpi | ^ { ( n / 6 - 2 ) ( l - k ) } \\ll | \\varpi | ^ { m + n k } + | \\varpi | ^ { m + n k } \\end{align*}"}
+{"id": "4708.png", "formula": "\\begin{align*} \\mathcal { A } \\vect u = z \\vect u , \\Gamma _ 1 \\vect u = \\mathcal { B } ( z ) \\Gamma _ 0 \\vect u , \\end{align*}"}
+{"id": "801.png", "formula": "\\begin{align*} \\left ( { D } ^ { \\alpha } _ { N } U ^ { n - \\sigma } , v _ h \\right ) \\ , - \\ , a \\big ( l ( U ^ { n , \\sigma } ) \\big ) \\ , ( \\Delta _ h U ^ { n , \\sigma } , v _ h ) \\ , = & \\ , ( f ^ { n - \\sigma } , v _ h ) , \\ ; \\ , \\forall v _ h \\in V _ h . \\\\ \\end{align*}"}
+{"id": "5076.png", "formula": "\\begin{align*} \\mathcal { R } ( v , \\gamma , \\gamma _ t , \\sigma _ t ) & = - \\gamma _ t v - \\frac { 1 } { N } \\sigma _ t v + \\beta \\mathcal { J } \\left [ \\frac { \\gamma _ { x x } } { \\left ( 1 - \\gamma _ x \\right ) ^ 2 } v - \\frac { \\gamma _ x ^ 2 } { 1 - \\gamma _ x } \\phi ' \\right ] , \\\\ \\mathcal { P } ( v , \\gamma ) & = - \\beta \\mathcal { J } \\left [ \\gamma _ x + \\frac { \\gamma _ x } { 1 - \\gamma _ x } \\right ] v , \\end{align*}"}
+{"id": "299.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } \\sum _ { n = 1 } ^ { \\infty } ( - n ^ 3 x ^ { n + 2 } y ^ { n + 1 } + 2 n ^ 3 x ^ { n + 2 } y ^ { n + 2 } - n ^ 3 x ^ { n + 2 } y ^ { n + 3 } ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "3586.png", "formula": "\\begin{gather*} a _ i ( n ) \\ : = \\ a _ i , \\\\ L ( n ) \\ : = \\ L , \\end{gather*}"}
+{"id": "6467.png", "formula": "\\begin{align*} \\beta _ { \\epsilon } ( t ) = \\frac { 1 } { 1 + 2 i t \\epsilon ^ 2 } , \\end{align*}"}
+{"id": "6806.png", "formula": "\\begin{align*} \\dim W ^ { \\mathrm { u } } ( p ^ - ) = 2 n - k ^ - , \\dim W ^ { \\mathrm { s } } ( p ^ + ) = k ^ + . \\end{align*}"}
+{"id": "7281.png", "formula": "\\begin{align*} x = \\frac { m } { ( n , m ) } u , y = \\frac { n } { ( n , m ) } u , \\end{align*}"}
+{"id": "6234.png", "formula": "\\begin{align*} \\psi ^ - ( x ) & + \\psi ^ - ( y ) = \\int _ { B _ 2 } g ( x - z ) + g ( y - z ) \\ , d \\mu ( z ) + \\int _ { B _ 3 } g ( x - w ) + g ( y - w ) \\ , d \\mu ( w ) \\\\ & \\geq - 2 \\mu ( B _ 2 \\cup B _ 3 ) + \\frac { C '' } 4 \\ , | y - x | ^ 2 ( k _ 2 + k _ 3 ) \\geq - 2 \\mu ( B _ 2 \\cup B _ 3 ) + \\frac { C '' } { 1 0 } \\ , | y - x | ^ 2 \\ , . \\end{align*}"}
+{"id": "2477.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle G ' ( t ) = - 4 I m \\int _ { \\mathbb { R } ^ N } \\left ( \\dfrac { a _ 2 } { 2 m _ 1 } x \\phi \\nabla \\overline { \\phi } + \\dfrac { a _ 1 } { 2 m _ 2 } x \\psi \\nabla \\overline { \\psi } \\right ) d x , \\end{array} \\right . \\end{align*}"}
+{"id": "3945.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ \\lambda ( v , s ^ { \\prime } ) & = f ( s ^ { \\prime } ) - \\lambda _ 1 \\boldsymbol { d } _ { \\mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ { \\prime } ) ^ { p _ 1 } - \\lambda _ 2 \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ { \\prime } ) ^ { p _ 2 } \\\\ & \\geq B + ( B - \\lambda _ 1 ) \\boldsymbol { d } _ { \\mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \\prime ) ^ { p _ 1 } + ( B - \\lambda _ 2 ) \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \\prime ) ^ { p _ 2 } \\geq B . \\end{aligned} \\end{align*}"}
+{"id": "9334.png", "formula": "\\begin{align*} & \\tau ^ * ( 2 ^ 2 \\cdot 3 ^ 2 \\cdot 5 ) = \\tau ( 2 ^ 2 \\cdot 3 ^ 2 \\cdot 5 ) - 2 ^ { 1 2 } \\tau ( 3 ^ 2 \\cdot 5 ) \\equiv 0 \\pmod { 2 3 } , \\\\ & \\tau ^ * ( 3 ^ 2 \\cdot 5 ) = \\tau ( 3 ^ 2 \\cdot 5 ) \\equiv 0 \\pmod { 2 3 } , \\\\ & \\tau ^ * ( 2 ^ 2 \\cdot 5 ) = \\tau ( 2 ^ 2 \\cdot 5 ) - 2 ^ { 1 2 } \\tau ( 5 ) \\equiv 0 \\pmod { 2 3 } , \\\\ & \\tau ^ * ( 5 ) = \\tau ( 5 ) \\equiv 0 \\pmod { 2 3 } . \\end{align*}"}
+{"id": "3843.png", "formula": "\\begin{align*} c _ { \\ell } ( s _ \\ell , s _ \\ell ' ) = ( s _ \\ell - s _ \\ell ' ) ^ { \\top } V _ \\ell ^ { - 1 } ( s _ \\ell - s _ \\ell ' ) , \\end{align*}"}
+{"id": "196.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b ) = 1 \\\\ a , b \\geq 1 } } \\left ( 1 - ( - 1 ) ^ a z ^ b \\right ) ^ { \\frac { b ^ { 2 } } { a ^ 3 } } = \\exp \\left \\{ \\frac { 3 z ( 1 + z ) \\zeta ( 3 ) } { 4 ( 1 - z ) ^ 3 } \\right \\} , \\end{align*}"}
+{"id": "3896.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { I } \\left ( \\lambda \\delta + ( 1 - \\lambda ) \\delta ^ \\prime \\right ) & \\geq \\lambda \\sup _ { \\gamma \\in \\Sigma ( \\delta ) } \\int _ { \\mathcal { S } } f ( s ) d \\gamma ( s ) + ( 1 - \\lambda ) \\sup _ { \\gamma ^ \\prime \\in \\Sigma ( \\delta ^ \\prime ) } \\int _ { \\mathcal { S } } f ( s ) d \\gamma ^ { \\prime } ( s ) \\\\ & \\geq \\lambda \\mathcal { I } ( \\delta ) + ( 1 - \\lambda ) \\mathcal { I } ( \\delta ^ \\prime ) . \\end{aligned} \\end{align*}"}
+{"id": "9209.png", "formula": "\\begin{align*} \\lvert A ( D ) \\rvert & = \\sum _ { i \\mbox { e v e n } } n _ i ( n _ 1 + n _ 3 + \\ldots + n _ { i + 1 } ) + \\sum _ { i \\mbox { o d d } } n _ i ( n _ 2 + n _ 4 + \\ldots + n _ { i + 1 } ) \\\\ & = n _ 0 n _ 1 + \\sum _ { i \\ge 2 \\mbox { e v e n } } n _ i ( n _ o + n _ { i - 1 } + n _ { i + 1 } ) \\\\ & = ( n _ e - 1 ) n _ o + \\sum _ { i \\ge 0 } n _ i n _ { i + 1 } . \\end{align*}"}
+{"id": "5470.png", "formula": "\\begin{align*} G ( \\theta ) \\approx & \\sum _ { b _ 1 , \\cdots , b _ M } { b \\choose b _ 1 , \\cdots , b _ M } \\prod _ { m = 1 } ^ { M } \\left ( C _ { M } ^ { m } ( - 1 ) ^ { m + 1 } \\right ) ^ { b _ m } \\\\ & \\frac { ( R _ { m a x } \\ ! \\ ! - \\ ! \\ ! R _ { m i n } ) \\pi } { 2 N } \\ ! \\sum _ { k = 1 } ^ { K } \\ ! \\sqrt { 1 \\ ! \\ ! - \\ ! \\psi _ k ^ 2 } \\exp \\bigg ( \\ ! \\ ! - \\ ! \\ ! \\ ! \\sum _ { m = 1 } ^ M m b _ m \\eta \\theta d _ k ^ { \\alpha } / \\rho \\\\ & - Q ( d _ k , \\theta ) \\bigg ) f _ { r _ 1 | \\Phi ( \\mathcal { A } ) > 0 } ( d _ k ) , \\end{align*}"}
+{"id": "1048.png", "formula": "\\begin{align*} \\frac { A _ y ( f + T ) } { A ( f + T ) } = \\frac { 1 } { T } + \\frac { Q ' ( T ) } { Q ( T ) } \\ , \\cdot \\end{align*}"}
+{"id": "251.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( 1 - z ^ n \\right ) ^ { \\frac { m ^ 2 } { n ^ 3 } } = \\sqrt { 1 - z } \\ ; \\exp \\left \\{ \\frac { - 1 } { 6 } L i _ 2 ( z ) - \\frac { 1 } { 3 } \\frac { z } { 1 - z } \\right \\} . \\end{align*}"}
+{"id": "8757.png", "formula": "\\begin{align*} F _ m ( p ) : = \\begin{cases} \\min _ { v } & c \\displaystyle \\sum _ { i = 1 } ^ n \\sum _ { k = 1 } ^ m p _ i ( \\phi _ { k , i } ( x _ i ) - d _ { i , k } ) \\cdot \\P ( \\omega _ k ) + \\sum _ { i \\neq j } \\alpha _ { i j } v _ { i j } \\\\ s . t \\quad & \\begin{cases} - v \\leq 0 \\\\ \\sum _ { j \\neq i } v _ { i j } - x _ { 0 i } \\leq 0 , \\forall i \\in I . \\end{cases} \\end{cases} \\end{align*}"}
+{"id": "1547.png", "formula": "\\begin{align*} \\int _ { k _ \\epsilon } ^ k | s | ^ { p - 2 } ( s - k ) _ - \\ , d s = \\tfrac { 1 } { p - 1 } \\ , \\mathfrak g _ - ( k _ \\epsilon , k ) \\geq \\tfrac { 1 } { \\boldsymbol \\gamma ( p ) } \\big ( | k _ \\epsilon | + | k | \\big ) ^ { p - 2 } ( k - k _ \\epsilon ) ^ 2 \\geq \\tfrac { 1 } { \\boldsymbol \\gamma ( p ) } M ^ p . \\end{align*}"}
+{"id": "5334.png", "formula": "\\begin{align*} \\| D _ { S , x } f \\| _ p \\le \\sum _ { T \\subseteq S } \\| [ E _ { T } [ f ] ] _ { S \\to x } \\| _ p \\leq \\sum _ { T \\subseteq S } r ^ { | S | } \\gamma = ( 2 r ) ^ { | S | } \\gamma , \\end{align*}"}
+{"id": "3772.png", "formula": "\\begin{align*} X _ S ( k , n , \\omega ) : = { \\rm G r } _ { ( k \\omega , \\dots , k \\omega ) } ( U _ { \\omega n , S } ) . \\end{align*}"}
+{"id": "8739.png", "formula": "\\begin{align*} & E \\{ ( \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { s } \\} + E \\{ ( \\| Y _ { 1 } - Y _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { s } \\} - 2 E \\{ ( \\| X _ { 1 } - Y _ { 1 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { s } \\} \\\\ & = \\sum ^ { s } _ { a = 0 } \\binom { s } { a } ( - p \\tau ) ^ { s - a } \\left ( E \\| X _ 1 - X _ 2 \\| ^ { 2 a } _ 2 + E \\| Y _ 1 - Y _ 2 \\| ^ { 2 a } _ 2 - 2 E \\| X _ 1 - Y _ 1 \\| ^ { 2 a } _ 2 \\right ) \\\\ & = \\sum ^ { s } _ { a = l } \\binom { s } { a } ( - p \\tau ) ^ { s - a } \\left ( E \\| X _ 1 - X _ 2 \\| ^ { 2 a } _ 2 + E \\| Y _ 1 - Y _ 2 \\| ^ { 2 a } _ 2 - 2 E \\| X _ 1 - Y _ 1 \\| ^ { 2 a } _ 2 \\right ) , \\end{align*}"}
+{"id": "8143.png", "formula": "\\begin{align*} \\bar \\Psi _ n = ( ( X _ \\bullet \\triangleright X _ \\bullet ) \\cdots ) \\triangleright X _ \\bullet . \\end{align*}"}
+{"id": "5727.png", "formula": "\\begin{align*} \\begin{pmatrix} f & a e + b f \\\\ c e + d f & p e + q f \\end{pmatrix} \\end{align*}"}
+{"id": "7623.png", "formula": "\\begin{align*} z f ^ { \\prime } ( z ) = \\frac { 1 } { 2 } ( p ( z ) + 1 ) f ( z ) , z \\in \\mathbb { D } \\end{align*}"}
+{"id": "3265.png", "formula": "\\begin{align*} { \\rm I n t } ( \\gamma ) = \\left \\{ w \\in \\C \\setminus \\{ 0 \\} ; | w | < R \\left ( \\frac { w } { | w | } \\right ) \\right \\} \\cup \\{ 0 \\} . \\end{align*}"}
+{"id": "2089.png", "formula": "\\begin{align*} g ( A ) & = \\frac { ( m ( 2 ^ n - 1 ) - 1 ) ( m ( 2 ^ n - 1 ) + d - 1 ) } { 2 } + \\frac { m ( m - 1 ) ( 2 ^ n - 1 ) } { 2 } + m ( 2 ^ { n - 1 } n - 1 ) \\\\ & = m ( 2 ^ { n - 1 } n - 1 ) + \\frac { 1 } { 2 } ( m ^ 2 ( 2 ^ n - 1 ) ^ 2 + ( m ^ 2 + d m - 3 m ) ( 2 ^ n - 1 ) - d + 1 ) . \\end{align*}"}
+{"id": "5684.png", "formula": "\\begin{align*} f ( z ) - P _ { \\infty } ( z ) = ( e ^ { - \\varepsilon y } ) y \\rightarrow \\infty \\epsilon > 0 . \\end{align*}"}
+{"id": "8214.png", "formula": "\\begin{align*} P ^ \\omega ( C _ t ^ c \\mid S ) & = \\frac { 1 } { P ^ \\omega ( S ) } \\left [ P ^ \\omega \\left ( C _ t ^ c \\cap S \\cap A _ { m ( t ) } \\right ) + P ^ \\omega \\left ( C _ t ^ c \\cap S \\cap A _ { m ( t ) } ^ c \\right ) \\right ] \\\\ & \\leq c ( \\omega ) \\left [ P ^ \\omega \\left ( C _ t ^ c \\mid A _ { m ( t ) } \\right ) + P ^ \\omega \\left ( A _ { m ( t ) } ^ c \\mid S \\right ) \\right ] . \\end{align*}"}
+{"id": "3273.png", "formula": "\\begin{align*} \\rho ( \\xi , w ) = \\rho ( \\overline { \\xi } , \\overline { w } ) \\end{align*}"}
+{"id": "8276.png", "formula": "\\begin{align*} G _ { k , Y } ( x , t _ 1 , t _ 2 ) = \\det \\left ( g _ { i + j + 1 + l _ { k - j } } ( x , t _ 1 , t _ 2 ) \\right ) _ { i , j = 0 , \\ldots , k - 1 } . \\end{align*}"}
+{"id": "8661.png", "formula": "\\begin{align*} & \\varphi _ 1 ( p ) = - \\frac { p } { \\log ( q ) } \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } ( 1 - y ) d y = - \\frac { p + q \\log ( q ) } { p \\log ( q ) } . \\end{align*}"}
+{"id": "8418.png", "formula": "\\begin{align*} \\sqrt { \\alpha } \\left ( \\frac { \\gamma } { 4 } - \\frac { \\log ( 4 \\beta ) } { 4 } + \\sum _ { n = 1 } ^ { \\infty } d ( n ) \\ , K _ { 0 } ( 2 n \\alpha ) \\right ) = \\sqrt { \\beta } \\left ( \\frac { \\gamma } { 4 } - \\frac { \\log ( 4 \\alpha ) } { 4 } + \\sum _ { n = 1 } ^ { \\infty } d ( n ) \\ , K _ { 0 } ( 2 n \\beta ) \\right ) . \\end{align*}"}
+{"id": "2866.png", "formula": "\\begin{align*} ( R \\otimes 1 _ L ) ( \\alpha \\otimes \\varphi _ L ) \\Delta ( l ) & = ( \\alpha \\otimes \\varphi _ L ) ( R \\otimes 1 _ L ) \\Delta ( l ) , \\\\ \\tau \\circ ( 1 _ L \\otimes R ) ( \\alpha \\otimes \\varphi _ L ) \\Delta ( l ) & = ( \\varphi _ L \\otimes \\alpha ) ( \\tau \\circ ( 1 _ L \\otimes R ) ) \\Delta ( l ) . \\end{align*}"}
+{"id": "3690.png", "formula": "\\begin{align*} \\begin{cases} \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) u = 0 & \\mathbb { R } ^ n , \\\\ u = 0 & W \\end{cases} \\end{align*}"}
+{"id": "387.png", "formula": "\\begin{align*} ( U ^ T P U / 2 ) _ t + ( U ^ T A _ i U ) _ { x _ i } = ( U _ { x _ i } ^ T A _ i U - U ^ T B _ i U _ { x _ i } ) - U ^ T C U . \\end{align*}"}
+{"id": "4070.png", "formula": "\\begin{align*} \\mathcal { S } ( I L , J M , c ) = \\sum _ { ( a , c ) = 1 } e \\left ( \\frac { a I L + \\bar { a } J M } { c } \\right ) , \\end{align*}"}
+{"id": "3653.png", "formula": "\\begin{align*} \\sum _ { k \\in [ n ] , k \\neq j } t _ k v _ k & = \\lambda v _ j , \\\\ t _ { k } ^ * v _ j & = \\lambda v _ { k } , \\forall k \\in [ n ] , k \\neq j . \\end{align*}"}
+{"id": "6596.png", "formula": "\\begin{align*} & E [ \\alpha _ { l , { \\hat { n } _ t } } e ^ { - j \\omega } ] = 0 , \\\\ & V a r [ ( \\alpha _ { l , { \\hat { n } _ t } } e ^ { - j \\omega } ) _ \\Re ] = 1 / 2 , \\\\ & V a r [ ( \\alpha _ { l , { \\hat { n } _ t } } e ^ { - j \\omega } ) _ \\Im ] = 1 / 2 . \\end{align*}"}
+{"id": "1604.png", "formula": "\\begin{align*} \\| S _ { \\xi \\chi } f \\| & = \\sup _ { \\| g \\| = 1 } \\bigg | \\langle S _ { \\xi \\chi } f , g \\rangle \\bigg | \\\\ & \\leq \\sup _ { \\| g \\| = 1 } \\sqrt { D _ { 2 } } \\| g \\| \\bigg ( \\int _ { \\Theta } v ^ { 2 } ( w ) \\| \\chi _ { w } \\pi _ { F ( w ) } f \\| ^ { 2 } d \\mu ( w ) \\bigg ) ^ { \\frac { 1 } { 2 } } \\\\ & \\leq \\sqrt { D _ { 2 } } \\bigg ( \\int _ { \\Theta } v ^ { 2 } ( w ) \\| \\chi _ { w } \\pi _ { F ( w ) } f \\| ^ { 2 } d \\mu ( w ) \\bigg ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
+{"id": "4399.png", "formula": "\\begin{align*} \\mathbf { c } \\cdot \\mathbf { p } ( u _ j ) = - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ d c _ k { \\rm R e } ( i u _ j , \\partial _ k u _ j ) _ { L ^ 2 ( \\R ^ d ) } = \\sum _ { k = 1 } ^ d I _ { j , k } - A _ j \\| \\nabla u _ j \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 - \\frac { | \\mathbf { c } | ^ 2 } { 1 6 A _ j } \\| u _ j \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 . \\end{align*}"}
+{"id": "4723.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} \\frac { 1 } { 1 - a ^ d / b ^ d } f _ 0 ( x ) & | x | \\leq b \\\\ 1 & | x | \\geq b \\end{cases} \\end{align*}"}
+{"id": "9166.png", "formula": "\\begin{align*} \\begin{aligned} x & = F _ { x } ( y ^ { 1 } , y ^ { 2 } , \\dots , y _ { [ 2 ] } ^ { 1 } , y _ { [ 1 ] } ^ { 2 } ) \\\\ \\bar { u } & = F _ { \\bar { u } } ( y ^ { 1 } , y ^ { 2 } , \\dots , y _ { [ 3 ] } ^ { 1 } , y _ { [ 2 ] } ^ { 2 } ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1872.png", "formula": "\\begin{align*} - D _ u f ( u ^ * ) \\in \\overline { L ( \\mathcal { K } , u ^ * ) } ^ \\circ = L ^ \\circ ( \\mathcal { K } , u ^ * ) . \\end{align*}"}
+{"id": "9122.png", "formula": "\\begin{align*} y _ { [ a ^ { j } ] } ^ { j } = v ^ { j } \\ , , j = 1 , \\ldots , m \\end{align*}"}
+{"id": "1543.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { M ^ { p - 2 } } { 2 ^ { p ( n + 3 ) } } \\int _ { \\widetilde { K } _ n } ( u - \\tilde { k } _ n ) _ - ^ 2 \\ , d x + \\underset { \\widetilde { Q } _ n } { \\iint } \\frac { | ( u - \\tilde { k } _ n ) _ { - } ( x , t ) - ( u - \\tilde { k } _ n ) _ { - } | ^ p } { | x - y | ^ { N + s p } } \\ , d x \\ , d y \\ , d t \\leq \\boldsymbol \\gamma \\frac { 2 ^ { n ( N + 2 p ) } } { \\theta \\varrho ^ { s p } } M ^ { p } | A _ n | . \\end{aligned} \\end{align*}"}
+{"id": "605.png", "formula": "\\begin{align*} s _ - = \\sqrt { \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\sqrt { \\frac { C - 4 } { C } } } \\end{align*}"}
+{"id": "8706.png", "formula": "\\begin{align*} & \\lim _ { x \\rightarrow 0 ^ - } \\bigg | \\frac { r ( x ) } { x ^ { l + 1 } } \\bigg | = \\bigg | \\frac { h ^ { ( l + 1 ) } ( 0 ) } { ( l + 1 ) ! } \\bigg | . \\end{align*}"}
+{"id": "8932.png", "formula": "\\begin{align*} \\mathsf { d } _ { c c } \\Bigg ( \\prod _ { l = 1 } ^ j y \\big ( \\bigl \\{ X _ { n i } ^ { ( l ) } \\bigr \\} \\big ) \\Bigg ) \\leq 2 ^ { k - 1 } \\mathsf { d } _ { \\mathrm { c o m } } ^ { ( j ) } \\big ( \\bigl \\{ X _ { n i } ^ { ( 1 ) } \\bigr \\} , \\dots , \\bigl \\{ X _ { n i } ^ { ( j ) } \\bigr \\} \\big ) . \\end{align*}"}
+{"id": "9174.png", "formula": "\\begin{align*} v _ { 1 } ^ { 1 } & = y _ { 1 , [ 2 ] } ^ { 1 , d } - a _ { 1 } ^ { 1 , 1 } ( y _ { 1 , [ 1 ] } ^ { 1 } - y _ { 1 , [ 1 ] } ^ { 1 , d } ) - a _ { 1 } ^ { 1 , 0 } ( y _ { 1 } ^ { 1 } - y _ { 1 } ^ { 1 , d } ) \\\\ v _ { 2 } ^ { 1 } & = y _ { 2 , [ 2 ] } ^ { 1 , d } - a _ { 2 } ^ { 1 , 1 } ( y _ { 2 , [ 1 ] } ^ { 1 } - y _ { 2 , [ 1 ] } ^ { 1 , d } ) - a _ { 2 } ^ { 1 , 0 } ( y _ { 2 } ^ { 1 } - y _ { 2 } ^ { 1 , d } ) \\end{align*}"}
+{"id": "3958.png", "formula": "\\begin{align*} \\widehat { \\pi } _ { 1 , 2 } = ( S _ 1 , S _ 2 ) = \\gamma ^ { \\eta _ 0 , \\delta } , \\widehat { \\pi } _ { 1 , 3 } = ( S _ 1 , \\widetilde { S } _ 1 ) = \\nu \\in \\Pi ( \\gamma _ 1 ^ { \\eta , \\delta } , \\gamma _ 1 ^ { \\star } ) , \\end{align*}"}
+{"id": "3692.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) u _ j + q _ j u _ j = 0 & \\Omega , \\\\ u _ j = f _ j & \\Omega ^ c . \\end{cases} \\end{align*}"}
+{"id": "836.png", "formula": "\\begin{align*} \\textbf { y } = \\sqrt { \\rho _ { f } } \\textbf { G } ^ T \\textbf { P } \\textbf { x } + \\textbf { w } , \\end{align*}"}
+{"id": "639.png", "formula": "\\begin{align*} \\sup _ { \\omega \\in P } \\omega ( a ^ * a ) = \\| a \\| ^ 2 , a \\in A . \\end{align*}"}
+{"id": "6697.png", "formula": "\\begin{align*} S A = - a S \\nabla E a - A S \\nabla \\tilde \\eta a - \\tilde a S \\nabla \\tilde \\eta A - R a - \\tilde r A - A , \\end{align*}"}
+{"id": "9258.png", "formula": "\\begin{align*} \\Psi _ { * , F _ 2 '' } ^ { \\alpha , \\beta } f ( x ) & \\simeq \\sup _ { t \\ge 3 x } t ^ { - \\beta } \\int _ { ( t - x ) / 2 } ^ { t - x } ( t + x - z ) ^ { - \\alpha - 1 / 2 } ( t - x - z ) ^ { \\alpha + \\beta - 1 / 2 } f ( z ) \\ , d z \\\\ & \\simeq \\sup _ { t \\ge 2 x } t ^ { - \\beta } \\int _ { t / 2 } ^ { t } ( t - z + x ) ^ { - \\alpha - 1 / 2 } ( t - z ) ^ { \\alpha + \\beta - 1 / 2 } f ( z ) \\ , d z \\\\ & = : J _ 1 f ( x ) + J _ 2 f ( x ) , \\end{align*}"}
+{"id": "8833.png", "formula": "\\begin{align*} v _ { 0 , \\lambda } ( t , x ) : = ( \\tau , z ) \\mapsto K _ { t - \\tau } ( x , z ) \\ , \\sigma ( u _ \\lambda ( \\tau , z ) ) \\ , 1 _ { [ 0 , t ] } ( \\tau ) . \\end{align*}"}
+{"id": "1466.png", "formula": "\\begin{align*} & \\int _ { \\mathcal { A } ^ l _ m } \\bigg | f _ 0 ^ { ' } \\Big ( U _ { \\mu ^ i _ m , \\xi ^ i _ m } ( x ) \\Big ) \\psi _ { \\mu ^ i _ m , \\xi ^ i _ m } ^ h ( x ) u _ m ( x ) \\bigg | d x \\\\ \\leq & C | \\psi _ { \\mu ^ i _ m , \\xi ^ i _ m } ^ h | _ { \\frac { 2 n } { n - 2 } } | u _ m | _ { \\frac { 2 n } { n - 2 } } \\Big ( \\int _ { \\mathcal { A } ^ l _ m } U _ { \\mu ^ i _ m , \\xi ^ i _ m } ^ { \\frac { 2 n } { n - 2 } } d x \\Big ) ^ { \\frac { 2 } { n } } = O \\bigg ( \\Big ( \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big ) ^ { \\frac { n } { n - 2 } } \\bigg ) . \\end{align*}"}
+{"id": "9091.png", "formula": "\\begin{align*} B = \\sum _ { i = 1 } ^ { n - r } b _ i Y _ i ^ 2 , \\end{align*}"}
+{"id": "4721.png", "formula": "\\begin{align*} \\begin{aligned} \\psi ( \\beta , \\mu ) & = \\sup _ { \\tilde \\rho } \\left [ \\tilde \\rho \\mu - \\phi ( \\beta , \\tilde \\rho ) \\right ] \\geq \\rho _ 0 \\mu - \\phi ( \\beta , \\rho _ 0 ) \\\\ \\psi _ 0 ( \\beta , \\mu ) & = \\sup _ { \\tilde \\rho } \\left [ \\tilde \\rho \\mu - \\phi _ 0 ( \\beta , \\tilde \\rho ) \\right ] = \\rho _ 0 \\mu - \\phi _ 0 ( \\beta , \\rho _ 0 ) \\end{aligned} \\end{align*}"}
+{"id": "8294.png", "formula": "\\begin{align*} A = A ( \\alpha ) = \\langle x _ 1 , x _ 2 , x _ 3 \\rangle / ( \\alpha x _ 3 x _ 1 + x _ 1 x _ 3 , \\alpha x _ 3 x _ 2 - x _ 2 x _ 3 , x _ 1 ^ 2 - x _ 2 ^ 2 ) . \\end{align*}"}
+{"id": "169.png", "formula": "\\begin{align*} L i _ 2 ( 1 ) = - \\frac { \\pi ^ 2 } { 6 } , \\end{align*}"}
+{"id": "5901.png", "formula": "\\begin{align*} T _ 0 : = L _ 0 + C \\hbox { a n d } \\widetilde { T } _ 0 : = \\widetilde { L } _ 0 + C ^ * \\ ; . \\end{align*}"}
+{"id": "2769.png", "formula": "\\begin{align*} \\| v _ t \\| _ { L ^ 2 ( \\mathrm { M } _ 0 ) } ^ 2 & = ( u | \\partial _ \\nu w _ t ) _ { L ^ 2 ( \\partial \\mathrm { M } _ 0 ) } - ( \\partial _ \\nu u | w _ t ) _ { L ^ 2 ( \\partial \\mathrm { M } _ 0 ) } \\\\ & \\le \\| u \\| _ { L ^ 2 ( \\partial \\mathrm { M } _ 0 ) } \\| \\partial _ \\nu w _ t \\| _ { L ^ 2 ( \\partial \\mathrm { M } _ 0 ) } + \\| \\partial _ \\nu u \\| _ { L ^ 2 ( \\partial \\mathrm { M } _ 0 ) } \\| w _ t \\| _ { L ^ 2 ( \\partial \\mathrm { M } _ 0 ) } . \\end{align*}"}
+{"id": "2448.png", "formula": "\\begin{align*} e ^ { r X ^ M } m = ( e ^ { r X } \\tau ( m ) ) \\cdot m & & r \\in \\R , m \\in M . \\end{align*}"}
+{"id": "1968.png", "formula": "\\begin{align*} n ^ { - 1 } g u ^ { p \\bar q } u _ { p \\bar q j } = g _ j + g _ v v _ j , \\end{align*}"}
+{"id": "6601.png", "formula": "\\begin{align*} \\begin{aligned} \\bar P _ b \\approx & \\frac { 1 } { 1 2 } \\int _ 0 ^ \\infty \\exp \\left ( - { \\frac { \\rho \\zeta ^ 2 x } { 4 ( 1 + \\rho ( 1 - \\zeta ^ 2 ) \\sigma _ e ^ 2 L ) } } \\right ) f ( x ) d x \\\\ & + \\frac { 1 } { 4 } \\int _ 0 ^ \\infty \\exp \\left ( - { \\frac { \\rho \\zeta ^ 2 x } { 3 ( 1 + \\rho ( 1 - \\zeta ^ 2 ) \\sigma _ e ^ 2 L ) } } \\right ) f ( x ) d x . \\end{aligned} \\end{align*}"}
+{"id": "748.png", "formula": "\\begin{align*} \\bar { \\alpha } ( \\Omega ) = \\inf \\left \\lbrace \\dfrac { \\mathrm { V o l } \\left ( \\Omega \\Delta B _ \\rho ( x ) \\right ) } { \\mathrm { V o l } ( B _ { \\rho } ) } : x \\in \\mathbb { R } ^ { n + 1 } , \\mathrm { V o l } ( \\Omega ) = \\mathrm { V o l } ( B _ { \\rho } ) \\right \\rbrace \\end{align*}"}
+{"id": "5678.png", "formula": "\\begin{align*} & U ( p ) \\left ( \\sum _ { n \\in \\mathbb { Z } } C ( n ) q ^ { n } \\right ) : = \\sum _ { n \\in \\mathbb { Z } } C ( p n ) q ^ { n } \\\\ & V ( p ) \\left ( \\sum _ { n \\in \\mathbb { Z } } C ( n ) q ^ { n } \\right ) : = \\sum _ { n \\in \\mathbb { Z } } C ( n / p ) q ^ { n } \\\\ & D ^ { k - 1 } \\left ( \\sum _ { n \\in \\mathbb { Z } } C ( n ) q ^ { n } \\right ) : = \\sum _ { n \\in \\mathbb { Z } } n ^ { k - 1 } C ( n ) q ^ { n } . \\end{align*}"}
+{"id": "7596.png", "formula": "\\begin{align*} | H _ { 2 , 1 } ( F _ { f } / 2 ) | & \\leq \\frac { 1 } { 1 2 } \\tau _ { 1 } ( 1 - \\tau ^ 2 _ { 1 } ) \\left ( | A | + | B | + | C | \\right ) \\\\ & = \\frac { 1 } { 1 9 2 } \\left ( 1 2 - 4 \\tau ^ 2 _ { 1 } - 5 \\tau ^ 4 _ { 1 } \\right ) \\\\ & \\leq \\frac { 1 } { 1 6 } . \\end{align*}"}
+{"id": "7955.png", "formula": "\\begin{align*} - \\mathrm { d i v } \\Big ( a \\big ( H ( \\nabla u ) \\big ) \\ , \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla u ) \\Big ) = f \\quad \\Omega \\ , . \\end{align*}"}
+{"id": "7071.png", "formula": "\\begin{align*} \\beta = \\alpha \\left ( \\left \\{ \\left . \\tilde { \\beta } _ j \\ \\right | \\ j \\in I ' \\right \\} \\right ) . \\end{align*}"}
+{"id": "1855.png", "formula": "\\begin{align*} \\begin{aligned} & Q _ { 1 } ( f , g ) - Q _ { 1 } ( h , h _ { - } ) + Q _ { 3 } ( f , g ) - Q _ { 3 } ( h , h _ { - } ) \\\\ & \\qquad = Q _ { 1 } ( f - h , g - h _ { - } ) + Q _ { 1 } ( f , h _ { - } ) + Q _ { 1 } ( g , h ) - 2 Q _ { 1 } ( h , h _ { - } ) \\\\ & \\qquad + Q _ { 3 } ( f - h , g - h _ { - } ) + Q _ { 3 } ( f , h _ { - } ) + Q _ { 3 } ( g , h ) - 2 Q _ { 3 } ( h , h _ { - } ) \\\\ & \\qquad \\stackrel { \\eqref { q e q } } { = } Q _ { 1 } ( f - h , g - h _ { - } ) + Q _ { 3 } ( f - h , g - h _ { - } ) . \\end{aligned} \\end{align*}"}
+{"id": "6517.png", "formula": "\\begin{align*} F _ X ( x ) = 1 & - \\frac { ( 1 - \\beta ^ 2 / \\alpha ^ 2 ) ^ { \\nu + 1 / 2 } } { 2 \\sqrt { \\pi } \\Gamma ( \\nu + 1 / 2 ) } \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { k ! } \\bigg ( \\frac { 2 \\beta } { \\alpha } \\bigg ) ^ k \\Gamma \\bigg ( \\frac { k + 1 } { 2 } \\bigg ) \\Gamma \\bigg ( \\nu + \\frac { k + 1 } { 2 } \\bigg ) \\tilde { G } _ { \\nu + k , \\nu } ( \\alpha ( x - \\mu ) ) , \\end{align*}"}
+{"id": "7577.png", "formula": "\\begin{align*} H _ { 2 , 1 } ( F _ { f } / 2 ) = \\gamma _ { 1 } \\gamma _ { 3 } - \\gamma ^ 2 _ { 2 } = \\dfrac { 1 } { 4 8 } \\left ( a ^ 4 _ 2 - 1 2 a ^ 2 _ 3 + 1 2 a _ 2 a _ 4 \\right ) . \\end{align*}"}
+{"id": "889.png", "formula": "\\begin{align*} S _ 1 = | \\varpi | ^ { 3 } \\left ( \\frac { | \\varpi | ^ { n - 1 } } { ( 1 - | \\varpi | ^ { - 1 } ) ^ 2 } \\mathcal { N } _ 1 - \\frac { | \\varpi | ^ { n - 1 } } { ( 1 - | \\varpi | ^ { - 1 } ) ^ 2 } \\mathcal { N } _ 2 \\right ) + O \\left ( | \\varpi | ^ { n + 3 } \\right ) . \\end{align*}"}
+{"id": "1287.png", "formula": "\\begin{align*} Z _ q ( K _ { n , m } ) & = \\left \\{ \\begin{aligned} & \\min ( n , m ) & \\mbox { i f } q = 0 \\\\ & n + m - 2 & \\mbox { f o r } q \\geq 1 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5567.png", "formula": "\\begin{align*} L _ { I } = \\left \\{ \\left [ \\begin{array} { c c c c } M _ { 1 } \\\\ & M _ { 2 } \\\\ & & \\ddots \\\\ & & & M _ { \\ell } \\end{array} \\right ] , \\ M _ { j } \\in G L ( d _ { j } , \\mathbb { R } ) , \\ \\det ( M _ { 1 } ) \\ldots \\det ( M _ { \\ell } ) = 1 \\right \\} . \\end{align*}"}
+{"id": "1556.png", "formula": "\\begin{align*} \\Lambda _ f ( s , \\alpha ) = \\gamma ( s ) \\sum _ { n \\geq 1 } \\frac { f _ n e ( n \\alpha ) } { n ^ s } . \\end{align*}"}
+{"id": "3170.png", "formula": "\\begin{align*} ( \\underbar { x } ' , \\underbar { y } ) = ( x _ 1 , \\ldots , x _ m + c ' , \\ldots , x _ N , y _ 1 , \\ldots , y _ N ) . \\end{align*}"}
+{"id": "725.png", "formula": "\\begin{align*} A _ { S } [ \\partial _ { t } \\gamma _ { S } ] ( e ) = G _ { V , 1 } ( e ) \\ , . \\end{align*}"}
+{"id": "440.png", "formula": "\\begin{align*} \\dfrac { d } { d t } \\| U \\| ^ 2 _ { I _ 2 \\otimes P _ x } + \\vec U ^ T ( I _ 2 \\otimes B ) { \\bf A } \\vec U = ( { \\bf D _ x } \\vec U ) ^ T ( I _ 2 \\otimes P _ x ) { \\bf A } \\vec U - ( { \\bf A } \\vec U ) ^ T ( I _ 2 \\otimes P _ x ) { \\bf D _ x } \\vec U , \\end{align*}"}
+{"id": "9186.png", "formula": "\\begin{align*} u = F _ { u } \\circ \\phi ( q ^ { 1 } , q ^ { 2 } , q ^ { 3 } , \\omega ^ { 1 } , \\omega ^ { 2 } , \\omega ^ { 3 } , v _ { 1 } ^ { 1 } , v _ { 1 , [ 1 ] } ^ { 1 } , v _ { 1 , [ 2 ] } ^ { 1 } , v _ { 2 } ^ { 1 } ) \\end{align*}"}
+{"id": "5397.png", "formula": "\\begin{align*} S ( n , \\alpha ) = \\sum _ { i = 0 } ^ { n } S ( i , \\alpha \\cdot \\chi _ { 0 } ) \\alpha ( T ) ^ { n - i } = \\sum _ { i = 0 } ^ { n } S ( i , \\iota _ \\alpha ) \\alpha ( T ) ^ { n - i } . \\end{align*}"}
+{"id": "7384.png", "formula": "\\begin{align*} X _ N ( L , \\alpha ) : = R _ N ^ 2 ( L , \\alpha , \\Delta ) - \\langle R _ N ^ 2 ( L , \\alpha , \\Delta ) \\rangle = \\sum _ { k \\ne 0 } b _ { k , N } ( L ) e ( k \\alpha ) . \\end{align*}"}
+{"id": "3726.png", "formula": "\\begin{align*} E _ p = F _ 1 ( E _ { p } ) \\supsetneq F _ 2 ( E _ { p } ) \\supsetneq \\cdots \\supsetneq F _ { l _ p } ( E _ { p } ) \\supsetneq F _ { l _ { p + 1 } } ( E _ { p } ) = 0 \\end{align*}"}
+{"id": "402.png", "formula": "\\begin{align*} W ^ T \\Lambda W + 2 U ^ T ( J ^ - T ^ { - 1 } ) ^ T \\Sigma ( \\sqrt { | \\Lambda ^ - | } W ^ - - R \\sqrt { \\Lambda ^ + } W ^ + ) ) = W ^ T \\Lambda W + 2 ( W ^ - ) ^ T \\Sigma ( \\sqrt { | \\Lambda ^ - | } W ^ - - R \\sqrt { \\Lambda ^ + } W ^ + ) . \\end{align*}"}
+{"id": "8043.png", "formula": "\\begin{align*} \\left \\| \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} \\right \\| _ { ( k ) } & \\le \\left \\| \\frac { A + B } { 2 } + ( r + \\omega ) I \\right \\| _ { ( k ) } + \\left \\| \\frac { A + B } { 2 } - r I \\right \\| _ { ( k ) } \\\\ & = \\left \\| A + B + \\omega I \\right \\| _ { ( k ) } . \\end{align*}"}
+{"id": "8230.png", "formula": "\\begin{align*} \\left < \\phi , Y _ t ^ N \\right > = \\left < \\phi ( \\cdot , 1 ) , Y _ { t , 1 } ^ N \\right > + \\left < \\phi ( \\cdot , - 1 ) , Y _ { t , - 1 } ^ N \\right > . \\end{align*}"}
+{"id": "3793.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) : = \\sup _ { \\gamma \\in \\Sigma _ { \\mathrm { D } } ( \\delta ) } \\int _ { \\mathcal { V } } g \\ , d \\gamma , \\delta \\in \\mathbb { R } _ { + } ^ 2 , \\end{align*}"}
+{"id": "5137.png", "formula": "\\begin{align*} \\| F \\| _ X \\leq K _ X \\| F - F _ N \\| _ X + K _ X \\sum _ { n = 1 } ^ \\infty K _ X ^ n \\| f _ n \\| _ X \\end{align*}"}
+{"id": "1430.png", "formula": "\\begin{align*} u ( x ) = V ( x ) + \\phi ( x ) , V ( x ) = V _ { \\bar { d } , \\bar { \\sigma } , \\xi } ( x ) = \\sum _ { i = 1 } ^ k ( - 1 ) ^ i P U _ { \\mu _ i , \\xi _ i } ( x ) , \\end{align*}"}
+{"id": "714.png", "formula": "\\begin{align*} \\partial _ t \\phi _ { A } + \\frac { 1 } { 2 } \\vert \\nabla \\phi _ { A } \\vert ^ 2 = - \\frac { p _ { A } } { \\rho _ { A } } - g x _ { 2 } \\ , . \\end{align*}"}
+{"id": "1997.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c w ) ^ n = f ^ n \\omega ^ n & \\textnormal { i n } & \\Omega , \\\\ w = 0 & \\textnormal { o n } & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "7909.png", "formula": "\\begin{align*} \\Lambda _ k : = \\bigcup _ { \\lambda \\vdash k } 2 \\lambda + 1 . \\end{align*}"}
+{"id": "5341.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ q \\leq \\gamma ' \\rho ^ { - d } = \\gamma \\left ( 9 9 \\cdot 2 \\ , r \\sqrt { q } \\sqrt { \\frac { \\log ( 1 / \\gamma ) } { d } } \\right ) ^ d \\leq \\gamma \\left ( 2 0 0 r \\sqrt { q } \\cdot \\max \\left \\{ \\sqrt { q } , \\sqrt { \\log ( 1 / \\gamma ) } \\right \\} \\right ) ^ d , \\end{align*}"}
+{"id": "6125.png", "formula": "\\begin{align*} A D _ { \\lambda } ^ { d } = \\left \\{ \\mathcal { K } \\times I \\in \\mathcal { A } \\ , : E _ { p , \\tau } \\left ( u \\ , ; \\ , { K } ^ { d } \\times I \\right ) \\leq \\frac { \\lambda } { 1 6 } \\right \\} \\end{align*}"}
+{"id": "7944.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) \\Pi _ { 4 e _ { ( \\xi , 0 ) } + e _ { ( 0 , 2 e _ 0 + 2 e _ { ( 0 , 1 ) } ) } } & = \\Pi _ { e _ { ( \\xi , 0 ) } } ^ 2 ( \\partial _ x \\Pi _ { e _ { ( \\xi , 0 ) } } ) ^ 2 \\\\ & + c _ { 4 e _ { ( \\xi , 0 ) } + e _ { ( 0 , 2 e _ 0 + 2 e _ { ( 0 , 1 ) } ) } } \\\\ & + c _ { 2 e _ { ( \\xi , 0 ) + e _ { ( 0 , 2 e _ { ( 0 , 1 ) } ) } } } \\Pi _ { e ( \\xi , 0 ) } ^ 2 , \\end{align*}"}
+{"id": "668.png", "formula": "\\begin{align*} V _ k ^ { j , \\tau } ( T , \\bar { s } ) & = O ( e ^ { ( \\max \\{ \\lambda _ { i } , \\lambda _ 1 a \\} - \\lambda _ 1 + 8 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k ) } ) \\\\ W _ k ^ { j , \\tau } ( T , \\bar { s } ) & = O ( e ^ { ( \\max \\{ \\lambda _ { i } - \\lambda _ 1 , \\lambda _ 1 a - \\lambda _ 1 , - \\lambda _ j \\} + 9 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k ) } ) . \\end{align*}"}
+{"id": "2966.png", "formula": "\\begin{align*} V _ I = \\{ e _ i \\ | \\ i \\in I \\} \\subset \\C ^ n \\ . \\end{align*}"}
+{"id": "3441.png", "formula": "\\begin{align*} g = ( d \\psi + \\hat \\sigma ) ^ 2 + \\hat { g } \\end{align*}"}
+{"id": "7905.png", "formula": "\\begin{align*} | \\mathcal { P } _ k | & = ( 2 k - 1 ) ! ! . \\end{align*}"}
+{"id": "831.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\omega } \\| u ^ k _ { g h } - u ^ k _ g u ^ k _ h \\| & = 0 , g , h \\in G , \\intertext { a n d } \\lim _ { k \\rightarrow \\omega } \\mathrm { t r } _ G ( u ^ k _ g ) & = 0 , g \\in G \\setminus \\{ 1 \\} . \\end{align*}"}
+{"id": "8702.png", "formula": "\\begin{align*} h ( x ) = \\sum ^ { l } _ { s = 0 } \\frac { h ^ { ( s ) } ( 0 ) x ^ s } { s ! } + c _ { l + 1 , a } ( x ) x ^ { l + 1 } , \\sup _ { x \\geq - a } | c _ { l + 1 , a } ( x ) | \\leq C , \\end{align*}"}
+{"id": "1776.png", "formula": "\\begin{align*} & \\vec { \\mathcal { A } } = \\mathcal { A } , \\\\ & \\overleftrightarrow { \\mathcal { A } } = \\overleftarrow { \\mathcal { A } } = \\coprod _ { n \\geq 0 } \\{ f : [ n ] \\to [ n ] \\mid f \\in \\mathcal { A } \\} . \\end{align*}"}
+{"id": "8626.png", "formula": "\\begin{align*} P \\left ( \\tau _ N < s _ N ( \\rho ) \\right ) = O \\big ( N ^ { 2 ( \\rho - 1 ) } \\big ) . \\end{align*}"}
+{"id": "9206.png", "formula": "\\begin{align*} W ( D ) & \\ge W ( H ) + \\sum _ { i = n _ 0 + 1 } ^ { n } \\left ( 2 i - 2 + a \\right ) \\\\ & > 2 \\binom { n _ 0 } { 2 } + 2 \\sum _ { i ' = n _ 0 } ^ { n - 1 } i ' + ( n - n _ 0 ) a \\\\ & \\ge 2 \\binom { n } { 2 } + a n - a n _ 0 . \\end{align*}"}
+{"id": "2132.png", "formula": "\\begin{align*} e ^ { - \\phi } = 1 - \\phi + O ( \\phi ^ 2 ) . \\end{align*}"}
+{"id": "6650.png", "formula": "\\begin{align*} \\mathcal { L } ^ s ( \\R ^ N ) : = \\Big \\{ f : \\R ^ N \\to \\R : \\ , \\| u \\| _ { 1 , s } : = \\int _ { \\R ^ N } \\frac { | f ( x ) | } { 1 + | x | ^ { N + 2 s } } \\ , d x < \\infty \\Big \\} . \\end{align*}"}
+{"id": "5386.png", "formula": "\\begin{align*} \\begin{bmatrix} x _ { 1 , k + 1 } \\\\ y _ { 1 , k } ^ 1 \\end{bmatrix} = \\begin{bmatrix} A _ 1 & B _ 1 \\\\ C _ 1 & D _ 1 \\end{bmatrix} \\begin{bmatrix} x _ { 1 , k } \\\\ u _ { 1 , k } ^ 1 \\end{bmatrix} , \\begin{bmatrix} x _ { 2 , k + 1 } \\\\ y _ { 2 , k } ^ 1 \\end{bmatrix} = \\begin{bmatrix} A _ 2 & B _ 2 \\\\ C _ 2 & D _ 2 \\end{bmatrix} \\begin{bmatrix} x _ { 2 , k } \\\\ u _ { 2 , k } ^ 1 \\end{bmatrix} , \\end{align*}"}
+{"id": "7681.png", "formula": "\\begin{align*} \\Lambda _ { , 2 } ( k , p ) = \\left ( \\frac { n - p } { p } \\right ) ^ p \\prod _ { s = 1 } ^ k \\left ( \\frac { n } { p } - 2 s - 1 \\right ) ^ p \\left ( \\frac { n ( p - 1 ) } { p } + 2 s - 1 \\right ) ^ p . \\end{align*}"}
+{"id": "3394.png", "formula": "\\begin{align*} p _ i ( x _ i ) : = \\frac { 1 } { 1 + e ^ { \\alpha _ i h _ i + \\beta _ i x _ i + \\gamma _ i } } , \\end{align*}"}
+{"id": "562.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial ^ { 2 } _ { t } v ( t , x ) + a ( t ) \\mathcal { H } _ { V } v ( t , x ) + q ( t ) v ( t , x ) = f ( t , x ) , ( t , x ) \\in ( 0 , T ] \\times \\mathbb { R } ^ { n } , \\\\ v ( 0 , x ) = u _ { 0 } ( x ) , x \\in \\mathbb { R } ^ { n } , \\\\ \\partial _ { t } v ( 0 , x ) = u _ 1 ( x ) , x \\in \\mathbb { R } ^ { n } , \\end{array} \\right . \\end{align*}"}
+{"id": "7739.png", "formula": "\\begin{align*} y ( Z , t ) = y ( Z , t _ { m } ) + \\alpha _ { m } ( Z ^ { t _ { m } } ) ( t - t _ { m } ) . \\end{align*}"}
+{"id": "4064.png", "formula": "\\begin{align*} S _ { \\mathrm { m a i n } } \\geq \\sum _ { L = 1 } ^ { \\infty } \\phi ( L ) | y _ L | ^ 2 \\bigg ( { \\frac { ( ( k - 1 ) ! ) ^ 2 } { 2 } } - \\frac { 1 } { 2 } + \\frac { \\delta } { 2 } \\bigg ) \\log p + \\sum _ { L = 1 } ^ { \\infty } \\phi ( L ) | y _ L | ^ 2 ( c _ 0 - c _ 1 ) . \\end{align*}"}
+{"id": "7179.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 \\left ( J \\cap B _ { 2 s \\rho } \\right ) \\geq \\mathcal { H } ^ 1 \\left ( J \\cap B _ { 2 r _ x } ( x ) \\right ) = \\mathcal { H } ^ 1 \\left ( J \\cap B _ { \\lambda _ x } ( x ) \\right ) \\geq \\eta \\lambda _ x > \\eta ( 1 - s ) \\rho , \\end{align*}"}
+{"id": "3031.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T T ^ * + x ^ 2 S S ^ * ) \\leq \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T T ^ * ) + \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( x ^ 2 S S ^ * ) \\end{align*}"}
+{"id": "6215.png", "formula": "\\begin{align*} G = \\bigsqcup _ { i = 1 } ^ d g _ { i } H . \\end{align*}"}
+{"id": "2555.png", "formula": "\\begin{align*} \\mathbf { v } \\cdot \\mathbf { n } = 0 \\ \\ \\ \\ \\ , \\partial D , \\end{align*}"}
+{"id": "144.png", "formula": "\\begin{align*} \\tilde { \\chi } _ j \\Pi _ \\lambda ( 1 - \\chi _ j ) u = 0 . \\end{align*}"}
+{"id": "7857.png", "formula": "\\begin{align*} f \\left ( \\sigma ( w ) \\right ) : = \\delta _ { \\sigma K } - \\frac { | \\mathcal { S } | } { | \\Omega | } \\sum _ { x K \\in G / K } \\delta _ { \\sigma x K } . \\end{align*}"}
+{"id": "6862.png", "formula": "\\begin{align*} \\mathcal { R } \\big ( a \\mid b \\big ) = a \\log \\tfrac { a } { b } + ( 1 - a ) \\log \\tfrac { 1 - a } { 1 - b } \\end{align*}"}
+{"id": "1364.png", "formula": "\\begin{align*} | \\Theta _ h ( t , y ^ * ) - \\Theta _ h ( s , z ^ * ) | & = | y ^ * ( h ( t ) ) - z ^ * ( h ( s ) ) | \\\\ & \\leq | y ^ * ( h ( t ) ) - y ^ * ( h ( s ) ) | + | J _ Y ( h ( s ) ) ( y ^ * ) - J _ Y ( h ( s ) ) ( z ^ * ) | \\\\ & \\leq \\| h ( t ) - h ( s ) \\| + | J _ Y ( h ( s ) ) ( y ^ * ) - J _ Y ( h ( s ) ) ( z ^ * ) | , \\end{align*}"}
+{"id": "5915.png", "formula": "\\begin{align*} \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 ^ { \\alpha } , g _ 2 ^ { \\alpha } ) = \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) \\nu ( \\alpha , g _ 1 ) ^ { - 1 } \\nu ( \\alpha , g _ 2 ) ^ { - 1 } \\nu ( \\alpha , g _ 1 g _ 2 ) . \\end{align*}"}
+{"id": "3049.png", "formula": "\\begin{align*} \\left ( 1 + \\log ( n - 1 ) \\right ) \\frac { 1 } { n } \\leq 1 , \\quad \\sum _ { k = 1 } ^ { n - 2 } \\frac { 1 } { k } \\leq \\log n , \\quad \\sum _ { k = 1 } ^ { n - 2 } \\log k \\leq n \\log n . \\end{align*}"}
+{"id": "5278.png", "formula": "\\begin{align*} ( 1 + y ) ^ q = 1 + q y + \\binom { q } { 2 } ( 1 + y ' ) ^ { q - 2 } ( 1 + y ' ) y ^ 2 \\end{align*}"}
+{"id": "4651.png", "formula": "\\begin{align*} X = \\left \\{ f \\in L ^ { \\infty } ( \\Omega \\times ( 0 , T ) ) \\ , \\bigr | \\ , f \\leq K + 1 \\ , \\mbox { a . e . i n } \\ , \\Omega \\times ( 0 , T ) \\right \\} . \\end{align*}"}
+{"id": "421.png", "formula": "\\begin{align*} U ^ T _ n \\tilde A U _ n - \\epsilon ( U ^ T _ n \\tilde F + \\tilde F ^ T U _ n ) \\tilde F = ( \\tilde F _ n , \\tilde F _ { \\tau } , 0 ) ^ T \\end{align*}"}
+{"id": "1925.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow + \\infty } \\langle D _ u \\theta ( u ^ { k + 1 } ) , u ^ * - u ^ { k + 1 } \\rangle _ \\mathcal { U } = 0 . \\end{align*}"}
+{"id": "7998.png", "formula": "\\begin{align*} \\int _ \\Omega f \\ , d x = 0 . \\end{align*}"}
+{"id": "9087.png", "formula": "\\begin{align*} a ^ - = n - z - \\overline { \\mathfrak { S } } ( f _ k ) . \\end{align*}"}
+{"id": "3203.png", "formula": "\\begin{align*} u _ n ( x \\Omega ) = \\sigma _ n ( x ) \\Omega , ~ ~ x \\in M . \\end{align*}"}
+{"id": "7872.png", "formula": "\\begin{align*} \\tau ( ( \\sigma ( 1 ) , \\sigma ( 2 ) , \\ldots , \\sigma ( k ) ) ) = ( 1 , 2 , \\ldots , k ) = \\hat { A } . \\end{align*}"}
+{"id": "7992.png", "formula": "\\begin{align*} \\nabla w = ( \\partial _ \\nu w ) \\ , \\nu \\quad \\partial \\Omega , \\end{align*}"}
+{"id": "5690.png", "formula": "\\begin{align*} \\widehat { S _ { k } } ^ { \\# , 0 } ( \\Gamma _ { 0 } ( N ) ) : = \\dfrac { S _ { k } ^ { \\# , 0 } ( \\Gamma _ { 0 } ( N ) ) } { D ^ { k - 1 } ( M _ { 2 - k } ^ { \\# } ( \\Gamma _ { 0 } ( N ) ) ) \\oplus S _ { k } ( \\Gamma _ { 0 } ( N ) ) } . \\end{align*}"}
+{"id": "4595.png", "formula": "\\begin{align*} Q = \\left \\{ \\begin{array} { l r } \\min \\{ S _ k - x , B \\} & s _ { k - 1 } < x \\leq s _ k , \\\\ 0 & x > s _ { m } ; \\end{array} \\right . \\end{align*}"}
+{"id": "6058.png", "formula": "\\begin{align*} m ( x _ 1 - x _ 3 ) + ( n + m ) ( x _ 2 - x _ 3 ) = 0 \\end{align*}"}
+{"id": "2221.png", "formula": "\\begin{align*} \\psi ( x , y ) = \\psi ( x , z ) \\frac { Q ( z ) } { Q ( y ) } + \\sum _ { y < p \\le z } \\psi ( x / p , p ^ - ) \\cdot \\frac { 1 } { p } \\prod _ { y < q < p } \\left ( 1 - \\frac { 1 } { q } \\right ) , \\end{align*}"}
+{"id": "5085.png", "formula": "\\begin{align*} x = A _ t ( A _ t ^ { - 1 } ( x ) ) = A _ t ^ { - 1 } ( x ) - \\gamma ( A _ t ^ { - 1 } ( x ) , t ) - \\frac { 1 } { N } \\sigma ( t ) \\end{align*}"}
+{"id": "2955.png", "formula": "\\begin{align*} \\frac { \\partial V _ { n + k + 1 } } { \\partial \\bar { c } _ { n - i } } = ( - 1 ) ^ { n - i + 1 } c _ { k + i + 1 } \\ . \\end{align*}"}
+{"id": "7034.png", "formula": "\\begin{align*} \\tilde Q _ \\ell = \\sum \\limits _ { j = 1 } ^ { s _ \\ell } b _ { \\ell i j } \\tilde { \\textbf { Q } } ^ { \\lambda _ j } \\end{align*}"}
+{"id": "5702.png", "formula": "\\begin{align*} \\xi _ { 2 - k } ( H ) = \\xi _ { 2 - k } ( f ) = \\dfrac { g } { \\left \\| g \\right \\| ^ { 2 } } \\not = 0 . \\end{align*}"}
+{"id": "3786.png", "formula": "\\begin{align*} f _ { \\mathcal J } ( a ) = \\begin{cases} a , & n \\notin J _ a , \\\\ n + a + 1 - x , & n \\in J _ a , J _ { a + 1 } = \\tau _ 1 ( J _ a \\setminus \\{ n \\} ) \\cup \\{ x \\} . \\end{cases} \\end{align*}"}
+{"id": "5210.png", "formula": "\\begin{align*} X \\qquad \\Leftrightarrow \\qquad \\mathcal { C } = \\mathsf { C } \\mathbb { P } \\qquad \\Leftrightarrow \\qquad \\mathcal { C } \\mathsf { C } \\mathbb { P } . \\end{align*}"}
+{"id": "8404.png", "formula": "\\begin{align*} ( E _ 1 = M \\oplus K _ X ^ { n - 1 } \\oplus K _ X ^ { n - 2 } \\oplus \\cdots \\oplus K _ X ^ { 2 - n } \\oplus K _ X ^ { 1 - n } \\oplus M ^ { - 1 } , \\quad \\theta _ 1 = \\begin{pmatrix} 0 & & & & & \\\\ 0 & 0 & & & & \\\\ & 1 & 0 & & & \\\\ & & \\ddots & \\ddots & \\\\ & & & 1 & 0 & \\\\ & & & & 0 & 0 \\end{pmatrix} ) . \\end{align*}"}
+{"id": "6920.png", "formula": "\\begin{align*} J ( m ' ) : = [ ( 1 + T ^ { - 1 } ) ^ { j } , ( 1 + T ^ { - 1 } ) ^ { j + 1 } ) , \\end{align*}"}
+{"id": "5911.png", "formula": "\\begin{align*} \\pi _ { \\psi } [ ( x , 0 ) + ( x ^ { \\ast } , 0 ) + ( 0 , k ) ] f ( y ) = \\psi ( k + \\langle x + y , x ^ { \\ast } \\rangle ) f ( x + y ) , \\end{align*}"}
+{"id": "2047.png", "formula": "\\begin{align*} D = 1 - \\Delta . \\end{align*}"}
+{"id": "3183.png", "formula": "\\begin{align*} M _ n ( f ) : = \\frac { 1 } { n ^ d } \\sum _ { ( j _ 1 , \\ldots , j _ d ) \\in \\Z ^ d _ + \\cap Q _ n } S _ 1 ^ { j _ 1 } \\cdots S _ d ^ { j _ d } ( M _ 1 ( f ) ) \\end{align*}"}
+{"id": "618.png", "formula": "\\begin{align*} K \\subset \\bigcup _ { i = 1 } ^ r \\{ \\psi \\in \\S ( B ) : \\psi ( f _ { \\lambda _ i } ) < 1 - 1 / { m _ i } \\} \\subset \\{ \\psi \\in \\S ( B ) : \\psi ( f ) < \\gamma \\} . \\end{align*}"}
+{"id": "5244.png", "formula": "\\begin{align*} \\langle h _ 3 \\rangle = W _ 1 , \\langle h _ 2 , h _ 3 \\rangle = W _ 2 , \\langle h _ 1 , h _ 2 , h _ 3 \\rangle = W _ 3 . \\end{align*}"}
+{"id": "3703.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) _ j u _ j + q u _ j : = \\sum _ { i = 1 } ^ M \\alpha _ i ^ j ( - \\Delta ) ^ { s _ i } u _ j + q u _ j = 0 & \\Omega , \\\\ u _ j = f _ j & \\Omega ^ c \\end{cases} \\end{align*}"}
+{"id": "1429.png", "formula": "\\begin{align*} \\xi _ i = \\xi + \\mu _ i \\sigma _ i , \\quad \\mbox { f o r \\ s o m e \\ p o i n t s } \\ \\sigma _ i \\in \\R ^ n , \\end{align*}"}
+{"id": "8195.png", "formula": "\\begin{align*} P \\left ( M _ t > \\gamma t \\right ) = P \\left ( \\exists \\ , u \\ : \\in \\mathcal { N } _ t : \\sup _ { 0 \\le s \\le t } | Y _ u ( s ) | > \\gamma t \\right ) \\le E [ N _ t ] \\ : \\mathbf { P } _ 0 \\left ( \\sup _ { 0 \\le s \\le t } | X _ s | > \\gamma t \\right ) . \\end{align*}"}
+{"id": "8764.png", "formula": "\\begin{align*} \\beta _ i \\in \\begin{cases} \\{ 0 \\} & \\mbox { i f } d _ i - x _ i < y _ i \\\\ \\{ p _ i \\} & \\mbox { i f } d _ i - x _ i > y _ i \\\\ [ 0 , p _ i ] & \\mbox { i f } d _ i - x _ i = y _ i \\end{cases} . \\end{align*}"}
+{"id": "7866.png", "formula": "\\begin{align*} - \\frac { \\binom { n - k } { k } \\binom { n - 1 } { k - 1 } } { \\binom { n - 1 } { k } } . \\end{align*}"}
+{"id": "4964.png", "formula": "\\begin{align*} n ( 1 - \\frac { 1 } { n } ) ^ t \\sum _ { k = 0 } ^ { n - 1 } \\Pr ( \\widetilde { X } _ t = k \\ , \\vert \\ , n - 1 ) = n ( 1 - \\frac { 1 } { n } ) ^ t . \\end{align*}"}
+{"id": "2463.png", "formula": "\\begin{align*} m _ 2 = 2 m _ 1 . \\end{align*}"}
+{"id": "8852.png", "formula": "\\begin{align*} u _ T ( x ) \\prod _ { k = 1 } ^ { { T \\over 2 } } ( f ( x ) - v _ { 2 k - 2 } ^ 2 ( x ) ) & \\sim u _ 0 ( x ) \\prod _ { k = 1 } ^ { { T \\over 2 } } ( f ( x ) - v _ { 2 k - 1 } ^ 2 ( x ) ) \\\\ u _ T ( x ) u _ 0 ( x ) \\prod _ { k = 1 } ^ { \\lfloor { T \\over 2 } \\rfloor } ( f ( x ) - v _ { 2 k - 1 } ^ 2 ( x ) ) & \\sim \\prod _ { k = 0 } ^ { \\lfloor { T \\over 2 } \\rfloor } ( f ( x ) - v _ { 2 k } ^ 2 ( x ) ) . \\end{align*}"}
+{"id": "2632.png", "formula": "\\begin{align*} \\begin{cases} u = - \\widetilde { P } _ 1 ( t _ 1 ) + \\widetilde { P } _ 1 ( t _ 2 ) - \\widetilde { P } _ 1 ( t _ 3 ) , \\\\ v = \\widetilde { P } _ 2 ( t _ 1 ) - \\widetilde { P } _ 2 ( t _ 2 ) + \\widetilde { P } _ 2 ( t _ 3 ) , \\\\ w = t _ 1 . \\end{cases} \\end{align*}"}
+{"id": "5224.png", "formula": "\\begin{align*} \\Tilde { z } _ { k , \\mathrm { a v g } } & = \\sum _ { k ' = k - K } ^ { k + K } \\left ( \\sum _ { l = 0 } ^ { L - 1 } z ' _ { k ' , l } \\right ) ^ { \\mu } \\\\ \\varphi _ { k , \\mathrm { e s t } , \\mathrm { m o d } } & = \\frac { 1 } { \\mu } \\mathrm { u n w r a p } \\left ( \\arg \\left ( \\Tilde { z } _ { k , \\mathrm { a v g } } \\right ) \\right ) . \\end{align*}"}
+{"id": "3026.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T ^ * T + x ^ 2 S ^ * S ) \\leq \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( T ^ * T ) + \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\frac { p } { 2 } } ( x ^ 2 S ^ * S ) \\end{align*}"}
+{"id": "5685.png", "formula": "\\begin{align*} f = \\sum _ { n \\gg - \\infty } C _ { f } ^ { + } ( n ) q ^ { n } + \\sum _ { n < 0 } C _ { f } ^ { - } ( n ) \\Gamma ( k - 1 , 4 \\pi \\left | n \\right | y ) q ^ { n } . \\end{align*}"}
+{"id": "1128.png", "formula": "\\begin{align*} \\delta _ r ( z ) = E _ I ( 0 , \\delta _ r ( L o g ( z ) ) ) . \\end{align*}"}
+{"id": "7069.png", "formula": "\\begin{align*} \\mathcal S = \\{ i \\in I \\mid \\alpha \\neq \\alpha ( \\{ \\beta _ j \\mid j \\geq i \\} ) \\} . \\end{align*}"}
+{"id": "2897.png", "formula": "\\begin{align*} ( f \\sharp g ) _ { m + m ' } ( \\xi ) = f _ m ( \\xi ) g _ { m ' } ( \\xi ) . \\end{align*}"}
+{"id": "2273.png", "formula": "\\begin{align*} w ( z ) = \\Phi ( z ) + T ( f ) ( z ) , \\end{align*}"}
+{"id": "8244.png", "formula": "\\begin{align*} W _ { h , q } ( i ) = \\begin{cases} h + 1 & i = q ; \\\\ 1 & k - j + h \\leq i \\leq k - 2 i \\neq q ; \\\\ l - j + 1 & i = k - 1 ; \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "3768.png", "formula": "\\begin{align*} v \\cdot u = & \\sinh ( \\phi - \\theta ) = \\left ( \\begin{array} { c } \\cosh ( \\phi - \\theta ) \\\\ \\sinh ( \\phi - \\theta ) \\\\ \\end{array} \\right ) \\cdot \\left ( \\begin{array} { c } 0 \\\\ 1 \\\\ \\end{array} \\right ) , \\end{align*}"}
+{"id": "4430.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( \\Psi _ { \\omega } ) = \\omega ^ { 2 - \\frac { d } { 2 } } S _ { 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } } ( \\Psi ) . \\end{align*}"}
+{"id": "3070.png", "formula": "\\begin{align*} r ^ \\delta u ' ( r ) ^ { p - 1 - \\alpha } = \\frac { \\delta } { n - 1 } \\int _ 0 ^ r s ^ \\delta f _ 1 ( s ) g _ 1 ( v ( s ) ) \\ , d s \\leq \\frac { \\delta } { n - 1 } \\ , f _ 1 ( r ) g _ 1 ( v ( r ) ) \\int _ 0 ^ r s ^ \\delta \\ , d s \\end{align*}"}
+{"id": "1649.png", "formula": "\\begin{align*} g ( \\nabla _ { X } \\ , \\xi _ i , \\ \\xi _ k ) = 0 , X \\in T M , \\ 1 \\le i , k \\le p . \\end{align*}"}
+{"id": "2360.png", "formula": "\\begin{align*} u _ i - u _ j + u _ j - u _ k = u _ i - u _ k \\rlap { . } \\end{align*}"}
+{"id": "4907.png", "formula": "\\begin{align*} \\left [ P \\right ] _ { i , j } ^ { } = \\begin{dcases} q _ i ^ { } & \\mbox { i f } \\ , j = i ; \\\\ p _ i ^ { } & \\mbox { i f } \\ , j = i + 1 ; \\\\ 0 & \\mbox { o t h e r w i s e } ; \\end{dcases} \\end{align*}"}
+{"id": "6987.png", "formula": "\\begin{align*} f = f _ 0 + f _ 1 q + \\ldots + f _ r q ^ r \\mbox { w i t h } \\deg ( f _ i ) < \\deg ( q ) \\mbox { f o r e v e r y } i , 0 \\leq i \\leq r , \\end{align*}"}
+{"id": "6745.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\frac { 1 } { \\ell _ i } \\sum _ { n \\leq \\ell _ i } \\mathbf { 1 } _ { \\underline { C } _ K } ( R ^ n ( \\Delta ( 0 ) ) ) \\mathbf { 1 } _ { C _ K } ( R ^ n ( \\Delta ( 0 ) ) ) = m _ H ( \\underline { C } _ K \\cap C _ K ) . \\end{align*}"}
+{"id": "1756.png", "formula": "\\begin{align*} i \\frac { \\partial \\psi _ 1 } { \\partial t } & = \\Big ( - \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial x ^ 2 } + \\frac { \\lambda _ 1 ^ 2 } { 2 } x ^ 2 + b _ { 1 1 } | \\psi _ 1 | ^ 2 + b _ { 1 2 } | \\psi _ 2 | ^ 2 ) \\Big ) \\psi _ 1 , \\ \\\\ i \\frac { \\partial \\psi _ 2 } { \\partial t } & = \\Big ( - \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial x ^ 2 } + \\frac { \\lambda _ 2 ^ 2 } { 2 } x ^ 2 + b _ { 2 1 } | \\psi _ 1 | ^ 2 + b _ { 2 2 } | \\psi _ 2 | ^ 2 ) \\Big ) \\psi _ 2 , \\ \\end{align*}"}
+{"id": "4915.png", "formula": "\\begin{align*} \\Pr ( \\widetilde { \\mathbf { X } } _ { t } = \\widetilde { \\mathbf { x } } ^ * _ { t } ) = p _ 0 \\cdot p _ 1 \\cdots p _ { k - 1 } \\cdot q _ 0 ^ { c _ 0 ^ { * } } \\cdot q _ 1 ^ { c _ 1 ^ { * } } \\cdots q _ k ^ { c _ k ^ { * } } . \\end{align*}"}
+{"id": "2924.png", "formula": "\\begin{align*} \\frac { t e ^ { x t } } { e ^ t - 1 } = \\sum _ { n = 0 } ^ \\infty B _ n ( x ) \\cdot \\frac { t ^ n } { n ! } . \\end{align*}"}
+{"id": "8660.png", "formula": "\\begin{align*} \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } ( 1 - y ) d y & = - \\frac { \\log ( q ) } p - \\frac { 1 } { p ^ 2 } \\big ( - \\log ( q ) - p \\big ) \\\\ & = \\frac { q } { p ^ 2 } \\log ( q ) + \\frac { 1 } { p } . \\end{align*}"}
+{"id": "8174.png", "formula": "\\begin{align*} \\Omega _ s = \\{ \\omega \\in \\Omega : \\exists \\ : \\ell _ 1 = \\ell _ 1 ( \\omega ) , \\ : \\forall \\ : \\ell \\geq \\ell _ 1 , \\ : [ - \\ell , \\ell ] ^ d \\ : \\ : R _ \\ell \\} . \\end{align*}"}
+{"id": "4523.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } S _ { \\omega , \\mathbf { c } } ( V ^ { \\lambda } ) = ( 2 L ( V ) + 2 N ( V ) ) \\lambda + \\mathbf { c } \\cdot P ( V ) = - ( \\lambda - 1 ) \\mathbf { c } \\cdot P ( V ) \\end{align*}"}
+{"id": "3666.png", "formula": "\\begin{align*} \\mathcal { M } _ { q _ 1 , \\alpha } f = \\mathcal { M } _ { q _ 2 , \\alpha } f \\end{align*}"}
+{"id": "7499.png", "formula": "\\begin{align*} \\hat { \\nabla } = D _ { \\theta } + \\sum ^ r _ { i = 1 } g _ i ( z , \\theta ) \\check { \\alpha } _ i + \\sum ^ r _ { i = 1 } { a _ i ( z , \\theta ) } e _ i , \\end{align*}"}
+{"id": "3909.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta _ 1 , 0 ) = \\widehat { \\mathcal { I } } _ { \\mathrm { D } } ( \\delta _ 1 , \\delta _ 2 ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\varpi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\int _ { \\mathcal { V } } g _ \\lambda \\left ( s _ 1 , s _ 2 \\right ) d \\varpi ( s _ 1 , s _ 2 ) \\right ] , \\end{align*}"}
+{"id": "8601.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { - 1 } S _ j ( \\tau _ N ) = \\nu \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s , \\end{align*}"}
+{"id": "1447.png", "formula": "\\begin{align*} \\liminf _ { m \\rightarrow \\infty } \\int _ \\Omega f _ \\epsilon ^ { ' } ( V _ m ) u _ m ^ 2 d x = C > 0 , \\end{align*}"}
+{"id": "8414.png", "formula": "\\begin{align*} \\zeta _ { Q } \\left ( s \\right ) = \\frac { 2 \\pi } { \\sqrt { | d | } } \\ , \\frac { 1 } { s - 1 } + \\frac { 2 \\pi } { \\sqrt { | d | } } \\ , \\left ( 2 \\gamma - \\log \\left ( \\frac { | d | } { a } \\right ) - 4 \\log \\ , ( | \\eta ( \\tau ) | ) \\right ) + O ( s - 1 ) , \\end{align*}"}
+{"id": "7834.png", "formula": "\\begin{align*} s _ { 1 } ( A ) = \\sup _ { x \\in \\mathbb { S } ^ { n - 1 } } \\Vert A x \\Vert _ { 2 } , s _ { n } ( A ) = \\inf _ { x \\in \\mathbb { S } ^ { n - 1 } } \\Vert A x \\Vert _ { 2 } . \\end{align*}"}
+{"id": "6269.png", "formula": "\\begin{align*} D ( t , \\mu ) : = \\Sigma _ t \\setminus R ( \\Sigma _ t ' , \\mu ) \\end{align*}"}
+{"id": "2709.png", "formula": "\\begin{align*} T _ 5 \\geq & - C s ^ 2 \\lambda ^ 4 \\iint _ Q \\xi \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d t - C \\lambda ^ 2 \\iint _ { Q } \\xi \\left | A \\nabla u \\cdot \\nabla \\eta \\right | ^ 2 d x d t \\\\ & - C s ^ 2 \\lambda ^ 3 \\int _ 0 ^ T \\int _ { \\omega } \\xi | u | ^ 2 d x d t - C \\lambda \\iint _ Q \\xi A \\nabla u \\cdot \\nabla u d x d t . \\end{align*}"}
+{"id": "6412.png", "formula": "\\begin{align*} \\widetilde { M } _ i : = M _ i \\bar { \\rtimes } _ { \\sigma ^ { \\varphi _ i } } \\mathbb { R } , \\widetilde { M _ 1 \\bar { \\otimes } M _ 2 } : = ( M _ 1 \\bar { \\otimes } M _ 2 ) \\bar { \\rtimes } _ { \\sigma ^ { \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 } } \\mathbb { R } \\end{align*}"}
+{"id": "9245.png", "formula": "\\begin{align*} ( - 1 ) ^ n \\frac { E _ { 2 n } } { ( 2 n ) ! } \\left ( \\frac { \\pi } { 2 } \\right ) ^ { 2 n } = \\frac { 4 } { \\pi } \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { ( 2 k + 1 ) ^ { 2 n + 1 } } \\end{align*}"}
+{"id": "7432.png", "formula": "\\begin{align*} \\partial _ t u _ \\varepsilon - \\varepsilon \\ , \\Delta _ x u _ \\varepsilon + \\mathrm { d i v } _ x \\big ( \\overrightarrow { V _ \\varepsilon } \\ , u _ \\varepsilon \\big ) = 0 , \\end{align*}"}
+{"id": "4506.png", "formula": "\\begin{align*} i \\partial _ t u + \\partial _ x ^ 2 u + i \\partial _ x ( | u | ^ 2 u ) = 0 , \\ \\ \\ ( t , x ) \\in \\R \\times \\R , \\end{align*}"}
+{"id": "6824.png", "formula": "\\begin{align*} \\beta _ n = \\sum _ { j \\leq n } \\frac { \\alpha _ j } { ( q ) _ { n - j } ( a q ) _ { n + j } } \\ ; \\ ; \\ ; \\ ; \\forall \\ , n \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "5225.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k a _ i \\leq \\sum _ { i = 1 } ^ k b _ i \\end{align*}"}
+{"id": "3033.png", "formula": "\\begin{align*} \\phi ( E _ { i i } \\otimes E _ { j j } ) = E _ { i i } \\otimes E _ { j j } \\end{align*}"}
+{"id": "7891.png", "formula": "\\begin{align*} \\mathcal { S } = \\left \\{ \\left ( A ^ \\pm , \\underline { A } ^ \\pm \\right ) : \\ A \\in \\binom { [ n ] } { k } \\mbox { a n d } n \\in A \\right \\} . \\end{align*}"}
+{"id": "2428.png", "formula": "\\begin{align*} y ( q x ) = h ( x ) y ( x ) \\quad h ( x ) \\in \\mathcal O ( \\C ) \\backslash \\{ 0 \\} \\end{align*}"}
+{"id": "1332.png", "formula": "\\begin{align*} \\frac { 1 } { e _ i ^ * ( u ) } = \\frac { \\theta _ i } { e ^ { \\rho _ i ( u ) } } - \\frac { \\theta _ i Z _ i ( u ) } { T _ i ^ { \\infty } } . \\end{align*}"}
+{"id": "5803.png", "formula": "\\begin{align*} m ( r _ n , \\frac { 1 } { A } ) & = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\log \\left | \\frac { 1 } { A ( r e ^ { \\iota \\theta } ) } \\right | d \\theta \\\\ & = 0 . \\end{align*}"}
+{"id": "2605.png", "formula": "\\begin{align*} E ^ * : = \\{ y : y \\cdot x \\leq h \\ \\ x \\in E \\} . \\end{align*}"}
+{"id": "694.png", "formula": "\\begin{align*} & p _ a ( S ( l ) D \\varphi ) \\leq C p _ a ( D \\varphi ) 0 < a < 1 , \\\\ & p _ a ( S ( l ) D \\varphi ) \\leq C ( 1 + \\log \\| Q ( l ) \\| ) p _ a ( D \\varphi ) a = 0 . \\end{align*}"}
+{"id": "9339.png", "formula": "\\begin{align*} \\begin{aligned} & \\big ( 1 + | x | ^ 2 + | y | ^ 2 + | z | ^ 2 + | \\tilde { z } | ^ 2 + | | \\gamma | | ^ 2 + | u | ^ 2 \\big ) ^ { - 1 } | f ( x , y , z , \\tilde { z } , \\gamma , u ) | \\\\ & + \\big ( 1 + | x | + | y | + | z | + | \\tilde { z } | + | | \\gamma | | + | u | \\big ) ^ { - 1 } | f _ { ( x , y , z , \\tilde { z } , \\gamma , u ) } ( x , y , z , \\tilde { z } , \\gamma , u ) | \\leq C , \\\\ & \\big ( 1 + | y | ^ 2 \\big ) ^ { - 1 } | \\phi ( y ) | + \\big ( 1 + | y | \\big ) ^ { - 1 } | \\phi _ y ( y ) | \\leq C . \\end{aligned} \\end{align*}"}
+{"id": "1984.png", "formula": "\\begin{align*} ( d d ^ c u ) ^ n = \\psi ( \\cdot , u ) \\omega ^ n . \\end{align*}"}
+{"id": "2629.png", "formula": "\\begin{align*} \\Omega ( \\varepsilon ) : = \\{ \\mathbf { t } \\in I ^ 3 : | F ( \\mathbf { t } ) | \\leq \\varepsilon \\} \\textrm { f o r a n y $ \\varepsilon > 0 $ } . \\end{align*}"}
+{"id": "7079.png", "formula": "\\begin{align*} \\nu ( g ' ) = \\nu _ i ( g ' ) = \\nu ( g ' ( a _ i ) ) \\mbox { f o r e v e r y } i \\in I ^ * \\mbox { w i t h } i \\geq i _ 0 . \\end{align*}"}
+{"id": "5787.png", "formula": "\\begin{align*} F = \\frac { f ' } { f } , \\end{align*}"}
+{"id": "1121.png", "formula": "\\begin{align*} f ( g ( x y ) ) + ( f ( x y ) ) g + \\alpha _ 1 \\big ( f ( g ( x y ) ) + f ( ( x y ) g ) \\big ) + \\alpha _ 2 \\big ( g ( f ( x y ) ) + g ( ( x y ) f ) \\big ) \\\\ = ( f ( g x ) ) y + ( ( f x ) g ) y + \\alpha _ 1 \\big ( ( f ( g x ) ) y + ( f ( x g ) ) y \\big ) + \\alpha _ 2 \\big ( ( g ( f x ) ) y + ( g ( x f ) ) y \\big ) \\\\ - ( f ( g y ) ) x + ( ( f y ) g ) x + \\alpha _ 1 \\big ( ( f ( g y ) ) x + ( f ( y g ) ) x \\big ) + \\alpha _ 2 \\big ( ( g ( f y ) ) x + ( g ( y f ) ) x \\big ) . \\end{align*}"}
+{"id": "5816.png", "formula": "\\begin{align*} { { \\hat V } _ { \\alpha , \\beta } } \\left ( { { \\boldsymbol { X } } , { \\boldsymbol { Y } } } \\right ) = { { \\hat V } _ { \\alpha , \\beta } } \\left ( { \\boldsymbol { e } } \\right ) = \\frac { 1 } { { { N ^ 2 } } } \\sum \\limits _ { i = 1 } ^ N { \\sum \\limits _ { j = 1 } ^ N { { G _ { \\alpha , \\beta } } \\left ( { { e _ i } - { e _ j } } \\right ) } } , \\end{align*}"}
+{"id": "4155.png", "formula": "\\begin{align*} \\vec { v } ( 1 ) = \\overline { A _ { 1 , 0 , 1 } A _ { 2 , 2 , 1 } A _ { 3 , 0 , 1 } A _ { 1 , 1 , 2 } A _ { 2 , 1 , 0 } A _ { 3 , 1 , 2 } } . \\end{align*}"}
+{"id": "7829.png", "formula": "\\begin{align*} \\textsf { E } \\Vert B A \\Vert ^ { p } = \\textsf { E } \\Vert B A \\Vert ^ { p } \\mathbb { I } _ { ( M \\le C _ { 1 } ( \\varepsilon ) ) } + \\sum _ { k = 1 } ^ { \\infty } \\textsf { E } \\Vert B A \\Vert ^ { p } \\mathbb { I } _ { ( 2 ^ { k - 1 } C _ { 1 } ( \\varepsilon ) < M \\le 2 ^ { k } C _ { 1 } ( \\varepsilon ) ) } . \\end{align*}"}
+{"id": "7808.png", "formula": "\\begin{align*} \\textsf { E } \\Vert B A \\Vert ^ { p } & = \\textsf { E } \\Vert B ( A - \\textsf { E } \\tilde { A } ) \\Vert ^ { p } = \\textsf { E } \\Vert \\textsf { E } ( B ( A - \\tilde { A } ) | A ) \\Vert ^ { p } \\le \\textsf { E } \\Vert B ( A - \\tilde { A } ) \\Vert ^ { p } \\\\ & = \\textsf { E } \\Vert B ( \\varepsilon _ { i j } ( a _ { i j } - \\tilde { a } _ { i j } ) ) _ { N \\times n } \\Vert ^ { p } \\le 2 ^ { p } \\textsf { E } \\Vert B ( \\varepsilon _ { i j } a _ { i j } ) _ { N \\times n } \\Vert ^ { p } . \\end{align*}"}
+{"id": "4406.png", "formula": "\\begin{align*} \\mu _ { \\omega , \\mathbf { c } } \\le \\frac { 1 } { 6 } L _ { \\omega , \\mathbf { c } } ( \\lambda U ) = \\frac { \\lambda ^ 2 } { 6 } L _ { \\omega , \\mathbf { c } } ( U ) < \\frac { 1 } { 6 } L _ { \\omega , \\mathbf { c } } ( U ) . \\end{align*}"}
+{"id": "6572.png", "formula": "\\begin{align*} \\begin{bmatrix} A ^ * A & A ^ * B \\\\ B ^ * A & B ^ * B \\end{bmatrix} \\end{align*}"}
+{"id": "4493.png", "formula": "\\begin{align*} \\Psi _ { \\omega } ( x ) : = \\omega ^ { \\frac { 1 } { 2 } } \\Psi ( \\omega ^ { \\frac { 1 } { 2 } } x ) . \\end{align*}"}
+{"id": "6852.png", "formula": "\\begin{align*} F _ { \\lambda , k , r } ( c _ 1 , c _ 2 ; b _ 1 , \\dots , b _ \\lambda ; a ; q ) = G _ { \\lambda , k , r } ( c _ 1 , c _ 2 ; b _ 1 , \\dots , b _ \\lambda ; a ; q ) , \\end{align*}"}
+{"id": "6036.png", "formula": "\\begin{align*} h _ 1 ( z ) : = A _ 1 z ^ { n + m } + B _ 1 \\overline { z } ^ m + C _ 1 \\textrm { f o r a l l } z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "6044.png", "formula": "\\begin{align*} g ( z ) : = | A | z ^ { n + m } - | B | \\overline { z } ^ m - | C | , z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "6339.png", "formula": "\\begin{align*} \\left | \\frac { \\partial z } { \\partial \\phi } \\det ( M _ 2 ) \\right | ( \\phi , \\omega , r ; t ) , \\ , \\left | \\frac { \\partial z } { \\partial r } \\det ( M _ 3 ) \\right | ( \\phi , \\omega , r ; t ) = o ( t ^ 5 ) , t \\to 0 . \\end{align*}"}
+{"id": "4240.png", "formula": "\\begin{align*} D _ Y ( X ) & = X + c _ { Y _ 2 } f _ { Y _ 1 } \\Omega ( X , Y _ 2 ) Y _ 2 + c _ { Y _ 1 } f _ { Y _ 2 } \\Omega ( X , Y _ 1 ) Y _ 1 \\\\ & = X - Y _ 2 + 2 n Y _ 1 = X - Y \\\\ D _ Y ( Y ) & = Y \\end{align*}"}
+{"id": "8714.png", "formula": "\\begin{align*} f ( p ^ { - 1 } \\| X _ { i _ 1 } - X _ { i _ 2 } \\| _ { 2 } ^ 2 ) = f ( \\widetilde { X } _ { i _ 1 , i _ 2 } + \\tau _ 1 ) = \\sum ^ { l } _ { s = 0 } \\frac { f ^ { ( s ) } ( \\tau _ 1 ) \\widetilde { X } _ { i _ 1 , i _ 2 } ^ s } { s ! } + c _ { l + 1 , \\tau _ 1 } ( \\widetilde { X } _ { i _ 1 , i _ 2 } ) \\widetilde { X } _ { i _ 1 , i _ 2 } ^ { l + 1 } . \\end{align*}"}
+{"id": "824.png", "formula": "\\begin{gather*} \\# \\mathcal { Z } _ { M , M } = \\sum _ { k + l = 2 } ^ n p _ k ( M ) p _ l ( M ) = \\sum _ { k = 1 } ^ { n - 1 } p _ k ( M ) \\sum _ { l = 1 } ^ { n - k } p _ l ( M ) \\\\ \\leq C \\sum _ { k = 1 } ^ { n - 1 } M ^ { k - 1 } M ^ { n - k } \\leq C n M ^ { n - 1 } \\end{gather*}"}
+{"id": "9036.png", "formula": "\\begin{align*} b _ { i , n + 1 } = b _ { \\eta ( i ) , \\eta ( n + 1 ) } = b _ { i , n } . \\end{align*}"}
+{"id": "6518.png", "formula": "\\begin{align*} F _ X ( x ) & = \\frac { ( 1 - \\beta ^ 2 / \\alpha ^ 2 ) ^ { \\nu + 1 / 2 } } { 2 \\sqrt { \\pi } \\Gamma ( \\nu + 1 / 2 ) } \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { k ! } \\bigg ( \\frac { 2 \\beta } { \\alpha } \\bigg ) ^ k \\Gamma \\bigg ( \\frac { k + 1 } { 2 } \\bigg ) \\Gamma \\bigg ( \\nu + \\frac { k + 1 } { 2 } \\bigg ) \\tilde { G } _ { \\nu + k , \\nu } ( - \\alpha ( x - \\mu ) ) . \\end{align*}"}
+{"id": "3229.png", "formula": "\\begin{align*} H ^ { q } ( P _ { 5 } ) \\simeq \\begin{cases} \\Q & q = 0 \\\\ 0 & q = 1 \\\\ \\Q ^ { 4 } & q = 2 \\\\ 0 & q = 3 \\\\ \\Q ^ { 5 } & q = 2 k \\geq 4 \\\\ \\Q ^ { 2 } & q = 2 k + 1 \\geq 5 . \\end{cases} \\end{align*}"}
+{"id": "8057.png", "formula": "\\begin{align*} f _ A ( x ) = \\mu _ H ( ( [ \\psi ^ { - 1 } ( x ) ] ^ { - 1 } \\cdot A ) \\cap H ) \\ \\ \\ \\ x \\in \\psi ( A ) , \\end{align*}"}
+{"id": "5615.png", "formula": "\\begin{align*} { \\rm I I } _ { x } = - \\sum _ { s \\in G } \\mu ( s ) \\frac { d s \\nu } { d \\nu } ( \\pi ( x ) ) H \\left ( \\xi _ { n - 1 } ^ { s ^ { - 1 } . x } \\right ) . \\end{align*}"}
+{"id": "8551.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n \\left ( e ^ { 2 \\alpha n } + 1 \\right ) } + 2 \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { ( 2 n - 1 ) } \\ , \\frac { 1 } { \\left ( e ^ { \\beta ( 2 n - 1 ) } - 1 \\right ) } = \\frac { \\alpha } { 4 } - \\log ( 2 ) - \\frac { 1 } { 4 } \\log \\left ( \\frac { \\alpha } { \\beta } \\right ) . \\end{align*}"}
+{"id": "3500.png", "formula": "\\begin{align*} \\begin{cases} L _ { \\lambda } [ X ] = - X '' + ( R - \\lambda I ) X = 0 , \\\\ X ( a ) = X ( t ) = 0 \\end{cases} \\end{align*}"}
+{"id": "399.png", "formula": "\\begin{align*} W ^ T \\Lambda W = ( W ^ + ) ^ T \\Lambda ^ + W ^ + - ( W ^ - ) ^ T | \\Lambda ^ - | W ^ - = ( \\sqrt { \\Lambda ^ + } W ^ + ) ^ T ( I - R ^ T R ) ( \\sqrt { \\Lambda ^ + } W ^ + ) \\geq 0 \\end{align*}"}
+{"id": "4838.png", "formula": "\\begin{align*} a ( j ) = \\begin{cases} b , & \\mbox { i f } j = 1 \\\\ f ( r + t ( j - 1 ) ) , & \\mbox { i f } j \\in \\{ 2 , \\cdots , m + 1 \\} . \\end{cases} \\end{align*}"}
+{"id": "4608.png", "formula": "\\begin{align*} \\bar { \\rm P } _ { m , i } = { m \\choose i } \\mathbb { P } ( E _ i ) , \\end{align*}"}
+{"id": "8550.png", "formula": "\\begin{align*} \\zeta _ { p , p ^ { \\prime } } ( s , \\ , c ) = c ^ { - s } \\ , \\zeta _ { p ^ { \\prime } , p } \\left ( s , \\ , \\frac { 1 } { c } \\right ) . \\end{align*}"}
+{"id": "632.png", "formula": "\\begin{align*} S _ x = \\{ \\psi \\in \\S ( B ) : \\psi ( x ^ * x ) = 1 \\} . \\end{align*}"}
+{"id": "8128.png", "formula": "\\begin{align*} R _ { \\mathbf { \\varepsilon } \\bullet } = ( - 1 ) ^ { l ( I ) - 1 } R _ I ( R _ \\emptyset = 1 , \\ R _ { \\bullet } = R _ 1 ) \\end{align*}"}
+{"id": "3046.png", "formula": "\\begin{align*} \\phi \\left ( \\bigotimes \\limits _ { i = 1 } ^ { m - 1 } X _ i E _ { j _ i j _ i } X _ i ^ * \\otimes B \\right ) = W _ X \\left ( \\bigotimes \\limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \\otimes \\varphi _ { j _ 1 , \\ldots , j _ { m - 1 } , X } ( B ) \\right ) W _ X ^ * \\end{align*}"}
+{"id": "380.png", "formula": "\\begin{align*} P U _ t + ( A _ i ( V ) U ) _ { x _ i } + B _ i ( V ) U _ { x _ i } + C ( V ) U = \\epsilon ( D _ i ( U ) ) _ { x _ i } , t \\geq 0 , \\vec x = ( x _ 1 , x _ 2 , . . , x _ k ) \\in \\Omega \\end{align*}"}
+{"id": "3589.png", "formula": "\\begin{align*} L ( m n ) \\ = \\ L ( m ) + L ( n ) - 1 \\ = \\ L ( m ) + L ( n ) - [ m n \\ < \\ 1 0 ^ { L ( m ) + L ( n ) - 1 } ] . \\end{align*}"}
+{"id": "5277.png", "formula": "\\begin{align*} \\| 1 + d Z \\| _ { q } ^ { q } \\ge 1 + \\binom { \\lfloor q \\rfloor } { 2 } d ^ 2 \\ge 1 + \\frac { q ^ 2 } { 6 } d ^ 2 . \\end{align*}"}
+{"id": "4712.png", "formula": "\\begin{align*} \\mathcal { A } = \\Re \\mathcal { A } + i \\Im \\mathcal { A } , \\Re \\mathcal { A } = \\frac { \\mathcal { A } + \\mathcal { A } ^ * } { 2 } , \\Im \\mathcal { A } = \\frac { \\mathcal { A } - \\mathcal { A } ^ * } { 2 i } , \\end{align*}"}
+{"id": "8185.png", "formula": "\\begin{align*} r ( t ) = \\frac { 1 } { 3 } \\frac { R _ 0 } { 5 ^ { 1 / d } } \\left ( \\frac { 2 } { 3 } \\right ) ^ { 1 / d } ( \\log \\log t ) ^ { 1 / d } , t > e . \\end{align*}"}
+{"id": "9316.png", "formula": "\\begin{align*} x _ j + \\log \\left ( \\sum _ { k = 1 } ^ n a _ { j k } x _ k \\right ) , j = 1 , \\ldots , n . \\end{align*}"}
+{"id": "8741.png", "formula": "\\begin{align*} \\| \\mathcal { T } _ { 1 , a } ^ { ( a _ 1 , a _ 2 ) } - \\mathcal { T } _ { 2 , a } ^ { ( a _ 1 , a _ 2 ) } \\| _ { \\rm F } ^ 2 & = O ( p ^ { a - a _ 1 - a _ 2 - c _ 1 + 1 + 2 ( a _ 1 - c _ 2 ) } ) = O ( p ^ { a + a _ 1 - a _ 2 - c _ 1 - 2 c _ 2 + 1 } ) \\\\ & = O ( p ^ { a + a _ 1 - a _ 2 - l + 1 } ) = O ( p ^ { a + a _ 1 - a _ 2 - 2 r + 2 } ) , \\end{align*}"}
+{"id": "803.png", "formula": "\\begin{align*} \\| { w - R _ h w } \\| \\ , + h \\ , \\| { \\nabla ( w - R _ h w ) } \\| \\ , \\le & \\ , C h ^ 2 \\ , \\| { \\Delta w } \\| , \\forall w \\in H ^ 2 \\cap H ^ 1 _ 0 . \\\\ \\end{align*}"}
+{"id": "4098.png", "formula": "\\begin{align*} 0 = ( ( \\lambda _ 3 v _ i ) _ { \\vec { v } } ) _ \\ell \\end{align*}"}
+{"id": "7807.png", "formula": "\\begin{align*} \\exp ( - \\sum _ { i \\le n } x _ { i } ) = \\exp ( ( x _ { n } - x _ { n - 1 } ) + 2 ( x _ { n - 1 } - x _ { n - 2 } ) + \\cdots + n x _ { 1 } ) . \\end{align*}"}
+{"id": "9026.png", "formula": "\\begin{align*} \\begin{cases} 2 \\lambda _ 2 + \\frac { 3 } { 2 } \\lambda _ 2 \\lambda _ { 2 , 3 } = \\lambda _ 1 \\\\ 2 \\lambda _ 2 + \\lambda _ 2 \\lambda _ { 2 , 4 } = \\lambda _ 1 \\\\ \\lambda _ 1 + \\lambda _ 2 = \\lambda _ { 2 , 3 } + \\lambda _ { 2 , 4 } = 1 \\end{cases} , \\end{align*}"}
+{"id": "5446.png", "formula": "\\begin{align*} f _ { Z } ( z ) = \\frac { z ^ { \\kappa - 1 } ( 1 - z ) ^ { \\beta - 1 } } { B ( \\kappa , \\beta ) } , \\end{align*}"}
+{"id": "29.png", "formula": "\\begin{align*} \\mathbf { T } _ { K ^ \\circ } ( \\widehat { \\mathbb { Z } } ) = \\mathbf { T } _ { K ^ \\circ } ( \\mathbb { A } _ f ) \\cap \\mathbf { G } ( \\widehat { \\mathbb { Z } } ) , \\mathbf { T } _ { K ^ \\circ } ^ 1 ( \\widehat { \\mathbb { Z } } ) = \\mathbf { T } _ { K ^ \\circ } ( \\widehat { \\mathbb { Z } } ) \\cap \\mathbf { G } ^ 1 ( \\widehat { \\mathbb { Z } } ) . \\end{align*}"}
+{"id": "9032.png", "formula": "\\begin{align*} b _ { i , n } b _ { \\tau ( i ) , n } = b _ { i , n } b _ { \\tau ( i ) , \\tau ( n ) } = b _ { i , n } b _ { i , n } \\ge 0 . \\end{align*}"}
+{"id": "6998.png", "formula": "\\begin{align*} \\Gamma = \\bigcup _ { i = 1 } ^ { \\epsilon } \\left ( \\gamma _ i + \\Delta \\right ) \\mbox { a n d } 0 = \\gamma _ 1 < \\ldots < \\gamma _ \\epsilon < \\Delta _ { > 0 } . \\end{align*}"}
+{"id": "3227.png", "formula": "\\begin{align*} F _ { 0 } & = \\{ P \\in F ~ | ~ \\} \\\\ F _ { 1 2 3 } & = \\{ P \\in F ~ | ~ \\} \\end{align*}"}
+{"id": "9204.png", "formula": "\\begin{align*} \\sum _ { x \\in V _ 1 } \\left ( d ( y , x ) - 1 \\right ) & \\ge \\sum _ { i = 2 } ^ r ( d ( y , w _ i ) - 1 ) + \\sum _ { i = 2 } ^ r ( d ( y , u _ i ) - 1 ) \\\\ & \\ge \\sum _ { i = 2 } ^ r ( i - 1 ) + \\sum _ { i = 2 } ^ r ( i - 2 ) \\\\ & = ( r - 1 ) ^ 2 . \\end{align*}"}
+{"id": "6704.png", "formula": "\\begin{align*} \\{ p _ 2 , \\Psi \\} ( x _ 0 , \\xi ) = H _ { p _ 2 } ( \\Psi ) ( x _ 0 , \\xi ) = \\frac { d } { d s } \\Psi \\circ x _ s ( x _ 0 , \\xi ) | _ { s = 0 } , \\\\ \\{ p _ 2 , \\{ p _ 2 , \\Psi \\} \\} ( x _ 0 , \\xi ) = H _ { p _ 2 } \\big ( H _ { p _ 2 } ( \\Psi ) \\big ) ( x _ 0 , \\xi ) = \\frac { d ^ 2 } { d s ^ 2 } \\Psi \\circ x _ s ( x _ 0 , \\xi ) | _ { s = 0 } . \\end{align*}"}
+{"id": "928.png", "formula": "\\begin{align*} E ( t ) = P _ s ( \\hat { L } _ p + \\hat { L } _ t ) + \\alpha P _ s ( \\hat { L } _ p + \\hat { L } _ { \\mathrm { f } } ) , \\end{align*}"}
+{"id": "1874.png", "formula": "\\begin{align*} \\min f ( u ^ * ) + \\langle p _ { u ^ * } , u - u ^ * \\rangle _ { \\mathcal { U } } + \\frac { c } { 2 } \\| u - u ^ * \\| _ { \\mathcal { U } } ^ 2 \\mbox { s . t . } \\ \\ S _ { u ^ * } u \\in K = \\mathcal { K } - G ( u ^ * ) + S _ { u ^ * } u ^ * \\end{align*}"}
+{"id": "1735.png", "formula": "\\begin{align*} \\nu ^ { ( n ) } _ t ( x ) = e ^ { - \\alpha t } \\nu _ 0 ( x ) + \\int _ 0 ^ t \\alpha e ^ { - \\alpha ( t - s ) } \\Psi ( \\nu ^ { ( n - 1 ) } _ s , \\mu ^ { ( n - 1 ) } _ s ) ( x ) \\mathrm { d } s , \\end{align*}"}
+{"id": "4670.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow + \\infty } \\left | v _ n \\right | _ p ^ p \\leq C \\left ( \\liminf _ { n \\rightarrow + \\infty } \\left | v _ n \\right | _ 2 \\right ) ^ { \\left ( 1 - \\beta _ p \\right ) p } K ^ { \\beta _ p p } \\beta _ p = N \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { p } \\right ) \\end{align*}"}
+{"id": "7030.png", "formula": "\\begin{align*} \\nu _ i ( Q _ j ) - \\nu _ i ( Q ' _ j ) = \\alpha _ i . \\end{align*}"}
+{"id": "8813.png", "formula": "\\begin{align*} P _ { { \\cal F } ^ \\perp } { \\cal S } = P _ { { \\cal F } ^ \\perp } { \\cal S } P _ { { \\cal F } ^ \\perp } \\end{align*}"}
+{"id": "5456.png", "formula": "\\begin{align*} Q ( r _ 1 , \\theta ) \\approx & \\frac { \\pi ^ 2 \\lambda R _ S ( R _ { m a x } - r _ 1 ) } { N R _ E } \\sum _ { n = 1 } ^ N \\sqrt { 1 - \\phi _ n ^ 2 } c _ n \\\\ & \\times \\left ( 1 - \\prod _ { m = 1 } ^ { M } { \\left ( 1 + \\frac { m \\eta \\theta r _ 1 ^ { \\alpha } } { M c _ n ^ { \\alpha } } \\right ) ^ { - M b _ m } } \\right ) \\end{align*}"}
+{"id": "4675.png", "formula": "\\begin{align*} \\vect u ( x ) = ( 2 \\pi \\varepsilon ) ^ { - 3 / 2 } \\int _ { Y ' } { \\rm e } ^ { { \\rm i } \\chi \\cdot x / \\varepsilon } ( \\mathcal { G } _ \\varepsilon \\vect u ) ( x / \\varepsilon , \\chi ) d \\chi . \\end{align*}"}
+{"id": "8517.png", "formula": "\\begin{align*} \\zeta _ { \\infty , 0 } ( s , \\ , c ) = \\frac { \\pi } { \\sqrt { c } } \\ , \\frac { 1 } { s - 1 } + \\frac { \\pi } { \\sqrt { c } } \\left ( 2 \\gamma - \\log \\left ( \\frac { c } { 4 } \\right ) + 8 \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { 2 n - 1 } \\cdot \\frac { 1 } { e ^ { \\pi \\sqrt { c } ( 2 n - 1 ) } - 1 } \\right ) + O \\left ( s - 1 \\right ) . \\end{align*}"}
+{"id": "2882.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 2 - 2 t _ { 2 } , & ~ ~ t _ 2 \\leq 0 , \\\\ 4 , & ~ ~ t _ 2 = 1 , \\\\ 6 , & ~ ~ t _ 2 = 2 , \\\\ 8 , & ~ ~ t _ 2 \\geq 3 . \\end{array} \\right . \\end{align*}"}
+{"id": "9284.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } w ^ { k } = \\bar w . \\end{align*}"}
+{"id": "2239.png", "formula": "\\begin{align*} T _ D ( f ) ( z ) = - \\frac { 1 } { \\pi } \\iint _ D \\frac { f ( \\zeta ) } { \\zeta - z } \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "5566.png", "formula": "\\begin{align*} \\limsup _ { \\ell \\to \\infty } h _ { \\mu } ( Z _ { 0 } , \\lambda _ { \\ell , 0 } ) = 0 \\end{align*}"}
+{"id": "3308.png", "formula": "\\begin{align*} b ( \\xi ) = | \\xi - 1 | ^ { \\delta _ 1 } | \\xi + 1 | ^ { \\delta _ { - 1 } } \\widetilde { b } ( \\xi ) \\end{align*}"}
+{"id": "6118.png", "formula": "\\begin{align*} \\mathcal { R } _ { 1 , 2 } = 2 ^ { - j _ { 0 } - 3 - 5 s ^ { - 1 } } ( s ( r _ { 2 } - r _ { 1 } ) ) ^ { \\frac { 1 } { s } } . \\end{align*}"}
+{"id": "3904.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { \\ell < 3 } K _ { \\ell } \\right ) \\cap K _ { 3 } = ( K _ 1 \\cup K _ 2 ) \\cap K _ 3 = \\{ 4 , 5 \\} \\in \\bigcup _ { \\ell < 3 } 2 ^ { K _ { \\ell } } = 2 ^ { K _ 1 } \\cup 2 ^ { K _ 2 } . \\end{align*}"}
+{"id": "4286.png", "formula": "\\begin{align*} \\rho e _ \\theta \\theta _ t + \\left ( \\rho u e _ \\theta - \\frac { 2 a ( \\theta ) } { Z ( \\theta ) } q \\right ) \\theta _ x + R \\rho \\theta u _ x + q _ x = \\frac { 2 a ( \\theta ) } { \\tau _ 1 ( \\theta ) } q ^ 2 + \\frac { 1 } { \\mu } S ^ 2 . \\end{align*}"}
+{"id": "2037.png", "formula": "\\begin{align*} u _ { t _ \\alpha t _ \\beta } ( 0 ) = - u _ { x _ n } ( 0 ) \\rho _ { t _ \\alpha t _ \\beta } ( 0 ) , ~ ~ \\alpha , \\beta < 2 n . \\end{align*}"}
+{"id": "7370.png", "formula": "\\begin{align*} \\Sigma _ N ^ 2 ( L , \\alpha ) = L + o ( L ) \\end{align*}"}
+{"id": "5461.png", "formula": "\\begin{align*} f _ { Z } ( z ) = \\frac { z ^ { \\kappa - 1 } ( 1 - z ) ^ { \\beta - 1 } } { B ( \\kappa , \\beta ) } , \\end{align*}"}
+{"id": "6548.png", "formula": "\\begin{align*} \\frac { i } { z ^ 2 } \\left ( \\frac { 1 } { z - \\gamma } + \\frac { 1 } { \\gamma } \\right ) = \\int _ { 0 } ^ { \\infty } \\frac { e ^ { - i t \\gamma } - 1 } { \\gamma ^ 2 } \\ , e ^ { i z t } \\ , d t , \\Im ( z ) > \\Im ( \\gamma ) \\end{align*}"}
+{"id": "9299.png", "formula": "\\begin{align*} J ( f ; x ) = J ( f _ 1 , \\ldots , f _ n ; x _ 1 , \\ldots , x _ n ) : = \\left | \\begin{array} { c c c } \\frac { \\partial f _ 1 } { \\partial x _ 1 } & \\ldots & \\frac { \\partial f _ n } { \\partial x _ 1 } \\\\ \\ldots & \\ldots & \\ldots \\\\ \\frac { \\partial f _ 1 } { \\partial x _ n } & \\ldots & \\frac { \\partial f _ n } { \\partial x _ n } \\\\ \\end{array} \\right | \\in \\C ^ { * } = \\C \\setminus \\{ 0 \\} . \\end{align*}"}
+{"id": "3771.png", "formula": "\\begin{align*} ( M _ i ) _ { k , l } : = \\delta _ { k , l + s _ { i + 1 } - s _ i } = \\begin{cases} 1 & { \\rm i f } \\ k = l + s _ { i + 1 } - s _ i , \\\\ 0 & { \\rm o t h e r w i s e } \\end{cases} \\end{align*}"}
+{"id": "8258.png", "formula": "\\begin{align*} \\Big ( \\frac { d } { d t } \\Big ) ^ { s } \\det ( p _ { i , j } ( t ) ) _ { i , j = 1 , \\ldots , k } = \\sum _ { \\substack { l _ 1 + \\cdots + l _ k = s \\\\ l _ 1 \\geq 0 , \\ldots , l _ k \\geq 0 } } \\binom { s } { l _ 1 , \\ldots , l _ k } \\det \\Big ( p _ { i , j } ^ { ( l _ { i } ) } ( t ) \\Big ) _ { i , j = 1 , \\ldots , k } , \\end{align*}"}
+{"id": "5526.png", "formula": "\\begin{align*} \\mathcal { T } : = \\cap _ { n = 1 } ^ { \\infty } \\sigma \\left ( \\xi _ { n } , \\xi _ { n + 1 } , \\xi _ { n + 2 } , \\ldots \\right ) . \\end{align*}"}
+{"id": "1659.png", "formula": "\\begin{align*} & 2 \\ , g ( ( \\nabla _ { X } { f } ) Y , Z ) = N ^ { \\ , ( 5 ) } ( X , Y , Z ) , \\\\ & 0 = N ^ { \\ , ( 5 ) } ( X , Y , Z ) + N ^ { \\ , ( 5 ) } ( Y , Z , X ) + N ^ { \\ , ( 5 ) } ( Z , X , Y ) , \\\\ 0 & = N ^ { \\ , ( 5 ) } ( { f } X , Y , Z ) + N ^ { \\ , ( 5 ) } ( { f } Y , Z , X ) + N ^ { \\ , ( 5 ) } ( { f } Z , X , Y ) . \\end{align*}"}
+{"id": "4029.png", "formula": "\\begin{align*} L ( f , s ) = \\prod _ { p | N } \\left ( 1 - a _ p ( f ) \\ , p ^ { - s } \\right ) ^ { - 1 } \\prod _ { p \\nmid N } \\left ( 1 - a _ p ( f ) \\ , p ^ { - s } + p ^ { k - 1 } p ^ { - 2 s } \\right ) ^ { - 1 } . \\end{align*}"}
+{"id": "5906.png", "formula": "\\begin{align*} \\theta _ { 3 / 2 } ( z , \\epsilon ) & = \\sum _ { n \\in \\Z } n e ^ { i \\epsilon \\pi n ^ 2 z } , \\end{align*}"}
+{"id": "4302.png", "formula": "\\begin{align*} f = f _ 0 * _ x K _ \\nu ( \\cdot , t ) - ( f u 1 _ { t \\in [ 0 , T ] } ) * _ { x , t } ( \\nabla K _ \\nu 1 _ { t > 0 } ) . \\end{align*}"}
+{"id": "9350.png", "formula": "\\begin{align*} \\begin{aligned} - d y _ { ( n , t ) } & = \\bigg [ G \\big ( y _ { ( n , t ) } , z _ { ( n , t ) } , \\tilde { z } _ { ( n , t ) } , \\gamma _ { ( n , t , e ) } \\big ) + \\varphi _ n ( t ) \\bigg ] d t - z _ { ( n , t ) } d W _ t - \\tilde { z } _ { ( n , t ) } d \\xi _ t \\\\ & \\quad - \\int _ { \\mathcal { E } } \\gamma _ { ( n , t , e ) } \\tilde { N } ( d e , d t ) , \\ t \\in [ 0 , \\infty ) . \\end{aligned} \\end{align*}"}
+{"id": "6363.png", "formula": "\\begin{align*} \\P _ x ( \\tau _ { \\Gamma } > t ) & = \\P _ { t ^ { - 1 / \\alpha } x } ( \\tau _ { \\Gamma } > 1 ) \\\\ & \\leq c \\left ( 1 + \\left ( t ^ { - 1 / \\alpha } | x | \\right ) ^ { \\alpha - \\beta } \\right ) M _ { \\Gamma } ( t ^ { - 1 / \\alpha } x ) \\\\ & = c \\left ( t ^ { - \\beta / \\alpha } + t ^ { - 1 } | x | ^ { \\alpha - \\beta } \\right ) M _ { \\Gamma } ( x ) . \\end{align*}"}
+{"id": "8246.png", "formula": "\\begin{align*} T _ { h } H _ { k , \\{ X ; Y _ { l - h , j - h } \\} } = \\sum _ { s = 1 } ^ { k } H _ { k , \\{ ( l _ { 1 } , \\ldots , l _ { s } + h , \\ldots , l _ { k } ) ; Y _ { l - h , j - h } \\} } , T _ { l } H _ { k , X } = \\sum _ { s = 1 } ^ { k } H _ { k , ( l _ 1 , \\ldots , l _ { s } + l , \\ldots , l _ { k } ) } . \\end{align*}"}
+{"id": "6944.png", "formula": "\\begin{align*} x _ i - x _ { m - j _ { 2 k } } & = E _ i - E _ { m - j _ { 2 k } } - \\frac 1 2 ( m - j _ { 2 k } - i + 1 ) \\\\ & = \\frac 1 4 ( - 2 i + 2 ( m - j _ { 2 k } ) ) - \\frac 1 2 ( m - j _ { 2 k } - i + 1 ) = - \\frac 1 2 \\end{align*}"}
+{"id": "7663.png", "formula": "\\begin{align*} { \\bf c t } _ \\kappa ( t ) = \\begin{cases} \\frac { 1 } { t } , & \\kappa = 0 , \\\\ \\kappa \\coth ( \\kappa t ) , & \\kappa > 0 . \\end{cases} \\end{align*}"}
+{"id": "4219.png", "formula": "\\begin{align*} T _ n ( \\phi ) = T _ n ( \\phi _ 0 ) T _ n ( \\phi _ 1 ) \\cdots T _ n ( \\phi _ R ) + P _ n K _ 1 P _ n + W _ n K _ 2 W _ n + C _ n \\end{align*}"}
+{"id": "7322.png", "formula": "\\begin{align*} z _ { i p } = z _ { i p } ^ 0 + \\sum _ { k = 1 } ^ { N _ 1 } u _ { k p } e _ { k i } , i = 1 , 2 , \\dots , n , z _ p ^ 0 = ( z _ { 1 p } ^ 0 , \\dots , z _ { n p } ^ 0 ) \\in S _ p ^ 1 , p \\in { \\cal P } _ 1 , \\end{align*}"}
+{"id": "1913.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { + \\infty } \\left ( \\mathcal { D } _ { \\eta ^ { k } } ( \\eta ^ { k + 1 } , \\eta ^ { k } ; \\lambda ^ { k } ) + \\beta \\| r ^ { k + 1 } \\| ^ 2 \\right ) \\leq \\mathcal { D } _ { \\eta ^ { 1 } } ( \\hat { \\eta } , \\eta ^ { 1 } ; \\lambda ^ { 1 } ) < + \\infty , \\end{align*}"}
+{"id": "5959.png", "formula": "\\begin{align*} \\overline { i } : P ^ { \\pm } _ { > 0 } ( \\R ) \\longrightarrow \\mathcal { H } ^ { \\pm } ; g = \\begin{pmatrix} a & b \\\\ 0 & ( \\det g ) a ^ { - 1 } \\end{pmatrix} \\longmapsto a b \\det g + i a ^ 2 \\det g . \\end{align*}"}
+{"id": "7458.png", "formula": "\\begin{align*} w _ { \\alpha } ^ { ( i ) } ( 0 , t ) = \\dfrac { 1 } { 2 \\ , \\mathrm { v } _ i \\ , h _ i ^ 2 } \\Big ( \\boldsymbol { d _ { \\alpha } } ( t ) - \\mathrm { v } _ 1 \\ , h _ 1 ^ 2 \\ , w _ { \\alpha } ^ { ( 1 ) } ( 0 , t ) \\Big ) , \\ \\ t \\in [ 0 , T ] , i \\in \\{ 2 , 3 \\} . \\end{align*}"}
+{"id": "6969.png", "formula": "\\begin{align*} \\overline S : = \\{ s _ i + \\Delta \\in \\Gamma / \\Delta \\mid i \\in I _ 1 \\} \\end{align*}"}
+{"id": "1066.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\partial _ t u + ( - \\Delta + \\varrho | x | ^ 2 ) ^ { \\beta } u & = & f ( u ) , ( x , t ) \\in \\R ^ d \\times ( 0 , \\infty ) , \\\\ u ( x , 0 ) & = & u _ 0 ( x ) , \\end{array} \\right . \\end{align*}"}
+{"id": "3427.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } | F ( B \\sum _ { j = 1 } ^ \\ell w _ j ) - F ( \\sum _ { j = 1 } ^ \\ell w _ j ) | \\leq & | B - 1 | \\int _ { \\R ^ N } \\sum _ { j = 1 } ^ \\ell w _ j | f ( \\theta \\sum _ { j = 1 } ^ \\ell w _ j ) | \\leq C e ^ { - \\frac { \\sigma \\xi ( \\boldsymbol p ) ^ 2 } 8 } , \\end{align*}"}
+{"id": "1626.png", "formula": "\\begin{align*} 0 = 2 \\ , d \\eta ^ j ( { f } X , \\xi _ i ) = ( { f } X ) ( \\eta ^ j ( \\xi _ i ) ) - \\xi _ i ( \\eta ^ j ( { f } X ) ) - \\eta ^ j ( [ { f } X , \\xi _ i ] ) = \\eta ^ j ( [ \\xi _ i , { f } X ] ) . \\end{align*}"}
+{"id": "5398.png", "formula": "\\begin{align*} \\xi _ { k , \\psi } ( f ) & : = \\psi ( p _ k ( f ) ) , \\end{align*}"}
+{"id": "3274.png", "formula": "\\begin{align*} { \\rm I m } ( f ( t ) ) = 0 { \\rm \\ f o r \\ } t \\le 0 \\end{align*}"}
+{"id": "4614.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } p _ k ( M , \\omega ) = 1 \\end{align*}"}
+{"id": "2141.png", "formula": "\\begin{align*} \\phi ( x ) = \\int _ { \\Omega } G ( x - y ) ( \\rho _ { + } - \\rho _ { - } ) ( y ) \\ , d y , \\end{align*}"}
+{"id": "6381.png", "formula": "\\begin{align*} x ^ { 2 } = \\frac { - 6 \\cdot 2 0 r ^ { 2 } \\pm 4 \\cdot 2 0 r ^ { 2 } \\sqrt { 3 } } { 6 } . \\end{align*}"}
+{"id": "7182.png", "formula": "\\begin{align*} w _ 1 : = \\begin{cases} \\phi ( u ) & \\mbox { o n } \\cup _ { i \\in \\N } B ^ i _ 1 , \\\\ u & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
+{"id": "454.png", "formula": "\\begin{align*} \\chi ( 0 , r ) = \\chi _ 0 ( r ) > 0 , \\ \\ \\chi _ r ( 0 , r ) = \\chi _ 1 ( r ) , \\end{align*}"}
+{"id": "7272.png", "formula": "\\begin{align*} a y ^ m = b x ^ n + c . \\end{align*}"}
+{"id": "8243.png", "formula": "\\begin{align*} T _ { h } Y : = ( t _ { 1 } + h , \\ldots , t _ { s } + h , h , \\ldots , h ) \\in \\mathbb { N } ^ k . \\end{align*}"}
+{"id": "3303.png", "formula": "\\begin{align*} { \\rm I m } ( f ( \\xi ) ) = 0 { \\rm \\ f o r \\ } \\xi \\in L \\end{align*}"}
+{"id": "5680.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\dfrac { ( D ^ { k - 1 } \\widetilde { \\mathcal { F } } _ { \\gamma } ) | U ( p ^ { 2 m + 1 } ) } { d _ { \\gamma } ( p ^ { 2 m + 1 } ) } = g . \\end{align*}"}
+{"id": "8535.png", "formula": "\\begin{align*} \\zeta _ { \\infty , p ^ { \\prime } } ( s , c ) : = \\sum _ { m , n \\neq 0 } \\frac { p ^ { \\prime 2 } + \\lambda _ { n } ^ { \\prime 2 } } { p ^ { \\prime } ( p ^ { \\prime } + \\frac { 1 } { \\pi } ) + \\lambda _ { n } ^ { \\prime 2 } } \\ , \\frac { 1 } { \\left ( m ^ { 2 } + c \\lambda _ { n } ^ { \\prime 2 } \\right ) ^ { s } } , \\ , \\ , \\ , \\ , ( s ) > 1 . \\end{align*}"}
+{"id": "360.png", "formula": "\\begin{align*} \\frac { c ^ { \\sigma } _ i } { c ^ { \\mu } _ i } = \\frac { c ^ { \\sigma } _ i } { c ^ { \\tau } _ i } \\cdot \\frac { c ^ { \\tau } _ i } { c ^ { \\mu } _ i } = \\frac { c ^ { \\tau } _ i } { c ^ { \\mu } _ i } \\cdot \\frac { c ^ { \\sigma } _ i } { c ^ { \\tau } _ i } \\end{align*}"}
+{"id": "8355.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\| h _ j \\| _ { p ^ * , q , \\mu } = 0 . \\end{align*}"}
+{"id": "8098.png", "formula": "\\begin{align*} - \\Delta \\xi _ i = \\gamma _ i \\xi _ i \\ ; \\mathcal { D } _ a , \\xi _ i = 0 \\ ; \\Gamma _ a , \\end{align*}"}
+{"id": "8808.png", "formula": "\\begin{align*} & ( 4 + t + 2 u ) p _ { i , j } + ( 3 + 2 t + 2 u ) p _ { j , i } \\\\ & = ( 2 + t ) p _ { i , j } + ( 1 + 2 t ) p _ { j , i } + p _ { i + 1 , j } + p _ { j , i + 1 } + p _ { i , j + 1 } + p _ { j + 1 , i } \\\\ & + u p _ { i - 1 , j } + u p _ { j , i - 1 } + u p _ { i , j - 1 } + u p _ { j - 1 , i } \\end{align*}"}
+{"id": "8811.png", "formula": "\\begin{align*} & p _ { 1 , 3 } = \\frac { ( 5 u t + u + t - 1 ) u } { 3 ( 1 + t ) ( \\displaystyle \\sum _ { k = 0 } ^ { n - 1 } u ^ k ) ^ 2 } \\end{align*}"}
+{"id": "8866.png", "formula": "\\begin{align*} \\mathbf r _ m = & \\mathbf S \\mathbf B \\mathbf { G } ^ { \\frac { 1 } { 2 } } \\mathbf h _ m + \\mathbf n _ m , m \\in \\mathcal { M } , \\end{align*}"}
+{"id": "7145.png", "formula": "\\begin{align*} \\partial _ t u & = \\Delta w , \\\\ w & = \\mathcal { L } _ \\varepsilon ^ \\Omega c _ \\varepsilon + \\Delta c + f ^ \\prime ( c _ \\varepsilon ) - f ^ \\prime ( c ) . \\end{align*}"}
+{"id": "8871.png", "formula": "\\begin{align*} a _ { 1 1 } = a _ { 1 2 } , a _ { i i } & = \\max \\{ a _ { i - 1 , i } , a _ { i , i + 1 } \\} i = 2 , \\ldots , n - 1 , a _ { n n } = a _ { n - 1 , n } , \\\\ a _ { i k } & = \\min \\{ a _ { i , k - 1 } , a _ { i + 1 , k } \\} i , k 1 \\leq i < k - 1 \\leq n - 1 , \\\\ a _ { k i } & = a _ { i k } i , k i > k . \\end{align*}"}
+{"id": "6147.png", "formula": "\\begin{align*} x _ 1 ^ j y _ 1 ^ { n - j } - x _ 0 x _ 1 ^ k y _ 1 ^ { n - k - 1 } = x _ 1 ^ j y _ 1 ^ { n - k - 1 } ( y _ 1 ^ { k - j + 1 } - x _ 0 ^ { k - j + 1 } ) + x _ 1 ^ j y _ 1 ^ { n - k - 1 } x _ 0 ( x _ 0 ^ { k - j } - x _ 1 ^ { k - j } ) , \\end{align*}"}
+{"id": "118.png", "formula": "\\begin{align*} \\chi e ^ { - i t h ^ { - 1 } \\tilde { P } _ h ( 0 ) } q _ 1 e ^ { - ( t _ 0 - t ) X } e ^ { - i s h ^ { - 1 } \\tilde { P } _ h ( z ) } \\chi f = \\chi e ^ { - t X } q _ 1 e ^ { - ( t _ 0 - t ) X } e ^ { - i s h ^ { - 1 } \\tilde { P } _ h ( z ) } \\chi f = 0 . \\end{align*}"}
+{"id": "4542.png", "formula": "\\begin{align*} \\varphi _ 1 , \\ \\varphi _ 2 , \\ \\varphi _ 3 \\in \\left ( \\bigcap _ { m = 1 } ^ { \\infty } H ^ m ( \\R ^ d ) \\right ) ^ d . \\end{align*}"}
+{"id": "4672.png", "formula": "\\begin{align*} \\tilde { S } ( a ) = \\left \\{ u \\in S ( a ) : I _ \\epsilon ( u ) \\leq \\Upsilon _ { 0 , a } + h ( \\epsilon ) \\right \\} . \\end{align*}"}
+{"id": "1081.png", "formula": "\\begin{align*} & t ^ { \\sigma } \\| \\Phi ( u ) ( t ) - \\Phi ( v ) ( t ) \\| _ { L ^ a } \\leq t ^ { \\sigma } \\int _ 0 ^ t \\left \\| e ^ { - ( t - \\tau ) H ^ { \\beta } } ( f ( u ( \\tau ) ) - f ( v ( \\tau ) ) ) \\ , \\right \\| _ { L ^ { a } } \\ , d \\tau \\\\ & \\leq C \\sum _ { k = 0 } ^ { \\infty } \\frac { \\lambda ^ k } { k ! } t ^ { \\sigma } \\int _ 0 ^ t ( t - \\tau ) ^ { - \\frac d { 2 \\beta } ( \\frac 1 r - \\frac 1 a ) } \\| ( u - v ) ( | u | ^ { p k + m - 1 } + | v | ^ { p k + m - 1 } ) \\| _ { L ^ r } \\ , d \\tau , \\end{align*}"}
+{"id": "312.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { x } { n ( n - x ) } : = - \\Psi ( 1 - x ) - \\gamma , \\textmd { w h e r e } \\gamma : = \\lim _ { N \\rightarrow \\infty } \\left ( \\sum _ { n = 1 } ^ { N } \\frac { 1 } { n } - \\log N \\right ) \\end{align*}"}
+{"id": "3435.png", "formula": "\\begin{align*} \\Delta X _ i + \\left ( \\lambda + \\frac { c ^ 2 } { m } \\right ) X _ i = 0 . \\end{align*}"}
+{"id": "8656.png", "formula": "\\begin{align*} a - b = \\int _ 0 ^ 1 \\int _ 0 ^ x ( 1 - p y ) ^ { - 1 } ( 1 - p x ) ^ { - 1 } y ^ { j - 1 } x ^ { j - 1 } ( y - x ) \\big ( ( 1 - p y ) ^ { - 1 } y - ( 1 - p x ) ^ { - 1 } x \\big ) d y d x . \\end{align*}"}
+{"id": "3781.png", "formula": "\\begin{align*} \\sigma _ J = \\sigma _ { j _ 1 + 1 } \\sigma _ { j _ 2 + 2 } \\dots \\sigma _ { j _ k + k } ( a ) . \\end{align*}"}
+{"id": "4564.png", "formula": "\\begin{align*} H ( \\mathcal { S } ) _ i = \\# \\{ k : \\nu _ k \\le i < \\nu _ k + p _ k \\} . \\end{align*}"}
+{"id": "3792.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { M } } ( \\mu _ 1 , \\mu _ 2 ) : = \\sup _ { \\gamma \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\int _ { \\mathcal { V } } g \\ , d \\gamma . \\end{align*}"}
+{"id": "9189.png", "formula": "\\begin{align*} e _ { 1 , [ 2 ] } ^ { 1 } + a _ { 1 } ^ { 1 , 1 } e _ { 1 , [ 1 ] } ^ { 1 } + a _ { 1 } ^ { 1 , 0 } e _ { 1 } ^ { 1 } & = 0 \\\\ e _ { 2 , [ 4 ] } ^ { 1 } + \\sum _ { \\beta = 0 } ^ { 3 } a _ { 2 } ^ { 1 , \\beta } e _ { 2 , [ \\beta ] } ^ { 1 } & = 0 \\end{align*}"}
+{"id": "7631.png", "formula": "\\begin{align*} y = - \\frac { c x ( 1 + x ) } { 2 ( 8 - x ) ( 1 - x ) } = y _ 0 ( c , x ) \\end{align*}"}
+{"id": "6649.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) & = C _ { N , s } \\cdot \\mathrm { P . V . } \\int _ { \\R ^ N } \\frac { u ( y ) - u ( x ) } { | x - y | ^ { N + 2 s } } \\ , d y \\\\ & = C _ { N , s } \\cdot \\lim _ { \\varepsilon \\to 0 ^ + } \\int _ { \\{ | x - y | \\geq \\varepsilon \\} } \\frac { u ( y ) - u ( x ) } { | x - y | ^ { N + 2 s } } \\ , d y , \\end{align*}"}
+{"id": "8662.png", "formula": "\\begin{align*} V ^ { \\perp } : = \\{ v \\in V | ( u , v ) = 0 \\ \\forall u \\in V \\} , \\qquad \\mathrm { a s \\ w e l l \\ a s \\ s e t } \\qquad \\overline { V } : = V / V ^ { \\perp } . \\end{align*}"}
+{"id": "5763.png", "formula": "\\begin{align*} M ^ { \\sharp } f ( x ) & = \\sup _ { Q \\ni x } \\inf _ { c } \\dfrac { 1 } { | Q | } \\int _ { Q } | f ( y ) - c | \\mathrm { d } y \\approx \\sup _ { Q \\ni x } \\dfrac { 1 } { | Q | } \\int _ { Q } | f ( y ) - f _ { Q } | \\mathrm { d } y , \\end{align*}"}
+{"id": "5654.png", "formula": "\\begin{align*} \\aligned m _ { r , \\nu } ( a , b ) & \\leq m _ { r , \\nu } ( | \\phi | _ 2 , | \\phi | _ 2 ) \\leq \\max _ { t \\in \\mathbb { R } } J _ \\nu ( t \\star ( \\phi , \\phi ) ) \\\\ & = \\max _ { t \\in \\mathbb { R } } \\Bigl [ 2 E ( t \\star \\phi ) - \\nu e ^ { ( \\gamma _ p + \\gamma _ q ) t } \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | \\phi | ^ { p } ) | \\phi | ^ { q } \\Bigr ] : = \\max _ { t \\in \\mathbb { R } } g ( t ) , \\endaligned \\end{align*}"}
+{"id": "7429.png", "formula": "\\begin{align*} U _ { n , \\beta } : = \\sum _ { x \\in \\Z ^ 2 } U _ { n , \\beta } ( x ) = \\sum _ { k = 1 } ^ \\infty ( \\sigma _ \\beta ^ 2 ) ^ { k } \\ ! \\sum _ { 0 = : n _ 0 < n _ 1 < \\dots < n _ k : = n } \\ \\prod _ { i = 1 } ^ k q _ { 2 ( n _ i - n _ { i - 1 } ) } ( 0 ) \\ , . \\end{align*}"}
+{"id": "6200.png", "formula": "\\begin{align*} | V _ e | \\le ( n - 2 ) | V _ i | < ( n - 2 ) | V _ i | + 2 = | V _ e | , \\end{align*}"}
+{"id": "6358.png", "formula": "\\begin{align*} p _ t ^ { T \\Gamma } ( T x , T y ) = p _ t ^ \\Gamma ( x , y ) , x , y \\in \\Gamma , \\ ; t > 0 . \\end{align*}"}
+{"id": "1932.png", "formula": "\\begin{align*} \\psi ( v _ n ) = \\frac { \\gamma _ 1 \\pi ^ n + \\gamma _ 2 \\pi ^ l } { 2 } , \\end{align*}"}
+{"id": "8901.png", "formula": "\\begin{align*} \\mathsf { d } _ { \\mathrm { c o m } } ( \\{ X _ { n i } \\} ) = \\sum _ { n = 1 } ^ N \\sum _ { i = 1 } ^ j \\| X _ { n i } \\| _ 1 . \\end{align*}"}
+{"id": "6070.png", "formula": "\\begin{align*} \\sup _ { x \\in G \\setminus \\{ e \\} } \\sup _ { r > 0 } \\frac { \\| \\delta _ { r } ^ { A } x \\| } { \\| \\delta _ { r } ^ { B } x \\| } = \\sup _ { x \\in G \\setminus \\{ e \\} } \\sup _ { r > 0 } \\frac { \\| \\delta _ { 1 / r } ^ { A } x \\| } { \\| \\delta _ { 1 / r } ^ { B } x \\| } \\leq R , \\end{align*}"}
+{"id": "5454.png", "formula": "\\begin{align*} f _ { r _ 1 | \\Phi ( \\mathcal { A } ) > 0 } ( r ) = \\upsilon ( \\lambda , R _ S ) r e ^ { - \\lambda \\pi \\frac { R _ S } { R _ E } r ^ 2 } , R _ { m i n } \\leq r \\leq R _ { m a x } , \\end{align*}"}
+{"id": "8472.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n } } { ( x ^ { 2 } + n ^ { 2 } ) ^ { s } } = - \\frac { x ^ { - 2 s } } { 2 } + \\frac { 2 ^ { \\frac { 3 } { 2 } - s } \\pi ^ { s } x ^ { \\frac { 1 } { 2 } - s } } { \\Gamma ( s ) } \\sum _ { n = 1 } ^ { \\infty } \\left ( 2 n - 1 \\right ) ^ { s - \\frac { 1 } { 2 } } \\ , K _ { s - \\frac { 1 } { 2 } } \\left ( \\pi ( 2 n - 1 ) x \\right ) . \\end{align*}"}
+{"id": "4265.png", "formula": "\\begin{align*} \\tilde { x } _ { t } = \\tilde { x } _ { 0 } + \\int _ { 0 } ^ { t } \\tilde { b } _ { r } \\mathrm { d } r + \\int _ { 0 } ^ { t } \\tilde { \\sigma } _ { r } \\mathrm { d } W _ { r } , 0 \\leq t \\leq T . \\end{align*}"}
+{"id": "3415.png", "formula": "\\begin{align*} \\ell E _ { \\ell ^ { - 1 } \\alpha } = \\inf _ { \\gamma \\in \\Gamma } \\sup _ { \\boldsymbol s \\in S } \\mathbb { J } ( \\gamma ( \\boldsymbol { s } ) ) . \\end{align*}"}
+{"id": "2196.png", "formula": "\\begin{align*} & d = 2 , ~ \\Omega = ( 0 , 1 ) \\times ( 0 , 1 ) , ~ T = 1 , ~ \\psi _ 1 ( { x } ) = \\psi _ 0 ( { x } ) = x _ 2 ( 1 - x _ 2 ) x _ 1 ( 1 - x _ 1 ) , \\\\ & a ( { x } ) = ( 3 0 + \\sin ( x _ 1 ) ^ 2 ) ( 3 0 + \\sin ( x _ 2 ) ^ 2 ) , \\\\ & f ( { x } , t ) = \\exp ( t ) [ x _ 1 ( 1 - x _ 1 ) x _ 2 ( 1 - x _ 2 ) - \\sin ( 2 x _ 1 ) ( 3 0 + \\sin ( x _ 2 ) ^ 2 ) ( 1 - 2 x _ 1 ) x _ 2 ( 1 - x _ 2 ) \\\\ & \\quad \\quad \\quad - \\sin ( 2 x _ 2 ) ( 3 0 + \\sin ( x _ 1 ) ^ 2 ) ( 1 - 2 x _ 2 ) x _ 1 ( 1 - x _ 1 ) + 2 a ( { x } ) ( x _ 1 ( 1 - x _ 1 ) + x _ 2 ( 1 - x _ 2 ) ) ] . \\end{align*}"}
+{"id": "2916.png", "formula": "\\begin{align*} B _ v : = \\{ [ v + L _ n ] \\} \\cup \\{ [ v + w + L _ { n m } ] \\mid w \\in ( L _ n / L _ { n m } ) ' \\} . \\end{align*}"}
+{"id": "7579.png", "formula": "\\begin{align*} p ( z ) = 1 + \\sum _ { n = 1 } ^ { \\infty } c _ n z ^ n \\end{align*}"}
+{"id": "2274.png", "formula": "\\begin{align*} & \\int _ 0 ^ { 2 \\pi } | \\Phi ( r e ^ { i \\theta } ) | ^ p \\ , d \\theta \\\\ & = \\int _ 0 ^ { 2 \\pi } | w ( r e ^ { i \\theta } ) - T ( f ) ( r e ^ { i \\theta } ) | ^ p \\ , d \\theta \\\\ & \\leq \\int _ 0 ^ { 2 \\pi } | w ( r e ^ { i \\theta } ) | ^ p \\ , d \\theta + \\int _ 0 ^ { 2 \\pi } | T ( f ) ( r e ^ { i \\theta } ) | ^ p \\ , d \\theta \\\\ & \\leq | | w | | ^ p _ { H ^ p ( D ) } + \\int _ 0 ^ { 2 \\pi } | T ( f ) ( r e ^ { i \\theta } ) | ^ p \\ , d \\theta . \\end{align*}"}
+{"id": "9081.png", "formula": "\\begin{align*} c _ 0 + c _ 1 + c _ 2 = | \\mathcal { F } | , \\end{align*}"}
+{"id": "8245.png", "formula": "\\begin{align*} \\widetilde { W } _ { h , q } ( i ) = \\begin{cases} h _ { 0 } + h + 1 & i = q ; \\\\ 1 & h - 1 \\leq i \\leq k - 2 i \\neq q \\\\ l - h _ { 0 } - k & i = k - 1 ; \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "8875.png", "formula": "\\begin{align*} P ( n ) & \\ge ( n / 2 ) ! \\cdot \\frac { ( n / 2 ) ! } { ( n / 2 ) ^ { n / 2 } } \\cdot ( n / 2 - 2 ) ^ { n / 2 } \\\\ & \\ge \\frac { 1 } { e ^ 2 } \\left ( ( n / 2 ) ! \\right ) ^ 2 ( 1 - o ( 1 ) ) \\\\ & = \\frac { \\sqrt { 2 \\pi } } { e ^ 2 } n ^ { 3 / 2 } \\frac { ( n - 1 ) ! } { 2 ^ { n + 1 } } ( 1 - o ( 1 ) ) \\ ; . \\end{align*}"}
+{"id": "9185.png", "formula": "\\begin{align*} \\delta ^ { 4 } ( \\varphi _ { r e s t _ { 1 } } ^ { 1 } ) = \\varphi _ { r e s t _ { 1 } , [ 4 ] } ^ { 1 } ( q ^ { 1 } , q ^ { 2 } , q ^ { 3 } , \\omega ^ { 1 } , \\omega ^ { 2 } , \\omega ^ { 3 } , v _ { 1 } ^ { 1 } , v _ { 1 , [ 1 ] } ^ { 1 } , v _ { 1 , [ 2 ] } ^ { 1 } , u ^ { 2 } ) \\ , . \\end{align*}"}
+{"id": "136.png", "formula": "\\begin{align*} A = { \\rm d i a g } \\left ( A _ 1 , \\cdots , A _ k \\right ) , A _ k = \\begin{pmatrix} B _ k & 1 & 0 & \\cdots & 0 \\\\ 0 & B _ k & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & B _ k \\end{pmatrix} , \\end{align*}"}
+{"id": "1324.png", "formula": "\\begin{align*} \\theta _ i ^ 2 T _ i ^ * ( u ) & = ( T _ i ^ { \\infty } ) ^ 2 \\int _ 0 ^ u \\frac { e ^ { 2 \\rho _ i ( v ) } } { ( T _ i ^ { \\infty } - T _ i ( v ) ) ^ 2 } d v \\\\ & = ( T _ i ^ { \\infty } ) ^ 2 \\left [ \\frac { 1 } { T _ i ^ { \\infty } - T _ i ( v ) } \\right ] _ 0 ^ u \\\\ & = ( T _ i ^ { \\infty } ) ^ 2 \\left ( \\frac { 1 } { T _ i ^ { \\infty } - T _ i ( u ) } - \\frac { 1 } { T _ i ^ { \\infty } } \\right ) . \\end{align*}"}
+{"id": "4906.png", "formula": "\\begin{align*} \\Delta ^ k \\left [ f ( x ) \\right ] _ { x = \\alpha } = \\sum _ { j = 0 } ^ k ( - 1 ) ^ { k - j } \\binom { k } { j } f ( \\alpha + j ) . \\end{align*}"}
+{"id": "2885.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 4 - 4 t _ 1 , & ~ ~ t _ 0 \\geq 4 , \\\\ 4 - 3 t _ 1 , & ~ ~ t _ 0 = 3 , \\\\ 4 - 2 t _ 1 , & ~ ~ t _ 0 = 2 , \\\\ t _ 0 t _ 1 - 2 t _ 1 , & ~ ~ t _ 0 \\leq 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "181.png", "formula": "\\begin{align*} 0 = L i _ 2 ( \\alpha _ 1 ^ { - 6 3 0 } ) - 2 L i _ 2 ( \\alpha _ 1 ^ { - 3 1 5 } ) - 3 L i _ 2 ( \\alpha _ 1 ^ { - 2 1 0 } ) - 1 0 L i _ 2 ( \\alpha _ 1 ^ { - 1 2 6 } ) \\end{align*}"}
+{"id": "5403.png", "formula": "\\begin{align*} B _ { L , k , 1 } & = \\sum _ { L \\le d < 2 L } \\sum _ { \\substack { p \\mid k , \\ , \\phi _ e ( q ) \\\\ e \\mid d \\\\ e > d / T } } \\log \\gcd ( p ^ { \\nu _ p ( k ) } , \\phi _ e ( q ) ) , \\\\ B _ { L , k , 2 } & = \\sum _ { L \\le d < 2 L } \\sum _ { \\substack { p \\mid k , \\ , \\phi _ e ( q ) \\\\ e \\mid d \\\\ e \\le d / T } } \\log \\gcd ( p ^ { \\nu _ p ( k ) } , \\phi _ e ( q ) ) . \\end{align*}"}
+{"id": "3300.png", "formula": "\\begin{align*} \\widetilde { \\rho } _ w ( \\theta , \\kappa ( \\theta ) ) = \\xi ^ { - N } e ^ { u + i v } ( \\theta ) , \\end{align*}"}
+{"id": "5310.png", "formula": "\\begin{align*} \\sqrt { \\frac { 1 - p } { p } } \\hat f ( \\{ i \\} ) = \\mu _ p ( f _ { i \\to 1 } ) - \\mu _ p ( f ) \\geq 2 \\mu _ p ( f ) . \\end{align*}"}
+{"id": "2595.png", "formula": "\\begin{align*} \\Omega ^ * _ 0 = H ^ * \\times \\omega ^ * \\end{align*}"}
+{"id": "3155.png", "formula": "\\begin{align*} \\sigma _ n ( x ) : = \\frac { 1 } { \\abs { W _ n } } \\sum _ { a \\in W _ n } \\phi ( a ) ( x ) , ~ n \\in \\N , \\end{align*}"}
+{"id": "1955.png", "formula": "\\begin{align*} \\gamma _ 1 ( a ) = \\gamma _ 1 ( L _ a , \\Omega ) = \\sup \\{ \\gamma \\geq 0 : ~ ~ \\exists \\phi < 0 , L _ a \\phi + \\gamma \\phi f \\geq 0 \\} , \\end{align*}"}
+{"id": "4006.png", "formula": "\\begin{align*} \\sup _ { y _ 2 ' \\in \\mathcal { Y } _ 2 } \\{ - y _ 2 ' d ( x ' ) - \\lambda _ 2 | y _ 2 - y _ 2 ' | \\} = \\begin{cases} \\infty & 0 \\le \\lambda _ 2 < 1 \\\\ - y _ 2 d ( x ' ) & \\lambda _ 2 \\ge 1 \\end{cases} , \\end{align*}"}
+{"id": "6403.png", "formula": "\\begin{align*} [ D \\psi : D \\varphi ] _ t = s ( \\psi ) [ D \\chi : D \\varphi ] _ t , t \\in \\mathbb { R } . \\end{align*}"}
+{"id": "3731.png", "formula": "\\begin{align*} \\ast ( d x ^ 0 \\wedge d x ^ 1 ) = d x ^ 2 \\wedge d x ^ 3 , \\ast ( d x ^ 0 \\wedge d x ^ 2 ) = d x ^ 3 \\wedge d x ^ 1 , \\ast ( d x ^ 0 \\wedge d x ^ 3 ) = d x ^ 1 \\wedge d x ^ 2 . \\end{align*}"}
+{"id": "8520.png", "formula": "\\begin{align*} Q _ { \\mu } ( s ) = \\intop _ { \\mu } ^ { \\infty } t ^ { s - 1 } e ^ { - t } d t \\end{align*}"}
+{"id": "1225.png", "formula": "\\begin{align*} a = \\varliminf _ { m \\to \\infty } \\big ( d ( m S , \\N ^ n \\setminus m S ) \\big ) ^ { 1 / { m } } > 0 . \\end{align*}"}
+{"id": "426.png", "formula": "\\begin{align*} ( \\vec U , \\vec V ) _ { \\Omega } = \\vec U ^ T P _ { \\Omega } \\vec V \\approx \\int \\limits _ { \\Omega } U ^ T V d \\Omega , ( \\vec U , \\vec U ) _ { \\Omega } = \\| \\vec U \\| ^ 2 _ { \\Omega } = \\vec U ^ T P _ { \\Omega } \\vec U \\approx \\int \\limits _ { \\Omega } U ^ T U d \\Omega = \\| U \\| ^ 2 _ { \\Omega } . \\end{align*}"}
+{"id": "9040.png", "formula": "\\begin{align*} y _ i ' = y _ { \\sigma ( i ) } \\quad b _ { i , j } ' = b _ { \\sigma ( i ) , \\sigma ( j ) } \\quad 1 \\le i , j \\le n . \\end{align*}"}
+{"id": "5355.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ 2 ^ 2 \\le ( e / \\rho ) ^ { d } \\gamma _ 1 \\cdot \\gamma . \\end{align*}"}
+{"id": "3982.png", "formula": "\\begin{align*} \\varphi _ \\ell ( x , \\lambda _ \\ell ) = \\frac { Q _ { \\ell , Y Y } ^ { - 1 } } { 4 \\lambda _ { \\ell } } + a _ \\ell x ^ { \\top } V _ { \\ell , X X } ^ { - 1 } V _ { \\ell , X Y } - \\lambda _ { \\ell } x ^ { \\top } V _ { \\ell , X X } ^ { - 1 } x . \\end{align*}"}
+{"id": "6250.png", "formula": "\\begin{align*} F _ { k + 1 } ( O [ g _ 1 ] ) = \\partial ( F _ k ( O [ g _ 1 ] ) ) + \\mathrm { S p a n } _ { \\mathbb C } ( 1 , g _ 1 , \\ldots , g _ { k + 1 } ) . \\end{align*}"}
+{"id": "5294.png", "formula": "\\begin{align*} 0 = \\langle A h , g \\rangle _ { \\mu _ p } = \\sum _ { d = 0 } ^ n \\left ( - \\frac { p } { 1 - p } \\right ) ^ d \\langle h ^ { = d } , g \\rangle \\geq \\mu _ { p } ( h ) \\mu _ { p } ( g ) - \\sum _ { d = 1 } ^ n \\left ( \\frac { p } { 1 - p } \\right ) ^ d | \\langle h ^ { = d } , g \\rangle | . \\end{align*}"}
+{"id": "7357.png", "formula": "\\begin{align*} \\Pr [ \\hat { M } = M ] & \\leq \\sum _ k \\binom { n } { k } \\frac { s ! } { ( k + s ) ! } ( 1 - y ) ^ { n - k } ( 1 - x ) ^ { k ( k + s ) } e ^ { k \\theta } \\\\ & \\leq \\sum _ k \\binom { n } { k } ( 1 - y ) ^ { n - k } e ^ { - k ^ 2 x } \\frac { e ^ { k \\theta } } { k ! } \\\\ & \\leq \\sum _ k \\binom { n } { k } ( 1 - y ) ^ { n - k } e ^ { - ( 2 \\ell k - \\ell ^ 2 ) x } e ^ { k \\theta } \\\\ & = e ^ { \\ell ^ 2 x } ( 1 - y + e ^ { \\theta - 2 \\ell x } ) ^ n , \\end{align*}"}
+{"id": "7881.png", "formula": "\\begin{align*} \\mbox { $ \\mathcal { S } _ + = \\left \\{ A ^ + \\in \\mathcal { S } : \\ A \\in \\binom { [ n ] } { k } \\mbox { a n d } a \\in A \\right \\} $ a n d $ \\mathcal { S } _ - = \\left \\{ A ^ - \\in \\mathcal { S } : \\ A \\in \\binom { [ n ] } { k } \\mbox { a n d } b \\in A \\right \\} $ . } \\end{align*}"}
+{"id": "7163.png", "formula": "\\begin{align*} \\limsup _ { \\rho \\to 0 } \\omega ( \\rho ) \\log \\left ( \\frac { 1 } { \\rho } \\right ) = 0 . \\end{align*}"}
+{"id": "9234.png", "formula": "\\begin{align*} \\texttt { w p t c o e f [ k ] } : = \\max \\Bigl \\{ \\texttt { m a g } \\Bigl ( 6 8 \\frac { ( L + 2 ) A ( \\sigma ) } { \\varepsilon _ 4 } \\frac { \\Gamma ( k + 1 / 2 ) ^ { 1 / 2 } } { ( B _ 1 a \\sqrt { \\pi } ) ^ k } \\Bigr ) , \\texttt { m a g } ( 4 0 ( L + 2 ) ) \\Bigr \\} . \\end{align*}"}
+{"id": "4261.png", "formula": "\\begin{align*} \\mathcal { V } \\left ( t , \\mu \\right ) = V \\left ( t , \\xi \\right ) , \\mathbb { P } \\end{align*}"}
+{"id": "6926.png", "formula": "\\begin{align*} \\sum _ { \\substack { j , l \\in \\mathcal { J } \\\\ j \\not = l } } r ( m _ j ) r ( n _ l ) \\Phi \\left ( | j - l | - 1 \\right ) \\le \\sum _ { \\substack { j , l \\in \\mathcal { J } \\\\ j \\not = l } } r ( m _ j ) ^ { 2 } \\Phi \\left ( | j - l | - 1 \\right ) . \\end{align*}"}
+{"id": "3547.png", "formula": "\\begin{align*} F _ i ( t , f _ 1 ( t ) , \\dots , f _ { n } ( t ) ) = P _ i [ & t , f _ 1 ( t ) , \\dots , f _ { n } ( t ) , \\\\ & j ( i B ( t ) ) , j ( i B ( f _ 1 ( t ) ) ) , \\dots , j ( i B ( f _ { n } ( t ) ) ) , \\\\ & i j ' ( i B ( t ) ) , i j ' ( i B ( f _ 1 ( t ) ) ) , \\dots , i j ' ( i B ( f _ { n } ( t ) ) ) , \\\\ & j '' ( i B ( t ) ) , j '' ( i B ( f _ 1 ( t ) ) ) , \\dots , j '' ( i B ( f _ { n } ( t ) ) ) ] = 0 . \\end{align*}"}
+{"id": "7216.png", "formula": "\\begin{align*} \\abs { \\lambda ^ { ( s ) } _ h - \\lambda ^ { ( 1 / 2 ) } _ h } \\stackrel { \\eqref { e q : d e f - l a m b d a - h - s } } { = } t _ h \\abs { \\widetilde { \\lambda } ^ { ( s ) } _ h - \\widetilde { \\lambda } ^ { ( 1 / 2 ) } _ h } \\leq t _ h \\left ( \\abs { \\widetilde { \\lambda } ^ { ( s ) } _ h - 1 } + \\abs { 1 - \\widetilde { \\lambda } ^ { ( 1 / 2 ) } _ h } \\right ) \\stackrel { \\eqref { e q : C a L e - t r i c k - c o n d i t i o n s } } { \\lesssim } 1 . \\end{align*}"}
+{"id": "8446.png", "formula": "\\begin{align*} I _ { m , p } ( s , x ) = I _ { m , p } ^ { ( 1 ) } ( s , x ) + I _ { m , p } ^ { ( 2 ) } ( s , x ) , \\end{align*}"}
+{"id": "2284.png", "formula": "\\begin{align*} f = \\sum _ { n } c _ n a _ n \\end{align*}"}
+{"id": "1420.png", "formula": "\\begin{align*} \\nabla f ( x ) = ( T _ { y , s } ^ { - 1 } ) ^ * \\nabla \\hat f ( s ) = ( T _ { y , s } ^ { - 1 } ) ^ * g \\end{align*}"}
+{"id": "5657.png", "formula": "\\begin{align*} | \\eta _ \\epsilon | ^ p _ p = \\left \\{ \\begin{array} { l l l } O ( \\epsilon ^ { N - \\frac { N - 2 } { 2 } p } ) , \\ & \\ N \\geq 4 \\ \\ p \\in ( 2 , 2 ^ * ) \\ \\ \\ N = 3 \\ \\ p \\in ( 3 , 6 ) , \\\\ O ( \\epsilon ^ { \\frac p 2 } ) , \\ & \\ N = 3 \\ \\ p \\in ( 2 , 3 ) , \\\\ O ( \\epsilon ^ { \\frac { 3 } { 2 } } | \\ln \\epsilon | ) , \\ & \\ N = 3 \\ \\ p = 3 . \\end{array} \\right . \\end{align*}"}
+{"id": "3840.png", "formula": "\\begin{align*} ( f _ \\ell ) _ { \\lambda _ \\ell } ( y _ \\ell ) = \\sup _ { y _ \\ell ^ \\prime \\in \\mathcal { Y } _ \\ell } \\left \\{ f _ \\ell ( y _ \\ell ^ { \\prime } ) - \\lambda _ \\ell c _ { Y _ \\ell } ( y _ { \\ell } , y _ { \\ell } ' ) \\right \\} . \\end{align*}"}
+{"id": "6212.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\to \\begin{pmatrix} - b & a \\\\ - d & c \\end{pmatrix} . \\end{align*}"}
+{"id": "4854.png", "formula": "\\begin{align*} ( \\prod _ { i = 1 } ^ m b ( i ) \\ast f ( t ( i ) ) ) \\ast b ( m + 1 ) \\in A \\cap \\sigma ( W ) . \\end{align*}"}
+{"id": "660.png", "formula": "\\begin{align*} \\int _ { I ^ { ( l ) } } S ( k , l ) \\varphi ( x ) \\ , d x = \\int _ { I ^ { ( k ) } } \\varphi ( x ) \\ , d x . \\end{align*}"}
+{"id": "4239.png", "formula": "\\begin{align*} \\Theta _ r & = ( n - 1 ) X _ 2 + ( ( 2 n - 3 ) n + 2 ) X _ 1 + n Y _ 2 + ( n - 1 ) Y _ 1 \\\\ \\Theta _ g & = X _ 2 + 2 ( n - 1 ) X _ 1 + Y _ 2 + 2 Y _ 1 . \\end{align*}"}
+{"id": "3343.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } ( 1 + b ^ * ) ( 1 + b ) = \\frac { 1 } { 4 } ( 2 + 2 \\Re ( b ) ) = \\frac { 1 } { 2 } ( 1 + \\Re ( b ) ) . \\end{align*}"}
+{"id": "9246.png", "formula": "\\begin{align*} M _ n = \\sum _ { k = 0 } ^ n ( n - k + 4 ) A _ k B _ { n - k } , N _ n = \\sum _ { k = 0 } ^ n A _ k B _ { n - k } \\end{align*}"}
+{"id": "1960.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c \\bar u ) ^ n \\leq \\psi ( z , \\bar u ) \\omega ^ n & \\textnormal { o n } & \\Omega , \\\\ \\bar u = 0 & \\textnormal { o n } & \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "1319.png", "formula": "\\begin{align*} \\langle \\eta , \\partial _ u ( H ^ { ( u ) } ) ^ { - 1 } \\eta \\rangle & = \\langle T ( u ) ^ { - 1 } ( H ^ { ( u ) } ) ^ { - 1 } \\eta , e ^ { 2 \\rho ( u ) } T ( u ) ^ { - 1 } ( H ^ { ( u ) } ) ^ { - 1 } \\eta \\rangle \\\\ & = \\langle \\widetilde \\eta ^ { ( u ) } , e ^ { 2 \\rho ( u ) } \\widetilde \\eta ^ { ( u ) } \\rangle = \\sum _ { i \\in V } ( \\widetilde \\eta _ i ^ { ( u ) } ) ^ 2 e ^ { 2 \\rho _ i ( u ) } . \\end{align*}"}
+{"id": "3402.png", "formula": "\\begin{align*} \\begin{aligned} A ( t , \\rho ) & \\leq S ( t , \\rho ) + \\int _ 0 ^ t \\Big [ S ( \\tau , \\rho ) + \\int _ 0 ^ \\tau A ( \\tau ' , \\rho ) \\ , d \\tau ' \\Big ] \\ , d \\tau \\\\ & \\leq S ( t , \\rho ) + \\int _ 0 ^ t S ( \\tau , \\rho ) \\ , d \\tau + \\int _ 0 ^ t ( t - \\tau ) A ( \\tau , \\rho ) \\ , d \\tau . \\end{aligned} \\end{align*}"}
+{"id": "6311.png", "formula": "\\begin{align*} \\mathcal N ( p ) : = \\inf \\{ s > 0 : C _ s ( p ) = + \\infty \\} = \\sup \\{ s > 0 : C _ s ( p ) = 0 \\} , \\end{align*}"}
+{"id": "3716.png", "formula": "\\begin{align*} W ^ { \\rm K M S } : = \\{ m \\in M \\ : \\alpha _ { i t } ( m ) \\in \\Xi \\mbox { f o r } 0 < t < \\pi \\} . \\end{align*}"}
+{"id": "3714.png", "formula": "\\begin{align*} & ( \\cosh ( t ) x _ 0 + \\sinh ( t ) x _ 1 - y _ 0 ) ^ 2 - ( \\cosh ( t ) x _ 1 + \\sinh ( t ) x _ 0 - y _ 1 ) ^ 2 \\\\ & \\sim \\Big ( e ^ { - t } \\frac { x _ 0 - x _ 1 } { 2 } - y _ 0 \\Big ) ^ 2 - \\Big ( e ^ { - t } \\frac { x _ 1 - x _ 0 } { 2 } - y _ 1 \\Big ) ^ 2 \\sim e ^ { - t } ( x _ 0 - x _ 1 ) ( - y _ 0 ) - e ^ { - t } ( x _ 1 - x _ 0 ) ( - y _ 1 ) \\\\ & = e ^ { - t } ( x _ 1 - x _ 0 ) ( y _ 0 + y _ 1 ) \\to \\infty . \\end{align*}"}
+{"id": "7677.png", "formula": "\\begin{align*} L ( t ) = \\frac { n - 1 } { t } , W ( t ) = \\frac { C } { t ^ { ( 2 - \\gamma ) p } } , w = t ^ { \\gamma p } , G ( t ) = \\frac { a } { t ^ { 2 p - 2 } } \\quad \\mbox { a n d } H ( t ) = \\frac { b } { t } , \\qquad \\forall t \\in ( 0 , \\sup _ \\Omega \\rho ) , \\end{align*}"}
+{"id": "8982.png", "formula": "\\begin{align*} v = 1 - u \\ \\ \\in C ^ \\infty ( \\overline \\Omega ) \\end{align*}"}
+{"id": "7607.png", "formula": "\\begin{align*} A : = \\frac { - \\tau ^ 3 _ { 1 } } { 8 ( 1 - \\tau ^ 2 _ { 1 } ) } , \\ ; \\ ; B : = \\frac { \\tau _ { 1 } } { 2 } \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; C : = \\frac { - ( 2 + \\tau ^ 2 _ { 1 } ) } { 3 \\tau _ { 1 } } . \\end{align*}"}
+{"id": "1798.png", "formula": "\\begin{align*} \\rho _ t ( a _ p ^ * a _ q ) = \\delta ( p - q ) \\Big ( \\chi ( p ) + \\big [ \\chi ( p ) - \\chi ^ \\perp ( p ) \\big ] f _ t ( p ) \\Big ) \\end{align*}"}
+{"id": "5286.png", "formula": "\\begin{align*} \\binom { q } { 3 } | 1 + y ' | ^ { q - 3 } ( 1 + y ' ) y ^ 3 \\le \\binom { q } { 3 } | y | ^ 3 ( 1 + | y | ) ^ { q - 3 } . \\end{align*}"}
+{"id": "8259.png", "formula": "\\begin{align*} F _ 1 ( M , k ) = \\frac { G ^ 2 ( k + 1 ) } { G ( 2 k + 1 ) } \\frac { X _ { M } ( k ) } { Y _ { M } ( k ) } , \\end{align*}"}
+{"id": "1483.png", "formula": "\\begin{align*} \\int _ \\Omega f _ 0 ( V ) \\psi ^ h _ { \\mu _ j , \\xi _ j } d x & = \\int _ { \\Omega \\setminus B ( \\xi , \\rho ) } f _ 0 ( V ) \\psi ^ h _ { \\mu _ j , \\xi _ j } d x + \\sum _ { l = 1 } ^ k \\int _ { \\mathcal { A } _ l } f _ 0 ( V ) \\psi ^ h _ { \\mu _ j , \\xi _ j } d x . \\end{align*}"}
+{"id": "1176.png", "formula": "\\begin{align*} T ( f ^ { 2 } ) = 2 f T ( f ) + 2 A ( f ) ^ { 2 } . \\end{align*}"}
+{"id": "6093.png", "formula": "\\begin{align*} \\int _ { s } ^ { t } S _ { t - \\tau } Y _ { \\tau } \\circ \\mathrm { d } \\mathbf { X } _ { \\tau } : = \\lim _ { \\substack { | \\pi | \\rightarrow 0 , \\\\ \\pi = \\lbrace \\tau _ 0 = s , \\tau _ { 1 } , . . . \\tau _ { m } = t \\rbrace } } \\sum _ { 0 \\leq j < m } \\big { [ } S _ { t - \\tau _ j } Y _ { \\tau _ j } \\circ ( \\delta X ) _ { \\tau _ j , \\tau _ { j + 1 } } + S _ { t - \\tau _ j } Y ^ { \\prime } _ { \\tau _ j } \\circ \\mathbb { X } _ { \\tau _ j , \\tau _ { j + 1 } } \\big { ] } . \\end{align*}"}
+{"id": "2017.png", "formula": "\\begin{align*} \\lambda _ 1 ^ n = \\frac { \\int _ \\Omega ( - \\varphi _ 1 ) ( d d ^ c \\varphi _ 1 ) ^ n } { \\int _ \\Omega ( - \\varphi _ 1 ) ^ { n + 1 } d \\nu _ { f } } = \\inf \\left \\{ \\frac { \\int _ \\Omega ( - \\phi ) ( d d ^ c \\phi ) ^ n } { \\int _ \\Omega ( - \\phi ) ^ { n + 1 } d \\nu _ { f } } ; \\phi \\in \\mathcal E ^ 1 ( \\Omega ) , \\phi \\neq 0 \\right \\} \\cdot \\end{align*}"}
+{"id": "1478.png", "formula": "\\begin{align*} | f ^ { ' } _ \\epsilon ( V ) - f ^ { ' } _ 0 ( V ) | _ { \\frac { n } { 2 } } = O \\bigg ( \\epsilon \\ln \\Big | \\ln \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big | \\bigg ) . \\end{align*}"}
+{"id": "6491.png", "formula": "\\begin{align*} T _ { n , m } ( x ) : & = \\left [ \\binom { x + m } { j - i + m } - \\binom { x + m } { m - i - j - 1 } \\right ] _ { i , j = 0 } ^ { n - 1 } . \\end{align*}"}
+{"id": "3624.png", "formula": "\\begin{align*} a _ 0 \\ \\equiv \\ b _ 0 + 2 \\ \\equiv \\ \\begin{cases} 4 , & , \\\\ 9 , & . \\end{cases} \\end{align*}"}
+{"id": "2415.png", "formula": "\\begin{align*} x = L + K L ^ \\theta + O ( 1 ) = L + K \\big ( x + O ( x ^ \\theta ) \\big ) ^ \\theta + O ( 1 ) = L + K x ^ \\theta + O ( 1 + x ^ { 2 \\theta - 1 } ) . \\end{align*}"}
+{"id": "292.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } A ( n , x ) B ( n , y ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "8761.png", "formula": "\\begin{align*} \\begin{cases} a _ { i , k } + b _ { i , k } + c _ { i , k } = 1 \\\\ d _ { i , k } - x _ i = r _ { i , k } + s _ { i , k } + t _ { i , k } \\\\ - C a _ { i , k } \\leq r _ { i , k } \\leq 0 \\\\ 0 \\leq s _ { i , k } \\leq \\bar { y } b _ { i , k } \\\\ \\bar { y } c _ { i , k } \\leq t _ { i , k } \\leq C c _ { i , k } \\\\ \\beta _ { i , k } = - c \\P ( \\omega _ k ) \\left ( \\frac { s _ { i , k } } { \\bar { y } } + c _ { i , k } \\right ) \\end{cases} \\end{align*}"}
+{"id": "5250.png", "formula": "\\begin{align*} v ( \\psi _ 0 ( a ) ) & = v \\left ( \\sum _ { j = 1 } ^ { n } \\psi _ j ( a ) \\varphi ( a ) ^ j \\right ) \\\\ & = v ( \\varphi ( a ) ) + v \\left ( \\sum _ { j = 1 } ^ { n } \\psi _ j ( a ) \\varphi ( a ) ^ { j - 1 } \\right ) \\\\ & \\geq v ( \\varphi ( a ) ) > 0 \\end{align*}"}
+{"id": "4020.png", "formula": "\\begin{align*} [ P , g ] = g ( P \\otimes \\lbrace 0 , i \\infty \\rbrace ) . \\end{align*}"}
+{"id": "9089.png", "formula": "\\begin{align*} \\mathfrak { S } ( f _ k ) = \\mathfrak { S } ( f _ k | _ { V \\setminus \\mathcal { F } } ) = \\tilde { k } + \\tilde { r } - 1 , \\ , \\ , \\ , \\ , \\ , \\overline { \\mathfrak { S } } ( f _ k ) = \\overline { \\mathfrak { S } } ( f _ k | _ { V \\setminus \\mathcal { F } } ) = n - | \\mathcal { F } | - \\tilde { k } + 1 . \\end{align*}"}
+{"id": "364.png", "formula": "\\begin{align*} \\bigvee ^ k _ { s = 1 } c ^ { \\sigma } _ { i _ s } \\imath _ { i _ s } \\colon \\bigvee ^ k _ { s = 1 } \\Sigma X _ { i _ s } \\longrightarrow ( \\underline { \\Sigma X } , \\underline { \\ast } ) ^ { \\partial \\sigma , c } . \\end{align*}"}
+{"id": "8397.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\triangle ( w _ 1 + \\cdots + w _ { n - 1 } ) \\geq \\frac { 1 } { \\max _ { i = 1 , \\cdots , n - 1 } r _ i } \\cdot \\frac { n ( n ^ 2 - 1 ) } { 1 2 } \\big ( \\prod _ { i = 1 } ^ { n - 1 } | \\gamma _ i | ^ { i ( n - i ) } \\big ) ^ { \\frac { 1 2 } { n ( n ^ 2 - 1 ) } } \\cdot e ^ { \\frac { 1 2 } { n ( n ^ 2 - 1 ) } ( w _ 1 + \\cdots + w _ { n - 1 } ) } . \\end{align*}"}
+{"id": "8308.png", "formula": "\\begin{align*} g = \\left ( { \\begin{array} { c c } a & b \\\\ c & d \\\\ \\end{array} } \\right ) = ( a b | c d ) = ( a , b | c , d ) \\ , , a , b , c , d \\in \\Z _ q \\ , , \\det ( g ) = a d - b c = 1 \\ , , \\end{align*}"}
+{"id": "5007.png", "formula": "\\begin{align*} \\Pr ( X _ t = k \\ , \\vert \\ , X _ 0 = r ) = \\frac { ( n - r ) _ { k - r + 1 } } { n ^ t } \\frac { 1 } { ( k - r ) ! } \\Delta ^ { k - r } \\left [ \\frac { x ^ t } { n - x } \\right ] _ { x = r } ^ { } . \\end{align*}"}
+{"id": "2765.png", "formula": "\\begin{align*} \\| \\varphi _ t \\| _ { H _ \\Gamma ^ { 3 / 2 } ( \\partial \\mathrm { M } ) } ^ 2 = \\sum _ { N _ t ^ 1 } \\tau _ j ^ { - 2 } | a _ j | ^ 2 \\le t ^ { - 2 } \\sum _ j | a _ j | ^ 2 = t ^ { - 2 } \\| u \\| _ { L ^ 2 ( \\mathrm { M } _ 0 ) } ^ 2 . \\end{align*}"}
+{"id": "5554.png", "formula": "\\begin{align*} \\nu _ { g } = \\lim _ { i \\to \\infty } \\sum _ { s \\in F } \\delta _ { \\left \\{ H _ { 0 } g s \\xi \\right \\} } \\mu ^ { ( n _ { i } ) } ( s ) = \\lim _ { i \\to \\infty } \\sum _ { s \\in F } \\delta _ { \\left \\{ g H _ { 0 } s \\xi \\right \\} } \\mu ^ { ( n _ { i } ) } ( s ) = g . \\nu _ { e } . \\end{align*}"}
+{"id": "570.png", "formula": "\\begin{align*} \\gamma ^ { c } ( x _ 1 , x _ 2 ) = \\int _ { \\R } \\int _ { \\R _ { + } } c \\left ( \\frac { - x _ { 1 } + y _ { 1 } } { 2 \\sqrt { x _ { 2 } } } , \\frac { y _ { 2 } } { 2 x _ { 2 } } \\right ) ( h _ 0 ( y _ { 1 } ) ) ^ { 2 } N ( y _ { 2 } ) [ N ( y _ { 2 } ) ] ^ T d y _ { 2 } d y _ { 1 } . \\end{align*}"}
+{"id": "9218.png", "formula": "\\begin{align*} & a \\oplus b = ( a + b ) ( 1 + \\eta _ 1 ) , & | \\eta _ 1 | \\le 2 ^ { - d } \\\\ & a \\ominus b = ( a - b ) ( 1 + \\eta _ 1 ) , & | \\eta _ 1 | \\le 2 ^ { - d } . \\end{align*}"}
+{"id": "5519.png", "formula": "\\begin{align*} ( g . \\lambda ) _ { x } = \\theta ( x , \\cdot ) _ { \\ast } \\left ( \\overline { \\mathbb { P } } _ { \\mu , x , g } \\right ) . \\end{align*}"}
+{"id": "851.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ { t t } + \\alpha f ( u _ { t } ) + \\beta u = \\lambda g ( u ) & \\mbox { f o r $ t > 0 $ } , \\\\ u ( 0 ) = u _ { 0 } , \\\\ u _ { t } ( 0 ) = v _ { 0 } , \\end{array} \\right . \\end{align*}"}
+{"id": "377.png", "formula": "\\begin{align*} \\mathfrak { d } ( \\tau ) _ { \\sigma } = \\begin{cases} p & \\tau = \\sigma \\\\ 1 & \\mbox { e l s e } . \\end{cases} \\end{align*}"}
+{"id": "7616.png", "formula": "\\begin{align*} H _ { q } ( n ) ( f ) : = \\begin{vmatrix} a _ { n } & a _ { n + 1 } & \\cdots & a _ { n + q - 1 } \\\\ a _ { n + 1 } & a _ { n + 2 } & \\cdots & a _ { n + q } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ a _ { n + q - 1 } & a _ { n + q } & \\cdots & a _ { n + 2 q - 2 } \\end{vmatrix} . \\end{align*}"}
+{"id": "8668.png", "formula": "\\begin{align*} \\mathcal { T } ( V , q ) : = \\{ \\alpha \\in \\mathcal { C } | \\alpha V \\alpha ^ { * } \\subseteq V , \\ N ( \\alpha ) \\in \\mathbb { F } \\} . \\end{align*}"}
+{"id": "6714.png", "formula": "\\begin{align*} \\chi _ 1 e ^ { - \\frac { | D _ t | ^ 2 } { \\lambda } } ( \\chi _ 2 u ) ( t , x ) & = \\left ( \\frac { \\lambda } { 4 \\pi } \\right ) ^ { \\frac { 1 } { 2 } } \\int _ { \\R _ s } \\chi _ 1 ( t , x ) e ^ { - \\frac { \\lambda } { 4 } | s - t | ^ 2 } ( \\chi _ 2 u ) ( s , x ) ~ d s \\\\ & = \\left ( \\frac { \\lambda } { 4 \\pi } \\right ) ^ { \\frac { 1 } { 2 } } \\chi _ 1 ( t , x ) \\int _ { s , | t - s | \\geq d } e ^ { - \\frac { \\lambda } { 4 } | s - t | ^ 2 } ( \\chi _ 2 u ) ( s , x ) ~ d s \\end{align*}"}
+{"id": "4586.png", "formula": "\\begin{align*} & \\beta \\circ \\psi _ V = \\psi _ V \\circ \\beta , \\\\ & T \\circ \\psi _ V = \\psi _ L \\circ T ' , \\\\ & \\psi _ V ( \\rho ( x , y ) u ) = \\rho ( \\psi _ L ( x ) , \\psi _ L ( y ) ) \\psi _ V ( u ) . \\end{align*}"}
+{"id": "1640.png", "formula": "\\begin{align*} 2 \\ , g ( ( \\nabla _ { \\xi _ i } { f } ) Y , Z ) & = N ^ { \\ , ( 5 ) } ( \\xi _ i , Y , Z ) , 1 \\le i \\le p , \\end{align*}"}
+{"id": "6999.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ r a _ i \\tilde { \\textbf { Q } } ^ { \\lambda _ i } . \\end{align*}"}
+{"id": "8378.png", "formula": "\\begin{align*} C _ { m } = E \\left \\{ \\left ( \\frac { X _ { 1 } + \\ldots + X _ { m } } { m } - \\theta ( m ) \\right ) \\otimes \\left ( \\frac { X _ { 1 } + \\ldots + X _ { m } } { m } - \\theta ( m ) \\right ) \\right \\} . \\end{align*}"}
+{"id": "9024.png", "formula": "\\begin{align*} C _ 1 ( \\lambda _ 1 \\rho _ 0 ) = C _ 2 ( \\lambda _ 2 \\rho _ 0 ) = C _ 3 ( \\lambda _ 3 \\rho _ 0 ) \\end{align*}"}
+{"id": "5855.png", "formula": "\\begin{align*} \\psi ( x ) = - \\sum \\limits _ { \\substack { ( y , y ' ) \\in \\partial \\Lambda \\\\ y \\in \\Lambda , y ' \\notin \\Lambda } } G _ { \\omega , \\Lambda , E } ( x , y ) \\psi ( y ' ) . \\end{align*}"}
+{"id": "2596.png", "formula": "\\begin{align*} v _ h ( y ) : = \\frac { 1 } { h } v ( T _ h ^ { - 1 } y ) . \\end{align*}"}
+{"id": "6941.png", "formula": "\\begin{align*} \\mathcal { O } ( a ) = \\{ & ( \\ell + a - \\frac 1 2 , \\ell + a , - 2 ( \\ell + a ) + \\frac 1 2 ) , ( \\ell + a + \\frac 1 2 , \\ell + a , - 2 ( \\ell + a ) - \\frac 1 2 ) , \\\\ & ( \\ell + a , \\ell + a - \\frac 1 2 , - 2 ( \\ell + a ) + \\frac 1 2 ) , ( \\ell + a , \\ell + a + \\frac 1 2 , - 2 ( \\ell + a ) - \\frac 1 2 ) \\mid \\ell \\in \\Z \\} \\end{align*}"}
+{"id": "7469.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { 3 } \\sum \\limits _ { k = 0 } ^ { + \\infty } \\varepsilon ^ { \\alpha _ i + k - 1 } \\ , \\left ( w _ { \\alpha _ i + k - 1 } ^ { ( i ) } ( x _ i ) + u _ { \\alpha _ i + k - 1 } ^ { ( i ) } \\big ( x _ i , \\tfrac { \\overline { x } _ i } { \\varepsilon } , t \\big ) \\right ) + \\sum \\limits _ { k = 0 } ^ { + \\infty } \\varepsilon ^ { k } \\ , \\left ( w _ { k } ^ { ( i ) } ( x _ i ) + u _ { k } ^ { ( i ) } \\big ( x _ i , \\tfrac { \\overline { x } _ i } { \\varepsilon } , t \\big ) \\right ) . \\end{align*}"}
+{"id": "4597.png", "formula": "\\begin{align*} \\begin{array} { l } C _ n ( x ) = \\min \\left \\{ \\begin{array} { l } L _ n ( x ) + \\int _ 0 ^ \\infty C _ { n - 1 } ( x - \\xi ) f _ n ( \\xi ) \\mathrm { d } \\xi , \\\\ \\min _ { x < y \\leq x + B } \\{ K + v ( y - x ) + L _ n ( y ) + \\int _ 0 ^ \\infty C _ { n - 1 } ( y - \\xi ) f _ n ( \\xi ) \\mathrm { d } \\xi \\} \\end{array} \\right \\} , \\\\ G _ n ( x ) = v x + L _ n ( x ) + \\int _ 0 ^ \\infty C _ { n - 1 } ( x - \\xi ) f _ n ( \\xi ) \\mathrm { d } \\xi , \\\\ C _ n ( x ) = - v x + \\min \\{ G _ n ( x ) , K + \\min _ { x \\leq y \\leq x + B } G _ n ( y ) \\} . \\end{array} \\end{align*}"}
+{"id": "9102.png", "formula": "\\begin{align*} B ^ { k + 1 } = \\{ p \\} \\cup B ^ k \\setminus \\{ q \\} \\end{align*}"}
+{"id": "2808.png", "formula": "\\begin{align*} \\gathered a _ { i k } \\in \\mathbb C , a _ { i k } = 0 , k > i + r , i > k + q , \\\\ a _ { i \\ , i + r } = 1 , a _ { i + q \\ , i } \\ne 0 , i \\ge 0 ; \\endgathered \\end{align*}"}
+{"id": "5271.png", "formula": "\\begin{align*} T _ { \\rho } ^ { [ m - 1 ] } = T _ { \\rho } ^ { ( m - 1 ) } \\circ \\cdots \\circ T _ { \\rho } ^ { ( 1 ) } . \\end{align*}"}
+{"id": "438.png", "formula": "\\begin{align*} \\vec U _ t + { \\bf D _ x } ( { \\bf A } \\vec U ) + { \\bf A } ^ T { \\bf D _ x } ( \\vec U ) = 0 \\vec U = \\begin{bmatrix} \\vec U _ 1 \\\\ [ 0 . 0 5 c m ] \\vec U _ 2 \\\\ [ 0 . 0 5 c m ] \\end{bmatrix} { \\bf A } = \\begin{bmatrix} { \\bf a _ { 1 1 } } & { \\bf a _ { 1 2 } } \\\\ [ 0 . 0 5 c m ] { \\bf a _ { 2 1 } } & { \\bf a _ { 2 2 } } \\\\ [ 0 . 0 5 c m ] \\end{bmatrix} . \\end{align*}"}
+{"id": "849.png", "formula": "\\begin{align*} L _ { C F } = 3 M K \\end{align*}"}
+{"id": "3590.png", "formula": "\\begin{align*} L ( m n ) \\ = \\ L ( m ) + L ( n ) \\ = \\ L ( m ) + L ( n ) - [ m n \\ < \\ 1 0 ^ { L ( m ) + L ( n ) - 1 } ] . \\end{align*}"}
+{"id": "4017.png", "formula": "\\begin{align*} \\mathbb { M } _ k : = \\mathbb { Z } [ X , Y ] _ { k - 2 } \\otimes _ { \\mathbb { Z } } \\mathbb { M } _ 2 , \\end{align*}"}
+{"id": "5794.png", "formula": "\\begin{align*} \\log ( A _ 1 ) & = \\log ( A + h ) \\\\ & = \\log A + o ( 1 ) \\\\ & \\geq r ^ \\gamma + o ( 1 ) . \\end{align*}"}
+{"id": "5616.png", "formula": "\\begin{align*} \\int _ { X } { \\rm I I } _ { x } d \\eta ( x ) = - \\sum _ { s \\in G } \\mu ( s ) \\int _ { Y } \\int _ { \\pi ^ { - 1 } ( y ) } H \\left ( \\xi _ { n - 1 } ^ { s ^ { - 1 } . x } \\right ) d \\eta ^ { y } ( x ) \\frac { d s \\nu } { d \\nu } ( y ) d \\nu ( y ) = - \\int _ { X } H ( \\xi _ { n - 1 } ^ { x } ) d \\eta ( x ) . \\end{align*}"}
+{"id": "4041.png", "formula": "\\begin{align*} S _ { \\mathrm { o f f } } : = \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\sum _ { p | c } \\frac { \\mathcal { S } ( i , j , c ) } { c } \\ , J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { i j } } { c } \\right ) \\ , \\big | L ( f , n + 1 ) \\big | ^ 2 . \\end{align*}"}
+{"id": "1412.png", "formula": "\\begin{align*} y _ { k } = \\cosh ( \\Delta _ { k - 1 } ) y _ { k - 1 } + \\sinh ( \\Delta _ { k - 1 } ) e _ { k } , e _ k ^ { ( k ) } = \\sinh ( \\Delta _ { k - 1 } ) y _ { k - 1 } + \\cosh ( \\Delta _ { k - 1 } ) e _ { k } . \\end{align*}"}
+{"id": "3064.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac { u ( r ) } { u _ 0 ( r ) } = \\lim _ { r \\to \\infty } \\frac { v ( r ) } { v _ 0 ( r ) } = \\lim _ { r \\to \\infty } \\frac { u ' ( r ) } { u _ 0 ' ( r ) } = \\lim _ { r \\to \\infty } \\frac { v ' ( r ) } { v _ 0 ' ( r ) } = 1 . \\end{align*}"}
+{"id": "2386.png", "formula": "\\begin{align*} f ( \\{ A _ { 0 } \\} ) = \\sum _ { i = 1 } ^ { 4 } \\sqrt { ( x - x _ { i } ) ^ 2 + ( y - y _ { i } ) ^ 2 + ( z - z _ { i } ) ^ 2 } \\to m i n . \\end{align*}"}
+{"id": "5169.png", "formula": "\\begin{align*} \\xi ^ { ( 0 ) } = \\sup \\bigg \\{ a \\ge 0 \\colon \\lim _ { n \\to \\infty } \\sup _ { v \\in \\mathbb { Z } ^ d , \\ , \\| v \\| \\ge n } \\mathbb { P } \\big ( \\gamma ( 0 , v ) \\subset \\mathrm { c y l } ( 0 , v , \\| v \\| ^ a ) \\big ) < 1 \\bigg \\} , \\end{align*}"}
+{"id": "578.png", "formula": "\\begin{align*} a _ { - } = \\lim _ { y \\rightarrow - \\infty } a ( y ) a _ { + } = \\lim _ { y \\rightarrow + \\infty } a ( y ) \\end{align*}"}
+{"id": "483.png", "formula": "\\begin{align*} t ^ { k _ r - k _ { r - 1 } + k _ n - 2 } t _ { 2 r - 1 } t _ { 2 r } t _ { 2 n + 1 } t _ { 2 n + 2 } t _ { 2 n + 3 } t _ { 2 n + 4 } = p q \\end{align*}"}
+{"id": "3874.png", "formula": "\\begin{align*} c _ { \\ell } ( s _ \\ell , s _ \\ell ' ) = | y _ \\ell - y _ \\ell ' | + \\| x _ { \\ell } - x _ \\ell ' \\| _ { 2 } . \\end{align*}"}
+{"id": "1315.png", "formula": "\\begin{align*} \\prod _ { i \\in V } \\phi _ i ( u ) ^ { 3 / 2 } \\sqrt { | H _ \\beta | } = \\prod _ { i \\in V } \\phi _ i ( u ) ^ 2 \\sqrt { | \\widetilde { H } ^ { ( u ) } | } \\sqrt { | K ^ { ( u ) } | } , \\end{align*}"}
+{"id": "5575.png", "formula": "\\begin{align*} S _ { y ' } : = \\{ ( y , r ) \\} \\times { \\rm T r e e } _ { F , H _ { 0 } } & \\to \\mathbb { R } _ { \\ge 0 } \\\\ x & \\mapsto H _ { \\alpha _ { x , g } \\parallel \\alpha _ { x , e } } \\left ( \\mathcal { P } _ { x , n } \\right ) \\end{align*}"}
+{"id": "7770.png", "formula": "\\begin{align*} \\begin{aligned} & z ^ { \\vec x ^ 1 } _ { r _ 1 } - z ^ { \\vec x ^ 2 } _ { r _ 1 } \\geq 0 , \\\\ & z ^ { \\vec x ^ 1 } _ { r _ 2 } - z ^ { \\vec x ^ 2 } _ { r _ 2 } \\leq 0 , \\end{aligned} \\end{align*}"}
+{"id": "6577.png", "formula": "\\begin{align*} \\Psi ( X ) = \\sum _ { i = 1 } ^ 3 \\sum _ { j = 1 } ^ 2 Z _ { i , j } ^ * X Z _ { i , j } \\end{align*}"}
+{"id": "8925.png", "formula": "\\begin{align*} y \\big ( \\bigl \\{ X _ { n i } ^ { ( j + 1 ) } \\bigr \\} \\big ) = Z _ { j + 1 } + \\sum _ { l = j + 2 } ^ k A _ l , \\end{align*}"}
+{"id": "226.png", "formula": "\\begin{align*} \\omega ( s _ 1 , . . . , s _ { M } \\mid t _ 1 , . . . , t _ { N } ) = \\sum _ { \\substack { m _ 1 , . . . , m _ { M } , \\ , n _ 1 , . . . , n _ { N } > 0 \\\\ \\sum _ { j = 1 } ^ { M } m _ j = \\sum _ { k = 1 } ^ { N } n _ k } } \\prod _ { j = 1 } ^ { M } \\frac { 1 } { { m _ j } ^ { s _ j } } \\prod _ { k = 1 } ^ { N } \\ ; \\frac { 1 } { { n _ k } ^ { t _ k } } \\end{align*}"}
+{"id": "1817.png", "formula": "\\begin{align*} \\ < \\Psi , [ \\N ^ \\ell , V _ B ( t ) ] \\Psi \\ > = \\mathrm { I m } \\int \\hat V ( k ) \\ < \\Psi , [ \\N ^ \\ell , b _ k ( t ) b _ { - k } ( t ) ] \\Psi \\ > \\d k \\forall t \\in \\R \\ . \\end{align*}"}
+{"id": "494.png", "formula": "\\begin{align*} p ( s _ { n } ) = q _ { s } ^ { s _ { n } } ( 1 - q _ { s } ) ^ { 1 - s _ { n } } . \\end{align*}"}
+{"id": "50.png", "formula": "\\begin{align*} \\Omega ( b ^ \\ast ) = ( J _ n \\Omega ( b ) J _ n ^ { - 1 } ) ^ { \\mathrm { t } } , \\forall b \\in \\mathcal { R } _ p . \\end{align*}"}
+{"id": "1929.png", "formula": "\\begin{align*} f ( w , z ) = \\exp \\left ( \\prod _ { k = 1 } ^ n P _ k ( w , z ) \\right ) \\end{align*}"}
+{"id": "7337.png", "formula": "\\begin{align*} v ^ 3 z + 1 = u ^ 3 . \\end{align*}"}
+{"id": "3743.png", "formula": "\\begin{align*} { \\rm I m } \\ W _ { \\vec { k } , \\vec { l } } ( \\vec { x } , \\vec { y } ) + { \\rm I m } \\ W _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) = 0 , \\end{align*}"}
+{"id": "71.png", "formula": "\\begin{align*} \\prod _ { \\ell \\in \\mathcal { S } } \\prod _ { i = 1 } ^ { r ( \\ell ) } \\mathbf { G } ^ 1 [ \\ell , \\mathcal { T } ( \\ell ) _ i ] \\backslash \\left ( \\mathbf { G } ^ 1 [ \\ell , \\mathcal { T } ( \\ell ) _ i ] \\Delta ^ { \\mathcal { T } ( \\ell ) _ i } ( \\mathbf { G } ^ 1 ( \\mathbb { Q } _ p ) ) \\mathbf { K } ^ 1 [ \\ell ] / \\mathbf { K } ^ 1 [ \\ell ] \\right ) , \\end{align*}"}
+{"id": "4666.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta u + \\mu u = \\lambda u + u \\log u ^ 2 , \\hbox { i n } \\ , \\ , \\mathbb { R } ^ N , \\\\ & \\int _ { \\mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } , \\end{aligned} \\right . \\end{align*}"}
+{"id": "2140.png", "formula": "\\begin{align*} \\frac { d } { d t } E ^ { \\mbox { t o t a l } } = - \\triangle , \\end{align*}"}
+{"id": "4121.png", "formula": "\\begin{align*} T _ \\lambda ^ { \\vec { v } } T _ { \\widehat { \\lambda } } ^ { \\vec { v } } = T _ { \\widehat { \\lambda } } ^ { \\vec { v } } T _ \\lambda ^ { \\vec { v } } \\ ; \\ ; T _ { \\lambda ^ m } ^ { \\vec { v } } = \\left ( T _ \\lambda ^ { \\vec { v } } \\right ) ^ m , \\end{align*}"}
+{"id": "6719.png", "formula": "\\begin{align*} \\mathcal { P } ( \\widetilde { X } _ \\eta ) = \\{ M _ \\ast ( \\nu _ \\eta \\vee \\kappa ) : \\kappa \\in \\mathcal { P } ( \\{ 0 , 1 \\} ^ \\Z ) \\} , \\end{align*}"}
+{"id": "4860.png", "formula": "\\begin{align*} W _ 2 ^ 2 ( T _ \\# \\mu , \\delta _ { T ( a ) } ) & = \\int _ { X } \\| x - T ( a ) \\| ^ 2 \\ , d ( T _ \\# \\mu ) ( x ) \\\\ & = \\int _ { X } \\| T ( x ) - T ( a ) \\| ^ 2 \\ , d \\mu ( x ) \\\\ & = \\int _ { X } \\| x - a \\| ^ 2 \\ , d \\mu ( x ) = W _ 2 ^ 2 ( \\mu , \\delta _ a ) . \\end{align*}"}
+{"id": "5653.png", "formula": "\\begin{align*} \\aligned m _ \\nu ( a , b ) = & \\inf _ { \\mathcal { P } _ \\nu ( a , b ) } \\Bigl \\{ \\frac { \\gamma _ p + \\gamma _ q - 2 } { 2 ( \\gamma _ p + \\gamma _ q ) } ( | \\nabla u | ^ 2 _ 2 + | \\nabla v | ^ 2 _ 2 ) \\\\ & + \\frac { 2 2 ^ * _ \\mu - \\gamma _ p - \\gamma _ q } { 2 2 ^ * _ \\mu ( \\gamma _ p + \\gamma _ q ) } \\int _ { \\mathbb { R } ^ N } [ ( I _ \\mu \\ast | u | ^ { 2 ^ * _ \\mu } ) | u | ^ { 2 ^ * _ \\mu } + ( I _ \\mu \\ast | v | ^ { 2 ^ * _ \\mu } ) | v | ^ { 2 ^ * _ \\mu } ] \\Bigr \\} > 0 . \\endaligned \\end{align*}"}
+{"id": "8677.png", "formula": "\\begin{align*} _ { n , m } ^ { 2 } = \\Delta _ { 0 } + \\Delta _ { 1 } + \\widetilde { \\Delta } _ { 2 } , \\end{align*}"}
+{"id": "1966.png", "formula": "\\begin{align*} B _ 1 : = e ^ { - C _ 0 ^ 2 - 2 } \\sup _ { \\bar \\Omega } ( 1 + | \\nabla v | ^ 2 ) . \\end{align*}"}
+{"id": "7457.png", "formula": "\\begin{align*} \\frac { d ^ n q _ 1 } { d t ^ n } \\Big | _ { t = 0 } = 0 \\frac { \\partial ^ n \\varphi ^ { ( 0 ) } } { \\partial t ^ n } \\Big | _ { t = 0 } = 0 . \\end{align*}"}
+{"id": "6675.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\R ^ N } \\rho ( x ) G _ \\alpha [ u ( x , \\tau ) ] v ( x , \\tau ) \\ , d x & \\leq \\int _ \\epsilon ^ \\tau \\int _ { \\R ^ N } G _ \\alpha ( u ) \\big [ \\Delta v - ( - \\Delta ) ^ s v + \\rho v _ t \\big ] \\ , d x d t \\\\ & + \\int _ { \\R ^ N } \\rho ( x ) v ( x , \\epsilon ) G _ \\alpha [ u ( x , \\epsilon ) ] \\ , d x \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2950.png", "formula": "\\begin{align*} u ( f ) ( \\tau ) & = \\prod _ { n = 1 } ^ \\infty \\frac { ( 1 - q ^ n ) ^ 2 } { ( 1 - q ^ { p n } ) ^ 2 } . \\end{align*}"}
+{"id": "6448.png", "formula": "\\begin{align*} f ( x ) = e ^ { - z | x | ^ 2 + \\xi \\cdot x } \\end{align*}"}
+{"id": "2566.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = g ( u ) , \\ \\ u \\ge 0 , & \\ \\ B _ R , \\\\ u = 0 , \\ \\ & \\ \\ \\partial B _ R , \\end{cases} \\end{align*}"}
+{"id": "3535.png", "formula": "\\begin{align*} \\wp '' ( z ) = 6 \\wp ^ 2 ( z ) - \\frac { g _ 2 } { 2 } . \\end{align*}"}
+{"id": "2385.png", "formula": "\\begin{align*} g ( x _ 1 , \\dots , x _ { k + 1 } ) = \\sum _ { \\emptyset \\neq I \\subsetneq [ k + 1 ] } a _ I ( x _ 1 , \\dots , x _ { k + 1 } ) \\left ( \\prod _ { i \\in I } x _ i \\right ) ' \\end{align*}"}
+{"id": "2192.png", "formula": "\\begin{align*} | | \\mathcal { E } _ { \\alpha } | | _ 2 & \\leq \\alpha \\big ( ( 1 + \\sqrt { 2 } \\alpha n ) ^ 2 + [ 2 ( 1 + \\sqrt { 2 } \\alpha n ) ^ 2 + 4 \\sqrt { 2 } n ( 1 + \\sqrt { 2 } \\alpha n ) ] \\\\ & \\quad ~ + [ 8 \\sqrt { 2 } n ( 1 + \\sqrt { 2 } \\alpha n ) + 8 n ^ 2 ] + 1 6 n ^ 2 \\big ) \\\\ & = 3 \\alpha [ \\sqrt { 2 } n ( \\alpha + 2 ) + 1 ] ^ 2 . \\end{align*}"}
+{"id": "349.png", "formula": "\\begin{align*} T ^ m _ j & = \\left \\{ ( A _ 0 \\subsetneq \\dots \\subsetneq A _ m ) \\in ( K ^ n _ k ) _ m \\mid \\begin{array} { l } A _ m = [ n ] \\\\ k \\not \\in A _ { j - 1 } \\\\ A _ { j } \\supsetneq A _ { j - 1 } \\cup \\{ k \\} \\end{array} \\right \\} . \\end{align*}"}
+{"id": "8929.png", "formula": "\\begin{gather*} \\Bigg ( \\sum _ { l = 1 } ^ { j } Z _ l + \\sum _ { l = j + 1 } ^ k B _ l ^ { ( j ) } \\Bigg ) \\Bigg ( Z _ { j + 1 } + \\sum _ { l = j + 2 } ^ k A _ l \\Bigg ) \\\\ { } = \\Bigg ( \\sum _ { i = 1 } ^ k S _ p \\Bigg ) \\Bigg ( \\sum _ { i = 1 } ^ { k } S _ { p + i } \\Bigg ) = \\sum _ { i = 1 } ^ { 2 k } S _ i + \\sum _ { q \\geq 2 } \\sum _ { ( p _ 1 , \\dots , p _ q ) \\in \\mathbb { I } _ { 2 k } ^ q } \\alpha _ { ( p _ 1 , \\dots , p _ q ) } [ S _ { p _ 1 } , \\dots , S _ { p _ q } ] . \\end{gather*}"}
+{"id": "4703.png", "formula": "\\begin{align*} \\Theta _ \\chi \\left ( { \\vect u } ^ { \\rm s o f t } \\oplus { \\vect u } ^ { \\rm s t i f f } \\right ) = { \\vect u } ^ { \\rm s o f t } \\oplus P _ { \\widehat { \\mathcal { H } } _ \\chi ^ { \\rm s t i f f } } \\vect u ^ { \\rm s t i f f } . \\end{align*}"}
+{"id": "5565.png", "formula": "\\begin{align*} h _ { \\mu } ( M , \\bar { \\lambda } _ { \\ell , p } ) = h _ { \\mu } ( M , { \\bold \\alpha } _ { p } ) . \\end{align*}"}
+{"id": "474.png", "formula": "\\begin{align*} V _ { \\mu } \\otimes V _ { \\nu } = \\bigoplus _ { \\substack { 0 \\leq k _ 1 \\leq \\cdots \\leq k _ n \\\\ [ 1 p t ] \\mu + \\nu - \\sum _ { i \\in I } k _ i \\alpha _ i \\in P _ { + } } } V _ { \\mu + \\nu - \\sum _ { i = 1 } ^ n k _ i \\alpha _ i } . \\end{align*}"}
+{"id": "7017.png", "formula": "\\begin{align*} f = f _ 0 + f _ 1 Q _ i + \\ldots + f _ r Q _ i ^ r \\end{align*}"}
+{"id": "6869.png", "formula": "\\begin{align*} C _ r = \\| d _ r \\| _ \\infty . \\end{align*}"}
+{"id": "4646.png", "formula": "\\begin{align*} \\alpha = q \\quad \\mbox { a n d } \\beta = \\frac { q } { q - 1 } , \\end{align*}"}
+{"id": "7963.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } b ( t ) = 0 . \\end{align*}"}
+{"id": "7463.png", "formula": "\\begin{align*} \\chi _ \\delta ^ { ( i ) } ( x _ i ) = \\left \\{ \\begin{array} { l l } 1 , & \\ \\ x _ i \\ge \\ell _ i - \\delta , \\\\ 0 , & \\ \\ x _ i \\le \\ell _ i - 2 \\delta , \\end{array} \\right . i \\in \\{ 2 , 3 \\} , \\end{align*}"}
+{"id": "272.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { y ( y + 1 ) ( y ^ 2 + 1 0 y + 1 ) } { ( 1 - y ) ^ 5 } ( L i _ 5 ( y z ) - L i _ 5 ( z ) ) \\right \\} \\end{align*}"}
+{"id": "8147.png", "formula": "\\begin{align*} \\sum _ F \\chi _ F ( \\alpha ) X _ F = \\lambda _ 1 ( A ) ^ \\alpha = \\exp \\left \\{ \\alpha \\sum _ { n \\ge 1 } ( - 1 ) ^ { n - 1 } \\varphi _ n \\right \\} \\end{align*}"}
+{"id": "3563.png", "formula": "\\begin{align*} J _ { T _ { * } } ^ { ( 1 ) } = ( - 1 ) ^ { 1 + \\tfrac { 3 } { m } } \\frac { 2 \\pi \\sigma ^ { 3 } } { m T _ { * } ^ { 3 / m } } \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { k ! } \\Gamma \\left ( \\frac { n k - 3 } { m } \\right ) ( - 1 ) ^ { ( m - n ) k / m } T _ { * } ^ { ( n - m ) k / m } , \\end{align*}"}
+{"id": "352.png", "formula": "\\begin{align*} d ( u , v ) = d ( u ^ { ( 0 ) } , v ) \\leq n _ 0 + \\frac { t + 1 } { t - 2 } \\cdot \\frac { n } { \\log n _ 0 } = \\big ( 1 + o ( 1 ) \\big ) \\frac { n } { \\log n } , \\end{align*}"}
+{"id": "3642.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N 1 \\cdot { \\rm P r o b } ( T _ n ) . \\end{align*}"}
+{"id": "1621.png", "formula": "\\begin{align*} d \\eta ^ i = \\Phi , 1 \\le i \\le p . \\end{align*}"}
+{"id": "1918.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow + \\infty } u ^ k = \\bar { u } \\mbox { a n d } \\lim \\limits _ { k \\rightarrow + \\infty } \\zeta ^ k = \\bar { \\zeta } . \\end{align*}"}
+{"id": "6605.png", "formula": "\\begin{align*} \\mathcal { O } _ k ( \\mathsf { G L } _ { m | n } ) = k [ t _ { i j } : 1 \\le i , j \\le m + n ] _ { \\det _ 0 } . \\end{align*}"}
+{"id": "2297.png", "formula": "\\begin{align*} h = \\sum _ { n } c _ n a _ n . \\end{align*}"}
+{"id": "5912.png", "formula": "\\begin{align*} \\pi _ { \\psi } [ \\begin{pmatrix} 1 & b \\\\ 0 & 1 \\end{pmatrix} , t ] f ( y ) = t \\psi ( \\tfrac { 1 } { 2 } \\langle y , y b \\rangle ) f ( y ) , \\end{align*}"}
+{"id": "5234.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c c c } x = 0 : & 0 & e _ 1 & e _ 2 \\\\ x = 1 : & 0 & e _ 3 & e _ 4 \\\\ x = \\infty : & e _ 5 & e _ 6 & e _ 7 \\end{array} \\right ) \\to \\left ( \\begin{array} { c c c } 0 & - e _ 1 & - e _ 2 \\\\ 0 & - e _ 3 & - e _ 4 \\\\ 2 - e _ 5 & 2 - e _ 6 & 2 - e _ 7 \\end{array} \\right ) , \\end{align*}"}
+{"id": "8725.png", "formula": "\\begin{align*} & \\quad \\sum _ { \\substack { a _ 1 > a _ 2 \\\\ a _ 1 + a _ 2 \\le a } } \\frac { ( - 2 ) ^ { a - a _ 1 - a _ 2 } a ! } { a _ 1 ! a _ 2 ! ( a - a _ 1 - a _ 2 ) ! } E \\{ \\| Y _ 1 \\| _ { 2 } ^ { 2 a _ 1 } \\| Y _ 2 \\| _ { 2 } ^ { 2 a _ 2 } ( Y _ 1 ^ { \\top } Y _ 2 ) ^ { a - a _ 1 - a _ 2 } \\} \\\\ & = \\sum _ { \\substack { a _ 1 > a _ 2 \\\\ a _ 1 + a _ 2 \\le a } } \\frac { ( - 2 ) ^ { a - a _ 1 - a _ 2 } a ! } { a _ 1 ! a _ 2 ! ( a - a _ 1 - a _ 2 ) ! } E \\{ \\| Y _ 1 \\| _ { 2 } ^ { 2 a _ 1 } \\| X _ 1 \\| _ { 2 } ^ { 2 a _ 2 } ( Y _ 1 ^ { \\top } X _ 1 ) ^ { a - a _ 1 - a _ 2 } \\} . \\end{align*}"}
+{"id": "3261.png", "formula": "\\begin{align*} \\| h _ \\beta \\| _ { L ^ q _ \\omega } & \\lesssim \\| g ^ { 1 / r } \\| _ { L ^ q _ \\omega } = \\| g \\| ^ { 1 / r } _ { L ^ { \\tilde q } _ \\omega } \\\\ & \\lesssim \\| ( h _ \\alpha ) ^ r \\| _ { L ^ { \\tilde p } _ \\omega } \\\\ & = \\| h _ \\alpha \\| _ { L ^ p _ \\omega } , \\end{align*}"}
+{"id": "6913.png", "formula": "\\begin{align*} r ( m _ j ) : = \\left ( \\sum _ { \\substack { ( 1 - T ^ { - 1 } ) ^ { j - 1 } \\le n \\le ( 1 + T ^ { - 1 } ) ^ { j + 2 } \\\\ n \\in \\mathcal { M } } } f ( n ) ^ 2 \\right ) ^ { 1 / 2 } , \\textrm { f o r e v e r y } j \\in \\mathcal { J } \\end{align*}"}
+{"id": "5847.png", "formula": "\\begin{align*} \\pi _ 1 ( \\tilde { \\mathcal { X } } _ { j _ 0 + 1 } ) = \\pi _ 1 \\left ( X ^ { \\mathcal { X } _ { j _ 0 } , \\mathcal { N } _ { j _ 0 } } _ { \\tau _ { H , w _ { j _ 0 } } } \\right ) \\leqslant \\pi _ 1 ( w _ { j _ 0 } ) + ( v _ + - 2 \\delta ) H . \\end{align*}"}
+{"id": "4486.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( U ) & : = E ( U ) + \\omega Q ( U ) + \\mathbf { c } \\cdot \\mathbf { P } ( U ) , \\\\ K _ { \\omega , \\mathbf { c } } ( U ) & : = \\partial _ { \\lambda } S _ { \\omega , \\mathbf { c } } ( \\lambda U ) | _ { \\lambda = 1 } = 2 L ( U ) + 3 N ( U ) + 2 \\omega Q ( U ) + 2 \\mathbf { c } \\cdot \\mathbf { P } ( U ) , \\\\ L _ { \\omega , \\mathbf { c } } ( U ) & : = K _ { \\omega , \\mathbf { c } } ( U ) - 3 N ( U ) = 2 L ( U ) + 2 \\omega Q ( U ) + 2 \\mathbf { c } \\cdot \\mathbf { P } ( U ) . \\end{align*}"}
+{"id": "6094.png", "formula": "\\begin{align*} ( \\delta Z ) _ { s , t } = S _ { t - s } Z _ { s } + \\int _ { s } ^ { t } S _ { t - \\tau } F ( Z _ { \\tau } ) \\mathrm { d } \\tau + \\int _ { s } ^ { t } S _ { t - \\tau } G ( Z _ { \\tau } ) \\circ \\mathrm { d } \\mathbf { X } _ { \\tau } , \\ \\ Z _ { 0 } = z _ 0 , \\ s , t \\in \\mathbb { R } , \\end{align*}"}
+{"id": "5227.png", "formula": "\\begin{align*} \\tilde { b } . e _ i \\otimes t & = \\sum _ { j } e _ j b _ { j , i } e _ i . e _ i \\otimes t \\\\ & = ( e _ 1 + \\dots + e _ m ) \\otimes ( \\sum _ { j } e _ j b _ { j , i } e _ i . t ) \\\\ & = ( e _ 1 + \\dots + e _ m ) \\otimes e _ \\ell t ' \\\\ & = e _ \\ell \\otimes t ' . \\end{align*}"}
+{"id": "6514.png", "formula": "\\begin{align*} \\det \\mathbf { T } _ { n , m , 0 , 0 } ( x ; q ) = \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ m \\frac { ( 1 - q ^ { x + i - j } ) ( 1 - q ^ { x + 2 i + j + a - b - 2 } ) } { ( 1 - q ^ { x + 2 i - j } ) ( 1 - q ^ { i + j - 1 } ) } . \\end{align*}"}
