diff --git "a/data_tmp/process_18/tokenized_finally.jsonl" "b/data_tmp/process_18/tokenized_finally.jsonl"
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-{"id": "4295.png", "formula": "\\begin{align*} \\boldsymbol { \\lambda } ^ { [ k + 1 ] } : = \\boldsymbol { \\lambda } ^ { [ k ] } + \\frac { \\rho } { N } ( \\sum _ { i = 1 } ^ N \\mathbf { p } _ i ^ { [ k ] } - \\mathbf { P } ^ { [ k + 1 ] } ) . \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} \\mu ( Z ) = \\Phi \\left \\{ \\left [ \\mu _ 1 ( s ^ { - a } ( Z ) ) : a \\in \\mathbb N ^ d \\right ] \\right \\} \\ , . \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} \\kappa _ 1 ^ 2 : = & \\sum \\limits _ { i = 1 } ^ m \\int _ { \\mathbb { R } ^ n } ( \\Pi \\nabla \\mathcal { L } \\xi _ i ) \\cdot ( a \\Pi \\nabla \\mathcal { L } \\xi _ i ) \\ , d \\mu < + \\infty \\ , , \\\\ \\kappa _ 2 ^ 2 : = & \\sum \\limits _ { 1 \\le i , j \\le m } \\int _ { \\mathbb { R } ^ n } ( \\Pi \\nabla A _ { i j } ) \\cdot ( a \\Pi \\nabla A _ { i j } ) \\ , d \\mu < + \\infty \\ , . \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} \\| e ^ { \\delta ( t ) O _ q } e ^ { - t K _ { \\nu , \\alpha } } \\| _ { \\mathcal { L } ( L ^ 2 ( \\mathbb { R } ^ 2 ) ) } = \\| e ^ { - i Q _ 2 } \\| _ { \\mathcal { L } ( L ^ 2 ( \\mathbb { R } ^ 2 ) ) } \\leq 1 . \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} ( n , k , \\lambda , \\mu ) = & ( 4 ^ m , 2 ^ { m - 1 } ( 2 ^ m \\pm 1 ) , 2 ^ { m - 1 } ( 2 ^ { m - 1 } \\pm 1 ) , 2 ^ { m - 1 } ( 2 ^ { m - 1 } \\pm 1 ) ) \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} ( \\mathbf { I } - \\mathbf { K } - \\mathbf { K _ { e } } ) ^ { - 1 } = ( \\mathbf { I } - \\mathbf { K } ) ^ { - 1 } \\sum _ { i = 0 } ^ { \\infty } \\left [ \\mathbf { K _ { e } } ( \\mathbf { I } - \\mathbf { K } ) ^ { - 1 } \\right ] ^ { i } . \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } h \\nabla ^ * \\Psi \\ , d \\mu = \\int _ { \\mathbb { R } ^ n } \\Psi \\cdot \\nabla h \\ , d \\mu \\ , \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\det ( D ^ \\alpha { } _ \\beta ) = + 1 \\end{align*}"}
-{"id": "9804.png", "formula": "\\begin{align*} \\varphi ( q , p ) = \\frac { e ^ { - \\frac { ( q ^ 2 + p ^ 2 ) } { 2 } } } { \\sqrt { \\pi } } \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} \\left \\{ \\| e _ i \\| / \\| e _ i \\| \\mid 1 \\le i , j \\le N \\right \\} \\cap | F ^ \\times | = \\{ 1 \\} . \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} p _ { _ 1 } ( y ) = \\frac { \\eta _ 0 p _ { _ 0 } ( y ) } { \\eta _ 1 ^ b + \\eta _ 2 ^ b y ^ 2 } . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} { \\rm { H } } ^ j ( \\overline { k } ^ { \\otimes _ k n } , G ) = 0 \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} I ( \\varepsilon ) : = \\varepsilon \\limsup _ { \\lambda \\to \\infty } \\lambda \\mathcal { L } ^ d \\left ( \\{ \\mathbf { M } ^ { \\varepsilon } ( \\mu ) > \\lambda \\} \\cap B _ { r ' } \\right ) = \\circ ( 1 ) ~ ~ ~ ~ \\varepsilon \\to 0 . \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} \\gamma = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n ( n - i + 1 ) ( n - i + 2 ) \\alpha _ i . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} \\varepsilon _ w = \\frac { 1 } { 1 0 0 N ( w ) } . \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{gather*} u : \\mathbb { R } \\times S ^ 1 \\longrightarrow M \\\\ \\partial _ s u ( s , t ) + J ( t , u ( s , t ) ) ( \\partial _ t u ( s , t ) - X _ { H _ t } ( u ( s , t ) ) ) = 0 \\end{gather*}"}
-{"id": "9556.png", "formula": "\\begin{align*} \\begin{array} { l } | P ^ { \\flat } _ m | \\leq ( k + m - 1 ) / ( k + m - 1 + k + m + 3 / 2 + W _ m / 3 ) \\leq 6 / m ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\ , { \\rm a n d } \\\\ \\\\ | ( P ^ { \\flat } _ m ) ' | \\leq ( W _ m / 3 + k + m + 3 / 2 ) | P _ m ^ { \\flat } / q | + ( k + m - 1 ) q ^ { k + m - 2 + k + m + 3 / 2 + W _ m / 3 } \\\\ \\\\ \\leq 2 + k + m - 1 = k + m + 1 ~ . \\end{array} \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} - \\varphi _ 1 ( \\epsilon _ { f , i j } ) + d _ 2 ( \\sigma _ { f , i j } ) = x _ i \\alpha _ 1 ( e _ { f , j } ) - x _ j \\alpha _ 1 ( e _ { f , i } ) \\subset ( x _ i , x _ j ) G _ 1 . \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{gather*} R _ k : \\mathcal { L } ( M ) \\rightarrow \\mathcal { L } ( M ) \\\\ R _ k ( x ) ( t ) = x \\Big ( t + \\frac { 1 } { k } \\Big ) \\end{gather*}"}
-{"id": "2799.png", "formula": "\\begin{align*} b ( k , \\theta ) \\leq 2 \\left ( \\sum _ { i = 0 } ^ { t - 4 } ( k - 1 ) ^ { i } + \\frac { ( k - 1 ) ^ { t - 3 } } { c } + \\frac { ( k - 1 ) ^ { t - 2 } } { c } \\right ) . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} \\int _ G f ( x ) d \\tilde { \\mu } ( x ) = \\int _ { G / / H } Q ( f ) ( \\ddot { x } ) d \\mu ( \\ddot { x } ) \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} \\nabla _ { \\partial _ r } Y = \\frac { \\partial Y } { \\partial r } + \\frac { Y } { r } \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} \\Delta _ p ( u _ 1 , u _ 2 ) \\geq \\frac { C _ p } { p } \\| u _ 1 - u _ 2 \\| ^ p , \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} h _ n ( t ) = \\int _ { ( 0 , + \\infty ) } { t ( 1 + s ) \\over t + s } \\ , d \\nu _ n ( s ) , t \\in [ 0 , + \\infty ) , \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( \\partial _ t - \\partial _ x ^ 2 - x ^ 2 \\partial _ y ^ 2 ) f _ { } ( t , x , y ) = \\mathbf 1 _ { \\omega _ { } } u _ { } ( t , x , y ) & t \\in [ 0 , T ] , ( x , y ) \\in \\Omega \\\\ f _ { } ( t , x , y ) = 0 & t \\in [ 0 , T ] , ( x , y ) \\in \\partial \\Omega \\\\ f _ { } ( 0 , x , y ) = f _ 0 ( x , y ) , ~ f _ { } ( T , x , y ) = 0 & ( x , y ) \\in \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} n _ s = s ^ 3 + d _ s \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} f ( x ) & = 1 + x - ( A - B x ) ( 1 + x ) ( x ^ 3 - 1 ) + m \\ ; h ( x ) , \\\\ g ( x ) & = x - x ( C - D x ) ( 1 + x ) ( x ^ 3 - 1 ) + m \\ ; h ( x ) , \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} \\underset { n \\rightarrow \\infty } { \\lim } \\frac { 2 l } { n - k _ 0 } \\log \\left ( { B } ( \\frac { n - k _ 0 } { 2 } , 0 . 5 ) \\right ) = 0 . \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} \\alpha ( y ) = e ^ { - G ( y ) } \\left [ C - \\int _ { 1 } f ( y ) e ^ { G ( y ) } \\right ] \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } C _ { i j } } { \\mathrm { d } t } = \\left ( \\frac { Q _ { i j } [ C ] ^ 2 } { C _ { i j } ^ 2 } - \\nu C _ { i j } ^ { \\gamma - 1 } \\right ) C _ { i j } L _ { i j } , \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} v ( w ) = \\int _ { C ( 0 , r ) ^ + } e ^ { z w } u ( z ) d z , \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} \\varepsilon \\dot { S } = - l ( S , N , F ) \\ , S = \\left [ f - c _ 1 ( S , N , F ) \\right ] N \\le 0 \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} \\sqrt { | D | } L _ D ( 1 ) \\frac { 1 } { 2 \\pi } \\sum _ { Q \\in \\mathcal { Q } _ { d } / \\mathrm { S L } _ 2 ( \\mathbb { Z } ) } \\int _ { \\mathrm { S L } _ 2 ( \\mathbb { Z } ) _ Q \\backslash S _ Q } \\frac { \\sqrt { d } d z } { Q ( z , 1 ) } = \\sqrt { | D | } L _ D ( 1 ) \\frac { h ( d ) \\mathrm { l o g } \\varepsilon _ d } { \\pi } , \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} \\zeta ( j , s ) \\coloneqq \\left ( \\prod _ { m = m ( j , s ) + 1 } ^ { N _ { k ( j , s ) } } p _ { k ( j , s ) , m } \\right ) \\left ( \\prod _ { k = k ( j , s ) + 1 } ^ j \\prod _ { l = 1 } ^ { N _ k } p _ { k , l } \\right ) . \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\big \\langle ( \\mathbf { E } , \\mathbf { H } ) ^ { T } , ( \\tilde { \\mathbf { E } } , \\tilde { \\mathbf { H } } ) ^ { T } \\big \\rangle _ { \\mathcal { H } } : = \\int _ { G } \\boldsymbol { \\varepsilon } \\mathbf { E } \\cdot \\tilde { \\mathbf { E } } \\ , \\mathrm { d } \\mathbf { x } + \\int _ { G } \\boldsymbol { \\mu } \\mathbf { H } \\cdot \\tilde { \\mathbf { H } } \\ , \\mathrm { d } \\mathbf { x } . \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} u \\delta _ { g _ S } d \\theta = 3 d \\theta ( \\nabla u ) . \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} \\left ( I _ n + \\beta a a ^ T \\right ) ^ { \\frac { 1 } { 2 } } = I _ n + \\frac { \\beta a a ^ T } { 1 + \\sqrt { 1 + \\beta a ^ T a } } . \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} K ( z , Z ) = \\left \\langle R _ H ( e _ Z ) , R _ H ( e _ z ) \\right \\rangle _ H . \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} ( b , s ) \\cdot ( z ' , z _ n ) = ( z ' + b , z _ n + 2 i \\langle z ' , b \\rangle + s + i | b | ^ 2 ) \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} G _ t = \\bigl \\{ s \\in ( 0 , 1 ) : \\tfrac { B ( s ) } { s } \\le \\tfrac { D ( t ) } { t } \\bigr \\} . \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} H ^ 5 = 1 , H ^ 4 E = H ^ 3 E ^ 2 = H ^ 2 E ^ 3 = 0 , H E ^ 4 = - \\delta , \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} \\bigoplus _ { \\substack { r + s = p } } U ^ { r , s } \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} \\theta _ { l , p } = \\zeta ^ { p ( l + p ) } . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} W _ { 1 } ( r ) = \\int _ { \\wp } r ^ { Q - 1 } \\phi _ { 1 } ( r y ) d \\sigma ( y ) , \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 x ^ k P _ n ( x ) d x = \\begin{cases} 0 , & \\\\ ( - 1 ) ^ n \\frac { k ( k - 1 ) \\dots ( k - n + 1 ) } { n ! } B ( k + 1 , n + 1 ) , & \\end{cases} \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} u _ { k + 1 } ( t ) = u _ k ( t ) { \\int } _ { \\Sigma } \\frac { a ( s , t ) y ( s ) } { ( A u _ k ) ( s ) } \\ , d s , t \\in \\Omega , k = 0 , \\dots . \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} \\mu ( I - e ^ { - \\alpha } P _ 1 ) ^ { - 1 } + \\sum _ { j = 1 } ^ { \\lfloor b / \\alpha \\rfloor } e ^ { - \\alpha j } \\sum _ { k = 1 } ^ { j } P _ 1 ^ { k - 1 } ( P _ k - P _ 1 ) P _ 1 ^ { j - k } , \\end{align*}"}
-{"id": "9895.png", "formula": "\\begin{align*} ( \\boldsymbol { D } ^ { s * } u , \\boldsymbol { D } ^ t \\phi ) = \\lim _ { n \\rightarrow \\infty } ( \\boldsymbol { D } ^ { s * } \\psi _ n , \\boldsymbol { D } ^ t \\phi ) = \\lim _ { n \\rightarrow \\infty } ( \\psi _ n , \\boldsymbol { D } ^ { s + t } \\phi ) , \\forall \\phi \\in C _ 0 ^ \\infty ( \\mathbb { R } ) , \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} S _ { Q } : = | \\wp | ^ { \\frac { 2 } { Q } } Q ^ { \\frac { Q - 2 } { Q } } ( Q - 2 ) \\left ( \\frac { \\Gamma ( Q / 2 ) \\Gamma ( 1 + Q / 2 ) } { \\Gamma ( Q ) } \\right ) ^ { \\frac { 2 } { Q } } . \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} X \\ = \\ X _ 0 \\ ; \\sqcup \\ ; X _ 1 \\ ; \\sqcup \\ ; \\dotsb \\ ; \\sqcup \\ ; X _ n \\ , , \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} \\chi ( \\Gamma \\backslash ( S \\backslash B ) ) = \\chi ( \\Gamma \\backslash S ) + \\chi ( B ) - \\chi ( B \\backslash S ) \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} R ( x ) = \\epsilon \\ln { x } , \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} - v _ { n , m } = - v _ { n , m - 1 } + D _ { n , m } + m - 1 . \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} \\mathcal { C } = \\{ \\mathbf { c } = ( \\mathbf { c } _ { 1 } , \\dots , \\mathbf { c } _ { n } ) : \\mathbf { H } \\cdot \\mathbf { c } = 0 \\} \\subseteq \\mathbb { B } ^ { n \\ell } . \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} \\mathcal { R _ I } = \\max ( X _ { 1 1 } , X _ { 1 1 } ^ { \\frac { \\mu } { 2 } } \\left ( \\frac { P } { N } \\right ) ^ { \\frac { 2 - \\mu } { 2 \\alpha } } \\left ( \\frac { 1 } { M } \\right ) ^ { \\frac { 1 } { 2 \\alpha } } ) , \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} u _ { - , a } ( x ) = \\frac { \\phi _ 3 ( x ) } { \\phi _ 3 ( a ) } \\frac { \\int _ x ^ \\infty \\phi _ 3 ^ { - 2 } ( y ) d y } { \\int _ a ^ \\infty \\phi _ 3 ^ { - 2 } ( y ) d y } , \\ x \\ge a ; \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} \\sup \\left \\{ | \\psi ^ { \\zeta , ( i ) } _ { x } ( x ) | + | \\psi ^ { \\zeta , ( i ) } _ { x x } ( x ) | \\ , \\mid \\ , x \\leq 0 \\right \\} & \\leq C ' \\\\ \\psi ^ { \\zeta , ( i ) } ( 0 ) & = u _ { 0 } ( 0 ) \\\\ \\psi ^ { \\zeta , ( i ) } _ { x } ( 0 ) & = u _ { 0 , x _ { i } } ( 0 ) \\\\ \\psi ^ { \\zeta , ( i ) } _ { x x } ( 0 ) & = b _ { i } \\\\ \\sup \\left \\{ | \\psi ^ { \\zeta , ( i ) } ( x ) - u _ { 0 } ( x ) | \\ , \\mid \\ , x \\in \\overline { I _ { i } } \\right \\} & \\leq C ' \\zeta ^ { 2 } \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} U _ \\lambda ( o , \\ , o \\ , | \\ , z ) & = \\sum _ { n = 1 } ^ \\infty f ^ { ( 2 n ) } _ \\lambda ( o , \\ , o ) z ^ { 2 n } = \\sum _ { n = 1 } ^ \\infty c _ { n - 1 } \\left ( \\frac { d - 1 } { d - 1 + \\lambda } \\right ) ^ { n - 1 } \\left ( \\frac { \\lambda } { d - 1 + \\lambda } \\right ) ^ n z ^ { 2 n } \\\\ & = \\frac { \\lambda } { d - 1 + \\lambda } z ^ 2 \\ , \\mathcal { C } \\left ( \\frac { \\lambda ( d - 1 ) z ^ 2 } { ( d - 1 + \\lambda ) ^ 2 } \\right ) , \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} \\min _ { \\substack { V _ { X \\widetilde { X } Y } \\ , : \\ , V _ { X Y } = P _ { X Y } , P _ { \\widetilde { X } } = P , \\\\ q ( V _ { \\widetilde { X } Y } ) \\ge q ( P _ { X Y } ) , d ( P _ { X \\widetilde { X } } ) \\ge \\Delta } } D \\big ( V _ { X \\widetilde { X } Y } \\| P \\times P _ { X Y } \\big ) , \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} \\mathcal { Q } _ { b } ( y ) = \\left \\{ \\begin{array} { c c c c c c c c c c } a _ { j } , & \\mbox { i f } y \\in ( a ^ - _ { j } , a ^ + _ { j } ] \\\\ X , & \\mbox { i f } y \\in ( a ^ + _ j , a ^ - _ { j + 1 } ] \\mbox { f o r } 1 \\leq j \\leq 2 ^ { b } - 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} d _ { \\mathcal { H } } ( X , Z ) = \\sup _ { h \\in \\mathcal { H } } \\mathbb { E } \\left [ \\Delta f _ h ( X ) - X \\cdot \\nabla f _ h ( X ) \\right ] , \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} x + 2 y + m x - x y + x y m = 0 . \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} \\delta A ^ \\flat = 0 . \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} C _ d = ( 1 - 2 \\beta \\sigma ^ 2 ) ^ { - \\frac { d } { 2 } } , \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} & \\hat { \\delta } ( g ) ( m \\otimes \\sigma ) : = d _ N g ( m \\otimes \\sigma ) - ( - 1 ) ^ { | g | } g ( d _ M m \\otimes \\sigma ) - ( - 1 ) ^ { | m | + | g | } g ( m \\otimes \\partial ( \\sigma ) ) \\\\ & + \\sum _ { ( \\sigma ) } ( - 1 ) ^ { | m | + | g | } g ( ( m \\cdot \\tau ( \\sigma ' ) ) \\otimes \\sigma '' ) + \\sum _ { ( \\sigma ) } ( - 1 ) ^ { | g | + | m | + | \\sigma ' | } g ( m \\otimes \\sigma ' ) \\cdot \\tau ( \\sigma '' ) , \\end{align*}"}
-{"id": "9984.png", "formula": "\\begin{align*} \\mathbb P ( B ) & = \\int _ 0 ^ { \\Delta x } u ( s , \\tau - \\Delta x ) [ ( \\mathbb P ( ( \\hbox { 1 j u m p } ) \\cap B | R ( \\tau - \\Delta x ) = s ) + \\mathbb P ( ( > \\hbox { 1 j u m p } ) \\cap B | R ( \\tau - \\Delta x ) = s ) ] d s \\\\ & = \\int _ 0 ^ { \\Delta x } u ( s , \\tau - \\Delta x ) \\left ( \\int _ { x + \\Delta x - s } ^ { x + 2 \\Delta x - s } p ( y ) d y \\right ) d s + o \\big ( ( \\Delta x ) ^ 2 \\big ) = u ( 0 , \\tau ) p ( x ) ( \\Delta x ) ^ 2 + o \\big ( ( \\Delta x ) ^ 2 \\big ) . \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} \\| z ^ k - x \\| ^ 2 & \\leq \\| x ^ k - x \\| ^ 2 + 2 \\lambda _ k f ( x ^ k , x ) \\\\ & - \\| y ^ k - x ^ k \\| ^ 2 - \\| z ^ k - y ^ k \\| ^ 2 + 2 \\beta _ k \\left ( \\| y ^ k - x ^ k \\| + \\| z ^ k - x ^ k \\| \\right ) \\\\ & = \\| x ^ k - x \\| ^ 2 + 2 \\lambda _ k f ( x ^ k , x ) \\\\ & + 2 \\beta _ k ^ 2 - \\left ( \\| y ^ k - x ^ k \\| - \\beta _ k \\right ) ^ 2 - \\left ( \\| z ^ k - x ^ k \\| - \\beta _ k \\right ) ^ 2 \\\\ & \\leq \\| x ^ k - x \\| ^ 2 + 2 \\lambda _ k f ( x ^ k , x ) + 2 \\beta _ k ^ 2 . \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} d ( - x _ 1 , - x _ 2 , - x _ 3 ) = - x _ 4 . \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} u ( t , x ) & = u _ 0 ( \\overline { x } ) \\exp \\left ( - \\int _ { 0 } ^ { t } \\left ( \\operatorname { d i v } ( \\mathbf { B } ) - G \\right ) ( s , X ( s , \\overline { x } ) ) d s \\right ) \\\\ & + \\int _ { 0 } ^ { t } F ( \\tau , X ( \\tau , \\overline { x } ) ) \\exp \\left ( - \\int _ { \\tau } ^ { t } \\left ( \\operatorname { d i v } ( \\mathbf { B } ) - G \\right ) ( s , X ( s , \\overline { x } ) ) d s \\right ) d \\tau , \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} G _ { n + r } H _ { m + n } - G _ n H _ { m + n + r } = ( - 1 ) ^ n ( G _ r H _ m - G _ 0 H _ { m + r } ) \\ , , \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} \\tau ^ { ( 1 ) } _ { s + 1 } = n - i , \\ \\tau ^ { ( 2 ) } _ { s + n - k } = n + 1 - i \\ \\ \\tau ^ { ( 2 ) } _ { s + n - k + 1 } < n - i . \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} \\bar { \\theta } ^ { ( 4 ) } _ { 3 } ( l ) = \\Theta ^ { - 1 } \\left ( \\phi ( \\hat { \\theta _ { 3 } } ( l ) ) \\ominus \\Theta ( h _ { 1 3 } ( l ) + e ^ { ( 1 ) } _ { 3 } ( l ) ) \\right ) \\in \\mathcal { A } , \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} \\frac { 1 } { r } + \\frac { c } { n } = \\delta \\left ( \\frac { 1 } { p } + \\frac { a - 1 } { n } \\right ) + ( 1 - \\delta ) \\left ( \\frac { 1 } { q } + \\frac { b } { n } \\right ) , \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} \\ell _ n & = \\lfloor \\sqrt [ 3 ] { n - d + d } \\rfloor = \\left \\lfloor \\sqrt [ 3 ] { n - d } \\ , \\sqrt [ 3 ] { 1 + \\frac { d } { n - d } } \\right \\rfloor \\\\ & = \\left \\lfloor s \\ , \\sqrt [ 3 ] { 1 + \\frac { d } { n - d } } \\right \\rfloor = \\left \\lfloor s \\ , \\Big ( 1 + \\frac { 1 } { 3 } n ^ { - 1 / 3 } \\ , + \\ , { \\rm o } ( n ^ { - 1 / 3 } ) \\Big ) \\right \\rfloor \\\\ & = s + \\left \\lfloor \\frac { 1 } { 3 } \\ , + \\ , { \\rm o } ( 1 ) \\right \\rfloor \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} \\nabla f ( \\mathbf { x } ^ { k + 1 } ) + \\sqrt { \\eta } \\mathbf { E } _ { } ^ T \\boldsymbol { \\nu } ^ k + \\frac { \\rho } { 2 } \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } \\mathbf { x } ^ k \\\\ + \\rho ( \\mathbf { D } + \\epsilon \\mathbf { P } ) ( \\mathbf { x } ^ { k + 1 } - \\mathbf { x } ^ k ) = \\mathbf { 0 } , \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} - \\frac { \\epsilon _ m ^ 2 \\beta _ m ^ 2 } { \\alpha _ m } & \\Delta u _ m ( x _ m + \\epsilon _ m \\beta _ m x ) + \\frac { \\beta _ m ^ 2 } { \\alpha _ m } \\lambda u _ m ( x _ m + \\epsilon _ m \\beta _ m x ) \\\\ & + \\beta _ m ^ 2 \\rho ( x _ m + \\epsilon _ m \\beta _ m x ) \\phi _ { u _ m } ( x _ m + \\epsilon _ m \\beta _ m x ) \\frac { 1 } { \\alpha _ m } u _ m ( x _ m + \\epsilon _ m \\beta _ m x ) = \\frac { \\beta _ m ^ 2 } { \\alpha _ m } u _ m ^ p ( x _ m + \\epsilon _ m \\beta _ m x ) . \\\\ \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} K ( \\beta , q ) : = \\left \\{ ( \\delta , p ) \\in \\mathbb R ^ 2 : \\ ; \\beta < \\delta < 1 - \\beta , \\ , \\frac 1 \\delta < p < q , 2 \\leq p \\right \\} , \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} \\mathcal B : T \\mathcal S \\times C ^ { m , \\alpha } _ { \\delta } ( M ) & \\rightarrow \\mathbb B \\\\ D \\mathcal B ( h ^ { ( 4 ) } , G ) = ( ~ & h _ { \\partial M } \\\\ & ( H _ { \\partial M } ) ' _ { h ^ { ( 4 ) } } + O _ 0 ( G ) \\\\ & t r _ { \\partial M } K ' _ { h ^ { ( 4 ) } } + O _ 0 ( G ) \\\\ & ( \\omega _ { \\mathbf n } ) ' _ { h ^ { ( 4 ) } } + N d _ { \\partial M } G + O _ 0 ( G ) \\\\ & \\beta _ { g ^ { ( 4 ) } } h ^ { ( 4 ) } ~ ) . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} & \\underset { \\{ \\mathbf { p } _ i \\} _ { i = 1 } ^ N , \\ , \\{ \\mathbf { q } _ i \\} _ { i = 1 } ^ N } { \\min } & \\sum _ { i = 1 } ^ { M } C _ i ( \\mathbf { p } _ i , k ) - \\sum _ { i = M + 1 } ^ N U _ i ( \\mathbf { q } _ i , k ) \\\\ & & \\underline { \\mathbf { p } } _ i ^ { [ k ] } \\leq \\mathbf { p } _ i \\leq \\overline { \\mathbf { p } } _ i ^ { [ k ] } , \\ , i = 1 , \\hdots , N \\\\ & \\sum _ { i = 1 } ^ N \\mathbf { q } _ i = \\mathbf { P } ^ { [ k ] } & \\mathbf { q } _ i = \\mathbf { p } _ i , \\ , i = 1 , \\hdots , N . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} & \\| u u _ h ^ \\kappa \\| _ { p ^ * } \\leq c _ \\Omega \\| u u _ h ^ { \\kappa } \\| _ { 1 , p } \\leq M _ 5 ( \\kappa , u ) \\left [ \\| u u _ h ^ { \\kappa } \\| _ { p } ^ p + 1 \\right ] ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} \\mathcal { H } W _ { l } ( s ; y l ^ c ) = & l ^ { \\frac { c } { 2 } } \\sum _ { \\substack { \\mu _ l \\in { } _ l \\mathfrak { X } \\setminus \\{ 1 \\} , \\\\ c = - 2 a ( \\mu ) } } \\mu _ l ( y ^ { - 1 } ) \\epsilon ( \\frac { 1 } { 2 } , \\mu _ l ^ { - 1 } \\pi _ { 0 , l } ) [ \\mathcal { M } W _ l ( s ; \\cdot ) ] ( \\mu _ l ) \\\\ & + l ^ { \\frac { c } { 2 } } B _ { \\pi _ { 0 , l } } ( l ^ { - c } ) [ \\mathcal { M } W _ l ( s ; \\cdot ) ] ( 1 ) . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ \\tau \\phi = - ( - \\Delta ) ^ s \\phi + p U ^ { p - 1 } ( y ) \\phi + h ( y , \\tau ) , ~ y \\in \\mathbb { R } ^ n , ~ \\tau \\geq \\tau _ 0 , \\\\ & \\phi ( y , \\tau _ 0 ) = e _ { 0 } Z _ 0 ( y ) , ~ y \\in \\mathbb { R } ^ n . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} \\widehat { m } _ { \\beta , \\Delta } ^ { \\pm } ( \\xi ) = \\pi \\left ( \\dfrac { e ^ { 2 \\pi \\beta ( \\Delta - \\xi ) } - e ^ { - 2 \\pi \\beta ( \\Delta - \\xi ) } } { \\left ( e ^ { \\pi \\beta \\Delta } \\mp e ^ { - \\pi \\beta \\Delta } \\right ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} { \\mathrm d } M ^ { \\varphi } _ { t } \\ , : = \\ , \\varphi ^ { \\prime } ( \\widetilde { X } _ { t } ) { \\mathrm d } B _ { t , 2 } \\ , , { \\mathrm d } A _ { t } ^ { \\varphi } \\ , : = \\ , \\varphi ^ { \\prime } ( \\widetilde { X } _ { t } ) b ( t , \\widetilde { X } _ { t } , F _ { t , 2 } ) { \\mathrm d } t + \\frac { 1 } { \\ , 2 \\ , } \\varphi ^ { \\prime \\prime } ( \\widetilde { X } _ { t } ) { \\mathrm d } t \\ , , \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} \\sum _ { \\ell = m } ^ { \\infty } e ^ { 2 \\tau \\phi _ { \\ell } } \\norm { u _ \\ell } ^ 2 \\leq \\frac { C } { \\kappa \\tau } \\sum _ { \\ell = m + 1 } ^ { \\infty } e ^ { 2 \\tau \\phi _ \\ell } \\norm { ( X u ) _ { \\ell } } ^ 2 \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} A ^ { W , W } _ t ( l ) = \\phi ( t , W _ t ) - \\phi ( 0 , W _ 0 ) - \\int _ 0 ^ t \\nabla \\phi ^ * ( r , W _ r ) \\mathrm d W _ r , \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} S _ { N , r } ^ { ( 4 , b ) } = \\left \\{ \\begin{array} { c c c } 5 S _ { N , r - 1 } ^ { ( 4 , b ) } - 2 S _ { N , r - 2 } ^ { ( 4 , b ) } + S _ { N , r - 3 } ^ { ( 4 , b ) } - 1 & i f & b = 1 , \\\\ 5 S _ { N , r - 1 } ^ { ( 4 , b ) } - 2 S _ { N , r - 2 } ^ { ( 4 , b ) } + S _ { N , r - 3 } ^ { ( 4 , b ) } + 1 & i f & b = 2 , 4 , \\\\ 5 S _ { N , r - 1 } ^ { ( 4 , b ) } - 2 S _ { N , r - 2 } ^ { ( 4 , b ) } + S _ { N , r - 3 } ^ { ( 4 , b ) } + 2 & i f & b = 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} \\hat { R } _ { i j } = \\begin{cases} 0 , & \\max \\{ | i | , | j | \\} \\leq C _ 0 K , i \\neq \\pm j , \\\\ ( 1 - \\Gamma _ K ) ( - \\mathbf { i } \\tilde { \\Omega } _ { i j } F _ { i j } + R _ { i j } ) , & \\max \\{ | i | , | j | \\} > C _ 0 K , i \\neq \\pm j , \\\\ 0 , & i = - j , | j | \\leq \\Pi , \\\\ R _ { ( - j ) j } , & i = - j , | j | > \\Pi . \\end{cases} \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} \\begin{pmatrix} \\delta \\mathrm { d } - 2 \\ , \\mathrm { R i c } & - \\frac 1 2 \\mathrm { d } \\\\ \\delta & 0 \\end{pmatrix} \\begin{pmatrix} A ^ \\flat \\\\ p \\end{pmatrix} = 0 . \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} \\norm { F } _ { A ^ 2 ( \\mathcal { U } _ p ) } ^ 2 = \\int _ { \\C ^ n } \\int _ { S ( p ( w ) , \\infty ) } \\abs { F ( z , w ) } ^ 2 \\d V ( z ) \\d V ( w ) < \\infty . \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} g \\big ( \\nabla \\phi ( z ( s ) ) , V ( s ) \\big ) = 0 , \\phi \\big ( z ( s ) \\big ) = 0 . \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} B = \\frac { 1 } { 4 \\pi } \\sum _ { j \\neq 0 } j ^ 2 | q _ j | ^ 4 + \\frac { 1 } { 2 \\pi } \\sum _ { j , l \\atop { j \\neq l } } | j l | | q _ j | ^ 2 | q _ l | ^ 2 . \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} \\sum _ { x \\in \\mathbb { Z } _ N } f ( x ) = 1 , \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} \\alpha _ n ( t , q ) = \\sum _ { 0 \\leq i + j \\leq n - 1 } t ^ i ( q - t ) ^ j ( 1 - t ) ^ { n - i - j - 1 } 2 ^ i ( i + j + 1 ) ! \\binom { i + j } { j } S ( n , i + j + 1 ) . \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} f ( t ) - \\sum _ { k = 0 } ^ { p - 1 } b _ k \\frac { t ^ { k r } } { ( k r ) ! } & = - \\frac { 1 } { 2 \\pi i } \\int _ { \\delta _ { \\beta _ 1 } } e ^ { t / u } \\left ( \\varphi ( u ) - \\sum _ { k = 0 } ^ { p - 1 } b _ k u ^ { k r } \\right ) \\ , \\frac { d u } { u } \\\\ & = - \\frac { 1 } { 2 \\pi i } \\sum _ { j = 1 } ^ 3 \\int _ { \\delta _ j } e ^ { t / u } \\left ( \\varphi ( u ) - \\sum _ { k = 0 } ^ { p - 1 } b _ k u ^ { k r } \\right ) \\ , \\frac { d u } { u } . \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} \\mathrm { j } _ L ( E ) = - \\frac { 1 } { 2 } E E ^ \\top \\mathbb { J } . \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} d \\bar { x } _ i ( s ) = \\frac { \\partial ( G ^ { - 1 } ) _ i } { \\partial \\phi _ j } \\mathcal { L } \\phi _ j \\ , d s + \\frac { 1 } { \\beta } ( \\nabla \\phi a \\nabla \\phi ^ T ) _ { j l } \\frac { \\partial ^ 2 ( G ^ { - 1 } ) _ { i } } { \\partial \\phi _ j \\partial \\phi _ l } d s + \\sqrt { 2 \\beta ^ { - 1 } } \\frac { \\partial ( G ^ { - 1 } ) _ i } { \\partial \\phi _ j } ( \\nabla \\phi \\ , \\sigma ) _ { j l } \\ , d w _ l ( s ) \\ , , \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} F ( X , Y + Z ) + F ( Y , Z ) - F ( X + Y , Z ) - F ( X , Y ) = 0 \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} W _ { p , n } ( i , j ) = \\square _ p [ j - i - 1 ] \\ , \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u + \\lambda ^ 2 e ^ { \\ , u } & = 0 , \\quad \\mbox { i n } \\ \\Omega \\subset \\R ^ 2 , \\\\ u & = 0 , \\quad \\mbox { o n } \\ , \\partial \\Omega , \\end{aligned} \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} 3 \\epsilon & < K ( R - | x - x _ 0 | ) - t \\leq - t + \\psi ( x ) + 2 \\epsilon = \\Phi ( t , x ) + 2 \\epsilon , \\\\ \\epsilon & < \\Phi ( t , x ) . \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} \\frac { 1 } { x _ { k - 1 } } & \\leq \\frac { 1 } { x _ { k } \\big [ 1 - 2 ( q - 1 ) x _ { k } \\big ] } \\\\ & = \\frac { 1 } { x _ { k } } + \\frac { 2 ( q - 1 ) } { 1 - 2 ( q - 1 ) x _ { k } } \\\\ & \\leq \\frac { 1 } { x _ { k } } + 1 0 ( q - 1 ) . \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} K _ i E _ j & = q ^ { \\langle \\alpha _ i , \\alpha _ j \\rangle } E _ j K _ i , & K _ i F _ j & = q ^ { - \\langle \\alpha _ i , \\alpha _ j \\rangle } F _ j K _ i , \\\\ L _ i E _ j & = q ^ { \\langle \\alpha _ i , \\alpha _ j \\rangle } E _ j L _ i , & L _ i F _ j & = q ^ { - \\langle \\alpha _ i , \\alpha _ j \\rangle } F _ j L _ i , \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} { n ; b ; 1 \\brack k } = \\sum _ { i \\ge 0 } \\binom { n } { i } \\binom { n - i } { i + k } = | B _ { n , k } | . \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} | | \\mathbf { T } ^ { \\alpha _ 1 } ( \\mu ) | | _ { L ^ { q _ 0 } ( \\mathbb { R } ^ d ) } \\lesssim ( c _ 1 + c _ 2 ) | \\mu | ( \\mathbb { R } ^ d ) , ~ ~ q _ 0 = \\frac { d } { d - \\frac { 1 } { 4 } \\min \\{ \\alpha , \\alpha _ 0 , \\alpha _ 1 \\} } > 1 . \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathbf { S } \\mathbf { x } + \\mathbf { n } , \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} \\sum _ { x = 1 } ^ n L ( y , x ) \\ & = \\ \\sum _ { x ' = y } ^ n \\sum _ { x = 1 } ^ n \\widehat L ( x , x ' ) - \\widehat L ( x - 1 , x ' ) \\\\ & = \\ \\sum _ { x ' = y } ^ n \\widehat L ( n , x ' ) \\ \\leq \\ \\sum _ { x ' = 1 } ^ n \\widehat L ( n , x ' ) \\ = \\ 0 \\ , \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} | | X _ { P } | | _ { s , r , p - 1 , \\mathbf { a } ; \\Xi _ r } = O ( r ^ { \\frac 7 4 } ) . \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\sigma _ \\gamma ( A ) = \\int _ { \\Omega \\times \\Omega } \\mathcal H ^ 1 ( [ x , y ] \\cap A ) \\ , \\mathrm { d } \\gamma ( x , y ) \\ ; \\ ; \\ ; \\mbox { f o r e v e r y B o r e l s e t } \\ ; A \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} \\theta = k _ \\alpha ( c _ + ^ { \\alpha } ( 1 + \\kappa ) + c _ { - } ^ { \\alpha } ( 1 - \\kappa ) ) , \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ j \\binom k j \\left ( { \\frac { { F _ r } } { { F _ { r + n } } } } \\right ) ^ j G _ { m + r k + n j } } = ( - 1 ) ^ { r k } \\left ( { \\frac { { F _ n } } { { F _ { r + n } } } } \\right ) ^ k G _ m , n + r \\ne 0 \\ , , \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} \\begin{bmatrix} I & V \\\\ U & I \\end{bmatrix} \\begin{bmatrix} \\frac { \\phi } { \\sigma _ { - } } \\\\ \\frac { \\psi } { \\sigma _ { + } } \\end{bmatrix} = \\begin{bmatrix} - P _ { - } \\left ( \\frac { q } { \\sigma { - } } \\right ) \\\\ - P _ { + } \\left ( \\frac { q } { z ^ { n } \\sigma _ { + } } \\right ) \\end{bmatrix} . \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n ( x _ i ( A ) - x _ i ( B ) ) ^ 2 \\leq \\sum _ { i , j = 1 } ^ n | A _ { i j } - B _ { i j } | ^ 2 . \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\# R _ N } { \\# S _ N } = \\frac { \\# \\{ X \\in \\mathbf { M } _ 4 : \\star ( X ) = 1 \\} } { \\# \\mathbf { M } _ 4 } = \\frac { 1 + m _ K n } { 2 ^ n } . \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} H _ { i , j } & = K _ { i , j } - K _ { j , j } , \\\\ K _ { i , j } & = ( Z B ) _ { i , j } , \\\\ Z _ { i , k } & = ( I - P + \\Pi ) ^ { - 1 } _ { i , k } , \\\\ B _ { k , j } & = H ( P _ { k , \\cdot } ) - \\dfrac { H ( X ) } { \\pi _ k } \\ 1 _ { k = j } , \\\\ \\Pi _ { k , j } & = \\pi _ j , \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} \\Psi ( z , w ) = \\left ( z , \\frac { w _ 1 } { z ^ { 1 / 2 m _ 1 } } , \\cdots , \\frac { w _ n } { z ^ { 1 / 2 m _ n } } \\right ) = \\left ( z , \\widehat { \\rho } _ { 1 / z } ( w ) \\right ) , ( z , w ) \\in \\mathcal { U } _ p , \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} y ( r + 0 ) = 0 \\ \\ \\mbox { a n d } \\ \\ \\lim _ { x \\rightarrow r _ { + } } \\{ y ^ { \\prime } ( x ) Y ( x ) - y ( x ) Y ^ { \\prime } ( x ) \\} = 0 ; \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} S _ 2 ( t , q ) & = - \\frac { 1 } { q ^ 2 } \\sum _ { j = 0 } ^ { n - 1 } j \\sum _ { r \\ge 1 } ( r - 1 ) ( - t q ^ j ) ^ { r } \\\\ & = - \\frac { 1 } { q ^ 2 } \\sum _ { r \\ge 1 } ( r - 1 ) ( - t ) ^ r \\sum _ { j = 0 } ^ { n - 1 } j q ^ { j r } \\implies \\\\ \\omega _ n ^ { 2 i } [ t ^ r ] S _ 2 ( t , \\omega _ n ^ i ) & = - ( r - 1 ) ( - 1 ) ^ r \\sum _ { j = 0 } ^ { n - 1 } j = - \\binom { n } { 2 } ( r - 1 ) ( - 1 ) ^ r . \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} X ^ A = X ^ A _ + + X ^ A _ - , X ^ A _ { \\pm } : \\Omega _ m \\to \\Omega _ { m \\pm 1 } . \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j > ( \\log ( 1 / \\epsilon ) ) ^ { 1 + \\delta } } j \\mu \\tilde { B } ^ j \\tilde { r } \\ , \\right \\| = O ( \\epsilon ^ 2 ) , \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} H _ { k , n - 1 } : = \\sum _ { a \\in A _ { + , k } } \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ { n - 1 } ) \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} \\Upsilon ^ 1 { } _ { 4 2 } = 2 m , \\Upsilon ^ 2 { } _ { 4 1 } = - 2 m . \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} \\mathrm { G } = \\sum _ { i = 1 } ^ n \\partial ^ 2 _ { x _ i } - n \\sum _ { i = 1 } ^ n x _ i \\partial _ { x _ i } + \\frac { \\beta } { 2 } \\sum _ { i \\neq j } \\frac { 1 } { x _ i - x _ j } ( \\partial _ { x _ i } - \\partial _ { x _ j } ) , \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} & \\frac { 1 } { 1 - \\theta } \\| u _ h ^ { n } \\| ^ 2 + 4 \\Delta t \\sum _ { k = 2 } ^ n \\| \\nabla u _ h ^ { k - \\theta } \\| ^ 2 + 4 \\gamma \\Delta t \\sum _ { k = 2 } ^ n \\| \\sigma _ h ^ { k - \\theta } \\| ^ 2 \\\\ \\leq & C \\Delta t \\sum _ { k = 0 } ^ n \\| u _ h ^ { k } \\| ^ 2 + 2 \\Delta t \\sum _ { k = 1 } ^ n \\| g ^ { k } \\| ^ 2 . \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\left ( f _ { M , M ' } \\circ ( E _ i \\otimes L _ i ^ { - 1 } ) \\right ) _ { | _ { M _ { \\lambda , \\mu } \\otimes M ' _ { \\lambda ' , \\mu ' } } } = q ^ { \\langle \\lambda + \\alpha _ i , \\mu ' \\rangle - \\langle \\alpha _ i , \\mu ' \\rangle } ( E _ i \\otimes 1 ) _ { | _ { M _ { \\lambda , \\mu } \\otimes M ' _ { \\lambda ' , \\mu ' } } } . \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\binom { n } j ( - 1 ) ^ { j - 1 } \\frac { a _ j } { j ^ k } = \\sum _ { i _ 1 = 1 } ^ { n } \\frac 1 { i _ 1 } \\sum _ { i _ { 2 } = 1 } ^ { i _ { 1 } } \\frac { 1 } { i _ { 2 } } \\dots \\sum _ { i _ { k } = 1 } ^ { i _ { k - 1 } } \\frac 1 { i _ { k } } \\sum _ { j = 1 } ^ { i _ { k } } \\binom { i _ { k } } { j } ( - 1 ) ^ { j - 1 } a _ j . \\end{align*}"}
-{"id": "9907.png", "formula": "\\begin{align*} o r _ { M / X } ( M ) = H ^ { l } _ { M } ( T _ { M } X ; \\Z _ { T _ { M } X } ) \\simeq H ^ { l } _ { M } ( X ; \\Z _ { X } ) . \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} \\hat I ( t ) = \\frac { 1 } { 2 } \\left [ ( m f ^ { - 1 } \\dot \\rho \\hat x _ 1 - \\rho \\hat p _ 1 ) ^ 2 + \\frac { \\nu ^ 2 } { \\rho ^ 2 } \\hat x _ 1 ^ 2 + ( m f ^ { - 1 } \\dot \\rho \\hat x _ 2 - \\rho \\hat p _ 2 ) ^ 2 + \\frac { \\nu ^ 2 } { \\rho ^ 2 } \\hat x _ 2 ^ 2 \\right ] , \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n c ^ k k ! c ^ { n - k } ( n - k ) ! \\le c ^ { n } ( n + 1 ) ! , \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} \\| \\psi ( x ) \\| = \\frac { 1 } { \\sqrt { 2 } } \\sqrt { \\psi ^ 2 ( \\lambda _ 1 ( x ) ) + \\psi ^ 2 ( \\lambda _ 2 ( x ) ) } . \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} \\Lambda = \\Big \\{ ( A _ 1 , \\ldots , A _ m ) \\colon A _ k \\colon H _ k \\to H _ k Q + \\sum _ { k = 1 } ^ m c _ k B _ k ^ \\ast A _ k B _ k \\colon H \\to H \\Big \\} . \\end{align*}"}
-{"id": "9931.png", "formula": "\\begin{align*} ( \\bar { \\partial } \\varphi ) _ \\beta = \\bar { \\partial } \\varphi _ { \\beta _ 1 } \\wedge \\dots \\wedge \\bar { \\partial } \\varphi _ { \\beta _ k } . \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & ( u ^ 3 ) ^ 2 & - u ^ 2 u ^ 3 \\\\ 0 & - u ^ 2 u ^ 3 & 1 + ( u ^ 2 ) ^ 2 \\end{pmatrix} , \\\\ w _ { 1 2 } = w _ { 3 1 } = 0 , w _ { 2 3 } = \\frac { 1 } { \\sqrt { 1 + ( u ^ 2 ) ^ 2 } } . \\end{gather*}"}
-{"id": "9357.png", "formula": "\\begin{align*} y \\coloneqq \\Im \\left ( \\frac { \\rho } { \\rho - \\tau } \\right ) = - \\frac { \\Im ( \\rho ) \\tau } { | \\rho - \\tau | ^ 2 } . \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} { \\sum _ { i = 1 } ^ { d } \\Big ( \\| \\ ; | D _ { q _ i } | ^ { \\frac { 2 } { 3 } } u \\| ^ 2 + \\| \\ ; | \\partial _ { q _ i } V ( q _ i ) | ^ { \\frac { 2 } { 3 } } u \\| ^ 2 _ { L ^ 2 } \\Big ) } \\le c \\| ( \\sqrt { A } + K _ { V } ) u \\| ^ 2 _ { L ^ 2 } \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} T : z \\in [ 0 , 1 ] \\mapsto T ( z ) = \\sum ^ { \\infty } _ { k = 1 } \\mathcal { S } ( a _ k ) / N ^ K \\in [ 0 , 1 ] , \\\\ \\mathcal { S } ( i ) = i + 1 \\mathcal { S } ( N - 1 ) = 0 . \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} W \\tilde { Q } ^ 2 = \\tilde { Q } ^ { ( 1 ) } , \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} { \\mathrm d } \\widetilde { Z } _ { t } ^ { \\varepsilon } \\ , = \\ , \\frac { Z _ { t } ^ { - 1 } b ( t , X _ { t } , F _ { t } ) } { \\ , ( 1 + \\varepsilon Z _ { t } ^ { - 1 } ) ^ { 2 } \\ , } { \\mathrm d } X _ { t } - \\frac { \\varepsilon Z _ { t } ^ { - 2 } \\lvert b ( t , X _ { t } , F _ { t } ) \\rvert ^ { 2 } \\ , } { ( 1 + \\varepsilon Z _ { t } ^ { - 1 } ) ^ { 3 } \\ , } { \\mathrm d } t \\ , ; t \\ge 0 \\ , . \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} \\mathcal { N } \\varphi _ { _ { 0 } } = \\mathcal { P } ( { } _ { 0 } I _ { T } ^ { 1 - q } \\Theta ( 0 ) ) . \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X _ { s } ^ { m } & = [ b ( s , X _ { s } ^ { m } , Y _ { s } ^ { m } , Z _ { s } ^ { m } , u _ { s } ^ { m } ) + l _ { s } ^ { 1 } ] d s + [ \\sigma ( s , X _ { s } ^ { m } , Y _ { s } ^ { m } , Z _ { s } ^ { m } , u _ { s } ^ { m } ) + l _ { s } ^ { 2 } ] d B _ { s } , \\\\ d Y _ { s } ^ { m } & = - g ( s , X _ { s } ^ { m } , Y _ { s } ^ { m } , Z _ { s } ^ { m } , u _ { s } ^ { m } ) d s + Z _ { s } ^ { m } d B _ { s } , \\\\ X _ { t } ^ { m } & = x , \\ Y _ { T } ^ { m } = \\phi \\left ( X _ { T } ^ { m } \\right ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} p _ { a } + p _ { a + 2 } & = ( N _ { a } + 3 N _ { a - 2 } ) + ( N _ { a + 2 } + 3 N _ { a } ) \\\\ & = ( N _ { a } + N _ { a + 2 } ) + 3 ( N _ { a - 2 } + N _ { a } ) \\\\ & = N _ { a + 3 } + 3 N _ { a + 1 } = p _ { a + 3 } . \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} [ t ^ { b n } ] \\prod _ { i = 0 } ^ { n - 1 } ( 1 - t \\omega _ n ^ { k i } ) ^ a = [ t ^ { b ( n , k ) } ] ( 1 - t ) ^ { a ( n , k ) } . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} R ^ { ( i ) } ( D ) = \\min \\limits _ { p ( \\hat { u } _ i | u _ i ) : \\mathbb { E } [ d ( U _ i , \\hat { U } _ i ) ] \\le D } I ( U _ i ; \\hat { U } _ i ) \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} \\mathbf { H } \\times \\boldsymbol { \\nu } + \\mathbf { g } ( \\mathbf { E } \\times \\boldsymbol { \\nu } ) \\times \\boldsymbol { \\nu } = \\mathbf { 0 } ( 0 , \\infty ) \\times \\Gamma \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} \\prod _ { k = \\ell ( s ) + 1 } ^ \\infty \\left ( 1 + c _ 1 e ^ { c _ 1 | s | - c _ 2 k } \\right ) \\leq \\prod _ { m = 1 } ^ \\infty \\left ( 1 + e ^ { - c _ 2 m } \\right ) , \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} S \\left [ x _ 1 ( t ) , x _ 2 ( t ) , \\frac { d x _ 1 ( t ) } { d t } , \\frac { d x _ 2 ( t ) } { d t } \\right ] = \\int _ { t _ 1 } ^ { t _ 2 } L ( x _ 1 , x _ 2 , \\dot x _ 1 , \\dot x _ 2 ) d t , \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} ( t , y ) + ( t ' , y ' ) = ( t t ' h ( y , y ' ) , y + y ' ) \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} [ L ^ t _ \\theta , u _ i [ f _ i ] ] & = \\sum _ { k \\geq - N } ( - m a _ { m k + 1 } ) [ L ^ t _ k , u _ i [ f _ i ] ] \\\\ & = \\sum _ { k \\geq - N } ( - a _ { m k + 1 } ) u _ i [ - t ^ { m k + 1 } \\partial _ t ( f _ i ) ] \\\\ & = u _ i [ \\theta ( f _ i ) ] , \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} \\psi ( t , z ) = \\exp \\left ( \\frac { t } { i } \\left ( - \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial z ^ 2 } + z \\right ) \\right ) \\phi ( z ) . \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} \\left [ \\begin{array} { l } \\lambda _ { n + 1 } - \\lambda _ n \\\\ \\lambda _ { n + 1 } - \\lambda _ { n - 1 } \\end{array} \\right ] = \\left [ \\begin{array} { l } 0 \\\\ 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} K ( z , z '' ) = & \\Big ( \\int _ { r ' < \\frac { r '' } { 2 } } + \\int _ { \\frac { r '' } { 2 } \\leq r ' \\leq 2 r '' } + \\int _ { r ' > 2 r '' } \\Big ) G ( z , z ' ) Q ( z ' , z '' ) \\ ; d \\mu ( z ' ) \\\\ = & K _ { 2 , 1 } ( z , z '' ) + K _ { 2 , 2 } ( z , z '' ) + K _ { 2 , 3 } ( z , z '' ) . \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} F _ i ( 0 ) = 0 P _ i ( x ) \\leqslant C ( 1 + F _ i ( x ) ) . \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} \\begin{cases} u \\in \\textup { P S H } ( X , \\theta ) , \\\\ \\theta _ u ^ n = f \\omega ^ n , \\\\ [ u ] = [ \\phi ] , \\end{cases} \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} u \\nabla _ { \\nabla u } Y ^ T = u ( \\nabla _ { g _ S } ) _ { \\nabla u } Y ^ T - \\frac { 1 } { 2 } u d \\theta ( \\nabla u , Y ^ T ) \\cdot \\partial _ t \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} D ^ { \\varphi } _ f ( x ) & : = D _ f ^ { \\phi } ( f ( z _ 1 ) \\dots f ( z _ n ) x ) \\\\ & = 1 + c ( f ) p _ 1 \\prod _ { i = 1 } ^ n f ( z _ i ) x + c ( f ) p _ 2 f \\prod _ { i = 1 } ^ n f ( z _ i ) ^ 2 x ^ 2 + \\dots + c ( f ) f ^ { r _ 0 - 1 } \\prod _ { i = 1 } ^ n f ( z _ i ) ^ { r _ 0 } x ^ { r _ 0 } . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} T ( n , k ) = \\sum _ { j = 0 } ^ { n - 1 - k } K ( n , k , j ) . \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} H u = - u ^ { \\prime \\prime } + ( V ( x ) + V _ 0 ( x ) ) u = E u , \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} \\sigma _ { 1 } ^ { j - 1 } \\cdot H _ { j , a _ { 1 , 1 } } \\cdot c ^ { 1 } _ { 1 } + \\sum _ { q _ i \\in Q } \\sigma _ { q _ i } ^ { j - 1 } \\cdot H _ { j , a _ { q _ i , 1 } } \\cdot c ^ { q _ i } _ { 1 } + \\sum _ { v _ i \\in V } \\sigma _ { v _ i } ^ { j - 1 } \\cdot H _ { j , a _ { v _ i , 1 } } \\cdot c ^ { v _ i } _ { 1 } = 0 j \\in [ r ] . \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} \\{ g _ 1 , \\dots , g _ m \\} = \\{ \\chi _ { D _ 1 } , \\dots , \\chi _ { D _ N } , x _ 1 \\chi _ { D _ 1 } , \\dots , x _ 1 \\chi _ { D _ N } , x _ 2 \\chi _ { D _ 2 } , \\dots , x _ 2 \\chi _ { D _ 2 } , \\dots , x _ n \\chi _ { D _ N } , \\dots , x _ n \\chi _ { D _ N } \\} . \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\frac 1 2 ) _ k ( r ) _ k ( 1 - r ) _ k } { k ! ^ 3 } z ^ k \\end{gather*}"}
-{"id": "3202.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { \\partial u } { \\partial t } ( t , x ) & = \\frac { 1 } { 2 } \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } ( t , x ) + b ( u ( t , x ) ) \\\\ & + \\sigma ( u ( t , x ) ) \\dot { W } ( t , x ) + \\eta ( d t d x ) , \\ t > 0 , \\ x \\in ( 0 , 1 ) , \\\\ u ( t , 0 ) & = u ( t , 1 ) = 0 , \\ t \\geq 0 , \\\\ u ( 0 , x ) & = h ( x ) , \\ x \\in [ 0 , 1 ] , \\\\ u ( t , x ) & \\geq 0 , \\ t > 0 , \\ x \\in [ 0 , 1 ] \\ a . s . , \\end{aligned} \\right . \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} z = \\left ( \\frac { t } { 2 s } \\right ) ^ { 2 s } , \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} q _ { n } ( x ) \\stackrel { \\mathrm { d e f } } { = } \\sup _ { | z - x | \\leq 3 h _ { n } / 2 } \\left \\vert f ( z ) - f ( x ) \\right \\vert . \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} G _ { g , n } ( \\omega ) = \\sum _ { d \\mid \\frac { n } { \\mathrm { o r d } ( \\omega ) } } \\sum _ { e \\mid d } \\mu ( \\frac { d } { e } ) f ( \\omega _ n ^ e ) . \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} E _ F = \\{ e \\in E : s ( e ) , r ( e ) \\in F \\} . \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} U _ n = S _ n ^ { - \\frac 1 2 } ( D U _ A ^ * U _ A ) ^ 2 S _ n ^ { - \\frac 1 2 } \\ \\mbox { a n d } \\ V _ n = S _ n ^ { - \\frac 1 2 } D D ^ * S _ n ^ { - \\frac 1 2 } \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} \\varphi ( 2 t ^ 2 - 5 t + 2 ) = \\varphi ( 2 - t ) = \\varphi ( 1 - 2 t ) = 0 . \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} d \\mu _ z ( x ) = \\frac { 1 } { Z \\ , Q ( z ) } \\exp \\Big [ - \\beta \\Big ( V _ 0 ( x ) + \\frac { 1 } { \\epsilon } V _ 1 ( x ) \\Big ) \\Big ] \\Big [ \\mbox { d e t } ( \\nabla \\xi \\nabla \\xi ^ T ) ( x ) \\Big ] ^ { - \\frac { 1 } { 2 } } \\ , d \\nu _ z ( x ) \\ , , \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} \\phi _ { i , n } > 0 , \\ ; \\phi _ { i , n } \\to u _ i \\ ; \\ ; W ^ { 1 , 1 } _ { l o c } ( \\Omega ) \\ ; \\ ; \\Delta \\phi _ { i , n } \\to V u _ i \\ ; \\ ; L ^ { 1 } _ { l o c } ( \\Omega ) \\ ; \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\left ( \\frac { m ^ 2 } { \\tau _ h ^ 2 } - ( h - \\tau _ h ) ^ { 2 b } \\right ) \\tau _ h ' = \\tau _ h - t _ h , \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} \\dim T o p ( \\phi ) = q ^ { ( e - 1 ) ( m - 1 ) } ( q ^ m + ( - 1 ) ^ { m + 1 } ) / ( q + 1 ) . \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} & \\eta = C \\exp \\big ( C M + C M ^ 4 \\big ) \\Big ( X _ n ( 0 ) + M ^ 2 \\Big ) \\Big \\{ \\nu ^ { - \\frac 1 2 } M + \\nu ^ { - 1 } \\Big ( X _ n ( 0 ) + M ^ 2 + 1 \\Big ) \\Big \\} . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} \\Phi ( \\mu _ f ^ \\star ) = \\mu _ f ^ \\star , \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} V ^ { a _ { \\theta ^ * _ { k - 1 } } \\eta ^ * _ { n } } ( X _ { \\theta ^ * _ { k } } ) - V ^ { a _ { \\theta ^ * _ { k - 1 } } \\eta ^ * _ { n - 1 } } ( X _ { \\theta ^ * _ { k } } ) + \\chi ( \\eta ^ * _ { n - 1 } , \\eta ^ * _ n ) = 0 \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} - i \\sigma ^ \\alpha { } _ { \\dot a b } \\ , \\partial _ { x ^ \\alpha } \\xi ^ b = m \\ , \\eta _ { \\dot a } \\ , , - i \\sigma ^ { \\alpha \\dot b a } \\ , \\partial _ { x ^ \\alpha } \\eta _ { \\dot b } = m \\ , \\xi ^ a \\ , . \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} W ^ { ( p ) } = e ^ { \\Phi ( p ) \\cdot } W _ { \\Phi ( p ) } , \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} x _ { t + 1 } = x _ { t } + f ( x _ { t } , \\theta _ { t } ) , t = 0 , \\dots , T - 1 . \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\frac { 1 } { B } - 1 = \\frac { ( d - 1 ) X _ \\alpha X _ \\beta + Y X _ \\beta } { Y ( X _ \\alpha - X _ \\beta ) } . \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} \\Psi ( v ) = 0 \\Leftrightarrow \\psi ( v ) = 0 \\Leftrightarrow \\psi ^ { \\prime } ( v ) = 0 \\Leftrightarrow v = e \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} D = \\{ x < x ^ { \\star } | x , x ^ { \\star } \\in S \\} . \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} { 1 \\over \\alpha } \\ , D _ \\alpha ( \\rho \\| \\sigma ) = { 1 \\over 1 - \\alpha } \\ , D _ { 1 - \\alpha } ( \\sigma \\| \\rho ) + { 1 \\over \\alpha ( 1 - \\alpha ) } \\log { \\rho ( 1 ) \\over \\sigma ( 1 ) } . \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} ( a y ^ 2 + z ^ 2 - ( a { + } 1 ) x ^ 2 - a ^ 2 - b ^ 2 - a ) ^ 2 \\ + \\ b ^ 2 ( x ^ 2 - y ^ 2 + 1 ) ^ 2 \\ - \\ 4 b ^ 2 x ^ 2 \\ = \\ 0 \\ , . \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} ( \\mathcal { L } ^ \\omega _ X f ) ( x ) = \\sum _ { y \\sim x } \\omega ( x , y ) ( f ( y ) - f ( x ) ) . \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\mbox { V a r } _ { \\mu _ z } ( f ) = \\int _ { \\Sigma _ z } f ^ 2 \\ , d \\mu _ z - \\Big ( \\int _ { \\Sigma _ z } f \\ , d \\mu _ z \\Big ) ^ 2 \\le \\frac { C \\epsilon \\delta } { c _ 2 K } \\mathcal { E } _ z ( f , f ) \\ , , \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{align*} \\overline { \\phi } _ { \\theta } = \\sum _ { i = 0 } ^ { r _ 0 } \\overline { \\phi _ { \\theta , i } } \\tau ^ i \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { \\beta } + 1 \\right ) u = \\frac { \\alpha } { \\beta } , \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} N ( \\varphi , \\sigma ) = \\frac { \\chi _ { \\sigma } } { 2 } | \\sigma | ^ 2 + \\chi _ { \\varphi } \\sigma ( 1 - \\varphi ) , \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} u ( \\alpha ( i _ 1 , 0 ) ) = \\left \\lceil i _ 1 \\left ( \\frac { 1 + \\sqrt { 5 } } { 2 } \\right ) ^ 2 \\right \\rceil \\ , . \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} D ^ b ( C ^ { [ n + g - 1 ] } ) = \\langle \\overbrace { D ^ b ( J _ C ) , \\ldots , D ^ b ( J _ C ) } ^ { n } , D ^ b ( C ^ { [ - n + g - 1 ] } ) \\rangle . \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} c _ 0 = 1 , c _ 1 = \\frac { 1 } { 2 } \\ , , c _ 2 = \\frac { 1 } { 1 2 } \\ , , c _ 3 = 0 , c _ 4 = \\frac { - 1 } { 7 2 0 } \\ , , \\qquad \\dots . \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} q a _ n ' ( q ) & = \\sum _ { k , \\ell } { n \\brack k } _ q q ^ { f ( n , k , \\ell ) } \\Big ( { n \\brack k } _ q \\big ( { n \\brack \\ell } _ q { k \\brack \\ell } _ q { k + \\ell \\brack n } _ q \\big ) ' \\\\ & + \\big ( { n \\brack k } _ q f ( n , k , \\ell ) + 2 { n \\brack k } _ q ' q \\big ) { n \\brack \\ell } _ q { k \\brack \\ell } _ q { k + \\ell \\brack n } _ q \\Big ) . \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} F = F ^ x + \\langle F ^ y , y \\rangle + \\langle F ^ z , z \\rangle + \\langle F ^ { \\bar { z } } , \\bar { z } \\rangle + \\langle F ^ { z z } z , z \\rangle + \\langle F ^ { z \\bar { z } } z , \\bar { z } \\rangle + \\langle F ^ { \\bar { z } \\bar { z } } \\bar { z } , \\bar { z } \\rangle , \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} c ^ \\tau ( x * y ) & = \\tau ( g _ 1 , g _ 2 ) c ^ \\tau ( x y ) \\\\ & = \\tau ( g _ 1 , g _ 2 ) \\tau ( h , g _ 1 g _ 2 ) c ( x y ) \\\\ & = \\tau ( h , g _ 1 ) \\tau ( h g _ 1 , g _ 2 ) c ( x ) y \\\\ & = \\tau ( h g _ 1 , g _ 2 ) c ^ \\tau ( x ) y = c ^ \\tau ( x ) * y \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} s = ( \\frac { 1 } { 2 } - \\frac { 1 } { p } ) d - \\frac { a } { q } . \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} i \\partial _ t v ( t ) & = H v + P _ c F ( v ( s ) + a ( s ) \\phi _ 0 ) , \\\\ i \\partial _ t a ( t ) & = - \\tfrac 1 2 { q ^ 2 } a ( t ) + \\langle \\phi _ 0 , F ( v ( t ) + a ( t ) \\phi _ 0 ) \\rangle . \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} ( \\sigma _ h ^ { n - \\theta } , w _ h ) + ( \\nabla u _ h ^ { n - \\theta } , \\nabla w _ h ) = 0 , ~ \\forall w _ h \\in L _ h , \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } W _ t f ( n ) = f ( n ) , \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} \\limsup _ { \\epsilon \\to 0 } \\limsup _ { L \\to \\infty } \\frac 1 { L ^ d } \\sum _ { i \\in \\mathbb T _ L ^ d } \\sum _ { k = 1 } ^ d \\sum _ { \\eta \\in \\Omega _ L } \\mu ^ L _ { [ 0 , T ] } ( \\eta ) \\ ; \\ ! \\Bigl | \\hat \\chi _ { i , i + e _ k } ^ { \\lfloor \\epsilon L \\rfloor } ( \\delta _ \\eta ) - \\hat \\chi _ { i , i + e _ k } ( \\nu _ { \\eta ^ { \\lfloor \\epsilon L \\rfloor } ( i ) } ) \\Bigr | = 0 , \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} \\int _ 0 ^ { t \\wedge \\tau _ M } g ( s , W _ s ) \\mathrm d s = \\int _ 0 ^ { t \\wedge \\tau _ M } g _ M ( s , W _ s ) \\mathrm d s . \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} \\mathbf { K } ^ { \\varepsilon , n } _ { e , \\rho } ( z ' ) = \\frac { \\varepsilon ^ { - d + 1 } } { \\rho ^ \\alpha } \\int _ { \\mathbb { R } ^ d } \\mathbf { K } _ { n } ( z ) \\varphi _ { \\zeta \\rho } ( z ' - z ) \\frac { \\langle \\phi ^ { e , \\varepsilon } _ \\rho ( z ' - z ) , \\eta ^ { \\kappa } _ { y _ { \\tau } } \\rangle } { | z ' - z | ^ { d - \\alpha } } d z - \\mathbf { c } ( \\varepsilon , \\kappa , \\tau , \\zeta ) \\varphi _ { \\rho } ( z ' ) \\mathbf { K } _ { n } ( z ' ) ~ ~ \\forall ~ ~ z ' \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} \\frac { \\log ( 3 . 7 0 8 \\cdot 1 0 ^ { - 5 } ) - \\log ( 1 . 0 1 4 \\cdot 1 0 ^ { - 6 } ) } { \\log ( 5 4 1 ) - \\log ( 8 7 8 0 ) } = - 1 . 2 9 1 \\ldots \\approx - 1 . 3 , \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} \\norm { \\widetilde { T } _ V f } _ { L ^ 2 ( \\mathcal { C } _ p ) } ^ 2 = \\int _ { \\mathbb { B } _ p } \\int _ { V _ { \\zeta } } \\abs { \\widetilde { T } _ V f ( \\gamma , \\zeta ) } ^ 2 \\d V ( \\gamma ) \\d V ( \\zeta ) = \\int _ { \\mathbb { B } _ p } \\int _ { \\R } \\abs { f ( t , w ) } ^ 2 \\lambda ( p ( \\zeta ) , t ) \\d t \\d V ( \\zeta ) = \\norm { f } _ { \\mathcal { X } _ p } ^ 2 , \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N \\lambda _ n t ^ { 1 / p ^ n } = a w \\in k . \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} \\vartheta \\left ( \\sum _ { n = 0 } ^ { \\infty } a _ n z ^ n \\right ) = - \\sum _ { n = 1 } ^ { \\infty } a _ { n + 1 } z ^ n . \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} F ( \\xi ^ { ( 1 ) } _ n , \\xi ^ { ( n ) } _ { 2 n } , . . . , \\xi _ { \\ell n } ^ { ( \\ell ) } ) = V _ { n + 1 } - V _ n , \\ , P - \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} & f ' ( h _ 1 ) ( \\xi ) \\\\ & = \\sum _ j f ( g _ j ) ( \\pi ( \\xi ) | _ { X _ j } ) \\\\ & = \\sum _ j [ \\sum _ { g } \\theta ( g , g _ j ) ( g ( \\pi ( \\xi ) | _ { X _ j } ) ) - \\sum _ { \\substack { g , \\bar { g } \\\\ g \\bar { g } = g _ j } } \\theta ( g , \\bar { g } ) ( \\pi ( \\xi ) | _ { X _ j } ) + \\sum _ { g } \\theta ( g _ j , g ) ( \\pi ( \\xi ) | _ { X _ j } ) ] . \\\\ \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} X _ n = \\sum _ { i = - \\infty } ^ { \\infty } c _ i \\varepsilon _ { n - i } , n \\geq 1 . \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} 2 \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 3 n ) q ^ n & = \\dfrac { f _ 3 ^ 3 } { f _ 1 ^ 5 f _ 2 ^ 2 } a ^ 2 ( q ) - \\dfrac { f _ 3 ^ 3 } { f _ 1 ^ { 5 } f _ 2 ^ 2 } a ( q ^ 2 ) \\\\ & \\equiv \\dfrac { 1 } { f _ 4 ^ 2 } \\cdot \\dfrac { f _ 3 ^ 3 } { f _ 1 } - \\dfrac { a ( q ^ 2 ) } { f _ 4 ^ 2 } \\cdot \\dfrac { f _ 3 ^ 3 } { f _ 1 } \\\\ & \\equiv \\left ( \\dfrac { 1 } { f _ 4 ^ 2 } - \\dfrac { a ( q ^ 2 ) } { f _ 4 ^ 2 } \\right ) \\left ( \\dfrac { f _ 4 ^ 3 f _ 6 ^ 2 } { f _ 2 ^ 2 f _ { 1 2 } } + q \\dfrac { f _ { 1 2 } ^ 3 } { f _ 4 } \\right ) ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} b ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { | T ' ( x ) | } \\left [ b ( x , y , t ) + \\frac { 1 } { 4 } h ( x ) \\left ( 2 x ^ { 2 } - 1 \\right ) \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } p _ { 0 } \\left ( y , t \\right ) \\right ] \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} f ( n + 1 ) ^ 2 + d f ( n + 1 ) & = \\left ( \\sum _ { i = 1 } ^ n ( k _ { i - 1 } ( f ) - f ( i ) ) \\right ) ^ 2 + d \\sum _ { i = 1 } ^ { n } ( k _ { i - 1 } ( f ) - f ( i ) ) \\\\ & \\qquad \\qquad \\qquad \\qquad + 2 d \\eta \\sum _ { i = 1 } ^ { n } ( k _ { i - 1 } ( f ) - f ( i ) ) + d ^ 2 \\underbrace { \\eta ( 1 + \\eta ) } _ { \\equiv 0 \\mod 2 } \\\\ & \\equiv \\left ( \\sum _ { i = 1 } ^ n ( k _ { i - 1 } ( f ) - f ( i ) ) \\right ) ^ 2 + d \\sum _ { i = 1 } ^ { n } ( k _ { i - 1 } ( f ) - f ( i ) ) \\mod 2 d . \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} \\mathbb { T } _ { f } = \\theta _ { L _ { \\xi } ( \\lambda _ f , \\mu _ f ) } \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} P f ( z ) = f ( ( 1 + \\sigma ) z ) . \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} \\begin{aligned} & - \\frac { \\log { R } } { 2 } \\left ( n - \\frac { 1 } { 2 } \\right ) \\exp \\left ( - \\frac { \\log { R } } { 2 } \\left ( n - \\frac { 1 } { 2 } \\right ) \\right ) \\\\ & \\quad \\le - \\frac { \\log { R } } { 2 } \\sqrt { \\frac { \\mathcal { N } _ F ( T ) } { K _ { F , 3 } ( T , \\tau ) } } \\exp \\left ( \\frac { \\log { R } } { 4 } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} & \\bigg | \\frac { \\partial ^ { p + 2 } f } { \\prod _ { j = 1 } ^ { p + 2 } \\partial x _ { i _ j } } ( x ) - \\frac { \\partial ^ { p + 2 } f } { \\prod _ { j = 1 } ^ { p + 2 } \\partial x _ { i _ j } } ( y ) \\bigg | \\leq - [ h ] _ { \\alpha , p } \\log \\eta + [ h ] _ { \\alpha , p } 2 ^ { \\alpha / 2 + 1 } \\frac { \\alpha + d + 1 } { \\alpha } \\frac { \\Gamma ( \\frac { \\alpha + d } { 2 } ) } { \\Gamma ( d / 2 ) } \\eta ^ { \\alpha / 2 } . \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ^ { k + 1 } - q \\| ^ 2 & \\leq \\beta _ k \\| v ^ { k } - q \\| ^ 2 + ( 1 - \\beta _ k ) \\| u ^ k - q \\| ^ 2 . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} g \\cdot f = \\sum _ { j _ 1 , \\dots , j _ n , j _ { n + 1 } \\geq 0 } ( z _ 1 ^ { j _ 1 } \\cdot f ) \\dots ( z _ n ^ { j _ n } \\cdot f ) ( t ^ { j _ { n + 1 } } \\cdot f ) . \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} c ( x , y ) = \\pi ( x ) p ( x , \\ , y ) , \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} W = \\sum _ { n \\ge 1 } \\sum _ { \\begin{subarray} { c } \\{ 1 , \\ldots , n + 1 \\} \\stackrel { \\psi } { \\to } V ( Q ) , \\\\ \\psi ( n + 1 ) = \\psi ( 1 ) \\end{subarray} } \\sum _ { e _ i \\in E _ { \\psi ( i ) , \\psi ( i + 1 ) } } a _ { \\psi , e _ { \\bullet } } \\cdot e _ 1 e _ 2 \\ldots e _ { n } \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} v = x _ { i _ 1 } x _ { i _ 2 + { 1 } } \\cdots x _ { { i _ { l - 1 } } + ( l - 2 ) } x _ k x _ { { i _ { l + 1 } } + l } \\cdots x _ { { i _ d } + ( d - 1 ) } \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} u _ 2 \\nabla u _ { 1 } = u _ 1 \\nabla u _ { 2 } \\ ; \\ ; \\Omega . \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} \\widetilde { b } _ l ( z ) = & \\int _ { \\Sigma _ z } ( \\mathcal { L } \\xi _ l ) ( x ) \\ , d \\mu _ z ( x ) = \\mathbf { E } _ \\mu \\bigg [ ( \\mathcal { L } \\xi _ l ) ( x ) \\ , \\Big | \\ , \\xi ( x ) = z \\bigg ] \\ , , 1 \\le l \\le m \\ , , \\\\ \\widetilde { \\sigma } ( z ) = & \\bigg [ \\int _ { \\Sigma _ z } \\big ( \\nabla \\xi a \\nabla \\xi ^ T \\big ) ( x ) \\ , d \\mu _ z ( x ) \\bigg ] ^ { \\frac { 1 } { 2 } } \\ , , \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} W _ 2 ^ 2 ( \\rho , \\rho _ t ^ \\xi ) \\le t \\int _ 0 ^ t \\| \\nabla \\xi \\| _ { \\rho _ s ^ \\xi } ^ 2 \\ ; \\ ! \\mathrm d s = t ^ 2 ( \\| \\nabla \\xi \\| _ \\rho ^ 2 + o ( 1 ) ) . \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} \\phi _ { X _ 0 } ( 1 ) = \\phi _ { X _ 1 } ( 1 ) = 1 . \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} \\begin{aligned} & r _ - ( x ) - \\hat r _ - ( x ) = \\frac D 2 \\lambda ( l + 1 ) ( \\gamma ( \\lambda ) + x ^ 2 ) ^ { \\frac { l - 3 } 2 } \\times \\\\ & \\Big [ \\frac { 2 c _ 1 } { D \\lambda ( l + 1 ) } ( \\gamma ( \\lambda ) + x ^ 2 ) ^ { \\frac { 3 - l } 2 } ( \\gamma _ 1 + x ^ 2 ) ^ l - ( l + 1 ) \\lambda x ^ 2 ( \\gamma ( \\lambda ) + x ^ 2 ) ^ { \\frac { l + 1 } 2 } - \\gamma ( \\lambda ) - l x ^ 2 \\Big ] . \\end{aligned} \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} K _ { i \\bar j } = - \\frac 1 2 \\mu _ { i k } ^ r \\mu _ { \\bar j \\bar k } ^ { \\bar r } + \\frac 1 4 \\mu _ { \\bar k \\bar r } ^ { \\bar i } \\mu _ { k r } ^ j + \\frac 1 2 \\mu _ { i k } ^ k \\mu _ { \\bar j \\bar r } ^ { \\bar r } - \\frac { 1 } { 2 } \\left ( \\mu _ { r k } ^ k \\mu _ { \\bar r \\bar j } ^ { \\bar i } + \\mu _ { r i } ^ j \\mu _ { \\bar r \\bar k } ^ { \\bar k } \\right ) . \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} { \\rm g c d } ( p _ 1 , p _ 2 ) = { \\rm g c d } ( r _ 1 , s _ 1 ) = { \\rm g c d } ( r _ 2 , s _ 2 ) = 1 \\ , . \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\left | ( w _ { m } ) _ { B _ { r _ j } } - ( w _ { m } ) _ { B _ { r _ 0 } } \\right | \\leq C \\sum _ { i = 1 } ^ { j } \\left \\| w _ { m } - ( w _ { m } ) _ { B _ { r _ i } } \\right \\| _ { \\underline { L } ^ 2 \\left ( B _ { r _ i } \\right ) } \\leq C r _ j r _ { m } ^ { - \\beta } . \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} \\nabla ^ 2 _ { x x } L ( x , y ) = \\nabla ^ 2 ( y c ) ( x ) = \\sum _ { i = 1 } ^ m y _ i \\nabla ^ 2 c _ i ( x ) . \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} & B _ q ( x , \\epsilon ) = \\{ y \\in X : q ( y - x ) < \\epsilon \\} , \\\\ & B _ q [ x , \\epsilon ] = \\{ y \\in X : q ( y - x ) \\le \\epsilon \\} . \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} \\Psi _ { \\tau , \\alpha } ( G ' ) = \\bigcup _ { G } \\Psi _ { \\tau , \\alpha } ( G , G ' ) \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} \\int _ { \\R } \\int _ { \\mathbb { B } _ p } g \\left ( t , \\frac { \\zeta _ 1 } { z ^ { 1 / 2 m _ 1 } } , \\cdots , \\frac { \\zeta _ n } { z ^ { 1 / 2 m _ n } } \\right ) \\overline { \\phi ( t , \\zeta ) } \\lambda ( p ( \\zeta ) , t ) \\d V ( \\zeta ) \\d t = 0 , \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} \\partial _ t u ( t ) & = \\partial _ t \\int _ 0 ^ t w ( t - y ) \\left ( \\int _ 0 ^ \\infty e ^ { - \\lambda s } p ^ \\beta _ s ( y ) \\ , d s \\right ) d y \\\\ & = w ( 0 ) \\int _ 0 ^ \\infty e ^ { - \\lambda s } p ^ \\beta _ s ( t ) \\ , d s + \\int _ 0 ^ t w ' ( t - y ) \\int _ 0 ^ \\infty e ^ { - \\lambda s } p ^ \\beta _ s ( y ) \\ , d s \\ , d y , & t > 0 , \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} \\lambda _ Q ( f ) = Q ^ { k / 2 - 1 } \\frac { G ( \\chi _ Q ) } { a _ Q } , \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} U _ \\lambda ( o , \\ , o \\ , | \\ , z ) = \\frac { ( d - 1 + \\lambda ) - \\sqrt { ( d - 1 + \\lambda ) ^ 2 - 4 \\lambda ( d - 1 ) z ^ 2 } } { 2 ( d - 1 ) } . \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} f ( \\psi ) = \\sum _ { j = 1 } ^ k \\left \\{ \\big [ - ( - \\Delta ) ^ { \\frac { s } { 2 } } \\eta _ { j , R } , - ( - \\Delta ) ^ { \\frac { s } { 2 } } \\tilde { \\phi } _ j \\big ] + \\tilde { \\phi } _ j \\big ( - ( - \\Delta ) ^ { s } - \\partial _ t \\big ) \\eta _ { j , R } \\right \\} + \\tilde { N } _ { \\mu , \\xi } ( \\tilde { \\phi } ) + S _ { o u t } . \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\in \\mathbb { Z } } a ( n , y ) e ^ { 2 \\pi i n x } . \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} M _ p ( 2 m ) \\sim C _ m \\cdot p ^ { m + 2 } = \\frac { ( 2 m ) ! } { m ! ( m + 1 ) ! } \\cdot p ^ { m + 2 } . \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} \\dot { q } _ a = \\frac { \\delta h _ \\textit { \\tiny R E G } } { \\delta p _ a } = u ( q _ a ( t ) , t ) \\quad \\hbox { a n d } \\dot { p } _ a = - \\ , \\frac { \\delta h _ \\textit { \\tiny R E G } } { \\delta q _ a } \\ , , \\quad \\hbox { f o r } a = 1 , 2 , \\dots , N , \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} & c _ k ( f _ { 0 . 9 , \\ , 0 . 4 } ) = O ( 1 / | k | ^ 2 ) + O \\left ( 1 / | k | ^ { ( 1 + 0 . 9 + 0 . 2 ) / ( 0 . 4 + 1 ) } \\right ) \\\\ & = O \\left ( 1 / | k | ^ { 1 . 5 } \\right ) . \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} | c | ^ { l i p } = \\frac { 2 n } { 2 n - 1 } , \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} ( X _ { t } ^ { ( 0 ) } ) \\ , = \\ , ( X _ { t } ^ { \\bullet } ) \\ , = \\ , \\frac { \\ , 1 - e ^ { - 2 t } \\ , } { 2 } \\ , , ( X _ { t } ^ { ( 1 ) } ) = ( X _ { t } ^ { \\dagger } ) \\ , = \\ , t \\ , e ^ { - 2 t } ( I _ { 0 } ( 2 t ) + I _ { 1 } ( 2 t ) ) \\ , . \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} ( \\Sigma _ h ^ { \\frac 1 2 } , w _ h ) + ( \\nabla U _ h ^ { \\frac 1 2 } , \\nabla w _ h ) = 0 , ~ \\forall w _ h \\in L _ h , \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} c ( X ) = \\vartheta ^ { n - 1 } \\mathfrak { e } _ n ( d ) H + \\vartheta ^ { n - 2 } \\mathfrak { e } _ n ( d ) H ^ 2 + \\cdots + \\vartheta \\mathfrak { e } _ n ( d ) H ^ { n - 1 } + \\mathfrak { e } _ n ( d ) H ^ { n } . \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} \\square ^ n : = \\{ 0 < 1 \\} ^ { n } \\ . \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} T _ { n , \\ell } \\triangleq \\sum _ { { \\mathbf x } \\in \\mathcal X _ g : x _ { n , \\ell } = 1 } p _ { \\mathbf x } , n \\in \\mathcal N , \\ell \\in \\mathcal L . \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} E ( \\Lambda ) = \\{ j k \\in E | j k \\cap \\Lambda \\not = \\emptyset \\} . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} I _ s = I _ s ( n ) = \\int _ 0 ^ 1 \\dots \\int _ 0 ^ 1 \\frac { P _ n ( x _ 1 ) Q _ n ( x _ 2 ) T _ n ( x _ 3 ) } { 1 - x _ 1 x _ 2 x _ 3 \\dots x _ s } d x _ 1 d x _ 2 d x _ 3 \\dots d x _ s . \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} \\frac { \\partial v } { \\partial t } + ( v \\cdot \\nabla ) v = \\nu \\Delta v - \\nabla p , \\div ( v ) = 0 , \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} f ' _ t ( t , x ) + m f ' _ x ( t , x ) + 2 x f '' _ { x , x } ( t , x ) = 0 , ( t , x ) \\in ( 0 , \\infty ) ^ 2 . \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} E _ 2 ^ { 1 , 2 } = \\frac { \\ker ( f _ 1 ) } { { \\rm { i m } } ( f _ 0 ) } . \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\eta _ { \\ast } \\mathcal { O } _ { Z _ { \\mathbf i } } = \\mathcal { O } _ { Z _ { \\mathcal I } } . \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} \\prod _ { k = 0 } ^ { d ' - 1 } \\left ( 1 + \\zeta ^ { 2 k } X \\right ) = \\zeta ^ { d ' ( d ' - 1 ) } ( X ^ { d ' } - ( - 1 ) ^ { d ' } ) = \\zeta ^ { d ' ( d ' + 1 ) } ( X ^ { d ' } + ( - 1 ) ^ { d ' + 1 } ) . \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} g \\big ( \\nabla \\phi ( x ( 0 ) ) , V ( 0 ) \\big ) = g \\big ( \\nabla \\phi ( x ( 1 ) ) , V ( 1 ) \\big ) = 0 , \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - \\lambda t } \\int _ { x } ^ { \\infty } e ^ { - \\xi y } g ( x , y , t ) \\ , \\mathrm { d } y \\ , \\mathrm { d } t = { } & e ^ { - \\xi x } \\int _ { 0 } ^ { \\infty } e ^ { - \\lambda t } E _ { \\frac { 1 } { 2 } } \\bigl ( - \\xi t ^ { \\frac { 1 } { 2 } } \\bigr ) \\ , \\mathrm { d } t \\\\ = { } & e ^ { - \\xi x } \\frac { \\lambda ^ { \\frac { 1 } { 2 } - 1 } } { \\xi + \\lambda ^ { \\frac { 1 } { 2 } } } \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} J _ \\gamma ( u ) \\ ; & : = \\ ; \\frac { \\mu \\lambda } { | \\cdot | ^ \\gamma } * | u | ^ 2 , \\\\ J _ \\gamma ^ { ( \\alpha ) } ( u ) \\ ; & : = \\ ; \\frac { \\mu \\lambda } { | \\cdot | ^ \\gamma + \\alpha } * | u | ^ 2 . \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} \\tau _ { \\theta } ( z , w ) = ( z + \\theta , w ) , \\ ; ( z , w ) \\in \\mathcal { U } _ p \\subset \\C \\times \\C ^ n \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} 1 = w _ { n , m } ( x _ { n , m } ) \\leq w _ { n , m } ( x ) x \\in \\overline { B } _ 0 . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} \\dim U _ k & = 2 + ( \\mathbf { v } _ k , \\mathbf { v } _ k ) \\\\ & = 2 ( g - k ^ 2 - k n ) . \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} I ( \\alpha u ^ + + \\beta u ^ - ) & < \\liminf _ { n \\rightarrow \\infty } \\left ( I ( u _ n ) - \\frac { 1 } { 4 } I ' ( u _ n ) [ u _ n ] \\right ) \\\\ & = \\liminf _ { n \\rightarrow \\infty } I ( u _ n ) \\\\ & = c \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} ( F ^ { - 1 } g ) ( x ) : = r ^ { - Q / 2 } g ( \\ln r , y ) . \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} ( I _ { \\mu } ^ { \\infty } ) ' ( u _ n ^ 1 ( x + y _ n ^ 1 ) ) ( h ) & = ( I _ { \\mu } ^ { \\infty } ) ' ( u _ n ^ 1 ) ( h _ n ) , \\\\ \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} \\{ f _ \\lambda g \\} = \\sum _ { i , j = 1 } ^ n \\sum _ { l \\geq 0 } \\sum _ { m \\geq 0 } \\frac { \\partial g } { \\partial u ^ j _ { ( m ) } } \\left ( \\lambda + \\partial \\right ) ^ m \\{ u ^ i _ { \\lambda + \\partial } u ^ j \\} \\left ( - \\lambda - \\partial \\right ) ^ l \\frac { \\partial f } { \\partial u ^ i _ { ( l ) } } \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} \\| a \\| & \\leq C _ M e ^ { C \\epsilon ^ { - \\mu } } \\sup \\limits _ { l \\in \\{ 1 , \\dots , m \\} } \\| ( \\Lambda _ { q _ 1 } - \\Lambda _ { q _ 2 } ) \\tilde { f } _ l ^ { ( 1 ) } \\| _ { H ^ { - s } ( W _ 2 ) } \\| h _ { l } ^ { ( 1 ) } \\| _ { L ^ 2 ( \\Omega ) } \\| h _ { l } ^ { ( 2 ) } \\| _ { L ^ 2 ( \\Omega ) } \\\\ & + C _ M \\epsilon \\sup \\limits _ { l \\in \\{ 1 , \\dots , m \\} } \\| h _ { l } ^ { ( 1 ) } \\| _ { H ^ { s } _ { \\overline { \\Omega } } } \\| h _ { l } ^ { ( 2 ) } \\| _ { H ^ { s } _ { \\overline { \\Omega } } } \\| a \\| . \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} R _ { \\mathcal { H } _ p } \\left ( e _ { ( z , w ) } \\circ T _ S \\right ) ( t , \\zeta ) = \\overline { \\chi _ { z , w } ( t , \\zeta ) } = 4 \\pi t \\overline { H _ p ( t ; w , \\zeta ) } e ^ { - 2 \\pi i \\overline { z } t } . \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} \\P \\otimes \\P \\left ( ( x , y ) , M _ n ( x , y ) \\leq k _ n \\right ) = \\mathcal { O } ( ( \\log n ) ^ { 1 + { b \\over 2 } } ) . \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 3 / 4 } ^ 0 \\bigg [ \\frac { b _ { 1 / 2 , 0 } ( d , s ) } { d ^ { 1 / 2 } \\Gamma ( s + 1 / 4 ) } \\bigg ] = \\frac { 3 } { \\sqrt { | d D | } L _ D ( 1 ) } \\mathrm { T r } _ { d , D } ( - \\log ( y | \\eta ( z ) | ^ 4 ) ) , \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} z ( t ) = C _ 1 \\cos \\left ( \\frac { \\omega _ 0 e ^ { - \\gamma t } } { \\gamma } \\right ) + C _ 2 \\sin \\left ( \\frac { \\omega _ 0 e ^ { - \\gamma t } } { \\gamma } \\right ) , \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} \\frac { \\partial } { \\partial x } & - \\frac { \\partial } { \\partial y } \\\\ \\frac { \\partial } { \\partial y } & \\frac { \\partial } { \\partial x } \\\\ \\end{matrix} \\right ) \\left ( \\begin{matrix} u \\\\ v \\\\ \\end{matrix} \\right ) = 0 \\Leftrightarrow \\overline { \\partial } w = 0 \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Q ^ { \\times } } \\psi ( \\gamma \\zeta ) W _ { \\phi } \\left ( \\left ( \\begin{matrix} \\gamma & 0 \\\\ 0 & 1 \\end{matrix} \\right ) \\right ) = \\sum _ { \\substack { m \\in \\N } } \\lambda _ { \\pi } \\left ( \\frac { m } { ( m , l ^ { \\infty } ) } \\right ) \\psi _ { \\infty } ( - \\zeta _ 0 m ) W _ { \\infty } ( m ) W _ { l } ( m ) . \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} n _ b ( W ^ n ( w ) ) = \\lvert ( 0 , 0 ) \\rvert _ { W _ n ( w ) } = \\lvert l _ 0 \\rvert _ S + \\lvert ( 0 , 0 ) \\rvert _ { W _ n ( w _ k ) } = \\lvert m _ 0 \\rvert + \\lVert ( \\phi w _ k ) _ n \\rVert _ S . \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} \\mathcal { M } \\phi ( \\tau , X ( \\tau ) ) : = \\inf _ { z \\in \\mathcal { Z } } [ \\phi ( \\tau , \\Gamma ( X ( \\tau ^ - ) , z ) ) + c ( \\tau , z ) \\cdot 1 _ { \\{ \\tau \\leq T \\} } ] , \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} \\begin{cases} ^ { C } _ { 0 } D _ { t } ^ { 0 . 6 } y ( x , t ) = \\displaystyle \\frac { \\partial ^ { 2 } } { \\partial x ^ { 2 } } y ( x , t ) & \\hbox { i n } [ 0 , 1 ] \\times [ 0 , T ] \\\\ y ( 0 , t ) \\ ; = \\ ; y ( 1 , t ) = 0 & \\hbox { i n } [ 0 , T ] \\\\ y ( x , 0 ) = y _ { 0 } ( x ) & \\hbox { i n } [ 0 , 1 ] , \\end{cases} \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} I ' ( \\theta v ^ + + T _ 0 v ^ - ) [ T _ 0 v ^ - ] \\leq H _ { T _ 0 } ( u ) = H _ { T _ 0 } ( e _ { T _ 0 } ) < 0 . \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} k ^ * ( y _ 0 ) = \\tilde { k } ^ * ( z ) \\ \\ \\ \\ \\forall y _ 0 \\in Y _ z . \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 v '' - r ( x ) v = - 1 , \\ - \\infty < x < a ; \\\\ & v ( a ) = 0 ; \\\\ & v \\ge 0 \\ \\ v \\ . \\end{aligned} \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} ( J _ 0 u ) ( n ) : = a _ { n + 1 } u ( { n + 1 } ) + a _ n u ( { n - 1 } ) + b _ { n + 1 } u ( n ) = E u ( n ) , n \\geq 0 , \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} u _ { k + 1 } = P _ C \\left [ u _ k + \\tau A ^ * ( y - A u _ k ) \\right ] , k = 0 , \\dots , \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} v _ 0 ( x ) = U \\left ( \\lambda \\mu _ \\lambda t \\right ) + 2 \\log \\mu _ \\lambda - \\log h _ \\gamma , x = X _ { \\gamma } ^ { - 1 } ( s , t ) . \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} T _ j \\bar { u } ( \\omega , x ) : = \\bar { u } ( \\tau _ { \\hat { e } _ j } \\omega , x ) = \\overline { T _ j u } ( \\omega , x ) . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} \\psi _ { \\bar X } : = \\psi ( \\bar X , \\cdot ) ( \\bar X \\in \\R ^ { n \\times m } ) . \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t ( u - u _ h ) ^ { \\frac 1 2 } , v _ h \\Big { ) } & - \\gamma ( \\nabla ( \\sigma - \\sigma _ h ) ^ { \\frac 1 2 } , \\nabla v _ h ) + ( \\nabla ( u - u _ h ) ^ { \\frac 1 2 } , \\nabla v _ h ) \\\\ & + ( f ^ { \\frac 1 2 } ( u ) - f ^ { \\frac 1 2 } ( u _ { h } ) , v _ h ) = 0 , \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} \\overline { ( \\kappa ( t ) ) ^ { - 1 } } \\ ; \\kappa ( t ) : = \\overline { e ^ { - i t H _ { K _ { \\nu , 0 } } } } e ^ { i t H _ { K _ { \\nu , 0 } } } = e ^ { i t E } \\Big ( a + b I - c J \\Big ) \\Big ( a + b I + c J \\Big ) ~ , \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} N ( r , f ( z + c ) ) = ( 1 + o ( 1 ) ) N ( r , f ( z ) ) , \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} \\dim N ^ + \\ge a + g - 1 - b = n + g - 1 . \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} \\delta > \\delta ( p ) = n \\left ( { 1 \\over p } - { 1 \\over 2 } \\right ) - { 1 \\over 2 } . \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} \\chi ( \\Gamma \\backslash ( S \\backslash e ) ) = \\chi ( \\Gamma \\backslash S ) + \\chi ( e ) - \\chi ( e \\backslash S ) \\leq \\chi ( \\Gamma \\backslash S ) \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} L ( \\psi , \\tilde { \\mathbb { A } } ) = \\prod _ { f } \\frac { \\big [ \\tilde { \\mathbb { A } } / f \\tilde { \\mathbb { A } } \\big ] _ { \\tilde { \\mathbb { A } } } } { \\big [ \\psi ( \\tilde { \\mathbb { A } } / f \\tilde { \\mathbb { A } } ) \\big ] _ { \\tilde { \\mathbb { A } } } } , \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 4 8 n + 1 2 ) q ^ n & \\equiv 2 f _ 2 ^ 3 ~ ( \\textup { m o d } ~ 4 ) \\intertext { a n d } \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 4 8 n + 3 6 ) q ^ n & \\equiv 2 f _ 6 ^ 3 ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} \\xi _ k F _ { k , 0 , r } ( z ) & = ( 1 - k ) G _ { 2 - k , 0 , r } ( z ) + G _ { 2 - k , 0 , r - 1 } ( z ) , \\\\ \\xi _ k G _ { k , 0 , r } ( z ) & = F _ { 2 - k , 0 , r - 1 } ( z ) . \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} | C ^ g _ { n , k } | = \\begin{cases} | C _ { ( g , n ) , k ( g , n ) / n } | & k g \\equiv 0 \\bmod n , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} u _ { n + 1 } = u _ n + \\Delta t \\varphi _ { 1 } ( \\Delta t J _ n ) F ( u _ n ) + \\Delta t \\tfrac { 2 } { 3 } \\varphi _ { 2 } ( \\Delta t J _ n ) R _ { n - 1 } , \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} \\langle z ( t ) , \\phi \\rangle = \\frac 1 2 \\int _ 0 ^ t \\langle z ( s ) , \\phi '' \\rangle d s + \\int _ 0 ^ t \\int _ 0 ^ 1 \\phi ( x ) \\eta ( d s d x ) . \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} \\alpha _ 3 : = \\lambda _ { \\min } ^ { \\Theta _ 2 } \\min \\left \\{ 1 , \\frac { 1 } { 2 } \\vartheta ^ * \\lambda _ { \\min } ^ { \\underline { Q } } \\right \\} s _ 1 . \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} S _ { 0 , \\sigma } ( t ) & \\geq - \\left [ \\Big ( M _ { 1 , \\sigma } ^ - ( t ) + M _ { 1 , \\sigma } ^ + ( t ) \\Big ) \\ , \\frac { 1 } { \\lambda _ { \\sigma } ( t ) } + \\dfrac { M _ { - 1 , \\sigma } ^ - ( t ) } { 2 } \\ , \\lambda _ { \\sigma } ( t ) \\right ] \\ell _ { 1 , \\sigma } ( t ) + O _ { c } \\left ( \\frac { \\mu _ { d , \\sigma } \\ , r _ { 2 , \\sigma } ( t ) } { \\lambda _ { \\sigma } ( t ) } \\right ) + O _ c ( \\mu _ { d , \\sigma } \\ , \\lambda _ { \\sigma } ( t ) \\ , r _ { 2 , \\sigma } ( t ) ) . \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} f \\overset { t \\to \\infty } \\sim g & \\Leftrightarrow \\lim _ { t \\to \\infty } \\frac { f ( t ) } { g ( t ) } = 1 . \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\omega = \\frac { i } { 2 } \\sum _ { p = 1 } ^ n d z _ p \\wedge d \\bar { z } _ p , \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} \\pi _ { t } ( \\varphi ) \\ , : = \\ , \\mathbb E [ \\varphi ( \\widetilde { X } _ { t } ) \\vert \\mathcal F ^ { X } _ { t } ] \\ , = \\ , \\frac { \\rho _ { t } ( \\varphi ) } { \\rho _ { t } ( { \\bf 1 } ) } \\ , , \\rho _ { t } ( \\varphi ) \\ , : = \\ , { \\mathbb E } _ { 0 } [ Z _ { t } ^ { - 1 } \\varphi ( \\widetilde { X } _ { t } ) \\vert \\mathcal F ^ { X } _ { T } ] \\ , ; 0 \\le t \\le T \\ , \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} \\{ f _ { i , j } & ( \\beta ^ { r ^ t } ) \\} _ { j \\in [ l ] } = \\{ \\beta ^ a : a _ i = 0 \\} \\bigcup \\\\ & \\Big ( \\bigcup _ { u = 0 } ^ { r - 2 } \\{ \\beta ^ { l + a } : a _ n = \\dots = a _ { t + 2 } = 0 , a _ { t + 1 } = u , a _ i = 0 \\} \\Big ) . \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} \\norm { a } _ { \\ell ^ 2 _ r } ^ 2 : = \\pi \\sum _ { k = 0 } ^ { \\infty } \\frac { r ^ { 2 k + 2 } } { k + 1 } \\abs { a _ k } ^ 2 < \\infty . \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} \\| \\psi ( x ) \\| = \\frac { 1 } { 2 } \\sqrt { 2 \\psi ^ 2 ( \\lambda _ 1 ( x ) ) + \\psi ^ 2 ( \\lambda _ 2 ( x ) ) + \\psi ^ 2 ( \\lambda _ 4 ( x ) ) } . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} a _ \\alpha ( z ) = \\sum _ { \\beta \\le \\alpha } \\frac { ( - 1 ) ^ { | \\alpha - \\beta | } z ^ { \\alpha - \\beta } } { ( \\alpha - \\beta ) ! \\beta ! } F z ^ \\beta . \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { F } _ q } ( A _ u h _ { u , j } ) = l / r , u = 1 , 2 , \\dots , r . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} C ( t ) = W \\log _ { 2 } \\left ( 1 + \\Gamma | h ( t ) | ^ { 2 } \\right ) , \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} 0 \\in 2 \\beta _ { \\ell } - 2 \\alpha _ { \\ell } + \\lambda \\mu _ { \\ell } \\partial | \\beta _ { \\ell } | , \\ell = 0 , 1 , \\ldots , L , \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} \\mathcal { R } _ j ^ 2 ( ( \\chi _ { | x | } - \\chi _ { | x | / 2 } ) \\nu ) ( x ) = \\frac { \\tilde { \\Omega } _ j ( x ) } { | x | ^ 2 } = \\frac { \\tilde { \\Omega } _ j ( . ) } { | . | ^ 2 } \\star \\delta _ { 0 } ( x ) ~ ~ \\forall ~ x \\in \\mathbb { R } ^ 2 \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} \\binom { n - p ^ j } { ( p - 1 ) p ^ j } \\beta _ { p ^ j , \\ , n - p ^ j } = 0 . \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} \\bar { g } ( x ) : = - \\min _ { y \\in D } \\bar { \\phi } ( x , y ) \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\mathbb { E } \\left [ X _ i \\right ] \\le _ { c x } \\sum _ { i = 1 } ^ { n } X _ i . \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} \\mathcal { W } : = \\lbrace \\bold { x } = ( x , y , z ) \\in \\R ^ 3 ~ : ~ \\vert z \\vert < \\delta \\rbrace , 0 < \\delta \\ll 1 . \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} \\tau = \\begin{pmatrix} & 1 & \\\\ 1 _ { n - 2 } & & \\\\ & & 1 \\end{pmatrix} \\begin{pmatrix} \\xi _ { n } ^ { - 1 } & & & \\\\ & \\ddots & & \\\\ & & \\xi _ { 3 } ^ { - 1 } & \\\\ & & & - y \\zeta _ { v } ^ { - 1 } \\xi _ { 2 } ^ { - 1 } & \\\\ & & & & - \\zeta _ { v } \\xi _ { 1 } ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} \\psi _ { \\theta } = \\sum _ { i = 1 } ^ r \\psi _ { \\theta , i } \\tau ^ i = \\sum _ { i = 1 } ^ r t ^ i \\ell _ i ( z _ 1 ) \\dots \\ell _ i ( z _ n ) \\phi _ { \\theta , i } \\tau ^ i . \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} d ^ n f ( g _ 0 , . . . , g _ { n + 1 } ) = \\sum _ { i = 0 } ^ { n + 1 } ( - 1 ) ^ i f ( g _ 0 , . . . , \\hat { g _ i } , . . . , g _ { n + 1 } ) . \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} u _ { x x } + 2 ( u - u ^ 3 ) - \\frac { 1 } { 2 } \\Omega ^ 2 x ^ 2 u = 0 \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} D _ { r } ( x ) = r x : = ( r ^ { \\nu _ { 1 } } x _ { 1 } , \\ldots , r ^ { \\nu _ { n } } x _ { n } ) , \\ ; \\ ; x = ( x _ { 1 } , \\ldots , x _ { n } ) \\in \\mathbb { G } , \\ ; \\ ; r > 0 , \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} U ^ * ( \\psi \\otimes ( g _ 1 \\otimes _ s \\dots \\otimes _ s g _ n ) ) & \\in \\textup { S p a n } ( D ) . \\\\ \\lVert ( H _ \\mu ( \\xi - k ) - H _ \\mu ( \\xi ) ) \\Gamma ( Q _ A ) \\psi \\lVert & = \\lVert ( H _ \\mu ( \\xi - k , A ) - H _ { \\mu } ( \\xi , A ) ) \\psi \\lVert . \\\\ \\lVert ( H _ { \\mu } ( \\xi ) - \\lambda ) \\Gamma ( Q _ A ) \\psi \\lVert & = \\lVert ( H _ \\mu ( \\xi , A ) - \\lambda ) \\psi \\lVert . \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} \\exists e \\forall n \\left [ \\begin{gathered} \\neg \\exists m \\exists p ( T _ 1 ( e , m , p ) \\wedge U ( p ) = n ) \\rightarrow n \\not \\in X \\\\ \\wedge \\\\ \\exists m \\exists p ( T _ 1 ( e , m , p ) \\wedge U ( p ) = n ) \\rightarrow n \\in X \\end{gathered} \\right ] \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} L ^ r { } _ s = \\begin{pmatrix} \\sqrt { - b c } & b \\\\ c & - \\sqrt { - b c } \\end{pmatrix} \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\widehat b _ { T , h } ( x ) : = \\frac { \\displaystyle { \\int _ { 0 } ^ { T } K \\left ( \\frac { X ( s ) - x } { h } \\right ) \\delta X ( s ) } } { \\displaystyle { \\int _ { 0 } ^ { T } K \\left ( \\frac { X ( s ) - x } { h } \\right ) d s } } \\textrm { $ ; $ } x \\in \\mathbb R \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} m \\left ( r , \\frac { f ( z + c ) } { f ( z ) } \\right ) = o \\left ( \\frac { T ( r , f ) } { r ^ { 1 - \\varsigma - \\varepsilon } } \\right ) , \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} \\| f g \\| _ { \\mathcal { B } _ { \\alpha } ^ { 0 } } = \\sum _ { k \\in \\mathbb { Z } ^ { n } } \\sup _ { t \\in [ 0 , \\infty ) } e ^ { \\alpha t | k | } | \\widehat { f g } ( t , k ) | . \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} \\| u ^ n - u _ h ^ { n } \\| + \\gamma ^ { \\frac 1 2 } \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| ( \\sigma - \\sigma _ h ) ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } + \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\nabla ( u - u _ { h } ) ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } \\leq C h ^ { m + 1 } , \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} H ' ( \\bar { \\theta } _ { v ' } - \\bar { \\theta } _ v , v , v ' ) = H ' ( \\bar { \\theta } _ { v } - \\bar { \\theta } _ { v ' } , v ' , v ) \\geq 0 \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} B = ( B _ 1 , \\ldots , B _ m ) \\colon H \\to H _ 1 \\times \\cdots \\times H _ m \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} [ \\mathbb { K } : F _ { [ h ] } ] = r ! \\prod _ { i = 1 } ^ h p _ i . \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} \\Phi ^ * \\circ T _ S ( f ) ( z ) & = \\Phi ^ * \\left ( \\int _ { \\mathbb { R } } f ( t ) e ^ { i 2 \\pi z t } \\d t \\right ) = \\left ( \\int _ { \\mathbb { R } } f ( t ) e ^ { i 2 \\pi \\Phi ( z ) t } \\d t \\right ) \\cdot \\Phi ' ( z ) = \\int \\limits _ { \\mathbb { R } } f ( t ) \\dfrac { z ^ { i 2 \\pi t } } { z } \\d t , \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} F ( x _ 1 , . . . , x _ \\ell ) = \\prod _ { i = 1 } ^ \\ell f _ i ( x _ i ) . \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} X ( r ) : = A _ 0 + A _ 2 r ^ 2 + A _ 4 r ^ 4 , & & Y ( r ) : = C _ 2 + D _ 2 + r ^ 2 ( C _ 4 + D _ 4 ) \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} b _ { 1 / 2 , 0 } ( 0 , s ) = \\pi ^ { 1 / 2 } 2 ^ { 5 / 2 - 6 s } \\Gamma ( 2 s ) \\frac { \\zeta ( 4 s - 2 ) } { \\zeta ( 4 s - 1 ) } . \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} \\max _ { 1 \\leq i \\leq n } | c _ { i r _ 0 } | = \\alpha = \\inf _ { r \\geq 0 } \\{ \\max _ { 1 \\leq i \\leq n } | c _ { i r } | \\} . \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} \\mathbf { E } _ \\mu \\big \\| A - \\widetilde { \\sigma } \\circ \\xi \\big \\| ^ 2 _ F = \\mathbf { E } _ \\mu \\big \\| A - ( \\mathbf { E } _ { \\mu _ z } A ) \\circ \\xi \\big \\| _ F ^ 2 + \\mathbf { E } _ \\mu \\big \\| \\big ( \\widetilde { \\sigma } - \\mathbf { E } _ { \\mu _ z } A \\big ) \\circ \\xi \\big \\| _ F ^ 2 \\ , , \\end{align*}"}
-{"id": "9820.png", "formula": "\\begin{align*} N ( W ) = \\mathrm { c o n j } ( W ) W = \\mathrm { d e t } ( M ) = a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 ~ . \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} & ( \\widetilde { X } _ { \\cdot } ) \\ , = \\ , ( X _ { \\cdot } ) \\ , = \\ , ( Y _ { \\cdot } ) \\ , , \\\\ & ( X _ { 0 } , \\widetilde { X } _ { 0 } , Y _ { 0 } ) \\ , = \\ , ( X _ { 0 } ) \\otimes ( \\widetilde { X } _ { 0 } ) \\otimes ( Y _ { 0 } ) \\ , , \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} \\| Q \\| ^ 2 _ { \\dot { H } ^ 1 _ c } + \\| Q \\| ^ 2 _ { L ^ 2 } = \\| Q \\| ^ { \\frac { 4 } { d } + 2 } _ { L ^ { \\frac { 4 } { d } + 2 } } . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} - n _ i = \\chi ( E _ i ) = - a _ i + b _ i \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} w ( b ) = w ( \\alpha \\ell + \\beta ) = \\beta + \\alpha \\ell - \\sum _ { i = 1 } ^ d x _ i \\left \\lfloor \\frac { \\alpha } { s _ i } \\right \\rfloor = \\beta + u ( \\alpha ) \\ , . \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} - \\frac { \\ , 1 \\ , } { \\ , 2 \\ , } \\int ^ { t } _ { 0 } f ^ { \\prime \\prime } ( \\langle \\mathrm m _ { s , n } , g \\rangle ) \\frac { 1 } { \\ , n ^ { 2 } \\ , } \\sum _ { i = 1 } ^ { n } \\lvert g ^ { \\prime } ( X _ { s , i } ^ { ( u ) } ) \\rvert ^ { 2 } { \\mathrm d } s \\ , = \\ , \\int ^ { t } _ { 0 } f ^ { \\prime } ( \\langle \\mathrm m _ { s , n } , g \\rangle ) \\frac { \\ , 1 \\ , } { \\ , n \\ , } \\sum _ { i = 1 } ^ { n } g ^ { \\prime } ( X _ { s , i } ^ { ( u ) } ) { \\mathrm d } W _ { s , i } \\ , , \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} ( a \\cdot x \\cdot b ) ( e ) = a ( r ( e ) ) x ( e ) b ( s ( e ) ) , e \\in E \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} d _ { \\rm T V } ( D _ n , P o ( \\lambda _ n ) ) \\leq \\frac { 1 - e ^ { - \\lambda _ n } } { \\lambda _ n } \\sum _ { i = 1 } ^ { n - 1 } p _ { i , n } ^ 2 \\leq \\min \\{ 1 , \\frac { 1 } { \\lambda _ n } \\} \\sum _ { i = 1 } ^ { n - 1 } p _ { i , n } ^ 2 . \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} \\theta ^ \\prime ( x , E ) = \\gamma ^ \\prime ( x , E ) + \\frac { C } { 2 \\gamma ^ \\prime ( x , E ) ( 1 + x ) } \\sin 2 \\theta \\sin ^ 2 \\theta . \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} f ( x ) & = 2 ( x ^ 2 + 1 ) - ( x ^ 4 + x ^ 2 + 1 ) + m \\ ; h ( x ) , \\\\ g ( x ) & = ( x ^ 3 + 1 ) - ( x ^ 4 + x ^ 2 + 1 ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} \\tilde { E } ^ { ( 2 ) } _ { a , b } ( \\tau ) = a _ 0 ( \\tilde { E } ^ { ( 2 ) } _ { a , b } ) + \\sum _ { \\substack { m , n \\geq 1 \\\\ m \\equiv a ( N ) } } \\zeta _ N ^ { b n } n q ^ { m n / N } + \\sum _ { \\substack { m , n \\geq 1 \\\\ m \\equiv - a ( N ) } } \\zeta _ N ^ { - b n } n q ^ { m n / N } - 2 \\sum _ { m , n \\geq 1 } n q ^ { m n } . \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} \\binom { n - i + p ^ j } { p ^ j } \\beta _ { i - p ^ j , \\ , n - i + p ^ j } = \\binom { i } { p ^ j } \\beta _ { i , \\ , n - i } . \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} \\nabla \\phi _ { j + 1 } = \\nabla \\phi _ { j } - \\varepsilon \\left ( D ^ { 2 } \\phi _ { j } - K _ { f } ^ { - 1 } \\frac { \\nabla \\phi _ { j } \\otimes \\nabla f } { f ^ { 2 } } \\right ) u _ { j } + \\varepsilon K _ { f } ^ { - 1 } J \\nabla \\phi _ { j } , \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} \\rho ( \\lambda ) ^ { - 1 } = z _ 0 = \\frac { 2 ( m + \\lambda - 1 ) } { m - 2 + [ ( m _ 1 - m _ 2 ) ^ 2 + 4 \\lambda ( \\sqrt { m _ 1 } + \\sqrt { m _ 2 } ) ^ 2 ] ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} \\rho _ { t , k } ( \\varphi ) \\ , = \\ , \\pi _ { 0 , k } ( \\varphi ) + \\int ^ { t } _ { 0 } \\rho _ { s , k } ( \\varphi b ) { \\mathrm d } X _ { s } + \\int ^ { t } _ { 0 } \\rho _ { s , k + 1 } ( \\widetilde { A } _ { s } \\varphi ) { \\mathrm d } s \\ , , \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} | | \\tilde { \\lambda } ( a , w ) - \\tilde { \\lambda } ( a , w ' ) | | _ { \\mathbb { R } ^ n } \\leq C _ m \\cdot | | ( a , w ) - ( a , w ' ) | | _ m = C _ m \\cdot | | w - w ' | | _ m . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{gather*} v _ B ^ { - 1 } \\overset { I } { A } = \\overset { I } { v } \\overset { I } { B } { ^ { - 1 } } \\overset { I } { A } v _ B ^ { - 1 } , v _ B ^ { - 1 } \\overset { I } { B } = \\overset { I } { B } v _ B ^ { - 1 } \\end{gather*}"}
-{"id": "146.png", "formula": "\\begin{align*} \\left | Q _ 2 ( z , z ' ) \\right | \\lesssim r ^ { - n } r ^ { s } ( r ' / r ) ^ { 1 - \\frac n 2 + \\nu _ 0 } . \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} \\mathbf { K } _ { i , n } ( r \\theta ) = \\frac { \\varepsilon ^ { - d + 1 } } { \\rho ^ \\alpha } \\int _ { \\mathbb { R } ^ d } \\mathbf { K } _ { n } ( r \\theta - y ) \\varphi _ { \\zeta \\rho } ( y ) \\frac { \\langle \\phi ^ { e , \\varepsilon } ( y / \\rho ) , \\eta ^ { \\kappa } _ { y _ { \\tau } } \\rangle } { | y | ^ { d - \\alpha } } \\left ( \\psi ( 2 ^ { - i } \\rho ^ { - 1 } y ) - \\psi ( 2 ^ { - i + 1 } \\rho ^ { - 1 } y ) \\right ) d y . \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} I _ s = I _ s ( n ) = \\theta _ s ( n ) = \\theta _ s . \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} \\gamma = \\gamma _ { \\ref { T h e o r e m E x i s t e n c e A n d U n i q u e n e s s O n D o m a i n } , 0 } ( \\eta ( \\chi ) , R ( \\chi , \\sigma , m , r , \\mathcal { U } _ 1 ) , T ' ) , \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} n _ b ( W ^ n ( w ) ) = \\max ( \\lvert m _ 0 \\rvert + \\lVert ( \\phi w _ k ) _ n \\rVert _ S , \\lVert ( \\phi w _ 0 ) _ n \\rVert _ S + \\lvert m _ k \\rvert ) . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} [ L _ 1 x ] _ { i , j } = \\sum _ { i ' , j ' } ( x _ { i ' , j ' } - x _ { i , j } ) , \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} \\frac { m ^ 2 } { t _ h } + \\frac { h ^ { 2 b + 1 } - ( h - t _ h ) ^ { 2 b + 1 } } { 2 b + 1 } = \\frac { m ^ 2 } { \\tau _ h } + \\frac { h ^ { 2 b + 1 } - ( h - \\min \\{ h , \\tau _ h \\} ) ^ { 2 b + 1 } } { 2 b + 1 } , \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} H = \\pi ^ { - 1 } ( U + \\ker \\beta _ 0 ) = \\tilde { U } + V + \\ker B _ 0 = \\tilde { U } + \\ker B _ 0 \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} [ W , Y ] _ t & = \\left [ W , \\int _ 0 ^ \\cdot Z _ r \\mathrm d W _ r \\right ] _ t = \\int _ 0 ^ t Z _ r ^ * \\mathrm d r , \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} B : = & ( B _ 1 \\times B _ 2 ) \\circ q ^ { - 1 } \\\\ & : [ 0 , \\infty ) ^ { s _ 1 + s _ 2 } \\times \\mathbb { R } ^ { k _ 1 - s _ 1 + k _ 2 - s _ 2 } \\times \\mathbb { W } _ 1 \\times \\mathbb { W } _ 2 \\rightarrow \\mathbb { W } _ 1 \\times \\mathbb { W } _ 2 \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta u = \\left ( \\frac { 1 } { | x | ^ { n - 2 } } * | u | ^ p \\right ) | u | ^ { p - 2 } u , & x \\in \\Omega , \\\\ u ( x ) = 0 , & x \\in \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} A : = \\Biggl ( \\left [ \\begin{array} { c c } 4 & 1 \\\\ 0 & 4 \\end{array} \\right ] , e ^ { \\frac { \\pi i } { 4 } } \\biggr ) , B _ { \\nu } : = \\left [ \\begin{array} { c c } 1 & 0 \\\\ 4 \\nu & 1 \\end{array} \\right ] , \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} a \\Delta _ \\kappa b - b \\Delta _ \\kappa a = 0 , \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} M : = \\max \\{ I \\in \\N ^ n \\mid C _ I \\neq 0 \\} . \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{gather*} A _ { K _ k ^ { ( k ) } } ( z ) = A _ { K _ k ^ { ( k ) } } ( z _ j ) \\end{gather*}"}
-{"id": "8331.png", "formula": "\\begin{align*} x \\cdot ( n \\otimes \\phi ) = \\sum x _ { ( 1 ) } n \\otimes x _ { ( 2 ) } \\phi = \\sum x _ { ( 1 ) } n \\otimes \\epsilon ( x _ { ( 2 ) } ) \\phi = ( \\sum x _ { ( 1 ) } \\epsilon ( x _ { ( 2 ) } ) n ) \\otimes \\phi ) = x n \\otimes \\phi , \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} \\sigma ( x _ i ) = \\epsilon ^ { s _ i } x _ i , \\ , \\sigma ( y _ i ) = \\epsilon ^ { - s _ i } y _ i \\ , . \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} K ^ { x } ( g ) = \\lambda \\ , g \\ , , s \\leq 0 , \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} \\dim V = \\sum _ { i = 0 } ^ { m ^ + } \\dim B _ i V . \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} \\textbf { T r } \\left ( ( x \\diamond s ) \\diamond t \\right ) = \\textbf { T r } \\left ( x \\diamond ( s \\diamond t ) \\right ) . \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} \\tilde { P } _ { q } ( h ) : = \\left \\{ x \\in V \\mid \\langle x , u _ { F } \\rangle + q ( y + 1 ) \\lambda _ { F } + h _ { F } \\geq 0 \\ \\right \\} . \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} h ^ { ( \\alpha , \\beta ) } _ { t } ( k ) : = \\int _ { - 1 } ^ 1 e ^ { - ( 1 - x ) t } p _ k ^ { ( \\alpha , \\beta ) } ( x ) \\ , d \\mu _ { \\alpha , \\beta } ( x ) = e ^ { - t } w _ { k } ^ { ( \\alpha , \\beta ) } \\int _ { - 1 } ^ 1 e ^ { x t } P _ k ^ { ( \\alpha , \\beta ) } ( x ) \\ , d \\mu _ { \\alpha , \\beta } ( x ) \\end{align*}"}
-{"id": "9857.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } a ( 5 n + 2 ) q ^ { n } & = 3 \\dfrac { E _ { 5 } E _ { 1 0 } } { E _ { 1 } ^ { 2 } E _ { 2 } ^ { 2 } } + 2 5 q \\dfrac { E _ { 5 } ^ { 3 } E _ { 1 0 } ^ { 3 } } { E _ { 1 } ^ { 4 } E _ { 2 } ^ { 4 } } + 1 2 5 q ^ { 2 } \\dfrac { E _ { 5 } ^ { 5 } E _ { 1 0 } ^ { 5 } } { E _ { 1 } ^ { 6 } E _ { 2 } ^ { 6 } } . \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} \\mathcal A _ { s } ( \\mathrm M ) g \\ , : = \\ , u \\int _ { \\mathbb R ^ { 2 } } \\widetilde { b } ( s , y _ { 1 } , y _ { 2 } ) g ^ { \\prime } ( y _ { 1 } ) \\mathrm M _ { s } ( { \\mathrm d } y _ { 1 } { \\mathrm d } y _ { 2 } ) + ( 1 - u ) \\int _ { \\mathbb R ^ { 2 } } \\widetilde { b } ( s , y _ { 1 } , y _ { 2 } ) g ^ { \\prime } ( y _ { 1 } ) \\mathrm m _ { s } ( { \\mathrm d } y _ { 1 } ) \\mathrm m _ { s } ( { \\mathrm d } y _ { 2 } ) \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} \\mathrm { p r } _ k ^ + ( g ) : = ( - 1 ) ^ { \\lfloor \\frac { \\lambda _ k + 1 } { 2 } \\rfloor } \\frac { 1 } { 2 \\sqrt { 2 } } \\biggl ( \\sum _ { \\nu \\ ( \\mathrm { m o d } \\ 4 ) } \\bigl ( g | _ k A \\bigr ) | _ k B _ { \\nu } \\biggr ) + \\frac { 1 } { 2 } g \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} \\eta _ { \\dot a } = \\pm \\sqrt { \\frac a 2 } \\ , \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} e ^ { i ( x ^ 3 + x ^ 4 ) / 2 } . \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} \\mathbf { x } = \\sum _ { j = 1 } ^ { N _ g } \\mathbf { \\Pi } ^ { ( j ) } \\sum _ { i \\in \\mathcal { G } _ j } \\sqrt { p _ i ^ { ( j ) } } \\mathbf { w } _ i ^ { ( j ) } s _ i ^ { ( j ) } , \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} \\sum _ { i \\in I } v _ { i } ^ { 2 } \\Vert \\pi _ { u W _ { i } } C f \\Vert ^ { 2 } & \\leq \\Vert u \\Vert ^ { 2 } \\sum _ { i \\in I } v _ { i } ^ { 2 } \\Vert \\pi _ { W _ { i } } u ^ { - 1 } C f \\Vert ^ { 2 } \\\\ & = \\Vert u \\Vert ^ 2 \\sum _ { i \\in I } v _ i ^ 2 \\Vert \\pi _ { W _ i } C u ^ { - 1 } f \\Vert ^ 2 \\\\ & \\leq B \\Vert u ^ { - 1 } \\Vert ^ { 2 } \\Vert u \\Vert ^ { 2 } \\Vert f \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} \\mathcal { E } _ { p } ( \\gamma , \\zeta _ { p } ) = \\abs { \\gamma } _ { p } ^ { - \\frac { 1 } { 2 } } W _ { \\xi ^ { - 1 } _ { p } \\tilde { \\pi } _ { p } } ( g _ { t , l , u ^ { - 1 } v } ) . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} u ( \\omega , - b ) = - u ( \\tau _ { - b } \\omega , b ) , b \\in \\hat { E } . \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} B _ { \\overline { \\nu } , 0 } ( t , \\overline { a } ) e ^ { a _ j t } = \\sum _ { N = 0 } ^ \\infty r _ { N , j } t ^ N , j = 1 , \\ldots , m , \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} n _ s ( I \\cup J ) = \\max ( n _ s ( I ) , n _ s ( J ) ) . \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} \\Re \\left ( 1 - \\left ( \\frac { \\rho } { \\rho - \\tau } \\right ) ^ n \\right ) & = \\Re \\left ( 1 - \\left ( 1 + \\frac { \\tau } { \\rho - \\tau } \\right ) ^ n \\right ) \\\\ & = - n \\tau \\Re \\left ( \\frac { 1 } { \\rho - \\tau } \\right ) - \\frac { n ( n - 1 ) \\tau ^ 2 } { 2 } \\Re \\left ( \\frac { 1 } { ( \\rho - \\tau ) ^ 2 } \\right ) \\\\ & \\quad - \\sum \\limits _ { j = 3 } ^ n \\binom { n } { j } \\Re \\left ( \\left ( \\frac { \\tau } { \\rho - \\tau } \\right ) ^ j \\right ) , \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} \\mu \\left ( { R } - \\log _ 2 \\left [ \\det \\left ( \\mathbf { I } _ { N _ R } + \\left ( 1 - \\rho \\right ) \\sigma ^ { - 2 } \\mathbf { H } \\mathbf { S } \\mathbf { H } ^ { \\rm H } \\right ) \\right ] \\right ) = 0 , \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} \\Pi ^ { ( \\mathfrak { c } , ( \\tau , Z ) ) } [ t _ 0 , x _ 0 ] = \\mathbb { E } \\left [ \\int _ { t _ 0 } ^ { \\tau _ S } e ^ { - \\delta r } R ( X ^ { t _ 0 , x _ 0 , ( \\tau , Z ) } _ r ) d r + \\sum _ { j \\geq 1 } e ^ { - \\delta \\tau _ { j } } c ( \\tau _ { j } ^ - , z _ j ) \\cdot 1 _ { \\{ \\tau _ { j } \\leq \\tau _ S \\} } \\right ] & , \\ ; \\\\ \\forall ( t _ 0 , x _ 0 ) \\in [ 0 , T ] \\times S , \\forall ( \\tau , Z ) \\in & U , \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} [ f ] _ { \\kappa } ^ { \\bar { n } } = \\prod _ { j = 0 } ^ { n - 1 } f ( z + j \\kappa ) = \\prod _ { j = 0 } ^ { n - 1 } \\exp \\left ( \\pi ( z ) \\omega ^ { z / \\kappa } \\omega ^ { j } \\right ) = \\exp \\left ( \\pi ( z ) \\omega ^ { z / \\kappa } \\sum _ { j = 0 } ^ { n - 1 } \\omega ^ j \\right ) \\equiv e ^ 0 = 1 . \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} | z ( u ) | _ { \\alpha ; \\bar { \\Omega } } = | | z ( u ) | | _ { C ( \\bar { \\Omega } ) } + [ z ( u ) ] _ { \\alpha ; \\bar { \\Omega } } \\leq C . \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{gather*} M _ { 1 2 } = M _ 1 R _ { 2 1 } M _ 2 R _ { 2 1 } ^ { - 1 } \\end{gather*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\tilde { S } _ { ( \\lambda , \\mu ) , ( \\lambda ' , \\mu ' ) } = i ^ { - n - \\lvert \\Phi ^ + \\rvert } \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho + \\lambda , ( w \\bullet ( \\lambda ' , \\mu ' ) ) _ 2 \\rangle + \\langle \\mu , ( w \\bullet ( \\lambda ' , \\mu ' ) ) _ 1 + 2 \\rho \\rangle } } { d ^ n } \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} L _ { n } ( \\phi _ { n } ) - L ( \\phi _ { B } ) & \\leq 2 \\sum _ { i = 0 } ^ { 1 } E \\left [ \\Big | p ( \\textbf { Z } , i ) P ( Y = i \\vert \\textbf { Z } ) - f _ { i } ( \\textbf { Z } ) \\Big | \\Bigg | D _ { n } \\right ] \\\\ & + 2 \\sum _ { i = 0 } ^ { 1 } E \\left [ \\Big | ( 1 - q ( \\textbf { X } , i ) ) P ( Y = i \\vert \\textbf { X } ) - g _ { i } ( \\textbf { X } ) \\Big | \\Bigg | D _ { n } \\right ] . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} X = \\mathrm { T o t } _ C ( L _ 1 \\oplus L _ 2 ) \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} \\sum _ { x \\in \\varpi ^ j { \\mathcal O } _ k ^ \\times } \\lambda ( x ) = \\left \\{ \\begin{array} { l l } 0 & \\ ; j < k - 1 \\\\ - 1 & \\ ; j = k - 1 \\\\ 1 & \\ ; j = k . \\end{array} \\right . \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} \\gamma _ v \\cup z = \\gamma _ v \\cup \\widehat { j } ( x ) = j ( \\gamma _ v ) \\cup x = ( \\beta + d f _ v ) \\cup x = ( \\beta \\cup x ) + ( d f _ v \\cup x ) , \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} E ( v ^ l _ n ) = \\frac { E ( \\tilde { v } ^ l _ n ) } { ( \\lambda ^ l _ n ) ^ 2 } + \\frac { ( \\lambda ^ l _ n ) ^ \\alpha - 1 } { \\alpha + 2 } \\| v ^ l _ n \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } \\geq \\frac { E ( \\tilde { v } ^ l _ n ) } { ( \\lambda ^ l _ n ) ^ 2 } . \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} \\frac { d } { d t } ( p ^ * _ t \\cdot v _ t ) = \\dot { p } ^ * _ t \\cdot v _ t + \\dot { v } _ t \\cdot p ^ * _ t = 0 \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} L ( X , \\dot { X } , A , \\dot { A } ) = I ( X ) \\dot { X } ^ 2 - U ( X ) + 2 F ( X ) A + K ( X ) A ^ 2 + Q ( X ) \\dot { A } ^ 2 + 2 C ( X ) \\dot { A } \\dot { X } . \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} ( E _ \\sigma ^ { \\frac 1 2 } , w ) + ( \\nabla E _ u ^ { \\frac 1 2 } , \\nabla w ) = ( R _ 4 ^ { \\frac 1 2 } , w ) , ~ \\forall w \\in H _ 0 ^ 1 , \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} k + h ^ \\vee = \\frac { u } { v } \\ , \\in \\ , \\mathbb Q _ { > 0 } , \\ \\ ( u , v ) = 1 , \\ \\ u , v > 0 \\ \\ \\ \\ \\begin{cases} u \\geq h ^ \\vee , \\ \\ v \\geq h & \\ \\ ( v , r ^ \\vee ) = 1 \\\\ u \\geq h , \\ \\ v \\geq r ^ \\vee h ^ \\vee & \\ \\ ( v , r ^ \\vee ) = r ^ \\vee \\end{cases} \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} L ^ { A } _ { t - d } \\circ \\theta _ { d } = \\frac { h _ { J _ t } ^ { A } { ( \\theta ) } } { h _ { J _ d } ^ { A } { ( \\theta ) } } e ^ { \\theta A ( d , t ) - ( t - d ) \\kappa ^ { A } ( \\theta ) } , \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { | X _ n | } { n } = \\frac { d - 1 - \\lambda } { d - 1 + \\lambda } = : \\mathcal { S } _ { \\mathbb { T } _ d } ( \\lambda ) \\hbox { \\rm a . s . } \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} \\Upsilon ^ \\prime : = \\{ w \\in \\Upsilon \\vert \\mathfrak { r } ^ + ( w ) \\subset \\mathfrak { r } _ W \\} . \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} r ( G _ { \\lambda } ) \\geq r ( G _ { \\lambda ' } ) & \\geq \\frac { 1 } { 2 } ( d _ 1 / 2 - 1 + d _ k / 2 - 1 + \\sum _ { j \\neq 1 , k } d _ j - 1 ) \\\\ & = \\frac { 1 } { 2 } ( \\dim ( V _ { \\lambda } ) - k - d _ 1 / 2 - d _ k / 2 ) . \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} o ( g ) & : = \\{ f : \\R _ + \\to \\R _ + \\mid \\limsup _ { t \\to \\infty } \\tfrac { f ( t ) } { g ( t ) } = 0 \\} , \\\\ O ( g ) & : = \\{ f : \\R _ + \\to \\R _ + \\mid \\limsup _ { t \\to \\infty } \\tfrac { f ( t ) } { g ( t ) } < \\infty \\} , \\\\ \\Omega ( g ) & : = \\{ f : \\R _ + \\to \\R _ + \\mid g \\in O ( f ) \\} . \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} ( \\tfrac { \\partial } { \\partial t } u _ d ) ( t , x ) + \\tfrac { 1 } { 2 } ( \\Delta _ x u _ d ) ( t , x ) + f _ d ( t , x , u _ d ( t , x ) ) = 0 \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { \\| u ^ { k } - x ^ k \\| } { \\rho _ k } = 0 . \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} & f ( P _ x ) \\triangleq \\mathcal { D } ( p _ { _ 1 } | | p _ { _ 0 } ) - \\mathcal { D } ( p _ { _ 0 } | | p _ { _ 1 } ) \\\\ & = \\frac { P _ y ^ 2 - \\sigma _ w ^ 4 } { \\sigma _ w ^ 2 P _ y } - \\log \\frac { P _ y ^ 2 } { \\sigma _ w ^ 4 } \\\\ & = \\frac { P _ x + \\sigma _ w ^ 2 } { \\sigma _ w ^ 2 } - \\frac { \\sigma _ w ^ 2 } { P _ x + \\sigma _ w ^ 2 } - \\frac { 1 } { 2 } \\log ( P _ x + \\sigma _ w ^ 2 ) + \\log \\sigma _ w . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} { D _ f } = - \\nu { { \\cal L } } ^ 2 , \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { 0 } ( \\textbf { x } ) & = I \\left \\lbrace \\textbf { x } \\in \\mathcal { C } ^ { ' } _ { 1 } \\cap \\mathcal { C } ^ { ' } _ { 0 } , ( 1 - q ( \\textbf { x } , 1 ) ) P \\{ Y = 1 \\vert \\textbf { X } = \\textbf { x } \\} > ( 1 - q ( \\textbf { x } , 0 ) ) P \\{ Y = 0 \\vert \\textbf { X } = \\textbf { x } \\} \\right \\rbrace \\\\ & + I \\left \\lbrace \\textbf { x } \\in \\mathcal { C } ^ { ' } _ { 1 } - \\mathcal { C } ^ { ' } _ { 0 } , q ( \\textbf { x } , 1 ) < 1 \\right \\rbrace . \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} F ( \\nabla ^ { g _ r } ) = ( r ^ 2 A _ 2 + A _ 0 + r ^ { - 2 } A _ { - 2 } ) + ( C _ 0 + r ^ { - 2 } C _ { - 2 } ) + ( D _ 0 + r ^ { - 2 } D _ { - 2 } ) \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} W ( z ) = U ( z ) ^ { - 1 } ( X ( z ) - Y ( z ) ) V ( z ) ^ { - 1 } \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} ( I I ) & \\geq c \\ , \\int \\eta \\ , \\abs { h } ^ { - 2 } \\abs { \\tau _ h V _ \\lambda ( \\nabla \\Q _ \\lambda ) } ^ 2 \\ , d x = : ( I I I ) . \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} \\varphi _ 1 + \\varphi _ { n } = \\varphi _ { n + 1 } . \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} v _ i ( \\tau , \\xi ) = & \\int _ 0 ^ \\tau \\sum _ { j = 1 } ^ n \\Big ( C _ { i j } \\big ( x _ i ( s , \\tau , \\xi ) \\big ) v _ j \\big ( s , x _ i ( s , \\tau , \\xi ) \\big ) + D _ { i j } \\big ( x _ i ( s , \\tau , \\xi ) \\big ) v _ j ( s , 0 ) \\\\ [ 6 p t ] & + f _ i \\big ( s , x _ i ( s , \\tau , \\xi ) \\big ) \\Big ) \\ , d s + v _ { 0 , i } \\big ( x _ i ( 0 , \\tau , \\xi ) \\big ) , \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} \\int \\vert x \\vert ^ 2 \\ , \\mu ( \\d x ) = \\mathbb { E } ( \\mathrm { T r a c e } ( H ^ 2 ) ) = 1 + \\frac { \\beta } { n } \\sum _ { k = 1 } ^ { n - 1 } k = 1 + \\frac { ( n - 1 ) \\beta } { 2 } . \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} \\mathbf { S } _ { f , g } = ( - 1 ) ^ { \\sum _ { i = 0 } ^ { n - 1 } ( k _ i ( f ) + k _ i ( g ) ) } \\frac { ( - 1 ) ^ { n d ( d - 1 ) / 2 } \\tau ( d ) ^ n } { d ^ n } \\prod _ { i = 0 } ^ { n - 1 } \\zeta ^ { - k _ i ( f ) k _ i ( g ) } \\sum _ { \\sigma \\in \\mathfrak { S } _ { n + 1 } } ( - 1 ) ^ { l ( \\sigma ) } \\prod _ { i = 1 } ^ { n + 1 } \\zeta ^ { f ( i ) g ( \\sigma ( i ) ) } . \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} \\int _ G f ( n x h ^ { - 1 } ) d \\tilde { \\mu } ( x ) = \\Delta _ H ( h ) \\int _ G f ( x ) \\cdot \\lambda \\big ( n , q ( x ) \\big ) d \\tilde { \\mu } ( x ) . \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} d _ s = s ^ 2 - s \\lfloor s ^ { 1 / 2 } \\rfloor - \\lfloor s / 4 \\rfloor < s ^ 2 \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} \\mathcal { R _ { S E } } _ { n e t } = \\mathbb { P \\left [ A _ { U E } \\right ] } \\mathcal { R _ { S E } } , \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} Y ( X _ \\alpha - X _ \\beta ) & = 2 ( k - c ) ( k + c - 2 ) ( c - 1 ) ( k - 1 ) ( \\cos 2 v - \\cos 2 w ) . \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\Delta _ { k } ( n ) q ^ { n } = \\dfrac { ( q ^ { 2 } ; q ^ { 2 } ) _ { \\infty } ( q ^ { 2 k + 1 } ; q ^ { 2 k + 1 } ) _ { \\infty } } { ( q ; q ) _ { \\infty } ^ 3 ( q ^ { 2 ( 2 k + 1 ) } ; q ^ { 2 ( 2 k + 1 ) } ) _ { \\infty } } . \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} \\mathcal { V } \\overline { c } = \\overline { 0 } , \\overline { c } : = ( c _ 0 , c _ 1 , \\ldots , c _ L ) ^ T , \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} | | u _ n ^ 1 | | _ { H ^ 1 } ^ 2 = | | u _ n - v _ 0 | | _ { H ^ 1 } ^ 2 = | | u _ n | | _ { H ^ 1 } ^ 2 - | | v _ 0 | | _ { H ^ 1 } ^ 2 + o ( 1 ) . \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} [ X , Y ] = [ J X , J Y ] \\quad [ X , J Y ] = - [ J X , Y ] \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} \\P \\left ( v _ c ( t ) \\le \\max _ { 0 \\le x \\le t } \\{ \\mu ( x ) \\} \\le u _ \\rho ( t ) \\textrm { e v e n t u a l l y } \\right ) = \\begin{cases} 1 , & \\rho > 0 c \\in ( 0 , 1 ) , \\\\ 0 , & \\rho \\le 0 c \\ge 1 , \\\\ \\end{cases} \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} \\Delta _ h t ^ { ( \\gamma ) _ h } = \\gamma h t ^ { ( \\gamma - 1 ) _ h } . \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } u ^ { \\epsilon } _ { t } - \\epsilon u ^ { \\epsilon } _ { x _ { i } x _ { i } } + H _ { i } ( t , x , u ^ { \\epsilon } _ { x _ { i } } ) = 0 & \\ , \\ , I _ { i } \\times ( 0 , T ) \\\\ \\sum _ { i = 1 } ^ { K } u ^ { \\epsilon } _ { x _ { i } } = B & \\ , \\ , \\{ 0 \\} \\times ( 0 , T ) \\\\ u ^ { \\epsilon } = u _ { 0 } & \\ , \\ , \\mathcal { I } \\times \\{ 0 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} F _ \\lambda ( x ) : = \\inf _ { y } \\left \\{ F ( y ) + \\tfrac { 1 } { 2 \\lambda } \\| y - x \\| ^ 2 _ 2 \\right \\} , \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} l = ( d - k + 1 ) ( d - k + 2 ) e ^ { ( 1 + o ( 1 ) ) n \\log n } . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} D ^ \\alpha { } _ \\beta = \\delta ^ \\alpha { } _ \\beta + \\operatorname { R e } \\left [ i u ^ \\alpha p _ \\beta \\ , e ^ { i p _ \\gamma x ^ \\gamma } \\right ] . \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} \\nabla f ( \\mathbf { x } ^ { \\star } ) + \\frac { \\rho ( 1 - \\eta ) } { 2 } \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } \\mathbf { x } ^ { \\star } + \\sqrt { \\eta } \\mathbf { E } _ { } ^ T \\boldsymbol { \\nu } ^ { \\star } = \\mathbf { 0 } \\\\ \\mathbf { E } _ { } \\mathbf { x } ^ { \\star } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} I ^ { + } f = \\int _ { 0 } ^ { t } e ^ { - \\Delta ^ { 2 } ( t - s ) } \\Delta f ( s , \\cdot ) \\ d s , \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} \\tilde { F } _ { k , m , r - 1 } ( z ) : = F _ { k , m , r - 1 } ( z ) + \\sum _ { \\substack { A _ k < n < 0 \\\\ ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) } } a _ k ( - m , n ) F _ { k , n , r - 1 } ( z ) . \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} < w _ \\gamma , \\xi > = \\int _ { \\Omega \\times \\Omega } \\mathrm { d } \\gamma ( x , y ) \\int _ { 0 } ^ { 1 } \\xi ( \\omega _ { x , y } ( t ) ) \\cdot { \\omega } ^ \\prime _ { x , y } ( t ) \\ , \\mathrm { d } t \\ ; \\ ; \\ ; \\mbox { f o r a l l } \\ ; \\ ; \\xi \\ ; \\in \\ ; C ( \\Omega , \\R ^ d ) . \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} c ( x _ 1 , \\ldots , x _ N ) = \\sum _ { 1 \\le i < j \\le N } f ( d ( x _ i , x _ j ) ) , \\ , ( x _ 1 , \\ldots , x _ N ) \\in X ^ N , \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} \\| \\pi ^ { ( d ) } ( S ) \\| \\geq \\| \\tau ^ { ( d ) } ( S ) U \\xi \\| = \\| U \\sigma ( S ) \\xi \\| = \\| S \\| \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} \\exp _ { \\psi } \\bigg ( \\sum \\limits _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) t ^ { \\deg _ { \\theta } ( a ) } } { a } \\bigg ) = 1 . \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\| x u ( t ) \\| ^ 2 _ { L ^ 2 } = 1 6 E ( u _ 0 ) , \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} \\norm { g } _ { \\mathcal { L } _ { \\mathrm { T r a n s } } ( p ) } ^ 2 : = \\int _ { \\C ^ n } \\int _ 0 ^ { \\infty } \\abs { g ( t , w ) } ^ 2 \\frac { e ^ { - 4 \\pi p ( w ) t } } { 4 \\pi t } \\d V ( w ) \\d t < \\infty . \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} | F ( x ) - F ( z ) | \\leq K [ 1 + \\sum _ { i = 1 } ^ \\ell ( | x _ i | ^ \\lambda + | z _ i | ^ \\lambda ) ] \\sum _ { i = 1 } ^ \\ell | x _ j - z _ j | ^ \\kappa \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} \\lambda _ { \\pi } ( m ) = \\chi _ l ( m ) ^ { - 1 } \\lambda _ { \\pi _ 0 } ( m ) . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} \\ln R ( n + 1 ) - \\ln R ( n ) = \\frac { o ( 1 ) } { n } . \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} D _ { t } ( \\mu _ { 1 } , \\mu _ { 2 } ) \\ , : = \\ , \\inf \\Big \\{ \\int ( \\sup _ { 0 \\le s \\le t } \\lvert X _ { s } ( \\omega _ { 1 } ) - X _ { s } ( \\omega _ { 2 } ) \\rvert \\wedge 1 ) { \\mathrm d } \\mu ( \\omega _ { 1 } , \\omega _ { 2 } ) \\Big \\} \\ , \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} \\eqref { e s t : i n h } \\lesssim \\| u _ 0 \\| _ { \\dot H ^ { s } ( X ) } + \\| u _ 1 \\| _ { \\dot H ^ { s - 1 } ( X ) } . \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} & T _ 1 = U _ 1 , \\\\ & T _ i = U _ i \\setminus \\bigcup \\limits _ { r = 1 } ^ { i - 1 } U _ r , 2 \\leq i \\leq N , \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} u _ { m + k - 1 } ( s , 0 ) = F _ { m + k - 1 } ( s ) + \\int _ { 0 } ^ s { \\cal G } _ { m + k - 1 , m + k } ( \\xi ) u _ { m + k } ( \\xi , 0 ) \\ , d \\xi \\mbox { f o r } 0 \\le s \\le t _ { k - 1 } , \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = s } \\varphi _ t ^ * \\o _ t & = \\varphi _ s ^ * \\left ( \\mathcal { L } _ { X _ s } \\o _ s \\right ) + \\varphi _ s ^ * ( d \\mu ) \\\\ & = \\varphi _ s ^ * \\left ( d ( \\iota _ { X _ s } \\o _ s ) - \\iota _ { X _ s } d \\o _ s \\right ) + \\varphi _ s ^ * d \\mu = - \\varphi _ s ^ * ( d \\mu ) + \\varphi _ s ^ * ( d \\mu ) = 0 . \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} S _ n ^ { ( k ) } = \\{ f \\in C ^ { k - 2 } [ 0 , 1 ] : & f \\mathcal F _ n \\} , \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\Pi = \\ , & I - \\sum _ { 1 \\le i , j \\le m } ( \\Phi ^ { - 1 } ) _ { i j } \\nabla \\xi _ { i } \\otimes ( a \\nabla \\xi _ { j } ) \\ , , \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{gather*} T _ { t _ { - \\alpha _ 2 } } ( E _ { i , i } ( s ) ) = \\begin{cases} E _ { 1 , 1 } ( s ) & , \\\\ E _ { 2 , 2 } ( s ) + \\delta _ { s , 0 } c & , \\\\ E _ { 3 , 3 } ( s ) - \\delta _ { s , 0 } c & , \\\\ E _ { i , i } ( s ) & \\end{cases} \\end{gather*}"}
-{"id": "5187.png", "formula": "\\begin{align*} + ( - 1 ) ^ { s - 4 } \\sum _ { r _ 1 = 1 } ^ { n } a _ { r _ 1 } b _ { r _ 1 } c _ { r _ 1 } \\frac { H ^ { ( 3 ) } _ { r _ 1 } } { r _ 1 ^ { s - 3 } } . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} | L _ G f ( v ) | & \\leq \\frac { 1 } { m ( v ) } \\sum _ { v \\sim v ' } w ( v , v ' ) ( | f ( v ' ) | + | f ( v ) | ) \\\\ & = | f ( v ) | + \\sum _ { v \\sim v ' } \\frac { w ( v , v ' ) } { m ( v ) } | f ( v ' ) | , \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} \\Vert V \\Vert _ { * } = \\max \\big \\{ \\Vert V ( 0 ) \\Vert , \\Vert V ( 1 ) \\Vert \\big \\} + \\Big ( \\int _ a ^ b \\big \\Vert V ' \\big \\Vert ^ 2 \\ , \\mathrm d s \\Big ) ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} \\overline { \\widetilde { S } ^ { R , R } _ { 1 , Y } } = \\frac { \\sqrt { \\dim ^ R ( \\bar { \\ 1 } ) } \\dim ^ R ( Y ^ * ) } { \\sqrt { \\dim ( \\mathcal { C } ) } } , \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\eta \\circ \\varphi = \\varphi \\circ \\xi . \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} \\mathbb { B } _ L = : \\mathbb { T } _ L \\cdot \\mathbb { A } , \\ ; \\ ; \\ ; \\ ; \\mathbb { B } _ R = : \\mathbb { A } \\cdot \\mathbb { T } _ R . \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} \\dim T _ x M ^ { H _ 1 } + \\dim T _ x M ^ { H _ 2 } - \\dim ( T _ x M ^ { H _ 1 } \\cap T _ x M ^ { H _ 2 } ) = \\dim T _ x M ^ H \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} [ H ( U ^ \\perp ) : H _ t ] = [ U ^ \\perp / U : \\Gamma ( \\Omega ( \\j ) ) ] \\leq | N \\cap ( 1 + \\j ) / N \\cap ( 1 + \\i ) | . \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} \\lim _ { | \\tilde { h } - h | \\to 0 } P _ t \\Phi ( \\tilde { h } ) = P _ t \\Phi ( h ) , \\ h \\in K _ 0 , t > 0 , \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} z ( t ) = y ( b , t ) . \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} x ^ { \\star } & = m ^ \\star ( \\kappa , \\lambda ) \\\\ \\hat { x } & = \\hat { m } ( \\kappa , \\lambda ) , \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\tilde { u } ( t , x ) - D \\Delta \\tilde { u } ( t , x ) = \\left ( \\lambda ( S , N , F ) + \\mu \\right ) \\tilde { u } ( t , x ) \\le 0 \\ & ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu \\tilde { u } ( t , x ) = 0 & ( 0 , T ) \\times \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{gather*} \\mathcal { M } ( z _ - , z _ + , [ 0 , 1 ] , \\rho , H ^ { ( k ) } , J ) \\\\ = \\Bigg \\{ ( \\tau , u ) \\in [ 0 , 1 ] \\times C ^ { \\infty } ( \\mathbb { R } \\times S ^ 1 \\rightarrow M ) \\ \\Bigg | \\ \\begin{matrix} u \\textrm { \\ s a t i s f i e s \\ } ( \\textrm { C } ) \\\\ \\lim _ { s \\rightarrow \\pm \\infty } u ( s , t ) = x _ { \\pm } ( t ) \\\\ [ u _ - \\sharp u , x _ - ] = R ^ { - \\frac { 1 } { k } \\tau } ( z _ + ) \\end{matrix} \\Bigg \\} \\end{gather*}"}
-{"id": "5363.png", "formula": "\\begin{align*} & \\ x ^ * \\in C \\\\ & x ^ * = T _ i ( x ^ * ) f ( x ^ * , y ) \\geq 0 y \\in C . \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} & \\| u u _ h ^ { \\kappa } \\| _ { 1 , p } ^ p \\leq M _ { 1 2 } ( \\kappa , u ) \\left [ \\| u u _ h ^ { \\kappa } \\| _ { p , \\partial \\Omega } ^ p + 1 \\right ] . \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\ell ^ { 2 } ( I ) = \\Big \\lbrace \\lbrace c _ { i } \\rbrace _ { i \\in I } : \\ c _ { i } \\in \\Bbb C , \\ \\sum _ { i \\in I } \\vert c _ { i } \\vert ^ { 2 } < \\infty \\Big \\rbrace \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} D + ( I - U _ A ^ * U _ A ) Y _ 0 = D ^ * + Y _ 0 ^ * ( I - U _ A ^ * U _ A ) . \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u _ k & = P _ C A ^ * w _ k , \\\\ w _ { k + 1 } & = w _ k + \\tau ( y - A u _ k ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 v '' - r ( x ) v = - 1 , \\ x \\in ( - L _ 1 , a ) ; \\\\ & v ( a ) = 0 ; \\\\ & v ( - L _ 1 ) = v ( 0 ) . \\end{aligned} \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathcal { V } } S _ i = 0 , \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} \\mu = \\frac { 1 } { \\log 2 } \\frac { d x } { 1 + x } \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} U _ 2 ( \\omega ) = - \\frac { \\mathrm { o r d } ( \\omega ) ^ 2 } { 6 } ( \\mathrm { o r d } ( \\omega ) ^ 2 - 1 ) a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , ( x - y ) ^ 2 } . \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} M = \\sup \\{ \\Vert S _ n ^ { - 1 } \\Vert : n \\in \\mathbb { N } \\} < + \\infty . \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} g ^ { M _ 1 } - g ^ { P _ 1 } = ( g ^ M - g ^ P ) + g '' , \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} \\beta _ { \\tilde g ^ { ( 4 ) } } \\delta ^ * _ { g _ { \\epsilon } ^ { ( 4 ) } } Y = \\beta _ { \\tilde g ^ { ( 4 ) } } \\delta ^ * _ { \\tilde g ^ { ( 4 ) } } Y + \\frac { \\epsilon } { 2 } \\beta _ { \\tilde g ^ { ( 4 ) } } L _ Y \\alpha ^ 2 , \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} \\mathcal T ^ * J _ { \\Pi _ 1 } ^ { X _ 1 } = J _ { \\Pi _ 2 } ^ { X _ 2 } , \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} \\| g _ 1 ( a _ n ) \\| = \\| \\sum _ { k = 1 } ^ n \\alpha _ { 1 , k } h _ { k } ( a _ n ) \\| \\geq \\| \\alpha _ { 1 , n } h _ n ( a _ n ) \\| - \\| \\sum _ { k = 1 } ^ { n - 1 } \\alpha _ { 1 , k } h _ k ( a _ n ) \\| \\geq n . \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} J _ 1 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & - 1 \\\\ 0 & 1 & 0 \\end{pmatrix} , J _ 2 = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ - 1 & 0 & 0 \\end{pmatrix} , J _ 3 = \\begin{pmatrix} 0 & - 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} { \\rm C a p } \\big ( \\{ \\tilde u = 0 \\} \\big ) = 0 . \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} M _ 0 ^ 2 = \\sup _ { x \\in { \\mathfrak C } } \\int _ 0 ^ 1 g ( \\dot x , \\dot x ) \\ , \\mathrm d t . \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } v _ { \\theta , t } ( 0 , t _ { 0 } ) + H _ { i } ( t _ { 0 } , x , v _ { \\theta , x _ { i } } ( \\cdot , t _ { 0 } ) ) + D T _ { \\theta } + \\zeta = 0 & \\ , \\ , I _ { i } ^ { \\nu } \\\\ \\sum _ { i = 1 } ^ { K } v _ { \\theta , x _ { i } } ( \\cdot , t _ { 0 } ) = 0 & \\ , \\ , \\{ 0 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} \\begin{aligned} & \\big ( \\int _ { - \\infty } ^ x d y \\phi _ 3 ^ { - 2 } ( y ) \\int _ y ^ a \\phi _ 3 ( z ) d z \\big ) \\big ( \\int _ x ^ a \\phi _ 3 ^ { - 2 } ( t ) d t \\big ) - \\\\ & \\big ( \\int _ x ^ a d t \\phi _ 3 ^ { - 2 } ( t ) \\int _ t ^ a \\phi _ 3 ( z ) d z \\big ) \\big ( \\int _ { - \\infty } ^ x \\phi _ 3 ^ { - 2 } ( y ) d y \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ T a _ t { ( b _ t - b _ { t + 1 } ) } & = \\sum _ { t = 0 } ^ T a _ t { [ ( b _ { t } - b ^ * ) - ( b _ { t + 1 } - b ^ * ) ] } \\\\ & = a _ 0 ( b _ 0 - b ^ * ) - a _ T ( b _ { T + 1 } - b ^ * ) + \\sum _ { t = 0 } ^ { T - 1 } \\left ( a _ { t + 1 } - a _ t \\right ) ( b _ { t + 1 } - b ^ * ) \\\\ & \\leq a _ 0 ( b _ 0 - b ^ * ) , \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} \\bigg | \\frac { \\partial ^ { p + 2 } f } { \\prod _ { j = 1 } ^ { p + 2 } \\partial x _ { i _ j } } ( x ) - \\frac { \\partial ^ { p + 2 } f } { \\prod _ { j = 1 } ^ { p + 2 } \\partial x _ { i _ j } } ( y ) \\bigg | \\leq & \\ ; [ h ] _ { \\alpha , p } | x - y | ^ \\alpha \\left ( A - 2 \\log | x - y | \\right ) , & & | x - y | \\leq 1 \\\\ \\leq & \\ ; A \\ ; [ h ] _ { \\alpha , p } & & | x - y | > 1 , \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} k : = \\frac { n \\ell - \\ell ^ 2 - b ^ 2 ( n - 1 ) } { a ^ 2 - b ^ 2 } . \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} \\begin{aligned} z = x _ { \\lambda } & = \\left [ \\lambda | x | ^ { - \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } } + ( 1 - \\lambda ) R ^ { - \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } } \\right ] ^ { - \\frac { p } { n - p } \\frac { \\theta - 1 } { \\theta } } \\frac { x } { | x | } \\\\ & = R \\exp _ { q } \\left [ \\lambda \\log _ { q } \\frac { | x | } { R } \\right ] \\frac { x } { | x | } . \\end{aligned} \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} \\mathfrak { I } _ t ^ { ( a , b , A , B , \\alpha , \\beta ) } ( n , m ) = \\int _ { - 1 } ^ { 1 } e ^ { - t ( 1 - x ) } P _ n ^ { ( a , b ) } ( x ) P _ m ^ { ( A , B ) } ( x ) ( 1 - x ) ^ \\alpha ( 1 + x ) ^ \\beta \\ , d x , \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} r _ { 1 3 } r _ { 1 2 } - r _ { 1 2 } r _ { 2 3 } + r _ { 2 3 } r _ { 1 3 } = \\lambda r _ { 1 3 } . \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} w _ { \\sigma } ^ 2 = s _ { \\rho _ { 1 , \\sigma } + \\rho _ { 2 , \\sigma } } s _ { \\rho _ { 1 , \\sigma } } , \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} I _ { n , d , t } = B _ t ( \\prod _ { i = 0 } ^ { d - 1 } x _ { n - i t } ) . \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} \\sum _ { \\lambda = m + i } ^ { j - 1 } = \\sum _ { \\lambda = m + i } ^ { j - 1 } ( - 1 ) ^ \\lambda ~ h ( a ^ { m + n - 1 } _ { 1 , i - 1 } , ~ f ^ n , ~ a ^ { m + n - 1 } _ { i + m , \\lambda - 1 } , ~ \\alpha ^ { m + n - 2 } ( a _ \\lambda \\cdot a _ { \\lambda + 1 } ) , ~ a ^ { m + n - 1 } _ { \\lambda + 2 , j } , ~ g ^ m , ~ a ^ { m + n - 1 } _ { j + n + 1 , m + n + p - 1 } ) \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\begin{aligned} U ( g ) & = P _ { + } ( z ^ { - n } u g ) \\\\ V ( g ) & = P _ { - } ( z ^ { n } v g ) . \\end{aligned} \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} Q [ z ] = - i D Q [ z ] i z . \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} \\mathcal L _ { \\beta , \\Omega } u = \\mathcal L _ { \\beta , \\Omega } ^ { } \\bar u + \\mathcal L _ { \\beta , \\Omega } \\phi _ 0 = - ( f + \\mathcal L _ \\Omega \\phi _ 0 ) + \\mathcal L _ { \\Omega } \\phi _ 0 = - f . \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} & f _ R ( t ) = \\begin{cases} f ( t ) & t < R \\\\ f ( R ) & \\end{cases} ~ ~ ~ \\\\ & f _ R ^ { - 1 } ( t ) = \\inf \\{ s ~ | ~ f _ R ( s ) = t \\} ; \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} \\boldsymbol { D } = \\boldsymbol { E } + \\vartheta | \\boldsymbol { E } | ^ 2 \\boldsymbol { E } , \\boldsymbol { B } = \\boldsymbol { H } , \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } \\mu ( f ^ i ) x ^ i = D _ f ^ { \\phi } ( x ) ^ { - 1 } . \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ r \\alpha _ j \\kappa ( T _ j ) = 0 . \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} V ( Q ^ { \\star } ) = \\{ 0 \\} \\sqcup V ( Q ) . \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} s _ { X } ^ R ( Y ^ * ) = s _ { \\bar { X } } ^ R ( Y ) . \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} \\mathbb { Q } : = \\left \\{ \\mathcal { T } ( R , p ) \\in S E ( 3 ) | R = \\mathcal { R } _ a ( \\theta ^ * , u ) , \\theta ^ * \\in ( 0 , \\pi ] , p = ( I _ 3 - \\mathcal { R } _ a ( \\theta ^ * , u ) ) b d ^ { - 1 } , u \\in \\mathbb { U } \\right \\} . \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} f ^ { ( p ) } ( 0 ) : = \\lim _ { z \\in S , z \\to 0 } f ^ { ( p ) } ( z ) \\in \\C . \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{gather*} i , j \\geq 0 , \\ j + i v \\leq k \\ \\ \\Rightarrow \\beta ( j + i v ) = \\beta ( j ) + \\beta ( i v ) = \\beta ( j ) + i \\beta ( v ) . \\end{gather*}"}