+{"id": "3328.png", "formula": "\\begin{align*} \\rho _ { w } ( j , w ) = \\overline { w } - 2 R _ j \\left ( - \\frac { 1 } { 2 } \\frac { \\overline { w } ^ 2 } { | w | ^ 3 } ( R _ j ) _ { \\overline { z } } + \\frac { 1 } { 2 } \\frac { 1 } { | w | } ( R _ j ) _ { z } \\right ) \\end{align*}"}
+{"id": "6516.png", "formula": "\\begin{align*} G _ { \\mu , \\nu } ( x ) & = x \\big ( K _ { \\nu } ( x ) \\tilde { t } _ { \\mu - 1 , \\nu - 1 } ( x ) + K _ { \\nu - 1 } ( x ) \\tilde { t } _ { \\mu , \\nu } ( x ) \\big ) , \\\\ \\tilde { G } _ { \\mu , \\nu } ( x ) & = 1 - G _ { \\mu , \\nu } ( x ) , \\end{align*}"}
+{"id": "8791.png", "formula": "\\begin{align*} ( u ^ 2 + 8 t u + 1 2 t ^ 2 + 6 u + 3 0 t + 1 6 ) c _ 2 & = ( u ^ 2 + 4 t u + 6 t ^ 2 + 1 0 u + 2 4 t + 2 4 ) b _ 2 \\\\ & + ( u ^ 2 + 4 t u + 6 u ) c _ 3 - ( u ^ 2 + 2 t u + 4 u ) b _ 3 \\end{align*}"}
+{"id": "4100.png", "formula": "\\begin{align*} Q _ { 1 , \\vec { v } } = \\left ( \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} , \\begin{pmatrix} 0 & 0 & 2 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ \\end{pmatrix} , \\begin{pmatrix} 0 & 2 & 0 \\\\ 0 & 0 & 2 \\\\ 1 & 0 & 0 \\end{pmatrix} \\right ) . \\end{align*}"}
+{"id": "5212.png", "formula": "\\begin{align*} c \\not \\precsim E _ d \\setminus \\{ d \\} \\ \\Rightarrow \\ d \\leq c \\qquad c \\precsim d \\ \\Rightarrow \\ c = d c ^ \\precsim \\in \\mathsf { S } \\mathbb { P } , \\end{align*}"}
+{"id": "1185.png", "formula": "\\begin{align*} \\overline { C } = \\max _ { p _ n } & \\sum _ { n } c _ { n } \\ , p _ { n } \\\\ & \\sum _ n p _ n = \\sum _ n d _ n \\end{align*}"}
+{"id": "4581.png", "formula": "\\begin{align*} & \\rho ( \\alpha ( x ) , \\alpha ( y ) ) \\circ \\beta = \\beta \\circ \\rho ( x , y ) , \\\\ & \\rho ( \\alpha ( x ) , \\alpha ( y ) ) \\rho ( a , b ) - \\rho ( \\alpha ( a ) , \\alpha ( b ) ) \\rho ( x , y ) = ( \\rho ( [ x , y , a ] , \\alpha ( b ) ) - \\rho ( [ x , y , b ] , \\alpha ( a ) ) ) \\circ \\beta , \\\\ & \\rho ( [ x , y , a ] , \\alpha ( b ) ) \\circ \\beta - \\rho ( \\alpha ( y ) , \\alpha ( a ) ) \\rho ( x , b ) = \\rho ( \\alpha ( a ) , \\alpha ( x ) ) \\rho ( y , b ) + \\rho ( \\alpha ( x ) , \\alpha ( y ) ) \\rho ( a , b ) , \\end{align*}"}
+{"id": "3735.png", "formula": "\\begin{align*} \\sum _ { \\beta = 1 } ^ { N ( \\gamma ) } d _ \\beta ^ \\gamma \\phi ^ { \\alpha _ 1 ^ { \\gamma , \\beta } , n } ( F _ 1 ) \\cdots \\phi ^ { \\alpha _ m ^ { \\gamma , \\beta } , n } ( F _ m ) 1 & = \\left ( S , \\prod _ { i = 1 } ^ m f _ i \\otimes \\rho _ n ( E ^ \\gamma ) \\right ) . \\end{align*}"}
+{"id": "5272.png", "formula": "\\begin{align*} \\| T _ { \\rho } ^ { ( m ) } T _ { \\rho } ^ { [ m - 1 ] } f \\| _ q ^ q \\le \\| G _ { m } T _ { \\rho } ^ { [ m - 1 ] } f \\| _ q ^ q + \\beta ^ q \\| L _ m T _ { \\rho } ^ { [ m - 1 ] } f \\| _ q ^ q = \\| T _ { \\rho } ^ { [ m - 1 ] } G _ { m } f \\| _ q ^ q + \\beta ^ q \\| T _ { \\rho } ^ { [ m - 1 ] } L _ m f \\| _ q ^ q . \\end{align*}"}
+{"id": "986.png", "formula": "\\begin{align*} ( \\psi _ 1 \\circ \\theta _ 1 ) \\cdot \\chi _ { f _ 1 } = ( \\psi _ 2 \\circ \\theta _ 2 ) \\cdot \\chi _ { f _ 2 } . \\end{align*}"}
+{"id": "4654.png", "formula": "\\begin{align*} \\hat { d } ( x , t ) = \\begin{cases} d ( x , t ) & \\mbox { i f } \\ , x \\in \\Omega \\ , \\mbox { a n d } \\ , t \\in ( 0 , T ) , \\\\ 0 & \\mbox { i f } \\ , x \\in \\Omega \\ , \\mbox { a n d } \\ , t \\geq T , \\end{cases} \\end{align*}"}
+{"id": "7529.png", "formula": "\\begin{align*} y ( x ) & = \\sum _ { n , j \\geq 0 } \\sum _ { k \\geq 1 } ( - 1 ) ^ { k } a _ { k , j } B _ { n , k } ( f _ { 1 , 0 } , \\ldots , f _ { n - k + 1 , 0 } ) \\frac { x ^ { n + j } } { n ! j ! } \\\\ & = \\sum _ { m \\geq n \\geq 0 } \\binom { m } { n } \\left \\{ \\sum _ { k \\geq 1 } ( - 1 ) ^ { k } a _ { k , m - n } B _ { n , k } ( f _ { 1 , 0 } , \\ldots , f _ { n - k + 1 , 0 } ) \\right \\} \\frac { x ^ m } { m ! } . \\end{align*}"}
+{"id": "5259.png", "formula": "\\begin{align*} \\| D _ { S , x } f \\| _ p \\le \\sum _ { T \\subseteq S } \\| [ E _ { T } [ f ] ] _ { S \\to x } \\| _ p \\leq \\sum _ { T \\subseteq S } r ^ { | S | } \\gamma = ( 2 r ) ^ { | S | } \\gamma , \\end{align*}"}
+{"id": "2736.png", "formula": "\\begin{align*} A _ q u = - \\Delta u + q , u \\in D ( A ) = H _ \\Delta ( \\mathrm { M } ) \\cap H _ 0 ^ 1 ( \\mathrm { M } ) . \\end{align*}"}
+{"id": "8182.png", "formula": "\\begin{align*} \\Omega _ s = \\{ \\omega \\in \\Omega : \\exists \\ : t _ 0 \\ : \\ : \\forall \\ : t \\geq t _ 0 , \\ : \\ : \\forall \\ : x \\in C ( 0 , k t ) \\ : \\ : \\exists \\ : y \\in B ( x , \\rho ( t ) ) \\ : \\ : \\ : B \\left ( y , R _ { \\rho ( t ) } + b \\right ) \\subseteq K ^ c \\} , \\end{align*}"}
+{"id": "41.png", "formula": "\\begin{align*} \\lambda ' = \\varphi _ { A ' } ^ \\vee Y _ q ^ \\vee \\overline { \\lambda } _ 0 Y _ q \\varphi _ { A ' } . \\end{align*}"}
+{"id": "4852.png", "formula": "\\begin{align*} ( \\prod _ { i = 1 } ^ u d ( i ) \\ast h _ l ( q ( i ) ) ) \\ast d ( u + 1 ) = ( \\prod _ { i = 1 } ^ u d ( i ) \\ast f _ l ( \\delta ( q ( i ) ) ) \\ast d ( u + 1 ) \\notin A _ 1 \\cap \\sigma ( L _ 1 ) . \\end{align*}"}
+{"id": "3602.png", "formula": "\\begin{align*} L ( q ^ { \\beta } ) \\ = \\ m + 1 - L ( f ) + [ r ( p ) \\ < \\ 1 0 ^ { L ( f ) + L ( q ^ { \\beta } ) - 1 } ] \\ \\le \\ 1 + [ r ( p ) \\ < \\ 1 0 ^ { L ( f ) + L ( q ^ { \\beta } ) - 1 } ] \\ \\le \\ 2 . \\end{align*}"}
+{"id": "8585.png", "formula": "\\begin{align*} p _ j ( t ) = P ( Z _ 0 ( t ) = j ) = \\sum _ { k = 1 } ^ \\infty \\big ( p _ { j , + } ^ k ( t ) - p _ { j , - } ^ k ( t ) \\big ) , \\end{align*}"}
+{"id": "4879.png", "formula": "\\begin{align*} v ( x ) = ( 1 - | x | ^ 2 ) _ + ^ s \\end{align*}"}
+{"id": "4955.png", "formula": "\\begin{align*} F ( k , t + 1 , n ) = \\begin{dcases} \\frac { n - k } { n } F ( k - 1 , t , n ) + \\frac { k } { n } F ( k , t , n ) & \\textnormal { i f } \\ ; k < t ; \\\\ 1 & \\textnormal { i f } \\ ; k \\ge t ; \\end{dcases} \\end{align*}"}
+{"id": "4272.png", "formula": "\\begin{align*} g ( \\kappa ) = \\left \\langle \\beta , \\kappa \\right \\rangle , \\beta , \\kappa \\in \\mathbb { R } ^ { d } . \\end{align*}"}
+{"id": "1891.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow + \\infty } \\langle \\lambda ^ k , S v \\rangle = \\langle \\lambda ^ * , S v \\rangle \\forall \\ v \\in \\mathcal { U } . \\end{align*}"}
+{"id": "7383.png", "formula": "\\begin{align*} \\begin{aligned} b _ { k , N } ( L ) & = \\int _ 0 ^ 1 R _ N ^ 2 ( L , \\alpha , \\Delta ) e ( - k \\alpha ) \\ , d \\alpha \\\\ & = \\frac { L } { N ^ 2 } \\sum _ { n \\in \\mathbb { Z } } \\widehat { \\Delta } \\left ( \\frac { L n } { N } \\right ) \\sum _ { 1 \\leq i \\not = j \\leq N } \\int _ 0 ^ 1 e ( ( n a _ i - n a _ j - k ) \\alpha ) \\ , d \\alpha \\\\ & = \\frac { L } { N ^ 2 } \\sum _ { n \\not = 0 } \\sum _ { \\substack { 1 \\leq i \\not = j \\leq N \\\\ n ( a _ i - a _ j ) = k } } \\widehat { \\Delta } \\left ( \\frac { L n } { N } \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "5504.png", "formula": "\\begin{align*} D _ { \\varrho } = \\left \\{ s \\in \\cap _ { \\alpha \\in \\varrho } \\ker ( \\alpha ) : \\alpha ( s ) < 0 \\mbox { f o r a l l } \\alpha \\in \\Delta - \\varrho \\right \\} . \\end{align*}"}
+{"id": "5861.png", "formula": "\\begin{align*} ( \\mathcal { Y } ^ { ( E ) } _ { x _ 0 , l , L _ 1 , L _ 2 } ) ^ c & = \\{ \\omega : \\Lambda _ { L _ 2 , L _ 1 } \\} \\\\ & = \\{ \\omega : \\partial \\Lambda ^ + _ { L _ 1 + l + 2 } \\partial \\Lambda ^ - _ { L _ 2 - l - 2 } \\} \\end{align*}"}
+{"id": "2052.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { \\nu _ k } { \\sigma ^ 2 _ k } = 0 . \\end{align*}"}
+{"id": "6329.png", "formula": "\\begin{align*} \\frac { \\partial z } { \\partial \\omega } ( \\phi , \\omega , r ; t ) = ( 1 + O ( \\varepsilon ) ) k t ^ 3 , \\forall \\ , t \\in [ - \\rho , \\rho ] . \\end{align*}"}
+{"id": "2292.png", "formula": "\\begin{align*} h = \\sum _ n c _ n a _ n . \\end{align*}"}
+{"id": "8359.png", "formula": "\\begin{align*} S _ { v e } = \\left \\{ \\begin{array} { c c } 1 , & s ( e ) = v , \\\\ 0 , & s ( e ) \\neq v , \\end{array} \\right . , T _ { v e } = \\left \\{ \\begin{array} { c c } 1 , & t ( e ) = v , \\\\ 0 , & t ( e ) \\neq v . \\end{array} \\right . \\end{align*}"}
+{"id": "3448.png", "formula": "\\begin{align*} ( a , b , t ) \\longmapsto \\left \\{ \\begin{bmatrix} ( t - b ) / 2 & & a \\\\ a & & ( t + b ) / 2 \\end{bmatrix} \\right \\} _ { \\Gamma _ { p } } , \\end{align*}"}
+{"id": "6991.png", "formula": "\\begin{align*} \\Gamma = \\bigcup _ { i = 1 } ^ \\epsilon \\left ( \\gamma _ i + v K \\right ) . \\end{align*}"}
+{"id": "7232.png", "formula": "\\begin{align*} \\Psi ( u , \\beta , A ) : = \\mathcal { G } ( u , \\beta , A ) - \\Phi ( u , \\beta , A ) , \\end{align*}"}
+{"id": "6555.png", "formula": "\\begin{align*} T ( f ) ( x ) & = \\int _ { \\mathbb H ^ n } K ( e , y ) | y | _ h ^ { - Q / p } d y | x | _ h ^ { - Q / p } \\\\ \\| T ( f ) ( x ) \\| _ { L _ { | x | _ h } ^ p L _ \\theta ^ { \\bar { p } _ 2 } ( \\mathbb H ^ n ) } & = \\omega _ Q ^ { \\frac { 1 } { \\bar { p } _ 2 } - \\frac { 1 } { \\bar { p } _ 1 } } \\int _ { \\mathbb H ^ n } K ( e , y ) | y | _ h ^ { - Q / p } d y \\| f _ j \\| _ { L _ { | x | _ h } ^ p L _ \\theta ^ { \\bar { p } _ 2 } ( \\mathbb H ^ n ) } \\end{align*}"}
+{"id": "2963.png", "formula": "\\begin{align*} A _ I = \\bigcap _ { i \\in I } A _ i \\hat { A } _ I = q ^ { - 1 } ( A _ I ) \\end{align*}"}
+{"id": "4348.png", "formula": "\\begin{align*} \\left . \\begin{array} { c } \\left ( x , x ^ { \\ast } \\right ) , \\left ( y , y ^ { \\ast } \\right ) \\in \\mathrm { g p h } T \\\\ \\left \\langle x - y , x ^ { \\ast } - y ^ { \\ast } \\right \\rangle = 0 \\end{array} \\right \\} \\Rightarrow h ( x , y ^ { \\ast } ) + h ( y , x ^ { \\ast } ) = h ( x , x ^ { \\ast } ) + h ( y , y ^ { \\ast } ) . \\end{align*}"}
+{"id": "2185.png", "formula": "\\begin{align*} c _ 1 = \\sqrt { \\frac { 3 } { 2 } } - 1 . \\end{align*}"}
+{"id": "6404.png", "formula": "\\begin{align*} \\widetilde { \\psi } : = \\widehat { \\psi } \\circ T _ M \\end{align*}"}
+{"id": "6089.png", "formula": "\\begin{align*} \\mathrm { d } Z _ t = A Z _ t \\ , \\mathrm { d } t + F ( Z _ t ) \\ , \\mathrm { d } t + G ( Z _ t ) \\circ \\mathrm { d } \\mathbf { X } _ t , \\ \\ \\ \\ Z _ { 0 } = z _ 0 \\in \\mathcal { B } _ { \\alpha } \\end{align*}"}
+{"id": "670.png", "formula": "\\begin{align*} \\| S ( k ) \\varphi \\| _ { \\sup } = O ( e ^ { ( - \\rho - \\lambda _ 1 + 2 \\tau ( \\lambda _ 1 + \\rho + 1 ) ) r ( 0 , k ) } ) c ( D \\varphi ) . \\end{align*}"}
+{"id": "5105.png", "formula": "\\begin{align*} \\int _ { B _ { \\rho } } \\left \\vert w - ( w ) _ { \\rho } \\right \\vert ^ { p _ 0 } & \\leq \\int _ { B _ { \\rho } } \\left \\vert w - w ( 0 ) \\right \\vert ^ { p _ 0 } \\\\ & \\leq \\int _ { 0 } ^ { \\rho } \\int _ { \\mathbb { S } ^ { n - 1 } } \\tau ^ { p _ 0 } \\left \\vert D w \\right \\vert ^ { p _ 0 } \\tau ^ { n - 1 } d \\tau d \\phi \\\\ & = C _ { 8 } ( n , p _ 0 , \\lambda ) \\rho ^ { n + p _ 0 } \\left \\| D w \\right \\| _ { L ^ { \\infty } ( B _ { 1 / 4 } ) } ^ { p _ 0 } . \\end{align*}"}
+{"id": "5095.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { P } } ( \\gamma , \\sigma , t ) = \\mathcal { P } \\left ( V \\left ( \\gamma , \\sigma , t \\right ) , \\gamma \\right ) , \\widetilde { \\mathcal { Q } } ( \\gamma , \\sigma , t ) = \\mathcal { Q } \\left ( V \\left ( \\gamma , \\sigma , t \\right ) , \\gamma \\right ) , \\end{align*}"}
+{"id": "8812.png", "formula": "\\begin{align*} T = W P _ { { \\cal F } ^ \\perp } { \\cal S } W ^ { - 1 } , \\varphi = W P _ { { \\cal F } ^ \\perp } 1 _ { \\mathbb D } . \\end{align*}"}
+{"id": "6878.png", "formula": "\\begin{align*} v _ j = \\left ( \\frac { 1 } { N } \\sum _ { i \\in [ N ] \\setminus j } A _ N ( i , j ) - \\lambda \\right ) ^ { - 1 } \\frac { 1 } { N } \\sum _ { i \\in [ N ] \\setminus j } A _ N ( i , j ) v _ i . \\end{align*}"}
+{"id": "8036.png", "formula": "\\begin{align*} Z = \\sum _ { j = 1 } ^ k A _ j \\otimes B _ j \\end{align*}"}
+{"id": "8424.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } ( s , \\lambda k ) _ { k } = \\frac { 1 } { \\left ( 1 + \\frac { 2 } { \\lambda } \\right ) ^ { s } } , \\end{align*}"}
+{"id": "5923.png", "formula": "\\begin{align*} \\overline { C } _ { X ^ { \\ast } } ( h , y I _ { 2 m } ) & = \\overline { C } _ { X ^ { \\ast } } ( [ \\lambda _ h , g _ h ] , [ y , g _ y ] ) \\\\ & = m _ { X ^ { \\ast } } ( g _ h ^ { y } g _ y ) ^ { - 1 } m _ { X ^ { \\ast } } ( g _ h ) m _ { X ^ { \\ast } } ( g _ y ) \\widetilde { C } _ { X ^ { \\ast } } ( [ \\lambda _ h , g _ h ] , [ y , g _ y ] ) \\\\ & = m _ { X ^ { \\ast } } ( g _ h ^ { y } g _ y ) ^ { - 1 } m _ { X ^ { \\ast } } ( g _ h ) m _ { X ^ { \\ast } } ( g _ y ) \\\\ & = ( x ( g _ h ^ y ) , x ( g _ y ) ) _ F = ( x ( g _ h ^ y ) , y ^ { m } ) _ F = 1 . \\end{align*}"}
+{"id": "6748.png", "formula": "\\begin{align*} \\lim _ { K \\to \\infty } \\overline { d } _ { ( \\ell _ i ) } ( \\{ n \\in \\N : \\eta _ { K } ( n ) \\neq \\eta ( n ) \\} ) = 0 \\end{align*}"}
+{"id": "6768.png", "formula": "\\begin{align*} \\nu _ p \\left ( \\sum _ { n = 0 } ^ { p ^ { \\ell - t + k } - 1 } [ c ] ^ s ( n ) \\right ) = \\begin{cases} \\ell - 1 & a _ k = a _ { k - 1 } = \\cdots = a _ 0 = p - 1 \\\\ \\geq \\ell & \\end{cases} \\end{align*}"}
+{"id": "1835.png", "formula": "\\begin{align*} \\nu \\Big ( a ^ * _ { p _ 1 } ( t ) a _ { p _ 1 - k } ( t ) a ^ * _ { h _ 2 } ( s ) a _ { h _ 2 - k } ( s ) \\Big ) & = | \\Lambda | \\delta ( k ) f _ 0 ( p _ 1 ) f _ 0 ( h _ 2 ) \\\\ & + | \\Lambda | \\delta ( k ) \\delta ( p _ 1 - h _ 2 ) e ^ { i ( t - s ) ( E _ { p _ 1 } - E _ { h _ 2 } ) } f _ 0 ( p _ 1 ) \\widetilde f _ 0 ( h _ 2 ) . \\end{align*}"}
+{"id": "1396.png", "formula": "\\begin{align*} \\int _ M d d ^ c \\omega _ 0 ^ { n - k - 1 } \\wedge \\omega ^ k \\stackrel { \\eqref { t h m : v a i s m a n - p p - e q 1 } } { = } - ( n - k - 1 ) \\int _ M ( - d J \\theta ) ^ { n - k } \\wedge \\omega ^ k < 0 . \\end{align*}"}
+{"id": "532.png", "formula": "\\begin{align*} \\left ( f , g \\right ) : = \\sum _ { k \\in \\hbar \\mathbb { Z } ^ { n } } f ( k ) \\overline { g ( k ) } , \\end{align*}"}
+{"id": "2499.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\xi ^ * = r ^ \\frac { N - 1 } { 2 } \\xi , \\eta ^ * = r ^ \\frac { N - 1 } { 2 } \\eta . \\end{array} \\right . \\end{align*}"}
+{"id": "6781.png", "formula": "\\begin{align*} t _ \\mu = \\pi s _ { j _ 1 } \\dots s _ { j _ l } , \\pi \\in \\Pi , \\ j _ 1 , \\dotsc , j _ l \\in I \\cup \\{ 0 \\} , \\ l = \\ell ( t _ \\mu ) \\end{align*}"}
+{"id": "131.png", "formula": "\\begin{align*} K = \\bigcap \\limits _ { t \\in \\mathbb { R } } \\varphi ^ t ( V ) . \\end{align*}"}
+{"id": "2803.png", "formula": "\\begin{align*} \\tilde { \\mathbf { n } } _ \\lambda = \\max \\left ( \\mathbf { n } _ \\lambda ^ 2 , \\sqrt { \\lambda } \\mathbf { b } _ \\lambda \\right ) . \\end{align*}"}
+{"id": "3737.png", "formula": "\\begin{align*} U ( \\vec { a } , \\Lambda ) & \\phi ^ { \\alpha , n } ( f ) U ( \\vec { a } , \\Lambda ) ^ { - 1 } \\left ( S , g _ \\beta \\otimes \\rho _ n ( E ^ \\beta ) \\right ) \\\\ & = \\left ( S , f ( \\Lambda ^ { - 1 } ( \\cdot - \\vec { a } ) ) ^ { S ^ \\flat } ( \\cdot ) ( \\bar { \\Lambda } [ \\hat { H } ( \\rho ) \\hat { f } _ 0 \\pm \\hat { P } ( \\rho ) \\hat { f } _ 1 ] ) \\cdot g _ \\beta \\otimes \\rho _ n ( E ^ \\beta ) \\right ) , \\end{align*}"}
+{"id": "8251.png", "formula": "\\begin{align*} Z _ { h , q } ( i ) = \\begin{cases} h - 1 & i = q ; \\\\ 1 & k - j + h \\leq i \\leq k - 2 i \\neq q ; \\\\ l - j + 1 & i = k - 1 ; \\\\ 0 & , \\end{cases} \\end{align*}"}
+{"id": "4107.png", "formula": "\\begin{align*} \\left ( T _ \\lambda ^ { \\vec { v } } \\right ) _ { i , j } = x _ { j - i + 1 } + x _ { j - i + 2 } \\alpha _ { n - 1 } + \\dots + x _ j \\alpha _ { n - i + 1 } \\end{align*}"}
+{"id": "5536.png", "formula": "\\begin{align*} p \\mapsto H _ { n , K } ( p ) : = \\int _ { Y } \\int _ { \\pi ^ { - 1 } ( y ) } H ( \\theta _ { L _ { x } \\ast } \\mathbb { P } _ { \\mu , n | K } ^ { y } ) d \\eta _ { p } ^ { y } ( x ) d \\nu ( y ) . \\end{align*}"}
+{"id": "4918.png", "formula": "\\begin{align*} \\Pr ( X _ t = j ) = \\Pr ( \\widetilde { X } _ t = j ) \\end{align*}"}
+{"id": "1718.png", "formula": "\\begin{align*} \\sqrt { | E _ 2 | ^ 2 + | X ^ 2 | ^ 2 } = | E _ 2 | = ( I + | X | ^ 2 ) ^ { 1 / 2 } | E | = \\Big [ \\begin{smallmatrix} | A | & 0 \\\\ 0 & 0 \\end{smallmatrix} \\Big ] . \\end{align*}"}
+{"id": "1258.png", "formula": "\\begin{align*} \\mu _ { X _ 1 } ( x _ 1 , x _ 1 ) = \\mu _ { X _ 2 } ( x _ 2 , x _ 2 ) = 1 \\end{align*}"}
+{"id": "3756.png", "formula": "\\begin{align*} G ( a ) = \\left \\{ \\begin{array} { l l } a + \\bar { k } , & \\hbox { $ 1 \\leq a \\leq \\bar { l } $ ; } \\\\ a + \\bar { k } + \\underline { k } , & \\hbox { $ \\bar { l } + 1 \\leq a \\leq \\bar { k } $ ; } \\\\ a + \\underline { k } + \\bar { k } - \\bar { l } \\equiv a + \\underline { l } , & \\hbox { $ r + 1 \\leq a \\leq r + \\underline { k } $ . } \\end{array} \\right . \\end{align*}"}
+{"id": "7371.png", "formula": "\\begin{align*} R _ N ^ 2 ( L , f ) : = \\frac { 1 } { N } \\sum _ { 1 \\leq i \\neq j \\leq N } \\sum _ { m \\in \\mathbb { Z } } f \\left ( \\frac { x _ i - x _ j + m } { \\ell } \\right ) . \\end{align*}"}
+{"id": "6736.png", "formula": "\\begin{align*} \\{ N _ \\ast ( \\nu _ { \\eta * } \\vee \\nu _ \\eta \\vee \\kappa ) : \\kappa \\in \\mathcal { P } ( \\{ 0 , 1 \\} ^ { \\Z } ) \\} = \\{ ( M _ H ) _ \\ast ( m _ H \\vee \\kappa ) : \\kappa \\in \\mathcal { P } ( \\{ 0 , 1 \\} ^ \\Z ) \\} . \\end{align*}"}
+{"id": "4494.png", "formula": "\\begin{align*} Q ( \\Psi _ { \\omega } ) = \\omega ^ { 1 - \\frac { d } { 2 } } Q ( \\Psi ) , \\ \\ E ( \\Psi _ { \\omega } ) = \\omega ^ { 2 - \\frac { d } { 2 } } E ( \\Psi ) , \\ \\ \\mathbf { P } ( \\Psi _ { \\omega } ) = \\omega ^ { \\frac { 3 - d } { 2 } } \\mathbf { P } ( \\Psi ) . \\end{align*}"}
+{"id": "3207.png", "formula": "\\begin{align*} \\overline { \\psi } _ { y _ 1 , y _ 2 } = \\norm { \\cdot } _ 1 - \\lim _ { n \\to \\infty } S _ n ( \\psi _ { y _ 1 , y _ 2 } ) . \\end{align*}"}
+{"id": "4894.png", "formula": "\\begin{align*} 0 & \\ge \\frac { c _ { 1 } \\rho _ 0 ^ { n - 1 } } { 2 } \\int _ { - 1 } ^ 1 \\phi ( \\tau ) \\ , d \\tau \\int _ { \\partial B _ 1 } G ( \\rho _ 0 e , \\rho _ 0 \\omega ) \\ , d H ^ { n - 1 } _ \\omega - C \\\\ & \\ge \\frac { c \\ , c _ { 1 } } { 2 \\rho _ 0 ^ { 1 - 2 s } } \\int _ { - 1 } ^ 1 \\phi ( \\tau ) \\ , d \\tau \\int _ { \\partial B _ 1 \\cap \\{ | e - \\omega | < \\eta \\} } \\ , \\frac { d H ^ { n - 1 } _ \\omega } { | e - \\omega | ^ { n - 2 s } } - C \\\\ & = + \\infty . \\end{align*}"}
+{"id": "7399.png", "formula": "\\begin{align*} R _ { N _ m } ^ 2 ( ( 1 + C / m ) L _ m , \\alpha , \\Delta ) = ( 1 + C / m ) L _ m + o ( 1 ) = L + o ( 1 ) . \\end{align*}"}
+{"id": "508.png", "formula": "\\begin{align*} \\frac { c _ { k - 1 } - 2 c _ { k } + c _ { k + 1 } } { h ^ 2 } & = f ( x _ k ) , k = 1 , \\ldots , N - 1 , \\\\ c _ 0 & = 0 , c _ N = 0 . \\end{align*}"}
+{"id": "4063.png", "formula": "\\begin{align*} S _ { \\mathrm { m a i n } } \\geq \\sum _ { L = 1 } ^ { \\infty } \\phi ( L ) | y _ L | ^ 2 \\left ( \\log \\left ( \\frac { p ^ { \\frac { ( ( k - 1 ) ! ) ^ 2 } { 2 } } } { D } \\right ) + c _ 0 - c _ 1 \\right ) , \\end{align*}"}
+{"id": "2035.png", "formula": "\\begin{align*} n \\psi ^ { - 1 \\slash n } \\Delta F = L ( H ) + \\frac { 1 } { n } \\sum _ { \\ell = 1 } ^ n u ^ { j \\bar k } u ^ { p \\bar q } u _ { \\ell j \\bar k } u _ { \\bar \\ell p \\bar q } - \\sum _ { \\ell = 1 } ^ n u ^ { j \\bar q } u ^ { p \\bar k } u _ { \\ell j \\bar k } u _ { \\bar \\ell p \\bar q } . \\end{align*}"}
+{"id": "5128.png", "formula": "\\begin{align*} \\partial _ i V & = \\frac { \\lambda _ i } { 1 + \\lambda _ i ^ 2 } V \\\\ \\partial _ { i i } V & = \\frac { 1 } { ( 1 + \\lambda _ i ^ 2 ) ^ 2 } V \\\\ \\partial _ { i j } V & = \\frac { \\lambda _ j \\lambda _ i } { ( 1 + \\lambda _ j ^ 2 ) ( 1 + \\lambda _ i ^ 2 ) } V . \\end{align*}"}
+{"id": "4382.png", "formula": "\\begin{align*} & ( \\mathcal { N } _ 1 , \\mathcal { N } _ 1 ' ) , & \\mathcal C _ 1 = \\left ( \\begin{matrix} \\mathfrak { s } ^ 2 & \\rho \\\\ \\rho & \\mathfrak { s } ^ 2 \\end{matrix} \\right ) , \\\\ & ( \\mathcal { N } _ 2 , \\mathcal { N } _ 2 ' ) , & \\mathcal C _ 2 = \\left ( \\begin{matrix} \\mathfrak { s } ^ 2 + | \\rho | & 0 \\\\ 0 & \\mathfrak { s } ^ 2 + | \\rho | \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "8799.png", "formula": "\\begin{align*} & 3 b _ 2 + c _ 2 = ( 4 u + 2 t + 5 ) b _ 3 - ( 4 t + 3 ) c _ 3 \\end{align*}"}
+{"id": "9011.png", "formula": "\\begin{align*} ( \\ast ) & = w _ j C ^ \\varphi _ { k j i } R ^ \\varphi _ { i k } + \\frac { 1 } { 2 } \\langle \\nabla w , \\nabla | T ^ \\varphi | ^ 2 \\rangle + \\frac { 1 } { 2 } ( 1 + w ) | C ^ \\varphi | ^ 2 \\\\ & w _ { t i } R ^ \\varphi _ { i k , t k } - \\alpha w _ { t i } R ^ \\varphi _ { i j } \\varphi ^ a _ j \\varphi ^ a _ t - \\alpha w _ t R ^ \\varphi _ { j k } \\varphi ^ a _ { j t } \\varphi ^ a _ k \\ , . \\end{align*}"}
+{"id": "9015.png", "formula": "\\begin{align*} \\int _ M | T ^ \\varphi | ^ 2 | \\nabla w | ^ 2 = 0 \\ , . \\end{align*}"}
+{"id": "8733.png", "formula": "\\begin{align*} & E \\{ ( \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { l } \\} + E \\{ ( \\| Y _ { 1 } - Y _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { l } \\} - 2 E \\{ ( \\| X _ { 1 } - Y _ { 1 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { l } \\} \\\\ & = \\sum ^ { l } _ { a = 0 } \\binom { l } { a } ( - p \\tau ) ^ { l - a } \\left ( E \\| X _ 1 - X _ 2 \\| ^ { 2 a } _ 2 + E \\| Y _ 1 - Y _ 2 \\| ^ { 2 a } _ 2 - 2 E \\| X _ 1 - Y _ 1 \\| ^ { 2 a } _ 2 \\right ) . \\end{align*}"}
+{"id": "8216.png", "formula": "\\begin{align*} \\underset { t \\to \\infty } { \\lim } P ^ \\omega ( D _ t ^ c \\mid C _ t ) = 0 . \\end{align*}"}
+{"id": "6267.png", "formula": "\\begin{align*} H _ { \\Sigma _ t } = t | \\lambda | \\phi + O ( t ^ 2 ) , \\end{align*}"}
+{"id": "3761.png", "formula": "\\begin{align*} H ^ n \\left ( x ^ 0 , \\bar { P } _ n ( \\vec { a } ) , 0 \\right ) & = \\left [ k _ { 1 2 } ^ n \\pm k _ { 2 1 } ^ n \\right ] \\left ( \\{ x ^ 0 \\} _ { \\tau = 1 } ^ { r + s } , \\bar { P } _ n ( \\vec { a } ) \\right ) \\\\ & = \\left [ h _ { 1 2 } ^ n \\pm h _ { 2 1 } ^ n \\right ] \\left ( \\{ x ^ 0 \\} _ { \\tau = 1 } ^ { r + s } , \\bar { P } _ n ( \\vec { a } ) \\right ) . \\end{align*}"}
+{"id": "2270.png", "formula": "\\begin{align*} w ( z ) = \\Phi ( z ) + T ( f ) ( z ) , \\end{align*}"}
+{"id": "1113.png", "formula": "\\begin{align*} \\langle ( \\varphi _ \\alpha , \\varphi _ \\beta ) , ( \\varphi _ { \\alpha ' } , \\varphi _ { \\beta ' } ) \\rangle = ( \\varphi _ \\alpha \\circ \\varphi _ { \\alpha ' } , \\varphi _ { \\beta ' } \\circ \\varphi _ \\beta ) = ( \\varphi _ { \\alpha \\alpha ' } , \\varphi _ { \\beta \\beta ' } ) . \\end{align*}"}
+{"id": "7686.png", "formula": "\\begin{align*} \\Theta _ L \\left ( \\tau , g ; \\begin{pmatrix} \\alpha \\\\ \\beta \\end{pmatrix} ; p \\right ) = \\sum _ { \\gamma \\in D _ L } \\theta _ { L + \\gamma } \\left ( \\tau , g ; \\begin{pmatrix} \\alpha \\\\ \\beta \\end{pmatrix} ; p \\right ) \\mathfrak { e } _ { \\gamma } . \\end{align*}"}
+{"id": "3391.png", "formula": "\\begin{align*} | p ^ { - \\sigma - i t } - 1 | ^ 2 & = p ^ { - 2 \\sigma } - 2 p ^ { - \\sigma } \\cos ( t \\log p ) + 1 \\\\ & \\leq p ^ { - 2 \\sigma } - 2 p ^ { - \\sigma } + 1 + 2 p ^ { - \\sigma } t ^ 2 ( \\log p ) ^ 2 \\\\ & \\leq ( p ^ { - \\sigma } - 1 ) ^ 2 + 2 p ^ { - \\sigma } t ^ 2 ( \\log p ) ^ 2 \\\\ & \\leq \\sigma ^ 2 ( \\log p ) ^ 2 + 2 t ^ 2 ( \\log p ) ^ 2 , \\end{align*}"}
+{"id": "7806.png", "formula": "\\begin{align*} f ( x _ { 1 } , \\cdots , x _ { n } ) = n ! \\cdot \\exp \\big ( - \\sum _ { i \\le n } x _ { i } \\big ) \\mathbf { 1 } _ { \\{ x _ { 1 } \\le \\cdots \\le x _ { n } \\} } . \\end{align*}"}
+{"id": "5465.png", "formula": "\\begin{align*} F _ { | g _ i | ^ 2 } ( x ) = \\gamma ( M , M x ) / \\Gamma ( M ) = 1 - \\sum _ { i = 0 } ^ M \\frac { ( M x ) ^ i } { i ! } e ^ { - M x } , \\end{align*}"}
+{"id": "9030.png", "formula": "\\begin{align*} \\tau ( i ) & = 2 n - i \\end{align*}"}
+{"id": "5429.png", "formula": "\\begin{align*} | r | ^ 2 \\omega \\ ^ n & = - c _ 1 ^ 2 \\wedge \\omega \\ ^ { n - 2 } + s ^ 2 \\omega \\ ^ n , \\\\ | R | ^ 2 \\omega \\ ^ n & = ( 2 c _ 2 - c _ 1 ^ 2 ) \\wedge \\omega \\ ^ { n - 2 } + | c _ 1 | ^ 2 \\omega \\ ^ n = 2 ( c _ 2 - c _ 1 ^ 2 ) \\wedge \\omega \\ ^ { n - 2 } + s ^ 2 \\omega \\ ^ n . \\end{align*}"}
+{"id": "9054.png", "formula": "\\begin{align*} ( \\theta , p _ \\theta , z , s ) \\mapsto ( x _ 1 , x _ 2 , y _ 1 , y _ 2 ) = ( e ^ s \\cos \\theta , e ^ s \\sin \\theta , z \\cos \\theta - p _ \\theta \\sin \\theta , z \\sin \\theta + p _ \\theta \\cos \\theta ) . \\end{align*}"}
+{"id": "5627.png", "formula": "\\begin{align*} C ( s , N , \\mu , r ) = C ( N , \\mu ) = \\pi ^ { \\frac { \\mu } { 2 } } \\frac { \\Gamma ( \\frac { N - \\mu } { 2 } ) } { \\Gamma ( N - \\frac { \\mu } { 2 } ) } \\Bigl ( \\frac { \\Gamma ( \\frac N 2 ) } { \\Gamma ( N ) } \\Bigr ) ^ { - 1 + \\frac { \\mu } { N } } . \\end{align*}"}
+{"id": "1758.png", "formula": "\\begin{align*} \\big [ \\frac { d ^ 2 } { d x ^ 2 } + a _ 1 ( | \\phi _ 1 | ^ 2 + | \\phi _ 2 | ^ 2 ) + U _ 1 ( x ) \\big ] \\phi _ 1 = \\nu _ 1 \\phi _ 1 \\ , \\\\ \\big [ \\frac { d ^ 2 } { d x ^ 2 } + a _ 2 ( | \\phi _ 1 | ^ 2 + | \\phi _ 2 | ^ 2 ) + U _ 2 ( x ) \\big ] \\phi _ 2 = \\nu _ 2 \\phi _ 2 \\ , \\end{align*}"}
+{"id": "5423.png", "formula": "\\begin{align*} y _ n ( h ) - y ( t ^ * ) = \\mathbf { e } _ p ( t ^ * ) \\cdot h ^ p + \\mathbf { e } _ { p + 1 } ( t ^ * ) \\cdot h ^ { p + 1 } + \\mathbf { e } _ { p + 2 } ( t ^ * ) \\cdot h ^ { p + 2 } + \\mathbf { E } ( t ^ * , h ) h ^ { p + 3 } , \\end{align*}"}
+{"id": "2683.png", "formula": "\\begin{align*} M = ( c ( m ) \\frac { q _ n } { \\vert \\vert p - a \\vert \\vert ^ { m + 2 } } ) ( I - ( m + 2 ) v v ^ T ) , \\end{align*}"}
+{"id": "2000.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c v _ \\lambda ) ^ n \\leq C ^ n f ^ n \\omega ^ n \\leq ( d d ^ c ( C u _ 0 ) ) ^ n & \\textnormal { i n } & \\Omega \\\\ v _ \\lambda = 0 & \\textnormal { o n } & \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "4395.png", "formula": "\\begin{align*} J _ { 0 } \\le \\| \\theta _ { \\epsilon } \\| _ { L ^ { \\infty } ( | x | < R ) } \\| \\nabla \\cdot \\varphi _ 3 \\| _ { L ^ 2 } \\| \\varphi _ 1 \\| _ { L ^ 4 } \\| \\varphi _ 2 \\| _ { L ^ 4 } \\le C \\theta _ { \\epsilon } ( R ) \\prod _ { j = 1 } ^ 3 \\| \\varphi _ j \\| _ { H ^ 1 } . \\end{align*}"}
+{"id": "8527.png", "formula": "\\begin{align*} \\lim _ { p , p ^ { \\prime } \\rightarrow \\infty } \\tilde { \\zeta } _ { p , p ^ { \\prime } } ( s , c ) = \\sum _ { m , n \\neq 0 } \\frac { 1 } { \\left ( m ^ { 2 } + c n ^ { 2 } \\right ) ^ { s } } \\end{align*}"}
+{"id": "3110.png", "formula": "\\begin{align*} S _ B [ n , k ] & = \\sum _ { j = k } ^ n \\binom { n } { j } [ 2 ] _ { q } ^ { j - k } q ^ { j - k } S [ j , k ] _ { q ^ 2 } , \\\\ S _ B [ n , k ] & = S _ D [ n , k ] + n \\cdot [ 2 ] _ q ^ { n - k - 1 } q ^ { n - k - 1 } S [ n - 1 , k ] _ { q ^ 2 } \\end{align*}"}
+{"id": "3565.png", "formula": "\\begin{align*} J _ { T _ { * } } = J _ { T _ { * } } ^ { ( 1 ) } \\textnormal { o r } J _ { T _ { * } } ^ { ( 2 ) } , \\end{align*}"}
+{"id": "1849.png", "formula": "\\begin{align*} - 3 F ( 2 \\pi / 3 ) + \\sum _ { i = 1 } ^ { 3 } F ( \\theta _ { i } ) \\leq - \\frac { 2 . 4 4 } { \\pi ^ { 2 } } \\lambda _ { 1 , n } ^ { r , s } \\leq - \\frac { 7 . 3 2 } { 2 \\pi ^ { 2 } } \\lambda _ { 1 , n } ^ { r , s } \\sum _ { i = 1 } ^ { 3 } \\Big ( \\frac { \\theta _ { i } } { 2 \\pi } - 1 / 3 \\Big ) ^ { 2 } . \\end{align*}"}
+{"id": "1079.png", "formula": "\\begin{align*} t ^ { \\sigma } \\| e ^ { - t H ^ { \\beta } } u _ 0 \\| _ { L ^ a } \\leq { C } t ^ { \\sigma } t ^ { - \\frac d { 2 \\beta } ( \\frac { 2 \\beta } { d ( m - 1 ) } - \\frac 1 a ) } \\| u _ 0 \\| _ { L ^ { \\frac { d ( m - 1 ) } { 2 \\beta } } } = { C } \\| u _ 0 \\| _ { L ^ { \\frac { d ( m - 1 ) } { 2 \\beta } } } \\leq { C } \\| u _ 0 \\| _ { \\exp L ^ { p } } , \\end{align*}"}
+{"id": "2260.png", "formula": "\\begin{align*} | F _ k ( x , y ) | = | f ( x _ i ( y + y _ k ) ) | \\leq \\frac { C } { | y + y _ k | ^ N } , \\end{align*}"}
+{"id": "8956.png", "formula": "\\begin{align*} \\phi _ l = \\Tilde { \\phi } _ { e } + \\sum _ { m = 1 } ^ { k } D _ { p _ m } ( r _ x ^ { l - 1 } \\Delta x ) ^ { p _ m } \\end{align*}"}
+{"id": "6571.png", "formula": "\\begin{align*} \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} \\quad { \\mathrm { a n d } } \\begin{bmatrix} B & X ^ * \\\\ X & A \\end{bmatrix} \\end{align*}"}
+{"id": "4938.png", "formula": "\\begin{align*} \\frac { i } { n } \\binom { n - i } { n - k } + \\frac { ( n - i ) } { n } \\binom { n - i - 1 } { n - k } = \\frac { k } { n } \\binom { n - i } { n - k } \\end{align*}"}
+{"id": "5019.png", "formula": "\\begin{align*} \\theta _ \\tau : \\mathcal { H } \\ni h \\mapsto ( \\langle h , \\tau _ j \\rangle ) _ { j = 1 } ^ n \\in \\mathbb { C } ^ n , \\theta _ \\omega : \\mathcal { H } \\ni h \\mapsto ( \\langle h , \\omega _ j \\rangle ) _ { j = 1 } ^ n \\in \\mathbb { C } ^ n . \\end{align*}"}
+{"id": "5543.png", "formula": "\\begin{align*} h _ { \\mu } \\left ( M , \\boldsymbol { \\alpha } _ { p } \\right ) & : = h _ { \\mu } ( Y , \\nu ) + \\int _ { G } \\int _ { X } D \\left ( \\alpha _ { x , g } \\parallel \\alpha _ { x , e } \\right ) \\varphi _ { g } ( \\pi ( x ) ) d \\eta _ { p } ( x ) d \\mu ( g ) \\\\ & = h _ { \\mu } ( Y , \\nu ) + \\int _ { G } \\int _ { Y } \\int _ { S _ { y } } D \\left ( \\alpha _ { x , g } \\parallel \\alpha _ { x , e } \\right ) d \\eta _ { p } ^ { y } d g \\nu ( y ) d \\mu ( g ) \\end{align*}"}
+{"id": "7867.png", "formula": "\\begin{align*} \\{ a _ 1 , a _ 2 , a _ 3 \\ldots , a _ k \\} ^ + : = \\overline { ( a _ 1 , a _ 2 , a _ 3 \\ldots , a _ k ) } \\mbox { a n d } \\{ a _ 1 , a _ 2 , a _ 3 \\ldots , a _ k \\} ^ - : = \\overline { ( a _ 2 , a _ 1 , a _ 3 , \\ldots , a _ k ) } . \\end{align*}"}
+{"id": "4806.png", "formula": "\\begin{align*} \\chi = \\begin{bmatrix} \\chi _ 1 , \\dots , \\chi _ r \\end{bmatrix} ^ T , \\end{align*}"}
+{"id": "1350.png", "formula": "\\begin{align*} \\forall x \\geq R : F ^ { \\mathrm { i n } } ( \\pm x ) = \\pm \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "2730.png", "formula": "\\begin{align*} \\begin{aligned} J _ 3 \\ge & - C s ^ 2 \\lambda ^ 2 \\iint _ { Q } \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 u ^ 2 d x d y d t - C s ^ 2 \\lambda ^ 2 \\int _ { 0 } ^ { T } \\int _ { \\omega _ 0 } \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 u ^ 2 d x d y d t , \\end{aligned} \\end{align*}"}
+{"id": "7970.png", "formula": "\\begin{align*} ( \\partial _ j \\partial _ i V ^ i ) \\ , V ^ j = ( \\partial _ j \\partial _ i V ^ j ) \\ , V ^ i . \\end{align*}"}
+{"id": "8384.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\triangle _ { g _ X } v _ k + ( e ^ { v _ { k - 1 } + v _ { k + 1 } - 2 v _ k } - 1 ) \\cdot | \\gamma _ k | ^ 2 \\frac { \\det ( h _ 1 | _ { F _ { k - 1 } } ) \\det ( h _ 1 | _ { F _ { k + 1 } } ) } { \\det ( h _ 1 | _ { F _ k } ) ^ 2 } / g _ X \\geq 0 , k = 1 , \\cdots , n - 1 . \\end{align*}"}
+{"id": "3501.png", "formula": "\\begin{gather*} 0 = \\int _ a ^ t \\| X ' \\| ^ 2 + \\langle ( R - \\lambda I ) X , X \\rangle d s . \\end{gather*}"}
+{"id": "7863.png", "formula": "\\begin{align*} a _ i = b _ { \\sigma ( i ) } , \\mbox { f o r } i \\in \\{ 1 , 2 , \\ldots , k \\} . \\end{align*}"}
+{"id": "6154.png", "formula": "\\begin{align*} & 0 _ 0 \\neq \\top _ 0 \\\\ & \\forall \\ , x : n ( x = 0 _ 0 \\vee x = \\top _ 0 ) \\end{align*}"}
+{"id": "7427.png", "formula": "\\begin{align*} q _ { n ^ i _ j - n ^ i _ { j - 1 } } ( x ^ i _ j - x ^ i _ { j - 1 } ) = \\sum _ { z \\in \\Z ^ 2 } q _ { n - n ^ i _ { j - 1 } } ( z - x ^ i _ { j - 1 } ) \\cdot q _ { n ^ i _ j - n } ( x ^ i _ j - z ) \\ , . \\end{align*}"}
+{"id": "985.png", "formula": "\\begin{align*} ( \\psi _ 1 ' \\circ \\theta _ 1 ' ) \\cdot \\chi _ { f _ 1 ' } = ( \\psi _ 2 ' \\circ \\theta _ 2 ' ) \\cdot \\chi _ { f _ 2 ' } \\end{align*}"}
+{"id": "6206.png", "formula": "\\begin{align*} \\Psi ( v _ 1 , \\ldots , v _ { 2 f } ) = \\Psi ( w _ 1 , \\ldots , w _ { 2 f } ) . \\end{align*}"}
+{"id": "1668.png", "formula": "\\begin{align*} g \\varrho g ^ { - 1 } = \\ \\left [ \\begin{array} { c c c c } \\varrho _ 1 & a _ { 1 2 } & \\cdots & a _ { 1 n } \\\\ 0 & \\varrho _ 2 & \\cdots & a _ { 2 n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & \\varrho _ n \\end{array} \\right ] \\end{align*}"}
+{"id": "6507.png", "formula": "\\begin{align*} \\det A _ n = \\prod _ { i = 1 } ^ n t _ i = \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ m \\frac { t _ i ( j , x ) } { t _ i ( j - 1 , x ) } = \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ m \\frac { ( x + i - j ) ( x + 2 i + j - 2 ) } { ( x + 2 i - j ) ( i + j - 1 ) } . \\square \\end{align*}"}
+{"id": "342.png", "formula": "\\begin{align*} \\| T u \\| _ { ( L ^ p ( V ) ) ^ \\ast } \\geq \\langle T u , f \\rangle = \\| u \\| _ q . \\end{align*}"}
+{"id": "6934.png", "formula": "\\begin{align*} \\log { \\left ( \\frac { 1 + b f ( p ) ^ 2 } { 1 + f ( p ) ^ 2 } \\right ) } = \\log { \\left ( \\frac { 1 - ( 1 - b ) f ( p ) ^ 2 } { 1 + f ( p ) ^ 2 } \\right ) } . \\end{align*}"}
+{"id": "3360.png", "formula": "\\begin{align*} ( 1 + B ^ * ) ( 1 + B ) = 2 + B ^ * + B = 2 + 2 \\Re ( B ) . \\end{align*}"}
+{"id": "4212.png", "formula": "\\begin{align*} \\phi ( t ) & = \\phi _ 0 ( t ) u _ { B _ 1 , \\tau _ 1 } ( t ) \\cdots u _ { B _ R , \\tau _ R } ( t ) \\end{align*}"}
+{"id": "2357.png", "formula": "\\begin{align*} \\left ( \\prod _ { x : X } M _ x \\right ) _ f = \\left ( \\prod _ { x : X } M _ x \\right ) ^ { D ( f ) } \\end{align*}"}
+{"id": "6170.png", "formula": "\\begin{align*} \\Gamma ( f ^ \\sharp ) \\circ \\phi _ F & = [ e _ { R } \\circ ( f ^ \\sharp _ 1 ) \\circ e ^ { - 1 } _ { k ( F ) } ] \\circ [ e _ { k ( F ) } \\circ \\rho _ F ] = e _ { R } \\circ ( f ^ \\sharp _ 1 ) \\circ \\rho _ F \\\\ & = e _ { R } \\circ [ l _ R \\circ f \\circ ( \\phi _ F ) ^ { - 1 } \\circ e _ { k ( F ) } ] \\circ \\rho _ F \\\\ & = f \\circ ( \\phi _ F ) ^ { - 1 } \\circ [ e _ { k ( F ) } \\circ \\rho _ F ] \\\\ & = f \\circ ( \\phi _ F ) ^ { - 1 } \\circ \\phi _ F = f . \\end{align*}"}
+{"id": "7730.png", "formula": "\\begin{align*} y ^ * ( N , u ) = \\lim _ { n \\rightarrow \\infty } \\sum _ { i = 0 } ^ { 2 ^ n - 1 } y ^ * ( N , \\frac { i } { 2 ^ n } ) { \\mathbf 1 } _ { [ \\frac { ( i - 1 ) } { 2 ^ n } , \\frac { i } { 2 ^ n } ) } ( u ) , \\end{align*}"}
+{"id": "8578.png", "formula": "\\begin{align*} & \\tau _ { j , + } ^ i ( k ) : = \\inf \\{ s > \\tau _ { j , - } ^ i ( k - 1 ) : C ^ i ( s ) = j \\} , \\\\ & \\tau _ { j , - } ^ i ( k ) : = \\inf \\{ s > \\tau _ { j , + } ^ i ( k ) : C ^ i ( s ) \\neq j \\} . \\end{align*}"}
+{"id": "7256.png", "formula": "\\begin{align*} \\Delta _ { n , k } = \\varphi _ { n - k } ( \\tilde S _ { k - 1 } + Z _ k ) - \\varphi _ { n - k } ( \\tilde S _ { k - 1 } + Y _ k ) \\ , . \\end{align*}"}
+{"id": "469.png", "formula": "\\begin{align*} D = \\sum D _ N ( x - x _ \\ast ) ^ N , \\ \\ W = \\sum W _ N ( x - x _ \\ast ) ^ N \\end{align*}"}
+{"id": "6057.png", "formula": "\\begin{align*} m ( x _ 1 - x _ 3 ) + ( n + m ) ( x _ 2 - x _ 3 ) = 2 \\pi k \\textrm { f o r s o m e } k \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "4278.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } 0 & = & \\psi ^ { \\prime } \\left ( t \\right ) - \\frac { 1 } { 4 } \\Pi _ { 5 } ^ { \\top } \\left ( t \\right ) \\Pi _ { 2 } ^ { - 1 } \\left ( t \\right ) \\Pi _ { 5 } \\left ( t \\right ) \\\\ \\varphi \\left ( T \\right ) & = & 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "587.png", "formula": "\\begin{align*} \\lim _ \\lambda \\| e ^ * _ \\lambda ( b - \\omega ( b ) I ) e _ \\lambda \\| = 0 , b \\in B . \\end{align*}"}
+{"id": "6739.png", "formula": "\\begin{align*} C : = \\varphi ^ { - 1 } ( A ) = \\bigcap _ { s \\in S } R ^ { - s } W ^ c . \\end{align*}"}
+{"id": "2758.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q ) u = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } , u _ { | \\mathrm { M } _ 0 } = 0 . \\end{align*}"}
+{"id": "47.png", "formula": "\\begin{align*} \\mathrm { E n d } _ { \\mathbb { C } } ( A _ g ) = \\{ x \\in K \\mid x g \\Lambda _ 0 \\subset g \\Lambda _ 0 \\} . \\end{align*}"}
+{"id": "5708.png", "formula": "\\begin{align*} C _ { h } ( n ) = C _ { f } ^ { + } ( n ) \\end{align*}"}
+{"id": "4722.png", "formula": "\\begin{align*} \\varGamma _ J = \\frac { Z } { Z _ J } F \\varGamma F , F = \\bigoplus _ { n = 0 } ^ \\infty F _ n , F _ n = \\prod _ { 1 \\leq i < j \\leq n } f ( x _ i - x _ j ) , \\end{align*}"}
+{"id": "8171.png", "formula": "\\begin{align*} c ( d , \\nu ) : = \\lambda _ d \\left ( \\frac { d } { \\nu \\omega _ d } \\right ) ^ { - 2 / d } . \\end{align*}"}
+{"id": "2498.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle \\xi _ { r r } ( 0 ) = - \\dfrac { 2 m _ 1 Q _ 1 ( 0 ) } { N } , \\quad \\eta _ { r r } ( 0 ) = - \\dfrac { 2 m _ 2 Q _ 2 ( 0 ) } { N } . \\end{array} \\right . \\end{align*}"}
+{"id": "5665.png", "formula": "\\begin{align*} \\ J _ \\nu ( u ^ * , v ^ * ) \\leq J _ \\nu ( u , v ) , \\ P _ \\nu ( u ^ * , v ^ * ) \\leq P _ \\nu ( u , v ) = 0 . \\end{align*}"}
+{"id": "416.png", "formula": "\\begin{align*} S ^ { - 1 } \\left ( \\begin{bmatrix} \\frac { 1 } { \\sqrt { 2 | U _ n | \\sqrt { U _ 1 } } } & 0 \\\\ 0 & \\frac { 1 } { \\sqrt { 2 | U _ n | \\sqrt { U _ 1 } } } \\end{bmatrix} \\begin{bmatrix} U _ 1 ^ 2 + U _ n ^ 2 \\\\ U _ n U _ { \\tau } \\end{bmatrix} - \\begin{bmatrix} R _ 1 \\\\ R _ 2 \\end{bmatrix} \\frac { 1 } { \\sqrt { 2 | U _ n | \\sqrt { U _ 1 } } } U _ 1 ^ 2 \\right ) = G . \\end{align*}"}
+{"id": "5884.png", "formula": "\\begin{align*} \\| u _ 0 \\| _ { L ^ 2 } = r \\ , . \\end{align*}"}
+{"id": "4436.png", "formula": "\\begin{align*} 2 L ( \\Phi ) + \\left ( \\frac { d } { 2 } + 1 \\right ) N ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) = 0 \\end{align*}"}
+{"id": "1446.png", "formula": "\\begin{align*} & \\sum \\limits _ { i = 1 } ^ k \\sum \\limits _ { l = 0 } ^ n c ^ { i l } _ m \\langle P \\psi ^ l _ { \\mu _ m ^ i , \\xi _ m ^ i } , P \\psi ^ h _ { \\mu _ m ^ j , \\xi _ m ^ j } \\rangle \\\\ = & c ^ { j h } _ m \\Big ( c _ h ( 1 + o ( 1 ) ) \\Big ) + O ( 1 ) \\sum \\limits _ { l = 0 , l \\neq h } ^ n c ^ { j l } _ m + o \\bigg ( \\Big ( \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big ) ^ { \\frac { n } { n - 2 } } \\bigg ) \\sum \\limits _ { i = 1 , i \\neq j } ^ k \\sum \\limits _ { l = 0 } ^ n c ^ { i l } _ m . \\end{align*}"}
+{"id": "8913.png", "formula": "\\begin{align*} \\| [ Z _ p , Z _ q ] \\| _ { p + q } = \\bigg \\| \\bigg [ \\sum _ { n = 1 } ^ { d _ 1 ^ p } \\big [ X _ { n 1 } ^ { ( p ) } , \\dots , X _ { n p } ^ { ( p ) } \\big ] , \\sum _ { m = 1 } ^ { d _ 1 ^ q } \\big [ X _ { m 1 } ^ { ( q ) } , \\dots , X _ { m q } ^ { ( q ) } \\big ] \\bigg ] \\bigg \\| _ { p + q } \\\\ \\leq \\sum _ { n = 1 } ^ { d _ 1 ^ p } \\sum _ { m = 1 } ^ { d _ 1 ^ q } \\big \\| \\big [ \\big [ X _ { n 1 } ^ { ( p ) } , \\dots , X _ { n p } ^ { ( p ) } \\big ] , \\big [ X _ { m 1 } ^ { ( q ) } , \\dots , X _ { m q } ^ { ( q ) } \\big ] \\big ] \\big \\| _ { p + q } . \\end{align*}"}
+{"id": "8606.png", "formula": "\\begin{align*} E [ X _ { \\ell , \\Delta } ^ 2 ] - E [ X _ { \\ell , \\Delta } ] = \\Delta ^ 2 \\nu ^ 2 E [ Z _ 0 ( \\ell \\Delta ) ^ 2 ] ( 1 + O ( \\Delta ) ) . \\end{align*}"}
+{"id": "6695.png", "formula": "\\begin{align*} S \\coloneqq \\Lambda ^ { 1 . 5 + \\delta } = ( I _ 2 - \\Delta _ 2 ) ^ { 1 / 2 } ; \\end{align*}"}
+{"id": "2003.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u ) ^ n = H ^ n ( z , u ) \\ , \\omega ^ n & \\textnormal { i n } & \\in \\Omega \\\\ \\\\ u = 0 & \\textnormal { o n } & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "1403.png", "formula": "\\begin{align*} \\forall \\ , p \\leq r < q , \\ ; \\ ; \\ ; \\ ; & \\sharp \\{ p \\leq s \\leq r \\ , | \\ , D _ { \\lambda } ( s ) = \\times \\} > \\sharp \\{ p \\leq s \\leq r \\ , | \\ , D _ { \\lambda } ( s ) = \\circ \\} \\\\ & \\sharp \\{ p \\leq s \\leq q \\ , | \\ , D _ { \\lambda } ( s ) = \\times \\} = \\sharp \\{ p \\leq s \\leq q \\ , | \\ , D _ { \\lambda } ( s ) = \\circ \\} . \\\\ \\end{align*}"}
+{"id": "5360.png", "formula": "\\begin{align*} L _ S [ f ] = \\sum _ { T \\subseteq S } ( - 1 ) ^ { | T | } E _ T [ f ] . \\end{align*}"}
+{"id": "1408.png", "formula": "\\begin{align*} M _ i - ( \\times _ \\dagger - \\overline { \\lambda } _ i - 1 ) & = M _ i - ( k _ i - \\overline { \\lambda } _ i - 1 ) + k _ i - \\times _ \\dagger \\le M _ { i ' } - ( k _ { i ' } - \\overline { \\lambda } _ { i ' } - 1 ) + k _ { i ' } - \\times _ \\dagger \\\\ & = M _ { i ' } - ( \\times _ \\dagger - \\overline { \\lambda } _ { i ' } - 1 ) \\end{align*}"}
+{"id": "5219.png", "formula": "\\begin{align*} f _ r ( t ) & = \\frac { t ^ { 2 r } } { ( 1 + t ^ 2 / k ) ^ { r + 1 / 2 + k / 2 } } \\times \\frac { \\Gamma ( 1 / 2 ) } { k ^ { r + 1 / 2 } \\pi ^ { 1 / 2 } B ( r + 1 / 2 , k / 2 ) } , \\end{align*}"}
+{"id": "4211.png", "formula": "\\begin{align*} u _ { B , 1 } ( e ^ { i x } ) & = \\exp ( ( x - \\pi ) i B ) , 0 < x < 2 \\pi , \\end{align*}"}
+{"id": "32.png", "formula": "\\begin{align*} \\mathcal { R } = \\mathrm { E n d } _ { \\kappa } ( \\overline { A } _ 0 ) = \\mathrm { M a t } _ n ( R ) , \\mathcal { B } = \\mathcal { R } \\otimes _ { \\mathbb { Z } } \\mathbb { Q } = \\mathrm { M a t } _ n ( B ) . \\end{align*}"}
+{"id": "211.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c ) = 1 \\\\ a , b , c \\geq 1 } } \\left ( \\frac { 1 } { 1 - x ^ a y ^ b z ^ c } \\right ) ^ { \\frac { c ^ 5 } { a ^ 3 b ^ 3 } } = \\exp \\left \\{ L i _ 3 ( x ) L i _ 3 ( y ) L i _ { - 5 } ( z ) \\right \\} \\end{align*}"}
+{"id": "7966.png", "formula": "\\begin{align*} \\l _ { { \\rm m i n } } \\ , \\mathrm { t r } ( M ) \\leq \\mathrm { t r } ( X M ) = \\mathrm { t r } ( M X ) \\leq \\l _ { { \\rm m a x } } \\ , \\mathrm { t r } ( M ) \\ , , \\end{align*}"}
+{"id": "7553.png", "formula": "\\begin{align*} P _ { i + 1 } = P _ i + \\begin{cases} ( 1 , 0 ) , & w i t h \\ p r o b a b i l i t y \\ \\alpha , \\\\ ( 0 , 1 ) , & w i t h \\ p r o b a b i l i t y \\ 1 - \\alpha , \\end{cases} \\end{align*}"}
+{"id": "5984.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } [ \\omega ] f ( [ \\epsilon , x ] ) = e ^ { - \\tfrac { \\pi i } { 4 } } \\nu ( \\epsilon , \\omega ) \\int _ { \\R } e ^ { 2 \\pi i \\epsilon x y } f ( [ \\epsilon , - y ] ) d y ; \\end{align*}"}
+{"id": "9033.png", "formula": "\\begin{align*} b _ { n , j } b _ { \\tau ( n ) , j } = b _ { n , j } b _ { n , j } \\ge 0 . \\end{align*}"}
+{"id": "1155.png", "formula": "\\begin{align*} \\mathcal { Q } ( f ) = \\mathcal { A } ( f , f ) = T ( f ^ { 2 } ) - 2 f T ( f ) - 2 B ( A ( f ) , A ( f ) ) = 0 \\end{align*}"}
+{"id": "2483.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle u _ { \\lambda } ( x ) = \\lambda ^ { \\frac { N } { 2 } } u ( \\lambda x ) , v _ { \\lambda } ( x ) = \\lambda ^ { \\frac { N } { 2 } } v ( \\lambda x ) . \\end{array} \\right . \\end{align*}"}
+{"id": "7571.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n = 1 } ^ { \\infty } a _ n z ^ n , \\ ; \\mbox { f o r } \\ ; z \\in \\mathbb { D } . \\end{align*}"}
+{"id": "1374.png", "formula": "\\begin{align*} x _ i v _ j = - v _ { 3 ( i + j ) \\bmod 5 } , i , j \\in \\{ 1 , 2 , 3 , 4 , 5 \\} . \\end{align*}"}
+{"id": "3492.png", "formula": "\\begin{align*} { \\Pr } _ S \\left [ | C _ S ( v ) | \\ge x \\mid V _ j \\cap S = W \\right ] \\le { \\Pr } _ S \\left [ | S ' ( v ) | \\ge \\frac { x } { \\Delta + 1 } \\mid V _ j \\cap S = W \\right ] \\le { \\Pr } _ { S } \\left [ | C _ { S _ 2 } ( v ) | \\ge \\frac { x } { \\Delta + 1 } \\right ] . \\end{align*}"}
+{"id": "7437.png", "formula": "\\begin{align*} \\overrightarrow { V _ \\varepsilon } ^ { ( 0 ) } ( x ) = \\nabla _ { \\xi } p ( \\xi ) \\big | _ { \\xi = \\frac { x } { \\varepsilon } } = \\varepsilon \\nabla _ x \\big ( p ( \\tfrac { x } { \\varepsilon } ) \\big ) , x \\in \\Omega ^ { ( 0 ) } _ \\varepsilon , \\end{align*}"}
+{"id": "2638.png", "formula": "\\begin{align*} \\sum _ { n _ 1 , n _ 2 \\in \\mathbb { Z } } \\prod _ { j = 1 , 2 } ( 1 + | n _ j | ) ^ { - N } \\sum _ { 2 ^ { k _ 1 - 1 } \\leq | q | \\leq 2 ^ { k _ 1 + 1 } } \\int _ { I _ { q , k _ 1 } } \\ ! H _ 1 ( t , x , y ) H _ 2 ( t , x , y ) \\ , \\mathrm { d } t , \\end{align*}"}
+{"id": "541.png", "formula": "\\begin{align*} A ( t ) : = \\left ( \\begin{array} { c c } 0 & 1 \\\\ a ( t ) & 0 \\end{array} \\right ) , Q ( t ) : = \\left ( \\begin{array} { c c } 0 & 0 \\\\ q ( t ) - a ( t ) & 0 \\end{array} \\right ) F ( t , \\xi ) : = \\left ( \\begin{array} { c } 0 \\\\ \\widehat { f } ( t , \\xi ) \\end{array} \\right ) . \\end{align*}"}
+{"id": "6420.png", "formula": "\\begin{align*} \\Vert x _ 1 \\bar { \\otimes } x _ 2 \\Vert _ p = \\Vert x _ 1 \\Vert _ p \\ , \\Vert x _ 2 \\Vert _ p \\end{align*}"}
+{"id": "5312.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\hat f ( \\{ i \\} ) ^ 2 \\leq 7 5 0 \\ , \\alpha ^ 2 \\log ( 1 / \\alpha ) . \\end{align*}"}
+{"id": "9039.png", "formula": "\\begin{align*} x _ i ' = x _ { \\sigma ( i ) } b _ { i , j } ' = b _ { \\sigma ( i ) , \\sigma ( j ) } 1 \\le i \\le m , 1 \\le j \\le n , \\end{align*}"}
+{"id": "4521.png", "formula": "\\begin{align*} 2 L ( V ) + 3 N ( V ) = \\frac { 3 } { 2 } ( 2 L ( V ) + 2 N ( V ) ) - L ( V ) = - \\frac { 3 } { 2 } \\mathbf { c } \\cdot \\mathbf { P } ( V ) - L ( V ) < 0 \\end{align*}"}
+{"id": "7445.png", "formula": "\\begin{align*} \\partial _ { \\bar { \\nu } _ { \\bar { \\xi } _ i } } u ^ { ( i ) } _ { p \\alpha + k } = \\big ( w ^ { ( i ) } _ { p \\alpha + k - 1 } + u ^ { ( i ) } _ { p \\alpha + k - 1 } \\big ) \\ , \\overline { V } ^ { ( i ) } \\boldsymbol { \\boldsymbol { \\cdot } } \\bar { \\nu } _ { \\bar { \\xi } _ i } \\ , - \\ , \\delta _ { p \\alpha + k - 1 , \\ , \\alpha - 1 } \\ , \\varphi ^ { ( i ) } \\big ( \\bar { \\xi } _ i , x _ i , t \\big ) , \\bar { \\xi } _ i \\in \\partial \\Upsilon _ i , \\end{align*}"}
+{"id": "8296.png", "formula": "\\begin{align*} d _ 1 = & \\ ; \\mu _ { 1 1 } + 2 + 3 + \\cdots + ( l - 1 ) \\\\ d _ 2 = & \\ ; \\mu _ { 1 1 } + 1 + 2 + \\cdots + ( l - 2 ) \\\\ d _ { l - 1 } = & \\ ; ( l - 2 ) + ( l - 3 ) + \\cdots + 1 + \\mu _ { ( l - 1 ) 1 } \\\\ d _ r = & \\ ; ( l - 1 ) + ( l - 2 ) + \\cdots + 2 + \\mu _ { ( l - 1 ) 1 } \\end{align*}"}
+{"id": "4429.png", "formula": "\\begin{align*} \\Psi _ { \\omega } ( x ) : = \\omega ^ { \\frac { 1 } { 2 } } \\Psi ( \\omega ^ { \\frac { 1 } { 2 } } x ) . \\end{align*}"}
+{"id": "7856.png", "formula": "\\begin{align*} \\delta _ { x K } ( \\sigma ) : = \\begin{cases} 1 & \\mbox { i f } \\sigma \\in x K \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "7840.png", "formula": "\\begin{align*} & \\textsf { P } \\{ \\inf _ { z \\in _ { J } ( K _ { 1 } , K _ { 2 } ) } \\Vert \\tilde { W } z \\Vert < t \\sqrt { d } \\} \\\\ & \\le ( C _ { 1 } t ) ^ { d } + \\sum _ { s = 2 ^ { k } , k \\le k _ { 1 } } ( ( C _ { 2 } t ) ^ { d } e ^ { - c s } + e ^ { - c _ { 1 } \\sqrt { N } } ) + \\textsf { P } \\{ \\Vert \\tilde { W } \\Vert > C _ { 0 } \\sqrt { N } \\} \\\\ & \\le ( C _ { 3 } t ) ^ { d } + k _ { 1 } e ^ { - c _ { 1 } \\sqrt { N } } + \\textsf { P } \\{ \\Vert A \\Vert > C _ { 0 } \\sqrt { N } \\} . \\end{align*}"}
+{"id": "4055.png", "formula": "\\begin{align*} S _ { \\mathrm { m a i n } } = \\sum _ { d _ 1 , d _ 2 < D } \\frac { 1 } { \\sqrt { d _ 1 d _ 2 } } \\sum _ { I d _ 1 , J d _ 2 < D } \\alpha _ { I d _ 1 } \\bar { \\alpha } _ { J d _ 2 } \\sum _ { L , M \\geq 1 } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } \\ , \\ \\delta _ { I L \\ , J M } \\end{align*}"}
+{"id": "3661.png", "formula": "\\begin{align*} \\begin{cases} w \\in H ^ { s _ M } ( \\mathbb { R } ^ n ) ( P ( ( - \\Delta ) ^ s ) + q ) w = 0 \\Omega w | _ { \\Omega ^ c } = 0 , \\\\ w \\equiv 0 . \\end{cases} \\end{align*}"}
+{"id": "3893.png", "formula": "\\begin{align*} \\left ( \\mathord { \\operatorname { p r o j } } _ { K _ N \\cap Q } \\circ { \\operatorname { p r o j } _ { K _ N } } ^ { - 1 } \\right ) \\# \\mu _ N = \\left ( \\mathord { \\operatorname { p r o j } } _ { K _ N \\cap Q } \\circ { \\mathrm { p r o j } _ { Q } } ^ { - 1 } \\right ) \\# \\gamma , \\end{align*}"}
+{"id": "6415.png", "formula": "\\begin{align*} \\widetilde { M } _ 1 \\bar { \\otimes } \\widetilde { M } _ 2 = ( M _ 1 \\bar { \\otimes } M _ 2 ) \\bar { \\rtimes } _ \\sigma G \\supset ( M _ 1 \\bar { \\otimes } M _ 2 ) \\bar { \\rtimes } _ \\sigma H = \\widetilde { M _ 1 \\bar { \\otimes } M _ 2 } , \\end{align*}"}
+{"id": "1270.png", "formula": "\\begin{align*} M _ q ( G ) \\geq \\eta ( B ) \\geq T + \\max _ { 1 \\le i _ 1 < \\ldots < i _ q \\le s } \\sum _ { j = 1 } ^ q a _ { i _ j } , \\end{align*}"}
+{"id": "2015.png", "formula": "\\begin{align*} \\eta _ 1 ^ { n } = \\inf \\{ E ( w ) \\ , ; \\ , w \\in \\mathcal E ^ 1 ( \\Omega ) , \\ , I _ g ( w ) = 1 \\} . \\end{align*}"}
+{"id": "1827.png", "formula": "\\begin{align*} \\Phi _ t ( p _ 1 , p _ 2 , p _ 3 , p _ 4 ) = - \\Phi _ t ( p _ 2 , p _ 1 , p _ 3 , p _ 4 ) = + \\Phi _ t ( p _ 2 , p _ 1 , p _ 4 , p _ 3 ) = - \\Phi _ t ( p _ 1 , p _ 2 , p _ 3 , p _ 4 ) \\end{align*}"}
+{"id": "2109.png", "formula": "\\begin{align*} \\mathcal { N } ( A , \\| \\cdot \\| , \\delta ) : = \\min \\Big \\{ K \\in \\mathbb { N } _ { > 0 } \\ , \\ , \\Big | \\ , \\ , \\exists z _ 1 , \\ldots z _ K \\in \\R ^ d \\ , \\ , \\colon \\ , \\ , A \\subset \\bigcup _ { k = 1 } ^ K B ^ { \\rm c l } ( z _ k , \\delta ) \\Big \\} . \\end{align*}"}
+{"id": "3235.png", "formula": "\\begin{align*} d ( T _ 1 \\gamma ) & = T _ 1 ^ 3 T _ 2 + T _ 1 ^ 2 T _ 2 ^ 2 , & & d ( T _ 1 ^ 2 \\beta ) = T _ 1 ^ 4 + T _ 1 ^ 2 T _ 2 ^ 2 + T _ 1 ^ 3 T _ 2 \\\\ d ( T _ 2 \\gamma ) & = T _ 1 ^ 2 T _ 2 ^ 2 + T _ 1 T _ 2 ^ 3 , & & d ( T _ 2 ^ 2 \\beta ) = T _ 1 ^ 2 T _ 2 ^ 2 + T _ 2 ^ 4 + T _ 1 T _ 2 ^ 3 \\\\ d ( T _ 3 \\gamma ) & = ( a ^ 2 b + a b ^ 2 ) T _ 3 ^ 4 , & & d ( T _ 1 T _ 2 \\beta ) = T _ 1 ^ 3 T _ 2 + T _ 1 T _ 2 ^ 3 + T _ 1 ^ 2 T _ 2 ^ 2 \\end{align*}"}
+{"id": "1458.png", "formula": "\\begin{align*} \\nabla \\tilde { u } _ m ^ i ( y ) = ( \\mu _ m ^ i ) ^ { \\frac { n } { 2 } } \\bigg [ \\Big ( \\nabla u _ m ( x ) \\Big ) \\chi _ m ^ i ( x ) + u _ m ( x ) \\Big ( \\nabla \\chi _ m ^ i ( x ) \\Big ) \\bigg ] , \\end{align*}"}
+{"id": "2831.png", "formula": "\\begin{align*} S _ k ^ { l } = \\tilde S _ k ^ { l } = 0 , l = 1 , \\dots , p ; k \\ne l - 1 + ( p + 1 ) k ' , \\ ; k ' \\in \\mathbb { Z } _ + . \\end{align*}"}
+{"id": "6078.png", "formula": "\\begin{align*} \\limsup _ { j \\to + \\infty } \\big \\| \\exp ( A ) ^ j \\exp ( B ) ^ { - \\lfloor \\varepsilon j \\rfloor } \\big \\| = \\infty , \\mbox { o r } \\limsup _ { j \\to - \\infty } \\big \\| \\exp ( A ) ^ j \\exp ( B ) ^ { - \\lfloor \\varepsilon j \\rfloor } \\big \\| = \\infty , \\end{align*}"}
+{"id": "4990.png", "formula": "\\begin{align*} \\begin{aligned} \\Pr ( X _ n > k ) & = \\frac { 1 } { 2 ^ n } \\sum _ { j = k + 1 } ^ n \\binom { n } { j } & \\\\ & = \\frac { 1 } { 2 } \\sum _ { j = k } ^ { n - 1 } \\frac { 1 } { 2 ^ j } \\binom { j } { k } , & \\end{aligned} \\end{align*}"}
+{"id": "8322.png", "formula": "\\begin{align*} a u = b v , a x = b y , c x = d y , \\end{align*}"}
+{"id": "1222.png", "formula": "\\begin{align*} \\left \\langle \\boldsymbol { \\nu } ; { \\boldsymbol { M } } ( \\boldsymbol { a } , - \\log ( 1 - x ) ) \\right \\rangle & = \\frac { 1 } { 1 - q } H ( A ( - \\log ( 1 - x ) ) ) F ( A ( - \\log ( 1 - x ) ) ) \\\\ & = \\frac { 1 } { 1 - q } H ( I ^ { - 1 } _ { \\boldsymbol { a } } ( q x ) / q ) ) F ( I ^ { - 1 } _ { \\boldsymbol { a } } ( q x ) / q ) ) . \\end{align*}"}
+{"id": "3397.png", "formula": "\\begin{align*} p ' _ k ( x ) & = \\frac { - \\beta _ k e ^ { \\alpha _ k h _ k + \\beta _ k x + \\gamma _ k } } { ( 1 + e ^ { \\alpha _ k h _ k + \\beta _ k x + \\gamma _ k } ) ^ 2 } = - \\beta _ k \\frac { 1 } { ( 1 + e ^ { \\alpha _ k h _ k + \\beta _ k x + \\gamma _ k } ) } \\frac { e ^ { \\alpha _ k h _ k + \\beta _ k x + \\gamma _ k } } { ( 1 + e ^ { \\alpha _ k h _ k + \\beta _ k x + \\gamma _ k } ) } = - \\beta _ k p _ k ( x ) ( 1 - p _ k ( x ) ) . \\end{align*}"}
+{"id": "5692.png", "formula": "\\begin{align*} F = - q ^ { - 1 } + \\sum ^ { \\infty } _ { n = 2 } C _ { F } ( n ) q ^ { n } \\in S _ { k } ^ { \\# , 0 } ( \\Gamma _ { 0 } ( N ) ) \\cap \\mathbb { Z } [ [ q ] ] [ q ^ { - 1 } ] \\end{align*}"}
+{"id": "1678.png", "formula": "\\begin{align*} \\phi _ 2 \\phi _ 1 = \\psi _ 2 \\tau \\widehat { \\tau } \\widehat { \\psi _ 1 } = \\psi _ 2 \\widehat { \\psi _ 1 } [ \\# G ] \\end{align*}"}
+{"id": "5273.png", "formula": "\\begin{align*} | 1 + y | ^ q = 1 + q y + \\binom { q } { 2 } y ^ 2 + \\binom { q } { 3 } | 1 + y ' | ^ { q - 3 } ( 1 + y ' ) y ^ 3 , \\end{align*}"}
+{"id": "4929.png", "formula": "\\begin{align*} \\left [ C \\right ] _ { i , j } ^ { } = \\begin{cases} 1 & \\textnormal { i f } j = i = 0 ; \\\\ 1 - p _ j ^ { } & \\textnormal { i f } j = i \\neq 0 ; \\\\ p _ j ^ { } & \\textnormal { i f } j = i + 1 ; \\\\ 0 & \\textnormal { o t h e r w i s e } ; \\end{cases} \\end{align*}"}
+{"id": "7836.png", "formula": "\\begin{align*} H _ { J } : = ( X _ { k } , k \\in J ) \\subset \\mathbb { R } ^ { N } . \\end{align*}"}
+{"id": "5721.png", "formula": "\\begin{align*} \\langle F , g \\rangle = 1 . \\end{align*}"}
+{"id": "3224.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { x \\in e M _ { + } e , ~ x \\neq 0 } \\abs { \\frac { ( S _ n ( \\mu ) - \\bar { \\mu } ) ( x ) } { \\tau ( x ) } } = 0 \\end{align*}"}
+{"id": "9343.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\mathcal { Z } _ t ^ u & = e ^ { - \\frac { \\beta } { 2 } t } \\mathcal { Z } _ t h ( x _ t ^ u , u _ t ) d \\xi _ t , \\ t \\in [ 0 , \\infty ) , \\\\ \\mathcal { Z } _ 0 ^ u & = 1 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2074.png", "formula": "\\begin{align*} ( \\rho \\otimes _ 0 \\varphi ) ^ { \\sigma ' , s ' } _ { \\sigma , s } ( a ' g a ) = \\rho _ { a ' g a \\sigma } ^ { \\sigma ' } \\cdot \\varphi _ s ^ { s ' } ( a ' g a ) = & \\rho _ { g \\sigma } ^ { ( a ' ) ^ { - 1 } \\sigma ' } \\cdot \\varphi _ s ^ { s ' } ( g ) \\\\ = & \\rho _ { g \\sigma } ^ { \\sigma ' } \\cdot \\varphi _ s ^ { s ' } ( g ) = ( \\rho \\otimes _ 0 \\varphi ) ^ { \\sigma ' , s ' } _ { \\sigma , s } ( g ) , \\end{align*}"}
+{"id": "6532.png", "formula": "\\begin{align*} \\tilde { t } _ { \\mu , \\nu } ( x ) & = \\frac { 1 } { 2 ^ { \\mu - 1 } \\Gamma \\big ( \\frac { \\mu - \\nu + 1 } { 2 } \\big ) \\Gamma \\big ( \\frac { \\mu + \\nu + 1 } { 2 } \\big ) } t _ { \\mu , \\nu } ( x ) \\\\ & = \\frac { 1 } { 2 ^ { \\mu + 1 } \\Gamma \\big ( \\frac { \\mu - \\nu + 3 } { 2 } \\big ) \\Gamma \\big ( \\frac { \\mu + \\nu + 3 } { 2 } \\big ) } { } _ 1 F _ 2 \\bigg ( 1 ; \\frac { \\mu - \\nu + 3 } { 2 } , \\frac { \\mu + \\nu + 3 } { 2 } ; \\frac { x ^ 2 } { 4 } \\bigg ) . \\end{align*}"}
+{"id": "7023.png", "formula": "\\begin{align*} \\nu _ i ( a _ j ) = \\nu _ i ( f _ j Q _ i ' ) < \\nu _ i ( b _ j Q _ i ) . \\end{align*}"}
+{"id": "8269.png", "formula": "\\begin{align*} b _ { q , m } ^ { ( l ) } = \\frac { 2 ( q + m ) - k ^ 2 + 2 ( l - 1 ) ( 2 k - l + 1 ) } { 2 l } b _ { q , m } ^ { ( l - 1 ) } + \\frac { 1 } { l } b _ { q , m - 1 } ^ { ( l - 1 ) } + \\frac { l - k - 2 } { l } b _ { q - 1 , m } ^ { ( l - 2 ) } . \\end{align*}"}
+{"id": "6445.png", "formula": "\\begin{align*} | \\langle v ( t ) \\rangle _ { K , \\sigma } | ^ 2 & = \\frac { \\sigma ^ 4 \\varepsilon ^ 2 } { ( \\sigma ^ 2 + h ^ 2 ) ^ 2 } | \\eta ( K ) | + \\frac { 2 \\sigma ^ 2 \\varepsilon ^ 4 } { \\sigma ^ 2 + h ^ 2 } \\Im \\big ( \\overline { \\eta ( K ) } V _ K ^ 1 ( t ) e ^ { - i t \\theta _ K } \\big ) \\\\ & \\quad + \\varepsilon ^ 6 \\Big ( | V _ K ^ 1 ( t ) | ^ 2 + \\frac { 2 \\sigma ^ 2 } { \\sigma ^ 2 + h ^ 2 } \\Im \\big ( \\overline { \\eta ( K ) } V _ K ^ 2 ( t ) e ^ { - i t \\theta _ K } \\big ) \\Big ) + \\mathcal { O } ( \\varepsilon ^ 8 ) . \\end{align*}"}
+{"id": "111.png", "formula": "\\begin{align*} { \\rm W F } _ h ' ( \\chi e ^ { - i t _ 0 h ^ { - 1 } \\tilde { P } _ h ( z ) } \\tilde { R } _ h ( z ) \\chi ) \\cap S ^ * ( \\mathcal { M } \\times \\mathcal { M } ) & \\subset \\\\ \\kappa \\big ( \\{ ( x , \\xi , y , \\eta ) \\ , : \\ , ( e ^ { - t _ 0 H _ p } ( x , \\xi ) , y , \\eta ) \\in \\Delta ( T ^ * \\mathcal { U } ) \\cup & \\ ; \\Omega _ + \\cup E ^ * _ + \\times E ^ * _ - \\setminus \\{ 0 \\} \\xi = 0 , \\eta \\neq 0 \\} \\big ) . \\end{align*}"}
+{"id": "1618.png", "formula": "\\begin{align*} [ { f } , { f } ] ( X , Y ) = ( { f } \\nabla _ Y { f } - \\nabla _ { { f } Y } { f } ) X - ( { f } \\nabla _ X { f } - \\nabla _ { { f } X } { f } ) Y ; \\end{align*}"}
+{"id": "1226.png", "formula": "\\begin{align*} \\| u _ m \\| _ { \\psi _ m } ^ 2 = \\int _ { \\C ^ n } | u _ m | ^ 2 ( 1 + | z | ^ 2 ) ^ { - a _ m } e ^ { - 2 m V } \\ , d \\lambda \\\\ \\leq \\dfrac 1 { a _ m } \\| f \\bar \\partial \\chi \\| _ { \\eta _ m } ^ 2 . \\end{align*}"}
+{"id": "7752.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\frac { 1 } { r } { \\rm d i s t } \\ , ( ( x + \\pi ) \\cap \\overline { B } _ r ( x ) , ( x + \\pi ) \\cap F \\cap \\overline { B } _ r ) = 0 \\ , . \\end{align*}"}
+{"id": "6195.png", "formula": "\\begin{align*} | V _ e ( T _ 1 ) | = | V _ i ( T _ 1 ) | ( n - 2 ) + 2 | V _ e ( T _ 2 ) | = | V _ i ( T _ 2 ) | ( n - 2 ) + 2 . \\end{align*}"}
+{"id": "2528.png", "formula": "\\begin{align*} t \\cdot A = ( \\mu _ 1 , \\ldots , \\mu _ n , \\mu _ { n ^ * } , \\ldots , \\mu _ { 1 ^ * } ) \\cdot A . \\end{align*}"}
+{"id": "49.png", "formula": "\\begin{align*} K = \\mathrm { E n d } _ { \\overline { K } _ 0 } ( A _ 0 ) \\otimes _ { \\mathbb { Z } } \\mathbb { Q } \\hookrightarrow \\mathcal { B } = \\mathrm { E n d } _ { \\overline { \\mathbb { F } } _ \\ell } ( \\overline { A } _ 0 ) \\otimes _ { \\mathbb { Z } } \\mathbb { Q } . \\end{align*}"}
+{"id": "8165.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 2 } ^ { \\infty } M _ j \\big ( \\frac { M - M _ 1 } { 1 - M _ 1 } \\big ) ^ { 2 - j } \\right ) \\bigg ( 1 - M _ 1 \\bigg ) ^ { - 1 } \\geq \\frac { M ( \\sum _ { j = 1 } ^ { \\infty } M _ j M ^ { 1 - j } - M _ 1 ) } { 1 - M _ 1 } > 1 . \\end{align*}"}
+{"id": "7415.png", "formula": "\\begin{align*} \\begin{aligned} X _ { N , M } ^ { ( i ) } = \\frac { \\sqrt { \\theta _ N } } { N } \\ , \\sum _ { k = 1 } ^ \\infty \\ \\sum _ { \\substack { \\frac { i - 1 } { M } N < n _ 1 < \\ldots < n _ k \\le \\frac { i } { M } N \\\\ x _ 1 , \\ldots , x _ k \\in \\Z ^ 2 } } \\ , & q _ { n _ 1 } ^ { \\phi _ N } ( x _ 1 ) \\ , \\xi _ { \\beta _ N } ( n _ 1 , x _ 1 ) \\ , \\times \\\\ & \\times \\prod _ { j = 2 } ^ { k } q _ { n _ j - n _ { j - 1 } } ( x _ j - x _ { j - 1 } ) \\ , \\xi _ { \\beta _ N } ( n _ j , x _ j ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "7939.png", "formula": "\\begin{align*} 1 & = \\sum _ { k _ 0 \\in \\N _ 0 } ( 1 - k _ 0 ) \\beta ( \\xi , k _ 0 e _ 0 ) + \\sum _ { k _ 0 \\in \\N _ 0 } ( 1 - k _ 0 ) \\beta ( 0 , k _ 0 e _ 0 ) \\\\ & - \\sum _ { k _ 0 \\in \\N _ 0 } k _ 0 \\beta ( 0 , k _ 0 e _ 0 + e _ { ( 0 , 1 ) } ) - \\sum _ { k _ 0 \\in \\N _ 0 } ( 1 + k _ 0 ) \\beta ( 0 , k _ 0 e _ 0 + 2 e _ { ( 0 , 1 ) } ) \\\\ & + \\sum _ { \\mathbf { n } \\in \\N _ 0 ^ d } \\beta ( \\mathbf { n } ) . \\end{align*}"}
+{"id": "8865.png", "formula": "\\begin{align*} & \\beta _ { ( n - 1 ) P + 1 } = . . . = \\beta _ { n P } , \\ n \\in \\mathcal { N } , \\\\ & \\beta _ i \\in [ 0 , 1 ] , i \\in \\mathcal { I } , \\\\ & \\alpha _ n = \\frac { \\sum _ { p \\in \\mathcal { P } } \\beta _ { ( n - 1 ) P + p } } { P } , n \\in \\mathcal { N } . \\end{align*}"}
+{"id": "6493.png", "formula": "\\begin{align*} D _ n ( a , b ) : & = \\left [ \\binom { 2 i + 2 a } { i - j + a - b } - \\binom { 2 i + 2 a } { i - j + a - b - 1 } \\right ] _ { i , j = 0 } ^ { n - 1 } \\end{align*}"}
+{"id": "467.png", "formula": "\\begin{align*} \\R _ F ( x ) = \\frac 1 3 , \\ \\ W _ F ( x ) = \\frac 1 3 , \\end{align*}"}
+{"id": "264.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { - ( n + 1 ) ^ 3 y ^ { n + 1 } + y ^ 3 + 4 y ^ 2 + y } { ( 1 - y ) ^ 4 } \\right ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "1981.png", "formula": "\\begin{align*} \\det ( u _ { j \\bar k } ( 0 ) ) = a u _ { n \\bar n } ( 0 ) + b , \\end{align*}"}
+{"id": "2774.png", "formula": "\\begin{align*} \\| \\partial _ \\nu v \\| _ { L ^ 2 ( \\Sigma ) } & = \\| ( \\Lambda _ { q _ 1 , \\lambda } ^ 0 - \\Lambda _ { q _ 2 , \\lambda } ^ 0 ) ( v _ 2 { _ { | \\partial M } } ) \\| _ { L ^ 2 ( \\Sigma ) } \\\\ & \\le \\| \\Lambda _ { q _ 1 , \\lambda } ^ 0 - \\Lambda _ { q _ 2 , \\lambda } ^ 0 \\| \\| v _ 2 \\| _ { H ^ { 3 / 2 } ( \\Sigma ) } \\\\ & \\le \\lambda ^ 2 \\mathbf { e } _ \\lambda e ^ { c / \\epsilon } \\| \\Lambda _ { q _ 1 , \\lambda } ^ 0 - \\Lambda _ { q _ 2 , \\lambda } ^ 0 \\| \\| u _ 2 \\| _ { L ^ 2 ( \\mathrm { M } _ 0 ) } , \\end{align*}"}
+{"id": "5818.png", "formula": "\\begin{align*} J = \\lambda \\sum \\limits _ { i = 1 } ^ N { { G _ { { \\alpha _ 1 } , { \\beta _ 1 } } } \\left ( { { e _ i } } \\right ) } + \\left ( { 1 - \\lambda } \\right ) \\sum \\limits _ { i = 1 } ^ N { \\sum \\limits _ { j = 1 } ^ N { { G _ { { \\alpha _ 2 } , { \\beta _ 2 } } } \\left ( { { e _ i } - { e _ j } } \\right ) } } , \\end{align*}"}
+{"id": "5670.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | u _ 0 | ^ { p } ) | v _ 0 | ^ { q } = D _ 0 ( | \\nabla u _ 0 | ^ 2 _ 2 + | \\nabla v _ 0 | ^ 2 _ 2 ) ^ { \\frac { \\gamma _ p + \\gamma _ q } { 2 } } . \\end{align*}"}
+{"id": "2212.png", "formula": "\\begin{align*} \\sum _ { y < p \\le x ^ { 1 / 3 } } \\frac { 1 } { p \\log p } \\omega \\left ( \\frac { \\log x } { \\log p } - 1 \\right ) & = \\frac { 1 } { \\log x } \\int _ { 3 ^ - } ^ { u } v \\omega ( v - 1 ) \\ , d G ( v ) \\\\ & = \\frac { 1 } { \\log x } \\left ( \\int _ { 3 } ^ { u } \\omega ( v - 1 ) \\ , d v + \\int _ { 3 ^ - } ^ { u } v \\omega ( v - 1 ) \\ , d \\left ( G ( v ) - \\log \\frac { v } { 2 } \\right ) \\right ) . \\end{align*}"}
+{"id": "4090.png", "formula": "\\begin{align*} \\vec { v } = \\left ( A ^ { ( 0 ) } , A ^ { ( 1 ) } , \\dots , A ^ { ( N - 1 ) } , \\overline { A ^ { ( N ) } , A ^ { ( N + 1 ) } , \\dots , A ^ { ( N + p - 1 ) } } \\right ) . \\end{align*}"}
+{"id": "4504.png", "formula": "\\begin{align*} \\mathbf { c } _ 0 \\cdot \\mathbf { P } ( \\Phi ) = - 2 E ( \\Phi ) \\end{align*}"}
+{"id": "7710.png", "formula": "\\begin{align*} \\vartheta _ L ( f ) ( g _ 0 ) = | a | ^ { r + 1 } \\Phi ^ { L _ r } _ { 0 _ { L _ { r + 1 } } } ( g _ { r + 1 } ) + \\sum _ { j = 0 } ^ r \\sum _ { \\lambda \\in \\Lambda _ { j + 1 } ^ + } | a | ^ j \\Phi ^ { L _ j } _ \\lambda ( ( g _ 0 ) _ j ) , \\end{align*}"}
+{"id": "6812.png", "formula": "\\begin{align*} d t - \\left ( \\textstyle \\sum _ { i = 1 } ^ { n - 2 } y _ i \\ , d x _ i \\right ) - y ' \\ , d x ' + \\phi ( x ' ) \\ , d y ' - y '' \\ , d x '' , \\end{align*}"}
+{"id": "5833.png", "formula": "\\begin{align*} F _ H ( w ) = \\bigcap _ { y \\in I _ H ( w ) } \\left \\{ X ^ y _ { [ 0 , \\tau _ { H , w } ] } \\subseteq B _ H ( w ) \\right \\} . \\end{align*}"}
+{"id": "176.png", "formula": "\\begin{align*} L i _ 2 \\left ( - \\frac { 1 } { 2 } \\right ) + \\frac { 1 } { 6 } L i _ 2 \\left ( \\frac { 1 } { 9 } \\right ) = - \\frac { \\pi ^ 2 } { 1 8 } + \\log 2 \\log 3 - \\frac { 1 } { 2 } ( \\log 2 ) ^ 2 - \\frac { 1 } { 3 } ( \\log 3 ) ^ 2 \\end{align*}"}
+{"id": "6296.png", "formula": "\\begin{align*} \\vec { H } ( \\lambda _ t ) = \\vec { h } _ { \\bar u ( t ) } ( \\lambda _ t ) , \\qquad \\forall \\ , t \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "3834.png", "formula": "\\begin{align*} \\Psi _ 2 \\left ( \\rho _ 1 ( y _ 1 , y _ 1 ^ \\prime ) ^ { p } , \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \\prime ) ^ p \\right ) = \\Psi _ 2 \\circ \\psi _ 2 \\left ( \\boldsymbol { d } _ { \\mathcal { Y } _ 1 } ( y _ 1 , y _ 1 ^ \\prime ) ^ { p } , \\boldsymbol { d } _ { \\mathcal { Y } _ 2 } ( y _ 2 , y _ 2 ^ \\prime ) ^ p , \\boldsymbol { d } _ { \\mathcal { X } } ( x , x ^ \\prime ) ^ p \\right ) . \\end{align*}"}
+{"id": "7282.png", "formula": "\\begin{align*} k x _ p + l y _ p = n x _ p \\leq m y _ p \\end{align*}"}
+{"id": "1711.png", "formula": "\\begin{align*} P _ n ^ { ( \\alpha , \\beta ) } ( 1 ) = \\frac { ( \\alpha + 1 ) _ n } { n ! } , P _ n ^ { ( \\alpha , \\beta ) } ( - 1 ) = ( - 1 ) ^ n \\frac { ( \\beta + 1 ) _ n } { n ! } , \\end{align*}"}
+{"id": "58.png", "formula": "\\begin{align*} \\mathbf { G } ^ 1 [ \\ell , \\sigma ] : = \\epsilon ( \\sigma ^ { - 1 } ) \\mathbf { G } ^ 1 [ \\ell ] \\epsilon ( \\sigma ) \\end{align*}"}
+{"id": "8275.png", "formula": "\\begin{align*} F ( n _ 1 , n _ 2 ) = \\left ( \\frac { \\partial } { \\partial t _ 1 } \\right ) ^ { n _ 1 } \\left ( \\frac { \\partial } { \\partial t _ 2 } \\right ) ^ { n _ 2 } \\det \\left ( g _ { i + j + 1 } ( x , t _ 1 , t _ 2 ) \\right ) _ { i , j = 0 , \\ldots , k - 1 } \\Bigg | _ { t _ 1 = t _ 2 = 0 } . \\end{align*}"}
+{"id": "6975.png", "formula": "\\begin{align*} \\alpha = \\alpha \\left ( \\left \\{ \\left . \\alpha _ i \\ \\right | \\ Q _ i \\in I _ { n _ 0 } \\right \\} \\right ) . \\end{align*}"}
+{"id": "5005.png", "formula": "\\begin{align*} F ( k , t , n ) = \\frac { 1 } { n ^ t } \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ { k - j } \\binom { n - j - 1 } { n - k } \\binom { n } { j } j ^ t \\\\ = \\frac { ( n ) _ { k + 1 } ^ { } } { n ^ t } \\frac { 1 } { k ! } \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ { k - j } \\binom { k } { j } \\frac { j ^ t } { n - j } . \\end{align*}"}
+{"id": "5889.png", "formula": "\\begin{align*} \\varphi ' ( t ) - N c = - N ^ 2 \\ , . \\end{align*}"}
+{"id": "8657.png", "formula": "\\begin{align*} \\frac { y } { 1 - p y } - \\frac { x } { 1 - p x } = \\frac { y - x } { ( 1 - p y ) ( 1 - p x ) } , \\end{align*}"}
+{"id": "2716.png", "formula": "\\begin{align*} J _ 5 \\geq & - C s ^ 2 \\lambda ^ 4 \\iint _ Q \\xi \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d y d t - C \\lambda ^ 2 \\iint _ { Q } \\xi \\left | A \\nabla u \\cdot \\nabla \\eta \\right | ^ 2 d x d y d t \\\\ & - C s ^ 2 \\lambda ^ 3 \\int _ 0 ^ T \\int _ { \\omega _ 0 } \\xi | u | ^ 2 d x d y d t - C \\lambda \\iint _ Q \\xi A \\nabla u \\cdot \\nabla u d x d y d t . \\end{align*}"}
+{"id": "7647.png", "formula": "\\begin{align*} \\max \\{ M ( c , x , 0 ) \\} \\leq 2 5 6 \\sqrt { 6 } \\ ; \\ ; \\mbox { f o r } \\ ; { c = 0 , x = \\sqrt { { 2 } / { 3 } } } . \\end{align*}"}
+{"id": "5521.png", "formula": "\\begin{align*} P _ { \\mu , x } ^ { n } \\left ( L _ { x } g ^ { - 1 } , A \\right ) = \\int _ { G } { \\bf 1 } _ { A } \\left ( x , L _ { x } g ^ { - 1 } s \\right ) \\frac { \\varphi _ { g ^ { - 1 } s } ( \\pi ( x ) ) } { \\varphi _ { g ^ { - 1 } } ( \\pi ( x ) ) } d \\mu ^ { ( n ) } ( s ) , \\end{align*}"}
+{"id": "840.png", "formula": "\\begin{align*} R _ { c } = \\log _ { 2 } \\left ( \\det \\left [ \\left ( \\rho _ { f } \\hat { \\textbf { G } } _ { c c } ^ T \\textbf { P } _ { c } \\textbf { P } _ { c } ^ { H } \\hat { \\textbf { G } } _ { c c } ^ * \\right ) \\textbf { R } _ { c } ^ { - 1 } + \\textbf { I } _ { K _ c } \\right ] \\right ) \\end{align*}"}
+{"id": "7486.png", "formula": "\\begin{align*} \\Phi ( \\pi ) = \\tilde { \\delta } \\mu _ k \\tilde { \\alpha } _ k \\mu _ { k - 1 } \\tilde { \\alpha } _ { k - 1 } \\cdots \\mu _ 1 \\tilde { \\alpha } _ 1 \\mu _ 0 . \\end{align*}"}
+{"id": "2433.png", "formula": "\\begin{align*} y ( q x ) = \\sin ( x ) y ( x ) . \\end{align*}"}
+{"id": "1027.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { k - 1 } \\frac { ( a ) _ k ( 1 - a ) _ k } { ( 1 ) _ { k } ( b ) _ { k } } \\{ H _ k ( a - 1 ) - H _ k ( - a ) \\} = \\frac { \\Gamma ( \\frac { b } { 2 } ) \\Gamma ( \\frac { 1 + b } { 2 } ) } { \\Gamma ( \\frac { a + b } { 2 } ) \\Gamma ( \\frac { 1 - a + b } { 2 } ) } \\\\ [ 1 m m ] & \\ : \\ : \\times \\bigg \\{ \\psi \\bigg ( \\frac { 1 - a + b } { 2 } \\bigg ) - \\psi \\bigg ( \\frac { a + b } { 2 } \\bigg ) \\bigg \\} . \\end{align*}"}
+{"id": "1481.png", "formula": "\\begin{align*} & \\sum _ { i \\neq l } ^ k \\int _ { \\mathcal { A } _ l } U _ { \\mu _ i , \\xi _ i } \\bigg | \\ln \\ln \\Big ( e + ( - 1 ) ^ l U _ { \\mu _ l , \\xi _ l } + \\sum _ { i \\neq l } ^ k ( - 1 ) ^ i U _ { \\mu _ i , \\xi _ i } \\Big ) \\bigg | ^ { \\frac { 2 n } { n + 2 } } d x = O \\Big ( \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big ) . \\end{align*}"}
+{"id": "5578.png", "formula": "\\begin{align*} \\limsup _ { \\ell \\to \\infty } h _ { \\mu } ( Z , \\lambda _ { \\ell , 0 } ) \\le h _ { \\mu } ( Z , \\lambda _ { \\kappa } ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { 2 } h _ { \\mu } \\left ( Z , \\lambda _ { \\delta _ { A _ { i } } } \\right ) . \\end{align*}"}
+{"id": "7709.png", "formula": "\\begin{align*} & \\varGamma _ L \\cap N _ { U _ r } \\\\ & = \\left \\{ [ \\mathrm { i d } _ { U _ r } , \\mathrm { i d } _ { ( L _ { r + 1 } ) _ \\R } , \\mathbf { x } , \\psi ] \\mid \\mathbf { x } \\in ( L _ { r + 1 } ) ^ { r + 1 } \\hbox { a n d } b _ { i j } + \\frac { ( x _ i , x _ j ) } { 2 } \\in \\Z \\hbox { f o r } 0 \\le i , j \\le r \\right \\} . \\end{align*}"}
+{"id": "6351.png", "formula": "\\begin{align*} A _ 1 : = a ( s _ 1 ) > \\frac { d ^ 2 ( s _ 1 ) } { d ^ 2 ( s _ 2 ) } a ( s _ 2 ) = : A _ 2 . \\end{align*}"}
+{"id": "1641.png", "formula": "\\begin{align*} & ( \\pounds _ { \\xi _ i } { f } ) X \\overset { \\eqref { 3 . 3 B } } = \\nabla _ { \\xi _ i } ( { f } X ) - \\nabla _ { { f } X } \\ , \\xi _ i - { f } ( \\nabla _ { \\xi _ i } X - \\nabla _ { X } \\ , \\xi _ i ) \\\\ & \\ = ( \\nabla _ { \\xi _ i } { f } ) X - \\nabla _ { { f } X } \\ , \\xi _ i + { f } \\nabla _ X \\ , \\xi _ i . \\end{align*}"}
+{"id": "7764.png", "formula": "\\begin{align*} ( M \\vec x ) _ { i ^ * } = M _ { i ^ * , j ^ * } + x _ { j ^ * } . \\end{align*}"}
+{"id": "2354.png", "formula": "\\begin{align*} \\varphi ( x ) = \\frac { m _ i ( x ) } { f ( x ) ^ { k } } \\rlap { . } \\end{align*}"}
+{"id": "3548.png", "formula": "\\begin{align*} Z : = \\frac { P } { R T \\rho } = A + B \\rho + C \\rho ^ { 2 } + \\cdots \\end{align*}"}
+{"id": "7203.png", "formula": "\\begin{align*} \\sup _ { y \\in B _ 1 } \\abs { p _ h ( y ) - p _ 0 } \\leq \\sup _ { y \\in B _ 1 } \\left \\{ \\abs { p _ h ( y ) - p _ h ^ 0 } + \\abs { p _ h ^ 0 - p _ 0 } \\right \\} = \\omega ( \\sigma _ h ) + { \\rm o } ( 1 ) , \\end{align*}"}
+{"id": "7532.png", "formula": "\\begin{align*} D ^ { j _ { \\nu } } ( \\varphi _ { \\nu } ^ { r _ { \\nu } } ) ( 0 ) & = r _ { \\nu } ! B _ { j _ \\nu , r _ \\nu } ( D ^ 1 ( \\varphi _ { \\nu } ) ( 0 ) , \\ldots , D ^ { j _ \\nu - r _ \\nu + 1 } ( \\varphi _ { \\nu } ) ( 0 ) ) \\\\ & = r _ { \\nu } ! B _ { j _ \\nu , r _ \\nu } ( f _ { 1 , \\nu } , f _ { 2 , \\nu } , \\ldots , f _ { j _ \\nu - r _ \\nu + 1 , \\nu } ) . \\end{align*}"}
+{"id": "8170.png", "formula": "\\begin{align*} R _ 0 = R _ 0 ( d , \\nu ) : = \\left ( \\frac { d } { \\nu \\omega _ d } \\right ) ^ { 1 / d } \\end{align*}"}
+{"id": "4161.png", "formula": "\\begin{align*} \\lambda _ s ( b ) \\lambda _ t ( b ' ) \\xi ( r ) & = b \\lambda _ t ( b ' ) \\xi ( s ^ { - 1 } r ) = b b ' \\xi ( t ^ { - 1 } s ^ { - 1 } r ) = b b ' \\xi ( ( s t ) ^ { - 1 } r ) \\\\ & = \\lambda _ { s t } ( b b ' ) \\xi ( r ) , \\end{align*}"}
+{"id": "4904.png", "formula": "\\begin{align*} h _ { t - k } ( X _ 0 , X _ 1 , . . . , X _ k ) = \\sum _ { j = 0 } ^ { k } X _ j ^ t \\prod _ { \\substack { i = 0 \\\\ i \\neq j } } ^ { k } \\frac { 1 } { X _ j - X _ i } \\end{align*}"}
+{"id": "2006.png", "formula": "\\begin{align*} ( L _ a - c ) \\phi _ 1 & = L _ a \\phi _ 1 - c \\phi _ 1 \\\\ & = - ( \\gamma _ 1 + c ) \\phi _ 1 \\\\ & \\leq 0 . \\end{align*}"}
+{"id": "192.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c ) = 1 \\\\ a , b , c \\geq 1 } } \\left ( \\frac { 1 } { 1 - x ^ a y ^ b z ^ c } \\right ) ^ { \\frac { 1 } { a ^ s b ^ t c ^ u } } = \\exp \\left \\{ \\left ( \\sum _ { i = 1 } ^ { \\infty } \\frac { x ^ i } { i ^ s } \\right ) \\left ( \\sum _ { j = 1 } ^ { \\infty } \\frac { y ^ j } { j ^ t } \\right ) \\left ( \\sum _ { k = 1 } ^ { \\infty } \\frac { z ^ k } { k ^ u } \\right ) \\right \\} . \\end{align*}"}
+{"id": "2119.png", "formula": "\\begin{align*} X ( K ) \\cap \\tilde { \\Gamma } = \\bigcup _ { i = 1 } ^ \\ell \\left ( S _ i + \\Gamma _ i \\right ) , \\end{align*}"}
+{"id": "709.png", "formula": "\\begin{align*} \\mu _ { S } ( e ) = ( \\phi _ { F } - \\phi _ { A } ) ( z _ { S } ( e ) ) \\ , , \\mu _ { B } ( e ) = ( \\phi _ { B } - \\phi _ { F } ) ( z _ { B } ( e ) ) \\ , , \\end{align*}"}
+{"id": "3216.png", "formula": "\\begin{align*} S _ n ' \\Big ( \\sigma ' _ 1 \\circ \\sigma ' _ k ( y ' ) - \\sigma _ { k - 1 } ' ( y ' ) \\Big ) & = S _ n ' \\Big ( \\sigma ' _ 1 \\circ \\sigma ' _ k ( y ' ) - \\sigma ' _ 1 ( y ' ) + \\sigma ' _ 1 ( y ' ) - y ' + y ' - \\sigma _ { k - 1 } ' ( y ' ) \\Big ) \\\\ & = \\sigma ' _ 1 ( S _ n ' ( \\sigma ' _ k ( y ' ) - y ' ) ) + S _ n ' ( \\sigma ' _ 1 ( y ' ) - y ' ) + S _ n ' ( y ' - \\sigma _ { k - 1 } ' ( y ' ) ) . \\end{align*}"}
+{"id": "948.png", "formula": "\\begin{align*} Z _ * : C H _ i ( X ) \\to C H _ i ( X ) , Z _ * ( [ Y ] ) = [ { \\pi _ { 1 , X } } _ * ( Z \\cdot \\pi _ { 2 , X } ^ * ( Y ) ) ] \\end{align*}"}
+{"id": "4810.png", "formula": "\\begin{align*} u ( t ) = A \\sin ( t + B ) , \\end{align*}"}
+{"id": "8760.png", "formula": "\\begin{align*} \\begin{aligned} & - M w _ i \\leq \\gamma _ i \\leq M w _ i \\\\ & - M ( 1 - w _ i ) \\leq \\sum _ { j \\neq i } v _ { i j } - x _ { 0 , i } \\leq M ( 1 - w _ i ) \\end{aligned} \\end{align*}"}
+{"id": "8552.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { e ^ { 2 \\pi \\lambda _ { n } } } { \\left ( \\sigma ( \\lambda _ { n } ) e ^ { 2 \\pi \\lambda _ { n } } - 1 \\right ) ^ { 2 } } \\cdot \\left \\{ \\pi \\ , \\sigma \\left ( \\lambda _ { n } \\right ) + \\frac { p } { \\left ( p - \\lambda _ { n } \\right ) ^ { 2 } } \\right \\} = \\frac { \\pi } { 2 4 } \\cdot \\frac { 1 + \\frac { 3 } { \\pi p } ( 1 + \\frac { 1 } { \\pi p } ) } { \\left ( 1 + \\frac { 1 } { \\pi p } \\right ) ^ { 3 } } - \\frac { 1 } { 8 } . \\end{align*}"}
+{"id": "7766.png", "formula": "\\begin{align*} ( M \\vec x ) _ i = \\max _ { j \\in J _ i } \\{ M _ { i j } + x _ j \\} . \\end{align*}"}
+{"id": "4213.png", "formula": "\\begin{align*} T ( u _ { B , \\tau } ) & = ( S \\otimes I ) T ( u _ { J , \\tau } ) ( S ^ { - 1 } \\otimes I ) , \\\\ T _ n ( u _ { B , \\tau } ) & = ( S \\otimes I _ n ) T _ n ( u _ { J , \\tau } ) ( S ^ { - 1 } \\otimes I _ n ) . \\end{align*}"}
+{"id": "4069.png", "formula": "\\begin{align*} S _ { \\mathrm { o f f } } ^ { c \\geq p ^ 2 } : = \\sum _ { d _ 1 , \\ , d _ 2 < D } \\ , \\ , \\ , \\ & \\sum _ { I d _ 1 , \\ , J d _ 2 < D } x _ { I d _ 1 } \\bar { x } _ { J d _ 2 } \\sqrt { I J } \\sum _ { L , M \\geq 1 } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } \\\\ & \\times \\left ( 2 \\pi i ^ { 2 k } \\sum _ { \\substack { p | c \\\\ c \\geq p ^ 2 } } \\frac { \\mathcal { S } ( I L , J M , c ) } { c } \\ , J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { I L J M } } { c } \\right ) \\right ) . \\end{align*}"}
+{"id": "4052.png", "formula": "\\begin{align*} 0 = \\sum _ { d _ 1 , d _ 2 < D } \\frac { 1 } { \\sqrt { d _ 1 d _ 2 } } \\sum _ { I d _ 1 , J d _ 2 < D } \\alpha _ { I d _ 1 } \\bar { \\alpha } _ { J d _ 2 } & \\sum _ { L , M \\geq 1 } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } \\\\ & \\times \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( I d _ 1 ) \\lambda _ f ( J d _ 2 ) \\lambda _ f ( L d _ 1 ) \\lambda _ f ( M d _ 2 ) . \\end{align*}"}
+{"id": "7888.png", "formula": "\\begin{align*} S & = \\left \\{ ( X _ i ^ \\pm , \\underline { X } _ i ^ \\pm ) : 0 \\leq i \\leq 1 \\right \\} \\cup \\left \\{ ( X _ i ^ + , \\underline { X } _ i ^ + ) : \\ 2 \\leq i \\leq k \\right \\} \\cup \\left \\{ ( X _ i ^ + , \\underline { X } _ i ^ - ) : \\ 2 \\leq i \\leq k \\right \\} , \\end{align*}"}
+{"id": "8105.png", "formula": "\\begin{align*} C ( T ) \\cap \\lbrace 1 , 2 \\rbrace ^ 4 = \\lbrace 2 2 2 2 , 2 2 1 2 , 1 2 2 2 , 1 1 2 2 , 1 2 1 2 , 1 1 1 2 , 1 1 1 1 \\rbrace \\end{align*}"}
+{"id": "6916.png", "formula": "\\begin{align*} \\hat { \\Phi } ( y ) : = \\int _ { - \\infty } ^ { \\infty } \\Phi ( t ) e ^ { - i t y } d t = \\sqrt { 2 \\pi } \\Phi ( y ) . \\end{align*}"}
+{"id": "4103.png", "formula": "\\begin{align*} = \\begin{pmatrix} 1 9 + 0 + 0 & 0 + 0 + 2 \\cdot 1 2 & 0 + 2 \\cdot 1 5 + 0 \\\\ 0 + 1 5 + 0 & 1 \\cdot 1 9 + 0 + 0 & 0 + 0 + 2 \\cdot 1 2 \\\\ 0 + 0 + 1 2 & 0 + 1 \\cdot 1 5 + 0 & 1 \\cdot 1 9 + 0 + 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "7426.png", "formula": "\\begin{align*} t = \\tfrac { 1 } { \\sqrt { N } } \\ , , s = \\tfrac { 1 } { \\sqrt { L } } = \\sqrt { \\tfrac { M } { N } } \\ , . \\end{align*}"}
+{"id": "7609.png", "formula": "\\begin{align*} | H _ { 2 , 1 } ( F _ { f } / 2 ) | & \\leq \\frac { 1 } { 9 6 } \\tau _ { 1 } ( 1 - \\tau ^ 2 _ { 1 } ) \\left ( | A | + | B | + | C | \\right ) \\\\ & = \\frac { 1 } { 2 3 0 4 } \\left ( 1 6 + 4 \\tau ^ 2 _ { 1 } - 1 7 \\tau ^ 4 _ { 1 } \\right ) \\\\ & \\leq \\frac { 2 3 } { 3 2 6 4 } . \\end{align*}"}
+{"id": "623.png", "formula": "\\begin{align*} \\{ b \\in B : b p = 0 \\} \\end{align*}"}
+{"id": "9021.png", "formula": "\\begin{align*} & \\underline m ( \\tau _ i , \\varepsilon ) : = \\max \\{ n \\in \\mathbb N : n \\varepsilon \\leq \\tau _ i \\} , \\\\ & \\underline m ( \\tau _ i , \\varepsilon ) \\varepsilon = , \\\\ & \\overline m ( \\tau _ i , \\varepsilon ) : = \\min \\{ n \\in \\mathbb N : n \\varepsilon \\geq \\tau _ i \\} , \\\\ & \\overline m ( \\tau _ i , \\varepsilon ) \\varepsilon = . \\end{align*}"}
+{"id": "2301.png", "formula": "\\begin{align*} T ( f ) ( z ) : = - \\frac { 1 } { \\pi } \\iint _ D \\frac { f ( \\zeta ) } { \\zeta - z } \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "2340.png", "formula": "\\begin{align*} \\sigma _ { R } ^ 2 ( t ) = \\sum _ { n \\geq 1 } \\frac { 1 } { n ! } \\int _ { - R } ^ { R } \\int _ { - R } ^ R \\gamma _ n ( t , x - y ) d x d y . \\end{align*}"}
+{"id": "2205.png", "formula": "\\begin{align*} \\Phi ( x , y ) = \\mu _ y ( u ) e ^ { \\gamma } x \\log y \\prod _ { p \\leq y } \\left ( 1 - \\frac { 1 } { p } \\right ) + O \\left ( \\frac { x R ( y ) } { \\log y } \\right ) . \\end{align*}"}
+{"id": "878.png", "formula": "\\begin{align*} \\widehat { P } ^ { 2 n / 3 - 2 / 3 - \\delta ( 1 - n / 8 ) + \\varepsilon } \\widehat { Y } \\ll \\widehat { P } ^ { 2 n / 3 + 1 / 3 - \\eta - \\delta ( 1 - n / 8 ) + \\varepsilon } = \\widehat { P } ^ { n - 5 - \\eta + \\varepsilon } , \\end{align*}"}
+{"id": "3418.png", "formula": "\\begin{align*} \\xi ( \\boldsymbol { p } ) = \\min _ { 1 \\leq i \\neq j \\leq \\ell } | p _ i - p _ j | . \\end{align*}"}
+{"id": "6458.png", "formula": "\\begin{align*} \\frac { 1 } { 4 z ( s ) } & = \\frac { 1 } { 2 \\sigma ^ 2 + 6 h ^ 2 + \\mathcal { O } ( s \\sigma ^ 4 ) } \\\\ & = \\frac { 1 } { 2 \\sigma ^ 2 } \\big ( 1 - \\frac { 3 h ^ 2 } { \\sigma ^ 2 } + \\mathcal { O } ( s \\sigma ^ 2 ) \\big ) \\\\ & = \\frac { 1 } { 2 \\sigma ^ 2 } - \\frac { 3 h ^ 2 } { 2 \\sigma ^ 4 } + \\mathcal { O } ( s ) . \\end{align*}"}
+{"id": "5748.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } p ( q + a ) = 0 \\\\ q ^ 2 = a ^ 2 , \\end{array} \\right . \\end{align*}"}
+{"id": "2979.png", "formula": "\\begin{align*} s _ { ( k - ( n - 1 ) , 0 , 0 , \\dots , 0 ) } \\cdot e _ 1 = s _ { ( k + 1 - ( n - 1 ) , 0 , 0 , \\dots , 0 ) } + s _ { ( k - ( n - 1 ) , 1 , 0 , \\dots , 0 ) } \\end{align*}"}
+{"id": "8321.png", "formula": "\\begin{align*} 0 \\leq a \\leq \\sum _ { j = 1 } ^ k w _ j a _ j , \\ 0 \\leq a _ j , w _ j \\ \\forall j \\implies a ^ p \\leq ( \\sum _ j w _ j ) ^ p \\sum _ { j = 1 } ^ k a _ j ^ p \\ \\ \\forall p \\in ( 0 , \\infty ) . \\end{align*}"}
+{"id": "4422.png", "formula": "\\begin{align*} G ( V ) = G ( V _ n ) - ( G ( V _ n ) - G ( V ) ) = G ( U _ n ) - ( G ( V _ n ) - G ( V ) ) . \\end{align*}"}
+{"id": "9336.png", "formula": "\\begin{align*} \\sigma _ { k - 3 } ( \\ell ) = \\sigma _ { \\frac { p - 1 } { 2 } } ( \\ell ) \\equiv 0 \\pmod { p } \\chi _ { - p } ( \\ell ) = - 1 . \\end{align*}"}
+{"id": "2592.png", "formula": "\\begin{align*} \\nu ( x ) = \\frac { D u ( x ) - x } { | D u ( x ) - x | } \\forall \\ x \\in \\mathcal F . \\end{align*}"}
+{"id": "157.png", "formula": "\\begin{align*} L i _ 2 ( - z ) - L i _ 2 ( 1 - z ) + \\frac { 1 } { 2 } L i _ 2 ( 1 - z ^ 2 ) = - \\frac { \\pi ^ 2 } { 1 2 } - \\log ( z ) \\log ( 1 + z ) . \\end{align*}"}
+{"id": "2332.png", "formula": "\\begin{align*} \\big \\| v ^ { ( r , z ) } ( s , x ) \\big \\| _ p = 2 G _ { s - r } ( x - z ) \\big \\| v ^ { ( r , z ) } ( s , x ) \\big \\| _ p \\leq C G _ { s - r } ( x - z ) . \\end{align*}"}
+{"id": "4238.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & - 1 & 0 & 0 \\\\ - 1 & - 1 & 0 & 0 \\\\ \\end{pmatrix} \\end{align*}"}
+{"id": "1435.png", "formula": "\\begin{align*} & E _ { \\mu , \\xi } = \\mbox { s p a n } \\left \\{ P \\psi ^ h _ { \\mu , \\xi } : h = 0 , 1 , \\cdots , n , \\ i = 1 , \\cdots , k \\right \\} , \\\\ & E _ { \\mu , \\xi } ^ \\perp = \\left \\{ \\phi \\in H ^ 1 _ 0 ( \\Omega ) : \\langle \\phi , P \\psi ^ h _ { \\mu , \\xi } \\rangle = 0 : h = 0 , 1 , \\cdots , n , \\ i = 1 , \\cdots , k \\right \\} . \\end{align*}"}
+{"id": "9198.png", "formula": "\\begin{align*} W ( G _ { n , r , s } ) & = W ( C _ { 2 r } ) + W ( K _ { n - 2 r } ) + \\sum _ { u \\in C _ { 2 r } , v \\in K _ { n - 2 r } } d ( u , v ) \\\\ & = r ^ 3 + \\binom { n - 2 r } { 2 } + ( n - 2 r ) ( r ^ 2 + 1 ) \\\\ & = \\binom { n } { 2 } + ( r - 1 ) ^ 2 n - r ( r - 1 ) ^ 2 \\end{align*}"}
+{"id": "1073.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\left ( \\exp \\left | \\frac { e ^ { - t H ^ { \\beta } } g } { \\lambda } \\right | ^ p - 1 \\right ) \\ , d x & \\leq \\sum _ { k = 1 } ^ { \\infty } \\frac { { C ^ { p k } } t ^ { - \\frac { d } { 2 \\beta } ( \\frac 1 q - \\frac { 1 } { p k } ) { p k } } \\| g \\| _ { L ^ q } ^ { p k } } { k ! \\lambda ^ { p k } } \\\\ & = t ^ { \\frac { d } { 2 \\beta } } \\left ( \\exp \\left ( \\frac { { C } t ^ { - \\frac { d } { 2 \\beta q } } \\| f \\| _ { L ^ q } } { \\lambda } \\right ) ^ p - 1 \\right ) . \\end{align*}"}
+{"id": "9105.png", "formula": "\\begin{align*} B ^ { k + 2 } = B ^ { k + 1 } \\cup \\{ s \\} \\setminus \\{ t \\} = B ^ { k } \\cup \\{ p , s \\} \\setminus \\{ q , t \\} . \\end{align*}"}
+{"id": "781.png", "formula": "\\begin{align*} 0 = \\int _ 0 ^ { \\rho } r \\phi ^ n ( r ) \\mathrm { d } r \\int _ { \\mathbb { S } ^ n } x _ l \\mathrm { d } A + \\rho ^ 2 \\phi ^ n ( \\rho ) \\int _ { \\mathbb { S } ^ n } u x _ l \\mathrm { d } A + O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } . \\end{align*}"}
+{"id": "8907.png", "formula": "\\begin{gather*} \\mathsf { d } _ { \\mathrm { c o m } } ( \\{ X _ { n i } \\} ) = \\sum _ { n = 1 } ^ { d _ 1 ^ 2 } \\| X _ { n 1 } \\| _ 1 + \\| X _ { n 2 } \\| _ 1 = 2 \\sum _ { n = 1 } ^ { d _ 1 ^ 2 } \\| X _ { n 1 } \\| _ 1 , \\\\ \\nu = \\sqrt { \\sum _ { n = 1 } ^ { d _ 1 ^ 2 } \\| X _ { n 1 } \\| _ 1 ^ 2 \\| X _ { n 2 } \\| _ 1 ^ 2 } = \\sqrt { \\sum _ { n = 1 } ^ { d _ 1 ^ 2 } \\| X _ { n 1 } \\| _ 1 ^ 4 } . \\end{gather*}"}
+{"id": "7547.png", "formula": "\\begin{align*} | ( \\partial u ) _ { B _ { j _ 1 + 1 } } | \\leq | ( \\partial u ) _ { B _ { j _ 0 } } | + \\left ( \\frac { \\sigma } { 2 } \\right ) ^ { - \\frac { d } { p } } \\sum _ { j = j _ 0 } ^ { j _ 1 } \\mathfrak { E } _ p ( \\partial u , B _ j ) \\leq \\frac { 1 } { 1 6 } \\lambda . \\end{align*}"}
+{"id": "6950.png", "formula": "\\begin{align*} \\partial _ { v _ 1 \\cdots v _ d } : = \\partial _ { v _ 1 } \\cdots \\partial _ { v _ d } v _ 1 , \\ldots , v _ d \\in W . \\end{align*}"}
+{"id": "4864.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\overline { c } } ^ { o p } \\left ( b \\right ) : = \\mathcal { F } ^ { o p } \\left ( \\overline { c } , b \\right ) . \\end{align*}"}
+{"id": "4776.png", "formula": "\\begin{align*} M _ { [ j + 1 ] } = S _ { [ j + 1 ] } + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "6375.png", "formula": "\\begin{align*} b ( u \\circ v ) + b ( v \\circ u ) = 0 , \\forall u , v \\in T , \\end{align*}"}
+{"id": "2465.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } & - \\dfrac { 1 } { 2 m _ 1 } \\Delta u ( x ) + \\omega _ 1 u ( x ) = - a _ 1 v ( x ) u ( x ) , ( 1 . 5 a ) \\\\ \\\\ & - \\dfrac { 1 } { 2 m _ 2 } \\Delta v ( x ) + \\omega _ 2 v ( x ) = - a _ 2 u ^ 2 ( x ) , ~ ( 1 . 5 b ) \\end{array} \\right . \\end{align*}"}
+{"id": "5502.png", "formula": "\\begin{align*} \\tilde { \\xi } _ { \\ast } \\left ( \\bar { m } _ { K } \\times \\left ( p _ { \\ast } \\left ( m _ { K \\cap Q } \\right ) \\times \\lambda \\right ) \\right ) = \\bar { m } _ { K } \\times \\xi _ { \\ast } \\left ( p _ { \\ast } \\left ( m _ { K \\cap Q } \\right ) \\times \\lambda \\right ) = \\bar { m } _ { K } \\times \\eta . \\end{align*}"}
+{"id": "7994.png", "formula": "\\begin{align*} W = h \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla w ) \\end{align*}"}
+{"id": "8605.png", "formula": "\\begin{align*} E [ X _ { \\ell , \\Delta } ^ 2 | { \\cal F } _ { ( \\ell + 1 ) \\Delta } ] - E [ X _ { \\ell , \\Delta } | { \\cal F } _ { ( \\ell + 1 ) \\Delta } ] = E [ X _ { \\ell , \\Delta } | { \\cal F } _ { ( \\ell + 1 ) \\Delta } ] ^ 2 , \\end{align*}"}
+{"id": "8071.png", "formula": "\\begin{align*} \\left \\langle F ^ { \\prime } ( u ) , u \\right \\rangle = 0 , \\enspace u \\in \\overline { \\operatorname { c o n v } } \\mathcal { G } \\Longrightarrow F ( u ) = 0 , \\end{align*}"}
+{"id": "8734.png", "formula": "\\begin{align*} & E \\| X _ 1 - X _ 2 \\| ^ { 2 l } _ 2 + E \\| Y _ 1 - Y _ 2 \\| ^ { 2 l } _ 2 - 2 E \\| X _ 1 - Y _ 1 \\| ^ { 2 l } _ 2 \\\\ & = \\sum _ { 0 \\leq a _ 1 + a _ 2 \\leq l } \\frac { ( - 2 ) ^ { l - a _ 1 - a _ 2 } l ! } { a _ 1 ! a _ 2 ! ( l - a _ 1 - a _ 2 ) ! } \\Big [ E \\{ \\| X _ 1 \\| _ { 2 } ^ { 2 a _ 1 } \\| X _ 2 \\| _ { 2 } ^ { 2 a _ 2 } ( X _ 1 ^ \\top X _ 2 ) ^ { l - a _ 1 - a _ 2 } \\} \\\\ & \\quad + E \\{ \\| Y _ 1 \\| _ { 2 } ^ { 2 a _ 1 } \\| Y _ 2 \\| _ { 2 } ^ { 2 a _ 2 } ( Y _ 1 ^ \\top Y _ 2 ) ^ { l - a _ 1 - a _ 2 } \\} - 2 E \\{ \\| X _ 1 \\| _ { 2 } ^ { 2 a _ 1 } \\| Y _ 1 \\| _ { 2 } ^ { 2 a _ 2 } ( X _ 1 ^ \\top Y _ 1 ) ^ { l - a _ 1 - a _ 2 } \\} \\Big ] . \\end{align*}"}
+{"id": "3610.png", "formula": "\\begin{align*} f \\ = \\ \\frac { r ( p ) } { q } \\ < \\ \\frac { 1 0 ^ { m + 1 } } { 1 1 } \\ \\le \\ 1 0 ^ m - 1 0 0 \\end{align*}"}
+{"id": "109.png", "formula": "\\begin{align*} T ( z ) = { \\rm t r } ^ \\flat ( \\chi e ^ { - i t _ 0 h ^ { - 1 } \\tilde { P } _ h ( z ) } \\tilde { R } _ h ( z ) \\chi ) \\end{align*}"}
+{"id": "3393.png", "formula": "\\begin{align*} x = ( 1 - \\varepsilon / 2 ) \\Biggl ( \\frac { \\log _ 2 T _ k \\log _ 4 T _ k } { \\sum _ { T _ { k - 1 } < p \\leq T _ k } \\frac { 1 } { 2 p ^ { 1 + 2 \\sigma _ k } } \\bigl ( \\frac { 2 \\log T _ k } { \\log p } \\bigr ) ^ 2 \\sin ^ 2 \\bigl ( \\frac { \\log p } { 2 \\log T _ k } \\bigr ) + \\frac { 1 } { 8 p ^ { 2 + 4 \\sigma _ k } } } \\Biggr ) ^ { 1 / 2 } \\ , , \\end{align*}"}
+{"id": "4711.png", "formula": "\\begin{align*} P _ { \\widehat { \\mathcal { H } } _ \\chi ^ { \\rm s t i f f } } \\left ( \\mathcal { A } _ { \\chi , \\varepsilon } ^ { \\rm a p p } - z I \\right ) ^ { - 1 } | _ { \\widehat { \\mathcal { H } } _ \\chi ^ { \\rm s t i f f } } = \\left ( \\frac { 1 } { \\varepsilon ^ 2 } \\mathcal { A } _ \\chi - \\mathcal { B } _ \\chi ( z ) \\right ) ^ { - 1 } . \\end{align*}"}
+{"id": "5435.png", "formula": "\\begin{align*} F _ { | g _ i | ^ 2 } ( x ) = \\gamma ( M , M x ) / \\Gamma ( M ) = 1 - \\sum _ { i = 0 } ^ M \\frac { ( M x ) ^ i } { i ! } e ^ { - M x } , \\end{align*}"}
+{"id": "2759.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q ) w = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } _ 1 , w _ { | \\Gamma } = \\partial _ \\nu w _ { | \\Gamma } = 0 . \\end{align*}"}
+{"id": "3019.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } \\lambda _ { i } ^ { \\gamma } ( A + \\alpha B ) \\leq \\sum _ { i = 1 } ^ { k } \\lambda _ i ^ { \\gamma } ( A ) + \\sum _ { i = 1 } ^ { k } \\lambda _ i ^ { \\gamma } ( \\alpha B ) \\hbox { f o r a l l $ 0 < \\alpha < 1 $ } \\end{align*}"}
+{"id": "3634.png", "formula": "\\begin{align*} p & \\ = \\ ( 4 , \\{ 9 \\} ^ { I } , a _ { m - I - 1 } , \\ldots , a _ { I + 1 } , \\{ 9 \\} ^ { I + 1 } ) _ { 1 0 } , \\\\ r ( p ) & \\ = \\ ( \\{ 9 \\} ^ { I + 1 } , a _ { I + 1 } , \\ldots , a _ { m - I - 1 } , \\{ 9 \\} ^ { I } , 4 ) _ { 1 0 } , \\\\ p - 2 & \\ = \\ ( 4 , \\{ 9 \\} ^ { I } , a _ { m - I - 1 } , \\ldots , a _ { I + 1 } , \\{ 9 \\} ^ { I } , 7 ) _ { 1 0 } , \\end{align*}"}
+{"id": "81.png", "formula": "\\begin{align*} { \\rm t r } ^ \\flat B = \\int _ \\mathcal { M } \\iota ^ * K _ B ( x ) d x = \\langle \\iota ^ * K _ B , 1 \\rangle . \\end{align*}"}
+{"id": "6099.png", "formula": "\\begin{align*} W ^ { \\theta , q } _ 0 \\coloneqq \\lbrace g \\in W ^ { \\theta , q } : g ( 0 ) = 0 \\rbrace . \\end{align*}"}
+{"id": "6126.png", "formula": "\\begin{align*} N D _ { \\lambda } ^ { d } = \\left \\{ \\mathcal { K } \\times I \\in \\mathcal { A } \\ , : E _ { p , \\tau } \\left ( u \\ , ; \\ , { K } ^ { d } \\times I \\right ) > \\frac { \\lambda } { 1 6 } \\right \\} . \\end{align*}"}
+{"id": "7406.png", "formula": "\\begin{align*} \\partial _ t v ^ { ( \\mathsf { s } , \\mathsf { c } ) } ( t , x ) \\ , = \\ , \\frac { \\mathsf { s } } { 2 } \\ , \\Delta _ x v ^ { ( \\mathsf { s } , \\mathsf { c } ) } ( t , x ) \\ , + \\ , \\mathsf { c } \\ , \\dot { W } ( t , x ) \\ , , \\end{align*}"}
+{"id": "6214.png", "formula": "\\begin{align*} A ^ { - 1 } \\cdot c _ { i - 1 } = w = | a _ { i - 1 } b _ { i + 1 } - b _ { i - 1 } a _ { i + 1 } | . \\end{align*}"}
+{"id": "1615.png", "formula": "\\begin{align*} g ( { f } X , { f } Y ) = - g ( X , Q \\ , Y ) + \\sum \\nolimits _ { i } \\eta ^ i ( X ) \\ , \\eta ^ i ( Y ) , X , Y \\in \\mathfrak { X } _ M , \\end{align*}"}
+{"id": "6379.png", "formula": "\\begin{align*} E _ r : Y ^ 2 = X ^ 3 + 2 0 r ^ 2 X . \\end{align*}"}
+{"id": "8534.png", "formula": "\\begin{align*} \\left ( \\frac { \\pi } { \\sqrt { c } } \\right ) ^ { - s } \\Gamma ( s ) \\ , \\tilde { \\zeta } _ { p , p ^ { \\prime } } \\left ( s , c \\right ) = \\left ( \\frac { \\pi } { \\sqrt { c } } \\right ) ^ { - ( 1 - s ) } \\Gamma \\left ( 1 - s \\right ) \\ , \\tilde { \\zeta } _ { p ^ { \\prime } , p } \\left ( 1 - s , c \\right ) . \\end{align*}"}
+{"id": "54.png", "formula": "\\begin{align*} h = Z _ \\mathbb { Q } ^ { - 1 } Y _ \\mathbb { Q } X Y _ f Z _ f ^ { - 1 } . \\end{align*}"}
+{"id": "4374.png", "formula": "\\begin{align*} \\mathcal { G } _ k ( h ) : = \\sum _ { n _ 0 \\leq j \\leq k } \\mathcal { N } _ j \\mathcal { G } _ k ( h ' ) : = \\sum _ { n _ 0 \\leq j \\leq k } \\mathcal { N } _ j ' , \\end{align*}"}
+{"id": "1016.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } \\binom { 4 k } { 2 k } } { x ^ k } ( 2 H _ { 4 k } - H _ { 2 k } ) = \\frac { 1 } { 2 } \\bigg ( \\log \\frac { x } { x - 6 4 } \\bigg ) \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } \\binom { 4 k } { 2 k } } { x ^ k } , \\end{align*}"}
+{"id": "7846.png", "formula": "\\begin{align*} \\Psi : = \\sum _ { \\phi \\in \\Lambda ( G , H ) } \\Phi _ \\phi . \\end{align*}"}
+{"id": "6563.png", "formula": "\\begin{align*} \\| H ( f ) \\| _ { L _ { | x | _ h } ^ p L _ { \\theta } ^ { \\bar { p } _ 1 } ( \\mathbb H ^ n ) \\rightarrow L _ { | x | _ h } ^ p L _ { \\theta } ^ { \\bar { p } _ 2 } ( \\mathbb H ^ n ) } = G \\| f \\| _ { L _ { | x | _ h } ^ p L _ { \\theta } ^ { \\bar { p } _ 2 } ( \\mathbb H ^ n ) } \\end{align*}"}
+{"id": "3201.png", "formula": "\\begin{align*} \\sigma _ 1 \\circ \\sigma _ n = w \\sigma _ { n + 1 } + ( 1 - w ) \\sigma _ { n - 1 } , ~ n \\in \\N , \\end{align*}"}
+{"id": "7509.png", "formula": "\\begin{align*} \\sum ^ n _ { k = 1 } \\frac { d _ k } { w _ j - z _ k } = 0 , j = 1 , \\dots \\deg ( p _ + ( z ) ) , \\end{align*}"}
+{"id": "8574.png", "formula": "\\begin{align*} & \\lim _ { N \\to \\infty } N ^ { - 1 } M ( \\tau _ N ) = \\begin{cases} \\dfrac { \\nu } { \\lambda } , & p = 0 , \\\\ - \\dfrac { \\nu q \\log ( q ) } { \\lambda p } , & 0 < p < 1 . \\end{cases} \\end{align*}"}
+{"id": "642.png", "formula": "\\begin{align*} \\rho ( a ) ^ * \\rho ( a ) = \\rho ( a ^ * a ) = \\rho ( a a ^ * ) \\end{align*}"}
+{"id": "3237.png", "formula": "\\begin{align*} \\| f \\| _ { \\Lambda ^ \\beta _ d } : = \\sup _ { d ( x , y ) \\ne 0 } \\frac { | f ( x ) - f ( y ) | } { d ( x , y ) ^ \\beta } < \\infty . \\end{align*}"}
+{"id": "2110.png", "formula": "\\begin{align*} | \\phi ' ( t ) | & = \\Big | \\frac 1 n \\sum _ { j = 1 } ^ n \\alpha ( y _ j ) q _ t ( x , y _ j ) \\Big | \\leqslant \\| \\alpha \\| _ { L ^ 2 ( \\hat \\nu ) } \\Big ( \\frac 1 n \\sum _ { j = 1 } ^ n q _ t ( x , y _ j ) ^ 2 \\Big ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "5156.png", "formula": "\\begin{align*} \\| g \\| _ { X ^ \\dag } = \\sup _ { \\| f \\| _ X = 1 } \\int _ \\Omega \\ ! | f g | w \\ , \\mathrm { d } \\mu = \\| g w \\| _ { X ' } = \\| g \\| _ { X ' ( w ) } , \\end{align*}"}
+{"id": "6501.png", "formula": "\\begin{align*} c _ { n , n } & = 1 ( n \\geq 1 ) , \\\\ \\sum _ { j = 1 } ^ n a _ { i , j } c _ { n , j } & = 0 ( 1 \\leq i < n ) , \\end{align*}"}
+{"id": "6033.png", "formula": "\\begin{align*} \\int V f _ 1 \\cdot J g \\ , d \\mu = \\lim \\int T ^ { k _ j } f _ 1 \\cdot J g \\ , d \\mu & = \\lim \\int T ^ { k _ j } f _ 1 \\otimes g \\ , d \\eta \\\\ & = \\lim \\int f _ 1 \\otimes S ^ { - k _ j } g \\ , d \\eta = \\lim \\int J ^ { \\ast } f _ 1 \\cdot S ^ { - k _ j } g \\ , d \\nu . \\end{align*}"}
+{"id": "6637.png", "formula": "\\begin{align*} I _ { \\tilde { Y } } = \\langle x _ 0 \\vartheta + y _ 0 \\zeta + \\alpha x _ 0 y _ 0 , ( x _ 0 , y _ 0 ) ^ { k + 2 } \\rangle . \\end{align*}"}
+{"id": "563.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial ^ { 2 } _ { t } w ( t , k ) + a ( t ) \\mathcal { H } _ { \\hbar , V } w ( t , k ) + q ( t ) w ( t , k ) = a ( t ) \\left ( \\mathcal { H } _ { V } - \\mathcal { H } _ { \\hbar , V } \\right ) v ( t , k ) , ~ ~ k \\in \\hbar \\mathbb { Z } ^ { n } , \\\\ w ( 0 , k ) = 0 , k \\in \\hbar \\mathbb { Z } ^ { n } , \\\\ \\partial _ { t } w ( 0 , k ) = 0 , k \\in \\hbar \\mathbb { Z } ^ { n } . \\end{array} \\right . \\end{align*}"}
+{"id": "5551.png", "formula": "\\begin{align*} D \\left ( \\alpha _ { x , e } \\parallel \\alpha _ { x , g } \\right ) = \\sup _ { n \\in \\mathbb { N } } \\left \\{ H _ { q _ { x , e } ^ { n } \\parallel q _ { x , g } ^ { n } } \\left ( \\mathcal { P } _ { x , n } \\right ) - \\Delta _ { x , g } ( n ) \\right \\} . \\end{align*}"}
+{"id": "3863.png", "formula": "\\begin{align*} g _ { \\lambda , A } ( v ) = \\sup _ { v ' \\in \\mathcal { V } } \\left \\{ g ( v ' ) - \\sum _ { \\ell = 1 } ^ { L } \\lambda _ \\ell \\tilde { c } _ { \\ell } \\left \\{ s _ { \\ell } , s _ { \\ell } ' \\right \\} \\right \\} \\end{align*}"}
+{"id": "5897.png", "formula": "\\begin{align*} T _ \\gamma ( x ) = \\left \\{ \\begin{array} { l l } x ( 1 + 2 ^ \\gamma x ^ \\gamma ) & i f \\ , x \\in [ 0 , 1 / 2 [ \\\\ 2 x - 1 & i f \\ , x \\in [ 1 / 2 , 1 ] \\end{array} \\right . . \\end{align*}"}
+{"id": "3503.png", "formula": "\\begin{align*} 0 = \\int _ a ^ t \\| X ' \\| ^ 2 + \\langle ( R - \\lambda I ) X , X \\rangle d s \\geq \\int _ a ^ t \\| X ' \\| ^ 2 + \\mu \\langle X , X \\rangle d s \\geq 0 , \\end{align*}"}
+{"id": "9267.png", "formula": "\\begin{align*} \\sup _ { \\lambda > 0 } \\lambda \\int _ { \\{ ( b - a ) / 2 < x < b / 3 \\ , : \\ , ( b - a ) / b > \\lambda \\} } x ^ { - 1 } \\ , d x \\lesssim \\int _ a ^ b x ^ { - 1 } \\ , d x = \\log \\frac { b } a \\simeq \\frac { b - a } a \\simeq \\frac { b - a } b , \\end{align*}"}
+{"id": "9307.png", "formula": "\\begin{align*} \\nu _ 0 = { \\rm I d } , \\nu _ { j } = \\nu _ { j } [ A , \\varphi ] ( f ) : = f - \\varphi ( A \\nu _ { j - 1 } ( f ) ) , \\ , j = 1 , 2 , 3 , \\ldots \\end{align*}"}
+{"id": "678.png", "formula": "\\begin{align*} & D ^ { n - 1 } \\mathfrak { r } _ { a , n } ( \\varphi ) = D ^ { n - 1 } \\varphi - \\sum _ { 1 \\leq i < i _ a } d ^ + _ { i , 0 } ( D ^ n \\varphi ) h _ { i , 1 } \\\\ & - \\sum _ { 1 \\leq i \\leq g } d ^ + _ { i , 1 } ( D ^ { n - 1 } \\varphi ) h _ { i } - \\sum _ { 1 \\leq s < \\gamma } d ^ 0 _ { s , 1 } ( D ^ { n - 1 } \\varphi ) c _ { s } - \\sum _ { j _ a < j \\leq g } d ^ - _ { - j , 1 } ( D ^ { n - 1 } \\varphi ) h _ { - j } . \\end{align*}"}
+{"id": "7914.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\beta ! } \\big ( \\partial ^ \\beta \\big ) | _ { z = 0 } u _ x = \\left \\{ \\begin{array} { l l } ( \\cdot - x ) ^ \\mathbf { n } & \\mbox { i f } \\beta = e _ \\mathbf { n } , \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\ ; \\end{align*}"}
+{"id": "5992.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } ( [ \\omega , 1 ] ) f _ 2 ( x ) & = \\overline { \\Pi } _ { \\psi } ( [ \\omega , 1 ] ) f ( [ - 1 , x ] ) \\\\ & = e ^ { - \\tfrac { \\pi i } { 4 } } \\nu ( - 1 , \\omega ) \\int _ { \\R } e ^ { - 2 \\pi i x y } f ( [ - 1 , - y ] ) d y \\\\ & = e ^ { - \\tfrac { \\pi i } { 4 } } \\nu ( - 1 , \\omega ) \\int _ { \\R } e ^ { - 2 \\pi i x y } f _ 2 ( - y ) d y \\\\ & = e ^ { - \\tfrac { \\pi i } { 4 } } \\nu ( - 1 , \\omega ) \\widehat { f _ 2 } ( - x ) . \\end{align*}"}
+{"id": "7588.png", "formula": "\\begin{align*} w ( z ) = \\frac { h ( z ) - 1 } { h ( z ) + 1 } . \\end{align*}"}
+{"id": "7540.png", "formula": "\\begin{align*} - \\Delta _ p v = 0 \\ ; \\ ; B _ r ( x _ 0 ) . \\end{align*}"}
+{"id": "1218.png", "formula": "\\begin{align*} H ( x ) = \\frac { q - I _ { \\boldsymbol { a } } ( q x ) } { q \\Pi _ { \\boldsymbol { a } } ( q x ) } ; \\end{align*}"}
+{"id": "607.png", "formula": "\\begin{align*} g ( 1 ) = - \\frac { 1 } { 2 } \\sqrt { C - 4 } < g ( 0 ) = 0 < g ( s _ - ) . \\end{align*}"}
+{"id": "1566.png", "formula": "\\begin{align*} c ^ { s + \\frac { \\l - 1 } { 2 } } \\L _ { f } \\left ( s , \\frac { a } { c } \\right ) = \\int _ { 0 } ^ { \\infty } f \\left ( \\frac { a } { c } - \\frac { t } { i c } \\right ) t ^ { s + \\frac { \\l - 1 } { 2 } } \\ , d t . \\end{align*}"}
+{"id": "4966.png", "formula": "\\begin{align*} p _ { k } ^ { } = \\frac { n - k } { n } p . \\end{align*}"}
+{"id": "2263.png", "formula": "\\begin{align*} P ( x , y ) = \\frac { y } { x ^ 2 + y ^ 2 } \\end{align*}"}
+{"id": "1318.png", "formula": "\\begin{align*} \\partial _ u ( H ^ { ( u ) } ) ^ { - 1 } = ( H ^ { ( u ) } ) ^ { - 1 } T ( u ) ^ { - 1 } \\partial _ u ( T ( u ) ) T ( u ) ^ { - 1 } ( H ^ { ( u ) } ) ^ { - 1 } , \\end{align*}"}
+{"id": "4771.png", "formula": "\\begin{align*} M _ { [ 1 ] } \\diamond \\cdots \\diamond M _ { [ j ] } & = S _ { [ 1 ] } \\diamond \\cdots \\diamond S _ { [ j ] } + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "2941.png", "formula": "\\begin{align*} C ( I d , \\gamma ) ( [ U ] \\mid U L ( p ) ) = C ( U L ( p ) , U L ( p ) \\gamma ) ( [ U ] ) . \\end{align*}"}
+{"id": "5919.png", "formula": "\\begin{align*} ( y _ 1 , g _ 1 , \\epsilon _ 1 ) \\cdot ( y _ 2 , g _ 2 , \\epsilon _ 2 ) & = ( y _ 1 y _ 2 , [ g _ 1 , \\epsilon _ 1 ] ^ { y _ 2 } [ g _ 2 , \\epsilon _ 2 ] ) \\\\ & = ( y _ 1 y _ 2 , [ g _ 1 ^ { y _ 2 } , \\nu ( y _ 2 , g _ 1 ) \\epsilon _ 1 ] [ g _ 2 , \\epsilon _ 2 ] ) \\\\ & = ( y _ 1 y _ 2 , [ g _ 1 ^ { y _ 2 } g _ 2 , \\nu ( y _ 2 , g _ 1 ) \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 ^ { y _ 2 } , g _ 2 ) \\epsilon _ 1 \\epsilon _ 2 ] ) . \\end{align*}"}
+{"id": "4326.png", "formula": "\\begin{gather*} \\psi : \\left ( - \\frac { W } { 2 } , \\frac { W } { 2 } \\right ) \\times \\left ( - \\frac { H } { 2 } , \\frac { H } { 2 } \\right ) \\subset \\mathbb { R } ^ 2 \\to \\mathbb { R } , \\\\ \\psi ( ( x _ 1 , x _ 2 ) ) = \\min \\{ W , H \\} \\cdot \\max \\left \\{ \\left ( \\frac { x _ 1 } { W } \\right ) ^ 2 , \\left ( \\frac { x _ 2 } { H } \\right ) ^ 2 \\right \\} . \\end{gather*}"}
+{"id": "3339.png", "formula": "\\begin{align*} ( D - C A ^ { - 1 } B ) D ^ { - 1 } C v = C v - C A ^ { - 1 } B D ^ { - 1 } C v = C A ^ { - 1 } ( A v - B D ^ { - 1 } C v ) = 0 . \\end{align*}"}
+{"id": "6971.png", "formula": "\\begin{align*} I _ n = I _ \\alpha . \\end{align*}"}
+{"id": "799.png", "formula": "\\begin{align*} \\left ( { D } ^ { \\alpha } _ { N } U ^ { n - \\sigma } , U ^ { n , \\sigma } \\right ) \\ , + \\ , a \\big ( l ( U ^ { n , \\sigma } ) \\big ) \\ , ( \\nabla U ^ { n , \\sigma } , \\nabla U ^ { n , \\sigma } ) \\ , = & \\ , ( f ^ { n - \\sigma } , U ^ { n , \\sigma } ) , \\ ; \\ , \\forall v _ h \\in V _ h . \\\\ \\end{align*}"}
+{"id": "742.png", "formula": "\\begin{align*} \\eta _ 1 ( m ) : = - \\frac { A } { m } \\frac { E ( m ) } { K ( m ) } , \\eta _ 2 ( m ) : = \\frac { A } { m } \\Big ( 1 - m - \\frac { E ( m ) } { K ( m ) } \\Big ) , \\quad \\eta _ 3 ( m ) : = \\frac { A } { m } \\Big ( 1 - \\frac { E ( m ) } { K ( m ) } \\Big ) . \\end{align*}"}
+{"id": "5327.png", "formula": "\\begin{align*} \\| T _ { \\rho } f \\| _ q ^ q = \\| T _ { \\rho ' \\frac { 1 } { \\sqrt { q } } } f ' \\| _ q ^ q \\le \\sum _ { S \\subseteq [ n ] } \\left ( \\frac { \\beta } { \\sqrt { q } } \\right ) ^ { q | S | } \\mathbb { E } _ { x \\sim \\Omega ^ S } \\| D _ { S , x } [ f ' ] \\| _ 2 ^ q . \\end{align*}"}
+{"id": "1724.png", "formula": "\\begin{align*} \\begin{cases} \\nu ( x ) = \\Psi ( \\nu , \\mu ) ( x ) \\\\ \\mu ( y ) = \\Phi ( \\nu , \\mu ) ( y ) . \\end{cases} \\end{align*}"}
+{"id": "1394.png", "formula": "\\begin{align*} \\omega _ 0 = \\theta \\wedge J \\theta - d J \\theta . \\end{align*}"}
+{"id": "2547.png", "formula": "\\begin{align*} M = \\sigma _ { j _ r | j _ r } \\circ \\dots \\circ \\sigma _ { j _ 1 | j _ 1 } ( N ) , \\end{align*}"}
+{"id": "2337.png", "formula": "\\begin{align*} \\Delta _ { h } ( r , z , t , x ) = & u ( r , z + h ) v ^ { ( r , z + h ) } ( t , x ) - u ( r , z ) v ^ { ( r , z ) } ( t , x ) \\\\ = & \\big ( u ( r , z + h ) - u ( r , z ) ) v ^ { ( r , z ) } ( t , x ) + u ( r , z + h ) \\big ( v ^ { ( r , z + h ) } ( t , x ) - v ^ { ( r , z ) } ( t , x ) \\big ) . \\end{align*}"}
+{"id": "5249.png", "formula": "\\begin{align*} R _ { D \\ ! F } = \\left ( \\begin{array} { c c c c } x = 0 & 0 & a + c + 1 & 2 a + 2 c + g + 2 \\\\ x = 1 & 0 & b + c + 1 & 2 b + 2 c + g + 2 \\\\ x = \\infty & \\ - 2 c \\ & \\ - a - b - 2 c - g - 1 \\ & \\ - 2 a - 2 b - 2 c - g - 2 \\end{array} \\right ) . \\end{align*}"}
+{"id": "1477.png", "formula": "\\begin{align*} & \\int _ \\Omega f ^ { ' } _ 0 ( U _ { \\mu _ i , \\xi _ i } ) \\psi ^ l _ { \\mu _ i , \\xi _ i } ( P \\psi ^ h _ { \\mu _ j , \\xi _ j } - \\psi ^ h _ { \\mu _ j , \\xi _ j } ) d x \\\\ & \\leq | f ^ { ' } _ 0 ( U _ { \\mu _ i , \\xi _ i } ) | _ { \\frac { n } { 2 } } | \\psi ^ l _ { \\mu _ i , \\xi _ i } | _ { \\frac { 2 n } { n - 2 } } | P \\psi ^ h _ { \\mu _ j , \\xi _ j } - \\psi ^ h _ { \\mu _ j , \\xi _ j } | _ { \\frac { 2 n } { n - 2 } } = o \\bigg ( \\Big ( \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big ) \\bigg ) . \\end{align*}"}
+{"id": "8680.png", "formula": "\\begin{align*} \\Delta _ { 0 } = 0 , \\Delta _ { 1 , 2 } = \\Delta _ { 1 , 3 } = 0 \\implies \\Delta _ { 1 } = \\Delta _ { 1 , 1 } . \\end{align*}"}
+{"id": "9167.png", "formula": "\\begin{align*} \\begin{array} { c c l } y _ { [ 3 ] } ^ { 1 } & = & v ^ { 1 } \\\\ y _ { [ 2 ] } ^ { 2 } & = & v ^ { 2 } \\end{array} \\end{align*}"}
+{"id": "2620.png", "formula": "\\begin{align*} f _ { 1 , m , \\sharp } ( x , y ) = \\sum _ { n = 1 } ^ { \\mathcal { N } _ 1 } h _ { 1 , n , m } ( x , y ) e ^ { i \\alpha _ { n , m } ( y ) x } \\end{align*}"}
+{"id": "3683.png", "formula": "\\begin{align*} \\begin{cases} A ( ( - \\Delta ) ^ \\alpha ) u = F & \\Omega , \\\\ u = f & \\Omega ^ c \\end{cases} \\end{align*}"}
+{"id": "3272.png", "formula": "\\begin{align*} \\rho ( j , w ) = | w | ^ 2 - R _ { j } ^ 2 \\left ( \\frac { w } { | w | } \\right ) \\end{align*}"}
+{"id": "6468.png", "formula": "\\begin{align*} f ( x ) = e ^ { - z | x | ^ 2 + \\xi \\cdot x } , \\end{align*}"}
+{"id": "1323.png", "formula": "\\begin{align*} \\frac { T _ i ^ { \\infty } } { T _ i ^ { \\infty } - T _ i ( u ) } \\frac { e ^ { \\rho _ i ( u ) } } { \\theta _ i } = e ^ * _ i ( u ) . \\end{align*}"}
+{"id": "6890.png", "formula": "\\begin{align*} w \\mapsto ( ( D F ) ( r , \\lambda ' , 0 ) ) ( 0 , 0 , w ) = F ( r , \\lambda ' , w ) \\end{align*}"}
+{"id": "3276.png", "formula": "\\begin{align*} \\rho _ { \\overline { w } } ( 0 , 0 ) = - i \\lambda e ^ { i \\pi \\beta _ 1 } \\end{align*}"}
+{"id": "7959.png", "formula": "\\begin{align*} H _ 0 ( \\nabla _ \\xi H ( \\xi ) ) = 1 \\end{align*}"}
+{"id": "5120.png", "formula": "\\begin{align*} \\left \\| D ^ 2 v \\right \\| ^ { p _ 0 } _ { L ^ { q _ 0 } ( B _ \\rho ) } = \\left \\| D ^ 2 v \\cdot \\mathbf { 1 } _ { B _ \\rho } \\right \\| ^ { p _ 0 } _ { L ^ { q _ 0 } ( B _ r ) } . \\end{align*}"}
+{"id": "825.png", "formula": "\\begin{gather*} \\# \\mathcal { Z } _ { M , N } = \\sum _ { k + l = 2 } ^ n p _ k ( M ) p _ l ( N ) = \\sum _ { k = 1 } ^ { n - 1 } \\left ( p _ k ( M ) \\sum _ { l = 1 } ^ { n - k } p _ l ( N ) \\right ) \\\\ \\leq C \\sum _ { k = 1 } ^ { n - 1 } M ^ { k - 1 } N ^ { n - k } \\leq C ( M + N ) ^ { n } \\end{gather*}"}
+{"id": "5200.png", "formula": "\\begin{align*} C \\in \\mathsf { C } \\mathbb { P } \\qquad \\Rightarrow \\qquad \\bigcup C = X . \\end{align*}"}
+{"id": "2464.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } & \\dfrac { 1 } { 2 c ^ 2 m _ 1 } \\partial _ t ^ 2 \\phi - \\dfrac { 1 } { 2 m _ 1 } \\Delta \\phi + \\dfrac { m _ 1 c ^ 2 } { 2 } \\phi = - a _ 1 \\psi \\overline { \\phi } , \\\\ \\\\ & \\dfrac { 1 } { 2 c ^ 2 m _ 2 } \\partial _ t ^ 2 \\psi - \\dfrac { 1 } { 2 m _ 2 } \\Delta \\psi + \\dfrac { m _ 2 c ^ 2 } { 2 } \\psi = - a _ 2 \\phi ^ 2 . \\end{array} \\right . \\end{align*}"}
+{"id": "6955.png", "formula": "\\begin{align*} b _ { \\delta , T } ( \\ell ) = b _ { \\delta , T } ( \\ell _ { \\sf p } ) + b _ { \\delta , T } ( \\ell _ { \\sf q } ) \\end{align*}"}
+{"id": "758.png", "formula": "\\begin{align*} \\mathrm { d } \\mu _ g = \\sqrt { \\mathrm { d e t } ( g _ { i j } ) } \\mathrm { d } A = \\phi ^ { n - 1 } D \\mathrm { d } A , \\\\ \\end{align*}"}
+{"id": "5428.png", "formula": "\\begin{align*} y ' _ 1 ( t ) = 0 . 1 y _ 1 ( t ) - 0 . 3 y _ 1 ( t ) y _ 2 ( t ) , y ' _ 2 ( t ) = 0 . 5 ( y _ 1 ( t ) - 1 ) y _ 2 ( t ) \\end{align*}"}
+{"id": "3278.png", "formula": "\\begin{align*} \\rho ( t , w ) = A t + { \\rm I m } ( e ^ { i \\pi ( 1 - \\beta _ 1 ) } w ) + B t ^ 2 + C | w | ^ 2 + \\end{align*}"}
+{"id": "8736.png", "formula": "\\begin{align*} \\| \\mathcal { T } _ { 1 , l } ^ { ( a _ 1 , a _ 2 ) } - \\mathcal { T } _ { 2 , l } ^ { ( a _ 1 , a _ 2 ) } \\| _ { \\rm F } ^ 2 = \\sum _ { k _ 1 , k _ 2 } s _ { k _ 1 , k _ 2 } ^ { l - 2 a _ 1 } s _ { k _ 1 , k _ 1 } ^ { a _ 1 } s _ { k _ 2 , k _ 2 } ^ { a _ 1 } ( \\mu _ { k _ 1 , l } ^ { ( 1 ) } - \\mu _ { k _ 1 , l } ^ { ( 2 ) } ) ( \\mu _ { k _ 2 , l } ^ { ( 1 ) } - \\mu _ { k _ 2 , l } ^ { ( 2 ) } ) . \\end{align*}"}
+{"id": "2671.png", "formula": "\\begin{align*} \\delta '' ( x ) & = \\sigma '' ( x ) - t _ { \\alpha } '' ( x ) \\\\ & = w _ { 1 } ( x ) \\delta '' _ { j - 1 } + w _ { 2 } ( x ) \\delta '' _ { j } + z _ { 1 } ( x ) \\sigma '' _ { j - 1 } + z _ { 2 } ( x ) \\sigma '' _ { j } , \\end{align*}"}
+{"id": "4280.png", "formula": "\\begin{align*} \\begin{cases} \\rho _ t + ( \\rho u ) _ x = 0 , \\\\ \\rho u _ t + \\rho u u _ x + p _ x = S _ x , \\\\ E _ t + ( u E + p u + q - S u ) _ x = 0 . \\end{cases} \\end{align*}"}
+{"id": "3689.png", "formula": "\\begin{align*} \\begin{cases} \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) u = 0 & \\mathbb { R } ^ n , \\\\ u = 0 & \\Omega ^ c , \\end{cases} \\end{align*}"}
+{"id": "5795.png", "formula": "\\begin{align*} \\frac { f ' } { f } = c A ( z ) ^ { 1 / k } - \\frac { k - 1 } { 2 k } \\frac { A ' ( z ) } { A ( z ) } + O ( r ^ { - 2 } ) , z \\in \\Omega , c ^ k = - 1 , \\end{align*}"}
+{"id": "550.png", "formula": "\\begin{align*} U _ { \\varepsilon } ( t , \\xi ) : = \\left ( \\begin{array} { c } i \\langle \\xi \\rangle \\widehat { u } _ { \\varepsilon } ( t , \\xi ) \\\\ \\partial _ { t } \\widehat { u } _ { \\varepsilon } ( t , \\xi ) \\end{array} \\right ) , U _ { 0 } ( \\xi ) : = \\left ( \\begin{array} { c } i \\langle \\xi \\rangle \\widehat { u } _ { 0 } ( \\xi ) \\\\ \\widehat { u } _ { 1 } ( \\xi ) \\end{array} \\right ) , \\end{align*}"}
+{"id": "5458.png", "formula": "\\begin{align*} \\Pr ( \\Phi ( \\mathcal { A } ) > 0 ) = 1 - \\exp \\left ( - 2 \\pi \\lambda R _ { m i n } R _ S \\right ) . \\end{align*}"}
+{"id": "8823.png", "formula": "\\begin{align*} d u _ n + A u _ n \\ , d t = f _ n ( u _ n ) \\ , d t + \\sigma ( u _ n ) B \\ , d W , u _ n ( 0 ) = u _ 0 , \\end{align*}"}
+{"id": "4665.png", "formula": "\\begin{align*} 1 < l _ 1 \\leq \\frac { F ' _ 1 ( s ) s } { F _ 1 ( s ) } \\leq m _ 1 : = \\sup _ { 0 < s < \\delta } \\left ( 2 + \\frac { 1 } { \\log s } \\right ) \\leq 2 , \\end{align*}"}
+{"id": "2591.png", "formula": "\\begin{align*} \\bar { u } ( x ) : = \\sup \\{ L ( x ) : L \\ , \\ L \\leq u \\ \\ U , \\ D L \\in \\Omega ^ * \\} . \\end{align*}"}
+{"id": "7002.png", "formula": "\\begin{align*} \\nu _ i ( f ) = \\min _ { 1 \\leq j \\leq r } \\left \\{ v ( b _ j ) + \\sum _ { k \\in I } \\lambda _ j ( Q _ k ) \\nu ( Q _ k ) \\right \\} . \\end{align*}"}
+{"id": "3509.png", "formula": "\\begin{align*} Y ' ( t ) = A ( t , \\lambda ) Y ( t ) , A ( t , \\lambda ) = \\begin{bmatrix} 0 & I \\\\ R ( t ) - \\lambda I & 0 \\end{bmatrix} \\end{align*}"}
+{"id": "4715.png", "formula": "\\begin{align*} \\nabla \\vect u | _ { \\partial \\Omega } = \\partial _ { \\vec { \\vect n } } \\vect u | _ { \\partial \\Omega } \\otimes \\vec { \\vect n } . \\end{align*}"}
+{"id": "3601.png", "formula": "\\begin{align*} L ( f ) & \\ = \\ m + 1 - L ( q ^ { \\beta } ) + [ r ( p ) \\ < \\ 1 0 ^ { L ( f ) + L ( q ^ { \\beta } ) - 1 } ] \\\\ & \\ \\le \\ m + 1 - \\beta ( \\ell - 1 ) - 1 + 1 \\ = \\ m + 1 - \\beta ( \\ell - 1 ) . \\end{align*}"}
+{"id": "7346.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { m + 1 } \\sum _ { i = 0 } ^ m w _ { n + i } - w \\right | & < C \\left | \\frac { 1 } { m + 1 } \\sum _ { i = 0 } ^ m x _ { n + i } - x + \\frac { 1 } { m + 1 } \\sum _ { i = 0 } ^ m y _ { n + i } - y \\right . \\\\ & + \\left . \\frac { 1 } { m + 1 } \\sum _ { i = 0 } ^ m z _ { n + i } - z \\right | , \\end{align*}"}
+{"id": "6592.png", "formula": "\\begin{align*} \\begin{aligned} P _ b = & \\Pr ( - \\mu _ D / \\sigma _ D ) = Q \\left ( \\sqrt { \\frac { P _ s \\zeta ^ 2 | \\hat \\eta - \\eta | ^ 2 } { 2 ( N _ 0 + P _ s ( 1 - \\zeta ^ 2 ) \\sigma _ e ^ 2 L ) } } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "2920.png", "formula": "\\begin{align*} \\gamma = \\begin{bmatrix} a \\beta & x \\\\ c \\beta & y \\end{bmatrix} \\begin{bmatrix} ( \\alpha \\beta ) ^ { - 1 } & 0 \\\\ 0 & D \\beta \\end{bmatrix} \\begin{bmatrix} 1 & \\alpha ( b y - x d ) \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} \\alpha & 0 \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "493.png", "formula": "\\begin{align*} Q _ { B } ^ { c } ( Y _ { ( m , n ) } ) = Q _ { B } ( R e \\{ Y _ { ( m , n ) } \\} ) + j Q _ { B } ( I m \\{ Y _ { ( m , n ) } \\} ) . \\end{align*}"}
+{"id": "7497.png", "formula": "\\begin{align*} \\Phi ( \\sigma ) = \\psi _ { S } ( \\Phi ( \\pi ) ) = ( \\psi _ { x _ { k } } \\circ \\cdots \\circ \\psi _ { x _ { 2 } } \\circ \\psi _ { x _ { 1 } } ) ( \\Phi ( \\pi ) ) . \\end{align*}"}
+{"id": "6662.png", "formula": "\\begin{align*} - ( - \\Delta ) ^ s \\psi ( r ) = - \\check C F ( a , b , c , - r ^ 2 ) \\textrm { f o r a l l } \\ ; \\ ; r > 1 \\ , , \\end{align*}"}
+{"id": "5807.png", "formula": "\\begin{align*} \\frac { f ^ { ( k ) } } { f } = F ^ k + \\frac { k ( k - 1 ) } { 2 } F ^ { k - 2 } F ' + P _ { k - 2 } ( F ) = - A , \\end{align*}"}
+{"id": "2201.png", "formula": "\\begin{align*} \\Lambda _ { \\alpha } = { \\rm d i a g } \\big ( \\lambda _ i ^ { ( k , \\alpha ) } \\big ) _ { i = 1 } ^ { n } \\textrm { w i t h } \\lambda _ i ^ { ( k , \\alpha ) } = \\lambda _ k - 2 \\alpha ^ { \\frac { 1 } { n } } \\theta _ n ^ { i } + \\lambda _ k \\alpha ^ { \\frac { 2 } { n } } \\theta _ n ^ { 2 i } \\textrm { a n d } \\theta _ n = \\exp \\Big ( \\frac { 2 \\pi { \\bf i } } { n } \\Big ) . \\end{align*}"}
+{"id": "4124.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } \\alpha _ j y ^ j + y ^ { n } = 0 . \\end{align*}"}
+{"id": "4065.png", "formula": "\\begin{align*} S _ { \\mathrm { m a i n } } \\gg \\log p \\sum _ { L = 1 } ^ { \\infty } \\phi ( L ) | y _ L | ^ 2 , \\end{align*}"}
+{"id": "7443.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { + \\infty } \\sum \\limits _ { p = 0 } ^ { 1 } \\varepsilon ^ { p \\alpha + k - 1 } \\ , \\mathfrak { b } _ { p \\alpha + k - 1 } \\end{align*}"}
+{"id": "8351.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ m \\alpha _ j \\nabla u _ j \\Big \\| _ { p , q , \\mu } ^ q & \\leq p \\sum _ { j = 1 } ^ m \\int _ 0 ^ \\infty t ^ { q - 1 } ( \\nabla u _ j ) _ { * \\mu } \\Big ( \\frac { t } { | \\alpha _ j | } \\Big ) ^ \\frac { q } { p } \\dd t \\\\ & = p \\sum _ { j = 1 } ^ m | \\alpha _ j | ^ q \\int _ 0 ^ \\infty t ^ { q - 1 } ( \\nabla u _ j ) _ { * \\mu } ( t ) ^ \\frac { q } { p } \\dd t = \\sum _ { j = 1 } ^ m | \\alpha _ j | ^ q \\| \\nabla u _ j \\| _ { p , q , \\mu } ^ q \\end{align*}"}
+{"id": "4184.png", "formula": "\\begin{align*} m ( t ; \\lambda ) = \\frac { 4 \\pi } { 3 } \\cdot \\begin{cases*} ( \\frac { 1 + t } { 1 + \\lambda } ) ^ 3 & i f $ 0 < \\lambda < \\frac { 1 } { t } $ , \\\\ ( \\frac { 1 - t } { \\lambda - 1 } ) ^ 3 & i f $ \\frac { 1 } { t } \\leq \\lambda < 2 $ , \\\\ 0 & o t h e r w i s e . \\end{cases*} \\end{align*}"}
+{"id": "1837.png", "formula": "\\begin{align*} \\chi ^ \\perp ( p , p - k ) 2 \\mathrm { R e } \\ , \\int _ 0 ^ t \\int _ 0 ^ { t _ 1 } G _ k ( t _ 2 ) e ^ { i t _ 2 ( E _ p - E _ { p - k } ) } \\d t _ 2 \\d t _ 1 = 2 \\pi t \\ , \\alpha ^ P _ t ( p , k ) \\end{align*}"}
+{"id": "3105.png", "formula": "\\begin{align*} S ^ { o } _ B [ n , k ] : = [ 2 k ] _ q \\ , S ^ { o } _ B [ n - 1 , k - 1 ] + [ 2 k + 1 ] _ q \\ , S ^ { o } _ B [ n - 1 , k ] \\end{align*}"}
+{"id": "3852.png", "formula": "\\begin{align*} c _ { \\ell } ( s _ \\ell , s _ \\ell ' ) & = | y _ \\ell - y _ \\ell ' | + \\| x _ \\ell - x _ \\ell ' \\| _ 2 . \\end{align*}"}
+{"id": "3254.png", "formula": "\\begin{align*} I _ 1 & \\le \\Bigg ( \\int _ { I ( x _ 0 , C r ) } \\bigg \\{ \\sum _ { k = k _ 0 + 1 } ^ \\infty { \\mathfrak R } ^ { - k \\beta } \\big | D _ k ( a ) ( x ) \\big | \\bigg \\} ^ { p ' } d \\omega ( x ) \\Bigg ) ^ { 1 / p ' } \\\\ & \\lesssim \\Bigg ( \\int _ { I ( x _ 0 , C r ) } \\bigg \\{ \\sum _ { k = k _ 0 + 1 } ^ \\infty { \\mathfrak R } ^ { - k \\beta } \\bigg \\} ^ { p ' } d \\omega ( x ) \\Bigg ) ^ { 1 / p ' } \\\\ & \\leq C r ^ \\beta \\omega ( I ) ^ { 1 / p ' } . \\end{align*}"}
+{"id": "6258.png", "formula": "\\begin{align*} Q _ { \\Sigma } [ \\phi ] = \\int _ { \\Sigma } | \\nabla \\phi | ^ 2 - \\left ( | A | ^ 2 + \\mathrm { R i c } ( \\nu , \\nu ) \\right ) \\phi ^ 2 - \\int _ { \\partial \\Sigma } h ( \\nu , \\nu ) \\phi ^ 2 , \\end{align*}"}
+{"id": "205.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c ) = 1 \\\\ a , b , c \\geq 1 } } \\left ( \\frac { 1 } { 1 - x ^ a y ^ b z ^ c } \\right ) ^ { \\frac { c ^ 3 } { a b ^ 3 } } = \\exp \\left \\{ L i _ 1 ( x ) L i _ 3 ( y ) L i _ { - 3 } ( z ) \\right \\} \\end{align*}"}
+{"id": "1040.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\bigg ( \\frac { - 1 } { 4 } \\bigg ) ^ k \\frac { ( \\frac { 1 } { 2 } ) _ k ( d ) _ k ^ 2 ( 1 - d ) _ k ^ 2 } { ( 1 ) _ { k } ^ 3 ( \\frac { 1 } { 2 } + d ) _ { k } ( \\frac { 3 } { 2 } - d ) _ { k } } ( d - d ^ 2 + 2 k + 5 k ^ 2 ) = \\frac { 1 - 2 d } { \\pi } \\tan ( d \\pi ) . \\end{align*}"}
+{"id": "5345.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ q \\leq ( 1 / \\rho ) ^ d \\gamma ' = \\gamma \\left ( 9 9 \\cdot 2 \\ , r \\sqrt { q } \\sqrt { \\frac { \\log ( 1 / \\gamma ) } { d } } \\right ) ^ d \\leq \\gamma \\left ( 2 0 0 r \\sqrt { q } \\cdot \\max \\left \\{ \\sqrt { q } , \\sqrt { \\log ( 1 / \\gamma ) } \\right \\} \\right ) ^ d , \\end{align*}"}
+{"id": "6141.png", "formula": "\\begin{align*} \\d ( x , y ) = \\sum _ { j = 0 } ^ { n - 1 } ( j + 1 ) ( x - y ) ^ 2 \\cdot y ^ j \\cdot x ^ { n - 1 - j } . \\end{align*}"}
+{"id": "5140.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = 1 } ^ \\infty g _ n f \\Big \\| _ { L ^ 1 ( \\Omega ) } \\leq \\sum _ { n = 1 } ^ \\infty \\| g _ n f \\| _ { L ^ 1 ( \\Omega ) } \\leq \\| f \\| _ X \\sum _ { n = 1 } ^ \\infty \\| g _ n \\| _ { X ' } . \\end{align*}"}
+{"id": "5509.png", "formula": "\\begin{align*} \\int _ { \\tilde { X } } f d \\tilde { \\boldsymbol { \\beta } } _ { \\tilde { \\nu } } ( w ) = \\int _ { G / \\Gamma } \\int _ { X } f _ { z } d \\boldsymbol { \\beta } _ { \\nu } \\left ( \\tau ( z ) ^ { - 1 } . w \\right ) d m _ { G / \\Gamma } ( z ) , \\mbox { w h e r e } w \\in G / P , f \\in L ^ { \\infty } \\left ( \\tilde { X } \\right ) , \\end{align*}"}
+{"id": "1803.png", "formula": "\\begin{align*} \\delta _ t ( x + \\lambda y ) = \\ , \\frac { 2 t } { \\pi } \\delta _ { x , 0 } \\ , + \\mathcal { O } ( 1 / t x ^ 2 ) + \\mathcal { O } ( t ^ 3 \\lambda ^ 2 | y | ^ 3 ) \\ . \\end{align*}"}
+{"id": "2938.png", "formula": "\\begin{align*} S = \\begin{bmatrix} 0 & - 1 \\\\ 1 & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "531.png", "formula": "\\begin{align*} \\left ( u _ { \\xi } , u _ { \\eta } \\right ) : = \\left \\{ \\begin{array} { c c } 1 , & \\xi = \\eta , \\\\ 0 , & \\xi \\neq \\eta , \\end{array} \\right . \\end{align*}"}
+{"id": "3144.png", "formula": "\\begin{align*} \\sum _ { j = i + 1 } ^ { m - k } \\binom { j } { i } p ( m , k , j ) = ( k + 1 ) p ( m - 1 , k + 1 , i ) + ( m + k ) p ( m - 1 , k , i ) . \\end{align*}"}
+{"id": "2116.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ { m } a ^ { ( i - 1 ) } _ n \\cdot R _ i \\colon n \\ge 0 \\right \\} . \\end{align*}"}
+{"id": "912.png", "formula": "\\begin{align*} S & \\ll \\widehat { Y } ^ { ( n + 3 ) / 2 + \\varepsilon } \\sum _ { d = d _ 1 d _ 2 } | d _ 1 | \\sum _ { d _ 2 = e f } | e | | f | \\sum _ { | t _ 2 | \\leq \\widehat { Y } } \\sum _ { | t _ 1 | \\leq \\frac { \\widehat { Y } } { | t _ 2 | } } | t _ 1 | ^ { - 1 / 2 } \\left ( \\widehat { V } ^ n + \\widehat { Y } ^ { n / 3 } \\right ) \\\\ & \\ll \\widehat { Y } ^ { ( n + 3 ) / 2 + \\varepsilon } | d | \\left ( \\widehat { V } ^ n + \\widehat { Y } ^ { n / 3 } \\right ) , \\end{align*}"}
+{"id": "7354.png", "formula": "\\begin{align*} \\Pr [ \\hat { M } = M ] \\leq \\inf _ { 0 \\leq \\gamma \\leq 1 } \\sum _ k \\binom { n } { k } \\frac { s ! } { ( k + s ) ! } ( 1 - Q ( ( 1 - \\gamma ) \\mu ) ) ^ { n - k } ( 1 - Q ( \\gamma \\mu ) ) ^ { k ( k + s ) } e ^ { k ( 2 \\gamma - 1 ) \\mu ^ 2 / 2 } \\end{align*}"}
+{"id": "8537.png", "formula": "\\begin{align*} \\tilde { \\zeta } _ { \\infty , p ^ { \\prime } } ( s , c ) : = \\lim _ { p \\rightarrow \\infty } \\tilde { \\zeta } _ { p , p ^ { \\prime } } ( s , c ) = \\sum _ { m , n \\neq 0 } \\frac { p ^ { \\prime 2 } + \\lambda _ { n } ^ { \\prime 2 } } { p ^ { \\prime } \\left ( p ^ { \\prime } + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { \\prime 2 } } \\frac { 1 } { \\left ( m ^ { 2 } + c \\lambda _ { n } ^ { \\prime 2 } \\right ) ^ { s } } = \\zeta _ { \\infty , p ^ { \\prime } } ( s , c ) , \\ , \\ , \\ , \\ , \\ , ( s ) > 1 . \\end{align*}"}
+{"id": "7536.png", "formula": "\\begin{align*} L \\log L ( \\Omega ) : = \\left \\{ f \\in L _ { l o c } ^ 1 ( \\Omega ) : \\abs { f } _ { L \\log L ( \\Omega ) } < \\infty \\right \\} , \\end{align*}"}
+{"id": "2446.png", "formula": "\\begin{align*} ( f * g ) ( \\gamma _ 0 ) = \\int _ { G _ { t ( \\gamma _ 0 ) } } f ( \\gamma ^ { - 1 } ) g ( \\gamma \\gamma _ 0 ) \\ d \\lambda _ { t ( \\gamma _ 0 ) } = \\int _ { G _ { s ( \\gamma _ 0 ) } } f ( \\gamma _ 0 \\gamma ^ { - 1 } ) g ( \\gamma ) \\ d \\lambda _ { s ( \\gamma _ 0 ) } . \\end{align*}"}
+{"id": "4843.png", "formula": "\\begin{align*} \\phi ( w ) = \\begin{cases} 1 & \\mbox { i f } ( \\prod _ { i = 1 } ^ m a ( i ) \\ast g _ w ( t ( i ) ) ) \\ast a ( m + 1 ) \\in A _ 1 \\\\ 2 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
+{"id": "3196.png", "formula": "\\begin{align*} \\lim _ { a \\to \\infty } \\sup _ { x \\in e M _ { + } e , ~ x \\neq 0 } \\abs { \\frac { ( M _ a ( \\mu ) - \\bar { \\mu } ) ( x ) } { \\tau ( x ) } } = 0 . \\end{align*}"}
+{"id": "3032.png", "formula": "\\begin{align*} \\phi ( E _ { i i } \\otimes Y E _ { j j } Y ^ * ) = U _ Y ( E _ { i i } \\otimes E _ { j j } ) V _ Y ^ * \\end{align*}"}
+{"id": "4076.png", "formula": "\\begin{align*} \\langle T ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) , f \\rangle = \\langle z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace , T f \\rangle = 0 . \\end{align*}"}
+{"id": "4180.png", "formula": "\\begin{align*} m ( \\lambda ) = \\int _ 0 ^ \\infty m ( t ; \\lambda ) \\ , d t . \\end{align*}"}
+{"id": "8845.png", "formula": "\\begin{align*} \\norm [ \\big ] { ( M _ T ^ * ) ^ { 1 / q } } _ { L ^ p ( \\Omega ) } = \\norm [ \\big ] { ( M _ T ^ * ) } ^ { 1 / q } _ { L ^ { p / q } ( \\Omega ) } \\lesssim q ^ { 1 / q } \\norm [ \\big ] { \\overline { v } _ \\lambda - \\overline { v } _ \\mu } _ { L ^ p ( \\Omega ; L ^ { 2 q } ( 0 , t ; L ^ q _ x H ) ) } , \\end{align*}"}
+{"id": "4440.png", "formula": "\\begin{align*} K _ { \\omega , \\mathbf { c } } ( U _ 0 ) & = 2 L ( U _ 0 ) + 3 N ( U _ 0 ) + 2 \\omega Q ( U _ 0 ) + \\mathbf { c } \\cdot \\mathbf { P } ( U _ 0 ) \\\\ & = 2 \\omega Q ( U _ 0 ) + \\sqrt { \\omega } \\ \\mathbf { c } _ 0 \\cdot \\mathbf { P } ( U _ 0 ) + 2 L ( U _ 0 ) + 3 N ( U _ 0 ) \\end{align*}"}
+{"id": "2379.png", "formula": "\\begin{align*} ( 1 - u ) ( 1 - r s ) & = \\underbrace { ( 1 - u ) ( 1 - e _ 1 r e _ 1 s e _ 1 ) } _ { = 1 } - ( 1 - \\underbrace { u } _ { = u e _ 1 } ) ( \\sum _ { i > 1 } e _ i r e _ 1 s e _ 1 ) \\\\ & = 1 - \\sum _ { i > 1 } e _ i r e _ 1 s e _ 1 . \\end{align*}"}
+{"id": "8386.png", "formula": "\\begin{align*} \\bigl | f _ 1 ^ j ( v ) - f ^ j ( v ) \\bigr | _ h \\leq \\sum _ { k = 1 } ^ j C ( j , k ) | f | ^ { j - k } _ h | f - f _ 1 | ^ k _ h | v | _ h \\leq | f - f _ 1 | _ h ( 1 + | f | _ h ) ^ j | v | _ h . \\end{align*}"}
+{"id": "6281.png", "formula": "\\begin{align*} H _ { \\xi } \\cap H _ { \\xi ' } \\subseteq & \\big ( G _ \\xi \\cap G _ { \\xi ' } \\cap [ \\beta , \\gamma ) \\big ) \\times X \\\\ \\subseteq & \\big ( R \\cap [ \\beta , \\gamma ) \\big ) \\times X = \\emptyset . \\end{align*}"}
+{"id": "6015.png", "formula": "\\begin{align*} \\theta _ { Y ^ { \\ast } , L } ( f ) ( y ) = \\int _ { Y ^ { \\ast } / Y ^ { \\ast } \\cap L } f ( [ \\dot { y } ^ { \\ast } , 0 ] + [ y , 0 ] ) d \\dot { y } ^ { \\ast } , \\end{align*}"}
+{"id": "3327.png", "formula": "\\begin{align*} \\rho ( j , w ) = | w | ^ 2 - R _ { j } ^ 2 \\left ( \\frac { w } { | w | } \\right ) \\end{align*}"}
+{"id": "5127.png", "formula": "\\begin{align*} V ( D ^ 2 u ( x _ 0 ) ) \\equiv V ( \\lambda _ 1 , \\dots , \\lambda _ n ) = \\Bigg [ ( 1 + \\lambda _ 1 ^ 2 ) . . . ( 1 + \\lambda _ n ^ 2 ) \\bigg ] ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "8783.png", "formula": "\\begin{align*} & ( 4 u + 2 t + 5 ) b _ 3 = ( 4 t + 3 ) c _ 3 + 3 b _ 2 + c _ 2 \\end{align*}"}
+{"id": "4911.png", "formula": "\\begin{align*} \\overline { F } ( k , t , \\mathbf { p } _ { n + 1 } ) \\overset { } { = } \\Pr ( X _ t > k \\ , \\vert \\ , X _ 0 = 0 , \\mathbf { p } _ { n + 1 } ) , k \\in S , \\end{align*}"}
+{"id": "695.png", "formula": "\\begin{align*} \\int _ { ( T ^ { ( l ) } ) ^ i J _ l } | S ( l ) D \\varphi ( x ) | d x = O ( e ^ { ( - \\lambda _ 1 ( 1 - a ) + \\tau ) k } ) ( c _ 0 + p _ a ) ( D \\varphi ) . \\end{align*}"}
+{"id": "9074.png", "formula": "\\begin{align*} - M _ { x y } f ( x ) f ( y ) ( a ( x ) - a ( y ) ) ^ 2 = 0 . \\end{align*}"}
+{"id": "8770.png", "formula": "\\begin{align*} \\begin{aligned} & \\lambda _ { i j } v _ { i j } = 0 , \\forall i , j \\in \\{ 1 , \\ldots , n \\} , \\ i \\neq j . \\\\ & \\gamma _ i \\left ( \\sum _ { j \\neq i } v _ { i j } - x _ 0 { _ i } \\right ) = 0 , \\forall i \\in \\{ 1 , \\ldots , n \\} . \\end{aligned} \\end{align*}"}
+{"id": "1791.png", "formula": "\\begin{align*} \\pi _ k ( X _ F , x ) ^ 0 = \\varprojlim _ { F ' / F } \\pi _ k ( X _ { F ' } , x _ { F ' } ) \\ ; . \\end{align*}"}
+{"id": "1658.png", "formula": "\\begin{align*} g ( h _ i Z , Q X ) = \\sum \\nolimits _ { j } \\eta ^ j ( X ) \\ , \\eta ^ j ( Z ) - \\eta ^ i ( X ) \\ , \\bar \\eta ( Z ) , \\end{align*}"}
+{"id": "645.png", "formula": "\\begin{align*} \\frac { d x } { d t } = \\frac { \\frac { \\partial H } { \\partial y } ( x , y ) } { V ( x , y ) } , \\frac { d y } { d t } = - \\frac { \\frac { \\partial H } { \\partial x } ( x , y ) } { V ( x , y ) } \\end{align*}"}
+{"id": "3755.png", "formula": "\\begin{align*} P _ 0 U ( \\vec { a } , 1 ) B _ s ^ n 1 = & P _ 0 \\psi ^ { \\beta _ 1 , n } ( g _ 1 ) _ { U ( \\vec { a } ) } \\psi ^ { \\beta _ 2 , n } ( g _ 2 ) _ { U ( \\vec { a } ) } \\cdots \\psi ^ { \\beta _ { s - 1 } , n } ( g _ { s - 1 } ) _ { U ( \\vec { a } ) } U ( \\vec { a } , 1 ) \\psi ^ { \\beta _ s , n } ( g _ s ) 1 \\\\ = & e ^ { i [ a ^ 0 \\hat { H } - a ^ 1 \\hat { P } ] } P _ 0 B _ s ^ { n , \\vec { a } } 1 = e ^ { i [ a ^ 0 \\hat { H } - a ^ 1 \\hat { P } ] } \\left [ B _ s ^ { n , \\vec { a } } 1 - \\left \\langle B _ s ^ n 1 , 1 \\right \\rangle 1 \\right ] , \\end{align*}"}
+{"id": "5147.png", "formula": "\\begin{align*} C : = \\sup _ { k \\geq 1 } \\| h _ k \\| _ Y < \\infty . \\end{align*}"}
+{"id": "6915.png", "formula": "\\begin{align*} \\Phi ( t ) : = e ^ { - t ^ 2 / 2 } \\end{align*}"}
+{"id": "336.png", "formula": "\\begin{align*} \\prod _ { \\substack { \\gcd ( j _ 1 , j _ 2 , j _ 3 , k ) = 1 \\\\ j _ 1 , j _ 2 , j _ 3 < k \\\\ j _ 1 , j _ 2 , j _ 3 \\geq 1 ; k \\geq 2 } } \\left ( \\frac { 1 } { 1 - y ^ { j _ 1 + j _ 2 + j _ 3 } z ^ k } \\right ) ^ { \\frac { 1 } { k } } , \\end{align*}"}
+{"id": "6189.png", "formula": "\\begin{align*} P ( E ) - P ( F ) = P ( E ) - P ( H ) + P ( H ) - P ( F ) \\geq 2 \\lambda \\Bigl ( f \\big ( | E \\Delta H | \\big ) - C ( n ) | E \\Delta H | ^ { 3 } \\Big ) . \\end{align*}"}
+{"id": "997.png", "formula": "\\begin{align*} ( \\delta - n _ { F ^ i } ) ( w ) = \\sum _ { i = 1 } ^ p a _ i r - \\langle n _ { F ^ i } , g \\rangle \\le r . \\end{align*}"}
+{"id": "445.png", "formula": "\\begin{align*} s = \\frac { t } { \\lambda ^ { \\frac { 1 } { 2 - \\gamma } } } , \\ \\ y = \\frac { x } { \\lambda } . \\end{align*}"}
+{"id": "4808.png", "formula": "\\begin{align*} \\ddot \\theta = \\sin ( \\theta ) - \\varepsilon \\dot \\theta - \\cos ( \\theta ) u , \\end{align*}"}
+{"id": "8989.png", "formula": "\\begin{align*} \\zeta _ \\delta = 0 \\ , \\partial \\Omega , \\zeta _ \\delta w _ \\delta = v \\ \\ \\forall \\ , \\delta \\end{align*}"}
+{"id": "6198.png", "formula": "\\begin{align*} r ( i ) = \\max \\{ | S | : S \\subseteq V _ e ^ { ( i ) } d ( v , w ) \\neq 2 v , w \\in S \\} . \\end{align*}"}
+{"id": "6858.png", "formula": "\\begin{align*} A _ N ( i , j ) = \\left \\{ \\begin{array} { l l } 1 , & , \\\\ 0 , & . \\end{array} \\right . \\end{align*}"}
+{"id": "3508.png", "formula": "\\begin{align*} Y ( t ) = ( x ^ 1 ( t ) , \\ldots , x ^ n ( t ) , ( x ^ 1 ) ' ( t ) , \\ldots , ( x ^ n ) ' ( t ) ) . \\end{align*}"}
+{"id": "35.png", "formula": "\\begin{align*} \\mathbf { K } _ { { \\overline { \\psi } _ 0 } } ^ 1 = \\mathbf { K } _ { { \\overline { \\psi } _ 0 } } \\cap \\mathcal { G } ^ 1 ( \\widehat { \\mathbb { Z } } ) . \\end{align*}"}
+{"id": "8389.png", "formula": "\\begin{align*} f = \\sum _ { U _ 1 , U _ 2 = V ^ { ( \\nu _ 1 ) } , W ^ { ( \\nu _ 1 ) } , \\kappa ( W ^ { ( \\nu _ 1 ) } ) } f _ { U _ 1 , U _ 2 } , \\end{align*}"}
+{"id": "4633.png", "formula": "\\begin{align*} \\mbox { w e d e n o t e b y $ \\Phi $ t h e p r i m i t i v e f u n c t i o n o f $ D $ , n a m e l y } \\ , \\Phi ( u ) = \\int _ 0 ^ u D ( s ) \\ , \\dd s . \\end{align*}"}
+{"id": "3380.png", "formula": "\\begin{align*} \\frac { Q _ \\eta } { Q _ n } = \\frac { Q _ { \\eta - 1 } } { Q _ n } + \\frac { q _ \\eta } { Q _ n } \\leq \\kappa + \\frac { q _ \\eta } { Q _ \\eta } \\frac { Q _ \\eta } { Q _ n } = \\kappa ( 1 + o ( 1 ) ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\frac { Q _ \\eta } { Q _ n } \\geq \\left ( 1 + \\frac { \\gamma } { 2 } \\right ) = \\frac { \\kappa + 1 } { 2 } \\end{align*}"}
+{"id": "2382.png", "formula": "\\begin{align*} \\hat { f } ( x _ 1 , \\dots , x _ k ) : = | I | ^ { - k ( k - 1 ) } \\sum _ { a _ 1 , \\dots , a _ k \\in I ^ { k - 1 } } f ( x _ 1 + \\sum _ { j = 2 } ^ k a _ { 1 , j - 1 } x _ j , x _ 2 + \\sum _ { j = 2 } ^ k a _ { 2 , j - 1 } x _ j , \\\\ \\dots , x _ k + \\sum _ { j = 2 } ^ k a _ { k , j - 1 } x _ j ) \\end{align*}"}
+{"id": "5293.png", "formula": "\\begin{align*} A h = A ( \\sum _ { S \\subset [ n ] } \\hat h ( S ) \\chi _ S ) = \\sum _ S \\left ( - \\frac { p } { 1 - p } \\right ) ^ { | S | } \\hat h ( S ) \\chi _ S = \\sum _ { d = 0 } ^ n \\left ( - \\frac { p } { 1 - p } \\right ) ^ d h ^ { = d } . \\end{align*}"}
+{"id": "4962.png", "formula": "\\begin{align*} \\overline { F } ( k - 1 , t , n ) - \\overline { F } ( k , t , n ) \\overset { ( 1 ) } { = } f ( k , t , n ) \\overset { ( 2 ) } { = } \\frac { n } { n - k } \\Big ( \\overline { F } ( k , t + 1 , n ) - \\overline { F } ( k , t , n ) \\Big ) , \\end{align*}"}
+{"id": "4426.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( \\Psi ) = \\mu _ { \\omega , \\mathbf { c } } \\le S _ { \\omega , \\mathbf { c } } ( \\Theta ) . \\end{align*}"}
+{"id": "4871.png", "formula": "\\begin{align*} a ( \\theta ) = a ( - \\theta ) \\end{align*}"}
+{"id": "8002.png", "formula": "\\begin{align*} | \\nabla ^ 2 \\psi ^ i _ m ( x ' ) | & \\leq c \\ , \\sum _ { j = 1 } ^ N \\bigg \\{ \\big | \\nabla ^ 2 M _ m ( \\phi ^ j ) ( H ^ { i j } _ m x ' ) \\big | \\ , \\xi _ j \\big ( x ' , \\psi ^ i _ m ( x ' ) \\big ) + 1 \\bigg \\} \\\\ & \\leq c \\ , \\sum _ { j = 1 } ^ N \\bigg \\{ M _ m ( | \\nabla ^ 2 \\phi ^ j | ) ( H ^ { i j } _ m x ' ) \\ , \\xi _ j \\big ( x ' , \\psi ^ i _ m ( x ' ) \\big ) + 1 \\bigg \\} \\end{align*}"}
+{"id": "4999.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { F } ( k , t + 1 ; p , n ) & = p _ { k } ^ { } \\overline { F } ( k - 1 , t ; p , n ) + ( 1 - p _ { k } ^ { } ) \\overline { F } ( k , t ; p , n ) \\\\ & = \\frac { ( n - k ) p } { n } \\overline { F } ( k - 1 , t ; p , n ) + \\frac { n - ( n - k ) p } { n } \\overline { F } ( k , t ; p , n ) , \\end{aligned} \\end{align*}"}
+{"id": "6772.png", "formula": "\\begin{align*} \\tau ( \\Sigma ^ s f ) = \\mathrm { l c m } \\left ( \\tau ( f ) , \\tau ( \\Sigma ^ { s - 1 } [ 1 ] ) \\right ) = 3 \\cdot 2 ^ { 2 + k } \\end{align*}"}
+{"id": "2025.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ n u ^ { p \\bar p } u _ { p \\bar p j } = f _ j + f _ u u _ j , \\end{align*}"}
+{"id": "3674.png", "formula": "\\begin{align*} \\begin{cases} \\nabla \\cdot ( \\sigma \\nabla u ) = 0 & \\Omega , \\\\ u = f & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "1680.png", "formula": "\\begin{align*} \\psi ( G ) \\subseteq \\ker \\widehat { \\phi ^ { ( p ) } } = \\pi ( \\ker \\widehat { \\phi } ) . \\end{align*}"}
+{"id": "8553.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { e ^ { 2 \\pi n } } { \\left ( e ^ { 2 \\pi n } - 1 \\right ) ^ { 2 } } = \\frac { 1 } { 2 4 } - \\frac { 1 } { 8 \\pi } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { e ^ { \\pi ( 2 n - 1 ) } } { \\left ( e ^ { \\pi ( 2 n - 1 ) } + 1 \\right ) ^ { 2 } } = \\frac { 1 } { 8 \\pi } . \\end{align*}"}
+{"id": "3098.png", "formula": "\\begin{align*} S _ B [ n , k ] : = S _ B [ n - 1 , k - 1 ] + [ 2 k + 1 ] _ q \\ , S _ B [ n - 1 , k ] \\end{align*}"}
+{"id": "8146.png", "formula": "\\begin{align*} \\left . \\frac { d } { d \\alpha } \\right | _ { \\alpha = 0 } \\Gamma _ T ( \\alpha ) . \\end{align*}"}
+{"id": "8588.png", "formula": "\\begin{align*} \\bar { S } _ { j , + } ^ k ( t ) \\approx \\nu \\int _ 0 ^ t Y e ^ { \\lambda s } p _ { j , + } ^ k ( t - s ) d s = : \\hat { S } _ { j , + } ^ k ( t ) . \\end{align*}"}
+{"id": "944.png", "formula": "\\begin{align*} J ^ 0 = \\{ ( p , q ) \\in X \\times X \\setminus \\Delta _ X \\ , | \\ , X \\cdot l _ { p , q } = 3 p + q \\in Z _ 0 ( X ) \\} \\end{align*}"}
+{"id": "306.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } \\sum _ { n = 1 } ^ { \\infty } ( - x y ^ { n + 1 } + 2 n x y ^ { n + 2 } - x y ^ { n + 2 } + x y ^ 2 + x y ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "6345.png", "formula": "\\begin{align*} T : = \\left \\{ ( s , t ) \\in [ 0 , 1 ] ^ 2 \\ , : \\ , t \\in [ s - \\lambda ( s ) , s + \\lambda ( s ) ] \\right \\} . \\end{align*}"}
+{"id": "8896.png", "formula": "\\begin{align*} [ X _ 1 , \\dots , X _ j ] _ c = [ X _ 1 , \\dots , X _ j ] + \\sum _ { p \\geq j + 1 } \\sum _ { ( i _ 1 , \\dots , i _ p ) \\in \\mathbb { I } _ j ^ p } \\gamma _ { ( i _ 1 , \\dots , i _ p ) } [ X _ { i _ 1 } , \\dots , X _ { i _ p } ] . \\end{align*}"}
+{"id": "4988.png", "formula": "\\begin{align*} \\overline { S } ( k , n ) = \\begin{dcases} 2 \\ , \\overline { S } ( k , n - 1 ) & k = 0 ; \\\\ \\overline { S } ( k , n - 1 ) \\ , + \\ , \\overline { S } ( k - 1 , n - 1 ) & 1 < k < n ; \\\\ 1 \\ , & k = n ; \\end{dcases} \\end{align*}"}
+{"id": "7413.png", "formula": "\\begin{align*} Z _ { ( A , B ] , \\beta } ^ \\omega ( z ) = 1 + \\sum _ { k = 1 } ^ \\infty \\ \\sum _ { \\substack { A < n _ 1 < \\ldots < n _ k \\le B \\\\ x _ 1 , \\ldots , x _ k \\in \\Z ^ 2 } } \\ , & q _ { n _ 1 } ( x _ 1 - z ) \\ , \\xi _ \\beta ( n _ 1 , x _ 1 ) \\times \\\\ & \\times \\prod _ { j = 2 } ^ { k } q _ { n _ j - n _ { j - 1 } } ( x _ j - x _ { j - 1 } ) \\ , \\xi _ \\beta ( n _ j , x _ j ) \\ , , \\end{align*}"}
+{"id": "925.png", "formula": "\\begin{align*} g _ { k , N } ( u , v ) & = u ^ N \\ , P _ k ^ { ( - N - 2 \\nu _ 1 , - N - 2 \\nu _ 2 ) } \\left ( 1 + 2 \\frac { v } { u } \\right ) , \\\\ h _ { k , N } ^ { + } ( u , v ) & = u ^ { N - 1 } ( N + 2 \\nu _ 1 - k ) P _ { k - 1 } ^ { ( - N - 2 \\nu _ 1 , - N - 2 \\nu _ 2 + 1 ) } \\left ( 1 + 2 \\frac { v } { u } \\right ) , \\\\ h _ { k , N } ^ { - } ( u , v ) & = ( k + 1 - 2 N - 2 \\nu _ { 1 2 } ) \\ , u ^ { N - 1 } P _ k ^ { ( - N - 2 \\nu _ 1 , - N - 2 \\nu _ 2 + 1 ) } \\left ( 1 + 2 \\frac { v } { u } \\right ) . \\end{align*}"}
+{"id": "8395.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\triangle u = | \\alpha | ^ 2 e ^ { 2 u } , \\end{align*}"}
+{"id": "6105.png", "formula": "\\begin{align*} L ^ { - 1 } \\leq A ( x , y , t ) \\leq L \\quad A ( x , y , t ) = A ( y , x , t ) \\quad ( x , y , t ) \\in \\mathbb { R } ^ { n } \\times \\mathbb { R } ^ { n } \\times ( 0 , T ) \\end{align*}"}
+{"id": "2456.png", "formula": "\\begin{align*} \\varphi = \\pi ( f _ 0 ) \\varphi + \\pi ( f _ 1 ) P ( X ) \\varphi . \\end{align*}"}
+{"id": "1830.png", "formula": "\\begin{align*} M _ p ( k _ 1 k _ 2 k _ 2 & k _ 3 , k _ 3 k _ 4 k _ 1 k _ 4 ) \\\\ & = 2 \\cos \\big [ ( t - s ) ( E _ 1 - E _ 3 ) \\big ] \\big ( \\delta ( p - k _ 3 ) - \\delta ( p - k _ 1 ) \\big ) \\Phi ( k _ 1 k _ 2 k _ 3 k _ 2 ) \\Phi ( k _ 1 k _ 4 k _ 3 k _ 4 ) \\end{align*}"}
+{"id": "5996.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } ( [ h ( - 1 ) , - i ] ) A ( [ \\epsilon , x ] ) & = ( - i ) e ^ { \\tfrac { \\pi i } { 4 } [ 1 + 1 ] } ( - 1 , \\epsilon ) _ \\R A ( [ \\epsilon , - x ] ) \\\\ & = ( - 1 , \\epsilon ) _ \\R A ( [ \\epsilon , - x ] ) . \\end{align*}"}
+{"id": "2087.png", "formula": "\\begin{align*} & F ( A ) = m ^ 2 \\cdot 2 ^ { 2 n } - ( m ^ 2 + m - m d ) 2 ^ n - d m - d + m , \\\\ & g ( A ) = m ( 2 ^ { n - 1 } n - 1 ) + \\frac { 1 } { 2 } ( m ^ 2 ( 2 ^ n - 1 ) ^ 2 + ( m ^ 2 + d m - 3 m ) ( 2 ^ n - 1 ) - d + 1 ) . \\end{align*}"}
+{"id": "8501.png", "formula": "\\begin{align*} \\left [ \\sigma ^ { m } \\left ( - i x \\right ) e ^ { - 2 \\pi i m x } \\right ] = \\cos \\left ( 2 \\pi m x + 2 m \\arctan \\left ( \\frac { x } { p } \\right ) \\right ) . \\end{align*}"}
+{"id": "4022.png", "formula": "\\begin{align*} T _ p ( P \\otimes \\lbrace \\alpha , \\beta \\rbrace ) = \\sum _ { g \\in R _ p } g ( P \\otimes \\lbrace \\alpha , \\beta \\rbrace ) . \\end{align*}"}
+{"id": "51.png", "formula": "\\begin{align*} ( K \\cap \\mathrm { E n d } _ { { \\kappa } } ( \\overline { A } _ g ) ) \\otimes _ { \\mathbb { Z } } \\mathbb { Z } _ \\ell = \\mathcal { O } _ { K , \\ell } . \\end{align*}"}
+{"id": "7055.png", "formula": "\\begin{align*} \\frac { d } { d x } ( \\tilde Q _ i ) = \\frac { d } { d x } \\left ( \\frac { Q _ i } { a _ i } \\right ) = \\frac { Q _ i ' } { a _ i } \\end{align*}"}
+{"id": "8971.png", "formula": "\\begin{align*} \\Tilde { \\varepsilon } = \\phi _ 1 - \\Tilde { \\phi _ { e } } = \\frac { \\phi _ 2 - \\phi _ 1 } { r ^ { \\Tilde { q } } - 1 } . \\end{align*}"}
+{"id": "8685.png", "formula": "\\begin{align*} f ( p ^ { - 1 } \\| X _ { i _ 1 } - X _ { i _ 2 } \\| _ { 2 } ^ 2 ) = f ( \\tau _ 1 ) + \\sum ^ { l - 1 } _ { s = 1 } \\frac { f ^ { ( s ) } ( \\tau _ 1 ) } { s ! } \\widetilde { X } _ { i _ 1 , i _ 2 } ^ s + c _ { l , \\tau _ 1 } ( \\widetilde { X } _ { i _ 1 , i _ 2 } ) \\widetilde { X } _ { i _ 1 , i _ 2 } ^ { l } , \\end{align*}"}
+{"id": "3827.png", "formula": "\\begin{align*} \\boldsymbol { d } _ { \\mathcal { V } } ( ( s _ 1 , s _ 2 ) , ( s _ 1 ^ \\prime , s _ 2 ^ \\prime ) ) = \\boldsymbol { d } _ { \\mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \\prime ) + \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \\prime ) . \\end{align*}"}
+{"id": "6406.png", "formula": "\\begin{align*} h _ \\psi ^ { i t } = [ D \\psi : D \\varphi ] _ t \\ , \\lambda ^ { \\varphi } ( t ) , t \\in \\mathbb { R } , \\end{align*}"}
+{"id": "7878.png", "formula": "\\begin{align*} \\alpha ( \\mathcal { K } ^ \\pm ( n , k ) ) \\leq \\frac { 2 \\binom { n } { k } } { 1 - \\frac { 2 \\binom { n - k } { k } } { - 2 \\binom { n - k - 1 } { k - 1 } } } = \\frac { 2 \\binom { n } { k } } { 1 + \\frac { n - k } { k } } = 2 \\binom { n - 1 } { k - 1 } = | \\mathcal { S } | . \\end{align*}"}
+{"id": "2444.png", "formula": "\\begin{align*} ( R _ \\gamma ) _ * X = X & & \\gamma \\in G , \\end{align*}"}
+{"id": "7741.png", "formula": "\\begin{align*} Z ( t ) & = N \\Bigl ( y ( Z , t ) \\Bigr ) . \\end{align*}"}
+{"id": "2146.png", "formula": "\\begin{align*} ( \\varphi ( d ( q , \\gamma ( s ) ) ) ) '' & \\le \\frac 1 r + \\frac { 4 \\epsilon \\sigma } b = \\frac 1 r + \\frac { 4 \\epsilon } { r } \\le \\frac 2 r . \\end{align*}"}
+{"id": "6804.png", "formula": "\\begin{gather*} \\dim W ^ { \\mathrm { u } } ( p ^ - ) = 2 n - k ^ - , \\dim W ^ { \\mathrm { s } } ( p ^ + ) = k ^ + + 1 . \\end{gather*}"}
+{"id": "1558.png", "formula": "\\begin{align*} g ( z ) = \\sum _ { n \\ge 1 } \\rho _ { g } ( n ) n ^ { \\frac { \\l - 1 } { 2 } } e ( n z ) . \\end{align*}"}
+{"id": "5710.png", "formula": "\\begin{align*} h ^ { \\sigma } = h \\end{align*}"}
+{"id": "3092.png", "formula": "\\begin{align*} ( \\lambda C _ \\lambda ) ^ { ( p - 1 - \\alpha ) ( p - 1 - k _ 2 ) - k _ 1 k _ 3 } & = \\left ( \\frac { c _ 1 } { \\mu ^ { k _ 1 } B _ \\lambda } \\right ) ^ { p - 1 - k _ 2 } \\left ( \\frac { c _ 2 } { \\mu ^ { k _ 2 } B _ \\mu } \\right ) ^ { k _ 1 } , \\\\ ( \\mu C _ \\mu ) ^ { ( p - 1 - \\alpha ) ( p - 1 - k _ 2 ) - k _ 1 k _ 3 } & = \\left ( \\frac { c _ 2 } { \\mu ^ { k _ 2 } B _ \\mu } \\right ) ^ { p - 1 - \\alpha } \\left ( \\frac { c _ 1 } { \\mu ^ { k _ 1 } B _ \\lambda } \\right ) ^ { k _ 3 } . \\end{align*}"}
+{"id": "8879.png", "formula": "\\begin{align*} H _ 2 ( n , p ) & \\ge \\left ( \\frac { p k } { ( k - 1 ) ( 1 + 2 / n ) } \\right ) ^ n e ^ { - 2 / p } ( 1 - o ( 1 ) ) c _ k n ^ { 1 / 2 } E ( n , { \\textstyle \\frac { k - 1 } { k } } ) \\\\ & = \\left ( \\frac { 1 } { ( 1 + 2 / n ) } \\right ) ^ n e ^ { - 2 / p } ( 1 - o ( 1 ) ) c _ k n ^ { 1 / 2 } E ( n , p ) \\\\ & = ( 1 + o ( 1 ) ) c _ p n ^ { 1 / 2 } E ( n , p ) \\\\ \\end{align*}"}
+{"id": "2915.png", "formula": "\\begin{align*} [ v + L _ n ] = \\sum _ { w \\in L _ n / L _ { n m } } [ v + w + L _ { n m } ] . \\end{align*}"}
+{"id": "4173.png", "formula": "\\begin{align*} \\| \\psi _ { n , t } ( b ) ^ * \\psi _ { n , t } ( b ) & + ( \\psi _ { n , e } ( c ) + \\alpha 1 _ { k _ n } ) ^ * ( \\psi _ { n , e } ( c ) + \\alpha 1 _ { k _ n } ) - \\| b \\| ^ 2 1 _ { k _ n } \\| \\\\ & \\leq \\| \\psi _ { n , t } ( b ) ^ * \\psi _ { n , t } ( b ) - \\psi _ { n , e } ( b ^ * b ) \\| + \\| \\psi _ { n , e } ( c ) ^ * \\psi _ { n , e } ( c ) - \\psi _ { n , e } ( c ^ * c ) \\| \\\\ & = \\| \\psi _ { n , t ^ { - 1 } } ( b ^ * ) \\psi _ { n , t } ( b ) - \\psi _ { n , e } ( b ^ * b ) \\| + \\| \\psi _ { n , e } ( c ^ * ) \\psi _ { n , e } ( c ) - \\psi _ { n , e } ( c ^ * c ) \\| < 2 \\varepsilon _ n \\end{align*}"}
+{"id": "2972.png", "formula": "\\begin{align*} \\kappa _ I ( \\rho \\cdot x ) = \\varphi _ I ( \\rho ) \\cdot \\kappa _ I ( x ) \\end{align*}"}
+{"id": "70.png", "formula": "\\begin{align*} \\mathcal { T } ( \\ell ) = \\bigsqcup _ { i = 1 } ^ { r ( \\ell ) } \\mathcal { T } ( \\ell ) _ i \\sigma , \\sigma ' \\in \\mathcal { T } ( \\ell ) _ i \\Leftrightarrow \\pi ( \\sigma ) = \\pi ( \\sigma ' ) . \\end{align*}"}
+{"id": "4716.png", "formula": "\\begin{align*} \\vect u = \\vect u _ 0 + \\sum _ { n = 1 } ^ \\infty \\vect u _ { n } , \\end{align*}"}
+{"id": "164.png", "formula": "\\begin{align*} L ( \\beta ) + \\frac { 1 } { 2 } L ( \\beta ^ 2 ) = \\frac { 5 } { 7 } , \\end{align*}"}
+{"id": "7359.png", "formula": "\\begin{align*} - \\frac { 1 } { x } \\log ( 1 - x ) & = \\frac { 1 } { x } \\sum _ { k = 1 } ^ \\infty \\frac { x ^ k } { k } \\\\ & < \\frac { 1 } { x } \\sum _ { k = 1 } x ^ k = \\frac { 1 } { 1 - x } \\end{align*}"}
+{"id": "3888.png", "formula": "\\begin{align*} \\Sigma = \\left [ \\begin{array} { c c c } 2 & - 1 & - 2 \\\\ - 1 & 4 & - 2 \\\\ - 2 & - 2 & 4 \\\\ \\end{array} \\right ] . \\end{align*}"}
+{"id": "1868.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} K - G ' ( u ^ * ) u ^ * & \\ \\ = \\ \\ \\mathcal { K } - G ( u ^ * ) ; \\\\ G ' ( u ^ * ) u ^ * \\in K & \\Longleftrightarrow G ( u ^ * ) \\in \\mathcal { K } \\end{aligned} \\right . \\end{align*}"}
+{"id": "5537.png", "formula": "\\begin{align*} P _ { \\mu , x } ^ { \\zeta } \\left ( L _ { x } g , A \\right ) = \\sum _ { s \\in G } { \\bf 1 } _ { A } \\left ( L _ { x } g s \\right ) \\frac { d g s . \\bar { \\lambda } } { d g . \\bar { \\lambda } } ( x , \\zeta ) d \\mu ( s ) . \\end{align*}"}
+{"id": "5150.png", "formula": "\\begin{align*} \\sup _ { n \\geq 1 } \\| g _ n \\| _ X = \\sup _ { n \\geq 1 } \\| g _ n \\| _ Y \\leq \\| f \\| _ Y . \\end{align*}"}
+{"id": "915.png", "formula": "\\begin{align*} F = F _ o + F _ e , \\end{align*}"}
+{"id": "9073.png", "formula": "\\begin{align*} \\lambda = \\frac { \\langle g , M g \\rangle } { \\langle g , g \\rangle } = \\lambda + \\frac { 1 } { \\langle g , g \\rangle } \\sum _ { \\{ x , y \\} \\in E } ( - M _ { x y } ) f ( x ) f ( y ) ( a ( x ) - a ( y ) ) ^ 2 . \\end{align*}"}
+{"id": "3699.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) u ^ { ( 1 ) } ( x ) + F ^ { ( 1 ) } ( x ) u ^ { ( 1 ) } ( x ) = 0 & \\Omega , \\\\ u ^ { ( 1 ) } = f ^ 1 & \\Omega ^ c . \\end{cases} \\end{align*}"}
+{"id": "2622.png", "formula": "\\begin{align*} f _ j & = \\sum _ { m \\in \\mathbb { Z } ^ 2 } \\widetilde { \\eta } _ m \\eta _ m f _ j \\\\ & = \\sum _ { m \\in \\mathbb { Z } ^ 2 } \\widetilde { \\eta } _ m f _ { j , m } + \\sum _ { m \\in \\mathbb { Z } ^ 2 } \\widetilde { \\eta } _ m \\left [ \\eta _ m \\left ( \\psi ^ { ( j ) } \\ast _ j f _ j \\right ) - \\psi ^ { ( j ) } \\ast _ j \\left ( \\eta _ m f _ j \\right ) \\right ] \\\\ & = : f _ { j , \\flat } + f _ { j , \\sharp } + f _ { j , } \\end{align*}"}
+{"id": "5506.png", "formula": "\\begin{align*} \\nu = \\int _ { G / P } g . \\lambda d \\nu _ { 0 } ( g ) = \\int _ { G / P _ { \\mathsf { f } ( I ) } } g . \\lambda d \\bar { \\nu } _ { 0 } ( g ) . \\end{align*}"}
+{"id": "5638.png", "formula": "\\begin{align*} \\l ( a , b ) = \\inf _ { ( u , v ) \\in \\mathcal { L } ( a , b ) } K ( u , v ) , \\ \\ \\mathcal { L } ( a , b ) : = \\{ ( u , v ) \\in \\mathcal { T } ( a , b ) : L ( u , v ) = 0 \\} . \\end{align*}"}
+{"id": "859.png", "formula": "\\begin{align*} \\frac { d \\Phi } { d U } ( U ; \\lambda , \\alpha ) = \\frac { V _ { s } } { U _ { s } } = \\alpha + \\frac { H _ { \\lambda } \\left ( \\phi _ { 2 } ( \\lambda ) - U \\right ) } { \\Phi ( U ; \\lambda , \\alpha ) } . \\end{align*}"}
+{"id": "6629.png", "formula": "\\begin{align*} \\| \\varphi ( T ) \\| _ { L ^ { p _ i , \\infty } } \\leq C _ i \\| T \\| _ { \\mathcal { B } _ { p _ i } ( \\mathcal { H } ) } \\ \\forall \\ T \\in \\mathcal { B } _ { p _ i } ( \\mathcal { H } ) , i = 0 , 1 \\end{align*}"}
+{"id": "5564.png", "formula": "\\begin{align*} h _ { \\mu } ( Z _ { p } , \\lambda _ { \\ell , p } ) = h _ { \\mu } ( M , \\bar { \\lambda } _ { \\ell , p } ) . \\end{align*}"}
+{"id": "8564.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { - 1 } E [ Z _ 0 ( r _ N ) | \\Omega _ \\infty ] = 1 . \\end{align*}"}
+{"id": "8830.png", "formula": "\\begin{align*} f _ \\lambda = \\frac { 1 } { \\lambda } \\bigl ( I - ( I - \\lambda f ) ^ { - 1 } \\bigr ) , \\end{align*}"}
+{"id": "1205.png", "formula": "\\begin{align*} G ( 0 , s ) = \\sum _ j \\frac { { a } _ j } { 1 - s { a } _ j } \\nu ( j ) \\end{align*}"}
+{"id": "8541.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow \\infty } \\mathcal { K } _ { \\nu , p } ( x ) = \\frac { \\Gamma \\left ( \\frac { 1 } { 2 } - \\nu \\right ) ( 2 x ) ^ { \\nu } } { \\sqrt { \\pi } } \\sum _ { n = 1 } ^ { \\infty } n ^ { \\nu } \\ , K _ { \\nu } ( x n ) , \\end{align*}"}
+{"id": "3379.png", "formula": "\\begin{align*} Q _ \\eta = \\frac { Q _ \\eta } { Q _ { \\eta - 1 } } Q _ { \\eta - 1 } \\leq \\left ( 1 + \\frac { \\gamma } { 4 } \\right ) Q _ n \\left ( 1 + \\frac { \\gamma } { 2 } \\right ) \\leq Q _ n ( 1 + \\gamma ) = \\kappa Q _ n \\end{align*}"}
+{"id": "6981.png", "formula": "\\begin{align*} f = f _ 0 + \\frac { f _ 1 } { a } \\cdot ( a q ) + \\ldots + \\frac { f _ n } { a ^ n } \\cdot ( a q ) ^ n \\end{align*}"}
+{"id": "1404.png", "formula": "\\begin{align*} \\forall \\ , p < r \\leq q , \\ ; \\ ; \\ ; \\ ; & \\sharp \\{ r \\leq s \\leq q \\ , | \\ , D _ { \\lambda } ( s ) = \\times \\} < \\sharp \\{ r \\leq s \\leq q \\ , | \\ , D _ { \\lambda } ( s ) = \\circ \\} \\\\ & \\sharp \\{ p \\leq s \\leq q \\ , | \\ , D _ { \\lambda } ( s ) = \\times \\} = \\sharp \\{ p \\leq s \\leq q \\ , | \\ , D _ { \\lambda } ( s ) = \\circ \\} . \\end{align*}"}
+{"id": "8447.png", "formula": "\\begin{align*} I _ { m , p } ^ { ( 1 ) } ( s , x ) = \\frac { ( - 1 ) ^ { m } } { 2 \\pi i } \\ , \\intop _ { \\mu - i \\infty } ^ { \\mu + i \\infty } \\Gamma \\left ( z \\right ) \\ , \\Gamma \\left ( s + z - \\frac { 1 } { 2 } \\right ) \\ , ( \\pi \\ , x \\ , m ) ^ { - 2 z } \\ , d z = 2 \\ , ( - 1 ) ^ { m } \\ , \\left ( \\pi x \\ , m \\right ) ^ { s - \\frac { 1 } { 2 } } \\ , K _ { s - \\frac { 1 } { 2 } } \\left ( 2 \\pi x \\ , m \\right ) \\end{align*}"}
+{"id": "366.png", "formula": "\\begin{align*} \\mbox { c o l i m } _ { \\omega \\subsetneq \\tau } ( \\underline { X } , \\underline { A } ) ^ { \\omega , c } = ( \\underline { X } , \\underline { A } ) ^ { \\partial \\tau , c } \\overset { \\imath ^ { \\mu , c } _ { \\partial \\tau , c } } { \\longrightarrow } ( \\underline { X } , \\underline { A } ) ^ { \\mu , c } \\end{align*}"}
+{"id": "9171.png", "formula": "\\begin{align*} \\delta ^ { 2 } ( \\varphi _ { r e s t _ { 1 } } ^ { 1 } ) = \\left ( x ^ { 1 } + \\bar { u } ^ { 1 } \\cos \\left ( \\tfrac { x ^ { 3 } + v _ { 1 } ^ { 1 } } { 2 } \\right ) \\right ) \\sin \\left ( \\tfrac { v _ { 1 } ^ { 1 } + v _ { 1 , [ 1 ] } ^ { 1 } } { 2 } \\right ) - \\left ( x ^ { 2 } + \\bar { u } ^ { 1 } \\sin \\left ( \\tfrac { x ^ { 3 } + v _ { 1 } ^ { 1 } } { 2 } \\right ) \\right ) \\cos \\left ( \\tfrac { v _ { 1 } ^ { 1 } + v _ { 1 , [ 1 ] } ^ { 1 } } { 2 } \\right ) \\ , . \\end{align*}"}
+{"id": "6838.png", "formula": "\\begin{align*} \\sum _ { s _ 1 \\geq \\dots \\geq s _ { r - 1 } \\geq 0 } \\frac { q ^ { s _ 1 ^ 2 + \\dots + s _ { r - 1 } ^ 2 + s _ { i } + \\dots + s _ { r - 1 } } } { ( q ) _ { s _ 1 - s _ 2 } \\dots ( q ) _ { s _ { r - 2 } - s _ { r - 1 } } ( q ^ 2 ; q ^ 2 ) _ { s _ { r - 1 } } } = \\frac { ( q ^ { 2 r } , q ^ { i } , q ^ { 2 r - i } ; q ^ { 2 r } ) _ \\infty } { ( q ) _ \\infty } , \\end{align*}"}
+{"id": "1842.png", "formula": "\\begin{align*} \\alpha ^ H ( h , k ) & = \\frac { ( 2 / \\pi ) } { ( 2 \\pi ) ^ 3 } \\sum _ { r \\in \\Z ^ 3 } \\chi ( r ) \\chi ^ \\perp ( r + k ) \\delta _ { h \\cdot k , r \\cdot k } \\ . \\end{align*}"}
+{"id": "3419.png", "formula": "\\begin{align*} d ( u , P ) = \\psi \\left ( \\int _ { B ( P , R _ 0 ) } u ^ 2 \\right ) , \\end{align*}"}
+{"id": "4218.png", "formula": "\\begin{align*} K _ 1 & = T ( \\phi ) - T ( \\phi _ 0 ) T ( \\phi _ 1 ) \\cdots T ( \\phi _ R ) \\\\ K _ 2 & = T ( \\tilde { \\phi } ) - T ( \\tilde { \\phi } _ 0 ) T ( \\tilde { \\phi } _ 1 ) \\cdots T ( \\tilde { \\phi } _ R ) \\end{align*}"}
+{"id": "533.png", "formula": "\\begin{align*} \\mathrm { H } ^ { - \\infty } _ { \\mathcal { H } _ { \\hbar , V } } : = \\mathcal { L } \\left ( \\mathrm { H } ^ { \\infty } _ { \\mathcal { H } _ { \\hbar , V } } , \\mathbb { C } \\right ) , \\end{align*}"}
+{"id": "4061.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { t = 0 } \\left ( \\frac { \\Gamma ( k + t ) ^ 2 } { ( 2 \\pi ) ^ { 2 t } } \\frac { X ^ t \\zeta ( 2 t + 1 ) } { t } \\right ) & = \\lim _ { t \\to 0 } \\left ( \\frac { d } { d t } \\left ( \\frac { t ^ 2 \\Gamma ( k + t ) ^ 2 } { ( 2 \\pi ) ^ { 2 t } } \\frac { X ^ t \\zeta ( 2 t + 1 ) } { t } \\right ) \\right ) \\\\ & = \\frac { \\left ( ( k - 1 ) ! \\right ) ^ 2 } { 2 } \\log X + c _ 0 ; \\end{align*}"}
+{"id": "1236.png", "formula": "\\begin{align*} g _ { \\gamma , 1 } = f _ { \\gamma , 1 } ^ { - 1 } g _ { \\gamma , 2 } = f _ { \\gamma , 2 } ^ { - 1 } . \\end{align*}"}
+{"id": "132.png", "formula": "\\begin{align*} \\mathcal { F } \\sqcup \\mathcal { K } = K _ 1 \\sqcup K _ 2 \\sqcup \\cdots \\sqcup K _ N \\end{align*}"}
+{"id": "3447.png", "formula": "\\begin{align*} ( a , b , c , t ) \\longmapsto \\left \\{ \\begin{bmatrix} ( t - b ) / 2 & & a \\\\ c & & ( t + b ) / 2 \\end{bmatrix} \\right \\} _ { \\Gamma _ { p } } . \\end{align*}"}
+{"id": "5972.png", "formula": "\\begin{align*} \\overline { \\overline { C } } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) & = \\overline { \\overline { C } } _ { X ^ { \\ast } } ( \\omega , \\omega ^ { - 1 } g _ 1 ) ^ { - 1 } \\overline { \\overline { C } } _ { X ^ { \\ast } } ( \\omega , \\omega ^ { - 1 } g _ 1 g _ 2 ) \\overline { \\overline { C } } _ { X ^ { \\ast } } ( \\omega ^ { - 1 } g _ 1 , g _ 2 ) . \\end{align*}"}
+{"id": "7495.png", "formula": "\\begin{align*} \\sigma = \\hat { \\varphi } _ { S } ( \\pi ) = ( \\hat { \\varphi } _ { x _ { k } } \\circ \\cdots \\circ \\hat { \\varphi } _ { x _ { 2 } } \\circ \\hat { \\varphi } _ { x _ { 1 } } ) ( \\pi ) . \\end{align*}"}
+{"id": "8556.png", "formula": "\\begin{align*} \\tau _ N : = \\inf \\{ t \\geq 0 : Z _ 0 ( t ) \\geq N \\} , \\end{align*}"}
+{"id": "3204.png", "formula": "\\begin{align*} D _ w : = \\{ z \\in \\C : \\abs { \\sqrt { z + 4 w - 4 w ^ 2 } + \\sqrt { z - 4 w + 4 w ^ 2 } } \\leq 2 \\sqrt { w } ~ ~ \\\\ \\abs { \\sqrt { z + 4 w - 4 w ^ 2 } - \\sqrt { z - 4 w + 4 w ^ 2 } } \\leq 2 \\sqrt { w } \\} . \\end{align*}"}
+{"id": "1425.png", "formula": "\\begin{align*} \\frac { 2 } { n } \\sum _ { i = 1 } ^ n \\left ( \\boldsymbol { \\hat { \\tilde { H } } _ { k 2 } } ( t _ p ) - \\boldsymbol { \\hat { \\tilde { H } } _ { k 2 } } ( t _ o ) \\right ) ^ T \\ ! \\hat { \\mathbf { \\Sigma } } _ \\mathbf { k } ^ { - 1 } \\tilde { \\mathbf { \\Sigma } } _ \\mathbf { k i } \\hat { \\mathbf { \\Sigma } } _ \\mathbf { k } ^ { - 1 } \\left ( \\boldsymbol { \\hat { \\tilde { H } } _ { k 2 } } ( t _ p ) - \\boldsymbol { \\hat { \\tilde { H } } _ { k 2 } } ( t _ o ) \\right ) , \\end{align*}"}
+{"id": "8639.png", "formula": "\\begin{align*} p _ 0 ( t ) & = \\frac { p ( e ^ { \\lambda t } - 1 ) } { e ^ { \\lambda t } - p } , \\\\ p _ j ( t ) & = \\frac { q ^ 2 e ^ { \\lambda t } } { ( e ^ { \\lambda t } - p ) ^ 2 } \\cdot \\left ( \\frac { e ^ { \\lambda t } - 1 } { e ^ { \\lambda t } - p } \\right ) ^ { j - 1 } , j \\geq 1 , \\end{align*}"}
+{"id": "2899.png", "formula": "\\begin{align*} \\varphi ( f ( \\xi ) ) = \\frac { 1 } { 2 \\pi i } \\int _ \\Gamma \\varphi ( z ) ( z - f ( \\xi ) ) ^ { - 1 } d z . \\end{align*}"}
+{"id": "7398.png", "formula": "\\begin{align*} \\begin{aligned} R _ N ^ 2 ( L , \\alpha , \\Delta ) & \\leq \\frac { N _ m } { N } R _ { N _ m } ^ 2 ( L N _ m / N , \\alpha , \\Delta ) \\\\ & \\leq ( 1 + C / m ) R _ { N _ m } ^ 2 ( ( 1 + C / m ) L _ m , \\alpha , \\Delta ) . \\end{aligned} \\end{align*}"}
+{"id": "23.png", "formula": "\\begin{align*} \\theta _ A \\colon K / \\Lambda \\simeq A ( \\mathbb { C } ) _ \\mathrm { t o r } = A ( \\overline { \\mathbb { Q } } ) _ { \\mathrm { t o r } } . \\end{align*}"}
+{"id": "5025.png", "formula": "\\begin{align*} & q _ 1 = \\frac { 1 } { a _ 1 } & \\\\ & q _ i = \\sum _ { j = i } ^ N \\frac { ( - 1 ) ^ { j - i } } { a _ 1 \\cdots a _ j } & \\\\ & q _ m = ( - 1 ) ^ { N - 1 } \\frac { 1 - a _ N + \\sum _ { k = 2 } ^ { N - 1 } ( - 1 ) ^ k a _ N \\prod _ { l = 2 } ^ k a _ { N - l + 1 } } { \\prod _ { k = 1 } ^ N a _ k - ( - 1 ) ^ N } & , \\end{align*}"}
+{"id": "564.png", "formula": "\\begin{align*} C _ { T } = c _ { 0 } ^ { - 1 } ( 1 + \\left \\| a \\right \\| _ { L ^ { \\infty } } ) e ^ { c _ { 0 } ^ { - 1 } \\left ( 1 + \\| a ^ { \\prime } \\| _ { L ^ { \\infty } } + \\| q \\| _ { L ^ { \\infty } } + 2 \\left \\| a \\right \\| _ { L ^ { \\infty } } \\right ) T } , \\end{align*}"}
+{"id": "9088.png", "formula": "\\begin{align*} \\tilde { \\omega } ^ + = n - | \\mathcal { F } | - ( \\tilde { k } + \\tilde { r } - 1 ) , \\ , \\ , \\ , \\ , \\ , \\tilde { \\omega } ^ - = \\tilde { k } - 1 . \\end{align*}"}
+{"id": "3457.png", "formula": "\\begin{align*} \\eta ^ { - 1 } S \\eta = \\gamma _ { p } ^ { m } \\end{align*}"}
+{"id": "721.png", "formula": "\\begin{align*} D _ { V } [ \\partial _ { t } \\gamma _ { S } ] ( e ) + B _ { B } [ \\partial _ { t } \\gamma _ { B } ] ( e ) = G _ { V , 2 } ( e ) \\end{align*}"}
+{"id": "5974.png", "formula": "\\begin{align*} \\overline { \\overline { \\Pi } } _ { \\psi } ( g ) = \\Pi _ { \\psi } ( g ) \\widetilde { s } ( g ) . \\end{align*}"}
+{"id": "3089.png", "formula": "\\begin{align*} \\lambda & = \\frac { ( p - \\alpha + m _ 1 ) ( p - 1 - k _ 2 ) + k _ 1 ( p - k _ 3 + m _ 2 ) } { ( p - 1 - \\alpha ) ( p - 1 - k _ 2 ) - k _ 1 k _ 3 } > 1 , \\\\ \\mu & = \\frac { ( p - 1 - \\alpha ) ( p + m _ 2 ) + k _ 3 ( 1 + m _ 1 ) } { ( p - 1 - \\alpha ) ( p - 1 - k _ 2 ) - k _ 1 k _ 3 } > 1 . \\end{align*}"}
+{"id": "9236.png", "formula": "\\begin{align*} F _ m < \\varepsilon _ 5 ( m ) : = \\frac { ( \\pi ^ 2 B _ 1 a ) ^ { m / 3 } } { 3 1 6 \\ , A ( \\sigma ) } \\Bigl ( \\frac { m ! } { \\Gamma ( m / 3 + 2 ) } \\Bigr ) ^ { 1 / 2 } \\ , \\varepsilon _ 4 \\end{align*}"}
+{"id": "769.png", "formula": "\\begin{align*} \\sigma _ k ( \\kappa ) = \\sum \\limits _ { m = 0 } ^ k \\dfrac { ( - 1 ) ^ m \\phi '^ { k - m } } { D ^ { k + 2 } \\phi ^ m } { n - m \\choose k - m } \\rho ^ m \\left [ \\phi ^ 2 \\sigma _ m ( D ^ 2 u ) + \\dfrac { k + n - 2 m } { n - m } \\rho ^ 2 u ^ i u _ j [ T _ m ] _ i ^ j ( D ^ 2 u ) \\right ] . \\end{align*}"}
+{"id": "5040.png", "formula": "\\begin{align*} \\deg ( [ x _ 1 ^ { \\alpha _ 1 } \\cdots x _ N ^ { \\alpha _ N } ] ) : = \\alpha _ 1 q _ 1 + \\dots + \\alpha _ N q _ N . \\end{align*}"}
+{"id": "4817.png", "formula": "\\begin{align*} \\log Z _ { n } = S _ n + \\log W _ { n } , \\end{align*}"}
+{"id": "8215.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } P ^ \\omega ( C _ t ^ c \\mid S ) = 0 . \\end{align*}"}
+{"id": "9269.png", "formula": "\\begin{align*} \\int _ { a } ^ b x ^ { \\lambda } \\ , d x \\simeq \\begin{cases} ( b - a ) b ^ { \\lambda } & \\lambda > - 1 , \\\\ \\log ( b / a ) & \\lambda = - 1 , \\\\ ( b - a ) b ^ { - 1 } a ^ { \\lambda + 1 } & \\lambda < - 1 , \\end{cases} 0 < a \\le b < \\infty . \\end{align*}"}
+{"id": "98.png", "formula": "\\begin{align*} \\partial _ t w = - i h ^ { - 1 } ( - i h X - i ( q _ 1 + W ) ) w , w | _ { t = 0 } = f , \\end{align*}"}
+{"id": "7701.png", "formula": "\\begin{align*} \\vartheta _ L ( f ) _ P ( g ) = \\int _ { \\Gamma _ L \\cap N \\backslash N } \\vartheta _ L ( f ) ( n g ) d n \\end{align*}"}
+{"id": "6875.png", "formula": "\\begin{align*} C _ r ^ 1 = \\widehat \\psi _ r ( 1 ) = \\inf _ { x \\in [ 0 , 1 ] } \\int _ { [ 0 , 1 ] } \\d y \\ , \\log \\frac { 1 } { r ( x , y ) } . \\end{align*}"}
+{"id": "3993.png", "formula": "\\begin{align*} \\delta _ 1 ^ { 1 / 2 } V _ { 1 , Y Y } ^ { - 1 / 2 } V _ { 1 , X Y } + \\delta _ 2 ^ { 1 / 2 } V _ { 2 , Y Y } ^ { - 1 / 2 } V _ { 2 , X Y } = 0 . \\end{align*}"}
+{"id": "5686.png", "formula": "\\begin{align*} f ^ { + } ( z ) : = \\sum _ { n \\gg - \\infty } C _ { f } ^ { + } ( n ) q ^ { n } , \\end{align*}"}
+{"id": "4576.png", "formula": "\\begin{align*} \\Phi _ N ( t ) = U _ N ( t ) \\Psi _ N ( t ) . \\end{align*}"}
+{"id": "4117.png", "formula": "\\begin{align*} \\left | \\frac { N ( y ) } { N ( 1 ) } \\right | = | \\alpha _ 0 | = \\left | \\frac { c _ 5 } { b _ 3 } \\right | . \\end{align*}"}
+{"id": "7960.png", "formula": "\\begin{align*} \\xi \\cdot \\nabla _ \\xi H ( \\xi ) = H ( \\xi ) \\end{align*}"}
+{"id": "3053.png", "formula": "\\begin{align*} n - \\delta ( G ) - 1 \\leq n - \\kappa ( G ) - 1 \\leq n - ( n - k ) - 1 = k - 1 \\end{align*}"}
+{"id": "5914.png", "formula": "\\begin{align*} \\pi _ { \\psi } ( [ \\omega , t ] ) f ( y ) = t \\int _ { X ^ { \\ast } } \\psi ( \\langle y , y ^ { \\ast } \\rangle ) f ( y ^ { \\ast } \\omega ^ { - 1 } ) d y ^ { \\ast } , \\end{align*}"}
+{"id": "3286.png", "formula": "\\begin{align*} \\widetilde { \\rho } ( t , w ) = A t ^ { 1 - | \\beta _ 1 | } + { \\rm I m } ( e ^ { i \\pi ( 1 - \\beta _ 1 ) } w ) + B t ^ { 2 - | \\beta _ 1 | } + C t ^ { | \\beta _ 1 | } | w | ^ 2 + \\end{align*}"}
+{"id": "8278.png", "formula": "\\begin{align*} \\det ( d _ { i , j } ) _ { i , j = 1 , \\ldots , n } = \\sum _ { j = 0 } ^ { 4 M } Z _ { k , n _ 1 , n _ 2 } ^ { ( j ) } ( \\i N \\beta ) ^ j + O _ N ( \\beta ^ { 4 M + 1 } ) , \\end{align*}"}
+{"id": "8498.png", "formula": "\\begin{align*} \\intop _ { 0 } ^ { \\infty } \\frac { x ^ { s + \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( b x ) } { x ^ { 2 } + a ^ { 2 } } \\ , d x = a ^ { s - \\frac { 1 } { 2 } } \\ , K _ { s - \\frac { 1 } { 2 } } ( a b ) , \\ , \\ , \\ , \\ , a , b > 0 . \\end{align*}"}
+{"id": "6452.png", "formula": "\\begin{align*} z ( s ) & = h ^ 2 \\beta _ h ( s ) + \\frac { h ^ 2 } { 2 } \\overline { \\beta _ h ( s ) } + \\frac { \\sigma ^ 2 } { 2 } \\overline { \\beta _ \\sigma ( s ) } , \\\\ \\zeta ( s ) & = i \\beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \\overline { \\beta _ h ( s ) } K _ 2 - i \\overline { \\beta _ \\sigma ( s ) } K , \\\\ \\gamma ( s ) & = - i s \\beta _ h ( s ) ( | K _ 1 | ^ 2 + | K _ 3 | ^ 2 ) + i s \\overline { \\beta _ h ( s ) } | K _ 2 | ^ 2 + i s \\overline { \\beta _ \\sigma ( s ) } | K | ^ 2 . \\end{align*}"}
+{"id": "5253.png", "formula": "\\begin{align*} L _ S T _ { \\rho } f = T _ { \\rho } L _ S f , \\mbox { a n d } D _ { S , x } T _ \\rho f = \\rho ^ { | S | } T _ { \\rho } D _ { S , x } f . \\end{align*}"}
+{"id": "705.png", "formula": "\\begin{align*} & \\lim _ { j \\to \\infty } \\frac { 1 } { j } \\log \\big ( \\| S ( j ) ( { \\xi } _ { [ ( \\sigma , k , l ) ] } ) \\| _ { L ^ 1 ( I ^ { ( j ) } ) } / | I ^ { ( j ) } | \\big ) = - \\lambda _ 1 ( n _ { \\sigma , k } - a _ { \\sigma , k } ) = - \\lambda _ 1 \\mathfrak { o } ( \\sigma , k ) \\\\ & \\lim _ { j \\to \\infty } \\frac { 1 } { j } \\log \\| S ( j ) ( { \\xi } _ { [ ( \\sigma , k , l ) ] } ) \\| _ { \\sup } = - \\lambda _ 1 \\mathfrak { o } ( \\sigma , k ) \\mathfrak { o } ( \\sigma , k ) > 0 . \\end{align*}"}
+{"id": "4271.png", "formula": "\\begin{align*} \\mathcal { E } _ { g } ^ { t } ( \\chi ) = \\zeta ( t ) , 0 \\leq t \\leq T . \\end{align*}"}
+{"id": "8912.png", "formula": "\\begin{align*} P _ l ( y ( \\{ X _ { n i } \\} ) ) = \\begin{cases} 0 , & l \\leq j - 1 , \\\\ Z _ j = Y ( \\{ X _ { n i } \\} ) , & l = j , \\\\ , & l \\geq j + 1 . \\end{cases} \\end{align*}"}
+{"id": "6917.png", "formula": "\\begin{align*} M _ 1 ( R , T ) & : = \\int _ { T ^ { \\beta } } ^ { T \\log { T } } \\left ( \\int _ { - ( \\log { t } ) ^ 2 } ^ { ( \\log { t } ) ^ 2 } K ( u ) d u \\right ) | R ( t ) | ^ { 2 } \\Phi \\left ( \\frac { t } { T } \\right ) d t , \\\\ M _ 2 ( R , T ) & : = \\int _ { T ^ { \\beta } } ^ { T \\log { T } } \\left ( \\int _ { - ( \\log { t } ) ^ 2 } ^ { ( \\log { t } ) ^ 2 } e ^ { - i \\theta } \\log \\zeta ( \\sigma + i ( t + u ) ) K ( u ) d u \\right ) | R ( t ) | ^ { 2 } \\Phi \\left ( \\frac { t } { T } \\right ) I ( \\sigma , t ) d t , \\end{align*}"}
+{"id": "4756.png", "formula": "\\begin{align*} \\tilde { S } = \\oplus ^ { \\operatorname { s } } _ i \\begin{pmatrix} \\tilde { S } _ { \\alpha _ i \\alpha _ i } & \\tilde { S } _ { \\alpha _ i \\beta _ i } \\\\ - \\tilde { S } _ { \\alpha _ i \\beta _ i } & \\tilde { S } _ { \\alpha _ i \\alpha _ i } \\end{pmatrix} + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "7403.png", "formula": "\\begin{align*} \\varphi _ R ( z ) : = \\int \\limits _ { [ z , z + ( 1 , 1 ) ) } \\varphi \\big ( \\tfrac { y } { \\sqrt R } \\big ) \\ , \\dd y \\end{align*}"}
+{"id": "2288.png", "formula": "\\begin{align*} w _ b = \\sum _ { n } c _ n a _ n + T ( f ) _ b , \\end{align*}"}
+{"id": "6327.png", "formula": "\\begin{align*} \\det ( M _ 2 ) = \\frac { \\partial x } { \\partial r } \\frac { \\partial y } { \\partial \\omega } - \\frac { \\partial y } { \\partial r } \\frac { \\partial x } { \\partial \\omega } . \\end{align*}"}
+{"id": "8947.png", "formula": "\\begin{align*} \\Tilde { \\phi } _ { e } & = \\phi _ 1 - C _ { p _ 1 } h ^ { p _ 1 } - C _ { p _ 2 } h ^ { p _ 2 } , \\\\ C _ { p _ 1 } h ^ { p _ 1 } & = \\frac { r ^ { p _ 2 } e _ { 2 1 } - e _ { 3 2 } } { r ^ { p _ 1 } ( 1 - r ^ { p _ 1 } ) ( 1 - r ) } , \\\\ C _ { p _ 2 } h ^ { p _ 2 } & = \\frac { e _ { 3 2 } - r ^ { p _ 1 } e _ { 2 1 } } { r ^ { p _ 1 } ( 1 - r ^ { p _ 2 } ) ( 1 - r ) } , \\end{align*}"}
+{"id": "7515.png", "formula": "\\begin{align*} p ^ i ( z ) = \\frac { p ^ i _ { - } ( z ) } { p ^ i _ { + } ( z ) } , q ^ i ( z ) = \\frac { q ^ i _ { - } ( z ) } { q ^ i _ { + } ( z ) } , i \\in I . \\end{align*}"}
+{"id": "7379.png", "formula": "\\begin{align*} \\widehat { \\Delta } ( x ) = \\int _ \\mathbb { R } \\Delta ( y ) e ( - x y ) \\ , d y = \\int _ { - 1 } ^ 1 \\left ( 1 - | y | \\right ) e ^ { - 2 \\pi i x y } \\ , d y = \\begin{cases} 1 , & x = 0 \\\\ \\frac { \\sin ^ 2 { \\pi x } } { \\pi ^ 2 x ^ 2 } , & x \\neq 0 \\end{cases} \\end{align*}"}
+{"id": "1760.png", "formula": "\\begin{align*} \\frac { 1 8 + W _ j ^ 2 - 9 V _ j } { 9 a _ j } = \\sum \\limits _ { n = 1 , 2 } A _ n ^ 2 , j = 1 , 2 \\ , \\end{align*}"}
+{"id": "1578.png", "formula": "\\begin{align*} \\sum _ { q \\geq 1 } \\frac { 1 } { 2 q + | x | } \\frac { 1 } { 2 ^ { 2 q + | x | } } \\frac { ( 2 q + | x | ) ! } { q ! ( q + | x | ) ! } \\leq C _ { 1 } \\sum _ { q \\geq 1 } \\frac { 1 } { 2 q + | x | } \\frac { 1 } { 2 ^ { 2 q + | x | } } \\frac { 2 ^ { 2 q + | x | } } { \\sqrt { q } } \\leq \\frac { C _ { 1 } } { 2 } \\sum _ { q \\geq 1 } \\frac { 1 } { q ^ { \\frac { 3 } { 2 } } } = : C _ { 1 } ' < \\infty , \\end{align*}"}
+{"id": "7195.png", "formula": "\\begin{align*} d = \\avg { w _ 0 \\cdot { \\bf e } _ k } _ { B } = \\lim _ { h \\to + \\infty } \\left ( \\avg { w _ h \\cdot { \\bf e } _ k } _ { B } - \\lambda _ h ^ { ( 1 / 2 ) } \\right ) = 0 , \\end{align*}"}
+{"id": "906.png", "formula": "\\begin{align*} \\sum _ { | d | \\leq \\widehat { K } } \\sum _ { d = e _ 1 f _ 1 f _ 2 ^ 2 d _ 3 } \\sum _ { d _ 3 = d _ 3 ' g _ 1 g _ 2 } | e _ 1 f _ 1 | ^ { - 1 } | f _ 2 | ^ { - 1 / 2 } | g _ 2 | | m ( g _ 2 ) | ^ { - 1 } | d _ 3 ' | ^ { - 1 / 2 } & \\leq \\sum _ { | d | \\leq \\widehat { K } } | h _ 1 \\cdots h _ k | ^ { - 1 } \\sum _ { d = e f _ 1 f _ 2 ^ 2 d _ 2 } \\sum _ { d _ 3 = d _ 3 ' g _ 1 g _ 2 } 1 \\\\ & \\ll \\sum _ { | d | \\leq \\widehat { K } } | d | ^ \\varepsilon | h _ 1 \\cdots h _ k | ^ { - 1 } \\\\ & \\ll \\widehat { K } ^ { 1 / k + \\varepsilon } . \\end{align*}"}
+{"id": "2004.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } L _ a ( u - v ) = H ( \\cdot , u ) - H ( \\cdot , v ) & \\textnormal { i n } & \\Omega \\\\ u - v = 0 & \\textnormal { i n } & \\partial \\Omega \\\\ u - v \\in C ^ { 2 } ( \\Omega ) \\cap C ^ 0 ( \\bar \\Omega ) . & & \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "1550.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { k _ \\epsilon } | \\tau | ^ { p - 2 } ( \\tau - k ) _ - \\ , d \\tau \\leq M \\int _ { 0 } ^ { k _ \\epsilon } | \\tau | ^ { p - 2 } \\ , d \\tau = M | s | ^ { p - 2 } s \\Big | _ { 0 } ^ { k _ \\epsilon } \\leq \\boldsymbol \\gamma ( p ) M ^ p \\epsilon . \\end{align*}"}
+{"id": "3452.png", "formula": "\\begin{align*} = U ^ { j _ { 1 } - 1 } \\alpha _ { p } U ^ { j _ { 2 } - 1 } \\alpha _ { p } \\dots U ^ { j _ { r } - 1 } \\alpha _ { p } \\iota \\end{align*}"}
+{"id": "6391.png", "formula": "\\begin{align*} \\sigma ^ { 5 } - 2 ^ { 4 } \\cdot 3 ^ { 6 } \\cdot \\tau ^ { 5 } = 5 r ^ { 2 } \\end{align*}"}
+{"id": "6949.png", "formula": "\\begin{align*} \\partial _ v ( a b ) : = \\partial _ v ( a ) b + ( - 1 ) ^ { | v | \\cdot | a | } a \\partial _ v ( b ) . \\end{align*}"}
+{"id": "7875.png", "formula": "\\begin{align*} 2 ( - 1 ) ^ j \\binom { n - k - j } { k - j } , \\end{align*}"}
+{"id": "6410.png", "formula": "\\begin{align*} [ D \\widetilde { \\psi } : D \\widetilde { \\varphi } ] _ t = s ( \\psi ) [ D \\widetilde { \\chi } : D \\widetilde { \\varphi } ] _ t = s ( \\psi ) [ D \\chi : D \\varphi ] _ t = [ D \\psi : D \\varphi ] _ t \\end{align*}"}
+{"id": "3295.png", "formula": "\\begin{align*} \\widetilde { \\rho } _ j ( \\xi , w ) = \\frac { 1 } { T _ j ( \\xi ) ^ { \\delta _ j } } \\rho \\left ( \\xi , ( \\xi - 1 ) ^ { \\delta _ 1 } ( \\xi + 1 ) ^ { \\delta _ { - 1 } } w + w _ 1 \\frac { 1 + \\psi ( \\xi ) } { 2 } + w _ { - 1 } \\frac { 1 - \\psi ( \\xi ) } { 2 } \\right ) \\end{align*}"}
+{"id": "3551.png", "formula": "\\begin{align*} A = \\left ( \\frac { n } { n - m } \\right ) \\left ( \\frac { n } { m } \\right ) ^ { \\frac { m } { n - m } } . \\end{align*}"}
+{"id": "2416.png", "formula": "\\begin{align*} f ( x ) & = \\frac { 1 } { 2 K } \\sum _ { j = L + 1 } ^ M j ^ { - \\theta } = \\frac { 1 } { 2 K } \\sum _ { x - K x ^ \\theta < j \\leq x + K x ^ \\theta } j ^ { - \\theta } + O ( x ^ { - \\theta } + x ^ { \\theta - 1 } ) \\\\ & = \\frac { 1 } { 2 K ( 1 - \\theta ) } \\big ( ( x + K x ^ \\theta ) ^ { 1 - \\theta } - ( x - K x ^ \\theta ) ^ { 1 - \\theta } \\big ) + O ( x ^ { - \\theta } + x ^ { \\theta - 1 } ) \\\\ & = 1 + O ( x ^ { - \\theta } + x ^ { \\theta - 1 } ) , \\end{align*}"}
+{"id": "6635.png", "formula": "\\begin{align*} \\phi ( h _ 1 + h _ 2 ) = ( d + 2 ) l - \\sum _ { i = 1 } ^ { q } e _ i . \\end{align*}"}
+{"id": "7015.png", "formula": "\\begin{align*} \\nu _ i ( f ' ) - \\nu _ i ( f ) = \\alpha _ { i } . \\end{align*}"}
+{"id": "8616.png", "formula": "\\begin{align*} & E \\left [ \\left ( Y e ^ { \\lambda s } - Z _ 0 ( s ) \\right ) ^ 2 \\right ] \\\\ & = E [ Z _ 0 ( s ) ^ 2 ] - 2 e ^ { \\lambda s } E [ Y Z _ 0 ( s ) ] + e ^ { 2 \\lambda s } E [ Y ^ 2 ] \\\\ & = e ^ { 2 \\lambda s } E [ Y ^ 2 ] - E [ Z _ 0 ( s ) ^ 2 ] . \\end{align*}"}
+{"id": "8149.png", "formula": "\\begin{align*} \\varphi _ n = ( - 1 ) ^ { n - 1 } \\left . \\frac { d } { d \\alpha } \\right | _ { \\alpha = 0 } \\sum _ { | T | = n } \\chi _ T ( \\alpha ) X _ T \\end{align*}"}
+{"id": "5759.png", "formula": "\\begin{align*} \\| T _ { \\alpha } ( \\vec { f } ) \\| _ { L ^ { q , \\infty } ( v _ { \\vec { w } } ^ { q } ) } & \\le C \\prod _ { j = 1 } ^ { m } \\| f _ { j } \\| _ { L ^ { p _ { j } } ( w _ { j } ^ { p _ { j } } ) } . \\end{align*}"}
+{"id": "3476.png", "formula": "\\begin{align*} C _ { \\mathrm { K P F } } = ( \\Delta \\log n ) ^ { \\kappa } ( \\log \\log n ) ^ { 1 + \\kappa } \\cdot O \\left ( \\frac { e ^ { 1 3 \\kappa } } { b ^ { 5 + 6 \\kappa } } \\right ) . \\end{align*}"}
+{"id": "2705.png", "formula": "\\begin{align*} \\begin{aligned} T _ 1 \\geq C s ^ 3 \\lambda ^ 4 \\iint _ Q \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d t - C s ^ 3 \\lambda ^ 3 \\int _ 0 ^ T \\int _ { \\omega } \\xi ^ 3 | u | ^ 2 d x d t . \\end{aligned} \\end{align*}"}
+{"id": "2639.png", "formula": "\\begin{align*} H _ 1 ( t , x , y ) = 2 ^ { \\sigma _ 1 l + k _ 1 } \\int _ { n _ 1 2 ^ { - ( \\sigma _ 1 l + k _ 1 ) } + P _ 1 ( 2 ^ { - l } t ) } ^ { ( n _ 1 + 1 ) 2 ^ { - ( \\sigma _ 1 l + k _ 1 ) } + P _ 1 ( 2 ^ { - l } t ) } | f _ 1 | \\left ( x + u , y \\right ) \\ , \\mathrm { d } u \\end{align*}"}
+{"id": "8735.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i _ 1 , \\ldots , i _ { l - a _ 1 - a _ 2 } \\leq p } ( \\mu _ { i _ 1 , \\ldots , i _ { l - a _ 1 - a _ 2 } } ^ { ( 1 ) } - \\mu _ { i _ 1 , \\ldots , i _ { l - a _ 1 - a _ 2 } } ^ { ( 2 ) } ) ^ 2 = \\| \\mathcal { T } _ { 1 , l } ^ { ( a _ 1 , a _ 2 ) } - \\mathcal { T } _ { 2 , l } ^ { ( a _ 1 , a _ 2 ) } \\| _ { \\rm F } ^ 2 , \\end{align*}"}
+{"id": "4413.png", "formula": "\\begin{align*} 2 N ( V ) = 2 { \\rm R e } ( v _ 3 , \\nabla ( v _ 1 \\cdot v _ 2 ) ) _ { L ^ 2 } = \\| \\nabla \\cdot v _ 3 - v _ 1 \\cdot \\overline { v _ 2 } \\| _ { L ^ 2 } ^ 2 - \\| \\nabla \\cdot v _ 3 \\| _ { L ^ 2 } ^ 2 - \\| v _ 1 \\cdot \\overline { v _ 2 } \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "3169.png", "formula": "\\begin{align*} \\kappa ( ( \\underbar { x } , \\underbar { y } ) ) : = \\sum _ { n = 1 } ^ { N } n \\Big [ S _ n ( \\mu ' ) ( x _ n ) - \\rho ( x _ n ) - S _ n ( \\mu ' ) ( y _ n ) - \\rho ( y _ n ) \\Big ] , ( \\underbar { x } , \\underbar { y } ) \\in R , \\end{align*}"}
+{"id": "3482.png", "formula": "\\begin{align*} \\mu ( \\sigma ) = \\frac { 1 } { Z _ H } e ^ { - H ( \\sigma ) } , \\end{align*}"}
+{"id": "5033.png", "formula": "\\begin{align*} \\overline g = E _ f ^ { - 1 } s ^ T . \\end{align*}"}
+{"id": "5547.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\sup _ { x \\in X ' } \\varepsilon _ { x , n } ( t ) = 0 . \\end{align*}"}
+{"id": "9282.png", "formula": "\\begin{align*} A _ { I } x ^ * = b _ { I } , \\mathcal { L } ( x ^ { * } , y ^ { * } ) \\leq \\mathcal { L } ( x , y ^ { * } ) \\forall x \\in X . \\end{align*}"}
+{"id": "9138.png", "formula": "\\begin{align*} \\begin{array} { r c l } x & = & F _ { x } ( \\varphi _ { [ 0 , R - 1 ] } ) \\\\ \\varphi _ { c } & = & \\varphi _ { c } \\\\ \\varphi _ { [ A , R ] } & = & \\varphi _ { [ A , R ] } \\ , , \\end{array} \\end{align*}"}
+{"id": "4032.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ D \\alpha _ i \\ , T _ i ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) = 0 , \\ , \\ \\forall \\ , \\ 0 \\leq n \\leq 2 k - 2 . \\end{align*}"}
+{"id": "521.png", "formula": "\\begin{align*} C _ { T } = c _ { 0 } ^ { - 1 } ( 1 + \\left \\| a \\right \\| _ { L ^ { \\infty } } ) e ^ { c _ { 0 } ^ { - 1 } \\left ( 1 + \\| a ^ { \\prime } \\| _ { L ^ { \\infty } } + \\| q \\| _ { L ^ { \\infty } } + 2 \\left \\| a \\right \\| _ { L ^ { \\infty } } \\right ) T } , \\end{align*}"}
+{"id": "3489.png", "formula": "\\begin{align*} \\pi ^ + _ v \\cdot \\left ( \\prod _ { i : v _ i \\in V _ E } P _ i \\right ) = \\pi ^ + _ v , \\end{align*}"}
+{"id": "5620.png", "formula": "\\begin{align*} P _ { \\mu , x } ^ { \\zeta , n } \\left ( L _ { x } , L _ { x } \\omega _ { n } \\right ) \\le e ^ { - ( \\delta - \\epsilon ) n } = e ^ { - 2 n \\epsilon } . \\end{align*}"}
+{"id": "3677.png", "formula": "\\begin{align*} \\begin{cases} \\nabla \\cdot ( y ^ { 1 - 2 \\gamma } \\nabla U ) = 0 & \\mathbb { R } ^ { n + 1 } _ + , \\\\ U ( x , 0 ) = u ( x ) & \\mathbb { R } ^ n , \\end{cases} \\end{align*}"}
+{"id": "2798.png", "formula": "\\begin{align*} \\| \\varphi _ t \\| _ { H _ \\Gamma ^ 1 ( \\partial \\mathrm { M } ) } ^ 2 = \\sum _ { N _ t ^ 1 } \\tau _ j ^ { - 2 } | a _ j | ^ 2 \\le t ^ { - 2 } \\sum _ j | a _ j | ^ 2 = t ^ { - 2 } \\| u \\| _ { L ^ 2 ( \\mathrm { M } _ 0 ) } ^ 2 , \\end{align*}"}
+{"id": "6319.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\dot h _ 1 ( t ) = - \\omega \\ , u _ 2 ( t ) , \\\\ \\displaystyle \\dot h _ 2 ( t ) = \\omega \\ , u _ 1 ( t ) . \\end{cases} \\end{align*}"}
+{"id": "93.png", "formula": "\\begin{align*} \\log | f ( 0 ) | + \\int _ 0 ^ r \\frac { n ( r ) } { r } d r = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\log | f ( r e ^ { i \\theta } ) | d \\theta , \\end{align*}"}
+{"id": "5372.png", "formula": "\\begin{align*} \\beta \\begin{bmatrix} \\tfrac { \\beta } { \\alpha } M _ 1 ^ H + M _ 2 ^ H \\\\ \\tfrac { \\beta } { \\alpha } M _ 1 ^ H - M _ 2 ^ H \\end{bmatrix} = \\begin{bmatrix} \\tfrac { \\beta } { \\alpha } I _ n & I _ n \\\\ \\tfrac { \\beta } { \\alpha } I _ n & - I _ n \\end{bmatrix} \\begin{bmatrix} M _ 1 ^ H \\\\ M _ 2 ^ H \\end{bmatrix} . \\end{align*}"}
+{"id": "4767.png", "formula": "\\begin{align*} \\left ( S ^ { - 1 } \\tilde { S } \\right ) _ { \\gamma _ i \\gamma _ i } = N _ { [ i ] } + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "7186.png", "formula": "\\begin{align*} w _ l : = \\begin{cases} \\phi ( w _ { l - 1 } ) & \\mbox { o n } \\cup _ { i \\in \\N } B ^ i _ l , \\\\ w _ { l - 1 } & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
+{"id": "5016.png", "formula": "\\begin{align*} \\left ( \\sum _ { j \\in M ^ c } | a _ j | ^ p \\right ) ^ \\frac { 1 } { p } \\leq \\varepsilon \\left ( \\sum _ { j = 1 } ^ n | a _ j | ^ p \\right ) ^ \\frac { 1 } { p } . \\end{align*}"}
+{"id": "1588.png", "formula": "\\begin{align*} F _ { 2 3 1 } ( t , x ) = \\dfrac { 1 } { 1 - ( t - 1 ) C _ 0 x - \\dfrac { x } { 1 - ( t - 1 ) C _ 1 x ^ 2 - \\dfrac { x } { 1 - ( t - 1 ) C _ 2 x ^ 3 - \\dfrac { x } { \\ddots } } } } . \\end{align*}"}
+{"id": "8638.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { - 1 } M ( \\tau _ N ) = \\nu \\int _ 0 ^ \\infty e ^ { - \\lambda s } ( 1 - p _ 0 ( s ) ) d s , \\end{align*}"}
+{"id": "5560.png", "formula": "\\begin{align*} \\mho _ { H } ( v ) : = { \\rm S h d } ( v ) \\cap \\partial \\left ( H \\backslash F \\right ) . \\end{align*}"}
+{"id": "3555.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { n = 0 } ^ { \\infty } C ( n ) x ^ { \\alpha n + \\beta - 1 } \\end{align*}"}
+{"id": "608.png", "formula": "\\begin{align*} \\sigma ( a ) = P _ H \\pi ( a ) | _ H , a \\in M . \\end{align*}"}
+{"id": "2097.png", "formula": "\\begin{align*} & F ( T ( n ) ) = 9 \\cdot 2 ^ { 2 n } - 3 \\cdot 2 ^ n - 1 , \\\\ & g ( T ( n ) ) = 9 \\cdot 2 ^ { 2 n - 1 } + ( 3 n - 5 ) 2 ^ { n - 1 } . \\end{align*}"}
+{"id": "8980.png", "formula": "\\begin{align*} I ( R ) = \\int _ { B _ R } \\psi _ + ^ 2 w ^ 2 , \\end{align*}"}
+{"id": "5971.png", "formula": "\\begin{align*} r _ { J ^ - } [ I ( f ) ] ( g ) & = r _ { J ^ - } [ g ^ { - 1 } I ( f ) ] ( \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} ) \\\\ & = g ^ { - 1 } r _ { J ^ - } [ I ( f ) ] ( \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} ) \\\\ & = 2 \\tfrac { \\partial } { \\partial \\overline { z } } f ( g i ) . \\end{align*}"}
+{"id": "2952.png", "formula": "\\begin{align*} g l _ 1 ( K U ) ^ 1 ( X ) : = [ X , B G L _ 1 ( K U ) ] \\ . \\end{align*}"}
+{"id": "3719.png", "formula": "\\begin{align*} \\left ( \\sigma \\circ ( \\sigma \\circ f ) \\ , , \\ , M \\ , , \\ , ( ( \\sigma \\circ f ) ^ * h ) \\circ ( f ^ * h \\circ \\phi ) \\ , , \\ , t \\right ) & = \\left ( f \\ , , \\ , M \\ , , \\ , ( f ^ * \\sigma ^ * h ) \\circ f ^ * h \\circ \\phi \\ , , \\ , t \\right ) \\\\ & = \\left ( f \\ , , \\ , M \\ , , \\ , f ^ * ( \\sigma ^ * h \\circ h ) \\circ \\phi \\ , , \\ , t \\right ) \\\\ & = \\left ( f \\ , , \\ , M \\ , , \\ , f ^ * ( \\textnormal { I d } _ L ) \\circ \\phi \\ , , \\ , t \\right ) , \\ , \\ , \\ , \\end{align*}"}
+{"id": "457.png", "formula": "\\begin{align*} t ^ \\ast : = \\min _ { r \\in [ 0 , 1 ] } t ^ \\ast ( r ) . \\end{align*}"}
+{"id": "5869.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Theta _ k ^ { n _ 1 } } | \\phi _ E ( x ) | ^ 2 = 1 - \\sum _ { x \\in \\Lambda _ { L _ { k - 1 } } \\setminus \\Theta _ k ^ { n _ 1 } } | \\phi _ E ( x ) | ^ 2 \\geq 1 - C L ^ { \\beta d } e ^ { - m ' L ^ { \\rho ^ { 2 n _ 1 - 1 } } } \\geq 1 / 2 \\end{align*}"}
+{"id": "2058.png", "formula": "\\begin{align*} \\widehat { P _ { \\leq N } u } ( k ) : = \\widehat { u _ { \\leq N } } ( k ) : = \\psi \\left ( \\frac { k } { N } \\right ) \\hat { u } ( k ) , \\end{align*}"}
+{"id": "727.png", "formula": "\\begin{align*} A _ { B , N } ^ * = \\frac 1 2 ( { \\rm I } _ { N } - R _ { B , N } ) , \\| R _ { B , N } \\| < 1 \\ , , \\end{align*}"}
+{"id": "2380.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 ) = & f ( 0 , 0 ) \\\\ & + 2 ^ { - 1 } [ f ( x _ 1 , 0 ) + f ( x _ 1 + x _ 2 , 0 ) - f ( 0 , 0 ) - f ( x _ 2 , 0 ) \\\\ & + f ( 0 , x _ 2 ) + f ( 0 , x _ 1 + x _ 2 ) - f ( 0 , 0 ) - f ( 0 , x _ 1 ) \\\\ & + f ( x _ 1 , x _ 1 ) + f ( x _ 2 , x _ 2 ) - f ( 0 , 0 ) - f ( x _ 1 + x _ 2 , x _ 1 + x _ 2 ) ] . \\end{align*}"}
+{"id": "8643.png", "formula": "\\begin{align*} \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } ( 1 - y ) y ^ { j - 1 } d y & = \\sum _ { k = 0 } ^ \\infty p ^ k \\left ( \\int _ 0 ^ { 1 } ( 1 - y ) y ^ { j + k - 1 } d y \\right ) \\\\ & = \\sum _ { k = 0 } ^ \\infty \\frac { p ^ k } { ( j + k ) ( j + k + 1 ) } . \\end{align*}"}
+{"id": "1157.png", "formula": "\\begin{align*} T ( f \\cdot g ) = T ( f ) \\cdot g + f \\cdot T ( g ) + 2 A ( f ) \\cdot A ( g ) \\end{align*}"}
+{"id": "2906.png", "formula": "\\begin{align*} 1 2 \\cdot \\mu _ { D R } ^ { p \\delta } = - [ ( \\mu _ { D D , \\delta } ) _ c ] \\end{align*}"}
+{"id": "65.png", "formula": "\\begin{align*} \\Omega _ 1 ( \\mathcal { G } _ 1 ^ 1 ( \\mathbb { Q } ) ) = \\Omega _ 2 ( \\mathcal { G } _ 2 ^ 1 ( \\mathbb { Q } ) ) . \\end{align*}"}
+{"id": "6205.png", "formula": "\\begin{align*} \\Psi ( v _ 1 , \\ldots , v _ { 2 f } ) = \\{ v _ 1 , v _ 2 \\} \\cup \\cdots \\cup \\{ v _ { 2 f - 1 } , v _ { 2 f } \\} . \\end{align*}"}
+{"id": "2932.png", "formula": "\\begin{align*} ( - ) ^ \\delta : M & \\to M \\\\ m & \\mapsto m ^ \\delta : = \\delta \\ast m \\end{align*}"}
+{"id": "1043.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } ( - 1 ) ^ k \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 4 ( d ) _ k ^ 3 ( 1 - d ) _ k ^ 3 } { ( 1 ) _ { 2 k } ^ 3 ( \\frac { 1 } { 2 } + d ) _ { 2 k } ( \\frac { 3 } { 2 } - d ) _ { 2 k } } \\Omega _ k ( d ) = \\frac { 1 - 2 d } { \\pi } \\tan ( d \\pi ) , \\end{align*}"}
+{"id": "413.png", "formula": "\\begin{align*} S ^ { - 1 } \\left ( p / \\sqrt { | u _ n | } - ( R _ 1 , R _ 2 ) \\begin{bmatrix} \\sqrt { | u _ n | } & 0 \\\\ 0 & \\sqrt { | u _ n | } \\end{bmatrix} \\begin{bmatrix} u _ n + p / u _ n \\\\ u _ { \\tau } \\end{bmatrix} \\right ) = G . \\end{align*}"}
+{"id": "5276.png", "formula": "\\begin{align*} 1 + q \\rho d X + \\binom { q } { 2 } \\rho ^ 2 d ^ 2 X ^ 2 + e \\frac { q ( q - 2 ) } { 6 } \\rho ^ 3 | d X | ^ 2 , \\end{align*}"}
+{"id": "5779.png", "formula": "\\begin{align*} \\| \\mathcal { M } _ { \\alpha } ( \\vec { f } ) \\| _ { L ^ { q } ( v _ { \\vec { w } } ^ { q } ) } & \\le C \\prod _ { j = 1 } ^ { m } \\| f _ { j } \\| _ { L ^ { p _ { j } } ( w _ { j } ^ { p _ { j } } ) } . \\end{align*}"}
+{"id": "2394.png", "formula": "\\begin{align*} \\vec { u } ( A _ { 0 } , A _ { 1 } ) + \\vec { u } ( A _ { 0 } , A _ { 3 } ) = - ( \\vec { u } ( A _ { 0 } , A _ { 2 } ) + \\vec { u } ( A _ { 0 } , A _ { 4 } ) ) \\end{align*}"}
+{"id": "6543.png", "formula": "\\begin{align*} \\xi _ F ( s ) = \\omega \\ , \\overline { \\xi _ F ( 1 - \\bar { s } ) } , \\end{align*}"}
+{"id": "4067.png", "formula": "\\begin{align*} S _ { \\mathrm { o f f } } = \\sum _ { d _ 1 , \\ , d _ 2 < D } \\ , \\ , \\ , \\ & \\sum _ { I d _ 1 , \\ , J d _ 2 < D } x _ { I d _ 1 } \\bar { x } _ { J d _ 2 } \\sqrt { I J } \\sum _ { L , M \\geq 1 } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } \\\\ & \\times \\left ( 2 \\pi i ^ { 2 k } \\sum _ { p | c } \\frac { \\mathcal { S } ( I L , J M , c ) } { c } \\ , J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { I L J M } } { c } \\right ) \\right ) . \\end{align*}"}
+{"id": "9098.png", "formula": "\\begin{align*} p _ { ( M - \\lambda _ k I ) } = p _ { ( N - \\lambda _ k I ) } + | \\mathcal { F } ^ c | , \\end{align*}"}
+{"id": "6685.png", "formula": "\\begin{align*} a _ { i j } = \\frac 1 2 \\epsilon _ { i m n } \\epsilon _ { j k l } \\partial _ { m } \\eta _ k \\partial _ { n } \\eta _ l , \\end{align*}"}
+{"id": "1359.png", "formula": "\\begin{align*} \\eta ( F _ { R } ) - \\eta ( F _ { L } ) & = \\int _ { F _ { L } } ^ { F _ { R } } \\eta ' ( y ) d y = - \\int _ { F _ { L } } ^ { F _ { R } } \\eta '' ( y ) ( y - F _ { L } ) d y + \\left . ( y - F _ { L } ) \\eta ' \\right | _ { F _ { L } } ^ { F _ { R } } \\\\ & = - \\int _ { F _ { L } } ^ { F _ { R } } \\eta '' ( y ) ( y - F _ { L } ( t ) ) d y + ( F _ { R } - F _ { L } ) \\eta ' ( F _ { R } ) . \\end{align*}"}
+{"id": "96.png", "formula": "\\begin{align*} e ^ { - i t h ^ { - 1 } \\tilde { P } _ h ( 0 ) } f = e ^ { - i t h ^ { - 1 } ( - i h X - i ( q _ 1 + W ) ) } f \\end{align*}"}
+{"id": "7044.png", "formula": "\\begin{align*} \\nu _ i \\left ( ( f _ \\gamma ) _ { \\tilde { \\bf Q } } \\right ) = \\nu \\left ( ( f _ \\gamma ) _ { \\tilde { \\bf Q } } \\right ) . \\end{align*}"}
+{"id": "777.png", "formula": "\\begin{align*} a _ 1 ^ 2 = O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 . \\end{align*}"}
+{"id": "7194.png", "formula": "\\begin{align*} \\frac { \\abs { w _ h - \\lambda _ h ^ { ( 1 / 2 ) } } ^ 2 } { t _ h } + 2 \\left ( w _ h - \\lambda _ h \\right ) \\cdot \\frac { \\lambda _ h ^ { ( 1 / 2 ) } } { t _ h } = t _ h \\left ( 1 - \\frac { \\lambda _ h ^ { ( 1 / 2 ) } } { t _ h } \\right ) \\left ( 1 + \\frac { \\lambda _ h ^ { ( 1 / 2 ) } } { t _ h } \\right ) . \\end{align*}"}
+{"id": "9240.png", "formula": "\\begin{align*} \\frac { 1 } { m ! } | P ^ { ( m ) } ( p ) - Q ^ { ( m ) } ( p ) | \\le \\sum _ { k = m } ^ { J - 1 } 2 ^ k \\delta _ k < \\frac 1 3 \\frac { \\varepsilon _ m } { m ! } \\end{align*}"}
+{"id": "2381.png", "formula": "\\begin{align*} \\begin{array} { l l } g _ M ( x _ 1 , \\dots , x _ k ) = & | R _ 1 | ^ { 1 - k } \\left ( \\sum _ { a _ 2 , \\dots , a _ k \\in R _ 1 } f ( x _ 1 + \\sum _ { i = 2 } ^ k a _ i x _ i , x _ 2 , \\dots , x _ k ) \\right . \\\\ & \\left . - \\sum _ { a _ 2 , \\dots , a _ k \\in R _ 1 } f ( e x _ 1 + \\sum _ { i = 2 } ^ k a _ i x _ i , x _ 2 , \\dots , x _ k ) \\right ) . \\end{array} \\end{align*}"}
+{"id": "4567.png", "formula": "\\begin{align*} A ^ \\vee = R \\circ F = \\langle F , \\{ Y ^ i , 0 \\in [ 0 , 5 ] \\} , \\{ X ^ i Z , i \\in [ 0 , 3 ] \\} , \\{ X ^ i , i \\in [ 1 , 4 ] \\} , W , W ^ 2 \\rangle , \\end{align*}"}
+{"id": "9338.png", "formula": "\\begin{align*} J ( u ) = \\bar { \\mathbb { E } } \\bigg [ \\int _ 0 ^ \\infty e ^ { - \\beta t } f \\Big ( x _ t ^ u , y _ t ^ u , z _ t ^ u , \\tilde { z } _ t ^ u , \\int _ { \\mathcal { E } } \\gamma _ { ( t , e ) } ^ u \\nu ( d e ) , u _ t \\Big ) d t + \\phi ( y _ 0 ^ u ) \\bigg ] , \\ \\beta \\ , \\end{align*}"}
+{"id": "6727.png", "formula": "\\begin{align*} X _ \\varphi = \\overline { [ \\underline { \\varphi } , \\varphi ] } . \\end{align*}"}
+{"id": "8073.png", "formula": "\\begin{align*} \\int _ { \\Omega } | X u | ^ { p - 2 } X u \\cdot X \\varphi d x = \\lambda \\int _ { \\Omega } | u | ^ { p - 2 } u \\varphi d x \\end{align*}"}
+{"id": "8579.png", "formula": "\\begin{align*} I _ { j , + } ^ i ( t ) : = \\sum _ { \\ell = 1 } ^ \\infty 1 _ { \\{ \\tau _ { j , + } ^ i ( \\ell ) \\leq t \\} } , I _ { j , - } ^ i ( t ) : = \\sum _ { \\ell = 1 } ^ \\infty 1 _ { \\{ \\tau _ { j , - } ^ i ( \\ell ) \\leq t \\} } , \\end{align*}"}
+{"id": "2628.png", "formula": "\\begin{align*} | \\mathcal { E } _ 1 | & = \\int _ { \\mathcal { E } _ 1 ' } \\ ! \\int _ { I } \\ ! \\mathbf { 1 } _ { \\mathcal { E } _ 1 } ( \\mathbf { z } - \\mathbf { p } ( t ) , t ) \\ , \\mathrm { d } t \\mathrm { d } \\mathbf { z } + \\int _ { \\mathbb { R } ^ 2 \\setminus \\mathcal { E } _ 1 ' } \\ ! \\int _ { I } \\ ! \\mathbf { 1 } _ { \\mathcal { E } _ 1 } ( \\mathbf { z } - \\mathbf { p } ( t ) , t ) \\ , \\mathrm { d } t \\mathrm { d } \\mathbf { z } \\\\ & \\leq 2 | \\mathcal { E } _ 1 ' | + c _ 1 C _ { P _ 1 , P _ 2 } | \\mathcal { E } _ 1 | \\end{align*}"}
+{"id": "3841.png", "formula": "\\begin{align*} c _ { Y _ { \\ell } } \\left ( y _ { \\ell } , y _ { \\ell } ^ { \\prime } \\right ) = \\inf _ { x _ { \\ell } , x _ { \\ell } ^ { \\prime } \\in \\mathcal { X } } c _ { \\ell } \\left ( \\left ( y _ { \\ell } , x _ { \\ell } \\right ) , \\left ( y _ { \\ell } ^ { \\prime } , x _ { \\ell } ^ { \\prime } \\right ) \\right ) . \\end{align*}"}
+{"id": "6347.png", "formula": "\\begin{align*} ( \\bar x , y ) = Q _ { \\psi _ 0 + ( y - y _ 0 ) \\bar x } , y \\in I . \\end{align*}"}
+{"id": "1376.png", "formula": "\\begin{align*} v _ 1 , \\ , v _ 2 = - g _ 4 v _ 1 , \\ , v _ 3 = - g _ 7 v _ 1 , \\ , v _ 4 = - g _ 3 v _ 1 , \\ , v _ 5 = - g _ 6 v _ 1 , \\ , v _ 6 = - g _ 2 v _ 1 , \\ , v _ 7 = - g _ 5 v _ 1 , \\end{align*}"}
+{"id": "7147.png", "formula": "\\begin{align*} \\frac { } { t } \\frac { 1 } { 2 } \\| u \\| _ { H ^ { - 1 } ( \\Omega ) } ^ 2 = \\int _ { \\Omega } \\Delta w ( - \\Delta _ N ) ^ { - 1 } u \\ : x = - \\int _ { \\Omega } w u \\ : x . \\end{align*}"}
+{"id": "2281.png", "formula": "\\begin{align*} w ( z ) = \\Phi ( z ) + \\widetilde { T } ( f ) ( z ) , \\end{align*}"}
+{"id": "791.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ n } u \\mathrm { d } A = - \\int _ { \\mathbb { S } ^ n } \\dfrac { n - j - 1 } { 2 } u ^ 2 \\mathbb { d } A - \\int _ { \\mathbb { S } ^ n } \\dfrac { j + 1 } { 2 n } | \\nabla u | ^ 2 \\mathrm { d } A + O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 + O ( \\varepsilon ) \\| \\nabla u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 . \\end{align*}"}
+{"id": "1338.png", "formula": "\\begin{align*} \\dot { x } _ { i } ^ { N } ( t ) = \\frac { 1 } { N } \\stackrel [ j = 1 ] { N } { \\sum } \\nabla V ( x _ { j } ( t ) - x _ { i } ( t ) ) , x _ { i } ^ { N } ( 0 ) = x _ { i } ^ { 0 , N } \\end{align*}"}
+{"id": "6562.png", "formula": "\\begin{align*} \\| H ( f ) \\| _ { L _ { | x | _ h } ^ p L _ { \\theta } ^ { \\bar { p } _ 1 } ( \\mathbb H ^ n ) \\rightarrow L _ { | x | _ h } ^ p L _ { \\theta } ^ { \\bar { p } _ 2 } ( \\mathbb H ^ n ) } = \\omega _ Q ^ { 1 / \\bar { p } _ 2 - 1 / \\bar { p } _ 1 } \\frac { \\omega _ Q Q } { ( Q - Q / p ) Q / p } . \\end{align*}"}
+{"id": "4129.png", "formula": "\\begin{align*} { T _ { \\varepsilon _ i } ^ { \\vec { v } } } ^ T = \\begin{pmatrix} x _ i & - \\alpha _ 1 y _ i - \\alpha _ 0 z _ i & - \\alpha _ 0 y _ i \\\\ y _ i & x _ i + \\alpha _ 2 y _ i & - \\alpha _ 0 z _ i \\\\ z _ i & y _ i + \\alpha _ 2 z _ i & x _ i + \\alpha _ 2 y _ i + \\alpha _ 1 z _ i \\end{pmatrix} . \\end{align*}"}
+{"id": "8775.png", "formula": "\\begin{align*} \\Gamma ( t ) = \\begin{cases} \\{ 0 \\} & \\mbox { i f } t < 0 \\\\ \\{ 1 \\} & \\mbox { i f } t > 0 \\\\ [ 0 , 1 ] & \\mbox { i f } t = 0 \\end{cases} . \\end{align*}"}
+{"id": "6316.png", "formula": "\\begin{align*} \\xi : M \\times \\R ^ 2 \\to T M , \\xi ( x , y , z ; u _ 1 , u _ 2 ) = \\bigg ( x , y , z ; u _ 1 , u _ 2 , \\frac 1 2 ( u _ 2 x - u _ 1 y ) \\bigg ) . \\end{align*}"}
+{"id": "3516.png", "formula": "\\begin{align*} \\lambda \\in \\Lambda ^ { l } ( \\sigma ) \\iff \\dim \\ker S _ 1 = l . \\end{align*}"}
+{"id": "3856.png", "formula": "\\begin{align*} f _ { \\theta , \\lambda } ( v ) & = \\sup _ { y _ 1 ' , y _ 2 ' , x ' } \\left [ f ( y _ 1 ' , y _ 2 ' , y ; \\theta ) - \\lambda _ 1 c _ 1 ( ( y _ 1 , x _ 1 ) , ( y _ 1 ' , x ' ) ) - \\lambda _ 2 c _ 2 ( ( y _ 2 , x _ 2 , y _ 2 ' , x ' ) ) \\right ] \\end{align*}"}
+{"id": "7410.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M _ N } X _ { N , M _ N } ^ { ( i ) } \\xrightarrow [ N \\to \\infty ] { d } \\ , \\mathcal { N } \\big ( 0 \\ , , \\sigma _ { \\varphi } ^ 2 \\big ) \\ , . \\end{align*}"}
+{"id": "7894.png", "formula": "\\begin{align*} \\sum _ { h \\in H } \\chi ^ { [ n - 1 , 1 ] } ( x h ) = - \\frac { | H | } { \\binom { n - k } { k } } \\binom { n - k - 1 } { k - 1 } . \\end{align*}"}
+{"id": "3851.png", "formula": "\\begin{align*} h _ 0 ( x ) = \\inf _ { u \\in \\mathcal { X } : d ( u ) = 0 } \\| x - u \\| _ 2 h _ 1 ( x ) = \\inf _ { u \\in \\mathcal { X } : d ( u ) = 1 } \\| x - u \\| _ 2 . \\end{align*}"}
+{"id": "7712.png", "formula": "\\begin{align*} \\vartheta _ L ( f ) _ P ( g ) = \\vartheta _ L ( f ) _ { P _ { \\le r } } ( g ) . \\end{align*}"}
+{"id": "2029.png", "formula": "\\begin{align*} G ( z ) : = H ( z ) e ^ { \\alpha \\vert z \\vert ^ 2 } , \\ , \\ , z \\in \\Omega , \\end{align*}"}
+{"id": "2177.png", "formula": "\\begin{align*} \\left \\Vert \\partial _ { t } ^ { s } \\widetilde { u } \\right \\Vert _ { H ^ { 2 , 1 } \\left ( Q _ { \\varepsilon , T } \\right ) } , \\left \\Vert \\partial _ { t } ^ { s } \\widetilde { m } \\right \\Vert _ { H ^ { 2 , 1 } \\left ( Q _ { \\varepsilon , T } \\right ) } \\leq C \\delta ^ { 1 - \\rho } , \\forall \\delta \\in \\left ( 0 , \\delta _ { 0 } \\right ) , s = 0 , 1 , 2 . \\end{align*}"}
+{"id": "3618.png", "formula": "\\begin{align*} p & \\ = \\ ( a _ m \\cdots a _ 0 ) _ { 1 0 } , \\\\ r ( p ) & \\ = \\ ( a _ 0 \\cdots a _ m ) _ { 1 0 } , \\\\ q & \\ = \\ ( b _ m \\cdots b _ 0 ) _ { 1 0 } , \\end{align*}"}
+{"id": "2188.png", "formula": "\\begin{align*} \\Re ( g _ { \\lambda _ i } ( z ) ) = \\Re ( \\lambda _ i - 2 z + \\lambda _ i z ^ 2 ) = \\lambda _ i + \\Re ( - 2 z + \\lambda _ i z ^ 2 ) \\geq \\lambda _ i - | - 2 z + \\lambda _ i z ^ 2 | \\geq \\frac { \\lambda _ i } { 2 } , \\end{align*}"}
+{"id": "1539.png", "formula": "\\begin{align*} ( x _ 0 , t _ 0 ) + Q _ { \\rho } ( \\theta ) = K _ { \\rho } ( x _ 0 ) \\times ( t _ 0 - \\theta \\rho ^ { s p } , t _ 0 ] . \\end{align*}"}
+{"id": "6540.png", "formula": "\\begin{align*} \\nu ( \\{ 0 \\} ) = 0 , \\int _ { - \\infty } ^ { \\infty } \\min ( 1 , \\lambda ^ 2 ) \\ , \\nu ( d \\lambda ) < \\infty \\end{align*}"}
+{"id": "773.png", "formula": "\\begin{align*} \\begin{aligned} A ^ m = & \\frac { \\partial } { \\partial \\left ( | \\nabla u | ^ 2 \\right ) } \\left . \\left [ ( - 1 ) ^ m \\rho ^ m { n - m \\choose k - m } \\frac { \\phi '^ { k - m } } { D ^ { k + 1 } } \\phi ^ { n - m + 1 } \\right ] \\right | _ { u = | \\nabla u | ^ 2 = 0 } \\\\ = & ( - 1 ) ^ { m + 1 } \\rho ^ { m + 2 } \\frac { k + 1 } { 2 } { n - m \\choose k - m } \\phi '^ { k - m } ( \\rho ) \\phi ^ { n - m - k - 2 } ( \\rho ) , \\end{aligned} \\end{align*}"}
+{"id": "6641.png", "formula": "\\begin{align*} Y ^ { ( i ) } _ { \\bar { a } } = Y ^ { ( i ) } _ { ( a _ 1 , \\dots , a _ m ) } \\end{align*}"}
+{"id": "2040.png", "formula": "\\begin{align*} u _ { \\alpha \\bar \\beta } ( 0 ) = - u _ { x _ n } ( 0 ) \\rho _ { \\alpha \\bar \\beta } ( 0 ) , ~ ~ \\alpha , \\beta \\leq n - 1 . \\end{align*}"}
+{"id": "1125.png", "formula": "\\begin{align*} \\zeta = \\exp ( \\sum _ j h _ j X _ j ) ( z ) , \\end{align*}"}
+{"id": "3808.png", "formula": "\\begin{align*} c _ 1 ( ( y _ 1 , x ) , ( y _ 1 ' , x ' ) ) = \\| x - x ' \\| _ { p } + \\kappa _ 1 | y _ 1 - y _ 1 ' | \\end{align*}"}
+{"id": "1123.png", "formula": "\\begin{align*} \\L u : = \\sum _ { i , j = 1 } ^ { m } a _ { i j } X _ i X _ j u - X _ 0 u , \\end{align*}"}
+{"id": "5791.png", "formula": "\\begin{align*} \\frac { f ^ { ( k ) } } { f } = F ^ k + \\frac { k ( k - 1 ) } { 2 } F ^ { k - 2 } F ' + P _ { k - 2 } ( F ) \\end{align*}"}
+{"id": "8226.png", "formula": "\\begin{align*} L _ N Y _ { s } ^ N ( \\phi ) & = \\frac { 1 } { \\sqrt { N } } \\sum _ { ( x , \\sigma ) \\in V } ( \\eta _ s ( x , \\sigma ) - \\rho ) \\cdot ( A \\phi ) ( \\tfrac { x } { N } , \\sigma ) + R _ 1 ( \\phi , N , s ) , \\end{align*}"}
+{"id": "1298.png", "formula": "\\begin{align*} \\psi ( t ) = K _ { t \\wedge \\tau } ^ { - 1 } ( X ( t ) + ( t \\wedge \\tau ) \\eta ) . \\end{align*}"}
+{"id": "5410.png", "formula": "\\begin{align*} \\sum _ { k > L } \\frac { R ^ k } { k } \\min \\{ q ^ { k / 2 } , d ( \\chi ) \\} \\le d ( \\chi ) \\sum _ { k > L } \\frac { R ^ k } { k } \\le \\frac { d ( \\chi ) } { L + 1 } \\sum _ { k > L } R ^ k = \\frac { d ( \\chi ) } { L + 1 } \\frac { R ^ { L + 1 } } { 1 - R } , \\end{align*}"}
+{"id": "6837.png", "formula": "\\begin{align*} \\sum _ { s _ 1 \\geq \\dots \\geq s _ { r - 1 } \\geq 0 } \\frac { q ^ { s _ 1 ^ 2 + \\dots + s _ { r - 1 } ^ 2 - s _ 1 - \\dots - s _ i } } { ( q ) _ { s _ 1 - s _ 2 } \\dots ( q ) _ { s _ { r - 1 } } } = \\sum _ { k = 0 } ^ { i } \\frac { ( q ^ { 2 r + 1 } , q ^ { r - i + k } , q ^ { r + i - k + 1 } ; q ^ { 2 r + 1 } ) _ \\infty } { ( q ) _ \\infty } . \\end{align*}"}
+{"id": "506.png", "formula": "\\begin{align*} L p _ { \\sigma ( j ) } ( y _ j ) = f ( y _ j ) , j = 1 , \\ldots , M , \\end{align*}"}
+{"id": "5059.png", "formula": "\\begin{align*} \\sigma _ { L ^ 2 _ { \\rm p e r } ( 0 , N T ) } \\left ( \\mathcal { A } [ \\phi ] \\right ) \\cap B ( 0 , r _ N ) = \\{ 0 \\} , \\end{align*}"}
+{"id": "5058.png", "formula": "\\begin{align*} \\mathcal A [ \\phi ] = - I + \\mathcal { J } \\mathcal { L } [ \\phi ] , \\end{align*}"}
+{"id": "6441.png", "formula": "\\begin{align*} T _ { \\varphi } ( x , i ) : = \\begin{cases} ( x , i + 1 ) & 0 \\leq i < h - 1 \\\\ ( \\varphi ( x ) , 0 ) & i = h - 1 \\end{cases} \\end{align*}"}
+{"id": "1060.png", "formula": "\\begin{align*} N _ 1 \\coloneqq ( d + 1 ) \\cdot \\min \\left \\{ \\prod _ { i = 1 } ^ n ( h _ i + 1 ) , \\binom { n + h } { n } \\right \\} \\end{align*}"}
+{"id": "2343.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\int _ { - R } ^ { R } \\int _ { - R } ^ R \\widehat { \\gamma } _ n ( t , s , x - y ) d x d y = 4 \\pi C _ H ^ n R \\int _ { T _ n ( s ) } \\big ( \\ell _ R * \\widehat { g } _ { \\pmb { t } _ n } ^ { ( n ) } \\big ) ( 0 ) d \\pmb { t } _ n , \\end{align*}"}
+{"id": "6451.png", "formula": "\\begin{align*} F _ { K _ 1 K _ 2 K _ 3 K } ( s , x ) = e ^ { - z ( s ) | x | ^ 2 + \\zeta ( s ) \\cdot x + \\gamma ( s ) } \\end{align*}"}
+{"id": "3290.png", "formula": "\\begin{align*} \\widetilde { \\rho } _ t ( t , \\kappa ( t ) ) + 2 { \\rm R e } ( \\widetilde { \\rho } _ w ( t , \\kappa ( t ) ) \\kappa ' ( t ) ) = 0 { \\rm \\ f o r \\ } t > 0 . \\end{align*}"}
+{"id": "5983.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } [ u ( b ) ] f ( [ \\epsilon , x ] ) = e ^ { \\pi i \\epsilon x ^ 2 b } f ( [ \\epsilon , x ] ) ; \\end{align*}"}
+{"id": "8061.png", "formula": "\\begin{align*} | A + \\{ a _ { 1 , 1 } , \\dots , a _ { r _ 0 , 3 } \\} | & \\geq \\sum _ { i = 1 } ^ { r _ 0 } | p _ 1 + \\{ a _ { i , 1 } , \\dots , a _ { i , 3 } \\} | \\\\ & \\geq \\sum _ { i = 1 } ^ { r _ 0 } ( | p _ 1 | + | p _ i | - 1 ) \\geq r _ 0 | p _ 1 | \\\\ & > ( d + 1 ) ^ 2 | A | ( 1 - 1 0 0 ^ { - d ^ 2 } ) / d \\\\ & \\geq ( d + 2 ) | A | , \\end{align*}"}
+{"id": "1527.png", "formula": "\\begin{align*} \\left ( \\sum _ { \\theta = 1 } ^ { d } ( h _ { p \\theta + p - 1 } t ^ { p \\theta + p - 1 } - g _ { p \\theta + p - 1 } ( a t + b ) ^ { p \\theta + p - 1 } ) \\right ) + f _ 0 ( t ) , \\end{align*}"}
+{"id": "9161.png", "formula": "\\begin{align*} \\begin{aligned} x ^ { 1 , + } & = x ^ { 1 } + u ^ { 1 } T \\cos \\left ( x ^ { 3 } + u ^ { 2 } \\tfrac { T } { 2 } \\right ) \\tfrac { \\sin \\left ( u ^ { 2 } \\tfrac { T } { 2 } \\right ) } { u ^ { 2 } \\tfrac { T } { 2 } } \\\\ x ^ { 2 , + } & = x ^ { 2 } + u ^ { 1 } T \\sin \\left ( x ^ { 3 } + u ^ { 2 } \\tfrac { T } { 2 } \\right ) \\tfrac { \\sin \\left ( u ^ { 2 } \\tfrac { T } { 2 } \\right ) } { u ^ { 2 } \\tfrac { T } { 2 } } \\\\ x ^ { 3 , + } & = x ^ { 3 } + u ^ { 2 } T \\ , , \\end{aligned} \\end{align*}"}
+{"id": "8046.png", "formula": "\\begin{align*} \\mu _ { j + 1 } ^ 2 ( A ) - \\mu _ { j } ^ 2 ( A _ { \\mathcal { S } } ) \\le \\mu _ 1 ^ 2 ( R ) = \\| A h \\| ^ 2 - | \\langle h , A h \\rangle | ^ 2 \\end{align*}"}
+{"id": "8441.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\intop _ { \\mu - i \\infty } ^ { \\mu + i \\infty } \\Gamma ( z ) \\ , \\Gamma ( s - z ) \\ , x ^ { - 2 z } d z = \\frac { \\Gamma ( s ) } { ( 1 + x ^ { 2 } ) ^ { s } } , \\ , \\ , \\ , \\ , x > 0 . \\end{align*}"}
+{"id": "1145.png", "formula": "\\begin{align*} ( T f ) ( x ) = \\Delta _ { h } f ( x ) = f ( x + h ) - f ( x ) \\end{align*}"}
+{"id": "8069.png", "formula": "\\begin{align*} F ^ { \\prime } ( u ) = \\mu G ^ { \\prime } ( u ) \\quad \\enspace \\mathcal { G } \\end{align*}"}
+{"id": "2494.png", "formula": "\\begin{align*} \\xi ^ { \\tau } ( x ) = \\frac { 1 } { \\tau ^ { 2 } } \\xi \\left ( \\frac { x } { \\tau } \\right ) , \\eta ^ { \\tau } ( x ) = \\frac { 1 } { \\tau ^ { 2 } } \\eta \\left ( \\frac { x } { \\tau } \\right ) , \\quad \\tau > 0 , \\end{align*}"}
+{"id": "3420.png", "formula": "\\begin{align*} \\Upsilon _ { j } ( u ) = \\frac { \\int _ { B _ j } d ( u , P ) P d P } { \\int _ { B _ j } d ( u , P ) d P } \\in B _ j . \\end{align*}"}
+{"id": "6013.png", "formula": "\\begin{align*} \\mathcal { F } _ { Y ^ { \\ast } X ^ { \\ast } } ( f ) ( x e _ { 1 } ^ { \\ast } ) & = \\int _ { t c e _ 1 \\in X } f ( [ t c e _ { 1 } , 0 ] [ x e ^ { \\ast } _ { 1 } , 0 ] ) | A _ { Y ^ { \\ast } X ^ { \\ast } } | ^ { 1 / 2 } d t \\\\ & = \\int _ { \\R } \\psi ( t c x ) f ( [ t c e _ { 1 } , 0 ] ) c ^ { \\tfrac { 1 } { 2 } } d t \\\\ & = \\int _ { \\R } \\psi ( t x ) f ( [ t e _ { 1 } , 0 ] ) c ^ { - \\tfrac { 1 } { 2 } } d t . \\end{align*}"}
+{"id": "7213.png", "formula": "\\begin{align*} \\forall s , t \\in [ 1 / 2 , 1 ) , z _ h ^ { ( s ) } = z _ h ^ { ( t ) } = v _ h \\mbox { a . e . o n } B _ 1 \\setminus \\left ( \\cup _ { \\mathcal { F } ^ { ( s ) } _ h } B \\bigcup \\cup _ { \\mathcal { F } ^ { ( t ) } _ h } B \\right ) \\end{align*}"}
+{"id": "1613.png", "formula": "\\begin{align*} N ^ { \\ , ( 1 ) } ( X , Y ) = [ { f } , { f } ] ( X , Y ) - 2 \\sum \\nolimits _ { i } d \\eta ^ i ( X , Y ) \\ , \\xi _ i . \\end{align*}"}
+{"id": "2225.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial h } \\lambda ( y ^ u , y ^ h ) & = e ^ { \\gamma } \\log y \\left ( \\int _ { 1 } ^ { u / h } y ^ { h t - u } \\omega ( t ) \\ , d t - y ^ { h - u } \\right ) - h ^ { - 1 } \\lambda ( y ^ { u - h } , y ^ h ) \\\\ & = h ^ { - 1 } \\lambda ( y ^ u , y ^ h ) - e ^ { \\gamma } y ^ { h - u } \\log y - h ^ { - 1 } \\lambda ( y ^ { u - h } , y ^ h ) . \\end{align*}"}
+{"id": "1289.png", "formula": "\\begin{align*} \\sigma ( C ) & = \\left \\{ \\sqrt { n m } - 1 ^ { ( n + m ) } , 1 ^ { ( n + m ) } \\right \\} . \\end{align*}"}
+{"id": "6590.png", "formula": "\\begin{align*} \\eta = \\sum _ { l = 1 } ^ L \\alpha _ { l , { n _ t } } \\hat \\beta _ l , \\ \\ \\hat \\eta = \\sum _ { l = 1 } ^ L \\alpha _ { l , { \\hat { n } _ t } } \\hat \\beta _ l e ^ { - j ( \\theta _ { l , n _ t } - \\theta _ { l , { \\hat { n } _ t } } ) } . \\end{align*}"}
+{"id": "5704.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ) = c F = c ^ { \\sigma } F . \\end{align*}"}
+{"id": "507.png", "formula": "\\begin{align*} u '' ( x ) = f ( x ) , x \\in ( a , b ) , u ( a ) = u ( b ) = 0 , \\end{align*}"}
+{"id": "739.png", "formula": "\\begin{align*} \\cal A _ { V , N } : = A _ { S , N } - C _ { V , N } B ^ { - 1 } _ { B , N } D _ { V , N } \\end{align*}"}
+{"id": "600.png", "formula": "\\begin{align*} C s ^ 4 - C s ^ 2 + 1 = 0 \\end{align*}"}
+{"id": "4247.png", "formula": "\\begin{align*} \\Upsilon _ { 0 , T } ^ { u , u ^ { \\prime } ; p } = \\mathbb { E } \\left [ \\int _ { 0 } ^ { T } \\left \\vert u _ { s } - u _ { s } ^ { \\prime } \\right \\vert ^ { p } \\mathrm { d } \\theta \\right ] . \\end{align*}"}
+{"id": "9111.png", "formula": "\\begin{align*} y _ { [ \\alpha ] } & = ( \\underbrace { y _ { [ \\alpha ] } ^ { 1 } , \\ldots , y _ { [ \\alpha ] } ^ { m _ { 1 } } } _ { y _ { 1 , [ \\alpha ] } } , \\underbrace { y _ { [ \\alpha ] } ^ { m _ { 1 } + 1 } , \\ldots , y _ { [ \\alpha ] } ^ { m } } _ { y _ { 2 , [ \\alpha ] } } ) \\end{align*}"}
+{"id": "903.png", "formula": "\\begin{align*} u = \\prod b _ i ^ i , ( b _ i , b _ j ) = 1 i \\neq j , b _ i . \\end{align*}"}
+{"id": "5776.png", "formula": "\\begin{align*} T _ { _ { \\alpha } } ( \\vec { f } ) ( z ) & = T _ { \\alpha } ( f _ { 1 } ^ { 0 } , \\dots , f _ { m } ^ { 0 } ) ( z ) + \\sum _ { ( \\rho _ { 1 } , \\dots , \\rho _ { m } ) \\in \\rho } T _ { \\alpha } ( f _ { 1 } ^ { \\rho _ { 1 } } , \\dots , f _ { m } ^ { \\rho _ { m } } ) ( z ) . \\end{align*}"}
+{"id": "8200.png", "formula": "\\begin{align*} \\widehat { P } ^ \\omega \\left ( ( \\log t ) ^ { 2 / d } \\left ( \\frac { \\log N _ t } { t } - \\beta \\right ) + c ( d , \\nu ) < - \\varepsilon \\right ) = \\widehat { P } ^ \\omega \\left ( N _ t < \\exp \\left [ t \\left ( \\beta - \\frac { c ( d , \\nu ) + \\varepsilon } { ( \\log t ) ^ { 2 / d } } \\right ) \\right ] \\right ) . \\end{align*}"}
+{"id": "6576.png", "formula": "\\begin{align*} | \\Psi ( H ) | = \\frac { 1 } { 4 } \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} + \\Psi ( | H | ) \\end{align*}"}
+{"id": "4220.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\det T _ n ( \\phi ) } { \\prod _ { k = 0 } ^ R \\det T _ n ( \\phi _ k ) } = E \\end{align*}"}
+{"id": "6186.png", "formula": "\\begin{align*} f ' ( t ) = \\frac { 1 - \\log t } { \\log ^ 2 t } \\in \\big ( 0 , 2 | \\log t | ^ { - 1 } \\big ) \\quad f '' ( t ) = \\frac { - 2 + \\log t } { t ( \\log t ) ^ 3 } > 0 t \\in ( 0 , 1 / e ) . \\end{align*}"}
+{"id": "4402.png", "formula": "\\begin{align*} \\mu _ { \\omega , \\mathbf { c } } = \\frac { 1 } { 6 } \\inf \\{ L _ { \\omega , \\mathbf { c } } ( \\Psi ) | \\ \\Psi \\in \\mathcal { H } ^ 1 \\backslash \\{ ( \\mathbf { 0 } , \\mathbf { 0 } , \\mathbf { 0 } ) \\} , \\ K _ { \\omega , \\mathbf { c } } ( \\Psi ) = 0 \\} . \\end{align*}"}
+{"id": "296.png", "formula": "\\begin{align*} = \\left ( \\frac { 1 } { 1 - x y z } \\right ) ^ { \\frac { x y } { ( 1 - x ) ^ 2 ( 1 - y ) ^ 3 } } \\times \\exp \\left \\{ \\frac { - x y ( 2 x + y - 3 ) } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 4 } L i _ 2 ( x y z ) \\right \\} \\end{align*}"}
+{"id": "7269.png", "formula": "\\begin{align*} P ( x _ 1 , \\dots , x _ n ) = 0 . \\end{align*}"}
+{"id": "1865.png", "formula": "\\begin{align*} L ( \\mathcal { K } , \\bar { u } ) = \\{ t v \\in \\mathcal { U } : \\ G ( \\bar { u } ) + G ' ( \\bar { u } ) v \\in \\mathcal { K } , \\ \\forall \\ t > 0 \\} . \\end{align*}"}
+{"id": "9337.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\xi _ t ^ u & = h ( x _ t ^ u , u _ t ) d t + d \\tilde { W } _ t , \\ t \\in [ 0 , \\infty ) , \\\\ \\xi _ 0 & = 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "5394.png", "formula": "\\begin{align*} S _ { x , \\chi } = \\sum _ { \\substack { n \\le x \\\\ p \\mid n \\implies p \\le y } } \\chi ( n ) + O ( x y ^ { - 1 / 2 } ( \\log m ) ^ 4 ) . \\end{align*}"}
+{"id": "8955.png", "formula": "\\begin{align*} \\phi = \\Tilde { \\phi } _ { e } + \\sum _ { m = 1 } ^ { k } D _ { p _ m } \\Delta x ^ { p _ m } \\end{align*}"}
+{"id": "5709.png", "formula": "\\begin{align*} h ^ { \\sigma } = h \\end{align*}"}
+{"id": "1385.png", "formula": "\\begin{align*} \\{ \\dim V , \\dim W \\} = \\{ \\{ 1 , 3 \\} , \\{ 1 , 4 \\} , \\{ 2 \\} , \\{ 2 , 3 \\} , \\{ 2 , 4 \\} \\} . \\end{align*}"}
+{"id": "7973.png", "formula": "\\begin{align*} ( \\mathcal { L } _ h V ) ^ 2 = \\mathrm { t r } \\big ( ( \\nabla V ) ^ 2 \\big ) + e ^ h \\mathrm { d i v } \\Big ( e ^ { - h } \\ , V \\ , \\mathcal { L } _ h V - e ^ { - h } \\ , \\nabla V \\ , V \\Big ) + \\nabla ^ 2 h \\ , V \\cdot V \\quad \\end{align*}"}
+{"id": "7727.png", "formula": "\\begin{align*} \\bigl ( y ( \\omega , t ) < s \\bigr ) & = \\bigl ( y ^ { * } ( \\omega , \\ , s ^ { - } ) > t \\bigr ) \\\\ \\bigl ( y ^ { * } ( \\omega , \\ , t ) < s \\bigr ) & = \\bigl ( y ( \\omega , \\ s ^ { - } ) > t \\bigr ) . \\end{align*}"}
+{"id": "2921.png", "formula": "\\begin{align*} \\sum _ { \\substack { w \\in V / L \\\\ w t = v } } \\sigma ( w ) = \\sigma ( v ) \\mid t ^ { - 1 } . \\end{align*}"}
+{"id": "3979.png", "formula": "\\begin{align*} \\begin{bmatrix} V _ { \\ell , Y Y } & V _ { \\ell , Y X } \\\\ V _ { \\ell , X Y } & V _ { \\ell , X X } \\end{bmatrix} & = \\begin{bmatrix} ( Q _ { \\ell } / Q _ { \\ell , X X } ) ^ { - 1 } & - Q _ { \\ell , Y Y } ^ { - 1 } Q _ { \\ell , Y X } ( Q _ { \\ell } / Q _ { \\ell , Y Y } ) ^ { - 1 } \\\\ - ( Q _ { \\ell } / Q _ { \\ell , Y Y } ) ^ { - 1 } Q _ { \\ell , X Y } Q _ { \\ell , Y Y } ^ { - 1 } & ( Q _ { \\ell } / Q _ { \\ell , Y Y } ) ^ { - 1 } \\end{bmatrix} , \\end{align*}"}
+{"id": "9251.png", "formula": "\\begin{align*} h ( m ) : = \\frac { \\Gamma ( ( m + 1 ) / 2 ) } { f _ { m / 2 } \\sqrt { m ! } } \\Bigl \\{ 1 . 3 2 5 \\sqrt { m + 1 } + 1 . 3 2 5 \\frac { \\sqrt { ( m - 1 ) ( m - 2 ) } } { \\sqrt { m } } + 1 . 5 9 \\frac { | 1 - 2 \\sigma | } { \\sqrt { m } } \\Bigr \\} . \\end{align*}"}
+{"id": "6840.png", "formula": "\\begin{align*} \\alpha ' _ n = \\frac { 1 + a } { 1 + a q ^ { 2 n } } q ^ n \\alpha _ n ( a ^ 2 , q ^ 2 ) \\quad \\mbox { a n d } \\quad \\beta ' _ n = \\sum _ { j \\leq n } \\frac { ( - a ) _ { 2 j } } { ( q ^ 2 ; q ^ 2 ) _ { n - j } } q ^ { j } \\beta _ j ( a ^ 2 , q ^ 2 ) . \\end{align*}"}
+{"id": "8388.png", "formula": "\\begin{align*} V = V ^ { ( \\nu _ 1 ) } \\oplus W ^ { ( \\nu _ 1 ) } \\oplus \\kappa ( W ^ { ( \\nu _ 1 ) } ) . \\end{align*}"}
+{"id": "7332.png", "formula": "\\begin{align*} m = \\frac { s \\prod _ { k = 1 } ^ { N _ 1 } U _ k ^ { e _ k } } { q } = s \\frac { \\prod _ { k = 1 } ^ { N _ 1 } d _ k ^ { e _ k } } { q } \\prod _ { k = 1 } ^ { N _ 1 } ( U ' _ k ) ^ { e _ k } d \\in { \\cal M } . \\end{align*}"}
+{"id": "5955.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( g ) f ( [ \\epsilon , w ] ) = ( ( \\det g ) \\epsilon , a ) _ { \\R } f ( [ ( \\det g ) \\epsilon , w g ^ { s ( ( \\det g ) \\epsilon ) } ] ) . \\end{align*}"}
+{"id": "1541.png", "formula": "\\begin{align*} \\sigma = B ^ { - 1 } \\frac { \\alpha ^ { p + 1 } } { \\gamma ^ p } \\end{align*}"}
+{"id": "1596.png", "formula": "\\begin{align*} \\tau _ { n - 1 } \\le \\tau _ n \\le C _ r \\ , \\tau _ { n - 1 } , \\mbox { f o r } \\ ; 2 \\le n \\le N . \\\\ \\end{align*}"}
+{"id": "8798.png", "formula": "\\begin{align*} & b _ 2 + c _ 2 = u ( b _ 3 + c _ 3 ) \\end{align*}"}
+{"id": "8998.png", "formula": "\\begin{align*} \\int _ M ( 1 + w ) | T ^ \\varphi | ^ 2 = - \\alpha \\int _ M ( 1 + w ) | \\tau ( \\varphi ) | ^ 2 \\ , . \\end{align*}"}
+{"id": "2657.png", "formula": "\\begin{align*} f ( x ) - t _ \\alpha ( x ) & = \\\\ & \\left ( f ' ( x _ { j - 1 } ) - \\alpha h _ j ( \\coth ( \\alpha h _ j ) + \\tanh ( \\alpha x _ j ) ) f [ x _ { j - 1 } , x _ j ] \\right ) ( x - x _ { j - 1 } ) \\\\ & \\quad + O ( \\overline { h } ^ 2 ) \\\\ & = ( f ' ( x _ { j - 1 } ) - f [ x _ { j - 1 } , x _ j ] ) ( x - x _ { j - 1 } ) + O ( \\overline { h } ^ 2 ) \\\\ & = O ( \\overline { h } ^ 2 ) . \\end{align*}"}
+{"id": "6139.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 3 } u _ { t } + \\mathcal { L } ^ { \\Phi } _ { A } u & = ( - \\Delta ) ^ { \\frac { s } { 2 } } f + g & & \\mbox { i n $ \\Omega \\times ( 0 , T ) $ } , \\\\ u & = h & & \\mbox { i n $ \\mathbb { R } ^ { n } \\setminus \\Omega \\times [ 0 , T ] $ } , \\\\ u ( \\cdot , 0 ) & = h ( \\cdot , 0 ) & & \\mbox { i n $ \\Omega $ } \\end{alignedat} \\right . \\end{align*}"}
+{"id": "8207.png", "formula": "\\begin{align*} \\underset { t \\to \\infty } { \\lim } P ^ \\omega ( A _ { m ( t ) } ^ c \\mid S ) = 0 , \\end{align*}"}
+{"id": "2791.png", "formula": "\\begin{align*} ( \\Delta + \\mu - q ) v = f \\ ; \\mathrm { i n } \\ ; M , \\partial _ \\nu v = \\mp a w + \\varphi . \\end{align*}"}
+{"id": "8159.png", "formula": "\\begin{align*} f ( q t ) = f ( t ) \\sigma _ { q t } ( A ) \\end{align*}"}
+{"id": "2214.png", "formula": "\\begin{align*} S ' ( u ) & = \\frac { 1 } { u - 1 } \\left ( \\log ( u - 2 ) + 1 + \\frac { u } { u - 2 } - \\frac { u ( \\log ( u - 2 ) + 1 ) } { u - 1 } - ( \\log ( u - 2 ) + 1 ) \\right ) \\\\ & = \\frac { u ( 1 - ( u - 2 ) \\log ( u - 2 ) ) } { ( u - 2 ) ( u - 1 ) ^ 2 } , \\end{align*}"}
+{"id": "8263.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 0 } ^ { 2 k + 1 } \\frac { ( x / 2 ) ^ i } { i ! } \\right ) \\left ( \\sum _ { i = 0 } ^ { k } \\frac { 1 } { i ! } \\left ( - \\sum _ { n = 1 } ^ k \\frac { c _ { 2 n } ( 4 x ) ^ { 2 n } } { 2 n } \\right ) ^ i \\right ) . \\end{align*}"}
+{"id": "7378.png", "formula": "\\begin{align*} \\ell \\sum _ { i = 1 } ^ N \\sum _ { m \\in \\mathbb { Z } } \\Delta \\left ( \\frac { m } { \\ell } \\right ) = L . \\end{align*}"}
+{"id": "1631.png", "formula": "\\begin{align*} ( \\pounds _ { \\xi _ i } \\ , \\Phi ) ( X , Y ) = ( \\pounds _ { \\xi _ i } \\ , g ) ( X , { f } Y ) + g ( X , ( \\pounds _ { \\xi _ i } { f } ) Y ) . \\end{align*}"}
+{"id": "740.png", "formula": "\\begin{align*} \\eta ( t , x ) = A \\ , \\cos ( k x - \\omega t ) \\ , , \\Phi ( t , x , y ) = A \\ , \\frac { \\omega } { k } \\ , \\frac { \\cosh k ( y + h _ 0 ) } { \\sinh k h _ 0 } \\ , \\sin ( k x - \\omega t ) \\ , , \\end{align*}"}
+{"id": "4179.png", "formula": "\\begin{align*} m ( \\lambda ) = \\frac { 4 \\pi } { 3 \\vphantom { \\lambda ^ 3 } } \\left ( \\frac { 1 } { 2 \\lambda ^ 3 } - \\frac { 2 } { ( \\lambda + 2 ) ^ 3 } \\right ) 1 _ { ( 0 , 2 ) } ( \\lambda ) . \\end{align*}"}
+{"id": "2999.png", "formula": "\\begin{align*} \\prod _ 0 ^ t \\exp f ( s ) \\ , d s = \\exp _ G ( t - t _ j ) f ( t _ j ) \\exp _ G ( t _ j - t _ { j - 1 } ) f ( t _ { j - 1 } ) \\ldots \\exp _ G t _ 1 f ( 0 ) . \\end{align*}"}
+{"id": "8158.png", "formula": "\\begin{align*} \\Psi _ n \\left ( \\frac { A } { 1 - q } \\right ) = \\frac 1 { 1 - q t } \\left . S _ n \\left ( \\frac { 1 - q t | } { | 1 - q } A \\right ) \\right | _ { t = \\frac 1 q } . \\end{align*}"}
+{"id": "7394.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq r \\leq N ^ 2 } W _ N ( w _ r , \\mathcal { A } ) ^ 2 = \\# \\{ ( i , j , k , l ) \\in [ 1 , N ] ^ 4 : i \\neq j , k \\neq l , a _ i - a _ j = a _ k - a _ l \\} \\leq E _ N ( \\mathcal { A } ) , \\end{align*}"}
+{"id": "3828.png", "formula": "\\begin{align*} \\boldsymbol { d } _ { \\mathcal { V } } ( ( s _ 1 , s _ 2 ) , ( s _ 1 ^ \\prime , s _ 2 ^ \\prime ) ) = \\left [ \\boldsymbol { d } _ { \\mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \\prime ) ^ p + \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \\prime ) ^ p \\right ] ^ { 1 / p } . \\end{align*}"}
+{"id": "4410.png", "formula": "\\begin{align*} L _ { \\omega , \\mathbf { c } } ( U _ n ) = K _ { \\omega , \\mathbf { c } } ( U _ n ) - 3 N ( U _ n ) \\ \\rightarrow \\ 0 . \\end{align*}"}
+{"id": "7641.png", "formula": "\\begin{align*} M ( 0 , x , y ) & = 1 1 5 2 - 5 7 6 x ^ 3 + ( 1 0 2 4 - 1 1 5 2 x - 8 9 6 x ^ 2 + 1 1 5 2 x ^ 3 - 1 2 8 x ^ 4 ) y ^ 2 \\\\ & = 5 7 6 ( 2 - x ^ 2 ) + 1 2 8 ( 8 - x ) ( 1 - x ) ^ 2 ( 1 + x ) y ^ 2 \\\\ & \\leq 5 7 6 ( 2 - x ^ 2 ) + 1 2 8 ( 8 - x ) ( 1 - x ) ^ 2 ( 1 + x ) \\\\ & = 1 0 2 4 - 8 9 6 x ^ 2 + 5 7 6 x ^ 3 - 1 2 8 x ^ 4 \\ ; \\leq 1 0 2 4 , ( x , y ) \\in ( 0 , 1 ) \\times ( 0 , 1 ) \\end{align*}"}
+{"id": "780.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\rho } \\int _ { \\mathbb { S } ^ n } r ( 1 + u ( x ) ) ^ 2 \\phi ^ n ( r ( 1 + u ( x ) ) ) x _ l \\mathrm { d } A \\mathrm { d } r = 0 , \\ \\ l = 1 , 2 , \\cdots , n + 1 . \\end{align*}"}
+{"id": "9208.png", "formula": "\\begin{align*} d ( v , V ) + d ( V , v ) = 2 r , \\deg ^ + ( v ) = n - d ( v , V ) \\mbox { a n d } \\deg ^ - ( v ) = n - d ( V , v ) \\end{align*}"}
+{"id": "2807.png", "formula": "\\begin{align*} A = A 1 = A 1 ( t ) = ( A 1 _ { i j } ) _ { i , j = 0 } ^ { \\infty } = \\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & \\dots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\dots \\\\ 0 _ { r 0 } & 0 & 0 & 0 & 0 & 0 & \\dots \\\\ a _ 0 \\cdots a _ r & 0 & 0 & 0 & 0 & 0 & \\dots \\\\ 0 & a _ 1 \\cdots a _ { r + 1 } & 0 & 0 & 0 & 0 & \\dots \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\dots \\end{pmatrix} . \\end{align*}"}
+{"id": "3600.png", "formula": "\\begin{align*} L ( q ^ \\beta ) \\ \\geq \\ \\beta L ( q ) - ( \\beta - 1 ) \\ = \\ \\beta \\ell - ( \\beta - 1 ) \\ = \\ \\beta ( \\ell - 1 ) + 1 . \\end{align*}"}
+{"id": "975.png", "formula": "\\begin{align*} g \\left ( { A } _ { \\xi } X , Y \\right ) = g \\left ( h ( X , Y ) , \\xi \\right ) . \\end{align*}"}
+{"id": "2191.png", "formula": "\\begin{align*} \\left | \\sum \\limits _ { j = 0 } ^ { n - 1 } g _ { \\lambda _ i } ^ { - \\frac { 1 } { 2 } } ( \\alpha \\theta _ n ^ { j } ) \\theta _ n ^ { - j k } \\right | & = \\left | \\lambda _ i ^ { - 1 / 2 } \\sum \\limits _ { j = 0 } ^ { n - 1 } \\theta _ n ^ { - j k } + \\sum \\limits _ { j = 0 } ^ { n - 1 } ( g _ { \\lambda _ i } ^ { - \\frac { 1 } { 2 } } ( \\alpha \\theta _ n ^ { j } ) - \\lambda _ i ^ { - 1 / 2 } ) \\theta _ n ^ { - j k } \\right | . \\end{align*}"}
+{"id": "6301.png", "formula": "\\begin{align*} ( q _ n , \\lambda _ n ) \\neq ( q _ n ' , \\lambda _ n ' ) E ( q _ n , \\lambda _ n ) = E ( q _ n ' , \\lambda _ n ' ) , \\qquad H ( \\lambda _ n ) , \\ , H ( \\lambda _ n ' ) \\to 0 . \\end{align*}"}
+{"id": "511.png", "formula": "\\begin{align*} L p ( y _ j ) = \\sum _ { x _ i \\in X _ j ^ \\mathrm { n d f } } w _ { j i } p ( x _ i ) , \\forall p \\in P _ j ^ \\mathrm { n d f } . \\end{align*}"}
+{"id": "6672.png", "formula": "\\begin{align*} \\gamma _ R ( x ) : = \\gamma \\left ( \\frac { | x | } { R } \\right ) \\textrm { f o r a l l } \\ ; \\ ; x \\in \\R ^ N \\ , ; \\end{align*}"}
+{"id": "2495.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\dfrac { d } { d \\tau } \\mathcal { S } \\left ( \\xi ^ { \\tau } , - \\eta ^ { \\tau } \\right ) \\Big | _ { \\tau = 1 } = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "3505.png", "formula": "\\begin{align*} \\omega ( X , Y ) = - \\langle X ( t ) , Y ' ( t ) \\rangle + \\langle X ' ( t ) , Y ( t ) \\rangle \\end{align*}"}
+{"id": "4989.png", "formula": "\\begin{align*} S ( k , n ) = \\begin{dcases} 1 & k = 0 ; \\\\ S ( k , n - 1 ) \\ , + \\ , S ( k - 1 , n - 1 ) & 1 < k < n ; \\\\ 2 S ( k , n - 1 ) & k = n ; \\end{dcases} \\end{align*}"}
+{"id": "6431.png", "formula": "\\begin{align*} f ( | x | \\bar { \\otimes } | y | ) = \\int _ { [ 0 , \\infty ) } f ( \\lambda ) \\ , e _ { | x | \\bar { \\otimes } | y | } ( d \\lambda ) = \\int _ { [ 0 , \\infty ) ^ 2 } f ( \\lambda _ 1 \\lambda _ 2 ) \\ , e _ { | x | , | y | } ( d \\lambda _ 1 , d \\lambda _ 2 ) \\end{align*}"}
+{"id": "6710.png", "formula": "\\begin{align*} p _ \\Psi ( x _ 0 , \\xi , \\tau ) = \\{ p _ \\Psi , \\Psi \\} ( x _ 0 , \\xi ) = 0 \\implies \\tau = 0 . \\end{align*}"}
+{"id": "4461.png", "formula": "\\begin{align*} | S _ { \\omega , \\mathbf { c } } ( U _ n ( t _ n ) ) - \\mu _ { \\omega , \\mathbf { c } } | = | S _ { \\omega , \\mathbf { c } } ( U _ { n , 0 } ) - S _ { \\omega , \\mathbf { c } } ( \\Phi _ { \\omega , \\mathbf { c } , n } ) | \\ \\rightarrow \\ 0 \\end{align*}"}
+{"id": "1710.png", "formula": "\\begin{align*} P _ \\gamma ^ { ( \\alpha , \\beta ) } ( 1 ) = \\frac { \\Gamma ( \\alpha + \\gamma + 1 ) } { \\Gamma ( \\alpha + 1 ) \\Gamma ( \\gamma + 1 ) } , \\end{align*}"}
+{"id": "7925.png", "formula": "\\begin{align*} R ^ * ( \\sigma \\curvearrowright \\tau ) = \\sigma \\curvearrowright R ^ * \\tau , \\end{align*}"}
+{"id": "3952.png", "formula": "\\begin{align*} h ( s ) = M \\left [ 1 + \\boldsymbol { d } _ { \\mathcal { S } _ 1 } ( s _ 1 ^ \\star , s _ 1 ) ^ { p _ 1 ' } + \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( s _ 2 ^ \\star , s _ 2 ) ^ { p _ 2 ' } \\right ] , \\end{align*}"}
+{"id": "9067.png", "formula": "\\begin{align*} z _ j = z _ 0 + \\frac { \\zeta _ j } { \\sqrt { n } } , j = 1 , 2 , \\dotsc , m , \\end{align*}"}
+{"id": "4387.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ 3 \\int _ { \\R ^ d } e ^ { p | x | } \\left ( | \\varphi _ j ( x ) | ^ 2 + | \\nabla \\varphi _ j ( x ) | ^ 2 \\right ) d x < \\infty \\end{align*}"}
+{"id": "6632.png", "formula": "\\begin{align*} d _ { \\Phi _ { T } } ( \\alpha ) = & \\mu \\left \\{ ( x , \\chi ) \\in G \\times \\widehat { G } : | \\Phi _ T ( x , \\chi ) | > \\alpha \\right \\} \\leq \\left ( \\frac { \\| T \\| _ { \\mathcal { B } _ 2 ( L ^ 2 ( G ) ) } } { \\alpha } \\right ) ^ 2 , \\\\ & \\mu \\left \\{ ( x , \\chi ) \\in G \\times \\widehat { G } : | \\Phi _ T ( x , \\chi ) | > \\alpha \\right \\} \\lesssim \\frac { M _ \\psi \\| T \\| _ { \\mathcal { B } _ 1 ( L ^ 1 ( G ) ) } } { \\alpha } . \\end{align*}"}
+{"id": "5027.png", "formula": "\\begin{align*} f - f _ { a d d } = \\sum _ { k = 1 } ^ N c _ k x _ 1 ^ { E _ { k 1 } } \\cdots x _ N ^ { E _ { k N } } . \\end{align*}"}
+{"id": "9180.png", "formula": "\\begin{align*} \\begin{aligned} x & = F _ { x } ( y ^ { 1 } , y ^ { 2 } , \\dots , y _ { [ 3 ] } ^ { 1 } , y _ { [ 3 ] } ^ { 2 } ) \\\\ u & = F _ { u } ( y ^ { 1 } , y ^ { 2 } , \\dots , y _ { [ 4 ] } ^ { 1 } , y _ { [ 4 ] } ^ { 2 } ) \\end{aligned} \\end{align*}"}
+{"id": "7503.png", "formula": "\\begin{align*} u ( z ) = \\zeta + \\ln ' \\big [ p ( z ) \\big ] \\end{align*}"}
+{"id": "5450.png", "formula": "\\begin{align*} F _ { | g _ i | ^ 2 } ( x ) = \\gamma ( M , M x ) / \\Gamma ( M ) = 1 - \\sum _ { i = 0 } ^ M \\frac { ( M x ) ^ i } { i ! } e ^ { - M x } , \\end{align*}"}
+{"id": "1187.png", "formula": "\\begin{align*} \\Delta ^ M = \\frac { \\sum _ { n m \\in \\mathcal { L } _ S } \\delta ^ M _ { n m } } { | \\mathcal { L } _ S | } \\end{align*}"}
+{"id": "8295.png", "formula": "\\begin{align*} x ^ t L _ e x = & \\ ; ( \\mu _ { 1 1 } + 2 ) x ^ 2 _ 1 - \\mu _ { 1 1 } x _ 1 x _ 2 - 2 x _ 1 x _ 3 - \\mu _ { 1 1 } x _ 1 x _ 2 + ( \\mu _ { 1 1 } + \\mu _ { 2 1 } ) x ^ 2 _ 2 - \\mu _ { 2 1 } x _ 2 x _ 3 \\\\ & - 2 x _ 1 x _ 3 - \\mu _ { 2 1 } x _ 2 x _ 3 + ( \\mu _ { 2 1 } + 2 ) x ^ 2 _ 3 \\\\ = & \\ ; \\mu _ { 1 1 } ( x ^ 2 _ 1 - 2 x _ 1 x _ 2 + x ^ 2 _ 2 ) + 2 ( x ^ 2 _ 1 - 2 x _ 1 x _ 3 + x ^ 2 _ 3 ) + \\mu _ { 2 1 } ( x ^ 2 _ 2 - 2 x _ 2 x _ 3 + x ^ 2 _ 3 ) \\\\ = & \\ ; \\mu _ { 1 1 } ( x _ 1 - x _ 2 ) ^ 2 + 2 ( x _ 1 - x _ 3 ) ^ 2 + \\mu _ { 2 1 } ( x _ 2 - x _ 3 ) ^ 2 \\\\ x ^ t L _ e x = & \\ ; \\sum _ { j = 1 } ^ { 3 - 1 } \\sum _ { i = 1 } ^ { 3 - j } \\mu _ { j i } ( x _ j - x _ { j + i } ) ^ 2 \\end{align*}"}
+{"id": "6371.png", "formula": "\\begin{align*} \\dot { x } = f ( x ) , \\dot { y } = f ( y ) , \\dot { z } = \\frac { f ( x ) - f ( y ) } { \\epsilon } . \\end{align*}"}
+{"id": "3209.png", "formula": "\\begin{align*} \\frac { 1 } { n + 1 } \\sum _ { l = 0 } ^ { n } \\kappa _ l ( x ) \\leq C _ w \\frac { 1 } { 3 n + 1 } \\sum _ { l = 0 } ^ { 3 n } \\kappa _ 1 ^ l ( x ) . \\end{align*}"}
+{"id": "1470.png", "formula": "\\begin{align*} H _ 2 = & \\Big | \\Big ( f _ \\epsilon ^ { ' } ( V ) - \\sum _ { i = 1 } ^ k ( - 1 ) ^ i f _ 0 ^ { ' } ( P U _ { \\mu _ i , \\xi _ i } ) \\Big ) \\phi \\Big | _ { \\frac { 2 n } { n + 2 } } \\\\ \\leq & \\Big | f _ \\epsilon ^ { ' } ( V ) - \\sum _ { i = 1 } ^ k ( - 1 ) ^ i f _ 0 ^ { ' } ( P U _ { \\mu _ i , \\xi _ i } ) \\Big | _ { \\frac { n } { 2 } } | \\phi | _ { \\frac { 2 n } { n - 2 } } \\leq C \\epsilon \\ln \\Big | \\ln \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big | \\| \\phi \\| . \\end{align*}"}
+{"id": "4572.png", "formula": "\\begin{align*} H _ N = \\sum _ { j = 1 } ^ N ( - \\Delta _ j ) + \\frac { 1 } { N - 1 } \\sum _ { 1 \\leq j < k \\leq N } w _ N ( x _ j - x _ k ) \\ , , \\end{align*}"}
+{"id": "6248.png", "formula": "\\begin{align*} \\forall n \\geq 0 , F _ n ( O [ g _ 1 ] ) = \\sum _ { k = 0 } ^ n ( F _ k O ) \\cdot g _ 1 ^ { n - k } , \\mathrm { g r } _ n ( O [ g _ 1 ] ) \\simeq \\mathbb C \\cdot X ^ n \\oplus Y ^ 2 \\cdot \\mathbb C [ X , Y ] _ { n - 2 } , \\end{align*}"}
+{"id": "7217.png", "formula": "\\begin{align*} \\lim _ { h \\to + \\infty } F _ h ( v _ h , \\gamma _ h , B _ s ) = : \\alpha ( s ) \\end{align*}"}
+{"id": "2781.png", "formula": "\\begin{align*} \\xi _ 1 = ( \\eta / 2 + \\eta _ 1 ) + i \\eta _ 2 , \\xi _ 2 = ( \\eta / 2 - \\eta _ 1 ) - i \\eta _ 2 . \\end{align*}"}
+{"id": "5625.png", "formula": "\\begin{align*} \\aligned \\left \\{ \\begin{array} { l l l } - \\Delta u + \\lambda _ 1 u = | u | ^ { 2 r _ 1 - 2 } u + \\nu p | v | ^ q | u | ^ { p - 2 } u \\ & \\mathbb { R } ^ N , \\\\ - \\Delta v + \\lambda _ 2 v = | v | ^ { 2 r _ 2 - 2 } v + \\nu q | u | ^ p | v | ^ { q - 2 } v \\ & \\mathbb { R } ^ N , \\\\ \\int _ { \\mathbb { R } ^ N } u ^ 2 = a ^ 2 , \\ \\int _ { \\mathbb { R } ^ N } v ^ 2 = b ^ 2 . \\end{array} \\right . \\endaligned \\end{align*}"}
+{"id": "7274.png", "formula": "\\begin{align*} x ^ 4 + a x y + y ^ 3 = 0 \\end{align*}"}
+{"id": "4655.png", "formula": "\\begin{align*} \\vartheta _ { m } ^ { i + 1 } = - \\frac { \\gamma } { ( \\gamma - 1 ) \\Delta x } \\big ( \\widetilde { m } _ { i + 1 } ^ { \\gamma - 1 } - \\widetilde { m } _ { i } ^ { \\gamma - 1 } \\big ) + \\frac 1 2 \\chi \\left ( \\frac { ( c _ x ) ^ i } { 1 + \\widetilde { m } _ { i } } + \\frac { ( c _ x ) ^ { i + 1 } } { 1 + \\widetilde { m } _ { i + 1 } } \\right ) , \\end{align*}"}
+{"id": "3211.png", "formula": "\\begin{align*} \\nu _ { y ' } ^ n ( x ) = \\langle y ' \\sigma _ n ( x ) \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } . \\end{align*}"}
+{"id": "7146.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\nabla v \\cdot \\nabla \\varphi \\ : x = \\int _ { \\Omega } u \\varphi \\ : x \\end{align*}"}
+{"id": "8354.png", "formula": "\\begin{align*} \\| h _ j \\| _ { p ^ * , q , \\mu } \\geq \\| u _ j \\| _ { p ^ * , q , \\mu } - \\| u _ j - h _ j \\| _ { p ^ * , q , \\mu } \\geq \\lambda - r = \\varepsilon _ 0 . \\end{align*}"}
+{"id": "2539.png", "formula": "\\begin{align*} B ( M ) = B ( N ) \\cup \\{ \\{ i , i ^ * \\} , \\{ j , j ^ * \\} \\} . \\end{align*}"}
+{"id": "1634.png", "formula": "\\begin{align*} ( \\pounds _ { \\xi _ i } \\ , d \\eta ^ j ) ( X , Y ) = \\xi _ i ( d \\eta ^ j ( X , Y ) ) - d \\eta ^ j ( [ \\xi _ i , X ] , Y ) - d \\eta ^ j ( X , [ \\xi _ i , Y ] ) , \\end{align*}"}
+{"id": "87.png", "formula": "\\begin{align*} \\widetilde { m } ( x , \\xi ) = ( 1 - \\chi _ m ( x , \\xi ) ) m ( x , \\xi ) \\log | \\xi | \\end{align*}"}
+{"id": "3540.png", "formula": "\\begin{align*} F _ i ( z , f _ 1 ( z ) , \\dots , f _ n ( z ) ) = P _ i ( & z , f _ 1 ( z ) , \\dots , f _ n ( z ) , \\\\ & g _ 1 ( z ) , g _ 1 ( f _ 1 ( z ) ) , \\dots , g _ 1 ( f _ n ( z ) ) , \\dots , \\\\ & g _ l ( z ) , g _ l ( f _ 1 ( z ) ) , \\dots , g _ l ( f _ n ( z ) ) ) \\end{align*}"}
+{"id": "1347.png", "formula": "\\begin{align*} \\mathbf { S } [ F _ { N } ] ( t , x ) & = \\frac { 1 } { N ^ { 2 } } \\underset { l , r } { \\sum } H ( x - x _ { l } ( t ) ) m _ { l } ( t ) m _ { r } ( t ) S ( x _ { l } ( t ) - x _ { r } ( t ) ) \\\\ & = \\frac { 1 } { N ^ { 2 } } \\stackrel [ r = 1 ] { N } { \\sum } \\stackrel [ l = 1 ] { ( i - 1 ) ^ { \\ast } ( j ) } { \\sum } m _ { l } ( t ) m _ { r } ( t ) S ( x _ { l } ( t ) - x _ { r } ( t ) ) , \\end{align*}"}
+{"id": "7591.png", "formula": "\\begin{align*} | H _ { 2 , 1 } ( F _ { f } / 2 ) | = \\frac { 1 } { 6 4 } . \\end{align*}"}
+{"id": "8827.png", "formula": "\\begin{align*} v _ n ( t ) = v _ 0 ( t ) + \\int _ 0 ^ t S ( t - s ) F _ n ( s ) v _ n ( s ) \\ , d s + \\int _ 0 ^ t S ( t - s ) \\sigma ' ( u ( s ) ) v _ n ( s ) B \\ , d W ( s ) , \\end{align*}"}
+{"id": "6896.png", "formula": "\\begin{align*} \\widehat \\psi _ r ( \\beta _ 2 ) = & \\inf _ { x \\in [ 0 , 1 ] } J _ r ( x , \\beta _ 2 ) \\\\ \\geq \\ , & \\inf _ { x \\in [ 0 , 1 ] } J _ r ( x , \\beta _ 1 ) + \\theta ( x , \\beta _ 1 ) ( \\beta _ 2 - \\beta _ 1 ) \\\\ \\geq \\ , & \\widehat \\psi _ r ( \\beta _ 1 ) + \\inf _ { x \\in [ 0 , 1 ] } \\theta ( x , \\beta _ 1 ) ( \\beta _ 2 - \\beta _ 1 ) > \\widehat \\psi _ r ( \\beta _ 1 ) . \\end{align*}"}
+{"id": "7751.png", "formula": "\\begin{align*} H : = \\bigcup _ { Q ' \\in \\mathcal { F } } Q ' \\ , . \\end{align*}"}
+{"id": "1253.png", "formula": "\\begin{align*} P _ k = Q _ k ^ { ( 1 ) } + Q _ k ^ { ( 2 ) } + R _ k ^ { ( 1 ) } + R _ k ^ { ( 2 ) } , \\end{align*}"}
+{"id": "858.png", "formula": "\\begin{align*} \\frac { d } { d s } \\left [ \\begin{array} { c c } U \\\\ V \\end{array} \\right ] = \\left [ \\begin{array} { c c } V \\\\ \\alpha V + H _ { \\lambda } \\left ( \\phi _ { 2 } ( \\lambda ) - U \\right ) \\end{array} \\right ] \\end{align*}"}
+{"id": "7762.png", "formula": "\\begin{align*} x _ { j _ 1 } - x _ { k _ 1 } & = ( x _ { j _ 1 } - x _ { j _ 0 } ) + ( x _ { j _ 0 } - x _ { k _ 0 } ) + ( x _ { k _ 0 } - x _ { k _ 1 } ) \\\\ & = ( M _ { i j _ 0 } - M _ { i j _ 1 } ) + ( M _ { 1 k _ 0 } - M _ { i j _ 0 } + y _ i - y _ 1 ) + ( M _ { 1 k _ 1 } - M _ { 1 k _ 0 } ) \\\\ & = M _ { 1 k _ 1 } - M _ { i j _ 1 } + y _ i - y _ 1 , \\end{align*}"}
+{"id": "892.png", "formula": "\\begin{align*} r _ 1 = \\begin{cases} \\prod \\limits _ { \\varpi ^ k \\in P _ 1 ( r ) } \\varpi ^ k & c _ 2 \\neq 0 \\\\ \\prod \\limits _ { \\varpi ^ k \\in P _ 2 ( r ) } \\varpi ^ k & \\end{cases} \\end{align*}"}
+{"id": "1714.png", "formula": "\\begin{align*} z _ 2 ^ { - 2 ( \\gamma + \\alpha + \\beta + 1 ) } { \\sf f } ( z _ 2 ^ { - 1 } ) = z _ 2 ^ { 2 \\gamma } { \\sf g } ( z _ 2 ^ { - 1 } ) . \\end{align*}"}
+{"id": "7122.png", "formula": "\\begin{align*} \\mathcal { R } _ \\varepsilon c ( x ) = \\int _ { \\mathbb { R } ^ n _ - } J _ \\varepsilon ( | x - y | ) ( c ( x ) - \\tilde { c } ( y ) ) y = \\int _ { \\mathbb { R } ^ n _ + } J _ \\varepsilon ( | x - \\hat { y } | ) ( c ( x ) - c ( \\hat { y } ) ) y \\end{align*}"}
+{"id": "3262.png", "formula": "\\begin{align*} \\| h _ { 2 1 } \\| _ { \\dot F _ { q , { \\rm D } } ^ { \\beta , \\infty } } + \\sum _ { j = 1 } ^ { | G | - 1 } \\| h _ { \\sigma _ j 1 } \\| _ { \\dot F _ { q , { \\rm D } } ^ { \\beta , \\infty } } \\lesssim \\| b \\| _ { \\Lambda ^ \\beta } \\Big ( \\| f _ 2 \\| _ { L ^ p _ \\omega } + \\sum _ { j = 1 } ^ { | G | - 1 } \\| f _ { \\sigma _ j } \\| _ { L ^ p _ \\omega } \\Big ) . \\end{align*}"}
+{"id": "99.png", "formula": "\\begin{align*} v ( t , x ) = e ^ { h ^ { - 1 } \\int _ 0 ^ t q _ 1 ( \\varphi ^ s ( x ) ) d s } w ( t , \\varphi ^ t ( x ) ) , \\end{align*}"}
+{"id": "4819.png", "formula": "\\begin{align*} \\frac { \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } \\geq x \\right ) } { 1 - \\Phi ( x ) } = 1 + o ( 1 ) \\quad \\frac { \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } \\leq - x \\right ) } { \\Phi ( - x ) } = 1 + o ( 1 ) \\end{align*}"}
+{"id": "1124.png", "formula": "\\begin{align*} \\omega _ f ( r ) : = \\sup _ { \\substack { z , \\zeta \\in H \\\\ d ( z , \\zeta ) < r } } | f ( z ) - f ( \\zeta ) | . \\end{align*}"}
+{"id": "4953.png", "formula": "\\begin{align*} \\Pr ( X _ t \\le k \\ , \\vert \\ , X _ 0 = r ) = \\frac { ( n - r ) \\cdots ( n - k ) } { n ^ t } \\frac { 1 } { ( k - r ) ! } \\Delta ^ { k - r } \\left [ \\frac { x ^ t } { n - x } \\right ] _ { x = r } ^ { } , \\end{align*}"}
+{"id": "1485.png", "formula": "\\begin{align*} ( 2 ^ * - 1 ) \\alpha _ n \\sum _ { i = 1 } ^ k \\mu _ i ^ { \\frac { n - 2 } { 2 } } \\int _ \\Omega U _ { \\mu _ i , \\xi _ i } ^ { 2 ^ * - 2 } H ( x , \\xi _ i ) \\psi ^ 0 _ { \\mu _ j , \\xi _ j } d x = o \\Big ( \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big ) . \\end{align*}"}
+{"id": "7926.png", "formula": "\\begin{align*} \\Psi _ + \\left ( \\Xi _ { \\mathfrak { l } } \\prod _ { i } \\mathcal { I } _ { \\mathbf { m } _ i } ( \\tau _ i ) \\right ) & = \\Psi _ + \\left ( \\prod _ { i } \\mathcal { I } _ { \\mathbf { m } _ i } ( \\tau _ i ) \\curvearrowright \\Xi _ { \\mathfrak { l } } \\right ) \\\\ & = \\rho \\left ( \\Psi \\left ( \\prod _ { i } \\mathcal { I } _ { \\mathbf { m } _ i } ( \\tau _ i ) \\right ) \\right ) \\Psi \\left ( \\Xi _ { \\mathfrak { l } } . \\right ) \\end{align*}"}
+{"id": "7373.png", "formula": "\\begin{align*} \\Delta ( x ) : = \\int _ \\mathbb { R } \\chi ( x + x _ 0 ) \\chi ( x _ 0 ) \\ , d x _ 0 = \\int _ { - \\frac { 1 } { 2 } } ^ \\frac { 1 } { 2 } \\chi ( x + x _ 0 ) \\ , d x _ 0 = \\max \\{ 1 - | x | , 0 \\} . \\end{align*}"}
+{"id": "5503.png", "formula": "\\begin{align*} \\phi _ { \\ast } \\left ( \\bar { m } _ { K } \\times \\eta \\right ) & = \\int _ { G / Q } \\int _ { K \\cap Q } \\tau ( y ) k . \\lambda d m _ { K \\cap Q } ( k ) d \\bar { m } _ { K } ( y ) \\\\ & = \\left ( \\int _ { G / Q } \\tau ( y ) . m _ { K \\cap Q } d \\bar { m } _ { K } ( y ) \\right ) \\ast \\lambda = m _ { K } \\ast \\lambda = \\nu . \\end{align*}"}
+{"id": "866.png", "formula": "\\begin{align*} \\overline { E } = \\{ \\ , ( x , y ) \\in V ^ 2 : \\ , \\} . \\end{align*}"}
+{"id": "7566.png", "formula": "\\begin{align*} \\mathbb { V } ( \\overline { S } _ n ) = \\mathbb { E } ( \\overline { S } _ n ^ 2 ) - \\mathbb { E } ( \\overline { S } _ n ) ^ 2 \\end{align*}"}
+{"id": "5729.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } p q + p c = p b ^ 2 + a ^ 2 b + a b c \\\\ p ( c - a ) = ( b - d ) \\{ p ( b + d ) - q ( a + c ) \\} \\\\ p ( d - b ) = a ^ 2 - c ^ 2 \\\\ q ^ 2 + p d = a ^ 2 + q b ^ 2 + a b ^ 2 + a b d \\\\ q ( d - b ) = a b - c d . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "2884.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 8 , & ~ ~ t _ { 0 } \\geq 6 , \\\\ 7 , & ~ ~ t _ { 0 } = 5 , \\\\ 6 , & ~ ~ t _ 0 = 4 , \\\\ 4 - t _ 0 , & ~ ~ t _ 0 \\leq 3 , \\end{array} \\right . \\end{align*}"}
+{"id": "8052.png", "formula": "\\begin{align*} f ( Z ) = f ( 0 ) P _ 0 + \\sum _ { j = 1 } ^ m f ( z _ j ^ { \\uparrow } ) P _ j , \\end{align*}"}
+{"id": "7792.png", "formula": "\\begin{align*} ( \\Psi \\circ \\Phi ) ( A , B ) = \\Psi ( \\Phi ( A , B ) ) = \\Psi ( A ^ r \\otimes B ^ { 1 - r } ) = A ^ r \\circ B ^ { 1 - r } . \\end{align*}"}
+{"id": "2200.png", "formula": "\\begin{align*} \\lambda _ k B _ 1 ^ { ( \\alpha ) } - 2 B _ 2 ^ { ( \\alpha ) } = D _ { \\alpha } ^ { - 1 } \\mathbb { F } ^ { * } \\Lambda _ { \\alpha } \\mathbb { F } D _ { \\alpha } , \\end{align*}"}
+{"id": "7396.png", "formula": "\\begin{align*} R _ { N _ m } ^ 2 ( L _ m , \\alpha , \\Delta ) = L _ m + o ( 1 ) \\end{align*}"}
+{"id": "8343.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\nabla u _ n \\| _ { p , q , \\mu } = \\| \\nabla u _ \\mu ^ \\bigstar \\| _ { p , q , \\mu } . \\end{align*}"}
+{"id": "6612.png", "formula": "\\begin{align*} \\varphi _ \\lambda ( x , y ) = - ( - 1 ) ^ { | x | | y | } \\varphi _ { - \\partial - \\lambda } ( y , x ) , \\end{align*}"}
+{"id": "7833.png", "formula": "\\begin{align*} W = B A = B _ { I } A _ { I } + B _ { I ^ { c } } A _ { I ^ { c } } . \\end{align*}"}
+{"id": "1797.png", "formula": "\\begin{align*} \\ < \\Psi _ { \\mathrm { S l a t e r } } , a _ p ^ * a _ q \\Psi _ { \\mathrm { S l a t e r } } \\ > = \\delta ( p - q ) \\chi ( p ) \\ , \\forall p , q \\in \\Lambda ^ * \\ . \\end{align*}"}
+{"id": "3134.png", "formula": "\\begin{align*} Q _ { m , k } ( x ) = ( x - k + 1 ) Q _ { m - 1 , k } ( x + 1 ) + ( m + k - 2 ) Q _ { m - 1 , k - 1 } ( x + 1 ) , \\end{align*}"}
+{"id": "4405.png", "formula": "\\begin{align*} \\lambda : = - \\frac { L _ { \\omega , \\mathbf { c } } ( U ) } { 3 N ( U ) } , \\end{align*}"}
+{"id": "5853.png", "formula": "\\begin{align*} ( P _ \\omega ( E ) ) = \\widetilde \\Theta _ { \\omega , E } , \\mu _ \\omega E . \\end{align*}"}
+{"id": "1421.png", "formula": "\\begin{align*} \\langle \\ddot { s } , \\nabla ^ 2 f ( x ) \\ddot { s } \\rangle = \\langle T _ { y , s } ^ { - 1 } \\ddot { s } , \\nabla ^ 2 \\hat { f } ( s ) T _ { y , s } ^ { - 1 } \\ddot { s } \\rangle - \\langle \\nabla f ( x ) , c _ { y , s , \\dot { s } } '' ( 0 ) \\rangle = - \\langle g , T _ { y , s } ^ { - 1 } c _ { y , s , \\dot { s } } '' ( 0 ) \\rangle . \\end{align*}"}
+{"id": "5299.png", "formula": "\\begin{align*} \\langle h , A g \\rangle _ { \\mu _ p } = 0 \\end{align*}"}
+{"id": "5829.png", "formula": "\\begin{align*} & \\{ X _ t = y , \\ , N _ t = \\Gamma , \\ , \\tau = t \\} \\\\ & = \\bigcup _ { o = y _ 0 , \\ldots , y _ t = y \\atop 1 = n _ 0 , \\ldots , n _ { t - 1 } = \\Gamma ( y _ { t - 1 } ) } \\begin{array} { c } \\{ X _ 1 = y _ 1 , \\ldots , X _ t = y _ t \\} \\cap \\{ N _ 1 ( y _ 0 ) = n _ 0 , \\ldots N _ t ( y _ { t - 1 } ) = n _ { t - 1 } \\} \\\\ \\cap \\ , \\{ N _ t = \\Gamma \\} \\cap \\{ f _ t ( U ( y _ 0 , n _ 0 ) , \\ldots , U ( y _ { t - 1 } , n _ { t - 1 } ) ) = 1 \\} . \\end{array} \\end{align*}"}
+{"id": "5177.png", "formula": "\\begin{align*} t ( z _ 0 , z _ 1 , z _ 2 , \\dots , z _ m ) = ( t ^ { b _ 0 } z _ 0 , t ^ { b _ 1 } z _ 1 , t ^ { b _ 2 } z _ 2 , \\dots , t ^ { b _ m } z _ m ) . \\end{align*}"}
+{"id": "7191.png", "formula": "\\begin{align*} w : = \\begin{cases} u & \\mbox { i n } B _ \\rho \\setminus \\overline { B _ { s \\rho } } \\\\ w _ s & \\mbox { i n } B _ { s \\rho } , \\end{cases} \\end{align*}"}
+{"id": "837.png", "formula": "\\begin{align*} R _ { C F } = \\log _ { 2 } \\left ( \\det \\left [ \\textbf { R } + \\textbf { I } _ K \\right ] \\right ) \\end{align*}"}
+{"id": "4246.png", "formula": "\\begin{align*} \\mathcal { P } _ { r } \\left ( \\mathbb { R } ^ { n } \\right ) = \\left \\{ \\mathbb { P } _ { \\xi } \\left \\vert \\xi \\in L ^ { r } \\left ( \\Omega ^ { 1 } , \\mathcal { G } , \\mathbb { P } ^ { 1 } \\right ) \\right . \\right \\} , \\end{align*}"}
+{"id": "6272.png", "formula": "\\begin{align*} \\Lambda ^ C _ { h , t } = \\{ \\exp _ x ( h \\phi _ t ( x ) \\nu ( x ) ) \\in M \\ ; | \\ ; x \\in B _ { t } \\setminus B _ { t ^ 2 } \\} , \\end{align*}"}
+{"id": "2038.png", "formula": "\\begin{align*} 0 \\leq u _ { n \\bar n } ( 0 ) = u _ { x _ n x _ n } + u _ { y _ n y _ n } ( 0 ) \\leq C . \\end{align*}"}
+{"id": "7706.png", "formula": "\\begin{align*} ( g _ 0 ) _ j = [ u _ { j + 1 } , \\ldots u _ { r + 1 } , a _ j , \\ldots , a _ r , g _ { r + 1 } ] \\in Q _ { \\langle z _ j \\rangle } \\end{align*}"}
+{"id": "241.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { 1 } { 2 } \\log \\left ( \\frac { 1 } { 1 - z } \\right ) + \\frac { 1 } { 2 } \\frac { z } { 1 - z } \\right \\} , \\end{align*}"}
+{"id": "3453.png", "formula": "\\begin{align*} S ^ { - 1 } r S = r ^ { - 1 } . \\end{align*}"}
+{"id": "2721.png", "formula": "\\begin{align*} \\begin{aligned} T _ 1 \\geq C s ^ 3 \\lambda ^ 4 \\iint _ Q \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d t - C s ^ 3 \\lambda ^ 3 \\int _ 0 ^ T \\int _ { \\omega } \\xi ^ 3 | u | ^ 2 d x d t . \\end{aligned} \\end{align*}"}
+{"id": "3574.png", "formula": "\\begin{align*} k \\cdot r _ { b } ( n ) \\ = \\ n , \\end{align*}"}
+{"id": "7644.png", "formula": "\\begin{align*} M ( c , x , 0 ) = ( 4 - c ^ 2 ) ^ 2 \\left ( 7 2 x ( 1 - x ^ 2 ) + x ^ 2 ( 2 c ^ 2 + ( 3 6 - 1 3 c ^ 2 ) x + 2 c ^ 2 x ^ 2 ) \\right ) . \\end{align*}"}
+{"id": "3455.png", "formula": "\\begin{align*} \\iota \\beta ^ { - 1 } r \\beta \\iota = \\beta ^ { - 1 } r ^ { - 1 } \\beta . \\end{align*}"}
+{"id": "5676.png", "formula": "\\begin{align*} | \\nabla u _ \\nu | ^ 2 _ 2 + | \\nabla v _ \\nu | ^ 2 _ 2 = ( 1 + o _ \\nu ( 1 ) ) [ ( \\gamma _ p + \\gamma _ q ) \\nu D _ 0 ] ^ { \\frac { 2 } { 2 - \\gamma _ p - \\gamma _ q } } . \\end{align*}"}
+{"id": "3911.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ^ { \\star } ( \\lambda ) = \\sup _ { \\pi \\in \\mathcal { G } _ { \\mathrm { D } , \\lambda } } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda d \\pi = \\sup _ { \\pi \\in \\bar { \\Gamma } } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ { \\lambda } d \\pi . \\end{align*}"}
+{"id": "5020.png", "formula": "\\begin{align*} \\| ( a _ j ) _ { j = 1 } ^ n \\| _ p \\coloneqq \\sum _ { j = 1 } ^ { n } | a _ j | ^ p , \\forall ( a _ j ) _ { j = 1 } ^ n \\in \\mathbb { K } ^ n , \\end{align*}"}
+{"id": "3791.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D M R } } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } } \\left \\{ \\lambda \\delta + \\int _ { \\mathcal { X } } \\sup _ { x ' \\in \\mathcal { X } } [ f ( x ' ) - \\lambda c ( x , x ' ) ] \\ , d \\mu ( x ) \\right \\} , \\end{align*}"}
+{"id": "7518.png", "formula": "\\begin{align*} \\nabla = \\partial _ z + Z - \\sum ^ r _ { i = 1 } \\ln ' \\Big [ q ^ i _ + ( z ) \\Big ] \\check { \\alpha } _ i + \\sum ^ { r } _ { i = 1 } \\Lambda _ i ( z ) e _ i . \\end{align*}"}
+{"id": "7240.png", "formula": "\\begin{align*} \\mathcal { E } ( u , \\overline { J _ u } , \\Omega ) = \\mathcal { G } ( u , \\Omega ) \\leq \\mathcal { G } ( v , \\Omega ) \\leq \\mathcal { E } ( v , \\Gamma , \\Omega ) . \\end{align*}"}
+{"id": "1769.png", "formula": "\\begin{align*} X _ T ^ { ( \\ast , n ) } : = \\sup _ { s \\in [ 0 , T ] } X _ s ^ { ( n ) } . \\end{align*}"}
+{"id": "2193.png", "formula": "\\begin{align*} | | \\tilde { \\bf r } _ k | | _ 2 & \\leq 2 \\left ( \\frac { \\sqrt { ( 1 + \\nu ^ 2 ) ^ 2 } - \\sqrt { ( 1 - \\nu ^ 2 ) ^ 2 } } { \\sqrt { ( 1 + \\nu ^ 2 ) ^ 2 } + \\sqrt { ( 1 - \\nu ^ 2 ) ^ 2 } } \\right ) ^ { \\lfloor k / 2 \\rfloor } | | \\tilde { \\bf r } _ 0 | | _ 2 \\\\ & = 2 \\left ( \\nu ^ 2 \\right ) ^ { \\lfloor k / 2 \\rfloor } | | \\tilde { \\bf r } _ 0 | | _ 2 \\\\ & \\leq 2 \\left ( \\nu ^ 2 \\right ) ^ { \\frac { k } { 2 } - \\frac { 1 } { 2 } } | | \\tilde { \\bf r } _ 0 | | _ 2 = 2 \\nu ^ { k - 1 } | | \\tilde { \\bf r } _ 0 | | _ 2 . \\end{align*}"}
+{"id": "6776.png", "formula": "\\begin{align*} \\mathcal Z _ 5 = ( 3 2 , 4 8 , 6 4 , & 8 8 , 6 4 , 8 0 , 8 8 , 9 2 , 6 4 , 8 0 , 8 8 , 1 0 4 , 9 2 , 1 0 4 , 1 0 8 , 9 4 , \\\\ & 7 8 , 8 8 , 9 6 , 1 0 8 , 9 6 , 1 0 4 , 1 0 8 , 1 1 0 , 1 0 2 , 1 0 8 , 1 1 2 , 1 1 8 , 1 1 4 , 1 1 8 , 1 2 0 , 6 4 ) . \\end{align*}"}
+{"id": "1492.png", "formula": "\\begin{align*} & \\tau _ { \\Lambda _ { t - \\cdot } } = \\inf \\{ s \\in [ 0 , t + 1 ] \\ ! : ( R _ s - \\Lambda _ { t - s } ) ( R _ 0 - \\Lambda _ t ) \\leq 0 \\} , \\end{align*}"}
+{"id": "6103.png", "formula": "\\begin{align*} \\mathrm { d } Z _ t = A Z _ t \\mathrm { d } t + F ( Z _ t ) \\mathrm { d } t + G ( Z _ t ) \\circ \\mathrm { d } \\mathbf { X } _ t , \\ \\ \\ \\ Z _ { 0 } = \\xi \\in \\mathcal { B } _ { \\alpha } . \\end{align*}"}
+{"id": "4295.png", "formula": "\\begin{align*} ( \\rho ( \\cdot , t ) , u ( \\cdot , t ) , \\theta ( \\cdot , t ) , q ( \\cdot , t ) , S ( \\cdot , t ) = ( 1 , 0 , 1 , 0 , 0 ) : = ( \\bar \\rho , \\bar u , \\bar \\theta , \\bar q , \\bar S ) \\end{align*}"}
+{"id": "1018.png", "formula": "\\begin{align*} { _ { 2 } F _ { 1 } } \\left [ \\begin{array} { c c c c c c c c } a , b \\\\ c \\end{array} ; x \\right ] = ( 1 - x ) ^ { c - a - b } { _ { 2 } F _ { 1 } } \\left [ \\begin{array} { c c c c c c c c } c - a , c - b \\\\ c \\end{array} ; x \\right ] , \\end{align*}"}
+{"id": "5107.png", "formula": "\\begin{align*} D ^ { 3 } \\bar { w } = D ^ { 3 } w \\end{align*}"}
+{"id": "8681.png", "formula": "\\begin{align*} E ( \\widetilde { \\Delta } _ { 2 } ) = E ( _ { n , m } ^ { 2 } ) - \\Delta _ { 0 } - E ( \\Delta _ { 1 } ) = 0 , \\end{align*}"}
+{"id": "822.png", "formula": "\\begin{align*} A ( x _ 1 t , \\ldots , x _ m t ) \\tilde { A } ( x _ 1 , \\ldots , x _ m , t ) + B ( x _ 1 t , \\ldots , x _ m t ) \\tilde { B } ( x _ 1 , \\ldots , x _ m , t ) = G . \\end{align*}"}
+{"id": "2548.png", "formula": "\\begin{align*} B ( M ) = B ( N ) \\cup \\left \\{ \\{ j , j ^ * \\} \\ , | \\ , j \\in A ( M ) \\right \\} . \\end{align*}"}
+{"id": "8116.png", "formula": "\\begin{align*} \\varphi ( M _ I ) = \\begin{cases} a ^ n & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "4206.png", "formula": "\\begin{align*} S _ \\alpha ( X ) = \\lim _ { \\epsilon \\to 0 } \\frac { 1 } { 4 \\pi i } \\int _ { \\Gamma } f _ \\alpha ( 1 + \\epsilon , \\lambda ) \\frac { d \\log D _ X ( \\lambda ) } { d \\lambda } d \\lambda , \\end{align*}"}
+{"id": "152.png", "formula": "\\begin{align*} L i _ 2 ( z ) = \\sum _ { r = 1 } ^ { \\infty } \\frac { z ^ r } { r ^ 2 } , \\ ; \\ ; | z | \\leq 1 . \\end{align*}"}
+{"id": "3584.png", "formula": "\\begin{align*} \\iota ( \\alpha ) \\ : = \\ \\begin{cases} 0 , & , \\\\ \\alpha , & . \\end{cases} \\end{align*}"}
+{"id": "1234.png", "formula": "\\begin{align*} T _ { \\alpha , \\beta } ( x ) = \\begin{cases} x ( 1 + 2 ^ { \\alpha } x ^ { \\alpha } ) x \\in [ 0 , \\tfrac 1 2 ) , \\\\ 2 ^ { \\beta } ( x - \\tfrac 1 2 ) ^ { \\beta } , x \\in [ \\tfrac 1 2 , 1 ] . \\end{cases} \\end{align*}"}
+{"id": "5731.png", "formula": "\\begin{align*} \\mathcal { E C S } _ { { \\rm { ( r a n k \\ , 2 ) } } } = \\{ S ( p , q , a , b , c , d ) \\in \\mathcal { E C } : { \\rm { r a n k } } \\ , S ( p , q , a , b , c , d ) = 2 \\} . \\end{align*}"}
+{"id": "5813.png", "formula": "\\begin{align*} T ( r , M ) & = T \\left ( r , \\frac { P ' } { P } - \\frac { Q ' } { Q } + P _ 0 \\right ) \\\\ & = m \\left ( r , \\frac { P ' } { P } - \\frac { Q ' } { Q } + P _ 0 \\right ) + N \\left ( r , \\frac { P ' } { P } - \\frac { Q ' } { Q } + P _ 0 \\right ) \\\\ & \\leq S ( r , P ) + S ( r , Q ) + O ( \\log r ) + N \\left ( r , \\frac { P ' } { P } \\right ) + N \\left ( r , \\frac { Q ' } { Q } \\right ) + N ( r , P _ 0 ) \\\\ & = N ( r , \\frac { 1 } { P } ) + N ( r , \\frac { 1 } { Q } ) + S ( r , W ) \\\\ & \\leq T ( r , P ) + T ( r , Q ) + S ( r , W ) , \\end{align*}"}
+{"id": "1251.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { I I } { \\delta } = \\frac { X _ { \\beta , 2 } } { f _ { \\beta , 2 } ' ( g _ { \\beta , 2 } ( x ) ) } \\cdot \\frac { f _ { \\beta , 2 } '' ( g _ { \\beta , 2 } ( x ) ) } { ( f _ { \\beta , 2 } ' ( g _ { \\beta , 2 } ( x ) ) ) ^ 2 } . \\end{align*}"}
+{"id": "1.png", "formula": "\\begin{align*} b _ t : = d i a g ( e ^ { t / m } , \\ldots , e ^ { t / m } , e ^ { - t / n } , \\ldots , e ^ { - t / n } ) , \\ t > 0 \\end{align*}"}
+{"id": "6522.png", "formula": "\\begin{align*} F _ X ( x ) & = M \\int _ { - \\infty } ^ x \\mathrm { e } ^ { \\beta t } ( - t ) ^ \\nu K _ { \\nu } ( - \\alpha t ) \\ , \\mathrm { d } t = M \\sum _ { k = 0 } ^ \\infty \\frac { \\beta ^ k } { k ! } \\int _ { - \\infty } ^ x ( - 1 ) ^ k ( - t ) ^ { \\nu + k } K _ \\nu ( - \\alpha t ) \\ , \\mathrm { d } t \\\\ & = M \\sum _ { k = 0 } ^ \\infty \\frac { ( - \\beta ) ^ k } { k ! } \\int _ { - x } ^ \\infty y ^ { \\nu + k } K _ \\nu ( \\alpha y ) \\ , \\mathrm { d } y , \\end{align*}"}
+{"id": "1514.png", "formula": "\\begin{align*} & \\overline { R } ^ { R ^ x _ { t } } _ s = R ^ x _ { t - s } , \\ ; \\ ; s \\in [ 0 , t ] , \\\\ & R ^ { \\overline { R } ^ x _ { t } } _ s = \\overline { R } ^ x _ { t - s } , \\ ; \\ ; s \\in [ 0 , t ] \\quad \\overline { R } ^ x _ { t } > 0 . \\end{align*}"}
+{"id": "5267.png", "formula": "\\begin{align*} | 1 + y | ^ q = 1 + q y + \\binom { q } { 2 } | 1 + y ' | ^ { q - 2 } y ^ 2 , \\end{align*}"}
+{"id": "3150.png", "formula": "\\begin{align*} \\| w \\| & = \\| v - v _ h \\| = \\| v - J v _ h + J v _ h - v _ h \\| \\\\ & \\leq \\| v - J v _ h \\| + \\| J v _ h - v _ h \\| = \\| v - J v _ h \\| + \\| ( J - 1 ) v _ h \\| . \\end{align*}"}
+{"id": "3372.png", "formula": "\\begin{align*} \\lim _ { \\substack { \\lambda \\to 1 ^ + \\\\ \\kappa \\to 1 ^ + } } \\limsup _ { m , n \\to \\infty } \\max _ { \\substack { P _ m \\leq P _ i \\leq \\lambda P _ { m } \\\\ Q _ n \\leq Q _ j \\leq \\kappa Q _ { n } } } \\left | u _ { i j } - u _ { m n } \\right | = 0 ; \\end{align*}"}
+{"id": "8238.png", "formula": "\\begin{align*} T _ h H _ { k , Y _ { l - h , j - h } } = ( I _ { i + j + h - 2 + t _ { k - j } } ) \\cdot ( F _ { i j } ) - \\frac { 1 } { \\sqrt { x } } ( ( i + j + h - 1 + t _ { k - j } ) I _ { i + j + h - 1 + t _ { k - j } } ) \\cdot ( F _ { i j } ) , \\end{align*}"}
+{"id": "4010.png", "formula": "\\begin{align*} \\varphi _ { \\lambda , \\ell } ( x _ 1 , x _ 2 ) = ( \\lambda _ 1 + \\lambda _ 2 ) h _ \\ell ( x _ 1 ) , \\forall \\ell = 1 , 2 . \\end{align*}"}
+{"id": "4688.png", "formula": "\\begin{align*} - \\partial ^ { \\rm s t i f f ( s o f t ) } _ \\nu \\bigl ( \\mathcal { A } _ { 0 , \\chi } ^ { \\rm s t i f f ( s o f t ) } \\bigr ) ^ { - 1 } = : \\Xi _ \\chi ^ { \\rm s t i f f ( s o f t ) } : \\mathcal { H } ^ { \\rm s t i f f ( s o f t ) } \\to { \\mathcal E } . \\end{align*}"}
+{"id": "9016.png", "formula": "\\begin{align*} \\int _ M | T ^ \\varphi | ^ 2 | \\nabla w | ^ 2 = 0 \\ , . \\end{align*}"}
+{"id": "7423.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb E \\big [ ( X _ { N , M } ^ { ( l ) } ) ^ 4 \\big ] \\le \\frac { \\theta _ N ^ 2 } { N ^ 4 } \\ , \\bigg ( \\ , \\Xi ( 1 ) + \\Xi ( 2 ) + \\sum _ { r = 3 } ^ \\infty \\Xi ( r ) \\ , \\bigg ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "5315.png", "formula": "\\begin{align*} \\| f ^ { = 1 } \\| _ 2 = \\sigma , \\end{align*}"}
+{"id": "6825.png", "formula": "\\begin{align*} \\alpha _ n = \\frac { 1 - a q ^ { 2 n } } { 1 - a } \\sum _ { j \\leq n } \\frac { ( a ) _ { n + j } } { ( q ) _ { n - j } } ( - 1 ) ^ { n - j } q ^ { \\left ( { n - j } \\atop { 2 } \\right ) } \\beta _ j \\ ; \\ ; \\ ; \\ ; \\forall \\ , n \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "6416.png", "formula": "\\begin{align*} ( \\pi _ { \\varphi _ 1 } ( a ) \\otimes \\pi _ { \\varphi _ 2 } ( b ) ) ( \\lambda ^ { \\varphi _ 1 } ( t _ 1 ) \\otimes \\lambda ^ { \\varphi _ 2 } ( t _ 2 ) ) = \\pi _ \\sigma ( a _ 1 \\otimes a _ 2 ) \\lambda ^ \\sigma ( t _ 1 , t _ 2 ) , a \\in M _ 1 , b \\in M _ 2 , ( t _ 1 , t _ 2 ) \\in G \\end{align*}"}
+{"id": "862.png", "formula": "\\begin{align*} ( x ^ { ( ( p - 1 ) - m ) } y ^ m ) ( x ^ { ( ( p - 1 ) - m ) - 2 } y ^ { m + 2 } ) = ( x ^ { ( ( p - 1 ) - m ) - 1 } y ^ { m + 1 } ) ^ 2 \\end{align*}"}
+{"id": "1380.png", "formula": "\\begin{align*} x _ i v _ j = - v _ { 3 i + 5 j \\bmod 7 } , i , j \\in \\{ 1 , 2 , \\dots , 7 \\} . \\end{align*}"}
+{"id": "7072.png", "formula": "\\begin{align*} I ' = I _ { < i _ 0 } : = \\{ i \\in I \\mid i < i _ 0 \\} . \\end{align*}"}
+{"id": "2414.png", "formula": "\\begin{align*} \\sum _ { J = 1 } ^ { \\infty } \\sum _ { k = 1 } ^ { \\infty } P ( A _ { J , k } ) \\le \\sum _ { J = 1 } ^ { \\infty } \\sum _ { k = 1 } ^ { \\infty } \\frac { 4 } { ( x _ { J } + 1 ) ^ { C ^ { 2 } / 2 } ( k + 1 ) ^ { C ^ { 2 } / 2 } } < \\infty . \\end{align*}"}
+{"id": "5767.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { m } A _ { p _ { j } } & \\subsetneq A _ { \\vec { P } } . \\end{align*}"}
+{"id": "8063.png", "formula": "\\begin{align*} \\begin{aligned} - \\operatorname { d i v } \\left ( | \\nabla u | ^ { p - 2 } \\nabla u \\right ) & = \\lambda | u | ^ { p - 2 } u & & \\Omega , \\\\ u & = 0 & & \\partial \\Omega . \\end{aligned} \\end{align*}"}
+{"id": "4622.png", "formula": "\\begin{align*} \\Delta f < n ^ { 1 / ( p + 1 ) ( - p + \\epsilon ) } - n ^ { ( 1 / ( p + 1 ) + \\epsilon ) ( - p - \\epsilon ) } \\\\ = n ^ { \\frac { - p } { p + 1 } } ( n ^ { \\epsilon / ( p + 1 ) } - n ^ { - \\epsilon ( p + 1 / ( p + 1 ) + \\epsilon ) } ) . \\end{align*}"}
+{"id": "4511.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u + \\frac { 1 } { 2 m } \\Delta u = \\overline { u } v , \\\\ i \\partial _ t v + \\frac { 1 } { 2 M } \\Delta v = u ^ 2 , \\end{cases} ( t , x ) \\in \\R \\times \\R ^ d , \\end{align*}"}
+{"id": "4487.png", "formula": "\\begin{align*} S _ { \\omega , \\mathbf { c } } ( U ) = \\frac { 1 } { 2 } L _ { \\omega , \\mathbf { c } } ( U ) + N ( U ) = \\frac { 1 } { 3 } K _ { \\omega , \\mathbf { c } } ( U ) + \\frac { 1 } { 6 } L _ { \\omega , \\mathbf { c } } ( U ) \\end{align*}"}
+{"id": "2302.png", "formula": "\\begin{align*} w _ b = \\sum _ { \\ell } c _ \\ell a _ \\ell + \\Psi _ b . \\end{align*}"}
+{"id": "2856.png", "formula": "\\begin{align*} ( \\widetilde { \\varphi } \\otimes \\widetilde { \\alpha } ) \\circ \\widetilde { \\Delta } ( x , m ) = ( \\varphi _ L \\otimes \\alpha ) \\Delta ( x ) + ( \\varphi _ M \\otimes \\alpha ) ( \\rho _ { \\beta } ( m ) ) - ( \\varphi _ M \\otimes \\alpha ) ( \\tau \\circ ( \\rho _ { \\beta } ( m ) ) ) . \\end{align*}"}
+{"id": "1284.png", "formula": "\\begin{align*} D = \\begin{bmatrix} - \\frac { \\sqrt { n } } { 2 } I _ n & \\left ( \\frac { - \\sqrt { n } } { 2 } + 1 \\right ) I _ n \\\\ \\left ( - \\frac { \\sqrt { n } } { 2 } + 1 \\right ) I _ n & - \\frac { \\sqrt { n } } { 2 } I _ n \\end{bmatrix} = \\begin{bmatrix} - \\frac { \\sqrt { n } } { 2 } & \\left ( - \\frac { \\sqrt { n } } { 2 } + 1 \\right ) \\\\ \\left ( - \\frac { \\sqrt { n } } { 2 } + 1 \\right ) & - \\frac { \\sqrt { n } } { 2 } \\end{bmatrix} \\otimes I _ n , \\end{align*}"}
+{"id": "5328.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ 2 ^ 2 = \\langle f , f ^ { = d } \\rangle \\le \\| f ^ { = d } \\| _ { q } \\| f \\| _ { q ' } . \\end{align*}"}
+{"id": "8679.png", "formula": "\\begin{align*} ( \\widetilde { \\Delta } _ { 2 } ) = O ( N ^ { - 1 } p ^ { - 2 } ) , E ( \\widetilde { \\Delta } _ { 2 } ) - T _ { 1 } = O ( p ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "8502.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } n ^ { s - \\frac { 1 } { 2 } } K _ { s - \\frac { 1 } { 2 } } ( 2 \\pi n \\ , x ) + \\frac { ( \\pi x ) ^ { \\frac { 1 } { 2 } - s } } { 4 } \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) - \\frac { \\pi ^ { - s } x ^ { - s - \\frac { 1 } { 2 } } \\Gamma ( s ) } { 4 } . \\end{align*}"}
+{"id": "2324.png", "formula": "\\begin{align*} D _ { r , z } u ( t , x ) = u ( r , z ) v ^ { ( r , z ) } ( t , x ) , \\end{align*}"}
+{"id": "458.png", "formula": "\\begin{align*} \\Gamma : = \\{ ( t , r ) \\ , | \\ , 1 - g ( r ) t = 0 \\} , \\end{align*}"}
+{"id": "4679.png", "formula": "\\begin{align*} \\left ( I - z \\mathcal { A } _ 0 ^ { - 1 } \\right ) ^ { - 1 } = I + z \\mathcal { A } _ 0 ^ { - 1 } \\left ( I - z \\mathcal { A } _ 0 ^ { - 1 } \\right ) ^ { - 1 } = I + z ( \\mathcal { A } _ 0 - z I ) ^ { - 1 } , \\end{align*}"}
+{"id": "6948.png", "formula": "\\begin{align*} \\langle v ^ { \\otimes d } , ( v ^ * ) ^ { \\otimes d } \\rangle = \\langle v , v ^ * \\rangle ^ d . \\end{align*}"}
+{"id": "4700.png", "formula": "\\begin{align*} \\Pi _ \\chi ^ { \\rm s o f t } = { \\rm e } ^ { - { \\rm i } \\chi y } \\Pi _ 0 ^ { \\rm s o f t } { \\rm e } ^ { { \\rm i } \\chi y } , \\Lambda _ \\chi ^ { \\rm s o f t } = { \\rm e } ^ { - { \\rm i } \\chi y } \\Lambda _ 0 ^ { \\rm s o f t } { \\rm e } ^ { { \\rm i } \\chi y } , \\Gamma _ { \\chi , 1 } ^ { \\rm s o f t } = { \\rm e } ^ { - { \\rm i } \\chi y } \\Gamma _ { 0 , 1 } ^ { \\rm s o f t } { \\rm e } ^ { { \\rm i } \\chi y } . \\end{align*}"}
+{"id": "1235.png", "formula": "\\begin{align*} f _ { \\gamma , 1 } ( x ) = x ( 1 + 2 ^ { \\alpha } x ^ { \\alpha } ) , f _ { \\gamma , 2 } ( x ) = 2 ^ { \\beta } ( x - \\tfrac 1 2 ) ^ { \\beta } . \\end{align*}"}
+{"id": "1695.png", "formula": "\\begin{align*} \\frac { \\kappa _ { \\ell + 1 } - \\kappa _ \\ell } { \\kappa _ \\ell - \\kappa _ { \\ell - 1 } } \\ , = \\ , \\frac { \\mu ( 1 ) + \\nu _ 1 ( \\ell ) } { \\mu ( 1 ) + \\nu _ 3 ( \\ell ) } \\ , \\ , \\ , \\textrm { f o r $ \\ell \\geq a $ } \\ , . \\end{align*}"}
+{"id": "1890.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta u ^ * & = \\lambda ^ * u ^ * \\mbox { i n } \\ \\ \\Omega ; \\\\ u ^ * & = 0 \\quad \\ \\ \\ \\mbox { o n } \\ \\ \\partial \\Omega ; \\\\ \\| u ^ * \\| _ { L ^ 2 ( \\Omega ) } & = 1 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7938.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) u = f ( u ) + g ( u ) \\partial _ x u + h ( u ) ( \\partial _ x u ) ^ 2 + \\sigma ( u ) \\xi . \\end{align*}"}
+{"id": "5061.png", "formula": "\\begin{align*} \\tilde { v } ( t ) : = \\psi \\left ( t \\right ) - \\phi . \\end{align*}"}
+{"id": "8410.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { e ^ { n ^ { 2 } x } - 1 } = \\frac { 1 } { 4 } + \\frac { \\pi ^ { 2 } } { 6 x } + \\frac { 1 } { 2 } \\sqrt { \\frac { \\pi } { x } } \\ , \\zeta \\left ( \\frac { 1 } { 2 } \\right ) + \\frac { 1 } { 2 } \\ , \\sqrt { \\frac { \\pi } { x } } \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { \\sqrt { n } } \\left ( \\frac { \\cos ( 2 \\pi \\sqrt { \\pi n / x } ) - \\sin ( 2 \\pi \\sqrt { \\pi n / x } ) - e ^ { - 2 \\pi \\sqrt { \\pi n / x } } } { \\cosh ( 2 \\pi \\sqrt { \\pi n / x } ) - \\cos ( 2 \\pi \\sqrt { \\pi n / x } ) } \\right ) , \\ , \\ , \\ , \\ , x > 0 . \\end{align*}"}
+{"id": "1742.png", "formula": "\\begin{align*} F ( m ' ) - F ( m ) = \\int _ { 0 } ^ { 1 } \\int _ { \\mathcal { M } } \\frac { \\delta F } { \\delta m } ( m + \\lambda ( m ' - m ) , x ) \\left ( m ' - m \\right ) ( \\mathrm { d } x ) \\ , \\mathrm { d } \\lambda . \\end{align*}"}
+{"id": "5908.png", "formula": "\\begin{align*} \\theta _ { 3 / 2 } ( \\tfrac { a z + b } { c z + d } , \\epsilon ) = \\lambda ^ { \\pm } ( \\gamma , \\epsilon ) ( \\sqrt { \\det \\gamma ( c z + d ) } ) ^ 3 \\sum _ { n \\in \\Z } n e ^ { i ( \\det \\gamma ) \\epsilon \\pi n ^ 2 z } , \\end{align*}"}
+{"id": "1570.png", "formula": "\\begin{align*} \\mathcal { W } _ n ^ { S A W } ( x _ 0 , x _ { 0 } ) = \\emptyset = \\mathcal { W } _ 0 ^ { S A W } ( x _ 0 , x ) \\forall x _ { 0 } \\neq x , n \\geq 1 . \\end{align*}"}
+{"id": "5475.png", "formula": "\\begin{align*} \\bar { F } _ { P _ s } ( \\theta , x ) \\overset { \\Delta } { = } \\Pr \\left ( P _ s ( \\theta ) > x \\right ) , x \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "1433.png", "formula": "\\begin{align*} \\psi ^ 0 ( y ) = \\frac { ( n - 2 ) \\alpha _ n } { 2 } \\frac { | y | ^ 2 - 1 } { ( 1 + | y | ^ 2 ) ^ { \\frac { n } { 2 } } } , \\psi ^ h ( y ) = ( n - 2 ) \\alpha _ n \\frac { y _ h } { ( 1 + | y | ^ 2 ) ^ { \\frac { n } { 2 } } } , \\quad \\mbox { f o r } \\ h = 1 , \\cdots , n . \\end{align*}"}
+{"id": "9223.png", "formula": "\\begin{align*} 3 L < \\frac { 2 a ^ 2 } { 2 5 } = \\frac { t } { 2 5 \\pi } . \\end{align*}"}
+{"id": "3568.png", "formula": "\\begin{align*} B ( T _ { * } ) = - \\frac { 2 \\pi \\sigma ^ { 3 } } { 9 T _ { * } ^ { 1 / 3 } } \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { k ! } \\Gamma \\left ( \\frac { 2 k - 1 } { 3 } \\right ) \\frac { 1 } { T _ { * } ^ { k / 3 } } . \\end{align*}"}
+{"id": "7103.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { - 1 } : \\{ c \\in ( H ^ 1 ( \\Omega ) ) ^ \\prime : c _ \\Omega = 0 \\} \\rightarrow \\{ c \\in H ^ 1 ( \\Omega ) : c _ \\Omega = 0 \\} . \\end{align*}"}
+{"id": "5401.png", "formula": "\\begin{align*} F ( u ) : = \\prod _ { \\deg P \\in S ^ c } ( 1 - u ^ { \\deg P } ) ^ { - 1 } = \\exp \\left ( \\sum _ { \\deg P \\in S ^ c } \\sum _ { k \\ge 1 } \\frac { u ^ { k \\deg P } } { k } \\right ) = : \\sum _ { i = 0 } ^ { \\infty } \\widetilde { A } _ i u ^ i . \\end{align*}"}
+{"id": "5904.png", "formula": "\\begin{align*} T _ 0 : = L _ 0 + C \\hbox { a n d } \\widetilde { T } _ 0 : = \\widetilde { L } _ 0 + C ^ * \\end{align*}"}
+{"id": "4011.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ^ \\star ( \\lambda ) = \\sup _ { \\gamma \\in \\mathcal { P } _ { \\mathrm { D } } } I _ { \\mathrm { D } , \\lambda } [ \\gamma ] = \\sup _ { \\pi \\in \\Gamma ( \\Pi , \\varphi _ \\lambda ) } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda \\ , d \\pi = \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\ldots , \\mu _ L ) } \\int _ { \\mathcal { V } } \\ , g _ \\lambda d \\pi . \\end{align*}"}
+{"id": "518.png", "formula": "\\begin{align*} \\mathcal { H } = - \\mathcal { L } + \\frac { 1 } { | x | ^ { 2 } } . \\end{align*}"}
+{"id": "4539.png", "formula": "\\begin{align*} \\| \\varphi _ 1 \\| _ { W ^ { 2 , p _ 0 } } = \\| ( - \\alpha \\Delta + 2 \\omega + i \\mathbf { c } \\cdot \\nabla ) ^ { - 1 } \\{ ( \\nabla \\cdot \\varphi _ 3 ) \\varphi _ 2 \\} \\| _ { W ^ { 2 , p _ 0 } } \\lesssim \\| ( \\nabla \\cdot \\varphi _ 3 ) \\varphi _ 2 \\| _ { L ^ { p _ 0 } } < \\infty \\end{align*}"}
+{"id": "7367.png", "formula": "\\begin{align*} S _ N ( \\ell ) = \\sum _ { j = 1 } ^ N \\chi _ \\ell ( x _ j ) . \\end{align*}"}
+{"id": "5978.png", "formula": "\\begin{align*} [ g , \\epsilon ] [ k ( t ) , 1 ] & = [ g k ( t ) , \\overline { \\overline { C } } _ { X ^ { \\ast } } ( g , k ( t ) ) \\epsilon ] \\\\ & = [ g k ( t ) , \\epsilon ] \\\\ & = [ p _ g , \\epsilon ] [ k _ g , 1 ] [ k ( t ) , 1 ] . \\end{align*}"}
+{"id": "7640.png", "formula": "\\begin{align*} M ( 2 , x , y ) = 0 , x \\in ( 0 , 1 ) , y \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "4364.png", "formula": "\\begin{align*} \\mathcal { A } \\left ( C , u \\right ) & : = \\arg \\min \\sum \\limits _ { i = 1 } ^ { m } \\left ( u _ { i } \\right ) ^ { p } d _ { S _ { i } } ^ { p } \\left ( x \\right ) + I _ { C } \\left ( x \\right ) , \\bigskip \\\\ \\mathcal { A } \\left ( l \\right ) & : = \\arg \\min \\sum \\limits _ { i = 1 } ^ { m } \\left ( l _ { i } \\right ) ^ { p } d _ { S _ { i } } ^ { p } \\left ( x \\right ) . \\end{align*}"}
+{"id": "6315.png", "formula": "\\begin{align*} ( x , y , z ) \\star ( x ' , y ' , z ' ) = \\bigg ( x + x ' , y + y ' , z + z ' + \\frac 1 2 ( x y ' - x ' y ) \\bigg ) , \\qquad \\forall \\ , ( x , y , z ) , ( x ' , y ' , z ' ) \\in \\R ^ 3 , \\end{align*}"}
+{"id": "1063.png", "formula": "\\begin{align*} f \\circ _ { } g \\coloneqq \\sum _ { i = 0 } ^ { \\infty } \\left ( \\sum _ { j + k + \\ell = i } \\frac { i ! } { j ! k ! \\ell ! } \\alpha ^ { j } \\beta ^ k \\gamma ^ \\ell a ( j + k ) b ( \\ell + k ) \\right ) t ^ { i } \\in k [ [ t ] ] \\ , , \\end{align*}"}
+{"id": "3650.png", "formula": "\\begin{align*} x _ 2 ^ 2 = y _ 2 ^ 2 , x _ 1 ^ 2 + x _ 2 ^ 2 = y _ 1 ^ 2 + y _ 2 ^ 2 , \\mathrm { a n d } x _ 1 ^ 2 + x _ 1 x _ 2 + x _ 2 ^ 2 = y _ 1 ^ 2 + y _ 1 y _ 2 + y _ 2 ^ 2 . \\end{align*}"}
+{"id": "245.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { 1 } { 4 } L i _ 2 ( z ) + \\frac { 1 } { 3 } \\log \\left ( \\frac { 1 } { 1 - z } \\right ) + \\frac { 1 } { 4 } \\frac { z } { 1 - z } \\right \\} , \\end{align*}"}
+{"id": "1070.png", "formula": "\\begin{align*} \\| e ^ { - t H ^ \\beta } g \\| _ { L ^ q } \\le \\begin{cases} C e ^ { - t d ^ \\beta } \\| g \\| _ { L ^ p } & t \\geq 1 \\\\ C t ^ { - \\sigma _ \\beta } \\| g \\| _ { L ^ p } & 0 < t \\leq 1 . \\end{cases} \\end{align*}"}
+{"id": "3953.png", "formula": "\\begin{align*} \\begin{aligned} \\boldsymbol { W } _ { p } ( \\nu ( t _ 0 ) , \\nu ( t _ 1 ) ) & = \\boldsymbol { W } _ { p } \\left ( \\nu ( t _ 0 ) , \\gamma + \\Delta \\pi ^ \\prime \\right ) \\\\ & \\leq ( 1 - \\Delta ) \\boldsymbol { W } _ { p } \\left ( \\nu ( t _ 0 ) , ( 1 - \\Delta ) ^ { - 1 } \\gamma \\right ) + \\underbrace { \\Delta \\boldsymbol { W } _ { p } \\left ( \\nu ( t _ 0 ) , \\pi ^ \\prime \\right ) } _ { = O ( \\Delta ) } . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "6864.png", "formula": "\\begin{align*} \\sigma ( S ) = \\{ \\lambda \\in \\mathbb C \\colon \\ , S - \\lambda I \\} . \\end{align*}"}
+{"id": "2121.png", "formula": "\\begin{align*} \\tau _ a ( \\alpha ) & = \\frac { 1 } { q } \\sum _ { u \\in \\mathbb { F } _ { q } } \\psi _ 1 \\left ( _ { \\mathbb { F } _ { q ^ m } / \\mathbb { F } _ { q } } ( u \\alpha ) - u a \\right ) \\\\ & = \\frac { 1 } { q } \\sum _ { u \\in \\mathbb { F } _ { q } } \\hat { \\psi _ 1 } ( u \\alpha ) \\psi _ 1 ( - u a ) , \\end{align*}"}
+{"id": "7578.png", "formula": "\\begin{align*} H _ { 2 , 1 } ( F _ { f _ { \\theta } } / 2 ) = \\frac { e ^ { 4 i \\theta } } { 4 8 } \\left ( a ^ 4 _ 2 - 1 2 a ^ 2 _ 3 + 1 2 a _ 2 a _ 4 \\right ) = e ^ { 4 i \\theta } H _ { 2 , 1 } ( F _ { f } / 2 ) . \\end{align*}"}
+{"id": "7340.png", "formula": "\\begin{align*} 1 + e v _ y ^ 2 v _ z ^ 2 - e u _ z ^ 2 = 0 , \\end{align*}"}
+{"id": "3995.png", "formula": "\\begin{align*} \\mathrm { R } _ { \\mathrm { D } } ( \\lambda , \\delta ) = \\langle \\lambda , \\delta \\rangle + \\sup _ { x ^ { \\prime } \\in \\mathbb { R } ^ d } \\left [ \\sum _ { 1 \\leq \\ell \\leq 2 } \\varphi _ \\ell ( x ^ { \\prime } , \\lambda _ \\ell ) \\right ] \\mathrm { R } _ { \\mathrm { D } } ( \\lambda , \\delta ) = \\langle \\lambda , \\delta \\rangle + \\sum _ { 1 \\leq \\ell \\leq 2 } \\sup _ { x ^ { \\prime } \\in \\mathbb { R } ^ d } \\varphi _ \\ell ( x ^ { \\prime } , \\lambda _ \\ell ) . \\end{align*}"}
+{"id": "2187.png", "formula": "\\begin{align*} | - 2 z + \\lambda _ i z ^ 2 | & \\leq 2 c _ 1 + \\lambda _ i c _ 1 ^ 2 \\\\ & \\leq 2 \\left ( \\sqrt { \\frac { 1 } { 2 } + \\frac { 1 } { \\lambda _ i ^ 2 } } - \\frac { 1 } { \\lambda _ i } \\right ) + \\lambda _ i \\left ( \\sqrt { \\frac { 1 } { 2 } + \\frac { 1 } { \\lambda _ i ^ 2 } } - \\frac { 1 } { \\lambda _ i } \\right ) ^ 2 = \\frac { \\lambda _ i } { 2 } . \\end{align*}"}
+{"id": "362.png", "formula": "\\begin{align*} ( \\underline { X } , \\underline { A } ) ^ { K , c } = \\mbox { c o l i m } _ { \\sigma \\subseteq K } ( \\underline { X } , \\underline { A } ) ^ { \\sigma , c } . \\end{align*}"}
+{"id": "7784.png", "formula": "\\begin{align*} x ^ { r ^ 2 + r + 1 } + k x ^ { r + 1 } - 1 = 0 , \\end{align*}"}
+{"id": "2005.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( L _ a - c ) w = 0 & \\textnormal { i n } & \\Omega \\\\ w = 0 & \\textnormal { o n } & \\partial \\Omega \\\\ w \\in C ^ { 2 } ( \\Omega ) \\cap C ^ 0 ( \\bar \\Omega ) . & & \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "6876.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i , j \\leq k } | \\alpha _ i \\alpha _ j | \\ , 2 d _ { \\square } ( f _ n , f ) , \\end{align*}"}
+{"id": "5144.png", "formula": "\\begin{align*} \\| f \\| _ { X '' } = \\sup _ { \\| g \\| _ { X ' } = 1 } \\| f g \\| _ { L ^ 1 ( \\Omega ) } \\leq \\sup _ { \\| g \\| _ { X ' } = 1 } \\| f \\| _ X \\| g \\| _ { X ' } = \\| f \\| _ X \\end{align*}"}
+{"id": "3068.png", "formula": "\\begin{align*} \\delta = \\frac { ( n - 1 ) ( p - 1 - \\alpha ) } { p - 1 } > 0 . \\end{align*}"}
+{"id": "580.png", "formula": "\\begin{align*} \\phi ^ { \\chi _ + } ( t _ { 1 } , t _ { 2 } ) = \\int _ { t _ 1 } ^ { \\infty } H ( s _ { 1 } ) [ H ( s _ { 1 } ) ] ^ T d s _ { 1 } . \\end{align*}"}
+{"id": "8089.png", "formula": "\\begin{align*} j - \\frac { 3 } { p } = \\left ( m - \\frac { 3 } { r } \\right ) \\alpha + ( 1 - \\alpha ) \\left ( - \\frac { 3 } { q } \\right ) , \\frac { j } { m } \\leq \\alpha \\leq 1 . \\end{align*}"}
+{"id": "5888.png", "formula": "\\begin{align*} \\varphi ' ( t ) - N c = - N ^ 2 \\ , , \\end{align*}"}
+{"id": "4335.png", "formula": "\\begin{align*} u _ n ( x , t ) = \\begin{cases} u _ { \\mathcal { K } _ n } ^ { \\{ L _ \\mathcal { P } \\} _ { \\mathcal { P } \\in \\mathcal { N } _ { \\mathcal { K } _ n } } } ( x , t ) & t \\in [ 0 , 5 0 ] , \\\\ - u _ { \\mathcal { K } _ n } ^ { \\{ L _ \\mathcal { P } \\} _ { \\mathcal { P } \\in \\mathcal { N } _ { \\mathcal { K } _ n } } } ( x , 1 0 0 - t ) & t \\in [ 5 0 , 1 0 0 ] . \\end{cases} \\end{align*}"}
+{"id": "4289.png", "formula": "\\begin{align*} ( \\rho \\eta ) _ t + \\left ( \\rho u \\eta + \\frac { q } { \\theta } \\right ) _ x = \\frac { q ^ 2 } { \\kappa ( \\theta ) \\theta ^ 2 } + \\frac { S ^ 2 } { \\mu \\theta } . \\end{align*}"}
+{"id": "3521.png", "formula": "\\begin{align*} \\det ( \\cos t I - \\sin t S ) = ( - \\sin t ) ^ n \\det ( S - \\cot t I ) = 0 . \\end{align*}"}
+{"id": "7279.png", "formula": "\\begin{align*} ( x , y , z , t ) = ( u v , w r , u w , v r ) \\ , \\ , u , v , w , r . \\end{align*}"}
+{"id": "283.png", "formula": "\\begin{align*} \\prod _ { \\substack { l , m , n \\geq 1 \\\\ l , m \\leq n ; \\ , \\gcd ( l , m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { l m ^ 2 } { n ^ 4 } } = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { n } { 2 } + \\frac { n ^ 2 } { 2 } \\right ) \\left ( \\frac { n } { 6 } + \\frac { n ^ 2 } { 2 } + \\frac { n ^ 3 } { 3 } \\right ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "3526.png", "formula": "\\begin{gather*} R ( t ) = \\begin{bmatrix} R _ 1 ( t ) _ { k \\times k } & R _ 2 ( t ) _ { k \\times ( n - k ) } \\\\ R _ 3 ( t ) _ { ( n - k ) \\times k } & R _ 4 ( t ) _ { ( n - k ) \\times ( n - k ) } \\end{bmatrix} , \\\\ G ( t ) = \\begin{bmatrix} G _ 1 ( t ) _ { k \\times k } & G _ 2 ( t ) _ { k \\times ( n - k ) } \\\\ G _ 3 ( t ) _ { ( n - k ) \\times k } & G _ 4 ( t ) _ { ( n - k ) \\times ( n - k ) } \\end{bmatrix} , \\end{gather*}"}
+{"id": "655.png", "formula": "\\begin{align*} & K ^ { a , i , \\tau } _ { k } ( T ) : = \\sum _ { l \\geq k } \\| Q | _ { U ^ { ( k ) } _ i } ( k , l + 1 ) ^ { - 1 } \\| \\| Z ( l + 1 ) \\| \\langle \\| Q ( l ) \\| \\rangle ^ { a } \\| Q ( l + 1 ) \\| ^ \\tau , \\\\ & C ^ { a , i , \\tau } _ { k } ( T ) : = \\sum _ { 0 \\leq l < k } \\| Q | _ { E ^ { ( l + 1 ) } _ i } ( l + 1 , k ) \\| \\| Z ( l + 1 ) \\| \\langle \\| Q ( l ) \\| \\rangle ^ { a } \\| Q ( l + 1 ) \\| ^ \\tau . \\end{align*}"}
+{"id": "4026.png", "formula": "\\begin{align*} \\mathrm { d i m } _ { \\mathbb { C } } \\ , \\mathbb { S } _ k ( \\Gamma , \\mathbb { C } ) = 2 \\ , \\mathrm { d i m } _ { \\mathbb { C } } \\ , { S } _ k ( \\Gamma ) . \\end{align*}"}
+{"id": "2390.png", "formula": "\\begin{align*} 1 + \\cos \\alpha _ { 1 0 2 } + \\cos \\alpha _ { 1 0 3 } + \\cos \\alpha _ { 1 0 4 } = 0 . \\end{align*}"}
+{"id": "2954.png", "formula": "\\begin{align*} ( 1 + \\overline { c } _ 1 + \\dots + \\overline { c } _ n ) \\cdot ( 1 + c _ 1 + \\dots + c _ { k + n } ) = 1 \\ . \\end{align*}"}
+{"id": "6631.png", "formula": "\\begin{align*} M _ \\psi : = \\| \\psi \\| _ { L ^ { 1 , \\infty } ( G \\times \\widehat { G } ) } = \\sup _ { s > 0 } \\ s \\underset { \\underset { | \\psi ( x , \\chi ) | > s } { ( x , \\chi ) \\in G \\times \\widehat { G } } } { \\int } d x d \\chi . \\end{align*}"}
+{"id": "5830.png", "formula": "\\begin{align*} \\frac { \\pi _ 1 ( X _ n ) } { \\pi _ 2 ( X _ n ) } = \\frac { \\pi _ 1 ( X _ { \\tau _ { H _ n } } ) } { H _ n } + \\frac { \\pi _ 1 ( X _ n ) - \\pi _ 1 ( X _ { \\tau _ { H _ n } } ) } { H _ n } + \\pi _ 1 ( X _ n ) \\left ( \\frac { 1 } { \\pi _ 2 ( X _ n ) } - \\frac { 1 } { H _ n } \\right ) . \\end{align*}"}
+{"id": "5624.png", "formula": "\\begin{align*} \\aligned \\left \\{ \\begin{array} { l l l } - \\Delta u + \\lambda _ 1 u = { ( K ( x ) \\ast | u | ^ { r _ 1 } ) | u | ^ { r _ 1 - 2 } u } + \\nu p ( K ( x ) \\ast | v | ^ q ) | u | ^ { p - 2 } u \\ & \\mathbb { R } ^ N , \\\\ - \\Delta v + \\lambda _ 2 v = ( K ( x ) \\ast | v | ^ { r _ 2 } ) | v | ^ { r _ 2 - 2 } v + \\nu q ( K ( x ) \\ast | u | ^ p ) | v | ^ { q - 2 } v \\ & \\mathbb { R } ^ N , \\end{array} \\right . \\endaligned \\end{align*}"}
+{"id": "2760.png", "formula": "\\begin{align*} ( u | v ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } = ( u | ( \\Delta + \\lambda - q ) w ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } = ( ( \\Delta + \\lambda - q ) u | w ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } = 0 . \\end{align*}"}
+{"id": "3616.png", "formula": "\\begin{align*} r ( p ) \\ = \\ f q \\ \\geq \\ 1 5 ( 1 0 ^ m - 3 1 ) \\ \\geq \\ 1 0 ^ { m + 1 } \\end{align*}"}
+{"id": "1629.png", "formula": "\\begin{align*} \\pounds _ { \\xi _ i } = \\iota _ { { \\xi _ i } } \\ , d + d \\ , \\iota _ { { \\xi _ i } } , \\end{align*}"}
+{"id": "5693.png", "formula": "\\begin{align*} v _ { p } ( C _ { F } ( p ^ { 2 m + 1 } ) ) = ( k - 1 ) m . \\end{align*}"}
+{"id": "2080.png", "formula": "\\begin{align*} O _ { B } ^ { H } ( M ) & = \\min \\left \\{ \\sum _ { i = 1 } ^ k 2 ^ i x _ i \\mid \\sum _ { i = 1 } ^ k ( 2 ^ i - 1 ) x _ i = M , \\ x _ i \\in \\mathbb { N } , 1 \\leq i \\leq k \\right \\} \\\\ & = \\min \\left \\{ M + \\sum _ { i = 1 } ^ k x _ i \\mid \\sum _ { i = 1 } ^ k ( 2 ^ i - 1 ) x _ i = M , \\ x _ i \\in \\mathbb { N } , 1 \\leq i \\leq k \\right \\} . \\end{align*}"}
+{"id": "7544.png", "formula": "\\begin{align*} F _ j : = F ( f , B _ { 4 \\rho _ j } ( x _ 0 ) ) . \\end{align*}"}
+{"id": "6127.png", "formula": "\\begin{align*} \\bigcup \\limits _ { \\mathcal { H } \\times J \\in N H } \\mathcal { H } \\times J = \\bigcup _ { \\mathcal { K } \\times I \\in P ^ { 1 } N D _ { \\lambda } } \\mathcal { K } \\times I = \\bigcup _ { \\mathcal { K } \\times I \\in P ^ { 2 } N D _ { \\lambda } } \\mathcal { K } \\times I . \\end{align*}"}
+{"id": "3937.png", "formula": "\\begin{align*} \\mathcal { I } ^ \\star ( \\lambda ) = \\sup _ { \\pi \\in \\Gamma \\left ( \\Pi \\left ( \\mu _ { 1 3 } , \\mu _ { 2 3 } \\right ) , \\phi _ \\lambda \\right ) } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\phi _ \\lambda \\ , d \\pi . \\end{align*}"}
+{"id": "8022.png", "formula": "\\begin{align*} M _ a = \\begin{pmatrix} 1 & 0 \\\\ a & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "3154.png", "formula": "\\begin{align*} \\sum _ { k = l } ^ { l + n } \\delta _ { k , k + 1 } ^ 2 & \\leq 4 e _ l ^ 2 + 2 ^ { 6 } \\varepsilon ^ { - 1 } \\Lambda _ 6 ^ { 2 } \\mu _ { l } ^ 2 + ( \\frac { 1 } { 2 } \\varepsilon C _ { \\rm { R E L } } ^ { - 2 } - 4 ) e _ { l + n + 1 } ^ 2 + ( \\frac { \\varepsilon } { 4 } - 2 ^ { 6 } \\varepsilon ^ { - 1 } \\Lambda _ 6 ^ 2 ) \\mu _ { l + n + 1 } ^ 2 \\\\ & + \\frac { \\varepsilon } { 2 } \\sum _ { k = l } ^ { l + n } \\mu _ k ^ 2 + \\frac { \\varepsilon } { 2 } C _ { \\rm { R E L } } ^ { - 2 } \\sum _ { k = l } ^ { l + n - 1 } e _ { k + 1 } ^ 2 . \\end{align*}"}
+{"id": "8604.png", "formula": "\\begin{align*} & E [ X _ { \\ell , \\Delta } | { \\cal F } _ { ( \\ell + 1 ) \\Delta } ] = \\nu \\int _ { \\ell \\Delta } ^ { ( \\ell + 1 ) \\Delta } Z _ 0 ( s ) d s = \\Delta \\nu Z _ 0 ( \\ell \\Delta ) ( 1 + O ( \\Delta ) ) \\end{align*}"}
+{"id": "2634.png", "formula": "\\begin{align*} G ( w ) : & = F ( \\mathbf { t } ( u _ 0 , v _ 0 , w ) ) \\\\ & = \\alpha ( \\bar { \\mathbf { z } } ) H ( t _ 1 ) ^ { - 1 } H ( t _ 2 ) H ( t _ 3 ) ^ { - 1 } - \\beta ( \\bar { \\mathbf { z } } + ( u _ 0 , v _ 0 ) ) , \\end{align*}"}
+{"id": "9049.png", "formula": "\\begin{align*} b _ { i , j } > 0 \\iff \\varepsilon ( i ) = + \\varepsilon ( j ) = - . \\end{align*}"}
+{"id": "4283.png", "formula": "\\begin{align*} E = \\frac { 1 } { 2 } \\rho u ^ 2 + \\frac { \\tau _ 2 } { 2 \\mu } \\rho S ^ 2 + \\rho e ( \\theta , q ) , \\end{align*}"}
+{"id": "3230.png", "formula": "\\begin{align*} \\sigma _ 2 & = - e _ 1 e _ 2 - e _ 2 e _ 3 - e _ 3 e _ 1 = t _ 1 ^ 2 + t _ 2 ^ 2 + t _ 1 t _ 2 \\\\ \\sigma _ 3 & = - e _ 1 e _ 2 e _ 3 = t _ 1 ^ 2 t _ 2 + t _ 1 t _ 2 ^ 2 . \\end{align*}"}
+{"id": "6771.png", "formula": "\\begin{align*} = [ 0 , 1 , 3 , 0 , 1 , 3 , 0 , 1 , 2 , 1 , 1 , 2 , 3 , 3 , 2 , 3 ] . \\end{align*}"}
+{"id": "2865.png", "formula": "\\begin{align*} R \\circ \\alpha = \\alpha \\circ R , \\end{align*}"}
+{"id": "8371.png", "formula": "\\begin{align*} D _ a = a - d , D _ b = b - d , D _ c = c - d \\end{align*}"}
+{"id": "4315.png", "formula": "\\begin{align*} \\begin{aligned} t _ { k , m } + 2 ^ { - 2 k } & = t _ { k + 1 , 2 m } , \\\\ t _ { k , m } + 2 ^ { 1 - 2 k } & = t _ { k , m + 1 } . \\end{aligned} \\end{align*}"}
+{"id": "9329.png", "formula": "\\begin{align*} X _ { 1 4 } : = E _ { 4 , \\mathbb { H } } X _ { 1 0 } . \\end{align*}"}
+{"id": "8471.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } I _ { m , p } ^ { \\star } ( s , x ) = \\sqrt { \\pi } 2 ^ { \\frac { 1 } { 2 } - s } x ^ { s + \\frac { 1 } { 2 } } \\intop _ { 0 } ^ { \\infty } \\ , \\frac { y ^ { s - \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( x y ) } { \\sigma \\left ( \\frac { y } { 2 \\pi } \\right ) e ^ { y } - 1 } \\ , d y . \\end{align*}"}
+{"id": "5531.png", "formula": "\\begin{align*} h _ { \\mu } ( Z , \\lambda ) & = h _ { \\mu } ( Y , \\nu ) + \\lim _ { n \\to \\infty } \\int _ { X } \\left ( H \\left ( \\xi _ { n } ^ { x } \\right ) - H \\left ( \\xi _ { n - 1 } ^ { x } \\right ) \\right ) d \\eta ( x ) \\\\ & = h _ { \\mu } ( Y , \\nu ) + \\lim _ { n \\to \\infty } \\frac { 1 } { n } H \\left ( \\xi _ { n } | X , \\eta \\right ) . \\end{align*}"}
+{"id": "7615.png", "formula": "\\begin{align*} \\mathcal { S } ^ * ( 1 / 2 ) = \\bigg \\{ f \\in \\mathcal { A } : { \\rm R e } \\left ( \\frac { z f ^ { \\prime } ( z ) } { f ( z ) } \\right ) > 1 / 2 , \\ ; z \\in \\mathbb { D } \\bigg \\} \\end{align*}"}
+{"id": "1455.png", "formula": "\\begin{align*} | \\nabla \\chi _ m ^ i ( x ) | \\leq \\frac { 2 } { \\sqrt { \\mu _ m ^ i \\mu _ m ^ { i - 1 } } } { \\rm a n d } | \\nabla ^ 2 \\chi _ m ^ i ( x ) | \\leq \\frac { 4 } { \\mu _ m ^ i \\mu _ m ^ { i - 1 } } , \\quad \\mbox { f o r \\ a n y } \\ i = 1 , \\cdots , k . \\end{align*}"}
+{"id": "2617.png", "formula": "\\begin{align*} B _ \\mathbf { m } ( s ) = \\int _ { \\mathbb { R } ^ 3 } \\ ! & \\mathcal { D } _ { \\widetilde { P } _ 1 ' ( \\bar { t } ) s } ^ { ( 1 ) } f _ { 1 , m _ 1 } \\left ( x + \\widetilde { P } _ 1 ( t ) , y \\right ) \\mathcal { D } _ { \\widetilde { P } _ 2 ' ( \\bar { t } ) s } ^ { ( 2 ) } f _ { 2 , m _ 2 } \\left ( x , y + \\widetilde { P } _ 2 ( t ) \\right ) \\\\ & \\widetilde { \\zeta } _ \\mathbf { m } ( x , y , t , s ) \\ , \\mathrm { d } x \\mathrm { d } y \\mathrm { d } t . \\end{align*}"}
+{"id": "4319.png", "formula": "\\begin{align*} \\begin{gathered} f ^ { 2 K } ( \\cdot , t ) \\xrightarrow { K \\to \\infty } f _ 0 \\circ \\big ( \\tilde { y } ^ \\mathrm { e v e n } _ t \\big ) ^ { - 1 } , \\\\ f ^ { 2 K + 1 } ( \\cdot , t ) \\xrightarrow { K \\to \\infty } f _ 0 \\circ \\big ( \\tilde { y } ^ \\mathrm { o d d } _ t \\big ) ^ { - 1 } , \\end{gathered} \\end{align*}"}
+{"id": "9126.png", "formula": "\\begin{align*} \\begin{array} { c c l } y _ { [ 2 ] } ^ { 1 } & = & v ^ { 1 } \\\\ y _ { [ 2 ] } ^ { 2 } & = & v ^ { 2 } \\ , . \\end{array} \\end{align*}"}
+{"id": "7507.png", "formula": "\\begin{align*} \\nabla = \\partial _ z + u ( z ) \\check { \\alpha } . \\end{align*}"}
+{"id": "8306.png", "formula": "\\begin{align*} \\frac { a } { q } = [ 0 ; c _ 1 , \\dots , c _ s ] = \\cfrac { 1 } { c _ 1 + \\cfrac { 1 } { c _ 2 + \\cfrac { 1 } { c _ 3 + \\cdots + \\cfrac { 1 } { c _ s } } } } \\ , , c _ j \\le M \\ , , \\forall j \\in [ s ] \\ , . \\end{align*}"}
+{"id": "5209.png", "formula": "\\begin{align*} \\mathsf { S } \\mathbb { P } & = \\{ S \\subseteq \\mathbb { P } : S \\} \\\\ & = \\{ S \\subseteq \\mathbb { P } : S \\} \\\\ & = \\{ S \\subseteq \\mathbb { P } : S \\} . \\end{align*}"}
+{"id": "4174.png", "formula": "\\begin{align*} \\psi _ s ( a ) \\psi _ t ( a ) = \\psi _ { s t } ( a b ) , \\ \\ \\psi _ t ( b ) ^ * = \\psi _ { t ^ { - 1 } } ( b ^ * ) . \\end{align*}"}
+{"id": "6474.png", "formula": "\\begin{align*} & E _ 2 ( t , \\eta ) \\\\ & = \\sum _ { K = K _ 1 - K _ 2 + K _ 3 } | \\eta _ K | ^ 2 | \\eta _ { K _ 1 } | ^ 2 | \\eta _ { K _ 2 } | ^ 2 | \\eta _ { K _ 3 } | ^ 2 \\Big ( \\frac { 1 } { | \\eta _ { K } | ^ 2 } - \\frac { 1 } { | \\eta _ { K _ 1 } | ^ 2 } + \\frac { 1 } { | \\eta _ { K _ 2 } | ^ 2 } - \\frac { 1 } { | \\eta _ { K _ 3 } | ^ 2 } \\Big ) \\Big | \\frac { \\sin ( \\frac { 1 } { 2 } t \\Delta \\omega _ { K K _ 1 K _ 2 K _ 3 } ) } { \\frac { 1 } { 2 } \\Delta \\omega _ { K K _ 1 K _ 2 K _ 3 } } \\Big | ^ 2 \\\\ & \\quad + o ( t ^ 2 L ^ 2 \\log ( L ) + t L ^ 4 ) \\end{align*}"}
+{"id": "2854.png", "formula": "\\begin{align*} \\widetilde { \\Delta } ( x , m ) & = \\Delta ( x ) + \\rho _ { \\beta } ( m ) - \\tau \\circ ( \\rho _ { \\beta } ( m ) ) , \\\\ \\widetilde { \\varphi } ( x , m ) & = ( \\varphi _ L ( x ) , \\varphi _ M ( m ) ) , \\\\ \\widetilde { \\alpha } ( x , m ) & = ( \\alpha ( x ) , \\beta ( m ) ) , \\end{align*}"}
+{"id": "1133.png", "formula": "\\begin{align*} T ( f \\cdot g ) = f T ( g ) + T ( f ) g + 2 B ( A ( f ) , A ( g ) ) \\end{align*}"}
+{"id": "3817.png", "formula": "\\begin{align*} f _ \\lambda ( v ) : = \\sup _ { s ^ \\prime \\in \\mathcal { S } } \\phi _ \\lambda ( v , s ^ { \\prime } ) . \\end{align*}"}