-{"id": "1355.png", "formula": "\\begin{align*} \\phi _ 3 ( x ) = e ^ { \\psi ( x ) } , \\ \\ \\psi ( x ) = \\lambda ( \\gamma + x ^ 2 ) ^ { \\frac { l + 1 } 2 } , \\ \\ \\gamma , \\lambda > 0 . \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} d _ { C E } ( x _ { 1 } Y _ { 1 } \\wedge \\dots \\wedge x _ { k } Y _ { k } ) = \\sum _ { i < j } \\pm x _ { 1 } Y _ { 1 } \\wedge \\dots \\wedge x _ { i } x _ { j } [ Y _ { i } , Y _ { j } ] \\wedge \\dots \\wedge \\widehat { x _ { j } Y _ { j } } \\wedge \\dots \\wedge x _ { k } Y _ { k } . \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} \\frac 1 t ( \\log ( X _ t ) - \\log ( X _ 0 ) ) & = \\frac 1 t \\int _ 0 ^ t \\frac { \\dot X _ s } { X _ s } d s = \\frac 1 t \\int _ 0 ^ t \\frac { \\alpha ( X _ s ) } { X _ s } d s \\leq \\alpha ' _ 0 < 0 . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} C ' = \\frac { 2 ^ { n - 1 } C } { 1 - 4 n ^ \\frac 1 2 ( e \\tau p ) ^ \\frac { 1 } { p } \\varepsilon } . \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} \\bar { C } ^ \\gamma & = c _ 0 ^ 2 \\bar { C } | \\mathcal { t } \\cdot \\nabla p | ^ 2 = c _ 0 ^ 2 \\bar { C } | \\partial _ s p | ^ 2 = c _ 0 ^ 2 / \\bar C . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} \\lambda _ { 1 , k } ^ { j - 1 } c ^ { I } _ { 1 , b ( 1 , k ) } + \\sum _ { i = 2 } ^ { i = n } \\lambda _ { i , b _ { i } } ^ { j - 1 } c ^ { I } _ { i , b ( 1 , k ) } = 0 j \\in [ r ] , k \\in [ 0 , s - 1 ] . \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} \\left \\{ x \\in M _ { \\mathbb R } \\ , \\ , \\ , \\ , \\large { | } \\ , \\ , \\ , \\ , \\ \\begin{array} { l l } d _ j - 1 < \\langle x , n _ j \\rangle < d _ j & \\mbox { f o r } j \\in \\Omega \\\\ \\langle x , n _ j \\rangle = d _ j & \\mbox { f o r } j \\not \\in \\Omega \\end{array} \\right \\} . \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} K _ 1 K _ 1 ^ \\top = \\frac { n } { 4 } I _ { \\ell } + \\frac { n - 4 } { 4 } J _ { \\ell } , K _ 1 ^ \\top K _ 1 = \\frac { \\ell + a } { 4 } I _ { n - 1 } + \\frac { a } { 2 } A _ 1 + \\frac { \\ell + a } { 4 } J _ { n - 1 } , \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} \\mathcal { E } _ z ( f , h ) = - \\int _ { \\Sigma _ z } ( \\mathcal { L } _ 0 f ) \\ , h \\ , d \\mu _ z \\ , = - \\int _ { \\Sigma _ z } f ( \\mathcal { L } _ 0 h ) \\ , d \\mu _ z \\ , = \\frac { 1 } { \\beta } \\int _ { \\Sigma _ z } ( a \\Pi \\nabla f ) \\cdot ( \\Pi \\nabla h ) \\ , d \\mu _ z \\ , , \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} L _ n v & = f _ { \\alpha , n } - f _ { \\alpha , \\infty } , & & x \\in \\R ^ 3 _ + , & & t \\in J ; \\\\ B v & = \\partial ^ \\alpha g _ { n } - \\partial ^ \\alpha g _ \\infty , & & x \\in \\partial \\R ^ 3 _ + , & & t \\in J ; \\\\ v ( 0 ) & = w _ { 0 , n } - w _ { 0 , \\infty } , & & x \\in \\R ^ 3 _ + . \\end{aligned} \\right . \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} n _ { t , s , b , a } \\cdot z _ 0 = \\frac { 1 } { \\frac { 1 } { 2 } ( - i s + | b | ^ 2 ) + 1 } \\left ( \\frac { t _ 1 } { \\sqrt { 2 k } } , \\ldots , \\frac { t _ k } { \\sqrt { 2 k } } , - i b _ 1 , \\ldots , - i b _ { n - 1 } , \\frac { 1 } { 2 } ( i s - | b | ^ 2 ) \\right ) . \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} \\mathcal G _ { \\mathcal Z } H ( x , . ) ( z ) & = \\Big ( b x - c x ^ 2 - \\int _ 0 ^ \\infty ( e ^ { - x y } - 1 + x y ) N ( d y ) \\Big ) \\frac { \\partial H } { \\partial x } ( x , z ) + H ( x p , z ) - H ( x , z ) \\end{align*}"}
-{"id": "9875.png", "formula": "\\begin{align*} \\mathcal { F } ( \\boldsymbol { D } ^ { \\mu } u ) = ( 2 \\pi i \\xi ) ^ { \\mu } \\mathcal { F } ( u ) \\mathcal { F } ( \\boldsymbol { D } ^ { \\mu * } u ) = ( - 2 \\pi i \\xi ) ^ { \\mu } \\mathcal { F } ( u ) , \\xi \\ne 0 , \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} \\mathcal { N } _ { 1 } \\varphi _ { _ { 0 } } = \\displaystyle \\mathcal { P } ( { } _ { 0 } I _ { T } ^ { 0 . 4 } \\ , \\Theta ( 0 ) ) , \\end{align*}"}
-{"id": "10001.png", "formula": "\\begin{align*} f ^ \\dagger ( q ) = \\inf \\{ s \\in \\mathbb R ^ + | f ( s ) > q \\} . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} y _ 1 \\cdot ( y _ 2 \\cdot \\Psi ) - y _ 2 \\cdot ( y _ 1 \\cdot \\Psi ) = [ y _ 1 , y _ 2 ] \\cdot \\Psi \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{gather*} T _ { t _ { - \\alpha _ 2 } } \\big ( E _ { 2 , k + 2 } \\big ) = - T _ { t _ { - \\alpha _ 2 } } T _ { k + 1 } \\big ( E _ { 2 , k + 1 } \\big ) = - T _ { k + 1 } T _ { t _ { - \\alpha _ 2 } } \\big ( E _ { 2 , k + 1 } \\big ) \\\\ \\hphantom { T _ { t _ { - \\alpha _ 2 } } \\big ( E _ { 2 , k + 2 } \\big ) } { } = T _ { k + 1 } \\big ( E _ { 2 , k + 1 } \\big ) ( 1 ) = - E _ { 2 , k + 2 } ( 1 ) . \\end{gather*}"}
-{"id": "1451.png", "formula": "\\begin{align*} P _ { k } = \\begin{cases} [ U _ { k , 3 k + 3 } , U _ { k , 3 k + 2 } , U _ { k , 3 k + 1 } , U _ { k , 3 k } ] & \\mbox { i f $ k $ i s o d d , } \\\\ [ U _ { k , 3 k } , U _ { k , 3 k + 1 } , U _ { k , 3 k + 2 } , U _ { k , 3 k + 3 } ] & \\mbox { i f $ k $ i s e v e n . } \\\\ \\end{cases} \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| r _ n \\| ^ 2 _ { \\dot { H } ^ 1 _ c } = 0 , \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ \\infty f ( t ) z ^ t = \\frac { \\sum _ { j = 0 } ^ n h _ j ^ * z ^ j } { ( 1 - z ) ^ { n + 1 } } \\ , . \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} & { { \\alpha } _ { 1 2 } } = { { \\alpha } _ { 2 1 } } = { { \\beta } _ { 1 1 } } = - { { \\beta } _ { 2 2 } } = { { k } _ { 1 } } \\\\ & { { \\beta } _ { 1 2 } } = { { \\beta } _ { 2 1 } } = { { \\alpha } _ { 2 2 } } = - { { \\alpha } _ { 1 1 } } = - { { k } _ { 2 } } \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} \\tau \\partial _ { t } \\mathbf { Z } ( t , s , \\cdot ) + \\partial _ { s } \\mathbf { Z } ( t , s , \\cdot ) = \\mathbf { 0 } ( t , s ) \\in ( 0 , \\infty ) \\times ( 0 , 1 ) . \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} D ^ { + } U ( m ) = \\frac { U ( m - 1 ) - U ( m ) } { \\Delta x } , D ^ { - } U ( m ) = \\frac { U ( m ) - U ( m + 1 ) } { \\Delta x } , \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} A _ { 5 } : = \\sup _ { R > 0 } \\left ( \\int _ { | x | \\geq R } \\phi _ { 5 } ( x ) d x \\right ) ^ { 1 / q } \\left ( \\int _ { 0 } ^ { R } \\left ( \\int _ { \\wp } r ^ { Q - 1 } \\psi _ { 5 } ( r y ) d \\sigma ( y ) \\right ) ^ { 1 - p ^ { \\prime } } d r \\right ) ^ { 1 / p ^ { \\prime } } < \\infty , \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} \\omega _ u ^ n = f \\omega ^ n \\ \\textup { o n } \\ X . \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} \\mathcal { C } _ { n e t } \\approx \\frac { \\pi \\lambda _ b } { A _ 2 } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( - 1 \\right ) ^ { n } R ^ { \\frac { n } { 2 } } \\Gamma \\left ( \\frac { n \\mu + 2 } { 2 } \\right ) } { n ! } , \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} \\theta ^ \\ast = \\left \\{ \\theta : y { \\dot \\kappa } ^ { - S } ( \\theta ) = - ( y - 1 ) { \\dot \\kappa } ^ { A } ( \\theta ) \\right \\} . \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} \\kappa _ { \\lambda } \\kappa _ { w \\mu } = \\sum _ { \\nu \\in P } a _ { e , w , \\lambda , \\mu } ^ { \\nu } \\kappa _ { \\nu } . \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} \\mathfrak I ^ * \\Phi ( x ) = \\int _ { ( X _ \\emptyset ) _ x } \\Phi ( v ) \\mu ( v ) , \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} ( E ^ { ( k ) } _ { a , b } | g ) ^ { \\sigma } = ( E ^ { ( k ) } _ { a , b } ) ^ \\sigma | g _ \\lambda , \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} \\| F ' ( u _ 1 ) - F ' ( u _ 2 ) \\| \\leq L \\| u _ 1 - u _ 2 \\| , \\forall \\ u _ 1 , u _ 2 \\in B , \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} \\left ( \\frac { \\sqrt { \\frac { \\pi } { 2 } } \\rho ( y ) } { e ^ { - 2 y ^ { 2 } } } - 1 \\right ) \\frac { 1 } { \\tau } = - y ^ { 4 } + \\frac { 7 y ^ { 2 } } { 2 } - \\frac { 1 1 } { 1 6 } + O ( \\tau ^ { 1 / 2 } ) \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} M _ \\varphi f ( z ) = \\varphi ( z ) f ( z ) . \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} \\alpha _ n ( u _ n ) = \\sum _ { v \\in I } \\alpha ( v ) v _ n ( u ) = \\begin{cases} \\alpha ( v _ 0 ) , & u _ n \\in W ^ { + } _ { n } ( v _ 0 ) \\\\ - \\alpha ( v _ 0 ) , & u _ n \\in W ^ { - } _ { n } ( v _ 0 ) \\end{cases} . \\end{align*}"}
-{"id": "9898.png", "formula": "\\begin{align*} f = \\boldsymbol { D } ^ { - s } u + \\boldsymbol { D } ^ { s * } u , \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X _ { t } = & b ( t , X _ { t } , y _ { t } , z _ { t } ) d t + \\sigma ( t , X _ { t } , y _ { t } , z _ { t } ) d B _ { t } , \\\\ d Y _ { t } = & - g ( t , X _ { t } , Y _ { t } , Z _ { t } ) d t + Z _ { t } d B _ { t } , \\\\ X _ { 0 } = & x , \\ Y _ { T ^ { \\prime } } = \\phi ( X _ { T ^ { \\prime } } ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ n \\lambda _ i ^ { s _ m } c _ { i j _ 0 } \\right | > \\alpha \\mbox { f o r a l l } m \\geq 0 . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} \\mathrm { T r } _ { d , D } ( f ) : = \\frac { 1 } { 2 \\pi } \\sum _ { Q \\in \\mathcal { Q } _ { d D } / \\mathrm { S L } _ 2 ( \\mathbb { Z } ) } \\chi _ D ^ { } ( Q ) \\int _ { \\mathrm { S L } _ 2 ( \\mathbb { Z } ) _ Q \\backslash S _ Q } f ( z ) \\frac { \\sqrt { d D } d z } { Q ( z , 1 ) } , \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} E _ 0 ^ { ( r ) } T _ a \\wedge T _ { - n } = A _ n ( 0 ) . \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} P _ { 1 / 2 , 0 } ( z , s ) = \\sum _ { r = - 1 } ^ { \\infty } F _ { 1 / 2 , 0 , r } ( z ) ( s - 3 / 4 ) ^ r . \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} \\mathbb E \\big [ { } - \\log Z _ { T } \\vert \\mathcal F _ { T } ^ { X } \\big ] \\ , = \\ , \\mathbb E \\Big [ \\int ^ { T } _ { 0 } b ( s , X _ { s } , F _ { s } ) { \\mathrm d } X _ { s } - \\frac { 1 } { 2 } \\int ^ { T } _ { 0 } \\lvert b ( s , X _ { s } , F _ { s } ) \\rvert ^ { 2 } { \\mathrm d } s \\Big \\vert \\mathcal F _ { T } ^ { X } \\Big ] \\ , . \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} \\mathbf { A } g ( x ) = e ^ { ( \\frac { 1 } { 2 } + \\beta ) x } g \\left ( n ( e ^ { x } - 1 ) \\right ) . \\end{align*}"}
-{"id": "9972.png", "formula": "\\begin{align*} \\omega ' _ \\beta = \\sum _ { \\lambda \\in \\Lambda } \\varphi _ \\lambda \\omega _ { \\lambda \\ , \\beta } ( \\beta \\in \\Lambda ^ { q _ 1 } ) . \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} \\mathcal G _ { \\mathcal Y } g ( y ) & = \\big ( \\mathcal G _ { \\mathcal X } ( g \\circ ( - \\log ) ) \\big ) ( e ^ { - y } ) \\\\ [ 1 e m ] & = g \\big ( - \\log ( p e ^ { - y } ) \\big ) - g ( y ) + \\alpha ( e ^ { - y } ) \\cdot \\big ( - \\tfrac 1 x g ' ( - \\log x ) \\big ) \\big | _ { x = e ^ { - y } } \\\\ [ 1 e m ] & = g ( y + \\log \\tfrac 1 p ) - g ( y ) - \\beta ( y ) g ' ( y ) . \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} F ( x , y ) = ( A _ 0 + A _ 1 ( x & - 1 ) + A _ 2 ( x - 1 ) ^ 2 ) & \\\\ & + ( B _ 0 + B _ 1 ( x - 1 ) + B _ 2 ( x - 1 ) ^ 2 ) ( y - 1 ) & \\\\ & + ( C _ 0 + C _ 1 ( x - 1 ) + C _ 2 ( x - 1 ) ^ 2 ) ( y - 1 ) ^ 2 . & \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} \\partial _ t C = \\frac { c _ 0 ^ 2 Q [ C ] ^ 2 } { C } - C ^ \\gamma , \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} G ( X ) = \\prod _ { i = 0 } ^ m ( X + 1 ) ^ { a _ i p ^ i } = \\prod _ { i = 0 } ^ m ( X ^ { p ^ i } + 1 ) ^ { a _ i } . \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} \\mu _ \\mathcal { K } ( x , y , z ) = \\left \\{ \\begin{array} { l l } x , & z \\in \\mathcal { K } , \\\\ y , & . \\end{array} \\right . \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} ( \\sigma _ h ^ { \\frac 1 2 } , w _ h ) + ( \\nabla u _ h ^ { \\frac 1 2 } , \\nabla w _ h ) = 0 , ~ \\forall w _ h \\in L _ h . \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} \\Vert \\sigma ( D _ x ) f \\Vert _ { B ^ s _ { \\infty , \\infty } ( G ) } : = \\sup _ { l \\geq 0 } 2 ^ { l s } \\Vert \\psi _ l ( \\mathcal { R } ) \\sigma ( D _ x ) f \\Vert _ { L ^ \\infty ( G ) } . \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} 2 \\mathsf { R e } ( P ^ * u , i B u ) = \\sum _ { j = 1 } ^ N \\underset { ( \\ref { G P 1 } . 1 ) } { \\underbrace { 2 \\mathsf { R e } ( X _ j ^ * f X _ j u , i B u ) } } + \\underset { ( \\ref { G P 1 } . 2 ) } { \\underbrace { 2 \\mathsf { R e } \\big ( ( X ^ * _ { N + 1 } - i X ^ * _ 0 ) u , i B u \\big ) } } . \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} \\lim _ { | x | \\to + \\infty } \\ \\frac { | \\nabla U ( x ) | ^ 2 } { | \\Delta U ( x ) | } = + \\infty , \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} 3 | x | = | x | + 2 | x | \\geq | x | + | y | \\geq | y ^ { - 1 } x | , \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} \\textrm { d e t } \\ ; { \\bf M } = \\textrm { d e t } \\ ; \\left | \\left | \\begin{array} { c c } f ^ { - 1 } m & 0 \\\\ 0 & f ^ { - 1 } ( m ) \\end{array} \\right | \\right | = m ^ 2 f ^ { - 2 } ( t ) \\neq 0 \\ ; . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} \\mathcal { N } u ( t ) = \\exp \\big ( [ S ^ * ( \\log u ) ] ( t ) \\big ) \\qquad t \\in \\Omega . \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} v ^ { - 1 } _ A \\overset { I } { A } = \\overset { I } { A } v ^ { - 1 } _ A , v ^ { - 1 } _ A \\overset { I } { B } = \\overset { I } { v } \\ ! ^ { - 1 } \\overset { I } { B } \\overset { I } { A } v ^ { - 1 } _ A v ^ { - 1 } _ B \\overset { I } { A } = \\overset { I } { v } \\overset { I } { B } \\ ! ^ { - 1 } \\overset { I } { A } v ^ { - 1 } _ B , v ^ { - 1 } _ B \\overset { I } { B } = \\overset { I } { B } v ^ { - 1 } _ B . \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { s _ 1 - 1 } \\beta ^ t \\alpha _ { i _ 1 } ^ { u _ 1 + p _ { i _ 1 } - 1 } = \\Big ( \\sum _ { t = 0 } ^ { s _ 1 - 1 } \\beta ^ t \\alpha _ { i _ 1 } ^ { p _ { i _ 1 } - 1 } \\Big ) \\alpha _ { i _ 1 } ^ { u _ 1 } \\in W _ { i _ 1 } ^ { ( 2 ) } \\alpha _ { i _ 1 } ^ { u _ 1 } \\subseteq K . \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} X ( v ) = \\sum _ { \\mathbf { v } \\in V } X ^ { ( \\mathbf { v } ) } \\partial _ { \\mathbf { v } } = \\sum _ { \\mathbf { v } \\in V } \\sum _ { ( k , i , \\alpha , \\beta ) \\in \\mathbb { I } } X ^ { ( \\mathbf { v } ) } _ { k , i , \\alpha , \\beta } e ^ { \\mathbf { i } k \\cdot x } y ^ i z ^ { \\alpha } \\bar { z } ^ { \\beta } \\partial _ { \\mathbf { v } } . \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} B _ k ( x + j ( y ) ) = B _ k x + B _ k j ( y ) = b _ k x + \\rho _ k y + j _ k ( \\beta _ k y ) , \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} \\nu ( s ( n , k ) ) = ( \\sigma ( k - 1 ) - \\sigma ( n - 1 ) ) / ( p - 1 ) \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} \\| K _ { V } ^ { - 1 } \\| _ { \\mathcal { L } ( L ^ 2 ( \\mathbb { R } ^ { 2 d } ) ) } = \\| \\displaystyle \\int _ 0 ^ { + \\infty } e ^ { - t K _ { V } } d t \\| _ { \\mathcal { L } ( L ^ 2 ( \\mathbb { R } ^ { 2 d } ) ) } & \\le \\displaystyle \\int _ 0 ^ { + \\infty } \\| e ^ { - t K _ { V } } \\| _ { \\mathcal { L } ( L ^ 2 ( \\mathbb { R } ^ { 2 d } ) ) } d t \\\\ & \\le \\frac { c } { \\sqrt { B } } ~ . \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} P = \\sum _ { \\alpha } \\frac { \\sigma ^ { | \\alpha | } z ^ { \\alpha } } { \\alpha ! } \\partial _ z ^ { \\alpha } . \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} \\Phi ( e , h ) = ( \\eqref { s y s t 1 } , \\eqref { s y s t 2 } ) . \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\binom { n } { p ^ t } b _ n = b _ { p ^ t } b _ { n - p ^ t } + \\sum _ { 0 < i , j < N } a _ { i j } b _ { p ^ t - i } b _ { n - p ^ t - j } . \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} \\Phi _ \\delta ^ { \\gamma } ( t ) = \\frac { 1 } { 2 } \\int _ { B _ r } \\log \\left ( 1 + \\frac { | X _ { 1 t } - X _ { 2 t } | ^ 2 + \\gamma \\langle \\eta ( X _ { 1 t } ) , X _ { 1 t } - X _ { 2 t } | \\rangle ^ 2 } { \\delta ^ 2 } \\right ) d x . \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} P _ x & = n _ y u _ z - n _ z u _ y , \\\\ P _ y & = n _ z u _ x - n _ x u _ z , \\\\ P _ z & = n _ x u _ y - n _ y u _ x , \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } P h _ n ( x _ n ) \\leq \\lim _ { n \\to + \\infty } P h _ m ( x _ n ) = P h _ m ( x ) , \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} ( \\mathbb { F } _ - ) ^ { \\alpha \\beta } = - i \\sigma ^ \\alpha { } _ { \\dot a b } \\ , \\zeta ^ b { } _ c \\ , \\sigma ^ { \\beta \\dot a c } \\ , , \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} T _ i ( K _ \\lambda ) = K _ { s _ i ( \\lambda ) } T _ i ( L _ \\lambda ) = L _ { s _ i ( \\lambda ) } , \\ \\forall \\lambda \\in Q , \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } u ^ { \\epsilon } - \\epsilon u ^ { \\epsilon } _ { x _ { i } x _ { i } } + H _ { i } ( x , u ^ { \\epsilon } _ { x _ { i } } ) = 0 & \\ , \\ , I _ { i } \\\\ \\sum _ { i = 1 } ^ { K } u ^ { \\epsilon } _ { x _ { i } } = B & \\ , \\ , \\{ 0 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} \\sqrt { R R ^ * } f = \\chi _ { - \\lambda } \\cdot ( \\widetilde { f } * \\omega _ H ) \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mathbb { P } ( \\mathcal { A } _ 1 ) = \\mathbb { P } \\left ( \\frac { Z _ 1 } { Z _ 1 + Z _ 2 } > \\Gamma _ { R R T } ^ { \\alpha } ( k _ 0 ) \\right ) = \\mathbb { P } \\left ( \\frac { 1 - \\Gamma _ { R R T } ^ { \\alpha } ( k _ 0 ) } { \\Gamma _ { R R T } ^ { \\alpha } ( k _ 0 ) } \\frac { Z _ 1 } { n \\sigma ^ 2 } > \\frac { Z _ 2 } { n \\sigma ^ 2 } \\right ) \\end{array} \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } - \\tfrac 1 t \\log \\mathbb P ( Z _ t > 0 ) & = \\begin{cases} 1 + \\max \\{ 0 , - k ( h ' ( 1 ) - 1 ) \\} & p = 0 , \\\\ 1 - p - \\lambda ( \\mu - 1 ) & \\nu \\leq p , \\\\ 1 - \\nu ( 1 + \\log \\tfrac 1 \\nu ) & p < \\nu \\leq 1 . \\end{cases} \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} w ' ( E P ) & = \\frac { - ( n - 0 + ( m + 1 ) - 0 - 1 ) } { 2 } + w ' ( P ) = - \\frac { n + m } { 2 } + v _ { n , m } \\\\ & = - \\frac { n + m } { 2 } - \\sum _ { k = D _ { n , m } } ^ { D _ { n , m } + m - 1 } k = - \\sum _ { k = D _ { n , m } } ^ { D _ { n , m } + m } k = - \\sum _ { k = D _ { n , m + 1 } } ^ { D _ { n , m + 1 } + ( m + 1 ) - 1 } k = v _ { n , m + 1 } . \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} S ( \\alpha _ 1 , \\ldots , \\alpha _ n ) = ( 0 , 0 , S ( \\alpha _ 1 ) , \\ldots , S ( \\alpha _ n ) ) . \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} \\mathbb { A } ^ { - 1 } _ L = \\Delta ^ { - 1 } \\left ( \\begin{matrix} - \\Delta ( b d ^ { - 1 } c - a ) ^ { - 1 } & \\Delta ( c - d b ^ { - 1 } a ) ^ { - 1 } \\\\ - c - [ a , c ] ( a - b d ^ { - 1 } c ) ^ { - 1 } & a - [ c , a ] ( d b ^ { - 1 } a - c ) ^ { - 1 } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} h ( x ) = h ( - x ) \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} C _ i = \\{ ( x _ 1 , \\cdots , x _ d ) \\in [ 0 , 1 ] ^ d \\colon | x _ 1 - i _ 1 \\Delta x | \\le \\Delta x / 2 , \\cdots , | x _ d - i _ d \\Delta x | \\le \\Delta x / 2 \\} . \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} P _ j - P _ 1 = ( j - 1 ) \\epsilon \\tilde { P } ^ { ( 1 ) } \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { \\dot { H } ^ 1 _ c } : = \\| \\sqrt { P _ c } u \\| ^ 2 _ { L ^ 2 } = \\int | \\nabla u ( x ) | ^ 2 - c | x | ^ { - 2 } | u ( x ) | ^ 2 d x , \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} \\mathsf { r e c t s } | _ \\mathcal { P } ( T ) = \\mathsf { r e c t s } | _ \\mathcal { P } ( T _ \\mathcal { C } ) . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\d X _ t = \\sqrt { 2 } \\ , \\d B _ t - \\nabla U ( X _ t ) \\ , \\d t - \\mathrm { n } _ t \\ , L ( \\d t ) . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} \\mathcal { J } ( \\varphi ) : = \\int _ M \\mathcal { L } \\bigl ( e _ 1 ( \\varphi ) , e _ 2 ( \\varphi ) , e _ 3 ( \\varphi ) , e _ 4 ( \\varphi ) \\bigr ) \\ , \\sqrt { - \\det g _ { \\mu \\nu } ( x ) } \\ d x \\ , , \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} w = x _ { i _ 1 } x _ { i _ 2 } \\cdots x _ { { i _ { l - 1 } } } x _ { k - ( l - 1 ) } x _ { { i _ { l + 1 } } } \\cdots x _ { { i _ d } } , \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} V ( q ) = \\lambda _ 1 q _ 1 + \\sum _ { i = 2 } ^ d \\frac { \\nu _ i } { 2 } q ^ 2 _ i \\ , , \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} \\bigl | \\chi ( t ) - \\dot \\gamma ( t ) \\bigr | ~ = ~ \\bigl | \\eta _ i - \\dot \\gamma ( t ) \\bigr | ~ < ~ \\delta \\qquad \\forall t \\in [ a _ i , b _ i ] , ~ ~ i = 1 , \\ldots , N , \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} & Q ^ 1 _ { i \\bar j } = g ^ { k \\bar l } g ^ { m \\bar n } T _ { i k \\bar n } T _ { \\bar j \\bar l m } \\ , , Q ^ 2 _ { i \\bar j } = g ^ { \\bar l k } g ^ { \\bar n m } T _ { \\bar l \\bar n i } T _ { k m \\bar j } \\ , , \\\\ & Q ^ 3 _ { i \\bar j } = g ^ { \\bar l k } g ^ { \\bar n m } T _ { i k \\bar l } T _ { \\bar j \\bar n m } \\ , , Q ^ 4 _ { i \\bar j } = \\frac 1 2 g ^ { \\bar l k } g ^ { \\bar n m } ( T _ { m k \\bar l } T _ { \\bar n \\bar j i } + T _ { \\bar n \\bar l k } T _ { m i \\bar j } ) \\ , , \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} C _ 1 = \\frac { 1 } { A ' _ 1 } ; C _ 2 = \\frac { 1 } { A _ 2 ' } ; t _ \\tau = B ( T ) ; \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} - \\frac 1 t \\log u _ 1 ( t ) & \\xrightarrow { t \\to \\infty } 1 - \\frac 1 \\lambda + \\frac 1 \\lambda \\log \\frac 1 \\lambda = 1 - \\tfrac 1 \\lambda ( 1 + \\log \\lambda ) = : A _ { \\lambda } . \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} \\hat { \\Omega } _ j : = R _ { j j } + \\sigma _ j \\langle \\partial _ x \\Omega _ j , F ^ y \\rangle = R _ { j j } + \\sigma _ j \\langle \\partial _ x \\tilde { \\Omega } _ j , F ^ y \\rangle . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} u ^ { \\theta } ( x , T _ { \\theta } ) & = u ( x , s ) - \\frac { ( T _ { \\theta } - s ) ^ { 2 } } { 2 \\theta } \\\\ & \\leq u _ { 0 } ( x ) + ( u ( x , T _ { \\theta } ) - u _ { 0 } ( x ) ) + ( u ( x , s ) - u ( x , T _ { \\theta } ) ) \\\\ & \\leq u _ { 0 } ( x ) + \\omega ( T _ { \\theta } ) + \\omega ( | T _ { \\theta } - s | ) \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} U ( x ) = ( a + b | x | ^ { \\frac { p } { p - 1 } } ) ^ { 1 - \\frac { n } { p } } , a , b > 0 \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} M _ { n _ 1 } ( w ) & = \\{ m _ { n _ 1 } \\} , \\\\ M _ { n _ 2 } ( w ) & = \\{ m _ { n _ 2 } \\} . \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\partial _ s v ( s , t ) - \\tau \\cdot \\rho ' ( s ) \\partial _ t v ( s , t ) + J _ { t + \\rho ( s ) } ( v ) ( \\partial _ t v ( s , t ) - X _ { H _ { t + \\rho ( s ) } ^ { ( k ) } } ) = 0 \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta _ { 1 } ( x ) & = \\sum _ { m = 0 } ^ { M } c _ { m } x ^ { - 1 + 2 \\beta - m } \\\\ \\zeta _ { 2 } ( - x ) & = \\sum _ { m = 0 } ^ { M } c _ { m } ' x ^ { - 1 - 2 \\beta - m } \\end{aligned} \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{gather*} \\mathbf { Q } ( \\hat { m } , m ^ \\star , \\kappa _ 0 , \\lambda _ 0 ) = \\left [ \\begin{array} { c } Q _ 1 ( \\hat { m } , m ^ \\star , \\kappa _ 0 , \\lambda _ 0 ) \\\\ Q _ 2 ( \\hat { m } , m ^ \\star , \\kappa _ 0 , \\lambda _ 0 ) \\\\ \\end{array} \\right ] = 0 \\end{gather*}"}
-{"id": "8481.png", "formula": "\\begin{align*} u _ { L _ \\xi ( \\lambda , \\mu ) } ( v _ { \\lambda , \\mu } ) = \\left ( \\varphi \\in L _ { \\xi } ( \\lambda , \\mu ) ^ * \\mapsto \\xi ^ { - \\langle \\lambda , \\mu \\rangle } \\varphi ( v _ { \\lambda , \\mu } ) \\right ) , \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} v _ { - , a } ( x ) = 2 \\phi _ 3 ( x ) \\frac { \\int _ x ^ \\infty d y \\thinspace \\phi ^ { - 2 } _ 3 ( y ) \\int _ a ^ x d t \\phi ^ { - 2 } _ 3 ( t ) \\int _ t ^ y \\phi _ 3 ( z ) d z } { \\int _ a ^ \\infty \\phi ^ { - 2 } _ 3 ( y ) d y } , \\ x \\ge a . \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} U ( t , x ) = \\int _ { - \\infty } ^ { x } u \\left ( t , \\xi \\right ) \\ , d \\xi , U ^ { \\sharp } ( t , x ) = \\int _ { - \\infty } ^ { x } u ^ { \\sharp } \\left ( t , \\xi \\right ) \\ , d \\xi . \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} \\binom { n } { p ^ t } b _ n = b _ { n - p ^ t } b _ { p ^ t } + \\sum _ { 0 < i , j < N } a _ { i j } b _ { n - p ^ t - i } b _ { p ^ t - j } \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} | \\sum _ { b = 1 } ^ n k _ b j _ b ^ 2 + j ^ 2 | < \\frac { | j | } { 5 0 n } , \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} t _ \\infty ( x _ 1 ) ^ * t _ \\infty ( x _ 2 ) = \\rho _ \\infty ( \\langle x _ 1 , x _ 2 \\rangle ) \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} \\bar R _ { x , y } \\ , z ^ { \\bot } = 0 , \\qquad ( x , \\ , y , \\ , z \\in T M ) . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} \\square ^ n \\otimes \\square ^ m = \\square ^ { n + m } \\ , \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} a ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } \\biggl [ a ( x , y , t ) - x \\frac { \\partial } { \\partial y } \\varphi ( x , y , t ) \\biggr ] \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} R ( n , b _ 1 , \\ldots , b _ g ; t ) : = t + \\sum _ { k = 1 } ^ g t ^ { \\frac { r _ k } { R _ k } } \\frac { 1 - t } { 1 - t ^ { \\frac { 1 } { R _ k } } } - \\sum _ { k = 0 } ^ g t ^ { \\frac { r ' _ k } { R ' _ k } } \\frac { 1 - t } { 1 - t ^ { \\frac { 1 } { { R ' _ k } } } } , \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} Y = Y ^ T - \\frac { Y ^ { \\perp } } { u } \\partial t , ~ ~ Y ^ T \\in T S Y ^ { \\perp } = \\frac { 1 } { u } \\langle Y , \\partial _ t \\rangle . \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\mathbf { E } _ { x ' } \\Big ( \\sup _ { t _ 1 \\le s \\le t } \\big | z ( s ) - \\bar { z } ( s ) \\big | ^ 2 \\Big ) \\le & \\ , 3 \\mathbf { E } _ { x ' } \\Big ( \\big | z ( t _ 1 ) - \\bar { z } ( t _ 1 ) \\big | ^ 2 \\Big ) e ^ { L _ 1 ( t - t _ 1 ) } \\\\ \\le & \\ , 3 \\mathbf { E } _ { x ' } \\Big ( \\sup _ { 0 \\le s \\le t _ 1 } \\big | \\xi ( x ( s ) ) - z ( s ) \\big | ^ 2 \\Big ) e ^ { L _ 1 ( t - t _ 1 ) } \\ , , \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} \\begin{aligned} & c _ { F , 1 } ( T _ 0 ) \\le 2 C _ { F , 1 } ( T _ 0 ) , \\\\ & c _ { F , 2 } \\le 2 C _ { F , 2 } ( T _ 0 ) + C _ { F , 1 } ( T _ 0 ) \\log { 2 } , \\\\ & c _ { F , 3 } ( T _ 0 ) \\le \\frac { 3 C _ { F , 3 } ( T _ 0 ) } { 2 } . \\end{aligned} \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} n _ b ( W ^ n ( w ) ) & = \\max ( \\lvert ( 0 , n ) \\rvert _ { W _ n ( w ) } , \\lvert ( n , 0 ) \\rvert _ { W _ n ( w ) } ) \\\\ & = \\max ( \\lvert m _ 0 \\rvert + \\lVert ( \\phi w _ k ) _ n \\rVert _ S , \\lVert ( \\phi w _ 0 ) _ n \\rVert _ S + \\lvert m _ k \\rvert ) . \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} C _ { t } = \\{ ( x , z ) , 0 \\leq x \\leq t , z \\geq t - x \\} . \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} I & = \\int _ 0 ^ \\infty e ^ { - L _ s } b _ s d s \\geq I _ { T ' } + \\int _ { T ' } ^ \\infty e ^ { - ( 1 + \\varepsilon ) h ( s ) } b _ s d s = \\infty . \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} \\pi ( k ) = \\begin{cases} n & 1 \\leq k \\leq n + 1 \\\\ \\left \\lfloor \\frac { k - n - 2 } { d - 1 } \\right \\rfloor & n + 2 \\leq k \\leq n d + 1 \\end{cases} . \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} \\Z _ K ( \\Omega \\underline { X } ) \\xrightarrow { \\widetilde { w } } D J _ K ( \\underline { X } ) \\to \\prod _ { i = 1 } ^ m X _ i \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} \\theta _ 1 ( z \\mid \\tau ' ) = - i ( - i \\tau ) ^ { \\frac { 1 } { 2 } } e ^ { \\frac { i \\tau z ^ 2 } { \\pi } } \\theta _ 1 ( z \\tau \\mid \\tau ) . \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} \\delta _ { g p } = 2 B _ x L _ p \\sqrt { n } ( ( 1 + x _ { \\infty } ^ { m a x } ) n + ( 1 + \\lambda _ { \\infty } ^ { m a x } ) m + 1 ) 2 ^ { - ( f l + 1 ) } \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\mathrm { G } P _ { k _ 1 , \\ldots , k _ n } = - n ( k _ 1 + \\cdots + k _ n ) P _ { k _ 1 , \\ldots , k _ n } . \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { B ^ s _ { \\infty , \\infty } ( G ) } : = \\sup _ { l \\geq 0 } 2 ^ { l s } \\Vert \\psi _ l ( \\mathcal { R } ) f \\Vert _ { L ^ \\infty ( G ) } \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} \\mathbf E \\left [ \\int _ 0 ^ { \\tau _ { t , x } } f _ n \\left ( - X ^ { t , \\beta } ( s ) , X ^ { x , \\alpha } ( s ) \\right ) d s \\right ] & = \\int _ 0 ^ \\infty P _ s ^ { \\beta , } P _ s ^ \\Omega f _ n ( t , x ) \\ , d s \\\\ & \\to \\int _ 0 ^ \\infty P _ s ^ { \\beta , } P _ s ^ \\Omega f ( t , x ) \\ , d s \\\\ & = \\mathbf E \\left [ \\int _ 0 ^ { \\tau _ { t , x } } f \\left ( - X ^ { t , \\beta } ( s ) , X ^ { x , \\alpha } ( s ) \\right ) d s \\right ] , \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} f ( x ) & = x ( x ^ 2 + x + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( 1 + x ) + m \\ : h ( x ) ) , \\\\ g ( x ) & = x ( x + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( 1 + x ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} F _ { i j } ( d x ) = \\mathbb { P } _ { i , 0 } ( J _ 1 = j , Y _ 1 \\in { d x } ) , \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} \\mathbf { M } f ( x ) = \\int _ { 0 } ^ { \\infty } M ( x , y ) d y . \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} \\theta '' ( g _ 0 , \\ldots , g _ { n - 1 } ) ( \\eta ) : = \\sum _ { s _ 0 , \\ldots , s _ { n - 1 } } \\theta ' ( h _ { s _ 0 } , \\ldots , h _ { s _ { n - 1 } } ) ( L ( \\eta ) | _ { [ s _ 0 , \\ldots , s _ { n - 1 } ] } ) . \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} I _ \\rho \\big ( Q ( f ) \\big ) = \\int _ G f ( x ) \\rho ( x ) d x . \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{gather*} \\langle M ' , a ^ { m } \\rangle = \\langle a ^ { m } , b ^ { m + 1 } \\rangle = \\langle b ^ { m + 1 } , M ' \\rangle = \\langle a ^ { m + 1 } \\rangle ^ { \\perp } \\\\ \\langle M , b ^ { m } \\rangle = \\langle b ^ { m } , a ^ { m } \\rangle = \\langle a ^ { m } , M \\rangle = \\langle b ^ { m + 1 } \\rangle ^ { \\perp } . \\end{gather*}"}
-{"id": "6692.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ { r j } \\binom k j \\left ( { \\frac { { F _ r } } { { F _ n } } } \\right ) ^ j G _ { m - r k + ( n + r ) j } } = ( - 1 ) ^ { r k } \\left ( { \\frac { { F _ { n + r } } } { { F _ n } } } \\right ) ^ k G _ m , n \\ne 0 \\ , , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} & ( a + b w ) ( c + d w ) = ( a c + \\alpha \\overline { d } b ) + ( d a + b \\overline { c } ) w , \\\\ & n \\bigl ( a + b w \\bigr ) = n ( a ) - \\alpha n ( b ) . \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\Delta ^ r ( k ^ n ) = \\sum _ { i = 0 } ^ { k - 1 } \\binom { k + n - i - 1 } { n - r } A ( n , i ) . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} f ( T ( v ) ) = \\| v \\| _ 2 ^ 2 - e ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} & 4 i ^ 2 ( d n ) ^ 2 - 4 0 i ^ 2 d n - 4 i ( d n ) ^ 2 - 1 6 i ^ 2 d ^ 3 n - 8 i ^ 2 \\Delta d ^ 2 n \\\\ & = 4 i ^ 2 ( d n ) ^ 2 - 2 i ( d n ) ^ 3 \\left ( 2 0 i / ( d ^ 2 n ^ 2 ) + 2 / ( d n ) + 8 i / n ^ 2 + 4 i \\Delta / ( d n ^ 2 ) \\right ) \\\\ & \\geq 4 i ^ 2 ( d n ) ^ 2 - 2 i ( d n ) ^ 3 \\left ( 1 3 \\Delta ^ 2 / n ^ 2 + 2 / n + 6 d \\Delta / n ^ 2 \\right ) . \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} W _ { \\phi , p } ( a ( y ) w n ( \\zeta _ { l } ) ) = \\abs { y } _ { p } ^ { \\frac { 1 } { 2 } } \\mathcal { H } W _ { l } ( y ) . \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} h \\left ( c ( x , \\xi ) , \\xi \\right ) - h \\left ( c ( x , \\xi ) + \\nabla c ( x , \\xi ) ( y - x ) , \\xi \\right ) & = L _ 1 ( \\xi ) \\| \\nabla c ( x , \\xi ) ( y - x ) \\| _ 2 \\\\ & \\leq L _ 1 ( \\xi ) L _ 3 ( \\xi ) \\cdot \\sqrt { q ( \\| x \\| _ 2 ) } \\| y - x \\| _ 2 \\\\ & \\leq \\sqrt { 2 } L _ 1 ( \\xi ) L _ 3 ( \\xi ) \\cdot \\sqrt { D _ { \\Phi } ( y , x ) } , \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} v _ { 1 , 1 } , \\dots , v _ { r _ 1 , 1 } , v _ { 1 , 2 } , \\dots , v _ { r _ 2 , 2 } , v _ { 1 , 3 } , \\dots , v _ { r _ { L - 1 } , L - 1 } , v _ { 1 , L } , \\dots , v _ { r _ L , L } , v _ { 1 , L + 1 } : = v , \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} \\pi ( [ p _ 1 , \\dots , p _ { r - 1 } , p _ r ] ) = [ p _ 1 , \\dots , p _ { r - 1 } ] . \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} L ( z _ 1 , \\dots , z _ n , s ) = \\sum _ { d \\geq 0 } \\sum _ { a \\in A _ { + , d } } a ( z _ 1 ) \\dots a ( z _ n ) a ^ { - s } \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} k _ { t , a } \\cdot z _ 0 = \\frac { 1 } { \\sqrt { 2 n } } \\ , ( t _ 1 , \\ldots , t _ n ) \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} \\begin{aligned} & \\Re \\left ( 1 - \\left ( \\frac { \\rho } { \\rho - \\tau } \\right ) ^ n \\right ) \\\\ & \\quad > - \\frac { n ( n - 1 ) \\tau ^ 2 } { \\Im ( \\rho ) ^ 2 } \\left ( \\frac { \\tau ^ 2 - \\Im ( \\rho ) ^ 2 } { 2 ( \\frac { \\tau ^ 2 } { | \\Im ( \\rho ) | } + | \\Im ( \\rho ) | ) ^ 2 } + \\frac { 0 . 1 8 4 ( n - 2 ) \\tau } { | \\Im ( \\rho ) | } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} & \\| \\mathbf { u } ^ { [ k + 1 ] } - \\mathbf { u } ^ { \\star [ k + 1 ] } \\| _ { \\mathbf { G } } \\leq \\left ( \\frac { 1 } { \\sqrt { 1 + \\delta _ { } } } \\right ) ^ { k + 1 } \\| \\mathbf { u } ^ { [ 0 ] } - \\mathbf { u } ^ { \\star [ 0 ] } \\| \\\\ + & \\sum _ { i = 0 } ^ { k } \\left ( \\frac { 1 } { \\sqrt { 1 + \\delta _ { } } } \\right ) ^ { k - i + 1 } \\| \\mathbf { u } ^ { \\star [ i ] } - \\mathbf { u } ^ { \\star [ i + 1 ] } \\| _ { \\mathbf { G } } , \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} & \\Psi _ 1 ( a , b , c ) = \\frac { \\rho ( ( a . b ) c ) \\psi \\left ( \\frac { | a - ( a . b ) b | } { 4 ( a . b ) ( 1 - ( a . b ) c ) } \\right ) } { 4 ^ { d - 1 } ( a . b ) ^ { d } ( 1 - ( a . b ) c ) ^ { d - 1 } } , ~ ~ \\\\ & \\Psi _ 2 ( a , b , c ) = \\frac { \\rho ( ( a . b ) c ) \\psi \\left ( \\frac { | a - ( a . b ) b | } { 4 ( a . b ) ( 1 - ( a . b ) c ) } \\right ) } { 4 ^ { d - 1 } ( a . b ) ^ { d - 1 } ( 1 - ( a . b ) c ) ^ { d } } c ( a - ( a . b ) b ) . \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} u _ { k + m } ( T _ 2 - \\delta , x ) = \\dots = u _ { k + 1 } ( T _ 2 - \\delta , x ) \\mbox { f o r } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} \\mathbb { E } _ x : = \\Psi ^ { - 1 } ( ( \\mathbb { R } ^ s \\times \\mathbb { E } ' ) _ { \\Psi ( x ) } ) , \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} ( \\chi ) & = \\frac { 2 } { \\sqrt { \\pi } } \\sum _ { k = 1 } ^ { \\infty } ( - 1 ) ^ { k + 1 } \\frac { \\chi ^ { 2 k - 1 } } { ( 2 k - 1 ) ( k - 1 ) ! } . \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} h _ 0 ( x ) = \\pi ^ { - 1 / 4 } e ^ { - x ^ 2 / 2 } , h _ 1 ( x ) = \\sqrt { 2 } x h _ 0 ( x ) ~ , \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\ , \\frac { 1 } { \\sigma ^ 2 _ { \\mathcal P _ { l } } ( S ^ { n _ l } h _ l ) } \\int \\left ( \\omega ^ { n _ l } ( h _ l , 2 \\epsilon _ l , p ) \\right ) ^ 2 d \\nu _ { \\mathcal P _ { l } } ( p ) = 0 \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} - D ^ { \\beta } _ 0 \\bar u = \\mathcal L _ \\beta u - D ^ { \\beta } _ 0 u _ 0 = \\lambda u - w = \\lambda \\bar u - ( w + \\lambda u _ 0 ) , \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\epsilon ^ 2 \\Delta u + u + \\rho ( x ) \\phi u = | u | ^ { p - 1 } u , & x \\in \\R ^ 3 , \\\\ \\ , \\ , \\ , - \\Delta \\phi = \\rho ( x ) u ^ 2 , \\ & x \\in \\R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} w ^ { ( n ) } ( x ) : = ( w \\otimes \\mathrm { I d } ^ { \\otimes n - 1 } ) \\circ \\cdots \\circ ( w \\otimes \\mathrm { I d } ) \\circ w ( x ) = 0 \\ . \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} \\sum _ { k , l } G ^ { k l } _ \\lambda ( \\Q ) \\xi _ k \\xi _ l & = \\abs { \\xi } ^ 2 + \\frac { \\abs { \\Q \\xi } ^ 2 } { \\abs { \\Q } ^ 2 } \\ , \\omega _ \\lambda ( \\abs { \\Q } ) . \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} & \\nabla _ { \\nabla _ { \\partial _ t } \\partial _ t } Y = \\nabla _ { u \\nabla u } Y = \\nabla _ { u \\nabla u } Y ^ T - \\nabla _ { u \\nabla u } ( \\frac { Y ^ { \\perp } } { u } \\partial _ t ) \\\\ & = \\nabla _ { u \\nabla u } Y ^ T - \\langle u \\nabla u , \\nabla \\frac { Y ^ { \\perp } } { u } \\rangle \\cdot \\partial _ t - \\frac { Y ^ { \\perp } } { u } \\nabla _ { u \\nabla u } \\partial _ t \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} \\lVert k \\cdot B \\psi \\lVert ^ 2 & = \\sum _ { i , j = 1 } ^ { \\nu } \\langle k _ i B _ i \\psi , k _ j B _ j \\psi \\rangle \\leq \\sum _ { i , j = 1 } ^ { \\nu } \\lvert k _ i \\lvert \\lvert k _ j \\lvert \\lVert B _ i \\psi \\lVert \\lVert B _ j \\psi \\lVert \\\\ & \\leq \\sum _ { i , j = 1 } ^ { \\nu } \\frac { 1 } { 2 } \\lvert k _ i \\lvert ^ 2 \\lVert B _ j \\psi \\lVert ^ 2 + \\frac { 1 } { 2 } \\lvert k _ j \\lvert \\lVert B _ i \\psi \\lVert ^ 2 = \\lvert k \\lvert ^ 2 \\lVert B \\psi \\lVert ^ 2 \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} w = \\alpha ( a _ 0 ^ z - r ^ z ) + \\beta _ 0 ( a _ 0 ^ x - r ^ x ) + \\gamma _ 0 ( a _ 0 ^ y - r ^ y ) \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac 1 r \\left \\| \\phi - \\ell _ \\phi \\right \\| _ { \\underline { L } ^ 2 \\left ( B _ { r } \\right ) } = 0 . \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} \\sup _ { \\pi \\in \\Pi } E _ { \\mu } ^ \\pi \\left ( \\sum _ { n = N + 1 } ^ \\infty \\beta ^ { n - 1 } r _ i ^ { l } ( s _ n , a _ n ) \\right ) \\le \\sup _ { \\pi \\in \\Pi } E _ { \\mu } ^ \\pi \\left ( \\sum _ { n = N + 1 } ^ \\infty \\beta ^ { n - 1 } w ( s _ n ) \\right ) . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} \\left \\{ ( x , y ) : m _ n ( x , y ) < r _ n \\right \\} = \\left \\{ ( x , y ) : S _ n ( x , y ) > 0 \\right \\} . \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} \\begin{aligned} \\chi ( S ) & = \\chi ( S \\cap U ) + \\chi \\bigg ( S \\cap \\bigg ( \\bigcup _ i e _ i \\bigg ) \\bigg ) - \\chi \\bigg ( S \\cap U \\cap \\bigg ( \\bigcup _ i e _ i \\bigg ) \\bigg ) \\\\ & = \\chi ( S \\cap U ) + \\sum _ i \\chi ( S \\cap e _ i ) - \\sum _ i \\chi ( S \\cap U \\cap e _ i ) \\ . \\end{aligned} \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{gather*} S ^ { \\delta _ n ^ k + v - \\alpha _ { k + 1 - v } } \\left ( \\alpha \\right ) = \\alpha \\iff \\begin{array} { c } \\{ \\alpha _ { j + ( k + 1 - v ) } = \\alpha _ j + \\alpha _ { k + 1 - v } \\} _ { j = 1 } ^ { v - 1 } \\\\ \\mbox { a n d } \\ \\ \\delta _ { n } ^ k = \\alpha _ { v } + \\alpha _ { k + 1 - v } \\\\ \\mbox { a n d } \\ \\ \\{ \\alpha _ { j + v } = \\alpha _ { j } + \\alpha _ { v } \\} _ { j = 1 } ^ { k - v } . \\end{array} \\end{gather*}"}
-{"id": "653.png", "formula": "\\begin{align*} M _ 1 ^ \\top M _ 1 & = \\begin{pmatrix} r _ 1 ^ \\top r _ 1 \\\\ r _ 1 ^ \\top r _ 2 \\\\ \\vdots \\\\ r _ 1 ^ \\top r _ m \\end{pmatrix} ( r _ 1 ^ \\top r _ 1 , r _ 2 ^ \\top r _ 1 , \\ldots , r _ m ^ \\top r _ 1 ) = \\begin{pmatrix} m J _ m & O & \\cdots & O \\\\ O & m J _ m & \\cdots & O \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ O & O & \\cdots & m J _ m \\\\ \\end{pmatrix} , \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} a ( \\phi ) & = \\frac { \\phi - 4 } { ( k - 1 ) \\phi - k ^ 2 } , \\\\ b ( \\phi ) & = \\frac { ( c - 1 ) ( k - 1 ) \\phi + ( k - c ) ^ 2 } { ( d - 1 ) ( c - 1 ) ( k - 1 ) \\phi + d ( k - c ) ^ 2 + 2 ( c - 1 ) ( k - c ) } \\\\ & = \\frac { X _ \\phi } { ( d - 1 ) X _ \\phi + Y } , \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} \\mathrm { T } ^ n ( \\theta ) = - \\frac { q _ n \\theta - p _ n } { q _ { n - 1 } \\theta - p _ { n - 1 } } = \\frac { | \\Delta _ n | } { | \\Delta _ { n - 1 } | } \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta U ^ \\pm _ \\gamma + U ^ \\pm _ \\gamma & = 0 , \\quad \\mbox { i n } \\Omega ^ \\pm , \\\\ \\partial _ n U ^ \\pm _ \\gamma & = 0 , \\mbox { o n } \\partial \\Omega , \\\\ U ^ \\pm _ \\gamma & = 1 , \\mbox { o n } \\gamma . \\end{aligned} \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} * \\mathbb { F } _ \\pm = \\pm i \\mathbb { F } _ \\pm \\ , . \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} [ i \\hbar D \\xi , \\psi \\psi ^ { \\dagger } ] & = \\frac { 1 } { 2 } \\left ( \\frac { \\delta F } { \\delta \\psi } \\psi ^ { \\dagger } - \\psi \\frac { \\delta F } { \\delta \\psi } ^ { \\dagger } \\right ) \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} D _ t ^ { ( \\alpha , \\beta ) } ( n , m ) = w _ { n } ^ { ( \\alpha + 1 , \\beta ) } w _ m ^ { ( \\alpha , \\beta ) } \\mathfrak { I } _ t ^ { ( \\alpha + 1 , \\beta , \\alpha , \\beta , \\alpha + 1 , \\beta ) } ( n , m ) . \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} U ( X ) = u ( \\P _ X ) , \\forall X \\in L ^ 2 ( \\Omega , \\R ^ { d + l } ) . \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} \\abs { z } < q \\left ( \\frac { s } { p ( \\omega ) } \\right ) ^ { 1 / M } : = q r ( \\omega ) . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} [ L _ 2 x ] _ { i , j } = \\sum _ { i ' , j } \\cos ( \\bar { \\theta } _ { i ' , j ' } - \\bar { \\theta } _ { i , j } ) ( x _ { i ' , j ' } - x _ { i , j } ) , \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} f ( 0 ^ + ) & = a - b + c + ( + \\infty ) d + \\int _ { ( 0 , + \\infty ) } s ^ { - 1 } \\ , d \\mu ( s ) , \\\\ f ' ( + \\infty ) & = b + ( + \\infty ) c + d + \\int _ { ( 0 , + \\infty ) } d \\mu ( s ) . \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} \\pi ( \\rho _ \\infty ( a ) ) = \\rho ( a ) , \\pi ( t _ \\infty ( x ) ) = t ( x ) \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} \\overline { r } ^ { ( k ) } ( x ) : = \\begin{pmatrix} \\left ( \\frac { \\mathrm { d } } { \\mathrm { d } x } \\right ) ^ k p _ 0 ( x ) & \\left ( \\frac { \\mathrm { d } } { \\mathrm { d } x } \\right ) ^ k p _ 1 ( x ) & \\cdots & \\left ( \\frac { \\mathrm { d } } { \\mathrm { d } x } \\right ) ^ k p _ { n - 1 } ( x ) \\end{pmatrix} ^ T , \\overline { r } _ j ^ { ( k ) } : = \\overline { r } ^ { ( k ) } ( x _ j ) , \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\sup _ { 0 \\le u \\le s } \\| Q ( \\epsilon ; t - u ) - ( \\tilde { Q } - \\epsilon \\tilde { Q } ^ { ( 1 ) } u ) \\| = O ( \\epsilon ^ 2 s ^ 2 ) \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} \\frac { N ^ N } { ( N - 1 ) ^ { N - 1 } } = \\frac { N ^ { N - 1 } N } { ( N - 1 ) ^ { N - 1 } } = N \\left ( 1 + \\frac { 1 } { N - 1 } \\right ) ^ { N - 1 } \\le N e . \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { E } } [ C , Q ] & = \\sup \\left \\{ \\int _ { \\mathbb { S } _ + ^ 1 } \\int _ \\Omega ( 2 c _ 0 ^ 2 a + 1 ) \\d C ( x , \\theta ) \\right . \\\\ & + \\int _ { \\mathbb { S } _ + ^ 1 } \\int _ \\Omega b \\d Q ( x , \\theta ) ; \\ ; \\left . ( a , b ) \\in C _ b ( \\Omega \\times \\mathbb { S } _ + ^ 1 , K _ 2 ) \\right \\} . \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} ( \\pi _ 1 ^ { \\mu _ 1 } ) ^ * \\omega _ { M _ { \\mu _ 1 } } = ( \\pi _ 1 ^ { \\mu _ 1 } ) ^ * ( \\chi ) ^ * \\omega _ { \\mathcal { O } _ { \\mu _ 2 } } ^ + , \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} & T ^ { * } _ { W } : \\mathcal { K } _ { 2 , W } \\rightarrow H \\\\ & T ^ { * } _ { W } ( v _ i ( C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f ) = \\sum _ { i \\in I } v _ { i } ^ { 2 } C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } f . \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} 2 \\rho ( n _ 1 ) - \\rho ( 2 n _ 1 ) = \\ln n _ 1 , \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathcal { V } } S _ j = 0 . \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} F ( x , y ) = a _ 1 + & a _ 2 y + a _ 3 x ^ 2 y x + a _ 4 x y x ^ 2 + a _ 5 x + a _ 6 y x + a _ 7 x ^ 2 y x ^ 2 \\\\ & + a _ 8 x y + a _ 9 x ^ 2 + a _ { 1 0 } y x ^ 2 + a _ { 1 1 } x ^ 2 y + a _ { 1 2 } x y x , \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} \\mathcal { B } _ v ( \\lambda ) \\otimes \\mathcal { B } _ w ( \\mu ) = \\bigsqcup _ { \\pi \\in \\mathcal { B } _ w ( \\mu ) ^ \\lambda } C ( \\pi , v ) \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} \\mathbb I _ m \\ , : = \\ , \\Bigl \\{ ( s , p ) \\ , \\colon \\ , \\Omega _ m ( s , p ) \\neq \\emptyset \\Bigr \\} , \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} V ' ( t ) = f ( 0 ) V ( t ) + \\int _ 0 ^ t f ' ( t - s ) V ( s ) \\ , d s - g ( 0 ) V ( t ) - \\int _ t ^ T g ' ( s - t ) V ( s ) \\ , d s . \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} \\mathcal { B } _ w ( \\lambda ) : = \\left \\{ \\left . \\tilde { f } _ { i _ 1 } ^ { a _ 1 } \\cdots \\tilde { f } _ { i _ l } ^ { a _ l } ( b _ \\lambda ) \\ \\right | \\ a _ 1 , \\ldots , a _ l \\geq 0 \\right \\} \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} C _ 0 = \\frac { 1 } { \\eta } \\ , \\sum _ { j = 1 } ^ 3 \\| A _ j \\| _ { L ^ \\infty ( \\Omega ) } . \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} \\bigcup _ { a \\geq 0 } \\tilde { f } _ i ^ a ( C ( \\pi , v ) ) \\setminus \\{ 0 \\} = \\widetilde { C } ( \\pi , v , i ) . \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} \\phi ( x , y ) : = \\langle \\tilde { B _ 1 } x - a , y - x \\rangle + y ^ T B _ 1 y - x ^ T B _ 1 x + h ( y ) - h ( x ) \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} I H _ { \\mathcal S } ( t ) = H _ { \\widetilde { \\mathcal S _ 1 } } ( t ) = Q _ { k - i } ^ { l - j } Q _ { k } ^ { k + j - i } = \\frac { P _ { l - j } } { P _ { k - i } P _ { l - j - k + i } } \\cdot \\frac { P _ { k + j - i } } { P _ { k } P _ { j - i } } . \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\left ( | X _ n | - \\sum _ { i = 1 } ^ { n - 1 } f ( X _ i ) \\right ) = 0 \\hbox { \\rm a . s . } \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} Q ^ * \\Bigl ( ( \\pi _ t ) _ { t \\in [ 0 , T ] } \\in \\mathcal D ( [ 0 , T ] ; \\mathcal M _ + ( \\Lambda ) ) : \\pi _ t ( \\mathrm d u ) = \\rho _ t ( u ) \\ ; \\ ! \\mathrm d u \\textrm { f o r a . a . } ~ t \\in [ 0 , T ] \\Bigr ) = 1 . \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} & & C ^ { - 1 } \\sigma ^ k \\| u \\| _ { L ^ q } \\le \\| \\nabla ^ k u \\| _ { L ^ q } \\le C \\sigma ^ { k + \\frac n p - \\frac n q } \\| u \\| _ { L ^ p } \\mbox { w h e n } \\mathrm { S u p p } \\ , \\mathcal { F } u \\subset \\sigma \\mathcal { C } \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} \\begin{cases} \\delta ^ 0 ( 1 ) = ( 0 ) \\\\ \\delta ^ 1 ( 1 ) = ( 1 ) \\\\ \\sigma ( 0 ) = \\sigma ( 1 ) = 1 \\\\ \\sigma ( 0 1 ) = 0 \\\\ \\gamma ( ( 0 ) \\otimes x ) = \\gamma ( x \\otimes ( 0 ) ) = x \\\\ \\gamma ( ( 1 ) \\otimes ( 1 ) ) = ( 1 ) \\\\ \\gamma ( ( 0 1 ) \\otimes ( 1 ) ) = \\gamma ( ( 1 ) \\otimes ( 0 1 ) ) = 0 \\ . \\end{cases} \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} m _ X ^ { N _ 0 , N _ 1 } ( x , \\xi ) : = N _ 1 \\left ( 2 - m _ X \\left ( x , \\frac { \\xi } { \\| \\xi \\| _ x } \\right ) - \\tilde { m } _ X \\left ( x , \\frac { \\xi } { \\| \\xi \\| _ x } \\right ) \\right ) - 2 N _ 0 \\tilde { m } _ X \\left ( x , \\frac { \\xi } { \\| \\xi \\| _ x } \\right ) , \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{gather*} T _ { w ( 1 , 1 ) } \\big ( E _ { k , 1 } ( s + 1 ) \\big ) = - E _ { k + 1 , 1 } ( s + 1 ) \\end{gather*}"}
-{"id": "2916.png", "formula": "\\begin{align*} B _ 1 ( t ) = \\begin{cases} \\displaystyle B ( t _ k ) + \\frac { B ( \\tau _ k ) - B ( t _ k ) } { \\tau _ k - t _ k } \\ , ( t - t _ k ) , & t \\in ( t _ k , \\tau _ k ) , \\ ; k \\in \\N , \\\\ B ( t ) , & . \\end{cases} \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\frac { d F } { d t } = & \\ , 2 C _ 1 \\mathbf { E } \\Big ( f ( t ) \\int _ 0 ^ t f ( s ) d s \\Big ) + C _ 2 \\mathbf { E } \\big ( f ( t ) \\big ) ^ 2 \\\\ \\le & \\ , C _ 1 \\mathbf { E } \\Big ( \\int _ 0 ^ t f ( s ) d s \\Big ) ^ 2 + ( C _ 1 + C _ 2 ) \\mathbf { E } \\big ( f ( t ) \\big ) ^ 2 \\\\ \\le & \\ , F ( t ) + ( C _ 1 + C _ 2 ) \\big ( g ( t ) + F ( t ) \\big ) \\\\ = & \\ , ( C _ 1 + C _ 2 + 1 ) F ( t ) + ( C _ 1 + C _ 2 ) g ( t ) \\ , . \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} l _ { \\sigma } = \\left \\{ \\begin{array} { c c } h _ { \\sigma } \\ & \\mbox { i f } \\ \\sigma \\notin I \\\\ - h _ { \\sigma } - 2 \\ & \\mbox { i f } \\ \\sigma \\in I \\end{array} \\right . , \\ g ^ \\prime = g - \\sum \\limits _ { \\sigma \\in I } ( h _ { \\sigma } + 1 ) . \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} \\xi ( t ) : = \\phi ^ \\prime ( t ) t - p \\phi ( t ) \\textrm { f o r a n y } t \\geq 0 . \\end{align*}"}
-{"id": "9744.png", "formula": "\\begin{align*} g = \\sum _ { j _ 1 , \\dots , j _ n , j _ { n + 1 } \\geq 0 } g _ { j _ 1 \\dots j _ { n + 1 } } z _ 1 ^ { j _ 1 } \\dots z _ n ^ { j _ { n } } t ^ { j _ { n + 1 } } , \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} \\Lambda h ( x ) = ( Q ^ f h ) ( x ) \\geq \\alpha _ n \\eta _ n ( h ) , \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} K f = \\int _ I u ( t ) f ( t ) \\ , d t . \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} \\eta = \\frac { \\mu ^ { 2 } } { 4 } . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} ( F ( v ) ) ( t , x ) = f ( t , x , v ( t , x ) ) . \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} K ( \\xi + a - k ) = & K ( \\xi - k ) + a \\cdot \\nabla K ( \\xi - k ) \\\\ & + \\int _ 0 ^ 1 \\int _ 0 ^ 1 t a \\cdot D ^ 2 K ( \\xi + s t a - k ) a d s d t \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} \\binom { \\bar { \\partial } } { - \\rho _ { K _ { \\varepsilon } } } \\underline { \\omega } + \\binom { 0 } { \\omega } = \\binom { \\bar { \\partial } \\underline { \\omega } } { 0 } . \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} H _ 1 ^ \\top H _ 1 = \\begin{pmatrix} \\ell & a { \\bf 1 } ^ \\top \\\\ a { \\bf 1 } & \\ell I _ { n - 1 } - a ( J _ { n - 1 } - I _ { n - 1 } ) \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{gather*} \\int _ { \\Omega } f _ { | \\tau } ( x ) I _ { \\tau < \\infty } \\ , \\mu ( d x ) = \\int _ { \\Omega } f ( x ) I _ { \\tau < \\infty } \\ , \\mu ( d x ) , \\\\ [ 1 0 p t ] \\int _ { \\Omega } f _ { | \\tau } ( x ) \\ , \\mu ( d x ) = \\int _ { \\Omega } f ( x ) \\ , \\mu ( d x ) . \\end{gather*}"}
-{"id": "9296.png", "formula": "\\begin{align*} M = G / P = U / K . \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\eta ^ { * } = \\arg m i n _ { \\eta } ~ S E R ( \\mathcal { Q } _ { b } ) . \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} \\underset { \\mathbf { x } \\in \\mathbb { R } ^ { n p } , \\mathbf { z } \\in \\mathbb { R } ^ { m p } } { } f ( \\mathbf { x } ) + \\frac { \\rho ( 1 - \\eta ) } { 2 } \\| \\mathbf { A x } + \\mathbf { B z } \\| ^ 2 \\\\ \\sqrt { \\eta } ( \\mathbf { A x } + \\mathbf { B z } ) = \\mathbf { 0 } . \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} & \\kappa _ 2 : ( \\kappa _ 2 + 1 ) p = ( \\kappa _ 1 + 1 ) p _ * , \\\\ & \\kappa _ 3 : ( \\kappa _ 3 + 1 ) p = ( \\kappa _ 2 + 1 ) p _ * , \\\\ & \\vdots \\vdots \\ , . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} V ^ { ( n + 1 ) } ( t ) = & f ( 0 ) V ^ { ( n ) } ( t ) + \\int _ 0 ^ t f ' ( t - s ) V ^ { ( n ) } ( s ) \\ , d s - g ( 0 ) V ^ { ( n ) } ( t ) - \\int _ t ^ T g ' ( s - t ) V ^ { ( n ) } ( s ) \\ , d s \\\\ [ 6 p t ] & + \\sum _ { k = 0 } ^ { n - 1 } ( - 1 ) ^ { n - k + 1 } g ^ { ( n - k ) } ( T - t ) V ^ { ( k ) } ( T ) \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} \\Sigma ' & \\le \\frac { c ( \\delta _ 2 / \\delta _ 1 ) ^ d ( \\delta _ 1 / \\delta ) ^ d } { ( \\delta _ 1 ^ { - 1 } \\rho ( x , y ) ) ^ \\sigma } = \\frac { c ( \\delta _ 1 / \\delta ) ^ d } { ( \\delta _ 2 ^ { - 1 } \\rho ( x , y ) ) ^ d ( \\delta _ 1 ^ { - 1 } \\rho ( x , y ) ) ^ { \\sigma - d } } \\\\ & \\le \\frac { c ( \\delta _ 1 / \\delta ) ^ d } { ( \\delta _ 2 ^ { - 1 } \\rho ( x , y ) ) ^ \\sigma } \\le \\frac { c 2 ^ \\sigma ( \\delta _ 1 / \\delta ) ^ d } { ( 1 + \\delta _ 2 ^ { - 1 } \\rho ( x , y ) ) ^ \\sigma } . \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} \\left | G _ { [ i ] } \\right | & = | G _ { i - 1 } | s _ i p _ i = \\frac { i - 1 } { d + i - 1 - k } \\prod _ { j = 1 } ^ i s _ j p _ j , \\\\ \\left | W _ { [ i ] } \\right | & = | W _ i | \\prod _ { j = 1 } ^ { i - 1 } s _ j p _ j = p _ i \\prod _ { j = 1 } ^ { i - 1 } s _ j p _ j = \\frac { 1 } { d + i - k } \\prod _ { j = 1 } ^ i s _ j p _ j . \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} P ^ { \\omega , \\mu } _ 0 ( X _ t < v t ) \\le P ^ { \\omega , \\mu } _ 0 ( H ( u t ) < t , X _ t < v t ) + P ^ { \\omega , \\mu } _ 0 ( H ( u t ) \\ge t ) . \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} \\mathcal { G } _ { D , 0 } ( z , s ) : = - 2 ^ { s - 2 } \\pi ^ { - \\frac { s } { 2 } - 1 } | D | ^ { \\frac { s } { 2 } } L _ D ( s ) \\Gamma \\biggl ( \\frac { s } { 2 } + 1 \\biggr ) P _ { \\frac { 3 } { 2 } , 0 } \\biggl ( z , \\frac { s } { 2 } + \\frac { 1 } { 4 } \\biggr ) . \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} \\psi ( \\nu ) = \\bar { \\mathcal { A } } _ { I } ( \\mathcal { I } ( \\mbox { r e a l } ( \\nu ) ) ) + i \\bar { \\mathcal { A } } _ { Q } ( \\mathcal { I } ( \\mbox { i m a g } ( \\nu ) ) ) , \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} \\bigoplus \\limits _ { k \\geq 2 d } R ^ { k } i _ 1 ^ * i ^ * j _ * \\mu _ { \\ell } ^ K ( V _ { \\lambda } ) [ - w ( \\lambda ) - k ] _ { \\vert _ { Z ^ \\prime } } \\simeq \\bigoplus \\limits _ { k \\geq 2 d } ( \\bigoplus _ { \\substack { p + q = k } } \\mu _ { \\ell } ^ { \\pi _ 1 ( K _ 1 ) } ( V ^ { p , q } ) ) [ - w ( \\lambda ) - k ] , \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} J _ { \\mathcal Q _ + , \\mathcal Q _ - } ( f _ 1 , \\ldots , f _ m ) = \\frac { \\int _ H \\prod _ { k = 0 } ^ { m + 1 } f _ k ^ { c _ k } ( B _ k x ) \\ , d x } { \\prod _ { k = 1 } ^ m \\Big ( \\int _ { H _ k } f _ k \\Big ) ^ { c _ k } } , \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} W _ \\lambda ( z ) ( s ) = W ( z ) ( s ) - \\big ( \\chi _ \\lambda ( s ) + \\lambda c ( s ) \\big ) \\theta \\big ( \\phi ( z ( s ) \\big ) \\ , \\nu \\big ( z ( s ) \\big ) . \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} \\varphi \\nabla ^ { \\tilde { g } } \\varphi ^ { - 1 } \\bigr | _ { r = 0 } = \\varphi \\nabla ^ { { g } } \\varphi ^ { - 1 } \\bigr | _ { r = 0 } . \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} \\mathfrak { A } = \\{ f \\in \\mathfrak { B } : f ( 0 ) \\ \\mbox { a n d } \\ f ( 1 ) \\mbox { \\ a r e b o t h \\ d i a g o n a l } \\} , \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} ( r _ 1 ^ { x _ 1 } , r _ 2 ^ { x _ 2 } , \\ldots , r _ d ^ { x _ d } ) : = ( \\underbrace { r _ 1 , r _ 1 , \\ldots , r _ 1 } _ { x _ 1 } , \\underbrace { r _ 2 , r _ 2 , \\ldots , r _ 2 } _ { x _ 2 } , \\ldots , \\underbrace { r _ d , r _ d , \\ldots , r _ d } _ { x _ d } ) \\ , . \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} T _ { 0 } \\geq \\max \\{ T _ 1 , T _ 2 \\} , \\quad \\left [ \\frac { r _ 1 } { T _ { 0 } } , \\frac { r _ 4 } { T _ { 0 } } \\right ] \\cap [ r _ 1 , r _ 4 ] = \\emptyset , \\quad \\lambda \\phi _ u ( T _ { 0 } r _ 1 ) < 1 , \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} F ( \\alpha , 0 ) = 0 , F ( \\alpha , 1 ) = h _ 1 , \\alpha \\in \\mathbb { R } \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\left ( - \\frac { 3 } { 1 6 } + \\frac { 7 } { 2 } \\cdot \\frac { 1 } { 4 } + C \\right ) = 0 \\Rightarrow C = - \\frac { 1 1 } { 1 6 } \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} u _ t + F ( v ( t , x ) , u ) _ x ~ = ~ \\varepsilon \\ , u _ { x x } , \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} \\omega ( \\xi ) & = \\sup \\left \\{ | u ( x , t ) - u ( y , s ) | \\vee | v ( x , t ) - v ( y , s ) | \\ , \\mid \\ , ( x , t ) , ( y , s ) \\in \\right . \\\\ & \\left . \\mathcal { I } \\times [ 0 , T ] , \\ , \\ , d ( x , y ) + | t - s | \\leq \\xi \\right \\} . \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} p ^ { ( 2 n ) } _ \\lambda ( o , o ) \\sim \\begin{cases} \\frac { ( d - 1 - \\lambda ) ^ 2 } { 1 6 ( \\pi \\lambda ) ^ { 1 / 2 } ( d - 1 ) ^ { 3 / 2 } } \\rho _ { \\mathbb { T } _ d } ( \\lambda ) ^ { 2 n } n ^ { - 3 / 2 } & \\hbox { i f } \\ \\lambda \\in ( 0 , \\ , d - 1 ) , \\\\ \\frac { 1 } { \\sqrt { \\pi n } } & \\hbox { i f } \\ \\lambda = d - 1 . \\end{cases} \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} T = T ^ \\mathrm { a x } + T ^ \\mathrm { v e c } + T ^ \\mathrm { t e n } , \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} \\int _ { Y \\times U } ( \\phi ( y _ 0 ) - \\phi ( y ) ) \\gamma ( d u , d y ) + \\int _ { Y \\times U } \\nabla \\phi ( y ) ^ T f ( u , y ) \\xi ( d u , d y ) = 0 \\ \\ \\forall \\phi ( \\cdot ) \\in C ^ 1 \\} . \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} y _ t + u ^ 2 y _ x + 3 y u _ x u = 0 . \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} \\Delta _ { \\rho , \\sigma } ^ p \\xi = \\Delta _ { \\rho , \\sigma } ^ p ( \\xi s ( \\sigma ) ) = \\eta s ( \\sigma ) . \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} \\begin{gathered} r _ { 1 3 } r _ { 1 2 } = \\sum \\limits _ { p } s _ { i j } ^ { k l } s _ { p i } ^ { s t } e _ { p j } \\otimes e _ { k l } \\otimes e _ { s t } , \\\\ r _ { 1 2 } r _ { 2 3 } = \\sum \\limits _ { p } s _ { i j } ^ { k l } s _ { l p } ^ { s t } e _ { i j } \\otimes e _ { k p } \\otimes e _ { s t } , \\\\ r _ { 2 3 } r _ { 1 3 } = \\sum \\limits _ { p } s _ { i j } ^ { k l } s _ { p r } ^ { l t } e _ { p r } \\otimes e _ { i j } \\otimes e _ { k t } . \\end{gathered} \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} \\varphi ( f ) = \\langle f , R _ H ( \\varphi ) \\rangle _ H , f \\in H . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} & Q _ { - \\nu , \\ell } \\subseteq Q _ { - j , 0 } Q _ { - \\nu , \\ell } = \\bigcup _ { i = 1 } ^ { 2 ^ { d \\nu } } Q _ { 0 , m _ i } , \\\\ & \\ ; k _ \\nu \\ ; \\ ; \\lambda ^ { ( j ) } _ { 0 , k _ i } \\ ; \\ ; 1 . \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} x = ( x - y ) + y \\in \\ker S + \\ker T . \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} \\mbox { V a r } _ { \\mu _ z } ( f ) = \\int _ { \\Sigma _ z } f ^ 2 \\ , d \\mu _ z - \\Big ( \\int _ { \\Sigma _ z } f \\ , d \\mu _ z \\Big ) ^ 2 \\le \\frac { \\delta } { c _ 2 \\rho _ 0 } \\mathcal { E } _ z ( f , f ) \\ , , \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} u ( t , x ) = w ( t , x ) - \\int _ 0 ^ x K ( x , y ) w ( t , y ) \\ , d y , \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} \\mathbf { i } u _ t + u _ { x x } + \\mathbf { i } \\Big ( f ( x , u , \\bar { u } ) \\Big ) _ x = 0 , x \\in \\mathbb { T } , \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\hat g = - u ^ 2 ( d f ) ^ 2 + X _ i d x ^ i \\odot d f + g _ { i j } d x ^ i d x ^ j . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} \\Theta ( ( p _ 1 , \\dots , p _ r ) , ( b _ 1 , \\dots , b _ r ) ) = ( p _ 1 b _ 1 , b _ 1 ^ { - 1 } p _ 2 b _ 2 , \\dots , b _ { r - 1 } ^ { - 1 } p _ r b _ r ) \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} 6 \\sum _ { j = 1 } ^ 3 \\sum _ { i = 1 } ^ q f ^ j _ i \\geq | X | \\geq \\sum _ { j = 1 } ^ 3 \\sum _ { i = 1 } ^ q ( a ^ j _ i + f ^ j _ i ) ^ 2 . \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} x _ 1 \\big ( [ 0 , 1 ] \\big ) = x _ 2 \\big ( [ 0 , 1 ] \\big ) \\Longrightarrow \\mathcal F ( x _ 1 ) = \\mathcal F ( x _ 2 ) . \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} \\varphi _ { f } ( \\gamma g _ { \\infty } k ) = f ( g _ { \\infty } ) \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} \\frac { p _ { \\beta , \\lambda } } { p _ { 0 } } = \\sqrt { 1 + \\lambda ^ { 2 } } \\exp ^ { - \\frac { y ^ { 2 } + ( 1 + \\lambda ^ { 2 } ) ( y - \\lambda \\beta ^ { \\tau } x ) ^ { 2 } } { 2 ( 1 + \\lambda ^ { 2 } ) } } \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\xi ( x ( t ) ) - z ( t ) = & \\int _ 0 ^ t \\varphi ( x ( s ) ) \\ , d s + \\int _ 0 ^ t \\Big ( \\widetilde { b } \\big ( \\xi ( x ( s ) ) \\big ) - \\widetilde { b } ( z ( s ) ) \\Big ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } M ( t ) \\ , , \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} \\mu ( w 0 ) + \\mu ( w 1 ) = \\mu ( w ) . \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\nabla f ( \\mathbf { x } ^ { k + 1 } ) + \\mathbf { E } _ { } ^ T \\boldsymbol { \\alpha } ^ k + \\frac { \\rho } { 2 } \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } \\mathbf { x } ^ k + \\frac { \\rho } { 2 } ( 2 \\mathbf { D } + \\frac { 2 } { \\rho } \\mathbf { P } ) ( \\mathbf { x } ^ { k + 1 } - \\mathbf { x } ^ k ) = \\mathbf { 0 } , \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} D ^ b ( C ^ { [ 2 g - 3 ] } ) = \\langle \\overbrace { D ^ b ( J _ C ) , \\ldots , D ^ b ( J _ C ) } ^ { g - 2 } , D ^ b ( C ) \\rangle . \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} \\dot { \\theta } _ { i , j } = \\omega + \\sin ( \\theta _ { i + 1 , j } - \\theta _ { i , j } ) + \\sin ( \\theta _ { i - 1 , j } - \\theta _ { i , j } ) + \\sin ( \\theta _ { i , j + 1 } - \\theta _ { i , j } ) + \\sin ( \\theta _ { i , j - 1 } - \\theta _ { i , j } ) , \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} f _ * ( x _ 1 , \\ldots , x _ n ) = \\frac { 1 } { n ! } \\sum _ { \\sigma \\in \\Sigma _ n } f ( x _ { \\sigma ( 1 ) } , \\ldots , x _ { \\sigma ( n ) } ) \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} b ( i , n ) = \\sum _ { p } \\left [ \\binom { n - 2 p + 1 } { p } \\binom { p } { i - p } + \\binom { n - 2 p - 1 } { p } \\binom { p } { i - p - 1 } \\right ] \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} N = \\sum _ { i = 1 } ^ d \\binom { m _ i + i } { i } , \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} \\begin{aligned} S _ { \\lambda } ( T _ { \\alpha , q } v ) ( x ) & = v \\left ( \\left ( - \\lambda ( q - 1 ) \\log _ { q } \\frac { | x | } { R } \\right ) ^ { - \\alpha } \\right ) \\\\ & = ( D _ { \\lambda ^ { - \\alpha } } v ) \\left ( \\left ( - ( q - 1 ) \\log _ { q } \\frac { | x | } { R } \\right ) ^ { - \\alpha } \\right ) = T _ { \\alpha , q } ( D _ { \\lambda ^ { - \\alpha } } v ) ( x ) \\end{aligned} \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} X _ t ^ { - 1 } & = \\tfrac 1 x e ^ { - L _ t } + \\int _ 0 ^ t e ^ { - L _ s } b _ s d s , \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} \\hom _ { [ n ] } ( i , j ) = \\begin{cases} * i \\leq j \\ , \\\\ \\emptyset \\end{cases} \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} \\textstyle \\prod \\limits _ { k = 1 } ^ { r } \\left ( \\frac { \\beta \\ , [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } { \\beta + [ \\boldsymbol { \\Lambda } ] _ { 1 , 1 } ^ 2 - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } \\right ) = 2 ^ { { R } } \\left ( \\frac { \\sigma ^ 2 } { p _ 1 ^ { ( \\rm i d , a ) } } \\right ) ^ { r } . \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} a _ n \\cdot a _ { n - 2 } & = ( 3 a _ { n - 1 } - a _ { n - 2 } ) a _ { n - 2 } \\\\ & = 3 a _ { n - 1 } a _ { n - 2 } - a _ { n - 2 } ^ 2 \\\\ & = 3 a _ { n - 1 } a _ { n - 2 } - ( a _ { n - 1 } a _ { n - 3 } - 1 ) \\\\ & = a _ { n - 1 } ( 3 a _ { n - 2 } - a _ { n - 3 } ) + 1 \\\\ & = a _ { n - 1 } ^ 2 + 1 \\ , . \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} A _ { j , i _ j } & : = \\sum _ { h = 0 } ^ L \\left | \\binom { L + i _ j } { h } a _ j ^ { L - h } \\right | < \\frac { 1 } { | a _ j | ^ { i _ j } } \\sum _ { h = 0 } ^ { L + i _ j } \\binom { L + i _ j } { h } \\left | a _ j \\right | ^ { L + i _ j - h } \\\\ & = \\frac { 1 } { | a _ j | ^ { i _ j } } \\left ( | a _ j | + 1 \\right ) ^ { L + i _ j } = \\left ( | a _ j | + 1 \\right ) ^ L \\left ( 1 + \\frac { 1 } { | a _ j | } \\right ) ^ { i _ j } , \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} T _ { \\widehat { \\mathcal { C } } } = \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & \\zeta _ 7 ^ 3 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & \\zeta _ 7 ^ 5 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & \\zeta _ 7 ^ 6 \\end{pmatrix} \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} \\rho \\phi _ { u _ n } u _ n & - \\rho _ { \\infty } \\bar \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) - \\rho \\phi _ { v _ 0 } v _ 0 \\\\ & = \\rho \\phi _ { u _ n } u _ n - \\rho \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) - \\rho \\phi _ { v _ 0 } v _ 0 + o ( 1 ) \\\\ & = o ( 1 ) \\textrm { i n } \\ , \\ , H ^ { - 1 } ( \\R ^ 3 ) . \\\\ \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} \\mathbb { A } = \\left ( \\begin{matrix} a & b \\\\ c & d \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} \\hat f ( t , x , \\omega ) ~ = ~ \\begin{cases} f ( t , x _ 1 , \\omega ) & \\hbox { i f } ~ ~ x < x _ 1 \\ , , \\\\ f ( t , x , \\omega ) & \\hbox { i f } ~ ~ x \\in [ x _ 1 , x _ 2 ] \\ , , \\\\ f ( t , x _ 2 , \\omega ) & \\hbox { i f } ~ ~ x > x _ 2 \\ , , \\end{cases} \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\omega _ { f } ^ \\sharp ( \\kappa ^ j r ) \\lesssim _ { d , \\gamma , R _ 0 , \\varrho _ 0 , \\kappa } \\int _ 0 ^ r \\frac { \\omega _ { f } ^ \\sharp ( t ) } { t } \\ , d t < \\infty , 0 < r \\le R _ 1 , \\end{align*}"}
-{"id": "9917.png", "formula": "\\begin{align*} \\beta _ n ^ { 0 } = C ' _ n \\frac { \\sum _ { i = 1 } ^ n \\overline { \\varPhi _ i ( z ) } } { \\| z \\| ^ { 2 n } } , C ' _ n = ( - 1 ) ^ { \\frac { n ( n - 1 ) } 2 } \\ , \\frac { ( n - 1 ) ! } { ( 2 \\pi \\sqrt { - 1 } ) ^ n } \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} H _ { N } \\ ; : = \\ ; & \\sum _ { i = 1 } ^ { N } S _ i + \\frac { \\mu \\lambda } { N - 1 } \\sum _ { i < j } \\frac 1 { | x _ i - x _ j | ^ \\gamma } \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} 2 \\delta ^ * _ { g ^ { ( 4 ) } } Y = L _ { Y ^ T } g ^ { ( 4 ) } - L _ { \\frac { Y ^ { \\perp } } { u } \\partial _ t } g ^ { ( 4 ) } . \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} D ^ b ( C ^ { [ 4 ] } ) = \\langle D ^ b ( J _ C ) , D ^ b ( C ^ { [ 2 ] } ) \\rangle . \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} I ( \\eta ( 1 , v ) ) \\leq I ( \\eta ( 0 , v ) ) = I ( v ) , \\ v \\in H ^ 1 _ { r a d } ( \\mathbb R ^ 3 ) . \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} \\rho _ { X _ { n + 1 } } ( \\psi _ 1 ( t ) , \\psi _ 2 ( t ) ) : = \\sup _ { t \\in [ 0 , n + 1 ] } \\| \\psi _ 1 ( t ) - \\psi _ 2 ( t ) \\| _ 2 . \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} & \\| u \\| ^ 2 _ { L ^ 2 } + \\| b \\| ^ 2 _ { L ^ 2 } + \\nu \\int _ 0 ^ T \\| \\nabla u \\| _ { L ^ 2 } ^ 2 d t = \\| u \\| ^ 2 _ { L ^ 2 } + \\| b \\| ^ 2 _ { L ^ 2 } \\leq \\| u _ 0 \\| ^ 2 _ { B ^ { s - 1 } _ { 2 , 2 } } + \\| b _ 0 \\| ^ 2 _ { B ^ s _ { 2 , 2 } } ) = : M _ 0 . \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\le \\mu - \\beta } \\frac { ( - 1 ) ^ \\gamma ( \\mu - \\beta ) ! } { \\gamma ! ( \\mu - \\beta - \\gamma ) ! } = 0 \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} D ^ b ( C ^ { [ n + g - 1 ] } ) = \\langle \\overbrace { D ^ b ( J _ C ) , \\ldots , D ^ b ( J _ C ) } ^ { n } , D ^ b ( C ^ { [ - n + g - 1 ] } ) \\rangle . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} K \\backslash G / H = \\{ K x H ; \\ x \\in G \\} , \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} A ^ { W , W } _ \\cdot ( l ) & = \\lim _ { n \\to \\infty } A ^ { W , W } _ \\cdot ( l _ n ) \\\\ & = \\lim _ { n \\to \\infty } \\left ( \\phi _ n ( \\cdot , W ) - \\phi _ n ( 0 , W _ 0 ) - \\int _ 0 ^ \\cdot \\nabla \\phi _ n ^ * ( r , W _ r ) \\mathrm d W _ r \\right ) . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{gather*} \\ ! \\sum _ { i = 0 } ^ p \\ ! ( - 1 ) ^ i \\nabla _ { a _ i } \\eta ( a _ 0 , \\dots , \\widehat { a _ i } , \\dots , a _ p ) + \\ ! \\sum _ { i < j } \\ ! ( - 1 ) ^ { i + j } \\eta ( [ a _ i , a _ j ] , a _ 0 , \\dots , \\widehat { a _ i } , \\dots , \\widehat { a _ j } , \\dots , a _ p ) ; \\ ! \\end{gather*}"}
-{"id": "8231.png", "formula": "\\begin{align*} 4 f ^ 2 | d _ 0 | = \\det M = - p v ^ 2 + 4 D ( n + s u v - t u ^ 2 ) . \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\tilde { \\cal V } \\subseteq 4 \\log L \\cdot { \\cal U } \\right ) \\geq 1 - \\sum _ { \\ell = 1 } ^ L \\frac { 1 } { L ^ 2 } = 1 - \\frac { 1 } { L } . \\\\ \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} \\left [ \\begin{array} { l l } p _ { n - 1 } & p _ { n } \\\\ q _ { n - 1 } & q _ { n } \\end{array} \\right ] \\cdot \\left [ \\begin{array} { l l } 1 & 0 \\\\ 0 & a _ { n + 1 } \\end{array} \\right ] \\cdot \\left [ \\begin{array} { l } \\lambda _ { n + 1 } - \\lambda _ n \\\\ \\lambda _ { n + 1 } - \\lambda _ { n - 1 } \\end{array} \\right ] = \\left [ \\begin{array} { l } 0 \\\\ 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} P _ { \\infty } ^ - \\ , = \\ , \\left [ A - ( A ^ 2 + R S ) ^ { 1 / 2 } \\right ] S ^ { - 1 } < 0 < P _ { \\infty } = \\left [ A + ( A ^ 2 + R S ) ^ { 1 / 2 } \\right ] S ^ { - 1 } \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} \\tilde { \\Phi } _ v ^ { \\epsilon } ( x ) = \\chi _ l ^ { - 1 } ( \\alpha + \\frac { 1 } { 2 } l ^ { 2 v } x + \\epsilon l ^ v ( \\alpha x + \\frac { 1 } { 4 } l ^ { 2 v } x ^ 2 ) _ { \\frac { 1 } { 2 } } ) \\psi _ l ( - \\frac { 1 } { l ^ c } ( \\frac { 1 } { 2 } l ^ v x + \\epsilon ( \\alpha x + \\frac { 1 } { 4 } l ^ { 2 v } x ^ 2 ) _ { \\frac { 1 } { 2 } } ) ) . \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} \\begin{cases} Y ' ( t ) \\geq C Y ( t ) ^ { 1 / ( 1 - \\alpha ) } \\\\ Y ( 0 ) > 0 \\end{cases} \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { 1 } ( \\textbf { z } ) & = I \\left \\lbrace \\textbf { z } \\in \\mathcal { C } _ { 1 } \\cap \\mathcal { C } _ { 0 } , p ( \\textbf { z } , 1 ) P \\{ Y = 1 \\vert \\textbf { Z } = \\textbf { z } \\} > p ( \\textbf { z } , 0 ) P \\{ Y = 0 \\vert \\textbf { Z } = \\textbf { z } \\} \\right \\rbrace \\\\ & + I \\left \\lbrace \\textbf { z } \\in \\mathcal { C } _ { 1 } - \\mathcal { C } _ { 0 } , p ( \\textbf { z } , 1 ) > 0 \\right \\rbrace . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} \\theta _ { n _ + , n _ - } ( t ) = - \\frac { \\nu } { 2 m } ( n _ + + n _ - + 1 ) \\int _ 0 ^ t \\frac { f ( t ' ) } { \\rho ^ 2 ( t ' ) } d t ' . \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} \\delta & = 1 + C ^ 2 + 2 A + 2 C - 2 C D + 4 D ^ 2 - 4 ( A ^ 2 - A B + B ^ 2 ) , \\\\ \\rho & = - 4 - 3 C + 6 ( 2 A + 2 A C + 4 B D - B C - 2 A D ) , \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} \\Delta _ p ^ { u _ 0 } ( u _ 1 , u _ 2 ) \\leq C _ F ^ p \\| F ( u _ 1 ) - F ( u _ 2 ) \\| ^ { \\frac { 1 + \\epsilon } { 2 } p } , u _ 1 , u _ 2 \\in B , \\ \\epsilon \\in ( 0 , 1 ] , \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} \\omega _ u ( X ) ( Y , Z ) = \\hat { S } _ u ( X , Y , Z ) - \\hat { S } _ u ( X , Z , Y ) - \\hat { \\Omega } _ u ( X , Z , Y ) + \\hat { \\Omega } _ u ( X , Y , Z ) - \\hat { \\Omega } _ u ( Y , Z , X ) . \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} \\alpha ( y , t ) = \\int d x \\ a ( x , y , t ) \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} & A = \\frac { 1 } { 4 } \\left ( a + d + \\sqrt { - 1 } c - \\sqrt { - 1 } b \\right ) , ~ ~ B = \\frac { 1 } { 4 } \\left ( a - d + \\sqrt { - 1 } c + \\sqrt { - 1 } b \\right ) \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} \\alpha ^ \\vee = \\frac { 2 \\alpha } { \\langle \\alpha , \\alpha \\rangle } . \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} \\omega _ \\eta ( y ( t ) ) - \\omega _ \\eta ( y ( t - t ' ) ) = \\int _ { t - t ' } ^ t ( \\mathcal { L } \\omega _ \\eta ) ( y ( s ) ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } \\int _ { t - t ' } ^ t \\big [ ( \\nabla \\omega _ \\eta ) ^ T \\sigma \\big ] ( y ( s ) ) \\ , d \\bar { w } ( s ) \\ , . \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} | K ( r , r ' , y , y ' ) | \\lesssim \\begin{cases} 0 , & r < r ' \\\\ r ^ { - \\alpha } r '^ { - \\beta } , & r \\geq r ' , \\end{cases} \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} e _ { n , k } ( q ) = \\sum _ { w \\in S ^ { ' } _ { n + 1 , k } } q ^ { W _ 3 ( w ) } = \\sum _ { w \\in C _ { n , k } } q ^ { W _ 1 ( w ) } . \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} P ^ { \\omega , \\mu } _ 0 \\left ( \\lim _ { t \\to \\infty } \\frac { 1 } { t } X _ t = v _ { \\mathbf { P } } \\right ) = 1 \\mathbf { P } \\otimes \\P \\textrm { - a l m o s t s u r e l y } . \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 0 , 1 ) = \\Phi ( 1 ) = \\int _ { 0 } ^ { \\infty } \\frac { \\cos ( \\pi x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x = \\frac { 2 - \\sqrt { 2 } } { 8 } , \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } _ + ^ 1 } \\int _ \\Omega s \\d Q ^ n = \\int _ { \\mathbb { S } _ + ^ 1 } \\int _ \\Omega \\d | Q ^ n | , \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} B ( x , \\rho ) = \\big \\{ z \\in \\mathfrak M : \\Vert z - x \\Vert _ * < \\rho \\big \\} \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} B = \\max \\left \\{ C , 2 \\left ( \\frac { 1 } { e \\tau p } \\right ) ^ \\frac { 1 } { p } \\right \\} , \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} \\partial _ i ( \\tilde { \\xi } _ n e ^ { F _ i } ) = \\tilde { g } ^ n _ i e ^ { F _ i } \\ ; a . e . \\ ; ( a _ i , b _ i ) \\ ; i = 1 , \\cdots , N . \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} \\beta _ { \\ell } = \\frac { 1 } { 1 + \\lambda \\mu _ { \\ell } ^ 2 } \\sum _ { j = 0 } ^ N \\omega _ j \\tilde { \\Phi } _ { \\ell } ( x _ j ) f ( x _ j ) , \\ell = 0 , 1 , \\ldots , L , \\quad \\lambda > 0 . \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} { \\rm V o l } ( V _ 0 ) : = \\sum _ { v \\in V _ 0 } m ( v ) . \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} d ( d f \\circ J ) ( u , v ) - d ( d f \\circ J ) ( J u , J v ) = d f ( J N _ J ( u , v ) ) . \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} \\ell _ q ( y _ m ) = \\sum _ { j = 0 } ^ m v _ j \\leq ( m + 1 ) ( q - 1 ) . \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} | D _ { \\lambda } ( S ) | = \\lambda ^ { Q } | S | { \\rm a n d } \\int _ { \\mathbb { G } } f ( \\lambda x ) d x = \\lambda ^ { - Q } \\int _ { \\mathbb { G } } f ( x ) d x . \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} M ( n , k ) \\ , \\le \\ , \\frac { 2 ^ n } { \\sum _ { t = 0 } ^ { \\lfloor ( k - 1 ) / 2 \\rfloor } \\binom { n } { t } } . \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\mathcal { R } _ { k l } ^ { \\nu } ( \\alpha _ { 1 , \\nu } ) = \\{ \\xi \\in \\mathcal { O } _ { \\nu } : | \\langle k , \\omega _ { \\nu } ( \\xi ) \\rangle + \\langle l , \\bar { \\Omega } _ { \\nu } ( \\xi ) \\rangle | < \\alpha _ { 1 , \\nu } \\frac { \\langle l \\rangle _ { \\infty } } { \\langle k \\rangle ^ { \\tau } } \\} , \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} L _ { 2 1 } = \\Delta ^ { - 1 } [ a , c ] ( a - b d ^ { - 1 } c ) ^ { - 1 } . \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta w _ 0 ^ \\pm & = 0 , \\quad \\mbox { i n } \\ \\Omega ^ \\pm , \\\\ w _ 0 ^ \\pm & = 0 , \\mbox { o n } \\ \\partial \\Omega ^ \\pm \\cap \\partial \\Omega . \\end{aligned} \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} E A _ \\lambda ( t ) = \\frac { 1 } { 2 } \\ln \\det [ D _ \\Lambda ( t ) ] - \\lambda t _ { j k } + \\sum _ { i l \\in E ( \\Lambda ) } \\left ( e ^ { t _ { i l } } - \\ln f _ \\alpha ( e ^ { t _ { i l } } ) - t _ { i l } \\right ) , \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} P _ 0 ( \\xi ) \\widehat { k } \\cdot \\nabla K ( \\xi - d \\Gamma ( m ) ) P _ 0 ( \\xi ) = \\widehat { k } \\cdot u ( \\xi ) P _ 0 ( \\xi ) \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} \\P ( X _ t \\neq Y _ t ) \\leq \\P ( \\tau \\geq t ) = e ^ { - t } \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} X ^ 2 - \\ell Y ^ 2 = - 3 \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{gather*} \\lambda ( \\epsilon , s + 1 ) \\chi ^ + _ p + \\delta ( \\epsilon , s + 1 ) G _ 1 + \\cdots \\\\ { } = \\big ( q ^ { - ( s + 1 ) } \\big ( q ^ s + q ^ { - s } \\big ) \\lambda ( \\epsilon , s ) - q ^ { - 2 s } \\lambda ( \\epsilon , s - 1 ) + ( - \\epsilon ) ^ { p - 1 } ( - 1 ) ^ { s - 1 } \\hat q q ^ { - ( s + 1 ) } \\delta ( \\epsilon , s ) \\big ) \\chi ^ + _ p \\\\ \\quad { } + \\big ( { - } q ^ { - ( s + 2 ) } \\big ( q ^ s + q ^ { - s } \\big ) \\delta ( \\epsilon , s ) - q ^ { - 2 s } \\delta ( \\epsilon , s - 1 ) \\big ) G _ 1 + \\cdots . \\end{gather*}"}
-{"id": "1762.png", "formula": "\\begin{align*} 2 \\ , \\lambda ( m ) = \\left \\langle h _ m ^ { - 1 } \\ , \\Phi _ G ( m ) \\ , h _ m , \\beta \\right \\rangle = \\big \\langle \\Phi _ G ( m ) , \\mathrm { A d } _ { h _ m } ( \\beta ) \\big \\rangle . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} D ^ b ( \\widehat { M } ^ { + } ) = \\langle \\overbrace { D ^ b ( \\widehat { U } ) , \\ldots , D ^ b ( \\widehat { U } ) } ^ { n } , D ^ b ( \\widehat { M } ^ { - } ) \\rangle . \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} \\left ( ( \\kappa + 1 ) ^ { M _ { 2 8 } } \\right ) ^ { \\frac { 1 } { \\sqrt { \\kappa + 1 } } } \\geq 1 \\lim _ { \\kappa \\to \\infty } \\left ( ( \\kappa + 1 ) ^ { M _ { 2 8 } } \\right ) ^ { \\frac { 1 } { \\sqrt { \\kappa + 1 } } } = 1 . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} \\langle \\psi _ { \\eta _ 1 } ^ \\ell | \\psi _ { \\eta _ 2 } ^ \\ell \\rangle = \\left [ ( 1 - | \\eta _ 1 | ^ 2 ) ( 1 - | \\eta _ 2 | ^ 2 ) \\right ] ^ { \\frac { \\ell + 1 } { 2 } } ( 1 - \\eta _ 1 \\eta _ 2 ^ * ) ^ { - \\ell - 1 } , \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} \\frac { \\dim ( \\mathbb { Z } ( \\mathcal { T } _ \\xi ) \\rtimes \\mathcal { S } ) } { n + 1 } = d ^ { 2 n } ( - 1 ) ^ { \\lvert \\Phi ^ + \\rvert } \\xi ^ { - \\langle 2 \\rho , 2 \\rho \\rangle } \\left ( \\sum _ { w \\in \\mathfrak { S } _ { n + 1 } } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho , w \\bullet 0 \\rangle } \\right ) ^ { - 2 } , \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ r \\prod _ { l = 1 } ^ { N _ k } e ^ { \\mathbf { a } _ k ( l ) \\varpi _ { i _ { k , l } } } ( b _ { k , l } ) = \\prod _ { s = 1 } ^ n ~ ~ ~ \\prod _ { \\substack { 1 \\leq k \\leq r , 1 \\leq l \\leq N _ k , \\\\ i _ { k , l } = s } } e ^ { \\mathbf { a } _ { k } ( l ) \\varpi _ s } ( b _ { k , l } ) . \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} | | u _ n ^ 2 | | _ { H ^ 1 } ^ 2 = | | u _ n ^ 1 | | _ { H ^ 1 } ^ 2 - | | v _ 1 | | _ { H ^ 1 } ^ 2 + o ( 1 ) = | | u _ n | | _ { H ^ 1 } ^ 2 - | | v _ 0 | | _ { H ^ 1 } ^ 2 - | | v _ 1 | | _ { H ^ 1 } ^ 2 + o ( 1 ) . \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} b = - \\frac i { 4 m a ^ 2 } * \\ ! ( p \\wedge u ^ \\flat \\wedge \\bar u ^ \\flat \\wedge v ^ \\flat ) . \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} \\big | \\partial _ \\lambda \\Psi [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] [ \\bar { \\lambda } ] ( x , t ) \\big | \\lesssim \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\| \\bar { \\lambda } ( t ) \\| _ { 1 + \\sigma } \\left ( \\sum _ { j = 1 } ^ k \\frac { \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } - 1 } ( t ) } { 1 + | y _ j | ^ { a - 2 s } } \\right ) , \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ n \\lambda _ i ^ r x _ i \\right | = \\left | \\lambda _ 1 ^ r \\sum _ { i = 1 } ^ n \\mu _ i ^ r x _ i \\right | = \\left | \\sum _ { i = 1 } ^ n \\mu _ i ^ r x _ i \\right | . \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} \\mathcal { L } _ { d , n } ( x , y _ m ) = \\frac { 1 } { \\theta ^ { d y _ m } } H _ { d , n + \\ell _ q ( y _ m ) } ( z _ 1 , \\dots , z _ n , \\theta , \\dots , \\theta , \\dots , \\theta ^ { q ^ m } , \\dots , \\theta ^ { q ^ m } ) = 0 . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} \\left < ( \\phi \\cdot v ) ^ { 2 n } \\right > _ { \\mu _ \\epsilon } = \\left < ( 2 n - 1 ) ! ! ( [ v ; G _ { d , \\epsilon } ( t ) v ] ^ { n } ) \\right > _ { \\tilde { \\mu } _ \\epsilon } \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} a _ { L _ \\xi ( \\lambda , \\mu ) } ( v _ { \\lambda , \\mu } ) = \\left ( \\varphi \\in L _ { \\xi } ( \\lambda , \\mu ) ^ * \\mapsto \\xi ^ { \\langle 2 \\rho , \\mu \\rangle } \\varphi ( v _ { \\lambda , \\mu } ) \\right ) . \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} f _ k ( x ) = \\tilde { g } _ k ( T _ k ( x ) ) \\det d T _ k ( x ) \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} f _ { 1 / 2 , m } ( z ) = F _ { 1 / 2 , - m , 0 } ( z ) , \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } k _ { 1 , \\sigma } + k _ { 2 , \\sigma } = \\kappa & \\forall \\sigma \\in I _ F ^ 0 \\\\ k _ { 1 , \\sigma } - k _ { 2 , \\sigma } - 2 = \\kappa & \\forall \\sigma \\in I _ F ^ 1 \\\\ - ( k _ { 1 , \\sigma } - k _ { 2 , \\sigma } + 4 ) = \\kappa & \\forall \\sigma \\in I _ F ^ 2 \\\\ - ( k _ { 1 , \\sigma } + k _ { 2 , \\sigma } + 6 ) = \\kappa & \\forall \\sigma \\in I _ F ^ 2 \\end{array} \\right . \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} \\int _ { \\partial S _ { \\varepsilon } ^ + } e ^ { z w } u _ { \\varepsilon ' } ( z ) d z = \\int _ { \\partial S _ { \\varepsilon } ^ + } e ^ { z w } u _ { \\varepsilon _ 1 ' } ( z ) d z . \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} A ( \\lambda ) ^ { - 1 } = \\left [ \\begin{array} { l l } a & - a a _ { 1 2 } a _ { 2 2 } ^ { - 1 } \\\\ - a _ { 2 2 } ^ { - 1 } a _ { 2 1 } a & a _ { 2 2 } ^ { - 1 } a _ { 2 1 } a a _ { 1 2 } a _ { 2 2 } ^ { - 1 } + a _ { 2 2 } ^ { - 1 } \\end{array} \\right ] . \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} u \\sim \\rho _ \\lambda , \\lambda ^ 2 e ^ { \\ , u } = o ( \\rho _ \\lambda ) \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\mathbf { \\widetilde { \\Phi } } _ { 0 , k } = \\left ( \\underset { \\bf 1 } { \\tfrac { N - k } { k } \\varphi ' _ { 0 } } , \\dots , \\underset { \\bf k } { \\tfrac { N - k } { k } \\varphi ' _ { 0 } } , \\underset { \\bf k + 1 } { - \\varphi ' _ { 0 } } , \\dots , \\underset { \\bf N } { - \\varphi ' _ { 0 } } \\right ) . \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} \\mathbf { H } ( p ) = \\mathbf { C } + \\int _ { [ 0 , \\infty [ } ( p + r ) ^ { - 1 } \\ , \\mathbf { G } ( r ) \\ , \\mu ( \\dd r ) . \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\Omega ( K ) = \\sum \\limits _ { i = 1 } ^ r n _ i \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} f \\cup _ \\alpha g = ( ( \\mu \\circ _ 0 f ) \\circ _ { m } g ) ~ ~ g \\cup _ \\alpha f = ( ( \\mu \\circ _ 1 f ) \\circ _ 0 g ) . \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} p _ { t , y , a } \\cdot ( z ' , 0 ) = \\left ( \\frac { 2 } { 2 - i y } t z ' , \\frac { i y } { 2 - i y } \\right ) \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} \\chi _ \\lambda ( ( h , x ) ) : = e ^ { 2 \\pi i \\lambda x } \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} \\lambda = \\| u \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} T _ { n } [ \\sigma ] p = q \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} \\mathsf { r e c t s } | _ { \\mathcal { P } ' } ( T _ S ) = \\mathsf { r e c t s } | _ { \\mathcal { P } ' } ( T ) , \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} \\int _ { H _ { e } } \\int _ { \\tilde { H } _ { e } } \\phi ( t + y _ 1 ) d | D ^ { s } f ^ e _ { y _ 1 } | ( t ) d \\mathcal { H } ^ { d - 1 } ( y _ 1 ) = \\int _ { \\mathbb { R } ^ d } \\phi ( x ) \\left | \\langle e , \\eta ( x ) \\rangle \\right | d | D ^ s f | ( x ) , \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} g ( w ) = \\sum _ { \\mathrm { w t } _ m \\alpha = \\frac { k } { M } } C _ { \\alpha } w ^ { \\alpha } , \\ ; w \\in \\C ^ n \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} X _ i & = \\{ x : c ( g _ 0 ^ { - 1 } , x ) = h _ i ^ { - 1 } \\} , \\\\ X _ j & = \\{ x : c ( g _ 1 ^ { - 1 } , x ) = h _ j ^ { - 1 } \\} , \\\\ X _ k & = \\{ x : c ( ( g _ 0 g _ 1 ) ^ { - 1 } , x ) = h _ k ^ { - 1 } \\} , \\\\ X _ s & = \\{ x : c ( ( g _ 1 g _ 2 ) ^ { - 1 } , x ) = h _ s ^ { - 1 } \\} , \\\\ X _ l & = \\{ x : c ( g _ 2 ^ { - 1 } , x ) = h _ l ^ { - 1 } \\} . \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\sum _ { i \\in { \\cal J } } w _ i Z _ i > \\frac { 1 } { 2 } \\right ) & = \\mathbb { P } \\left ( \\Xi > \\frac { 1 } { 2 } \\right ) \\\\ & \\geq \\mathbb { P } \\left ( \\Xi > \\frac { \\mathbb { E } ( \\Xi ) } { 2 } \\right ) \\\\ & \\geq 1 - e ^ { - \\frac { \\mathbb { E } ( \\Xi ) } { 8 } } \\geq 1 - e ^ { - \\frac { 1 } { 8 } } . \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} \\langle u ( t ) , \\phi \\rangle = & \\langle h , \\phi \\rangle + \\frac { 1 } { 2 } \\int _ 0 ^ t \\langle u ( s ) , \\phi '' \\rangle d s + \\int _ 0 ^ t \\langle b ( u ( s ) ) , \\phi \\rangle d s \\\\ & + \\int _ 0 ^ t \\int _ 0 ^ 1 \\sigma ( u ( s , x ) ) \\phi ( x ) W ( d s d x ) + \\int _ 0 ^ t \\int _ 0 ^ 1 \\phi ( x ) \\eta ( d s d x ) , \\ t \\geq 0 \\ a . s . , \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} K _ t ^ { ( - 1 / 2 , - 1 / 2 ) } ( m , n ) = \\frac { \\pi } { 2 } e ^ { - t } ( I _ { n - m } ( t ) + I _ { n + m } ( t ) ) . \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} \\Vert v _ { i } ( C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f \\Vert ^ { 2 } = \\sum _ { i \\in I } v _ { i } ^ { 2 } \\langle \\pi _ { W _ { i } } C ^ { \\prime } f , \\pi _ { W _ { i } } C f \\rangle \\leq B \\Vert f \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } d \\mu _ k ^ z ( x ) = \\sum _ { j } \\| s _ { k , j } ( z ) \\| ^ 2 \\ , \\delta _ { \\mu _ { k , j } } ( x ) , \\\\ \\\\ d \\mu _ k ^ { z , \\delta } ( x ) = \\sum _ { j } \\| s _ { k , j } ( z ) \\| ^ 2 \\ , \\delta _ { k ^ { \\delta } ( \\mu _ { k , j } - H ( z ) ) } ( x ) , \\\\ \\\\ d \\mu _ k ^ { ( z , u , \\epsilon ) , \\delta } ( x ) = \\sum _ { j } \\| s _ { k , j } ( z + k ^ { - \\epsilon } u ) \\| ^ 2 \\ , \\delta _ { k ^ { \\delta } ( \\mu _ { k , j } - H ( z ) ) } ( x ) . \\end{array} \\right . \\end{align*}"}
-{"id": "9929.png", "formula": "\\begin{align*} \\Z = \\Z _ M ( M ) \\simeq o r _ { M / X } ( M ) = H ^ { 1 } _ M ( X ; \\Z _ X ) \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\Psi ( x ) = 0 \\Leftrightarrow \\psi ( x ) = 0 \\Leftrightarrow \\psi ^ { \\prime } ( x ) = 0 \\Leftrightarrow x = e \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\| \\nabla _ h f \\| _ h ^ 2 = \\sum _ { 1 \\leq i , j \\leq n } \\phi ^ { - 1 } g ^ { i j } \\frac { \\partial f } { \\partial x _ i } \\frac { \\partial f } { \\partial x _ j } \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} { \\rm \\Lambda } _ L = & \\underset { x \\in [ - 1 , \\ , 1 ] } \\max \\sum _ { j = 1 } ^ { N } w _ j \\Big | \\sum _ { \\ell = 0 } ^ { L } \\frac { 1 } { 1 + \\lambda \\mu _ { \\ell } ^ 2 } \\tilde { \\Phi } _ { \\ell } ( x ) \\tilde { \\Phi } _ { \\ell } ( x _ j ) \\Big | \\\\ \\leq & \\underset { x \\in [ - 1 , \\ , 1 ] } \\max \\sum _ { j = 0 } ^ { N } w _ j \\sum _ { \\ell = 0 } ^ { L } \\frac { 1 } { 1 + \\lambda \\mu _ { \\ell } ^ 2 } \\Big | \\tilde { \\Phi } _ { \\ell } ( x ) \\tilde { \\Phi } _ { \\ell } ( x _ j ) \\Big | . \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{gather*} u = S ( b _ i ) a _ i = b _ i S ^ { - 1 } ( a _ i ) u ^ { - 1 } = S ^ { - 2 } ( b _ i ) a _ i = S ^ { - 1 } ( b _ i ) S ( a _ i ) = b _ i S ^ 2 ( a _ i ) . \\end{gather*}"}
-{"id": "4891.png", "formula": "\\begin{align*} \\dot { \\mu } _ 0 ( t ) = - \\frac { 2 s c _ 1 } { ( 2 s c _ 1 A + c _ 2 ) ( n - 2 s ) } \\mu _ 0 ^ { n - 4 s + 1 } ( t ) . \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} a ( x , y , t ) = h ( x ) \\alpha ( y , t ) \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} \\hat { \\mathfrak { p } } _ p : = \\mathfrak { g } [ \\pi ^ { - 1 } ( \\mathbb { D } _ p ) ] ^ \\Gamma \\oplus \\mathbb { C } C \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} \\| \\kappa ^ { ( d ) } ( A + \\delta K ) \\| = \\| \\kappa ^ { ( d ) } ( A ) \\| = \\| A \\| < \\| A + \\delta K \\| . \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { n - 1 } ( q + 1 - \\omega ^ i ) = \\frac { ( q + 1 ) ^ n - 1 } { q } = q ^ { n - 1 } + \\ldots + \\binom { n } { 3 } q ^ 2 + \\binom { n } { 2 } q + n . \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} \\partial _ t \\beta ( u ) + \\operatorname { d i v } ( \\mathbf { B } \\beta ( u ) ) + \\operatorname { d i v } ( \\mathbf { B } ) \\left ( u \\beta ' ( u ) - \\beta ( u ) \\right ) = G u \\beta ' ( u ) + F \\beta ' ( u ) \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} L ( g , p ) & = \\frac { 1 } { 2 } | D _ t g _ t ( x ) | ^ 2 + p ( t , g _ t ( x ) ) ( \\det \\nabla g _ t ( x ) - 1 ) \\\\ & : = L ^ 1 ( g , p ) + L ^ 2 ( g , p ) \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} \\frac { i ( e ) } { i ( \\overline { e } ) } = \\frac { N ( \\partial _ 0 e ) } { N ( \\partial _ 1 e ) . } \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{gather*} f : \\mathcal { L } ( M ) \\rightarrow \\mathcal { L } ( M ) \\\\ f ( x ) ( t ) = \\phi _ L ^ t ( x ( t ) ) \\end{gather*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u & = 0 , \\quad \\mbox { i n } \\Omega , \\\\ u & = \\ell , \\quad \\mbox { i n } \\partial \\Omega _ \\ell , \\ell = 0 , 2 , \\end{aligned} \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} f ' _ p ( t ) = p f _ p ( t ) - \\frac { ( p + 1 ) ^ 2 } { p + 2 } f _ { p + 1 } ( t ) , p \\ge 0 . \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} X : = \\C ^ k \\times ^ \\Lambda M = ( \\C ^ k \\times M ) / \\Lambda . \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} ( \\partial \\sigma ) ( i _ 0 , \\dots , i _ { n + 1 } ) = \\sum _ { r = 0 } ^ { n + 1 } ( - 1 ) ^ r \\sigma ( i _ 0 \\dots \\widehat { i _ r } \\dots i _ { n + 1 } ) | _ { U _ { i _ 0 \\dots i _ { n + 1 } } } \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Q ^ { \\times } } \\psi ( \\gamma \\zeta ) W _ { \\phi } \\left ( \\left ( \\begin{matrix} \\gamma & 0 \\\\ 0 & 1 \\end{matrix} \\right ) \\right ) = \\sum _ { \\gamma \\in \\Q ^ { \\times } } \\tilde { W } _ { \\phi } \\left ( \\left ( \\begin{matrix} \\gamma & 0 \\\\ 0 & 1 \\end{matrix} \\right ) \\left ( \\begin{matrix} 1 & 0 \\\\ - \\zeta & 1 \\end{matrix} \\right ) \\right ) . \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} \\Gamma = \\langle \\gamma _ 1 ^ { \\pm 1 } , \\ldots , \\gamma _ m ^ { \\pm 1 } \\rangle \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} & \\nabla _ g \\mathcal { U } ( g ) : = g \\mathbb { P } ( ( I _ 4 - g ^ { - 1 } ) \\mathbb { A } ) , \\\\ & \\Psi _ { \\mathcal { U } } ( g ) : = \\{ I _ 4 \\} { \\textstyle \\bigcup } \\left \\{ g = \\mathcal { T } ( R , p ) : R = \\mathcal { R } _ { a } ( \\pi , v ) , \\right . \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\left . p = ( I _ 3 - \\mathcal { R } _ { a } ( \\pi , v ) ) b d ^ { - 1 } , v \\in \\mathcal { E } ( Q ) \\right \\} . \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{gather*} T _ 0 T _ 1 \\cdots T _ { k - 1 } T _ { k } \\big ( x _ { k + 1 } ^ + \\big ) = T _ 0 T _ 1 \\cdots T _ { k - 1 } \\big ( \\big [ x _ { k } ^ + , x _ { k + 1 } ^ + \\big ] \\big ) = \\big [ T _ 0 T _ 1 \\cdots T _ { k - 1 } \\big ( x _ { k } ^ + \\big ) , x _ { k + 1 } ^ + \\big ] \\\\ \\hphantom { T _ 0 T _ 1 \\cdots T _ { k - 1 } T _ { k } \\big ( x _ { k + 1 } ^ + \\big ) } { } = [ E _ { N , k + 1 } ( 1 ) , E _ { k + 1 , k + 2 } ] = E _ { N , k + 2 } ( 1 ) . \\end{gather*}"}
-{"id": "8000.png", "formula": "\\begin{align*} A _ { T , t } ^ { ( i _ 1 i _ 2 ) } = \\frac { T - t } { 2 } \\sum _ { i = 1 } ^ { \\infty } \\frac { 1 } { \\sqrt { 4 i ^ 2 - 1 } } \\left ( \\zeta _ { i - 1 } ^ { ( i _ 1 ) } \\zeta _ { i } ^ { ( i _ 2 ) } - \\zeta _ i ^ { ( i _ 1 ) } \\zeta _ { i - 1 } ^ { ( i _ 2 ) } \\right ) . \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} d ( R | S ) = \\frac { 1 + n } { 2 ^ { n } } . \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { n _ 0 - 1 } \\sum _ { M _ i , \\Sigma ( M _ i ) = n } | M _ i | ^ k c ^ { \\Sigma ( M _ i ) } + \\sum _ { n = n _ 0 } ^ \\infty e ^ { 1 0 \\sqrt { n } } n ^ k c ^ { n } . \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} L _ n v & = f _ { \\alpha , \\infty } , & & x \\in \\R ^ 3 _ + , & & t \\in J ; \\\\ B v & = \\partial ^ \\alpha g _ \\infty , & & x \\in \\partial \\R ^ 3 _ + , & & t \\in J ; \\\\ v ( 0 ) & = w _ { 0 , \\infty } , & & x \\in \\R ^ 3 _ + ; \\end{aligned} \\right . \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} \\bar { \\theta } _ { i , j } = \\frac { 2 \\pi [ i ] _ { N _ 1 } } { N _ 1 } + \\frac { 2 \\pi [ j ] _ { N _ 2 } } { N _ 2 } , \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} \\left [ u \\frac { d ^ 2 } { d u ^ 2 } + ( \\ell - u + 1 ) \\frac { d } { d u } + n \\right ] L _ n ^ { \\ell } ( u ) = 0 , \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} \\frac { w _ { \\lambda _ j } ^ \\prime ( 1 ) } { w _ { \\lambda _ j } ( 1 ) } = \\frac { h _ { 1 , \\lambda _ j } ^ \\prime ( 1 ) } { h _ { 1 , \\lambda _ j } ( 1 ) } + \\frac { n - 1 } { 2 } \\sqrt { K _ 0 } , \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\int _ { \\C ^ n } \\int _ { S ( p ( w ) , \\infty ) } \\abs { T _ S f ( z , w ) } ^ 2 \\d V ( z ) \\d V ( w ) = \\int _ { \\C ^ n } \\int _ 0 ^ { \\infty } \\abs { f ( t , w ) } ^ 2 \\frac { e ^ { - 4 \\pi p ( w ) t } } { 4 \\pi t } \\d t \\d V ( w ) \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} & \\left \\{ \\widetilde { \\nu } _ n > m , M ( m ) \\le n ^ { 1 / 2 + \\varepsilon / 2 } \\right \\} \\\\ & \\subset \\left \\{ { \\rm d i s t } ( x + S ( j ) , \\partial K ) \\le \\frac { 1 } { 2 } \\left ( n ^ { 1 / 2 - \\varepsilon } + \\frac { | x + S ( j ) | } { n ^ { 2 \\varepsilon } } \\right ) , | x + S ( j ) | \\le n ^ { 1 / 2 + \\varepsilon / 2 } , j \\le m \\right \\} \\\\ & \\subset \\left \\{ { \\rm d i s t } ( x + S ( j ) , \\partial K ) \\le n ^ { 1 / 2 - \\varepsilon } , j \\le m \\right \\} = \\left \\{ \\nu _ n > m \\right \\} , \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} \\begin{aligned} S _ { n , p } \\left ( \\frac { n - p } { p - 1 } \\right ) ^ { - \\frac { n - 1 } { n } } \\left ( \\int _ { B _ { R } } \\frac { | u ( x ) | ^ { p ^ { * } } } { \\left [ \\log _ { { \\frac { n - 1 } { p - 1 } } } \\frac { R } { | x | } \\right ] ^ { \\frac { p ( n - 1 ) } { n - p } } } d x \\right ) ^ { \\frac { 1 } { p ^ { * } } } \\le \\Biggl ( \\int _ { { B _ { R } } } | L _ p u ( x ) | ^ { p } d x \\Biggr ) ^ { { \\frac { 1 } { p } } } \\end{aligned} \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} & \\beta _ { \\l } + \\sigma _ { \\l - 1 } \\lambda _ { \\l } \\left ( 1 - \\frac { 1 } { \\l } \\right ) ^ 2 \\\\ & = ( 2 \\l + d - 2 ) \\left ( 1 - \\frac { 1 } { \\l } \\right ) + \\frac { 2 \\delta } { \\l - 1 } \\lambda _ { \\l } \\left ( 1 - \\frac { 1 } { \\l } \\right ) ^ 2 \\\\ & = 2 ( 1 + \\delta ) \\l + O ( 1 ) \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} | \\Pi \\eta | \\ , | \\eta | \\ge ( \\Pi \\eta ) \\cdot \\eta = \\eta \\cdot ( \\Pi ^ T \\eta ) = | \\eta | ^ 2 \\ , , \\forall \\ , \\eta \\ , \\in T _ x \\Sigma _ z \\ , . \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} P \\{ Y = 0 \\vert \\textbf { X } = \\textbf { x } \\} = \\dfrac { f _ { \\textbf { X } \\vert Y = 0 } ( \\textbf { x } \\vert 0 ) P \\{ Y = 0 \\} } { f _ \\textbf { X } ( \\textbf { x } ) } = \\dfrac { \\dfrac { ( 1 - p ) } { \\mu { ( \\mathcal { C } _ { 0 } ) } } } { \\dfrac { p } { \\mu { ( \\mathcal { C } _ { 1 } ) } } + \\dfrac { 1 - p } { \\mu { ( \\mathcal { C } _ { 0 } ) } } } \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} \\chi ( M ^ { 2 n } ) ( t ) & = \\sum _ { p = 0 } ^ { 2 n } ( - 1 ) ^ p \\dim \\mathcal { H } ^ p ( M ^ { 2 n } , g _ J ) \\\\ & = \\sum _ { p = 0 } ^ { 2 n } ( - 1 ) ^ p b ^ p ( M ^ { 2 n } ) ( t ) , t \\in [ 0 , 1 ) . \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} \\mathcal { L } _ V = - \\partial ^ 2 _ r - \\frac { n - 1 } r \\partial _ r + \\frac 1 { r ^ 2 } \\big ( \\Delta _ h + V _ 0 ( y ) \\big ) . \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} ( f ^ { \\prime } r ^ { m - 1 } e ^ { - \\frac { r ^ 2 } { 4 } } ) ^ \\prime = r ^ { m - 1 } e ^ { - \\frac { r ^ 2 } { 4 } } ( - \\lambda e ^ { - \\frac { r ^ 2 } { 2 ( m - 2 ) } } + r ^ { - 2 } \\lambda _ k ) f . \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\mathbb { P } \\left \\lbrace \\left \\lvert p ^ { n , r } _ t \\right \\rvert \\ge \\varepsilon , - \\tau < t \\le \\tau ^ { n , r } _ R \\right \\rbrace = 0 \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} h ( k , \\ell + 1 ) = \\left ( \\frac { \\ell } { \\ell + 1 } - \\frac { f ( k ) } { \\ell + 1 } \\right ) h ( k , \\ell ) + \\frac { f ( k ) } { \\ell + 1 } h ( k - 1 , \\ell ) , \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} R _ 1 ^ { n - \\theta } = & \\mathcal { D } _ t ( u ( t _ { n - \\theta } ) - u ^ { n - \\theta } ) + \\gamma \\bigtriangleup ( \\sigma ( t _ { n - \\theta } ) - \\sigma ^ { n - \\theta } ) - \\bigtriangleup ( u ( t _ { n - \\theta } ) - u ^ { n - \\theta } ) + ( f ( u ( t _ { n - \\theta } ) ) - f ^ { n - \\theta } ( u ) ) \\\\ = & O ( \\Delta t ^ 2 ) , \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} \\lVert a ( k _ 1 , \\dots , k _ n ) \\psi _ { g , \\eta } \\lVert & \\leq \\sum _ { i = 1 } ^ { n } \\frac { \\omega ( k _ i ) } { \\omega ( k _ 1 ) + \\cdots + \\omega ( k _ n ) } g ^ { n } \\frac { \\lvert v ( k _ 1 ) \\lvert \\dots \\lvert v ( k _ n ) \\lvert } { \\omega ( k _ 1 ) \\dots \\omega ( k _ n ) } \\\\ & = g ^ { n } \\frac { \\lvert v ( k _ 1 ) \\lvert \\dots \\lvert v ( k _ n ) \\lvert } { \\omega ( k _ 1 ) \\dots \\omega ( k _ n ) } . \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} m \\ddot { \\boldsymbol { q } } _ a = - \\nabla V ( \\boldsymbol { q } _ a ) - \\frac { \\hbar ^ 2 } { 8 m } { \\frac { \\partial } { \\partial \\boldsymbol { q } _ a } \\int \\frac { \\sum _ { b } w _ b \\nabla K ( \\boldsymbol { y } - \\boldsymbol { q } _ a ) \\cdot \\nabla K ( \\boldsymbol { y } - \\boldsymbol { q } _ b ) } { \\sum _ { c } w _ c K ( \\boldsymbol { y } - \\boldsymbol { q } _ c ) } \\ , d ^ 3 y } \\ , . \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} { e ^ { - \\rho _ 0 - 1 } } = { \\sqrt { 1 + 2 \\rho _ 1 \\sigma _ w ^ 2 } } . \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { F } _ q } ( c _ { 1 , j } , c _ { 2 , j } , \\dots , c _ { l , j } ) & = l , \\\\ \\sum _ { i \\neq j } \\dim _ { \\mathbb { F } _ q } ( c _ { 1 , i } , c _ { 2 , i } , \\dots , c _ { l , i } ) & = \\frac { ( n - 1 ) l } { r } . \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} \\hat { \\mu } ^ p _ { \\Lambda , \\epsilon } ( d \\phi , d t ) : = \\frac { 1 } { Z ^ p _ { \\Lambda , \\epsilon } ( \\alpha ) } e ^ { - A ^ p ( \\phi , t ) } \\prod _ { j k \\in E ( \\mathbb { T } _ N ) } \\Big ( f _ \\alpha ( e ^ { t _ { j k } } ) e ^ { t _ { j k } } d t _ { j k } \\Big ) \\prod _ { j \\in \\Lambda } d \\phi _ j , \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} \\check f ( \\chi ) = \\int _ { A _ X } f ( \\xi ) \\chi ^ { - 1 } ( \\xi ) \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} \\begin{cases} h _ { \\partial M } = 0 \\\\ H ' _ h = 0 \\\\ t r _ { \\partial M } \\delta _ { g _ 0 } ^ * Y + 2 G = 0 \\\\ [ \\delta _ { g _ 0 } ^ * Y ( \\mathbf n ) ] ^ T + \\nabla _ { g _ 0 ^ T } G = 0 . \\end{cases} \\quad ~ \\partial M . \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} A ( x ) = \\operatorname { R e } \\left [ \\mathbb { A } ( x ) \\right ] \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} f ( 0 , 0 , z _ 3 , \\cdots , z _ N ) = g ( 0 , 0 , z _ 3 , \\cdots , z _ N ) = 0 , \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} u _ l ( T _ { o p t } , x ) = 0 \\mbox { f o r } x \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { s - 1 } \\lambda _ { 1 , k } ^ { j - 1 } c ^ { I } _ { 1 , b ( 1 , k ) } + \\sum _ { i = 2 } ^ { i = n } \\lambda _ { i , b _ { i } } ^ { j - 1 } \\mu _ { i , 1 } ^ { ( b ) } = 0 j \\in [ r ] . \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} - \\Delta Q _ c - c | x | ^ { - 2 } Q _ c + Q _ c = | Q _ c | ^ { \\alpha } Q _ c . \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} \\nu _ p ( n ! ) & = ( n - \\sigma _ p ( n ) ) / ( p - 1 ) . \\\\ \\nu _ p \\binom { n } { m } & = ( \\sigma _ p ( m ) + \\sigma _ p ( n - m ) - \\sigma _ p ( n ) ) / ( p - 1 ) . \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} \\delta = \\sup _ { f \\in \\mathcal { F } } \\sup _ { | z | < 1 } \\left | - \\overline { z } + \\frac { 1 } { 2 } ( 1 - | z | ^ 2 ) \\frac { f '' ( z ) } { f ' ( z ) } \\right | \\quad ( z \\in \\Delta ) . \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} \\psi ( - e ^ { - \\pi } ) = a 2 ^ { - 3 / 4 } e ^ { \\pi / 8 } \\quad \\psi ( - e ^ { - 2 \\pi } ) = a 2 ^ { - 1 5 / 1 6 } e ^ { \\pi / 4 } , \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta _ 0 & = \\frac { 1 } { 2 } ( d d ^ { * _ 0 } + d ^ { * _ 0 } d ) \\\\ & = \\Delta _ { \\partial _ 0 } \\coloneqq ( \\partial _ { J _ 0 } \\partial _ { J _ 0 } ^ { * _ 0 } + \\partial _ { J _ 0 } ^ { * _ 0 } \\partial _ { J _ 0 } ) \\\\ & = \\Delta _ { \\bar { \\partial } _ { J _ 0 } } \\coloneqq ( \\bar { \\partial } _ { J _ 0 } \\bar { \\partial } _ { J _ 0 } ^ { * _ 0 } + \\bar { \\partial } _ { J _ 0 } ^ { * _ 0 } \\bar { \\partial } _ { J _ 0 } ) \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} \\Delta ( a - [ c , a ] ( d b ^ { - 1 } a - c ) ^ { - 1 } ) & = \\Delta ( a ( d b ^ { - 1 } a - c ) - [ c , a ] ) ( d b ^ { - 1 } a - c ) ^ { - 1 } \\\\ & = \\Delta ( a d b ^ { - 1 } - c ) a ( d b ^ { - 1 } a - c ) ^ { - 1 } \\\\ & = \\Delta ( a d - c b ) b ^ { - 1 } a ( d b ^ { - 1 } a - c ) ^ { - 1 } \\\\ & = b ^ { - 1 } a ( d b ^ { - 1 } a - c ) ^ { - 1 } \\\\ & = ( d - c a ^ { - 1 } b ) ^ { - 1 } \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} \\phi _ * ( T ) = T + \\sum _ { j = 1 } ^ n ( T q _ j ) \\partial _ { v _ j } \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} \\norm { \\phi } ^ 2 _ r = \\sum _ { x \\in \\mathbb { Z } ^ d } \\phi ( x ) ^ 2 e ^ { - 2 r | x | } \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} ( S , u ) : = ( S , v ) \\in \\mathrm { C } ( [ 0 , T ] ; \\mathbb { R } \\times \\mathrm { d o m } ( - D \\Delta ) ) \\cap \\mathrm { C } ^ 1 ( ( 0 , T ] ; \\mathbb { R } \\times \\mathrm { L } ^ 2 ( \\Omega ) ) \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} \\Theta _ { 2 } ( \\widetilde { F } _ m ) & = 2 ( \\ell + 1 ) + 2 ( \\ell ' - k ^ 2 - 4 k ( k + r - 1 ) + 1 ) \\\\ & = 4 \\ell - 6 k ^ 2 - 4 s _ 1 k + 2 s _ 2 + 4 . \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} K _ t ^ { ( - 1 / 2 , - 1 / 2 ) } ( m , n ) = \\int _ 0 ^ \\pi e ^ { - t ( 1 - \\cos \\theta ) } \\cos ( m \\theta ) \\cos ( n \\theta ) \\ , d \\theta . \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} ( \\mathbf { a } ^ k ) _ i ( 0 ) & = \\int _ { \\Omega } \\varphi _ 0 w _ i \\d x ~ ~ \\forall 1 \\leq i \\leq k , \\\\ ( \\mathbf { c } ^ k ) _ i ( 0 ) & = \\int _ { \\Omega } \\sigma _ 0 w _ i \\d x ~ ~ \\forall 1 \\leq i \\leq k , \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} | u ( x ) - u ( x + y ) | & = | u ( x ) - u ( x + t _ 1 y ) + u ( x + t _ 2 y ) - u ( x + y ) | \\\\ & \\le | u ( x ) - u ( x + t _ 1 y ) | + | u ( x + t _ 2 y ) - u ( x + y ) | \\\\ & = \\left | \\int _ 0 ^ { t _ 1 } ( \\nabla u ( x + t y ) , y ) d t \\right | + \\left | \\int _ { t _ 2 } ^ 1 ( \\nabla u ( x + t y ) , y ) d t \\right | \\\\ \\mbox { ( b y \\eqref { u ' - b o u n d } ) } & \\le C _ 4 | y | \\left ( \\int _ 0 ^ { t _ 1 } + \\int _ { t _ 2 } ^ 1 \\right ) | x + t y | ^ { p - 1 } d t \\le C | y | \\big ( | x | ^ { p - 1 } + | y | ^ { p - 1 } \\big ) , \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} \\int _ \\Omega S \\phi \\ , \\d x = \\phi ( x ^ + ) - \\phi ( x ^ - ) , \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} & M _ G \\left ( ( x ^ 2 + 1 ) ( x ^ 3 + 1 ) + k h ( x ) + y \\left ( ( x ^ 2 + 1 ) + k \\ : h ( x ) \\right ) \\right ) = - 2 ^ 4 3 ^ 3 ( 1 + 2 k ) , \\\\ & M _ G \\left ( 1 + m \\ : h ( x ) + y \\left ( ( 1 - x - x ^ 5 ) + m \\ : h ( x ) \\right ) \\right ) = - 2 ^ 6 3 ^ 3 m . \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} [ [ x , y ] , \\varphi ( z ) ] + [ [ z , x ] , \\varphi ( y ) ] + [ [ y , z ] , \\varphi ( x ) ] = 0 \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} ( x \\cup z ) - ( d f _ v \\cup y _ v ) = ( x \\cup z ) + ( \\alpha \\cup y _ v ) . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} c _ j x _ 1 X _ 2 p _ { j } + X _ 2 r _ j = c _ j x _ 1 \\left ( \\frac { \\partial } { \\partial { x _ 2 } } p _ { j } + \\sum _ { k > 2 } p _ { k } \\frac { \\partial } { \\partial { x _ k } } p _ { j } \\right ) + \\frac { \\partial } { \\partial { x _ 2 } } r _ j + \\sum _ { \\ell > 2 } p _ { \\ell } \\frac { \\partial } { \\partial { x _ \\ell } } r _ j = 0 . \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} H ^ 2 ( X , \\mathbb { R } ) = \\mathbb { R } [ D ] + \\mathbb { R } [ H ] . \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} ( \\Lambda _ q [ f ] , [ g ] ) = B _ q ( u _ f , g ) . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} \\rho _ \\infty ( a ) t _ \\infty ( x ) = t _ \\infty ( \\phi _ X ( a ) x ) , t _ \\infty ( x ) \\rho _ \\infty ( a ) = t _ \\infty ( x a ) \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} [ e ^ v ] _ { A _ \\infty ( \\mathbb R ) } = \\left ( \\sup _ { I = ( a , b ) \\subset \\R } | I | ^ { - 1 } \\int _ I e ^ { v ( x ) } \\ , d x \\right ) \\exp \\left ( [ - v ] _ I \\right ) = \\sup _ { I = ( a , b ) \\subset \\R } | I | ^ { - 1 } \\int _ I e ^ { v ( x ) - v _ I } \\ , d x . \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} T ^ \\mathrm { a x } _ { \\alpha \\beta \\gamma } = \\frac 1 3 ( T _ { \\alpha \\beta \\gamma } + T _ { \\beta \\gamma \\alpha } + T _ { \\gamma \\alpha \\beta } ) , \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} & \\quad \\theta '' ( g _ 0 , \\ldots , g _ { n - 1 } ) ( \\eta ) \\\\ & = \\sum _ { s _ 0 , \\ldots , s _ { n - 1 } } \\theta ' ( h _ { s _ 0 } , \\ldots , h _ { s _ { n - 1 } } ) ( L ( \\eta ) | _ { [ s _ 0 , \\ldots , s _ { n - 1 } ] } ) \\\\ & = \\sum _ { s _ 0 , \\ldots , s _ { n - 1 } } \\sum _ { t _ 0 , \\ldots , t _ { n - 1 } } \\theta ( g _ { t _ 0 } , \\ldots , g _ { t _ { n - 1 } } ) ( \\pi ( L ( \\eta ) | _ { [ s _ 0 , \\ldots , s _ { n - 1 } ] } ) | _ { [ t _ 0 , \\ldots , t _ { n - 1 } ] } ) . \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{gather*} \\frac { L ( z , 1 ) } { \\Re F _ 1 ( z ) } = - \\frac { L ( z , 1 ) } { \\pi \\ , \\Re F _ 0 ( 1 - z ) } \\in \\mathbb Q , \\end{gather*}"}
-{"id": "1365.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 u '' - r ( x ) u = 0 , \\ x \\in ( - L _ 1 , a ) ; \\\\ & u ( a ) = 1 ; \\\\ & u ( - L _ 1 ) = u ( 0 ) . \\end{aligned} \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} & \\beta _ 0 + m _ 1 \\beta _ 1 + \\cdots + m _ k \\beta _ k = \\beta , \\\\ & n _ 0 + m _ 1 n _ 1 + \\cdots + m _ k n _ k = n \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} = ( - 1 ) ^ n \\sum _ { k = n } ^ { \\infty } \\frac { k ( k - 1 ) \\dots ( k - n + 1 ) } { n ! } \\frac { B ^ 2 ( k + 1 , n + 1 ) T ( k , n ) } { ( k + 1 ) ^ { s - 3 } } . \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} \\min \\bigg \\{ \\int _ { \\overline \\Omega } \\varphi ( \\nabla u ) \\ , : \\ , u \\in B V ( \\Omega ' ) , \\ ; u = \\tilde g \\ , \\mbox { o n } \\ , \\Omega ' \\setminus \\Omega \\bigg \\} . \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} G \\cong D _ 8 * G _ 1 \\in \\mathcal { C } & \\Longleftrightarrow n _ 2 ( D _ 8 * G _ 1 ) = 2 ^ { n - 1 } - 1 \\\\ & \\Longleftrightarrow 2 ^ { n - 2 } + 2 n _ 2 ( G _ 1 ) + 1 = 2 ^ { n - 1 } - 1 \\\\ & \\Longleftrightarrow n _ 2 ( G _ 1 ) = 2 ^ { n - 3 } - 1 \\\\ & \\Longleftrightarrow G _ 1 \\in \\cal { C } . \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} A _ \\mu ( t ) = \\int _ 0 ^ t \\mu ( S _ r ) \\dd r \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} ( ( q _ 1 - q _ 2 ) h _ j ^ { ( 1 ) } , h _ j ^ { ( 2 ) } ) _ { \\Omega } & = - ( ( q _ 1 - q _ 2 ) r _ j ^ { ( 1 ) } , h _ j ^ { ( 2 ) } ) _ { \\Omega } - ( ( q _ 1 - q _ 2 ) r _ j ^ { ( 2 ) } , h _ j ^ { ( 1 ) } ) _ { \\Omega } - ( ( q _ 1 - q _ 2 ) r _ j ^ { ( 2 ) } , r _ j ^ { ( 1 ) } ) _ { \\Omega } \\\\ & + ( ( \\Lambda _ { q _ 1 } - \\Lambda _ { q _ 2 } ) f _ j ^ { ( 1 ) } , f _ j ^ { ( 2 ) } ) _ { \\Omega _ e } . \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} g _ { ( 6 , 1 0 , 1 5 ) } ^ { ( 4 , 8 , 3 ) } ( z ) = ( 1 + z ^ 3 ) ( 1 + z ^ 2 + z ^ 4 ) ( 1 + z + z ^ 2 + z ^ 3 + z ^ 4 ) ^ 2 \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} K _ 2 = - \\frac { 2 } { 5 } K _ 1 '' + \\frac { 8 } { 2 5 } K _ 1 ^ 2 + \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 u '' - r ( x ) u = 0 , \\ a < x < \\infty ; \\\\ & u ( a ) = 1 ; \\\\ & 0 \\le u \\le 1 \\ \\ u \\ . \\end{aligned} \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} \\| T \\| _ { p \\to q } : = \\sup _ { 0 \\neq f \\in \\ell ^ p ( V ) } \\frac { \\| T f \\| _ q } { \\| f \\| _ p } . \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} - v _ { n , m } = - v _ { n - 1 , m } + ( D _ { n , m } + m - 1 ) - D _ { n - 1 , m } = - v _ { n - 1 , m } + m . \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} W _ { B ( y , r ) } ( x ) : = \\inf \\{ n \\geq 1 : T _ \\theta ^ n ( x ) \\in B ( y , r ) \\} \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} R _ { - 1 } \\bigl ( \\check { X } ( j ) \\bigr ) = - \\bigl ( \\check { X } ( j + 1 ) - \\check { X } ( j ) \\bigr ) \\bigl ( \\check { X } ( j ) - \\check { X } ( j - 1 ) \\bigr ) B ^ { } _ { \\mathcal { D } , j } \\ \\ ( j = 0 , 1 , \\ldots , N ) . \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} d X _ t = u ( t , X _ t ) d t + \\sqrt { 2 \\nu } d B _ t + \\beta _ j ( X _ t ) \\dot { Z } ^ j _ t d t , X _ 0 = x \\in \\R ^ d \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} R _ { 2 2 } & = ( d - c a ^ { - 1 } b ) ^ { - 1 } \\left ( [ d , a ] - [ c , b ] - c a ^ { - 1 } [ b , a ] \\right ) ( \\Delta ' ) ^ { - 1 } \\\\ & = \\left ( a - ( d - c a ^ { - 1 } b ) ^ { - 1 } \\Delta ' \\right ) ( \\Delta ' ) ^ { - 1 } . \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} T _ { \\vec { { \\bf f } } } ( \\xi ) : = \\nabla \\xi - \\vec { { \\bf f } } \\xi , \\ ; \\ ; \\xi \\in C _ c ^ { 1 } ( \\Omega ) \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} K _ { 1 , 3 } ( z , z '' ) = & \\Big ( \\int _ { 2 r < r ' < \\frac { r '' } 2 } + \\int _ { \\frac { r '' } 2 \\leq r ' \\leq 2 r '' } + \\int _ { r ' > 2 r '' } \\Big ) G ( z , z ' ) Q ( z ' , z '' ) \\ ; d \\mu ( z ' ) \\\\ = & K _ { 1 , 3 1 } ( z , z '' ) + K _ { 1 , 3 2 } ( z , z '' ) + K _ { 1 , 3 3 } ( z , z '' ) . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} m _ X ( x , \\xi ) : = \\frac { 1 } { 2 T _ { \\alpha _ 0 } ' } \\int _ { - T _ { \\alpha _ 0 } ' } ^ { T _ { \\alpha _ 0 } ' } m _ 0 \\circ \\tilde { \\Phi } ^ { X } _ t ( x , \\xi ) d t . \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} \\int _ G f ( x ) d \\tilde { \\mu } ( x ) = \\int _ { K \\backslash G / H } Q ( f ) ( \\ddot { x } ) d \\mu ( \\ddot { x } ) . \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} \\nu _ p ( s ( n , k ) ) = ( \\sigma ( k - 1 ) - \\sigma ( n - 1 ) ) ) / ( p - 1 ) . \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} \\sup _ { \\begin{smallmatrix} 0 \\le n \\le m \\\\ n , m \\in \\mathbb { N } \\end{smallmatrix} } \\frac { 1 } { m - n + 1 } \\Bigg ( \\sum _ { k = n } ^ m w ( k ) \\Bigg ) \\max _ { n \\le k \\le m } w ( k ) ^ { - 1 } < \\infty , \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} \\left | g ( z ) - \\sum _ { j = 0 } ^ { n - 1 } b _ j z ^ j \\right | \\le C _ 1 A _ 1 ^ n M _ n | z | ^ n . \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} & D _ 2 \\cos _ k w \\\\ & = 2 \\cos _ k \\left ( \\frac { w _ 1 + w _ 2 } 2 \\right ) \\left [ \\cos _ k \\left ( \\frac { w _ 1 - w _ 2 } 2 \\right ) - \\cos _ k \\left ( \\frac { w _ { - 1 } - w _ { - 2 } } 2 + \\pi j \\right ) \\right ] \\\\ & = 4 \\cos _ k \\left ( \\frac { w _ 1 + w _ 2 } 2 \\right ) \\sin _ k \\left ( \\frac { w _ 1 - w _ 2 - w _ { - 1 } + w _ { - 2 } + 2 \\pi j } { 4 } \\right ) \\sin _ k \\left ( \\frac { w _ 1 - w _ 2 + w _ { - 1 } - w _ { - 2 } + 2 \\pi j } { 4 } \\right ) . \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} ( f | _ k g ) ( \\tau ) = F \\Bigl ( \\frac { \\C } { 2 \\pi i ( \\Z + \\tau \\Z ) } , d z , \\beta ' _ \\tau \\Bigr ) \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} u ( y , r ) - \\varphi ( y , r + ( t - s ) ) & \\leq u ^ { \\theta } ( y , r + ( t - s ) ) + \\frac { ( t - s ) ^ { 2 } } { 2 \\theta } - \\varphi ( y , r + ( t - s ) ) \\\\ & \\leq u ^ { \\theta } ( x , t ) - \\varphi ( x , t ) + \\frac { ( t - s ) ^ { 2 } } { 2 \\theta } \\\\ & = u ( x , s ) - \\varphi ( x , t ) . \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} R _ k ^ i [ n ] ( \\mathbf { p } , \\mathbf { s } , \\mathbf { r } ) = s _ { k } ^ i [ n ] \\mathcal { B } \\log _ 2 \\Big ( 1 + \\frac { H _ k ^ i [ n ] p _ k ^ i [ n ] } { \\norm { \\mathbf { r } [ n ] - \\mathbf { r } _ k } ^ 2 } \\Big ) , \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} f ^ { \\nu } \\left ( t , x , 0 \\right ) = 0 , f ^ { \\nu } \\left ( t , x , 1 \\right ) = h ^ { \\nu } ( t ) \\ , \\doteq \\ , \\int _ 0 ^ T \\rho _ { \\delta _ { \\nu } } \\left ( t - s \\right ) h ( s ) \\ ; d s , \\forall ( t , x ) \\in \\Omega . \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} B ( x ) = \\begin{pmatrix} 0 & \\nu _ 3 ( x ) & - \\nu _ 2 ( x ) & 0 & 0 & 0 \\\\ - \\nu _ 3 ( x ) & 0 & \\nu _ 1 ( x ) & 0 & 0 & 0 \\\\ \\nu _ 2 ( x ) & - \\nu _ 1 ( x ) & 0 & 0 & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} \\nabla ^ 1 : = { } & \\varphi \\nabla ^ { \\tilde { g } } \\varphi ^ { - 1 } \\bigr | _ { r = 0 } , & \\nabla ^ 0 : = { } & \\varphi \\nabla ^ { { g } } \\varphi ^ { - 1 } \\bigr | _ { r = 0 } . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} \\epsilon ( h ) = \\frac { ( 4 + 7 m ) \\sqrt { \\log ( m + 1 ) } } { \\sqrt { \\log \\log h } } . \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} t _ i = \\sum _ { j = m _ { i - 1 } + 1 } ^ { m _ i } ( - 1 ) ^ { \\sigma ( j ) } \\ , s _ j . \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} | V ( x ) | \\leq \\frac { C _ { w + 1 } } { x - t T _ { w + 1 } } \\leq \\frac { C _ { w + 1 } } { ( J _ w + t T _ { w + 1 } ) - t T _ { w + 1 } } = \\frac { C _ { w + 1 } } { J _ w } . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} \\norm { \\psi _ N \\phi _ { \\epsilon } \\ast F _ y } _ { L ^ 2 ( \\mathbb { R } ) } \\leq \\norm { \\psi _ N \\phi _ { \\epsilon } } _ { L ^ 1 ( \\mathbb { R } ) } \\norm { F _ y } _ { L ^ 2 ( \\mathbb { R } ) } & \\leq \\norm { \\phi _ { \\epsilon } } _ { L ^ 1 ( \\R ) } \\norm { F _ y } _ { L ^ 2 ( \\R ) } = \\norm { \\phi } _ { L ^ 1 ( \\mathbb { R } ) } \\norm { F _ y } _ { L ^ 2 ( \\mathbb { R } ) } . \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} & b _ M ( m \\otimes \\{ a _ 1 | . . . | a _ n \\} ) : = d _ M m \\otimes \\{ a _ 1 | . . . | a _ n \\} - \\sum _ { i = 1 } ^ n m \\otimes ( - 1 ) ^ { \\epsilon _ i } \\{ a _ 1 | . . . | d _ A a _ i | . . . | a _ n \\} \\\\ & + ( - 1 ) ^ { | m | } ( m \\cdot a _ 1 ) \\otimes \\{ a _ 2 | . . . | a _ n \\} + \\sum _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ { \\epsilon _ i } m \\otimes \\{ a _ 1 | . . . | a _ i a _ { i + 1 } | . . . | a _ n \\} \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} { \\rm g c d } ( p _ 2 , a _ 1 r _ 1 + s _ 1 ) = 1 \\ , . \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} ( g \\cdot \\alpha ) ( a , b ) = \\alpha ( ( a , b ) g ) ( ( a , b ) \\in ( \\Z / N \\Z ) ^ 2 ) . \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} \\inf _ { ( \\gamma , \\xi ) \\in \\Omega ( y _ 0 ) } \\langle k , \\gamma \\rangle \\ : = k ^ * ( y _ 0 ) \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } p _ { 0 } ( y , t ) \\int d x \\ x h ( x ) = \\langle x \\rangle \\frac { \\partial } { \\partial y } p _ { 0 } ( y , t ) = 0 \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} | | \\Omega | | _ { W ^ { \\alpha _ 0 , 1 } ( B _ 2 \\backslash B _ 1 ) } : = \\int _ { B _ 2 \\backslash B _ 1 } | \\Omega | + \\int _ { B _ 2 \\backslash B _ 1 } \\int _ { B _ 2 \\backslash B _ 1 } \\frac { | \\Omega ( x ) - \\Omega ( y ) | } { | x - y | ^ { d + \\alpha _ 0 } } d x d y \\leq c _ 1 , \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } + u \\cdot \\nabla \\right ) \\left ( K _ { f } \\nabla \\phi \\right ) - f J ^ { 2 } u = J \\nabla \\phi , \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d \\| | D _ { q _ i } | e ^ { - t ( K _ V + \\sqrt { A } ) } \\| _ { { \\cal L } ( L ^ { 2 } ( \\mathbb { R } ^ { 2 d } ) ) } + \\| | \\partial _ { q _ i } V ( q _ i ) | e ^ { - t ( K _ V + \\sqrt { A } ) } \\| _ { \\mathcal { L } ( L ^ 2 ( \\mathbb { R } ^ { 2 d } ) ) } \\le \\frac { c } { t ^ { \\frac { 3 } { 2 } } } \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} F : = \\{ { f ' _ i } \\mid i \\in I \\} . \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } k _ { 1 , \\sigma } = \\iota _ 2 - 1 & \\forall \\sigma \\in I _ F ^ 2 \\\\ k _ { 2 , \\sigma } = \\iota _ 2 & \\forall \\sigma \\in I _ F ^ 3 \\end{array} \\right . \\end{align*}"}
-{"id": "9835.png", "formula": "\\begin{align*} u ^ 2 = & \\beta _ 2 ^ { q - 1 } \\\\ = & ( a u ^ q ) ^ { q - 1 } \\\\ = & u ^ { q ^ 2 - q } . \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\tilde { h } ( x _ n ) \\ge \\liminf _ { n \\to \\infty } \\frac { | x _ n | } { | g ' ( y _ n ) | } = \\liminf _ { n \\to \\infty } \\frac { | g ( y _ n ) | } { | g ' ( y _ n ) ) | } \\ge \\alpha , \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { ( - 1 ) ^ { n ( j + s ) + s } \\binom k j \\binom j s \\frac { { G _ { m + 2 k - ( n + 1 ) j + 2 n s } } } { { F _ n ^ j } } } } = ( - 1 ) ^ k G _ m , n \\ne 0 \\ , , \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} \\deg ( D | _ S ) = \\deg ( D ' | _ S ) - \\deg ( \\div ( \\phi ) | _ S ) \\geq p _ a ( S ) + \\psi ( S ) - p _ a ( S ) + 1 \\geq \\psi ( S ) + 1 \\ , , \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\sum _ { h _ 1 \\in H } h _ 1 ^ { - 1 } f ' ( h _ 1 ) & = \\sum _ { g \\in G } \\pi ^ * ( g ^ { - 1 } f ( g ) ) , \\\\ \\sum _ { h _ 1 \\in H } f ' ( h _ 1 ) & = \\sum _ { g \\in G } \\pi ^ * ( f ( g ) ) . \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} - \\Delta v - c | x | ^ { - 2 } v - \\frac { d _ M } { M } v - | v | ^ \\alpha v = 0 . \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} \\Im i \\langle v , H D _ j Q \\rangle & = \\Im \\bigl [ i \\langle v , E D _ j Q \\rangle - i \\mu \\langle v , D _ j ( | Q | ^ p Q ) \\rangle \\bigr ] \\\\ & = \\Im \\langle v , D _ j D Q i E z \\rangle - \\Im i \\mu \\langle v , D _ j ( | Q | ^ p Q ) \\rangle , \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} L _ { 1 1 } & = \\Delta ^ { - 1 } ( d ( b d ^ { - 1 } c - a ) + \\Delta ) ( b d ^ { - 1 } c - a ) ^ { - 1 } \\\\ & = - \\Delta ^ { - 1 } ( [ b , d ] d ^ { - 1 } c + [ d , a ] + [ c , b ] ) ( b d ^ { - 1 } c - a ) ^ { - 1 } . \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} \\ell = \\frac { 1 } { k _ 1 } N _ 1 ( \\delta ) + \\frac { 1 } { k _ 2 } N _ 2 ( \\delta ) + \\dotsb + \\frac { 1 } { k _ m } N _ m ( \\delta ) \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} \\ , \\Phi ( \\mathrm m ) \\ , : = \\ , ( X ^ { \\mathrm m } _ { t } , 0 \\le t \\le T ) \\ , , \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} \\alpha ( N _ k ) : = N _ k ( N _ k + 1 ) \\sqrt { 5 N _ k ^ 2 + 5 N _ k + 5 } \\in \\lambda \\Z . \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} \\varphi ( d ^ { \\mathrm { c l o u d } } ) = e ^ { - \\beta _ c d ^ { \\mathrm { c l o u d } } } , \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} M : = \\begin{pmatrix} ( g _ 1 , h _ { 1 } ^ { ( 1 ) } h _ { 1 } ^ { ( 2 ) } ) _ { \\Omega } & \\dots & ( g _ m , h _ { 1 } ^ { ( 1 ) } h _ { 1 } ^ { ( 2 ) } ) _ { \\Omega } \\\\ \\vdots & \\vdots & \\vdots \\\\ ( g _ 1 , h _ { m } ^ { ( 1 ) } h _ { m } ^ { ( 2 ) } ) _ { \\Omega } & \\dots & ( g _ m , h _ { m } ^ { ( 1 ) } h _ { m } ^ { ( 2 ) } ) _ { \\Omega } \\end{pmatrix} \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} P _ N ( A ^ { - s / 2 } h ) = A ^ { - s / 2 } ( P _ N h ) . \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ r \\exp \\left ( p _ i ( e ^ { t \\alpha _ i } - 1 ) \\right ) = \\prod _ { i = 1 } ^ r \\exp \\left ( p _ i ( ( 1 - \\delta ) ^ { \\alpha _ i } - 1 ) \\right ) \\leq \\prod _ { i = 1 } ^ r \\exp \\left ( - p _ i \\delta \\alpha _ i \\right ) , \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} ( \\sigma ^ { n - \\theta } , w ) + ( \\nabla u ^ { n - \\theta } , \\nabla w ) = 0 , ~ \\forall w \\in H _ 0 ^ 1 . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} \\lim _ { | v | \\rightarrow + \\infty } \\frac { j ( v ) } { | v | } = + \\infty . \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{gather*} a _ 1 = \\left [ \\frac { z } { 1 + \\lambda } + b \\right ] l _ 1 ^ { - 1 } z ^ { - l _ 1 } , \\\\ a _ 2 = \\left [ \\frac { z } { 1 + \\lambda } - b \\right ] l _ 2 ^ { - 1 } z ^ { - l _ 2 } , \\end{gather*}"}