+{"id": "5455.png", "formula": "\\begin{align*} & G ( \\theta ) \\approx \\sum _ { b _ 1 , \\cdots , b _ M } { b \\choose b _ 1 , \\cdots , b _ M } \\prod _ { m = 1 } ^ { M } \\left ( C _ { M } ^ { m } ( - 1 ) ^ { m + 1 } \\right ) ^ { b _ m } \\\\ & \\frac { ( R _ { m a x } \\ ! \\ ! - \\ ! \\ ! R _ { m i n } ) \\pi } { 2 N } \\ ! \\sum _ { k = 1 } ^ { K } \\ ! \\sqrt { 1 \\ ! \\ ! - \\ ! \\psi _ k ^ 2 } \\exp \\left ( - Q ( d _ k , \\theta ) \\right ) f _ { r _ 1 | \\Phi ( \\mathcal { A } ) > 0 } ( d _ k ) , \\end{align*}"}
+{"id": "5367.png", "formula": "\\begin{align*} z ^ H M _ 2 M _ 2 ^ H z = \\| M _ 2 ^ H z \\| ^ 2 = \\| w \\| ^ 2 \\leq \\| v \\| ^ 2 = \\| M _ 1 ^ H z \\| ^ 2 = z ^ H M _ 1 M _ 1 ^ H z \\end{align*}"}
+{"id": "295.png", "formula": "\\begin{align*} \\prod _ { \\substack { l , m , n \\geq 1 \\\\ l , m \\leq n ; \\ , \\gcd ( l , m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - x ^ l y ^ m z ^ n } \\right ) ^ { \\frac { l m ^ 2 } { n ^ 4 } } \\end{align*}"}
+{"id": "9009.png", "formula": "\\begin{align*} ( \\ast ) & = w _ { i j } R ^ \\varphi _ { i k } R ^ \\varphi _ { k j } + w _ j R ^ \\varphi _ { i k } R ^ \\varphi _ { k j , i } + w _ j R ^ \\varphi _ { i k , i } R ^ \\varphi _ { k j } \\\\ & \\phantom { = \\ ; } + w _ { t i } R _ { i j k t } R ^ \\varphi _ { j k } + w _ t R _ { i j k t , i } R ^ \\varphi _ { j k } + w _ t R _ { i j k t } R ^ \\varphi _ { j k , i } \\ , . \\end{align*}"}
+{"id": "2942.png", "formula": "\\begin{align*} 1 2 \\cdot \\mu ^ { p \\delta } _ { D R } = - [ ( \\mu _ { D D , \\delta } ) _ c ] . \\end{align*}"}
+{"id": "3494.png", "formula": "\\begin{align*} \\gamma ( a ) = p , \\gamma ( b ) = q , \\end{align*}"}
+{"id": "7505.png", "formula": "\\begin{align*} \\Lambda ( z ) = q _ { + } ^ 2 ( z ) p ^ 2 ( z ) L ( z ) \\end{align*}"}
+{"id": "3493.png", "formula": "\\begin{align*} J ( \\sigma ) = \\int _ a ^ b F ( \\sigma , \\dot \\sigma ) \\ , d t \\end{align*}"}
+{"id": "3767.png", "formula": "\\begin{align*} \\acute { \\rho } _ \\sigma ( \\hat { s } ) = & \\sum _ { 0 \\leq a < b \\leq 3 } \\acute { \\rho } _ \\sigma ^ { a b } ( \\hat { s } ) [ \\det \\acute { J } _ { a b } ^ \\sigma ] ( \\hat { s } ) , \\\\ | \\acute { \\rho } _ \\sigma | ( \\hat { s } ) = & \\left | \\sum _ { 0 \\leq a < b \\leq 3 } \\acute { \\rho } _ \\sigma ^ { a b } ( \\hat { s } ) [ \\det \\acute { J } _ { a b } ^ \\sigma ] ( \\hat { s } ) \\right | . \\end{align*}"}
+{"id": "8079.png", "formula": "\\begin{align*} \\langle G ^ { \\prime } ( u ) - G ^ { \\prime } ( v ) , u - v \\rangle = 0 . \\end{align*}"}
+{"id": "1733.png", "formula": "\\begin{align*} \\frac { \\delta F } { \\delta \\nu } ( \\nu _ t , \\mu _ t , x ) + \\frac { \\sigma ^ 2 } { 2 } \\log \\frac { \\Psi ( \\nu _ t , \\mu _ t ) ( x ) } { \\pi ( x ) } = C _ t , \\end{align*}"}
+{"id": "1097.png", "formula": "\\begin{align*} ( b b ' ) \\ast - = - \\langle b \\ast - , b ' \\ast - \\rangle , \\forall b , b ' \\in B , \\end{align*}"}
+{"id": "9129.png", "formula": "\\begin{align*} \\begin{array} { c c l } u ^ { 1 } & = & v ^ { 1 } - x ^ { 1 } \\\\ u ^ { 2 } & = & \\left ( 1 - v ^ { 1 } + v _ { [ 1 ] } ^ { 1 } \\right ) v ^ { 2 } \\ , . \\end{array} \\end{align*}"}
+{"id": "1232.png", "formula": "\\begin{align*} d ( S , ( \\delta , 1 ) ) \\leq d ( ( \\delta , f ( \\delta ) ) , ( \\delta , 1 ) ) = 1 - f ( \\delta ) \\end{align*}"}
+{"id": "3079.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 ^ + } \\frac { A ( r ) ^ 2 } { A ' ( r ) } \\cdot \\left [ \\mathcal W ( r ) ^ { p - 1 - \\alpha } \\right ] ' & = \\lim _ { r \\to 0 ^ + } r ^ \\delta u _ 0 ' ( r ) ^ { p - 1 - \\alpha } \\cdot \\frac { g _ 1 ( v ( r ) ) } { h _ 1 ( v _ 0 ( r ) ) } = 0 \\end{align*}"}
+{"id": "77.png", "formula": "\\begin{align*} { \\rm e l l } _ h ( A ) = \\{ ( x _ 0 , \\xi _ 0 ) \\in \\overline { T } ^ * \\mathcal { M } \\ , : \\ , ( \\langle \\xi \\rangle ^ { - k } \\sigma _ h ( A ) ) ( x _ 0 , \\xi _ 0 ) \\neq 0 \\} . \\end{align*}"}
+{"id": "2848.png", "formula": "\\begin{align*} { \\bf E } \\left ( { \\bf A } , { \\mathbb V } \\right ) = \\left \\{ { \\bf A } _ { \\theta } : \\theta \\left ( x , y \\right ) = \\sum _ { i = 1 } ^ { s } \\theta _ { i } \\left ( x , y \\right ) e _ { i } \\ \\ \\ \\ \\left \\langle \\left [ \\theta _ { 1 } \\right ] , \\left [ \\theta _ { 2 } \\right ] , \\dots , \\left [ \\theta _ { s } \\right ] \\right \\rangle \\in { \\bf T } _ { s } ( { \\bf A } ) \\right \\} . \\end{align*}"}
+{"id": "5148.png", "formula": "\\begin{align*} E : = \\bigcap _ { m = 1 } ^ \\infty \\bigcup _ { k = m } ^ \\infty E _ k \\end{align*}"}
+{"id": "2278.png", "formula": "\\begin{align*} \\left | \\langle T ( f ) _ b , \\varphi \\rangle - \\langle T ( f ) ( \\cdot ) , \\varphi \\rangle \\right | & = \\left | \\lim _ { r \\nearrow 1 } \\int _ 0 ^ { 2 \\pi } T ( f ) ( r e ^ { i \\theta } ) \\varphi ( \\theta ) \\ , d \\theta - \\int _ 0 ^ { 2 \\pi } T ( f ) ( e ^ { i \\theta } ) \\varphi ( \\theta ) \\ , d \\theta \\right | \\\\ & \\leq C _ \\varphi \\lim _ { r \\nearrow 1 } \\int _ 0 ^ { 2 \\pi } | T ( f ) ( r e ^ { i \\theta } ) - T ( f ) ( e ^ { i \\theta } ) | \\ , d \\theta = 0 , \\end{align*}"}
+{"id": "3812.png", "formula": "\\begin{align*} \\varphi _ \\lambda ( v , v ^ \\prime ) = g \\left ( s _ 1 ^ \\prime , s _ 2 ^ \\prime \\right ) - \\lambda _ 1 c _ 1 \\left ( s _ 1 , s _ 1 ^ \\prime \\right ) - \\lambda _ 2 c _ 2 \\left ( s _ 2 , s _ 2 ^ \\prime \\right ) , \\end{align*}"}
+{"id": "1829.png", "formula": "\\begin{align*} \\nu \\big ( [ [ a _ p ^ * a _ p , V _ F ( t ) ] , V _ F ( s ) ] \\big ) = \\nu \\big ( [ [ a _ p ^ * a _ p , : \\ ! V _ F ( t ) \\ ! : ] , V _ F ( s ) ] \\big ) \\end{align*}"}
+{"id": "8157.png", "formula": "\\begin{align*} S _ n \\left ( \\frac { 1 - q t | } { | 1 - q } A \\right ) = ( 1 - q t ) \\sum _ { k = 0 } ^ n ( - q t ) ^ k R _ { 1 ^ k , n - k } \\left ( \\frac { A } { 1 - q } \\right ) \\end{align*}"}
+{"id": "4341.png", "formula": "\\begin{align*} \\begin{array} { l l } \\underset { x \\in X } { } & \\sum \\limits _ { i = 1 } ^ { m } f _ { i } \\left ( x \\right ) , \\end{array} \\end{align*}"}
+{"id": "2788.png", "formula": "\\begin{align*} \\tilde { \\mathbf { m } } _ \\lambda = \\lambda \\mathbf { b } _ \\lambda \\mathbf { e } _ \\lambda . \\end{align*}"}
+{"id": "2742.png", "formula": "\\begin{align*} H ^ s ( \\Sigma ) = \\{ \\varphi = h _ { | \\Sigma } ; \\ ; h \\in H ^ { 1 / 2 } ( \\partial \\mathrm { M } ) \\} , \\end{align*}"}
+{"id": "6289.png", "formula": "\\begin{align*} \\frac \\nu 2 \\| \\bar u ( t ) \\| ^ 2 = \\max _ { v \\in \\R ^ k } \\frac \\nu 2 \\| v \\| ^ 2 . \\end{align*}"}
+{"id": "4587.png", "formula": "\\begin{align*} & \\psi _ V ( [ u , v , w ] _ { T ' } ) \\\\ = & \\psi _ V ( \\rho ( T ' u , T ' v ) w ) \\\\ = & \\rho ( \\psi _ L ( T ' u ) , \\psi _ L ( T ' v ) ) \\psi _ V ( w ) \\\\ = & \\rho ( T \\psi _ V ( u ) , T \\psi _ V ( v ) ) \\psi _ V ( w ) \\\\ = & [ \\psi _ V ( u ) , \\psi _ V ( v ) , \\psi _ V ( w ) ] _ T . \\end{align*}"}
+{"id": "2339.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 } \\sup _ { y \\in \\R } \\Big ( \\int _ { \\R ^ 2 } I _ k d x d z \\Big ) ^ 2 | \\hbar | ^ { 2 H - 2 } | h | ^ { 2 H - 2 } d h d \\hbar < C k = 2 , 3 , 4 . \\end{align*}"}
+{"id": "6104.png", "formula": "\\begin{align*} u _ { t } + \\mathcal { L } ^ { \\Phi } _ { A } u = ( - \\Delta ) ^ { \\frac { s } { 2 } } f + g \\quad \\Omega _ { T } \\equiv \\Omega \\times ( 0 , T ) , \\end{align*}"}
+{"id": "2246.png", "formula": "\\begin{align*} P _ r ( \\theta ) = \\frac { 1 - r ^ 2 } { 1 - 2 r \\cos ( \\theta ) + r ^ 2 } , \\end{align*}"}
+{"id": "1409.png", "formula": "\\begin{align*} l o n g t a i l ( \\lambda ) = \\max _ { \\Sigma \\in \\mathbb { S } } \\sharp \\Sigma \\cap C _ { \\lambda } \\end{align*}"}
+{"id": "4159.png", "formula": "\\begin{align*} \\bigg \\| \\lambda _ s ( b ) \\bigg ( \\sum _ { t \\in G } b _ t \\delta _ t \\bigg ) \\bigg \\| ^ 2 & = \\bigg \\| \\sum _ { t \\in G } b b _ t \\delta _ { s t } \\bigg \\| ^ 2 = \\bigg \\| \\sum _ { t \\in G } ( b b _ t ) ^ * b b _ t \\bigg \\| = \\bigg \\| \\sum _ { t \\in G } b _ t ^ * ( b ^ * b ) b _ t \\bigg \\| \\\\ & \\leq \\bigg \\| \\sum _ { t \\in G } \\| b \\| ^ 2 b _ t ^ * b _ t \\bigg \\| = \\| b \\| ^ 2 \\bigg \\| \\sum _ { t \\in G } b _ t ^ * b _ t \\bigg \\| = \\| b \\| ^ 2 \\bigg \\| \\sum _ { t \\in G } b _ t \\delta _ t \\bigg \\| ^ 2 \\end{align*}"}
+{"id": "5785.png", "formula": "\\begin{align*} g ^ { ( k ) } + A _ 1 g = 0 \\end{align*}"}
+{"id": "9127.png", "formula": "\\begin{align*} \\begin{array} { c c c c l } v ^ { 1 } & = & \\delta ( \\varphi ^ { 1 } ) & = & x ^ { 1 } + u ^ { 1 } \\\\ v ^ { 2 } & = & \\delta ^ { 2 } ( \\varphi ^ { 2 } ) & = & \\frac { u ^ { 2 } } { u _ { [ 1 ] } ^ { 1 } + 1 } \\end{array} \\end{align*}"}
+{"id": "5781.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\alpha } ( \\vec { f } ) ( x ) & \\le \\prod _ { i = 1 } ^ { m } M _ { \\alpha _ { i } } f _ { i } ( x ) \\ \\ \\ x \\in \\mathbb { R } ^ { n } . \\end{align*}"}
+{"id": "6534.png", "formula": "\\begin{align*} \\tilde { t } _ { \\nu , \\nu } ( x ) = \\mathbf { L } _ \\nu ( x ) . \\end{align*}"}
+{"id": "1152.png", "formula": "\\begin{align*} T ( f ^ { 2 } ) = 2 f T ( f ) + 2 B ( A ( f ) , A ( f ) ) \\left ( f \\in U \\subset P \\right ) \\end{align*}"}
+{"id": "3012.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } \\lambda _ i ^ { \\gamma } ( C + D ) + \\sum _ { i = 1 } ^ { k } \\lambda _ i ^ { \\gamma } ( C - D ) \\geq 2 \\sum _ { i = 1 } ^ k \\lambda _ i ^ { \\gamma } ( C ) . \\end{align*}"}
+{"id": "8648.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } M _ j ( t ) & = \\frac { \\nu q Y } { \\lambda } \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } ( 1 - y ) \\sum _ { k = j } ^ { \\infty } y ^ { k - 1 } d y \\\\ & = \\frac { \\nu q Y } { \\lambda } \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } y ^ { j - 1 } d y . \\end{align*}"}
+{"id": "8412.png", "formula": "\\begin{align*} \\zeta _ { Q } ( s ) = \\sum _ { m , n \\neq 0 } \\frac { 1 } { \\left ( a m ^ { 2 } + b m n + c n ^ { 2 } \\right ) ^ { s } } , \\ , \\ , \\ , \\ , \\ , ( s ) > 1 , \\end{align*}"}
+{"id": "1795.png", "formula": "\\begin{align*} ( \\partial _ T + P \\cdot \\nabla _ X ) F = 4 \\pi & \\int \\d P _ 2 \\d P _ 3 \\d P _ 4 \\delta ( P + P _ 2 - P _ 3 - P _ 4 ) \\delta ( P ^ 2 + P _ 2 ^ 2 - P _ 3 ^ 2 - P _ 4 ^ 2 ) \\\\ & \\times | \\hat V ( P - P _ 3 ) - \\hat V ( P - P _ 4 ) | ^ 2 ( F F _ 2 \\widetilde F _ 3 \\widetilde F _ 4 - F _ 4 F _ 3 \\widetilde F _ 2 \\widetilde F ) \\ . \\end{align*}"}
+{"id": "6268.png", "formula": "\\begin{align*} | { \\Sigma _ t } | = | { \\Sigma _ 0 } | - \\frac { 1 } { 2 } | \\lambda | t ^ 2 + O ( t ^ 3 ) . \\end{align*}"}
+{"id": "3377.png", "formula": "\\begin{align*} P _ \\mu = \\frac { P _ \\mu } { P _ { \\mu - 1 } } P _ { \\mu - 1 } \\leq \\left ( 1 + \\frac { \\delta } { 4 } \\right ) P _ m \\left ( 1 + \\frac { \\delta } { 2 } \\right ) \\leq P _ m ( 1 + \\delta ) = \\lambda P _ m , \\end{align*}"}
+{"id": "6356.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { \\frac { d + 2 \\beta } { \\alpha } \\frac { q - 1 } { q } } \\| P _ t ^ { \\Gamma } f - A \\Psi _ t \\| _ { q , M _ { \\Gamma } } = 0 . \\end{align*}"}
+{"id": "4285.png", "formula": "\\begin{align*} ( \\rho ( x , 0 ) , u ( x , 0 ) , \\theta ( x , 0 ) , S ( x , 0 ) , q ( x , 0 ) ) = ( \\rho _ 0 , u _ 0 , \\theta _ 0 , S _ 0 , q _ 0 ) . \\end{align*}"}
+{"id": "9103.png", "formula": "\\begin{align*} B ^ { k + 2 } = B ^ { k + 1 } \\cup \\{ s \\} \\setminus \\{ p \\} = B ^ { k } \\cup \\{ s \\} \\setminus \\{ q \\} . \\end{align*}"}
+{"id": "7933.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) ( u - u ( x ) ) = u ( x ) \\xi + ( u - u ( x ) ) \\xi . \\end{align*}"}
+{"id": "7845.png", "formula": "\\begin{align*} ( x , y ) \\sim _ { X \\boxtimes Y } ( u , v ) \\Leftrightarrow \\begin{cases} x = u \\mbox { a n d } y \\sim _ Y v \\\\ y = v \\mbox { a n d } x \\sim _ X u \\\\ x \\sim _ X u \\mbox { a n d } y \\sim _ Y v . \\end{cases} \\end{align*}"}
+{"id": "5706.png", "formula": "\\begin{align*} ( d - d ^ { \\sigma } ) g = D ^ { k - 1 } ( h ^ { \\sigma } - h ) \\end{align*}"}
+{"id": "1015.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } \\binom { 3 k } { k } } { x ^ k } ( 3 H _ { 3 k } - H _ k ) = \\bigg ( \\log \\frac { x } { x - 2 7 } \\bigg ) \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } \\binom { 3 k } { k } } { x ^ k } , \\end{align*}"}
+{"id": "2832.png", "formula": "\\begin{align*} \\frac { S _ k ^ l ( t ) } { S _ { l - 1 } ^ l ( t ) } = \\tilde S _ k ^ l ( t ) , l = 1 , \\dots , p ; k \\in \\mathbb { Z } _ + . \\end{align*}"}
+{"id": "1180.png", "formula": "\\begin{align*} { 1 \\over 2 } \\sum _ j E _ j \\R ^ \\bullet \\pi _ * ( d _ j ) = \\Theta ^ 2 ( V ) ^ { - 1 } \\end{align*}"}
+{"id": "1487.png", "formula": "\\begin{align*} \\int _ \\Omega \\Big | V ^ p \\Big ( \\ln \\ln ( e + V ) \\Big ) ^ 2 \\psi ^ h _ { \\mu _ j , \\xi _ j } \\Big | d x = O ( ( \\ln | \\ln \\mu _ i | ) ^ 2 ) = O \\bigg ( \\Big ( \\ln \\Big | \\ln \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big | \\Big ) ^ 2 \\bigg ) . \\end{align*}"}
+{"id": "8673.png", "formula": "\\begin{align*} _ { n , m } ^ { 2 } = \\frac { 1 } { n ( n - 1 ) } \\sum _ { i _ { 1 } \\neq i _ { 2 } } k ( X _ { i _ { 1 } } , X _ { i _ { 2 } } ) + \\frac { 1 } { m ( m - 1 ) } \\sum _ { j _ { 1 } \\neq j _ { 2 } } k ( Y _ { j _ { 1 } } , Y _ { j _ { 2 } } ) - \\frac { 2 } { n m } \\sum _ { i , j } k ( X _ { i } , Y _ { j } ) , \\end{align*}"}
+{"id": "6895.png", "formula": "\\begin{align*} \\left | \\lambda _ { \\min } ( T _ n ) - \\lambda _ { \\min } ( T ) \\right | = \\left | \\inf _ { \\| f \\| _ 2 = 1 } \\langle f , T _ n f \\rangle - \\inf _ { \\| f \\| _ 2 = 1 } \\langle f , T f \\rangle \\right | \\leq \\sup _ { \\| f \\| _ 2 = 1 } | \\langle f , T _ n f \\rangle - \\langle f , T f \\rangle | \\to 0 \\end{align*}"}
+{"id": "8960.png", "formula": "\\begin{align*} e _ { i j } = \\phi _ i - \\phi _ j = \\sum _ { m = 1 } ^ { \\infty } C _ { q _ m } h ^ { q _ m } \\big [ r ^ { ( i - j ) q _ m } - 1 \\big ] r ^ { ( j - 1 ) q _ m } \\end{align*}"}
+{"id": "5460.png", "formula": "\\begin{align*} \\bar { F } _ { P _ s } ( \\theta , x ) \\overset { \\Delta } { = } \\Pr \\left ( P _ s ( \\theta ) > x \\right ) , x \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "6836.png", "formula": "\\begin{align*} \\alpha _ n = ( - 1 ) ^ { n + m } q ^ { \\left ( { n + m \\atop 2 } \\right ) } \\frac { 1 - a q ^ { 2 n } } { 1 - a } \\frac { ( a ) _ { n - m } } { ( q ) _ { n + m } } \\mbox { a n d } \\quad \\beta _ n = \\delta _ { n , - m } , \\end{align*}"}
+{"id": "8922.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ 3 y \\big ( \\bigl \\{ X _ { n i } ^ { ( j ) } \\bigr \\} \\big ) = \\sum _ { l = 1 } ^ 3 Z _ l + \\sum _ { l = 4 } ^ k B _ l ^ { ( 3 ) } \\bigl ( \\bigl \\{ X _ { n i } ^ { ( 1 ) } \\bigr \\} , \\bigl \\{ X _ { n i } ^ { ( 2 ) } \\bigr \\} , \\bigl \\{ X _ { n i } ^ { ( 3 ) } \\bigr \\} \\bigr ) . \\end{align*}"}
+{"id": "7864.png", "formula": "\\begin{align*} \\mathcal { O } _ i = \\left \\{ ( A , B ) : \\ \\mbox { $ A , B \\in \\binom { [ n ] } { k } $ s u c h t h a t } | A \\cap B | = k - i \\right \\} , \\end{align*}"}
+{"id": "6009.png", "formula": "\\begin{align*} \\mathcal { F } ' _ { X ^ { \\ast } X } : & L ^ 2 ( X ^ { \\ast } ) \\longrightarrow L ^ 2 ( X ) ; \\\\ & g \\longmapsto \\mathcal { F } ' _ { X ^ { \\ast } X } ( g ) ( x ) = \\int _ { X ^ { \\ast } } \\psi ( \\langle x ^ { \\ast } , x \\rangle ) g ( x ^ { \\ast } ) d x ^ { \\ast } , \\end{align*}"}
+{"id": "5607.png", "formula": "\\begin{align*} { \\rm I I } = \\int _ { X } \\int _ { G ^ { \\mathbb { N } } } \\log \\frac { d \\left ( \\omega _ { 1 } \\lambda \\right ) _ { x } } { d \\lambda _ { x } } \\left ( \\psi _ { x } ( \\omega ) \\right ) d \\mathbb { P } _ { \\mu } ^ { \\pi ( x ) } ( \\omega ) d \\eta ( x ) , \\end{align*}"}
+{"id": "8763.png", "formula": "\\begin{align*} F ( p ) : = \\begin{cases} \\min _ { v } & \\sum _ { i = 1 } ^ n c p _ i \\max ( - x _ i - y _ i , - d _ i ) + \\sum _ { i \\neq j } \\alpha _ { i j } v _ { i j } \\\\ s . t . \\quad & \\begin{cases} - v \\leq 0 \\\\ \\sum _ { j \\neq i } v _ { i j } - x _ { 0 i } \\leq 0 , \\forall i \\in I \\end{cases} \\end{cases} \\end{align*}"}
+{"id": "5052.png", "formula": "\\begin{align*} \\| \\langle X , t \\rangle \\| _ { L _ k } \\leq L k ^ { 1 / \\alpha } \\| \\langle X , t \\rangle \\| _ { L _ 2 } = L k ^ { 1 / \\alpha } \\| t \\| _ 2 . \\end{align*}"}
+{"id": "1584.png", "formula": "\\begin{align*} F ( x ) : = \\frac { x ^ { N } } { x + m ^ 2 } . \\end{align*}"}
+{"id": "962.png", "formula": "\\begin{align*} | f ( x ) - f ( y ) | \\leqslant C _ n \\cdot \\frac { ( \\Vert Q \\Vert _ 1 ) ^ { \\frac { 1 } { q } } } { \\log ^ { \\frac { 1 } { n } } \\left ( 1 + \\frac { r _ 0 } { 2 | x - y | } \\right ) } \\ , , \\ , \\ , r _ 0 = d ( K , \\partial D ) , \\end{align*}"}
+{"id": "5264.png", "formula": "\\begin{align*} E _ S ( E _ { T ^ c } L _ T [ f ] ) _ { S \\rightarrow x } = E _ S E _ { T ^ c } L _ { T ' } ( L _ S [ f ] ) _ { S \\rightarrow x } = E _ { T '^ c } L _ { T ' } [ D _ { S , x } [ f ] ] = ( D _ { S , x } f ) ^ { = T \\setminus S } . \\end{align*}"}
+{"id": "7869.png", "formula": "\\begin{align*} \\begin{cases} \\tau _ 0 ( A ) & = \\sigma ( A ) \\\\ \\tau _ i ( A ) & = \\mu ( A , i ) \\left ( \\tau _ { i - 1 } ( A ) \\right ) , \\mbox { f o r } 1 \\leq i \\leq k , \\end{cases} \\end{align*}"}
+{"id": "596.png", "formula": "\\begin{align*} f ( \\alpha , s ) & = \\rho ( s ^ 2 t \\cos ( \\gamma + \\beta ) + s \\sqrt { 1 - s ^ 2 } \\cos ( \\alpha + \\beta ) ) . \\end{align*}"}
+{"id": "3477.png", "formula": "\\begin{align*} r = \\Omega \\left ( \\frac { b ^ { 5 + 6 \\kappa } } { \\chi \\beta \\Delta e ^ { \\beta \\Delta } \\cdot ( \\Delta \\log n ) ^ { \\kappa } ( \\log \\log n ) ^ { 1 + \\kappa } \\cdot e ^ { 1 3 \\kappa } } \\right ) . \\end{align*}"}
+{"id": "2081.png", "formula": "\\begin{align*} N _ { d r } = \\left ( \\sum _ { i = 1 } ^ k x _ i \\right ) a + r ( a + d ) = \\left ( \\sum _ { i = 1 } ^ k 2 ^ i x _ i \\right ) a + r d . \\end{align*}"}
+{"id": "1486.png", "formula": "\\begin{align*} J _ 3 = & \\int _ \\Omega \\Big [ f _ 0 ( V ) - f _ \\epsilon ( V ) \\Big ] \\psi ^ h _ { \\mu _ j , \\xi _ j } d x \\\\ = & \\epsilon \\int _ \\Omega V ^ p \\ln \\ln ( e + V ) \\psi ^ h _ { \\mu _ j , \\xi _ j } d x - \\epsilon ^ 2 \\int _ \\Omega V ^ p \\Big ( \\ln \\ln ( e + V ) \\Big ) ^ 2 \\psi ^ h _ { \\mu _ j , \\xi _ j } d x . \\end{align*}"}
+{"id": "2673.png", "formula": "\\begin{align*} \\| D ^ i ( \\sigma [ f ] - f ) \\| _ { L ^ { \\infty } [ a , b ] } & \\leq C _ { f , X } \\overline { h } ^ { 4 - i } ; i = 0 , 1 , 2 { } \\\\ \\| D ^ 3 ( \\sigma [ f ] - f ) \\| _ { L ^ { \\infty } [ a , b ] } & \\leq C _ { f , X } \\overline { h } \\end{align*}"}
+{"id": "5430.png", "formula": "\\begin{align*} r _ 1 ( u , \\bar v ) & = \\sum _ { j = 1 } ^ n R ( e _ j , \\bar e _ j , u , \\bar v ) , & r _ 2 ( u , \\bar v ) & = \\sum _ { j = 1 } ^ n R ( e _ j , u , \\bar v , \\bar e _ j ) , \\\\ r _ 3 ( u , \\bar v ) & = \\sum _ { j = 1 } ^ n R ( u , \\bar v , e _ j , \\bar e _ j ) , & r _ 4 ( u , \\bar v ) & = \\sum _ { j = 1 } ^ n R ( u , e _ j , \\bar e _ j , \\bar v ) , \\end{align*}"}
+{"id": "7563.png", "formula": "\\begin{align*} \\binom i s \\alpha ^ s ( 1 - \\alpha ) ^ { i - s } \\binom { j - i } r \\alpha ^ r ( 1 - \\alpha ) ^ { j - i - r } , \\end{align*}"}
+{"id": "9340.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\xi _ t ^ u & = e ^ { - \\frac { \\beta } { 2 } t } h ( x _ t ^ u , u _ t ) d t + d \\tilde { W } _ t , \\ t \\in [ 0 , \\infty ) , \\\\ \\xi _ 0 & = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "404.png", "formula": "\\begin{align*} S ^ { - 1 } ( I ^ - - R I ^ + ) \\sqrt { | \\Lambda | } T ^ { - 1 } U = G L = S ^ { - 1 } ( I ^ - - R I ^ + ) \\sqrt { | \\Lambda | } T ^ { - 1 } . \\end{align*}"}
+{"id": "3485.png", "formula": "\\begin{align*} K _ j = K _ j ^ 2 = \\left ( \\prod _ { i : v _ i \\in I _ j } P _ i \\right ) ^ 2 = \\left ( \\prod _ { i : v _ i \\in I _ j } P _ i \\prod _ { i : v _ i \\notin I _ j } I \\right ) ^ 2 = \\prod _ { i = 1 } ^ { n } \\hat { P } _ i \\prod _ { i = 0 } ^ { n - 1 } \\hat { P } _ { n - i } . \\end{align*}"}
+{"id": "1086.png", "formula": "\\begin{align*} [ ( d , D ) , ( d ' , D ' ) ] = ( d \\circ d ' - d ' \\circ d , D \\circ d ' - d ' \\circ D ) . \\end{align*}"}
+{"id": "5301.png", "formula": "\\begin{align*} A f ( x ) = f ( \\neg x ) , \\end{align*}"}
+{"id": "3522.png", "formula": "\\begin{align*} L ( t ) = \\begin{bmatrix} - S _ 1 ( t ) & - S _ 2 ( t ) \\\\ 0 & I \\\\ I & 0 \\\\ S _ 3 ( t ) & S _ 4 ( t ) \\\\ \\end{bmatrix} , \\end{align*}"}
+{"id": "9130.png", "formula": "\\begin{align*} \\begin{array} { c c l } y _ { [ 1 ] } ^ { 1 } & = & v ^ { 1 } \\\\ y _ { [ 2 ] } ^ { 2 } & = & v ^ { 2 } \\ , . \\end{array} \\end{align*}"}
+{"id": "1936.png", "formula": "\\begin{align*} f ^ * ( z _ 1 , \\ldots , z _ { m } ) = \\sum _ { n = m } ^ { \\infty } P _ n ( z _ 1 , \\ldots , z _ { m } ) = \\sum _ { i _ 1 \\geq 1 , \\ldots , i _ m \\geq 1 } c _ { i _ 1 , \\ldots , i _ m } z _ 1 ^ { i _ 1 } \\cdots z _ m ^ { i _ m } , \\end{align*}"}
+{"id": "6405.png", "formula": "\\begin{align*} [ D \\widetilde { \\psi } : D \\tau _ M ] _ t = [ D \\psi : D \\varphi ] _ t \\ , \\lambda ^ { \\varphi } ( t ) \\end{align*}"}
+{"id": "1303.png", "formula": "\\begin{align*} \\rho _ i ( u ) = \\log ( \\theta _ i ) + B ^ * _ i ( u ) + \\frac { 1 } { 2 } u + \\log \\left ( \\frac { T _ i ^ { \\infty } - T _ i ( u ) } { T _ i ^ { \\infty } } \\right ) . \\end{align*}"}
+{"id": "1101.png", "formula": "\\begin{align*} b \\ast ( b ' \\ast x ) = 0 , \\forall b , b ' \\in B , \\ ; \\forall x \\in X . \\end{align*}"}
+{"id": "8597.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } e ^ { - \\lambda t } S _ { j , + } ( t ) = \\nu Y \\int _ 0 ^ \\infty e ^ { - \\lambda s } \\left ( \\sum _ { k = 1 } ^ \\infty p _ { j , + } ^ k ( s ) \\right ) d s , \\\\ & \\lim _ { t \\to \\infty } e ^ { - \\lambda t } S _ { j , - } ( t ) = \\nu Y \\int _ 0 ^ \\infty e ^ { - \\lambda s } \\left ( \\sum _ { k = 1 } ^ \\infty p _ { j , - } ^ k ( s ) \\right ) d s , \\end{align*}"}
+{"id": "3525.png", "formula": "\\begin{align*} L ' ( t _ 0 ) & = \\Phi ' ( t _ 0 ) L ( t _ 0 ) G ( t _ 0 ) + \\Phi ( t _ 0 ) L ( t _ 0 ) G ' ( t _ 0 ) \\\\ & = A ( t _ 0 ) L ( t _ 0 ) + L ( t _ 0 ) G ' ( t _ 0 ) . \\end{align*}"}
+{"id": "6647.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + \\langle b , \\ , \\nabla u \\rangle + \\rho \\ , c u = 0 \\ , \\ , \\ , \\R ^ N , \\\\ \\end{align*}"}
+{"id": "8264.png", "formula": "\\begin{align*} I _ { i + j + \\alpha - 1 } ( 2 \\sqrt { x } ) = \\sqrt { x } \\frac { d } { d x } I _ { i + j + \\alpha } ( 2 \\sqrt { x } ) + \\frac { \\alpha + i + j } { 2 \\sqrt { x } } I _ { i + j + \\alpha } ( 2 \\sqrt { x } ) , \\end{align*}"}
+{"id": "3198.png", "formula": "\\begin{align*} \\bar { \\mu } : = \\norm { \\cdot } _ 1 - \\lim _ { a \\to \\infty } M _ a ( \\mu ) . \\end{align*}"}
+{"id": "8637.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } M _ - ( t ) = \\nu Y \\int _ 0 ^ { \\infty } e ^ { - \\lambda s } p _ 0 ( s ) d s , \\end{align*}"}
+{"id": "5645.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | u _ n | ^ { p } ) | v _ n | ^ { q } = \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | u | ^ { p } ) | v | ^ { q } + o _ n ( 1 ) . \\end{align*}"}
+{"id": "4181.png", "formula": "\\begin{align*} u _ \\mathrm { l i n } ( t , r ) = \\begin{cases*} 2 & i f $ 0 < r < t $ , \\\\ \\frac { 2 t } { r } & i f $ t \\leq r < 1 - t $ , \\\\ \\frac { 1 - r + t } { r } & i f $ 1 - t \\leq r < 1 + t $ , \\\\ 0 & o t h e r w i s e . \\end{cases*} \\end{align*}"}
+{"id": "6409.png", "formula": "\\begin{align*} [ D \\widetilde { \\psi } : D \\tau _ M ] _ t = s ( \\psi ) [ D \\widetilde { \\chi } : D \\tau _ M ] _ t & = s ( \\psi ) [ D \\widetilde { \\chi } : D \\widetilde { \\varphi } ] _ t \\ , [ D \\widetilde { \\varphi } : D \\tau _ M ] _ t \\\\ & = [ D \\widetilde { \\psi } : D \\widetilde { \\varphi } ] _ t \\ , \\lambda ^ { \\varphi } ( t ) \\end{align*}"}
+{"id": "5136.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = 1 } ^ N f _ n \\Big \\| _ X \\leq \\sum _ { n = 1 } ^ N K _ X ^ n \\| f _ n \\| _ X . \\end{align*}"}
+{"id": "3217.png", "formula": "\\begin{align*} \\norm { S _ n ' ( y ' - \\sigma _ 1 ' ( y ' ) ) } & = \\norm { \\frac { 1 } { n + 1 } \\sum _ { l = 0 } ^ { n } \\sigma _ l ' ( y ' - \\sigma _ 1 ' ( y ' ) ) } \\\\ & = \\norm { \\frac { w } { n + 1 } ( y ' - \\sigma _ { n + 1 } ' ( y ' ) ) + \\frac { 1 - w } { n + 1 } \\sigma _ n ' ( y ' ) } \\\\ & \\leq \\frac { 1 + w } { n + 1 } \\norm { y ' } \\end{align*}"}
+{"id": "6321.png", "formula": "\\begin{align*} t ^ * = \\begin{cases} \\displaystyle \\frac { 2 \\mathbb S ^ \\circ } { | \\omega | } , & \\omega \\neq 0 , \\\\ \\infty , & \\omega = 0 . \\end{cases} \\end{align*}"}
+{"id": "4311.png", "formula": "\\begin{align*} \\begin{gathered} f ^ { \\mathrm { e v e n } } ( \\cdot , 1 ) = f _ 0 , \\\\ f ^ { \\mathrm { o d d } } ( \\cdot , 1 ) = f _ 0 \\circ y _ { - 2 \\tau _ 1 } ^ { ( i _ 1 ; L _ 1 ) } . \\end{gathered} \\end{align*}"}
+{"id": "3869.png", "formula": "\\begin{align*} \\Sigma _ { \\mathrm { M } } ( \\delta _ 0 ) : = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\boldsymbol { K } ( \\mu _ { ( Y _ 1 , \\dotsc , Y _ L ) | X = x } , \\gamma _ { ( Y _ 1 , \\dotsc , Y _ L ) | X = x } ) \\le \\delta _ 0 x , \\ ; \\mu _ { X } = \\gamma _ X \\right \\} , \\end{align*}"}
+{"id": "8892.png", "formula": "\\begin{align*} \\| v \\| _ { V } = \\min \\{ \\| u \\| _ E \\mid f ( u ) = v \\} . \\end{align*}"}
+{"id": "4729.png", "formula": "\\begin{align*} \\begin{aligned} \\abs { \\xi _ { \\textnormal { ( i i i ) } } ( x _ 1 , x _ 2 ) } & \\leq \\abs { \\xi _ { \\textnormal { ( i ) } } ( x _ 2 , x _ 2 ) } + \\abs { \\xi _ { \\textnormal { ( i i ) } } ( x _ 2 , x _ 2 ) } + \\abs { \\xi _ { = 1 } ( x _ 2 , x _ 2 ) } + \\abs { \\xi _ { \\geq 3 } ( x _ 2 , x _ 2 ) } \\\\ & + C \\sup _ { \\mu , \\nu } \\sup _ { z _ 1 , z _ 2 } \\abs { \\partial _ { x _ 1 } ^ \\mu \\partial _ { x _ 1 } ^ \\nu \\xi _ { \\textnormal { ( i i i ) } } ( z _ 1 , z _ 2 ) } | x _ 1 - x _ 2 | ^ 2 , \\end{aligned} \\end{align*}"}
+{"id": "4345.png", "formula": "\\begin{align*} \\sigma \\left ( b \\right ) : = \\left \\{ g _ { i } ( x ) \\leq b _ { i } , \\ ; i = 1 , \\ldots , m \\right \\} , \\end{align*}"}
+{"id": "1763.png", "formula": "\\begin{align*} \\Gamma _ 0 : = \\Gamma \\setminus \\{ 0 \\} , \\varrho ( z ) : = 1 \\vee \\mathbf { d } \\left ( z , \\Gamma _ 0 ^ c \\right ) ^ { - 1 } \\end{align*}"}
+{"id": "5129.png", "formula": "\\begin{align*} \\partial _ { i i } V = \\bigg ( \\frac { 1 } { 1 + \\lambda _ i ^ 2 } - 2 e _ i ^ 2 \\bigg ) V + e _ i ^ 2 V \\\\ \\partial _ { i j } V = V e _ i e _ j . \\end{align*}"}
+{"id": "516.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ { t } ^ { 2 } u ( t , x ) + a ( t ) \\mathcal { H } _ { V } u ( t , x ) + q ( t ) u ( t , x ) = f ( t , x ) , \\quad ( t , x ) \\in ( 0 , T ] \\times \\mathbb { R } ^ { n } , \\\\ u ( 0 , x ) = u _ { 0 } ( x ) , x \\in \\mathbb { R } ^ { n } , \\\\ \\partial _ { t } u ( 0 , x ) = u _ { 1 } ( x ) , x \\in \\mathbb { R } ^ { n } , \\end{array} \\right . \\end{align*}"}
+{"id": "6101.png", "formula": "\\begin{align*} \\tau _ { n + 1 , \\omega } ^ { I } ( \\chi ) \\coloneqq \\sup \\big \\lbrace \\tau : \\ \\tau ^ { I } _ { n , \\delta _ 1 , \\omega } ( \\chi ) \\leq \\tau \\leq b \\ \\ \\ \\ W _ { \\mathbf { X } ( \\omega ) , \\gamma , \\delta _ 1 } ^ { { \\gamma - \\delta _ 1 } } ( \\tau _ { n , \\omega } ^ { I } ( \\chi ) , \\tau ) \\leq \\chi \\big \\rbrace . \\end{align*}"}
+{"id": "7614.png", "formula": "\\begin{align*} \\mathcal { S } ^ * ( \\alpha ) = \\bigg \\{ f \\in \\mathcal { A } : { \\rm R e } \\left ( \\frac { z f ^ { \\prime } ( z ) } { f ( z ) } \\right ) > \\alpha , \\ ; z \\in \\mathbb { D } \\bigg \\} \\end{align*}"}
+{"id": "3707.png", "formula": "\\begin{align*} W _ M ^ + ( h ) : = \\{ m \\in M \\ : X ^ M _ h ( m ) \\in V _ + ( m ) \\} \\end{align*}"}
+{"id": "5393.png", "formula": "\\begin{align*} e _ k ( t _ 1 , \\ldots , t _ n ) : = \\sum _ { 1 \\le j _ 1 < j _ 2 < \\cdots < j _ k \\le n } t _ { j _ 1 } t _ { j _ 2 } \\cdots t _ { j _ k } \\end{align*}"}
+{"id": "5528.png", "formula": "\\begin{align*} H \\left ( \\xi _ { n } | X , \\eta \\right ) = \\int _ { X } H \\left ( \\xi _ { n } ^ { x } \\right ) d \\eta ( x ) . \\end{align*}"}
+{"id": "3153.png", "formula": "\\begin{align*} \\sum _ { k = l } ^ { l + n } \\delta _ { k , k + 1 } ^ 2 + 4 ( e _ { l + n + 1 } ^ 2 - e _ l ^ 2 ) \\leq 2 ^ { 6 } \\varepsilon ^ { - 1 } \\Lambda _ 6 ^ { 2 } ( \\mu _ { l } ^ 2 - \\mu _ { l + n + 1 } ^ 2 ) + \\frac { \\varepsilon } { 4 } \\sum _ { k = l } ^ { l + n } ( \\mu _ { k + 1 } ^ 2 + \\mu _ { k } ^ 2 ) + \\frac { 1 } { 2 } \\varepsilon C _ { r l } ^ { - 2 } \\sum _ { k = l } ^ { l + n } e _ { k + 1 } ^ 2 . \\end{align*}"}
+{"id": "6357.png", "formula": "\\begin{align*} \\P _ x ( \\tau _ { \\Gamma } > t ) = \\P _ { t ^ { - 1 / \\alpha } x } ( \\tau _ { \\Gamma } > 1 ) , x \\in \\Gamma , \\ ; t > 0 . \\end{align*}"}
+{"id": "4432.png", "formula": "\\begin{align*} S _ { 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } } ( \\Theta ^ { \\omega } ) = \\omega ^ { \\frac { d } { 2 } - 2 } S _ { \\omega , \\mathbf { c } } ( \\Theta ) . \\end{align*}"}
+{"id": "5017.png", "formula": "\\begin{align*} \\| x \\| - \\| V \\theta _ g x \\| = \\| \\theta _ f x \\| - \\| V \\theta _ g x \\| \\leq \\| \\theta _ f x - V \\theta _ g x \\| \\leq ( \\varepsilon + \\delta ) \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "410.png", "formula": "\\begin{align*} W ^ - = \\begin{bmatrix} u _ n + p / u _ n \\\\ u _ { \\tau } \\end{bmatrix} , \\Lambda ^ - = \\begin{bmatrix} u _ n & 0 \\\\ 0 & u _ n \\end{bmatrix} , W ^ + = p , \\Lambda ^ + = - 1 / u _ n \\end{align*}"}
+{"id": "8076.png", "formula": "\\begin{align*} F ^ { \\prime } ( u ) = \\mu G ^ { \\prime } ( u ) \\mathcal { G } , \\end{align*}"}
+{"id": "2583.png", "formula": "\\begin{align*} \\mathcal { B } ( X ) = \\mathcal { B } ( X \\setminus \\{ x \\} ) \\cup \\left | \\operatorname { s t } ( \\mathcal { N } ( x ) ) \\right | . \\\\ \\end{align*}"}
+{"id": "4609.png", "formula": "\\begin{align*} \\mathbb { P } ( E _ i ) = \\mathbb { P } ( E _ { i , 1 } ) \\mathbb { P } ( E _ { i , 2 } ) . \\end{align*}"}
+{"id": "3312.png", "formula": "\\begin{align*} w \\rho _ w ( \\theta , w ) = e ^ { c ( \\theta , w ) + i d ( \\theta , w ) } \\end{align*}"}
+{"id": "1560.png", "formula": "\\begin{align*} h ( v ) = \\chi _ { [ 0 , 1 ] } ( v ) \\frac { g ( v ) } { \\sqrt { 1 + v } } . \\end{align*}"}
+{"id": "2175.png", "formula": "\\begin{align*} \\frac { 2 \\rho \\beta b ^ { 2 } } { d } = \\frac { 2 \\rho \\beta } { 3 / 2 + \\left ( 1 + \\beta \\right ) \\left ( T / 2 - \\varepsilon \\right ) ^ { 2 } / \\left ( \\varepsilon \\left ( T - \\varepsilon \\right ) \\right ) } . \\end{align*}"}
+{"id": "3676.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ \\gamma u + q u = 0 & \\Omega , \\\\ u = f & \\Omega ^ c , \\end{cases} \\end{align*}"}
+{"id": "690.png", "formula": "\\begin{align*} \\mathrm { o s c } ( v , T ^ N ( I _ \\alpha ^ { ( k ) } ) ) = \\mathrm { o s c } ( v , I _ \\alpha ^ { ( k ) } ) + O ( e ^ { ( - \\lambda _ 1 ( 1 - a ) + \\tau ) k } ) ( c _ 0 ( D \\varphi ) + p _ a ( D \\varphi ) ) . \\end{align*}"}
+{"id": "8058.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty ( f _ X ( x _ 0 + k \\lambda ( J ) ) & - f _ A ( x _ 0 + ( k - 1 ) \\lambda ( J ) ) ) \\geq \\sum _ { k = 0 } ^ { k _ 0 } ( f _ X ( x _ 0 + k \\lambda ( J ) ) - f _ A ( x _ 0 + ( k - 1 ) \\lambda ( J ) ) ) \\\\ & = f _ X ( x _ 0 + k _ 0 \\lambda ( J ) ) + \\sum _ { k = 0 } ^ { k _ 0 - 1 } ( ( f _ X ( x _ 0 + k \\lambda ( J ) ) - f _ A ( x _ 0 + k \\lambda ( J ) ) \\\\ & \\geq f _ X ( x _ 0 + k _ 0 \\lambda ( J ) ) . \\end{align*}"}
+{"id": "2327.png", "formula": "\\begin{align*} f _ n ( \\pmb { t } _ n , \\pmb { x } _ n , t , x ) = & f _ n ( t _ 1 , \\dots , t _ n , x _ 1 , \\dots , x _ n , t , x ) \\\\ \\coloneqq & G _ { t - t _ n } ( x - x _ n ) \\times \\dots \\times G _ { t _ 2 - t _ 1 } ( x _ 2 - x _ 1 ) \\mathbf { 1 } _ { T _ n ( t ) } ( \\pmb { t } _ n ) \\end{align*}"}
+{"id": "4601.png", "formula": "\\begin{align*} \\log \\left ( 1 + \\frac { P _ k } { 1 + ( M - 1 ) P _ { k + 1 } } \\right ) = R , 1 \\leq k \\leq K - 1 , \\end{align*}"}
+{"id": "8613.png", "formula": "\\begin{align*} P ( W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 , W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) = 1 ) - \\nu \\Delta p ^ k _ { j , + } ( t - \\Delta \\ell _ 2 ) E [ Z _ 0 ( \\Delta \\ell _ 2 ) ; W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) = 1 ] . \\end{align*}"}
+{"id": "5159.png", "formula": "\\begin{align*} X ( u ) ^ \\dag = X ( u ) ' ( w ) = X ' ( u ^ { - 1 } w ) = X ' ( v ) , \\end{align*}"}
+{"id": "4676.png", "formula": "\\begin{align*} \\left ( \\mathcal { A } _ \\varepsilon - z I \\right ) ^ { - 1 } = \\mathcal { G } _ \\varepsilon ^ { - 1 } \\left ( \\int _ { Y ' } ^ \\oplus \\left ( \\frac { 1 } { \\varepsilon ^ 2 } \\mathcal { A } _ { \\chi , \\varepsilon } - z I \\right ) ^ { - 1 } d \\chi \\right ) \\mathcal { G } _ \\varepsilon , z \\in \\rho ( \\mathcal { A } _ \\varepsilon ) , \\end{align*}"}
+{"id": "4681.png", "formula": "\\begin{align*} M ( z ) = \\Lambda + z \\Pi ^ * \\left ( I - z \\mathcal { A } _ 0 ^ { - 1 } \\right ) ^ { - 1 } \\Pi , z \\in \\rho ( \\mathcal { A } _ 0 ) . \\end{align*}"}
+{"id": "7908.png", "formula": "\\begin{align*} 2 \\lambda + 1 = \\left \\{ 2 \\lambda ^ { ( i ) } + 1 : i \\in \\mathcal { I } ( \\lambda ) \\cup \\left \\{ t + 1 \\right \\} \\right \\} . \\end{align*}"}
+{"id": "1089.png", "formula": "\\begin{align*} ( b , x ) \\cdot ( b ' , x ' ) = ( b b ' , x x ' + b \\ast x ' + x \\ast b ' ) \\end{align*}"}
+{"id": "481.png", "formula": "\\begin{align*} f [ k ] = f ( \\underbrace { 1 , \\dots , 1 } _ { k } ) . \\end{align*}"}
+{"id": "9268.png", "formula": "\\begin{align*} B _ 2 f ( x ) & \\simeq x ^ { \\eta } \\bigg ( \\sup _ { x < b } \\frac { 1 } { b } \\int _ { 0 } ^ { x } z ^ { - \\eta } \\abs { f ( z ) } \\ , d z + \\sup _ { x < b } \\frac { 1 } { b } \\int _ { x } ^ { b } z ^ { - \\eta } \\abs { f ( z ) } \\ , d z \\bigg ) \\\\ & \\lesssim x ^ { \\eta - 1 } \\int _ { 0 } ^ { x } z ^ { - \\eta } \\abs { f ( z ) } \\ , d z + x ^ { \\eta } \\int _ { x } ^ { \\infty } z ^ { - \\eta - 1 } \\abs { f ( z ) } \\ , d z = H _ { 1 - \\eta } | f | ( x ) + H ^ { \\infty } _ { \\eta } | f | ( x ) . \\end{align*}"}
+{"id": "363.png", "formula": "\\begin{align*} ( \\underline { D } ^ { 2 } , \\underline { S } ^ { 1 } ) ^ { \\{ 1 \\} , c } = \\begin{cases} D ^ 2 & k = 1 ; \\\\ P ^ 2 ( k ) & k > 1 . \\end{cases} \\end{align*}"}
+{"id": "7393.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ 1 \\big | X _ N ( L , \\alpha ) \\big | ^ 2 \\ , d \\alpha & \\ll \\frac { L } { N ^ 3 } \\sum _ { \\substack { 1 \\leq r \\leq N ^ 2 \\\\ 1 \\leq s \\leq N ^ 2 } } W _ N ( w _ r , \\mathcal { A } ) W _ N ( w _ s , \\mathcal { A } ) \\frac { \\gcd ( w _ r , w _ s ) } { \\sqrt { | w _ r w _ s | } } \\\\ & \\ll \\frac { L } { N ^ 3 } \\sum _ { 1 \\leq r \\leq N ^ 2 } W _ N ( w _ r , \\mathcal { A } ) ^ 2 \\exp \\left ( \\frac { 1 0 \\log r } { \\log \\log ( r + 1 ) } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "3899.png", "formula": "\\begin{align*} \\varphi ( x ) = \\varphi ( t x _ 0 + ( 1 - t ) x _ 1 ) \\geq t \\varphi ( x _ 0 ) + ( 1 - t ) \\varphi ( x _ 1 ) = \\infty . \\end{align*}"}
+{"id": "4216.png", "formula": "\\begin{align*} T _ n ( \\phi ) = T _ n ( \\phi _ 0 ) T _ n ( \\phi _ 1 ) \\cdots T _ n ( \\phi _ R ) + P _ n K _ 1 P _ n + W _ n K _ 2 W _ n + C _ n \\end{align*}"}
+{"id": "1620.png", "formula": "\\begin{align*} \\Phi ( X , Y ) = g ( X , { f } Y ) , X , Y \\in \\mathfrak { X } _ M . \\end{align*}"}
+{"id": "6731.png", "formula": "\\begin{align*} \\mathcal { P } ( X _ \\eta ) = \\mathcal { P } ( X _ \\varphi ) = \\{ ( M _ H ) _ \\ast ( m _ H \\vee \\kappa ) : \\kappa \\in \\mathcal { P } ( \\{ 0 , 1 \\} ^ \\Z ) \\} . \\end{align*}"}
+{"id": "5478.png", "formula": "\\begin{align*} \\kappa = \\frac { M _ 1 M _ 2 - M _ 1 ^ 2 } { M _ 1 ^ 2 - M _ 2 } , \\beta = \\frac { ( 1 - M _ 1 ) ( M _ 2 - M _ 1 ) } { M _ 1 ^ 2 - M _ 2 } , \\end{align*}"}
+{"id": "5530.png", "formula": "\\begin{align*} h ( B , \\nu _ { B } ) = h \\left ( Y , \\nu \\right ) + \\int _ { Y } { \\rm I } ( \\mathbb { P } _ { \\mu , 1 } ^ { y } , \\mathcal { T } _ { y } ) d \\nu ( y ) = h \\left ( Y , \\nu \\right ) + \\int _ { Y } \\inf _ { n \\in \\mathbb { N } } { \\rm I } ( \\mathbb { P } _ { \\mu , 1 } ^ { y } , \\mathbb { P } _ { \\mu , n } ^ { y } ) d \\nu ( y ) . \\end{align*}"}
+{"id": "1643.png", "formula": "\\begin{align*} g ( ( \\pounds _ { \\xi _ i } { f } ) X , Y ) & = N ^ { \\ , ( 5 ) } ( \\xi _ i , X , Y ) + g ( { f } \\nabla _ { X } \\ , \\xi _ i - \\nabla _ { { f } X } \\ , \\xi _ i , \\ Y ) . \\end{align*}"}
+{"id": "1053.png", "formula": "\\begin{align*} \\Delta ( f ) ( t ) \\coloneqq \\sum _ { i = 0 } ^ { + \\infty } a ( i , \\ldots , i ) t ^ i \\in k [ [ t ] ] \\ , . \\end{align*}"}
+{"id": "9028.png", "formula": "\\begin{align*} \\frac { \\bar C ( p _ 1 , \\bar p , \\rho ) } { ( T - \\varphi _ { p _ 2 , p _ 1 } ( t ) ) ^ 2 } - \\frac { \\bar C ( p _ 2 , p _ 1 , \\rho ) } { ( \\varphi _ { p _ 2 , p _ 1 } ( t ) - t ) ^ 2 } = 0 , \\end{align*}"}
+{"id": "3398.png", "formula": "\\begin{align*} u ( r ) = \\max \\Big \\{ 0 , ( n - 1 ) \\inf _ { r \\geq r _ 0 } \\log \\frac { f ( r ) } { f ( r _ 0 ) } \\Big \\} . \\end{align*}"}
+{"id": "2795.png", "formula": "\\begin{align*} \\mathcal { T } ^ \\ast \\mathcal { T } = \\mu _ j \\psi _ j , j \\ge 1 . \\end{align*}"}
+{"id": "526.png", "formula": "\\begin{align*} Q u ( k ) : = \\frac { 1 } { V ( k ) + 1 } u ( k ) , u \\in \\ell ^ { 2 } ( \\hbar \\mathbb { Z } ^ { n } ) . \\end{align*}"}
+{"id": "2508.png", "formula": "\\begin{align*} \\eta _ A = \\| | A | \\| , \\ , \\ , \\eta _ P = \\| | P | \\| , \\ , \\ , \\eta _ M = \\| M \\| , \\ , \\ , \\eta _ N = \\| N \\| _ A , \\end{align*}"}
+{"id": "4946.png", "formula": "\\begin{align*} \\left [ \\Sigma \\phantom { ^ { } } \\right ] _ { i , j } ^ { } = \\begin{cases} 1 & i \\le j ; \\\\ 0 & ; \\end{cases} \\end{align*}"}
+{"id": "6292.png", "formula": "\\begin{align*} \\bar u ( t ) = \\hat \\lambda _ t ^ * , \\qquad t \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "7238.png", "formula": "\\begin{align*} \\overline { u } _ h : = \\begin{cases} \\zeta ( v + m _ h ) + ( 1 - \\zeta ) z _ h & \\mbox { o n } B _ { \\sigma ' } \\\\ \\varphi z _ h + ( 1 - \\varphi ) u _ h & \\mbox { o n } B _ r \\setminus B _ { \\sigma ' } . \\end{cases} \\end{align*}"}
+{"id": "2708.png", "formula": "\\begin{align*} T _ { 5 1 } = & s \\lambda ^ 3 \\iint _ Q \\xi u A \\nabla u \\cdot \\nabla \\eta \\left ( A \\nabla \\eta \\cdot \\nabla \\eta \\right ) d x d t \\\\ \\geq & - C s ^ 2 \\lambda ^ 4 \\iint _ Q \\xi \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d t - C \\lambda ^ 2 \\iint _ Q \\xi | A \\nabla u \\cdot \\nabla \\eta | ^ 2 d x d t , \\end{align*}"}
+{"id": "7081.png", "formula": "\\begin{align*} g = g ( a _ i ) + g ' ( a _ i ) ( x - a _ i ) + \\ldots + ( x - a _ i ) ^ d . \\end{align*}"}
+{"id": "5811.png", "formula": "\\begin{align*} m ( r , k Y ) = m \\left ( r , - \\frac { R _ { k - 2 } } { M Y ^ { k - 2 } } + \\frac { h } { M Y ^ { k - 2 } } \\right ) \\\\ m ( r , Y ) \\leq m \\left ( r , - \\frac { R _ { k - 2 } } { M Y ^ { k - 2 } } \\right ) + m \\left ( r , \\frac { h } { M Y ^ { k - 2 } } \\right ) + \\log 2 . \\\\ & \\end{align*}"}
+{"id": "6667.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } u _ \\eta = \\eta . \\end{align*}"}
+{"id": "8941.png", "formula": "\\begin{align*} - \\Delta ( \\zeta \\overline v ) = { \\rm c u r l } ( \\zeta \\omega _ \\theta e _ \\theta ) - \\nabla ( \\overline v \\cdot \\nabla \\zeta ) + { \\rm c u r l } ( \\nabla \\zeta \\times \\overline v ) . \\end{align*}"}
+{"id": "8611.png", "formula": "\\begin{align*} & E \\left [ \\left ( S ^ k _ { j , + } ( t ) - \\bar { S } ^ k _ { j , + , \\Delta } ( t ) \\right ) ^ 2 \\right ] \\\\ & = \\nu \\Delta \\sum _ { \\ell = 0 } ^ { \\lfloor t / \\Delta \\rfloor } p ^ k _ { j , + } ( t - \\ell \\Delta ) E [ Z _ 0 ( \\ell \\Delta ) ] \\\\ & + 2 \\sum _ { \\ell _ 1 < \\ell _ 2 } E \\left [ W ^ k _ { \\ell _ 1 \\Delta , t } ( j ) \\left ( W ^ k _ { \\ell _ 2 \\Delta , t } ( j ) - \\nu \\Delta Z _ 0 ( \\Delta \\ell _ 2 ) p ^ k _ { j , + } ( t - \\Delta \\ell _ 2 ) \\right ) \\right ] . \\end{align*}"}
+{"id": "4872.png", "formula": "\\begin{align*} \\limsup _ { r \\to 0 ^ + } \\Phi ( r ) = + \\infty , \\qquad { \\mbox { w h e r e } } \\ ; \\Phi ( r ) : = \\frac { \\displaystyle \\inf _ { B _ { r / 2 } ( x _ r ) } | u | } { r ^ { 2 s } } . \\end{align*}"}
+{"id": "3416.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta v _ i = f ( v _ i ) - \\mu _ i v _ i + \\lambda v _ i \\ \\ \\ \\ \\mathbb R ^ N , \\\\ \\lim _ { | x | \\to \\infty } v _ i ( x ) = 0 , i = 1 , 2 , \\cdots , \\ell , \\\\ \\sum _ { i = 1 } ^ \\ell | v _ i | _ 2 ^ 2 = \\alpha . \\end{cases} \\end{align*}"}
+{"id": "6376.png", "formula": "\\begin{align*} \\tau _ j ( f ) _ i ( 0 ) = \\begin{cases} \\sigma ( \\tau _ j ) , & \\tau _ i = \\tau _ j , \\\\ 0 , & , \\end{cases} \\end{align*}"}
+{"id": "4878.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s u = \\lambda u & , \\\\ u = 0 & , \\end{cases} \\end{align*}"}
+{"id": "8633.png", "formula": "\\begin{align*} \\bar { M } _ + ( t ) : = \\nu \\int _ 0 ^ t Z _ 0 ( s ) d s , \\end{align*}"}
+{"id": "7257.png", "formula": "\\begin{align*} \\theta _ { Z , p , q } ( k ) \\leq 2 \\sup _ { ( a _ 1 , \\dots , a _ p ) \\in \\Gamma _ { p , q } } \\prod _ { i = 1 } ^ p 2 ^ { a _ i } \\theta _ { X , p , q } ( k ) \\ , , \\end{align*}"}
+{"id": "3708.png", "formula": "\\begin{align*} \\exp ( 2 z ) \\exp ( 2 x _ 1 ) \\exp ( 2 z ) = \\exp ( 2 x _ 1 ) , \\end{align*}"}
+{"id": "9176.png", "formula": "\\begin{align*} v _ { 1 } ^ { 1 } & = y _ { 1 , [ 2 ] } ^ { 1 , d } - a _ { 1 } ^ { 1 , 1 } ( \\varphi _ { 1 , [ 1 ] } ^ { 1 } - y _ { 1 , [ 1 ] } ^ { 1 , d } ) - a _ { 1 } ^ { 1 , 0 } ( \\varphi _ { 1 } ^ { 1 } - y _ { 1 } ^ { 1 , d } ) \\\\ v _ { 1 , [ 1 ] } ^ { 1 } & = y _ { 1 , [ 3 ] } ^ { 1 , d } - a _ { 1 } ^ { 1 , 1 } ( v _ { 1 } ^ { 1 } - y _ { 1 , [ 2 ] } ^ { 1 , d } ) - a _ { 1 } ^ { 1 , 0 } ( \\varphi _ { 1 , [ 1 ] } ^ { 1 } - y _ { 1 , [ 1 ] } ^ { 1 , d } ) \\\\ v _ { 2 } ^ { 1 } & = y _ { 2 , [ 2 ] } ^ { 1 , d } - a _ { 2 } ^ { 1 , 1 } ( \\varphi _ { 2 , [ 1 ] } ^ { 1 } - y _ { 2 , [ 1 ] } ^ { 1 , d } ) - a _ { 2 } ^ { 1 , 0 } ( \\varphi _ { 2 } ^ { 1 } - y _ { 2 } ^ { 1 , d } ) \\end{align*}"}
+{"id": "5492.png", "formula": "\\begin{align*} \\mathfrak { a } _ { I } = \\bigcap _ { \\alpha \\in I } \\ker \\alpha . \\end{align*}"}
+{"id": "1922.png", "formula": "\\begin{align*} \\begin{aligned} \\theta ( \\bar { u } ) + I _ K ( \\bar { \\zeta } ) = \\lim \\limits _ { k \\rightarrow + \\infty } \\theta ( u ^ k ) + I _ K ( \\zeta ^ k ) \\leq \\theta ( \\hat { u } ) + I _ K ( \\hat { \\zeta } ) + \\beta \\sum \\limits _ { i = n } ^ { + \\infty } \\| r ^ { i } \\| ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "7943.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) \\Pi _ { e _ { ( \\xi , 0 ) } + 3 e _ { ( \\xi , e _ 0 ) } } & = \\Pi _ { e _ { ( \\xi , 0 ) } + 2 e _ { ( \\xi , e _ 0 ) } } \\xi \\\\ & + c _ { e _ { ( \\xi , 0 ) } + 3 e _ { ( \\xi , e _ 0 ) } } \\\\ & + c _ { e _ { ( \\xi , 0 ) } + e _ { ( \\xi , e _ 0 ) } } \\Pi _ { e _ { ( \\xi , 0 ) } + e _ { ( \\xi , e _ 0 ) } } , \\end{align*}"}
+{"id": "5970.png", "formula": "\\begin{align*} e ^ { \\begin{bmatrix} 0 & \\tfrac { t } { 2 } \\\\ \\tfrac { t } { 2 } & 0 \\end{bmatrix} } & = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} + \\begin{bmatrix} 0 & \\tfrac { t } { 2 } \\\\ \\tfrac { t } { 2 } & 0 \\end{bmatrix} + o ( t ^ 2 ) , \\\\ e ^ { \\begin{bmatrix} t / 2 & 0 \\\\ 0 & - t / 2 \\end{bmatrix} } & = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} + \\begin{bmatrix} t / 2 & 0 \\\\ 0 & - t / 2 \\end{bmatrix} + o ( t ^ 2 ) ; \\end{align*}"}
+{"id": "146.png", "formula": "\\begin{align*} 1 = \\chi + \\sum \\limits _ j \\chi _ j \\end{align*}"}
+{"id": "4954.png", "formula": "\\begin{align*} F ( k , t , n ) = \\frac { ( n ) _ { k + 1 } ^ { } } { n ^ t } \\frac { 1 } { k ! } \\Delta ^ { k } \\left [ \\frac { x ^ t } { n - x } \\right ] _ { x = 0 } ^ { } . \\end{align*}"}
+{"id": "3374.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { P _ { \\lambda _ m } } { P _ m } = \\varphi ( \\lambda ) \\ , \\ , \\ , \\ , \\ , \\end{align*}"}
+{"id": "1005.png", "formula": "\\begin{align*} \\mathcal { D } _ x \\ : ( 1 + x ) _ r = ( 1 + x ) _ r H _ r ( x ) . \\end{align*}"}
+{"id": "6306.png", "formula": "\\begin{align*} \\delta ( E ( q , \\lambda ) ) = \\sqrt { 2 H ( \\lambda ) } , \\qquad \\forall \\ , ( q , \\lambda ) \\in D _ \\epsilon . \\end{align*}"}
+{"id": "8819.png", "formula": "\\begin{align*} \\Phi _ q ' ( u ) = q \\norm [ \\big ] { u } _ { L ^ q ( \\mu ; H ) } ^ { q - 2 } J ( u ) , \\norm [ \\big ] { \\Phi ' ( u ) } _ { L ^ { q ' } ( \\mu ; H ) } = q \\norm [ \\big ] { u } _ { L ^ q ( \\mu ; H ) } ^ { q - 1 } , \\end{align*}"}
+{"id": "4015.png", "formula": "\\begin{align*} g \\lbrace \\alpha , \\beta \\rbrace = \\lbrace g ( \\alpha ) , g ( \\beta ) \\rbrace , \\end{align*}"}
+{"id": "5149.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } f _ k = \\liminf _ { k \\to \\infty } g _ k = h \\end{align*}"}
+{"id": "1316.png", "formula": "\\begin{align*} \\frac { d \\widetilde \\beta _ i ^ { ( u ) } } { d \\beta _ i } = \\frac { 1 } { \\Big ( 1 - \\frac { \\beta _ i } { \\beta _ i ^ { ( u ) } } \\Big ) ^ 2 } = \\frac { 1 } { \\phi _ i ( u ) ^ 2 } . \\end{align*}"}
+{"id": "4687.png", "formula": "\\begin{align*} \\mathcal { H } = P _ { \\rm s o f t } \\mathcal { H } \\oplus P _ { \\rm s t i f f } \\mathcal { H } = \\mathcal { H } ^ { \\rm s o f t } \\oplus \\mathcal { H } ^ { \\rm s t i f f } . \\end{align*}"}
+{"id": "3960.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { I } _ { \\mathrm { D } } ( \\eta _ 0 , \\delta ) - \\mathcal { I } _ { \\mathrm { D } } ( \\eta , 0 ) & = \\mathcal { I } _ { \\mathrm { D } } ( \\eta _ 0 , \\delta ) - \\mathcal { I } _ { \\mathrm { D } } ( \\eta , \\delta ) + \\mathcal { I } _ { \\mathrm { D } } ( \\eta , \\delta ) - \\mathcal { I } _ { \\mathrm { D } } ( \\eta , 0 ) \\\\ & \\leq \\Psi ( \\eta _ 0 - \\eta , 0 ) + \\Psi ( 0 , \\delta ) = \\Psi ( | \\eta _ 0 - \\eta | , 0 ) + \\Psi ( 0 , \\delta ) . \\end{aligned} \\end{align*}"}
+{"id": "3202.png", "formula": "\\begin{align*} S _ n ( x ) = \\frac { 1 } { n + 1 } \\sum _ { 0 } ^ { n } \\sigma _ k ( x ) , ~ n \\in \\N . \\end{align*}"}
+{"id": "3929.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta _ 1 , 0 ) = \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\inf _ { \\lambda _ 1 \\in \\mathbb { R } _ { + } } \\left [ \\lambda _ 1 \\delta _ 1 + \\int g _ { \\lambda , 1 } \\ , d \\pi \\right ] = \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ { + } } \\left [ \\langle \\lambda , \\left ( \\delta _ 1 , 0 \\right ) \\rangle + \\int _ { \\mathcal { V } } g _ { \\lambda } \\ , d \\pi \\right ] . \\end{align*}"}
+{"id": "3886.png", "formula": "\\begin{align*} \\left ( \\mathord { \\operatorname { p r o j } } _ { K _ j \\cap K _ { j + 1 } } \\circ { \\operatorname { p r o j } _ { K _ j } } ^ { - 1 } \\right ) \\# \\mu _ j = \\mathcal { N } ( 0 , 1 ) , \\forall j \\in [ 3 ] . \\end{align*}"}
+{"id": "1728.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\nu _ t ( x ) = - a ( \\nu _ t , \\mu _ t , x ) \\nu _ t ( x ) , \\\\ \\partial _ t \\mu _ t ( y ) = b ( \\nu _ t , \\mu _ t , y ) \\mu _ t ( y ) , \\end{cases} \\end{align*}"}
+{"id": "5201.png", "formula": "\\begin{align*} ( p , q ) & = p ^ < \\cap q ^ > = \\{ r \\in \\mathbb { P } : p < r < q \\} , \\\\ [ p , q ] & = p ^ \\leq \\cap q ^ \\geq = \\{ r \\in \\mathbb { P } : p \\leq r \\leq q \\} . \\end{align*}"}
+{"id": "1774.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } D _ \\Theta X _ \\cdot ^ { ( n ) } = D _ \\Theta X _ \\cdot \\textrm { ~ ~ w e a k l y ~ ~ i n ~ ~ } L ^ p ( \\Omega \\times [ 0 , T ] ) . \\end{align*}"}
+{"id": "6510.png", "formula": "\\begin{align*} E _ { 2 n + 1 } ( f ) & = \\frac 1 2 \\det [ u _ { j - i } - u _ { j + i + 2 } ] _ { 0 } ^ { n - 1 } \\times \\det [ u _ { j - i } + u _ { j + i } ] _ 0 ^ n , \\\\ E _ { 2 n } ( f ) & = \\ , \\ , \\ , \\ , \\ , \\det [ u _ { j - i } - u _ { j + i + 1 } ] _ { 0 } ^ { n - 1 } \\times \\det [ u _ { j - i } + u _ { j + i } ] _ 0 ^ { n - 1 } . \\end{align*}"}
+{"id": "2230.png", "formula": "\\begin{align*} h \\log y \\int _ { 2 } ^ { u / h } y ^ { h t - u } t \\rho ( t ) \\ , d t & = u h ^ { - 1 } \\rho ( u h ^ { - 1 } ) - 2 \\rho ( 2 ) y ^ { 2 h - u } - \\int _ { 2 } ^ { u / h } y ^ { h t - u } ( t \\rho ( t ) ) ' \\ , d t \\\\ & \\le u h ^ { - 1 } \\rho ( u h ^ { - 1 } ) - 2 \\rho ( 2 ) y ^ { 2 h - u } + \\int _ { 2 } ^ { u / h } y ^ { h t - u } \\rho ( t - 1 ) \\ , d t . \\end{align*}"}
+{"id": "4647.png", "formula": "\\begin{align*} \\tilde { a } _ 1 = \\frac { \\frac { p + \\gamma - 1 } { 2 } - \\frac { p + \\gamma - 1 } { 4 q } } { \\frac { p + \\gamma - 1 } { 2 } + \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "6060.png", "formula": "\\begin{align*} x ' _ 1 - x ' _ 3 = ( n ' + m ' ) \\delta + 2 \\pi z ' _ 1 \\quad \\textrm { a n d } x ' _ 2 - x ' _ 3 = - m ' \\delta + 2 \\pi z ' _ 2 \\textrm { f o r } \\delta \\in \\mathbb { R } , \\end{align*}"}
+{"id": "2268.png", "formula": "\\begin{align*} T ( f ) ( z ) = - \\frac { 1 } { \\pi } \\iint _ D \\frac { f ( \\zeta ) } { \\zeta - z } \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "6556.png", "formula": "\\begin{align*} T _ { m } ( f _ 1 , . . . , f _ m ) ( x ) & = \\int _ { \\mathbb R ^ { n m } } \\frac { f _ 1 ( y _ 1 ) . . . f _ m ( y _ m ) } { ( 1 + | y _ 1 | ^ n + . . . + | y _ m | ^ n ) ^ m } d y _ 1 . . . d y _ m \\\\ & = \\int _ { \\mathbb R ^ n } . . . \\int _ { \\mathbb R ^ n } \\frac { f _ 1 ( | x | y _ 1 ) . . . f _ m ( y _ m ) } { ( 1 + | y _ 1 | ^ n + . . . + | y _ m | ^ n ) ^ m } d y _ 1 . . . d y _ m . \\end{align*}"}
+{"id": "5569.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c } 1 & 2 \\\\ 0 & 1 \\end{array} \\right ) , \\ B = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 2 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "1186.png", "formula": "\\begin{align*} \\delta ^ M _ { n m } = 1 0 0 \\frac { M _ { n m } + M _ { m n } } { 2 M ^ 0 _ { n m } } \\end{align*}"}
+{"id": "8777.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ n c \\beta _ i ( e _ { i \\bullet } - e _ { \\bullet i } ) + \\nabla \\left ( \\sum _ { i \\neq j } \\alpha _ { i j } v _ { i j } - \\langle \\lambda , v \\rangle + \\sum _ { i \\in I } \\gamma _ i ( \\sum _ { j \\neq i } v _ { i j } - x _ { 0 i } ) \\right ) . \\end{align*}"}
+{"id": "6444.png", "formula": "\\begin{align*} R _ k ( t , u , v , w ) : = \\int _ { k = \\ell - m + j } e ^ { i t \\Delta \\omega _ { k \\ell m j } } u _ \\ell \\bar v _ m w _ j . \\end{align*}"}
+{"id": "6972.png", "formula": "\\begin{align*} \\alpha = \\alpha \\left ( \\left \\{ \\left . \\min \\limits _ { \\ell \\in I _ 0 ( g , i ) } \\alpha _ \\ell \\ \\right | \\ Q _ i \\in I _ { n _ 0 } \\right \\} \\right ) . \\end{align*}"}
+{"id": "1343.png", "formula": "\\begin{align*} \\partial _ { t } F + \\partial _ { x } ( A ( F ) ) = \\mathbf { S } [ F ] ( t , x ) \\end{align*}"}
+{"id": "2208.png", "formula": "\\begin{align*} \\sum _ { y < p \\le \\sqrt { x } } \\frac { x } { p \\log ( x / p ) } & \\ge \\frac { x } { \\log y } \\left ( \\frac { \\log ( u - 1 ) } { u } - \\frac { 2 c _ 1 } { u \\log ^ 2 y } \\right ) \\\\ & = \\frac { x } { \\log y } \\left ( \\omega ( u ) - \\frac { 2 c _ 1 } { u \\log ^ 2 y } \\right ) - \\frac { x } { \\log x } . \\end{align*}"}
+{"id": "7301.png", "formula": "\\begin{align*} x ^ 2 y = z ^ 2 + 1 \\end{align*}"}
+{"id": "5801.png", "formula": "\\begin{align*} \\frac { y ' } { y } = c \\omega A ( z ) ^ { 1 / k } - \\frac { k - 1 } { 2 k } \\frac { A ' ( z ) } { A ( z ) } + O ( r ^ { - 2 } ) , w ^ k = 1 . \\end{align*}"}
+{"id": "1765.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\infty - } : = \\bigcap _ { p \\ge 1 } \\mathbb { H } _ p , \\mathbb { V } _ { \\infty - } : = \\bigcap _ { p \\ge 1 } \\mathbb { V } _ p . \\end{align*}"}
+{"id": "7223.png", "formula": "\\begin{align*} \\Theta ^ * ( J _ u , x ) : = \\limsup _ { r \\to 0 } \\frac { \\mathcal { H } ^ 1 \\left ( J _ u \\cap B _ r ( x ) \\right ) } { 2 \\pi r } \\geq \\theta _ \\delta > 0 . \\end{align*}"}
+{"id": "4812.png", "formula": "\\begin{align*} V ( x _ 1 , x _ 2 , x _ 3 ) = \\frac { 1 } { 2 } x _ 3 ^ 2 + 1 - x _ 1 + \\alpha ( 1 - x _ 1 ^ 3 ) , \\end{align*}"}
+{"id": "8101.png", "formula": "\\begin{align*} \\partial f ^ { \\lambda } : = \\frac { I - \\left ( I + \\lambda \\partial f \\right ) ^ { - 1 } } { \\lambda } , \\lambda \\in ( 0 , 1 ) , \\end{align*}"}
+{"id": "5161.png", "formula": "\\begin{align*} \\xi _ { \\theta } ( y ) = \\xi _ { \\theta _ 0 } ( y ) + ( \\theta - \\theta _ 0 ) ^ \\prime \\dot { \\xi } _ { \\theta _ 0 } ( y ) + r _ { \\theta } ( y ) , \\end{align*}"}
+{"id": "2996.png", "formula": "\\begin{align*} V ( t _ 1 , \\dots , t _ n ) = \\frac { 1 } { n + k } \\sum _ { i = 1 } ^ n t _ i ^ { n + k } = V _ { n + k } ( t _ 1 , \\dots , t _ n ) \\ . \\end{align*}"}
+{"id": "4048.png", "formula": "\\begin{align*} G _ k ( x ) \\ll \\int _ { \\mathrm { R e } ( t ) = \\sqrt { x } } \\frac { ( k + t ) ^ { 2 k + 2 t - 1 } } { ( 2 \\pi ) ^ { 2 t } x ^ t } \\ , e ^ { - 2 ( k + t ) } \\ , \\frac { d t } { t } \\ll e ^ { - c \\sqrt { x } } \\end{align*}"}
+{"id": "3715.png", "formula": "\\begin{align*} \\tau _ h ^ G : = ( \\kappa _ h ^ G ) ^ 2 \\end{align*}"}
+{"id": "6677.png", "formula": "\\begin{align*} & - \\int _ \\Omega \\langle \\nabla u , \\nabla \\varphi \\rangle \\ , d x - \\frac { C _ { N , s } } { 2 } \\iint _ { \\R ^ { 2 N } } \\frac { ( u ( x ) - u ( y ) ) ( \\hat { \\varphi } ( x ) - \\hat { \\varphi } ( y ) ) } { | x - y | ^ { N + 2 s } } \\ , d x \\ , d y - \\int _ \\Omega V ( x ) u \\varphi \\ , d x \\\\ & \\geq \\ , [ ] \\ , \\int _ \\Omega f ( x ) \\varphi \\ , d x . \\end{align*}"}
+{"id": "2460.png", "formula": "\\begin{align*} \\pi _ 1 ( f _ 1 ) \\cdots \\pi _ k ( f _ k ) \\psi = \\theta _ W ( f _ k \\otimes \\ldots \\otimes f _ 1 \\otimes \\rho ) * \\psi \\end{align*}"}
+{"id": "1259.png", "formula": "\\begin{align*} \\mu _ { p } ( ( x _ 1 , x _ 2 ) , ( y _ 1 , y _ 2 ) ) > 0 \\wedge \\mu _ { p } ( ( y _ 1 , y _ 2 ) , ( x _ 1 , x _ 2 ) ) > 0 \\Rightarrow ( x _ 1 , x _ 2 ) = ( y _ 1 , y _ 2 ) \\end{align*}"}
+{"id": "1274.png", "formula": "\\begin{align*} \\eta ( B ^ { \\prime } ) = \\eta ( B ) + 1 \\ge T _ 0 + \\max _ { 1 \\le i _ 1 < \\ldots < i _ { q _ 0 } \\le s } \\sum _ { j = 1 } ^ { q _ 0 } a _ { i _ j } . \\end{align*}"}
+{"id": "4156.png", "formula": "\\begin{align*} \\xi \\cdot b ( s ) = \\xi ( s ) b \\xi \\in X , \\ b \\in B _ e , \\end{align*}"}
+{"id": "9157.png", "formula": "\\begin{align*} v _ { i , [ \\gamma ] } ^ { j _ { i } } = y _ { i , [ \\kappa _ { i } ^ { j _ { i } } + \\gamma ] } ^ { j _ { i } , d } - \\sum _ { \\alpha = \\kappa _ { i } ^ { j _ { i } } } ^ { \\kappa _ { i } ^ { j _ { i } } - 1 + \\gamma } a _ { i } ^ { j _ { i } , \\alpha - \\gamma } ( v _ { i , [ \\alpha - \\kappa _ { i } ^ { j _ { i } } ] } ^ { j _ { i } } - y _ { i , [ \\alpha ] } ^ { j _ { i } , d } ) - \\sum _ { \\beta = \\gamma } ^ { \\kappa _ { i } ^ { j _ { i } } - 1 } a _ { i } ^ { j _ { i } , \\beta - \\gamma } ( y _ { i , [ \\beta ] } ^ { j _ { i } } - y _ { i , [ \\beta ] } ^ { j _ { i } , d } ) \\ , . \\end{align*}"}
+{"id": "888.png", "formula": "\\begin{align*} \\rho _ 2 ( \\varpi ) = | \\varpi | ^ { n - 1 } + O \\left ( | \\varpi | ^ { ( n + 1 ) / 2 } \\right ) . \\end{align*}"}
+{"id": "3497.png", "formula": "\\begin{align*} H _ b ( X , Y ) & = \\int _ a ^ b \\langle X ' , Y ' \\rangle + \\langle R ( \\dot { \\gamma } , X ) \\dot { \\gamma } , Y \\rangle d s \\\\ & = \\int _ a ^ b \\langle - X '' + R ( \\dot { \\gamma } , X ) \\dot { \\gamma } , Y \\rangle d s . \\end{align*}"}
+{"id": "3451.png", "formula": "\\begin{align*} r z ^ { 2 } - ( p - s ) z - q = 0 . \\end{align*}"}
+{"id": "1638.png", "formula": "\\begin{align*} g ( X , Z ) & = - \\Phi ( { f } X , Z ) - g ( X , \\widetilde { Q } Z ) + \\sum \\nolimits _ { i } \\big ( \\eta ^ i ( X ) \\ , \\eta ^ i ( Z ) + \\eta ^ i ( X ) \\ , \\eta ^ i ( \\widetilde { Q } Z ) \\big ) \\\\ & = - \\Phi ( { f } X , Z ) + \\sum \\nolimits _ { i } \\eta ^ i ( X ) \\ , \\eta ^ i ( Z ) - g ( X , \\widetilde { Q } Z ) . \\end{align*}"}
+{"id": "8687.png", "formula": "\\begin{align*} ( \\widetilde { \\Delta } _ { l } ) = O ( N ^ { - 1 } p ^ { - l } ) . \\end{align*}"}
+{"id": "7605.png", "formula": "\\begin{align*} | H _ { 2 , 1 } ( F _ { f } / 2 ) | = \\frac { 1 } { 1 4 4 } | \\tau _ { 2 } | ^ 2 \\leq \\frac { 1 } { 1 4 4 } . \\end{align*}"}
+{"id": "6620.png", "formula": "\\begin{align*} [ ( [ [ x _ \\mu { y _ 1 } ] _ { \\eta + \\mu } y _ 2 ] ) _ { \\mu + \\gamma } \\varphi _ { \\lambda } ( w , v ) ] + ( - 1 ) ^ { x y _ 1 } [ ( [ { y _ 1 } _ \\eta [ x _ \\mu y _ 2 ] ] ) _ { \\mu + \\gamma } \\varphi _ { \\lambda } ( w , v ) ] = [ ( \\varphi _ \\lambda ( x , [ { y _ 1 } _ \\eta y _ 2 ] ) ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] , \\end{align*}"}
+{"id": "1189.png", "formula": "\\begin{align*} \\Delta ^ L = \\frac { \\sum _ { n m \\in \\mathcal { L } } \\delta ^ L _ { n m } } { | \\mathcal { L } | } \\end{align*}"}
+{"id": "1296.png", "formula": "\\begin{align*} T ( u ) = \\int _ 0 ^ u \\exp ( 2 \\rho ( v ) ) d v . \\end{align*}"}
+{"id": "9031.png", "formula": "\\begin{align*} b _ { i , \\tau ( i ) } = 0 . \\end{align*}"}
+{"id": "3541.png", "formula": "\\begin{align*} \\frac { \\partial P _ { i + n + 1 } } { \\partial y _ j } ( y _ { j + n + 1 } , y _ { j + 2 n + 2 } ) & + a _ j \\wp ' ( a _ j f _ j ( x , y ) ) \\frac { \\partial P _ { i + n + 1 } } { \\partial y _ { j + n + 1 } } ( y _ { j + n + 1 } , y _ { j + 2 n + 2 } ) \\\\ & + a _ j \\wp '' ( a _ j f _ j ( x , y ) ) \\frac { \\partial P _ { i + n + 1 } } { \\partial y _ { j + 2 n + 2 } } ( y _ { j + n + 1 } , y _ { j + 2 n + 2 } ) = 0 . \\end{align*}"}
+{"id": "2747.png", "formula": "\\begin{align*} \\tilde { \\mathbf { m } } _ \\lambda = \\lambda \\mathbf { b } _ \\lambda \\mathbf { e } _ \\lambda . \\end{align*}"}
+{"id": "983.png", "formula": "\\begin{align*} Q = Q _ 1 \\oplus ^ { \\mathrm { f i n e } } _ P Q _ 2 \\end{align*}"}
+{"id": "3075.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\left [ r ^ \\delta u ' ( r ) ^ { p - 1 - \\alpha } \\right ] ' = \\frac { \\delta } { n - 1 } \\ , r ^ \\delta f _ 1 ( r ) g _ 1 ( v ( r ) ) & & \\quad , \\\\ & \\left [ r ^ \\delta u _ 0 ' ( r ) ^ { p - 1 - \\alpha } \\right ] ' = \\frac { \\delta } { n - 1 } \\ , r ^ \\delta f _ 1 ( r ) h _ 1 ( v _ 0 ( r ) ) & & \\quad \\end{aligned} \\right . \\end{align*}"}
+{"id": "1273.png", "formula": "\\begin{align*} \\eta ( B ) \\ge T _ 0 - 1 + \\max _ { 1 \\le i _ 1 < \\ldots < i _ { q _ 0 } \\le s } \\sum _ { j = 1 } ^ { q _ 0 } a _ { i _ j } , \\end{align*}"}
+{"id": "2957.png", "formula": "\\begin{align*} R _ F ( H ) & = R ( H ) [ F ( \\left . \\rho \\right | _ H ) ^ { - 1 } ] \\ . \\end{align*}"}
+{"id": "5008.png", "formula": "\\begin{align*} \\Sigma _ n ^ { - 1 } U _ n ^ { } = \\left [ \\begin{array} { c | c } U _ { n - 1 } & \\mathbf { 0 } \\\\ \\hline \\mathbf { 0 ' } & 1 \\end{array} \\right ] \\end{align*}"}
+{"id": "8563.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { - 1 } S _ j ( \\tau _ N ) = \\nu \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s , j \\geq 1 , \\end{align*}"}
+{"id": "4008.png", "formula": "\\begin{align*} ( f _ { \\mathcal { S } } ) _ { \\lambda } ( s _ 1 , s _ 2 ) & = \\sup _ { x ^ \\prime \\in \\mathcal { X } } \\left \\{ - y _ 2 d ( x ' ) - y _ 1 ( 1 - d ( x ' ) ) - \\sum _ { 1 \\leq \\ell \\leq 2 } \\lambda _ \\ell \\| x _ \\ell - x ' \\| \\right \\} \\\\ & = - \\min \\{ y _ 2 + \\varphi _ { \\lambda , 1 } ( x _ 1 , x _ 2 ) , y _ 1 + \\varphi _ { \\lambda , 0 } ( x _ 1 , x _ 2 ) \\} , \\end{align*}"}
+{"id": "8469.png", "formula": "\\begin{align*} I _ { m , p } ^ { \\star ( 1 ) } ( s , x ) = ( - 1 ) ^ { m } \\sqrt { \\pi } 2 ^ { \\frac { 1 } { 2 } - s } x ^ { s + \\frac { 1 } { 2 } } \\ , \\intop _ { 0 } ^ { \\infty } y ^ { s - \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( x y ) \\ , e ^ { - m y } d y \\end{align*}"}
+{"id": "513.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\hbar , V } u ( k ) : = \\left ( - \\hbar ^ { - 2 } \\mathcal { L } _ { \\hbar } + V \\right ) u ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\end{align*}"}
+{"id": "3823.png", "formula": "\\begin{align*} \\boldsymbol { d } _ { \\mathcal { S } _ \\ell } ( ( y _ \\ell , x ) , ( y ^ { \\prime } _ \\ell , x ^ \\prime ) ) = \\boldsymbol { d } _ { \\mathcal { Y } _ \\ell } ( y _ \\ell , y ^ { \\prime } _ \\ell ) + \\boldsymbol { d } _ { \\mathcal { X } } ( x , x ^ { \\prime } ) , \\end{align*}"}
+{"id": "1542.png", "formula": "\\begin{align*} \\nu = B _ 1 ^ { - 1 } \\theta ^ { \\beta } \\end{align*}"}
+{"id": "8908.png", "formula": "\\begin{align*} \\mathsf { d } _ { \\mathrm { c o m } } ( \\{ X _ { n i } \\} ) = 2 \\sum _ { n = 1 } ^ { d _ 1 ^ 2 } \\| X _ { n 1 } \\| _ 1 \\leq 2 d _ 1 ^ { \\frac { 3 } { 2 } } \\sqrt [ 4 ] { \\sum _ { n = 1 } ^ { d _ 1 ^ 2 } \\| X _ { n 1 } \\| _ 1 ^ 4 } = 2 d _ 1 ^ { \\frac { 3 } { 2 } } \\sqrt { \\nu } . \\end{align*}"}
+{"id": "1979.png", "formula": "\\begin{align*} u _ { t _ \\alpha t _ \\beta } ( 0 ) = - u _ { x _ n } ( 0 ) \\rho _ { t _ \\alpha t _ \\beta } ( 0 ) , ~ ~ \\alpha , \\beta < 2 n . \\end{align*}"}
+{"id": "2257.png", "formula": "\\begin{align*} Z = \\{ z = x + i y \\in \\mathbb { C } : x \\in I , 0 < y < \\gamma \\} \\end{align*}"}
+{"id": "1215.png", "formula": "\\begin{align*} G ( t _ 0 , 0 ) = A ' ( t _ 0 ) G ( 0 , q A ( t _ 0 ) ) . \\end{align*}"}
+{"id": "2633.png", "formula": "\\begin{align*} \\left | \\frac { \\partial ( u , v , w ) } { \\partial ( t _ 1 , t _ 2 , t _ 3 ) } \\right | ( \\mathbf { t } ) = \\left | \\widetilde { P } _ 1 ' ( t _ 2 ) \\widetilde { P } _ 2 ' ( t _ 3 ) - \\widetilde { P } _ 1 ' ( t _ 3 ) \\widetilde { P } _ 2 ' ( t _ 2 ) \\right | \\asymp 2 ^ { - \\mathfrak { c } | l | } | t _ 2 - t _ 3 | , \\end{align*}"}
+{"id": "6175.png", "formula": "\\begin{align*} ( a * b ) ( r ) = \\sum _ { s t = r } a ( s ) \\alpha _ s ( b ( t ) ) u ( s , t ) , a ^ * ( t ) = \\alpha _ t ( a ( t ^ { - 1 } ) ^ * ) u ( t , t ^ { - 1 } ) ^ * u ( 1 , 1 ) ^ * \\end{align*}"}
+{"id": "4529.png", "formula": "\\begin{align*} g ( 0 ) = 2 L ( U ) > 0 , \\ \\ g ( 1 ) = I _ { \\omega , \\mathbf { c } } ( U ) < 0 \\end{align*}"}
+{"id": "2881.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 4 - 4 t _ 1 , & ~ ~ t _ 0 \\geq 4 , \\\\ 4 - 3 t _ 1 , & ~ ~ t _ 0 = 3 , \\\\ 4 - 2 t _ 1 , & ~ ~ t _ 0 = 2 , \\\\ t _ 0 t _ 1 - 2 t _ 1 , & ~ ~ t _ 0 \\leq 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "4889.png", "formula": "\\begin{align*} \\Psi \\left ( - \\frac 1 2 , x ' , 0 \\right ) = 0 | x ' | < \\frac 1 4 . \\end{align*}"}
+{"id": "2874.png", "formula": "\\begin{align*} ( \\alpha \\otimes \\Delta _ c ) \\circ \\Delta _ c ( l ) & = \\alpha ( l _ 1 ) \\otimes T ( l _ { 1 2 } ) \\otimes T ( \\alpha ( l _ 2 ) ) - \\alpha ( l _ 1 ) \\otimes T ( \\alpha ( l _ 2 ) ) \\otimes T ( l _ { 1 2 } ) \\\\ & - T ( \\alpha ( l _ 2 ) ) \\otimes \\alpha ( l _ 1 ) \\otimes T ( l _ { 1 2 } ) + T ( \\alpha ( l _ 2 ) ) \\otimes T ( l _ { 1 2 } ) \\otimes \\alpha ( l _ 1 ) . \\end{align*}"}
+{"id": "8443.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\mu - i \\infty } ^ { \\mu + i \\infty } \\Gamma ( z ) \\ , \\Gamma ( s - z ) \\ , \\zeta _ { p } ( 2 z ) \\ , x ^ { 2 z } d z = \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\frac { 1 } { 2 } - \\mu - i \\infty } ^ { \\frac { 1 } { 2 } - \\mu + i \\infty } \\Gamma ( z ) \\ , \\Gamma ( s - z ) \\ , \\zeta _ { p } ( 2 z ) \\ , x ^ { 2 z } d z - \\frac { 1 } { 2 } \\ , \\frac { \\Gamma ( s ) } { 1 + \\frac { 1 } { \\pi p } } + \\frac { \\sqrt { \\pi } \\ , x } { 2 } \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
+{"id": "5063.png", "formula": "\\begin{align*} L ^ 1 _ N : = L ^ 1 _ { } ( 0 , N T ) , L ^ 2 _ N : = L ^ 2 _ { } ( 0 , N T ) , H ^ m _ N : = H ^ m _ { } ( 0 , N T ) . \\end{align*}"}
+{"id": "8201.png", "formula": "\\begin{align*} A _ t = \\{ \\exists \\ : z _ 0 = z _ 0 ( \\omega ) \\in [ - k t , k t ] ^ d \\ : \\ : \\ : \\ : Z _ t ( B ( z _ 0 , r ( t ) ) ) \\geq e ^ { \\delta t } \\} , \\end{align*}"}
+{"id": "7921.png", "formula": "\\begin{align*} | ( \\gamma , \\mathbf { n } ) | & : = | \\gamma | + \\eta - | \\mathbf { n } | , \\\\ | i | & : = | e _ i | ; \\end{align*}"}
+{"id": "2745.png", "formula": "\\begin{align*} & \\mathbf { b } _ \\mu = \\sqrt { 2 \\cosh \\left ( \\sqrt { \\mu } / 2 \\right ) } , \\mu > 0 , \\\\ & \\mathbf { m } _ \\lambda = \\max \\left ( \\lambda ^ 2 , \\mathbf { b } _ \\lambda \\right ) \\lambda ^ 5 \\mathbf { e } _ \\lambda ^ 2 . \\end{align*}"}
+{"id": "5371.png", "formula": "\\begin{align*} & ~ ~ ~ ~ ( \\beta M _ 1 ^ H + \\alpha M _ 2 ^ H ) ^ H ( \\beta M _ 1 ^ H - \\alpha M _ 2 ^ H ) + ( \\beta M _ 1 ^ H - \\alpha M _ 2 ^ H ) ^ H ( \\beta M _ 1 ^ H + \\alpha M _ 2 ^ H ) \\\\ & = 2 \\vert \\beta \\vert ^ 2 ( M _ 1 M _ 1 ^ H - \\tfrac { \\vert \\alpha \\vert ^ 2 } { \\vert \\beta \\vert ^ 2 } M _ 2 M _ 2 ^ H ) \\\\ & \\geq 2 \\vert \\beta \\vert ^ 2 ( 1 - \\tfrac { \\vert \\alpha \\vert ^ 2 } { \\vert \\beta \\vert ^ 2 } ) M _ 2 M _ 2 ^ H \\\\ & \\geq 0 . \\end{align*}"}
+{"id": "7890.png", "formula": "\\begin{align*} V ^ { + , + } & = \\left \\{ ( A ^ + , \\underline { A } ^ + ) : \\ A \\in \\binom { [ n ] } { k } \\right \\} , \\\\ V ^ { + , - } & = \\left \\{ ( A ^ + , \\underline { A } ^ - ) : \\ A \\in \\binom { [ n ] } { k } \\right \\} , \\\\ V ^ { - , + } & = \\left \\{ ( A ^ - , \\underline { A } ^ + ) : \\ A \\in \\binom { [ n ] } { k } \\right \\} , \\\\ V ^ { - , - } & = \\left \\{ ( A ^ - , \\underline { A } ^ - ) : \\ A \\in \\binom { [ n ] } { k } \\right \\} . \\end{align*}"}
+{"id": "79.png", "formula": "\\begin{align*} { \\rm W F } ' _ h ( B ) = \\{ ( x , \\xi , y , \\eta ) \\ , : \\ , ( x , \\xi , y , - \\eta ) \\in { \\rm W F } _ h ( \\mathcal { K } _ B ) \\} \\end{align*}"}
+{"id": "7271.png", "formula": "\\begin{align*} y ^ m = P ( x ) = a _ n x ^ n + \\dots + a _ 1 x + a _ 0 , \\end{align*}"}
+{"id": "4691.png", "formula": "\\begin{align*} S _ { \\chi , \\varepsilon } ( z ) & = S _ { \\chi } ^ { \\rm s o f t } ( z ) + S _ { \\chi } ^ { \\rm s t i f f } ( \\varepsilon ^ 2 z ) , \\\\ [ 0 . 3 e m ] M _ { \\chi , \\varepsilon } ( z ) & = M _ { \\chi } ^ { \\rm s o f t } ( z ) + \\varepsilon ^ { - 2 } M _ { \\chi } ^ { \\rm s t i f f } ( \\varepsilon ^ 2 z ) , z \\in \\rho ( \\mathcal { A } _ { \\chi , \\varepsilon , 0 } ) . \\end{align*}"}
+{"id": "1905.png", "formula": "\\begin{align*} L _ { \\beta } ( u , \\zeta ; \\lambda ) = \\theta ( u ) + I _ K ( \\zeta ) + \\langle \\lambda , S u - \\zeta \\rangle + \\frac { \\beta } { 2 } \\| S u - \\zeta \\| ^ 2 , \\end{align*}"}
+{"id": "7855.png", "formula": "\\begin{align*} P v ( x H ) = \\begin{cases} 1 - \\frac { | \\mathcal { S } | } { | \\Omega | } & \\mbox { i f } x H \\in \\mathcal { S } \\\\ - \\frac { | \\mathcal { S } | } { | \\Omega | } & \\mbox { o t h e r w i s e } \\end{cases} . \\end{align*}"}
+{"id": "8944.png", "formula": "\\begin{align*} \\phi _ l = \\Tilde { \\phi } _ { e } + \\sum _ { m = 1 } ^ { k } C _ { p _ m } ( r ^ { l - 1 } h ) ^ { p _ m } , \\end{align*}"}
+{"id": "6389.png", "formula": "\\begin{align*} a \\tau ^ { p } - b \\sigma ^ { p } = c r ^ { 2 } . \\end{align*}"}
+{"id": "1288.png", "formula": "\\begin{align*} \\sigma ( A ) & = \\left \\{ \\sqrt { n m } + 1 ^ { ( 1 ) } , - \\sqrt { n m } + 1 ^ { ( 1 ) } , 1 ^ { ( n + m - 2 ) } , \\sqrt { n m } - 1 ^ { ( 1 ) } , - \\sqrt { n m } - 1 ^ { ( 1 ) } , - 1 ^ { ( n + m - 2 ) } \\right \\} . \\end{align*}"}
+{"id": "3257.png", "formula": "\\begin{align*} \\begin{aligned} & \\mbox { ( i ) } B = B _ 1 \\cup B _ 2 , \\widetilde B = E _ 1 \\cup E _ 2 \\mbox { a n d } \\omega ( E _ i ) \\ge \\frac 1 2 \\omega ( \\widetilde B ) , i = 1 , 2 ; \\\\ & \\mbox { ( i i ) } b ( x ) - b ( y ) \\mbox { d o e s n o t c h a n g e s i g n f o r a l l } ( x , y ) \\in B _ i \\times E _ i , i = 1 , 2 ; \\\\ & \\mbox { ( i i i ) } | b ( x ) - m _ b ( \\tilde B ) | \\le | b ( x ) - b ( y ) | \\mbox { f o r a l l } ( x , y ) \\in B _ i \\times E _ i , i = 1 , 2 . \\end{aligned} \\end{align*}"}
+{"id": "6014.png", "formula": "\\begin{align*} \\theta _ { L , Y ^ { \\ast } } ( f ' ) ( w ) = \\sum _ { l \\in L / L \\cap Y ^ { \\ast } } f ' ( w + l ) \\psi ( \\tfrac { \\langle l , w \\rangle } { 2 } + \\tfrac { \\langle y _ { l } , y ^ { \\ast } _ l \\rangle } { 2 } ) , \\end{align*}"}
+{"id": "9310.png", "formula": "\\begin{align*} \\left ( A ^ { T } \\right ) ^ { \\odot ( d - 1 ) } \\ , { \\rm d i a g } \\ , A = 0 . \\end{align*}"}
+{"id": "9095.png", "formula": "\\begin{align*} Y _ 1 = 0 , \\ldots , Y _ { n - r } = 0 . \\end{align*}"}
+{"id": "3804.png", "formula": "\\begin{align*} \\mathrm { R W } _ 0 ( d ) : = \\inf _ { \\gamma \\in \\Sigma _ 0 ( \\delta _ 0 ) } \\mathbb { E } _ { \\gamma } [ Y _ 1 ( 1 - d ( X ) ) + Y _ 2 d ( X ) ] , \\end{align*}"}
+{"id": "1456.png", "formula": "\\begin{align*} \\tilde { u } _ m ^ i ( y ) = ( \\mu _ m ^ i ) ^ { \\frac { n - 2 } { 2 } } u _ m ( \\mu _ m ^ i y + \\xi _ m ) \\chi _ m ^ i ( \\mu _ m ^ i y + \\xi _ m ) . \\end{align*}"}
+{"id": "2518.png", "formula": "\\begin{align*} \\mu _ { T ' } ( \\overline { T ' \\cdot x } ) = \\{ e _ i + e _ j \\ : : \\ : \\{ i , j \\} \\in { \\mathcal B } ' ( x ) \\} . \\end{align*}"}
+{"id": "9037.png", "formula": "\\begin{align*} b _ { i , j + 2 } & = b _ { \\eta ( i ) , \\eta ( j + 2 ) } = b _ { i , j } . \\end{align*}"}
+{"id": "3517.png", "formula": "\\begin{align*} \\lambda ( t ) = J _ K \\lambda _ { S ( t ) } = e ^ { i t } J _ K \\lambda _ { S ( 0 ) } \\implies \\lambda _ { S ( t ) } = e ^ { i t } \\lambda _ { S ( 0 ) } . \\end{align*}"}
+{"id": "6256.png", "formula": "\\begin{align*} & G _ 2 ^ { \\mathrm { r e g } } ( \\xi , \\sigma _ 1 , \\sigma _ 2 , \\alpha _ 1 , \\alpha _ 2 , \\beta _ 1 , \\beta _ 2 ) = \\phi _ 2 ( \\sigma _ 1 , \\sigma _ 2 , \\alpha _ 1 , \\alpha _ 2 , \\beta _ 1 , \\beta _ 2 ) \\\\ & + H _ 2 ( \\xi , \\sigma _ 1 , \\sigma _ 2 , \\alpha _ 1 - \\tau \\beta _ 1 , \\alpha _ 2 - \\tau \\beta _ 2 , \\beta _ 1 + \\beta _ 2 , \\beta _ 2 ) + \\phi _ 1 ( \\sigma _ 1 , \\alpha _ 1 , \\beta _ 1 ) H _ 1 ( \\xi , \\sigma _ 2 , \\alpha _ 2 - \\tau \\beta _ 2 , \\beta _ 2 ) \\ , . \\end{align*}"}
+{"id": "4363.png", "formula": "\\begin{align*} d _ { p } \\left ( \\overline { b } , \\Theta _ { c } \\right ) ^ { p } \\geq \\inf _ { x \\in \\mathbb { R } ^ { n } } \\sum \\limits _ { i = 1 } ^ { m } \\left ( l _ { i } \\right ) ^ { p } d _ { S _ { i } } ^ { p } \\left ( x \\right ) . \\end{align*}"}
+{"id": "1931.png", "formula": "\\begin{align*} \\psi ( z _ 1 , \\ldots , z _ m ) = \\sum _ { ( k _ 1 , \\ldots , k _ m ) \\in \\Z ^ m _ { \\geq 0 } } \\Re ( c _ { k _ 1 , \\ldots , k _ m } ) z _ 1 ^ { k _ 1 } \\cdots z _ m ^ { k _ m } \\in \\Q [ [ z _ 1 , \\ldots , z _ m ] ] \\end{align*}"}
+{"id": "2752.png", "formula": "\\begin{align*} u _ { q , \\lambda } ( \\varphi ) = \\Phi - R _ q ( \\lambda ) ( ( \\Delta + \\lambda - q ) \\Phi ) . \\end{align*}"}
+{"id": "1012.png", "formula": "\\begin{align*} & \\quad \\sum _ { k = 0 } ^ { \\infty } ( 2 0 k ^ 2 + 8 k + 1 ) \\frac { \\binom { 2 k } { k } ^ 5 } { ( - 2 ^ { 1 2 } ) ^ k } = \\frac { 8 } { \\pi ^ 2 } , \\\\ [ 1 m m ] & \\sum _ { k = 0 } ^ { \\infty } ( 8 2 0 k ^ 2 + 1 8 0 k + 1 3 ) \\frac { \\binom { 2 k } { k } ^ 5 } { ( - 2 ^ { 2 0 } ) ^ k } = \\frac { 1 2 8 } { \\pi ^ 2 } . \\end{align*}"}
+{"id": "5842.png", "formula": "\\begin{align*} \\left ( X _ T , X _ { T + 1 } , \\ldots , X _ { T + T _ { n _ 1 } ^ { o u t } - T ' } \\right ) = \\left ( X ' _ { T ' } , X ' _ { T ' + 1 } , \\ldots , X ' _ { T _ { n _ 1 } ^ { o u t } } \\right ) . \\end{align*}"}
+{"id": "5698.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { m \\rightarrow \\infty } \\dfrac { \\left ( C _ { D ^ { k - 1 } ( f ) } ( p ^ { 2 m + 1 } ) - C _ { D ^ { k - 1 } ( f ' ) } ( p ^ { 2 m + 1 } ) \\right ) } { \\beta ^ { 2 m } } = 0 . \\end{align*}"}
+{"id": "8586.png", "formula": "\\begin{align*} \\{ Z _ 0 ( t ) = j \\} & = \\bigcup _ { k \\geq 1 } \\{ \\tau _ j ^ + ( k ) \\leq t , \\tau _ j ^ - ( k ) > t \\} \\\\ & = \\bigcup _ { k \\geq 1 } \\{ \\tau _ j ^ + ( k ) \\leq t \\} \\backslash \\{ \\tau _ j ^ - ( k ) \\leq t \\} . \\end{align*}"}
+{"id": "7676.png", "formula": "\\begin{align*} \\Delta _ g ^ 2 u _ \\delta & = u _ \\delta ^ { ( 4 ) } + 2 L u _ \\delta ^ { ( 3 ) } + L ^ 2 u _ \\delta '' + L '' u _ \\delta ' + 2 L ' u _ \\delta '' + L L ' u _ \\delta ' \\\\ & \\sim u _ \\delta ^ { ( 4 ) } + 2 L u _ \\delta ^ { ( 3 ) } + L ^ 2 u _ \\delta '' = s ^ 2 ( s - L ) ^ 2 e ^ { - s \\rho } \\sim s ^ 4 ( 1 - p ) ^ 2 e ^ { s \\rho } . \\end{align*}"}
+{"id": "5741.png", "formula": "\\begin{align*} ( K ^ * ) ^ 3 = \\{ k ^ 3 : k \\in K ^ * \\} , \\end{align*}"}
+{"id": "4835.png", "formula": "\\begin{align*} \\begin{array} { c c c } K ( T ) & = & \\bigcup \\{ L : L T \\} \\\\ & = & \\ , \\ , \\ , \\ , \\ , \\bigcup \\{ R : R T \\} . \\end{array} \\end{align*}"}
+{"id": "4909.png", "formula": "\\begin{align*} f ( k , t , \\mathbf { p } _ { n + 1 } ) \\overset { } { = } \\Pr ( X _ t = k \\ , \\vert \\ , X _ 0 = 0 , \\mathbf { p } _ { n + 1 } ) , k \\in S , \\end{align*}"}
+{"id": "2300.png", "formula": "\\begin{align*} \\Psi ( z ) : = \\sum _ { k = 1 } ^ { n - 1 } \\left ( T ^ k ( \\Phi _ k ) ( z ) \\right ) + T ^ n ( f ) ( z ) , \\end{align*}"}
+{"id": "7231.png", "formula": "\\begin{align*} \\rho _ j : = \\rho _ { h _ j } , s _ j : = s _ { h _ j } , u _ j : = u _ { \\rho _ { h _ j } } , w _ j : = w ^ { ( s _ { h _ j } ) } _ { \\rho _ { h _ j } } . \\end{align*}"}
+{"id": "3414.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta v _ i = f ( v _ i ) + \\lambda v _ i \\ \\ \\ \\ \\mathbb R ^ N , \\\\ v _ i ( x ) > 0 , \\ \\lim _ { | x | \\to \\infty } v _ i ( x ) = 0 , i = 1 , 2 , \\cdots , \\ell , \\\\ \\sum _ { i = 1 } ^ { \\ell } | v _ i | _ 2 ^ 2 = \\alpha . \\end{cases} \\end{align*}"}
+{"id": "6462.png", "formula": "\\begin{align*} \\partial _ s \\Big ( \\frac { 1 } { z } \\Big ) ( s ) & = - \\frac { z ' ( s ) } { z ( s ) ^ 2 } \\\\ & = - \\frac { h ^ 2 \\beta _ h ' ( s ) + \\frac { h ^ 2 } { 2 } \\overline { \\beta _ h ' ( s ) } + \\frac { \\sigma ^ 2 } { 2 } \\overline { \\beta _ \\sigma ' ( s ) } } { \\big ( 2 \\sigma ^ 2 + \\mathcal { O } ( h ^ 2 + s \\sigma ^ 4 ) \\big ) ^ 2 } \\\\ & \\sim \\frac { i \\sigma ^ 4 } { 4 \\sigma ^ 4 } \\end{align*}"}
+{"id": "250.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { m ^ 2 } { n ^ 3 } } = \\sqrt { \\frac { 1 } { 1 - z } } \\ ; \\exp \\left \\{ \\frac { 1 } { 6 } L i _ 2 ( z ) + \\frac { 1 } { 3 } \\frac { z } { 1 - z } \\right \\} , \\end{align*}"}
+{"id": "214.png", "formula": "\\begin{align*} L i _ { - n } ( z ) = \\sum _ { k = 0 } ^ { n } k ! S ( n + 1 , k + 1 ) \\left ( \\frac { z } { 1 - z } \\right ) ^ { k + 1 } ( n = 1 , 2 , 3 , \\ldots ) , \\end{align*}"}
+{"id": "5857.png", "formula": "\\begin{align*} \\begin{aligned} | G _ { \\omega , E , \\Lambda _ L ( x ) } ( a , b ) | & \\leq | G _ { \\omega , E _ 0 , \\Lambda _ L ( x ) } ( a , b ) | + | E - E _ 0 | \\cdot | G _ { \\omega , E , \\Lambda _ L ( x ) } ( a , b ) | \\cdot | G _ { \\omega , E _ 0 , \\Lambda _ L ( x ) } ( a , b ) | \\\\ & \\leq e ^ { - m _ 0 | a - b | } + e ^ { - m ' L } e ^ { L ^ { 1 - \\eta } + L ^ { 1 - \\eta } } \\\\ & \\leq e ^ { - m | a - b | } \\end{aligned} \\end{align*}"}
+{"id": "2968.png", "formula": "\\begin{align*} v \\ast a = v \\cdot \\alpha _ { ( e , \\sigma ) } ( a ) \\ . \\end{align*}"}
+{"id": "606.png", "formula": "\\begin{align*} g ( s _ - ) = \\frac { 1 } { 4 } ( - \\sqrt { C - 4 } + \\sqrt { C } ) \\end{align*}"}
+{"id": "3727.png", "formula": "\\begin{align*} r e s _ p ( \\nabla ) ( F _ i ( E _ p ) ) \\subseteq F _ i ( E _ p ) \\ , , \\ , \\ , \\textnormal { a n d } \\ , \\ , r e s _ p ( \\nabla ) | _ { F _ i ( E _ p ) / F _ { i + 1 } ( E _ p ) } = \\frac { k _ { i , p } } { r } \\cdot \\ , . \\end{align*}"}
+{"id": "2458.png", "formula": "\\begin{align*} A _ 0 & = \\{ \\varphi \\in C _ c ^ \\infty ( M ) ^ K : \\rho _ i \\cdot \\varphi \\in A _ 0 ^ { ( i ) } i = 1 , \\ldots , N \\} . \\end{align*}"}
+{"id": "2997.png", "formula": "\\begin{align*} \\frac { \\partial V } { \\partial \\bar { c } _ { n - ( j + 1 ) } } = ( - 1 ) ^ { n - j } c _ { k + j + 1 } \\ . \\end{align*}"}
+{"id": "48.png", "formula": "\\begin{align*} \\mathrm { E n d } _ { \\mathbb { C } } ( A _ g ) \\otimes _ { \\mathbb { Z } } \\mathbb { Z } _ \\ell = \\mathcal { O } _ { K , \\ell } . \\end{align*}"}
+{"id": "86.png", "formula": "\\begin{align*} \\widetilde { m } ( x , \\xi ) = ( 1 - \\chi _ m ( x , \\xi ) ) m ( x , \\xi ) \\log | \\xi | \\end{align*}"}
+{"id": "3754.png", "formula": "\\begin{align*} C _ r ^ n & = \\psi ^ { \\alpha _ 1 , n } ( g _ 1 ) \\cdots \\psi ^ { \\alpha _ s , n } ( g _ s ) \\psi ^ { \\alpha _ { s + 1 } , n } ( f _ 1 ) \\cdots \\psi ^ { \\alpha _ { r } , n } ( f _ { r - s } ) , \\\\ D _ s ^ n & = \\psi ^ { \\beta _ { 1 } , n } ( f _ { r - s + 1 } ) \\cdots \\psi ^ { \\beta _ { s } , n } ( f _ r ) . \\end{align*}"}
+{"id": "3058.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta _ p u & = c _ 1 | x | ^ { m _ 1 } \\cdot g _ 1 ( v ) \\cdot | \\nabla u | ^ { \\alpha } & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta _ p v & = c _ 2 | x | ^ { m _ 2 } \\cdot g _ 2 ( v ) \\cdot g _ 3 ( | \\nabla u | ) & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "5043.png", "formula": "\\begin{align*} h : \\xi _ g \\mapsto h ^ { * } ( \\xi _ g ) : = \\prod _ { k \\in I _ g ^ c } h ^ { - 1 } _ k \\ , \\cdot \\xi _ g \\end{align*}"}
+{"id": "6788.png", "formula": "\\begin{align*} | f | _ { L \\log L ( B _ \\rho ( x _ 0 ) ) } : = \\int _ { B _ { \\rho } ( x _ 0 ) } | f | \\log \\left ( e + \\frac { | f | } { \\int _ { B _ { \\rho } ( x _ 0 ) } | f | \\ , d x } \\right ) \\ , d x , \\end{align*}"}
+{"id": "489.png", "formula": "\\begin{align*} ( c d ) ^ 2 = c ^ { 4 - 2 n } t ^ { k _ 2 - k _ 1 - 1 } t _ 3 t _ 4 . \\end{align*}"}
+{"id": "6689.png", "formula": "\\begin{align*} \\Delta q & = \\partial _ j \\bigl ( ( \\delta _ { j k } - a _ { j i } a _ { k i } ) \\partial _ k q \\bigr ) + \\partial _ j ( a _ { j i } a _ { k i } \\partial _ k q ) = \\partial _ j \\bigl ( ( \\delta _ { j k } - a _ { j i } a _ { k i } ) \\partial _ k q \\bigr ) + \\partial _ t a _ { j i } \\partial _ j v _ i , \\end{align*}"}
+{"id": "5580.png", "formula": "\\begin{align*} h _ { \\mu } \\left ( G / P , \\nu _ { P } \\right ) = \\sum _ { 1 \\le i < j \\le d } \\lambda _ { i } - \\lambda _ { j } . \\end{align*}"}
+{"id": "8402.png", "formula": "\\begin{align*} F _ j ( E ) = F _ j ( V ) \\oplus F _ j ( W ) . \\end{align*}"}
+{"id": "2362.png", "formula": "\\begin{align*} \\tilde { c } _ { i j } = ( k \\mapsto c _ { k i } ) - ( k \\mapsto c _ { k j } ) = ( k \\mapsto c _ { k i } - c _ { k j } ) = ( k \\mapsto c _ { k j } + c _ { j i } - c _ { k j } ) = ( k \\mapsto c _ { j i } ) \\end{align*}"}
+{"id": "3083.png", "formula": "\\begin{align*} \\mathcal W _ \\infty ^ { p - 2 - \\alpha } \\ , \\lim _ { r \\to \\infty } \\frac { u '' ( r ) } { u _ 0 '' ( r ) } = \\lim _ { r \\to \\infty } \\frac { g _ 1 ( v ( r ) ) } { h _ 1 ( v _ 0 ( r ) ) } = \\lim _ { r \\to \\infty } Q _ 1 ( \\mathcal V ( r ) , v _ 0 ( r ) ) \\end{align*}"}
+{"id": "7531.png", "formula": "\\begin{align*} D ^ l ( \\varphi _ 1 ^ { s _ 1 } \\varphi _ 2 ^ { r _ 2 } \\dotsm \\varphi _ k ^ { r _ k } ) = \\sum _ { \\substack { j _ 1 + j _ 2 + \\cdots + j _ k = l \\\\ j _ 1 , j _ 2 , \\ldots , j _ k \\geq 0 } } \\ , \\frac { l ! } { j _ 1 ! j _ 2 ! \\cdots j _ k ! } \\ ; D ^ { j _ 1 } ( \\varphi _ 1 ^ { s _ 1 } ) D ^ { j _ 2 } ( \\varphi _ 2 ^ { r _ 2 } ) \\cdots D ^ { j _ k } ( \\varphi _ k ^ { r _ k } ) . \\end{align*}"}
+{"id": "4169.png", "formula": "\\begin{align*} \\varphi ( d ) = 0 & \\Rightarrow \\varphi ( d ^ * ) \\varphi ( d ) = 0 \\Rightarrow f ( \\varphi ( d ) ^ * \\varphi ( d ) ) = 0 \\Rightarrow f \\varphi ( d ^ * d ) = 0 \\\\ & \\Rightarrow g ( d ^ * d ) = 0 \\Rightarrow d = 0 . \\end{align*}"}
+{"id": "3925.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) & = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\int g _ { \\lambda } ( s _ 1 , s _ 2 ) \\ , d \\pi ( s _ 1 , s _ 2 ) \\right ] . \\end{align*}"}
+{"id": "4346.png", "formula": "\\begin{align*} \\begin{array} { l l } \\underset { x \\in X } { } & \\sum \\limits _ { i = 1 } ^ { m } [ g _ { i } \\left ( x \\right ) - b _ { i } ] _ { + } ^ { p } , \\end{array} \\end{align*}"}
+{"id": "5610.png", "formula": "\\begin{align*} \\mathrm { I } \\left ( \\xi _ { 1 } , \\xi _ { n } | X \\right ) = H \\left ( \\xi _ { n } | X \\right ) - H \\left ( \\xi _ { n - 1 } | X \\right ) . \\end{align*}"}
+{"id": "8291.png", "formula": "\\begin{gather*} A = \\langle x _ 1 , x _ 2 , x _ 3 , x _ 4 \\rangle / ( r _ 1 , r _ 2 , r _ 3 , r _ 4 , r _ 5 , r _ 6 ) \\ \\\\ r _ 1 = x _ 1 x _ 4 + x _ 4 x _ 1 - x _ 2 ^ 2 , r _ 2 = x _ 2 x _ 4 + x _ 4 x _ 2 - x _ 3 ^ 2 - x _ 4 ^ 2 , \\\\ r _ 3 = x _ 3 x _ 4 + x _ 4 x _ 3 - x _ 1 ^ 2 , r _ 4 = x _ 1 x _ 3 + x _ 3 x _ 1 - x _ 4 ^ 2 , \\\\ r _ 5 = x _ 2 x _ 3 + x _ 3 x _ 2 - x _ 1 ^ 2 , r _ 6 = x _ 1 x _ 2 + x _ 2 x _ 1 - x _ 3 ^ 2 . \\end{gather*}"}
+{"id": "7063.png", "formula": "\\begin{align*} h _ i d \\left ( \\sum \\limits _ { j = 1 } ^ { s _ \\ell } b _ { i _ { } i j } { \\textbf { X } } ^ { \\lambda _ j } \\right ) \\equiv h _ i g ' ( \\eta ) \\mod \\mathcal J _ 1 . \\end{align*}"}
+{"id": "753.png", "formula": "\\begin{align*} \\Phi ( r ) = \\left \\lbrace \\begin{array} { l l l } \\dfrac { 1 } { 2 } r ^ 2 , & r \\in [ 0 , + \\infty ) , & K = 0 , \\\\ 1 - \\cos r , & r \\in [ 0 , \\dfrac { \\pi } { 2 } ) , & K = 1 , \\\\ \\cosh r - 1 , & r \\in [ 0 , + \\infty ) , & K = - 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "8288.png", "formula": "\\begin{align*} A ( \\alpha ) = \\langle x _ 1 , x _ 2 , x _ 3 \\rangle / ( \\alpha x _ 3 x _ 1 + x _ 1 x _ 3 , \\alpha x _ 3 x _ 2 - x _ 2 x _ 3 , x _ 1 ^ 2 - x _ 2 ^ 2 ) . \\end{align*}"}
+{"id": "4997.png", "formula": "\\begin{align*} f ( k , t ; p , n ) = \\overline { F } ( k - 1 , t ; p , n ) - \\overline { F } ( k , t ; p , n ) . \\end{align*}"}
+{"id": "2812.png", "formula": "\\begin{align*} S ( \\mathfrak M ( \\lambda , L 1 ) ) ( t ) = \\begin{pmatrix} S ^ { 1 , 1 } _ k \\\\ \\vdots \\\\ S ^ { r , 1 } _ k \\end{pmatrix} _ { k = 0 } ^ \\infty ; S _ k ^ { m , 1 } = ( L 1 ^ k ) _ { m - 1 , \\ , 0 } ; \\end{align*}"}
+{"id": "5945.png", "formula": "\\begin{align*} \\phi ( h ^ { ( \\sigma , g ) } ) = \\phi ( [ \\sum ^ m _ { k = 1 } ( a _ k - i b _ k ) f _ k ] g , - t ) = ( [ \\sum ^ m _ { k = 1 } ( a _ k e _ k - i b _ k e ^ { \\ast } _ k ) ] g , - t ) ; \\end{align*}"}
+{"id": "348.png", "formula": "\\begin{align*} S ^ m _ j & : = \\left \\{ ( A _ 0 \\subsetneq \\dots \\subsetneq A _ m ) \\in ( K ^ n _ k ) _ m \\mid \\begin{array} { l } A _ m = [ n ] \\\\ k \\not \\in A _ { j - 1 } \\\\ A _ { j } = A _ { j - 1 } \\cup \\{ k \\} \\end{array} \\right \\} . \\end{align*}"}
+{"id": "2315.png", "formula": "\\begin{align*} \\beta = ( 1 + \\frac { c } { \\sqrt { 2 } \\alpha } ) ( \\alpha - \\frac { c } { \\sqrt { 2 } } ) > \\alpha - \\frac { c } { \\sqrt { 2 } } > \\alpha - c = \\eta \\rho > 0 \\end{align*}"}
+{"id": "4917.png", "formula": "\\begin{align*} \\overline { F } ( \\underline { k } , t , \\mathbf { p } _ { n + 1 } ) = h _ { t - k - 1 } ( q _ 0 , q _ 1 , . . . , q _ k , 1 ) \\prod _ { i = 0 } ^ k p _ i , \\end{align*}"}
+{"id": "3561.png", "formula": "\\begin{align*} J _ { \\Lambda } = - 2 \\pi \\int _ { 0 } ^ { \\infty } \\left [ \\exp \\left ( - \\frac { 1 } { \\Lambda } \\left [ \\left ( \\frac { \\sigma } { r } \\right ) ^ { n } - \\left ( \\frac { \\sigma } { r } \\right ) ^ { m } \\right ] \\right ) \\right ] \\ , r ^ { 2 } d r . \\end{align*}"}
+{"id": "3867.png", "formula": "\\begin{align*} f _ { \\lambda , A } ( v ) = \\sup _ { s ' \\in \\mathcal { S } } \\left \\{ f ( s ' ) - \\sum _ { \\ell = 1 } ^ { L } \\lambda _ \\ell \\tilde { c } _ { \\ell } ( s _ { \\ell } , s _ { \\ell } ' ) \\right \\} , \\end{align*}"}
+{"id": "5730.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } p ' y ^ 2 + ( a ' + c ' ) x y = z \\\\ x ^ 2 + q ' y ^ 2 + ( b ' + d ' ) x y = w \\\\ p ' w ^ 2 + ( a ' + c ' ) z w = p x + q z \\\\ z ^ 2 + q ' w ^ 2 + ( b ' + d ' ) z w = p y + q w \\\\ p ' y w + a ' x w + c ' y z = a x + b z \\\\ x z + q ' y w + b ' x w + d ' y z = a y + b w \\\\ p ' y w + a ' y z + c ' x w = c x + d z \\\\ x z + q ' y w + b ' y z + d ' x w = c y + d w \\end{array} \\right . \\end{align*}"}
+{"id": "6297.png", "formula": "\\begin{align*} \\xi _ { u ( t ) } ( q ) : = \\xi ( q , u ( t ) ) = \\sum _ { i = 1 } ^ k u _ i ( t ) X _ i ( q ) , \\qquad \\forall \\ , q \\in M , t \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "8335.png", "formula": "\\begin{align*} \\int _ E \\phi ( s ) ^ q \\dd s \\leq \\sum _ { j \\in \\mathcal J } \\int _ { a _ j } ^ { b _ j } \\phi ( s ) ^ q \\dd s & \\leq \\sum _ { j \\in \\mathcal J } \\int _ { \\{ x \\in \\Sigma : u ^ * _ \\mu ( b _ j ) < | u ( x ) | \\leq u ^ * _ \\mu ( a _ j ) \\} } | \\nabla u ( x ) | ^ q \\dd \\mu ( x ) \\\\ & = \\sum _ { j \\in \\mathcal J } \\int _ { \\{ x \\in \\Sigma : u ^ * _ \\mu ( b _ j ) < | u ( x ) | < u ^ * _ \\mu ( a _ j ) \\} } | \\nabla u ( x ) | ^ q \\dd \\mu ( x ) . \\end{align*}"}
+{"id": "2893.png", "formula": "\\begin{align*} \\partial _ { x _ j } \\big | _ { x = 0 } f ( \\xi + B x ) = \\sum _ { k = 1 } ^ n B _ { k j } \\partial _ { \\xi _ k } f ( \\xi ) = : \\partial _ { B , \\xi _ j } f ( \\xi ) , j = 1 , \\ldots , n . \\end{align*}"}
+{"id": "8350.png", "formula": "\\begin{align*} b _ m ( E ) = \\| E \\| . \\end{align*}"}
+{"id": "4512.png", "formula": "\\begin{align*} I _ { \\omega , \\mathbf { c } } ( \\Phi ) : = 2 L ( \\Phi ) + \\left ( \\frac { d } { 2 } + 1 \\right ) N ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) . \\end{align*}"}
+{"id": "5691.png", "formula": "\\begin{align*} \\dim \\widehat { S _ { k } } ^ { \\# , 0 } ( \\Gamma _ { 0 } ( N ) ) = \\dim S _ { k } ( \\Gamma _ { 0 } ( N ) ) . \\end{align*}"}
+{"id": "6901.png", "formula": "\\begin{align*} \\hat \\psi _ r ( \\beta ) = J _ r ( x _ * , \\beta ) \\geq \\frac { 1 } { 2 v _ r ( x _ * ) } ( \\beta - C _ r ) ^ 2 + O ( ( \\beta - C _ r ) ) ^ 3 ) , \\beta \\downarrow C _ r . \\end{align*}"}
+{"id": "6119.png", "formula": "\\begin{align*} M = ( c _ { 0 } c _ { q } + c _ { 0 } \\tilde { c } ) 2 ^ { 1 0 n ( j _ { 0 } + 4 + 5 s ^ { - 1 } ) } s ^ { - \\frac { 5 n } { s } } \\end{align*}"}
+{"id": "622.png", "formula": "\\begin{align*} A _ p = \\{ a \\in A : a p = p a = 0 \\} . \\end{align*}"}
+{"id": "1024.png", "formula": "\\begin{align*} & \\qquad \\qquad \\frac { ( \\frac { 1 } { 4 } ) _ k ( \\frac { 3 } { 4 } ) _ k } { ( 1 ) _ { k } ^ 2 } = \\frac { \\binom { 2 k } { k } \\binom { 4 k } { 2 k } } { 6 4 ^ k } , \\\\ [ 1 m m ] & H _ k ( - \\tfrac { 1 } { 4 } ) + H _ k ( - \\tfrac { 3 } { 4 } ) = 4 H _ { 4 k } - 2 H _ { 2 k } , \\end{align*}"}
+{"id": "3125.png", "formula": "\\begin{align*} \\frac { 1 } { \\prod _ { i = 0 } ^ n ( 1 - t q ^ { r i } ) } = \\sum _ { k = \\ell } ^ n \\frac { ( t q ^ { r \\cdot k } ) ^ { k - \\ell } } { \\prod _ { i = 0 } ^ k ( 1 - t q ^ { r i } ) } { n - \\ell \\brack k - \\ell } _ { q ^ r } \\end{align*}"}
+{"id": "8037.png", "formula": "\\begin{align*} \\lambda _ { 2 n - 1 } \\left ( \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} \\right ) & \\le \\lambda _ { n } \\left ( \\begin{bmatrix} A & 0 \\\\ 0 & 0 \\end{bmatrix} \\right ) + \\lambda _ { n } \\left ( \\begin{bmatrix} 0 & 0 \\\\ 0 & B \\end{bmatrix} \\right ) \\\\ & = \\lambda _ { n } ( A ) + \\lambda _ n ( B ) \\\\ & \\le \\lambda _ n ( A + B ) . \\end{align*}"}
+{"id": "2439.png", "formula": "\\begin{align*} U _ n = \\left ( ( - \\infty , 0 ) \\times \\{ 0 \\} \\right ) \\cup \\left ( [ 0 , \\infty ) \\times \\{ n \\} \\right ) & & \\chi _ n = \\mathrm { p r o j } _ 1 | _ { U _ n } & & n \\in \\Z . \\end{align*}"}
+{"id": "8959.png", "formula": "\\begin{align*} \\frac { \\partial \\phi } { \\partial t } + a _ x \\frac { \\partial \\phi } { \\partial x } + a _ y \\frac { \\partial \\phi } { \\partial y } = 0 , \\end{align*}"}
+{"id": "4053.png", "formula": "\\begin{align*} \\lambda _ f ( r ) \\lambda _ f ( s ) = \\sum _ { d | ( r , s ) } \\lambda _ f \\left ( \\frac { r s } { d } \\right ) , \\end{align*}"}
+{"id": "5800.png", "formula": "\\begin{align*} \\frac { y ' } { y } = d A ( z ) ^ { 1 / k } - \\frac { k - 1 } { 2 k } \\frac { A ' ( z ) } { A ( z ) } + O ( r ^ { - 2 } ) , d ^ k = - 1 . \\end{align*}"}
+{"id": "6927.png", "formula": "\\begin{align*} \\sum _ { \\substack { m , n \\in \\mathcal { M } ' \\\\ m \\not = n } } r ( m ) r ( n ) \\Phi \\left ( T \\log { \\frac { m } { n } } \\right ) \\ll \\sum _ { j \\in \\mathcal { J } } r ( m _ j ) ^ { 2 } . \\end{align*}"}
+{"id": "6537.png", "formula": "\\begin{align*} \\int _ { a + P _ { r , s , A } } | f | ^ 2 e ^ { - \\varphi } = | P _ { r , s , A } | e ^ { - \\varphi ( a ) } . \\end{align*}"}
+{"id": "5713.png", "formula": "\\begin{align*} \\dfrac { C _ { D ^ { k - 1 } ( f ) } ( p ^ { 2 m + 1 } ) } { \\beta ^ { 2 m } } = c \\dfrac { C _ { F } ( p ^ { 2 m + 1 } ) } { \\beta ^ { 2 m } } + \\dfrac { c _ { D ^ { k - 1 } ( h ) } ( p ^ { 2 m + 1 } ) } { \\beta ^ { 2 m } } \\end{align*}"}
+{"id": "520.png", "formula": "\\begin{align*} \\mathrm { H } _ { \\mathcal { H } _ { \\hbar , V } } ^ { s } : = \\left \\{ u \\in \\mathrm { H } ^ { - \\infty } _ { \\mathcal { H } _ { \\hbar , V } } : \\left ( I + \\mathcal { H } _ { \\hbar , V } \\right ) ^ { s / 2 } u \\in \\ell ^ { 2 } \\left ( \\hbar \\mathbb { Z } ^ { n } \\right ) \\right \\} , \\end{align*}"}
+{"id": "5817.png", "formula": "\\begin{align*} J = 2 \\sum \\limits _ { i = 1 } ^ N { { G _ { { \\alpha _ 1 } , { \\beta _ 1 } } } \\left ( { { e _ i } } \\right ) } + \\sum \\limits _ { i = 1 } ^ N { \\sum \\limits _ { j = 1 } ^ N { { G _ { { \\alpha _ 2 } , { \\beta _ 2 } } } \\left ( { { e _ i } - { e _ j } } \\right ) } } . \\end{align*}"}
+{"id": "5279.png", "formula": "\\begin{align*} ( 1 + \\rho | d X | ) ^ { q - 2 } \\leq ( 1 + \\omega ) ^ { q - 2 } \\leq e ^ { \\omega ( q - 2 ) } = \\rho ^ { - 1 / 2 } . \\end{align*}"}
+{"id": "5198.png", "formula": "\\begin{align*} S ^ \\sqsubset = \\mathop { \\sqsupset } [ S ] = \\{ b \\in B : \\exists s \\in S \\ ( s \\sqsubset b ) \\} . \\\\ \\intertext { L i k e w i s e , t h e \\emph { i m a g e } o f a n y $ T \\subseteq B $ i s t h e p r e i m a g e o f t h e o p p o s i t e r e l a t i o n $ \\sqsupset $ , i . e . } T ^ \\sqsupset = \\mathop { \\sqsubset } [ T ] = \\{ a \\in A : \\exists t \\in T \\ ( a \\sqsubset t ) \\} . \\end{align*}"}
+{"id": "3853.png", "formula": "\\begin{align*} \\mathrm { R W } ( d ) = \\sup _ { \\lambda \\ge 1 } \\left [ \\inf _ { \\pi \\in \\Pi ( \\mu _ { 1 3 } , \\mu _ { 2 3 } ) } \\int _ { \\mathcal { V } } \\min \\{ y _ 2 + \\varphi _ { \\lambda , 1 } ( x _ 1 , x _ 2 ) , y _ 1 + \\varphi _ { \\lambda , 0 } ( x _ 1 , x _ 2 ) \\} d \\pi ( v ) - \\langle \\lambda , \\delta \\rangle \\right ] , \\end{align*}"}
+{"id": "3619.png", "formula": "\\begin{align*} ( b _ m \\cdots b _ 0 ) _ { 1 0 } \\ = \\ ( a _ m \\cdots a _ 0 ) _ { 1 0 } - v ( f ) . \\end{align*}"}
+{"id": "6476.png", "formula": "\\begin{align*} \\hat { f } ( k ) = \\frac { \\pi } { z } e ^ { - \\frac { | k | ^ 2 } { 4 z } - \\frac { i ( a \\cdot k ) } { 2 z } + \\frac { | a | ^ 2 } { 4 z } } . \\end{align*}"}
+{"id": "3291.png", "formula": "\\begin{align*} \\widetilde { \\rho } _ t ( t , w ) = A \\beta _ 1 t ^ { \\beta _ 1 - 1 } + B ( 1 + \\beta _ 1 ) t ^ { \\beta _ 1 } + ( 1 - \\beta _ 1 ) C t ^ { - \\beta _ 1 } | w | ^ 2 + \\end{align*}"}
+{"id": "1162.png", "formula": "\\begin{align*} B ( A ( f ) , A ( 1 ) ) = B ( A ( f ) , 0 ) = 0 \\end{align*}"}
+{"id": "2445.png", "formula": "\\begin{align*} e ^ { r X ^ B } b = t ( e ^ { r X } b ) & & r \\in \\R , b \\in B . \\end{align*}"}
+{"id": "7703.png", "formula": "\\begin{align*} M _ { \\le r } = \\bigcap _ { 0 \\le j \\le r } M _ { U _ j } . \\end{align*}"}
+{"id": "6260.png", "formula": "\\begin{align*} \\omega _ \\Pi = \\inf _ { \\{ \\Gamma _ t \\} \\in \\Pi } \\sup _ { t \\in I ^ n } | \\Sigma _ t | , \\end{align*}"}
+{"id": "3832.png", "formula": "\\begin{align*} \\rho _ \\ell ( y _ \\ell , y _ \\ell ^ \\prime ) : = \\inf _ { x _ \\ell , x _ \\ell ^ \\prime \\in \\mathcal { X } } \\boldsymbol { d } _ { \\mathcal { S } _ \\ell } \\left ( ( y _ \\ell , x _ \\ell ) , ( y _ \\ell ^ { \\prime } , x _ \\ell ^ { \\prime } ) \\right ) ^ { p _ \\ell } . \\end{align*}"}
+{"id": "3910.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta _ 1 , 0 ) = \\inf _ { \\lambda _ 1 \\in \\mathbb { R } _ { + } } \\left [ \\lambda _ 1 \\delta _ 1 + \\sup _ { \\varpi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\int _ { \\mathcal { V } } g _ { \\lambda , 1 } ( v ) \\ , d \\varpi ( v ) \\right ] . \\end{align*}"}
+{"id": "7534.png", "formula": "\\begin{align*} \\begin{aligned} - \\Delta _ p u + ( - \\Delta _ p ) ^ s u = f \\ ; \\ ; \\Omega , \\end{aligned} \\end{align*}"}
+{"id": "183.png", "formula": "\\begin{align*} L i _ 3 ( z ) + L i _ 3 ( - z ) = \\frac { 1 } { 4 } L i _ 3 ( z ^ 2 ) \\end{align*}"}
+{"id": "2255.png", "formula": "\\begin{align*} T _ { \\C } ( f ) ( z ) = - \\frac { 1 } { \\pi } \\iint _ { \\C } \\frac { f ( \\zeta ) } { \\zeta - z } \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "3002.png", "formula": "\\begin{align*} \\norm { \\rho ( t ) - r ( t ) } _ u & = \\norm { r ( i / N ) - r ( t ) } _ u \\\\ & = \\norm { 1 - \\exp { ( t - i / N ) f _ i } } _ u \\\\ & \\leq ( t - i / N ) \\norm { f _ i } _ 2 \\\\ & \\leq N ^ { - 1 } \\norm { f _ i } _ 2 . \\end{align*}"}
+{"id": "1035.png", "formula": "\\begin{align*} & { _ { 5 } F _ { 4 } } \\left [ \\begin{array} { c c c c c c c c } a , 1 + \\frac { a } { 2 } , b , c , d \\\\ \\frac { a } { 2 } , 1 + a - b , 1 + a - c , 1 + a - d \\end{array} ; 1 \\right ] \\\\ [ 1 m m ] & \\ : \\ : = \\frac { \\Gamma ( 1 + a - b ) \\Gamma ( 1 + a - c ) \\Gamma ( 1 + a - d ) \\Gamma ( 1 + a - b - c - d ) } { \\Gamma ( 1 + a ) \\Gamma ( 1 + a - b - c ) \\Gamma ( 1 + a - b - d ) \\Gamma ( 1 + a - c - d ) } , \\end{align*}"}
+{"id": "1834.png", "formula": "\\begin{align*} \\nu \\Big ( a ^ * _ { p _ 1 } ( t ) a _ { q _ 1 } ( t ) a ^ * _ { p _ 2 } ( s ) a _ { q _ 2 } ( s ) \\Big ) & = \\delta ( q _ 1 - p _ 1 ) \\delta ( q _ 2 - p _ 2 ) f _ 0 ( p _ 1 ) f _ 0 ( p _ 2 ) \\\\ & + \\delta ( q _ 1 - p _ 2 ) \\delta ( q _ 2 - p _ 1 ) e ^ { i ( t - s ) ( E _ { p _ 1 } - E _ { p _ 2 } ) } f _ 0 ( p _ 1 ) \\widetilde f _ 0 ( p _ 2 ) \\ . \\end{align*}"}
+{"id": "8642.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\infty } e ^ { - \\lambda s } p _ j ( s ) d s = \\frac { q } { \\lambda } \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } ( 1 - y ) y ^ { j - 1 } d y . \\end{align*}"}
+{"id": "1363.png", "formula": "\\begin{align*} \\bigl \\{ \\lim \\langle A x _ n , y _ n \\rangle & \\colon x _ n , y _ n \\in S _ H \\ \\forall n \\in \\N , \\ \\lim \\langle T x _ n , y _ n \\rangle = \\| T \\| \\bigr \\} \\\\ & = \\{ \\lim \\langle A x _ n , T x _ n \\rangle \\colon x _ n \\in S _ H \\ \\forall n \\in \\N , \\ \\lim \\| T x _ n \\| = \\| T \\| \\bigr \\} \\end{align*}"}
+{"id": "2910.png", "formula": "\\begin{align*} ( \\gamma \\ast \\mu ) ( f ) ( \\tau ) = \\mu ( f \\mid \\gamma ) ( \\gamma ^ { - 1 } \\ast \\tau ) . \\end{align*}"}
+{"id": "706.png", "formula": "\\begin{align*} \\Delta \\Psi = f \\ , , \\lim _ { x _ { 2 } \\to + \\infty } \\partial _ { 2 } \\Psi = - \\lim _ { x _ { 2 } \\to - \\infty } \\partial _ { 2 } \\Psi \\ , , | \\Psi | \\leq C ( | x _ { 2 } | + 1 ) \\end{align*}"}
+{"id": "4921.png", "formula": "\\begin{align*} \\Pr ( \\widetilde { X } _ t = k + 1 ) = \\sum _ { j = k + 1 } ^ n \\Pr ( X _ t = j ) \\overset { } { = } \\overline { F } ( k , t , \\mathbf { p } _ { n + 1 } ) , \\end{align*}"}
+{"id": "5336.png", "formula": "\\begin{align*} r '' _ { d - 1 } : = \\sqrt { \\frac { d - 1 } { d } } r ' _ d , \\mbox { a n d } \\gamma '' _ { d - 1 } : = \\left ( \\frac { 3 3 r } { \\sqrt { d - 1 } } \\right ) ^ { d - 1 } r \\gamma _ 1 \\log ^ { ( d - 1 ) / 2 } \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) . \\end{align*}"}
+{"id": "3219.png", "formula": "\\begin{align*} \\nu ( x ) = \\langle y _ 1 ' x \\Omega _ { \\rho } , \\Omega _ { \\rho } \\rangle _ { \\rho } x \\in M . \\end{align*}"}
+{"id": "5988.png", "formula": "\\begin{align*} \\overline { C } _ { X ^ { \\ast } } ( \\omega ^ { - 1 } , u ( - t ) ) & = \\overline { c } _ { X ^ { \\ast } } ( \\omega ^ { - 1 } , u ( - t ) ) \\\\ & = ( x ( \\omega ^ { - 1 } ) , x ( u ( - t ) ) ) _ \\R ( - x ( \\omega ^ { - 1 } ) x ( u ( - t ) ) , x ( \\omega ^ { - 1 } u ( - t ) ) ) _ \\R \\\\ & = 1 ; \\end{align*}"}
+{"id": "1071.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\left ( \\exp \\left | \\frac { e ^ { - t H ^ { \\beta } } g } { \\lambda } \\right | ^ p - 1 \\right ) \\ , d x & \\leq \\sum _ { k = 1 } ^ { \\infty } \\frac { { C ^ { p k } } \\| g \\| _ { L ^ { p k } } ^ { p k } } { k ! \\lambda ^ { p k } } = \\int _ { \\R ^ d } \\left ( \\exp \\left | \\frac { { C } g } { \\lambda } \\right | ^ p - 1 \\right ) \\ , d x . \\end{align*}"}
+{"id": "1493.png", "formula": "\\begin{align*} & \\tau ^ - _ { \\Lambda _ { t - \\cdot } } : = \\inf \\{ s \\in [ 0 , t + 1 ] \\ ! : ( R _ s - \\Lambda _ { t - s } ) ( R _ 0 - \\Lambda _ { t - } ) \\leq 0 \\} , \\\\ & \\tau _ { \\Lambda _ { t + \\cdot } } : = \\inf \\{ s \\geq 0 \\ ! : ( R _ s - \\Lambda _ { t + s } ) ( R _ 0 - \\Lambda _ t ) \\leq 0 \\} , \\\\ & \\tau ^ - _ { \\Lambda _ { t + \\cdot } } : = \\inf \\{ s \\geq 0 \\ ! : ( R _ s - \\Lambda _ { t + s } ) ( R _ 0 - \\Lambda _ { t - } ) \\leq 0 \\} , \\end{align*}"}
+{"id": "1109.png", "formula": "\\begin{align*} \\langle ( f \\ast - , - \\ast f ) , ( g \\ast - , - \\ast g ) \\rangle = ( h \\ast - , - \\ast h ) , \\end{align*}"}
+{"id": "6473.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ 0 ^ s & e ^ { i s ( | K | ^ 2 - | K _ 1 | ^ 2 + | K _ 2 | ^ 2 ) } e ^ { - i s ' ( | K _ 3 | ^ 2 - | K _ 4 | ^ 2 + | K _ 5 | ^ 2 ) } e ^ { - i s | K - K _ 1 + K _ 2 | ^ 2 } e ^ { i s ' | K _ 3 - K _ 4 + K _ 5 | ^ 2 } \\dd s ' \\dd s \\\\ & = \\int _ 0 ^ t \\int _ 0 ^ s e ^ { i s ( | K | ^ 2 - | K _ 1 | ^ 2 + | K _ 2 | ^ 2 - | K _ 6 | ^ 2 ) } e ^ { - i s ' ( | K _ 3 | ^ 2 - | K _ 4 | ^ 2 + | K _ 5 | ^ 2 - | K _ 6 | ^ 2 ) } \\dd s ' \\dd s \\end{align*}"}
+{"id": "5937.png", "formula": "\\begin{align*} \\widetilde { C } _ { X ^ { \\ast } } ( s _ 1 ( - 1 ) \\otimes I _ 2 , I _ { 1 } \\otimes g _ 2 ) & = \\widetilde { C } _ { X ^ { \\ast } } ( [ - 1 , 1 ] , [ 1 , I _ { 1 } \\otimes g _ 2 ] ) \\\\ & = \\nu ( 1 , 1 ) \\widetilde { c } _ { X ^ { \\ast } } ( 1 , I _ { 1 } \\otimes g _ 2 ) = 1 ; \\end{align*}"}
+{"id": "8346.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ \\infty \\alpha _ j \\tilde { u } _ j \\Big \\| _ { p ^ * , q , \\mu } ^ q \\geq \\sum _ { j = 1 } ^ \\infty | \\alpha _ j | ^ q \\int _ { \\delta _ { j + 1 } } ^ { \\delta _ j } t ^ { \\frac { q } { p ^ * } - 1 } ( \\tilde { u } _ j ) _ \\mu ^ * ( t ) ^ q \\dd t \\geq \\sum _ { j = 1 } ^ \\infty \\frac { \\| u _ j \\| _ { p ^ * , q , \\mu } ^ q } { ( 1 + \\varepsilon _ 1 ) ^ q } | \\alpha _ j | ^ q . \\end{align*}"}
+{"id": "8729.png", "formula": "\\begin{align*} \\bigg ( \\frac { 1 } { n } \\sum _ { i } \\| X _ i \\| _ { 2 } ^ { 2 } \\bigg ) & = \\frac { 1 } { n } ( \\| X _ 1 \\| _ { 2 } ^ { 2 } ) \\\\ & \\le \\frac { C } { n } \\times [ ( \\| X _ 1 - \\mu _ 1 \\| _ { 2 } ^ { 2 } ) + \\{ \\mu _ 1 ^ { \\top } ( X _ 1 - \\mu _ 1 ) \\} ] \\\\ & = O ( p N ^ { - 1 } ) , \\end{align*}"}
+{"id": "4162.png", "formula": "\\begin{align*} x \\cdot y ( s ) = \\sum _ { t \\in G } x ( t ) y ( t ^ { - 1 } s ) & = \\sum _ { t \\in G } x ( s t ^ { - 1 } ) y ( t ) , s \\in G \\\\ x ^ * ( s ) & = x ( s ^ { - 1 } ) ^ * , s \\in G . \\end{align*}"}
+{"id": "6992.png", "formula": "\\begin{align*} \\mu \\left ( \\tilde { \\textbf { Q } } \\right ) : = \\left \\{ \\mu ( q ) \\ \\left | \\ q \\in \\tilde { \\textbf { Q } } \\right . \\right \\} \\subseteq \\{ \\gamma _ 1 , \\ldots , \\gamma _ \\epsilon \\} . \\end{align*}"}
+{"id": "6302.png", "formula": "\\begin{align*} \\lambda _ n = t _ n \\lambda ^ + ( q ) , \\lambda _ n ' = t _ n ' \\lambda ^ + ( q ) , \\end{align*}"}
+{"id": "6166.png", "formula": "\\begin{align*} l _ R ( d y ) l _ R ( d z ) & = l _ R ( d ) l _ R ( d ) + l _ R ( d ) l _ R ( z ) + l _ R ( d ) l _ R ( y ) + l _ R ( y ) l _ R ( z ) \\\\ & = l _ R ( d ) l _ R ( d ) + l _ R ( d ) [ l _ R ( z ) + l _ R ( y ) ] + l _ R ( y ) l _ R ( z ) \\\\ & = l _ R ( d ) l _ R ( d ) + l _ R ( d ) l _ R ( y z ) + l _ R ( y ) l _ R ( z ) \\\\ & = l _ R ( d ) l _ R ( d ) + l _ R ( d ) l _ R ( a b ) + l _ R ( a ) l _ R ( b ) \\\\ & = l _ R ( d ) l _ R ( d ) + l _ R ( d ) l _ R ( a ) + l _ R ( d ) l _ R ( b ) + l _ R ( a ) l _ R ( b ) \\\\ & = l _ R ( d a ) l _ R ( d b ) , \\end{align*}"}
+{"id": "5612.png", "formula": "\\begin{align*} { \\rm I I } _ { x } = \\sum _ { s \\in G } \\mu ( s ) \\frac { d s \\nu } { d \\nu } ( \\pi ( x ) ) \\sum _ { \\xi _ { n } ^ { x } \\in L _ { x } \\backslash G } \\log \\left ( P _ { \\mu , x } ^ { n - 1 } ( L _ { x } s , \\xi _ { n } ^ { x } ) \\right ) P _ { \\mu , x } ^ { n - 1 } ( L _ { x } s , \\xi _ { n } ^ { x } ) \\end{align*}"}
+{"id": "584.png", "formula": "\\begin{align*} f _ { ( x _ 1 , x _ 2 ) , v } ( \\gamma ^ { a _ { \\frac { \\alpha } { 4 \\sqrt { x _ 2 t _ 2 } } } ^ r } ) = f _ { ( t _ 1 , t _ 2 ) , w } ( \\gamma ^ { a _ { \\frac { \\alpha } { 4 \\sqrt { x _ 2 t _ 2 } } } ^ r } ) \\forall \\alpha > 0 , r \\in \\R , \\end{align*}"}
+{"id": "7107.png", "formula": "\\begin{align*} \\det \\Phi _ t ( x , y ) = \\det \\left ( \\begin{array} { c c } & - \\\\ t & ( 1 - t ) \\end{array} \\right ) = 1 \\end{align*}"}
+{"id": "235.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ 2 z ^ k = \\frac { - n ^ 2 z ^ { n + 3 } + ( 2 n ^ 2 + 2 n - 1 ) z ^ { n + 2 } - ( n ^ 2 + 2 n + 1 ) z ^ { n + 1 } + z ^ 2 + z } { ( 1 - z ) ^ 3 } , \\end{align*}"}
+{"id": "5590.png", "formula": "\\begin{align*} \\int _ { X } f ( x ) d \\lambda ( x ) & = \\int _ { X _ { 0 } } \\mathcal { E } _ { s } f ( e , e , x _ { 0 } ) d \\lambda ( x _ { 0 } ) = \\int _ { X ' } \\phi d \\lambda ' \\\\ & = \\int _ { X ' } \\phi \\left ( x ' \\right ) d u . \\lambda ' ( x ' ) = \\int _ { X _ { 0 } } \\mathcal { E } _ { s } f ( u , e , x _ { 0 } ) d \\lambda ( x _ { 0 } ) = \\int _ { X } f ( x ) d u . \\lambda ( x ) , \\end{align*}"}
+{"id": "371.png", "formula": "\\begin{align*} \\mbox { i f $ \\tau = \\{ i \\} $ a n d $ \\mathfrak { u } = \\{ j \\} $ t h e n $ \\eta ( \\sigma ) ^ { \\ast } ( y _ { i , j } ) = \\displaystyle \\frac { c ^ { \\sigma } _ { i } } { c ^ { \\{ i \\} } _ { i } } x _ { i , j } $ } . \\end{align*}"}
+{"id": "4351.png", "formula": "\\begin{align*} \\mathcal { A } \\mathcal { = } \\left \\{ x \\in X \\mid 0 \\in \\sum _ { i = 1 } ^ { m } \\partial f _ { i } \\left ( x \\right ) \\right \\} = \\arg \\min \\sum _ { i = 1 } ^ { m } f _ { i } . \\end{align*}"}
+{"id": "6609.png", "formula": "\\begin{align*} \\widetilde { A } : = A / K _ 1 A . \\end{align*}"}
+{"id": "3927.png", "formula": "\\begin{align*} \\inf _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\sup _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } F ( \\pi , \\lambda ) = \\sup _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } \\inf _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } F ( \\pi , \\lambda ) . \\end{align*}"}
+{"id": "5380.png", "formula": "\\begin{align*} V _ i ( E _ i x _ { i , k + 1 } ) - V _ i ( E _ i x _ { i , k } ) \\leq \\| u _ { i , k } \\| ^ 2 - \\| y _ { i , k } \\| ^ 2 , i = 1 , 2 . \\end{align*}"}
+{"id": "8203.png", "formula": "\\begin{align*} \\underset { t \\to \\infty } { \\lim } P ^ \\omega ( \\mathcal { R } ( \\alpha t ) \\cap \\widehat { \\Phi } _ { \\alpha t } ^ \\omega = \\emptyset \\mid S _ { \\alpha t } ) = \\underset { t \\to \\infty } { \\lim } P ^ \\omega ( E _ t ^ c \\mid S _ { \\alpha t } ) = 0 . \\end{align*}"}
+{"id": "3894.png", "formula": "\\begin{align*} \\begin{aligned} \\boldsymbol { K } _ \\ell \\left ( \\mu _ \\ell , \\gamma ^ { \\prime \\prime } _ { \\ell , 3 } \\right ) & \\leq \\lambda \\boldsymbol { K } _ \\ell ( \\mu _ \\ell , \\gamma _ \\ell ) + ( 1 - \\lambda ) \\boldsymbol { K } _ \\ell \\left ( \\mu _ \\ell , \\gamma _ \\ell ^ \\prime \\right ) \\leq \\lambda \\delta _ \\ell + ( 1 - \\lambda ) \\delta _ \\ell ^ \\prime . \\end{aligned} \\end{align*}"}
+{"id": "1632.png", "formula": "\\begin{align*} \\nabla _ { \\xi _ i } \\ , \\xi _ j = 0 , 1 \\le i , j \\le p ; \\end{align*}"}
+{"id": "4303.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { T } ^ d \\times [ 0 , T ) } \\bar { f } \\left ( \\frac { \\partial \\phi } { \\partial t } + u \\cdot \\nabla \\phi \\right ) \\ ; d x d t \\\\ & = \\lim _ { i \\to \\infty } \\int _ { \\mathbb { T } ^ d \\times [ 0 , T ) } f ^ { \\nu _ i } \\left ( \\frac { \\partial \\phi } { \\partial t } + u \\cdot \\nabla \\phi + \\nu _ i \\Delta \\phi \\right ) \\ ; d x d t \\\\ & = - \\int _ { \\mathbb { T } ^ d } f _ 0 \\phi _ 0 \\ ; d x , \\end{align*}"}
+{"id": "7345.png", "formula": "\\begin{align*} f \\left ( \\sum _ { l = 1 } ^ n \\| w _ l - w \\| _ \\infty \\right ) & \\leq f \\left ( \\sum _ { l = 1 } ^ n k \\| x _ l - x \\| _ \\infty + k \\| y _ l - y \\| _ \\infty + k \\| z _ l - z \\| _ \\infty \\right ) \\\\ & \\leq k f \\left ( \\sum _ { l = 1 } ^ n \\| x _ l - x \\| _ \\infty \\right ) + k f \\left ( \\sum _ { l = 1 } ^ n \\| y _ l - y \\| _ \\infty \\right ) \\\\ & + k f \\left ( \\sum _ { l = 1 } ^ n \\| z _ l - z \\| _ \\infty \\right ) , \\end{align*}"}
+{"id": "297.png", "formula": "\\begin{align*} \\prod _ { \\substack { l , m , n \\geq 1 \\\\ l , m \\leq n ; \\ , \\gcd ( l , m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - x ^ l y ^ m z ^ n } \\right ) ^ { \\frac { l m ^ 2 } { n ^ 4 } } \\end{align*}"}
+{"id": "463.png", "formula": "\\begin{align*} Y : = \\left \\{ y _ \\ast \\in [ 2 , 3 ] \\ , \\big | \\ \\exists \\ , y \\in ( s ( \\tilde y _ \\ast ) , y _ \\ast ) \\ \\omega ( y ; \\tilde y _ \\ast ) = \\frac 1 3 \\ \\ \\tilde y _ \\ast \\in [ y _ \\ast , 3 ] \\right \\} . \\end{align*}"}
+{"id": "547.png", "formula": "\\begin{align*} S _ { t } ( t ) : = \\left ( \\begin{array} { c c } a ^ { \\prime } ( t ) & 0 \\\\ 0 & 0 \\end{array} \\right ) \\left ( S Q - Q ^ { * } S \\right ) ( t ) : = \\left ( \\begin{array} { c c } 0 & a ( t ) - q ( t ) \\\\ q ( t ) - a ( t ) & 0 \\end{array} \\right ) , \\end{align*}"}
+{"id": "7352.png", "formula": "\\begin{align*} \\Pr [ \\hat { M } = M ] \\geq \\frac { \\Pr [ d ( \\hat { M } ' , M ) \\leq n ^ { 1 - \\beta } ] } { | \\{ m : d ( m , \\hat { M } ' ) \\leq n ^ { 1 - \\beta } \\} | } . \\end{align*}"}
+{"id": "1917.png", "formula": "\\begin{align*} \\begin{aligned} | \\beta \\sum \\limits _ { i = n } ^ { k - 1 } \\langle r ^ { i + 1 } , r ^ k \\rangle | \\leq \\frac { \\beta } { 2 } \\sum \\limits _ { i = n } ^ { k - 1 } ( \\| r ^ { i + 1 } \\| ^ 2 + \\| r ^ k \\| ^ 2 ) \\leq \\beta \\sum \\limits _ { i = n } ^ { k } \\| r ^ { i } \\| ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "2623.png", "formula": "\\begin{align*} f _ { j , \\flat } & = \\sum _ { m \\in \\mathbb { Z } ^ 2 } \\widetilde { \\eta } _ m f _ { j , m , \\flat } , \\\\ f _ { j , \\sharp } & = \\sum _ { m \\in \\mathbb { Z } ^ 2 } \\ ! \\widetilde { \\eta } _ m f _ { j , m , \\sharp } , \\\\ f _ { j , } & = \\sum _ { m \\in \\mathbb { Z } ^ 2 } \\widetilde { \\eta } _ m \\left [ \\eta _ m \\left ( \\psi ^ { ( j ) } \\ast _ j f _ j \\right ) - \\psi ^ { ( j ) } \\ast _ j \\left ( \\eta _ m f _ j \\right ) \\right ] . \\end{align*}"}
+{"id": "2840.png", "formula": "\\begin{align*} L 1 _ { N } ^ { N + 2 + k ' } + c _ 0 L 1 _ { N } ^ { N + 1 + k ' } + \\dots + c _ { N + 1 } L 1 _ { N } ^ { k ' } = O , k ' \\in \\mathbb { Z } _ + \\end{align*}"}
+{"id": "1482.png", "formula": "\\begin{align*} \\Big | f _ \\epsilon ( V ) - f _ 0 ( V ) \\Big | _ { \\frac { 2 n } { n + 2 } } = O \\bigg ( \\epsilon \\ln \\Big | \\ln \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big | \\bigg ) . \\end{align*}"}
+{"id": "4850.png", "formula": "\\begin{align*} \\begin{array} { r l } ( \\prod \\limits _ { i = 1 } ^ m a ( i ) \\ast g _ { w ( l ) } ( t ( i ) ) ) \\ast a ( m + 1 ) = & ( \\prod \\limits _ { i = 1 } ^ m a ( i ) \\ast ( \\prod \\limits _ { p = 1 } ^ r c _ { t ( i ) } ( p ) \\ast h _ l ( z _ { t ( i ) } ( p ) ) ) \\ast c _ { t ( i ) } ( r + 1 ) ) ) \\ast a ( m + 1 ) \\\\ = & ( \\prod \\limits _ { i = 1 } ^ u d ( i ) \\ast h _ l ( q ( i ) ) ) \\ast d ( u + 1 ) . \\end{array} \\end{align*}"}
+{"id": "3228.png", "formula": "\\begin{align*} F _ { 0 } & = \\{ P \\in F ~ | ~ \\} \\\\ F _ { i j k } & = \\{ P \\in F ~ | ~ p _ { i } , p _ { j } , p _ { k } \\} \\\\ F _ { 1 2 3 4 } & = \\{ P \\in F ~ | ~ \\} \\end{align*}"}
+{"id": "450.png", "formula": "\\begin{align*} \\ddot \\l ( t ) \\l ( t ) ^ 2 & = \\delta , \\ \\ \\delta \\in \\mathbb R , \\\\ \\l ( 0 ) = \\l _ 0 , \\ \\ \\dot \\l ( 0 ) & = \\l _ 1 \\end{align*}"}
+{"id": "5852.png", "formula": "\\begin{align*} & \\chi _ I ( H _ \\omega ) u = \\left ( \\int _ B P _ \\omega ( E ) d \\mu _ \\omega ( E ) \\right ) u , \\\\ & f ( H _ \\omega ) \\chi _ I ( H _ \\omega ) u = \\left ( \\int _ { B } f ( E ) P _ \\omega ( E ) d \\mu _ \\omega ( E ) \\right ) u , \\end{align*}"}
+{"id": "7813.png", "formula": "\\begin{align*} ( \\textsf { E } \\vert g _ { i j } g _ { i j } ^ { \\prime } \\vert ^ { p } ) ^ { 1 / p } = ( \\textsf { E } ( \\textsf { E } _ { ( g _ { i j } ) } \\vert g _ { i j } g _ { i j } ^ { \\prime } \\vert ^ { p } ) ) ^ { 1 / p } \\asymp p , \\end{align*}"}
+{"id": "4214.png", "formula": "\\begin{align*} H ( a ) H ( \\tilde { b } ) & = H ( a ) T ( \\tilde { f } ) H ( \\tilde { b } ) + H ( a ) T ( \\tilde { g } ) H ( \\tilde { b } ) \\\\ & = \\Big ( T ( a ) H ( \\tilde { f } ) - H ( a f ) \\Big ) H ( \\tilde { b } ) + H ( a ) \\Big ( H ( \\tilde { g } ) T ( b ) - H ( \\tilde { g } \\tilde { b } ) \\Big ) . \\end{align*}"}
+{"id": "7621.png", "formula": "\\begin{align*} & 2 c _ 2 = c _ 1 ^ 2 + \\delta ( 4 - c _ 1 ^ 2 ) , \\\\ \\ ; \\ ; & 4 c _ 3 = c _ 1 ^ 3 + 2 ( 4 - c _ 1 ^ 2 ) c _ 1 \\delta - ( 4 - c _ 1 ^ 2 ) c _ 1 \\delta ^ 2 + 2 ( 4 - c 1 ^ 2 ) ( 1 - | \\delta | ^ 2 ) \\eta , \\\\ & 8 c _ 4 = c _ 1 ^ 4 + ( 4 - c _ 1 ^ 2 ) \\delta ( c _ 1 ^ 2 ( \\delta ^ 2 - 3 \\delta + 3 ) + 4 \\delta ) - 4 ( 4 - c _ 1 ^ 2 ) ( 1 - | \\delta | ^ 2 ) ( c _ 1 ( \\delta - 1 ) \\eta \\\\ & \\quad + \\overline { \\delta } \\eta ^ 2 - ( 1 - | \\eta | ^ 2 ) \\rho ) \\end{align*}"}
+{"id": "5482.png", "formula": "\\begin{align*} \\sum _ { i + j = n } \\pi _ i \\circ \\pi _ j = 0 , f o r n \\geq 0 . \\end{align*}"}
+{"id": "1051.png", "formula": "\\begin{align*} N \\coloneqq d \\cdot \\min \\left \\{ \\prod _ { i = 1 } ^ n ( h _ i + 1 ) , \\binom { n + h } { n } \\right \\} \\end{align*}"}
+{"id": "5233.png", "formula": "\\begin{align*} R _ 2 = \\left ( \\begin{array} { l l l } x = 0 : & 0 & a _ 1 \\\\ x = 1 : & 0 & a _ 2 \\\\ x = \\infty : & a _ 3 & a _ 4 \\end{array} \\right ) = \\left ( \\begin{array} { l l c } x = 0 : & 0 & 1 - c \\\\ x = 1 : & 0 & c - a - b \\\\ \\xi = \\infty : & a & b \\end{array} \\right ) = R _ { a b c } , \\end{align*}"}
+{"id": "6191.png", "formula": "\\begin{align*} \\mathcal F ( s , z ) = ( 1 + s ) ^ { n - 1 } \\bigg ( 1 + \\frac { | z | ^ 2 } { 2 ( 1 + s ) ^ 2 } \\Biggr ) + O \\big ( | z | ^ 4 \\big ) = 1 + ( n - 1 ) s + \\frac { ( n - 1 ) ( n - 2 ) } { 2 } s ^ 2 + \\frac { | z | ^ 2 } { 2 } + O \\big ( | s | ^ 3 + | z | ^ 3 \\big ) , \\end{align*}"}
+{"id": "4024.png", "formula": "\\begin{align*} g ( P \\otimes \\lbrace \\alpha \\rbrace ) = ( g P ) \\otimes \\lbrace g ( \\alpha ) \\rbrace \\ , \\ \\ , \\ P \\lbrace \\alpha \\rbrace \\in \\mathbb { B } _ k , \\ , g \\in \\Gamma . \\end{align*}"}
+{"id": "2039.png", "formula": "\\begin{align*} \\det ( u _ { j \\bar k } ( 0 ) ) = a u _ { n \\bar n } ( 0 ) + b , \\end{align*}"}
+{"id": "6483.png", "formula": "\\begin{align*} Z = \\left ( \\begin{array} { c c } A & 0 \\\\ 0 & A \\end{array} \\right ) , T = \\left ( \\begin{array} { c c } B & 0 \\\\ 0 & B \\end{array} \\right ) , X = \\left ( \\begin{array} { c c } 0 & C \\\\ C & 0 \\end{array} \\right ) , Y = \\left ( \\begin{array} { c c } 0 & i C \\\\ - i C & 0 \\end{array} \\right ) , \\end{align*}"}
+{"id": "1451.png", "formula": "\\begin{align*} \\| u _ m \\| ^ 2 = \\int _ \\Omega f _ \\epsilon ^ { ' } ( V _ m ) u _ m ^ 2 d x + \\int _ \\Omega f _ \\epsilon ^ { ' } ( V _ m ) ( h _ m + \\omega _ m ) u _ m d x . \\end{align*}"}
+{"id": "8434.png", "formula": "\\begin{align*} R _ { p , p ^ { \\prime } } ( x ) = \\frac { 1 } { 4 \\left ( 1 + \\frac { 1 } { \\pi p } \\right ) } + \\frac { 1 } { 2 } \\sqrt { \\frac { \\pi } { x } } \\ , \\zeta _ { p ^ { \\prime } } \\left ( \\frac { 1 } { 2 } \\right ) + \\frac { \\pi ^ { 2 } } { 6 x } \\ , \\frac { 1 + \\frac { 3 } { \\pi p } ( 1 + \\frac { 1 } { \\pi p } ) } { \\left ( 1 + \\frac { 1 } { \\pi p ^ { \\prime } } \\right ) \\left ( 1 + \\frac { 1 } { \\pi p } \\right ) ^ { 2 } } , \\end{align*}"}
+{"id": "8062.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j = 1 } ^ { m } X _ { j } ^ { * } \\left ( \\left | X u \\right | ^ { p - 2 } X _ { j } u \\right ) & = - \\lambda | u | ^ { p - 2 } u & & \\Omega , \\\\ u & = 0 & & \\partial \\Omega , \\end{aligned} \\end{align*}"}
+{"id": "1233.png", "formula": "\\begin{align*} d ( m S , \\N ^ 2 \\setminus m S ) = m d ( S , ( 1 / m ) \\N ^ 2 \\setminus S ) \\leq m ( 1 - f ( 1 / m ) ) . \\end{align*}"}
+{"id": "5994.png", "formula": "\\begin{align*} d \\overline { \\Pi } _ { \\psi } ( e ^ - ) f ( [ \\epsilon , x ] ) & = \\frac { d } { d t } \\Big \\{ \\Pi _ { \\psi } ( \\exp ( t e ^ - ) ) f ( [ \\epsilon , x ] ) \\Big \\} \\Big | _ { t = 0 } \\\\ & = \\frac { \\epsilon i } { 4 \\pi } \\frac { d ^ 2 } { d x ^ 2 } f ( [ \\epsilon , x ] ) . \\end{align*}"}
+{"id": "3003.png", "formula": "\\begin{align*} & \\Phi _ j ( f _ 0 , f _ 1 , f _ 2 , \\ldots , f _ { N - 1 } ) \\\\ & = \\exp _ K ( - t _ 1 f _ 0 ) \\ldots \\exp _ K ( - ( t _ j - t _ { j - 1 } ) f _ { j - 1 } ) f _ j \\exp _ K ( t _ j - t _ { j - 1 } ) f _ { j - 1 } ) \\ldots \\exp _ K t _ 1 f _ 0 . \\end{align*}"}
+{"id": "4482.png", "formula": "\\begin{align*} \\mathbf { P } ( U ) : = \\mathbf { p } ( u _ 1 ) + \\mathbf { p } ( u _ 2 ) + \\mathbf { p } ( u _ 3 ) , \\end{align*}"}
+{"id": "2507.png", "formula": "\\begin{align*} A y = r , \\end{align*}"}
+{"id": "765.png", "formula": "\\begin{align*} [ T _ k ] _ i ^ j ( A _ 1 , A _ 2 , \\cdots , A _ k ) = \\dfrac { 1 } { k ! } \\delta _ { i i _ 1 i _ 2 \\cdots i _ k } ^ { j j _ 1 j _ 2 \\cdots j _ k } ( A _ 1 ) ^ { i _ 1 } _ { j _ 1 } ( A _ 2 ) ^ { i _ 2 } _ { j _ 2 } \\cdots ( A _ k ) ^ { i _ k } _ { j _ k } . \\end{align*}"}
+{"id": "1410.png", "formula": "\\begin{align*} P _ { x \\to y } w = \\sinh ( \\| v \\| ) x + \\cosh ( \\| v \\| ) w , P _ { x \\to y } u = u . \\end{align*}"}
+{"id": "2430.png", "formula": "\\begin{align*} y ( q x ) = m ( x ) y ( x ) \\end{align*}"}
+{"id": "8518.png", "formula": "\\begin{align*} \\zeta _ { 0 , p ^ { \\prime } } ( s , c ) : = \\sum _ { m , n \\neq 0 } \\frac { p ^ { \\prime 2 } + \\lambda _ { n } ^ { \\prime 2 } } { p ^ { \\prime } \\left ( p ^ { \\prime } + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { \\prime 2 } } \\ , \\frac { 1 } { \\left ( \\left ( m - \\frac { 1 } { 2 } \\right ) ^ { 2 } + c \\ , \\lambda _ { n } ^ { \\prime 2 } \\right ) ^ { s } } , \\ , \\ , \\ , \\ , ( s ) > 1 . \\end{align*}"}
+{"id": "4498.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\omega , \\mathbf { c } } : = \\{ \\Psi \\in \\mathcal { H } ^ 1 \\backslash \\{ ( \\mathbf { 0 } , \\mathbf { 0 } , \\mathbf { 0 } ) \\} \\ | \\ S _ { \\omega , \\mathbf { c } } ( \\Psi ) = \\mu _ { \\omega , \\mathbf { c } } , \\ K _ { \\omega , \\mathbf { c } } ( \\Psi ) = 0 \\} . \\end{align*}"}
+{"id": "1264.png", "formula": "\\begin{align*} | S ' | \\leq | N b d ( S ' ) | \\leq | N b d ( S ) | - 1 < | S | - 1 = | S ' | . \\end{align*}"}
+{"id": "1605.png", "formula": "\\begin{align*} \\langle S _ { \\xi \\chi } f , g \\rangle & = \\langle \\int _ { \\Theta } v ( w ) s ( w ) \\pi _ { G ( w ) } \\xi _ { w } ^ { \\ast } \\chi _ { w } \\pi _ { F ( w ) } f d \\mu ( w ) , g \\rangle \\\\ & = \\int _ { \\Theta } v ( w ) s ( w ) \\langle f , \\pi _ { F ( w ) } \\chi _ { w } ^ { \\ast } \\xi _ { w } \\pi _ { G ( w ) } g \\rangle d \\mu ( w ) \\\\ & = \\langle f , \\int _ { \\Theta } v ( w ) s ( w ) \\pi _ { F ( w ) } \\chi _ { w } ^ { \\ast } \\xi _ { w } \\pi _ { G ( w ) } g d \\mu ( w ) \\rangle \\\\ & = \\langle f , S _ { \\chi \\xi } g \\rangle . \\end{align*}"}
+{"id": "44.png", "formula": "\\begin{align*} \\mathbb { Q } ^ \\times \\cap ( \\widehat { \\mathbb { Z } } ^ p ) ^ \\times \\ni \\mu ( a ) = \\mu ( b ) ^ { - 1 } \\in \\mathbb { Z } _ p ^ \\times , \\end{align*}"}
+{"id": "8316.png", "formula": "\\begin{align*} M ( g a , g b ) = M ( g a _ 1 , g a _ 2 , g b _ 1 , g b _ 2 ) = M ( a , b ) \\ , . \\end{align*}"}
+{"id": "505.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ D c _ k L p _ { k , \\sigma ( j ) } ( y _ j ) = f ( y _ j ) , j = 1 , \\ldots , D . \\end{align*}"}
+{"id": "1255.png", "formula": "\\begin{align*} & \\sum _ { j = 0 } ^ \\infty \\int _ { T ^ { - m } [ b _ { \\ell + 1 } , 1 ] \\cap [ b _ { j + 1 } , b _ j ] } \\frac { | u ' ( x ) | } { ( T ^ m ) ' x } \\ , d x \\\\ & = \\sum _ { j = p } ^ \\infty \\int _ { T ^ { - m } [ b _ { \\ell + 1 } , 1 ] \\cap [ b _ { j + 1 } , b _ j ] } \\frac { | u ' ( x ) | } { ( T ^ m ) ' x } \\ , d x + \\int _ { T ^ { - m } [ b _ { \\ell + 1 } , 1 ] \\cap [ b _ { p } , \\frac 1 2 ] } \\frac { | u ' ( x ) | } { ( T ^ m ) ' x } \\ , d x \\end{align*}"}
+{"id": "5172.png", "formula": "\\begin{align*} \\gamma _ 0 : = \\gamma ( 0 , v ) , \\gamma _ h : = \\gamma ( h , v + h ) , T _ 0 : = T ( 0 , v ) , T _ h : = T ( h , v + h ) . \\end{align*}"}
+{"id": "1083.png", "formula": "\\begin{align*} & t ^ { \\sigma } \\left \\| \\int _ 0 ^ t e ^ { - ( t - \\tau ) H ^ { \\beta } } ( f ( u ) - f ( v ) ) \\ , d \\tau \\right \\| _ { L ^ { a } } \\\\ & \\leq C \\sum _ { k = 0 } ^ { \\infty } \\frac { \\lambda ^ k } { k ! } t ^ { \\sigma } \\int _ 0 ^ t ( t - \\tau ) ^ { - \\frac d { 2 \\beta } ( \\frac 1 r - \\frac 1 a ) } \\| ( u - v ) ( | u | ^ { p k + m - 1 } + | v | ^ { p k + m - 1 } ) \\| _ { L ^ r } \\ , d \\tau . \\end{align*}"}
+{"id": "7310.png", "formula": "\\begin{align*} A \\prod _ { i = 1 } ^ n x _ i ^ { \\alpha ' _ i } = B \\prod _ { i = 1 } ^ n x _ i ^ { \\gamma ' _ i } , \\end{align*}"}
+{"id": "6159.png", "formula": "\\begin{align*} x \\ast _ { n , n } x & = ( a _ { 1 } \\ast a _ { 2 } \\ast \\cdots \\ast a _ { n } ) \\ast ( a _ { 1 } \\ast a _ { 2 } \\ast \\cdots \\ast a _ { n } ) \\\\ & = ( a _ { 1 } \\ast a _ { 1 } ) \\ast ( a _ 2 \\ast a _ 2 ) \\ast \\cdots ( a _ n \\ast a _ { n } ) = ( \\top _ 1 \\ast a _ 1 ) \\ast ( \\top _ 1 \\ast a _ 2 ) \\ast \\cdots \\ast ( \\top _ 1 \\ast a _ n ) \\\\ & = ( \\top _ n ) \\ast ( a _ { 1 } \\ast a _ { 2 } \\ast \\cdots \\ast a _ { n } ) ; \\end{align*}"}
+{"id": "2036.png", "formula": "\\begin{align*} u = h \\rho , \\end{align*}"}
+{"id": "1068.png", "formula": "\\begin{align*} u ( t ) = e ^ { - t H ^ { \\beta } } u _ 0 + \\int _ 0 ^ t e ^ { - ( t - s ) H ^ { \\beta } } \\ , f ( u ( s ) ) \\ , d s . \\end{align*}"}
+{"id": "9260.png", "formula": "\\begin{align*} E _ { \\varepsilon } & = \\big \\{ ( t , x , z ) \\in E : t - | x - z | < \\varepsilon \\sqrt { x z } \\ ; \\ ; \\textrm { o r } \\ ; \\ ; x + z - t < \\varepsilon \\sqrt { x z } \\big \\} , \\\\ F _ { \\varepsilon } & = \\big \\{ ( t , x , z ) \\in F : t - ( x + z ) < \\varepsilon \\sqrt { x z } \\ ; \\ ; \\textrm { o r } \\ ; \\ ; t > \\varepsilon ^ { - 1 } \\sqrt { x z } \\big \\} . \\end{align*}"}
+{"id": "8945.png", "formula": "\\begin{align*} \\phi = \\Tilde { \\phi } _ { e } + C _ { p _ 1 } h ^ { p _ 1 } + C _ { p _ 2 } h ^ { p _ 2 } . \\end{align*}"}
+{"id": "201.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c ) = 1 \\\\ a , b , c \\geq 1 } } \\left ( \\frac { 1 } { 1 - x ^ a y ^ b z ^ c } \\right ) ^ { \\frac { 1 } { a ^ s b ^ t c ^ u } } = \\exp \\left \\{ \\left ( \\sum _ { i = 1 } ^ { \\infty } \\frac { x ^ i } { i ^ s } \\right ) \\left ( \\sum _ { j = 1 } ^ { \\infty } \\frac { y ^ j } { j ^ t } \\right ) \\left ( \\sum _ { k = 1 } ^ { \\infty } \\frac { z ^ k } { k ^ u } \\right ) \\right \\} . \\end{align*}"}
+{"id": "2608.png", "formula": "\\begin{align*} v _ { h _ 0 } ( y ) : = \\frac { 1 } { M h _ 0 } v ( A _ { h _ 0 } ^ { - 1 } y ) . \\end{align*}"}
+{"id": "8884.png", "formula": "\\begin{align*} H _ r ( n , p ) & \\ge ( p / q ) ^ n e ^ { - 2 / p } H ( T _ r ( n , k ) ) ( 1 - o ( 1 ) ) \\\\ & \\ge ( p / q ) ^ n \\frac { 1 } { 2 } \\left ( \\frac { k - r + 1 } { k } \\right ) ^ n ( n - 1 ) ! ( k - r + 1 ) ^ { - \\Theta ( \\sqrt { n } ) } \\\\ & = E ( n , p ) \\left ( \\frac { k - r + 1 } { q k } \\right ) ^ n ( k - r + 1 ) ^ { - \\Theta ( \\sqrt { n } ) } \\\\ & = E ( n , p ) \\left ( \\frac { ( k - r + 1 ) ! k ^ { r - 1 } } { k ! } \\right ) ^ n ( k - r + 1 ) ^ { - \\Theta ( \\sqrt { n } ) } \\\\ & = E ( n , p ) \\left ( \\frac { ( k - r + 1 ) ! k ^ { r - 1 } } { k ! } \\right ) ^ { n - o ( n ) } \\ ; . \\end{align*}"}
+{"id": "6749.png", "formula": "\\begin{align*} \\lim _ { K \\to \\infty } \\overline { d } _ { ( \\ell _ { i } ) } ( \\{ n \\in \\N : x _ K ( n ) \\neq x ( n ) \\} ) = 0 . \\end{align*}"}
+{"id": "8305.png", "formula": "\\begin{align*} M = O ( \\log q / \\log \\log q ) \\end{align*}"}
+{"id": "4731.png", "formula": "\\begin{align*} Q = \\begin{pmatrix} X & Y \\\\ - Y & X \\end{pmatrix} , \\end{align*}"}
+{"id": "7219.png", "formula": "\\begin{align*} \\lim _ { h \\to + \\infty } { ( _ h ) } = 0 , \\end{align*}"}
+{"id": "2194.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c } u _ { t t } ( { x } , t ) = \\nabla ( a ( { x } ) \\nabla u ( { x } , t ) ) + f ( { x } , t ) , & ( x , t ) \\in \\Omega \\times ( 0 , T ] , \\\\ u = 0 , & ( { x } , t ) \\in \\partial \\Omega \\times ( 0 , T ] , \\\\ u ( { x } , 0 ) = \\psi _ 0 , ~ u _ t ( { x } , 0 ) = \\psi _ 1 , & x \\in \\Omega \\subset \\mathbb { R } ^ { d } . \\end{array} \\right . \\ , \\end{align*}"}
+{"id": "3645.png", "formula": "\\begin{align*} \\mathcal { G } _ w \\subset \\bigcup _ { j = 1 } ^ { K } \\mathcal { S } _ j ( \\mathcal { U } ) . \\end{align*}"}
+{"id": "8800.png", "formula": "\\begin{align*} & 2 c _ 2 = ( 4 t + 3 u + 3 ) c _ 3 - ( u + 2 t + 5 ) b _ 3 \\end{align*}"}
+{"id": "6832.png", "formula": "\\begin{align*} \\beta ^ { ( N ) } _ n = \\frac { ( b _ 1 q , \\ldots , b _ N q ) _ n } { ( b _ 1 , \\ldots , b _ N ) _ n } \\beta _ n . \\end{align*}"}
+{"id": "8365.png", "formula": "\\begin{align*} L = B C _ E ^ { - 1 } B ^ T C _ V , B = S - T , \\end{align*}"}
+{"id": "13.png", "formula": "\\begin{align*} \\sum _ { p _ i \\in \\Z } \\int _ { [ 0 , N ] ^ n } \\chi _ { \\frac { \\vartheta _ i } { \\| N \\bar { q } + v '' \\| ^ { w _ i } } } ' ( N p _ i + v _ { m + i } + \\langle \\bar { u } _ i , N \\bar { q } + v '' \\rangle ) \\ , d \\bar { u } _ i = 2 N ^ { n - 1 } \\vartheta _ i \\| N \\bar { q } + v '' \\| ^ { - w _ i } . \\end{align*}"}
+{"id": "3085.png", "formula": "\\begin{align*} \\mathcal V _ \\infty ^ { p - 1 } = \\mathcal V _ \\infty ^ { k _ 2 } \\cdot \\mathcal W _ \\infty ^ { k _ 3 } = \\mathcal V _ \\infty ^ { k _ 2 } \\cdot \\mathcal U _ \\infty ^ { k _ 3 } \\end{align*}"}
+{"id": "1924.png", "formula": "\\begin{align*} \\begin{aligned} & | \\langle \\lambda ^ { k + 1 } , \\zeta ^ * - \\zeta ^ { k + 1 } \\rangle + \\langle D _ u \\theta ( u ^ { k + 1 } ) , u ^ * - u ^ { k + 1 } \\rangle _ \\mathcal { U } | \\\\ & \\leq \\beta \\sum \\limits _ { i = n } ^ { k + 1 } \\| r ^ { i } \\| ^ 2 + | \\langle \\lambda ^ n , \\zeta ^ * - \\zeta ^ { k + 1 } \\rangle + \\langle D _ u \\theta ( u ^ n ) , u ^ * - u ^ { k + 1 } \\rangle _ \\mathcal { U } | . \\end{aligned} \\end{align*}"}
+{"id": "5028.png", "formula": "\\begin{align*} { \\overline g = ( \\alpha _ 1 / r , \\dots , \\alpha _ N / r ) } { \\overline g = \\frac { 1 } { r } ( \\alpha _ 1 , \\dots , \\alpha _ N ) } \\end{align*}"}
+{"id": "782.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ n } x _ l \\mathrm { d } A = 0 , \\ \\ l = 1 , 2 , \\cdots , n + 1 . \\end{align*}"}
+{"id": "1879.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\langle \\tilde { \\lambda } , \\bar { \\zeta } _ 0 \\rangle > 0 , \\\\ & \\langle \\lambda ^ * , \\bar { \\zeta } _ 0 \\rangle > 0 . \\end{aligned} \\right . \\quad ( \\mbox { C o n s i s t e n c y \\ C o n d i t i o n } ) \\end{align*}"}
+{"id": "7999.png", "formula": "\\begin{align*} v = u . \\end{align*}"}
+{"id": "1877.png", "formula": "\\begin{align*} \\mathrm { K e r } ( \\tilde { \\lambda } ) \\cap \\overline { R ( S ) } = \\mathrm { K e r } ( \\lambda ^ * ) \\cap \\overline { R ( S ) } \\quad ( \\mbox { C o m p a t i b i l i t y \\ C o n d i t i o n } ) \\end{align*}"}
+{"id": "7949.png", "formula": "\\begin{align*} & B ( t ) = t ^ 3 ( 1 + ( \\ln t ) ^ 2 ) ^ { - \\frac 1 2 } \\exp ( \\ln t \\arctan ( \\ln t ) ) ) ; \\\\ & B ( t ) = t ^ { 4 + \\sin \\sqrt { 1 + ( \\ln t ) ^ 2 } } . \\end{align*}"}
+{"id": "1911.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\eta } ( \\hat { \\eta } , \\eta ; \\lambda ) : = \\theta ( \\hat { u } ) - \\theta ( u ) - \\langle D _ u \\theta ( u ) , \\hat { u } - u \\rangle _ \\mathcal { U } + I _ K ( \\hat { \\zeta } ) - I _ K ( \\zeta ) - \\langle \\lambda , \\hat { \\zeta } - \\zeta \\rangle \\geq 0 , \\end{align*}"}
+{"id": "2398.png", "formula": "\\begin{align*} \\vec { u } ( A _ { 0 } , A _ { 3 } ) = ( \\cos a _ { 3 , 1 0 2 } \\cos \\omega _ { 3 , 1 0 2 } , \\cos a _ { 3 , 1 0 2 } \\sin \\omega _ { 3 , 1 0 2 } , \\sin a _ { 3 , 1 0 2 } ) , \\end{align*}"}
+{"id": "6065.png", "formula": "\\begin{align*} \\delta _ { t } ^ { \\Lambda } : = \\exp \\circ \\exp ( \\ln ( t ) \\Lambda ) \\circ \\exp ^ { - 1 } , t > 0 , \\end{align*}"}
+{"id": "5783.png", "formula": "\\begin{align*} f ^ { ( k ) } + A ( z ) f = 0 \\end{align*}"}
+{"id": "5134.png", "formula": "\\begin{align*} \\| g \\| _ { X ' } : = \\sup _ { \\substack { \\| f \\| _ X = 1 } } \\| f g \\| _ { L ^ 1 ( \\Omega ) } , \\end{align*}"}
+{"id": "2095.png", "formula": "\\begin{align*} & F ( A ) = 9 \\cdot 2 ^ { 2 n } + 3 ( d - 2 ) \\cdot 2 ^ n - 2 d + 1 , \\\\ & g ( A ) = 9 \\cdot 2 ^ { 2 n - 1 } + ( 3 n - 8 ) \\cdot 2 ^ { n - 1 } + ( 3 \\cdot 2 ^ { n - 1 } - 1 ) d + 1 . \\end{align*}"}
+{"id": "4499.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\omega , \\mathbf { c } } ^ * ( \\eta ) : = \\{ \\Psi \\in \\mathcal { M } _ { \\omega , \\mathbf { c } } \\ | \\ G ( \\Psi ) \\ge \\eta \\} , \\end{align*}"}
+{"id": "4528.png", "formula": "\\begin{align*} f ' ( \\lambda ) = \\lambda \\left \\{ 2 L ( U ) + \\left ( \\frac { d } { 2 } + 1 \\right ) \\lambda ^ { \\frac { d } { 2 } - 1 } N ( U ) \\right \\} = : \\lambda g ( \\lambda ) . \\end{align*}"}
+{"id": "5064.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ 1 _ N } = \\int _ 0 ^ { N T } | f ( x ) | d x \\quad \\mbox { a n d } \\left \\langle f , g \\right \\rangle _ { L ^ 2 _ N } = \\int _ 0 ^ { N T } f ( x ) g ( x ) d x , \\end{align*}"}
+{"id": "4757.png", "formula": "\\begin{align*} \\tilde { S } _ { \\alpha _ i \\alpha _ i } + \\iota \\tilde { S } _ { \\alpha _ i \\beta _ i } = \\left ( x _ 1 + \\iota y _ 1 , \\ldots , x _ { | \\alpha _ i | } + \\iota y _ { | \\alpha _ i | } \\right ) . \\end{align*}"}
+{"id": "3739.png", "formula": "\\begin{align*} T ( \\rho _ n , \\vec { a } ) & = e ^ { - i [ \\vec { a } \\cdot ( \\hat { H } ( \\rho _ n ) \\hat { f } _ 0 + \\hat { P } ( \\rho _ n ) \\hat { f } _ 1 ) ] } , T ( \\rho _ n , \\vec { a } ) ^ { - 1 } = e ^ { i [ \\vec { a } \\cdot ( \\hat { H } ( \\rho _ n ) \\hat { f } _ 0 + \\hat { P } ( \\rho _ n ) \\hat { f } _ 1 ) ] } , \\end{align*}"}
+{"id": "9149.png", "formula": "\\begin{align*} \\delta ^ { k _ { 2 } ^ { j } - 1 } ( \\varphi _ { r e s t _ { 1 } } ^ { j } ) & = \\varphi _ { r e s t _ { 1 } , [ k _ { 2 } ^ { j } - 1 ] } ^ { j } ( x , v _ { 1 } , v _ { 1 , [ 1 ] } , \\ldots ) \\\\ \\delta ^ { k _ { 2 } ^ { j } } ( \\varphi _ { r e s t _ { 1 } } ^ { j } ) & = \\varphi _ { r e s t _ { 1 } , [ k _ { 2 } ^ { j } ] } ^ { j } ( x , v _ { 1 } , v _ { 1 , [ 1 ] } , \\ldots , u _ { r e s t _ { 1 } } ) \\end{align*}"}
+{"id": "4376.png", "formula": "\\begin{align*} \\begin{aligned} \\ 1 ( x \\in [ 0 , \\Delta ^ { - 1 } ] ) & \\geq | \\mathcal D ^ - _ { \\Delta , A } ( x ) | ^ 2 - C e ^ { - \\Delta ^ { A - 1 } } \\\\ | \\mathcal D ^ - _ { \\Delta , A } ( x ) | ^ 2 & \\geq ( 1 - C e ^ { - \\Delta ^ { A - 1 } } ) \\mathbf { 1 } ( x \\in [ \\Delta ^ { - A / 2 } , \\Delta ^ { - 1 } - \\Delta ^ { - A / 2 } ] ) , \\end{aligned} \\end{align*}"}
+{"id": "5605.png", "formula": "\\begin{align*} \\frac { d g . \\lambda } { d \\lambda } ( \\psi ( x , \\omega ) ) = \\frac { d g . \\eta } { d \\eta } ( x ) \\frac { d \\left ( g . \\lambda \\right ) _ { x } } { d \\lambda _ { x } } \\left ( \\psi ( x , \\omega ) \\right ) . \\end{align*}"}
+{"id": "9303.png", "formula": "\\begin{align*} \\mathcal { S } _ { j k } ( A ) : = \\sum \\limits _ { \\ell = p _ 1 + p _ 2 + \\ldots + p _ { k - 1 } + 1 } ^ { p _ 1 + p _ 2 + \\ldots + p _ k } a _ { p _ 1 + p _ 2 + \\ldots + p _ j , \\ell } \\ , , j , k = 1 , \\ldots , m . \\end{align*}"}
+{"id": "351.png", "formula": "\\begin{align*} d ( u ^ { ( 0 ) } , v ) \\leq d ( u ^ { ( L ) } , v ) + \\sum _ { i = 0 } ^ { L - 1 } d ( u ^ { ( i ) } , u ^ { ( i + 1 ) } ) \\leq n _ 0 + ( t + 1 ) L . \\end{align*}"}
+{"id": "1389.png", "formula": "\\begin{align*} c ( \\beta _ m \\otimes \\beta _ n ) = b \\sum _ { k = 0 } ^ m \\binom { m } { k } ( 1 - a ) ^ k a ^ n \\beta _ { n + k } \\otimes \\beta _ { m - k } . \\end{align*}"}
+{"id": "5672.png", "formula": "\\begin{align*} \\aligned m _ \\nu ( a , b ) \\leq J _ \\nu ( t _ \\nu \\star ( u _ 0 , v _ 0 ) ) = \\bar { f } \\Bigl ( \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | t _ \\nu \\star { u _ 0 } | ^ { p } ) | t _ \\nu \\star { v _ 0 } | ^ { q } \\Bigr ) . \\endaligned \\end{align*}"}
+{"id": "8126.png", "formula": "\\begin{align*} \\varphi ^ - ( M _ n ) = - P _ - ( a ^ n ) , \\ \\varphi ^ + ( M _ n ) = P _ + ( a ^ n ) , \\\\ \\varphi ^ - ( M _ { i j } ) = P _ - ( P _ - ( a ^ i ) a ^ j ) , \\ \\varphi ^ + ( M _ { i j } ) = - P _ + ( P _ - ( a ^ i ) a ^ j ) , \\ldots \\end{align*}"}
+{"id": "571.png", "formula": "\\begin{align*} V _ { 2 } V _ 1 M _ { \\tilde { c } \\circ \\kappa } V _ 1 ^ { - 1 } V _ { 2 } ^ { - 1 } = M _ { \\tilde { c } \\circ \\kappa \\circ \\psi _ 1 \\circ \\psi _ 2 } . \\end{align*}"}
+{"id": "8399.png", "formula": "\\begin{align*} C = \\begin{pmatrix} 0 _ n & A \\\\ B & 0 _ m \\end{pmatrix} . \\end{align*}"}
+{"id": "1184.png", "formula": "\\begin{align*} & M _ { n m } = b _ { n m } \\times \\underset { \\mathcal { R } ( \\mathbf { F } , \\mathbf { M } ) \\ , \\cap \\ , \\mathcal { X } ^ 0 _ { n m } } { \\max } ( \\theta _ n - \\theta _ m ) \\\\ & M _ { m n } = b _ { n m } \\times \\underset { \\mathcal { R } ( \\mathbf { F } , \\mathbf { M } ) \\ , \\cap \\ , \\mathcal { X } ^ 0 _ { n m } } { \\max } ( \\theta _ m - \\theta _ n ) \\end{align*}"}
+{"id": "904.png", "formula": "\\begin{align*} \\Sigma ( W ) \\ll \\widehat { Y } ^ { ( n + 5 ) / 2 + \\varepsilon } \\sum _ { | d | \\leq \\widehat { K } } \\sum _ { d = e _ 1 f _ 1 f _ 2 ^ 2 d _ 3 } \\sum _ { d _ 3 = d _ 3 ' g _ 1 g _ 2 } | e _ 1 f _ 1 | ^ { - 1 } | f _ 2 | ^ { - 1 / 2 } | g _ 2 | | m ( g _ 2 ) | ^ { - 1 } | d _ 3 ' | ^ { - 1 / 2 } \\left ( \\widehat { W } + \\widehat { Y } ^ { n / 3 } \\right ) . \\end{align*}"}
+{"id": "7267.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\limsup _ { n \\rightarrow \\infty } n ^ { - 1 } \\big \\Vert \\sum _ { i = 1 } ^ n Y _ { i , N } \\big \\Vert ^ 2 _ 2 = 0 \\ , . \\end{align*}"}
+{"id": "887.png", "formula": "\\begin{align*} \\rho _ 1 ( \\varpi ) = | \\varpi | ^ { n - 2 } + O \\left ( | \\varpi | ^ { ( n - 1 ) / 2 } \\right ) , \\end{align*}"}
+{"id": "1998.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c \\underline u ) ^ n \\geq ( 1 - ( \\delta + \\mu _ 1 ) \\underline u ) ^ n f ^ n \\omega ^ n & \\textnormal { i n } & \\Omega , \\\\ \\underline u = 0 & \\textnormal { o n } & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "4752.png", "formula": "\\begin{align*} J _ { 2 | \\alpha _ i | } \\tilde { S } _ { \\gamma _ i \\gamma _ i } J _ { 2 | \\alpha _ i | } ^ T = \\tilde { S } _ { \\gamma _ i \\gamma _ i } + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "285.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { 1 } { 1 2 } L i _ 2 ( z ) + \\frac { 1 } { 3 } L i _ 1 ( z ) + \\frac { 5 } { 1 2 } L i _ 0 ( z ) + \\frac { 1 } { 6 } L i _ { - 1 } ( z ) \\right \\} \\end{align*}"}
+{"id": "6244.png", "formula": "\\begin{align*} J _ { \\overrightarrow 0 } ( a ) = J _ { \\overrightarrow 0 } ( a ^ { ( 1 ) } ) ( z _ 0 ) \\cdot J _ { z _ 0 } ( a ^ { ( 2 ) } ) . \\end{align*}"}
+{"id": "7024.png", "formula": "\\begin{align*} f ' = F _ 0 + F _ 1 Q _ i + \\ldots + F _ r Q _ i ^ r \\end{align*}"}
+{"id": "4969.png", "formula": "\\begin{align*} p _ { i , j } ^ { ( t + 1 ) } \\overset { } { = } \\left [ P ^ { t + 1 } \\right ] _ { i , j } ^ { } = \\begin{dcases} ( 1 - p ) ^ { t + 1 } & \\mbox { i f } \\ ; i = j = 0 ; \\\\ q _ { j } ^ { } p _ { i , j } ^ { ( t ) } \\ , + \\ , p _ { j - 1 } ^ { } p _ { i , j - 1 } ^ { ( t ) } & \\mbox { i f } \\ ; 0 < i \\le j \\le n j < t ; \\\\ 0 & \\mbox { o t h e r w i s e } ; \\\\ \\end{dcases} \\end{align*}"}
+{"id": "5099.png", "formula": "\\begin{align*} F ( D ^ { 2 } u , D u , u ) = \\int _ { \\Omega } \\left ( G \\left ( \\det ( u _ { i j } ) \\right ) - g u \\right ) d x \\end{align*}"}
+{"id": "6654.png", "formula": "\\begin{align*} \\gamma ( r ) = \\begin{cases} 1 & \\textrm { i f } \\ , \\ , 0 \\leq r \\leq \\frac 1 2 \\\\ 0 & \\textrm { i f \\ \\ } r \\geq 1 \\ , \\end{cases} ; \\quad \\quad \\gamma ' ( r ) < 0 . \\end{align*}"}
+{"id": "3557.png", "formula": "\\begin{align*} I = \\int _ { 0 } ^ { \\infty } f ( x ) \\ , d x = \\int _ { 0 } ^ { \\infty } \\sum _ { n = 0 } ^ { \\infty } C ( n ) x ^ { \\alpha n + \\beta - 1 } = \\sum _ { n = 0 } ^ { \\infty } C ( n ) \\langle \\alpha n + \\beta \\rangle . \\end{align*}"}
+{"id": "7189.png", "formula": "\\begin{align*} \\widetilde { w } : = \\left ( \\mathcal { P } _ a \\vert _ { \\mathcal { M } } \\right ) ^ { - 1 } \\circ \\mathcal { P } _ a \\circ w \\end{align*}"}
+{"id": "9022.png", "formula": "\\begin{align*} P _ \\varepsilon ( p _ i , s ) = \\{ ( p _ j , F ( \\tau _ j ) ) : ( p _ j , \\tau _ j ) \\in P ( p _ i , s ) \\} . \\end{align*}"}
+{"id": "4550.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } S _ { \\omega , \\mathbf { c } } ( \\Phi ^ { \\lambda } ) \\big | _ { \\lambda = 1 } = 0 . \\end{align*}"}
+{"id": "3830.png", "formula": "\\begin{align*} g ( v ) - g ( v ^ { \\prime } ) \\leq \\Psi \\left ( \\boldsymbol { d } _ { \\mathcal { V } } ( v , v ^ \\prime ) , \\boldsymbol { d } _ { \\mathcal { V } } ( v , v ^ \\prime ) \\right ) = \\omega ( \\boldsymbol { d } _ { \\mathcal { V } } ( v , v ^ \\prime ) ) . \\end{align*}"}
+{"id": "3530.png", "formula": "\\begin{align*} T ( t ) = \\begin{bmatrix} - t I & 0 \\\\ 0 & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "7877.png", "formula": "\\begin{align*} 2 \\binom { n - k } { k } \\mbox { a n d } - 2 \\binom { n - k - 1 } { k - 1 } . \\end{align*}"}
+{"id": "5126.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\sqrt { \\det g } g ^ { i j } \\delta ^ { k l } u _ { i k } \\eta _ { j l } d x = 0 \\forall \\eta \\in C _ { 0 } ^ { \\infty } ( \\Omega ) \\end{align*}"}
+{"id": "6502.png", "formula": "\\begin{align*} s _ { i , n } : = \\sum _ { j = 1 } ^ n a _ { i , j } c _ { n , j } = \\sum _ { j = 1 } ^ n \\binom { m + x } { m - i + j } c _ { n , j } - \\sum _ { j = 1 } ^ n \\binom { m + x } { m - i - j + 1 } c _ { n , j } . \\end{align*}"}
+{"id": "2562.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) , \\ \\ u \\ge 0 , & \\ \\ B _ R , \\\\ u = 0 , \\ \\ & \\ \\ \\partial B _ R . \\end{cases} \\end{align*}"}
+{"id": "512.png", "formula": "\\begin{align*} Y & = ( x _ j , j \\in J _ 1 , x _ j , j \\in J _ 2 , \\ldots , x _ j , j \\in J _ m ) , \\\\ ( \\sigma ( 1 ) , \\ldots , \\sigma ( | Y | ) & = ( \\underbrace { 1 , \\ldots , 1 } _ { | J _ 1 | } , \\underbrace { 2 , \\ldots , 2 } _ { | J _ 2 | } , \\ldots , \\underbrace { m , \\ldots , m } _ { | J _ m | } ) . \\end{align*}"}
+{"id": "21.png", "formula": "\\begin{align*} K \\simeq \\mathrm { E n d } _ { \\overline { F } } ^ \\circ ( A ) : = \\mathrm { E n d } _ { \\overline { F } } ( A ) \\otimes _ { \\mathbb { Z } } \\mathbb { Q } \\end{align*}"}
+{"id": "4944.png", "formula": "\\begin{align*} \\frac { ( n - r ) \\cdot ( n - r - 1 ) \\cdots ( n - k + 1 ) } { n ^ { k - r } } = \\frac { ( n - r ) _ { ( k - r ) } ^ { } } { n ^ { k - r } } \\end{align*}"}
+{"id": "208.png", "formula": "\\begin{align*} = e x p \\left \\{ \\frac { z ( 1 + 4 z + z ^ 2 ) L i _ 2 ( x ) L i _ 2 ( y ) } { ( 1 - z ) ^ 4 } \\right \\} , \\end{align*}"}
+{"id": "8897.png", "formula": "\\begin{gather*} \\psi _ 2 ( x _ 1 , x _ 2 ) = [ x _ 1 , x _ 2 ] _ c , \\\\ \\psi _ n ( x _ 1 , \\dots , x _ n ) = [ x _ 1 , \\psi _ { n - 1 } ( x _ 2 , \\dots , x _ n ) ] _ c , i \\geq 3 . \\end{gather*}"}