-{"id": "8018.png", "formula": "\\begin{align*} I _ { T , t } ^ { ( i _ 1 i _ 2 ) q } = \\frac { T - t } { 2 } \\zeta _ { 0 } ^ { ( i _ 1 ) } \\zeta _ { 0 } ^ { ( i _ 2 ) } + { \\hat A } _ { T , t } ^ { ( i _ 1 i _ 2 ) q } . \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ N \\big ( 1 + | \\lambda _ j ( A ) | \\big ) \\leq \\prod _ { j = 1 } ^ N \\big ( 1 + \\mu _ j ( A ) \\big ) . \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S = \\lvert T ^ { - n } [ \\phi v _ 0 ] \\rvert _ S > \\lvert T ^ { - n } [ \\phi v ] \\rvert _ S + 2 \\lvert A ( I ) \\rvert = \\lvert ( \\phi v ) _ n \\rvert _ S + 2 \\lvert A ( I ) \\rvert . \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} w _ 0 = \\begin{cases} w _ 0 ^ - = \\rho _ \\lambda c _ \\Omega K _ \\gamma ^ - , \\mbox { i n } \\ \\Omega ^ - , \\medskip \\\\ w _ 0 ^ + = \\rho _ \\lambda c _ \\Omega K ^ + _ \\gamma , \\quad \\mbox { i n } \\Omega ^ + . \\\\ \\end{cases} \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} \\lvert u \\rvert _ S = \\lvert ( 0 , j ) \\rvert _ { W _ n ( w ) } = \\lvert ( \\phi w ) _ n \\rvert _ S - ( n - j ) . \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\nabla _ \\alpha A ^ \\alpha = \\rho ^ { - 1 } \\partial _ \\alpha ( \\rho A ^ \\alpha ) = - \\delta A ^ \\flat , \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta u + u + \\lambda \\phi u = f ( u ) & \\mathbb R ^ 3 , \\smallskip \\\\ - \\Delta \\phi = u ^ 2 & \\mathbb R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} | \\{ s \\in [ 2 r + 1 ] : i _ s = i _ { s _ 1 } \\} | = 1 | \\{ s \\in [ 2 r + 1 ] : i _ s = i _ { s _ 2 } \\} | = 1 . \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} H = \\ker B _ i + \\ker B _ k = V + \\ker B _ k , \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } J ( g _ { A _ 1 } , \\ldots , g _ { A _ { i - 1 } } , g _ { A ^ { ( t ) } _ i } , g _ { A _ { i + 1 } } , \\ldots , g _ { A _ m } ) = 0 , \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} < \\sigma _ \\gamma , \\phi > = \\int _ { \\Omega \\times \\Omega } \\mathrm { d } \\gamma ( x , y ) \\int _ { 0 } ^ { 1 } \\phi ( \\omega _ { x , y } ( t ) ) \\ , | | { \\omega } ^ \\prime _ { x , y } ( t ) | | \\ , \\mathrm { d } t \\ ; \\ ; \\ ; \\mbox { f o r a l l } \\ ; \\ ; \\phi \\ ; \\in \\ ; C ( \\Omega ) \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} L ( \\psi ^ { \\vee } , s - 1 ) : = \\prod \\limits _ { f } D ^ { \\psi } _ f ( f ^ { - s } ) ^ { - 1 } = \\sum _ { a \\in A _ { + } } \\frac { \\mu _ { \\psi } ( a ) } { a ^ s } , \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} \\mathcal { d } _ \\mathrm { F R } ( C _ 0 , C _ 1 ) ^ 2 & : = \\inf \\left \\{ \\int _ 0 ^ 1 \\int _ \\Omega \\int _ { \\mathbb { S } ^ 1 _ + } | r _ t ( x ) | ^ 2 \\ , \\d C _ t ( x , \\theta ) \\ , \\d t ; \\ ; { ( C , r ) \\in \\mathcal { A } [ C _ 0 , C _ 1 ] } \\right \\} , \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\omega ( X ) ( Y ) = \\omega ( Y ) ( X ) \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} \\mathcal { H } W _ { l } ( s ; y l ^ c ) = & \\omega _ { \\pi _ { 0 } , l } ( l ^ { - \\frac { c } { 2 } } ) l ^ { \\frac { c } { 2 } } \\sum _ { \\mu _ l \\in { } _ l \\mathfrak { X } _ { - \\frac { c } { 2 } } ' } \\mu _ l ( y ^ { - 1 } ) \\epsilon ( \\frac { 1 } { 2 } , \\mu _ l ^ { - 1 } ) ^ 2 [ \\mathcal { M } W _ l ( s ; \\cdot ) ] ( \\mu ) \\\\ = & \\omega _ { \\pi _ { 0 } , l } ( l ^ { - \\frac { c } { 2 } } ) l ^ { \\frac { c } { 2 } } \\mathcal { L } _ { s , - \\frac { c } { 2 } } ( y b ^ 2 ) , \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} B = \\begin{bmatrix} 0 & B ' \\\\ B '' & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} S ^ { R , R } T S ^ { R , R } = \\tau ^ { - } ( \\mathcal { C } ) T ^ { - 1 } S ^ { R , R } T ^ { - 1 } . \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} R ^ k = \\left [ 3 ^ k , 3 ^ k + 1 / 2 \\right ] L ^ k = \\left [ - 3 ^ k - 1 , - 3 ^ k \\right ] . \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} u ( x ) = | x | ^ p m _ 1 \\left ( \\frac { x } { | x | } \\right ) , x \\in K . \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} f ( 0 ) = { } & g ( J _ 1 ( 0 ) , J _ 2 ( 0 ) ) = 0 , \\\\ f ' ( 0 ) = { } & g ( J ' _ 1 ( 0 ) , J _ 2 ( 0 ) ) + g ( J _ 1 ( 0 ) , J _ 2 ' ( 0 ) ) = g ( \\gamma ' _ 1 ( 0 ) , W _ 2 ' ( 0 ) ) = 0 , \\\\ f '' ( 0 ) = { } & 2 g ( J _ 1 ' ( 0 ) , J _ 2 ' ( 0 ) ) = 0 { } & . \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} \\bar { \\alpha } = \\frac { 1 } { 2 \\sqrt { 2 } \\delta ( v ) } \\int _ { 2 \\sqrt { 2 } \\delta ( v ) } ^ { \\sqrt { 2 } \\delta ( v ) } \\rho ^ { \\prime } ( \\sigma ) d \\sigma = \\frac { 1 } { \\sqrt { 2 } \\delta ( v ) } \\int _ { \\sqrt { 2 } \\delta ( v ) } ^ { 2 \\sqrt { 2 } \\delta ( v ) } \\frac { d \\sigma } { \\psi ^ { \\prime \\prime } \\left ( \\rho ( \\sqrt { 2 } \\sigma ) \\right ) } , \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} \\langle \\psi \\mid T \\phi \\rangle = \\langle T ^ { \\dagger } \\psi \\mid \\phi \\rangle ; \\quad ~ ~ ~ \\phi , \\psi \\in V _ \\mathbb { H } ^ L . \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\langle \\epsilon ( h ) , U ^ * G _ A ( \\xi ) U \\epsilon ( f ) \\rangle = \\langle U \\epsilon ( h ) , G ( \\xi ) U \\epsilon ( f ) \\rangle = \\langle \\epsilon ( h ) , H _ \\mu ( \\xi ) \\epsilon ( f ) \\rangle \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } v _ T ( y _ 0 ) = k ^ * ( y _ 0 ) = d ^ * ( y _ 0 ) = \\tilde { k } ^ * ( z ) \\ \\ \\ \\ \\ \\forall y _ 0 \\in Y _ z , \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} \\| f \\| _ \\Phi = \\sup \\left \\{ \\int _ G | f ( s ) v ( s ) | \\ , d s : \\int _ G \\Psi ( | v ( s ) | ) \\ , d s \\le 1 \\right \\} , \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} \\mathcal { J } p _ n ( x ) = ( x - s ^ { + } ) p _ n ( x ) , x \\in X . \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta K _ \\gamma ^ \\pm & = 1 , \\mbox { i n } \\ \\quad \\Omega ^ \\pm , \\\\ \\partial _ n K _ \\gamma & = 0 \\mbox { o n } \\ \\quad \\partial \\Omega ^ \\pm \\cap \\partial \\Omega . \\end{aligned} \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} T ( U , V ) = \\alpha \\{ u ( V ) U - u ( U ) V \\} + \\beta \\{ u ( V ) \\varphi U - u ( U ) \\varphi Y \\} , \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} \\Vert f ' _ h - f ' \\Vert _ 1 & = \\int _ { - \\infty } ^ \\infty | g ' ( t ) | \\ , \\mathrm { d } t = \\sum _ { k = 1 } ^ M \\int _ { t _ { k - 1 } } ^ { t _ { k } } | g ' ( t ) | \\ , \\mathrm { d } t = \\sum _ { k = 1 } ^ M \\big | g ( t _ k ) - g ( t _ { k - 1 } ) \\big | \\\\ & = | g ( t _ 1 ) - g ( - \\infty ) | + | g ( \\infty ) - g ( t _ { M - 1 } ) | + \\sum _ { k = 2 } ^ { M - 1 } \\big | g ( t _ k ) - g ( t _ { k - 1 } ) \\big | , \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} ( H \\otimes K _ 1 ) ^ \\top ( H \\otimes K _ 1 ) & = H ^ \\top H \\otimes K _ 1 ^ \\top K _ 1 = k I _ k \\otimes ( m I _ m - J _ m ) , \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} \\Phi ^ * ( F ) = ( F \\circ \\Phi ) \\cdot ( \\Phi ' ) . \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} x ( t + 1 ) = A x ( t ) + B u ( t ) , \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} m _ n ( x , y ) > r = n ^ { - 1 / ( 1 - \\epsilon ) } . \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} e ' x e ' = E ( x ) e ' \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\Omega _ m \\ , = \\ , \\bigcup _ { ( s , p ) \\in \\mathbb I _ m } \\Omega _ m ( s , p ) . \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} \\mathcal C _ F ^ 2 = \\left \\{ C \\subseteq \\mathbb N ^ 2 : \\lim _ { n _ 1 \\wedge n _ 2 \\rightarrow \\infty } \\frac { \\# ( C \\cap ( [ n _ 1 ] \\times [ n _ 2 ] ) ) } { n _ 1 n _ 2 } \\right \\} \\ , , \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} \\mu ( x , \\theta , C ) & = \\delta _ \\Gamma ( x ) \\otimes \\delta ( \\theta - \\mathcal { t } ( x ) ) \\otimes \\delta ( C - \\bar C ) + \\eta ( x , \\theta ) \\otimes \\delta ( C ) \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} U ^ \\alpha { } _ \\beta = \\delta ^ \\alpha { } _ \\beta - \\frac 1 2 \\operatorname { R e } \\left [ i \\left ( p ^ \\alpha u _ \\beta - u ^ \\alpha p _ \\beta \\right ) e ^ { i p _ \\gamma x ^ \\gamma } \\ , \\right ] - \\frac { u _ \\gamma \\bar u ^ \\gamma } { 1 6 } p ^ \\alpha p _ \\beta \\ , , \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} \\beta _ { \\ell } = \\frac { 1 } { 2 } S _ { \\lambda \\mu _ { \\ell } } ( 2 \\alpha _ { \\ell } ) , \\ell = 0 , 1 , \\ldots , L , \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} f ( z _ k ) = \\prod _ { i = 1 } ^ { d } ( z _ k - m _ i ) \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} \\mu = \\frac { 1 } { \\log 2 } \\frac { d x } { 1 + x } \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} \\langle \\alpha _ { u _ { k , s } } , \\alpha _ { u _ { k , t } } ^ { \\vee } \\rangle = \\begin{cases} 2 & s = t , \\\\ - 1 & s - t = \\pm 1 , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} P _ { - n } ( X , \\beta ) \\times J _ C \\stackrel { \\cong } { \\to } M _ 1 ^ { \\star } \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} \\mu _ i = \\sum _ { j = 1 } ^ { i - 1 } k _ { j - 1 } ( f ) - \\sum _ { j = 1 } ^ i f ( j ) \\eta _ i = f ( i + 1 ) - f ( i ) - 1 . \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} u _ { k + 1 } ( T _ { o p t } , x ) = \\dots = u _ { k + m } ( T _ { o p t } , x ) = 0 \\mbox { f o r } x \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} P _ { i , \\epsilon } ( z ) = \\frac { u ^ { 2 i + 1 - \\epsilon } - 1 } { u ^ { i - \\epsilon } ( u + 1 ) ^ { \\epsilon } ( u - 1 ) } . \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\omega _ X g ( \\omega , \\cdot ) = 0 , ~ ~ ~ \\mathbb { P } - a . s . ~ \\omega . \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} 2 \\int _ 0 ^ { t ' } ( \\mathcal { L } \\omega _ \\eta ) ( x ( s ) ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } \\Big [ M ( t ' ) + \\overline { M } ( t ) - \\overline { M } ( t - t ' ) \\Big ] = 0 \\ , , \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} Q _ { \\nu , \\ell } ( R , M ) \\lesssim \\begin{cases} R ^ { 2 \\nu - n + 3 } M ^ { - n } \\| b _ { \\nu , \\ell } ( \\rho ) \\chi ( \\frac { \\rho } M ) \\rho ^ { \\frac { n - 1 } 2 } \\| ^ 2 _ { L ^ 2 } , ~ R \\lesssim 1 ; \\\\ R ^ { - ( n - 2 ) } M ^ { - n } \\| b _ { \\nu , \\ell } ( \\rho ) \\chi ( \\frac { \\rho } M ) \\rho ^ { \\frac { n - 1 } 2 } \\| ^ 2 _ { L ^ 2 } , ~ R \\gg 1 . \\end{cases} \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} u ^ q + \\frac { u } { \\beta ^ { q - 1 } } + ( \\frac { T r _ { q ^ 3 | q } ( \\beta ) } { \\beta ^ q } + 1 ) = 0 . \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} \\int _ { S } \\langle \\beta _ { \\tilde g ^ { ( 4 ) } } h , Y \\rangle _ { \\tilde g ^ { ( 4 ) } } \\cdot u \\cdot d v o l _ { g _ S } = \\int _ { S } \\langle h , \\delta _ { \\tilde g ^ { ( 4 ) } } ^ * Y + \\frac { 1 } { 2 } ( \\delta _ { \\tilde g ^ { ( 4 ) } } Y ) \\tilde g ^ { ( 4 ) } \\rangle _ { \\tilde g ^ { ( 4 ) } } u \\cdot d v o l _ { g _ S } . \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} \\dim F _ C = \\max \\{ \\dim F _ \\emptyset , \\dim C \\} , \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} \\chi ( S _ 1 \\cup S _ 2 ) = \\chi ( S _ 1 ) + \\chi ( S _ 2 ) - \\chi ( S _ 1 \\cap S _ 2 ) \\ , . \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} | | X _ { R } | | _ { s , r , p - 1 , \\mathbf { a } ; \\Xi _ r } = O ( r ^ { \\frac 7 4 } ) . \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} \\hat { q } ( i ) = \\sum _ { j = 0 } ^ { n } \\hat { \\sigma } ( i - j ) \\hat { p } ( j ) \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} \\mathbb { E } [ X ] = \\lim _ { n \\rightarrow \\infty } \\sum _ { i = 1 } ^ { n } \\mathbb { E } \\left [ X _ i \\right ] , \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} S ( x ) = \\delta ( x - { x ^ + } ) - \\delta ( x - { x ^ - } ) . \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} \\dim ( F ^ + \\cap F ^ - ) \\ge \\dim B - ( b - 1 ) - ( a - 1 ) = g - 1 . \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} ( f _ n , \\ , f _ n ) & = \\sum _ { k = 0 } ^ n \\sum _ { x \\in \\partial B _ G ( k ) } g ( k ) ^ 2 \\mu ( x ) = \\sum _ { k = 0 } ^ n \\sum _ { x \\in \\partial B _ G ( k ) } g ( k ) ^ 2 \\ , ( d _ x ^ + + d _ x ^ 0 + \\lambda d _ x ^ - ) \\ , \\lambda ^ { - | x | } \\\\ & \\ge ( \\lambda \\wedge 1 ) d \\ , \\sum _ { k = 0 } ^ n M _ k \\ , g ( k ) ^ 2 \\ , \\lambda ^ { - k } . \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} \\sum _ { i \\in P } | \\{ j \\in [ 1 , N ] : a _ { i , j } = a _ { 1 , j } \\} | \\ell + | \\{ j \\in [ 1 , N ] : a _ { i , j } \\neq a _ { 1 , j } \\} | \\dfrac { \\ell } { s } \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\frac { | \\Delta _ n | } { | \\Delta _ 0 | } = \\prod _ { m = 1 } ^ n \\mathrm { T } ^ m ( \\theta ) \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } - \\tfrac 1 t \\log \\mathbb P ( Z _ t > 0 ) & = \\kappa ( 1 - \\nu - \\nu \\log ( \\tfrac 1 \\nu ) ) . \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} 2 ( a - b ) a = a ^ 2 - b ^ 2 + ( a - b ) ^ 2 \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} Q _ { 1 2 } = 1 , Q _ { 3 2 } = 1 , \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} ( \\mathbb { E } _ 1 ) _ { x _ 1 } \\times ( \\mathbb { E } _ 2 ) _ { x _ 2 } = ( \\mathbb { E } _ 1 \\times \\mathbb { E } _ 2 ) _ { ( x _ 1 , x _ 2 ) } . \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} p _ 1 ^ 2 ( s ) \\prod _ { j = 1 } ^ { d - 1 } \\left ( s - \\frac 1 { \\alpha _ j } \\right ) - s \\prod _ { j = 1 } ^ d \\left ( s - \\frac 1 { a _ j } \\right ) = c . \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} C _ 1 \\oplus C _ 2 = \\{ ( c _ 1 , c _ 2 ) \\mid c _ 1 \\in C _ 1 , c _ 2 \\in C _ 2 \\} \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} \\N = F ( Q + v ) - F ( Q ) - i D Q ( \\dot z + i E z ) . \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\frac { \\# R _ N } { \\# S _ N } = \\frac { \\# R ' _ N } { \\# S ' _ N } \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} P _ I = \\bigcup _ { w \\in W _ I } B w B = \\overline { B w _ I B } \\subset G , \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} \\mathbf { p } _ i ^ { [ k + 1 ] } = \\underset { \\mathbf { p } _ i } { } \\ , f _ i ( \\mathbf { p } _ i , k + 1 ) + \\boldsymbol { \\lambda } _ i ^ { [ k ] T } \\mathbf { p } _ i \\\\ + \\frac { \\rho } { 2 } \\| \\mathbf { p } _ i - \\mathbf { q } _ i ^ { [ k ] } \\| ^ 2 \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} \\mathcal D = \\{ \\Phi _ { ( \\psi , f ) } \\in \\mathcal D _ 4 : \\mathbf n _ g ( f ) = 0 \\partial M \\} . \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} \\phi ( \\cdot , x ) = \\phi ( \\cdot , \\hat { x } ) + \\hat { z } . \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} \\widehat { \\alpha } = \\arg { \\underset { \\alpha } { \\min } \\left \\{ q ( \\alpha ) \\right \\} } , \\end{align*}"}
-{"id": "9912.png", "formula": "\\begin{align*} - \\frac 1 { 2 \\pi \\sqrt { - 1 } } \\frac 1 z = - \\frac 1 { 2 \\pi \\sqrt { - 1 } } \\frac { \\psi _ { + } - \\psi _ { - } } z , \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} E ^ { ( k ) } _ { a , b } ( \\tau ) = a _ 0 ( E ^ { ( k ) } _ { a , b } ) + \\sum _ { \\substack { m , n \\geq 1 \\\\ m \\equiv a ( N ) } } \\zeta _ N ^ { b n } n ^ { k - 1 } q ^ { m n / N } + ( - 1 ) ^ k \\sum _ { \\substack { m , n \\geq 1 \\\\ m \\equiv - a ( N ) } } \\zeta _ N ^ { - b n } n ^ { k - 1 } q ^ { m n / N } ( q = e ^ { 2 \\pi i \\tau } ) . \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} \\mathbf { w } _ t ( r , \\ , \\theta ) : = \\left ( \\cos ( t \\psi ( r ) ) \\cos ( j \\theta ) , \\ , \\cos ( t \\psi ( r ) ) \\sin ( j \\theta ) , \\ , \\sin ( t \\psi ( r ) ) , \\ , 0 , \\ , \\ldots , \\ , 0 \\right ) \\ ! . \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} ( \\partial ^ { p , q } _ { { \\rm { h o r } } } \\sigma ) ( i _ 0 , \\dots , i _ { p + 1 } , j _ 0 , \\dots , j _ q ) = \\sum _ { r = 0 } ^ { p + 1 } ( - 1 ) ^ r \\sigma ( i _ 0 , \\dots , \\widehat { i _ r } , \\dots , i _ { p + 1 } , j _ 0 , \\dots , j _ q ) | _ { U _ { i _ 0 \\dots i _ { p + 1 } j _ 0 \\dots j _ q } } , \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} \\mathrm { s p } _ S ( T ^ { - 1 } , [ \\phi w ' ] ) = \\mathrm { s p } _ S ( T ^ { - 1 } , [ \\phi w ] ) - 1 . \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} \\begin{array} { c c } a ^ 2 = \\sum a _ i ^ 2 & \\sum a _ i = - 3 a - 2 . \\end{array} \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} e _ { \\mathbb { Q } } ( \\psi ) e _ { \\mathbb { Q } } ( \\psi ) ^ { g } = e _ { \\mathbb { Q } } ( \\psi ) e _ { \\mathbb { Q } } ( \\varphi ) e _ { \\mathbb { Q } } ( \\varphi ) ^ { g } e _ { \\mathbb { Q } } ( \\psi ) ^ { g } . \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 1 + \\mu ( u ^ 3 ) ^ 2 & - 2 \\beta u ^ 3 & \\beta u ^ 2 - \\mu u ^ 1 u ^ 3 \\\\ - 2 \\beta u ^ 3 & \\mu & \\beta u ^ 1 \\\\ \\beta u ^ 2 - \\mu u ^ 1 u ^ 3 & \\beta u ^ 1 & \\mu ( u ^ 1 ) ^ 2 \\end{pmatrix} , \\\\ w _ { 1 2 } = 0 , w _ { 2 3 } = \\sqrt { \\frac { \\beta ^ 2 ( \\beta ^ 2 - \\mu ^ 2 ) } { \\det g } } u ^ 1 , w _ { 3 1 } = \\sqrt { \\frac { \\beta ^ 2 ( \\beta ^ 2 - \\mu ^ 2 ) } { \\det g } } u ^ 2 , \\end{gather*}"}
-{"id": "2975.png", "formula": "\\begin{align*} d _ M = \\liminf _ { n \\rightarrow \\infty } E ( v _ n ) & \\geq \\liminf _ { n \\rightarrow \\infty } \\left ( E ( \\tilde { V } ^ { j _ 0 } ) + E ( \\tilde { v } ^ { j _ 0 } _ n ) \\right ) \\\\ & \\geq E ( \\tilde { V } ^ { j _ 0 } ) + \\liminf _ { n \\rightarrow \\infty } E ( \\tilde { v } ^ { j _ 0 } _ n ) \\\\ & \\geq E ( \\tilde { V } ^ { j _ 0 } ) \\geq d _ M . \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} E _ 0 ^ { ( r ; [ - L _ 1 , L _ 2 ] ) } T _ a = \\begin{cases} \\frac { v _ { + , a ; - L _ 1 , L _ 2 } ( 0 ) } { u _ { + , a ; - L _ 1 , L _ 2 } ( 0 ) } , \\ 0 < a \\le L _ 2 ; \\\\ \\frac { v _ { - , a ; - L _ 1 , L _ 2 } ( 0 ) } { u _ { - , a ; - L _ 1 , L _ 2 } ( 0 ) } , \\ - L _ 1 \\le a < 0 . \\end{cases} \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} \\tau ( x ) : = \\inf \\{ n : g _ { | n } ( x ) > \\lambda \\} \\quad ( \\inf \\emptyset : = \\infty ) \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} f _ { r r } + ( \\frac { m - 1 } { r } - \\frac { r } { 2 } ) f _ r = ( - \\lambda e ^ { - \\frac { r ^ 2 } { 2 ( m - 2 ) } } + r ^ { - 2 } \\lambda _ k ) f . \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} - \\Delta _ { g } \\bigl ( h _ { 2 , \\lambda } ( r ) \\bigr ) = - \\left \\{ \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + ( n - 1 ) S ( r ) \\frac { \\partial } { \\partial r } \\right \\} h _ { 2 , \\lambda } ( r ) = \\left ( \\frac { ( n - 1 ) ^ 2 } { 4 } { K _ 0 } + \\lambda \\right ) h _ { 2 , \\lambda } ( r ) \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} \\| K _ t ( \\cdot , n ) \\| _ { \\ell ^ 2 ( \\mathbb { N } ) } = \\| e ^ { ( \\cdot - s ^ { + } ) t } p _ n \\| _ { L ^ 2 ( X , \\mu ) } \\le \\| p _ n \\| _ { L ^ 2 ( X , \\mu ) } = 1 . \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} \\mathcal { F } _ { D , 0 } ( z , s ) : = 2 ^ { s - 1 } \\pi ^ { - \\frac { s + 1 } { 2 } } | D | ^ { \\frac { s } { 2 } } L _ D ( s ) \\Gamma \\biggl ( \\frac { s + 1 } { 2 } \\biggr ) P _ { \\frac { 1 } { 2 } , 0 } \\biggl ( z , \\frac { s } { 2 } + \\frac { 1 } { 4 } \\biggr ) . \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( \\partial _ t - \\partial _ x ^ 2 - q ( x ) ^ 2 \\partial _ y ^ 2 ) f ( t , x , y ) = \\mathbf 1 _ \\omega u ( t , x , y ) & t \\in [ 0 , T ] , ( x , y ) \\in \\Omega , \\\\ f ( t , x , y ) = 0 & t \\in [ 0 , T ] , ( x , y ) \\in \\partial \\Omega , \\\\ f ( 0 , x , y ) = f _ 0 ( x , y ) & ( x , y ) \\in \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} \\eta ( t ) ~ \\doteq ~ \\begin{cases} \\delta \\quad & \\hbox { i f } t \\in [ a _ i , b _ i ] , ~ ~ i = 1 , \\ldots , N , \\\\ L & \\hbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} S & : H \\longmapsto H \\\\ S f & = T T ^ { * } f = \\sum _ { i \\in I } \\langle f , f _ { i } \\rangle f _ { i } . \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} \\alpha '' _ 3 x _ 1 ^ 2 + \\alpha '' _ 4 \\cdot 2 x _ 1 x _ 2 = 0 . \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} & \\lim _ { l \\to \\infty } \\nu _ { \\mathcal P _ l } \\left ( \\bigg \\{ \\frac { 1 } { k _ l ( n _ l + M _ l ) } \\big ( S ^ { k _ l ( n _ l + M _ l ) } h - \\mathbb { E } _ { \\mathcal P _ { l } } ( S ^ { k _ l ( n _ l + M _ l ) } h ) \\big ) \\leqslant k _ l ^ { - 1 / 2 + \\eta } \\bigg \\} \\right ) \\\\ & = \\lim _ { l \\to \\infty } \\mathcal N ( k _ l ^ { 1 / 2 + \\eta } ( n _ l + M _ l ) / s _ l ) = 1 \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} f _ { R } ( x ) = \\begin{cases} | x | ^ { - ( Q + \\lambda ) } , \\ ; 1 \\leq | x | \\leq R , \\\\ 0 , , \\end{cases} \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\P ^ 1 _ Q = \\P ( \\ker \\varphi _ Q ) . \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} | \\mathcal { B } _ { a } ( x ) | \\leq \\begin{cases} C | x | ^ { - ( Q - a ) } , \\ ; x \\in \\mathbb { G } \\backslash \\{ 0 \\} , \\\\ C | x | ^ { - Q } , \\ ; x \\in \\mathbb { G } \\ ; \\ ; | x | \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} J _ n = \\frac { \\partial F } { \\partial u } ( u _ n ) \\quad N _ n ( u ) = F ( u ) - J _ n u \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} & z ^ { - \\frac { 1 - 2 s } { s } } ( w '' ( z ) , w ( z ) ) _ H \\\\ & \\qquad \\ge ( \\delta v _ \\delta ( z ) - \\hat \\delta v _ { \\hat \\delta } ( z ) , v _ \\delta ( z ) - v _ { \\hat \\delta } ( z ) ) _ H \\\\ & \\qquad = \\delta \\norm { v _ \\delta ( z ) } ^ 2 _ H - \\delta \\ , ( v _ \\delta , v _ { \\hat \\delta } ( z ) ) _ { H } - \\hat \\delta \\ , ( v _ \\delta ( z ) , v _ { \\hat \\delta } ( z ) ) _ { H } + \\hat { \\delta } \\ , \\norm { v _ \\delta } ^ 2 _ H . \\end{align*}"}
-{"id": "9995.png", "formula": "\\begin{align*} \\frac d { d t } ( e ^ { - q \\tau ( t ) } \\bar F _ t ) = ( 1 - \\bar P ( q ) ) e ^ { - q \\tau ( t ) } , \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} \\lambda y + ( 1 - \\lambda ) y '' = Q _ { k _ j } c ' + b _ { k _ j } + A P _ j ( \\lambda \\mu ( c , y ) + ( 1 - \\lambda ) \\mu ( c '' , y '' ) ) . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} \\hat { e } = \\max _ { \\delta \\in G ^ * } \\frac { 1 } { m } \\left ( \\frac { 1 } { k _ 1 } N _ 1 ( \\delta ) + \\frac { 1 } { k _ 2 } N _ 2 ( \\delta ) + \\dotsb + \\frac { 1 } { k _ m } N _ m ( \\delta ) \\right ) , \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} K _ { i j } ( x , 0 ) = \\sum _ { ( r , s ) \\not \\in { \\cal J } } c _ { i j r s } ( B ) K _ { r s } ( x , 0 ) \\mbox { f o r } x \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} U _ { n 2 } & = u _ n + \\tfrac { 3 } { 4 } \\Delta t \\varphi _ { 1 } ( \\tfrac { 3 } { 4 } \\Delta t J _ n ) F ( u _ n ) , \\\\ u _ { n + 1 } & = u _ n + \\Delta t \\varphi _ { 1 } ( \\Delta t J _ n ) F ( u _ n ) \\\\ & + \\Delta t \\tfrac { 3 2 } { 9 } \\varphi _ 3 ( \\Delta t J _ n ) ( N _ n ( U _ { n 2 } ) - N _ n ( u _ n ) ) . \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} C _ 0 = \\frac { 1 } { \\eta } \\ , \\sum _ { j = 1 } ^ 3 \\| A _ j \\| _ { L ^ \\infty ( \\Omega ) } . \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} F ( x _ 1 , . . . , x _ \\ell ) = \\sum _ { i = 1 } ^ \\ell F _ i ( x _ 1 , . . . , x _ i ) \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} \\xi _ n \\to 0 \\ ; L ^ 1 _ { l o c } ( \\Omega ) \\ ; \\ ; \\ ; \\ ; T _ { \\vec { { \\bf f } } } ( \\xi _ n ) : = \\vec { { \\bf g } } _ n \\to \\vec { { \\bf g } } \\ ; ( L ^ 1 ( \\Omega ) ) ^ N . \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{gather*} \\overline { \\nabla } ^ { \\mathrm { b a s } , \\overline { g } } _ { \\mathrm { h o r } ( a ) } \\mathrm { h o r } ( b ) = \\mathrm { h o r } \\big ( \\nabla ^ { \\mathrm { b a s } , g } _ { a } b \\big ) , \\end{gather*}"}
-{"id": "2918.png", "formula": "\\begin{align*} \\frac { B ( \\tau _ k ) - B ( t _ k ) } { \\tau _ k - t _ k } \\le \\frac { 2 B ( \\tau _ k ) } { \\tau _ k } = \\frac { 2 D ( t _ k ) } { t _ k } \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} - \\log _ { q } \\frac { | x | } { R } = - \\frac { \\left ( \\frac { | x | } { R } \\right ) ^ { 1 - q } - 1 } { 1 - q } = \\left ( \\frac { | x | } { R } \\right ) ^ { 1 - q } \\frac { \\left ( \\frac { R } { | x | } \\right ) ^ { 1 - q } - 1 } { 1 - q } = \\left ( \\frac { | x | } { R } \\right ) ^ { 1 - q } \\left ( \\log _ { q } \\frac { R } { | x | } \\right ) \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} ( J u ) ( n ) = ( a _ { n + 1 } + a ' _ { n + 1 } ) u ( { n + 1 } ) + ( a _ n + a _ n ' ) u ( { n - 1 } ) + ( b _ { n + 1 } + b ' _ { n + 1 } ) u ( n ) = E u ( n ) , n \\geq 0 , \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} P _ { i , \\epsilon } ( z ) = ( z - 2 ) P _ { i - 1 , \\epsilon } ( z ) - P _ { i - 2 , \\epsilon } ( z ) \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} G _ { m + n } - ( - 1 ) ^ n G _ { m - n } = F _ n ( G _ { m - 1 } + G _ { m + 1 } ) \\ , , \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} L _ { 2 1 } & = ( \\Delta ' ) ^ { - 1 } ( [ a , c ] a ^ { - 1 } ( a c ^ { - 1 } d - b ) - ( [ b , c ] - [ a , c ] a ^ { - 1 } b ) ) ( a c ^ { - 1 } d - b ) ^ { - 1 } \\\\ & = ( \\Delta ' ) ^ { - 1 } ( [ a , c ] c ^ { - 1 } d - [ b , c ] ) ( a c ^ { - 1 } d - b ) ^ { - 1 } \\\\ & = ( ( a c ^ { - 1 } d - b ) ^ { - 1 } - ( \\Delta ' ) ^ { - 1 } c ) . \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} \\mathbf { F } = b _ { 0 + } ^ { - 1 } ( \\tilde { F _ 0 } \\times \\tilde { F } _ 1 \\times \\cdots \\times \\tilde { F } _ { m ^ + } ) . \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{gather*} \\overset { I } { A } _ 1 \\overset { J } { B } _ 2 = \\overset { I } { S ^ { - 1 } ( a _ i ) } _ 2 \\overset { I J } { ( R ' ) } _ { 1 2 } \\overset { J } { B } _ 2 \\overset { I J } { R } _ { 1 2 } \\overset { I } { A } _ 1 \\overset { I J } { ( R ' ) } _ { 1 2 } \\overset { J } { ( b _ i ) } _ 1 , \\end{gather*}"}
-{"id": "3112.png", "formula": "\\begin{align*} \\mathcal { I } _ { 2 , \\beta _ 1 , \\beta _ 2 , \\beta _ 3 } \\ ! \\ ! = \\ ! \\ ! 3 \\ ! \\ ! \\int _ { [ 0 , 1 ] ^ 2 } \\ ! \\check { f } _ { \\mathbf { t } } ( x , y ) ^ s x ^ { 3 \\beta _ 1 + 4 \\beta _ 2 + 1 2 \\beta _ 3 + 2 4 s } y ^ { 3 \\beta _ 1 + 4 \\beta _ 2 + 1 3 \\beta _ 3 + 1 + 2 6 s } q ( y ) \\frac { d x } { x } \\frac { d y } { y } , \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} ( u , v ) : = E _ \\nu \\left ( \\sum _ { b \\in \\hat { E } } \\omega _ b u ( \\omega , b ) \\cdot v ( \\omega , b ) \\right ) . \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} & \\left \\| \\frac { 1 } { r } \\left ( \\sum _ { i = 1 } ^ r { h ( x _ i ) \\rho _ { ( x _ i s ) ^ { - 1 } } ( f ) } \\right ) - \\frac { 1 } { r } \\left ( \\sum _ { i = 1 } ^ r { h ( y _ i ) \\rho _ { ( y _ i t ) ^ { - 1 } } ( f ) } \\right ) \\right \\| _ { L _ p ( m _ { X A S } ) } \\\\ & \\leq \\frac { 1 } { 6 } \\epsilon \\| \\nu \\| \\sup { \\left \\{ \\left \\| \\rho _ { ( a s ' ) ^ { - 1 } } ( f ) \\right \\| _ { L _ p ( m _ { X A S } ) } : s ' \\in S , a \\in A \\right \\} } \\leq \\frac { 1 } { 6 } \\epsilon \\| \\nu \\| \\| f \\| _ { L _ p ( m _ { X A S } ) } \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} U ( s ) = \\int _ { \\wp } s ^ { Q - 1 } \\left ( f ( s y ) ( \\psi _ { 1 } ( s y ) ) ^ { 1 / p } g ( s ) \\right ) ^ { p } d \\sigma ( y ) , \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} \\mathbf { M } _ { \\mathbf { m } , \\{ \\phi _ { k } \\} , \\{ \\psi _ { k } \\} } ( h ) = \\sum _ { k \\in I } m _ { k } \\left \\langle h | \\psi _ { k } \\right \\rangle \\phi _ { k } ; h \\in U _ { \\mathbb { H } } ^ { L } . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} \\| g \\| _ { B ^ { \\alpha } ( \\mathbb { R } \\times \\wp ) } : = \\sup _ { t > 0 } \\{ t ^ { - \\alpha / 2 } \\| F e ^ { - t A ^ { 2 } } F ^ { - 1 } f \\| _ { L ^ { \\infty } ( \\mathbb { R } \\times \\wp ) } \\} < \\infty . \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} F [ \\widetilde Q ] = 3 \\left ( ( 1 + q ) ^ 2 + ( 1 - q ) ^ 2 + q ^ 2 \\right ) = 6 + 3 q ^ 2 > F [ Q ] . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{gather*} \\langle s _ k , s _ o \\rangle = \\langle s _ { k o } , s _ k \\rangle = \\langle s _ { o } , s _ { k o } \\rangle = ^ { \\perp } \\langle s _ i , s _ j \\rangle = \\langle s _ { i o } , s _ { j o } \\rangle ^ { \\perp } \\end{gather*}"}
-{"id": "8983.png", "formula": "\\begin{align*} \\dot { x } _ t = \\tilde { f } ( x _ t , \\theta , \\P _ { x _ t } ) . \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} \\left \\langle f , R _ { \\mathcal { S } _ p ( 1 ) } ( e _ { \\zeta } \\circ D _ t ) \\right \\rangle _ { \\mathcal { S } _ p ( 1 ) } = D _ t f ( \\zeta ) = f ( \\widehat { \\rho } _ t ( \\zeta ) ) t ^ { 1 / 2 \\mu } = \\left \\langle f , t ^ { 1 / 2 \\mu } R _ { \\mathcal { S } _ p ( 1 ) } ( e _ { \\widehat { \\rho } _ t ( \\zeta ) } ) \\right \\rangle _ { \\mathcal { S } _ p ( 1 ) } . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} \\lvert ( i , j ) \\rvert _ { W _ n ( w ) } = \\lvert u \\rvert _ S = \\lvert l \\rvert _ S + \\lvert m \\rvert _ S + \\lvert r \\rvert _ S . \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} T _ S f ( z , w ) = \\int _ 0 ^ { \\infty } f ( t , w ) e ^ { i 2 \\pi z t } \\d t , \\textrm { f o r } ( z , w ) \\in \\mathcal { U } _ p \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{gather*} f _ 1 ( \\tau ) = \\sum _ { n = 1 } ^ \\infty a _ 1 ( n ) q ^ n = \\eta ( 4 \\tau ) ^ 6 = q \\prod _ { m = 1 } ^ \\infty \\big ( 1 - q ^ { 4 m } \\big ) ^ 6 , \\end{gather*}"}
-{"id": "6067.png", "formula": "\\begin{align*} b _ { 1 } ( s ) - b _ { 2 } ( s ) = \\tilde { \\beta } _ { 1 } ( s ) \\hat { Y } _ { s } + \\tilde { \\gamma } _ { 1 } ( s ) \\hat { Z } _ { s } , \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } Y _ { s } ^ { 1 , u } = Y _ { s } ^ { u } - \\varphi \\left ( s , X _ { s } ^ { u } \\right ) , \\ Z _ { s } ^ { 1 , u } { = Z _ { s } ^ { u } - { D \\varphi \\left ( s , X _ { s } ^ { u } \\right ) } ^ { \\intercal } \\sigma \\left ( s , X _ { s } ^ { u } , Y _ { s } ^ { u } , Z _ { s } ^ { u } , u _ { s } \\right ) . } \\end{array} \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\left \\langle \\frac { 1 } { ( \\zeta q ^ { x _ N ( t ) + N } ; q ) _ \\infty } \\right \\rangle = \\sum _ { l \\in \\mathbb { Z } } \\frac { 1 } { ( \\zeta q ^ l ; q ) _ { \\infty } } \\mathbb { P } [ x _ N ( t ) + N = l ] , \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} \\omega _ 0 : = \\sup _ { t \\in ( t _ 0 , T _ + ) } \\| u ( t ) \\| _ { W ^ { 1 , \\infty } ( G ) } < \\infty . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} \\rho = \\exp ( - h _ { \\mathrm { t o p } } ( f ) ) \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 2 4 ( 3 n + 1 ) + 1 2 ) q ^ n & \\equiv 2 f _ { 4 } ^ 3 - 2 q f _ { 1 2 } ^ 3 ~ ( \\textup { m o d } ~ 4 ) . \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} I H _ { \\mathcal S } ( t ) = \\frac { P _ { j } } { P _ { j - 1 } } \\cdot \\frac { P _ { l - j + 1 } } { P _ { k - j + 1 } P _ { l - k } } - \\left ( t ^ { 2 ( l - k ) } + t ^ { 2 ( l - k + 1 ) } + \\dots + t ^ { 2 ( j - 1 ) } \\right ) \\cdot \\frac { P _ { l - j } } { P _ { k - j } P _ { l - k } } , \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} F _ { 1 / 2 , 0 , 0 } ( z ) & = \\mathrm { L C } _ { s = 3 / 4 } ^ 0 [ P _ { 1 / 2 , 0 } ( z , s ) ] , \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} \\mu = \\zeta ^ 1 \\wedge \\zeta ^ 2 \\otimes Z _ 3 + \\bar \\zeta ^ 1 \\wedge \\bar \\zeta ^ 2 \\otimes \\bar Z _ 3 \\ , , \\end{align*}"}
-{"id": "9996.png", "formula": "\\begin{align*} \\bar F _ t ( q ) e ^ { - q \\tau ( t ) } - \\bar F _ 0 ( q ) = ( 1 - \\bar P ( q ) ) \\int _ 0 ^ t e ^ { - q \\tau ( s ) } d s . \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} \\sum _ { \\stackrel { ( \\{ \\{ g _ 1 , g _ 2 \\} , \\ldots , \\{ g _ { 2 r - 1 } , g _ { 2 r } \\} \\} , \\{ q _ 1 , \\ldots , q _ { k - 2 r } \\} ) } { { } _ { \\{ g _ 1 , g _ 2 , \\ldots , g _ { 2 r - 1 } , g _ { 2 r } , q _ 1 , \\ldots , q _ { k - 2 r } \\} = \\{ 1 , 2 , \\ldots , k \\} } } } a _ { g _ 1 g _ 2 , \\ldots , g _ { 2 r - 1 } g _ { 2 r } , q _ 1 \\ldots q _ { k - 2 r } } . \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{gather*} ( a ) _ n = \\frac { \\Gamma ( a + n ) } { \\Gamma ( a ) } = \\begin{cases} 1 & \\ n = 0 , \\\\ a ( a + 1 ) \\dotsb ( a + n - 1 ) & \\ n > 0 , \\\\ [ 1 m m ] \\dfrac { 1 } { ( a - 1 ) ( a - 2 ) \\dotsb ( a - ( - n ) ) } & \\ n < 0 . \\end{cases} \\end{gather*}"}
-{"id": "9197.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\mathrm { T } y ) ( t ) = \\sum _ { i = m } ^ { M } k _ i \\mu _ k ( t ) y ^ { ( i ) } ( t ) < 0 , . \\end{align*}"}
-{"id": "9971.png", "formula": "\\begin{align*} S _ \\alpha = S _ { \\alpha _ 1 } \\cap \\dots \\cap S _ { \\alpha _ k } . \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} \\begin{aligned} \\Theta _ j = & - \\mu _ j ^ { n - 2 s } \\left ( H ( x , q _ j ) - \\Phi _ j ^ * ( x , t ) \\right ) \\\\ & + \\sum _ { i \\neq j } \\left [ ( \\mu _ j \\mu _ i ^ { - 1 } ) ^ { \\frac { n - 2 s } { 2 } } U ( y _ i ) - ( \\mu _ j \\mu _ i ) ^ { \\frac { n - 2 s } { 2 } } \\left ( H ( x , q _ i ) - \\Phi _ i ^ * ( x , t ) \\right ) \\right ] . \\end{aligned} \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} \\varphi ( \\rho ) \\stackrel { \\mathrm { d e f } } { = } \\frac { 2 } { \\pi } \\left ( \\rho \\arcsin \\rho + \\sqrt { 1 - \\rho ^ { 2 } } - 1 \\right ) , \\rho \\in \\left [ - 1 , 1 \\right ] . \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} \\quad \\theta ( 0 ) = 0 , \\ \\theta ( 1 ) = 1 , \\theta ( 0 ) = 1 , \\ \\theta ( 1 ) = 0 . \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} L _ s = \\sum _ m \\lambda _ { \\pi _ 0 } ( m ) e \\left ( - \\frac { a \\overline { l ^ { \\frac { n _ l } { 2 } } } } { b } m \\right ) W _ { \\infty } \\left ( m \\right ) W _ l ( s ; m ) , \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} H ^ \\pm _ { C _ R } = a ^ \\pm + b ^ \\pm \\log r , \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} u ^ { * } ( x , t ) = \\limsup _ { \\delta \\to 0 ^ { + } } \\sup \\left \\{ U ^ { \\epsilon } ( m , s ) \\ , \\mid \\ , d ( - m \\Delta x , x ) + | t - s \\Delta t | + \\epsilon < \\delta \\right \\} , \\\\ u _ { * } ( x , t ) = \\liminf _ { \\delta \\to 0 ^ { + } } \\inf \\left \\{ U ^ { \\epsilon } ( m , s ) \\ , \\mid \\ , d ( - m \\Delta x , x ) + | t - s \\Delta t | + \\epsilon < \\delta \\right \\} . \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} s _ 2 = \\psi _ 4 , s _ 3 = \\psi _ 6 , s _ 5 = 2 ^ { 1 2 } 3 ^ 5 \\chi _ { 1 0 } , s _ 6 = 2 ^ { 1 2 } 3 ^ 6 \\chi _ { 1 2 } , \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} \\Gamma ( U ) d \\Gamma ( \\omega ) \\Gamma ( U ) ^ * & = d \\Gamma ( U \\omega U ^ * ) . \\\\ \\Gamma ( U ) W ( f , V ) \\Gamma ( U ) ^ * & = W ( U f , U V U ^ * ) . \\\\ \\Gamma ( U ) \\varphi ( f ) \\Gamma ( U ) ^ * & = \\varphi ( U f ) . \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} \\phi _ { B } ( \\textbf { Z } , \\delta ) = \\delta \\tilde { \\phi } _ 1 ( \\textbf { Z } ) + ( 1 - \\delta ) \\tilde { \\phi } _ 0 ( \\textbf { X } ) , \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} c ^ R ( x _ 1 , \\dots , x _ N ) = \\sum _ { 1 \\le i < j \\le N } \\min \\{ R , f ( d ( x _ i , x _ j ) \\} ) . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} \\eta _ n ( S ) = \\frac { | S \\cap K _ n | } { | K _ n | } , \\alpha _ n = | K _ n | \\Big ( \\inf _ { x , y \\in K _ n } p _ t ^ f ( x , y ) \\Big ) > 0 . \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} \\d X _ t ^ i = \\sqrt { 2 } \\ , \\d B _ t ^ i - n X _ t ^ i \\d t + \\beta \\sum _ { j \\colon j \\neq i } \\frac 1 { X ^ i _ t - X ^ j _ t } \\ , \\d t , 1 \\leq i \\leq n \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} \\Lambda ^ \\vee = \\{ \\varphi \\in \\overline { V } ^ \\vee \\mid \\langle \\varphi , \\lambda \\rangle \\in \\Z \\forall \\lambda \\in \\Lambda \\} . \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\Omega C = \\mathbf { k } \\oplus s ^ { - 1 } \\overline { C } \\oplus ( s ^ { - 1 } \\overline { C } \\otimes s ^ { - 1 } \\overline { C } ) \\oplus ( s ^ { - 1 } \\overline { C } \\otimes s ^ { - 1 } \\overline { C } \\otimes s ^ { - 1 } \\overline { C } ) \\oplus . . . \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{gather*} \\delta _ n ^ k + d - \\alpha ( k + 1 - d ) = \\delta _ n ^ k + d - \\alpha \\left ( \\left ( \\frac { k + 1 } { d } - 1 \\right ) d \\right ) = \\delta _ n ^ k + d - \\left ( \\frac { k + 1 } { d } - 1 \\right ) \\alpha \\left ( d \\right ) \\\\ = \\delta _ n ^ k + d - \\left ( \\frac { k + 1 } { d } - 1 \\right ) d \\frac { \\delta _ n ^ k } { k + 1 } = d + d \\frac { \\delta _ n ^ k } { k + 1 } = d + d \\frac { n - k + 1 } { k + 1 } = d \\frac { n + 2 } { k + 1 } . \\end{gather*}"}
-{"id": "4893.png", "formula": "\\begin{align*} I ( b ) : = \\frac { 1 } { n - 2 s } \\left [ \\sum _ { j = 1 } ^ k b _ j ^ { n - 2 s } H ( q _ j , q _ j ) - \\sum _ { i \\neq j } b _ j ^ { \\frac { n - 2 s } { 2 } } b _ i ^ { \\frac { n - 2 s } { 2 } } G ( q _ j , q _ i ) - \\sum _ { j = 1 } ^ k b _ j ^ { 2 s } \\right ] . \\end{align*}"}
-{"id": "9937.png", "formula": "\\begin{align*} b _ { \\Omega _ 1 \\times \\dots \\times \\Omega _ \\ell } ( f _ 1 f _ 2 \\cdots f _ \\ell ) = b _ { \\Omega _ 1 } ( f _ 1 ) \\times b _ { \\Omega _ 2 } ( f _ 2 ) \\times \\cdots \\times b _ { \\Omega _ \\ell } ( f _ \\ell ) . \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} H _ { i } ( x , p ) = H _ { i } ( p ) - f _ { i } ( x ) . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} \\mathcal { A } ( \\gamma , \\xi ) = ( 0 , { \\bf 0 } , { \\bf 0 } ) , \\ \\ \\ \\ \\langle ( k + 2 \\epsilon , M \\epsilon ) , ( \\gamma , \\xi ) \\rangle = 0 \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} x \\in K , \\ F ( x ) \\in K ^ * , \\ \\langle F ( x ) , x \\rangle = 0 . \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} v ^ { k } & = \\alpha _ k x ^ k + ( 1 - \\alpha _ k ) T x ^ k , \\\\ x ^ { k + 1 } & = \\beta _ k v ^ k + ( 1 - \\beta _ k ) T z ^ k , \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{gather*} \\Delta ( X ) = \\square ( X ) X = x _ { i } ^ { \\pm } , h _ i , \\\\ \\Delta \\big ( x _ { i , 1 } ^ + \\big ) = \\square ( x _ { i , 1 } ^ + ) - \\hbar \\big [ 1 \\otimes x _ i ^ + , \\Omega _ + \\big ] , \\\\ \\Delta \\big ( x _ { i , 1 } ^ - \\big ) = \\square ( x _ { i , 1 } ^ - ) + \\hbar \\big [ x _ i ^ - \\otimes 1 , \\Omega _ + \\big ] , \\\\ \\Delta \\big ( \\tilde { h } _ { i , 1 } \\big ) = \\square ( \\tilde { h } _ { i , 1 } ) + \\hbar [ h _ i \\otimes 1 , \\Omega _ + ] . \\end{gather*}"}
-{"id": "3073.png", "formula": "\\begin{align*} \\delta _ \\alpha [ f , g ] _ \\alpha = & ~ ( - 1 ) ^ { m + n } [ \\mu , [ f , g ] _ \\alpha ] _ \\alpha \\\\ = & ~ ( - 1 ) ^ { m + n } [ [ \\mu , f ] _ \\alpha , g ] _ \\alpha + ( - 1 ) ^ { n + 1 } [ f , [ \\mu , g ] _ \\alpha ] _ \\alpha \\\\ = & ~ ( - 1 ) ^ { n + 1 } [ \\delta _ \\alpha f , g ] _ \\alpha + [ f , \\delta _ \\alpha g ] _ \\alpha . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} ( T ^ 1 S ^ 1 f ) ( h ) ( \\nu ) & = \\sum _ i \\langle L ^ { * * } ( f ' ( g _ i ) ) , \\nu _ i \\rangle = \\sum _ i \\langle f ' ( g _ i ) , \\nu _ i L \\rangle \\\\ & = \\sum _ i \\sum _ j \\langle \\pi ^ { * * } ( f ( h _ j ) ) , ( \\nu _ i L ) _ j \\rangle \\\\ & \\quad \\mbox { ( H e r e , $ ( \\nu _ i L ) _ j : = ( \\nu _ i L ) | _ { X _ { g _ i , h _ j } } $ , $ X _ { g _ i , h _ j } = \\{ x : c ( g _ i ^ { - 1 } , x ) = h _ j ^ { - 1 } \\} . $ ) } \\\\ & = \\sum _ i \\sum _ j \\langle f ( h _ j ) , ( \\nu _ i L ) _ j \\pi \\rangle . \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} \\rho _ { t } ( { \\bf 1 } ) \\ , = \\ , 1 + \\int ^ { t } _ { 0 } \\rho _ { s , 2 } ( b ) { \\mathrm d } X _ { s } \\ , = \\ , 1 + \\int ^ { t } _ { 0 } \\rho _ { s } ( { \\bf 1 } ) \\pi _ { s , 2 } ( b ) { \\mathrm d } X _ { s } \\ , ; 0 \\le t \\le T \\ , . \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} \\langle x _ i , y _ i \\rangle = \\begin{cases} 1 \\\\ r \\end{cases} , \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} \\zeta _ k & = i \\frac { z _ k } { 1 + z _ n } , 1 \\leq k \\leq n - 1 , \\\\ \\zeta _ n & = i \\frac { 1 - z _ n } { 1 + z _ n } . \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} \\Pr [ Y _ { \\mathbf x , n , \\ell , m , j } = 0 ] = \\left ( 1 + \\frac { a _ { m } b _ { m , j } \\lambda _ u } { 3 . 5 ( \\sum _ { i = j } ^ L T _ { m , i } ) \\lambda _ b } \\right ) ^ { - 4 . 5 } . \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathbf { H } \\mathbf { x } + \\mathbf { n } , \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} I ( h ( \\alpha , \\beta ) ) = I ( \\alpha u ^ + + \\beta u ^ - ) < I ( u ) = c , \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{gather*} \\beta = j - i i = n - i + 2 j = n - i + 2 + \\beta \\end{gather*}"}
-{"id": "6893.png", "formula": "\\begin{align*} \\lambda ^ 2 e ^ { \\ , u } \\approx \\begin{cases} \\lambda ^ 2 \\exp \\left ( U \\left ( \\lambda \\mu _ \\lambda t \\right ) + 2 \\log \\mu _ \\lambda - \\log h _ \\gamma \\right ) , | t | < \\delta , \\medskip \\\\ \\lambda ^ 2 \\exp \\left ( ( \\beta + 2 \\log \\beta ) H _ \\gamma ^ \\pm + \\tilde H ^ \\pm \\right ) , \\mathop { d i s t } ( x , \\gamma ) > \\delta . \\end{cases} \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} S _ { n , p } \\left ( \\frac { n - p } { p - 1 } \\right ) ^ { \\frac 1 n - 1 } \\left ( \\int _ { B _ { R } } \\frac { | u ( x ) | ^ { p ^ { * } } } { \\left [ \\log _ { { \\frac { n - 1 } { p - 1 } } } \\frac { R } { | x | } \\right ] ^ { \\frac { p ( n - 1 ) } { n - p } } } d x \\right ) ^ { \\frac { 1 } { p ^ { * } } } \\le \\| \\nabla u \\| _ { L ^ p ( B _ R ) } . \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} c _ 1 ^ \\tau c _ 2 ^ \\tau ( x ) & = \\tau ( h _ 1 , h _ 2 g ) \\tau ( h _ 2 , g ) c _ 1 c _ 2 ( x ) \\\\ & = \\tau ( h _ 1 , h _ 2 ) \\tau ( h _ 1 h _ 2 , g ) c _ 1 c _ 2 ( x ) \\\\ & = \\tau ( h _ 1 , h _ 2 ) ( c _ 1 c _ 2 ) ^ \\tau ( x ) , \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} W ( f , 1 ) F _ { \\eta } ( v , \\omega ) W ( f , 1 ) ^ * = & \\eta W ( 2 f , - 1 ) + d \\Gamma ( \\omega ) + \\varphi ( v - \\omega f ) \\\\ & + \\lVert \\omega ^ { 1 / 2 } f \\lVert ^ 2 - 2 \\textup { R e } ( \\langle v , f \\rangle ) . \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} u = \\begin{pmatrix} \\sqrt b \\\\ - \\sqrt { - c } \\end{pmatrix} . \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { 1 } ( \\textbf { Z } ) = \\begin{cases} 1 p ( \\textbf { Z } , 1 ) P \\{ Y = 1 \\vert \\textbf { Z } \\} > p ( \\textbf { Z } , 0 ) P \\{ Y = 0 \\vert \\textbf { Z } \\} \\\\ 0 \\end{cases} \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} \\mbox { V a r } _ { \\mu } ( f ) = \\int _ { \\mathbb { R } ^ n } f ^ 2 \\ , d \\mu - \\Big ( \\int _ { \\mathbb { R } ^ n } f \\ , d \\mu \\Big ) ^ 2 \\le \\frac { 1 } { \\alpha } \\mathcal { E } ( f , f ) \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} \\wp : = \\{ x \\in \\mathbb { G } : \\ , | x | = 1 \\} , \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} H _ \\tau ( x _ { 1 , \\tau } , x _ { 2 , \\tau } , t _ \\tau , p _ { 1 , \\tau } , p _ { 2 , \\tau } , p _ \\tau ) = p _ { 1 , \\tau } \\dot { x } _ { 1 , \\tau } + p _ { 2 , \\tau } \\dot { x } _ { 2 , \\tau } + p _ \\tau \\dot { t } _ \\tau - L \\ ; , \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} h ( t ) ( g ( h ( t ) ) - \\log t ) & > \\frac { g ( h ( t ) ) } { g ( \\lambda h ( t ) ) } \\frac { t } { \\lambda } + h ( t ) \\log t \\left ( \\frac { g ( h ( t ) ) } { g ( \\lambda h ( t ) ) } - 1 \\right ) \\\\ & \\ge \\frac { g ( h ( t ) ) } { g ( \\lambda h ( t ) ) } \\frac { t } { \\lambda } + c t \\left ( \\frac { g ( h ( t ) ) } { g ( \\lambda h ( t ) ) } - 1 \\right ) , \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} \\sum _ { j \\in N ( i ) } \\widetilde Q _ { i j } = \\sum _ { j \\in N ( i ) } Q _ { i j } = S _ i . \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} a ' + b ' = - 2 b , a ' b ' = a ^ 3 + b ^ 2 . \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{gather*} \\nabla ^ { \\mathrm { a d } } _ a ( b , u ) : = ( [ a , b ] _ A , [ \\varrho _ A a , u ] ) \\end{gather*}"}
-{"id": "5092.png", "formula": "\\begin{align*} ( X _ { \\cdot , i } ^ { ( u ) } , X _ { \\cdot , i + 1 } ^ { ( u ) } ) \\ , = \\ , ( X _ { \\cdot , 1 } ^ { ( u ) } , X _ { \\cdot , 2 } ^ { ( u ) } ) \\ , ; i \\ , = \\ , 1 \\ldots , n - 1 \\ , , \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} \\langle k , \\omega ( \\xi ) \\rangle + \\langle l , \\Omega ( \\xi ) \\rangle = ( \\sum _ { b = 1 } ^ n k _ b j _ b ^ 2 + \\sum _ { j \\in \\mathbb { Z } _ * } l _ j j ^ 2 ) + \\sum _ { b = 1 } ^ n ( k _ b j _ b + \\sum _ { j \\in \\mathbb { Z } _ * } \\frac { l _ j j } { n - \\frac 1 2 } ) \\xi _ b . \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} \\lvert ( i , j ) \\rvert _ { W _ { n _ 1 } ( w ) } - \\lvert ( i , j ) \\rvert _ { W _ { n _ 2 } ( w ) } = \\lvert m _ { n _ 1 } \\rvert _ S - \\lvert m _ { n _ 2 } \\rvert _ S . \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\sum _ { i \\in N ( j ) } G _ i = S _ j . \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} \\| u \\| _ { ( \\kappa + 1 ) p ^ * } = \\| u ^ { \\kappa + 1 } \\| _ { p ^ * } ^ { { \\frac { 1 } { \\kappa + 1 } } } & \\leq M _ { 2 9 } ^ { \\frac { 1 } { \\kappa + 1 } } \\left ( ( \\kappa + 1 ) ^ { M _ { 2 8 } } \\right ) ^ { { \\frac { 1 } { \\kappa + 1 } } } \\left [ \\| u ^ { \\kappa + 1 } \\| _ { \\tilde { q } _ 1 } ^ p + 1 \\right ] ^ { \\frac { 1 } { ( \\kappa + 1 ) p } } . \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} & M \\left ( 1 + t \\ : h ( x ) + y t \\ : h ( x ) \\right ) = 1 + 1 2 t , \\\\ & M \\left ( 1 + x ( x ^ 3 + 1 ) + t \\ : h ( x ) + y \\left ( ( 1 + x ^ 3 ) + t \\ : h ( x ) \\right ) \\right ) = 5 + 1 2 t , \\\\ & M \\left ( 1 + t \\ ; h ( x ) + y \\left ( ( 1 + x ^ 3 ) + t \\ : h ( x ) \\right ) \\right ) = - 3 ^ 3 ( 1 + 4 t ) . \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{gather*} \\begin{array} { c } \\{ \\beta ( j + u ) = \\beta ( j ) + \\beta ( u ) \\} _ { j = 1 } ^ { k - u } \\mbox { a n d } \\ \\ \\{ \\beta ( j + v ) = \\beta ( j ) + \\beta ( v ) \\} _ { j = 1 } ^ { k - v } . \\end{array} \\end{gather*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\gamma ( y , t ) = \\int d x \\ c ( x , y , t ) \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} A _ \\cdot ^ { W , W } ( g ) = \\int _ 0 ^ \\cdot g ( s , W _ s ) \\mathrm d s . \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} \\partial _ s v ( s , t ) - \\rho ' ( s ) \\partial _ t v ( s , t ) + J _ { t + \\rho ( s ) } ( v ) ( \\partial _ t v ( s , t ) - X _ { H _ { t + \\rho ( s ) } ^ { ( k ) } } ( v ( s , t ) ) ) = 0 . \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} \\Gamma ( \\rho ) = \\left \\{ \\gamma \\in \\mathcal { P } ( X ^ N ) ~ \\big | ~ \\textrm { p r } ^ i _ \\sharp \\gamma = \\rho i \\right \\} , \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} H _ { N } ^ { ( \\alpha ) } \\ ; = \\ ; \\sum _ { i = 1 } ^ { N } S _ i + \\frac { \\mu \\lambda } { N - 1 } \\sum _ { i < j } \\frac 1 { | x _ i - x _ j | ^ \\gamma + \\alpha } , \\quad \\alpha > 0 . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} \\Phi _ \\delta ( t ) = \\int _ { 0 } ^ { t } \\Phi _ \\delta ' ( s ) d s & \\leq \\int _ { 0 } ^ { t } \\int _ { B _ R } \\frac { | \\mathbf { B } _ s ( X _ 1 ( s , x ) ) - \\mathbf { B } _ s ( X _ 2 ( s , x ) ) | } { \\delta + | X _ 1 ( s , x ) - X _ 2 ( s , x ) | } d x d s \\\\ & \\leq \\int _ { 0 } ^ { t } \\int _ { B _ R } \\min \\left \\{ \\frac { 2 | | \\mathbf { B } _ s | | _ { L ^ \\infty } } { \\delta } , \\frac { | \\mathbf { B } _ s ( X _ 1 ( s , x ) ) - \\mathbf { B } _ s ( X _ 2 ( s , x ) ) | } { | X _ 1 ( s , x ) - X _ 2 ( s , x ) | } \\right \\} d x . \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} X = X _ 0 + P Z P , \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} p _ i \\equiv 1 \\ ; \\ , s \\ ; \\ ; i = 1 , 2 , \\dots , n . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} g \\big ( W ( z ) ( 0 ) , \\nabla \\phi ( z ( 0 ) ) \\big ) = g \\big ( W ( z ) ( 1 ) , \\nabla \\phi ( z ( 1 ) ) \\big ) = 0 . \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ j \\binom k j \\left ( { \\frac { { F _ r } } { { F _ { r + n } } } } \\right ) ^ j G _ { n j } } = \\left ( { \\frac { { F _ n } } { { F _ { r + n } } } } \\right ) ^ k ( F _ { r k + 1 } G _ { 0 } - F _ { r k } G _ 1 ) \\ , , \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} \\nabla _ { e _ i } \\nabla _ { e _ i } Y = \\nabla _ { e _ i } \\nabla _ { e _ i } Y ^ T - \\nabla _ { e _ i } \\nabla _ { e _ i } ( \\frac { Y ^ { \\perp } } { u } \\partial _ t ) . \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} \\Pi _ 1 : \\mathcal E & \\rightarrow \\mathbf B ( \\partial M ) , \\\\ \\Pi _ 1 ( g ^ { ( 4 ) } ) & = ( g _ { \\partial M } , H _ { \\partial M } , t r _ { \\partial M } K , \\omega _ { \\mathbf n } ) . \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} M _ F = \\frac { 1 } { \\pi } \\log { \\frac { q } { 2 \\pi e } } + \\frac { 0 . 3 1 7 } { e } + \\frac { 0 . 3 1 7 \\log { q } + 6 . 4 0 1 } { 3 } \\end{align*}"}
-{"id": "9896.png", "formula": "\\begin{align*} \\widehat { f } ( \\xi ) = \\left ( ( 2 \\pi i \\xi ) ^ { - s } + ( - 2 \\pi i \\xi ) ^ s \\right ) \\widehat { u } ( \\xi ) . \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} \\left | \\nabla _ y u ( x ) \\right | ^ 2 = \\left | \\frac { \\partial x } { \\partial y } \\nabla _ x u ( x ) \\right | ^ 2 = \\frac { | x | ^ 2 } { | y | ^ 2 } \\left ( \\left | \\nabla _ { \\mathbb { S } ^ { n - 1 } } u ( x ) \\right | ^ 2 + \\left ( \\frac { | y | } { | x | } \\phi ' ( | y | ) \\right ) ^ 2 \\left | \\nabla _ r u ( x ) \\right | ^ 2 \\right ) . \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} V ( Q ^ { \\star } ) = \\{ 0 \\} \\sqcup V ( Q ) \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} \\mathsf { a } _ { \\alpha , p } ( \\lambda , k ) : = \\left ( \\prod _ { l = 0 } ^ k ( \\psi ( \\lambda - l \\alpha ) - p ) \\right ) ^ { - 1 } , k \\in \\mathbb { N } _ 0 , \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} | \\langle \\tilde { \\nabla } \\psi , \\psi \\rangle | ^ 2 | \\psi | ^ { 4 p - 4 } = & \\frac { 1 } { 4 } \\big | d | \\psi | ^ 2 \\big | ^ 2 | \\psi | ^ { 4 p - 4 } = \\frac { 1 } { 4 p ^ 2 } \\big | d | \\psi | ^ { 2 p } \\big | ^ 2 , \\\\ | \\langle \\nabla d \\phi , d \\phi \\rangle | ^ 2 | d \\phi | ^ { 2 p - 4 } = & \\frac { 1 } { 4 } \\big | d | d \\phi | ^ 2 \\big | ^ 2 | d \\phi | ^ { 2 p - 4 } = \\frac { 1 } { p ^ 2 } \\big | d | d \\phi | ^ p \\big | ^ 2 . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} \\tau ( R , \\omega ^ \\prime ) : = \\inf \\left \\lbrace t \\ge 0 ; \\left \\lvert X _ t ^ { \\omega ^ \\prime } \\right \\rvert > R \\left \\lvert Y _ t ^ { \\omega ^ \\prime } \\right \\rvert > R \\right \\rbrace \\ , . \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{gather*} T _ { 0 } ( E _ { j , j } ( s ) ) = \\begin{cases} E _ { 1 , 1 } ( s ) - \\delta _ { s , 0 } c & j = N , \\\\ E _ { N , N } ( s ) + \\delta _ { s , 0 } c & j = 1 , \\\\ E _ { j , j } ( s ) & \\end{cases} \\end{gather*}"}
-{"id": "9106.png", "formula": "\\begin{align*} h _ 4 ( x ^ 4 + 3 x y + y ^ 3 ) & = x ^ 4 + 3 s ^ 2 x y + s y ^ 3 , \\\\ h _ 2 ( x ^ 4 + 3 x y + y ^ 3 ) & = s ^ { - 2 } x ^ 4 + 3 x y + s ^ { - 1 } y ^ 3 , \\\\ h _ 1 ( d x ) & = s \\ , d x - x \\ , d s , \\\\ h _ 0 ( d x ) & = d x - \\tfrac x s \\ , d s , \\\\ h _ 1 ( y \\ , d x ) & = y \\ , d x - \\tfrac { x y } s \\ , d s , \\\\ h _ 1 ( y \\ , d x - x \\ , d y ) & = y \\ , d x - x \\ , d y , \\\\ h _ 1 ( d x \\wedge d y ) & = z \\ , d x \\wedge d y + x \\ , d y \\wedge d z + y \\ , d z \\wedge d x . \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\int _ 0 ^ t \\langle - \\Delta _ p u _ N , \\phi w _ j \\rangle _ p \\ , d \\tau = \\int _ 0 ^ t \\langle \\eta , \\phi w _ j \\rangle _ p \\ , d \\tau \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\mathbb M _ m : = \\left \\{ \\frac { 1 } { 2 ^ { m } } , \\dots , \\frac { 2 ^ { m } - 1 } { 2 ^ { m } } \\right \\} \\subset [ 0 , 1 ] . \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} \\hat { \\mu } ^ p _ { \\Lambda , \\epsilon } ( d t ) = \\frac { 1 } { Z ' _ { \\Lambda , \\epsilon } ( \\alpha ) } \\det [ D _ { \\Lambda , \\epsilon } ( t ) ] ^ { - 1 / 2 } \\prod _ { j k \\in E ( \\mathbb { T } _ N ) } \\Big ( \\exp ( - e ^ { t _ { j k } } + t _ { j k } ) f _ \\alpha ( e ^ { t _ { j k } } ) d t _ { j k } \\Big ) , \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} I _ { k _ 1 } \\oplus \\cdots \\oplus I _ { k _ m } \\oplus \\begin{bmatrix} 0 & I _ { k _ { m + 1 } } \\\\ I _ { k _ { m + 1 } } & 0 \\end{bmatrix} \\oplus \\cdots \\oplus \\begin{bmatrix} 0 & I _ { k _ l } \\\\ I _ { k _ l } & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} \\omega ^ { - 1 } \\tau ' \\omega = \\tau ' . \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} ( \\Phi , \\phi ) ^ ! \\nabla _ { a } \\phi ^ { \\dagger } ( s ) : = \\phi ^ { \\dagger } ( \\nabla _ { \\Phi ( a ) } s ) , a \\in \\Gamma ( \\phi ^ ! ( A ) ) , s \\in \\Gamma ( D ) , \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} \\lvert ( \\phi w ) _ { n + 1 } \\rvert _ S - \\lvert ( \\phi w ) _ { n } \\rvert _ S = s p ( w ) . \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} \\| u ^ n - u _ H ^ { n } \\| + \\gamma ^ { \\frac 1 2 } \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| ( \\sigma - \\sigma _ H ) ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } + \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\nabla ( u - u _ { H } ) ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } \\leq C H ^ { m + 1 } , \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} a y ^ 2 + z ^ 2 \\ - \\ ( a { + } 1 ) x ^ 2 - a ^ 2 { - } b ^ 2 { - } a \\ = \\ 2 b q \\quad \\mbox { a n d } x ^ 2 - y ^ 2 + 1 \\ = \\ 2 p \\ , . \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} = | \\{ v \\in G ( I ) : \\max ( v ) = n , \\deg v = \\deg u \\} | . \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} \\binom { y } { i } : = \\prod _ { j = 0 } ^ n \\binom { \\beta _ j } { i _ j } . \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} S u ( s ) = \\int _ \\Omega b ( s , t ) u ( t ) \\ , d t , \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} m _ { u , p } = \\bigg \\{ \\lambda = \\{ \\lambda _ k \\} _ { k \\in \\mathbb { Z } ^ d } & \\subset \\mathbb { C } : \\\\ \\| \\lambda | m _ { u , p } \\| = & \\sup _ { j \\in \\mathbb { N } _ 0 ; m \\in \\mathbb { Z } ^ d } | Q _ { - j , m } | ^ { \\frac { 1 } { u } - \\frac { 1 } { p } } \\Big ( \\sum _ { k : \\ , Q _ { 0 , k } \\subset Q _ { - j , m } } \\ ! \\ ! \\ ! | \\lambda _ k | ^ p \\Big ) ^ \\frac { 1 } { p } < \\infty \\bigg \\} , \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} & = B \\sum _ { n = 1 } ^ N \\theta _ { k , n } \\big ( \\log _ 2 \\big ( 1 + h _ { k , n } p _ { k , n } \\big ) - \\log _ 2 \\big ( 1 + { g _ { k , n } p _ { k , n } } \\big ) \\big ) ^ + , \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} R _ k \\mapsto \\sum _ { \\substack { 0 \\leq s _ 1 , \\dots , s _ q \\leq m + 1 \\\\ 0 \\leq s _ { q + 1 } \\leq r } } \\binom { r } { s _ { q + 1 } } \\left ( \\prod _ { i = 1 } ^ q \\binom { m + 1 } { s _ i } \\right ) \\left ( \\prod _ { i = 1 } ^ { q + 1 } ( - \\lambda _ i ) ^ { s _ i } \\right ) R _ { k - \\sum _ { i = 1 } ^ { q + 1 } s _ i } , \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} d Y _ { s } ^ { 1 , u } = - F _ { 1 } ( s , X _ { s } ^ { u } , Y _ { s } ^ { 1 , u } , Z _ { s } ^ { 1 , u } , u _ { s } ) d s + Z _ { s } ^ { 1 , u } d B _ { s } , \\ Y _ { t + \\delta } ^ { 1 , u } = 0 , \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{gather*} \\frac { L ( g , 1 ) } { F _ 1 ( 1 ) } = - \\frac 1 8 , \\frac { L ( g , 2 ) } { F _ 2 ( 1 ) } = \\frac 1 { 3 2 } , \\frac { L ( g , 3 ) } { F _ 3 ( 1 ) } = - \\frac 3 { 4 4 8 } , \\\\ \\frac { L ( g , 4 ) } { F _ 4 ( 1 ) } = \\frac 1 { 6 4 0 } \\qquad \\frac { L ( g , 5 ) } { F _ 5 ( 1 ) } = - \\frac 5 { 1 2 0 3 2 } . \\end{gather*}"}
-{"id": "1552.png", "formula": "\\begin{align*} k = ( m + 1 ) q + r , \\mbox { } 0 \\leq q < n , \\mbox { } 0 \\leq r < m + 1 . \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} Q _ y ( \\alpha ) = \\frac { \\alpha ( 1 + y e ^ { - \\alpha } ) } { 1 - e ^ { - \\alpha } } \\in \\mathbb { Q } [ y ] [ [ \\alpha ] ] \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} [ \\phi ( \\cdot , W ) - \\phi ( 0 , W _ 0 ) , N ] _ t = [ \\phi ( \\cdot , W ) , N ] _ t = \\int _ 0 ^ t \\nabla \\phi ^ * ( r , W _ r ) \\mathrm d [ W , N ] _ r , \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } g ( x ) { \\mathrm d } \\mathrm m _ { t } ( x ) - \\int ^ { t } _ { 0 } [ \\mathcal A _ { s } ( \\mathrm M _ { s } ) g ] { \\mathrm d } s \\ , = \\ , \\eta _ { t } \\ , = \\ , \\langle \\mathrm m _ { 0 } , g \\rangle \\ , \\ , = \\ , \\int _ { \\mathbb R } g ( x ) { \\mathrm d } \\mathrm m _ { 0 } ( x ) \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} F _ A ^ * ( b X ^ m d X ) = b ^ p X ^ m d X = d \\left ( b ^ p \\frac { X ^ { m + 1 } } { m + 1 } \\right ) , \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} D = \\{ x \\in \\mathbb { R } ^ n : x _ 1 > \\cdots > x _ n \\} . \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} \\lvert r _ { j _ 1 } \\rvert _ S - \\lvert r _ { j _ 2 } \\rvert _ S = j _ 1 ' - j _ 2 ' = j _ 1 - j _ 2 . \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} \\theta ( n , i , j , k , r ) : = \\binom { i } { r } \\binom { n - i } { n - k - r } \\binom { i - r } { j + i - k - 2 r } = \\binom { i } { r } \\binom { n - i } { k - i + r } \\binom { i - r } { k - j + r } . \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} 0 & = \\int _ N f ( x y ^ { - 1 } ) \\Delta _ N ( y ^ { - 1 } ) 1 _ { q ^ { - 1 } ( A ) \\cap N } ( x ) d \\omega ( x ) \\\\ & = \\int _ N f ( x ) 1 _ { q ^ { - 1 } ( A ) \\cap N } ( x y ) d \\omega ( x ) \\\\ & = \\int _ N f ( x ) 1 _ { A \\cap q ( N ) } \\big ( x \\cdot q ( y ) \\big ) d \\omega ( x ) . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} & c _ k ( f _ { 0 . 7 , \\ , 0 . 5 } ) = O ( 1 / | k | ^ 2 ) + O \\left ( 1 / | k | ^ { ( 1 + 0 . 7 + 0 . 2 5 ) / ( 0 . 5 + 1 ) } \\right ) \\\\ & = O \\left ( 1 / | k | ^ { 1 . 3 } \\right ) . \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} F _ r \\sum _ { j = 0 } ^ k { ( - 1 ) ^ { r j } \\left ( { \\frac { { F _ { n + r } } } { { F _ n } } } \\right ) ^ j G _ { m + n + r + r j } } = ( - 1 ) ^ { k r } F _ n \\left ( \\frac { { F _ { n + r } } } { { F _ n } } \\right ) ^ { k + 1 } G _ { m + ( k + 1 ) r } - ( - 1 ) ^ r F _ n G _ m , n \\ne 0 \\ , , \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} \\mathcal { L } _ { s , c } ( m ) = \\begin{cases} \\gamma l ^ { \\frac { c } { 2 } - \\frac { n _ l } { 4 } } \\chi _ l ^ { - 1 } ( \\overline { a } b ) \\sum _ { \\pm } \\Phi _ c ^ { \\pm } ( \\frac { m } { a b } ) & \\frac { \\alpha b m } { a } \\in ( \\Z _ l ^ { \\times } ) ^ 2 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} \\mathcal { P } ( \\mathcal { B } ) = \\{ y \\in \\R ^ n : y = \\sum _ { i = 1 } ^ { n } \\alpha _ { i } g _ { i } , \\ 0 \\leq \\alpha _ { i } < 1 \\} . \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} x + y + m + x y m = 0 . \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{gather*} S _ { i , j } ( p ) = E _ { i , j } ( p ) E _ { j , i } ( m - p ) . \\end{gather*}"}
-{"id": "5262.png", "formula": "\\begin{align*} P u = 0 , u | _ { \\partial \\Omega } = 0 . \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\Theta _ 0 ( & \\psi _ k ( g ) , \\varphi _ k ( b ) ) \\\\ & = \\Theta _ 0 ( ( e , \\dots , e , F _ k ( g ) , e , \\dots , e ) , ( e , \\dots , e , F _ k ( b ) , F _ k ( h ) , \\dots , F _ k ( h ) ) ) \\\\ & = ( e , \\dots , e , F _ k ( g ) F _ k ( b ) , \\Upsilon ( F _ k ( b ) ) ^ { - 1 } F _ k ( h ) , e , \\dots , e ) \\\\ & = ( e , \\dots , e , F _ k ( g b ) , e , e , \\dots , e ) \\\\ & = \\psi _ k ( g b ) . \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ n u _ \\alpha \\lambda _ \\alpha + \\Lambda \\mapsto \\left ( e ^ { 2 \\pi i u _ 1 } , \\ldots , e ^ { 2 \\pi i u _ n } \\right ) , \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} \\rho ( Y ) \\geq 4 + 2 = 6 . \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} { \\hat A } _ { T , t } ^ { ( i _ 1 i _ 2 ) } = \\frac { T - t } { 2 \\pi } \\sum _ { r = 1 } ^ { \\infty } \\frac { 1 } { r } \\left ( \\zeta _ { 2 r } ^ { ( i _ 1 ) } \\zeta _ { 2 r - 1 } ^ { ( i _ 2 ) } - \\zeta _ { 2 r - 1 } ^ { ( i _ 1 ) } \\zeta _ { 2 r } ^ { ( i _ 2 ) } + \\sqrt { 2 } \\left ( \\zeta _ { 2 r - 1 } ^ { ( i _ 1 ) } \\zeta _ { 0 } ^ { ( i _ 2 ) } - \\zeta _ { 0 } ^ { ( i _ 1 ) } \\zeta _ { 2 r - 1 } ^ { ( i _ 2 ) } \\right ) \\right ) . \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t U _ h ^ { \\frac 1 2 } , v _ h \\Big { ) } & - \\gamma ( \\nabla \\Sigma _ h ^ { \\frac 1 2 } , \\nabla v _ h ) + ( \\nabla U _ h ^ { \\frac 1 2 } , \\nabla v _ h ) \\\\ & + \\frac 1 2 ( \\mathfrak { F } ( U _ { h } ^ { 1 } , u _ { H } ^ { 1 } ) + f ( U _ { h } ^ { 0 } ) , v _ h ) = ( g ^ { \\frac 1 2 } , v _ h ) , ~ \\forall v _ h \\in L _ h , \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} \\overline { S } ( Y , U ) = ( n - 1 ) ( 1 - \\beta ) g ( Y , V ) - ( n - 1 ) { \\beta } ^ { 2 } \\eta ( Y ) \\eta ( U ) , \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} c _ k ( f ^ * ) = O \\left ( 1 / | k | ^ { 1 + \\mu } \\right ) . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} z ^ { \\ , [ j ] } _ i = \\left \\{ \\begin{array} { c l } 2 & \\hbox { i f } i = j \\\\ 1 & \\hbox { i f } i \\neq j \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} ( \\mathrm M _ { t } ^ { ( j ) } , t \\ge 0 ) \\ , : = \\ , ( ( \\overline { X } _ { t , 1 } , \\ldots , \\overline { X } _ { t , j } ) , t \\ge 0 ) \\ , ; j \\ , = \\ , 1 , \\ldots , k + 1 \\ , . \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} F _ { ( - j ) j } = 0 , \\end{align*}"}
-{"id": "9740.png", "formula": "\\begin{align*} \\sum \\limits _ { a \\in A _ { + , i } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) } { a } = \\gamma _ i \\ell _ i ( z _ 1 ) \\dots \\ell _ i ( z _ n ) . \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} ( V U ) _ { i , j } = \\sum _ { k = 1 } ^ { \\infty } \\hat { u } ( k + n - j ) \\hat { v } ( i - n - k ) , \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} \\forall n \\exists k ( a ^ * _ n ( k ) = 1 ) \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} 4 ( u v ) ^ 3 + ( u v ( u - v ) ) ^ 2 & = 0 ^ 3 + ( u v ( u + v ) ) ^ 2 . \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} \\tau _ { \\Omega } : = \\left \\{ x \\in M _ { \\mathbb R } \\ , \\ , \\ , \\ , \\Large { | } \\ , \\ , \\ , \\ , \\ \\begin{array} { l l } d _ j - 1 < \\langle x , n _ j \\rangle < d _ j & \\mbox { f o r } n _ j \\in \\Omega , \\\\ \\langle x , n _ j \\rangle = d _ j & \\mbox { f o r } n _ j \\not \\in \\Omega \\end{array} \\right \\} . \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} V ( t ) = V _ 0 ( t ) + g _ 0 ( t ) \\end{align*}"}
-{"id": "9986.png", "formula": "\\begin{align*} \\bar u ( q , \\tau ) : = \\int _ { \\mathbb { R ^ + } } e ^ { - q x } u ( x , \\tau ) d x , q \\in \\mathbb C _ + , \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} G ^ { k l } _ \\lambda ( \\Q ) : = \\delta _ { k , l } + \\frac { \\sum _ j ( \\Q _ { j k } \\ , \\Q _ { j l } ) } { \\abs { \\Q } ^ 2 } \\ , \\omega _ \\lambda ( \\abs { \\Q } ) . \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} S z = S ( x - v ) = S x \\textup { a n d } T z = T ( y + w ) = T y , \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} g ( r ) = \\left \\{ \\int _ { \\wp } \\int _ { 0 } ^ { r } s ^ { Q - 1 } ( \\psi _ { 1 } ( s y ) ) ^ { 1 - p ' } d s d \\sigma ( y ) \\right \\} ^ { 1 / ( p p ' ) } , \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} \\big \\langle \\omega , \\gamma \\big \\rangle _ \\C : = \\big \\langle c o m p ( \\omega ) , \\eta \\big \\rangle \\in \\C . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { m ^ + } \\Big ( \\sup _ { H _ k } f _ k \\Big ) ^ { c _ k } \\times \\prod _ { k = m ^ + + 1 } ^ m \\Big ( \\sup _ { ( t + 1 ) ( F _ k \\times G _ k ) } f _ k ^ { - 1 } \\Big ) ^ { - c _ k } , \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} p _ { N } ^ { \\ell _ 1 - } ( x ) = \\dfrac { \\sum \\limits _ { j = 0 } ^ N \\dfrac { \\Omega _ j } { x - x _ j } \\left ( f ( x _ j ) + \\sum \\limits _ { \\ell = 0 } ^ N c _ { \\ell } \\tilde { \\Phi } _ { \\ell } ( x _ j ) \\right ) } { \\sum \\limits _ { j = 0 } ^ N \\dfrac { \\Omega _ j } { x - x _ j } } . \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} \\xi _ j ( t ) = \\xi _ j ^ 0 ( t ) + \\int _ { t } ^ \\infty h ( s ) d s , \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} a _ { n + 1 } \\left ( \\lambda _ { n + 1 } p _ n - \\hat { q } _ { n - 1 } \\right ) + \\left ( \\lambda _ { n + 1 } p _ { n - 1 } - \\hat { q } _ { n - 2 } \\right ) & = 0 \\\\ a _ { n + 1 } \\left ( \\lambda _ { n + 1 } q _ n - a _ 1 \\hat { q } _ { n - 1 } - \\hat { p } _ { n - 1 } \\right ) + \\left ( \\lambda _ { n + 1 } q _ { n - 1 } - a _ 1 \\hat { q } _ { n - 2 } - \\hat { p } _ { n - 2 } \\right ) & = 0 \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} R _ { \\mathcal { X } _ p } \\left ( e _ { ( z , w ) } \\circ T _ V \\right ) ( t , \\zeta ) = \\overline { X _ p ( t ; w , \\zeta ) } \\frac { \\overline { z } ^ { - 2 \\pi i t } } { \\overline { z } ^ { 1 + 1 / 2 \\mu } } . \\end{align*}"}
-{"id": "9892.png", "formula": "\\begin{align*} ( v , \\phi ) = \\lim _ { n \\rightarrow \\infty } ( \\boldsymbol { D } ^ { - s } \\psi _ n , \\phi ) = \\lim _ { n \\rightarrow \\infty } ( \\psi _ n , \\boldsymbol { D } ^ { - s * } \\phi ) = ( u , \\boldsymbol { D } ^ { - s * } \\phi ) . \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\{ x _ 1 < x _ 2 < \\cdots < x _ k \\} & = X _ + ( D ) , \\\\ \\{ y _ 1 < y _ 2 < \\cdots < y _ k \\} & = \\{ 1 , \\dotsc , n - 1 \\} \\setminus X _ - ( D ) . \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} a _ \\lambda = g ( W ( z ) ( \\lambda ) , \\nu ( x ( \\lambda ) ) ) ^ + , b _ \\lambda = g ( W ( z ) ( 1 - \\lambda ) , \\nu ( x ( 1 - \\lambda ) ) ) ^ + \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} A _ { \\epsilon } = \\bigcup _ { x \\in A } \\{ z \\in X \\colon d ( z , x ) \\leq \\epsilon \\} \\ , . \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} \\mathfrak { M } ^ { - 1 } \\circ \\mathfrak { M } = \\mathfrak { M } \\circ \\mathfrak { M } ^ { - 1 } = 1 . \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} T ( X ^ * ) + M ^ T \\alpha - \\beta & = 0 , \\\\ \\beta ^ T X ^ * & = 0 , \\\\ \\beta \\geq 0 , \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} w _ 1 = \\begin{cases} q ^ s , & N = 1 , \\\\ q ^ { s } / w _ 2 \\cdots w _ N , & N \\ge 2 . \\end{cases} \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} \\Delta ( K _ i ) & = K _ i \\otimes K _ i & \\varepsilon ( K _ i ) & = 1 & S ( K _ i ) & = K _ i ^ { - 1 } , \\\\ \\Delta ( E _ i ) & = 1 \\otimes E _ i + E _ i \\otimes K _ i & \\varepsilon ( E _ i ) & = 0 & S ( E _ i ) & = - E _ i K _ i ^ { - 1 } . \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} x = \\lambda _ 1 ( x ) v _ 1 + \\lambda _ 2 ( x ) v _ 2 + \\lambda _ 4 ( x ) v _ 4 . \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} \\rho _ { t , k } ( \\varphi ) \\ , : = \\ , \\mathbb E _ { 0 } [ Z _ { t } ^ { - 1 } \\varphi ( t , \\overline { X } _ { t , 1 } , \\ldots , \\overline { X } _ { t , k } ) \\vert \\mathcal F _ { T } ^ { X } ] \\ , ; k \\ , = \\ , 2 , \\ldots , n \\ , , \\ , \\ , 0 \\le t \\le T \\ , , \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} \\lvert ( i , 2 \\lvert m _ k \\rvert ) \\rvert _ { W _ n ( w ) } - \\lvert ( i , j ) \\rvert _ { W _ n ( w ) } = \\lvert r _ { 2 \\lvert m _ k \\rvert } \\rvert _ S - \\lvert r _ j \\rvert _ S > 0 . \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} \\omega ( x , t + \\Re \\varphi ( x ) ) = e ^ { t + \\Re \\varphi ( x ) } \\rho ( x ) = e ^ { \\Re \\varphi ( x ) } \\omega ( x , t ) . \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} \\mathcal { A } = \\left \\{ \\mathbf { Q } \\in W ^ { 1 , \\phi } ( \\Omega ; S _ 0 ) \\colon f _ B ( \\Q ) \\in L ^ 1 ( \\Omega ) , \\ \\Q = \\Q _ b ~ \\textrm { o n } \\partial \\Omega \\right \\} \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} K : = I \\sqcup \\{ \\epsilon _ k \\mid 1 \\le k \\le \\ell \\} . \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} \\psi \\colon \\mathbb { C } ^ a \\otimes \\mathbb { C } ^ b \\to \\mathbb { C } ^ g , \\ \\psi ( \\vec { \\alpha } , \\vec { \\beta } ) = \\left ( \\sum _ { i , j } a _ { i j k } \\alpha _ i \\beta _ j \\right ) _ { 1 \\le k \\le g } \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} \\overline { a } ( i , j ) & = 2 \\ell - i - j + n + 1 + \\overline { \\lambda } _ j , \\\\ \\overline { b } ( i , j ) & = j - i + n - \\overline { \\lambda } _ j . \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} \\mu = ( 1 _ U \\lambda _ \\nu ) \\circ g ^ { - 1 } \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} ( H _ { 1 , 1 } \\otimes H _ { 2 , 1 } ) ^ \\top ( H _ { 1 , 1 } \\otimes H _ { 2 , 1 } ) & = H _ { 1 , 1 } ^ \\top H _ { 1 , 1 } \\otimes H _ { 2 , 1 } ^ \\top H _ { 2 , 1 } = n _ 1 n _ 2 I _ { n _ 1 } \\otimes J _ { n _ 1 } \\otimes I _ { n _ 2 } \\otimes J _ { n _ 2 } , \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} q _ j = c _ j x _ 1 p _ { j } + r _ j , \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\mathcal { C } ( x ) : = \\sum _ { k = 0 } ^ { \\infty } c _ k x ^ k = \\frac { 1 - \\sqrt { 1 - 4 x } } { 2 x } , x \\in \\left [ - \\frac { 1 } { 4 } , \\frac { 1 } { 4 } \\right ] . \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} \\mathbf { Q } ( \\hat { m } , m ^ \\star , \\kappa , \\lambda ) = \\left [ \\begin{array} { c } Q _ 1 ( \\hat { m } , m ^ \\star , \\kappa , \\lambda ) \\\\ Q _ 2 ( \\hat { m } , m ^ \\star , \\kappa , \\lambda ) \\\\ \\end{array} \\right ] = 0 \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} \\widetilde { w } \\left ( z \\right ) = \\widetilde { u } + \\sqrt { - 1 } \\widetilde { v } = u + \\alpha u - \\beta v + \\sqrt { - 1 } \\left ( v + \\alpha v + \\beta u \\right ) \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} w _ k = \\sum _ { l = 0 } ^ p \\varphi _ l ( \\rho _ k \\ , A ) v _ l , k = 1 , \\ldots , N _ s , \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} & \\| W _ { \\psi _ { i , 1 , * } , \\varphi _ { i , 1 , * } } - T ^ { - 1 } _ { ( V ^ * b ^ 0 ) _ { [ j ] } } C _ { U ^ * } W _ { \\chi , \\phi } C _ { V ^ * } \\| \\\\ = & \\ ; \\| T ^ { - 1 } _ { ( V ^ * b ^ 0 ) _ { [ j ] } } C _ { U ^ * } W _ { \\psi _ { i } , \\varphi _ { i } } C _ { V ^ * } - T ^ { - 1 } _ { ( V ^ * b ^ 0 ) _ { [ j ] } } C _ { U ^ * } W _ { \\chi , \\phi } C _ { V ^ * } \\| \\\\ \\leq & \\ ; \\| T ^ { - 1 } _ { ( V ^ * b ^ 0 ) _ { [ j ] } } \\| \\| W _ { \\psi _ i , \\varphi _ i } - W _ { \\chi , \\phi } \\| \\to 0 , i \\to \\infty . \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } p _ { n - 1 } & p _ { n } \\\\ q _ { n - 1 } & q _ { n } \\end{array} \\right ] = \\left [ \\begin{array} { c c } p _ { n - m - 2 } & p _ { n - m - 1 } \\\\ q _ { n - m - 2 } & q _ { n - m - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c c } B ^ { ( m ) } _ { n - m - 1 } & B ^ { ( m + 1 ) } _ { n - m - 1 } \\\\ A ^ { ( m ) } _ { n - m - 1 } & A ^ { ( m + 1 ) } _ { n - m - 1 } \\end{array} \\right ] \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} \\norm { F ( \\cdot , w ) } ^ 2 _ { A ^ 2 ( S ( p ( w ) , \\infty ) ) } & = \\int _ { S ( p ( w ) , \\infty ) } \\abs { F ( z , w ) } ^ 2 \\d V ( z ) & = \\int _ 0 ^ { \\infty } \\abs { f ( t , w ) } ^ 2 \\frac { e ^ { - 4 \\pi p ( w ) t } } { 4 \\pi t } \\d t . \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\beta < n ( \\frac { 1 } { s } - 1 ) , \\gamma _ { 1 } < \\frac { n } { q _ { 1 } ^ { \\prime } } , \\gamma _ { 2 } < \\frac { n } { q _ { 2 } ^ { \\prime } } , \\\\ & \\quad \\alpha + \\beta + \\gamma _ { 1 } + \\gamma _ { 2 } = n + \\frac { n } { t } - \\frac { n } { q _ { 1 } } - \\frac { n } { q _ { 2 } } , \\\\ & \\qquad \\qquad \\quad \\beta + \\gamma _ { 1 } + \\gamma _ { 2 } \\geq 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} \\mathbf { c } ( \\varepsilon , \\kappa , \\tau , \\zeta ) = \\frac { \\varepsilon ^ { - d + 1 } } { \\rho ^ \\alpha } \\int _ { \\mathbb { R } ^ d } \\varphi _ { \\zeta \\rho } ( z ' - z ) \\frac { \\langle \\phi ^ { e , \\varepsilon } ( ( z ' - z ) / \\rho ) , \\eta ^ { \\kappa } _ { y _ { \\tau } } \\rangle } { | z ' - z | ^ { d - \\alpha } } d z = \\varepsilon ^ { - d + 1 } \\int _ { \\mathbb { R } ^ d } \\varphi _ { \\zeta } ( z ) \\frac { \\langle \\phi ^ { e , \\varepsilon } ( z ) , \\eta ^ { \\kappa } _ { y _ { \\tau } } \\rangle } { | z | ^ { d - \\alpha } } d z . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} c ^ \\tau ( x * y ) & = \\tau ( g _ 1 , g _ 2 ) \\tau ( h , g _ 1 g _ 2 ) c ( x y ) \\\\ & = \\tau ( h , g _ 2 ) \\tau ( g _ 1 , h g _ 2 ) x c ( y ) = x * c ^ \\tau ( y ) . \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{gather*} \\big [ T _ { w ( 1 , m ) } ( E _ { N , 1 } ( 1 ) ) , E _ { 1 , 1 } ( \\pm s ) \\big ] = T _ { w ( 1 , m ) } \\big ( \\big [ E _ { N , 1 } ( 1 ) , E _ { 1 , 1 } ( \\pm s ) \\big ] \\big ) = T _ { w ( 1 , m ) } \\big ( E _ { N , 1 } ( 1 \\pm s ) \\big ) . \\end{gather*}"}
-{"id": "2536.png", "formula": "\\begin{align*} V _ h : = \\{ v : v | _ { D _ 0 } \\in \\mathbb { P } _ k ( D _ 0 ) \\ \\forall D \\in \\mathcal { T } _ h \\ { \\rm a n d } \\ v | _ { e } \\in \\mathbb { P } _ k ( e ) \\ \\forall e \\in \\mathcal { E } _ h \\} . \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} q _ 1 - q _ 2 = \\sum \\limits _ { j = 1 } ^ { m } a _ j g _ j , \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} d _ { \\rm T V } ( D _ n , P o ( \\lambda _ n ) ) \\leq C \\ , \\begin{cases} \\frac { 1 } { n + 1 } , & 0 < \\gamma < \\frac { 1 } { 2 } , \\\\ \\frac { \\log ( n ) } { n } , & \\gamma = \\frac { 1 } { 2 } , \\\\ n ^ { - 2 ( 1 - \\gamma ) } , & \\frac { 1 } { 2 } < \\gamma < 1 , \\\\ \\end{cases} , \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} h _ { \\sigma } = \\left \\{ \\begin{array} { c c } 2 \\ & \\mbox { i f } \\ \\sigma = \\hat { \\sigma } \\\\ 0 \\ & \\mbox { o t h e r w i s e } \\end{array} \\right . , \\ g = 1 . \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} \\mathbf { K } ( x ) = \\Omega ( x ) K ( x ) ~ ~ \\forall ~ x \\in \\mathbb { R } ^ d \\backslash \\{ 0 \\} , \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} N _ \\varepsilon ( t ) \\geq N _ { i n } \\exp ( - t ) \\geq N _ { i n } \\exp ( - T ) = : \\delta ( T ) , \\qquad \\forall t \\in [ 0 , T ] , \\ \\forall \\varepsilon > 0 . \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} \\mu _ l ( \\alpha ( a ) , \\mu _ r ( m , b ) ) = \\mu _ r ( \\mu _ l ( a , m ) , \\alpha ( b ) ) . \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } ( n , u _ { n } ) ^ { k } = \\sum _ { n \\leq x } \\sum _ { \\substack { d \\mid ( n , u _ n ) } } J _ { k } ( d ) = \\sum _ { d \\leq x } J _ { k } ( d ) \\sum _ { \\substack { n \\leq x \\\\ d \\mid ( n , u _ n ) } } 1 = \\sum _ { \\substack { d \\leq x \\\\ ( d , a _ 2 ) = 1 } } J _ { k } ( d ) \\sum _ { \\substack { n \\leq x \\\\ \\ell _ u ( d ) \\mid n } } 1 , \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} \\displaystyle \\beta = \\frac { 2 \\alpha + 1 } { 8 \\alpha - 2 } , B = - \\frac { ( 4 \\alpha - 1 ) ^ 2 } { 3 } A . \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} \\lvert ( \\phi w ) _ { n + 1 } \\rvert _ S - \\lvert ( \\phi w ) _ { n } \\rvert _ S & = \\lvert ( 0 , 0 ) \\rvert _ { W _ { n + 1 } ( w ) } - \\lvert ( 0 , 0 ) \\rvert _ { W _ n ( w ) } \\\\ & = \\lvert m _ { n + 1 } \\rvert _ S - \\lvert m _ { n } \\rvert _ S \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ s \\mu _ \\ell ^ r x _ \\ell = 0 \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} \\partial \\varphi ( A ) = \\bigcup _ { x \\in A } \\partial \\varphi ( x ) . \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} \\Phi \\cdot \\Psi = ( \\Phi ^ + \\cdot \\Psi ^ + + \\Phi ^ - \\cdot \\Psi ^ - , \\Phi ^ + \\cdot \\Psi ^ - + \\Phi ^ - \\cdot \\Psi ^ + ) . \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} - \\Delta w _ 0 + \\lambda w _ 0 = w _ 0 ^ p , x \\in \\R ^ 3 . \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} b : = \\Psi \\circ ( h - \\mathfrak { s } ) \\circ \\psi ^ { - 1 } : U ' \\rightarrow U ' \\lhd ( \\mathbb { R } ^ { k ' _ 1 + k ' _ 2 } \\times \\mathbb { W } _ 1 \\times \\mathbb { W } _ 2 ) \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} \\lim _ { | x | \\to + \\infty } \\bar { \\omega } _ k ( x ) = 0 . \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} S _ i ( q ^ k - 1 ) = \\sum _ { a \\in A _ { + , i } } a ^ { q ^ k - 1 } = \\begin{cases} L _ i ( \\theta ^ { q ^ k } - \\theta ) \\dots ( \\theta ^ { q ^ k } - \\theta ^ { q ^ { i - 1 } } ) & k \\geq i \\\\ 0 & k < i . \\end{cases} \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} S = \\frac { 1 } { 3 } \\begin{pmatrix} 1 - \\zeta _ 3 & 1 - \\zeta _ 3 ^ 2 & \\zeta _ 3 - \\zeta _ 3 ^ 3 \\\\ 1 - \\zeta _ 3 ^ 2 & 1 - \\zeta _ 3 & \\zeta _ 3 ^ 2 - \\zeta _ 3 \\\\ \\zeta _ 3 - \\zeta _ 3 ^ 2 & \\zeta _ 3 ^ 2 - \\zeta _ 3 & \\zeta _ 3 - \\zeta _ 3 ^ 2 \\end{pmatrix} T = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & \\zeta _ 3 ^ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} \\alpha x ^ 2 + \\alpha ^ \\sigma y ^ 2 = z ^ 2 . \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} W _ { \\phi , l } ( a ( \\gamma ) ) = \\omega _ { \\tilde { \\pi } , l } ( \\gamma ) \\abs { \\gamma } _ { l } ^ { \\frac { 1 } { 2 } } W _ { l } ( \\gamma ) , \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} x _ 2 ( s ) = x _ 1 \\big ( \\theta ( s ) \\big ) \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} \\int _ { t _ 2 } ^ { t _ 1 } f ( t ) \\ , d t = 0 . \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} \\mathcal { L } _ R ^ \\psi [ v ] ( x ) : = \\int _ { B _ R } J ( x - y ) v ( y ) \\d y + \\int _ { B _ R ^ C } J ( x - y ) \\psi ( y ) \\d y - v ( x ) . \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} H ^ q ( W , V _ { \\lambda } ) \\simeq \\bigoplus \\limits _ { w \\in \\Upsilon ^ \\prime \\vert l ( w ) = q } U _ { w . ( \\lambda + \\rho ) - \\rho } , \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} u _ t - \\mathcal { L } [ u ] + | D u | ^ m = f \\quad \\quad \\R ^ N . \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} A = C _ 0 \\otimes I + C _ 1 \\otimes T _ 1 + \\ldots + C _ r \\otimes T _ r \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} & { { D } _ { x } } u = { { D } _ { y } } v \\\\ & { { D } _ { y } } u = - { { D } _ { x } } v \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} { } & Q _ i ^ 2 \\sum _ { s = 1 } ^ r V _ s ^ { ( i ) } \\bar { V } _ l ^ { ( i ) } ( V _ p ^ { ( i ) } ) ^ t ( V _ s ^ { ( i ) } ) ^ * = \\delta _ { l p } I _ { n _ i } . \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 2 u ^ 3 + \\lambda ( u ^ 2 ) ^ 2 & - 1 - \\lambda u ^ 1 u ^ 2 & - u ^ 1 \\\\ - 1 - \\lambda u ^ 1 u ^ 2 & \\lambda ( u ^ 1 ) ^ 2 & 0 \\\\ - u ^ 1 & 0 & \\lambda \\end{pmatrix} , \\\\ w _ { 1 2 } = \\frac { \\lambda } { \\sqrt { - \\det g } } , w _ { 2 3 } = 0 , w _ { 3 1 } = \\frac { \\lambda u ^ 1 } { \\sqrt { - \\det g } } , \\end{gather*}"}
-{"id": "8528.png", "formula": "\\begin{align*} \\# ( Q ( M _ n ) \\setminus M _ n ) & = ( \\ell _ 1 + 1 ) ( \\ell _ 2 + 1 ) ( \\ell _ 3 + 1 ) - n \\\\ & \\le ( n ^ { 1 / 3 } + K n ^ { 1 / 1 2 } + { \\rm O } ( 1 ) ) ^ 3 - n = 3 K n ^ { 3 / 4 } + { \\rm o } ( n ^ { 3 / 4 } ) , \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} M _ { \\mu , \\nu } ( y ) & = y ^ { \\nu + \\frac { 1 } { 2 } } e ^ { \\frac { y } { 2 } } \\frac { \\Gamma ( 1 + 2 \\nu ) } { \\Gamma ( \\nu + \\mu + \\frac { 1 } { 2 } ) \\Gamma ( \\nu - \\mu + \\frac { 1 } { 2 } ) } \\int _ 0 ^ 1 t ^ { \\nu + \\mu - \\frac { 1 } { 2 } } ( 1 - t ) ^ { \\nu - \\mu - \\frac { 1 } { 2 } } e ^ { - y t } d t , \\\\ W _ { \\mu , \\nu } ( y ) & = y ^ { \\nu + \\frac { 1 } { 2 } } e ^ { \\frac { y } { 2 } } \\frac { 1 } { \\Gamma ( \\nu - \\mu + \\frac { 1 } { 2 } ) } \\int _ 1 ^ { \\infty } t ^ { \\nu + \\mu - \\frac { 1 } { 2 } } ( t - 1 ) ^ { \\nu - \\mu - \\frac { 1 } { 2 } } e ^ { - y t } d t . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} Q _ \\alpha ( \\rho \\| \\sigma ) : = \\| \\Delta _ { \\rho , \\sigma } ^ { \\alpha / 2 } \\xi _ \\sigma \\| ^ 2 \\ \\in [ 0 , + \\infty ) , \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} \\limsup _ { r \\to \\infty } \\frac { h ( r ) h ( r h ( r ) ) \\log T ( r ) } { r } = \\zeta , \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\zeta _ { f , g } ( z ) = \\prod _ { \\varpi \\ \\mbox { \\tiny p r i m e } } \\left ( 1 - z ^ { \\mathrm { p e r } ( \\varpi ) } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} L v = u \\ , \\omega ( u , v ) , \\forall v \\in V . \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} P _ { 0 } ^ { ( \\alpha , \\beta ) } ( x ) = 1 , \\quad P _ { 1 } ^ { ( \\alpha , \\beta ) } ( x ) = \\frac { \\alpha + \\beta + 2 } { 2 } x + \\frac { \\alpha - \\beta } { 2 } . \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} x _ v ( t ) = [ P _ t x _ 0 ] _ v = \\sum _ { v ' \\in V } p _ t ( v , v ' ) x _ { v ' , 0 } \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} \\Pi : & \\mathbb E \\rightarrow \\mathbf B \\\\ \\Pi ( [ g ^ { ( 4 ) } ] ) & = ( g _ { \\partial M } , H _ { \\partial M } , t r _ { \\partial M } K , \\omega _ { \\mathbf n } ) . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\begin{cases} u _ t + F ( v ( t , x ) , u ) _ x ~ = ~ 0 , \\\\ u ( 0 , x ) ~ = ~ u _ 0 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\int d x \\ x a ( x , y , t ) = \\frac { \\partial } { \\partial y } ( y p _ { 0 } ( y , t ) ) + \\frac { 1 } { 2 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\int d x \\ x ^ { 2 } \\varphi ( x , y , t ) - \\frac { \\partial } { \\partial t } p _ { 0 } ( y , t ) \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} E : = \\int _ { 0 } ^ \\pi L ( x , u , u _ x ) d x \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} ( s _ 1 , \\dotsc , s _ n ) \\mapsto \\{ n + 1 - i \\mid s _ i = - \\} . \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} L ' _ { i j , i j } = \\frac { 1 } { R _ { e } } \\sum _ { T ' \\in S ( G ' ) \\atop e ' \\in T ' } \\Pi ( T ' ) . \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} Z _ V & : = \\prod \\nolimits _ { i \\in V } Z _ i . \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } x ^ { k } \\ , f _ { Z _ { n } } ( x ) d x = n \\sum _ { j = 0 } ^ { n - 1 } \\binom { n - 1 } { j } ( - 1 ) ^ { j } \\int _ { 0 } ^ { \\infty } x ^ { k } e ^ { - ( 1 + j ) x } d x \\\\ & = n \\sum _ { j = 0 } ^ { n - 1 } \\binom { n - 1 } { j } ( - 1 ) ^ { j } \\frac { 1 } { ( 1 + j ) ^ { k + 1 } } \\int _ { 0 } ^ { \\infty } t ^ { k } e ^ { - t } d t = k ! \\sum _ { j = 1 } ^ { n } \\binom { n } { j } ( - 1 ) ^ { j - 1 } \\frac { 1 } { j ^ { k } } , \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} k ( x ) = \\int _ { \\left [ \\frac { x } { c } , \\infty \\right ) } h \\left ( \\frac { x } { r } \\right ) \\nu ( d r ) = \\int _ { \\left [ \\frac { x } { c } , \\infty \\right ) \\setminus \\{ \\frac { x } { r } \\in g ( T ) \\} } h \\left ( \\frac { x } { r } \\right ) \\nu ( d r ) + \\int _ { \\{ \\frac { x } { r } \\in g ( T ) \\} } h \\left ( \\frac { x } { r } \\right ) \\nu ( d r ) . \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} \\frac { L ' _ \\chi } { L _ \\chi } ( s ) = - \\sum _ \\mathfrak a \\frac { \\Lambda ( \\mathfrak a ) \\chi ( \\mathfrak a ) } { N ( \\mathfrak a ) ^ s } , \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} \\langle x _ i , y _ i \\rangle = \\begin{cases} 1 \\\\ 2 \\\\ 4 \\end{cases} . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} \\dot { \\psi } = \\frac { 1 } { M + 1 } \\mathcal { F } ( \\psi ) . \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} d ^ n f ( g _ 0 , . . . , g _ { n + 1 } ) = \\sum _ { i = 0 } ^ { n + 1 } ( - 1 ) ^ i f ( g _ 0 , . . . , \\hat { g _ i } , . . . , g _ { n + 1 } ) . \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} [ w ] _ { A _ \\infty ( \\mathbb R ) } \\ge w _ I \\exp \\left ( [ - \\log w ] _ I \\right ) \\ge \\exp \\left ( ( \\log w ) _ I \\right ) \\exp \\left ( [ - \\log w ] _ I \\right ) = 1 , \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} X Z ( x , v ) : = \\frac { D Z ( \\varphi _ { t } ( x , v ) ) } { d t } | _ { t = 0 } \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} \\left \\| \\prod _ { 0 \\le k \\le m } P _ { n - k } ( \\epsilon ) - \\left ( \\tilde { { P } } ^ { m + 1 } - \\epsilon \\sum _ { j = 1 } ^ { m } j \\tilde { P } ^ { m - j } \\tilde { P } ^ { ( 1 ) } \\tilde { P } ^ { j } \\right ) \\right \\| = O ( \\epsilon ^ 2 m ^ 3 ) \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { \\binom k j \\binom j s \\left ( { \\frac { { f _ 2 } } { { f _ 3 } } } \\right ) ^ j \\left ( { - \\frac { { 1 } } { { f _ 2 } } } \\right ) ^ s X _ { m - ( c - a ) k + ( c - b ) j + b s } } } = \\left ( - \\frac { f _ 1 } { f _ 3 } \\right ) ^ k X _ m \\ , , \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{gather*} T _ 1 ( E _ { 2 , 2 } ) = T _ 1 ( h _ 2 + E _ { 3 , 3 } ) = h _ 2 + h _ 1 + E _ { 3 , 3 } = E _ { 1 , 1 } , \\\\ T _ 1 ( E _ { 1 , 1 } ) = T _ 1 ( h _ 1 + E _ { 2 , 2 } ) = - h _ 1 + E _ { 1 , 1 } = E _ { 2 , 2 } . \\end{gather*}"}
-{"id": "8606.png", "formula": "\\begin{align*} \\Vert \\sigma ( D _ x ) f \\Vert _ { B ^ s _ { \\infty , \\infty } ( G ) } & = \\sup _ { l \\geq 0 } 2 ^ { l s } \\Vert \\psi _ l ( \\mathcal { R } ) \\sigma ( D _ x ) f \\Vert _ { L ^ \\infty ( G ) } \\\\ & \\lesssim \\sup \\{ C _ { \\alpha } : { | \\alpha | \\leq k } \\} \\sup _ { l \\geq 0 } 2 ^ { l s } \\Vert \\psi _ l ( \\mathcal { R } ) f \\Vert _ { L ^ \\infty ( G ) } \\\\ & \\asymp \\sup \\{ C _ { \\alpha } : { | \\alpha | \\leq k } \\} \\Vert f \\Vert _ { B ^ s _ { \\infty , \\infty } ( G ) } . \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} \\oplus _ { \\lambda \\in \\Lambda } ( p _ m | _ { E _ \\lambda } ) = \\pi ( p _ m ) \\neq 0 . \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} \\int F _ i ( y _ 1 , . . . , y _ { i - 1 } , z ) d \\mu ( z ) = 0 , \\ , \\ , \\forall \\ , y _ 1 , . . . , y _ { i - 1 } \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} \\rho _ { s , 2 } ( \\varphi b ) \\ , : = \\ , { \\mathbb E } _ { 0 } \\big [ Z _ { s } ^ { - 1 } \\varphi ( \\widetilde { X } _ { s } ) b ( s , X _ { s } , F _ { s } ) \\vert \\mathcal F _ { T } ^ { X } \\big ] \\ , , \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} u ^ { f } ( x , v ) = w ( 0 , x , v ) . \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} D _ \\alpha ( \\rho \\| \\sigma ) : = { 1 \\over \\alpha - 1 } \\log { Q _ \\alpha ( \\rho \\| \\sigma ) \\over \\rho ( 1 ) } . \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} \\ , ( X _ { s } ( \\omega _ { i } ) , 0 \\le s \\le T ) \\ , = \\ , \\mu _ { i } \\ , \\ , \\ , \\ , i \\ , = \\ , 1 , 2 \\ , \\ , ( X _ { s } ( \\omega _ { 1 } ) , X _ { s } ( \\omega _ { 2 } ) , 0 \\le s \\le T ) \\ , = \\ , \\mu \\ , . \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} ( K ^ * x ) ( t ) = \\big ( x , u ( t ) \\big ) _ X . \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} \\theta = \\theta ' _ t + \\iota _ t ( \\theta '' _ t ) = \\theta ' _ { t ' } + \\iota _ { t ' } ( \\theta '' _ { t ' } ) , \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\mu } t ^ { \\tilde { \\beta } _ i } = t + \\sum _ { i } ( \\delta _ i - 2 ) \\left ( t ^ { \\nu _ i / N _ i } \\frac { 1 - t } { 1 - t ^ { 1 / N _ i } } \\right ) , \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} \\norm { f } _ { \\mathcal { L } _ { \\mathrm { S c a l } } ( p ) } ^ 2 : = \\int _ { \\R } \\int _ { \\mathbb { B } _ p } \\abs { f ( t , \\zeta ) } ^ 2 \\lambda ( p ( \\zeta ) , t ) \\d V ( \\zeta ) \\d t < \\infty , \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} \\widehat b _ { T , h } ( x ) - b ( x ) = \\frac { B _ { T , h } ( x ) } { \\widehat f _ { T , h } ( x ) } + \\frac { S _ { T , h } ( x ) } { \\widehat f _ { T , h } ( x ) } , \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} | \\bar { \\Omega } _ { i j } | \\geq m | i ^ 2 - j ^ 2 | \\geq m \\max \\{ | i | , | j | \\} > m C _ 0 K = 2 | \\omega | _ { \\mathcal { O } } K , \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{gather*} \\begin{binom} { n + 1 } { k } \\end{binom} \\left ( \\prod _ { l = k } ^ { n } ( m - l ) \\right ) . \\end{gather*}"}
-{"id": "7335.png", "formula": "\\begin{align*} \\int _ G f ( k x h ^ { - 1 } ) d \\tilde { \\mu } ( x ) = \\Delta _ K ( k ) \\Delta _ H ( h ) \\int _ G f ( x ) d \\tilde { \\mu } ( x ) . \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta w _ k + \\lambda w _ k + \\rho ( x _ 0 + \\epsilon _ k x ) \\phi _ { u _ k } ( x _ 0 + \\epsilon _ k x ) w _ k = w _ k ^ p , & x \\in \\R ^ 3 \\\\ \\ , \\ , \\ , - \\Delta \\phi _ { u _ k } ( x _ 0 + \\epsilon _ k x ) = \\rho ( x _ 0 + \\epsilon _ k x ) w _ k ^ 2 , & x \\in \\R ^ 3 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} g ( X , Y ) = h ( C X , Y ) = h ( X , C Y ) , & & ( \\forall ) X , Y \\in T M \\bigr | _ { U ^ c } . \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} w _ { \\sigma } ^ 1 = s _ { \\rho _ { 2 , \\sigma } } , \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} { \\sf M } \\left \\{ \\left ( I _ { T , t } ^ { ( i _ 1 i _ 2 ) } - I _ { T , t } ^ { ( i _ 1 i _ 2 ) q } \\right ) ^ { 2 n } \\right \\} \\le C _ { n , 2 } \\left ( \\frac { 3 ( T - t ) ^ { 2 } } { 2 \\pi ^ 2 } \\left ( \\frac { \\pi ^ 2 } { 6 } - \\sum \\limits _ { r = 1 } ^ q \\frac { 1 } { r ^ 2 } \\right ) \\right ) ^ n \\ \\to 0 \\ \\ \\ \\hbox { i f } \\ q \\to \\infty , \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } u _ { t } ^ { \\theta } ( 0 , t _ { 0 } ) + H _ { i } ( t , x , u _ { x _ { i } } ^ { \\theta } ) - \\tilde { C } _ { \\zeta } = 0 & \\ , \\ , I _ { i } ^ { \\nu } \\times ( t _ { 0 } - \\nu , ( t _ { 0 } + \\nu ) \\wedge T ) \\\\ \\sum _ { i = 1 } ^ { K } u _ { x _ { i } } ^ { \\theta } = 0 & \\ , \\ , \\{ 0 \\} \\times ( t _ { 0 } - \\nu , ( t _ { 0 } + \\nu ) \\wedge T ) \\end{array} \\right . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} f ( 0 ) = { } & 0 = f ' ( 0 ) , \\\\ f '' ( 0 ) = { } & 2 g ( J _ 1 ' ( 0 ) , J _ 2 ' ( 0 ) ) = 2 g ( W _ 1 ' ( 0 ) , W _ 2 ' ( 0 ) ) , \\\\ f ''' ( 0 ) = { } & \\partial _ r [ g \\left ( R ^ g ( \\partial _ r , J _ 1 ( r ) ) \\partial _ r , J _ 2 ( r ) \\right ) + 2 g ( J _ 1 ' ( r ) , J _ 2 ' ( r ) ) + g \\left ( J _ 1 ( r ) , R ^ g ( \\partial _ r , J _ 2 ( r ) ) \\partial _ r \\right ) ] \\bigr | _ { r = 0 } = 0 \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} s : = \\min \\{ 2 \\le k \\le \\ell \\mid \\epsilon _ { k - 1 } + \\epsilon _ k \\in I \\} . \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} y + z = x \\ge y - z y + z = x \\ge z - y , \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} \\mathbf { L } _ { \\mathcal { G } } \\mathbf { X } = \\mathbf { 0 } , \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\left ( ( E _ i \\otimes 1 ) \\circ f _ { M , M ' } \\right ) _ { | _ { M _ { \\lambda , \\mu } \\otimes M ' _ { \\lambda ' , \\mu ' } } } = q ^ { \\langle \\lambda , \\mu ' \\rangle } ( E _ i \\otimes 1 ) _ { | _ { M _ { \\lambda , \\mu } \\otimes M ' _ { \\lambda ' , \\mu ' } } } , \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} \\frac { S ^ \\chi _ { ( \\lambda , \\lambda ' ) , ( \\mu , \\mu ' ) } } { S ^ \\chi _ { ( 0 , 0 ) , ( \\mu , \\mu ' ) } } = \\frac { e ^ { 2 \\pi i \\left ( ( \\lambda ' , \\mu + \\rho ) + ( \\lambda , \\mu ' + \\rho ) \\right ) } \\chi _ \\mu ( \\lambda ) \\chi ' _ { \\mu ' } ( \\lambda ' ) } { \\chi _ \\mu ( 0 ) \\chi ' _ { \\mu ' } ( 0 ) } . \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\mathcal { Q } _ N : = \\left \\{ q \\in L ^ { \\infty } ( \\Omega ) : \\ q \\mbox { i s p i e c e w i s e c o n s t a n t } , \\ \\| q \\| _ { L ^ { \\infty } ( \\Omega ) } \\leq \\frac { \\lambda _ 1 } { 2 } \\right \\} , \\ d _ { \\mathcal { Q } _ N } ( q _ 1 , q _ 2 ) = \\| q _ 1 - q _ 2 \\| _ { L ^ { \\infty } ( \\Omega ) } . \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} \\begin{array} { l l } \\underset { y } { \\min } & f ( x ^ k , y ) \\\\ s . a . & h ( x ^ k , y ) = 0 \\\\ & y \\geq 0 \\end{array} \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} v _ { + , a } ( x ) = 2 \\phi _ 1 ( x ) \\int _ x ^ a d y \\thinspace \\phi ^ { - 2 } _ 1 ( y ) \\int _ { - \\infty } ^ y \\phi _ 1 ( z ) d z , \\ x \\le a ; \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} H _ I \\coloneqq \\{ h \\in H \\mid \\alpha _ i ( h ) = 1 i \\in I \\} ^ 0 . \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sigma _ k \\| w ^ k \\| = 0 . \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n } ( i _ 0 ^ * i ^ * j _ * \\mathcal { R } _ { \\ell } ( ^ { \\lambda } \\mathcal { V } ) ) \\simeq R ^ { n } ( i _ 0 ^ * i ^ * j _ * \\mathcal { R } _ { \\ell } ( ^ { \\lambda } \\mathcal { V } ) ) = R ^ { n - w ( \\lambda ) } i _ 0 ^ * i ^ * j _ * \\mu _ { \\ell } ^ K ( V ) . \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} L ^ \\nu b _ { \\xi } ( x ) = \\sc | A _ { \\xi } | ^ { 1 / 2 } b ^ { - 2 j ( K - \\nu ) } g ( b ^ { - j } \\sqrt { L } ) ( x , \\xi ) . \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} a ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { | T ' ( x ) | } a ( x , y , t ) \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} - \\mathcal { L } [ \\nu ] + | D \\nu | ^ m + \\sigma \\nu - f = \\sigma \\Theta - \\sigma M \\mathcal { A } \\ , . \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} L ( \\xi ) = a ( \\xi ) I _ { 1 1 \\times 1 1 } \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} \\mathbb { A _ { U E } } = \\left [ X _ { 1 1 } \\leq M ^ { \\frac { 1 } { \\alpha \\mu } } \\left ( \\frac { N } { P } \\right ) ^ { \\frac { 2 - \\mu } { \\alpha \\mu } } X _ { 2 1 } ^ { \\frac { 2 } { \\mu } } \\right ] . \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} a ( v b ) = v ( \\bar { a } b ) , ( v b ) a = v ( a b ) , ( v a ) ( v b ) = b \\bar { a } , a , b \\in M _ 2 ( F ) . \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} \\int _ { Q } B _ { { l } ( Q ) } ( f , g ) ^ { t } v d x & \\leq \\left ( \\int _ { Q } B _ { l ( Q ) } ( f , g ) ( x ) d x \\right ) ^ { t } \\left ( \\int _ { Q } v ^ { \\frac { 1 } { 1 - t } } d x \\right ) ^ { 1 - t } \\\\ & = \\left ( \\int _ { Q } \\int _ { | y | _ { \\infty } \\leq l ( Q ) } f ( x - y ) g ( x + y ) d y d x \\right ) ^ { t } \\left ( \\int _ { Q } v ^ { \\frac { 1 } { 1 - t } } d x \\right ) ^ { 1 - t } . \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} \\frac D 2 \\phi '' ( x ) - r ( x ) \\phi ( x ) = 0 , \\ x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} \\displaystyle \\delta _ \\Gamma = \\underset { n \\to \\infty } { \\overline { \\lim } } \\frac { 1 } { 2 n } \\log \\frac { ( q - 1 ) q ^ { s _ n } } { q } = \\underset { n \\to \\infty } { \\overline { \\lim } } \\frac { s _ n } { 2 n } \\log q . \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} & ( 1 , ( 1 2 ) ( 3 4 ) , ( 1 3 ) ( 2 4 ) , ( 1 4 ) ( 2 3 ) , ( 1 2 3 ) , ( 2 4 3 ) , ( 1 4 2 ) , ( 1 3 4 ) , ( 1 3 2 ) , ( 1 4 3 ) , ( 2 3 4 ) , ( 1 2 4 ) ) \\\\ & \\ ; \\ ; \\ ; \\ ; = ( 1 , \\beta , \\alpha ^ 2 \\beta \\alpha , \\alpha \\beta \\alpha ^ 2 , \\alpha , \\beta \\alpha , \\alpha ^ 2 \\beta \\alpha ^ 2 , \\alpha \\beta , \\alpha ^ 2 , \\beta \\alpha ^ 2 , \\alpha ^ 2 \\beta , \\alpha \\beta \\alpha ) . \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} \\xi ^ a = \\eta _ { \\dot a } = \\pm \\sqrt { \\frac { m a } { \\sqrt c } \\ , \\operatorname { a r c t a n h } \\left ( \\frac { \\sqrt c } 2 \\ , \\right ) } \\begin{pmatrix} 0 \\\\ i \\end{pmatrix} e ^ { i m x ^ 4 } . \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} \\delta _ { i } ^ { ( j ) } = \\delta _ { 1 } ^ { ( j ) } ( 2 ^ { R ^ { t h } } - 1 + C ) ( 2 ^ { R ^ { t h } } + C ) ^ { i - 2 } . \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} \\Big ( \\left ( H ( x ) \\cap H ( z ) \\right ) \\setminus \\left ( H ( x ) \\cap H ( y ) \\right ) \\Big ) \\cap G = \\emptyset . \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} G ( t , q , x ) = 1 + \\frac { e ^ { x ( 1 - t ) } - 1 } { 1 + q - ( t + q ) e ^ { x ( 1 - t ) } } . \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ n \\mu _ i ^ { r _ e } x _ i \\right | _ { e \\geq 0 } \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} \\sigma ^ { n - \\theta } - \\bigtriangleup u ^ { n - \\theta } = 0 . \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} S _ f ( \\lambda \\rho \\| \\lambda \\sigma ) = \\lambda S _ f ( \\rho \\| \\sigma ) . \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} h _ { \\alpha \\beta } ( x ) = [ D ^ \\mu { } _ \\alpha ( x ) ] \\ , [ g _ { \\mu \\nu } ( x ) ] \\ , [ D ^ \\nu { } _ \\beta ( x ) ] \\ , . \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} \\{ u _ h > 0 \\} \\subset \\{ w > 0 \\} = \\{ x _ N < t _ h \\} . \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} \\kappa ( \\theta ) = \\lim _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log \\mathbb { E } e ^ { \\theta M ( t ) } , \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} \\frac { 1 } { a _ n + a _ n ^ { \\prime } } = O ( 1 ) . \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} a _ { N } ' = \\frac { a _ { N } } { ( 1 - C N ^ { - \\frac { 1 } { 3 } } ) ( 1 - N ^ { - \\frac { 1 } { 3 } } ) } < a _ { * } . \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} \\hat a _ { i , i ' } ( \\mu ) : = \\sum _ { \\eta \\in \\Omega _ L } 2 \\bigl [ \\mu ( \\eta ) \\hat r ^ V _ { \\eta , \\eta ^ { i , i ' } } \\mu ( \\eta ^ { i , i ' } ) \\hat r ^ V _ { \\eta ^ { i , i ' } , \\eta } \\bigr ] ^ { 1 / 2 } , \\hat \\chi ^ V _ { i , i ' } ( \\mu ) : = \\frac 1 2 \\sum _ { \\eta \\in \\Omega _ L } \\mu ( \\eta ) \\bigl ( \\hat r ^ V _ { \\eta , \\eta ^ { i , i ' } } + \\hat r ^ V _ { \\eta , \\eta ^ { i ' , i } } \\bigr ) , \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} \\frac { 1 } { ( r + k + 1 ) ( k + 1 ) ^ s } = \\sum _ { j = 1 } ^ { s } \\frac { ( - 1 ) ^ { j - 1 } } { r ^ j ( k + 1 ) ^ { s + 1 - j } } + \\frac { ( - 1 ) ^ s } { r ^ s ( r + k + 1 ) } , \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} N = ( 1 - e ^ { - t } ) + 2 S ( t ) \\Big ( S ( t ) + C ( t ) \\Big ) \\le 1 + \\frac { 4 } { r _ 1 } \\le 2 ~ . \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} p ^ { n , \\omega ^ \\prime } _ t = X ^ { n , \\omega ^ \\prime } _ { \\kappa ( n , t ) } - X ^ { n , \\omega ^ \\prime } _ { t ^ - } , t \\in ( - \\tau , \\infty ) \\ , . \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} \\Psi ( a ^ * t b ) = \\Psi ( a ) ^ * \\Psi ( t ) \\Psi ( b ) \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} ( f \\cup g ) ( a _ 1 , \\ldots , a _ { m + n } ) = \\mu ( f ( a _ 1 , \\ldots , a _ m ) , g ( a _ { m + 1 } , \\ldots , a _ { m + n } ) ) . \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} d ( q + _ o a , a , b ) = q + _ o b . \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} & T _ j \\big ( \\prod _ { i = 1 } ^ { n } ( x _ i + \\hat { x } _ i ) ( y _ i + \\hat { y } _ i ) ( z _ i + \\hat { z } _ i ) ( w _ i + \\hat { w } _ i ) \\big ) \\\\ = & \\ ( x _ j + \\hat { x } _ j + z _ j ) \\prod _ { k \\neq j } ( x _ k + \\hat { x } _ k ) \\times \\prod _ { i = 1 } ^ { n } ( y _ i + \\hat { y } _ i ) ( z _ i + \\hat { z } _ i ) ( w _ i + \\hat { w } _ i ) \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} ( a \\cdot \\Phi ) ( x ) = \\delta ^ { \\frac { 1 } { 2 } } ( a ) \\Phi ( a \\cdot x ) \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{gather*} T _ { i } ( E _ { j , j } ( s ) ) = \\begin{cases} E _ { i + 1 , i + 1 } ( s ) & j = i , \\\\ E _ { i , i } ( s ) & j = i + 1 , \\\\ E _ { j , j } ( s ) \\ & , \\end{cases} i \\neq 0 . \\end{gather*}"}
-{"id": "8709.png", "formula": "\\begin{align*} w : = u _ { \\eta \\bar \\eta } , \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { \\partial } { \\partial t } w _ n = \\mathcal { L } w _ n = \\frac D 2 w _ n '' + r ( x ) \\big ( w _ n ( 0 , t ) - w _ n ( x , t ) \\big ) , \\ x \\in ( - n , a ) ; \\\\ & w _ n ( x , 0 ) = 1 , \\ x \\in ( - n , a ) ; \\ \\ w _ n ( a , t ) = w _ n ( - n , t ) = 0 , \\ t > 0 . \\end{aligned} \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} \\tfrac 1 2 ( i + 1 ) ( i - 2 ) + \\tfrac 1 2 ( i + 3 ) i - ( i + 2 ) ( i - 1 ) - 1 = 0 , \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t h ^ i ( t ) + \\frac 1 2 \\Delta h ^ i ( t ) = ( \\nabla \\gamma ^ i ) ^ * ( t ) \\ , b ( t ) + f ( t , \\cdot , \\gamma ^ i , \\nabla \\gamma ^ i ) \\\\ h ^ i ( T ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} \\lvert ( n , 0 ) \\rvert _ { W _ n ( w ) } = \\lvert ( n , 0 ) \\rvert _ { W _ n ( w _ 0 ) } + \\lvert r _ 0 \\rvert _ S = \\lVert ( \\phi w _ 0 ) _ n \\rVert _ S + \\lvert m _ k \\rvert . \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} | ( i + j ) - \\frac { 2 } { 2 n - 1 } \\sum _ { b = 1 } ^ n j _ b | > \\frac { 1 } { 2 n - 1 } , \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} F ^ { ( p r ) } ( t ) & = b _ p + \\left ( f ( - t ) - \\sum _ { k = 1 } ^ { p } b _ k \\frac { t ^ { k r } } { ( k r ) ! } \\right ) ^ { ( p r ) } ( t ) \\\\ & = b _ p + \\frac { 1 } { 2 \\pi i } \\int _ { 1 - \\infty \\ , i } ^ { 1 + \\infty \\ , i } e ^ { - t z } ( - z ) ^ { p r } \\left ( \\frac { \\varphi ( z ) } { z } - \\sum _ { k = 1 } ^ { p } \\frac { ( - 1 ) ^ { k r } b _ k } { z ^ { k r + 1 } } \\right ) \\ , d z , \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} \\rho : = - \\sum _ { i = 1 } ^ N i \\epsilon _ i \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} \\frac { \\partial g } { \\partial \\overline { w } _ j } = 0 1 \\leq j \\leq n , \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} \\overline { \\mathcal { R } ( C P ) } = \\mathcal { R } ( C P ) = \\mathcal { R } ( C A ^ * ) . \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} \\Lambda ( T ) = \\underset { k \\to + \\infty } { \\lim } \\ \\big \\| T ^ k \\big \\| _ { \\mathcal { B } ( E ) } ^ { \\frac { 1 } { k } } = \\underset { k \\geq 1 } { \\inf } \\ \\big \\| T ^ k \\big \\| _ { \\mathcal { B } ( E ) } ^ { \\frac { 1 } { k } } . \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} ( v '' _ { \\lambda } ( z ) , v _ { \\lambda } ( z ) ) _ { H } & = z ^ { \\frac { 1 - 2 s } { s } } ( A _ { \\lambda } v _ { \\lambda } ( z ) , v _ { \\lambda } ( z ) ) _ { H } + z ^ { \\frac { 1 - 2 s } { s } } \\ , \\delta \\ , \\norm { v _ { \\lambda } ( z ) } ^ 2 _ { H } \\\\ & \\ge z ^ { \\frac { 1 - 2 s } { s } } \\ , \\delta \\ , \\norm { v _ { \\lambda } ( z ) } ^ 2 _ { H } \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} r ( B ) : = \\sum _ { i \\in B } r _ i . \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } k _ { 1 , \\sigma } = \\kappa & \\forall \\sigma \\in I _ F ^ 0 \\\\ k _ { 2 , \\sigma } - 1 = \\kappa & \\forall \\sigma \\in I _ F ^ 1 \\\\ - ( k _ { 2 , \\sigma } + 3 ) = \\kappa & \\forall \\sigma \\in I _ F ^ 2 \\\\ - ( k _ { 1 , \\sigma } + 4 ) = \\kappa & \\forall \\sigma \\in I _ F ^ 2 \\end{array} \\right . \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l i m } \\cr $ \\stackrel { } { { } _ { p _ 1 , \\ldots , p _ k \\to \\infty } } $ \\cr } } } \\Biggl \\Vert K ( t _ 1 , \\ldots , t _ k ) - \\sum _ { j _ 1 = 0 } ^ { p _ 1 } \\ldots \\sum _ { j _ k = 0 } ^ { p _ k } C _ { j _ k \\ldots j _ 1 } \\prod _ { l = 1 } ^ { k } \\phi _ { j _ l } ( t _ l ) \\Biggr \\Vert _ { L _ 2 ( [ t , T ] ^ k ) } = 0 , \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\lVert ( v _ n - v ) ( \\omega ^ { - 1 / 2 } + 1 ) \\lVert = 0 \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} \\tilde { T } ( F ) : = \\{ \\mbox { d i a g } ( \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 1 ^ { - 1 } \\nu , \\alpha _ 2 ^ { - 1 } \\nu ) \\ \\vert \\ \\alpha _ 1 , \\alpha _ 2 , \\nu \\in \\mathbb { G } _ m ( F ) \\} , \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} f ( t ) = a + b ( t - 1 ) + c ( t - 1 ) ^ 2 + \\int _ { [ 0 , + \\infty ) } { ( t - 1 ) ^ 2 \\over t + s } \\ , d \\mu ( s ) , t \\in ( 0 , + \\infty ) , \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} b _ { i } = \\epsilon ^ { - 1 } \\left ( \\epsilon - H _ { 1 } ( 0 , 0 , u _ { 0 , x _ { 1 } } ( 0 ) ) + H _ { i } ( 0 , 0 , u _ { 0 , x _ { i } } ( 0 ) ) \\right ) . \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\varepsilon \\Delta \\phi - \\mathcal { L } _ R ^ { \\psi } [ \\phi ] = f \\ , , & x \\in B _ R , \\\\ \\phi = g & x \\in \\partial B _ R \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} ( \\tau _ z \\omega ) ( x , y ) : = \\omega ( x + z , y + z ) , \\forall \\{ x , y \\} \\in E . \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ j \\binom k j G _ j } = F _ { k + 1 } G _ { 0 } - F _ { k } G _ 1 \\ , . \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} \\tilde E = \\{ ( x , a ' ) \\in E \\times A ' \\mid p ( x ) = f ( a ' ) \\} . \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} p _ i = \\begin{cases} [ a _ { 1 , i } , b _ { 1 , i } , a _ { 2 , i } ] & \\mbox { i f $ i \\equiv 0 \\pmod { 4 } $ } \\\\ [ a _ { 3 , i } , b _ { 2 , i } , a _ { 4 , i } ] & \\mbox { i f $ i \\equiv 1 \\pmod { 4 } $ } \\\\ [ a _ { 4 , i } , b _ { 2 , i } , a _ { 3 , i } ] & \\mbox { i f $ i \\equiv 2 \\pmod { 4 } $ } \\\\ [ a _ { 2 , i } , b _ { 1 , i } , a _ { 1 , i } ] & \\mbox { i f $ i \\equiv 3 \\pmod { 4 } $ } \\\\ \\end{cases} \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - \\lambda t } \\bigl ( D ^ { \\frac { 1 } { 2 } } f \\bigr ) ( t ) \\ , \\mathrm { d } t = \\lambda ^ { \\frac { 1 } { 2 } } \\tilde { f } ( \\lambda ) - \\lambda ^ { \\frac { 1 } { 2 } - 1 } f ( 0 ) , \\quad \\lambda > 0 . \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ d T _ k T _ k ^ * \\leq I . \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} \\frac { d } { d s } \\varphi _ s = X _ 0 ( \\varphi _ s ) = ( p \\partial _ q + q \\partial _ p ) \\varphi _ s { ~ . } \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} & \\{ R = I _ 3 , p = 0 \\} \\{ R = \\mathcal { R } _ a ( \\pi , v ) , p = ( I _ 3 - \\mathcal { R } _ { a } ( \\pi , v ) ) b d ^ { - 1 } \\} , \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} H _ { j , i } = H _ { i } ^ { j - 1 } , j \\in [ r ] , i \\in [ n ] . \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} \\mathfrak F _ \\chi = \\gamma ( \\chi , 0 , \\psi ) \\mathfrak R _ \\chi . \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} = \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { ( k + 1 ) ^ { s - 3 } } \\sum _ { r _ 1 = 0 } ^ { n } \\sum _ { r _ 2 = 0 } ^ { n } \\sum _ { r _ 3 = 0 } ^ { n } \\frac { a _ { r _ 1 } b _ { r _ 2 } c _ { r _ 3 } } { ( r _ 1 + k + 1 ) ( r _ 2 + k + 1 ) ( r _ 3 + k + 1 ) } . \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} u _ { 0 , l } = \\sum _ { \\nu \\in A } \\sum _ { \\ell = 1 } ^ { d ( \\nu ) } a _ { \\nu , \\ell } ( r ) \\varphi _ { \\nu , \\ell } ( y ) , A = \\{ \\nu \\in \\chi _ \\infty : \\nu \\leq 1 + \\nu _ 0 \\} . \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} \\nu ( p ) = \\frac { \\nabla \\phi ( p ) } { \\big \\Vert \\nabla \\phi ( p ) \\big \\Vert } , p \\in \\partial \\Omega , \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} I ( \\tilde { P } _ q ( h ) ) = I _ { \\varphi } ( \\tilde { P } _ q ( h ) ) : = \\int _ { { \\tilde { P } _ q ( h ) } } \\varphi ( x ) \\ , d x \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} \\lambda & = \\frac { n ( a ^ 2 - a ( b - 1 ) b + b ^ 3 - 2 b \\ell ) + 2 ( b - \\ell ) ( a ^ 2 + a b - b ( b + \\ell ) ) } { ( a - b ) ^ 2 ( a + b ) } , \\\\ \\mu & = \\frac { b n ( - a b + a + b ^ 2 + b - 2 \\ell ) + 2 b ( a - \\ell ) ( b - \\ell ) } { ( a - b ) ^ 2 ( a + b ) } . \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} g ^ * \\o _ { \\tilde { F } } & = \\frac { 1 } { \\mu ( G ) } \\int _ { G } g ^ * ( h ^ * \\o _ 1 ) d \\mu ( h ) \\\\ & = \\frac { 1 } { \\mu ( G ) } \\int _ { G } ( h g ) ^ * ( \\o _ 1 ) d \\mu ( h ) = \\frac { 1 } { \\mu ( G ) } \\int _ { G } k ^ * \\o _ 1 d \\mu ( k ) = \\o _ { \\tilde { F } } , \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} W ^ { ( \\alpha , \\beta ) } _ { \\ast } f ( n ) = \\sup _ { t > 0 } | W _ t ^ { ( \\alpha , \\beta ) } f ( n ) | \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} u = \\chi \\left ( \\log | f _ 1 | , \\ldots , \\log | f _ r | \\right ) \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} x \\mapsto \\ ! x ^ 3 , y \\mapsto x ^ 4 ( 1 - y ) \\Longrightarrow \\mathcal { I } _ { \\beta _ 1 , \\beta _ 2 , \\beta _ 3 } \\ ! \\ ! = \\ ! \\ ! 3 \\ ! \\ ! \\ ! \\int _ { [ 0 , 1 ] ^ 2 } \\ ! \\ ! \\ ! \\tilde { f } _ { \\mathbf { t } } ( x , y ) ^ s x ^ { 3 \\beta _ 1 + 4 \\beta _ 2 + 1 2 \\beta _ 3 + 2 4 s } y ^ { \\beta _ 3 + 1 } q ( y ) \\frac { d x } { x } \\ ! \\frac { d y } { y } \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} H _ 0 \\cap H _ 0 ^ { \\perp _ { \\mathcal Q } } \\subseteq \\{ x \\in H _ 0 \\colon \\mathcal Q ( x , x ) = 0 \\} = \\{ 0 \\} . \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} | k | < \\big ( \\frac { n } { 2 n - 1 } + \\frac { 1 } { 1 0 0 \\sum _ { b = 1 } ^ n | j _ b | } \\big ) | k | , \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} \\alpha _ j = \\int _ { a ^ - _ j } ^ { a ^ + _ j } \\int _ { a ^ - _ j } ^ { a ^ + _ j } P ( y _ { B } , y _ { C } ) d y _ B d y _ C . \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} \\inf _ { j \\geq 1 } \\lambda ^ \\alpha _ j = \\inf _ { j \\geq 1 } \\left ( \\frac { \\sqrt { M } } { \\| V ^ j \\| _ { L ^ 2 } } \\right ) ^ \\alpha = \\left ( \\frac { \\sqrt { M } } { \\| V ^ { j _ 0 } \\| _ { L ^ 2 } } \\right ) ^ \\alpha . \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} \\begin{aligned} & D \\lambda _ 1 c _ 1 + D \\lambda _ 1 c _ 2 x ^ 2 + D \\lambda _ 1 | x | ^ { m } \\ge \\\\ & D c _ 2 \\gamma _ 1 + D c _ 2 x ^ 2 + D ( \\lambda _ 1 - \\delta _ 1 ) \\gamma _ 1 | x | ^ { m - 2 } + D ( \\lambda _ 1 - \\delta _ 1 ) | x | ^ { m } . \\end{aligned} \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} X _ 1 = \\bigg \\{ \\psi ( t ) , \\ t \\in [ 0 , 1 ] \\bigg | \\ & \\psi ( 0 ) = \\psi _ 0 , \\ \\| \\psi ( t ) \\| _ 2 \\leq 2 C _ 1 ( 1 + t ) ^ { - \\frac { d } { 4 } } \\| \\psi _ 0 \\| _ 1 \\ \\ { \\rm a n d } \\ \\\\ & \\sqrt { Q ( \\psi ( t ) ) } \\leq 2 C _ 1 C _ Q ( 1 + t ) ^ { - \\frac { d } { 4 } - \\frac { \\eta } { 2 } } \\| \\psi _ 0 \\| _ 1 , \\ 0 \\leq t \\leq 1 \\bigg \\} . \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} \\min _ { \\beta _ { \\ell } \\in \\mathbb { R } } ~ ~ \\left \\{ \\sum _ { j = 0 } ^ N \\omega _ j \\left ( \\sum _ { \\ell = 0 } ^ { L } \\beta _ { \\ell } \\tilde { \\Phi } _ { \\ell } ( x _ j ) - f ( x _ j ) \\right ) ^ 2 + \\lambda \\sum _ { \\ell = 0 } ^ L | \\mu _ { \\ell } \\beta _ { \\ell } | \\right \\} , \\quad \\lambda > 0 , \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} \\big ( \\boldsymbol { \\varepsilon } ( \\mathbf { x } ) \\big ) ^ { T } = \\boldsymbol { \\varepsilon } ( \\mathbf { x } ) \\big ( \\boldsymbol { \\mu } ( \\mathbf { x } ) \\big ) ^ { T } = \\boldsymbol { \\mu } ( \\mathbf { x } ) \\mathbf { x } \\in \\bar { G } \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} b = \\pm \\sqrt { \\frac c { 4 m ^ 2 } - a ^ 2 } \\ , . \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { F } } [ \\overline { C } , q ] \\rightarrow \\min - \\nabla \\cdot q = S . \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} d \\mu _ { \\alpha , \\beta } ( x ) = ( 1 - x ) ^ \\alpha ( 1 + x ) ^ { \\beta } \\ , d x . \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} ( t , y ) \\cdot ( z ' , z _ n ) = ( t z ' , z _ n + y ) \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} Z ( v _ 1 ) \\in \\mathbb { R } _ { > 0 } Z ( v _ 2 ) , \\ v _ 1 + v _ 2 = v \\in \\Gamma _ X \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} f ( r ) = c _ { 1 } ( c _ { 2 } r ^ { q / p - 1 } + 1 ) ^ { \\frac { q } { p - q } } , \\ ; c _ { 1 } > 0 , \\ ; c _ { 2 } > 0 . \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} X = \\C ^ k \\times ^ \\pi M \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} \\lambda _ \\nu ( U \\cap B _ 1 ( 0 ) ) \\nu \\int _ { 0 } ^ { \\infty } f ( k e _ 1 ) k ^ { \\nu - 1 } d \\lambda _ 1 & = \\int _ { 0 } ^ { \\infty } f ( k e _ 1 ) d \\mu = \\int _ { U } f ( k ) d \\lambda _ \\nu \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} J = \\sum _ { t = 0 } ^ { N - 1 } \\| u ( t ) \\| _ 1 . \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} \\Sigma ( \\xi - k ) - \\Sigma ( \\xi ) = \\Sigma ( \\xi - k ) - \\Sigma \\left ( \\frac { \\lvert \\xi \\lvert } { \\lvert \\xi - k \\lvert } ( \\xi - k ) \\right ) \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } \\mathbb { E } \\left [ { \\int } _ { t } ^ { T } \\left \\vert \\tilde { Y } _ { s } ^ { m } - W \\left ( s , X _ { s } ^ { m } \\right ) \\right \\vert ^ { 2 } d s \\right ] & \\leq \\sum _ { i = 0 } ^ { m - 1 } \\int _ { t _ { i } ^ { m } } ^ { t _ { i + 1 } ^ { m } } \\mathbb { E } \\left [ \\left \\vert \\tilde { Y } _ { s } ^ { m } - W \\left ( s , X _ { s } ^ { m } \\right ) \\right \\vert ^ { 2 } \\right ] d s \\\\ & \\leq \\bar { C } ( 1 + | x | ^ { 2 } ) \\frac { T } { m } . \\end{array} \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} \\partial _ t ^ p u ( t _ 0 ) = S _ { \\chi , \\sigma , G , m , p } ( t _ 0 , f , u _ 0 ) p \\in \\{ 0 , \\ldots , m \\} \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\tilde { \\phi } = - ( - \\Delta ) ^ { s } \\phi + p ( u ^ * _ { \\mu , \\xi } ) ^ { p - 1 } \\tilde { \\phi } + \\tilde { N } ( \\tilde { \\phi } ) + S ( \\mu ^ * _ { \\mu , \\xi } ) , & \\Omega \\times ( t _ 0 , \\infty ) , \\\\ \\tilde { \\phi } = - u ^ * _ { \\mu , \\xi } , & ( \\mathbb { R } ^ n \\setminus \\Omega ) \\times ( t _ 0 , \\infty ) . \\end{cases} \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} d X _ t & = \\big [ A _ t x + B _ t \\alpha + \\beta _ t \\big ] d t + \\sigma d W _ t , X _ 0 = \\xi \\\\ d Y _ t & = \\bigg [ - \\frac { 1 } { 2 } P _ t X _ t ^ 2 + \\big ( A _ t X _ t + \\beta _ t \\big ) \\frac { Z _ t } { \\sigma } + \\frac { 1 } { 2 } \\frac { B _ t ^ 2 } { Q _ t } \\frac { Z _ t ^ 2 } { \\sigma ^ 2 } \\bigg ] d t + d W _ t , Y _ T = \\frac { 1 } { 2 } S X _ T ^ 2 . \\\\ \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} 0 = ( P u , u ) = ( - \\Delta u , u ) + ( V u , u ) = \\norm { \\nabla u } ^ 2 + ( V u , u ) \\geq ( ( \\lambda _ 1 + V ) u , u ) . \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} \\widetilde N ^ { [ q ] } _ \\kappa \\left ( r , \\frac { 1 } { f } \\right ) = \\int _ 0 ^ r \\frac { \\tilde n ^ { [ q ] } _ \\kappa ( t , 1 / f ) - \\tilde n ^ { [ q ] } _ \\kappa ( 0 , 1 / f ) } { t } \\ , d t + \\tilde n ^ { [ q ] } _ \\kappa ( 0 , 1 / f ) \\log r . \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} T _ g ( r ) \\leq \\sum _ { j = 1 } ^ { m + 1 } \\widetilde N ^ { [ m - 1 ] } _ \\kappa \\left ( r , \\frac { 1 } { g _ j } \\right ) + o \\left ( \\frac { T _ g ( r ) } { r ^ { 1 - \\varsigma ( g ) - \\varepsilon } } \\right ) , \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} \\Vert f _ h - f \\Vert _ p = \\left ( \\int _ { - \\infty } ^ { \\infty } | f ( t + h ) - f ( t ) | ^ p \\mathrm { d } t \\right ) ^ { 1 / p } \\leq C h ^ \\mu , \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} C _ { \\mu } = 2 c _ { \\mu , h } c ' \\mu ( W ) \\big ( 1 + \\mu _ f ^ \\star ( W ) \\big ) = \\frac { 2 } { \\mu _ h ( h ^ { - 1 } ) } \\big ( 1 + \\mu _ h ( W h ^ { - 1 } ) \\big ) \\big ( 1 + \\mu _ f ^ \\star ( W ) \\big ) \\frac { \\mu ( W ) } { \\mu ( h ) } . \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} \\begin{array} { c } \\nabla g = 0 , \\\\ c _ { i j k } + c _ { i k j } = 0 , \\\\ g _ { i j , k } + g _ { j k , i } + g _ { k i , j } = 0 , \\\\ w _ { i j } + w _ { j i } = 0 , \\\\ w _ { i j , l } - c ^ s _ { i j } w _ { s l } = 0 , \\\\ c _ { n m l , k } + c ^ s _ { m l } c _ { s n k } + w _ { m l } w _ { n k } = 0 . \\end{array} \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} \\Theta _ 2 : = \\begin{bmatrix} 1 & - \\frac { 1 } { 2 } | \\sin \\phi _ 2 | \\| b \\| \\\\ - \\frac { 1 } { 2 } | \\sin \\phi _ 2 | \\| b \\| & \\frac { 1 } { 4 } ( \\sin ^ 2 \\phi _ 2 \\| b \\| ^ 2 + d ^ 2 ) \\end{bmatrix} , \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} \\left \\vert \\rho ( t ) - \\rho _ { n , x , y } \\right \\vert \\ , f ( x ) \\leq 3 \\ , \\sup _ { | z - x | \\leq 3 h _ { n } / 2 } \\left \\vert f ( z ) - f ( x ) \\right \\vert = 3 \\ , q _ { n } ( x ) , \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow 0 } \\frac { x ^ { 1 - m / 2 } } { ( 2 t ) ^ { ( n - m ) / 2 } } \\psi \\left ( \\frac { x } { 2 t } \\right ) = 0 , t > 0 . \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} \\rho ( x h ) = \\frac { \\Delta _ H ( h ) } { \\Delta _ G ( h ) } \\rho ( x ) , ( x \\in G , \\ h \\in H ) . \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} \\widetilde { \\mathbf { S } } : = \\frac { 1 } { \\sqrt { \\dim ^ R ( \\bar { \\ 1 } ) } \\sqrt { \\dim ( \\mathcal { C } ) } } \\mathbf { S } . \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} \\sum ^ a _ { i = 1 } I _ { A _ i } - \\sum ^ b _ { i = 1 } I _ { B _ j } \\leq ( \\geq ) h I _ D . \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} ( F ( U ( t ) f ) ) ( s , y ) = ( U ( s ) ( U ( t ) f ) ) ( y ) = ( U ( s + t ) f ) ( y ) = ( F f ) ( s + t , y ) . \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} \\nabla _ v w = [ \\nabla _ v w ] ^ T + \\langle \\nabla _ v w , \\partial _ t \\rangle \\cdot \\frac { \\partial _ t } { - u ^ 2 } = ( \\nabla _ { g _ S } ) _ v w + \\frac { 1 } { 2 } d \\theta ( w , v ) \\cdot \\partial _ t . \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\# \\left ( \\mathcal { A } _ N \\cap \\Gamma _ \\alpha ^ { m , \\varepsilon } \\right ) } { \\mathcal { S } _ { \\mathcal { A } _ N , \\alpha } } = \\mathcal { R } ( \\Gamma _ \\alpha ^ { m , \\varepsilon } ) , 1 \\le m \\le M _ \\alpha ^ { ( \\varepsilon ) } \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} ( F - N c ) _ - = l ( c , N , F ) \\ , S , \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} f _ { r r } + ( \\frac { m - 1 } { r } - \\frac { r } { 2 } ) f _ r = ( - \\lambda + r ^ { - 2 } \\lambda _ k ) f . \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 7 2 n + 9 ) q ^ n \\equiv f _ 1 ^ 3 = f _ 3 a ( q ^ 3 ) - 3 q f _ 9 ^ 3 ~ ( \\textup { m o d } ~ 2 ) , \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} W _ { \\varphi _ { f } } = W _ { \\infty } \\otimes \\bigotimes _ { p < \\infty } W _ { p } . \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} \\underline { V } ^ { i j } ( x ) = \\sup \\limits _ { \\alpha \\in \\Gamma ^ i } \\inf \\limits _ { \\nu \\in \\mathcal { B } ^ j } J ( x , \\alpha ( \\nu ) , \\nu ) \\big ) . \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} K _ i K _ j = K _ j K _ i , K _ i K _ i ^ { - 1 } = 1 = K _ i ^ { - 1 } K _ i , \\forall 1 \\leq i , j \\leq n , \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} 1 + z \\frac { T _ g '' ( z ) } { T _ g ' ( z ) } = z \\frac { f ' ( z ) } { f ( z ) } + z \\frac { g '' ( z ) } { g ' ( z ) } + 1 . \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} ( b _ 1 , \\dots , b _ r ) \\cdot v = e ^ { \\lambda _ 1 } ( b _ 1 ) \\cdots e ^ { \\lambda _ r } ( b _ r ) v . \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} G ( z , z ' ) \\lesssim \\begin{cases} r '^ { - n } r ^ { - s } ( r / r ' ) ^ { s - \\frac n 2 + \\nu ' _ 0 } , r ' > 2 r ; \\\\ r ^ { - 1 } d ( z , z ' ) ^ { - ( n - 1 + s ) } , r \\sim r ' ; \\\\ r ^ { - n - s } ( r ' / r ) ^ { 1 - \\frac n 2 + \\nu ' _ 0 } , r ' < \\frac { r } { 2 } ; \\end{cases} \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } f ( x ) d x = \\int _ { 0 } ^ { \\infty } \\int _ { \\wp } f ( r y ) r ^ { Q - 1 } d \\sigma ( y ) d r \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\epsilon ^ 2 \\Delta u + \\lambda u + \\rho ( x ) \\phi u = | u | ^ { p - 1 } u , & x \\in \\R ^ 3 \\\\ \\ , \\ , \\ , - \\Delta \\phi = \\rho ( x ) u ^ 2 , & x \\in \\R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} \\Delta ( Z _ { \\mathcal I } , \\mathcal L _ { \\mathcal I , \\lambda _ 1 , \\dots , \\lambda _ r } , v , \\tau ) = \\Delta ( Z _ { \\mathbf i } , \\eta _ { \\mathbf i , \\mathcal I } ^ { \\ast } \\mathcal L _ { \\mathcal I , \\lambda _ 1 , \\dots , \\lambda _ r } , v , \\eta _ { \\mathbf i , \\mathcal I } ^ { \\ast } \\tau ) . \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} y ^ { \\prime \\prime } ( x ; \\tau ) + f ( x ; \\tau ) y ( x ; \\tau ) = 0 \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} u ( x , t ) = u _ { \\Omega } ( x ) ( u _ 0 ( x + X ) - u _ { 0 } ( x - X ) - 1 ) + A ( t ) ( \\chi _ 1 ( x + X ) - \\chi _ 1 ( x - X ) ) \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} \\varepsilon _ 1 = \\frac { \\varepsilon \\tilde { \\varepsilon } ( 1 - \\varepsilon / 3 ) | I \\cap V | } { 7 2 M } . \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\lvert f _ 2 ( g h ) - f _ 2 ( g ) - f _ 2 ( h ) \\rvert = & \\lvert f _ 2 ( g h ) - f _ 1 ( g h ) + f _ 1 ( g h ) \\\\ & - f _ 2 ( g ) - f _ 1 ( g ) + f _ 1 ( g ) \\\\ & - f _ 2 ( h ) - f _ 1 ( h ) + f _ 1 ( h ) \\rvert \\\\ \\leq & \\lvert f _ 1 ( g h ) - f _ 1 ( g ) - f _ 1 ( h ) \\rvert \\\\ & + \\lvert f _ 2 ( g h ) - f _ 1 ( g h ) \\rvert \\\\ & + \\lvert f _ 2 ( g ) - f _ 1 ( g ) \\rvert \\\\ & + \\lvert f _ 2 ( h ) - f _ 1 ( h ) \\rvert \\\\ \\leq & D ( f _ 1 ) + 3 \\lVert f _ 1 - f _ 2 \\rVert _ \\infty \\\\ < & \\infty . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} & { { a } _ { 1 } } = - { { b } _ { 2 } } = { { k } _ { 1 y } } + { { k } _ { 2 x } } \\\\ & { { b } _ { 1 } } = { { a } _ { 2 } } = { { k } _ { 1 x } } - { { k } _ { 2 y } } \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} d ^ * ( y _ 0 ) = k ^ * ( y _ 0 ) \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} a _ { n m } ( 1 ) & \\equiv f _ 0 ( 1 ) = G _ { g _ 0 } ( 1 ) = \\sum _ { d \\in D _ n } \\sum _ { e \\mid d } \\mu ( \\frac { d } { e } ) a _ { m e } ( 1 ) \\\\ & = \\sum _ { e \\mid n } a _ { m e } ( 1 ) \\sum _ { d : e \\mid d \\mid n , \\ , d \\neq n } \\mu ( \\frac { d } { e } ) \\bmod n ^ { 1 + \\min \\{ p - 2 , r - 1 \\} } . \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} i \\partial _ t \\mathbf { U } ( t , x ) - \\mathbf { H } ^ \\alpha _ \\delta \\mathbf { U } ( t , x ) + \\mathbf { U } ( t , x ) \\log | \\mathbf { U } ( t , x ) | ^ 2 = 0 , \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} \\Phi _ { M , R } \\circ \\Upsilon _ M ^ i = 0 , \\ \\ \\Upsilon _ { M , R } ^ i \\circ \\Upsilon _ M ^ j = 0 \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} d _ { 2 } ( v _ { 1 } \\vee \\dots \\vee v _ { k } ) = \\frac { 1 } { 2 } \\sum _ { \\substack { i + j = k \\\\ ( \\mu , \\nu ) \\in \\mathrm { S h } _ { i , j } } } \\pm [ v _ { \\mu ( 1 ) } \\vee \\dots \\vee v _ { \\mu ( i ) } , v _ { \\nu ( 1 ) } \\vee \\dots \\vee v _ { \\nu ( j ) } ] , \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} T _ n ( x ) = c _ 0 + c _ 1 x + \\dots + c _ n x ^ n ; \\ \\ \\ c ^ * = \\max _ { 0 \\leq i \\leq n } | c _ i | ; \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} \\Pi ( T ' ) = \\Pi ( G ' ) \\Pi ( \\Psi ( T ' ) ) . \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} \\beta _ { i , i + j } ( I ^ { \\tau ^ t } ) = \\sum _ { u \\in G ( I ^ { \\tau ^ t } ) _ j } \\binom { \\max ( u ) - 1 } { i } . \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\nu \\left ( \\mathrm { t r } \\left ( \\mathbf { S } \\right ) - P _ T \\right ) = 0 . \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} I ' ( u _ n ) ( v ) & = I ' _ { \\mu _ n } ( u _ n ) ( v ) + ( \\mu _ n - 1 ) \\int _ { \\R ^ 3 } u _ n ^ { p } v \\\\ & \\leq | \\mu _ n - 1 | \\ , | | u _ n | | ^ { p } _ { L ^ { p + 1 } } | | v | | _ { L ^ { p + 1 } } \\\\ & \\leq S _ { p + 1 } ^ { - \\frac { 1 } { 2 } } | \\mu _ n - 1 | \\ , | | u _ n | | ^ { p } _ { L ^ { p + 1 } } | | v | | _ { H ^ 1 } . \\\\ \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c r } p _ 1 = \\frac { \\partial L } { \\partial \\dot { x } _ 1 } = f ^ { - 1 } ( t ) m \\dot { x } _ 1 , \\\\ p _ 2 = \\frac { \\partial L } { \\partial \\dot { x } _ 2 } = f ^ { - 1 } ( t ) m \\dot { x } _ 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} \\sigma : G / H \\times G / H & \\longrightarrow H \\\\ ( \\bar g _ 1 , \\bar g _ 2 ) \\ & \\mapsto \\ s ( \\bar g _ 1 ) s ( \\bar g _ 2 ) s ( \\bar g _ 1 \\bar g _ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} J ( \\sigma ' ) = ( X \\cup B ) \\cap ( Y \\cap B _ 0 ) = E \\cap F \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} = \\| F ( u _ k ^ { \\delta } ) - v ^ { \\delta } \\| ^ { p } \\bigg [ - \\frac { \\mu } { 2 } + \\frac { \\mu } { 2 } L C _ F ^ 2 \\bigg ( \\frac { p } { C _ p } \\bigg ) ^ { 2 / p } \\| F ( u _ k ^ { \\delta } ) - v ^ { \\delta } \\| ^ { \\epsilon } \\bigg ] \\leq \\frac { \\mu } { 2 } L C _ F ^ 2 \\bigg ( \\frac { p } { C _ p } \\bigg ) ^ { 2 / p } \\delta ^ { \\epsilon } \\| F ( u _ k ^ { \\delta } ) - v ^ { \\delta } \\| ^ { p } . \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} H _ { \\infty } ( q , t ) = \\sum _ { p = 0 } ^ { \\infty } \\frac { q ^ { p ( p - 1 ) } t ^ p } { ( 1 - q ) ( 1 - q ^ 2 ) \\cdots ( 1 - q ^ { p } ) } . \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} L ( \\phi ^ { \\vee } , 0 ) = \\sum _ { a \\in A _ { + } } \\frac { \\mu ( a ) } { a } = L ( \\phi , A ) . \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} J ( f ) = J _ { S } ( f ^ n ) \\cup S . \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } p _ { n - 1 } \\left ( \\lambda _ { n + 1 } - \\lambda _ { n - 1 } \\right ) + a _ { n + 1 } p _ n \\left ( \\lambda _ { n + 1 } - \\lambda _ n \\right ) & = 0 \\\\ q _ { n - 1 } \\left ( \\lambda _ { n + 1 } - \\lambda _ { n - 1 } \\right ) + a _ { n + 1 } q _ n \\left ( \\lambda _ { n + 1 } - \\lambda _ n \\right ) & = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} f ' ( C _ \\infty ( x ) ) & = - ( 1 + \\gamma ) ( c _ 0 ^ 2 B ( x ) ^ 2 ) ^ { \\frac { \\gamma - 1 } { \\gamma + 1 } } \\leq 0 . \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} b u _ 4 ( t , 0 ) + \\int _ { t _ 2 } ^ { t _ 1 } \\lambda _ 4 \\alpha u _ 4 ( s , 0 ) \\ , d s = - a f ( t ) \\mbox { f o r } t \\in ( t _ 2 , t _ 1 ) . \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} e ^ { - V ( t ) } = \\int \\varrho ( d \\kappa ) \\exp \\big [ - \\frac { 1 } { 2 } \\kappa t ^ 2 \\big ] \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} \\kappa ( \\epsilon ) = \\mu ( I - e ^ { - \\alpha } \\tilde { P } ) ^ { - 1 } r + \\epsilon e ^ { - 2 \\alpha } \\mu \\tilde { P } ( I - e ^ { - \\alpha } \\tilde { P } ) ^ { - 2 } \\tilde { P } ^ { ( 1 ) } ( I - e ^ { - \\alpha } \\tilde { P } ) ^ { - 1 } r + O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ w ) \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} \\mathbb { B } _ L : = \\ , _ { c } \\mathbb { A } ^ { - 1 } _ L \\cdot \\mathbb { A } - \\mathbb { I } \\ ; \\ ; \\ ; \\ ; \\mathbb { B } _ R : = \\mathbb { A } \\cdot \\ , _ { c } \\mathbb { A } ^ { - 1 } _ R - \\mathbb { I } . \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{gather*} E _ z \\colon \\ y ^ 2 = x ( 1 - x ) ( x - z ) , z \\in \\mathbb C \\setminus \\{ 0 , 1 , \\infty \\} , \\end{gather*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\inf _ { \\gamma \\in W _ 1 ( y _ 0 ) } \\int _ { Y \\times U } k ( y , u ) \\gamma ( d y , d u ) = k ^ * ( y _ 0 ) , \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k Z ^ { \\phi } ( t _ i ) & = \\prod _ { i = 1 } ^ k \\phi _ 0 ( t _ i ) \\\\ & \\qquad + \\sum _ { \\emptyset \\neq H \\subset \\{ 1 , \\dots , k \\} } \\bigg ( \\prod _ { i \\notin H } \\phi _ 0 ( t _ i ) \\sum _ { \\substack { 1 \\le j _ i \\le \\xi _ 0 ( t _ i ) , \\\\ \\forall i \\in H } } \\ \\prod _ { i \\in H } Z ^ { \\phi } _ { j _ i } ( t _ i - \\sigma _ { j _ i } ) \\bigg ) . \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} U + { \\cal K } ( U ) = F \\mbox { i n } { \\cal X } . \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} I H _ { \\mathcal S } ( t ) = H _ { \\widetilde { \\mathcal S } } ( t ) - H _ { \\Delta } ( t ) g ( t ) . \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} [ c , d ] \\Delta ^ { - 1 } & = c a ^ { - 1 } ( [ c , b ] \\Delta ^ { - 1 } - b R _ { 2 1 } ) + d R _ { 2 1 } \\\\ & = c a ^ { - 1 } [ c , b ] \\Delta ^ { - 1 } + ( d - c a ^ { - 1 } b ) R _ { 2 1 } \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} & h _ k \\left ( f _ \\sigma ( x ) \\ , d x _ { \\sigma ( 1 ) } \\wedge \\ldots \\wedge d x _ { \\sigma ( k ) } \\right ) = \\\\ & \\int _ 0 ^ 1 t ^ { k - 1 } f _ \\sigma ( t x ) \\ , d t \\cdot \\sum _ { j = 1 } ^ k ( - 1 ) ^ { j + 1 } x _ { \\sigma ( j ) } d x _ { \\sigma ( 1 ) } \\wedge \\ldots \\wedge d x _ { \\sigma ( j - 1 ) } \\wedge d x _ { \\sigma ( j + 1 ) } \\wedge \\ldots \\wedge d x _ { \\sigma ( k ) } . \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} s - r = \\lfloor s ^ { 1 / 2 } \\rfloor + 1 . \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} \\int _ { Y \\times U } q ( y , u ) \\gamma _ { u ( \\cdot ) , T } ( d y , d u ) = \\frac { 1 } { T } \\int _ 0 ^ T q ( y ( t ) , u ( t ) ) d t \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} c _ { \\ast } ( \\Omega , p , \\nabla P _ { 0 } ) : = \\max \\left \\{ \\sup _ { \\mu \\leq M _ { \\ast } } c ( \\Omega , p , \\lambda _ { 0 } / 2 , \\mu ) , \\sup _ { \\mu \\leq M _ { \\ast } } c ( \\Omega , p , \\lambda _ { 0 } , \\mu ) \\right \\} , \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} \\mathbb { E } : = | x | \\R . \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} \\P ( X _ { n + 1 } = 1 | X _ { n } = 0 ) & = \\frac { f ( 0 ) } { n } \\times \\frac { n } { n + 1 } = \\frac { f ( 0 ) } { n + 1 } , \\\\ \\P ( X _ { n + 1 } = 0 | X _ { n } = 0 ) & = ( 1 - \\frac { f ( 0 ) } { n } ) \\times \\frac { n } { n + 1 } + \\frac { 1 } { n + 1 } = 1 - \\frac { f ( 0 ) } { n + 1 } . \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} I _ { T , t } ^ { ( i _ 1 i _ 2 ) q } = \\frac { T - t } { 2 } \\zeta _ { 0 } ^ { ( i _ 1 ) } \\zeta _ { 0 } ^ { ( i _ 2 ) } + A _ { T , t } ^ { ( i _ 1 i _ 2 ) q } , \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} \\{ X _ 1 , X _ 0 \\} = - X _ 2 , \\ , \\ , \\{ X _ 1 , X _ 2 \\} = ( 2 + x _ 1 ) \\xi _ 3 , \\ , \\ , \\{ X _ 2 , X _ 0 \\} = 0 . \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} g ^ { * } X = X , g ^ { * } Y = \\zeta Y , g ^ { * } X _ { i } = \\zeta ^ { i } X _ { i } \\ , ( i = 0 , \\cdots , 3 ) , \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} T _ m \\sim _ { c _ 1 , c _ 2 } \\begin{cases} \\frac { m } { \\sqrt { q } + m \\sqrt { | \\xi | } } , \\ \\ \\ \\ - m | \\xi | - \\sqrt { q | \\xi | } \\leq a ' \\leq \\sqrt { q | \\xi | } , \\\\ \\\\ \\sqrt { q } \\left ( \\frac { m | \\xi | } { a ' ( m | \\xi | + a ' ) } + \\frac { | \\gamma | } { \\sqrt { a ' ( m | \\xi | + a ' ) } } \\right ) , \\ \\ . \\end{cases} \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} p _ { 0 } ( y , t ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } e ^ { - 2 y ^ { 2 } } \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} W _ { i , n , r } = Y _ { i , n , r } + R _ { i , n , r } - R _ { i , n - 1 , r } . \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} X _ 0 = D + ( I - P ) D ^ * + ( I - P ) D ^ * ( D P ) ^ \\dag D ( I - P ) . \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{gather*} { } _ 4 F _ 3 \\biggl ( \\begin{matrix} \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 \\\\ 1 , \\ , 1 , \\ , 1 \\end{matrix} \\biggm | 1 \\biggr ) = \\frac { 1 6 L ( f , 2 ) } { \\pi ^ 2 } , \\end{gather*}"}