+{"id": "1723.png", "formula": "\\begin{align*} \\Phi ( \\nu , \\mu ) ( y ) = \\frac { 1 } { Z ' ( \\nu , \\mu ) } \\exp \\left ( { \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta \\mu } ( \\nu , \\mu , y ) - U ^ { \\rho } ( y ) } \\right ) , \\end{align*}"}
+{"id": "3.png", "formula": "\\begin{align*} \\int _ { U ^ - } \\theta ^ - = \\int _ { U ^ 0 } \\theta ^ 0 = 1 \\end{align*}"}
+{"id": "3797.png", "formula": "\\begin{align*} \\Sigma ( \\delta ) : = \\Sigma ( \\mu _ { 1 3 } , \\mu _ { 2 3 } , \\delta ) = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\ \\boldsymbol { K } _ 1 ( \\mu _ { 1 3 } , \\gamma _ { 1 3 } ) \\leq \\delta _ 1 , \\ \\boldsymbol { K } _ 2 ( \\mu _ { 2 3 } , \\gamma _ { 2 3 } ) \\leq \\delta _ 2 \\right \\} \\end{align*}"}
+{"id": "7620.png", "formula": "\\begin{align*} p ( z ) = 1 + \\sum _ { n = 1 } ^ { \\infty } c _ n z ^ n \\end{align*}"}
+{"id": "7447.png", "formula": "\\begin{align*} \\Delta _ { \\bar { \\xi } _ i } u ^ { ( i ) } _ { \\alpha - 1 } = 0 \\ \\ \\ \\ \\Upsilon _ i , \\partial _ { \\bar { \\nu } _ { \\bar { \\xi } _ i } } u ^ { ( i ) } _ { \\alpha - 1 } = 0 \\ \\ \\ \\ \\partial \\Upsilon _ i , \\langle u _ { \\alpha - 1 } ^ { ( i ) } \\rangle _ { \\Upsilon _ i } = 0 , \\end{align*}"}
+{"id": "784.png", "formula": "\\begin{align*} a _ 1 = \\int _ { \\mathbb { S } ^ n } u Y _ 1 \\mathrm { d } A = \\dfrac { \\sqrt { n + 1 } } { \\sqrt { \\mathrm { A r e a } ( \\mathbb { S } ^ n ) } } \\sum \\limits _ { l = 1 } ^ { n + 1 } v _ l \\int _ { \\mathbb { S } ^ n } u x _ l \\mathrm { d } A = O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } . \\end{align*}"}
+{"id": "4357.png", "formula": "\\begin{align*} \\alpha _ { 1 } ^ { \\frac { 1 } { p - 1 } } \\left ( x - P _ { 1 } \\left ( x \\right ) \\right ) + \\alpha _ { 2 } ^ { \\frac { 1 } { p - 1 } } \\left ( x - P _ { 2 } \\left ( x \\right ) \\right ) = 0 , \\end{align*}"}
+{"id": "291.png", "formula": "\\begin{align*} \\prod _ { \\substack { l , m , n \\geq 1 \\\\ l , m \\leq n ; \\ , \\gcd ( l , m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - x ^ l y ^ m z ^ n } \\right ) ^ { \\frac { l m ^ 2 } { n ^ 4 } } \\end{align*}"}
+{"id": "5038.png", "formula": "\\begin{align*} f ^ T = \\sum _ { k = 1 } ^ N c _ k x _ 1 ^ { E _ { 1 k } } \\cdots x _ N ^ { E _ { N k } } . \\end{align*}"}
+{"id": "3323.png", "formula": "\\begin{align*} \\widetilde { d } ( \\theta , w ) = d ( \\theta , w ) - \\chi _ 0 ( \\theta ) d _ 0 ( w ) - \\chi _ { \\pi } ( \\theta ) d _ { \\pi } ( w ) \\end{align*}"}
+{"id": "5497.png", "formula": "\\begin{align*} m _ { K } = \\int _ { G / Q } \\tau ( y ) . m _ { K \\cap Q } d \\bar { m } _ { K } ( y ) . \\end{align*}"}
+{"id": "5841.png", "formula": "\\begin{align*} \\left ( X _ T , X _ { T + 1 } , \\ldots , X _ { T + t } \\right ) = \\left ( X ' _ { T ' } , X ' _ { T ' + 1 } , \\ldots , X ' _ { T ' + t } \\right ) . \\end{align*}"}
+{"id": "5746.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } p ' y ^ 2 = z \\cdots \\mbox { ( 7 $ - $ 1 ) } \\\\ x ^ 2 = w \\cdots \\mbox { ( 7 $ - $ 2 ) } \\\\ p ' x ^ 4 = p x + q p ' y ^ 2 \\cdots \\mbox { ( 7 $ - $ 3 ) } \\\\ p '^ 2 y ^ 4 = p y + q x ^ 2 \\cdots \\mbox { ( 7 $ - $ 4 ) } \\\\ p ' y x = a \\cdots \\mbox { ( 7 $ - $ 5 ) } \\\\ p ' y x = c \\cdots \\mbox { ( 7 $ - $ 6 ) } \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "1523.png", "formula": "\\begin{align*} \\lambda ( r + s ) = a r s + \\lambda ( r ) + \\lambda ( s ) \\ , \\ , r , s \\in R . \\end{align*}"}
+{"id": "6031.png", "formula": "\\begin{align*} \\int \\bigl ( \\prod _ { i = 1 } ^ n V f _ i \\bigr ) \\otimes g \\ , d \\eta = \\int \\bigl ( \\prod _ { i = 1 } ^ n V f _ i \\bigr ) d \\mu \\int g \\ , d \\nu . \\end{align*}"}
+{"id": "6849.png", "formula": "\\begin{align*} \\left [ { N + 1 \\atop j } \\right ] = \\left [ { N \\atop j } \\right ] + q ^ { N + 1 - j } \\left [ { N \\atop j - 1 } \\right ] , \\end{align*}"}
+{"id": "6800.png", "formula": "\\begin{gather*} \\dim W ^ { \\mathrm { u } } ( p ^ - ) = 2 n - k ^ - , \\dim W ^ { \\mathrm { s } } ( p ^ + ) = k ^ + + 1 , \\end{gather*}"}
+{"id": "317.png", "formula": "\\begin{align*} \\zeta ( 2 , 1 ) = \\zeta ( 3 ) . \\end{align*}"}
+{"id": "7791.png", "formula": "\\begin{align*} ( A \\otimes B ^ { - 1 } ) ^ r = \\int ^ { \\infty } _ 0 ( s + ( A \\otimes B ^ { - 1 } ) ) ^ { - 1 } d \\mu ( s ) . \\end{align*}"}
+{"id": "3462.png", "formula": "\\begin{align*} s _ \\lambda ( X , Y ) = \\frac { X ^ { \\lambda _ 1 + 1 } Y ^ { \\lambda _ 2 } - X ^ { \\lambda _ 2 } Y ^ { \\lambda _ 1 + 1 } } { X - Y } . \\end{align*}"}
+{"id": "2030.png", "formula": "\\begin{align*} H _ \\ell = - \\alpha z _ { \\bar \\ell } H ; ~ ~ ~ \\ ; \\ ; ~ H _ { \\bar \\ell } = - \\alpha z _ { \\ell } H , ~ ~ \\ell = 1 , \\cdots , n , \\end{align*}"}
+{"id": "5381.png", "formula": "\\begin{align*} \\| y _ { 1 , k } ^ 1 \\| ^ 2 + \\| y _ { 2 , k } ^ 1 \\| ^ 2 = \\left \\| \\begin{bmatrix} y _ { 1 , k } ^ 1 \\\\ y _ { 2 , k } ^ 1 \\end{bmatrix} \\right \\| ^ 2 \\leq \\left \\| \\begin{bmatrix} u _ { 1 , k } ^ 1 \\\\ u _ { 2 , k } ^ 1 \\end{bmatrix} \\right \\| ^ 2 = \\| u _ { 1 , k } ^ 1 \\| ^ 2 + \\| u _ { 2 , k } ^ 1 \\| ^ 2 . \\end{align*}"}
+{"id": "1136.png", "formula": "\\begin{align*} T ( f ^ { n } ) = n f ^ { n - 1 } T ( f ) + n B ( A ( f ) , A ( f ^ { n - 1 } ) ) , \\end{align*}"}
+{"id": "1816.png", "formula": "\\begin{align*} [ \\N ^ \\ell , b _ { k } ( t ) ] = \\sum _ { n = 0 } ^ { \\ell - 1 } \\binom { \\ell } { n } ( - 2 ) ^ { \\ell - n } \\N ^ n b _ k ( t ) \\ , \\forall k \\in \\Lambda ^ * , \\ t \\in \\R \\ . \\end{align*}"}
+{"id": "4027.png", "formula": "\\begin{align*} \\big \\langle z ^ m \\lbrace 0 , \\infty \\rbrace , f \\big \\rangle = \\int _ 0 ^ { \\infty } f ( z ) z ^ m d z 0 \\leq m \\leq k - 2 . \\end{align*}"}
+{"id": "9328.png", "formula": "\\begin{align*} G _ { k , \\mathbb { H } } : = - \\frac { ( 2 ^ { k - 2 } - 1 ) \\ , B _ k \\ , B _ { k - 2 } } { 4 k ( k - 2 ) } \\ , E _ { k , \\mathbb { H } } . \\end{align*}"}
+{"id": "1163.png", "formula": "\\begin{align*} T ( f ^ { n } ) = n f ^ { n - 1 } T ( f ) + n B ( A ( f ) , A ( f ^ { n - 1 } ) ) \\end{align*}"}
+{"id": "6640.png", "formula": "\\begin{align*} C = Y _ 0 \\subset Y _ 1 \\subset \\dots \\subset Y _ k = Y \\end{align*}"}
+{"id": "2820.png", "formula": "\\begin{align*} \\overset \\cdot { S _ k ^ { 1 , n } } = S _ { k + q + 1 } ^ { 1 , n } - S _ { q + n } ^ { 1 , n } S _ k ^ { 1 , n } ; n = 1 , \\dots , q . \\end{align*}"}
+{"id": "7132.png", "formula": "\\begin{align*} R _ 2 ( \\hat { x } , \\bar { y } ) : = \\sum _ { | \\beta | = 2 } \\frac { 2 } { \\beta ! } \\left ( \\int _ { 0 } ^ 1 ( 1 - t ) ^ \\beta c ( \\bar { y } + t ( \\hat { x } - \\bar { y } ) ) t \\right ) ( \\hat { x } - \\bar { y } ) ^ \\beta . \\end{align*}"}
+{"id": "1031.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { k } \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 2 } { ( 1 ) _ { k } ( b ) _ { k } } \\bigg \\{ H _ { 2 k } ^ { ( 2 ) } - \\frac { 1 } { 4 } H _ { k } ^ { ( 2 ) } \\bigg \\} = \\frac { \\Gamma ( \\frac { b } { 2 } ) \\Gamma ( \\frac { 1 + b } { 2 } ) } { 1 6 \\Gamma ( \\frac { 1 + 2 b } { 4 } ) ^ 2 } \\psi \\ , ' \\bigg ( \\frac { 1 + 2 b } { 4 } \\bigg ) . \\end{align*}"}
+{"id": "2062.png", "formula": "\\begin{align*} u _ { \\leq N } = \\frac { 1 } { N ^ d } \\check { \\psi } \\left ( \\frac { \\cdot } { N } \\right ) \\ast u . \\end{align*}"}
+{"id": "1876.png", "formula": "\\begin{align*} \\lambda ^ * = \\bar { \\lambda } | _ { \\overline { R ( S ) } } , \\end{align*}"}
+{"id": "6790.png", "formula": "\\begin{align*} u ' ( t ) + \\mathcal { A } ( u ( t ) , t ) = f ( t ) , \\mathcal { V } ^ * , \\ , u ( t _ 0 ) = u _ 0 H , \\end{align*}"}
+{"id": "4518.png", "formula": "\\begin{align*} K _ { \\omega , \\mathbf { c } } ( V ^ { \\lambda } ) & = 2 L ( V ^ { \\lambda } ) + 3 N ( V ^ { \\lambda } ) + 2 \\omega M ( V ^ { \\lambda } ) + 2 \\mathbf { c } \\cdot \\mathbf { P } ( V ^ { \\lambda } ) \\\\ & = 2 \\lambda ^ 2 L ( V ) + 3 \\lambda ^ { \\frac { d } { 2 } + 1 } N ( V ) + 2 \\omega M ( V ) + 2 \\lambda \\mathbf { c } \\cdot \\mathbf { P } ( V ) , \\end{align*}"}
+{"id": "6624.png", "formula": "\\begin{align*} 0 = [ ( \\varphi _ { \\lambda } ( x , [ { y _ 1 } _ \\mu y _ 2 ] ) + ( - 1 ) ^ { y _ 2 ( x + y _ 1 ) } \\varphi _ { - \\partial - \\lambda } ( y _ 2 , [ x _ \\mu y _ 1 ] , ) - ( - 1 ) ^ { x y _ 1 } \\varphi _ \\lambda ( y _ 1 , [ x _ \\mu y _ 2 ] ) ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] . \\end{align*}"}
+{"id": "1677.png", "formula": "\\begin{align*} \\alpha = \\pi \\widehat { \\phi ^ { ( p ) } } \\psi \\phi \\end{align*}"}
+{"id": "3172.png", "formula": "\\begin{align*} U = \\sum _ { u \\in Z _ + ^ d } a ( u ) T _ 1 ^ { u _ 1 } \\cdots T _ d ^ { u _ d } \\end{align*}"}
+{"id": "1021.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( a ) _ k ( b ) _ k } { ( 1 ) _ { k } ( a + b ) _ { k } } \\frac { H _ k ( a - 1 ) + H _ k ( b - 1 ) } { x ^ k } = \\bigg ( \\log \\frac { x } { x - 1 } \\bigg ) \\sum _ { k = 0 } ^ { \\infty } \\frac { ( a ) _ k ( 1 - a ) _ k } { ( 1 ) _ { k } ( a + b ) _ { k } } \\frac { 1 } { x ^ k } . \\end{align*}"}
+{"id": "599.png", "formula": "\\begin{align*} 2 t s \\cos ( \\gamma + \\beta ) = \\frac { 2 s ^ 2 - 1 } { \\sqrt { 1 - s ^ 2 } } . \\end{align*}"}
+{"id": "8576.png", "formula": "\\begin{align*} \\varphi _ j ( p ) : = \\frac { \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s } { \\int _ 0 ^ \\infty e ^ { - \\lambda s } \\big ( \\sum _ { k = j } ^ \\infty p _ k ( s ) \\big ) d s } = \\begin{cases} \\dfrac { 1 } { j + 1 } , & p = 0 , \\\\ 1 - \\dfrac { \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } y ^ { j } d y } { \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } y ^ { j - 1 } d y } , & 0 < p < 1 , \\end{cases} \\end{align*}"}
+{"id": "4922.png", "formula": "\\begin{align*} f ( \\underline { k } , t , \\mathbf { p } _ { n + 1 } ) = \\left ( \\prod _ { j = 0 } ^ { k - 1 } p _ j ^ { } \\right ) \\sum _ { j = 0 } ^ { k } q _ j ^ { t - k } \\prod _ { \\substack { i = 0 \\\\ i \\neq j } } ^ { k } \\frac { 1 } { q _ j ^ { } - q _ i ^ { } } \\ , . \\end{align*}"}
+{"id": "2538.png", "formula": "\\begin{align*} B ( N ) = \\{ \\{ i , j \\} , \\{ i , j ^ * \\} , \\{ j , i ^ * \\} , \\{ j ^ * , i ^ * \\} \\} , \\end{align*}"}
+{"id": "8618.png", "formula": "\\begin{align*} E \\left [ \\left ( Y e ^ { \\lambda t } - Z _ 0 ( t ) \\right ) ^ 2 \\right ] = c _ 2 e ^ { \\lambda t } , \\end{align*}"}
+{"id": "6091.png", "formula": "\\begin{align*} \\mathcal { E } ^ { 0 , \\gamma } _ { \\tilde { \\alpha } - \\gamma ; I } : = C ( I ; \\mathcal { B } _ { \\tilde { \\alpha } - \\gamma } ) \\cap C ^ { \\gamma } ( I ; \\mathcal { B } _ { \\tilde { \\alpha } - 2 \\gamma } ) , \\ \\ \\ \\mathcal { E } ^ { \\gamma , 2 \\gamma } _ { \\tilde { \\alpha } ; I } : = C ^ { \\gamma } _ { 2 } ( I ; \\mathcal { B } _ { \\tilde { \\alpha } - \\gamma } ) \\cap C ^ { 2 \\gamma } _ { 2 } ( I ; \\mathcal { B } _ { \\tilde { \\alpha } - 2 \\gamma } ) . \\end{align*}"}
+{"id": "2115.png", "formula": "\\begin{align*} F ^ n ( P ) = \\sum _ { i = 0 } ^ { m - 1 } a ^ { ( i ) } _ n \\cdot F ^ i ( P ) . \\end{align*}"}
+{"id": "3131.png", "formula": "\\begin{align*} T ( z ) = \\sum _ { n = 1 } ^ { \\infty } n ^ { n - 1 } \\frac { z ^ n } { n ! } . \\end{align*}"}
+{"id": "5555.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( B \\cap \\left [ H _ { 0 } \\omega _ { n } ; \\omega _ { n } ^ { - 1 } \\omega _ { n + 1 } \\right ] \\neq \\emptyset \\mbox { i n f i n i t e l y o f t e n } \\right ) = 0 . \\end{align*}"}
+{"id": "3949.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { A _ 1 \\cap A _ 2 } | g - g _ { \\rho } | \\ , d \\pi & \\leq \\int _ { A _ 1 \\cap A _ 2 } M \\left [ \\boldsymbol { d } _ { \\mathcal { S } _ 1 } ( s ^ \\star _ 1 , s _ 1 ) ^ { p _ 1 ^ { \\prime } } + \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( s _ 2 , s ^ \\star _ 1 ) ^ { p _ 2 ^ { \\prime } } \\right ] d \\pi ( s _ 1 , s _ 2 ) \\\\ & \\leq B ( \\rho ^ { p _ 1 ^ \\prime - p _ 1 } + \\rho ^ { p _ 2 ^ \\prime - p _ 2 } ) . \\end{aligned} \\end{align*}"}
+{"id": "7574.png", "formula": "\\begin{align*} | \\gamma _ { 1 } | \\leq 1 , \\ ; \\ ; | \\gamma _ { 2 } | \\leq \\dfrac { 1 } { 2 } + \\dfrac { 1 } { e } = 0 . 6 3 5 \\ldots \\end{align*}"}
+{"id": "7718.png", "formula": "\\begin{align*} C _ n ^ \\lambda ( x ) & = \\frac { \\Gamma ( n + 2 \\lambda ) } { \\Gamma ( n + 1 ) \\Gamma ( 2 \\lambda ) } { } _ 2 F _ 1 \\left ( - n , n + 2 \\lambda ; \\lambda + \\frac { 1 } { 2 } ; \\frac { 1 } { 2 } - \\frac { x } { 2 } \\right ) \\\\ & = \\frac { \\Gamma ( n + 2 \\lambda ) } { \\Gamma ( n + 1 ) \\Gamma ( 2 \\lambda ) } { } _ 2 F _ 1 \\left ( - \\frac { n } { 2 } , \\frac { n } { 2 } + \\lambda ; \\lambda + \\frac 1 2 ; 1 - x ^ 2 \\right ) \\end{align*}"}
+{"id": "7425.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { N \\to \\infty } \\ , \\frac { \\theta _ N ^ 2 } { N ^ 4 } \\ , \\Xi ( 1 ) = 0 \\qquad \\lim _ { N \\to \\infty } \\ , \\frac { \\theta _ N ^ 2 } { N ^ 4 } \\ , \\sum _ { r = 3 } ^ \\infty \\Xi ( r ) = 0 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "9194.png", "formula": "\\begin{align*} h = \\psi ( z ^ { 1 } , \\ldots , z ^ { l } , g ^ { l + 1 } ( z ) , \\ldots , g ^ { k } ( z ) ) \\end{align*}"}
+{"id": "6149.png", "formula": "\\begin{align*} - ( a , b ) & = ( - a , - b ) , \\\\ ( a , b ) \\cdot ( c , d ) & = ( a \\cdot c , b \\cdot d ) , \\\\ ( a , b ) + ( c , d ) & = \\{ ( e , f ) \\in F _ 1 \\times F _ 2 : e \\in a + c \\mbox { a n d } f \\in b + d \\} \\cap ( F _ 1 \\times _ h F _ 2 ) . \\end{align*}"}
+{"id": "5039.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N q _ k = ( 1 , \\dots , 1 ) E _ f ^ { - 1 } \\boldsymbol { 1 } = ( 1 , \\dots , 1 ) ( E _ f ^ T ) ^ { - 1 } \\boldsymbol { 1 } = \\sum _ { k = 1 } ^ N q _ k ^ T . \\end{align*}"}
+{"id": "9170.png", "formula": "\\begin{align*} v _ { 1 } ^ { 1 } & = \\varphi _ { 1 , [ 2 ] } ^ { 1 } = 2 \\bar { u } ^ { 2 } - x ^ { 3 } \\\\ v _ { 1 , [ 1 ] } ^ { 1 } & = \\varphi _ { 1 , [ 3 ] } ^ { 1 } = 2 \\bar { u } _ { [ 1 ] } ^ { 2 } - 2 \\bar { u } ^ { 2 } + x ^ { 3 } \\ , , \\\\ & \\ : \\ : \\vdots \\end{align*}"}
+{"id": "6718.png", "formula": "\\begin{align*} h ( \\widetilde { X } _ \\eta ) = 0 \\iff \\mathcal { P } ( \\widetilde { X } _ \\eta ) = \\{ \\delta _ { \\boldsymbol { 0 } } \\} \\iff \\widetilde { X } _ \\eta \\end{align*}"}
+{"id": "2353.png", "formula": "\\begin{align*} \\prod _ { x : X } \\exists _ { k _ x : \\N , m _ x : M _ x } . \\varphi ( x ) = \\frac { m _ x } { f ( x ) ^ { k _ x } } \\rlap { . } \\end{align*}"}
+{"id": "7089.png", "formula": "\\begin{align*} \\gamma = \\sup _ { n \\in \\N } \\{ \\nu ( x - a _ n ) \\} = \\sup _ { a \\in K } \\{ \\nu ( x - a ) \\} = \\sup { \\rm d i s t } ( \\eta , K ) . \\end{align*}"}
+{"id": "5046.png", "formula": "\\begin{align*} \\Psi ( [ p ] \\xi _ g ) : = [ p ] \\xi _ { g ^ { - 1 } } . \\end{align*}"}
+{"id": "5333.png", "formula": "\\begin{align*} L _ S [ f ] = \\sum _ { T \\subseteq S } ( - 1 ) ^ { | T | } E _ T [ f ] . \\end{align*}"}
+{"id": "7409.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\ , \\bigg \\| \\ , X _ N \\ , - \\ , \\sum _ { i = 1 } ^ { M _ N } X _ { N , M _ N } ^ { ( i ) } \\ , \\bigg \\| _ { L ^ 2 } = 0 \\ , , \\end{align*}"}
+{"id": "8598.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } \\big ( { S } _ { j , + } ( t ) - { S } _ { j , - } ( t ) \\big ) = \\nu Y \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s . \\end{align*}"}
+{"id": "7738.png", "formula": "\\begin{align*} \\dot z ^ { * } ( Y , \\ , y ^ { * } ( N , t ) ) = \\frac { 1 } { \\dot y ^ { * } ( N , t ) } , \\ \\forall t \\ge 0 , \\ , \\pi - \\end{align*}"}
+{"id": "4036.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( i ) \\lambda _ f ( j ) \\big | \\langle ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) , f \\rangle \\big | ^ 2 . \\end{align*}"}
+{"id": "998.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\mathbf { u } ( E _ i ) = 0 \\end{align*}"}
+{"id": "8493.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { 1 } { \\left ( \\lambda _ { n } ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } = \\frac { \\sqrt { \\pi } x ^ { 1 - 2 s } } { 2 \\Gamma ( s ) } \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) - \\frac { 1 } { 2 } \\ , \\frac { x ^ { - 2 s } } { 1 + \\frac { 1 } { \\pi p } } + 2 \\sin ( \\pi s ) x ^ { 1 - 2 s } \\ , \\intop _ { 1 } ^ { \\infty } \\frac { ( t ^ { 2 } - 1 ) ^ { - s } } { \\sigma ( x t ) e ^ { 2 \\pi x t } - 1 } d t , \\end{align*}"}
+{"id": "7010.png", "formula": "\\begin{align*} f ' = \\frac { d } { d x } \\left ( \\sum _ { j = 1 } ^ r \\tilde b _ j \\tilde { \\textbf { Q } } ^ { \\lambda _ j } \\right ) = \\sum _ { j = 1 } ^ r \\sum _ { l \\leq i } \\tilde b _ j \\lambda _ j ( Q _ { l } ) \\tilde { \\textbf { Q } } ^ { \\lambda _ j } \\frac { Q ' _ { l } } { Q _ { l } } . \\end{align*}"}
+{"id": "1900.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\langle D _ u \\theta ( u ^ * ) , v \\rangle _ \\mathcal { U } + \\langle \\tilde { \\lambda } , S v \\rangle _ * = 0 \\forall \\ v \\in \\mathcal { U } ; \\\\ & - \\langle \\tilde { \\lambda } , \\zeta - S u ^ * \\rangle _ * \\geq 0 \\forall \\ \\zeta \\in K , \\end{aligned} \\right . \\end{align*}"}
+{"id": "3992.png", "formula": "\\begin{align*} \\mathrm { R } _ { \\mathrm { D } } ( \\lambda , \\delta ) = \\langle \\lambda , \\delta \\rangle + \\frac { V _ { 1 , Y Y } } { 4 \\lambda _ 1 } + \\frac { V _ { 2 , Y Y } } { 4 \\lambda _ 2 } . \\end{align*}"}
+{"id": "1561.png", "formula": "\\begin{align*} h ( v ) = \\begin{cases} 0 & v \\in [ - 1 , 0 ) , \\\\ 2 a v & v \\in [ 0 , 1 ] . \\end{cases} \\end{align*}"}
+{"id": "3179.png", "formula": "\\begin{align*} A _ n ( \\mu ) ( x ) \\leq \\frac { \\chi _ d } { n _ d } \\sum _ { j = 0 } ^ { n _ d - 1 } U ^ j ( \\mu ) ( x ) \\leq \\epsilon \\rho ( x ) x \\in ( e M e ) _ + . \\end{align*}"}
+{"id": "6037.png", "formula": "\\begin{align*} h _ 2 ( z ) : = A _ 2 z ^ { n + m } + B _ 2 \\overline { z } ^ m + C _ 2 \\textrm { f o r a l l } z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "2285.png", "formula": "\\begin{align*} H ( f ) ( \\theta ) = \\lim _ { \\epsilon \\to 0 } \\frac { 1 } { \\pi } \\int _ { \\epsilon \\leq | t | \\leq \\pi } \\frac { f ( \\theta - t ) } { 2 \\tan ( t / 2 ) } \\ , d t . \\end{align*}"}
+{"id": "1942.png", "formula": "\\begin{align*} \\Theta _ j : = a _ 0 + \\sum _ { S \\in \\mathcal { Q } _ m } \\left ( \\prod _ { i \\in S } \\alpha _ i ^ { ( j ) } \\right ) f _ S ( u _ { j , S } ) \\end{align*}"}
+{"id": "9027.png", "formula": "\\begin{align*} \\begin{cases} \\frac { C _ 2 ( \\lambda _ 2 \\rho _ 0 ) } { 2 } + \\frac { C _ 3 ( \\lambda _ 2 \\lambda _ { 2 , 3 } \\rho _ 0 ) } { 2 } = C _ 1 ( \\lambda _ 1 \\rho _ 0 ) \\\\ \\frac { C _ 2 ( \\lambda _ 2 \\rho _ 0 ) } { 2 } + \\frac { C _ 4 ( \\lambda _ 2 \\lambda _ { 2 , 4 } \\rho _ 0 ) } { 2 } = C _ 1 ( \\lambda _ 1 \\rho _ 0 ) \\\\ \\lambda _ 1 + \\lambda _ 2 = \\lambda _ { 2 , 3 } + \\lambda _ { 2 , 4 } = 1 \\end{cases} . \\end{align*}"}
+{"id": "8708.png", "formula": "\\begin{align*} \\textup { v a r } ( U ^ \\top A U ) = 2 \\textup { t r } ( A ^ 2 ) + \\sum _ { i = 1 } ^ { q } \\delta _ i a _ { i i } ^ 2 , \\end{align*}"}
+{"id": "8793.png", "formula": "\\begin{align*} & 2 ( u + 1 ) ( 4 + t + u ) b _ 1 \\\\ & = 2 ( u + 1 ) ( 1 + 2 t ) c _ 1 + u ( u + 1 ) ( b _ 1 + c _ 1 ) \\\\ & + u ( b _ 1 + c _ 1 + u b _ 3 + u c _ 3 ) + 2 u ( u + 1 ) b _ 2 \\end{align*}"}
+{"id": "869.png", "formula": "\\begin{align*} L _ 1 ( d \\underline { c } , k ) \\coloneqq \\{ \\underline { x } \\in K _ \\infty ^ 2 \\colon d \\underline { c } \\cdot \\underline { x } = k \\} \\end{align*}"}
+{"id": "7434.png", "formula": "\\begin{align*} \\varepsilon ^ { \\alpha - 1 } \\ , w _ { \\alpha - 1 } ^ { ( i ) } ( x _ i , t ) + \\varepsilon ^ { \\alpha } \\ , w _ { \\alpha } ^ { ( i ) } ( x _ i , t ) + \\sum \\limits _ { k = 2 } ^ { - \\lfloor \\alpha \\rfloor } \\varepsilon ^ { \\alpha + k - 1 } \\ , \\Big ( w _ { \\alpha + k - 1 } ^ { ( i ) } ( x _ i ) + u _ { \\alpha + k - 1 } ^ { ( i ) } \\big ( x _ i , \\tfrac { \\overline { x } _ i } { \\varepsilon } , t \\big ) \\Big ) . \\end{align*}"}
+{"id": "8738.png", "formula": "\\begin{align*} \\| \\mathcal { T } _ { 1 , l } ^ { ( a _ 1 , a _ 2 ) } - \\mathcal { T } _ { 2 , l } ^ { ( a _ 1 , a _ 2 ) } \\| _ { \\rm F } ^ 2 & = \\sum _ { k _ 1 , k _ 2 } s _ { k _ 1 , k _ 1 } ^ { r } s _ { k _ 2 , k _ 2 } ^ { r } ( \\mu _ { k _ 1 , l } ^ { ( 1 ) } - \\mu _ { k _ 1 , l } ^ { ( 2 ) } ) ( \\mu _ { k _ 2 , l } ^ { ( 1 ) } - \\mu _ { k _ 2 , l } ^ { ( 2 ) } ) \\\\ & = \\bigg \\{ \\sum _ { k } s _ { k , k } ^ { r } ( \\mu _ { k , l } ^ { ( 1 ) } - \\mu _ { k , l } ^ { ( 2 ) } ) \\bigg \\} ^ { 2 } = O ( p ^ 2 ) . \\end{align*}"}
+{"id": "7906.png", "formula": "\\begin{align*} \\mathcal { I } ( \\lambda ) : = \\left \\{ i \\in \\{ 1 , 2 , \\ldots , t \\} \\mid \\lambda _ { i - 1 } > \\lambda _ { i } \\right \\} . \\end{align*}"}
+{"id": "937.png", "formula": "\\begin{align*} \\mathbb { E } ( X ) = \\sum _ { \\substack { 0 \\leq i < j \\leq h \\\\ j - i \\leq t } } \\mathbb { P } ( s _ i = s _ j ) + \\sum _ { \\substack { 0 \\leq i \\leq h < j \\leq k \\\\ j - i \\leq t } } \\mathbb { P } ( s _ i = s _ j ) + \\sum _ { \\substack { h < i < j \\leq k \\\\ j - i \\leq t } } \\mathbb { P } ( s _ i = s _ j ) . \\end{align*}"}
+{"id": "1175.png", "formula": "\\begin{align*} r ^ { 4 } \\left \\{ \\left [ T ( f ^ { 2 } ) - 2 f T ( f ) - 2 A ( f ) ^ { 2 } \\right ] - 2 \\left [ A ( f ^ { 2 } ) ^ { 2 } - 4 f ^ { 2 } A ( f ) ^ { 2 } \\right ] \\right \\} + r ^ { 2 } \\left \\{ T ( f ^ { 2 } ) - 2 f T ( f ) - 2 A ( f ) ^ { 2 } \\right \\} = 0 . \\end{align*}"}
+{"id": "4082.png", "formula": "\\begin{align*} \\small g _ 3 & = 2 \\eta _ 1 + 1 \\eta _ 2 + 8 \\eta _ 3 \\\\ g _ 2 & = g _ 3 \\eta _ 1 + 2 \\eta _ 2 + 1 \\eta _ 3 \\\\ g _ 1 & = g _ 2 \\eta _ 1 + g _ 3 \\eta _ 2 + 2 \\eta _ 3 \\\\ g _ 0 & = g _ 1 \\eta _ 1 + g _ 2 \\eta _ 2 + g _ 3 \\eta _ 3 \\end{align*}"}
+{"id": "4545.png", "formula": "\\begin{align*} H ^ { m } ( \\R ^ d ) \\subset \\left \\{ f \\in C ^ r ( \\R ^ d ) \\left | \\ \\displaystyle \\lim _ { | x | \\rightarrow \\infty } \\partial ^ { \\alpha } f ( x ) = 0 , \\ \\alpha \\in \\Z ^ d : \\ | \\alpha | \\le r \\right . \\right \\} \\end{align*}"}
+{"id": "7912.png", "formula": "\\begin{align*} z _ { \\mathbf { n } } : = \\tfrac { 1 } { \\mathbf { n } ! } \\partial ^ \\mathbf { n } p ( 0 ) . \\end{align*}"}
+{"id": "1199.png", "formula": "\\begin{align*} [ ( B _ n + i ) ^ { - 1 } - ( B + i ) ^ { - 1 } ] \\psi = ( B _ n + i ) ^ { - 1 } [ ( B + i ) \\phi - ( B _ n + i ) \\phi _ n ] - \\phi + \\phi _ n \\longrightarrow 0 \\end{align*}"}
+{"id": "652.png", "formula": "\\begin{gather*} \\{ 0 \\} = E _ { 0 } ( \\pi , \\lambda ) \\subset E _ { - 1 } ( \\pi , \\lambda ) \\subset \\ldots \\subset E _ { - g } ( \\pi , \\lambda ) \\subset E _ { c s } ( \\pi , \\lambda ) \\\\ = E _ { g + 1 } ( \\pi , \\lambda ) \\subset E _ { g } ( \\pi , \\lambda ) \\subset \\ldots \\subset E _ { 1 } ( \\pi , \\lambda ) = \\Gamma : = \\R ^ { \\mathcal { A } } \\end{gather*}"}
+{"id": "5255.png", "formula": "\\begin{align*} \\| D _ { S , x } [ f ' ] \\| _ 2 ^ { q - 2 } & = \\| \\left ( \\frac { 1 } { \\sqrt { 2 } } \\right ) ^ { | S | } T _ { 1 / \\sqrt { 2 } } D _ { S , x } [ f ] \\| _ 2 ^ { q - 2 } = \\left ( \\frac { 1 } { \\sqrt { 2 } } \\right ) ^ { | S | ( q - 2 ) } \\| T _ { 1 / \\sqrt { 2 } } D _ { S , x } f \\| _ 2 ^ { q - 2 } \\\\ & \\le \\left ( \\frac { 1 } { \\sqrt { 2 } } \\right ) ^ { | S | ( q - 2 ) } \\| D _ { S , x } f \\| _ 2 ^ { q - 2 } \\le ( r ^ { | S | } \\gamma ) ^ { q - 2 } , \\end{align*}"}
+{"id": "1237.png", "formula": "\\begin{align*} g _ { \\gamma , 2 } ( x ) = \\tfrac { 1 } { 2 } ( x ^ { \\frac { 1 } { \\beta } } + 1 ) . \\end{align*}"}
+{"id": "1464.png", "formula": "\\begin{align*} \\mu _ m ^ i \\int _ { \\Omega } \\nabla P \\psi _ { \\mu ^ i _ m , \\xi ^ i _ m } ^ h ( x ) \\nabla u _ m ( x ) d x = o ( 1 ) . \\end{align*}"}
+{"id": "1047.png", "formula": "\\begin{align*} A ( f + T ) = T \\cdot Q ( T ) \\ ; \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\ ; Q ( 0 ) \\not = 0 \\ , . \\end{align*}"}
+{"id": "6433.png", "formula": "\\begin{align*} ( x \\bar { \\otimes } y ) ( x ' \\bar { \\otimes } y ' ) = ( x x ' ) \\bar { \\otimes } ( y y ' ) \\end{align*}"}
+{"id": "1368.png", "formula": "\\begin{align*} g _ i v _ j = - v _ { 2 i - j \\bmod 3 } , i , j \\in \\{ 1 , 2 , 3 \\} . \\end{align*}"}
+{"id": "3994.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) \\leq \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + \\mathrm { R } ( \\lambda _ { \\mathrm { D } } ^ \\star , \\delta ) \\leq \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + \\mathrm { R } _ { \\mathrm { D } } ( \\lambda _ { \\mathrm { D } } ^ { \\star } , \\delta ) = \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) . \\end{align*}"}
+{"id": "5836.png", "formula": "\\begin{align*} H _ k = h _ 0 L _ k . \\end{align*}"}
+{"id": "6295.png", "formula": "\\begin{align*} d _ { \\lambda _ t } H = d _ { \\lambda _ t } h _ { \\bar u ( t ) } , \\qquad \\forall \\ , t \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "2599.png", "formula": "\\begin{align*} \\frac { T _ h \\nu _ { \\Omega } ( 0 ) } { | T _ h \\nu _ { \\Omega } ( 0 ) | } = \\frac { T _ 2 ( T _ 1 \\nu _ { \\Omega } ( 0 ) ) } { | T _ 2 ( T _ 1 \\nu _ { \\Omega } ( 0 ) ) | } = \\frac { T _ 2 e _ 2 } { | T _ 2 e _ 2 | } + o ( 1 ) . \\end{align*}"}
+{"id": "2588.png", "formula": "\\begin{align*} ( D u ) _ { \\sharp } f \\chi _ { _ { \\{ u > \\frac { 1 } { 2 } | x | ^ 2 \\} } } = g , \\end{align*}"}
+{"id": "3091.png", "formula": "\\begin{align*} ( \\lambda C _ \\lambda ) ^ { p - 1 - \\alpha } = \\frac { c _ 1 } { \\mu ^ { k _ 1 } B _ \\lambda } \\ , ( \\mu C _ \\mu ) ^ { k _ 1 } , ( \\mu C _ \\mu ) ^ { p - 1 - k _ 2 } = \\frac { c _ 2 } { \\mu ^ { k _ 2 } B _ \\mu } \\ , ( \\lambda C _ \\lambda ) ^ { k _ 3 } . \\end{align*}"}
+{"id": "9005.png", "formula": "\\begin{align*} ( 1 + w ) R ^ \\varphi _ { k j , i } = w _ { k j , i } - w _ i R ^ \\varphi _ { j k } + \\frac { S ^ \\varphi } { m - 1 } w _ i \\delta _ { j k } \\ , . \\end{align*}"}
+{"id": "2091.png", "formula": "\\begin{align*} N _ { d r } & = \\left ( \\sum _ { i = 1 } ^ n 2 ^ i x _ i \\right ) a + \\sum _ { i = 1 } ^ n ( 2 ^ i - 1 ) x _ i d \\\\ & = \\left ( \\sum _ { i = 1 } ^ { n - 1 } 2 ^ i x _ i \\right ) a + \\sum _ { i = 1 } ^ { n - 1 } ( 2 ^ i - 1 ) x _ i d + ( 2 ^ n a + ( 2 ^ n - 1 ) d ) x _ n . \\end{align*}"}
+{"id": "5939.png", "formula": "\\begin{align*} \\phi ( [ \\sum ^ m _ { k = 1 } ( a _ k + i b _ k ) f _ k , t ] ^ { \\sigma } ) & = \\phi ( [ \\sum ^ m _ { k = 1 } ( a _ k - i b _ k ) f _ k , - t ] ) = [ \\sum ^ m _ { k = 1 } a _ k e _ k - b _ k e _ k ^ { \\ast } , - t ] \\\\ & = [ \\sum ^ m _ { k = 1 } a _ k e _ k + b _ k e _ k ^ { \\ast } , t ] ^ { s ( - 1 ) } = \\phi ( [ \\sum ^ m _ { k = 1 } ( a _ k + i b _ k ) f _ k , t ] ) ^ { \\phi ( \\sigma ) } . \\end{align*}"}
+{"id": "2744.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q ) u = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } , u _ { | \\partial \\mathrm { M } } = \\varphi . \\end{align*}"}
+{"id": "949.png", "formula": "\\begin{align*} Z _ * : Z _ i ( X ) \\to Z _ i ( X ) , Z _ * ( Y ) = { \\pi _ { 2 , Z } } _ * ( \\pi _ { 1 , Z } ^ * ( Y ) ) . \\end{align*}"}
+{"id": "5766.png", "formula": "\\begin{align*} \\| g \\| _ { L ( \\log L ) , Q } & = \\inf \\Big \\{ \\lambda > 0 : \\frac { 1 } { | Q | } \\int _ { Q } \\frac { | g ( x ) | } { \\lambda } \\log ( e + | f | / \\lambda ) \\mathrm { d } x \\le 1 \\Big \\} . \\end{align*}"}
+{"id": "752.png", "formula": "\\begin{align*} ( \\phi ' ) ^ 2 + K \\phi ^ 2 = 1 , \\ \\phi '' = - K \\phi . \\end{align*}"}
+{"id": "4635.png", "formula": "\\begin{align*} A = \\displaystyle { \\frac { \\frac { 1 } { Q } - \\frac { 1 } { P } } { \\frac { 1 } { Q } + \\frac { 1 } { n } - \\frac { 1 } { R } } } . \\end{align*}"}
+{"id": "853.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\Big ( \\left \\vert u ( t ) - \\phi _ { 1 } ( \\lambda ) \\right \\vert + \\left \\vert u _ { t } ( t ) \\right \\vert \\Big ) = 0 \\ . \\end{align*}"}
+{"id": "4035.png", "formula": "\\begin{align*} 0 = \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\ , \\bigg | \\sum _ { i = 1 } ^ D \\alpha _ i \\ , \\langle T _ i ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) , f \\rangle \\bigg | ^ 2 \\end{align*}"}
+{"id": "8609.png", "formula": "\\begin{align*} E [ Z _ 0 ( \\ell \\Delta ) W ^ k _ { \\ell \\Delta , t } ( j ) ] = \\Delta \\nu p _ { j , + } ^ k ( t - \\ell \\Delta ) E [ Z _ 0 ( \\ell \\Delta ) ^ 2 ] , \\end{align*}"}
+{"id": "7422.png", "formula": "\\begin{align*} s = \\tfrac { 1 } { \\sqrt { L } } \\ , . \\end{align*}"}
+{"id": "8636.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } M _ + ( t ) = \\nu Y \\int _ 0 ^ { \\infty } e ^ { - \\lambda s } d s = \\nu Y / \\lambda , \\end{align*}"}
+{"id": "5459.png", "formula": "\\begin{align*} G ( \\theta ) = & \\mathbb { E } _ { \\Phi | \\Phi ( \\mathcal { A } ) > 0 } \\left \\{ \\left ( \\sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } \\right . \\right . \\\\ & \\left . \\left . \\prod \\limits _ { x _ { i } \\in \\Phi \\backslash \\{ x _ 1 \\} \\cap { \\mathcal { A } } } \\frac { 1 } { \\left ( 1 + \\frac { m \\eta \\theta r _ 1 ^ { \\alpha } } { M r _ i ^ { \\alpha } } \\right ) ^ M } \\right ) ^ b \\right \\} , \\end{align*}"}
+{"id": "8583.png", "formula": "\\begin{align*} & \\tau _ { j } ^ + ( k ) : = \\inf \\{ s > \\tau _ { j } ^ - ( k - 1 ) : Z _ 0 ( s ) = j \\} , \\\\ & \\tau _ { j } ^ - ( k ) : = \\inf \\{ s > \\tau _ { j } ^ + ( k ) : Z _ 0 ( s ) \\neq j \\} , k \\geq 1 . \\end{align*}"}
+{"id": "1651.png", "formula": "\\begin{align*} g ( ( \\pounds _ { \\xi _ i } { f } ) X , \\xi _ j ) = - g ( \\nabla _ { { f } X } \\ , \\xi _ i , \\xi _ j ) = 0 . \\end{align*}"}
+{"id": "4290.png", "formula": "\\begin{align*} \\rho e _ t + \\rho u e _ x + p u _ x + q _ x = \\frac { 1 } { \\mu } S ^ 2 . \\end{align*}"}
+{"id": "6735.png", "formula": "\\begin{align*} \\mathcal { P } ( X _ \\varphi ) = \\mathcal { P } ( [ \\underline { \\varphi } , \\varphi ] ) . \\end{align*}"}
+{"id": "6187.png", "formula": "\\begin{align*} \\min \\left \\{ P ( G ) + 2 \\lambda f ( | E \\Delta G | \\big ) \\ , : \\ , | G | = | B | \\right \\} , \\end{align*}"}
+{"id": "91.png", "formula": "\\begin{align*} ( P _ 0 - z - i M h W ) ^ { - 1 } ( P _ 0 - z ) = I + i ( P _ 0 - z - i M h W ) ^ { - 1 } M h W . \\end{align*}"}
+{"id": "6252.png", "formula": "\\begin{align*} L ^ \\sigma _ { h _ { s _ 1 } , \\ldots , h _ { s _ k } } \\ , = \\ , I _ { z _ 0 } ( [ \\alpha _ { s _ 1 } | \\cdots | \\alpha _ { s _ k } ] ) , k \\geq 0 , ( s _ 1 , \\ldots , s _ k ) \\in S _ \\star ^ k , \\end{align*}"}
+{"id": "6873.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { - 1 } \\log \\mathbb { P } ^ * _ N ( \\lambda _ N / N \\geq \\beta ) = - \\widehat { \\psi } _ r ( \\beta ) , \\beta \\in [ C _ r , 1 ] , \\end{align*}"}
+{"id": "4599.png", "formula": "\\begin{align*} \\begin{array} { l l l } V _ { n + 1 } ( x ) & = [ K + \\min _ { x \\leq y \\leq x + B } & ( v y + L _ { n + 1 } ( y ) + \\int _ 0 ^ \\infty C _ { n } ( y - \\xi ) f _ { n + 1 } ( \\xi ) \\mathrm { d } \\xi ) \\\\ & & - ( v x + L _ { n + 1 } ( x ) + \\int _ 0 ^ \\infty C _ { n } ( x - \\xi ) f _ { n + 1 } ( \\xi ) \\mathrm { d } \\xi ) ] ^ - . \\end{array} \\end{align*}"}
+{"id": "3473.png", "formula": "\\begin{align*} \\Psi ^ \\tau _ \\mu ( ( u , a ) , ( v , b ) ) = \\mu ( \\sigma _ v = b \\mid \\sigma _ u = a , \\sigma _ \\Lambda = \\tau ) - \\mu ( \\sigma _ v = b \\mid \\sigma _ \\Lambda = \\tau ) \\end{align*}"}
+{"id": "7491.png", "formula": "\\begin{align*} \\delta = x d _ 2 \\cdots d _ l \\quad \\quad \\tilde { \\delta } = ( x , d _ l , d _ { l - 1 } \\dots , d _ 2 ) . \\end{align*}"}
+{"id": "8131.png", "formula": "\\begin{align*} f \\mapsto \\tilde f : = ( f ( 0 ) , f ( x _ 1 ) , f ( x _ 1 , x _ 2 ) , \\ldots ) \\end{align*}"}
+{"id": "4391.png", "formula": "\\begin{align*} J _ 1 : = ( ( \\nabla \\cdot \\varphi _ 3 ) \\varphi _ 2 , \\theta _ { \\epsilon } \\varphi _ 1 ) , \\ \\ J _ 2 : = ( ( \\nabla \\cdot \\overline { \\varphi _ 3 } ) \\varphi _ 1 , \\theta _ { \\epsilon } \\varphi _ 2 ) , \\ \\ J _ 3 : = - ( \\nabla ( \\varphi _ 1 \\cdot \\overline { \\varphi _ 2 } ) , \\theta _ { \\epsilon } \\varphi _ 3 ) . \\end{align*}"}
+{"id": "6422.png", "formula": "\\begin{align*} x _ 1 \\bar { \\otimes } x _ 2 = ( h _ { \\varphi _ 1 } ^ { \\eta / q } \\bar { \\otimes } h _ { \\varphi _ 2 } ^ { \\eta / q } ) ( x _ 1 ' \\bar { \\otimes } x _ 2 ' ) ( h _ { \\varphi _ 1 } ^ { ( 1 - \\eta ) / q } \\bar { \\otimes } h _ { \\varphi _ 2 } ^ { ( 1 - \\eta ) / q } ) = h _ { \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 } ^ { \\eta / q } ( x _ 1 ' \\bar { \\otimes } x _ 2 ' ) h _ { \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 } ^ { ( 1 - \\eta ) / q } , \\end{align*}"}
+{"id": "4208.png", "formula": "\\begin{align*} c ( t ) = \\frac { \\det \\phi ( t ) } { \\prod _ { k = 1 } ^ R u _ { \\beta _ k , \\tau _ k } ( t ) } \\end{align*}"}
+{"id": "696.png", "formula": "\\begin{align*} \\int _ { I _ \\alpha ^ { ( k ) } } \\Big | \\sum _ { i = 0 } ^ { N - 1 } D \\varphi ( T ^ i x ) \\Big | \\ , d x & = k \\| Z ( k + 1 ) \\| O ( e ^ { ( - \\lambda _ 1 ( 1 - a ) + \\tau ) k } ) ( c _ 0 + p _ a ) ( D \\varphi ) \\\\ & = O ( e ^ { ( - \\lambda _ 1 ( 1 - a ) + 3 \\tau ) k } ) ( c _ 0 + p _ a ) ( D \\varphi ) . \\end{align*}"}
+{"id": "3553.png", "formula": "\\begin{align*} \\frac { 1 } { T _ { * } } = \\frac { A \\varepsilon } { k T } . \\end{align*}"}
+{"id": "4210.png", "formula": "\\begin{align*} G ( 1 + z ) = ( 2 \\pi ) ^ { z / 2 } e ^ { - ( z + 1 ) z / 2 - \\gamma _ E z ^ 2 / 2 } \\prod _ { k = 1 } ^ \\infty \\left ( \\left ( 1 + \\frac { z } { k } \\right ) ^ k e ^ { - z + z ^ 2 / ( 2 k ) } \\right ) \\end{align*}"}
+{"id": "3662.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) _ j u _ j + q _ j u _ j : = \\sum _ { i = 1 } ^ M \\alpha _ i ^ j ( - \\Delta ) ^ { s _ i } u _ j + q _ j u _ j = 0 & \\Omega , \\\\ u _ j = f _ j & \\Omega ^ c \\end{cases} \\end{align*}"}
+{"id": "3009.png", "formula": "\\begin{align*} \\phi ( A \\otimes B ) = U ( \\varphi _ 1 ( A ) \\otimes \\varphi _ 2 ( B ) ) V \\hbox { f o r a l l $ A \\in M _ m $ a n d $ B \\in M _ n $ , } \\end{align*}"}
+{"id": "5417.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k \\alpha _ j y _ { n + j } = \\sum _ { j = 0 } ^ k h \\beta _ j f _ { n + j } \\end{align*}"}
+{"id": "5377.png", "formula": "\\begin{align*} u _ { i , k } = \\begin{bmatrix} u _ { i , k } ^ 1 \\\\ u _ { i , k } ^ 2 \\end{bmatrix} , y _ { i , k } = \\begin{bmatrix} y _ { i , k } ^ 1 \\\\ y _ { i , k } ^ 2 \\end{bmatrix} , B _ i = [ B _ i ^ 1 , B _ i ^ 2 ] , C _ i = \\begin{bmatrix} C _ i ^ 1 \\\\ C _ i ^ 2 \\end{bmatrix} , D _ i = \\begin{bmatrix} D _ i ^ { 1 1 } & D _ i ^ { 1 2 } \\\\ D _ i ^ { 2 1 } & D _ i ^ { 2 2 } \\end{bmatrix} , i = 1 , 2 , \\end{align*}"}
+{"id": "7638.png", "formula": "\\begin{align*} M ( 0 , 1 , y ) = 5 7 6 , \\ ; y \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "5771.png", "formula": "\\begin{align*} | \\{ x \\in \\mathbb { R } ^ { n } : | T _ { \\alpha } ( \\vec { f } ) ( x ) | > \\lambda \\} | & \\le \\sum _ { s = 1 } ^ { 2 ^ { m } } | E _ { s } | + C | \\Omega ^ { * } | \\\\ & \\le \\sum _ { s = 2 } ^ { 2 ^ { m } } | E _ { s } | + C B ^ { q } \\lambda ^ { - \\frac { n } { m n - \\alpha } } \\gamma ^ { q - \\frac { n } { m n - \\alpha } } + C ( \\lambda \\gamma ) ^ { \\frac { - n } { m n - \\alpha } } . \\end{align*}"}
+{"id": "5351.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ q \\le \\sqrt { \\gamma } ( 6 \\sqrt { 2 } r q ) ^ d \\le \\gamma ( 6 \\sqrt { 2 } e ^ 2 r q ) ^ { d } \\le \\gamma \\left ( 6 4 r \\sqrt { q } \\sqrt { \\max \\{ \\log ( 1 / \\gamma ) , \\ , q \\} } \\right ) ^ d , \\end{align*}"}
+{"id": "8005.png", "formula": "\\begin{align*} g _ k ( ( t + s ) / 2 ) & = \\| A _ k ^ { ( t + s ) / 2 } Z _ k B _ k ^ { t + s } Z ^ * _ k A _ k ^ { ( t + s ) / 2 } \\| _ { \\infty } ^ { 1 / 2 } \\\\ & = \\rho ^ { 1 / 2 } ( A _ k ^ { t } Z _ k B _ k ^ { t + s } Z ^ * _ k A _ k ^ { s } ) \\\\ & \\le \\| A _ k ^ { t } Z _ k B _ k ^ { t + s } Z ^ * _ k A _ k ^ { s } \\| _ { \\infty } ^ { 1 / 2 } \\\\ & \\le \\| A _ k ^ { t } Z _ k B _ k ^ { t } \\| _ { \\infty } ^ { 1 / 2 } \\| B _ k ^ { s } Z ^ * _ k A _ k ^ { s } \\| _ { \\infty } ^ { 1 / 2 } \\\\ & = \\{ g _ k ( t ) g _ k ( s ) \\} ^ { 1 / 2 } . \\end{align*}"}
+{"id": "7141.png", "formula": "\\begin{align*} \\partial _ { x _ n } u ( \\hat { x } ) = \\partial _ { x _ n } u ( \\hat { x } ) - \\partial _ { x _ n } u ( \\hat { x } ^ \\prime , 0 ) = \\Big ( \\int _ { 0 } ^ 1 \\partial _ { x _ n } ^ 2 u ( \\hat { x } ^ \\prime , t x _ n ) \\ : t \\Big ) x _ n . \\end{align*}"}
+{"id": "6741.png", "formula": "\\begin{align*} \\underline { C } : = \\underline { \\varphi } ^ { - 1 } ( \\underline { A } ) = \\Gamma _ { H , H ^ * } ^ { - 1 } ( ( \\varphi ^ * ) ^ { - 1 } ( \\underline { A } ) ) = \\Gamma _ { H , H ^ * } ^ { - 1 } \\bigcap _ { s \\in \\underline { S } } ( R ^ * ) ^ { - s } ( W ^ * ) ^ c = \\bigcap _ { s \\in \\underline { S } } \\Gamma _ { H , H ^ * } ^ { - 1 } ( R ^ * ) ^ { - s } ( W ^ * ) ^ c . \\end{align*}"}
+{"id": "2658.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 } s _ \\alpha [ Y ] ( x ) = l [ Y ] ( x ) \\lim _ { \\alpha \\to 0 } t _ \\alpha [ Y ] ( x ) = l [ Y ] ( x ) \\end{align*}"}
+{"id": "1825.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ { * 2 } } \\phi _ t ( p _ 1 , p _ 2 , q _ 2 , q _ 1 ) \\delta ( q _ 1 - p _ 2 ) \\d p _ 2 \\d q _ 1 = \\delta ( p _ 1 - p _ 3 ) g ( p _ 1 ) \\end{align*}"}
+{"id": "5026.png", "formula": "\\begin{align*} f = f _ 0 + f _ 1 + \\dots + f _ p + f _ { a d d } , \\end{align*}"}
+{"id": "8559.png", "formula": "\\begin{align*} t _ N : = \\inf \\{ t \\geq 0 : Y e ^ { \\lambda t } = N \\} . \\end{align*}"}
+{"id": "8411.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { \\left ( n ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } = \\frac { \\sqrt { \\pi } \\ , x ^ { 1 - 2 s } } { 2 \\Gamma ( s ) } \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) - \\frac { x ^ { - 2 s } } { 2 } + \\frac { 2 \\ , \\pi ^ { s } x ^ { \\frac { 1 } { 2 } - s } } { \\Gamma ( s ) } \\ , \\sum _ { n = 1 } ^ { \\infty } n ^ { s - \\frac { 1 } { 2 } } \\ , K _ { s - \\frac { 1 } { 2 } } ( 2 \\pi n x ) . \\end{align*}"}
+{"id": "7004.png", "formula": "\\begin{align*} \\nu _ i ( f ) = \\min _ { 1 \\leq j \\leq r } \\left \\{ v \\left ( \\tilde b _ j \\right ) \\right \\} . \\end{align*}"}
+{"id": "546.png", "formula": "\\begin{align*} \\inf \\limits _ { t \\in [ 0 , T ] } a ( t ) = a _ { 0 } \\sup \\limits _ { t \\in [ 0 , T ] } a ( t ) = a _ { 1 } . \\end{align*}"}
+{"id": "5314.png", "formula": "\\begin{align*} \\mu _ p ( g ) = \\mu _ p ( f _ { S \\to 0 } ) \\leq 8 \\exp \\left ( - \\frac { 0 . 0 0 0 1 } { p } \\right ) \\leq \\frac { 1 } { 4 } \\mu _ p ( f ) . \\end{align*}"}
+{"id": "5121.png", "formula": "\\begin{align*} \\tilde { C } = ( n , \\Lambda , p _ 0 , \\| D ^ 2 u \\| _ { L ^ { \\infty } ( B _ 1 ) } ) \\omega ^ { p _ 0 } . \\end{align*}"}
+{"id": "9243.png", "formula": "\\begin{align*} V _ n = ( - 1 ) ^ n E _ { 2 n } \\frac { \\pi ^ { 2 n } } { ( 2 n ) ! } , W _ n = \\frac { 1 } { n ! } \\Bigl ( \\frac { \\pi } { 2 } \\Bigr ) ^ n . \\end{align*}"}
+{"id": "1252.png", "formula": "\\begin{align*} P _ k : = \\frac { \\int _ 0 ^ 1 \\phi \\circ T ^ k _ { \\gamma + \\delta } \\ , d m - \\int _ 0 ^ 1 \\phi \\circ T ^ k _ \\gamma \\ , d m - D \\cdot \\delta } { \\Vert \\delta \\Vert } . \\end{align*}"}
+{"id": "1473.png", "formula": "\\begin{align*} \\int _ \\Omega f ^ { ' } _ 0 ( U _ { \\mu _ i , \\xi _ i } ) \\phi \\psi ^ h _ { \\mu _ j , \\xi _ j } d x = 0 . \\end{align*}"}
+{"id": "2738.png", "formula": "\\begin{align*} ( A _ q u | v ) = \\mathfrak { a } _ q ( u , v ) , u \\in D ( A ) , \\ ; v \\in H _ 0 ^ 1 ( \\mathrm { M } ) . \\end{align*}"}
+{"id": "1211.png", "formula": "\\begin{align*} A '' ( t ) = A ' ( t ) \\left ( A ' ( t ) F ( A ( t ) ) - 1 \\right ) A ( 0 ) = 0 A ' ( 0 ) = 1 . \\end{align*}"}
+{"id": "3549.png", "formula": "\\begin{align*} B = B ( T ) = - 2 \\pi \\int _ { 0 } ^ { \\infty } \\left [ e ^ { - u ( r ) / k T } - 1 \\right ] r ^ { 2 } \\ , d r , \\end{align*}"}
+{"id": "2729.png", "formula": "\\begin{align*} \\begin{aligned} J _ 1 \\geq C s ^ 3 \\lambda ^ 4 \\iint _ Q \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d y d t - C s ^ 3 \\lambda ^ 3 \\int _ 0 ^ T \\int _ { \\omega _ 0 } \\xi ^ 3 | u | ^ 2 d x d y d t , \\end{aligned} \\end{align*}"}
+{"id": "9092.png", "formula": "\\begin{align*} Y _ 1 = 0 , \\ldots , Y _ d = 0 , \\ , x _ i - x _ j = 0 , \\ , \\ , \\{ i , j \\} \\in E \\setminus E ( H ) , \\end{align*}"}
+{"id": "665.png", "formula": "\\begin{gather*} \\Delta u _ { k + 1 } = u _ { k + 1 } - Z ( k + 1 ) u _ k = P _ { U ^ { ( k + 1 ) } _ { - j } } \\Delta v _ { k + 1 } , \\\\ \\Delta e _ { k + 1 } = e _ { k + 1 } - Z ( k + 1 ) e _ k = P _ { E ^ { ( k + 1 ) } _ { - j } } \\Delta v _ { k + 1 } , \\\\ \\Delta u _ 0 = u _ 0 = P _ { U ^ { ( 0 ) } _ { - j } } \\Delta v _ { 0 } , \\ \\Delta e _ 0 = e _ 0 = P _ { E ^ { ( 0 ) } _ { - j } } \\Delta v _ { 0 } . \\end{gather*}"}
+{"id": "1987.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u ) ^ n = \\psi ( \\cdot , u ) \\ , \\omega ^ n & \\textnormal { o n } & \\Omega \\\\ u = 0 & \\textnormal { o n } & \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "130.png", "formula": "\\begin{align*} | \\hat { \\varphi } _ { l , d } ( \\zeta ) | = | l \\hat { \\varphi } ( l \\zeta ) e ^ { - i d \\zeta } | \\leq C l e ^ { ( d - l ) \\Im \\zeta } ( 1 + | l \\zeta | ) ^ { - N } . \\end{align*}"}
+{"id": "7296.png", "formula": "\\begin{align*} n m ' - k m ' - l k ' = ( n - k ) \\frac { m - l } { ( k , m - l ) } - l \\frac { k } { ( k , m - l ) } = \\frac { n m - n l - k m } { ( k , m - l ) } > 0 , \\end{align*}"}
+{"id": "4353.png", "formula": "\\begin{align*} D _ { i } ^ { p } ( x ) : = \\alpha _ { i } \\nabla d _ { S _ { i } } ^ { p } \\left ( x \\right ) . \\end{align*}"}
+{"id": "926.png", "formula": "\\begin{align*} & \\alpha ^ + _ { k } \\gamma _ k ^ + = \\frac { k ( 2 \\nu _ 1 + N - k ) ( 2 \\nu _ 2 + N - k ) ( 2 \\nu _ { 1 2 } + 2 N - k ) } { ( \\nu _ { 1 2 } + N - k ) ^ 2 } , \\\\ & \\alpha ^ - _ { k } \\gamma _ { k } ^ + = \\frac { 4 ( 2 \\nu _ 3 + k ) ( N - k ) ( 2 \\nu _ { 1 2 } + N - k - 1 ) ( 2 \\nu _ { 1 2 3 } + 2 N - k - 1 ) } { ( 2 \\nu _ { 1 2 } + 2 N - 2 k - 1 ) ^ 2 } , \\\\ & \\beta ^ \\pm _ k = \\mp \\frac { ( \\nu _ 1 - \\nu _ 2 ) ( \\nu _ { 1 2 } + N ) } { \\nu _ { 1 2 } + N - k } \\pm \\frac { ( 2 \\nu _ { 1 2 } - 1 ) ( 2 \\nu _ { 1 2 } + 4 \\nu _ 3 + 2 N + 1 ) } { 2 ( 2 \\nu _ { 1 2 } + 2 N - 2 k \\mp 1 ) } . \\end{align*}"}
+{"id": "1628.png", "formula": "\\begin{align*} \\eta ^ i ( [ Q X , { f } Y ] ) + ( { f } Y ) ( \\eta ^ i ( X ) ) - ( { f } X ) ( \\eta ^ i ( Y ) ) + \\eta ^ i ( [ { f } X , Y ] ) = 0 . \\end{align*}"}
+{"id": "3356.png", "formula": "\\begin{align*} S ^ * T T ^ * S = ( I \\otimes T ) Z ^ * Z ( I \\otimes T ^ * ) \\leq I \\otimes T T ^ * . \\end{align*}"}
+{"id": "2971.png", "formula": "\\begin{align*} ( k , \\sigma ) \\cdot f = \\psi ( k ) \\cdot ( \\sigma \\ast f ) \\ . \\end{align*}"}
+{"id": "4260.png", "formula": "\\begin{align*} V \\left ( t , \\xi \\right ) = \\inf _ { u \\in \\Lambda _ { t , \\xi } } J \\left ( t , \\xi ; u \\right ) \\leq \\inf _ { u ^ { \\prime } \\in \\Lambda _ { t , \\xi } } J \\left ( t , \\xi ^ { \\prime } ; u ^ { \\prime } \\right ) = V \\left ( t , \\xi ^ { \\prime } \\right ) . \\end{align*}"}
+{"id": "3932.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\int _ { \\mathcal { V } } g _ \\lambda \\ , d \\pi \\right ] = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\left [ \\langle \\lambda , \\delta \\rangle + \\int _ { \\mathcal { V } } g _ \\lambda \\ , d \\pi \\right ] . \\end{align*}"}
+{"id": "8241.png", "formula": "\\begin{align*} \\frac { \\partial G _ { k } } { \\partial t } ( x , t ) = \\sum _ { i = 0 } ^ { l - 1 } ( - 1 ) ^ { i + 1 } ( \\frac { 1 } { \\sqrt { x } } S _ { i + 1 } G _ { k } - T _ { i } G _ { k } ) ( \\frac { 2 t } { \\sqrt { x } } ) ^ { i } + ( - 1 ) ^ { l } T _ { l + 2 } G _ { k } ( \\frac { 2 t } { \\sqrt { x } } ) ^ { l } . \\end{align*}"}
+{"id": "2114.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ r \\sum _ { j = 1 } ^ s F ^ { k _ { i , j } n _ j } ( Q _ j ) \\colon n _ j \\ge 0 j = 1 , \\dots , s \\right \\} , \\end{align*}"}
+{"id": "6390.png", "formula": "\\begin{align*} D = ( x ^ 2 - x + 2 / 3 , 1 5 x - 1 0 ) , \\end{align*}"}
+{"id": "3738.png", "formula": "\\begin{align*} U ( \\vec { a } , \\Lambda ) \\phi ^ { \\alpha , n } & ( f ) U ( \\vec { a } , \\Lambda ) ^ { - 1 } 1 \\\\ & = ( \\Lambda S _ 0 + \\vec { a } , e ^ { - i [ \\vec { a } \\cdot ( \\hat { H } ( \\rho _ n ) \\hat { g } _ 0 + \\hat { P } ( \\rho _ n ) \\hat { g } _ 1 ) ] } f ^ { S _ 0 ^ \\flat } ( \\Lambda ^ { - 1 } ( \\cdot - \\vec { a } ) ) \\otimes \\rho _ n ( E ^ \\alpha ) ) . \\end{align*}"}
+{"id": "6354.png", "formula": "\\begin{align*} g ' ( t ) = f ' \\bigg ( \\alpha ( s _ 2 ) + \\frac { d ( s _ 2 ) } { d ( s _ 1 ) } ( t - \\alpha ( s _ 1 ) ) \\bigg ) \\leq f ' ( t ) . \\end{align*}"}
+{"id": "4358.png", "formula": "\\begin{align*} \\mathcal { A } ( \\alpha , p ) : = \\arg \\min \\sum \\limits _ { i = 1 } ^ { m } \\alpha _ { i } d _ { S _ { i } } ^ { p } , \\end{align*}"}
+{"id": "6588.png", "formula": "\\begin{align*} \\begin{aligned} y = & \\sqrt { P _ s } \\zeta \\sum _ { l = 1 } ^ L \\hat h _ l e ^ { j \\phi _ { l , n _ t } } g _ { l , n _ t } + \\\\ & \\sqrt { P _ s ( 1 - \\zeta ^ 2 ) } \\sum _ { l = 1 } ^ L \\Delta { h } e ^ { j \\phi _ { l , n _ t } } g _ { l , n _ t } + n _ 0 , \\end{aligned} \\end{align*}"}
+{"id": "8840.png", "formula": "\\begin{align*} F _ \\lambda ( t ) : = \\frac { f ' ( u ( t ) ) } { 1 - \\lambda f ' ( u ( t ) ) } = \\frac { F ( t ) } { 1 - \\lambda F ( t ) } , \\lambda > 0 , \\end{align*}"}
+{"id": "1164.png", "formula": "\\begin{align*} T ( f ^ { n } ) = n f ^ { n - 1 } T ( f ) + n B ( A ( f ) , A ( f ^ { n - 1 } ) ) \\end{align*}"}
+{"id": "8592.png", "formula": "\\begin{align*} E \\big | S _ { j , + } ^ k ( t ) - \\hat { S } _ { j , + } ^ k ( t ) \\big | = O ( \\theta ^ { k / 2 } t e ^ { \\lambda t / 2 } ) . \\end{align*}"}
+{"id": "3596.png", "formula": "\\begin{align*} q ( 2 ) \\ = \\ \\frac { 1 0 } { 3 } \\ = \\ 3 . \\overline { 3 } \\ > \\ 3 . 3 2 \\cdots \\ = \\ \\log _ 2 1 0 , \\end{align*}"}
+{"id": "8445.png", "formula": "\\begin{align*} I _ { m , p } ( s , x ) : = \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\mu - i \\infty } ^ { \\mu + i \\infty } \\left ( 2 z , 2 \\pi p m \\right ) _ { m } \\ , \\Gamma \\left ( z \\right ) \\ , \\Gamma \\left ( s + z - \\frac { 1 } { 2 } \\right ) \\ , ( \\pi \\ , x \\ , m ) ^ { - 2 z } \\ , d z . \\end{align*}"}
+{"id": "4888.png", "formula": "\\begin{align*} U _ m ( z ) : = \\frac { U ( z _ 0 + \\lambda _ m z ) } { \\lambda _ m ^ { k _ 0 + s } } , \\lambda _ m : = 1 - r _ m \\to 0 ^ + , \\end{align*}"}
+{"id": "337.png", "formula": "\\begin{align*} \\prod _ { \\substack { \\gcd ( j _ 1 , j _ 2 , j _ 3 , j _ 4 , k ) = 1 \\\\ j _ 1 , j _ 2 , j _ 3 , j _ 4 < k \\\\ j _ 1 , j _ 2 , j _ 3 , j _ 4 \\geq 1 ; k \\geq 2 } } \\left ( \\frac { 1 } { 1 - y ^ { j _ 1 + j _ 2 + j _ 3 + j _ 4 } z ^ k } \\right ) ^ { \\frac { 1 } { k } } , \\end{align*}"}
+{"id": "3266.png", "formula": "\\begin{align*} { \\rm R e } ( \\overline { a ( \\xi ) } f ( \\xi ) ) = 0 { \\rm \\ f o r \\ } \\xi \\in L \\end{align*}"}
+{"id": "1785.png", "formula": "\\begin{align*} \\psi ^ \\ell _ { i _ 1 , \\ldots , i _ n } ( f _ { i _ 1 , \\ldots , i _ n } ) = f _ { i _ 1 , \\ldots , i _ \\ell , i _ \\ell , \\ldots , i _ n } \\end{align*}"}
+{"id": "8449.png", "formula": "\\begin{align*} K _ { \\nu } ( z ) = \\frac { \\sqrt { \\pi } z ^ { \\nu } } { 2 ^ { \\nu } \\Gamma \\left ( \\nu + \\frac { 1 } { 2 } \\right ) } \\ , \\intop _ { 1 } ^ { \\infty } e ^ { - z t } \\left ( t ^ { 2 } - 1 \\right ) ^ { \\nu - \\frac { 1 } { 2 } } d t , \\ , \\ , \\ , \\ , \\ , ( \\nu ) > - \\frac { 1 } { 2 } , \\ , \\ , \\ , | \\arg ( z ) | < \\frac { \\pi } { 2 } . \\end{align*}"}
+{"id": "3941.png", "formula": "\\begin{align*} \\sup _ { \\gamma \\in \\bar { \\mathcal { P } } } \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 , \\gamma \\right ) } \\int _ { \\mathcal { V } \\times \\mathcal { S } } \\phi _ \\lambda \\ , d \\pi = \\sup _ { \\pi \\in \\mathcal { G } _ { \\lambda } } \\int _ { \\mathcal { V } \\times \\mathcal { S } } \\phi _ \\lambda \\ , d \\pi . \\end{align*}"}
+{"id": "1530.png", "formula": "\\begin{align*} ( a t + b ) ^ { p \\theta + p - 1 } & = ( a t + b ) ^ { p \\theta } ( a t + b ) ^ { p - 1 } \\\\ & = \\begin{cases} a ^ { p \\theta } t ^ { p \\theta + p - 1 } b = 0 , \\\\ \\sum _ { m = 0 } ^ { p - 1 } \\binom { p - 1 } { m } \\left ( a ^ { p \\theta - m } b ^ m t ^ { p \\theta + p - 1 - m } + a ^ { - m } b ^ { p \\theta + m } t ^ { p - 1 - m } \\right ) . \\end{cases} \\\\ & = a ^ { p \\theta } t ^ { p \\theta + p - 1 } + r _ { \\theta } ( t ) , \\end{align*}"}
+{"id": "7173.png", "formula": "\\begin{align*} C _ T : = { \\rm c o n v } \\left ( \\cup B \\left ( x ' , \\alpha \\delta _ h \\right ) \\right ) , \\end{align*}"}
+{"id": "9305.png", "formula": "\\begin{align*} Z _ { k \\ell } [ a ] : = \\left ( \\begin{array} { c c c c c } a _ 1 & a _ 2 & \\ldots & a _ { \\ell - 1 } & - a _ 1 - a _ 2 - \\ldots - a _ { \\ell - 1 } \\\\ \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\ a _ 1 & a _ 2 & \\ldots & a _ { \\ell - 1 } & - a _ 1 - a _ 2 - \\ldots - a _ { \\ell - 1 } \\\\ \\end{array} \\right ) , { \\rm i f \\ } \\ell \\geq 2 , \\end{align*}"}
+{"id": "9160.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } ^ { 1 } & = u ^ { 1 } \\cos ( x ^ { 3 } ) \\\\ \\dot { x } ^ { 2 } & = u ^ { 1 } \\sin ( x ^ { 3 } ) \\\\ \\dot { x } ^ { 3 } & = u ^ { 2 } \\ , , \\end{aligned} \\end{align*}"}
+{"id": "9215.png", "formula": "\\begin{align*} & x \\oslash y = \\frac { x } { y } ( 1 + \\eta _ 1 ) , & | \\eta _ 1 | \\le 2 ^ { - d } , \\end{align*}"}
+{"id": "544.png", "formula": "\\begin{align*} S A - A ^ { * } S = 0 . \\end{align*}"}
+{"id": "7868.png", "formula": "\\begin{align*} \\mathsf { M } ( \\sigma ( A ) , i ) : = \\min _ { i \\leq j \\leq k } \\left \\{ \\sigma ( a _ j ) : \\ \\sigma ( a _ j ) < \\sigma ( a _ i ) \\right \\} . \\end{align*}"}
+{"id": "4275.png", "formula": "\\begin{align*} Y _ { 0 } = \\mathbb { E } \\left [ \\Psi _ { T } \\left \\langle G X _ { T } , X _ { T } \\right \\rangle + \\int _ { 0 } ^ { T } \\Psi _ { s } ( \\left \\langle Q _ { s } X _ { s } , X _ { s } \\right \\rangle + \\left \\langle R _ { s } u _ { s } , u _ { s } \\right \\rangle ) \\mathrm { d } s \\right ] . \\end{align*}"}
+{"id": "5976.png", "formula": "\\begin{align*} \\overline { \\overline { \\Pi } } _ { \\psi } [ \\omega ] f ( [ \\epsilon , x ] ) = - i \\nu ( \\epsilon , \\omega ) \\int _ { \\R } e ^ { 2 \\pi i \\epsilon x y } f ( [ \\epsilon , - y ] ) d y ; \\end{align*}"}
+{"id": "4692.png", "formula": "\\begin{align*} \\mathcal { R } _ { \\chi , \\varepsilon } ( z ) = \\left ( \\mathcal { A } _ { 0 , \\chi , \\varepsilon } - z I \\right ) ^ { - 1 } - S _ { \\chi , \\varepsilon } ( z ) M _ { \\chi , \\varepsilon } ( z ) ^ { - 1 } S _ { \\chi , \\varepsilon } ( \\overline { z } ) ^ * , \\end{align*}"}
+{"id": "6461.png", "formula": "\\begin{align*} z ( s ) & = h ^ 2 \\beta _ h ( s ) + \\frac { h ^ 2 } { 2 } \\overline { \\beta _ h ( s ) } + \\frac { \\sigma ^ 2 } { 2 } \\overline { \\beta _ \\sigma ( s ) } , \\\\ \\zeta ( s ) & = i \\beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \\overline { \\beta _ h ( s ) } K _ 2 - i \\overline { \\beta _ \\sigma ( s ) } K , \\\\ \\widetilde \\gamma ( s ) & = - i s ( \\beta _ h ( s ) - 1 ) ( | K _ 1 | ^ 2 + | K _ 3 | ^ 2 ) + i s ( \\overline { \\beta _ h ( s ) } - 1 ) | K _ 2 | ^ 2 + i s ( \\overline { \\beta _ \\sigma ( s ) } - 1 ) | K | ^ 2 . \\end{align*}"}
+{"id": "1583.png", "formula": "\\begin{align*} \\mathrm { f } ( k ) : = \\sum _ { j = 1 } ^ { d } 2 ( 1 - \\cos k _ j ) . \\end{align*}"}
+{"id": "9175.png", "formula": "\\begin{align*} e _ { 1 , [ 2 ] } ^ { 1 } + a _ { 1 } ^ { 1 , 1 } e _ { 1 , [ 1 ] } ^ { 1 } + a _ { 1 } ^ { 1 , 0 } e _ { 1 } ^ { 1 } & = 0 \\\\ e _ { 2 , [ 2 ] } ^ { 1 } + a _ { 2 } ^ { 1 , 1 } e _ { 2 , [ 1 ] } ^ { 1 } + a _ { 2 } ^ { 1 , 0 } e _ { 2 } ^ { 1 } & = 0 \\end{align*}"}
+{"id": "650.png", "formula": "\\begin{gather*} I _ { \\alpha } : = [ l _ \\alpha , r _ \\alpha ) , l _ \\alpha = \\sum _ { \\pi _ 0 ( \\beta ) < \\pi _ 0 ( \\alpha ) } \\lambda _ \\beta , \\ ; \\ ; \\ ; r _ \\alpha = \\sum _ { \\pi _ 0 ( \\beta ) \\leq \\pi _ 0 ( \\alpha ) } \\lambda _ \\beta . \\end{gather*}"}
+{"id": "2943.png", "formula": "\\begin{align*} \\Theta ( u , \\tau ) : = q ^ { 1 / 1 2 } ( t ^ { 1 / 2 } - t ^ { - 1 / 2 } ) \\prod _ { n > 0 } ( 1 - q ^ n t ) ( 1 - q ^ n t ^ { - 1 } ) . \\end{align*}"}
+{"id": "8548.png", "formula": "\\begin{align*} \\sqrt { \\alpha } \\left ( \\frac { 1 } { 4 } \\gamma - \\frac { 1 } { 4 } \\log ( 4 \\beta ) + \\sum _ { m , n = 1 } ^ { \\infty } K _ { 0 } ( 2 m n \\ , \\alpha ) \\right ) = \\sqrt { \\beta } \\left ( \\frac { 1 } { 4 } \\gamma - \\frac { 1 } { 4 } \\log ( 4 \\alpha ) + \\sum _ { m , n = 1 } ^ { \\infty } K _ { 0 } ( 2 m n \\ , \\beta ) \\right ) , \\end{align*}"}
+{"id": "4175.png", "formula": "\\begin{align*} \\big | \\| \\varphi _ { n , t } ( b ) \\| ^ 2 - \\| b \\| ^ 2 \\big | & = \\big | \\| \\varphi _ { n , t } ( b ) ^ * \\varphi _ { n , t } ( b ) \\| - \\| b ^ * b \\| \\big | \\\\ & \\leq \\big | \\| \\varphi _ { n , t ^ { - 1 } } ( b ^ * ) \\varphi _ { n , t } ( b ) \\| - \\| \\varphi _ { n , e } ( b ^ * b ) \\| \\big | + \\big | \\| \\varphi _ { n , e } ( b ^ * b ) \\| - \\| b ^ * b \\| \\big | \\\\ & \\leq \\big | \\| \\varphi _ { n , t ^ { - 1 } } ( b ^ * ) \\varphi _ { n , t } ( b ) - \\varphi _ { n , e } ( b ^ * b ) \\| \\big | + \\big | \\| \\varphi _ { n , e } ( b ^ * b ) \\| - \\| b ^ * b \\| \\big | \\end{align*}"}
+{"id": "6012.png", "formula": "\\begin{align*} \\mathcal { G } _ { X ^ { \\ast } X } ( f ) ( x ) & = \\int _ { X ^ { \\ast } } f ( [ y ^ { \\ast } , 0 ] [ x , 0 ] ) d y ^ { \\ast } \\\\ & = \\int _ { X ^ { \\ast } } \\psi ( \\langle y ^ { \\ast } , x \\rangle ) f ( y ^ { \\ast } ) d y ^ { \\ast } \\\\ & = \\mathcal { F } ' _ { X ^ { \\ast } X } ( f ) ( x ) . \\end{align*}"}
+{"id": "7667.png", "formula": "\\begin{align*} \\phi ( t ) = \\begin{cases} t - \\frac { \\delta } { 2 } , & \\mbox { i f } t \\in \\left [ \\frac { \\delta } { 2 } , \\frac { \\delta } { 2 } + 1 \\right ] , \\\\ 1 , & \\mbox { i f } t \\in \\left [ \\frac { \\delta } { 2 } + 1 , \\delta - 1 \\right ] , \\\\ \\delta - t , & \\mbox { i f } t \\in \\left [ \\delta - 1 , \\delta \\right ] , \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "9019.png", "formula": "\\begin{align*} V ( p , t ) = \\inf _ { t ' \\in ] t , T ] } \\{ V ( \\bar p , t ' ) + C ( p , \\bar p , t , t ' ) \\} = \\inf _ { t ' \\in ] t , T ] } [ C ( p , \\bar p , t , t ' ) + \\tilde C ( \\bar p , t ' ) ] \\end{align*}"}
+{"id": "5930.png", "formula": "\\begin{align*} \\Pi _ { \\psi } [ \\begin{pmatrix} a & 0 \\\\ 0 & a ^ { \\ast - 1 } \\end{pmatrix} ] f ( [ \\epsilon , y ] ) = | \\det ( a ) | ^ { 1 / 2 } ( \\det a , \\epsilon ) _ F f ( [ \\epsilon , y a ] ) . \\end{align*}"}
+{"id": "3102.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ N ( 1 - z q ^ { i - 1 } ) & = \\sum _ { j = 0 } ^ N { N \\brack j } _ q ( - 1 ) ^ j z ^ j q ^ { j ( j - 1 ) / 2 } ; \\\\ \\frac { 1 } { \\prod _ { i = 1 } ^ N ( 1 - z q ^ { i - 1 } ) } & = \\sum _ { j = 0 } ^ { \\infty } { N + j - 1 \\brack j } _ q z ^ j . \\end{align*}"}
+{"id": "3679.png", "formula": "\\begin{align*} H ^ { - s } ( \\mathbb { R } ^ d ) : = \\{ \\xi \\in \\mathcal { S } ' ( \\mathbb { R } ^ d ) : \\{ 1 + | \\xi | ^ { - s } \\hat { \\xi } \\} \\in L ^ 2 ( \\mathbb { R } ^ d ) \\} , \\end{align*}"}
+{"id": "1249.png", "formula": "\\begin{align*} f _ { \\beta + \\delta , 2 } ' ( y ) = f _ { \\beta , 1 } ' ( y ) + \\partial _ 2 f ' _ { \\beta , 2 } ( y ) \\cdot \\delta + O ( \\delta ^ 2 ) , \\end{align*}"}
+{"id": "6794.png", "formula": "\\begin{align*} W ^ { \\mathrm { s } } ( p ) & = \\R ^ k _ { x _ 1 , \\dots , x _ k } \\times \\{ 0 \\} _ { x _ { k + 1 } , \\dots , x _ { 2 n } } , \\\\ W ^ { \\mathrm { u } } ( p ) & = \\{ 0 \\} _ { x _ 1 , \\dots , x _ k } \\times \\R ^ { 2 n - k } _ { x _ { k + 1 } , \\dots , x _ { 2 n } } . \\end{align*}"}
+{"id": "4549.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } S _ { \\omega , \\mathbf { c } } ( \\Phi ^ { \\lambda } ) \\big | _ { \\lambda = 1 } = \\sum _ { j = 1 } ^ 3 \\left . \\left \\langle ( D _ j S _ { \\omega , \\mathbf { c } } ) ( \\Phi ^ { \\lambda } ) , \\frac { d } { d \\lambda } \\Phi ^ { \\lambda } \\right \\rangle \\right | _ { \\lambda = 1 } = \\sum _ { j = 1 } ^ 3 \\left \\langle D _ j S _ { \\omega , \\mathbf { c } } ( \\Phi ) , \\frac { d } { 2 } \\Phi + \\nabla \\cdot \\Phi \\right \\rangle . \\end{align*}"}
+{"id": "2510.png", "formula": "\\begin{align*} \\| w \\| _ A = \\| A ^ \\frac { 1 } { 2 } w \\| \\le \\| w \\| = \\| A ^ { - \\frac { 1 } { 2 } } A ^ \\frac { 1 } { 2 } w \\| \\le \\| A ^ { - \\frac { 1 } { 2 } } \\| \\cdot \\| w \\| _ A = \\kappa ^ \\frac { 1 } { 2 } \\| w \\| _ A \\end{align*}"}
+{"id": "246.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { m ^ 4 } { n ^ 5 } } = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { - n } { 3 0 } + \\frac { n ^ 3 } { 3 } + \\frac { n ^ 4 } { 2 } + \\frac { n ^ 5 } { 5 } \\right ) \\frac { z ^ n } { n ^ 5 } \\right \\} \\end{align*}"}
+{"id": "9200.png", "formula": "\\begin{align*} W ( G ) & \\ge W ( G _ { n _ 0 } ) + \\sum _ { i = n _ 0 + 1 } ^ { n } \\left ( i + ( r - 1 ) ^ 2 \\right ) \\\\ & > \\binom { n _ 0 } { 2 } + \\sum _ { i ' = n _ 0 } ^ { n - 1 } i ' + ( n - n _ 0 ) + ( n - n _ 0 ) ( r - 1 ) ^ 2 \\\\ & \\ge \\binom { n } { 2 } + a n _ 0 + ( n - n _ 0 ) a \\\\ & = \\binom { n } { 2 } + a n \\end{align*}"}
+{"id": "6042.png", "formula": "\\begin{align*} g _ { u , v } ( z ) : = u | A | z ^ { n + m } + v | B | \\overline { z } ^ m + | C | , z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "953.png", "formula": "\\begin{align*} h ( x , y ) = \\frac { | x - y | } { \\sqrt { 1 + { | x | } ^ 2 } \\sqrt { 1 + { | y | } ^ 2 } } \\ , , \\ , \\ , x \\ne \\infty \\ne y , \\ , \\ , h ( x , \\infty ) = \\frac { 1 } { \\sqrt { 1 + { | x | } ^ 2 } } . \\end{align*}"}
+{"id": "2686.png", "formula": "\\begin{align*} F ( p ) = \\frac { 1 } { 2 } | | p | | ^ 2 + \\sum _ { i = 1 } ^ { n } \\frac { m _ i } { \\vert \\vert p - x _ i \\vert \\vert } . \\end{align*}"}
+{"id": "5637.png", "formula": "\\begin{align*} K ( u , v ) = \\frac 1 2 ( | \\nabla u | ^ 2 _ 2 + | \\nabla v | ^ 2 _ 2 ) - \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | u | ^ { p } ) | v | ^ { q } , \\end{align*}"}
+{"id": "3115.png", "formula": "\\begin{align*} t ^ n = \\sum _ { k = 0 } ^ { n } S [ n , k ] ( t ) _ { k , q } , \\end{align*}"}
+{"id": "8983.png", "formula": "\\begin{align*} c _ m \\Delta v = - S ^ \\varphi u + \\tilde S ^ { \\tilde \\varphi } u ^ { \\frac { m + 2 } { m - 2 } } \\ge - S ^ \\varphi u \\left ( 1 - u ^ { \\frac { 4 } { m - 2 } } \\right ) = - c _ m \\Lambda ( x ) v \\end{align*}"}
+{"id": "6845.png", "formula": "\\begin{align*} \\alpha ' _ n = \\alpha _ n ( a ^ 2 , q ^ 2 ) \\quad \\mbox { a n d } \\quad \\beta ' _ n = \\sum _ { j \\leq n } \\frac { ( - a q ) _ { 2 j } } { ( q ^ 2 ; q ^ 2 ) _ { n - j } } q ^ { n - j } \\beta _ j ( a ^ 2 , q ^ 2 ) . \\end{align*}"}
+{"id": "852.png", "formula": "\\begin{align*} \\beta \\phi = \\lambda g ( \\phi ) . \\end{align*}"}
+{"id": "3702.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) ( u ^ { ( \\ell ) } _ 1 - u ^ { ( \\ell ) } _ 2 ) + ( F ^ { ( \\ell ) } _ 1 - F ^ { ( \\ell ) } _ 2 ) u ^ { ( \\ell ) } _ 1 + F ^ { ( 1 ) } _ 2 ( u ^ { ( \\ell ) } _ 1 - u ^ { ( \\ell ) } _ 2 ) = 0 & \\Omega , \\\\ \\mathcal { L } ( u ^ { ( \\ell ) } _ 1 - u ^ { ( \\ell ) } _ 2 ) = 0 & \\Omega ^ c , \\\\ u ^ { ( \\ell ) } _ 1 - u ^ { ( \\ell ) } _ 2 = 0 & \\Omega ^ c . \\end{cases} \\end{align*}"}
+{"id": "8621.png", "formula": "\\begin{align*} P ( W ^ k _ { \\ell \\Delta , t } ( j ) = 1 ) = P ( X _ { \\ell , \\Delta } = 1 , D ^ j _ { \\ell \\Delta } ( t ) \\geq k ) ( 1 + O ( \\Delta ) ) . \\end{align*}"}
+{"id": "9142.png", "formula": "\\begin{align*} y _ { [ 0 , R ] } = \\Psi ( x , z , v _ { [ 0 , R - A ] } ) \\end{align*}"}
+{"id": "5821.png", "formula": "\\begin{align*} \\frac { { \\partial { J _ { G M E E F P } } } } { { \\partial { { \\boldsymbol { x } } _ k } } } = { \\boldsymbol { W } } _ k ^ { } { { \\boldsymbol { \\Lambda } } _ k } { { \\boldsymbol { d } } _ k } - { \\boldsymbol { W } } _ k ^ T { { \\boldsymbol { \\Lambda } } _ k } { { \\boldsymbol { W } } _ k } { { \\boldsymbol { x } } _ k } \\end{align*}"}
+{"id": "4930.png", "formula": "\\begin{align*} \\left [ P \\right ] _ { i , j } ^ { } = \\begin{dcases} \\frac { i } { n } & \\mbox { i f } \\ , j = i ; \\\\ \\frac { n - i } { n } & \\mbox { i f } \\ , j = i + 1 ; \\\\ 0 & \\mbox { o t h e r w i s e } ; \\end{dcases} \\end{align*}"}
+{"id": "89.png", "formula": "\\begin{align*} L = \\pi ^ { - 1 } ( L _ - ) \\cup \\pi ^ { - 1 } ( L _ + ) \\cup \\kappa ( E _ s ^ * ) \\cup \\kappa ( E _ u ^ * ) \\cup ( E _ 0 ^ * \\cap p ^ { - 1 } ( 1 ) ) , \\end{align*}"}
+{"id": "7225.png", "formula": "\\begin{align*} 0 < \\mathcal { H } ^ 1 \\left ( \\Omega \\cap \\left ( \\overline { J _ u } \\setminus J _ u \\right ) \\right ) \\leq \\sum _ { h = 1 } ^ \\infty \\mathcal { H } ^ 1 \\left ( \\Omega _ { \\delta _ h } \\cap \\left ( \\overline { J _ u } \\setminus J _ u \\right ) \\right ) = 0 , \\end{align*}"}
+{"id": "1978.png", "formula": "\\begin{align*} u = h \\rho , \\end{align*}"}
+{"id": "8767.png", "formula": "\\begin{align*} \\sum _ { j \\neq i _ 0 } v _ { i _ 0 j } = x _ { 0 , i _ 0 } \\quad v _ { i _ 0 j } = 0 , \\ , \\forall j \\neq 0 , \\end{align*}"}
+{"id": "9144.png", "formula": "\\begin{align*} \\begin{array} { r c l } x & = & F _ { x } ( y _ { [ 0 , R - 1 ] } ) \\\\ z & = & y _ { c } \\\\ v _ { [ 0 , R - A - 1 ] } & = & y _ { [ A , R - 1 ] } \\\\ v _ { [ R - A ] } & = & y _ { [ R ] } \\end{array} \\end{align*}"}
+{"id": "3806.png", "formula": "\\begin{align*} f ( y _ 1 , y _ 2 , x ; \\theta ) = \\log ( 1 + \\exp ( - y _ 1 \\langle \\theta , ( y _ 2 , x ) \\rangle ) ) , \\end{align*}"}
+{"id": "3764.png", "formula": "\\begin{align*} \\rho _ \\sigma ^ { a b } = & \\frac { 1 } { \\sqrt { \\det \\left [ 1 + W _ { a b } ^ { c d , T } W _ { a b } ^ { c d } \\right ] } } \\equiv \\frac { | J _ { a b } ^ \\sigma | } { \\sqrt { \\det \\left [ J _ { a b } ^ { \\sigma , T } J _ { a b } ^ \\sigma + J _ { c d } ^ { \\sigma , T } J _ { c d } ^ \\sigma \\right ] } } . \\end{align*}"}
+{"id": "4965.png", "formula": "\\begin{align*} { k ( n - k ) \\Pr ( X _ t = k \\ , \\vert \\ , n ) = k n ( 1 - \\frac { 1 } { n } ) ^ t \\Pr ( \\widetilde { X } _ t = k \\ , \\vert \\ , n - 1 ) } , \\end{align*}"}
+{"id": "7364.png", "formula": "\\begin{align*} & a _ 1 = i - \\lambda _ 1 , a _ 2 = \\lambda _ 1 - \\lambda _ 2 , \\ldots , a _ { r - i } = \\lambda _ { r - i - 1 } - \\lambda _ { r - i } . \\end{align*}"}
+{"id": "4342.png", "formula": "\\begin{align*} \\begin{array} { l l } \\underset { x \\in X } { } & \\sum \\limits _ { i = 1 } ^ { m } \\alpha _ { i } d \\left ( x , S _ { i } \\right ) ^ { p } , \\end{array} \\end{align*}"}
+{"id": "90.png", "formula": "\\begin{align*} \\partial _ t v = - i ( P _ 0 - i M h W - z ) v , v | _ { t = 0 } = u . \\end{align*}"}
+{"id": "46.png", "formula": "\\begin{align*} \\mathrm { E n d } _ { \\mathbb { C } } ( A ) \\otimes _ { \\mathbb { Z } } \\mathbb { Z } _ { \\ell } = \\mathcal { O } _ K \\otimes _ { \\mathbb { Z } } \\mathbb { Z } _ { \\ell } = : \\mathcal { O } _ { K , \\ell } . \\end{align*}"}
+{"id": "1064.png", "formula": "\\begin{align*} S _ { r } ( f \\circ _ { } g ) = \\sum _ { \\substack { 0 \\leq s , r \\\\ s + t \\leq r } } \\frac { r ! } { s ! t ! ( r - s - t ) ! } \\alpha ^ { s / p } \\beta ^ { t / p } \\gamma ^ { ( r - s - t ) / p } S _ { r - t } ( f ) \\circ _ { } S _ { r - s } ( g ) \\ , , \\end{align*}"}
+{"id": "7861.png", "formula": "\\begin{align*} \\mathcal { S } = \\left \\{ \\{ B , \\underline { B } \\} : \\ 2 k - 1 , 2 k \\in B \\right \\} . \\end{align*}"}
+{"id": "1299.png", "formula": "\\begin{align*} \\mathcal { F } ^ X _ t = \\sigma \\big ( ( X _ i ( s ) ) _ { 0 \\leq s \\leq t _ i } , i \\in V \\big ) . \\end{align*}"}
+{"id": "4561.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\ell , M } = \\bigl ( ( p _ 1 , \\nu _ 1 ) , \\ldots , ( p _ s , \\nu _ s ) \\bigr ) , \\end{align*}"}
+{"id": "8425.png", "formula": "\\begin{align*} \\zeta _ { p } ( 1 - s ) = \\frac { 2 \\ , \\cos \\left ( \\frac { \\pi s } { 2 } \\right ) \\Gamma ( s ) } { ( 2 \\pi ) ^ { s } } \\eta _ { p } ( s ) . \\end{align*}"}
+{"id": "356.png", "formula": "\\begin{align*} ( \\underline { X } , \\underline { A } ) ^ { K } = \\bigcup _ { \\sigma \\subseteq K } ( \\underline { X } , \\underline { A } ) ^ { \\sigma } \\subseteq \\prod ^ m _ { i = 1 } X _ i . \\end{align*}"}
+{"id": "1179.png", "formula": "\\begin{align*} \\lim _ { y \\to - 1 ^ + } \\phi ( y ) = { 1 \\over 2 } \\sum _ j E _ j d _ j . \\end{align*}"}
+{"id": "8134.png", "formula": "\\begin{align*} \\sigma _ a ^ + = \\sum _ { I } \\varphi ^ + ( M _ I ) S ^ I , \\end{align*}"}
+{"id": "3114.png", "formula": "\\begin{align*} t ^ n = \\sum _ { k = 0 } ^ { n } S _ D ( n , k ) ( t ) _ k ^ D + n \\left ( ( t - 1 ) ^ { n - 1 } - ( t ) _ { n - 1 } ^ D \\right ) , \\end{align*}"}
+{"id": "2817.png", "formula": "\\begin{align*} \\overset \\cdot { S _ k ^ { m , 1 } } = S _ { k + r + 1 } ^ { m , 1 } - S _ { r + 1 } ^ { 1 , 1 } S _ k ^ { m , 1 } ; m = 1 , \\dots , r ; \\end{align*}"}
+{"id": "2375.png", "formula": "\\begin{align*} [ t ( f _ 1 ' , \\ldots , f _ k ' ) ] ( x _ 1 , \\ldots , x _ m ) & = t ( f _ 1 ' ( x _ 1 , \\ldots , x _ m ) , \\ldots , f _ k ' ( x _ 1 , \\ldots , x _ m ) ) \\\\ & = t ( f _ 1 ( x _ 1 / \\alpha , \\ldots , x _ m / \\alpha ) , \\ldots , f _ k ( x _ 1 / \\alpha , \\ldots , x _ m / \\alpha ) ) \\\\ & = [ t ( f _ 1 , \\ldots , f _ k ) ] ' ( x _ 1 , \\ldots , x _ m ) \\in \\varphi ( C ) . \\end{align*}"}
+{"id": "9271.png", "formula": "\\begin{align*} \\texttt { L H S } \\eqref { i d : 2 r e f 2 } & \\simeq \\int _ a ^ b x ^ { \\xi + \\lambda } \\ , d x \\simeq \\begin{cases} ( b - a ) b ^ { \\xi + \\lambda } & \\xi + \\lambda > - 1 , \\\\ \\log \\Big ( 1 + \\frac { b - a } { a } \\Big ) & \\xi + \\lambda = - 1 , \\\\ ( b - a ) b ^ { - 1 } a ^ { \\xi + \\lambda + 1 } & \\xi + \\lambda < - 1 \\end{cases} \\\\ & \\simeq \\texttt { R H S } \\eqref { i d : 2 r e f 2 } . \\end{align*}"}
+{"id": "2646.png", "formula": "\\begin{align*} & x ^ { \\gamma } y ^ { \\alpha } z ^ { \\beta } - x ^ { \\alpha } y ^ { \\gamma } z ^ { \\beta } \\\\ = & - ( x - y ) x ^ { \\gamma } y ^ { \\gamma } z ^ { \\beta } \\sum _ { i = 0 } ^ { \\alpha - \\gamma - 1 } x ^ i y ^ { \\alpha - \\gamma - 1 - i } \\\\ = & - ( x - y ) \\bigg ( x ^ { \\gamma } z ^ { \\beta } \\sum _ { i = 0 } ^ { \\alpha - \\beta - 1 } x ^ i y ^ { \\alpha - 1 - i } + x ^ { \\alpha - \\beta + \\gamma } z ^ { \\beta } \\sum _ { i = 0 } ^ { \\beta - \\gamma - 1 } x ^ i y ^ { \\beta - 1 - i } \\bigg ) . \\end{align*}"}
+{"id": "7700.png", "formula": "\\begin{align*} f _ { L _ 1 } \\left ( M \\tau ; M \\begin{pmatrix} r \\\\ t \\end{pmatrix} \\right ) = \\phi ( \\tau ) ^ { 2 \\nu } \\rho _ { L _ 1 } ( M , \\phi ) f _ { L _ 1 } \\left ( \\tau ; \\begin{pmatrix} r \\\\ t \\end{pmatrix} \\right ) \\end{align*}"}
+{"id": "140.png", "formula": "\\begin{align*} m = m _ - - m _ + + R ( \\chi _ - \\circ \\pi - \\chi _ + \\circ \\pi ) \\end{align*}"}
+{"id": "8232.png", "formula": "\\begin{align*} L _ N \\pi _ { s } ^ N ( \\phi ) = \\pi _ { s } ^ N ( A \\phi ) + R _ 3 ( \\phi , N , s ) . \\end{align*}"}
+{"id": "747.png", "formula": "\\begin{align*} \\mathrm { V o l } ( \\Omega ) = \\mathrm { V o l } ( B ) , \\ \\mathrm { b a r } ( \\Omega ) = O , \\end{align*}"}
+{"id": "8822.png", "formula": "\\begin{align*} v _ 0 ( t , x ) : = ( \\tau , z ) \\mapsto K _ { t - \\tau } ( x , z ) \\ , \\sigma ( u ( \\tau , z ) ) \\ , 1 _ { [ 0 , t ] } ( \\tau ) . \\end{align*}"}
+{"id": "4892.png", "formula": "\\begin{align*} a _ 0 ( z , e ) : = \\frac { 2 ^ s ( \\rho ^ 2 - | z | ^ 2 ) ^ s } { s \\rho ^ s \\ , | \\rho e - z | ^ n } , \\end{align*}"}
+{"id": "8509.png", "formula": "\\begin{align*} \\frac { 2 \\sqrt { \\pi } \\ , c ^ { \\frac { 1 } { 2 } - s } \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) } { \\Gamma ( s ) } \\zeta _ { p ^ { \\prime } } ( 2 s - 1 ) = \\frac { \\pi } { \\sqrt { c } } \\ , \\frac { 1 } { s - 1 } + \\frac { \\pi } { \\sqrt { c } } \\left ( 2 C _ { p ^ { \\prime } } ^ { ( 1 ) } - \\log \\left ( 4 c \\right ) \\right ) + O \\left ( s - 1 \\right ) . \\end{align*}"}
+{"id": "755.png", "formula": "\\begin{align*} \\phi = \\phi ( r ) , \\ \\phi ' = \\phi ' ( r ) , \\ r = \\rho ( 1 + u ) , \\end{align*}"}
+{"id": "4996.png", "formula": "\\begin{align*} \\overline { F } ( k , t + 1 ; p , n ) = \\begin{dcases} \\frac { ( n - k ) p } { n } \\overline { F } ( k - 1 , t ; p , n ) + \\frac { n - ( n - k ) p } { n } \\overline { F } ( k , t ; p , n ) & \\textnormal { i f } \\ ; k \\le t ; \\\\ 0 & \\textnormal { i f } \\ ; k > t . \\end{dcases} \\end{align*}"}
+{"id": "6108.png", "formula": "\\begin{align*} p _ { \\# } = p \\left ( 1 + \\frac { 2 s } { n } \\right ) . \\end{align*}"}
+{"id": "6621.png", "formula": "\\begin{align*} [ \\varphi _ \\lambda ( [ x _ \\mu y _ 1 ] , y _ 2 ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] + ( - 1 ) ^ { x y _ 1 } [ \\varphi _ \\lambda ( y _ 1 , [ x _ \\mu y _ 2 ] ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] = [ ( \\varphi _ \\lambda ( x , [ { y _ 1 } _ \\eta y _ 2 ] ) ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] . \\end{align*}"}
+{"id": "3490.png", "formula": "\\begin{align*} \\eta = \\frac { \\Theta ( n ^ { - 1 } ) } { 1 - \\left ( 1 - \\Omega ( n ^ { - 1 } ) \\right ) } = O ( 1 ) , \\end{align*}"}
+{"id": "4939.png", "formula": "\\begin{align*} \\begin{aligned} i \\frac { ( n - i ) ! } { ( n - k ) ! ( k - i ) ! } + ( n - i ) \\frac { ( n - i - 1 ) ! } { ( n - k ) ! ( k - i - 1 ) ! } & = i \\frac { ( n - i ) ! } { ( n - k ) ! ( k - i ) ! } + \\frac { ( n - i ) ! } { ( n - k ) ! ( k - i - 1 ) ! } \\\\ & = i \\binom { n - i } { n - k } + ( k - i ) \\binom { n - i } { n - k } \\\\ & = k \\binom { n - i } { n - k } , \\end{aligned} \\end{align*}"}
+{"id": "3226.png", "formula": "\\begin{align*} X _ { h G } : = E G \\times _ G X \\end{align*}"}
+{"id": "8868.png", "formula": "\\begin{align*} d ^ { ( 2 ) * } _ i = \\arg \\min \\limits _ { d \\in \\mathcal { D } ^ { ( 2 ) } _ i \\cup \\{ - \\beta _ i , 1 - \\beta _ i \\} } f ^ { ( 2 ) } _ { \\boldsymbol \\beta , i } ( d ) , \\end{align*}"}
+{"id": "8756.png", "formula": "\\begin{align*} E V S I = \\frac { 1 } { T } \\sum _ { t = 1 } ^ T \\varphi ( y ^ t , d _ 0 ^ t ) - \\frac { 1 } { T } \\sum _ { t = 1 } ^ T \\psi ( y ^ t , d _ 0 ^ t ) . \\end{align*}"}
+{"id": "4025.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { \\mathrm { p e t } } = \\int _ { \\Gamma \\backslash \\mathbb { H } } f ( z ) \\overline { g ( z ) } \\ , ( \\mathrm { I m } ( z ) ) ^ { k } \\ , \\mu _ { \\mathrm { h y p } } ( z ) \\ , \\ \\ , \\ f , g \\in S _ { k } ( \\Gamma ) . \\end{align*}"}
+{"id": "6194.png", "formula": "\\begin{align*} | V _ e | = ( n - 2 ) | V _ i | + 2 . \\end{align*}"}
+{"id": "2349.png", "formula": "\\begin{align*} 0 > v v ' = - k - 2 e r + \\frac { e ^ { 2 } - 1 } { k } + k r ^ { 2 } > k r ^ { 2 } - k - 2 r ^ { 2 } - \\frac { 1 } { k } \\geq 4 ( k - 2 ) - k - \\frac { 1 } { k } . \\end{align*}"}
+{"id": "4551.png", "formula": "\\begin{align*} L ( \\Phi ^ { \\lambda } ) = \\lambda ^ 2 L ( \\Phi ) , \\ \\ N ( \\Phi ^ { \\lambda } ) = \\lambda ^ { \\frac { d } { 2 } + 1 } N ( \\Phi ) , \\ \\ Q ( \\Phi ^ { \\lambda } ) = Q ( \\Phi ) , \\ \\ \\mathbf { P } ( \\Phi ^ { \\lambda } ) = \\lambda \\mathbf { P } ( \\Phi ) . \\end{align*}"}
+{"id": "7652.png", "formula": "\\begin{align*} & M ( 0 , 0 , 0 ) = M ( 2 , 0 , 0 ) = M ( 2 , 1 , 0 ) = M ( 2 , 1 , 1 ) = M ( 2 , 0 , 1 ) = 0 , \\\\ & M ( 0 , 1 , 0 ) = M ( 0 , 1 , 1 ) = 5 7 6 , M ( 0 , 0 , 1 ) = 1 0 2 4 . \\end{align*}"}
+{"id": "358.png", "formula": "\\begin{align*} \\rho _ a ( e ^ { t i } \\wedge x ) = e ^ { a t i } \\wedge x . \\end{align*}"}
+{"id": "3449.png", "formula": "\\begin{align*} h ^ { - 1 } g ^ { n } h = g ^ { - n } . \\end{align*}"}
+{"id": "1702.png", "formula": "\\begin{align*} C _ n ^ \\lambda ( \\cos \\theta ) : = \\frac { \\Gamma ( \\lambda + \\frac 1 2 ) \\Gamma ( 2 \\lambda + n ) } { \\Gamma ( 2 \\lambda ) \\Gamma ( \\lambda + \\frac 1 2 + n ) } P _ n ^ { ( \\lambda - \\frac 1 2 , \\lambda - \\frac 1 2 ) } ( \\cos \\theta ) , \\end{align*}"}
+{"id": "8470.png", "formula": "\\begin{align*} I _ { m , p } ^ { \\star } ( s , x ) = \\sqrt { \\pi } 2 ^ { \\frac { 1 } { 2 } - s } x ^ { s + \\frac { 1 } { 2 } } \\intop _ { 0 } ^ { \\infty } y ^ { s - \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( x y ) \\ , \\left ( \\frac { 2 \\pi p - y } { 2 \\pi p + y } \\right ) ^ { m } \\ , e ^ { - m y } d y \\end{align*}"}
+{"id": "8678.png", "formula": "\\begin{align*} T _ { 1 } = ( 2 p ^ { 2 } ) ^ { - 1 } [ & f ^ { ( 2 ) } ( \\tau _ { 1 } ) E \\{ ( \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau _ { 1 } ) ^ { 2 } \\} + f ^ { ( 2 ) } ( \\tau _ { 2 } ) E \\{ ( \\| Y _ { 1 } - Y _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau _ { 2 } ) ^ { 2 } \\} \\\\ & - 2 f ^ { ( 2 ) } ( \\tau _ { 3 } ) E \\{ ( \\| X _ { 1 } - Y _ { 1 } \\| _ { 2 } ^ { 2 } - p \\tau _ { 3 } ) ^ { 2 } \\} ] \\end{align*}"}
+{"id": "7527.png", "formula": "\\begin{align*} \\frac { \\varphi _ 0 ( x ) ^ k } { k ! } = \\sum _ { n \\geq k } B _ { n , k } ( f _ { 1 , 0 } , \\ldots , f _ { n - k + 1 , 0 } ) \\frac { x ^ n } { n ! } , \\end{align*}"}
+{"id": "4870.png", "formula": "\\begin{align*} L w ( x ) = P V \\int _ { \\R ^ n } \\frac { w ( x ) - w ( x + y ) } { | y | ^ { n + 2 s } } a \\left ( \\frac { y } { | y | } \\right ) d y , \\end{align*}"}
+{"id": "630.png", "formula": "\\begin{align*} x _ { n + 1 } = \\frac { 1 } { \\| x _ n + \\alpha _ n u \\| } ( x _ n + \\alpha _ n u ) , \\end{align*}"}
+{"id": "4428.png", "formula": "\\begin{align*} \\mu _ { \\omega , \\mathbf { c } } = \\omega ^ { 2 - \\frac { d } { 2 } } \\mu _ { 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } } . \\end{align*}"}
+{"id": "2496.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle \\dfrac { N } { 4 } \\Lambda a _ 1 a _ 2 \\int _ { \\mathbb { R } ^ N } \\eta \\xi ^ 2 d x = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "3367.png", "formula": "\\begin{align*} \\pi _ { \\mu } ( f ) \\bar { x } \\begin{pmatrix} f ( T ^ t ) & 0 \\\\ \\star & \\star \\end{pmatrix} \\begin{pmatrix} \\bar { x } \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} \\bar { x } \\\\ \\star \\end{pmatrix} . \\end{align*}"}
+{"id": "539.png", "formula": "\\begin{align*} \\| f \\| _ { \\mathrm { H } _ { \\mathcal { H } _ { \\hbar , V } } ^ { s } } : = \\left \\| ( I + \\mathcal { H } _ { \\hbar , V } ) ^ { s / 2 } f \\right \\| _ { \\ell ^ { 2 } ( \\hbar \\mathbb { Z } ^ { n } ) } = \\left ( \\sum \\limits _ { \\xi \\in \\mathcal { I } _ { \\hbar } } \\langle \\xi \\rangle ^ { 2 s } \\widehat { f } ( \\xi ) | ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } = \\left ( \\sum \\limits _ { \\xi \\in \\mathcal { I } _ { \\hbar } } \\left ( 1 + \\lambda _ { \\xi } \\right ) ^ { s } \\widehat { f } ( \\xi ) | ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "7816.png", "formula": "\\begin{align*} \\Vert B A ^ { \\prime } \\Vert ^ { 2 } = \\Vert ( B A ^ { \\prime } ) ^ { * } \\Vert ^ { 2 } \\le 4 \\max _ { x \\in \\mathcal { N } } \\Vert ( B A ^ { \\prime } ) ^ { * } x \\Vert _ { 2 } ^ { 2 } = 4 \\max _ { x \\in \\mathcal { N } } \\sum _ { j \\le n } \\langle W _ { j } ^ { \\prime } , x \\rangle ^ { 2 } . \\end{align*}"}
+{"id": "2158.png", "formula": "\\begin{align*} u \\mid _ { S _ { T } \\diagdown \\left ( \\Gamma _ { 1 T } ^ { + } { } \\right ) } = 0 . \\end{align*}"}
+{"id": "8460.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\left \\{ \\frac { 1 } { \\sqrt { n ^ { 2 } + x ^ { 2 } } } - \\frac { 1 } { n } \\right \\} + \\frac { 1 } { 2 x } + \\gamma + \\log \\left ( \\frac { x } { 2 } \\right ) = 2 \\ , \\sum _ { n = 1 } ^ { \\infty } K _ { 0 } ( 2 \\pi n x ) , \\end{align*}"}
+{"id": "6872.png", "formula": "\\begin{align*} J _ r ( x , \\beta ) = \\int _ { [ 0 , 1 ] } \\d y \\ , \\mathcal { R } \\big ( \\widehat { r } _ \\beta ( x , y ) \\mid r ( x , y ) \\big ) , x \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "8090.png", "formula": "\\begin{align*} \\boldsymbol { \\Omega } ( \\mathbf { x } ) = \\theta ( \\mathbf { x } ) \\mathbf { A } + \\phi ( \\mathbf { x } ) \\mathbf { B } + \\chi ( \\mathbf { x } ) \\mathbf { C } . \\end{align*}"}
+{"id": "5603.png", "formula": "\\begin{align*} \\mathrm { I } ( \\xi _ { 1 } ^ { x } , \\mathcal { T } _ { x } ) = \\int _ { G ^ { \\mathbb { N } } } \\log \\frac { d ( \\omega _ { 1 } \\lambda ) _ { x } } { d \\lambda _ { x } } \\left ( \\psi ( x , \\omega ) \\right ) d \\mathbb { P } _ { \\mu } ^ { \\pi ( x ) } ( \\omega ) . \\end{align*}"}
+{"id": "4827.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } \\leq x \\right ) - \\Phi ( x ) & = - \\left [ \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } > x \\right ) - ( 1 - \\Phi ( x ) ) \\right ] \\\\ & \\geq - \\left [ ( 1 - \\Phi ( x ) ) \\exp \\left \\{ C \\frac { 1 + x ^ { 3 } } { \\sqrt { n } } \\right \\} - ( 1 - \\Phi ( x ) ) \\right ] \\\\ & = - ( 1 - \\Phi ( x ) ) \\left ( \\exp \\left \\{ C \\frac { 1 + x ^ { 3 } } { \\sqrt { n } } \\right \\} - 1 \\right ) . \\end{align*}"}
+{"id": "1013.png", "formula": "\\begin{align*} & \\quad \\ : \\ : \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ 5 } { ( - 2 ^ { 1 2 } ) ^ k } \\Big \\{ ( 2 0 k ^ 2 + 8 k + 1 ) \\big [ 8 H _ { 2 k } ^ { ( 2 ) } - 3 H _ { k } ^ { ( 2 ) } \\big ] + 4 \\Big \\} = \\frac { 8 } { 3 } , \\\\ [ 1 m m ] & \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ 5 } { ( - 2 ^ { 2 0 } ) ^ k } \\Big \\{ ( 8 2 0 k ^ 2 + 1 8 0 k + 1 3 ) \\big [ 1 1 H _ { 2 k } ^ { ( 2 ) } - 3 H _ { k } ^ { ( 2 ) } \\big ] + 4 3 \\Big \\} = \\frac { 1 2 8 } { 3 } . \\end{align*}"}
+{"id": "7131.png", "formula": "\\begin{align*} K _ \\varepsilon ( | A _ \\gamma ( \\hat { x } , \\hat { x } ) ( \\hat { x } - \\bar { y } ) | ) : = J _ \\varepsilon ( | A _ \\gamma ( \\hat { x } , \\hat { y } ) ( \\hat { x } - \\bar { y } ) | ) \\big | \\det F _ \\gamma ( \\bar { y } ) \\big | - J _ \\varepsilon ( | A _ \\gamma ( \\hat { x } , \\hat { x } ) ( \\hat { x } - \\bar { y } ) | ) \\big | \\det F _ \\gamma ( \\hat { x } ) \\big | \\end{align*}"}
+{"id": "240.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { m ^ 1 } { n ^ 2 } } = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { n } { 2 } + \\frac { n ^ 2 } { 2 } \\right ) \\frac { z ^ n } { n ^ 2 } \\right \\} \\end{align*}"}
+{"id": "1749.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\textrm { o u t } } = 1 - \\sum ^ M _ { m = 1 } \\mathcal { V } ^ { ( M , m ) } \\xi ^ m \\mathcal { P } _ { \\textrm { o u t } } ^ { ( m ) } . \\end{align*}"}
+{"id": "6706.png", "formula": "\\begin{align*} \\frac { 1 } { i \\tau } \\{ \\overline { p _ \\Phi } , p _ \\Phi \\} & = \\frac { 1 } { i \\tau } \\{ \\Re p _ { \\Phi } - i \\Im p _ { \\Phi } , \\Re p _ { \\Phi } + i \\Im p _ { \\Phi } \\} \\\\ & = \\frac { 1 } { \\tau } \\{ \\Re p _ { \\Phi } , \\Im p _ { \\Phi } \\} - \\frac { 1 } { \\tau } \\{ \\Im p _ { \\Phi } , \\Re p _ { \\Phi } \\} = \\frac { 2 } { \\tau } \\{ \\Re p _ { \\Phi } , \\Im p _ { \\Phi } \\} . \\end{align*}"}
+{"id": "920.png", "formula": "\\begin{align*} h _ { k , N } ( u , v ) & = u ^ N P ^ { ( - N - 2 \\nu _ 1 - 1 , - N - 2 \\nu _ 2 ) } _ k \\left ( 1 + 2 \\frac { v } { u } \\right ) , \\\\ g _ { k , N } ^ { + } ( u , v ) & = \\frac { 2 N + 2 \\nu _ { 1 2 } - k } { N + 2 \\nu _ 1 - k } \\ u ^ N P ^ { ( - N - 2 \\nu _ 1 , - N - 2 \\nu _ 2 ) } _ k \\left ( 1 + 2 \\frac { v } { u } \\right ) , \\\\ g _ { k , N } ^ { - } ( u , v ) & = - u ^ N P ^ { ( - N - 2 \\nu _ 1 , - N - 2 \\nu _ 2 ) } _ { k - 1 } \\left ( 1 + 2 \\frac { v } { u } \\right ) . \\end{align*}"}
+{"id": "6326.png", "formula": "\\begin{align*} \\frac { \\partial x } { \\partial r } = O ( t ) \\qquad \\qquad \\frac { \\partial y } { \\partial r } = O ( t ) . \\end{align*}"}
+{"id": "3487.png", "formula": "\\begin{align*} \\Pr [ X _ t ^ + ( v ) \\neq X _ t ^ - ( v ) ] & \\le \\sum _ { c \\in [ q ] } \\Pr [ X _ t ^ + ( v ) \\ge c , X _ t ^ - ( v ) < c ] \\\\ & = \\sum _ { c \\in [ q ] } \\Pr [ X _ t ^ + ( v ) \\ge c ] - \\Pr [ X _ t ^ - ( v ) \\ge c ] . \\end{align*}"}
+{"id": "2715.png", "formula": "\\begin{align*} J _ { 5 1 } = & s \\lambda ^ 3 \\iint _ Q \\xi u A \\nabla u \\cdot \\nabla \\eta \\left ( A \\nabla \\eta \\cdot \\nabla \\eta \\right ) d x d y d t \\\\ \\geq & - C s ^ 2 \\lambda ^ 4 \\iint _ Q \\xi \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d y d t - C \\lambda ^ 2 \\iint _ Q \\xi | A \\nabla u \\cdot \\nabla \\eta | ^ 2 d x d y d t , \\end{align*}"}
+{"id": "5890.png", "formula": "\\begin{align*} L ^ { 1 } ( G ) = L ^ { 1 } ( G ) \\ast L ^ { 1 } ( G ) , \\end{align*}"}
+{"id": "6707.png", "formula": "\\begin{align*} c _ { \\Psi } ( \\xi , \\tau ) = \\frac { 1 } { i \\tau } \\{ \\overline { p _ { \\Psi } } , p _ { \\Psi } \\} ( x _ 0 , \\xi , \\tau ) , \\tau > 0 c _ { \\Psi } ( \\xi , 0 ) = 2 \\{ p _ 2 , \\{ p _ 2 , \\Psi \\} \\} ( x _ 0 , \\xi ) , \\end{align*}"}
+{"id": "1521.png", "formula": "\\begin{align*} H ( A _ 1 . . . A _ k - B _ 1 . . . B _ k ) \\leq H ( A _ 1 ) + \\sum _ { j = 3 } ^ k ( 2 j - 3 ) \\log L + k \\log K . \\end{align*}"}
+{"id": "6081.png", "formula": "\\begin{align*} \\| y \\| = \\| x _ 1 \\cdot x _ 1 ^ { - 1 } \\cdot y \\| \\geq \\big ( c \\| x _ 1 \\| - \\| x _ 1 ^ { - 1 } y \\| ^ { \\gamma } \\big ) ^ { 1 / \\gamma } \\geq \\big ( c - \\epsilon ) ^ { 1 / \\gamma } > \\theta , \\end{align*}"}
+{"id": "2106.png", "formula": "\\begin{align*} g ( A ) = & 2 ^ { k - 1 } ( 2 ^ k - 1 ) m ^ 2 + \\frac { 1 } { 2 } ( d - 1 ) ( 2 ^ k - 1 ) m + ( 2 ^ { 2 k - 1 } + k 2 ^ { k - 1 } - 2 ^ { k + 1 } ) m \\\\ & \\ \\ + 2 ^ { 2 k - 3 } + ( d + k ) 2 ^ { k - 2 } - 5 \\cdot 2 ^ { k - 2 } - d + 1 . \\end{align*}"}
+{"id": "5898.png", "formula": "\\begin{align*} \\nu ( f \\circ T . g ) = \\nu ( f . K g ) , \\ ; \\textrm { f o r a n y } f , g \\in \\mathbb L ^ 2 ( \\nu ) . \\end{align*}"}
+{"id": "3084.png", "formula": "\\begin{align*} \\mathcal U _ \\infty ^ { p - 1 - \\alpha } = \\mathcal V _ \\infty ^ { k _ 1 } . \\end{align*}"}
+{"id": "8755.png", "formula": "\\begin{align*} F ( p ) : = \\begin{cases} \\max _ { v } & \\sum _ { i = 1 } ^ n c p _ i \\mathbb { E } _ { \\xi } \\left [ \\min ( x _ i + y _ i , d _ i ) \\right ] - \\sum _ { i \\neq j } \\alpha _ { i j } v _ { i j } \\\\ s . t \\quad & \\begin{cases} v \\geq 0 \\\\ \\sum _ { k \\neq j } v _ { j k } \\leq x _ { 0 , j } , \\forall j \\in I \\\\ \\end{cases} \\end{cases} , \\end{align*}"}
+{"id": "735.png", "formula": "\\begin{align*} \\cal R : = { \\rm I } _ { N _ { S } } - 2 \\tilde { \\cal A } _ { - 1 } \\cal A _ { D , N } \\ , , \\end{align*}"}
+{"id": "4158.png", "formula": "\\begin{align*} \\lambda _ s ( b ) \\left ( \\sum _ { t \\in G } b _ t \\delta _ t \\right ) = \\sum _ { t \\in G } b b _ t \\delta _ { s t } . \\end{align*}"}
+{"id": "8196.png", "formula": "\\begin{align*} P ^ \\omega ( F _ { j , t } ^ c ) = P ^ \\omega ( M _ { j h ( t ) } > k j h ( t ) ) & \\leq P ( M _ { j h ( t ) } > k j h ( t ) ) \\\\ & \\leq E [ N _ { j h ( t ) } ] \\ : \\mathbf { P } _ 0 \\bigg ( \\sup _ { 0 \\le s \\le j h ( t ) } | X _ s | > k j h ( t ) \\bigg ) \\\\ & = \\exp [ j h ( t ) ( \\beta - k ^ 2 / 2 ) ( 1 + o ( 1 ) ) ] . \\end{align*}"}
+{"id": "7660.png", "formula": "\\begin{align*} u ^ i = g ^ { i j } \\frac { \\partial u } { \\partial x ^ j } , \\end{align*}"}
+{"id": "874.png", "formula": "\\begin{align*} F _ 1 = \\lambda _ 0 G _ 1 + \\lambda _ 1 x _ 1 G _ 2 + \\cdots + \\lambda _ n x _ n G _ n \\end{align*}"}
+{"id": "3972.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } \\left \\{ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\pi \\in \\Pi ( \\mu _ { 1 3 } , \\mu _ { 2 3 } ) } \\int _ { \\mathcal { S } _ 1 \\times \\mathcal { S } _ 2 } ( f _ \\mathcal { S } ) _ \\lambda \\ , d \\pi \\right \\} , \\end{align*}"}
+{"id": "8015.png", "formula": "\\begin{align*} p \\le s \\le t \\le q , \\ p + q = s + t \\Rightarrow f ( s ) + f ( t ) \\le f ( p ) + f ( q ) . \\end{align*}"}
+{"id": "4577.png", "formula": "\\begin{align*} \\| \\theta _ N ( t ) \\| _ { L ^ 4 } ^ 4 \\le \\frac { 2 \\delta ^ 2 } { a ^ * } \\| \\nabla \\theta _ N ( t ) \\| ^ 2 = \\frac { 1 } { 1 6 \\| w \\| _ { L ^ 1 } } \\| \\nabla \\theta _ N ( t ) \\| ^ 2 . \\end{align*}"}
+{"id": "8003.png", "formula": "\\begin{align*} | A \\tilde { Z } N \\tilde { Z } ^ * A | & = \\left | A \\tilde { Z } | N | ^ { 1 / 2 } U | N | ^ { 1 / 2 } \\tilde { Z } ^ * A \\right | \\\\ & \\prec _ { \\log } A \\tilde { Z } | N | \\tilde { Z } ^ * A \\end{align*}"}
+{"id": "3475.png", "formula": "\\begin{align*} \\mathcal { E } _ P ( f , g ) = \\langle f , ( I - P ) g \\rangle _ { \\mu } = \\frac { 1 } { 2 } \\sum _ { x , y \\in \\Omega } \\mu ( x ) P ( x , y ) ( f ( x ) - f ( y ) ) ( g ( x ) - g ( y ) ) , \\end{align*}"}
+{"id": "6398.png", "formula": "\\begin{align*} \\theta _ s ^ M \\circ \\pi _ { \\varphi } = \\pi _ { \\varphi } , \\theta _ s ^ M ( \\lambda ^ { \\varphi } ( t ) ) = e ^ { - i t s } \\lambda ^ { \\varphi } ( t ) \\end{align*}"}
+{"id": "3097.png", "formula": "\\begin{align*} 2 ^ k k ! \\ , S _ B ( n , k ) = \\sum _ { \\ell = 0 } ^ k B _ { n , \\ell } \\binom { n - \\ell } { k - \\ell } \\end{align*}"}
+{"id": "4903.png", "formula": "\\begin{align*} h _ t ( X _ 0 , . . . , X _ j , . . . , X _ k ) = \\sum _ { d = 0 } ^ { d ^ \\prime } X _ j ^ d h _ { t - d } ( X _ 0 , . . . , X _ { j - 1 } , X _ { j + 1 } . . . , X _ k ) , d ^ \\prime \\le t . \\end{align*}"}
+{"id": "978.png", "formula": "\\begin{align*} \\left ( \\overline { \\nabla } _ { X } F \\right ) Y = 0 , \\end{align*}"}
+{"id": "5648.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } ( | \\nabla \\tilde { u } _ n | ^ 2 _ 2 + | \\nabla \\tilde { v } _ n | ^ 2 _ 2 ) = 0 , \\ \\ \\lim _ { n \\rightarrow \\infty } ( | \\nabla \\tilde { u } _ n | ^ 2 _ 2 + | \\nabla \\tilde { v } _ n | ^ 2 _ 2 ) \\geq S ^ { \\frac { 2 ^ * _ \\mu } { 2 ^ * _ \\mu - 1 } } _ { H , L } . \\end{align*}"}
+{"id": "5306.png", "formula": "\\begin{align*} \\Rightarrow \\ | \\langle h ^ { = d } , g \\rangle | \\leq e \\ , \\mu _ { p } ( h ) \\mu _ { p } ( g ) \\left ( e ^ 6 \\ , r \\ , \\sqrt { q } \\cdot \\sqrt { A } \\right ) ^ { d } . \\end{align*}"}
+{"id": "8565.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } M ( t ) = \\nu Y \\int _ 0 ^ \\infty e ^ { - \\lambda s } ( 1 - p _ 0 ( s ) ) d s , \\end{align*}"}
+{"id": "19.png", "formula": "\\begin{align*} X _ { \\mathbf { K } } ( \\mathbb { C } ) = \\mathbf { G } ( \\mathbb { Q } ) \\backslash \\left ( \\mathbf { X } \\times \\mathbf { G } ( \\mathbb { A } _ f ) / \\mathbf { K } \\right ) \\end{align*}"}
+{"id": "2388.png", "formula": "\\begin{align*} \\cos \\alpha _ { 2 0 3 } = \\cos \\alpha _ { 1 0 4 } , \\end{align*}"}
+{"id": "61.png", "formula": "\\begin{align*} \\Omega ^ { - 1 } ( \\epsilon ( \\sigma ) ) C \\in \\mathrm { C e n t } _ { \\mathcal { G } ( \\mathbb { Q } _ p ) } ( \\mathcal { G } ^ 1 ( \\mathbb { Q } _ p ) ) = Z ( \\mathcal { G } ( \\mathbb { Q } _ p ) ) . \\end{align*}"}
+{"id": "6240.png", "formula": "\\begin{align*} T _ { 1 } g _ { n } \\ , = \\ , & g _ n \\ , , \\quad ( T _ { \\tau } - i d ) ( g _ n ) \\ , = \\ , \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( 2 \\pi \\mathrm { i } ) ^ { n - k } } { ( n - k ) ! } \\ , g _ k \\ , , \\\\ & g _ n ' \\ , = \\ , \\sum _ { k = 0 } ^ { n - 1 } ( - 1 ) ^ { n - k } \\ , E _ { n - k + 1 } \\ , g _ k \\ , . \\end{align*}"}
+{"id": "4786.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\lfloor 8 t ^ 2 \\rfloor } \\exp \\left ( j - \\frac { j ^ 2 } { 4 t ^ 2 } \\right ) \\leq \\sum _ { j = 0 } ^ { \\lfloor 8 t ^ 2 \\rfloor } \\exp \\left ( j \\right ) \\leq 8 t ^ 2 \\exp ( 8 t ^ 2 ) \\leq L \\exp ( L ) \\end{align*}"}
+{"id": "1007.png", "formula": "\\begin{align*} \\psi ^ { ( n ) } ( x ) = \\frac { d ^ { n + 1 } } { d x ^ { n + 1 } } \\big \\{ \\log \\Gamma ( x ) \\big \\} = \\frac { d ^ { n } } { d x ^ { n } } \\psi ( x ) . \\end{align*}"}
+{"id": "5376.png", "formula": "\\begin{align*} E _ 1 x _ { 1 , k + 1 } & = A _ 1 x _ { 1 , k } + B _ 1 u _ { 1 , k } , & y _ { 1 , k } & = C _ 1 x _ { 1 , k } + D _ 1 u _ { 1 , k } , \\\\ E _ 2 x _ { 2 , k + 1 } & = A _ 2 x _ { 2 , k } + B _ 2 u _ { 2 , k } & y _ { 2 , k } & = C _ 2 x _ { 2 , k } + D _ 2 u _ { 2 , k } \\end{align*}"}
+{"id": "5812.png", "formula": "\\begin{align*} = ( c _ 0 + 1 ) T ( r , M ) + T ( r , h ) + S ( r , Y ) . \\end{align*}"}
+{"id": "2909.png", "formula": "\\begin{align*} ( \\gamma \\ast \\mu ) ( f ) = \\mu ( f \\mid \\gamma ) \\mid \\gamma ^ { - 1 } . \\end{align*}"}
+{"id": "2256.png", "formula": "\\begin{align*} \\langle f _ b , \\varphi \\rangle : = \\lim _ { y \\searrow 0 } \\int _ { - \\infty } ^ { \\infty } f ( x + i y ) \\ , \\varphi ( x ) \\ , d x \\end{align*}"}
+{"id": "8127.png", "formula": "\\begin{align*} P ^ { 1 , 2 , 3 } _ { + , - , - } = P _ - ( P _ - ( P _ + ( a ) a ^ 2 ) a ^ 3 ) P _ { + , + , - } ( a ) = P _ - ( P _ + ( P _ + ( a ) a ) a ) . \\end{align*}"}
+{"id": "6602.png", "formula": "\\begin{align*} M _ X ( s ) = \\frac { 1 } { \\sqrt { 1 - 2 s \\sigma ^ 2 } } \\exp \\left ( \\frac { \\mu ^ 2 s } { 1 - 2 s \\sigma ^ 2 } \\right ) . \\end{align*}"}
+{"id": "6687.png", "formula": "\\begin{align*} q = 0 \\Gamma _ 1 \\times ( 0 , T ) . \\end{align*}"}
+{"id": "6798.png", "formula": "\\begin{align*} W ^ { \\mathrm { s } } ( p ) & = \\R ^ k _ { x _ 1 , \\dots , x _ k } \\times \\{ 0 \\} _ { x _ { k + 1 } , \\dots , x _ { 2 n } } , \\\\ W ^ { \\mathrm { u } } ( p ) & = \\{ 0 \\} _ { x _ 1 , \\dots , x _ k } \\times \\R ^ { 2 n - k } _ { x _ { k + 1 } , \\dots , x _ { 2 n } } . \\end{align*}"}
+{"id": "691.png", "formula": "\\begin{align*} \\mathrm { o s c } ( v , T ^ N ( I _ \\alpha ^ { ( k ) } ) ) \\leq \\mathrm { o s c } ( v , I _ \\alpha ^ { ( k ) } ) + \\int _ { I _ \\alpha ^ { ( k ) } } \\Big | \\sum _ { i = 0 } ^ { N - 1 } D \\varphi ( T ^ i x ) \\Big | \\ , d x . \\end{align*}"}
+{"id": "5753.png", "formula": "\\begin{align*} \\begin{aligned} | K _ { \\alpha } ( x , y _ { 1 } , \\dots , y _ { j } , \\dots , y _ { m } ) & - K _ { \\alpha } ( x , y _ { 1 } , \\dots , y _ { j } ' , \\dots , y _ { m } ) | \\\\ & \\le \\dfrac { A } { \\Big ( \\sum \\limits _ { j = 1 } ^ { m } | x - y _ { j } | \\Big ) ^ { m n - \\alpha } } \\omega \\Big ( \\frac { | y _ { j } - y _ { j } ' | } { | x - y _ { 1 } | + \\cdots + | x - y _ { m } | } \\Big ) \\end{aligned} \\end{align*}"}
+{"id": "8992.png", "formula": "\\begin{align*} \\left ( \\frac { m - 2 } { m } + w \\right ) \\nabla S ^ \\varphi = 0 \\ , M \\ , . \\end{align*}"}
+{"id": "6255.png", "formula": "\\begin{align*} G _ 1 ^ { \\mathrm { r e g } } ( \\xi , \\sigma , \\alpha , \\beta ) = H _ 1 ( \\xi , \\sigma , \\alpha - \\tau \\beta , \\beta ) \\ , + \\ , \\phi _ 1 ( \\sigma , \\alpha , \\beta ) \\ , . \\end{align*}"}
+{"id": "5057.png", "formula": "\\begin{align*} \\partial _ t \\left ( \\begin{array} { c } \\tilde { v } _ r \\\\ \\tilde { v } _ i \\end{array} \\right ) = \\mathcal { A } [ \\phi ] \\left ( \\begin{array} { c } \\tilde { v } _ r \\\\ \\tilde { v } _ i \\end{array} \\right ) + \\widetilde { \\mathcal { N } } [ \\phi ] ( \\tilde { v } ) , \\end{align*}"}
+{"id": "6274.png", "formula": "\\begin{align*} \\Gamma ' _ { s , t } : = R ( \\Sigma ' _ s , \\phi ( s , t ) ) \\cup R ( \\Sigma ' _ t , \\phi ( s , t ) ) \\cup N ( s , t , \\phi ( s , t ) ) . \\end{align*}"}
+{"id": "8724.png", "formula": "\\begin{align*} S _ { 1 } & = \\sum _ { 2 a _ 1 \\le a } \\frac { ( - 2 ) ^ { a - 2 a _ 1 } a ! } { a _ 1 ! a _ 1 ! ( a - 2 a _ 1 ) ! } E \\{ \\| X _ 1 \\| _ { 2 } ^ { 2 a _ 1 } \\| Y _ 1 \\| _ { 2 } ^ { 2 a _ 1 } ( X _ 1 ^ { \\top } Y _ 1 ) ^ { a - 2 a _ 1 } \\} \\\\ & = \\sum _ { 2 a _ 1 \\le a } \\frac { ( - 2 ) ^ { a - 2 a _ 1 } a ! } { a _ 1 ! a _ 1 ! ( a - 2 a _ 1 ) ! } E \\{ \\| Y _ 1 \\| _ { 2 } ^ { 2 a _ 1 } \\| Y _ 2 \\| _ { 2 } ^ { 2 a _ 1 } ( Y _ 1 ^ { \\top } Y _ 2 ) ^ { a - 2 a _ 1 } \\} , \\end{align*}"}
+{"id": "814.png", "formula": "\\begin{align*} y ^ { \\prime } \\left ( x \\right ) = f \\left ( x , y \\left ( x \\right ) \\right ) + \\int \\limits _ { x _ { 0 } } ^ { x } K \\left ( x , y \\left ( t \\right ) , t \\right ) d t \\end{align*}"}
+{"id": "5145.png", "formula": "\\begin{align*} \\| g \\| _ { X ' } = \\sup _ { \\| f \\| _ X \\leq 1 } \\| f g \\| _ { L ^ 1 ( \\Omega ) } \\leq \\sup _ { \\| f \\| _ E \\leq C } \\| f g \\| _ { L ^ 1 ( \\Omega ) } = C \\| g \\| _ { E ' } \\end{align*}"}
+{"id": "4434.png", "formula": "\\begin{align*} 2 \\omega Q ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) = ( 4 - d ) \\omega ^ { 2 - \\frac { d } { 2 } } \\mu _ { 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } } . \\end{align*}"}
+{"id": "427.png", "formula": "\\begin{align*} Q _ { x _ i } + Q _ { x _ i } ^ T = E ^ T P _ { \\partial \\Omega } N _ { i } E , \\end{align*}"}
+{"id": "2699.png", "formula": "\\begin{align*} i _ * I C _ N [ - 1 ] = i _ * i ^ ! p _ N ^ * ( I C _ N [ 1 ] ) \\to p _ N ^ * ( I C _ N [ 1 ] ) \\to R j _ * j ^ * p _ N ^ * ( I C _ N [ 1 ] ) = R j _ * I C _ { N _ \\eta } [ 1 ] \\rightsquigarrow \\end{align*}"}
+{"id": "8794.png", "formula": "\\begin{align*} & ( u ^ 2 + 2 t u + 2 t + 8 u + 8 ) b _ 1 \\\\ & = ( u ^ 2 + 4 t u + 4 t + 4 u + 2 ) c _ 1 + 2 u ( u + 1 ) b _ 2 + u ^ 2 b _ 3 + u ^ 2 c _ 3 \\end{align*}"}
+{"id": "3928.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta _ 1 , 0 ) = \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\inf _ { \\lambda _ 1 \\in \\mathbb { R } _ { + } } \\left [ \\lambda _ 1 \\delta _ 1 + \\int _ { \\mathcal { V } } g _ { \\lambda , 1 } \\ , d \\pi \\right ] . \\end{align*}"}
+{"id": "793.png", "formula": "\\begin{align*} ( ^ { c } _ { 0 } { D } ^ { \\alpha } _ { t } u , v ) \\ , + \\ , a \\big ( l ( u ) \\big ) \\ , ( \\nabla u , \\nabla v ) \\ , & = & \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\big ( f , v \\big ) , & \\forall v \\in H ^ 1 _ 0 ( \\Omega ) . & \\\\ u ( \\cdot , 0 ) & = & \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! u _ 0 ( \\cdot ) , \\ : & \\mbox { i n } \\ ; \\ , \\Omega . \\quad & \\end{align*}"}
+{"id": "2419.png", "formula": "\\begin{align*} d ( \\mathcal { N } _ { 1 } , \\mathcal { N } _ { 2 } ) = \\sup _ { j \\ge 1 } \\frac { \\abs { n _ { 1 , j } - n _ { 2 , j } } } { b _ { j } - a _ { j } } , \\mathcal { N } _ { i } = ( n _ { i , j } ) _ { j \\ge 1 } , i = 1 , 2 . \\end{align*}"}
+{"id": "3074.png", "formula": "\\begin{align*} \\mathcal U ( r ) = \\frac { u ( r ) } { u _ 0 ( r ) } , \\mathcal V ( r ) = \\frac { v ( r ) } { v _ 0 ( r ) } , \\mathcal W ( r ) = \\frac { u ' ( r ) } { u ' _ 0 ( r ) } , \\mathcal Y ( r ) = \\frac { v ' ( r ) } { v ' _ 0 ( r ) } \\end{align*}"}
+{"id": "7444.png", "formula": "\\begin{align*} \\mathcal { L } _ M ( \\varepsilon ) = \\sum \\limits _ { k = 0 } ^ { - \\lfloor \\alpha \\rfloor } \\varepsilon ^ { \\alpha + k - 1 } \\ , \\mathfrak { b } _ { \\alpha + k - 1 } + \\sum \\limits _ { k = 1 } ^ { M + \\lfloor \\alpha \\rfloor } \\varepsilon ^ { k - 1 } \\ , \\mathfrak { b } _ { k - 1 } + \\sum \\limits _ { k = - \\lfloor \\alpha \\rfloor + 1 } ^ { M } \\varepsilon ^ { \\alpha + k - 1 } \\ , \\mathfrak { b } _ { \\alpha + k - 1 } , \\end{align*}"}
+{"id": "4725.png", "formula": "\\begin{align*} I _ g \\leq C a ^ d \\log b / a , I _ \\gamma \\leq C \\zeta ^ { d / 2 } = C ( 1 + \\abs { \\log z } ) ^ { d / 2 } , I _ { | x | ^ n g } \\leq C a ^ d b ^ n . \\end{align*}"}
+{"id": "7387.png", "formula": "\\begin{align*} \\sum _ { n \\ne 0 } \\widehat { \\Delta } ( a n ) ^ { 2 } \\le \\sum _ { n \\in \\mathbb { Z } } \\widehat { \\Delta } ( a n ) - 1 = \\frac { 1 } { a } \\sum _ { m \\in \\mathbb { Z } } \\Delta \\left ( \\frac { m } { a } \\right ) - 1 = \\frac { 1 } { a } - 1 < \\frac { 1 } { a } . \\end{align*}"}
+{"id": "5013.png", "formula": "\\begin{align*} \\left ( \\sum _ { j \\in M ^ c } | a _ j | ^ 2 \\right ) ^ \\frac { 1 } { 2 } \\leq \\varepsilon \\left ( \\sum _ { j = 1 } ^ d | a _ j | ^ 2 \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "657.png", "formula": "\\begin{align*} & V ^ { j , \\tau } _ { k } ( T , \\bar { s } ) : = \\sum _ { l \\geq k } \\| Q | _ { U ^ { ( k ) } _ { - j } } ( k , l + 1 ) ^ { - 1 } \\| \\| Q ( l + 1 ) \\| ^ \\tau \\| Z ( l + 1 ) \\| s _ { l } , \\\\ & W ^ { j , \\tau } _ { k } ( T , \\bar { s } ) : = \\sum _ { 0 \\leq l < k } \\| Q | _ { E ^ { ( l + 1 ) } _ { - j } } ( l + 1 , k ) \\| \\| Q ( l + 1 ) \\| ^ \\tau \\| Z ( l + 1 ) \\| s _ l . \\end{align*}"}
+{"id": "6490.png", "formula": "\\begin{align*} \\det ( a _ { i j } ) = \\sum _ { \\pi \\in S _ n } \\operatorname { W e i g h t } ( \\pi ) \\ ; . \\end{align*}"}
+{"id": "4834.png", "formula": "\\begin{align*} \\overline { A } = \\{ p \\in \\beta S : A \\in p \\} . \\end{align*}"}
+{"id": "4936.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { n } \\left [ u _ { 0 , k } ^ { } , u _ { 0 , k } ^ { } + ( n { - } 1 ) u _ { 1 , k } ^ { } , . . . , \\ , i u _ { i , k } ^ { } + ( n { - } i ) u _ { i + 1 , k } ^ { } , \\ , . . . , ( k { - } 1 ) u _ { k - 1 , k } ^ { } + ( n { - } k { + } 1 ) u _ { k , k } , k u _ { k , k } ^ { } , \\mathbf { 0 } _ { n - k } ^ \\top \\right ] ^ \\top . \\end{aligned} \\end{align*}"}
+{"id": "17.png", "formula": "\\begin{align*} J _ 1 = \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} , J _ n = \\mathrm { d i a g } ( J _ 1 , \\cdots , J _ 1 ) \\in \\mathrm { G L } _ { 2 n } ( \\mathbb { Z } ) . \\end{align*}"}
+{"id": "4225.png", "formula": "\\begin{align*} E _ 1 & = \\det T ( \\phi ) T ( \\phi _ R ) ^ { - 1 } \\cdots T ( \\phi _ 0 ) ^ { - 1 } , \\\\ E _ 2 & = \\det T ( \\tilde { \\phi } ) T ( \\tilde { \\phi } _ R ) ^ { - 1 } \\cdots T ( \\tilde { \\phi } _ 0 ) ^ { - 1 } , \\\\ E _ 3 & = \\det T ( \\phi _ { 0 } ) T ( \\phi _ { 0 } ^ { - 1 } ) , \\end{align*}"}
+{"id": "8930.png", "formula": "\\begin{align*} B _ l ^ { ( j + 1 ) } = \\sum _ { q = 2 } ^ k \\sum _ { ( p _ 1 , \\dots , p _ q ) \\in \\mathbb { I } _ { 2 k } ^ q , \\ , \\deg ( S _ { p _ 1 } ) + \\cdots + \\deg ( S _ { p _ q } ) = l } \\alpha _ { ( p _ 1 , \\dots , p _ q ) } [ S _ { p _ 1 } , \\dots , S _ { p _ q } ] . \\end{align*}"}
+{"id": "8629.png", "formula": "\\begin{align*} \\omega _ 0 ( t ) = Z _ 0 ( t ) - Y e ^ { \\lambda t } . \\end{align*}"}
+{"id": "3067.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\left [ u ' ( r ) ^ { p - 1 - \\alpha } \\right ] ' + \\frac { \\delta } { r } \\ , u ' ( r ) ^ { p - 1 - \\alpha } = \\frac { \\delta } { n - 1 } \\ , f _ 1 ( r ) g _ 1 ( v ( r ) ) & & \\\\ & \\left [ v ' ( r ) ^ { p - 1 } \\right ] ' + \\frac { n - 1 } { r } \\ , v ' ( r ) ^ { p - 1 } = f _ 2 ( r ) g _ 2 ( v ( r ) ) \\cdot g _ 3 ( u ' ( r ) ) & & \\end{aligned} \\right . \\end{align*}"}
+{"id": "6341.png", "formula": "\\begin{align*} J ( \\phi , \\omega , r ; t ) = C ( 1 + O ( \\varepsilon ) ) | t | ^ 5 , \\forall \\ , t \\in [ - \\rho , \\rho ] . \\end{align*}"}
+{"id": "4505.png", "formula": "\\begin{align*} 0 < \\mu _ { 1 , \\mathbf { c } _ 0 } = E ( \\Phi ) + Q ( \\Phi ) + \\mathbf { c } _ 0 \\cdot \\mathbf { P } ( \\Phi ) = Q ( \\Phi ) - E ( \\Phi ) \\end{align*}"}
+{"id": "9290.png", "formula": "\\begin{align*} \\min \\limits _ { x \\in D _ { I } } \\to f ( x ) = \\sum \\limits ^ { m } _ { i = 1 } f _ { i } ( x _ { i } ) , \\end{align*}"}
+{"id": "5548.png", "formula": "\\begin{align*} \\max _ { A \\in \\mathcal { P } _ { x , n } } \\frac { q _ { x , n , 2 } ^ { t } ( A ) } { q _ { x , n , 1 } ^ { t } ( A ) } & \\le \\frac { 1 } { \\left ( 1 - \\varepsilon _ { x , n , 1 } ( t ) \\right ) \\left ( 1 - \\varepsilon _ { x , n , 2 } ( t ) \\right ) } \\max _ { A \\in \\mathcal { P } _ { x , n } } \\frac { \\beta _ { x , 2 } ( A ) } { \\beta _ { x , 1 } ( A ) } \\\\ & \\le \\frac { C } { \\left ( 1 - \\varepsilon _ { x , n , 1 } ( t ) \\right ) \\left ( 1 - \\varepsilon _ { x , n , 2 } ( t ) \\right ) } = : C _ { n , t } . \\end{align*}"}
+{"id": "4150.png", "formula": "\\begin{align*} Q _ { 1 , \\vec { v } _ { s , t , f , 0 } ^ { ( 2 ) } } = \\left ( \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 0 & 0 & 1 \\\\ 1 & 0 & - t \\\\ 0 & 1 & - s \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 0 & 1 & - s \\\\ 0 & - t & s t + 1 \\\\ 1 & - s & s ^ { 2 } - t \\end{array} \\right ) \\right ) , \\end{align*}"}
+{"id": "1921.png", "formula": "\\begin{align*} \\begin{aligned} \\theta ( u ^ k ) + I _ K ( \\zeta ^ k ) & \\leq \\theta ( \\hat { u } ) + I _ K ( \\hat { \\zeta } ) - [ \\langle \\lambda ^ k , \\hat { \\zeta } - \\zeta ^ k \\rangle + \\langle D _ u \\theta ( u ^ k ) , \\hat { u } - u ^ k \\rangle _ \\mathcal { U } ] \\\\ & \\leq \\theta ( \\hat { u } ) + I _ K ( \\hat { \\zeta } ) + \\beta \\sum \\limits _ { i = n } ^ { k } \\| r ^ { i } \\| ^ 2 + | \\langle \\lambda ^ n , r ^ k \\rangle | . \\end{aligned} \\end{align*}"}
+{"id": "535.png", "formula": "\\begin{align*} \\left ( \\mathcal { F } _ { \\mathcal { H } _ { \\hbar , V } } ^ { - 1 } g \\right ) ( k ) : = \\sum \\limits _ { \\xi \\in \\mathcal { I } _ { \\hbar } } g ( \\xi ) u _ { \\xi } ( k ) , g \\in \\mathcal { S } ( \\mathcal { I } _ { \\hbar } ) , \\end{align*}"}
+{"id": "2858.png", "formula": "\\begin{align*} H \\circ \\varphi _ L = \\varphi _ { K } \\circ H . \\end{align*}"}
+{"id": "1494.png", "formula": "\\begin{align*} \\partial _ t \\big ( x ^ 2 w ( t , x ) \\big ) = \\partial _ x \\ , \\Big ( \\frac { 1 } { 2 } \\partial _ x \\big ( x ^ 2 w ( t , x ) \\big ) - \\frac { 1 } { x } \\big ( x ^ 2 w ( t , x ) \\big ) \\ ! \\Big ) = \\Big ( \\frac { 1 } { 2 } \\partial _ { x x } + \\frac { 1 } { x } \\partial _ x \\Big ) ^ * \\big ( x ^ 2 w ( t , x ) \\big ) , \\end{align*}"}
+{"id": "5982.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } ( g _ 1 ) \\overline { \\Pi } _ { \\psi } ( g _ 2 ) & = \\overline { \\Pi } _ { \\psi } ( g _ 1 g _ 2 ) \\overline { C } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) . \\end{align*}"}
+{"id": "4709.png", "formula": "\\begin{align*} \\widehat { \\vect u } = \\sum _ { i = 1 } ^ 3 \\alpha _ i \\widehat { \\Pi } _ \\chi ^ { \\rm s t i f f } \\psi _ i ^ \\chi . \\end{align*}"}
+{"id": "8156.png", "formula": "\\begin{align*} \\frac { 1 - q t | } { | 1 - q } = ( 1 - q t ) \\times \\frac 1 { 1 - q } \\end{align*}"}
+{"id": "6618.png", "formula": "\\begin{align*} [ [ x _ \\mu y ] _ { \\mu + \\gamma } \\varphi _ { \\lambda } ( w , v ) ] = [ ( \\varphi _ \\lambda ( x , y ) ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] . \\end{align*}"}
+{"id": "6220.png", "formula": "\\begin{align*} \\psi _ \\mu ( x ) = \\int g ( x - y ) \\ , d \\mu ( y ) \\ , , \\end{align*}"}
+{"id": "5574.png", "formula": "\\begin{align*} \\alpha _ { x , g } = ( \\zeta _ { H } ) _ { \\ast } \\left ( r ^ { - 1 } \\left ( \\bar { p } _ { k } \\right ) _ { \\ast } \\left ( \\tau ( y ) ^ { - 1 } ( g \\nu _ { P } ) ^ { y } \\right ) \\right ) , \\ \\mbox { w h e r e } x = \\left ( y , r , H \\right ) . \\end{align*}"}
+{"id": "574.png", "formula": "\\begin{align*} \\gamma _ { j k } ^ { b } ( x _ { 1 } , x _ { 2 } ) = \\int _ { \\R _ { + } } b \\left ( \\frac { y _ { 2 } } { 2 x _ { 2 } } \\right ) \\ell _ { j - 1 } ( y _ { 2 } ) \\ell _ { k - 1 } ( y _ { 2 } ) d y _ { 2 } , j , k = 1 , . . . , n . \\\\ \\end{align*}"}
+{"id": "6634.png", "formula": "\\begin{align*} \\phi ( h _ 1 ) = ( d + 1 ) l - \\sum _ { i = 1 } ^ { q } { e _ i } \\phi ( h _ 2 ) = l . \\end{align*}"}
+{"id": "3187.png", "formula": "\\begin{align*} \\abs { \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } U ^ j ( M _ 1 ( \\mu ) ) ( x ) } \\leq \\frac { \\epsilon } { \\chi _ d } \\rho ( x ) x \\in ( e M e ) _ + n \\in \\{ 1 , \\ldots , ( N + 1 ) _ d \\} . \\end{align*}"}
+{"id": "238.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ 4 z ^ k = - \\frac { n ^ 4 z ^ { n + 5 } + ( - 4 n ^ 4 - 4 n ^ 3 + 6 n ^ 2 - 4 n + 1 ) z ^ { n + 4 } + ( 6 n ^ 4 + 1 2 n ^ 3 - 6 n ^ 2 - 1 2 n + 1 1 ) z ^ { n + 3 } } { ( 1 - z ) ^ 5 } \\end{align*}"}
+{"id": "5.png", "formula": "\\begin{align*} \\Tilde { \\theta } ( u ^ - u ^ 0 ) : = \\theta ^ - ( u ^ - ) \\theta ^ 0 ( u ^ 0 ) \\Delta ( u ^ 0 ) ^ { - 1 } . \\end{align*}"}
+{"id": "100.png", "formula": "\\begin{align*} \\partial _ t v = - h ^ { - 1 } e ^ { h ^ { - 1 } \\int _ 0 ^ t q _ 1 ( \\varphi ^ s ( x ) ) d s } W w ( t , \\varphi ^ t ( x ) ) . \\end{align*}"}
+{"id": "6002.png", "formula": "\\begin{align*} \\theta _ { 3 / 2 } ( p _ g , \\epsilon ) & = \\theta _ { L , X ^ { \\ast } } \\Big ( J ( p _ g , i ) ^ 3 \\overline { \\Pi } _ { \\psi } ( p _ g ) B \\Big ) [ \\epsilon , 0 ] \\\\ & = \\sum _ { n \\in \\Z } J ( p _ g , i ) ^ 3 \\overline { \\Pi } _ { \\psi } ( p _ g ) B ( \\epsilon , n ) \\\\ & = \\sum _ { n \\in \\Z } n e ^ { i \\epsilon \\pi n ^ 2 z _ g } . \\end{align*}"}
+{"id": "8820.png", "formula": "\\begin{align*} u ( t , x ) & = u _ 0 ( x ) + \\int _ 0 ^ t \\ ! \\ ! \\int _ G K _ { t - s } ( x , y ) f ( u ( s , y ) ) \\ , d y \\ , d s \\\\ & + \\int _ 0 ^ t \\ ! \\ ! \\int _ G K _ { t - s } ( x , y ) \\sigma ( u ( s , y ) ) \\ , \\bar { W } ( d y , d s ) , \\end{align*}"}
+{"id": "8561.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } S _ j ( t ) = \\nu Y \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s , j \\geq 1 , \\end{align*}"}
+{"id": "4183.png", "formula": "\\begin{align*} u _ \\mathrm { l i n } ( t , r ) = \\begin{cases*} 2 & i f $ 0 < r < 1 - t $ , \\\\ \\frac { 1 + r - t } { r } & i f $ 1 - t \\leq r < t $ , \\\\ \\frac { 1 - r + t } { r } & i f $ t \\leq r < 1 + t $ , \\\\ 0 & o t h e r w i s e . \\end{cases*} \\end{align*}"}
+{"id": "8099.png", "formula": "\\begin{align*} \\mathbf { S } _ { 6 k + i + j + ( i > 0 ) } : = \\xi _ k \\left ( \\mathbf { e } _ i \\otimes \\mathbf { e } _ j + \\mathbf { e } _ j \\otimes \\mathbf { e } _ i \\right ) , \\end{align*}"}
+{"id": "9151.png", "formula": "\\begin{align*} v _ { s } & = \\varphi _ { s , [ \\kappa _ { s } ] } ( x , v _ { 1 } , v _ { 1 , [ 1 ] } , \\ldots , v _ { s - 1 } , v _ { s - 1 , [ 1 ] } , \\ldots , u _ { r e s t _ { s - 1 } } ) \\ , , \\end{align*}"}
+{"id": "1056.png", "formula": "\\begin{align*} c _ 0 \\cdot \\Delta _ G ( f ) + c _ 1 \\cdot \\Delta _ G ( f ) ^ p + \\cdots + c _ N \\cdot \\Delta _ G ( f ) ^ { p ^ N } = 0 \\ , , \\end{align*}"}
+{"id": "6333.png", "formula": "\\begin{align*} \\frac { \\partial z } { \\partial \\omega } ( \\phi , \\omega , r ; t ) = ( 1 + O ( \\varepsilon ) ) ( 1 + \\varepsilon ) ^ { n } \\frac { r ^ 2 t ^ 3 } { 6 } , \\qquad \\forall \\ , t \\in ( 0 , \\rho ] , \\end{align*}"}
+{"id": "3822.png", "formula": "\\begin{align*} \\mathcal { P } _ p ( \\mathcal { X } ) = \\left \\{ \\mu \\in \\mathcal { P } ( \\mathcal { X } ) : \\int _ { \\mathcal { X } } \\boldsymbol { d } ( x _ 0 , x ) ^ p d \\mu ( x ) < \\infty \\right \\} , \\end{align*}"}
+{"id": "7880.png", "formula": "\\begin{align*} \\mbox { $ \\mathcal { S } _ + = \\left \\{ A ^ + \\in \\mathcal { S } : \\ A \\in \\binom { [ n ] } { k } \\right \\} $ a n d $ \\mathcal { S } _ - = \\left \\{ A ^ - \\in \\mathcal { S } : \\ A \\in \\binom { [ n ] } { k } \\right \\} $ . } \\end{align*}"}
+{"id": "631.png", "formula": "\\begin{align*} \\phi _ n ( x _ n ^ * x _ n ) = \\psi _ n ( x _ n ^ * x _ n ) = 1 \\end{align*}"}
+{"id": "6424.png", "formula": "\\begin{align*} \\begin{cases} \\mathrm { t r } \\big [ \\big ( h _ \\varphi ^ { ( 1 - \\alpha ) / 2 z } h _ \\psi ^ { \\alpha / z } h _ \\varphi ^ { ( 1 - \\alpha ) / 2 z } \\big ) ^ z \\big ] & ( ) , \\\\ \\Vert x \\Vert _ z ^ z & ( ) , \\\\ + \\infty & ( ) , \\end{cases} \\end{align*}"}
+{"id": "3539.png", "formula": "\\begin{align*} \\frac { j ''' } { j ' } - \\frac { 3 } { 2 } \\left ( \\frac { j '' } { j ' } \\right ) ^ 2 + \\left ( \\frac { j ^ 2 - 1 9 6 8 j + 2 6 5 4 2 0 8 } { 2 j ^ 2 ( j - 1 7 2 8 ) ^ 2 } \\right ) ( j ' ) ^ 2 = 0 . \\end{align*}"}
+{"id": "8963.png", "formula": "\\begin{align*} \\begin{bmatrix} ( r ^ { p _ 1 } - 1 ) & ( r ^ { p _ 2 } - 1 ) \\\\ ( r ^ { p _ 1 } - 1 ) r ^ { p _ 1 } & ( r ^ { p _ 2 } - 1 ) r ^ { p _ 2 } \\end{bmatrix} \\begin{bmatrix} C _ { p _ 1 } h ^ { p _ 1 } \\\\ C _ { p _ 2 } h ^ { p _ 2 } \\end{bmatrix} & = \\begin{bmatrix} e _ { 2 1 } \\\\ e _ { 3 2 } \\end{bmatrix} , \\end{align*}"}
+{"id": "8536.png", "formula": "\\begin{align*} \\left ( \\frac { \\pi } { \\sqrt { c } } \\right ) ^ { - s } \\Gamma ( s ) \\ , \\zeta _ { \\infty , p ^ { \\prime } } \\left ( s , c \\right ) = \\left ( \\frac { \\pi } { \\sqrt { c } } \\right ) ^ { - ( 1 - s ) } \\Gamma \\left ( 1 - s \\right ) \\ , \\tilde { \\zeta } _ { p ^ { \\prime } , \\infty } \\left ( 1 - s , c \\right ) . \\end{align*}"}
+{"id": "3710.png", "formula": "\\begin{align*} \\exp ( y f ) . x = \\frac { x } { 1 + x y } . \\end{align*}"}
+{"id": "8191.png", "formula": "\\begin{align*} M _ t : = \\inf \\{ r \\geq 0 : \\mathcal { R } ( t ) \\subseteq B ( 0 , r ) \\} \\end{align*}"}
+{"id": "1754.png", "formula": "\\begin{align*} J _ { + } ^ { ( m ) } & = \\sum _ { \\mathbf { k } \\in \\mathcal { S } _ m } \\binom { m } { \\mathbf { k } } \\frac { w _ 1 ^ { k _ 1 } w _ 2 ^ { k _ 2 } w _ 3 ^ { k _ 3 } } { 2 ^ { k _ 1 + k _ 2 + k _ 3 - 1 } } \\mathcal { J } _ { + } ^ { ( m , \\mathbf { k } ) } , \\end{align*}"}
+{"id": "8877.png", "formula": "\\begin{align*} P ( n ) & \\ge \\frac { n - 1 } { 2 } P ( n - 1 ) \\\\ & \\ge \\frac { n - 1 } { 2 } \\frac { \\sqrt { 2 \\pi } } { e ^ 2 } n ^ { 3 / 2 } \\frac { ( n - 2 ) ! } { 2 ^ { n } } ( 1 - o ( 1 ) ) \\\\ & = \\frac { \\sqrt { 2 \\pi } } { e ^ 2 } n ^ { 3 / 2 } \\frac { ( n - 1 ) ! } { 2 ^ { n + 1 } } ( 1 - o ( 1 ) ) \\ ; . \\end{align*}"}
+{"id": "4618.png", "formula": "\\begin{align*} \\sum _ i d ( k ) _ i ^ 2 + d ( k ) _ i = 2 k . \\end{align*}"}
+{"id": "3034.png", "formula": "\\begin{align*} \\phi ( X E _ { i i } X ^ * \\otimes B ) = W _ X ( E _ { i i } \\otimes \\varphi _ { i , X } ( B ) ) { W } _ X ^ * \\hbox { f o r a l l \\quad } B \\in M _ n , \\end{align*}"}
+{"id": "5261.png", "formula": "\\begin{align*} \\| f _ { [ \\ell ] \\rightarrow 1 } \\| _ 2 ^ 2 = ( \\ell ( 1 - p ) ) ^ 2 + ( n / d - \\ell ) ( p ( 1 - p ) ) \\le \\ell ^ 2 + 1 \\end{align*}"}
+{"id": "1937.png", "formula": "\\begin{align*} | c _ { i _ 1 , \\ldots , i _ m } | < s _ { i _ 1 + \\cdots + i _ m } : = \\frac { 1 } { \\binom { i _ 1 + \\cdots + i _ m - 1 } { m - 1 } ( i _ 1 + \\cdots + i _ m ) ! } , \\end{align*}"}
+{"id": "3989.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } \\left \\{ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\pi \\in \\widetilde { \\mathit { \\Pi } } } \\int _ { \\mathbb { R } ^ { 2 d + 2 } } \\left ( f _ { \\mathcal { S } } \\right ) _ \\lambda \\ , d \\pi \\right \\} \\end{align*}"}
+{"id": "2891.png", "formula": "\\begin{align*} ( J \\star T ) \\times \\Delta [ 1 ] & = \\big ( ( J \\times \\{ 0 \\} ) \\star ( T \\times \\Delta [ 1 ] ) \\big ) \\sqcup _ { ( J \\times \\{ 0 \\} ) \\star ( T \\times \\{ 1 \\} ) } \\big ( ( J \\times \\Delta [ 1 ] ) \\star ( T \\times \\{ 1 \\} ) \\big ) \\end{align*}"}
+{"id": "879.png", "formula": "\\begin{align*} N ( P ) = c \\widehat { P } ^ { n - 5 } + O \\left ( \\widehat { P } ^ { n - 5 - \\delta ' } \\right ) \\end{align*}"}
+{"id": "6567.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ n B ^ * _ i A _ i \\right | \\le \\left ( \\sum _ { i = 1 } ^ n A ^ * _ i A _ i \\right ) \\# V ^ * \\left ( \\sum _ { i = 1 } ^ n B _ i ^ * B _ i \\right ) V , \\end{align*}"}
+{"id": "7433.png", "formula": "\\begin{align*} \\partial _ t { w } ^ { ( i ) } _ 0 ( x _ i , t ) \\ , + \\ , \\partial _ { x _ i } \\big ( v _ i ^ { ( i ) } ( x _ i , t ) \\ , w ^ { ( i ) } _ 0 ( x _ i , t ) \\big ) = - \\widehat { \\varphi } ^ { ( i ) } , \\end{align*}"}
+{"id": "8854.png", "formula": "\\begin{align*} ( x - a ) ^ N \\prod _ { k = 1 } ^ { \\lfloor { T \\over 2 } \\rfloor } ( f ( x ) - v _ { 2 k - 1 } ^ 2 ( x ) ) \\sim \\prod _ { k = 0 } ^ { \\lfloor { T \\over 2 } \\rfloor } ( f ( x ) - v _ { 2 k } ^ 2 ( x ) ) \\end{align*}"}
+{"id": "8843.png", "formula": "\\begin{align*} d ( \\overline { v } _ \\lambda - \\overline { v } _ \\mu ) + A ( \\overline { v } _ \\lambda - \\overline { v } _ \\mu ) \\ , d t = ( F _ \\lambda \\overline { v } _ \\lambda - F _ \\mu \\overline { v } _ \\mu ) \\ , d t + \\sigma ' ( u ) ( \\overline { v } _ \\lambda - \\overline { v } _ \\mu ) B \\ , d W \\end{align*}"}
+{"id": "1126.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\frac { X _ i u ( ( s X _ 0 ) x , t - s ) - X _ i u ( x , t ) } { | s | ^ { 1 / 2 } } = 0 , \\end{align*}"}
+{"id": "7830.png", "formula": "\\begin{align*} C _ { 2 } ( \\varepsilon ) ^ { p } ( \\sqrt { n p } ) ^ { p } + \\sum _ { k = 1 } ^ { \\infty } C _ { 2 } ( \\varepsilon ) ^ { p } ( \\sqrt { n p } ) ^ { p } ( 2 ^ { k } ) ^ { p / 2 } \\exp ( - c 2 ^ { \\frac { 2 k } { 4 + \\varepsilon } } ) \\le C _ { 3 } ( \\varepsilon ) ^ { p } ( \\sqrt { n p } ) ^ { p } . \\end{align*}"}
+{"id": "2134.png", "formula": "\\begin{align*} \\int _ \\Omega f \\ , d x = 0 . \\end{align*}"}
+{"id": "3171.png", "formula": "\\begin{align*} ( \\underbar { x } , \\underbar { y } ' ) : = ( x _ 1 , \\ldots , x _ N , y _ 1 , \\ldots , y _ m + c ' , \\dots , y _ N ) . \\end{align*}"}
+{"id": "2937.png", "formula": "\\begin{align*} \\Delta _ \\delta ( \\tau ) : = \\prod _ { D \\mid N } \\Delta ( D \\tau ) ^ { n _ D } \\end{align*}"}
+{"id": "2577.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { 1 } { t } \\left ( \\int _ { B _ R } | \\nabla g ^ t | ^ 2 \\d x - \\int _ { B _ R } | \\nabla g ^ 0 | ^ 2 \\d x \\right ) = 0 . \\end{align*}"}
+{"id": "6387.png", "formula": "\\begin{align*} x ( x ^ 2 + 2 0 r ^ 2 ) = 3 ^ { p - 2 } w ^ p . \\end{align*}"}
+{"id": "6263.png", "formula": "\\begin{align*} Q _ { \\Sigma } [ \\phi ] = - \\int _ { \\Sigma } \\phi L _ { \\Sigma } \\phi + \\int _ { \\partial \\Sigma } \\phi ( \\nabla _ n \\phi - h ( \\nu , \\nu ) \\phi ) , \\end{align*}"}
+{"id": "4585.png", "formula": "\\begin{align*} & [ u , v , w ] _ T = \\rho ( T u , T v ) w , \\end{align*}"}
+{"id": "6673.png", "formula": "\\begin{align*} \\rho \\phi _ t + \\Delta \\phi - ( - \\Delta ) ^ s \\phi = 0 \\textrm { i n } \\ ; \\ ; S _ T \\ , , \\end{align*}"}
+{"id": "2969.png", "formula": "\\begin{align*} \\sigma \\cdot [ E , \\lambda ] = [ E _ { \\sigma ^ { - 1 } } , \\lambda _ { \\sigma ^ { - 1 } } ] \\end{align*}"}
+{"id": "6145.png", "formula": "\\begin{align*} b _ { j + 1 } - b _ j & = ( x - y ) ^ 2 \\cdot ( z - w ) \\cdot z ^ j \\cdot w ^ { n - 2 - j } \\\\ & = ( x - y ) \\cdot ( z - w ) \\cdot x \\cdot z ^ j \\cdot w ^ { n - 2 - j } - ( x - y ) \\cdot ( z - w ) \\cdot y \\cdot z ^ j \\cdot w ^ { n - 2 - j } . \\end{align*}"}
+{"id": "367.png", "formula": "\\begin{align*} c ^ { \\wedge \\sigma / \\tau } = \\bigwedge _ { i \\in \\tau } \\left ( \\frac { c ^ { \\sigma } _ i } { c ^ { \\tau } _ i } \\right ) : \\bigwedge _ { i \\in \\tau } X _ i \\to \\bigwedge _ { i \\in \\tau } X _ i . \\end{align*}"}
+{"id": "3660.png", "formula": "\\begin{align*} \\widehat { ( - \\Delta ) ^ \\gamma u } = | \\xi | ^ { 2 \\gamma } \\hat { u } , \\hat { u } ( \\xi ) = \\int _ { \\mathbb { R } ^ n } e ^ { - i \\xi \\cdot x } u ( x ) \\ , d x . \\end{align*}"}
+{"id": "4116.png", "formula": "\\begin{align*} \\left | \\frac { N ( y ) } { N ( 1 ) } \\right | = \\left | \\frac { c _ 5 } { b _ 3 } \\right | , \\left | \\frac { N ( 1 ) } { N ( x ) } \\right | = \\left | \\frac { \\widetilde { c _ 5 } } { \\widetilde { b _ 3 } } \\right | , \\left | \\frac { N ( x ) } { N ( y ) } \\right | = \\left | \\frac { \\widehat { c _ 5 } } { \\widehat { b _ 3 } } \\right | . \\end{align*}"}
+{"id": "736.png", "formula": "\\begin{align*} \\cal A _ { D , N } = \\frac 1 2 \\tilde { \\cal A } ( { \\rm I } _ { N _ { S } } - \\cal R ) \\ , , \\end{align*}"}
+{"id": "4527.png", "formula": "\\begin{align*} f ' ( 0 ) = \\mathbf { c } \\cdot \\mathbf { P } ( U ) \\ge 0 , \\ \\ f ' ( 1 ) = I _ { \\omega , \\mathbf { c } } ( U ) < 0 \\end{align*}"}
+{"id": "5382.png", "formula": "\\begin{align*} \\mathbf { V } ( \\mathbf { E } \\mathbf { x } _ { k + 1 } ) - \\mathbf { V } ( \\mathbf { E } \\mathbf { x } _ { k } ) \\leq \\sum _ { i , j = 1 } ^ 2 \\| u _ { i , k } ^ j \\| ^ 2 - \\| y _ { i , k } ^ j \\| ^ 2 \\leq \\| u _ { 1 , k } ^ 2 \\| ^ 2 + \\| u _ { 2 , k } ^ 2 \\| ^ 2 - \\| y _ { 1 , k } ^ 2 \\| ^ 2 - \\| y _ { 2 , k } ^ 2 \\| ^ 2 = \\| \\mathbf { u } _ k \\| ^ 2 - \\| \\mathbf { y } _ k \\| ^ 2 , \\end{align*}"}
+{"id": "8991.png", "formula": "\\begin{align*} \\liminf _ { r \\to \\infty } \\frac { 1 } { r ^ 2 } \\int _ { B _ r } { \\bar v } ^ 2 \\le \\liminf _ { r \\to \\infty } \\frac { | \\Omega \\cap B _ r | } { r ^ 2 } = 0 . \\end{align*}"}
+{"id": "3658.png", "formula": "\\begin{align*} \\mp _ s = \\begin{bmatrix} \\mp _ { s , 1 } & \\mp _ { s , 2 } & \\cdots & \\mp _ { s , 2 ^ { t - s } } \\end{bmatrix} . \\end{align*}"}
+{"id": "4890.png", "formula": "\\begin{align*} y _ { 1 , m } : = \\left ( \\frac { 1 } { 1 - r _ m } \\right ) \\left [ \\frac { 1 + r _ m } { 2 } \\sqrt { 1 - \\left ( \\frac { 2 | x ' | ( 1 - r _ m ) } { 1 + r _ m } \\right ) ^ 2 } - 1 \\right ] . \\end{align*}"}
+{"id": "8795.png", "formula": "\\begin{align*} & ( u ^ 2 + 2 t u + 2 t + 8 u + 8 ) ( [ 3 + t + u ] b _ 2 - [ 1 + 2 t ] c _ 2 - u b _ 3 ) \\\\ & = ( u ^ 2 + 4 t u + 4 t + 4 u + 2 ) ( [ 2 + 2 t + u ] c _ 2 - [ 2 + t ] b _ 2 - u c _ 3 ) \\\\ & + 2 u ( u + 1 ) b _ 2 + u ^ 2 b _ 3 + u ^ 2 c _ 3 \\end{align*}"}
+{"id": "1538.png", "formula": "\\begin{align*} K ( x , y , t ) = K ( y , x , t ) \\end{align*}"}
+{"id": "7873.png", "formula": "\\begin{align*} \\mathcal { O } _ i = \\left \\{ ( A ^ + , B ^ + ) \\in \\binom { [ n ] } { k } ^ \\pm \\times \\binom { [ n ] } { k } ^ \\pm : \\ | A \\cap B | = k - i \\right \\} , \\end{align*}"}
+{"id": "2459.png", "formula": "\\begin{align*} u : \\R ^ k \\times B \\to G & & u ( t _ 1 , \\ldots , t _ k , b ) = e ^ { t _ 1 X _ 1 } \\cdots e ^ { t _ k X _ k } b \\end{align*}"}
+{"id": "3121.png", "formula": "\\begin{align*} t ^ { n } = \\sum _ { k = 0 } ^ { n } S _ r [ n , k ] ( t ) _ { k , q } ^ r , \\end{align*}"}
+{"id": "265.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { - y ( y ^ 2 + 4 y + 1 ) ( y ^ n - 1 ) + 3 n ( y ^ 2 - 1 ) y ^ { n + 1 } - 3 n ^ 2 ( ( y - 1 ) ^ 2 y ^ { n + 1 } ) + n ^ 3 ( y - 1 ) ^ 3 y ^ { n + 1 } } { ( 1 - y ) ^ 4 } \\right ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "923.png", "formula": "\\begin{align*} g ( u , v ) = u ^ N \\phi ( v / u ) , \\end{align*}"}
+{"id": "7365.png", "formula": "\\begin{align*} E _ N ( \\mathcal { A } ) = \\# \\{ ( i , j , k , l ) \\in [ 1 , N ] ^ 4 : a _ i + a _ j = a _ k + a _ l \\} ; \\end{align*}"}
+{"id": "4343.png", "formula": "\\begin{align*} \\begin{array} { l l } & \\left \\Vert u \\right \\Vert _ { \\alpha , p } \\\\ & \\cap _ { i = 1 } ^ { m } \\left ( S _ { i } + u _ { i } \\right ) \\neq \\emptyset , \\\\ & u = \\left ( u _ { 1 } , . . . , u _ { m } \\right ) \\in X ^ { m } , \\end{array} \\end{align*}"}
+{"id": "8114.png", "formula": "\\begin{align*} \\ < X _ { T _ 1 \\cdots T _ r } , X _ T \\ > _ \\triangleright = \\sum _ { T ' } X _ { T ' } \\end{align*}"}
+{"id": "3858.png", "formula": "\\begin{align*} c _ 1 ( ( y _ 1 , x ) , ( y _ 1 ' , x ' ) ) = \\| x - x ' \\| _ { p } + \\kappa _ 1 | y _ 1 - y _ 1 ' | \\end{align*}"}
+{"id": "2655.png", "formula": "\\begin{align*} t _ { j } ( x ) = \\dfrac { \\tanh ( \\alpha x _ { j } ) - \\tanh ( \\alpha x ) } { \\tanh ( \\alpha x _ { j } ) - \\tanh ( \\alpha x _ { j - 1 } ) } y _ { j - 1 } + \\dfrac { \\tanh ( \\alpha x ) - \\tanh ( \\alpha x _ { j - 1 } ) } { \\tanh ( \\alpha x _ { j } ) - \\tanh ( \\alpha x _ { j - 1 } ) } y _ { j } . \\end{align*}"}
+{"id": "5997.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } ( [ h ( - 1 ) , i ] ) B ( [ \\epsilon , x ] ) & = i e ^ { \\tfrac { \\pi i } { 4 } [ 1 + 1 ] } ( - 1 , \\epsilon ) _ \\R B ( [ \\epsilon , - x ] ) \\\\ & = ( - 1 , - \\epsilon ) _ \\R B ( [ \\epsilon , - x ] ) \\\\ & = ( - 1 , \\epsilon ) _ \\R B ( [ \\epsilon , x ] ) . \\end{align*}"}
+{"id": "8394.png", "formula": "\\begin{align*} a _ { k , n } = \\prod _ { l = 1 } ^ { k - 1 } ( \\frac { l ( n - l ) } { 2 } ) ^ { \\frac { 1 } { 2 } } \\cdot \\prod _ { l = k } ^ { n - 1 } ( \\frac { l ( n - l ) } { 2 } ) ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "3981.png", "formula": "\\begin{align*} z _ \\ell ' = ( \\lambda _ \\ell Q _ { \\ell , Y Y } ) ^ { - 1 } \\left [ \\frac { a _ \\ell } { 2 } - \\lambda _ \\ell Q _ { \\ell , Y X } ( x ' - x _ \\ell ) \\right ] . \\end{align*}"}
+{"id": "8860.png", "formula": "\\begin{align*} \\mathbf r _ m = & \\sum _ { n \\in \\mathcal N } \\alpha _ n g _ n ^ { \\frac { 1 } { 2 } } \\mathbf H _ { n , m } \\mathbf s _ n + \\mathbf n _ m \\\\ = & \\sum _ { n \\in \\mathcal N } \\alpha _ n g _ n ^ { \\frac { 1 } { 2 } } \\mathbf H _ { n , m } \\mathbf { F } ^ { H } \\tilde { \\mathbf s } _ n + \\mathbf n _ m , m \\in \\mathcal { M } , \\end{align*}"}
+{"id": "8168.png", "formula": "\\begin{align*} S _ t = \\{ N _ t \\geq 1 \\} , S = \\bigcap _ { t \\geq 0 } S _ t \\end{align*}"}
+{"id": "6958.png", "formula": "\\begin{align*} f = f _ 0 + f _ 1 q + \\ldots + f _ n q ^ n . \\end{align*}"}
+{"id": "9053.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 , y _ 1 , y _ 2 , w ) = \\left ( r \\cos \\vartheta , r \\sin \\vartheta , p _ r \\cos \\vartheta - \\frac { \\sin \\vartheta } { r } p _ \\vartheta , p _ r \\sin \\vartheta + \\frac { \\cos \\vartheta } { r } p _ \\vartheta , w \\right ) . \\end{align*}"}
+{"id": "7176.png", "formula": "\\begin{gather*} \\mathcal { H } ^ 1 ( J \\cap \\partial B _ { r _ x } ( x ) ) = 0 , \\\\ \\eta r _ x \\leq \\mathcal { H } ^ 1 \\left ( J \\cap B _ { r _ x } ( x ) \\right ) \\leq \\mathcal { H } ^ 1 \\left ( J \\cap B _ { 2 r _ x ( x ) } \\right ) < 2 \\eta r _ x . \\end{gather*}"}
+{"id": "2023.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ n u ^ { p \\bar { p } } \\frac { \\beta _ { p \\bar { p } } } { \\beta } \\leq \\sum _ { p = 1 } ^ n u ^ { p \\bar { p } } \\left ( \\vert A u _ p + B \\bar z _ { p } \\vert ^ 2 - u u _ { p \\bar p } - \\vert u _ p \\vert ^ 2 - B \\right ) \\end{align*}"}
+{"id": "9216.png", "formula": "\\begin{align*} 1 + \\eta _ k = \\prod _ { j = 1 } ^ k ( 1 + \\xi _ j ) | \\xi _ j | \\le 2 ^ { - d } . \\end{align*}"}
+{"id": "8939.png", "formula": "\\begin{align*} | \\sigma ( r , x _ 3 , t ) | = r | v _ \\theta ( r , x _ 3 , t ) | \\leq C _ 1 e ^ { - c \\ln ^ \\frac 1 4 ( 1 / r ) } \\end{align*}"}
+{"id": "7958.png", "formula": "\\begin{align*} H _ 0 ( x ) = \\sup _ { \\xi \\neq 0 } \\frac { \\xi \\cdot x } { H ( \\xi ) } \\end{align*}"}
+{"id": "958.png", "formula": "\\begin{align*} Q _ { q } ( y ) : = \\begin{cases} \\frac { | D f ( x ) | ^ { q } } { | J ( x , f ) | } , \\ , \\ , & x = f ^ { - 1 } ( y ) \\in \\Omega \\setminus ( S \\cup Z ) , \\\\ \\ , \\ , 0 , \\ , \\ , & x = f ^ { - 1 } ( y ) \\in S \\cup Z . \\end{cases} \\end{align*}"}
+{"id": "1300.png", "formula": "\\begin{align*} Y _ i ( t ) = X _ i ( T _ i + t ) \\end{align*}"}
+{"id": "6711.png", "formula": "\\begin{align*} p _ 2 ( x _ 0 , \\xi _ 0 ) = \\{ p _ 2 , \\Psi \\} ( x _ 0 , \\xi _ 0 ) = 0 \\{ p _ 2 , \\{ p _ 2 , \\Psi \\} \\} ( x _ 0 , \\xi _ 0 ) < 0 . \\end{align*}"}
+{"id": "2061.png", "formula": "\\begin{align*} \\norm { u } _ { L ^ 2 } ^ 2 = \\sum _ { N \\in 2 ^ { \\mathbb Z ^ d } } \\norm { u _ N } _ { L ^ 2 } ^ 2 \\end{align*}"}
+{"id": "4846.png", "formula": "\\begin{align*} d _ i = \\begin{cases} b _ i , & \\mbox { i f } i \\in \\bigcup _ { p = 1 } ^ { r + 1 } L ( p ) \\\\ l , & \\mbox { i f } i \\in \\{ s ( 1 ) , s ( 2 ) , \\cdots , s ( r ) \\} . \\end{cases} \\end{align*}"}
+{"id": "7479.png", "formula": "\\begin{align*} \\Phi ( \\pi ) = ( 4 , 3 , 2 ) ( 5 ) ( 8 , 1 ) ( 9 , 6 , 7 ) ( 1 0 ) ( 1 1 ) \\end{align*}"}
+{"id": "6986.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ s a _ i \\textbf { Q } ^ { \\lambda _ i } \\end{align*}"}
+{"id": "3425.png", "formula": "\\begin{align*} - \\Delta w _ t = \\lambda w _ t + f ( u _ t ) - f ( u ) . \\end{align*}"}
+{"id": "8974.png", "formula": "\\begin{align*} \\Omega _ + = \\big \\{ x \\in M \\ : \\ w ( x ) > 0 \\big \\} , \\end{align*}"}
+{"id": "9121.png", "formula": "\\begin{align*} \\begin{array} { c c l l } x ^ { i } & = & F _ { x } ^ { i } ( y , \\dots , y _ { [ R - 1 ] } ) \\ , , & i = 1 , \\dots , n \\\\ u ^ { j } & = & F _ { u } ^ { j } ( y , \\dots , y _ { [ R ] } ) \\ , , & j = 1 , \\dots , m \\ , . \\end{array} \\end{align*}"}
+{"id": "3232.png", "formula": "\\begin{align*} d ( \\beta ) & = \\left ( a _ 1 T _ 1 + a _ 2 T _ 2 \\right ) ^ 2 + \\left ( b _ 1 T _ 1 + b _ 2 T _ 2 \\right ) ^ 2 + \\left ( a _ 1 T _ 1 + a _ 2 T _ 2 \\right ) \\left ( b _ 1 T _ 1 + b _ 2 T _ 2 \\right ) + ( a _ 3 ^ 2 + a _ 3 b _ 3 + b _ 3 ^ 2 ) T _ 3 ^ 2 , \\\\ d ( \\gamma ) & = \\left ( a _ 1 T _ 1 + a _ 2 T _ 2 \\right ) \\left ( b _ 1 T _ 1 + b _ 2 T _ 2 \\right ) ^ 2 + \\left ( a _ 1 T _ 1 + a _ 2 T _ 2 \\right ) ^ 2 \\left ( b _ 1 T _ 1 + b _ 2 T _ 2 \\right ) + ( a _ 3 ^ 2 b _ 3 + a _ 3 b _ 3 ^ 2 ) T _ 3 ^ 3 . \\end{align*}"}
+{"id": "2397.png", "formula": "\\begin{align*} \\vec { u } ( A _ { 0 } , A _ { 2 } ) = ( \\cos \\alpha _ { 1 0 2 } , \\sin \\alpha _ { 1 0 2 } , 0 ) , \\end{align*}"}
+{"id": "8768.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x _ i } \\phi _ { i , k } = \\begin{cases} 0 & \\mbox { i f } d _ { i , k } - x _ i \\leq 0 \\\\ \\frac { x _ i - d _ i ( \\omega _ k ) } { \\bar { y } } & \\mbox { i f } 0 \\leq d _ { i , k } - x _ i \\leq \\bar { y } \\\\ - 1 & \\mbox { i f } d _ { i , k } - x _ i \\geq \\bar { y } \\end{cases} \\end{align*}"}
+{"id": "6744.png", "formula": "\\begin{align*} \\frac { 1 } { \\ell _ i } \\sum _ { n \\leq \\ell _ i } \\mathbf { 1 } _ { \\underline { A } \\times A } ( \\underline { \\varphi } ( R ^ n ( \\Delta ( 0 ) ) ) , \\varphi ( R ^ n ( \\Delta ( 0 ) ) ) ) = \\frac { 1 } { \\ell _ i } \\sum _ { n \\leq \\ell _ i } \\mathbf { 1 } _ { \\underline { C } } ( R ^ n ( \\Delta ( 0 ) ) ) \\mathbf { 1 } _ { C } ( R ^ n ( \\Delta ( 0 ) ) ) . \\end{align*}"}
+{"id": "5373.png", "formula": "\\begin{align*} \\tfrac { d } { d t } E x ( t ) = ( J - R ) Q x ( t ) , J = - J ^ H , R = R ^ H \\geq 0 , Q ^ H E = E ^ H Q \\end{align*}"}
+{"id": "3407.png", "formula": "\\begin{align*} - \\varepsilon ^ 2 \\Delta w + V ( x ) w = w \\log w ^ 2 , w \\in H ^ 1 ( \\R ^ N ) . \\end{align*}"}
+{"id": "4612.png", "formula": "\\begin{align*} { \\rm P } _ { j , j } = & 1 - \\sum ^ { K } _ { i = 1 } \\bar { \\rm P } _ { j , j + i } \\\\ = & 1 - \\sum ^ { K } _ { i = 1 } \\sum ^ { M - j } _ { m = i + 1 } { M - j \\choose m } \\mathbb { P } _ { \\rm T X } ^ m \\left ( 1 - \\mathbb { P } _ { \\rm T X } \\right ) ^ { M - j - m } \\\\ & \\times { m \\choose i } \\mathbb { P } ( E _ { i , 1 } ) \\mathbb { P } ( E _ { i , 2 } ) . \\end{align*}"}
+{"id": "2171.png", "formula": "\\begin{align*} 2 \\lambda \\left ( \\delta \\right ) = \\ln \\left ( \\delta ^ { - 2 \\rho / d } \\right ) . \\end{align*}"}
+{"id": "4349.png", "formula": "\\begin{align*} h ( x , x ^ { \\ast } ) + h ( y , y ^ { \\ast } ) = \\left \\langle x , x ^ { \\ast } \\right \\rangle + \\left \\langle y , y ^ { \\ast } \\right \\rangle \\Rightarrow h ( x , x ^ { \\ast } ) + h ( y , y ^ { \\ast } ) = \\left \\langle x , y ^ { \\ast } \\right \\rangle + \\left \\langle y , x ^ { \\ast } \\right \\rangle . \\end{align*}"}
+{"id": "1700.png", "formula": "\\begin{align*} L ( \\tau ) = \\sum _ { i = 1 } ^ { R } V _ { i } \\end{align*}"}
+{"id": "7975.png", "formula": "\\begin{align*} ( \\partial _ j \\partial _ i V ^ i ) \\ , V ^ j = ( \\partial _ j \\partial _ i V ^ j ) \\ , V ^ i . \\end{align*}"}
+{"id": "7062.png", "formula": "\\begin{align*} h _ i d \\left ( \\sum \\limits _ { j = 1 } ^ { s _ \\ell } b _ { i _ { } i j } { \\textbf { X } } ^ { \\lambda _ j } \\right ) \\end{align*}"}
+{"id": "8270.png", "formula": "\\begin{align*} D _ { l , q } ( n ) = \\sum _ { m = 0 } ^ { l - 2 q } B _ { q , m } ^ { ( l ) } ( n ) = \\frac { n + ( l - 1 ) ( 2 k - l + 1 ) } { l } D _ { l - 1 , q } ( n ) + \\frac { l - k - 2 } { l } D _ { l - 2 , q - 1 } ( n - 1 ) . \\end{align*}"}
+{"id": "6769.png", "formula": "\\begin{align*} f ^ { p ^ k - \\gamma } = [ e _ { 0 } ] ^ { p ^ k - \\gamma } + [ e _ { 1 } ] ^ { 1 + p ^ k - \\gamma } + \\dots + [ e _ { \\gamma } ] ^ { \\gamma + p ^ k - \\gamma } + \\cdots + [ e _ { \\eta - 1 } ] ^ { \\eta - 1 + p ^ k - \\gamma } . \\end{align*}"}
+{"id": "1694.png", "formula": "\\begin{align*} Z _ { k + 1 } - Z _ k = ( X _ k + X _ k ' ) { \\bf 1 } _ { S _ k } + E _ k { \\bf 1 } _ { S _ k ^ c } \\ , , \\end{align*}"}
+{"id": "8436.png", "formula": "\\begin{align*} f _ { k } ( z , \\lambda ) : = \\frac { 1 } { 2 \\pi i } \\intop _ { \\sigma - i \\infty } ^ { \\sigma + i \\infty } \\Gamma ( s ) \\ , ( s , \\lambda ) _ { k } \\ , z ^ { - s } d s = \\left ( \\frac { \\lambda - z } { \\lambda + z } \\right ) ^ { k } \\ , e ^ { - z } , \\ , \\ , \\ , \\ , \\sigma > 0 , \\ , \\ , ( z ) > 0 . \\end{align*}"}
+{"id": "6265.png", "formula": "\\begin{align*} \\lambda : = \\inf \\left \\{ Q _ { \\Sigma } [ \\phi ] \\ , : \\ , \\int _ { \\Sigma } \\phi ^ 2 = 1 \\mbox { a n d } \\nabla _ n \\phi = h ( \\nu , \\nu ) \\phi \\mbox { o n } \\partial \\Sigma \\right \\} . \\end{align*}"}
+{"id": "113.png", "formula": "\\begin{align*} \\chi e ^ { - ( t _ 0 - t ) X } ( Q _ \\infty + q _ 1 ) e ^ { - i t h ^ { - 1 } \\tilde { P } _ h } \\tilde { R } _ h ( z ) \\chi = 0 . \\end{align*}"}
+{"id": "3593.png", "formula": "\\begin{align*} \\log _ 2 ( 1 0 ^ { x + 1 } - 1 ) \\ < \\ \\log _ 2 ( 1 0 ^ { x + 1 } ) \\ = \\ ( x + 1 ) \\log _ 2 1 0 . \\end{align*}"}
+{"id": "8212.png", "formula": "\\begin{align*} P ^ \\omega ( C _ t ^ c \\mid A _ { m ( t ) } ) \\leq ( 1 - p ( t ) ) ^ { I ( t ) } = e ^ { - p ( t ) I ( t ) } , \\end{align*}"}
+{"id": "821.png", "formula": "\\begin{align*} B ( x _ 1 t , \\ldots , x _ m t ) = 1 + t B _ 1 ( x _ 1 , \\ldots , x _ m ) + \\cdots + t ^ d B _ d ( x _ 1 , \\ldots , x _ m ) . \\end{align*}"}
+{"id": "8957.png", "formula": "\\begin{align*} \\phi = \\Tilde { \\phi } _ { e } + D _ { 2 } \\Delta x ^ { 2 } + D _ { 3 } \\Delta x ^ { 3 } . \\end{align*}"}
+{"id": "293.png", "formula": "\\begin{align*} A ( n , x ) = \\frac { x - ( 1 + n ) x ^ { n + 1 } + n x ^ { n + 2 } } { ( 1 - x ) ^ 2 } \\end{align*}"}
+{"id": "8857.png", "formula": "\\begin{align*} \\# \\bigg \\{ k : \\frac { t - 1 } { 2 } < k < \\frac { t j - 1 } { 2 } \\bigg \\} = \\frac { t j - 1 } { 2 } - \\frac { t - 1 } { 2 } - \\delta _ { j > 1 } \\end{align*}"}
+{"id": "8787.png", "formula": "\\begin{align*} & b _ 1 = ( 3 + t + u ) b _ 2 - ( 1 + 2 t ) c _ 2 - u b _ 3 \\end{align*}"}
+{"id": "510.png", "formula": "\\begin{align*} \\sum _ { x _ i \\in X _ j ^ \\mathrm { n d f } } w _ { j i } \\hat u _ i = f ( y _ j ) , j = 1 , \\ldots , M , \\end{align*}"}
+{"id": "6350.png", "formula": "\\begin{align*} \\{ y = s \\} \\cap { \\rm G r a p h } ( f ) = \\{ \\left ( \\alpha ( s ) , s \\right ) ; \\left ( \\beta ( s ) , s \\right ) \\} . \\end{align*}"}
+{"id": "3065.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta _ p u & = f _ 1 ( | x | ) \\cdot g _ 1 ( v ) \\cdot | \\nabla u | ^ { \\alpha } & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta _ p v & = f _ 2 ( | x | ) \\cdot g _ 2 ( v ) \\cdot g _ 3 ( | \\nabla u | ) & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "9043.png", "formula": "\\begin{align*} \\hat { y } _ { i ; t } = \\prod _ { j \\in [ m ] } x _ { j ; t } ^ { b ^ { ( t ) } _ { i , j } } \\end{align*}"}
+{"id": "1114.png", "formula": "\\begin{align*} \\Phi _ { k , i } ( \\alpha _ 1 , \\ldots , \\alpha _ k ) = 0 , i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "2043.png", "formula": "\\begin{align*} \\{ x = a _ 0 \\} , \\ldots , \\{ x \\ , = a _ n \\} , \\{ y = b _ 0 \\} , \\ldots , \\{ y = b _ n \\} . \\end{align*}"}
+{"id": "8969.png", "formula": "\\begin{align*} \\phi _ 1 & = \\Tilde { \\phi } _ { e } + C _ { \\Tilde { q } } h ^ { \\Tilde { q } } , \\\\ \\phi _ 2 & = \\Tilde { \\phi } _ { e } + C _ { \\Tilde { q } } ( r h ) ^ { \\Tilde { q } } , \\\\ \\phi _ 3 & = \\Tilde { \\phi } _ { e } + C _ { \\Tilde { q } } ( r ^ 2 h ) ^ { \\Tilde { q } } . \\end{align*}"}
+{"id": "1484.png", "formula": "\\begin{align*} ( 2 ^ * - 1 ) \\alpha _ n \\sum _ { i = 1 } ^ k \\mu _ i ^ { \\frac { n - 2 } { 2 } } \\int _ \\Omega U _ { \\mu _ i , \\xi _ i } ^ { 2 ^ * - 2 } ( x ) H ( x , \\xi _ i ) \\psi ^ h _ { \\mu _ j , \\xi _ j } d x = o \\Big ( \\frac { \\epsilon } { | \\ln \\epsilon | ^ 2 } \\Big ) . \\end{align*}"}
+{"id": "487.png", "formula": "\\begin{align*} n \\mapsto k _ 1 , \\ ; a \\mapsto t ^ { k _ 1 - 1 } t _ 1 t _ r , \\ ; b \\mapsto c ^ { 2 n - 2 } d ^ 2 t ^ { 1 - k _ 1 } t _ r / t _ 1 = t ^ { - k _ 0 } t _ 2 t _ r , \\end{align*}"}
+{"id": "8347.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ \\infty \\alpha _ j \\tilde { u } _ j \\Big \\| _ { p ^ * , q , \\mu } \\geq \\frac { \\lambda } { 1 + \\varepsilon _ 1 } \\Big ( \\sum _ { j = 1 } ^ \\infty | \\alpha _ j | ^ q \\Big ) ^ { \\frac 1 { q } } . \\end{align*}"}
+{"id": "4991.png", "formula": "\\begin{align*} \\overline { S } ( k , n ) = \\sum _ { j = k + 1 } ^ n \\binom { n } { j } = \\frac { 1 } { 2 } \\sum _ { j = k } ^ { n - 1 } { 2 ^ { n - j } } \\binom { j } { k } . \\end{align*}"}
+{"id": "2266.png", "formula": "\\begin{align*} w _ b = \\sum _ { n } c _ n a _ n , \\end{align*}"}
+{"id": "3139.png", "formula": "\\begin{align*} g ( m , u , x + 1 ) - g ( m , u , x ) = \\frac { \\partial } { \\partial u } g ( m - 1 , u , x ) . \\end{align*}"}
+{"id": "361.png", "formula": "\\begin{align*} \\imath ^ { \\sigma , c } _ { \\tau , c } \\circ \\imath ^ { \\tau , c } _ { \\mu , c } = ( \\imath ^ { \\sigma , c } _ { \\partial \\sigma , c } \\circ \\imath ^ { \\partial \\sigma , c } _ { \\tau , c } ) \\circ \\imath ^ { \\tau , c } _ { \\mu , c } = \\imath ^ { \\sigma , c } _ { \\partial \\sigma , c } \\circ ( \\imath ^ { \\partial \\sigma , c } _ { \\tau , c } \\circ \\imath ^ { \\tau , c } _ { \\mu , c } ) = \\imath ^ { \\sigma , c } _ { \\partial \\sigma , c } \\circ \\imath ^ { \\partial \\sigma , c } _ { \\mu , c } = \\imath ^ { \\sigma , c } _ { \\mu , c } . \\end{align*}"}
+{"id": "8716.png", "formula": "\\begin{align*} E \\{ ( U _ { 1 } ^ { \\top } \\Gamma _ { 1 } ^ { \\top } \\Gamma _ { 1 } U _ { 2 } ) ^ { 4 } \\} = O ( ^ { 2 } ( \\Sigma _ { 1 } ^ { 2 } ) ) = O ( p ^ { 2 } ) . \\end{align*}"}
+{"id": "1563.png", "formula": "\\begin{align*} G _ { \\d , \\nu , \\l } ( H ) = 0 . \\end{align*}"}
+{"id": "3606.png", "formula": "\\begin{align*} 1 0 ^ m - 6 \\ \\le \\ v ( f ) \\ = \\ v ( 2 ^ \\delta ) + v ( 3 ^ \\gamma ) \\ \\le \\ 2 + \\delta + 3 + \\gamma . \\end{align*}"}
+{"id": "738.png", "formula": "\\begin{align*} u _ { n + 1 } & = 2 \\sum _ { k = 0 } ^ { n + 1 } \\cal R ^ k \\tilde { \\cal A } _ { - 1 } { \\rm R H S } = \\cal R \\Big ( 2 \\sum _ { k = 0 } ^ { n } \\cal R ^ k \\tilde { \\cal A } _ { - 1 } { \\rm R H S } \\Big ) + 2 \\tilde { \\cal A } _ { - 1 } { \\rm R H S } \\\\ & = \\cal R u _ { n } + u _ { 0 } , u _ { 0 } = 2 \\tilde { \\cal A } _ { - 1 } { \\rm R H S } \\ , . \\end{align*}"}
+{"id": "3212.png", "formula": "\\begin{align*} \\nu _ { y ' } ^ n ( x ) & = \\langle y ' \\sigma _ n ( x ) \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } \\\\ & \\leq \\norm { y ' } \\langle \\sigma _ n ( x ) \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } \\\\ & = \\norm { y ' } \\rho ( \\sigma _ n ( x ) ) \\\\ & = \\norm { y ' } \\rho ( x ) \\\\ & = \\norm { y ' } \\langle x \\Omega _ \\rho , \\Omega _ \\rho \\rangle _ { \\rho } . \\end{align*}"}
+{"id": "6919.png", "formula": "\\begin{align*} R ( 0 ) ^ 2 & = \\sum _ { m , n \\in \\mathcal { M } ' } r ( m ) r ( n ) \\le | \\mathcal { M } ' | \\sum _ { m \\in \\mathcal { M } ' } r ( m ) ^ 2 \\end{align*}"}
+{"id": "4701.png", "formula": "\\begin{align*} { \\rm e } ^ { { \\rm i } \\chi y } \\bigl ( \\mathcal { A } _ { 0 , \\chi } ^ { \\rm s o f t } - z I \\bigr ) ^ { - 1 } { \\rm e } ^ { - { \\rm i } \\chi y } = \\bigl ( \\mathcal { A } _ { 0 , 0 } ^ { \\rm s o f t } - z I \\bigr ) ^ { - 1 } , \\end{align*}"}
+{"id": "3108.png", "formula": "\\begin{align*} S _ B ( n , k , q ) & = q ^ { 2 k - 1 } ( 1 + q ) \\sum \\limits _ { \\pi \\in B _ { \\subseteq } ( [ n - 1 ] , k - 1 ) } q ^ { m ( \\pi ) } + ( 1 + q ) \\sum _ { i = 1 } ^ k q ^ { 2 i - 1 } \\sum \\limits _ { \\tau ' \\in B _ { \\subseteq } ( [ n - 1 ] , k ) } q ^ { m ( \\tau ' ) } \\\\ & + \\sum \\limits _ { \\pi \\in B _ { \\subseteq } ( [ n - 1 ] , k ) } q ^ { m ( \\pi ) } \\\\ & = [ 2 k + 1 ] _ q S _ B ( n - 1 , k , q ) + q ^ { 2 k - 1 } ( 1 + q ) S _ B ( n - 1 , k - 1 , q ) . \\end{align*}"}
+{"id": "7689.png", "formula": "\\begin{align*} H ( x _ 1 ^ { k - l } h ( x ' ) ) = h ( x ' ) H x _ 1 ^ { k - l } = \\frac { ( k - l ) ! } { 2 ^ { k - l } } \\| x \\| ^ { k - l } C _ { k - l } ^ { \\frac { n - 2 } { 2 } + l } \\left ( \\frac { x _ 1 } { \\| x \\| } \\right ) h ( x ' ) \\end{align*}"}
+{"id": "288.png", "formula": "\\begin{align*} = \\sqrt [ 3 ] { \\left ( \\frac { 1 } { 1 - z } \\right ) } \\ ; \\exp \\left \\{ \\frac { 1 } { 1 2 } L i _ 2 ( z ) + \\frac { z ( 7 - 5 z ) ) } { 1 2 ( 1 - z ) ^ 2 } \\right \\} . \\end{align*}"}
+{"id": "7313.png", "formula": "\\begin{align*} a x _ i + P ( x _ 1 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ n ) = 0 , a \\neq 0 , \\end{align*}"}
+{"id": "2451.png", "formula": "\\begin{align*} ( \\psi * f ) ( x ) = \\int _ { G _ { \\sigma ( x ) } } \\psi ( x \\cdot \\gamma ^ { - 1 } ) g ( \\gamma ) \\ d \\lambda _ { \\sigma ( x ) } . \\end{align*}"}
+{"id": "7848.png", "formula": "\\begin{align*} \\xi _ \\phi ( X _ i ) : = \\frac { k _ i } { | H | } \\sum _ { h \\in H } \\phi ( x _ \\ell h ) , \\end{align*}"}
+{"id": "7979.png", "formula": "\\begin{align*} \\int _ \\Omega \\big ( \\mathrm { d i v } \\ , V \\big ) ^ 2 \\ , d x = \\int _ \\Omega \\mathrm { t r } \\big ( ( \\nabla \\ , V ) ^ 2 \\big ) d x + \\int _ { \\partial \\Omega } \\Big ( ( \\mathrm { d i v } _ T V ) \\ , V \\cdot \\nu - \\nabla _ T V \\ , V _ T \\cdot \\nu \\Big ) d \\mathcal { H } ^ { n - 1 } . \\end{align*}"}
+{"id": "350.png", "formula": "\\begin{align*} u & = \\left ( ( s _ m , f _ m ) , \\dots , ( s _ 1 , f _ 1 ) \\right ) \\\\ v & = \\left ( ( s ' _ n , f ' _ n ) , \\dots , ( s ' _ 1 , f ' _ 1 ) \\right ) , \\end{align*}"}
+{"id": "2853.png", "formula": "\\begin{align*} \\rho _ { a d _ { \\alpha } } = ( \\alpha \\otimes \\Delta ) - \\xi \\circ ( \\Delta \\otimes \\alpha ) : L \\otimes L \\rightarrow L \\otimes L \\otimes L \\otimes L . \\end{align*}"}
+{"id": "5318.png", "formula": "\\begin{align*} Y ( x , y ) = 1 \\Rightarrow y \\in \\tilde { \\mathcal { A } } \\Rightarrow x \\in \\tilde { \\mathcal { A } } ^ \\uparrow = \\mathcal { B } \\Rightarrow X ( x , y ) = 1 . \\end{align*}"}
+{"id": "1831.png", "formula": "\\begin{align*} M _ p ( k _ 1 k _ 2 & k _ 3 k _ 4 , k _ 3 k _ 4 k _ 1 k _ 2 ) \\\\ & = 2 \\cos \\big [ ( t - s ) \\Delta E ( \\vec k ) \\big ] | \\Phi ( \\vec k ) | ^ 2 \\big ( \\delta ( p - k _ 1 ) + \\delta ( p - k _ 2 ) - \\delta ( p - k _ 3 ) - \\delta ( p - k _ 4 ) \\big ) \\ . \\end{align*}"}
+{"id": "3190.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\sup _ { x \\in M _ + , x \\neq 0 } \\abs { ( B _ l ( \\nu ) - \\bar { \\nu } ) ( x ) } / \\rho ( x ) = 0 . \\end{align*}"}
+{"id": "8117.png", "formula": "\\begin{align*} \\alpha ( Y _ T ) = \\begin{cases} - a & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "411.png", "formula": "\\begin{align*} W ^ + = \\begin{bmatrix} u _ n + p / u _ n \\\\ u _ { \\tau } \\end{bmatrix} , \\Lambda ^ + = \\begin{bmatrix} u _ n & 0 \\\\ 0 & u _ n \\end{bmatrix} , W ^ - = p , \\Lambda ^ - = - 1 / u _ n . \\end{align*}"}
+{"id": "4862.png", "formula": "\\begin{align*} \\begin{array} { l l } & c ^ { \\prime } x \\\\ & a _ { t } ^ { \\prime } x \\leq b _ { t } , \\ , \\ , \\ , t \\in T : = \\left \\{ 1 , 2 , . . . , m \\right \\} , \\end{array} \\end{align*}"}
+{"id": "7229.png", "formula": "\\begin{align*} u : = \\begin{cases} { \\bf e } _ 1 & \\mbox { i n } \\Omega , \\\\ 0 & \\mbox { i n } B _ 1 \\setminus \\overline { \\Omega } . \\end{cases} \\end{align*}"}
+{"id": "7598.png", "formula": "\\begin{align*} g ( z ) = z \\exp \\left ( \\int _ { 0 } ^ { z } \\frac { x ^ 2 + \\sqrt { 1 + x ^ 4 } - 1 } { x } d x \\right ) = z + \\frac { z ^ 3 } { 2 } + \\frac { z ^ 5 } { 4 } + \\cdots , \\end{align*}"}
+{"id": "6974.png", "formula": "\\begin{align*} \\min \\limits _ { \\ell \\in I _ 0 ( g , i ) } \\alpha _ \\ell = \\alpha _ i , \\end{align*}"}
+{"id": "3086.png", "formula": "\\begin{align*} u ( r ) = C _ \\lambda r ^ \\lambda , v ( r ) = C _ \\mu r ^ \\mu . \\end{align*}"}
+{"id": "7832.png", "formula": "\\begin{align*} N _ { 0 } : = | I | \\le C ^ { - 2 } n \\log ^ { 5 } n \\le n ^ { 1 + \\varepsilon / 1 0 } , \\end{align*}"}
+{"id": "6331.png", "formula": "\\begin{align*} f _ \\psi ( s ) = C ' _ \\circ ( \\psi ) \\cdot s ^ 2 + o ( s ^ 2 ) , s \\to 0 . \\end{align*}"}
+{"id": "8262.png", "formula": "\\begin{align*} x ^ { - \\frac { k ^ 2 } { 2 } - l } f _ l ( x ) = ( - 1 ) ^ { \\frac { k ( k - 1 ) } { 2 } } \\frac { G ^ 2 ( k + 1 ) } { G ( 2 k + 1 ) } \\sum _ { i = 0 } ^ \\infty \\left ( \\sum _ { q = 0 } ^ { l } \\sum _ { m = 0 } ^ { l - q } \\Big ( \\prod _ { s = 0 } ^ { m - 1 } ( i + 2 l - q + \\frac { k ^ 2 } { 2 } - s ) \\Big ) c _ { q , m } ^ { ( l ) } a _ { i + 2 l - q } \\right ) x ^ i . \\end{align*}"}
+{"id": "1168.png", "formula": "\\begin{align*} T \\left ( \\frac { 1 } { f ^ { 2 } - 1 } \\right ) = - \\frac { 1 } { ( f ^ { 2 } - 1 ) ^ { 2 } } T ( f ^ { 2 } - 1 ) + \\frac { 2 } { ( f ^ { 2 } - 1 ) ^ { 3 } } A ( f ^ { 2 } - 1 ) ^ { 2 } \\end{align*}"}
+{"id": "261.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { - y ( y + 1 ) ( y ^ n - 1 ) + 2 n ( y - 1 ) y ^ { n + 1 } - n ^ 2 ( y - 1 ) ^ 2 y ^ { n + 1 } } { ( 1 - y ) ^ 3 } \\right ) \\frac { z ^ n } { n ^ 3 } \\right \\} \\end{align*}"}
+{"id": "5667.png", "formula": "\\begin{align*} e ^ { ( \\gamma _ p + \\gamma _ q - 2 ) t ( u , v ) } = \\frac { | \\nabla u | ^ 2 _ 2 + | \\nabla v | ^ 2 _ 2 } { ( \\gamma _ p + \\gamma _ q ) \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | u | ^ { p } ) | v | ^ { q } } . \\end{align*}"}
+{"id": "9068.png", "formula": "\\begin{align*} \\abs { \\frac { 1 } { \\sqrt { n } } \\frac { \\partial \\Re f _ r } { \\partial x } } & \\le \\abs { \\frac { 1 } { \\sqrt { n } } \\frac { \\partial f _ r } { \\partial x } } = \\abs { \\frac { \\partial ( f - f _ 0 ) } { \\partial x } } = \\abs { \\frac { \\partial ( \\log h - \\log h _ 0 ) } { \\partial x } } \\\\ & \\le \\abs { \\frac { 1 } { h _ 0 } \\cdot \\frac { \\partial h _ 0 } { \\partial x } - \\frac { 1 } { h } \\cdot \\frac { \\partial h } { \\partial x } } \\le \\frac { C } { \\sqrt { n } } , \\end{align*}"}
+{"id": "3629.png", "formula": "\\begin{align*} ( b _ m \\cdots 3 ) _ { 1 0 } \\ = \\ ( 2 \\cdots 7 ) _ { 1 0 } - 4 , \\end{align*}"}
+{"id": "6805.png", "formula": "\\begin{align*} ( k ^ + ) + ( 2 n - k ^ - - 1 ) - ( 2 n - 1 ) + ( 1 ) = k ^ + - k ^ - + 1 . \\end{align*}"}
+{"id": "8337.png", "formula": "\\begin{align*} ( \\nabla u _ \\mu ^ \\bigstar ) _ \\mu ^ * ( t ) = \\phi ^ * ( t ) . \\end{align*}"}
+{"id": "8700.png", "formula": "\\begin{align*} ^ 2 ( P _ X , P _ Y ) = \\omega ( \\max \\{ p ^ { - 1 / 2 } N ^ { - 1 } , p ^ { - l / 2 } N ^ { - 1 / 2 } \\} ) . \\end{align*}"}
+{"id": "6708.png", "formula": "\\begin{align*} c _ { \\Phi } ( \\xi , \\tau ) + C _ 1 \\lambda \\frac { \\left | p _ { \\Phi } ( x _ 0 , \\xi , \\tau ) \\right | ^ 2 } { | \\xi | ^ 2 + \\tau ^ 2 } & = \\lambda \\left ( c _ { \\Psi } ( \\xi , \\lambda \\tau ) + 2 \\lambda \\left | \\{ p _ { \\Psi } , \\Psi \\} ( x _ 0 , \\xi , \\lambda \\tau ) \\right | ^ 2 + C _ 1 \\frac { \\left | p _ { \\Psi } ( x _ 0 , \\xi , \\lambda \\tau ) \\right | ^ 2 } { | \\xi | ^ 2 + \\tau ^ 2 } \\right ) . \\end{align*}"}
+{"id": "2108.png", "formula": "\\begin{align*} O _ { B } ^ { H } ( M ) & = \\min \\left \\{ \\sum _ { i = 1 } ^ k h ^ i x _ i \\mid \\sum _ { i = 1 } ^ k ( h ^ i - h + 1 ) x _ i = M , \\ x _ i \\in \\mathbb { N } , 1 \\leq i \\leq k \\right \\} \\\\ & = \\min \\left \\{ M + ( h - 1 ) \\cdot \\sum _ { i = 1 } ^ k x _ i \\mid \\sum _ { i = 1 } ^ k ( h ^ i - h + 1 ) x _ i = M , \\ x _ i \\in \\mathbb { N } , 1 \\leq i \\leq k \\right \\} . \\end{align*}"}
+{"id": "7246.png", "formula": "\\begin{align*} \\sup _ { k \\leq n } | S _ k - T _ k | = o ( n ^ { 1 / p } ( \\log n ) ^ { \\eta } ) \\end{align*}"}
+{"id": "1672.png", "formula": "\\begin{align*} H = \\cap _ { \\varrho } \\ker \\sigma \\end{align*}"}
+{"id": "1912.png", "formula": "\\begin{align*} \\sum \\limits _ { k = n } ^ { m } \\left ( \\mathcal { D } _ { \\eta ^ { k } } ( \\eta ^ { k + 1 } , \\eta ^ { k } ; \\lambda ^ { k } ) + \\beta \\| r ^ { k + 1 } \\| ^ 2 \\right ) = \\mathcal { D } _ { \\eta ^ { n } } ( \\hat { \\eta } , \\eta ^ { n } ; \\lambda ^ { n } ) - \\mathcal { D } _ { \\eta ^ { m } } ( \\hat { \\eta } , \\eta ^ { m } ; \\lambda ^ { m } ) . \\end{align*}"}
+{"id": "3017.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\ell } s _ i ^ p ( A ) = \\sum _ { i = 1 } ^ { k } s _ i ^ p ( A ) \\hbox { a n d } \\sum _ { i = 1 } ^ { k - \\ell } s _ i ^ p ( B ) = \\sum _ { i = 1 } ^ { k } s _ i ^ p ( B ) . \\end{align*}"}
+{"id": "8899.png", "formula": "\\begin{align*} y ( \\{ X _ { n i } \\} ) = \\prod _ { n = 1 } ^ N [ X _ { n 1 } , \\dots , X _ { n j } ] _ c . \\end{align*}"}
+{"id": "7951.png", "formula": "\\begin{align*} \\Psi _ \\Omega ( r ) = \\begin{cases} \\sup \\limits _ { x \\in \\partial \\Omega } \\| \\mathcal B \\| _ { L ^ { n - 1 , \\infty } ( \\partial \\Omega \\cap B _ r ( x ) ) } \\quad n \\geq 3 , \\\\ \\\\ \\sup \\limits _ { x \\in \\partial \\Omega } \\| \\mathcal B \\| _ { L ^ { 1 , \\infty } \\log L ( \\partial \\Omega \\cap B _ r ( x ) ) } \\quad n = 2 \\end{cases} \\end{align*}"}
+{"id": "7385.png", "formula": "\\begin{align*} \\sum _ { n \\in \\mathbb { Z } } \\widehat { \\Delta } ( a n ) = \\frac { 1 } { a } \\sum _ { m \\in \\mathbb { Z } } \\Delta \\left ( \\frac { m } { a } \\right ) . \\end{align*}"}
+{"id": "5769.png", "formula": "\\begin{align*} f _ { j } & = g _ { j } + b _ { j } = g _ { j } + \\sum _ { k _ { j } } b _ { j , k _ { j } } \\end{align*}"}
+{"id": "3240.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\omega ( Q ) \\psi _ { { \\mathfrak R } ^ j } ( x , x _ { Q } ) q _ { { \\mathfrak R } ^ j } h ( x _ { Q } ) , \\end{align*}"}
+{"id": "3685.png", "formula": "\\begin{align*} \\norm { \\cdot } _ { ( H ^ \\alpha ( \\Omega ) ) ^ * } : = \\sum _ { i = 1 } ^ p \\norm { F } _ { ( H ^ { \\alpha _ i } ( \\Omega ) ) ^ * } . \\end{align*}"}
+{"id": "437.png", "formula": "\\begin{align*} U _ t + ( A U ) _ x + A ^ T U _ x = 0 A = \\begin{bmatrix} a _ { 1 1 } & a _ { 1 2 } \\\\ [ 0 . 0 5 c m ] a _ { 2 1 } & a _ { 2 2 } \\\\ [ 0 . 0 5 c m ] \\end{bmatrix} = \\begin{bmatrix} \\alpha \\frac { U _ 2 } { \\sqrt { U _ 1 } } & ( 1 - 3 \\alpha ) \\sqrt { U _ 1 } \\\\ [ 0 . 0 5 c m ] 2 \\alpha \\sqrt { U _ 1 } & \\frac { 1 } { 2 } \\frac { U _ 2 } { \\sqrt { U _ 1 } } \\\\ [ 0 . 0 5 c m ] \\end{bmatrix} \\end{align*}"}
+{"id": "6701.png", "formula": "\\begin{align*} \\nabla ( e ^ { - \\tau \\Phi } ) & = - \\tau \\nabla \\Phi e ^ { - \\tau \\Phi } \\\\ \\Delta ( e ^ { - \\tau \\Phi } ) & = \\div ( \\nabla ( e ^ { - \\tau \\Phi } ) ) = - \\tau \\div ( \\nabla \\Phi e ^ { - \\tau \\Phi } ) = - \\tau ( \\Delta \\Phi ) e ^ { - \\tau \\Phi } - \\tau \\nabla \\Phi \\cdot \\nabla ( e ^ { - \\tau \\Phi } ) \\\\ & = - \\tau ( \\Delta \\Phi ) e ^ { - \\tau \\Phi } + \\tau ^ 2 | \\nabla \\Phi | ^ 2 e ^ { - \\tau \\Phi } . \\end{align*}"}
+{"id": "2454.png", "formula": "\\begin{align*} ( \\pi ( f ) \\psi ) ( m ) = \\int _ \\R f ( - t ) \\psi ( e ^ { t X } m ) \\ d t . \\end{align*}"}
+{"id": "3191.png", "formula": "\\begin{align*} \\nu ( x ) = \\langle y _ 1 ' x \\Omega _ { \\rho } , \\Omega _ { \\rho } \\rangle _ { \\rho } x \\in M . \\end{align*}"}
+{"id": "1919.png", "formula": "\\begin{align*} S \\bar { u } = \\bar { \\zeta } . \\end{align*}"}
+{"id": "4714.png", "formula": "\\begin{align*} \\Re \\left \\langle \\mathcal { A } x , x \\right \\rangle = \\left \\langle \\Re \\mathcal { A } x , x \\right \\rangle , \\Im \\left \\langle \\mathcal { A } x , x \\right \\rangle = \\left \\langle \\Im \\mathcal { A } x , x \\right \\rangle . \\end{align*}"}
+{"id": "6173.png", "formula": "\\begin{align*} r _ n ( G ) \\left ( \\overline \\bigotimes ^ { s - 1 } _ { i = 0 } [ \\langle 1 , - g _ { 1 , i } \\rangle ] \\overline \\otimes . . . \\overline \\otimes [ \\langle 1 , - g _ { n , i } \\rangle ] \\overline \\otimes I ^ { n + 1 } ( G ) \\right ) : = \\\\ \\sum ^ { s - 1 } _ { i = 0 } l ( - 1 ) ^ { 2 ^ { n - 1 } - n } l ( g _ { 1 , i } ) \\otimes . . . \\otimes l ( g _ { n , i } ) + \\mathcal Q _ { 2 n - 1 } ( G ) \\end{align*}"}
+{"id": "6930.png", "formula": "\\begin{align*} \\mathcal { L } = \\textrm { s u p p } ( f ) \\setminus ( M \\cup L ) \\end{align*}"}
+{"id": "5556.png", "formula": "\\begin{align*} A _ { n } = \\left \\{ x \\in H _ { 0 } \\backslash F : - \\frac { 1 } { n } \\log P _ { \\mu , H _ { 0 } } ^ { n } ( o , x ) \\ge h - \\epsilon \\right \\} , \\end{align*}"}
+{"id": "5466.png", "formula": "\\begin{align*} P _ s ( \\theta ) & \\overset { \\Delta } { = } \\left ( > \\theta , \\Phi ( \\mathcal { A } ) > 0 | \\Phi \\right ) \\\\ & = \\left ( > \\theta | \\Phi , \\Phi ( \\mathcal { A } ) > 0 \\right ) \\mathbf { 1 } ( \\Phi ( \\mathcal { A } ) > 0 | \\Phi ) , \\end{align*}"}
+{"id": "7735.png", "formula": "\\begin{align*} \\tilde z ( Y ^ { t } ( \\omega ) , t ) = y ^ { * } ( \\omega , t ) , \\forall t \\in \\Q , \\forall \\omega \\in A . \\end{align*}"}
+{"id": "697.png", "formula": "\\begin{align*} | v ( y ) - v ( x ) | \\leq \\sum _ { q = 1 } ^ { q ( k ) } \\mathrm { o s c } ( v , J ^ { ( k ) } ( q ) ) + \\sum _ { l > k } \\sum _ { \\epsilon = \\pm } \\sum _ { q = 1 } ^ { q _ \\epsilon ( l ) } \\mathrm { o s c } ( v , J _ \\epsilon ^ { ( l ) } ( q ) ) \\end{align*}"}
+{"id": "4734.png", "formula": "\\begin{align*} \\begin{pmatrix} W _ { \\mathcal { I } \\mathcal { I } } & X _ { \\mathcal { I } \\mathcal { I } } \\\\ Y _ { \\mathcal { I } \\mathcal { I } } & Z _ { \\mathcal { I } \\mathcal { I } } \\end{pmatrix} . \\end{align*}"}
+{"id": "6775.png", "formula": "\\begin{align*} ( c _ 1 , c _ 2 , c _ 3 , c _ 4 ) : = & \\ ; 2 ^ { k - 5 } ( 4 8 , 3 2 , 4 0 , 4 4 ) \\\\ ( c _ 1 ' , c _ 2 ' , c _ 3 ' , c _ 4 ' ) : = & \\ ; 2 ^ { k - 5 } ( 4 8 , 4 0 , 4 4 , 4 8 ) \\\\ ( c _ 1 '' , c _ 2 '' , c _ 3 '' , c _ 4 '' ) : = & \\ ; 2 ^ { k - 5 } ( 3 2 , 3 2 , 4 8 , 6 4 ) \\\\ \\mathcal Z _ k : = & \\ ; \\big ( Z ( s ) \\big ) _ { 2 ^ k \\leq s < 2 ^ { k + 1 } } . \\end{align*}"}
+{"id": "1928.png", "formula": "\\begin{align*} f ( w , z ) = e ^ { w + z } \\quad \\mbox { a n d } g ( w , z ) = e ^ { w z } , \\end{align*}"}
+{"id": "2895.png", "formula": "\\begin{align*} f ^ { B , \\alpha } ( \\xi ) & : = \\partial _ x ^ \\alpha \\big | _ { x = 0 } \\alpha _ { - x } \\big ( f ( \\xi + B x ) \\big ) \\\\ & = \\sum _ { \\beta + \\gamma = \\alpha } \\binom \\alpha \\beta i ^ { | \\beta | } \\delta ^ \\beta \\big ( \\partial _ x ^ \\gamma \\big | _ { x = 0 } f ( \\xi + B x ) \\big ) \\\\ & = \\sum _ { \\beta + \\gamma = \\alpha } \\binom \\alpha \\beta i ^ { | \\beta | } \\delta ^ \\beta \\partial _ { B , \\xi } ^ \\gamma f ( \\xi ) . \\end{align*}"}
+{"id": "4865.png", "formula": "\\begin{align*} \\mathcal { L } _ { D } \\left ( b , d \\right ) : = \\left \\{ x \\in \\mathbb { R } ^ { n } : a _ { t } ^ { \\prime } x \\leq b _ { t } , \\ , t \\in T ; \\ ; - a _ { t } ^ { \\prime } x \\leq d _ { t } , \\ , t \\in D \\right \\} . \\end{align*}"}
+{"id": "4627.png", "formula": "\\begin{align*} \\tau \\partial _ t c = \\alpha \\Delta c - c + \\lambda d + \\beta m . \\end{align*}"}
+{"id": "8381.png", "formula": "\\begin{align*} C ^ { \\lor } ( u ^ { \\lor } , v ^ { \\lor } ) = C ( \\Psi _ C ^ { - 1 } ( u ^ { \\lor } ) , \\Psi _ C ^ { - 1 } ( v ^ { \\lor } ) ) . \\end{align*}"}
+{"id": "9286.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\| p ^ { k } - y ^ { k } \\| = \\lim \\limits _ { k \\rightarrow \\infty } \\| p ^ { k } - y ^ { k - 1 } \\| = \\lim \\limits _ { k \\rightarrow \\infty } \\| x ^ { k } - x ^ { k - 1 } \\| = 0 , \\end{align*}"}
+{"id": "3281.png", "formula": "\\begin{align*} \\frac { 1 } { t ^ s } \\rho ( t , t ^ s \\kappa ( s ) ) = A t ^ { 1 - s } + { \\rm I m } ( e ^ { i \\pi ( 1 - \\beta _ 1 ) } \\kappa ( t ) ) + B t ^ { 2 - s } + C t ^ s | \\kappa ( t ) | ^ 2 + \\end{align*}"}
+{"id": "4109.png", "formula": "\\begin{align*} b _ 3 = \\gamma _ 2 , c _ 3 - b _ 2 = \\gamma _ 1 , c _ 2 = - \\gamma _ 0 , \\alpha _ 2 = \\frac { 2 c _ 3 - b _ 2 } { b _ 3 } , \\\\ \\alpha _ 1 = \\frac { c _ 3 ^ 2 - b _ 1 - b _ 3 c _ 2 - b _ 2 c _ 3 } { b _ 3 ^ 2 } , \\alpha _ 0 = \\frac { - c _ 1 - c _ 3 c _ 2 } { b _ 3 ^ 2 } . \\end{align*}"}
+{"id": "8310.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ n b _ j ( a ' _ 1 a _ j - a _ 1 a ' _ j ) \\equiv \\lambda ( a _ 1 ' - a _ 1 ) \\pmod q \\ , , \\end{align*}"}
+{"id": "8234.png", "formula": "\\begin{align*} \\left | \\int _ s ^ t L _ N \\pi _ { r } ^ N ( \\phi _ j ) d r \\right | ^ 2 & = \\left ( \\int _ s ^ t \\left ( \\pi _ { r } ^ N ( A \\phi _ j ) + R _ 3 ( \\phi _ j , N , r ) \\right ) d r \\right ) ^ 2 \\\\ & \\leq 2 \\left ( \\int _ s ^ t \\pi _ { r } ^ N ( A \\phi _ j ) d r \\right ) ^ 2 + 2 \\left ( \\int _ s ^ t R _ 3 ( \\phi _ j , N , r , \\sigma ) d r \\right ) ^ 2 . \\end{align*}"}
+{"id": "627.png", "formula": "\\begin{align*} x _ { n + 1 } = \\frac { 1 } { \\| x _ n + \\alpha _ n u \\| } ( x _ n + \\alpha _ n u ) \\in U . \\end{align*}"}
+{"id": "2834.png", "formula": "\\begin{align*} \\hat S _ k ^ l ( t ) = \\frac { S _ k ^ l ( t ) } { S _ { l - 1 } ^ l ( t ) } , l = 1 , \\dots , p ; \\end{align*}"}
+{"id": "2409.png", "formula": "\\begin{align*} t \\log \\Big ( 1 + \\frac { \\delta _ j } { 3 j } \\Big ) = \\frac { t \\delta _ j } { 3 j } + O \\Big ( \\frac { t } { j ^ 2 } \\Big ) = \\frac { t \\delta _ j } { 3 j } + O \\Big ( \\frac { 1 } { K } \\Big ) , j \\ge \\beta K . \\end{align*}"}
+{"id": "332.png", "formula": "\\begin{align*} s _ h ( 3 , 2 ) = \\frac { 1 5 } { 2 } \\zeta ( 5 ) + \\zeta ( 2 ) \\zeta ( 3 ) , \\end{align*}"}
+{"id": "3905.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { \\ell < m } K _ { \\ell } \\right ) \\cap K _ { m } = \\bigcup _ { \\ell < m } \\left ( K _ { \\ell } \\cap K _ { m } \\right ) = K _ { 1 } \\cap K _ { m } \\in 2 ^ { K _ { 1 } } \\subset \\bigcup _ { \\ell < m } 2 ^ { K _ { \\ell } } . \\end{align*}"}
+{"id": "2930.png", "formula": "\\begin{align*} S = \\begin{bmatrix} 0 & - 1 \\\\ 1 & 0 \\end{bmatrix} ; T = \\begin{bmatrix} 1 & 1 \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "8921.png", "formula": "\\begin{align*} y \\bigl ( \\bigl \\{ X _ { n i } ^ { ( 1 ) } \\bigr \\} \\bigr ) \\cdot y \\bigl ( \\bigl \\{ X _ { n i } ^ { ( 2 ) } \\bigr \\} \\bigr ) = Z _ 1 + Z _ 2 + \\sum _ { l = 3 } ^ k B _ l ^ { ( 2 ) } \\bigl ( \\bigl \\{ X _ { n i } ^ { ( 1 ) } \\bigr \\} , \\bigl \\{ X _ { n i } ^ { ( 2 ) } \\bigr \\} \\bigr ) . \\end{align*}"}
+{"id": "6633.png", "formula": "\\begin{align*} x _ 0 y _ 0 + x _ 1 y _ 1 + x _ 2 y _ 2 = 0 \\end{align*}"}
+{"id": "4417.png", "formula": "\\begin{align*} L _ { \\omega , \\mathbf { c } } ( V ) & = L _ { \\omega , \\mathbf { c } } ( V _ n ) - L _ { \\omega , \\mathbf { c } } ( V _ n - V ) - ( L _ { \\omega , \\mathbf { c } } ( V _ n ) - L _ { \\omega , \\mathbf { c } } ( V _ n - V ) - L _ { \\omega , \\mathbf { c } } ( V ) ) \\\\ & < L _ { \\omega , \\mathbf { c } } ( V _ n ) - 6 \\mu _ { \\omega , \\mathbf { c } } - ( L _ { \\omega , \\mathbf { c } } ( V _ n ) - L _ { \\omega , \\mathbf { c } } ( V _ n - V ) - L _ { \\omega , \\mathbf { c } } ( V ) ) \\\\ & \\rightarrow 0 \\end{align*}"}
+{"id": "4685.png", "formula": "\\begin{align*} \\vect u = ( \\mathcal { A } _ 0 - z I ) ^ { - 1 } \\vect f + \\left ( I - z \\mathcal { A } _ 0 ^ { - 1 } \\right ) ^ { - 1 } \\Pi ( \\overline { \\beta _ 0 + \\beta _ 1 M ( z ) } ) ^ { - 1 } \\left ( \\vect g - \\beta _ 1 \\Pi ^ * \\left ( I - z \\mathcal { A } _ 0 ^ { - 1 } \\right ) ^ { - 1 } \\vect f \\right ) . \\end{align*}"}
+{"id": "3094.png", "formula": "\\begin{align*} S [ n , k ] : = S [ n - 1 , k - 1 ] + ( 1 + q + \\cdots + q ^ { k - 1 } ) \\ , S [ n - 1 , k ] \\end{align*}"}
+{"id": "7390.png", "formula": "\\begin{align*} \\delta ( n ) = \\begin{cases} 1 , & n = 0 \\\\ 0 , & n \\neq 0 . \\end{cases} \\end{align*}"}
+{"id": "6428.png", "formula": "\\begin{align*} \\widetilde { D } _ { \\alpha , z } ( \\psi _ 1 \\bar { \\otimes } \\psi _ 2 | | \\varphi _ 1 \\bar { \\otimes } \\varphi _ 2 ) = \\widetilde { D } _ { \\alpha , z } ( \\psi _ 1 | | \\varphi _ 1 ) + \\widetilde { D } _ { \\alpha , z } ( \\psi _ 2 | | \\varphi _ 2 ) , \\end{align*}"}
+{"id": "6169.png", "formula": "\\begin{align*} h _ n ( \\eta ) : = h _ n ( \\rho ( a _ 1 ) \\otimes . . . \\otimes \\rho ( a _ n ) ) = l _ R ( f ( a _ 1 ) ) \\ast . . . \\ast l _ R ( f ( a _ n ) ) = 0 \\in R _ n . \\end{align*}"}
+{"id": "1853.png", "formula": "\\begin{align*} \\gamma _ { n } ( \\Omega _ { i } ) = \\gamma _ { n } ( \\Omega _ { i } ' ) , \\qquad \\forall \\ , 1 \\leq i \\leq 3 . \\end{align*}"}
+{"id": "1591.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { m } \\ , \\mu _ s \\ , ^ { c } _ { 0 } { D } ^ { \\alpha _ s } _ { t } u ( x , t ) - a ( l ( u ) ) \\ : \\Delta u ( x , t ) & = f ( x , t ) & \\mbox { i n } & \\Omega \\times ( 0 , T ] , \\\\ u ( x , t ) & = 0 & \\mbox { o n } & \\partial \\Omega \\times ( 0 , T ] , \\\\ u ( x , 0 ) & = u _ 0 ( x ) & \\mbox { i n } & \\Omega , \\end{align*}"}
+{"id": "4047.png", "formula": "\\begin{align*} L ( f , k + t ) ^ 2 = \\sum _ { l , \\ , m \\geq 1 } \\frac { \\lambda _ f ( l ) \\lambda _ f ( m ) } { \\sqrt { l m } } ( l m ) ^ { - t } . \\end{align*}"}
+{"id": "7343.png", "formula": "\\begin{align*} x ^ a y ^ b + y ^ a z ^ b + z ^ a x ^ b = 0 , \\end{align*}"}
+{"id": "6196.png", "formula": "\\begin{align*} | V ( T ) | = ( n - 1 ) m + 2 . \\end{align*}"}
+{"id": "2093.png", "formula": "\\begin{align*} P F ( S ( n ) ) = \\{ F ( S ( n ) ) , F ( S ( n ) ) - 1 , . . . , F ( S ( n ) ) - ( n - 2 ) \\} . \\end{align*}"}
+{"id": "174.png", "formula": "\\begin{align*} L i _ 2 ( \\phi ^ { - 1 } ) = \\frac { \\pi ^ 2 } { 1 0 } - ( \\log \\phi ) ^ 2 , \\end{align*}"}
+{"id": "6925.png", "formula": "\\begin{align*} & \\sum _ { \\substack { m , n \\in \\mathcal { M } ' \\\\ m \\not = n } } r ( m ) r ( n ) \\Phi \\left ( T \\log { \\frac { m } { n } } \\right ) \\\\ & \\le \\sum _ { \\substack { j , l \\in \\mathcal { J } \\\\ j \\not = l } } r ( m _ j ) r ( n _ l ) \\Phi \\left ( T ( | j - l | - 1 ) \\log { ( 1 + T ^ { - 1 } ) } \\right ) \\\\ & \\ll \\sum _ { \\substack { j , l \\in \\mathcal { J } \\\\ j \\not = l } } r ( m _ j ) r ( n _ l ) \\Phi \\left ( | j - l | - 1 \\right ) \\end{align*}"}
+{"id": "6004.png", "formula": "\\begin{align*} \\theta _ { 3 / 2 } ( z , \\epsilon ) & = \\sum _ { n \\in \\Z } n e ^ { i \\epsilon \\pi n ^ 2 z } , \\end{align*}"}
+{"id": "5909.png", "formula": "\\begin{align*} \\widetilde { c } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) = \\gamma ( \\psi ( q ( g _ 1 , g _ 2 ) / 2 ) ) . \\end{align*}"}
+{"id": "3069.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\left [ r ^ \\delta u ' ( r ) ^ { p - 1 - \\alpha } \\right ] ' = \\frac { \\delta } { n - 1 } \\ , r ^ \\delta f _ 1 ( r ) g _ 1 ( v ( r ) ) & & \\\\ & \\left [ r ^ { n - 1 } v ' ( r ) ^ { p - 1 } \\right ] ' = r ^ { n - 1 } f _ 2 ( r ) g _ 2 ( v ( r ) ) \\cdot g _ 3 ( u ' ( r ) ) & & \\end{aligned} \\right . \\end{align*}"}
+{"id": "4324.png", "formula": "\\begin{align*} u ^ { ( i , k , n ; L ) } ( x , t ) = 0 . \\end{align*}"}
+{"id": "451.png", "formula": "\\begin{align*} \\Delta _ r w ( r ) + \\pi w ( r ) ^ 3 = - \\frac 3 4 \\delta . \\end{align*}"}
+{"id": "8094.png", "formula": "\\begin{align*} \\displaystyle \\mathbf { M } = 2 \\boldsymbol { \\Pi } \\mathbf { W } \\mathbf { R } ^ T - 2 \\boldsymbol { \\mathit { S } } , \\end{align*}"}
+{"id": "327.png", "formula": "\\begin{align*} s _ h ( m + 1 , m ) = \\sum _ { k = 1 } ^ { \\infty } \\left ( \\frac { 1 } { 1 } + \\frac { 1 } { 2 } + \\cdots + \\frac { 1 } { k } \\right ) ^ { m + 1 } \\frac { 1 } { ( k + 1 ) ^ m } . \\end{align*}"}
+{"id": "1648.png", "formula": "\\begin{align*} g ( [ { f } , { f } ] ( \\xi _ i , Z ) , { f } X ) & = g ( { f } ^ 2 \\ , [ \\xi _ i , Z ] - { f } [ \\xi _ i , { f } Z ] , { f } X ) = - g ( { f } ( \\pounds _ { \\xi _ i } { f } ) Z , { f } X ) \\\\ & = g ( ( \\pounds _ { \\xi _ i } { f } ) Z , Q X ) - \\sum \\nolimits _ { j } \\eta ^ j ( X ) \\ , \\eta ^ j ( ( \\pounds _ { \\xi _ i } { f } ) Z ) . \\end{align*}"}
+{"id": "8193.png", "formula": "\\begin{align*} P ^ \\omega ( E _ t , S _ t ) \\leq P ^ \\omega \\left ( E _ { 1 , t } , S _ { 1 , t } , F _ { 1 , t } \\right ) \\prod _ { j = 2 } ^ { t / h ( t ) } P ^ \\omega \\left ( E _ { j , t } , S _ { j , t } , F _ { j , t } \\ : \\bigg \\vert \\ : S _ { j - 1 , t } \\ , , \\ : \\bigcap _ { k = 1 } ^ { j - 1 } ( E _ { k , t } , F _ { k , t } ) \\right ) + \\sum _ { j = 1 } ^ { t / h ( t ) } P ^ \\omega ( F _ { j , t } ^ c ) \\end{align*}"}
+{"id": "667.png", "formula": "\\begin{align*} s _ k & = O ( e ^ { - \\lambda _ 1 k } e ^ { ( \\max \\{ \\lambda _ { i } , \\lambda _ 1 a \\} + 5 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k ) } ) \\\\ & = O ( e ^ { - \\lambda _ 1 ( 1 - \\tau ) r ( 0 , k + 1 ) } e ^ { ( \\max \\{ \\lambda _ { i } , \\lambda _ 1 a \\} + 5 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k ) } ) \\\\ & = O ( e ^ { ( \\max \\{ \\lambda _ { i } , \\lambda _ 1 a \\} - \\lambda _ 1 + 6 \\tau ( 1 + \\lambda _ 1 ) ) r ( 0 , k + 1 ) } ) . \\end{align*}"}
+{"id": "2864.png", "formula": "\\begin{align*} \\varphi _ L \\circ R = R \\circ \\varphi _ L \\end{align*}"}
+{"id": "5986.png", "formula": "\\begin{align*} d \\overline { \\Pi } _ { \\psi } ( h ) f ( [ \\epsilon , x ] ) & = \\frac { d } { d t } \\Big \\{ \\overline { \\Pi } _ { \\psi } ( [ e ^ { t h } , 1 ] ) f ( [ \\epsilon , x ] ) \\Big \\} \\Big | _ { t = 0 } \\\\ & = \\frac { d } { d t } \\Big \\{ e ^ { \\tfrac { t } { 2 } } f ( [ \\epsilon , x e ^ t ] ) \\Big \\} \\Big | _ { t = 0 } \\\\ & = \\{ \\tfrac { 1 } { 2 } + x \\tfrac { d } { d x } \\} f ( [ \\epsilon , x ] ) . \\end{align*}"}
+{"id": "4306.png", "formula": "\\begin{gather*} y _ { \\tau _ 2 } ^ { ( 2 ; L _ 2 ) } \\circ y _ { \\tau _ 1 } ^ { ( 1 ; L _ 1 ) } \\circ y _ { \\tau _ 2 } ^ { ( 2 ; L _ 2 ) } \\circ y _ { \\tau _ 1 } ^ { ( 1 ; L _ 1 ) } = \\mathrm { I d } , \\end{gather*}"}
+{"id": "1904.png", "formula": "\\begin{align*} I _ K ( \\zeta ) = \\left \\{ \\begin{aligned} & 0 , & & \\mbox { i f } \\ \\zeta \\in K , \\\\ & + \\infty , & & \\mbox { i f } \\ \\zeta \\not \\in K . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6076.png", "formula": "\\begin{align*} \\sum _ { I \\in \\mathbb { N } _ 0 ^ n } c _ I \\eta ^ I , \\eta ^ I : = \\eta ^ { i _ 1 } _ 1 \\dots \\eta ^ { i _ n } _ n , \\end{align*}"}
+{"id": "6945.png", "formula": "\\begin{align*} 2 x _ i + x _ { m + k } & = 2 ( E _ i + k - 1 ) + ( F _ k - m + i ) \\\\ & = \\frac 1 2 ( m - 2 n + 1 - 2 i ) + 2 k - 2 + \\frac 1 2 ( m + 2 n + 2 - 4 k ) - m + i \\\\ & = \\frac 1 2 - 2 + 1 = - \\frac 1 2 . \\end{align*}"}
+{"id": "8222.png", "formula": "\\begin{align*} A \\phi ( x , \\sigma ) = \\left ( \\tfrac { \\kappa } { 2 } \\partial _ { x x } + \\sigma \\lambda \\partial _ x \\right ) \\phi ( x , \\sigma ) + \\sum _ { \\sigma ' \\in S } c ( \\sigma , \\sigma ' ) \\big ( \\phi ( x , \\sigma ' ) - \\phi ( x , \\sigma ) \\big ) . \\end{align*}"}
+{"id": "8083.png", "formula": "\\begin{align*} \\mathcal { F } _ n : = \\left \\{ \\mathcal { M } \\subset \\mathcal { F } : \\mathcal { M } \\right . \\left . \\gamma ( \\mathcal { M } ) \\geq n \\right \\} \\end{align*}"}
+{"id": "8356.png", "formula": "\\begin{align*} V ( X / G ) = V ( X ) / G = \\{ v G : v \\in V ( X ) \\} , H ( X / G ) = H ( X ) / G = \\{ h G : h \\in H ( X ) \\} , \\end{align*}"}
+{"id": "2517.png", "formula": "\\begin{align*} G ( d , n ) = \\bigsqcup _ { M \\in M ^ d _ { S _ n } ( K ) } G _ M , \\end{align*}"}
+{"id": "2148.png", "formula": "\\begin{align*} \\tilde f ( x ) = f ( x ) + 2 0 \\delta \\cdot \\sum _ { i = 1 } ^ m \\left ( d ( x , p _ i ) - d ( p , p _ i ) \\right ) . \\end{align*}"}
+{"id": "3093.png", "formula": "\\begin{align*} \\left \\{ v _ S : S \\in \\binom { [ n + m ] } { n } \\right \\} \\end{align*}"}
+{"id": "6843.png", "formula": "\\begin{align*} a _ m = \\frac { 1 } { ( q ) _ \\infty } \\sum _ { j \\in \\mathbb { Z } } ( - 1 ) ^ j q ^ { r j ^ 2 - i j + m ( r - 1 ) j } \\frac { 1 - q ^ { ( m + 2 j ) ( i + 1 ) } } { 1 - q ^ { m + 2 j } } \\frac { ( - 1 ) ^ m q ^ { ( m + 2 j ) ( m + 1 ) } } { 1 + q ^ { m + 2 j } } . \\end{align*}"}
+{"id": "3889.png", "formula": "\\begin{align*} \\left ( \\mathord { \\operatorname { p r o j } } _ { Q } \\circ { \\operatorname { p r o j } _ { K _ i } } ^ { - 1 } \\right ) \\# \\mu _ i = \\left ( \\mathord { \\operatorname { p r o j } } _ { Q } \\circ { \\operatorname { p r o j } _ { K _ j } } ^ { - 1 } \\right ) \\# \\mu _ j . \\end{align*}"}
+{"id": "6542.png", "formula": "\\begin{align*} F ( s ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { a _ F ( n ) } { n ^ s } \\end{align*}"}
+{"id": "8075.png", "formula": "\\begin{align*} F ( u ) : = \\int _ { \\Omega } | u | ^ p d x G ( u ) : = \\int _ { \\Omega } | X u | ^ p d x . \\end{align*}"}
+{"id": "6487.png", "formula": "\\begin{align*} D ^ t = \\pm \\left ( \\begin{array} { c c } 2 \\pi a & s \\\\ 0 & 2 \\pi a \\end{array} \\right ) , \\end{align*}"}
+{"id": "1228.png", "formula": "\\begin{align*} { \\mathcal M } \\big ( e ^ { \\psi _ m } \\big ) ( z ) & = \\dfrac 1 { \\Omega _ { 2 n } \\delta ^ { 2 n } } \\int _ { B ( z , \\delta ) } ( 1 + | \\zeta | ^ 2 ) ^ { a _ m } e ^ { 2 m V ( \\zeta ) } \\ , d \\lambda ( \\zeta ) \\\\ & \\leq ( 1 + ( | z | + \\delta ) ^ 2 ) e ^ { 2 m ( V ( z ) + \\varepsilon ) } , \\end{align*}"}
+{"id": "1427.png", "formula": "\\begin{align*} - \\Delta u = u ^ { 2 ^ * - 1 } , \\ \\ u > 0 \\quad \\mbox { i n } \\ \\ \\mathbb { R } ^ n . \\end{align*}"}
+{"id": "6344.png", "formula": "\\begin{align*} \\bar \\rho : = \\rho \\big ( E _ t ^ { - 1 } \\big ( L _ { \\gamma ( \\bar s ) ^ { - 1 } } ( \\gamma ( t + \\bar s ) ) \\big ) \\big ) > 0 . \\end{align*}"}
+{"id": "6016.png", "formula": "\\begin{align*} \\theta _ { Y ^ { \\ast } , L } ( f ) ( x e _ Y ) & = \\int _ { Y ^ { \\ast } / Y ^ { \\ast } \\cap L } f ( [ t e _ { Y ^ { \\ast } } , 0 ] + [ x e _ Y , 0 ] ) d t \\\\ & = \\int _ { t \\in \\R / \\Z } f ( x e _ y + t e _ { Y ^ { \\ast } } ) \\psi ( - \\tfrac { x t } { 2 } ) d t . \\end{align*}"}
+{"id": "1753.png", "formula": "\\begin{align*} W _ m = \\mathcal { V } ^ { ( M , m ) } \\sum _ { \\mathbf { k } \\in \\mathcal { S } _ m } \\binom { m } { \\mathbf { k } } \\frac { w _ 1 ^ { k _ 1 } w _ 2 ^ { k _ 2 } w _ 3 ^ { k _ 3 } } { 2 ^ { k _ 1 + k _ 2 + k _ 3 - 1 } } \\end{align*}"}
+{"id": "2878.png", "formula": "\\begin{align*} Z _ k X _ { - \\beta _ k } ^ { \\pi ( \\beta _ k ) } \\otimes v _ \\lambda = 1 \\otimes v _ \\lambda . \\end{align*}"}
+{"id": "6023.png", "formula": "\\begin{align*} g = ( - I ) ^ { \\epsilon } n _ { 2 } ^ { k _ 1 } ( n ^ - _ { 2 } ) ^ { k _ 2 } \\cdots n _ { 2 } ^ { k _ l } ( n ^ - _ { 2 } ) ^ { k _ { l + 1 } } . \\end{align*}"}
+{"id": "1821.png", "formula": "\\begin{align*} [ \\mathcal { O } _ 2 ( k ) , D ( k ) ] & = \\int _ { \\Lambda ^ { * } } \\Big ( \\ 1 _ \\S ( p + k ) - \\ 1 _ \\S ( p ) \\Big ) \\chi ^ \\perp ( p , p + k ) a _ p ^ * a _ p \\d p \\\\ & - \\int _ { \\Lambda ^ { * } } \\Big ( \\ 1 _ \\S ( h - k ) - \\ 1 _ \\S ( h ) \\Big ) \\chi ( h , h - k ) a _ h ^ * a _ h \\d h \\ . \\end{align*}"}
+{"id": "115.png", "formula": "\\begin{align*} A \\chi e ^ { - i t _ 0 h ^ { - 1 } \\tilde { P } _ h ( 0 ) } \\tilde { R } _ h ( z ) \\chi B = & \\ ; A \\chi e ^ { - t _ 0 X } \\tilde { R } _ h ( z ) \\chi B \\\\ & \\ ; \\ ; - h ^ { - 1 } \\int _ 0 ^ { t _ 0 } A \\chi e ^ { - i t h ^ { - 1 } \\tilde { P } _ h ( 0 ) } ( W + Q _ \\infty + q _ 1 ) e ^ { - ( t _ 0 - t ) X } \\tilde { R } _ h ( z ) \\chi B d t . \\end{align*}"}
+{"id": "963.png", "formula": "\\begin{align*} \\widetilde { q } _ { y _ 0 } ( r ) = \\frac { 1 } { \\omega _ { n - 1 } r ^ { n - 1 } } \\int \\limits _ { S ( y _ 0 , r ) } \\widetilde { Q } ( y ) \\ , d \\mathcal { H } ^ { n - 1 } ( y ) \\end{align*}"}
+{"id": "523.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\hbar , V } u ( k ) : = \\left ( - \\hbar ^ { - 2 } \\mathcal { L } _ { \\hbar } + V \\right ) u ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\end{align*}"}
+{"id": "1759.png", "formula": "\\begin{align*} \\varphi _ j = \\frac { W _ j } { 3 } \\arctan [ \\sinh ( x ) ] , j = 1 , 2 \\ , \\end{align*}"}
+{"id": "4503.png", "formula": "\\begin{align*} \\R ^ 3 _ + : = \\left \\{ ( \\alpha , \\beta , \\gamma ) \\in \\R ^ 3 \\ | \\ \\alpha > 0 , \\ \\beta > 0 , \\ \\gamma > 0 \\right \\} , \\ \\ \\R ^ 3 _ - : = \\left \\{ ( \\alpha , \\beta , \\gamma ) \\in \\R ^ 3 \\ | \\ \\alpha < 0 , \\ \\beta < 0 , \\ \\gamma < 0 \\right \\} . \\end{align*}"}
+{"id": "2902.png", "formula": "\\begin{align*} ( f \\sharp g ) _ { m + m ' - j } ( \\xi , \\mu ) = \\sum _ { k + l + | \\alpha | = j } \\frac { ( - i ) ^ { | \\alpha | } } { \\alpha ! } \\partial _ \\xi ^ \\alpha f _ { m - k } ( \\xi , \\mu ) g _ { m ' - l } ^ { B , \\alpha } ( \\xi , \\mu ) , j \\geq 0 . \\end{align*}"}
+{"id": "2316.png", "formula": "\\begin{align*} y \\gets c _ 2 A r + c _ 1 r = c _ 2 ( A - I ) r + ( c _ 1 - c _ 2 ) r = - c _ 2 E r + ( c _ 1 - c _ 2 ) r \\end{align*}"}
+{"id": "6705.png", "formula": "\\begin{align*} & q _ { R } ( x , \\xi , \\tau ) = \\frac { p _ { \\Phi } + \\overline { p _ { \\Phi } } } { 2 } ( x , \\xi , \\tau ) = \\Re p _ { \\Phi } ( x , \\xi , \\tau ) = p _ 2 ( x , \\xi ) - \\tau ^ 2 p _ 2 ( x , d \\Phi ( x ) ) , \\\\ & q _ { I } ( x , \\xi , \\tau ) = \\frac { p _ { \\Phi } - \\overline { p _ { \\Phi } } } { 2 i } ( x , \\xi , \\tau ) = \\Im p _ { \\Phi } ( x , \\xi , \\tau ) = 2 \\tau \\widetilde { p _ 2 } ( x , \\xi , d \\Phi ( x ) ) . \\end{align*}"}
+{"id": "6883.png", "formula": "\\begin{align*} J _ r ( x , \\beta ) = \\sup _ { \\theta \\in \\R } [ \\theta \\beta - \\Lambda _ r ( x , \\theta ) ] . \\end{align*}"}
+{"id": "5499.png", "formula": "\\begin{align*} \\varphi \\circ \\xi \\left ( g P , x _ { 0 } \\right ) = \\varphi \\circ \\xi \\left ( g . ( P , \\alpha ( g , P ) ^ { - 1 } . x _ { 0 } ) \\right ) = g . \\left ( \\varphi \\circ \\xi \\left ( P , \\alpha ( g , P ) ^ { - 1 } . x _ { 0 } \\right ) \\right ) = g Q . \\end{align*}"}
+{"id": "7753.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\frac { g ( r ) } { h ( r ) } = 0 \\ , . \\end{align*}"}
+{"id": "2845.png", "formula": "\\begin{align*} \\overset \\cdot c _ i = c _ i ^ 2 \\left ( \\prod _ { j = 1 } ^ { r } c _ { i + j } - \\prod _ { j = 1 } ^ { r } c _ { i - j } \\right ) . \\end{align*}"}
+{"id": "2811.png", "formula": "\\begin{align*} H = ( \\alpha _ { i , j } ) _ { i , j = 0 } ^ \\infty . \\end{align*}"}
+{"id": "7243.png", "formula": "\\begin{align*} \\Big \\Vert \\sup _ { k \\leq n } | S _ k - T _ k | \\Big \\Vert _ 2 = O ( n ^ { 1 / p } ( \\log n ) ^ { 1 / 2 } ) \\ , . \\end{align*}"}
+{"id": "8792.png", "formula": "\\begin{align*} & 2 ( u + 1 ) ( 4 + t + u ) b _ 1 \\\\ & = 2 ( u + 1 ) ( 1 + 2 t ) c _ 1 + 2 ( u + 1 ) a _ 3 + 2 u ( u + 1 ) a _ 2 + 2 u ( u + 1 ) b _ 2 \\end{align*}"}
+{"id": "9331.png", "formula": "\\begin{align*} & a ( G _ { 1 4 , \\mathbb { H } } ; T ) = \\sum _ { d \\mid \\varepsilon ( T ) } d ^ { 1 3 } \\ , b _ { 1 4 } ^ * ( 2 ( T ) / d ^ 2 ) , \\\\ & b _ { 1 4 } ^ * ( \\ell ) = \\sigma _ { 1 1 } ( \\ell ) - 2 ^ { 1 2 } \\sigma _ { 1 1 } ( \\ell / 4 ) . \\end{align*}"}
+{"id": "6808.png", "formula": "\\begin{align*} ( k ^ + - 1 ) + ( 2 n - k ^ - - 1 ) - ( 2 n - 1 ) = k ^ + - k ^ - - 1 . \\end{align*}"}
+{"id": "1508.png", "formula": "\\begin{align*} & \\frac { ( r + x + \\varepsilon _ 2 ) ^ { 1 / 2 } ( r + x - \\varepsilon _ 2 ) ^ { 1 / 2 } ( r + \\varepsilon _ 2 - x ) ^ { 1 / 2 } ( x + \\varepsilon _ 2 - r ) ^ { 1 / 2 } } { 2 r } . \\end{align*}"}
+{"id": "8253.png", "formula": "\\begin{align*} \\begin{pmatrix} \\tau _ { k , Y _ { 2 , 1 } } \\\\ \\tau _ { k , Y _ { 2 , 2 } } \\end{pmatrix} = \\begin{pmatrix} 1 / 2 & 1 / 2 \\\\ 1 / 2 & - 1 / 2 \\end{pmatrix} \\begin{pmatrix} T _ 1 \\tau _ { k , Y _ { 1 , 1 } } \\\\ T _ 2 \\tau _ { k } \\end{pmatrix} , \\end{align*}"}
+{"id": "6598.png", "formula": "\\begin{align*} & E [ \\hat \\beta _ l ( \\alpha _ { l , { { n } _ t } } - \\alpha _ { l , { \\hat { n } _ t } } e ^ { - j \\omega } ) ] = \\frac { \\sqrt { \\pi } } { 2 } E ( \\hat \\beta _ l ) , \\\\ & V a r [ \\hat \\beta _ l ( \\alpha _ { l , { { n } _ t } } - \\alpha _ { l , { \\hat { n } _ t } } e ^ { - j \\omega } ) ] = 2 - \\frac { \\pi } { 4 } E ^ 2 ( \\hat \\beta _ l ) . \\end{align*}"}
+{"id": "7133.png", "formula": "\\begin{align*} A _ \\gamma ( \\hat { x } , \\hat { x } ) = F _ \\gamma ( \\hat { x } ) = U ( \\hat { x } ) \\left ( \\begin{array} { c c c | c } & & & 0 \\\\ & A ^ \\prime ( \\hat { x } ) & & \\vdots \\\\ & & & 0 \\\\ \\hline 0 & \\ldots & 0 & 1 \\end{array} \\right ) , \\end{align*}"}
+{"id": "2072.png", "formula": "\\begin{align*} ( \\partial _ n ) _ \\sigma ^ { \\sigma ' } : = \\begin{cases} ( - 1 ) ^ i & \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "7494.png", "formula": "\\begin{align*} d _ q < d _ { q + 1 } < \\cdots < d _ l < x \\qquad \\qquad \\tilde { \\delta } = ( x , d _ l , d _ { l - 1 } \\dots , d _ 2 ) ; \\end{align*}"}
+{"id": "1950.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u ) ^ n = \\psi ( \\cdot , u ) \\ , \\omega ^ n & \\textnormal { i n } & \\Omega , \\\\ u = \\varphi & \\textnormal { i n } & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "1106.png", "formula": "\\begin{align*} \\Phi _ { k , i } ( x _ 1 , \\ldots , x _ k ) = 0 , i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "4803.png", "formula": "\\begin{align*} \\tilde { \\mathcal { L } } ( \\varphi ) = \\frac { \\tilde { \\mathcal { K } } ( \\varphi ) - \\varphi } { \\tau } . \\end{align*}"}
+{"id": "1623.png", "formula": "\\begin{align*} N ^ { \\ , ( 2 ) } _ i ( X , Y ) = \\eta ^ i ( [ \\widetilde { Q } X , \\ , { f } Y ] ) ; \\end{align*}"}
+{"id": "8666.png", "formula": "\\begin{align*} \\xi \\eta ' + \\eta \\xi ' = \\xi ' \\eta + \\eta ' \\xi = - ( \\xi , \\eta ) _ { \\mathbb { F } } , \\qquad \\mathrm { w h e r e } \\qquad ( \\xi , \\eta ) _ { \\mathbb { F } } : = ( u , v ) - 2 a b \\end{align*}"}
+{"id": "6485.png", "formula": "\\begin{align*} D ^ t = \\left ( \\begin{array} { c c } \\sqrt { \\lambda } + b & b \\\\ - b & - ( \\sqrt { \\lambda } + b ) \\end{array} \\right ) \\end{align*}"}
+{"id": "6871.png", "formula": "\\begin{align*} v _ r ( x ) = \\int _ { [ 0 , 1 ] } \\d y \\ , r ( x , y ) [ 1 - r ( x , y ) ] . \\end{align*}"}
+{"id": "2059.png", "formula": "\\begin{align*} P _ { N } u : = u _ { N } : = u _ { \\leq 2 N } - u _ { \\leq N } \\\\ P _ { A \\leq \\cdot \\leq B } : = u _ { A \\leq \\cdot \\leq B } : = u _ { \\leq B } - u _ { \\leq A } . \\end{align*}"}
+{"id": "4308.png", "formula": "\\begin{align*} ( k _ 1 , m _ 1 ) < _ \\mathrm { t i m e } ( k _ 2 , m _ 2 ) \\begin{cases} m _ 1 2 ^ { - k _ 1 } < m _ 2 2 ^ { - k _ 2 } , \\\\ m _ 1 2 ^ { - k _ 1 } = m _ 2 2 ^ { - k _ 2 } k _ 1 < k _ 2 . \\end{cases} \\end{align*}"}
+{"id": "6785.png", "formula": "\\begin{align*} & \\frac { 1 } { ( q ; q ) _ \\infty ( x ^ \\alpha ; q ) _ \\infty ( q x ^ { - \\alpha } ; q ) _ \\infty } \\\\ & = \\frac { 1 } { \\big ( ( 1 - q ) ( 1 - q ^ 2 ) \\dotsm \\big ) \\big ( ( 1 - x ^ \\alpha ) ( 1 - q x ^ { \\alpha } ) \\dotsm \\big ) \\big ( ( 1 - q x ^ { - \\alpha } ) ( 1 - q ^ 2 x ^ { - \\alpha } ) \\dotsm \\big ) } \\\\ & = ( 1 + x ^ \\alpha + x ^ { 2 \\alpha } + x ^ { 3 \\alpha } + \\dotsm ) + q ( x ^ { - \\alpha } + 2 + 3 x ^ \\alpha + 3 x ^ { 2 \\alpha } + \\dotsm ) + q ^ 2 \\dotsm . \\end{align*}"}
+{"id": "4451.png", "formula": "\\begin{align*} h ( 0 ) & = \\mu ( \\omega , \\mathbf { c } ) = S _ { \\omega , \\mathbf { c } } ( \\Phi ) , \\\\ h ' ( 0 ) & = - \\frac { 1 } { \\sqrt { \\omega } } ( 2 \\omega Q ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) ) , \\\\ h '' ( 0 ) & = \\frac { 3 - d } { \\omega } ( 2 \\omega Q ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) ) . \\end{align*}"}
+{"id": "5197.png", "formula": "\\begin{align*} \\widetilde { { S } } _ { \\lambda ^ 3 } \\widetilde { { S } } _ { \\lambda ^ 3 } & = ( y _ { 4 } - y _ { 2 } ) ( y _ { 4 } - y _ { 3 } ) \\widetilde { { S } } _ { \\lambda ^ 3 } + ( y _ { 4 } - y _ { 3 } ) \\widetilde { { S } } _ { \\lambda ^ 1 } + \\widetilde { { S } } _ { \\lambda ^ 0 } \\\\ & = ( Y _ { \\lambda ^ 5 } - Y _ { \\lambda ^ 3 } ) ( Y _ { \\lambda ^ 2 } - Y _ { \\lambda ^ 1 } ) \\widetilde { { S } } _ { \\lambda ^ 3 } + ( Y _ { \\lambda ^ 2 } - Y _ { \\lambda ^ 1 } ) \\widetilde { { S } } _ { \\lambda ^ 1 } + \\widetilde { { S } } _ { \\lambda ^ 0 } . \\end{align*}"}
+{"id": "8923.png", "formula": "\\begin{align*} Z = \\sum _ { j = 1 } ^ k Z _ j = \\prod _ { j = 1 } ^ k y \\big ( \\bigl \\{ X _ { n i } ^ { ( j ) } \\bigr \\} \\big ) . \\end{align*}"}
+{"id": "9321.png", "formula": "\\begin{align*} { ( J _ \\varepsilon ( u ^ { \\varepsilon , \\alpha } ) ) ^ 2 - ( J _ \\varepsilon ( u ^ { \\varepsilon } ) ) ^ 2 \\over J _ \\varepsilon ( u ^ { \\varepsilon , \\alpha } ) + J _ \\varepsilon ( u ^ { \\varepsilon } ) } = J _ \\varepsilon ( u ^ { \\varepsilon , \\alpha } ) - J _ \\varepsilon ( u ^ { \\varepsilon } ) \\ge - \\sqrt { \\varepsilon } d ( u ^ { \\varepsilon , \\alpha } , u ^ { \\varepsilon } ) \\ge - C \\alpha \\sqrt { \\varepsilon } , \\end{align*}"}
+{"id": "6874.png", "formula": "\\begin{align*} \\widehat { \\psi } _ r ( \\beta ) = \\inf _ { x \\in [ 0 , 1 ] } J _ r ( x , \\beta ) . \\end{align*}"}
+{"id": "2219.png", "formula": "\\begin{align*} C _ 5 ( y _ 0 ) & = \\sup _ { t \\ge y _ 0 } \\frac { \\exp ( C _ 2 ( y _ 0 ) R ( t ) ) - 1 } { R ( t ) } = \\frac { \\exp ( C _ 2 ( y _ 0 ) R ( y _ 0 ) ) - 1 } { R ( y _ 0 ) } , \\\\ C _ 6 ( y _ 0 ) & = \\sup _ { t \\ge y _ 0 } \\frac { 1 - \\exp ( - C _ 1 ( y _ 0 ) R ( t ) ) } { R ( t ) } = C _ 1 ( y _ 0 ) . \\end{align*}"}
+{"id": "2170.png", "formula": "\\begin{align*} \\delta ^ { 2 } e ^ { 2 \\lambda \\left ( \\delta \\right ) d } = \\delta ^ { 2 - 2 \\rho } . \\end{align*}"}
+{"id": "6290.png", "formula": "\\begin{align*} H ( \\lambda ) = \\max _ { u \\in \\R ^ k } \\left ( - \\frac { \\norm { u } ^ 2 } { 2 } \\right ) = 0 . \\end{align*}"}
+{"id": "6792.png", "formula": "\\begin{align*} \\Sigma = W _ 1 \\cup W _ 2 \\cup _ Y W _ 3 \\cup W _ 4 . \\end{align*}"}
+{"id": "7172.png", "formula": "\\begin{align*} u ^ \\nu _ z : = u ( z + t \\nu ) , z : = \\left ( { \\rm I d } - \\nu \\otimes \\nu \\right ) x , \\nu : = \\frac { x - y } { \\abs { x - y } } . \\end{align*}"}
+{"id": "1947.png", "formula": "\\begin{align*} f ^ * _ n ( z _ 1 , \\ldots , z _ m ) : = f ^ * _ { n , L } ( z _ 1 , \\ldots , z _ m ) \\end{align*}"}
+{"id": "6074.png", "formula": "\\begin{align*} | I | = i _ 1 + \\dots + i _ n , d _ A ( I ) = v _ 1 i _ 1 + \\dots + v _ n i _ n . \\end{align*}"}
+{"id": "5979.png", "formula": "\\begin{align*} \\chi _ { n } ( [ g , \\epsilon ] [ k ( t ) , 1 ] ) & = \\chi _ { n } ( [ g k ( t ) , \\epsilon ] ) \\\\ & = \\chi _ { n } ( [ p _ g , \\epsilon ] [ k _ g k ( t ) , 1 ] ) \\\\ & = \\chi _ { n } ( [ p _ g , \\epsilon ] ) \\chi _ { n } ( [ k _ g , 1 ] ) \\chi _ { n } ( [ k ( t ) , 1 ] ) \\\\ & = \\chi _ { n } ( [ g , \\epsilon ] ) \\chi _ { n } ( [ k ( t ) , 1 ] ) . \\end{align*}"}
+{"id": "6961.png", "formula": "\\begin{align*} \\textbf { Q } ^ \\lambda : = \\prod _ { q \\in \\textbf { Q } } q ^ { \\lambda ( q ) } \\in K [ x ] . \\end{align*}"}
+{"id": "222.png", "formula": "\\begin{align*} \\omega ( s _ 1 , . . . , s _ { K + 1 } ) = \\sum _ { m _ 1 , . . . , m _ K > 0 } \\frac { 1 } { { m _ 1 } ^ { s _ 1 } { m _ 2 } ^ { s _ 2 } \\ldots { m _ K } ^ { s _ { K } } ( m _ 1 + m _ 2 + \\ldots + m _ { K } ) ^ { s _ { K + 1 } } } \\end{align*}"}
+{"id": "5469.png", "formula": "\\begin{align*} f _ { r _ 1 | \\Phi ( \\mathcal { A } ) > 0 } ( r ) = \\upsilon ( \\lambda , R _ S ) r e ^ { - \\lambda \\pi \\frac { R _ S } { R _ E } r ^ 2 } , R _ { m i n } \\leq r \\leq R _ { m a x } , \\end{align*}"}
+{"id": "331.png", "formula": "\\begin{align*} \\prod _ { \\substack { \\gcd ( j _ 1 , j _ 2 , j _ 3 , j _ 4 , j _ 5 , k ) = 1 \\\\ j _ 1 , j _ 2 , j _ 3 , j _ 4 , j _ 5 < k \\\\ j _ 1 , j _ 2 , j _ 3 , j _ 4 , j _ 5 \\geq 1 ; k \\geq 2 } } \\left ( \\frac { 1 } { 1 - y ^ { j _ 1 + j _ 2 + j _ 3 + j _ 4 + j _ 5 } z ^ k } \\right ) ^ { \\frac { 1 } { k } } = \\exp \\left [ \\sum _ { k = 1 } ^ { \\infty } \\left ( \\frac { y } { 1 } + \\frac { y ^ 2 } { 2 } + \\cdots + \\frac { y ^ k } { k } \\right ) ^ { 5 } \\frac { z ^ { k + 1 } } { ( k + 1 ) ^ 4 } \\right ] . \\end{align*}"}
+{"id": "771.png", "formula": "\\begin{align*} \\delta _ { i _ 1 i _ 2 \\cdots i _ { m + 1 } i _ { m + 2 } \\cdots i _ k } ^ { j _ 1 j _ 2 \\cdots j _ { m + 1 } j _ { m + 2 } \\cdots j _ k } \\delta ^ { i _ { m + 2 } } _ { j _ { m + 2 } } \\cdots \\delta ^ { i _ k } _ { j _ k } = \\delta _ { i _ 1 i _ 2 \\cdots i _ { m + 1 } } ^ { i _ 1 j _ 2 \\cdots j _ { m + 1 } } { n - m - 1 \\choose k - m - 1 } ( k - m - 1 ) ! , \\end{align*}"}
+{"id": "7110.png", "formula": "\\begin{align*} \\Big \\| \\widehat { \\mathcal { L } ^ { \\mathbb { R } ^ n } _ \\varepsilon c } + \\widehat { \\Delta c } \\Big \\| _ { L ^ 2 ( \\mathbb { R } ^ n ) } ^ 2 = \\frac { 1 } { ( 2 \\pi ) ^ n } \\int _ { \\mathbb { R } ^ n } \\big | \\big ( - \\mathcal { F } ( J _ \\varepsilon ) ( \\xi ) + \\mathcal { F } ( J _ \\varepsilon ) ( 0 ) - | \\xi | ^ 2 \\big ) \\mathcal { F } ( c ) ( \\xi ) \\big | ^ 2 \\xi , \\end{align*}"}
+{"id": "6167.png", "formula": "\\begin{align*} x \\cdot p = 1 \\cdot a & \\mbox { a n d } l _ R ( x ) l _ R ( p ) = l _ R ( 1 ) l _ R ( a ) = 0 \\\\ y \\cdot q = 1 \\cdot a & \\mbox { a n d } l _ R ( y ) l _ R ( q ) = l _ R ( 1 ) l _ R ( a ) = 0 . \\end{align*}"}
+{"id": "5882.png", "formula": "\\begin{align*} \\theta _ n ( t ) = & \\ , - \\nu _ n ^ 2 t + ( \\nu _ { n - 1 } + 1 ) ^ 2 t - c t + \\theta _ { n - 1 } ( t ) \\\\ = & \\ , - \\nu _ n ^ 2 \\ , t + ( \\nu _ { n - 1 } + 1 ) ^ 2 t - \\nu _ { n - 1 } ^ 2 \\ , t + ( \\nu _ { n - 2 } + 1 ) ^ 2 t - 2 c t + \\theta _ { n - 2 } ( t ) \\\\ = & \\quad \\ldots \\\\ = & \\ , - \\nu _ n ^ 2 \\ , t + ( n - \\ell ) t + \\nu _ { \\ell } ^ 2 \\ , t - ( n - \\ell ) c t + 2 t \\sum _ { k = \\ell } ^ { n - 1 } \\nu _ { k } + \\theta _ { \\ell } ( t ) \\ , , \\end{align*}"}
+{"id": "5711.png", "formula": "\\begin{align*} C _ { g } ( p ^ { 2 m + 1 } ) = 0 \\end{align*}"}
+{"id": "7067.png", "formula": "\\begin{align*} \\beta _ j < \\tilde { \\beta } _ j + \\epsilon < \\alpha _ { i _ - } + \\epsilon = \\alpha _ i . \\end{align*}"}
+{"id": "1619.png", "formula": "\\begin{align*} [ { f } , { f } ] ( X , \\xi _ i ) = { f } ( \\nabla _ { \\xi _ i } { f } ) X + \\nabla _ { { f } X } \\ , \\xi _ i - { f } \\ , \\nabla _ { X } \\ , \\xi _ i , X \\in \\mathfrak { X } _ M . \\end{align*}"}
+{"id": "5162.png", "formula": "\\begin{align*} u _ \\lambda = \\arg \\underset { u \\in H } { \\min } \\big ( \\lambda \\lVert u \\rVert _ H ^ 2 + \\frac { 1 } { n } \\sum _ { i = 1 } ^ n L ( y _ i , u ( x _ i ) \\big ) , \\end{align*}"}
+{"id": "2516.png", "formula": "\\begin{align*} \\| \\delta _ { d } \\| _ A \\le \\| P \\delta _ { d _ c } \\| _ A + \\| \\delta _ { p _ \\nu } \\| _ A = \\| \\delta _ { d _ c } \\| _ { A _ c } + \\| \\delta _ { p _ \\nu } \\| _ A \\le C _ 2 \\| A ^ { - 1 } r \\| _ A . \\end{align*}"}
+{"id": "7391.png", "formula": "\\begin{align*} \\frac { L } { N } \\sum _ { n _ 1 , n _ 2 \\neq 0 } \\widehat { \\Delta } \\left ( \\frac { L n _ 1 } { N } \\right ) \\widehat { \\Delta } \\left ( \\frac { L n _ 2 } { N } \\right ) \\delta ( n _ 1 w _ r - n _ 2 w _ s ) = \\frac { L } { N } \\sum _ { n _ 0 \\neq 0 } \\widehat { \\Delta } \\left ( \\frac { L w _ r n _ 0 } { N d } \\right ) \\widehat { \\Delta } \\left ( \\frac { L w _ s n _ 0 } { N d } \\right ) . \\end{align*}"}
+{"id": "8256.png", "formula": "\\begin{align*} T _ { h } G _ { k , Y } = T _ { h - 2 } G _ { k , Y } - \\frac { S _ { h - 1 } G _ { k , Y } } { \\sqrt { x } } - \\frac { 2 t } { \\sqrt { x } } T _ { h + 1 } G _ { k , Y } . \\end{align*}"}
+{"id": "1832.png", "formula": "\\begin{align*} \\mu \\big ( [ [ N , T + T ^ * ] , S ] \\big ) = 2 \\mathrm { R e } \\ , \\mu \\big ( [ [ N , T ] , S ] \\big ) \\ . \\end{align*}"}
+{"id": "786.png", "formula": "\\begin{align*} a _ 0 ^ 2 = O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 . \\end{align*}"}
+{"id": "3353.png", "formula": "\\begin{align*} U = \\begin{pmatrix} \\frac { 1 } { \\sqrt { 2 } } & 0 & \\frac { 1 } { \\sqrt { 2 } } \\\\ 0 & 1 & 0 \\\\ \\frac { 1 } { \\sqrt { 2 } } & 0 & - \\frac { 1 } { \\sqrt { 2 } } \\end{pmatrix} . \\end{align*}"}
+{"id": "4961.png", "formula": "\\begin{align*} \\overline { F } ( k , t + 1 , n ) = \\begin{dcases} \\frac { k } { n } \\overline { F } ( k , t , n ) + \\frac { n - k } { n } \\overline { F } ( k - 1 , t , n ) & \\textnormal { i f } \\ ; k \\le t ; \\\\ 0 & \\textnormal { i f } \\ ; k > t ; \\end{dcases} \\end{align*}"}
+{"id": "5066.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\| \\partial _ x ^ l \\partial _ t ^ j s _ { p , N } ( t ) \\partial ^ k _ x v \\right \\| _ { L _ N ^ 2 } \\leq C _ { j , l , k } ( 1 + t ) ^ { - \\frac { 1 } { 4 } - \\frac { l + j } { 2 } } \\| v \\| _ { L _ N ^ 1 } , & & v \\in H ^ k _ N , \\\\ & \\left \\| \\partial _ x ^ l \\partial _ t ^ j s _ { p , N } ( t ) \\partial ^ k _ x v \\right \\| _ { L _ N ^ 2 } \\leq C _ { j , l , k } ( 1 + t ) ^ { - \\frac { l + j } { 2 } } \\| v \\| _ { L _ N ^ 2 } , & & v \\in H ^ k _ N , \\end{aligned} \\end{align*}"}
+{"id": "7645.png", "formula": "\\begin{align*} \\quad \\quad \\quad \\quad \\frac { \\partial } { \\partial c } M ( c , x , 0 ) = 2 c ( 4 - c ^ 2 ) ( ( 8 - 6 c ^ 2 ) x ^ 4 + ( 3 9 c ^ 2 + 2 0 ) x ^ 3 + ( 8 - 6 c ^ 2 ) x ^ 2 - 1 4 4 x ) = 0 \\end{align*}"}
+{"id": "4496.png", "formula": "\\begin{align*} \\Lambda ( \\theta ) \\Phi : = ( e ^ { 2 i \\theta } \\varphi _ 1 , e ^ { i \\theta } \\varphi _ 2 , e ^ { i \\theta } \\varphi _ 3 ) . \\end{align*}"}
+{"id": "7268.png", "formula": "\\begin{align*} a x ^ n + b y ^ m = c z ^ k , \\end{align*}"}
+{"id": "8025.png", "formula": "\\begin{align*} Q + Q ^ * & \\simeq 2 P \\oplus \\begin{pmatrix} 2 I & R \\\\ R & 0 \\end{pmatrix} \\\\ & \\simeq 2 P \\oplus \\left \\{ \\begin{pmatrix} I & I \\\\ I & I \\end{pmatrix} + \\begin{pmatrix} R & 0 \\\\ 0 & - R \\end{pmatrix} \\right \\} . \\end{align*}"}
+{"id": "995.png", "formula": "\\begin{align*} \\Sigma ( \\psi ) ( \\omega _ r ) = \\{ m \\in \\bar \\omega \\ , | \\ , \\langle \\delta - n _ { F ^ i } , m \\rangle \\le r , 1 \\le i \\le p \\} . \\end{align*}"}
+{"id": "8213.png", "formula": "\\begin{align*} P ^ \\omega ( D _ t ^ c \\mid S ) & = \\frac { 1 } { P ^ \\omega ( S ) } \\left [ P ^ \\omega ( D _ t ^ c \\cap S \\cap C _ t ) + P ^ \\omega ( D _ t ^ c \\cap S \\cap C _ t ^ c ) \\right ] \\\\ & \\leq c ( \\omega ) \\left [ P ^ \\omega ( D _ t ^ c \\cap S _ t \\cap C _ t ) + P ^ \\omega ( D _ t ^ c \\cap S \\cap C _ t ^ c ) \\right ] \\\\ & \\leq c ( \\omega ) \\left [ P ^ \\omega ( D _ t ^ c \\mid C _ t ) + P ^ \\omega ( C _ t ^ c \\mid S ) \\right ] , \\end{align*}"}
+{"id": "5860.png", "formula": "\\begin{align*} ( E , \\sigma ( H _ { \\Lambda _ 2 } ) ) = \\Vert ( H _ { \\Lambda _ { L _ 2 } } - E ) ^ { - 1 } \\Vert ^ { - 1 } = \\inf _ { \\psi \\in \\ell ^ 2 } \\frac { \\Vert ( H _ { \\Lambda _ { L _ 2 } } - E ) \\psi \\Vert } { \\Vert \\psi \\Vert } . \\end{align*}"}
+{"id": "7363.png", "formula": "\\begin{align*} ( \\lambda , \\ell ) + \\rho = ( \\lambda _ 1 + n - d - 1 , \\lambda _ 2 + n - d - 2 , \\ldots , \\lambda _ { n - d - 1 } + 1 , \\ell ) . \\end{align*}"}
+{"id": "2734.png", "formula": "\\begin{align*} \\Delta u = \\frac { 1 } { \\sqrt { | g | } } \\sum _ { j , k = 1 } ^ n \\frac { \\partial } { \\partial x _ j } \\left ( \\sqrt { | g | } g ^ { j k } \\frac { \\partial } { \\partial x _ k } u \\right ) . \\end{align*}"}
+{"id": "4624.png", "formula": "\\begin{align*} p _ { Y _ k , 1 } ( X _ f ) = p _ { Z , k } ( X _ f ) . \\end{align*}"}
+{"id": "3479.png", "formula": "\\begin{align*} \\Gamma ( S _ i ) \\le \\left ( \\frac { 5 e ^ 2 \\Delta ( S _ i ) } { b ^ 2 } \\right ) ^ { \\kappa } b _ 1 ( 3 \\log _ 2 | S _ i | ) ^ { 1 + \\kappa } = b _ 2 \\Delta ( S _ i ) ^ \\kappa ( \\log _ 2 | S _ i | ) ^ { 1 + \\kappa } \\le b _ 2 | S _ i | ^ \\kappa ( \\log _ 2 | S _ i | ) ^ { 1 + \\kappa } , \\end{align*}"}
+{"id": "8026.png", "formula": "\\begin{align*} a = \\frac { 1 + r ^ 2 } { r \\sqrt { 1 - r ^ 2 } } . \\end{align*}"}
+{"id": "797.png", "formula": "\\begin{gather*} J \\begin{bmatrix} { { \\alpha } ^ { n } } \\\\ d \\end{bmatrix} = \\begin{bmatrix} A & b \\\\ c & - 1 \\end{bmatrix} \\begin{bmatrix} { { \\alpha } ^ { n } } \\\\ d \\end{bmatrix} = \\begin{bmatrix} \\bar { F } \\\\ F _ { M + 1 } \\end{bmatrix} , \\end{gather*}"}
+{"id": "1880.png", "formula": "\\begin{align*} \\| [ \\zeta ] \\| _ q = \\inf \\{ \\| \\zeta - \\zeta ' \\| : \\ \\zeta ' \\in \\mathrm { K e r } ( \\lambda ^ * ) \\cap \\overline { R ( S ) } \\} . \\end{align*}"}
+{"id": "4866.png", "formula": "\\begin{align*} \\overline { b } + \\widetilde { \\mu } _ { i } \\left ( \\widetilde { b } - \\overline { b } \\right ) = \\overline { b } + \\mu _ { i } \\left ( b - \\overline { b } \\right ) , i = 1 , . . . , N - 1 . \\end{align*}"}
+{"id": "638.png", "formula": "\\begin{align*} \\sup _ { \\omega \\in P } \\omega ( a ^ * a ) = \\| a \\| ^ 2 , a \\in A . \\end{align*}"}
+{"id": "7564.png", "formula": "\\begin{align*} \\sum _ { \\exp ( \\log ^ \\beta ( n - n ' ) ) \\le p \\le \\exp ( \\log ^ \\beta n ) } \\log ( 1 - p ^ { - 1 } ) \\ll & \\sum _ { \\exp ( \\log ^ \\beta ( n - n ' ) ) \\le p \\le \\exp ( \\log ^ \\beta n ) } p ^ { - 1 } \\\\ = & \\beta \\log \\log n - \\beta \\log \\log ( n - n ' ) + O ( \\log ^ { - \\beta } n ) \\\\ \\ll & \\log ^ { - \\beta } n . \\end{align*}"}
+{"id": "2253.png", "formula": "\\begin{align*} f _ 1 ( z ) = i c _ 0 - I _ 0 + \\frac { 1 } { 2 \\pi } \\langle ( h _ 0 ) _ b , P _ r ( \\theta - \\cdot ) \\rangle \\end{align*}"}
+{"id": "4248.png", "formula": "\\begin{align*} \\rho _ { s } ^ { t , \\mu ; u } = \\mathbb { P } _ { X _ { s } ^ { t , \\xi ; u } } ^ { W } . \\end{align*}"}
+{"id": "8669.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { T } } ( V , q ) : = \\{ \\alpha \\in \\mathcal { C } | \\alpha ( \\mathbb { F } \\oplus V ) \\alpha ^ { * } \\subseteq \\mathbb { F } \\oplus V , \\ N ( \\alpha ) \\in \\mathbb { F } \\} . \\end{align*}"}
+{"id": "8832.png", "formula": "\\begin{align*} D u _ \\lambda ( t , x ) & = v _ { 0 , \\lambda } ( t ) + \\int _ 0 ^ t \\ ! \\ ! \\int _ G K _ { t - s } ( x , y ) f ' _ \\lambda ( u _ \\lambda ( s , y ) ) D u _ \\lambda ( s , y ) \\ , d y \\\\ & + \\int _ 0 ^ t \\ ! \\ ! \\int _ G K _ { t - s } ( x , y ) \\sigma ' _ \\lambda ( u _ \\lambda ( s , y ) ) D u _ \\lambda ( s , y ) \\ , \\bar { W } ( d y , d s ) , \\end{align*}"}
+{"id": "4788.png", "formula": "\\begin{align*} \\eta : = \\frac { 1 } { t } \\sqrt { \\log \\left ( \\sqrt { \\frac e 2 } \\ , t ( b - a ) \\right ) } \\end{align*}"}
+{"id": "5189.png", "formula": "\\begin{align*} \\sigma _ r : = \\begin{pmatrix} 1 & 2 & 3 & \\cdots & n - 1 & n \\\\ n & n - 1 & n - 2 & \\cdots & 2 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "476.png", "formula": "\\begin{align*} t ^ { k _ r - k _ { r - 1 } + k _ n - 2 } t _ { 2 r - 1 } t _ { 2 r } t _ { 2 n + 1 } t _ { 2 n + 2 } t _ { 2 n + 3 } t _ { 2 n + 4 } = p q \\end{align*}"}
+{"id": "5326.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ { 2 } ^ { 2 } \\leq \\left ( \\frac { C r } { d } \\right ) ^ { 2 d } \\alpha ^ { 2 } \\log ^ { d } \\left ( 1 / \\alpha \\right ) . \\end{align*}"}
+{"id": "2722.png", "formula": "\\begin{align*} \\begin{aligned} T _ 3 \\ge & - C s ^ 2 \\lambda ^ 2 \\iint _ { Q } \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 u ^ 2 d x d t - C s ^ 2 \\lambda ^ 2 \\int _ { 0 } ^ { T } \\int _ { \\omega } \\xi ^ 3 u ^ 2 d x d t , \\end{aligned} \\end{align*}"}
+{"id": "1308.png", "formula": "\\begin{align*} Y = Y ^ { ( u ) } : t \\mapsto \\big ( X _ i ( T _ i ( u ) + t ) \\big ) _ { i \\in V } \\end{align*}"}
+{"id": "5100.png", "formula": "\\begin{align*} F ( D ^ { 2 } u , D u , u ) & = \\int _ \\Omega \\tilde { g } \\left ( \\det ( u _ { i j } ) \\right ) d x + \\int _ \\Omega \\Tilde { G } ( x ) \\det ( u _ { i j } ) d x \\\\ & + \\int _ \\Omega \\int _ \\Omega W ( x , y ) \\det ( u _ { i j } ( y ) ) \\det ( u _ { i j } ( x ) ) d y d x , \\end{align*}"}
+{"id": "3966.png", "formula": "\\begin{align*} \\begin{aligned} \\boldsymbol { K } _ 1 ( \\widehat { \\gamma } _ { 1 , 3 } , \\mu _ 1 ) & \\leq \\left ( \\frac { \\eta } { \\eta _ 0 } \\right ) \\boldsymbol { K } _ 1 ( \\gamma ^ { \\eta _ 1 , \\delta } _ { 1 , 3 } , \\mu _ 1 ) + \\left ( 1 - \\frac { \\eta } { \\eta _ 0 } \\right ) \\boldsymbol { K } _ 1 ( \\widetilde { \\gamma } _ { 1 , 3 } , \\mu _ 1 ) \\leq \\eta . \\end{aligned} \\end{align*}"}
+{"id": "1266.png", "formula": "\\begin{align*} Z _ 0 ( G ) = M _ 0 ( G ) = T . \\end{align*}"}
+{"id": "4998.png", "formula": "\\begin{align*} \\overline { F } ( k , t + 1 ; p , n ) = \\overline { F } ( k , t ; p , n ) + p _ { k } ^ { } f ( k , t ; p , n ) . \\end{align*}"}
+{"id": "4777.png", "formula": "\\begin{align*} M _ { [ 1 ] } \\diamond \\cdots \\diamond M _ { [ j + 1 ] } & = S _ { [ 1 ] } \\diamond \\cdots \\diamond S _ { [ j + 1 ] } + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "6844.png", "formula": "\\begin{align*} \\sum _ { s _ 1 \\geq \\dots \\geq s _ { r - 1 } \\geq 0 } \\frac { q ^ { s _ 1 ^ 2 + \\dots + s _ { r - 1 } ^ 2 - s _ 1 - \\cdots - s _ i + s _ { r - 1 } } } { ( q ) _ { s _ 1 - s _ 2 } \\dots ( q ) _ { s _ { r - 2 } - s _ { r - 1 } } ( q ^ 2 ; q ^ 2 ) _ { s _ { r - 1 } } } = \\sum _ { k = 0 } ^ { i } \\frac { ( q ^ { 2 r } , q ^ { r - i + 2 k - 1 } , q ^ { r + i - 2 k + 1 } ; q ^ { 2 r } ) _ \\infty } { ( q ) _ \\infty } , \\end{align*}"}
+{"id": "7576.png", "formula": "\\begin{align*} H _ { q , n } ( F _ { f } / 2 ) = \\begin{vmatrix} \\gamma _ { n } & \\gamma _ { n + 1 } & \\cdots & \\gamma _ { n + q - 1 } \\\\ \\gamma _ { n + 1 } & \\gamma _ { n + 2 } & \\cdots & \\gamma _ { n + q } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ \\gamma _ { n + q - 1 } & \\gamma _ { n + q } & \\cdots & \\gamma _ { n + 2 ( q - 1 ) } \\end{vmatrix} . \\end{align*}"}
+{"id": "6515.png", "formula": "\\begin{align*} p ( x ) = M \\mathrm { e } ^ { \\beta ( x - \\mu ) } | x - \\mu | ^ { \\nu } K _ { \\nu } ( \\alpha | x - \\mu | ) , x \\in \\mathbb { R } , \\end{align*}"}
+{"id": "4298.png", "formula": "\\begin{align*} F _ 0 > \\frac { 4 c _ 2 } { c _ 3 } = \\frac { 3 2 \\sigma \\max \\rho _ 0 M ^ 2 } { 3 - \\gamma } . \\end{align*}"}
+{"id": "5011.png", "formula": "\\begin{align*} \\mathbf { w _ { 0 : 7 } } = [ 0 , 0 , 1 , 0 , 0 , 2 , 0 , 0 ] ^ \\top , \\end{align*}"}
+{"id": "4640.png", "formula": "\\begin{align*} \\alpha = \\frac { n q } { n q - n + 2 - \\theta } \\quad \\mbox { a n d } \\beta = \\frac { n q } { n - 2 + \\theta } , \\end{align*}"}
+{"id": "4086.png", "formula": "\\begin{align*} N _ { \\Q ( \\beta ) | \\Q } ( \\gamma ) = \\det ( T _ \\gamma ) \\in \\Q . \\end{align*}"}
+{"id": "2786.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q _ 1 ) w = \\psi ( q _ 2 - q _ 1 ) u _ 2 + [ \\Delta , \\psi ] u . \\end{align*}"}
+{"id": "3126.png", "formula": "\\begin{align*} { n + m \\brack m } _ { q } & = \\sum _ { k = \\ell } ^ n { n - \\ell \\brack k - \\ell } _ { q } { k + m - ( k - \\ell ) \\brack m - ( k - \\ell ) } _ { q } q ^ { k ( k - \\ell ) } \\\\ & = \\sum _ { k = 0 } ^ { n - \\ell } { n - \\ell \\brack k } _ { q } { m + \\ell \\brack m - k } _ { q } q ^ { k ( k + \\ell ) } , \\end{align*}"}
+{"id": "6231.png", "formula": "\\begin{align*} \\psi _ \\infty ( w ) + \\psi _ \\infty ( v ) = \\int _ { \\R ^ 2 \\setminus ( Q _ 1 \\cup Q _ 2 \\cup Q _ 3 ) } g ( b - w ) + g ( b - v ) \\ , d \\mu \\geq - 6 \\eta \\ , . \\end{align*}"}
+{"id": "1846.png", "formula": "\\begin{align*} - 3 F ( 2 \\pi / 3 ) + \\sum _ { i = 1 } ^ { 3 } F ( \\theta _ { i } ) \\leq - \\frac { 3 } { 2 \\pi ^ { 2 } } \\Big ( \\lambda _ { 1 , n } ^ { r , s } + \\frac { 1 } { 2 5 } \\lambda _ { 5 , n } ^ { r , s } \\Big ) . \\end{align*}"}
+{"id": "1564.png", "formula": "\\begin{align*} M = \\begin{pmatrix} a & \\frac { a \\overline { a } - 1 } { c } \\\\ c & \\overline { a } \\end{pmatrix} , \\end{align*}"}
+{"id": "4948.png", "formula": "\\begin{align*} U _ n ^ { - 1 } \\Sigma = \\left [ \\begin{array} { c | c } U _ { n - 1 } ^ { - 1 } & \\mathbf { 0 } \\\\ \\hline \\mathbf { 0 } ^ \\top & 1 \\end{array} \\right ] . \\end{align*}"}
+{"id": "2801.png", "formula": "\\begin{align*} \\| v _ t \\| _ { L ^ 2 ( \\mathrm { M } _ 0 ) } ^ 2 = ( u | ( \\Delta + \\lambda - q ) w _ t ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } . \\end{align*}"}
+{"id": "711.png", "formula": "\\begin{align*} \\tilde \\gamma _ { S } ( e ) = \\partial _ { e } \\Big [ \\phi _ { F } ( z _ { S } ( e ) ) - \\phi _ { A } ( z _ { S } ( e ' ) ) \\Big ] = \\partial _ { e } \\mu _ { S } ( e ) \\tilde \\gamma _ { B } ( e ) = - \\partial _ { e } \\phi _ { F } ( z _ { B } ( e ) ) = \\partial _ { e } \\mu _ { B } ( e ) \\ , , \\end{align*}"}
+{"id": "1899.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { r } { 2 } \\| \\lambda ^ k \\| _ { ( \\tilde { X } , \\| \\cdot \\| _ * ) ' } & \\leq \\langle \\lambda ^ k , r x ^ k \\rangle _ * = - \\langle \\lambda ^ k , S v ^ k - t _ k ( \\zeta ^ k - \\zeta ^ * ) \\rangle _ * < + \\infty . \\end{aligned} \\end{align*}"}
+{"id": "6113.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } w _ t + \\mathcal { L } ^ { \\Phi } _ A w & = 0 \\quad \\mbox { i n } Q _ 3 , \\\\ w & = u \\quad \\mbox { i n } ( \\mathbb { R } ^ n \\setminus B _ 3 ) \\times \\Lambda _ 3 \\cup B _ 3 \\times \\{ - 3 ^ { 2 s } \\} . \\end{array} \\right . \\end{align*}"}
+{"id": "5170.png", "formula": "\\begin{align*} \\mathbb P \\Big ( \\big ( g ^ + _ { \\tau _ 1 } ( X _ 1 ) , \\dots , g ^ + _ { \\tau _ n } ( X _ n ) \\big ) \\in A \\Big ) \\ge \\exp \\Big ( - \\frac { p \\| \\tau \\| ^ 2 } { 2 ( p - 1 ) } \\Big ) \\cdot \\mathbb P \\big ( ( X _ 1 , \\dots , X _ n ) \\in A \\big ) ^ p , \\end{align*}"}
+{"id": "8056.png", "formula": "\\begin{align*} X = A \\cdot \\{ b _ 1 , b _ 2 \\} = ( A \\cdot b _ 1 ) \\cup ( A \\cdot b _ 2 ) . \\end{align*}"}
+{"id": "503.png", "formula": "\\begin{align*} L u ( x ) = f ( x ) , \\forall x \\in G , \\end{align*}"}
+{"id": "8188.png", "formula": "\\begin{align*} P ^ \\omega ( E _ t \\mid S _ t ) = \\frac { P ^ \\omega ( E _ t \\cap S _ t ) } { P ^ \\omega ( S _ t ) } \\leq c _ 1 P ^ \\omega ( E _ t \\cap S _ t ) . \\end{align*}"}
+{"id": "649.png", "formula": "\\begin{align*} & \\limsup _ { j \\to \\infty } \\frac { 1 } { j } \\log \\| S ( j ) \\mathfrak { r } _ r ( f ) \\| _ { \\sup } \\leq - \\lambda _ 1 r r > 0 \\\\ & \\limsup _ { j \\to \\infty } \\frac { 1 } { j } \\log \\| S ( j ) \\mathfrak { r } _ r ( f ) \\| _ { L ^ 1 } \\leq - \\lambda _ 1 r r \\leq 0 . \\end{align*}"}
+{"id": "1602.png", "formula": "\\begin{align*} \\int _ { \\Theta } v ^ { 2 } ( w ) \\| \\chi _ { w } \\pi _ { F ( w ) } ( I _ { U } + L ) ^ { \\ast } \\pi _ { ( L + G ) F ( w ) } f \\| ^ { 2 } d \\mu ( w ) & = \\int _ { \\Theta } v ^ { 2 } ( w ) \\| \\chi _ { w } \\pi _ { F ( w ) } ( I _ { U } + L ) ^ { \\ast } f \\| ^ { 2 } d \\mu ( w ) \\\\ & \\leq B \\| ( I _ { U } + L ) ^ { \\ast } f \\| ^ { 2 } \\\\ & \\leq B \\| I _ { U } + L \\| ^ { 2 } \\| f \\| ^ { 2 } \\end{align*}"}
+{"id": "6038.png", "formula": "\\begin{align*} \\overline { h } _ 2 ( z ) : = \\overline { A _ 2 } z ^ { n + m } + \\overline { B _ 2 } \\overline { z } ^ m + \\overline { C _ 2 } \\textrm { f o r a l l } z \\in \\mathbb { C } . \\end{align*}"}
+{"id": "4963.png", "formula": "\\begin{align*} \\left [ C \\right ] _ { i , j } ^ { } = \\begin{cases} \\dfrac { n } { n } & \\textnormal { i f } j = i = 1 ; \\\\ \\dfrac { j } { n } & \\textnormal { i f } j = i \\neq 0 ; \\\\ \\dfrac { n - j } { n } & \\textnormal { i f } j = i + 1 ; \\\\ 0 & \\textnormal { o t h e r w i s e } ; \\end{cases} \\end{align*}"}
+{"id": "7570.png", "formula": "\\begin{align*} H _ { 2 , 1 } ( F _ { f } / 2 ) : = \\begin{vmatrix} \\gamma _ 1 & \\gamma _ 2 \\\\ \\gamma _ 2 & \\gamma _ 3 \\end{vmatrix} , \\end{align*}"}
+{"id": "5622.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\overline { \\mathbb { P } } _ { \\mu } \\left ( \\exists m \\ge n : \\ L _ { x } \\omega _ { m } \\in A _ { n } ( x , \\zeta ) | ( x , \\zeta ) \\in U \\right ) = 0 . \\end{align*}"}
+{"id": "4305.png", "formula": "\\begin{gather*} \\left ( y _ { \\tau _ 2 } ^ { ( i _ 2 ; L _ 2 ) } \\right ) ^ 2 \\circ y _ { \\tau _ 1 } ^ { ( i _ 1 ; L _ 1 ) } \\circ \\left ( y _ { \\tau _ 2 } ^ { ( i _ 2 ; L _ 2 ) } \\right ) ^ 2 \\circ y _ { \\tau _ 1 } ^ { ( i _ 1 ; L _ 1 ) } = \\mathrm { I d } . \\end{gather*}"}
+{"id": "5187.png", "formula": "\\begin{align*} & \\lambda ^ 0 = ( 1 , 2 ) < \\lambda ^ 1 = ( 1 , 3 ) < \\lambda ^ 2 = ( 1 , 4 ) < \\lambda ^ 3 = ( 1 , 5 ) < \\lambda ^ 4 = ( 2 , 3 ) < \\\\ & \\lambda ^ 5 = ( 2 , 4 ) < \\lambda ^ 6 = ( 2 , 5 ) < \\lambda ^ 7 = ( 3 , 4 ) < \\lambda ^ 8 = ( 3 , 5 ) < \\lambda ^ 9 = ( 4 , 5 ) . \\end{align*}"}
+{"id": "9106.png", "formula": "\\begin{align*} B ^ { k + \\ell } = B ^ { k + \\ell - 1 } \\cup \\{ u \\} \\setminus \\{ v \\} = B ^ { k + \\ell - 2 } \\cup \\{ e , u \\} \\setminus \\{ f , v \\} . \\end{align*}"}
+{"id": "3292.png", "formula": "\\begin{align*} \\widetilde { \\rho } _ t ( t , w ) = A ( 1 - | \\beta _ 1 | ) t ^ { - | \\beta _ 1 | } + B ( 2 - | \\beta _ 1 | ) t ^ { 1 - | \\beta _ 1 | } + | \\beta _ 1 | C t ^ { | \\beta _ 1 | - 1 } | w | ^ 2 \\end{align*}"}
+{"id": "8210.png", "formula": "\\begin{align*} \\sup _ { [ z _ 0 - \\ell ( t ) , z _ 0 + \\ell ( t ) ] ^ d } V ( \\ : \\cdot \\ : , \\omega ) = o ( \\log \\ell ( t ) ) , t \\to \\infty . \\end{align*}"}
+{"id": "4934.png", "formula": "\\begin{align*} P \\mathbf { u } _ 0 = \\lambda _ 0 \\mathbf { u } _ 0 = \\mathbf { 0 } . \\end{align*}"}
+{"id": "7516.png", "formula": "\\begin{align*} \\tilde { \\Lambda } _ i ( z ) = p _ + ^ i ( z ) = q _ + ^ i ( z ) ~ { \\rm f o r } ~ i \\in I _ w \\cup I _ b . \\end{align*}"}
+{"id": "5463.png", "formula": "\\begin{align*} \\kappa = \\frac { M _ 1 M _ 2 - M _ 1 ^ 2 } { M _ 1 ^ 2 - M _ 2 } , \\beta = \\frac { ( 1 - M _ 1 ) ( M _ 2 - M _ 1 ) } { M _ 1 ^ 2 - M _ 2 } , \\end{align*}"}
+{"id": "1449.png", "formula": "\\begin{align*} \\liminf _ { m \\rightarrow \\infty } \\| u _ m \\| = C > 0 . \\end{align*}"}
+{"id": "635.png", "formula": "\\begin{align*} P _ V a | _ V = \\phi ( a ) I , a \\in M \\end{align*}"}
+{"id": "1227.png", "formula": "\\begin{align*} \\| ( f - p _ m ) e ^ { - m V } \\| _ K = \\| u _ m e ^ { - m V } \\| _ K \\leq \\dfrac { C _ { \\varepsilon , \\gamma } e ^ { m \\varepsilon } \\| f \\| _ { X _ { R - \\gamma / 2 } } } { d _ m ^ { 1 / 2 } ( R - \\gamma ) ^ m } . \\end{align*}"}
+{"id": "4683.png", "formula": "\\begin{align*} M ( z ) = \\Lambda + z \\left ( \\Pi ^ * \\Pi \\right ) + z ^ 2 \\left ( \\Pi ^ * ( \\mathcal { A } _ 0 - z I ) ^ { - 1 } \\Pi \\right ) . \\end{align*}"}
+{"id": "6182.png", "formula": "\\begin{align*} f ( t ) = \\left \\{ \\begin{array} { l l } \\frac { t } { | \\log t | } & \\ 0 < t < \\frac 1 e \\\\ \\frac 1 e & \\ t \\ge \\frac 1 e \\end{array} \\right . . \\end{align*}"}
+{"id": "927.png", "formula": "\\begin{align*} \\mathbf { y } _ t = \\beta _ t \\mathbf { x } _ t + \\mathbf { w } _ t , \\end{align*}"}
+{"id": "6933.png", "formula": "\\begin{align*} C : = \\inf _ { \\sigma \\in ( 1 / 2 , 0 . 8 8 ) } \\frac { 1 } { 1 + f ( p ) ^ 2 } = 0 . 2 1 5 3 3 \\dots . \\end{align*}"}
+{"id": "921.png", "formula": "\\begin{align*} & \\alpha ^ - _ { k } \\gamma _ k ^ + = \\frac { 4 ( k + 1 ) ( N - k ) ( 2 \\nu _ { 1 2 } + 2 N - k ) ( 2 \\nu _ { 1 2 } + N - k - 1 ) } { ( 2 \\nu _ { 1 2 } + 2 N - 2 k - 1 ) ^ 2 } , \\\\ & \\alpha ^ + _ { k } \\gamma _ k ^ - = \\frac { ( 2 \\nu _ 1 + N - k ) ( 2 \\nu _ 2 + N - k ) ( 2 \\nu _ 3 + k ) ( 2 \\nu _ { 1 2 3 } + 2 N - k ) } { ( \\nu _ { 1 2 } + N - k ) ^ 2 } , \\\\ & \\beta ^ \\pm _ k = \\pm \\frac { ( \\nu _ 1 - \\nu _ 2 ) ( \\nu _ { 1 2 } + 2 \\nu _ 3 + N ) } { \\nu _ { 1 2 } + N - k } \\mp \\frac { ( 2 \\nu _ { 1 2 } - 1 ) ( 2 \\nu _ { 1 2 } + 2 N + 1 ) } { 2 ( 2 \\nu _ { 1 2 } + 2 N - 2 k \\mp 1 ) } . \\end{align*}"}
+{"id": "7065.png", "formula": "\\begin{align*} \\nu _ i ( g ' ) = \\nu ( g ' ) . \\end{align*}"}
+{"id": "7932.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) u = u \\xi , \\end{align*}"}
+{"id": "6846.png", "formula": "\\begin{align*} \\alpha ^ { ( N ) } _ n = ( 1 - a q ^ { 2 n - N } ) ( a q ^ { 1 - N } ) _ N \\sum _ { j \\in \\mathbb { Z } } ( - 1 ) ^ j \\frac { a ^ j q ^ { ( 2 n - N ) j - j ( j + 1 ) / 2 } } { ( a q ^ { 2 n - N - j } ) _ { N + 1 } } \\left [ { N \\atop j } \\right ] \\alpha _ { n - j } , \\end{align*}"}
+{"id": "7931.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) \\Phi = - \\lambda \\Phi ^ 3 + \\xi - ( 6 c _ { 2 e _ { ( \\xi , 0 ) } + e _ { ( 0 , 2 ) } } + 3 c _ { 4 e _ { ( \\xi , 0 ) } + e _ { ( 0 , 2 ) } + e _ { ( 0 , 3 ) } } ) \\lambda \\Phi , \\end{align*}"}
+{"id": "8120.png", "formula": "\\begin{align*} a = a ( z ) = \\sum _ { n \\ge 0 } a _ n z ^ { n - 1 } . \\end{align*}"}
+{"id": "4497.png", "formula": "\\begin{align*} \\mu _ { \\omega , \\mathbf { c } } : = \\inf \\{ S _ { \\omega , \\mathbf { c } } ( \\Psi ) \\ | \\ \\Psi \\in \\mathcal { H } ^ 1 \\backslash \\{ ( \\mathbf { 0 } , \\mathbf { 0 } , \\mathbf { 0 } ) \\} , \\ K _ { \\omega , \\mathbf { c } } ( \\Psi ) = 0 \\} \\end{align*}"}
+{"id": "6670.png", "formula": "\\begin{align*} \\begin{cases} \\rho \\ , u _ t - \\Delta u + ( - \\Delta ) ^ s u = 0 & \\ , \\ , \\ , S _ T : = \\R ^ N \\times ( 0 , T ] \\\\ u = 0 & \\ , \\ , \\ , \\R ^ N \\times \\{ 0 \\} . \\end{cases} \\end{align*}"}
+{"id": "7098.png", "formula": "\\begin{align*} \\mathcal { L } _ \\varepsilon c ( x ) : = - \\int _ { \\Omega } J _ \\varepsilon ( | x - y | ) c ( y ) \\ : y + \\int _ { \\Omega } J _ \\varepsilon ( | x - y | ) c ( x ) \\ : y \\ ; \\ ; \\ ; x \\in \\Omega . \\end{align*}"}
+{"id": "1049.png", "formula": "\\begin{align*} \\frac { P ( f + T ) } { A ( f + T ) } = \\frac { g ( T ) } { T } + \\frac { g ( T ) Q ' ( T ) } { Q ( T ) } \\ , \\cdot \\end{align*}"}
+{"id": "8967.png", "formula": "\\begin{align*} C _ { \\Tilde { q } } h ^ { \\Tilde { q } } = \\sum _ { m = 1 } ^ { \\infty } C _ { q _ m } h ^ { q _ m } , \\end{align*}"}
+{"id": "2264.png", "formula": "\\begin{align*} h ( x + i y ) = \\frac { 1 } { \\pi } \\langle h _ b , P ( x - \\cdot , y ) \\rangle . \\end{align*}"}
+{"id": "3859.png", "formula": "\\begin{align*} c _ 2 ( ( y _ 2 , x ) , ( y _ 2 , x ' ) ) = \\| x - x ' \\| _ { p } + \\kappa _ 2 \\| y _ 2 - y _ 2 ' \\| _ { p ' } . \\end{align*}"}
+{"id": "8066.png", "formula": "\\begin{align*} Q = \\log _ { 2 } C \\end{align*}"}
+{"id": "918.png", "formula": "\\begin{align*} \\phi ( z ) = P ^ { ( - N - 2 \\nu _ 1 - 1 , - N - 2 \\nu _ 2 ) } _ k \\left ( 1 + 2 z \\right ) . \\end{align*}"}
+{"id": "9085.png", "formula": "\\begin{align*} \\omega ^ + = n - ( k + r - 1 ) , \\ , \\ , \\ , \\ , \\ , \\omega ^ - = k - 1 . \\end{align*}"}
+{"id": "6241.png", "formula": "\\begin{align*} \\omega _ { n , a + 1 } = \\omega _ { n , a } , \\omega _ { n , a + \\tau } = \\sum _ { k = 0 } ^ n { ( 2 \\pi \\mathrm { i } ) ^ { n - k } \\over ( n - k ) ! } \\omega _ { k , a } . \\end{align*}"}
+{"id": "2153.png", "formula": "\\begin{align*} \\Theta ^ { \\{ \\partial W ^ 1 , \\partial W ^ 2 , \\dots , \\partial W ^ { n - 2 } \\} } ( x ) = \\Theta ( x , \\partial W ^ { n - 1 } ) + \\Theta ( x , \\partial W ^ { n } ) \\end{align*}"}
+{"id": "707.png", "formula": "\\begin{align*} \\int ( \\omega + | z _ { S , e } | ^ { - 1 } \\gamma _ { S } \\delta _ { \\Gamma _ { S } } + | z _ { B , e } | ^ { - 1 } \\gamma _ { B } \\delta _ { \\Gamma _ { B } } ) = 0 \\ , , \\end{align*}"}
+{"id": "1866.png", "formula": "\\begin{align*} C ^ \\circ = \\{ v \\in \\mathcal { U } : \\ \\langle v , w \\rangle _ { \\mathcal { U } } \\leq 0 \\forall \\ w \\in C \\} . \\end{align*}"}
+{"id": "5173.png", "formula": "\\begin{align*} q _ e : = \\max \\left \\{ p _ { e ' } \\cdot \\exp \\bigg ( - \\frac { \\| e ' - e \\| } { \\log n } \\bigg ) \\colon e ' \\in E ( \\mathbb Z ^ d ) \\right \\} \\end{align*}"}
+{"id": "7000.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ r b _ j \\textbf { Q } ^ { \\lambda _ j } \\mbox { w i t h $ b _ j \\in K $ a n d } \\lambda _ j ( Q _ k ) = 0 \\mbox { i f } k \\geq i \\end{align*}"}
+{"id": "2948.png", "formula": "\\begin{align*} \\mu _ { K S } ^ \\delta ( f ) ( \\tau ) = \\prod _ { D \\mid N } \\mu _ { K S } ( f ) ( D \\tau ) ^ { n _ D } = \\prod _ { D \\mid N } \\zeta ^ { \\pi _ c ( f ) } \\cdot u _ c ( f ) ( D \\tau ) . \\end{align*}"}
+{"id": "1745.png", "formula": "\\begin{align*} y _ { } & = \\sqrt { P } \\sum _ { n = 1 } ^ N h _ { } ^ { ( n ) } \\exp ( j \\theta _ n ) h _ { } ^ { ( n ) } x + \\epsilon _ { } , \\end{align*}"}
+{"id": "6176.png", "formula": "\\begin{align*} \\psi ( b ) ( t ) = b ( t ) \\xi ( t ) , b \\in F ( \\alpha , u ) , t \\in G , \\end{align*}"}
+{"id": "8342.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| u _ n \\| _ { p ^ * , q , \\mu } = \\| u _ \\mu ^ \\bigstar \\| _ { p ^ * , q , \\mu } . \\end{align*}"}
+{"id": "8482.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\intop _ { 0 } ^ { \\infty } \\ , \\left ( J _ { 0 } ( 2 \\pi x y ) - 1 \\right ) e ^ { - 2 \\pi n y } \\ , d y = - \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { x } { n } \\ , \\intop _ { 0 } ^ { \\infty } e ^ { - 2 \\pi n y } \\ , J _ { 1 } ( 2 \\pi x y ) \\ , d y , \\end{align*}"}
+{"id": "6423.png", "formula": "\\begin{align*} \\widetilde { D } _ \\alpha ( \\psi | | \\varphi ) : = \\frac { 1 } { \\alpha - 1 } \\log \\frac { \\widetilde { Q } _ \\alpha ( \\psi | | \\varphi ) } { \\psi ( 1 ) } , \\end{align*}"}
+{"id": "8715.png", "formula": "\\begin{align*} E \\{ ( U _ { 1 } ^ { \\top } \\Gamma _ { 1 } ^ { \\top } \\Gamma _ { 1 } U _ { 1 } - ( \\Sigma _ { 1 } ) ) ^ { 4 } \\} = O ( \\{ ( \\Gamma _ { 1 } ^ { \\top } \\Gamma _ { 1 } \\Gamma _ { 1 } ^ { \\top } \\Gamma _ { 1 } ) \\} ^ { 2 } ) = O ( ^ { 2 } ( \\Sigma _ { 1 } ^ { 2 } ) ) = O ( p ^ { 2 } ) . \\end{align*}"}
+{"id": "269.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( - \\frac { ( 6 n ^ 4 + 1 2 n ^ 3 - 6 n ^ 2 - 1 2 n + 1 1 ) y ^ { n + 3 } } { ( 1 - y ) ^ 5 } \\right ) \\frac { z ^ n } { n ^ 5 } \\right \\} \\end{align*}"}
+{"id": "8961.png", "formula": "\\begin{align*} e _ { i j } ( x _ o ) & \\equiv \\Gamma _ a e _ { i j } ( x _ a ) + \\Gamma _ b e _ { i j } ( x _ b ) \\\\ & = \\sum _ { m = 1 } ^ { \\infty } \\big [ \\Gamma _ a C _ { q _ m } ( x _ a ) + \\Gamma _ b C _ { q _ m } ( x _ b ) \\big ] h ^ { q _ m } \\big [ r ^ { ( i - j ) q _ m } - 1 \\big ] r ^ { ( j - 1 ) q _ m } \\\\ & = \\sum _ { m = 1 } ^ { \\infty } \\big [ C _ { q _ m } ( x _ a ) + \\mathcal { O } ( h ^ 2 ) \\big ] h ^ { q _ m } \\big [ r ^ { ( i - j ) q _ m } - 1 \\big ] r ^ { ( j - 1 ) q _ m } , \\end{align*}"}
+{"id": "8211.png", "formula": "\\begin{align*} q _ u ( t ) \\geq \\exp \\left [ - \\frac { ( 2 \\ell ( t ) ) ^ 2 } { 2 \\ell ( t ) } ( 1 + o ( 1 ) ) - \\ell ( t ) \\log \\ell ( t ) \\right ] \\geq \\exp \\left [ - 2 \\ell ( t ) \\log \\ell ( t ) \\right ] = : p ( t ) , \\end{align*}"}
+{"id": "3107.png", "formula": "\\begin{align*} q ^ { k ^ 2 } [ 2 ] _ q ^ { k } S _ B [ n , k ] = \\sum _ { \\pi \\in B _ { \\subseteq } ( [ n ] , k ) } q ^ { m ( \\pi ) } \\end{align*}"}
+{"id": "4790.png", "formula": "\\begin{align*} \\bigg \\langle \\nabla V ( x ) , f ( x ) + \\sum _ { i = 1 } ^ m g _ i ( x ) u _ i \\bigg \\rangle < 0 , \\forall x \\in \\mathcal { D } \\setminus \\{ x _ * \\} . \\end{align*}"}
+{"id": "8000.png", "formula": "\\begin{align*} \\int _ \\Omega f \\ , d x = 0 . \\end{align*}"}
+{"id": "8776.png", "formula": "\\begin{align*} \\partial \\left ( \\sum _ { i = 1 } ^ n c p _ i \\max \\{ - x _ i - y _ i , - d _ i \\} \\right ) & = \\sum _ { i = 1 } ^ n c p _ i \\partial \\left ( \\max \\{ 0 , - x _ i - y _ i + d _ i \\} \\right ) \\\\ & = \\sum _ { i = 1 } ^ n c p _ i \\Gamma ( - x _ i - y _ i + d _ i ) \\nabla _ v ( - x _ i ) \\\\ & = \\sum _ { i = 1 } ^ n c p _ i \\Gamma ( - x _ i - y _ i + d _ i ) ( e _ { i \\bullet } - e _ { \\bullet i } ) \\end{align*}"}
+{"id": "2491.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle ( 1 + 2 \\Lambda ) \\dfrac { a _ 1 } { 4 m _ 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\eta | ^ 2 d x + \\dfrac { a _ 1 } { 2 } \\omega _ 2 \\int _ { \\mathbb { R } ^ N } \\eta ^ 2 d x - \\dfrac { 1 } { 2 } \\left ( 1 + \\dfrac { N } { 2 } \\Lambda \\right ) a _ 1 a _ 2 \\int _ { \\mathbb { R } ^ N } \\xi ^ 2 \\eta d x = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "3406.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u _ j = f ( u _ j ) + \\lambda u _ j \\ \\ \\ \\ \\mathbb R ^ N , \\\\ u _ j ( x ) > 0 , \\ \\lim _ { | x | \\to \\infty } u _ j ( x ) = 0 , i = 1 , 2 , \\cdots , \\ell \\\\ \\sum _ { i = 1 } ^ { \\ell } | u _ j | _ 2 ^ 2 = \\alpha . \\end{cases} \\end{align*}"}
+{"id": "7572.png", "formula": "\\begin{align*} f ( z ) = z + \\sum _ { n = 2 } ^ { \\infty } a _ n z ^ n , \\ ; \\mbox { f o r } \\ ; z \\in \\mathbb { D } . \\end{align*}"}
+{"id": "8600.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } e ^ { - \\lambda ( \\tau _ N - t _ N ) } \\cdot N ^ { - 1 } S _ j ( \\tau _ N ) & = Y ^ { - 1 } \\lim _ { N \\to \\infty } e ^ { - \\lambda \\tau _ N } S _ j ( \\tau _ N ) \\\\ & = \\nu \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s , \\end{align*}"}
+{"id": "3438.png", "formula": "\\begin{align*} 0 = - | A | ^ 2 + 4 c ^ 2 \\left ( \\lambda + \\frac { c ^ 2 } { m } \\right ) , \\end{align*}"}
+{"id": "4031.png", "formula": "\\begin{align*} \\big \\langle z ^ m \\lbrace 0 , \\infty \\rbrace , f \\big \\rangle = \\frac { m ! \\ , i ^ { m + 1 } } { ( 2 \\pi ) ^ { m + 1 } } \\ , L ( f , m + 1 ) \\ , \\ \\ , \\ 0 \\leq m \\leq k - 2 . \\end{align*}"}
+{"id": "3119.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } S _ B [ n , k ] ( t ) _ { k , q } ^ D & = \\sum _ { k = 0 } ^ { n - 1 } S _ B [ n , k ] ( t ) _ { k , q } ^ D + S _ B [ n , n ] ( t ) _ { n , q } ^ D - [ n ] _ { q } \\ , q ^ { n - 1 } ( t ) _ { n - 1 , q } ^ D + [ n ] _ { q } \\ , q ^ { n - 1 } ( t ) _ { n - 1 , q } ^ D \\\\ & = \\sum _ { k = 0 } ^ { n } S _ B [ n , k ] ( t ) _ { k , q } ^ B + [ n ] _ { q } \\ , q ^ { n - 1 } ( t ) _ { n - 1 , q } ^ D \\\\ & = t ^ n + [ n ] _ { q } \\ , q ^ { n - 1 } ( t ) _ { n - 1 , q } ^ D , \\end{align*}"}
+{"id": "2094.png", "formula": "\\begin{align*} R ( a - 1 ) & = \\Big \\{ ( x _ 1 , x _ 2 , . . . , x _ { n + 1 } ) \\mid ( x _ n , x _ { n + 1 } ) = ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) ; \\\\ & \\ \\ \\ 0 \\leq x _ i \\leq 2 \\ \\ \\ \\ 1 \\leq i \\leq n - 1 ; \\ \\ \\ \\ x _ i = 2 , \\ \\ \\ \\ x _ j = 0 \\ \\ \\ \\ j \\leq i - 1 \\Big \\} \\\\ & \\ \\ \\ \\bigcup \\Big \\{ ( 0 , 0 , . . . , 0 , x _ { n } , x _ { n + 1 } ) \\mid ( x _ n , x _ { n + 1 } ) = ( 2 , 0 ) , ( 1 , 1 ) \\Big \\} . \\end{align*}"}
+{"id": "956.png", "formula": "\\begin{align*} \\| u \\mid L ^ 1 _ p ( D \\| = \\| \\nabla u \\mid L _ p ( D ) \\| . \\end{align*}"}
+{"id": "407.png", "formula": "\\begin{align*} P U _ t + \\frac { 1 } { 2 } \\left [ ( A U ) _ x + A U _ x + ( B U ) _ y + B U _ y \\right ] = 0 . \\end{align*}"}
+{"id": "2021.png", "formula": "\\begin{align*} G ( z ) : = \\log \\beta ( z ) + u ^ 2 \\slash 2 + B \\vert z \\vert ^ 2 \\end{align*}"}
+{"id": "7942.png", "formula": "\\begin{align*} ( \\partial _ t - \\partial _ x ^ 2 ) \\Pi _ { 2 e _ { ( \\xi , 0 ) } + e _ { ( 0 , 2 e _ { ( 0 , 1 ) } ) } } = ( \\partial _ x \\Pi _ { e _ { ( \\xi , 0 ) } } ) ^ 2 + c _ { 2 e _ { ( \\xi , 0 ) } + e _ { ( 0 , 2 e _ { ( 0 , 1 ) } ) } } , \\end{align*}"}
+{"id": "4415.png", "formula": "\\begin{align*} K _ { \\omega , \\mathbf { c } } ( V _ n - V ) & = - K _ { \\omega , \\mathbf { c } } ( V ) + K _ { \\omega , \\mathbf { c } } ( V _ n ) - ( K _ { \\omega , \\mathbf { c } } ( V _ n ) - K _ { \\omega , \\mathbf { c } } ( V _ n - V ) - K _ { \\omega , \\mathbf { c } } ( V ) ) \\\\ & \\rightarrow - K _ { \\omega , \\mathbf { c } } ( V ) < 0 . \\end{align*}"}
+{"id": "4773.png", "formula": "\\begin{align*} W _ { [ j + 1 ] } = M _ { [ j + 1 ] } + \\mathcal { O } ( \\| H \\| ^ 2 ) , \\end{align*}"}
+{"id": "6172.png", "formula": "\\begin{align*} s _ n ( G ) \\left ( \\sum ^ { s - 1 } _ { i = 0 } l ( g _ { 1 , i } ) \\otimes . . . \\otimes l ( g _ { n , i } ) + \\mathcal Q _ n ( G ) \\right ) : = \\overline \\bigotimes ^ { s - 1 } _ { i = 0 } [ \\langle 1 , - g _ { 1 , i } \\rangle ] \\overline \\otimes . . . \\overline \\otimes [ \\langle 1 , - g _ { n , i } \\rangle ] \\overline \\otimes I ^ { n + 1 } ( G ) . \\end{align*}"}
+{"id": "8880.png", "formula": "\\begin{align*} \\mathbb { E } [ H ( G ) ] = \\frac { 1 } { 2 n } \\sum _ { \\pi \\in \\L } p ^ { f ( \\pi ) } \\ ; . \\end{align*}"}
+{"id": "6676.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( \\varphi ) & = \\frac { C _ { N , s } } { 2 } \\iint _ { \\R ^ { 2 N } } \\frac { ( u ( x ) - u ( y ) ) ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { N + 2 s } } \\ , d x \\ , d y . \\end{align*}"}
+{"id": "8722.png", "formula": "\\begin{align*} ( I _ 2 ) = \\frac { 4 } { n } \\{ f ^ { ( s ) } ( \\tau _ 1 ) E ( \\widetilde { X } _ { 1 , 0 } ^ s \\mid X _ 1 ) - f ^ { ( s ) } ( \\tau _ 3 ) E ( \\widetilde { Z } _ { 1 , 0 } ^ s \\mid X _ 1 ) \\} . \\end{align*}"}
+{"id": "4221.png", "formula": "\\begin{align*} E _ 1 & = \\det T ( \\phi ) T ( \\phi _ R ) ^ { - 1 } \\cdots T ( \\phi _ 0 ) ^ { - 1 } , \\\\ E _ 2 & = \\det T ( \\tilde { \\phi } ) T ( \\tilde { \\phi } _ R ) ^ { - 1 } \\cdots T ( \\tilde { \\phi } _ 0 ) ^ { - 1 } , \\end{align*}"}
+{"id": "2810.png", "formula": "\\begin{align*} a _ { i + q , i } = \\frac { \\Delta _ { i + q } \\Delta _ { i - 1 } } { \\Delta _ { i + q - 1 } \\Delta _ { i } } , i \\ge 0 ; \\end{align*}"}
+{"id": "905.png", "formula": "\\begin{align*} d = h ' h _ 1 h _ 2 ^ 2 \\cdots h _ k ^ k h _ { k + 1 } , ( h ' , h _ i ) = ( h _ i , h _ j ) = 1 , h ' \\mid ( N \\Delta _ { F _ 2 } ) ^ \\infty i \\neq j , \\end{align*}"}
+{"id": "4464.png", "formula": "\\begin{align*} \\displaystyle \\limsup _ { n \\rightarrow \\infty } G ( U _ n ( t _ n ) ) = \\displaystyle \\limsup _ { n \\rightarrow \\infty } G ( \\Phi _ { \\omega , \\mathbf { c } , n } ) \\ge \\eta . \\end{align*}"}
+{"id": "6484.png", "formula": "\\begin{align*} X Y Z = i \\left ( \\begin{array} { c c } A & 0 \\\\ 0 & - A \\end{array} \\right ) . \\end{align*}"}
+{"id": "6419.png", "formula": "\\begin{align*} | x _ 1 \\bar { \\otimes } x _ 2 | ^ p = | x _ 1 | ^ p \\bar { \\otimes } | x _ 2 | ^ p = h _ { \\psi _ 1 } \\bar { \\otimes } h _ { \\psi _ 2 } = h _ { \\psi _ 1 \\bar { \\otimes } \\psi _ 2 } . \\end{align*}"}
+{"id": "980.png", "formula": "\\begin{align*} F \\xi = B \\xi + C \\xi \\end{align*}"}
+{"id": "1008.png", "formula": "\\begin{align*} \\psi ^ { ( n ) } ( x + 1 ) = \\psi ^ { ( n ) } ( x ) + \\frac { ( - 1 ) ^ n n ! } { x ^ { n + 1 } } . \\end{align*}"}
+{"id": "2666.png", "formula": "\\begin{align*} b _ { j } = & \\left ( \\sinh ( 2 \\alpha x _ j ) - 2 \\alpha h _ j - 2 \\tanh ( \\alpha x _ { j - 1 } ) ( \\cosh ^ 2 ( \\alpha x _ j ) - \\alpha ^ 2 h _ { j } ^ 2 ) \\right ) / F _ { j - 1 } \\\\ & - \\left ( \\sinh ( 2 \\alpha x _ j ) + 2 \\alpha h _ { j + 1 } - 2 \\tanh ( \\alpha x _ { j + 1 } ) ( \\cosh ^ 2 ( \\alpha x _ j ) - \\alpha ^ 2 h _ { j + 1 } ^ 2 ) \\right ) / F _ { j } \\\\ = & \\dfrac { h _ { j } + h _ { j + 1 } } { 3 } + O \\left ( ( \\alpha \\overline { h } ) ^ 2 \\right ) ; \\alpha \\to 0 . \\end{align*}"}
+{"id": "8731.png", "formula": "\\begin{align*} & f ^ { ( 2 ) } ( \\tau _ 1 ) E ( \\widetilde { X } _ { 1 , 2 } ^ 2 \\mid X _ 1 ) - f ^ { ( 2 ) } ( \\tau _ 3 ) E ( \\widetilde { Z } _ { 1 , 1 } ^ 2 \\mid X _ 1 ) \\\\ & = p ^ { - 2 } f ^ { ( 2 ) } ( \\tau ) \\left \\{ E ( \\| X _ 1 - X _ 2 \\| _ 2 ^ 4 \\mid X _ 1 ) - E ( \\| X _ 1 - Y _ 1 \\| _ 2 ^ 4 \\mid X _ 1 ) \\right \\} \\\\ & = - p ^ { - 2 } f ^ { ( 2 ) } ( \\tau ) \\big \\{ 4 E ( \\| X _ 2 \\| _ 2 ^ 2 X _ 2 ^ \\top - \\| Y _ 1 \\| _ 2 ^ 2 Y _ 1 ^ \\top ) X _ 1 + E ( \\| X _ 2 \\| _ { 2 } ^ { 4 } - \\| Y _ 1 \\| _ { 2 } ^ { 4 } ) \\big \\} . \\end{align*}"}
+{"id": "7543.png", "formula": "\\begin{align*} B _ j = B _ { \\rho _ j } ( X ) , \\rho _ j = \\left ( \\frac { \\sigma } { 2 } \\right ) ^ { j - 1 } \\rho . \\end{align*}"}
+{"id": "8632.png", "formula": "\\begin{align*} \\hat { M } _ + ( t ) : = \\nu \\int _ 0 ^ t Y e ^ { \\lambda s } d s \\end{align*}"}
+{"id": "3329.png", "formula": "\\begin{align*} w \\rho _ { w } ( j , w ) = | w | ^ 2 - 2 i \\frac { R _ j } { | w | } { \\rm I m } ( w ( R _ j ) _ { z } ) \\end{align*}"}
+{"id": "6299.png", "formula": "\\begin{align*} \\langle \\lambda _ 0 , w \\rangle = 0 , \\forall \\ , w \\in T _ { \\gamma ( 0 ) } \\Sigma . \\end{align*}"}
+{"id": "381.png", "formula": "\\begin{align*} L ( V ) U = G ( \\vec x , t ) , t \\geq 0 , \\vec x = ( x _ 1 , x _ 2 , . . , x _ k ) \\in \\partial \\Omega . \\end{align*}"}
+{"id": "7470.png", "formula": "\\begin{align*} d : = \\left \\{ \\begin{array} { l l } \\frac { p } { 2 } & \\ p \\geq 2 , \\\\ \\frac { 2 p } { p ( n + 2 ) - 2 n } & \\ p \\leq 2 , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "7084.png", "formula": "\\begin{align*} \\alpha _ n = - \\frac { v ( a ) } { p ^ n } \\mbox { a n d } \\beta _ n = - \\frac { v ( a ) } { p ^ { n - 1 } } . \\end{align*}"}
+{"id": "1207.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty s ^ { k - 1 } R _ k ( t ) = \\sum _ j \\nu ( j ) \\sum _ { k = 1 } ^ \\infty s ^ { k - 1 } M _ j ( \\boldsymbol { a } , t ) ^ k = G ( t , s ) . \\end{align*}"}
+{"id": "2587.png", "formula": "\\begin{align*} B _ { \\mathrm { m } } \\ , ( 1 ; 0 ; 0 , 0 ; 0 , \\ldots ) ^ { \\mathrm { T } } & = ( 0 ; 1 ; 0 , 0 ; 0 , \\ldots ) ^ { \\mathrm { T } } , \\\\ B _ { \\mathrm { m } } ^ 2 ( 1 ; 0 ; 0 , 0 ; 0 , \\ldots ) ^ { \\mathrm { T } } & = ( 0 ; 0 ; 1 , 1 ; 0 , \\ldots ) ^ { \\mathrm { T } } , \\\\ B _ { \\mathrm { m } } ^ 3 ( 1 ; 0 ; 0 , 0 ; 0 , \\ldots ) ^ { \\mathrm { T } } & = ( 0 ; 0 ; 0 , 0 ; 1 , 1 , 2 , 1 , 1 ; 0 , \\ldots ) ^ { \\mathrm { T } } , \\end{align*}"}
+{"id": "3023.png", "formula": "\\begin{align*} { \\phi } ( X E _ { i i } X ^ * \\otimes B ) = ( Z _ X E _ { i i } Z _ X ^ * ) \\otimes \\varphi _ { i , X } ( B ) \\hbox { f o r a l l \\quad } i = 1 , \\ldots , m \\hbox { a n d } B \\in M _ n . \\end{align*}"}
+{"id": "7655.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ g u = - \\lambda _ u , & \\Omega , \\\\ u = 0 , & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "2529.png", "formula": "\\begin{align*} t \\cdot [ x _ { i , j } ] = [ t _ i t _ j x _ { i , j } ] , \\ ; t _ i = \\begin{cases} \\mu _ i & i \\in [ n ] , \\\\ \\mu _ { i ^ * } & i \\in [ n ^ * ] . \\end{cases} \\end{align*}"}
+{"id": "3282.png", "formula": "\\begin{align*} + t ^ s { \\rm R e } ( D \\kappa ( t ) ^ 2 ) + t { \\rm R e } ( E \\kappa ( t ) ) + \\sqrt { t ^ { 2 - 2 s } + | \\kappa ( t ) | ^ 2 } \\ , g ( t , t ^ s \\kappa ( t ) ) = 0 . \\end{align*}"}
+{"id": "8689.png", "formula": "\\begin{align*} | \\textup { v a r } \\{ f ^ { ( 2 ) } ( \\tau _ 1 ) E ( \\widetilde { X } _ { 1 , 2 } ^ 2 \\mid X _ 1 ) - f ^ { ( 2 ) } ( \\tau _ 3 ) E ( \\widetilde { Z } _ { 1 , 1 } ^ 2 \\mid X _ 1 ) \\} | = O \\bigg ( p ^ { - 4 } \\sum _ { k = 1 } ^ { q } \\{ s _ { k k } ( \\mu _ { k , 3 } ^ { ( 1 ) } - \\mu _ { k , 3 } ^ { ( 2 ) } ) \\} ^ 2 \\bigg ) , \\end{align*}"}
+{"id": "1926.png", "formula": "\\begin{align*} P ( z _ 1 , \\ldots , z _ m , f ( z _ 1 , \\ldots , z _ m ) ) = 0 , \\ \\forall ( z _ 1 , \\ldots , z _ m ) \\in \\Omega , \\end{align*}"}
+{"id": "1802.png", "formula": "\\begin{align*} w _ m ( p ) \\equiv \\begin{cases} \\ < p \\ > ^ m \\ , & p \\in \\S \\\\ 1 \\ , & p \\in \\Lambda ^ * \\backslash \\S \\end{cases} \\ . \\end{align*}"}
+{"id": "257.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { y L i _ 2 ( z ) } { ( 1 - y ) ^ 2 } - \\frac { y L i _ 2 ( y z ) } { ( 1 - y ) ^ 2 } - \\frac { y } { ( 1 - y ) ^ 2 } \\log \\left ( \\frac { 1 } { 1 - y z } \\right ) + \\frac { y ^ 2 } { ( 1 - y ) ^ 2 } \\log \\left ( \\frac { 1 } { 1 - y z } \\right ) \\right \\} \\end{align*}"}
+{"id": "9341.png", "formula": "\\begin{align*} \\frac { d \\bar { \\mathbb { P } } } { d \\mathbb { P } } \\bigg | _ { \\mathcal { F } _ t } = \\mathcal { Z } _ t , \\end{align*}"}
+{"id": "82.png", "formula": "\\begin{align*} { \\rm t r } ^ \\flat ( P ( h ) ) = \\mathcal { O } ( h ^ { - 2 n - m } ) . \\end{align*}"}
+{"id": "5323.png", "formula": "\\begin{align*} r \\gtrsim t ^ { 1 / s } \\frac { 1 } { p } \\approx ( n p ) ^ { p / \\log n } \\frac { 1 } { p } = e ^ { p ( 1 - \\frac { \\log ( 1 / p ) } { \\log ( n ) } ) } \\frac { 1 } { p } \\approx \\frac { 1 } { p } . \\end{align*}"}
+{"id": "7452.png", "formula": "\\begin{align*} \\Psi _ { p \\alpha + k } ^ { ( i ) } ( \\xi _ i , t ) = \\sum \\limits _ { j = 1 } ^ { k + 1 } \\frac { \\xi _ i ^ j } { j ! } \\ , \\dfrac { \\partial ^ j w _ { p \\alpha + k - j } ^ { ( i ) } } { \\partial x _ i ^ j } ( 0 , t ) i \\in \\{ 1 , 2 , 3 \\} . \\end{align*}"}
+{"id": "6780.png", "formula": "\\begin{align*} ( q ) _ \\lambda = \\prod _ { i = 0 } ^ n \\prod _ { j = 1 } ^ { \\lambda _ i - \\delta _ { i , \\sigma } } \\left ( 1 - q ^ j \\right ) \\end{align*}"}
+{"id": "3833.png", "formula": "\\begin{align*} \\Psi _ 1 \\left ( \\boldsymbol { d } _ { \\mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \\prime ) ^ p , \\rho _ 2 ( y _ 1 , y _ 1 ^ \\prime ) ^ { p } \\right ) = \\Psi _ 1 \\circ \\psi _ 1 \\left ( \\boldsymbol { d } _ { \\mathcal { Y } _ 1 } ( y _ 1 , y _ 1 ^ \\prime ) ^ { p } , \\boldsymbol { d } _ { \\mathcal { Y } _ 2 } ( y _ 2 , y _ 2 ^ \\prime ) ^ p , \\boldsymbol { d } _ { \\mathcal { X } } ( x , x ^ \\prime ) ^ p \\right ) , \\end{align*}"}
+{"id": "7417.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\ , \\sum _ { a N < n \\le b N } \\ , \\frac { q _ { 2 n } ^ { \\varphi _ N , \\varphi _ N } } { N ^ 2 } \\ , = \\ ! \\ ! \\int \\limits _ { \\R ^ 2 \\times \\R ^ 2 } \\varphi ( x ) \\ , \\varphi ( x ' ) \\ , \\bigg ( \\int _ { a } ^ { b } \\frac { 1 } { u } \\ , g \\bigg ( \\frac { x - x ' } { \\sqrt { u } } \\bigg ) \\ , \\mathrm { d } u \\bigg ) \\ , \\mathrm { d } x \\ , \\mathrm { d } x ' \\ , , \\end{align*}"}
+{"id": "1574.png", "formula": "\\begin{align*} j _ m : = \\max \\{ j \\in \\{ 0 , \\dotsc , n \\} | \\ | w _ { j } | < \\ell \\} . \\end{align*}"}
+{"id": "2071.png", "formula": "\\begin{align*} \\frac { x } { y } = t = \\frac { 1 } { 2 } \\left ( \\alpha - \\sqrt { \\alpha ^ 2 - 4 } \\right ) \\end{align*}"}
+{"id": "7952.png", "formula": "\\begin{align*} & B ( t ) = t ^ 3 ( 1 + ( \\ln t ) ^ 2 ) ^ { - \\frac 1 2 } \\exp ( \\ln t \\arctan ( \\ln t ) ) ) ; \\\\ & B ( t ) = t ^ { 4 + \\sin \\sqrt { 1 + ( \\ln t ) ^ 2 } } . \\end{align*}"}
+{"id": "5849.png", "formula": "\\begin{align*} \\chi _ { I , \\pm } ( H _ \\omega ) : = i _ - \\chi _ I ( H _ \\omega ) i _ + \\in \\mathcal B ( \\mathcal H _ + , \\mathcal H _ - ) , \\end{align*}"}
+{"id": "429.png", "formula": "\\begin{align*} ( \\vec U , { \\bf D _ { x _ i } } \\vec V ) = \\vec U ^ T ( I _ n \\otimes P _ { \\Omega } ) ( { \\bf D _ { x _ i } } \\vec V ) = - ( { \\bf D _ { x _ i } } \\vec U , \\vec V ) + ( E \\vec U ) ^ T ( I _ n \\otimes P _ { \\partial \\Omega } ) N _ { i } ( E \\vec V ) . \\end{align*}"}
+{"id": "702.png", "formula": "\\begin{align*} \\widehat { \\xi } _ { [ ( \\sigma , k , l ) ] } ( s ) & = \\frac { ( s - l _ \\alpha ) ^ { \\frac { k - ( m _ \\sigma - 2 ) } { m _ \\sigma } } } { m _ \\sigma ^ 2 k ! } k \\neq m _ \\sigma - 2 \\ \\operatorname { m o d } m _ \\sigma \\\\ \\widehat { \\xi } _ { [ ( \\sigma , k , l ) ] } ( s ) & = - \\frac { ( s - l _ \\alpha ) ^ { \\frac { k - ( m _ \\sigma - 2 ) } { m _ \\sigma } } \\log ( s - l _ \\alpha ) } { m _ \\sigma ^ 2 k ! } k = m _ \\sigma - 2 \\ \\operatorname { m o d } m _ \\sigma ; \\end{align*}"}
+{"id": "8712.png", "formula": "\\begin{align*} E ( \\| A ^ { \\top } U \\| _ { 2 } ^ { L } ) = E \\{ ( U ^ { \\top } A A ^ { \\top } U ) ^ { L / 2 } \\} = O ( \\{ \\textup { t r } ( A A ^ { \\top } ) \\} ^ { L / 2 } ) . \\end{align*}"}
+{"id": "6236.png", "formula": "\\begin{align*} E _ r ' \\ , = \\ , - r \\ , E _ { r + 1 } \\ , . \\end{align*}"}
+{"id": "5802.png", "formula": "\\begin{align*} \\omega \\left [ 2 \\pi \\iota n \\left ( r _ n , \\frac { 1 } { f } \\right ) + \\frac { k - 1 } { 2 k } 2 \\pi \\iota n \\left ( r _ n , \\frac { 1 } { A } \\right ) \\right ] + o ( 1 ) = 2 \\pi \\iota n \\left ( r _ n , \\frac { 1 } { y } \\right ) + \\frac { k - 1 } { 2 k } 2 \\pi \\iota n \\left ( r _ n , \\frac { 1 } { A } \\right ) . \\end{align*}"}
+{"id": "190.png", "formula": "\\begin{align*} L i _ 3 ( \\phi ^ { - 2 } ) = \\frac { 4 } { 5 } \\zeta ( 3 ) - \\frac { 2 } { 1 5 } \\pi ^ 2 \\log \\phi + \\frac { 2 } { 3 } ( \\log \\phi ) ^ 3 , \\end{align*}"}
+{"id": "7590.png", "formula": "\\begin{align*} H _ { 2 , 1 } ( F _ { f } / 2 ) & = \\frac { 1 } { 1 9 2 } \\left ( - 3 \\tau ^ 4 _ { 1 } + 4 ( 1 - \\tau ^ 2 _ { 1 } ) \\tau ^ 2 _ { 1 } \\tau _ { 2 } - 4 ( 1 - \\tau ^ 2 _ { 1 } ) ( 3 + \\tau ^ 2 _ { 1 } ) \\tau ^ 2 _ { 2 } \\right . \\\\ & \\left . + 1 6 \\tau _ { 1 } \\tau _ { 3 } ( 1 - \\tau ^ 2 _ { 1 } ) ( 1 - | \\tau ^ 2 _ { 2 } | ) \\right ) . \\end{align*}"}
+{"id": "651.png", "formula": "\\begin{align*} Q ( k , l ) = Z ( l ) \\cdot Z ( l - 1 ) \\cdot \\dotsc \\cdot Z ( k + 2 ) \\cdot Z ( k + 1 ) = A ^ { ( n _ { l } - n _ { k } ) } ( \\mathcal { R } ^ { n _ k } ( T ) ) ^ t . \\end{align*}"}
+{"id": "4386.png", "formula": "\\begin{align*} \\bar u _ s = ( d _ 0 + ( s - s _ 1 ) \\alpha s _ 1 ^ { \\alpha - 1 } ) \\mathbf { 1 } _ { s ^ \\alpha < d _ 0 } + ( d _ 0 - ( s - s _ 2 ) \\alpha ( t - s _ 2 ) ^ { \\alpha - 1 } ) \\mathbf { 1 } _ { ( t - s ) ^ \\alpha < d _ 0 } + \\min ( s , t - s ) ^ \\alpha \\mathbf { 1 } _ { s _ 1 < s < s _ 2 } . \\end{align*}"}
+{"id": "7666.png", "formula": "\\begin{align*} f ( a , b ) : = ( p - 1 ) \\left ( 2 a b \\kappa ( n - 1 ) - a ^ { \\frac { p } { p - 1 } } - a b ^ 2 p \\right ) \\ge C . \\end{align*}"}
+{"id": "534.png", "formula": "\\begin{align*} \\left ( \\mathcal { F } _ { \\mathcal { H } _ { \\hbar , V } } f \\right ) ( \\xi ) = \\widehat { f } ( \\xi ) : = \\sum \\limits _ { k \\in \\hbar \\mathbb { Z } ^ { n } } f ( k ) \\overline { u _ { \\xi } ( k ) } , \\xi \\in \\mathcal { I } _ { \\hbar } , \\end{align*}"}
+{"id": "4111.png", "formula": "\\begin{align*} Q _ { \\pi ( 1 ) , \\vec { v _ 2 } } = \\left ( P \\left ( Q _ { 1 , \\vec { v } } \\right ) _ 1 P ^ { - 1 } , P \\left ( Q _ { 1 , \\vec { v } } \\right ) _ 2 P ^ { - 1 } , P \\left ( Q _ { 1 , \\vec { v } } \\right ) _ 3 P ^ { - 1 } \\right ) P ^ T . \\end{align*}"}
+{"id": "6535.png", "formula": "\\begin{align*} \\int x ^ \\mu K _ \\nu ( x ) \\ , \\mathrm { d } x = - 2 ^ { \\mu - 1 } \\Gamma \\bigg ( \\frac { \\mu - \\nu + 1 } { 2 } \\bigg ) \\Gamma \\bigg ( \\frac { \\mu + \\nu + 1 } { 2 } \\bigg ) G _ { \\mu , \\nu } ( x ) , \\end{align*}"}
+{"id": "1345.png", "formula": "\\begin{align*} F _ { N } ( t , x ) \\coloneqq - \\frac { 1 } { 2 } + \\frac { 1 } { N } \\stackrel [ k = 1 ] { N } { \\sum } m _ { k } ( t ) H ( x - x _ { k } ( t ) ) , \\end{align*}"}
+{"id": "5152.png", "formula": "\\begin{align*} X ( w ) ' = X ' ( w ^ { - 1 } ) . \\end{align*}"}
+{"id": "1104.png", "formula": "\\begin{align*} \\langle f , g \\rangle = - f \\circ g . \\end{align*}"}
+{"id": "2365.png", "formula": "\\begin{align*} \\left ( ( \\mathcal G : ) \\times ( u : \\mathcal G ^ { L } ) \\times ( i , j : L ) \\to u _ i = u _ j \\right ) \\to \\rlap { . } \\end{align*}"}
+{"id": "311.png", "formula": "\\begin{align*} \\Gamma ( x ) : = \\int _ { 0 } ^ { \\infty } e ^ { - t } t ^ { x - 1 } d t , \\Psi ( x ) : = \\frac { \\Gamma ' ( x ) } { \\Gamma ( x ) } . \\end{align*}"}
+{"id": "2913.png", "formula": "\\begin{align*} \\gamma \\ast m ^ { \\delta } = \\sum _ { g } n _ g \\cdot ( \\gamma g ) \\ast m = \\sum _ g n _ g \\cdot ( g \\gamma _ g ) \\ast m = \\sum _ g n _ g \\cdot g \\ast m = m ^ \\delta . \\end{align*}"}
+{"id": "3466.png", "formula": "\\begin{align*} \\mathsf { F l } _ { 1 2 } = \\{ L _ 1 \\subset L _ 2 \\subset \\C ^ 4 | \\dim L _ i = i \\} , \\ \\ \\mathsf { F l } _ { 1 2 3 } = \\{ L _ 1 \\subset L _ 2 \\subset L _ 3 \\subset \\C ^ 4 | \\dim L _ i = i \\} \\end{align*}"}
+{"id": "2211.png", "formula": "\\begin{align*} \\Phi \\left ( x , x ^ { 1 / 3 } \\right ) \\ge \\frac { x } { \\log x ^ { 1 / 3 } } \\left ( \\omega \\left ( \\frac { \\log x } { \\log x ^ { 1 / 3 } } \\right ) + \\frac { \\Delta _ 3 ^ - } { \\log x ^ { 1 / 3 } } \\right ) = \\frac { 3 x } { \\log y } \\left ( \\frac { \\omega ( 3 ) } { u } + \\frac { 3 \\Delta _ 3 ^ - } { u ^ 2 \\log y } \\right ) . \\end{align*}"}
+{"id": "4105.png", "formula": "\\begin{align*} \\left ( T _ \\lambda ^ { \\vec { v } } \\right ) _ { i , j } = \\displaystyle \\beta _ { - i + j } + \\sum _ { k = j } ^ { n - 1 } \\beta _ k \\sum _ { r = 0 } ^ { \\min \\{ k - j , n - i \\} } \\alpha _ { n - j - r } \\sum _ { \\mathclap { \\substack { p _ 1 \\dots p _ m \\\\ m \\leq k - j - r \\\\ p _ s \\geq 1 \\\\ \\sum _ { s = 1 } ^ m p _ s = k - j - r } } } ( - 1 ) ^ { m + 1 } \\alpha _ { n - p _ 1 } \\dots \\alpha _ { n - p _ m } . \\end{align*}"}
+{"id": "5089.png", "formula": "\\begin{align*} \\mathcal { B } _ T \\left ( \\mathcal { A } [ \\phi ] v \\right ) ( \\xi , x ) = \\mathcal { A } _ \\xi [ \\phi ] \\mathcal { B } _ T ( v ) ( \\xi , x ) , v \\in L ^ 2 _ N , \\end{align*}"}
+{"id": "9291.png", "formula": "\\begin{align*} D _ { I } = X \\bigcap F _ { I } , \\ X = X _ { 1 } \\times \\dots \\times X _ { m } = \\prod \\limits _ { i = 1 } ^ { m } X _ { i } , \\ X _ { i } \\subseteq \\mathbb { R } ^ { n } , \\ i = 1 , \\dots , m ; \\end{align*}"}
+{"id": "4157.png", "formula": "\\begin{align*} \\langle \\xi , \\eta \\rangle : = \\sum _ { t \\in G } \\xi ( t ) ^ * \\eta ( t ) . \\end{align*}"}
+{"id": "2.png", "formula": "\\begin{align*} \\int _ { U ^ - U ^ 0 U } \\varphi ( g ) \\ , d m _ G ( g ) = \\int _ { U ^ - \\times U ^ 0 \\times U } \\varphi ( u ^ - u ^ 0 u ) \\Delta ( u ^ 0 ) \\ , d \\nu ^ - ( u ^ - ) d \\nu ^ 0 ( u ^ 0 ) d \\nu ( u ) \\end{align*}"}
+{"id": "7740.png", "formula": "\\begin{align*} \\Bigl ( y ( Z , t ) \\le s \\Bigr ) & = \\Bigl ( \\alpha _ { m } ( Z ^ { t _ { m } } ) \\le \\frac { s - y ( t _ { m } ) } { t - t _ { m } } \\Bigr ) \\\\ & = \\Bigl ( \\alpha _ { m } ( Z ^ { t _ { m } } ) \\le \\frac { s - y ( t _ { m } ) } { t - t _ { m } } \\Bigr ) \\cap ( y ( Z , t _ { m } ) \\le s ) \\end{align*}"}
+{"id": "4556.png", "formula": "\\begin{align*} H = ( 1 , 2 , 3 , 2 , 1 ) , P _ H = \\begin{array} { c c c } \\bullet & \\bullet & \\bullet \\\\ \\bullet & \\bullet & \\\\ \\bullet & \\bullet & \\\\ \\bullet & & \\\\ \\bullet & & \\end{array} H ^ \\vee = ( P _ H ) ^ \\vee = ( 5 , 3 , 1 ) : \\begin{array} { c c c c c } \\bullet & \\bullet & \\bullet & \\bullet & \\bullet \\\\ \\bullet & \\bullet & \\bullet & & \\\\ \\bullet & & & & \\end{array} \\end{align*}"}
+{"id": "0.png", "formula": "\\begin{align*} \\Lambda _ { u } = \\left \\{ \\left ( \\frac { N p _ 1 + v _ 1 + \\langle \\bar { u } _ 1 , N \\bar { q } + v '' \\rangle } { N } , \\ldots , \\frac { N p _ m + v _ m + \\langle \\bar { u } _ m , N \\bar { q } + v '' \\rangle } { N } , \\frac { N \\bar { q } + v '' } { N } \\right ) : ( \\bar { p } , \\bar { q } ) \\in \\Z ^ m \\times \\Z ^ n \\right \\} \\end{align*}"}
+{"id": "6863.png", "formula": "\\begin{align*} d _ h ( x ) = \\int _ { [ 0 , 1 ] } \\d y \\ , h ( x , y ) , x \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "6464.png", "formula": "\\begin{align*} \\sum _ { \\substack { K = K _ 1 - K _ 2 + K _ 3 \\\\ \\Delta \\omega _ { K K _ 1 K _ 2 K _ 3 } \\neq 0 } } & \\eta _ { K _ 1 } \\overline { \\eta _ { K _ 2 } } \\eta _ { K _ 3 } \\frac { 1 - e ^ { - i t \\Delta \\omega _ { K K _ 1 K _ 2 K _ 3 } } } { i \\Delta \\omega _ { K K _ 1 K _ 2 K _ 3 } } \\\\ & = L ^ 4 \\int _ { K = k _ 1 - k _ 2 + k _ 3 } \\eta ( k _ 1 ) \\overline { \\eta ( k _ 2 ) } \\eta ( k _ 3 ) \\frac { 1 - e ^ { - i t \\Delta \\omega _ { K k _ 1 k _ 2 k _ 3 } } } { i \\Delta \\omega _ { K k _ 1 k _ 2 k _ 3 } } \\dd k _ 1 \\dd k _ 2 \\dd k _ 3 + \\mathcal { O } ( L ^ 3 ) \\end{align*}"}
+{"id": "8554.png", "formula": "\\begin{align*} \\Omega _ \\infty : = \\{ Z _ 0 ( t ) > 0 \\ ; \\} . \\end{align*}"}
+{"id": "901.png", "formula": "\\begin{align*} \\sum _ { | d | \\leq \\widehat { T } ^ { - 1 } \\widehat { Y } ^ { 1 / 2 } } \\widehat { C } _ 1 \\widehat { C } _ 2 | d | = \\widehat { Y } \\sum _ { | d | \\leq \\widehat { T } ^ { - 1 } \\widehat { Y } ^ { 1 / 2 } } | d | ^ { - 1 } \\ll \\widehat { Y } ^ { 1 + \\varepsilon } . \\end{align*}"}
+{"id": "9253.png", "formula": "\\begin{align*} d = \\texttt { w p i n i t i a l } + 1 + \\texttt { m a g } \\Bigl \\{ 1 2 \\frac { t } { 2 \\pi } \\log \\frac { t } { 2 \\pi } \\Bigr \\} \\end{align*}"}
+{"id": "715.png", "formula": "\\begin{align*} A _ { t w } = \\frac { \\rho _ { F } - \\rho _ { A } } { \\rho _ { F } + \\rho _ { A } } \\end{align*}"}
+{"id": "938.png", "formula": "\\begin{align*} \\mathbb { E } ( X ) \\leq 0 + \\frac { 2 t ^ 2 } { \\ell + 2 } + \\sum _ { \\substack { h < i < j \\leq k \\\\ j - i \\leq t } } \\mathbb { P } ( s _ i = s _ j ) . \\end{align*}"}
+{"id": "3898.png", "formula": "\\begin{align*} \\varphi ( x ) = \\inf _ { \\lambda \\in \\mathbb { R } ^ n _ + } \\left \\{ \\langle \\lambda , x \\rangle + \\varphi ^ \\star ( \\lambda ) \\right \\} . \\end{align*}"}
+{"id": "3214.png", "formula": "\\begin{align*} & S _ n ' ( y ' - \\sigma _ { k + 1 } ' ( y ' ) ) \\\\ & = ( 1 - \\frac { 1 } { w } ) S _ n ' ( y ' - \\sigma _ { k - 1 } ' ( y ' ) ) - \\frac { 1 } { w } \\sigma ' _ 1 ( S _ n ' ( \\sigma ' _ k ( y ' ) - y ' ) ) - \\frac { 1 } { w } S _ n ' ( \\sigma ' _ 1 ( y ' ) - y ' ) . \\end{align*}"}
+{"id": "6887.png", "formula": "\\begin{align*} & \\psi _ r ( \\beta ) = I _ r ( \\bar { h } ) \\geq \\int _ { S _ r ( \\beta ) } \\d x \\ , J _ r ( x , \\beta ) . \\end{align*}"}
+{"id": "2128.png", "formula": "\\begin{align*} ( \\rho , u , \\mathbb { F } ) ( x , t ) \\mid _ { t = 0 } = ( \\rho _ 0 , u _ 0 , \\mathbb { F } _ 0 ) ( x ) , x \\in \\Omega \\end{align*}"}
+{"id": "453.png", "formula": "\\begin{align*} \\chi _ { t t } + \\frac { G ( r ) } { \\chi ^ 2 } = 0 , \\end{align*}"}
+{"id": "7471.png", "formula": "\\begin{align*} A _ { n } ( t ) = \\sum _ { k = 0 } ^ { \\left \\lfloor ( n - 1 ) / 2 \\right \\rfloor } \\gamma _ { k } t ^ { k } ( 1 + t ) ^ { n - 1 - 2 k } . \\end{align*}"}
+{"id": "2649.png", "formula": "\\begin{align*} ( D ^ 2 - \\alpha ^ 2 ) ^ k s = 0 \\quad [ a , b ] \\setminus X , \\end{align*}"}
+{"id": "5088.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { n l } } ( x , t ) = \\gamma ( x , t ) + \\frac 1 N \\sigma ( t ) , \\end{align*}"}
+{"id": "7341.png", "formula": "\\begin{align*} x ^ 2 + y ^ 3 = z ^ 5 . \\end{align*}"}
+{"id": "3724.png", "formula": "\\begin{align*} \\sigma _ Y ( g \\cdot y ) = g ^ { - 1 } \\sigma _ Y ( y ) \\ , \\ , \\forall \\ , y \\in Y \\ , . \\end{align*}"}
+{"id": "4323.png", "formula": "\\begin{align*} x _ i = \\sum _ { k = 1 } ^ \\infty x _ { i , k } 2 ^ { - k } , \\end{align*}"}
+{"id": "3117.png", "formula": "\\begin{align*} S _ B [ n , k ] = S _ D [ n , k ] + n \\cdot [ 2 ] _ q ^ { n - k - 1 } q ^ { n - k - 1 } S [ n - 1 , k ] _ { q ^ 2 } . \\end{align*}"}
+{"id": "6513.png", "formula": "\\begin{align*} \\det \\mathbf { T } _ { n , m , a , b } ( x ; q ) = q ^ { \\varphi ( n , a , b ) } \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ { m + b - a } \\frac { ( 1 - q ^ { x + m + i - j } ) ( 1 - q ^ { x + 2 i + j + a - b - 2 } ) } { ( 1 - q ^ { x + m + 2 i - j - 1 } ) ( 1 - q ^ { i + j - 1 } ) } . \\end{align*}"}
+{"id": "7779.png", "formula": "\\begin{align*} x ^ r \\cdot x ^ q + r y ^ { r - 1 } \\cdot ( y ^ q z - y z ^ q ) + y ^ r \\cdot ( - z ^ q ) = 0 \\end{align*}"}
+{"id": "9346.png", "formula": "\\begin{align*} - d y _ t = g ( x _ t , y _ t , z _ t , \\tilde { z } _ t , \\gamma _ { ( t , e ) } ) d t - z _ t d W _ t - \\tilde { z } _ t d \\xi _ t - \\int _ { \\mathcal { E } } \\gamma _ { ( t , e ) } \\tilde { N } ( d e , d t ) , \\ t \\in [ 0 , \\infty ) , \\end{align*}"}
+{"id": "1041.png", "formula": "\\begin{align*} \\lim _ { d \\to \\frac { 1 } { 2 } } \\bigg \\{ \\sec ^ 2 ( d \\pi ) - \\frac { 2 } { \\pi } \\frac { \\tan ( d \\pi ) } { 1 - 2 d } \\bigg \\} = \\frac { 2 } { 3 } . \\end{align*}"}
+{"id": "1807.png", "formula": "\\begin{align*} f _ t ( p ) = | \\Lambda | ^ { - 1 } { \\nu _ t ( a _ p ^ * a _ p ) } \\ . \\end{align*}"}
+{"id": "3324.png", "formula": "\\begin{align*} \\widetilde { d } ( \\theta , f ( \\theta ) ) = { \\rm R e } ( q ( \\theta , f ( \\theta ) ) ) + \\widetilde { e } ( \\theta , f ( \\theta ) ) \\end{align*}"}
+{"id": "2009.png", "formula": "\\begin{align*} \\lim _ { j \\to + \\infty } \\int _ \\Omega ( - u _ j ) ^ { n + 1 } d V _ g = \\int _ \\Omega ( - u ) ^ { n + 1 } d V _ g . \\end{align*}"}
+{"id": "6067.png", "formula": "\\begin{align*} \\rho _ { A } ( \\delta _ { 1 / r } ^ { A } \\delta _ { r } ^ { B } x ) & = r ^ { - 1 } \\rho _ { A } ( \\delta _ { r } ^ { B } x ) \\\\ & \\leq C r ^ { - 1 } \\rho _ { B } ( \\delta _ { r } ^ { B } x ) = C \\rho _ { B } ( x ) \\leq C \\sup \\{ \\rho _ { B } ( z ) \\colon \\| z \\| = 1 \\} \\leq D , \\end{align*}"}
+{"id": "310.png", "formula": "\\begin{align*} = \\left ( \\frac { 1 } { 1 - x y z } \\right ) ^ { \\frac { x y } { ( 1 - x ) ^ 2 ( 1 - y ) ^ 3 } } \\times \\exp \\left \\{ \\frac { - x y ( 2 x + y - 3 ) } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 4 } L i _ 2 ( x y z ) \\right \\} \\end{align*}"}
+{"id": "9287.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\| y ^ { k } - y ^ { k - 1 } \\| = 0 . \\end{align*}"}
+{"id": "5383.png", "formula": "\\begin{align*} \\ker \\begin{bmatrix} D _ 1 ^ { 1 1 } & - I \\\\ - I & D _ 2 ^ { 1 1 } \\end{bmatrix} = \\{ 0 \\} . \\end{align*}"}
+{"id": "4293.png", "formula": "\\begin{align*} \\theta \\mathrm { d } \\eta = \\mathrm { d } e + p \\mathrm { d } v - \\frac { Z ( \\theta ) } { \\theta } q \\mathrm { d } q , \\end{align*}"}
+{"id": "2227.png", "formula": "\\begin{align*} \\eta ( x , y ) = \\eta ( y ^ u , y ^ h ) ( 1 - H _ y ( h ) ) + \\int _ { 1 } ^ { h } \\eta ( y ^ { u - v } , ( y ^ v ) ^ - ) \\ , d H _ y ( v ) - E _ 3 ( h ; y , u ) , \\end{align*}"}
+{"id": "8915.png", "formula": "\\begin{align*} U _ n = [ X _ { n 1 } , \\dots , X _ { n j } ] + \\sum _ { m \\geq j + 1 } \\sum _ { ( i _ 1 , \\dots , i _ m ) \\in \\mathbb { I } _ j ^ m } \\gamma _ { ( i _ 1 , \\dots , i _ m ) } [ X _ { n i _ 1 } , \\dots , X _ { n i _ m } ] . \\end{align*}"}
+{"id": "835.png", "formula": "\\begin{gather*} g _ { r , s } ( x ) = r ^ { 2 - m } h ( x / r ) - s ^ { 2 - m } h ( x / s ) . \\end{gather*}"}
+{"id": "586.png", "formula": "\\begin{align*} f _ { ( x _ 1 , x _ 2 ) , w } ( \\gamma ^ { a _ { \\frac { \\alpha } { 2 \\sqrt { x _ 2 } } } ^ { r _ 1 } } \\gamma ^ { a _ { \\frac { \\alpha } { 2 \\sqrt { x _ 2 } } } ^ { r _ 2 } } ) = f _ { ( x _ 1 , x _ 2 ) , v } ( \\gamma ^ { a _ { \\frac { \\alpha } { 2 \\sqrt { x _ 2 } } } ^ { r _ 1 } } \\gamma ^ { a _ { \\frac { \\alpha } { 2 \\sqrt { x _ 2 } } } ^ { r _ 2 } } ) \\forall \\alpha > 0 , r _ 1 , r _ 2 \\in \\R . \\end{align*}"}
+{"id": "628.png", "formula": "\\begin{align*} \\| x _ n - x _ { n + 1 } \\| = \\frac { \\| ( 1 - \\| x _ n + \\alpha _ n u \\| ) x _ n + \\alpha _ n u \\| } { \\| x _ n + \\alpha _ n u \\| } < \\frac { \\delta } { 2 ^ { n + 1 } } \\end{align*}"}
+{"id": "7115.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } J _ \\varepsilon ( | x | ) x _ m ^ 2 \\ : x = \\int _ { \\mathbb { R } ^ n } J _ \\varepsilon ( | x | ) x _ 1 ^ 2 \\ : x \\end{align*}"}
+{"id": "1995.png", "formula": "\\begin{align*} C f - ( 1 - \\lambda \\underline u ) f & = \\lambda C f ( \\Vert u _ 0 \\Vert _ { C ^ 0 ( \\bar \\Omega ) } + u _ 0 ) \\\\ & \\geq 0 . \\end{align*}"}
+{"id": "4858.png", "formula": "\\begin{align*} \\| m ( x , y ) - z \\| ^ p & = \\| \\frac { 1 } { 2 } x + \\frac { 1 } { 2 } y - z \\| ^ p \\\\ & < 2 ^ { p - 1 } \\left ( \\| \\frac { 1 } { 2 } ( x - z ) \\| ^ p + \\| \\frac { 1 } { 2 } ( y - z ) \\| ^ p \\right ) \\\\ & = 2 ^ { p - 1 } \\left ( \\frac { 1 } { 2 ^ p } \\| x - z \\| ^ p + \\frac { 1 } { 2 ^ p } \\| y - z \\| ^ p \\right ) \\\\ & = \\frac { 1 } { 2 } \\| x - z \\| ^ p + \\frac { 1 } { 2 } \\| y - z \\| ^ p \\end{align*}"}
+{"id": "448.png", "formula": "\\begin{align*} \\ddot A ( t ) & = \\delta \\det ( A ( t ) ) ^ { 1 - \\gamma } A ( t ) ^ { - \\top } , \\\\ A ( 0 ) & = A _ 0 , \\ \\ \\dot A ( 0 ) = A _ 1 , \\\\ \\nabla _ x w & = - \\delta \\frac { \\gamma - 1 } { \\gamma } x . \\end{align*}"}
+{"id": "4642.png", "formula": "\\begin{align*} \\tilde { a } = \\frac { \\frac { p + \\gamma - 1 } { 2 } - \\frac { ( p + \\gamma - 1 ) ( q - 1 ) } { 2 q ( p - \\gamma + 1 ) } } { \\frac { p + \\gamma - 1 } { 2 } + \\frac { 1 } { 2 } } = \\frac { ( q ( p - \\gamma ) + 1 ) ( p + \\gamma - 1 ) } { q ( p + \\gamma ) ( p - \\gamma + 1 ) } . \\end{align*}"}
+{"id": "8567.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } \\frac { S _ j ( t ) } { M ( t ) } = \\lim _ { N \\to \\infty } \\frac { S _ j ( \\tau _ N ) } { M ( \\tau _ N ) } = \\frac { \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s } { \\int _ 0 ^ \\infty e ^ { - \\lambda s } ( 1 - p _ 0 ( s ) ) d s } , j \\geq 1 , \\end{align*}"}
+{"id": "7402.png", "formula": "\\begin{align*} Z _ { N , \\beta } ^ \\omega ( \\varphi ) : = \\int _ { \\R ^ 2 } Z _ { N , \\beta } ^ { \\omega } ( \\lfloor \\sqrt { N } x \\rfloor ) \\ , \\varphi ( x ) \\ , \\dd x = \\frac { 1 } { N } \\sum _ { z \\in \\Z ^ 2 } Z _ { N , \\beta } ^ { \\omega } ( z ) \\ , \\varphi _ N ( z ) \\ , , \\end{align*}"}
+{"id": "7930.png", "formula": "\\begin{align*} | \\beta | = ( - \\frac { 5 } { 2 } - ) \\beta ( \\xi , 0 ) + 2 \\beta ( 0 , 1 ) + 4 \\beta ( 0 , 2 ) + 6 \\beta ( 0 , 3 ) + \\sum _ { \\mathbf { n } \\in \\N _ 0 ^ 4 } ( | \\mathbf { n } | - 2 ) \\beta ( \\mathbf { n } ) . \\end{align*}"}
+{"id": "5962.png", "formula": "\\begin{align*} \\widetilde { s } ( g ) = \\widetilde { s } ( k _ g ) \\textrm { a n d } \\overline { s } ( g ) = \\overline { s } ( k _ g ) . \\end{align*}"}
+{"id": "8647.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - \\lambda s } ( 1 - p _ 0 ( s ) ) d s = \\dfrac 1 { \\lambda } \\int _ 0 ^ { 1 } \\frac { q } { 1 - p x } d x = \\begin{cases} \\dfrac 1 \\lambda , & p = 0 , \\\\ - \\dfrac { q \\log ( q ) } { \\lambda p } , & 0 < p < 1 . \\end{cases} \\end{align*}"}
+{"id": "3149.png", "formula": "\\begin{align*} 1 \\geq M _ k = \\max _ { 1 \\leq l \\leq k } u _ l m _ k = \\max _ { 1 \\leq l \\leq k - 1 } u _ l \\geq 0 . \\end{align*}"}
+{"id": "6221.png", "formula": "\\begin{align*} \\mathcal C = \\bigg \\{ z \\in \\R ^ N : \\ , 4 R \\leq | z | \\leq 5 R , \\ , \\frac { z \\cdot z ^ + } { | z | \\cdot | z ^ + | } \\geq \\cos ( \\pi / 1 5 ) \\bigg \\} \\ , , \\end{align*}"}
+{"id": "4166.png", "formula": "\\begin{align*} \\| T _ { s , v } ( x ) \\| & = \\| P _ s ( \\lambda ( x ) ( v \\delta _ e ) ) \\| \\leq \\| \\lambda ( x ) ( v \\delta _ e ) \\| _ 2 \\leq \\| \\lambda ( x ) \\| \\| v \\delta _ e \\| _ 2 \\\\ & = \\| x \\| _ \\lambda \\| v \\| \\leq \\| x \\| _ { \\lambda } . \\end{align*}"}
+{"id": "5358.png", "formula": "\\begin{align*} \\gamma '' : = \\left ( \\frac { 1 0 0 r } { \\sqrt { d - 1 } } \\right ) ^ { d - 1 } r \\gamma _ 1 \\log ^ { ( d - 1 ) / 2 } \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) \\end{align*}"}
+{"id": "6257.png", "formula": "\\begin{align*} E ( a , b , c ) : = \\left \\{ \\frac { x ^ 2 } { a ^ 2 } + \\frac { y ^ 2 } { b ^ 2 } + \\frac { z ^ 2 } { c ^ 2 } \\leq 1 \\right \\} \\subset \\mathbb { R } ^ 3 . \\end{align*}"}
+{"id": "8173.png", "formula": "\\begin{align*} R _ \\ell : = R _ 0 ( \\log \\ell ) ^ { 1 / d } - ( \\log \\log \\ell ) ^ 2 , \\ : \\ : \\ell > 1 . \\end{align*}"}
+{"id": "1304.png", "formula": "\\begin{align*} \\lim _ { t \\to \\tau _ i } U _ i ( t ) = + \\infty , \\end{align*}"}
+{"id": "5810.png", "formula": "\\begin{align*} k Y + \\frac { R _ { k - 2 } ( Y , M ) } { M Y ^ { k - 2 } } = \\frac { h } { M Y ^ { k - 2 } } . \\end{align*}"}
+{"id": "7849.png", "formula": "\\begin{align*} A ( X _ i / \\pi ) = \\begin{bmatrix} a _ i & k _ i - a _ i \\\\ \\frac { ( k _ i - a _ i ) | \\mathcal { S } | } { | \\Omega | - | \\mathcal { S } | } & \\frac { d | \\Omega | + a _ i | \\mathcal { S } | - 2 k _ i | \\mathcal { S } | } { | \\Omega | - | \\mathcal { S } | } \\end{bmatrix} . \\end{align*}"}
+{"id": "1474.png", "formula": "\\begin{align*} P U _ { \\mu , \\xi } ( x ) = U _ { \\mu , \\xi } ( x ) - \\alpha _ n \\mu ^ { \\frac { n - 2 } { 2 } } H ( x , \\xi ) + O ( \\mu ^ { \\frac { n + 2 } { 2 } } ) , \\end{align*}"}
+{"id": "2112.png", "formula": "\\begin{align*} X ( K ) \\cap \\Gamma = \\left \\{ p ^ n Q \\colon n \\ge 0 \\right \\} . \\end{align*}"}
+{"id": "7028.png", "formula": "\\begin{align*} I _ n = \\{ i \\in I \\mid Q _ i \\in \\textbf { Q } _ n \\} . \\end{align*}"}
+{"id": "4735.png", "formula": "\\begin{align*} T ' = \\begin{pmatrix} W ' & X ' \\\\ Y ' & Z ' \\end{pmatrix} , \\end{align*}"}
+{"id": "1983.png", "formula": "\\begin{align*} u _ { x _ n } ( 0 ) \\leq w _ { x _ n } ( 0 ) = : - \\delta _ 0 < 0 . \\end{align*}"}
+{"id": "4425.png", "formula": "\\begin{align*} D _ j S _ { \\omega , \\mathbf { c } } ( \\Psi ) = 0 , \\ \\ j = 1 , 2 , 3 . \\end{align*}"}
+{"id": "2144.png", "formula": "\\begin{align*} \\rho _ { + } = e ^ { - \\phi } , \\end{align*}"}
+{"id": "4412.png", "formula": "\\begin{align*} L _ { \\omega , \\mathbf { c } } ( V ) & = \\sum _ { j = 1 } ^ 3 \\left ( C _ { 1 , j } \\| \\nabla v _ j \\| _ { L ^ 2 } ^ 2 + C _ { 2 , j } \\| v _ j \\| _ { L ^ 2 } ^ 2 + \\sum _ { k = 1 } ^ d C _ { 3 , j , k } \\| \\partial _ k v _ j + c _ { j , k } v _ j \\| _ { L ^ 2 } ^ 2 \\right ) \\\\ & = \\sum _ { j = 1 } ^ 3 \\left ( C _ { 1 , j } \\sum _ { k = 1 } ^ d \\| \\partial _ k v _ j \\| _ { L ^ 2 } ^ 2 + C _ { 2 , j } \\| v _ j \\| _ { L ^ 2 } ^ 2 + \\sum _ { k = 1 } ^ d C _ { 3 , j , k } \\| \\partial _ k v _ j + c _ { j , k } v _ j \\| _ { L ^ 2 } ^ 2 \\right ) \\end{align*}"}
+{"id": "4336.png", "formula": "\\begin{align*} f ^ n ( x , t ) = \\begin{cases} f ^ { \\mathcal { K } _ n } ( x , t ) & t \\in [ 0 , 5 0 ] , \\\\ f ^ { \\mathcal { K } _ n } ( x , 1 0 0 - t ) & t \\in [ 5 0 , 1 0 0 ] . \\end{cases} \\end{align*}"}
+{"id": "4650.png", "formula": "\\begin{align*} M _ k \\leq C _ 8 ^ { k + \\sum _ { j = 1 } ^ { k - 1 } 2 ^ j ( k - j ) } M _ { 0 } ^ { 2 ^ k } \\leq C _ 8 ^ { 2 } C _ 8 ^ { 2 ^ { k } } M _ { 0 } ^ { 2 ^ k } . \\end{align*}"}
+{"id": "5245.png", "formula": "\\begin{align*} \\begin{array} { l c l } P _ { 0 } & = & - \\theta ^ 4 - ( r + 2 - e _ 1 - e _ 2 - e _ 3 ) \\theta ^ 3 - ( r ^ 2 + ( 2 - e _ 1 - e _ 2 - e _ 3 ) r - e _ 1 \\\\ & & - e _ 2 - e _ 3 + e _ 1 e _ 2 + e _ 1 e _ 3 + e _ 2 e _ 3 ) \\theta ^ 2 - ( r ^ 3 + ( 2 - e _ 1 - e _ 2 - e _ 3 ) r ^ 2 \\\\ & & - ( e _ 1 + e _ 2 + e _ 3 - e _ 1 e _ 2 - e _ 1 e _ 3 - e _ 2 e _ 3 ) r - 2 + e _ 1 \\\\ & & + e _ 2 + e _ 3 - e _ 1 e _ 2 e _ 3 ) \\theta - ( r - 1 ) ( r - e _ 1 + 1 ) ( r + 1 - e _ 2 ) ( r + 1 - e _ 3 ) . \\end{array} \\end{align*}"}
+{"id": "1970.png", "formula": "\\begin{align*} D u ^ { k \\bar q } = - u ^ { q \\bar q } u ^ { k \\bar k } D u _ { q \\bar k } . \\end{align*}"}
+{"id": "641.png", "formula": "\\begin{align*} \\rho ( a ^ * a ) = \\rho ( a ) ^ * \\rho ( a ) , a \\in X . \\end{align*}"}
+{"id": "2512.png", "formula": "\\begin{align*} \\| A ^ { - 1 } r _ \\mu \\| _ A = \\| y _ \\mu - A ^ { - 1 } r \\| _ A = \\| ( I - M A ) A ^ { - 1 } r \\| _ A \\le \\| A ^ { - 1 } r \\| _ A \\end{align*}"}
+{"id": "5614.png", "formula": "\\begin{align*} P _ { \\mu , x } ^ { n - 1 } ( L _ { x } s , L _ { x } h ) = P _ { \\mu , s ^ { - 1 } x } ^ { n - 1 } \\left ( L _ { s ^ { - 1 } . x } , L _ { s ^ { - 1 } . x } s ^ { - 1 } h \\right ) . \\end{align*}"}
+{"id": "5954.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( g ) f ( [ \\epsilon , w ] ) & = \\widetilde { C } _ { X ^ { \\ast } } ( g , h _ { \\epsilon } ) \\widetilde { \\beta } ( g ^ { s ( \\epsilon ) } ) ^ { - 1 } f ( [ \\epsilon , w g ^ { s ( \\epsilon ) } ] ) . \\end{align*}"}
+{"id": "7983.png", "formula": "\\begin{align*} \\mathrm { t r } \\ , \\mathcal { B } ^ H = \\mathrm { d i v } _ T \\big ( \\nabla _ \\xi H ( \\nu ) \\big ) \\ , . \\end{align*}"}
+{"id": "8538.png", "formula": "\\begin{align*} \\eta _ { p } ( 2 s ) = \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } \\cdot \\frac { 1 } { 2 s - 1 } + C _ { p } ^ { ( 2 ) } - \\frac { 2 e ^ { 2 \\pi p } } { 1 + \\frac { 1 } { \\pi p } } Q _ { 2 \\pi p } ( 0 ) + O \\left ( s - \\frac { 1 } { 2 } \\right ) , \\end{align*}"}
+{"id": "5205.png", "formula": "\\begin{align*} \\mathrm { c l } ( X ) = ( \\bigcup X ) ^ \\supseteq . \\end{align*}"}
+{"id": "8401.png", "formula": "\\begin{align*} F _ { 2 i } ( V ) = K _ X ^ { n - 1 } \\oplus \\cdots \\oplus K _ X ^ { n + 1 - 2 i } \\ , \\ , \\ , ( i = 1 , \\ldots , n ) . \\end{align*}"}
+{"id": "2096.png", "formula": "\\begin{align*} F ( A ) & = ( a + d - 2 + 2 ) a - d = 9 \\cdot 2 ^ { 2 n } + 3 ( d - 2 ) \\cdot 2 ^ n - 2 d + 1 . \\end{align*}"}
+{"id": "322.png", "formula": "\\begin{align*} a _ a ( m , n ) = \\sum _ { k = 1 } ^ { \\infty } \\left ( \\log 2 + \\frac { 1 } { 2 } ( - 1 ) ^ { k + 1 } \\left [ \\Psi ( \\frac { k } { 2 } + \\frac { 1 } { 2 } ) - \\Psi ( \\frac { k } { 2 } + 1 ) \\right ] \\right ) ^ m \\frac { ( - 1 ) ^ { k + 1 } } { ( k + 1 ) ^ n } , \\end{align*}"}
+{"id": "6861.png", "formula": "\\begin{align*} I _ r ( h ) = \\int _ { [ 0 , 1 ] ^ 2 } \\d x \\ , \\d y \\ , \\ , \\mathcal { R } \\big ( h ( x , y ) \\mid r ( x , y ) \\big ) , h \\in \\mathcal { W } , \\end{align*}"}
+{"id": "3255.png", "formula": "\\begin{align*} [ b , R _ { { \\rm D } , \\ell } ] f ( x ) & = b ( x ) R _ { { \\rm D } , \\ell } f ( x ) - R _ { { \\rm D } , \\ell } ( b f ) ( x ) \\\\ & = \\int _ { \\mathbb R ^ N } ( b ( x ) - b ( y ) ) R _ { { \\rm D } , \\ell } f ( y ) d \\omega ( y ) , \\end{align*}"}
+{"id": "5655.png", "formula": "\\begin{align*} | \\nabla \\eta _ \\epsilon | ^ 2 _ 2 = S ^ { \\frac { N } { 2 } } + O ( \\epsilon ^ { N - 2 } ) = C ( N , \\mu ) ^ { \\frac { N } { 2 } \\cdot \\frac { 1 } { 2 ^ * _ \\mu } } S ^ { \\frac { N } { 2 } } _ { H , L } + O ( \\epsilon ^ { N - 2 } ) , \\end{align*}"}
+{"id": "2964.png", "formula": "\\begin{align*} Y _ A = \\{ ( w , z ) \\in A \\times S ^ 1 \\setminus \\{ 1 \\} \\ | \\ z \\neq w _ i i \\in \\{ 1 , . . . , n \\} \\} \\ , \\end{align*}"}
+{"id": "7731.png", "formula": "\\begin{align*} Y ^ { y ( a ) } ( t ) & = Y \\left ( y ( a ) \\wedge t \\right ) \\\\ & = N \\left ( y ^ { * } ( N , \\ , y ( a ) \\wedge t ) \\right ) \\\\ & = N \\left ( a \\wedge y ^ { * } ( N , t ) \\right ) . \\end{align*}"}
+{"id": "5789.png", "formula": "\\begin{align*} = \\frac { P ' } { P } + U ' . \\end{align*}"}
+{"id": "7276.png", "formula": "\\begin{align*} a x ^ n y ^ q + b x ^ k y ^ l + c x ^ r y ^ m = 0 , \\end{align*}"}
+{"id": "5806.png", "formula": "\\begin{align*} F = \\frac { P ' } { P } + P _ 0 + Y - \\frac { Q ' } { Q } = Y + M , \\end{align*}"}
+{"id": "2050.png", "formula": "\\begin{align*} \\lim _ { \\nu , \\sigma \\to 0 } \\frac { \\nu } { \\sigma ^ 2 } = 0 ; \\end{align*}"}
+{"id": "5864.png", "formula": "\\begin{align*} & \\sigma ^ { ( \\mathcal { I } , r e d ) } ( H _ { \\omega , \\Lambda _ L ( x _ 0 ) } ) : = \\\\ & \\left \\{ E \\in \\sigma ^ { ( \\mathcal { I } ) } ( H _ { \\omega , \\Lambda _ L ( x _ 0 ) } ) : \\left ( E , \\sigma ^ { ( \\mathcal { I } ) } ( H _ { \\omega , \\Lambda _ { L _ n } ( x _ 0 ) } \\right ) \\leq 2 e ^ { - \\frac { \\hat { m } } { K } L _ n } , n = 1 , \\cdots , n _ 1 \\right \\} \\end{align*}"}
+{"id": "3766.png", "formula": "\\begin{align*} \\int _ S \\ d \\acute { \\rho } : = & \\sum _ { 0 \\leq a < b \\leq 3 } \\int _ { I ^ 2 } \\acute { \\rho } _ \\sigma ^ { a b } ( s , t ) [ \\det \\acute { J } _ { a b } ^ \\sigma ( \\hat { s } ) ] \\ d \\hat { s } , \\\\ \\int _ S d | \\acute { \\rho } | : = & \\int _ { I ^ 2 } \\left | \\sum _ { 0 \\leq a < b \\leq 3 } \\acute { \\rho } _ \\sigma ^ { a b } ( \\hat { s } ) [ \\det \\acute { J } _ { a b } ^ \\sigma ] ( \\hat { s } ) \\right | \\ d \\hat { s } . \\end{align*}"}
+{"id": "1276.png", "formula": "\\begin{align*} \\eta ( B ^ { \\prime } ) = \\eta ( B ) + 1 \\ge T _ 0 + \\max _ { 1 \\le i _ 1 < \\ldots < i _ { q _ 0 } \\le s - 1 } \\sum _ { j = 1 } ^ { q _ 0 } a _ { i _ j } \\end{align*}"}
+{"id": "101.png", "formula": "\\begin{align*} \\partial _ t w = - i h ^ { - 1 } \\tilde { P } _ h ( 0 ) w = - i h ^ { - 1 } ( - i h X - i ( q _ 1 + W + Q _ \\infty ) ) w , w | _ { t = 0 } = f . \\end{align*}"}
+{"id": "7886.png", "formula": "\\begin{align*} \\mathcal { O } ^ - _ i : = \\left \\{ \\left ( ( A ^ + , \\underline { A } ^ + ) , ( B ^ + , \\underline { B } ^ - ) \\right ) , \\left ( ( A ^ + , \\underline { A } ^ + ) , ( B ^ - , \\underline { B } ^ + ) \\right ) : A , B \\in \\binom { [ n ] } { k } \\mbox { a n d } | A \\cap B | = k - i \\right \\} . \\end{align*}"}
+{"id": "7823.png", "formula": "\\begin{align*} \\lambda = \\min ( \\frac { t } { 2 C _ { 1 } ^ { 2 } \\sum _ { i \\le N } \\Vert B _ { i } \\Vert _ { 2 } ^ { 4 } } , \\frac { 1 } { C _ { 1 } \\max _ { i } \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } } ) . \\end{align*}"}
+{"id": "4560.png", "formula": "\\begin{align*} P = \\big ( p _ 1 ^ { n _ 1 } , \\ldots , p _ c ^ { n _ c } , ( d - n ) ^ { d - n + k - 1 } \\big ) , \\end{align*}"}
+{"id": "4450.png", "formula": "\\begin{align*} h ' ( \\tau ) & = - ( 4 - d ) ( \\sqrt { \\omega } - \\tau ) ^ { 3 - d } \\mu \\left ( 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } \\right ) , \\\\ h '' ( \\tau ) & = ( 3 - d ) ( 4 - d ) ( \\sqrt { \\omega } - \\tau ) ^ { 2 - d } \\mu \\left ( 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } \\right ) , \\end{align*}"}
+{"id": "817.png", "formula": "\\begin{align*} h ( \\alpha _ 1 , \\ldots , \\alpha _ n ) = \\log H ( \\alpha _ 1 , \\ldots , \\alpha _ n ) . \\end{align*}"}
+{"id": "7070.png", "formula": "\\begin{align*} \\nu _ j ( g ' ) = \\nu ( g ' ) \\mbox { f o r e v e r y } j \\in I ' , j \\geq j _ 0 . \\end{align*}"}
+{"id": "2804.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q _ 1 ) \\tilde { v } _ 2 = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } ( \\partial _ \\nu - i a ) \\tilde { v } _ 2 = ( \\partial _ \\nu - i a ) v _ 2 . \\end{align*}"}
+{"id": "4976.png", "formula": "\\begin{align*} \\hat { p } _ { 1 , n } ^ { ( t + 1 ) } = \\hat { p } _ { 1 , n } ^ { ( t ) } \\ , + \\ , \\hat { p } _ { n - 1 , n } ^ { } \\hat { p } _ { 1 , n - 1 } ^ { ( t ) } . \\end{align*}"}
+{"id": "6791.png", "formula": "\\begin{align*} \\Sigma = W _ 1 \\cup _ { K _ 1 } \\cdots \\cup _ { K _ { 2 m - 1 } } W _ { 2 m } \\end{align*}"}
+{"id": "855.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left [ \\begin{array} { c c } u \\\\ u _ { t } \\end{array} \\right ] = \\left [ \\begin{array} { c c } u _ { t } \\\\ - \\alpha f ( u _ { t } ) - H _ { \\lambda } ( u ) \\end{array} \\right ] . \\end{align*}"}
+{"id": "2275.png", "formula": "\\begin{align*} & \\int _ 0 ^ { 2 \\pi } | T ( f ) ( r e ^ { i \\theta } ) | ^ p \\ , d \\theta \\\\ & = \\int _ { D _ { r , * } } | T ( f ) ( r e ^ { i \\theta } ) | ^ p \\ , d \\theta + \\int _ { D ^ { r , * } } | T ( f ) ( r e ^ { i \\theta } ) | ^ p \\ , d \\theta \\\\ & \\leq 2 \\pi + \\int _ 0 ^ { 2 \\pi } | T ( f ) ( r e ^ { i \\theta } ) | ^ \\gamma \\ , d \\theta \\\\ & \\leq 2 \\pi + C | | f | | _ { L ^ q ( D ) } < \\infty . \\end{align*}"}
+{"id": "8485.png", "formula": "\\begin{align*} \\sigma _ { p } ( 2 \\pi x ) = - \\frac { 1 } { 2 } \\ , \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } + \\frac { 1 } { 2 \\pi x } + 2 \\ , \\intop _ { 0 } ^ { \\infty } \\frac { \\sin ( x y ) } { \\sigma ( y ) \\ , e ^ { 2 \\pi y } - 1 } \\ , d y . \\end{align*}"}
+{"id": "4669.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\left ( \\left | \\nabla v _ n \\right | ^ 2 + ( V ( \\epsilon x ) + 1 ) \\left | v _ n \\right | ^ 2 \\right ) d x + \\int _ { \\mathbb { R } ^ N } F ' _ 1 ( v _ n ) v _ n d x - \\lambda _ \\epsilon \\int _ { \\mathbb { R } ^ N } \\left | v _ n \\right | ^ 2 d x = \\int _ { \\mathbb { R } ^ N } F ' _ 2 ( v _ n ) v _ n d x + o _ n ( 1 ) . \\end{align*}"}
+{"id": "5030.png", "formula": "\\begin{align*} E _ f ^ { - 1 } = \\left ( \\overline \\rho _ 1 | \\dots | \\overline \\rho _ N \\right ) . \\end{align*}"}
+{"id": "4112.png", "formula": "\\begin{align*} f ( y ) = y ^ 3 + \\alpha _ 2 y ^ 2 + \\alpha _ 1 y + \\alpha _ 0 = 0 \\end{align*}"}
+{"id": "6086.png", "formula": "\\begin{align*} \\| h \\| _ { H ^ 1 _ A } = \\sup _ { \\substack { f \\in ( H ^ 1 _ A ) ^ * \\\\ \\| f \\| _ { ( H ^ 1 _ A ) ^ * } \\leq 1 } } | f ( h ) | \\asymp \\sup _ { \\substack { f \\in ( H ^ 1 _ B ) ^ * \\\\ \\| f \\| _ { ( H ^ 1 _ B ) ^ * } \\leq 1 } } | f ( h ) | = \\| h \\| _ { H ^ 1 _ B } \\end{align*}"}
+{"id": "7918.png", "formula": "\\begin{align*} \\sigma \\curvearrowright ^ { \\mathbf { n } } X ^ { \\mathbf { n } ' } = \\sigma \\delta _ \\mathbf { n } ^ { \\mathbf { n } ' } \\end{align*}"}
+{"id": "1916.png", "formula": "\\begin{align*} \\| S u ^ k - \\zeta ^ k \\| = \\| r ^ k \\| \\ \\rightarrow 0 \\ \\ \\mbox { a s } \\ \\ k \\rightarrow + \\infty . \\end{align*}"}
+{"id": "7988.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } \\nabla _ \\xi H ^ 2 ( \\nabla v ) \\cdot ( \\partial _ \\nu v ) \\nu = H ^ 2 ( \\nabla v ) \\ , . \\end{align*}"}
+{"id": "8202.png", "formula": "\\begin{align*} P ^ \\omega \\left ( A _ t ^ c \\cap S _ { \\alpha t } \\right ) & = P ^ \\omega \\left ( A _ t ^ c \\cap S _ { \\alpha t } \\cap E _ t ^ c \\right ) + P ^ \\omega \\left ( A _ t ^ c \\cap S _ { \\alpha t } \\cap E _ t \\right ) \\\\ & \\leq P ^ \\omega \\left ( E _ t ^ c \\mid S _ { \\alpha t } \\right ) + P ^ \\omega \\left ( A _ t ^ c \\mid E _ t \\right ) . \\end{align*}"}
+{"id": "3687.png", "formula": "\\begin{align*} \\begin{cases} \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) u = 0 & \\mathbb { R } ^ n , \\\\ u = 0 & W , \\end{cases} \\end{align*}"}
+{"id": "4793.png", "formula": "\\begin{align*} & ( a ) \\ - \\bigg \\langle \\nabla V ( x ) , f ( x ) + \\sum _ { i = 1 } ^ m g _ i ( x ) u _ i ( x ) \\bigg \\rangle \\\\ & + \\sum _ { j = 1 } ^ J a _ j ( x ) \\sigma _ j ( x ) + \\sum _ { \\ell = 1 } ^ L b _ \\ell ( x ) \\rho _ \\ell ( x ) \\ \\mathrm { i s \\ S O S } , \\\\ & ( b ) \\ \\sigma _ j ( x ) \\ \\mathrm { i s \\ S O S } , \\end{align*}"}
+{"id": "8675.png", "formula": "\\begin{align*} X _ { i } = \\Gamma _ 1 U _ { i } + \\mu _ { 1 } \\ ( i = 1 , \\ldots , n ) , Y _ { j } = \\Gamma _ 2 V _ { j } + \\mu _ { 2 } \\ ( j = 1 , \\ldots , m ) , \\end{align*}"}
+{"id": "1857.png", "formula": "\\begin{align*} \\mu _ { i _ 1 i _ 2 } = \\min \\{ \\mu _ { i _ 1 j } \\ | \\ i _ 1 < j \\leq n , \\ b _ { n - j } ( x ) \\ne 0 \\} \\end{align*}"}
+{"id": "6168.png", "formula": "\\begin{align*} l _ R ( x y ) l _ R ( p q a ) & = l _ R ( x y ) l _ R ( p ) + l _ R ( x y ) l _ R ( q ) + l _ R ( x y ) l _ R ( a ) \\\\ & = l _ R ( y ) l _ R ( p ) + l _ R ( x ) l _ R ( q ) + l _ R ( x ) l _ R ( a ) + l _ R ( y ) l _ R ( a ) \\\\ & = l _ R ( y ) l _ R ( p a ) + l _ R ( x ) l _ R ( q a ) \\\\ & = l _ R ( y ) l _ R ( x ) + l _ R ( x ) l _ R ( y ) = 0 . \\end{align*}"}
+{"id": "3884.png", "formula": "\\begin{align*} \\left ( \\mathord { \\operatorname { p r o j } } _ { K _ i \\cap K _ j } \\circ { \\operatorname { p r o j } _ { K _ i } } ^ { - 1 } \\right ) \\# \\mu _ i = \\left ( \\mathord { \\operatorname { p r o j } } _ { K _ i \\cap K _ j } \\circ { \\operatorname { p r o j } _ { K _ j } } ^ { - 1 } \\right ) \\# \\mu _ j . \\end{align*}"}
+{"id": "4796.png", "formula": "\\begin{align*} \\mathcal { L } V = \\bigg \\langle \\nabla V ( x ) , f ( x ) + \\sum _ { i = 1 } ^ m g _ i ( x ) u _ i \\bigg \\rangle , \\end{align*}"}
+{"id": "8287.png", "formula": "\\begin{align*} w = x _ 1 x _ 3 x _ 1 + \\alpha x _ 3 x _ 1 ^ 2 + \\alpha ^ { - 1 } x _ 1 ^ 2 x _ 3 + x _ 2 x _ 3 x _ 2 - \\alpha x _ 3 x _ 2 ^ 2 - \\alpha ^ { - 1 } x _ 2 ^ 2 x _ 3 \\end{align*}"}
+{"id": "8292.png", "formula": "\\begin{align*} \\begin{pmatrix} y _ 1 \\\\ y _ 2 \\end{pmatrix} a = \\sigma ( a ) \\begin{pmatrix} y _ 1 \\\\ y _ 2 \\end{pmatrix} + \\delta ( a ) \\end{align*}"}
+{"id": "4131.png", "formula": "\\begin{align*} M _ 1 = \\begin{pmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\\\ 1 & 1 & - 1 \\end{pmatrix} M _ 2 = \\begin{pmatrix} - 1 & 1 & 1 \\\\ 1 & 0 & - 1 \\\\ - 1 & 0 & 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "9003.png", "formula": "\\begin{align*} \\varphi ^ a _ s w _ s = - ( 1 + w ) \\varphi ^ a _ { t t } = 0 \\end{align*}"}
+{"id": "1194.png", "formula": "\\begin{align*} \\left \\langle S , \\begin{pmatrix} \\hat { P } & \\ell \\end{pmatrix} F _ e \\begin{pmatrix} \\hat { P } & \\ell \\end{pmatrix} ^ \\top \\right \\rangle = 0 & e \\in \\hat { E } ^ \\circ \\end{align*}"}
+{"id": "7708.png", "formula": "\\begin{align*} Z ( N _ { U _ r } ) = \\{ [ \\mathrm { i d } _ { U _ r } , \\mathrm { i d } _ { L _ { r + 1 } } , 0 , \\psi ] \\} \\end{align*}"}
+{"id": "5774.png", "formula": "\\begin{align*} | E _ { s } ^ { ( \\ell ) } | & = | \\{ x \\in \\mathbb { R } ^ { n } \\setminus \\Omega ^ { * } : | T _ { \\alpha } ( b _ { 1 } , \\dots , b _ { \\ell } , g _ { \\ell + 1 } , \\dots , g _ { m } ) ( x ) | > \\lambda / 2 ^ { m } \\} | , \\end{align*}"}
+{"id": "8804.png", "formula": "\\begin{align*} & ( 2 u ^ 3 + 1 8 u ^ 2 t + 4 5 u t ^ 2 + 3 6 t ^ 3 + 1 6 u ^ 2 + 1 0 3 u t \\\\ & + 1 3 5 t ^ 2 + 5 4 u + 1 6 1 t + 6 0 ) ( t + 2 ) \\\\ & = ( 2 u ^ 3 + 1 2 u ^ 2 t + 2 7 u t ^ 2 + 1 8 t ^ 3 + 2 2 u ^ 2 + 9 5 u t \\\\ & + 9 9 t ^ 2 + 8 0 u + 1 7 5 t + 1 0 0 ) ( 2 t + 1 ) \\end{align*}"}
+{"id": "6888.png", "formula": "\\begin{align*} h _ { \\beta } ( x , y ) = \\begin{cases} r ( x , y ) , & x , y \\not \\in S _ r ( \\beta ) \\\\ \\hat { r } _ \\beta ( x , y ) , & x \\in S _ r ( \\beta ) y \\in S _ r ( \\beta ) ( x , y ) \\not \\in S _ r ( \\beta ) ^ 2 , \\\\ \\min \\{ \\hat { r } _ \\beta ( x , y ) , \\hat { r } _ \\beta ( y , x ) \\} , & x , y \\in S _ r ( \\beta ) . \\end{cases} \\end{align*}"}
+{"id": "8938.png", "formula": "\\begin{align*} { \\rm d i v } \\ , \\overline v = 0 , { \\rm c u r l } \\ , \\overline v = \\omega _ \\theta e _ \\theta \\end{align*}"}
+{"id": "2859.png", "formula": "\\begin{align*} \\varphi _ V \\circ \\rho _ A ( x ) - \\rho _ A ( \\alpha ( x ) ) \\circ \\varphi _ V = \\rho _ A ( \\varphi _ L ( x ) ) \\circ A \\end{align*}"}
+{"id": "1260.png", "formula": "\\begin{align*} \\mu _ { X _ 1 } ( x _ 1 , y _ 1 ) > 0 \\wedge \\mu _ { X _ 1 } ( y _ 1 , x _ 1 ) > 0 \\Rightarrow x _ 1 = y _ 1 \\end{align*}"}
+{"id": "6617.png", "formula": "\\begin{align*} [ ( \\varphi _ \\lambda ( x , y ) ) _ { \\lambda + \\mu } [ w _ \\gamma v ] ] = [ [ x _ \\mu y ] _ { \\mu + \\gamma } \\varphi _ { \\lambda } ( w , v ) ] , \\end{align*}"}
+{"id": "674.png", "formula": "\\begin{align*} \\mathfrak { h } _ { i _ a } ( \\varphi ) = \\sum _ { 1 \\leq i < i _ a } d ^ + _ { i , 0 } ( \\varphi ) h _ i . \\end{align*}"}
+{"id": "2181.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ 1 } = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } \\| f ( x ) \\| _ { } \\ , d x < \\infty , \\end{align*}"}
+{"id": "6503.png", "formula": "\\begin{align*} t _ n : = t _ n ( m , x ) : = \\sum _ { j = 1 } ^ n a _ { n , j } c _ { n , j } = \\sum _ { j = 1 } ^ n \\binom { m + x } { m - n + j } c _ { n , j } - \\sum _ { j = 1 } ^ n \\binom { m + x } { m - n - j + 1 } c _ { n , j } . \\end{align*}"}
+{"id": "1775.png", "formula": "\\begin{align*} ( K _ { \\leq n } ) _ p = \\begin{cases} K _ p & \\hbox { i f } p \\leq n \\\\ \\varnothing & \\hbox { i f } p > n . \\end{cases} \\end{align*}"}
+{"id": "3365.png", "formula": "\\begin{align*} \\pi _ { \\nu } ( f ) = f ( \\pi _ { \\nu } ( S ) ) = ( I \\otimes c ^ * ) \\left ( I - \\sum _ { j = 1 } ^ d \\pi _ { \\nu } ( S _ j ) \\otimes T _ { 0 , j } ^ * \\right ) ^ { - 1 } \\left ( \\sum _ { j = 1 } ^ d \\pi _ { \\nu } ( S _ j ) \\otimes T _ j ^ * x \\right ) . \\end{align*}"}
+{"id": "4758.png", "formula": "\\begin{align*} \\| z _ 1 \\| ^ 2 & = \\| x _ 1 \\| ^ 2 + \\| y _ 1 \\| ^ 2 \\\\ & = \\left \\| \\begin{pmatrix} x _ 1 \\\\ - y _ 1 \\end{pmatrix} \\right \\| ^ 2 = 1 + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "2420.png", "formula": "\\begin{align*} Y ( F ) = \\bigcup _ { M > 1 } \\bigcup _ { k > 1 } \\bigcup _ { l > 1 } Y ( F ; M , k , l ) , \\end{align*}"}
+{"id": "8082.png", "formula": "\\begin{align*} \\mathcal { F } : = \\{ u \\in S _ { 0 } ^ { 1 , p } ( \\Omega ) : \\enspace F ( u ) = 1 \\} . \\end{align*}"}
+{"id": "1684.png", "formula": "\\begin{align*} \\mu ^ 2 = \\frac { 1 } { ( \\deg \\tau ) ^ 2 } \\tau \\pi \\psi \\widehat { \\tau } \\tau \\pi \\psi \\widehat { \\tau } = \\frac { 1 } { \\deg \\tau } \\tau \\pi \\psi \\pi \\psi \\widehat { \\tau } = \\frac { - d p } { \\deg \\tau } \\tau \\widehat { \\tau } = - d p , \\end{align*}"}
+{"id": "7513.png", "formula": "\\begin{align*} \\partial _ z q ^ i ( z ) + \\langle Z , \\alpha _ i \\rangle q ^ i ( z ) = L _ i ( z ) \\Big [ { p ^ i ( z ) } \\Big ] ^ { 2 } \\prod _ { j \\neq i , j \\in I _ w \\cup I _ g } p ^ j ( z ) ^ { a _ { j i } } \\prod _ { j \\neq i , j \\in I _ b } p ^ j ( z ) ^ { a _ { j i } / 2 } . \\end{align*}"}
+{"id": "234.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ 1 z ^ k = \\frac { z - ( 1 + n ) z ^ { n + 1 } + n z ^ { n + 2 } } { ( 1 - z ) ^ 2 } , \\end{align*}"}
+{"id": "478.png", "formula": "\\begin{align*} c ^ 2 t ^ { k - 1 } t _ 2 t _ 3 t _ 4 t _ 5 = p q . \\end{align*}"}
+{"id": "1858.png", "formula": "\\begin{align*} v _ p ^ x ( b _ 0 a _ 0 ( x ) ) - v _ p ^ x ( b _ j a _ j ( x ) ) & = v _ p ^ x ( ( n + 1 ) ! a _ 0 ( x ) ) - v _ p ^ x \\bigg ( \\frac { ( n + 1 ) ! } { ( j + 1 ) ! } a _ j ( x ) \\bigg ) \\\\ & \\leq v _ p ( ( n + 1 ) ! ) - v _ p \\bigg ( \\frac { ( n + 1 ) ! } { ( j + 1 ) ! } \\bigg ) = v _ p ( ( j + 1 ) ! ) . \\end{align*}"}
+{"id": "4448.png", "formula": "\\begin{align*} h ( \\tau ) : = \\mu \\bigg ( ( \\sqrt { \\omega } - \\tau ) ^ 2 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } ( \\sqrt { \\omega } - \\tau ) \\bigg ) = ( \\sqrt { \\omega } - \\tau ) ^ { 4 - d } \\mu \\left ( 1 , \\frac { \\mathbf { c } } { \\sqrt { \\omega } } \\right ) , \\end{align*}"}
+{"id": "5636.png", "formula": "\\begin{align*} \\aligned \\left \\{ \\begin{array} { l l l } - \\Delta u + \\lambda _ 1 u = p ( I _ \\mu \\ast | v | ^ q ) | u | ^ { p - 2 } u \\ & \\mathbb { R } ^ N , \\\\ - \\Delta v + \\lambda _ 2 v = q ( I _ \\mu \\ast | u | ^ p ) | v | ^ { q - 2 } v \\ & \\mathbb { R } ^ N , \\\\ \\int _ { \\mathbb { R } ^ N } u ^ 2 = a ^ 2 , \\quad \\int _ { \\mathbb { R } ^ N } v ^ 2 = b ^ 2 . \\end{array} \\right . \\endaligned \\end{align*}"}
+{"id": "1949.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l c r } ( d d ^ c u ) ^ n = ( - \\lambda u ) ^ n f ^ n \\omega ^ n & \\textnormal { i n } & \\Omega \\\\ u = 0 & \\textnormal { o n } & \\partial \\Omega \\\\ \\Vert u \\Vert _ { C ^ 0 ( \\bar \\Omega ) } = 1 , & & \\end{array} \\right . \\end{align*}"}
+{"id": "1140.png", "formula": "\\begin{align*} \\Delta _ { y _ { 1 } , \\ldots , y _ { m } } A ^ { \\ast } ( x ) = \\left \\{ \\begin{array} { r c l } 0 & & m > n \\\\ n ! A ( y _ { 1 } , \\ldots , y _ { m } ) & & m = n . \\end{array} \\right . \\end{align*}"}
+{"id": "4649.png", "formula": "\\begin{align*} & K _ 1 = \\frac { k _ D ^ 2 ( n + 2 ) } { 2 n \\chi ^ 2 p ^ 2 C _ 5 ^ 2 } , \\end{align*}"}
+{"id": "6207.png", "formula": "\\begin{align*} | a _ i b _ { i + 1 } - b _ i a _ { i + 1 } | = 1 , i = 2 , 3 , \\ldots , n - 1 . \\end{align*}"}
+{"id": "3763.png", "formula": "\\begin{align*} J _ { a b } ^ \\sigma ( s , t ) = \\left ( \\begin{array} { c c } \\sigma _ a ' ( s , t ) & \\dot { \\sigma } _ a ( s , t ) \\\\ \\sigma _ b ' ( s , t ) & \\dot { \\sigma } _ b ( s , t ) \\\\ \\end{array} \\right ) , \\ a \\neq b , \\end{align*}"}
+{"id": "8994.png", "formula": "\\begin{align*} R ^ \\varphi _ { i j , i } = \\frac { 1 } { 2 } S ^ \\varphi _ j - \\alpha \\varphi ^ a _ { s s } \\varphi ^ a _ j \\end{align*}"}
+{"id": "3613.png", "formula": "\\begin{align*} q \\ = \\ p - 2 \\ \\geq \\ 1 0 ^ m + 3 - 2 \\ = \\ 1 0 ^ m + 1 , \\end{align*}"}
+{"id": "4802.png", "formula": "\\begin{align*} \\tilde { \\mathcal { K } } ( \\varphi ) : = \\langle c , A \\psi \\rangle + \\sum _ { i = 1 } ^ m \\langle c , B _ i \\psi \\rangle u _ i , \\end{align*}"}
+{"id": "1533.png", "formula": "\\begin{align*} & ( t ^ { p ( p - 1 ) i + p - 1 } , t ^ { p ( p - 1 ) j + p - 1 } ) \\\\ = & \\left ( \\sum _ { \\theta = 1 } ^ { d } ( h _ { p \\theta + p - 1 } - g _ { p \\theta + p - 1 } a ^ { p \\theta } ) t ^ { p \\theta + p - 1 } + f _ { h , g } ( t ) , \\right . \\\\ & \\ , \\ , \\left . \\sum _ { \\theta = 1 } ^ { d } ( g _ { p \\theta + p - 1 } - h _ { p \\theta + p - 1 } a ^ { p \\theta } ) t ^ { p \\theta + p - 1 } + f _ { g , h } ( t ) \\right ) . \\end{align*}"}
+{"id": "9148.png", "formula": "\\begin{align*} y _ { 1 , [ 0 , \\kappa _ { 1 } - 1 ] } & = \\varphi _ { 1 , [ 0 , \\kappa _ { 1 } - 1 ] } ( \\zeta _ { [ - q _ { 1 } , - 1 ] } , x ) \\\\ y _ { 1 , [ \\kappa _ { 1 } ] } & = v _ { 1 } \\\\ y _ { r e s t _ { 1 } , [ 0 , K _ { r e s t _ { 1 } } - 1 ] } & = \\varphi _ { r e s t _ { 1 } , [ 0 , K _ { r e s t _ { 1 } } - 1 ] } ( \\zeta _ { [ - q _ { 1 } , - 1 ] } , x ) \\\\ y _ { r e s t _ { 1 } , [ K _ { r e s t _ { 1 } } ] } & = \\varphi _ { r e s t _ { 1 } , [ K _ { r e s t _ { 1 } } ] } ( \\zeta _ { [ - q _ { 1 } , - 1 ] } , x , v _ { 1 } ) \\end{align*}"}
+{"id": "1075.png", "formula": "\\begin{align*} \\| e ^ { - t H ^ { \\beta } } g \\| _ { L ^ { \\phi } } & \\leq \\inf \\left \\{ \\lambda > 0 : t ^ { \\frac p { p - 1 } } \\ , \\left ( \\exp \\left \\{ \\left ( { C } t ^ { - \\frac p { p - 1 } } \\frac { \\| g \\| _ { L ^ 1 } } { \\lambda } \\right ) ^ { 2 p } \\right \\} - 1 \\right ) \\leq 1 \\right \\} \\\\ & = { C } t ^ { - \\frac p { p - 1 } } \\left ( \\log ( t ^ { - \\frac p { p - 1 } } + 1 ) \\right ) ^ { - \\frac 1 { 2 p } } \\| g \\| _ { L ^ 1 } . \\end{align*}"}
+{"id": "9173.png", "formula": "\\begin{align*} \\begin{array} { c c l } y _ { 1 , [ 2 ] } ^ { 1 } & = & v _ { 1 } ^ { 1 } \\\\ y _ { 2 , [ 2 ] } ^ { 1 } & = & v _ { 2 } ^ { 1 } \\ , , \\end{array} \\end{align*}"}
+{"id": "4126.png", "formula": "\\begin{align*} ( T _ \\varepsilon ^ { \\vec { v } } ) _ { n , j - 1 } = - ( T _ \\varepsilon ^ { \\vec { v } } ) _ { 1 , j } \\alpha _ 0 , \\end{align*}"}
+{"id": "1845.png", "formula": "\\begin{align*} 2 \\frac { \\int _ { b = \\theta _ { 2 } } ^ { b = \\theta _ { 1 } } e ^ { \\frac { \\rho r s \\cos ( b ) } { 1 - \\rho ^ { 2 } } } \\ , \\d b } { 2 \\pi \\int _ { 0 } ^ { 2 \\pi } e ^ { \\frac { \\rho r s \\cos a } { 1 - \\rho ^ { 2 } } } \\d a } \\leq 2 \\frac { \\int _ { b = 0 } ^ { b = \\theta _ { 1 } - \\theta _ { 2 } } e ^ { \\frac { \\rho r s \\cos ( b ) } { 1 - \\rho ^ { 2 } } } \\ , \\d b } { 2 \\pi \\int _ { 0 } ^ { 2 \\pi } e ^ { \\frac { \\rho r s \\cos a } { 1 - \\rho ^ { 2 } } } \\d a } < \\frac { \\theta _ { 1 } - \\theta _ { 2 } } { 2 \\pi ^ { 2 } } . \\end{align*}"}