-{"id": "8207.png", "formula": "\\begin{align*} \\mathrm { T } ( \\Theta _ A [ d ] ) _ { \\mathrm { g r } } : = \\bigoplus _ { m \\ge 0 } ( \\Theta _ A [ d ] \\{ 1 \\} ) ^ { \\otimes m } . \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} A _ { 1 - 2 s } u : = - \\frac { 1 - 2 s } { t } u _ { t } - u _ { t t } + A u \\ni 0 \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} k ^ 2 \\theta _ { r - 2 , q ^ 2 } + k ( 1 - \\alpha - \\beta ) \\theta _ { r - 1 , q ^ 2 } - k \\theta _ { r - 2 , q ^ 2 } + \\alpha \\beta \\theta _ { r , q ^ 2 } = 0 . \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} \\mathbb A \\bigl ( ( \\pi _ t ) _ { t \\in [ 0 , T ] } \\bigr ) : = \\frac 1 2 \\bigl [ \\mathcal F _ \\alpha ^ V ( \\rho _ T ) - \\mathcal F _ \\alpha ^ V ( \\rho _ 0 ) + \\mathcal E \\bigl ( ( \\rho _ t ) _ { t \\in [ 0 , T ] } \\bigr ) + \\mathcal E ^ \\star \\bigl ( ( \\rho _ t ) _ { t \\in [ 0 , T ] } \\bigr ) \\bigr ] . \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} 0 . 2 / T ^ \\leq \\| u ( t ) \\| _ 1 \\leq 1 / T ^ , T ^ = 7 . 5 . \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} \\lVert f \\circ X _ w ^ { - 1 } - f \\rVert _ \\infty = & \\sup _ { v \\in F _ n } \\lvert f ( X _ w ^ { - 1 } v ) - f ( v ) \\rvert \\\\ = & \\sup _ { v \\in F _ n } \\lvert f ( w v w ^ { - 1 } ) - f ( v ) \\rvert \\\\ \\leq & \\sup _ { v \\in F _ n } \\lvert f ( w ) + f ( v w ^ { - 1 } ) - f ( v ) \\rvert + D ( f ) \\\\ \\leq & \\sup _ { v \\in F _ n } \\lvert f ( w ) + f ( w ^ { - 1 } ) \\rvert + 2 D ( f ) \\\\ < & \\infty \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} \\phi ( \\cdot , x ) = \\mathcal { M } \\phi ( \\cdot , x ) = \\sup _ { z \\in \\mathcal { Z } } \\{ \\phi ( \\cdot , x - \\kappa - ( 1 + \\lambda ) z ) + z ) \\} . \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} A _ i ^ { ( n + 1 ) } = \\sum _ { j + k = i } A _ j ^ { ( n ) } \\otimes A _ k ^ { ( 1 ) } = A _ i ^ { ( n ) } \\otimes A _ 0 ^ { ( 1 ) } + A _ { i - 1 } ^ { ( n ) } \\otimes A _ 1 ^ { ( 1 ) } \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} \\rho _ \\lambda c _ \\Omega \\partial _ n K ^ - _ \\gamma = a _ 0 \\lambda \\mu _ \\lambda = - \\rho _ \\lambda c _ \\Omega \\partial _ n K ^ + _ \\gamma , a _ 0 = 2 , \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} d ( v , u ) = \\int \\left [ v ( t ) \\log \\frac { v ( t ) } { u ( t ) } - v ( t ) + u ( t ) \\right ] \\ , d t , \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} \\alpha '' ( x ) & = - 2 c - \\int _ 0 ^ \\infty x ^ 2 e ^ { - x y } N ( d y ) < 0 \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} \\lVert a f \\rVert _ S & = \\lVert \\sum _ { v \\in I } a \\alpha ( v ) \\# v \\rVert _ S \\\\ & = \\max \\{ \\lvert v \\rvert _ S \\mid v \\in I , a \\alpha ( v ) \\neq 0 \\} \\\\ & = \\max \\{ \\lvert v \\rvert _ S \\mid v \\in I , \\alpha ( v ) \\neq 0 \\} \\\\ & = \\lVert f \\rVert _ S \\\\ & = \\lvert a \\rvert _ 0 \\lVert f \\rVert _ S . \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} \\mathcal { U } _ { 1 } ( \\tilde { g } ) - \\min _ { g _ q \\in \\mathbb { Q } } \\mathcal { U } _ { 1 } ( \\tilde { g } g _ q ) & = ( 1 - \\cos \\theta ^ * ) \\max _ { u _ q \\in \\mathbb { U } } \\Delta _ Q ( u _ q , v ) , \\forall v \\in \\mathcal { E } ( Q ) \\\\ & \\geq ( 1 - \\cos \\theta ^ * ) \\min _ { v \\in \\mathcal { E } ( Q ) } \\max _ { u _ q \\in \\mathbb { U } } \\Delta _ Q ( u _ q , v ) \\\\ & \\textstyle = ( 1 - \\cos \\theta ^ * ) \\Delta ^ * _ Q > \\delta , \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} p \\in \\partial \\Omega , \\ ; \\overline \\exp _ p ^ { - 1 } ( \\partial \\Omega ) = T _ p ( \\partial \\Omega ) . \\end{align*}"}
-{"id": "9664.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - \\alpha } \\sum _ { j = 2 } ^ { \\infty } \\frac { ( \\| \\Delta f \\| _ { \\mathcal { B } _ { \\alpha } ^ { 0 } } ) ^ { j } } { j ! } \\leq r _ { 1 } . \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} y ^ 2 + y = x ^ 3 - 2 x ^ 2 + 1 \\end{align*}"}
-{"id": "9881.png", "formula": "\\begin{align*} | u | _ L : = \\| \\boldsymbol { D } ^ s u \\| _ { L ^ 2 ( \\mathbb { R } ) } ~ ~ ~ ~ \\widetilde { W } ^ { s } _ L ( \\mathbb { R } ) ~ ~ ~ ~ | u | _ R : = \\| \\boldsymbol { D } ^ { s * } u \\| _ { L ^ 2 ( \\mathbb { R } ) } ~ ~ ~ ~ \\widetilde { W } ^ { s } _ R ( \\mathbb { R } ) , \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} E _ 0 ^ { ( r ) } T _ a = \\frac { e ^ { \\sqrt { \\frac { 2 r } D } \\thinspace | a | } - 1 } r , \\ a \\in \\mathbb { R } . \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} \\mathbf { i } u _ t + u _ { x x } + \\mathbf { i } ( | u | ^ 2 u ) _ x = 0 , \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} \\tau _ 1 \\cdot \\tau _ 2 & = \\tau _ 1 \\cdot ( \\sigma _ 2 - \\sigma _ 0 - 4 e ) \\\\ & = \\tau _ 1 \\cdot \\sigma _ 2 \\\\ & = ( \\sigma _ 1 - \\sigma _ 0 - 4 e ) \\cdot \\sigma _ 2 \\\\ & = 8 - 0 - 4 \\\\ & = 4 . \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} M ' _ { k , k } = \\lambda _ { q + 1 } ^ f \\prod _ { i = 1 } ^ q \\left ( \\sum _ { s = 0 } ^ { m + 1 } \\binom { m + 1 } { s } ( - \\lambda _ i ) ^ s \\lambda _ { q + 1 } ^ { m + 1 - s } \\right ) = \\lambda _ { q + 1 } ^ f \\prod _ { i = 1 } ^ q ( \\lambda _ { q + 1 } - \\lambda _ i ) ^ { m + 1 } . \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{gather*} \\chi ( \\ , \\cdot \\ , ) = \\left ( \\frac { - 4 } { \\cdot } \\right ) , \\ ; \\left ( \\frac { - 3 } { \\cdot } \\right ) , \\ ; \\left ( \\frac { - 2 } { \\cdot } \\right ) \\ ; \\ ; \\left ( \\frac { - 4 } { \\cdot } \\right ) \\qquad r = \\frac 1 2 , \\ ; \\frac 1 3 , \\ ; \\frac 1 4 , \\ ; \\frac 1 6 , \\ ; . \\end{gather*}"}
-{"id": "7524.png", "formula": "\\begin{align*} \\lambda ( s , t ) = \\begin{cases} \\dfrac { 1 } { 4 \\pi t } \\left ( e ^ { - 4 \\pi \\sin ^ { - 1 } ( s ) t } - e ^ { - 4 \\pi ( \\pi - \\sin ^ { - 1 } ( s ) ) t } \\right ) , \\ ; t \\neq 0 \\\\ \\pi - 2 \\sin ^ { - 1 } s , t = 0 . \\end{cases} \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} \\left ( \\frac { | Q | } { | Q ^ { \\prime } | } \\right ) ^ { \\frac { 1 - s } { a s } } | Q ^ { \\prime } | ^ { \\frac { 1 } { r } } = | Q | ^ { \\frac { 1 - s } { a s } } | Q ^ { \\prime } | ^ { \\frac { 1 } { r } - \\frac { 1 - s } { a s } } \\leq | Q | ^ { \\frac { 1 } { r } } , \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} \\Pi ^ T a \\nabla V = \\mathcal { O } \\big ( \\frac { 1 } { \\epsilon } \\big ) \\ , , \\mbox { a n d } ( I - \\Pi ^ T ) a \\nabla V = \\mathcal { O } ( 1 ) \\ , . \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} a ( i , j ) & = 2 n - i - j - 1 + m + c _ j , \\\\ b ( i , j ) & = j - i + m - c _ j . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} \\mu _ j ^ { \\frac { n + 2 s } { 2 } } S ( u ^ * _ { \\mu , \\xi } ) = \\mu _ j ^ { \\frac { n + 2 s } { 2 } } S ( u _ { \\mu , \\xi } ) - \\mu _ { 0 j } E _ { 0 j } [ \\bar { \\mu } _ 0 , \\dot { \\mu } _ { 0 j } ] + A _ j ( y ) . \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} \\norm { F } _ { A ^ 2 ( \\mathcal { U } _ p ) } ^ 2 = \\int _ { \\C ^ n } \\norm { F ( \\cdot , w ) } ^ 2 _ { A ^ 2 ( S ( p ( w ) , \\infty ) ) } \\int _ { \\C ^ n } \\int _ 0 ^ { \\infty } \\abs { f ( t , w ) } ^ 2 \\frac { e ^ { - 4 \\pi p ( w ) t } } { 4 \\pi t } \\d t = \\norm { f } ^ 2 _ { \\mathcal { L } _ { \\mathrm { T r a n s } } ( p ) } . \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t ( u - U _ h ) ^ { n - \\theta } , v _ h \\Big { ) } & - \\gamma ( \\nabla ( \\sigma - \\Sigma _ h ) ^ { n - \\theta } , \\nabla v _ h ) + ( \\nabla ( u - U _ h ) ^ { n - \\theta } , \\nabla v _ h ) \\\\ & + ( f ^ { n - \\theta } ( u ) - ( 1 - \\theta ) \\mathfrak { F } ( U _ { h } ^ { n } , u _ { H } ^ { n } ) - \\theta f ( U _ { h } ^ { n - 1 } ) , v _ h ) = 0 , \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} s : = \\big ( m ( | A | - 1 ) \\big ) ^ { ( \\log _ 2 ( | A | ) - 1 ) } . \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} M _ G ( F ) & \\equiv \\prod _ { x , y = \\pm 1 } ( 1 - g ( x , y ) ^ 2 ) \\equiv 1 - \\sum _ { x , y = \\pm 1 } g ( x , y ) ^ 2 \\equiv 1 - \\left ( \\sum _ { x , y = \\pm 1 } g ( x , y ) \\right ) ^ 2 . \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} - \\log \\left \\vert \\det \\left ( 1 - \\mathcal { L } _ { s , \\lambda } ^ { N } \\right ) \\right \\vert = \\mathrm { R e } \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n } \\mathrm { T r } \\ , \\left ( \\mathcal { L } _ { s , \\lambda } ^ { n N } \\right ) \\leq \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n } \\left \\vert \\mathrm { T r } \\ , \\left ( \\mathcal { L } _ { s , \\lambda } ^ { n N } \\right ) \\right \\vert . \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} x y = q y x . \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} \\partial _ { \\omega } F ^ { z \\bar { z } } _ { i j } + \\mathbf { i } \\ ( \\Omega _ i - \\Omega _ j ) F ^ { z \\bar { z } } _ { i j } = R ^ { z \\bar { z } } _ { i j } ; \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} \\Sigma ( \\xi - k ) - \\Sigma \\left ( \\frac { \\lvert \\xi \\lvert } { \\lvert \\xi - k \\lvert } ( \\xi - k ) \\right ) & \\geq - \\omega \\left ( \\frac { \\lvert \\xi \\lvert } { \\lvert \\xi - k \\lvert } ( \\xi - k ) - \\xi - k \\right ) \\\\ & = - \\omega \\left ( ( \\lvert \\xi \\lvert - \\lvert \\xi - k \\lvert ) \\frac { \\xi - k } { \\lvert \\xi - k \\lvert } \\right ) \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} \\mu _ X ( \\chi ) = \\gamma ( \\chi , - \\frac { \\check \\alpha } { 2 } , \\frac { 1 } { 2 } , \\psi ^ { - 1 } ) \\gamma ( \\chi , - \\frac { \\check \\alpha } { 2 } , \\frac { 1 } { 2 } , \\psi ) \\gamma ( \\chi , \\check \\alpha , 0 , \\psi ) ; \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} D _ { t } U ( m , s ) = \\frac { U ( m , s + 1 ) - U ( m , s ) } { \\Delta t } , \\\\ D ^ { + } U ( m , s ) = \\frac { U ( m - 1 , s ) - U ( m , s ) } { \\Delta x } , \\\\ D ^ { - } U ( m , s ) = \\frac { U ( m , s ) - U ( m + 1 , s ) } { \\Delta x } , \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} A ( x ) = \\operatorname { R e } \\left [ \\mathbb { A } ( x ) \\right ] + ( p _ \\gamma x ^ \\gamma ) \\ , v \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} & d \\nu ^ 1 _ { k , z } ( y _ { d - k } , . . . , y _ 1 ) = d | D f ^ { \\eta _ { d - k } ^ { \\kappa } ( y _ { \\tau } ) } _ { \\sum _ { i = 1 } ^ { d - k - 1 } y _ i + z } | ( y _ { d - k } ) d \\mathcal { H } ^ 1 ( y _ { d - k - 1 } ) . . . d \\mathcal { H } ^ 1 ( y _ 1 ) , \\\\ & d \\nu ^ 2 _ { k , z } ( y _ { d - k } , . . . , y _ 1 ) = 1 _ { | \\sum _ { i = 1 } ^ { d - k } ( y _ { \\tau } ) _ { \\eta _ i ^ { \\kappa } ( y _ { \\tau } ) } - \\sum _ { i = 1 } ^ { d - k } y _ i | \\leq 4 \\tau } d \\nu ^ 1 _ { k , z } ( y _ { d - k } , . . . , y _ 1 ) , \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} B _ { n + 1 , p } = p B _ { n , p } - \\frac { ( p + 1 ) ^ 2 } { p + 2 } B _ { n , p + 1 } , p \\ge 0 , \\ n \\ge 0 . \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} \\int _ { x _ 0 } ^ x \\frac { 1 } { 2 \\gamma ^ { \\prime } ( y , \\hat { E } ) } \\frac { 1 } { 1 + y - b } \\sin 2 \\theta ( y , { E } ) \\sin 2 \\theta ( y , \\hat { E } ) d y = O ( \\frac { 1 } { x _ 0 - b } ) , \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } W ( m ) + F _ { i } ( D ^ { + } W ( m ) , D ^ { - } W ( m ) ) \\geq f _ { i } ( - m \\Delta x ) & \\ , \\ , m \\in J _ { i } \\setminus \\{ 0 \\} \\\\ W ( 0 ) \\geq \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } W ( 1 _ { i } ) \\end{array} \\right . \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} \\varphi ^ { \\prime } ( x ) = i | \\varphi ^ { \\prime } ( x ) | e ^ { i \\delta ( x ) } . \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} K ( \\xi - d \\Gamma _ A ( m ) ) = K ( \\xi ) \\oplus \\bigoplus _ { n = 0 } ^ \\infty K ( \\xi - d \\Gamma _ A ^ { ( n ) } ( m ) ) \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} \\begin{array} { c c l } \\Delta _ m & = & ( - 1 ) ^ m q ^ { A _ m + k + m - 1 / 2 } ( 1 - q ^ { 1 / 2 } ) ( ( 1 + q ^ { k + m + 3 / 2 } ) ( 1 - q ^ { k + m } ) \\\\ \\\\ & & - q ^ { k + m + 1 } ( 1 + q ^ { k + m - 1 / 2 } ) ( 1 - q ^ { k + m + 1 } ) ) / T _ m \\\\ \\\\ & = & U _ m ( 1 - q ^ { 1 / 2 } ) ( 1 - q ^ { k + m } ) ( ( 1 + q ^ { k + m + 3 / 2 } ) - q ^ { k + m + 1 } ( 1 + q ^ { k + m - 1 / 2 } ) ) \\\\ \\\\ & & - U _ m ( 1 - q ^ { 1 / 2 } ) q ^ { 2 k + 2 m + 1 } ( 1 + q ^ { k + m - 1 / 2 } ) ( 1 - q ) \\\\ \\\\ & = & U _ m ( 1 - q ^ { 1 / 2 } ) ( K _ m + L _ m + M _ m ) ~ , \\end{array} \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} 2 \\Psi ( v ) = 2 \\sum _ { j = 1 } ^ { N } \\Psi ( v ^ j ) = \\sum _ { j = 1 } ^ { N } \\left [ 2 \\psi ( \\lambda _ 1 ( v ^ j ) ) + \\psi ( \\lambda _ 2 ( v ^ j ) ) + \\psi ( \\lambda _ 2 ( v ^ j ) ) \\right ] : = \\sum _ { j = 1 } ^ { 4 N } \\psi ( z _ j ) , \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} 0 = \\frac { \\partial } { \\partial y } ( y \\beta ( y ) ) + \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\beta ( y ) + \\left ( - 4 y ^ { 4 } + 1 0 y ^ { 2 } - \\frac { 7 } { 4 } \\right ) \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } e ^ { - 2 y ^ { 2 } } . \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{gather*} R ( 1 ) R ( e _ { 1 i + 1 } ) = R ^ 2 ( e _ { 1 i + 1 } ) , \\\\ R ( e _ { 1 i + 1 } ) R ( 1 ) = R ^ 2 ( e _ { 1 i + 1 } ) + R ( e _ { 1 i + 2 } ) \\end{gather*}"}
-{"id": "1570.png", "formula": "\\begin{align*} \\left | \\sum _ { \\ell = 1 } ^ n \\mu _ \\ell ^ f x _ \\ell \\right | = \\max _ { \\substack { 1 \\leq i \\leq t \\\\ 0 \\leq j \\leq m } } \\{ | c _ { i , j } | \\} \\neq 0 . \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} \\int _ \\Omega \\mathbb { P } [ \\mu ] \\nabla p \\cdot \\nabla \\phi \\ , \\delta _ \\Gamma ( x ) \\ , \\d x & = \\int _ 0 ^ 1 \\bar { C } \\partial _ s p \\ , \\partial _ s \\phi \\ , \\d s \\\\ & = G \\int _ 0 ^ 1 \\partial _ s \\phi ( x ( s ) ) \\ , \\d s = G ( \\phi ( x ^ - ) - \\phi ( x ^ + ) ) . \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} M _ { \\lambda , \\mu } = \\left \\{ m \\in M \\ \\middle \\vert \\ K _ i \\cdot m = q _ i ^ { \\langle \\lambda , \\alpha _ i ^ \\vee \\rangle } m , \\ L _ i \\cdot m = q _ i ^ { \\langle \\mu , \\alpha _ i ^ \\vee \\rangle } m , \\ \\forall 1 \\leq i \\leq n \\right \\} . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} f ( x ) & = ( x ^ 2 + x + 1 ) - ( x ^ 4 + x ^ 2 + 1 ) + m \\ : h ( x ) , \\\\ g ( x ) & = ( x ^ 2 - x + 1 ) - ( x ^ 4 + x ^ 2 + 1 ) - m \\ : h ( x ) , \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} \\left \\| \\nabla v ( \\cdot , U , \\xi ' ) - \\nabla v ( \\cdot , U , \\xi ) - \\sum _ { i = 1 } ^ d ( \\xi _ i ' - \\xi _ i ) \\nabla \\tilde { w } _ { e _ i } ( \\cdot , U , \\xi ) \\right \\| _ { \\underline { L } ^ 2 ( U ) } \\leq C \\left | \\xi - \\xi ' \\right | ^ { 1 + \\beta } . \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} C = \\{ \\lambda \\in P ^ + \\mid \\langle \\lambda + \\rho , \\theta _ 0 \\rangle < l ' \\} . \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\sum _ { j = 0 } ^ N \\omega _ j \\tilde { \\Phi } _ { \\ell } ( x _ j ) f ( x _ j ) = \\int _ { - 1 } ^ 1 w ( x ) \\tilde { \\Phi } _ { \\ell } ( x ) f ( x ) d x , 0 \\leq \\ell \\leq L . \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} d w \\cup x = j ( \\beta ) \\cup x = \\alpha \\cup x . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} y ( t ) = h ( t ) x ( t ) + n ( t ) , \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} D _ i ^ { ( r ) } : = T _ { i , i ; i - 1 } ^ { ( r ) } & 1 \\le i \\le n , r > 0 \\\\ E _ i ^ { ( r ) } : = T _ { i , i + 1 ; i } ^ { ( r ) } & 1 \\leq i < j \\leq n , r > s _ { i , j } \\\\ F _ i ^ { ( r ) } : = T _ { i + 1 , i ; i } ^ { ( r ) } & 1 \\leq i < j \\leq n , r > s _ { j , i } . \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} \\mathbb A _ L ^ { \\tilde V } \\bigl ( Q _ L \\bigr ) : = \\mathcal H \\bigl ( Q _ L | Q _ L ^ { \\tilde V } \\bigr ) = \\mathcal H \\bigl ( P _ L | P _ L ^ { \\tilde V } \\bigr ) = \\frac 1 2 \\int _ 0 ^ T \\Phi _ L \\bigl ( \\mu ^ L _ t , \\jmath ^ L _ t , F ^ { \\tilde V _ t } ( \\mu ^ L _ t ) \\bigr ) \\ ; \\ ! \\mathrm d t . \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} w ( s x _ 1 \\otimes \\cdots \\otimes s x _ n ) = \\sum _ { i = 1 } ^ { n - 1 } ( s x _ 1 \\otimes \\cdots s x _ i ) \\otimes ( s x _ { i + 1 } \\otimes \\cdots \\otimes s x _ n ) \\ . \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} c _ k ( f _ { \\alpha , \\beta } ) = O \\left ( 1 / | k | ^ { ( 1 + \\alpha + \\frac { \\beta } { 2 } ) / ( \\beta + 1 ) } \\right ) + O \\big ( 1 / | k | ^ 2 \\big ) \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} B _ \\theta ( X , Y ) = d \\theta ( X , J Y ) \\end{align*}"}
-{"id": "9954.png", "formula": "\\begin{align*} \\int _ f a _ { X } = n ( a _ { \\tilde f } ) \\ , a _ { Y } \\in j ^ { - 1 } o r _ { Y } ( N ) . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} | k _ b | \\leq | k _ b j _ b | < \\sum _ { j \\in \\mathbb { Z } _ * } \\frac { | l _ j j | } { n - \\frac 1 2 } + \\frac { | k | } { 1 0 0 n \\sum _ { b = 1 } ^ n | j _ b | } \\leq \\frac { | k | } { 2 n - 1 } + \\frac { | k | } { 1 0 0 n \\sum _ { b = 1 } ^ n | j _ b | } . \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} R ( \\lambda + i 0 ; x , y ) = R _ 1 ( \\lambda ; x , y ) + R _ 2 ( \\lambda ; x , y ) , \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} A _ { \\overline { l } , 0 } ( t , \\overline \\alpha ) e ^ { \\alpha _ j t } - A _ { \\overline { l } , j } ( t , \\overline \\alpha ) = R _ { \\overline { l } , j } ( t , \\overline \\alpha ) , j = 1 , \\ldots , m , \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} m _ k : = \\frac { \\lambda _ + ^ { k + 1 } - \\lambda _ - ^ { k + 1 } } { \\lambda _ + - \\lambda _ - } , \\lambda _ + : = \\frac { m _ \\xi + \\sqrt { m _ \\xi ^ 2 + 4 m _ \\eta } } { 2 } , \\lambda _ - : = \\frac { m _ \\xi - \\sqrt { m _ \\xi ^ 2 + 4 m _ \\eta } } { 2 } \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} \\gamma _ 0 = \\gamma _ 0 ( \\chi , \\sigma , m , r , \\mathcal { U } _ 1 , \\tilde { T } ) = \\gamma _ { \\ref { T h e o r e m E x i s t e n c e A n d U n i q u e n e s s O n D o m a i n } ; 0 } ( \\eta ( \\chi ) , \\tilde { r } , \\tilde { T } ) \\geq 1 , \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} \\kappa _ { 0 0 } & = \\sum _ { n \\geq 0 } q _ 1 ( 0 ) ^ { n - 1 } \\prod _ { k = 0 } ^ { n - 1 } q _ { ( 0 1 ) ^ k 0 0 ( 1 ) } + \\sum _ { n \\geq 1 } q _ 1 ( 0 ) ^ { n - 2 } \\prod _ { k = 0 } ^ { n - 1 } q _ { ( 0 1 ) ^ k 0 0 ( 1 ) } \\\\ \\kappa _ { 1 1 } & = \\sum _ { n \\geq 0 } q _ 1 ( 0 ) ^ { n } \\prod _ { k = 1 } ^ { n - 1 } q _ { ( 0 1 ) ^ k 1 ( 1 ) } + \\sum _ { n \\geq 1 } q _ 1 ( 0 ) ^ { n } \\prod _ { k = 1 } ^ { n - 1 } q _ { ( 0 1 ) ^ k 1 ( 1 ) } , \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} \\xi = ( x _ 1 , x _ 2 , \\cdots , x _ m ) ^ T \\ , , \\phi = ( x _ { m + 1 } , \\cdots , x _ { n } ) ^ T \\ , , \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} \\forall \\ , g \\in H , \\forall \\ , \\psi \\in \\mathcal { O } ( H ) , \\widetilde { g } \\widetilde { h } = \\widetilde { g h } , g \\widetilde { h } = \\widetilde { h } g , \\widetilde { h } \\psi = \\psi \\big ( S ^ { - 1 } ( h '' ) ? \\big ) \\widetilde { h ' } . \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} g ( x ) = f ( b ( x - a ) ) , \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} \\dim H = \\sum _ { k = 1 } ^ m c _ k \\dim B _ k H . \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t } ^ { \\mathrm m } \\ , = \\ , b ( t , X _ { t } ^ { \\mathrm m } , u \\ , \\delta _ { \\widetilde { X } ^ { \\mathrm m } _ { t } } + ( 1 - u ) \\mathrm m _ { t } ) { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , ; 0 \\le t \\le T \\ , , \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} \\bigcup _ { n \\geq 0 } \\tilde { f } _ k ^ n ( \\mathcal { B } _ w ( \\lambda ) ) \\setminus \\{ 0 \\} = \\mathcal { B } _ { w } ( \\lambda ) . \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} i _ 0 & = \\min \\{ 1 \\leq j \\leq k \\mid s _ j \\notin \\{ s , s ^ { - 1 } \\} \\} \\\\ i _ 1 & = \\max \\{ 1 \\leq j \\leq k \\mid s _ j \\notin \\{ s , s ^ { - 1 } \\} \\} . \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} \\log \\Big | \\dfrac { \\partial } { \\partial x } \\phi ( t , x ) \\Big | & = \\int _ 0 ^ t \\dfrac { \\partial } { \\partial x } b ( s , \\phi ( s , x ) ) \\ , d s \\\\ & = \\int _ 0 ^ t \\lim _ { k \\to \\infty } v _ k ( s , \\phi _ k ( s , x ) ) \\ , d s \\\\ & = \\lim _ { k \\to \\infty } \\log \\Big | \\dfrac { \\partial } { \\partial x } \\phi _ k ( t , x ) \\Big | . \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } x b ^ { - 1 } \\sim \\phi b ^ { m _ 0 } x b ^ { n _ k } + \\sum _ { i = n _ n + 1 } ^ { - 1 } \\sum _ { s \\in S _ b } \\phi b ^ { m _ 0 } x b ^ { i } s . \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t } ^ { \\bullet } \\ , = \\ , - ( X _ { t } ^ { \\bullet } - \\mathbb E [ X _ { t } ^ { \\bullet } ] ) { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , , t \\ge 0 \\ , . \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} \\sum _ { h _ 1 } f ' ( h _ 1 ) ( \\xi ) & = \\sum _ { h _ 1 } \\sum _ j f ( g _ j ) ( \\pi ( \\xi ) | _ { X _ j } ) = \\sum _ g \\sum _ { h _ 1 , h _ 1 \\in c ( g , X ) } f ( g ) ( \\pi ( \\xi ) | _ { X _ { g , h _ 1 } } ) \\\\ & = \\sum _ g f ( g ) ( \\pi ( \\xi ) ) = \\sum _ { g } \\pi ^ * ( f ( g ) ) ( \\xi ) . \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} \\mathcal H = \\frac { 1 } { 2 } \\int _ { - \\infty } ^ \\infty u _ t ^ 2 + u _ x ^ 2 + ( u ^ 2 - 1 ) ^ 2 + \\frac { 1 } { 2 } \\Omega ^ 2 x ^ 2 u ^ 2 \\ , d x = \\mathcal T + \\mathcal V , \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} L _ d = L _ d ^ { 1 } \\cup L _ d ^ 2 , R _ d = R _ d ^ { 1 } \\cup R _ d ^ 2 . \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} ( \\mathrm { T } \\mathcal { K } _ X ^ { - 1 } ) _ { \\mathrm { g r } } : = \\mathrm { T } ( \\mathcal { K } _ X ^ { - 1 } \\{ 1 \\} ) = \\bigoplus _ { m \\ge 0 } \\mathcal { K } _ X ^ { - m } \\{ m \\} \\end{align*}"}
-{"id": "9958.png", "formula": "\\begin{align*} \\int _ { M _ 1 \\times M _ 2 } ( u _ 1 \\times u _ 2 ) \\otimes ( v _ 1 \\wedge v _ 2 ) = \\left ( \\int _ { M _ 1 } u _ 1 \\otimes v _ 1 \\right ) \\ , \\ , \\left ( \\int _ { M _ 2 } u _ 2 \\otimes v _ 2 \\right ) . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} U _ { 2 } ( t ) = A _ { 2 } V _ { m + 1 } ( t ) + \\int _ t ^ { t _ 0 } G _ { 2 } ( t , s ) V _ { m + 1 } ( s ) \\ , d s + \\int _ t ^ { t _ 0 } H _ { 2 } ( t , s ) U _ { 2 } ( s ) \\ , d s \\mbox { f o r } t _ 2 \\le t < t _ 1 , \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} \\tilde g ^ { ( 4 ) } = - u ^ 2 ( d t + \\theta ) ^ 2 + g _ S . \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} L ( x _ 1 , x _ 2 , \\dot { x } _ 1 , \\dot { x } _ 2 , t ) = f ^ { - 1 } ( t ) \\left ( \\frac { m } { 2 } ( \\dot { x } _ 1 ^ 2 + \\dot { x } _ 2 ^ 2 ) - \\frac { m \\omega ^ 2 ( t ) } { 2 } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) \\right ) , \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} C _ { j , j + 1 } ( u _ 1 , v _ 1 , u _ 2 , 1 ) = C _ { X ^ 2 _ j , X ^ 1 _ j } \\star C _ { X ^ 1 _ j , X ^ 1 _ { j + 1 } } ( v _ 1 , u _ 1 , u _ 2 ) , \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} \\Phi _ G ( m ) = \\imath \\ , h _ m \\begin{pmatrix} \\lambda ( m ) & 0 \\\\ 0 & - \\lambda ( m ) \\end{pmatrix} \\ , h _ m ^ { - 1 } ; \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { m = 0 } ^ { \\infty } c _ m ( f ) p _ m ( x ) , \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} A = V \\Sigma W ^ { * } . \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} & \\frac { 0 . 4 3 2 \\tau ^ 2 } { T } \\left ( \\frac { 1 } { 2 \\pi } \\log { ( 2 T ) } + \\frac { 1 } { 2 \\pi } \\log { \\frac { 2 0 0 } { \\pi e } } \\right . \\\\ & \\left . - \\frac { 0 . 6 3 4 } { 3 T } \\log ( 2 ^ \\frac { 1 } { 3 } T ) - \\frac { 1 } { 3 T } ( 0 . 3 1 7 \\log { 2 } + 0 . 6 3 4 \\log { 1 0 0 } + 1 2 . 8 0 2 ) \\right ) \\\\ & \\coloneqq K _ { F , 3 } ( T , \\tau ) \\\\ & > 0 . \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { 1 } ( \\textbf { z } ) = I \\left \\lbrace \\textbf { z } \\in \\mathcal { C } _ { 1 } \\cap \\mathcal { C } _ { 0 } , p ( \\textbf { z } , 1 ) \\dfrac { p } { \\mu { ( \\mathcal { C } _ { 1 } ) } } > p ( \\textbf { z } , 0 ) \\dfrac { 1 - p } { \\mu { ( \\mathcal { C } _ { 0 } ) } } \\right \\rbrace + I \\left \\lbrace \\textbf { z } \\in \\mathcal { C } _ { 1 } - \\mathcal { C } _ { 0 } \\right \\rbrace \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} A _ { \\overline { l } , 0 } ( t , \\overline \\alpha ) e ^ { \\alpha _ j t } - A _ { \\overline { l } , j } ( t , \\overline \\alpha ) = R _ { \\overline { l } , j } ( t , \\overline \\alpha ) , j = 1 , \\ldots , m , \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} m _ j ^ - = m _ j - \\# a ( s _ j ) + \\# a ^ { - 1 } ( s _ { j + 1 } ) . \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} \\begin{array} { l l } \\log ( \\rho ( n , l ) ) = \\frac { 2 l } { n - k _ 0 } \\log \\left ( { \\frac { n - k _ 0 } { 2 p } } \\right ) + \\frac { 2 l } { n - k _ 0 } \\log ( { B } ( \\frac { n - k _ 0 } { 2 } , 0 . 5 ) ) \\\\ + \\frac { 2 l } { n - k _ 0 } \\log ( \\alpha ) \\end{array} \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} \\rho _ H ( \\Gamma _ 1 , \\Gamma _ 2 ) : = \\max \\{ \\sup _ { \\gamma \\in \\Gamma _ 1 } \\rho ( \\gamma , \\Gamma _ 2 ) , \\sup _ { \\gamma \\in \\Gamma _ 2 } \\rho ( \\gamma , \\Gamma _ 1 ) \\} , \\ \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} V ^ { \\bar { i } \\bar { j } } ( x ) - U ^ { \\bar { i } \\bar { j } } ( x ) = V ^ { i _ 0 j _ 0 } ( x ) - U ^ { i _ 0 j _ 0 } ( x ) \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ \\Omega ( D _ \\alpha u \\cdot \\phi _ \\alpha + p \\psi ) \\ , d x \\\\ & = \\int _ { \\Omega _ r ( x _ 0 ) } \\big ( D _ \\alpha v - ( D _ \\alpha v ) _ { \\Omega _ r ( x _ 0 ) } \\big ) \\cdot f _ \\alpha + \\big ( \\pi - ( \\pi ) _ { \\Omega _ r ( x _ 0 ) } \\big ) g \\ , d x . \\end{aligned} \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} - 2 \\sin 2 \\theta ( y , { E } ) \\sin 2 \\theta ( y , \\hat { E } ) = \\cos ( 2 \\theta ( y , { E } ) + 2 \\theta ( y , \\hat { E } ) ) - \\cos ( 2 \\theta ( y , { E } ) - 2 \\theta ( y , \\hat { E } ) ) . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} | \\Theta ( C _ n ) | & = \\sum _ { i = 1 } ^ n \\left ( 4 - b ( x _ i ) \\right ) = 4 n - \\sum _ { i = 1 } ^ n b ( x _ i ) \\\\ & = 4 n \\ , - \\ , 2 | B ( C _ n ) | = 4 n \\ , + \\ , 2 E ( C _ n ) \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 3 / 4 } ^ 0 \\bigg [ \\frac { b _ { 1 / 2 , 0 } ( d , s ) } { d ^ { 1 / 2 } \\Gamma ( s + 1 / 4 ) } \\bigg ] = \\frac { \\pi } { \\sqrt { d } } \\frac { 2 } { B ( 1 ) } \\mathrm { T r } _ { d , 1 } ( 3 / 2 \\pi ) = \\frac { 3 } { \\sqrt { d } } \\mathrm { T r } _ { d , 1 } ( 1 ) , \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} G _ { 3 / 2 , m , 0 } ( z ) = \\frac { 2 } { \\sqrt { \\pi } } \\biggl ( q ^ m + O ( q ) \\biggr ) , \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} \\sum _ { N \\ge n _ 1 \\ge \\cdots \\ge n _ k \\ge 1 } \\frac 1 { n _ 1 ^ { i _ 1 } \\cdots n _ k ^ { i _ k } } & = \\sum _ { N - 1 \\ge n _ 1 \\ge \\cdots \\ge n _ k \\ge 1 } \\frac 1 { n _ 1 ^ { i _ 1 } \\cdots n _ k ^ { i _ k } } \\\\ & \\quad + \\sum _ { N = n _ 1 \\ge n _ 2 \\ge \\cdots \\ge n _ k \\ge 1 } \\frac 1 { n _ 1 ^ { i _ 1 } \\cdots n _ k ^ { i _ k } } . \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} ( - 1 ) { ! } { ! } : = 1 , & & ( 2 n - 1 ) { ! } { ! } : = 1 \\cdot 3 \\cdot \\ldots \\cdot ( 2 n - 1 ) . \\end{align*}"}
-{"id": "9904.png", "formula": "\\begin{align*} \\| u \\| _ { \\widehat { H } ^ \\mu ( \\mathbb { R } ) } : = \\left ( \\| u \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } + | u | ^ 2 _ { \\widehat { H } ^ \\mu ( \\mathbb { R } ) } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\# \\mathcal { A } _ N \\cap \\Gamma _ \\alpha ^ { m , \\varepsilon } } { \\mathcal { S } _ { \\mathcal { A } _ N , \\alpha } } = \\mathcal { R } ( \\Gamma _ \\alpha ^ { m , \\varepsilon } ) , 1 \\le m \\le M _ \\alpha ^ { ( \\varepsilon ) } \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} \\phi ( t _ { n - \\theta } ) = & ( 1 - \\theta ) \\phi ( t _ { n } ) + \\theta \\phi ( t _ { n - 1 } ) + O ( \\Delta t ^ 2 ) \\\\ = & ( 1 - \\theta ) \\phi ^ { n } + \\theta \\phi ^ { n - 1 } \\\\ \\triangleq & \\phi ^ { n - \\theta } \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} & \\ x \\in C \\\\ & f ( x , y ) \\geq 0 y \\in C \\langle a ^ i , x \\rangle \\leq b _ i \\ ( i = 1 , \\ldots , m ) . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} \\mathcal { H } _ { 0 } \\stackrel { \\mathrm { d e f } } { = } \\left \\{ K , K ^ { 2 } , \\left | K \\right | ^ { 3 } , \\mathbf { 1 } \\{ x : | x | \\leq 1 / 2 \\} \\right \\} . \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} \\mathcal { I } _ { s , c } ^ { \\pm } ( m ) = \\left ( \\frac { \\mathcal { M } l ^ { 3 \\epsilon n _ l } } { m } \\right ) ^ { \\frac { 1 } { 4 } } \\min \\left ( M , \\frac { B Z l ^ { \\frac { n _ l } { 2 } } } { A } \\right ) e ( \\theta _ { s , c } m ) W _ { s , c } ( m ) + O ( l ^ { - 1 0 0 n _ l } ) , \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\underset { \\{ \\mathbf { x } _ i \\} \\in \\mathbb { R } ^ p , \\{ \\mathbf { z } _ { a ( i , j ) } \\} \\in \\mathbb { R } ^ p } { } \\sum _ { i = 1 } ^ n f _ i ( \\mathbf { x } _ i ) \\\\ \\mathbf { x } _ i = \\mathbf { z } _ { a ( i , j ) } , \\ , \\mathbf { x } _ j = \\mathbf { z } _ { a ( i , j ) } , \\ , \\forall ( i , j ) \\in \\mathcal { A } , \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} W _ { \\phi , p } ( a ( y ) w n ( \\zeta _ p ) ) = \\sum _ { \\mu _ p \\in { } _ p \\mathfrak { X } } \\mu _ p ( y ^ { - 1 } ) B _ { \\mu _ p ^ { - 1 } \\pi _ { p } , \\frac { 1 } { 2 } } ( \\varpi _ { p } ^ { - a ( \\mu _ p \\tilde { \\pi } _ { p } ) } y ^ { - 1 } ) \\epsilon ( \\frac { 1 } { 2 } , \\mu _ p ^ { - 1 } \\pi _ { p } ) [ \\mathfrak { M } W _ l ^ { \\omega _ { \\pi , p } } ] ( \\mu _ p ) . \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} \\mathcal { P } _ { 2 } = \\frac { 3 2 } { 9 } y ^ { 6 } - \\frac { 3 1 } { 3 } y ^ { 4 } + \\frac { 1 5 } { 2 } y ^ { 2 } - \\frac { 3 7 } { 4 8 } \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} d ( z , z ' ) = \\begin{cases} \\sqrt { r ^ 2 + r '^ 2 - 2 r r ' \\cos ( d _ Y ( y , y ' ) ) } , & d _ Y ( y , y ' ) \\leq \\pi ; \\\\ r + r ' , & d _ Y ( y , y ' ) \\geq \\pi , \\ \\end{cases} \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\binom k j \\left ( { \\frac { f _ 2 } { f _ 1 } } \\right ) ^ j X _ { m - a k + ( a - b ) j } } = \\frac { { X _ m } } { { f _ 1 { } ^ k } } \\ , , \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} \\tau = 3 \\ , e ^ { 4 5 } - 3 \\ , e ^ { 6 7 } . \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ N c _ { \\ell } \\tilde { \\Phi } _ { \\ell } ( x ) = \\dfrac { \\sum \\limits _ { j = 0 } ^ N \\dfrac { \\Omega _ j } { x - x _ j } \\left ( \\sum \\limits _ { \\ell = 0 } ^ N c _ { \\ell } \\tilde { \\Phi } _ { \\ell } ( x _ j ) \\right ) } { \\sum \\limits _ { j = 0 } ^ N \\dfrac { \\Omega _ j } { x - x _ j } } . \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} e ^ { - \\delta ( t ) O _ q } e ^ { - i Q _ 2 } = e ^ { - i \\delta ( t ) ( i ^ { - 1 } O _ q ) } e ^ { - i Q _ 2 } = \\pm e ^ { - i t ( i ^ { - 1 } K _ { \\nu , \\alpha } ) } = \\pm e ^ { - t K _ { \\nu , \\alpha } } \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} \\limsup _ { \\lambda \\to \\infty } A _ 1 ( \\lambda , \\varepsilon ) = 0 . \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} \\theta ( \\psi ( G ) ) ( t ) ( m \\otimes \\sigma ) = \\sum _ { ( E Z ( t \\otimes \\sigma ) ) } ( - 1 ) ^ { \\epsilon } G ( [ E Z ( t \\otimes \\sigma ) ' ] ) ( m \\otimes E Z ( t \\otimes \\sigma ) '' ) , \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} u ( x , t ) = u _ { \\Omega } ( \\phi _ + + \\phi _ { - } - 1 ) + A ( \\chi _ { + } + \\chi _ { - } ) . \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} g ( z ) = & \\ \\ \\ \\ 1 + z ^ { 2 c _ 1 - 1 } + z ^ { 2 ( 2 c _ 1 - 1 ) } + \\dots + z ^ { ( k - 2 ) ( 2 c _ 1 - 1 ) } \\\\ & + z ^ { c _ 1 + c _ 2 } \\left ( 1 + z ^ { 2 c _ 1 - 1 } + z ^ { 2 ( 2 c _ 1 - 1 ) } + \\dots + z ^ { ( k - 2 ) ( 2 c _ 1 - 1 ) } \\right ) \\\\ & + \\frac { z ^ { ( k - 1 ) ( 2 c _ 1 - 1 ) } + z ^ { c _ 2 + 1 } } { z ^ { c _ 1 } + 1 } . \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} T ( x ) = T ( - x ) \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} C ( c _ 0 ^ 2 | \\theta \\cdot \\nabla p | ^ 2 - 1 ) = 0 , \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} \\xi ^ a = \\pm \\sqrt { \\frac a 2 } \\ , \\begin{pmatrix} 0 \\\\ i \\end{pmatrix} e ^ { i ( x ^ 3 + x ^ 4 ) / 2 } , \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} \\Upsilon ^ n : = \\left \\{ \\mathbf { u } \\in \\R ^ n \\ | \\ u _ 1 ^ 2 \\geq \\sum \\limits _ { i = 3 } ^ { n } u _ i ^ 2 , \\ u _ 1 \\geq 0 , \\ u _ 2 \\geq 0 \\right \\} . \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} F ( X ; Y ) = F ( Y \\ , | \\ , X ) - F ( Y ) \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} \\frac 1 { | B _ \\rho | } \\int _ { B _ \\rho } w - \\inf _ { B _ { 2 \\rho } } w \\le C \\big ( \\sum _ { k = 2 } ^ N \\nu _ { \\eta _ k } ( 2 \\rho ) - \\nu _ { \\eta _ k } ( \\rho ) + \\rho ( \\rho + 1 ) \\big ) . \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} \\nabla \\psi _ { \\epsilon } ( y ) = d _ { \\epsilon } + h _ { \\epsilon } , \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} y \\bullet \\varphi ( x _ 1 , \\dots , x _ n ) = [ \\varphi ( x _ 1 , \\dots , x _ n ) , y ] - \\sum _ { i = 1 } ^ n \\varphi ( x _ 1 , \\dots , x _ { i - 1 } , [ x _ i , y ] , x _ { i + 1 } , \\dots , x _ n ) , \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} f _ m ( \\alpha _ m ) = \\left ( \\frac { C _ m } { C _ m + B _ m } \\right ) ^ { C _ m / B _ m } \\frac { B _ m } { C _ m + B _ m } \\left \\{ \\begin{array} { c l } \\leq & \\frac { B _ m } { C _ m + B _ m } = \\frac { g m + h } { a m ^ 2 + ( b + g ) m + c + h } \\\\ \\\\ = & ( e ^ { - 1 } g / a m ) ( 1 + o ( 1 ) ) \\end{array} \\right . \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t F + v \\cdot \\nabla _ { x } F = Q ^ { \\epsilon } ( F , F ) , ~ ~ t > 0 , x \\in \\mathbb { T } ^ { 3 } , v \\in \\R ^ 3 ; \\\\ & F | _ { t = 0 } = F _ { 0 } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} d X _ t = u ( t , X _ t ) d t + \\beta _ j ( X _ t ) d \\Z _ t ^ j + d B _ t , X _ 0 = x \\in \\R ^ d \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} c _ 4 ( \\mathcal T _ { \\widetilde X } ) = \\widetilde X \\cdot c _ 4 ( \\mathcal T _ { \\mathbb P } ) - \\widetilde X ^ 2 \\cdot c _ 3 ( \\mathcal T _ { \\mathbb P } ) + \\widetilde X ^ 3 \\cdot c _ 2 ( \\mathcal T _ { \\mathbb P } ) - \\widetilde X ^ 4 \\cdot c _ 1 ( \\mathcal T _ { \\mathbb P } ) + \\widetilde X ^ 5 . \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( n - \\frac { 1 } { 2 } \\right ) \\exp \\left ( - \\frac { \\log { R } } { 2 } \\left ( n - \\frac { 1 } { 2 } \\right ) \\right ) \\ge \\sqrt { \\frac { \\mathcal { N } _ F ( T ) } { K _ { F , 3 } ( T , \\tau ) } } \\exp \\left ( \\frac { \\log { R } } { 4 } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} \\norm { T _ S f ( \\cdot , w ) } _ { A ^ 2 ( S ( p ( w ) , \\infty ) ) } ^ 2 = \\int _ { S ( p ( w ) , \\infty ) } \\abs { T _ S f ( z , w ) } ^ 2 \\d V ( z ) = \\int _ 0 ^ { \\infty } \\abs { f ( t , w ) } ^ 2 \\frac { e ^ { - 4 \\pi p ( w ) t } } { 4 \\pi t } \\d t . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} w _ \\mu ( t ) = \\sum _ { k = 1 } ^ \\infty b ^ { - k \\mu } \\cos ( b ^ k \\pi t ) , \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} K ( \\xi + a - d \\Gamma _ A ( m ) ) = & K ( \\xi - d \\Gamma _ A ( m ) ) + a \\cdot \\nabla K ( \\xi - d \\Gamma _ A ( m ) ) \\\\ & + E _ { \\xi , A } ( a ) \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} \\dot { \\theta } _ { i , j } = \\omega + \\sum _ { i ' , j ' } \\sin ( \\theta _ { i ' , j ' } - \\theta _ { i , j } ) , \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} E ( r _ n ) = \\frac { E ( \\tilde { r } _ n ) } { \\lambda _ n ^ 2 } + \\frac { \\lambda _ n ^ \\alpha - 1 } { \\alpha + 2 } \\| r _ n \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } \\geq \\frac { E ( \\tilde { r } _ n ) } { \\lambda _ n ^ 2 } . \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} \\rho _ { s , k } ( \\varphi b ) \\ , : = \\ , { \\mathbb E } _ { 0 } \\big [ Z _ { s } ^ { - 1 } \\varphi ( s , \\overline { X } _ { s , 1 } , \\ldots , \\overline { X } _ { s , k } ) b ( s , X _ { s , 1 } , F _ { s , 1 } ) \\ , \\vert \\ , \\mathcal F _ { T } ^ { X } \\ , \\big ] \\ , \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} A _ j : = ( 2 ^ { j } \\rho ) ^ d \\int _ { S ^ { d - 1 } } \\sup _ { r \\in [ 2 ^ { j } \\rho , 2 ^ { j + 1 } \\rho ] } | \\mathbf { K } ^ { \\varepsilon , n } _ { e , \\rho } ( r \\theta ) | \\leq C | \\log ( \\varepsilon ) | 2 ^ { j \\frac { 1 } { 2 } \\min \\{ \\alpha , 1 \\} } . \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} P ' ( V ) & : = T ^ d ( U \\oplus V ) = \\bigoplus _ { J \\subseteq [ d ] } \\left ( \\bigotimes _ { j \\in [ d ] \\setminus J } U _ j \\otimes \\bigotimes _ { j \\in J } V _ j \\right ) , \\\\ X ' ( V ) & : = X ( U \\oplus V ) \\subseteq P ' ( V ) , \\\\ Z ' ( V ) & : = \\left \\{ q \\in X ' ( V ) ~ \\middle | ~ h ( ( T ^ d ( 1 _ { U _ i } \\oplus 0 _ { V _ i \\to U _ i } ) _ i ) q ) \\neq 0 \\right \\} . \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } v _ T ( y _ 0 ) = \\tilde { k } ^ * ( z ) \\ \\ \\ \\ \\forall y _ 0 \\in Y _ z . \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} & t _ 1 ^ { n _ 1 } = 1 = t _ 2 ^ { n _ 2 } , & & t _ 1 \\theta _ 1 = \\theta _ 1 t _ 1 , & & t _ 1 \\theta _ 2 = \\omega \\ , \\theta _ 2 t _ 1 , & & t _ 1 t _ 2 = t _ 2 t _ 1 , \\\\ & \\theta _ 1 ^ 2 = 0 = \\theta _ 2 ^ 2 , & & t _ 2 \\theta _ 2 = \\theta _ 2 t _ 2 , & & t _ 2 \\theta _ 1 = \\omega ^ { - 1 } \\theta _ 1 t _ 2 , & & \\theta _ 1 \\theta _ 2 = - \\omega \\ , \\theta _ 2 \\theta _ 1 . \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 7 2 n + 4 8 ) q ^ n & \\equiv 2 \\dfrac { f _ 2 f _ 6 ^ 3 } { f _ 1 ^ 4 } \\\\ & \\equiv 2 \\dfrac { f _ 6 ^ 3 } { f _ 2 } ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} N _ { f , g } ^ h = \\sum _ { k \\in \\Psi ^ { \\# } ( Y , \\pi ) } \\frac { \\mathbb { S } _ { f , k } \\mathbb { S } _ { g , k } \\overline { \\mathbb { S } _ { h , k } } } { \\mathbb { S } _ { f _ { \\mathrm { s p } } , k } } \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} \\left | 2 - i y - \\frac { 1 } { n - 1 } \\sum _ { i = 1 } ^ { n - 1 } t _ i \\right | & = \\sqrt { ( 2 - \\frac { 1 } { n - 1 } \\operatorname { R e } \\sum _ { i = 1 } ^ { n - 1 } t _ i ) ^ 2 + ( y + \\frac { 1 } { n - 1 } \\operatorname { I m } \\sum _ { i = 1 } ^ { n - 1 } t _ i ) ^ 2 } \\\\ & \\geq \\sqrt { 1 + ( | y | - 1 ) ^ 2 } , \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} z ( t ) = C y ( x , t ) = y ( b , t ) , \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} \\| \\chi _ { A _ n } \\| _ k \\le m _ k \\phi _ { X _ 1 } ( t _ n ) = \\frac { m _ k } { m _ n } m _ n \\phi _ { X _ 1 } ( t _ n ) = \\frac { m _ k } { m _ n } \\| \\chi _ { A _ n } \\| _ n . \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} c _ k ( f ) = O ( 1 / | k | ^ { m + \\mu } ) \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} L _ { 2 2 } = - \\Delta ^ { - 1 } [ a , c ] ( a - b d ^ { - 1 } c ) ^ { - 1 } b d ^ { - 1 } . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} Q \\overline { \\nabla } ^ { ' } _ { X } Y = \\{ ( 1 - \\beta ) g ( X , \\phi Y ) - \\alpha g ( X , Y ) \\} Q \\xi + B h ( X , \\phi Y ) . \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} \\psi _ b ( \\mathbb { A } ) : = ( \\mathbb { P } ( \\mathbb { A } ) ) = \\begin{bmatrix} \\psi _ a ( A ) \\\\ b \\end{bmatrix} , \\psi _ a ( A ) : = \\frac { 1 } { 2 } \\begin{bmatrix} a _ { 3 2 } - a _ { 2 3 } \\\\ a _ { 1 3 } - a _ { 3 1 } \\\\ a _ { 2 1 } - a _ { 1 2 } \\end{bmatrix} , \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} \\mu _ p ( 1 + \\varpi _ p ^ { \\kappa } x ) = \\psi _ p \\left ( \\frac { \\alpha _ { \\mu _ p } x } { \\varpi _ p ^ n } \\right ) . \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} { n ( t - 1 ) ^ 2 \\over t + n } & = 1 + n t - ( 1 + n ) { t ( 1 + n ) \\over t + n } , \\\\ { ( t - 1 ) ^ 2 \\over t + ( 1 / n ) } & = n + t - ( 1 + n ) { t \\bigl ( 1 + { 1 \\over n } \\bigr ) \\over t + { 1 \\over n } } , \\\\ { ( t - 1 ) ^ 2 \\over t + s } & = { 1 \\over s } + t - { 1 + s \\over s } \\cdot { t ( 1 + s ) \\over t + s } . \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} | Z _ i | \\leq 2 \\hat \\ell \\beta n = ( 4 \\ell \\hat \\ell \\beta ) \\frac { n } { 2 \\ell } \\leq \\frac { \\varepsilon } { 2 \\Delta ^ 2 } | S _ { \\min } | . \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} \\sum _ { j = - \\infty } ^ { \\infty } ( 2 ^ { j } \\rho ) ^ d \\int _ { S ^ { d - 1 } } \\sup _ { r \\in [ 2 ^ { j } \\rho , 2 ^ { j + 1 } \\rho ] } | \\mathbf { K } ^ { \\varepsilon , n } _ { e , \\rho } ( r \\theta ) | d \\mathcal { H } ^ { d - 1 } ( \\theta ) \\lesssim | \\log ( \\varepsilon ) | . \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} \\phi ( q ) : = 1 + 2 \\sum _ { j = 1 } ^ { \\infty } ( - 1 ) ^ j q ^ { j ^ 2 / 2 } ~ . \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} { \\hat A } _ { T , t } ^ { ( i _ 1 i _ 2 ) q } = \\frac { T - t } { 2 \\pi } \\sum _ { r = 1 } ^ { q } \\frac { 1 } { r } \\left ( \\zeta _ { 2 r } ^ { ( i _ 1 ) } \\zeta _ { 2 r - 1 } ^ { ( i _ 2 ) } - \\zeta _ { 2 r - 1 } ^ { ( i _ 1 ) } \\zeta _ { 2 r } ^ { ( i _ 2 ) } + \\sqrt { 2 } \\left ( \\zeta _ { 2 r - 1 } ^ { ( i _ 1 ) } \\zeta _ { 0 } ^ { ( i _ 2 ) } - \\zeta _ { 0 } ^ { ( i _ 1 ) } \\zeta _ { 2 r - 1 } ^ { ( i _ 2 ) } \\right ) \\right ) . \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} S ( n , k ) = \\binom { n } { k } B _ { n - k } ^ { ( - k ) } \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} H _ { n , \\mu } ( \\xi , A ) = \\int ^ { \\oplus } _ { ( A ^ c ) ^ n } H _ \\mu ( \\xi - k _ 1 - \\dots - k _ n , A ) + \\omega ( k _ 1 ) + \\dots + \\omega ( k _ n ) d \\lambda _ { \\nu } ^ { \\otimes n } \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} ( e _ { ( z , w ) } \\circ T ) a & = \\left \\langle a , R _ { \\mathcal { Y } _ p } ( e _ { ( z , w ) } \\circ T ) \\right \\rangle _ { \\mathcal { Y } _ p } = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\pi } { k + 1 } \\int _ { \\mathbb { B } _ p } a _ k ( \\zeta ) \\overline { R _ { \\mathcal { Y } _ p } ( e _ { ( z , w ) } \\circ T ) ( k , \\zeta ) } ( 1 - p ( \\zeta ) ) ^ { k + 1 } \\d V ( \\zeta ) \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} \\bigg \\{ { \\bf x } = ( x _ { k , l } ) _ { 2 \\le k \\le r , 1 \\le l \\le N _ k } & \\in \\widehat { \\Delta } _ { { \\bf i } , \\lambda _ 1 , \\ldots , \\lambda _ r } \\cap \\mathbb { Z } ^ { N _ 2 + \\cdots + N _ r } ~ \\bigg | \\\\ & \\lambda _ 1 + \\cdots + \\lambda _ r - \\sum _ { 2 \\le k \\le r , 1 \\le l \\le N _ k } x _ { k , l } \\alpha _ { i _ { k , l } } = \\nu \\bigg \\} . \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} u _ t = a ( t , x ) u _ { x x } + b ( t , x ) u _ x + c ( t , x ) u \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } x _ { 1 , \\tau } \\\\ x _ { 2 , \\tau } \\\\ t _ \\tau \\\\ p _ { 1 , \\tau } \\\\ p _ { 2 , \\tau } \\\\ p _ \\tau \\end{array} \\right ] = \\left [ \\begin{array} { c c } A _ 1 ( Q _ 1 , T ) \\\\ A _ 2 ( Q _ 2 , T ) \\\\ B ( T ) \\\\ C _ 1 ( Q _ 1 , T ) P _ 1 + D _ 1 ( Q _ 1 , T ) \\\\ C _ 2 ( Q _ 2 , T ) P _ 2 + D _ 2 ( Q _ 2 , T ) \\\\ F ( Q _ 1 , Q _ 2 , T , P _ 1 , P _ 2 , P _ T ) \\end{array} \\right ] \\ ; . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} \\langle X \\cdot \\psi , \\xi \\rangle _ { \\Sigma M } = - \\langle \\psi , X \\cdot \\xi \\rangle _ { \\Sigma M } \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} \\frac { v _ { 1 } / l } { u _ { 1 } / l } = \\frac { v _ { 1 } } { u _ { 1 } } = \\frac { t } { s } = \\frac { q _ { 1 } } { p _ { 1 } } = \\frac { q _ { 1 } / l } { p _ { 1 } / l } \\| I _ { \\alpha } ( | f | ^ { l } ) ^ { 1 / l } \\| _ { \\mathcal { M } _ { v _ { 1 } } ^ { u _ { 1 } } } = \\| I _ { \\alpha } ( | f | ^ { l } ) \\| _ { \\mathcal { M } _ { v _ { 1 } / l } ^ { u _ { 1 } / l } } ^ { 1 / l } . \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} L _ { m , n } ^ { I , \\alpha } ( x ) = \\begin{vmatrix} L _ { m } ^ { ( \\alpha ) } ( - x ) & - L _ { n - m - 1 } ^ { ( \\alpha ) } ( x ) \\\\ L _ { m } ^ { ( \\alpha - 1 ) } ( - x ) & L _ { n - m } ^ { ( \\alpha - 1 ) } ( x ) \\end{vmatrix} , \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} \\partial _ n H ^ - _ \\gamma = | \\partial _ n \\psi | \\frac { b ^ - } { \\sqrt { R _ 1 R _ 2 } } , \\partial _ n H ^ + _ \\gamma = | \\partial _ n \\psi | \\frac { b ^ + } { \\sqrt { R _ 1 R _ 2 } } . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} S ( r ) : = & \\frac { f _ 1 ' ( r ) } { f _ 1 ( r ) } = \\sqrt { K _ 0 } + f ( r ) , \\\\ K ( r ) : = & - \\frac { f _ 1 '' ( r ) } { f _ 1 ( r ) } = - ( \\sqrt { K _ 0 } + f ( r ) ) ^ 2 - f ^ { \\prime } ( r ) \\\\ = & - { K _ 0 } - 2 \\sqrt { K _ 0 } f ( r ) - f ^ 2 ( r ) - f ^ { \\prime } ( r ) , \\\\ q ( r ) : = & \\frac { ( n - 1 ) ( n - 3 ) } { 4 } S ( r ) ^ 2 - \\frac { ( n - 1 ) } { 2 } K ( r ) = \\frac { ( n - 1 ) ^ 2 } { 4 } ( \\sqrt { K _ 0 } + f ( r ) ) ^ 2 + \\frac { n - 1 } { 2 } f ^ { \\prime } ( r ) . \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { ( - 1 ) ^ s \\binom k j \\binom j s \\frac { { G _ { m + ( n + 1 ) k - 2 j + ( n + 1 ) s } } } { { F _ n ^ s } } } } = ( - 1 ) ^ { ( n + 1 ) k } \\frac { { G _ m } } { { F _ n ^ k } } , n \\ne 0 \\ , , \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} \\ell ( G , \\mathcal M ^ 2 _ S ) = \\ell ( G , \\mathcal M ^ 2 _ R ) = 0 \\ , , \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} u ( 0 ) + \\max _ { i } \\max _ { \\tilde { \\theta } \\in [ 0 , 1 ] } H _ { i } \\left ( 0 , u _ { x _ { i } } ( 0 ) - \\tilde { \\theta } \\left ( \\sum _ { j = 1 } ^ { K } u _ { x _ { j } } ( 0 ) \\right ) ^ { + } \\right ) \\geq 0 . \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { K _ L } ( t ) : = K _ L ^ { - p } \\phi ( K _ L t ) \\qquad \\textrm { f o r } t \\geq 0 \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ j \\binom k j G _ { 2 j } } = ( - 1 ) ^ k G _ k \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} 2 \\left \\langle \\nabla _ { \\partial _ r } Y , Z \\right \\rangle _ h = \\left \\langle { \\partial _ r Y } , Z \\right \\rangle _ h - \\left \\langle { \\partial _ r Z } , Y \\right \\rangle _ h = 0 . \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} Z ( K ; ( P \\underline { X } , \\Omega \\underline { X } ) ) \\to Z ( K ; ( \\underline { X } ^ { [ 0 , 1 ] } , P \\underline { X } ) ) \\to \\prod _ { i = 1 } ^ m X _ i . \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} | q \\phi \\rangle = | \\phi \\rangle \\overline { q } , ~ ~ \\langle \\phi | = | \\phi \\rangle ^ { \\dagger } , ~ ~ \\langle q \\phi | = q \\langle \\phi | \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} B _ { \\overline { \\nu } , 0 } ( t , \\overline \\alpha ) = \\sum _ { h = 0 } ^ L c _ h \\frac { L ! } { h ! } t ^ h , \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} g ( C _ n , x ) = \\sum _ { k = 0 } ^ m \\frac { 1 } { n - k + 1 } \\binom { n } { k } \\binom { 2 n - 2 k } { n } ( x - 1 ) ^ k , m = \\lfloor n / 2 \\rfloor . \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} p _ { _ 1 } ( y ) = \\frac { 1 } { \\sqrt { 2 \\pi P _ y } } e ^ { - \\frac { y ^ 2 } { 2 P _ y } } , \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} \\Delta ( X ^ n ) = ( X + 1 ) ^ n - X ^ n . \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( x , y , t ) = { } & \\varphi ( x , y , t ) + \\tau ^ { 1 / 2 } a ( x , y , t ) + \\tau b ( x , y , t ) \\\\ & + \\tau ^ { 3 / 2 } c ( x , y , t ) + \\tau ^ { 2 } d ( x , y , t ) , \\end{aligned} \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} | u _ j - u | = 2 \\max ( u _ j , u ) - u _ j - u \\leq 2 ( \\tilde { u } _ j - u ) + ( u - u _ j ) . \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} m \\left ( r , \\frac { f ( z + c ) } { f ( z ) } \\right ) = o \\left ( \\frac { T ( r , f ) } { ( \\log r ) ^ { \\nu - \\varepsilon } } \\right ) , \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} p _ k ^ { ( \\rm i d , a ) } = \\frac { \\beta \\ , p _ 1 ^ { ( \\rm i d , a ) } } { \\beta + [ \\boldsymbol { \\Lambda } ] _ { 1 , 1 } ^ 2 - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } . \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} \\sum _ { d \\mid n } \\mu ( d ) a _ { n m / d } ( \\omega _ n ^ { i d } ) = 0 . \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } F _ { i } ( t , x , u , u _ { x _ { i } } ) = 0 & \\ , \\ , I _ { i } ^ { \\delta } \\times ( 0 , T ) \\\\ \\sum _ { i = 1 } ^ { K } u _ { x _ { i } } = B & \\ , \\ , \\{ 0 \\} \\times ( 0 , T ) \\end{array} \\right . \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} H ^ p _ { \\Lambda , \\epsilon } ( \\phi ) = \\sum _ { j k \\in E ( \\mathbb { T } _ N ) } V ( \\phi _ j - \\phi _ k ) + \\epsilon \\sum _ { j \\in \\mathbb { T } _ N } \\phi _ j ^ 2 \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} \\mu \\sum _ { j = 0 } ^ { n - 1 } \\tilde { P } ^ j r = ( n - 1 ) \\tilde { \\pi } r + \\mu ( I - \\tilde { P } + \\tilde { \\Pi } ) ^ { - 1 } r + O ( \\epsilon ^ k ) \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} a ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } a ( x , y , t ) \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} L _ i \\phi _ j = 0 \\ , , \\mbox { f o r } ~ 1 \\le i \\le m \\ , , \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} ( u < \\phi - 2 t ) = ( t ^ { - 1 } u + \\phi - t ^ { - 1 } \\phi < \\phi - 2 ) \\subset ( u _ t < v - 1 ) \\subset ( u _ t < \\phi - 1 ) = ( u < \\phi - t ) . \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} q _ { P , H ' } ( z ) = p _ { P , H ' } ( z ) \\cdot \\prod _ { i = 1 } ^ n | X _ i | \\ , \\mbox { . } \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} d X _ t & = b \\left ( t , X _ t , \\mathcal { L } ( X _ t ) , \\hat { \\alpha } \\left ( t , X _ t , \\mathcal { L } ( X _ t ) , \\sigma ^ { - 1 } Z _ t \\right ) \\right ) d t + \\sigma d W _ t \\\\ X _ 0 & = \\xi , \\\\ d Y _ t & = - f \\left ( t , X _ t , \\mathcal { L } ( X _ t ) , \\hat { \\alpha } \\left ( t , X _ t , \\mathcal { L } ( X _ t ) , \\sigma ^ { - 1 } Z _ t \\right ) \\right ) d t + Z _ t d W _ t \\\\ Y _ T & = g ( X _ T , \\mathcal { L } ( X _ T ) ) , \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} r ( n ) = \\begin{dcases*} 0 & \\\\ 2 ^ { - k } \\phantom { - } & \\end{dcases*} \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} x ( b ) = & \\arg \\min f ( x ) \\\\ & s . t . A x = b \\\\ & x \\in \\mathcal { X } \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 0 & 1 & 1 \\\\ 1 & - 2 u ^ 3 & u ^ 2 + u ^ 3 \\\\ 1 & u ^ 2 + u ^ 3 & - 2 u ^ 2 \\end{pmatrix} , \\\\ w _ { 1 2 } = w _ { 3 1 } = 0 , w _ { 2 3 } = \\frac { 2 } { \\sqrt { 4 u ^ 2 + 4 u ^ 3 } } . \\end{gather*}"}
-{"id": "1786.png", "formula": "\\begin{align*} \\Pi _ { k \\boldsymbol { \\nu } } \\big ( x _ { 1 } , x _ { 2 } \\big ) \\sim \\Pi _ { k \\boldsymbol { \\nu } } \\big ( x _ { 1 } , x _ { 2 } \\big ) _ 1 = \\Pi _ { k \\boldsymbol { \\nu } } \\big ( x _ { 1 } , x _ { 2 } \\big ) _ { 1 1 } + \\Pi _ { k \\boldsymbol { \\nu } } \\big ( x _ { 1 } , x _ { 2 } \\big ) _ { 1 2 } , \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} M _ { n } ^ { ( N , r ) } = \\sum _ { m = 1 } ^ n W _ { m , r } ^ { ( N ) } = \\sum _ { i = 1 } ^ { \\ell } M _ { i , n , r } ^ { ( N ) } . \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { \\mathcal F ( \\rho _ t ^ \\xi ) - \\mathcal F ( \\rho ) } t = \\int _ \\Lambda \\nabla [ L _ f ( \\rho ) ] \\cdot \\nabla \\xi \\ ; \\ ! \\mathrm d u \\quad \\textrm { a n d } \\lim _ { t \\to 0 } \\frac { \\mathcal V ( \\rho _ t ^ \\xi ) - \\mathcal V ( \\rho ) } t = \\int _ \\Lambda \\rho \\nabla V \\cdot \\nabla \\xi \\ ; \\ ! \\mathrm d u . \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} \\mathcal A _ v = \\big \\{ A _ v ^ { S _ 1 } ( w ) - \\alpha ^ { S _ 2 } ( w , v ) : w \\in T _ p S _ 1 \\cap T _ p S _ 2 \\big \\} . \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} u _ { \\mu } ( x ) = \\mu ^ { \\frac { n - p } { p } } u ( \\mu x ) , \\mu > 0 , \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} \\{ F \\in d g C a t _ { \\mathbf { k } } ( \\Lambda ( J ^ n \\times ( X , b ) ) , C h _ { \\mathbf { k } } ) : F | _ { \\Lambda ( \\{ x \\} \\times ( X , b ) ) } = P , F | _ { \\Lambda ( \\{ y \\} \\times ( X , b ) ) } ) = Q \\} \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} J ^ { ( \\alpha , \\beta ) } f ( n ) = a _ { n - 1 } ^ { ( \\alpha , \\beta ) } f ( n - 1 ) + b _ n ^ { ( \\alpha , \\beta ) } f ( n ) + a _ { n } ^ { ( \\alpha , \\beta ) } f ( n + 1 ) , n \\ge 1 , \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} \\left | \\begin{array} { l l l l } C _ 2 & C _ 3 & \\dots & C _ { m + 1 } \\\\ C _ 3 & C _ 4 & \\dots & C _ { m + 2 } \\\\ & & \\dots \\\\ C _ { m + 1 } & C _ { m + 2 } & \\dots & C _ { 2 m } \\end{array} \\right | = 0 . \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} \\mathbb { P } ( D > d ) = \\mathbb { P } \\left \\{ \\sup _ { t \\ge { d } } \\left ( A ( d , t ) - S ( 0 , t ) \\right ) > 0 \\right \\} . \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} w _ 0 ^ \\pm - b _ 0 - 2 \\log \\mu _ \\lambda + \\log h _ \\gamma = \\mathcal O \\left ( \\frac { \\log \\log \\frac { 1 } { \\lambda } } { \\log \\frac { 1 } { \\lambda } } \\right ) \\ \\hbox { o n } \\ \\gamma \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\Re ( \\xi ^ + _ i ) = - \\Re ( \\xi ^ - _ i ) \\in \\mathbb { Z } , \\ \\sum _ { i \\in V ( Q ) } m _ i \\cdot \\Re ( \\xi ^ { \\pm } _ i ) = 0 . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\leq C _ 1 R ^ { - d / q } \\| w _ { \\bar { L } } \\| _ { L ^ q ( B _ R ) } & \\leq C _ 1 R ^ { - d / q } \\| w _ { \\bar { L } } \\| ^ { 1 / q } _ { L ^ 1 ( B _ R ) } \\| w _ { \\bar { L } } \\| ^ { 1 - 1 / q } _ { L ^ \\infty ( B _ R ) } \\\\ & \\stackrel { \\eqref { e q : e 6 } } { \\leq } C _ 1 R ^ { - d / q } 2 ^ { p - p / q } \\| w _ { \\bar { L } } \\| ^ { 1 / q } _ { L ^ 1 ( B _ R ) } , \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} H ( X , Y + Z ) + H ( Y , Z ) - H ( X + Y , Z ) - H ( X , Y ) = 0 . \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} \\| I _ { \\alpha } ( | f | ^ { l } ) ^ { 1 / l } \\| _ { \\mathcal { M } _ { v _ { 1 } } ^ { u _ { 1 } } } \\lesssim \\| f ^ { l } \\| _ { \\mathcal { M } _ { q _ { 1 } / l } ^ { p _ { 1 } / l } } ^ { \\frac { 1 } { l } } = \\| f \\| _ { \\mathcal { M } _ { q _ { 1 } } ^ { p _ { 1 } } } . \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} J _ i ( \\pi ) : = E _ { \\mu } ^ \\pi \\left ( \\sum _ { n = 1 } ^ \\infty \\beta ^ { n - 1 } r _ i ( s _ n , a _ n ) \\right ) , i \\in I _ 0 . \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} \\nu _ 2 ( S ( c 2 ^ h , k ) ) = \\sigma _ 2 ( k ) - 1 \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} ( - 1 ) ^ { k + \\frac { k d } { n } } & \\left ( \\left ( n \\frac { ( 3 d ^ 2 + 1 ) n ^ 2 - 6 d ^ 2 n + 2 d ^ 2 } { 1 2 d } - 2 \\binom { n } { 3 } \\right ) \\left ( n \\binom { d - 2 } { \\frac { k d } { n } - 2 } - k \\binom { d - 1 } { \\frac { k d } { n } - 1 } \\right ) \\right . \\\\ & \\left . - \\binom { n } { 2 } \\left ( n \\binom { d - 2 } { \\frac { k d } { n } - 2 } - ( k - 1 ) \\binom { d - 1 } { \\frac { k d } { n } - 1 } \\right ) + \\binom { n } { 2 } ^ 2 \\binom { d - 1 } { \\frac { k d } { n } - 1 } \\right ) . \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} h _ { t , s , a } \\cdot ( z ' , 0 ) = \\left ( \\frac { t } { \\cosh s } z ' , \\tanh s \\right ) . \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} K ( n , i , j ) = \\sum _ { k = 0 } ^ { n - 1 } \\binom { k } { i } \\binom { n - 1 - k } { i + j - k } A ( n , k ) . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} | | u | ^ { p - 2 } u | _ { \\alpha ; \\bar { \\Omega } } = | | | u | ^ { p - 2 } u | | _ { C ( \\bar { \\Omega } ) } + [ | u | ^ { p - 2 } u ] _ { \\alpha ; \\bar { \\Omega } } \\leq C ( \\Omega ) + C ( \\Omega , p ) [ u ] _ { \\alpha ; \\bar { \\Omega } } \\leq C . \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} \\lim _ { q \\to + \\infty } \\deg ( p ) / q = 0 \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} t = E ( \\theta \\ , \\beta ) = e ^ { \\theta \\beta } = \\begin{pmatrix} e ^ { \\imath \\theta } & 0 \\\\ 0 & e ^ { - \\imath \\theta } \\end{pmatrix} . \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| v ^ k - x ^ k \\| = \\lim _ { k \\to \\infty } ( 1 - \\alpha _ k ) \\| x ^ k - T x ^ k \\| = 0 . \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} \\begin{aligned} U _ R ( x ) & = \\left [ a + b \\left \\{ \\frac { 1 } { | x | ^ { \\frac { n - p } { p - 1 } } } - \\frac { 1 } { R ^ { \\frac { n - p } { p - 1 } } } \\right \\} ^ { - \\frac { p } { n - p } } \\right ] ^ { 1 - \\frac { n } { p } } \\\\ & = \\left [ a + b R ^ { \\frac { p } { p - 1 } } \\left \\{ - \\frac { n - p } { p - 1 } \\log _ { \\frac { n - 1 } { p - 1 } } \\frac { | x | } { R } \\right \\} ^ { - \\frac { p } { n - p } } \\right ] ^ { 1 - \\frac { n } { p } } \\end{aligned} \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} C ( \\pi , s _ i v ) = ( C ( \\pi , v ) \\cup \\varDelta ^ \\prime ( \\pi , v , , i ) ) \\setminus E ( \\pi , v , i ) . \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} \\lvert ( i , j ) \\rvert _ { W _ { n _ 1 } ( w ) } - \\lvert ( i , j ) \\rvert _ { W _ { n _ 2 } ( w ) } = & ( \\lvert l _ i \\rvert _ S + \\lvert m _ { n _ 1 } \\rvert _ S + \\lvert r _ j \\rvert _ S ) \\\\ & - ( \\lvert l _ i \\rvert _ S + \\lvert m _ { n _ 2 } \\rvert _ S + \\lvert r _ j \\rvert _ S ) \\\\ = & \\lvert m _ { n _ 1 } \\rvert _ S - \\lvert m _ { n _ 2 } \\rvert _ S . \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} \\partial _ { \\omega } F ^ { z \\bar { z } } _ { ( - j ) j } + \\mathbf { i } ( \\Omega _ { - j } - \\Omega _ j ) F _ { ( - j ) j } ^ { z \\bar { z } } = R _ { ( - j ) j } ^ { z \\bar { z } } ; \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } Q ( f ) ( \\ddot { x } ) d \\mu _ \\rho ( \\ddot { x } ) = \\int _ G f ( x ) \\rho ( x ) d x . \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} \\tilde { O } ( \\Lambda _ { 2 d } ) : = \\{ g | _ { \\Lambda _ { 2 d } } : g \\in O ( \\Lambda _ { \\rm K 3 } ) , \\ , g ( \\lambda _ { 2 d } ) = \\lambda _ { 2 d } \\} . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} M _ { \\mathbb Z _ { 2 n } } \\Big ( g ( x ) g ( x ^ { - 1 } ) - x ^ n & f ( x ) f ( x ^ { - 1 } ) \\Big ) \\\\ & = ( - 1 ) ^ n M _ { \\mathbb Z _ { 2 n } } \\Big ( f ( x ) f ( x ^ { - 1 } ) - x ^ n g ( x ) g ( x ^ { - 1 } ) \\Big ) , \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} z = ( z _ { i j } ) _ { \\begin{subarray} { l } 0 \\leq i \\leq k \\\\ 0 \\leq j \\leq k + n + 1 \\end{subarray} } = \\bordermatrix { & 0 & 1 & 2 & \\cdots & k + n + 1 \\cr 0 & 1 & z _ { 0 1 } & z _ { 0 2 } & \\cdots & z _ { 0 , k + n + 1 } \\cr 1 & 0 & z _ { 1 1 } & z _ { 1 2 } & \\cdots & z _ { 1 , k + n + 1 } \\cr \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\cr k & 0 & z _ { k 1 } & z _ { k 2 } & \\cdots & z _ { k , k + n + 1 } \\cr } , z _ { i j } \\in \\C . \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} \\mathcal { H } W _ { l } ( y ) = \\abs { y } _ { p } ^ { - \\frac { 1 } { 2 } } \\sum _ { \\mu _ p \\in { } _ p \\mathfrak { X } } \\mu _ p ( y ^ { - 1 } ) B _ { \\mu _ p ^ { - 1 } \\pi _ { p } , \\frac { 1 } { 2 } } ( \\varpi _ { p } ^ { - a ( \\mu _ p \\tilde { \\pi } _ { p } ) } y ^ { - 1 } ) \\epsilon ( \\frac { 1 } { 2 } , \\mu _ p ^ { - 1 } \\pi _ { p } ) [ \\mathfrak { M } W _ l ] ( \\mu _ p ) . \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} m \\left ( r , \\frac { f ( z + c ) } { f ( z ) } \\right ) = o \\left ( \\frac { T ( e ^ { \\epsilon } r , f ) } { r ^ { 1 - \\varepsilon / 2 } } \\right ) = o \\left ( \\frac { r ^ { \\sigma + \\varepsilon / 2 } } { r ^ { 1 - \\varepsilon / 2 } } \\right ) = o ( r ^ { \\sigma - 1 + \\varepsilon } ) , \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{gather*} L \\big ( \\eta ( 3 \\tau ) ^ 8 , 2 \\big ) = \\frac { \\Gamma ( 1 / 3 ) ^ 9 } { 9 6 \\pi ^ 4 } \\qquad L \\big ( \\eta ( 3 \\tau ) ^ 8 , 3 \\big ) = \\frac { \\Gamma ( 1 / 3 ) ^ 9 } { 1 4 4 \\sqrt 3 \\pi ^ 3 } \\end{gather*}"}
-{"id": "975.png", "formula": "\\begin{gather*} \\alpha _ 1 = \\delta _ n ^ k - \\alpha _ k , \\{ \\alpha _ j = \\delta _ n ^ k + \\alpha _ { j - 1 } - \\alpha _ k \\} _ { j = 2 } ^ k \\\\ \\{ \\alpha _ j = \\alpha _ { j + ( i - 1 ) } - \\alpha _ { i - 1 } \\} _ { j = 1 } ^ { k + 1 - i } , \\ \\ \\delta _ { n } ^ k = \\alpha _ { k + 2 - i } + \\alpha _ { i - 1 } , \\ \\{ \\delta _ n ^ k + \\alpha _ { j } - \\alpha _ { i - 1 } = \\alpha _ { j + k + 2 - i } \\} _ { j = 1 } ^ { i - 2 } \\ \\ k \\geq i \\geq 2 \\end{gather*}"}
-{"id": "2781.png", "formula": "\\begin{align*} \\phi ' ( \\cdot , x - \\kappa - ( 1 + \\lambda ) \\hat { z } ) = \\frac { 1 } { 1 + \\lambda } . \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} d z ( s ) = \\widetilde { b } ( z ( s ) ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } \\widetilde { \\sigma } ( z ( s ) ) \\ , d \\widetilde { w } ( s ) \\ , , s \\ge 0 \\ , , \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} W _ { t , \\alpha } \\overset { \\textrm { d e f } } = \\begin{bmatrix} S _ { 1 , t , \\alpha } & M _ { \\psi _ { t , \\alpha } } \\\\ ( M _ { \\psi _ { t , \\alpha } } ) ^ \\circ & S _ { 2 , t , \\alpha } \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} & M \\left ( ( 1 + x ^ 2 ) ( 1 + x ( x ^ 3 + 1 ) ) + t \\ : h ( x ) + y \\left ( ( 1 + x ^ 3 ) ( 1 + x ^ 2 ) + t \\ : h ( x ) \\right ) \\right ) = 2 ^ 4 ( 5 + 6 t ) , \\\\ & M \\left ( ( 1 - x ) + t \\ : h ( x ) + y \\left ( ( 1 + x ^ 2 ) ( 1 + x ^ 3 ) + t \\ : h ( x ) \\right ) \\right ) = - 2 ^ 6 ( 1 + 3 t ) , \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} \\langle ( \\phi _ 0 , s _ 0 ) , \\phi _ 1 , \\ldots , \\phi _ n \\rangle = \\langle ( \\phi _ 1 , s _ 1 ) , \\phi _ 0 , \\phi _ 2 , \\ldots , \\phi _ n \\rangle \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} R ^ * = U _ \\lambda \\sqrt { R R ^ * } . \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\binom k j F _ { s + j } } = F _ { s + 2 k } \\mbox { a n d } \\sum _ { j = 0 } ^ k { \\binom k j L _ { s + j } } = L _ { s + 2 k } \\ , , \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\left ( y \\alpha ( y , t ) \\right ) + \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) - \\frac { \\partial } { \\partial t } \\alpha ( y , t ) = - \\frac { 1 } { 8 } \\frac { \\partial ^ { 3 } } { \\partial y ^ { 3 } } p _ { 0 } ( y , t ) \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} \\mathcal T ^ * \\Theta _ \\Pi = J _ \\Pi . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { L } g _ i \\circ T ^ { n _ i } = \\big ( g _ 1 \\cdot G \\circ T ^ { n _ 2 - n _ 1 } \\big ) \\circ T ^ { n _ 1 } \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { c c } \\textrm { s i g n } ( \\zeta _ 1 ) & 0 \\\\ 0 & \\textrm { s i g n } ( \\zeta _ 2 ) \\end{array} \\right ) , \\ , \\ , \\ , \\ , \\ , v = \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) : = \\left ( \\begin{array} { c c } \\eta _ 1 & 0 \\\\ 0 & \\eta _ 2 \\end{array} \\right ) B . \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} \\bigg | \\sum _ { j \\geq 0 } \\bigg ( \\binom { y } { j } - \\binom { y _ m } { j } \\bigg ) ( \\langle a \\rangle - 1 ) ^ { j } \\bigg | _ { \\infty } \\leq q ^ { - q ^ { m } } . \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} S ( n , k ) \\nu ( S ( n , k ) ) = ( \\sigma ( k ) - \\sigma ( n ) ) / ( p - 1 ) . \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} \\omega ( x ) & = P ^ \\omega ( X _ { n + 1 } = x + 1 | X _ n = x ) \\\\ & = 1 - P ^ \\omega ( X _ { n + 1 } = x - 1 | X _ n = x ) . \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} \\begin{aligned} \\gamma _ { 1 } ( y , t ) = & - \\frac { 1 } { 2 4 } \\frac { \\partial ^ { 3 } } { \\partial y ^ { 3 } } p _ { 0 } ( y , t ) \\\\ = & \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( \\frac { 8 } { 3 } y ^ { 3 } - 2 y \\right ) e ^ { - 2 y ^ { 2 } } \\end{aligned} \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} & M \\left ( 1 + x ^ 2 + t \\ : h ( x ) + y t \\ : h ( x ) \\right ) = 2 ^ 4 ( 1 + 6 t ) , \\\\ & M \\left ( ( 1 + x ^ 2 ) + t \\ : h ( x ) + y \\left ( ( 1 + x ^ 2 ) ( 1 + x ^ 3 ) + t \\ : h ( x ) \\right ) \\right ) = - 3 ^ 3 2 ^ 4 ( 1 + 2 t ) , \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} \\tilde { \\bf X } : = s ^ { - 1 } ( \\tilde { X } ) = t ^ { - 1 } ( \\tilde { X } ) , \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} f _ * \\O _ { \\overline { Y } } ( A ) = f _ * \\O _ { \\overline { Y } } ( k f ^ * D - N + P ) = \\O _ Y ( k D ) \\otimes _ { \\O _ Y } f _ * \\O _ { \\overline { Y } } ( - N + P ) . \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} h ( \\phi X , Y ) = C h ( X , Y ) + \\phi Q \\overline { \\nabla } ^ { ' } _ { Y } X . \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} P \\{ Y = 1 \\vert \\textbf { X } = \\textbf { x } \\} = \\dfrac { f _ { \\textbf { X } \\vert Y = 1 } ( \\textbf { x } \\vert 1 ) P \\{ Y = 1 \\} } { f _ \\textbf { X } ( \\textbf { x } ) } = \\dfrac { \\dfrac { p } { \\mu { ( \\mathcal { C } _ { 1 } ) } } } { \\dfrac { p } { \\mu { ( \\mathcal { C } _ { 1 } ) } } + \\dfrac { 1 - p } { \\mu { ( \\mathcal { C } _ { 0 } ) } } } \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} E = \\sum _ { c \\geq - 2 } L _ { s , c } . \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\lvert u _ n \\rvert _ S & = \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - ( n - i ) - ( n - j ) \\\\ & = \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - 2 n + i + j \\\\ & \\geq \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - 2 n + ( 2 n - 2 \\lvert I \\rvert + 1 ) \\\\ & = \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - 2 \\lvert I \\rvert + 1 \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { 2 k } \\sum _ { j = 1 } ^ k { ( \\omega _ j f ( x _ j ) + \\omega _ j ^ { - 1 } f ( x _ j ^ { - 1 } ) ) } \\right | & \\geq \\frac { 1 } { 2 k } \\left ( \\left | \\sum _ { j : \\omega _ j \\in \\R } ^ k { | f ( x _ j ) | } \\right | ^ 2 + \\left | \\sum _ { j : \\omega _ j \\not \\in \\R } ^ k { | f ( x _ j ) | } \\right | ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } \\geq \\frac { 1 } { 8 } . \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\omega _ { p } ( y ) = \\chi ( p ) ^ { - v _ { p } ( y ) } . \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} x ^ { \\tau , \\epsilon } _ t = x _ 0 + \\int _ { 0 } ^ { t } f ( x ^ { \\tau , \\epsilon } _ s , \\theta ^ { \\tau , \\epsilon } _ s ) d s . \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\lambda ^ 2 _ { n } \\int _ \\Omega e ^ { \\ , u _ { n } } = \\infty . \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } t _ n ^ 2 = \\dfrac { 4 \\pi } { \\alpha _ 0 } . \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} \\lvert u \\rvert _ S = \\lvert ( i , 0 ) \\rvert _ { W _ n ( w ) } = \\lvert ( \\phi w ) _ n \\rvert _ S - ( n - i ) . \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\begin{aligned} & \\varrho ( s ) = t \\Leftrightarrow \\psi ( t ) = s s \\geq 0 , t \\geq 1 , \\\\ & \\rho ( s ) = t \\Leftrightarrow - \\psi ^ { \\prime } ( t ) = 2 s s \\geq 0 , 0 < t \\leq 1 . \\end{aligned} \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} F G _ \\sigma ^ - = \\left \\{ f \\in F G \\ ; | \\ ; h ( f ( x ) , y ) + h ( x , f ( y ) ) = 0 \\ ; \\ ; x , y \\in M \\right \\} . \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} \\mathrm { G } = \\Delta - \\langle \\nabla U , \\nabla \\rangle = \\sum _ { i = 1 } ^ n \\partial ^ 2 _ { x _ i } - \\sum _ { i = 1 } ^ n ( \\partial _ { x _ i } V ) ( x ) \\partial _ { x _ i } - \\sum _ { i \\neq j } W ' ( x _ i - x _ j ) \\partial _ { x _ i } \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} V _ K = ( K ^ T K ) ^ { 1 / 2 } U _ K = K ( K ^ T K ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} B : = \\ln ( ( y ) ^ { 2 s ^ 2 } \\varphi _ { ( 4 s + 1 ) / 2 } ) = - ( \\beta / 2 ) \\eta - \\beta - \\beta ^ 2 / 4 - K - K _ 1 ~ , \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} X ( t ) = a ( t ) - C ( t ) . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} S ( u _ { \\mu , \\xi } ) = - \\sum _ { i = 1 } ^ k \\partial _ t u _ i + \\left ( \\sum _ { i = 1 } ^ k u _ i \\right ) ^ p - \\sum _ { i = 1 } ^ k U ^ p _ { \\mu _ i , \\xi _ i } - \\sum _ { i = 1 } ^ k \\mu _ i ^ { \\frac { n - 2 s } { 2 } } ( - \\Delta ) ^ s \\Phi _ i ^ * ( x , t ) . \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} L _ { 0 } : = \\sup \\limits _ { l \\in \\{ 1 , \\dots , m \\} } \\| h _ { l } ^ { ( 1 ) } \\| _ { H ^ { s } _ { \\overline { \\Omega } } } \\| h _ { l } ^ { ( 2 ) } \\| _ { H ^ { s } _ { \\overline { \\Omega } } } , \\ L _ { 1 } : = \\sup \\limits _ { l \\in \\{ 1 , \\dots , m \\} } \\| h _ { l } ^ { ( 1 ) } \\| _ { L ^ 2 ( \\Omega ) } \\| h _ { l } ^ { ( 2 ) } \\| _ { L ^ 2 ( \\Omega ) } , \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} G \\in \\mathcal { C } & \\Longleftrightarrow \\alpha ( G ) = \\frac { 3 } { 4 } \\\\ & \\Longleftrightarrow \\frac { 1 + n _ 2 ( G ) + n _ 4 ( G ) } { 2 ^ n } = \\frac { 3 } { 4 } \\\\ & \\Longleftrightarrow \\frac { 2 ^ n - n _ 4 ( G ) } { 2 ^ n } = \\frac { 3 } { 4 } \\\\ & \\Longleftrightarrow n _ 4 ( G ) = 2 ^ { n - 2 } \\\\ & \\Longleftrightarrow n _ 2 ( G ) = 2 ^ { n - 1 } - 1 , \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} L _ { 1 1 } & = \\Delta ^ { - 1 } ( [ d , b ] d ^ { - 1 } + b + \\Delta ( c - d b ^ { - 1 } a ) ^ { - 1 } ) d b ^ { - 1 } \\\\ & = \\Delta ^ { - 1 } ( d b d ^ { - 1 } + \\Delta ( c - d b ^ { - 1 } a ) ^ { - 1 } ) d b ^ { - 1 } \\\\ & = \\Delta ^ { - 1 } ( d + \\Delta ( b d ^ { - 1 } c - a ) ^ { - 1 } ) , \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} Q _ 1 ^ k & = E \\bigg [ e ^ { - \\beta n } \\Big ( M ^ { \\phi } ( n ) - \\max _ { 0 \\le i \\le n } D _ i ^ { \\phi } ( n - \\tau _ i ) \\Big ) \\bigg ] ^ { \\ ! k } \\\\ & \\le E \\bigg [ \\sum _ { i > n } e ^ { - k \\beta n } \\big ( D _ i ^ { \\phi } ( t - \\tau _ n ) \\big ) ^ k \\bigg ] \\\\ & = E \\bigg [ \\sum _ { i > n } e ^ { - k \\beta \\tau _ i } \\zeta _ i ^ k ( n - \\tau _ i ) \\bigg ] \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} \\dot \\kappa ( \\theta ) = \\lim _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\left [ \\frac { 1 } { \\mathbb { E } e ^ { \\theta M ( t ) } } \\mathbb { E } \\left [ e ^ { \\theta M ( t ) } M ( t ) \\right ] \\right ] , \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\int _ { B _ { 2 R } } H _ j [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ( y , t ( \\tau ) ) Z _ l ( y ) d y = 0 , \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} \\omega ( E _ a , E _ b ) = \\omega ( S E _ a , S E _ b ) , \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} \\frac { 1 } { \\alpha _ { \\beta } Q ' e } = \\widetilde { A } ^ { Q ' } = \\widetilde { B } ^ { Q ' } , \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} q _ 1 - q _ 2 = \\sum \\limits _ { j = 1 } ^ { m } a _ j g _ j , \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} \\langle \\epsilon ( g _ 1 ) , W ( v _ g , - 1 ) \\epsilon ( g _ 2 ) \\rangle & = e ^ { - \\lVert v _ g \\lVert ^ 2 / 2 + \\langle v _ g , g _ 2 \\rangle } \\langle \\epsilon ( g _ 1 ) , \\epsilon ( v _ g - g _ 2 ) \\rangle \\\\ & = e ^ { - \\lVert v _ g \\lVert ^ 2 / 2 + \\langle v _ g , g _ 2 \\rangle + \\langle g _ 1 , v _ g \\rangle - \\langle g _ 1 , g _ 2 \\rangle } , \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { \\binom k j \\binom { k - j } s \\left ( { \\frac { { f _ 3 } } { { f _ 1 } } } \\right ) ^ j \\left ( { \\frac { { f _ 2 } } { { f _ 1 } } } \\right ) ^ s X _ { m - a k - ( c - a ) j - ( b - a ) s } } } = \\frac { { X _ m } } { { f _ 1 ^ k } } \\ , , \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} w = s _ { i _ 1 } s _ { i _ 2 } \\cdots s _ { i _ \\ell } , \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} Z _ k ^ { \\phi } \\equiv S _ { k } ^ { \\phi } & \\equiv \\sum \\limits _ { a \\in A _ { + , k } } \\frac { \\mu ( a ) C _ a ( X _ 1 ) \\dots C _ a ( X _ n ) } { a } \\pmod { X _ n ^ q } \\\\ & \\equiv \\sum \\limits _ { a \\in A _ { + , k } } \\frac { \\mu ( a ) C _ a ( X _ 1 ) \\dots C _ a ( X _ { n - 1 } ) a X _ n } { a } \\pmod { X _ n ^ q } \\\\ & \\equiv X _ n \\sum _ { a \\in A _ { + , k } } \\mu ( a ) C _ a ( X _ 1 ) \\dots C _ a ( X _ { n - 1 } ) \\pmod { X ^ q _ n } . \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} | \\nabla _ { x ' } \\chi ( x _ 0 ' ) | = 0 , \\Omega _ { R _ 0 } ( x _ 0 ) = \\{ x \\in B _ { R _ 0 } ( x _ 0 ) : x _ 1 > \\chi ( x ' ) \\} . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} \\norm { f ( \\cdot , w ) } _ { L ^ 2 ( \\R , \\omega _ { p ( w ) , \\infty } ) } ^ 2 = \\int _ 0 ^ { \\infty } \\abs { f ( t , w ) } ^ 2 \\frac { e ^ { - 4 \\pi p ( w ) t } } { 4 \\pi t } \\d t < \\infty . \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} x \\mapsto x ^ { 3 } , y \\mapsto x ^ 4 y \\Longrightarrow \\mathcal { I } _ { 2 , \\beta _ 1 , \\beta _ 2 } = 3 \\int _ { \\mathcal { D } _ 2 } { f } ^ * _ { t , s } ( x , y ) ^ s x ^ { 3 \\beta _ 1 + 4 \\beta _ 2 + 2 1 s } y ^ { \\beta _ 2 } \\frac { d x } { x } \\frac { d y } { y } . \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} \\vec { m } = ( m _ i ) _ { i \\in V ( Q ) } , \\ m _ i = \\dim V _ i \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} \\theta ^ \\ast = \\left \\{ \\theta : y \\left ( \\dot \\kappa ^ { A } ( \\theta ) + \\dot \\kappa ^ { - S } ( \\theta ) \\right ) = 1 \\right \\} . \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} u ^ 2 + ( \\beta ^ { q ^ 2 - 1 } + \\beta ^ { q - 1 } + 1 ) u + \\beta ^ { q - 1 } = 0 \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} U \\mid _ { \\mathcal { F } ( \\mathcal { H } _ A ) } = \\Gamma ( Q _ A ) . \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} \\ddot { \\rho } + \\eta \\dot { \\rho } + \\omega ^ 2 \\rho = \\frac { \\nu ^ 2 f ^ 2 } { m ^ 2 \\rho ^ 3 } . \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} \\begin{aligned} a ^ + + \\frac { 1 } { 4 } R ^ 2 & = 1 \\\\ a ^ - + b ^ - \\log R + \\frac { 1 } { 4 } R ^ 2 & = 1 , \\\\ a ^ - + b ^ - \\log R _ 1 + \\frac { 1 } { 4 } R _ 1 ^ 2 & = 1 , \\\\ R ^ 2 + b ^ - & = 0 . \\end{aligned} \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} \\sum _ { T \\in S ( G ) \\atop e \\in T } \\Pi ( T ) = ( \\ell _ { j j } - \\ell ^ e _ { j j } ) ( L _ { i i } ) _ { j j } = \\frac { L _ { i j , i j } } { R _ { e } } . \\tag * { $ \\Box $ } \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} \\psi ( - e ^ { - \\pi / 2 } ) = a 2 ^ { - 7 / 1 6 } e ^ { \\pi / 1 6 } ( \\sqrt { 2 } - 1 ) ^ { 1 / 4 } . \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} s _ 1 = d + 1 - k , s _ 2 = d + 2 - k . \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} ( \\sigma _ H ^ { \\frac 1 2 } , w _ h ) + ( \\nabla u _ H ^ { \\frac 1 2 } , \\nabla w _ H ) = 0 , ~ \\forall w _ H \\in L _ H . \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} \\ , \\mathrm M _ { t } ( \\mathbb R \\times { \\mathrm d } y ) \\ , = \\ , \\mathrm M _ { t } ( { \\mathrm d } y \\times \\mathbb R ) \\ , = : \\ , \\mathrm m _ { t } ( { \\mathrm d } y ) \\ , ; \\ , 0 \\le t \\le T \\ , , \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} [ ( p _ { k , l } ) _ { 1 \\leq k \\leq r , 1 \\leq l \\leq N _ k } ] \\mapsto \\left [ \\prod _ { l = 1 } ^ { N _ 1 } p _ { 1 , l } , \\dots , \\prod _ { l = 1 } ^ { N _ r } p _ { r , l } \\right ] . \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} r _ j ( x _ 3 , \\dots , x _ n ) = \\sum _ { k > 2 } a _ k ^ j x _ k + \\sum _ { \\ell > 2 } b _ \\ell ^ j x _ \\ell ^ 2 , \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 2 4 n + 2 2 ) q ^ n & \\equiv 3 6 \\dfrac { f _ 2 ^ 2 f _ { 6 } ^ 4 } { f _ 3 ^ 2 } \\equiv 3 6 f _ 2 ^ 2 f _ { 6 } ^ 3 ~ ( \\textup { m o d } ~ 8 ) , \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} \\begin{aligned} \\cos ( x ) + \\cos ( y ) = & 2 \\cos \\left ( \\frac { x + y } { 2 } \\right ) \\cos \\left ( \\frac { x - y } { 2 } \\right ) \\\\ \\Rightarrow & \\cos \\left ( \\pi u _ { 0 } + \\frac { 2 \\pi } { N } \\cdot j \\right ) + \\cos \\left ( \\pi u _ { 0 } - \\frac { 2 \\pi } { N } \\cdot j \\right ) = 2 \\cos \\left ( \\pi u _ { 0 } \\right ) \\cos \\left ( \\frac { 2 \\pi } { N } \\cdot j \\right ) \\end{aligned} \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} \\delta = \\sum _ { j = 0 } ^ { \\infty } \\mu \\left ( \\prod _ { k = 1 } ^ { j } B _ k \\right ) r _ { j + 1 } , \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} N _ \\Phi ( f ) = \\inf \\left \\{ k > 0 : \\int _ G \\Phi \\left ( \\frac { | f ( s ) | } { k } \\right ) \\ , d s \\le 1 \\right \\} . \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} a ( k ^ { 2 } - k ) = \\sum _ { r \\geq 2 } ( r ^ { 2 } - r ) t _ { r } , \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} \\Delta ( Y ) = \\sum _ { i } ( \\Sigma ^ 2 \\rho _ { i } ) \\bigl ( v _ { i } , \\Sigma X _ { i } \\bigr ) = \\sum _ { j } ( \\Sigma ^ 2 \\rho ' _ { j } ) \\bigl ( \\Sigma X ' _ { j } , v ' _ { j } \\bigr ) , \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\Delta _ { \\rho , \\sigma } = \\int _ { [ 0 , + \\infty ) } t \\ , d E _ { \\rho , \\sigma } ( t ) . \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\R ^ 3 } ( | \\nabla \\bar { u } | ^ 2 + \\bar { u } ^ 2 ) + \\frac { 1 } { 4 } \\int _ { \\R ^ 3 } \\rho \\phi _ { \\bar { u } } \\bar { u } ^ 2 - \\frac { \\mu } { p + 1 } \\int _ { \\R ^ 3 } \\bar { u } ^ { p + 1 } = \\bar { c } , \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} 2 \\ , \\mathsf { R e } ( P ^ * u , - i X _ 0 u ) \\geq c _ K \\sum _ { j = 0 } ^ N | \\ ! | X _ j u | \\ ! | _ 0 ^ 2 + \\frac 3 2 | \\ ! | X _ 0 u | \\ ! | _ 0 ^ 2 - C _ K | \\ ! | u | \\ ! | _ 0 ^ 2 . \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} U _ \\alpha ( \\phi ) = \\left ( \\frac { \\sin \\alpha \\phi } { \\alpha \\sin \\phi } \\right ) ^ { \\alpha / ( 1 - \\alpha ) } \\frac { \\sin ( 1 - \\alpha ) \\phi } { \\alpha \\sin \\phi } , \\phi \\in ( - \\pi , \\pi ) \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} [ f ; D _ { \\Lambda , \\epsilon } ( t ) f ] = \\sum _ { j k \\in E ( \\mathbb { T } _ N ) } \\beta ( f _ j - f _ k ) ^ 2 e ^ { t _ { j k } } + \\epsilon \\sum _ { j \\in \\Lambda } f _ j ^ 2 \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{gather*} T : C F ( H ^ { ( k ) } , J ) \\longrightarrow C F ( H ^ { ( k ) } , J ^ { + \\frac { 1 } { k } } ) \\\\ T ( z _ - ) = \\sum _ { z _ + \\in \\widetilde { P } ( H ^ { ( k ) } ) } \\sharp \\mathcal { N } ( z _ - , z _ + , [ 0 , 1 ] , \\rho , H ^ { ( k ) } , J ) \\cdot z _ + \\end{gather*}"}
-{"id": "1160.png", "formula": "\\begin{align*} \\limsup _ k \\max ( a _ k , b _ k ) = & \\sup ( \\limsup _ k a _ k , \\limsup _ k b _ k ) \\\\ \\liminf _ k \\max ( a _ k , b _ k ) \\geq & \\sup ( \\liminf _ k a _ k , \\liminf _ k b _ k ) \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} X _ H G _ X ^ { N _ 0 , N _ 1 } ( x , \\xi ) = X _ H ( m _ { X } ^ { N _ 0 , N _ 1 } ) ( x , \\xi ) \\ln ( 1 + f ( x , \\xi ) ) + m _ { X } ^ { N _ 0 , N _ 1 } ( x , \\xi ) \\frac { X _ H f ( x , \\xi ) } { 1 + f ( x , \\xi ) } . \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} T _ 1 ( q ) & = 2 \\sum _ { k , \\ell , \\ , \\mathrm { o r d } ( \\omega ) \\mid \\ell } { n \\brack k } _ q ^ 2 q ^ { f ( n , k , \\ell ) } q ^ 2 { n \\brack \\ell } ' _ q { k \\brack \\ell } ' _ q { k + \\ell \\brack n } _ q , \\\\ T _ 2 ( q ) & = 2 \\sum _ { k , \\ell , \\ , \\mathrm { o r d } ( \\omega ) \\nmid \\ell } { n \\brack k } _ q ^ 2 q ^ { f ( n , k , \\ell ) } q ^ 2 { n \\brack \\ell } ' _ q { k \\brack \\ell } ' _ q { k + \\ell \\brack n } _ q . \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} t \\ge \\frac { n - n ' ( \\beta _ 0 ) } { \\sum _ { i = 1 } ^ k m _ i ( \\omega \\cdot \\beta _ i ) } . \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { \\sqrt { t - x } } { \\sqrt { t } } & = \\lim _ { t \\rightarrow \\infty } \\sqrt { \\frac { t - x } { t } } = \\lim _ { t \\rightarrow \\infty } \\sqrt { 1 - \\frac { x } { t } } = \\sqrt { \\lim _ { t \\rightarrow \\infty } \\left ( 1 - \\frac { x } { t } \\right ) } = 1 . \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{gather*} n ! R ^ n ( 1 ) = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { n - k } \\lambda ^ { n - k } s ( n , k ) ( R ( 1 ) ) ^ k , \\\\ ( R ( 1 ) ) ^ n = \\sum \\limits _ { k = 1 } ^ n k ! \\lambda ^ { n - k } S ( n , k ) R ^ k ( 1 ) , \\end{gather*}"}
-{"id": "3059.png", "formula": "\\begin{align*} - \\hat { \\Delta } Y \\varpi ^ i + \\varpi ^ l Z = ( \\varpi ^ { i + j + 1 } \\beta ' - \\varpi ^ { r - i } e ^ 2 ) / d . \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} \\eta _ { { \\bf i } , \\mathcal I } ^ { \\ast } \\mathcal { L } _ { \\mathcal I , \\lambda _ 1 , \\dots , \\lambda _ r } & = \\bigg \\{ ( \\left [ ( p _ { k , l } ) _ { 1 \\leq k \\leq r , 1 \\leq l \\leq N _ k } ] , [ p _ 1 , \\dots , p _ r , w ] \\right ) ~ \\bigg | ~ \\\\ & \\left [ \\prod _ { l = 1 } ^ { N _ 1 } p _ { 1 , l } , \\dots , \\prod _ { l = 1 } ^ { N _ { r } } p _ { r , l } \\right ] = [ p _ 1 , \\dots , p _ r ] Z _ { \\mathcal I } \\bigg \\} . \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} G _ { \\tau } ( \\chi ) & = \\sum _ { ( x _ 1 , \\ldots , x _ r ) \\in X _ { \\tau } } \\prod _ { i = 1 } ^ r \\chi ( 1 - x _ i x ) \\\\ & = ( \\sum _ { a \\in { { \\bf { F } } _ q } } \\chi ( 1 + a x ) ) ^ { c _ 1 } ( \\sum _ { a \\in { { \\bf { F } } _ q } } \\chi ^ 2 ( 1 + a x ) ) ^ { c _ 2 } \\cdots ( \\sum _ { a \\in { { \\bf { F } } _ q } } \\chi ^ { r } ( 1 + a x ) ) ^ { c _ { r } } \\\\ & = \\prod _ { i = 1 } ^ { r } ( \\sum _ { a \\in { { \\bf { F } } _ q } } \\chi ^ i ( 1 + a x ) ) ^ { c _ i } . \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} f _ { l } ( 0 ) = f _ { r } ( 0 ) = 0 , f _ { l } ( 1 ) = f _ { r } ( 1 ) . \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} f ' ( g _ 0 , g _ 1 , g _ 2 ) ( \\tau ) : = \\sum _ { i , j , l } \\langle f ( h _ i , h _ j , h _ l ) , \\tau | _ { ( g _ 0 g _ 1 ) X _ l \\cap g _ 0 X _ j \\cap X _ i } \\pi \\rangle . \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} ` t = \\max _ { u _ V \\in ` R _ + ^ V : u _ i + u _ j \\leq 1 , \\forall i , j \\in V } u ( V ) = \\frac { \\abs { V } } { 2 } . \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} L ^ t _ \\theta : = L ^ t _ { \\theta ' } + L ^ t _ { \\theta '' } . \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} m _ { v , u } = - m _ { u , v } , \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} E _ { \\bullet } ^ { \\star } = ( E _ 0 , E _ 1 , \\ldots , E _ k ) . \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} - \\log p _ { _ 1 } ( y ) - 1 + \\eta _ 1 ^ a + \\eta _ 2 ^ a y ^ 2 = 0 , \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} \\begin{array} { c } p _ t = \\left ( \\frac { p _ x } { ( 1 + p ^ 2 + q ^ 2 ) ^ { 3 / 2 } } \\right ) _ { x x } \\\\ \\ \\\\ q _ t = \\left ( \\frac { q _ x } { ( 1 + p ^ 2 + q ^ 2 ) ^ { 3 / 2 } } \\right ) _ { x x } \\end{array} \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} & f | _ { ] 0 , \\infty [ } \\\\ & \\lim _ { t \\to 0 + } f ( t ) = + \\infty . \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} m _ { C ( \\mathbf i , \\mathbf a ) } = \\rho ( \\alpha ) | d \\alpha | , \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} K _ { \\rho _ 0 } = 2 \\Vert { D F ( x ^ * ) } ^ { - 1 } \\Vert _ U . \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} \\beta _ { 1 } ( y , t ) = - \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } p _ { 0 } ( y , t ) \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} \\varepsilon _ { n } \\stackrel { \\mathrm { d e f } } { = } \\sup _ { H \\in \\mathcal { H } _ { 0 } } \\ ; \\sup _ { x \\in E _ { n } } \\left | h _ { n } ^ { - 1 } \\int _ { \\mathbf { R } } f ( z ) \\ , H \\left ( \\frac { x - z } { h _ { n } } \\right ) \\ , d z - f ( x ) \\int _ { \\mathbf { R } } H ( z ) \\ , d z \\right | \\rightarrow 0 , \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} q ( w ) = q ( ( - j + n ) y + z _ j - c ) \\ge q ( z _ j - c ) - q ( ( j - n ) y ) = q ( z _ j - c ) . \\end{align*}"}
-{"id": "9938.png", "formula": "\\begin{align*} \\tau _ Y = ( \\tau ^ { n - 1 , - } - \\tau ^ { n - 1 , + } ) | _ Y \\\\ = \\left ( ( \\tau _ 1 ^ { n - 1 , - } - \\tau _ 1 ^ { n - 1 , + } ) | _ Y , \\ , ( \\tau _ { 0 1 } ^ { n - 2 , - } - \\tau _ { 0 1 } ^ { n - 2 , + } ) | _ Y \\right ) . \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} u _ { 0 , n } ( x ) : = \\mu _ n \\lambda _ n ^ { \\frac { d } { 2 } } Q ( \\lambda _ n x ) . \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ { n _ 1 } \\nu ( k ) ( n _ 1 n _ 2 / k ^ 2 ) - q _ { n _ 1 , n _ 2 } \\right | = & \\left | \\sum _ { k = 1 } ^ { n _ 1 } \\nu ( k ) \\left ( n _ 1 n _ 2 / k ^ 2 - \\lfloor n _ 1 / k \\rfloor \\lfloor n _ 2 / k \\rfloor \\right ) \\right | \\\\ \\le & ( n _ 1 + n _ 2 ) \\sum _ { k = 1 } ^ { n _ 1 } ( 1 / k ) = o ( n _ 1 n _ 2 ) \\ , . \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} & M _ G \\left ( ( x ^ 2 + 1 ) + m \\ : h ( x ) + y m \\ : h ( x ) \\right ) = 2 ^ 4 ( 6 m + 1 ) , \\\\ & M _ G \\left ( ( x ^ 4 + x ^ 2 + 1 ) + m \\ : h ( x ) + y \\left ( 1 + m \\ : h ( x ) \\right ) \\right ) = 2 ^ 6 ( 3 m + 1 ) . \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} z ( t ) = \\displaystyle \\int _ { D } y ( x , t ) f ( x ) d x . \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\begin{array} { l l } \\min & L ( v , \\mu ^ k ) \\\\ s . a . & C ' _ { z ^ { k } } ( v - z ^ { k } ) = 0 , \\\\ & | | v - z ^ k | | _ { \\infty } \\leq \\delta _ { k , i } . \\end{array} \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} S : V ^ { L } _ { \\mathbb { H } } \\longrightarrow V ^ { L } _ { \\mathbb { H } } , ~ S \\psi = T T ^ { \\dagger } \\psi = \\displaystyle \\sum _ { k \\in I } \\left \\langle \\psi | \\phi _ { k } \\right \\rangle \\phi _ { k } . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} & \\frac { \\kappa p + 1 } { 2 ( \\kappa + 1 ) ^ p } \\| u u _ h ^ { \\kappa } \\| _ { 1 , p } ^ p \\\\ & \\leq \\left [ \\frac { \\kappa p + 1 } { ( \\kappa + 1 ) ^ p } + M _ 9 ( \\kappa p + 1 ) L ( \\kappa , u ) \\right ] \\| u u _ h ^ { \\kappa } \\| _ { p } ^ p \\\\ [ 1 e x ] & + M _ { 1 0 } G ( \\kappa , u ) \\| u u _ h ^ { \\kappa } \\| _ { p , \\partial \\Omega } ^ p + M _ { 1 1 } \\kappa . \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} \\mathcal { L } \\phi ( s , x ) = \\Gamma x \\frac { \\partial \\phi } { \\partial x } ( s , x ) & + \\frac { 1 } { 2 } \\sigma ^ 2 x ^ 2 \\frac { \\partial ^ 2 \\phi } { \\partial x ^ 2 } ( s , x ) \\\\ & \\begin{aligned} + \\int _ \\mathbb { R } \\Big \\{ \\phi ( s , x ( 1 + \\gamma ( z ) ) - \\phi ( s , x ) - x \\gamma ( z ) \\frac { \\partial \\phi } { \\partial x } \\Big \\} \\nu ( d z ) , & \\\\ \\forall ( s , x ) \\in \\mathbb { R } _ { > 0 } \\times \\mathbb { R } . & \\end{aligned} \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} C _ 3 : = & | R ^ N | _ { L ^ \\infty } + \\frac { 1 + r } { 2 } ( 3 | A | ^ 2 _ { L ^ \\infty } + \\frac { 3 } { 2 } | B | ^ 2 _ { L ^ \\infty } ) , C _ 4 : = \\frac { 1 + r } { 2 } ( 3 | C | ^ 2 _ { L ^ \\infty } + \\frac { 3 } { 2 } | B | ^ 2 _ { L ^ \\infty } ) . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} E ^ * ( z , \\tau ) = \\hat \\zeta ( - z ) y ^ { ( 1 + z ) / 2 } + \\hat \\zeta ( z ) y ^ { ( 1 - z ) / 2 } + \\sum _ { n \\in \\Z - \\{ 0 \\} } | n | ^ { z / 2 } \\sigma _ { - z } ( | n | ) \\ , 2 y ^ { 1 / 2 } K _ { z / 2 } ( 2 \\pi | n | y ) e ^ { 2 \\pi i n x } \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} E ( \\varphi ) = 0 . \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\int d x \\ x b ( x , y , t ) = \\frac { \\partial } { \\partial y } ( y \\alpha ( y , t ) ) + \\frac { 1 } { 2 } \\langle x ^ { 2 } \\rangle \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) - \\frac { \\partial } { \\partial t } \\alpha ( y , t ) \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} \\lvert ( i _ 1 , j _ 1 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i _ 2 , j _ 2 ) \\rvert _ { W _ n ( w ) } = j _ 1 - j _ 2 . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} \\int _ M p _ t ( x , y ) d \\mu ( y ) = 1 \\quad \\hbox { f o r $ x \\in M $ a n d $ t > 0 $ . } \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} \\tilde { \\mathfrak { s } } & : = \\mathfrak { s } | _ { \\tilde { U } } : \\tilde { U } \\rightarrow \\tilde { U } \\lhd \\mathbb { F } , \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} 1 + ( c _ 1 ( k a - 1 ) - k ) a + ( c _ 2 a + 1 ) ( k a - 1 ) = ( c _ 1 + c _ 2 ) a ( k a - 1 ) \\ , . \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} \\Psi _ { j } ( x , t , y , s ) & = u ^ { \\epsilon , \\theta } ( x , t ) - u _ { \\theta } ( y , s ) - \\frac { ( x - y ) ^ { 2 } } { 2 \\eta } - \\frac { ( t - s ) ^ { 2 } } { 2 \\eta } - b t \\\\ & - \\sqrt { \\theta } ( T + 1 - t ) ^ { - 1 } + g _ { \\eta , \\nu } ( - \\nu y ) - \\alpha x ^ { 2 } \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} b ( x , y , t ) = h ( x ) \\beta _ { 0 } ( y , t ) + x h ( x ) \\beta _ { 1 } ( y , t ) \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} x _ 1 + x _ 2 a _ 1 = 2 x _ 2 . \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} \\begin{cases} ^ { C } _ { 0 } D _ { t } ^ { q } \\varphi ( x , t ) = A \\varphi ( x , t ) & \\hbox { i n } Q _ { T } \\\\ \\varphi ( \\xi , t ) = 0 & \\hbox { o n } \\Sigma _ { T } \\\\ \\varphi ( x , 0 ) = \\varphi _ { _ { 0 } } ( x ) & \\hbox { i n } \\Omega . \\end{cases} \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} \\mathbf { v } _ f = ( v _ { f ( 1 ) } \\wedge \\cdots \\wedge v _ { f ( n + 1 ) } ) \\otimes ( v _ { f ( n + 2 ) } \\wedge \\cdots \\wedge v _ { f ( n + d ) } ) \\otimes \\cdots \\otimes ( v _ { f ( n + 2 + ( n - 1 ) ( d - 1 ) ) } \\wedge \\cdots \\wedge v _ { f ( n d + 1 ) } ) \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} \\mathcal { L } _ V = \\Delta _ { g } + \\frac { V _ 0 ( y ) } { r ^ 2 } . \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} \\pi _ F ( \\rho _ \\infty ( a ) ) = \\rho ^ F _ \\infty ( \\gamma _ F ( a ) ) , \\pi _ F ( t _ \\infty ( x ) ) = t ^ F _ \\infty ( \\delta _ F ( x ) ) \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} \\psi ( \\overline { G ( X , Y ) d \\pi } ) = \\psi \\left ( \\overline { ( \\tilde { H } ( X + Y ) - \\tilde { H } ( X ) - \\tilde { H } ( Y ) ) d \\pi } \\right ) = f _ 0 \\left ( \\psi \\left ( \\overline { \\tilde { H } ( X ) d \\pi } \\right ) \\right ) , \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} \\limsup _ { \\lambda \\to \\infty } A _ { 1 4 } ( \\lambda , \\varepsilon ) = 0 . \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} v : = x _ 2 , \\ , y : = x _ 1 - _ o x _ 2 , \\ , w : = y _ 2 , \\ , z : = y _ 1 - _ o y _ 2 . \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} \\ell ^ p ( V ) = \\{ f : V \\to \\mathbb { C } \\ | \\ \\sum _ { v \\in V } | f ( v ) | ^ p < \\infty \\} , \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{gather*} y _ { t - 1 } ^ 2 = R ( y _ t ) R ( y _ t ) = 2 R ( y _ { t - 1 } y _ t ) - R \\big ( y _ t ^ 2 \\big ) , \\\\ y _ { t - 1 } R ( 1 ) = R ( y _ t ) R ( 1 ) = R ( y _ { t - 1 } + y _ t R ( 1 ) ) - y _ { t - 1 } , \\end{gather*}"}
-{"id": "9944.png", "formula": "\\begin{align*} H _ j = \\{ \\ , y \\in \\R ^ n _ y \\mid \\ , - c \\langle y ' , \\tilde { \\eta } _ j \\rangle < y _ 1 < c \\langle y ' , \\tilde { \\eta } _ j \\rangle \\ , \\} . \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} P _ { \\beta } ( X ) = \\sum _ { n \\in \\mathbb { Z } } P _ { n , \\beta } q ^ n \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} a _ { \\sigma } , b _ { \\sigma } , c _ { \\sigma } , d _ { \\sigma } \\in k \\ \\mbox { s u c h t h a t } \\ a _ { \\sigma } d _ { \\sigma } - b _ { \\sigma } c _ { \\sigma } = a _ { \\hat { \\sigma } } d _ { \\hat { \\sigma } } - b _ { \\hat { \\sigma } } c _ { \\hat { \\sigma } } \\in k ^ { \\times } \\ \\mbox { f o r e v e r y } \\ \\sigma , \\hat { \\sigma } \\in I _ F \\} . \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} F ( X , Y ) = G ( X + Y ) - G ( X ) - G ( Y ) + \\sum _ { i = 0 } ^ N r _ i S ( X ^ { p ^ i } , Y ^ { p ^ i } ) , \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} \\mathbb { G } ( x , \\ , y \\ , | \\ , z ) : = \\mathbb { G } _ \\lambda ( x , \\ , y \\ , | \\ , z ) = \\sum _ { n = 0 } ^ \\infty p ^ { ( n ) } ( x , \\ , y ) z ^ n , ~ x , ~ y \\in V ( G ) , ~ z \\in \\mathbb { C } , \\ | z | < R _ { \\mathbb { G } } \\ , , \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 3 / 4 } ^ 0 \\bigg [ \\frac { b _ { 1 / 2 , 0 } ( d , s ) } { d ^ { 1 / 2 } \\Gamma ( s + 1 / 4 ) } \\bigg ] & = \\mathrm { L C } _ { s = 3 / 4 } ^ 0 \\bigg [ \\frac { 2 ^ { 3 / 2 - 2 s } \\pi ^ { s + 1 / 4 } | D | ^ { - s + 1 / 4 } } { d ^ { 1 / 2 } \\Gamma ( s + 1 / 4 ) } \\frac { \\widetilde { \\mathrm { T r } } _ { d , D } ( G _ 0 ( z , 2 s - 1 / 2 ) ) } { L _ D ( 2 s - 1 / 2 ) } \\bigg ] . \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} \\frac { \\partial \\widehat { w } ^ + ( \\vec { u } ) } { \\partial u _ k } = \\sum _ { i , j } x _ i y _ j \\frac { \\partial w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) } { \\partial u _ k } + \\sum _ { i , i ' , j , j ' } x _ i x _ { i ' } y _ j y _ { j ' } \\frac { \\partial w _ { i i ' j j ' } ^ { ( 2 ) } ( \\vec { u } ) } { \\partial u _ k } + \\cdots = 0 \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\frac 1 { L ^ d } \\mathcal F _ { L , \\alpha } ^ V \\bigl ( \\mu ^ L _ 0 \\bigr ) = \\mathcal F _ \\alpha ^ V ( \\rho _ 0 ) . \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} \\begin{aligned} & A x = b , & x \\in \\Upsilon , \\\\ & A ^ T y + s = c , & s \\in \\Upsilon , \\\\ & R ( x ) s = \\mu e . \\end{aligned} \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\mathrm { E x t } ^ 1 _ A ( T , T ) = 0 . \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} \\lim _ { | p | \\to \\infty } H _ { i } ( t , x , p ) = \\infty \\ , \\ , \\ , \\ , ( x , t ) . \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} & \\frac { \\kappa p + 1 } { 2 ( \\kappa + 1 ) ^ p } \\| u u _ h ^ { \\kappa } \\| _ { 1 , p } ^ p \\\\ & \\leq \\left [ \\frac { \\kappa p + 1 } { ( \\kappa + 1 ) ^ p } + M _ 1 ( \\kappa p + 1 ) L ( \\kappa , u ) \\right ] \\| u u _ h ^ { \\kappa } \\| _ { p } ^ p + M _ 2 G ( \\kappa , u ) \\| u u _ h ^ { \\kappa } \\| _ { p , \\partial \\Omega } ^ p + M _ 3 \\kappa , \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} \\gamma ^ { ( d ) } _ { n } ( A ) = \\Gamma _ 1 ' \\oplus \\Gamma ' _ 2 \\oplus \\ldots \\oplus \\Gamma ' _ r \\oplus C _ 0 I _ { n - 2 r } . \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} & q _ { g , n , \\ell } ( \\mathbf p ) \\triangleq \\Pr \\left [ C _ { g , n , \\ell } \\geq R _ { g , \\ell } \\right ] = \\Pr \\left [ \\frac { 1 } { K _ { g , n , \\ell } } \\log _ 2 ( 1 + { \\rm S I R } _ { g , n , \\ell } ) \\geq \\theta _ g \\right ] , g \\in \\{ { \\rm S V C } , { \\rm D A S H } \\} , \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} u ( t ) = Q [ z ( t ) ] + v ( t ) , \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} \\rho _ \\lambda = \\rho ( \\lambda ) = \\frac { 1 } { R _ \\mathbb { G } } = \\limsup _ { n \\to \\infty } p ^ { ( n ) } ( x , x ) ^ { 1 / n } = \\limsup _ { n \\to \\infty } p ^ { ( n ) } ( o , o ) ^ { 1 / n } \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} U _ { i , j } = \\begin{cases} \\left \\{ g + 3 i + 5 , \\frac { 1 } { 2 } ( 2 g - 3 j + 6 i + 7 ) , \\frac { 1 } { 2 } ( 2 g + 3 j - 1 2 i - 2 3 ) \\right \\} _ { g - 1 } & \\mbox { i f $ j \\equiv 1 \\pmod { 2 } $ , } \\\\ \\left \\{ g + 3 i + 5 , \\frac { 1 } { 2 } ( 2 g - 3 j + 6 i + 1 0 ) \\frac { 1 } { 2 } ( 2 g + 3 j - 1 2 i - 2 0 ) \\right \\} _ g & \\mbox { i f $ j \\equiv 0 \\pmod { 2 } $ . } \\\\ \\end{cases} \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} u ( t , x ) = \\int _ { \\mathbb { R } } G ^ { \\varepsilon } ( t , x - y ) \\ , u _ 0 ( y ) \\ , d y - \\int _ 0 ^ t \\int _ { \\mathbb { R } } G ^ { \\varepsilon } ( t - s , x - y ) f \\left ( s , y , u ( s , y ) \\right ) _ { y } \\ , d y \\ , d s \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} \\iota ( \\alpha ) x = \\theta ( \\alpha ) ^ { - k ' } x \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u + \\lambda ^ 2 e ^ { \\ , u } & = \\rho _ \\lambda c _ \\Omega , c _ \\Omega = \\frac { 1 } { | \\Omega | } , \\\\ \\partial _ n u & = 0 , \\mbox { o n } \\partial \\Omega . \\end{aligned} \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} \\beta ( h , d ) = \\frac { h d l } { h + d - k } , \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial { { z } ^ { i } } } = \\frac { 1 } { 2 } \\left ( \\frac { \\partial } { \\partial { { x } ^ { i } } } - \\sqrt { - 1 } \\frac { \\partial } { \\partial { { y } ^ { i } } } \\right ) \\\\ \\frac { \\partial } { \\partial { { \\overline { z } } ^ { i } } } = \\frac { 1 } { 2 } \\left ( \\frac { \\partial } { \\partial { { x } ^ { i } } } + \\sqrt { - 1 } \\frac { \\partial } { \\partial { { y } ^ { i } } } \\right ) \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} J ^ { ( \\alpha , \\beta ) } p ^ { ( \\alpha , \\beta ) } _ n ( x ) = x p _ n ^ { ( \\alpha , \\beta ) } ( x ) , x \\in \\ [ - 1 , 1 ] , \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} h = \\sum _ { j = - 1 } ^ \\infty \\sum _ { m \\in \\mathbb Z } \\bar \\mu _ { j , m } h _ { j , m } . \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} F ( \\gamma , \\zeta ) = \\int _ { \\R } f ( t , \\zeta ) \\frac { \\gamma ^ { i 2 \\pi t } } { \\gamma } \\d t , \\gamma \\in V _ { \\zeta } \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} \\mathbf { W } = \\mathbf { I } _ n - \\frac { \\xi \\rho } { 2 } \\mathbf { V } \\\\ \\tilde { \\mathbf { W } } _ n = \\mathbf { I } - \\frac { \\xi \\rho } { 2 } ( 1 - \\eta ) \\mathbf { V } . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { K } \\tilde { \\xi } _ { i } = \\left ( \\sum _ { i = 1 } ^ { K } \\xi _ { i } \\right ) + \\left ( \\sum _ { j = 1 } ^ { K } \\xi _ { j } - B \\right ) ^ { - } \\geq B . \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} D _ t < D _ t g _ t . h > = < D _ t D _ t g _ t . h > + < D _ t g _ t , D _ t h > + < d D _ t g _ t , d h > . \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} f _ { \\varepsilon , \\varepsilon ' } ( z ) = \\begin{cases} \\Re \\langle z , \\varepsilon ' \\xi _ 0 \\rangle \\ , \\ , & \\ , \\ , z \\in S _ { \\varepsilon } , \\\\ + \\infty \\ , \\ , & . \\end{cases} \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} \\delta S ^ 2 & = E \\int _ 0 ^ T \\int \\varphi ( t , g _ t ( x ) ) ( \\det \\nabla g _ t ( x ) - 1 ) d t d x \\\\ & + E \\int _ 0 ^ T \\int ( \\nabla p ( t , g _ t ( x ) ) . h ( t , g _ t ( x ) ) ( \\det \\nabla g _ t ( x ) - 1 ) d t d x \\\\ & + E \\int _ 0 ^ T \\int p ( t , g _ t ( x ) ) \\left . \\frac { d } { d \\varepsilon } \\right | _ { \\varepsilon = 0 } \\det \\nabla ( g _ t ( x ) + \\epsilon h ( t , g _ t ( x ) ) d t d x \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} \\gamma ^ i ( t , W _ t ) + h ^ i ( t , W _ t ) = E \\left [ \\Phi ( W _ T ) \\vert \\mathcal F _ t \\right ] . \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} h ( C \\omega ( X ) ( Y ) , Z ) = \\frac { 1 } { 2 } \\left ( h ( ( \\nabla _ X ^ h C ) Y , Z ) + h ( ( \\nabla _ Y ^ h C ) X , Z ) - h ( ( \\nabla ^ h _ Z C ) X , Y \\right ) . \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} \\rho _ { F _ { p ^ r q ^ s } } = \\frac { 1 } { \\limsup _ { n \\to \\infty } | p _ n | ^ { r / n } | q _ n | ^ { s / n } } \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} ( t , r ) \\cdot ( z ' , z _ n ) = ( r t z ' , r ^ 2 z _ n ) \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} I ( t _ 1 u _ 1 + t _ 2 u _ 2 + t _ 3 u _ 3 ) & = I ( t _ 1 u _ 1 + t _ 2 u _ 2 + t _ 3 u _ 3 ) - \\sum _ { i = 1 } ^ 3 \\frac { t _ i ^ 4 } { 4 } I ' ( u ) [ u _ i ] \\\\ & = \\sum _ { i = 1 } ^ 3 \\left ( I ( t _ i u _ i ) - \\frac { t _ i ^ 4 } { 4 } I ' ( u _ i ) [ u _ i ] \\right ) - \\frac { \\lambda } { 4 } \\sum _ { i < j } ( t _ i ^ 2 - t _ j ^ 2 ) ^ 2 \\int _ { \\mathbb R ^ 3 } \\phi _ i u _ j ^ 2 d x . \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} U = ( i , j ) \\to ( i + 1 , j + 1 ) , H = ( i , j ) \\to ( i + 1 , j ) , D = ( i , j ) \\to ( i + 1 , j - 1 ) , \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} \\mathrm { O u t } ( F _ n ) \\curvearrowright H ^ k ( F _ n , \\mathbb { R } ) = \\begin{cases} \\mathbb { R } , & k = 0 \\\\ \\mathbb { R } ^ n , & k = 1 \\\\ 0 , & k > 1 \\\\ \\end{cases} . \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} U _ 2 ( \\omega ) = 2 n ^ 2 \\sum _ { \\ell ' = 0 } ^ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } \\sum _ { j = 0 } ^ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } - 1 } \\binom { \\frac { n } { \\mathrm { o r d } ( \\omega ) } - 1 } { j } ^ 2 \\binom { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } { \\ell ' } \\binom { j } { \\ell ' } \\binom { j + \\ell ' } { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } \\sum _ { i = 1 } ^ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } - 1 } \\frac { \\omega ^ i } { ( \\omega ^ i - 1 ) ^ 2 } . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} B _ { i j , l ' } = ( a \\nabla \\xi _ i ) _ l \\frac { \\partial ( a \\nabla \\xi _ j ) _ { l ' } } { \\partial x _ l } - ( a \\nabla \\xi _ j ) _ l \\frac { \\partial ( a \\nabla \\xi _ i ) _ { l ' } } { \\partial x _ l } \\ , , 1 \\le l ' \\le n \\ , . \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} \\Delta _ k P _ { k , m } ( z , s ) = \\bigl ( s - \\frac { k } { 2 } \\bigr ) \\bigl ( 1 - \\frac { k } { 2 } - s \\bigr ) P _ { k , m } ( z , s ) . \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\sup \\limits _ { \\textbf { t } \\in \\mathcal { C } } \\inf \\limits _ { \\textbf { y } \\in \\widehat { \\mathcal { C } } _ \\ell } \\| \\textbf { t } - \\textbf { y } \\| \\leq \\max \\limits _ { i = 1 } ^ { k } \\sup \\limits _ { \\textbf { t } \\in B ( \\textbf { t } _ i , \\frac { \\epsilon } { 2 } ) } \\inf \\limits _ { \\textbf { y } \\in \\widehat { \\mathcal { C } } _ \\ell } \\| \\textbf { t } - \\textbf { y } \\| . \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} L _ { \\Gamma } ( s , \\varrho ) & = \\prod _ { \\overline { \\gamma } \\in \\overline { \\Gamma } _ { p } } \\prod _ { k = 0 } ^ { \\infty } \\det \\left ( 1 - \\varrho ( \\gamma ) e ^ { - ( s + k ) \\ell ( \\gamma ) } \\right ) \\\\ & = \\prod _ { \\overline { \\gamma } \\in \\overline { \\Gamma } _ { p } } \\prod _ { k = 0 } ^ { \\infty } \\det \\left ( 1 - \\varrho ( \\gamma ) N _ G ( \\gamma ) ^ { - ( s + k ) } \\right ) \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} K _ { i \\bar j } = \\frac 1 2 \\left ( - \\mu _ { \\bar k i } ^ r \\mu _ { k \\bar j } ^ { \\bar r } + \\mu _ { k \\bar r } ^ { \\bar i } \\mu _ { \\bar k r } ^ j - \\mu _ { i \\bar r } ^ { \\bar k } \\mu _ { \\bar j r } ^ { k } + \\mu _ { k \\bar k } ^ { \\bar i } \\mu _ { \\bar r r } ^ j - \\mu _ { k \\bar k } ^ { \\bar r } \\mu _ { \\bar j i } ^ { r } - \\mu _ { \\bar k k } ^ { r } \\mu _ { i \\bar j } ^ { \\bar r } \\right ) \\ , . \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} \\mu = \\frac { 1 } { 3 } \\left ( \\mu _ L + \\mu _ C + \\mu _ R \\right ) , \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\tilde { N } _ { \\mu , \\xi } ( \\tilde { \\phi } ) = ( u _ { \\mu , \\xi } + \\tilde { \\phi } ) ^ p - u ^ p _ { \\mu , \\xi } - p u _ { \\mu , \\xi } ^ { p - 1 } \\tilde { \\phi } . \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} R ( \\underline { X } ) = \\begin{bmatrix} [ c c c c c ] X _ { n + i } & X _ { n + j } & & & \\\\ - X _ i & - X _ j & & & \\\\ & & \\ddots & & \\\\ & & & X _ { n + i } & X _ { n + j } \\\\ & & & - X _ { i } & - X _ { j } \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} \\dim _ { \\mathbb R } \\Delta ( Z _ { \\mathbf i } , \\eta _ { { \\bf i } , \\mathcal I } ^ { \\ast } \\mathcal L , v ) = \\dim _ { \\mathbb R } \\Delta ( Z _ { \\mathcal I } , \\mathcal L , v ) = \\dim _ { \\mathbb C } Z _ { \\mathcal I } = \\dim _ { \\mathbb C } Z _ { \\mathbf i } \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{gather*} 0 \\leq \\beta < \\frac { n } { 2 } \\ \\ \\mbox { a n d } \\ \\ i = \\frac { n } { 2 } + 1 \\ \\ \\mbox { a n d } \\ \\ j = \\frac { n } { 2 } + 1 + \\beta . \\end{gather*}"}
-{"id": "9093.png", "formula": "\\begin{align*} \\lambda _ k = \\begin{cases} \\pm 2 ^ { - j _ \\ell \\frac { d } { u } } & Q _ { 0 , k } \\subset Q _ { - j _ \\ell , m _ \\ell } , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{gather*} \\mathcal { D } _ { h ( u ) } v = [ h ( u ) , v ] , \\mathcal { E } _ { h ( u ) } \\mathrm { h o r } ( a ) = \\mathrm { h o r } ( \\nabla _ u a ) , \\end{gather*}"}
-{"id": "136.png", "formula": "\\begin{align*} J _ { \\nu } ( r ) = \\frac { ( r / 2 ) ^ { \\nu } } { \\Gamma \\left ( \\nu + \\frac 1 2 \\right ) \\Gamma ( 1 / 2 ) } \\int _ { - 1 } ^ { 1 } e ^ { i s r } ( 1 - s ^ 2 ) ^ { ( 2 \\nu - 1 ) / 2 } \\mathrm { d } s . \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} & \\sum _ { x _ { 1 } \\in H } \\cdots \\sum _ { x _ { n + 1 } \\in H , \\ [ x _ { 1 } , \\ldots , x _ { n + 1 } ] \\in H \\cap Z ^ { \\otimes } ( G ) } \\vert C ^ { \\otimes } _ { G } ( [ x _ { 1 } , \\ldots , x _ { n + 1 } ] ) \\vert = \\\\ & \\vert H \\vert ^ { n + 1 } d _ { n } ^ { \\otimes } ( \\overline { H } ) \\vert G \\vert . \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} | Y _ { t _ { i } ^ { m } } ^ { i , m } - W \\left ( t _ { i } ^ { m } , X _ { t _ { i } ^ { m } } ^ { i - 1 , m } \\right ) | \\leq \\frac { 1 } { m ^ { 2 } ( 2 C + 1 ) ^ { m } } i = 0 , \\ldots , m - 1 , \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} w = \\beta _ 1 \\beta _ 2 \\dots \\beta _ { p _ w } \\alpha _ w s _ w , \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} \\Delta v ^ { k } = \\lambda _ k v ^ { k } , \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} a _ \\alpha ( z ) = \\sum _ { \\beta \\le \\alpha } \\frac { ( - 1 ) ^ { | \\alpha - \\beta | } z ^ { \\alpha - \\beta } } { ( \\alpha - \\beta ) ! \\beta ! } F z ^ \\beta , \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} \\lambda ^ 2 \\chi = \\chi '' - V '' ( u _ { \\Omega } ) \\chi \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} \\gamma ( x , E ) = k ( E ) x + \\eta ( x , E ) , \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} \\begin{cases} W _ { t } ^ { \\varepsilon } + g ( t , x , W _ x ^ { \\varepsilon } ) ~ = ~ \\varepsilon W ^ { \\varepsilon } _ { x x } \\ , , \\\\ W ^ { \\varepsilon } ( 0 , x ) ~ = ~ 0 . \\end{cases} \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} T - \\sum _ { i = 1 } ^ N ( b _ i - a _ i ) ~ < ~ \\delta . \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} Z ( u ) = - ( X _ { \\perp } u ) _ 0 i v = ( i ( \\eta _ + u _ { - 1 } - \\eta _ - u _ 1 ) ) i v , \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } F _ { i } ( x , u , u _ { x _ { i } } ) = 0 & \\ , \\ , I _ { i } \\\\ \\sum _ { i = 1 } ^ { K } u _ { x _ { i } } = B & \\ , \\ , \\{ 0 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} R _ \\lambda ( h h _ 1 , k k _ 1 ) = \\overline { \\chi _ \\lambda ( h _ 1 ) } R _ \\lambda ( h , k ) \\chi _ \\lambda ( k _ 1 ) . \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} \\frac { a ( \\beta ) } { a ( \\alpha ) } = 1 + \\frac { \\cos 2 v - \\cos 2 w } { 1 - \\cos 2 v } \\cdot \\frac { ( k - 2 ) ^ 2 } { ( k - 2 ) ^ 2 + 2 ( k - 1 ) ( 1 - \\cos 2 w ) } \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} E _ { N } ^ { \\rm Q } = E _ { a _ { N } } ^ { \\rm H } \\big ( 1 + o ( 1 ) _ { N \\to \\infty } \\big ) = ( a _ { * } - a _ { N } ) ^ { \\frac { q } { q + 1 } } \\Big ( \\frac { q + 1 } { q } \\cdot \\frac { \\Lambda } { a _ { * } } + o ( 1 ) _ { N \\to \\infty } \\Big ) \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} : \\ a = - N , : \\ a = q ^ { - N } , \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} \\bigl [ \\mathcal { H } ^ { X _ 1 \\ , } _ { \\mathcal { D } } , \\mathcal { H } ^ { X _ 2 \\ , } _ { \\mathcal { D } } \\bigr ] = 0 . \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} u _ \\rho ( t ) & = ( t \\log t \\log \\log t ) ^ { 1 / \\alpha } ( \\log \\log \\log t ) ^ { 1 / \\alpha + \\rho } , \\\\ v _ c ( t ) & = c ( t / \\log \\log t ) ^ { 1 / \\alpha } . \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} b ( k , \\theta ) \\leq M ( k , t , c ) = 2 \\left ( \\sum _ { i = 0 } ^ { t - 4 } ( k - 1 ) ^ { i } + \\frac { ( k - 1 ) ^ { t - 3 } } { c } + \\frac { ( k - 1 ) ^ { t - 2 } } { c } \\right ) , \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} \\phi ( \\tau ) = \\sum _ { n \\in \\Z - \\{ 0 \\} } \\lambda _ { \\phi } ( n ) \\ , y ^ { 1 / 2 } K _ { \\nu _ \\infty / 2 } ( 2 \\pi | n | y ) \\ , e ^ { 2 \\pi i n x } , \\lambda _ { \\phi } ( 1 ) = 1 . \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{gather*} T _ 0 T _ 1 \\cdots T _ { N - 2 } \\big ( x _ { N - 1 } ^ + \\big ) = T _ 0 T _ 1 \\cdots T _ { N - 3 } \\big ( \\big [ x _ { N - 2 } ^ + , x _ { N - 1 } ^ + \\big ] \\big ) = \\big [ T _ 0 T _ 1 \\cdots T _ { N - 3 } \\big ( x _ { N - 2 } ^ + \\big ) , T _ 0 \\big ( x _ { N - 1 } ^ + \\big ) \\big ] \\\\ \\hphantom { T _ 0 T _ 1 \\cdots T _ { N - 2 } \\big ( x _ { N - 1 } ^ + \\big ) } { } = [ E _ { N , N - 1 } ( 1 ) , - E _ { N - 1 , 1 } ( 1 ) ] = - E _ { N , 1 } ( 2 ) . \\end{gather*}"}