+{"id": "3007.png", "formula": "\\begin{align*} \\| \\phi ( A \\otimes B ) \\| _ { ( p , k ) } = \\| A \\otimes B \\| _ { ( p , k ) } \\hbox { f o r a l l \\quad } A \\in M _ m \\hbox { a n d } B \\in M _ n . \\end{align*}"}
+{"id": "4424.png", "formula": "\\begin{align*} 0 = K _ { \\omega , \\mathbf { c } } ( \\Psi ) = \\partial _ { \\lambda } S _ { \\omega , \\mathbf { c } } ( \\lambda \\Psi ) \\big | _ { \\lambda = 1 } = \\sum _ { j = 1 } ^ 3 \\langle D _ j S _ { \\omega , \\mathbf { c } } ( \\Psi ) , \\psi _ j \\rangle = \\eta \\sum _ { j = 1 } ^ 3 \\langle D _ j K _ { \\omega , \\mathbf { c } } ( \\Psi ) , \\psi _ j \\rangle = \\eta \\partial _ { \\lambda } K _ { \\omega , \\mathbf { c } } ( \\lambda \\Psi ) \\big | _ { \\lambda = 1 } . \\end{align*}"}
+{"id": "6117.png", "formula": "\\begin{align*} U _ { \\lambda } : = \\left \\{ ( x , y , t ) \\in \\mathcal { Q } _ { r _ { 1 } } \\ , : \\ , | D ^ { \\tau } d _ { s } u ( x , y , t ) | \\geq \\lambda \\right \\} \\subset \\left ( \\bigcup \\limits _ { i \\in \\mathbb { N } } \\mathcal { Q } _ { 5 \\rho _ { { i } } } \\left ( z _ { i } \\right ) \\right ) \\bigcup \\left ( \\bigcup \\limits _ { \\mathcal { K } \\times I \\in \\mathcal { A } } \\mathcal { K } \\times I \\right ) \\end{align*}"}
+{"id": "9018.png", "formula": "\\begin{align*} V ( p , t , \\rho ) = \\inf _ { ( r , \\sigma , \\pi ) } J ( p , t , ( r , \\sigma , \\pi ) , \\rho ) \\end{align*}"}
+{"id": "6850.png", "formula": "\\begin{align*} f _ { N , j , n } ( b _ 1 , \\ldots , b _ N ) = \\sum _ { M \\in \\Z } \\left ( ( - 1 ) ^ M \\sum _ { u \\in \\mathbb { Z } } a ^ { j - u } q ^ { ( M - u ) ( n - u ) + ( j - u ) ( n - N ) } \\begin{bmatrix} M \\\\ u \\end{bmatrix} \\begin{bmatrix} N - M \\\\ j - u \\end{bmatrix} \\right ) \\sum _ { \\substack { I \\subseteq \\{ 1 , \\dots , N \\} \\\\ s . t . | I | = M } } \\prod _ { i \\in I } b _ { i } . \\end{align*}"}
+{"id": "5031.png", "formula": "\\begin{align*} g = \\prod _ { k \\in I _ g ^ c } \\rho _ k ^ { s _ k } , \\end{align*}"}
+{"id": "2489.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\delta _ { \\xi } \\left [ \\mathcal { S } ( \\xi , - \\eta ) + \\Lambda \\mathcal { Q } ( \\xi , - \\eta ) \\right ] = 0 , \\quad \\delta _ { - \\eta } \\left [ \\mathcal { S } ( \\xi , - \\eta ) + \\Lambda \\mathcal { Q } ( \\xi , - \\eta ) \\right ] = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "2694.png", "formula": "\\begin{align*} \\Phi _ s ( n ) = n \\prod \\limits _ { \\substack { p ^ s \\mid n \\\\ p } } \\left ( 1 - \\frac { 1 } { p ^ s } \\right ) . \\end{align*}"}
+{"id": "5907.png", "formula": "\\begin{align*} \\theta _ { 1 / 2 } ( \\tfrac { a z + b } { c z + d } , \\epsilon ) = \\lambda ^ { \\pm } ( \\gamma , \\epsilon ) \\sqrt { \\det \\gamma ( c z + d ) } \\sum _ { n \\in \\Z } e ^ { i ( \\det \\gamma ) \\epsilon \\pi n ^ 2 z } , \\end{align*}"}
+{"id": "558.png", "formula": "\\begin{align*} E _ { \\varepsilon } ( t , \\xi ) : = ( S _ { \\varepsilon } ( t ) W _ { \\varepsilon } ( t , \\xi ) , W _ { \\varepsilon } ( t , \\xi ) ) , \\end{align*}"}
+{"id": "5135.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = 1 } ^ N f _ n \\Big \\| _ X \\leq \\sum _ { n = 1 } ^ N K _ X ^ n \\| f _ n \\| _ X . \\end{align*}"}
+{"id": "5640.png", "formula": "\\begin{align*} \\aligned \\left \\{ \\begin{array} { l l l } - \\Delta u + \\mu _ 1 u = ( I _ \\mu \\ast H _ { 1 1 } u ) K _ { 1 1 } + ( I _ \\mu \\ast H _ { 1 2 } v ) K _ { 1 2 } \\ & \\mathbb { R } ^ N , \\\\ - \\Delta v + \\mu _ 2 v = ( I _ \\mu \\ast H _ { 2 1 } v ) K _ { 2 1 } + ( I _ \\mu \\ast H _ { 2 2 } u ) K _ { 2 2 } \\ & \\mathbb { R } ^ N , \\end{array} \\right . \\endaligned \\end{align*}"}
+{"id": "441.png", "formula": "\\begin{align*} R _ { \\mu \\nu } - \\frac 1 2 R g _ { \\mu \\nu } = 8 \\pi T _ { \\mu \\nu } , \\ \\ \\mu , \\nu = 0 , 1 , 2 , 3 . \\end{align*}"}
+{"id": "8070.png", "formula": "\\begin{align*} \\mathcal { G } = \\{ u \\in X : G ( u ) = 1 \\} . \\end{align*}"}
+{"id": "8274.png", "formula": "\\begin{align*} g _ \\beta ( x , t _ 1 , t _ 2 ) = \\sum _ { n = 0 } ^ \\infty \\sum _ { m = 0 } ^ \\infty \\frac { t _ 1 ^ n t _ 2 ^ m } { n ! m ! } I _ { 2 n + 3 m + \\beta } ( 2 \\sqrt { x } ) . \\end{align*}"}
+{"id": "4690.png", "formula": "\\begin{align*} \\lambda _ \\chi ^ { \\rm s t i f f ( s o f t ) } ( \\vect g , \\vect h ) : = a _ { \\chi } ^ { \\rm s t i f f ( s o f t ) } \\bigl ( \\Pi _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\vect g , \\Pi _ \\chi ^ { \\rm s t i f f ( s o f t ) } \\vect h \\bigr ) , \\vect g , \\vect h \\in H ^ { 1 / 2 } ( \\Gamma ; \\C ^ 3 ) . \\end{align*}"}
+{"id": "5258.png", "formula": "\\begin{align*} L _ S [ f ] = \\sum _ { T \\subseteq S } ( - 1 ) ^ { | T | } E _ T [ f ] . \\end{align*}"}
+{"id": "9308.png", "formula": "\\begin{align*} x _ j + \\left ( \\sum _ { k = 1 } ^ n a _ { j k } x _ k \\right ) ^ d , j = 1 , \\ldots , n . \\end{align*}"}
+{"id": "509.png", "formula": "\\begin{align*} \\Delta u ( x ) = f ( x ) , x \\in \\Omega = ( 0 , 1 ) ^ 2 , u | _ { \\partial \\Omega } = 0 , \\end{align*}"}
+{"id": "5235.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c c } x = 0 : & 0 & e _ 1 \\\\ x = 1 : & 0 & e _ 2 \\\\ x = \\infty : & e _ 3 & e _ 4 \\end{array} \\right ) \\to \\left ( \\begin{array} { c c c } 0 & - e _ 1 \\\\ 0 & - e _ 2 \\\\ 1 - e _ 3 & 1 - e _ 4 \\end{array} \\right ) . \\end{align*}"}
+{"id": "27.png", "formula": "\\begin{align*} \\mathbf { T } _ { K ^ \\circ } ( R ) & = \\{ x \\in K \\otimes _ { \\mathbb { Q } } R \\mid c ( x ) x \\in R ^ \\times \\} , \\\\ \\mathbf { T } _ { K ^ \\circ } ^ 1 ( R ) & = \\{ x \\in K \\otimes _ { \\mathbb { Q } } R \\mid c ( x ) x = 1 \\} . \\end{align*}"}
+{"id": "3788.png", "formula": "\\begin{align*} \\boldsymbol { W } _ { p } ( \\mu , \\nu ) & = \\left [ \\inf _ { \\pi \\in \\Pi ( \\mu , \\nu ) } \\int _ { \\mathcal { X } \\times \\mathcal { X } } \\boldsymbol { d } ^ p \\ , d \\pi \\right ] ^ { 1 / p } . \\end{align*}"}
+{"id": "7928.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) \\Phi = - \\lambda \\Phi ^ 3 + \\xi ; \\end{align*}"}
+{"id": "1674.png", "formula": "\\begin{align*} ( \\phi _ 1 ^ { ( p ) } \\circ \\pi ) ( x , y ) & = \\phi _ 1 ^ { ( p ) } ( x ^ p , y ^ p ) \\\\ & = ( f ^ { ( p ) } ( x ^ p , y ^ p ) , g ^ { ( p ) } ( x ^ p , y ^ p ) ) \\\\ & = ( ( f ( x , y ) ) ^ p , ( g ( x , y ) ) ^ p ) \\\\ & = ( \\pi \\circ \\phi _ 1 ) ( x , y ) . \\end{align*}"}
+{"id": "4782.png", "formula": "\\begin{align*} w ( x ) : = \\frac { \\phi ( x ) } { \\sqrt { x + 4 } } , \\check { h } ( x ) : = 4 \\cosh \\left ( \\tfrac { x } { 2 } \\right ) \\int _ { 0 } ^ { \\infty } w \\left ( 4 \\sinh ^ 2 \\left ( \\tfrac { x } { 2 } \\right ) + y ^ 2 \\right ) \\d y , \\end{align*}"}
+{"id": "1898.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\langle D _ u \\theta ( u ^ * ) , v \\rangle _ \\mathcal { U } + \\langle \\tilde { \\lambda } , S v \\rangle _ { * } = 0 \\forall \\ v \\in \\mathcal { U } ; \\\\ & - \\langle \\tilde { \\lambda } , \\zeta - S u ^ * \\rangle _ * \\geq 0 \\forall \\ \\zeta \\in K , \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "5524.png", "formula": "\\begin{align*} \\mathbb { \\overline { P } } _ { \\mu , x , g ^ { - 1 } } ( A ) = \\overline { \\mathbb { P } } _ { \\mu , g . x , i d } \\left ( g . A \\right ) . \\end{align*}"}
+{"id": "4914.png", "formula": "\\begin{align*} \\begin{aligned} \\Pr ( T _ k > t ) = \\Pr ( X _ t \\le k - 1 ) , \\Pr ( T _ k \\le t ) = \\Pr ( X _ t > k - 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "447.png", "formula": "\\begin{align*} \\ddot A ( t ) x + \\frac \\gamma { \\gamma - 1 } A ( t ) ^ { - \\top } \\det ( A ( t ) ) ^ { 1 - \\gamma } \\nabla _ x w & = 0 , \\ \\ \\gamma > 1 , \\end{align*}"}
+{"id": "5416.png", "formula": "\\begin{align*} y ' ( t ) = f ( t , y ( t ) ) , y ( t _ 0 ) = y _ 0 , \\end{align*}"}
+{"id": "6477.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } e ^ { - z | x | ^ 2 + \\xi \\cdot x } \\dd x = \\frac { \\pi } { z } e ^ { \\frac { \\xi \\cdot \\xi } { 4 z } } . \\end{align*}"}
+{"id": "5958.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( g ) f ( [ \\epsilon , w ] ) = \\Upsilon ( g ) f ( [ ( \\det g ) \\epsilon , w g ^ { s ( ( \\det g ) \\epsilon ) } ] ) , \\end{align*}"}
+{"id": "277.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( 1 - y ^ m z ^ n \\right ) ^ { \\frac { m ^ 2 } { n ^ 3 } } = \\left ( 1 - y z \\right ) ^ { \\frac { y } { 1 - y } } \\end{align*}"}
+{"id": "3915.png", "formula": "\\begin{align*} \\sup _ { \\gamma \\in \\mathcal { P } _ { \\mathrm { D } } } I _ { \\mathrm { D } , \\lambda } [ \\gamma ] = \\sup _ { \\gamma \\in \\mathcal { P } _ { \\mathrm { D } } } \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 , \\gamma ) } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda \\ , d \\pi . \\end{align*}"}
+{"id": "5246.png", "formula": "\\begin{align*} \\begin{array} { r l } s _ { 1 1 } & = e _ 1 + e _ 2 + e _ 3 , s _ { 1 2 } = e _ 4 + e _ 5 + e _ 6 , s _ { 1 3 } = e _ 7 + e _ 8 + e _ 9 , \\\\ s _ { 2 1 } & = e _ 1 e _ 2 + e _ 1 e _ 3 + e _ 2 e _ 3 , s _ { 2 2 } = e _ 4 e _ 5 + e _ 4 e _ 6 + e _ 5 e _ 6 , \\\\ s _ { 2 3 } & = e _ 7 e _ 8 + e _ 7 e _ 9 + e _ 8 e _ 9 , s _ { 3 1 } = e _ 1 e _ 2 e _ 3 , s _ { 3 2 } = e _ 4 e _ 5 e _ 6 , \\\\ s _ { 3 3 } & = e _ 7 e _ 8 e _ 9 , s = - ( s _ { 1 1 } + s _ { 1 2 } + s _ { 1 3 } - 6 ) / 3 . \\end{array} \\end{align*}"}
+{"id": "2713.png", "formula": "\\begin{align*} \\begin{aligned} J _ 1 \\geq C s ^ 3 \\lambda ^ 4 \\iint _ Q \\xi ^ 3 \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d y d t - C s ^ 3 \\lambda ^ 3 \\int _ 0 ^ T \\int _ { \\omega _ 0 } \\xi ^ 3 | u | ^ 2 d x d y d t , \\end{aligned} \\end{align*}"}
+{"id": "6338.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { 2 } c x ^ 2 + o ( x ^ 2 ) f ' ( x ) = c x + o ( x ) . \\end{align*}"}
+{"id": "2584.png", "formula": "\\begin{align*} B _ \\vee \\Biggl ( \\sum _ { | \\tau | = \\ell } \\omega ( \\tau ) \\cdot \\tau \\Biggr ) = \\sum _ { | \\tau | = \\ell + 1 } \\omega ( \\tau ) \\cdot \\tau , \\end{align*}"}
+{"id": "5510.png", "formula": "\\begin{align*} \\int _ { \\tilde { X } } g ^ { - 1 } . f d \\tilde { \\lambda } = \\int _ { G / \\Gamma } \\int _ { X } f _ { z } d \\boldsymbol { \\beta } ( \\tau ( z ) ^ { - 1 } g P ) d m _ { G / \\Gamma } ( z ) . \\end{align*}"}
+{"id": "3798.png", "formula": "\\begin{align*} Y = D Y _ 2 + ( 1 - D ) Y _ 1 . \\end{align*}"}
+{"id": "6384.png", "formula": "\\begin{align*} C : 9 x ( x ^ 2 + 2 0 r ^ 2 ) = y ^ 3 . \\end{align*}"}
+{"id": "5406.png", "formula": "\\begin{align*} L ( u , \\chi ) = \\exp \\left ( \\sum _ { k \\ge 1 } \\frac { u ^ k } { k } \\sum _ { f \\in \\mathcal { M } _ { k , q } } \\chi ( f ) \\Lambda ( f ) \\right ) . \\end{align*}"}
+{"id": "3480.png", "formula": "\\begin{align*} A ( u , v ) = \\max _ { ( \\sigma , \\tau ) \\in S _ { u , v } } d _ { T V } ( \\mu _ { v } ( \\cdot \\mid \\sigma ) , \\mu _ v ( \\cdot \\mid \\tau ) ) , \\end{align*}"}
+{"id": "5845.png", "formula": "\\begin{align*} \\pi _ 2 ( \\mathcal { X } _ { j _ 0 } ) & = \\pi _ 2 ( y ) + j _ 0 H \\in \\pi _ 2 ( w ) + j _ 0 H + [ 0 , H ' ) = \\pi _ 2 ( w _ { j _ 0 } ) + [ 0 , H ' ) \\end{align*}"}
+{"id": "1144.png", "formula": "\\begin{align*} f _ { 1 } ( x ) = f _ { 2 } ( x ) A f _ { 1 } ( x ) \\neq A f _ { 2 } ( x ) \\end{align*}"}
+{"id": "3248.png", "formula": "\\begin{align*} & U _ 1 = \\{ z \\in \\mathbb R ^ N : \\| z - x \\| > 5 { \\mathfrak R } ^ k , \\| z - \\sigma _ 1 ( x ) \\| \\le 5 { \\mathfrak R } ^ k \\} , \\\\ & U _ { j + 1 } = \\{ z \\in \\mathbb R ^ N : \\| z - x \\| > 5 { \\mathfrak R } ^ k , \\| z - \\sigma _ { j + 1 } ( x ) \\| \\le 5 { \\mathfrak R } ^ k \\} \\setminus \\Big ( \\bigcup _ { i = 1 } ^ j U _ i \\Big ) \\quad \\mbox { f o r } 1 \\leq j \\leq | G | - 1 . \\end{align*}"}
+{"id": "9254.png", "formula": "\\begin{align*} \\Delta _ x u - u _ { t t } = 0 , u ( x , 0 ) = f ( x ) , u _ t ( x , 0 ) = g ( x ) , \\end{align*}"}
+{"id": "6307.png", "formula": "\\begin{align*} \\tilde D _ \\epsilon : = \\{ ( \\bar q , \\lambda ) : \\bar q \\in V _ q , \\ , 0 < \\sqrt { 2 H ( \\lambda ) } < \\epsilon \\} , \\end{align*}"}
+{"id": "314.png", "formula": "\\begin{align*} \\zeta ( s , t ) : = \\sum _ { n > m > 0 } \\frac { 1 } { n ^ s m ^ t } = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ s } \\sum _ { m = 1 } ^ { n - 1 } \\frac { 1 } { m ^ t } \\Re ( s ) > 1 , \\ ; \\Re ( t ) > 1 . \\end{align*}"}
+{"id": "8885.png", "formula": "\\begin{align*} \\operatorname { s y s } ( M , g ) = \\inf \\{ l ( c ) \\mid c \\colon \\} , \\end{align*}"}
+{"id": "4204.png", "formula": "\\begin{align*} S ( \\rho _ A ) = \\lim _ { \\epsilon \\to 0 } \\frac 1 { 4 \\pi i } \\int _ { \\Gamma ( \\epsilon ) } e ( 1 + \\epsilon , \\lambda ) \\frac { d \\log D _ L ( \\lambda ) } { d \\lambda } d \\lambda , \\end{align*}"}
+{"id": "6286.png", "formula": "\\begin{align*} d _ \\lambda h = \\sigma ( \\cdot , \\vec h ( \\lambda ) ) , \\qquad \\forall \\ , \\lambda \\in T ^ * M , \\end{align*}"}
+{"id": "5744.png", "formula": "\\begin{align*} ( \\sharp _ 1 ) \\left \\{ \\begin{array} { @ { \\ , } l l l } p = 0 \\\\ q = - a \\\\ a \\ne 0 \\\\ b = 0 \\\\ c = a \\\\ d = 0 \\end{array} \\right . \\ , \\ , { \\rm { o r } } \\ , \\ , \\ , \\ , ( \\sharp _ 2 ) \\left \\{ \\begin{array} { @ { \\ , } l l l } p \\ne 0 \\\\ q = - a \\\\ a \\ne 0 \\\\ b = 0 \\\\ c = a \\\\ d = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1593.png", "formula": "\\begin{align*} ( ^ { c } _ { 0 } { D } _ { t } \\ , u , v ) \\ , + \\ , a \\big ( l ( u ) \\big ) \\ , ( \\nabla u , \\nabla v ) & = \\big ( f , v \\big ) , & & \\forall v \\in H ^ 1 _ 0 ( \\Omega ) , & & & \\\\ \\big ( u ( \\cdot , 0 ) , v \\big ) & = \\big ( u _ 0 ( \\cdot ) , v \\big ) , & & \\mbox { i n } \\ ; \\ , \\Omega . & & & \\end{align*}"}
+{"id": "8924.png", "formula": "\\begin{gather*} Q _ { l j } ( 0 , \\dots , 0 ) = 0 \\bigl \\| B _ l ^ { ( j ) } \\bigr \\| _ l \\leq Q _ { l j } \\big ( \\nu _ 1 , \\sqrt { \\nu _ 2 } , \\dots , \\sqrt [ k ] { \\nu _ k } \\big ) . \\end{gather*}"}
+{"id": "4468.png", "formula": "\\begin{align*} \\frac { 1 } { 3 \\mu _ { \\omega , 0 } } | c P ( \\Phi _ { \\omega } ) | \\le \\frac { 1 } { 3 \\mu _ { \\omega , 0 } } | c | \\cdot 3 \\| \\Phi _ { \\omega } \\| _ { \\mathcal { H } ^ 1 } ^ 2 = \\frac { | c | } { \\mu _ { \\omega , 0 } } \\| \\Phi _ { \\omega } \\| _ { \\mathcal { H } ^ 1 } ^ 2 , \\end{align*}"}
+{"id": "76.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ s _ h } = \\| \\langle h D \\rangle ^ s u \\| _ { L ^ 2 } . \\end{align*}"}
+{"id": "8405.png", "formula": "\\begin{align*} V = N \\oplus N ^ { - 1 } K _ X , \\gamma = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , \\beta = \\begin{pmatrix} \\nu & q _ 2 \\\\ q _ 2 & \\mu \\end{pmatrix} . \\end{align*}"}
+{"id": "2470.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\phi ( 0 , x ) = \\phi _ 0 ( x ) , \\quad \\psi ( 0 , x ) = \\psi _ 0 ( x ) , x \\in \\mathbb { R } ^ N , \\end{array} \\right . \\end{align*}"}
+{"id": "4699.png", "formula": "\\begin{align*} \\widehat { P } _ \\chi - \\widehat { P } _ 0 = \\widehat { P } _ \\chi ( \\widehat { P } _ \\chi - \\widehat { P } _ 0 ) + ( \\widehat { P } _ \\chi - \\widehat { P } _ 0 ) \\widehat { P } _ 0 . \\end{align*}"}
+{"id": "7603.png", "formula": "\\begin{align*} H _ { 2 , 1 } ( F _ { f } / 2 ) & = \\frac { 1 } { 2 3 0 4 } \\left ( - 3 \\tau ^ 4 _ { 1 } + 1 2 ( 1 - \\tau ^ 2 _ { 1 } ) \\tau ^ 2 _ { 1 } \\tau _ { 2 } - 8 ( 1 - \\tau ^ 2 _ { 1 } ) ( 2 + \\tau ^ 2 _ { 1 } ) \\tau ^ 2 _ { 2 } \\right . \\\\ & \\left . + 2 4 \\tau _ { 1 } \\tau _ { 3 } ( 1 - \\tau ^ 2 _ { 1 } ) ( 1 - | \\tau ^ 2 _ { 2 } | ) \\right ) . \\end{align*}"}
+{"id": "832.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\omega } \\| \\phi _ k ( a b ) - \\phi _ k ( a ) \\phi _ k ( b ) \\| = 0 \\lim _ { k \\rightarrow \\omega } \\mathrm { t r } _ \\mathcal Q ( \\phi _ k ( a ) ) = \\mathrm { t r } _ G ( a ) . \\end{align*}"}
+{"id": "2690.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { 1 \\leq m _ 1 , m _ 2 , \\ldots , m _ k \\leq n ^ s \\\\ ( m _ 1 , n ^ s ) _ s = 1 \\\\ ( m _ 2 , n ^ s ) _ s = 1 \\\\ \\cdots \\\\ ( m _ k , n ^ s ) _ s = 1 \\\\ 1 \\leq b _ 1 , b _ 2 , \\ldots , b _ r \\leq n ^ s } } ( m _ 1 - a _ 1 , m _ 2 - a _ 2 , & \\ldots , m _ k - a _ k , b _ 1 , b _ 2 , \\cdots , b _ r , n ^ s ) _ s \\\\ & = \\Phi _ s ( n ^ s ) ^ k \\sum \\limits _ { \\substack { d ^ s | n ^ s } } \\frac { ( d ^ s ) ^ r } { \\Phi _ s ( \\frac { n ^ s } { d ^ s } ) ^ { k - 1 } } . \\end{align*}"}
+{"id": "640.png", "formula": "\\begin{align*} \\omega ( b ) = \\langle \\sigma ( b ) \\xi , \\xi \\rangle , b \\in B . \\end{align*}"}
+{"id": "5532.png", "formula": "\\begin{align*} \\sup \\left \\{ n : H _ { k } \\cap K _ { n } = \\bar { H } \\cap K _ { n } \\right \\} \\underset { k \\to \\infty } { \\longrightarrow } \\infty . \\end{align*}"}
+{"id": "7743.png", "formula": "\\begin{align*} y \\bigl ( \\sum _ { j = 1 } ^ { q - 1 } \\epsilon _ { T _ { j } ( X ) } , \\ , T _ { q } ( X ) \\bigr ) = T _ { q } ( N ) = y \\bigl ( \\sum _ { j = 1 } ^ { q - 1 } \\epsilon _ { T _ { j } ( X ) } , \\ , T _ { q } ( Y ) \\bigr ) . \\end{align*}"}
+{"id": "1535.png", "formula": "\\begin{align*} h ( t ) = t ^ { \\ell + y } f ( t ) - a t ^ { 2 k + \\ell + x } f ( t ^ { - 1 } ) . \\end{align*}"}
+{"id": "5991.png", "formula": "\\begin{align*} d \\overline { \\Pi } _ { \\psi } ( e ^ - ) f ( [ 1 , x ] ) & = d \\overline { \\Pi } _ { \\psi } ( e ^ - ) f _ 1 ( x ) \\\\ & = \\frac { d } { d t } \\Big \\{ \\overline { \\Pi } _ { \\psi } ( \\exp ( t e ^ - ) ) f _ 1 ( x ) \\Big \\} \\Big | _ { t = 0 } \\\\ & = \\int _ { \\R } ( - \\pi i y ^ 2 ) \\widehat { f _ 1 } ( y ) e ^ { 2 \\pi i x y } d y \\\\ & = - \\frac { 1 } { 4 \\pi i } \\frac { d ^ 2 } { d x ^ 2 } \\int _ { \\R } \\widehat { f _ 1 } ( y ) e ^ { 2 \\pi i x y } d y \\\\ . & = - \\frac { 1 } { 4 \\pi i } \\frac { d ^ 2 } { d x ^ 2 } f _ 1 ( x ) . \\end{align*}"}
+{"id": "1743.png", "formula": "\\begin{align*} \\frac { \\delta F } { \\delta m } ( m ' , x ) - \\frac { \\delta F } { \\delta m } ( m , x ) = \\int _ { 0 } ^ { 1 } \\int _ { \\mathcal { M } } \\frac { \\delta ^ 2 F } { \\delta m ^ 2 } ( m + \\lambda ( m ' - m ) , x , x ' ) \\left ( m ' - m \\right ) ( \\mathrm { d } x ' ) \\ , \\mathrm { d } \\lambda . \\end{align*}"}
+{"id": "7714.png", "formula": "\\begin{align*} & n _ 0 ( u ) z _ 0 = z _ 0 , n _ 0 ( u ) x = - ( u , x ) z _ 0 + x , n _ 0 ( u ) z _ 0 ^ \\ast = - Q ( u ) z _ 0 + u + z _ 0 ^ \\ast , \\\\ & m _ 0 ( a , g _ 1 ) z _ 0 = a z _ 0 , m _ 0 ( a , g _ 1 ) x = g _ 1 x , m _ 0 ( a , g _ 1 ) z _ 0 ^ \\ast = a ^ { - 1 } z _ 0 ^ \\ast \\end{align*}"}
+{"id": "3567.png", "formula": "\\begin{align*} B ( T _ { * } ) = - \\frac { 2 \\pi \\sigma ^ { 3 } } { n } T _ { * } ^ { - 3 / n } \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { k ! } \\Gamma \\left ( \\frac { k m - 3 } { n } \\right ) T _ { * } ^ { - ( n - m ) k / n } , \\end{align*}"}
+{"id": "7758.png", "formula": "\\begin{gather*} \\Phi ^ - ( \\nu , m ) \\coloneqq \\{ \\beta \\in P ( \\R ^ m ) : \\beta = \\phi ( \\nu ) \\phi \\in \\mathcal { M } \\} , \\\\ \\Phi ^ + ( \\mu , n ) \\coloneqq \\{ \\alpha \\in P ( \\R ^ n ) : \\mu = \\phi ( \\alpha ) \\phi \\in \\mathcal { M } \\} , \\end{gather*}"}
+{"id": "7078.png", "formula": "\\begin{align*} \\alpha _ i = \\nu \\left ( \\frac { d } { d x } ( x - a _ i ) \\right ) - \\nu ( x - a _ i ) = \\nu ( 1 ) - \\nu ( x - a _ i ) = - \\nu ( x - a _ i ) . \\end{align*}"}
+{"id": "3039.png", "formula": "\\begin{align*} \\| A _ s ( \\mathrm { i , j } ) + x B _ s ( \\mathrm { i , j } ) \\| _ { ( p , k ) } ^ p = 2 ^ { s - 1 } + a _ s x ^ p \\hbox { f o r } 0 < x < 1 , \\end{align*}"}
+{"id": "2487.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\xi = ( \\xi _ { \\infty } ) _ { \\beta } , \\eta = ( \\eta _ { \\infty } ) _ { \\beta } , \\end{array} \\right . \\end{align*}"}
+{"id": "8364.png", "formula": "\\begin{align*} \\xi = \\prod _ { v \\in V ( X ) } c ( v ) \\sum _ { T \\subset X } \\prod _ { e \\in E ( T ) } c ( e ) ^ { - 1 } , \\end{align*}"}
+{"id": "1952.png", "formula": "\\begin{align*} L _ a = \\frac { 1 } { n } \\sum _ { j , k = 1 } ^ n a _ { j \\bar k } \\frac { \\partial ^ 2 } { \\partial z _ j \\partial \\bar z _ k } . \\end{align*}"}
+{"id": "256.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - y ^ m z ^ n } \\right ) ^ { \\frac { m ^ 1 } { n ^ 2 } } = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { y - ( 1 + n ) y ^ { n + 1 } + n y ^ { n + 2 } } { ( 1 - y ) ^ 2 } \\right ) \\frac { z ^ n } { n ^ 2 } \\right \\} \\end{align*}"}
+{"id": "8562.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { - 1 } S _ j ( r _ N ) = \\nu X \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s , j \\geq 1 , \\end{align*}"}
+{"id": "8218.png", "formula": "\\begin{align*} p _ s = \\exp \\left [ - \\frac { \\lambda _ d s } { \\widehat { R } ^ 2 ( s ) } ( 1 + o ( 1 ) ) \\right ] = \\exp \\left [ - \\frac { c ( d , \\nu ) ( t - ( m ( t ) + \\ell ( t ) ) ) } { ( \\log \\ell ( t ) ) ^ { 2 / d } } ( 1 + o ( 1 ) ) \\right ] . \\end{align*}"}
+{"id": "6671.png", "formula": "\\begin{align*} \\rho \\ , u _ t - \\Delta u + ( - \\Delta ) ^ s u = \\ , 0 \\textrm { i n } \\ ; \\ ; S _ T \\ , , \\end{align*}"}
+{"id": "3326.png", "formula": "\\begin{align*} e ^ { \\pm H \\widetilde { d } } = e ^ { \\pm H { \\rm R e } ( q ) } e ^ { \\pm H \\widetilde { e } } \\end{align*}"}
+{"id": "5104.png", "formula": "\\begin{align*} L _ { 0 } w = 0 B _ { R } \\implies \\left \\Vert D w \\right \\Vert _ { L ^ { \\infty } ( B _ { R / 4 } ) } \\leq C _ { 3 } ( \\lambda , n , p _ 0 , R ) \\left \\Vert w \\right \\Vert _ { L ^ { p _ 0 } ( B _ { R } ) } . \\end{align*}"}
+{"id": "7102.png", "formula": "\\begin{align*} \\mathcal { F } ( f ) ( \\xi ) : = \\int _ { \\mathbb { R } ^ n } e ^ { - i x \\cdot \\xi } f ( x ) \\ : x . \\end{align*}"}
+{"id": "2970.png", "formula": "\\begin{align*} \\sigma \\cdot ( x , w , z ) = ( \\sigma \\cdot x , \\sigma \\cdot w , z ) \\ . \\end{align*}"}
+{"id": "8973.png", "formula": "\\begin{align*} \\int _ M \\frac { S } { m - 1 } w ^ 2 = \\int _ M | \\nabla w | ^ 2 \\ , . \\end{align*}"}
+{"id": "2550.png", "formula": "\\begin{align*} \\lambda _ t = \\lambda _ { j _ \\ell } = \\lambda _ { j _ \\ell ^ * } = \\pm 1 . \\end{align*}"}
+{"id": "6813.png", "formula": "\\begin{align*} ( \\Lambda _ { 2 , 0 } ) ^ { \\sharp } = S ^ { n - 1 } _ 0 \\times C ' \\end{align*}"}
+{"id": "4390.png", "formula": "\\begin{align*} I _ { 1 , j } = \\sum _ { k = 1 } ^ d ( \\nabla \\varphi _ { j } ^ { ( k ) } \\cdot \\nabla \\theta _ { \\epsilon } , \\varphi _ { j } ^ { ( k ) } ) , \\ \\ \\ I _ { 2 , j } = ( i ( \\mathbf { c } \\cdot \\nabla ) \\varphi _ j , \\theta _ { \\epsilon } \\varphi _ j ) , \\ \\ ( \\varphi _ j = ( \\varphi _ j ^ { ( 1 ) } , \\cdots , \\varphi _ j ^ { ( d ) } ) ) \\end{align*}"}
+{"id": "3316.png", "formula": "\\begin{align*} \\rho _ w ( 1 + , w ) = \\overline { \\rho _ w ( 1 - , \\overline { w } ) } = \\overline { \\rho _ w ( 1 - , w ) } \\end{align*}"}
+{"id": "5910.png", "formula": "\\begin{align*} \\overline { c } _ { X ^ { \\ast } } ( g _ 1 , g _ 2 ) = ( x ( g _ 1 ) , x ( g _ 2 ) ) _ F ( - x ( g _ 1 ) x ( g _ 2 ) , x ( g _ 1 g _ 2 ) ) _ F ( ( - 1 ) ^ t , \\det ( 2 q ) ) _ F ( - 1 , - 1 ) _ F ^ { \\tfrac { t ( t - 1 ) } { 2 } } \\epsilon ( 2 q ) , \\end{align*}"}
+{"id": "5317.png", "formula": "\\begin{align*} - \\frac { 1 } { 8 e ^ 7 } \\frac { 1 } { p } < - | S | \\log r = - | S | \\end{align*}"}
+{"id": "3305.png", "formula": "\\begin{align*} \\widetilde { B } ( \\xi ) = e ^ { - i \\frac { \\pi + 2 \\theta } { 4 } } \\end{align*}"}
+{"id": "7298.png", "formula": "\\begin{align*} x ^ 4 + x y + 2 y ^ 3 = 0 . \\end{align*}"}
+{"id": "2336.png", "formula": "\\begin{align*} D _ { r , z } u ( t , x ) = & \\sum _ { j \\geq 1 } \\sum _ { n \\geq j } I _ { n - 1 } ( f _ j ^ { ( n ) } \\big ( \\cdot , r , z , t , x ) \\big ) \\\\ = & \\sum _ { j \\geq 1 } I _ { j - 1 } \\big ( f _ { j - 1 } ( \\cdot , r , z ) \\big ) \\sum _ { n \\geq j } I _ { n - j } \\big ( g _ { n - j } ( \\cdot , r , z , t , x ) \\big ) . \\end{align*}"}
+{"id": "3876.png", "formula": "\\begin{align*} \\varphi _ { \\lambda , \\ell } ( x _ 1 , \\dotsc , x _ L ) = \\min _ { x ' , d ( x ' ) = \\ell } \\sum _ { \\ell = 1 } ^ { L } \\lambda _ \\ell \\| x _ \\ell - x ' \\| _ { 2 } . \\end{align*}"}
+{"id": "1811.png", "formula": "\\begin{align*} [ b _ \\ell ( s ) , D _ k ^ * ( t ) ] & = \\int _ { \\Lambda ^ * } \\chi ^ \\perp ( p ) \\chi ^ \\perp ( p - k ) \\chi ( p - \\ell ) e ^ { i ( t - s ) E _ p } a _ { p - \\ell } ( s ) a _ { p - k } ( t ) \\d p \\\\ & + \\int _ { \\Lambda ^ * } \\chi ( h ) \\chi ( h + k ) \\chi ^ \\perp ( h + \\ell ) e ^ { i ( t - s ) E _ h } a _ { h + \\ell } ( s ) a _ { h + k } ( t ) \\d h \\ . \\end{align*}"}
+{"id": "5309.png", "formula": "\\begin{align*} \\| D _ { i , x } [ f ^ { = 1 } ] \\| _ 2 = | \\mathbb { E } [ D _ { i , x } f ] | = | \\hat { f } ( \\{ i \\} ) | \\ , | \\chi _ i ( x _ i ) | \\leq \\delta \\sqrt { \\frac { 1 - p } { p } } \\leq \\frac { \\sqrt { 2 } \\sigma } { \\sqrt { m p } } , \\end{align*}"}
+{"id": "3611.png", "formula": "\\begin{align*} p \\ = \\ v ( f ) + q \\ \\le \\ f + 9 7 \\ < \\ 1 0 ^ m - 1 0 0 + 9 7 \\ = \\ 1 0 ^ m - 3 . \\end{align*}"}
+{"id": "3622.png", "formula": "\\begin{align*} a _ 0 \\ = \\ \\begin{cases} 3 , & , \\\\ 8 , & . \\end{cases} \\end{align*}"}
+{"id": "2252.png", "formula": "\\begin{align*} w _ 0 ( r e ^ { i \\theta } ) : = i c - I + \\frac { 1 } { 2 \\pi } \\langle h _ b , P _ r ( \\theta - \\cdot ) \\rangle . \\end{align*}"}
+{"id": "1582.png", "formula": "\\begin{align*} \\sum _ { | q | _ { 1 } = n } \\frac { ( n + | x | _ { 1 } ) ! } { \\prod _ { i = 1 } ^ { d } ( q _ { i } + | x _ { i } | ) ! } \\leq d ^ { n + | x | _ { 1 } } \\end{align*}"}
+{"id": "1524.png", "formula": "\\begin{align*} s = \\left ( \\begin{smallmatrix} a _ { 1 , 1 } & 0 \\\\ 0 & a _ { 2 , 2 } \\end{smallmatrix} \\right ) = \\left ( \\begin{smallmatrix} a _ { 1 , 1 } & 0 \\\\ 0 & a _ { 1 , 1 } ^ { - 1 } \\end{smallmatrix} \\right ) \\ , \\ , s = \\left ( \\begin{smallmatrix} 0 & a _ { 1 , 2 } \\\\ - a _ { 2 , 1 } & 0 \\end{smallmatrix} \\right ) = \\left ( \\begin{smallmatrix} 0 & a _ { 1 , 2 } \\\\ - a _ { 1 , 2 } ^ { - 1 } & 0 \\end{smallmatrix} \\right ) . \\end{align*}"}
+{"id": "1282.png", "formula": "\\begin{align*} B = \\begin{bmatrix} A & I _ n \\\\ I _ n & A \\end{bmatrix} , \\end{align*}"}
+{"id": "3930.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } \\left ( 0 , \\delta _ 2 \\right ) = \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\inf _ { \\lambda _ 2 \\in \\mathbb { R } _ { + } } \\left [ \\lambda _ 2 \\delta _ 2 + \\int _ { \\mathcal { V } } g _ { \\lambda , 2 } \\ , d \\pi \\right ] = \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ { + } } \\left [ \\langle \\lambda , \\left ( 0 , \\delta _ 2 \\right ) \\rangle + \\int _ { \\mathcal { V } } g _ { \\lambda } \\ , d \\pi \\right ] . \\end{align*}"}
+{"id": "6721.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } d \\left ( \\bigcup _ { s \\leq r } \\operatorname { P e r } ( x , s ) \\right ) = 1 . \\end{align*}"}
+{"id": "8850.png", "formula": "\\begin{align*} y ^ h _ \\lambda ( t ) - v ^ h _ \\lambda ( t ) = \\int _ 0 ^ t S ( t - s ) ( - f ' _ \\lambda ( u _ \\lambda ) ) v ^ h _ \\lambda \\ , d s + \\int _ 0 ^ t S ( t - s ) \\sigma ' ( u _ \\lambda ) ( y ^ h _ \\lambda ( t ) - v ^ h _ \\lambda ( t ) ) B \\ , d W ( s ) , \\end{align*}"}
+{"id": "1501.png", "formula": "\\begin{align*} & \\ , \\cos ^ 3 \\alpha \\ , \\bigg ( \\ ! ( 1 + \\Delta / R ) ^ 3 \\bigg ( 1 + \\frac { 3 } { 2 } \\tan ^ 2 \\alpha \\ , ( \\Delta / R ) \\big ( 2 + O ( \\Delta / R ) \\big ) + \\tan ^ 4 \\alpha \\ , O ( \\Delta ^ 2 / R ^ 2 ) \\ ! \\bigg ) - 1 \\bigg ) \\\\ & = \\cos ^ 3 \\alpha \\ , \\Big ( \\ ! ( 1 + \\Delta / R ) ^ 3 - 1 + ( 1 + \\Delta / R ) ^ 3 ( \\Delta / R ) \\tan ^ 2 \\alpha \\ , \\big ( 3 + O ( \\Delta / R ) + \\tan ^ 2 \\alpha \\ , O ( \\Delta / R ) \\big ) \\ ! \\Big ) . \\end{align*}"}
+{"id": "8353.png", "formula": "\\begin{align*} B _ { V ^ 1 L ^ { p , q } ( \\Sigma , \\mu ) } \\subseteq \\bigcup _ { k = 1 } ^ m \\big ( g _ k + r B _ { L ^ { p ^ * , q } ( \\Sigma , \\mu ) } \\big ) . \\end{align*}"}
+{"id": "6096.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\sum _ { 0 \\leq n < 2 ^ { m } } ( 1 - \\frac { 2 n } { 2 ^ { m + 1 } } ) ^ { \\frac { \\gamma ( i + 1 ) - 1 - \\delta - \\frac { \\epsilon } { 2 } } { 1 - \\gamma + \\delta + \\epsilon } } \\frac { 1 } { 2 ^ { m + 1 } } = \\frac { 1 } { 2 } \\int _ { 0 } ^ { 1 } ( 1 - x ) ^ { \\frac { \\gamma ( i + 1 ) - \\delta - 1 - \\frac { \\epsilon } { 2 } } { 1 - \\gamma + \\delta + \\epsilon } } \\mathrm { d } x , \\end{align*}"}
+{"id": "1665.png", "formula": "\\begin{align*} \\mu ( \\bigcap _ { s \\in U } B _ s ) & = \\int \\prod _ s \\chi _ { B _ s } ( \\{ i _ j \\} _ { j \\in s } ) \\ , d \\mu ( i _ 1 , \\ldots , i _ n ) \\\\ & = \\int \\chi _ { B _ t } ( \\{ i _ j \\} _ { j \\in t } ) \\int \\prod _ { s \\neq t } \\chi _ { B _ s } ( \\{ i _ j \\} _ { j \\in s } ) \\ , d \\mu \\ , d \\mu ( \\{ i _ j \\} _ { j \\in [ n ] \\setminus t } ) . \\end{align*}"}
+{"id": "7826.png", "formula": "\\begin{align*} \\textsf { E } ( \\Xi ^ { 2 } ) ^ { p } & \\le ( C \\log n \\sum _ { i \\le N } \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } ) ^ { p } \\int _ { 0 } ^ { \\infty } \\exp ( - C _ { 1 } z \\log n \\sum _ { i \\le N } \\Vert B _ { i } \\Vert _ { 2 } ^ { 2 } ) p z ^ { p - 1 } \\ , d z \\\\ & = C _ { 2 } ^ { p } p \\int _ { 0 } ^ { \\infty } e ^ { - t } t ^ { p - 1 } \\ , d t \\lesssim C _ { 2 } ^ { p } p ^ { p + 1 } . \\end{align*}"}
+{"id": "7659.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { g } ^ 2 u = - \\lambda _ { } \\Delta _ g u , & \\Omega , \\\\ u = \\frac { \\partial u } { \\partial \\bf n } = 0 , & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "7139.png", "formula": "\\begin{align*} z _ t & : = M _ t ( \\hat { x } , \\bar { y } ) ( \\hat { x } - \\bar { y } ) , \\\\ M _ t ( \\hat { x } , \\bar { y } ) & : = A _ \\gamma ( \\hat { x } , \\hat { x } ) + t \\big ( A _ \\gamma ( \\hat { x } , \\bar { y } ) - A _ \\gamma ( \\hat { x } , \\hat { x } ) \\big ) \\end{align*}"}
+{"id": "3205.png", "formula": "\\begin{align*} \\sigma _ n = \\frac { 1 } { \\abs { W _ n } } \\sum _ { a \\in W _ n } \\phi _ a . \\end{align*}"}
+{"id": "7416.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Z ^ 2 } q _ { n } ^ { f } ( x ) ^ 2 = q _ { 2 n } ^ { f , f } q _ { m } ^ { f , f } : = \\sum _ { z , z ' \\in \\Z ^ 2 } q _ { m } ( z - z ' ) \\ , f ( z ) \\ , f ( z ' ) \\ , . \\end{align*}"}
+{"id": "5296.png", "formula": "\\begin{align*} | \\langle h ^ { = d } , g \\rangle | \\leq e \\ , \\mu _ { p } ( h ) \\mu _ { p } ( g ) \\left ( 4 0 0 \\ , r \\ , \\sqrt { q } \\cdot \\sqrt { A } \\right ) ^ { d } . \\end{align*}"}
+{"id": "2392.png", "formula": "\\begin{align*} \\vec { u } ( A _ { 0 } , A _ { 1 } ) + \\vec { u } ( A _ { 0 } , A _ { 2 } ) = - ( \\vec { u } ( A _ { 0 } , A _ { 3 } ) + \\vec { u } ( A _ { 0 } , A _ { 4 } ) ) , \\end{align*}"}
+{"id": "6353.png", "formula": "\\begin{align*} \\tilde t = \\alpha ( s _ 2 ) + \\frac { d ( s _ 2 ) } { d ( s _ 1 ) } ( \\tilde t - \\alpha ( s _ 1 ) ) . \\end{align*}"}
+{"id": "3066.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\left [ r ^ { n - 1 } u ' ( r ) | u ' ( r ) | ^ { p - 2 } \\right ] ' = r ^ { n - 1 } f _ 1 ( r ) g _ 1 ( v ( r ) ) \\cdot | u ' ( r ) | ^ { \\alpha } & & \\\\ & \\left [ r ^ { n - 1 } v ' ( r ) | v ' ( r ) | ^ { p - 2 } \\right ] ' = r ^ { n - 1 } f _ 2 ( r ) g _ 2 ( v ( r ) ) \\cdot g _ 3 ( | u ' ( r ) | ) & & \\\\ & u ' ( 0 ) = v ' ( 0 ) = 0 , \\ : u ( r ) > 0 , v ( r ) > 0 & & \\end{aligned} \\right . \\end{align*}"}
+{"id": "7897.png", "formula": "\\begin{align*} \\sigma ( A ) = C \\mbox { a n d } \\sigma ( B ) = D . \\end{align*}"}
+{"id": "1549.png", "formula": "\\begin{align*} \\frac { \\int _ { 0 } ^ k | s | ^ { p - 2 } ( s - k ) _ { - } \\ , d s } { \\int _ { k _ \\epsilon } ^ k | s | ^ { p - 2 } ( s - k ) _ { - } \\ , d s } = 1 + \\frac { \\int _ { 0 } ^ { k _ \\epsilon } | s | ^ { p - 2 } ( s - k ) _ { - } \\ , d s } { \\int _ { k _ \\epsilon } ^ k | s | ^ { p - 2 } ( s - k ) _ { - } \\ , d s } = 1 + I _ { \\epsilon } . \\end{align*}"}
+{"id": "7805.png", "formula": "\\begin{align*} \\textsf { E } \\eta _ { ( n ) } = \\textsf { E } \\sum _ { i = 1 } ^ { n } \\frac { T _ { i } } { 2 ( n - i + 1 ) } = \\sum _ { i = 1 } ^ { n } \\frac { 1 } { n - i + 1 } = \\sum _ { i = 1 } ^ { n } \\frac { 1 } { i } \\asymp \\log n . \\end{align*}"}
+{"id": "171.png", "formula": "\\begin{align*} L i _ 2 ( - \\phi ) = - \\frac { \\pi ^ 2 } { 1 0 } - ( \\log \\phi ) ^ 2 , \\end{align*}"}
+{"id": "8627.png", "formula": "\\begin{align*} P \\left ( \\tau _ N < s _ N ( \\rho ) \\right ) = P \\left ( \\sup _ { t \\leq s _ N ( \\rho ) } Z _ 0 ( t ) \\geq N \\right ) \\leq \\frac { 1 } { N ^ 2 } E \\left [ Z _ 0 \\left ( s _ N ( \\rho ) \\right ) ^ 2 \\right ] = O \\left ( N ^ { 2 ( \\rho - 1 ) } \\right ) . \\end{align*}"}
+{"id": "2237.png", "formula": "\\begin{align*} \\varphi ( q _ { n * } ( \\xi , \\ldots , \\xi , a , b ) ) = q _ { n * } ( \\varphi ( \\xi ) , \\ldots , \\varphi ( \\xi ) , \\varphi ( a ) , \\varphi ( b ) ) , \\end{align*}"}
+{"id": "1411.png", "formula": "\\begin{align*} \\cos ( \\theta ) = \\frac { \\tanh ( \\Delta _ { k - 1 } ) } { \\tanh ( r _ { k - 1 } ) } , \\sin ( \\theta ) = \\frac { \\sinh ( r _ k ) } { \\sinh ( r _ { k - 1 } ) } , \\cosh ( r _ { k - 1 } ) = \\cosh ( r _ k ) \\cosh ( \\Delta _ { k - 1 } ) . \\end{align*}"}
+{"id": "418.png", "formula": "\\begin{align*} \\Psi ( M _ n ) = 1 - \\frac { 2 ( \\gamma - 1 ) } { \\gamma ( 2 - \\gamma ) } \\frac { 1 } { M _ n ^ 2 } , \\end{align*}"}
+{"id": "4023.png", "formula": "\\begin{align*} \\mathbb { B } _ k = \\mathbb { Z } [ X , Y ] _ { k - 2 } \\otimes \\mathbb { B } . \\end{align*}"}
+{"id": "3912.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } \\left \\{ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\pi \\in \\bar { \\Gamma } } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda \\ , d \\pi \\right \\} . \\end{align*}"}
+{"id": "7488.png", "formula": "\\begin{align*} \\delta ^ { \\prime } = x \\alpha _ { m + 1 } \\cdots \\alpha _ k \\delta \\pi _ j \\pi _ { j + 1 } \\cdots \\pi _ { l - 1 } \\end{align*}"}
+{"id": "2243.png", "formula": "\\begin{align*} \\mathcal { T } _ D ^ n ( f ) ( z ) = - \\frac { 1 } { 2 \\pi } \\iint _ { | \\zeta | < 1 } \\left ( \\frac { f ( \\zeta ) } { \\zeta } \\ , \\frac { \\zeta + z } { \\zeta - z } + \\frac { \\overline { f ( \\zeta ) } } { \\overline { \\zeta } } \\ , \\frac { 1 + z \\overline { \\zeta } } { 1 - z \\overline { \\zeta } } \\right ) ( \\zeta - z + \\overline { \\zeta - z } ) ^ { n - 1 } \\ , d \\xi \\ , d \\eta , \\end{align*}"}
+{"id": "3998.png", "formula": "\\begin{align*} \\underset { x ^ { \\prime } \\in \\mathbb { R } ^ d } { \\arg \\max } \\ \\varphi _ { \\ell } ( x ^ { \\prime } , \\lambda _ { \\ell } ) = \\frac { 1 } { 2 } a _ \\ell { \\lambda _ { \\mathrm { D } , \\ell } ^ { \\star } } ^ { - 1 } V _ { 2 , X Y } , \\ell = 1 , 2 . \\end{align*}"}
+{"id": "7058.png", "formula": "\\begin{align*} \\Psi ( \\tilde h d X _ i + \\mathcal J ) = \\frac { a _ i h } { Q ' _ i } \\frac { Q ' _ i } { a _ i } + I _ \\beta = h + I _ \\beta . \\end{align*}"}
+{"id": "648.png", "formula": "\\begin{align*} v ( T x ) - v ( x ) = \\varphi _ f ( x ) I , \\end{align*}"}
+{"id": "8260.png", "formula": "\\begin{align*} \\tilde { c } ^ { ( 1 ) } _ { i , j } = \\begin{cases} c ^ { ( 1 ) } _ { i , j } & j \\leq l _ 0 - 1 ; \\\\ ( - 1 ) ^ { i + j } \\frac { 2 k - 2 j + l } { l _ 0 } & l _ 0 \\leq j \\leq l _ 1 . \\end{cases} \\end{align*}"}
+{"id": "396.png", "formula": "\\begin{align*} L _ C ( U ) = L _ C ( 2 ( I ^ - T ^ { - 1 } ) ^ T \\Sigma ( \\sqrt { | \\Lambda ^ - | } W ^ - - R \\sqrt { \\Lambda ^ + } W ^ + - S G ) ) , \\end{align*}"}
+{"id": "1200.png", "formula": "\\begin{align*} \\mathcal { H } = R ( B + i ) = R ( A + i ) + R ( B \\vert _ V + i ) . \\end{align*}"}
+{"id": "30.png", "formula": "\\begin{align*} t ( [ z _ 0 , g ] _ { \\mathbf { K } } ) = [ z _ 0 , t g ] _ { \\mathbf { K } } . \\end{align*}"}
+{"id": "6492.png", "formula": "\\begin{align*} \\det T _ { n , n } ( x ) = n ! ^ { - n } \\prod _ { i = 0 } ^ { n - 1 } \\binom { n + i } i ^ { - 1 } \\cdot \\prod _ { s = 1 - n } ^ { 3 n - 2 } ( x + s ) ^ { C T _ y ( y ^ { - s } \\varphi _ n ( y ) ) } ; \\end{align*}"}
+{"id": "1093.png", "formula": "\\begin{align*} r ' ( \\varphi ( b ) , x ) = i ( x ) s ' ( \\varphi ( b ) ) = i ( x ) s ( b ) = r ( b , x ) , \\end{align*}"}
+{"id": "6152.png", "formula": "\\begin{align*} & \\forall \\ , x : n \\forall \\ , y : n \\forall \\ , z : n ( ( x + _ n y ) + _ n z = x + _ n ( y + _ n z ) ) \\\\ & \\forall \\ , x : n ( x + _ n 0 _ n = x ) \\\\ & \\forall \\ , x : n \\forall \\ , y : n ( x + _ n y = y + _ n x ) \\\\ & \\forall \\ , x : n ( x + _ n x = 0 _ n ) \\\\ & \\forall \\ , x : n ( 1 \\cdot _ n x = x ) \\\\ & \\forall \\ , x : n \\forall \\ , y : n \\forall \\ , a : 0 ( a \\cdot _ n ( x + _ n y ) = a \\cdot _ n x + _ n a \\cdot _ n y ) \\\\ & \\forall \\ , x : n \\forall \\ , a : 0 \\forall \\ , b : 0 ( ( a + _ 0 b ) \\cdot _ n x = a \\cdot _ n x + _ n b \\cdot _ n x ) \\end{align*}"}
+{"id": "4233.png", "formula": "\\begin{align*} E _ 1 & = \\det T ( \\phi ) T ( \\phi _ R ) ^ { - 1 } \\cdots T ( \\phi _ 1 ) ^ { - 1 } T ( \\psi _ S ) ^ { - 1 } \\cdots T ( \\psi _ 1 ) ^ { - 1 } T ( \\psi _ 0 ) ^ { - 1 } \\\\ E _ 2 & = \\det T ( \\tilde { \\phi } ) T ( \\tilde { \\phi } _ R ) ^ { - 1 } \\cdots T ( \\tilde { \\phi } _ 1 ) ^ { - 1 } T ( \\tilde { \\psi } _ S ) ^ { - 1 } \\cdots T ( \\tilde { \\psi } _ 1 ) ^ { - 1 } T ( \\tilde { \\psi } _ 0 ) ^ { - 1 } \\\\ E _ 3 & = \\prod _ { j = 0 } ^ S \\det T ( \\psi _ j ) T ( \\psi _ j ^ { - 1 } ) . \\end{align*}"}
+{"id": "6751.png", "formula": "\\begin{align*} \\{ \\nu _ \\eta \\} = \\mathcal { P } ( X _ \\eta ) = \\mathcal { P } ( X _ { \\eta ^ * } ) = \\{ \\nu _ { \\eta ^ * } \\} . \\end{align*}"}
+{"id": "3880.png", "formula": "\\begin{align*} \\sup _ { \\pi \\in \\Pi ( \\mu _ 1 , \\dotsc , \\mu _ L ) } \\int _ { \\mathcal { X } } f d \\pi = \\inf _ { ( \\phi _ \\ell ) _ { \\ell \\in [ L ] } \\in \\Phi _ f } \\left \\{ \\sum _ { \\ell = 1 } ^ { L } \\int _ { \\mathcal { X } _ \\ell } \\phi _ \\ell d \\mu _ \\ell \\right \\} . \\end{align*}"}
+{"id": "8862.png", "formula": "\\begin{align*} \\tilde { \\mathbf r } _ m = & \\mathbf F \\mathbf r _ m = \\sum _ { n \\in \\mathcal N } \\alpha _ n g _ n ^ { \\frac { 1 } { 2 } } \\mathbf F \\mathbf H _ { n , m } \\mathbf F ^ H \\tilde { \\mathbf s } _ n + \\tilde { \\mathbf n } _ m \\\\ = & \\sum _ { n \\in \\mathcal N } \\alpha _ n g _ n ^ { \\frac { 1 } { 2 } } { \\rm d i a g } ( \\tilde { \\mathbf { s } } _ n ) \\mathbf { F } ( \\mathbf H _ { n , m } ) _ { : , 1 } + \\tilde { \\mathbf n } _ m , m \\in \\mathcal { M } , \\end{align*}"}
+{"id": "6660.png", "formula": "\\begin{align*} \\Delta [ G _ { \\alpha } ( u ) ] = G _ { \\alpha } ' ( u ) \\Delta u + G _ { \\alpha } '' ( u ) | \\nabla u | ^ 2 \\ , \\ , \\R ^ N . \\end{align*}"}
+{"id": "3350.png", "formula": "\\begin{align*} T _ 1 = \\dfrac { 1 } { 2 } \\begin{pmatrix} - 1 & 0 & - 1 \\\\ - 1 & 0 & 1 \\\\ - 1 & 0 & - 1 \\end{pmatrix} , T _ 2 = \\dfrac { 1 } { 2 } \\begin{pmatrix} 1 & - 1 & 0 \\\\ - 1 & - 1 & 0 \\\\ - 1 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "9330.png", "formula": "\\begin{align*} & a ( X _ { 1 4 } ; T ) = \\sum _ { d \\mid \\varepsilon ( T ) } d ^ { 1 3 } \\tau ^ * ( 2 ( T ) / d ^ 2 ) , \\\\ & \\tau ^ * ( \\ell ) = \\tau ( \\ell ) - 2 ^ { 1 2 } \\tau ( \\ell / 4 ) , \\end{align*}"}
+{"id": "8838.png", "formula": "\\begin{align*} \\widetilde { u } _ n : = \\sum _ { k = n } ^ { N ( n ) } \\alpha _ { n , k } u _ { \\lambda _ k } . \\end{align*}"}
+{"id": "5239.png", "formula": "\\begin{align*} \\begin{array} { l } - \\{ \\ ( - w - 1 ) ( - w + b - a - 1 ) y ^ { - 1 } - ( - w + c - a - 1 ) ( - ) ( w + a ) \\ \\} \\\\ = ( w + a ) ( w - c + a + 1 ) - ( w - b + a + 1 ) ( w + 1 ) y ^ { - 1 } . \\end{array} \\end{align*}"}
+{"id": "6831.png", "formula": "\\begin{align*} f _ { N , j , n } ( b _ 1 , \\ldots , b _ N ) : = \\sum _ { M \\in \\Z } \\left ( ( - 1 ) ^ M \\sum _ { u \\in \\mathbb { Z } } a ^ { j - u } q ^ { ( M - u ) ( n - u ) + ( j - u ) ( n - N ) } \\begin{bmatrix} M \\\\ u \\end{bmatrix} \\begin{bmatrix} N - M \\\\ j - u \\end{bmatrix} \\right ) e _ M ( b _ 1 , \\dots , b _ N ) , \\end{align*}"}
+{"id": "175.png", "formula": "\\begin{align*} L i _ 2 \\left ( \\frac { 1 } { 3 } \\right ) - \\frac { 1 } { 6 } L i _ 2 \\left ( \\frac { 1 } { 9 } \\right ) = \\frac { \\pi ^ 2 } { 1 8 } - \\frac { 1 } { 6 } ( \\log 3 ) ^ 2 , \\end{align*}"}
+{"id": "6608.png", "formula": "\\begin{align*} L = A \\otimes _ P \\overline { L } . \\end{align*}"}
+{"id": "3900.png", "formula": "\\begin{align*} \\psi ( x ) : = \\begin{cases} - \\varphi ( x ) & x \\in \\mathbb { R } ^ n _ + \\\\ \\infty & x \\notin \\mathbb { R } ^ n _ + . \\end{cases} \\end{align*}"}
+{"id": "281.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( 1 - y ^ m z ^ n \\right ) ^ { \\frac { m ^ 4 } { n ^ 5 } } = \\left \\{ \\frac { \\left ( 1 - y z \\right ) ^ { y ^ 5 - 4 y ^ 4 + 6 y ^ 3 + y } } { \\left ( 1 - z \\right ) ^ { 4 } } \\right \\} ^ { \\frac { 1 } { ( 1 - y ) ^ 5 } } \\end{align*}"}
+{"id": "1490.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty e ^ { \\mu b - \\frac { \\mu ^ 2 } { 2 } } \\ , \\frac { 1 } { \\sqrt { 2 \\pi } } \\ , e ^ { - \\frac { b ^ 2 } { 2 } } \\ , \\mathrm { d } b = 1 . \\end{align*}"}
+{"id": "1137.png", "formula": "\\begin{align*} A ^ { \\ast } ( x ) = A \\left ( x , \\ldots , x \\right ) \\left ( x \\in G \\right ) . \\end{align*}"}
+{"id": "4696.png", "formula": "\\begin{align*} \\lambda _ \\chi ^ { \\rm s t i f f } : H ^ { 1 / 2 } ( \\Gamma ; \\C ^ 3 ) \\times H ^ { 1 / 2 } ( \\Gamma ; \\C ^ 3 ) \\to \\R , \\lambda _ \\chi ^ { \\rm s t i f f } ( \\vect g , \\vect h ) : = a _ { 0 , \\chi } ^ { \\rm s t i f f } ( \\Pi _ \\chi ^ { \\rm s t i f f } \\vect g , \\Pi _ \\chi ^ { \\rm s t i f f } \\vect h ) . \\end{align*}"}
+{"id": "4197.png", "formula": "\\begin{align*} a \\overline { b } = \\frac { 1 } { 3 } \\sum _ { t = 0 } ^ 2 e ^ { \\frac { 2 \\pi i t } { 3 } } \\vert a + e ^ { \\frac { - 2 \\pi i t } { 3 } } b \\vert ^ 2 , ~ ~ a , b \\in \\mathbb { C } \\end{align*}"}
+{"id": "5808.png", "formula": "\\begin{align*} \\frac { y ^ { ( k ) } } { y } = Y ^ k + \\frac { k ( k - 1 ) } { 2 } Y ^ { k - 2 } Y ' + P _ { k - 2 } ( G ) = - A - h . \\end{align*}"}
+{"id": "1712.png", "formula": "\\begin{align*} P _ 0 ^ { ( \\alpha , \\beta ) } ( z ) = 1 , \\end{align*}"}
+{"id": "7987.png", "formula": "\\begin{align*} \\nabla v = ( \\partial _ \\nu v ) \\ , \\nu \\quad \\partial \\Omega , \\end{align*}"}
+{"id": "9252.png", "formula": "\\begin{align*} d = \\texttt { w p i n i t i a l } + 1 + \\texttt { m a g } \\Bigl \\{ 3 \\Bigl ( \\frac { t } { 2 \\pi } \\Bigr ) ^ { 3 / 2 } \\log \\frac { t } { 2 \\pi } \\Bigr \\} \\end{align*}"}
+{"id": "2067.png", "formula": "\\begin{align*} \\mu _ C \\big ( Q _ 1 ( x , y , z , w ) \\big ) = Q _ 2 ( x , \\alpha x - y , z , \\alpha z - w ) \\end{align*}"}
+{"id": "6157.png", "formula": "\\begin{align*} & \\forall \\ , x : n ( h _ n ( x ) = \\top _ 1 \\ast _ { 1 n } x ) \\end{align*}"}
+{"id": "355.png", "formula": "\\begin{align*} ( \\underline { X } , \\underline { A } ) ^ { \\sigma } = \\{ ( x _ 1 , \\ldots , x _ m ) \\in \\prod ^ m _ { i = 1 } X _ i \\mid x _ i \\in A _ i i \\notin \\sigma \\} . \\end{align*}"}
+{"id": "68.png", "formula": "\\begin{align*} \\mathcal { T } = \\bigsqcup _ { i = 1 } ^ { r ' } \\mathcal { T } _ i ' . \\end{align*}"}
+{"id": "6715.png", "formula": "\\begin{align*} \\P ( \\omega _ 1 ^ { \\bar w } = \\omega _ 1 ) \\le \\ell / ( n - \\ell ) . \\end{align*}"}
+{"id": "4682.png", "formula": "\\begin{align*} M ( z ) \\Gamma _ 0 \\vect u = \\Gamma _ 1 \\vect u . \\end{align*}"}
+{"id": "9225.png", "formula": "\\begin{align*} \\texttt { s u m } & = v ' _ 1 ( 1 + \\eta _ N ) + v ' _ 2 ( 1 + \\eta _ { N - 1 } ) + \\cdots + v ' _ n ( 1 + \\eta _ 1 ) \\\\ & = v _ 1 ( 1 + \\eta _ { N + 1 } ) + v _ 2 ( 1 + \\eta _ { N } ) + \\cdots + v _ N ( 1 + \\eta _ 2 ) \\\\ & = ( a _ 1 + \\alpha _ 1 ) ( 1 + \\eta _ { N + 1 } ) + ( a _ 2 + \\alpha _ 2 ) ( 1 + \\eta _ { N } ) + \\cdots + ( a _ N + \\alpha _ N ) ( 1 + \\eta _ 2 ) \\end{align*}"}
+{"id": "2978.png", "formula": "\\begin{align*} \\frac { 1 } { \\Delta } \\ , a _ { ( p , \\lambda _ 2 , \\dots , \\lambda _ n ) } = s _ { ( k - ( n - 1 ) , \\lambda _ 2 , \\dots , \\lambda _ n ) } \\ . \\end{align*}"}
+{"id": "688.png", "formula": "\\begin{align*} \\int _ J | \\varphi ( x ) | d x \\leq \\left \\{ \\begin{array} { c l } \\frac { \\norm { \\varphi } _ { L ^ 1 ( I ) } | J | } { | I | } + \\frac { 2 ^ { a + 3 } p _ a ( \\varphi ) | J | ^ { 1 - a } } { a ( 1 - a ) } & 0 < a < 1 , \\\\ \\frac { \\norm { \\varphi } _ { L ^ 1 ( I ) } | J | } { | I | } + 4 p _ a ( \\varphi ) | J | ( 1 + \\log \\frac { | I | } { | J | } ) & a = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "36.png", "formula": "\\begin{align*} H ^ 1 ( A , q ) : = H ^ 1 ( A _ { \\overline { \\mathbb { F } } _ \\ell } ^ \\vee , \\mathbb { Z } _ q ( 1 ) ) , \\ , H ^ 1 ( A , \\ell ) : = H ^ 1 _ \\mathrm { c r y s } ( A ^ \\vee / W ( { \\kappa } ) ) \\end{align*}"}
+{"id": "2155.png", "formula": "\\begin{align*} \\varphi _ { \\lambda } \\left ( x _ { 1 } , t \\right ) = \\exp \\left [ 2 \\lambda \\left ( x _ { 1 } ^ { 2 } - \\alpha \\left ( t - T / 2 \\right ) ^ { 2 } \\right ) \\right ] . \\end{align*}"}
+{"id": "8664.png", "formula": "\\begin{align*} \\widetilde { Z } ( \\mathcal { C } ) : = \\{ \\alpha \\in \\mathcal { C } | \\alpha v = v \\alpha ' \\ \\forall v \\in M \\} \\quad \\mathrm { a n d } \\quad \\Gamma ( M , q ) : = \\{ \\alpha \\in \\mathcal { C } ^ { \\times } | \\alpha V \\alpha '^ { - 1 } = V \\} \\end{align*}"}
+{"id": "4763.png", "formula": "\\begin{align*} Q _ { [ i ] } = \\begin{pmatrix} \\tilde { S } _ { \\alpha _ i \\alpha _ i } & \\tilde { S } _ { \\alpha _ i \\beta _ i } \\\\ - \\tilde { S } _ { \\alpha _ i \\beta _ i } & \\tilde { S } _ { \\alpha _ i \\alpha _ i } \\end{pmatrix} + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "7639.png", "formula": "\\begin{align*} M ( 2 , 0 , y ) = M ( 2 , 1 , y ) = M ( 2 , x , 0 ) = M ( 2 , x , 1 ) = M ( c , 0 , 0 ) = 0 \\end{align*}"}
+{"id": "4762.png", "formula": "\\begin{align*} Q _ { [ i ] } \\coloneqq \\begin{pmatrix} U & V \\\\ - V & U \\end{pmatrix} \\end{align*}"}
+{"id": "5087.png", "formula": "\\begin{align*} \\partial _ x \\left ( A _ t ^ { - 1 } ( x ) \\right ) = \\frac { 1 } { 1 - \\gamma _ x \\left ( A _ t ^ { - 1 } ( x ) , t \\right ) } , \\partial _ x ^ 2 \\left ( A _ t ^ { - 1 } ( x ) \\right ) = \\frac { \\gamma _ { x x } \\left ( A _ t ^ { - 1 } ( x ) , t \\right ) } { \\left ( 1 - \\gamma _ x \\left ( A _ t ^ { - 1 } ( x ) , t \\right ) \\right ) ^ 3 } , \\end{align*}"}
+{"id": "6052.png", "formula": "\\begin{align*} g _ { - 1 , - 1 } \\textrm { i s E g e r v \\ ' a r y e q u i v a l e n t t o } \\begin{cases} g _ { - 1 , 1 } & \\textrm { i f a n d o n l y i f } n + m \\textrm { i s a n e v e n n u m b e r } , \\\\ g _ { 1 , 1 } & \\textrm { i f a n d o n l y i f } n \\textrm { i s a n e v e n n u m b e r } , \\\\ g _ { 1 , - 1 } & \\textrm { i f a n d o n l y i f } m \\textrm { i s a n e v e n n u m b e r } , \\\\ \\end{cases} \\end{align*}"}
+{"id": "4193.png", "formula": "\\begin{align*} \\hat { w _ 1 } ( \\xi ) & = \\int _ 0 ^ \\infty \\frac { d } { d \\tau \\vphantom { ( e ^ { - \\tau } + 1 ) ^ 3 } } \\left ( \\frac { e ^ { - 3 \\tau } } { 2 \\vphantom { ( e ^ { - \\tau } + 1 ) ^ 3 } } - \\frac { 4 e ^ { - 6 \\tau } } { ( e ^ { - \\tau } + 1 ) ^ 3 } \\right ) \\frac { e ^ { - i \\xi \\tau } } { i \\xi \\vphantom { ( e ^ { - \\tau } + 1 ) ^ 3 } } \\ , d \\tau . \\end{align*}"}
+{"id": "4698.png", "formula": "\\begin{align*} \\Lambda ^ { \\rm h o m } \\vect \\xi : \\vect \\zeta , \\Lambda ^ { \\rm h o m } \\vect \\xi : \\vect \\xi \\geq \\eta | \\vect \\xi | ^ 2 , \\forall \\vect \\xi , \\vect \\zeta \\in \\R ^ { 3 \\times 3 } , \\vect \\xi ^ \\top = \\vect \\xi , \\vect \\zeta ^ \\top = \\vect \\zeta , \\end{align*}"}
+{"id": "1337.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } _ { i } ^ { N } ( t ) = \\frac { 1 } { N } \\stackrel [ j = 1 ] { N } { \\sum } m _ { j } ^ { N } ( t ) \\partial _ { x } V ( x _ { j } ( t ) - x _ { i } ( t ) ) , & x _ { i } ^ { N } ( 0 ) = x _ { i } ^ { 0 , N } \\\\ \\dot { m } _ { i } ^ { N } ( t ) = \\psi _ { i } ^ { N } ( \\mathbf { x } _ { N } ( t ) , \\mathbf { m } _ { N } ( t ) ) , & m _ { i } ^ { N } ( 0 ) = m _ { i } ^ { 0 , N } . \\end{cases} \\end{align*}"}
+{"id": "8699.png", "formula": "\\begin{align*} \\frac { \\textup { M M D } _ { n , m } ^ { 2 } } { \\surd { \\textup { v a r } ( \\Delta _ { 1 } ) } } = \\frac { \\Delta _ 1 + \\sum ^ { l - 1 } _ { s = 2 } \\{ \\Delta _ s - E ( \\Delta _ s ) \\} + \\{ \\widetilde { \\Delta } _ l - E ( \\widetilde { \\Delta } _ l ) \\} + ^ 2 ( P _ X , P _ Y ) } { \\surd { \\textup { v a r } ( \\Delta _ { 1 } ) } } . \\end{align*}"}
+{"id": "2240.png", "formula": "\\begin{align*} \\lim _ { r \\nearrow 1 } \\int _ 0 ^ { 2 \\pi } | T _ D ( f ) ( e ^ { i \\theta } ) - T _ D ( f ) ( r e ^ { i \\theta } ) | ^ \\gamma \\ , d \\theta = 0 , \\end{align*}"}
+{"id": "5931.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( g ) f ( [ 1 , y ] ) = \\pi _ { \\psi } ( g ) f _ 1 ( y ) = \\int _ { X ^ { \\ast } } \\psi ( \\langle y , y ^ { \\ast } \\rangle ) f _ 1 ( y ^ { \\ast } \\omega ^ { - 1 } ) d y ^ { \\ast } . \\end{align*}"}
+{"id": "8068.png", "formula": "\\begin{align*} \\begin{aligned} u _ n & \\rightharpoonup u & & X , \\\\ T u _ n & \\rightarrow v \\enspace & & X ^ { * } , \\\\ \\end{aligned} \\enspace \\Longrightarrow \\enspace u _ n \\rightarrow u X . \\end{align*}"}
+{"id": "8408.png", "formula": "\\begin{align*} ( E _ 0 = L K _ X ^ { \\frac { 1 } { 2 } } \\oplus L K _ X ^ { - \\frac { 1 } { 2 } } \\oplus L ^ { - 1 } K _ X ^ { \\frac { 1 } { 2 } } \\oplus L ^ { - 1 } K _ X ^ { - \\frac { 1 } { 2 } } , \\theta _ 0 = \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & \\mu & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} ) , \\end{align*}"}
+{"id": "4898.png", "formula": "\\begin{align*} r _ 0 ( x , z ) : = \\frac { ( \\rho ^ 2 - | x | ^ 2 ) ( \\rho ^ 2 - | z | ^ 2 ) } { \\rho ^ 2 | x - z | ^ 2 } . \\end{align*}"}
+{"id": "659.png", "formula": "\\begin{align*} C _ { \\alpha , n } ^ { a , + } ( \\varphi ) & = ( - 1 ) ^ { n } C _ { \\alpha } ^ + ( D ^ n \\varphi ) : = ( - 1 ) ^ { n + 1 } \\lim _ { x \\searrow l _ \\alpha } D ^ { n + 1 } \\varphi ( x ) ( x - l _ \\alpha ) ^ { 1 + a } , \\\\ C _ { \\alpha , n } ^ { a , - } ( \\varphi ) & = C _ { \\alpha } ^ - ( D ^ n \\varphi ) : = \\lim _ { x \\nearrow r _ \\alpha } D ^ { n + 1 } \\varphi ( x ) ( r _ \\alpha - x ) ^ { 1 + a } \\end{align*}"}
+{"id": "3280.png", "formula": "\\begin{align*} { \\rm I m } ( e ^ { i \\pi s } \\kappa ( t ) ) = - { \\rm I m } ( e ^ { i \\pi ( 1 + s ) } \\kappa ( t ) ) = 0 . \\end{align*}"}
+{"id": "2310.png", "formula": "\\begin{align*} | | \\sum _ { j } c _ j a _ j | | _ { a t } : = \\left ( \\inf \\sum _ { j } | c _ j | ^ p \\right ) ^ { 1 / p } , \\end{align*}"}
+{"id": "6730.png", "formula": "\\begin{align*} \\nu ( A ) = \\int _ { H \\times \\{ 0 , 1 \\} ^ \\Z } \\mathbf { 1 } _ A ( \\underline { \\varphi } ( h ) + x \\cdot ( \\varphi ( h ) - \\underline { \\varphi } ( h ) ) ) \\ , d \\rho ( h , x ) . \\end{align*}"}
+{"id": "8154.png", "formula": "\\begin{align*} e _ n ^ { ( k ) } = \\frac 1 { k ! } \\sum _ { I \\vDash n } \\varphi ^ I \\end{align*}"}
+{"id": "6921.png", "formula": "\\begin{align*} \\sum _ { \\substack { m , v \\in \\mathcal { M } \\\\ m p = v } } f ( m ) f ( v ) \\le \\sum _ { \\substack { m ' , v ' \\in \\mathcal { M } ' \\\\ | p m ' / v ' - 1 | \\le 3 / T } } r ( m ' ) r ( v ' ) . \\end{align*}"}
+{"id": "6990.png", "formula": "\\begin{align*} f = \\sum _ { i = 0 } ^ r \\left ( \\sum _ { j = 1 } ^ { s _ i } a _ { i , j } \\textbf { Q } ^ { \\lambda _ { i , j } } \\right ) q ^ i = \\sum _ { i = 0 } ^ r \\sum _ { j = 1 } ^ { s _ i } a _ { i , j } \\textbf { Q } ^ { \\lambda ' _ { i , j } } , \\end{align*}"}
+{"id": "8692.png", "formula": "\\begin{align*} \\mu _ { i _ 1 , \\ldots , i _ { a - a _ 1 - a _ 2 } } ^ { ( 1 ) } = \\sum _ { j _ 1 , \\ldots , j _ { a _ 1 } } E \\bigg [ \\prod ^ { a - a _ 1 - a _ 2 } _ { s = 1 } \\{ x _ { i _ s } - E ( x _ { i _ s } ) \\} \\prod ^ { a _ 1 } _ { s = 1 } \\{ x _ { j _ s } - E ( x _ { j _ s } ) \\} ^ 2 \\bigg ] , \\end{align*}"}
+{"id": "7575.png", "formula": "\\begin{align*} H _ { q , n } ( f ) : = \\begin{vmatrix} a _ { n } & a _ { n + 1 } & \\cdots & a _ { n + q - 1 } \\\\ a _ { n + 1 } & a _ { n + 2 } & \\cdots & a _ { n + q } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ a _ { n + q - 1 } & a _ { n + q } & \\cdots & a _ { n + 2 ( q - 1 ) } \\end{vmatrix} . \\end{align*}"}
+{"id": "6435.png", "formula": "\\begin{align*} c = c ( \\theta ) : = \\inf _ { j , n } \\left \\{ \\lvert { \\theta ^ n } _ j ( \\mathcal A ) \\rvert : 0 \\le j < \\ell ^ n \\right \\} . \\end{align*}"}
+{"id": "1812.png", "formula": "\\begin{align*} [ D _ k ( t ) , \\N ] = [ D _ k ^ * ( t ) , \\N ] = 0 \\end{align*}"}
+{"id": "8821.png", "formula": "\\begin{align*} D u ( t , x ) & = v _ 0 ( t , x ) + \\int _ 0 ^ t \\ ! \\ ! \\int _ G K _ { t - s } ( x , y ) f ' ( u ( s , y ) ) D u ( s , y ) \\ , d y \\\\ & + \\int _ 0 ^ t \\ ! \\ ! \\int _ G K _ { t - s } ( x , y ) \\sigma ' ( u ( s , y ) ) D u ( s , y ) \\ , \\bar { W } ( d y , d s ) , \\end{align*}"}
+{"id": "3533.png", "formula": "\\begin{align*} H _ b ( X , Y ) = \\langle \\langle P X , Y \\rangle \\rangle = \\int _ a ^ b \\langle P X , Y \\rangle d s . \\end{align*}"}
+{"id": "7493.png", "formula": "\\begin{align*} \\beta ^ { \\prime } = \\underset { \\delta ^ { \\prime } } { \\underbrace { d _ p d _ { p + 1 } \\cdots d _ { q - 1 } } } d _ q d _ { q + 1 } \\cdots d _ l x \\pi _ j \\cdots \\pi _ n . \\end{align*}"}
+{"id": "4137.png", "formula": "\\begin{align*} ( v _ 1 ^ { ( i ) } , v _ 2 ^ { ( i ) } , v _ 0 ^ { ( i ) } ) ^ T \\mapsto \\left ( v _ 1 ^ { ( i ) } , v _ 2 ^ { ( i ) } - j v _ 1 ^ { ( i ) } , v _ 0 ^ { ( i ) } - k v _ 1 ^ { ( i ) } \\right ) ^ T = ( v _ 1 ^ { ( i + 1 ) } , v _ 2 ^ { ( i + 1 ) } , v _ 0 ^ { ( i + 1 ) } ) , \\end{align*}"}
+{"id": "1875.png", "formula": "\\begin{align*} \\langle D _ u \\theta ( u ^ * ) , v \\rangle _ \\mathcal { U } + \\langle \\bar { \\lambda } ^ * , S v \\rangle = 0 \\forall \\ v \\in \\mathcal { U } . \\end{align*}"}
+{"id": "6801.png", "formula": "\\begin{align*} ( k ^ + ) + ( 2 n - k ^ - - 1 ) - ( 2 n - 1 ) = k ^ + - k ^ - . \\end{align*}"}
+{"id": "8587.png", "formula": "\\begin{align*} S _ { j , + } ^ k ( t ) \\approx \\nu \\int _ 0 ^ t Z _ 0 ( s ) p _ { j , + } ^ k ( t - s ) d s = : \\bar { S } _ { j , + } ^ k ( t ) . \\end{align*}"}
+{"id": "1330.png", "formula": "\\begin{align*} \\mathcal { L } \\left ( \\frac { 1 } { 2 \\theta _ i ^ 2 T _ i ^ * ( u ) } \\Bigg | \\mathcal { Z } _ u , Z _ i ^ * ( u ) = z _ i / \\theta _ i \\right ) = I G \\left ( \\frac { 1 } { 2 \\theta _ i z _ i } , \\frac { 1 } { 2 z _ i ^ 2 } \\right ) . \\end{align*}"}
+{"id": "1720.png", "formula": "\\begin{align*} \\nu ^ * ( x ) = \\frac { 1 } { Z ( \\nu ^ * , \\mu ^ * ) } \\exp { \\left ( - \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta \\nu } ( \\nu ^ * , \\mu ^ * , x ) - U ^ { \\pi } ( x ) \\right ) } , \\end{align*}"}
+{"id": "5757.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p , \\infty } ( w ) } & = \\sup _ { t > 0 } t w \\big ( \\{ x \\in \\mathbb { R } ^ { n } : | f ( x ) | > t \\} \\big ) ^ { 1 / p } , \\end{align*}"}
+{"id": "9059.png", "formula": "\\begin{align*} \\beta = q _ { i + } ( \\beta _ 1 ) q _ { i - } ( \\beta _ 2 ) . \\end{align*}"}
+{"id": "6061.png", "formula": "\\begin{align*} \\delta _ r ^ A = \\exp ( \\ln ( r ) A ) , r > 0 , \\end{align*}"}
+{"id": "137.png", "formula": "\\begin{align*} A = { \\rm d i a g } \\left ( A _ 1 , \\cdots , A _ k \\right ) , A _ k = \\begin{pmatrix} B _ k & \\epsilon & 0 & \\cdots & 0 \\\\ 0 & B _ k & \\epsilon & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & B _ k \\end{pmatrix} . \\end{align*}"}
+{"id": "6809.png", "formula": "\\begin{align*} \\Sigma \\times \\{ 0 \\} = W _ 1 \\cup W _ 2 \\cup _ Y W _ 3 \\cup W _ 4 \\end{align*}"}
+{"id": "3138.png", "formula": "\\begin{align*} R _ m ( u , 0 ) = u \\cdot R _ m ( u , 1 ) ( m \\geq 1 ) . \\end{align*}"}
+{"id": "3704.png", "formula": "\\begin{align*} \\begin{cases} P ( ( - \\Delta ) ^ s ) _ 1 \\tilde { u } + q \\tilde { u } = \\left ( \\alpha _ m ^ 2 - \\alpha _ m ^ 1 \\right ) ( - \\Delta ) ^ { s _ m } u _ 2 & \\Omega , \\\\ \\tilde { u } = 0 & \\Omega ^ c \\end{cases} \\end{align*}"}
+{"id": "1927.png", "formula": "\\begin{align*} f ( z _ 1 , \\ldots , z _ m ) = \\sum _ { ( k _ 1 , \\ldots , k _ m ) \\in \\Z ^ m _ { \\geq 0 } } c _ { k _ 1 , \\ldots , k _ m } z _ 1 ^ { k _ 1 } \\cdots z _ m ^ { k _ m } , \\end{align*}"}
+{"id": "4066.png", "formula": "\\begin{align*} S _ { \\mathrm { o f f } } = \\sum _ { d _ 1 , \\ , d _ 2 < D } \\frac { 1 } { \\sqrt { d _ 1 d _ 2 } } & \\sum _ { I d _ 1 , \\ , J d _ 2 < D } \\alpha _ { I d _ 1 } \\bar { \\alpha } _ { J d _ 2 } \\sum _ { L , M \\geq 1 } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } \\\\ & \\times \\left ( 2 \\pi i ^ { 2 k } \\sum _ { p | c } \\frac { \\mathcal { S } ( I L , J M , c ) } { c } \\ , J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { I L J M } } { c } \\right ) \\right ) \\end{align*}"}
+{"id": "5899.png", "formula": "\\begin{align*} m = m _ n \\xrightarrow [ ] { } + \\infty , \\frac { m } { \\sqrt n } \\xrightarrow [ ] { } 0 \\end{align*}"}
+{"id": "4284.png", "formula": "\\begin{align*} e ( \\theta ) = C _ v \\theta + a ( \\theta ) q ^ 2 , \\qquad p ( \\rho , \\theta ) = R \\rho \\theta , \\end{align*}"}
+{"id": "3175.png", "formula": "\\begin{align*} ( \\overline { \\xi } ( T ) ) ^ j = \\sum _ { \\lambda = 0 } ^ { \\infty } \\alpha _ { j + \\lambda } ^ { ( j ) } T ^ \\lambda \\end{align*}"}
+{"id": "5929.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( g ) f ( [ 1 , y ] ) = \\pi _ { \\psi } ( g ) f _ 1 ( y ) = | \\det ( a ) | ^ { 1 / 2 } f _ 1 ( y a ) = | \\det ( a ) | ^ { 1 / 2 } f ( [ 1 , y a ] ) . \\end{align*}"}
+{"id": "8839.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } z _ n ( t , x ) = v ( t , x ) L ^ r ( \\Omega ; H ) \\end{align*}"}
+{"id": "4883.png", "formula": "\\begin{align*} \\begin{dcases} L \\psi \\le - \\frac { \\alpha _ r } { r ^ { 2 s } } & B _ r ( x _ r ) \\setminus \\overline { B _ { r / 2 } ( x _ r ) } , \\\\ \\psi \\ge \\frac { \\alpha _ r } { r ^ { 2 s } } ( r ^ 2 - | x - x _ r | ^ 2 ) _ + ^ s & \\\\ \\psi \\le \\alpha _ r C & B _ { r / 2 } ( x _ r ) , \\\\ \\psi = 0 & \\R ^ n \\setminus B _ r ( x _ r ) . \\end{dcases} \\end{align*}"}
+{"id": "7472.png", "formula": "\\begin{align*} \\pi = \\alpha _ 1 \\cdots \\alpha _ k \\beta \\end{align*}"}
+{"id": "8676.png", "formula": "\\begin{align*} \\tau _ { i } = 2 \\frac { ( \\Sigma _ { i } ) } { p } ~ i = 1 , 2 , \\tau _ { 3 } = \\frac { ( \\Sigma _ { 1 } ) + ( \\Sigma _ { 2 } ) + \\| \\mu _ { 1 } - \\mu _ { 2 } \\| ^ { 2 } _ { 2 } } { p } . \\end{align*}"}
+{"id": "6907.png", "formula": "\\begin{align*} f ( p ) : = \\frac { 1 } { \\sqrt { | \\log { ( 2 \\sigma - 1 ) } | } } , \\ ; p \\in P . \\end{align*}"}
+{"id": "6425.png", "formula": "\\begin{align*} h _ \\psi ^ { \\alpha / z } = h _ \\varphi ^ { ( \\alpha - 1 ) / 2 z } x h _ \\varphi ^ { ( \\alpha - 1 ) / 2 z } . \\end{align*}"}
+{"id": "2794.png", "formula": "\\begin{align*} ( \\mathcal { T } \\varphi , v ) _ { L ^ 2 ( \\mathrm { M } _ 0 ) } = ( \\varphi | \\overline { \\mathfrak { I } \\overline { w } } ) _ { H ^ { 1 / 2 } _ \\Gamma ( \\mathrm { M } ) } . \\end{align*}"}
+{"id": "7253.png", "formula": "\\begin{align*} W _ 2 ( P _ { B _ n } , P _ { \\sigma B } ) = O \\bigl ( n ^ { - 1 / 4 } ( \\log n ) ^ { 1 / 4 } \\bigr ) \\ , . \\end{align*}"}
+{"id": "2085.png", "formula": "\\begin{align*} & F ( A ) = \\left ( a + d - 2 + \\left ( \\sum _ { i = 1 } ^ k x _ i \\right ) _ { a - 1 } \\right ) a - d , \\\\ & g ( A ) = \\sum _ { r = 1 } ^ { a - 1 } \\left ( \\sum _ { i = 1 } ^ k x _ i \\right ) _ r + \\frac { ( a - 1 ) ( a + d - 1 ) } { 2 } , \\end{align*}"}
+{"id": "7123.png", "formula": "\\begin{align*} c ( x ) - c ( \\hat { y } ) & = \\nabla c ( x ) \\cdot ( x - \\hat { y } ) + R _ 2 ( x , \\hat { y } ) \\\\ & = \\nabla _ { x ^ \\prime } c ( x ) \\cdot ( x ^ \\prime - y ^ \\prime ) + \\partial _ { x _ n } c ( x ) ( x _ n + y _ n ) + R _ 2 ( x , \\hat { y } ) , \\end{align*}"}
+{"id": "4411.png", "formula": "\\begin{align*} & V _ n = ( v _ { 1 n } , v _ { 2 n } , v _ { 3 n } ) : = U _ n ( \\cdot - y _ n ) \\ \\ ( v _ { j n } \\in ( H ^ 1 ( \\R ^ d ) ) ^ d ) , \\\\ & V = ( v _ { 1 } , v _ { 2 } , v _ { 3 } ) \\ \\ ( v _ { j } \\in ( H ^ 1 ( \\R ^ d ) ) ^ d ) . \\end{align*}"}
+{"id": "2619.png", "formula": "\\begin{align*} R _ { s , j } & = \\left ( 1 + 2 ^ { \\mathfrak { c } | l | } \\lambda ^ { \\gamma + \\epsilon _ 3 - \\epsilon _ 4 } \\right ) \\sum _ { m _ j \\in \\mathcal { M } _ j } \\sum _ { k _ j \\in \\mathbb { Z } ^ 2 } \\left | a _ { j , m _ j , k _ j , s } \\right | ^ 2 \\\\ & \\lesssim 2 ^ { \\mathfrak { c } | l | } \\lambda ^ { \\gamma + \\epsilon _ 3 - \\epsilon _ 4 } \\sum _ { m _ j \\in \\mathcal { M } _ j } F _ { j , m _ j } ( s ) . \\end{align*}"}
+{"id": "992.png", "formula": "\\begin{align*} \\partial \\big ( ( n _ v ) _ { v \\in V ( G ) } \\big ) = \\big ( n _ { v _ E } - n _ { v ' _ E } \\big ) _ { E \\in E ( G ) } . \\end{align*}"}
+{"id": "3152.png", "formula": "\\begin{align*} \\begin{aligned} \\ 6 + \\ 8 + & \\ 7 + \\ 9 \\leq C _ { \\rm { d } 2 } \\\\ & ( \\ 1 + \\ 2 + \\ 4 + \\ 3 + \\ 8 ) \\end{aligned} \\end{align*}"}
+{"id": "4255.png", "formula": "\\begin{align*} \\tilde { y } _ { s } = \\zeta + \\int _ { s } ^ { T } \\tilde { f } \\left ( s , \\tilde { y } _ { s } , \\tilde { z } _ { s } , \\mathbb { P } _ { \\left ( \\tilde { y } _ { s } , \\tilde { z } _ { s } \\right ) } \\right ) \\mathrm { d } s - \\int _ { s } ^ { T } \\tilde { z } _ { s } \\mathrm { d } W _ { s } ^ { t _ { 0 } } . \\end{align*}"}
+{"id": "4377.png", "formula": "\\begin{align*} \\mathcal J _ { \\ell } = \\{ n _ 0 + 1 , n _ 0 + 2 , \\dots , \\ell - 1 , \\ell \\} . \\end{align*}"}
+{"id": "4322.png", "formula": "\\begin{align*} L _ { n + 1 } = 2 L _ { n - 1 } ( 2 M + 1 ) , \\end{align*}"}
+{"id": "2306.png", "formula": "\\begin{align*} w ( z ) = \\Phi _ 0 ( z ) + \\widetilde { \\Psi } ( z ) , \\end{align*}"}
+{"id": "2406.png", "formula": "\\begin{align*} \\vec { \\delta } _ { 1 0 2 } \\cdot \\vec { \\delta } _ { 2 0 3 } = 1 + \\cos \\alpha _ { 1 0 2 } + \\cos \\alpha _ { 1 0 3 } + \\cos \\alpha _ { 2 0 3 } . \\end{align*}"}
+{"id": "4044.png", "formula": "\\begin{align*} \\big | L ( f , k ) \\big | ^ 2 = \\frac { 2 } { \\left ( ( k - 1 ) ! \\right ) ^ 2 } \\sum _ { l , \\ , m \\geq 1 } G _ k \\left ( \\frac { l m } { p } \\right ) \\frac { \\lambda _ f ( l ) \\lambda _ f ( m ) } { \\sqrt { l m } } , \\end{align*}"}
+{"id": "7974.png", "formula": "\\begin{align*} \\big ( \\mathrm { d i v } \\ , V \\big ) ^ 2 = \\mathrm { t r } \\big ( ( \\nabla V ) ^ 2 \\big ) + \\mathrm { d i v } \\Big ( V \\ , \\mathrm { d i v } V - \\nabla V \\ , V \\Big ) \\ , . \\end{align*}"}
+{"id": "1281.png", "formula": "\\begin{align*} Z _ q ( B _ n ) = \\left \\{ \\begin{aligned} 2 & \\mbox { i f } q = 0 \\\\ n & \\mbox { f o r } q \\geq 1 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2611.png", "formula": "\\begin{align*} P _ 1 ( t ) = a _ { \\sigma _ 1 } t ^ { \\sigma _ 1 } + a _ { \\sigma _ 1 + 1 } t ^ { \\sigma _ 1 + 1 } + \\cdots + a _ { d _ 1 } t ^ { d _ 1 } \\end{align*}"}
+{"id": "7611.png", "formula": "\\begin{align*} h _ { 0 } ( z ) & = z \\exp \\left ( \\sqrt { \\frac { 6 9 } { 6 8 } } \\int _ { 0 } ^ { z } \\frac { x ^ 2 + \\sqrt { 1 + x ^ 4 } - 1 } { x } d x \\right ) \\\\ & = z + \\frac { \\sqrt { 6 9 } } { 4 \\sqrt { 1 7 } } z ^ 3 + \\frac { 1 } { 4 } \\left ( \\frac { 6 9 } { 1 3 6 } + \\frac { \\sqrt { 6 9 } } { 4 \\sqrt { 1 7 } } \\right ) z ^ 5 + \\cdots , . \\end{align*}"}
+{"id": "720.png", "formula": "\\begin{align*} A _ { S } [ \\partial _ { t } \\gamma _ { S } ] ( e ) + C _ { V } [ \\partial _ { t } \\gamma _ { B } ] ( e ) = G _ { V , 1 } ( e ) \\end{align*}"}
+{"id": "2665.png", "formula": "\\begin{align*} F _ { j - 1 } = 4 \\alpha ^ 2 h _ { j } ( \\tanh ( \\alpha x _ j ) - \\tanh ( \\alpha x _ { j - 1 } ) - \\alpha h _ j \\tanh ( \\alpha x _ { j - 1 } ) \\tanh ( \\alpha x _ j ) ) , \\end{align*}"}
+{"id": "7362.png", "formula": "\\begin{align*} \\lambda ^ { ( i ) } = ( \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\lambda _ j , i , \\lambda _ { j + 1 } + 1 , \\ldots , \\lambda _ { n - d - 1 } + 1 ) \\in B _ { n - d , k } \\ , \\backslash \\ , B _ { n - d - 1 , k } . \\end{align*}"}
+{"id": "300.png", "formula": "\\begin{align*} \\times \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } \\sum _ { n = 1 } ^ { \\infty } ( + 3 n ^ 2 x ^ { n + 1 } y ^ { n + 1 } - 4 n ^ 2 x ^ { n + 1 } y ^ { n + 2 } + n ^ 2 x ^ { n + 1 } y ^ { n + 3 } ) \\frac { z ^ n } { n ^ 4 } \\right \\} \\end{align*}"}
+{"id": "6334.png", "formula": "\\begin{align*} \\frac { \\omega ^ 2 } { r ^ 2 } \\ , \\frac { \\partial z } { \\partial \\phi } ( \\phi , \\omega , r ; t ) = \\frac { \\partial } { \\partial s _ 1 } F ( 0 , p ( \\phi + \\omega t ) ) \\cdot p ' ( \\phi ) + \\frac { \\partial } { \\partial s _ 2 } F ( 0 , p ( \\phi + \\omega t ) ) \\cdot p ' ( \\phi + \\omega t ) . \\end{align*}"}
+{"id": "7136.png", "formula": "\\begin{align*} \\partial _ { x _ n } u ( \\hat { x } ) _ { \\big | \\{ x _ n = 0 \\} } = \\partial _ { x _ n } \\big ( c \\circ F _ \\gamma \\big ) ( \\hat { x } ) _ { \\big | \\{ x _ n = 0 \\} } = \\nabla c ( x ^ \\prime , \\gamma ( x ^ \\prime ) ) \\cdot ( - \\mathbf { n } ( x ^ \\prime , \\gamma ( x ^ \\prime ) ) ) = 0 , \\end{align*}"}
+{"id": "6942.png", "formula": "\\begin{align*} x _ { m - j _ { 2 k } } = \\lambda _ { m - j _ { 2 k } } + E _ { m - j _ { 2 k } } = E _ { m - j _ { 2 k } } + k - 1 . \\end{align*}"}
+{"id": "4317.png", "formula": "\\begin{align*} \\tilde { y } _ { t _ { k , m } + 2 ^ { 2 - 2 k } } ^ { K } \\circ \\big ( \\tilde { y } _ { t _ { k , m } } ^ { K } \\big ) ^ { - 1 } \\begin{aligned} [ t ] & = \\begin{cases} y _ { 2 \\tau _ k } ^ { ( i _ k ; L _ k ) } & k = K \\ ; ( \\mathrm { m o d } \\ ; 2 ) , \\\\ \\mathrm { I d } & . \\end{cases} \\\\ & = : Y _ { k , m } ^ K , \\end{aligned} \\end{align*}"}
+{"id": "7244.png", "formula": "\\begin{align*} \\sup _ { k \\leq n } | S _ k - T _ k | = o ( n ^ { 1 / p } ( \\log n ) ^ { 1 / 2 - 1 / p } ) \\end{align*}"}
+{"id": "7715.png", "formula": "\\begin{align*} \\int _ { S ^ { m - 1 } } e ^ { i t ( u , \\sigma ) } \\phi ( \\sigma ) d \\sigma = ( 2 \\pi ) ^ { \\frac { m } { 2 } } i ^ { l } \\phi ( u ) t ^ { 1 - \\frac { m } { 2 } } J _ { \\frac { m } { 2 } - 1 + l } ( t ) , u \\in S ^ { m - 1 } , \\end{align*}"}
+{"id": "617.png", "formula": "\\begin{align*} K \\subset \\bigcup _ { i = 1 } ^ r \\{ \\psi \\in \\S ( B ) : \\psi ( f _ { \\lambda _ i } ) < 1 - 1 / { m _ i } \\} . \\end{align*}"}
+{"id": "6059.png", "formula": "\\begin{align*} x _ 1 - x _ 3 = ( n + m ) \\delta + 2 \\pi z _ 1 \\quad \\textrm { a n d } x _ 2 - x _ 3 = - m \\delta + 2 \\pi z _ 2 \\textrm { f o r } \\delta \\in \\mathbb { R } , \\end{align*}"}
+{"id": "8236.png", "formula": "\\begin{align*} \\Lambda _ A ( s ) = \\prod _ { n = 1 } ^ N ( 1 - s e ^ { - \\i \\theta _ n } ) , \\end{align*}"}
+{"id": "6073.png", "formula": "\\begin{align*} \\| f \\| _ { H ^ { p } _ { A } } ^ { p } : = \\| M ^ 0 _ { \\phi , A } f \\| _ { p } ^ { p } . \\end{align*}"}
+{"id": "8690.png", "formula": "\\begin{align*} T _ { s - 1 } = \\frac { f ^ { ( s ) } ( \\tau ) } { s ! p ^ { s } } \\left [ E \\{ ( \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { s } \\} + E \\{ ( \\| Y _ { 1 } - Y _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { s } \\} - 2 E \\{ ( \\| X _ { 1 } - Y _ { 1 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { s } \\} \\right ] \\end{align*}"}
+{"id": "92.png", "formula": "\\begin{align*} F ( z ) = I + i ( P _ 0 - z - i M h W ) ^ { - 1 } M h W , \\end{align*}"}
+{"id": "9177.png", "formula": "\\begin{align*} \\bar { u } ^ { 1 } & = \\eta ^ { 1 } ( \\zeta _ { [ - 1 ] } ^ { 1 } , x ^ { 1 } , x ^ { 2 } , x ^ { 3 } , y _ { 1 , [ 0 , 3 ] } ^ { 1 , d } , y _ { 2 , [ 0 , 2 ] } ^ { 1 , d } ) \\\\ \\bar { u } ^ { 2 } & = \\eta ^ { 2 } ( \\zeta _ { [ - 1 ] } ^ { 1 } , x ^ { 1 } , x ^ { 2 } , x ^ { 3 } , y _ { 1 , [ 0 , 3 ] } ^ { 1 , d } , y _ { 2 , [ 0 , 2 ] } ^ { 1 , d } ) \\ , . \\end{align*}"}
+{"id": "2244.png", "formula": "\\begin{align*} \\langle f _ b , \\varphi \\rangle : = \\lim _ { r \\nearrow 1 } \\int _ 0 ^ { 2 \\pi } f ( r e ^ { i \\theta } ) \\ , \\varphi ( \\theta ) \\ , d \\theta \\end{align*}"}
+{"id": "4147.png", "formula": "\\begin{align*} Q _ { 1 , \\vec { v } _ { s , t , f , 0 } ^ { ( 1 ) } } = \\left ( \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 0 & t & 1 \\\\ 1 & t ^ { 2 } + s & t \\\\ 0 & s t + 1 & s \\end{array} \\right ) , \\left ( \\begin{array} { r r r } 0 & 1 & 0 \\\\ 0 & t & 1 \\\\ 1 & s & 0 \\end{array} \\right ) \\right ) , \\end{align*}"}
+{"id": "2898.png", "formula": "\\begin{gather*} \\partial _ \\xi ^ \\alpha f ( \\xi ) \\sim \\sum _ { j \\geq 0 } \\partial _ \\xi ^ \\alpha f _ { m - j } ( \\xi ) \\\\ g ^ { B , \\alpha } ( \\xi ) : = \\partial _ x ^ \\alpha \\big | _ { x = 0 } \\Big ( \\alpha _ { - x } \\big ( g ( \\xi + B x ) \\big ) \\Big ) \\sim \\sum _ { j \\geq 0 } g _ { m ' - j } ^ { B , \\alpha } ( \\xi ) , \\end{gather*}"}
+{"id": "684.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq j < N ^ + } \\varphi ( T ^ i y ) = \\sum _ { l = 0 } ^ k \\sum _ { 0 \\leq j < q ( l ) } S ( l ) \\varphi ( ( T ^ { ( l ) } ) ^ j ( y ( l ) ) ) q ( l ) \\leq \\| Z ( l + 1 ) \\| . \\end{align*}"}
+{"id": "3556.png", "formula": "\\begin{align*} \\langle b \\rangle = \\int _ { 0 } ^ { \\infty } x ^ { b - 1 } \\ , d x \\end{align*}"}
+{"id": "4753.png", "formula": "\\begin{align*} \\tilde { S } _ { \\gamma _ i \\gamma _ i } ^ T \\tilde { S } _ { \\gamma _ i \\gamma _ i } = I _ { 2 | \\alpha _ i | } + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "213.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c , d ) = 1 \\\\ a , b , c , d \\geq 1 } } \\left ( \\frac { 1 } { 1 - w ^ a x ^ b y ^ c z ^ d } \\right ) ^ { \\frac { 1 } { a ^ r b ^ s c ^ t d ^ u } } = \\exp \\left \\{ L i _ r ( w ) L i _ { s } ( x ) L i _ { t } ( y ) L i _ u ( z ) \\right \\} . \\end{align*}"}
+{"id": "1934.png", "formula": "\\begin{align*} f ( z _ 1 , \\ldots , z _ { m } ) = a _ 0 + \\left ( \\sum _ { S \\in \\mathcal { Q } _ m } \\left ( \\prod _ { i \\in S } z _ i \\right ) f _ S ( z _ S ) \\right ) + f ^ * ( z _ 1 , \\ldots , z _ { m } ) , \\end{align*}"}
+{"id": "1373.png", "formula": "\\begin{align*} v _ 1 , \\ , v _ 2 = - x _ 3 v _ 1 , \\ , v _ 3 = - x _ 5 v _ 1 , \\ , v _ 4 = - x _ 2 v _ 1 , \\ , v _ 5 = - x _ 4 v _ 1 \\end{align*}"}
+{"id": "7073.png", "formula": "\\begin{align*} \\beta = \\alpha \\left ( \\left \\{ \\left . \\tilde { \\beta } _ j \\ \\right | \\ j \\in I ' \\right \\} \\right ) . \\end{align*}"}
+{"id": "1652.png", "formula": "\\begin{align*} ( \\nabla _ { X } { f } ) \\ , \\xi _ i = - { f } \\ , \\nabla _ { X } \\ , \\xi _ i . \\end{align*}"}
+{"id": "5472.png", "formula": "\\begin{align*} M _ b ( \\theta ) = \\mathbb { E } _ { \\Phi | \\Phi ( \\mathcal { A } ) > 0 } \\{ P _ s ^ b ( \\theta ) \\} \\Pr ( \\Phi ( \\mathcal { A } ) > 0 ) . \\end{align*}"}
+{"id": "7127.png", "formula": "\\begin{align*} a _ \\varepsilon ( x _ n ) : = \\int _ { B _ { \\delta } ( x ^ \\prime ) \\times ( 0 , \\delta ) } \\rho _ \\varepsilon ( | x - \\hat { y } | ) y . \\end{align*}"}
+{"id": "6579.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ k A _ { i } \\right | \\le \\frac { k } { 4 } I + \\sum _ { i = 1 } ^ k | A _ { i } | . \\end{align*}"}
+{"id": "2505.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\phi _ { \\mu } = \\mu ^ { \\frac { N } { 2 } } \\phi ( \\mu x ) , \\psi _ { \\mu } = \\mu ^ { \\frac { N } { 2 } } \\psi ( \\mu x ) , \\end{array} \\right . \\end{align*}"}
+{"id": "1375.png", "formula": "\\begin{align*} \\langle x _ 1 , x _ 2 , \\dots , x _ 7 : x _ i x _ j x _ i ^ { - 1 } = x _ { 5 i + 3 j \\bmod 7 } \\rangle . \\end{align*}"}
+{"id": "1736.png", "formula": "\\begin{align*} \\mu ^ { ( n ) } _ t ( y ) = e ^ { - \\alpha t } \\mu _ 0 ( y ) + \\int _ 0 ^ t \\alpha e ^ { - \\alpha ( t - s ) } \\Phi ( \\nu ^ { ( n - 1 ) } _ s , \\mu ^ { ( n - 1 ) } _ s ) ( y ) \\mathrm { d } s . \\end{align*}"}
+{"id": "4733.png", "formula": "\\begin{align*} \\begin{pmatrix} W _ { \\mathcal { I } \\mathcal { J } } & X _ { \\mathcal { I } \\mathcal { J } } \\\\ Y _ { \\mathcal { I } \\mathcal { J } } & Z _ { \\mathcal { I } \\mathcal { J } } \\end{pmatrix} . \\end{align*}"}
+{"id": "5895.png", "formula": "\\begin{align*} \\beta _ { 1 , X } ( k ) = \\norm { \\sup _ { \\norm { f } _ \\infty \\leq 1 } \\abs { P _ { X _ k \\lvert \\mathcal F _ 0 } ( f ) - P _ { X _ k } ( f ) } } _ 1 . \\end{align*}"}
+{"id": "4074.png", "formula": "\\begin{align*} \\mathrm { A n n } ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) = \\lbrace T \\in \\mathbb { T } \\mid T ( z ^ n \\otimes \\lbrace 0 , \\infty \\rbrace ) = 0 \\rbrace \\end{align*}"}
+{"id": "9232.png", "formula": "\\begin{align*} \\widetilde { \\varepsilon _ 5 } ( m ) : = \\min \\Bigl \\{ 4 F _ m , \\frac { ( \\pi ^ 2 B _ 1 a ) ^ { m / 3 } } { 3 1 6 \\ , A ( \\sigma ) } \\Bigl ( \\frac { m ! } { \\Gamma ( m / 3 + 2 ) } \\Bigr ) ^ { 1 / 2 } \\ , \\varepsilon _ 4 \\Bigr \\} . \\end{align*}"}
+{"id": "4304.png", "formula": "\\begin{align*} \\epsilon _ n = \\min _ { k \\in \\{ 0 , . . . , n - 1 \\} } \\{ \\delta _ { n - k } 2 ^ { - k - 1 } \\} . \\end{align*}"}
+{"id": "8954.png", "formula": "\\begin{align*} \\frac { ( r _ t ) ^ l \\Delta t } { [ ( r _ x ) ^ { l } \\Delta x ] ^ s } = \\bigg [ \\frac { r _ t } { ( r _ x ) ^ s } \\bigg ] ^ l \\frac { \\Delta t } { \\Delta x ^ s } = \\frac { \\Delta t } { \\Delta x ^ s } . \\end{align*}"}
+{"id": "8166.png", "formula": "\\begin{align*} V ( x , \\omega ) = \\sum _ i W ( x - x _ i ) . \\end{align*}"}
+{"id": "2549.png", "formula": "\\begin{align*} \\lambda _ t = \\lambda _ { j _ \\ell } = \\lambda _ { j _ \\ell ^ * } \\end{align*}"}
+{"id": "480.png", "formula": "\\begin{align*} p _ r = x _ 1 ^ r + x _ 2 ^ r + \\cdots , \\end{align*}"}
+{"id": "3104.png", "formula": "\\begin{align*} [ 2 ] _ q ^ k [ k ] _ { q ^ 2 } ! S _ B [ n , k ] = \\sum _ { \\ell = 0 } ^ { k } q ^ { ( n - k ) ( 2 \\ell - n - k ) } B _ { n , n - \\ell } ( q ) { n - \\ell \\brack k - \\ell } _ { q ^ 2 } . \\end{align*}"}
+{"id": "6580.png", "formula": "\\begin{align*} \\mathbf { h } = \\zeta \\mathbf { \\hat h } + \\sqrt { 1 - \\zeta ^ 2 } \\Delta \\mathbf { h } , \\end{align*}"}
+{"id": "5913.png", "formula": "\\begin{align*} \\pi _ { \\psi } [ \\begin{pmatrix} a & 0 \\\\ 0 & a ^ { \\ast - 1 } \\end{pmatrix} , t ] f ( y ) = t | \\det ( a ) | ^ { 1 / 2 } f ( y a ) , \\end{align*}"}
+{"id": "6383.png", "formula": "\\begin{align*} x = A ^ { 2 } - 5 B ^ { 2 } , y = 3 ( A ^ { 2 } - 5 B ^ { 2 } ) ( ( A ^ { 2 } + 5 B ^ { 2 } ) ^ 2 + 2 0 A ^ 2 B ^ 2 ) , \\end{align*}"}
+{"id": "2149.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { d _ { M \\cap \\partial W _ 0 } ( x , z ) } { d _ X ( x , z ) } = 1 \\end{align*}"}
+{"id": "4928.png", "formula": "\\begin{align*} F ( k , t + 1 , \\mathbf { p } _ { n + 1 } ) = \\begin{dcases} p _ { k } ^ { } F ( k - 1 , t , \\mathbf { p } _ { n + 1 } ) + ( 1 - p _ { k } ^ { } ) F ( k , t , \\mathbf { p } _ { n + 1 } ) & \\textnormal { i f } \\ ; k < t ; \\\\ 1 & \\textnormal { i f } \\ ; k \\ge t ; \\end{dcases} \\end{align*}"}
+{"id": "2296.png", "formula": "\\begin{align*} w _ b = H _ b + ( \\widetilde { T } ( f ) ) _ b \\end{align*}"}
+{"id": "7478.png", "formula": "\\begin{align*} \\Phi ( \\pi ) = \\tilde { \\delta } \\mu _ k \\tilde { \\alpha } _ k \\mu _ { k - 1 } \\tilde { \\alpha } _ { k - 1 } \\cdots \\mu _ 1 \\tilde { \\alpha } _ 1 \\mu _ 0 \\end{align*}"}
+{"id": "1709.png", "formula": "\\begin{align*} B _ \\gamma ^ { ( \\alpha , \\beta ) } w _ { \\gamma } ^ { ( \\alpha , \\beta ) } ( z ) + A _ \\gamma ^ { ( \\alpha , \\beta ) } ( z ) w _ { \\gamma + 1 } ^ { ( \\alpha , \\beta ) } ( z ) + w _ { \\gamma + 2 } ^ { ( \\alpha , \\beta ) } ( z ) = 0 , \\end{align*}"}
+{"id": "8169.png", "formula": "\\begin{align*} \\{ T > t \\} T = \\inf \\{ s \\geq 0 : Z _ t ( K ) \\geq 1 \\} . \\end{align*}"}
+{"id": "8765.png", "formula": "\\begin{align*} \\bar { d } _ { 0 , i } = D _ 0 h _ { 0 , i } , \\forall i \\in I . \\end{align*}"}
+{"id": "1881.png", "formula": "\\begin{align*} [ \\zeta ] = [ \\zeta _ 0 ] , \\ \\ \\| [ \\zeta ] \\| _ q = \\| \\zeta _ 0 \\| \\ \\ \\mbox { a n d } \\ \\ \\| \\zeta _ 0 \\| \\leq \\| \\zeta \\| . \\end{align*}"}
+{"id": "1170.png", "formula": "\\begin{align*} T \\left ( \\frac { 2 } { f ^ { 2 } - 1 } \\right ) = T \\left ( \\frac { 1 } { f - 1 } - \\frac { 1 } { f + 1 } \\right ) = T \\left ( \\frac { 1 } { f - 1 } \\right ) - T \\left ( \\frac { 1 } { f + 1 } \\right ) \\end{align*}"}
+{"id": "286.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { 1 } { 1 2 } L i _ 2 ( z ) + \\frac { 1 } { 3 } \\log \\left ( \\frac { 1 } { 1 - z } \\right ) + \\frac { 5 } { 1 2 } \\frac { z } { 1 - z } + \\frac { 1 } { 6 } \\frac { z } { ( 1 - z ) ^ 2 } \\right \\} \\end{align*}"}
+{"id": "8010.png", "formula": "\\begin{align*} f ( \\Phi ( A ) ) & = f ( ( \\pi ( A ) ) _ { \\mathcal { S } } ) \\\\ & \\le \\frac { U ( f ( \\pi ( A ) ) _ { \\mathcal { S } } U ^ * + V ( f ( \\pi ( A ) ) _ { \\mathcal { S } } V ^ * } { 2 } \\\\ & = \\frac { U ( \\pi ( f ( A ) ) _ { \\mathcal { S } } U ^ * + V ( \\pi ( f ( A ) ) _ { \\mathcal { S } } V ^ * } { 2 } \\\\ & = \\frac { U \\Phi ( f ( A ) ) U ^ * + V \\Phi ( f ( A ) ) V ^ * } { 2 } , \\end{align*}"}
+{"id": "6261.png", "formula": "\\begin{align*} \\omega _ \\Pi = k | \\Sigma | . \\end{align*}"}
+{"id": "6538.png", "formula": "\\begin{align*} L _ { \\varphi } ( a , r , s , A ) : & = \\frac { 1 } { | P _ { r , s , A } | K _ { P _ { r , s , A } , \\varphi } ( a ) } \\\\ & = \\inf \\left \\{ \\frac { \\int _ { a + P _ { r , s , A } } | f | ^ 2 e ^ { - \\varphi } } { | P _ { r , s , A } | e ^ { - \\varphi ( a ) } } ~ \\middle | ~ f \\in A ^ 2 ( a + P _ { r , s , A } , \\varphi ) \\And f ( a ) = 1 \\right \\} , \\end{align*}"}
+{"id": "5542.png", "formula": "\\begin{align*} \\eta \\times \\widetilde { \\mathbb { P } } _ { \\mu } \\left ( \\left \\{ L _ { x } \\omega _ { n } \\in S ( x , \\omega ) \\mbox { f o r i n f i n i t e l y m a n y } n \\in \\mathbb { N } \\right \\} \\right ) = 0 . \\end{align*}"}
+{"id": "8284.png", "formula": "\\begin{align*} M _ N ( 2 k ) = \\frac { G ^ 2 ( 1 + k ) G ( N + 1 ) G ( N + 1 + 2 k ) } { G ( 1 + 2 k ) G ^ 2 ( N + 1 + k ) } . \\end{align*}"}
+{"id": "7086.png", "formula": "\\begin{align*} \\nu ( g ' ) = v ( p ) + ( p - 1 ) \\nu ( x ) = v ( p ) . \\end{align*}"}
+{"id": "6367.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } t ^ { \\frac { d + 2 \\beta } { \\alpha } \\frac { q - 1 } { q } } \\| P _ t ^ { \\Gamma } f \\| _ { q , M _ \\Gamma } = 0 . \\end{align*}"}
+{"id": "7533.png", "formula": "\\begin{align*} \\begin{aligned} - \\Delta _ p u + ( - \\Delta _ p ) ^ s u = f \\ ; \\ ; \\Omega , \\end{aligned} \\end{align*}"}
+{"id": "3968.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } \\left [ \\rho _ 2 ( Y _ 2 , y _ 2 ) \\right ] - 1 & \\leq \\mathbb { E } \\left [ \\boldsymbol { d } _ { \\mathcal { S } _ 2 } ( S _ 2 , s _ 2 ) ^ { p _ 2 } \\right ] = \\boldsymbol { W } _ { p _ 2 } \\left ( \\gamma _ { 2 , 3 } ^ { \\eta _ 1 , \\delta } , \\delta _ { s _ 2 } \\right ) ^ { p _ 2 } \\leq \\left [ \\delta ^ { 1 / p _ 2 } + \\boldsymbol { W } _ { p _ 2 } \\left ( \\mu _ 2 , \\delta _ { s _ 2 } \\right ) \\right ] ^ { p _ 2 } , \\\\ \\end{aligned} \\end{align*}"}
+{"id": "6980.png", "formula": "\\begin{align*} f = f _ 0 + f _ 1 q + \\ldots + f _ n q ^ n \\end{align*}"}
+{"id": "7234.png", "formula": "\\begin{align*} u : = \\overline { u } + \\overline { m } \\end{align*}"}
+{"id": "5675.png", "formula": "\\begin{align*} \\aligned \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | { u _ \\nu } | ^ { p } ) | { v _ \\nu } | ^ { q } = & ( 1 + o _ \\nu ( 1 ) ) [ ( \\gamma _ p + \\gamma _ q ) \\nu D ^ { \\frac { 2 } { \\gamma _ p + \\gamma _ q } } _ 0 ] ^ { \\frac { \\gamma _ p + \\gamma _ q } { 2 - \\gamma _ p - \\gamma _ q } } \\\\ = & ( 1 + o _ \\nu ( 1 ) ) \\nu ^ { \\frac { \\gamma _ p + \\gamma _ q } { 2 - \\gamma _ p - \\gamma _ q } } \\int _ { \\mathbb { R } ^ N } ( I _ \\mu * | { u _ 0 } | ^ { p } ) | { v _ 0 } | ^ { q } . \\endaligned \\end{align*}"}
+{"id": "3572.png", "formula": "\\begin{align*} r _ b ( n ) \\ : = \\ ( a _ 0 a _ 1 \\cdots a _ { L - 1 } ) _ b . \\end{align*}"}
+{"id": "4092.png", "formula": "\\begin{gather*} \\begin{aligned} \\left [ T _ { i j } , T _ { j k } \\right ] & = T _ { i k } & & i \\neq k , \\\\ \\left [ T _ { i j } , T _ { k \\ell } \\right ] & = 1 & & i \\neq \\ell , j \\neq k , \\\\ & & & i , j , k \\in \\{ 1 , \\dots , n \\} \\\\ ( T _ { 1 2 } T _ { 2 1 } ^ { - 1 } T _ { 1 2 } ) ^ 4 & = 1 . \\end{aligned} \\end{gather*}"}
+{"id": "5631.png", "formula": "\\begin{align*} \\aligned \\min \\{ p , q \\} \\left \\{ \\begin{array} { l l l } > \\frac { 6 - \\mu } { 2 } 2 < \\mu < 3 , \\ & N = 3 , \\\\ < 2 , \\ & N = 4 , \\end{array} \\right . \\endaligned \\end{align*}"}
+{"id": "9014.png", "formula": "\\begin{align*} \\int _ M ( 1 + w ) | T ^ \\varphi | ^ 2 = 0 , \\int _ M w ( 1 + w ) | T ^ \\varphi | ^ 2 = \\int _ M ( 1 + w ) ^ 2 | T ^ \\varphi | ^ 2 . \\end{align*}"}
+{"id": "3717.png", "formula": "\\begin{align*} \\exp ( t x ) H = \\exp ( t y ) H \\mbox { f o r a l l } t \\in \\R \\end{align*}"}
+{"id": "7950.png", "formula": "\\begin{align*} \\Theta ( x _ 1 , \\dots , x _ n ) = \\frac { c | x _ 1 | } { \\log | x _ 1 | } \\end{align*}"}
+{"id": "6336.png", "formula": "\\begin{align*} p ( \\phi + \\vartheta ) = \\frac { 1 } { y _ 0 } \\vartheta + o ( \\vartheta ) p ' ( \\phi + \\vartheta ) = \\frac { 1 } { y _ 0 } + o ( 1 ) , \\qquad \\vartheta \\to 0 . \\end{align*}"}
+{"id": "772.png", "formula": "\\begin{align*} \\begin{aligned} \\displaystyle \\int _ M \\sigma _ k ( \\kappa ) \\mathrm { d } \\mu _ g = & \\int _ { \\mathbb { S } ^ n } \\sum \\limits _ { m = 0 } ^ k \\left [ \\left ( A _ 0 ^ m + A _ 1 ^ m u + A _ 2 ^ m u ^ 2 + A ^ m | \\nabla u | ^ 2 \\right ) \\sigma _ m ( D ^ 2 u ) + B ^ m u ^ i u _ j [ T _ m ] _ i ^ j ( D ^ 2 u ) \\right ] \\mathrm { d } A \\\\ & + O ( \\varepsilon ) \\| u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 + O ( \\varepsilon ) \\| \\nabla u \\| _ { L ^ 2 ( \\mathbb { S } ^ n ) } ^ 2 , \\end{aligned} \\end{align*}"}
+{"id": "3877.png", "formula": "\\begin{align*} \\int _ { \\mathcal { T } } \\left [ \\sup _ { x \\in \\mathcal { X } } \\varphi ( t , x ) \\right ] d \\mu ( t ) = \\sup _ { \\pi \\in \\Gamma ( \\mu , \\varphi ) } \\int _ { \\mathcal { T } \\times \\mathcal { X } } \\varphi \\left ( t , x \\right ) d \\pi ( t , x ) . \\end{align*}"}
+{"id": "4115.png", "formula": "\\begin{align*} x = \\gamma _ 2 y ^ 2 + \\gamma _ 1 y + \\gamma _ 0 . \\end{align*}"}
+{"id": "8780.png", "formula": "\\begin{align*} ( 4 u ^ 2 + 2 t u + 2 t + 1 0 u + 6 ) b _ 3 & = ( 4 t u + 4 t + 2 u + 2 ) c _ 3 + 2 ( u + 1 ) b _ 2 \\\\ & + b _ 1 + c _ 1 + u ( b _ 3 + c _ 3 ) + ( u + 1 ) ( b _ 3 + c _ 3 ) \\end{align*}"}
+{"id": "1555.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\ge 1 } f _ { n } n ^ { \\frac { \\lambda - 1 } { 2 } } e ( n z ) . \\end{align*}"}
+{"id": "5374.png", "formula": "\\begin{align*} ( z ( t ) , - \\dot z ( t ) ) \\in \\mathcal { M } ^ { - 1 } \\mathcal { L } = \\{ ( z , w ) ~ \\vert ~ ( w , v ) \\in \\mathcal { M } , ~ ( z , v ) \\in \\mathcal { L } \\} , \\end{align*}"}
+{"id": "4236.png", "formula": "\\begin{align*} X & = X _ 2 - 2 X _ 1 \\\\ Y & = Y _ 2 - 2 n Y _ 1 \\end{align*}"}
+{"id": "9263.png", "formula": "\\begin{align*} R f _ N ( x ) \\gtrsim x ^ { - 1 } \\int _ N ^ { 2 x + N } f _ N ( z ) \\ , d z = x ^ { - 1 } , 1 / 2 < x < N , N > 1 . \\end{align*}"}
+{"id": "2468.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { M } : = \\left \\{ ( u , v ) \\in H ^ 1 ( \\mathbb { R } ^ N ) \\times H ^ 1 ( \\mathbb { R } ^ N ) \\backslash \\{ ( 0 , 0 ) \\} , Q ( u , v ) = 0 \\right \\} , \\end{array} \\right . \\end{align*}"}
+{"id": "770.png", "formula": "\\begin{align*} \\delta _ { i _ 1 i _ 2 \\cdots i _ m i _ { m + 1 } \\cdots i _ k } ^ { j _ 1 j _ 2 \\cdots j _ m j _ { m + 1 } \\cdots j _ k } \\delta ^ { i _ { m + 1 } } _ { j _ { m + 1 } } \\cdots \\delta ^ { i _ k } _ { j _ k } = \\delta _ { i _ 1 i _ 2 \\cdots i _ m } ^ { j _ 1 j _ 2 \\cdots j _ m } { n - m \\choose k - m } ( k - m ) ! , \\end{align*}"}
+{"id": "3698.png", "formula": "\\begin{align*} \\begin{cases} \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) \\tilde { u } = 0 & \\Omega , \\\\ \\mathcal { L } \\tilde { u } = 0 & \\Omega ^ c , \\\\ \\tilde { u } = 0 & Q , \\end{cases} \\end{align*}"}
+{"id": "8904.png", "formula": "\\begin{gather*} \\mathsf { d } _ { c c } ( [ X _ { n 1 } , \\dots , X _ { n j } ] _ c ) \\leq \\sum _ { i = 1 } ^ { j - 1 } 2 ^ i \\mathsf { d } _ { c c } ( X _ { n i } ) + 2 ^ { j - 1 } \\mathsf { d } _ { c c } ( X _ { n j } ) . \\end{gather*}"}
+{"id": "8905.png", "formula": "\\begin{align*} u = \\sum _ { s , t = 1 } ^ { d _ 1 } \\alpha _ { \\mathrm { s t } } X _ s \\otimes X _ t = \\sum _ { s , t = 1 } ^ { d _ 1 } \\big ( \\alpha _ { \\mathrm { s t } } ^ 1 X _ s \\big ) \\otimes \\big ( \\alpha _ { \\mathrm { s t } } ^ 2 X _ t \\big ) = : \\sum _ { n = 1 } ^ { d _ 1 ^ 2 } X _ { n 1 } \\otimes X _ { n 2 } , \\end{align*}"}
+{"id": "4977.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { p } _ { 2 , n } ^ { ( t ) } & = \\hat { p } _ { 1 , n } ^ { ( t ) } \\ , + \\ , ( n - 1 ) \\hat { p } _ { 1 , n - 1 } ^ { ( t ) } , \\end{aligned} \\end{align*}"}
+{"id": "7600.png", "formula": "\\begin{align*} 1 + \\frac { z f ^ { \\prime \\prime } ( z ) } { f ^ { \\prime } ( z ) } = w ( z ) + \\sqrt { 1 + w ^ 2 ( z ) } , \\end{align*}"}
+{"id": "497.png", "formula": "\\begin{align*} p ( \\bar { H } _ { ( m , n ) } ) & = \\int _ { - \\infty } ^ { \\infty } p ( \\bar { H } _ { ( m , n ) } | \\gamma _ { ( m , n ) } ) p ( \\gamma _ { ( m , n ) } ) d \\gamma _ { ( m , n ) } \\\\ & = \\frac { a } { \\pi b } \\left ( \\frac { | | \\bar { H } _ { ( m , n ) } | | ^ { 2 } } { b } + 1 \\right ) ^ { - ( a + 1 ) } . \\end{align*}"}
+{"id": "280.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - y ^ m z ^ n } \\right ) ^ { \\frac { m ^ 4 } { n ^ 5 } } = \\left \\{ \\frac { \\left ( 1 - z \\right ) ^ { 4 } } { \\left ( 1 - y z \\right ) ^ { y ^ 5 - 4 y ^ 4 + 6 y ^ 3 + y } } \\right \\} ^ { \\frac { 1 } { ( 1 - y ) ^ 5 } } \\end{align*}"}
+{"id": "8573.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda t } M ( t ) = \\begin{cases} \\dfrac { \\nu Y } { \\lambda } , & p = 0 , \\\\ - \\dfrac { \\nu q \\log ( q ) Y } { \\lambda p } , & 0 < p < 1 . \\end{cases} \\end{align*}"}
+{"id": "1826.png", "formula": "\\begin{align*} V _ F ( t ) \\ = \\ : \\ ! V _ F ( t ) \\ ! : \\ + \\ N ( g ) \\ . \\end{align*}"}
+{"id": "7025.png", "formula": "\\begin{align*} \\nu _ i \\left ( l a _ l Q _ i ^ { l - 1 } \\right ) = \\nu _ i ( f ) + \\alpha _ i . \\end{align*}"}
+{"id": "5664.png", "formula": "\\begin{align*} m _ \\nu ( a , b ) = \\inf _ { \\mathcal { O } _ \\nu ( a , b ) } \\max _ { t \\in \\mathbb { R } } J _ \\nu ( t \\star ( u , v ) ) . \\end{align*}"}
+{"id": "5179.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ j z _ { r _ j } z _ { s _ j } = 0 , \\end{align*}"}
+{"id": "8995.png", "formula": "\\begin{align*} \\Delta w + \\frac { S ^ \\varphi } { m - 1 } w = 0 \\ , . \\end{align*}"}
+{"id": "2693.png", "formula": "\\begin{align*} c _ r ^ s ( n ) & = \\sum \\limits _ { \\substack { { ( h , r ^ s ) _ s = 1 } \\\\ h = 1 } } ^ { r ^ s } e ^ { \\frac { 2 \\pi i n h } { r ^ s } } . \\end{align*}"}
+{"id": "4980.png", "formula": "\\begin{align*} \\hat { p } _ { k - 1 , n } ^ { ( t + 1 ) } = \\hat { p } _ { k - 1 , n } ^ { ( t ) } + \\hat { p } _ { n - 1 , n } ^ { \\phantom { ( ) } } \\hat { p } _ { k - 1 , n - 1 } ^ { ( t ) } . \\end{align*}"}
+{"id": "289.png", "formula": "\\begin{align*} \\prod _ { \\substack { l , m , n \\geq 1 \\\\ l , m \\leq n ; \\ , \\gcd ( l , m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { l m ^ 2 } { n ^ 4 } } = \\sqrt [ 3 ] { \\left ( \\frac { 1 } { 1 - z } \\right ) } \\ ; \\exp \\left \\{ \\frac { 1 } { 1 2 } \\left ( L i _ 2 ( z ) + \\frac { z ( 7 - 5 z ) } { ( 1 - z ) ^ 2 } \\right ) \\right \\} , \\end{align*}"}
+{"id": "3759.png", "formula": "\\begin{align*} T _ 1 ^ n \\left ( \\{ x _ \\tau ^ 0 \\} _ { \\tau = 1 } ^ { r } , \\vec { a } \\right ) & : = U ( \\vec { a } , 1 ) T _ 1 ^ n \\left ( \\{ x _ \\tau ^ 0 \\} _ { \\tau = 1 } ^ { r } \\right ) U ( \\vec { a } , 1 ) ^ { - 1 } , \\\\ T _ 2 ^ n \\left ( \\{ x _ \\theta ^ 0 \\} _ { \\theta = r + 1 } ^ { r + s } , \\vec { a } \\right ) & : = U ( \\vec { a } , 1 ) T _ 2 ^ n \\left ( \\{ x _ \\tau ^ 0 \\} _ { \\tau = r + 1 } ^ { r + s } \\right ) U ( \\vec { a } , 1 ) ^ { - 1 } . \\end{align*}"}
+{"id": "116.png", "formula": "\\begin{align*} \\chi e ^ { - i t h ^ { - 1 } \\tilde { P } _ h ( 0 ) } q _ 1 e ^ { - ( t _ 0 - t ) X } \\tilde { R } _ h ( z ) \\chi = \\chi e ^ { - t X } q _ 1 e ^ { - ( t _ 0 - t ) X } \\tilde { R } _ h ( z ) \\chi = 0 . \\end{align*}"}
+{"id": "7656.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\lambda _ ( B _ R ) = \\frac { ( n - 1 ) ^ 2 \\kappa ^ 2 } { 4 } , \\end{align*}"}
+{"id": "4095.png", "formula": "\\begin{align*} M = \\mathcal { Q } _ { \\ell , \\vec { v } } \\left ( M _ { \\bullet , \\ell } \\right ) \\end{align*}"}
+{"id": "5411.png", "formula": "\\begin{align*} \\frac { R } { ( L + 1 ) ( 1 - R ) } \\le \\frac { 1 } { L + 1 } \\frac { 1 } { 1 - R } \\le \\frac { \\log q } { 2 \\log d ( \\chi ) } \\frac { 4 \\log d ( \\chi ) } { \\log q \\log ( d ( \\chi ) \\log d ( \\chi ) / n ) } = \\frac { 2 } { \\log ( d ( \\chi ) \\log d ( \\chi ) / n ) } \\le 1 \\end{align*}"}
+{"id": "9228.png", "formula": "\\begin{align*} \\frac { C _ k ( p ) } { a ^ k } = \\frac { 1 } { ( \\pi ^ 2 a ) ^ { k } } \\sum _ { j = 0 } ^ { \\lfloor 3 k / 2 \\rfloor } \\Bigl ( \\frac { \\pi } { 2 i } \\Bigr ) ^ j d ^ { ( k ) } _ j F ^ { ( 3 k - 2 j ) } ( p ) \\end{align*}"}
+{"id": "2838.png", "formula": "\\begin{align*} \\alpha _ { i , j } = \\sum _ { v = 0 } ^ { N + q } C _ v \\alpha _ { i , j - v - 1 } , C _ v \\in \\mathbb { C } , j \\ge N + q + 1 , \\end{align*}"}
+{"id": "6372.png", "formula": "\\begin{align*} \\dot { \\widetilde { x } } = \\widetilde { f } ( \\widetilde { x } ) , \\dot { \\widetilde { y } } = \\widetilde { f } ( \\widetilde { y } ) , \\dot { \\widetilde { z } } = \\frac { \\widetilde { f } ( \\widetilde { x } ) - \\widetilde { f } ( \\widetilde { y } ) } { \\epsilon } , \\end{align*}"}
+{"id": "4288.png", "formula": "\\begin{align*} \\eta : = C _ v \\ln \\theta - R \\ln \\rho - \\left ( \\frac { Z ( \\theta ) } { 2 \\theta } \\right ) ^ \\prime q ^ 2 . \\end{align*}"}
+{"id": "2341.png", "formula": "\\begin{align*} \\int _ { - R } ^ R \\int _ { - R } ^ R e ^ { - i \\xi ( x - y ) } d x d y = \\left | \\int _ { - R } ^ { R } e ^ { - i \\xi x } d x \\right | ^ 2 = \\frac { 4 \\sin ^ 2 ( R | \\xi | ) } { | \\xi | ^ 2 } = 4 \\pi R \\ell _ R ( \\xi ) \\end{align*}"}
+{"id": "5494.png", "formula": "\\begin{align*} \\mathfrak { p } _ { I } = \\mathfrak { m } _ { I } \\oplus \\mathfrak { a } _ { I } \\oplus \\mathfrak { n } ^ { I } . \\end{align*}"}
+{"id": "5352.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ 2 ^ 2 \\le \\left ( \\frac { 1 0 0 r } { d } \\right ) ^ d \\gamma _ 1 \\cdot \\gamma \\log ^ { d } \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) . \\end{align*}"}
+{"id": "5283.png", "formula": "\\begin{align*} \\left | 1 + \\rho d X \\right | ^ { q } \\le 1 + q \\rho d X + \\binom { q } { 2 } \\rho ^ 2 d ^ 2 X ^ 2 + \\binom { q } { 3 } \\rho ^ 3 | d X | ^ 3 ( 1 + | \\rho d X | ) ^ { q - 3 } . \\end{align*}"}
+{"id": "7763.png", "formula": "\\begin{align*} ( M \\vec x ) _ { i ^ * } - ( M \\vec x ) _ 1 = y _ { i ^ * } - y _ 1 . \\end{align*}"}
+{"id": "8513.png", "formula": "\\begin{align*} \\zeta _ { p , 0 } ( s , c ) = \\frac { \\pi } { \\sqrt { c } } \\ , \\frac { 1 } { s - 1 } + \\frac { \\pi } { \\sqrt { c } } \\left ( 2 \\gamma - \\log \\left ( \\frac { c } { 4 } \\right ) + 8 \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { 2 n - 1 } \\cdot \\frac { 1 } { \\sigma \\left ( \\sqrt { c } \\lambda _ { n } ^ { \\prime } \\right ) e ^ { ( 2 n - 1 ) \\pi \\sqrt { c } } - 1 } \\right ) + O ( s - 1 ) . \\end{align*}"}
+{"id": "8303.png", "formula": "\\begin{align*} \\mu ( \\Gamma _ X ) = \\delta _ { x _ 0 } ( \\Gamma _ X ) + \\int _ { \\bf X } \\int _ { \\bf A } p ( \\Gamma _ X | y , a ) \\sigma ^ s ( d a | y ) \\mu ( d y ) , ~ ~ \\forall \\Gamma _ X \\in { \\cal B } ( { \\bf X } ) : \\end{align*}"}
+{"id": "3594.png", "formula": "\\begin{align*} q ( n ) \\ = \\ \\frac { 1 0 ^ { n - 1 } } { n + 1 } , \\quad . \\end{align*}"}
+{"id": "153.png", "formula": "\\begin{align*} L i _ 2 ( z ) = - \\int _ { 0 } ^ { z } \\frac { \\log ( 1 - t ) } { t } d t . \\end{align*}"}
+{"id": "8368.png", "formula": "\\begin{align*} \\big | \\Im L ' / \\Im \\widetilde { L } \\big | = \\left | \\frac { \\langle u _ 1 ' \\cdots u _ n ' \\rangle } { \\langle u _ 1 ' \\cdots u _ { n - 1 } ' \\rangle } \\right | = \\frac { c _ v } { c ( v _ n ) } . \\end{align*}"}
+{"id": "6400.png", "formula": "\\begin{align*} [ D \\widetilde { \\varphi } : D \\tau _ M ] _ t = \\lambda ^ { \\varphi } ( t ) , t \\in \\mathbb { R } , \\end{align*}"}
+{"id": "4750.png", "formula": "\\begin{align*} J _ { 2 | \\alpha _ i | } = \\tilde { S } _ { \\gamma _ i \\gamma _ i } ^ T J _ { 2 | \\alpha _ i | } \\tilde { S } _ { \\gamma _ i \\gamma _ i } + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "8223.png", "formula": "\\begin{align*} A ^ * \\phi ( x , \\sigma ) = \\left ( \\tfrac { \\kappa } { 2 } \\partial _ { x x } - \\sigma \\lambda \\partial _ x \\right ) \\phi ( x , \\sigma ) + \\sum _ { \\sigma ' \\in S } c ( \\sigma , \\sigma ' ) \\big ( \\phi ( x , \\sigma ' ) - \\phi ( x , \\sigma ) \\big ) . \\end{align*}"}
+{"id": "7178.png", "formula": "\\begin{align*} I _ x : = \\left \\{ \\lambda _ x / 2 ^ k : k \\in \\N , \\ , \\ , \\mathcal { H } ^ 1 \\left ( J \\cap B _ { \\lambda _ x / 2 ^ k } ( x ) \\right ) \\geq \\eta \\lambda _ x / 2 ^ k \\right \\} \\end{align*}"}
+{"id": "8773.png", "formula": "\\begin{align*} \\lambda _ { i j } = \\sum _ { k = 1 } ^ m \\left ( \\beta _ { i , k } - \\beta _ { j , k } \\right ) + \\gamma _ i + \\alpha _ { i j } \\leq 4 m p _ { \\max } , \\forall j \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
+{"id": "5476.png", "formula": "\\begin{align*} f _ { Z } ( z ) = \\frac { z ^ { \\kappa - 1 } ( 1 - z ) ^ { \\beta - 1 } } { B ( \\kappa , \\beta ) } , \\end{align*}"}
+{"id": "3581.png", "formula": "\\begin{align*} 5 \\cdot 1 0 ^ m - 1 \\ = \\ 4 \\underbrace { 9 \\cdots 9 } _ { m } , \\end{align*}"}
+{"id": "7333.png", "formula": "\\begin{align*} 1 = \\sum _ { i = 1 } ^ n ( \\gamma _ i - \\alpha _ i ) z _ i = \\sum _ { i = 1 } ^ n ( \\gamma _ i - \\alpha _ i ) \\sum _ { k = 1 } ^ { N _ 1 } u _ k e _ { k i } = \\sum _ { k = 1 } ^ { N _ 1 } u _ k \\sum _ { i = 1 } ^ n ( \\gamma _ i - \\alpha _ i ) e _ { k i } = \\sum _ { k = 1 } ^ { N _ 1 } u _ k e _ k , \\end{align*}"}
+{"id": "8530.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow \\infty , p ^ { \\prime } \\rightarrow 0 ^ { + } } \\tilde { \\zeta } _ { p , p ^ { \\prime } } ( s , c ) = \\sum _ { m \\in \\mathbb { Z } , n \\neq 0 } \\frac { 1 } { \\left ( m ^ { 2 } + c \\ , \\left ( n - \\frac { 1 } { 2 } \\right ) ^ { 2 } \\right ) ^ { s } } . \\end{align*}"}
+{"id": "75.png", "formula": "\\begin{align*} T _ x ^ \\ast \\mathcal { M } = E ^ * _ 0 ( x ) \\oplus E ^ * _ s ( x ) \\oplus E ^ * _ u ( x ) , \\end{align*}"}
+{"id": "5559.png", "formula": "\\begin{align*} \\bar { \\lambda } = \\int _ { { \\rm T r e e } _ { F } ^ { H _ { 0 } } } \\left ( \\zeta _ { H } \\right ) _ { \\ast } \\nu d \\rho ( H ) . \\end{align*}"}
+{"id": "4809.png", "formula": "\\begin{align*} V ( \\theta , \\dot \\theta ) = \\frac { 1 } { 2 } \\dot \\theta ^ 2 + 1 - \\cos ( \\theta ) + \\alpha ( 1 - \\cos ^ 3 ( \\theta ) ) , \\end{align*}"}
+{"id": "7777.png", "formula": "\\begin{align*} x ^ r \\cdot ( x ^ q y - x y ^ q ) + y ^ r \\cdot ( y ^ q z - y z ^ q ) + ( z ^ r + k x ^ r ) \\cdot ( z ^ q x - z x ^ q ) = 0 . \\end{align*}"}
+{"id": "5359.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ 2 ^ 2 \\le \\left ( \\frac { 1 0 0 r ' } { d } \\right ) ^ d \\gamma _ 1 \\| f ^ { = d } \\| _ 2 \\log ^ d \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) . \\end{align*}"}
+{"id": "5695.png", "formula": "\\begin{align*} E _ { g } ( z ) : = \\sum _ { n > 0 } n ^ { 1 - k } C _ { g } ( n ) q ^ { n } \\end{align*}"}
+{"id": "5488.png", "formula": "\\begin{align*} \\psi _ { \\mu } & : \\mathcal { P } _ { \\mu } ( X ) \\to \\mathcal { P } _ { H } ( X ) \\\\ \\eta & \\mapsto \\lambda = \\boldsymbol { \\beta } _ { \\eta } ( H ) , \\end{align*}"}
+{"id": "6201.png", "formula": "\\begin{align*} \\tilde { \\varphi } ( v ) = \\begin{cases} \\varphi ( v ) , & v \\in T , \\\\ w _ { i } , & v = v _ { i } . \\end{cases} \\end{align*}"}
+{"id": "5206.png", "formula": "\\begin{align*} \\bigcup p ^ \\in = p ^ \\wedge . \\end{align*}"}
+{"id": "6603.png", "formula": "\\begin{align*} A B E P \\leq \\frac { 1 } { \\log _ 2 N _ t } \\sum _ { \\hat n _ t = 1 } ^ { N _ t } \\sum _ { n _ t \\neq \\hat n _ t } ^ { N _ t } \\bar P _ i N ( \\hat n _ t \\to n _ t ) , \\ \\ \\ i \\in \\{ a , b \\} \\end{align*}"}
+{"id": "2732.png", "formula": "\\begin{align*} J _ 5 \\geq & - C s ^ 2 \\lambda ^ 4 \\iint _ Q \\xi \\left | A \\nabla \\eta \\cdot \\nabla \\eta \\right | ^ 2 | u | ^ 2 d x d y d t - C \\lambda ^ 2 \\iint _ { Q } \\xi \\left | A \\nabla u \\cdot \\nabla \\eta \\right | ^ 2 d x d y d t \\\\ & - C s ^ 2 \\lambda ^ 3 \\int _ 0 ^ T \\int _ { \\omega _ 0 } \\xi | u | ^ 2 d x d y d t - C \\lambda \\iint _ Q \\xi A \\nabla u \\cdot \\nabla u d x d y d t . \\end{align*}"}
+{"id": "7775.png", "formula": "\\begin{align*} g ( x , y ) & = g _ m ( x , y ) + g _ { m + 1 } ( x , y ) + \\cdots + g _ s ( x , y ) \\\\ h ( x , y ) & = h _ n ( x , y ) + h _ { n + 1 } ( x , y ) + \\cdots + h _ t ( x , y ) \\end{align*}"}
+{"id": "7475.png", "formula": "\\begin{align*} \\pi = \\cdots y z \\cdots \\beta _ 1 ( \\pi ) \\cdots x \\cdots \\qquad \\phi ( x , \\pi ) = \\cdots y x z \\cdots \\beta _ 1 ( \\pi ) \\cdots . \\end{align*}"}
+{"id": "5750.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\omega ( 2 ^ { - j } ) & \\approx \\int _ { 0 } ^ { 1 } \\omega ( t ) \\frac { \\mathrm { d } t } { t } < \\infty . \\end{align*}"}
+{"id": "681.png", "formula": "\\begin{align*} | S ( k ) D ^ j \\varphi ( x ) | & = | D ^ j v ( T ^ { Q _ \\alpha ( k ) } ( x ) ) - D ^ j v ( x ) | \\leq \\left \\{ \\begin{array} { l l } \\| D ^ { j } v \\| _ { C ^ 1 } | I ^ { ( k ) } | & 0 \\leq j < m \\\\ \\| D ^ { m } v \\| _ { C ^ b } | I ^ { ( k ) } | ^ b & j = m . \\end{array} \\right . \\end{align*}"}
+{"id": "841.png", "formula": "\\begin{align*} \\begin{aligned} & \\underset { S _ { n } } { } ~ R _ { M M S E } \\left ( S _ { n } \\right ) \\\\ & \\ \\left \\| \\textbf { P } _ c \\left ( S _ { n } \\right ) \\right \\| ^ { 2 } \\leq P . \\end{aligned} \\end{align*}"}
+{"id": "8943.png", "formula": "\\begin{align*} \\phi = \\Tilde { \\phi } _ { e } + \\sum _ { m = 1 } ^ { k } C _ { p _ m } h ^ { p _ m } , \\end{align*}"}
+{"id": "6134.png", "formula": "\\begin{align*} \\tau = \\min \\left \\{ \\frac { s } { 2 } , \\frac { 1 - s } { 2 } \\right \\} . \\end{align*}"}
+{"id": "6308.png", "formula": "\\begin{align*} \\phi _ t ( p , q ) : = e _ t ( \\gamma _ { p , q } ) = \\exp _ q ( ( t - 1 ) \\lambda _ p ) , \\lambda _ p \\in T ^ * _ q M p = \\exp _ q ( - \\lambda _ p ) . \\end{align*}"}
+{"id": "5980.png", "formula": "\\begin{align*} J ( g p _ z , i ) & = \\sqrt { ( c y i + c x + d y ^ { - 1 } \\det p _ z ) \\det ( g p _ z ) } \\\\ & = \\sqrt { y ^ { - 1 } \\det g ( c z + d ) } \\\\ & = J ( g , z ) J ( p _ z , i ) . \\end{align*}"}
+{"id": "5462.png", "formula": "\\begin{align*} \\mathbb { E } \\{ Z \\} = \\frac { \\kappa } { \\kappa + \\beta } , \\mathbb { E } \\{ Z ^ 2 \\} = \\frac { \\kappa ( \\kappa + 1 ) } { \\kappa + \\beta ( \\kappa + \\beta + 1 ) } . \\end{align*}"}
+{"id": "3573.png", "formula": "\\begin{align*} 8 7 1 2 \\ = \\ 4 \\cdot r ( 8 7 1 2 ) , \\ \\ 9 8 0 1 \\ = \\ 9 \\cdot r ( 9 8 0 1 ) . \\end{align*}"}
+{"id": "6814.png", "formula": "\\begin{align*} H _ { i , t _ 0 + \\epsilon } = H _ { i , t _ 0 - \\epsilon } \\cup N ( L _ i ) \\end{align*}"}
+{"id": "7450.png", "formula": "\\begin{align*} \\widehat { \\varphi } ^ { ( i ) } ( x _ i , t ) : = \\frac { 1 } { \\pi h ^ 2 _ i } \\int _ { \\partial \\Upsilon _ i } \\varphi ^ { ( i ) } ( x _ i , \\bar { \\xi } _ i , t \\big ) \\ , d \\sigma _ { \\bar { \\xi } _ i } . \\end{align*}"}
+{"id": "5324.png", "formula": "\\begin{align*} W ^ { = 1 } [ f ] = p ( 1 - p ) \\approx p \\end{align*}"}
+{"id": "3148.png", "formula": "\\begin{align*} \\norm { \\widetilde { \\psi } _ m } _ { X _ { \\mu _ 0 , \\xi _ 0 } } \\leq \\frac { C } { \\mu _ 0 } \\norm { g } _ { Y _ { \\mu _ 0 , \\xi _ 0 } } , m = 0 , 1 , \\ldots , N . \\end{align*}"}
+{"id": "3878.png", "formula": "\\begin{align*} \\sup _ { \\pi \\in \\Gamma ( \\mathcal { G } , \\varphi ) } \\int _ { \\mathcal { T } \\times \\mathcal { X } } \\varphi ( t , x ) d \\pi ( t , x ) = \\sup _ { \\mu \\in \\mathcal { G } } \\left \\{ \\int _ { \\mathcal { T } } \\left [ \\sup _ { x \\in \\mathcal { X } } \\varphi ( t , x ) \\right ] d \\mu ( t ) \\right \\} . \\end{align*}"}
+{"id": "229.png", "formula": "\\begin{align*} = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\prod _ { j = 1 } ^ { M } { L i _ { s _ j } } ^ { ( d _ j ) } \\left ( e ^ { i \\theta } \\right ) \\prod _ { k = 1 } ^ { N } { L i _ { t _ k } } ^ { ( e _ k ) } \\left ( e ^ { - i \\theta } \\right ) d \\theta , \\end{align*}"}
+{"id": "3627.png", "formula": "\\begin{align*} a _ 0 \\ \\equiv \\ b _ 0 + 4 \\ \\equiv \\ \\begin{cases} 7 , & , \\\\ 1 2 , & . \\end{cases} \\end{align*}"}
+{"id": "5409.png", "formula": "\\begin{align*} \\sum _ { 1 \\le k \\le L } \\frac { R ^ { k } } { k } \\min \\{ q ^ { k / 2 } , d ( \\chi ) \\} = \\sum _ { 1 \\le k \\le L } \\frac { ( R \\sqrt { q } ) ^ k } { k } \\le \\sum _ { 1 \\le k \\le L } ( R \\sqrt { q } ) ^ k \\le \\frac { ( R \\sqrt { q } ) ^ L } { 1 - ( R \\sqrt { q } ) ^ { - 1 } } \\le 6 d ( \\chi ) R ^ L \\end{align*}"}
+{"id": "8282.png", "formula": "\\begin{align*} f ( \\theta ) = ( - 1 ) ^ k e ^ { - \\i k \\theta } \\prod _ { j = 1 } ^ { 2 k } ( e ^ { \\i \\theta } - e ^ { \\i \\alpha _ j } ) . \\end{align*}"}
+{"id": "1946.png", "formula": "\\begin{align*} f ^ * _ { n , l } ( z _ 1 , \\ldots , z _ m ) : = f ^ * _ { n , l - 1 } ( z _ 1 , \\ldots , z _ m ) + \\delta _ { n , l } z _ 1 ^ { j _ 1 } \\cdots z _ m ^ { j _ m } A _ n ( z _ 1 , \\ldots , z _ m ) \\end{align*}"}
+{"id": "9075.png", "formula": "\\begin{align*} 0 = & - \\lambda h ( x _ 0 ) = M h ( x _ 0 ) = \\sum _ { y \\sim x _ 0 } M _ { x _ 0 y } h ( y ) \\\\ = & \\sum _ { \\substack { y \\sim x _ 0 \\\\ y \\in D _ j \\setminus D _ i } } M _ { x _ 0 y } h ( y ) = \\left ( 1 - \\frac { a _ j } { a _ i } \\right ) \\sum _ { \\substack { y \\sim x _ 0 \\\\ y \\in D _ j \\setminus D _ i } } M _ { x _ 0 y } f ( y ) . \\end{align*}"}
+{"id": "2493.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle \\dfrac { 6 } { N } \\left ( \\dfrac { a _ 2 } { 2 m _ 1 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\xi | ^ 2 d x + \\dfrac { a _ 1 } { 4 m _ 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\eta | ^ 2 d x \\right ) - \\dfrac { 3 } { 2 } a _ 1 a _ 2 \\int _ { \\mathbb { R } ^ N } \\eta \\xi ^ 2 d x = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "5090.png", "formula": "\\begin{align*} \\sigma _ { L ^ 2 _ N } \\left ( \\mathcal { A } [ \\phi ] \\right ) = \\bigcup _ { \\xi \\in \\Omega _ N } \\sigma _ { L ^ 2 _ { \\rm p e r } ( 0 , T ) } \\left ( \\mathcal { A } _ \\xi [ \\phi ] \\right ) , \\end{align*}"}
+{"id": "9135.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\mathrm { d } x ^ { 1 } \\\\ \\mathrm { d } x ^ { 2 } \\\\ \\mathrm { d } x ^ { 3 } \\\\ \\mathrm { d } \\varphi ^ { 1 } & = & \\mathrm { d } x ^ { 1 } \\\\ \\mathrm { d } \\varphi ^ { 2 } & = & \\mathrm { d } x ^ { 2 } \\\\ \\mathrm { d } \\delta ( \\varphi ^ { 1 } ) & = & \\mathrm { d } x ^ { 1 } + \\mathrm { d } u ^ { 1 } \\\\ \\mathrm { d } \\delta ( \\varphi ^ { 2 } ) & = & \\frac { 1 } { u ^ { 1 } + 1 } \\mathrm { d } x ^ { 3 } - \\frac { x ^ { 3 } } { ( u ^ { 1 } + 1 ) ^ { 2 } } \\mathrm { d } u ^ { 1 } \\end{array} \\end{align*}"}
+{"id": "5792.png", "formula": "\\begin{align*} \\frac { y ^ { ( k ) } } { y } = Y ^ k + \\frac { k ( k - 1 ) } { 2 } Y ^ { k - 2 } Y ' + P _ { k - 2 } ( Y ) , \\end{align*}"}
+{"id": "2701.png", "formula": "\\begin{align*} \\phi ( ( \\overline { \\mathbb { Q } } _ { \\ell } ) _ { \\widetilde { M } } ) = \\phi ( \\mathcal { I C } _ { \\widetilde { M } } ) = 0 . \\end{align*}"}
+{"id": "6275.png", "formula": "\\begin{align*} \\mathcal { G } ^ M _ { r } ( K , \\gamma ) \\cap B _ { \\rho ( r ) } ( p ) = C N _ { r } ( \\zeta ) \\cap B _ { \\rho ( r ) } ( p ) , \\end{align*}"}
+{"id": "7456.png", "formula": "\\begin{align*} N _ { p \\alpha + k } ( \\xi , t ) = w ^ { ( i ) } _ { p \\alpha + k } ( 0 , t ) + \\mathcal { O } ( \\exp ( - \\beta _ 0 \\xi _ i ) ) \\mbox { a s } \\ \\ \\xi _ i \\to + \\infty , \\ \\ \\xi \\in \\Xi ^ { ( i ) } , i = \\{ 1 , 2 , 3 \\} ( \\beta _ 0 > 0 ) \\end{align*}"}
+{"id": "8473.png", "formula": "\\begin{align*} 2 ^ { \\frac { 1 } { 2 } - s } \\sum _ { n = 1 } ^ { \\infty } \\left ( 2 n - 1 \\right ) ^ { s - \\frac { 1 } { 2 } } \\ , K _ { s - \\frac { 1 } { 2 } } \\left ( \\pi ( 2 n - 1 ) x \\right ) = \\frac { \\Gamma ( s ) \\pi ^ { - s } x ^ { - s - \\frac { 1 } { 2 } } } { 4 } + \\pi \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n } \\ , \\intop _ { 0 } ^ { \\infty } \\ , y ^ { s - \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( 2 \\pi x y ) \\ , e ^ { - 2 \\pi n y } \\ , d y . \\end{align*}"}
+{"id": "6048.png", "formula": "\\begin{align*} \\frac { A ^ m B ^ { n + m } } { C ^ { n + 2 m } } = \\frac { 1 } { r ^ { 2 m ( n + m ) } } ( - 1 ) ^ { n } \\frac { \\sin ^ m ( \\frac { m \\theta } { 2 } ) \\sin ^ { n + m } ( \\frac { ( n + m ) \\theta } { 2 } ) } { \\sin ^ { n + 2 m } ( \\frac { ( n + 2 m ) \\theta } { 2 } ) } , \\end{align*}"}
+{"id": "2471.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle \\int _ { \\mathbb { R } ^ N } \\left ( a _ 2 | \\phi ( t , x ) | ^ 2 + a _ 1 | \\psi ( t , x ) | ^ 2 \\right ) d x = \\int _ { \\mathbb { R } ^ N } \\left ( a _ 2 | \\phi _ 0 ( x ) | ^ 2 + a _ 1 | \\psi _ 0 ( x ) | ^ 2 \\right ) d x ; \\end{array} \\right . \\end{align*}"}
+{"id": "8030.png", "formula": "\\begin{align*} f ( V A V ^ * ) = V f ( A ) V ^ * \\end{align*}"}
+{"id": "8180.png", "formula": "\\begin{align*} f ( \\ell ) = \\left \\lceil e ^ { a \\ell ^ { 3 / 2 } } \\right \\rceil \\leq \\left \\lceil e ^ { a ( n + 1 ) ^ { 3 / 2 } } \\right \\rceil = f ( n + 1 ) . \\end{align*}"}
+{"id": "7649.png", "formula": "\\begin{align*} \\quad \\quad \\quad \\frac { \\partial } { \\partial c } M ( c , x , 1 ) = & ( 4 - c ^ 2 ) [ 3 2 x ( 1 - x ) ( 1 + x ) ^ 2 - 4 0 c ^ 2 x ( 1 - x ) ( 1 + x ) ^ 2 \\\\ & - 6 c ^ 3 x ^ 2 ( 2 - 1 3 x + 2 x ^ 2 ) + 8 c ( - 3 2 + 3 0 x ^ 2 - 3 1 x ^ 3 + 6 x ^ 4 ) ] = 0 \\end{align*}"}
+{"id": "12.png", "formula": "\\begin{align*} \\sigma ^ 2 = \\sum _ { s = - \\infty } ^ { \\infty } \\Theta _ { \\infty } ( s ) = \\frac { 2 } { \\zeta _ N ( m + n ) } \\sum _ { s = - \\infty } ^ { \\infty } \\left ( \\sum _ { \\substack { p \\geq 1 \\\\ g c d ( p , N ) = 1 } } \\sum _ { \\substack { q \\geq 1 \\\\ q = p ( m o d \\ { N } ) } } \\int _ { \\R ^ { m + n } } \\chi ( p a ^ s x ) \\chi ( q x ) \\ , d x \\right ) . \\end{align*}"}
+{"id": "6359.png", "formula": "\\begin{align*} M _ { \\Gamma } ( x ) = | x | ^ { \\beta } M _ { \\Gamma } ( x / | x | ) , x \\in \\Gamma . \\end{align*}"}
+{"id": "7.png", "formula": "\\begin{align*} \\alpha ( x ) = \\alpha _ { d } ( \\pi ( x ) ) , \\end{align*}"}
+{"id": "6818.png", "formula": "\\begin{align*} ( W , \\lambda _ 1 , \\phi _ 1 ) = | W _ 1 ^ { \\flat } : = W _ 0 ^ { \\flat } \\cup h _ { n - 1 } ; \\ , \\mathcal { L } | \\ , \\cup \\ , h _ n \\end{align*}"}
+{"id": "5473.png", "formula": "\\begin{align*} \\Pr ( \\Phi ( \\mathcal { A } ) > 0 ) = 1 - \\exp \\left ( - 2 \\pi \\lambda R _ { m i n } R _ S \\right ) . \\end{align*}"}
+{"id": "7124.png", "formula": "\\begin{align*} R _ 2 ( x , \\hat { y } ) : = \\sum _ { | \\beta | = 2 } \\frac { 2 } { \\beta ! } \\left ( \\int _ { 0 } ^ 1 ( 1 - t ) ^ \\beta c ( \\hat { y } + t ( x - \\hat { y } ) ) t \\right ) ( x - \\hat { y } ) ^ \\beta . \\end{align*}"}
+{"id": "1341.png", "formula": "\\begin{align*} \\psi _ { i } ^ { N } ( \\mathbf { x } _ { N } , \\mathbf { m } _ { N } ) = \\frac { 1 } { N } \\stackrel [ j = 1 ] { N } { \\sum } m _ { i } m _ { j } S ( x _ { j } - x _ { i } ) \\end{align*}"}
+{"id": "9163.png", "formula": "\\begin{align*} \\begin{aligned} x ^ { 1 , + } & = x ^ { 1 } + \\bar { u } ^ { 1 } \\cos ( \\bar { u } ^ { 2 } ) \\\\ x ^ { 2 , + } & = x ^ { 2 } + \\bar { u } ^ { 1 } \\sin ( \\bar { u } ^ { 2 } ) \\\\ x ^ { 3 , + } & = 2 \\bar { u } ^ { 2 } - x ^ { 3 } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "8732.png", "formula": "\\begin{align*} T _ { l - 1 } = \\frac { f ^ { ( l ) } ( \\tau ) } { l ! p ^ { l } } \\times \\left [ E \\{ ( \\| X _ { 1 } - X _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { l } \\} + E \\{ ( \\| Y _ { 1 } - Y _ { 2 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { l } \\} - 2 E \\{ ( \\| X _ { 1 } - Y _ { 1 } \\| _ { 2 } ^ { 2 } - p \\tau ) ^ { l } \\} \\right ] . \\end{align*}"}
+{"id": "6132.png", "formula": "\\begin{align*} \\begin{aligned} I _ { 2 , 2 } \\leq \\frac { c \\lambda ^ { \\tilde { p } - 1 } } { \\mathcal { M } ^ { \\frac { 2 s } { n } } } \\sum _ { \\mathcal { K } \\times I \\in \\mathcal { A } } \\int _ { \\lambda _ { 0 } } ^ { N \\mathcal { M } ^ { - 1 } } \\mu _ { \\tau , t } \\left ( \\mathcal { K } \\times I \\right ) \\ , d \\lambda \\end{aligned} \\end{align*}"}
+{"id": "1009.png", "formula": "\\begin{align*} & \\psi \\ , ' \\bigg ( \\frac { 1 } { 4 } \\bigg ) = \\pi ^ 2 + 8 G , \\\\ [ 2 m m ] & \\psi \\ , ' \\bigg ( \\frac { 3 } { 4 } \\bigg ) = \\pi ^ 2 - 8 G , \\end{align*}"}
+{"id": "4869.png", "formula": "\\begin{align*} a _ { t } ^ { \\prime } x ^ { \\mu } - b _ { t } ^ { \\mu } = \\left ( 1 - \\mu \\right ) \\left ( a _ { t } ^ { \\prime } \\overline { x } - \\overline { b } _ { t } \\right ) + \\mu \\left ( a _ { t } ^ { \\prime } x - b _ { t } \\right ) \\leq 0 , \\end{align*}"}
+{"id": "4726.png", "formula": "\\begin{align*} n _ g ^ * : = \\sum _ { \\lambda = 1 } ^ \\kappa n _ \\lambda ^ * , n _ g : = \\sum _ { \\ell = 1 } ^ k n _ \\ell - 2 k . \\end{align*}"}
+{"id": "6050.png", "formula": "\\begin{align*} H ( z ) : = A _ 1 z ^ { n ' + m ' } + B _ 1 \\overline { z } ^ { m ' } + C _ 1 \\textrm { f o r a l l } z \\in \\mathbb { C } \\end{align*}"}
+{"id": "389.png", "formula": "\\begin{align*} \\epsilon \\int \\limits _ { \\Omega } U ^ T ( D _ i ) _ { x _ i } d \\Omega = \\epsilon \\oint \\limits _ { \\partial \\Omega } U ^ T ( n _ i D _ i ) d s - \\epsilon \\int \\limits _ { \\Omega } U _ { x _ i } ^ T D _ i d \\Omega . \\end{align*}"}
+{"id": "5289.png", "formula": "\\begin{align*} \\| 1 + d Z \\| _ { q } ^ { q } \\ge 1 + \\binom { \\lfloor q \\rfloor } { 2 } d ^ 2 \\ge 1 + \\frac { q ^ 2 } { 6 } d ^ 2 . \\end{align*}"}
+{"id": "2007.png", "formula": "\\begin{align*} E ( \\phi ) : = \\frac { 1 } { n + 1 } \\int _ \\Omega ( - \\phi ) ( d d ^ c \\phi ) ^ n . \\end{align*}"}
+{"id": "7870.png", "formula": "\\begin{align*} \\mathsf { s o r t } ( \\sigma ( A ) ) : = \\mu ( A , k ) \\ldots \\mu ( A , 2 ) \\mu ( A , 1 ) . \\end{align*}"}
+{"id": "5238.png", "formula": "\\begin{align*} E _ y = ( - w + a ) ( - w + b ) - ( - w + c ) ( - y ) w . \\end{align*}"}
+{"id": "626.png", "formula": "\\begin{align*} w ( b ) = \\sup _ { \\psi \\in \\S ( B ) } | \\psi ( b ) | \\end{align*}"}
+{"id": "6388.png", "formula": "\\begin{align*} a w _ { 2 } ^ { p } - b w _ { 1 } ^ { 2 p } = c r ^ 2 \\end{align*}"}
+{"id": "8293.png", "formula": "\\begin{gather*} y _ 2 y _ 1 = p y _ 1 y _ 2 , x _ 2 x _ 1 = p x _ 1 x _ 2 , \\\\ y _ 1 x _ 1 = - x _ 1 y _ 1 + p ^ 2 x _ 2 y _ 1 + x _ 1 y _ 2 - p x _ 2 y _ 2 , \\\\ y _ 1 x _ 2 = - p x _ 1 y _ 1 + x _ 2 y _ 1 + x _ 1 y _ 2 - p x _ 2 y _ 2 , \\\\ y _ 2 x _ 1 = - p x _ 1 y _ 1 - 2 p ^ 2 x _ 2 y _ 1 + p x _ 1 y _ 2 - p x _ 2 y _ 2 , \\\\ y _ 2 x _ 2 = - p x _ 1 y _ 1 + p ^ 2 x _ 2 y _ 1 + x _ 1 y _ 2 - x _ 2 y _ 2 , \\end{gather*}"}
+{"id": "2940.png", "formula": "\\begin{align*} C ( \\gamma _ 1 , \\gamma _ 2 ) + C ( \\gamma _ 2 , \\gamma _ 3 ) = C ( \\gamma _ 1 , \\gamma _ 3 ) . \\end{align*}"}
+{"id": "1026.png", "formula": "\\begin{align*} & { _ { 2 } F _ { 1 } } \\left [ \\begin{array} { c c c c c c c c } a , 1 - a \\\\ b \\end{array} ; \\frac { 1 } { 2 } \\right ] = \\frac { \\Gamma ( \\frac { b } { 2 } ) \\Gamma ( \\frac { 1 + b } { 2 } ) } { \\Gamma ( \\frac { a + b } { 2 } ) \\Gamma ( \\frac { 1 - a + b } { 2 } ) } . \\end{align*}"}
+{"id": "6817.png", "formula": "\\begin{align*} W _ i ^ { \\natural } = W ^ { \\flat } \\cup h _ { i , 1 } \\cup \\cdots \\cup h _ { i , k _ i } , \\end{align*}"}
+{"id": "4235.png", "formula": "\\begin{align*} \\iota _ \\pm = \\left ( \\frac { \\sqrt { \\lambda ^ { 2 } - \\cos ^ { 2 } ( \\Delta \\xi / 2 ) } \\pm \\sin ( \\Delta \\xi / 2 ) } { \\sqrt { \\lambda ^ { 2 } - 1 } } \\right ) ^ { 2 } \\end{align*}"}
+{"id": "7314.png", "formula": "\\begin{align*} U _ k = 1 e _ k = 0 \\prod _ { k = 1 } ^ { N } U _ k ^ { e _ k } \\leq q ^ { 1 + e _ M } , \\end{align*}"}
+{"id": "594.png", "formula": "\\begin{align*} g = \\begin{bmatrix} g _ { 1 1 } & g _ { 1 2 } \\\\ \\ 0 & g _ { 2 2 } \\end{bmatrix} \\end{align*}"}
+{"id": "6348.png", "formula": "\\begin{align*} \\big \\{ l ( s ) = \\{ y = s + k x \\} \\ , : \\ , s \\in \\R \\big \\} \\end{align*}"}
+{"id": "3037.png", "formula": "\\begin{align*} \\| \\phi ( A _ 1 \\otimes \\cdots \\otimes A _ m ) \\| _ { ( p , k ) } = \\| A _ 1 \\otimes \\cdots \\otimes A _ m \\| _ { ( p , k ) } \\end{align*}"}
+{"id": "1377.png", "formula": "\\begin{align*} x _ i v _ j = - v _ { 5 i + 3 j \\bmod 7 } , i , j \\in \\{ 1 , 2 , \\dots , 7 \\} . \\end{align*}"}
+{"id": "6692.png", "formula": "\\begin{align*} Q = ( \\Vert \\eta \\Vert _ { { 4 . 5 } } + \\Vert v \\Vert _ { { 4 } } + 1 ) ^ 2 \\end{align*}"}
+{"id": "7526.png", "formula": "\\begin{align*} g ( y ) : = \\sum _ { n \\geq 1 } \\varphi _ { n } \\frac { y ^ n } { n ! } = - \\varphi _ 0 . \\end{align*}"}
+{"id": "9000.png", "formula": "\\begin{align*} \\begin{cases} C ^ \\varphi _ { i j k } = - C ^ \\varphi _ { i k j } \\ , C ^ \\varphi _ { i k k } = 0 \\\\ C ^ \\varphi _ { k k i } = \\alpha \\varphi ^ a _ { k k } \\varphi ^ a _ i \\\\ C ^ \\varphi _ { i j k } + C ^ \\varphi _ { j k i } + C ^ \\varphi _ { k i j } = 0 \\ , . \\end{cases} \\end{align*}"}
+{"id": "5969.png", "formula": "\\begin{align*} h = \\begin{bmatrix} 1 & 0 \\\\ 0 & - 1 \\end{bmatrix} , e ^ + = \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix} , e ^ - = \\begin{bmatrix} 0 & 0 \\\\ 1 & 0 \\end{bmatrix} , J ^ + = \\tfrac { 1 } { 2 } \\begin{bmatrix} - i & 1 \\\\ 1 & i \\end{bmatrix} , J ^ - = \\tfrac { 1 } { 2 } \\begin{bmatrix} i & 1 \\\\ 1 & - i \\end{bmatrix} , J ^ 0 = \\begin{bmatrix} 0 & 1 \\\\ - 1 & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "732.png", "formula": "\\begin{align*} \\tilde A = { \\rm I } _ { N _ { S } } + ( a _ { j } ) _ { i , j } . \\end{align*}"}
+{"id": "1028.png", "formula": "\\begin{align*} H _ k ( a - 1 ) - H _ k ( - a ) = \\sum _ { i = 1 } ^ k \\frac { 1 } { a - 1 + i } - \\sum _ { i = 1 } ^ k \\frac { 1 } { - a + i } = \\sum _ { i = 1 } ^ k \\frac { 1 - 2 a } { ( a - 1 + i ) ( - a + i ) } , \\end{align*}"}
+{"id": "3803.png", "formula": "\\begin{align*} \\Theta ( \\delta ) = \\left [ \\min _ { \\gamma \\in \\Sigma ( \\delta ) } \\int _ \\mathcal { S } f ( y _ 1 , y _ 2 ) \\ , d \\gamma ( s ) , \\max _ { \\gamma \\in \\Sigma ( \\delta ) } \\int _ \\mathcal { S } f ( y _ 1 , y _ 2 ) \\ , d \\gamma ( s ) \\right ] , \\end{align*}"}
+{"id": "2279.png", "formula": "\\begin{align*} w _ b = \\sum _ { n } c _ n a _ n + T ( f ) _ b , \\end{align*}"}
+{"id": "7874.png", "formula": "\\begin{align*} A ^ \\pm _ { n , k , i } = \\begin{bmatrix} A _ { n , k , i } & A _ { n , k , i } \\\\ A _ { n , k , i } & A _ { n , k , i } \\end{bmatrix} = \\begin{bmatrix} 1 & 1 \\\\ 1 & 1 \\end{bmatrix} \\otimes A _ { n , k , i } . \\end{align*}"}
+{"id": "6539.png", "formula": "\\begin{align*} \\widehat { \\mu } ( t ) = \\exp \\left [ - \\frac { 1 } { 2 } a t ^ 2 + i b t + \\int _ { - \\infty } ^ { \\infty } \\left ( e ^ { i t \\lambda } - 1 - \\frac { i t \\lambda } { 1 + \\lambda ^ 2 } \\right ) \\nu ( d \\lambda ) \\right ] , \\end{align*}"}
+{"id": "6497.png", "formula": "\\begin{align*} \\det \\left [ \\binom { 2 m } { j - i + m } - \\binom { 2 m } { m - i - j - 1 } \\right ] _ { i , j = 0 } ^ { n - 1 } = \\prod _ { 1 \\leq i \\leq j \\leq m - 1 } \\frac { 2 n + i + j } { i + j } . \\end{align*}"}
+{"id": "5820.png", "formula": "\\begin{align*} { { \\boldsymbol { d } } _ k } = { { \\boldsymbol { W } } _ k } { { \\boldsymbol { x } } _ k } + { { \\boldsymbol { e } } _ k } \\end{align*}"}
+{"id": "2437.png", "formula": "\\begin{align*} y ( q ^ 2 x ) - ( q x \\sin ( 2 q x ) + \\cos ( x ) ) y ( q x ) + x \\cos ( x ) \\sin ( 2 x ) y ( x ) = 0 . \\end{align*}"}
+{"id": "3432.png", "formula": "\\begin{align*} \\left \\langle X , \\nabla _ X X \\right \\rangle = \\frac { 1 } { 2 } \\nabla _ X \\left ( | X | ^ 2 \\right ) . \\end{align*}"}
+{"id": "4546.png", "formula": "\\begin{align*} \\lim _ { | x | \\rightarrow \\infty } \\nabla \\cdot \\varphi _ 3 ( x ) = 0 . \\end{align*}"}
+{"id": "2318.png", "formula": "\\begin{align*} \\langle A ( v - s d ) , v - s d \\rangle = \\langle A v , v \\rangle - 2 s \\langle A v , d \\rangle + s ^ 2 \\langle A d , d \\rangle = \\langle A v , v \\rangle - 2 s \\| d \\| ^ 2 + s ^ 2 \\langle A d , d \\rangle . \\end{align*}"}
+{"id": "5568.png", "formula": "\\begin{align*} p _ { k } : Q \\to G _ { 2 } : = \\left \\{ M \\in G L ( 2 , \\mathbb { R } ) , \\left | \\det ( M ) \\right | = 1 \\right \\} \\end{align*}"}
+{"id": "502.png", "formula": "\\begin{align*} \\ell _ { j i } ( x ) = \\frac { 1 } { h ^ 2 } \\begin{cases} \\frac 1 2 ( x - x _ { j - 2 } ) ( x - x _ { j - 1 } ) , & i = j - 1 , \\\\ - ( x - x _ { j - 1 } ) ( x - x _ { j + 1 } ) , & i = j , \\\\ \\frac 1 2 ( x - x _ { j + 1 } ) ( x - x _ { j + 2 } ) , & i = j + 1 , \\\\ 0 , & \\end{cases} \\end{align*}"}
+{"id": "7549.png", "formula": "\\begin{align*} \\sum _ { a \\in \\mathcal { A } } \\left \\lvert \\sum _ { l \\equiv v ( \\bmod a ) } \\binom n l \\alpha ^ l ( 1 - \\alpha ) ^ { n - l } - \\frac 1 a \\right \\rvert = O \\left ( \\log ^ { - A } n \\right ) \\end{align*}"}
+{"id": "8164.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { y ^ { ( i ) } _ n ( I _ 1 , J _ 1 ) } { y ^ { ( i + 1 ) } _ n ( I _ 1 , J _ 1 ) } = \\infty , \\end{align*}"}
+{"id": "1815.png", "formula": "\\begin{align*} \\ < \\Psi , [ \\N ^ \\ell , V _ { F B } ( t ) ] \\Psi \\ > = 2 \\mathrm { I m } \\int _ { \\Lambda ^ * } \\hat V ( k ) \\langle \\Psi , [ \\N ^ \\ell , D ^ * _ k ( t ) b _ k ( t ) ] \\Psi \\rangle \\ . \\end{align*}"}
+{"id": "2597.png", "formula": "\\begin{align*} \\det \\ , D ^ 2 v _ 0 = c _ 0 \\chi _ { _ { \\Omega ^ * _ 0 } } \\quad \\R ^ n \\end{align*}"}
+{"id": "6725.png", "formula": "\\begin{align*} X _ \\varphi : = \\overline { \\varphi ( H ) } \\end{align*}"}
+{"id": "8109.png", "formula": "\\begin{align*} \\sum _ I \\ < Y _ F , S ^ I \\ > M _ I ( X ) = \\Gamma _ F ( X ) \\end{align*}"}
+{"id": "2613.png", "formula": "\\begin{align*} \\widetilde { T } _ l ( f _ 1 , f _ 2 ) ( x , y ) = \\int _ \\mathbb { R } \\ ! f _ 1 \\left ( x + \\widetilde { P } _ 1 ( t ) , y \\right ) f _ 2 \\left ( x , y + \\widetilde { P } _ 2 ( t ) \\right ) \\zeta ( x , y , t ) \\ , \\mathrm { d } t . \\end{align*}"}
+{"id": "2389.png", "formula": "\\begin{align*} \\cos \\alpha _ { 1 0 3 } = \\cos \\alpha _ { 2 0 4 } \\end{align*}"}
+{"id": "1757.png", "formula": "\\begin{align*} i \\frac { \\partial \\psi _ 1 } { \\partial t } & = \\Big ( - \\frac { \\partial ^ 2 } { \\partial x ^ 2 } - a _ 1 ( | \\psi _ 1 | ^ 2 + | \\psi _ 2 | ^ 2 ) - U _ 1 ( x ) \\Big ) \\psi _ 1 , \\ \\\\ i \\frac { \\partial \\psi _ 2 } { \\partial t } & = \\Big ( - \\frac { \\partial ^ 2 } { \\partial x ^ 2 } - a _ 2 ( | \\psi _ 1 | ^ 2 + | \\psi _ 2 | ^ 2 ) - U _ 2 ( x ) \\Big ) \\psi _ 2 , \\ \\end{align*}"}
+{"id": "1069.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ { \\phi } } = \\inf \\left \\{ \\lambda > 0 : \\int _ { \\R ^ d } \\phi \\left ( \\frac { | u ( x ) | } { \\lambda } \\right ) d x \\leq 1 \\right \\} . \\end{align*}"}
+{"id": "3318.png", "formula": "\\begin{align*} f = e ^ g . \\end{align*}"}
+{"id": "4775.png", "formula": "\\begin{align*} W _ { [ j + 1 ] } & = S _ { [ j + 1 ] } + \\mathcal { O } ( \\| H \\| ^ 2 ) . \\end{align*}"}
+{"id": "9297.png", "formula": "\\begin{align*} \\hat \\phi ( 0 ) - \\phi ( 0 ) + \\int _ { - \\infty } ^ { \\infty } \\hat \\phi ( x ) | x | d x = \\int _ { - \\infty } ^ { \\infty } \\phi ( x ) W _ { \\rm U } ^ 1 ( x ) d x = \\int _ { - \\infty } ^ { \\infty } \\phi ( x ) \\left ( 1 - \\frac { \\sin ^ 2 ( \\pi x ) } { ( \\pi x ) ^ 2 } \\right ) d x . \\end{align*}"}
+{"id": "1839.png", "formula": "\\begin{align*} | \\delta _ t ( \\lambda y ) - 2 t / \\pi | = t | \\delta _ 1 ( t \\lambda y ) - \\delta _ 1 ( 0 ) | \\leq C t ( t \\lambda | y | ) ^ 2 \\ . \\end{align*}"}
+{"id": "3004.png", "formula": "\\begin{align*} & \\Phi _ j ( f _ 0 , f _ 1 , f _ 2 , \\ldots , f _ { N - 1 } ) \\\\ & = f _ j + \\exp _ K ( - t _ 1 f _ 0 ) \\ldots \\exp _ K ( - ( t _ j - t _ { j - 1 } ) f _ { j - 1 } ) g _ j \\exp _ K ( t _ j - t _ { j - 1 } ) f _ { j - 1 } ) \\ldots \\exp _ K t _ 1 f _ 0 , \\end{align*}"}
+{"id": "8802.png", "formula": "\\begin{align*} & ( u ^ 2 + 8 t u + 1 2 t ^ 2 + 6 u + 3 0 t + 1 6 ) ( [ 4 t + 3 u + 3 ] c _ 3 - [ u + 2 t + 5 ] b _ 3 ) \\\\ & = ( u ^ 2 + 4 t u + 6 t ^ 2 + 1 0 u + 2 4 t + 2 4 ) ( [ 3 u + 2 t + 5 ] b _ 3 - [ u + 4 t + 3 ] c _ 3 ) \\\\ & + 2 ( u ^ 2 + 4 t u + 6 u ) c _ 3 - 2 ( u ^ 2 + 2 t u + 4 u ) b _ 3 \\end{align*}"}
+{"id": "4205.png", "formula": "\\begin{align*} \\phi ( \\theta ) = \\begin{pmatrix} i \\lambda & g ( \\theta ) \\\\ - g ( \\theta ) ^ { - 1 } & i \\lambda \\end{pmatrix} , \\end{align*}"}
+{"id": "4438.png", "formula": "\\begin{align*} \\mu _ { \\omega , \\mathbf { c } } = S _ { \\omega , \\mathbf { c } } ( \\Phi ) & = L ( \\Phi ) + N ( \\Phi ) + \\omega Q ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) = \\frac { 1 } { 4 - d } \\left ( 2 \\omega Q ( \\Phi ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi ) \\right ) . \\end{align*}"}
+{"id": "5732.png", "formula": "\\begin{align*} \\mathcal { E C S } _ { { \\rm { I } } } = \\mathcal { E C S } _ { 0 0 1 } \\cup \\mathcal { E C S } _ { 0 1 0 } \\cup \\mathcal { E C S } _ { 1 0 0 } , \\end{align*}"}
+{"id": "1044.png", "formula": "\\begin{align*} \\Omega _ k ( d ) & = 2 k ( 1 + 6 k ) + \\frac { ( d + k ) ( 1 - d + k ) ( 2 - d + 3 k ) } { 3 - 2 d + 4 k } \\\\ [ 1 m m ] & \\quad + \\frac { ( d + k ) ( 1 - d + k ) ^ 3 } { ( 1 + 2 d + 4 k ) ( 3 - 2 d + 4 k ) } . \\end{align*}"}
+{"id": "1939.png", "formula": "\\begin{align*} \\pi ( n , j ) : \\mu _ { n , 1 } ^ { ( j ) } z _ 1 + \\cdots + \\mu _ { n , m } ^ { ( j ) } z _ m - \\lambda _ n ^ { ( j ) } = 0 \\end{align*}"}
+{"id": "1166.png", "formula": "\\begin{align*} T \\left ( \\frac { 1 } { f } \\right ) = - \\frac { 1 } { f ^ { 2 } } T ( f ) + \\frac { 2 } { f ^ { 3 } } A ( f ) ^ { 2 } \\end{align*}"}
+{"id": "1861.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\langle D _ u \\theta ( u ^ * ) , v \\rangle _ \\mathcal { U } + \\langle \\bar { \\lambda } , S v \\rangle = 0 \\forall \\ v \\in \\mathcal { U } ; \\\\ & - \\langle \\bar { \\lambda } , \\zeta - S u ^ * \\rangle \\geq 0 \\forall \\ \\zeta \\in K . \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "6811.png", "formula": "\\begin{align*} \\Lambda _ 2 : = ( \\Lambda _ { 2 , + } ) ^ { \\sharp } \\cup ( \\Lambda _ { 2 , 0 } ) ^ { \\sharp } \\cup ( \\Lambda _ { 2 , - } ) ^ { \\sharp } \\subset \\Gamma , \\end{align*}"}
+{"id": "2198.png", "formula": "\\begin{align*} { \\mathcal { C } } _ { \\alpha } = B _ 1 ^ { ( \\alpha ) } \\otimes L + B _ 2 ^ { ( \\alpha ) } \\otimes ( - 2 I _ { m } ) . \\end{align*}"}
+{"id": "4574.png", "formula": "\\begin{align*} K _ 1 ( t ) = q ( t ) \\tilde { K } _ 1 ( t ) q ( t ) \\ , , K _ 2 ( t ) = q ( t ) \\otimes q ( t ) \\tilde { K } _ 2 ( t ) \\end{align*}"}
+{"id": "1037.png", "formula": "\\begin{align*} \\alpha _ k ( a , b , c , d , e ) & = \\frac { ( 1 + 2 a - b - c - d + 2 k ) ( a - e + k ) } { 1 + 2 a - b - c - d - e + k } \\\\ [ 1 m m ] & \\quad + \\frac { ( 1 + a - b - c + k ) ( 1 + a - b - d + k ) ( e + k ) } { ( 1 + a - b + 2 k ) ( 1 + 2 a - b - c - d - e + k ) } . \\end{align*}"}
+{"id": "1174.png", "formula": "\\begin{align*} ( f ^ 2 - 1 ) \\left [ T ( f ^ { 2 } ) - 2 f T ( f ) - 2 A ( f ) ^ { 2 } \\right ] = 2 \\left [ A ( f ^ { 2 } ) ^ { 2 } - 4 f ^ { 2 } A ( f ) ^ { 2 } \\right ] . \\end{align*}"}
+{"id": "7197.png", "formula": "\\begin{align*} \\lim _ { h \\to + \\infty } \\gamma _ h \\mathcal { H } ^ 1 \\left ( \\left \\{ \\widehat { f _ h } \\neq \\widehat { g _ h } \\right \\} \\cap \\partial B _ \\rho \\right ) = 0 \\end{align*}"}
+{"id": "8420.png", "formula": "\\begin{align*} \\tan ( \\pi y ) = - \\frac { y } { p } , \\ , \\ , \\ , \\ , \\ , p > 0 . \\end{align*}"}
+{"id": "4519.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow + 0 } K _ { \\omega , \\mathbf { c } } ( V ^ { \\lambda } ) = 2 \\omega M ( V ) > 0 . \\end{align*}"}
+{"id": "7184.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 \\left ( J _ { w _ 1 } \\cap \\cup _ { i \\in \\N } B ^ i _ 1 \\right ) \\leq \\liminf _ { h \\to + \\infty } \\mathcal { H } ^ 1 \\left ( J _ { w ^ h _ 1 } \\cap \\cup _ { i \\in \\N } B ^ i _ 1 \\right ) = 0 \\end{align*}"}
+{"id": "8557.png", "formula": "\\begin{align*} p _ { i , j } ( t ) : = P ( Z _ 0 ( t ) = j | Z _ 0 ( 0 ) = i ) \\end{align*}"}
+{"id": "1219.png", "formula": "\\begin{align*} i _ { \\boldsymbol { a } } = q . \\end{align*}"}
+{"id": "2779.png", "formula": "\\begin{align*} \\Delta _ g w = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } , w _ { | \\partial \\mathrm { M } } = 1 . \\end{align*}"}
+{"id": "3678.png", "formula": "\\begin{align*} H ^ s ( \\mathbb { R } ^ n ) = \\{ u \\in \\mathcal { S } ' ( \\mathbb { R } ^ n ) : \\{ \\xi \\mapsto ( 1 + | \\xi | ^ 2 ) ^ { s / 2 } \\hat { u } ( \\xi ) \\} \\in L ^ 2 ( \\mathbb { R } ^ n ) \\} , \\end{align*}"}
+{"id": "4459.png", "formula": "\\begin{align*} \\omega _ { n \\pm } : = \\omega _ { \\pm } ( 1 / n ) = \\left ( \\sqrt { \\omega } \\pm \\frac { 1 } { n } \\right ) ^ 2 , \\ \\ \\mathbf { c } _ { n \\pm } : = \\mathbf { c } _ { \\pm } ( 1 / n ) = \\frac { \\mathbf { c } } { \\sqrt { \\omega } } \\left ( \\sqrt { \\omega } \\pm \\frac { 1 } { n } \\right ) . \\end{align*}"}
+{"id": "6039.png", "formula": "\\begin{align*} ( A _ 2 - \\overline { A _ 2 } ) z ^ { n + m } + ( B _ 2 - \\overline { B _ 2 } ) \\overline { z } ^ m + ( C _ 2 - \\overline { C _ 2 } ) = 0 \\textrm { f o r a l l } z \\in \\mathbb { C } . \\end{align*}"}
+{"id": "9083.png", "formula": "\\begin{align*} \\mathfrak { S } ( f _ k ) = k + r - 1 - | \\mathcal { F } | + c _ 1 , \\ , \\ , \\ , \\ , \\ , \\overline { \\mathfrak { S } } ( f _ k ) = n - k + 1 - | \\mathcal { F } | + c _ 2 , \\end{align*}"}
+{"id": "3588.png", "formula": "\\begin{align*} L ( m n ) \\ = \\ L ( m ) + L ( n ) - [ m n \\ < \\ 1 0 ^ { L ( m ) + L ( n ) - 1 } ] . \\end{align*}"}
+{"id": "7035.png", "formula": "\\begin{align*} v ( b _ { \\ell i } ) = - \\min \\limits _ j v ( b _ { \\ell i j } ) . \\end{align*}"}
+{"id": "8968.png", "formula": "\\begin{align*} \\phi = \\Tilde { \\phi } _ { e } + C _ { \\Tilde { q } } h ^ { \\Tilde { q } } . \\end{align*}"}
+{"id": "9217.png", "formula": "\\begin{align*} ( 1 + \\eta _ r ) ( 1 + \\eta _ s ) = ( 1 + \\eta _ { r + s } ) . \\end{align*}"}
+{"id": "9326.png", "formula": "\\begin{align*} \\overline { W } _ t = W _ t + \\int _ 0 ^ t C _ { s } \\ , d s . \\end{align*}"}
+{"id": "5174.png", "formula": "\\begin{align*} \\Big ( \\sum _ { e \\in E ( \\mathbb Z ^ d ) } p _ e ^ 2 \\Big ) ^ d \\le \\Big ( \\sum _ { e \\in E ( \\mathbb Z ^ d ) } q _ e p _ e \\Big ) ^ d \\le C n ( \\log n ) ^ { \\frac { d ( d + 1 ) } { 2 } } \\Big ( \\sum _ { e \\in E ( \\mathbb Z ^ d ) } p _ e ^ 2 \\Big ) ^ { \\frac { d - 1 } { 2 } } . \\end{align*}"}
+{"id": "1698.png", "formula": "\\begin{align*} \\hat Z _ { s , \\ell } = \\Phi ^ { s } _ { 2 } ( y ) - \\Phi ^ { \\ell } _ { 2 } ( x ) . \\end{align*}"}
+{"id": "189.png", "formula": "\\begin{align*} L i _ 3 ( 1 ) = \\zeta ( 3 ) , \\end{align*}"}
+{"id": "3059.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta _ { p } u & = v ^ { k _ 1 } \\cdot | \\nabla u | ^ { \\alpha } & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta _ { p } v & = v ^ { k _ 2 } \\cdot | \\nabla u | ^ { k _ 3 } & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "313.png", "formula": "\\begin{align*} \\zeta ( s ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ s } \\Re ( s ) > 1 , \\end{align*}"}
+{"id": "6247.png", "formula": "\\begin{align*} J _ { \\overrightarrow 0 } ( t ) = J _ { \\overrightarrow 0 } ( t ^ { ( 1 ) } ) ( \\mathrm { s e c t } ( \\delta / 2 ) ) \\cdot J _ { \\mathrm { s e c t } ( \\delta / 2 ) } ( t ^ { ( 2 ) } ) \\end{align*}"}
+{"id": "3733.png", "formula": "\\begin{align*} \\phi ^ { \\alpha , n } ( g ) ^ \\ast & \\left ( S , f _ \\beta \\otimes \\rho _ { n } ( E ^ \\beta ) \\right ) \\\\ = & - \\left ( S , \\overline { g ^ { S ^ \\flat } } \\cdot f _ \\beta \\otimes \\rho _ n ( [ E ^ \\alpha , E ^ \\beta ] ) \\right ) + \\left \\langle ( S , f _ \\beta \\otimes \\rho _ { n } ( E ^ \\beta ) ) , \\phi ^ { \\alpha , n } ( g ) 1 \\right \\rangle 1 . \\end{align*}"}
+{"id": "3309.png", "formula": "\\begin{align*} { \\rm I m } ( f ( \\xi ) ) = 0 { \\rm \\ f o r \\ } \\xi \\in L \\end{align*}"}
+{"id": "3865.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta ) = \\sup _ { \\gamma \\in \\Sigma ( \\delta ) } \\int _ { \\mathcal { S } } f \\ , d \\gamma , \\end{align*}"}
+{"id": "1110.png", "formula": "\\begin{align*} x ( y z ) & = ( x y ) z - ( x z ) y , \\\\ ( y z ) x & = ( y x ) z - y ( x z ) , \\end{align*}"}
+{"id": "1092.png", "formula": "\\begin{align*} l ' ( \\varphi ( b ) , x ) = s ' ( \\varphi ( b ) ) i ' ( x ) = s ( b ) i ( x ) = l ( b , x ) \\end{align*}"}
+{"id": "2429.png", "formula": "\\begin{align*} \\varphi _ m ' ( x ) & = - \\widetilde { E } _ m ' ( x ) \\\\ & = - \\left ( f _ 1 ' ( x ) + f _ 1 ( x ) \\sum _ { k = 0 } ^ { m - 1 } \\frac { x ^ k } { q ^ { k + 1 } - 1 } \\right ) \\exp \\left ( \\sum _ { k = 1 } ^ { m } \\frac { x ^ k } { k ( q ^ k - 1 ) } \\right ) \\\\ & = - \\left ( \\sum _ { n \\geq 0 } \\frac { ( n + 1 ) x ^ n } { ( q ; q ) _ { n + 1 } } + \\sum _ { k = 0 } ^ { m - 1 } \\left ( \\sum _ { n \\geq k } \\frac { x ^ n } { ( q ^ { k + 1 } - 1 ) ( q ; q ) _ { n - k } } \\right ) \\right ) \\exp \\left ( \\sum _ { k = 1 } ^ { m } \\frac { x ^ k } { k ( q ^ k - 1 ) } \\right ) . \\end{align*}"}
+{"id": "3370.png", "formula": "\\begin{align*} \\sigma _ { m n } : = \\frac { 1 } { P _ { m n } } \\sum _ { i = 0 } ^ m \\sum _ { j = 0 } ^ { n } p _ { i j } u _ { i j } \\end{align*}"}
+{"id": "4203.png", "formula": "\\begin{align*} H _ { \\alpha } ^ { \\mathrm { X Y } } = - \\frac \\alpha 2 \\sum _ { j = 0 } ^ { M - 1 } \\left ( ( 1 + \\gamma ) \\sigma ^ x _ j \\sigma ^ x _ { j + 1 } + ( 1 - \\gamma ) \\sigma ^ y _ j \\sigma ^ y _ { j + 1 } \\right ) - \\sum _ { j = 0 } ^ { M - 1 } \\sigma ^ z _ j , \\end{align*}"}
+{"id": "5699.png", "formula": "\\begin{align*} D ^ { k - 1 } ( f ) = c F + D ^ { k - 1 } ( h ) + d g . \\end{align*}"}
+{"id": "8914.png", "formula": "\\begin{gather*} \\bigl [ \\big [ X _ { n 1 } ^ { ( p ) } , \\dots , X _ { n p } ^ { ( p ) } \\big ] , \\big [ X _ { m 1 } ^ { ( q ) } , \\dots , X _ { m q } ^ { ( q ) } \\big ] \\bigr ] \\\\ { } = \\sum _ { \\sigma \\in S } \\epsilon _ { \\sigma } \\big [ X _ { n \\sigma ( 1 ) } ^ { ( p ) } , \\big [ X _ { n \\sigma ( 2 ) } ^ { ( p ) } , \\dots , \\big [ X _ { n \\sigma ( p ) } ^ { ( p ) } , \\big [ X _ { m 1 } ^ { ( q ) } , \\dots , X _ { m q } ^ { ( q ) } \\big ] \\big ] , \\ldots \\big ] \\big ] , \\end{gather*}"}
+{"id": "2704.png", "formula": "\\begin{align*} \\left \\| P _ 1 u \\right \\| ^ 2 + \\left \\| P _ 2 u \\right \\| ^ 2 + 2 \\left ( P _ 1 u , P _ 2 u \\right ) = \\left \\| e ^ { - s \\sigma } f \\right \\| ^ 2 . \\end{align*}"}
+{"id": "2815.png", "formula": "\\begin{align*} S _ { m - 1 } ^ { m , 1 } = a _ 0 \\dots a _ { m - 2 } , \\ ; m \\ge 2 ; \\ ; S _ { r + 1 } ^ { 1 , 1 } = a _ 0 \\dots a _ { r - 1 } ; S _ { n - 1 } ^ { 1 , n } = 1 , \\ ; S _ { n + q } ^ { 1 , n } = \\sum _ { l = 0 } ^ { n - 1 } b _ l ; \\end{align*}"}
+{"id": "9323.png", "formula": "\\begin{align*} J ( \\overline { u } ) = { 1 \\over 2 } \\overline { Y } _ 0 x + { 1 \\over 2 } \\int _ 0 ^ T L _ { t } \\ , d \\overline { \\mu } _ { t } . \\end{align*}"}
+{"id": "224.png", "formula": "\\begin{align*} \\omega ( s _ 1 , s _ 2 , s _ { 3 } ) = \\sum _ { m _ 1 , \\ , m _ 2 > 0 } \\frac { 1 } { { m _ 1 } ^ { s _ 1 } { m _ 2 } ^ { s _ 2 } ( m _ 1 + m _ 2 ) ^ { s _ { 3 } } } , \\end{align*}"}
+{"id": "8882.png", "formula": "\\begin{align*} \\beta = \\begin{cases} \\frac { 1 } { 2 } - \\delta & { \\rm i f ~ } r \\ge 4 \\\\ \\frac { 1 } { 3 } - \\delta & { \\rm i f ~ } r = 3 \\ ; . \\end{cases} \\end{align*}"}
+{"id": "2371.png", "formula": "\\begin{align*} U \\cup V & = D ( ( X - a _ 1 ) \\dots ( X - a _ k ) ) \\rlap { , } \\\\ U & = D ( ( X - a _ 1 ) \\dots ( X - a _ k ) ( X - b _ 1 ) \\dots ( X - b _ l ) ) \\rlap { , } \\\\ V & = D ( ( X - a _ 1 ) \\dots ( X - a _ k ) ( X - c _ 1 ) \\dots ( X - c _ m ) ) \\rlap { , } \\\\ U \\cap V & = D ( ( X - a _ 1 ) \\dots ( X - a _ k ) ( X - b _ 1 ) \\dots ( X - b _ l ) ( X - c _ 1 ) \\dots ( X - c _ m ) ) \\rlap { , } \\end{align*}"}
+{"id": "984.png", "formula": "\\begin{align*} g ^ { \\flat } ( p , t ) = ( \\bar g ^ { \\flat } ( p ) , \\chi _ g ( p ) \\cdot t ) \\end{align*}"}
+{"id": "1295.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( u ) = \\log ( \\theta ) + B ^ * ( u ) + \\frac { 1 } { 2 } u + \\log \\left ( \\frac { T ^ { \\infty } - T ( u ) } { T ^ { \\infty } } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "4039.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\sum _ { f \\in \\mathcal { B } _ { 2 k } ( p ) } \\omega _ f \\lambda _ f ( i ) \\lambda _ f ( j ) \\big | L ( f , n + 1 ) \\big | ^ 2 . \\end{align*}"}
+{"id": "3499.png", "formula": "\\begin{align*} \\begin{cases} L _ { \\lambda } [ X ] = - X '' + ( R - \\lambda I ) X = 0 , \\\\ X ( a ) = X ( b ) = 0 \\end{cases} \\end{align*}"}
+{"id": "525.png", "formula": "\\begin{align*} \\mathrm { D o m } \\left ( \\mathcal { H } _ { \\hbar , V } \\right ) : = \\left \\{ u \\in \\ell ^ { 2 } ( \\hbar \\mathbb { Z } ^ { n } ) : ( I + \\mathcal { H } _ { \\hbar , V } ) u \\in \\ell ^ { 2 } ( \\hbar \\mathbb { Z } ^ { n } ) \\right \\} . \\end{align*}"}
+{"id": "6881.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } \\frac { 2 } { N ^ 2 } \\log \\P _ { N , r _ N } ( U _ \\varepsilon ^ x ) \\geq - \\lim _ { N \\to \\infty } \\frac { 2 } { N ^ 2 } \\int _ { U _ \\varepsilon ^ x } \\left ( \\log \\frac { \\d \\P _ { N , h _ N } } { \\d \\P _ { N , r _ N } } \\right ) \\d \\P _ { N , h _ N } = - I _ r ( h ) . \\end{align*}"}
+{"id": "8788.png", "formula": "\\begin{align*} & c _ 1 = ( 2 + 2 t + u ) c _ 2 - ( 2 + t ) b _ 2 - u c _ 3 \\end{align*}"}
+{"id": "1864.png", "formula": "\\begin{align*} T _ w ( M , \\bar { u } ) = \\{ v \\in \\mathcal { U } : \\ \\exists \\{ u _ n \\} \\subset M , \\ \\{ t _ n \\} \\subset \\mathbb { R } ^ + , \\ u _ n \\rightarrow \\bar { u } , \\ t _ n \\rightarrow 0 ^ + , \\frac { 1 } { t _ n } ( u _ n - \\bar { u } ) \\rightharpoonup v \\} \\end{align*}"}
+{"id": "8012.png", "formula": "\\begin{align*} \\lambda _ { - k } ( f ( A _ { \\cal S } ) ) = & \\min _ { \\{ h \\in { \\cal F } \\ , : \\ , \\Vert h \\Vert = 1 \\} } \\langle h , f ( A _ { \\cal S } ) h \\rangle \\\\ = & \\min _ { \\{ h \\in { \\cal F } \\ , : \\ , \\Vert h \\Vert = 1 \\} } f ( \\langle h , A _ { \\cal S } h \\rangle ) \\\\ \\le & \\inf _ { \\{ h \\in { \\cal F } \\ , : \\ , \\Vert h \\Vert = 1 \\} } \\langle h , f ( A ) h \\rangle \\le \\lambda _ { - k } ( f ( A ) _ { \\cal S } ) \\end{align*}"}
+{"id": "1223.png", "formula": "\\begin{align*} 1 / m _ { \\boldsymbol { \\nu } , q } = \\int _ 0 ^ { 1 / ( q \\mathrm { k } ^ * ) } \\Pi ( q t ) \\dd t , \\end{align*}"}
+{"id": "28.png", "formula": "\\begin{align*} \\mathbf { T } _ { K ^ \\circ } = \\mathrm { R e s } _ { K / \\mathbb { Q } } ( \\mathbb { G } _ m ) \\cap \\mathbf { G } _ { \\mathbb { Q } } , \\mathbf { T } _ { K ^ \\circ } ^ 1 = \\mathbf { T } _ { K ^ \\circ } \\cap \\mathbf { G } ^ 1 _ { \\mathbb { Q } } . \\end{align*}"}