-{"id": "4614.png", "formula": "\\begin{align*} K ( n , 0 , j ) = ( j + 1 ) ! S ( n , j + 1 ) , \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} \\Vert ( v _ i ( C ^ { * } \\pi _ { W _ { i } } C ' ) ^ { \\frac { 1 } { 2 } } f \\Vert _ { 2 } ^ { 2 } = \\sum _ { i \\in I } v _ { i } ^ { 2 } \\langle \\pi _ { W _ { i } } C ' f , \\pi _ { W _ { i } } C f \\rangle . \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\mathbb { E } _ { p _ { 0 } } [ \\frac { p _ { \\beta _ { 1 } , \\lambda } } { p _ { 0 } } \\frac { p _ { \\beta _ { 2 } , \\lambda } } { p _ { 0 } } ] = \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\sup _ { A \\in \\alpha _ l } \\limsup _ { j \\in \\mathbb N } \\frac { \\mu ( B _ { \\epsilon ^ * } ^ j ( A ) \\cap P _ j ) } { \\mu ( A \\cap P _ j ) } = 1 . \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} \\| U _ h ^ { n } \\| ^ 2 + \\Delta t \\sum _ { k = 2 } ^ n \\| \\nabla U _ h ^ { k - \\theta } \\| ^ 2 + \\gamma \\Delta t \\sum _ { k = 2 } ^ n \\| \\Sigma _ h ^ { k - \\theta } \\| ^ 2 \\leq & C ( \\| U _ h ^ { 0 } \\| ^ 2 + \\| u _ H ^ { 0 } \\| ^ 2 + \\Delta t \\sum _ { k = 1 } ^ n \\| g ^ { k } \\| ^ 2 ) . \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} H _ k ( x ) = ( x , x + 3 ^ k , - 2 x - 3 ^ k ) . \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} g ^ { ( i ) } ( X ) & = \\sum _ { i ' = 0 } ^ i ( X ^ { q } ) ^ { ( i ' ) } \\cdot f ^ { ( i - i ' ) } ( X ) \\\\ & = \\sum _ { i ' = 0 } ^ i { q \\choose i ' } X ^ { q - i ' } \\cdot f ^ { ( i - i ' ) } ( X ) \\\\ & = X ^ { q } f ^ { ( i ) } ( X ) . \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} L _ { \\Gamma } ( s , \\varrho ) = \\det ( 1 - \\mathcal { L } _ { s , \\varrho } ) . \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} F = b _ + ^ { - 1 } ( \\tilde { F } _ 1 \\times \\cdots \\times \\tilde { F } _ { m ^ + } ) \\subseteq V . \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } v ' ( s ) ^ 2 \\frac { ( B ( s ) - 1 ) \\sinh ( F _ n ( s ) ) ^ { 4 ( n - 1 ) } } s = 0 . \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 1 } ^ r [ ( \\ref { t r p a r t } ) ] ^ { \\mathrm { h o l } } & = \\sum _ { \\substack { 0 < d \\equiv 0 , 1 ( 4 ) \\\\ d D \\neq \\square } } \\frac { 1 } { \\sqrt { d } } \\mathrm { L C } _ { s = 1 } ^ r \\biggl [ \\mathrm { T r } _ { d , D } ( G _ m ( z , s ) ) \\biggr ] q ^ d \\\\ & = \\sum _ { \\substack { 0 < d \\equiv 0 , 1 ( 4 ) \\\\ d D \\neq \\square } } \\frac { 1 } { \\sqrt { d } } \\mathrm { T r } _ { d , D } ( F _ { 0 , m , r } ) q ^ d . \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} \\frac { \\left | \\langle \\eta ( x _ 1 ) , \\mathbf { B } ( x _ 1 ) - \\mathbf { B } ( x _ 2 ) \\rangle \\right | } { | x _ 1 - x _ 2 | } \\lesssim \\sum _ { l = 1 , 2 } | | \\nabla \\eta | | _ { L ^ \\infty } \\mathbf { I } _ 1 ( \\mu ) ( x _ l ) + \\mathbf { M } ( | D ^ a \\mathbf { B } | ) ( x _ l ) , \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| u ^ k - x ^ k \\| = 0 . \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| y ^ k - z ^ k \\| = 0 . \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} \\mathcal F ^ V ( \\rho ) : = \\mathcal F ( \\rho ) + \\mathcal V ( \\rho ) , \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} B ( t ) = | t | ^ { - s } ( 1 - 2 ^ { - 2 s - 2 } ) ( 1 - 2 ^ { - 2 s - 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } a _ { i i } ^ { ( k ) } = 1 . \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} M ( V ) = \\begin{pmatrix} V & A ^ T \\\\ A & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} \\Theta ^ 1 _ { \\alpha } = \\bigcup _ { \\nu \\geq 0 } \\bigcup _ { | k | > J _ { \\nu } , | l | \\leq 2 \\atop { l \\neq e _ { - j } - e _ j } } \\mathcal { R } _ { k l } ^ { \\nu } ( \\alpha _ { 1 , \\nu } ) . \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} \\begin{array} { l l } \\Phi ^ { i _ \\epsilon j _ \\epsilon } _ { \\epsilon } ( x _ \\epsilon , y _ \\epsilon ) = \\max \\limits _ { x , y \\in B _ R } \\max \\limits _ { i , j \\in \\mathcal { I } } \\Phi ^ { i j } _ { \\epsilon } ( x , y ) . \\end{array} \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\begin{aligned} \\log _ q r & : = \\frac { r ^ { 1 - q } - 1 } { 1 - q } , \\\\ \\exp _ q ( r ) & : = [ 1 + ( 1 - q ) r ] ^ { \\frac { 1 } { 1 - q } } , \\end{aligned} \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} R [ J ' ( u ) ] = R [ \\Psi ' ( u ) ] - R [ \\Phi ' ( u ) ] = u - R [ f ( u ) ] , \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} G _ m ( z , s ) = \\sum _ { r = - 1 } ^ { \\infty } F _ { 0 , m , r } ( z ) ( s - 1 ) ^ r . \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} \\{ f , g \\} _ { P B } = \\frac { \\partial f } { \\partial x _ 1 } \\frac { \\partial g } { \\partial p _ 1 } + \\frac { \\partial f } { \\partial x _ 2 } \\frac { \\partial g } { \\partial p _ 2 } - \\frac { \\partial f } { \\partial p _ 1 } \\frac { \\partial g } { \\partial x _ 1 } - \\frac { \\partial f } { \\partial p _ 2 } \\frac { \\partial g } { \\partial x _ 2 } \\ ; , \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} T ^ { v i r } _ { y * } ( X ) - T _ { y * } ( X ) = ( i _ { \\Sigma , X } ) _ * M _ y ( X ) \\in \\mathbf { H } _ { \\bullet } ( X ) [ y ] , \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\begin{aligned} & x ^ + = x + \\alpha \\Delta x , \\\\ & y ^ + = y + \\alpha \\Delta y , \\\\ & s ^ + = s + \\alpha \\Delta s . \\end{aligned} \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} t ^ i \\ell _ { i } ( z _ 1 ) \\dots \\ell _ { i } ( z _ n ) \\cdot ( Y _ 1 Y _ 2 \\dots Y _ { j - 1 } Y _ j Y _ { j + 1 } \\dots Y _ n ) = \\tau ^ i ( Y _ 1 \\dots Y _ n W ) . \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} K _ 0 = \\max _ { x \\in \\overline \\Omega } \\Vert \\nabla \\phi ( x ) \\Vert . \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} \\tilde { f } ( ( p _ { k , l } ) _ { k , l } ) = C ' C f \\left ( \\prod _ { l = 1 } ^ { N _ 1 } p _ { 1 , l } , \\dots , \\prod _ { l = 1 } ^ { N _ r } p _ { r , l } \\right ) \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} \\lim \\limits _ { \\delta \\downarrow 0 } ( f _ \\delta , ( - \\mathcal { L } _ X ^ \\omega ) ^ { - 1 } f _ \\delta ) = ( f , ( - Q ) ^ { - 1 } f ) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\tilde { \\mu } - a . s . ~ \\omega . \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{gather*} T _ { t _ { - \\alpha _ 2 } } \\big ( E _ { 2 , 3 } ( - s ) \\big ) = ( - 1 ) ^ N E _ { 2 , 3 } ( - s + 2 ) , \\\\ T _ { t _ { - \\alpha _ 2 } } \\big ( E _ { 2 , k + 1 } ( - s ) \\big ) = - E _ { 2 , k + 1 } ( - s + 1 ) . \\end{gather*}"}
-{"id": "534.png", "formula": "\\begin{align*} - v _ { n , m } = - v _ { n , m - 1 } + D _ { n , m } . \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} P _ { G ^ { \\prime } } ( x ) = x ^ { 3 i } ( x + 1 ) ^ { 6 i } ( x + 2 i + 2 ) ( x ^ 4 - ( 8 i + 2 ) x ^ 3 - ( - 8 i ^ 2 + 4 i + 3 ) x ^ 2 - ( - 8 i ^ 3 - 2 0 i ^ 2 - 8 i ) x - 8 i ^ 4 - 1 2 i ^ 3 - 4 i ^ 2 ) \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} \\mathfrak F \\Phi ( v ) = \\int \\mathfrak R ( a \\cdot \\Phi ) ( v ) \\psi ^ { - 1 } ( a ) | a | d ^ \\times a . \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} \\Psi : \\mathbb { E } \\rightarrow \\mathbb { R } ^ s \\times \\mathbb { E } ' \\Psi ( C ) = [ 0 , \\infty ) ^ s \\times \\mathbb { E } ' s \\geq 0 . \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} s _ { \\alpha } ( v ) = v - \\langle v , \\alpha ^ \\vee \\rangle \\alpha = v - \\langle v , \\alpha \\rangle \\alpha ^ \\vee , \\quad \\forall v \\in \\mathfrak { h } ^ * . \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } \\tfrac 1 t \\log \\mathbb P ( P _ s \\leq x s s \\leq t ) = - ( 1 - x + x \\log x ) . \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} C ( x , \\theta ) = \\alpha ( x ) \\delta \\left ( \\theta - { \\hat { \\theta } } ( x ) \\right ) , \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\frac { T ( r + 1 , f ) } { T ( r , f ) } = 1 , \\end{align*}"}
-{"id": "9666.png", "formula": "\\begin{align*} \\| \\mathcal { T } h - \\mathcal { T } \\tilde { h } \\| _ { \\mathcal { B } _ { \\alpha } ^ { 2 } } \\leq \\frac { 1 } { 1 - \\alpha } \\sum _ { j = 2 } ^ { \\infty } \\| F _ { j } - \\tilde { F } _ { j } \\| _ { \\mathcal { B } _ { \\alpha } ^ { 0 } } . \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} U _ 1 ( t ) = A _ 1 V _ m ( t ) + \\int _ t ^ { t _ 0 } G _ { 1 } ( t , s ) V _ m ( s ) \\ , d s + \\int _ t ^ { t _ 0 } H _ { 1 } ( t , s ) U _ 1 ( s ) \\ , d s \\mbox { f o r } t _ 1 \\le t \\le t _ 0 , \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} v _ n ( x ) = V ^ { j _ 0 } ( x - x ^ { j _ 0 } _ n ) + v ^ l _ n ( x ) . \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { B ^ r _ { p , q } } : = \\left ( \\sum _ { m = 0 } ^ { \\infty } 2 ^ { m r q } \\Vert \\sum _ { 2 ^ m \\leq \\langle \\xi \\rangle < 2 ^ { m + 1 } } d _ { \\xi } [ \\xi ( x ) \\widehat { f } ( \\xi ) ] \\Vert ^ q _ { L ^ p ( G ) } \\right ) ^ { \\frac { 1 } { q } } < \\infty . \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} A ^ { \\ast } ( t ) = A ( t ) - B ( t ) , \\end{align*}"}
-{"id": "9647.png", "formula": "\\begin{align*} h _ { t } = \\Delta e ^ { - \\Delta h } , \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - \\alpha t } \\mu ( \\d t ) = 1 . \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} \\left ( \\left ( \\textstyle { \\bigwedge } ^ { n + 1 } \\mathcal { S } \\right ) \\otimes \\left ( \\textstyle { \\bigwedge } ^ { d - 1 } \\mathcal { S } \\right ) ^ { \\otimes n } \\right ) ( \\mathbf { v } _ f ) = \\sum _ { g \\in \\Psi ( Y , \\pi ) } \\mathbf { S } _ { g , f } \\mathbf { v } _ g . \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} \\lim _ { \\eta \\to 0 } \\sin \\biggl ( ( 1 - \\alpha ) \\arctan \\frac { | \\gamma | } { \\eta } \\biggr ) = \\cos \\biggl ( \\frac { \\pi \\alpha } { 2 } \\biggr ) . \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\to 0 } \\frac { 1 } { | \\log ( \\delta ) | } \\int _ { 0 } ^ { T } \\int _ { B _ r } \\frac { \\left | \\langle \\eta ( X _ { 1 t } ) , \\mathbf { B } ( X _ { 1 t } ) - \\mathbf { B } ( X _ { 2 t } ) \\rangle \\right | } { \\delta + | X _ { 1 t } - X _ { 2 t } | } d x d t = 0 . \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} \\lambda _ n ( f ) = \\int _ K f ( n k n ^ { - 1 } ) d k . \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} \\mathcal { I } _ { s , c } ^ { \\pm } ( m ) = \\mathcal { H } _ { \\pm } W _ { \\infty } \\left ( \\frac { m } { b ^ 2 l ^ { 2 c } } \\right ) . \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} c _ h ( x _ 1 , \\dots , x _ N ) = h ( x _ 1 + x _ 2 + \\cdots + x _ N ) , h : \\R ^ d \\to \\R ^ { + } \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} u _ { + , a } ( x ) = \\frac { \\phi _ 3 ( x ) } { \\phi _ 3 ( a ) } \\frac { \\int _ { - \\infty } ^ x \\phi _ 3 ^ { - 2 } ( y ) d y } { \\int _ { - \\infty } ^ a \\phi _ 3 ^ { - 2 } ( y ) d y } , \\ x \\le a ; \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} ( K \\otimes _ A B ) ^ { \\times } = K ^ { \\times } \\cdot B ^ { \\times } : = \\{ a b \\mid a \\in K ^ { \\times } , b \\in B ^ { \\times } \\} . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} \\int _ s ^ t \\delta \\tilde { B } _ { s r } ^ i d Z _ r ^ j - \\int _ s ^ t \\delta B _ { s r } ^ i d Z _ r ^ j & = ( \\delta \\tilde { B } _ { s t } ^ i - \\delta \\tilde { B } _ { s t } ^ i ) \\delta Z _ { s t } ^ j + \\int _ s ^ t \\delta Z ^ j _ { s r } d B ^ i _ r - \\int _ s ^ t \\delta Z ^ j _ { s r } d \\tilde { B } ^ i _ r \\\\ & = - \\int _ s ^ t u ( r , X _ r ) ^ i d r \\delta Z _ { s t } ^ j + \\int _ s ^ t \\delta Z _ { s r } ^ i u ( r , X _ s ) ^ j d r \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\partial _ { t } W ( t , x ) + \\inf \\limits _ { u \\in U } H ( t , x , W ( t , x ) , D W ( t , x ) , D ^ { 2 } W \\left ( t , x \\right ) , u ) = 0 , \\\\ W ( T , x ) = \\phi ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\left \\{ ( g _ j , g , \\bar { g } , h , \\bar { h } ) \\begin{tabular} { | l } $ h \\bar { h } = h _ 1 , g \\bar { g } = g _ j $ \\\\ $ g _ j \\in c ' ( h _ 1 , Y ) , h \\in c ( g , X ) , \\bar { h } \\in c ( \\bar { g } , X ) $ \\\\ $ g X _ { \\bar { g } , \\bar { h } } \\cap X _ { g , h } \\neq \\emptyset $ \\end{tabular} \\right \\} . \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} V ( x _ 1 , x _ 2 , \\cdot \\cdot \\cdot , x _ N ) : = ( \\rho ( x _ 1 ) + f ( x ) ) / ( w ( x ) + k ) . \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} \\frac { \\phi } { \\sigma _ { - } } = - ( I - V U ) ^ { - 1 } \\left ( P ^ { + } \\left ( \\frac { z ^ { i } } { \\sigma _ { - } } \\right ) \\right ) - \\frac { z ^ { i } } { \\sigma _ { - } } \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} G _ 0 ( x , t ) = \\int _ 0 ^ t \\mathcal { A } _ 0 ( x , s ) s d s , \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} P _ 0 Q ( k ) P _ 0 & = P _ 0 Q _ 0 ( k ) P _ 0 + \\frac { \\lvert k \\lvert } { \\omega ( k ) } P _ 0 Q _ 0 ( k ) ( k \\cdot \\nabla K ( \\xi - d \\Gamma ( m ) ) Q ( k ) P _ 0 + P _ 0 o _ 1 ( k ) P _ { 0 } \\\\ & = P _ 0 + \\frac { \\lvert k \\lvert } { \\omega ( k ) } P _ 0 ( \\widehat { k } \\cdot \\nabla K ( \\xi - d \\Gamma ( m ) ) ) ( 1 - P _ 0 ) Q ( k ) P _ 0 \\\\ & + \\left ( \\frac { \\lvert k \\lvert } { \\omega ( k ) } - C _ \\omega \\right ) \\widehat { k } \\cdot u P _ 0 Q ( k ) P _ 0 + C _ \\omega \\widehat { k } \\cdot u P _ 0 Q ( k ) P _ 0 + P _ 0 o _ 1 ( k ) P _ { 0 } \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} w = \\sum _ { m ^ + < j \\le m + 1 } | c _ j | B _ j ^ * y _ j . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} \\Delta _ Q ^ * = \\min _ { v \\in \\mathcal { E } ( Q ) } \\max _ { u _ q \\in \\mathbb { U } } \\Delta ( u _ q , v ) \\geq \\frac { 2 } { 3 } \\lambda _ 1 ^ Q . \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} \\mathcal { G } = \\left \\{ \\ , g \\in L ^ { 2 } ( \\Omega ) \\ ; | \\ ; g = 0 \\hbox { i n } \\ ; L ^ { 2 } ( \\Omega ) \\backslash \\mathcal { E } \\ , \\right \\} . \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} \\deg _ { \\theta } ( \\phi _ { \\theta , i ( \\phi ) } ) + n ( 1 + q + \\dots + q ^ { i ( \\phi ) - 1 } ) & \\leq \\frac { q - 1 } { r } - ( n + 1 ) + \\frac { n } { q } + \\frac { 1 } { q } + n + n ( q + \\dots + q ^ { i ( \\phi ) - 1 } ) \\\\ & < q + n ( q + \\dots + q ^ { i ( \\phi ) - 1 } ) \\\\ & \\leq q + ( q - 1 ) ( q + \\dots + q ^ { i ( \\phi ) - 1 } ) = q ^ { i ( \\phi ) } . \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} K _ \\lambda ( z , w ) = ( 1 - z ^ t \\overline { w } ) ^ { - \\lambda } = ( 1 - \\langle z , w \\rangle ) ^ { - \\lambda } \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} A _ 1 ^ 2 - A _ 2 ^ { \\mu } = \\left ( \\pi ^ 2 \\lambda _ b ^ 2 \\beta ^ { \\frac { 2 - \\mu } { 2 } } \\right ) - \\left ( \\frac { \\pi ^ 2 \\lambda _ b \\mathbb { P } \\left [ \\mathbb { A _ { U E } } \\right ] \\sqrt { \\Theta } } { 2 } \\right ) ^ { \\mu } = 0 . \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} B _ { n } = \\sum _ { j = 1 } ^ { n } Y _ { j } \\ , h _ { j } ^ { n } , \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} G _ { k , 0 , 0 } ( z ) & = P _ { k , 0 } ( z , k / 2 ) \\\\ & = 1 + \\sum _ { ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) , n > 0 } b _ { k , 0 } ( n , k / 2 ) \\frac { n ^ { k - 1 } } { \\Gamma ( k ) } q ^ n , \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} t _ { 2 } = \\frac { a k ^ { 2 } - 2 1 a k - 1 0 b k } { 1 2 } t _ { 6 } = \\frac { a k ^ { 2 } + 3 a k + 2 b k } { 3 6 } . \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\begin{cases} \\rho ^ { n + 1 } _ i - \\rho ^ n _ i + \\Delta t \\cdot \\textrm { d i v } ( m ^ { n + 1 } ) | _ i = 0 \\ , , \\\\ \\rho ^ { N } _ i = \\rho _ i \\ . & \\end{cases} \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} t _ \\tau = \\frac { ( t _ 2 - t _ 1 ) ( \\tau - \\tau _ 1 ) } { \\tau _ 2 - \\tau _ 1 } + t _ 1 \\ ; , \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} [ \\hat { x } _ 1 , \\hat { p } _ 1 ] = i \\hbar ; [ \\hat { x } _ 2 , \\hat { p } _ 2 ] = i \\hbar ; [ \\hat { x } _ 1 , \\hat { p } _ 2 ] = 0 ; [ \\hat { x } _ 2 , \\hat { p } _ 1 ] = 0 . \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} \\kappa ( L ( p z ) - L ( z ) ) - b z L ' ( z ) + c z L '' ( z ) + z \\int _ 0 ^ \\infty L ( z + y ) - L ( z ) + y L ' ( z ) N ( d y ) = 0 , \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} \\begin{aligned} V ( u ) & = V ( u _ { a } + u _ { b } ) \\\\ & = V ( u _ { a } ) + V ^ { \\prime } ( u _ { a } ) u _ { b } + \\frac { V ^ { \\prime \\prime } ( u _ { a } ) } { 2 ! } u _ { b } ^ { 2 } + \\frac { V ^ { ^ { \\prime \\prime \\prime } } ( u _ { a } ) } { 3 ! } u _ { b } ^ { 3 } + \\frac { V ^ { ( 4 ) } ( u _ { a } ) } { 4 ! } u _ { b } ^ { 4 } \\end{aligned} \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{gather*} \\big [ \\big [ x _ i ^ + , x _ j ^ + \\big ] , \\big \\{ h _ i , x _ i ^ { - } \\big \\} \\big ] = \\big \\{ \\big [ \\big [ x _ i ^ + , x _ j ^ + \\big ] , h _ i \\big ] , x _ i ^ - \\big \\} + \\big \\{ h _ i , \\big [ \\big [ x _ i ^ + , x _ j ^ + \\big ] , x _ i ^ - \\big ] \\big \\} \\\\ \\hphantom { \\big [ \\big [ x _ i ^ + , x _ j ^ + \\big ] , \\big \\{ h _ i , x _ i ^ { - } \\big \\} \\big ] } { } = - \\big \\{ \\big [ x _ i ^ + , x _ j ^ + \\big ] , x _ i ^ - \\big \\} - \\big \\{ h _ i , x _ j ^ { + } \\big \\} = - \\big \\{ T _ i ( x _ j ^ + ) , x _ i ^ - \\big \\} - \\big \\{ h _ i , x _ j ^ { + } \\big \\} . \\end{gather*}"}
-{"id": "6925.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta _ g K ^ \\pm _ \\gamma & = 1 , \\mbox { i n } \\Omega ^ \\pm , \\\\ K ^ \\pm _ \\gamma & = 1 , \\mbox { o n } \\gamma , \\\\ \\partial _ n K ^ - _ \\gamma + \\partial _ n K ^ + _ \\gamma & = 0 , \\mbox { o n } \\gamma , \\end{aligned} \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} \\mathcal O _ r = \\bigcup _ { i = 1 } ^ k \\mathcal U _ { \\gamma _ i } ( r ) , \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} H ^ k ( F _ n , \\mathbb { R } ) = \\begin{cases} \\mathbb { R } , & k = 0 \\\\ \\mathbb { R } ^ n , & k = 1 \\\\ 0 , & k > 1 \\\\ \\end{cases} . \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} ( \\mathrm { d } p ) _ \\alpha ( x ) = \\frac { \\partial p } { \\partial x ^ \\alpha } ( \\varphi ( x ) ) \\ , , \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} \\frac { | { \\mathcal W } | } { | { \\mathcal W ' } | } = O ( d ^ 4 / ( d n ) ^ 2 ) = O ( d ^ 2 / n ^ 2 ) . \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} C ^ { - 1 } \\Big \\| \\sum _ { k = 1 } ^ { \\infty } c _ k f _ { k } \\Big \\| _ E \\le \\Big \\| \\sum _ { k = 1 } ^ { \\infty } c _ k x _ { k } \\Big \\| _ W \\le C \\Big \\| \\sum _ { k = 1 } ^ { \\infty } c _ k f _ { k } \\Big \\| _ E . \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} w ( r ) = \\log \\frac { 8 } { ( 1 + r ^ 2 ) ^ 2 } \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} d ( v , u ) = f ( v ) - f ( u ) - f ' ( u ; v - u ) , \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} & C ^ { 0 + \\alpha } ( \\mathbb R ^ d ) = \\{ h \\in C _ b ^ { 0 , 0 } ( \\mathbb R ^ d ) : \\| h \\| _ { C ^ { 0 + \\alpha } ( \\mathbb R ^ d ) } < \\infty \\} \\\\ & C ^ { 1 + \\alpha } ( \\mathbb R ^ d ) = \\{ h \\in C _ b ^ { 1 , 0 } ( \\mathbb R ^ d ) : \\| h \\| _ { C ^ { 1 + \\alpha } ( \\mathbb R ^ d ) } < \\infty \\} , \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} \\ln R ( x , E ) - \\ln R ( a , E ) = - \\int _ { a } ^ x \\frac { C } { 2 \\gamma ^ { \\prime } ( y , E ) } \\frac { 1 } { 1 + x } \\sin ^ 2 2 \\theta ( y ) d y \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} n _ b ( W ^ n ( w ) ) = \\lvert m _ 0 \\rvert + \\lVert ( \\phi w _ k ) _ n \\rVert _ S . \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t U _ h ^ { n - \\theta } , v _ h \\Big { ) } & - \\gamma ( \\nabla \\Sigma _ h ^ { n - \\theta } , \\nabla v _ h ) + ( \\nabla U _ h ^ { n - \\theta } , \\nabla v _ h ) \\\\ & + ( ( 1 - \\theta ) \\mathfrak { F } ( U _ { h } ^ { n } , u _ { H } ^ { n } ) + \\theta f ( U _ { h } ^ { n - 1 } ) , v _ h ) = ( g ^ { n - \\theta } , v _ h ) , ~ \\forall v _ h \\in L _ h , \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} \\Psi \\left ( \\frac { x + s } { 2 } \\right ) & \\leq \\frac { 1 } { 2 } [ \\lambda _ 1 ( x ) + \\lambda _ 1 ( s ) ] + \\frac { 1 } { 4 } [ \\lambda _ 2 ( x ) + \\lambda _ 2 ( s ) + \\lambda _ 4 ( x ) + \\lambda _ 4 ( s ] \\\\ & = \\frac { 1 } { 2 } \\Psi ( x ) + \\frac { 1 } { 2 } \\Psi ( s ) \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} f _ { M , M ' } ( m \\otimes m ' ) = q ^ { \\langle \\lambda , \\mu ' \\rangle } m \\otimes m ' , \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} & \\lim _ { \\varepsilon \\to + 0 } \\sum _ { \\beta > 0 , n \\in \\mathbb { Z } } L _ { n , \\beta } ^ { t _ i + \\varepsilon } q ^ n t ^ { \\beta } \\\\ & = \\exp \\left ( \\sum _ { n / \\omega \\cdot \\beta = t _ i } ( - 1 ) ^ { n - 1 } n N _ { n , \\beta } q ^ n t ^ { \\beta } \\right ) \\cdot \\left ( \\lim _ { \\varepsilon \\to + 0 } \\sum _ { \\beta > 0 , n \\in \\mathbb { Z } } L _ { n , \\beta } ^ { t _ i - \\varepsilon } q ^ n t ^ { \\beta } \\right ) . \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} | | z ( u ) | u | ^ { p - 1 } u | | _ q \\cdot | | u | | _ { q ' } \\leq C ( \\Omega , n , p ) | | u | | _ { p q } ^ { 2 p - 1 } | | u | | _ { p q } = C ( \\Omega , n , p ) | | u | | _ { p q } ^ { 2 p } . \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} T _ x ( M ^ { H _ 1 } \\cap M ^ { H _ 2 } ) = T _ x M ^ { H _ 1 } \\cap T _ x M ^ { H _ 2 } \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} f ( x ) = \\widetilde V \\left ( x , \\lambda _ { t + 1 } , B _ { k + 1 } \\backslash \\{ \\lambda _ { t + 1 } \\} , J _ k + t T _ { k + 1 } , J _ k + ( t + 1 ) T _ { k + 1 } , t T _ { k + 1 } , \\frac { w _ { \\lambda _ { t + 1 } } ^ \\prime ( J _ k + t T _ { k + 1 } ) } { w _ { \\lambda _ { t + 1 } } ( J _ k + t T _ { k + 1 } ) } \\right ) , \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} g ^ { ( 4 ) } = - N ^ 2 d \\tau ^ 2 + g _ { i j } ( d x ^ i + X ^ i d \\tau ) ( d x ^ j + X ^ j d \\tau ) . \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} T _ x M ^ G & = f ( U ^ G ) = f ( M ^ G \\cap U ) = f ( M ^ { H _ 1 } \\cap M ^ { H _ 2 } \\cap U ) \\\\ & = f ( ( M ^ { H _ 1 } \\cap U ) \\cap ( M ^ { H _ 2 } \\cap U ) ) = f ( U ^ { H _ 1 } \\cap U ^ { H _ 2 } ) \\\\ & = f ( U ^ { H _ 1 } ) \\cap f ( U ^ { H _ 2 } ) = T _ x M ^ { H _ 1 } \\cap T _ x M ^ { H _ 2 } . \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} c l _ { X } ( V ' ( x ) \\cap \\tilde { X } ) = c l _ { \\tilde { X } } ( V ' ( x ) \\cap \\tilde { X } ) . \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} \\Omega _ n : = \\{ x \\in \\Omega : d ( x , \\partial \\Omega ) > 1 / n \\} ; \\ ; \\phi _ { i , n } : = u _ i \\star \\rho _ n \\ ; \\ ; \\Omega _ n , \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} \\div \\left ( \\psi ^ \\prime ( | \\nabla \\Q | ) 2 \\nabla \\Q \\right ) = \\frac { 1 } { L } \\left ( - A Q _ { i j } - B \\left ( Q _ { i p } Q _ { p j } - | \\mathbf { Q } | ^ 2 \\delta _ { i j } / 3 \\right ) + C Q _ { p q } Q _ { p q } Q _ { i j } \\right ) \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { s - 1 } \\lambda _ { 1 , k } ^ { j - 1 } c ^ { I } _ { 1 , b ( 1 , k ) } + \\sum _ { k = 0 } ^ { s - 1 } \\sum _ { i = 2 } ^ { i = n } \\lambda _ { i , b _ { i } } ^ { j - 1 } c ^ { I } _ { i , b ( 1 , k ) } = 0 j \\in [ r ] . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} S ( u ^ * _ { \\mu , \\xi } ) ( x , t ) = \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } - 1 } \\sum _ { j = 1 } ^ k \\dot { \\mu } _ j f _ { j } + \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } } \\sum _ { j = 1 } ^ k \\dot { \\xi } _ j \\cdot \\vec { f } _ { j } + \\mu _ 0 ^ { \\frac { n + 2 s } { 2 } } f , \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} \\varphi ( \\mu ) : = f ( \\delta _ { \\omega } ( \\mu ) ) = \\sum _ { k = 0 } ^ { \\infty } f _ k ( \\delta _ { \\omega } ( \\mu ) ) = \\sum _ { k = 0 } ^ { \\infty } f _ k ( \\omega ) ( \\mu ^ M ) ^ { k / M } = \\sum _ { k = 0 } ^ { \\infty } f _ k ( \\omega ) \\mu ^ k , \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} \\mu ^ * = 2 ^ { \\frac { { R } } { r _ s } } { \\sigma ^ 2 } \\left ( \\left ( 1 - \\rho ^ * \\right ) \\left ( \\textstyle \\prod _ { k = 1 } ^ { r _ s } \\frac { [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } { \\nu ^ * - \\rho ^ * [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } \\right ) ^ { \\frac { 1 } { r _ s } } \\right ) ^ { - 1 } \\ln 2 . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} \\left ( \\frac { \\sqrt { \\frac { \\pi } { 2 } } \\rho ( y ) } { e ^ { - 2 y ^ { 2 } } } - 1 \\right ) \\frac { 1 } { \\sqrt { \\tau } } = \\frac { 8 } { 3 } y ^ { 3 } - 2 y + \\sqrt { \\tau } \\left ( \\frac { 3 2 } { 9 } y ^ { 6 } - \\frac { 3 1 } { 3 } y ^ { 4 } + \\frac { 1 5 } { 2 } y ^ { 2 } - \\frac { 3 7 } { 4 8 } \\right ) + O ( \\tau ) \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} u ^ \\omega ( x ) : = \\sum _ { y \\sim x } \\omega ( x , y ) v ^ \\omega ( x ) : = \\sum _ { y \\sim x } \\frac { 1 } { \\omega ( x , y ) } . \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} \\chi _ n = \\mu P _ 1 P _ 2 \\cdots P _ n r . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} \\frac { | G | } { | C | } I _ j & = \\sum _ { \\chi } \\overline { \\chi } ( g ) J _ j ( \\chi ) \\\\ & = k _ j ( 1 ) - k _ j ( 0 ) \\sum _ { \\chi } \\overline { \\chi } ( g ) \\left \\{ a ( \\chi ) - \\delta ( \\chi ) \\right \\} - \\sum _ { \\chi } \\overline { \\chi } ( g ) \\left ( \\sum _ { \\rho _ { \\chi } \\in Z ( \\chi ) } k _ j ( \\rho _ { \\chi } ) \\right ) - \\sum _ { \\chi } \\overline { \\chi } ( g ) V _ j ( \\chi ) \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} f ^ { * * } ( t ) = \\frac { 1 } { t } \\int _ { 0 } ^ { t } f ^ * ( s ) \\ , \\d s \\quad . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} \\eta ( \\theta ) = \\limsup _ { n \\to \\infty } { \\log q _ { n + 1 } \\over \\log q _ n } . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} \\deg ( D | _ { S _ 1 } ) + \\deg ( D | _ { S _ 2 } ) = \\deg ( D | _ { S _ 1 \\cap S _ 2 } ) + \\deg ( D | _ { S _ 1 \\cup S _ 2 } ) \\ , . \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} \\mu = \\zeta ^ 1 \\wedge \\zeta ^ 2 \\otimes Z _ 2 + \\lambda \\ , \\zeta ^ 1 \\wedge \\zeta ^ 3 \\otimes Z _ 3 + \\bar \\zeta ^ 1 \\wedge \\bar \\zeta ^ 2 \\otimes \\bar Z _ 2 + \\bar \\lambda \\ , \\bar \\zeta ^ 1 \\wedge \\bar \\zeta ^ 3 \\otimes \\bar Z _ 3 \\ , , \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} I = L _ \\xi ( ( d - ( n + 1 ) ) \\varpi _ 1 - d \\varpi _ n , ( d - ( n + 1 ) ) \\varpi _ 1 + d \\varpi _ n ) \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ \\tau \\phi = - ( - \\Delta ) ^ s \\phi + p U ^ { p - 1 } ( y ) \\phi + h ( y , \\tau ) , & y \\in \\mathbb { R } ^ n , ~ \\tau \\geq \\tau _ 0 , \\\\ \\phi ( y , \\tau _ 0 ) = e _ { 0 } Z _ 0 ( y ) , & y \\in \\mathbb { R } ^ n . \\end{cases} \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} \\mathcal { F } \\left \\{ e ^ { - a | t | } \\right \\} ( \\nu ) = \\frac { 2 a } { a ^ 2 + ( 2 \\pi \\nu ) ^ 2 } . \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} G ( t , q , x ) = \\sum _ { n \\geq 0 } \\alpha _ n ( t , q ) \\frac { x ^ n } { n ! } . \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} N = ( n + 1 ) d ^ n ( - 1 ) ^ { \\lvert \\Phi ^ + \\rvert } \\prod _ { \\alpha \\in \\Phi ^ + } \\frac { 1 } { ( \\xi ^ { \\langle \\alpha , \\rho \\rangle } - \\xi ^ { - \\langle \\alpha , \\rho \\rangle } ) ^ 2 } . \\end{align*}"}
-{"id": "9798.png", "formula": "\\begin{align*} \\frac { 1 } { Q _ t ( e ' _ q ) } = \\frac { N } { D } \\le \\frac { c } { \\nu t ^ 3 } \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; t \\le \\frac { 4 } { r _ 1 } { ~ . } \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} \\ < D _ { n } f ( w ) , \\ , h \\ > _ { H _ { n } } = \\left . \\frac { d } { d \\varepsilon } f ( w + \\varepsilon h ) \\right | _ { \\varepsilon = 0 } . \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 0 , n ) = \\Phi ( n ) = \\frac { 1 } { n } \\sqrt { \\left ( \\frac { 2 } { n } \\right ) } ~ \\Upsilon \\left ( \\frac { 1 } { n } \\right ) - \\Upsilon ( n ) , \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} \\gamma \\cup z = \\pi ( w ) \\cup z = w \\cup \\widehat { \\pi } ( z ) = w \\cup d x = d w \\cup x - d ( w \\cup x ) . \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} \\ , _ { c } \\mathbb { A ' } ^ { - 1 } _ L = { \\Delta ' } ^ { - 1 } \\left ( \\begin{matrix} d & - b \\\\ - c & a \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} \\mathbb { I } : = \\mathbb { Z } ^ n \\times \\mathbb { N } ^ n \\times \\mathbb { N } ^ { \\mathbb { Z } _ * } \\times \\mathbb { N } ^ { \\mathbb { Z } _ * } , \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{gather*} A = \\langle a ^ m , a ^ { m + 1 } \\rangle B = \\langle b ^ m , b ^ { m + 1 } \\rangle C = \\langle c ^ m , c ^ { m + 1 } \\rangle D = \\langle d ^ m , d ^ { m + 1 } \\rangle . \\end{gather*}"}
-{"id": "10007.png", "formula": "\\begin{align*} \\bar F _ t ( q ) = e ^ { q } \\left ( \\frac { q } { q + 1 } \\int _ t ^ \\infty e ^ { - q \\tau ( s ) } d s \\right ) = e ^ { q } \\left ( \\frac { q } { q + 1 } \\right ) \\left ( ( 1 - t ) e ^ { - q } + \\int _ 1 ^ \\infty e ^ { - q s } d s \\right ) = \\frac { t } { q + 1 } + ( 1 - t ) . \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} \\mathcal { U } [ 0 ] = F _ m \\cdot \\left ( \\prod _ { i = 1 } ^ { m } \\alpha _ i ^ { l _ 0 l _ i } \\right ) \\prod _ { 1 \\le i < j \\le m } ( \\alpha _ i - \\alpha _ j ) ^ { l _ i l _ j } . \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} a _ 2 \\triangleright a _ 1 = ( 1 - t ) \\cdot a _ 1 + t \\cdot a _ 2 . \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} j = 4 ^ k l + t , \\ , \\ , \\ , \\ , \\ , \\ , \\ , 0 \\leq l < l _ 0 , \\ , 0 \\leq t \\leq 4 ^ k - 1 ; \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} T ( \\alpha ( r + | c | ) , f ) = T \\left ( r + | c | + \\frac { r + | c | } { T ( r + | c | , f ) ^ { \\varepsilon } } , f \\right ) \\leq C T ( r + | c | , f ) \\leq C ^ 2 T ( r , f ) \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} C ( \\mathbf i , \\mathbf a ) \\coloneqq \\{ x = ( x _ 1 , \\dots , x _ N ) \\in \\R ^ N & \\mid A _ j ( x ) \\leq x _ j \\leq 0 0 < x _ j < A _ j ( x ) \\\\ & 1 \\leq j \\leq N \\} . \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} \\dfrac { 1 } { \\epsilon _ 0 } \\overline { \\rho } H = ( 2 \\pi R ) H \\cdot E ( R ) , \\quad E ( R ) = \\dfrac { 1 } { 2 \\pi \\epsilon _ 0 } \\dfrac { 1 } { R } . \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} \\gamma _ - ( s v ^ + + v ^ - ) & = I ' ( v ^ - ) [ v ^ - ] + s ^ 2 \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { v ^ - } ( v ^ + ) ^ 2 d x \\\\ & \\leq I ' ( v ^ - ) [ v ^ - ] + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { v ^ - } ( v ^ + ) ^ 2 d x = \\gamma _ - ( v ) < 0 . \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} \\langle & \\phi , A \\psi ( k ) \\rangle _ + \\\\ & = \\langle ( H _ \\mu ( \\xi - k ) + \\omega ( k ) - \\Sigma ( \\xi ) ) ( H _ \\mu ( \\xi - k ) + \\omega ( k ) - \\Sigma ( \\xi ) ) ^ { - 1 } \\phi , A \\psi ( k ) \\rangle _ + \\\\ & = \\langle ( H _ \\mu ( \\xi - k ) + \\omega ( k ) - \\Sigma ( \\xi ) ) ^ { - 1 } \\phi , h ( k ) \\rangle \\\\ & = \\langle \\phi , ( H _ \\mu ( \\xi - k ) + \\omega ( k ) - \\Sigma ( \\xi ) ) ^ { - 1 } h ( k ) \\rangle _ + . \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} i \\partial _ t u + \\Delta u + c | x | ^ { - 2 } u = - | u | ^ \\alpha u , u ( 0 ) = u _ 0 \\in H ^ 1 , ( t , x ) \\in \\R ^ + \\times \\R ^ d , \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} \\sup _ { ( \\psi ( \\cdot ) , \\eta ( \\cdot ) ) \\in \\mathcal { Q } ( \\epsilon ) } \\psi ( y _ 0 ) : = \\bar d ^ * ( y _ 0 , \\epsilon ) \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} H ^ { ( n ) } ( \\xi ) & = K ( \\xi - d \\Gamma ( m ) ) + d \\Gamma ( \\omega ) + \\mu \\varphi ( v _ n ) \\geq - \\mu ^ 2 \\lVert \\omega ^ { - 1 / 2 } v _ n \\lVert ^ 2 \\geq - \\mu ^ 2 \\lVert \\omega ^ { - 1 / 2 } v \\lVert ^ 2 \\\\ \\Sigma _ n ( \\xi ) & = \\inf ( \\sigma ( H ^ { ( n ) } ( \\xi ) ) ) \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} M : = \\begin{pmatrix} ( g _ 1 , h _ { 1 } ^ { ( 1 ) } h _ { 1 } ^ { ( 2 ) } ) _ { \\Omega } & \\dots & ( g _ m , h _ { 1 } ^ { ( 1 ) } h _ { 1 } ^ { ( 2 ) } ) _ { \\Omega } \\\\ \\vdots & \\vdots & \\vdots \\\\ ( g _ 1 , h _ { m } ^ { ( 1 ) } h _ { m } ^ { ( 2 ) } ) _ { \\Omega } & \\dots & ( g _ m , h _ { m } ^ { ( 1 ) } h _ { m } ^ { ( 2 ) } ) _ { \\Omega } \\end{pmatrix} \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} \\left | \\int _ 0 ^ \\infty \\rho ^ { - \\frac { n - 2 } 2 } \\rho ^ { \\nu _ 0 } e ^ { i t \\rho } \\chi ( \\rho ) \\rho ^ { n - 1 } d \\rho \\right | \\geq \\frac 1 2 \\int _ 0 ^ \\infty \\rho ^ { - \\frac { n - 2 } 2 } \\rho ^ { \\nu _ 0 } \\chi ( \\rho ) \\rho ^ { n - 1 } d \\rho \\geq c . \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} \\lvert f _ 1 + f _ 2 \\rvert _ S & = \\min \\{ \\lVert h \\rVert _ S \\mid f _ 1 + f _ 2 \\sim h \\} \\\\ & \\leq \\min \\{ \\lVert h _ 1 + h _ 2 \\rVert _ S \\mid f _ 1 \\sim h _ 1 , f _ 2 \\sim h _ 2 \\} \\\\ & \\leq \\lVert g _ 1 + g _ 2 \\rVert _ S \\\\ & \\leq \\max ( \\lVert g _ 1 \\rVert _ S , \\lVert g _ 2 \\rVert _ S ) \\\\ & = \\max ( \\lvert [ f _ 1 ] \\rvert _ S , \\lvert [ f _ 2 ] \\rvert _ S ) . \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} { D } \\ { \\rm i s \\ s t a r \\ s h a p e d \\ w i t h \\ r e s p e c t \\ t o \\ a \\ d i s c / b a l l \\ } \\mathfrak { B } _ D \\subset D { \\rm \\ w i t h \\ r a d i u s \\ = \\ } \\rho _ D h _ D , \\ 0 < \\rho _ D < 1 . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} 0 \\le & n _ 1 n _ 2 / k ^ 2 - \\lfloor n _ 1 / k \\rfloor \\lfloor n _ 2 / k \\rfloor \\\\ = & ( n _ 2 / k - \\lfloor n _ 2 / k \\rfloor ) ( n _ 1 / k ) + ( n _ 1 / k - \\lfloor n _ 1 / k \\rfloor ) \\lfloor n _ 2 / k \\rfloor \\le ( n _ 1 + n _ 2 ) / k \\ , , \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} Q ( x ) = \\left ( \\tfrac { ( p + 2 ) | q | ^ 2 } { \\mu ( p | q | | x | + 2 ) ^ 2 } \\right ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} \\nabla ^ * \\nabla Y = \\frac { 1 } { u ^ 2 } [ \\nabla _ { \\partial _ t } \\nabla _ { \\partial _ t } Y - \\nabla _ { \\nabla _ { \\partial _ t } \\partial _ t } Y ] - \\Sigma _ i [ \\nabla _ { e _ i } \\nabla _ { e _ i } Y - \\nabla _ { \\nabla _ { e _ i } e _ i } Y ] , \\end{align*}"}
-{"id": "9856.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } a ( n ) q ^ { n } = \\dfrac { 1 } { E _ { 1 } E _ { 2 } } . \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} \\mu ( \\mathrm d x ) = \\frac { \\mathrm { e } ^ { - U ( x _ 1 , \\ldots , x _ n ) } } { Z _ { \\mu } } \\d x . \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} \\overline W _ t & = \\frac { p ^ { \\overline P ^ t _ t } e ^ { \\delta t } } { X _ 0 ^ { - 1 } + \\vartheta \\int _ 0 ^ t p ^ { \\overline P _ s ^ t } e ^ { \\delta s } d s } = \\frac 1 { X _ 0 ^ { - 1 } p ^ { - P _ t } e ^ { - \\delta t } + \\vartheta \\int _ 0 ^ t p ^ { - P _ { t - s } } e ^ { - \\delta ( t - s ) } d s } \\\\ [ 1 e m ] & = \\Big ( X _ 0 ^ { - 1 } p ^ { - P _ t } e ^ { - \\delta t } + \\vartheta \\int _ 0 ^ t p ^ { - P _ s } e ^ { - \\delta s } d s \\Big ) ^ { - 1 } . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} \\langle \\nabla _ X ^ P Y , Z \\rangle - \\langle \\pi ^ * \\nabla _ X Y , Z \\rangle = - \\frac { 1 } { 2 } \\langle [ Y , Z ] , X \\rangle \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} H _ { i , j } & = ( n - 1 ) \\log ( n - 1 ) , H ^ c _ { i , j } = 2 ( n - 1 ) \\log ( n - 1 ) , \\\\ H ^ { a v } & = \\left ( \\dfrac { ( n - 1 ) ^ 2 } { n } + 1 \\right ) \\log ( n - 1 ) , \\\\ ( t ^ { r e l } + 1 ) H ( X ) & = \\left ( \\dfrac { n - 1 } { n } + 1 \\right ) \\log ( n - 1 ) \\leq H ^ { a v } \\leq \\left ( \\dfrac { ( n - 1 ) ^ 2 } { n } + 1 \\right ) \\log n = ( ( n - 1 ) t ^ { r e l } + 1 ) \\log n . \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} \\beta _ { 1 } ( y , t ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } ( 1 - 4 y ^ { 2 } ) e ^ { - 2 y ^ { 2 } } \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} v ( k , \\theta ) \\leq 1 + \\sum _ { i = 0 } ^ { t - 3 } k ( k - 1 ) ^ i + \\frac { k ( k - 1 ) ^ { t - 2 } } { c } . \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t } ^ { \\dagger \\ ! \\dagger } \\ , = \\ , ( X _ { t } ^ { \\dagger \\ ! \\dagger } - \\widetilde { X } _ { t } ^ { \\dagger \\ ! \\dagger } ) { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , ; t \\ge 0 \\ , \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} W ^ k _ i = W ^ k _ i ( t , \\hat { g } _ i ( t ) , g _ 0 ) = \\hat { g _ i } ( t ) ^ { p q } ( \\Gamma ( \\hat { g _ i } ( t ) ) _ { p q } ^ k - \\Gamma ( g _ 0 ) _ { p q } ^ k ) \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} \\nu _ { \\gamma , C } ( s ) = e ^ \\frac { C \\tau ( s ) } { ( \\ln ( 1 + \\tau ( s ) ) ) ^ \\gamma } \\ \\ \\ \\ ( s \\in G ) . \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u + u & = \\lambda e ^ { \\ , u } , \\mbox { i n } \\Omega , \\\\ \\partial _ n u & = 0 , \\mbox { o n } \\partial \\Omega . \\end{aligned} \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 v '' - r ( x ) v = - 1 , \\ a < x < \\infty ; \\\\ & v ( a ) = 0 ; \\\\ & v \\ge 0 \\ \\ v \\ . \\end{aligned} \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} \\widehat { w } = w ^ { ( 0 ) } ( \\vec { u } ) + \\sum _ { i , j } x _ i y _ j w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) + \\sum _ { i , i ' , j , j ' } x _ i x _ { i ' } y _ j y _ { j ' } w _ { i i ' j j ' } ^ { ( 2 ) } ( \\vec { u } ) + \\cdots \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} \\begin{aligned} \\det X & = ( - 1 ) ^ { p } \\mathbf { G } [ \\tau ] ^ { - p } \\det Y \\\\ & = ( - 1 ) ^ { p } \\mathbf { G } [ \\tau ] ^ { - p } c n ^ { - p ^ { 2 } + 2 \\beta p } \\left ( 1 + o ( 1 ) \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} [ t _ 0 , \\ldots , t _ { n - 1 } ] = X _ { g _ { t _ 0 } , h _ { s _ 0 } } \\cap g _ { t _ 0 } X _ { g _ { t _ 1 } , h _ { s _ 1 } } \\cap \\cdots \\cap ( g _ { t _ 0 } \\ldots g _ { t _ { n - 2 } } ) X _ { g _ { t _ { n - 1 } } h _ { s _ { n - 1 } } } , \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} \\rho ( m _ t ) R _ 1 = R _ 1 - \\epsilon ( [ g _ t g _ 1 ^ { - 1 } ] ) R _ t = R _ 1 - \\epsilon ( [ g _ t ] ) R _ t . \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\psi ( \\varphi ( a / \\alpha , q ) ) & = \\psi ( q + _ o r ( a ) ) \\\\ & = \\big ( ( q + _ o r ( a ) ) / \\alpha , \\ , ( q + _ o r ( a ) ) - _ o r ( q + _ o r ( a ) ) \\big ) . \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} [ T _ { a ^ 2 } , T _ { b ^ 2 } ] = 4 ( T _ b T _ a T _ b T _ a - T _ a T _ b T _ a T _ b + T _ a T _ { a b } T _ b - T _ b T _ { a b } T _ a ) \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} e _ 1 = 1 , e _ 2 = \\frac { 1 + J } 2 , e _ 3 = \\frac { I + I J } 2 , e _ 4 = \\frac { s D J + I J } p \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} L _ s = \\sum _ m \\lambda _ { \\pi _ 0 } ( m ) W _ l ( s ; m ) e \\left ( \\frac { a \\overline { b } } { l ^ { \\frac { n _ l } { 2 } } } m \\right ) F \\left ( \\frac { m } { M } \\right ) . \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} s _ i = s _ j + a _ { i j } X _ i \\cap X _ j . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} V _ { \\varphi _ R } ( u ( t ) ) : = \\int \\varphi _ R ( x ) | u ( t , x ) | ^ 2 d x . \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} \\varphi ( n , E ) = p ( n ) e ^ { i \\frac { k ( E ) } { q } n } , \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ j \\kappa ^ { \\gamma ( i - 1 ) } \\omega _ { \\bullet } ( \\kappa ^ { j - i } r ) \\le \\kappa ^ { - \\gamma } \\tilde { \\omega } _ { \\bullet } ( \\kappa ^ j r ) . \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} K _ n ( q ) & = \\frac { ( q + 1 ) \\prod _ { i = 0 } ^ { n - 1 } ( q ^ { 2 n + 1 - i } + 1 ) \\prod _ { i = 1 } ^ { \\lfloor n / 2 \\rfloor - 1 } ( q ^ { 2 n + 1 - 2 i } + 1 ) } { \\prod _ { i = 1 } ^ { \\lfloor n / 2 \\rfloor } ( q ^ { 2 i + 2 } + 1 ) } \\\\ & = ( q ^ { 2 n + 1 } + 1 ) ( q ^ { n + 2 } + 1 ) ( q + 1 ) \\prod _ { i = 1 } ^ { \\lfloor n / 2 \\rfloor } \\frac { ( q ^ { 2 n + 1 - i } + 1 ) ( q ^ { 2 n - i } + 1 ) ^ 2 } { ( q ^ { 2 i + 2 } + 1 ) } . \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} A _ k + A _ { 2 - k } = \\left \\{ \\begin{array} { l l } - 1 & k = \\lambda _ k + 1 / 2 \\lambda _ k \\equiv \\ell _ k ( 2 ) , \\\\ - 3 & k = \\lambda _ k + 1 / 2 \\lambda _ k \\not \\equiv \\ell _ k ( 2 ) , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} 1 \\ > \\ > 2 \\ > \\ > = \\ > \\ > \\ > \\ > = \\ > \\ > \\sum _ L | X _ L ( \\bar k ) | \\ > \\ > = \\ > \\ > \\sum _ L | L ( \\Z ) _ { \\Z } | , \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} \\begin{aligned} & \\dot X = Y \\\\ & \\dot Y = - \\frac { 1 } { 2 } \\frac { a ' _ 0 ( X ) } { a _ 0 ( X ) } Y ^ 2 - \\frac { 1 } { 2 } \\frac { a _ 1 ' ( X ) } { a _ 0 ( X ) } . \\end{aligned} \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} I _ { \\mu } ^ { \\infty } ( u _ n ^ 1 ) & = \\frac { 1 } { 2 } ( | | u _ n | | _ { H ^ 1 } ^ 2 - | | v _ 0 | | _ { H ^ 1 } ^ 2 ) + \\frac { 1 } { 4 } \\int _ { \\R ^ 3 } \\rho _ { \\infty } \\bar \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) ^ 2 \\\\ & - \\frac { \\mu } { p + 1 } \\left ( | | ( u _ n ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } - | | ( v _ 0 ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } \\right ) + o ( 1 ) . \\\\ \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} c _ 2 ( S , N , F ) = f . \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} h ( x ) = \\begin{cases} - 1 & v ( x ) = 0 \\\\ - \\frac { v ( x ) } { \\lvert v ( x ) \\lvert } & v ( x ) \\neq 0 \\end{cases} \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} \\mathcal J = ( J _ { 1 , 1 } , \\dots , J _ { 1 , N _ 1 } , \\dots , J _ { r , 1 } , \\dots , J _ { r , N _ r } ) \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{gather*} \\partial _ s u ( s , t ) + J _ 1 ( u ( s , t ) ) ( \\partial _ t u ( s , t ) - X _ { H ^ { ( k ) } } ( u ( s , t ) ) ) \\\\ = ( \\phi _ L ^ t ) _ * ( \\partial _ s v ( s , t ) ) + J _ 1 ( u ( s , t ) ) ( ( \\phi _ L ^ t ) _ * ( \\partial _ t v ( s , t ) ) - ( \\phi _ L ^ t ) _ * X _ { K ^ { ( k ) } } ( v ( s , t ) ) ) \\end{gather*}"}
-{"id": "9667.png", "formula": "\\begin{align*} x _ { 1 } ^ { j } - x _ { 2 } ^ { j } = ( x _ { 1 } - x _ { 2 } ) \\sum _ { \\ell = 0 } ^ { j - 1 } x _ { 1 } ^ { j - 1 - \\ell } x _ { 2 } ^ { \\ell } . \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { k - 1 } | g _ { l } | ^ { - 2 - \\delta } \\mathbf { E } [ | S ( l ) | ^ p ; \\tau _ x > l ] \\le C ( 1 + | x | ^ { p - \\gamma } ) . \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb R ^ 3 } F ( v _ n ) d x = 0 . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{gather*} \\tilde { f } _ { t , s } ( x , y ) : = t x ^ { 6 } y + s x y ^ { 1 0 } + x ^ { 7 } + x ^ { 3 } + y ^ { 1 1 } . \\end{gather*}"}
-{"id": "1155.png", "formula": "\\begin{align*} v _ { i , i + k } = \\tau _ b ( T ^ { - 1 } ( \\tau _ b ( w ) ) ) \\equiv _ S ( s _ 1 , m _ i ^ - , . . . , s _ k ) . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\delta _ { \\beta } = \\frac { \\rho ^ 2 } { 1 - 2 \\beta \\sigma ^ 2 } . \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} F _ { m , \\ , j } ( X ) = F _ { m - 1 , \\ , j } ( X ) - F _ { m - 1 , \\ , j } ( X - 1 ) . \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} \\langle \\tilde { J } _ \\mu ^ { ( \\alpha ) } ( X ) , & \\ s _ { \\mu } ( X ) \\rangle = \\ell ! \\alpha ^ \\ell ( ( \\ell + 1 ) \\alpha + ( n - 1 ) ) ( \\alpha + ( n - 2 ) ) \\cdots ( \\alpha + 1 ) \\alpha \\\\ = & \\ell \\cdot \\ell ! ( \\alpha + ( n - \\ell - 2 ) ) \\cdots ( \\alpha + 1 ) \\alpha ^ { \\ell + 2 } + \\ell ! ( \\alpha + ( n - \\ell - 1 ) ) \\cdots ( \\alpha + 1 ) \\alpha ^ { \\ell + 1 } , \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} \\tilde { A } _ { 1 - 2 s } v : = - z ^ { - \\frac { 1 - 2 s } { s } } v '' + A v \\ni 0 . \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} \\tilde { U } = \\pi ^ { - 1 } ( U ) \\subseteq \\pi ^ { - 1 } ( \\ker \\beta _ { m + 1 } ) = V + \\ker B _ { m + 1 } = \\ker B _ { m + 1 } . \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( \\ker \\beta _ { m + 1 } ) = V + \\ker B _ { m + 1 } . \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} \\begin{array} { l l } v _ { q p } ^ * = v _ { s d p } ^ * , & \\mathrm { i f } ~ ~ L \\leq 3 , \\\\ v _ { q p } ^ * \\leq 8 L \\cdot v _ { s d p } ^ * , & \\mathrm { i f } ~ ~ L > 3 . \\end{array} \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} \\partial _ n K ^ + _ \\gamma + \\partial _ n K ^ - _ \\gamma = 0 . \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} c ^ \\omega _ { - \\epsilon t } ( y ) : = \\sum _ { x \\in ( y , u t ] } G _ { ( - \\epsilon t , x ) } ^ { \\omega } ( x - 1 , y ) \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} s & = x _ 1 + x _ 2 + \\dotsb + x _ { n + 1 } . \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} 0 & \\le \\sum _ { x \\in ( y , u t ] } G ^ \\omega _ { ( - \\epsilon t , x ) } ( x - 1 , y ) ( 1 + \\eta ( \\epsilon ) ) ^ { x - y } - c ^ \\omega _ { - \\epsilon t } ( y ) \\\\ & \\le \\sum _ { x \\in ( y , \\infty ) } c _ 1 e ^ { - c _ 2 ( x - y ) } ( ( 1 + \\eta ( \\epsilon ) ) ^ { x - y } - 1 ) \\to 0 . \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} \\sum _ { m \\in \\Z _ { \\neq 0 } } e \\bigg ( \\frac { a m } { \\ell q } \\bigg ) \\frac { A _ { f } ( m , c _ { 2 } , \\ldots , c _ { n - 1 } ) } { \\abs { m } ^ { \\frac { n - 1 } { 2 } } \\prod _ { i = 2 } ^ { n - 1 } \\abs { c _ { i } } ^ { \\frac { i ( n - i ) } { 2 } } } \\phi _ { \\infty } ( \\gamma ) \\prod _ { p \\mid M } \\phi _ { p } ( \\gamma ) . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} { n ; b ; q \\brack a } _ 2 = \\sum _ { i = 0 } ^ { \\lfloor \\frac { n - a } { 2 } \\rfloor } q ^ { i ( i + b ) } { n \\brack i } _ q { n - i \\brack i + a } _ q , \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} i \\partial _ t \\Lambda _ { m } ( \\mu , \\widehat { u } ) = 2 \\Lambda _ { m } \\left ( \\mu D _ m \\cos , \\widehat { u } \\right ) + \\nu \\Lambda _ { m + 1 } ( S _ { m } \\mu , \\widehat { u } ) . \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} 1 0 \\sum _ { k = k _ 0 } ^ \\infty \\beta ^ { - \\epsilon _ 2 k } = { 1 0 \\beta ^ { - \\epsilon _ 2 k _ 0 } \\over 1 - \\beta ^ { - \\epsilon _ 2 } } < 1 . \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} h _ { 1 , \\sigma } = \\left \\{ \\begin{array} { c c } k _ { 2 , \\sigma } \\ & \\mbox { i f } \\ \\sigma = \\hat { \\sigma } \\\\ k _ { 1 , \\sigma } \\ & \\mbox { o t h e r w i s e } \\end{array} \\right . , \\ h _ { 2 , \\sigma } = \\left \\{ \\begin{array} { c c } k _ { 1 , \\sigma } \\ & \\mbox { i f } \\ \\sigma = \\hat { \\sigma } \\\\ k _ { 2 , \\sigma } \\ & \\mbox { o t h e r w i s e } \\end{array} \\right . \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} K _ { \\nu , 0 } = \\frac { 1 } { 2 } ( - \\partial _ p ^ 2 + p ^ 2 ) + \\sqrt { \\nu } \\Big ( p \\partial _ q + q \\partial _ p \\Big ) = O _ p + \\sqrt { \\nu } X _ { 0 } ~ . \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} \\big | \\partial _ \\xi \\Psi [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] [ \\bar { \\xi } ] ( x , t ) \\big | \\lesssim \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\| \\bar { \\xi } ( t ) \\| _ { 1 + \\sigma } \\left ( \\sum _ { j = 1 } ^ k \\frac { \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } - 1 } ( t ) } { 1 + | y _ j | ^ { a - 2 s } } \\right ) , \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} \\{ x _ { i } , x _ { j } \\} & = 0 & \\{ \\xi _ { i } , \\xi _ { j } \\} & = 0 & \\{ x _ { i } , \\xi _ { j } \\} & = \\delta _ { i j } . \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} \\phi ' ( \\cdot , \\hat { x } ) = \\frac { 1 } { 1 + \\lambda } . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\sup _ { n \\ge 1 } \\frac { \\ , 1 \\ , } { \\ , \\sqrt { n } \\ , } \\sum _ { i = 1 } ^ { n } \\mathbb E [ \\sup _ { 0 \\le s \\le t } \\lvert X _ { s , i } ^ { ( u ) } - \\overline { X } _ { s , i } \\rvert ] < \\infty \\ , . \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} \\zeta _ { f , g } ( z ) = \\exp \\left ( \\sum _ { n \\geq 1 } \\frac { z ^ n } { n } \\sum _ { x \\in \\mathrm { F i x } f ^ n } \\prod _ { m = 1 } ^ { n } g ( f ^ m x ) \\right ) \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{gather*} { { \\phi _ 1 } ^ { - 1 } _ * } _ { \\phi _ 1 ( x ) } \\left ( R _ { \\phi _ 1 ( x ) } \\right ) = { { \\phi _ 2 } ^ { - 1 } _ * } _ { \\phi _ 2 ( x ) } ( R _ { \\phi _ 2 ( x ) } ) \\end{gather*}"}
-{"id": "3115.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = s } \\varphi _ t ^ * \\o _ t & = \\frac { d } { d t } \\Big | _ { t = s } \\varphi _ t ^ * \\o _ s + \\varphi _ s ^ * \\left ( \\frac { d } { d t } \\Big | _ { t = s } \\o _ t \\right ) = \\varphi _ s ^ * ( \\mathcal { L } _ { X _ s } \\o _ s ) + \\varphi _ s ^ * ( d \\mu ) \\\\ & = \\varphi _ s ^ * \\left ( d ( \\iota _ { X _ s } \\o _ s ) + \\iota _ { X _ s } d \\o _ s \\right ) + \\varphi _ s ^ * ( d \\mu ) = - \\varphi _ s ^ * ( d \\mu ) + \\varphi _ s ^ * ( d \\mu ) = 0 , \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} B _ { 0 + } : = ( B _ 0 , B _ 1 , \\ldots , B _ { m ^ + } ) \\colon H \\to H _ 0 \\times H _ 1 \\times \\cdots \\times H _ { m ^ + } \\textup { i s a l i n e a r i s o m o r p h i s m . } \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} = \\int \\limits _ t ^ T \\left ( { \\bf 1 } _ { \\{ \\tau < s \\} } - \\sum _ { j = 0 } ^ { q } \\int \\limits _ t ^ s \\phi _ j ( \\tau ) d \\tau \\cdot \\phi _ j ( \\tau ) \\right ) ^ 2 d \\tau \\to 0 \\ \\ \\ \\hbox { i f } \\ q \\to \\infty . \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} V = B ^ * \\left ( \\begin{array} { c c } \\eta _ 1 & 0 \\\\ 0 & \\eta _ 2 \\end{array} \\right ) U \\left ( \\begin{array} { c c } \\eta _ 1 & 0 \\\\ 0 & \\eta _ 2 \\end{array} \\right ) B = v ^ * U v , \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ r \\prod _ { l = 1 } ^ { N _ k } e ^ { \\mathbf { a } _ k ( l ) \\varpi _ { i _ { k , l } } } ( b _ { k , l } ) = \\prod _ { s = 1 } ^ n \\prod _ { j = 1 } ^ r e ^ { d _ { j , s } \\varpi _ s } ( b _ { k ( j , s ) , m ( j , s ) } ) . \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} p ( \\delta _ { \\omega } ( \\mu ) ) = \\abs { \\mu } ^ M p ( \\omega ) = \\abs { \\mu } ^ M p ( \\delta _ z ( 1 / \\abs { z } ) ) = \\left ( \\frac { \\mu } { \\abs { z } } \\right ) ^ M p ( z ) , \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} T = \\bigl ( ( 5 , 4 , 3 ) , ( 5 , 4 ) , ( 5 , 4 , 3 , 2 ) , ( 5 , 4 , 3 , 2 , 1 ) \\bigr ) \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} \\begin{bmatrix} a \\\\ n \\end{bmatrix} _ i : = \\frac { [ a ] _ i [ a - 1 ] _ i \\cdots [ a - n + 1 ] _ i } { [ n ] _ i [ n - 1 ] _ i \\cdots [ 1 ] _ i } \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} \\mathbf { x } ^ { k + 1 } : = ( \\mathbf { W } \\otimes \\mathbf { I } _ p ) \\mathbf { x } ^ { k } - \\xi \\nabla f ( \\mathbf { x } ^ { k + 1 } ) + \\sum _ { t = 0 } ^ { k } ( \\mathbf { W } - \\tilde { \\mathbf { W } } ) \\otimes \\mathbf { I } _ p \\mathbf { x } ^ t , \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} \\rho = \\rho _ \\lambda \\longrightarrow + \\infty , \\lambda \\searrow 0 , \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} \\partial _ { \\otimes \\tau } ( m \\otimes c ) = d _ M m \\otimes c + ( - 1 ) ^ { | m | } m \\otimes d _ C c + \\sum _ { ( c ) } ( - 1 ) ^ { | m | } ( m \\cdot \\tau ( c ' ) ) \\otimes c '' \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} \\varphi _ { 0 , m } - h | _ X = \\varphi _ { 0 , m } - ( f | _ X - g | _ X ) = \\varphi _ { 0 , m } - \\tau + \\sigma \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} \\sup _ { s \\le T } \\mathbb { E } \\left \\lvert X _ { s \\wedge \\tau ( R ) } ^ { \\omega ^ \\prime } - Y _ { s \\wedge \\tau ( R ) } ^ { \\omega ^ \\prime } \\right \\rvert ^ 2 = 0 , \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} u ( s , x _ 0 ) & = \\mathbb E \\bigg [ \\Phi ( x _ 0 + W _ { T - s } ) \\\\ & + \\int _ s ^ T f ( r , W _ r + x _ 0 , u ( r , W _ r + x _ 0 ) , \\nabla u ( r , W _ r + x _ 0 ) ) \\mathrm d r \\\\ & + A _ T ^ { W , W } ( ( \\nabla u ^ * b ) ( x _ 0 + \\cdot ) ) - A _ s ^ { W , W } ( ( \\nabla u ^ * b ) ( x _ 0 + \\cdot ) ) \\bigg ] \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} \\widetilde { S } ^ { R , R } : = \\frac { 1 } { \\sqrt { \\dim ^ R ( \\bar { \\ 1 } ) } \\sqrt { \\dim ( \\mathcal { C } ) } } S ^ { R , R } . \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c c c c c } \\frac { 1 } { 2 } \\alpha _ n & + & \\frac { 1 } { 4 } \\gamma _ n & - & \\frac { 1 } { p + 1 } \\delta _ n & = & c _ { \\mu _ n } , \\\\ \\alpha _ n & + & \\gamma _ n & - & \\delta _ n & = & 0 , \\\\ \\frac { 1 } { 2 } \\alpha _ n & + & \\left ( \\frac { 5 + 2 k } { 4 } \\right ) \\gamma _ n & - & \\frac { 3 } { p + 1 } \\delta _ n & \\leq & 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\prod _ { \\alpha \\in \\Phi ^ + } \\xi ^ { \\langle \\alpha , \\rho \\rangle } - \\xi ^ { - \\langle \\alpha , \\rho \\rangle } = \\xi ^ { - 2 \\langle \\rho , \\rho \\rangle } \\sum _ { w \\in \\mathfrak { S } _ { n + 1 } } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho , w \\bullet 0 \\rangle } . \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} \\| \\vartheta \\| _ { - 1 , \\pi } ^ 2 : = \\sup _ { g \\in H _ \\pi ^ 1 ( \\Lambda ; \\mathbb R ) } \\bigl ( 2 \\langle \\vartheta , g \\rangle - \\| g \\| _ { 1 , \\pi } ^ 2 \\bigr ) , \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} \\hat { \\mu } ^ \\psi _ { \\Lambda , \\epsilon } ( d \\phi , d t ) : = \\frac { 1 } { Z ^ \\psi _ { \\Lambda , \\epsilon } ( \\alpha ) } e ^ { - A ^ \\psi ( \\phi , t ) } \\prod _ { j k \\in E ( \\Lambda ) } \\Big ( f _ \\alpha ( e ^ { t _ { j k } } ) e ^ { t _ { j k } } d t _ { j k } \\Big ) \\prod _ { j \\in \\Lambda } d \\phi _ j , \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} \\lambda _ n ( L _ { t ^ { - 1 } } f ) = \\int _ K L _ { t ^ { - 1 } } f ( n k n ^ { - 1 } ) d k = \\int _ K f ( t n k n ^ { - 1 } ) d k = \\int _ K f ( n k n ^ { - 1 } ) d k . \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 t ^ n v ( t ) \\ , d t & = \\int _ 0 ^ r t ^ n v ( t ) \\ , d t + \\int _ r ^ 1 t ^ n v ( t ) \\ , d t \\\\ & \\le r ^ n \\int _ 0 ^ 1 v ( t ) \\ , d t + \\sup _ { x \\in [ r , 1 ] } \\frac { v ( x ) } { w ( x ) } \\int _ r ^ 1 t ^ n w ( t ) \\ , d t . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} S = \\{ \\tilde { e } _ i ^ n ( b ) \\ | \\ n \\geq 0 \\} \\cup \\{ \\tilde { f } _ i ^ n ( b ) \\ | \\ n \\geq 0 \\} \\setminus \\{ 0 \\} \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} \\kappa ^ { A } ( \\theta ) + \\kappa ^ { - S } ( \\theta ) = 0 , \\ \\theta > 0 , \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} \\exp \\left ( { - \\big ( 1 + \\beta x ^ 2 \\big ) ^ \\alpha } \\right ) = \\int _ 0 ^ \\infty \\exp { \\left ( - \\frac { 1 } { 2 } \\omega x ^ 2 \\right ) } \\rho ( \\omega ) d \\omega . \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\partial _ t u - D \\Delta u & = \\lambda ( N , F , S ) \\ , u & & [ 0 , T ] \\times \\Omega , \\\\ \\partial _ \\nu u & = 0 & & \\ [ 0 , T ] \\times \\partial \\Omega , \\\\ u ( 0 ) & = { u _ { i n } } & & \\ \\Omega , \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} ( b \\cdot f ) ( p _ 1 , \\dots , p _ r ) & \\coloneqq f ( b ^ { - 1 } p _ 1 , p _ 2 , \\dots , p _ r ) , \\\\ ( b \\cdot \\tilde { f } ) ( ( p _ { k , l } ) _ { k , l } ) & \\coloneqq \\tilde { f } ( b ^ { - 1 } p _ { 1 , 1 } , p _ { 1 , 2 } , \\dots , p _ { 1 , N _ 1 } , \\dots , p _ { r , N _ r } ) . \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} ( z , w _ 1 , \\ldots , w _ n , y ) \\mapsto \\left ( z , \\sum _ { p = 1 } ^ n \\left ( w _ p + \\frac { i } { 2 } y \\bar { z } _ p \\right ) d z _ p , y \\right ) . \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} P _ { \\phi } ( z _ 1 , \\dots , z _ n ) : = \\exp _ { \\varphi } \\bigg ( \\sum _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) } { a } \\bigg ) . \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} ( T _ j h ) : = h \\circ \\tau _ { \\hat { e } _ j } , j = 1 , . . . , d . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\frac 1 { s _ l ^ 2 } \\max _ { A \\in \\alpha _ l } \\sum _ { 1 \\leqslant i \\leqslant k _ l } \\int _ A ( \\omega _ { a _ i } ^ { n _ l } ( h _ l , \\epsilon ^ * , p ) ) ^ 2 d \\mu _ A ( p ) = 0 , \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} - \\Delta u + u = u ^ { p } , x \\in \\R ^ 3 , \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} \\dot { \\lambda } _ j + \\frac { 1 } { t } \\left ( P ^ T d i a g \\left ( \\frac { ( 2 s - 1 ) \\bar { \\sigma } _ r b _ r ^ { 2 - 2 s } + 1 } { n - 4 s } \\right ) P \\lambda \\right ) _ j = \\Pi _ 1 [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ( t ) \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} \\mathbb { S } _ { f , g } = \\frac { ( - 1 ) ^ { \\sum _ { i = 0 } ^ { n - 1 } ( k _ i ( f ) + k _ i ( g ) ) } } { d ^ n } \\prod _ { i = 0 } ^ { n - 1 } \\zeta ^ { k _ i ( f ) k _ i ( \\sigma \\cdot g ) } \\sum _ { \\sigma \\in \\mathfrak { S } _ { n + 1 } } \\varepsilon ( \\sigma ) \\prod _ { i = 1 } ^ { n + 1 } \\zeta ^ { - f ( i ) ( \\sigma \\cdot g ) ( i ) } , \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} 0 \\rightarrow \\mathrm { H o m } _ A ( T \\oplus A , & \\ \\mathrm { r a d } ( P _ i ) ) \\xrightarrow { f _ i } \\mathrm { H o m } _ A ( T \\oplus A , P _ i ) \\rightarrow \\mathrm { H o m } _ A ( T \\oplus A , S _ i ) \\rightarrow \\\\ & \\rightarrow \\mathrm { E x t } ^ 1 _ A ( T \\oplus A , \\mathrm { r a d } ( P _ i ) ) \\rightarrow \\mathrm { E x t } ^ 1 _ A ( T \\oplus A , P _ i ) = 0 . \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} \\begin{cases} w _ { t } ^ { \\varepsilon } + g ( t , x , w ^ { \\varepsilon } ) _ { x } ~ = ~ \\varepsilon w ^ { \\varepsilon } _ { x x } \\ , , \\\\ w ^ { \\varepsilon } ( 0 , x ) ~ = ~ 0 , \\end{cases} \\end{align*}"}
-{"id": "9900.png", "formula": "\\begin{align*} \\widehat { \\boldsymbol { D } ^ { - s } u } = ( 2 \\pi i \\xi ) ^ { - s } \\widehat { u } \\ , \\ , , \\widehat { \\boldsymbol { D } ^ { s * } u } = ( - 2 \\pi i \\xi ) ^ { - s } \\widehat { u } \\ , \\ , \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} a _ 0 - b _ 0 = c _ 0 - d _ 0 \\qquad \\qquad \\epsilon _ 0 = \\zeta _ 0 . \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} \\liminf _ { r \\to 0 } \\frac { \\int _ { [ - r , r ] } x ^ 2 \\ , \\nu ( d x ) } { r ^ 2 \\log ( \\frac { 1 } { r } ) } = : C > \\frac { 1 } { 2 p } \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| v ( t ) - e ^ { - i t H } v _ + \\| _ { H ^ 1 } = 0 . \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} \\begin{cases} u _ t + f ^ { \\nu } \\left ( t , x , u \\right ) _ x ~ = ~ \\varepsilon u _ { x x } \\ , , \\\\ u ( 0 ) = u _ 0 ^ { \\nu } \\ , , \\end{cases} \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\| \\nabla V ^ { j _ 0 } \\| ^ 2 _ { L ^ 2 } - \\frac { 1 } { \\alpha + 2 } \\| V ^ { j _ 0 } \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } = d _ M . \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} u ( m a - 1 , a - 1 ) = u ( a ^ 2 - 1 , m - 1 ) . \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} \\sigma ( \\oplus _ { n = 1 } ^ \\infty t _ n ) = \\oplus _ { n = 1 } ^ \\infty P _ n \\sigma _ n ( t _ n ) P _ n \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} A _ 1 \\odot A _ 2 : = \\{ \\gamma _ 1 \\gamma _ 2 : \\gamma _ 1 \\in A _ 1 , \\gamma _ 2 \\in A _ 2 \\} . \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} Q ( L _ n f ) = L _ n Q ( f ) , f \\in C _ c ( G ) . \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} x ^ n + y ^ n = z ^ n , \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} F _ { i _ 1 } : = \\mathbb { F } _ p ( \\{ \\alpha _ j : j \\neq i _ 1 \\} ) . \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} E _ { \\bullet } = ( E _ 1 , E _ 2 , \\ldots , E _ k ) . \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} - \\frac { T - t } { \\pi \\sqrt { 2 } } \\zeta _ { 0 } ^ { ( i _ 1 ) } \\sum _ { r = q + 1 } ^ { \\infty } \\frac { 1 } { r } \\zeta _ { 2 r - 1 } ^ { ( i _ 2 ) } . \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} \\| f \\| _ p = \\inf _ { g \\ge 0 } \\frac { \\int f g } { \\| g \\| _ { p ' } } . \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} U = e ^ F , \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} J ( \\tau ) = B \\cap ( A \\cup C ) \\cap ( A \\cup D ) = B \\cap ( A \\cup ( C \\cap D ) ) = B \\cap A , \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} ( u _ n ) _ + ^ p - ( u _ n - v _ 0 ) _ + ^ p - ( v _ 0 ) ^ p = o ( 1 ) , \\textrm { i n } \\ , \\ , H ^ { - 1 } ( \\R ^ 3 ) . \\\\ \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { ( - 1 ) ^ { n s + j } \\binom k j \\binom { k - j } s \\frac { { G _ 0 ^ j G _ n ^ s } } { { G _ r ^ { j + s } } } H _ { m - n k + r j + ( n + r ) s } } } = ( - 1 ) ^ { n k } \\left ( { \\frac { { G _ { n + r } } } { { G _ r } } } \\right ) ^ k H _ m , G _ r \\ne 0 \\ , , \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} D _ t F ( t , \\xi _ t ( x ) ) = \\lim _ { \\epsilon \\rightarrow 0 } \\frac { 1 } { \\epsilon } E _ t [ F ( t + \\epsilon , \\xi _ { t + \\epsilon } ( x ) ) - F ( t , \\xi _ t ( x ) ) ] \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} \\widetilde { T } _ V f ( \\gamma , \\zeta ) = \\int _ { \\R } f ( t , \\zeta ) \\frac { \\gamma ^ { 2 i \\pi t } } { \\gamma } \\d t , \\gamma \\in V _ { \\zeta } \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\frac { \\dim ( \\mathbb { Z } ( \\mathcal { T } _ \\xi ) \\rtimes \\mathcal { S } ) } { n + 1 } = d ^ { 2 n } ( - 1 ) ^ { \\lvert \\Phi ^ + \\rvert } \\xi ^ { - \\langle 2 \\rho , 2 \\rho \\rangle } \\left ( \\sum _ { w \\in \\mathfrak { S } _ { n + 1 } } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho , w \\bullet 0 \\rangle } \\right ) ^ { - 2 } . \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} \\gamma _ i = \\{ ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) \\in A ^ 4 \\mid ( x _ 1 , x _ 2 ) \\in \\alpha _ { i } , \\big ( d ( x _ 1 , x _ 2 , x _ 3 ) , x _ 4 \\big ) \\in \\alpha _ { i - 1 } \\} . \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} \\| O _ p u \\| ^ 2 _ { L ^ 2 } & \\le \\frac { 1 } { L ^ 2 } \\int _ 0 ^ L e ^ { 4 s } d s = \\frac { 1 } { 4 L ^ 2 } ( e ^ { 4 L } - 1 ) \\\\ & \\le \\frac { 1 } { L ^ 2 } e ^ { 4 L } { ~ . } \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} \\Psi _ { j } ( k , t , y , s ) & = U ^ { \\theta } ( k , t ) - u _ { \\theta } ( y , s ) - \\frac { ( - k \\Delta x - y ) ^ { 2 } } { 2 \\eta } - \\frac { ( t - s ) ^ { 2 } } { 2 \\eta } - b t \\\\ & \\quad - \\sqrt { \\theta } ( T + 1 - t ) ^ { - 1 } + g _ { \\eta , \\nu } ( - \\nu y ) - \\alpha ( k \\Delta x ) ^ { 2 } . \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} a = \\sqrt { \\frac { u _ \\alpha \\bar u ^ \\alpha } 2 } \\ , , \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} \\varepsilon ( s , \\pi _ { v } , \\psi _ { v } ) = \\varepsilon ( 1 / 2 , \\pi _ { v } , \\psi _ { v } ) \\ , q _ { v } ^ { a ( \\pi _ { v } ) ( \\frac { 1 } { 2 } - s ) } \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\int _ { D } p q \\phi \\ { \\rm d } x = \\int _ { D } ( v - w ) q \\ { \\rm d } x = \\int _ { D } ( \\Pi _ { k , D } ^ 0 v - w ) q \\ { \\rm d } x , \\ \\forall q \\in \\mathbb { P } _ { k } . \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} \\int _ X p _ n ( x ) p _ m ( x ) \\ , d \\mu ( x ) = \\delta _ { n m } , \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\int _ 0 ^ t e ^ { - i ( t - s ) H } P _ c F ( s ) \\ , d s & = \\int _ \\R e ^ { - i ( t - s ) H } P _ c F ( s ) \\ , d s \\\\ & + \\int _ { 0 } ^ \\infty e ^ { - i ( t - s ) H } P _ c F ( s ) \\ , d s \\\\ & - \\int _ { - \\infty } ^ 0 e ^ { - i ( t - s ) H } P _ c F ( s ) \\ , d s . \\end{aligned} \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} A _ { s - 3 , 4 } = a _ { 0 } b _ { 0 } c _ { 0 } , \\ \\ \\ A _ { s - 3 , 3 } = \\sum _ { r = 1 } ^ { n } \\frac { S _ { 0 0 r } - a _ { r } b _ { r } c _ { r } } { r } , \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} \\rho _ t ( y ) = \\frac { 1 } { ( 2 \\pi \\tau _ \\alpha ( t ) ) ^ { \\frac { n } { 2 } } } \\int _ { \\R ^ n } \\rho _ 0 ( x ) e ^ { - \\frac { \\| y - e ^ { - \\alpha t } x \\| ^ 2 } { 2 \\tau _ \\alpha ( t ) } } \\ , d x \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} \\beta ( \\theta ) = \\log \\left ( \\lim _ { n \\to \\infty } | q _ n | ^ { 1 / n } \\right ) = - \\log \\rho \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} \\dot { x } _ v = \\frac { 1 } { m ( v ) } \\sum _ { v ' \\in V } w ( v , v ' ) ( x _ { v ' } - x _ v ) , \\ \\ \\ \\ \\ v \\in V . \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} R ^ { ( i ) } _ { } ( D ) = \\begin{cases} \\frac { 1 } { 2 } \\log \\frac { 1 - \\rho ^ 2 } { D } & 0 < D \\le 1 - \\rho ^ 2 , \\\\ 0 & D > 1 - \\rho ^ 2 . \\end{cases} \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } f ( z ^ k , x ^ k ) = \\lim _ { k \\to \\infty } [ \\sigma _ k \\| w ^ k \\| ] \\| w ^ k \\| = 0 . \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} H ( U , V ) = \\frac { 1 } { 2 } [ T ( U , V ) + T ^ { ' } ( U , V ) + T ^ { ' } ( V , U ) ] , \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} \\Phi ^ + = \\left \\{ s _ { i _ 1 } s _ { i _ 2 } \\cdots s _ { i _ k } ( \\alpha _ { i _ { k + 1 } } ) \\ \\middle \\vert \\ 0 \\leq k \\leq r - 1 \\right \\} . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} v \\lambda & = \\lfloor v \\rfloor ^ \\lambda \\lambda \\\\ & = s _ { i _ 1 } \\cdots s _ { i _ { k - 1 } } s _ { i _ k } s _ { i _ { k + 1 } } \\cdots s _ { i _ l } \\lambda \\\\ & = s _ { i _ 1 } \\cdots s _ { i _ { k - 1 } } s _ { i _ { k + 1 } } \\cdots s _ { i _ l } \\lambda , \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} \\sigma ( z ) = ( - z ) ^ { \\beta } \\tau ( z ) . \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} \\sigma ^ { \\pm } = \\sigma _ { B \\pm \\varepsilon _ B , \\omega \\pm \\varepsilon _ { \\omega } } \\in U ( X ) \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} K _ { F } ( y ) = ( 1 + y ) ^ { \\dim F } \\sum _ { k = 0 } ^ { m _ F } \\frac { 1 } { \\dim F - k + 1 } \\binom { \\dim F } { k } \\binom { 2 \\dim F - 2 k } { \\dim F } ( - y - 1 ) ^ k . \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} R _ \\Gamma ( u , \\lambda ) = 0 . \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} \\phi _ m ( z , s ) : = \\left \\{ \\begin{array} { l l } 2 \\pi | m | ^ { \\frac { 1 } { 2 } } y ^ { \\frac { 1 } { 2 } } I _ { s - \\frac { 1 } { 2 } } ( 2 \\pi | m | y ) e ^ { 2 \\pi i m x } & m \\neq 0 , \\\\ y ^ s & m = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} \\dot { p } _ { 1 , \\tau } = \\{ p _ { 1 , \\tau } , { H _ { \\tau } } _ T \\} _ { P B } = - \\lambda m \\omega ^ 2 ( t _ \\tau ) f ^ { - 1 } ( t _ \\tau ) x _ { 1 , \\tau } ; \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} \\mu _ p ( 1 + \\varpi _ p ^ { \\kappa } x ) = \\psi _ p \\left ( \\frac { \\alpha _ { \\mu _ p } } { \\varpi _ p ^ n } \\log _ p ( 1 + \\varpi _ p ^ { \\kappa } x ) \\right ) \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} & W _ { 0 } ( - e ^ { - 1 } ) = - 1 , \\qquad \\qquad \\lim \\limits _ { x \\to \\infty } W _ { 0 } ( x ) = \\infty , \\\\ & W _ { - 1 } ( - e ^ { - 1 } ) = - 1 \\lim \\limits _ { x \\to 0 ^ { - } } W _ { - 1 } ( x ) = - \\infty . \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} z \\mapsto - \\frac { i } { 2 } \\sum _ { p = 1 } ^ n \\bar { z } _ p \\ , d z _ p . \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} \\overline { \\left ( \\frac { \\widetilde { S } ^ { R , R } _ { X , Y } } { \\widetilde { S } ^ { R , R } _ { 1 , Y } } \\right ) } = \\frac { \\widetilde { S } ^ { R , R } _ { X ^ * , Y } } { \\widetilde { S } ^ { R , R } _ { 1 , Y } } \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} f ( a _ 3 ) = ( 1 - t _ { \\kappa ( a _ 2 ) } ) \\cdot f ( a _ 1 ) + t _ { \\kappa ( a _ 1 ) } \\cdot f ( a _ 2 ) . \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 0 + } v _ { \\lambda } ( z ) = v _ { \\delta } ( z ) \\qquad \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} \\left ( b ^ { ( 2 ) } _ i \\right ) _ { i = 1 } ^ { 1 3 } = \\left ( \\frac { 7 } { 2 0 } , \\frac { 2 4 } { 3 5 } , \\frac { 2 7 5 } { 3 3 6 } , \\frac { 7 2 8 } { 8 2 5 } , \\frac { 5 2 5 } { 5 7 2 } , \\frac { 2 9 9 2 } { 3 1 8 5 } , \\frac { 5 1 8 7 } { 5 4 4 0 } , \\frac { 2 8 0 0 } { 2 9 0 7 } , \\frac { 1 2 9 0 3 } { 1 3 3 0 0 } , \\frac { 1 9 0 0 0 } { 1 9 4 8 1 } , \\frac { 9 0 0 9 } { 9 2 0 0 } , \\frac { 3 7 3 5 2 } { 3 8 0 2 5 } , \\frac { 5 0 3 7 5 } { 5 1 1 5 6 } \\right ) \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} C _ 1 : = n | R ^ N | _ { L ^ \\infty } + \\frac { n | E | ^ 2 _ { L ^ \\infty } } { 4 \\delta _ 1 } , C _ 2 : = \\frac { n | F | ^ 2 _ { L ^ \\infty } } { 4 \\delta _ 2 } . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} \\mathrm { G } = \\sum _ { i = 1 } ^ n \\partial ^ 2 _ { x _ i } - n \\sum _ { i = 1 } ^ n x _ i \\partial _ { x _ i } + \\beta \\sum _ { i \\neq j } \\frac { 1 } { x _ i - x _ j } \\partial _ { x _ i } , \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} \\kappa ^ A ( \\theta ) + \\kappa ^ { - S } ( \\theta ) = 0 . \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} & \\left | N ( f , r ) - { q \\choose r } q ^ { k - r } \\displaystyle { \\left ( \\sum _ { j = 0 } ^ { d - r } ( - 1 ) ^ { j } { q - r \\choose j } q ^ { - j } \\right ) } \\right | \\\\ & \\leq \\sum _ { j = k + 1 } ^ d { j \\choose r } { \\frac { q } p + m \\sqrt { q } + j \\choose j } { m - 1 \\choose d - j } \\sqrt { q } ^ { d - j } . \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} E ( n ) _ { z _ 0 } & = \\{ k _ { ( t , a ) } \\mid a ^ { n + 1 } = 1 \\} \\simeq \\Z _ n , \\\\ \\widetilde { E ( n ) } _ { z _ 0 } & = \\left \\{ \\left . ( 1 , \\ldots , 1 , \\frac { k } { n + 1 } ) \\ , \\right | \\ , k \\in \\Z \\right \\} \\cong \\Z . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} S = - \\nabla \\cdot w \\quad \\mbox { i n } \\Omega , w \\cdot \\nu = 0 \\quad \\mbox { o n } \\partial \\Omega , \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\big ( \\mathrm M _ { t , n } , 0 \\le t \\le T \\big ) \\ , = \\ , \\delta _ { ( \\mathrm M _ { t } , 0 \\le t \\le T ) } \\mathcal M ( \\Omega _ { 2 } ) \\ , . \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} \\begin{array} { l l } { \\min } & f ( x ) \\\\ s . t . & C ( x ) = 0 , \\\\ & x \\in \\Omega , \\end{array} \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} N _ { i n } = \\int _ { \\Omega } u _ { i n } \\ , d x > 0 . \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ n { \\mu _ \\ell } ^ r x _ \\ell = \\sum _ { i = 1 } ^ t \\sum _ { j = 0 } ^ m c _ { i , j } \\binom { r } { j } \\mu _ i ^ { r - j } + \\sum _ { \\ell > s } \\mu _ \\ell ^ r x _ \\ell + C _ m , \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} \\mathrm { t r } ( P _ { N } ) = r ^ { N / 2 } \\ , U _ { N } \\bigl ( \\frac { n } { 2 \\sqrt { r } } \\bigr ) . \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} \\mathbb { B ' } _ R & = \\begin{pmatrix} 0 & [ b , a ] \\\\ [ c , d ] & [ d , a ] - [ c , b ] \\end{pmatrix} ( \\Delta ' ) ^ { - 1 } . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} d _ { m i n } ( \\Lambda ) = \\min _ { x \\ne x ' } \\| x - x ' \\| = 2 \\rho , \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} X _ { \\langle F z , \\bar { z } \\rangle } = ( 0 , - ( \\sigma _ { j _ b } \\partial _ { x _ b } \\langle F z , \\bar { z } \\rangle ) _ { 1 \\leq b \\leq n } , - \\mathbf { i } ( \\sigma _ j \\partial _ { \\bar { z } _ j } \\langle F z , \\bar { z } \\rangle ) _ { j \\in \\mathbb { Z } _ * } , \\mathbf { i } ( \\sigma _ j \\partial _ { z _ j } \\langle F z , \\bar { z } \\rangle ) _ { j \\in \\mathbb { Z } _ * } ) ^ T \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma _ { + } ( z ) & = ( 1 - z ) ^ { \\beta } \\tau _ { + } ( z ) \\\\ \\sigma _ { - } ( z ) & = ( 1 - z ^ { - 1 } ) ^ { - \\beta } \\tau _ { - } ( z ) \\end{aligned} \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} L ( \\phi ^ { \\vee } , s - 1 ) = \\sum _ { a \\in A _ { + } } \\frac { \\mu ( a ) } { a ^ s } , \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} & \\mathcal { B } _ { \\mathcal { I } , \\lambda _ 1 , \\ldots , \\lambda _ r } \\\\ & = \\left \\{ \\tilde { f } _ { i _ { 1 , 1 } } ^ { x _ 1 } \\cdots \\tilde { f } _ { i _ { 1 , N _ 1 } } ^ { x _ { N _ 1 } } ( b _ { \\lambda _ 1 } \\otimes b ) ~ \\Big | ~ x _ 1 , \\ldots , x _ { N _ 1 } \\in \\mathbb { Z } _ { \\ge 0 } , \\ b \\in \\mathcal { B } _ { ( I _ 2 , \\ldots , I _ r ) , \\lambda _ 2 , \\ldots , \\lambda _ r } \\right \\} \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} x _ { n + 1 } ^ { ( i ) } = ( 1 - \\alpha ) T _ N ( x _ n ^ { ( i ) } ) + \\frac { 1 } { 2 } \\alpha ( T _ N ( x _ n ^ { ( i - 1 ) } ) + T _ N ( x _ n ^ { ( i + 1 ) } ) ) , \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} - \\nabla \\cdot ( \\mathbb { P } [ \\hat { C } ] \\nabla p ) = S , \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\tilde { j } ( v _ 1 ( 0 ) - \\varphi ) + \\frac { 1 } { 2 } \\tilde { j } ( v _ 2 ( 0 ) - \\varphi ) & > \\tilde { j } \\Big ( \\frac { v _ 1 ( 0 ) + v _ 2 ( 0 ) } { 2 } - \\varphi \\Big ) \\\\ & \\ge \\tilde { j } ( v _ 1 ( 0 ) - \\varphi ) + \\Big ( v _ 1 ' ( 0 ) , \\frac { v _ 2 ( 0 ) - v _ 1 ( 0 ) } { 2 } \\Big ) \\\\ & = \\frac { 1 } { 2 } \\tilde { j } ( v _ 1 ( 0 ) - \\varphi ) + \\frac { 1 } { 2 } \\tilde { j } ( v _ 2 ( 0 ) - \\varphi ) , \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} \\tilde { f } _ i ^ a ( \\pi _ 1 \\otimes \\pi _ 2 ) = \\tilde { f } _ i ^ M ( \\pi _ 1 ) \\otimes \\tilde { f } _ i ^ { a - M } ( \\pi _ 2 ) \\in \\mathcal { B } _ { s _ i v } ( \\lambda ) \\otimes \\mathcal { B } _ w ( \\mu ) . \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ { \\lambda } ( x ) : = \\lambda ^ { - \\frac { \\theta - 1 } { \\theta } } u ( x _ { \\lambda } ) , \\\\ & x _ { \\lambda } : = \\left [ \\lambda | x | ^ { - \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } } + ( 1 - \\lambda ) R ^ { - \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } } \\right ] ^ { - \\frac { p } { n - p } \\frac { \\theta - 1 } { \\theta } } \\frac { x } { | x | } = R \\exp _ { q } \\left [ \\lambda \\log _ { q } \\frac { | x | } { R } \\right ] \\frac { x } { | x | } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} & \\left ( \\prod _ { k = 1 } ^ r \\prod _ { l = 1 } ^ { N _ k } e ^ { \\mathbf { a } _ { k } ( l ) \\varpi _ { i _ { k , l } } } ( b _ { k , l } ) \\right ) \\left ( \\prod _ { s = 1 } ^ n \\prod _ { j = 1 } ^ r e ^ { d _ { j , s } \\varpi _ s } ( \\zeta ( j , s ) ) ^ { - 1 } \\right ) \\\\ & = \\left ( \\prod _ { j = 1 } ^ r \\prod _ { s = 1 } ^ n e ^ { d _ { j , s } \\varpi _ s } ( \\zeta ( j , s ) ' ) ^ { - 1 } \\right ) \\left ( \\prod _ { k = 1 } ^ r e ^ { \\lambda _ k } ( b _ { k , N _ k } ) \\right ) . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} | k _ b j _ b + \\frac { j } { n - \\frac 1 2 } | < \\frac { | j | } { 5 0 n \\sum _ { b = 1 } ^ n | j _ b | } , b = 1 , \\cdots , n . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} \\begin{aligned} & \\mu _ 0 ^ { - ( n - 2 s - 1 ) } ( t ) \\int _ { \\mathbb { R } ^ n } E _ { 0 j } ( y , t ) Z _ { n + 1 } ( y ) d y \\\\ & \\approx c _ 1 \\left [ b _ j ^ { n - 2 s - 1 } H ( q _ j , q _ j ) - \\sum _ { i \\neq j } b _ j ^ { \\frac { n - 2 s } { 2 } - 1 } b _ i ^ { \\frac { n - 2 s } { 2 } } G ( q _ j , q _ i ) \\right ] \\\\ & - \\frac { 2 s c _ 1 A + c _ 2 } { ( n - 4 s ) c _ { n , s } ^ { n - 4 s } } b _ j ^ { 2 s - 1 } \\end{aligned} \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{gather*} a _ { m + n } \\Delta _ g v _ 1 + s _ { g + h } v _ 1 = \\lambda _ { 1 } ^ f ( g _ u ) u ^ { p _ { m + n } - 2 } v _ 1 , \\\\ a _ { m + n } \\Delta _ g v _ 2 + s _ { g + h } v _ 2 = \\lambda _ { 2 } ^ f ( g _ u ) u ^ { p _ { m + n } - 2 } v _ 2 . \\end{gather*}"}
-{"id": "9891.png", "formula": "\\begin{align*} ( \\boldsymbol { D } ^ { - s } u , \\phi ) = ( u , \\boldsymbol { D } ^ { - s * } \\phi ) , \\forall \\phi \\in C _ 0 ^ \\infty ( \\mathbb { R } ) . \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} \\delta _ F ( C _ i ) = \\delta _ { F ' } ( \\tilde { C _ i } ) . \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} \\Gamma ( q / p ' + 2 ) ^ { 1 / q } & = \\left ( ( 1 + o ( 1 ) ) \\sqrt { 2 \\pi \\left ( q / p ' + 1 \\right ) } \\left ( \\frac { q / p ' + 1 } { e } \\right ) ^ { q / p ' + 1 } \\right ) ^ { 1 / q } \\\\ & = ( 1 + o ( 1 ) ) \\left ( \\frac { q } { e p ' } \\right ) ^ { 1 / p ' } . \\end{align*}"}
-{"id": "9948.png", "formula": "\\begin{align*} \\tau ^ { n - 1 , + } = ( 0 , \\ , f \\ , \\kappa ^ { n - 2 } _ { 0 1 } ) . \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} n \\mathrm { W } _ 2 ( L _ A , L _ B ) ^ 2 = \\sum _ { i = 1 } ^ n ( x _ i ( A ) - x _ i ( B ) ) ^ 2 \\leq \\mathrm { T r a c e } ( ( A - B ) ^ 2 ) = \\left \\Vert A - B \\right \\Vert _ { \\mathrm { H S } } ^ 2 . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} V ' ( t ) = \\int _ 0 ^ t f ( t - s ) V ' ( s ) + \\int _ t ^ T g ( t - s ) V ' ( s ) \\ , d s - g ( T - t ) V ( T ) . \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} \\mathcal { H } W _ { l } ( s ; y l ^ c ) = \\begin{cases} \\omega _ { \\pi _ { 0 } , l } ( l ^ { - \\frac { c } { 2 } } ) l ^ { \\frac { c } { 2 } } \\mathcal { L } _ { s , - \\frac { c } { 2 } } ( y b ^ 2 ) & , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} F = \\{ j _ 1 , \\ldots , j _ 1 + ( t - 1 ) , j _ 2 , \\ldots , j _ 2 + ( t - 1 ) , \\ldots , j _ { d - 1 } , \\ldots , j _ { d - 1 } + ( t - 1 ) \\} \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} K ( n , i , j ) = K ( n , n - j - i , j ) . \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} \\chi _ y ( X ) : = \\sum _ { j , p \\ge 0 } ( - 1 ) ^ { j - p } \\dim _ { \\mathbb { C } } \\mathrm { G r } _ F ^ p H _ c ^ j ( X ; \\mathbb { C } ) \\cdot y ^ p , \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} p _ { _ 0 } ( y ) & = \\mathcal { N } ( 0 , P _ x ) , \\\\ p _ { _ 1 } ( y ) & = \\mathcal { N } ( 0 , P _ x + \\sigma _ w ^ 2 ) , \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} \\min \\left \\{ \\min _ { i } F _ { i } ( t _ { \\epsilon } , 0 , u ( 0 , t _ { \\epsilon } ) , \\varphi _ { x _ { i } } ( 0 ) ) , \\sum _ { i = 1 } ^ { K } \\varphi _ { x _ { i } } ( 0 ) - B \\right \\} \\leq 0 . \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} p = \\sum _ { J \\subseteq \\{ 1 , 2 , \\ldots , n \\} } H _ J ( p ) . \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{gather*} T _ 0 T _ 1 T _ 0 ( E _ { 2 , 1 } ( 1 ) ) = T _ 0 T _ 1 T _ 0 \\big ( T _ 2 T _ 3 \\cdots T _ { N - 1 } \\big ( x _ 0 ^ + \\big ) \\big ) = T _ 0 T _ 1 T _ 2 T _ 3 \\cdots T _ { N - 2 } \\big ( x _ { N - 1 } ^ + \\big ) = - E _ { N , 1 } ( 2 ) . \\end{gather*}"}
-{"id": "3734.png", "formula": "\\begin{align*} - \\nabla \\cdot [ ( m \\otimes m ) \\nabla p ] & = S , \\\\ \\frac { \\partial m } { \\partial t } - c _ 0 ^ 2 ( m \\cdot \\nabla p ) \\nabla p + | m | ^ { 2 ( \\gamma - 1 ) } m & = 0 , \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} a = \\frac { 1 + \\langle X , \\mathbf n \\rangle \\mathbf n ( f ) } { \\sqrt { [ 1 + \\langle X , \\mathbf n \\rangle \\mathbf n ( f ) ] ^ 2 - N ^ 2 | \\mathbf n ( f ) | ^ 2 } } . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} \\lvert m _ { n } \\rvert _ S = & k + \\sum _ { j = 1 } ^ { k - 1 } \\lvert m _ j + n ( \\# a ( s _ j ) - \\# a ^ { - 1 } ( s _ j ) ) \\rvert \\\\ = & k + \\sum _ { j \\in A ^ + } \\lvert m _ j + n \\rvert + \\sum _ { j \\in A ^ - } \\lvert m _ j - n \\rvert + \\sum _ { j \\in A ^ 0 } \\lvert m _ j \\rvert \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} L ( \\varphi ^ { \\vee } , s - 1 ) : = \\prod _ { f } D ^ { \\varphi } _ f ( f ^ { - s } ) ^ { - 1 } = \\sum _ { a \\in A _ { + } } \\frac { \\mu _ { \\varphi } ( a ) } { a ^ s } , \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{gather*} \\partial _ s u ( s , t ) + J _ t ( \\partial _ t u ( s , t ) - X _ { H ^ { ( k ) } } ( u ( s , t ) ) ) \\\\ = \\partial _ s v ( s , t - \\rho ( s ) ) - \\rho ' ( s ) \\partial _ t v ( s , t - \\rho ( s ) ) \\\\ + J _ t ( v ( s , t - \\rho ( s ) ) ) ( \\partial _ t v ( s , t - \\rho ( s ) ) - X _ { H ^ { ( k ) } } ( v ( s , t - \\rho ( s ) ) ) ) \\end{gather*}"}
-{"id": "9599.png", "formula": "\\begin{align*} \\{ x _ 1 , p _ 1 \\} _ { P B } = 1 ; \\{ x _ 2 , p _ 2 \\} _ { P B } = 1 ; \\{ x _ 1 , p _ 2 \\} _ { P B } = \\{ x _ 2 , p _ 1 \\} _ { P B } = 0 , \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} V _ 0 = \\begin{pmatrix} I _ j & 0 \\\\ 0 & V _ 1 \\end{pmatrix} U _ 0 = \\begin{pmatrix} I _ j & 0 \\\\ 0 & U _ 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} E ( \\{ y _ 1 , \\dots , y _ n \\} ) : = \\frac { 1 } { 2 } \\sum _ { i \\neq j } v _ 2 ( | y _ i - y _ j | ) \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} \\| v \\| _ { H ^ { \\beta / 2 , \\lambda } ( \\mathbb { R } ^ n ) } = \\left ( \\| v \\| _ { L ^ 2 ( \\mathbb { R } ^ n ) } ^ 2 + | v | ^ 2 _ { H ^ { \\beta / 2 , \\lambda } ( \\mathbb { R } ^ n ) } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} K = ( f ) \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} [ p , q , r , s , t ] \\mapsto [ x , y , z , w ] = [ p r ^ 7 , q r ^ { 1 1 } , r s , r t ] \\sim [ r ^ { - 1 } p , r ^ { - 1 } q , s , t ] \\end{align*}"}
-{"id": "9862.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\Delta _ { 2 } ( 5 n + 4 ) q ^ { n } & \\equiv 4 1 \\dfrac { E _ { 1 0 } ^ { 3 } } { E _ { 1 } E _ { 2 } ^ { 3 } E _ { 5 } } = 4 1 \\dfrac { E _ { 2 } ^ { 2 } } { E _ { 1 } } \\dfrac { E _ { 1 0 } } { E _ { 2 } ^ { 5 } } \\dfrac { E _ { 1 0 } ^ { 2 } } { E _ { 5 } } \\equiv \\psi ( q ) \\psi ( q ^ { 5 } ) \\pmod { 5 } , \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} d \\xi _ t ( x ) = d M _ t ( x ) + D _ t \\xi ( x ) d t , \\xi _ 0 ( x ) = x , \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} \\varphi _ k = 1 - q ^ k + \\Delta _ 2 + \\Delta _ 4 + \\Delta _ 6 + \\cdots ~ . \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} f _ { \\varepsilon } ( t _ 0 ) < \\tilde { c } \\quad \\Longrightarrow \\dot { p } _ { \\varepsilon } ( t ) = \\begin{cases} \\frac { d } { d t } u ( f _ \\varepsilon ) ( t ) & \\ u ( f _ \\varepsilon ( t ) ) \\leq S _ \\varepsilon ( t ) , \\\\ 0 & \\ u ( f _ \\varepsilon ( t ) ) > S _ \\varepsilon ( t ) , \\end{cases} \\quad [ t _ 0 , t _ 0 + \\delta ] . \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} G _ \\nu ^ f ( \\rho ) & = \\int _ { \\R ^ n } \\rho ( x ) \\left \\langle \\nabla \\log \\frac { \\rho ( x ) } { \\nu ( x ) } , \\big ( \\alpha I \\big ) \\ , \\nabla \\log \\frac { \\rho ( x ) } { \\nu ( x ) } \\right \\rangle \\ , d x \\\\ & = \\alpha \\int _ { \\R ^ n } \\rho ( x ) \\left \\| \\nabla \\log \\frac { \\rho ( x ) } { \\nu ( x ) } \\right \\| ^ 2 \\ , d x \\\\ & = \\alpha \\ , J _ \\nu ( \\rho ) \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } k _ { 2 , \\sigma } = \\iota _ 1 & \\forall \\sigma \\in I _ F ^ 2 \\\\ k _ { 1 , \\sigma } = \\iota _ 1 - 1 & \\forall \\sigma \\in I _ F ^ 3 \\end{array} \\right . \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} d X _ t = d \\tilde { B } _ t + \\beta _ j ( X _ t ) d \\Z ^ j _ t \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} p _ { c , m } ( \\mathcal { Q } _ { b } ) = \\sum _ { j = 1 } ^ { 2 ^ b } \\sum _ { k \\neq j } ^ { 2 ^ b } \\int _ { a ^ - _ j } ^ { a ^ + _ j } \\int _ { a ^ - _ k } ^ { a ^ + _ k } P ( y _ { B } , y _ { C } ) d y _ B d y _ C , \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\alpha _ i = \\sum _ { j \\in J } \\beta _ j \\mbox { a n d f o r a l l } S \\subset I ; \\ ; \\sum _ { i \\in S } \\alpha _ i \\le \\sum _ { j ; \\ ; S \\sim j } \\beta _ j \\ , . \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} \\Sigma : = \\bigcap _ { j = 0 } ^ N \\Sigma _ j \\subset T ^ * \\Omega \\setminus 0 , \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} g ( x ) = - \\min _ { y \\in D } \\Big \\{ \\langle \\tilde { B _ 1 } x - \\alpha , y - x \\rangle + \\beta \\sum _ { i = 1 } ^ N y ^ 2 _ i + \\sum _ { i = 1 } ^ N h _ i ( y _ i ) \\Big \\} + \\beta \\sum _ { i = 1 } ^ N x _ i ^ 2 + \\sum _ { i = 1 } ^ N h _ i ( x _ i ) \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} V _ 0 ^ T ( f ) = \\sup _ \\mathcal { P } \\sum _ { k = 1 } ^ N | f ( t _ k ) - f ( t _ { k - 1 } ) | < \\infty , \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} \\alpha _ { \\pm } \\bigl ( \\check { X } ( j ) \\bigr ) = \\check { X } ( j \\pm 1 ) - \\check { X } ( j ) \\ \\ ( j = 0 , 1 , \\ldots , N ) , \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} a ( g ) ( f _ 1 \\otimes _ s \\cdots \\otimes _ s f _ n ) & = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } \\langle g , f _ i \\rangle f _ 1 \\otimes _ s \\cdots \\otimes _ s \\widehat { f } _ i \\otimes _ s \\cdots \\otimes _ s f _ n \\\\ a ^ \\dagger ( g ) ( f _ 1 \\otimes _ s \\cdots \\otimes _ s f _ n ) & = \\sqrt { n + 1 } g \\otimes _ s f _ 1 \\otimes _ s \\cdots \\otimes _ s f _ n \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 1 } ^ r \\biggl [ B ( s ) \\Gamma \\biggl ( \\frac { s + 1 } { 2 } \\biggr ) \\varphi _ { \\frac { 1 } { 2 } , n ^ 2 D } \\biggl ( z , \\frac { s } { 2 } + \\frac { 1 } { 4 } \\biggr ) \\biggr ] ^ { } = \\mathrm { L C } _ { s = 1 } ^ r [ B ( s ) e ^ { - \\frac { \\pi i s } { 2 } } ] q ^ { n ^ 2 D } . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} \\int | x | ^ { - 2 } | V ^ { j _ 0 } ( x - x ^ { j _ 0 } _ n ) | ^ 2 d x = \\int _ { ( V ^ { j _ 0 } ) } | x + x ^ { j _ 0 } _ n | ^ { - 2 } | V ^ { j _ 0 } ( x ) | ^ 2 d x . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} \\iota \\left ( h \\otimes ( X _ { 1 } \\ldots X _ { n } \\cdot \\tilde { y } _ i ^ { n _ { \\mu , i } + 1 } \\cdot v _ { + } ) \\right ) = 0 , \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} u ( t ) = & P ( T - t ) \\Phi - \\int _ t ^ T P ( r - t ) \\left ( \\nabla u ^ * ( r ) b ( r ) \\right ) \\mathrm d r \\\\ & - \\int _ t ^ T P ( r - t ) f ( r , \\cdot , u ( r ) , \\nabla u ( r ) ) \\mathrm d r \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} J u = E _ j u \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} W _ { \\psi _ { 1 } , \\varphi _ { 1 } } f ( z ) & = \\psi _ { 1 , * } ( z ' _ { [ j ] } ) e ^ { - \\left \\langle z _ { [ j ] } , ( V ^ * b ^ 0 ) _ { [ j ] } \\right \\rangle } f \\big ( z _ { [ j ] } + ( V ^ * b ^ 0 ) _ { [ j ] } , G z ' _ { [ j ] } + ( b ^ 1 ) ' _ { [ j ] } \\big ) \\\\ & = ( T _ { ( V ^ * b ^ 0 ) _ { [ j ] } } W _ { \\psi _ { 1 , * } , \\varphi _ { 1 , * } } ) f ( z ) , \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{gather*} T _ 2 T _ 3 \\cdots T _ { N - 1 } \\big ( x _ 1 ^ - \\big ) = T _ 2 \\big ( x _ 1 ^ - \\big ) = - E _ { 3 , 1 } . \\end{gather*}"}
-{"id": "7780.png", "formula": "\\begin{align*} E _ { \\tilde { \\mu } } ( e ^ { i \\phi ( f ) } | \\mathcal { F } ) ( \\omega ) = e ^ { - 1 / 2 ( f , ( - \\mathcal { L } ^ \\omega _ X ) ^ { - 1 } f ) } . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} \\phi \\circ \\mu _ E ( ( 0 , a ) , ( 0 , b ) ) = \\mu ' _ { E } ( \\phi ( 0 , a ) , \\phi ( 0 , b ) ) \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} \\chi ( X ) = \\frac { S _ { L _ { \\xi } ( - 2 \\rho , 2 \\rho ) , X } } { \\dim ^ + ( L _ { \\xi } ( - 2 \\rho , 2 \\rho ) ) } . \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} & \\sum _ { m = 1 } ^ { \\infty } N _ { X \\underset { x , y } { \\star } Y } ( m ) u ^ m \\\\ = & \\left ( \\sum _ { m = 1 } ^ { \\infty } N _ X ( m ) u ^ m + \\sum _ { m = 1 } ^ { \\infty } N _ X ( m ) u ^ m \\sum _ { m = 1 } ^ { \\infty } N _ Y ( m ) u ^ m + \\cdots \\right ) \\\\ + & \\left ( \\sum _ { m = 1 } ^ { \\infty } N _ Y ( m ) u ^ m + \\sum _ { m = 1 } ^ { \\infty } N _ Y ( m ) u ^ m \\sum _ { m = 1 } ^ { \\infty } N _ X ( m ) u ^ m + \\cdots \\right ) \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} \\pi _ r ^ \\delta ( \\lambda ) = 1 - \\exp \\Big ( - \\lambda c _ d \\int _ 0 ^ \\infty a ^ { d - 1 } \\mu [ | a - 2 \\delta r | \\vee | r - 2 \\delta r - a | , \\infty ] d a \\Big ) . \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} \\Delta _ d f ( n ) = f ( n - 1 ) - 2 f ( n ) + f ( n + 1 ) , n \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} \\ell _ 0 ( X ) = \\min _ { \\overline g \\in \\overline \\Gamma } \\ell ( g ) . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} M \\mathrm { { \\overline { S N R } } } _ { i } ^ { \\mu } \\geqslant \\max _ { j \\neq i } \\mathrm { \\mathrm { { \\overline { I N R } } } } _ { i j } \\max _ { k \\neq i } \\mathrm { \\mathrm { { \\overline { I N R } } } } _ { k i } \\quad \\forall i = 1 , . . . , n , \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} E \\bigg [ \\int _ 0 ^ { \\infty } e ^ { - k \\beta t } V ( d t ) \\bigg ] & = E \\bigg [ \\int _ 0 ^ { \\infty } \\int _ t ^ { \\infty } k \\beta e ^ { - k \\beta s } d s \\ , V ( d t ) \\bigg ] \\\\ & = E \\bigg [ \\int _ 0 ^ { \\infty } k \\beta e ^ { - k \\beta s } \\int _ 0 ^ s V ( d t ) \\ , d s \\bigg ] \\\\ & = E \\bigg [ \\int _ 0 ^ { \\infty } k \\beta e ^ { - k \\beta s } V ( s ) \\ , d s \\bigg ] \\\\ & = \\int _ 0 ^ { \\infty } k \\beta e ^ { - k \\beta s } E V ( s ) \\ , d s . \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} & y _ { m , t } ( n ) = \\\\ & \\sqrt { P _ { m , k } } \\sqrt { \\alpha _ m } s _ { m , t } ( n ) \\otimes f _ { m , l } \\otimes g _ { m , l } X _ m ( n ) + w _ { m , t } ( n ) , \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} \\partial _ t ^ p u ( t _ 0 ) = S _ { \\chi , \\sigma , G , m , p } ( t _ 0 , f , u _ 0 ) \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\lambda _ c ^ * & : = \\sup \\{ \\lambda \\ge 0 : \\theta ^ * ( \\lambda ) > 0 \\} . \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} E _ P [ F ( X _ { \\cdot } ) ] & = E _ Q \\left [ F ( X ^ Y _ { \\cdot } ) \\exp \\left ( \\int _ 0 ^ T u _ j ( s , X _ s ^ Y ) d Y ^ j _ s - \\frac 1 2 \\int _ 0 ^ T | u ( s , X _ s ^ Y ) | ^ 2 d s \\right ) \\right ] \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} \\int _ { ( 0 , + \\infty ) } f ( t ) \\ , d \\| E _ { \\rho , \\sigma } ( t ) \\xi _ \\sigma \\| ^ 2 & \\ge \\int _ { ( 0 , + \\infty ) } ( a + b t ) \\ , d \\| E _ { \\rho , \\sigma } ( t ) \\xi _ \\sigma \\| ^ 2 \\\\ & = a \\| s _ M ( \\rho ) s _ { M ' } ( \\sigma ) \\xi _ \\sigma \\| ^ 2 + b \\| \\Delta _ { \\rho , \\sigma } ^ { 1 / 2 } \\xi _ \\sigma \\| ^ 2 \\\\ & = a \\| s _ M ( \\rho ) \\xi _ \\sigma \\| ^ 2 + b \\| s _ M ( \\sigma ) \\xi _ \\rho \\| ^ 2 \\\\ & = a \\sigma ( s _ M ( \\rho ) ) + b \\rho ( s _ M ( \\sigma ) ) > - \\infty , \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} \\sum _ { g \\in G } | \\xi ' _ x ( g ) - \\xi ' _ { x ' } ( g ) | & = \\sum _ { g \\in K } | \\xi ' _ x ( g ) - \\xi ' _ { x ' } ( g ) | + \\sum _ { g \\not \\in K } | \\xi ' _ x ( g ) - \\xi ' _ { x ' } ( g ) | \\\\ & \\leq \\sum _ { h \\in F } | \\xi _ { \\phi ( x ) } ( h ) - \\xi _ { \\phi ( x ' ) } ( h ) | + \\sum _ { h \\not \\in F } | \\xi _ { \\phi ( x ) } ( h ) | + \\sum _ { h \\not \\in F } | \\xi _ { \\phi ( x ' ) } ( h ) | \\\\ & \\leq \\epsilon / 4 + \\epsilon / 4 + \\epsilon / 2 = \\epsilon . \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} E _ { U } : = \\bigcup _ { i = 1 } ^ { n } B _ { u _ { i } } . \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} \\sum _ { j \\in N ( j ) } Q _ { i j } = S _ i \\mbox { f o r a l l } j \\in \\mathcal { V } . \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} ( I _ { \\mu } ^ { \\infty } ) ' ( u _ n ^ 2 ) ( h ) & = ( I _ { \\mu } ^ { \\infty } ) ' ( u _ n ^ 1 ( \\cdot + y _ n ^ 1 ) - v _ 1 ) ( h _ n ) . \\\\ \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} U ( t ) = - a _ 0 | t | + b _ 0 + \\mathcal { O } ( e ^ { \\ , - a _ 0 | t | } ) , | t | \\to \\infty , \\quad \\mbox { w h e r e } \\ a _ 0 = 2 \\ \\hbox { a n d } \\ b _ 0 = \\log 2 . \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} \\tilde { \\mathfrak I } _ \\chi ^ * \\Phi ( x ) = \\int _ { X _ \\emptyset } \\Phi ( v ) \\tilde \\chi \\circ p ( ( x , v ) ) d v , \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} S ^ { R , R } _ { \\delta _ k , \\delta _ l } = \\xi ^ { k + l } \\zeta ^ { 2 k l } \\quad \\theta _ { \\delta _ k } = \\xi ^ k \\zeta ^ { k ^ 2 } . \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } \\partial _ { t } W ( t , x ) + \\inf \\limits _ { u \\in U } H ( t , x , W ( t , x ) , D W ( t , x ) , D ^ { 2 } W \\left ( t , x \\right ) , u ) = 0 , \\ W ( T , x ) = \\phi ( x ) . \\end{array} \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} p ( x ) = & \\det \\left ( x I - ( M - I ) \\right ) = \\det \\begin{pmatrix} x + w _ 1 & - w _ 1 & 0 & \\cdots \\\\ 0 & x + w _ 2 & - w _ 2 & \\vdots \\\\ \\vdots & \\ddots & \\ddots & \\vdots \\\\ - w _ n & \\cdots & 0 & x + w _ n \\end{pmatrix} \\\\ = & \\prod \\limits _ { i = 1 } ^ n \\left ( x + w _ i \\right ) + ( - 1 ) ^ { n - 1 } \\prod \\limits _ { i = 1 } ^ n ( - w _ i ) \\\\ = & \\prod \\limits _ { i = 1 } ^ n \\left ( x + w _ i \\right ) - \\prod \\limits _ { i = 1 } ^ n w _ i \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} \\mathrm { L } H _ { k _ 1 , \\ldots , k _ n } = - ( k _ 1 + \\cdots + k _ n ) H _ { k _ 1 , \\ldots , k _ n } \\quad \\mathrm { L } = \\Delta - \\langle x , \\nabla \\rangle \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} H _ p ( t ; w , W ) = t ^ { 1 / \\mu } H _ p \\left ( 1 ; \\widehat { \\rho } _ t ( w ) , \\widehat { \\rho } _ t ( W ) \\right ) , \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} \\langle \\Delta _ r u , \\varphi _ k \\rangle & = \\Delta _ r \\langle u , \\varphi _ k \\rangle , \\\\ \\langle \\Delta _ \\theta u , \\varphi _ k \\rangle & = \\langle u , \\Delta _ \\theta \\varphi _ k \\rangle = - \\lambda _ k \\langle u , \\varphi _ k \\rangle , \\\\ \\langle \\frac { \\partial u } { \\partial r } , \\varphi _ k \\rangle & = \\frac { \\partial \\langle u , \\varphi _ k \\rangle } { \\partial r } . \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} N ( K - k + 1 ) \\leq D _ k = - \\lim _ { P _ t \\rightarrow \\infty } \\frac { \\log \\Pr ( R _ k < R ^ { t h } ) } { \\log P _ t } \\leq N ( K - k + 1 ) . \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} r & : = f \\circ ( r _ 1 \\times r _ 2 ) \\circ f ^ { - 1 } : U \\rightarrow U , \\\\ U & : = f ( U _ 1 \\times U _ 2 ) \\subset C , \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} f ( \\zeta ) = \\sum _ { k = 0 } ^ { \\infty } f _ k ( \\zeta ) , \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} h ' ( s ) = 2 \\int _ { \\mathbb { R } ^ n } p _ s \\ , \\mathcal { L } p _ s \\ , d \\mu = - 2 \\mathcal { E } ( p _ s , p _ s ) \\le - 2 \\alpha h ( s ) \\ , , \\end{align*}"}
-{"id": "9952.png", "formula": "\\begin{align*} a _ { X } = a _ { \\tilde f } \\otimes f ^ { - 1 } a _ { Y } . \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} \\binom { n } { m } \\equiv \\prod _ i \\binom { n _ i } { m _ i } \\mod p . \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} ( a ; q ) _ 0 = 1 , ( a ; q ) _ n = \\prod _ { i = 0 } ^ { n - 1 } ( 1 - a q ^ i ) , ( a ; q ) _ { \\infty } = \\lim _ { n \\to \\infty } ( a ; q ) _ n . \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} \\begin{aligned} & < K _ { F , 1 } ( \\tau ) n \\log { n } + e ^ { \\frac { 5 \\tau ^ 2 M _ F } { 2 } } N ^ { \\frac { 5 \\tau ^ 2 A _ F } { 2 } } \\sum _ { \\substack { N < | \\Im ( \\rho ) | \\le T ( n ) } } \\left ( e ^ { - \\frac { 5 \\tau ^ 2 M _ F } { 2 } } N ^ { - \\frac { 5 \\tau ^ 2 A _ F } { 2 } } + 1 \\right ) \\\\ & \\quad + \\sum \\limits _ { \\substack { \\rho \\\\ | \\Im ( \\rho ) | \\le N } } 1 - \\frac { 1 } { 2 0 } R ^ n \\end{aligned} \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} a _ { m , i , j } = \\sum _ { k = 1 } ^ { m - i } \\sum _ { \\substack { i _ 1 , \\ldots , i _ j \\\\ i _ 1 + \\ldots + i _ j = k } } \\binom { m - i } { k } \\frac { ( - 1 ) ^ k } { i _ 1 \\ldots i _ j } \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} \\mathbb { P \\left [ A _ { U E } \\right ] } = \\mathbb { E } _ { X _ { 1 1 } , X _ { 2 1 } } \\left [ \\mathbf { 1 } \\left ( X _ { 1 1 } \\leq X _ { 2 1 } \\right ) \\times \\mathbf { 1 } \\left ( X _ { 1 1 } \\leq \\left ( M ^ { \\frac { 1 } { \\alpha \\mu } } \\left ( \\frac { N } { P } \\right ) ^ { \\frac { 2 - \\mu } { \\alpha \\mu } } X _ { 2 1 } ^ { \\frac { 2 } { \\mu } } \\right ) \\right ) \\right ] , \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} u _ { n + 1 } = u _ n + \\Delta t \\varphi _ 1 ( \\Delta t J _ n ) F ( u _ n ) \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} ( r [ F ] + [ l ] , 1 ) = ( r [ F ] , 1 ) + ( [ l ] , 0 ) . \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} \\begin{aligned} & | D v ( x ) - D v ( y ) | + | \\pi ( x ) - \\pi ( y ) | \\\\ & \\le C R ^ { - d / 2 } \\| | D v | + | \\pi | \\| _ { L ^ 2 ( \\Omega _ { R } ( x _ 0 ) ) } \\bigg ( \\left ( \\frac { r } { R } \\right ) ^ { \\gamma } + \\int _ 0 ^ { 2 r } \\frac { \\varrho ^ \\sharp _ 0 ( t ) + \\omega ^ \\sharp _ { A ^ { \\alpha \\beta } } ( t ) } { t } \\ , d t \\bigg ) , \\end{aligned} \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} \\mathcal Q ( \\lambda x + y ) = \\lambda ^ 2 \\mathcal Q ( x ) + 2 \\lambda \\mathcal Q ( x , y ) + \\mathcal Q ( y ) = \\lambda ^ 2 \\mathcal Q ( x ) + \\mathcal Q ( y ) > 0 \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { A } [ \\overline { u } ] ( x ) & \\geq f ( a x ) - f ( x ) - a ^ N ( a - 1 ) \\| D J \\| _ { \\infty } \\mu \\sup _ { | z | < 1 } \\Psi ( a ( x + z ) ) \\\\ & \\geq a f ( x ) - f ( x ) - a ^ N ( a - 1 ) \\| D J \\| _ { \\infty } \\mu f ( x ) o _ x ( 1 ) \\end{aligned} \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} V _ \\mathrm { n e w } = [ \\ , V \\ , j \\ , ] ~ V _ { Q } , j = h / p , W _ \\mathrm { n e w } = W _ u W _ { Q } , W _ u = \\begin{bmatrix} W & 0 \\\\ 0 & \\delta _ { s + 1 } ^ { - 1 / 2 } \\end{bmatrix} , \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} ( \\partial ^ { p , q } _ { { \\rm { v e r } } } \\sigma ) ( i _ 0 , \\dots , i _ p , j _ 0 , \\dots , j _ { q + 1 } ) = \\sum _ { r = 0 } ^ { q + 1 } ( - 1 ) ^ r \\sigma ( i _ 0 , \\dots , i _ p , j _ 0 , \\dots , \\widehat { j _ r } , \\dots , j _ { q + 1 } ) | _ { U _ { i _ 0 \\dots i _ p j _ 0 \\dots j _ { q + 1 } } } . \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} \\psi _ \\alpha { } ^ \\beta ( x ) \\ \\frac { \\partial \\varphi ^ \\gamma } { \\partial x ^ \\beta } ( x ) = \\delta _ \\alpha { } ^ \\gamma . \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} m x = 1 - x - m - 2 y . \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\begin{array} { c } m i n \\Big \\{ m a x \\Big [ r V ^ { i j } ( x ) - \\mathcal { A } V ^ { i j } ( x ) - f ^ { i j } ( x ) ; \\qquad \\qquad \\qquad \\qquad \\\\ V ^ { i j } ( x ) - N ^ { i j } [ V ] ( x ) \\Big ] ; V ^ { i j } ( x ) - M ^ { i j } [ V ] ( x ) \\Big \\} = 0 . \\end{array} \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} h _ a ( l ) + h _ b ( x ) = 0 , \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} A W = V \\Sigma , A ^ { * } V = W \\Sigma , \\Sigma = \\mathrm { d i a g } ( \\sigma _ 1 , \\ldots , \\sigma _ k ) . \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\begin{bmatrix} b _ { 1 1 } & b _ { 1 2 } & b _ { 1 3 } \\\\ 0 & b _ { 2 2 } & b _ { 2 3 } \\\\ 0 & 0 & b _ { 3 3 } \\end{bmatrix} \\mapsto \\begin{bmatrix} b _ { 1 1 } & t b _ { 1 2 } & t ^ 2 b _ { 1 3 } \\\\ 0 & b _ { 2 2 } & t b _ { 2 3 } \\\\ 0 & 0 & b _ { 3 3 } \\end{bmatrix} . \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} \\binom { Y + 1 } { k } = \\binom { Y } { k } + \\binom { Y } { k - 1 } \\in \\Q [ Y ] , \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} a ^ \\prime = - \\frac { a _ 0 ^ 2 } { 2 b _ 0 c _ 0 } + \\frac { b _ 0 } { 2 c _ 0 } + \\frac { c _ 0 } { 2 b _ 0 } = : A _ 0 , b ^ \\prime = \\frac 1 2 \\frac { a _ 0 } { c _ 0 } - \\frac 1 2 \\frac { b _ 0 ^ 2 } { a _ 0 c _ 0 } + \\frac 1 2 \\frac { c _ 0 } { a _ 0 } = : B _ 0 \\ , , c ^ \\prime = \\frac 1 2 \\frac { a _ 0 } { b _ 0 } + \\frac 1 2 \\frac { b _ 0 } { a _ 0 } - \\frac 1 2 \\frac { c _ 0 ^ 2 } { a _ 0 b _ 0 } = : C _ 0 \\ , , \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} \\binom { n + r } { r } \\equiv 1 \\mod p . \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} \\liminf _ { r \\to 0 } { \\log W _ { B ( y , r ) } ( x ) \\over - \\log r } = 1 \\limsup _ { r \\to 0 } { \\log W _ { B ( y , r ) } ( x ) \\over - \\log r } = \\eta . \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{gather*} \\varphi \\ast \\psi = \\varphi \\ ! \\left ( ? b _ j b _ i \\right ) \\psi \\big ( S ^ { - 1 } ( a _ i ) ? a _ j \\big ) , x _ m \\mapsto \\varphi ( x _ m ' b _ j b _ i ) \\psi \\big ( S ^ { - 1 } ( a _ i ) x _ m '' a _ j \\big ) . \\end{gather*}"}
-{"id": "3884.png", "formula": "\\begin{align*} L ' _ { i j , i j } = \\frac { \\Pi ( G ' ) } { R _ { e } } \\Bigl ( L _ { i i } - \\frac { L _ { i j , i j } } { R _ { e } } \\Bigr ) . \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} \\beta _ { \\tilde g _ 0 ^ { ( 4 ) } } h ^ { ( 4 ) } = 0 . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} M ( X ) = c ^ { F } ( X ) - c _ { \\textup { S M } } ( X ) \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} P & = \\{ \\lambda \\in \\mathfrak { h } ^ * \\mid \\langle \\lambda , \\alpha ^ \\vee \\rangle \\in \\mathbb { Z } , \\ \\forall \\alpha \\in \\Pi \\} \\\\ P ^ + & = \\{ \\lambda \\in \\mathfrak { h } ^ * \\mid \\langle \\lambda , \\alpha ^ \\vee \\rangle \\in \\mathbb { N } , \\ \\forall \\alpha \\in \\Pi \\} \\\\ P ^ \\vee & = \\{ \\lambda \\in \\mathfrak { h } ^ * \\mid \\langle \\lambda , \\alpha \\rangle \\in \\mathbb { Z } , \\ \\forall \\alpha \\in \\Pi \\} \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} \\mathbf { G } [ \\sigma ] = \\exp \\left ( \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } | \\log \\sigma ( \\theta ) | d \\theta \\right ) . \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} ( e \\sigma e ) ( ( e - y ( s ) ) ^ * ( e - y ( s ) ) ) & = \\sigma ( e ( 1 - x ( s ) ) ^ * e ( 1 - x ( s ) ) e ) \\le \\sigma ( ( 1 - x ( s ) ) ^ * ( 1 - s ( s ) ) ) , \\\\ ( e \\rho e ) ( y ( s ) y ( s ) ^ * ) & = \\sigma ( e x ( s ) e x ( s ) ^ * e ) \\le \\sigma ( x ( s ) x ( s ) ^ * ) , \\end{align*}"}
-{"id": "9897.png", "formula": "\\begin{align*} \\widehat { f } ( \\xi ) = \\left ( ( 2 \\pi i \\xi ) ^ { - s } + ( 2 \\pi i \\xi ) ^ s \\right ) \\widehat { u } ( \\xi ) . \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{gather*} R _ { 1 2 } M _ 1 R _ { 2 1 } M _ 2 = M _ 2 R _ { 1 2 } M _ 1 R _ { 2 1 } . \\end{gather*}"}
-{"id": "7022.png", "formula": "\\begin{align*} \\tfrac d { d t } W _ t & = \\frac { p ^ { P _ t } \\delta e ^ { \\delta t } \\eta ( t ) - p ^ { P _ t } e ^ { \\delta t } \\eta ' ( t ) } { \\eta ( t ) ^ 2 } = W _ t \\cdot \\frac { \\delta \\eta ( t ) - \\vartheta p ^ { P _ t } e ^ { \\delta t } } { \\eta ( t ) } = W _ t ( \\delta - \\vartheta W _ t ) = w ( W _ t ) . \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} R = R ^ x + \\langle R ^ y , y \\rangle + \\langle R ^ z , z \\rangle + \\langle R ^ { \\bar { z } } , \\bar { z } \\rangle + \\langle R ^ { z z } z , z \\rangle + \\langle R ^ { z \\bar { z } } z , \\bar { z } \\rangle + \\langle R ^ { \\bar { z } \\bar { z } } \\bar { z } , \\bar { z } \\rangle , \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} \\epsilon ( s , \\mu _ p \\tilde { \\pi } _ { p } ) = p ^ { ( \\frac { 1 } { 2 } - s ) a ( \\mu _ p \\tilde { \\pi } _ p ) } \\epsilon ( \\frac { 1 } { 2 } , \\mu _ p \\tilde { \\pi } _ p ) . \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} d ( x ^ \\alpha d x _ \\sigma ) & : = \\sum _ { i = 1 } ^ n \\left ( \\frac { \\partial x ^ \\alpha } { \\partial x _ i } d x _ i \\right ) \\wedge d x _ { \\sigma ( 1 ) } \\wedge \\dots \\wedge d x _ { \\sigma ( k ) } , \\\\ \\kappa ( x ^ \\alpha d x _ \\sigma ) & : = \\sum _ { i = 1 } ^ k \\left ( ( - 1 ) ^ { i + 1 } x ^ \\alpha x _ { \\sigma ( i ) } \\right ) d x _ { \\sigma ( 1 ) } \\wedge \\cdots \\wedge \\widehat { d x _ { \\sigma ( i ) } } \\wedge \\cdots \\wedge d x _ { \\sigma ( k ) } . \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} A & = ( 0 : 0 : 1 ) , & B & = ( 1 : 0 : 0 ) , & C & = ( 0 : 1 : 0 ) , \\\\ A _ 1 & = ( 1 : 1 : 0 ) , & B _ 1 & = ( 0 : 1 : 1 ) , & C _ 1 & = ( - 1 : 0 : 1 ) , \\\\ A _ 2 & = ( 1 : m : 0 ) , & B _ 2 & = ( 0 : y : 1 ) , & C _ 2 & = ( x : 0 : 1 ) . \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} \\gamma = \\kappa _ 0 \\kappa _ 1 \\kappa _ 2 \\kappa _ 3 \\Delta _ 2 ^ 2 = \\sigma _ 2 \\sigma _ 1 ^ 2 \\sigma _ 2 \\sigma _ 3 \\sigma _ 4 w _ 1 \\sigma _ 2 \\sigma _ 3 \\sigma _ 2 ^ { - 1 } w _ 2 \\sigma _ 2 \\sigma _ 4 ^ { - 1 } \\sigma _ 3 ^ { - 1 } \\sigma _ 1 ^ 2 . \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} F ( x _ 1 , . . . , x _ \\ell ) = \\prod _ { i = 1 } ^ \\ell f _ i ( x _ i ) \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} W : = W _ T = \\left \\{ \\mbox { c o n t i n u o u s m a p p i n g s f r o m $ [ 0 , T ] $ t o $ \\R $ } \\right \\} , \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} \\mathcal { N } \\varphi _ { _ { 0 } } = \\mathcal { P } ( { } _ { 0 } I _ { T } ^ { 1 - q } \\Theta ( 0 ) ) . \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} v ( w ) = \\int _ { C ( 0 , r ) ^ + } e ^ { z w } u ( z ) d z , \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} R ( 1 ) = a + b , R ( e ' ) = a _ 1 + b _ 1 , R \\big ( e _ k ^ 1 \\big ) = a _ 2 , \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} \\widetilde { H } ^ { s } ( W ) = Z _ 1 \\perp H _ 1 , \\ L ^ 2 ( \\Omega ) = Z _ 2 \\perp H _ 2 . \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} \\chi ( X ) = - \\sum _ { k = 0 } ^ { n - 1 } \\binom { n + 1 } { k } ( - d ) ^ { n - k } , \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} \\o _ { M , K } ( u , u ' ) & \\ge K | \\pi ^ * \\o _ B ( w , w ' ) | - | | \\mu | | \\cdot | w | \\ , | w ' | \\\\ & \\ge K c _ 0 | \\pi _ * w | | \\pi _ * w ' | - | | \\mu | | \\cdot | w | \\ , | w ' | \\\\ & \\ge K c _ 0 c ^ 2 _ 1 | w | | w ' | - | | \\mu | | \\cdot | w | \\ , | w ' | \\\\ & = | w | ( K c ^ 2 _ 1 c _ 0 - | | \\mu | | ) > 0 \\ , , \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ t \\langle \\sqrt { H } P _ c v ( s ) \\sqrt { H } P _ c \\N \\rangle \\ , d s & = \\int _ 0 ^ t \\langle \\partial _ x v ( s ) , \\partial _ x \\N \\rangle \\ , d s \\\\ & + \\int _ 0 ^ t \\langle m ( \\partial _ x ) \\partial _ x v ( s ) , \\partial _ x \\N \\rangle \\ , d s \\end{aligned} \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} b _ k x \\in F _ k + \\big ( - r \\rho _ k ( \\mathbf { G } ) \\big ) \\subseteq F _ k + \\rho _ k ( - \\mathbf { G } ) : = \\tilde { F } _ k \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} W ( t , x ) = \\underset { u \\in \\mathcal { U } [ t , T ] } { e s s \\inf } Y _ { t } ^ { t , x ; u } . \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} \\partial R _ s F + \\frac 1 2 \\vert \\nabla R _ s F \\vert ^ 2 = 0 , \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} ( \\sigma \\circ \\pi ) ( \\mu ( a ) ) & = \\sigma ( \\phi ( a ) ) = \\sigma ( \\oplus _ { n = 1 } ^ \\infty \\mu _ n ( E _ n a E _ n ) ) \\\\ & = \\oplus _ { n = 1 } ^ \\infty P _ n \\sigma _ n ( \\mu _ n ( E _ n a E _ n ) ) P _ n = \\oplus _ { n = 1 } ^ \\infty P _ n \\psi _ n ( E _ n a E _ n ) P _ n \\\\ & = \\oplus _ { n = 1 } ^ \\infty P _ n \\mu ( E _ n a E _ n ) P _ n = \\oplus _ { n = 1 } ^ \\infty \\mu ( E _ n a E _ n ) \\\\ & = \\lim _ { N \\to \\infty } \\mu \\left ( \\oplus _ { n = 1 } ^ N E _ n a E _ n \\right ) \\\\ & = \\mu ( a ) \\end{align*}"}
-{"id": "9879.png", "formula": "\\begin{align*} \\widetilde { W } ^ { s } _ L ( \\mathbb { R } ) = \\{ u \\in L ^ 2 ( \\mathbb { R } ) , \\boldsymbol { D } ^ s u \\in L ^ 2 ( \\mathbb { R } ) \\} , \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} \\tfrac { 1 } { D _ { n + 1 } } T r _ { d _ i } F ( d _ j ) \\stackrel { ( F o n ) } { = } \\tfrac { 1 } { D _ { n + 1 } } T r _ { p r _ 1 } T r _ { b a } F ( b a ) F ( p r _ 2 ) & \\stackrel { ( D e g ) } { = } \\tfrac { 1 } { D _ { n } } T r _ { p r _ 1 } F ( p r _ 2 ) \\\\ & \\stackrel { ( C d B ) } { = } \\tfrac { 1 } { D _ { n } } F ( d _ { j - 1 } ) T r _ { d _ i } \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} \\mathcal { L } = & - a \\nabla V \\cdot \\nabla + \\frac { 1 } { \\beta } ( \\nabla \\cdot a ) \\cdot \\nabla + \\frac { 1 } { \\beta } a : \\nabla ^ 2 \\\\ = & \\frac { e ^ { \\beta V } } { \\beta } \\sum _ { 1 \\le i , j \\le n } \\frac { \\partial } { \\partial x _ i } \\Big ( a _ { i j } e ^ { - \\beta V } \\frac { \\partial } { \\partial x _ j } \\Big ) \\ , , \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} \\left \\lfloor \\frac { i _ 1 - i _ 2 } { a } \\right \\rfloor = \\left \\{ \\begin{array} { c c } 0 & i _ 1 \\geq i _ 2 \\\\ - 1 & i _ 1 < i _ 2 \\end{array} \\right . \\ , . \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} 0 = - 0 = - \\gamma ( v _ 0 v _ { 1 } ) - \\ldots - \\gamma ( v _ { \\ell - 1 } v _ \\ell ) = \\gamma ( v _ { \\ell } v _ { \\ell - 1 } ) + \\ldots + \\gamma ( v _ 1 v _ 0 ) = \\gamma ( P ' ) . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( \\sqrt { \\mathbf { U } } + \\sqrt { \\mathbf { V } } ) \\otimes \\mathbf { I } _ p \\mathbf { x } ^ { \\star } = \\frac { 1 } { 2 } ( \\sqrt { \\mathbf { U } } - \\sqrt { \\mathbf { V } } ) \\otimes \\mathbf { I } _ p \\mathbf { x } ^ { \\star } , \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } ( y \\alpha ( y , t ) + \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) - \\frac { \\partial } { \\partial t } \\alpha ( y , t ) = 0 \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} M = Q \\cdot { \\rm { d i a g } } ( \\lambda _ 1 , \\dots , \\lambda _ n ) \\cdot Q ^ { - 1 } \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} C _ 1 A _ 1 ' = 1 ; \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} + \\sum _ { k = 0 } ^ { \\infty } \\sum _ { \\substack { { r _ 1 , r _ 2 , r _ 3 = 1 } \\\\ r _ 1 \\ne r _ 2 \\ne r _ 3 } } ^ { n } \\frac { a _ { r _ 1 } b _ { r _ 2 } c _ { r _ 3 } } { ( r _ 3 - r _ 2 ) ( r _ 1 - r _ 2 ) } \\left ( \\frac { 1 } { r _ 2 + k + 1 } - \\frac { 1 } { r _ 1 + k + 1 } \\right ) . \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} - \\nabla \\cdot ( \\mathbb { P } [ \\mu ] \\nabla p ) = S \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} \\chi _ { ( \\lambda , \\lambda ) } = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } e ^ { ( w ( \\lambda + \\rho ) , w ( \\lambda + \\rho ) ) } } { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } e ^ { ( w ( \\rho ) , w ( \\rho ) ) } } . \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} C _ 1 \\dot { A } _ 1 + \\dot { B } \\frac { \\partial F } { \\partial P _ 1 } = 0 ; \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { i _ 0 } \\Phi ( x , \\kappa ^ j \\rho ) \\lesssim \\sum _ { j = 0 } ^ { i _ 0 } \\Psi ( y _ 0 , 5 \\kappa ^ j \\rho ) + \\| D u \\| _ { L ^ \\infty ( \\Omega _ { 5 \\rho } ( y _ 0 ) ) } \\int _ 0 ^ { 5 \\rho } \\frac { \\varrho ^ \\sharp _ 0 ( t ) } { t } \\ , d t . \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} \\gamma ( t u ) = I ' ( t u ) [ t u ] < t ^ 4 \\left [ \\norm { u } ^ 2 + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ u u ^ 2 d x - \\int _ { \\mathbb R ^ 3 } \\frac { f ( t u ) } { t ^ 3 } u \\ , d x \\right ] < t ^ 4 \\gamma ( u ) = 0 \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} q = r _ 1 \\wedge s _ 1 + \\dots + r _ k \\wedge s _ k \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { 0 } ( \\textbf { x } ) & = I \\left \\lbrace \\textbf { x } \\in \\mathcal { C } ^ { ' } _ { 1 } \\cap \\mathcal { C } ^ { ' } _ { 0 } , ( 1 - q ( \\textbf { x } , 1 ) ) \\dfrac { p } { \\mu { ( \\mathcal { C } ^ { ' } _ { 1 } ) } } > ( 1 - q ( \\textbf { x } , 0 ) ) \\dfrac { 1 - p } { \\mu { ( \\mathcal { C } ^ { ' } _ { 0 } ) } } \\right \\rbrace \\\\ & + I \\left \\lbrace \\textbf { x } \\in \\mathcal { C } ^ { ' } _ { 1 } - \\mathcal { C } ^ { ' } _ { 0 } , q ( \\textbf { x } , 1 ) < 1 \\right \\rbrace . \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} - \\epsilon b _ { i } + H { i } ( 0 , 0 , u _ { 0 , x _ { i } } ( 0 ) ) = - \\epsilon b _ { 1 } + H _ { 1 } ( 0 , 0 , u _ { 0 , x _ { 1 } } ( 0 ) ) . \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} f _ 2 = & \\phi b ^ { - 2 } a - \\phi a b ^ { - 2 } a - \\phi a b ^ { - 3 } a \\\\ & - \\phi b ^ { - 1 } a + \\phi a b ^ { - 1 } a + \\phi a b ^ { - 2 } a \\\\ & + \\phi a b ^ { - 3 } a \\\\ = & \\phi b ^ { - 2 } a - \\phi b ^ { - 1 } a + \\phi a b ^ { - 1 } a . \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\| f g \\| _ { L ^ { p , r } } \\leq C \\| f \\| _ { L ^ { p _ 0 , r _ 0 } } \\| g \\| _ { L ^ { p _ 1 , r _ 1 } } , \\frac 1 p = \\frac { 1 } { p _ 0 } + \\frac { 1 } { p _ 1 } , ~ ~ \\frac 1 r = \\frac { 1 } { r _ 0 } + \\frac { 1 } { r _ 1 } . \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} + \\Biggl . { \\bf 1 } _ { \\{ i _ 1 = i _ 4 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 1 = j _ 4 \\} } { \\bf 1 } _ { \\{ i _ 2 = i _ 3 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 2 = j _ 3 \\} } \\Biggr ) , \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} \\varphi _ + ( x ) & = \\sum _ { 1 \\le i \\le m ^ + } c _ i \\varphi _ i ( B _ i x ) , \\\\ \\varphi _ - ( x ) & = \\sum _ { m ^ + < j \\le m } ( - c _ j ) \\varphi _ j ( B _ j x ) , \\\\ \\varphi ( x ) & = \\varphi _ 0 ( B _ 0 x ) + \\varphi _ + ( x ) - \\varphi _ - ( x ) - \\varphi _ { m + 1 } ( B _ { m + 1 } x ) = \\sum _ { k = 0 } ^ { m + 1 } c _ k \\varphi _ k ( B _ k x ) . \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} \\partial ^ 2 _ { i i } \\phi ( x ) = \\partial _ i \\phi ( x ) - ( R R ^ \\ast ) ^ 2 _ { i i } = ( R R ^ \\ast ) _ { i i } - ( R R ^ \\ast ) ^ 2 _ { i i } . \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial \\overline { \\zeta } _ j } = 0 \\ , 1 \\leq j \\leq n , \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} f _ m = f _ m ( \\epsilon ) = O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ w ) , \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} C _ { s t } = \\alpha ( s , t ) W + ( 1 - \\alpha ( s , t ) - \\beta ( s , t ) ) P + \\beta ( s , t ) M , \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} \\tilde p : = - \\frac 1 2 p \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} \\zeta ( t ) & = t ^ { \\frac { 1 } { 2 } } + \\frac { 9 L _ \\sigma ^ 4 } { 4 } t + \\frac { 3 L _ b ^ 2 } { 2 } t ^ 2 + \\frac { 1 8 L _ b ^ 2 L _ \\sigma ^ 2 } { 5 \\sqrt { \\pi } } t ^ { \\frac 5 2 } , \\ t \\geq 0 . \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} \\sum _ { a \\in A _ { + , i } } \\mu ( a ) a ^ { q ^ k - 1 } = \\begin{cases} \\gamma _ i ( \\theta ^ { q ^ k } - \\theta ) \\dots ( \\theta ^ { q ^ k } - \\theta ^ { q ^ { i - 1 } } ) & k \\geq i \\\\ 0 & k < i \\end{cases} \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} z = u ( t , . ) - \\frac { 1 } { | \\Omega | } N ( u ( t ) ) . \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} e ^ { - t A ^ { 2 } } = F ^ { - 1 } \\circ \\mathcal { F } ^ { - 1 } ( e ^ { - t \\tau ^ { 2 } } \\mathcal { F } \\circ F ) . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{gather*} f ( \\tau ) = \\sum _ { n = 1 } ^ \\infty a ( n ) q ^ n = \\eta ( 2 \\tau ) ^ 4 \\eta ( 4 \\tau ) ^ 4 = q \\prod _ { m = 1 } ^ \\infty \\big ( 1 - q ^ { 2 m } \\big ) ^ 4 \\big ( 1 - q ^ { 4 m } \\big ) ^ 4 . \\end{gather*}"}
-{"id": "5208.png", "formula": "\\begin{align*} \\leq c ^ * \\sum _ { j = 0 } ^ { \\infty } \\frac { B ( n + 1 + j , n + 1 ) } { ( 2 n + j + 1 ) ( n + 1 + j ) ^ { s - 3 } } \\frac { \\Gamma ^ 2 ( n + j + 1 ) } { \\Gamma ( 2 n + j + 1 ) \\Gamma ( j + 1 ) } . \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} s ( r ) : = \\lambda ^ { - 1 } ( r - 1 + \\mu ) \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} D = N - K + 1 . \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} \\sigma ^ { \\alpha \\dot b a } \\ , \\sigma ^ \\beta { } _ { \\dot b c } + \\sigma ^ { \\beta \\dot b a } \\ , \\sigma ^ \\alpha { } _ { \\dot b c } = - 2 g ^ { \\alpha \\beta } \\delta ^ a { } _ c \\ , , \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} \\tau _ 1 + \\tau _ 4 = 0 \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { q + 2 } x _ i / y _ i = 0 . \\end{align*}"}
-{"id": "9790.png", "formula": "\\begin{align*} \\| e ^ { - t M z \\partial _ z } z _ q u \\| ^ 2 _ { H _ { \\phi } } & = \\int _ { \\mathbb { C } ^ 2 } | v ( e ^ { - t M } z ) | ^ 2 | ( e ^ { - t M } z ) _ { q } | ^ { 2 } e ^ { - 2 \\phi ( z ) } L ( d z ) \\\\ & = e ^ { 2 t \\operatorname { T r } M } \\int _ { \\mathbb { C } ^ 2 } | v ( z ' ) | ^ 2 | z ' _ q | ^ 2 e ^ { - \\phi ( z ' ) } e ^ { - 2 [ \\phi ( e ^ { t M } z ' ) - \\phi ( z ' ) ] } L ( d z ' ) { ~ . } \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} F _ k & = b _ k ( \\mathbf { F } ) + \\rho _ k ( \\mathbf { G } ) , \\\\ G _ k & = \\beta _ k ( \\mathbf { G } ) \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} \\tilde { B } _ i = P _ { ( \\R v ) ^ \\bot } B _ i \\colon H \\to \\tilde { H } _ i . \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} R _ { - 1 } \\bigl ( \\check { X } ( j ) \\bigr ) = - R _ 0 \\bigl ( \\check { X } ( j ) \\bigr ) B ^ { } _ { \\mathcal { D } , j } \\ \\ ( j = 0 , 1 , \\ldots , N ) . \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} b _ f \\cdot b _ g = \\sum _ { k \\in \\Psi ^ { \\# } ( Y , \\pi ) } N _ { f , g } ^ k b _ k . \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} s : ( 0 , \\infty ) \\ni w \\mapsto \\begin{cases} w ^ { 1 - n / 2 } , & ; \\\\ - \\log ( w ) , & . \\end{cases} \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} k _ { 1 , \\sigma } = \\left \\{ \\begin{array} { c c } 1 \\ & \\mbox { i f } \\ \\sigma = \\hat { \\sigma } \\\\ 0 \\ & \\mbox { o t h e r w i s e } \\end{array} \\right . , \\ k _ { 2 , \\sigma } = \\left \\{ \\begin{array} { c c } - 1 \\ & \\mbox { i f } \\ \\sigma = \\hat { \\sigma } \\\\ 0 \\ & \\mbox { o t h e r w i s e } \\end{array} \\right . , \\ c = 0 , \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} \\lambda [ x , y ] - [ x , \\varphi ( y ) ] - i \\lambda ( y ) x = 0 \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} L ^ { \\vec { t } } _ \\theta ( h _ 1 \\otimes \\cdots \\otimes h _ s ) : = \\sum _ i h _ 1 \\otimes \\cdots \\otimes L ^ { t _ i } _ { \\theta _ i } \\cdot h _ i \\otimes \\cdots \\otimes h _ s , \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} \\eta _ i = \\displaystyle \\sum _ { s = i } ^ { \\ell - 1 } t _ s \\ \\ ( 1 \\le i \\le \\ell - 1 ) \\mu _ j = \\eta _ j + \\ell ( \\tau ^ { ( j ) } ) \\ \\ ( 1 \\le j \\le \\ell ) . \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\| u _ N - Q \\| _ { L ^ 2 } = 0 . \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} b : = \\Psi \\circ ( h - \\mathfrak { s } ) \\circ \\psi ^ { - 1 } : U ' \\rightarrow U ' \\lhd \\mathbb { R } ^ { k ' } \\times ( \\tilde { \\mathbb { W } } \\times \\mathbb { R } ^ n ) \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} \\int _ { \\Omega } | f ( u _ n ) u _ n | \\leq \\int _ { \\mathbb { R } ^ 2 } A ( x ) f ( u _ n ) u _ n = m ( \\left \\| u _ n \\right \\| ^ 2 ) \\left \\| u _ n \\right \\| ^ 2 - I ' ( u _ n ) u _ n \\leq C _ 1 . \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} K ( g ) = \\lambda g + \\mathcal { L } _ X g , \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} \\mathcal T _ \\emptyset ^ * M _ \\chi ^ { Y _ \\emptyset } = \\frac { \\mu _ X ( \\chi ) } { \\mu _ Y ( \\chi ) } \\cdot M _ \\chi ^ { X _ \\emptyset } . \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} J ( f _ 1 , \\ldots , f _ m ) = \\sqrt { \\frac { \\det Q _ - } { \\det Q _ + } } \\int _ H \\prod _ { k = 0 } ^ { m + 1 } f _ k ^ { c _ k } ( B _ k x ) \\ , d x . \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} \\widehat { H } & = \\underbrace { - \\frac { \\hbar ^ 2 } { 2 M } \\Delta _ { \\boldsymbol { r } } } _ { \\hbox { $ : = \\ , \\widehat { T } _ n $ } } - \\ , \\underbrace { \\frac { \\hbar ^ 2 } { 2 m } \\Delta _ { \\boldsymbol { x } } + V ( \\boldsymbol { r } , \\boldsymbol { x } ) } _ { \\hbox { $ : = \\ , \\widehat { H } _ e $ } } \\ , , \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} L ^ { - S } _ { t } = \\frac { h _ { J _ t } ^ { - S } { ( \\theta ) } } { h _ { J _ 0 } ^ { - S } { ( \\theta ) } } e ^ { - \\theta S ( 0 , t ) - t \\kappa ^ { - S } ( \\theta ) } . \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} V ( x ) = \\mathcal { U } ( \\tilde { g } ) + \\frac { 1 } { 2 } \\tilde { b } _ a \\Gamma ^ { - 1 } \\tilde { b } _ a , \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} m ( v ) : = \\sum _ { v ' \\in V } w ( v , v ' ) = \\sum _ { v \\sim v ' } w ( v , v ' ) . \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} p _ { 1 } = \\frac { n } { \\alpha } < \\frac { l n } { \\alpha } , p _ { 2 } < \\frac { l ^ { \\prime } n } { \\alpha } , v > q _ { 2 } , \\quad \\frac { v } { u } = \\frac { q _ { 1 } } { p _ { 1 } } , \\quad \\frac { 1 } { u } = \\frac { 1 } { p _ { 1 } } - \\frac { \\alpha } { l n } = \\frac { \\alpha } { l ^ { \\prime } n } . \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} L _ D ( 2 s - 1 / 2 ) ^ { - 1 } & = \\left \\{ \\begin{array} { l l } 2 ( s - 3 / 4 ) + O ( ( s - 3 / 4 ) ^ 2 ) & D = 1 , \\\\ L _ D ( 1 ) ^ { - 1 } + O ( s - 3 / 4 ) & D \\neq 1 , \\\\ \\end{array} \\right . \\\\ G _ 0 ( z , 2 s - 1 / 2 ) & = \\frac { 3 } { 2 \\pi } \\frac { 1 } { s - 3 / 4 } - \\frac { 3 } { \\pi } \\log ( y | \\eta ( z ) | ^ 4 ) + C + O ( s - 3 / 4 ) , \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} \\langle ( Q ^ k ) ^ { \\infty } ( x ^ k ) , x ^ k \\rangle = 0 , \\ \\langle ( Q ^ k ) ^ { \\infty } ( x ^ k ) , y \\rangle \\geq 0 \\ \\forall y \\in K ^ { \\infty } . \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} P _ { N , m } = N _ { m } + N _ { - m } \\ \\textrm { a n d } \\ Q _ { N , m } = N _ { m } - N _ { - m } . \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} { 1 \\over N _ k ^ \\tau } \\leq { 1 \\over q _ k ^ { ( \\eta - \\epsilon ) \\tau } } = { 1 \\over q _ k ^ { ( \\eta - \\epsilon ) ( { 1 \\over \\eta } + \\delta ) } } = { 1 \\over q _ k ^ { 1 + \\delta \\eta - \\epsilon ( { 1 \\over \\eta } + \\delta ) } } = { 1 \\over q _ k ^ { 1 + \\epsilon _ 1 } } . \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} \\partial _ t g _ t = - K ( g _ t ) \\ , , g _ { | t = 0 } = g _ 0 , \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} u + C = \\{ u + c \\mid c \\in C \\} \\end{align*}"}