+{"id": "2772.png", "formula": "\\begin{align*} ( \\Delta + \\lambda - q _ 1 ) w = \\chi ( q _ 2 - q _ 1 ) v _ 2 + [ \\Delta , \\psi ] v . \\end{align*}"}
+{"id": "4960.png", "formula": "\\begin{align*} \\frac { n } { n - k } F ( k , t + 1 , n ) = \\frac { k } { n - k } F ( k , t , n ) + F ( k - 1 , t , n ) , \\end{align*}"}
+{"id": "5606.png", "formula": "\\begin{align*} h ( Z , \\lambda ) = \\int _ { G ^ { \\mathbb { N } } } \\int _ { X } \\log \\frac { d \\omega _ { 1 } . \\eta } { d \\eta } \\left ( x \\right ) d \\eta _ { \\omega } ( x ) d \\mathbb { P } _ { \\mu } ( \\omega ) + & \\int _ { G ^ { \\mathbb { N } } } \\int _ { X } \\log \\frac { d \\left ( \\omega _ { 1 } \\lambda \\right ) _ { x } } { d \\lambda _ { x } } \\left ( \\psi ( x , \\omega ) \\right ) d \\eta _ { \\omega } ( x ) d \\mathbb { P } _ { \\mu } ( \\omega ) = { \\rm I } + { \\rm I I } . \\end{align*}"}
+{"id": "4178.png", "formula": "\\begin{align*} \\| e _ t \\varphi ( a ) e _ t - \\varphi ( a ) \\| & = \\| v _ t v _ t ^ * \\varphi ( a ) v _ t v _ t ^ * - \\varphi ( a ) \\| \\\\ & \\leq \\| v _ t v _ t ^ * \\varphi ( a ) v _ t v _ t ^ * - v _ t \\varphi ( \\alpha _ { t ^ { - 1 } } ( a ) ) v _ t ^ * \\| + \\| v _ t \\varphi ( \\alpha _ { t ^ { - 1 } } ( a ) ) v _ t ^ * - \\varphi ( a ) \\| \\\\ & \\leq \\| v _ t ^ * \\varphi ( a ) v _ t - \\varphi ( \\alpha _ { t ^ { - 1 } } ( a ) ) \\| \\end{align*}"}
+{"id": "6796.png", "formula": "\\begin{align*} W ^ { \\mathrm { s } } ( p ) & = \\R ^ k _ { x _ 1 , \\dots , x _ k } \\times \\{ 0 \\} _ { x _ { k + 1 } , \\dots , x _ { 2 n - 1 } } \\times \\{ x _ { 2 n } \\leq 0 \\} , \\\\ W ^ { \\mathrm { u } } ( p ) & = \\{ 0 \\} _ { x _ 1 , \\dots , x _ k } \\times \\R ^ { 2 n - k - 1 } _ { x _ { k + 1 } , \\dots , x _ { 2 n - 1 } } \\times \\{ x _ { 2 n } \\geq 0 \\} . \\end{align*}"}
+{"id": "2104.png", "formula": "\\begin{align*} g ( A ) & = \\sum _ { r = 1 } ^ { a - 1 } \\left ( \\sum _ { i = 1 } ^ { k } x _ i \\right ) _ r + \\frac { ( a - 1 ) ( a + d - 1 ) } { 2 } \\\\ & = \\sum _ { i = 1 } ^ { m - 1 } i ( 2 ^ k - 1 ) + m ( 2 ^ { k - 1 } - 1 ) + \\sum _ { r = 1 } ^ { a - 1 } \\left ( \\sum _ { i = 1 } ^ { k - 1 } x _ i \\right ) _ r + \\frac { ( a - 1 ) ( a + d - 1 ) } { 2 } . \\end{align*}"}
+{"id": "8975.png", "formula": "\\begin{align*} \\Lambda = \\frac { m - 2 } { 2 } \\lambda \\end{align*}"}
+{"id": "2469.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } K = \\inf \\limits _ { ( u , v ) \\in \\mathcal { M } } S ( u , v ) . \\end{array} \\right . \\end{align*}"}
+{"id": "9057.png", "formula": "\\begin{align*} R _ { \\theta _ 0 } ( z _ 1 , z _ 2 , u ) = ( z _ 1 \\cos \\theta _ 0 - z _ 2 \\sin \\theta _ 0 , z _ 1 \\sin \\theta _ 0 + z _ 2 \\cos \\theta _ 0 , u ) . \\end{align*}"}
+{"id": "4481.png", "formula": "\\begin{align*} L ( U ) : = & \\frac { \\alpha } { 2 } \\| \\nabla u _ 1 \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 + \\frac { \\beta } { 2 } \\| \\nabla u _ 2 \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 + \\frac { \\gamma } { 2 } \\| \\nabla u _ 3 \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 , \\\\ N ( U ) : = & { \\rm R e } \\left ( u _ 3 , \\nabla ( u _ 1 \\cdot \\overline { u _ 2 } ) \\right ) _ { L ^ 2 ( \\R ^ d ) } . \\end{align*}"}
+{"id": "1528.png", "formula": "\\begin{align*} f _ 0 ( t ) = & \\phantom { a } ( h _ { p - 1 } t ^ { p - 1 } - g _ { p - 1 } ( a t + b ) ^ { p - 1 } ) \\\\ & + \\sum _ { \\ell = 0 } ^ { p - 2 } \\sum _ { \\theta = 0 } ^ { d } ( h _ { p \\theta + \\ell } t ^ { p \\theta + \\ell } - g _ { p \\theta + \\ell } ( a t + b ) ^ { p \\theta + \\ell } ) . \\end{align*}"}
+{"id": "2635.png", "formula": "\\begin{align*} & \\left | \\left \\lbrace ( x , y , t ) \\in \\ ! K : \\left | \\widetilde { P } _ 1 ' ( t ) \\alpha ( x + \\widetilde { P } _ 1 ( t ) , y ) - \\widetilde { P } _ 2 ' ( t ) \\beta ( x , y + \\widetilde { P } _ 2 ( t ) ) \\right | \\leq \\ ! \\varepsilon \\ ! \\right \\rbrace \\right | \\\\ \\lesssim & 2 ^ { c | l | } \\varepsilon ^ { 1 / 3 5 } \\end{align*}"}
+{"id": "5579.png", "formula": "\\begin{align*} \\bigcup _ { \\ell \\in \\mathbb { N } } \\left \\{ h _ { \\mu } \\left ( Z , \\lambda _ { \\ell , p } \\right ) : p \\in [ 0 , 1 ] \\right \\} = \\left ( h _ { \\mu } \\left ( G / P _ { I } \\right ) , h _ { \\mu } \\left ( G / P _ { I ' } \\right ) \\right ] . \\end{align*}"}
+{"id": "2841.png", "formula": "\\begin{align*} \\begin{cases} \\overset \\cdot a _ 0 = a _ 0 a _ 1 a _ 2 , \\overset \\cdot a _ 1 = a _ 1 a _ 2 a _ 3 , \\\\ \\overset \\cdot a _ 2 = a _ 2 a _ 3 a _ 4 \\ , - \\ , a _ 2 a _ 1 a _ 0 , \\\\ \\overset \\cdot a _ 3 = - a _ 3 a _ 2 a _ 1 , \\ ; \\overset \\cdot a _ 4 = - a _ 4 a _ 3 a _ 2 ; \\end{cases} \\end{align*}"}
+{"id": "5934.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( [ w ' , t ] ) f ( \\epsilon , w ) = \\psi ( t + \\epsilon \\tfrac { \\langle w , w ' \\rangle } { 2 } ) f ( \\epsilon , w + w ' ) , \\end{align*}"}
+{"id": "9063.png", "formula": "\\begin{align*} \\beta ( a , b , c ) & = \\sigma _ 2 \\sigma _ 1 ^ { a + 1 } \\sigma _ 2 \\sigma _ 1 ^ { b + 1 } \\sigma _ 2 \\sigma _ 1 ^ { c + 1 } = \\sigma _ 2 \\sigma _ 1 ( \\sigma _ 2 \\sigma _ 1 ) \\sigma _ 2 ^ a \\sigma _ 1 ^ { b - 1 } \\sigma _ 2 ^ c ( \\sigma _ 1 \\sigma _ 2 ) \\sigma _ 1 = \\Delta _ 3 \\sigma _ 1 \\sigma _ 2 ^ a \\sigma _ 1 ^ { b - 1 } \\sigma _ 2 ^ c \\Delta _ 3 . \\end{align*}"}
+{"id": "4901.png", "formula": "\\begin{align*} h _ { d } ( X _ 0 , X _ 1 , . . . , X _ k ) \\overset { } { = } \\sum \\limits _ { 0 \\le i _ 1 \\leqslant i _ 2 \\leqslant \\cdots \\leqslant i _ d \\le k } X _ { i _ 1 } \\cdot X _ { i _ 2 } \\cdots X _ { i _ d } \\ , , \\end{align*}"}
+{"id": "6756.png", "formula": "\\begin{align*} \\Psi ( h , x ) = ( h , \\widehat { x } _ { \\varphi ( h ) - \\underline { \\varphi } ( h ) } ) \\end{align*}"}
+{"id": "5186.png", "formula": "\\begin{align*} \\frac { l _ { t - 1 } \\overline { \\bf { b } } _ { ( i ) } } { l _ { t - 1 } { \\bf { b } } _ { ( i ) } } = b _ i . \\end{align*}"}
+{"id": "8011.png", "formula": "\\begin{align*} \\sigma _ k \\left [ f \\left ( \\sum _ { i = 1 } ^ m Z ^ * _ i A _ i Z _ i \\right ) \\right ] \\le \\sigma _ k \\left [ \\sum _ { i = 1 } ^ m Z ^ * _ i f ( A ) _ i Z _ i \\right ] , k = 1 , \\ldots , n , \\end{align*}"}
+{"id": "8778.png", "formula": "\\begin{align*} 0 = c ( \\beta _ i - \\beta _ j ) + \\alpha _ { i j } - \\lambda _ { i j } + \\gamma _ { i } , \\quad \\forall i \\neq j . \\end{align*}"}
+{"id": "4531.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } S _ { \\omega , \\mathbf { c } } ( \\Phi _ 0 ^ { \\lambda } ) = - ( \\sqrt { \\lambda } - 1 ) ( 2 L ( \\Phi _ 0 ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi _ 0 ) ) \\left ( \\lambda + \\frac { \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi _ 0 ) } { 2 L ( \\Phi _ 0 ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi _ 0 ) } \\sqrt { \\lambda } + \\frac { \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi _ 0 ) } { 2 L ( \\Phi _ 0 ) + \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi _ 0 ) } \\right ) \\end{align*}"}
+{"id": "7075.png", "formula": "\\begin{align*} \\nu ( g ' ) - \\nu _ j ( g ' ) = \\beta _ j - \\tilde { \\beta } _ j = ( \\beta _ j - \\alpha _ { i _ 0 } ) - ( \\tilde { \\beta } _ j - \\alpha _ { i _ 0 } ) \\in \\Delta . \\end{align*}"}
+{"id": "4678.png", "formula": "\\begin{align*} S ( z ) = \\left ( I - z \\mathcal { A } _ 0 ^ { - 1 } \\right ) ^ { - 1 } \\Pi , M ( z ) = \\Gamma _ 1 \\left ( I - z \\mathcal { A } _ 0 ^ { - 1 } \\right ) ^ { - 1 } \\Pi . \\end{align*}"}
+{"id": "7790.png", "formula": "\\begin{align*} x ^ r & = \\int ^ { \\infty } _ 0 ( s + x ) ^ { - 1 } d \\mu ( s ) , \\end{align*}"}
+{"id": "763.png", "formula": "\\begin{align*} \\sigma _ k ( \\lambda ) = \\sum \\limits _ { 1 \\leqslant i _ 1 < i _ 2 < \\cdots < i _ k \\leqslant n } \\lambda _ { i _ 1 } \\lambda _ { i _ 2 } \\cdots \\lambda _ { i _ k } \\end{align*}"}
+{"id": "2778.png", "formula": "\\begin{align*} \\Lambda _ { q _ \\mathfrak { c } , 0 } ^ 0 = \\Lambda _ { 0 , 0 } ^ 0 . \\end{align*}"}
+{"id": "4021.png", "formula": "\\begin{align*} R _ p = \\left \\lbrace \\left ( \\begin{array} { c c c } 1 & r \\\\ 0 & p \\end{array} \\right ) \\mid r = 0 , 1 , \\ldots , p - 1 \\right \\rbrace \\cup \\left \\lbrace \\left ( \\begin{array} { c c c } p & 0 \\\\ 0 & 1 \\end{array} \\right ) \\right \\rbrace . \\end{align*}"}
+{"id": "8331.png", "formula": "\\begin{align*} C _ D = \\mu ( B _ 1 \\cap \\Sigma ) , \\end{align*}"}
+{"id": "2644.png", "formula": "\\begin{align*} 2 ^ l I & \\gtrsim _ \\tau \\int _ { [ 0 , 1 ] ^ 3 } \\ ! \\ ! f ( x , y ) f \\left ( x + P _ 1 ( t ) , y \\right ) f \\left ( x , y + P _ 2 ( t ) \\right ) \\tau _ l ( t ) \\ , \\mathrm { d } t \\mathrm { d } x \\mathrm { d } y \\\\ & = I _ 1 + I _ 2 + I _ 3 , \\end{align*}"}
+{"id": "8546.png", "formula": "\\begin{align*} \\zeta _ { p } ( s ) = - \\frac { 1 } { 2 } \\ , \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } + \\left ( \\frac { 1 } { 2 } C _ { p } ^ { ( 2 ) } - \\frac { \\gamma } { 2 } - \\frac { e ^ { 2 \\pi p } } { 1 + \\frac { 1 } { \\pi p } } Q _ { 2 \\pi p } ( 0 ) - \\frac { \\log ( 2 \\pi ) } { 2 } \\right ) \\ , s + O \\left ( s ^ { 2 } \\right ) \\end{align*}"}
+{"id": "1087.png", "formula": "\\begin{align*} ( x , y ) & \\mapsto [ x , y ] = x \\ast y - y \\ast x \\\\ ( x , y ) & \\mapsto x \\circ y = x \\ast y + y \\ast x \\end{align*}"}
+{"id": "1054.png", "formula": "\\begin{align*} N \\coloneqq ( d + 1 ) \\cdot \\binom { n + h } { n } \\ , . \\end{align*}"}
+{"id": "8442.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { 1 } { \\left ( \\lambda _ { n } ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } = \\frac { x ^ { - 2 s } } { \\Gamma ( s ) } \\ , \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\mu - i \\infty } ^ { \\mu + i \\infty } \\Gamma ( z ) \\ , \\Gamma ( s - z ) \\ , \\zeta _ { p } ( 2 z ) \\ , x ^ { 2 z } d z \\end{align*}"}
+{"id": "6521.png", "formula": "\\begin{align*} F _ X ( x ) = 1 - M \\int _ x ^ \\infty \\mathrm { e } ^ { \\beta t } t ^ \\nu K _ { \\nu } ( \\alpha t ) \\ , \\mathrm { d } t = 1 - M \\sum _ { k = 0 } ^ \\infty \\frac { \\beta ^ k } { k ! } \\int _ x ^ \\infty t ^ { \\nu + k } K _ \\nu ( \\alpha t ) \\ , \\mathrm { d } t . \\end{align*}"}
+{"id": "4951.png", "formula": "\\begin{align*} V ^ { - 1 } = \\left [ \\begin{array} { c | c } U _ { n - 1 } ^ { - 1 } & \\mathbf { 0 } \\\\ \\hline \\mathbf { 0 ' } & 1 \\end{array} \\right ] , \\end{align*}"}
+{"id": "5835.png", "formula": "\\begin{align*} X ^ { y ' } _ { \\tau _ { n _ k } } & \\geqslant \\pi _ 1 ( y ) + v \\ , ( m _ k - \\pi _ 2 ( y ) ) - ( \\beta + 1 ) n _ k '' \\\\ & \\geqslant \\pi _ 1 ( y ' ) - n _ k '' + v \\ , ( m _ k - n _ k '' - n _ k ' ) - ( \\beta + 1 ) n _ k '' \\\\ & = \\pi _ 1 ( y ' ) + ( v - \\epsilon _ k ) \\ , n _ k . \\end{align*}"}
+{"id": "8456.png", "formula": "\\begin{align*} \\frac { 1 } { \\Gamma ( s ) } = \\frac { 1 } { \\sqrt { \\pi } } + \\frac { \\gamma + 2 \\log ( 2 ) } { \\sqrt { \\pi } } \\left ( s - \\frac { 1 } { 2 } \\right ) + O \\left ( \\left ( s - \\frac { 1 } { 2 } \\right ) ^ { 2 } \\right ) , \\end{align*}"}
+{"id": "8720.png", "formula": "\\begin{align*} ( P _ { 1 } ) & = \\frac { 4 ( n - 2 ) } { n ( n - 1 ) } Q _ { 1 } + \\frac { 2 } { n ( n - 1 ) } Q _ { 2 } , \\\\ ( P _ { 2 } ) & = \\frac { 4 ( m - 2 ) } { m ( m - 1 ) } Q _ { 1 } + \\frac { 2 } { m ( m - 1 ) } Q _ { 2 } , \\\\ ( P _ { 3 } ) & = \\frac { 4 ( n + m - 2 ) } { n m } Q _ { 1 } + \\frac { 4 } { n m } Q _ { 2 } . \\end{align*}"}
+{"id": "6613.png", "formula": "\\begin{align*} \\varphi _ \\lambda ( x , [ y _ \\mu z ] ) = [ \\varphi _ \\lambda ( x , y ) _ { \\lambda + \\mu } z ] + ( - 1 ) ^ { | x | | y | } [ y _ \\mu \\varphi _ \\lambda ( x , z ) ] \\end{align*}"}
+{"id": "1587.png", "formula": "\\begin{align*} A _ { 2 3 1 } ( x ) = \\dfrac { x } { 1 + C _ 1 x ^ 2 ( x + 1 ) - \\dfrac { x } { 1 + C _ 2 x ^ 3 ( x + 1 ) - \\dfrac { x } { 1 + C _ 3 x ^ 4 ( x + 1 ) - \\dfrac { x } { \\ddots } } } } . \\end{align*}"}
+{"id": "3988.png", "formula": "\\begin{align*} \\mathcal { I } ( \\delta _ 1 , 0 ) & = \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + ( V _ { 1 } / V _ { 1 , X X } ) ^ { 1 / 2 } \\delta _ 1 ^ { 1 / 2 } = \\mathcal { I } _ { \\mathrm { D } } ( \\delta _ 1 , 0 ) , \\\\ \\mathcal { I } ( 0 , \\delta _ 2 ) & = \\mathbb { E } [ Y _ 2 ] - \\mathbb { E } [ Y _ 1 ] + ( V _ { 2 } / V _ { 2 , X X } ) ^ { 1 / 2 } \\delta _ 2 ^ { 1 / 2 } = \\mathcal { I } _ { \\mathrm { D } } ( 0 , \\delta _ 2 ) . \\end{align*}"}
+{"id": "679.png", "formula": "\\begin{align*} D ^ { n } \\mathfrak { r } _ { a , n } ( \\varphi ) = D ^ { n } \\varphi - \\sum _ { 1 \\leq i < i _ a } d ^ + _ { i , 0 } ( D ^ n \\varphi ) h _ { i } . \\end{align*}"}
+{"id": "1976.png", "formula": "\\begin{align*} 0 \\geq \\frac { b } { 2 } \\sum _ { p = 1 } ^ n u ^ { p \\bar p } - \\frac { A _ 1 ( 1 + \\Vert \\nabla v \\Vert ^ 2 + \\Delta v ) } { g \\Delta u } . \\end{align*}"}
+{"id": "4600.png", "formula": "\\begin{align*} & { \\textstyle \\min _ { x \\leq y \\leq x + B } \\{ K + G _ n ( y ) - G _ n ( x ) - V _ n ( x ) + V _ n ( y ) \\} } \\\\ [ - 5 p t ] & { \\textstyle = K + \\min _ { x \\leq y \\leq x + B } G _ n ( y ) - G _ n ( x ) > 0 . } \\end{align*}"}
+{"id": "6807.png", "formula": "\\begin{align*} ( k ^ + - 1 ) + ( 2 n - k ^ - - 1 ) - ( 2 n - 1 ) = k ^ + - k ^ - - 1 . \\end{align*}"}
+{"id": "8480.png", "formula": "\\begin{align*} \\Gamma \\left ( s \\right ) = \\sqrt { \\pi } - \\sqrt { \\pi } \\left ( 2 \\log ( 2 ) + \\gamma \\right ) \\left ( s - \\frac { 1 } { 2 } \\right ) + O \\left ( s - \\frac { 1 } { 2 } \\right ) ^ { 2 } . \\end{align*}"}
+{"id": "7428.png", "formula": "\\begin{align*} \\frac { 1 } { n ^ { \\gamma } } \\ , \\bigg \\| \\frac { f } { w _ t } \\bigg \\| _ { \\ell ^ { r } } < \\frac { 1 } { n ^ { \\frac { 1 } { r } } } \\ , \\bigg \\| \\frac { f } { w _ t } \\bigg \\| _ { \\ell ^ { p } } \\iff n > \\tilde { n } : = \\Bigg ( \\frac { \\| \\frac { f } { w _ { t } } \\| _ { \\ell ^ r } } { \\| \\frac { f } { w _ { t } } \\| _ { \\ell ^ p } } \\Bigg ) ^ { \\frac { p r } { p - r } } \\ , . \\end{align*}"}
+{"id": "7817.png", "formula": "\\begin{align*} \\langle W _ { j } ^ { \\prime } , x \\rangle = \\sum _ { i \\le N } g _ { i j } \\left \\langle \\tilde { g } _ { i j } B _ { i } , x \\right \\rangle \\end{align*}"}
+{"id": "6703.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\frac { d } { d s } \\chi _ s ( x _ 0 , \\xi _ 0 ) = H _ { p _ 2 } \\big ( \\chi _ s ( x _ 0 , \\xi _ 0 ) \\big ) , \\\\ \\chi _ 0 ( x _ 0 , \\xi _ 0 ) = ( x _ 0 , \\xi _ 0 ) , \\end{array} \\right . \\end{align*}"}
+{"id": "5874.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } U ( t ) = B _ { u ( t ) } \\ , U ( t ) \\\\ U ( 0 ) = \\operatorname { I d } \\end{cases} \\ , . \\end{align*}"}
+{"id": "8506.png", "formula": "\\begin{align*} 2 ^ { 4 - 2 s } c ^ { \\frac { 1 } { 2 } - s } \\sin ( \\pi s ) \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { \\prime 2 } + \\lambda _ { n } ^ { \\prime 2 } } { p ^ { \\prime } \\left ( p ^ { \\prime } + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { \\prime 2 } } \\ , \\lambda _ { n } ^ { \\prime 1 - 2 s } \\ , \\intop _ { 0 } ^ { \\infty } \\ , \\frac { y ^ { - s } ( y + 1 ) ^ { - s } } { \\sigma \\left ( \\lambda _ { n } ^ { \\prime } \\ , ( 2 y + 1 ) \\ , \\sqrt { c } \\right ) e ^ { 2 \\pi \\lambda _ { n } ^ { \\prime } ( 2 y + 1 ) \\sqrt { c } } - 1 } d y , \\end{align*}"}
+{"id": "496.png", "formula": "\\begin{align*} p ( \\gamma _ { ( m , n ) } ) = \\Gamma ( \\gamma _ { ( m , n ) } ; a , b ) \\end{align*}"}
+{"id": "1892.png", "formula": "\\begin{align*} \\min \\frac { 1 } { 2 } x ^ 2 \\mbox { s . t . } \\ \\left \\{ \\begin{aligned} x & = 1 ; \\\\ 2 x & = 2 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "195.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\left ( \\sum _ { g = 1 } ^ { \\infty } \\frac { v ^ g } { g ^ q } \\right ) \\left ( \\sum _ { h = 1 } ^ { \\infty } \\frac { w ^ h } { h ^ r } \\right ) \\left ( \\sum _ { i = 1 } ^ { \\infty } \\frac { x ^ i } { i ^ s } \\right ) \\left ( \\sum _ { j = 1 } ^ { \\infty } \\frac { y ^ j } { j ^ t } \\right ) \\left ( \\sum _ { k = 1 } ^ { \\infty } \\frac { z ^ k } { k ^ u } \\right ) \\right \\} . \\end{align*}"}
+{"id": "6003.png", "formula": "\\begin{align*} \\theta _ { 1 / 2 } ( z , \\epsilon ) & = \\sum _ { n \\in \\Z } e ^ { i \\epsilon \\pi n ^ 2 z } , \\end{align*}"}
+{"id": "4442.png", "formula": "\\begin{align*} \\mathbf { c } _ 0 \\cdot \\mathbf { P } ( \\Phi ) = - 2 ( L ( \\Phi ) + N ( \\Phi ) ) = - 2 E ( \\Phi ) \\end{align*}"}
+{"id": "2594.png", "formula": "\\begin{align*} q _ 1 = D v ( p ) \\cdot e _ 1 \\geq \\frac { v ( p ) - v ( z ) } { | p - z | } = \\frac { h } { p _ 1 } \\geq C \\frac { h } { s ^ 2 } . \\end{align*}"}
+{"id": "7882.png", "formula": "\\begin{align*} \\Omega _ { n , k } = \\left \\{ ( A ^ \\pm , \\underline { A } ^ \\pm ) : \\ A \\in \\binom { [ n ] } { k } \\right \\} . \\end{align*}"}
+{"id": "1851.png", "formula": "\\begin{align*} - 3 F ( 2 \\pi / 3 ) + \\sum _ { i = 1 } ^ { 3 } F ( \\theta _ { i } ) \\leq - \\frac { 1 } { \\pi ^ { 2 } } \\frac { 1 3 } { 7 2 } \\lambda _ { 1 , n } ^ { r , s } . \\leq - \\frac { 1 } { \\pi ^ { 2 } } \\frac { 1 3 } { 9 } \\lambda _ { 1 , n } ^ { r , s } \\sum _ { i = 1 } ^ { 3 } \\Big ( \\frac { \\theta _ { i } } { 2 \\pi } - 1 / 3 \\Big ) ^ { 2 } . \\end{align*}"}
+{"id": "7386.png", "formula": "\\begin{align*} \\Delta \\left ( \\frac { m } { a } \\right ) = \\begin{cases} 1 , & m = 0 \\\\ 0 , & m \\neq 0 . \\end{cases} \\end{align*}"}
+{"id": "8059.png", "formula": "\\begin{align*} \\mu ( A \\cdot \\{ b _ 1 , b _ 2 , b _ 3 \\} ) = \\mu ( X ) + \\mu ( ( A \\cdot b _ 3 ) \\setminus X ) \\geq \\mu ( A ) + \\mu ( B ) , \\end{align*}"}
+{"id": "7528.png", "formula": "\\begin{align*} A _ { k , 1 } ( \\varphi _ 1 ( x ) , \\ldots , \\varphi _ k ( x ) ) = \\sum _ { j \\geq 0 } a _ { k , j } \\frac { x ^ j } { j ! } . \\end{align*}"}
+{"id": "1311.png", "formula": "\\begin{align*} f ( \\Phi ( t ) ) - f ( \\Phi ( 0 ) ) = f ( \\rho ( v ) , T ( v ) ) - f ( \\rho ( u ) , T ( u ) ) , \\end{align*}"}
+{"id": "3948.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow \\infty } \\pi ^ { n } _ 1 ( U _ 1 ) = \\liminf _ { n \\rightarrow \\infty } \\pi ^ { n } ( U _ 1 \\times \\mathcal { S } _ 2 ) \\geq \\pi ^ \\infty ( U _ 1 \\times \\mathcal { S } _ 2 ) = \\pi ^ \\infty _ 1 ( U _ 1 ) . \\end{align*}"}
+{"id": "8004.png", "formula": "\\begin{align*} \\Phi ( X ) = \\sum _ { i = 1 } ^ m \\sum _ { j = 1 } ^ n Z _ { i , j } ^ * X Z _ { i , j } \\end{align*}"}
+{"id": "22.png", "formula": "\\begin{align*} x _ 0 = [ z _ 0 , g _ 0 ] _ { \\mathbf { K } } \\in X _ { \\mathbf { K } } ( \\mathbb { C } ) . \\end{align*}"}
+{"id": "3063.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta _ p u & = c _ 1 | x | ^ { m _ 1 } \\cdot v ^ { k _ 1 } \\cdot | \\nabla u | ^ { \\alpha } & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta _ p v & = c _ 2 | x | ^ { m _ 2 } \\cdot v ^ { k _ 2 } \\cdot | \\nabla u | ^ { k _ 3 } & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8704.png", "formula": "\\begin{align*} h ^ { ( l + 1 ) } ( x ) = \\sum ^ { R } _ { i = 1 } \\frac { b _ i ( x ) } { ( a + x ) ^ { c _ i } } , \\end{align*}"}
+{"id": "4979.png", "formula": "\\begin{align*} \\hat { p } _ { k - 1 , n } ^ { ( t + 1 ) } = \\hat { p } _ { k - 1 , k - 1 } ^ { \\phantom { ( ) } } \\hat { p } _ { k - 1 , n } ^ { ( t ) } + \\hat { p } _ { k - 1 , k } ^ { \\phantom { ( ) } } \\hat { p } _ { k , n } ^ { ( t ) } . \\end{align*}"}
+{"id": "3100.png", "formula": "\\begin{align*} [ 2 ] ^ k [ k ] _ { q ^ 2 } ! S _ B [ n , k ] = \\sum _ { \\ell = 0 } ^ { k } q ^ { k ( k - 2 \\ell ) } B _ { n , \\ell } ( q ) { n - \\ell \\brack k - \\ell } _ { q ^ 2 } \\end{align*}"}
+{"id": "2824.png", "formula": "\\begin{align*} b _ i = a _ i \\cdots a _ { i + p - 1 } , \\end{align*}"}
+{"id": "4254.png", "formula": "\\begin{align*} y _ { s } = \\zeta + \\int _ { s } ^ { T } \\tilde { f } \\left ( s , y _ { s } , z _ { s } , \\mathbb { P } _ { \\left ( y _ { s } , z _ { s } \\right ) } \\right ) \\mathrm { d } s - \\int _ { s } ^ { T } z _ { s } \\mathrm { d } W _ { s } . \\end{align*}"}
+{"id": "2442.png", "formula": "\\begin{align*} \\tfrac { d } { d t } e ^ { t X } m = X ( m ) & & m \\in M \\end{align*}"}
+{"id": "5092.png", "formula": "\\begin{align*} V ( \\gamma , \\sigma , t ) [ x ] & = \\psi \\left ( x - \\gamma ( x ) - \\frac { 1 } { N } \\sigma , t \\right ) - \\phi ( x ) , \\end{align*}"}
+{"id": "4001.png", "formula": "\\begin{align*} \\frac { \\partial \\mathrm { R } _ { \\mathrm { D } } ( \\lambda , \\delta ) } { \\partial \\lambda _ \\ell } = \\delta _ { \\ell } - \\frac { V _ \\ell / V _ { \\ell , X X } } { 4 \\lambda _ \\ell ^ 2 } - \\frac { 1 } { 4 } V _ o ^ { \\top } \\Lambda _ \\lambda ^ { - 1 } V _ { \\ell , X X } ^ { - 1 } \\Lambda _ \\lambda ^ { - 1 } V _ o . \\end{align*}"}
+{"id": "2737.png", "formula": "\\begin{align*} \\mathfrak { a } _ q ( u , v ) = \\int _ { \\mathrm M } ( \\nabla u | \\overline { \\nabla v } ) _ g d \\mu + \\int _ { \\mathrm { M } } q u \\overline { v } d \\mu , u , v \\in H _ 0 ^ 1 ( \\mathrm { M } ) , \\end{align*}"}
+{"id": "2669.png", "formula": "\\begin{align*} \\| \\delta '' \\| _ { \\infty } = \\| ( I + M ) ^ { - 1 } b \\| _ { \\infty } \\leq 3 \\| b \\| _ { \\infty } . \\end{align*}"}
+{"id": "3218.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { x \\in M _ + , x \\neq 0 } \\abs { ( S _ n ( \\nu ) - \\bar { \\nu } ) ( x ) } / \\rho ( x ) = 0 . \\end{align*}"}
+{"id": "6526.png", "formula": "\\begin{align*} S _ 1 = \\sqrt { \\pi } \\Gamma ( \\nu + 1 / 2 ) \\sum _ { k = 0 } ^ \\infty \\frac { ( \\nu + 1 / 2 ) _ k } { k ! } \\bigg ( \\frac { \\beta } { \\alpha } \\bigg ) ^ { 2 k } = \\frac { \\sqrt { \\pi } \\Gamma ( \\nu + 1 / 2 ) } { ( 1 - \\beta ^ 2 / \\alpha ^ 2 ) ^ { \\nu + 1 / 2 } } , \\end{align*}"}
+{"id": "1130.png", "formula": "\\begin{align*} I ( z ) & \\leq | \\partial ^ 2 u _ { k } ( z ) - \\partial ^ 2 u _ { k } ( 0 ) | + | \\partial ^ 2 u _ { k } ( 0 ) - \\partial ^ 2 u ( 0 ) | + \\\\ & \\qquad \\qquad \\quad + | \\partial ^ 2 u ( z ) - \\partial ^ 2 u _ { k } ( z ) | = : I _ { 1 } ( z ) + I _ { 2 } ( z ) + I _ { 3 } ( z ) . \\end{align*}"}
+{"id": "4908.png", "formula": "\\begin{align*} p _ { i , j } ^ { ( t + 1 ) } = \\begin{dcases} ( 1 - p _ 0 ) p _ { 0 , 0 } ^ { ( t ) } & \\mbox { i f } \\ ; i = j = 0 ; \\\\ ( 1 - p _ j ^ { } ) p _ { i , j } ^ { ( t ) } \\ , + \\ , p _ { j - 1 } ^ { } p _ { i , j - 1 } ^ { ( t ) } & \\mbox { i f } 0 < i \\le j \\le n j < t ; \\\\ 0 & \\mbox { o t h e r w i s e } ; \\\\ \\end{dcases} \\end{align*}"}
+{"id": "3578.png", "formula": "\\begin{align*} v ( n ) \\ = \\ \\sum _ { \\substack { 1 \\leq i \\leq k , \\\\ \\alpha _ i = 1 } } p _ i + \\sum _ { \\substack { 1 \\leq i \\leq k , \\\\ \\alpha _ i \\geq 2 } } \\left ( p _ i + \\alpha _ i \\right ) . \\end{align*}"}
+{"id": "7801.png", "formula": "\\begin{align*} \\Xi _ { 1 } ^ { 2 } = \\sum _ { i \\le N } ( \\max _ { j \\le n } g ^ { \\prime 2 } _ { i j } ) \\Vert B _ { i } \\Vert ^ { 2 } _ { 2 } , \\end{align*}"}
+{"id": "4746.png", "formula": "\\begin{align*} \\mu _ j \\tilde { S } _ { \\gamma _ i \\gamma _ j } & = \\mu _ i J _ { 2 | \\alpha _ i | } \\tilde { S } _ { \\gamma _ i \\gamma _ j } J ^ T _ { 2 | \\alpha _ j | } + \\mathcal { O } ( \\| H \\| ) . \\end{align*}"}
+{"id": "259.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - y ^ m z ^ n } \\right ) ^ { \\frac { m ^ 2 } { n ^ 3 } } \\end{align*}"}
+{"id": "4502.png", "formula": "\\begin{align*} \\sigma : = \\max \\left \\{ \\frac { 1 } { 2 | \\alpha | } , \\frac { 1 } { | \\beta | } , \\frac { 1 } { | \\gamma | } \\right \\} = \\left ( \\min \\{ 2 | \\alpha | , | \\beta | , | \\gamma | \\} \\right ) ^ { - 1 } > 0 \\end{align*}"}
+{"id": "2189.png", "formula": "\\begin{align*} | g _ { \\lambda _ i } ( z ) ^ { - \\frac { 1 } { 2 } } - \\lambda _ i ^ { - \\frac { 1 } { 2 } } | & = \\frac { | g _ { \\lambda _ i } ( z ) - \\lambda _ i | } { | \\lambda _ i g _ { \\lambda _ i } ( z ) | ^ { \\frac { 1 } { 2 } } | g _ { \\lambda _ i } ( z ) ^ { \\frac { 1 } { 2 } } + \\lambda _ i ^ { \\frac { 1 } { 2 } } | } \\\\ & = \\frac { | \\lambda _ i z ^ 2 - 2 z | } { | \\lambda _ i g _ { \\lambda _ i } ( z ) | ^ { \\frac { 1 } { 2 } } | g _ { \\lambda _ i } ( z ) ^ { \\frac { 1 } { 2 } } + \\lambda _ i ^ { \\frac { 1 } { 2 } } | } . \\end{align*}"}
+{"id": "216.png", "formula": "\\begin{align*} = \\exp \\left \\{ L i _ q ( v ) L i _ r ( w ) L i _ { s } ( x ) L i _ { t } ( y ) L i _ u ( z ) \\right \\} . \\end{align*}"}
+{"id": "6753.png", "formula": "\\begin{align*} Z _ \\infty : = \\{ z \\in \\{ 0 , 1 \\} ^ \\Z : | z \\cap ( - \\infty , 0 ] | = | z \\cap [ 0 , \\infty ) | = \\infty \\} . \\end{align*}"}
+{"id": "6996.png", "formula": "\\begin{align*} 0 < v \\left ( a ^ { - 1 } \\right ) \\leq \\mu ( q ) = \\gamma _ i , \\mbox { f o r s o m e } i , 1 \\leq i \\leq \\epsilon , \\end{align*}"}
+{"id": "899.png", "formula": "\\begin{align*} A & \\ll | d _ 3 | | r _ 3 | ^ { n / 2 + 3 / 2 + \\varepsilon } \\widehat { C } _ 1 \\widehat { C } _ 2 \\left ( | r _ 3 | ^ { n / 2 - 1 } + \\widehat { V } ^ { n - 2 } \\right ) \\end{align*}"}
+{"id": "3076.png", "formula": "\\begin{align*} \\frac { A ( r ) } { A ' ( r ) } \\cdot \\left [ \\mathcal W ( r ) ^ { p - 1 - \\alpha } \\right ] ' + \\mathcal W ( r ) ^ { p - 1 - \\alpha } = Q _ 1 ( \\mathcal V ( r ) , v _ 0 ( r ) ) , \\end{align*}"}
+{"id": "3680.png", "formula": "\\begin{align*} \\langle A ( ( - \\Delta ) ^ \\alpha ) u , v \\rangle = \\left \\langle \\alpha _ 1 , ( ( - \\Delta ) ^ { a _ 1 } u ) v \\right \\rangle + \\left \\langle \\alpha _ 2 , ( ( - \\Delta ) ^ { a _ 2 } u ) v \\right \\rangle + \\cdots + \\left \\langle \\alpha _ p , ( ( - \\Delta ) ^ { a _ p } u ) v \\right \\rangle . \\end{align*}"}
+{"id": "886.png", "formula": "\\begin{align*} v _ \\varpi = \\nu _ { \\varpi } ( \\Delta _ { F _ 2 } ) . \\end{align*}"}
+{"id": "3336.png", "formula": "\\begin{align*} \\widetilde { \\gamma } _ { \\xi } = \\frac { 1 - \\overline { a } \\xi } { \\xi - a } \\ , \\frac { 1 - a \\xi } { \\xi - \\overline { a } } \\ , \\gamma _ { \\xi } . \\end{align*}"}
+{"id": "5176.png", "formula": "\\begin{align*} \\Big ( \\sum _ { e \\in E ( \\mathbb Z ^ d ) } q _ e p _ e \\Big ) ^ d = \\mu ^ d \\le C ( 8 / 3 ^ d ) ^ k \\| q \\| ^ { d - 1 } \\sum _ { e \\in E ( \\mathbb Z ^ d ) } q _ e \\le C \\| q \\| ^ { d - 1 } \\sum _ { e \\in E ( \\mathbb Z ^ d ) } q _ e , \\end{align*}"}
+{"id": "7466.png", "formula": "\\begin{align*} - \\varepsilon \\ , \\partial _ { \\boldsymbol { \\nu } _ \\varepsilon } \\mathfrak { U } _ { M } ^ { ( \\varepsilon ) } = \\varepsilon ^ { \\alpha } \\varphi ^ { ( 0 ) } _ \\varepsilon \\ \\Big ( \\partial \\Omega ^ { ( 0 ) } _ { \\varepsilon , \\gamma } \\setminus \\Big \\{ \\bigcup _ { i = 1 } ^ 3 \\overline { \\Upsilon _ \\varepsilon ^ { ( i ) } ( 2 \\ell _ 0 \\varepsilon ^ \\gamma ) } \\Big \\} \\Big ) \\times ( 0 , T ) . \\end{align*}"}
+{"id": "248.png", "formula": "\\begin{align*} \\prod _ { \\substack { m , n \\geq 1 \\\\ m \\leq n ; \\ , ( m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - z ^ n } \\right ) ^ { \\frac { m ^ 1 } { n ^ 2 } } = \\sqrt { \\frac { 1 } { 1 - z } } \\ ; \\exp \\left \\{ \\frac { 1 } { 2 } \\frac { z } { 1 - z } \\right \\} , \\end{align*}"}
+{"id": "452.png", "formula": "\\begin{align*} A ( t ) = ( \\l ( t ) , \\l ( t ) , \\l ( t ) ) , \\end{align*}"}
+{"id": "9077.png", "formula": "\\begin{align*} \\mathfrak { W } ( f _ k ) = m \\leq k . \\end{align*}"}
+{"id": "6243.png", "formula": "\\begin{align*} & J ( a , z ) ( t ) = J _ { \\mathrm { s e c t } ( t ) } ( a ) ( z ) = J _ { \\mathrm { s e c t } ( t ) } ( a ^ { ( 1 ) } ) ( \\mathrm { s e c t } ( \\delta / 2 ) ) J _ { \\mathrm { s e c t } ( \\delta / 2 ) } ( a ^ { ( 2 ) } ) ( z ) \\\\ & = J ( a ^ { ( 1 ) } , \\mathrm { s e c t } ( \\delta / 2 ) ) ( t ) \\cdot J _ { \\mathrm { s e c t } ( \\delta / 2 ) } ( a ^ { ( 2 ) } ) ( z ) \\end{align*}"}
+{"id": "2625.png", "formula": "\\begin{align*} & \\left | \\left \\lbrace ( x , y , t ) \\in K : \\left | \\widetilde { P } _ 1 ' ( t ) \\alpha ( x + \\widetilde { P } _ 1 ( t ) , y ) - \\widetilde { P } _ 2 ' ( t ) \\beta ( x , y + \\widetilde { P } _ 2 ( t ) ) \\right | \\leq \\varepsilon \\right \\rbrace \\right | \\\\ \\lesssim & 2 ^ { c | l | } \\varepsilon ^ { \\frac { 1 } { 7 ( 8 ( d _ 1 + d _ 2 ) - 1 7 ) } } \\end{align*}"}
+{"id": "3571.png", "formula": "\\begin{align*} ( a _ { L - 1 } \\cdots a _ 1 a _ 0 ) _ b \\ : = \\ \\sum ^ { L - 1 } _ { i = 0 } a _ i b ^ i . \\end{align*}"}
+{"id": "826.png", "formula": "\\begin{gather*} \\Z ^ { \\# S - 1 } = \\bigsqcup _ { M , N \\in \\Z ^ 2 } \\mathcal { Z } _ { M , N } \\\\ : = \\bigsqcup _ { M , N \\in \\Z ^ 2 } \\left \\{ ( v _ \\nu ) _ { \\nu \\in S _ \\sigma } \\in \\Z ^ { \\# S - 1 } \\ , : \\ , \\sum _ { \\nu \\in S _ \\sigma } \\min ( 0 , v _ \\nu ) = - M , \\sum _ { \\nu \\in S _ \\sigma } \\max ( 0 , v _ \\nu ) = N \\right \\} \\end{gather*}"}
+{"id": "3918.png", "formula": "\\begin{align*} \\sup _ { \\gamma \\in \\mathcal { P } _ { \\mathrm { D } } } \\sup _ { \\pi \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 , \\gamma \\right ) } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda \\ , d \\pi = \\sup _ { \\pi \\in \\mathcal { G } _ { \\mathrm { D } , \\lambda } } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ \\lambda \\ , d \\pi . \\end{align*}"}
+{"id": "4475.png", "formula": "\\begin{align*} U _ { \\lambda } ( t , x ) : = \\lambda ^ { - 1 } U ( \\lambda ^ { - 2 } t , \\lambda ^ { - 1 } x ) , \\ \\ \\ ( U = ( u _ 1 , u _ 2 , u _ 3 ) , \\ \\lambda > 0 ) . \\end{align*}"}
+{"id": "8377.png", "formula": "\\begin{align*} C _ { m , n } = \\frac { 1 } { { n \\choose m } } \\sum _ { 1 \\leq i _ { 1 } < i _ { 2 } < \\ldots < i _ { m } \\leq n } \\left ( \\frac { X _ { i _ { 1 } } + \\ldots + X _ { i _ { m } } } { m } - \\theta ( m ) \\right ) \\otimes \\left ( \\frac { X _ { i _ { 1 } } + \\ldots + X _ { i _ { m } } } { m } - \\theta ( m ) \\right ) , \\end{align*}"}
+{"id": "7542.png", "formula": "\\begin{align*} & H _ 1 \\geq ( 3 ) ( 6 4 ) \\left ( \\frac { \\sigma } { 2 } \\right ) ^ { - \\frac { d } { p } } , \\\\ & H _ 2 \\geq \\left ( 2 ^ { 1 0 0 } C _ 0 \\left ( \\frac { \\sigma } { 2 } \\right ) ^ { - \\frac { d } { p } ( k + 1 ) } \\right ) ^ { \\max \\{ 1 , \\frac { 1 } { p - 1 } \\} } \\\\ & H _ 3 \\geq 2 ^ { 1 0 0 } C _ 0 \\left ( \\frac { \\sigma } { 2 } \\right ) ^ { - \\frac { d } { p } ( k + 1 ) } \\end{align*}"}
+{"id": "1247.png", "formula": "\\begin{align*} \\partial _ i g _ { \\gamma , i } ( x ) = - \\frac { X _ { \\gamma , i } ( x ) } { f _ { \\gamma , i } ' ( g _ { \\gamma , i } ( x ) ) } , i \\in \\{ 1 , 2 \\} . \\end{align*}"}
+{"id": "2887.png", "formula": "\\begin{align*} r _ 1 = ( \\frac { - t _ 0 } { 1 } \\ominus \\frac { - 1 } { 0 } ) \\bullet \\frac { 1 } { 0 } - ( \\frac { 1 } { 0 } \\ominus \\frac { 1 } { 1 } ) \\bullet \\frac { 1 } { 0 } - ( \\frac { 1 } { 1 } \\ominus \\frac { 1 } { 2 } ) \\bullet \\frac { 1 } { 0 } - \\cdots - ( \\frac { 1 } { - t _ 1 - 1 } \\ominus \\frac { 1 } { - t _ { 1 } } ) \\bullet \\frac { 1 } { 0 } = t _ { 1 } - 1 . \\end{align*}"}
+{"id": "7716.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( a , b ; c ; z ) = ( 1 - z ) ^ { - a } { } _ 2 F _ 1 \\left ( a , c - b ; c ; \\frac { z } { z - 1 } \\right ) \\end{align*}"}
+{"id": "2368.png", "formula": "\\begin{align*} M ( f ) _ f = 0 & \\ ; \\Leftrightarrow \\ ; \\exists k : \\N . \\ ; \\\\ & \\ ; \\Leftrightarrow \\ ; \\exists k : \\N . \\ ; \\forall n : \\N . \\ ; f ^ k \\in ( f ^ n ) \\subseteq R \\\\ & \\ ; \\Leftrightarrow \\ ; \\exists k : \\N . \\ ; f ^ k \\in ( f ^ { k + 1 } ) \\subseteq R \\rlap { . } \\end{align*}"}
+{"id": "6047.png", "formula": "\\begin{align*} g ( z ) : = u | A | z ^ { n + m } + v | B | \\overline { z } ^ m + | C | \\textrm { f o r a l l } z \\in \\mathbb { C } . \\end{align*}"}
+{"id": "2904.png", "formula": "\\begin{gather*} g _ { - m } ( \\xi , \\mu ) = ( f _ m ( \\xi ) - \\mu ^ m ) ^ { - 1 } , \\\\ g _ { - m - j } ( \\xi , \\mu ) = - \\sum _ { \\substack { k + l + | \\alpha | = j \\\\ l < j } } \\frac { ( - i ) ^ { | \\alpha | } } { \\alpha ! } ( f _ m ( \\xi ) - \\mu ^ m ) ^ { - 1 } \\partial _ \\xi ^ \\alpha f _ { m - k } ( \\xi ) g _ { - m - l } ^ { B , \\alpha } ( \\xi , \\mu ) , j \\geq 1 . \\end{gather*}"}
+{"id": "8916.png", "formula": "\\begin{align*} \\prod _ { n = 1 } ^ { d _ 1 ^ j } U _ n = Z _ j + \\sum _ { q \\geq 2 } \\sum _ { ( n _ 1 , \\dots , n _ q ) \\in \\mathbb { I } _ { d _ 1 ^ j } ^ q } \\beta _ { ( n _ 1 , \\dots , n _ q ) } [ U _ { n _ 1 } , \\dots , U _ { n _ q } ] . \\end{align*}"}
+{"id": "6528.png", "formula": "\\begin{align*} \\mathbb { P } ( \\overline { Z } _ n \\leq 0 ) = \\frac { 1 } { 2 } - \\frac { \\Gamma ( ( n + 1 ) / 2 ) } { \\sqrt { \\pi } \\Gamma ( n / 2 ) } \\rho ( 1 - \\rho ^ 2 ) ^ { n / 2 } { } _ 2 F _ 1 \\bigg ( 1 , \\frac { n + 1 } { 2 } ; \\frac { 3 } { 2 } ; \\rho ^ 2 \\bigg ) . \\end{align*}"}
+{"id": "930.png", "formula": "\\begin{align*} E _ o = P _ s \\hat { L } _ p + \\alpha P _ s \\hat { L } _ p + \\alpha P _ s \\hat { L } _ { \\mathrm { f } } , \\end{align*}"}
+{"id": "2413.png", "formula": "\\begin{align*} P \\bigl ( S - E ( S ) \\ge v \\bigr ) \\le \\exp \\Biggl ( - \\frac { 2 v ^ { 2 } } { \\sum _ { j = 1 } ^ { J } ( b _ { j } - a _ { j } ) ^ { 2 } } \\Biggr ) . \\end{align*}"}
+{"id": "2896.png", "formula": "\\begin{align*} ( f \\sharp g ) _ { m + m ' - j } ( \\xi ) = \\sum _ { k + l + | \\alpha | = j } \\frac { ( - i ) ^ { | \\alpha | } } { \\alpha ! } ( \\partial _ \\xi ^ \\alpha f _ { m - k } ) ( \\xi ) g _ { m ' - l } ^ { B , \\alpha } ( \\xi ) , j \\geq 0 . \\end{align*}"}
+{"id": "974.png", "formula": "\\begin{align*} \\overline { \\nabla } _ { X } \\xi = - A _ { \\xi } X + \\nabla _ { X } ^ { \\bot } \\xi \\end{align*}"}
+{"id": "603.png", "formula": "\\begin{align*} g ( s _ + ) = \\frac { 1 } { 4 } ( \\sqrt { C - 4 } + \\sqrt { C } ) \\end{align*}"}
+{"id": "8110.png", "formula": "\\begin{align*} \\sum _ I \\ < Y _ F , \\Lambda ^ I \\ > M _ I ( X ) = ( - 1 ) ^ { | F | } \\Gamma _ F ( - X ) = : \\chi _ F ( X ) . \\end{align*}"}
+{"id": "6574.png", "formula": "\\begin{align*} T = \\begin{pmatrix} t _ { 1 , 1 } & t _ { 1 , 2 } & t _ { 1 , 3 } \\\\ t _ { 2 , 1 } & t _ { 2 , 2 } & t _ { 2 , 3 } \\\\ t _ { 3 , 1 } & t _ { 3 , 2 } & t _ { 3 , 3 } \\end{pmatrix} \\mapsto \\Psi ( T ) = \\begin{pmatrix} t _ { 1 , 1 } & \\frac { 1 } { 2 } t _ { 1 , 2 } \\\\ \\frac { 1 } { 2 } t _ { 2 , 1 } & \\frac { 1 } { 4 } t _ { 2 , 2 } + \\frac { 3 } { 4 } t _ { 3 , 3 } \\end{pmatrix} . \\end{align*}"}
+{"id": "8977.png", "formula": "\\begin{align*} f = - \\log w , \\mu = 1 , \\lambda ( x ) = - \\frac { S ^ \\varphi } { m - 1 } \\frac { e ^ f } { m } . \\end{align*}"}
+{"id": "3933.png", "formula": "\\begin{align*} \\widehat { \\Sigma } ( \\delta _ 1 , \\delta _ 2 ) = \\left \\{ \\gamma \\in \\mathcal { P } ( \\mathcal { S } ) : \\boldsymbol { K } _ 1 ( \\gamma _ { 1 3 } , \\mu _ { 1 3 } ) \\leq \\delta _ 1 , \\widehat { \\boldsymbol { K } } _ 2 ( \\gamma _ { 2 3 } , \\mu _ { 2 3 } ) \\leq \\delta _ 2 \\right \\} . \\end{align*}"}
+{"id": "1313.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde { W } ^ { ( u ) } = W ( K ^ { ( u ) } ) ^ { - 1 } = W + W ( H ^ { ( u ) } ) ^ { - 1 } W , \\\\ \\widetilde \\eta ^ { ( u ) } = \\widetilde { W } ^ { ( u ) } T ( u ) \\eta + \\eta , \\\\ \\widetilde \\theta _ i ^ { ( u ) } = e ^ { \\rho _ i ( u ) } i \\in V \\end{cases} \\end{align*}"}
+{"id": "7554.png", "formula": "\\begin{align*} \\omega _ i ( r ) = \\prod \\limits _ { g \\mid \\rho _ i ( r ) } \\frac { - \\kappa _ i ( ( g , r ) ) } { g } . \\end{align*}"}
+{"id": "53.png", "formula": "\\begin{align*} ( h _ p ^ { - 1 } ) ^ \\vee \\overline { \\lambda } _ 0 h _ p ^ { - 1 } = ( Y _ \\mathbb { Q } ^ { - 1 } ) ^ \\vee Y _ p ^ \\vee \\overline { \\lambda } _ 0 Y _ p Y _ \\mathbb { Q } ^ { - 1 } , ( h _ q ^ { - 1 } ) ^ \\vee \\overline { \\lambda } _ 0 h _ q ^ { - 1 } = ( Y _ \\mathbb { Q } ^ { - 1 } ) ^ \\vee Y _ q ^ \\vee \\overline { \\lambda } _ 0 Y _ q Y _ \\mathbb { Q } ^ { - 1 } , \\ , \\forall q \\ne p . \\end{align*}"}
+{"id": "4641.png", "formula": "\\begin{align*} a = \\frac { \\frac { p + \\gamma - 1 } { 2 } - \\frac { ( p + \\gamma - 1 ) ( n q - n + 2 - \\theta ) } { 2 n q ( p - \\gamma + 1 ) } } { \\frac { p + \\gamma - 1 } { 2 } + \\frac { 1 } { n } - \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "1773.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } D _ \\Theta X _ \\cdot ^ { ( n ) } = D _ \\Theta X _ \\cdot \\textrm { ~ ~ w e a k l y ~ i n ~ } L ^ p ( \\Omega \\times [ 0 , T ] ) . \\end{align*}"}
+{"id": "287.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { 1 } { 1 2 } L i _ 2 ( z ) + \\frac { z ( 7 - 5 z ) ) } { 1 2 ( 1 - z ) ^ 2 } + \\frac { 1 } { 3 } \\log \\left ( \\frac { 1 } { 1 - z } \\right ) \\right \\} \\end{align*}"}
+{"id": "5546.png", "formula": "\\begin{align*} \\max _ { A \\in \\mathcal { P } _ { x , n } } \\left | 1 - \\frac { \\beta _ { x } \\left ( A \\right ) } { q _ { x , n } ^ { t } ( A ) } \\right | \\le \\varepsilon _ { x , n } ( t ) \\quad \\textrm { w i t h } \\quad \\lim _ { t \\to \\infty } \\varepsilon _ { x , n } ( t ) = 0 . \\end{align*}"}
+{"id": "966.png", "formula": "\\begin{align*} h ( \\overline { f } _ m ( x _ m ) , \\overline { f } _ m ( x _ 0 ) ) \\geqslant \\varepsilon _ 0 , m = 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "4849.png", "formula": "\\begin{align*} q ( 1 ) < q ( 2 ) < \\cdots < q ( u ) = z _ { t ( m ) } ( r ) = n t ( m ) + s ( r ) < n t ( m ) + n \\leq n \\beta + n . \\end{align*}"}
+{"id": "7653.png", "formula": "\\begin{align*} \\rho ( \\gamma ) g ( \\rho , y ) = g ( \\rho , \\gamma y ) \\rho _ 0 ( y ) \\end{align*}"}
+{"id": "7076.png", "formula": "\\begin{align*} \\alpha = \\alpha ( \\{ \\beta _ j \\mid j \\in I ' \\} ) = \\alpha ( \\{ \\tilde { \\beta } _ j \\mid j \\in I ' \\} ) , \\end{align*}"}
+{"id": "7622.png", "formula": "\\begin{align*} f _ 0 ( z ) = \\frac { z } { ( 1 - z ^ 3 ) ^ { 1 / 3 } } . \\end{align*}"}
+{"id": "6136.png", "formula": "\\begin{align*} U _ { \\lambda } = \\left \\{ ( x , y , t ) \\in \\mathcal { Q } _ { r _ { 1 } } \\ , : \\ , | D ^ { \\tau } d _ { s } u ( x , y , t ) | \\geq \\lambda \\right \\} \\subset \\left ( \\bigcup \\limits _ { i \\in \\mathbb { N } } \\mathcal { Q } _ { 5 \\rho _ { { i } } } \\left ( z _ { i } \\right ) \\right ) \\bigcup \\left ( \\bigcup \\limits _ { \\mathcal { K } \\times I \\in \\mathcal { A } } \\mathcal { K } \\times I \\right ) \\end{align*}"}
+{"id": "2691.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { m = 1 \\\\ ( m . n ^ s ) _ s = 1 } } ^ { n ^ s } ( m - 1 , n ^ s ) _ s = \\Phi _ s ( n ^ s ) \\tau _ s ( n ^ s ) . \\end{align*}"}
+{"id": "8797.png", "formula": "\\begin{align*} ( u ^ 2 + 4 t u + 6 t ^ 2 + 1 0 u + 2 4 t + 2 8 ) b _ 2 & = ( u ^ 2 + 8 t u + 1 2 t ^ 2 + 6 u + 3 0 t + 1 2 ) c _ 2 \\\\ & + ( u ^ 2 + 2 t u + 8 u ) b _ 3 - ( u ^ 2 + 4 t u + 2 u ) c _ 3 \\end{align*}"}
+{"id": "4379.png", "formula": "\\begin{align*} \\mathfrak { s } _ j ^ 2 = \\frac { 1 } { 2 } + { \\rm O } ( e ^ { - { c \\sqrt { j } } } ) . \\end{align*}"}
+{"id": "7487.png", "formula": "\\begin{align*} \\sigma = \\alpha _ 1 \\alpha _ 2 \\cdots \\alpha _ m \\underset { \\beta ^ { \\prime } } { \\underbrace { x \\alpha _ { m + 1 } \\cdots \\alpha _ k \\delta \\pi _ j \\cdots \\pi _ { l - 1 } \\pi _ { l + 1 } \\cdots \\pi _ n } . } \\end{align*}"}
+{"id": "7160.png", "formula": "\\begin{align*} u \\in C ^ 1 \\left ( \\Omega \\setminus \\overline { J _ u } \\right ) . \\end{align*}"}
+{"id": "6584.png", "formula": "\\begin{align*} & E ( \\hat \\beta _ l ) = \\sqrt { \\frac { \\pi } { 4 \\kappa + 4 } } e ^ { - \\frac { \\kappa } { 2 } } \\left [ ( 1 + \\kappa ) I _ 0 \\left ( \\frac { \\kappa } { 2 } \\right ) + \\kappa I _ 1 \\left ( \\frac { \\kappa } { 2 } \\right ) \\right ] , \\\\ & V a r ( \\hat \\beta _ l ) = 1 - { E } ^ 2 ( \\hat \\beta _ { l } ) , \\end{align*}"}
+{"id": "7033.png", "formula": "\\begin{align*} \\mu _ 0 ( f ) = \\min \\limits _ { j \\in \\{ 1 , \\dots , s \\} } \\left \\{ \\nu \\left ( b _ j \\textbf { Q } ^ { \\lambda _ j } \\right ) \\right \\} . \\end{align*}"}
+{"id": "3691.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } u = 0 & \\Omega ^ c , \\\\ u = 0 & W \\subset \\Omega ^ c , \\end{cases} \\end{align*}"}
+{"id": "847.png", "formula": "\\begin{align*} S _ { n _ { b } } = \\underset { S _ { n } \\in S _ { n \\left ( j \\right ) } } { \\textup { a r g m a x } } \\left \\{ R _ { c } \\left ( S _ { n } \\right ) \\right \\} \\end{align*}"}
+{"id": "3088.png", "formula": "\\begin{align*} ( p - 1 - \\alpha ) \\lambda - k _ 1 \\mu & = p - \\alpha + m _ 1 , \\\\ ( p - 1 - k _ 2 ) \\mu - k _ 3 \\lambda & = p - k _ 3 + m _ 2 . \\end{align*}"}
+{"id": "5168.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( \\mathbb Z ^ d ) } \\mathbb P \\big ( e \\in \\gamma ( 0 , v ) \\big ) ^ 2 \\le C \\| v \\| ^ { 1 + \\xi ( d - 1 ) } \\| v \\| ^ { - 2 \\xi ( d - 1 ) } = C \\| v \\| ^ { 1 - \\xi ( d - 1 ) } . \\end{align*}"}
+{"id": "6774.png", "formula": "\\begin{align*} | E _ { s } | & = 2 \\cdot b _ { k - 2 } \\cdot 2 ^ { \\mathfrak z ( \\lfloor b _ { k - 3 } \\cdots b _ 0 \\rfloor _ 2 ) } \\\\ & = b _ { k - 2 } \\cdot 2 ^ { \\mathfrak z ( s ) } = \\begin{cases} 2 ^ { \\mathfrak z ( s ) } & b _ { k - 2 } = 1 , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "2949.png", "formula": "\\begin{align*} \\mu _ { K S } ^ \\delta ( f ) ( \\tau ) = \\prod _ { D \\mid N } u ( f ) ( D \\tau ) ^ { c ^ 2 - 1 } . \\end{align*}"}
+{"id": "4297.png", "formula": "\\begin{align*} G ( t ) = G ( 0 ) > 0 . \\end{align*}"}
+{"id": "2754.png", "formula": "\\begin{align*} \\Delta \\Phi = 0 \\ ; \\mathrm { i n } \\ ; \\mathrm { M } , \\Phi _ { | \\partial M } = u _ { | \\partial M } . \\end{align*}"}
+{"id": "1169.png", "formula": "\\begin{align*} T \\left ( \\frac { 2 } { f ^ { 2 } - 1 } \\right ) = - \\frac { 2 } { ( f ^ { 2 } - 1 ) ^ { 2 } } T ( f ^ { 2 } - 1 ) + \\frac { 4 } { ( f ^ { 2 } - 1 ) ^ { 3 } } A ( f ^ { 2 } - 1 ) ^ { 2 } \\end{align*}"}
+{"id": "7096.png", "formula": "\\begin{align*} \\mathcal { E } ^ { N L } _ \\varepsilon ( c ) : = \\frac { 1 } { 4 } \\int _ { \\Omega } \\int _ { \\Omega } J _ \\varepsilon ( x - y ) \\big | c ( x ) - c ( y ) \\big | ^ 2 \\ : y x + \\int _ { \\Omega } f ( c ( x ) ) \\ : x . \\end{align*}"}
+{"id": "6661.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s w \\ , = \\ , 0 \\textrm { i n } \\ ; \\ ; \\R ^ N \\ , . \\end{align*}"}
+{"id": "1828.png", "formula": "\\begin{align*} \\Phi _ t ^ { ( 1 ) } ( \\vec p ) = \\ , \\frac { 1 } { 2 } \\ , e ^ { i t \\Delta E ( \\vec p ) } \\delta ( p _ 1 + p _ 2 - p _ 3 - p _ 4 ) \\big ( \\hat V ( p _ 1 - p _ 4 ) - \\hat V ( p _ 1 - p _ 3 ) \\big ) \\big ( \\chi _ { 1 2 3 4 } + \\chi ^ \\perp _ { 1 2 3 4 } \\big ) \\end{align*}"}
+{"id": "1286.png", "formula": "\\begin{align*} \\sigma ( A ) = \\left \\{ \\sqrt { n m } ^ { ( 1 ) } , - \\sqrt { n m } ^ { ( 1 ) } , 0 ^ { ( n + m - 2 ) } \\right \\} . \\end{align*}"}
+{"id": "3103.png", "formula": "\\begin{align*} [ z ^ { n - k } ] \\prod \\limits _ { i = 1 } ^ { n - \\ell } \\left ( 1 + \\frac { z q } { q ^ { 2 i } } \\right ) = q ^ { ( n - k ) ( 2 \\ell - n - k ) } { n - \\ell \\brack n - k } _ { q ^ 2 } , \\end{align*}"}
+{"id": "7682.png", "formula": "\\begin{align*} L ( t ) = \\frac { n - 1 } { t } , G ( t ) = \\frac { n ( n - 4 ) } { 4 t ^ 2 } , H ( t ) = \\frac { n - 4 } { 2 t } \\quad \\mbox { a n d } W ( t ) = \\frac { n ( n - 8 ) } { 4 } , \\forall t > 0 , \\end{align*}"}
+{"id": "4463.png", "formula": "\\begin{align*} G ( U _ n ( t _ n ) ) & = ( 4 - 2 d ) \\omega Q ( U _ n ( t _ n ) ) + ( 3 - d ) \\mathbf { c } \\cdot \\mathbf { P } ( U _ n ( t _ n ) ) \\\\ & \\ = ( 4 - 2 d ) \\omega Q ( U _ n ( 0 ) ) + ( 3 - d ) \\mathbf { c } \\cdot \\mathbf { P } ( U _ n ( 0 ) ) \\\\ & \\ = ( 4 - 2 d ) \\omega Q ( \\Phi _ { \\omega , \\mathbf { c } , n } ) + ( 3 - d ) \\mathbf { c } \\cdot \\mathbf { P } ( \\Phi _ { \\omega , \\mathbf { c } , n } ) + O ( \\delta _ n ) . \\end{align*}"}
+{"id": "1673.png", "formula": "\\begin{align*} ( f ( g ( x , y ) , h ( x , y ) ) ) ^ { ( p ) } = f ^ { ( p ) } ( g ^ { ( p ) } ( x , y ) , h ^ { ( p ) } ( x , y ) ) . \\end{align*}"}
+{"id": "1617.png", "formula": "\\begin{align*} { f } ^ 3 X = { f } ( { f } ^ 2 X ) = { f } \\ , Q X - \\sum \\nolimits _ { i } \\eta ^ i ( X ) \\ , { f } \\xi _ i = { f } \\ , Q X , \\\\ { f } ^ 3 X = { f } ^ 2 ( { f } X ) = Q \\ , { f } X - \\sum \\nolimits _ { i } \\eta ^ i ( { f } X ) \\ , \\xi _ i = Q \\ , { f } X \\end{align*}"}
+{"id": "4110.png", "formula": "\\begin{align*} c _ 4 = \\frac { c _ 1 c _ 3 - c _ 1 b _ 2 + c _ 2 b _ 1 } { b _ 3 } , c _ 5 = \\frac { c _ 3 c _ 2 + c _ 1 } { b _ 3 } c _ 6 = \\frac { c _ 3 ^ 2 + b _ 3 c _ 2 - b _ 1 - b _ 2 c _ 3 } { b _ 3 } . \\end{align*}"}
+{"id": "7642.png", "formula": "\\begin{align*} M ( c , 0 , y ) = 6 4 ( 4 - c ^ 2 ) ^ 2 y ^ 2 \\leq 1 0 2 4 , c \\in ( 0 , 2 ) , y \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "761.png", "formula": "\\begin{align*} h _ { i j } = \\dfrac { 1 } { D } \\left [ 2 \\phi ' \\rho ^ 2 u _ i u _ j + \\phi ^ 2 \\phi ' s _ { i j } - \\phi \\rho u _ { i j } \\right ] , \\end{align*}"}
+{"id": "308.png", "formula": "\\begin{align*} = \\exp \\left \\{ \\frac { 1 } { ( 1 - x ) ^ 3 ( 1 - y ) ^ 5 } ( x y L i _ 1 ( x y z ) - 2 x y ^ 2 L i _ 1 ( x y z ) + x y ^ 3 L i _ 1 ( x y z ) ) \\right \\} \\end{align*}"}
+{"id": "1033.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ 2 } { 3 2 ^ k ( 1 + k ) } \\bigg \\{ H _ { 2 k } ^ { ( 2 ) } - \\frac { 1 } { 4 } H _ { k } ^ { ( 2 ) } \\bigg \\} = \\frac { \\Gamma ( \\frac { 3 } { 2 } ) } { 1 6 \\Gamma ( \\frac { 5 } { 4 } ) ^ 2 } \\psi \\ , ' \\bigg ( \\frac { 5 } { 4 } \\bigg ) . \\end{align*}"}
+{"id": "6110.png", "formula": "\\begin{align*} \\frac { \\mu _ { \\tau , t } \\left ( \\mathcal { Q } _ { R } ( x _ { 0 } , y _ { 0 } ) ) \\right ) } { \\mu _ { \\tau , t } \\left ( \\mathcal { Q } _ { \\rho } ( x _ { 0 } , y _ { 0 } ) \\right ) } = \\left ( \\frac { R } { \\rho } \\right ) ^ { n + 2 s + 2 \\tau } . \\end{align*}"}
+{"id": "6777.png", "formula": "\\begin{align*} \\oplus _ { \\mu \\colon \\rho ( \\mu ) = \\lambda } \\Delta _ { \\mu } ^ { m _ \\mu } \\twoheadrightarrow \\imath _ { \\leq \\lambda } \\imath ^ { * } _ { \\leq \\lambda } M / \\imath _ { < \\lambda } \\imath ^ { * } _ { < \\lambda } M = ( M ) _ { \\not < \\lambda } / ( M ) _ { \\not \\leq \\lambda } , \\end{align*}"}
+{"id": "5871.png", "formula": "\\begin{align*} ( E , \\sigma ^ { ( \\mathcal { I } , r e d ) } ( H _ { \\omega , \\Lambda _ { L } ( x _ 0 ) } ) ) \\leq e ^ { - \\frac { \\hat { m } } { K } L } = e ^ { - \\frac { \\hat m } { 9 } \\frac { L } { 1 0 0 } } \\end{align*}"}
+{"id": "1042.png", "formula": "\\begin{align*} & \\theta _ k ( a , b , c , d ) \\\\ [ 1 m m ] & \\ : = \\frac { 2 k ( 1 + 2 a - b - c - d + 3 k ) } { a } + \\frac { ( a + k ) ( 1 + a - b - c + k ) } { a ( 1 + a - b + 2 k ) } \\\\ [ 1 m m ] & \\quad \\times \\frac { ( 1 + a - b - d + k ) ( 1 + a - c - d + k ) ( 2 + a - b - d + 3 k ) } { ( 1 + a - d + 2 k ) ( 2 + a - b - c - d + 2 k ) } \\\\ [ 1 m m ] & \\ : + \\frac { ( a + k ) ( c + k ) ( 1 - b + k ) ( 1 - d + k ) } { a ( 1 + 2 k ) ( 1 + a - b + 2 k ) ( 1 + a - c + 2 k ) } \\\\ [ 1 m m ] & \\quad \\times \\frac { ( 1 + a - b - c + k ) ( 1 + a - b - d + k ) ( 1 + a - c - d + k ) } { ( 1 + a - d + 2 k ) ( 2 + a - b - c - d + 2 k ) } . \\end{align*}"}
+{"id": "66.png", "formula": "\\begin{align*} G ^ { \\mathcal { T } ' } = \\left \\{ ( g _ { \\sigma } ) \\in \\prod _ { \\sigma \\in \\mathcal { T } } \\mid g _ { \\sigma } = 1 , \\ , \\forall \\sigma \\notin \\mathcal { T } ' \\right \\} \\end{align*}"}
+{"id": "6488.png", "formula": "\\begin{align*} n ! = \\sum _ { \\pi \\in S _ n } 1 \\ ; . \\end{align*}"}
+{"id": "6026.png", "formula": "\\begin{align*} d f _ { i j } + \\sum _ { k } f _ { k j } \\circ f _ { i k } = 0 \\end{align*}"}
+{"id": "7161.png", "formula": "\\begin{align*} \\mathcal { M } \\subset \\mathcal { B } ^ k _ M : = \\left \\{ y \\in \\R ^ k : \\abs { y } < M \\right \\} . \\end{align*}"}
+{"id": "78.png", "formula": "\\begin{align*} A u = \\mathcal { O } ( h ^ \\infty ) _ { C ^ \\infty } . \\end{align*}"}
+{"id": "7050.png", "formula": "\\begin{align*} \\nu ( h ) = \\nu _ i ( h ) \\mbox { a n d } \\nu _ i ( f _ { \\tilde { \\textbf { Q } } } ) \\geq 0 . \\end{align*}"}
+{"id": "7798.png", "formula": "\\begin{align*} \\textsf { P } \\{ g _ { i } ^ { 2 } > t \\} = \\textsf { P } \\{ \\vert g _ { i } \\vert > \\sqrt { t } \\} \\sim e ^ { - t / 2 } , \\end{align*}"}
+{"id": "8432.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\cdot \\frac { 1 } { \\sigma ^ { \\prime } \\left ( \\frac { \\lambda _ { n } ^ { 2 } x } { 2 \\pi } \\right ) e ^ { \\lambda _ { n } ^ { 2 } x } - 1 } = \\frac { 1 } { 2 \\pi i } \\ , \\intop _ { \\mu - i \\infty } ^ { \\mu + i \\infty } \\frac { \\zeta _ { p ^ { \\prime } } ( 1 - s ) \\ , \\zeta _ { p } ( 2 s ) } { 2 \\cos \\left ( \\frac { \\pi s } { 2 } \\right ) } \\ , \\left ( \\frac { x } { 2 \\pi } \\right ) ^ { - s } \\ , d s . \\end{align*}"}
+{"id": "4241.png", "formula": "\\begin{align*} D _ \\Theta = \\begin{pmatrix} 2 & 1 \\\\ - 1 & 0 \\end{pmatrix} D _ Y = \\begin{pmatrix} 1 & 0 \\\\ - 1 & 1 \\end{pmatrix} \\end{align*}"}
+{"id": "6582.png", "formula": "\\begin{align*} \\mathbf { \\hat h } = \\sqrt { \\frac { \\kappa } { \\kappa + 1 } } \\mathbf { \\hat h } ^ { L o S } + \\sqrt { \\frac { 1 } { \\kappa + 1 } } \\mathbf { \\hat h } ^ { N L o S } , \\end{align*}"}
+{"id": "4043.png", "formula": "\\begin{align*} S _ { \\mathrm { o f f } } \\ll p ^ { - \\frac { 1 } { 2 } } \\sum _ { i = 1 } ^ D \\sum _ { j = 1 } ^ D \\alpha _ i \\bar { \\alpha } _ j \\ , ( i , j ) ^ { \\frac { 1 } { 2 } } \\ , ( i j ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "7312.png", "formula": "\\begin{align*} a \\prod _ { i = 1 } ^ { n _ 1 } x _ i ^ { \\alpha _ i } + b \\prod _ { i = 1 } ^ { n _ 2 } y _ i ^ { \\beta _ i } + c \\prod _ { i = 1 } ^ { n _ 3 } z _ i ^ { \\gamma _ i } = 0 , \\end{align*}"}
+{"id": "2156.png", "formula": "\\begin{align*} Q _ { \\varepsilon , T } = \\Omega \\times \\left ( \\varepsilon , T - \\varepsilon \\right ) . \\end{align*}"}
+{"id": "8115.png", "formula": "\\begin{align*} \\ < X _ F , \\ < X _ { F ' } , X _ T \\ > \\ > = \\ < X _ F X _ { F ' } , X _ T \\ > . \\end{align*}"}
+{"id": "4673.png", "formula": "\\begin{align*} L ^ { \\rm s t i f f ( s o f t ) } _ \\varepsilon : = \\bigl \\{ \\vect f \\in L ^ 2 ( \\R ^ 3 ; \\R ^ 3 ) : \\ \\vect f \\equiv 0 \\mbox { o n } \\Omega _ { \\rm s o f t ( s t i f f ) } ^ { \\varepsilon } \\bigr \\} . \\end{align*}"}
+{"id": "5905.png", "formula": "\\begin{align*} \\theta _ { 1 / 2 } ( z , \\epsilon ) & = \\sum _ { n \\in \\Z } e ^ { i \\epsilon \\pi n ^ 2 z } , \\end{align*}"}
+{"id": "5927.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( g ) f ( [ 1 , y ] ) & = \\pi _ { \\psi } ( g ) f _ 1 ( y ) = \\psi ( \\tfrac { 1 } { 2 } \\langle y , y b \\rangle ) f _ 1 ( y ) \\\\ & = \\psi ( \\tfrac { 1 } { 2 } \\langle y , y b \\rangle ) f ( [ 1 , y ] ) , \\end{align*}"}
+{"id": "4967.png", "formula": "\\begin{align*} q _ { k } ^ { } = 1 - p _ { k } ^ { } = 1 - \\frac { n - k } { n } p = \\frac { n - ( n - k ) p } { n } \\ , . \\end{align*}"}
+{"id": "4886.png", "formula": "\\begin{align*} { \\mbox { $ u \\ge w = \\psi $ i n $ B _ r ( x _ r ) \\setminus \\overline { B _ { r / 2 } ( x _ r ) } $ . } } \\end{align*}"}
+{"id": "2008.png", "formula": "\\begin{align*} \\frac { d } { d t } E ( \\phi _ t ) = \\int _ \\Omega ( - \\dot { \\phi } _ t ) ( d d ^ c \\phi _ t ) ^ n , \\ , t \\in I . \\end{align*}"}
+{"id": "890.png", "formula": "\\begin{align*} S _ 1 = O ( | \\varpi | ^ { n + 3 } ) . \\end{align*}"}
+{"id": "5412.png", "formula": "\\begin{align*} D _ { T } ( n ) : = \\max _ { f \\in B _ T ( n ) } \\{ \\deg ( f ) \\} . \\end{align*}"}
+{"id": "1193.png", "formula": "\\begin{align*} & - \\sum _ { e \\in \\delta _ { } ( i ) } \\omega ( e ) v _ e + \\sum _ { e \\in \\delta _ { } ( i ) } \\omega ( e ) v _ e = 0 & ( i \\in \\hat { V } ) , \\\\ & \\sum _ { e \\in \\hat { E } ^ \\circ } \\omega ( e ) z ( e ) v _ e = 0 , \\end{align*}"}
+{"id": "2870.png", "formula": "\\begin{align*} T \\circ \\alpha = \\alpha \\circ T . \\end{align*}"}
+{"id": "3450.png", "formula": "\\begin{align*} c z ^ { 2 } - ( a - d ) z - b = 0 . \\end{align*}"}
+{"id": "2847.png", "formula": "\\begin{align*} { \\bf T } _ { s } ( { \\bf A } ) = \\left \\{ W = \\left \\langle \\left [ \\theta _ { 1 } \\right ] , \\left [ \\theta _ { 2 } \\right ] , \\dots , \\left [ \\theta _ { s } \\right ] \\right \\rangle \\in G _ { s } \\left ( { \\rm H ^ { 2 } } \\left ( { \\bf A } , \\mathbb C \\right ) \\right ) : \\bigcap \\limits _ { i = 1 } ^ { s } \\operatorname { A n n } ( \\theta _ { i } ) \\cap \\operatorname { A n n } ( { \\bf A } ) = 0 \\right \\} , \\end{align*}"}
+{"id": "8349.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ \\infty \\alpha _ j u _ j \\chi _ { B _ { R _ { j + 1 } } } \\Big \\| _ { p ^ * , q , \\mu } \\leq \\varepsilon _ 2 \\| \\{ \\alpha _ j \\} _ { j = 1 } ^ \\infty \\| _ { \\ell _ q } . \\end{align*}"}
+{"id": "2696.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { b = 1 } } ^ { n ^ s } ( b , n ^ s ) _ s e ^ { \\frac { 2 \\pi i b j } { n ^ s } } & = \\sum \\limits _ { \\substack { d ^ s \\mid n ^ s \\\\ \\frac { n ^ s } { d ^ s } \\mid j } } \\Phi _ s ( { d ^ s } ) { \\frac { n ^ s } { d ^ s } } \\\\ & = \\sum \\limits _ { \\substack { d ^ s \\mid n ^ s \\\\ { d ^ s } \\mid j } } \\Phi _ s ( { \\frac { n ^ s } { d ^ s } } ) { { d ^ s } } \\\\ & = \\sum \\limits _ { \\substack { { d ^ s } \\mid ( j , n ^ s ) _ s } } \\Phi _ s ( { \\frac { n ^ s } { d ^ s } } ) { { d ^ s } } . \\end{align*}"}
+{"id": "9093.png", "formula": "\\begin{align*} x _ i - x _ j = 0 , \\ , \\ , \\{ i , j \\} \\in E . \\end{align*}"}
+{"id": "6976.png", "formula": "\\begin{align*} F = a _ { Q 0 } + a _ { Q 1 } Q + \\ldots + a _ { Q D } Q ^ D \\end{align*}"}
+{"id": "3722.png", "formula": "\\begin{align*} ( \\sigma _ U ) _ * ( ( f _ \\sigma ^ * \\widehat { \\sigma _ L } ) \\circ ( \\sigma _ U ^ * \\phi ) ) = ( \\sigma _ U ) _ * ( f _ \\sigma ^ * \\widehat { \\sigma _ L } ) \\circ ( \\sigma _ U ) _ * ( \\sigma _ U ^ * \\phi ) . \\end{align*}"}
+{"id": "964.png", "formula": "\\begin{align*} I = \\int \\limits _ { R _ * } ^ { R _ 0 } \\frac { d t } { t ^ { \\frac { n - 1 } { q - 1 } } \\widetilde { q } _ { z _ 1 } ^ { 1 / ( q - 1 ) } ( t ) } \\ , . \\end{align*}"}
+{"id": "2012.png", "formula": "\\begin{align*} \\eta _ 1 ^ n : = \\inf \\left \\{ \\frac { E ( \\phi ) } { I _ g ( \\phi ) } ; \\phi \\in \\mathcal E ^ 1 ( \\Omega ) , w \\neq 0 \\right \\} , \\end{align*}"}
+{"id": "6163.png", "formula": "\\begin{align*} l _ R ( b ) l _ R ( - c ) & = l _ R ( b ) l _ R ( - a b d ) = l _ R ( b ) l _ R ( a d ) = l _ R ( b ) l _ R ( a ) + l _ R ( b ) l _ R ( d ) \\\\ & = l _ R ( a ) l _ R ( b ) + l _ R ( b ) l _ R ( d ) = l _ R ( c ) l _ R ( d ) + l _ R ( b ) l _ R ( d ) \\\\ & = l _ R ( b c ) l _ R ( d ) = l _ R ( a d ) l _ R ( d ) . \\end{align*}"}
+{"id": "184.png", "formula": "\\begin{align*} L i _ 3 ( - z ) - L i _ 3 ( - z ^ { - 1 } ) = - \\frac { 1 } { 6 } ( \\log z ) ^ 3 - \\frac { 1 } { 6 \\pi ^ 2 } \\log z \\end{align*}"}
+{"id": "9219.png", "formula": "\\begin{align*} x \\otimes a & = x a ( 1 + \\eta _ 1 ) , & a \\otimes x & = a x ( 1 + \\eta _ 1 ) . \\end{align*}"}
+{"id": "1355.png", "formula": "\\begin{align*} \\partial _ { t } F + \\partial _ { x } ( A ( t , F ) ) = \\mathbf { S } [ F ] ( t , x ) . \\end{align*}"}
+{"id": "7211.png", "formula": "\\begin{align*} \\widetilde \\lambda _ h ^ { ( s ) } : = \\abs { \\avg { \\widetilde { z } _ h ^ { ( s ) } } _ { B _ s } } \\end{align*}"}
+{"id": "4483.png", "formula": "\\begin{align*} \\mathbf { p } ( f ) : = ( P _ 1 ( f ) , \\cdots P _ d ( f ) ) , \\ \\ P _ k ( f ) : = - \\frac { 1 } { 2 } { \\rm R e } ( i f , \\partial _ k f ) _ { L ^ 2 ( \\R ^ d ) } \\ \\ ( k = 1 , \\cdots d ) \\end{align*}"}
+{"id": "8486.png", "formula": "\\begin{align*} \\varphi _ { p } ( s , x ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { 1 } { \\left ( \\lambda _ { n } ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } , \\ , \\ , \\ , \\ , ( s ) > \\frac { 1 } { 2 } , \\ , \\ , \\ , \\ , x > 0 , \\end{align*}"}
+{"id": "2182.png", "formula": "\\begin{align*} g ( x ) = L - 2 I _ { m } e ^ { \\mathbf { i } x } + L e ^ { 2 \\mathbf { i } x } . \\end{align*}"}
+{"id": "5096.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { R } } ( \\gamma , \\gamma _ t , \\sigma , \\sigma _ t , t ) = \\mathcal { R } \\left ( V \\left ( \\gamma , \\sigma , t \\right ) , \\gamma , \\gamma _ t , \\sigma _ t \\right ) , \\end{align*}"}
+{"id": "5221.png", "formula": "\\begin{align*} f _ r ( t ) = \\frac { \\Gamma ( 1 / 2 ) } { k ^ { r + 1 / 2 } \\pi ^ { 1 / 2 } B ( r + 1 / 2 , k / 2 ) } \\cdot \\frac { t ^ { 2 r } } { ( 1 + t ^ 2 / k ) ^ { r + 1 / 2 + k / 2 } } . \\end{align*}"}
+{"id": "3695.png", "formula": "\\begin{align*} 0 = P ( ( - \\Delta ) ^ s ) \\tilde { u } + ( q _ 1 - q _ 2 ) u _ 1 + q _ 2 \\tilde { u } = ( q _ 1 - q _ 2 ) u _ 1 \\Omega . \\end{align*}"}
+{"id": "1444.png", "formula": "\\begin{align*} \\phi _ m - i ^ * [ f _ \\epsilon ^ { ' } ( V _ m ) \\phi _ m ] = h _ m + \\omega _ m , \\end{align*}"}
+{"id": "9113.png", "formula": "\\begin{align*} \\mathrm { r a n k } ( \\partial _ { ( x , u ) } f ) = n \\ , , \\end{align*}"}
+{"id": "5736.png", "formula": "\\begin{align*} \\mathcal { E C S } _ { { \\rm { I I I } } } = \\{ S ( p , q , a , b , c , d ) \\in \\mathcal { E C } : p , a , c \\ne 0 \\} . \\end{align*}"}
+{"id": "7657.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { g } ^ 2 u = \\lambda _ { } u , & \\Omega , \\\\ u = \\frac { \\partial u } { \\partial \\bf n } = 0 , & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "8272.png", "formula": "\\begin{align*} D _ { k + 1 , 0 } ( i ) = \\frac { i ( 2 k - 1 + i ) } { 2 } \\prod _ { s = 3 } ^ { k + 1 } \\frac { ( 2 k - s + 1 ) ( s - 1 ) + i } { s } . \\end{align*}"}
+{"id": "5716.png", "formula": "\\begin{align*} \\alpha _ { g } \\not = 0 . \\end{align*}"}
+{"id": "3913.png", "formula": "\\begin{align*} \\sup _ { \\pi \\in \\bar { \\Gamma } } \\int _ { \\mathcal { V } \\times \\mathcal { V } } \\varphi _ { \\lambda } \\ , d \\pi = \\sup _ { \\gamma \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\int _ { \\mathcal { V } } g _ \\lambda ( v ) \\ , d \\gamma ( v ) , \\end{align*}"}
+{"id": "6828.png", "formula": "\\begin{align*} \\alpha ' _ n = ( 1 - a ) \\left ( \\frac { 1 - b q ^ n } { 1 - b } \\frac { \\alpha _ n } { 1 - a q ^ { 2 n } } - \\frac { q ^ { n - 1 } ( a q ^ { n - 1 } - b ) } { 1 - b } \\frac { \\alpha _ { n - 1 } } { 1 - a q ^ { 2 n - 2 } } \\right ) \\end{align*}"}
+{"id": "1436.png", "formula": "\\begin{align*} \\Pi ^ { \\bot } _ { \\mu , \\xi } \\Big ( V + \\phi - i ^ * [ f _ \\epsilon ( V + \\phi ] \\Big ) = 0 , \\end{align*}"}
+{"id": "4829.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } \\leq x \\right ) - \\Phi ( x ) & = - \\left [ \\mathbb { P } \\left ( \\frac { \\log Z _ { n } - n \\mu } { \\sigma \\sqrt { n } } > x \\right ) - ( 1 - \\Phi ( x ) ) \\right ] \\\\ & \\leq - \\left [ ( 1 - \\Phi ( x ) ) \\exp \\left \\{ - C \\frac { 1 + x ^ { 3 } } { \\sqrt { n } } \\right \\} - ( 1 - \\Phi ( x ) ) \\right ] \\\\ & = - ( 1 - \\Phi ( x ) ) \\left ( \\exp \\left \\{ - C \\frac { 1 + x ^ { 3 } } { \\sqrt { n } } \\right \\} - 1 \\right ) . \\end{align*}"}
+{"id": "3141.png", "formula": "\\begin{align*} g _ m ( u , x ) = \\frac { 1 } { ( 1 - u ) ^ { m + 1 } } p _ m \\left ( \\frac { u } { 1 - u } , x \\right ) . \\end{align*}"}
+{"id": "762.png", "formula": "\\begin{align*} h ^ i _ j = \\dfrac { \\phi ' \\delta ^ i _ j } { D } - \\dfrac { \\rho u ^ i _ j } { D \\phi } + \\dfrac { \\phi ' \\rho ^ 2 u ^ i u _ j } { D ^ 3 } + \\dfrac { \\rho ^ 3 u ^ i u _ k u ^ k _ j } { D ^ 3 \\phi } , \\end{align*}"}
+{"id": "6021.png", "formula": "\\begin{align*} \\chi ( g n _ 2 ^ - ) & = ( \\tfrac { c ' + d } { d } ) ( - i ) e ^ { \\tfrac { - \\pi i } { 1 2 } [ - \\tfrac { c + 2 d } { d } + \\tfrac { b } { d } + 1 2 s ( c + 2 d , d ) ] } \\\\ & = ( \\tfrac { c ' } { | d | } ) ( - i ) e ^ { \\tfrac { - \\pi i } { 1 2 } [ - \\tfrac { c } { d } + \\tfrac { b } { d } + 1 2 s ( c , d ) ] } e ^ { \\tfrac { \\pi i } { 6 } } \\\\ & = \\chi ( g ) \\chi ( n _ 2 ^ - ) . \\end{align*}"}
+{"id": "5490.png", "formula": "\\begin{align*} h _ { \\mu } \\left ( X , \\eta \\right ) = - \\int _ { G } \\int _ { X } \\log \\frac { d g ^ { - 1 } \\eta ' } { d \\eta ' } ( x ) d \\eta ( x ) d \\mu ( g ) . \\end{align*}"}
+{"id": "5659.png", "formula": "\\begin{align*} \\aligned & \\frac { 2 ^ * _ \\mu - 1 } { 2 2 ^ * _ \\mu } S ^ { \\frac { 2 ^ * _ \\mu } { 2 ^ * _ \\mu - 1 } } _ { H , L } > m _ { \\nu _ n } ( a , b ) = J _ { \\nu _ n } ( u _ n , v _ n ) - \\frac { 1 } { 2 2 ^ * _ \\mu } P _ { \\nu _ n } ( u _ n , v _ n ) \\\\ \\geq & \\bigl ( \\frac 1 2 - \\frac 1 { 2 2 ^ * _ \\mu } \\bigr ) ( | \\nabla u _ n | ^ 2 _ 2 + | \\nabla v _ n | ^ 2 _ 2 ) - C \\nu _ n = \\bigl ( \\frac 1 2 - \\frac 1 { 2 2 ^ * _ \\mu } \\bigr ) ( | \\nabla u _ n | ^ 2 _ 2 + | \\nabla v _ n | ^ 2 _ 2 ) + o _ n ( 1 ) . \\endaligned \\end{align*}"}
+{"id": "3664.png", "formula": "\\begin{align*} \\mathcal { M } _ { q _ j , \\alpha ^ j } : H ^ { s _ M } ( \\Omega ^ c ) \\to H ^ { - s _ M } ( \\Omega ) , f \\mapsto \\left . \\tilde { P } ( ( - \\Delta ) ^ { \\tilde { s } } ) u _ f \\right | _ { \\Omega } , j = 1 , 2 , \\end{align*}"}
+{"id": "9325.png", "formula": "\\begin{align*} 0 \\leq J ( u ^ { n } ) \\le J _ n ( u ^ { n } ) \\le J _ n ( u ^ { a } ) = J ( u ^ { a } ) \\le M , \\end{align*}"}
+{"id": "8558.png", "formula": "\\begin{align*} Y \\stackrel { \\mathcal { D } } { = } p \\delta _ 0 + q \\xi , \\end{align*}"}
+{"id": "3461.png", "formula": "\\begin{align*} X ^ 4 + E _ 1 X ^ 3 + E _ 2 X ^ 2 + E _ 3 X + E _ 4 = \\prod _ { i = 1 } ^ 4 ( X + X _ i ) . \\end{align*}"}
+{"id": "4660.png", "formula": "\\begin{align*} F _ 2 ( s ) - F _ 1 ( s ) = \\frac { 1 } { 2 } s ^ { 2 } \\log s ^ { 2 } , s \\in \\mathbb { R } , \\end{align*}"}
+{"id": "5330.png", "formula": "\\begin{align*} \\| f ^ { = d } \\| _ 2 ^ 2 \\leq \\| f ^ { = d } \\| _ { q } \\| f \\| _ { q ' } \\leq \\rho ^ { - d } \\gamma \\cdot \\gamma _ 1 \\left ( \\frac { \\gamma _ 2 } { \\gamma _ 1 } \\right ) ^ \\theta . \\end{align*}"}
+{"id": "9141.png", "formula": "\\begin{align*} \\begin{array} { r c l } x ^ { + } & = & f ( x , F _ { u } \\circ \\Psi ( x , z , v _ { [ 0 , R - A ] } ) ) \\\\ z ^ { + } & = & \\psi _ { c , [ 1 ] } ( x , z , v _ { [ 0 , R - A ] } ) \\\\ v _ { [ 0 , R - A - 1 ] } ^ { + } & = & v _ { [ 1 , R - A ] } \\end{array} \\end{align*}"}
+{"id": "5790.png", "formula": "\\begin{align*} Y = \\frac { Q ' } { Q } + V ' . \\end{align*}"}
+{"id": "3628.png", "formula": "\\begin{align*} a _ 0 \\ = \\ \\begin{cases} 7 , & , \\\\ 2 , & . \\end{cases} \\end{align*}"}
+{"id": "2914.png", "formula": "\\begin{align*} \\mu ^ \\delta ( f ) = \\sum n _ D \\cdot ( \\mu ( f \\mid L R ( D ) ) \\mid U L ( D ) ) . \\end{align*}"}
+{"id": "7408.png", "formula": "\\begin{align*} X _ { N , M } ^ { ( i ) } & = \\frac { \\sqrt { \\theta _ N } } { N } \\sum _ { z \\in \\Z ^ 2 } \\big \\{ Z _ { ( \\frac { i - 1 } { M } N , \\frac { i } { M } N ] , \\beta _ N } ^ { \\omega } ( z ) - 1 \\big \\} \\ , \\varphi _ N \\big ( \\tfrac { z } { \\sqrt { N } } \\big ) \\ , . \\end{align*}"}
+{"id": "6855.png", "formula": "\\begin{align*} d _ { \\square } ( h _ 1 , h _ 2 ) = \\sup _ { S , T \\subset [ 0 , 1 ] } \\bigg | \\int _ { S \\times T } \\d x \\ , \\d y \\ , \\big [ h _ 1 ( x , y ) - h _ 2 ( x , y ) \\big ] \\bigg | , \\end{align*}"}
+{"id": "8440.png", "formula": "\\begin{align*} \\varphi _ { p } ( s , x ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { p ^ { 2 } + \\lambda _ { n } ^ { 2 } } { p \\left ( p + \\frac { 1 } { \\pi } \\right ) + \\lambda _ { n } ^ { 2 } } \\ , \\frac { 1 } { \\left ( \\lambda _ { n } ^ { 2 } + x ^ { 2 } \\right ) ^ { s } } , \\ , \\ , \\ , \\ , ( s ) > \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "4648.png", "formula": "\\begin{align*} \\frac { p a } { p + \\gamma - 1 } = \\frac { p } { p + \\gamma - 1 } \\frac { ( p + \\gamma - 1 ) ( 1 - \\frac { 1 } { p } ) } { p + \\gamma - 1 + \\frac { 2 } { n } - 1 } < 1 , \\end{align*}"}
+{"id": "3870.png", "formula": "\\begin{align*} c ( ( y _ 1 , \\dotsc , y _ L ) , ( y _ 1 ' , \\dotsc , y _ L ' ) ) = \\sum _ { \\ell = 1 } ^ L | y _ \\ell - y _ \\ell ' | . \\end{align*}"}
+{"id": "4401.png", "formula": "\\begin{align*} A _ 1 = \\frac { 1 } { 4 } \\left ( \\alpha + \\frac { | \\mathbf { c } | ^ 2 } { 8 \\omega } \\right ) , \\ \\ A _ 2 = \\frac { 1 } { 4 } \\left ( \\beta + \\frac { | \\mathbf { c } | ^ 2 } { 4 \\omega } \\right ) , \\ \\ A _ 3 = \\frac { 1 } { 4 } \\left ( \\gamma + \\frac { | \\mathbf { c } | ^ 2 } { 4 \\omega } \\right ) , \\end{align*}"}
+{"id": "789.png", "formula": "\\begin{align*} \\dfrac { \\| \\phi ^ n ( \\rho ( 1 + u ) ) \\| _ { L ^ { \\infty } ( \\mathbb { S } ^ n ) } } { \\phi ^ n ( \\rho ) } = 1 + O ( \\varepsilon ) . \\end{align*}"}
+{"id": "5366.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\alpha , \\beta } ( \\mathcal { M } ) : = \\{ ( v , w ) ~ \\vert ~ ( \\alpha ( v + w ) , - \\beta ( v - w ) ) \\in \\mathcal { M } \\} . \\end{align*}"}
+{"id": "8087.png", "formula": "\\begin{align*} \\mathbf { A } : \\mathbf { B } = \\sum _ { i , j = 1 } ^ 3 \\mathbf { A } _ { i j } \\mathbf { B } _ { i j } , \\end{align*}"}
+{"id": "7619.png", "formula": "\\begin{align*} \\begin{cases} & A _ 2 = - a _ 2 , \\\\ & A _ 3 = 2 a _ 2 ^ 2 - a _ 3 , \\\\ & A _ 4 = 5 a _ 2 a _ 3 - 5 a _ 2 ^ 3 - a _ 4 , \\\\ & A _ 5 = 1 4 a _ 2 ^ 4 - 2 1 a _ 3 a _ 2 ^ 2 + 6 a _ 2 a _ 4 + 3 a _ 3 ^ 2 - a _ 5 . \\end{cases} \\end{align*}"}
+{"id": "135.png", "formula": "\\begin{align*} \\chi _ K \\varphi ^ { - t * } \\chi _ K = \\chi _ K \\varphi _ K ^ { - t * } \\chi _ K : C ^ \\infty ( M ) \\to C ^ \\infty ( M ) , \\end{align*}"}
+{"id": "4525.png", "formula": "\\begin{align*} f ( \\lambda ) : = \\lambda ^ 2 L ( U ) + \\lambda ^ { \\frac { d } { 2 } + 1 } N ( U ) + \\omega M ( U ) + \\lambda \\mathbf { c } \\cdot \\mathbf { P } ( U ) . \\end{align*}"}
+{"id": "5875.png", "formula": "\\begin{align*} \\nu _ { n + 1 } \\geq \\ , \\min \\{ \\langle L _ { u } g \\mid g \\rangle \\ ; \\| g \\| _ { L ^ 2 } = 1 \\ , , \\ ; g \\in E ^ { \\bot } \\cap H ^ \\frac 1 2 _ + \\} \\ , . \\end{align*}"}
+{"id": "3545.png", "formula": "\\begin{align*} i j ''' ( i t ) = \\frac { - 3 } { 2 } \\frac { ( j '' ( i t ) ) ^ 2 } { i j ' ( i t ) } + \\left ( \\frac { j ^ 2 ( i t ) - 1 9 6 8 j ( i t ) + 2 6 5 4 2 0 8 } { 2 j ^ 2 ( i t ) ( j ( i t ) - 1 7 2 8 ) ^ 2 } \\right ) ( i j ' ( i t ) ) ^ 3 \\end{align*}"}
+{"id": "1901.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta y & = u \\ \\mbox { i n } \\ \\Omega , \\\\ y & = 0 \\ \\mbox { o n } \\ \\partial \\Omega , \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "3016.png", "formula": "\\begin{align*} \\| A + B \\| _ { ( p , k ) } ^ p = \\sum _ { i = 1 } ^ { \\ell } s _ i ^ p ( A ) + \\sum _ { i = 1 } ^ { k - \\ell } s _ i ^ p ( B ) \\leq \\sum _ { i = 1 } ^ { k } s _ i ^ p ( A ) + \\sum _ { i = 1 } ^ { k } s _ i ^ p ( B ) . \\end{align*}"}
+{"id": "8336.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathcal J } \\int _ { \\{ x \\in \\Sigma : u ^ * _ \\mu ( b _ j ) < | u ( x ) | < u ^ * _ \\mu ( a _ j ) \\} } | \\nabla u ( x ) | ^ q \\dd \\mu ( x ) & = \\int _ { \\bigcup _ { j \\in \\mathcal J } \\{ x \\in \\Sigma : u ^ * _ \\mu ( b _ j ) < | u ( x ) | < u ^ * _ \\mu ( a _ j ) \\} } | \\nabla u ( x ) | ^ q \\dd \\mu ( x ) \\\\ & \\leq \\int _ 0 ^ { \\sum _ { j \\in \\mathcal J } ( b _ j - a _ j ) } ( \\nabla u ) _ \\mu ^ * ( s ) ^ q \\dd s \\\\ & = \\int _ 0 ^ t ( \\nabla u ) _ \\mu ^ * ( s ) ^ q \\dd s . \\end{align*}"}
+{"id": "7439.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { + \\infty } \\sum \\limits _ { p = 0 } ^ { 1 } \\varepsilon ^ { p \\alpha + k - 1 } \\ , \\left ( w _ { p \\alpha + k - 1 } ^ { ( i ) } ( x _ i ) + u _ { p \\alpha + k - 1 } ^ { ( i ) } \\Big ( x _ i , \\frac { \\overline { x } _ i } { \\varepsilon } , t \\Big ) \\right ) \\end{align*}"}
+{"id": "5241.png", "formula": "\\begin{align*} \\tilde { Q } _ 1 \\circ \\partial & = \\partial \\circ Q _ 1 \\circ f , \\\\ \\tilde { Q } _ 2 \\circ \\partial & = \\partial \\circ \\frac 1 { f } \\circ Q _ 2 , \\\\ \\tilde { Q } _ 1 \\circ \\tilde { Q } _ 2 \\circ \\partial & = \\partial \\circ Q _ 1 \\circ Q _ 2 . \\end{align*}"}
+{"id": "7483.png", "formula": "\\begin{align*} \\alpha _ i = a _ 1 \\cdots a _ p x a _ { p + 1 } \\cdots a _ l , \\quad \\quad \\tilde { \\alpha } _ i = ( a _ l , \\dots , a _ { p + 1 } , x , a _ p , \\dots , a _ 1 ) . \\end{align*}"}
+{"id": "6963.png", "formula": "\\begin{align*} e ( L / K , v ) = ( v L : v K ) . \\end{align*}"}
+{"id": "5232.png", "formula": "\\begin{align*} \\begin{array} { r l } s _ { 1 1 } & = e _ 1 + e _ 2 + e _ 3 , s _ { 1 2 } = e _ 4 + e _ 5 + e _ 6 , s _ { 1 3 } = e _ 7 + e _ 8 + e _ 9 , \\\\ s _ { 2 1 } & = e _ 1 e _ 2 + e _ 1 e _ 3 + e _ 2 e _ 3 , s _ { 2 2 } = e _ 4 e _ 5 + e _ 4 e _ 6 + e _ 5 e _ 6 , \\\\ s _ { 2 3 } & = e _ 7 e _ 8 + e _ 7 e _ 9 + e _ 8 e _ 9 , s _ { 3 1 } = e _ 1 e _ 2 e _ 3 , s _ { 3 2 } = e _ 4 e _ 5 e _ 6 , \\\\ s _ { 3 3 } & = e _ 7 e _ 8 e _ 9 , s = - ( s _ { 1 1 } + s _ { 1 2 } + s _ { 1 3 } - 6 ) / 3 . \\end{array} \\end{align*}"}
+{"id": "6740.png", "formula": "\\begin{align*} C _ K : = \\bigcap _ { s \\in S } R ^ { - s } W _ K ^ c . \\end{align*}"}
+{"id": "7435.png", "formula": "\\begin{align*} \\sum _ { i } \\mathrm { v } _ i \\ , h _ i ^ 2 \\ , w _ { \\alpha } ^ { ( i ) } ( 0 , t ) = \\boldsymbol { d _ { \\alpha } } ( t ) \\end{align*}"}
+{"id": "671.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\log \\norm { Q ( k ) h _ i } } { k } = \\lambda _ i 1 \\leq | i | \\leq g , \\ \\lim _ { k \\rightarrow \\infty } \\frac { \\log \\norm { Q ( k ) c _ s } } { k } = 0 1 \\leq s < \\gamma . \\end{align*}"}
+{"id": "2490.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle ( 1 + 2 \\Lambda ) \\dfrac { a _ 2 } { 2 m _ 1 } \\int _ { \\mathbb { R } ^ N } | \\nabla \\xi | ^ 2 d x + a _ 2 \\omega _ 1 \\int _ { \\mathbb { R } ^ N } \\xi ^ 2 d x - \\left ( 1 + \\dfrac { N } { 2 } \\Lambda \\right ) a _ 1 a _ 2 \\int _ { \\mathbb { R } ^ N } \\xi ^ 2 \\eta d x = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "6204.png", "formula": "\\begin{align*} V _ { e } = B \\cup R _ 0 \\cup _ i F _ i , \\end{align*}"}
+{"id": "7814.png", "formula": "\\begin{align*} \\textsf { E } \\xi = \\int _ { 0 } ^ { \\infty } \\textsf { P } \\{ \\xi > t \\} \\ , d t . \\end{align*}"}
+{"id": "8849.png", "formula": "\\begin{align*} d y _ \\lambda ^ h + A y _ \\lambda ^ h = \\bigl ( \\sigma ( u _ \\lambda ) Q h + \\sigma ' ( u _ \\lambda ) y _ \\lambda ^ h B \\ , d W , y _ \\lambda ^ h ( 0 ) = 0 . \\end{align*}"}
+{"id": "1841.png", "formula": "\\begin{align*} F _ 0 ( p ) = \\sum _ { q \\in U } \\delta ( p - q ) \\ . \\end{align*}"}
+{"id": "7016.png", "formula": "\\begin{align*} S _ i ( f ' ) = \\{ l - 1 \\mid l \\in S _ i ( f ) \\setminus \\{ 0 \\} \\mbox { a n d } p \\nmid l \\} . \\end{align*}"}
+{"id": "2511.png", "formula": "\\begin{align*} \\| ( I - P B _ c ( P ^ t A P ) ^ { - 1 } P ^ t A ) ( I - M A ) \\| _ A ^ 2 & \\le \\| ( I - P B _ c ( P ^ t A P ) ^ { - 1 } P ^ t A ) \\| _ A ^ 2 \\| I - M A \\| _ A ^ 2 \\\\ & \\le \\| ( I - P B _ c ( P ^ t A P ) ^ { - 1 } P ^ t A ) \\| _ A ^ 2 \\\\ & = \\| T \\| _ A ^ 2 + \\| P ( B _ c - I _ c ) ( P ^ t A P ) ^ { - 1 } P ^ t A \\| _ A ^ 2 \\\\ & \\le \\| T \\| _ A ^ 2 + \\| B _ c - I _ c \\| _ { A _ c } ^ 2 \\| ( P ^ t A P ) ^ { - 1 } P ^ t A \\| _ { A _ c } ^ 2 \\\\ & \\le \\| T \\| _ A ^ 2 + \\| I - T \\| _ A ^ 2 = 1 . \\end{align*}"}
+{"id": "9211.png", "formula": "\\begin{align*} \\lvert A ( D _ i ) \\rvert - \\lvert A ( D _ { i + 1 } ) \\rvert = \\deg _ { D _ i } ( v ) \\le 2 \\left \\lfloor \\frac { n - i } 2 \\right \\rfloor - ( r - 2 ) \\end{align*}"}
+{"id": "8468.png", "formula": "\\begin{align*} \\frac { 1 } { \\left ( x ^ { 2 } + t ^ { 2 } \\right ) ^ { s } } = \\frac { \\sqrt { \\pi } \\ , 2 ^ { \\frac { 1 } { 2 } - s } } { \\Gamma ( s ) x ^ { s - \\frac { 1 } { 2 } } } \\intop _ { 0 } ^ { \\infty } y ^ { s - \\frac { 1 } { 2 } } J _ { s - \\frac { 1 } { 2 } } ( x y ) \\ , e ^ { - t y } \\ , d y , \\end{align*}"}
+{"id": "3471.png", "formula": "\\begin{align*} H ( \\sigma ) = - \\beta \\sum _ { \\{ u , v \\} \\in E } \\ 1 ( \\sigma _ u = \\sigma _ v ) . \\end{align*}"}
+{"id": "3782.png", "formula": "\\begin{align*} \\sigma _ J ( a ) = \\begin{cases} a - k , & a - k \\notin J , \\\\ a + n - k , & a - k \\in J . \\end{cases} . \\end{align*}"}
+{"id": "7153.png", "formula": "\\begin{align*} \\partial _ { \\mathbf { n } } c & = 0 \\ ; \\ ; \\ ; \\ ; \\partial \\Omega \\times ( 0 , T ) , \\\\ c | _ { t = 0 } & = c _ 0 \\ ; \\ ; \\ ; \\Omega . \\end{align*}"}
+{"id": "4466.png", "formula": "\\begin{align*} A _ { \\omega } : = \\sup _ { \\Phi \\in \\widetilde { \\mathcal { M } } _ { \\omega , c _ 1 ( \\omega ) } } \\| \\Phi \\| _ { \\mathcal { H } ^ 1 } \\in ( 0 , \\infty ) . \\end{align*}"}
+{"id": "3177.png", "formula": "\\begin{align*} U = \\overline { \\xi } ( U _ 1 ) \\circ \\overline { \\xi } ( U _ 2 ) , \\end{align*}"}
+{"id": "1312.png", "formula": "\\begin{align*} \\frac { d } { d s } U _ i ( T _ i ( u ) + s ) = \\frac { 1 } { X _ i ( T _ i ( u ) + s ) ^ 2 } = \\frac { 1 } { Z _ i ( s ) ^ 2 } , \\end{align*}"}
+{"id": "1059.png", "formula": "\\begin{align*} \\Delta ( f ) = \\varphi \\left ( f \\odot \\frac { 1 } { 1 - t _ 1 \\cdots t _ n } \\right ) \\ , , \\end{align*}"}
+{"id": "6109.png", "formula": "\\begin{align*} \\mu _ { \\tau , t } \\left ( \\mathcal { Q } _ { R } ( x _ { 0 } ) \\right ) = c _ { n } \\frac { R ^ { n + 2 s + 2 \\tau } } { \\tau } . \\end{align*}"}
+{"id": "4394.png", "formula": "\\begin{align*} | J _ j | & \\le \\int _ { \\R ^ d } | \\nabla \\cdot \\varphi _ 3 | \\theta _ { \\epsilon } | \\varphi _ 1 | | \\varphi _ 2 | d x = \\int _ { | x | \\ge R } | \\nabla \\cdot \\varphi _ 3 | \\theta _ { \\epsilon } | \\varphi _ 1 | | \\varphi _ 2 | d x + \\int _ { | x | < R } | \\nabla \\cdot \\varphi _ 3 | \\theta _ { \\epsilon } | \\varphi _ 1 | | \\varphi _ 2 | d x = : J _ { \\infty } + J _ { 0 } . \\end{align*}"}
+{"id": "9050.png", "formula": "\\begin{align*} p _ x & = \\partial _ x z ( x , y ) , & p _ y & = \\partial _ y z ( x , y ) . \\end{align*}"}
+{"id": "2203.png", "formula": "\\begin{align*} \\Phi ( x , y ) = \\Phi ( x , z ) + \\sum _ { y < p \\le z } \\sum _ { v \\ge 1 } \\Phi ( x / p ^ { v } , p ) \\end{align*}"}
+{"id": "8032.png", "formula": "\\begin{align*} W = \\bigoplus _ { j = 1 } ^ { \\beta } \\left \\{ Q _ j \\otimes I _ n \\right \\} \\end{align*}"}
+{"id": "6028.png", "formula": "\\begin{align*} \\theta _ \\alpha ^ j + \\theta _ \\beta ^ j = \\rho _ j . \\end{align*}"}
+{"id": "6928.png", "formula": "\\begin{align*} \\frac { 1 } { \\sum _ { j \\in \\N } f ( j ) ^ 2 } \\sum _ { v \\in M } f ( v ) ^ 2 = o ( 1 ) \\textrm { a s } N \\to \\infty . \\end{align*}"}
+{"id": "2011.png", "formula": "\\begin{align*} \\int _ { \\{ h _ j \\geq k ^ { n + 1 } \\} } h _ j d V _ g = \\int _ { \\{ u _ j \\leq - k \\} } ( - u _ j ) ^ { n + 1 } g d V \\end{align*}"}
+{"id": "8650.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { S _ j ( t ) } { M _ j ( t ) } = \\lim _ { N \\to \\infty } \\frac { S _ j ( \\tau _ N ) } { M _ j ( \\tau _ N ) } & = \\frac { \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } ( 1 - y ) y ^ { j - 1 } d y } { \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } y ^ { j - 1 } d y } \\\\ & = 1 - \\frac { \\int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } y ^ { j } d y } { \\int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } y ^ { j - 1 } d y } = : \\varphi _ j ( p ) . \\end{align*}"}
+{"id": "6842.png", "formula": "\\begin{align*} a _ m = \\frac { 1 } { ( q ) _ \\infty } \\sum _ { j \\in \\mathbb { Z } } ( - 1 ) ^ j q ^ { r j ^ 2 - i j + m ( r - 1 ) j } \\frac { 1 - q ^ { ( m + 2 j ) ( i + 1 ) } } { 1 - q ^ { m + 2 j } } \\frac { 1 } { 1 + q ^ { m + 2 j } } . \\end{align*}"}
+{"id": "693.png", "formula": "\\begin{align*} \\int _ { I _ \\alpha ^ { ( k ) } } \\Big | \\sum _ { i = 0 } ^ { N - 1 } D \\varphi ( T ^ i x ) \\Big | \\ , d x \\leq \\sum _ { l = 0 } ^ { k } \\sum _ { 0 \\leq i < q ( l ) } \\int _ { ( T ^ { ( l ) } ) ^ i J _ l } | S ( l ) D \\varphi ( x ) | d x . \\end{align*}"}
+{"id": "282.png", "formula": "\\begin{align*} \\prod _ { \\substack { l , m , n \\geq 1 \\\\ l , m \\leq n ; \\ , \\gcd ( l , m , n ) = 1 } } \\left ( \\frac { 1 } { 1 - x ^ l y ^ m z ^ n } \\right ) ^ { \\frac { 1 } { l ^ a m ^ b n ^ c } } = \\exp \\left \\{ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\sum _ { l = 1 } ^ { n } \\frac { x ^ l } { l ^ a } \\right ) \\left ( \\sum _ { m = 1 } ^ { n } \\frac { y ^ m } { m ^ b } \\right ) \\frac { z ^ n } { n ^ c } \\right \\} . \\end{align*}"}
+{"id": "3626.png", "formula": "\\begin{align*} ( b _ m \\cdots 7 ) _ { 1 0 } \\ = \\ ( 4 \\cdots 9 ) _ { 1 0 } - 2 , \\end{align*}"}
+{"id": "8459.png", "formula": "\\begin{align*} \\zeta _ { p } ( 2 s ) = \\frac { 1 } { 2 s - 1 } + C _ { p } ^ { ( 1 ) } + O \\left ( s - \\frac { 1 } { 2 } \\right ) , \\end{align*}"}
+{"id": "4749.png", "formula": "\\begin{align*} J _ { 2 | \\alpha _ i | } & = \\tilde { S } ^ T _ { \\gamma _ i } J _ { 2 n } \\tilde { S } _ { \\gamma _ i } \\\\ & = \\sum _ { k = 1 } ^ r \\tilde { S } _ { \\gamma _ k \\gamma _ i } ^ T J _ { 2 | \\alpha _ k | } \\tilde { S } _ { \\gamma _ k \\gamma _ i } \\\\ & = \\tilde { S } _ { \\gamma _ i \\gamma _ i } ^ T J _ { 2 | \\alpha _ i | } \\tilde { S } _ { \\gamma _ i \\gamma _ i } + \\sum _ { k \\neq i , k = 1 } ^ r \\tilde { S } _ { \\gamma _ k \\gamma _ i } ^ T J _ { 2 | \\alpha _ k | } \\tilde { S } _ { \\gamma _ k \\gamma _ i } . \\end{align*}"}
+{"id": "2443.png", "formula": "\\begin{align*} & \\sum _ { m _ 1 , \\ldots , m _ n \\geq 0 } c _ { m _ 1 } \\cdots c _ { m _ n } a _ { m _ 1 + \\ldots + m _ n + r } \\\\ \\leq & \\sum _ { m _ 1 , \\ldots , m _ n \\geq n } c _ { m _ 1 } \\cdots c _ { m _ n } a _ { m _ 1 + \\ldots + m _ n + r } + n \\sum _ { m = 0 } ^ { n - 1 } c _ m \\sum _ { m _ 1 , \\ldots , m _ { n - 1 } \\geq 0 } c _ { m _ 1 } \\cdots c _ { m _ { n - 1 } } a _ { m _ 1 + \\ldots + m _ { n - 1 } + m + r } . \\end{align*}"}
+{"id": "4176.png", "formula": "\\begin{align*} \\| u _ t - v _ t \\| & \\leq \\| u _ t - w _ t \\| + \\| w _ t - v _ t \\| = \\| w _ t x _ t - w _ t \\| + \\| v _ t p _ t - v _ t \\| \\\\ & \\leq \\| w _ t ( x _ t - p _ t ) \\| + \\| v _ t p _ t - v _ t q _ t \\| + \\| v _ t q _ t - v _ t \\| \\\\ & \\leq \\| x _ t - p _ t \\| + \\| p _ t - q _ t \\| + \\| v _ t v _ t ^ * v _ t - v _ t \\| < 1 0 \\eta . \\end{align*}"}
+{"id": "524.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\hbar } u ( k ) : = \\sum \\limits _ { j = 1 } ^ { n } \\left ( u \\left ( k + \\hbar v _ { j } \\right ) + u \\left ( k - \\hbar v _ { j } \\right ) \\right ) - 2 n u ( k ) , k \\in \\hbar \\mathbb { Z } ^ { n } , \\end{align*}"}
+{"id": "1616.png", "formula": "\\begin{align*} g ( { f } X , Y ) = - g ( X , { f } Y ) , g ( Q X , Y ) = g ( X , Q Y ) . \\end{align*}"}
+{"id": "3609.png", "formula": "\\begin{align*} m \\ = \\ L ( f ) \\ \\le \\ m + 1 - \\beta , \\end{align*}"}
+{"id": "5782.png", "formula": "\\begin{align*} \\| T _ { \\alpha } ( \\vec { f } ) \\| _ { q ( \\cdot ) } & \\le C \\prod _ { i = 1 } ^ { m } \\| M _ { \\alpha _ { i } } f _ { i } \\| _ { q _ { i } ( \\cdot ) } \\le C \\prod _ { i = 1 } ^ { m } \\| f _ { i } \\| _ { p _ { i } ( \\cdot ) } . \\end{align*}"}
+{"id": "8864.png", "formula": "\\begin{align*} \\mathcal S ( t ) \\triangleq \\bigg \\{ ( \\mathbf { x } , \\mathbf { y } ) | & \\mathbf { x } , \\mathbf { y } \\in \\{ 0 , 1 \\} ^ { P - 1 } , x _ p + y _ p \\leq 1 , p \\in \\mathcal { P } \\backslash \\{ P \\} , \\\\ & \\sum _ { p \\in \\mathcal { P } \\backslash \\{ P \\} } ( x _ p + 2 y _ p ) = t \\bigg \\} , \\\\ h ( \\mathbf { z } , t ) \\triangleq & \\sum _ { ( \\mathbf { x } , \\mathbf { y } ) \\in \\mathcal S ( t ) } \\prod _ { p = 1 } ^ { P - 1 } 2 ^ { x _ p } z _ p ^ { x _ p + 2 y _ p } , \\ \\mathbf { z } \\in \\mathbb { R } ^ { P - 1 } _ { + + } , \\end{align*}"}
+{"id": "6183.png", "formula": "\\begin{align*} e ^ \\xi S = \\frac { n - 1 } { \\sqrt { 1 + | D _ \\tau \\xi | ^ 2 } } - { \\rm d i v } _ \\tau \\bigg ( \\frac { D _ \\tau \\xi } { \\sqrt { 1 + | D _ \\tau \\xi | ^ 2 } } \\bigg ) , \\end{align*}"}
+{"id": "7366.png", "formula": "\\begin{align*} \\chi _ \\ell ( x ) = \\sum _ { n \\in \\mathbb { Z } } \\chi \\left ( \\frac { x - x _ 0 + n } { \\ell } \\right ) . \\end{align*}"}
+{"id": "1153.png", "formula": "\\begin{align*} \\mathcal { A } ( f , g ) = T ( f \\cdot g ) - f T ( g ) - T ( f ) g - 2 B ( A ( f ) , A ( g ) ) \\left ( f , g \\in P \\right ) . \\end{align*}"}
+{"id": "4613.png", "formula": "\\begin{align*} { \\rm P } _ { j , j + i } = & \\sum ^ { M - j } _ { m = i + 1 } { M - j \\choose m } \\mathbb { P } _ { \\rm T X } ^ m \\left ( 1 - \\mathbb { P } _ { \\rm T X } \\right ) ^ { M - j - m } \\\\ & \\times \\frac { M - j - i } { M - j } \\bar { \\rm P } _ { m , i } . \\end{align*}"}
+{"id": "8481.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n } } { \\sqrt { x ^ { 2 } + n ^ { 2 } } } = - \\frac { 1 } { 2 x } + 2 \\sum _ { n = 1 } ^ { \\infty } K _ { 0 } \\left ( \\pi ( 2 n - 1 ) x \\right ) , \\ , \\ , \\ , \\ , \\ , x > 0 , \\end{align*}"}
+{"id": "5544.png", "formula": "\\begin{align*} \\int _ { S _ { y } } f _ { n } ( y ) d \\eta _ { p } ^ { y } = \\lim _ { m \\to \\infty } \\int _ { S _ { y } } f _ { n } ( y ) d \\eta _ { p _ { m } } ^ { y } \\le \\liminf _ { m \\to \\infty } \\int _ { S _ { y } } D \\left ( \\alpha _ { x , g } \\parallel \\alpha _ { x , e } \\right ) d \\eta _ { p _ { m } } ^ { y } . \\end{align*}"}
+{"id": "922.png", "formula": "\\begin{align*} u h _ u + v h _ v & = ( - \\nu _ { 1 2 3 } + \\mu / 2 - 3 / 4 ) h , \\\\ u g _ u + v g _ v & = ( - \\nu _ { 1 2 3 } + \\mu / 2 + 1 / 4 ) g . \\end{align*}"}
+{"id": "6784.png", "formula": "\\begin{align*} \\frac { x ^ \\lambda } { ( q ; q ) _ \\infty ^ { \\mathrm { r k } ( \\mathfrak { g } ) } \\prod _ { \\alpha \\in \\Delta _ + } ( 1 - x ^ \\alpha ) \\prod _ { \\alpha \\in \\Delta } ( q x ^ \\alpha ; q ) _ \\infty } = \\sum _ { \\mu \\in P } m _ { \\lambda \\mu } ( q ) \\frac { E _ \\mu ( x , q , 0 ) } { ( q ) _ \\mu } . \\end{align*}"}
+{"id": "2621.png", "formula": "\\begin{align*} f _ { 2 , m , \\sharp } ( x , y ) = \\sum _ { n = 1 } ^ { \\mathcal { N } _ 2 } h _ { 2 , n , m } ( x , y ) e ^ { i \\beta _ { n , m } ( x ) y } \\end{align*}"}
+{"id": "33.png", "formula": "\\begin{align*} ( - ) ^ \\ast \\colon B \\rightarrow B , b \\mapsto b ^ \\ast : = \\mathrm { T r d } ( b ) - b . \\end{align*}"}
+{"id": "324.png", "formula": "\\begin{align*} s _ h ( 3 , 2 ) = \\frac { 1 5 } { 2 } \\zeta ( 5 ) + \\zeta ( 2 ) \\zeta ( 3 ) , \\end{align*}"}
+{"id": "4030.png", "formula": "\\begin{align*} \\Lambda ( f , s ) = \\left ( \\frac { \\sqrt { N } } { 2 \\pi } \\right ) ^ s \\Gamma ( s ) \\ , L ( f , s ) \\end{align*}"}
+{"id": "1783.png", "formula": "\\begin{align*} \\begin{pmatrix} f _ { 0 0 } & f _ { 0 1 } & f _ { 0 2 } & \\ldots \\\\ f _ { 1 0 } & f _ { 1 1 } & f _ { 1 2 } & \\ldots \\\\ \\vdots & \\vdots & & \\end{pmatrix} \\end{align*}"}
+{"id": "4056.png", "formula": "\\begin{align*} S _ { \\mathrm { o f f } } = \\sum _ { d _ 1 , \\ , d _ 2 < D } \\frac { 1 } { \\sqrt { d _ 1 d _ 2 } } & \\sum _ { I d _ 1 , \\ , J d _ 2 < D } \\alpha _ { I d _ 1 } \\bar { \\alpha } _ { J d _ 2 } \\sum _ { L , M \\geq 1 } \\frac { G _ k ( L M d _ 1 d _ 2 \\slash p ) } { \\sqrt { L M } } \\\\ & \\times \\left ( 2 \\pi i ^ { 2 k } \\sum _ { p | c } \\frac { \\mathcal { S } ( I L , J M , c ) } { c } \\ , J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { I L J M } } { c } \\right ) \\right ) \\end{align*}"}
+{"id": "2692.png", "formula": "\\begin{align*} c _ r ( n ) & = \\sum \\limits _ { \\substack { { m = 1 } \\\\ ( m , r ) = 1 } } ^ { r } e ^ { \\frac { 2 \\pi i m n } { r } } \\end{align*}"}
+{"id": "7039.png", "formula": "\\begin{align*} \\mathcal I = \\mathcal I _ 1 + \\mathcal I _ 2 . \\end{align*}"}
+{"id": "5313.png", "formula": "\\begin{align*} \\exp \\left ( - \\frac { 0 . 0 0 0 1 } { p } \\right ) \\leq \\mu _ p ( f ) = \\frac { \\mu _ p ( f ) } { r ^ { 0 } } \\leq \\frac { \\mu _ p ( f _ { S \\rightarrow x } ) } { r ^ { | S | } } \\leq e ^ { - | S | } , \\end{align*}"}
+{"id": "6056.png", "formula": "\\begin{align*} H _ 2 ( z ) : = A _ 2 z ^ { n ' + m ' } + B _ 2 \\overline { z } ^ { m ' } + C _ 2 \\textrm { f o r a l l } z \\in \\mathbb { C } \\end{align*}"}
+{"id": "1122.png", "formula": "\\begin{align*} ( f ( g h ) ) x & = ( ( f g ) h ) x - ( ( f h ) g ) x \\\\ x ( f ( g h ) ) & = x ( ( f g ) h ) - x ( ( f h ) g ) . \\end{align*}"}
+{"id": "7765.png", "formula": "\\begin{align*} x _ j - x _ k & = M _ { 1 k } - M _ { i j } + ( x _ j + M _ { i j } ) - ( M _ { 1 k } + x _ k ) \\\\ & = M _ { 1 k } - M _ { i j } + ( M \\vec x ) _ i - ( M \\vec x ) _ 1 \\\\ & = M _ { 1 k } - M _ { i j } + y _ i - y _ 1 , \\end{align*}"}
+{"id": "2026.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j \\leq n } u _ { j p } u _ { \\bar j } + u _ p u _ { p \\bar p } = - \\beta ( u u _ p + B \\bar z _ p ) , \\end{align*}"}
+{"id": "1352.png", "formula": "\\begin{align*} \\chi ( t , x , s , y ) & = b _ { \\varepsilon } \\left ( \\frac { x - y } { 2 } \\right ) b _ { \\varepsilon } \\left ( \\frac { t - s } { 2 } \\right ) g \\left ( \\frac { x + y } { 2 } \\right ) h _ { \\delta } \\left ( \\frac { t + s } { 2 } \\right ) \\\\ & = b _ { \\varepsilon } \\left ( \\overline { y } \\right ) b _ { \\varepsilon } \\left ( \\overline { s } \\right ) g \\left ( \\overline { x } \\right ) h _ { \\delta } \\left ( \\overline { t } \\right ) . \\end{align*}"}
+{"id": "7012.png", "formula": "\\begin{align*} \\nu _ i ( f ) = \\min _ { 1 \\leq j \\leq r } \\left \\{ v \\left ( \\tilde { b } _ j \\right ) \\right \\} \\mbox { a n d } \\alpha _ \\ell \\geq \\gamma \\mbox { f o r e v e r y } \\ell \\in I _ 0 ( f , i ) , \\end{align*}"}
+{"id": "9312.png", "formula": "\\begin{align*} G ( 2 , d ) : = \\left ( \\begin{array} { c c } s t ^ d & - s ^ { d + 1 } \\\\ t ^ { d + 1 } & - s ^ d t \\\\ \\end{array} \\right ) \\equiv \\left ( \\begin{array} { c c } 1 & - 1 \\\\ 1 & - 1 \\\\ \\end{array} \\right ) \\odot \\left ( \\begin{array} { c c } s t ^ d & s ^ { d + 1 } \\\\ t ^ { d + 1 } & s ^ d t \\\\ \\end{array} \\right ) . \\end{align*}"}
+{"id": "5588.png", "formula": "\\begin{align*} \\mathcal { E } _ { s } & : C ( X ) \\to \\tilde { L } ^ { \\infty } ( X ) \\\\ \\mathcal { E } _ { s } f \\left ( u , v , x _ { 0 } \\right ) & = \\lim _ { N \\to \\infty } \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ { N } s ^ { n } . \\left ( f \\circ \\xi \\right ) \\left ( u , v , x _ { 0 } \\right ) = \\mathbb { E } _ { \\lambda } \\left ( \\tilde { f } \\left ( u , \\cdot \\right ) | \\mathcal { F } ^ { s } \\right ) ( x _ { 0 } ) . \\end{align*}"}
+{"id": "8050.png", "formula": "\\begin{align*} A = \\lambda _ 1 ^ { \\uparrow } ( A ) E _ 1 + \\lambda _ 2 ^ { \\uparrow } ( A ) ( E _ 2 - E _ 1 ) + \\lambda _ 3 ^ { \\uparrow } ( A ) ( E _ 3 - E _ 2 ) + . . . . + \\lambda _ n ^ { \\uparrow } ( A ) ( E _ n - E _ { n - 1 } ) \\end{align*}"}
+{"id": "8163.png", "formula": "\\begin{align*} E _ Q ( x ) = { x + 3 \\choose 3 } + { x + 2 \\choose 3 } = \\frac { ( x + 1 ) ( x + 2 ) ( 2 x + 3 ) } { 6 } \\end{align*}"}
+{"id": "6554.png", "formula": "\\begin{align*} \\| T ( f ) ( x ) \\| _ { L _ { | x | _ h } ^ p L _ \\theta ^ { \\bar { p } _ 2 } ( \\mathbb H ^ n ) } = \\omega _ Q ^ { \\frac { 1 } { \\bar { p } _ 2 } - \\frac { 1 } { \\bar { p } _ 1 } } \\int _ { \\mathbb H ^ n } K ( e , y ) | y | _ h ^ { - Q / p } d y \\| f _ j \\| _ { L _ { | x | _ h } ^ p L _ \\theta ^ { \\bar { p } _ 2 } ( \\mathbb H ^ n ) } . \\end{align*}"}
+{"id": "5190.png", "formula": "\\begin{align*} ( t _ 1 , t _ 2 , \\dots , t _ n , t ) \\sum _ { i = 0 } ^ m a _ { i } e _ { \\lambda ^ i } = \\sum _ { i = 0 } ^ m t { \\cdot } t _ { \\lambda ^ i } a _ { i } e _ { \\lambda ^ i } , \\end{align*}"}
+{"id": "8042.png", "formula": "\\begin{align*} \\begin{bmatrix} A & X \\\\ X ^ * & B \\end{bmatrix} = U \\begin{bmatrix} \\frac { A + B } { 2 } + { \\mathrm { R e } \\ , } X & 0 \\\\ 0 & 0 \\end{bmatrix} U ^ * + V \\begin{bmatrix} 0 & 0 \\\\ 0 & \\frac { A + B } { 2 } - { \\mathrm { R e } \\ , } X \\end{bmatrix} V ^ * \\end{align*}"}
+{"id": "1230.png", "formula": "\\begin{align*} s = \\lambda _ 0 a _ 0 + \\cdots + \\lambda _ n a _ n , \\lambda _ j \\geq 0 , \\ j = 0 , \\dots , n , \\sum _ { k = 0 } ^ n \\lambda _ k = n + 1 . \\end{align*}"}
+{"id": "8628.png", "formula": "\\begin{align*} & E \\left [ \\left ( Z _ 0 ( a ) e ^ { - \\lambda a } - Y \\right ) ^ 2 \\right ] = O \\left ( e ^ { - \\lambda a } \\right ) , \\\\ & E \\left [ \\left ( Z _ 0 ( a ) e ^ { - \\lambda a } - Z _ 0 ( b ) e ^ { - \\lambda b } \\right ) ^ 2 \\right ] = O \\left ( e ^ { - \\lambda a } \\right ) , \\end{align*}"}
+{"id": "4804.png", "formula": "\\begin{align*} \\tilde { \\mathcal { L } } ( \\varphi ) = \\tau ^ { - 1 } \\langle c , A \\psi - \\phi \\rangle + \\tau ^ { - 1 } \\sum _ { i = 1 } ^ m \\langle c , B _ i \\psi \\rangle u _ i , \\end{align*}"}
+{"id": "7907.png", "formula": "\\begin{align*} \\begin{cases} 2 \\lambda ^ { ( i ) } + 1 & : = [ 2 \\lambda _ 1 , 2 \\lambda _ 2 , \\ldots , 2 \\lambda _ i + 1 , \\ldots , 2 \\lambda _ t ] , \\ \\mbox { f o r } i \\in \\mathcal { I } ( \\lambda ) \\\\ 2 \\lambda ^ { ( t + 1 ) } + 1 & : = [ 2 \\lambda _ 1 , 2 \\lambda _ 2 , \\ldots , 2 \\lambda _ t , 1 ] . \\end{cases} \\end{align*}"}
+{"id": "3575.png", "formula": "\\begin{align*} n \\ = \\ p _ { 1 } ^ { \\alpha _ 1 } \\cdots p _ { k } ^ { \\alpha _ k } , \\end{align*}"}
+{"id": "3709.png", "formula": "\\begin{align*} h = \\frac { 1 } { 2 } \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\theta ( x ) = - x ^ \\top , \\mbox { w e h a v e } \\tau = \\theta \\tau _ h , \\begin{pmatrix} a & b \\\\ c & - a \\end{pmatrix} \\mapsto \\begin{pmatrix} - a & c \\\\ b & a \\end{pmatrix} . \\end{align*}"}
+{"id": "843.png", "formula": "\\begin{align*} k _ { e x \\left ( 1 \\right ) } = \\underset { k \\in S _ { n \\left ( 1 \\right ) } } { { \\arg \\min } } ~ \\textbf { g } _ { k } ^ { H } \\textbf { g } _ { k } \\end{align*}"}
+{"id": "3231.png", "formula": "\\begin{align*} d ( \\beta ) & = \\left ( a _ 1 T _ 1 + a _ 2 T _ 2 \\right ) ^ 2 + \\left ( b _ 1 T _ 1 + b _ 2 T _ 2 \\right ) ^ 2 + \\left ( a _ 1 T _ 1 + a _ 2 T _ 2 \\right ) \\left ( b _ 1 T _ 1 + b _ 2 T _ 2 \\right ) , \\\\ d ( \\gamma ) & = \\left ( a _ 1 T _ 1 + a _ 2 T _ 2 \\right ) \\left ( b _ 1 T _ 1 + b _ 2 T _ 2 \\right ) ^ 2 + \\left ( a _ 1 T _ 1 + a _ 2 T _ 2 \\right ) ^ 2 \\left ( b _ 1 T _ 1 + b _ 2 T _ 2 \\right ) . \\end{align*}"}
+{"id": "2515.png", "formula": "\\begin{align*} \\| y + \\delta _ y - A ^ { - 1 } r \\| _ A \\le \\rho _ { t g } \\| A ^ { - 1 } r \\| _ A , \\rho _ { t g } = \\rho _ { t g } ^ * + \\delta _ { \\rho _ { t g } } , \\end{align*}"}
+{"id": "5252.png", "formula": "\\begin{align*} T _ \\rho f = \\sum _ { S \\subseteq [ n ] } \\rho ^ { | S | } f ^ { = S } . \\end{align*}"}
+{"id": "2842.png", "formula": "\\begin{align*} \\begin{cases} \\overset \\cdot b _ 0 = b _ 0 ( b _ 1 + b _ 2 ) , \\\\ \\overset \\cdot b _ 1 = b _ 1 ( b _ 2 + b _ 3 - b _ 1 ) , \\\\ \\overset \\cdot b _ 2 = b _ 2 ( b _ 3 - b _ 1 - b _ 0 ) , \\\\ \\overset \\cdot b _ 3 = b _ 3 ( - b _ 2 - b _ 1 ) . \\end{cases} \\end{align*}"}
+{"id": "4127.png", "formula": "\\begin{align*} M = \\pm \\left ( { T _ { \\varepsilon _ 1 } ^ { \\vec { v } } } ^ T \\right ) ^ { m _ 1 } \\end{align*}"}
+{"id": "8315.png", "formula": "\\begin{align*} M ( g a , g b ) = M ( g a _ 1 , \\dots , g a _ n , g b _ 1 , \\dots , g b _ m ) = M ( a , b ) \\end{align*}"}
+{"id": "5933.png", "formula": "\\begin{align*} \\Pi _ { \\psi } ( h _ 1 ) \\Pi _ { \\psi } ( h _ 2 ) f = \\widetilde { C } _ { X ^ { \\ast } } ( h _ 1 , h _ 2 ) \\Pi _ { \\psi } ( h _ 1 h _ 2 ) f . \\end{align*}"}
+{"id": "5538.png", "formula": "\\begin{align*} h \\left ( Z , \\lambda \\right ) - h \\left ( M , \\bar { \\lambda } \\right ) = \\lim _ { n \\to \\infty } - \\frac { 1 } { n } \\log P _ { \\mu , x } ^ { \\zeta , n } \\left ( L _ { x } , L _ { x } \\omega _ { n } \\right ) . \\end{align*}"}
+{"id": "5965.png", "formula": "\\begin{align*} \\overline { \\overline { C } } _ { X ^ { \\ast } } ( p , g ) & = \\overline { C } _ { X ^ { \\ast } } ( p , g ) \\overline { s } ( p ) \\overline { s } ( g ) \\overline { s } ( p g ) ^ { - 1 } = 1 . \\end{align*}"}
+{"id": "2720.png", "formula": "\\begin{align*} \\left \\| P _ 1 u \\right \\| ^ 2 + \\left \\| P _ 2 u \\right \\| ^ 2 + 2 \\left ( P _ 1 u , P _ 2 u \\right ) = \\left \\| e ^ { - s \\sigma } f \\right \\| ^ 2 . \\end{align*}"}
+{"id": "8150.png", "formula": "\\begin{align*} \\Sigma : \\ t ^ p \\mapsto \\Sigma _ 0 ^ t s ^ p \\delta s = \\frac { B _ { p + 1 } ( t ) - B _ { p + 1 } ( 0 ) } { p + 1 } \\end{align*}"}
+{"id": "6335.png", "formula": "\\begin{align*} A ( q ) = \\frac 1 2 y _ 0 q + O ( q ^ 2 ) , \\qquad q \\to 0 . \\end{align*}"}
+{"id": "8271.png", "formula": "\\begin{align*} D _ { l , 0 } ( n ) = \\frac { n ( 2 k - 1 + n ) } { 2 } \\prod _ { s = 3 } ^ l \\frac { ( 2 k - s + 1 ) ( s - 1 ) + n } { s } . \\end{align*}"}
+{"id": "8273.png", "formula": "\\begin{align*} F ( n _ 1 , n _ 2 ) = \\sum _ { \\substack { \\sum _ { j = 0 } ^ { k - 1 } h _ { 1 j } = n _ 1 \\\\ \\sum _ { j = 0 } ^ { k - 1 } h _ { 2 j } = n _ 2 } } \\binom { n _ 1 } { h _ { 1 1 } , \\ldots , h _ { 1 k } } \\binom { n _ 2 } { h _ { 2 1 } , \\ldots , h _ { 2 k } } \\det \\Big ( I _ { 2 h _ { 1 j } + 3 h _ { 2 j } + i + j + 1 } ( 2 \\sqrt { x } ) \\Big ) _ { i , j = 0 , \\ldots , k - 1 } . \\end{align*}"}
+{"id": "3729.png", "formula": "\\begin{gather*} 1 : x ^ { 3 } = 0 , 2 : x ^ { 2 } y = 0 , 3 : x y ( x - y ) = 0 , 4 : x y z = 0 , \\\\ 5 : z ( x ^ { 2 } + y z ) = 0 , 6 : x ( x ^ { 2 } + y z ) = 0 , 7 : x ^ { 3 } - y ^ { 2 } z = 0 , 8 : x ^ { 3 } + y ^ { 3 } - x y z = 0 , \\end{gather*}"}
+{"id": "1624.png", "formula": "\\begin{align*} d \\eta ^ j ( \\xi _ i , \\ , \\cdot ) = 0 ; \\end{align*}"}
+{"id": "5114.png", "formula": "\\begin{align*} \\int _ { B _ { 2 s } ( x _ 0 ) } b ^ { i j , k l } v _ { i j } \\tau ^ { 4 } v _ { k l } d x & = \\int _ { B _ { 2 s } ( x _ 0 ) } b ^ { i j , k l } w _ { i j } ( \\tau ^ 4 v ) _ { k l } d x \\\\ & \\qquad - \\int _ { B _ { 2 s } ( x _ 0 ) } b ^ { i j , k l } v _ { i j } [ ( \\tau ^ { 4 } ) _ { k l } v + ( \\tau ^ { 4 } ) _ { l } v _ { k } + ( \\tau ^ { 4 } ) _ { k } v _ { l } ] d x . \\end{align*}"}
+{"id": "2939.png", "formula": "\\begin{align*} C ( \\gamma _ 1 , \\gamma _ 2 ) ( f ) = \\tilde { \\mu } _ \\delta \\{ \\gamma _ 1 i \\infty \\to \\gamma _ 2 i \\infty \\} ( \\phi ^ \\ast f ) . \\end{align*}"}
+{"id": "6068.png", "formula": "\\begin{align*} \\delta _ { 1 / r } ^ { A } \\delta _ { r } ^ { B } \\{ x \\in G \\colon \\| x \\| = 1 \\} \\subseteq \\mathcal { B } ^ { A } ( e , D ) . \\end{align*}"}
+{"id": "5124.png", "formula": "\\begin{align*} \\Delta _ g \\Theta = 0 , \\end{align*}"}
+{"id": "2254.png", "formula": "\\begin{align*} f _ \\nu ( z ) : = \\frac { 1 } { | z | ^ { \\nu } } \\ , f \\left ( \\frac { 1 } { z } \\right ) . \\end{align*}"}
+{"id": "2082.png", "formula": "\\begin{align*} N _ { d r } ( m + 1 ) - N _ { d r } ( m ) & = a ^ 2 + a d + \\left ( \\sum _ { i = 1 } ^ k x _ i ^ { \\prime } - \\sum _ { i = 1 } ^ k x _ i \\right ) a \\\\ & \\geq \\left ( a + d + \\sum _ { i = 1 } ^ { k - 1 } x _ i ^ { \\prime } - \\sum _ { i = 1 } ^ { k - 1 } x _ i \\right ) a \\\\ & \\geq ( a + d - k ) a \\geq 0 . \\end{align*}"}
+{"id": "2682.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\Big ( q _ i ( p - x _ { i } ) \\ss _ i ^ { m + 2 } \\Big ) = 0 . \\end{align*}"}
+{"id": "583.png", "formula": "\\begin{align*} h _ k ( s ) & = e ^ { - s ^ 2 / 2 } \\sum _ { m = 0 } ^ { [ k / 2 ] } d _ { k m } \\ s ^ { k - 2 m } , d _ { k m } = \\frac { 1 } { \\sqrt { 2 ^ k k ! \\sqrt { \\pi } } } \\frac { ( - 1 ) ^ m k ! 2 ^ { k - 2 m } } { m ! ( k - 2 m ) ! } \\\\ & = e ^ { - s ^ 2 / 2 } \\sum _ { m = 0 } ^ { k } c _ { k m } \\ s ^ { m } , \\end{align*}"}
+{"id": "5432.png", "formula": "\\begin{align*} s _ 1 ( T _ v ) & = \\sum _ { j , k = 1 } ^ n T _ v ( e _ j , \\bar e _ j , e _ k , \\bar e _ k ) = \\sum _ { j , k = 1 } ^ n R ( e _ j , \\bar e _ j , v , \\bar v ) R ( e _ k , \\bar e _ k , v , \\bar v ) = r ( v , \\bar v ) ^ 2 , \\\\ s _ 2 ( T _ v ) & = \\sum _ { j , k = 1 } ^ n T _ v ( e _ j , \\bar e _ k , e _ k , \\bar e _ j ) = \\sum _ { j , k = 1 } ^ n | R ( e _ j , \\bar e _ k , v , \\bar v ) | ^ 2 . \\end{align*}"}
+{"id": "5639.png", "formula": "\\begin{align*} \\mathcal { S } _ a : = \\{ u \\in H ^ 1 ( \\mathbb { R } ^ N ) : | u | _ 2 = a \\} , E ( u ) = \\frac 1 2 | \\nabla u | ^ 2 _ 2 - \\frac { 1 } { 2 2 ^ * _ \\mu } \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | u | ^ { 2 ^ * _ \\mu } ) | u | ^ { 2 ^ * _ \\mu } . \\end{align*}"}
+{"id": "1761.png", "formula": "\\begin{align*} L _ 1 = - \\nu _ 1 + \\partial _ x ^ 2 + U _ 1 + 2 a _ 1 \\phi _ 1 \\phi _ 1 ^ * + a _ 1 \\phi _ 2 \\phi _ 2 ^ * \\ , \\\\ L _ 2 = - \\nu _ 2 + \\partial _ x ^ 2 + U _ 2 + a _ 2 \\phi _ 1 \\phi _ 1 ^ * + 2 a _ 2 \\phi _ 2 \\phi _ 2 ^ * \\ , \\end{align*}"}
+{"id": "9302.png", "formula": "\\begin{align*} M : = \\left ( \\begin{array} { c c r c c r } 0 & 1 3 & - 1 3 & 9 & 1 0 & - 1 9 \\\\ 1 2 & 1 1 & - 1 1 & 7 & 8 & - 1 5 \\\\ 1 2 & 1 1 & - 1 1 & 7 & 8 & - 1 5 \\\\ 6 & 4 & 5 & 1 & 2 & - 3 \\\\ 6 & 4 & 5 & 1 & 2 & - 3 \\\\ 6 & 4 & 5 & 1 & 2 & - 3 \\\\ \\end{array} \\right ) . \\end{align*}"}
+{"id": "2312.png", "formula": "\\begin{align*} \\begin{aligned} - \\Delta u & = f , \\\\ u & = 0 , \\end{aligned} \\end{align*}"}
+{"id": "1115.png", "formula": "\\begin{align*} \\Phi _ { k , i } ( \\alpha _ 1 , \\ldots , \\alpha _ k ) = 0 , \\forall i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "5617.png", "formula": "\\begin{align*} { \\rm I } ( \\xi _ { 1 } , \\mathcal { T } | X ) = \\int _ { X } { \\rm I } ( \\xi _ { 1 } ^ { x } , \\mathcal { T } _ { x } ) d \\eta ( x ) = \\int _ { X } \\lim _ { n \\to \\infty } { \\rm I } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) d \\eta ( x ) = \\lim _ { n \\to \\infty } \\int _ { X } { \\rm I } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) d \\eta ( x ) \\end{align*}"}
+{"id": "3923.png", "formula": "\\begin{align*} \\pi _ n ( \\cdot ) = \\frac { \\pi \\left ( \\cdot \\cap \\left ( A _ { 1 n } \\right ) \\right ) } { \\pi \\left ( A _ { 1 n } \\right ) } . \\end{align*}"}
+{"id": "2228.png", "formula": "\\begin{align*} \\frac { \\lambda ( y ^ u , y ^ h ) - \\lambda ( y ^ { u - h } , y ^ h ) } { h e ^ { \\gamma } \\log y } & = \\int _ { 1 } ^ { u / h } y ^ { h t - u } \\omega ( t ) \\ , d t - \\int _ { 2 } ^ { u / h } y ^ { h t - u } \\omega ( t - 1 ) \\ , d t \\\\ & = \\int _ { 1 } ^ { 2 } y ^ { h t - u } \\omega ( t ) \\ , d t + \\int _ { 2 } ^ { u / h } y ^ { h t - u } ( \\omega ( t ) - \\omega ( t - 1 ) ) \\ , d t \\\\ & = \\int _ { 1 } ^ { 2 } t ^ { - 1 } y ^ { h t - u } \\ , d t - \\int _ { 2 } ^ { u / h } y ^ { h t - u } t \\omega ' ( t ) \\ , d t . \\end{align*}"}
+{"id": "8599.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } e ^ { - \\lambda \\tau _ N } S _ j ( \\tau _ N ) = \\nu Y \\int _ 0 ^ \\infty e ^ { - \\lambda s } p _ j ( s ) d s , \\end{align*}"}
+{"id": "1039.png", "formula": "\\begin{align*} \\beta _ k ( a , b , c , d ) & = \\frac { k ( 1 + 2 a - b - c - d + 2 k ) } { a } \\\\ [ 1 m m ] & \\quad + \\frac { ( a + k ) ( 1 + a - b - c + k ) ( 1 + a - b - d + k ) } { a ( 1 + a - b + 2 k ) } . \\end{align*}"}
+{"id": "6779.png", "formula": "\\begin{align*} \\langle E _ { \\lambda } ( x , q , 0 ) , E _ { \\mu } ( x ^ { - 1 } , q ^ { - 1 } , \\infty ) \\rangle _ 0 = ( q ) _ \\lambda \\delta _ { \\lambda , \\mu } , \\end{align*}"}
+{"id": "7612.png", "formula": "\\begin{align*} h ( z ) = \\int _ { 0 } ^ { z } \\frac { h _ { 0 } ( x ) } { x } d x = z + \\frac { \\sqrt { 6 9 } } { 1 2 \\sqrt { 1 7 } } z ^ 3 + \\frac { 1 } { 2 0 } \\left ( \\frac { 6 9 } { 1 3 6 } + \\frac { \\sqrt { 6 9 } } { 4 \\sqrt { 1 7 } } \\right ) z ^ 5 + \\cdots , \\end{align*}"}
+{"id": "9227.png", "formula": "\\begin{align*} \\texttt { w p r s s u m } = \\max \\Bigl \\{ \\texttt { m a g } ( 1 0 ( L + 1 ) ) , \\texttt { m a g } \\Bigl ( \\frac { 4 . 4 c \\sqrt { \\pi } } { \\varepsilon _ 2 b a } ( L + 3 ) ^ 2 \\Bigr ) \\Bigr \\} + 1 . \\end{align*}"}
+{"id": "3208.png", "formula": "\\begin{align*} & \\kappa _ 1 \\circ \\kappa _ n = w \\kappa _ { n + 1 } + ( 1 - w ) \\kappa _ { n - 1 } ~ a n d \\\\ & \\kappa _ 0 ( x ) = x ~ x \\in E . \\end{align*}"}
+{"id": "3682.png", "formula": "\\begin{align*} \\langle \\varphi , \\psi \\rangle : = & \\int _ { \\mathbb { R } ^ n } \\varphi \\psi \\ , d x , \\varphi , \\psi \\in L ^ 2 ( \\mathbb { R } ^ n ) , \\\\ ( \\varphi , \\psi ) _ { \\Omega } : = & \\int _ { \\Omega } \\varphi \\psi \\ , d x , \\varphi , \\psi \\in L ^ 2 ( \\Omega ) . \\end{align*}"}
+{"id": "4133.png", "formula": "\\begin{align*} M _ S = \\begin{pmatrix} 2 & 3 & 1 \\\\ 1 & 3 & 1 \\\\ 1 & 2 & 1 \\end{pmatrix} = M _ 2 ^ { - 2 } . \\end{align*}"}
+{"id": "5747.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { @ { \\ , } l l l } p ' y ^ 2 = z \\cdots \\mbox { ( 8 $ - $ 1 ) } \\\\ x ^ 2 = w \\cdots \\mbox { ( 8 $ - $ 2 ) } \\\\ p '^ 2 x ^ 6 = p p ' x ^ 3 + q a ^ 2 \\cdots \\mbox { ( 8 $ - $ 3 ) } \\\\ a ^ 4 = p p ' a x ^ 3 + q p '^ 2 x ^ 6 \\cdots \\mbox { ( 8 $ - $ 4 ) } \\\\ p ' x y = a \\cdots \\mbox { ( 8 $ - $ 5 ) } \\\\ a = c . \\cdots \\mbox { ( 8 $ - $ 6 ) } \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "6146.png", "formula": "\\begin{align*} \\d ( x _ 0 , x _ 1 ) - \\d ( x _ 0 , y _ 1 ) & = n ( x _ 1 ^ { n + 1 } - y _ 1 ^ { n + 1 } ) - ( n + 1 ) x _ 0 ( x _ 1 ^ n - y _ 1 ^ n ) \\\\ & = ( x _ 1 - y _ 1 ) \\sum _ { j , k } ( x _ 1 ^ j y _ 1 ^ { n - j } - x _ 0 x _ 1 ^ k y _ 1 ^ { n - k - 1 } ) . \\end{align*}"}
+{"id": "165.png", "formula": "\\begin{align*} L ( \\gamma ) + \\frac { 1 } { 2 } L ( \\gamma ^ 2 ) = \\frac { 4 } { 7 } . \\end{align*}"}
+{"id": "25.png", "formula": "\\begin{align*} z _ 0 = ( \\Phi ( b _ { n + 1 } ) \\cdots \\Phi ( b _ { 2 n } ) ) ^ { - 1 } ( \\Phi ( b _ 1 ) \\cdots \\Phi ( b _ n ) ) \\in \\mathbf { X } . \\end{align*}"}
+{"id": "6866.png", "formula": "\\begin{align*} \\lambda _ N = \\| L _ N \\| = \\sup _ { \\substack { u \\in L ^ 2 ( [ 0 , 1 ] ) \\\\ \\| u \\| _ 2 = 1 } } \\| L _ N u \\| _ 2 . \\end{align*}"}
+{"id": "8081.png", "formula": "\\begin{align*} \\mu _ n = \\mu _ n G \\left ( u _ n \\right ) = \\mu _ n \\left \\langle G ^ { \\prime } ( u _ n ) , u _ n \\right \\rangle = \\left \\langle F ^ { \\prime } ( u _ n ) , u _ n \\right \\rangle = F \\left ( u _ n \\right ) = \\beta _ n . \\end{align*}"}
+{"id": "8572.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { - 1 } S _ j ( \\tau _ N ) = \\frac { \\nu } { \\lambda } \\frac 1 { j ( j + 1 ) } . \\end{align*}"}
+{"id": "6049.png", "formula": "\\begin{align*} A r ^ { n + m } + B r ^ m + C = 0 \\textrm { a n d } ( n + m ) A r ^ { n + m - 1 } + m B r ^ { m - 1 } = 0 , \\end{align*}"}
+{"id": "2359.png", "formula": "\\begin{align*} ( N / M ) _ f = ( N / M ) ^ { D ( f ) } . \\end{align*}"}
+{"id": "6097.png", "formula": "\\begin{align*} \\sup _ { \\substack { \\pi , \\\\ \\pi = \\lbrace \\kappa _ 0 = s , \\kappa _ { 1 } , . . . \\kappa _ { m } = t \\rbrace } } \\big { \\lbrace } \\sum _ { j } \\big { [ } | h _ { \\kappa _ { j } , \\kappa _ { j + 1 } } | ^ { \\frac { 1 } { { \\gamma } ^ { \\prime } } } \\big { ] } \\big { \\rbrace } < \\infty . \\end{align*}"}
+{"id": "6688.png", "formula": "\\begin{align*} \\partial _ { l } \\div \\eta & = ( \\delta _ { k j } - a _ { k j } ) \\partial _ { l k } \\eta _ j + \\int _ 0 ^ t ( \\partial _ t a _ { k j } \\partial _ { l k } \\eta _ j + \\partial _ l a _ { k j } \\partial _ k v _ j ) \\ , \\d s , \\end{align*}"}
+{"id": "7953.png", "formula": "\\begin{align*} I [ u ] = \\int _ \\Omega B \\big ( | \\nabla u | \\big ) \\ , d x - \\int _ \\Omega f u \\ , d x \\end{align*}"}
+{"id": "2068.png", "formula": "\\begin{align*} \\mu _ D \\Big ( \\mu _ C \\big ( Q _ 1 ( x , y , z , w ) \\big ) \\Big ) & = Q _ 1 \\big ( \\alpha ( \\alpha x - y ) - x , \\alpha x - y , \\alpha ( \\alpha z - w ) - z , \\alpha z - w \\big ) \\\\ & : = Q _ 1 ( x ' , y ' , z ' , w ' ) \\end{align*}"}
+{"id": "4028.png", "formula": "\\begin{align*} L ( f , s ) = \\sum _ { n \\geq 1 } \\frac { a _ f ( n ) } { n ^ s } \\mathrm { R e } ( s ) \\gg 0 . \\end{align*}"}
+{"id": "8545.png", "formula": "\\begin{align*} \\gamma _ { p } : = C _ { p } ^ { ( 2 ) } + \\frac { C _ { p } ^ { ( 1 ) } } { 1 + \\frac { 1 } { \\pi p } } - \\gamma - \\frac { 2 e ^ { 2 \\pi p } \\ , Q _ { 2 \\pi p } ( 0 ) } { 1 + \\frac { 1 } { \\pi p } } - \\log ( 2 \\pi ) \\left ( 1 - \\frac { 1 } { 1 + \\frac { 1 } { \\pi p } } \\right ) , \\end{align*}"}
+{"id": "7485.png", "formula": "\\begin{align*} \\pi = \\alpha _ 1 \\alpha _ 2 \\cdots \\alpha _ k \\underset { \\beta } { \\underbrace { \\delta \\pi _ j \\cdots \\pi _ { l - 1 } x \\pi _ { l + 1 } \\cdots \\pi _ n } } . \\end{align*}"}
+{"id": "4340.png", "formula": "\\begin{align*} \\mathrm { d o m } \\left ( T _ { 1 } \\cap \\left ( - T _ { 2 } \\right ) \\right ) = \\left \\{ x \\in X \\mid 0 \\in \\left ( T _ { 1 } + T _ { 2 } \\right ) \\left ( x \\right ) \\right \\} , \\end{align*}"}
+{"id": "8464.png", "formula": "\\begin{align*} K _ { n - \\frac { 1 } { 2 } } ( x ) = \\sqrt { \\frac { \\pi } { 2 x } } \\ , e ^ { - x } \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( n - 1 + k ) ! } { k ! \\ , ( n - 1 - k ) ! \\ , ( 2 x ) ^ { k } } , \\ , \\ , \\ , \\ , n \\in \\mathbb { N } , \\end{align*}"}
+{"id": "5368.png", "formula": "\\begin{align*} \\alpha ( v + w ) = M _ 1 ^ H z , - \\beta ( v - w ) = M _ 2 ^ H z \\end{align*}"}
+{"id": "7674.png", "formula": "\\begin{align*} \\Phi ( t ) = \\left ( s ( 2 \\coth ( \\kappa t ) - 2 ) \\phi ' ( t ) + s ^ 2 ( 1 - 2 \\coth ( \\kappa t ) ) \\phi ( t ) \\right ) ^ 2 \\left ( e ^ { - \\kappa t } \\sinh ( \\kappa t ) \\right ) ^ { n - 1 } , \\end{align*}"}
+{"id": "329.png", "formula": "\\begin{align*} \\prod _ { \\substack { \\gcd ( j _ 1 , j _ 2 , j _ 3 , k ) = 1 \\\\ j _ 1 , j _ 2 , j _ 3 < k \\\\ j _ 1 , j _ 2 , j _ 3 \\geq 1 ; k \\geq 2 } } \\left ( \\frac { 1 } { 1 - y ^ { j _ 1 + j _ 2 + j _ 3 } z ^ k } \\right ) ^ { \\frac { 1 } { k } } = \\exp \\left [ \\sum _ { k = 1 } ^ { \\infty } \\left ( \\frac { y } { 1 } + \\frac { y ^ 2 } { 2 } + \\cdots + \\frac { y ^ k } { k } \\right ) ^ { 3 } \\frac { z ^ { k + 1 } } { ( k + 1 ) ^ 2 } \\right ] . \\end{align*}"}
+{"id": "7380.png", "formula": "\\begin{align*} R _ N ^ 2 ( L , \\alpha , \\Delta ) = \\frac { L } { N ^ 2 } \\sum _ { n \\in \\mathbb { Z } } \\widehat { \\Delta } \\left ( \\frac { L n } { N } \\right ) \\sum _ { 1 \\leq i \\neq j \\leq N } e ( n \\alpha ( a _ i - a _ j ) ) ; \\end{align*}"}
+{"id": "3368.png", "formula": "\\begin{align*} U \\xi _ { \\nu } = U \\pi _ { \\nu } ( f ) \\xi _ { \\nu } = \\pi _ { \\mu } U \\xi _ { \\nu } . \\end{align*}"}
+{"id": "6399.png", "formula": "\\begin{align*} \\lambda ^ { \\varphi } ( t ) a = \\sigma ^ { \\varphi } _ t ( a ) \\lambda ^ { \\varphi } ( t ) , t \\in \\mathbb { R } , a \\in M . \\end{align*}"}
+{"id": "1731.png", "formula": "\\begin{align*} \\mu _ t ( y ) = e ^ { - \\alpha t } \\mu _ 0 ( y ) + \\int _ 0 ^ t \\alpha e ^ { - \\alpha ( t - s ) } \\Phi ( \\nu _ s , \\mu _ s ) ( y ) \\mathrm { d } s . \\end{align*}"}
+{"id": "5003.png", "formula": "\\begin{align*} \\begin{aligned} \\Pr ( X _ t \\le k \\ , \\vert \\ , X _ 0 = r ) & = \\frac { n - k } { n ^ t } \\binom { n - r } { n - k } \\sum _ { j = 0 } ^ { k - r } ( - 1 ) ^ { k - r - j } \\binom { k - r } { j } \\frac { ( r + j ) ^ t } { n - r - j } \\\\ \\end{aligned} \\end{align*}"}
+{"id": "3136.png", "formula": "\\begin{align*} R _ { m } ( u , x ) = \\sum _ { k = 0 } ^ m Q _ { m + 1 , k } ( x - m - 1 ) ( 1 - u ) ^ { m - k } . \\end{align*}"}
+{"id": "624.png", "formula": "\\begin{align*} \\lim _ \\lambda \\| b e _ \\lambda \\| = 0 , b \\in L _ \\omega . \\end{align*}"}
+{"id": "3814.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } _ { + } ^ 2 } \\left [ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\varpi \\in \\Pi ( \\mu _ 1 , \\mu _ 2 ) } \\int _ { \\mathcal { V } } g _ \\lambda \\ , d \\varpi \\right ] \\end{align*}"}
+{"id": "3706.png", "formula": "\\begin{align*} \\alpha _ t ( g H ) = \\exp ( t h ) g H . \\end{align*}"}
+{"id": "56.png", "formula": "\\begin{align*} X _ { t g } = t X _ g . \\end{align*}"}
+{"id": "7850.png", "formula": "\\begin{align*} V = \\{ ( x H , y H ) : \\ x H \\in \\Omega \\setminus \\mathcal { S } \\mbox { a n d } y H \\in \\mathcal { S } \\} . \\end{align*}"}
+{"id": "7106.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n - 1 } } \\int _ { \\mathbb { R } ^ { n - 1 } } a ( x - y ) \\ : b ( ( 1 - t ) y + t x ) \\ : \\textup { d } x \\textup { d } y = \\| a \\| _ { L ^ 1 ( \\mathbb { R } ^ { n - 1 } ) } \\| b \\| _ { L ^ 1 ( \\mathbb { R } ^ { n - 1 } ) } \\end{align*}"}
+{"id": "2553.png", "formula": "\\begin{align*} \\overline { \\operatorname { S p G } _ N } = \\overline { G _ M \\cap \\mathcal H } , \\end{align*}"}
+{"id": "5649.png", "formula": "\\begin{align*} \\nu _ 0 : = \\frac { 1 } { \\gamma _ p + \\gamma _ q } \\frac { S ^ { \\frac { ( 2 - \\gamma _ p - \\gamma _ q ) 2 ^ * _ \\mu } { 2 2 ^ * _ \\mu - 2 } } _ { H , L } } { C _ { N , p , q } } \\frac { ( 2 2 ^ * _ \\mu - 2 ) ( 2 - \\gamma _ p - \\gamma _ q ) ^ { \\frac { 2 - \\gamma _ p - \\gamma _ q } { 2 2 ^ * _ \\mu - 2 } } } { ( 2 2 ^ * _ \\mu - \\gamma _ p - \\gamma _ q ) ^ { \\frac { 2 2 ^ * _ \\mu - \\gamma _ p - \\gamma _ q } { 2 2 ^ * _ \\mu - 2 } } ( a ^ 2 + b ^ 2 ) ^ { \\frac { p + q - \\gamma _ p - \\gamma _ q } { 2 } } } , \\end{align*}"}
+{"id": "1305.png", "formula": "\\begin{align*} d X ( t ) = d B ( t ) + \\frac { 1 } { X ( t ) } d t - \\frac { X ( t ) } { \\tau - t } d t , \\end{align*}"}
+{"id": "8320.png", "formula": "\\begin{align*} \\sup _ { j , k \\leq n } \\| S _ k - S _ j \\| \\leq 2 \\sup _ j \\| S _ j \\| = 2 U _ n . \\end{align*}"}
+{"id": "3914.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mathrm { D } } ( \\delta ) = \\inf _ { \\lambda \\in \\mathbb { R } ^ 2 _ + } \\left \\{ \\langle \\lambda , \\delta \\rangle + \\sup _ { \\gamma \\in \\Pi \\left ( \\mu _ 1 , \\mu _ 2 \\right ) } \\int _ { \\mathcal { V } } g _ \\lambda \\ , d \\gamma \\right \\} . \\end{align*}"}
+{"id": "6095.png", "formula": "\\begin{align*} & Z ^ { \\prime } _ { s } = G ( Z _ s ) Z ^ { \\# } _ { s , t } = Z _ { s , t } - G ( Z _ s ) \\circ ( \\delta X ) _ { s , t } . \\end{align*}"}
+{"id": "7947.png", "formula": "\\begin{align*} i _ b = \\inf _ { t > 0 } \\frac { t \\ , b ' ( t ) } { b ( t ) } \\quad s _ b = \\sup _ { t > 0 } \\frac { t \\ , b ' ( t ) } { b ( t ) } , \\end{align*}"}
+{"id": "5142.png", "formula": "\\begin{align*} \\| \\iota ( g ) \\| _ { X ^ \\ast } = \\sup _ { \\| f \\| _ X = 1 } \\left | \\int _ \\Omega \\ ! f g \\ , \\mathrm { d } \\mu \\right | \\leq \\sup _ { \\| f \\| _ X = 1 } \\int _ \\Omega \\ ! | f g | \\ , \\mathrm { d } \\mu = \\| g \\| _ { X ' } , \\end{align*}"}
+{"id": "5602.png", "formula": "\\begin{align*} \\overline { \\mathbb { P } } _ { \\mu , x , g } | _ { \\mathcal { I } _ { x } } = \\theta ( x , \\cdot ) _ { \\ast } \\overline { \\mathbb { P } } _ { \\mu , x , g } = ( g \\lambda ) _ { x } . \\end{align*}"}
+{"id": "5408.png", "formula": "\\begin{align*} | S ( n , \\chi ) | = q ^ { n / 2 } | [ u ^ n ] L ( u / \\sqrt { q } , \\chi ) | & \\le q ^ { n / 2 } [ u ^ n ] \\exp \\bigg ( \\sum _ { k \\ge 1 } \\frac { u ^ k } { k } \\min \\{ q ^ { k / 2 } , d ( \\chi ) \\} \\bigg ) \\\\ & \\le q ^ { n / 2 } \\exp \\bigg ( \\sum _ { k \\ge 1 } \\frac { R ^ k } { k } \\min \\{ q ^ { k / 2 } , d ( \\chi ) \\} \\bigg ) R ^ { - n } \\end{align*}"}
+{"id": "8855.png", "formula": "\\begin{align*} N _ k = \\sum _ { i = 1 } ^ { k - 2 } N _ { i } + 2 \\end{align*}"}
+{"id": "2790.png", "formula": "\\begin{align*} ( \\Delta + \\mu - q ) u = f \\ ; \\mathrm { i n } \\ ; M , ( \\partial _ \\nu u \\mp i a ) u = \\varphi \\ ; \\mathrm { o n } \\ ; \\partial M , \\end{align*}"}
+{"id": "3006.png", "formula": "\\begin{align*} { \\ , } ^ g \\phi ( _ h f ) & = \\phi \\left ( ^ { g ^ { - 1 } } ( _ h f ) \\right ) \\\\ & = \\phi \\left ( _ h \\left ( ^ { g ^ { - 1 } } f \\right ) \\right ) \\\\ & = \\phi \\left ( ^ { g ^ { - 1 } } f \\right ) \\\\ & = { \\ , } ^ g \\phi ( f ) . \\end{align*}"}
+{"id": "8533.png", "formula": "\\begin{align*} \\frac { 2 \\sqrt { \\pi } \\ , c ^ { \\frac { 1 } { 2 } - s } } { \\Gamma ( s ) } \\cdot \\frac { \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) } { 1 + \\frac { 1 } { \\pi p } } \\ , \\zeta _ { p ^ { \\prime } } ( 2 s - 1 ) = \\frac { \\pi } { \\sqrt { c } \\left ( 1 + \\frac { 1 } { \\pi p } \\right ) } \\ , \\frac { 1 } { s - 1 } + \\frac { \\pi } { \\sqrt { c } \\left ( 1 + \\frac { 1 } { \\pi p } \\right ) } \\left ( 2 C _ { p ^ { \\prime } } ^ { ( 1 ) } - \\log \\left ( 4 c \\right ) \\right ) + O \\left ( s - 1 \\right ) . \\end{align*}"}
+{"id": "5883.png", "formula": "\\begin{align*} ( n _ 2 - n _ 1 ) \\ , \\sum _ { k = \\ell } ^ { n _ 1 - 1 } \\nu _ k \\ , = \\ , ( n _ 1 - \\ell ) \\ , \\sum _ { k = n _ 1 } ^ { n _ 2 - 1 } \\nu _ k \\ , . \\end{align*}"}
+{"id": "3644.png", "formula": "\\begin{align*} Q ( G , Z , \\lambda , \\mathcal { G } _ w ) : = \\sum _ { ( w _ { \\mathbf { U } } , \\mathbf { U } ) \\in \\mathcal { G } _ w } e ^ { - \\lambda \\mathfrak { l } ( \\mathbf { U } ) } < p < 1 . \\end{align*}"}
+{"id": "9311.png", "formula": "\\begin{align*} H _ { n , d } ( s ) = ( s ^ { \\mathcal { I } - e _ j + d e _ k } ) , j , k = 1 , \\ldots , n , \\end{align*}"}
+{"id": "2719.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } z - ( y ^ { \\alpha _ y } \\partial _ { x x } z + x ^ { \\alpha _ x } \\partial _ { y y } z ) = \\chi _ { \\omega } g , & \\mbox { i n } \\ Q , \\\\ z ( x , y , t ) = 0 \\ \\mbox { o r } \\ A \\nabla z \\cdot \\nu = 0 , & \\mbox { o n } \\ \\Sigma , \\\\ z ( x , y , 0 ) = z _ { 0 } ( x , y ) , & \\mbox { i n } \\ \\Omega , \\end{cases} \\end{align*}"}
+{"id": "7523.png", "formula": "\\begin{align*} \\Theta ( i ) = W _ F ( i , 0 ) + W _ F ( i , 1 ) + \\hdots + W _ F ( i , i - 1 ) . \\end{align*}"}
+{"id": "8205.png", "formula": "\\begin{align*} P ^ \\omega \\left ( A _ t ^ c \\mid E _ t \\right ) \\leq P \\left ( \\big | Z _ s ^ { B _ s } \\big | < e ^ { \\delta t } \\right ) \\leq P \\left ( \\big | Z _ s ^ { B _ s } \\big | < e ^ { - \\sqrt { \\beta / 2 } \\ , \\hat { r } ( s ) } p _ s e ^ { \\beta s } \\right ) = e ^ { - \\sqrt { \\beta / 2 } \\ , \\hat { r } ( s ) ( 1 + o ( 1 ) ) } , \\end{align*}"}
+{"id": "8697.png", "formula": "\\begin{align*} \\frac { \\textup { M M D } _ { n , m } ^ { 2 } } { \\surd { \\textup { v a r } ( \\Delta _ { 1 } ) } } = \\frac { \\Delta _ 0 + T _ 1 + \\Delta _ 1 + \\{ \\widetilde { \\Delta } _ { 2 } - E ( \\widetilde { \\Delta } _ { 2 } ) \\} + \\{ E ( \\widetilde { \\Delta } _ { 2 } ) - T _ 1 \\} } { \\surd { ( \\Delta _ 1 ) } } . \\end{align*}"}
+{"id": "7802.png", "formula": "\\begin{align*} \\textsf { P } \\{ \\sum _ { i = 1 } ^ { n } a _ { i } \\eta _ { i } \\ge t \\} \\le \\exp \\big ( - c \\min ( \\frac { t ^ { 2 } } { K ^ { 2 } \\Vert a \\Vert _ { 2 } ^ { 2 } } , \\frac { t } { K \\Vert a \\Vert _ { \\infty } } ) \\big ) , \\end{align*}"}
+{"id": "8528.png", "formula": "\\begin{align*} \\lim _ { p , p ^ { \\prime } \\rightarrow 0 ^ { + } } \\tilde { \\zeta } _ { p , p ^ { \\prime } } ( s , c ) = \\sum _ { m \\ne 0 , n \\neq 0 } \\frac { ( - 1 ) ^ { m } } { \\left ( m ^ { 2 } + c \\left ( n - \\frac { 1 } { 2 } \\right ) ^ { 2 } \\right ) ^ { s } } . \\end{align*}"}
+{"id": "2606.png", "formula": "\\begin{align*} \\frac { b _ n } { b _ i } \\geq C _ \\epsilon h _ 0 ^ { \\epsilon - \\eta } \\rightarrow \\infty \\ \\ h _ 0 \\rightarrow 0 \\ \\ i = 1 , \\cdots , n - 1 . \\end{align*}"}
+{"id": "7465.png", "formula": "\\begin{align*} \\Phi _ { \\mathfrak { p } } ^ { ( i ) } ( x _ i , \\bar { \\xi } _ i , t ) : = \\big ( w ^ { ( i ) } _ { \\mathfrak { p } - 1 } + u ^ { ( i ) } _ { \\mathfrak { p } - 1 } \\big ) \\ , \\overline { V } ^ { ( i ) } \\boldsymbol { \\boldsymbol { \\cdot } } \\bar { \\nu } _ { \\bar { \\xi } _ i } , \\end{align*}"}
+{"id": "9193.png", "formula": "\\begin{align*} h = \\psi ( g ^ { 1 } , \\ldots , g ^ { l } ) \\ , . \\end{align*}"}
+{"id": "5561.png", "formula": "\\begin{align*} p _ { v } ( B ) : = \\mathbb { P } _ { \\mu } \\left ( B \\cap \\left [ v \\omega _ { t } ; \\omega _ { t } ^ { - 1 } \\omega _ { t + 1 } \\right ] = \\emptyset \\mbox { f o r a l l } t \\in \\mathbb { N } \\right ) > 0 . \\end{align*}"}
+{"id": "5407.png", "formula": "\\begin{align*} \\left | \\sum _ { f \\in \\mathcal { M } _ { k , q } } \\chi ( f ) \\Lambda ( f ) \\right | \\le \\sum _ { f \\in \\mathcal { M } _ { k , q } } \\Lambda ( f ) = q ^ k \\end{align*}"}
+{"id": "3042.png", "formula": "\\begin{align*} U A _ { s } ( \\mathrm { i , j } ) V = I _ { 2 ^ { s - 1 } } \\oplus 0 _ { N - 2 ^ { s - 1 } } \\hbox { a n d } A _ { s } ( \\mathrm { i , j } ) \\perp B _ s ( \\mathrm { i , j } ) . \\end{align*}"}
+{"id": "4972.png", "formula": "\\begin{align*} \\overline { F } ( k , t + 1 ; p , n ) = \\begin{dcases} \\frac { ( n - k ) p } { n } \\overline { F } ( k - 1 , t ; p , n ) + \\frac { n - ( n - k ) p } { n } \\overline { F } ( k , t ; p , n ) & \\textnormal { i f } \\ ; k \\le t ; \\\\ 0 & \\textnormal { i f } \\ ; k > t . \\end{dcases} \\end{align*}"}
+{"id": "3112.png", "formula": "\\begin{align*} t ^ n = \\sum _ { k = 0 } ^ { n } S ( n , k ) ( t ) _ k , \\end{align*}"}
+{"id": "8796.png", "formula": "\\begin{align*} & ( u ^ 3 + 4 t u ^ 2 + 6 t ^ 2 u + 1 1 u ^ 2 + 2 8 t u + 6 t ^ 2 + 3 8 u + 2 4 t + 2 8 ) b _ 2 \\\\ & = ( u ^ 3 + 8 t u ^ 2 + 1 2 t ^ 2 u + 7 u ^ 2 + 3 8 t u + 1 2 t ^ 2 + 1 8 u + 3 0 t + 1 2 ) c _ 2 \\\\ & + ( u ^ 3 + 2 t u ^ 2 + 2 t u + 9 u ^ 2 + 8 u ) b _ 3 - ( u ^ 3 + 4 t u ^ 2 + 4 t u + 3 u ^ 2 + 2 u ) c _ 3 \\end{align*}"}
+{"id": "7916.png", "formula": "\\begin{align*} \\Pi _ { x e _ \\mathbf { n } } = ( \\cdot - x ) ^ \\mathbf { n } . \\end{align*}"}
+{"id": "5071.png", "formula": "\\begin{align*} \\tilde { v } ( t ) : = \\psi \\left ( t \\right ) - \\phi , \\end{align*}"}
+{"id": "8825.png", "formula": "\\begin{align*} f _ n ( x ) : = f ( 0 ) + \\int _ 0 ^ x f ' ( y ) \\chi _ n ( y ) \\ , d y , \\end{align*}"}
+{"id": "9109.png", "formula": "\\begin{align*} y _ { [ A , B ] } = ( y _ { [ a ^ { 1 } , b ^ { 1 } ] } ^ { 1 } , \\ldots , y _ { [ a ^ { m } , b ^ { m } ] } ^ { m } ) \\end{align*}"}
+{"id": "4309.png", "formula": "\\begin{align*} t _ { k , m } = \\sum _ { ( k ' , m ' ) < _ \\mathrm { t i m e } ( k , m ) } 2 ^ { - 2 k ' } < 1 , \\end{align*}"}
+{"id": "8707.png", "formula": "\\begin{align*} r ( - a ) = \\frac { ( - 1 ) ^ { l + 1 } } { l ! } \\int ^ { 0 } _ { - a } h ^ { ( l + 1 ) } ( t ) ( a + t ) ^ l d t . \\end{align*}"}
+{"id": "4924.png", "formula": "\\begin{align*} \\overline { F } ( k , t + 1 , \\mathbf { p } _ { n + 1 } ) = \\begin{dcases} p _ { k } ^ { } \\overline { F } ( k - 1 , t , \\mathbf { p } _ { n + 1 } ) + ( 1 - p _ { k } ^ { } ) \\overline { F } ( k , t , \\mathbf { p } _ { n + 1 } ) & \\textnormal { i f } \\ ; k \\le t ; \\\\ 0 & \\textnormal { i f } \\ ; k > t ; \\end{dcases} \\end{align*}"}
+{"id": "7898.png", "formula": "\\begin{align*} ( S , \\underline { S } ) ^ + & : = \\overline { ( ( a _ 1 , a _ 2 , \\ldots , a _ k ) , ( b _ 1 , b _ 2 , \\ldots , b _ { n - k } ) ) } , \\\\ ( S , \\underline { S } ) ^ - & : = \\overline { ( ( a _ 2 , a _ 1 , \\ldots , a _ k ) , ( b _ 1 , b _ 2 , \\ldots , b _ { n - k } ) ) } . \\end{align*}"}
+{"id": "7031.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ s b _ j \\textbf { X } ^ { \\lambda _ j } . \\end{align*}"}
+{"id": "6083.png", "formula": "\\begin{align*} \\exp ( Q _ { j } ) = \\exp ( A ) ^ j \\exp ( B ) ^ { - \\lfloor \\varepsilon j \\rfloor - d _ { j } } , j \\geq 1 , \\end{align*}"}
+{"id": "7473.png", "formula": "\\begin{align*} \\varphi _ x ( \\pi ) \\coloneqq \\begin{cases} w _ 1 w _ 4 x w _ 2 w _ 5 , & x \\pi , \\\\ \\pi , & x \\pi . \\end{cases} \\end{align*}"}
+{"id": "5647.png", "formula": "\\begin{align*} \\aligned | \\nabla \\tilde { u } _ n | ^ 2 _ 2 + | \\nabla \\tilde { v } _ n | ^ 2 _ 2 = \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | \\tilde { u } _ n | ^ { 2 ^ * _ \\mu } ) | \\tilde { u } _ n | ^ { 2 ^ * _ \\mu } + \\int _ { \\mathbb { R } ^ N } ( I _ \\mu \\ast | \\tilde { v } _ n | ^ { 2 ^ * _ \\mu } ) | \\tilde { v } _ n | ^ { 2 ^ * _ \\mu } + o _ n ( 1 ) . \\endaligned \\end{align*}"}
+{"id": "7511.png", "formula": "\\begin{align*} v ( z ) | _ { W ^ i } = \\begin{pmatrix} p ^ i ( z ) ^ { - 1 } & 0 \\\\ 0 & p ^ i ( z ) \\prod _ { j \\neq i , j \\in I _ w \\cup I _ g } p ^ j ( z ) ^ { a _ { j i } } \\prod _ { j \\neq i , j \\in I _ b } p ^ j ( z ) ^ { a _ { j i } / 2 } \\end{pmatrix} \\begin{pmatrix} 1 & - \\frac { q ^ i _ { - } ( z ) } { q ^ i _ { + } ( z ) } \\\\ 0 & 1 \\end{pmatrix} \\end{align*}"}
+{"id": "194.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b , c , d , e ) = 1 \\\\ a , b , c , d , e \\geq 1 } } \\left ( \\frac { 1 } { 1 - v ^ a w ^ b x ^ c y ^ d z ^ e } \\right ) ^ { \\frac { 1 } { a ^ q b ^ r c ^ s d ^ t e ^ u } } \\end{align*}"}
+{"id": "9139.png", "formula": "\\begin{align*} \\begin{array} { c c l } z ^ { + } & = & \\psi _ { c , [ 1 ] } ( x , z , v _ { [ 0 , R - A ] } ) \\\\ u & = & F _ { u } \\circ \\Psi ( x , z , v _ { [ 0 , R - A ] } ) \\end{array} \\end{align*}"}
+{"id": "2509.png", "formula": "\\begin{align*} \\| w \\| & = \\| A A ^ { - 1 } w \\| \\le \\| A ^ \\frac { 1 } { 2 } \\| \\cdot \\| A ^ \\frac { 1 } { 2 } A ^ { - 1 } w \\| = \\| A ^ { - 1 } w \\| _ A , \\\\ \\| w _ c \\| & = \\| A _ c ^ { - \\frac { 1 } { 2 } } A _ c ^ \\frac { 1 } { 2 } w _ c \\| \\le \\| A _ c ^ { - \\frac { 1 } { 2 } } \\| \\cdot \\| A _ c ^ \\frac { 1 } { 2 } w _ c \\| = \\kappa _ c ^ \\frac { 1 } { 2 } \\| w _ c \\| _ { A _ c } , \\end{align*}"}
+{"id": "1938.png", "formula": "\\begin{align*} | P _ n ( z _ 1 , \\ldots , z _ m ) | < \\frac { \\binom { n - 1 } { m - 1 } } { \\binom { n - 1 } { m - 1 } n ! } \\max \\{ 1 , R \\} ^ { n } = \\frac { \\max \\{ 1 , R \\} ^ { n } } { n ! } , \\end{align*}"}
+{"id": "5780.png", "formula": "\\begin{align*} \\| \\mathcal { M } _ { \\alpha } ( \\vec { f } ) \\| _ { L ^ { q , \\infty } ( v _ { \\vec { w } } ^ { q } ) } & \\le C \\prod _ { j = 1 } ^ { m } \\| f _ { j } \\| _ { L ^ { p _ { j } } ( w _ { j } ^ { p _ { j } } ) } . \\end{align*}"}
+{"id": "5981.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { \\psi } ( g ) = \\Pi _ { \\psi } ( g ) m _ { X ^ { \\ast } } ( g ) . \\end{align*}"}
+{"id": "3392.png", "formula": "\\begin{align*} \\int _ { \\frac { - 1 } { 2 \\log T _ k } } ^ { \\frac { 1 } { 2 \\log T _ k } } p ^ { - i t } \\ , d t & = \\frac { 2 \\sin \\bigl ( \\frac { \\log p } { 2 \\log T _ k } \\bigr ) } { \\log p } , \\ , \\ , \\int _ { \\frac { - 1 } { 2 \\log T _ k } } ^ { \\frac { 1 } { 2 \\log T _ k } } p ^ { - 2 i t } \\ , d t = \\frac { 1 } { \\log T _ k } + O \\biggl ( \\frac { ( \\log p ) ^ 2 } { ( \\log T _ k ) ^ 3 } \\biggr ) . \\end{align*}"}
+{"id": "957.png", "formula": "\\begin{align*} \\int \\limits _ E u \\circ f ( x ) | J ( x , f ) | ~ d m ( x ) = \\int \\limits _ { \\mathbb R ^ n \\setminus \\varphi ( S ) } u ( y ) N _ f ( E , y ) ~ d m ( y ) \\end{align*}"}
+{"id": "4558.png", "formula": "\\begin{align*} P _ { x , C } & = ( 4 ^ 2 , 1 ^ 2 ) , & P _ { y + z , C } & = ( 4 , 2 ^ 3 ) , & P _ { y , C } & = ( 3 ^ 2 , 2 ^ 2 ) , & P _ { x ^ 2 , C } & = ( 2 ^ 4 , 1 ^ 2 ) , & P _ { z , C } & = ( 2 ^ 3 , 1 ^ 4 ) . \\end{align*}"}
+{"id": "5420.png", "formula": "\\begin{align*} r _ n ^ { [ 3 ] } ( h ) : = \\frac { 2 ^ { 3 p + 3 } \\cdot y _ { 8 n } \\left ( \\frac { h } { 8 } \\right ) - 7 \\cdot 2 ^ { 2 p + 1 } \\cdot y _ { 4 n } \\left ( \\frac { h } { 4 } \\right ) + 7 \\cdot 2 ^ { p } \\cdot y _ { 2 n } \\left ( \\frac { h } { 2 } \\right ) - y _ n ( h ) } { \\left ( 2 ^ p - 1 \\right ) \\left ( 2 ^ { p + 1 } - 1 \\right ) \\left ( 2 ^ { p + 2 } - 1 \\right ) } . \\end{align*}"}
+{"id": "4513.png", "formula": "\\begin{align*} I _ { \\omega , \\mathbf { c } } ( \\Phi ) = \\frac { d } { d \\lambda } S _ { \\omega , \\mathbf { c } } ( \\Phi ^ { \\lambda } ) | _ { \\lambda = 1 } \\end{align*}"}
+{"id": "7060.png", "formula": "\\begin{align*} \\mathcal J _ 1 = \\left ( \\left . b _ { \\ell i } \\left ( X _ \\ell - \\sum \\limits _ { j = 1 } ^ { s _ { \\ell i } } b _ { \\ell i j } { \\textbf { X } } ^ { \\lambda _ j } \\right ) \\ \\right | \\ i , \\ell \\in I \\setminus \\{ i _ { } \\} , i < \\ell \\right ) \\end{align*}"}
+{"id": "3484.png", "formula": "\\begin{align*} P _ { \\phi } = \\prod _ { i : v _ i \\in V _ E } P _ i \\prod _ { i : v _ i \\in V _ O } P _ i \\prod _ { i : v _ i \\in V _ O } P _ i \\prod _ { i : v _ i \\in V _ E } P _ i . \\end{align*}"}
+{"id": "7799.png", "formula": "\\begin{align*} \\| \\xi \\| _ { \\psi _ { 1 } } = \\inf \\{ K > 0 : \\textsf { E } \\exp ( \\frac { | \\xi | } { K } ) \\le 2 \\} . \\end{align*}"}
+{"id": "2404.png", "formula": "\\begin{align*} \\vec { \\delta } _ { 2 0 3 } = ( \\cos \\alpha _ { 1 0 2 } + \\cos a _ { 3 , 1 0 2 } \\cos \\omega _ { 3 , 1 0 2 } , \\sin \\alpha _ { 1 0 2 } + \\cos a _ { 3 , 1 0 2 } \\sin \\omega _ { 3 , 1 0 2 } , \\sin a _ { 3 , 1 0 2 } ) \\end{align*}"}
+{"id": "1061.png", "formula": "\\begin{align*} N _ 2 \\coloneqq ( d ' + 1 ) \\cdot \\min \\left \\{ \\prod _ { i = 1 } ^ n ( h ' _ i + 1 ) , \\binom { n + h ' } { n } \\right \\} \\ , . \\end{align*}"}
+{"id": "2377.png", "formula": "\\begin{align*} & r ( x , z ) + s ( x , z ) : = m ( r ( x , z ) , z , s ( x , z ) ) , \\\\ & - r ( x , z ) : = m ( z , r ( x , z ) , z ) \\\\ & r ( x , z ) \\cdot s ( x , z ) : = r ( s ( x , z ) , z ) \\end{align*}"}
+{"id": "3749.png", "formula": "\\begin{align*} \\hat { P } \\sum _ { n = 0 } ^ \\infty v _ n : = \\sum _ { n = 1 } ^ \\infty \\hat { P } ( \\rho _ n ) v _ n , \\hat { H } \\sum _ { n = 0 } ^ \\infty v _ n : = \\sum _ { n = 1 } ^ \\infty \\hat { H } ( \\rho _ n ) v _ n , \\end{align*}"}
+{"id": "4848.png", "formula": "\\begin{align*} ( \\prod _ { j = 1 } ^ m a ( j ) \\ast g _ { w ( l ) } t ( j ) ) \\ast a ( m + 1 ) = ( \\prod _ { i = 1 } ^ u d ( i ) \\ast h _ l ( q ( i ) ) ) \\ast d ( u + 1 ) . \\end{align*}"}
+{"id": "4659.png", "formula": "\\begin{align*} - \\Delta u + V ( x ) u = \\lambda u + g ( u ) , \\mathbb { R } ^ N , \\end{align*}"}
+{"id": "8640.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\infty } e ^ { - \\lambda s } p _ j ( s ) d s = \\frac 1 { \\lambda } \\int _ 0 ^ { \\infty } \\frac { q ^ 2 e ^ { - \\lambda s } } { ( 1 - p e ^ { - \\lambda s } ) ^ 2 } \\cdot \\left ( \\frac { 1 - e ^ { - \\lambda s } } { 1 - p e ^ { - \\lambda { s } } } \\right ) ^ { j - 1 } \\cdot \\lambda e ^ { - \\lambda s } d s . \\end{align*}"}
+{"id": "5421.png", "formula": "\\begin{align*} r _ n ^ { \\mathrm { m u l t i p l e } } ( h ) : = \\frac { 2 ^ { p + 1 } \\cdot r _ { 2 n } ^ { [ 1 ] } \\left ( \\frac { h } { 2 } \\right ) - r _ n ^ { [ 1 ] } ( h ) } { 2 ^ { p + 1 } - 1 } . \\end{align*}"}
+{"id": "1418.png", "formula": "\\begin{align*} \\mathcal { I } _ { k + 1 } ^ s = \\{ T \\in \\mathcal { I } _ k : s \\langle g _ k , \\log _ { x _ k } ( y ) \\rangle \\geq 0 , \\forall y \\in B ( c _ T , \\epsilon r ) \\} . \\end{align*}"}
+{"id": "57.png", "formula": "\\begin{align*} \\mathbf { G } ^ 1 [ \\ell ] : = \\Omega ( \\mathcal { G } _ { \\overline { \\psi } _ 0 } ^ { 1 , \\mathbf { K } } ) , \\mathbf { K } ^ 1 [ \\ell ] : = \\Omega ( \\mathbf { K } _ { \\overline { \\psi } _ 0 , p } ^ 1 ) . \\end{align*}"}
+{"id": "7053.png", "formula": "\\begin{align*} \\nu \\left ( \\frac { d } { d x } ( f _ { \\tilde { \\textbf { Q } } } ) \\right ) \\geq \\nu ( g ' ) - b = \\nu ( g ' ) - \\nu _ i ( g ) . \\end{align*}"}
+{"id": "9344.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t & = \\tilde { b } ( x _ t ) d t + \\sigma ( x _ t ) d W _ t + \\tilde { \\sigma } ( x _ t ) d \\xi _ t + \\int _ \\mathcal { E } l ( x _ { t - } , e ) \\tilde { N } ( d e , d t ) , \\ t \\in [ 0 , \\infty ) , \\\\ x _ 0 & = x _ 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8789.png", "formula": "\\begin{align*} & ( u ^ 2 + 4 t u + 4 t + 6 + 6 ) ( [ 2 + 2 t + u ] c _ 2 - [ 2 + t ] b _ 2 - u c _ 3 ) \\\\ & = ( u ^ 2 + 2 t u + 2 t + 6 u + 4 ) ( [ 3 + t + u ] b _ 2 - [ 1 + 2 t ] c _ 2 - u b _ 3 ) \\\\ & + 2 u ( u + 1 ) c _ 2 + u ^ 2 b _ 3 + u ^ 2 c _ 3 \\end{align*}"}
+{"id": "4135.png", "formula": "\\begin{align*} T _ { 1 2 } = \\left ( \\begin{array} { r r r } 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , T _ { 1 3 } = \\left ( \\begin{array} { r r r } 1 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , T _ { 2 1 } = \\left ( \\begin{array} { r r r } 1 & 0 & 0 \\\\ 1 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , \\end{align*}"}
+{"id": "597.png", "formula": "\\begin{align*} g ( s ) = f ( \\alpha _ * , s ) , 0 \\leq s \\leq 1 . \\end{align*}"}
+{"id": "5037.png", "formula": "\\begin{align*} \\overline g = E _ f ^ { - 1 } s . \\end{align*}"}
+{"id": "6181.png", "formula": "\\begin{align*} B + v = \\big \\{ y \\in \\mathbb R ^ n \\colon y = t \\omega ( 1 + v ( \\omega ) ) \\omega \\in \\mathbb S ^ { n - 1 } , \\ , 0 \\le t < 1 \\big \\} . \\end{align*}"}
+{"id": "5989.png", "formula": "\\begin{align*} \\overline { C } _ { X ^ { \\ast } } ( \\omega ^ { - 1 } u ( - t ) , \\omega ) & = \\overline { c } _ { X ^ { \\ast } } ( \\omega ^ { - 1 } u ( - t ) , \\omega ) \\\\ & = ( x ( \\omega ^ { - 1 } u ( - t ) ) , x ( \\omega ) ) _ \\R ( - x ( \\omega ^ { - 1 } u ( - t ) ) x ( \\omega ) , x ( u _ - ( t ) ) _ \\R \\\\ & = 1 ; \\end{align*}"}
+{"id": "3345.png", "formula": "\\begin{align*} T Z _ j ^ * w = T Z _ j ^ * p ( Z ^ * ) v = Y _ j ^ * p ( Y ^ * ) T v = Y _ j ^ * T p ( Z ^ * ) v = Y _ j ^ * T w . \\end{align*}"}
+{"id": "6472.png", "formula": "\\begin{align*} a ( s , s ' ) & = z ( s ) \\big ( 1 + \\mathcal { O } ( s \\sigma ^ 2 ) \\big ) + \\widetilde z ( s ' ) \\big ( 1 + \\mathcal { O } ( s ' h ^ 2 ) \\big ) \\\\ & = \\frac { 1 } { 2 } \\sigma ^ 2 + \\frac { 5 } { 2 } h ^ 2 + \\mathcal { O } ( s \\sigma ^ 2 + s ' h ^ 2 ) \\\\ b ( s , s ' ) & = \\zeta ( s ) \\big ( 1 + \\mathcal { O } ( s \\sigma ^ 2 ) \\big ) + \\widetilde \\zeta ( s ' ) \\big ( 1 + \\mathcal { O } ( s \\sigma ^ 2 ) \\big ) \\\\ & = - i ( K - K _ 1 + K _ 2 - K _ 3 + K _ 4 - K _ 5 ) + \\mathcal { O } ( s \\sigma ^ 2 ) , \\\\ c ( s , s ' ) & = \\mathcal { O } ( s ) . \\end{align*}"}
+{"id": "7687.png", "formula": "\\begin{align*} \\vartheta _ L ( f ; p ) ( g ) = ( f ( \\tau ) , \\Theta _ L ( \\tau , g ; p ) ) _ \\tau \\end{align*}"}
+{"id": "9096.png", "formula": "\\begin{align*} \\langle g , B g \\rangle = \\sum _ { \\{ i , j \\} \\in E } ( - M _ { i j } ) f _ k ( i ) f _ k ( j ) ( g ( i ) - g ( j ) ) ^ 2 = \\sum _ { \\{ i , j \\} \\in E } a _ { i j } ( g ( i ) - g ( j ) ) ^ 2 \\end{align*}"}
+{"id": "1882.png", "formula": "\\begin{align*} F ( [ \\zeta ] ) = \\langle f , \\zeta \\rangle \\forall \\ [ \\zeta ] \\in [ \\mathcal { X } ] . \\end{align*}"}
+{"id": "4975.png", "formula": "\\begin{align*} \\hat { p } _ { 1 , n } ^ { ( t + 1 ) } = \\hat { p } _ { 1 , 1 } ^ { \\phantom { ( ) } } \\hat { p } _ { 1 , n } ^ { ( t ) } \\ , + \\ , \\hat { p } _ { 1 , 2 } ^ { \\phantom { ( ) } } \\hat { p } _ { 2 , n } ^ { ( t ) } . \\end{align*}"}
+{"id": "1885.png", "formula": "\\begin{align*} \\mathcal { N } ( [ \\zeta ^ * ] , [ K ] ) = \\{ \\tilde { F } \\in [ \\mathcal { X } ] ' : \\ - \\tilde { F } ( [ \\zeta ] - [ \\zeta ^ * ] ) \\geq 0 \\forall \\ [ \\zeta ] \\in [ K ] \\} . \\end{align*}"}
+{"id": "5576.png", "formula": "\\begin{align*} \\left \\Vert \\frac { d \\tau ( y ) ^ { - 1 } ( g \\nu _ { P } ) ^ { y } } { d \\bar { m } _ { K \\cap Q } } \\right \\Vert _ { \\infty } = \\left \\Vert \\frac { d ( g \\nu _ { P } ) ^ { y } } { d \\left ( \\tau ( y ) . \\bar { m } _ { K } \\right ) ^ { y } } \\right \\Vert _ { \\infty } \\le \\left \\Vert \\frac { d ( g \\nu _ { P } ) } { d \\left ( \\tau ( y ) . \\bar { m } _ { K } \\right ) } \\right \\Vert _ { \\infty } \\left \\Vert \\frac { d \\left ( \\tau ( y ) . \\bar { m } _ { K } \\right ) } { d \\left ( g \\nu _ { P } \\right ) } \\right \\Vert _ { \\infty } . \\end{align*}"}
+{"id": "3147.png", "formula": "\\begin{align*} \\norm { \\phi } _ X \\leq C \\norm { g - \\sum _ { j = 0 } ^ { N } \\dfrac { \\displaystyle \\int _ { \\R ^ N } g Z _ j } { \\displaystyle \\int _ { \\R ^ N } H _ j Z _ j } H _ j } _ Y . \\end{align*}"}
+{"id": "5444.png", "formula": "\\begin{align*} G ( \\theta ) = & \\mathbb { E } _ { \\Phi | \\Phi ( \\mathcal { A } ) > 0 } \\left \\{ \\left ( \\sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } \\right . \\right . \\\\ & \\left . \\left . \\prod \\limits _ { x _ { i } \\in \\Phi \\backslash \\{ x _ 1 \\} \\cap { \\mathcal { A } } } \\frac { 1 } { \\left ( 1 + \\frac { m \\eta \\theta r _ 1 ^ { \\alpha } } { M r _ i ^ { \\alpha } } \\right ) ^ M } \\right ) ^ b \\right \\} , \\end{align*}"}
+{"id": "1329.png", "formula": "\\begin{align*} \\left ( Z _ i ( u ) , \\frac { 1 } { T _ i ( v ) } \\right ) _ { u \\geq 0 , v \\geq 0 , i \\in V } = \\left ( \\theta _ i Z ^ * _ i ( u ) , \\frac { 1 } { T _ i ^ { \\infty } } + \\frac { 1 } { \\theta _ i ^ 2 T _ i ^ * ( v ) } \\right ) _ { u \\geq 0 , v \\geq 0 , i \\in V } . \\end{align*}"}
+{"id": "2927.png", "formula": "\\begin{align*} \\log ( 1 - x ) = - \\sum _ { n = 1 } ^ \\infty \\frac { x ^ n } { n } . \\end{align*}"}
+{"id": "8949.png", "formula": "\\begin{align*} \\frac { \\partial \\phi } { \\partial t } + a \\frac { \\partial \\phi } { \\partial x } = 0 , \\end{align*}"}
+{"id": "4223.png", "formula": "\\begin{align*} L _ 1 & = K _ 1 T ( \\phi _ R ) ^ { - 1 } \\cdots T ( \\phi _ 0 ) ^ { - 1 } = T ( \\phi ) T ( \\phi _ R ) ^ { - 1 } \\cdots T ( \\phi _ 0 ) ^ { - 1 } - I , \\\\ L _ 2 & = K _ 2 T ( \\tilde { \\phi } _ R ) ^ { - 1 } \\cdots T ( \\tilde { \\phi } _ 0 ) ^ { - 1 } = T ( \\tilde { \\phi } ) T ( \\tilde { \\phi } _ R ) ^ { - 1 } \\cdots T ( \\tilde { \\phi } _ 0 ) ^ { - 1 } - I , \\end{align*}"}
+{"id": "3770.png", "formula": "\\begin{align*} X _ S ( k , n , \\omega ) = { \\rm G r } _ { k \\omega , \\dots , k \\omega } ( U _ { n \\omega , S } ) \\end{align*}"}
+{"id": "1716.png", "formula": "\\begin{align*} \\textrm { $ ( $ { \\sc w o t } $ ) $ } \\lim _ { n \\to \\infty } T ^ n = \\begin{bmatrix} P \\oplus 0 & 0 \\\\ 0 & A \\end{bmatrix} . \\end{align*}"}
+{"id": "6161.png", "formula": "\\begin{align*} - a & : = ( - 1 ) \\cdot a , \\\\ a \\cdot 0 & = 0 \\cdot a : = 0 , \\\\ a + 0 & = 0 + a = \\{ a \\} , \\\\ a + ( - a ) & = \\Gamma ( R ) , \\\\ \\mbox { f o r } & a , b \\ne 0 , a \\ne - b \\mbox { d e f i n e } \\\\ a + b & : = \\{ c \\in \\Gamma ( R ) : \\mbox { t h e r e e x i s t } d \\in G ( R ) \\mbox { s u c h t h a t } \\\\ & a \\cdot b = c \\cdot d \\in G ( R ) \\mbox { a n d } l _ R ( a ) l _ R ( b ) = l _ R ( c ) l _ R ( d ) \\in R _ 2 \\} . \\end{align*}"}
+{"id": "5069.png", "formula": "\\begin{align*} \\psi ( 0 ) = \\phi + v _ 0 , \\end{align*}"}
+{"id": "9025.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\sum _ { i = 1 } ^ 3 \\left ( C _ i ( \\lambda ' _ i \\rho _ 0 ) - C _ i ( \\lambda _ i '' \\rho _ 0 ) \\right ) ( \\lambda _ i ' - \\lambda _ i '' ) > 0 \\\\ \\displaystyle ( \\lambda ' _ 1 , \\lambda ' _ 2 , \\lambda ' _ 3 ) \\neq ( \\lambda '' _ 1 , \\lambda '' _ 2 , \\lambda '' _ 3 ) \\ \\mbox { c o n v e x t r i p l e s a n d f o r a n y } \\rho _ 0 > 0 . \\end{array} \\end{align*}"}
+{"id": "5618.png", "formula": "\\begin{align*} H ( \\xi _ { 1 } ^ { x } ) = { \\rm I } ( \\xi _ { 1 } ^ { x } , \\xi _ { 1 } ^ { x } ) \\ge \\inf _ { n } { \\rm I } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) = \\lim _ { n \\to \\infty } { \\rm I } ( \\xi _ { 1 } ^ { x } , \\xi _ { n } ^ { x } ) . \\end{align*}"}
+{"id": "3741.png", "formula": "\\begin{align*} { \\rm R e } \\ W _ { \\vec { k } , \\vec { l } } ( \\vec { x } , \\vec { y } ) - { \\rm R e } \\ W _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) & = 0 , \\\\ { \\rm I m } \\ W _ { \\vec { k } , \\vec { l } } ( \\vec { x } , \\vec { y } ) + { \\rm I m } \\ W _ { \\vec { k } , \\vec { l } } ( \\vec { y } , \\vec { x } ) & = 0 , \\end{align*}"}
+{"id": "6120.png", "formula": "\\begin{align*} \\left \\{ Q _ { 2 ^ { j _ { 0 } } \\rho _ { z _ { i } } } ( z _ { i } ) \\right \\} _ { i \\in \\mathbb { N } } D _ { \\kappa \\lambda } \\subset \\bigcup _ { z _ { 0 } \\in D _ { \\kappa \\lambda } } Q _ { \\rho _ { z _ { 0 } } } ( z _ { 0 } ) \\subset \\bigcup \\limits _ { i = 1 } ^ { \\infty } Q _ { 5 \\times 2 ^ { j _ { 0 } } \\rho _ { z _ { i } } } ( z _ { i } ) . \\end{align*}"}
+{"id": "1239.png", "formula": "\\begin{align*} v _ { \\gamma , 2 } ( x ) = ( \\log 2 + \\log ( x - \\tfrac 1 2 ) ) g _ { \\beta , 2 } ( x ) \\forall x \\in ( \\tfrac 1 2 , 1 ] . \\end{align*}"}
+{"id": "198.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( a , b ) = 1 \\\\ a , b \\geq 1 } } \\left ( 1 - \\frac { z ^ b } { \\phi ^ { 2 a } } \\right ) ^ { \\frac { b ^ { 2 } } { a ^ 3 } } = \\exp \\left \\{ \\left ( \\frac { 4 } { 5 } \\zeta ( 3 ) - \\frac { 2 } { 1 5 } \\pi ^ 2 \\log \\phi + \\frac { 2 } { 3 } ( \\log \\phi ) ^ 3 \\right ) \\left ( \\frac { z ( 1 + z ) } { ( 1 - z ) ^ 3 } \\right ) \\right \\} , \\end{align*}"}
+{"id": "636.png", "formula": "\\begin{align*} I = P _ { V } x ^ * x | _ V & = P _ V x ^ * P _ V x | _ V + P _ V x ^ * P _ { V ^ \\perp } x | V \\\\ & = | \\phi ( x ) | ^ 2 I + z ^ * z \\end{align*}"}
+{"id": "7601.png", "formula": "\\begin{align*} w ( z ) = \\frac { h ( z ) - 1 } { h ( z ) + 1 } . \\end{align*}"}