-{"id": "9941.png", "formula": "\\begin{align*} b _ \\Omega ( f ) | _ N = b _ { \\Omega _ Y } ( f | _ Y ) . \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} g ^ { ( < s ) } ( x ) = y . \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X _ { s } = & \\bar { b } ( s , X _ { s } ) d s + \\bar { \\sigma } ( s , X _ { s } ) d B _ { s } , \\\\ d Y _ { s } = & - \\bar { g } ( s , X _ { s } , Y _ { s } , Z _ { s } ) d s + Z _ { s } d B _ { s } , \\\\ X _ { 0 } = & x _ { 0 } , \\ Y _ { T ^ { \\prime } } = \\bar { \\phi } ( X _ { T ^ { \\prime } } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} \\abs { f _ k ( z ) } = \\abs { f _ k ( \\delta _ { \\omega } ( \\abs { z } ) ) } = \\abs { f _ k ( \\omega ) } \\left ( \\abs { z } ^ M \\right ) ^ { k / M } \\leq \\abs { f _ k ( \\omega ) } r ^ k ( \\omega ) q ^ k . \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} \\| O _ p u \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } & = \\| O _ p \\Big ( \\frac { 1 } { L } \\int _ 0 ^ L \\varphi _ s ( q , p ) d s \\Big ) \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } \\le \\frac { 1 } { L ^ 2 } \\int _ 0 ^ L \\| O _ p \\varphi _ s \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } d s { ~ . } \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} v ( w ) = \\int _ { C ( 0 , r ) ^ + } e ^ { z w } u ( z ) d z , \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} f _ { m + 1 } & \\le f _ m + d \\ , \\epsilon ^ { 2 } n ^ 2 + \\epsilon f _ m m \\| \\tilde { P } ^ { ( 1 ) } \\| + \\epsilon ^ 2 m \\sum _ { i = 1 } ^ { m } i \\| \\tilde { P } ^ { ( 1 ) } \\| ^ 2 \\\\ & \\le f _ m ( 1 + \\epsilon m \\| \\tilde { P } ^ { ( 1 ) } \\| ) + d \\ , \\epsilon ^ { 2 } n ^ 2 + \\epsilon ^ 2 n ^ 3 \\| \\tilde { P } ^ { ( 1 ) } \\| ^ 2 \\\\ & \\le f _ m ( 1 + \\epsilon n \\| \\tilde { P } ^ { ( 1 ) } \\| ) + d '' \\epsilon ^ { 2 } n ^ 3 \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} ( \\phi w ) _ n = \\sum _ { u \\in W _ { n } ( w ) } w _ n ( u ) \\phi u . \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} w ( v , v ' ) = \\left \\{ \\begin{array} { c l } H ' ( \\bar { \\theta } _ { v ' } - \\bar { \\theta } _ { v } ) & : v ' \\in N ( v ) \\\\ 0 & : v ' \\notin N ( v ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} r ^ - _ c ( \\alpha , \\gamma ) = \\frac { \\gamma - \\sqrt { \\alpha ^ 2 + \\gamma ^ 2 - 1 } } { 1 - \\alpha } . \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} \\sup _ { a \\in K } \\| P _ N a - a \\| _ { H ^ s _ r ( \\mathbb R ^ d ) } = \\sup _ { 0 \\leq t \\leq T } \\| P _ N g ( t , \\cdot ) - g ( t , \\cdot ) \\| _ { H ^ s _ r ( \\mathbb R ^ d ) } \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} I _ p ( h ) , p = 1 , \\ldots , N ( h ) , \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - \\lambda x } f _ \\alpha ( x ) d x = e ^ { - \\lambda ^ \\alpha } . \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} \\begin{array} { l c } \\min & \\ ; F ( d , X ) = \\eta _ 1 F _ 1 ( R X ) + \\eta _ 2 F _ 2 ( d ) \\\\ s . a . & C ( s ) = 0 , \\\\ & s \\in \\Omega \\times \\Delta . \\\\ \\end{array} \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} K ( m , n , c ) : = \\sum _ { \\substack { d ( c ) ^ * \\\\ a d \\equiv 1 ( c ) } } e \\biggl ( \\frac { a m + d n } { c } \\biggr ) , e ( x ) : = e ^ { 2 \\pi i x } \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} \\sigma \\equiv \\begin{pmatrix} I _ { m \\times m } & 0 \\\\ 0 & \\frac { 1 } { \\sqrt { \\delta } } I _ { ( n - m ) \\times ( n - m ) } \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{gather*} s _ 2 = \\kappa ( s _ 1 ) , s _ 3 = \\kappa ( s _ 2 ) , s _ { 2 o } = \\kappa ( s _ { 1 o } ) , s _ { 3 o } = \\kappa ( s _ { 2 o } ) , s _ { 2 3 } = \\kappa ( s _ { 1 2 } ) , s _ { 1 3 } = \\kappa ( s _ { 2 3 } ) , \\end{gather*}"}
-{"id": "6095.png", "formula": "\\begin{align*} \\pi ( k , \\alpha , \\beta ) = \\sum _ { b = 1 } ^ n k _ b j _ b + \\sum _ { j \\in \\mathbb { Z } _ * } ( \\alpha _ j - \\beta _ j ) j , \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} W _ l ( x + y l ^ { \\kappa } ) = W _ l ( x ) x \\in \\Z _ l ^ { \\times } , y \\in \\Z _ l , \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} J = \\frac { \\partial F } { \\partial u } . \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} H ^ s ( \\Gamma , V ) = \\begin{cases} V ^ { \\Gamma } & \\mbox { i f } s = 0 \\\\ V _ { \\Gamma } & \\mbox { i f } s = 1 \\\\ \\{ 0 \\} & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} P _ { \\phi } ( x ) : = x ^ { r _ 0 } + p _ { r _ 0 - 1 } x ^ { r _ 0 - 1 } + \\dots + p _ 1 x + p _ 0 \\in A [ x ] \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } V ( m ) + F _ { i } ( D ^ { + } V ( m ) , D ^ { - } V ( m ) ) \\leq f _ { i } ( - m \\Delta x ) & \\ , \\ , m \\in J _ { i } \\setminus \\{ 0 \\} \\\\ V ( 0 ) \\leq \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } V ( 1 _ { i } ) \\end{array} \\right . \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} C _ { j , R , m } ( x , y ) = \\emptyset . \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} \\kappa \\sum _ { \\l = m } ^ { \\infty } \\lambda _ { \\l } \\gamma _ { \\l } ^ 2 \\norm { u _ { \\l } } ^ 2 \\leq \\sum _ { \\l = m + 1 } ^ { \\infty } \\frac { \\alpha _ { \\l } \\gamma _ { \\l + 1 } ^ 2 \\beta _ { \\l } \\gamma _ { \\l - 1 } ^ 2 } { \\alpha _ { \\l } \\gamma _ { \\l + 1 } ^ 2 - \\beta _ { \\l } \\gamma _ { \\l - 1 } ^ 2 } \\norm { ( X u ) _ { \\l } } ^ 2 \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} L ( z _ 1 , \\dots , z _ n ; x , y ) = \\sum _ { d \\geq 0 } x ^ { - d } \\sum \\limits _ { a \\in A _ { + , d } } a ( z _ 1 ) \\dots a ( z _ n ) \\langle a \\rangle ^ y \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} \\sum _ { \\substack { d \\mid n \\\\ ( \\frac { n } { d } , i ) = f ' } } \\mu ( d ) = \\sum _ { \\substack { d \\mid \\frac { n } { f ' } \\\\ ( \\frac { n } { d f ' } , \\frac { i } { f ' } ) = 1 } } \\mu ( d ) . \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\tilde { \\phi } = - ( - \\Delta ) ^ { s } \\tilde { \\phi } + p ( u ^ * _ { \\mu , \\xi } ) ^ { p - 1 } \\tilde { \\phi } + \\tilde { N } ( \\tilde { \\phi } ) + S ( u ^ * _ { \\mu , \\xi } ) & \\Omega \\times ( t _ 0 , \\infty ) , \\\\ \\tilde { \\phi } = - u ^ * _ { \\mu , \\xi } & ( \\mathbb { R } ^ n \\setminus \\Omega ) \\times ( t _ 0 , \\infty ) , \\end{cases} \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} \\dim B _ 0 V = \\dim V - \\dim ( V \\cap \\ker B _ 0 ) = \\dim V - \\dim \\big ( V \\cap ( \\ker B _ + ) ^ { \\perp _ { \\mathcal Q } } \\big ) . \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\mu _ n } ( f g ) = \\mathbb { E } _ { \\mu _ n } \\left ( g ( \\psi ) \\mathbb { E } _ { \\mu _ { \\Lambda , 1 / n } ^ \\psi } ( f ) \\right ) \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} V ( x ^ + ) - V ( x ) & = \\mathcal { U } ( \\tilde { g } ^ + ) - \\mathcal { U } ( \\tilde { g } ) \\\\ & \\leq \\min _ { g _ q \\in \\mathbb { Q } } \\mathcal { U } ( \\tilde { g } g _ q ) - \\mathcal { U } ( \\tilde { g } ) \\leq - \\delta , \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} L _ g ( u ) = \\alpha _ s ^ f ( M , g ) u ^ { s - 1 } . \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} L ( \\phi ) - L ( \\phi _ { B } ) & \\leq 2 \\sum _ { i = 0 } ^ { 1 } E \\left [ \\Big | p ( \\textbf { Z } , i ) P ( Y = i \\vert \\textbf { Z } ) - f _ { i } ( \\textbf { Z } ) \\Big | \\right ] \\\\ & + 2 \\sum _ { i = 0 } ^ { 1 } E \\left [ \\Big | ( 1 - q ( \\textbf { X } , i ) ) P ( Y = i \\vert \\textbf { X } ) - g _ { i } ( \\textbf { X } ) \\Big | \\right ] . \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} \\begin{aligned} & \\underset { ( p _ 1 , p _ 2 , \\dots , p _ { n - 1 } ) } { } & & \\mathbf { E } [ C ] \\\\ & & & \\mathbf { E } \\left [ L _ { n - 1 } \\right ] \\le \\varepsilon _ { \\mathrm { t r e e } } , \\\\ & & & \\textstyle \\sum _ { i = 1 } ^ { n - 1 } \\log _ 2 \\left ( { 1 } / { p _ i } \\right ) = M - B , \\\\ & & & p _ i \\in \\left [ { 1 } / { 2 ^ { J } } , 1 \\right ] ~ \\forall ~ i \\in [ 1 : n - 1 ] . \\end{aligned} \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} H ^ \\pm _ { \\gamma } = H ^ \\pm _ { C _ R } \\circ \\psi . \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} L _ i L _ j u - L _ j L _ i u = c _ { i j } ^ k L _ k u = c _ { i j } ^ k ( a \\nabla \\xi _ k ) _ { l ' } \\frac { \\partial u } { \\partial x _ { l ' } } \\ , , \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} D _ { n } [ \\sigma ] = \\mathbf { G } [ \\tau ] ^ { n + 1 } n ^ { - \\beta ^ { 2 } } c \\left ( 1 + o ( 1 ) \\right ) \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} \\limsup _ { r \\rightarrow \\infty } \\frac { T ( r + | c | , f ) } { T ( r - | c | , f ) } = \\infty \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} H _ 1 ^ \\top H _ 1 = \\ell I _ n + a A + b ( J _ n - A - I _ n ) . \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} \\sum _ { \\lambda = i } ^ { m + i - 1 } = & ~ ( - 1 ) ^ i ~ h ( a ^ { m + n - 1 } _ { 1 , i - 1 } , ~ a _ i ^ { m + n - 2 } \\cdot f ^ { n - 1 } , ~ a ^ { m + n - 1 } _ { i + m + 1 , j } , ~ g ^ m , ~ a ^ { m + n - 1 } _ { j + n + 1 , m + n + p - 1 } ) \\\\ & ~ + ( - 1 ) ^ { m + i - 1 } ~ h ( a ^ { m + n - 1 } _ { 1 , i - 1 } , ~ f ^ { n - 1 } \\cdot a ^ { m + n - 2 } _ { i + m } , ~ a ^ { m + n - 1 } _ { i + m + 1 , j } , ~ g ^ m , ~ a ^ { m + n - 1 } _ { j + n + 1 , m + n + p - 1 } ) . \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} \\P _ 1 : & = \\{ p \\equiv 1 \\bmod 1 2 \\ ; : \\ ; p = ( 6 k + 2 ) ^ 2 + ( 6 t + 3 ) ^ 2 \\} , \\\\ \\P _ 2 : & = \\{ p \\equiv 1 \\bmod 1 2 \\ ; : \\ ; p = ( 6 k ) ^ 2 + ( 6 t + 1 ) ^ 2 \\} . \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} & \\mathbb { J } _ { s , n } ( 0 ) = { \\begin{bmatrix} 0 & I _ n & 0 & \\ldots & 0 & 0 \\\\ 0 & 0 & I _ n & \\ldots & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\ldots & 0 & I _ n \\\\ 0 & 0 & 0 & \\ldots & 0 & 0 \\\\ \\end{bmatrix} , } \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{gather*} x \\mapsto x y ^ 3 , y \\mapsto y ^ 4 \\Longrightarrow \\mathcal { I } _ { 1 , \\beta _ 1 , \\beta _ 2 } = 4 \\int _ { [ 0 , 1 ] ^ 2 } \\tilde { f } _ { t , s } ( x , y ) ^ s x ^ { \\beta _ 1 } y ^ { 3 \\beta _ 1 + 4 \\beta _ 2 + 2 1 s } \\frac { d x } { x } \\frac { d y } { y } \\end{gather*}"}
-{"id": "2622.png", "formula": "\\begin{align*} p _ { L , N + 1 } ^ { \\ell _ 2 } ( x ) = \\sum _ { \\ell = 0 } ^ L \\frac { \\tilde { \\Phi } _ { \\ell } ( x ) } { 1 + \\lambda \\mu _ { \\ell } ^ 2 } \\sum _ { j = 0 } ^ N \\omega _ j \\tilde { \\Phi } _ { \\ell } ( x _ j ) f ( x _ j ) . \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\lambda : = \\frac { \\sqrt { M } } { \\| V \\| _ { L ^ 2 } } \\geq 1 , \\lambda _ n : = \\frac { \\sqrt { M } } { \\| r _ n \\| _ { L ^ 2 } } \\geq 1 . \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} \\chi _ { D _ j } ( x ) : = \\left \\{ \\begin{array} { l l } 1 \\mbox { i f } x \\in D _ j , \\\\ 0 \\mbox { e l s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} u _ t = A u + g ( u ) \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} - \\Delta Q _ 0 + Q _ 0 = | Q _ 0 | ^ { \\alpha } Q _ 0 . \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} \\| \\phi _ k \\| _ { a , \\tau _ 1 ^ k } = 1 , \\| h _ k \\| _ { 2 s + a , \\tau _ 1 ^ k } \\to 0 . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} C ' \\coloneqq \\prod _ { s = 1 } ^ n \\prod _ { j = 1 } ^ r e ^ { d _ { j , s } \\varpi _ s } ( \\zeta ( j , s ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} C ' = e ^ { d _ { 1 , 2 } \\varpi _ 2 } ( p _ { 1 , 3 } ) ^ { - 1 } e ^ { d _ { 1 , 3 } \\varpi _ 3 } ( p _ { 1 , 1 } p _ { 1 , 2 } p _ { 1 , 3 } ) ^ { - 1 } e ^ { d _ { 2 , 1 } \\varpi _ 1 } ( p _ { 2 , 1 } ) ^ { - 1 } e ^ { d _ { 2 , 2 } \\varpi _ 2 } ( p _ { 1 , 3 } p _ { 2 , 1 } ) ^ { - 1 } , \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} \\mod ( \\alpha ) = 2 d ( \\Theta ^ { - 1 } ( \\alpha ) ) , \\mbox { w h e r e } \\Theta ( R ) : = d ( R ) / R . \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} \\mathbb { F } : = - \\frac 1 2 \\mathrm { d } \\mathbb { A } ^ \\flat = - \\frac i 2 ( p \\wedge u ^ \\flat ) \\ , e ^ { i p _ \\gamma x ^ \\gamma } , \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\sum _ { r _ 1 = 1 } ^ { n } \\frac { a _ { r _ 1 } b _ { r _ 1 } c _ { r _ 1 } } { ( r _ 1 + k + 1 ) ^ 3 } = \\sum _ { r _ 1 = 1 } ^ { n } a _ { r _ 1 } b _ { r _ 1 } c _ { r _ 1 } \\left ( \\zeta ( 3 ) - H ^ { ( 3 ) } _ { r _ 1 } \\right ) ; \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} \\phi _ v ( y ) = \\phi _ v ( y ) - \\phi _ v ( \\theta _ { \\lambda } v ) \\leq \\max _ { z \\geq \\varepsilon _ 0 v } \\left \\{ \\phi _ v '' ( z ) \\right \\} \\frac { ( y - \\theta _ { \\lambda } v ) ^ 2 } { 2 } \\leq - c _ 2 \\frac { ( y - \\theta _ { \\lambda } v ) ^ 2 } { v } . \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} f ( t , x ) = f ( t , 0 ) - \\frac { x ^ { 1 - m / 2 } } { ( 2 t ) ^ { ( n - m ) / 2 } } \\psi \\left ( \\frac { x } { 2 t } \\right ) , t > 0 , x > 0 , \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} \\gamma _ - ( u ) & \\leq \\liminf _ { n \\rightarrow \\infty } \\left [ \\norm { u _ n ^ - } ^ 2 + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u _ n ^ - } u _ n ^ 2 d x - \\int _ { \\mathbb R ^ 3 } f ( u _ n ^ - ) u _ n ^ - d x \\right ] \\\\ & = \\liminf _ { n \\rightarrow \\infty } \\gamma _ - ( u _ n ) = 0 , \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} | B _ i | = \\frac { r ! } { t _ { i + 1 } } \\prod _ { j = i + 1 } ^ h p _ j | G _ i | i \\in [ h ] . \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} B _ { \\overline { \\nu } , 0 } ( t , \\overline { \\alpha } ) e ^ { \\alpha _ j t } - B _ { \\overline { \\nu } , j } ( t , \\overline { \\alpha } ) = S _ { \\overline { \\nu } , j } ( t , \\overline { \\alpha } ) , j = 1 , \\ldots , m , \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} \\cot ( \\eta ( n + 1 ) + \\gamma ( n ) ) = \\cot ( \\eta ( n ) + \\gamma ( n ) ) - \\frac { 2 } { \\omega } b ' _ { n + 1 } \\vert \\varphi ( n ) \\vert ^ 2 \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} 0 = F _ { p - 1 } A \\subseteq F _ p A = A . \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} 0 & = \\nabla ^ 2 ( \\hat y c ) ( \\hat x ) [ x _ 2 - x _ 1 ] + \\nabla c ( \\hat x ) ^ \\top ( y _ 2 - y _ 1 ) \\\\ y _ 2 - y _ 1 & = Q _ { k _ j } \\nabla c ( \\hat x ) [ x _ 2 - x _ 1 ] + A P _ j ( \\mu _ { j _ 2 } - \\mu _ { j _ 1 } ) \\\\ 0 & = A ^ \\top \\nabla c ( \\hat x ) [ x _ 2 - x _ 1 ] . \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} g _ { P } ( x ) = f _ { 0 } + ( f _ { 1 } - f _ { 0 } ) x + ( f _ { 2 } - f _ { 1 } ) x ^ { 2 } + \\dotsb + ( f _ { m } - f _ { m - 1 } ) x ^ { m } , \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\hat { y } ^ { N } \\cdot V = 0 . \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} E _ { \\Omega } [ T ] : = \\frac { 1 } { 2 } \\int _ { \\Omega } \\left ( ( x _ { 1 } - T _ { 1 } ( x ) ) ^ { 2 } + ( x _ { 2 } - T _ { 2 } ( x ) ) ^ { 2 } - 2 x _ { 3 } T _ { 3 } ( x ) \\right ) \\ , d x , \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} h ( t ) + u _ 4 ( t , 0 ) + \\alpha u _ 3 ( t , 0 ) \\chi _ { [ T _ { o p t } - \\tau _ 1 , T - \\tau _ 1 ] } = \\int _ { T - \\tau _ 2 } ^ t g ( t - s ) u _ 4 ( s , 0 ) \\ , d s + \\int _ t ^ { T - \\tau _ 1 } f ( s - t ) u _ 4 ( s , 0 ) \\ , d s , \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} \\ \\| { \\bf r } ^ k \\| _ 2 ^ 2 / \\sigma ^ 2 = \\chi ^ 2 _ { n - k } k \\geq k _ 0 \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\sin \\pi ( s _ j - s _ i ) = \\sin \\pi ( n _ j - n _ i ) = 0 . \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} \\tilde w _ { i , j } & = \\int ^ { T } _ { 0 } \\tilde \\chi _ j ( t ) f _ { i } ( t ) \\ , d t , \\\\ f _ i ( t ) & = \\sum _ { \\ell = 1 } ^ s \\tilde w _ { i , \\ell } \\ , \\tilde \\chi _ \\ell ( t ) = \\sum _ { \\ell = 1 } ^ s \\delta ^ { - 1 / 2 } _ \\ell \\tilde w _ { i , \\ell } \\ , \\chi _ \\ell ( t ) . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} E _ 0 ^ { ( r ; [ - L _ 1 , L _ 2 ] ) } T _ a = \\begin{cases} \\frac { a ( a + L _ 1 ) } D , \\ 0 < a \\le L _ 2 ; \\\\ \\frac { | a | ( | a | + L _ 2 ) } D , \\ - L _ 1 \\le a < 0 . \\end{cases} \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} \\forall N > 0 \\ \\exists \\ n \\ge N : 2 \\rho ( n ) - \\rho ( 2 n ) = \\ln n , \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} g _ 1 '' ( \\omega ) & = a '' _ { n m } ( \\omega ) - ( m ^ 2 \\omega ^ { m ^ 2 - 1 } ) ^ 2 a _ n '' ( \\omega ^ { m ^ 2 } ) - ( m ^ 2 ( m ^ 2 - 1 ) \\omega ^ { m ^ 2 - 2 } ) a _ n ' ( \\omega ^ { m ^ 2 } ) \\\\ & = \\frac { m ^ 4 b _ n + m ^ 2 c _ n } { \\omega ^ 2 } - \\frac { m ^ 4 ( b _ n + c _ n ) } { \\omega ^ 2 } - \\frac { m ^ 2 ( m ^ 2 - 1 ) a _ n ' ( 1 ) } { \\omega ^ 2 } = \\frac { - m ^ 2 ( m ^ 2 - 1 ) ( c _ n + a _ n ' ( 1 ) ) } { \\omega ^ 2 } \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} \\widetilde { f } ( r ) : = \\left ( \\frac { 1 } { | \\wp | } \\int _ { \\wp } | f ( r y ) | ^ { p } d \\sigma ( y ) \\right ) ^ { 1 / p } , \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} ( z , u ) \\leftrightarrow \\left ( z , \\sum _ { p = 1 } ^ n u _ p \\ , d z _ p \\right ) , \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} \\widehat { u } _ { m } \\ , : = \\ , \\Big ( \\int ^ { T } _ { 0 } X _ { t } ^ { 2 } { \\mathrm d } t \\Big ) ^ { - 1 } \\cdot \\Big ( \\int ^ { T } _ { 0 } X _ { t } ^ { 2 } { \\mathrm d } t + \\int ^ { T } _ { 0 } X _ { t } { \\mathrm d } X _ { t } \\Big ) \\ , = \\ , 1 - \\Big ( 2 \\int ^ { T } _ { 0 } X _ { t } ^ { 2 } { \\mathrm d } t \\Big ) ^ { - 1 } \\big ( T - X _ { T } ^ { 2 } \\big ) \\ , . \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} \\Delta _ { X } = \\Delta _ { 0 } + \\cdots + \\Delta _ { s } + \\Delta ^ { s + 1 } \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} = y ^ { - \\upsilon } \\sum _ { k = 0 } ^ { \\infty } \\bigg [ ~ \\frac { ( \\lambda ) _ { k } } { k ! } \\sum _ { \\ell = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ { \\ell } ( \\lambda b + b k ) ^ { \\ell } \\Gamma \\left ( \\upsilon + \\frac { \\ell } { 2 } \\right ) } { y ^ { \\frac { \\ell } { 2 } } ~ \\ell ! } \\cos \\left ( \\frac { \\upsilon \\pi } { 2 } + \\frac { \\ell \\pi } { 4 } \\right ) \\bigg ] , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} \\bigg \\langle \\psi \\bigg | \\frac { \\delta F } { \\delta \\psi } \\bigg \\rangle = 0 , \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} R ( q ) : = \\dfrac { ( q ; q ^ { 5 } ) _ { \\infty } ( q ^ { 4 } ; q ^ { 5 } ) _ { \\infty } } { ( q ^ { 2 } ; q ^ { 5 } ) _ { \\infty } ( q ^ { 3 } ; q ^ { 5 } ) _ { \\infty } } . \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} \\left \\{ m \\in M \\ \\middle \\vert \\ K _ i m = \\xi ^ { \\langle \\lambda , \\alpha _ i \\rangle } m , L _ i m = \\xi ^ { \\langle \\mu , \\alpha _ i \\rangle } m , \\ \\forall 1 \\leq i \\leq n \\right \\} \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } W _ t f ( n ) = \\sum _ { m = 0 } ^ { \\infty } f ( m ) \\int _ X e ^ { ( x - s ^ { + } ) t } ( x - s ^ { + } ) p _ m ( x ) p _ n ( x ) \\ , d \\mu ( x ) . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} \\psi = i d _ { [ 0 , \\infty ) ^ s } \\times \\overline { \\psi } : [ 0 , \\infty ) ^ s \\times \\mathbb { E } ' & \\rightarrow [ 0 , \\infty ) ^ s \\times \\mathbb { R } ^ { k - s } \\times \\mathbb { W } , \\\\ ( v , e ) & \\mapsto ( v , \\overline { \\psi } ( e ) ) \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} a _ n ^ { ( \\alpha , \\beta ) } = \\frac { 2 } { 2 n + \\alpha + \\beta + 2 } \\sqrt { \\frac { ( n + 1 ) ( n + \\alpha + 1 ) ( n + \\beta + 1 ) ( n + \\alpha + \\beta + 1 ) } { ( 2 n + \\alpha + \\beta + 1 ) ( 2 n + \\alpha + \\beta + 3 ) } } , n \\geq 1 , \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} \\rho ( n , m ) = \\rho _ { \\ell } ( n , m ) = \\min _ { 1 \\leq i , j \\leq \\ell } | i n - j m | . \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} & ~ h _ { i , j } + ( - 1 ) ^ { m - 1 } h ' _ { i + 1 , j } + ( - 1 ) ^ { ( m - 1 ) + ( n - 1 ) } h '' _ { i + 1 , j + 1 } \\\\ = & ~ ( \\delta _ \\alpha h ) ( a ^ { m + n - 2 } _ { 1 , i } , ~ f ^ { n - 1 } , ~ a ^ { m + n - 2 } _ { i + m + 1 , j } , ~ g ^ { m - 1 } , ~ a ^ { m + n - 2 } _ { j + n + 1 , m + n + p - 1 } ) = 0 , \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} & u \\in C _ { \\partial \\Omega } ( [ 0 , T ] \\times \\Omega ) \\cap C ^ { 1 , 2 } ( ( 0 , T ) \\times \\Omega ) , \\\\ & | \\partial _ t u ( t , x ) | \\le C t ^ { - \\gamma } , ( t , x ) \\in ( 0 , T ] \\times \\Omega , \\gamma \\in ( 0 , 1 ) , \\ C > 0 , \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} P ^ { k - 1 } ( z ) + Q ^ { k - 1 } ( z ) + \\overline Q ^ { k } ( z ) = 0 . \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} p ^ s \\sum _ { \\substack { n = 1 \\\\ p \\nmid n } } ^ { p r } \\frac { 1 } { n ^ s } = \\sum _ { k = 0 } ^ \\infty ( - 1 ) ^ k { r + k \\choose k + 1 } r ^ { k + 1 } \\zeta _ p ( s + k + 1 ) , \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} \\lvert b ( s , X _ { s } , F _ { s } ^ { ( u ) } ) \\rvert \\ , = \\ , \\Big \\lvert \\int _ { \\mathbb R } \\widetilde { b } ( s , X _ { s } , y ) { \\mathrm d } F _ { s } ^ { ( u ) } ( y ) \\Big \\rvert \\le C _ { T } \\big ( 1 + u ( \\lvert X _ { s } \\rvert + \\lvert \\widetilde { X } _ { s } \\rvert ) + ( 1 - u ) ( \\lvert X _ { s } \\rvert + \\mathbb E [ \\lvert X _ { s } \\rvert ] ) \\big ) \\ , \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} F [ \\widetilde Q ] = 3 \\left ( ( 2 - q ) ^ 2 + ( 1 + q ) ^ 2 + q ^ 2 \\right ) = 3 ( 5 - 2 q + 3 q ^ 2 ) , \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\phi _ \\delta ( f ) : = \\phi ( f _ \\delta ( x ) ) . \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} 1 = \\norm { \\phi _ v } _ { D } ^ { 2 } = \\int _ { \\mathbb R ^ 3 } \\abs { \\nabla \\phi _ v } ^ 2 d x = \\int _ { \\mathbb R ^ 3 } \\phi _ v v ^ 2 d x \\leq C \\norm { v } ^ 4 \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} u ( x ) = R ( x ) | \\varphi ( x ) | \\sin \\theta ( x ) \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} \\overline { v } = \\sum \\limits _ { j = 1 } ^ { \\infty } \\beta _ j \\psi _ j + \\sum \\limits _ { l = 1 } ^ m \\tilde { \\beta } _ l z _ l , \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} \\overline { \\nabla } _ { U } V = \\nabla _ { U } V + \\alpha \\{ \\eta ( V ) U - g ( U , V ) \\xi \\} + \\beta \\{ \\eta ( V ) \\phi U - g ( \\phi U , V ) \\xi \\} . \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} v ( w ) = \\int _ { \\partial S _ { \\varepsilon } ^ + } e ^ { z w } u _ { \\varepsilon ' } ( z ) d z , \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} 2 ^ { m ^ + } D \\prod _ k \\lambda _ k ^ { c _ k } \\ge \\det \\Big ( \\sum _ { i \\le m ^ + } c _ i \\lambda _ i \\ , u _ i \\otimes u _ i \\Big ) = \\det ( u _ 1 , \\ldots , u _ { m ^ + } ) ^ 2 \\prod _ { i \\le m ^ + } c _ i \\lambda _ i . \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} \\chi _ D ^ { } ( Q ) : = \\left \\{ \\begin{array} { l l } \\bigl ( \\frac { D } { r } \\bigr ) , & ( a , b , c , D ) = 1 ( r , D ) = 1 Q r , \\\\ 0 , & ( a , b , c , D ) > 1 \\end{array} \\right . \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} \\| \\lambda | \\ell _ { u ' } \\| \\leq \\sum _ { j = 0 } ^ \\infty \\left \\| \\lambda ^ { ( j ) } | \\ell _ { u ' } \\right \\| \\leq \\sum _ { j = 0 } ^ \\infty \\| \\lambda ^ { ( j ) } \\| ^ { ( j ) } _ { u , p } , \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} Y _ { j } ( x ) = \\frac { \\partial } { \\partial x _ { j } } - x _ { j } x . \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} \\varphi ( [ x , y ] ) = [ \\varphi ( x ) , \\varphi ( y ) ] \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 2 u ^ 3 - 2 \\lambda ( u ^ 3 ) ^ 2 & - 1 - 2 \\lambda u ^ 3 & - u ^ 1 + \\lambda u ^ 2 + 2 \\lambda u ^ 1 u ^ 3 \\\\ - 1 - 2 \\lambda u ^ 3 & - 2 \\lambda & \\lambda u ^ 1 \\\\ - u ^ 1 + \\lambda u ^ 2 + 2 \\lambda u ^ 1 u ^ 3 & \\lambda u ^ 1 & - 2 \\lambda ( u ^ 1 ) ^ 2 \\end{pmatrix} , \\\\ \\omega _ { 1 2 } = 0 , \\omega _ { 2 3 } = \\frac { \\sqrt { 3 } \\lambda ^ 2 u ^ 1 } { \\sqrt { - \\det g } } , \\omega _ { 3 1 } = \\frac { \\sqrt { 3 } \\lambda ^ 2 u ^ 2 } { \\sqrt { - \\det g } } , \\end{gather*}"}
-{"id": "4109.png", "formula": "\\begin{align*} h _ \\mathrm { t o p } ( f ) = \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\# \\mathrm { F i x } ( f ^ n ) \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\rho _ { \\infty } \\bar \\phi _ { ( u _ n ^ 1 - v _ 1 ( \\cdot - y _ n ^ 1 ) ) } ( u _ n ^ 1 - v _ 1 ( x - y _ n ^ 1 ) ) ^ 2 & = \\int _ { \\R ^ 3 } \\rho _ { \\infty } \\bar \\phi _ { u _ n ^ 1 } ( u _ n ^ 1 ) ^ 2 - \\int _ { \\R ^ 3 } \\rho _ { \\infty } \\bar \\phi _ { v _ 1 } v _ 1 ^ 2 + o ( 1 ) . \\\\ \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} \\mathcal F \\left ( A ^ { - s / 2 } ( P _ N h ) \\right ) ( \\xi ) & = \\left ( 1 + \\frac { \\xi ^ 2 } { 2 } \\right ) ^ { - s / 2 } \\mathcal F ( P _ N h ) ( \\xi ) \\\\ & = \\left ( 1 + \\frac { \\xi ^ 2 } { 2 } \\right ) ^ { - s / 2 } \\mathcal F ( h ) ( \\xi ) \\mathcal F ( \\phi _ N ) ( \\xi ) \\\\ & = \\mathcal F \\left ( A ^ { - s / 2 } h \\right ) ( \\xi ) \\mathcal F ( \\phi _ N ) ( \\xi ) . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} V ( x ) = \\frac { O ( 1 ) } { 1 + x } \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} p _ k ^ { ( \\rm i d ) } = \\left ( \\frac { \\mu _ 2 ^ * } { \\ln 2 \\left ( \\nu _ 2 ^ * - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 \\right ) } - \\frac { \\sigma ^ 2 } { [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } \\right ) ^ + \\quad \\forall \\ , k = 1 , 2 , \\ldots , r . \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} W _ { \\xi ^ { - 1 } _ p \\tilde { \\pi } _ p } ( g _ { t , k , v } ) = \\sum _ { \\mu _ p \\in { } _ p \\mathfrak { X } _ k } c _ { t , k } ( \\mu _ p ) \\mu _ p ( v ) , \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} \\partial _ i K ( \\xi - d \\Gamma ( m ) ) & Q _ 0 ( k , \\xi ) = \\partial _ i K ( \\xi - d \\Gamma ( m ) ) \\frac { \\omega ( k ) } { H _ \\mu ( \\xi ) - \\Sigma ( \\xi ) + \\omega ( k ) + 1 } \\\\ & + \\partial _ i K ( \\xi - d \\Gamma ( m ) ) \\frac { 1 } { H _ \\mu ( \\xi ) - \\Sigma ( \\xi ) + \\omega ( k ) + 1 } Q _ 0 ( k , \\xi ) \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} N = \\sum _ { i = 1 } ^ d \\rho _ q ( i , m _ i ) = \\sum _ { i = 1 } ^ d \\rho _ q ( i , n _ i ) \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} B ^ \\kappa = - 2 \\left [ A ^ \\kappa ( \\nabla _ \\gamma A ^ \\gamma ) - A ^ \\gamma ( \\nabla _ \\gamma A ^ \\kappa ) \\right ] . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} \\ell _ { N } = \\Lambda ( a _ { * } - a _ { N } ) ^ { - \\frac { 1 } { q + 1 } } \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} T _ V ^ { - 1 } F ( t ) = ( T _ S ^ { - 1 } \\circ \\left ( \\Phi ^ * \\right ) ^ { - 1 } ) F ( t ) = T ^ { - 1 } _ S \\left ( \\left ( \\Phi ^ * \\right ) ^ { - 1 } F \\right ) ( t ) = \\int _ { \\R } \\left ( \\Phi ^ * \\right ) ^ { - 1 } F ( x + i c ) e ^ { 2 \\pi c t } e ^ { - i 2 \\pi x t } \\d x , \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x _ k } \\left [ 2 \\psi ^ \\prime ( | \\nabla \\Q | ^ 2 ) Q _ { i j , k } \\right ] = \\frac { 1 } { L } \\left ( - A Q _ { i j } - B \\left ( Q _ { i p } Q _ { p j } - | \\mathbf { Q } | ^ 2 \\delta _ { i j } / 3 \\right ) + C Q _ { p q } Q _ { p q } Q _ { i j } \\right ) \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} \\chi _ { \\ell } ( K _ n \\square K _ { 1 , s } ) = \\begin{cases} n & s < n ! \\\\ n + 1 & s \\geq n ! . \\end{cases} \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} m = \\left ( \\begin{array} { c c } e ^ b \\left ( a e ^ { - d } p + e ^ { - d } \\right ) & e ^ { - b - d } p - e ^ b \\left ( a e ^ { - d } p + e ^ { - d } \\right ) q \\\\ e ^ b \\left ( a \\left ( e ^ d - c e ^ d p \\right ) - c e ^ d \\right ) & e ^ { - b } \\left ( e ^ d - c e ^ d p \\right ) - e ^ b \\left ( a \\left ( e ^ d - c e ^ d p \\right ) - c e ^ d \\right ) q \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} w _ 0 ^ \\pm \\circ X ^ { - 1 } _ \\gamma ( s , t ) & = w _ 0 ^ \\pm \\circ X ^ { - 1 } _ \\gamma ( s , 0 ) + t \\partial _ t w _ 0 ^ \\pm \\circ X ^ { - 1 } _ \\gamma ( s , t ) \\mid _ { t = 0 } + \\dots \\\\ & = w _ 0 ^ \\pm ( s , 0 ) + t \\partial _ n w _ 0 ^ \\pm \\circ X ^ { - 1 } _ \\gamma ( s , 0 ) + \\dots \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\beta _ { 1 } ^ { \\tau } x & = \\frac { 1 } { \\sqrt { 2 } } u + \\frac { 1 } { \\sqrt { 2 } } v \\\\ \\beta _ { 1 } ^ { \\tau } x & = \\frac { 1 } { \\sqrt { 2 } } u - \\frac { 1 } { \\sqrt { 2 } } v , \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} K _ 1 & = 2 ^ { 3 / 2 } \\frac { 2 d + 1 } { d } \\frac { \\Gamma ( \\frac { 1 + d } { 2 } ) } { \\Gamma ( d / 2 ) } \\\\ K _ 2 & = 2 \\sqrt { \\frac 2 \\pi } \\sqrt d . \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} \\sum \\limits _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) } { a } = \\log _ { \\varphi } ( 1 ) = \\frac { \\log _ { \\phi } ( \\omega _ n ) } { \\omega _ n } . \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} \\chi ( M ^ { 2 n } ) & = \\sum _ { p = 0 } ^ { 2 n } ( - 1 ) ^ p b ^ p ( M ^ { 2 n } , g _ J ) \\\\ & = \\sum _ { p = 0 } ^ { 2 n } ( - 1 ) ^ p b ^ p ( M ^ { 2 n } , g _ t ) , t \\in [ 0 , 1 ) . \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} \\lVert & ( H _ { \\eta } ( v _ g , \\omega ) + C _ g + i ) ^ { - 1 } - ( H _ { 0 } ( v _ g , \\omega ) + C _ g + i ) ^ { - 1 } \\lVert \\\\ & = \\lVert ( V _ g H _ { \\eta } ( v _ g , \\omega ) V _ g ^ * + C _ g + i ) ^ { - 1 } - ( V _ g H _ { 0 } ( v _ g , \\omega ) V _ g ^ * + C _ g + i ) ^ { - 1 } \\lVert \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} \\rho _ { s , 3 } ( \\widetilde { \\mathcal A } _ { s } \\varphi ) \\ , : = \\ , { \\mathbb E } _ { 0 } \\Big [ Z _ { s } ^ { - 1 } \\Big ( \\varphi ^ { \\prime } ( \\widetilde { X } _ { s } ) b ( s , \\widetilde { X } _ { s } , { F } _ { s , 2 } ) + \\frac { 1 } { \\ , 2 \\ , } \\varphi ^ { \\prime \\prime } ( \\widetilde { X } _ { s } ) \\Big ) \\Big \\vert \\mathcal F _ { T } ^ { X } \\Big ] \\ , , 0 \\le s \\le T \\ , . \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} \\mathcal { B } ( \\lambda ) : = \\left \\{ \\left . \\tilde { f } _ { i _ 1 } \\cdots \\tilde { f } _ { i _ r } ( b _ \\lambda ) \\ \\right | \\ r \\geq 0 , \\ i _ 1 , \\ldots , i _ r \\in I \\right \\} \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } { \\log m _ n ( x , y ) \\over - \\log n } = { 1 \\over \\eta } \\limsup _ { n \\to \\infty } { \\log m _ n ( x , y ) \\over - \\log n } = 1 . \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} G _ n ( x ) & = \\frac { 1 } { \\norm { u _ n } _ \\infty } ( f ( x , u _ n ( x ) ) + h ( x ) + U _ 2 ( x ) u _ n ( x ) + \\rho _ n \\Phi _ 1 ( x ) ) , \\\\ I _ n ( x ) & = \\frac { 1 } { \\norm { u _ n } _ \\infty } ( f ( x , - u ^ - _ n ( x ) ) + h ( x ) - U _ 2 ( x ) u ^ - _ n ( x ) - C + ( \\rho _ n \\wedge 0 ) \\Phi _ 1 ( x ) ) . \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} d z _ i ( s ) = & - \\frac { \\partial V } { \\partial z _ i } \\big ( z ( s ) , y ( s ) \\big ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } d w _ i ( s ) \\ , , 1 \\le i \\le m \\ , , \\\\ d y _ j ( s ) = & - \\frac { 1 } { \\delta } \\frac { \\partial V } { \\partial y _ j } \\big ( z ( s ) , y ( s ) \\big ) \\ , d s + \\sqrt { \\frac { 2 \\beta ^ { - 1 } } { \\delta } } \\ , d w _ j ( s ) \\ , , m + 1 \\le j \\le n \\ , . \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} & 1 + \\sum _ { \\beta > 0 , n \\in \\mathbb { Z } } P _ { n , \\beta } q ^ n t ^ { \\beta } \\\\ & = \\exp \\left ( \\sum _ { \\beta > 0 , n > 0 } ( - 1 ) ^ { n - 1 } n N _ { n , \\beta } q ^ n t ^ { \\beta } \\right ) \\cdot \\left ( \\sum _ { \\beta > 0 , n \\in \\mathbb { Z } } L _ { n , \\beta } q ^ n t ^ { \\beta } \\right ) . \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} \\inf \\{ \\theta \\ | \\ ( 1 , { \\bf 0 } , { \\bf 0 } , \\theta ) \\in \\bar { H } \\} : = k _ { s u b } ^ * ( y _ 0 ) . \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} B _ 1 ( t ) = \\begin{cases} | b | & ( | t ^ { - 1 } b ^ 2 | \\le 1 ) , \\\\ ( 2 ^ { - 1 } + 2 ^ { s } ) | t | ^ { s + 1 } | b | ^ { - 2 s - 1 } & ( | t ^ { - 1 } b ^ 2 | > 1 ) . \\end{cases} \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { m = 0 } ^ \\infty f ( m ) p _ m ( x ) , \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} 0 & = k ( 1 , s ) = \\cdots = k ( j _ 1 - 1 , s ) , \\\\ j _ u & = k ( j _ u , s ) = \\cdots = k ( j _ { u + 1 } - 1 , s ) 1 \\leq u \\leq x . \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} \\mathcal { R } _ { k ( - j ) j } ^ { \\nu } ( \\alpha _ { 2 , \\nu } ) = \\{ \\xi \\in \\mathcal { O } _ { \\nu } : | \\langle k , \\omega _ { \\nu } ( \\xi ) \\rangle + \\bar { \\Omega } _ { \\nu , ( - j ) } ( \\xi ) - \\bar { \\Omega } _ { \\nu , j } ( \\xi ) | < \\alpha _ { 2 , \\nu } \\frac { | j | } { \\langle k \\rangle ^ { \\tau } } \\} . \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} \\| \\mathbf { E } _ { } ^ T ( \\boldsymbol { \\alpha } ^ { k + 1 } - \\boldsymbol { \\alpha } ^ { \\star } ) \\| ^ 2 = \\| \\nabla f ( \\mathbf { x } ^ { k + 1 } ) - \\nabla f ( \\mathbf { x } ^ { \\star } ) \\\\ + \\frac { \\rho ( 1 - \\eta ) } { 2 } \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } ( \\mathbf { x } ^ { k + 1 } - \\mathbf { x } ^ { \\star } ) + \\mathbf { M } ( \\mathbf { x } ^ { k + 1 } - \\mathbf { x } ^ k ) \\| ^ 2 , \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} \\zeta ^ { s } ( s _ 1 , \\ldots , s _ k ) : = \\sum _ { i = 0 } ^ k ( - 1 ) ^ { s _ 1 + \\ldots + s _ k } \\zeta ( s _ i , \\ldots , s _ 1 ) \\zeta ( s _ { i + 1 } , \\ldots , s _ k ) \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} H _ 0 \\varphi = - \\varphi ^ { \\prime \\prime } + V _ 0 ( x ) \\varphi = E \\varphi . \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} S ( a , b ) = \\{ z \\in \\C \\ ; | \\ ; a < \\mathrm { I m } \\ , z < b \\} . \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} I _ 2 \\xleftarrow { \\begin{pmatrix} f _ 1 & f _ 2 \\end{pmatrix} } R _ 2 ^ 2 \\xleftarrow { \\begin{pmatrix} - 2 x _ 1 \\\\ x _ 0 \\end{pmatrix} } R _ 2 \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} \\min \\left \\{ \\int _ { \\overline { \\Omega } \\times \\overline { \\Omega } } | | x - y | | \\ , \\mathrm { d } \\gamma : \\ ; \\gamma \\in \\mathcal { M } ^ + ( \\overline { \\Omega } \\times \\overline { \\Omega } ) , \\ , ( \\Pi _ x ) _ { \\# } \\gamma = f ^ + \\ , \\ , \\mbox { a n d } \\ , \\ , \\ , ( \\Pi _ y ) _ { \\# } \\gamma = f ^ - \\right \\} , \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} \\kappa _ { v \\lambda } \\kappa _ { w _ \\circ \\mu } = \\sum _ { \\nu \\in P } a _ { v , w _ \\circ , \\lambda , \\mu } ^ { \\nu } \\kappa _ { \\nu } . \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} \\mu _ k = \\frac { 1 } { 1 + f ( k ) } \\prod \\limits _ { i = 0 } ^ { k - 1 } \\frac { f ( i ) } { 1 + f ( i ) } , k \\in \\N _ 0 . \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} & \\mathbb { H } [ \\phi ^ { n } ] = ( 3 - 2 \\theta ) { \\| \\phi ^ { n } \\| } ^ { 2 } - ( 1 - 2 \\theta ) { \\| \\phi ^ { n - 1 } \\| } ^ { 2 } + ( 2 - \\theta ) ( 1 - 2 \\theta ) { \\| \\phi ^ { n } - \\phi ^ { n - 1 } \\| } ^ { 2 } , ~ n \\geq 1 , \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} d _ 1 h _ { i + 1 } + ( - 1 ) ^ { i } h _ { i + 1 } d _ 1 & = d _ 0 h _ i + ( - 1 ) ^ i h _ i d _ 0 , \\qquad 1 \\leq i \\leq r - 1 , \\\\ 0 & = ( - 1 ) ^ r d _ 0 h _ r + h _ r d _ 0 , \\\\ d _ 1 h _ 1 + h _ 1 d _ 1 & = g - f . \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} \\nabla _ v \\partial _ t = ( \\nabla _ v \\partial _ t ) ^ T - u ^ { - 2 } \\langle \\nabla _ v \\partial _ t , \\partial _ t \\rangle \\cdot \\partial _ t \\\\ = - \\frac { 1 } { 2 } u ^ 2 d \\theta ( v ) + u ^ { - 1 } v ( u ) \\cdot \\partial _ t \\quad \\forall v \\in T S . \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} \\Big ( \\beta ( b ) 1 - \\beta ( 1 ) b \\Big ) D ( a ) + \\Big ( \\beta ( a ) 1 - \\beta ( 1 ) a \\Big ) D ( b ) = 0 . \\end{align*}"}
-{"id": "9930.png", "formula": "\\begin{align*} M = M _ 1 \\times \\dots \\times M _ \\ell , X = X _ 1 \\times \\dots \\times X _ \\ell \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} R ( x , \\xi ) = ( r ( x ) , \\Gamma ( x , \\xi ) ) , \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} \\sigma = \\tau \\epsilon ^ { { \\rm a d } h } , \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( s , t ) = - ( \\Delta + s ) \\dot \\varphi ( s , t ) . \\end{aligned} \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} x _ { u _ 1 v _ 1 , \\theta _ 1 } \\leq \\sum _ { j = 1 } ^ { k - 1 } \\sum _ { v \\notin \\{ u _ 1 , v _ 1 , \\ldots , v _ j \\} } x _ { v _ j v , \\theta _ { j + 1 } } \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\mathbb { E } = \\mathbb { K } \\oplus L . \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} \\lVert \\widehat { k } \\cdot \\nabla K ( \\xi - d \\Gamma ( m ) ) Q _ 0 ( k , \\xi ) \\psi \\lVert ^ 2 \\leq \\sum _ { i = 1 } ^ { \\nu } \\lVert \\partial _ i K ( \\xi - d \\Gamma ( m ) ) Q _ 0 ( k , \\xi ) \\psi \\lVert ^ 2 \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} v = ( x , y , z , \\bar { z } ) \\in \\mathcal { P } ^ { a , p } , | | v | | _ { s , r , p } = \\frac { | x | } { s } + \\frac { | y | _ 1 } { r ^ 2 } + \\frac { | | z | | _ { a , p } } { r } + \\frac { | | \\bar { z } | | _ { a , p } } { r } , \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} \\mu _ \\lambda = \\frac { \\rho _ \\lambda c _ \\Omega | \\partial _ n K ^ \\pm _ \\gamma | } { a _ 0 \\lambda } \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} Q _ { 2 1 } = \\sum _ { i , j , k , l } E ^ k _ l \\otimes E ^ i _ j \\otimes Q ^ { i k } _ { j l } \\in \\mathrm { M a t } _ n ( \\mathbb { C } ) \\otimes \\mathrm { M a t } _ m ( \\mathbb { C } ) \\otimes A . \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} G _ 1 ( q , y ) = \\sum _ { p = 1 } ^ n L _ p ( q ) y ^ p . \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} c _ m : = \\frac { \\mu ( ( m , N ) ) } { m \\cdot a _ m } \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{gather*} \\overset { I } { \\widetilde { L } } { ^ { ( \\pm ) } } = \\bigl ( \\overset { I } { a _ i ^ { ( \\pm ) } } \\bigr ) \\widetilde { b _ i ^ { ( \\pm ) } } \\in \\mathrm { M a t } _ { \\dim ( I ) } \\bigl ( \\mathcal { H } ( \\mathcal { O } ( H ) ) \\bigr ) \\end{gather*}"}
-{"id": "6868.png", "formula": "\\begin{align*} \\partial _ n H _ \\gamma ^ + + \\partial _ n H ^ - _ \\gamma = 0 \\mbox { o n } \\ \\gamma . \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\Phi _ { m + 1 } & = & \\Phi _ m + \\Psi _ m ~ , \\\\ \\\\ { \\rm w h e r e } & & \\\\ \\\\ \\Psi _ m & : = & ( - 1 ) ^ m q ^ { A _ m } + ( - 1 ) ^ { m + 1 } D _ { m + 1 } - ( - 1 ) ^ m D _ m \\\\ \\\\ & = & ( - 1 ) ^ m q ^ { A _ m } ( q ^ { k + m - 1 / 2 } ( 1 + q ^ { k + m + 1 / 2 } ) - q ^ { k + m } ( 1 + q ^ { k + m - 1 / 2 } ) ) / S _ m \\\\ \\\\ & = & ( - 1 ) ^ m q ^ { A _ m + k + m - 1 / 2 } ( 1 - q ^ { 1 / 2 } ) ( 1 - q ^ { k + m } ) / S _ m ~ . \\end{array} \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} ( B ^ t B ) A & = - J _ 0 B ^ { - 1 } J _ 0 B A = - J _ 0 B ^ { - 1 } J _ 0 C B = - J _ 0 B ^ { - 1 } C J _ 0 B \\\\ & = - J _ 0 A B ^ { - 1 } J _ 0 B = - A J _ 0 B ^ { - 1 } J _ 0 B = A ( B ^ t B ) . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} W ( z ) ( s ) = \\big ( \\mathrm d \\ , \\overline \\exp _ { z ( s ) } ( w ( s ) ) \\big ) ^ { - 1 } \\big ( V _ x ( s ) \\big ) , \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} \\gamma ( t ) = 2 \\lambda t - { \\lambda \\over \\mu } ( 1 - e ^ { - \\mu t } ) , t \\ge 0 . \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } \\int _ 0 ^ \\pi e ^ { z \\cos \\theta } \\cos ( m \\theta ) \\ , d \\theta = I _ m ( z ) , | \\arg ( z ) | < \\pi , \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} \\alpha _ 4 : = \\max \\left \\{ ( \\frac { 1 } { 2 } \\| b \\| ^ 2 + \\frac { 1 } { 4 } d ^ 2 ) , \\frac { 1 } { 2 } \\lambda _ { \\max } ^ { \\underline { Q } } \\right \\} s _ 2 . \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} m _ i = \\dim V _ i , \\ 1 \\le i \\le k \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} \\mathcal { C } ^ { I } = \\{ ( C ^ { I } _ i ) _ { i \\in [ n ] } : \\sum _ { i = 1 } ^ { i = n } H _ { j , i } \\cdot C ^ { I } _ i = 0 , j \\in [ r ] \\} , \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} u _ { s } ^ { 1 , m } = \\sum _ { j = 1 } ^ { \\infty } I _ { A _ { j } ^ { m } } ( X _ { t _ { 1 } ^ { m } } ^ { 0 , m } ) u _ { s } ^ { 1 , j , m } \\in \\mathcal { U } ^ { t } [ t _ { 1 } ^ { m } , t _ { 2 } ^ { m } ] \\ ; s \\in \\lbrack t _ { 1 } ^ { m } , t _ { 2 } ^ { m } ] . \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} d B _ t = d \\tilde { B } _ t - u ( t , X _ t ) d t = d X _ t - \\beta _ j ( X _ t ) d \\Z ^ j _ t - u ( t , X _ t ) d t \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} f _ 2 \\sum _ { j = 0 } ^ k { f _ 1 ^ j X _ { m - b - a j } } = X _ m - f _ 1 ^ { k + 1 } X _ { m - ( k + 1 ) a } \\ , , \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} 0 = ( E ' _ c ) ^ \\dagger \\left ( \\sum _ { i = 1 } ^ k \\alpha _ i E ' _ { a _ i } \\right ) = E _ c ^ \\dagger \\left ( \\sum _ { i = 1 } ^ k \\alpha _ i E _ { a _ i } \\right ) , \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} ( h _ { \\lambda , \\mu } , h _ { 0 , 0 } ) _ { \\mathbb { Z } ( \\mathcal { T } ) _ \\xi \\rtimes \\mathcal { S } } = \\sum _ { ( \\lambda ' , \\mu ' ) \\in \\overline { C } } \\frac { \\dim ^ - ( L _ \\xi ( \\lambda ' , \\mu ' ) ) } { \\dim ^ + ( L _ \\xi ( \\lambda ' , \\mu ' ) ) } S _ { ( \\lambda , \\mu ) , ( \\lambda ' , \\mu ' ) } S _ { ( \\lambda ' , \\mu ' ) , ( 0 , 0 ) } \\neq 0 . \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} H = \\Lambda + B + Q _ 2 + R , \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} \\widetilde { B } = \\limsup _ { q \\rightarrow \\infty } \\sup _ { f \\in L ^ { Q } _ { 1 } ( B ( x _ { 0 } , r ) ) \\backslash \\{ 0 \\} } \\frac { \\left \\| \\frac { f } { | \\cdot | ^ { \\frac { \\beta } { q } } } \\right \\| _ { L ^ { q } ( B ( x _ { 0 } , r ) ) } } { q ^ { 1 - 1 / Q } \\| \\nabla _ { H } f \\| _ { L ^ { Q } ( B ( x _ { 0 } , r ) ) } } . \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} N _ { X \\underset { x , y } { \\star } Y } ( m ) = & \\left ( N _ X ( m ) + \\sum N _ { X } ( k _ 1 ) N _ { Y } ( k _ 2 ) + \\sum N _ { X } ( k _ 1 ) N _ { Y } ( k _ 2 ) N _ { X } ( k _ 3 ) + \\cdots \\right ) \\\\ + & \\left ( N _ Y ( m ) + \\sum N _ { Y } ( k _ 1 ) N _ { X } ( k _ 2 ) + \\sum N _ { Y } ( k _ 1 ) N _ { X } ( k _ 2 ) N _ { Y } ( k _ 3 ) + \\cdots \\right ) \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} \\begin{cases} { } ^ { C } _ { 0 } D _ { t } ^ { q } y ( x , t ) = A y ( x , t ) & \\hbox { i n } Q _ { T } \\\\ y ( \\xi , t ) = 0 & \\hbox { o n } \\Sigma _ { T } \\\\ y ( x , 0 ) = y _ { 0 } ( x ) & \\hbox { i n } \\Omega . \\end{cases} \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} D _ 1 ( \\rho \\| \\sigma ) = { D ( \\rho \\| \\sigma ) \\over \\rho ( 1 ) } , \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} \\frac { u ^ { \\prime } ( 0 , E _ j ) } { u ( 0 , E _ j ) } = \\tan \\theta _ j , \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} & \\Pi ^ 2 = \\ , \\Pi \\ , , \\Pi ^ T a = a \\Pi \\ , , \\Pi P = \\Pi \\ , , \\\\ & \\Pi \\nabla \\xi _ i = 0 \\ , , \\Pi ^ T \\eta = \\eta \\ , , | \\Pi \\eta | \\ge | \\eta | \\ , , \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} \\vartheta ( r ) : = \\left \\{ \\begin{array} { c l } 2 r & 0 \\leq r \\leq 1 , \\\\ 2 [ r - ( r - 1 ) ^ 3 ] & 1 < r \\leq 1 + 1 / \\sqrt { 3 } , \\\\ \\vartheta ' < 0 & 1 + 1 / \\sqrt { 3 } < r < 2 , \\\\ 0 & r \\geq 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} D _ { \\widetilde \\Phi } ( y , x ) \\geq \\frac { 1 } { 2 } \\sum _ { i = 0 } ^ n a _ i \\| y \\| _ 2 ^ i \\cdot \\| x - y \\| _ 2 ^ 2 \\qquad \\forall x , y . \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} T _ { w + 1 } = T _ w C _ { w + 1 } \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} \\norm { g } _ { \\mathcal { H } _ p } ^ 2 : = \\int _ { \\C ^ n } \\int _ 0 ^ { \\infty } \\abs { g ( t , w ) } ^ 2 \\frac { e ^ { - 4 \\pi p ( w ) t } } { 4 \\pi t } \\d V ( w ) \\d t < \\infty , \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} \\begin{aligned} ( c _ 0 ^ 2 | \\nabla p | ^ 2 - 1 ) \\bar { C } & = 0 \\\\ - \\nabla \\cdot ( \\bar { C } \\nabla p ) & = S . \\end{aligned} \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\o _ { M , K } ( u , u ' ) & = ( K \\pi ^ * \\o _ B ) ( n + w , n ' + w ' ) + \\mu ( n + w , n ' + w ' ) \\\\ & = ( K \\pi ^ * \\o _ B ) ( w , w ' ) + \\mu ( n , n ' ) + \\mu ( w , w ' ) \\ , . \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} \\lambda _ F ( n , \\tau ) = \\lim _ { t \\to \\infty } \\sum \\limits _ { \\substack { \\rho \\\\ | \\Im ( \\rho ) | \\le t \\\\ 0 \\le \\Re ( \\rho ) \\le \\tau } } \\left ( 1 - \\left ( \\frac { \\rho } { \\rho - \\tau } \\right ) ^ n \\right ) . \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} \\Delta _ { g _ 0 } u - ( \\nabla _ { g _ 0 } h , \\nabla _ { g _ 0 } u ) = - \\lambda u . \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} { \\bf w } _ t ^ { ( i ) } - \\frac { t } { \\Delta } { \\bf w } _ { \\Delta } ^ { ( i ) } = \\frac { 1 } { 2 } a _ { i , 0 } + \\sum _ { r = 1 } ^ { \\infty } \\left ( a _ { i , r } { \\rm c o s } \\frac { 2 \\pi r t } { \\Delta } + b _ { i , r } { \\rm s i n } \\frac { 2 \\pi r t } { \\Delta } \\right ) , \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left \\{ \\frac { k _ n ^ 1 } { n } \\ , \\frac { 1 } { 1 - p _ 1 } + \\frac { k _ n ^ 2 } { n } \\ , \\frac { 1 } { 1 - p _ 2 } \\right \\} = 1 \\hbox { \\rm a . s . } \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} k ( x ) \\ge \\int _ { ( x , 1 ) } ( \\psi ^ { - 1 } ) ' ( - \\log ( x ) ) \\nu ( d r ) = ( \\psi ^ { - 1 } ) ' ( - \\log ( x ) ) \\nu ( ( x , 1 ) ) . \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} f ( x ) = \\lambda \\langle u , x \\rangle + c , \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} \\begin{pmatrix} \\nabla \\xi \\\\ \\nabla \\phi \\end{pmatrix} a \\begin{pmatrix} \\nabla \\xi ^ T & \\nabla \\phi ^ T \\end{pmatrix} = \\begin{pmatrix} \\nabla \\xi a \\nabla \\xi ^ T & 0 \\\\ 0 & \\nabla \\phi a \\nabla \\phi ^ T \\end{pmatrix} = \\begin{pmatrix} \\mathcal { O } ( 1 ) & 0 \\\\ 0 & \\mathcal { O } ( \\frac { 1 } { \\delta } ) \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} \\frac { \\delta H } { \\delta \\tilde { \\rho } } = \\widehat { H } _ e - \\frac { \\hbar ^ 2 } { 2 M } \\left [ K * \\left ( \\bar { D } ^ { - 1 } \\nabla \\bar { \\rho } \\right ) \\right ] = : \\widehat { H } _ \\textit { e f f } \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} F _ { \\eta } = \\left \\{ r \\in \\mathbb { R } ^ + : \\frac { T ( r + s ) - T ( r ) } { T ( r ) } \\cdot h ( r ) \\geq \\eta \\right \\} \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 1 9 2 n + 7 2 ) q ^ n \\\\ & \\equiv 2 f _ 3 ^ 3 + 3 \\dfrac { f _ 3 ^ 3 f _ 4 ^ 5 } { f _ 1 ^ 4 f _ 8 ^ 2 } \\\\ & \\equiv 2 f _ 3 ^ 3 + 3 f _ 3 ^ 3 \\cdot \\dfrac { f _ 4 } { f _ 2 ^ 2 } \\\\ & \\equiv 2 f _ 3 ^ 3 + 3 f _ 3 ^ 3 \\left ( \\dfrac { f _ { 1 2 } ^ 4 f _ { 1 8 } ^ 6 } { f _ 6 ^ 8 f _ { 3 6 } ^ 3 } + 2 q ^ 2 \\dfrac { f _ { 1 2 } ^ 3 f _ { 1 8 } ^ 3 } { f _ 6 ^ 7 } + 4 q ^ 4 \\dfrac { f _ { 1 2 } ^ 2 f _ { 3 6 } ^ 3 } { f _ 6 ^ 6 } \\right ) ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} f _ { K _ { \\varepsilon } } ( z ) = \\begin{cases} 0 \\ , \\ , & \\ , \\ , z \\in K _ { \\varepsilon } , \\\\ + \\infty \\ , \\ , & . \\end{cases} \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} \\lefteqn { 2 h ^ { \\varphi } \\left ( C ^ { \\varphi } \\omega ( X ) ^ { \\varphi } ( Y ' ) , Z ' \\right ) = h ^ { \\varphi } \\left ( ( \\nabla ^ { \\varphi } _ X C ^ { \\varphi } ) ( Y ' ) , Z ' \\right ) + } & \\\\ & + h ^ { \\varphi } \\left ( ( \\nabla _ { \\varphi ^ { - 1 } ( Y ' ) } ^ { \\varphi } C ^ { \\varphi } ) ( \\varphi ( X ) ) ) , Z ' \\right ) - h ^ { \\varphi } \\left ( ( \\nabla _ { \\varphi ^ { - 1 } ( Z ' ) } ^ { \\varphi } C ^ { \\varphi } ) ( \\varphi ( X ) ) ) , Y ' \\right ) . \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} & \\frac { 1 } { \\mu _ 1 } = e ^ { t } ( a ^ 2 - b ^ 2 + c ^ 2 + \\sqrt { - N ( v ) } ) \\\\ & \\mu _ 1 = e ^ { - t } ( a ^ 2 - b ^ 2 + c ^ 2 - \\sqrt { - N ( v ) } ) \\\\ & \\frac { 1 } { \\mu _ 2 } = e ^ { t } ( a ^ 2 - b ^ 2 + c ^ 2 - \\sqrt { - N ( v ) } ) \\\\ & \\mu _ 2 = e ^ { - t } ( a ^ 2 - b ^ 2 + c ^ 2 + \\sqrt { - N ( v ) } ) ~ . \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} \\sum _ { \\eta , \\eta ' \\in \\Omega _ L } A _ { \\eta , \\eta ' } = 2 \\sum _ { i \\in \\mathbb T _ L ^ d } \\sum _ { k = 1 } ^ d \\sum _ { \\eta \\in \\Omega _ L } A _ { \\eta , \\eta ^ { i , i + e _ k } } \\mathbf 1 _ { \\{ \\eta ( i ) > 0 \\} } . \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} \\hat U = \\exp \\left [ - \\frac { i m f ^ { - 1 } \\dot { \\rho } } { 2 \\rho } ( \\hat x _ 1 ^ 2 + \\hat x _ 2 ^ 2 ) \\right ] , \\ , \\ , \\ , \\ , \\ , \\hat U ^ \\dag \\hat U = \\hat U \\hat U ^ \\dag = \\mathbf { I } . \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} 3 | x | = | x | + 2 | x | \\geq | x | + | y | \\geq | y ^ { - 1 } x | , \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} L ' _ { i j } = \\sum _ { T ' \\in S ( G ' ) } \\Pi ( T ' ) & = \\sum _ { T ' \\in S ( G ' ) } \\Pi ( G ' ) \\Pi ( \\Psi ( T ' ) ) = \\\\ & = \\Pi ( G ' ) \\sum _ { T \\in S ( G ) } \\Pi ( T ) = \\Pi ( G ' ) L _ { i j } , \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} \\gamma ( t u ) = I ' ( t u ) [ t u ] > t ^ 4 \\left [ \\norm { u } ^ 2 + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ u u ^ 2 d x - \\int _ { \\mathbb R ^ 3 } \\frac { f ( t u ) } { t ^ 3 } u \\ , d x \\right ] > t ^ 4 \\gamma ( u ) = 0 , \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} { { u } _ { y } } = - { { v } _ { x } } + f \\left ( u , v \\right ) , { { u } _ { x } } = { { v } _ { y } } + g \\left ( u , v \\right ) \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} M ( t ' ) = \\int _ 0 ^ { t ' } \\big [ ( \\nabla \\omega _ \\eta ) ^ T \\sigma \\big ] ( x ( s ) ) \\ , d w ( s ) \\ , , \\mbox { a n d } \\ , \\overline { M } ( t ' ) = \\int _ 0 ^ { t ' } \\big [ ( \\nabla \\omega _ \\eta ) ^ T \\sigma \\big ] ( y ( s ) ) \\ , d \\bar { w } ( s ) \\ , . \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} q _ { n _ 1 , n _ 2 } = & n _ 1 n _ 2 - \\sum _ p \\lfloor n _ 1 / p \\rfloor \\lfloor n _ 2 / p \\rfloor + \\sum _ { p _ 1 \\le p _ 2 } \\lfloor n _ 1 / ( p _ 1 p _ 2 ) \\rfloor \\lfloor n _ 2 / ( p _ 1 p _ 2 ) \\rfloor - . . . \\\\ = & \\sum _ { k = 1 } ^ { n _ 1 } \\nu ( k ) \\lfloor n _ 1 / k \\rfloor \\lfloor n _ 2 / k \\rfloor \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\sum _ { i = 1 } ^ l \\| v ^ 1 _ { i , t } \\| _ { L ^ 2 ( \\rho _ { i , t } ^ 1 ) } \\ , d t + \\int _ 0 ^ T \\sum _ { i = 1 } ^ l \\| v ^ 2 _ { i , t } \\| _ { L ^ 2 ( \\rho _ { i , t } ^ 2 ) } \\ , d t < + \\infty , \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\psi ( a _ 1 , a _ 2 , e _ 1 , e _ 2 ) & = q \\circ ( \\psi _ 1 \\times \\psi _ 2 ) ( a _ 1 , e _ 1 , a _ 2 , e _ 2 ) \\\\ & = q \\circ ( i d _ { [ 0 , \\infty ) ^ { s _ 1 } } \\times \\overline { \\psi } _ 1 \\times i d _ { [ 0 , \\infty ) ^ { s _ 2 } } \\times \\overline { \\psi } _ 2 ) ( a _ 1 , e _ 1 , a _ 2 , e _ 2 ) \\\\ & = ( a _ 1 , a _ 2 , d _ 1 , d _ 2 , w _ 1 , w _ 2 ) . \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{gather*} \\begin{array} { c } \\left \\{ \\beta ( j + u ) = \\beta ( j ) + \\beta ( u ) \\right \\} _ { j = 1 } ^ { k - u } \\\\ \\mbox { a n d } \\ \\ \\beta ( 0 ) = 0 \\ \\ \\mbox { a n d } \\ \\ \\delta _ { n } ^ k = \\beta ( v ) + \\beta ( u ) \\\\ \\mbox { a n d } \\ \\ \\left \\{ \\beta ( j + v ) = \\beta ( j ) + \\beta ( v ) \\right \\} _ { j = 1 } ^ { k - v } \\end{array} \\end{gather*}"}
-{"id": "8726.png", "formula": "\\begin{align*} \\nabla \\phi ( p ) \\not = 0 p \\in \\phi ^ { - 1 } \\big ( [ - \\delta _ 0 , \\delta _ 0 ] \\big ) ; \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} \\frac { n - p } { p - 1 } \\log _ { { \\frac { n - 1 } { p - 1 } } } \\frac { R } { | x | } = 1 - \\left ( \\frac { | x | } { R } \\right ) ^ { \\frac { n - p } { p - 1 } } < 1 \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{gather*} \\langle \\delta , s _ { i j } \\rangle = \\langle s _ { k o } , \\delta \\rangle = \\langle s _ { i j } , s _ { k o } \\rangle = \\langle s _ { i k } , s _ { j k } \\rangle ^ { \\perp } = ^ { \\perp } \\langle s _ { i o } , s _ { j o } \\rangle \\end{gather*}"}
-{"id": "4918.png", "formula": "\\begin{align*} ( 1 + | y | ) | \\nabla _ y \\phi _ j ( y , t ) | \\chi _ { B _ { 2 R } ( 0 ) } ( y ) + | \\phi _ j ( y , t ) | \\leq M \\frac { \\mu _ 0 ^ { n - 2 s + \\sigma } } { 1 + | y | ^ a } , j = 1 , \\cdots , k \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} \\left \\{ ( h , \\bar { h } , g _ s , g _ t ) \\begin{tabular} { | l } $ h \\bar { h } = h _ 1 $ \\\\ $ g _ s \\in c ' ( h , Y ) , g _ t \\in c ' ( \\bar { h } , Y ) $ \\\\ $ g _ s X _ { g _ t , \\bar { h } } \\cap X _ { g _ s , h } \\neq \\emptyset $ \\end{tabular} \\right \\} \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} r _ n ( G ) & = | \\{ \\\\ & \\phantom { r ( G ) } G \\} | , \\\\ c _ n ( G ) & = | \\{ n \\} | . \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} ( \\psi _ k , v ) _ { \\Omega } = 0 \\mbox { f o r a l l } k \\in \\N , \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} { } \\theta ' ( h _ 0 , \\dots , h _ { n - 1 } ) ( \\xi ) : = \\sum _ { t _ 0 , \\dots , t _ { n - 1 } } \\theta ( g _ { t _ 0 } , \\dots , g _ { t _ { n - 1 } } ) ( \\pi ( \\xi ) | _ { [ t _ 0 , \\ldots , t _ { n - 1 } ] } ) . \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} \\begin{cases} R i c ' _ h - D ^ 2 v = 0 \\\\ \\Delta _ { g _ 0 } v = 0 \\\\ \\delta _ { g _ 0 } \\delta _ { g _ 0 } ^ * Y - d \\delta _ { g _ 0 } Y = 0 \\\\ \\Delta G = 0 . \\end{cases} \\quad ~ M . \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} a _ { \\xi } = L ^ { K } b _ { \\xi } , \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ z } \\mathcal { L } _ 0 ( f h ) \\ , d \\mu _ z = 0 \\ , , \\mbox { a n d } \\int _ { \\Sigma _ z } ( \\mathcal { L } _ 0 f ) h \\ , d \\mu _ z = \\int _ { \\Sigma _ z } f ( \\mathcal { L } _ 0 h ) \\ , d \\mu _ z \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} h ( k , \\ell + 1 ) & = \\left ( \\frac { \\ell } { \\ell + 1 } - \\frac { f ( k ) } { \\ell + 1 } \\right ) h ( k , \\ell ) + \\frac { f ( k ) } { \\ell + 1 } h ( k - 1 , \\ell ) \\\\ & \\leq \\left ( \\frac { \\ell } { \\ell + 1 } - \\frac { f ( k ) } { \\ell + 1 } \\right ) \\frac { C } { \\ell } + \\frac { f ( k ) } { \\ell + 1 } \\frac { C } { \\ell } = \\frac { C } { \\ell + 1 } , \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} X = D + ( I - U _ A ^ * U _ A ) D ^ * + ( I - U _ A ^ * U _ A ) Y ( I - U _ A ^ * U _ A ) , \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 1 & - 2 \\lambda u ^ 3 & \\lambda u ^ 2 \\\\ - 2 \\lambda u ^ 3 & 4 & \\lambda u ^ 1 \\\\ \\lambda u ^ 2 & \\lambda u ^ 1 & 0 \\end{pmatrix} , \\\\ w _ { 1 2 } = 0 , w _ { 2 3 } = \\frac { \\lambda ^ 2 u ^ 1 } { \\sqrt { \\det g } } , w _ { 3 1 } = \\frac { \\lambda ^ 2 u ^ 2 } { \\sqrt { \\det g } } , \\end{gather*}"}
-{"id": "1227.png", "formula": "\\begin{align*} w _ 0 & = b ^ { m _ 0 } s _ 1 . . . s _ k , \\\\ w _ k & = s _ 1 . . . s _ k b ^ { m _ k } . \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} | | X | | _ { s , r , q , \\mathbf { a } } : = \\sup _ { ( y , z , \\bar { z } ) \\in D ( r ) } | | ( \\sum _ { k , i , \\alpha , \\beta } e ^ { \\mathbf { a } | \\pi ( k , \\alpha , \\beta ; v ) | } | X _ { k , i , \\alpha , \\beta } ^ { ( \\mathbf { v } ) } | e ^ { | k | s } | y | ^ i | z ^ { \\alpha } | | \\bar { z } ^ { \\beta } | ) _ { \\mathbf { v } \\in V } | | _ { s , r , q } , \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} \\phi w \\circ T ^ { n + 1 } & = \\phi w \\circ T ^ n \\circ T \\\\ & = \\sum _ { u \\in W ^ { \\star } _ { n } ( w ) } \\phi u \\circ T \\\\ & = \\sum _ { u \\in W ^ { \\star } _ { n } ( w ) } \\sum _ { v \\in W ^ { \\star } _ { 1 } ( u ) } \\phi v \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} \\dot { Y } = - \\gamma Y + \\tau ^ { 1 / 2 } \\sum _ { n = 1 } ^ { \\infty } x _ { n - 1 } \\delta ( t - n \\tau ) \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} \\mathbb { P \\left [ A _ { U E } \\right ] } = \\int _ 0 ^ \\infty \\int _ 0 ^ { \\min \\left ( X _ { 2 1 } , M ^ { \\frac { 1 } { \\alpha \\mu } } \\left ( \\frac { N } { P } \\right ) ^ { \\frac { 2 - \\mu } { \\alpha \\mu } } X _ { 2 1 } ^ { \\frac { 2 } { \\mu } } \\right ) } { \\left ( 2 \\pi \\lambda _ b \\right ) ^ 2 x _ { 1 1 } x _ { 2 1 } \\mathrm { e } ^ { - \\pi \\lambda _ b x _ { 2 1 } ^ 2 } \\mathrm { d } x _ { 1 1 } \\mathrm { d } x _ { 2 1 } } . \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} \\lim _ { ( \\rho , \\delta ) \\to ( 0 , 0 ) } \\sup _ { z \\in B ( x , \\rho ) } \\Vert W _ \\lambda ( z ) - V _ x \\Vert _ * = 0 \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} f ( s ( \\phi ) ) = f ( t ( \\phi ) ) \\phi \\in { \\bf X } . \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} { \\mathrm d } \\overline { X } _ { t , n } \\ , = \\ , b ( t , \\overline { X } _ { t , n } , u \\cdot \\delta _ { \\overline { X } _ { t , n + 1 } } + ( 1 - u ) \\cdot \\mathcal L _ { \\overline { X } _ { t , n } } ) { \\mathrm d } t + { \\mathrm d } W _ { t , n } \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} \\Delta ( Z _ { \\mathbf i } , \\mathcal L _ { \\bf i , \\bf a } , v _ { \\bf i } ^ { \\rm h i g h } , \\tau _ { \\bf i , \\bf a } ) = - \\Delta _ { { \\bf i } , { \\bf a } } . \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} - \\Delta u + u + \\rho ( x ) \\phi _ u u = | u | ^ { p - 1 } u , \\\\ \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} h : & = ( H _ { \\oplus } ( \\xi ) - \\Sigma ( \\xi ) ) A \\psi \\\\ & = - \\mu z ( v ) \\psi + A ( H _ { + } ( \\xi ) - \\Sigma ( \\xi ) ) \\psi \\\\ & = - \\mu z ( v ) \\psi + A ( H _ \\mu ( \\xi ) - \\Sigma ( \\xi ) ) \\psi \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} C = \\left [ 0 , 1 / 2 \\right ] , \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\nu } & \\sum _ { n = 0 } ^ { \\infty } \\int _ { A ^ n } \\lvert \\psi ^ { ( n ) } ( k ) \\partial _ i K ( \\xi - m ^ { ( n ) } ( k ) ) \\lvert ^ 2 d \\lambda _ { n \\nu } \\\\ & \\leq \\sum _ { n = 0 } ^ { \\infty } \\int _ { A ^ n } C _ K \\lvert \\psi ^ { ( n ) } ( k ) K ( \\xi - m ^ { ( n ) } ( k ) ) \\lvert ^ 2 d \\lambda _ { n \\nu } + C _ K \\lVert \\psi \\lVert ^ 2 \\\\ & = C _ K \\lVert K ( \\xi - d \\Gamma _ A ( m ) ) \\psi \\lVert ^ 2 + C _ K \\lVert \\psi \\lVert ^ 2 . \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} \\delta ( y , t ) = \\int d x \\ d ( x , y , t ) \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} f ( x ) = \\inf \\{ \\tau \\in I : \\ , x \\in K _ \\tau \\} = \\min \\{ \\tau \\in I : \\ , x \\in K _ \\tau \\} ; \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} a - d \\leq 1 { \\rm i f } \\delta > 0 { \\rm a n d } \\frac { 1 } { r } + \\frac { c } { n } = \\frac { 1 } { p } + \\frac { a - 1 } { n } . \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} \\sum _ { l = - k } ^ k | \\nabla u _ { k , l } | ^ 2 = & \\sum _ { l = - k + 1 } ^ { k - 1 } \\Big [ 2 \\left \\{ \\frac { ( k + l ) ( k + l + 1 ) ( 2 k + 1 ) } { 4 ( 2 k - 1 ) } + \\frac { ( k - l ) ( k - l + 1 ) ( 2 k + 1 ) } { 4 ( 2 k - 1 ) } \\right \\} \\\\ & \\quad \\quad + \\frac { ( k + l ) ( k - l ) ( 2 k + 1 ) } { 2 k - 1 } \\Big ] | Y _ { k - 1 , l } | ^ 2 \\\\ = & \\sum _ { l = - k + 1 } ^ { k - 1 } \\frac { k ( 2 k + 1 ) ^ 2 } { 2 k - 1 } | Y _ { k - 1 , l } | ^ 2 \\\\ = & \\frac { k ( 2 k + 1 ) ^ 2 } { 2 k - 1 } \\cdot \\frac { 2 k - 1 } { 4 \\pi } = \\frac { k ( 2 k + 1 ) ^ 2 } { 4 \\pi } . \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} 0 = t ! \\binom { t + 2 } { 2 } R ^ { t + 1 } ( 1 ) + ( t - 1 ) ! \\left ( \\binom { t + 2 } { 3 } + 6 \\binom { t + 2 } { 4 } \\right ) R ^ t ( 1 ) + \\ldots = 0 . \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} y ( t _ { k + 1 } ) = \\varphi _ 0 ( \\tau _ k A ) y ( t _ k ) + \\sum _ { i = 1 } ^ { p } \\tau ^ i _ k \\varphi _ i ( \\tau _ k A ) \\sum _ { j = 0 } ^ { p - i } \\frac { t ^ j _ k } { j ! } v _ { i + j } . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} [ a \\cup b , c ] = [ a , c ] \\cup b + ( - 1 ) ^ { ( | c | - 1 ) | a | } a \\cup [ b , c ] . \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} \\begin{aligned} x _ { e , 0 } - x _ { e , 0 } & = 0 & & & x _ { e , e } - x _ { e , e } & = 0 \\\\ x _ { e , e } + x _ { e , 0 } & = 0 & & & x _ { 0 , e } + x _ { e , e } & = 0 \\\\ x _ { 0 , e } - x _ { e , 0 } & = 0 & & & x _ { 0 , e } - x _ { 0 , e } & = 0 \\end{aligned} \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} M ( t ) = \\int _ 0 ^ t \\Big ( A ( x ( s ) ) - \\widetilde { \\sigma } ( z ( s ) ) \\Big ) \\ , d \\widetilde { w } ( s ) \\ , , \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} S ^ \\alpha { } _ \\beta ( x ) : = [ g ^ { \\alpha \\gamma } ( x ) ] \\ , [ h _ { \\gamma \\beta } ( x ) ] - \\delta ^ \\alpha { } _ \\beta \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { B ^ r _ { p , q } } = \\Vert \\{ 2 ^ { m r } \\Vert \\textnormal { O p } ( \\chi _ { m } ) f \\Vert _ { L ^ p ( G ) } \\} _ { m = 0 } ^ { \\infty } \\Vert _ { l ^ q ( \\mathbb { N } ) } , \\ , \\ , 0 < p , q \\leq \\infty , \\ , r \\in \\mathbb { R } . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} \\kappa _ { 1 1 } = \\sum _ { m \\geq 0 } \\prod _ { j = 0 } ^ { m - 1 } q _ { 0 ^ j 1 1 } ( 0 ) { \\rm ~ ~ a n d ~ ~ } \\kappa _ { 1 0 ^ k 1 } = 1 + \\sum _ { m \\geq k + 1 } \\prod _ { j = 0 } ^ { m - 1 } q _ { 0 ^ j 1 0 ^ k 1 } ( 0 ) , ~ \\forall k \\geq 1 . \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} \\underset { | x | \\to + \\infty } { \\lim } \\ \\Big ( - \\beta ( 1 - \\beta ) | \\nabla U | ^ 2 + \\beta \\Delta U + f \\Big ) = - \\infty . \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} \\| \\mathrm { d } A ^ \\flat \\| _ g ^ 2 = - 4 m ^ 2 \\left ( \\frac 1 2 u _ \\alpha \\bar u ^ \\alpha + v _ \\beta v ^ \\beta \\right ) . \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} \\nabla f ( \\mathbf { x } ) + \\mathbf { A } ^ T \\boldsymbol { \\lambda } ^ { \\star } = \\mathbf { 0 } \\\\ \\mathbf { B } ^ T \\boldsymbol { \\lambda } ^ { \\star } = \\mathbf { 0 } \\\\ \\mathbf { A } \\mathbf { x } ^ { \\star } + \\mathbf { B } \\mathbf { z } ^ { \\star } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} w _ k ( x ) \\leq \\left ( \\frac { \\lambda } { 2 } \\right ) ^ { \\frac { 1 } { p - 1 } } = \\omega ( x ) , | x | = R _ 0 , \\ , \\ , \\forall k \\geq K . \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} \\mathcal { H } _ { E } & = \\begin{bmatrix} \\mathcal { H } _ { e _ { 1 } } & \\mathcal { H } _ { e _ { 2 } } & \\dots & \\mathcal { H } _ { e _ { r } } \\end{bmatrix} . \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} ( \\nabla h ) _ j : = T _ j h - h . \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{gather*} R ( 1 ) R ( e _ { | i | + 1 1 } ) = R ^ 2 ( e _ { | i | + 1 1 } ) + R ( e _ { | i | 1 } ) , \\\\ R ( e _ { | i | + 1 1 } ) R ( 1 ) = R ^ 2 ( e _ { | i | + 1 1 } ) + R ( e _ { | i | + 1 2 } ) , \\end{gather*}"}
-{"id": "8522.png", "formula": "\\begin{align*} E \\bigg [ \\sum _ { i > n } e ^ { - k \\beta \\tau _ i } \\zeta _ i ^ k ( t - \\tau _ i ) \\bigg ] & = E \\bigg [ \\sum _ { i > n } e ^ { - k \\beta \\tau _ i } E \\big ( \\zeta _ i ^ k ( t - \\tau _ i ) \\bigm | \\tau _ i \\big ) \\bigg ] \\\\ & \\le \\sup _ { t \\ge 0 } E \\big ( \\zeta _ i ^ k ( t ) \\big ) \\ , E \\bigg [ \\sum _ { i > n } e ^ { - k \\beta \\tau _ i } \\bigg ] . \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} \\Psi _ { D } ( \\tau _ \\infty ) \\gamma _ { D , \\infty } = \\gamma _ { D , \\infty } \\left [ \\begin{smallmatrix} t & 0 \\\\ 0 & t ^ { - 1 } \\end{smallmatrix} \\right ] \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} ( M , s ) , \\ M = \\{ d w = 0 \\} , \\ s = w + ( d w ) ^ 2 . \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} \\Vert D F _ N ( \\omega ) ( x ^ * ) ^ { - 1 } \\Vert _ U \\leq \\frac { \\Vert D F ( x ^ * ) ^ { - 1 } \\Vert _ U } { 1 - \\frac { 1 } { 2 } } = 2 \\Vert D F ( x ^ * ) ^ { - 1 } \\Vert _ U . \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} \\binom { ( p - r ) p ^ n } { p ^ n } b _ { ( p - r ) p ^ n } = b _ { p ^ n } b _ { ( p - r - 1 ) p ^ n } + \\sum _ { 0 < i , j < N } a _ { i j } b _ { p ^ n - i } b _ { ( p - r - 1 ) p ^ n - j } . \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta u + u + \\phi u = | u | ^ { p - 1 } u , & x \\in \\R ^ 3 , \\\\ \\ , \\ , \\ , - \\Delta \\phi = u ^ 2 , & x \\in \\R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} L u = \\lambda u \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} d ( d f \\circ J ) ( u , v ) - d ( d f \\circ J ) ( J u , J v ) = d f ( J N _ J ( u , v ) ) . \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} \\widetilde { \\Psi } _ { S S S a } ( \\tau _ { n } , \\tau _ { n } ' , \\tau _ { n } '' ) = \\max \\{ \\widetilde { \\Psi } _ { S S S } , \\psi _ { 3 } ( \\tau _ { n } '' ) \\} , \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} I ' ( u ) v = m ( \\left \\| u \\right \\| ^ 2 ) \\int _ { \\mathbb { R } ^ 2 } ( \\nabla u \\cdot \\nabla v + b ( x ) u v ) - \\int _ { \\mathbb { R } ^ 2 } A ( x ) f ( u ) v , \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\frac { \\hbar ^ 2 D } { 4 M } \\bigg \\| \\nabla \\ ! \\bigg ( \\frac { \\tilde \\rho } { D } \\bigg ) \\bigg \\| ^ 2 = \\frac { \\hbar ^ 2 D } { 4 M } \\bigg \\| \\frac { \\nabla \\tilde \\rho } { D } - \\frac { \\nabla D } { D ^ 2 } \\tilde \\rho \\bigg \\| ^ 2 = \\frac { \\hbar ^ 2 } { 4 M } \\left ( \\frac { \\| \\nabla \\tilde \\rho \\| ^ 2 } { D } - \\frac { | \\nabla D | ^ 2 } { D } \\right ) . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} \\mathbf { E } \\left [ \\sum _ { l = 0 } ^ { \\infty } | f ( x + g _ l + S ( l ) ) | \\mathbb { I } \\{ \\tau > l \\} \\right ] \\le C \\left ( 1 + | x | ^ { p - \\gamma } + \\frac { | x | ^ p } { ( { \\rm d i s t } ( x , \\partial K ) ) ^ \\gamma } \\right ) \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} [ S ] _ { \\mathrm { P G L } ( 2 , \\mathbb { Z } ) } = [ S ] _ { \\mathrm { P S L } ( 2 , \\mathbb { Z } ) } \\cup R \\cdot [ S ] _ { \\mathrm { P S L } ( 2 , \\mathbb { Z } ) } \\cdot R \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} \\lambda - \\mathcal { L } _ R ^ { \\underline { u } } [ w _ R ] + | D w _ R | ^ m = f + \\mu _ R , x \\in B _ R , \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} b _ j ^ { n - 2 s } H ( q _ j , q _ j ) - \\sum _ { i \\neq j } ( b _ i b _ j ) ^ { \\frac { n - 2 s } { 2 } } G ( q _ j , q _ i ) - \\frac { 2 s c _ 1 A + c _ 2 } { ( n - 4 s ) c _ { n , s } ^ { n - 4 s } c _ 1 } b _ j ^ { 2 s } = 0 j = 1 , \\cdots , k . \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} \\frac { d \\mu _ n ( \\ddot { y } ) } { d \\mu ( \\ddot { y } ) } = \\lambda ( n , \\ddot { y } ) . \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} \\int _ { G } f ( y ) \\rho ( y ) d y & = \\int _ { K \\backslash G / H } \\int _ { K } \\int _ { H } f ( k ^ { - 1 } y h ) d h d k d \\mu _ \\rho ( \\ddot { y } ) . \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} H ( x _ 1 , x _ 2 , p _ 1 , p _ 2 , t ) = p _ 1 \\dot { x } _ 1 + p _ 2 \\dot { x } _ 2 - L ( x _ 1 , x _ 2 , \\dot { x } _ 1 , \\dot { x } _ 2 , t ) , \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} R & : = ( f \\times i d _ { \\mathbb { F } _ 1 \\times \\mathbb { F } _ 2 } ) \\circ l \\circ ( R _ 1 \\times R _ 2 ) \\circ l ^ { - 1 } \\circ ( f \\times i d _ { \\mathbb { F } _ 1 \\times \\mathbb { F } _ 2 } ) ^ { - 1 } \\\\ & : U \\lhd \\mathbb { F } _ 1 \\times \\mathbb { F } _ 2 \\rightarrow U \\lhd \\mathbb { F } _ 1 \\times \\mathbb { F } _ 2 \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} q = 1 + \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } . \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} \\left < \\prod _ { k = 1 } ^ n \\exp ( - t _ { i _ k j _ k } ) \\right > _ { \\alpha , \\Lambda , \\epsilon } \\le \\prod _ { k = 1 } ^ { n } \\left < \\exp ( - n t _ { i _ k j _ k } ) \\right > _ { \\alpha , \\Lambda , \\epsilon } ^ { 1 / n } \\le C ( n , \\alpha ) . \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} \\Sigma ( 1 ) : = \\bigcup _ { \\sigma \\in \\Sigma } \\sigma ( 1 ) \\ , . \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} x ^ { \\alpha } d x _ { \\sigma } : = \\left ( x _ 1 ^ { \\alpha _ 1 } x _ 2 ^ { \\alpha _ 2 } \\dots x _ n ^ { \\alpha _ n } \\right ) d x _ { \\sigma ( 1 ) } \\wedge \\dots \\wedge d x _ { \\sigma ( k ) } , \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} f ^ { - 1 } : \\begin{array} { l l } x _ { n } = T ^ { - 1 } ( x _ { n + 1 } ) \\\\ y _ { n } = \\lambda ^ { - 1 } ( y _ { n + 1 } - \\tau ^ { 1 / 2 } T ^ { - 1 } ( x _ { n + 1 } ) ) \\end{array} \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{gather*} T _ i T _ j \\big ( x _ { i , 1 } ^ + \\big ) = T _ i \\big ( \\big [ x _ { j } ^ + , x _ { i , 1 } ^ + \\big ] \\big ) = \\big [ \\big [ x _ i ^ + , x _ j ^ + \\big ] , - x _ { i , 1 } ^ - + \\tfrac { \\hbar } { 2 } \\big \\{ h _ i , x _ i ^ { - } \\big \\} \\big ] , \\end{gather*}"}
-{"id": "5678.png", "formula": "\\begin{align*} [ K _ - , K _ + ] \\psi _ n ^ { \\ell } ( u ) = ( 2 n + \\ell + 1 ) \\psi _ n ^ { \\ell } ( u ) . \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t } ^ { ( u ) } \\ , = \\ , - \\big ( u \\ , ( X _ { t } ^ { ( u ) } - \\widetilde { X } _ { t } ^ { ( u ) } ) + ( 1 - u ) ( X _ { t } ^ { ( u ) } - \\mathbb E [ X _ { t } ^ { ( u ) } ] ) \\big ) \\ , { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , ; t \\ge 0 \\ , \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} \\ddot Y + R _ { \\dot \\gamma } Y = 0 , \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} \\mathcal { C } _ { n e t } \\approx = \\frac { \\pi \\lambda _ b } { A _ 1 + A _ 2 } . \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } H ( t , x , v , p , A , u ) \\\\ = \\frac { 1 } { 2 } \\mathrm { t r } [ \\sigma \\sigma ^ { \\intercal } ( t , x , v , V ( t , x , v , p , u ) , u ) A ] + p ^ { \\intercal } b ( t , x , v , V ( t , x , v , p , u ) , u ) \\\\ \\ \\ + g ( t , x , v , V ( t , x , v , p , u ) , u ) , \\\\ ( t , x , v , p , A , u ) \\in \\lbrack 0 , T ] \\times \\mathbb { R } ^ { n } \\times \\mathbb { R } \\times \\mathbb { R } ^ { n } \\times \\mathbb { S } ^ { n } \\times U , \\end{array} \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} \\rho _ 1 = - \\frac { 1 } { 2 \\sigma _ w ^ 2 } + \\frac { 1 } { 2 P _ y } . \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} \\tilde { T } = T _ { n - p } [ z ^ { - p } \\sigma ] = ( - 1 ) ^ { - ( n - p + 1 ) p } T _ { n - p } [ ( - z ) ^ { - p } \\sigma ] , \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} u _ t = a ( x , u , u _ x ) u _ { x x } + f ( x , u , u _ x ) \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} u _ { t } - \\bigtriangleup u = & \\sigma ( u ) | \\nabla \\phi | ^ 2 , \\\\ - \\nabla \\cdot ( \\sigma ( u ) \\nabla \\phi ) = & 0 . \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} \\varphi ( x , E ) = p ( x , E ) e ^ { i k ( E ) x } \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} \\nabla \\cdot E ( \\bold { x } ) = \\dfrac { 1 } { \\epsilon _ 0 } \\rho ( \\bold { x } ) . \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\limsup _ { n , m \\to \\infty } I ^ { n , m } _ { R , T } = 0 \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} & ( f ^ { 1 / 2 } e _ P ) ( f ^ { 1 / 2 } e _ P ) = f e _ P \\\\ & = e e _ P = ( e ^ { 1 / 2 } e _ P ) ( e ^ { 1 / 2 } e _ P ) . \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} Z ^ \\phi ( t ) = \\phi _ 0 ( t ) + \\sum _ { i = 1 } ^ { \\xi _ 0 ( t ) } Z _ i ^ { \\phi } ( t - \\sigma _ i ) , \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} M ( t ) = \\begin{cases} g ^ { - 1 } ( ( 1 + \\epsilon ) \\log t ) , & \\textrm { o r } \\\\ u _ \\rho ( t ) , & \\textrm { w h e n ~ \\eqref { e q : s t a b i l i t y } h o l d s , } \\end{cases} \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} s ^ + ( \\mathcal { Q } ) = \\dim H _ 0 , s ^ - ( \\mathcal { Q } ) = \\dim H _ { m + 1 } . \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} C _ { 1 , 0 } & = C _ { 1 , 0 } ( \\chi , \\sigma , r , \\omega _ 0 , \\mathcal { U } _ 1 ) = C _ { \\ref { L e m m a C e n t r a l E s t i m a t e I n N o r m a l D i r e c t i o n } , 1 , 0 } ( \\eta ( \\chi ) , R _ 1 ( \\chi , \\sigma , r , \\omega _ 0 , \\mathcal { U } _ 1 ) ) , \\\\ C _ 1 & = C _ 1 ( \\chi , \\sigma , r , \\omega _ 0 , \\mathcal { U } _ 1 , T ^ * ) = C _ { \\ref { L e m m a C e n t r a l E s t i m a t e I n N o r m a l D i r e c t i o n } , 1 } ( \\eta ( \\chi ) , R _ 1 ( \\chi , \\sigma , r , \\omega _ 0 , \\mathcal { U } _ 1 ) , T ^ * ) . \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} \\frac { 2 n + \\alpha + \\beta + 2 } { 2 } ( 1 - x ) P _ n ^ { ( \\alpha + 1 , \\beta ) } ( x ) = ( n + \\alpha + 1 ) P _ n ^ { ( \\alpha , \\beta ) } ( x ) - ( n + 1 ) P _ { n + 1 } ^ { ( \\alpha , \\beta ) } ( x ) , \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} = \\int \\limits _ t ^ T \\left ( ( s - t ) - \\sum _ { j = 0 } ^ { q } \\left ( \\int \\limits _ t ^ s \\phi _ j ( \\tau ) d \\tau \\right ) ^ 2 \\right ) d s . \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} \\mathbf { a } _ { k } ( l ) = \\begin{cases} d _ { j _ u , s } + d _ { j _ u + 1 , s } + \\cdots + d _ { j _ { u + 1 } - 1 , s } & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} \\dot { \\tilde { R } } ~ ~ & = \\tilde { R } ( - k _ \\beta \\psi _ \\omega + \\hat { R } \\tilde { b } _ \\omega ) ^ \\times , \\\\ \\dot { \\tilde { p } } ~ ~ & = \\tilde { R } ( - k _ \\beta \\psi _ v + \\hat { p } ^ \\times \\hat { R } \\tilde { b } _ \\omega + \\hat { R } \\tilde { b } _ v ) , \\\\ \\dot { \\psi } _ { \\tilde { R } } & = E ( \\tilde { R } ) ( - k _ \\beta \\psi _ \\omega + \\hat { R } \\tilde { b } _ \\omega ) , \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} \\mathfrak M = \\Big \\{ x \\in H ^ 1 \\big ( [ 0 , 1 ] , \\overline \\Omega \\big ) : x ( 0 ) , x ( 1 ) \\in \\partial \\Omega \\Big \\} , \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} \\Phi ( x ) = & r \\int _ 0 ^ 1 [ ( \\bar { \\tau } ^ 2 _ s ) ^ * ( \\iota _ { \\frac { \\partial } { \\partial r } } \\Psi ) ] ( x ) d s \\\\ = & \\int _ 0 ^ { | | x | | _ h } ( \\tau ^ 2 _ { - t } ) ^ * ( \\iota _ v \\Psi ) ( x ) d t . \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} u ( t ) = v ( t ) + a ( t ) \\phi _ 0 : = P _ c u ( t ) + \\langle \\phi _ 0 , u ( t ) \\rangle \\phi _ 0 . \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} u ( - \\delta _ { i } ) - \\psi ( - \\delta _ { i } ) + \\alpha ( \\delta ) \\delta & \\leq \\frac { u ( - \\delta _ { i } ) - \\psi ( - \\delta _ { i } ) } { 2 } \\\\ & < 0 = u ( 0 ) - \\psi ( 0 ) < u ( x _ { \\delta } ) - \\psi ( x _ { \\delta } ) - \\alpha ( \\delta ) x _ { \\delta } . \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{gather*} h \\triangleright \\psi = \\psi ( ? h ) = \\psi '' ( h ) \\psi ' , \\varphi \\triangleright \\psi = \\varphi \\psi h \\in H , \\psi , \\varphi \\in \\mathcal { O } ( H ) . \\end{gather*}"}
-{"id": "6015.png", "formula": "\\begin{align*} \\phi '' ( x _ 0 ) = \\beta ( \\beta + 1 ) x _ 0 ^ { - \\beta - 2 } = \\beta ( \\beta + 1 ) \\Big ( \\frac { k } { \\beta } \\Big ) ^ { \\frac { \\beta + 2 } { \\beta + 1 } } = \\beta ^ { \\frac { - 1 } { \\beta + 1 } } ( \\beta + 1 ) k ^ { \\frac { \\beta + 2 } { \\beta + 1 } } . \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} E _ \\xi ( t ) : = \\frac { 1 } { 2 } \\int _ { G } \\big | \\mathbf { E } ( t , \\cdot ) \\big | ^ { 2 } \\mathrm { d } \\mathbf { x } + \\frac { 1 } { 2 } \\int _ { G } \\big | \\mathbf { H } ( t , \\cdot ) \\big | ^ { 2 } \\mathrm { d } \\mathbf { x } + \\xi \\tau \\int _ { 0 } ^ { 1 } \\int _ { \\Gamma } \\big | \\mathbf { E } ( t - \\tau s , \\mathbf { x } ) \\times \\boldsymbol { \\nu } \\big | ^ { 2 } \\mathrm { d } \\mathbf { x } \\mathrm { d } s , \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} \\mathcal { I } _ h ( v _ t ) = g _ h ( t ) . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} \\pi _ \\lambda ( g ) K _ z = \\overline { j _ \\lambda ( g ^ { - 1 } , z ) } K _ { g ^ { - 1 } \\cdot z } . \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} \\mu ( x , \\theta , C ) = \\lambda ( x , C ) \\otimes \\delta \\left ( \\theta - { \\frac { \\nabla p } { | \\nabla p | } ( x ) } \\right ) , \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} U _ { \\mu , \\xi } ( x ) = \\mu ^ { - \\frac { n - 2 s } { 2 } } U _ 0 \\left ( \\frac { x - \\xi } { \\mu } \\right ) = \\alpha _ { n , s } \\left ( \\frac { \\mu } { \\mu ^ 2 + | x - \\xi | ^ 2 } \\right ) ^ { \\frac { n - 2 s } { 2 } } , \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} V _ n ( s ) = - \\chi _ n ( s \\ , ) \\nabla \\phi ( z ( s ) ) , \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} H ( X , Y ) - H ( Y , X ) = 0 , \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} [ u [ f ] , L ^ t _ \\theta ] = - u [ \\theta ( f ) ] , \\ , [ u [ f ] , L ^ { t ' } _ \\theta ] = - u [ \\theta ( f ) ] . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{gather*} \\langle s _ { 0 , n } , S ( s _ { i , n } ) \\rangle = S \\left ( \\langle s _ { n , n } , s _ { i , n } \\rangle \\right ) = S \\left ( \\langle s _ { i , n - 1 } , s _ { n , n } \\rangle \\right ) = \\langle s _ { i + 1 , n } , s _ { 0 , n } \\rangle = \\langle s _ { 0 , n } , s _ { 0 , i } \\rangle \\Rightarrow S ( s _ { i , n } ) \\sim s _ { 0 , i } . \\end{gather*}"}
-{"id": "2858.png", "formula": "\\begin{align*} | y ^ { - 1 } x | \\geq | x | - | y | > | x | - \\frac { | x | } { 2 } = \\frac { | x | } { 2 } , \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} \\frac { \\ , 1 \\ , } { \\ , n \\ , } \\sum _ { i = 1 } ^ { n } \\mathbb E [ ( \\lvert X _ { t + \\delta , i } ^ { ( u ) } - X _ { t , i } ^ { ( u ) } \\rvert ^ { 2 } + \\lvert X _ { t + \\delta , i + 1 } ^ { ( u ) } - X _ { t , i + 1 } ^ { ( u ) } \\rvert ^ { 2 } ) ^ { 1 / 2 } \\vert \\mathcal F _ { t } ] \\le \\mathbb E \\big [ \\ , \\mathfrak f ( \\delta ) \\ , \\vert \\ , \\mathcal F _ { t } \\big ] \\ , ; 0 \\le t \\le T - \\delta \\ , , \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} \\dot { \\psi } = \\tilde { L } _ { \\bar { \\theta } } \\psi + \\mathcal { G } ( \\psi ) , \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} a _ 1 ( n ) & = 2 n ( n - 2 ) ^ 3 ( n - 4 ) ( 6 n ^ 3 - n ^ 2 - 5 n + 2 ) \\geq 0 . \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| T z ^ k - z ^ k \\| = 0 . \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\mathbb { K } = \\mathbb { F } ( \\beta ) \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} T _ { n + s } = T _ n + ( \\zeta + o ( 1 ) ) \\left ( \\frac { T _ n } { h ( n ) } \\right ) , \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} \\begin{aligned} & \\big ( \\int _ x ^ a d y \\phi _ 3 ^ { - 2 } ( y ) \\int _ y ^ a \\phi _ 3 ( z ) d z \\big ) \\big ( \\int _ { - \\infty } ^ a \\phi _ 3 ^ { - 2 } ( t ) d t \\big ) = \\\\ & \\big ( \\int _ x ^ a d y \\phi _ 3 ^ { - 2 } ( y ) \\int _ y ^ a \\phi _ 3 ( z ) d z \\big ) \\big ( \\int _ { - \\infty } ^ x \\phi _ 3 ^ { - 2 } ( t ) d t \\big ) + \\\\ & \\big ( \\int _ x ^ a d y \\phi _ 3 ^ { - 2 } ( y ) \\int _ y ^ a \\phi _ 3 ( z ) d z \\big ) \\big ( \\int _ x ^ a \\phi _ 3 ^ { - 2 } ( t ) d t \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} f ( X ) = \\prod _ { i = 1 } ^ { d } ( X - m _ i ) = \\prod _ { i = 1 } ^ d b _ { i i } . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { Q } \\ , ^ { C } _ { t } D _ { T } ^ { q } \\Psi ( x , t ) = A ^ { * } \\mathcal { Q } \\Psi ( x , t ) + C ^ { * } C \\mathcal { Q } \\varphi ( x , t ) & \\hbox { i n } Q _ { T } \\\\ \\Psi ( \\xi , t ) = 0 & \\hbox { o n } \\Sigma _ { T } \\\\ \\Psi ( x , T ) = 0 & \\hbox { i n } \\Omega . \\end{cases} \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} \\| u \\| _ { ( \\kappa _ n + 1 ) p ^ * } \\leq M _ { 3 2 } \\| u \\| _ { ( k _ 0 + 1 ) p ^ * } < \\infty , \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} \\mathcal { H } : \\begin{cases} \\dot { x } ~ ~ \\in F ( x ) & x \\in \\mathcal { F } _ c : = \\{ x \\in \\mathcal { S } : ( \\hat { g } , \\hat { b } _ a ) \\in \\mathcal { F } _ o \\} \\\\ x ^ + \\in G ( x ) & x \\in \\mathcal { J } _ c : = \\{ x \\in \\mathcal { S } : ( \\hat { g } , \\hat { b } _ a ) \\in \\mathcal { J } _ o \\} \\end{cases} \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\nabla \\cdot \\left ( D _ p L ( \\nabla v ( \\cdot , U , \\xi ) , x ) \\right ) = 0 & \\mbox { i n } & \\ U , \\\\ & v ( \\cdot , U , \\xi ) = \\ell _ \\xi & \\mbox { o n } & \\ \\partial U . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} A _ j u = y _ j , j = 0 , \\dots , N - 1 . \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\chi _ i ( t , x ) \\ , \\doteq \\ , \\begin{cases} \\alpha _ { i , 0 } , & \\hbox { i f } ~ x < \\gamma _ { i , 1 } ( t ) , \\\\ \\alpha _ { i , k } , & \\hbox { i f } ~ \\gamma _ { i , k } ( t ) < x < \\gamma _ { i , k + 1 } ( t ) , k = 1 , 2 , \\ldots , N ( i ) - 1 , \\\\ \\alpha _ { i , N ( i ) } , & \\hbox { i f } ~ \\gamma _ { i , N ( i ) } ( t ) < x , \\end{cases} \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} \\phi _ p = \\sqrt { \\ell ! } U _ { u _ p } ^ * ( \\Gamma ( P _ { B _ { u _ p } ^ c } ) \\psi _ p \\otimes P _ { B _ { u _ p } } g ^ { ( 1 ) } _ { u _ p } \\otimes _ s \\dots \\otimes _ s P _ { B _ { u _ p } } g ^ { ( \\ell ) } _ { u _ p } ) \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} b ' _ { n + 1 } = b ' _ { n + 1 } ( E , A , n _ 0 , n _ 1 , v , \\theta _ 0 ) = \\frac { C } { n - v } \\sin ( 2 \\eta ( n ) + 2 \\gamma ( n ) ) . \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} \\lim _ \\alpha S _ f ( e _ \\alpha \\rho e _ \\alpha \\| e _ \\alpha \\sigma e _ \\alpha ) = + \\infty . \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} \\| \\eta \\| & = \\sum _ x | \\eta ( x ) | ; \\\\ \\| f \\| & = \\max _ x | f ( x ) | ; \\\\ \\| A \\| & = \\max _ x \\sum _ y | A ( x , y ) | . \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} & \\| u ( t ) \\| ^ 2 _ { \\widetilde { L } ^ { \\infty } ( [ 0 , t ] ; \\ , { B } ^ { s - 1 } _ { 2 , 2 } ) } + \\| u \\| ^ 2 _ { \\widetilde { L } ^ 2 ( [ 0 , t ] ; \\ , { B } ^ s _ { 2 , 2 } ) } \\\\ \\leq & C _ 1 ( \\nu ) M _ 0 + C _ 1 ( \\nu ) \\int ^ t _ 0 \\| u \\| ^ { 4 + \\frac 4 { r - 2 } } _ { \\widetilde { L } ^ { \\infty } ( [ 0 , t ^ { \\prime } ] ; \\ , B ^ { s - 1 } _ { 2 , 2 } ) } d t ^ { \\prime } + C _ 2 \\int ^ t _ 0 \\| b \\| ^ 4 _ { \\widetilde { L } ^ { \\infty } ( [ 0 , t ^ { \\prime } ] ; \\ , B ^ { s } _ { 2 , 2 } ) } d t ^ { \\prime } . \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} \\nu _ 2 ( S ( 2 ^ h , k ) ) = \\sigma _ 2 ( k ) - 1 \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} { \\mathrm d } X ^ { \\dagger } _ { t } \\ , = \\ , b ( t , X _ { t } ^ { \\dagger } , \\ , \\delta _ { \\widetilde { X } _ { t } ^ { \\dagger } } ) \\ , { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} f ( x ) & = ( x ^ 2 + 1 ) + A ( x - 1 ) ( x ^ 4 + x ^ 2 + 1 ) + m \\ : h ( x ) , \\\\ g ( x ) & = B x ( x - 1 ) ( x ^ 4 + x ^ 2 + 1 ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} ( I - P _ n + \\Pi _ n ) h _ { n 1 } & = P _ n r , \\\\ ( I - P _ n + \\Pi _ n ) h _ { n 2 } & = h _ { n 1 } , \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} | \\sum _ { b = 1 } ^ n k _ b j _ b ^ 2 + \\sum _ { j \\in \\mathbb { Z } _ * } l _ j j ^ 2 | < \\frac { 1 } { 5 0 n } \\sum _ { j \\in \\mathbb { Z } _ * } | j l _ j | , \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} X \\ & = \\ \\ S _ 1 \\ d _ 1 \\ S _ 2 \\ \\ldots d _ { i - 1 } \\ ! - \\ ! 1 \\ S _ i \\ d _ { i } - 1 \\ldots \\ d _ { j - 1 } \\ \\ \\sigma _ j \\ \\ \\ d _ j \\ \\ \\ldots \\ S _ k \\\\ Y \\ & = \\ \\ S _ 1 \\ d _ 1 \\ S _ 2 \\ \\ldots \\ \\ d _ { i - 1 } \\ \\ \\ \\sigma _ i \\ \\ \\ d _ i \\ldots \\ d _ { j - 1 } \\ ! - \\ ! 1 \\ S _ j \\ d _ { j } - 1 \\ \\ \\ldots \\ S _ k \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} w _ n + z _ n = \\partial ^ \\alpha u _ n . \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} U _ \\lambda ^ * T ^ { ( \\lambda ) } _ \\varphi U _ \\lambda f & = ( \\sqrt { R R ^ * } ) ^ { - 1 } R T _ \\varphi ^ { ( \\lambda ) } R ^ * ( \\sqrt { R R ^ * } ) ^ { - 1 } f \\\\ & = ( R R ^ * ) ^ { - 1 } R T _ \\varphi ^ { ( \\lambda ) } R ^ * f , \\end{align*}"}
-{"id": "9951.png", "formula": "\\begin{align*} i ^ { - 1 } { o r } _ { X } = i ^ { - 1 } ( o r _ { \\tilde f } \\otimes _ { { } _ { \\Z _ { X } } } \\tilde f ^ { - 1 } o r _ Y ) = i ^ { - 1 } o r _ { \\tilde f } \\otimes _ { { } _ { \\Z _ { M } } } f ^ { - 1 } j ^ { - 1 } { o r } _ { Y } . \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} k _ { 1 , \\sigma } = 0 \\ \\ \\forall \\sigma , \\ k _ { 2 , \\sigma } = \\left \\{ \\begin{array} { c c } 2 \\ & \\mbox { i f } \\ \\sigma = \\hat { \\sigma } \\\\ 0 \\ & \\mbox { o t h e r w i s e } \\end{array} \\right . , \\ c = 0 . \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} \\ell ^ { 1 , \\infty } ( \\mathbb { N } , w ) = \\left \\{ f = \\{ f ( n ) \\} _ { n \\ge 0 } : \\| f \\| _ { \\ell ^ { 1 , \\infty } ( \\mathbb { N } , w ) } : = \\sup _ { t > 0 } t \\sum _ { \\{ m \\in \\mathbb { N } : | f ( m ) | > t \\} } w ( m ) < \\infty \\right \\} , \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} \\Phi _ t = - \\int _ 0 ^ t \\mathrm { n } _ s \\ , L ( \\d s ) \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\{ ( \\left ( \\begin{array} { c c c c } 1 & & & \\\\ & \\sigma ( t ) & & \\\\ & & 1 & \\\\ & & & \\sigma ( t ^ { - 1 } ) \\end{array} \\right ) ) _ { \\sigma \\in I _ F } \\ \\vert t = \\omega _ 1 ^ { p _ 1 } \\dots \\omega _ { d - 1 } ^ { p _ { d - 1 } } , p _ 1 , \\dots , p _ { d - 1 } \\in \\mathbb { Z } \\} \\hookrightarrow T _ { 1 , L } ( L ) , \\end{align*}"}
-{"id": "9940.png", "formula": "\\begin{align*} \\Omega _ Y = \\Omega \\cap Y , V _ Y = V \\cap Y . \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\pm } W _ { \\infty } ( y ) = \\int _ { \\R _ { > 0 } } \\mathcal { J } _ { \\infty , \\kappa } ^ { \\pm } ( 4 \\pi \\sqrt { x y } ) W _ { \\infty } ( x ) d x , \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} \\dot { t } _ \\tau = \\{ t _ \\tau , { H _ { \\tau } } _ T \\} _ { P B } = \\lambda ; \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} [ a _ 1 , a _ 2 , \\ldots , a _ n ] = \\frac { 1 } { a _ 1 + } \\frac { 1 } { a _ 2 + } \\cdots \\frac { 1 } { a _ { n - 1 } + } \\frac { 1 } { a _ n } \\ . \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} C = \\begin{bmatrix} C _ { 1 1 } & C _ { 1 2 } & C _ { 1 3 } & C _ { 1 4 } \\\\ 0 & C _ { 2 2 } & 0 & C _ { 2 4 } \\\\ C _ { 3 1 } & C _ { 3 2 } & C _ { 3 3 } & C _ { 3 4 } \\\\ 0 & 0 & 0 & C _ { 4 4 } \\end{bmatrix} . \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} [ \\alpha _ { j , p } \\omega _ 1 , \\alpha _ { j , p } \\omega _ 2 ] = [ \\eta _ 1 , \\eta _ 2 ] k _ { j , p } ( \\forall \\ , p ) . \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} \\| p ( v ) - p ( x ) \\| _ { \\infty } & = | v _ 1 - x _ 1 | = | v _ 1 | , \\\\ \\| p ( v ) - p ( y ) \\| _ { \\infty } & = | v _ 2 - y _ 2 | = | v _ 2 - 3 | , \\\\ \\| p ( v ) - p ( z ) \\| _ { \\infty } & = | v _ 3 - z _ 3 | = | v _ 3 + 3 | . \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} \\widetilde { F } _ { \\eta , c , g } & : = \\eta W ( 2 \\omega ^ { - 1 } P _ { c } v _ g , - 1 ) + d \\Gamma ( \\omega ) + \\varphi ( \\overline { P } _ c v _ g ) + \\lVert \\omega ^ { - 1 / 2 } P _ { \\omega } ( ( c , \\widetilde { m } ] ) v _ g \\lVert ^ 2 \\\\ A _ { \\eta , c , g } & : = \\eta W ( 2 \\omega ^ { - 1 } P _ { c } v _ g , - 1 ) + d \\Gamma ( \\omega ) + \\lVert \\omega ^ { - 1 / 2 } P _ { \\omega } ( ( c , \\widetilde { m } ] ) v _ g \\lVert ^ 2 \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta u + u + \\rho ( x ) \\phi u = | u | ^ { p - 1 } u , & x \\in \\R ^ 3 , \\\\ \\ , \\ , \\ , - \\Delta \\phi = \\rho ( x ) u ^ 2 , \\ & x \\in \\R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\mathcal L _ { \\overline { I } _ { n , \\ell } } ( s ) = \\exp \\left ( - \\frac { 2 \\pi } { \\alpha } \\Big ( 1 - \\sum _ { j = \\ell } ^ L T _ { n , j } \\Big ) \\lambda _ h s ^ { \\frac { 2 } { \\alpha } } B \\left ( \\frac { 2 } { \\alpha } , 1 - \\frac { 2 } { \\alpha } \\right ) \\right ) . \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} \\partial _ t C = - C \\partial _ C \\mathcal { E } [ C ] , \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} T _ { \\mathcal { C } } = \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & \\zeta _ 5 ^ 3 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & \\zeta _ 5 ^ 2 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & - 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} \\begin{cases} u _ t + f ^ \\sharp ( t , x , u ) _ x ~ = ~ \\varepsilon u _ { x x } \\ , , \\\\ u ( 0 , x ) ~ = ~ u _ 0 ^ \\sharp ( x ) . \\end{cases} \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} \\varphi _ { 1 , * } ( z ) = A ^ 1 z + \\big ( 0 _ { [ j ] } , ( b ^ 1 ) ' _ { [ j ] } \\big ) = \\big ( z _ { [ j ] } , G z ' _ { [ j ] } + ( b ^ 1 ) ' _ { [ j ] } \\big ) \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} \\min _ { \\beta _ { \\ell } \\in \\mathbb { R } } ~ ~ \\left \\{ \\sum _ { j = 0 } ^ N \\omega _ j \\left ( \\sum _ { \\ell = 0 } ^ { L } \\beta _ { \\ell } \\tilde { \\Phi } _ { \\ell } ( x _ j ) - f ( x _ j ) \\right ) ^ 2 + \\lambda \\sum _ { \\ell = 0 } ^ L ( \\mu _ { \\ell } \\beta _ { \\ell } ) ^ 2 \\right \\} , \\quad \\lambda > 0 , \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} Z ( W , s , \\mu _ p ) = \\int _ { \\Q _ { p } ^ { \\times } } W ( a ( y ) ) \\mu _ p ( y ) \\abs { y } _ { p } ^ { s - \\frac { 1 } { 2 } } d ^ { \\times } y , \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} \\theta ^ { ( 3 ) } _ { 2 } ( l ) = h _ { 2 3 } ( l ) + + e ^ { ( 3 ) } _ { 2 } ( l ) \\mbox { a n d } \\theta ^ { ( 3 ) } _ { 1 } ( l ) = h _ { 1 3 } ( l ) + e ^ { ( 3 ) } _ { 1 } ( l ) , \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{gather*} f \\overset { I } { M } = \\overset { J } { M } f , \\end{gather*}"}
-{"id": "4051.png", "formula": "\\begin{align*} \\beta ( \\theta ) = \\frac { \\pi ^ 2 } { 1 2 \\log 2 } \\mbox { f o r L e b e s g u e a l m o s t e v e r y } \\theta \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { \\lambda } _ j + \\frac { 1 } { t } \\left ( P ^ T d i a g \\left ( \\frac { \\frac { n - 2 s } { 2 s } \\bar { \\sigma } _ r b _ r ^ { 2 - 2 s } + 1 } { n - 4 s } \\right ) P \\lambda \\right ) _ j = \\Pi _ 1 [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ( t ) , \\\\ & \\dot { \\xi } _ j = \\Pi _ { 2 , j } [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ( t ) , j = 1 , \\cdots , k . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} \\eta = \\frac { \\mu ^ { 2 } } { 4 } . \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\frac { \\dim ^ L ( X ) } { \\dim ^ R ( X ) } S ^ { R , R } _ { X , Y } = S ^ { L , R } _ { X , Y } = \\frac { \\dim ^ R ( Y ) } { \\dim ^ L ( Y ) } S ^ { L , L } _ { X , Y } . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} \\alpha = r ^ { \\frac 8 5 } \\gamma ^ { - 1 } , \\beta = \\frac { 1 } { 1 3 } , \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} H A H ^ \\top = ( \\underbrace { \\theta , \\ldots , \\theta } _ { \\ell } , \\underbrace { \\tau , \\ldots , \\tau } _ { n - \\ell - 1 } , k ) \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } k ^ * ( y _ 0 , \\epsilon ) = k ^ * ( y _ 0 ) . \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\partial _ { t } F ( t , x ) + \\inf \\limits _ { u \\in U } H ( t , x , F ( t , x ) , D F ( t , x ) , D ^ { 2 } F ( t , x ) , u ) = 0 , \\\\ F ( T , x ) = \\phi ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} ( \\Delta d \\phi ) ( e _ i ) = d \\phi ( ^ M ( e _ i ) ) + R ^ N ( d \\phi ( e _ j ) , d \\phi ( e _ i ) ) d \\phi ( e _ j ) + \\nabla _ { e _ i } \\tau ( \\phi ) . \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} u _ { k } ( T _ { 2 } - \\delta , x ) = 0 \\mbox { f o r } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} [ \\hat x _ 1 , \\hat p _ 1 ] = i \\hbar ; [ \\hat x _ 2 , \\hat p _ 2 ] = i \\hbar ; [ \\hat x _ 1 , \\hat p _ 2 ] = 0 ; [ \\hat x _ 2 , \\hat p _ 1 ] = 0 \\ ; , \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} T ^ { - 1 } _ S F ( t ) = \\int _ { \\R } F ( x + i c ) e ^ { 2 \\pi c t } e ^ { - i 2 \\pi x t } \\d x , \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} \\theta _ u ^ n = \\mu , \\ u \\in \\mathcal { E } ( X , \\theta , \\phi ) , \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} \\O _ { \\tilde F , \\ , \\l } ( u , J _ { \\tilde F } u ) = ( \\l \\rho \\o _ { \\tilde F } + ( 1 - \\l \\rho ) \\tilde \\o _ 1 + \\l d \\rho \\wedge \\eta ) ( u , J _ { \\tilde F } u ) \\ge \\frac { c } { 2 } | u | ^ 2 \\ , . \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } { \\log m _ n ( x , y ) \\over - \\log n } \\geq \\limsup _ { k \\to \\infty } { \\log m _ { n _ k } ( x , y ) \\over - \\log n _ k } \\geq \\lim _ { k \\to \\infty } { \\log r _ k \\over - \\log W _ { B ( y , r _ k ) } ( x ) } = 1 . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} C _ { j , j + 1 } ( u _ 1 , v _ 1 , 1 , v _ 2 ) = C _ { X ^ 1 _ j , X ^ 2 _ j } \\star C _ { X ^ 2 _ j , X ^ 2 _ { j + 1 } } ( u _ 1 , v _ 1 , u _ 2 ) . \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} \\left | f \\left ( t , x , \\omega \\right ) - f _ { i } \\left ( t , x , \\omega \\right ) \\right | < \\delta \\quad \\left ( t , x , \\omega \\right ) \\in \\left [ a _ { i } , b _ { i } \\right ] \\times \\mathbb { R } \\times \\left [ 0 , 1 \\right ] \\ , . \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} J = \\left [ \\begin{array} { c c c c c c } 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\\\ - 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & - 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & - 1 & 0 & 0 & 0 \\end{array} \\right ] \\ ; . \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} F ^ { z \\bar { z } } _ { ( - j ) j } = 0 \\end{align*}"}
-{"id": "9676.png", "formula": "\\begin{align*} L ( \\tilde { C } , \\mathbb { A } ) = \\sum _ { a \\in A _ { + } } \\frac { a ( z _ 1 ) \\dots a ( z _ n ) } { a } . \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} W _ { ( \\rho ( h ) \\varphi ) } ( g ^ \\iota ) = \\rho ( h ^ \\iota ) W _ { \\tilde { \\varphi } } ( g ) \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} \\psi _ { n _ 1 , n _ 2 } ( x _ 1 , x _ 2 , t ) = e ^ { i \\theta _ { n _ 1 , n _ 2 } ( t ) } \\phi _ { n _ 1 , n _ 2 } ( x _ 1 , x _ 2 , t ) . \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\left \\| \\nabla u - ( \\nabla u ) _ { B _ { R / 2 } } \\right \\| _ { \\underline { L } ^ 2 ( B _ { R / 2 } ) } = \\inf _ { \\ell \\in \\mathcal { P } _ 1 } \\left \\| \\nabla u - \\nabla \\ell \\right \\| _ { \\underline { L } ^ 2 ( B _ { R / 2 } ) } \\leq \\frac { C } { R } \\inf _ { \\ell \\in \\mathcal { P } _ 1 } \\left \\| u - \\ell \\right \\| _ { \\underline { L } ^ 2 ( B _ { R } ) } . \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} \\log | L ( \\sigma + i t , \\pi ) | \\leq \\bigg ( \\dfrac { 3 } { 4 } - \\dfrac { \\sigma } { 2 } \\bigg ) \\log C ( t , \\pi ) - \\dfrac { 1 } { 2 } \\displaystyle \\sum _ { \\gamma } g ^ { - } _ { \\sigma , \\Delta } ( t - \\gamma ) + O ( d ) . \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} Y ^ + _ n : = Y _ { \\tau _ n } = \\frac 1 { F ( \\tau _ n ) } \\sum \\limits _ { k = 1 } ^ n f ( \\tau _ k ) \\quad Y ^ - _ n : = Y _ { \\tau _ { n + 1 } - } = \\frac 1 { F ( \\tau _ { n + 1 } ) } \\sum \\limits _ { k = 1 } ^ n f ( \\tau _ k ) \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} c _ 0 ^ 2 C | \\mathcal { t } \\cdot \\nabla p | ^ 2 - C ^ \\gamma = 0 \\qquad \\mbox { o n } \\Gamma , \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} N _ { m - ( a - 2 ) } & = N _ { m - ( a - 1 ) } + N _ { m - ( a + 1 ) } \\\\ N _ { m - 2 ( a - 2 ) } & = N _ { m - 2 ( a - 1 ) } + 2 N _ { m - 2 a } + N _ { m - 2 ( a + 1 ) } \\\\ N _ { m - 3 ( a - 2 ) } & = 4 N _ { m - 3 ( a - 1 ) } - 3 N _ { m - 3 a } + N _ { m - 3 ( a + 1 ) } . \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} K ^ d & = z ^ d = 1 , \\\\ E ^ d & = F ^ d = 0 , \\\\ [ z , E ] & = [ z , F ] = [ z , K ] = 0 , \\\\ K E & = \\zeta E K , \\\\ K F & = \\zeta ^ { - 1 } F K , \\\\ [ E , F ] & = K - z K ^ { - 1 } . \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} b ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } \\left [ b ( x , y , t ) + h ( x ) \\frac { 1 } { 2 } ( x ^ { 2 } - \\langle x ^ { 2 } \\rangle ) \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } p _ { 0 } ( y , t ) \\right ] \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} \\mathbf { D } = \\boldsymbol { \\varepsilon } \\mathbf { E } \\mathbf { B } = \\boldsymbol { \\mu } \\mathbf { H } . \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{gather*} \\forall i \\in I \\alpha ( d _ i ) = d _ i \\frac { \\delta _ n ^ k } { k + 1 } \\end{gather*}"}
-{"id": "4138.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } ( n - 3 i ) x _ i f _ { n + 1 - i } = 0 . \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} | B _ { n , k } ^ g | = \\begin{cases} B _ { ( g , n ) , k ( g , n ) / n } & k g \\equiv 0 \\bmod n , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} f ( x ) & = - 1 - ( A - B x ) ( x ^ 3 - 1 ) ( 1 + x ) + ( 2 m + 1 ) h ( x ) \\\\ g ( x ) & = ( x ^ 2 + x + 1 ) - ( C - 1 - D x ) ( x ^ 3 - 1 ) ( 1 + x ) + 2 m \\ : h ( x ) . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} \\overline \\exp _ { z ( s ) } \\big ( w ( s ) \\big ) = x ( s ) , \\forall \\ , s . \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} \\mathrm M _ { t , n } ^ { ( j ) } \\ , : = \\ , \\frac { 1 } { \\ , n \\ , } \\sum _ { i = 1 } ^ { n } \\delta _ { ( X _ { t , i } ^ { ( u ) } , \\ , \\ldots , \\ , X _ { t , i + j - 1 } ^ { ( u ) } ) } \\ , ; j \\ , = \\ , 2 , \\ldots , k \\ , \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} \\mu ( h ( v , v ) ) = \\mu ( h ( w , w ) ) . \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} \\mu ( e _ 1 ) = \\mathrm { d i a g } \\left ( - \\frac 1 2 , - \\frac 1 2 , \\frac 1 2 , \\frac 1 2 \\right ) , ~ \\mu ( e _ 2 ) = \\left ( \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & - 1 & 0 & 0 \\\\ - 1 & 0 & 0 & 0 \\\\ \\end{array} \\right ) , ~ \\mu ( e _ 3 ) = \\left ( \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ - 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} K _ { i j } ( x , x ) = C _ { i j } ( x ) / \\big ( a _ { i j } ( x ) - b _ { i j } ( x ) \\big ) \\mbox { f o r } x \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ m } u ^ 2 d V _ g = \\int _ { \\mathbb { R } ^ m } u ^ 2 e ^ { - \\frac { | x | ^ 2 } { 4 ( m - 2 ) } m } d V _ { g _ 0 } ; \\\\ & \\int _ { \\mathbb { R } ^ m } | \\nabla _ g u | _ g d V _ g = \\int _ { \\mathbb { R } ^ m } | \\nabla _ { g _ 0 } u | _ { g _ 0 } ^ 2 e ^ { - \\frac { | x | ^ 2 } { 4 } } d V _ { g _ 0 } . \\\\ \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} A _ m ( s ) : = \\# \\Big \\{ \\ell \\in \\{ 1 , \\dots , d \\} \\ , \\colon \\ , s _ \\ell < 2 ^ m - 1 \\Big \\} \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} U W ( f _ 1 \\oplus f _ 2 , V _ 1 \\oplus V _ 2 ) U ^ * & = W ( f _ 1 , V _ 1 ) \\otimes W ( f _ 2 , V _ 2 ) \\\\ U d \\Gamma ( \\omega _ 1 \\oplus \\omega _ 2 ) U ^ * & = d \\Gamma ( \\omega _ 1 ) \\otimes 1 + 1 \\otimes d \\Gamma ( \\omega _ 2 ) \\\\ U \\varphi ( f _ 1 , f _ 2 ) U ^ * & = \\varphi ( f _ 1 ) \\otimes 1 + 1 \\otimes \\varphi ( f _ 2 ) \\\\ U a ( f _ 1 , f _ 2 ) U ^ * & = a ( f _ 1 ) \\otimes 1 + 1 \\otimes a ( f _ 2 ) \\\\ U a ^ { \\dagger } ( f _ 1 , f _ 2 ) U ^ * & = a ^ { \\dagger } ( f _ 1 ) \\otimes 1 + 1 \\otimes a ^ { \\dagger } ( f _ 2 ) . \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} \\mathbb P \\Big ( \\int ^ { t } _ { 0 } \\lvert \\pi _ { s , k } ( b ) \\rvert ^ { 2 } { \\mathrm d } s < \\infty \\Big ) \\ , = \\ , 1 \\ , ; k \\ , = \\ , 2 , \\ldots , n \\ , , \\ , \\ , 0 \\le t \\le T \\ , . \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} \\| v _ n \\| ^ 2 _ { L ^ 2 } = M n \\geq 1 \\lim _ { n \\rightarrow \\infty } E ( v _ n ) = d _ M , \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} \\int _ { A _ { r _ 2 , r _ 3 } } ( \\mathfrak u ^ 2 - \\lambda \\phi _ { \\mathfrak u } ) \\mathfrak u ^ 2 d x > \\delta \\quad \\quad \\int _ { A _ { r _ i , r _ { i + 1 } } } ( \\mathfrak u ^ 2 + \\lambda \\phi _ { \\mathfrak u } ) \\mathfrak u ^ 2 d x < \\frac { \\delta } { 4 } , \\quad i = 1 , 3 . \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} \\boldsymbol { A } = \\begin{pmatrix} \\boldsymbol { O } & \\boldsymbol { N } \\\\ \\boldsymbol { N } ^ { \\top } & \\boldsymbol { O } \\end{pmatrix} , \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} p _ * = \\begin{cases} \\frac { ( N - 1 ) p } { N - p } & p < N , \\\\ q \\in ( 1 , \\infty ) & p \\geq N . \\end{cases} \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} f ( \\widetilde { \\varphi _ t \\ ; } ( z ) ) - f ( \\widetilde { \\varphi _ { t _ 0 } } ( z ) ) & = f \\big ( z _ { [ j ] } , t \\widetilde { G } z ' _ { [ j ] } \\big ) - f \\big ( z _ { [ j ] } , t _ 0 \\widetilde { G } z ' _ { [ j ] } \\big ) \\\\ & = ( t - t _ 0 ) \\sum _ { i = j + 1 } ^ n \\widetilde { g } _ { i i } z _ i \\dfrac { \\partial f } { \\partial z _ i } \\big ( z _ { [ j ] } , \\tau \\widetilde { G } z ' _ { [ j ] } \\big ) , \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} | S _ { \\alpha } ^ 2 f ( z ) | \\leq & \\sum _ { k : 2 ^ k \\geq \\gamma } \\frac { 1 } { 2 ^ { k \\alpha } } \\int _ { d ( z , z ' ) \\sim 2 ^ k } | f ( z ' ) | \\ ; d \\mu ( z ' ) \\\\ \\lesssim & \\sum _ { k : 2 ^ k \\geq \\gamma } \\frac { 1 } { 2 ^ { k \\alpha } } \\Big ( \\int _ { d ( z , z ' ) \\sim 2 ^ k } \\ ; d \\mu ( z ' ) \\Big ) ^ \\frac { 1 } { p ' } \\| f \\| _ { L ^ p } \\\\ \\lesssim & \\sum _ { k : 2 ^ k \\geq \\gamma } \\frac { 2 ^ \\frac { k n } { p ' } } { 2 ^ { k \\alpha } } \\leq C _ 1 \\gamma ^ { - \\frac { n } { q } } , \\frac 1 { p ' } + \\frac 1 q = \\frac \\alpha n \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} \\phi _ { n } \\stackrel { \\mathrm { d e f } } { = } \\int _ { \\mathbf { R } \\backslash E _ { n } } f ( x ) \\ , d x \\rightarrow 0 , \\mbox { { a s } } n \\rightarrow \\infty , \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( X ) } ^ p = \\int _ { X } | f ( r , y ) | ^ p d \\mu ( r , y ) = \\int _ 0 ^ \\infty \\int _ Y | f ( r , y ) | ^ p r ^ { n - 1 } d r d \\mu _ Y ( y ) . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} { \\bf B } ^ { S } ( \\nu ; \\Delta ) = \\prod _ { \\substack { p | f \\\\ p \\not \\in S } } \\left \\{ \\frac { \\zeta _ p ( - \\nu _ p ) } { L _ p \\left ( \\frac { - \\nu _ p + 1 } { 2 } , \\chi _ { D } \\right ) } | f | _ p ^ { \\frac { \\nu _ p - 1 } { 2 } } + \\frac { \\zeta _ p ( \\nu _ p ) } { L _ p \\left ( \\frac { \\nu _ p + 1 } { 2 } , \\chi _ { D } \\right ) } | f | _ p ^ { \\frac { - \\nu _ p - 1 } { 2 } } \\right \\} , \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} \\delta A = - \\nabla ^ \\alpha A _ \\alpha , \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} \\left [ \\begin{array} { l l } B ^ { ( m ) } _ { n - m - 1 } & B ^ { ( m + 1 ) } _ { n - m - 1 } \\\\ A ^ { ( m ) } _ { n - m - 1 } & A ^ { ( m + 1 ) } _ { n - m - 1 } \\end{array} \\right ] = \\left [ \\begin{array} { l l } B ^ { ( m - 1 ) } _ { n - m - 1 } & B ^ { ( m ) } _ { n - m - 1 } \\\\ A ^ { ( m - 1 ) } _ { n - m - 1 } & A ^ { ( m ) } _ { n - m - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n } \\end{array} \\right ] \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} \\rho ( x ) : = \\sqrt { - \\det g _ { \\alpha \\beta } ( x ) } \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} c ( g _ j ^ { - 1 } , X ) & : = \\{ h _ { t _ j } ^ { - 1 } | ~ t _ j \\} , \\\\ X _ { t _ j } & : = \\{ x \\in X : c ( g _ j ^ { - 1 } , x ) = h _ { t _ j } ^ { - 1 } \\} , \\\\ c ( ( g _ i g _ { i + 1 } ) ^ { - 1 } , X ) & : = \\{ h _ { s _ i } ^ { - 1 } | ~ s _ i \\} , \\\\ X _ { s _ i } & : = \\{ x \\in X : c ( ( g _ i g _ { i + 1 } ) ^ { - 1 } , x ) = h _ { s _ i } ^ { - 1 } \\} . \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} \\underset { \\mathbf { x } \\in \\mathbb { R } ^ { n p } , \\mathbf { z } \\in \\mathbb { R } ^ { m p } } { } f ( \\mathbf { x } ) \\ , \\ , \\bar { \\mathbf { A } } \\mathbf { x } + \\mathbf { B } \\mathbf { z } = \\mathbf { 0 } , \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} \\lim _ { | N | \\to \\infty } G ^ { \\epsilon } ( N + i y ) = \\lim _ { | N | \\to \\infty } G ^ { \\epsilon } _ y ( N ) = \\lim _ { | N | \\to \\infty } \\int _ { \\mathbb { R } } \\widehat { G ^ { \\epsilon } _ c } ( t ) e ^ { - 2 \\pi ( y - c ) t } e ^ { i 2 \\pi N t } \\d t = 0 , \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} W _ { \\psi _ { i , 1 , * } , \\varphi _ { i , 2 , * } } f ( z ' _ { [ j ] } ) = \\psi _ { i , 1 , * } ( z ' _ { [ j ] } ) f \\big ( G ^ i z ' _ { [ j ] } + ( b ^ { i , 1 } ) ' _ { [ j ] } \\big ) = W _ { \\psi _ { i , 1 , * } , \\varphi _ { i , 1 , * } } f ( z ) , \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} 1 = R ( x ) R ( x ) = 2 R ( R ( x ) ) - R ( x ^ 2 ) = 2 - R ( x ^ 2 ) , \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} d X _ t = \\mu ( t , X _ t , \\alpha _ t ) \\mathrm d t + \\sigma ( t , X _ t , \\alpha _ t ) \\mathrm d W _ t , \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} \\gamma _ + ( v ) = - \\int _ { \\mathbb R ^ 3 } \\phi _ 3 u _ 1 ^ 2 d x < 0 \\quad \\quad \\gamma _ - ( v ) = - \\int _ { \\mathbb R ^ 3 } \\phi _ 3 u _ 2 ^ 2 d x < 0 . \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} l ( x ) : = \\begin{cases} \\frac { u _ + ( x ) - u _ - ( x ) } { v _ + ( \\xi ( x ) ) - v _ - ( \\xi ( x ) ) } & , v _ + ( \\xi ( x ) ) \\neq v _ - ( \\xi ( x ) ) \\\\ \\frac { \\partial _ x ( u _ { + } ( x ) - u _ { - } ( x ) ) } { \\partial _ x ( v _ { + } ( \\xi ( x ) ) - v _ { - } ( \\xi ( x ) ) ) } & , v _ + ( \\xi ( x ) ) = v _ - ( \\xi ( x ) ) \\end{cases} \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} B ( z , w ; Z , W ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { k + 1 } { \\pi } Y _ p ( k ; w , W ) z ^ { k } \\overline { Z } ^ k . \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} p _ { L , N + 1 } ^ { \\ell _ 2 } ( x ) = \\dfrac { \\sum \\limits _ { j = 0 } ^ N \\left ( \\omega _ j \\sum \\limits _ { \\ell = 0 } ^ N \\dfrac { \\tilde { \\Phi } _ { \\ell } ( x _ j ) \\tilde { \\Phi } _ { \\ell } ( x ) } { 1 + \\lambda \\mu _ { \\ell } ^ 2 } \\right ) f ( x _ j ) } { ( 1 + \\lambda \\mu _ 0 ^ 2 ) \\sum \\limits _ { j = 0 } ^ N \\omega _ j \\sum \\limits _ { \\ell = 0 } ^ N \\dfrac { \\tilde { \\Phi } _ { \\ell } ( x _ j ) \\tilde { \\Phi } _ { \\ell } ( x ) } { 1 + \\lambda \\mu _ { \\ell } ^ 2 } } . \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} | \\ ! | u | \\ ! | _ { 1 / r } ^ 2 \\leq C _ K \\Bigl ( \\sum _ { j = 0 } ^ N | \\ ! | X _ j u | \\ ! | _ 0 ^ 2 + | \\ ! | u | \\ ! | _ 0 ^ 2 \\Bigr ) , \\ , \\ , \\ , \\ , \\forall u \\in C _ 0 ^ \\infty ( K ) . \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} ( H _ 1 \\otimes H _ 1 ) ^ \\top ( H _ 1 \\otimes H _ 1 ) & = H _ 1 ^ \\top H _ 1 \\otimes H _ 1 ^ \\top H _ 1 = ( m I _ m - J _ m ) \\otimes ( m I _ m - J _ m ) , \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} f _ j ( z ) = \\Im \\bigl \\langle u - Q [ z ] , D _ j Q [ z ] \\bigr \\rangle \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} & \\lambda _ 1 \\left ( \\frac { x + s } { 2 } \\right ) = \\frac { 1 } { 2 } \\lambda _ 1 ( x ) + \\frac { 1 } { 2 } \\lambda _ 1 ( s ) , \\\\ & \\lambda _ 4 \\left ( \\frac { x + s } { 2 } \\right ) \\leq \\frac { 1 } { 2 } \\lambda _ 4 ( x ) + \\frac { 1 } { 2 } \\lambda _ 4 ( s ) , \\\\ & \\lambda _ 2 \\left ( \\frac { x + s } { 2 } \\right ) \\geq \\frac { 1 } { 2 } \\lambda _ 2 ( x ) + \\frac { 1 } { 2 } \\lambda _ 2 ( s ) . \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} P _ N h : = \\sum _ { j = - 1 } ^ N \\sum _ { m = - N } ^ { N } \\bar \\mu _ { j , m } h _ { j , m } , \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} H u = E _ j u \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} \\begin{cases} x + y + z + w & = \\ell , \\\\ x + y - z - w & = a , \\\\ x - y + z - w & = a , \\\\ x - y - z + w & = - a . \\end{cases} \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} B _ 1 ( t ) = | 2 b | \\max ( 1 , | t ^ { - 1 } b ^ 2 | ) ^ { - s - 1 } + 2 ^ { - 1 } | b | . \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} c = \\inf _ { u \\in \\mathcal N } I ( u ) , \\mathcal N = \\{ u \\in E \\setminus \\{ 0 \\} \\ , \\ , | \\ , \\ , I ' ( u ) u = 0 \\} , \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{gather*} \\hom ( s _ { i , n } , s _ { 0 , i } ) = 1 \\hom ^ j ( s _ { i , n } , s _ { 0 , i } ) = 0 , j \\neq 0 \\end{gather*}"}
-{"id": "4595.png", "formula": "\\begin{align*} \\frac { p _ { \\beta _ { 1 } , \\lambda } } { p _ { 0 } } \\frac { p _ { \\beta _ { 2 } , \\lambda } } { p _ { 0 } } = ( 1 + \\lambda ^ { 2 } ) \\exp ^ { - \\frac { y ^ { 2 } + ( 1 + \\lambda ^ { 2 } ) ( y - \\lambda \\beta _ { 1 } ^ { \\tau } x ) ^ { 2 } } { 2 ( 1 + \\lambda ^ { 2 } ) } - \\frac { y ^ { 2 } + ( 1 + \\lambda ^ { 2 } ) ( y - \\lambda \\beta _ { 2 } ^ { \\tau } x ) ^ { 2 } } { 2 ( 1 + \\lambda ^ { 2 } ) } } \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} s ( u K ) : = u L . \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} [ K ^ 2 , K _ \\pm ] = 0 = [ K ^ 2 , K _ 0 ] . \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} \\underset { \\sigma ^ 2 \\rightarrow 0 } { \\lim } \\mathbb { P } ( \\mathcal { A } _ 1 ) \\leq \\underset { \\sigma ^ 2 \\rightarrow 0 } { \\lim } \\mathbb { P } ( \\{ R R ( k _ 0 ) > \\Gamma _ { R R T } ^ { \\alpha } ( k _ 0 ) \\} ) = 0 . \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} b ( x , y , t ) = h ( x ) \\beta ( y , t ) \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} f ^ \\prime & : = e _ { 1 2 } + e _ { 1 3 } + e _ { 2 3 } + e _ { 2 5 } + e _ { 3 5 } + e _ { 3 6 } \\\\ e ^ \\prime & : = e _ { 1 2 } + e _ { 1 3 } + e _ { 1 4 } + e _ { 2 3 } + e _ { 2 5 } + e _ { 3 6 } , \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} \\delta = \\frac { 1 } { 2 } , a = 2 , \\mu = e ^ { 2 \\tau } \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} P ^ { ( \\alpha , \\beta ) } _ { \\ast } f ( n ) = \\sup _ { t > 0 } | P _ t ^ { ( \\alpha , \\beta ) } f ( n ) | . \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} u _ { g _ 1 } u _ { g _ 2 } = \\gamma ( g _ 1 , g _ 2 ) u _ { g _ 1 g _ 2 } g _ 1 , g _ 2 \\in G , \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} D _ n = \\sum _ { i = 1 } ^ { n - 1 } X _ { i , n } , \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} \\rho _ { \\infty } \\bar \\phi _ { u _ n ^ 1 } ( \\cdot + y _ n ^ 1 ) u _ n ^ 1 ( \\cdot + y _ n ^ 1 ) - \\rho _ { \\infty } \\bar \\phi _ { ( u _ n ^ 1 ( \\cdot + y _ n ^ 1 ) - v _ 1 ) } ( u _ n ^ 1 ( \\cdot + y _ n ^ 1 ) - v _ 1 ) - \\rho _ { \\infty } \\bar \\phi _ { v _ 1 } v _ 1 = o ( 1 ) , \\textrm { i n } \\ , \\ , H ^ { - 1 } ( \\R ^ 3 ) . \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} [ - \\int _ 0 ^ \\cdot \\nabla \\phi ^ * ( r , W _ r ) \\mathrm d W _ r , N ] _ t = - \\int _ 0 ^ t \\nabla \\phi ^ * ( r , W _ r ) \\mathrm d [ W , N ] _ r , \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} \\langle f , \\overline { \\chi _ { z , w } } \\rangle _ { \\mathcal { H } _ p } & = \\int _ 0 ^ { \\infty } \\int _ { \\C ^ n } f ( t , \\zeta ) \\left ( 4 \\pi t H _ p ( t , w , \\zeta ) e ^ { 2 \\pi i z t } \\right ) \\frac { e ^ { - 4 \\pi p ( \\zeta ) t } } { 4 \\pi t } \\d V ( \\zeta ) \\d t = \\int _ 0 ^ { \\infty } f ( t , w ) e ^ { i 2 \\pi z t } \\d t , \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } k _ { 1 , \\sigma } = \\kappa & \\forall \\sigma \\in I _ F ^ 0 \\\\ k _ { 2 , \\sigma } = \\kappa + 1 & \\forall \\sigma \\in I _ F ^ 1 \\end{array} \\right . \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} \\mathcal { D } ( p _ { _ 1 } | | p _ { _ 0 } ) = \\frac { 1 } { 2 } \\left ( \\frac { P _ y } { \\sigma _ w ^ 2 } - 1 + \\log \\frac { \\sigma _ w ^ 2 } { P _ y } \\right ) . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} \\sum _ { g \\ge 0 } \\sum _ { n \\in \\mathbb { Z } } P _ { n , g } z ^ n q ^ { g - 1 } = \\left ( \\sqrt { z } - \\frac { 1 } { \\sqrt { z } } \\right ) ^ { - 2 } \\frac { 1 } { \\Delta ( z , q ) } . \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} Q _ \\alpha ( \\rho \\| \\sigma ) = Q _ { 1 - \\alpha } ( \\rho \\| \\sigma ) , \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} \\bigcap _ { c \\in I ( C ; u ) } N [ c ] = \\{ u \\} \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} c _ 0 ^ 2 C | \\theta \\cdot \\nabla p | ^ 2 - C ^ \\gamma = 0 \\qquad \\mbox { o n s u p p } ( \\mu ) \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} \\dot { p } ^ * _ t = - \\nabla _ x f ( x ^ * _ s , \\theta ^ * _ s ) p ^ * _ t , p ^ * _ T = - \\nabla _ x \\Phi ( x ^ * _ T , y _ 0 ) . \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} ( X + A + \\Phi ) u = 0 . \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} \\eta ( u ^ { \\varepsilon , \\nu } ) _ t + q ^ { \\nu } ( t , x , u ^ { \\varepsilon , \\nu } ) _ x ~ = ~ a ^ { \\varepsilon , \\nu } + b ^ { \\varepsilon , \\nu } + c ^ { \\varepsilon , \\nu } \\ , , \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} y ^ 2 = - x ^ 3 + h \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} u ( t ) = \\sum ^ { s } _ { j = 1 } \\tilde { u } _ { j } \\ , \\tilde { \\chi } _ j ( t ) , \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} d = \\inf \\{ J ( u ) : u \\in H _ 0 ^ 1 ( \\Omega ) \\setminus \\{ 0 \\} , I ( u ) = 0 \\} . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} Y ( t ) & = \\frac { \\partial f } { \\partial s } ( t , 0 ) = d \\tau ^ 2 _ t ( \\frac { d \\lambda } { d s } ( 0 ) ) \\\\ & = d \\tau ^ 2 _ t ( Y ( 0 ) ) = d \\tau ^ 2 _ t ( X ) . \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} ( I - \\lambda _ n ^ { - 1 } \\Delta _ L ) w _ n = \\lambda _ n ^ { - 1 } ( \\lambda _ n I - \\Delta _ L ) w _ n = \\lambda _ n ^ { - 1 } \\frac { f _ n ( x ) } { \\| R _ { \\lambda _ n } f _ n \\| _ { C ^ { 0 , \\alpha } } } . \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} W _ n ( B ) = B ^ { n + 1 } , \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} \\begin{aligned} & \\ge \\frac { n ( n - 1 ) \\tau ^ 2 } { \\Im ( \\rho ) ^ 2 } \\min \\left \\{ \\frac { - 1 + ( 2 e ) ^ 2 } { 2 \\left ( \\frac { 1 } { 2 e } + 2 e \\right ) ^ 2 } , \\lim _ { n \\to \\infty } \\left ( \\frac { - 1 + ( n e ) ^ 2 } { 2 \\left ( \\frac { 1 } { n e } + n e \\right ) ^ 2 } - \\frac { 0 . 1 8 4 ( n - 2 ) } { n e } \\right ) \\right \\} \\\\ & > \\frac { 0 . 4 3 2 n ( n - 1 ) \\tau ^ 2 } { \\Im ( \\rho ) ^ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\mathbb { S } _ { f , g } = \\frac { ( - 1 ) ^ { n ( d - 1 ) } } { \\tau ( d ) ^ n } \\overline { \\mathbf { S } _ { f , g } } . \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} e _ i b & = b _ L \\otimes \\cdots \\otimes b _ { j _ + + 1 } \\otimes e _ i b _ { j _ + } \\otimes b _ { j _ + - 1 } \\otimes \\cdots \\otimes b _ 1 , \\\\ f _ i b & = b _ L \\otimes \\cdots \\otimes b _ { j _ - + 1 } \\otimes f _ i b _ { j _ - } \\otimes b _ { j _ - - 1 } \\otimes \\cdots \\otimes b _ 1 . \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} \\wp : = \\{ x \\in \\mathbb { G } : \\ , | x | = 1 \\} , \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} \\infty = \\limsup _ { t \\to T ^ - } Y ( t ) = \\limsup _ { t \\to T ^ - } \\left ( G ( t ) ^ { 1 - \\alpha } + N ' ( t ) \\right ) . \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} S _ { \\varepsilon _ k } ( \\tilde { t } ) = u \\bigl ( f _ { \\varepsilon _ k } ( \\tilde { t } ) \\bigr ) S _ { \\varepsilon _ k } ( \\tilde { t } ) = l \\bigl ( f _ { \\varepsilon _ k } ( \\tilde { t } ) \\bigr ) . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} e ^ { \\langle z , \\varepsilon ' \\xi _ 0 \\rangle } \\omega = e ^ { \\langle z , ( \\varepsilon ' - \\varepsilon ' _ 1 ) \\xi _ 0 \\rangle } e ^ { \\langle z , \\varepsilon ' _ 1 \\xi _ 0 \\rangle } \\omega \\end{align*}"}
-{"id": "9943.png", "formula": "\\begin{align*} \\bigcap _ { 1 \\le j \\le n - 1 } \\{ \\ , y ' \\in \\R _ { y ' } ^ { n - 1 } \\mid \\langle y ' , \\ , \\tilde { \\eta } _ j \\rangle > 0 \\ , \\} \\ , \\ , \\subset \\ , \\ , \\Gamma \\cap \\{ y _ 1 = 0 \\} . \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} S _ { k , r } : = \\sum _ { i = 1 } ^ k ( 1 + S _ i ) ^ r \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} a _ { V , \\Omega } ( u , h ) = \\Phi _ f ( h ) , \\forall h \\in W ^ { 1 , \\infty } _ c ( \\Omega ) . \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} L ^ t _ { \\theta } : = \\sum _ { k \\geq - N } ( - m a _ { m k + 1 } ) L ^ t _ k \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} f = \\sum _ { i , j } u _ { i j } \\pi ^ { v ( a _ { i j } ) } x ^ i y ^ j , u _ { i j } = \\frac { a _ { i j } } { \\pi ^ { v ( a _ { i j } ) } } . \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} & C o l o r _ 1 = \\{ 1 , 2 , 3 , 4 3 , 4 4 , 4 5 , 4 6 , 4 9 , 5 1 \\} , \\\\ & C o l o r _ 2 = \\{ 4 , 5 , 6 , 8 , 1 1 , 1 4 , 1 8 , 2 0 , 2 1 , 2 3 , 2 8 , 3 3 , 3 6 , 4 8 \\} , \\\\ & C o l o r _ 3 = \\{ 7 , 1 3 , 1 5 , 1 7 , 2 5 , 2 6 , 2 9 , 3 1 , 3 4 , 3 7 , 3 8 , 4 0 , 4 2 , 4 7 \\} , \\\\ & C o l o r _ 4 = \\{ 9 , 1 0 , 1 2 , 1 6 , 1 9 , 2 2 , 2 4 , 2 7 , 3 0 , 3 2 , 3 5 , 3 9 , 4 1 , 5 0 \\} . \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} \\Lambda : = \\left ( \\begin{array} { c c c } \\mbox { l o g } \\vert \\sigma _ 1 ( \\gamma _ 1 ) \\vert & \\dots & \\mbox { l o g } \\vert \\sigma _ d ( \\gamma _ 1 ) \\vert \\\\ \\vdots & \\dots & \\vdots \\\\ \\mbox { l o g } \\vert \\sigma _ 1 ( \\gamma _ { d - 1 } ) \\vert & \\dots & \\mbox { l o g } \\vert \\sigma _ d ( \\gamma _ { d - 1 } ) \\vert \\end{array} \\right ) \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} \\alpha _ t ( g ) = \\lambda ( f ( g t , t ) ) = \\mu ( h ( g t , t ) ) , g \\in \\Omega ( \\i ) . \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} \\hat { \\tilde { g } } ( y , \\xi , \\lambda ) & = \\int _ { 0 } ^ { \\infty } e ^ { - \\lambda t } \\int _ { - \\infty } ^ { + \\infty } e ^ { i \\xi x } g ( x , y , t ) \\ , \\mathrm { d } x \\ , \\mathrm { d } t \\\\ & = \\int _ { 0 } ^ { \\infty } e ^ { - \\lambda t } e ^ { i \\xi y \\ , - \\ , \\xi ^ { 2 } t } \\ , \\mathrm { d } t \\\\ & = \\frac { e ^ { i \\xi y } } { \\lambda + \\xi ^ { 2 } } . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} u _ \\tau = e ^ { \\tau \\Phi } u , f _ \\tau = e ^ { \\tau \\Phi } f , g _ \\tau = e ^ { \\tau \\Phi ( \\cdot , 0 ) } g , u _ { 0 , \\tau } = e ^ { \\tau \\Phi ( 0 , \\cdot ) } u _ 0 \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} \\kappa ^ { S } ( \\theta ) + \\kappa ^ { - A } ( \\theta ) = 0 , \\ \\theta < 0 , \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} x = \\sum _ { i } ( x , v _ i ^ { \\mu } ) u _ i ^ { \\mu } \\quad \\mathrm { a n d } y = \\sum _ { i } ( u _ i ^ { \\mu } , y ) v _ i ^ { \\mu } . \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} a _ { m + n } = & f ( g ^ { m + n } ) + ( m + n + 1 ) D ( f ) \\\\ \\leq & f ( g ^ m ) + f ( g ^ n ) + D ( f ) + ( m + n + 1 ) D ( f ) \\\\ = & f ( g ^ m ) + ( m + 1 ) D ( f ) + f ( g ^ n ) + ( n + 1 ) D ( f ) \\\\ = & a _ m + a _ n . \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} R i c _ { g ^ { ( 4 ) } } = 0 . \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} M ' = \\frac { N _ C } { \\gcd ( C , N ) } \\cdot m _ { \\overline { C } } , \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} | X | = X / \\sim \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} a ( i , j ) & = 2 n - i - j + m + c _ j , \\\\ b ( i , j ) & = j - i + m - c _ j . \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} S ( t , \\omega ^ \\prime ) : = \\sup _ { l \\in \\mathbb { N } } \\left \\lvert X ^ { n _ l , \\omega ^ \\prime } _ { \\kappa ( n _ l , t ) } \\right \\rvert \\ , , \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} F _ p ( z ) = \\sum _ { n \\geq 0 } p _ n z ^ n F _ q ( z ) = \\sum _ { n \\geq 0 } q _ n z ^ n \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} ( M v ) ( s ) : = \\sup _ { a \\in A ( s ) } \\int _ S v ( t ) p ( d t | s , a ) , \\ \\ s \\in S . \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} E ( v _ n ) = E ( \\tilde { V } ^ { j _ 0 } ) + E ( \\tilde { v } ^ { j _ 0 } _ n ) + o _ n ( 1 ) . \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| \\langle \\phi _ 0 , v ( t ) \\rangle \\phi _ 0 \\| _ { H ^ 1 } = 0 . \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} \\int _ { \\mathbb { B } _ p } q ( \\zeta ) \\phi ( t , \\zeta ) \\lambda ( p ( \\zeta ) , t ) \\d V ( \\zeta ) = 0 , \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} | c _ { i r } | < | a _ 1 | = \\alpha = \\inf _ { r \\geq 0 } \\{ \\max _ { 1 \\leq i \\leq n } | c _ { i r } | \\} , \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} \\Phi _ L ( \\mu , j , F ) : = \\Psi _ L ( \\mu , j ) - \\langle j , F \\rangle _ L + \\Psi ^ \\star _ L ( \\mu , F ) \\ge 0 , \\end{align*}"}
-{"id": "9956.png", "formula": "\\begin{align*} a _ { X } = a _ { \\tilde f } \\otimes f ^ { - 1 } a _ { Y } \\quad \\int _ f a _ { X } = 1 . \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} x _ 1 + x _ 2 + x _ 3 = x + x + 3 ^ k - 2 x - 3 ^ k = 0 , \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} u _ { k + 1 } = u _ k + \\tau A ^ * ( y - A u _ k ) , k = 0 , \\dots . \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} R f ( t , x ) = e ^ { - 2 \\pi i \\lambda x } f \\left ( \\frac { t _ 1 } { \\sqrt { 2 n } } , \\ldots , \\frac { t _ n } { \\sqrt { 2 n } } \\right ) . \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\int d x \\ x ^ { 2 } b ( x , y , t ) = \\frac { 1 } { 2 } \\langle x ^ { 2 } \\rangle \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\beta ( y , t ) = \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\beta ( y , t ) \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 , \\ t \\in T } f ^ { ( m ) } ( t ) = \\begin{cases} b _ p & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} U _ \\lambda ( o , \\ , o \\ , | \\ , z ) & = \\sum _ { n = 1 } ^ \\infty f ^ { ( 2 n ) } _ \\lambda ( o , \\ , o ) z ^ { 2 n } = \\sum _ { n = 1 } ^ \\infty c _ { n - 1 } \\left ( \\frac { d - 1 } { d - 1 + \\lambda } \\right ) ^ { n - 1 } \\left ( \\frac { \\lambda } { d - 1 + \\lambda } \\right ) ^ n z ^ { 2 n } \\\\ & = \\frac { \\lambda } { d - 1 + \\lambda } z ^ 2 \\ , \\mathcal { C } \\left ( \\frac { \\lambda ( d - 1 ) z ^ 2 } { ( d - 1 + \\lambda ) ^ 2 } \\right ) , \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} E ( v _ \\lambda ) = \\frac { \\lambda ^ 2 } { 2 } \\| v _ \\lambda \\| ^ 2 _ { \\dot { H } ^ 1 _ c } - \\frac { \\lambda ^ { \\frac { d \\alpha } { 2 } } } { \\alpha + 2 } \\| v \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} d X _ t & = b \\left ( t , X _ t , \\mu _ { t } , \\hat { \\alpha } \\left ( t , X _ t , \\mu _ { t } , \\sigma ^ { - 1 } Z _ t \\right ) \\right ) d t + \\sigma d W _ t \\\\ X _ 0 & = \\xi , \\\\ d Y _ t & = - f \\left ( t , X _ t , \\mu _ { t } , \\hat { \\alpha } \\left ( t , X _ t , \\mu _ { t } , \\sigma ^ { - 1 } Z _ t \\right ) \\right ) d t + Z _ t d W _ t \\\\ Y _ T & = g ( X _ T , \\mu _ { T } ) , \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t u + \\frac 1 2 \\Delta u = { - \\nabla u ^ * \\ , b - f ( \\cdot , u , \\nabla u ) } \\\\ u ( T ) = \\Phi , \\end{array} \\right . \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{gather*} ( \\psi \\cdot \\varphi ) ( x ) = ( \\psi \\otimes \\varphi ) ( \\Delta ( x ) ) , \\eta ( 1 ) = \\varepsilon , \\\\ \\Delta ( \\psi ) ( x \\otimes y ) = \\psi ( x y ) , \\varepsilon ( \\psi ) = \\psi ( 1 ) , S ( \\psi ) = \\psi \\circ S \\end{gather*}"}
-{"id": "4553.png", "formula": "\\begin{align*} \\dim H _ k ( T _ z , u _ z ^ { \\pm 1 } ) = \\begin{cases} \\binom { k + n } { k } & ( z \\in Z ^ { ( 0 ) } ) , \\\\ \\binom { k + n } { k } - 1 & ( z \\in Z ^ { ( 1 ) } ) . \\end{cases} \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} M _ 3 = \\frac { 1 } { 1 0 0 n \\sum _ { b = 1 } ^ n | j _ b | } . \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} \\{ x _ { 2 , \\tau } , p _ \\tau \\} _ { D B } = - \\frac { f ( t _ \\tau ) } { m } p _ { 2 , \\tau } ; \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} G _ k ( \\bar { X } , \\bar { T } ) = ( \\nabla _ { \\ ! \\ ! \\bar { X } } ^ { k } \\boldsymbol { \\cdot } \\bar { T } ) ( F ( \\bar { X } ) ) = \\sum _ { \\bar { w } \\in D _ k } P _ { k , \\bar { w } } ( \\bar { T } ) X _ 1 ^ { w _ 1 } \\cdots X _ n ^ { w _ n } . \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} \\lvert ( i _ 1 , j _ 1 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i _ 2 , j _ 2 ) \\rvert _ { W _ n ( w ) } = i _ 1 - i _ 2 . \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} \\sup _ { 1 \\le i \\le n } \\| P _ i ( \\epsilon ) - ( \\tilde { P } + \\epsilon ( i - 1 ) \\tilde { P } ^ { ( 1 ) } ) \\| = O ( \\epsilon ^ { 2 } n ^ 2 ) \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} \\begin{cases} \\nabla u _ 1 ( x _ 1 ) = \\sum ^ N _ { i = 2 } x _ i \\\\ \\cdots \\\\ \\nabla u _ j ( x _ j ) = \\sum ^ N _ { i = 1 , i \\neq j } x _ i \\\\ \\cdots \\\\ \\nabla u _ N ( x _ N ) = \\sum ^ { N - 1 } _ { i = 1 } x _ i \\\\ \\end{cases} \\begin{cases} x _ 1 + \\nabla u _ 1 ( x _ 1 ) = \\sum ^ N _ { i = 1 } x _ i \\\\ \\cdots \\\\ x _ j + \\nabla u _ j ( x _ j ) = \\sum ^ N _ { i = 1 } x _ i \\\\ \\cdots \\\\ x _ N + \\nabla u _ N ( x _ N ) = \\sum ^ N _ { i = 1 } x _ i \\\\ \\end{cases} , \\medskip \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} y ( s - 0 ) = 0 \\ \\ \\mbox { a n d } \\ \\ \\lim _ { x \\rightarrow s _ { - } } \\{ y ^ { \\prime } ( x ) Y ( x ) - y ( x ) Y ^ { \\prime } ( x ) \\} = 0 . \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} \\| f \\| _ { B ( G ) } : = \\inf \\{ \\| x \\| _ H \\| y \\| _ H : f ( t ) = \\langle \\pi ( t ) x , y \\rangle _ H t \\in G \\} , \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ p ( - 1 ) ^ l { a \\choose { p - l } } & = { { a - 1 } \\choose p } \\\\ \\sum _ { l = 0 } ^ p ( - 1 ) ^ l { a \\choose { p - l } } l & = - { { a - 2 } \\choose { p - 1 } } \\\\ \\sum _ { l = 0 } ^ p ( - 1 ) ^ l { a \\choose { p - l } } l ( l - 1 ) & = 2 { { a - 3 } \\choose { p - 2 } } . \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} \\Phi ( u ( t ) ) = \\dot { K } ( t ) u _ 0 + K ( t ) u _ 1 + \\int _ { 0 } ^ t K ( t - s ) ( | u | ^ { \\frac { 4 } { n - 2 } } u ( s ) ) d s \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} \\| u \\| _ { ( \\kappa _ 1 + 1 ) p ^ * } = \\| u ^ { \\kappa _ 1 + 1 } \\| _ { p ^ * } ^ { \\frac { 1 } { \\kappa _ 1 + 1 } } & \\leq M _ 7 ( \\kappa _ 1 , u ) \\left [ \\| u \\| _ { p ^ * } ^ { p ^ * } + 1 \\right ] ^ { \\frac { 1 } { ( \\kappa _ 1 + 1 ) p } } < \\infty . \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} \\partial ( f \\circ X ^ { - 1 } ) = ( \\partial f ) \\circ X ^ { - 1 } , \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| T u ^ k - u ^ k \\| = 0 . \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} t _ 0 = T _ { o p t } , t _ 1 = t _ 0 - \\tau _ 1 , \\cdots , t _ k = t _ 0 - \\tau _ k , \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} F ( \\sqrt { L } ) ( x , y ) = f ( \\sqrt { L } ) [ g ( \\sqrt { L } ) ( \\cdot , y ) ] ( x ) = f ( \\sqrt { L } ) [ g ( \\sqrt { L } ) ( x , \\cdot ) ] ( y ) , \\forall x , y \\in M . \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ \\infty \\mathbb { E } _ p \\big ( \\Phi _ A U _ { k - 2 } ( a _ E , p ) \\big ) t ^ k & = \\mathbb { E } _ p \\Big ( \\Phi _ A \\sum _ { k = 2 } ^ \\infty U _ { k - 2 } ( a _ E , p ) t ^ k \\Big ) \\\\ & = \\mathbb { E } _ p \\Big ( \\frac { \\Phi _ A t ^ 2 } { p t ^ 2 - a _ E t + 1 } \\Big ) \\\\ & = \\frac { t ^ 2 } { ( 1 + p t ^ 2 ) } \\cdot \\mathbb { E } _ p \\left ( \\Phi _ A \\sum _ { \\ell = 0 } ^ \\infty \\left ( \\frac { a _ E t } { 1 + p t ^ 2 } \\right ) ^ \\ell \\right ) . \\\\ \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} \\log _ { \\phi } ( \\omega _ n ) & = \\gamma _ 0 \\omega _ n + \\gamma _ 1 \\tau ( \\omega _ n ) + \\gamma _ 2 \\tau ^ 2 ( \\omega _ n ) + \\dots \\\\ & = \\gamma _ 0 \\omega _ n + \\gamma _ 1 \\ell _ 1 ( z _ 1 ) \\ell _ 1 ( z _ 2 ) \\dots \\ell _ 1 ( z _ n ) \\omega _ n + \\gamma _ 2 \\ell _ 2 ( z _ 1 ) \\ell _ 2 ( z _ 2 ) \\dots \\ell _ 2 ( z _ n ) \\omega _ n + \\dots \\\\ & = \\omega _ n \\log _ { \\varphi } ( 1 ) , \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} \\mathbb { E } _ { z \\sim \\mu } L ( g ( \\theta _ t , z ) , \\frac { d } { d t } g ( \\theta _ t , z ) ) = \\int _ { \\Omega } L ( x , v ( t , x ) ) \\rho ( \\theta _ t , x ) d x . \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} \\varphi ( n ) = \\vert \\varphi ( n ) \\vert e ^ { i \\gamma ( n ) } . \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { r } \\binom { d } { s } ( - 1 ) ^ s = ( - 1 ) ^ r \\binom { d - 1 } { r } , \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} = \\frac { 1 } { \\mathcal { T } } \\int _ 0 ^ { \\mathcal { T } } \\left ( \\int _ 0 ^ t \\nabla \\phi ( y _ { \\mathcal { T } } ( s ) ) ^ T f ( y _ { \\mathcal { T } } ( s ) , u _ { \\mathcal { T } } ( s ) ) d s \\right ) d t . \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} \\deg _ { x _ k } P ( x _ k ) \\le n _ k n - \\sum _ { i = 1 } ^ { n _ k } i - \\sum _ { i = 0 } ^ { n _ k - 1 } i = n _ k ( n - n _ k ) . \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} ( A - B ) ^ 2 - ( C - D ) ^ 2 & = ( A + B ) ^ 2 - ( C + D ) ^ 2 - 4 A B + 4 C D \\\\ & \\equiv ( A + B ) ^ 2 - ( C + D ) ^ 2 + 4 C D . \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} s ( n , k ) = s ( n - 1 , k - 1 ) - ( n - 1 ) s ( n - 1 , k ) . \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} B _ { \\overline { \\nu } , 0 } ( t , \\overline { a } ) = \\sum _ { h = 0 } ^ L c _ h \\frac { L ! } { h ! } t ^ h , \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} \\left \\langle \\psi \\bigg | \\frac { \\delta h } { \\delta \\psi } \\right \\rangle = \\left ( \\chi ^ * \\frac { \\delta h } { \\delta \\chi } \\right ) . \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} M _ { 2 , 0 } \\otimes M _ { l ' , p } \\simeq \\begin{cases} M _ { 2 , p } & l ' = 1 , \\\\ M _ { l ' + 1 , p } \\oplus M _ { l ' - 1 , p + 1 } & 1 < l ' < d - 1 , \\\\ M _ { d - 2 , p + 1 } & l ' = d - 1 . \\end{cases} \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} K _ t ^ { ( \\alpha , \\beta ) } ( m , n ) = \\sum _ { k = | m - n | } ^ { m + n } c ( k , n , m , \\alpha , \\beta ) \\int _ { - 1 } ^ 1 e ^ { - ( 1 - x ) t } p _ k ^ { ( \\alpha , \\beta ) } ( x ) \\ , d \\mu _ { \\alpha , \\beta } ( x ) , \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} & \\Big ( \\big ( A _ 0 - \\sum _ { j = 1 } ^ 3 \\partial _ j \\psi A _ j \\big ) \\xi , \\xi \\Big ) _ { \\R ^ 6 \\times \\R ^ 6 } \\geq \\eta | \\xi | ^ 2 - \\sum _ { j = 1 } ^ 3 \\| \\partial _ j \\psi \\| _ { L ^ \\infty ( \\R ^ 3 ) } \\| A _ j \\| _ { L ^ \\infty ( \\Omega ) } | \\xi | ^ 2 \\\\ & \\geq \\eta | \\xi | ^ 2 - \\eta K C _ 0 | \\xi | ^ 2 = 0 \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} C _ p \\int | \\nabla u _ f - \\nabla f | ^ 2 d \\mu & = C _ p \\int | \\nabla u _ f | ^ 2 d \\mu + C _ p \\int | \\nabla f | ^ 2 d \\mu - 2 C _ p \\int \\nabla u _ f \\cdot \\nabla f d \\mu \\\\ & = C _ p \\int | \\nabla u _ f | ^ 2 d \\mu + C _ p \\int | \\nabla f | ^ 2 d \\mu - 2 \\int | f | ^ 2 d \\mu \\\\ & \\leq \\epsilon ^ 2 C _ p \\int | \\nabla f | ^ 2 d \\mu . \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} \\lvert u \\rvert _ S = \\lvert ( i , j ) \\rvert _ { W _ n ( w ) } = \\lvert ( \\phi w ) _ n \\rvert _ S - ( n - i ) - ( n - j ) . \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} \\iota ^ { \\pm } \\colon \\widehat { M } _ p ^ { \\pm } \\stackrel { \\cong } { \\to } \\{ d \\widehat { w } ^ { \\pm } = 0 \\} \\subset \\widehat { Y } _ U ^ { \\pm } . \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} u _ { t t } - \\Delta _ p u - \\Delta u _ t = f ( u ) \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} V ( x ) = \\frac { n } { 2 } \\vert x \\vert ^ 2 = \\frac n 2 ( x _ 1 ^ 2 + \\dotsb + x _ n ^ 2 ) , \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} \\hat { k _ 0 } = \\underset { k = 1 , \\dotsc , p } { \\min } n \\log ( \\sigma ^ 2 _ k ) + h ( k , \\sigma ^ 2 _ k ) , \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} f _ 1 ^ { \\prime \\prime } ( \\alpha ) & \\leq \\psi ^ { \\prime \\prime } \\left ( \\lambda _ 2 ( v ) - 2 \\sqrt { 2 } \\alpha \\delta ( v ) \\right ) ( \\| d _ x \\| ^ 2 + \\| d _ s \\| ^ 2 ) = \\\\ & = 2 \\delta ( v ) ^ 2 \\psi ^ { \\prime \\prime } \\left ( \\lambda _ 2 ( v ) - 2 \\sqrt { 2 } \\alpha \\delta ( v ) \\right ) , \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} L ^ r { } _ s = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} . \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} p : | W | \\to | \\O _ X | = X \\times \\C \\to \\C . \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} \\mathcal { I } ( ( f _ i ) , ( f _ j ) ) ( x ) = e ^ { - \\mathcal { Q _ + } ( B _ 0 x ) } e ^ { \\mathcal { Q _ - } ( B _ { m + 1 } x ) } \\prod _ { i = 1 } ^ { m ^ + } f _ i ^ { c _ i } ( B _ i x ) \\prod _ { j = 1 + m ^ + } ^ m f _ j ^ { c _ j } ( B _ j x ) \\quad \\textup { f o r $ x \\in H $ . } \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} \\bigcup _ { a _ 1 , \\ldots , a _ l \\geq 0 } \\tilde { f } _ { i _ 1 } ^ { a _ 1 } \\cdots \\tilde { f } _ { i _ l } ^ { a _ l } ( C ( \\pi , e ) ) \\setminus \\{ 0 \\} = C ( \\pi , v ) . \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} \\psi ( \\overline { G _ i ( X , Y ) d \\pi _ i } ) = \\psi ( \\overline { ( H _ i ( X + Y ) - H _ i ( X ) - H _ i ( Y ) ) d \\pi _ i } ) . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} \\Delta \\hat { \\Omega } _ j = \\Delta R _ { j j } + \\sigma _ j \\langle \\partial _ x \\Delta \\tilde { \\Omega } _ j , F ^ y \\rangle + \\sigma _ j \\langle \\partial _ x \\tilde { \\Omega } _ j , \\Delta F ^ y \\rangle . \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } Q ( f ) ( \\ddot { x } ) d \\mu _ \\rho ( \\ddot { x } ) = \\int _ G f ( x ) \\rho ( x ) d x \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} P _ G ( x ) = x ^ { 4 i } ( x + 1 ) ^ { 5 i + 1 } ( x + 2 i + 1 ) ( x ^ 3 - ( 7 i + 2 ) x ^ 2 - ( 7 i + 3 ) x + 1 2 i ^ 3 + 1 8 i ^ 2 + 6 i ) \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} r = \\sum _ { i = 0 } ^ { p - 1 } \\pi ^ i \\lambda _ i ^ p \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} \\Upsilon ^ 1 { } _ { 4 2 } = \\pm 1 , \\Upsilon ^ 2 { } _ { 4 1 } = \\mp 1 . \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} \\hat { p } _ { d + 1 } ^ 2 ( s ) - \\hat { q } _ 1 ^ 2 ( s ) s \\prod _ { j = 1 } ^ d \\left ( s - \\frac 1 { a _ j } \\right ) \\prod _ { j = 1 } ^ { d - 1 } \\left ( s - \\frac 1 { \\alpha _ j } \\right ) = c ' . \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} \\det \\left ( \\frac { \\partial x } { \\partial y } \\right ) = \\frac { | x | ^ n } { | y | ^ n } \\det \\left ( \\begin{array} { c c c c } a \\frac { y _ 1 ^ 2 } { | y | ^ 2 } + 1 & a \\frac { y _ 1 y _ 2 } { | y | ^ 2 } & \\cdots & a \\frac { y _ 1 y _ n } { | y | ^ 2 } \\\\ a \\frac { y _ 2 y _ 1 } { | y | ^ 2 } & a \\frac { y _ 2 ^ 2 } { | y | ^ 2 } + 1 & \\cdots & \\vdots \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a \\frac { y _ n y _ 1 } { | y | ^ 2 } & \\cdots & \\cdots & a \\frac { y _ n ^ 2 } { | y | ^ 2 } + 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} | D | \\approx h _ D ^ n { \\ \\rm a n d \\ } | \\partial D | \\approx h _ D ^ { n - 1 } , \\ n = 2 , 3 , \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} ( \\mathbb { A ' } ^ { - 1 } _ L ) _ { 1 2 } & = ( \\Delta ' ) ^ { - 1 } \\left ( - b - ( [ d , b ] - [ d , a ] a ^ { - 1 } b ) ( d - c a ^ { - 1 } b ) ^ { - 1 } \\right ) \\\\ & = - ( \\Delta ' ) ^ { - 1 } \\left ( b ( d - c a ^ { - 1 } b ) + ( [ d , b ] - [ d , a ] a ^ { - 1 } b ) \\right ) ( d - c a ^ { - 1 } b ) ^ { - 1 } \\\\ & = - ( \\Delta ' ) ^ { - 1 } \\left ( - b c + a d \\right ) a ^ { - 1 } b ( d - c a ^ { - 1 } b ) ^ { - 1 } \\\\ & = - ( d b ^ { - 1 } a - c ) ^ { - 1 } \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\int _ X \\theta _ { V _ { \\theta } } ^ n = c _ k \\int _ X \\theta _ { \\phi } ^ n + a \\left ( \\int _ X \\theta _ { V _ \\theta } ^ n - \\int _ { \\{ \\phi > V _ { \\theta } - k \\} } \\theta _ { \\max ( \\phi , V _ { \\theta } - k ) } ^ n \\right ) . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t u ( t ) + \\frac 1 2 \\Delta u ( t ) = - \\nabla u ^ * ( t ) \\ , b ( t ) - f ( t , \\cdot , u ( t ) , \\nabla u ( t ) ) \\\\ u ( T ) = \\Phi . \\end{array} \\right . \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} \\Big \\{ \\sum _ { i = 1 } ^ h u _ i t _ i : u _ i = 0 , 1 , \\dots , s _ i - 1 ; i = 1 , 2 , \\dots , h \\Big \\} & = \\{ 0 , 1 , 2 , \\dots , t _ { h + 1 } - 1 \\} , \\\\ \\Big \\{ \\sum _ { i = 1 } ^ { h + 1 } u _ i t _ i : u _ i = 0 , 1 , \\dots , s _ i - 1 i = 1 , 2 , \\dots , h + 1 \\Big \\} & = \\{ 0 , 1 , 2 , \\dots , r ! - 1 \\} . \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} \\psi ( k , v ) = \\psi ( v , v ) = \\left ( \\frac { \\sqrt { \\lambda } } { 1 + \\lambda } \\right ) ^ v \\lambda ^ { - v / 2 } \\frac { v ! } { v ! 0 ! } = \\frac { 1 } { ( 1 + \\lambda ) ^ v } \\leq c _ 4 v ^ { - \\frac { 1 } { 2 } } = c _ 4 k ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ j \\binom k j \\frac { { G _ { n j } } } { { L _ n { } ^ j } } } = \\frac { F _ { n k + 1 } G _ { 0 } - F _ { n k } G _ 1 } { L _ n { } ^ k } \\ , , \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} E _ { i , i + 1 } ^ { ( r ) } & : = E _ { i } ^ { ( r ) } , \\\\ F _ { i , i + 1 } ^ { ( r ) } & : = F _ i ^ { ( r ) } \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} X \\cdot X = - \\ > \\frac { m _ 1 + . . . + m _ n } { m } ( \\in - \\N ) . \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} e ^ { t \\mathcal { J } } f = \\int _ { X } e ^ { - t ( s ^ + - \\lambda ) } \\ , d E _ { J } ( \\lambda ) f \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} \\tau _ U ( s ) = \\inf \\{ n \\in \\N : s \\in U ^ n \\} \\ \\ \\ \\ s \\neq e , \\ \\ \\tau _ U ( e ) = 0 . \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} \\tilde { f } _ { i _ k } ^ { n + 1 } ( \\pi ^ { s _ { i _ { k + 1 } } \\cdots s _ { i _ l } \\lambda } \\otimes \\pi ^ { w \\mu } ) = \\tilde { f } _ { i _ k } ^ { n } ( \\pi ^ { s _ { i _ { k + 1 } } \\cdots s _ { i _ l } \\lambda } ) \\otimes \\tilde { f } _ { i _ k } ( \\pi ^ { w \\mu } ) \\not = 0 . \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} \\widehat { w } = \\sum _ { i , j } x _ i y _ j w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) + \\sum _ { i , i ' , j , j ' } x _ i x _ { i ' } y _ j y _ { j ' } w _ { i i ' j j ' } ^ { ( 2 ) } ( \\vec { u } ) + \\cdots \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} \\left ( C _ t ^ 1 , \\ldots , { C } _ t ^ { { M } } \\right ) \\mapsto f _ t \\left ( C _ t ^ 1 , \\ldots , { C } _ t ^ { { M } } \\right ) = \\sum _ { m = 1 } ^ { { M } } { C } _ t ^ m . \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} C _ { t _ 1 \\ldots { t _ n } } = C _ { t _ 1 t _ 2 } \\star C _ { t _ 2 t _ 3 } \\star \\ldots \\star C _ { t _ { n - 1 } t _ n } , \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} \\chi ( C ) = \\chi ( S ' ) + \\chi ( \\Gamma \\backslash S ) - \\chi ( \\Gamma ) \\ , . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} \\Psi _ k ^ + ( u , \\mathbf { v } _ 1 , \\mathbf { v } _ 2 , \\vartheta , g \\ , T ) : = u \\ , \\psi \\left ( \\widetilde { \\mu } _ { g e ^ { - \\imath \\vartheta B / \\sqrt { k } } g ^ { - 1 } } ( x _ { 1 k } ) , x _ { 2 k } \\right ) - \\frac { \\vartheta } { \\sqrt { k } } \\ , \\nu . \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} T ^ { * } _ { W } Q T _ { \\tilde { W } } = I _ { H } . \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} | x | ^ { \\frac { n - p } { p } } \\lambda ^ { - \\frac { \\theta - 1 } { \\theta } } = | z | ^ { \\frac { n - p } { p } } \\left [ ( q - 1 ) \\log _ { q } \\frac { R } { | z | } + \\lambda \\left ( \\frac { R } { | z | } \\right ) ^ { 1 - q } \\right ] ^ { - \\frac { \\theta - 1 } { \\theta } } . \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} \\| T \\| _ { H S } : = \\left ( \\underset { \\underset { k = 1 , . . , n } { j _ { k } \\in J _ { k } } } { \\sum } \\left \\Vert T \\left ( e _ { j _ { 1 } } ^ { 1 } , . . . , e _ { j _ { n } } ^ { n } \\right ) \\right \\Vert ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } < \\infty \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{gather*} \\partial \\overline { S } = \\partial _ 1 S \\bigsqcup \\partial _ 2 S \\\\ f ( \\partial _ 1 S ) \\subset \\partial B ( r ) , \\ \\ \\ f ( \\partial _ 2 S ) \\subset \\partial B ( 2 r ) \\\\ \\partial _ s f ( s , t ) + i \\partial _ t f ( s , t ) = 0 \\end{gather*}"}
-{"id": "9860.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } c ( 5 n + 4 ) q ^ { n } & \\equiv \\dfrac { E _ { 1 0 } ^ { 3 } } { E _ { 1 } ^ 2 E _ { 2 } ^ 2 E _ { 5 } } \\equiv \\dfrac { E _ { 1 } ^ { 3 } E _ { 2 } ^ { 3 } E _ { 1 0 } ^ { 2 } } { E _ { 5 } } \\pmod { 5 } . \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} K _ 1 = { p . \\partial _ q - \\lambda _ 1 \\partial _ p } + \\frac { 1 } { 2 } ( - \\partial ^ 2 _ p + p ^ 2 - 1 ) \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} \\limsup _ { R \\to \\infty } \\limsup _ { n \\to \\infty } \\mathbb { P } \\left \\lbrace \\tau ^ { n , r } _ R \\le T \\right \\rbrace & \\le \\limsup _ { R \\to \\infty } \\limsup _ { n \\to \\infty } \\mathbb { P } \\left \\lbrace \\sup _ { t \\in [ 0 , \\tau ^ { n , r } _ R ] } \\left \\lvert X ^ { n , r } _ t \\right \\rvert \\ge \\frac { R } { 4 } ; \\tau ^ { n , r } _ R \\le T \\right \\rbrace = 0 \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} A u = y \\quad u \\in C \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{gather*} h _ a ( l ) = \\frac { 1 } { 2 } \\sigma ^ 2 l ( l - 1 ) + l \\Gamma - \\delta + \\int _ \\mathbb { R } \\Big \\{ ( 1 + \\gamma ( z ) ) ^ l - 1 - l \\gamma ( z ) \\Big \\} \\nu ( d z ) \\\\ h _ b ( x ) = \\epsilon \\ln { x } - \\delta ( b \\ln { x } + c ) + b \\Gamma - \\frac { 1 } { 2 } \\sigma ^ 2 b + \\int _ \\mathbb { R } \\big \\{ b \\ln ( 1 + \\gamma ( z ) ) - b \\gamma ( z ) \\big \\} \\nu ( d z ) , \\end{gather*}"}
-{"id": "9206.png", "formula": "\\begin{align*} \\begin{cases} - D \\Delta z ( x ) = h ( x ) & x \\in \\Omega \\\\ \\partial _ \\nu z = 0 & x \\in \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} | ( R _ m ^ { \\flat } ) ' | \\leq 2 ( k + m + 1 ) + ( k + m + 1 ) = 3 ( k + m + 1 ) ~ . \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} I _ { T , t } ^ { ( i _ 1 i _ 2 ) } = \\frac { 1 } { 2 } I _ { T , t } ^ { ( i _ 1 ) } I _ { T , t } ^ { ( i _ 2 ) } + A _ { T , t } ^ { ( i _ 1 i _ 2 ) } \\ \\ \\ \\hbox { w . p . 1 } , \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} \\sigma _ u \\cdot \\sigma _ v = \\sum _ { w \\in W ^ P } c _ { u , v } ^ w \\sigma _ w . \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ { 0 } ^ { 2 s } \\partial _ t \\phi = & - ( - \\Delta ) ^ s _ y \\phi + p U ^ { p - 1 } ( y ) \\phi \\\\ & + \\Bigg \\{ p \\mu _ { 0 } ^ { \\frac { n - 2 s } { 2 } } \\frac { \\mu _ { 0 } ^ { 2 s } } { \\mu ^ { 2 s } } U ^ { p - 1 } \\left ( \\frac { \\mu _ { 0 } } { \\mu } y \\right ) \\psi ( \\xi + \\mu _ { 0 } y , t ) + \\cdots \\Bigg \\} \\chi _ { B _ { 2 R } ( 0 ) } ( y ) , y \\in \\mathbb { R } ^ n . \\end{aligned} \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\ln f _ \\alpha ( e ^ { t } ) = g '' / g - ( g ' / g ) ^ 2 = - 3 c ^ 2 z + O ( 1 ) \\to - \\infty . \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} \\int ( x _ 1 + \\cdots + x _ n ) \\ , \\mu ( \\d x ) = 0 , \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} ( a b ) \\varphi ( c ) = ( c a ) \\varphi ( b ) , \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} \\nabla _ { \\partial _ t } \\nabla _ { \\partial _ t } Y & = \\nabla _ { \\partial _ t } \\nabla _ Y \\partial _ t = \\nabla _ { \\nabla _ Y \\partial _ t } \\partial _ t = \\nabla _ { \\nabla _ { Y ^ T } \\partial _ t } \\partial _ t - \\frac { Y ^ { \\perp } } { u } \\nabla _ { \\nabla _ { \\partial _ t } \\partial _ t } \\partial _ t \\\\ & = \\nabla _ { \\nabla _ { Y ^ T } \\partial _ t } \\partial _ t - \\frac { Y ^ { \\perp } } { u } \\nabla _ { u \\nabla u } \\partial _ t . \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} A ^ \\alpha ( x ) = \\begin{pmatrix} a \\cos ( 2 m x ^ 4 ) \\\\ a \\sin ( 2 m x ^ 4 ) \\\\ 2 m b x ^ 4 \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} \\theta ^ { ( 1 ) } _ { 2 } ( l ) = h _ { 1 2 } ( l ) + e ^ { ( 1 ) } _ { 2 } ( l ) \\mbox { a n d } \\theta ^ { ( 1 ) } _ { 3 } ( l ) = h _ { 1 3 } ( l ) + e ^ { ( 1 ) } _ { 3 } ( l ) , \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} f _ 1 ( r ) = e ^ { \\sqrt { K _ 0 } ( r - 1 ) } . \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} 2 \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 4 8 n ) q ^ n & \\equiv \\dfrac { f _ 2 ^ 4 f _ 6 ^ 2 } { f _ 4 f _ { 1 2 } } \\cdot \\dfrac { 1 } { f _ 1 ^ 4 } - \\dfrac { f _ 6 ^ 2 } { f _ 3 ^ 4 } + 2 ~ ( \\textup { m o d } ~ 8 ) . \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} ( T _ i - q ) ( T _ i + 1 ) & = 0 \\\\ T _ i T _ j & = T _ j T _ i a _ { i j } ( 1 ) = 0 \\\\ T _ i T _ j T _ i & = T _ j T _ i T _ j a _ { i j } ( 1 ) = - 1 . \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} \\det ( R ) = f ( X ) + c ( f ) p _ 1 ( X ) f ( z _ 1 ) + c ( f ) p _ 2 ( X ) f ( z _ 1 ) ^ 2 + \\dots + c ( f ) f ( z _ 1 ) ^ { r _ 0 } . \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { N - 1 } \\cos ( 4 j k ( E ) + \\phi ) = 0 . \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} ( \\nabla _ g u , \\nabla _ g \\varphi ) _ g = ( f , \\varphi ) _ g . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} r _ ! ( f \\times d n ) ( t ) = d t \\cdot \\left ( \\int _ F \\Phi ( c ) | c | ^ { - 1 } \\psi ( \\frac { t } { c } ) d c \\right ) = \\int f ( \\frac { t } { z } ) \\psi ( z ) | z | d ^ \\times z . \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} \\beta _ k ( x + V ) = B _ k x + V _ k . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} \\frac { d \\mu } { d x } = \\frac { 1 } { Z } e ^ { - \\beta V } \\ , , \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} \\Lambda + \\frac { 1 } { t } \\left ( P ^ T d i a g \\left ( \\frac { \\frac { n - 2 s } { 2 s } \\bar { \\sigma } _ r b _ r ^ { 2 - 2 s } + 1 } { n - 4 s } \\right ) \\right ) P \\int _ { t } ^ \\infty \\Lambda ( s ) d s = h ( t ) , \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} \\tilde { \\Phi } & : = \\Phi | _ { \\tilde { \\rho } ^ { - 1 } ( \\tilde { V } ) } : \\tilde { \\rho } ^ { - 1 } ( \\tilde { V } ) \\rightarrow \\Phi ( \\tilde { \\rho } ^ { - 1 } ( \\tilde { V } ) ) . \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} \\mathfrak { S } _ \\mathbb { Z } = \\left \\{ N \\in \\mathrm { P S L } ( 2 , \\mathbb { Z } ) : N ( \\gamma _ M ^ \\pm ) = \\gamma _ M ^ \\pm , \\ N ( \\theta _ \\pm ) = \\theta _ \\pm \\right \\} \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} \\underset { \\bar { \\mathbf { x } } \\in \\mathbb { R } ^ p } { } \\bar { f } ( \\bar { \\mathbf { x } } ) = \\sum _ { i = 1 } ^ n f _ i ( \\bar { \\mathbf { x } } ) . \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} \\tilde { f } _ { i _ l } ^ { a _ l } \\left ( b _ \\lambda \\otimes b \\right ) = \\tilde { f } _ { i _ l } ^ { c _ l } ( b _ \\lambda ) \\otimes \\tilde { f } _ { i _ l } ^ { d _ l } ( b ) . \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} \\langle F _ A u , Z \\rangle = \\sum _ { \\gamma , \\delta } \\langle ( F _ A ) _ { \\gamma \\delta } u _ \\gamma , Z _ \\delta \\rangle = \\sum _ { \\gamma , \\delta } ( f _ A ) _ { \\gamma \\delta } ( v , u _ \\gamma Z _ \\delta ) \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} f ( O z + v , t - 2 \\langle O z , J v \\rangle ) & = f ( O ( z + O ^ T v ) , t - 2 \\langle O z , J O O ^ T v \\rangle ) \\\\ & = f ( O ( z + O ^ T v ) , t - 2 \\langle O z , O J O ^ T v \\rangle ) \\\\ & = f ( O ( z + O ^ T v ) , t - 2 \\langle z , J O ^ T v \\rangle ) \\\\ & = f ( z + O ^ T v , t - 2 \\langle z , J O ^ T v \\rangle ) \\\\ & \\ge f ( z , t ) + \\langle z , O ^ T v \\rangle \\\\ & = f ( O z , t ) + \\langle O z , v \\rangle , \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} t ( r ) : = R ( r - 1 + R ^ { - 1 } ) \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( s , 0 ) = 0 , s \\in ( - \\infty , 0 ] . \\end{aligned} \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} I _ { \\mu } ^ { \\infty } ( v _ 1 ) & = \\max _ { t \\in \\R } I _ { \\mu } ^ { \\infty } ( \\gamma ( t ) ) \\\\ & = \\max _ { t \\in [ 0 , M ] } I _ { \\mu } ^ { \\infty } ( \\gamma ( t ) ) \\\\ & = \\max _ { t \\in [ 0 , 1 ] } I _ { \\mu } ^ { \\infty } ( \\gamma _ 0 ( t ) ) . \\\\ \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} \\mathcal { L } ' ( x ' ) = \\mathcal { L } _ R + \\mathcal { L } _ p = \\| \\zeta \\| ^ 2 . \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} g ( y ) = 4 y \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} \\tilde { G } _ { k , m , r } ( z ) : = m ^ { k - 1 } G _ { k , m , r } ( z ) - \\sum _ { \\substack { 0 < n \\leq A _ k \\\\ ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) } } a _ k ( - n , m ) n ^ { k - 1 } G _ { k , n , r } ( z ) , \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} \\int _ \\Omega \\int _ { \\mathbb { S } _ + ^ 1 } \\theta \\cdot \\nabla \\phi \\d Q ( x , \\theta ) = \\int _ \\Omega S \\phi \\d x \\mbox { f o r a l l } \\phi \\in C ^ \\infty ( \\Omega ) , \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| u _ { 0 , n } \\| _ { L ^ 2 } = \\lim _ { n \\rightarrow \\infty } \\mu _ n \\| Q \\| _ { L ^ 2 } = \\| Q \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} g ( \\omega , 0 ) = 0 ; \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{gather*} \\overset { I \\otimes J } { T } \\ ! \\ ! \\ ! _ { 1 2 } = \\overset { I } { T _ 1 } \\overset { J } { T _ 2 } , \\eta ( 1 ) = \\overset { \\mathbb { C } } { T } , \\Delta ( \\overset { I } { T ^ { a } _ { b } } ) = \\overset { I } { T ^ { a } _ { i } } \\otimes \\overset { I } { T ^ { i } _ { b } } , \\\\ \\varepsilon ( \\overset { I } { T } ) = \\mathbb { I } _ { \\dim ( I ) } , S ( \\overset { I } { T } ) = \\overset { I } { T } { ^ { - 1 } } , \\end{gather*}"}
-{"id": "4524.png", "formula": "\\begin{align*} \\min _ { u \\in L ^ 1 _ + ( \\Omega ) } d ( \\cdot , u _ 0 ) \\quad A u = y . \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} ( \\tau _ 1 + \\tau _ 4 ) ^ 2 = 0 \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} 1 = \\sum _ { j \\in \\Z } \\varphi ( 2 ^ { - j } \\lambda ) , \\quad \\lambda > 0 . \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} \\mathcal { T } _ \\xi \\rtimes \\mathcal { S } = \\bigoplus _ { \\nu \\in G } ( \\mathcal { T } _ \\xi \\rtimes \\mathcal { S } ) _ \\nu . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} h _ { \\alpha \\beta } ( x ) = [ V ^ \\mu { } _ \\alpha ( x ) ] \\ , [ g _ { \\mu \\nu } ( x ) ] \\ , [ V ^ \\nu { } _ \\beta ( x ) ] \\ , . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{gather*} F ( \\lambda p ) = \\lambda F ( p ) , \\end{gather*}"}
-{"id": "1278.png", "formula": "\\begin{align*} h _ i ^ \\pi : [ 0 , 1 ] \\rightarrow \\mathbb { R } , \\ h _ i ^ \\pi ( t ) : = \\langle \\pi ( t ) , \\alpha _ i ^ \\vee \\rangle ; \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\left ( \\big ( d \\varphi ^ { 0 } _ { t } ( x _ { 0 } ) \\big ) ^ { - 1 } \\cdot \\frac { \\partial } { \\partial \\tau } \\varphi ^ { \\tau } _ { t } ( x _ { 0 } ) _ { | \\tau = 0 } \\right ) = \\big ( d \\varphi ^ { 0 } _ { t } ( x _ { 0 } ) \\big ) ^ { - 1 } \\cdot \\left ( \\frac { \\partial X _ \\tau } { \\partial \\tau } \\big ( \\varphi ^ { 0 } _ { t } ( x _ { 0 } ) \\big ) \\right ) _ { | \\tau = 0 } . \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} E ( \\tilde { V } ^ j _ n ) = \\frac { \\lambda ^ 2 _ j } { 2 } \\| V ^ j ( \\cdot - x ^ j _ n ) \\| ^ 2 _ { \\dot { H } ^ 1 _ c } - \\frac { \\lambda _ j ^ { \\alpha + 2 } } { \\alpha + 2 } \\| V ^ j \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} \\left ( 5 \\beta , \\left ( 5 \\beta \\right ) ^ \\sigma \\right ) _ 2 = ( \\beta , \\beta ^ \\sigma ) _ 2 \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{gather*} f ( \\lambda v ) = \\lambda f ( v ) , \\end{gather*}"}
-{"id": "2860.png", "formula": "\\begin{align*} \\left ( \\int _ { \\{ | x | < R \\} } \\frac { d x } { | x | ^ { b } } \\right ) ^ { \\frac { 1 } { q } } & \\left ( \\int _ { \\{ 2 R < | x | \\} } \\left ( \\frac { | x | } { 2 } \\right ) ^ { ( a - Q ) p ^ { \\prime } } d x \\right ) ^ { \\frac { 1 } { p ^ { \\prime } } } \\\\ & \\leqslant C R ^ { \\frac { Q - b } { q } } R ^ { \\frac { ( a - Q ) p ' + Q } { p ' } } \\\\ & \\leqslant C , \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} F ( t ) = C _ 1 \\mathbf { E } \\Big ( \\int _ 0 ^ t f ( s ) d s \\Big ) ^ 2 + C _ 2 \\int _ 0 ^ t \\mathbf { E } \\big ( f ( s ) \\big ) ^ 2 d s \\ , . \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} | Z ( E ) | = | \\sum _ { i } ^ m Z ( A _ i ) | > \\sum _ { i } ^ m \\cos ( 2 \\epsilon ) | Z ( A _ i ) | \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{gather*} T _ i T _ j \\big ( x _ i ^ + \\big ) = x _ j ^ + , T _ i T _ j \\big ( x _ i ^ - \\big ) = x _ j ^ - , T _ i T _ j ( h _ i ) = h _ j . \\end{gather*}"}
-{"id": "2677.png", "formula": "\\begin{align*} \\mathbf { B } _ k = \\begin{pmatrix} B _ k & \\nabla c ( x ^ k ) ^ \\top \\\\ - \\nabla c ( x ^ k ) & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{gather*} T _ i ( E _ { N , N } ) = \\begin{cases} E _ { N - 1 , N - 1 } & , \\\\ E _ { N , N } & . \\end{cases} \\end{gather*}"}
-{"id": "9652.png", "formula": "\\begin{align*} h ( t , \\cdot ) = e ^ { - \\Delta ^ { 2 } t } h _ { 0 } + \\sum _ { j = 2 } ^ { \\infty } \\int _ { 0 } ^ { t } e ^ { - \\Delta ^ { 2 } ( t - s ) } \\Delta F _ { j } ( s , \\cdot ) \\ d s . \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} q = r _ 1 \\otimes s _ 1 + \\dots + r _ k \\otimes s _ k \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} { \\mu } _ E ( ( m , a ) , ( n , b ) ) = ~ & ( \\mu ( m , b ) + \\mu ( a , n ) + f ( a , b ) , ~ \\mu ( a , b ) ) , \\\\ { \\alpha } _ E ( ( m , a ) ) = ~ & ( \\alpha _ M ( a ) , \\alpha ( a ) ) . \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} \\xi ( \\rho ) & = [ \\hat \\xi , \\rho ] \\ , . \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} L _ { u } ( \\beta _ 1 ) = & u \\left ( u ^ q \\right ) ^ { q ^ 2 } + ( u + 1 ) \\left ( u ^ q \\right ) ^ q + ( u ^ 2 + u ) u ^ q \\\\ = & u ^ 2 + 1 + ( u + 1 ) ^ 2 \\\\ = & 0 , \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} - \\Delta u + u + \\rho ( x ) \\phi _ u u = \\mu | u | ^ { p - 1 } u , \\mu \\in \\left [ \\frac { 1 } { 2 } , 1 \\right ] , \\\\ \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} & | \\langle \\bar { L } ( . , \\theta ) , \\ , \\mu \\rangle - \\langle \\bar { L } ( . , \\theta ) , \\ , \\hat { \\mu } \\rangle | \\\\ \\leq & K _ L \\times \\inf \\Big \\{ \\| X - Y \\| _ { L ^ 2 } \\ , \\Big | \\ , X , Y \\in L ^ 2 ( \\Omega , \\R ^ { d + l } ) \\P _ X = \\mu , \\ , \\P _ Y = \\nu \\Big \\} \\\\ \\leq & K _ L W _ 2 ( \\mu , \\hat { \\mu } ) . \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} & \\Pr [ K _ { { \\rm S V C } , n , \\ell } = k | B _ { { \\rm S V C } , n , \\ell , 0 } \\ \\ \\mathbf x ] = \\sum _ { \\mathbf k \\in \\mathcal { S Q } _ { \\mathbf x , n , \\ell } ( k ) } \\prod _ { m \\in \\mathcal N _ { \\mathbf x } } \\Pr [ v _ m = k _ m | B _ { { \\rm S V C } , n , \\ell , 0 } \\ \\ \\mathbf x ] . \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} \\dot { \\theta } _ i = \\omega + \\sin ( \\theta _ { i + 1 } - \\theta _ i ) + \\sin ( \\theta _ { i - 1 } - \\theta _ i ) , \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} \\widehat f _ { T , h } ( x ) : = \\frac { 1 } { T h } \\int _ { 0 } ^ { T } K \\left ( \\frac { X ( s ) - x } { h } \\right ) d s . \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} \\Big | \\psi ( z ) - \\sum _ { k = 0 } ^ { p - 1 } ( - 1 ) ^ { k r } b _ k z ^ k \\Big | \\le C A ^ p M _ { p } | z | ^ p , z \\in S _ \\alpha . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} c _ { V , W } ( v \\otimes w ) = ( - 1 ) ^ { | v | | w | } w \\otimes v , \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( { \\mathrm { M } } _ { t , n } ^ { ( k ) } , 0 \\le t \\le T ) \\ , = \\ , \\delta _ { ( { \\mathrm { M } } _ { t } ^ { ( k ) } , \\ , \\ , 0 \\ , \\le \\ , t \\ , \\le \\ , T ) } \\mathcal M ( \\Omega _ { k } ) \\ , . \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} c ( x _ 1 , \\ldots , x _ N ) = \\sum _ { 1 \\le i < j \\le N } f ( d ( x _ i , x _ j ) ) , \\ , ( x _ 1 , \\ldots , x _ N ) \\in X ^ N , \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} \\dfrac { s _ { y + 1 } } { s _ { y } } & = \\dfrac { \\binom { N } { j + 1 } q ^ { K - ( j + 1 ) } } { \\binom { N } { j } q ^ { K - j } } = \\dfrac { N - j } { ( j + 1 ) q } \\le \\dfrac { N - ( K - g + 1 ) } { ( K - g + 2 ) q } < 1 . \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ { g ^ { ( 4 ) } } \\delta ^ * _ { g ^ { ( 4 ) } _ 0 } \\big ( \\beta _ { g ^ { ( 4 ) } } g ^ { ( 4 ) } _ 0 \\big ) = 0 \\quad M , \\\\ - \\beta _ { g ^ { ( 4 ) } } g ^ { ( 4 ) } _ 0 = 0 \\quad \\partial M . \\end{cases} \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} & Q ^ 1 _ { 1 \\bar 1 } = \\mu _ { k \\bar r } ^ { \\bar 1 } \\mu _ { \\bar k r } ^ { 1 } \\ , , Q ^ 2 _ { 1 \\bar 1 } = \\mu _ { \\bar k \\bar r } ^ { \\bar 1 } \\mu _ { k r } ^ 1 \\ , , \\\\ & Q ^ 3 _ { 1 \\bar 1 } = \\mu _ { k \\bar k } ^ { \\bar 1 } \\mu _ { \\bar r r } ^ 1 \\ , , Q ^ 4 _ { 1 \\bar 1 } = 0 \\ , . \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} 2 \\mathsf { R e } ( P ^ * u , i B u ) \\geq \\frac { c _ 0 } { 2 } \\sum _ { j = 1 } ^ N | \\ ! | X _ j u | \\ ! | _ 0 ^ 2 - | \\ ! | f | \\ ! | _ { L ^ \\infty ( K ) } \\sum _ { j = 0 } ^ N | \\ ! | [ X _ j , X _ 0 ] u | \\ ! | _ 0 ^ 2 \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} \\begin{array} { c } m a x \\Big \\{ m i n \\Big [ r V ^ { i j } ( x ) - \\mathcal { A } V ^ { i j } ( x ) - f ( x , i , j ) ; \\qquad \\qquad \\qquad \\qquad \\\\ V ^ { i j } ( x ) - M ^ { i j } [ V ] ( x ) \\Big ] ; V ^ { i j } ( x ) - N ^ { i j } [ V ] ( x ) \\Big \\} = 0 , \\end{array} \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} 0 = \\frac { \\partial } { \\partial y } ( y p _ { 0 } ( y , t ) ) + \\frac { 1 } { 2 } \\langle x ^ { 2 } \\rangle \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } p _ { 0 } ( y , t ) - \\frac { \\partial } { \\partial t } p _ { 0 } ( y , t ) \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{gather*} \\big [ x _ { i } ^ + , T _ { w ( i , m ) } \\big ( x _ { i - 1 } ^ + \\big ) \\big ] = ( - 1 ) ^ { m - 1 } \\times \\begin{cases} h _ { - \\theta } ( m ) & , \\\\ h _ i ( m ) & . \\end{cases} \\end{gather*}"}
-{"id": "4559.png", "formula": "\\begin{align*} \\begin{array} { | c | c | c | c | } \\hline & i = 1 , \\ldots , k & i = k + 1 , \\ldots , k + n & i = k + n + 1 \\\\ \\hline \\arg L _ i & - \\pi & 0 & 0 \\\\ \\hline \\end{array} \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} K ( t , s ) = \\int _ { \\R ^ d } e ^ { i ( t - s ) | \\xi | ^ a } ( 1 + | \\xi | ^ 2 ) ^ { \\frac { a - d } { 2 } } e ^ { i [ X ( t ) - X ( s ) ] \\xi } \\hat h ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} \\Delta _ k : = - y ^ 2 \\biggl ( \\frac { \\partial ^ 2 } { \\partial x ^ 2 } + \\frac { \\partial ^ 2 } { \\partial y ^ 2 } \\biggr ) + i k y \\biggl ( \\frac { \\partial } { \\partial x } + i \\frac { \\partial } { \\partial y } \\biggr ) . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} - \\sum _ { k = 1 } ^ { p } \\frac 1 k \\frac { x ^ { p - k } } { ( x - 1 ) ^ { p - k + 1 } } & = \\sum _ { \\ell = 0 } ^ { p - 1 } \\binom { p } \\ell \\frac { 1 } { ( x - 1 ) ^ { \\ell + 1 } } ( H _ \\ell - H _ p ) \\\\ & = \\sum _ { \\ell = 0 } ^ { p } \\binom { p } \\ell \\frac { 1 } { ( x - 1 ) ^ { \\ell + 1 } } H _ \\ell - H _ p \\frac { x ^ p } { ( x - 1 ) ^ { p + 1 } } . \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} M ^ i _ t = & \\gamma ^ i ( t , W _ t ) - \\gamma ^ i ( 0 , W _ 0 ) + h ^ i ( t , W _ t ) - h ^ i ( 0 , W _ 0 ) \\\\ & - \\int _ 0 ^ t ( \\nabla h ^ i ) ^ * ( r , W _ r ) \\mathrm d W _ r . \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} \\| b \\| _ { \\widetilde { L } ^ { \\infty } _ t ( { B } ^ s _ { 2 , 2 } ) } \\leq \\| b _ 0 \\| _ { { B } ^ s _ { 2 , 2 } } + C _ 3 \\| u \\| _ { \\widetilde { L } ^ { 1 } _ t ( B ^ { s + 1 } _ { 2 , 2 } ) } \\| b \\| _ { \\widetilde { L } ^ { \\infty } _ t ( B ^ s _ { 2 , 2 } ) } . \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} f ( z ) - \\sum _ { | \\mu | \\le m - 1 } \\frac { f _ { \\mu } } { \\mu ! } z ^ { \\mu } = \\frac { 1 } { ( m - 1 ) ! } \\int _ { 0 } ^ { 1 } ( 1 - t ) ^ { m - 1 } \\partial _ { t } ^ { m } f ( t z ) d t . \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} \\frac { 1 } { p _ \\gamma } = \\frac { 1 - \\gamma } { p _ 0 } + \\frac { \\gamma } { p _ 1 } . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} S _ N - M _ { N \\ell } ^ { ( N , r ) } = \\sum _ { i = 1 } ^ { \\ell } \\sum _ { n = 1 } ^ { N } ( Y _ { i , i n } - Y _ { i , i n , r } ) + \\sum _ { i = 1 } ^ \\ell ( R _ { i , N \\ell , r } - R _ { i , 0 , r } ) \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { i } \\left \\langle \\frac { \\lambda - \\mu } { 2 } + \\rho , \\alpha _ j \\right \\rangle & = f ( i + 1 ) - \\frac { n ( n + 1 ) } { 2 } + \\mu _ { i + 1 } - \\mu _ i \\\\ & = k _ { i - 1 } ( f ) - \\frac { n ( n + 1 } { 2 } + \\frac { ( n - i ) ( n - i + 1 ) } { 2 } - \\frac { ( n - i + 1 ) ( n - i + 2 ) } { 2 } \\\\ & = k _ { i - 1 } ( f ) + i - 1 - \\frac { n ( n + 3 ) } { 2 } \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} \\widehat { u } _ { M } \\ , : = \\ , \\Big [ 1 - \\Big ( \\frac { \\ , 2 \\ , } { \\ , T \\ , } \\int ^ { T } _ { 0 } X _ { t } ^ { 2 } { \\mathrm d } t \\Big ) ^ { - 2 } \\Big ] ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} t _ 1 ~ < ~ t _ 2 ~ < ~ ~ \\cdots ~ ~ < ~ t _ m ~ < ~ t _ { m + 1 } ~ = ~ T \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} H = \\Lambda + G + K , \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} p _ { L , N + 1 } ^ { \\ell _ { 1 } } ( x ) & = \\sum \\limits _ { \\ell = 0 } ^ N \\frac { S _ { \\lambda \\mu _ { \\ell } } \\left ( 2 \\sum \\limits _ { j = 0 } ^ N \\omega _ j \\tilde { \\Phi } _ { \\ell } ( x _ j ) f ( x _ j ) \\right ) } { 2 } \\tilde { \\Phi } _ { \\ell } ( x ) \\\\ & = \\sum \\limits _ { \\ell = 0 } ^ N \\left ( \\sum _ { j = 0 } ^ N \\omega _ j \\tilde { \\Phi } _ { \\ell } ( x _ j ) f ( x _ j ) \\right ) \\tilde { \\Phi } _ { \\ell } ( x ) + \\sum _ { \\ell = 0 } ^ N c _ { \\ell } \\tilde { \\Phi } _ { \\ell } ( x ) , \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\left ( | X _ n | - \\frac { m _ 2 - \\lambda } { m - 1 + \\lambda } \\sum _ { k = 0 } ^ { n - 1 } I _ { \\{ \\langle X _ k \\rangle = 1 \\} } - \\frac { m _ 1 - \\lambda } { m - 1 + \\lambda } \\sum _ { k = 0 } ^ { n - 1 } I _ { \\{ \\langle X _ k \\rangle = 2 \\} } \\right ) = 0 \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} ( t , s , b , x ) \\cdot ( t ' , s ' , b ' , x ' ) = ( t t ' , s + s ' , b ' + b ' , x + x ' ) . \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} \\max ( \\| X _ n \\| _ t , \\| X _ { n , r } \\| _ t ) \\leq 2 K ( 1 + \\ell \\tau _ { \\lambda t } ^ \\lambda ) = \\varrho _ { t } \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} Z _ 1 ^ * : = \\bigsqcup _ { \\substack { Z ^ \\prime \\ \\mbox { \\tiny { s t r a t u m o f } } \\ \\partial S _ K ^ * \\\\ \\mbox { \\tiny { c o n t r i b u t i n g t o } } \\ Z _ 1 } } S _ { K , Z ^ \\prime } ^ * , \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} \\pi _ r ^ \\alpha ( \\lambda ) = f ( \\alpha , r ) r ^ { - c } , \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} & \\underset { \\{ \\mathbf { p } _ { i } \\} _ { i = 1 } ^ N } { \\min } & \\sum _ { i = 1 } ^ { M } C _ i ( \\mathbf { p } _ i , k ) - \\sum _ { i = M + 1 } ^ { N } U _ i ( \\mathbf { p } _ i , k ) \\\\ & & \\underline { \\mathbf { p } } _ i ^ { [ k ] } \\leq \\mathbf { p } _ i \\leq \\overline { \\mathbf { p } } _ i ^ { [ k ] } , \\ , i = 1 , \\hdots , N \\\\ & & \\sum _ { i = 1 } ^ N \\mathbf { p } _ i = \\mathbf { P } ^ { [ k ] } , \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} \\bar { T } ( t ) : = t ^ { p / 2 - 2 } T ( t ) , \\bar { H } ( t ) : = t ^ { p / 2 - 2 } H ( t ) . \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} \\begin{cases} z ^ { k , m } = ( 1 - \\eta ^ m ) x ^ k + \\eta ^ m y ^ m \\\\ f ( z ^ { k , m } , x ^ k ) - f ( z ^ { k , m } , y ^ k ) \\geq \\frac { \\mu } { 2 \\rho _ k } \\| x ^ k - y ^ k \\| ^ 2 . \\end{cases} \\end{align*}"}
-{"id": "9591.png", "formula": "\\begin{align*} \\Delta ^ n = \\left \\{ ( x _ 0 , \\dots , x _ n ) \\in \\mathbb { R } ^ n \\mid \\sum _ { i = 0 } ^ n x _ i = 1 \\ , , 0 \\leq x _ i \\leq 1 i \\right \\} \\ , . \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} F _ { m , \\ , m } ( X + 1 ) - F _ { m , \\ , m } ( X ) = \\sum _ { s = 0 } ^ m ( - 1 ) ^ s \\binom { m } { s } \\binom { X - s } { m - 1 } = F _ { m , \\ , m - 1 } ( X ) . \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} \\lambda \\rho _ { n + 1 } ( x ' , y ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } \\rho _ { n } ( x , \\lambda ^ { - 1 } ( y - \\tau ^ { 1 / 2 } x ) ) \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\kappa _ { v \\lambda } \\kappa _ { w \\mu } = \\sum _ { \\nu \\in P } a _ { v , w , \\lambda , \\mu } ^ { \\nu } \\kappa _ { \\nu } . \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} F ^ + \\cap F ^ - = \\widehat { M } ^ { + } \\times _ { \\widehat { U } } \\widehat { M } ^ { - } . \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} F ^ { \\ast } ( \\chi _ { j } ) & = F ^ { \\ast } ( \\alpha _ { u _ j } + \\cdots + \\alpha _ { u _ m } ) \\\\ & = \\sum _ { l = 1 } ^ m \\langle \\alpha _ { u _ j } + \\cdots + \\alpha _ { u _ m } , \\alpha _ { u _ l } ^ { \\vee } \\rangle \\varpi _ l \\\\ & = - \\varpi _ { j - 1 } + \\varpi _ { j } + \\varpi _ m , \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} \\langle \\nu , \\nu \\rangle = l '^ 2 \\left ( 2 \\nu _ \\alpha ^ 2 - 2 \\nu _ \\alpha \\nu _ \\beta + \\frac { 2 } { 3 } \\nu _ \\beta ^ 2 \\right ) = 2 l ' \\frac { l ' } { 3 } ( 3 \\nu _ \\alpha ^ 2 - 3 \\nu _ \\alpha \\nu _ \\beta + \\nu _ \\beta ^ 2 ) . \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} \\left | \\sum \\limits _ { j = 1 } ^ { m } a _ j ( g _ j , h _ { l } ^ { ( 1 ) } h _ { l } ^ { ( 2 ) } ) _ { \\Omega } \\right | & \\leq C d ( \\mathcal { C } _ 1 ( f ^ { ( 1 ) } ) , \\mathcal { C } _ 2 ( f ^ { ( 2 ) } ) ) e ^ { C \\epsilon ^ { - \\mu } } \\| h _ l ^ { ( 1 ) } \\| _ { H ^ s _ { \\overline { \\Omega } } } \\| h _ l ^ { ( 2 ) } \\| _ { H ^ s _ { \\overline { \\Omega } } } \\\\ & + \\epsilon \\| q _ 1 - q _ 2 \\| _ { L ^ { \\infty } ( \\Omega ) } \\| h _ l ^ { ( 1 ) } \\| _ { H ^ s _ { \\overline { \\Omega } } } \\| h _ l ^ { ( 2 ) } \\| _ { H ^ s _ { \\overline { \\Omega } } } . \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} \\sum _ { | \\gamma | > 1 } | k _ 2 ( \\rho ) | & \\leq x ^ 2 \\sum _ { m = 1 } ^ { \\infty } \\{ n _ L ( 2 m ) + n _ L ( - 2 m ) \\} x ^ { - ( 2 m - 1 ) ^ 2 } \\\\ & \\leq 5 . 4 4 x ^ 2 \\sum _ { m = 1 } ^ { \\infty } \\{ \\log d _ L + n _ L \\log ( 2 m + 2 ) \\} x ^ { - ( 2 m - 1 ) ^ 2 } { \\mbox { b y ( \\ref { n L t - W i t h o u t G R H S h o r t } ) } } \\\\ & \\leq c _ { 2 0 } x \\log d _ L , \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} F _ { \\gamma _ { e 1 } } ( \\gamma ) & = \\bigg \\{ \\begin{array} { l l } 1 - e ^ { - \\frac { \\gamma } { \\rho _ { e } ( \\phi _ { 2 } - \\phi _ { 1 } \\gamma ) } } , & \\gamma \\leq \\frac { \\phi _ { 2 } } { \\phi _ { 1 } } , \\\\ 1 , & \\gamma > \\frac { \\phi _ { 2 } } { \\phi _ { 1 } } , \\end{array} \\bigg . \\\\ F _ { \\gamma _ { e 2 } } ( \\gamma ) & = 1 - e ^ { - \\frac { \\gamma } { \\phi _ 2 \\rho _ e } } . \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\frac 1 { s _ l ^ 2 } \\sum _ { i = 1 } ^ { k _ l } L _ { \\mathcal P _ l } ( S _ { a _ i } ^ { n _ l } h _ l , \\eta s _ l ) = 0 , \\forall \\eta > 0 \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} e _ 1 ( \\varphi ) + e _ 2 ( \\varphi ) + e _ 3 ( \\varphi ) + e _ 4 ( \\varphi ) = 0 \\ , . \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} \\left | f ( w ) - \\sum _ { j = 0 } ^ { n - 1 } a _ j w ^ j \\right | & = \\left | g ( w ^ { 1 / q } ) - \\sum _ { j = 0 } ^ { n - 1 } b _ { q j } ( w ^ { 1 / q } ) ^ { q j } \\right | = \\left | g ( w ^ { 1 / q } ) - \\sum _ { k = 0 } ^ { q n - 1 } b _ { k } ( w ^ { 1 / q } ) ^ { k } \\right | \\\\ & \\le C _ 1 A _ 1 ^ { q n } M _ { q n } | w ^ { 1 / q } | ^ { q n } . \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} \\gamma _ { + } ( u ) = I ' ( u ^ { + } ) [ u ^ { + } ] + \\lambda \\int _ { \\mathbb R ^ { 3 } } \\phi _ { u ^ { + } } ( u ^ { - } ) ^ { 2 } d x = \\gamma ( u ^ { + } ) + \\lambda \\int _ { \\mathbb R ^ { 3 } } \\phi _ { u ^ { + } } ( u ^ { - } ) ^ { 2 } d x \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} \\phi _ { P ( n ) } ( t , y ) = \\left ( 1 - \\frac { 1 } { 2 } i y - \\frac { 1 } { 2 ( n - 1 ) } \\sum _ { i = 1 } ^ { n - 1 } t _ i \\right ) ^ { - \\lambda } = 2 ^ { \\lambda } \\left ( 2 - i y - \\frac { 1 } { n - 1 } \\sum _ { i = 1 } ^ { n - 1 } t _ i \\right ) ^ { - \\lambda } \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} \\sqrt { \\ell ! } \\biggl ( & \\int _ { B _ { u _ p } ^ { \\ell } } \\lVert ( H _ { B _ { u _ p } ^ c } ( \\xi - x _ 1 - \\dots - x _ \\ell ) + \\omega ^ { ( \\ell ) } ( x _ 1 , \\dots . , x _ \\ell ) \\\\ & - \\Sigma ( \\xi - k _ 0 ) - \\omega ^ { ( \\ell ) } ( k _ 1 , \\dots , k _ \\ell ) ) \\Gamma ( P _ { B _ { u _ p } ^ c } ) \\psi _ p \\lVert ^ 2 \\lvert g _ { u _ p } ( x ) \\lvert ^ 2 d \\lambda _ \\nu ( x ) \\biggl ) ^ { 1 / 2 } : = \\sqrt { \\ell ! } \\gamma _ p \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} \\# \\{ \\alpha \\in \\mathbf { M } _ 4 : \\star ( \\alpha ) = 1 \\} \\geq 1 . \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\sum _ { r _ 1 = 1 } ^ { n } \\frac { S _ { 0 r _ 1 r _ 1 } } { ( k + 1 ) ( r _ 1 + k + 1 ) ^ 2 } = \\sum _ { r _ 1 = 1 } ^ n \\frac { S _ { 0 r _ 1 r _ 1 } H _ { r _ 1 } } { r _ 1 ^ 2 } - \\sum _ { r _ 1 = 1 } ^ n \\frac { S _ { 0 r _ 1 r _ 1 } } { r _ 1 } \\left ( \\zeta ( 2 ) - H ^ { ( 2 ) } _ { r _ 1 } \\right ) ; \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} \\lambda _ { n , m } : = \\inf _ { \\phi \\in H ^ 1 _ 0 ( \\Omega _ m ) \\setminus \\{ 0 \\} } \\frac { \\int _ { \\Omega _ m } | \\nabla \\phi | ^ 2 + ( V ^ + - V _ n ^ - ) \\phi ^ 2 } { \\int _ { \\Omega _ m } \\phi ^ 2 } . \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} c \\prod _ { i = 0 } ^ { p - 1 } i ^ { k _ { 1 , i } } \\det \\left [ \\left ( j ^ { k _ { 2 , i } } \\right ) _ { 0 \\le i , j < p } \\right ] n ^ { - p + 2 \\beta p - k _ { 0 } - \\cdots - k _ { p - 1 } } . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} A = \\bigoplus _ { X \\in \\mathcal I } X \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} d _ { s v } = \\frac { f ( i _ { s v 2 } ) - f ( i _ { s v 1 } ) } { i _ { s v 2 } - i _ { s v 1 } } \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} \\int _ \\mathbb { K } r ( s , a ) Q _ \\mu ^ \\pi ( d s \\times d a ) = E _ { \\mu } ^ \\pi \\left ( \\sum _ { n = 1 } ^ \\infty \\beta ^ { n - 1 } r ( s _ n , a _ n ) \\right ) , \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 0 , 4 ) = \\Phi ( 4 ) = \\int _ { 0 } ^ { \\infty } \\frac { \\cos ( 4 \\pi x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x = \\frac { 3 - \\sqrt { 2 } } { 3 2 } , \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{gather*} b = \\epsilon \\delta ^ { - 1 } , \\\\ c = \\epsilon \\delta ^ { - 2 } \\left ( \\Gamma - \\frac { 1 } { 2 } \\sigma ^ 2 \\right ) + \\epsilon \\delta ^ { - 2 } \\int _ \\mathbb { R } \\big \\{ \\ln ( 1 + \\gamma ( z ) ) - \\gamma ( z ) \\big \\} \\nu ( d z ) . \\end{gather*}"}
-{"id": "5053.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\ , \\frac { 1 } { s _ l ^ j } \\sum _ { i = 1 } ^ { k _ l } \\int \\left ( \\omega _ { a _ i } ^ { n _ l } ( h _ l , 4 \\epsilon _ l , p ) \\right ) ^ j d \\nu _ { \\mathcal P _ { l } } ( p ) = 0 , j = 1 , 2 , \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} 1 & > \\sum _ { i = 1 } ^ r \\frac { ( 1 - \\lambda ) + ( m _ i - 1 ) + [ ( ( 1 - \\lambda ) + ( m _ i - 1 ) ) ^ 2 + 4 \\lambda m _ i ] ^ { 1 / 2 } } { 2 m } \\\\ & = \\sum _ { i = 1 } ^ r \\frac { ( m _ i - \\lambda ) + ( m _ i + \\lambda ) } { 2 m } = 1 , \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} Q _ t ( z _ t ) - \\lambda _ t = 0 , \\ \\langle \\lambda _ t , z _ t \\rangle = 0 , \\ \\langle \\lambda _ t , y \\rangle \\geq 0 \\ \\forall y \\in K . \\end{align*}"}
-{"id": "9885.png", "formula": "\\begin{align*} f = p \\boldsymbol { D } ^ { - s } u + q \\boldsymbol { D } ^ { s * } u , \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} i \\frac { \\partial } { \\partial t } \\psi ( x _ 1 , x _ 2 , t ) = \\hat H ( t ) \\psi ( x _ 1 , x _ 2 , t ) , \\ , \\ , \\ , \\ , \\ , \\ , \\psi ( x _ 1 , x _ 2 , t ) \\in \\mathcal { H } \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\Lambda ^ { - \\frac { 3 } { 2 } } ( a _ { * } - a _ { k } ) ^ { \\frac { 3 } { 4 } } u _ { k } ( y _ k + \\Lambda ^ { - 1 } ( a _ { * } - a _ { k } ) ^ { \\frac { 1 } { 2 } } x ) = Q ( x ) . \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} ( c \\theta _ - + d ) ( c \\theta _ + + d ) = \\det M \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} \\mathbf { x } = \\mathbf { \\lambda } \\odot \\mathbf { \\alpha } \\triangleq ( \\lambda _ 1 \\alpha _ 1 , \\lambda _ 2 \\alpha _ 2 , \\ldots , \\lambda _ N \\alpha _ N ) ^ T , \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} V ( \\nabla \\phi ) = [ 1 + \\beta ( \\nabla \\phi ) ^ 2 ] ^ \\alpha \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} E ( m T ) \\leq \\gamma E ( ( m - 1 ) T ) \\leq \\ldots \\leq \\gamma ^ { m } E ( 0 ) , \\ m = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} \\begin{aligned} a ^ + = - \\frac { \\log R _ 1 } { \\log \\left ( \\sqrt { \\frac { R _ 2 } { R _ 1 } } \\right ) } , & a ^ - = - \\frac { \\log R _ 2 } { \\log \\left ( \\sqrt { \\frac { R _ 1 } { R _ 2 } } \\right ) } , \\\\ b ^ + = \\frac { 1 } { \\log \\left ( \\sqrt { \\frac { R _ 2 } { R _ 1 } } \\right ) } , & b ^ - = \\frac { 1 } { \\log \\left ( \\sqrt { \\frac { R _ 1 } { R _ 2 } } \\right ) } . \\end{aligned} \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} c _ j x _ 1 \\left ( \\frac { \\partial } { \\partial { x _ 2 } } p _ { j } + \\sum _ { k > 2 } p _ { k } \\frac { \\partial } { \\partial { x _ k } } p _ { j } \\right ) + \\sum _ { \\ell > 2 } p _ { \\ell } \\frac { \\partial } { \\partial { x _ \\ell } } r _ j = 0 \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\ , \\frac 1 { s ^ 2 _ l } \\ \\sigma ^ 2 _ { \\mathcal P _ { l } } \\left ( \\sum _ { i = 1 } ^ { k _ l } S _ { a _ { i } + n _ l } ^ { M _ l } h _ l \\right ) = 0 . \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} A _ j = \\mu _ 0 ^ { n + 4 s } f ( \\mu _ 0 ^ { - 1 } \\mu , \\xi , \\mu _ j y ) + \\frac { \\mu _ 0 ^ { 2 n - 4 s } } { 1 + | y _ j | ^ { 2 s } } g ( \\mu _ 0 ^ { - 1 } \\mu , \\xi , \\mu _ j y ) , y _ j = \\frac { x - \\xi _ j } { \\mu _ j } , \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} V ( r ) = \\int _ { 0 } ^ { r } \\int _ { \\wp } s ^ { Q - 1 } \\left ( ( \\psi _ { 1 } ( s y ) ) ^ { 1 / p } g ( s ) \\right ) ^ { - p ' } d \\sigma ( y ) d s , \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} H _ \\tau = \\dot { t } _ \\tau \\phi \\ ; \\sim 0 \\ ; . \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} x y = x + y . \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{gather*} T _ { i } T _ { i + 1 } \\cdots T _ { j - 1 } \\big ( x _ { j } ^ + \\big ) = E _ { i , j + 1 } , T _ { i } T _ { i + 1 } \\cdots T _ { j - 1 } \\big ( x _ { j } ^ - \\big ) = E _ { j + 1 , i } , \\\\ T _ { j } T _ { j - 1 } \\cdots T _ { i + 1 } \\big ( x _ { i } ^ + \\big ) = ( - 1 ) ^ { j - i } E _ { i , j + 1 } , T _ { j } T _ { j - 1 } \\cdots T _ { i + 1 } \\big ( x _ { i } ^ - \\big ) = ( - 1 ) ^ { j - i } E _ { j + 1 , i } . \\end{gather*}"}
-{"id": "7564.png", "formula": "\\begin{align*} \\mathcal { S } _ p ( t ) = A ^ 2 \\left ( \\C ^ n , e ^ { - 4 \\pi p t } \\right ) . \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} X u = - f , u | _ { \\partial ( S M ) } = 0 , \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} W ( t , x ) = \\underset { u \\in \\mathcal { U } [ t , T ] } { e s s \\inf } Y _ { t } ^ { t , x ; u } . \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow \\infty } E ( V ^ { j _ 0 } ( \\cdot - x ^ { j _ 0 } _ n ) ) & \\leq \\liminf _ { n \\rightarrow \\infty } E ( V ^ { j _ 0 } ( \\cdot - x ^ { j _ 0 } _ n ) ) + \\liminf _ { n \\rightarrow \\infty } E ( v ^ { j _ 0 } _ n ) \\\\ & \\leq \\liminf _ { n \\rightarrow \\infty } \\left ( E ( V ^ { j _ 0 } ( \\cdot - x ^ { j _ 0 } _ n ) ) + E ( v ^ { j _ 0 } _ n ) \\right ) \\\\ & = \\liminf _ { n \\rightarrow \\infty } E ( v _ n ) = d _ M . \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} \\alpha _ 1 ( x _ 1 , x _ 2 ) = x _ 1 , \\alpha _ 2 ( x _ 1 , x _ 2 ) = x _ 2 , \\alpha _ 3 ( x _ 1 , x _ 2 ) = - x _ 1 + x _ 2 . \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} \\rho _ q ( d , m _ d ) \\le N = \\sum _ { i = e } ^ d \\rho _ q ( i , n _ i ) \\le \\sum _ { j = 0 } ^ { a - 1 } \\sum _ { \\ell = 0 } ^ { q - 2 } \\rho _ q ( d - j ( q - 1 ) - \\ell , m _ d - j - 1 ) + \\sum _ { i = 1 } ^ b \\rho _ q ( e + i - 1 , m _ d - a - 1 ) . \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} \\mathcal { O } _ r : = U _ { - \\alpha } \\Xi _ r , \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} x \\mapsto x , y \\mapsto x y \\Longrightarrow \\mathcal { I } _ { 1 , \\beta _ 1 , \\beta _ 2 , \\beta _ 3 } \\ ! \\ ! = \\ ! \\ ! 3 \\ ! \\ ! \\int _ { [ 0 , 1 ] ^ 2 } \\ ! \\hat { f } _ { \\mathbf { t } } ( x , y ) ^ s x ^ { 3 \\beta _ 1 + 4 \\beta _ 2 + 1 3 \\beta _ 3 + 1 + 2 6 s } y ^ { \\beta _ 3 + 1 } q ( x y ) \\frac { d x } { x } \\frac { d y } { y } \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} H _ { 0 , 1 } & = H _ { 1 , 0 } = \\dfrac { 1 } { p } \\left ( - p \\log p - ( 1 - p ) \\log ( 1 - p ) \\right ) , \\\\ H ^ c _ { 0 , 1 } & = \\dfrac { 2 } { p } \\left ( - p \\log p - ( 1 - p ) \\log ( 1 - p ) \\right ) , \\\\ ( t ^ { r e l } + 1 ) H ( X ) & = H ^ { a v } = \\left ( \\dfrac { 1 } { p } + 1 \\right ) \\left ( - p \\log p - ( 1 - p ) \\log ( 1 - p ) \\right ) \\leq ( t ^ { r e l } + 1 ) \\log 2 . \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} \\begin{array} { c c c c c c c c c } \\theta ( q , x ) & = & \\theta ( - v , x ) & = & \\psi _ 1 + \\psi _ 2 & , & { \\rm w h e r e } & & \\\\ \\\\ \\psi _ 1 ( v , x ) & : = & \\theta ( v ^ 4 , - x ^ 2 / v ) & & { \\rm a n d } & & \\psi _ 2 ( v , x ) & : = & - v x \\theta ( v ^ 4 , - v x ^ 2 ) ~ ; \\end{array} \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} x p _ n ( x ) = a _ { n - 1 } p _ { n - 1 } ( x ) + b _ { n } p _ n ( x ) + a _ { n } p _ { n + 1 } ( x ) , x \\in X , \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\Phi ( G ) = \\left \\{ f : G \\to \\C : f \\ \\ \\int _ G \\Phi ( | f ( s ) | ) \\ , d s < \\infty \\right \\} . \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 0 , 1 / 2 ) = \\Phi \\left ( \\frac { 1 } { 2 } \\right ) = \\int _ { 0 } ^ { \\infty } \\frac { \\cos \\left ( \\frac { \\pi x } { 2 } \\right ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x = \\frac { 1 } { 4 \\pi } , \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} \\pi ( k , \\alpha , \\beta ; \\mathbf { v } ) : = \\begin{cases} \\pi ( k , \\alpha , \\beta ) , & \\mbox { i f } \\ \\ \\mathbf { v } \\in \\{ \\tilde { x } _ 1 , \\cdots , \\tilde { x } _ n , \\tilde { y } _ 1 , \\cdots , \\tilde { y } _ n \\} , \\\\ \\pi ( k , \\alpha , \\beta ) - j , & \\mbox { i f } \\ \\ \\mathbf { v } = \\tilde { z } _ j , \\\\ \\pi ( k , \\alpha , \\beta ) + j , & \\mbox { i f } \\ \\ \\mathbf { v } = \\bar { \\tilde { z } } _ j . \\end{cases} \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} \\mathcal { R } _ { k ( - j ) j } ^ { \\nu } ( \\alpha _ { 2 , \\nu } ) = \\{ \\xi \\in \\mathcal { O } _ { \\nu } : | \\langle k , \\omega _ { \\nu } ( \\xi ) \\rangle + \\bar { \\Omega } _ { \\nu , - j } ( \\xi ) - \\bar { \\Omega } _ { \\nu , j } ( \\xi ) | < \\alpha _ { 2 , \\nu } \\frac { | j | } { \\langle k \\rangle ^ { \\tau } } \\} . \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} u _ { 0 , n } ( x ) : = \\mu _ n Q _ { } ( x ) , \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} \\mathrm { u s p } _ S ( T ^ { - 1 } , [ f ] ) = & \\mathrm { u s p } _ S ( T ^ { - 1 } , [ \\sum _ { v \\in I } \\alpha ( v ) \\phi v ] ) \\\\ \\leq & \\sup _ { v \\in I } \\mathrm { u s p } _ S ( T ^ { - 1 } , [ \\alpha ( v ) \\phi v ] ) \\\\ = & \\sup _ { v \\in I } \\mathrm { u s p } _ S ( T ^ { - 1 } , [ \\phi v ] ) . \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} S _ { G , m , p } ( t _ 0 , \\chi ( \\hat { u } ) , A _ 1 ^ { \\operatorname { c o } } , A _ 2 ^ { \\operatorname { c o } } , A _ 3 ^ { \\operatorname { c o } } , \\sigma ( \\hat { u } ) , f , u _ 0 ) = S _ { \\chi , \\sigma , G , m , p } ( t _ 0 , f , u _ 0 ) \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} h ( 1 ) & = \\gamma _ - ( \\theta ( 1 ) u ^ + + u ^ - ) = I ' ( u ^ - ) [ u ^ - ] + \\theta ( 1 ) ^ 2 \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } ( u ^ - ) ^ 2 d x \\\\ & < I ' ( u ^ - ) [ u ^ - ] + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } ( u ^ - ) ^ 2 d x = \\gamma _ - ( u ) \\leq 0 . \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} \\kappa ( \\hat { x } , x ^ { \\star } , \\lambda ) = z \\left [ l _ 1 ^ { - 1 } + l _ 2 ^ { - 1 } - 1 \\right ] + b \\left ( 1 + \\lambda \\right ) \\left [ l _ 1 ^ { - 1 } - l _ 2 ^ { - 1 } + \\ln { \\hat { x } } - \\ln { x ^ \\star } \\right ] , \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} a ( x , y ) \\frac { \\partial \\tilde { g } } { \\partial x } = \\sqrt { \\lambda } \\tilde { g } . \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} I _ { \\mu } ' ( u _ n ) ( u _ n ) _ - & = \\int _ { \\R ^ 3 } \\left ( \\nabla u _ n \\nabla ( ( u _ n ) _ - ) + u _ n ( u _ n ) _ - \\right ) + \\int _ { \\R ^ 3 } \\rho \\phi _ { u _ n } u _ n ( u _ n ) _ - - { \\mu } \\int _ { \\R ^ 3 } ( u _ n ) _ + ^ p ( u _ n ) _ - \\\\ & = | | ( u _ n ) _ - | | ^ 2 _ { H ^ 1 } + \\int _ { \\R ^ 3 } \\rho \\phi _ { u _ n } ( u _ n ) _ - ^ 2 . \\\\ \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} - \\varphi _ t - ( - \\Delta ) ^ s \\varphi - \\frac { n - 2 s } { 2 } \\alpha _ { n , s } \\frac { \\dot { \\mu } _ j } { \\mu _ j } \\frac { \\mu ^ { - ( n - 2 s ) } _ j } { \\left ( 1 + \\left | \\frac { x - \\xi _ j } { \\mu _ j } \\right | ^ 2 \\right ) ^ { \\frac { n - 2 s } { 2 } } } = 0 \\mathbb { R } ^ n \\times ( t _ 0 , + \\infty ) . \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} \\tilde { \\sigma } _ q ( U ) \\tilde { t } _ U = \\epsilon _ q ^ { - 1 } \\tilde { t } _ U . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { B ^ s _ { \\infty , \\infty } ( G ) } = \\sup _ { \\ell \\geq 0 } 2 ^ { l s } \\Vert \\psi _ { l } ( \\mathcal { R } ^ { \\frac { 1 } { 2 } } ) \\Vert _ { L ^ \\infty ( G ) } \\asymp \\sup _ { \\ell \\geq 0 } 2 ^ { l s / 2 } \\Vert \\psi _ { l } ( \\mathcal { R } ) \\Vert _ { L ^ \\infty ( G ) } \\lesssim \\Vert f \\Vert _ { \\Lambda ^ s ( G ) } . \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} \\int _ 0 ^ t f ( s ) d D _ s & = \\sum _ { k = 1 } ^ { D _ t } f ( \\tau _ k ) = \\sum _ { k = 1 } ^ { P _ { \\Lambda ( t ) } } f ( \\Lambda ^ { - 1 } ( \\sigma _ k ) ) \\sim \\int _ 0 ^ { \\Lambda ( t ) } f ( \\Lambda ^ { - 1 } ( s ) ) d s = \\int _ 0 ^ t f ( s ) \\kappa _ s d s . \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} p ( y , t ) = \\int d x \\ \\rho ( x , y , t ) \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} p ( \\gamma _ i - \\epsilon _ i ) = p \\gamma _ i - \\delta _ i = 0 , \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} f _ n ( 0 ^ + ) & = a - b + c + n d + \\int _ { [ 1 / n , n ] } s ^ { - 1 } \\ , d \\mu ( s ) , \\\\ f _ n ' ( + \\infty ) & = b + n c + d + \\int _ { [ 1 / n , n ] } d \\mu ( s ) . \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} ( x ; q ) _ n = ( - x ) ^ n q ^ { \\frac { n ( n - 1 ) } { 2 } } \\left ( x ^ { - 1 } q ^ { 1 - n } ; q \\right ) _ n , \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} ^ R _ { \\mathcal { C } } ( x , y ) _ n = \\{ f : J ^ n \\to \\mathcal { C } \\in S e t _ { \\Delta } ( J ^ n , \\mathcal { C } ) : f ( x ) = a , f ( y ) = b \\} \\end{align*}"}
-{"id": "9991.png", "formula": "\\begin{align*} u ( 0 , \\tau ) = u _ 0 ( \\tau ) * \\sum _ { i = 0 } ^ \\infty p ^ { * ( i ) } ( \\tau ) , \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} \\mathsf { r e c t s } | _ { \\mathcal { P } } ( T _ S ) = \\mathsf { r e c t s } | _ { \\mathcal { P } } ( T ) . \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} J [ t _ 0 , x _ 0 ; u ] = \\mathbb { E } \\left [ \\int _ { t _ 0 } ^ { \\tau _ S } h ( s , X _ s ^ { t _ { 0 } , x _ 0 , u } ) + \\sum _ { j \\geq 1 } c ( \\tau _ { j } , z _ j ) \\cdot 1 _ { \\{ \\tau _ j \\leq \\tau _ S \\} } + \\phi ( X _ { \\tau _ S } ^ { t _ { 0 } , x _ 0 , u } ) \\cdot 1 _ { \\{ \\tau _ S < \\infty \\} } \\right ] , \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} c _ 0 | \\theta \\cdot \\nabla p ( x ) | = 1 \\mbox { o n s u p p } ( C ( x , \\cdot ) ) \\mbox { f o r a l l } x \\in \\Omega . \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} K _ { 2 , 3 } ( z , z '' ) = & \\Big ( \\int _ { 2 r '' < r ' < \\frac { r } 2 } + \\int _ { \\frac { r } 2 \\leq r ' \\leq 2 r } + \\int _ { r ' > 2 r } \\Big ) G ( z , z ' ) Q ( z ' , z '' ) \\ ; d \\mu ( z ' ) \\\\ = & K _ { 2 , 3 1 } ( z , z '' ) + K _ { 2 , 3 2 } ( z , z '' ) + K _ { 2 , 3 3 } ( z , z '' ) . \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} \\mathrm { { { S N R } } } _ { i } \\geqslant \\max _ { j \\neq i } \\mathrm { \\mathrm { { { I N R } } } } _ { i j } \\max _ { k \\neq i } \\mathrm { \\mathrm { { { I N R } } } } _ { k i } \\quad \\forall i = 1 , . . . , n , \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} M ( m ) : = \\max _ { \\substack { 1 \\leq i \\leq t \\\\ 0 \\leq j \\leq m } } \\{ | c _ { i , j } | \\} \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} G ^ { \\epsilon } ( z ) = T \\left ( \\widehat { G ^ { \\epsilon } _ c } e ^ { 2 \\pi c ( \\cdot ) } \\right ) ( z ) = \\int _ { \\mathbb { R } } \\widehat { G ^ { \\epsilon } _ c } ( t ) e ^ { 2 \\pi c t } e ^ { i 2 \\pi z t } \\d t , \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} \\{ \\Phi _ { ( \\psi , f ) } ^ * ( \\hat g ^ { ( 4 ) } ) : ~ \\Phi _ { ( \\psi , f ) } \\in \\mathcal D _ 4 , ~ \\mathbf n ( f ) | _ { \\partial M } = \\hat F | _ { \\partial M } \\} . \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{gather*} F _ 1 ( z ) = - \\Gamma ( r ) \\Gamma ( 1 - r ) F _ 0 ( 1 - z ) = - \\frac \\pi { \\sin \\pi r } F _ 0 ( 1 - z ) . \\end{gather*}"}
-{"id": "9258.png", "formula": "\\begin{align*} \\beta _ { i } ( t ) = u _ { 0 } ( - a _ { i } ) + \\left ( \\epsilon u _ { 0 , x _ { i } x _ { i } } ( - a _ { i } ) - \\tilde { H } ^ { ( R ) } _ { i } ( 0 , - a _ { i } , u _ { 0 , x _ { i } } ( - a _ { i } ) ) \\right ) t . \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} \\gamma ^ i ( t , W _ t ) + h ^ i ( t , W _ t ) = & - ( M ^ i _ T - M ^ i _ t ) - \\int _ t ^ T ( \\nabla h ^ i ) ^ * ( r , W _ r ) \\mathrm d W _ r \\\\ & + \\gamma ^ i ( T , W _ T ) + h ^ i ( T , W _ T ) \\\\ = & \\Phi ( W _ T ) - ( \\tilde M ^ i _ T - \\tilde M ^ i _ t ) , \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} u ( t , x ) & = \\mathbf E \\left [ q ( Y ^ { x } ( t ) ) \\mathbf 1 _ { \\{ W ( t ) > 1 \\} } \\right ] \\\\ & = \\mathbf E \\left [ q ( Y ^ { x } ( t ) ) | Y ^ x ( t ) \\right ] \\\\ \\Big ( & = \\mathbf E \\left [ q ( Y ^ { x } ( t ) ) | Y ^ x ( t ) \\right ] \\Big ) . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} g _ v ' \\cup z = ( x + j ( f _ v ) ) \\cup z = ( x \\cup z ) + ( f _ v \\cup \\widehat { j } ( z ) ) = ( x \\cup z ) + ( f _ v \\cup d y _ v ) = ( x \\cup z ) - ( d f _ v \\cup y _ v ) + d ( f _ v \\cup y _ v ) . \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} B ( t ) \\begin{cases} t ^ { q _ 0 } \\ , e ^ { - \\sqrt { \\log 1 / t } } & , \\\\ t ^ { q _ \\infty } \\ , e ^ { \\sqrt { \\log t } } & . \\end{cases} \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\sigma ' : = \\sigma _ { ( A _ 0 , B _ 0 ) } ^ { ( X , Y ) } = \\{ ( A _ 0 \\cup X , B _ 0 \\cap Y ) \\} \\cup \\{ ( A _ i \\cap Y , B _ i \\cup X ) \\colon 1 \\leq i \\leq n \\} . \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} \\nu ( y ) = \\frac { \\nabla \\phi ( y ) } { \\sqrt { g ( \\nabla \\phi ( y ) , \\nabla \\phi ( y ) ) } } ; \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} \\nu _ 2 ( s ( n , k ) ) = h - \\sigma _ 2 ( n - 1 ) . \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} \\mathbf { i } \\partial _ { \\omega } ( \\Delta F _ { i j } ) + \\bar { \\Omega } _ { i j } \\Delta F _ { i j } + \\tilde { \\Omega } _ { i j } \\Delta F _ { i j } = - \\mathbf { i } \\partial _ { \\Delta \\omega } F _ { i j } - ( \\Delta \\Omega _ { i j } ) F _ { i j } + \\mathbf { i } \\Delta R _ { i j } : = Q _ { i j } , \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} ( f | g ) ^ { \\sigma } = f ^ { \\sigma } | g _ \\lambda , \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} \\mathbb { V } = \\{ ( V _ i , u _ e ) \\} _ { i \\in V ( Q ) , e \\in E ( Q ) } \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} F ( z ) = T _ S ( \\widehat { F } _ c e ^ { 2 \\pi c ( \\cdot ) } ) ( z ) = \\int _ { \\R } \\widehat { F } _ c ( t ) e ^ { 2 \\pi c t } e ^ { i 2 \\pi z t } \\d t = \\int _ { \\R } \\widehat { F } _ c ( t ) e ^ { i 2 \\pi ( z - i c ) t } \\d t , \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\log \\frac { 1 } { 3 } \\bigg | \\frac { \\eta \\bigl ( \\frac { - 1 + 2 \\sqrt { 2 } i } { 3 } \\bigr ) } { \\eta ( 2 \\sqrt { 2 } i ) } \\bigg | ^ 4 = 2 \\cdot \\frac { \\pi } { 4 } \\cdot \\frac { \\log ( 3 + 2 \\sqrt { 2 } ) } { \\pi } = \\log ( 1 + \\sqrt { 2 } ) , \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\tilde { f } _ n ^ { ( z ) } = \\mathcal { F } ^ { - 1 } \\{ \\textbf { G } \\} _ n . \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} X _ { t } ^ { \\mathrm m _ { i } } \\ , = \\ , X _ { 0 } ^ { ( u ) } + \\int ^ { t } _ { 0 } b ( s , X _ { s } ^ { \\mathrm m _ { i } } , u \\ , \\delta _ { \\widetilde { X } ^ { \\mathrm m _ { i } } _ { s } } + ( 1 - u ) \\mathrm m _ { i , s } ) { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , ; 0 \\le t \\le T \\ , , i \\ , = \\ , 1 , 2 \\ , . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} M ' _ { k , \\ell } = \\sum _ { \\substack { 0 \\leq s _ 1 , \\dots , s _ q \\leq m + 1 \\\\ 0 \\leq s _ { q + 1 } \\leq r } } \\binom { r } { s _ { q + 1 } } \\left ( \\prod _ { i = 1 } ^ q \\binom { m + 1 } { s _ i } \\right ) \\left ( \\prod _ { i = 1 } ^ { q + 1 } ( - \\lambda _ i ) ^ { s _ i } \\right ) \\binom { k - \\sum _ { i = 1 } ^ { q + 1 } s _ i + f } { r _ \\ell } \\lambda _ { q _ \\ell + 1 } ^ { k - \\sum _ { i = 1 } ^ { q + 1 } s _ i + f - r _ { \\ell } } . \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} | \\nabla f | ( x ) : = \\limsup _ { y \\to x } \\frac { | f ( x ) - f ( y ) | } { d ( x , y ) } \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} [ [ x , y ] , \\varphi ( \\psi ( z ) ) ] \\otimes a b \\alpha ( c ) + [ [ z , x ] , \\varphi ( \\psi ( y ) ) ] \\otimes c a \\alpha ( b ) + [ [ y , z ] , \\varphi ( \\psi ( x ) ) ] \\otimes b c \\alpha ( a ) = 0 \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} \\nabla ^ { \\mathrm { b a s } } _ a ( b , u ) : = \\big ( \\nabla _ { \\varrho _ A b } a + [ a , b ] _ A , \\varrho _ A \\nabla _ { u } a + [ \\varrho _ A a , u ] \\big ) ; \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} p ^ \\omega ( t , x , y ) : = P ^ { \\omega } _ x [ X _ t = y ] . \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} K ( z , Z ) = \\int _ { \\Omega } K ( z , \\zeta ) \\overline { K ( Z , \\zeta ) } \\lambda ( \\zeta ) \\d V ( \\zeta ) . \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} \\mathcal R _ 0 ( \\lambda ) = ( D _ m + \\lambda ) R _ 0 ( \\lambda ^ 2 - m ^ 2 ) . \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} a _ 1 + \\ldots + a _ m = a _ { m + 1 } , \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} \\sum _ { p = p _ { 1 } ; p _ { 2 } , \\ ; h \\colon T \\to Y } ^ { \\sim } \\mathbf { P o l y } _ { c } \\left ( \\mathcal { E } \\right ) _ { X , Y } \\left ( \\left ( s , p _ { 1 } , h \\right ) , \\left ( a , i , b \\right ) \\right ) \\times \\mathbf { P o l y } _ { c } \\left ( \\mathcal { E } \\right ) _ { Y , Z } \\left ( \\left ( h , p _ { 2 } , t \\right ) , \\left ( u , j , v \\right ) \\right ) \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} \\Pi ^ T a \\nabla V = & \\ , \\Pi ^ T a \\nabla V _ 0 + \\frac { 1 } { \\epsilon } a \\nabla V _ 1 \\ , \\\\ ( I - \\Pi ) ^ T a \\nabla V = & \\ , ( I - \\Pi ) ^ T a \\nabla V _ 0 \\ , , \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} A _ { I _ 1 , J _ 1 } = & \\{ 0 \\} ^ { I _ 1 } \\times \\{ 1 \\} ^ { J _ 1 } \\times \\{ 0 , 1 \\} ^ { T \\backslash ( I _ 1 \\cup J _ 1 ) } \\times \\{ 0 , 1 \\} ^ { T ^ c } \\\\ A _ { I _ 2 , J _ 2 } = & \\{ 0 \\} ^ { I _ 2 } \\times \\{ 1 \\} ^ { J _ 2 } \\times \\{ 0 , 1 \\} ^ { T \\backslash ( I _ 2 \\cup J _ 2 ) } \\times \\{ 0 , 1 \\} ^ { T ^ c } \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} I _ { \\mu } ^ { \\infty } ( u _ n ^ 2 ) & = | | u _ n ^ 1 | | _ { H ^ 1 } ^ 2 - | | v _ 1 | | _ { H ^ 1 } ^ 2 + \\frac { 1 } { 4 } \\int _ { \\R ^ 3 } \\rho _ { \\infty } \\bar \\phi _ { ( u _ n ^ 1 - v _ 1 ( \\cdot - y _ n ^ 1 ) ) } ( u _ n ^ 1 - v _ 1 ( x - y _ n ^ 1 ) ) ^ 2 \\\\ & \\qquad - \\frac { \\mu } { p + 1 } \\left ( | | ( u _ n ^ 1 ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } - | | ( v _ 1 ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } \\right ) + o ( 1 ) . \\\\ \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\psi _ l ( \\mathcal { R } ) \\sigma ( D _ x ) = \\sigma ( D _ x ) \\psi _ l ( \\mathcal { R } ) = \\sigma _ { l } ( D _ x ) \\psi _ l ( \\mathcal { R } ) \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} \\pi ( s _ { \\mu , \\nu } ) \\xi _ m = \\rho ( a _ { \\mu , \\nu } ) \\xi _ m + \\sum _ { j = 1 } ^ { r _ { \\mu , \\nu } } t ( x ^ { ( \\mu , \\nu ) } _ { j , 1 } ) t ( x ^ { ( \\mu , \\nu ) } _ { j , 2 } ) \\cdots t ( x ^ { ( \\mu , \\nu ) } _ { j , N _ j } ) \\xi _ m = \\tau ( s _ { \\mu , \\nu } ) \\xi _ m \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } p _ { n - 1 } & p _ { n } \\\\ q _ { n - 1 } & q _ { n } \\end{array} \\right ] = \\left [ \\begin{array} { c c } p _ { n - m - 2 } & p _ { n - m - 1 } \\\\ q _ { n - m - 2 } & q _ { n - m - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n - m } \\end{array} \\right ] \\cdots \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n } \\end{array} \\right ] \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} \\Psi ( a ) \\Psi ( a ) ^ * \\leq \\Psi ( a a ^ * ) \\leq \\Psi ( a ^ * a ) = \\Psi ( a ) ^ * \\Psi ( a ) . \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} ( k , l , m ) : = \\begin{cases} { \\rm f o r } \\ 2 \\leq k \\leq 3 , \\ l = 0 , \\ m = 2 , \\\\ { \\rm f o r } \\ 4 \\leq k , \\ l = k - 3 , \\ m = 2 + l . \\end{cases} \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} \\begin{gathered} \\big | \\partial _ \\lambda ^ \\alpha b ( \\lambda , z , z ' ) \\big | \\leq C _ { \\alpha , X } \\lambda ^ { - \\alpha } ( 1 + \\lambda d ( z , z ' ) ) ^ { - K } K > 0 , \\end{gathered} \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} I _ 2 ( k ) & = O \\Big ( k ^ { \\frac { \\beta + 2 } { \\beta + 1 } } \\Big ) ^ { - 1 / 2 } \\cdot O \\big ( x _ 0 ^ \\alpha \\big ) \\\\ & = O \\Big ( k ^ { - \\frac { \\beta / 2 + 1 } { \\beta + 1 } } \\Big ) \\cdot O \\Big ( k ^ { - \\frac { \\alpha } { \\beta + 1 } } \\Big ) \\\\ & = O \\Big ( k ^ { - \\frac { 1 + \\alpha + \\beta / 2 } { \\beta + 1 } } \\Big ) . \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} B _ { \\xi _ { n } } ( t ) = \\sum _ { j = 1 } ^ { n } \\xi _ { j , n } \\ Y _ { j } \\ , h _ { j } ^ { n } ( t ) . \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} U _ k ( t ) = A _ { k } V _ { m + k - 1 } ( t ) + \\int _ t ^ { t _ 0 } G _ { k } ( t , s ) V _ { m + k - 1 } ( s ) \\ , d s + \\int _ t ^ { t _ 0 } H _ { k } ( t , s ) U _ k ( s ) \\ , d s \\mbox { f o r } t _ k \\le t < t _ { k - 1 } , \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} \\frac { ` b ( ` m ) } { ` g ( ` l , ` p , ` r ) } = 1 - \\max _ { e \\in E } \\frac { u ^ * ( ` x ( e ) ) } { u ^ * ( V ) } = 1 - \\frac { 1 } { ` t } \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} [ E _ i , F _ i ^ { ( k + 1 ) } ] = F _ i ^ { ( k ) } \\frac { q _ i ^ { - k } K _ i - q _ i ^ { k } L _ i ^ { - 1 } } { q _ i - q _ i ^ { - 1 } } , \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} F _ \\Psi ( C _ \\ell , q ) & : = \\# \\{ \\textbf { z } \\in ( \\Z / q \\Z ) ^ \\ell \\mid \\textbf { z } \\cdot T _ \\Psi + \\textbf { g } \\cdot S _ \\Psi \\in ( ( \\Z / q \\Z ) ^ \\times ) ^ { \\# \\Psi } \\} , \\\\ \\textbf { g } & : = ( \\overline 0 , \\overline 0 , \\ldots , \\overline 1 ) \\in ( \\Z / q \\Z ) ^ \\ell . \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} P _ { \\varphi } w = 0 , w | _ { \\partial \\Omega } = \\partial _ { \\nu } w | _ { \\Omega } = 0 \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & L ( \\chi ( u ) , A _ 1 , A _ 2 , A _ 3 , \\sigma ( u ) ) v = f _ { \\alpha } , & & x \\in \\R ^ 3 _ + , & & t \\in ( 0 , T ) ; \\\\ & B v = \\partial ^ \\alpha g , & & x \\in \\partial \\R ^ 3 _ + , & & t \\in ( 0 , T ) ; \\\\ & v ( 0 ) = \\partial ^ { ( 0 , \\alpha _ 1 , \\alpha _ 2 , 0 ) } S _ { \\chi , \\sigma , \\R ^ 3 _ + , m , \\alpha _ 0 } ( 0 , f , u _ 0 ) , & & x \\in \\R ^ 3 _ + . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} f _ a ( x ) = ( 2 \\pi ) ^ { - d } \\int _ { \\R ^ d } e ^ { i x \\xi } e ^ { - | \\xi | ^ a } d \\xi . \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} K ( z , w ; Z , W ) = \\int _ { \\R } X _ p ( t ; w , W ) \\frac { z ^ { 2 \\pi i t } \\cdot \\overline { Z } ^ { - 2 \\pi i t } } { ( z \\overline { Z } ) ^ { 1 + 1 / 2 \\mu } } \\d t . \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } \\int _ { \\Omega } \\bigl | f ^ { \\nu } ( t , x , u ( t , x ) ) - f ( t , x , u ( t , x ) ) \\bigr | \\ , d t \\ , d x ~ = ~ 0 . \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} H _ 0 \\oplus H _ 0 ^ { \\perp _ { \\mathcal Q } } = H . \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} + \\Biggl . { \\bf 1 } _ { \\{ i _ 2 = i _ 5 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 2 = j _ 5 \\} } { \\bf 1 } _ { \\{ i _ 3 = i _ 4 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 3 = j _ 4 \\} } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\Biggr ) , \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} E ^ { ( k ) } _ { a , b } ( \\tau ) = \\frac { ( k - 1 ) ! } { ( - 2 \\pi i ) ^ k } \\sum _ { \\substack { \\omega \\in \\Z \\tau + \\Z \\\\ \\omega \\neq - ( \\tilde { a } \\tau + \\tilde { b } ) / N } } \\frac { 1 } { ( \\omega + \\frac { \\tilde { a } \\tau + \\tilde { b } } { N } ) ^ k | \\omega + \\frac { \\tilde { a } \\tau + \\tilde { b } } { N } | ^ { 2 s } } \\Biggl . \\Biggr | _ { s = 0 } \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} \\sigma _ A ( x , \\xi ) = \\xi ( x ) ^ { * } ( A \\xi ) ( x ) , \\end{align*}"}
-{"id": "9945.png", "formula": "\\begin{align*} H _ \\pm = \\{ \\ , y \\in \\R ^ n _ y \\mid \\ , \\pm y _ 1 > c ^ 2 | y ' | \\ , \\} . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} S _ { \\varepsilon _ k } = p _ { \\varepsilon _ k } \\dot { p } _ { \\varepsilon _ k } = 0 \\qquad J \\varepsilon _ k < \\varepsilon _ 0 . \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} K = \\deg ( G ) - g + 1 . \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} Q _ \\alpha ( \\rho \\| \\sigma ) = \\begin{cases} \\int _ { ( 0 , + \\infty ) } t ^ \\alpha \\ , d \\| E _ { \\rho , \\sigma } ( t ) \\xi _ \\sigma \\| ^ 2 & , \\\\ + \\infty & . \\end{cases} \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} \\omega ' ( t ) & : = \\sqrt { \\phi ' ( t ) \\ , t \\ , } . \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} N ( 0 , r ) = { q \\choose r } q ^ { k - r - 1 } ( q - 1 ) \\left ( \\sum _ { j = 0 } ^ { k - r - 1 } ( - 1 ) ^ j { q - r - 1 \\choose j } q ^ { - j } \\right ) . \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} D _ { n i } = N _ n ( U _ { n i } ) - N _ n ( u _ n ) , \\ \\ 2 \\leq i \\leq s \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} \\int _ { \\tilde { H } _ { e _ 1 } } . . . \\int _ { \\tilde { H } _ { e _ { d - k } } } 1 \\wedge \\left ( \\frac { \\rho } { | \\sum _ { i = 1 } ^ { d - k } ( x _ i - y _ i ) | } \\right ) ^ { d - k + \\frac { 1 } { 4 } } d \\omega ( y _ { d - k } , . . . , y _ 1 ) \\lesssim \\rho ^ { d - k } \\mathbf { M } ( \\omega , \\bigotimes _ { i = 1 } ^ { d - k } \\tilde { H } _ { e _ i } ) ( \\sum _ { i = 1 } ^ { d - k } x _ i ) , \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} \\alpha _ { n e w } = \\underset { a > \\alpha } { \\min } \\{ a : \\{ k : R R ( k ) \\leq \\Gamma _ { R R T } ^ { \\alpha } ( k ) \\} \\neq \\phi \\} . \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} \\bar { v } ^ \\omega _ q ( x ) : = \\limsup _ { n \\to \\infty } \\norm { v ^ \\omega } _ { q , \\tilde { B } ( x , n ) } \\le \\sum _ { i = 1 } ^ { d } \\mathbb { E } _ { \\tilde { \\mu } } ( \\omega ( 0 , e _ i ) ^ { - q } ) ^ { 1 / q } . \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} M ( r ) = \\sup _ { j \\in \\{ 1 , \\ldots , m + 1 \\} } N \\left ( r , \\frac { 1 } { f _ j } \\right ) , \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} \\widehat { \\Lambda } \\otimes _ \\Z \\C = \\C ^ { 2 n } . \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} \\mathcal { F } _ { D , m } ( z , s ) : = \\Gamma \\biggl ( \\frac { s + 1 } { 2 } \\biggr ) \\sum _ { n | m } \\biggl ( \\frac { D } { n } \\biggr ) P _ { \\frac { 1 } { 2 } , \\frac { m ^ 2 D } { n ^ 2 } } \\biggl ( z , \\frac { s } { 2 } + \\frac { 1 } { 4 } \\biggr ) . \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} E _ b ( W _ n ( w ) ) & = \\{ v \\in W _ n ( w ) \\mid \\lvert v \\rvert _ S > n _ b ( W _ n ( w ) ) \\} \\\\ & = \\{ v \\in W _ n ( w ) \\mid \\lvert v \\rvert _ S > \\max \\{ \\lvert v \\rvert _ S \\mid v \\in W _ n ( w ) , \\tau _ b ( v ) \\neq v \\} \\} . \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} \\mathcal F _ \\alpha ^ V ( \\pi ) : = \\sup _ { h \\in C ( \\Lambda ; \\mathbb R ) } \\biggl [ \\langle \\pi , h \\rangle - \\int _ \\Lambda \\log \\biggl ( \\frac { Z _ 1 ( f ' ( a ) + h ( u ) - V ( u ) ) } { Z _ 1 ( f ' ( a ) - V ( u ) ) } \\biggr ) \\ ; \\ ! \\mathrm d u \\biggr ] . \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} | T ^ { ( 2 ) } _ { a } ( x ) | \\leq \\widetilde { B } _ { a } ( x ) : = C _ { 2 } \\begin{cases} | x | ^ { a - Q } , \\ ; x \\in \\mathbb { G } \\backslash \\{ 0 \\} , \\\\ | x | ^ { - Q } , \\ ; x \\in \\mathbb { G } \\ ; \\ ; | x | \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} \\left ( \\alpha _ { \\l } - \\sigma _ { \\l + 1 } \\lambda _ { \\l } \\left ( 1 + \\frac { 1 } { \\l + d - 2 } \\right ) ^ 2 \\right ) \\gamma _ { \\l + 1 } ^ 2 & = ( 2 ( 1 - \\delta ) + O ( \\l ^ { - 1 } ) ) \\mu ^ { \\l + 1 } , \\\\ \\left ( \\beta _ { \\l } + \\sigma _ { \\l - 1 } \\lambda _ { \\l } \\left ( 1 - \\frac { 1 } { \\l } \\right ) ^ 2 \\right ) \\gamma _ { \\l - 1 } ^ 2 & = ( 2 ( 1 + \\delta ) + O ( \\l ^ { - 1 } ) ) \\mu ^ { \\l - 1 } \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} P _ 0 ^ \\omega ( H ( u t ) \\ge A _ \\mu ^ { - 1 } ( t ) ) = P _ 0 ^ \\omega ( A _ \\mu ( H ( u t ) ) \\ge t ) . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} s : = \\big ( m ( q - 1 ) \\big ) ^ { h - 1 } . \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} \\lambda < \\frac { { \\norm { \\phi _ \\lambda } } _ { D } ^ { - 2 } \\displaystyle { \\int _ { \\mathbb R ^ 3 } u _ { \\lambda } ^ 4 d x } } { { \\norm { \\phi _ \\lambda } } _ { D } ^ { - 2 } \\displaystyle { \\int _ { \\mathbb R ^ 3 } \\phi _ \\lambda u _ { \\lambda } ^ 2 d x } } = \\frac { \\displaystyle { \\int _ { \\mathbb R ^ 3 } v ^ 4 d x } } { \\displaystyle { \\int _ { \\mathbb R ^ 3 } \\phi _ { v } v ^ 2 d x } } = \\int _ { \\mathbb R ^ 3 } v ^ 4 d x \\leq C \\norm { v } ^ 4 \\end{align*}"}
-{"id": "9987.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } \\bar u - q \\bar u = ( \\bar p ( q ) - 1 ) u ( 0 , s ) . \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} ( M , g ) = \\Bigl ( \\R ^ + \\times S ^ { n - 1 } ( 1 ) , d r ^ 2 + f _ 1 ^ { \\ , 2 } ( r ) g _ { S ^ { n - 1 } ( 1 ) } \\Bigr ) . \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} \\dfrac { \\Gamma _ { \\R } ' } { \\Gamma _ { \\R } } ( s ) = \\dfrac { 1 } { 2 } \\log s + O ( 1 ) , \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} W _ { t } + \\mathcal { H } ^ { \\nu } \\left ( t , x , W _ { x } \\right ) = \\varepsilon W _ { x x } , \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\tilde { \\omega } _ { f } ( \\kappa ^ j r ) \\lesssim _ { d , R _ 0 , \\varrho _ 0 , \\kappa } \\int _ 0 ^ { r } \\frac { \\tilde { \\omega } _ { f } ( t ) } { t } \\ , d t < \\infty \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} F _ \\Psi ( D _ \\ell , q ) & : = \\# \\{ \\textbf { z } \\in ( \\Z / q \\Z ) ^ \\ell \\mid \\textbf { z } \\cdot T _ \\Psi + \\textbf { g } \\cdot S _ \\Psi \\in ( ( \\Z / q \\Z ) ^ \\times ) ^ { \\# \\Psi } \\} , \\\\ \\textbf { g } & : = ( \\overline 0 , \\overline 0 , \\ldots , \\overline 1 ) \\in ( \\Z / q \\Z ) ^ \\ell . \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} \\phi _ { E ( n ) } ( t ) = \\left ( 1 - \\frac { 1 } { 2 n } \\sum _ { i = 1 } ^ { n } t _ i \\right ) ^ { - \\lambda } \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} & \\mathcal { H } ' : \\begin{cases} \\dot { x } ' ~ ~ \\in F ( x ' ) & x ' \\in \\mathcal { F } _ c ' : = \\{ x ' \\in \\mathcal { S } : ( \\hat { g } , \\hat { b } _ a ) \\in \\mathcal { F } _ o \\} \\\\ x '^ + \\in G ( x ' ) & x ' \\in \\mathcal { J } _ c ' : = \\{ x ' \\in \\mathcal { S } : ( \\hat { g } , \\hat { b } _ a ) \\in \\mathcal { J } _ o \\} \\end{cases} \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } & \\sim \\phi b - \\sum _ { s \\in S _ b } \\sum _ { i = 1 } ^ { m _ 0 - 1 } \\phi b ^ i s \\\\ & \\sim \\phi b - \\sum _ { s \\in S _ b } \\sum _ { i = 1 } ^ { m _ 0 - 1 } ( \\phi b s - \\sum _ { j = 1 } ^ { i - 1 } \\sum _ { s ' \\in S _ b } \\phi s ' b ^ j s ) \\\\ & = \\phi b - \\sum _ { s \\in S _ b } ( m _ 0 - 1 ) \\phi b s + \\sum _ { s \\in S _ b } \\sum _ { s ' \\in S _ b } \\sum _ { k = 1 } ^ { m _ 0 - 2 } ( m _ 0 - 1 - k ) \\phi s ' b ^ k s = f _ { m _ 0 } . \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} d ( \\mathcal { C } _ 1 , \\mathcal { C } _ 2 ) = \\max \\left \\{ \\sup \\limits _ { h \\in \\mathcal { C } _ 2 , h \\neq 0 } \\inf \\limits _ { k \\in \\mathcal { C } _ 1 } \\frac { \\| h - k \\| _ { H } } { \\| h \\| _ { H } } , \\sup \\limits _ { h \\in \\mathcal { C } _ 1 , h \\neq 0 } \\inf \\limits _ { k \\in \\mathcal { C } _ 2 } \\frac { \\| h - k \\| _ { H } } { \\| h \\| _ { H } } \\right \\} . \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( \\ker \\beta _ { 0 + } ) = \\pi ^ { - 1 } \\Big ( \\bigcap _ { i = 0 } ^ { m ^ + } \\ker \\beta _ i \\Big ) = \\bigcap _ { i = 0 } ^ { m ^ + } \\pi ^ { - 1 } ( \\ker \\beta _ i ) = \\bigcap _ { i = 0 } ^ { m ^ + } ( V + \\ker B _ i ) , \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\mathbf { A } ( t ) = \\big ( A _ 1 ^ { 1 / 2 } \\exp ( t Y _ 1 ) A _ 1 ^ { 1 / 2 } , \\ldots , A _ m ^ { 1 / 2 } \\exp ( t Y _ m ) A _ m ^ { 1 / 2 } \\big ) . \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} H _ Q & : = \\begin{pmatrix} \\mathbf { q } '' _ { \\xi x } & \\mathbf { q } '' _ { \\xi \\xi } \\\\ \\\\ - \\mathbf { q } '' _ { x x } & - \\mathbf { q } '' _ { x \\xi } \\end{pmatrix} \\ , , \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} F _ { [ i ] } : = \\mathbb { F } _ p ( \\{ \\alpha _ j : j \\in [ n ] \\backslash [ i ] \\} ) . \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} \\omega a _ n ' ( \\omega ) = \\mathrm { o r d } ( \\omega ) ^ 2 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } ' ( 1 ) . \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} P _ R \\triangleq \\mathbb { E } \\left \\{ \\mathbf { x } ^ { \\rm H } \\mathbf { H } ^ { \\rm H } \\mathbf { H } \\mathbf { x } \\right \\} = \\mathrm { t r } \\left ( \\mathbf { H } \\mathbf { S } \\mathbf { H } ^ { \\rm H } \\right ) . \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} c _ { 1 } \\ , \\norm { x } _ { H } \\le \\inf _ { \\xi \\in W ^ { 1 , 2 } _ { \\frac { 1 } { 2 } , \\frac { 3 s - 1 } { 2 s } } ( H ) : \\xi ( 0 ) = x } \\norm { \\xi } _ { W ^ { 1 , 2 } _ { \\frac { 1 } { 2 } , \\frac { 3 s - 1 } { 2 s } } ( H ) } \\le c _ { 2 } \\norm { x } _ { H } \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} d \\xi ( x ( s ) ) = & \\ , ( \\mathcal { L } \\xi ) ( x ( s ) ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } ( \\nabla \\xi \\sigma ) \\big ( x ( s ) \\big ) \\ , d w ( s ) \\ , , \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} & \\| u u _ h ^ { \\kappa } \\| _ { 1 , p } ^ p \\leq M _ 4 ( \\kappa , u ) \\left [ \\| u u _ h ^ { \\kappa } \\| _ { p } ^ p + 1 \\right ] \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} \\mu = \\zeta ^ 1 \\wedge \\zeta ^ 2 \\otimes Z _ 2 - \\ , \\zeta ^ 1 \\wedge \\zeta ^ 3 \\otimes Z _ 3 + \\bar \\zeta ^ 1 \\wedge \\bar \\zeta ^ 2 \\otimes \\bar Z _ 2 - \\bar \\zeta ^ 1 \\wedge \\bar \\zeta ^ 3 \\otimes \\bar Z _ 3 \\ , . \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} c = \\frac { 2 \\sum _ { b = 1 } ^ n \\xi _ b } { 2 n - 1 } . \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} | \\sum _ { b = 1 } ^ n k _ b j _ b ^ 2 + i ^ 2 - j ^ 2 | < \\frac { 1 } { 5 0 n } | i + j | , \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} \\widetilde Q _ { 1 2 } : = 1 + q , \\widetilde Q _ { 3 2 } = 2 - q , \\widetilde Q _ { 3 1 } = q , \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} n ( n + \\alpha + \\beta + 1 ) - m ( m + \\alpha + \\beta + 1 ) = ( n - m ) ( n + m + \\alpha + \\beta + 1 ) \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\lbrace X _ s ^ { \\omega ^ \\prime } = Y _ s ^ { \\omega ^ \\prime } \\right \\rbrace = \\lim _ { R \\to \\infty } \\left [ \\mathbb { P } \\left \\lbrace X _ s ^ { \\omega ^ \\prime } = Y _ s ^ { \\omega ^ \\prime } , s \\le \\tau ( R ) \\right \\rbrace + \\mathbb { P } \\left \\lbrace s > \\tau ( R ) \\right \\rbrace \\right ] = 1 \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} \\lambda ^ { \\theta } r ^ { \\frac { n } { p } \\theta - 1 } \\phi _ { \\lambda } '^ { \\theta - 1 } = \\phi _ { \\lambda } ^ { \\frac { n } { p } \\theta - 1 } , \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} a _ n \\Delta _ g u + s _ g u = c | u | ^ { p _ n - 2 } u , \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} ( E ' _ d ) ^ \\dagger \\left ( E ' _ c - \\sum _ { i = 1 } ^ k \\beta _ i E ' _ { a _ i } \\right ) = E _ d ^ \\dagger \\left ( E _ c - \\sum _ { i = 1 } ^ k \\beta _ i E _ { a _ i } \\right ) = 0 d = 1 , \\ldots , m , \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\| \\gamma _ n ^ { ( d ) } ( A ) \\| = \\| A \\| . \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} a _ n ( \\omega ) & = a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } ( 1 ) , \\\\ \\omega a ' _ n ( \\omega ) & = \\mathrm { o r d } ( \\omega ) ^ 2 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , x z - y ^ 2 - z ^ 2 + \\frac { 3 x y + y z - x ^ 2 } { 2 } + f ( x , y , z ) } , \\\\ \\omega ^ 2 a '' _ n ( \\omega ) & = \\mathrm { o r d } ( \\omega ) ^ 4 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , Q _ 1 } + \\mathrm { o r d } ( \\omega ) ^ 2 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , Q _ 2 } , \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} m _ { \\xi } = L ^ { K } b _ { \\xi } , \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} R _ { 2 } ^ { s } ( \\phi _ { 2 } , R _ { E } ) \\ ! = \\ ! [ C _ { 2 } - R _ { E } ] ^ { + } = [ \\log _ { 2 } ( 1 \\ ! + \\ ! \\phi _ { 2 } \\tilde { \\rho _ { 2 } } ) \\ ! - \\ ! R _ { E } ] ^ { + } , \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} \\sum _ { \\omega \\in Z \\left ( \\zeta _ L \\right ) \\atop \\omega \\neq \\omega _ 0 } \\frac { \\Re s - 1 } { | s - \\omega | ^ 2 } & \\leq \\sum _ { \\omega \\in Z \\left ( \\zeta _ L \\right ) } \\Re \\frac { 1 } { s - \\omega } \\\\ & = \\frac { 1 } { 2 } \\log d _ L + \\Re \\left ( \\frac { 1 } { s } + \\frac { 1 } { s - 1 } \\right ) + \\Re \\frac { \\gamma _ L ^ { \\prime } } { \\gamma _ L } ( s ) + \\Re \\frac { \\zeta _ L ^ { \\prime } } { \\zeta _ L } ( s ) . \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{gather*} F _ 0 ( z ) = { } _ 2 F _ 1 \\biggl ( \\begin{matrix} \\frac 1 2 , \\ , \\frac 1 2 \\\\ 1 \\end{matrix} \\biggm | z \\biggr ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( \\frac 1 2 ) _ k ^ 2 } { k ! ^ 2 } z ^ k . \\end{gather*}"}
-{"id": "2808.png", "formula": "\\begin{align*} b ( k , \\theta ) \\leq M ( k , t , c ) = 2 \\left ( \\sum _ { i = 0 } ^ { t - 4 } ( k - 1 ) ^ { i } + \\frac { ( k - 1 ) ^ { t - 3 } } { c } + \\frac { ( k - 1 ) ^ { t - 2 } } { c } \\right ) . \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} - \\hat { \\Delta } ' d \\varpi ^ { i + k } + \\varpi ^ l Z = ( \\varpi ^ { i + j + 1 } \\beta ' - \\varpi ^ { r - i } e ^ 2 ) / d . \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} M ( t ) = \\int _ 0 ^ t \\big ( \\xi ( x ( s ) ) - z ( s ) \\big ) ^ T \\Big ( A \\big ( x ( s ) \\big ) - \\widetilde { \\sigma } \\big ( z ( s ) \\big ) \\Big ) \\ , d \\widetilde { w } _ s \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} d _ { \\mathbb { R } ^ n \\times C } ( p , x ) = d _ C ( x ) ( p , x ) \\in \\mathbb { R } ^ n \\times C . \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { m ^ + } \\Big ( \\sup _ { H _ k } f _ k \\Big ) ^ { c _ k } \\times \\prod _ { k = m ^ + + 1 } ^ m \\Big ( \\sup _ { ( ( t + 1 ) F _ k ) \\times G _ k } f _ k ^ { - 1 } \\Big ) ^ { - c _ k } . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} \\mathbb { A } ^ \\alpha ( x ) = a \\begin{pmatrix} 1 \\\\ \\mp i \\\\ 0 \\\\ 0 \\end{pmatrix} e ^ { i ( x ^ 3 + x ^ 4 ) } , \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} \\widetilde { X } ^ { ( u ) } _ { t } \\ , = \\ , \\mathbb E ^ { M } \\Big [ \\int ^ { t } _ { 0 } \\sum _ { k = 0 } ^ { \\infty } { \\bf 1 } _ { \\{ M ( t - s ) \\ , = \\ , k \\} } { \\mathrm d } W _ { s , k } \\vert M ( 0 ) \\ , = \\ , 0 \\Big ] \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} \\phi _ { \\lambda } ( r ) = \\left ( \\lambda ^ { - \\frac { \\theta } { \\theta - 1 } } r ^ { - \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } } + \\left ( 1 - \\lambda ^ { - \\frac { \\theta } { \\theta - 1 } } \\right ) R ^ { - \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } } \\right ) ^ { - \\frac { p } { n - p } \\frac { \\theta - 1 } { \\theta } } . \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} I _ V ( u ) - \\lambda _ 1 \\int _ { \\Omega } u ^ 2 \\geq 0 \\ ; \\ ; \\forall u \\in H ( V , \\Omega ) \\hbox { a n d } I _ V ( \\Phi ) - \\lambda _ 1 \\int _ { \\Omega } \\Phi ^ 2 = 0 . \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} \\Delta \\rho _ \\varphi ( x ) = \\rho _ { \\boldsymbol \\varphi } ( x ) \\left [ \\frac 1 2 g ^ { \\mu \\nu } ( \\boldsymbol \\varphi ( x ) ) \\frac { \\partial g _ { \\mu \\nu } } { \\partial y ^ \\gamma } ( \\boldsymbol \\varphi ( x ) ) \\Delta \\varphi ^ \\gamma + \\frac { \\partial \\Delta \\varphi ^ \\mu } { \\partial y ^ \\mu } ( x ) \\right ] . \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} \\psi ( \\zeta \\gamma ) = e ( a \\gamma / \\ell q ) . \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} \\varphi ( x , E ) = | \\varphi ( x , E ) | e ^ { i \\gamma ( x , E ) } . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{gather*} R _ k : C F ( H ^ { ( k ) } , J ) \\rightarrow C F ( H ^ { ( k ) } , J ^ { + \\frac { 1 } { k } } ) \\\\ R _ k \\bigg ( \\sum _ { z \\in \\widetilde { P } ( H ^ { ( k ) } ) } a _ z \\cdot z \\bigg ) = \\sum _ { z \\in \\widetilde { P } ( H ^ { ( k ) } ) } a _ z \\cdot R _ k ( z ) \\end{gather*}"}
-{"id": "8160.png", "formula": "\\begin{align*} \\langle d \\Phi _ f ( \\mathbf { \\hat N } ) , d \\Phi _ f ( \\partial _ { x ^ i } ) \\rangle _ { g ^ { ( 4 ) } } = \\langle \\mathbf { \\hat N } , \\partial _ { x ^ i } \\rangle _ { \\Phi ^ * g ^ { ( 4 ) } } = 0 , ~ \\forall i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} \\partial _ l \\mathbf { B } ^ i = \\sum _ { j = 1 } ^ { m } \\left ( \\frac { \\Omega _ { j R } ^ i ( \\cdot ) } { | \\cdot | ^ { d } } \\right ) \\star \\mu _ { j R } ^ l ~ ~ \\mathcal { D } ' ( B _ { 2 R } ) , \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\varphi ( K x H ) = \\begin{cases} \\frac { \\rho _ 1 ( x ) } { \\rho _ 2 ( x ) } & \\ x \\in N \\\\ 0 & \\ x \\not \\in N \\end{cases} , \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} \\mathcal { L } _ { s , c } ( m ) = \\frac { \\zeta _ l ( 1 ) } { l ^ { \\frac { n _ l } { 2 } - q - \\abs { r } - k } } \\sum _ { \\substack { \\xi \\mod l ^ c , \\\\ \\xi } } \\frac { \\tau ( \\xi ) ^ 2 } { l ^ c } \\xi ( b ^ 2 \\overline { m } ) \\hat { f } _ s ( \\xi ) . \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} \\sup _ { \\pi \\in \\Pi } \\left | E _ { \\mu } ^ \\pi \\left ( \\sum _ { n = 1 } ^ N \\beta ^ { n - 1 } r _ i ( s _ n , a _ n ) \\right ) - E _ { \\mu } ^ \\pi \\left ( \\sum _ { n = 1 } ^ N \\beta ^ { n - 1 } r _ i ^ { l } ( s _ n , a _ n ) \\right ) \\right | \\to 0 \\quad \\mbox { a s } l \\to \\infty . \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} G = G L _ n \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} \\widehat { K _ 1 } = \\frac { 1 } { 2 } \\Big ( 2 i p \\sqrt { D _ q ^ 2 + { \\lambda _ 1 ^ 2 } } + ( - \\partial ^ 2 _ p + p ^ 2 - 1 ) \\Big ) . \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} \\P ( N _ { Y } = k ) & = \\prod _ { i = 0 } ^ { k - 1 } \\frac { f ( i ) } { 1 + f ( i ) } - \\prod _ { i = 0 } ^ { k } \\frac { f ( i ) } { 1 + f ( i ) } \\\\ & = \\left ( 1 - \\frac { f ( k ) } { 1 + f ( k ) } \\right ) \\prod _ { i = 0 } ^ { k - 1 } \\frac { f ( i ) } { 1 + f ( i ) } = \\frac { 1 } { 1 + f ( k ) } \\prod _ { i = 0 } ^ { k - 1 } \\frac { f ( i ) } { 1 + f ( i ) } = \\mu _ k . \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} \\dim \\big ( V \\cap ( \\ker B _ + ) ^ { \\perp _ { \\mathcal Q } } \\big ) = \\sum _ { i = 1 } ^ { m ^ + } \\dim B _ i V , \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} & d \\nu ^ 1 _ { k , \\sum _ { i = d - k + 1 } ^ { d } x _ i } ( y _ { d - k } , . . . , y _ 1 ) = d | D f ^ { e _ { d - k } } _ { \\sum _ { i = 1 } ^ { d - k - 1 } y _ i + \\sum _ { i = d - k + 1 } ^ { d } x _ i } | ( y _ { d - k } ) d \\mathcal { H } ^ 1 ( y _ { d - k - 1 } ) . . . d \\mathcal { H } ^ 1 ( y _ 1 ) , \\\\ & d \\nu ^ 2 _ { k , \\sum _ { i = d - k + 1 } ^ { d } x _ i } ( y _ { d - k } , . . . , y _ 1 ) = \\mathbf { 1 } _ { | \\sum _ { i = 1 } ^ { d - k } y _ { 0 i } - \\sum _ { i = 1 } ^ { d - k } y _ i | \\leq 2 \\varepsilon } d \\nu ^ 1 _ { k , \\sum _ { i = d - k + 1 } ^ { d } x _ i } ( y _ { d - k } , . . . , y _ 1 ) . \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} b _ n ^ { \\prime } = \\frac { O ( 1 ) } { 1 + n } \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} f \\circ T ^ n = \\sum _ { v \\in I } \\alpha ( v ) \\phi v \\circ T ^ n \\sim \\sum _ { v \\in I } \\alpha ( v ) ( \\phi v ) _ n = f _ n . \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} \\tilde { S } ( x + V ) = S x , \\tilde { T } ( x + V ) = T x \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 2 ^ b } \\sum _ { k \\neq j } ^ { 2 ^ b } \\int _ { a ^ - _ j } ^ { a ^ + _ j } \\int _ { a ^ - _ k } ^ { a ^ + _ k } P ( y _ { B } , y _ { C } ) d y _ B d y _ C \\leq \\eta . \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} t _ \\infty ( y ) h & = \\lim _ { \\lambda \\in \\Lambda } \\rho _ \\infty ( e _ \\lambda ) t _ \\infty ( y ) h = \\lim _ { \\lambda \\in \\Lambda } t _ \\infty ( e _ \\lambda \\cdot y ) h \\\\ & = \\lim _ { \\lambda \\in \\Lambda } t _ \\infty ( \\gamma _ F ( e _ \\lambda ) \\cdot y ) h = \\lim _ { \\lambda \\in \\Lambda } \\rho _ \\infty ( \\gamma _ F ( e _ \\lambda ) ) t _ \\infty ( y ) h \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ m a _ i - \\prod _ { i = 1 } ^ m b _ i = \\sum _ { i = 1 } ^ m \\prod _ { 1 \\leq j < i } a _ j ( a _ i - b _ i ) \\prod _ { i < j \\leq m } b _ j . \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} w = p _ 0 B _ 1 C _ 1 p _ 1 B _ 2 C _ 2 \\cdots B _ k C _ k p _ k ( k \\in \\mathbb { N } ) , \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} J _ w = \\sum _ { i } ^ w N ( i ) T _ i . \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} & \\P ( Z _ n \\in J n ) \\\\ & \\ge \\P ( \\phi ( X _ n ) \\in J n , \\ \\tau ^ X = \\infty ) > 0 . \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} \\begin{cases} u _ t = - ( - \\Delta ) ^ { s } u + u ^ { \\frac { n + 2 s } { n - 2 s } } \\Omega \\times ( 0 , \\infty ) , \\\\ u = 0 ( \\mathbb { R } ^ n \\setminus \\Omega ) \\times ( 0 , \\infty ) , \\\\ u ( \\cdot , 0 ) = u _ 0 \\mathbb { R } ^ n , \\end{cases} \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} \\Sigma = \\begin{bmatrix} \\sigma _ 1 & 0 & \\ldots & 0 \\\\ 0 & \\sigma _ 2 & \\ldots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\ldots & \\sigma _ m \\\\ 0 & 0 & \\ldots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\end{bmatrix} \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} p ' ( x ) & = \\frac { B ( x ) } { C ( x ) } \\qquad \\mbox { w i t h } B ( x ) = \\int _ 0 ^ x S ( y ) \\ , \\d y . \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ r \\exp \\left ( p _ i ( ( 1 + \\delta ) ^ { \\alpha _ i } - 1 ) \\right ) & \\leq \\prod _ { i = 1 } ^ r \\exp \\left ( p _ i \\delta \\alpha _ i \\right ) \\\\ & = \\exp \\left ( \\delta \\cdot \\mathbb { E } ( \\Xi ) \\right ) \\leq e ^ { \\delta s } , \\\\ \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} Y ^ 2 ( M , g ) : = \\inf _ { h \\in [ g ] } \\lambda _ 2 ( h ) v o l ( M , h ) ^ { \\frac { 2 } { n } } , \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ n w _ 0 ^ + & = - a _ 0 \\lambda \\mu _ \\lambda , \\\\ \\partial _ n w _ 0 ^ - & = a _ 0 \\lambda \\mu _ \\lambda , \\end{aligned} \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} \\underset { \\mathbf { x } \\in \\mathbb { R } ^ { n p } } { } \\ , \\ , f ( \\mathbf { x } ) + \\frac { \\rho ( 1 - \\eta ) } { 4 } \\| \\mathbf { x } \\| ^ 2 _ { \\mathbf { V } \\otimes \\mathbf { I } _ p } \\ , \\ , \\ , \\ , \\ , \\sqrt { \\eta } \\sqrt { \\mathbf { V } } \\mathbf { x } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} \\| f \\| _ { p , \\tau } : = \\sup _ { z \\in { \\mathbb C } ^ n } | f ( z ) | \\exp ( - \\tau | z | ^ p ) < \\infty . \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} m _ 1 = n _ 1 < m _ 2 . \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\ell ( \\gamma ) = \\ell ( g ) = \\log N _ G ( g ) , \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} A _ 0 = I _ { ( \\ell + 1 ) n } , A _ 1 = I _ { \\ell + 1 } \\otimes A , A _ 2 = I _ { \\ell + 1 } \\otimes ( J _ n - A - I _ n ) , \\tilde { L } = A _ 3 - A _ 4 . \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} V ( \\vec { \\mu } ) : = V ( \\mu _ { 1 } ) \\otimes \\cdots \\otimes V ( \\mu _ { a } ) , \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\| { \\Delta } _ j b \\| ^ 2 _ { L ^ 2 } \\leq & \\| [ { \\Delta } _ j , \\ , u \\cdot \\nabla ] b \\| _ { L ^ 2 } \\| { \\Delta } _ j b \\| _ { L ^ 2 } + \\| { \\Delta } _ j ( b \\cdot \\nabla u ) \\| _ { L ^ 2 } \\| { \\Delta } _ j b \\| _ { L ^ 2 } \\| { \\Delta } _ j b \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} \\rho _ N = \\frac { U _ { N - 1 } ( Q / 2 ) } { U _ { N } ( Q / 2 ) } . \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} \\rho ( \\lambda ) = \\rho ( \\lambda , a , b ) = \\frac { \\int _ 0 ^ { \\min \\{ b , \\lambda \\} } \\frac { d t } { \\sqrt { ( \\lambda - t ) ( b - t ) ( a - t ) } } } { 2 \\int ^ a _ { \\max \\{ b , \\lambda \\} } \\frac { d t } { \\sqrt { ( \\lambda - t ) ( b - t ) ( a - t ) } } } \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} \\bullet \\ ; & ( B _ 0 , B _ + ) \\colon H \\to H _ 0 \\times \\cdots \\times H _ { m ^ + } \\mbox { i s b i j e c t i v e } , \\\\ \\bullet \\ ; & \\ker B _ + \\subset \\ker B _ { m + 1 } , \\\\ \\bullet \\ ; & \\mbox { f o r a l l } x \\in H , \\mathcal Q ( x ) = \\mathcal Q _ + ( B _ 0 x ) - \\mathcal Q _ - ( B _ { m + 1 } x ) . \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} f ( t , x , \\omega ) ~ = ~ \\begin{cases} f _ { l } ( \\omega ) & ~ x \\le \\gamma \\left ( t \\right ) , \\\\ f _ { r } ( \\omega ) & ~ x > \\gamma \\left ( t \\right ) , \\end{cases} \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } U ^ { \\theta } ( k , t + \\Delta t ) = U ( k , s _ { 0 } + 1 ) - \\frac { ( t - s _ { 0 } \\Delta t ) ^ { 2 } } { 2 \\theta } \\\\ U ^ { \\theta } ( k ' , t ) \\geq U ( k ' , s _ { 0 } ) - \\frac { ( t - s _ { 0 } \\Delta t ) ^ { 2 } } { 2 \\theta } & \\ , \\ , k ' \\in J _ { i } \\end{array} \\right . \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\tilde { Q } _ 1 ( F ) : = \\{ \\left ( \\begin{array} { c c c c } \\alpha & * & * & * \\\\ & a & * & b \\\\ & & \\beta & \\\\ & c & * & d \\end{array} \\right ) \\vert a d - b c = \\alpha \\beta \\in \\mathbb { G } _ { m , F } ( F ) \\} \\cap \\mbox { G S p } _ { 4 , F } ( F ) \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} ( f ^ { \\pm } ) ^ { - 1 } ( 0 ) \\subset \\{ d w ^ { \\pm } = 0 \\} . \\end{align*}"}
-{"id": "9924.png", "formula": "\\begin{align*} \\check { \\chi } _ { H _ { { n + 1 } } } ( z ) = \\left \\{ \\begin{matrix} 0 & z \\in H _ { { n + 1 } } , \\\\ 1 & . \\end{matrix} \\right . \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} \\phi '' ( x ) = \\beta ( \\beta + 1 ) x ^ { - \\beta - 2 } \\geq d k ^ { \\frac { \\beta + 2 } { \\beta + 1 } } . \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } p \\\\ q \\end{array} \\right ) _ t = A \\left ( \\begin{array} { c } \\delta H / \\delta p \\\\ \\delta H / \\delta q \\end{array} \\right ) = B \\left ( \\begin{array} { c } \\delta G / \\delta p \\\\ \\delta G / \\delta q \\end{array} \\right ) , \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} \\| Q \\| _ Z : = \\| \\langle x \\rangle ^ { \\frac { 5 } { 2 } } Q \\| _ { L _ x ^ 1 L _ t ^ \\infty \\cap L _ { t , x } ^ \\infty } + \\| \\partial _ x Q \\| _ { L _ { t , x } ^ \\infty \\cap L _ t ^ \\infty L _ x ^ 2 } \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} & \\int \\limits _ { \\mathbb { R } _ + } \\left | | v _ { j , n } | ^ 2 \\log | v _ { j , n } | ^ 2 - | v _ { j } | ^ 2 \\log | v _ { j } | ^ 2 \\right | d x \\leq \\int \\limits _ { \\{ | v _ { j , n } | \\geq | v _ { j } | \\} } \\left ( 1 + | \\log | v _ { j , n } | ^ 2 | \\right ) \\left | | v _ { j , n } | ^ 2 - | v _ { j } | ^ 2 \\right | d x \\\\ & + \\int \\limits _ { \\{ | v _ { j , n } | \\leq | v _ { j } | \\} } \\left ( 1 + | \\log | v _ { j } | ^ 2 | \\right ) \\left | | v _ { j , n } | ^ 2 - | v _ { j } | ^ 2 \\right | d x . \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} - \\Delta Q _ { } - c | x | ^ { - 2 } Q _ { } + Q _ { } = | Q _ { } | ^ { \\frac { 4 } { d } } Q _ { } . \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} & M _ G \\left ( 1 + m ( 1 + x ) ( 1 + y ) ( 1 + z ) \\right ) = 8 m + 1 , \\\\ & M _ G \\left ( 2 + m ( 1 + x ) ( 1 + y ) ( 1 + z ) \\right ) = 2 ^ 8 ( 4 m + 1 ) , \\\\ & M _ G \\left ( 3 + z + k ( 1 + x ) ( 1 + y ) ( 1 + z ) \\right ) = 2 ^ { 1 2 } ( 2 k + 1 ) , \\\\ & M _ G \\left ( x + y + z - 3 + ( 1 - x ) ( 1 - y ) ( 1 - z ) + k ( 1 + x ) ( 1 + y ) ( 1 + z ) \\right ) = 2 ^ { 1 2 } ( 2 k ) , \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ { \\infty } p _ { _ 1 } ( y ) \\log \\frac { 1 } { p _ { _ 1 } ( y ) } d y + \\eta _ 0 \\left [ \\mathcal { D } ( p _ { _ 0 } | | p _ { _ 1 } ) - 2 \\epsilon ^ 2 \\right ] \\\\ & + \\eta _ 1 \\left [ \\int _ { - \\infty } ^ { \\infty } p _ { _ 1 } ( y ) d y - 1 \\right ] + \\eta _ 2 \\left [ \\int _ { - \\infty } ^ { \\infty } y ^ 2 p _ { _ 1 } ( y ) d y - P _ y \\right ] \\\\ & = \\int _ { - \\infty } ^ { \\infty } \\overline { \\mathcal { L } } ( y , p _ { _ 1 } ( y ) ) d y - c , \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} P _ 1 \\tau _ a ( v _ 1 ) = \\tau _ b ( P _ 1 v _ 1 ) = \\tau _ b ( P _ 1 v _ 2 ) = P _ 1 \\tau _ a ( v _ 1 ) . \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} L = \\sum _ { a , b , t } \\sum _ { \\substack { m \\in \\Z , \\\\ m \\equiv \\frac { b } { a } \\alpha _ { \\chi _ l } \\ ( l ^ t ) } } \\lambda _ { \\pi } ( m ) F ( \\frac { m } { M } ) , \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} \\| f \\| _ \\infty : = \\sup _ { v \\in V } | f ( v ) | . \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{gather*} \\tau = \\tau ( z ) = - \\frac { i F _ 1 ( z ) } { 2 \\pi F _ 0 ( z ) } = \\frac { i F _ 0 ( 1 - z ) } { 2 \\sin \\pi r \\ , F _ 0 ( z ) } \\end{gather*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\mathbf { z } [ i ] = \\mathbf { x } [ i ] + \\mathbf { n } _ b [ i ] . \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} E = F \\times G / \\left \\langle \\bigl ( f ( x ) , \\iota ( x ) ^ { - 1 } \\bigr ) \\mid x \\in H \\right \\rangle . \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} u _ { n } ^ { \\prime \\prime } ( x ) + \\lambda _ { n , 4 } ( x ) u _ { n } ( x ) = 0 , \\ \\ \\ u _ { n } ( x ) = \\frac { e ^ { - x / 2 } x ^ { ( \\alpha + 1 ) / 2 } } { x + \\alpha } \\widehat { L } _ { n } ^ { ( \\alpha ) } ( x ) , \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} & { \\rm S I R } _ { g , n , \\ell } = \\frac { d _ { 0 } ^ { - \\alpha } | h _ { 0 } | ^ 2 } { \\sum _ { b \\in \\Phi _ b \\setminus \\{ B _ { g , n , \\ell , 0 } \\} } d _ { b } ^ { - \\alpha } | h _ { b } | ^ 2 } , g \\in \\{ { \\rm S V C } , { \\rm D A S H } \\} . \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} ( U \\Delta ) ^ { * } v _ { i } = \\sigma _ { i } w _ { i } . \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} \\int _ X f \\ , d \\mu = \\int _ 0 ^ { + \\infty } \\mu ( f > t ) d t . \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} \\mu ^ { \\psi } _ { \\Lambda } : = \\frac { 1 } { Z _ { \\Lambda } ( \\psi _ { \\Lambda ^ c } ) } e ^ { - H _ { \\Lambda } ( \\phi \\vee \\psi ) } \\prod _ { j \\in \\Lambda } d \\phi _ j . \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} J _ i ( \\pi ) = J _ i ^ - ( \\pi ) + J _ i ^ + ( \\pi ) . \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} u _ { r r } + \\frac { m - 1 } { r } u _ r + \\frac { 1 } { r ^ 2 } \\Delta _ \\theta u - \\frac { r } { 2 } \\frac { \\partial u } { \\partial r } = - \\lambda u . \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} B = \\frac { Y ( X _ \\alpha - X _ \\beta ) } { ( d - 1 ) X _ \\alpha X _ \\beta + Y X _ \\alpha } . \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} \\Upsilon ^ n : = \\left \\{ \\mathbf { x } \\in \\R ^ n \\ | \\ ( x _ 1 + x _ 2 ) ^ 2 \\geq 2 \\sum \\limits _ { i = 3 } ^ { n } x _ i ^ 2 , \\ x _ 1 \\geq x _ 2 , \\ x _ 1 + x _ 2 \\geq 0 \\right \\} . \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} f = - ( X + A + \\Phi ) u \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ { n } \\binom { n } j ( - 1 ) ^ { j - 1 } \\frac { a _ j } { j ^ k } = \\sum _ { j = 1 } ^ { n } \\sum _ { i = j } ^ { n } \\binom { i - 1 } { j - 1 } ( - 1 ) ^ { j - 1 } \\frac { a _ j } { j ^ k } = \\sum _ { i = 1 } ^ { n } \\frac 1 i \\sum _ { j = 1 } ^ { i } \\binom { i } { j } ( - 1 ) ^ { j - 1 } \\frac { a _ j } { j ^ { k - 1 } } . \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} p ( x ) = \\frac { 1 } { \\omega \\sqrt { 2 \\pi } } e ^ { - \\frac { ( x - \\mu ) ^ 2 } { 2 \\omega ^ 2 } } \\left [ 1 + \\left ( \\frac { \\theta ( x - \\mu ) } { \\omega \\sqrt { 2 } } \\right ) \\right ] , \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} \\chi _ { \\ell } ( M \\square K _ { 1 , s } ) = \\begin{cases} k & s < P _ { \\ell } ( M , k ) \\\\ k + 1 & s \\geq P _ { \\ell } ( M , k ) . \\end{cases} \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} f \\sim g _ 0 = \\sum _ { v \\in I \\setminus \\{ v _ 0 \\} } \\alpha ( v ) \\phi v + \\alpha ( v _ 0 ) \\beta _ 0 ( b ) \\phi b + \\sum _ { y \\in J _ 0 } \\alpha ( v _ 0 ) \\beta _ 0 ( y ) \\phi y . \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} \\theta _ { 1 2 } ( t ) = \\theta \\left ( \\begin{bmatrix} 0 & t & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} \\right ) , \\theta _ { 1 3 } ( t ) = \\theta \\left ( \\begin{bmatrix} 0 & 0 & t \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} \\right ) , \\theta _ { 2 3 } ( t ) = \\theta \\left ( \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & t \\\\ 0 & 0 & 0 \\end{bmatrix} \\right ) \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} d \\hat { Y } _ { s } = - \\left ( \\Pi _ { 1 } ( s ) + \\Pi _ { 2 } ( s ) \\right ) d s + \\hat { Z } _ { s } d B _ { s } , \\ \\hat { Y } _ { T } = 0 , \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} N _ { m } & = N _ { m - 1 } + N _ { m - 3 } \\\\ & = ( N _ { m - 3 } + 2 N _ { m - 5 } + N _ { m - 7 } ) + ( N _ { m - 5 } + 2 N _ { m - 7 } + N _ { m - 9 } ) \\\\ & = ( N _ { m - 3 } + N _ { m - 5 } ) + 2 ( N _ { m - 5 } + N _ { m - 7 } ) + ( N _ { m - 7 } + N _ { m - 9 } ) \\\\ & = N _ { m - 2 } + 2 N _ { m - 4 } + N _ { m - 6 } . \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} \\left \\langle \\frac { 1 } { ( \\zeta q ^ { \\l _ N } ; q ) _ { \\infty } } \\right \\rangle = \\det \\left ( 1 - f K \\right ) _ { L ^ 2 ( \\R ) } . \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{gather*} \\mathrm { c s } ( \\nabla ) = \\operatorname { T r } _ s ( \\exp ( i R _ { \\nabla } ) ) = \\sum \\frac { i ^ q } { q ! } \\mathrm { c s } ^ q ( \\nabla ) \\in \\Omega ( A ; \\C ) . \\end{gather*}"}
-{"id": "5062.png", "formula": "\\begin{align*} \\mu \\left ( \\bigcup _ { U \\in \\mathcal U _ l } U \\right ) = 1 - o \\left ( \\frac 1 { k _ l } \\right ) , \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 0 , n ) = \\Phi ( n ) = \\int _ { 0 } ^ { \\infty } \\frac { \\cos ( \\pi n x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x , \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - t } + 2 \\sum _ { n \\geq 1 } \\frac { t \\alpha _ n ( t , 1 ) } { ( 1 - t ) ^ { n + 1 } } \\frac { x ^ n } { n ! } = 1 + t \\frac { e ^ x } { 2 - ( 1 + t ) e ^ x } . \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} A _ \\lambda - ( 1 - p ^ k - \\hat \\alpha k ) & = p ^ k + \\frac 1 \\lambda \\log \\frac 1 { p ^ k } - \\frac 1 \\lambda ( 1 + \\log \\lambda ) \\\\ [ 1 e m ] & = h ( p ^ k ) - h ( \\lambda ^ { - 1 } ) > 0 . \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} & \\lambda _ 2 ( v ) = \\lambda ^ { \\frac { r } { 2 } } _ 2 ( x ) \\lambda ^ { \\frac { r } { 2 } } _ 2 ( s ) \\lambda ^ { \\frac { 1 - r } { 2 } } _ 4 ( x ) \\lambda ^ { \\frac { 1 - r } { 2 } } _ 4 ( s ) \\\\ & \\lambda _ 4 ( v ) = \\lambda ^ { \\frac { 1 - r } { 2 } } _ 2 ( x ) \\lambda ^ { \\frac { 1 - r } { 2 } } _ 2 ( s ) \\lambda ^ { \\frac { r } { 2 } } _ 4 ( x ) \\lambda ^ { \\frac { r } { 2 } } _ 4 ( s ) \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} W _ \\alpha \\overset { \\textrm { d e f } } = \\begin{bmatrix} S _ { 1 , \\alpha } & M _ { \\psi _ \\alpha } \\\\ ( M _ { \\psi _ \\alpha } ) ^ \\circ & S _ { 2 , \\alpha } \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} \\langle F ^ { \\ast } \\lambda , \\beta _ { l } ^ { \\vee } \\rangle = \\langle \\lambda , F _ { \\ast } \\beta _ { l } ^ { \\vee } \\rangle = \\langle \\lambda , \\alpha _ { u _ l } ^ { \\vee } \\rangle \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} D = \\frac { \\det A } { \\prod _ { i = k } ^ m ( \\det A _ k ) ^ { c _ k } } \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} \\frac { ( \\Delta t ) ^ { 2 } ( s _ { 1 } - s ) ^ { 2 } } { 2 \\theta } \\leq \\frac { ( t + \\Delta t - s \\Delta t ) ^ { 2 } } { 2 \\theta } & = U ( k , s ) - U ^ { \\theta } ( k , t + \\Delta t ) \\\\ & \\leq U ( k , s ) - U ( k , s _ { 1 } ) + \\frac { ( \\Delta t ) ^ { 2 } } { 2 \\theta } \\\\ & \\leq L _ { 2 } \\tilde { L } _ { c } \\Delta t | s - s _ { 1 } | + \\frac { ( \\Delta t ) ^ { 2 } } { 2 \\theta } . \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} Q ( z ) = & \\frac { 1 } { Z } \\int _ { \\Sigma _ z } e ^ { - \\beta V ( x ) } \\Big [ \\det ( \\nabla \\xi \\nabla \\xi ^ T ) ( x ) \\Big ] ^ { - \\frac { 1 } { 2 } } \\ , d \\nu _ z ( x ) \\\\ = & \\frac { 1 } { Z } \\int _ { \\mathbb { R } ^ n } \\delta ( \\xi ( x ) - z ) \\ , e ^ { - \\beta V ( x ) } \\ , d x \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} f ( \\tilde { x } _ k ) - & f ( x ^ \\star ) + \\langle \\lambda , r ( \\tilde { x } _ k ) \\rangle \\\\ & \\leq \\langle \\lambda - \\lambda _ k , r ( \\tilde { x } _ k ) \\rangle - \\frac { \\rho } { 2 } \\| r ( \\tilde { x } _ k ) \\| ^ 2 + \\epsilon _ { i n } ^ k . \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} A _ 3 ^ { \\operatorname { c o } } = \\frac { 1 } { 2 } \\Big ( C ^ { \\operatorname { c o } T } B ^ { \\operatorname { c o } } + B ^ { \\operatorname { c o } T } C ^ { \\operatorname { c o } } \\Big ) . \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} A \\ , x \\ , = \\ , y \\ , , \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} \\xi = u t _ { 1 } ^ { e _ { 1 } } t _ { 2 } ^ { e _ { 2 } } + x ( t _ { 1 } , t _ { 2 } ) \\end{align*}"}
-{"id": "9982.png", "formula": "\\begin{align*} \\mathbb P ( R ( \\tau ) \\in [ x , x + \\Delta x ] ) = u ( x , \\tau ) \\Delta x + \\frac 1 2 \\partial _ x u ( x , \\tau ) ( \\Delta x ) ^ 2 + o ( ( \\Delta x ) ^ 2 ) . \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} \\oint \\frac { z ^ { m - j } d z } { ( 1 - e ^ { - z } ) ^ { m + 1 - i } } & = \\oint \\frac { z ^ { m - j } ( e ^ z ) ^ { m - i } d e ^ z } { ( e ^ z - 1 ) ^ { m + 1 - i } } \\\\ & = \\oint \\frac { ( \\ln ( 1 + x ) ) ^ { m - j } ( x + 1 ) ^ { m - i } d x } { x ^ { m + 1 - i } } \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} I _ { \\mu } ( u ) & \\geq \\frac { 1 } { 2 } | | u | | _ { H ^ 1 } ^ 2 + \\frac { 1 } { 1 6 \\pi } \\left ( | | u | | ^ 2 _ E - | | u | | _ { H ^ 1 } ^ 2 \\right ) ^ 2 - C \\mu | | u | | _ { H ^ 1 } ^ { p + 1 } \\\\ & = \\frac { 1 } { 2 } | | u | | _ { H ^ 1 } ^ 2 + \\frac { 1 } { 4 \\pi } \\left ( \\frac { 1 } { 4 } | | u | | ^ 4 _ E - \\frac { 1 } { 2 } | | u | | ^ 2 _ E | | u | | _ { H ^ 1 } ^ 2 + \\frac { 1 } { 4 } | | u | | _ { H ^ 1 } ^ 4 \\right ) - C \\mu | | u | | _ { H ^ 1 } ^ { p + 1 } . \\\\ \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} \\tfrac { d } { d t } g _ t = - { \\rm M } ( g _ t ) \\ , , g _ { | t = 0 } = g _ 0 \\ , , \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} S ( U , \\xi ) = ( n - 1 ) \\eta ( U ) , \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} \\mathrm { i f } \\ ; \\Sigma \\le P , \\ ; \\mathrm { t h e n } & \\mathbb E \\left ( \\prod _ { i = 1 } ^ m f _ i ( X _ i ) \\right ) \\le \\prod _ { i = 1 } ^ m \\mathbb E \\left ( f _ i ( X _ i ) ^ { p _ i } \\right ) ^ { \\frac { 1 } { p _ i } } , \\\\ \\mathrm { i f } \\ ; \\Sigma \\ge P , \\ ; \\mathrm { t h e n } & \\mathbb E \\left ( \\prod _ { i = 1 } ^ m f _ i ( X _ i ) \\right ) \\ge \\prod _ { i = 1 } ^ m \\mathbb E \\left ( f _ i ( X _ i ) ^ { p _ i } \\right ) ^ { \\frac { 1 } { p _ i } } , \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} ( q r ^ { 1 1 } ) ^ 2 = - ( p r ^ 7 ) ^ { 3 } + h ( r s , r t ) , \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} \\Re ( s ) = 1 - \\frac { 1 } { 5 . 6 0 \\log { ( q \\max \\{ 1 , | \\Im ( s ) | \\} ) } } . \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} \\square _ p ^ n \\otimes \\square _ p ^ m = \\square _ p ^ { n + m } \\ . \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} \\int ^ { \\infty } _ { 0 } | h ' ( r ) | ^ { 2 } r ^ { Q + 1 - 2 \\beta } d r & = \\int ^ { \\infty } _ { 0 } | ( Q - 2 \\beta ) s ^ { \\frac { Q - 2 \\beta - 1 } { Q - 2 \\beta } } h ' ( s ) | ^ { 2 } s ^ { \\frac { Q - 2 \\beta + 1 } { Q - 2 \\beta } } \\frac { s ^ { \\frac { 1 } { Q - 2 \\beta } - 1 } d s } { Q - 2 \\beta } \\\\ & = ( Q - 2 \\beta ) \\int ^ { \\infty } _ { 0 } s ^ { 2 } | h ' ( s ) | ^ { 2 } d s . \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} C _ { w } = C ( E _ 1 , E _ 2 , \\cdots , E _ { N ( w ) } ) . \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} A _ { \\overline { l } , 0 } ( t , \\overline \\alpha ) = \\sum _ { h = 0 } ^ L ( L _ 0 - h ) ! \\sigma _ { L _ 0 - h } \\ ! \\left ( \\overline l , \\overline { \\alpha } \\right ) t ^ h . \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} [ \\mathfrak { M } W ] ( \\mu _ p , s ) = [ \\mathfrak { M } W _ { l } ] ( \\mu _ p , 0 ) s \\in \\C . \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} \\nu = r + e ^ { - \\alpha } P _ 1 \\nu . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} C _ { j _ k \\ldots j _ 1 } = \\int \\limits _ { [ t , T ] ^ k } K ( t _ 1 , \\ldots , t _ k ) \\prod _ { l = 1 } ^ { k } \\phi _ { j _ l } ( t _ l ) d t _ 1 \\ldots d t _ k \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} \\widetilde { \\mu } _ g \\big ( x + ( \\theta , \\mathbf { v } ) \\big ) = x + \\big ( \\theta , \\mathrm { d } _ { m _ x } \\widetilde { \\mu } _ g ( \\mathbf { v } ) \\big ) ( g \\in G _ x ) . \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} \\varphi _ \\pm : \\begin{pmatrix} x ^ 1 \\\\ x ^ 2 \\\\ x ^ 3 \\\\ x ^ 4 \\\\ \\end{pmatrix} \\mapsto \\begin{pmatrix} x ^ 1 \\\\ x ^ 2 \\\\ x ^ 3 \\\\ x ^ 4 \\\\ \\end{pmatrix} + a \\begin{pmatrix} \\cos ( x ^ 3 + x ^ 4 ) \\\\ \\pm \\sin ( x ^ 3 + x ^ 4 ) \\\\ 0 \\\\ 0 \\end{pmatrix} \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} \\mathcal { F } \\left \\{ e ^ { - \\pi t ^ 2 } | t | ^ \\mu \\right \\} ( \\nu ) & = \\left ( \\mathcal { F } \\left \\{ e ^ { - \\pi t ^ 2 } \\right \\} \\ast \\mathcal { F } \\left \\{ | t | ^ \\mu \\right \\} \\right ) ( \\nu ) \\\\ & = \\left ( e ^ { - \\pi \\nu ^ 2 } \\ast \\frac { - 2 \\sin \\left ( \\frac { \\pi \\mu } { 2 } \\right ) \\Gamma ( \\mu + 1 ) } { | 2 \\pi \\nu | ^ { 1 + \\mu } } \\right ) ( \\nu ) , \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} | \\psi ( x , t ) | \\lesssim \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\sum _ { j = 1 } ^ k \\frac { \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } + \\sigma } ( t ) } { 1 + | y _ j | ^ { a - 2 s } } + e ^ { - \\delta ( t - t _ 0 ) } \\| \\psi _ 0 \\| _ { L ^ { \\infty } ( \\mathbb { R } ^ n ) } , \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} 5 N ( A _ F \\log { N } + M _ F ) \\ge 1 5 \\left ( A _ F \\cdot \\frac { 1 - 1 5 M _ F } { 1 5 A _ F } + M _ F \\right ) = 1 . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} j _ \\lambda ( g h , z ) = j _ \\lambda ( g , h \\cdot z ) j _ \\lambda ( h , z ) . \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} K _ 0 ( m , n ) = \\delta _ { m n } , \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} \\mu ( e _ 1 ) = \\mathrm { d i a g } ( 1 , 1 , - 1 , - 1 ) , \\mu ( e _ 2 ) = \\mathrm { d i a g } ( 1 , - 1 , 1 , - 1 ) , \\mu ( e _ 3 ) = \\mathrm { d i a g } ( 1 , - 1 , - 1 , 1 ) . \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} \\lim \\limits _ { \\delta \\downarrow 0 } ( f _ \\delta , ( - \\mathcal { L } _ X ^ \\omega ) ^ { - 1 } f _ \\delta ) = \\int _ { 0 } ^ { \\infty } d t ( f , e ^ { t Q } f ) = ( f , - Q ^ { - 1 } f ) . \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} - \\Delta u + V ^ + u = \\alpha \\chi _ { K _ 1 } - \\chi _ { K _ 2 } , u \\in H ^ 1 _ 0 ( \\Omega _ * ) , \\ ; \\ ; \\alpha \\in \\R , \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} \\mathcal { F } _ { D , m } ( z , s ) : = B ( s ) \\Gamma \\biggl ( \\frac { s + 1 } { 2 } \\biggr ) \\sum _ { n | m } \\biggl ( \\frac { D } { n } \\biggr ) P _ { \\frac { 1 } { 2 } , \\frac { m ^ 2 D } { n ^ 2 } } \\biggl ( z , \\frac { s } { 2 } + \\frac { 1 } { 4 } \\biggr ) , \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} G ^ { \\epsilon } ( z ) = G ^ { \\epsilon } _ y ( x ) = \\left ( \\phi _ { \\epsilon } \\ast F _ y \\right ) ( x ) . \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} z ( x ) \\equiv 1 + ( - 6 ) \\left ( \\sum _ { k = 1 } ^ { t - 1 } ( - 2 ) ^ { k - 1 } y _ k \\right ) \\pmod { n } . \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} \\pi _ { t , k } ( \\varphi ) \\ , = \\ , \\frac { \\ , \\rho _ { t , k } ( \\varphi ) \\ , } { \\ , \\rho _ { t } ( { \\bf 1 } ) \\ , } \\ , ; k \\ , = \\ , 2 , \\ldots , n \\ , , \\ , \\ , 0 \\le t \\le T \\ , . \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} | | X _ B | | _ { \\tilde { r } , p - 1 , \\mathbf { a } } , | | X _ { Q _ 1 } | | _ { \\tilde { r } , p - 1 , \\mathbf { a } } , | | X _ { Q _ 2 } | | _ { \\tilde { r } , p - 1 , \\mathbf { a } } = O ( \\tilde { r } ^ 2 ) . \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} f ( a x + b ) = \\delta g ( x ) ( q u ) ^ 2 p ^ { 2 v + 2 \\epsilon } . \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} F _ \\ell ( x _ 1 , . . . , x _ \\ell ) = F ( x _ 1 , . . . , x _ \\ell ) - \\int F ( x _ 1 , . . . , x _ { \\ell - 1 } , z ) d \\mu ( z ) \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { S } \\int _ { { B _ { R } } } \\frac { | x | ^ { \\frac { n \\theta } { p ^ { * } } } | u ( x ) | ^ { { \\theta } } } { \\left [ ( q - 1 ) \\log _ { q } \\frac { R } { | x | } \\right ] ^ { \\theta } } \\frac { d x } { | x | ^ { { n } } } \\le \\int _ { { B _ { R } } } | x | ^ { \\frac { n \\theta } { p } } | \\nabla _ r u | ^ { \\theta } \\frac { d x } { | x | ^ { { n } } } \\end{aligned} \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} S ^ { R , R } _ { ( l , p ) , ( d - 1 , 1 ) } = - \\dim ^ R ( M _ { l , p } ) , \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} \\epsilon \\mu \\sum _ { k = 1 } ^ { \\infty } a _ { k 1 } \\tilde { P } ^ { k - 1 } \\tilde { P } ^ { ( 1 ) } e ^ { - \\alpha k } \\sum _ { l = k } ^ { \\infty } e ^ { - \\alpha l } \\tilde { P } ^ l r = \\epsilon \\mu \\sum _ { k = 1 } ^ { \\infty } a _ { k 1 } \\tilde { P } ^ { k - 1 } \\tilde { P } ^ { ( 1 ) } e ^ { - \\alpha k } ( I - e ^ { - \\alpha } \\tilde { P } ) ^ { - 1 } r , \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} \\xi ( [ x , y ] , \\psi ( t ) ) + \\xi ( [ t , x ] , \\psi ( y ) ) + \\xi ( [ y , t ] , \\psi ( x ) ) = 0 \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} y = \\varphi ( x ) . \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} & \\frac { U ( o , o \\vert z ) } { r } \\\\ & = \\frac { - \\left ( \\left ( m + \\lambda - 1 \\right ) - m U ( o , o | z ) - \\left ( d - 1 \\right ) z \\right ) + \\sqrt { \\left ( \\left ( m + \\lambda - 1 \\right ) - m U ( o , o | z ) - \\left ( d - 1 \\right ) z \\right ) ^ 2 + 4 \\lambda d z ^ 2 } } { 2 m } ; \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} \\mathbf { K } _ { i , n , l } ( r \\theta ) & = \\frac { \\varepsilon ^ { - d + 1 } } { \\rho ^ \\alpha } \\int _ { \\mathbb { R } ^ d } \\mathbf { 1 } _ { 2 ^ { i - l - 1 } \\rho < | r \\theta - y | \\leq 2 ^ { i - l } \\rho } \\mathbf { K } _ { n } ( r \\theta - y ) \\varphi _ { \\zeta \\rho } ( y ) \\frac { \\langle \\phi ^ { e , \\varepsilon } ( y / \\rho ) , \\eta ^ { \\kappa } _ { y _ { \\tau } } \\rangle } { | y | ^ { d - \\alpha } } \\\\ & ~ ~ ~ ~ \\times \\left ( \\psi ( 2 ^ { - i } \\rho ^ { - 1 } y ) - \\psi ( 2 ^ { - i + 1 } \\rho ^ { - 1 } y ) \\right ) d y . \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } v _ T ( y _ 0 ) : = v ^ * ( y _ 0 ) , \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} A _ 9 ( \\lambda , \\varepsilon ) \\leq 4 \\sum _ { i = 1 2 } ^ { 1 4 } A _ i ( \\lambda / 4 , \\varepsilon ) . \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} \\nu = \\gamma _ { \\rm h } \\frac { \\left ( \\overline { \\Delta x } \\right ) ^ { n _ { \\gamma } } } { \\Delta t _ e } , \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} \\langle v ( t ) , \\phi _ 0 \\rangle = 0 , \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} ( \\mathbb { A } ^ { - 1 } _ R ) _ { 2 1 } & = - \\left ( c + ( d - c a ^ { - 1 } b ) ^ { - 1 } [ c , d ] \\right ) ( \\Delta ' ) ^ { - 1 } \\\\ & = - ( d - c a ^ { - 1 } b ) ^ { - 1 } \\left ( ( d - c a ^ { - 1 } b ) c + [ c , d ] \\right ) ( \\Delta ' ) ^ { - 1 } \\\\ & = - ( d - c a ^ { - 1 } b ) ^ { - 1 } c \\left ( d - a ^ { - 1 } b c \\right ) ( \\Delta ' ) ^ { - 1 } \\\\ & = - ( d - c a ^ { - 1 } b ) ^ { - 1 } c a ^ { - 1 } \\\\ & = - ( a c ^ { - 1 } d - b ) ^ { - 1 } . \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} f ( U ^ K ) = T _ x M ^ K \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} C _ { K , S } : = \\frac { m _ K n + 1 } { \\left ( \\sqrt { 2 } \\right ) ^ { 3 n - 1 } } . \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} \\norm { T _ V ^ { - 1 } F } _ { L ^ 2 ( \\R , \\omega _ { a , b } ) } = \\norm { T ^ { - 1 } _ S \\circ \\left ( \\Phi ^ * \\right ) ^ { - 1 } F } _ { L ^ 2 ( \\R , \\omega _ { a , b } ) } = \\norm { \\left ( \\Phi ^ * \\right ) ^ { - 1 } F } _ { A ^ 2 ( S ( a , b ) ) } = \\norm { F } _ { A ^ 2 ( V ( a , b ) ) } . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ m c _ n \\theta _ { 1 2 } ( v _ n ) = 0 . \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} \\mu ^ p w ^ { p ^ n } X ^ { p ^ n } d w / w = \\mu w ^ { p ^ { n - 1 } } X ^ { p ^ { n - 1 } } d w / w \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} R ( \\xi , U ) V = g ( U , V ) \\xi - \\eta ( V ) U , \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} M ( x , y ) = V U _ { \\{ - x \\} , \\{ - y \\} } . \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} \\| f \\| _ { L _ { a } ^ { p } ( \\Omega ) } = \\left ( \\int _ { \\Omega } ( | \\R ^ { \\frac { a } { \\nu } } f ( x ) | ^ { p } + | f ( x ) | ^ { p } ) d x \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} \\frac { r _ { k _ 1 , k _ 2 } } { k _ 1 k _ 2 } = \\prod _ { p \\in { \\rm c d } ( k _ 1 , k _ 2 ) } ( 1 - p _ j ^ { - 2 } ) \\ , . \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} \\Delta ( x ) = \\sum _ { \\substack { 0 \\leq \\nu \\leq \\mu \\\\ i , j } } \\left ( x , v _ i ^ { \\nu } v _ j ^ { \\mu - \\nu } \\right ) u _ i ^ \\nu \\otimes K _ \\nu u _ j ^ { \\mu - \\nu } , \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} c ( x _ 1 , \\dots , x _ N ) = \\sum _ { 1 \\leq i < j \\leq N } \\dfrac { 1 } { \\vert x _ i - x _ j \\vert ^ { s } } . \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} E _ { \\bullet } = ( E _ 1 , E _ 2 , \\ldots , E _ k ) \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} A _ 1 A & = [ 1 : - 1 : 0 ] , & A _ 1 B _ 2 & = [ 1 : - 1 : y ] , & A _ 1 C _ 2 & = [ 1 : - 1 : - x ] , \\\\ B _ 1 B & = [ 0 : 1 : - 1 ] , & B _ 1 C _ 2 & = [ 1 : x : - x ] , & B _ 1 A _ 2 & = [ m : - 1 : 1 ] , \\\\ C _ 1 C & = [ 1 : 0 : 1 ] , & C _ 1 A _ 2 & = [ m : - 1 : m ] , & C _ 1 B _ 2 & = [ y : - 1 : y ] . \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} \\phi ^ * ( t ) : = \\int _ 0 ^ t ( \\phi ^ \\prime ) ^ { - 1 } ( s ) \\ , \\d s \\textrm { f o r a n y } t \\geq 0 . \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} F = T _ S ( \\widehat { F } _ c e ^ { 2 \\pi c ( \\cdot ) } ) \\norm { \\widehat { F } _ c e ^ { 2 \\pi c ( \\cdot ) } } _ { L ^ 2 ( \\mathbb { R } ; \\omega _ { a , b } ) } = \\norm { F } _ { A ^ 2 ( S ( a , b ) ) } , \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} S _ { f _ n } ( \\rho \\| \\sigma ) & = f _ n ( 0 ^ + ) \\sigma ( 1 ) + f _ n ' ( + \\infty ) \\rho ( 1 ) + S _ { - h _ n } ( \\rho \\| \\sigma ) \\\\ & = f _ n ( 0 ^ + ) \\sigma ( 1 ) + f _ n ' ( + \\infty ) \\rho ( 1 ) - \\int _ { ( 0 , + \\infty ) } h _ n ( t ) \\ , d \\| E _ { \\rho , \\sigma } ( t ) \\xi _ \\sigma \\| ^ 2 . \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } x ^ 2 e ^ { - q ^ 2 x ^ 2 } d x = \\frac { \\sqrt { \\pi } } { 4 q ^ 3 } \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} A _ { N } ^ { \\ast } ( w _ { 1 , r } ) & = A _ { N } ^ { \\ast } ( a _ { 1 } , \\{ 1 \\} _ { b _ { 1 } - 1 } , w _ { 2 , r } ) = \\sum _ { n = 1 } ^ { N } \\binom { N } { n } \\frac { ( - 1 ) ^ { k - 1 } } { n ^ { k } } \\zeta ^ { \\ast } _ { n } ( \\{ 1 \\} _ { b _ 1 - 1 } , w _ { 2 , r } ) \\\\ & = \\sum _ { i _ 1 = 1 } ^ { N } \\frac { 1 } { i _ 1 } \\dots \\sum _ { i _ { k } = 1 } ^ { i _ { k - 1 } } \\frac 1 { i _ { k } } \\sum _ { n = 1 } ^ { i _ { k } } \\binom { i _ { k } } { n } ( - 1 ) ^ { n - 1 } \\zeta ^ { \\ast } _ { n } ( \\{ 1 \\} _ { b _ 1 - 1 } , w _ { 2 , r } ) . \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} u _ { - , a } ( x ) = \\frac { \\phi _ 1 ( x ) } { \\phi _ 1 ( a ) } \\frac { \\int _ x ^ \\infty \\phi _ 1 ^ { - 2 } ( y ) d y } { \\int _ a ^ \\infty \\phi _ 1 ^ { - 2 } ( y ) d y } , \\ x \\ge a ; \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} \\xi _ y : = \\xi + b _ a . \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} H ( X ) = \\bigoplus _ { \\boldsymbol { \\nu } \\in \\widehat { G } } H ( X ) _ { \\boldsymbol { \\nu } } , \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\int _ { K / / H } f \\ , d ( \\delta _ { [ x ] } * \\delta _ { [ y ] } ) = \\int _ K f \\circ \\pi ( u ) \\ , d ( \\delta _ x * \\delta _ y ) ( u ) . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} X ^ { ( \\mathbf { v } ) } ( v ) = \\sum _ { ( k , i , \\alpha , \\beta ) \\in \\mathbb { I } } X ^ { ( \\mathbf { v } ) } _ { k , i , \\alpha , \\beta } e ^ { \\mathbf { i } k \\cdot x } y ^ { i } z ^ { \\alpha } \\bar { z } ^ { \\beta } \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 2 8 8 n + 2 1 6 ) q ^ n & \\equiv f _ 6 ^ 3 \\cdot \\dfrac { f _ 1 ^ 2 } { f _ 2 } \\\\ & \\equiv f _ 6 ^ 3 \\left ( \\dfrac { f _ 9 ^ 2 } { f _ { 1 8 } } - 2 q \\dfrac { f _ 3 f _ { 1 8 } ^ 2 } { f _ 6 f _ 9 } \\right ) ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\sup _ { s \\in [ t , T ] } \\int _ K | X _ n ( s , x ) - X ( s , x ) | \\wedge 1 d x = 0 . \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{gather*} d a = a d , d b = q ^ 2 b d , d c = q ^ { - 2 } c d , \\\\ b a = a b + q ^ { - 1 } \\hat q b d , c b = b c + q ^ { - 1 } \\hat q \\big ( d a - d ^ 2 \\big ) , c a = a c - q ^ { - 1 } \\hat q d c . \\end{gather*}"}
-{"id": "184.png", "formula": "\\begin{gather*} c _ 1 = \\frac { 1 } { b _ 1 } , \\ \\dots , \\ c _ { 2 d - 1 } = \\frac { 1 } { b _ { 2 d - 1 } } , \\ c _ { 2 d } = 0 , \\\\ \\hat { \\mathcal { P } } _ { 2 d } ( s ) = \\prod _ { j = 1 } ^ { 2 d } ( s - c _ j ) . \\end{gather*}"}
-{"id": "5807.png", "formula": "\\begin{align*} M ^ i _ t = Y ^ i _ t - Y ^ i _ 0 + A _ t ^ { W , W } ( ( \\nabla \\gamma ^ i ) ^ * \\ , b + \\tilde f ^ i ) . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} \\lvert ( i , j ) \\rvert _ { W _ n ( w ) } = \\lvert l _ i \\rvert _ S + \\lvert m \\rvert _ S + \\lvert r _ j \\rvert _ S . \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} \\int _ D \\frac { \\partial \\xi } { \\partial x _ i } \\ { \\rm d } x = \\int _ { \\partial D } { \\xi } n _ i \\ { \\rm d } s . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} E ^ { ( 1 ) } = 2 \\delta \\mathrm { d } - 4 \\ , \\mathrm { R i c } . \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} Q f : = \\sum ^ d _ { i , j = 1 } q _ { i j } \\frac { \\partial ^ 2 } { \\partial _ i \\partial _ j } f , \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} A ( x [ f ] ) \\cdot ( h \\otimes \\sum _ { \\mu \\in D _ { c , p '' } } \\ , I _ \\mu ) - h \\otimes \\beta ( x ) = ( A ( x [ f ] ) \\cdot h ) \\otimes \\sum _ { \\mu \\in D _ { c , p '' } } \\ , I _ \\mu , \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} y _ { n + 1 } = R ( n , y _ n ) = \\frac { P ( n , y _ n ) } { Q ( n , y _ n ) } , \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} \\{ x _ { 1 , \\tau } , p _ \\tau \\} _ { D B } = - \\frac { f ( t _ \\tau ) } { m } p _ { 1 , \\tau } ; \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} \\| Y _ t - 1 \\| _ { \\mathcal L ^ 2 } = \\frac { \\ell ( t ) } t \\in O ( t ^ { - 1 + \\varepsilon } ) . \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\log \\left | \\dfrac { \\partial } { \\partial x } \\ , \\phi ( t , x ) \\right | = \\int _ 0 ^ t \\frac { \\ , \\partial } { \\ , \\partial x } b ( s , \\phi ( s , x ) ) \\ , d s . \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} W _ \\ast f ( n ) = \\sup _ { t > 0 } | W _ t f ( n ) | , f \\in \\ell ^ 2 ( \\mathbb { N } ) , \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} d \\Gamma ( \\omega ) & = ( d \\Gamma ( \\omega _ 1 ) , \\dots , d \\Gamma ( \\omega _ p ) ) \\\\ d \\Gamma ^ { ( n ) } ( \\omega ) & = ( d \\Gamma ^ { ( n ) } ( \\omega _ 1 ) , \\dots , d \\Gamma ^ { ( n ) } ( \\omega _ p ) ) \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} \\Lambda _ t = M _ t \\left [ \\begin{array} { c } 1 \\\\ 0 \\end{array} \\right ] \\mathbb { Z } \\oplus M _ t \\left [ \\begin{array} { c } \\Re \\tilde \\tau \\\\ \\Im \\tilde \\tau \\end{array} \\right ] \\mathbb { Z } \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} \\norm { u _ k ^ \\delta - u _ k } _ X \\leq \\tau \\norm { A } \\delta k , k = 0 , \\dots . \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} a _ { n + 1 } \\varphi ( n + 1 ) + b _ { n + 1 } \\varphi ( n ) + a _ n \\varphi ( n - 1 ) = E \\varphi ( n ) , \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} \\sum _ { v > 2 } a _ v p _ { v } + \\sum _ { \\ell > 2 } p _ { \\ell } \\frac { \\partial } { \\partial x _ \\ell } r _ j = \\sum _ { \\ell > 2 } \\left ( a _ \\ell + \\frac { \\partial } { \\partial x _ \\ell } r _ j \\right ) p _ { \\ell } = 0 \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} \\mathrm { g r } ( G ) = \\frac { 1 } { z _ * } , \\ \\mbox { w h e r e $ z _ * $ i s t h e u n i q u e p o s t i v e n u m b e r s a t i s f y i n g } \\ \\sum _ { i = 1 } ^ r \\frac { \\psi _ i ( z _ * ) } { 1 + \\psi _ i ( z _ * ) } = 1 . \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} h _ M ( x ) : = \\begin{cases} 1 & 0 \\leq x < \\frac 1 2 , \\\\ - 1 & \\frac 1 2 \\leq x < 1 , \\\\ 0 & x \\notin [ 0 , 1 ) , \\end{cases} \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} \\langle - \\Delta _ p u , \\phi \\rangle _ { X ' , X } = \\int _ 0 ^ T \\langle - \\Delta _ p u ( \\tau ) , \\phi ( \\tau ) \\rangle _ p \\ , d \\tau ; u , \\phi \\in X , \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( { \\{ t _ { j k } < \\left < t _ { j k } \\right > + a \\} } \\right ) > 1 - a ^ { - 2 } c ( \\alpha ) = \\frac { 3 } { 4 } . \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} V _ k = \\frac { 1 } { \\sqrt { | z _ 1 | ^ 2 + | z _ 2 | ^ 2 } } \\ , \\bigl ( z _ 1 E ^ { ( n ) } _ { 1 , k + 1 } + z _ 2 E ^ { ( n ) } _ { k + 1 , 1 } \\bigr ) . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} ( ( ( \\Pi _ { k , D } ^ \\Delta \\xi , q ) ) ) _ D = ( ( ( \\xi , q ) ) ) _ D , \\ \\forall q \\in \\mathbb { P } _ k ( D ) , \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} f _ i ( q ) = \\frac { 1 } { i ! n ^ i } \\Big ( ( q - 1 ) ^ i G _ { h _ i , n } ( q ) - \\sum _ { m _ 1 = 0 } ^ { i - 1 } \\sum _ { m _ 2 = m _ 1 } ^ { i } \\binom { i } { m _ 2 } f _ { m _ 1 } ^ { ( i - m _ 2 ) } ( q ) R _ { n , m _ 1 , m _ 2 } ( q ) ( q - 1 ) ^ { i - m _ 2 } q ^ { i - m _ 2 } \\Big ) , \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{gather*} F _ 0 ( z ) = { } _ 2 F _ 1 \\biggl ( \\begin{matrix} r , \\ , 1 - r \\\\ 1 \\end{matrix} \\biggm | z \\biggr ) \\end{gather*}"}
-{"id": "8076.png", "formula": "\\begin{align*} ( ( U \\lhd \\mathbb { F } ) [ i ] ) _ m : = U _ m \\oplus F _ { m + i } , \\ , \\ , m \\geq 0 . \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\Psi _ D ( a + b t ^ { - 1 } \\sqrt { D } ) \\gamma _ { D , 2 } = \\gamma _ { D , 2 } \\left [ \\begin{smallmatrix} t ^ { - 1 } & 0 \\\\ 0 & 1 \\end{smallmatrix} \\right ] \\left [ \\begin{smallmatrix} a & b \\\\ 5 b & a \\end{smallmatrix} \\right ] \\left [ \\begin{smallmatrix} t & 0 \\\\ 0 & 1 \\end{smallmatrix} \\right ] \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} \\lambda ( \\hat { x } , x ^ { \\star } ) & = \\left ( \\frac { z } { b } \\right ) \\frac { l _ 2 ^ { - 1 } z ^ { - l _ 2 } - l _ 1 ^ { - 1 } z ^ { - l _ 1 } } { l _ 1 ^ { - 1 } z ^ { - l _ 1 } + l _ 2 ^ { - 1 } z ^ { - l _ 2 } } - 1 \\\\ \\kappa ( \\hat { x } , x ^ { \\star } ) & = z \\left [ l _ 1 ^ { - 1 } + l _ 2 ^ { - 1 } - 1 \\right ] - z \\frac { l _ 1 ^ { - 1 } z ^ { - l _ 1 } - l _ 2 ^ { - 1 } z ^ { - l _ 2 } } { l _ 1 ^ { - 1 } z ^ { - l _ 1 } + l _ 2 ^ { - 1 } z ^ { - l _ 2 } } \\left [ l _ 1 ^ { - 1 } - l _ 2 ^ { - 1 } + \\ln { \\hat { x } } - \\ln { x ^ \\star } \\right ] , \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} J ( f _ 1 , \\ldots , f _ m ) = \\frac { \\int _ H e ^ { - \\mathcal Q ( x ) } \\prod _ { k = 1 } ^ m f _ k ^ { c _ k } ( B _ k x ) \\ , d x } { \\prod _ { k = 1 } ^ m \\left ( \\int _ { H _ k } f _ k \\right ) ^ { c _ k } } \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} | \\mu | _ { s , \\tau + 1 } : = \\sum _ { k \\in \\mathbb { Z } ^ n } | \\hat { \\mu } _ k | | k | ^ { \\tau + 1 } e ^ { | k | s } \\leq C \\tilde { \\gamma } , \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} = \\frac { T - t } { 2 } \\left ( \\zeta _ 0 ^ { ( i _ 1 ) } \\zeta _ 0 ^ { ( i _ 2 ) } + \\sum _ { j = 1 } ^ { \\infty } \\frac { 1 } { \\sqrt { 4 j ^ 2 - 1 } } \\left ( \\zeta _ { j - 1 } ^ { ( i _ 1 ) } \\zeta _ { j } ^ { ( i _ 2 ) } - \\zeta _ j ^ { ( i _ 1 ) } \\zeta _ { j - 1 } ^ { ( i _ 2 ) } \\right ) \\right ) . \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} \\widehat { \\rho } _ { \\gamma } ( w _ 1 , \\dots , w _ n ) = \\left ( \\gamma ^ { 1 / 2 m _ 1 } w _ 1 , \\dots , \\gamma ^ { 1 / 2 m _ n } w _ n \\right ) , \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} T f ( n ) = \\sum _ { m = 0 } ^ { \\infty } K ( n , m ) \\cdot f ( m ) , \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} \\dot \\lambda + \\lambda ^ 2 + 2 \\ , { \\lambda } \\ , g ( X / n , \\dot \\gamma ) + k = 0 . \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { Q } \\ , ^ { C } _ { t } D _ { T } ^ { q } \\Theta ( x , t ) = A ^ { * } \\mathcal { Q } \\Theta ( x , t ) + C ^ { * } \\mathcal { Q } z ( t ) & \\hbox { i n } Q _ { T } \\\\ \\Theta ( \\xi , t ) = 0 & \\hbox { o n } \\Sigma _ { T } \\\\ \\Theta ( x , T ) = 0 & \\hbox { i n } \\Omega . \\end{cases} \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} \\frac { 1 } { 4 \\Delta t } ( \\mathbb { H } [ u _ h ^ { n } ] - \\mathbb { H } [ u _ h ^ { n - 1 } ] ) + \\| \\nabla u _ h ^ { n - \\theta } \\| ^ 2 + \\gamma \\| \\sigma _ h ^ { n - \\theta } \\| ^ 2 \\leq & - ( f ^ { n - \\theta } ( u _ h ) , u _ h ^ { n - \\theta } ) + ( g ^ { n - \\theta } , u _ h ^ { n - \\theta } ) \\\\ \\leq & \\frac 1 2 \\| f ^ { n - \\theta } ( u _ h ) \\| ^ 2 + \\| u _ h ^ { n - \\theta } \\| ^ 2 + \\frac 1 2 \\| g ^ { n - \\theta } \\| ^ 2 . \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} Q ( x ) = \\sum _ { v \\in V } \\sum _ { v ' \\in N ( v ) } | x _ { v ' } - x _ v | ^ 2 . \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} X _ 0 ^ { ( 1 ) } = \\{ x = ( x _ { i j } ) \\in X ^ { ( 1 ) } \\mid \\textrm { e x a c t l y o n e $ x _ { \\imath \\jmath } $ i s z e r o } \\} . \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } c ( 5 n + 4 ) q ^ { n } & = 4 1 \\dfrac { E _ { 1 0 } ^ { 3 } } { E _ { 1 } ^ 2 E _ { 2 } ^ 2 E _ { 5 } } + 8 6 0 q \\dfrac { E _ { 1 0 } ^ { 6 } } { E _ { 1 } ^ { 5 } E _ { 2 } E _ { 5 } ^ { 2 } } + 6 8 0 0 q ^ { 2 } \\dfrac { E _ { 1 0 } ^ { 9 } } { E _ { 1 } ^ { 8 } E _ { 5 } ^ { 3 } } \\\\ & \\quad + 2 4 0 0 0 q ^ { 3 } \\dfrac { E _ { 2 } E _ { 1 0 } ^ { 1 2 } } { E _ { 1 } ^ { 1 1 } E _ { 5 } ^ { 4 } } + 3 2 0 0 0 q ^ { 4 } \\dfrac { E _ { 2 } ^ { 2 } E _ { 1 0 } ^ { 1 5 } } { E _ { 1 } ^ { 1 4 } E _ { 5 } ^ { 5 } } . \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} u _ { k + 1 } ( t , 0 ) = \\dots = u _ { k + m } ( t , 0 ) = 0 \\mbox { f o r } t \\ge \\tau _ { k + 1 } . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} \\mathbb { A } ^ \\alpha ( x ) = u ^ \\alpha \\ , e ^ { i p _ \\beta x ^ \\beta } . \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} p _ { _ 1 } ( y ) = \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma _ w } \\int _ { - \\infty } ^ { \\infty } e ^ { - \\frac { ( y - x ) ^ 2 } { 2 \\sigma _ w ^ 2 } } p ( x ) d x , \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} f ^ { ( i ) } ( \\omega ) = r ^ { ( i ) } ( \\omega ) \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} x ^ { \\star } - \\kappa - ( 1 + \\lambda ) \\hat { z } = \\hat { x } . \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} L _ { 1 1 } & = ( \\Delta ' ) ^ { - 1 } ( [ d , a ] a ^ { - 1 } ( a c ^ { - 1 } d - b ) - ( [ d , b ] - [ d , a ] a ^ { - 1 } b ) ) ( a c ^ { - 1 } d - b ) ^ { - 1 } \\\\ & = ( \\Delta ' ) ^ { - 1 } ( [ d , a ] c ^ { - 1 } d - [ d , b ] ) ( a c ^ { - 1 } d - b ) ^ { - 1 } \\\\ & = ( \\Delta ' ) ^ { - 1 } ( d ( a - b d ^ { - 1 } c ) - \\Delta ' ) ( a - b d ^ { - 1 } c ) ^ { - 1 } \\\\ & = ( ( \\Delta ' ) ^ { - 1 } d - ( a - b d ^ { - 1 } c ) ^ { - 1 } ) . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\displaystyle \\int _ { \\mathbb { R } ^ 2 \\backslash B _ { R _ 0 } ( 0 ) } A ( x ) ^ r v ^ { 2 m } & \\leq & C _ 5 \\left \\| v \\right \\| ^ { 2 m } + C _ 6 \\left \\| v \\right \\| ^ { 2 r / \\beta _ 0 } \\left \\| v \\right \\| ^ { 2 ( m \\beta _ 0 - r ) / \\beta _ 0 } \\\\ & = & C _ 7 \\left \\| v \\right \\| ^ { 2 m } , \\end{array} \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} L ( t ) = \\frac { h _ { J _ t } { ( \\theta ) } } { h _ { J _ 0 } { ( \\theta ) } } e ^ { \\theta { S ( t ) } - t \\kappa ( \\theta ) } , \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} Q \\overline { \\nabla } ^ { ' } _ { X } Y = Q \\nabla ^ { ' } _ { X } Y + \\alpha \\eta ( Y ) Q X - \\alpha g ( X , Y ) Q \\xi - \\beta g ( \\phi X , Y ) Q \\xi \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} \\sigma ^ + = \\tfrac 1 2 ( | s | + s ) , \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} \\overline { v } ( k ) a ( k ) ( \\overline { F } _ { \\eta } ( v , \\omega ) - \\lambda ) ^ { - 1 } v & = \\lvert v ( k ) \\lvert ^ 2 ( F _ { - \\eta } ( v , \\omega ) + \\omega ( k ) + \\lambda ) ^ { - 1 } \\Omega \\\\ & - \\lvert v ( k ) \\lvert ^ 2 ( F _ { - \\eta } ( v , \\omega ) + \\omega ( k ) - \\lambda ) ^ { - 1 } ( \\overline { F } _ { \\eta } ( v , \\omega ) - \\lambda ) ^ { - 1 } v . \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} T _ 1 : = \\tau _ k + \\sum _ { l = 1 } ^ m \\tau _ { k + l } . \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 1 + \\lambda ( u ^ 2 ) ^ 2 + \\mu ( u ^ 3 ) ^ 2 & - \\lambda ( u ^ 3 + u ^ 1 u ^ 2 ) & - \\lambda u ^ 2 - \\mu u ^ 1 u ^ 3 \\\\ - \\lambda ( u ^ 3 + u ^ 1 u ^ 2 ) & \\lambda ( u ^ 1 ) ^ 2 + \\mu & 2 \\lambda u ^ 1 \\\\ - \\lambda u ^ 2 - \\mu u ^ 1 u ^ 3 & 2 \\lambda u ^ 1 & \\lambda + \\mu ( u ^ 1 ) ^ 2 \\end{pmatrix} , \\\\ w _ { 1 2 } = \\sqrt { \\frac { \\lambda ( \\mu ^ 2 - \\lambda ^ 2 ) } { \\det g } } u ^ 1 , w _ { 2 3 } = 0 , w _ { 3 1 } = \\sqrt { \\frac { \\lambda ( \\mu ^ 2 - \\lambda ^ 2 ) } { \\det g } } , \\end{gather*}"}
-{"id": "7260.png", "formula": "\\begin{align*} C _ p ( \\nu _ 1 ) - C _ p ( \\nu _ 2 ) & \\geq \\frac { C _ p ( \\nu _ 2 ) ( C _ p ( \\nu _ 2 ) - 1 ) ^ 2 } { 3 ( C _ p ( \\nu _ 2 ) - 1 ) ^ 2 + 4 C _ p ( \\nu _ 2 ) } \\\\ & \\geq \\begin{cases} \\frac { 1 } { 6 } ( C _ p ( \\nu _ 2 ) - 1 ) ^ 2 & 1 \\leq C _ p ( \\nu _ 2 ) < 2 \\\\ \\frac { 1 } { 3 } ( C _ p ( \\nu _ 2 ) - 1 ) - 1 / 6 & C _ p ( \\nu _ 2 ) \\geq 1 , \\end{cases} \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} \\begin{aligned} \\rho _ { n + 1 } ( x , y ) = { } & \\rho ( x , y , n \\tau + \\tau ) \\\\ & = \\rho ( x , y , n \\tau ) + \\tau \\frac { \\partial } { \\partial t } \\rho ( x , y , n \\tau ) + \\frac { 1 } { 2 } \\tau ^ { 2 } \\frac { \\partial ^ { 2 } } { \\partial t ^ { 2 } } \\rho ( x , y , n \\tau ) \\end{aligned} \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} \\dot { N } ( t ) \\geq - \\| \\lambda ( N ( t ) , F ( t ) , S ( t ) , \\cdot ) \\| _ { \\mathrm { L } ^ \\infty ( \\Omega ) } \\int _ { \\Omega } u ( x , t ) \\ , d x = - \\| \\lambda ( N ( t ) , F ( t ) , S ( t ) , \\cdot ) \\| _ { \\mathrm { L } ^ \\infty ( \\Omega ) } N ( t ) , \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} \\beta ( y ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( \\frac { 1 } { 3 } y ^ { 4 } + \\frac { 3 } { 2 } y ^ { 2 } - \\frac { 7 } { 1 6 } \\right ) e ^ { - 2 y ^ { 2 } } . \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial \\overline { \\zeta } _ j } = 0 \\ , 1 \\leq j \\leq n . \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} [ X _ 1 , [ X _ 2 , Y _ 2 ] ] = [ [ X _ 1 , X _ 2 ] , Y _ 2 ] + [ X _ 2 , [ X _ 1 , Y _ 2 ] ] = 0 , \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} \\| S u _ i - f _ i \\| _ E < \\delta _ i , \\ ; \\ ; i = 1 , 2 , \\dots \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} u ^ 2 + ( \\beta ^ { q ^ 2 - 1 } + \\beta ^ { q - 1 } + 1 ) u + \\beta ^ { q - 1 } = 0 . \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} \\bar { \\alpha } _ { j } = \\max \\{ a _ j , \\beta _ { j } - \\delta _ { j } \\} , \\bar { \\beta } _ { j } = \\min \\{ \\alpha _ { j } + \\delta _ { j } , b _ j \\} . \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} \\limsup _ { t \\rightarrow \\infty } \\frac { G ( e _ t ) } { t ^ 4 } < \\frac { \\delta } { 4 } - \\delta + \\frac { \\delta } { 4 } = - \\frac { \\delta } { 2 } , \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} \\mathcal { L } ( e _ 1 , e _ 2 , e _ 3 , e _ 4 ) = e _ 1 . \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} \\frac { \\pi ( - \\zeta ) ^ y } { \\sin \\pi y } = \\int _ { - \\infty } ^ { \\infty } d x \\frac { - \\zeta e ^ { x y } } { - \\zeta + e ^ x } , \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} L _ i L _ j = L _ j L _ i , L _ i L _ i ^ { - 1 } = 1 = L _ i ^ { - 1 } L _ i , \\forall 1 \\leq i , j \\leq n , \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} y \\cdot \\Psi : = \\hat { y } \\cdot \\Psi , \\ , C \\cdot \\Psi : = c \\Psi . \\\\ \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} \\frac { \\lambda _ n ^ 2 } { 2 \\log \\frac { 1 } { \\lambda _ n } } \\int _ \\Omega e ^ { \\ , u _ { n } } \\ , d x \\longrightarrow \\frac { 4 \\pi } { \\log { \\sqrt { \\frac { R _ 1 } { R _ 2 } } } } \\mbox { a s } \\ \\lambda _ n \\to 0 \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} 0 & = \\Im \\biggl [ i \\langle v , H D _ j Q \\rangle + i \\langle F ( Q + v ) - F ( Q ) , D _ j Q \\rangle \\\\ & - \\langle D Q ( \\dot z + i E z ) , D _ j Q \\rangle + \\langle v , D _ j D Q \\dot z \\rangle \\biggr ] . \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} ( \\widetilde { F } _ { \\eta } & ( v _ g , \\omega ) - i ) ^ { - 1 } - ( d \\Gamma ( \\omega ) - i ) ^ { - 1 } \\\\ = & \\eta ( \\widetilde { F } _ { \\eta } ( v _ g , \\omega ) - i ) ^ { - 1 } W ( v _ g , - 1 ) ( d \\Gamma ( \\omega ) - i ) ^ { - 1 } \\\\ = & \\eta ^ 2 ( \\widetilde { F } _ { \\eta } ( v _ g , \\omega ) - i ) ^ { - 1 } W ( v _ g , - 1 ) ( d \\Gamma ( \\omega ) - i ) ^ { - 1 } W ( v _ g , - 1 ) ( d \\Gamma ( \\omega ) - i ) ^ { - 1 } \\\\ & + \\eta ( d \\Gamma ( \\omega ) - i ) ^ { - 1 } W ( v _ g , - 1 ) ( d \\Gamma ( \\omega ) - i ) ^ { - 1 } . \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} [ r _ { 1 2 } , r _ { 1 3 } ] + [ r _ { 1 2 } , r _ { 2 3 } ] + [ r _ { 1 3 } , r _ { 2 3 } ] = 0 , \\end{align*}"}
-{"id": "9997.png", "formula": "\\begin{align*} \\bar F _ 0 ( q ) = ( 1 - \\bar P ( q ) ) \\int _ 0 ^ \\infty e ^ { - q \\tau ( s ) } d s , \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} \\eta _ n h _ \\sigma ^ p = h _ \\rho ^ p x _ n h _ \\sigma ^ { 1 / 2 } . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} c _ { \\l } = \\frac { l - 1 } { l + 1 } = 1 - \\frac { 2 } { l + 1 } \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} z ^ { - n } \\sigma _ { - } p _ { i } = z ^ { i - n } \\sigma _ { + } ^ { - 1 } + z ^ { - n } u \\frac { \\phi } { \\sigma _ { - } } + \\frac { \\psi } { \\sigma _ { + } } . \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} \\mu = \\sum _ { j = 1 } ^ r \\sum _ { s \\in [ n ] \\setminus \\{ i _ { k , l } \\mid 1 \\leq k \\leq j , 1 \\le l \\le N _ k \\} } d _ { j , s } \\varpi _ s . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} ( M e ^ { - t A ^ { 2 } } f ) ( \\tau , y ) = e ^ { - t \\tau ^ { 2 } } ( M f ) ( \\tau , y ) . \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} \\{ f ( 1 ) , \\ldots , f ( n + 1 ) \\} = \\left \\{ \\left ( - \\langle \\mu , \\varpi _ 1 \\rangle + \\sum _ { j = 1 } ^ { i - 1 } \\left \\langle \\frac { \\lambda + \\mu } { 2 } + \\rho , \\alpha _ j \\right \\rangle \\right ) ^ { } \\ \\middle \\vert \\ 1 \\leq i \\leq n + 1 \\right \\} \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} \\omega _ { b } = j _ b ^ 2 + j _ b \\xi _ b , \\Omega _ { j } = j ^ 2 + j \\frac { \\sum _ { b = 1 } ^ n \\xi _ b } { n - \\frac 1 2 } , \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} \\begin{array} { l l } \\Phi _ { \\epsilon } ( x , y ) = U ^ { i _ 0 j _ 0 } ( x ) - V ^ { i _ 0 j _ 0 } ( y ) - \\displaystyle \\frac { 1 } { 2 \\epsilon } | x - y | ^ { 2 } \\qquad ( x , y ) \\in \\mathbb { R } ^ m \\times \\mathbb { R } ^ m . \\end{array} \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ { k - 1 } \\| x _ { n _ k } \\| _ { j } e _ j \\Big \\| _ E \\le \\sum _ { j = 1 } ^ { k - 1 } \\| x _ { n _ k } \\| _ { j } \\le \\frac { 2 } { m _ k } \\sum _ { j = 1 } ^ { k - 1 } m _ j \\le 2 ^ { - k } , \\ ; \\ ; k = 1 , 2 , \\dots \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} \\mathbf { G } = \\beta _ { 0 + } ^ { - 1 } ( G _ 0 \\times G _ 1 \\times \\cdots \\times G _ { m ^ + } ) \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} V _ k = B _ k V . \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} I _ { \\mu } ^ { \\infty } ( v _ j ) & \\geq c _ { \\mu } ^ { \\infty } \\\\ & = \\inf _ { \\gamma \\in \\Gamma ^ { \\infty } } \\max _ { t \\in [ 0 , 1 ] } I _ { \\mu } ^ { \\infty } ( \\gamma ( t ) ) , \\\\ & \\geq \\inf _ { \\gamma \\in \\Gamma ^ { \\infty } } \\max _ { t \\in [ 0 , 1 ] } I _ { \\mu } ( \\gamma ( t ) ) \\\\ & \\geq \\inf _ { \\gamma \\in \\Gamma } \\max _ { t \\in [ 0 , 1 ] } I _ { \\mu } ( \\gamma ( t ) ) \\\\ & = c _ { \\mu } . \\\\ \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} \\begin{aligned} h ( 0 , 2 ) & = f ( 0 ) \\P ( X _ 2 = 0 ) - \\P ( X _ 2 \\geq 1 ) \\\\ & = f ( 0 ) \\bigg ( 1 - \\frac { f ( 0 ) } { 2 } \\bigg ) - \\frac { f ( 0 ) } { 2 } = \\frac { f ( 0 ) ( 1 - f ( 0 ) ) } { 2 } , \\end{aligned} \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} \\mathbf { S } _ { } ^ * \\triangleq \\mathbf { Q } ^ { - \\frac { 1 } { 2 } } \\Tilde { \\mathbf { V } } \\Tilde { \\boldsymbol { \\Lambda } } _ { \\rm o } \\Tilde { \\mathbf { V } } ^ { \\rm H } \\mathbf { Q } ^ { - \\frac { 1 } { 2 } } = \\Tilde { \\mathbf { V } } \\left ( \\frac { \\ln 2 } { \\mu _ 2 } \\left ( \\nu _ 2 \\mathbf { I } _ { r } - \\boldsymbol { \\Lambda } ^ { \\rm H } \\boldsymbol { \\Lambda } \\right ) \\right ) ^ { - 1 } \\Tilde { \\boldsymbol { \\Lambda } } _ { \\rm o } \\Tilde { \\mathbf { V } } ^ { \\rm H } , \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\int _ N f ( x ) 1 _ { A \\cap q ( N ) } ( x \\cdot \\ddot { y } ) d \\omega ( x ) = 0 \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} & T ^ { * } _ { W } : \\mathcal { K } _ { 2 , W } \\rightarrow H \\\\ & T ^ { * } _ { W } ( v _ i ( C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f ) = \\sum _ { i \\in I } v _ { i } ^ { 2 } C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } f . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} \\xi ( t , x ) = E [ \\xi _ 0 ( X _ t ^ x ) ] , \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} N - \\rho _ q ( d , m _ d ) < \\rho _ q ( d , m _ d + 1 ) - \\rho _ q ( d , m _ d ) = \\sum _ { i = 1 } ^ { \\min \\{ d , q - 1 \\} } \\rho _ q ( d - i , m _ d ) . \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} ( \\varrho , u ) | _ { t = 0 } = ( \\varrho _ 0 , u _ 0 ) , \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} g ' _ h ( t ) = - \\frac { m ^ { 2 } } { t ^ { 2 } } + h - t . \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} \\phi ^ * f = ( f \\circ \\phi ) \\cdot \\det \\phi ' , f \\in A ^ 2 ( \\Omega _ 2 ) . \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { D _ t } p _ { \\tau _ k } ^ { - 1 } & = \\exp \\Big ( \\sum _ { s \\leq t } \\log \\tfrac 1 { p _ s } \\cdot ( D _ s - D _ { s - } ) \\Big ) = \\exp \\Big ( \\int _ 0 ^ t \\log \\tfrac 1 { p _ s } d D _ s \\Big ) , \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} A ^ { W , Y } ( l ) = A ^ { W , W } ( \\nabla \\gamma ^ * \\ , l ) , \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} b ( t , x , \\mu ) \\ , = \\ , \\int _ { \\mathbb R } \\widetilde { b } ( t , x , y ) \\mu ( { \\mathrm d } y ) \\ , ; t \\in [ 0 , \\infty ) \\ , , \\ , \\ , x \\in \\mathbb R \\ , , \\ , \\ , \\mu \\in \\mathcal M ( \\mathbb R ) \\ , , \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{gather*} \\beta ( j + i v ) = \\beta ( j ) + i \\beta ( v ) = \\beta ( j ) + \\beta ( i v ) \\beta ( j + i u ) = \\beta ( j ) + i \\beta ( u ) = \\beta ( j ) + \\beta ( i u ) . \\end{gather*}"}
-{"id": "9126.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { i = M } \\sigma _ { i } ^ { j - 1 } \\cdot H _ { j , a _ { i , 1 } } \\cdot c ^ { i } _ { 1 } = 0 j \\in [ r ] . \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} A & : = \\prod _ { i = 1 } ^ { s - 1 } \\left ( q - p _ i - 1 \\right ) , \\\\ B & : = \\sum _ { k = 1 } ^ { s - 2 } ( q - p _ 1 ) \\ldots ( q - p _ { k - 1 } ) ( q - p _ { k + 1 } - 1 ) \\ldots ( q - p _ { s - 1 } - 1 ) , \\\\ C & = \\sum _ { k = s - 1 } ^ { \\ell } C _ k , \\mbox { w i t h } C _ k : = \\prod _ { n = 1 } ^ { s - 2 } \\left ( q - p _ n + \\tau _ { n , k } \\right ) , \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} - \\nabla _ x \\log \\rho _ { 0 | t } & ( x \\ , | \\ , y ) = - \\nabla _ x \\log \\rho _ 0 ( x ) - \\nabla _ x \\log \\rho _ { t | 0 } ( y \\ , | \\ , x ) \\\\ & ~ ~ ~ ~ ~ ~ = - \\nabla _ x \\log \\rho _ 0 ( x ) + \\frac { \\alpha e ^ { - \\alpha t } } { 1 - e ^ { - 2 \\alpha t } } ( e ^ { - \\alpha t } x - y ) . \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} \\mathcal { F } \\left \\{ f ^ { ( m ) } \\right \\} = ( i 2 \\pi \\nu ) ^ m \\widehat { f } . \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} S _ { \\mathfrak { C } } \\psi = \\displaystyle \\sum _ { k \\in I } \\langle \\psi | \\phi _ { k } \\rangle \\mathfrak { C } \\phi _ { k } . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} \\phi \\in ( \\mathcal L _ { \\beta , \\Omega } ^ \\infty ) , \\quad \\quad \\mathcal L _ { \\beta , \\Omega } ^ \\infty \\phi = ( - D ^ \\beta _ { \\infty } + \\mathcal L _ \\Omega ) \\phi . \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} f _ { n , m } : = ( V _ n ^ - + \\lambda _ { n , m } ) w _ { n , m } . \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} \\int _ { ( 0 , + \\infty ) } f ( t ) \\ , d \\| E _ { \\rho , \\sigma } ( t ) \\xi _ \\sigma \\| ^ 2 & = \\int _ { ( 0 , + \\infty ) } f ( t ) t ^ { - 1 } \\ , d \\| E _ { \\sigma , \\rho } ( t ^ { - 1 } ) \\xi _ \\rho \\| ^ 2 \\\\ & = \\int _ { ( 0 , + \\infty ) } t f ( t ^ { - 1 } ) \\ , d \\| E _ { \\sigma , \\rho } ( t ) \\xi _ \\rho \\| ^ 2 , \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} \\begin{cases} u ^ k \\in C f ( u ^ k , y ) + \\frac { 1 } { \\rho _ k } \\langle y - u ^ k , u ^ k - x ^ k \\rangle \\geq 0 , y \\in C \\\\ x ^ { k + 1 } = \\beta _ k v ^ k + ( 1 - \\beta _ k ) T u ^ k . \\end{cases} \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} T ^ \\beta \\vdash \\bigwedge _ { i = 1 } ^ m ( \\Box _ { T ^ \\beta } \\psi _ i \\rightarrow \\psi _ i ) \\rightarrow [ 1 ] _ S \\varphi . \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} B _ n ( \\omega _ n ^ k ) = B _ { n / ( n , k ) } ^ { ( n , k ) } ( \\omega _ { n / ( n , k ) } ^ { k / ( n , k ) } ) . \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\in \\mathbb { Z } } \\sum _ { j = 0 } ^ { r - 1 } \\biggl ( c _ { n , j } ^ - u _ { k , n } ^ { [ j ] , - } ( y ) e ^ { 2 \\pi i n x } + c _ { n , j } ^ + u _ { k , n } ^ { [ j ] , + } ( y ) e ^ { 2 \\pi i n x } \\biggr ) , \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} \\int _ { \\Sigma } a ( s , t ) \\ , d s = 1 \\qquad t \\in \\Omega . \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 1 + \\lambda ( u ^ 2 ) ^ 2 & \\lambda ( - u ^ 1 u ^ 2 + u ^ 3 ) & \\lambda u ^ 2 \\\\ \\lambda ( - u ^ 1 u ^ 2 + u ^ 3 ) & 4 + \\lambda ( u ^ 1 ) ^ 2 & - 2 \\lambda u ^ 1 \\\\ \\lambda u ^ 2 & - 2 \\lambda u ^ 1 & \\lambda \\end{pmatrix} , \\\\ w _ { 1 2 } = \\frac { \\lambda \\sqrt { \\lambda } u ^ 1 } { \\sqrt { - \\det g } } , w _ { 2 3 } = 0 , w _ { 3 1 } = - \\frac { \\lambda \\sqrt { \\lambda } } { \\sqrt { - \\det g } } , \\end{gather*}"}
-{"id": "3095.png", "formula": "\\begin{align*} \\gcd ( q , n _ 1 ) = 1 \\gcd ( q , m ) = 1 . \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = P _ { - } \\left ( \\frac { q } { \\sigma _ { - } } \\right ) + \\frac { \\phi } { \\sigma _ { - } } + P _ { - } \\left ( \\frac { z ^ { n } \\psi } { \\sigma _ { - } } \\right ) \\\\ 0 & = P _ { + } \\left ( \\frac { q } { z ^ { n } \\sigma _ { + } } \\right ) + P _ { + } \\left ( \\frac { \\phi } { z ^ { n } \\sigma _ { + } } \\right ) + \\frac { \\psi } { \\sigma _ { + } } . \\end{aligned} \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} x _ 1 \\frac { \\partial } { \\partial { x _ 2 } } p _ { 2 3 } = - x _ 1 \\frac { \\partial } { \\partial { x _ 2 } } x _ 1 x _ 2 x _ 4 = - x _ 1 ^ 2 x _ 4 = - x _ 4 p _ { 4 } . \\\\ \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} \\sum ^ a _ { i = 1 } \\mu ( A _ i ) \\leq \\sum ^ b _ { j = 1 } \\mu ( B _ j ) , \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} \\alpha _ { i _ 1 } ^ t \\in K t = 0 , 1 , \\dots , p _ { i _ 1 } - 2 . \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} \\int \\limits _ t ^ s \\phi _ j ( \\tau ) d \\tau = \\frac { T - t } { 2 \\pi r } \\left \\{ \\begin{matrix} \\phi _ { 2 r - 1 } ( s ) , \\ & \\ j = 2 r \\cr \\cr \\cr \\sqrt { 2 } \\phi _ 0 ( s ) - \\phi _ { 2 r } ( s ) , \\ & \\ j = 2 r - 1 \\end{matrix} \\right . \\ , \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} N _ 1 ( \\delta ) + N _ 2 ( \\delta ) + \\dotsb + N _ m ( \\delta ) = \\lambda \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} \\Psi ( \\beta v ) = \\sum _ { j = 1 } ^ { N } \\Psi ( \\beta v ^ j ) & = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { 4 N } \\psi ( \\beta z _ j ) \\\\ & \\leq \\frac { 4 N } { 2 } \\psi \\left ( \\beta \\varrho \\left ( \\frac { 1 } { 4 N } \\sum _ { i = 1 } ^ { 4 N } \\psi ( z _ i ) \\right ) \\right ) \\\\ & = 2 N \\psi \\left ( \\beta \\varrho \\left ( \\frac { \\Psi ( v ) } { 4 N } \\right ) \\right ) . \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} W ( z ) = U ( z ) ^ { - 1 } ( X ( z ) - Y ( z ) ) V ( z ) ^ { - 1 } \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ \\infty c _ k ( f ) e ^ { i 2 \\pi k t / T } , \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} E _ 1 ( T _ { ( 2 , 8 ) } ) = ( \\Delta _ 1 ( T _ { ( 2 , 8 ) } ) ) \\cdot ( t _ 1 - 1 , t _ 2 - 1 ) = ( t _ { 1 } ^ { 3 } + t _ { 1 } ^ { 2 } t _ { 2 } + t _ { 1 } t _ { 2 } ^ { 2 } + t _ { 2 } ^ { 3 } ) \\cdot ( t _ 1 - 1 , t _ 2 - 1 ) . \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} u _ { m + k - 1 } ( t , 1 ) = M _ { k - 1 } \\Big ( u _ { k + 1 } \\big ( t , x _ { k + 1 } ( - \\tau _ { m + k - 1 } , 0 , 0 ) \\big ) , \\dots , u _ { k + m - 2 } \\big ( t , x _ { k + m - 2 } ( - \\tau _ { m + k - 1 } , 0 , 0 ) \\big ) \\Big ) , \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} e _ { n , k } ( 1 ) = | C _ { n , k } | . \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} \\ , ( X _ { \\cdot } , \\widetilde { X } _ { \\cdot } ) \\ , = \\ , ( X _ { \\cdot } ^ { ( u ) } , \\widetilde { X } _ { \\cdot } ^ { ( u ) } ) \\ , : = \\ , ( \\overline { X } _ { \\cdot , 1 } , \\overline { X } _ { \\cdot , 2 } ) \\ , \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } p _ s ^ { \\beta } ( t , r ) \\ , d s & = ( t - r ) ^ { \\beta - 1 } \\int _ 0 ^ { \\infty } u ^ { - 1 / \\beta } p ^ { \\beta } _ 1 ( u ^ { - 1 / \\beta } ) \\ , d u = ( t - r ) ^ { \\beta - 1 } \\frac { 1 } { \\Gamma ( \\beta ) } , \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} \\mathcal { P } _ { 0 } = 1 \\end{align*}"}
-{"id": "9961.png", "formula": "\\begin{align*} V _ U = U \\times \\sqrt { - 1 } \\R ^ n \\subset X . \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} Y _ t = \\xi + \\int _ t ^ T \\hat f \\left ( r , \\omega , Y _ r , Z _ r \\right ) \\mathrm d r - \\int _ t ^ T Z _ r \\mathrm d W _ r . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} \\{ n \\in \\mathbb N \\ , : \\ , K \\cap K _ n \\not = \\emptyset \\} \\hbox { i s a f i n i t e s e t . } \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} & \\left ( 5 \\beta \\gamma ^ 2 , \\left ( 5 \\beta \\gamma ^ 2 \\right ) ^ \\sigma \\right ) _ 2 = 1 \\\\ \\implies & \\left ( 5 \\beta , \\left ( 5 \\beta \\right ) ^ \\sigma \\right ) _ 2 = 1 \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} \\Delta ^ { - 1 } [ a , c ] = L _ { 2 1 } ( a - b d ^ { - 1 } c ) \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} \\mathbf { I } _ \\alpha ^ k ( \\mu , X ) ( x ) = \\int _ X \\frac { 1 } { | x - z | ^ { k - \\alpha } } d | \\mu | ( z ) ~ ~ \\forall ~ x \\in X , 0 < \\alpha < k . \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} \\frac { \\log { \\log { N } } } { N } \\le \\frac { 3 } { 8 } \\frac { \\log { N } } { N } < \\frac { 3 } { 8 } \\left ( \\frac { 5 \\tau ^ 2 A _ F } { 4 } + 1 \\right ) \\frac { \\log { N } } { N } \\le \\frac { 3 } { 8 } \\cdot \\frac { 1 } { 3 } \\log { R } = \\frac { 1 } { 8 } \\log { R } . \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} C ^ { - 1 } ( f d g ) = f ^ p g ^ { p - 1 } d g \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} [ s _ 0 , \\ldots , s _ { n - 1 } ] : = Y _ { s _ 0 } \\cap h _ { s _ 0 } Y _ { s _ 1 } \\cap \\cdots \\cap ( h _ { s _ 0 } \\cdots h _ { s _ { n - 2 } } ) Y _ { s _ { n - 1 } } . \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} \\begin{aligned} & g _ 1 h _ 1 \\exp ( \\alpha X ) = \\exp \\left ( - \\sum t _ { k l } X _ l ^ k \\right ) \\tilde { x } \\gamma \\tilde { x } ^ { - 1 } \\exp \\left ( \\sum t _ { k l } X _ l ^ k \\right ) , \\\\ & g _ 2 h _ 2 \\exp ( \\alpha X ) = \\exp \\left ( - \\sum s _ { k l } V _ l ^ k \\right ) \\tilde { x } \\gamma \\tilde { x } ^ { - 1 } \\exp \\left ( \\sum s _ { k l } X _ l ^ k \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} t _ { 0 } = \\left ( 2 n \\right ) ^ { - 1 / 2 } \\sum _ { i = 1 } ^ { n } \\left ( y _ { i } ^ { 2 } - 1 \\right ) \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} C ( \\rho ^ 0 , \\rho ^ 1 ) : = \\inf _ { v _ t } ~ \\int _ 0 ^ 1 \\int _ \\Omega L ( x , v ( t , x ) ) \\rho ( t , x ) d x d t , \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} { \\rm I } ( \\mathcal { S } _ R ) \\geq | V \\cdot \\coprod \\limits _ { i = 1 } ^ r b _ i ^ { n _ i - 1 } | + 1 = ( | V | + 1 ) + \\sum \\limits _ { i = 1 } ^ r ( n _ i - 1 ) = { \\rm D } ( { \\rm U } ( \\mathcal { S } _ R ) ) + \\Omega ( K ) - \\omega ( K ) . \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} f : \\begin{array} { l l } x _ { n + 1 } = T \\left ( x _ { n } \\right ) \\\\ y _ { n + 1 } = \\lambda y _ { n } + \\tau ^ { 1 / 2 } x _ { n } \\end{array} \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} \\frac { \\tilde { h } } { h } = \\mu _ h \\left ( \\frac { \\tilde { h } } { h } \\right ) , \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} ( V ( t ) ^ 2 ) ' = 2 V ' ( t ) V ( t ) , V ( t ) = | y ( t ) | \\cdot | \\widetilde { y ' } ( t ) | , \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} v _ A ( { t _ 0 , x _ 0 } ) = \\sup _ { u \\in \\mathcal { U } } \\Pi ^ { ( \\mathfrak { c } , u ) } [ t _ 0 , x _ 0 ] , v _ P ( { t _ 0 , x _ 0 } ) = \\sup _ { u \\in \\mathcal { U } } Q ^ { ( u ) } [ t _ 0 , x _ 0 ] , \\ ; \\forall ( t _ 0 , x _ 0 ) \\in [ 0 , T ] \\times S . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} h ( P , t ) : = f _ { n } ( P ) ( t - 1 ) ^ { n } + f _ { n - 1 } ( P ) ( t - 1 ) ^ { n - 1 } + \\dotsb + f _ { 0 } ( P ) \\ , . \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} \\sum _ { j = n } ^ { \\infty } e ^ { - \\alpha j } \\| r \\| = e ^ { - \\alpha n } ( 1 - e ^ { - \\alpha } ) ^ { - 1 } S . \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } I ^ { n , \\omega ^ \\prime } _ { T , R } = 0 \\ , . \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} \\deg ( D | _ V ) = \\deg ( D | _ { C ' } ) = \\deg ( D | _ C ) = k < k + 1 \\leq p _ a ( C ' ) = p _ a ( V ) \\ , , \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} i _ A = \\lim _ { t \\to \\infty } \\frac { \\log t } { \\log h _ A ( t ) } \\quad I _ A = \\lim _ { t \\to 0 ^ + } \\frac { \\log t } { \\log h _ A ( t ) } . \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { h ( n ) h ( n h ( n ) ) \\log T _ n } { n } = \\zeta , \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} & - y ^ 2 \\left ( \\tfrac { \\partial ^ 2 } { \\partial x ^ 2 } + \\tfrac { \\partial ^ 2 } { \\partial x ^ 2 } \\right ) \\phi ( \\tau ) = \\tfrac { 1 - \\nu _ \\infty ^ 2 } { 4 } \\ , \\phi ( \\tau ) , \\\\ & \\phi ( p \\tau ) + \\sum _ { j = 0 } ^ { p - 1 } \\phi \\left ( \\tfrac { \\tau + j } { p } \\right ) = ( p ^ { ( \\nu _ p + 1 ) / 2 } + p ^ { ( - \\nu _ p + 1 ) / 2 } ) \\ , \\phi ( \\tau ) . \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} \\begin{aligned} d ( \\pm \\exp ( ( 1 - \\lambda ^ { - 2 } ) \\alpha U ) , e ) & = d ( n \\exp ( s X ) n ^ { - 1 } \\exp ( - s X ) , e ) \\\\ & \\le d ( n \\exp ( s X ) n ^ { - 1 } , e ) + d ( \\exp ( - s X ) , e ) \\\\ & = d ( m , e ) + \\abs { s } \\\\ & \\le 1 6 0 C j \\delta + 1 6 0 j \\delta + 4 . \\end{aligned} \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} \\mathcal H \\bigl ( P _ L | P _ L ^ { \\tilde V } \\bigr ) = \\mathcal H \\bigl ( \\mu ^ L _ 0 | ( X _ 0 ) _ \\# P _ L ^ { \\tilde V } \\bigr ) + \\frac 1 2 \\int _ 0 ^ T \\Phi _ L \\bigl ( \\mu ^ L _ t , \\jmath ^ L _ t , F ^ { \\tilde V _ t } ( \\mu ^ L _ t ) \\bigr ) \\ ; \\ ! \\mathrm d t . \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} \\phi _ { B } ( \\textbf { Z } , \\delta ) = \\delta \\tilde { \\phi } _ { 1 } ( \\textbf { z } ) + ( 1 - \\delta ) \\tilde { \\phi } _ { 0 } ( \\textbf { x } ) \\\\ [ 1 0 p t ] \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} \\frac { k } { c _ 1 } = - \\frac { \\lambda G _ { t - 3 } ( \\theta ) } { F _ { t - 2 } ( \\theta ) } + 1 \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} E _ { \\bullet } = ( E _ 1 , E _ 2 , \\ldots , E _ k ) . \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta u + \\lambda \\phi u = g ( u ) & \\mathbb R ^ 3 \\\\ - \\Delta \\phi = \\lambda u ^ 2 & \\mathbb R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} s _ { h + 1 } : = \\frac { r ! } { t _ { h + 1 } } . \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} x ^ d + a _ { d - 1 } x ^ { d - 1 } + \\cdots + a _ { k } x ^ { k } + g ( x ) = ( x - x _ 1 ) \\cdots ( x - x _ r ) w ( x ) . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} P _ X ( m ) & \\leq \\binom { m + k - 1 } { k - 1 } + \\binom { m + k - 2 } { k - 1 } P _ { X / T } ( 1 ) + \\binom { m + k - 3 } { k - 1 } P _ { X / T } ( 2 ) + \\cdots \\\\ & + \\binom { k + 1 } { k - 1 } P _ { X / T } ( m - 2 ) + k P _ { X / T } ( m - 1 ) + P _ { X / T } ( m ) \\\\ & = \\binom { m + k - 1 } { k - 1 } + \\sum _ { i = 1 } ^ m \\binom { m + k - i - 1 } { k - 1 } P _ { X / T } ( i ) . \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} Y : = \\C ^ k \\times ^ \\Xi M \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} \\omega _ { \\xi \\eta } ( \\beta , \\beta ) = \\omega _ { \\xi \\eta } ( \\beta ) , \\quad \\hbox { a n d } \\omega _ { \\xi \\eta } ( \\varepsilon ) \\le \\omega _ { \\xi \\eta } ( \\beta , \\gamma ) , \\ ; \\ ; \\hbox { i f $ 0 < \\beta , \\gamma \\le \\varepsilon $ . } \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} e ^ { i t d \\Gamma ( \\omega ) } & = \\Gamma ( e ^ { i t \\omega } ) = W ( 0 , e ^ { i t \\omega } ) \\\\ e ^ { i t \\varphi ( i f ) } & = W ( t f , 1 ) . \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\beta _ 1 , \\beta _ 2 , \\beta _ 3 } : = \\int _ { \\mathcal { D } } f _ { \\mathbf { t } } ( x , y ) ^ s x ^ { \\beta _ 1 } y ^ { \\beta _ 2 } ( x ^ 4 - y ^ 3 ) ^ { \\beta _ 3 } \\frac { d x } { x } \\frac { d y } { y } \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} \\mathcal { C } _ r = \\mathcal { O } _ r ^ - \\cap \\Big ( U _ { - \\alpha } ( \\Xi _ r \\setminus \\Xi _ { \\frac r 2 } ) \\Big ) . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} & 0 < y _ { 0 } ^ { ( m ) } = \\langle y _ 0 , \\phi _ { m } \\rangle , 0 < y _ { 0 } ^ { ( M ) } = \\langle y _ 0 , \\phi _ { M } \\rangle \\\\ & 0 \\le y _ { 0 } ^ { ( k ) } = \\langle y _ 0 , \\phi _ { k } \\rangle m < k < M , \\quad 0 = y _ { 0 } ^ { ( k ) } \\quad \\quad 0 \\leq k < m . \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\begin{aligned} ( a b ) ^ \\prime = c \\ , , { ( a c ) } ^ \\prime = b \\ , , { ( b c ) } ^ \\prime = a \\ , . \\end{aligned} \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} m _ { 1 , \\Phi } ( g \\otimes f ) ( t ) = \\int _ G f ( s ) g ( s ^ { - 1 } t ) L ( s , s ^ { - 1 } t ) d t \\ \\ ( s \\in G ) , \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} \\mathcal { E } ( f , h ) : = - \\langle \\mathcal { L } f , h \\rangle _ \\mu = - \\langle f , \\mathcal { L } h \\rangle _ \\mu = \\frac { 1 } { \\beta } \\int _ { \\mathbb { R } ^ n } ( a \\nabla f ) \\cdot \\nabla h \\ , d \\mu \\ , , \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} \\Gamma ^ { \\rm { a b } } \\simeq \\widehat { \\rm { H } } ^ { - 2 } ( \\Gamma , \\Z ) \\xlongrightarrow { \\cup u } \\widehat { \\rm { H } } ^ 0 ( \\Gamma , L ^ { \\times } ) = k ^ { \\times } / N _ { L / k } ( L ^ { \\times } ) . \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} & d _ { n + 1 } ^ { \\otimes } ( H , G ) = \\\\ & \\frac { 1 } { \\vert H \\vert ^ { n + 1 } \\vert G \\vert } \\vert \\lbrace { ( x _ { 1 } , \\ldots , x _ { n + 1 } , y ) : [ x _ { 1 } , \\ldots , x _ { n + 1 } ] \\otimes y = 1 _ { H \\otimes G } , x _ { i } \\in H , y \\in G } \\rbrace \\vert . \\end{align*}"}
-{"id": "9888.png", "formula": "\\begin{align*} \\begin{aligned} ( \\boldsymbol { D } ^ { - s } \\psi , \\boldsymbol { D } ^ { s * } \\psi ) & = ( \\widehat { \\boldsymbol { D } ^ { - s } \\psi } , \\overline { \\widehat { \\boldsymbol { D } ^ { s * } \\psi } } ) \\\\ & = ( \\mathcal { F } ( \\boldsymbol { D } ^ { - s } \\psi ) , \\overline { \\mathcal { F } ( \\boldsymbol { D } ^ { s * } \\psi ) } ) \\\\ & = ( ( 2 \\pi i \\xi ) ^ { - s } \\widehat { \\psi } , \\overline { ( - 2 \\pi i \\xi ) ^ { s } \\widehat { \\psi } } ) . \\end{aligned} \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} f ( t ) & = t ^ \\beta a ( t ) \\exp \\Big ( \\int _ 1 ^ t \\frac { \\varepsilon ( y ) } y d y \\Big ) . \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} 0 = e ^ { - \\delta s } \\epsilon \\ln { x } + \\frac { \\partial \\phi } { \\partial s } ( s , x ) & + \\Gamma x \\frac { \\partial \\phi } { \\partial x } ( s , x ) + \\frac { 1 } { 2 } \\sigma ^ 2 x ^ 2 \\frac { \\partial ^ 2 \\phi } { \\partial x ^ 2 } ( s , x ) \\\\ & + \\int _ \\mathbb { R } \\Big \\{ \\phi ( s , x ( 1 + \\gamma ( z ) ) - \\phi ( s , x ) - x \\gamma ( z ) \\frac { \\partial \\phi } { \\partial x } \\Big \\} \\nu ( d z ) . \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} s _ i = d + i - k . \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} A _ u ^ 4 ( a _ 1 , a _ 2 ) ( \\xi _ 1 , \\xi _ 2 ) = \\pi ^ * g ^ B ( \\tau _ 0 ' ( { \\omega } _ u ^ h ( a _ 2 ) ) \\xi _ 1 , \\tau _ 0 ' ( { \\omega } _ u ^ h ( a _ 1 ) ) \\xi _ 2 ) - \\pi ^ * g ^ B ( \\tau _ 0 ' ( { \\omega } _ u ^ h ( a _ 1 ) ) \\xi _ 1 , \\tau _ 0 ' ( { \\omega } _ u ^ h ( a _ 2 ) ) \\xi _ 2 ) \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} u _ 2 ( T , x ) = 0 \\mbox { f o r } x \\in [ x _ { 1 2 } , 1 ] \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} \\Omega = \\left \\{ \\frac { e ^ { i ( \\mu + \\pi ) / \\ell } } { | \\zeta | q ^ n } ; \\ell \\in \\mathbb { Z } _ { > 0 } , L , n \\in \\mathbb { N } \\right \\} . \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} - \\mathbf { i } \\partial _ { \\omega } F _ { i j } + \\bar { \\Omega } _ { i j } F _ { i j } + \\Gamma _ K ( \\tilde { \\Omega } _ { i j } F _ { i j } ) = - \\mathbf { i } \\ \\Gamma _ K R _ { i j } , \\Gamma _ K F _ { i j } = F _ { i j } , \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} \\mbox { t r } \\Big ( ( \\mathbf { E } _ { \\mu _ z } A ) ( \\mathbf { E } _ { \\mu _ z } A ^ 2 ) ^ { \\frac { 1 } { 2 } } \\Big ) \\ge \\mathbf { E } _ { \\mu _ z } \\mbox { t r } \\Big ( ( \\mathbf { E } _ { \\mu _ z } A ) A ( \\cdot ) \\Big ) = \\mbox { t r } \\Big ( \\big ( \\mathbf { E } _ { \\mu _ z } A \\big ) ^ 2 \\Big ) \\ , , \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} d \\cdot \\iota _ { ( t + 1 ) / 2 , d } = \\tau ' _ t ( d ) \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} \\eta _ d : = \\min _ { E _ d \\in \\mathcal { C } ^ { 2 } _ d } \\# \\Theta _ { 2 } ( E _ d ) , \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} Y ^ { ( i ) } = \\bigcap _ { 1 \\leq t \\leq i } s _ t ^ { - 1 } ( 0 ) \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} w = \\sum _ { i = 1 } ^ a \\sum _ { j = 1 } ^ b x _ i y _ j w _ { i j } ( \\vec { u } ) \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} a _ 2 ( n ) & = \\frac { 2 ( n - 2 ) ^ 2 } n ( 1 8 n ^ 7 - 1 2 3 n ^ 6 + 3 7 0 n ^ 5 - 5 5 1 n ^ 4 + 2 5 8 n ^ 3 + 3 6 0 n ^ 2 - 5 2 8 n + 1 9 2 ) > 0 , \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} \\begin{gathered} \\forall v \\in E _ s ^ * ( X , x ) , \\ \\| ( d \\varphi _ { t } ^ X ( x ) ^ T ) ^ { - 1 } v \\| \\leq C e ^ { - \\lambda t } \\| v \\| , \\\\ \\forall v \\in E _ u ^ * ( X , x ) , \\ \\| ( d \\varphi _ { - t } ^ X ( x ) ^ T ) ^ { - 1 } v \\| \\leq C e ^ { - \\lambda t } \\| v \\| , \\end{gathered} \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} 0 = \\varepsilon ( x ) = \\sum _ { \\substack { \\lambda + \\nu = \\mu \\\\ i , j } } \\left ( x , v _ i ^ { \\lambda } v _ j ^ { \\nu } \\right ) u _ i ^ \\lambda S ( u _ j ^ { \\nu } ) K _ { - \\lambda } \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\hat { u } ( n - i + k ) \\sum _ { l = 0 } ^ { \\infty } ( n - j ) ^ { - 1 - l } h _ { l } \\left ( \\frac { j + k } { n - j } \\right ) , \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} \\big \\langle \\mu ^ r ( h ? ) \\psi , x \\big \\rangle & = \\mu ^ r ( h x ' ) \\psi ( x '' ) = \\mu ^ r ( h ' x ' ) \\psi \\big ( S ^ { - 1 } ( h ''' ) h '' x '' \\big ) = \\psi \\big ( S ^ { - 1 } ( h ''' ) \\mu ^ r ( h ' x ' ) h '' x '' \\big ) \\\\ & = \\psi \\big ( S ^ { - 1 } ( h '' ) \\mu ^ r ( ( h ' x ) ' ) ( h ' x ) '' \\big ) = \\mu ^ r ( h ' x ) \\psi \\big ( S ^ { - 1 } ( h '' ) \\big ) , \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} v ( 0 ) + \\lambda \\Theta _ { s } v ( 0 ) = \\varphi . \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} \\frac { { \\rm { D } } w } { \\partial \\overline { z ^ { i } } } = 0 \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} L _ t = \\int _ 0 ^ t ( b _ s - d _ s ) d s - \\int _ 0 ^ t \\log \\Big ( \\frac 1 { p _ s } \\Big ) d D _ s & \\sim \\iota h ( t ) . \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} { \\rm { d e t } } ( \\mathbf { M } ) = \\left ( \\prod _ { q = 0 } ^ { n - 1 } \\lambda _ { q + 1 } ^ f \\prod _ { 1 \\leq i < q + 1 } ( \\lambda _ { q + 1 } - \\lambda _ i ) ^ { m + 1 } \\right ) ^ { m + 1 } = \\left ( \\prod _ { i = 1 } ^ n \\lambda _ i \\right ) ^ { f ( m + 1 ) } \\left ( \\prod _ { 1 \\leq i < j \\leq n } ( \\lambda _ j - \\lambda _ i ) \\right ) ^ { ( m + 1 ) ^ 2 } , \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} g ^ \\pm ( \\lambda ) & = - \\frac 1 { 2 \\pi } \\big ( \\log ( \\lambda / 2 ) + \\gamma \\big ) \\pm \\frac { i } 4 \\\\ g _ 1 ^ \\pm ( \\lambda ) & = - \\frac { \\lambda ^ 2 } 4 g ^ \\pm ( \\lambda ) - \\frac { \\lambda ^ 2 } { 8 \\pi } \\\\ G _ 0 f ( x ) & = - \\frac { 1 } { 2 \\pi } \\int _ { \\R ^ 2 } \\log | x - y | f ( y ) \\ , d y , \\\\ G _ 1 f ( x ) & = \\int _ { \\R ^ 2 } | x - y | ^ 2 f ( y ) \\ , d y , \\\\ G _ 2 f ( x ) & = \\frac 1 { 8 \\pi } \\int _ { \\R ^ 2 } | x - y | ^ 2 \\log | x - y | f ( x ) \\ , d y . \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} \\int _ { D } \\Pi _ { k , D } ^ \\Delta \\xi \\ { \\rm d } s = \\int _ { D } \\xi \\ { \\rm d } x . \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} A = \\{ ( e , h ) \\in \\R \\times P _ c H ^ 1 : | e | \\leq | z | , \\| h \\| _ { H ^ 1 } \\leq | z | ^ 2 \\} , \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} c _ 0 ^ 2 C _ i | \\mathcal { t } _ i \\cdot \\nabla p | ^ 2 - C _ i ^ \\gamma = 0 \\qquad \\mbox { o n } \\Gamma _ i , \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} ( g \\otimes g ' ) \\circ ( f \\otimes f ' ) = ( - 1 ) ^ { | g ' | | f | } ( g \\circ f ) \\otimes ( g ' \\circ f ' ) , \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} - \\mathcal { L } _ R ^ { \\psi } [ \\chi ] = M + 1 \\quad B _ R , \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} f ( x , y ) & = ( P x + Q y + r ) ^ T ( y - x ) , \\\\ T x & = ( I + U ) ^ { - 1 } x , \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} R ^ { ( i ) } _ { } ( D ) = \\min \\limits _ { p ( w | u _ i ) } \\min \\limits _ { { g _ j : \\mathcal { U } _ j \\times \\mathcal { W } \\rightarrow \\mathcal { U } _ i } \\atop { \\mathbb { E } [ d ( U _ i , g _ j ( U _ j , W ) ) ] \\le D } } I ( U _ i ; W | U _ j ) \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} \\mathcal { L } _ P ( \\mathbb { R } ^ n ) \\circ L = \\{ \\psi \\in \\mathcal { L } _ P ( ( \\mathbb { C } ^ * ) ^ n ) \\ , : \\ , \\psi \\ ; \\rm { i s } \\ ; ( S ^ 1 ) ^ n { - \\rm i n v a r i a n t } \\} . \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} \\ell _ { \\star } ^ { 2 } = \\left \\{ u ( n , t ) : \\sup _ { t > 0 } | u ( n , t ) | \\in \\ell ^ 2 ( \\mathbb { N } ) \\right \\} . \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} \\epsilon _ { \\nu } ^ { 1 - 2 \\beta ' } \\frac { 2 ^ { \\nu } B _ { \\nu } } { \\alpha _ { 0 } } \\leq ( \\epsilon _ { 0 } \\prod _ { \\mu = 0 } ^ { \\infty } ( \\frac { 2 ^ { \\mu } B _ { \\mu } } { \\alpha _ { 0 } } ) ^ { \\frac { 1 } { 3 \\kappa ^ { \\mu + 1 } } } ) ^ { \\kappa ^ { \\nu } ( 1 - 2 \\beta ' ) } \\leq ( \\frac { \\gamma _ 0 } { 8 0 } ) ^ { \\kappa ^ { \\nu } } , \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} F = y ^ 2 - ( x ^ 5 - x ) ( x ^ 4 + 2 ) \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} H \\circ \\varphi = \\varphi \\circ H . \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} \\mathcal { T } _ \\xi = \\bigoplus _ { \\nu \\in P } \\mathcal { T } _ { \\xi , \\nu } \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} ( A \\wedge B ) _ { \\alpha \\beta } = A _ \\alpha B _ \\beta - A _ \\beta B _ \\alpha \\ , . \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} f _ B \\left ( \\mathbf { Q } \\right ) : = - \\frac { A } { 2 } \\textrm { t r } \\mathbf { Q } ^ 2 - \\frac { B } { 3 } \\textrm { t r } \\mathbf { Q } ^ 3 + \\frac { C } { 4 } \\left ( \\textrm { t r } \\mathbf { Q } ^ 2 \\right ) ^ 2 + M \\left ( A , B , C \\right ) \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} \\Delta ( y ) = \\sum _ { \\substack { 0 \\leq \\nu \\leq \\mu \\\\ i , j } } \\left ( u _ i ^ { \\nu } u _ j ^ { \\mu - \\nu } , y \\right ) L _ \\nu ^ { - 1 } v _ j ^ { \\mu - \\nu } \\otimes v _ i ^ { \\nu } . \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} b ' : = \\Psi ' \\circ ( h ' - \\mathfrak { s } ' ) \\circ \\psi '^ { - 1 } : \\hat { U } '' \\rightarrow \\hat { U } '' \\lhd \\mathbb { R } ^ { k ' } \\times ( \\tilde { \\mathbb { W } } \\times \\mathbb { R } ^ n ) \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} T ^ { \\beta } V _ { T , h } ( x ) \\xrightarrow [ T \\rightarrow \\infty ] { \\mathbb P } 0 \\textrm { $ ; $ } \\forall \\beta \\in [ 0 , 1 - H [ , \\mbox { w h e r e } \\ ; V _ { T , h } ( x ) : = \\left | \\frac { S _ { T , h } ( x ) } { \\widehat f _ { T , h } ( x ) } \\right | . \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} \\ \\mbox { s u c h t h a t } \\ \\rho = a _ { \\sigma } d _ { \\sigma } - b _ { \\sigma } c _ { \\sigma } \\ \\mbox { f o r e v e r y } \\ \\sigma \\in I _ F \\} = \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} \\inf _ { \\gamma \\in \\Gamma _ { p e r } ( y _ 0 ) } \\int _ { Y \\times U } k ( y , u ) \\gamma ( d y , d u ) = v _ { p e r } ( y _ 0 ) , \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} f = p _ 1 ^ { n _ { 1 } } p _ 2 ^ { n _ { 2 } } \\cdots p _ r ^ { n _ r } , \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\widehat { F _ y } ( t ) & = e ^ { - 2 \\pi ( y - c ) t } \\widehat { F _ c } ( t ) , \\ : t \\in \\mathbb { R } \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} \\mathcal { W } _ { k , n } ( y , s ) : = \\left \\{ \\begin{array} { l l } \\Gamma ( s + \\mathrm { s g n } ( n ) \\frac { k } { 2 } ) ^ { - 1 } | n | ^ { k - 1 } ( 4 \\pi | n | y ) ^ { - k / 2 } W _ { \\mathrm { s g n } ( n ) \\frac { k } { 2 } , s - 1 / 2 } ( 4 \\pi | n | y ) & n \\neq 0 , \\\\ \\dfrac { ( 4 \\pi ) ^ { 1 - k } y ^ { 1 - s - k / 2 } } { ( 2 s - 1 ) \\Gamma ( s - k / 2 ) \\Gamma ( s + k / 2 ) } & n = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} L _ n ^ { + } ( X , \\beta ) = _ K L _ n ^ { - } ( X , \\beta ) . \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\binom k j G _ j } = G _ { 2 k } \\ , . \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} z ( h ( v , v ) - h ( w , w ) ) = 0 . \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} u _ { N , j } ( 0 ) = u ^ 0 _ { N , j } , \\ , \\ , \\ , u ' _ { N , j } ( 0 ) = u ^ 1 _ j , \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{gather*} T _ { w ( i , m ) } ( x _ { i - 1 } ^ + ) = ( - 1 ) ^ { m - 1 } \\times \\begin{cases} E _ { 1 , N } ( m - 1 ) & , \\\\ E _ { i + 1 , i } ( m ) & . \\end{cases} \\end{gather*}"}
-{"id": "789.png", "formula": "\\begin{align*} \\dim X = 3 , \\ K _ X = 0 . \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} K ( \\beta , q ) : = \\left \\{ ( \\delta , p ) \\in \\mathbb R ^ 2 : \\ ; \\beta < \\delta < 1 - \\beta , \\ , \\frac d \\delta < p < q \\right \\} , \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} & f '' ( g _ 1 ) ( \\eta ) \\\\ & = \\sum _ j f ' ( h _ j ) ( L ( \\eta ) | _ { Y _ j } ) ~ ( \\mbox { d e f . o f $ T _ 1 $ } ) \\\\ & = \\sum _ j \\sum _ i f ( g _ i ) ( \\pi ( L ( \\eta ) | _ { Y _ j } ) | _ { X _ { g _ i , h _ j } } ) ~ ( \\mbox { d e f . o f $ S _ 1 $ } ) \\\\ & = \\sum _ { j , i } f ( g _ i ) ( \\eta | _ { X _ { g _ i , h _ j } \\cap \\psi ( Y _ j ) } ) . ~ ( \\mbox { L e m m a \\ref { l e m : c o m p o s i t i o n o f p i a n d L u n d e r r e s t r i c t i o n s } } ) \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} r _ e = \\frac { L _ { i j , i j } } { L _ { 1 1 } } \\ , . \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} \\left ( \\prod _ { 1 \\le j \\le m } a _ j ^ { \\binom { \\nu _ j } { 2 } } \\right ) \\prod _ { 1 \\le i < j \\le m } ( a _ i - a _ j ) ^ { \\min \\left \\{ \\nu _ i ^ 2 , \\ , \\nu _ j ^ 2 \\right \\} } \\underset { \\mathbb { Z } } { \\bigg | } D ( \\overline { a } ) , \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} \\tau | _ x = c _ 1 \\left ( e ^ { 4 5 } - e ^ { 6 7 } \\right ) + c _ 2 \\left ( e ^ { 4 6 } + e ^ { 5 7 } \\right ) + c _ 3 \\left ( e ^ { 4 7 } - e ^ { 5 6 } \\right ) , \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} \\psi _ \\varepsilon ( x ) = \\psi ( \\varepsilon ) + \\psi ' ( \\varepsilon ) ( x - \\varepsilon ) , x \\in [ 0 , \\epsilon ] . \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} v _ \\lambda \\cdot v _ \\mu = v _ { \\lambda + \\mu } \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} h _ \\mathrm { t o p } ( f ) = \\sum _ { \\lambda \\in \\mathrm { s p e c } ( M _ f ) : | \\lambda | > 1 } \\log | \\lambda | \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} \\sum _ i \\langle h _ i f ( h _ j , h _ l ) , \\tau | _ { ( g _ 0 g _ 1 ) X _ l \\cap g _ 0 X _ j \\cap X _ i } \\pi \\rangle = \\langle f ( h _ j , h _ l ) , ( g _ 0 ^ { - 1 } \\tau ) | _ { g _ 1 X _ l \\cap X _ j } \\pi \\rangle . \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} \\phi = e ^ { 1 2 3 } + e ^ { 1 4 5 } + e ^ { 1 6 7 } + e ^ { 2 4 6 } - e ^ { 2 5 7 } - e ^ { 3 4 7 } - e ^ { 3 5 6 } , \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} T \\Vert f ' _ h - f ' \\Vert _ 1 & = \\int _ 0 ^ T | f ' ( t + h ) - f ' ( t ) | \\mathrm { d } t = \\int _ 0 ^ T | g _ 1 ' ( t ) - g _ 2 ' ( t ) | \\mathrm { d } t \\\\ & \\leq \\int _ 0 ^ T g _ 1 ' ( t ) \\mathrm { d } t + \\int _ 0 ^ T g _ 2 ' ( t ) \\mathrm { d } t \\\\ & = g _ 1 ( T ) - g _ 1 ( 0 ) + g _ 2 ( T ) - g _ 2 ( 0 ) \\\\ & = V _ 0 ^ T ( g ) - V _ 0 ^ 0 ( g ) + V _ 0 ^ T ( g ) - g ( T ) - V _ 0 ^ 0 ( g ) + g ( 0 ) \\\\ & = 2 V _ 0 ^ T ( g ) - g ( T ) + g ( 0 ) \\\\ & = 2 V _ 0 ^ T ( g ) , \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} & { \\gamma } = C \\exp \\big ( C M + C M ^ 4 \\big ) \\Big ( X _ n ( 0 ) + M ^ 2 + 1 \\Big ) , \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} \\mu _ 1 ( \\rho _ 1 ) & \\ge \\nu \\left ( \\partial ( \\rho _ 1 B ) \\cap \\left ( { 1 \\over \\sqrt { 3 } } B + { \\bf v } _ 1 \\right ) \\right ) \\\\ & = 2 \\pi \\sqrt { { 1 \\over 3 } + { 4 \\over { 5 7 } } \\sqrt { { 2 0 3 3 } \\over 3 } } \\left ( \\sqrt { { 1 \\over 3 } + { 4 \\over { 5 7 } } \\sqrt { { 2 0 3 3 } \\over 3 } } - { { \\sqrt { 2 0 3 3 } } \\over { 5 7 } } - { 1 \\over \\sqrt { 3 } } \\right ) . \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} \\prod _ { i , j = 1 } ^ N \\frac { ( q ^ { s _ i - a _ j + 1 } ; q ) _ { \\infty } } { \\sin \\pi ( s _ i - a _ j ) } \\prod _ { 1 \\le i < j \\le N } \\frac { \\sin \\pi ( s _ j - s _ i ) \\sin \\pi ( a _ j - a _ i ) ( q ^ { s _ j } - q ^ { s _ i } ) ( q ^ { a _ j } - q ^ { a _ i } ) } { ( q ^ { a _ i - a _ j } ; q ) _ { \\infty } ( q ^ { a _ j - a _ i } ; q ) _ { \\infty } } \\end{align*}"}
-{"id": "9913.png", "formula": "\\begin{align*} \\Bigl ( \\frac { \\partial f } { \\partial z _ { i } } \\ , d \\bar z _ { 1 } \\wedge \\cdots \\wedge d \\bar z _ { n } , \\ , \\sum _ { j = 1 } ^ { n } \\frac { \\partial g _ { j } } { \\partial z _ { i } } \\ , d \\bar z _ { 1 } \\wedge \\cdots \\wedge \\widehat { d \\bar z _ { j } } \\wedge \\cdots \\wedge d \\bar z _ { n } \\Bigr ) . \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} w _ p = u _ p - \\tfrac { i } { 2 } \\bar { z } _ p . \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{gather*} \\phi _ * ( T _ x N ) + \\varrho _ A ( A _ { \\phi ( x ) } ) = T _ { \\phi ( x ) } M , x \\in N , \\end{gather*}"}
-{"id": "5025.png", "formula": "\\begin{align*} \\dim T o p ( \\phi ) = ( q ^ { ( e - 1 ) ( m - 1 ) } + ( - 1 ) ^ m q ^ { ( e - 2 ) ( m - 1 ) } ) ( q ^ m + ( - 1 ) ^ { m + 1 } ) / ( q + 1 ) . \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} \\int \\limits _ t ^ s \\phi _ j ( \\tau ) d \\tau = \\frac { T - t } { 2 } \\left ( \\frac { \\phi _ { j + 1 } ( s ) } { \\sqrt { ( 2 j + 1 ) ( 2 j + 3 ) } } - \\frac { \\phi _ { j - 1 } ( s ) } { \\sqrt { 4 j ^ 2 - 1 } } \\right ) \\ \\ \\ \\hbox { f o r } \\ \\ \\ j \\ge 1 . \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { Z } } 2 ^ { j s } \\left \\| [ u \\cdot \\nabla , \\dot { \\Delta } _ j ] v \\right \\| _ { L ^ 2 } \\lesssim \\| \\nabla u \\| _ { \\dot { B } _ { 2 , 1 } ^ { \\frac { n } { 2 } } } \\| v \\| _ { \\dot { B } _ { 2 , 1 } ^ { s } } , \\ s = \\frac n 2 , \\ - 1 + \\frac n 2 \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} h _ { t } = - \\Delta ^ { 2 } h + \\sum _ { j = 2 } ^ { \\infty } \\Delta F _ { j } , \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} V : = H ^ 0 ( \\overline { Y } , A ) \\hookrightarrow H ^ 0 ( Y , \\O _ Y ( k D ) ) . \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} \\nabla ^ * \\Psi ^ { ( i ) } = \\beta \\nabla V \\cdot \\Psi ^ { ( i ) } - \\mbox { d i v } \\ , \\Psi ^ { ( i ) } = \\beta \\nabla V \\cdot ( a \\Pi \\nabla u _ i ) - \\mbox { d i v } ( a \\Pi \\nabla u _ i ) = - \\beta \\mathcal { L } _ 0 u _ i \\ , . \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} \\widehat { q } _ { y } ( \\textbf { x } ) = \\sum _ { j : Y _ j = y } \\delta _ { j } K \\left ( \\dfrac { \\textbf { X } _ j - \\textbf { x } } { h _ { n } } \\right ) \\Big / \\sum _ { j : Y _ j = y } K \\left ( \\dfrac { \\textbf { X } _ j - \\textbf { x } } { h _ { n } } \\right ) , \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} \\chi _ { ( \\lambda , \\mu ) } = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } e ^ { ( w \\bullet ( \\lambda , \\mu ) ) } } { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } e ^ { ( w \\bullet ( 0 , 0 ) ) } } . \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} \\rho ( m _ t ) R _ j = R _ j - \\epsilon ( [ g _ t g _ j ^ { - 1 } ] ) R _ t , \\rho ( m _ t ^ { - 1 } ) R _ j = R _ j + \\epsilon ( [ m ^ { - 1 } g _ t g _ j ^ { - 1 } ] ) R _ t , \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\Sigma = \\left \\{ ( w _ i ) \\in \\{ a , b \\} ^ \\mathbb { Z } : \\ \\mbox { w o r d } \\ b b \\ \\mbox { i s f o r b i d d e n } \\right \\} \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} A ( G ) _ 1 = \\Z [ G ^ 2 ] , \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} u _ 1 ( x , t ) = u _ \\Omega ( x ) \\tanh ( x - X ( t ) ) , \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} \\binom { \\bar { \\partial } \\underline { \\omega } } { 0 } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\binom { 0 } { \\omega } \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} u _ { + , a } ( x ) = \\frac { \\phi _ 1 ( x ) } { \\phi _ 1 ( a ) } , \\ x \\le a ; \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} f _ { \\geq 4 } ( x , u , \\bar { u } ) = \\frac { \\partial F _ { \\geq 5 } } { \\partial { \\bar { u } } } ( x , u , \\bar { u } ) , \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} p _ 1 ^ { ( \\rm i d , a ) } = { P _ T } \\left ( { 1 + \\textstyle \\sum \\limits _ { k = 2 } ^ { r } \\left ( \\frac { \\beta } { \\beta + [ \\boldsymbol { \\Lambda } ] _ { 1 , 1 } ^ 2 - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } \\right ) } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} L _ i F _ j = q ^ { - \\langle \\alpha _ i , \\alpha _ j \\rangle } F _ j L _ i , \\forall 1 \\leq i , j \\leq n , \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} w = p _ 0 B _ 1 C _ 1 p _ 1 B _ 2 C _ 2 \\cdots B _ k C _ k p _ k ( k \\in \\mathbb { N } _ 0 ) , \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} S ( x ) : = K ( x , 0 ) \\Sigma ( 0 ) Q . \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{gather*} S \\left ( \\langle G _ + \\rangle ^ { \\perp } \\right ) = \\langle F _ - \\rangle ^ { \\perp } S \\left ( \\langle F _ - \\rangle ^ { \\perp } \\right ) = \\langle G _ + \\rangle ^ { \\perp } S \\left ( \\langle G _ - \\rangle ^ { \\perp } \\right ) = \\langle F _ + \\rangle ^ { \\perp } \\theta \\left ( \\langle F _ + \\rangle ^ { \\perp } \\right ) = \\langle F _ - \\rangle ^ { \\perp } \\end{gather*}"}
-{"id": "1615.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\nabla \\cdot D _ p \\overline { L } ( \\nabla \\overline { u } ) = 0 = \\nabla \\cdot D _ p \\overline { L } ( \\nabla \\overline { v } ) & \\mbox { i n } & \\ B _ { R / 2 } , \\\\ & \\overline { u } = u , \\ \\overline { v } = v & \\mbox { o n } & \\ \\partial B _ { R / 2 } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} \\chi _ j ( t ) = \\begin{cases} 1 , & t _ { j } < t < t _ { j + 1 } , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} \\ell _ { n _ s } & = \\lfloor \\sqrt [ 3 ] { n _ s - d _ s + d _ s } \\rfloor = \\left \\lfloor \\sqrt [ 3 ] { n _ s - d _ s } \\ , \\sqrt [ 3 ] { 1 + \\frac { d _ s } { n _ s - d _ s } } \\right \\rfloor \\\\ & = \\left \\lfloor s \\ , \\sqrt [ 3 ] { 1 + \\frac { d _ s } { s ^ 3 } } \\right \\rfloor \\le \\left \\lfloor s \\ , \\Big ( 1 + \\frac { 1 } { 3 } \\frac { d _ s } { s ^ 3 } \\Big ) \\right \\rfloor = s \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} G _ { \\left ( y ^ { p } + c \\right ) ^ \\alpha } ( p s _ 1 ) + G _ { \\left ( 1 + c x ^ { p } \\right ) ^ \\alpha } ( p s _ 2 ) = \\frac { c ^ { - s _ 2 } } { p } \\boldsymbol { B } \\left ( s _ 1 , s _ 2 \\right ) \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} \\begin{aligned} | \\sin \\left ( N \\pi \\left ( u _ { 0 } + \\frac { 2 } { N } \\cdot j \\right ) \\right ) | = & | \\sin \\left ( N \\pi u _ { 0 } + 2 \\pi j \\right ) | \\\\ = & | \\sin \\left ( N \\pi u _ { 0 } \\right ) | \\ \\ \\ \\end{aligned} \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} \\widehat { \\mathrm m } _ { t } \\ , = \\ , ( Y _ { t } ) \\ , = \\ , ( X _ { t } ) \\ , = \\ , ( \\widetilde { X } _ { t } ) \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} 0 = E _ 0 \\subset E _ 1 \\subset \\cdots \\subset E _ n = E \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} h _ j ( x _ j ) = a _ j x _ j + l n ( 1 + \\gamma _ j x _ j ) , \\ ( j = 1 , \\ldots , n ) , \\ h _ i ( x _ j ) : = \\mu _ i x _ i \\ ( i = n + 1 , \\ldots , N ) . \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\mbox { a n a l y t i c d - c r i t i c a l f l i p } & \\mbox { i f } h ^ 1 ( F ) > 1 \\\\ \\mbox { a n a l y t i c d - c r i t i c a l d i v i s o r i a l c o n t r a c t i o n } & \\mbox { i f } h ^ 1 ( F ) = 1 \\\\ \\mbox { a n a l y t i c d - c r i t i c a l M F S } & \\mbox { i f } h ^ 1 ( F ) = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ { g ( x ) } \\mathbf { E } \\left [ ( { \\rm d i s t } ( x + g _ { l } + S ( l ) ) , \\partial K ) ^ { - 2 - \\delta } ; \\tau _ x > l \\right ] \\le C g ^ { - \\gamma } ( x ) . \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} L : = \\sum _ { n \\in \\Z } \\lambda _ { \\pi } ( m ) F \\left ( \\frac { n } { M } \\right ) \\ll _ { \\pi , \\epsilon } Z ^ { \\frac { 5 } { 2 } } l ^ { \\frac { 7 } { 6 } } M ^ { \\frac { 1 } { 2 } } l ^ { ( \\frac { 1 } { 6 } + \\epsilon ) n _ l } , \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mu _ t = - \\pi \\big ( P _ { \\mu _ t } \\big ) \\mu _ t , \\mu _ { t | = 0 } = \\mu _ 0 \\ , . \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} \\mathcal { H } = \\bigg \\{ f ~ \\Big | ~ f : \\mathbb { R } ^ n \\rightarrow \\mathbb { R } , ~ | f | _ \\mu < + \\infty \\bigg \\} \\ , , \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} A ^ { W , W } _ t ( ( \\nabla \\gamma ^ i ) ^ * \\ , b + \\tilde f ^ i ) = h ^ i ( t , W _ t ) - h ^ i ( 0 , W _ 0 ) - \\int _ 0 ^ t ( \\nabla h ^ i ) ^ * ( r , W _ r ) \\mathrm d W _ r . \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} \\mathcal { L } ( G - \\sum _ { i = 1 } ^ { N } P _ i ) \\setminus \\bigcup _ { j = 1 } ^ { N } \\mathcal { L } ( G - \\sum _ { i = 1 } ^ { N } P _ i + P _ j ) . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} \\left | f ( t ) - \\sum _ { k = 0 } ^ { p - 1 } b _ k \\frac { t ^ { k r } } { ( k r ) ! } \\right | \\le C _ 2 A _ 2 ^ p \\frac { M _ p } { ( p r ) ! } | t | ^ { p r } . \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} { \\bf A } _ { k , n } ( \\phi ) = \\sum _ { f \\in H _ k } \\mu _ { f } ( \\phi ) \\ , \\lambda _ { f } ( n ) , n \\in \\N \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} K _ { 3 } ( U , V ) W = \\{ ( \\alpha ^ { 2 } + \\beta ) g ( V , W ) + \\alpha \\beta \\Phi ( V , W ) \\} \\eta ( U ) . \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} \\nabla \\cdot ( \\mathbb { P } [ \\mu ] \\nabla p ) & = \\partial _ s ( \\bar { C } \\partial _ s p ) \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} H _ 1 ^ \\top H _ 1 = \\frac { 1 } { \\theta - \\tau } ( n ^ 2 A - \\tau n I _ n - ( k - \\tau ) J _ n ) . \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} G \\simeq { \\rm { H } } _ 0 ( A ( G ) ) , \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} u \\in L ^ r _ { l o c } ( \\Omega ) \\hbox { a s : } u \\equiv u _ m \\hbox { o n } \\Omega _ m , m = 1 , 2 , 3 , \\cdot \\cdot \\cdot . \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} N ^ { + } \\hookrightarrow \\widehat { M } ^ { + } , \\ ( \\vec { x } , \\vec { u } ) \\mapsto ( \\vec { x } , \\vec { y } = 0 , \\vec { u } ) . \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} a ( k ) x = & ( F _ { - \\eta } ( v , \\omega ) + \\omega ( k ) - \\lambda ) ^ { - 1 } a ( k ) ( F _ { \\eta } ( v , \\omega ) - \\lambda ) x \\\\ & - v ( k ) ( F _ { - \\eta } ( v , \\omega ) + \\omega ( k ) - \\lambda ) ^ { - 1 } . \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} R _ { \\alpha } \\tilde { v } = \\sum \\limits _ { \\sigma _ j \\geq \\alpha } \\frac { \\beta _ j } { \\sigma _ j } \\varphi _ j \\in H _ 1 \\subset H ^ { s } _ { \\overline { W } } . \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} D _ { n } [ \\sigma ] = \\mathbf { G } [ \\tau ] ^ { n + 1 } n ^ { - \\beta ^ { 2 } } G ( 1 + \\beta ) G ( 1 - \\beta ) E [ \\tau ] \\left ( 1 + o ( 1 ) \\right ) , \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} \\int _ { G } | z | ^ s | k _ { l } ( z ) | d z & = \\int _ { G } | z | ^ { s - \\frac { n } { 2 } - \\varepsilon } | z | ^ { \\frac { n } { 2 } + \\varepsilon } | k _ { l } ( z ) | d z \\\\ & \\leq I _ { s , n , \\varepsilon } \\Vert \\psi _ { l } \\Vert _ { { H } ^ { \\frac { n } { 2 } + \\varepsilon } ( \\widehat { G } ) } , \\ , \\textnormal { w h e r e , } \\ , I _ { s , n , \\varepsilon } : = \\left ( \\int _ { G } | z | ^ { 2 ( s - \\varepsilon ) - n } d z \\right ) ^ { \\frac { 1 } { 2 } } < \\infty . \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} \\xi \\circ \\varphi = \\varphi \\circ \\xi , \\qquad \\forall \\xi \\in H , \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi ( x , y ) & = \\langle \\tilde { B } _ 1 x - a , y \\rangle + y ^ T B _ 1 y + h ( y ) - x ^ T B _ 1 x + a ^ T x - h ( x ) \\\\ & \\leq \\langle \\tilde { B } _ 1 u ^ I - a , y \\rangle + y ^ T B _ 1 y + h ( y ) - ( l ^ I ) ^ T B _ 1 l ^ I + a ^ T u ^ I - h ( l ^ I ) . \\end{aligned} \\end{align*}"}
-{"id": "9864.png", "formula": "\\begin{align*} ( \\phi , \\boldsymbol { D } ^ { - \\sigma } \\psi ) = ( \\boldsymbol { D } ^ { - \\sigma * } \\phi , \\psi ) , \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} H = \\{ i _ 1 , \\ldots , i _ 1 + ( t - 1 ) , \\ldots , i _ { d - 2 } , \\ldots , i _ { d - 2 } + ( t - 1 ) , i _ { d - 1 } , \\ldots , i _ { d - 1 } + s \\} \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} \\alpha _ 1 | \\Q - \\Pi ( \\Q ) | & \\leq \\alpha _ 2 \\sqrt { f _ B ( \\Q ) } \\leq \\left ( \\frac { \\partial f _ B } { \\partial Q _ { i j } } ( \\Q ) + \\frac { B } { 3 } | \\Q | ^ 2 \\delta _ { i j } \\right ) \\nu _ { i j } ( \\Q ) \\\\ & \\leq \\alpha _ 3 \\sqrt { f _ B ( \\Q ) } \\leq \\alpha _ 4 | \\Q - \\Pi \\left ( \\Q \\right ) | \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} V _ { \\zeta } = V \\left ( a ( \\zeta ) , b ( \\zeta ) \\right ) = \\{ \\gamma \\in \\C \\ ; | \\ ; \\sin ^ { - 1 } p ( \\zeta ) < \\arg \\gamma < \\pi - \\sin ^ { - 1 } p ( \\zeta ) \\} . \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} \\ln R ( n ) \\geq \\ln R ( n _ 0 ) - \\frac { 1 } { 3 } \\sum _ { k = n _ 0 } ^ n \\frac { 1 } { k } . \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} \\| u \\| _ { p _ { m + n } } \\leq \\liminf _ { i \\to + \\infty } \\| u _ i \\| _ { p _ { m + n } } = 1 . \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} | \\sum _ { b = 1 } ^ n k _ b j _ b ^ 2 + i ^ 2 + j ^ 2 | < \\frac { 1 } { 5 0 n } ( | i | + | j | ) , \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow \\infty } \\frac { q ^ { 1 - 1 / p } } { ( \\Gamma ( q / p ' + 2 ) ) ^ { \\frac { 1 } { q } } } = \\lim _ { q \\rightarrow \\infty } \\frac { q ^ { 1 - 1 / p } } { ( 1 + o ( 1 ) ) \\left ( \\frac { q } { e p ' } \\right ) ^ { 1 / p ' } } = ( e p ' ) ^ { 1 / p ' } , \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} \\| \\sqrt { H } P _ c v ( t ) \\| _ { L _ x ^ 2 } ^ 2 = \\| \\sqrt { H } P _ c v ( 0 ) \\| _ { L ^ 2 } ^ 2 + \\Im \\int _ 0 ^ t \\langle \\sqrt { H } P _ c v ( s ) , \\sqrt { H } P _ c \\ , \\N \\rangle \\ , d s , \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } \\int _ K \\int _ H f ( k ^ { - 1 } n h ) d h d k & = \\int _ N f ( n ) \\rho _ 1 ( n ) d n \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} - \\Delta Q _ { c , \\emph { r a d } } - c | x | ^ { - 2 } Q _ { c , \\emph { r a d } } + Q _ { c , \\emph { r a d } } = | Q _ { c , \\emph { r a d } } | ^ { \\alpha } Q _ { c , \\emph { r a d } } . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 8 6 4 n + 2 1 6 ) q ^ n & \\equiv \\dfrac { f _ 3 ^ 2 } { f _ { 6 } } \\cdot f _ 2 ^ 3 \\\\ & \\equiv \\dfrac { f _ 3 ^ 2 } { f _ { 6 } } \\left ( f _ 6 a ( q ^ 6 ) - 3 q ^ 2 f _ { 1 8 } ^ 3 \\right ) ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} \\{ x _ { 2 , \\tau } , p _ { 2 , \\tau } \\} _ { D B } = 1 ; \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} \\xi ( t ) : = \\left ( \\frac { t ^ 2 } { 2 } - \\frac { t ^ 4 } { 4 } \\right ) \\left ( \\abs { \\nabla u } ^ 2 + u ^ 2 \\right ) + \\frac { t ^ 4 } { 4 } f ( u ) u - F ( t u ) , t \\geq 0 \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} C ( \\pi , s _ i v ) \\subset C ( \\pi , v ) \\cup \\left \\{ \\tilde { f } _ i ^ a ( \\pi _ 1 \\otimes \\pi _ 2 ) \\left | \\begin{array} { l } \\pi _ 1 \\otimes \\pi _ 2 \\in C ( \\pi , v ) , \\\\ \\tilde { e } _ i ( \\pi _ 1 \\otimes \\pi _ 2 ) = 0 , \\\\ \\tilde { f } _ i ^ a ( \\pi _ 1 \\otimes \\pi _ 2 ) \\not \\in D ( \\pi , ) \\sqcup \\{ 0 \\} , \\\\ a \\geq 0 . \\end{array} \\right . \\right \\} \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} \\sum _ { \\beta \\in Q ^ { \\vee } _ + , \\vec { n } \\in \\Z ^ r _ { \\ge 0 } } f _ { \\beta , \\vec { n } } ( q ) \\otimes _ { R ( G ) [ q ^ { \\pm 1 } ] } \\left ( \\prod _ { j = 1 } ^ r ( p _ i ^ { - 1 } q ^ { Q _ i \\partial _ { Q _ i } } ) ^ { n _ i } \\right ) Q ^ { \\beta } J ( Q , q ) = 0 , \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} \\partial _ t g _ t = - S ( g _ t ) + Q ^ 1 ( g _ t ) \\ , , g _ { | t = 0 } = g _ 0 . \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{gather*} \\abs { \\bigcup _ { p \\in P D _ d } \\left \\{ \\alpha \\in X _ n ^ k : \\alpha \\ \\mbox { i s $ \\frac { d } { p } $ - a d d i t i v e } \\right \\} } = \\sum _ { \\{ B \\subset P D _ d : B \\neq \\emptyset \\} } ( - 1 ) ^ { \\abs { B } + 1 } \\left ( \\begin{array} { c } \\frac { d } { \\prod _ { p \\in B } p } \\frac { n + 2 } { k + 1 } - 1 \\\\ \\frac { d } { \\prod _ { p \\in B } p } - 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "7436.png", "formula": "\\begin{align*} \\nabla _ r u ( x ) : = \\left ( \\frac { x } { | x | } \\cdot \\nabla u ( x ) \\right ) \\frac { x } { | x | } , \\nabla _ { \\mathbb { S } ^ { n - 1 } } u ( x ) : = \\nabla u ( x ) - \\nabla _ r u ( x ) . \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} L _ { u } ( \\beta _ 2 ) = & u \\beta _ 2 ^ { q ^ 2 } + ( u + 1 ) \\beta _ 2 ^ q + ( u ^ 2 + u ) \\beta _ 2 \\\\ = & \\beta _ 2 \\left [ u \\beta _ 2 ^ { q ^ 2 - 1 } + ( u + 1 ) \\beta _ 2 ^ { q - 1 } + ( u ^ 2 + u ) \\right ] \\\\ = & \\beta _ 2 \\left [ u \\left ( u ^ 2 \\right ) ^ { q + 1 } + ( u + 1 ) u ^ 2 + ( u ^ 2 + u ) \\right ] \\\\ = & \\beta _ 2 \\left [ u ( u + 1 ) ^ 2 + ( u + 1 ) u ^ 2 + ( u ^ 2 + u ) \\right ] \\\\ = & 0 . \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} - \\nabla \\cdot \\left ( D _ p \\overline { L } \\left ( \\nabla \\overline { u } _ r \\right ) \\right ) = - \\nabla \\cdot \\left ( D _ p \\overline { L } \\left ( \\nabla \\overline { v } _ r \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { y } ^ { ( k ) } ( t ) & = \\mu _ { k } ( t ) \\ , y ^ { ( k ) } ( t ) , \\mu _ { k } ( t ) : = R ( t ) - D \\lambda _ k \\qquad t > 0 , \\\\ y ^ { ( k ) } ( 0 ) & = \\langle y _ 0 , \\phi _ k \\rangle . \\end{aligned} \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} \\ln f _ \\alpha ( e ^ { t } ) = \\int _ { t _ 0 } ^ t g ( s ) d s + \\ln f _ \\alpha ( e ^ { t _ 0 } ) < ( - \\alpha - 1 - \\epsilon ) t + \\ln f _ \\alpha ( e ^ { t _ 0 } ) \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} d _ { M , } = \\liminf _ { n \\rightarrow \\infty } E ( v _ n ) \\geq \\liminf _ { n \\rightarrow \\infty } \\left ( E ( V ) + E ( r _ n ) \\right ) & \\geq E ( V ) + \\liminf _ { n \\rightarrow \\infty } E ( r _ n ) \\\\ & \\geq E ( V ) \\geq d _ { M , } . \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) = \\sum _ { k = 1 } ^ g a _ { i j k } u _ k + ( \\mbox { h i g h e r o r d e r t e r m s i n } \\vec { u } ) \\end{align*}"}
-{"id": "9975.png", "formula": "\\begin{align*} e ( G ' ) & \\leq \\binom { ( n - i ) - ( k - i ) + 1 } { 2 } + \\frac { ( k - i ) ^ 2 - 3 ( k - i ) + 4 } { 2 } \\\\ [ 5 p t ] & = \\binom { n - k + 1 } { 2 } + \\frac { ( k - i ) ^ 2 - 3 ( k - i ) + 4 } { 2 } . \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} S _ f ^ k ( \\rho \\| \\sigma ) & : = \\ < k \\xi _ \\sigma , f ( \\Delta _ { \\rho , \\sigma } ) k \\xi _ \\sigma \\ > = \\int _ { [ 0 , + \\infty ) } f ( t ) \\ , d \\| E _ { \\rho , \\sigma } ( t ) k \\xi _ \\sigma \\| ^ 2 \\\\ & \\ = \\int _ { ( 0 , + \\infty ) } f ( t ) \\ , d \\| E _ { \\rho , \\sigma } ( t ) k \\xi _ \\sigma \\| ^ 2 + f ( 0 ^ + ) \\ < k \\xi _ \\sigma , ( 1 - s _ M ( \\rho ) s _ { M ' } ( \\sigma ) ) k \\xi _ \\sigma \\ > . \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{gather*} ( z _ B \\triangleright \\psi ) ( x ) = \\mathcal { D } ^ { - 1 } ( z ) \\big ( \\big ( v ^ { - 1 } \\big ) ' x ' \\big ) \\psi \\big ( v \\big ( v ^ { - 1 } \\big ) '' x '' \\big ) = \\big ( \\mathcal { D } ^ { - 1 } ( z ) \\psi ^ v \\big ) \\big ( v ^ { - 1 } x \\big ) = \\big ( \\mathcal { D } ^ { - 1 } ( z ) \\psi ^ v \\big ) ^ { v ^ { - 1 } } ( x ) \\end{gather*}"}
-{"id": "985.png", "formula": "\\begin{gather*} \\beta ( d ) = d \\frac { \\delta _ n ^ k } { k + 1 } i \\geq 0 , j \\geq 0 , j + i d \\leq k \\Rightarrow \\beta ( j + i d ) = \\beta ( j ) + \\beta ( i d ) = \\beta ( j ) + i \\beta ( d ) . \\end{gather*}"}
-{"id": "9165.png", "formula": "\\begin{align*} H ( x ) = \\left ( G ( x ) + \\frac { \\rho } { K + \\rho } \\right ) e ^ { - ( x - x _ 0 ) ( K + \\rho ) } . \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} \\mathcal { E } _ { a _ { k } } ^ { \\rm H } ( u _ { k } ) = \\mathcal { E } _ { a _ { k } } ^ { \\rm H } ( \\ell _ { k } ^ { \\frac { 3 } { 2 } } \\tilde { u } _ k ( \\ell _ { k } \\cdot ) ) \\geq \\ell _ { k } \\frac { a _ { * } - a _ { k } } { a _ { * } } \\| ( - \\Delta ) ^ { \\frac { 1 } { 4 } } \\tilde { u } _ k \\| _ { L ^ { 2 } } ^ 2 + \\frac { 1 } { \\ell _ { k } ^ { p } } \\int _ { \\mathbb { R } ^ { 3 } } V ( x ) | \\tilde { u } _ k ( x ) | ^ { 2 } { \\rm d } x . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} r _ i > \\max _ { C _ I \\neq 0 } \\left \\{ \\sum _ { j = i + 1 } ^ n r _ j I _ j \\right \\} , \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} Z = \\C ^ r \\times ^ { \\Lambda , \\rho } W \\xrightarrow { \\sim } Q \\times ^ S W \\xrightarrow { \\Phi _ f \\times ^ S _ W } ( V \\times S ) \\times ^ S W \\xrightarrow { \\sim } V \\times W ; \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} ( \\Lambda + \\tilde { \\mu } ) \\hat { u } = \\hat { p } , \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} \\nu _ * ( \\eta ) \\ ; \\ ! \\hat r _ { \\eta , \\eta ^ { i , i + e _ k } } ^ 0 = \\nu _ * ( \\eta ^ { i , i + e _ k } ) \\ ; \\ ! \\hat r _ { \\eta ^ { i , i + e _ k } , \\eta } ^ 0 \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} \\ , ( X _ { t } ^ { ( u ) } ) \\ , = \\ , \\int ^ { t } _ { 0 } e ^ { - 2 v } I _ { 0 } ( 2 u v ) { \\mathrm d } v \\ , = \\ , \\left \\{ \\begin{array} { c c } O ( 1 ) \\ , , & \\ , u \\in [ 0 , 1 ) \\ , , \\\\ O ( \\sqrt { t } ) \\ , , & \\ , u \\ , = \\ , 1 \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} \\frac { p _ { \\beta _ { 1 } , \\lambda } } { p _ { 0 } } \\frac { p _ { \\beta _ { 2 } , \\lambda } } { p _ { 0 } } & = ( 1 + \\lambda ^ { 2 } ) \\exp ^ { \\frac { y ^ { 2 } } { 1 + \\lambda ^ { 2 } } - \\frac { 1 } { 2 } ( ( y - \\frac { \\lambda } { \\sqrt { 2 } } ( u + v ) ) ^ { 2 } + ( y - \\frac { \\lambda } { \\sqrt { 2 } } ( u - v ) ) ^ { 2 } ) } \\\\ & = ( 1 + \\lambda ^ { 2 } ) \\exp ^ { \\frac { y ^ { 2 } } { 1 + \\lambda ^ { 2 } } - ( y - \\frac { \\lambda } { \\sqrt { 2 } } u ) ^ { 2 } - \\frac { \\lambda ^ { 2 } } { 2 } v ^ { 2 } } \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} \\sum _ { i \\in N ( j ) } G _ i \\phi ( x _ j ) & = S _ j \\phi ( x _ j ) , \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} | A | = | \\mathcal { L } ( G - \\sum _ { i = 1 } ^ { N } P _ i ) | - \\sum _ { j = 1 } ^ { N } | A _ j | + \\dots + ( - 1 ) ^ { K - g + 1 } \\sum _ { j _ 1 , \\dots , j _ { K - g + 1 } \\in [ N ] , j _ 1 \\neq \\dots \\neq j _ { K - g + 1 } } | \\cap _ { \\alpha = 1 } ^ { K - g + 1 } A _ { j _ \\alpha } | . \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} d g _ t ( x ) = \\sqrt { 2 \\nu } d W _ t + v ( t , g _ t ( x ) ) d t , g _ 0 ( x ) = x \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} U ( A \\otimes 1 + 1 \\otimes B ) U ^ * & = A + B ^ { ( 0 ) } \\oplus \\bigoplus _ { n = 1 } ^ \\infty ( A \\otimes 1 + 1 \\otimes B ^ { ( n ) } ) \\\\ U A \\otimes B U ^ * & = A \\otimes B = B ^ { ( 0 ) } A \\oplus \\bigoplus _ { n = 1 } ^ \\infty A \\otimes B ^ { ( n ) } . \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} d \\mathcal { U } ( g ) \\cdot g X & = \\langle \\nabla _ g \\mathcal { U } ( g ) , g X \\rangle _ g = \\langle \\langle g ^ { - 1 } \\nabla _ g \\mathcal { U } ( g ) , X \\rangle \\rangle . \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} u _ { k + 1 } ^ \\delta = \\begin{cases} u _ k ^ \\delta \\int _ \\Sigma \\frac { a _ { [ k ] } ( s , \\cdot ) \\ , y _ { [ k ] } ^ \\delta ( \\cdot ) } { ( A _ { [ k ] } u _ k ^ \\delta ) ( s ) } \\ , d s & d ( y _ j ^ \\delta , A _ j u _ k ^ \\delta ) > \\tau \\gamma \\delta _ { [ k ] } , \\\\ u _ k ^ \\delta & , \\end{cases} \\end{align*}"}
-{"id": "9914.png", "formula": "\\begin{align*} \\int _ { M } u = \\int _ { R _ { 1 } } \\tau _ { 1 } + \\int _ { R _ { 0 1 } } \\tau _ { 0 1 } . \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} \\begin{aligned} & = \\gamma _ { 0 } ( y , t ) \\left ( \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { h ( x ) } { \\vert T ' ( x ) \\vert } \\right ) + \\gamma _ { 1 } ( y , t ) \\sum _ { x \\in T ^ { - 1 } ( x ' ) } ( 4 x ^ { 3 } - 3 x ) \\frac { h ( x ) } { \\vert T ' ( x ) \\vert } \\\\ & = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { h ( x ) } { \\vert T ' ( x ) \\vert } \\left ( \\gamma _ { 0 } ( y , t ) + 4 x ^ { 3 } \\gamma _ { 1 } ( y , t ) \\right ) \\end{aligned} \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} u ( x , y , t ) = \\frac { e ^ { - \\frac { ( y - x - \\mu t ) ^ { 2 } } { 4 t } } } { \\sqrt { 4 \\pi t } } = \\frac { e ^ { - \\frac { ( y - x ) ^ { 2 } } { 4 t } } } { \\sqrt { 4 \\pi t } } e ^ { - \\mu ^ { 2 } \\frac { t } { 4 } + \\frac { \\mu } { 2 } ( y - x ) } , t > 0 , \\ ; x , y \\in \\mathbb { R } . \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} D _ { n } [ \\sigma ] = \\mathbf { G } [ \\tau ] ^ { n + 1 } n ^ { - \\beta ^ { 2 } } G ( 1 + \\beta ) G ( 1 - \\beta ) e ^ { - i n \\beta \\theta _ { 1 } } E [ \\tau ] \\left ( 1 + o ( 1 ) \\right ) . \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} \\tilde { C } = \\{ ( \\lambda , \\mu ) \\in P \\times Q \\mid \\lambda + \\mu \\in 2 C \\} . \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} \\max _ { 1 \\leq i \\leq m + 1 } \\{ \\deg a _ i \\} \\leq \\sum _ { i = 1 } ^ { m + 1 } \\deg \\mathrm { r \\tilde { a } d } _ { \\kappa } ( a _ i ) - \\frac { 1 } { 2 } m ( m - 1 ) . \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\begin{aligned} c ( x , y , t ) & = h ( x ) \\gamma _ { 0 } ( y , t ) + x h ( x ) \\gamma _ { 1 } ( y , t ) \\\\ \\Rightarrow & l = h ( x ' ) \\gamma _ { 0 } ( y , t ) + x ' h ( x ' ) \\gamma _ { 1 } ( y , t ) \\end{aligned} \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} u ^ { q + 1 } + u + 1 = 0 . \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} h _ { \\mathrm { t o p } } ( f _ \\theta ) = \\ell \\beta ( \\theta ) \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} \\frac { { { \\rm { D } } ^ { 2 } } } { \\partial { { z } ^ { i } } \\partial \\overline { { { z } ^ { j } } } } = 0 \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} | F ^ { ( p r ) } ( t ) | & = \\left | \\left ( f ( t ) - \\sum _ { k = 1 } ^ { p - 1 } b _ k \\frac { t ^ { k r } } { ( k r ) ! } \\right ) ^ { ( p r ) } \\right | \\le ( p r ) ! \\left ( \\frac { 1 + \\varepsilon } { \\varepsilon } \\right ) ^ { p r } \\frac { C _ 2 A _ 2 ^ p M _ p } { ( p r ) ! } = C _ 3 A _ 3 ^ p M _ p . \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} L _ 1 + L _ 3 = 0 , \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} f \\sim g _ 1 = \\sum _ { v \\in I \\setminus \\{ v _ 1 \\} } \\alpha ( v ) \\phi v + \\alpha ( v _ 1 ) \\beta _ 1 ( w _ 1 ) \\phi w _ 1 + \\sum _ { y \\in J _ 1 } \\alpha ( v _ 1 ) \\beta ( y ) \\phi y . \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} e ^ { \\frac { \\mu ^ { 2 } } { 4 } t } v ( x , x , t ) = \\frac { 1 } { \\sqrt { 4 \\pi t } } + \\frac { 1 } { \\sqrt { 4 \\pi t } } . \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } \\log _ q r = \\log r , \\lim _ { q \\to 1 } \\exp _ q r = e ^ r \\ r > 0 . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} M _ X ( \\sigma , T ) = O _ { \\sigma } ( T ^ { \\delta } ) \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} K _ { 2 } = \\left ( \\int _ { \\wp } \\int _ { 0 } ^ { \\infty } s ^ { Q - 1 } ( \\psi _ { 3 } ( s y ) ) ^ { 1 - p ' } \\left ( \\frac { \\int _ { \\wp } \\int _ { 0 } ^ { s } r ^ { Q - 1 } h ( r w ) ( \\psi _ { 3 } ( r w ) ) ^ { 1 - p ' } d r d \\sigma ( w ) } { \\int _ { \\wp } \\int _ { 0 } ^ { s } r ^ { Q - 1 } ( \\psi _ { 3 } ( r w ) ) ^ { 1 - p ' } d r d \\sigma ( w ) } \\right ) ^ { p } d s d \\sigma ( y ) \\right ) ^ { \\frac { q - 1 } { p } } \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} \\mathcal { N } ( U _ A ) = \\mathcal { N } ( A ) \\ \\mbox { a n d } \\ A ( I - U _ A ^ * U _ A ) = 0 . \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ H \\mathcal { I } ( ( f _ { i , n } ) , ( f _ j ) ) = \\int _ H \\mathcal { I } ( ( f _ i ) , ( f _ j ) ) . \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} \\mathbb { F } : = \\mathbb { F } _ p ( \\alpha _ 1 , \\dots , \\alpha _ n ) . \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} r _ 1 = \\sqrt { 4 \\nu - 1 } ~ ; \\ ; \\ ; \\ ; Q _ t ( z ) = \\ ; ^ t \\bar { z } \\Big [ ( e ^ { t M } ) ^ * ( e ^ { t M } ) - \\Big ] z \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; S _ t ( z _ 1 , z _ 2 ) = \\ ; ^ t \\bar { z _ 1 } \\Big [ ( e ^ { t M } ) ^ * ( e ^ { t M } ) - \\Big ] z _ 2 \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} \\max ( 1 , | t ^ { - 1 } b ^ 2 | ) ^ { - s - 1 } & ( | x | < | b | ) , \\\\ \\max ( 1 , | 2 t ^ { - 1 } b ^ 2 | ) ^ { - s - 1 } & ( | x | = | b | ) . \\end{cases} \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} P f : = \\sum _ \\alpha a _ \\alpha ( z ) \\partial _ z ^ \\alpha f \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} F _ { C } \\{ g ( x ) ; y \\} = \\int _ { 0 } ^ { \\infty } g ( x ) \\cos ( x y ) d x = G _ { C } ( y ) , ~ ~ ~ ~ ~ ( y > 0 ) , \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { x \\in T _ { N } ^ { - 1 } ( x ' ) } T _ { 2 } ( x ) = & \\\\ = & \\cos \\left ( 2 \\pi u _ { 0 } \\right ) + \\cos \\left ( 2 \\pi u _ { 0 } + 2 \\pi \\right ) + \\sum _ { j = 1 } ^ { \\frac { N } { 2 } - 1 } \\left ( \\cos \\left ( 2 \\pi u _ { 0 } + \\frac { 4 \\pi } { N } j \\right ) + \\cos \\left ( 2 \\pi u _ { 0 } - \\frac { 4 \\pi } { N } j \\right ) \\right ) \\end{aligned} \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} \\partial _ t \\mu _ t + \\partial _ C \\left ( ( c _ 0 ^ 2 | \\theta \\cdot \\nabla p | ^ 2 - C ^ { \\gamma - 1 } ) C \\mu _ t \\right ) = 0 , \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\mathsf { I } ^ { ( q ) } _ { \\alpha , p } ( y , d ; r ) : = \\int _ y ^ { d } r ( z ) \\frac { ( \\mathcal { W } ^ { ( q ( z / r ( z ) ) ^ { \\alpha } ) } _ { \\alpha , p } ) ' _ + ( r ( z ) ) } { \\mathcal { W } ^ { ( q ( z / r ( z ) ) ^ { \\alpha } ) } _ { \\alpha , p } ( r ( z ) ) } \\frac { \\dd z } { z } , \\{ y , d \\} \\subset ( 0 , \\infty ) , \\ , y \\leq d , \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} \\widehat { \\textbf { F } } _ t [ \\theta ] = \\widehat { \\textbf { F } } [ \\theta ] ^ t , \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{gather*} \\{ s _ i : i = 1 , 2 , 3 \\} \\ \\cup \\ \\{ s _ { i o } : i = 1 , 2 , 3 \\} \\ \\cup \\ \\{ s _ { i j } : 1 \\leq i < j \\leq 3 \\} \\ \\cup \\ \\{ s _ { 1 2 3 } , s _ o , \\delta \\} \\end{gather*}"}
-{"id": "6583.png", "formula": "\\begin{align*} n _ 1 ^ { n _ 1 } \\cdots n _ d ^ { n _ d } & = \\exp \\left ( n _ 1 \\log n _ 1 + \\cdots + n _ d \\log n _ d \\right ) \\\\ & \\leq \\exp \\left ( \\frac { n } { d } \\log \\frac { n } { d } + \\left ( 1 + \\log \\frac { n } { d } \\right ) \\sum \\limits _ { j = 1 } ^ d \\left ( n _ j - \\frac { n } { d } \\right ) + d ^ 2 \\right ) \\\\ & = \\exp \\{ d ^ 2 \\} d ^ { - n } n ^ n . \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} s = \\lim _ { k \\rightarrow \\infty } s _ k , \\end{align*}"}
-{"id": "9669.png", "formula": "\\begin{align*} \\| f \\| _ { \\mathcal { B } _ { \\alpha , T } ^ { j } } = \\int _ { \\mathbb { R } ^ { n } } | \\xi | ^ { j } \\sup _ { t \\in [ 0 , T ] } e ^ { \\alpha t | \\xi | } | \\hat { f } ( t , \\xi ) | \\ d \\xi . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} \\bigcap _ { c \\in I ( u ) } N [ c ] = \\{ u \\} \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} d X _ t = u ( t , X _ t ) d t + V _ j ( X _ t ) d \\tilde { \\Z } ( B ) ^ j _ t , X _ 0 = x \\in \\R ^ d \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} \\begin{aligned} K _ { F , 4 } ( \\tau ) & \\coloneqq 2 \\left ( \\vphantom { \\frac { C _ { F , 3 } ( T _ 0 ) } { e ^ 3 \\tau } } A _ F e \\tau \\log { ( e ^ 2 \\tau ) } + | B _ F | e \\tau \\right . \\\\ & \\left . + \\frac { C _ { F , 1 } ( T _ 0 ) \\log { ( e ^ 2 \\tau ) } + C _ { F , 2 } ( T _ 0 ) } { 3 } + \\frac { C _ { F , 3 } ( T _ 0 ) } { 9 e \\tau } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} & { { v } _ { x } } + { { u } _ { y } } + v \\left ( { { \\alpha } _ { x } } - { { \\beta } _ { y } } \\right ) + u \\left ( { { \\beta } _ { x } } + { { \\alpha } _ { y } } \\right ) = 0 \\\\ & { { u } _ { x } } - { { v } _ { y } } + u \\left ( { { \\alpha } _ { x } } - { { \\beta } _ { y } } \\right ) - v \\left ( { { \\beta } _ { x } } + { { \\alpha } _ { y } } \\right ) = 0 \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\left \\| { x } \\right \\| _ q = \\inf \\{ q ( y ) + q ( y - x ) : y \\in X \\} . \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} \\kappa _ { 0 0 } & = \\sum _ { n \\geq 0 } \\prod _ { k = 1 } ^ { n - 1 } q _ { ( 1 0 ) ^ k 0 } ( 0 ) \\prod _ { k = 0 } ^ { n - 1 } q _ { ( 0 1 ) ^ k 0 0 ( 1 ) } + \\sum _ { n \\geq 1 } \\prod _ { k = 1 } ^ { n - 2 } q _ { ( 1 0 ) ^ k 0 } ( 0 ) \\prod _ { k = 0 } ^ { n - 1 } q _ { ( 0 1 ) ^ k 0 0 ( 1 ) } \\\\ \\kappa _ { 1 1 } & = \\sum _ { n \\geq 0 } \\prod _ { k = 0 } ^ { n - 1 } q _ { ( 1 0 ) ^ k 1 1 } ( 0 ) \\prod _ { k = 1 } ^ { n - 1 } q _ { ( 0 1 ) ^ k 1 ( 1 ) } + \\sum _ { n \\geq 1 } \\prod _ { k = 0 } ^ { n - 1 } q _ { ( 1 0 ) ^ k 1 1 } ( 0 ) \\prod _ { k = 1 } ^ { n - 2 } q _ { ( 0 1 ) ^ k 1 ( 1 ) } . \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} \\textstyle \\mathbf { S _ { _ \\mathrm { S M } } } = \\mathbf { V } \\left ( \\frac { \\mu _ { _ { \\rm S M } } } { \\ln 2 } \\left ( \\nu _ { _ { \\rm S M } } \\ , \\mathbf { I } _ r - \\rho _ { _ { \\rm S M } } \\ , \\boldsymbol { \\Lambda } ^ { \\rm H } \\boldsymbol { \\Lambda } \\right ) ^ { - 1 } - \\frac { \\sigma ^ 2 } { 1 - \\rho _ { _ { \\rm S M } } } \\ , \\mathbf { \\Lambda } ^ { - 2 } \\right ) \\mathbf { V } ^ { \\rm H } . \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} \\| \\mathcal { T } h - \\mathcal { T } \\tilde { h } \\| _ { \\mathcal { B } _ { \\alpha } ^ { 2 } } \\leq \\left ( \\frac { 1 } { 1 - \\alpha } \\sum _ { j = 2 } ^ { \\infty } \\frac { ( r _ { 0 } + r _ { 1 } ) ^ { j - 1 } } { ( j - 1 ) ! } \\right ) \\| h - \\tilde { h } \\| _ { \\mathcal { B } _ { \\alpha } ^ { 2 } } . \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\frac { { X _ { m + a - ( b - a ) j } } } { { ( - f _ 1 / f _ 2 ) ^ j } } } = f _ 1 X _ m + \\frac { f _ 2 } { { ( - f _ 1 / f _ 2 ) ^ k } } X _ { m - ( k + 1 ) ( b - a ) } \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} \\overline { \\nabla } ^ { ' } _ { X } Y = \\nabla ^ { ' } _ { X } Y + \\alpha \\eta ( Y ) X - \\alpha g ( X , Y ) \\xi - \\beta g ( \\phi X , Y ) \\xi . \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} \\begin{cases} x ^ 0 \\in C , \\\\ x ^ { k + 1 } \\in C f ( x ^ { k + 1 } , y ) + \\frac { 1 } { r _ k } \\langle y - x ^ { k + 1 } , x ^ { k + 1 } - x ^ k \\rangle \\geq 0 , \\forall y \\in C , \\end{cases} \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} d ^ * ( y _ 0 ) = \\tilde { k } ^ * ( z ) \\ \\ \\ \\ \\forall y _ 0 \\in Y _ z . \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} ( \\nu _ i L ) _ j \\pi ( \\xi ) & = ( \\nu _ i L ) ( \\pi ( \\xi ) | _ { X _ { g _ i , h _ j } } ) = \\nu ( [ L ( \\pi ( \\xi ) | _ { X _ { g _ i , h _ j } } ) ] | _ { Y _ i } ) \\\\ & = \\nu ( \\xi | _ { \\phi ( X _ { g _ i , h _ j } ) \\cap Y _ i } ) \\qquad ( \\mbox { L e m m a \\ref { l e m : c o m p o s i t i o n o f p i a n d L u n d e r r e s t r i c t i o n s } } ) \\\\ & = ( \\nu | _ { \\phi ( X _ { g _ i , h _ j } ) \\cap Y _ i } ) ( \\xi ) . \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} \\lambda ^ i \\mu ^ { d - i } & & i & \\in \\mathbb { Z } , \\\\ \\lambda ^ { d + i } ( \\lambda + \\mu ) ^ { - i } & & i & = \\{ 1 , 2 , \\ldots \\} . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} w _ \\epsilon ( x ) = \\frac { x ^ { \\epsilon - 1 / 2 } \\sqrt { 4 q ^ 2 - x } } { k ^ 2 - x } \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} M '' ( t ) & = \\int _ { \\Omega } u u _ t \\ , d x = - I ( u ( t ) ) \\\\ & = - 2 p J ( u ( t ) ) + ( p - 1 ) \\int _ { \\Omega } | \\nabla u ( t ) | ^ 2 \\ , d x \\\\ & \\geq - 2 p J ( u _ 0 ) + ( p - 1 ) \\lambda \\int _ { \\Omega } u ( t ) ^ 2 \\ , d x . \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} \\bar { \\omega } _ k ( x ) \\leq 0 , | x | = R _ 0 , \\ , \\ , \\forall k \\geq K . \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} G _ x = \\left \\{ h _ { m _ x } \\ , \\begin{pmatrix} e ^ { \\imath \\vartheta _ j } & 0 \\\\ 0 & e ^ { - \\imath \\vartheta _ j } \\end{pmatrix} \\ , h _ { m _ x } ^ { - 1 } \\ , : \\ , j = 1 , \\ldots , N _ x \\right \\} . \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} \\varphi ( S _ i ( S _ j ( v _ 0 ) ) ) = \\frac { \\partial } { \\partial x _ i } \\left ( \\varphi ( S _ j ( v _ 0 ) ) \\right ) = \\frac { \\partial f _ j } { \\partial x _ i } \\\\ \\varphi ( S _ j ( S _ i ( v _ 0 ) ) ) = \\frac { \\partial } { \\partial x _ j } \\left ( \\varphi ( S _ i ( v _ 0 ) ) \\right ) = \\frac { \\partial f _ i } { \\partial x _ j } \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} c ^ \\tau ( x ) = \\tau ( h , g ) c ( x ) \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} \\| O _ p \\varphi _ s \\| _ { L ^ 2 ( \\Bbb { R } ^ 2 ) } = \\| e ^ { - s X _ 0 } O _ p e ^ { s X _ 0 } \\varphi _ 0 \\| _ { L ^ 2 ( \\Bbb { R } ^ 2 ) } \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} \\xi _ X : = \\xi _ M ^ \\sharp - \\langle \\Phi _ G \\circ p , \\xi \\rangle \\ , \\partial _ \\theta . \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} \\Psi ' _ { s i r } = \\max \\{ \\psi _ { 1 } , \\psi _ { 2 } ' \\} \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} u ' ( t ) = L u ( t ) + N ( u ( t ) ) = J _ n u ( t ) + N _ n ( u ( t ) ) , \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} \\partial _ l \\mathbf { B } ^ i = \\sum _ { j = 1 } ^ { m } \\left ( \\frac { \\Omega _ j ^ i ( . ) } { | . | ^ 2 } \\right ) \\star \\mu _ { j R } ^ { l } ~ ~ ~ ~ \\mathcal { D } ' ( B _ R ) \\end{align*}"}
-{"id": "9946.png", "formula": "\\begin{align*} \\kappa ^ n = ( 0 , \\ , \\kappa ^ { n - 1 } _ { 0 1 } ) = ( 0 , \\ , \\ , ( - 1 ) ^ n ( n - 1 ) ! \\ , \\check { \\chi } _ { H _ { n } \\cup H _ - } \\ , \\bar { \\partial } \\varphi _ + \\wedge \\bar { \\partial } \\varphi _ 1 \\wedge \\cdots \\wedge \\bar { \\partial } \\varphi _ { n - 2 } ) , \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{gather*} \\forall \\ , \\psi \\in \\mathcal { O } ( H ) , \\psi \\mu ^ l = \\varepsilon ( \\psi ) \\mu ^ l \\big ( \\ \\mu ^ r \\psi = \\varepsilon ( \\psi ) \\mu ^ r \\big ) . \\end{gather*}"}
-{"id": "718.png", "formula": "\\begin{align*} [ \\tau \\omega _ 1 , \\tau \\omega _ 2 ] = [ \\omega _ 1 , \\omega _ 2 ] \\Psi _ D ( \\tau ) \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\widetilde { \\mathbb { P } } _ { n } \\left ( \\left | \\frac { \\mathfrak { S } ( d , n - k ) } { n } - \\widetilde { \\mu } \\right | > \\eta \\right ) = 0 , \\ \\forall \\eta > 0 . \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} \\sum _ { i \\neq j } ^ n t _ i = \\frac { ( n - 1 ) l } { r } . \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { P } \\left \\lbrace \\sup _ { t \\in [ 0 , T ] } \\left \\lvert X ^ { n , \\omega ^ \\prime } _ t - X _ t ^ { \\omega ^ \\prime } \\right \\rvert > \\varepsilon \\right \\rbrace = 0 \\qquad \\forall \\ , \\varepsilon > 0 \\ , . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\left | N _ i ( u ) - \\mu _ { d + i - q } { q \\choose i } q ^ { k + i - q } \\right | \\leq \\sum _ { j = k + 1 } ^ d { j \\choose q - i } { \\frac { q } p + m \\sqrt { q } + j \\choose j } { m - 1 \\choose d - j } \\sqrt { q } ^ { d - j } . \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} p ' ( w ) = \\begin{cases} p ( w ) + \\epsilon ( 0 , - 1 ) & w \\sim y , \\\\ p ( w ) & \\end{cases} \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} | \\psi _ l ( \\mathcal { R } ) f ( x ) | & = \\left | \\int _ { G } f ( x z ^ { - 1 } ) k _ { l } ( z ) d z \\right | \\\\ & = \\left | \\int _ { G } ( f ( x z ^ { - 1 } ) - f ( x ) ) k _ { l } ( z ) d z \\right | \\\\ & \\leq \\int _ { G } | z | ^ s | k _ { l } ( z ) | d z \\Vert f \\Vert _ { \\Lambda ^ s ( G ) } . \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} V _ \\lambda ( z ) ( s ) = \\frac { W _ \\lambda ( z ) ( s ) } { \\Vert W _ \\lambda ( z ) \\Vert _ * } . \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} \\Delta _ q ( u _ 1 ^ * , u _ 2 ^ * ) \\leq \\frac { G _ q } { q } \\| u _ 1 ^ * - u _ 2 ^ * \\| ^ q , \\forall \\ u _ 1 ^ * , u _ 2 ^ * \\in U ^ * , \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} \\mathcal { R } : = \\{ r _ { \\Omega } P _ q f + z : \\ f \\in H _ 1 , \\ z \\in Z _ 2 \\} \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} | F | _ { c _ \\lambda } & = \\sum \\limits _ { e \\in F } c _ \\lambda ( e ) , \\ F \\subset E \\left ( \\mathbb { Z } ^ d \\right ) , \\\\ | K | _ \\pi & = \\sum \\limits _ { x \\in K } \\pi ( x ) , \\ K \\subset \\mathbb { Z } ^ d , \\\\ \\psi ( \\mathbb { Z } ^ d , \\ , t ) & = \\inf \\left \\{ | \\partial _ E K | _ { c _ \\lambda } : \\ t \\leq \\vert K \\vert _ \\pi < \\infty \\right \\} , \\ t > 0 ; \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} G ( v , Q ) & : = \\tfrac { p + 2 } { 2 } \\mu v \\int _ 0 ^ 1 \\bigl [ | Q + \\theta v | ^ p - | Q | ^ p \\bigr ] \\ , d \\theta \\\\ & + \\tfrac { p } { 2 } \\mu \\bar v \\int _ 0 ^ 1 \\bigl [ | Q + \\theta v | ^ { p - 2 } ( Q + \\theta v ) ^ 2 - | Q | ^ { p - 2 } Q ^ 2 \\bigr ] \\ , d \\theta \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} \\ , _ { c } \\mathbb { A } ^ { - 1 } _ L = { ( \\det \\mathbb { A } ) } ^ { - 1 } \\left ( \\begin{matrix} d & - b \\\\ - c & a \\end{matrix} \\right ) \\ ; \\ ; \\ ; \\ ; \\ , _ { c } \\mathbb { A } ^ { - 1 } _ R = \\left ( \\begin{matrix} d & - b \\\\ - c & a \\end{matrix} \\right ) { ( \\det \\mathbb { A } ) } ^ { - 1 } , \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} \\sum _ { 0 \\le t < n , t \\neq i - 1 } \\ ! \\ ! \\ ! \\ ! { \\dim } _ F ( \\{ f _ { i , j } ( \\beta ^ { r ^ t } ) \\} _ { j \\in [ l ] } ) & \\le ( n - 1 ) \\frac { l } { r } + ( r - 1 ) \\sum _ { t = 0 } ^ { i - 2 } \\frac { l } { r ^ { i - t } } + ( r - 1 ) \\sum _ { t = i } ^ { n - 1 } \\frac { l } { r ^ { n - t + 1 } } \\\\ & = l \\Big ( \\frac { n - 1 } { r } + \\frac { r ^ { i - 1 } - 1 } { r ^ i } + \\frac { r ^ { n - i } - 1 } { r ^ { n - i + 1 } } \\Big ) \\\\ & < \\ , l \\frac { n + 1 } { n - k } . \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} \\operatorname { L i n } : \\Omega ^ 1 ( M ) \\oplus \\Omega ^ 0 ( M ) \\to \\Omega ^ 1 ( M ) \\oplus \\Omega ^ 0 ( M ) , \\begin{pmatrix} v \\\\ f \\end{pmatrix} \\mapsto \\begin{pmatrix} \\delta \\mathrm { d } - 2 \\ , \\mathrm { R i c } & \\mathrm { d } \\\\ \\delta & 0 \\end{pmatrix} \\begin{pmatrix} v \\\\ f \\end{pmatrix} . \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} { u ( t ) \\choose \\dot { u } ( t ) } = V _ 0 ( t ) { u _ 0 \\choose u _ 1 } - \\int ^ { t } _ { 0 } V _ 0 ( t - s ) { 0 \\choose F ( u ( s ) ) } d s , \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} \\phi _ a ( t ) : = \\int _ 0 ^ t \\phi _ a ' ( s ) \\ , \\d s \\qquad \\phi ' _ a ( t ) : = \\phi ' ( a + t ) \\frac { t } { a + t } . \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} \\tilde { P } _ { 1 } ( h ) = \\left \\{ x \\in V \\mid \\langle x , u _ { F } \\rangle + ( y + 1 ) \\lambda _ { F } + h _ { F } \\geq 0 \\ \\right \\} . \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} c _ k ( f ) = O \\left ( \\frac { r ( | k | ) } { | k | ^ { 1 . 5 } } \\right ) , \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} E = E _ 0 \\oplus ( V \\otimes F [ - 1 ] ) \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} I _ { j k } ( x ; z ) : = \\int _ { - \\infty } ^ { + \\infty } \\mathrm { d } \\theta \\ , \\int _ 0 ^ { + \\infty } \\mathrm { d } u \\left [ e ^ { \\imath \\ , k \\ , \\Psi _ k ( x ; u , \\theta , z ) } \\cdot A _ { j k } ( x ; u , \\theta , z ) \\right ] ; \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} \\gamma _ { k + 1 } \\leq - K _ 8 \\gamma _ k ^ { \\frac { 2 } { 1 + \\epsilon } } + \\alpha _ k \\gamma _ k + K _ { 1 1 } \\beta _ k , \\end{align*}"}
-{"id": "9942.png", "formula": "\\begin{align*} V = \\{ \\ , z \\in X \\mid \\ , | x _ 1 | < \\epsilon , \\ , | y | < \\epsilon \\ , \\} \\quad \\Omega = ( M \\times \\sqrt { - 1 } \\Gamma ) \\cap V \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} A _ j ^ { \\operatorname { c o } } = \\begin{pmatrix} 0 & - J _ j \\\\ J _ j & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} \\frac { S ^ \\chi _ { ( \\lambda , 0 ) , ( \\mu , \\mu ' ) } } { S ^ \\chi _ { ( 0 , 0 ) , ( \\mu , \\mu ' ) } } & = \\frac { e ^ { 2 \\pi i \\left ( ( \\lambda , \\mu ' + \\rho ) \\right ) } \\chi _ \\mu ( \\lambda ) } { \\chi _ \\mu ( 0 ) } \\ \\in \\ S _ v ^ { - 1 } R _ v . \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} \\mathrm { T r } ( B _ n ( \\omega _ n ^ k ) ) = \\mathrm { T r } ( B ^ { ( n , k ) } _ 1 ( 1 ) ) . \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} a _ { n + 1 } + \\frac { 1 } { 6 } ( a _ { n + 1 } - 1 ) ^ 2 = a _ n , ~ ~ ~ ~ n \\geq 0 . \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} \\nabla K ( \\xi - d \\Gamma _ A ( m ) ) & = ( \\partial _ 1 K ( \\xi - d \\Gamma _ A ( m ) ) , \\dots , \\partial _ { \\nu } K ( \\xi - d \\Gamma _ A ( m ) ) ) \\\\ \\Sigma _ A ( \\xi ) & = \\inf ( \\sigma ( H _ \\mu ( \\xi , A ) ) ) \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} \\int \\limits _ t ^ T \\left ( { \\bf w } _ s ^ { ( i _ 1 ) } - { \\bf w } _ t ^ { ( i _ 1 ) } \\right ) ^ { ( q ) } d { \\bf w } _ s ^ { ( i _ 2 ) } = \\sum _ { j = 0 } ^ { q } \\int \\limits _ t ^ T \\phi _ j ( \\tau ) d { \\bf w } _ { \\tau } ^ { ( i _ 1 ) } \\int \\limits _ t ^ T \\int \\limits _ t ^ s \\phi _ j ( \\tau ) d \\tau d { \\bf w } _ { s } ^ { ( i _ 2 ) } \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} Z _ { t } \\ , : = \\ , \\exp \\Big ( - \\int ^ { t } _ { 0 } b ( s , X _ { s } , F _ { s } ) { \\mathrm d } B _ { s } - \\frac { 1 } { \\ , 2 \\ , } \\int ^ { t } _ { 0 } \\lvert b ( s , X _ { s } , F _ { s } ) \\rvert ^ { 2 } { \\mathrm d } s \\Big ) \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} \\binom { - a } { r } = ( - 1 ) ^ r \\binom { a + r - 1 } { r } , \\binom { - ( n + 1 ) } { r } = ( - 1 ) ^ r \\binom { n + r } { r } . \\end{align*}"}
-{"id": "9990.png", "formula": "\\begin{align*} [ \\bar u ( 0 , \\cdot ) ] ( q ) = \\bar u _ 0 ( q ) \\sum _ { i = 0 } ^ \\infty ( \\bar p ( q ) ) ^ i . \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} \\sup _ { x = ( x _ 1 , \\dots , x _ \\ell ) \\in A _ n } \\lVert ( H _ \\mu ( \\xi - x _ 1 - \\dots - x _ \\ell ) - H _ \\mu ( \\xi - k _ 0 ) ) \\psi _ p \\lVert \\leq \\frac { 1 } { p } . \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} \\nabla f ( \\mathbf { x } ^ { \\star } ) + \\mathbf { E } _ { } ^ T ( \\sqrt { \\eta } \\boldsymbol { \\nu } ^ { \\star } ) = \\mathbf { 0 } \\\\ \\mathbf { E } _ { } \\mathbf { x } ^ { \\star } = \\mathbf { 0 } , \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} M _ p ( A ; m ) = ( p ^ 2 - p ) \\mathbb { E } _ p \\big ( \\Phi _ A a _ { E } ( p ) ^ m \\big ) , \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} g _ 1 ( r ) ( W _ 1 , W _ 2 ) = { } & g ( W _ 1 , W _ 2 ) - 2 r \\mathrm { I I } _ { W _ 0 } ( W _ 1 , W _ 2 ) + O ( r ^ 2 ) ; \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} \\exp ( - \\mathcal { Q } ( x ) ) = \\sqrt { \\frac { \\det Q _ - } { \\det Q _ + } } f _ 0 ^ { c _ 0 } ( B _ 0 x ) f _ { m + 1 } ^ { c _ { m + 1 } } ( B _ { m + 1 } x ) , \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} \\iota ^ { \\pm } \\colon \\widehat { M } _ p \\stackrel { \\cong } { \\to } \\{ d w ^ { \\pm } = 0 \\} \\subset Y _ { \\widehat { U } _ p } ^ { \\pm } . \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} C _ z = \\big \\{ s \\in [ 0 , 1 ] \\ , : \\ , \\phi ( z ( s ) ) = 0 \\big \\} , \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} R _ { 1 1 } & = - a ^ { - 1 } b ( d - c a ^ { - 1 } b ) ^ { - 1 } [ c , d ] ( \\Delta ' ) ^ { - 1 } \\\\ & = - ( d b ^ { - 1 } a - c ) ^ { - 1 } [ c , d ] ( \\Delta ' ) ^ { - 1 } . \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } - \\tfrac 1 t \\log \\mathbb P ( Z _ t > 0 ) & = ( 1 - p ) \\kappa - \\lambda ( \\mu - 1 ) . \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\| \\gamma _ n ^ { ( d ) } ( A ) \\xi \\| = \\| \\gamma _ n ^ { ( d ) } ( A ) \\| . \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D O } _ \\textup { t } ( 2 n + 1 ) q ^ n & = \\dfrac { f _ 2 ^ 4 f _ 6 ^ 4 } { f _ 4 f _ { 1 2 } } \\cdot \\dfrac { 1 } { f _ 1 ^ 4 } \\\\ & = \\dfrac { f _ 2 ^ 4 f _ 6 ^ 4 } { f _ 4 f _ { 1 2 } } \\cdot \\left ( \\dfrac { f _ 4 ^ { 1 4 } } { f _ 2 ^ { 1 4 } f _ 8 ^ 4 } + 4 q \\dfrac { f _ 4 ^ 2 f _ 8 ^ 4 } { f _ 2 ^ { 1 0 } } \\right ) , \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} \\ , _ { c } \\mathbb { A ' } ^ { - 1 } _ R = \\left ( \\begin{matrix} d & - b \\\\ - c & a \\end{matrix} \\right ) { \\Delta ' } ^ { - 1 } , \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ \\infty | k | ^ { 2 s } | c _ k ( g ) | ^ 2 = \\sum _ { k = - \\infty } ^ \\infty O \\left ( | k | ^ { 2 s - 4 - 2 \\mu } \\right ) < \\infty , \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} \\| \\mathcal P ( t ) - \\mathcal P ( t _ 0 ) \\| & = \\| C _ U T _ { ( V ^ * b ^ 0 ) _ { [ j ] } } \\mathcal P _ { 1 } ( t ) C _ V - C _ U T _ { ( V ^ * b ^ 0 ) _ { [ j ] } } \\mathcal P _ { 1 } ( t _ 0 ) C _ V \\| \\\\ & \\leq \\| T _ { ( V ^ * b ^ 0 ) _ { [ j ] } } \\| \\| \\mathcal P _ { 1 } ( t ) - \\mathcal P _ { 1 } ( t _ 0 ) \\| \\to 0 , t \\to t _ 0 . \\end{align*}"}
-{"id": "9687.png", "formula": "\\begin{align*} \\omega _ n : = \\xi \\prod _ { j = 0 } ^ { \\infty } \\frac { y ^ { q ^ j } } { \\tau ^ j ( \\alpha ) } . \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} M ^ q & = C ^ q e ^ { 2 q } \\left ( \\dfrac { q } { p } \\right ) ^ n \\left ( \\dfrac { q } { 2 \\pi } \\right ) ^ { n - j } \\\\ \\times & \\sum _ { i = j + 1 } ^ n \\int _ { \\mathbb C ^ { n - j } } | \\widetilde { g } _ { i i } z _ i | ^ q \\big ( 1 + | \\widetilde { g } _ { i i } z _ i | \\big ) ^ q e ^ { \\frac { q \\left | \\widetilde { G } z ' _ { [ j ] } \\right | ^ 2 } { 2 } } e ^ { - \\frac { q \\left | z ' _ { [ j ] } \\right | ^ 2 } { 2 } } \\ ; d A ( z ' _ { [ j ] } ) < \\infty , \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} \\lambda ^ { - 1 } \\alpha ( \\exp ( X ) ) - \\sigma \\le d _ G ( e , \\exp ( X ) ) = \\norm { X } \\le \\lambda \\alpha ( \\exp ( X ) ) + \\sigma . \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{gather*} \\overline { \\nabla } ^ { \\mathrm { b a s } , \\overline { g } } = \\nabla ^ { \\mathbb { V } } \\oplus p ^ ! \\big ( \\nabla ^ { \\mathrm { b a s } , g } \\big ) . \\end{gather*}"}
-{"id": "1639.png", "formula": "\\begin{align*} ( P _ { N } - P _ { N } ' ) ^ 4 = \\frac { \\rho _ { N - 1 } } { \\rho _ { N } } \\ , ( P _ { N } - P _ { N } ' ) ^ 2 . \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} P \\left \\lbrace \\textbf { X } \\in \\widehat { \\mathcal { C } } _ { N } \\Delta \\mathcal { C } \\Big | D _ { n } \\right \\rbrace \\leq I \\{ N > 0 \\} \\frac { \\mu ( \\widehat { \\mathcal { C } } _ { N } \\Delta \\mathcal { C } ) } { \\mu ( \\mathcal { C } ) } + I \\{ N = 0 \\} . \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} L _ i u = ( a \\nabla \\xi _ i ) _ l \\frac { \\partial u } { \\partial x _ { l } } \\ , . \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} & 2 \\psi ^ \\prime ( | \\nabla \\Q | ^ 2 ) Q _ { i j , k k } + 4 Q _ { i j , k } \\psi ^ { \\prime \\prime } ( | \\nabla \\Q | ^ 2 ) Q _ { p q , r } Q _ { p q , r k } \\\\ & \\qquad = \\frac { 1 } { L } \\left ( - A Q _ { i j } - B \\left ( Q _ { i p } Q _ { p j } - | \\mathbf { Q } | ^ 2 \\delta _ { i j } / 3 \\right ) + C | \\mathbf { Q } | ^ 2 Q _ { i j } \\right ) . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} & \\mathbf { E } \\Big ( \\sup _ { 0 \\le t ' \\le t } \\big | \\xi ( x ( t ' ) ) - z ( t ' ) \\big | ^ 2 \\Big ) \\\\ \\le & \\ , 3 \\mathbf { E } \\bigg [ \\sup _ { 0 \\le t ' \\le t } \\Big | \\int _ 0 ^ { t ' } \\varphi ( x ( s ) ) \\ , d s \\Big | ^ 2 \\bigg ] + 3 L _ b ^ 2 \\ , \\mathbf { E } \\Big ( \\int _ 0 ^ t \\big | \\xi ( x ( s ) ) - z ( s ) \\big | \\ , d s \\Big ) ^ 2 + \\frac { 6 } { \\beta } \\mathbf { E } \\sup _ { 0 \\le s \\le t } \\big | M ( s ) \\big | ^ 2 \\ , . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X _ { s } = & b ( s , X _ { s } , Y _ { s } , Z _ { s } ) d s + \\sigma ( s , X _ { s } , Y _ { s } , Z _ { s } ) d B _ { s } , \\\\ d Y _ { s } = & - g ( s , X _ { s } , Y _ { s } , Z _ { s } ) d s + Z _ { s } d B _ { s } , s \\in \\lbrack 0 , T ^ { \\prime } ] , \\\\ X _ { 0 } = & x , \\ Y _ { T ^ { \\prime } } = \\phi ( X _ { T ^ { \\prime } } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} \\mathcal { E } _ { a } ^ { \\rm H } ( u ) = \\| ( - \\Delta + m ^ { 2 } ) ^ { \\frac { 1 } { 4 } } u \\| _ { L ^ { 2 } } ^ 2 + \\int _ { \\mathbb { R } ^ { 3 } } V ( x ) | u ( x ) | ^ { 2 } { \\rm d } x - \\frac { a } { 2 } \\iint _ { \\mathbb { R } ^ { 3 } \\times \\mathbb { R } ^ { 3 } } \\frac { | u ( x ) | ^ { 2 } | u ( y ) | ^ { 2 } } { | x - y | } { \\rm d } x { \\rm d } y . \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} - \\Delta u + b ( x ) u = \\dfrac { g ( x ) f ( u ) } { | x | ^ a } + h ( x ) , \\ \\ x \\in \\mathbb { R } ^ 2 , \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} \\mathcal C = \\left \\{ \\bigcup _ { k = 1 } ^ K A _ { I _ k , J _ k } : K \\in \\mathbb N , I _ k \\cap J _ k = \\emptyset , | I _ k | , | J _ k | < \\infty \\right \\} \\bigcup \\{ \\emptyset \\} \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} X = \\mathrm { T o t } _ S ( K _ S ) . \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} \\Lambda _ { - } = \\underset { \\textrm { I m } \\ ; \\lambda < 0 } \\bigoplus \\ker ( H _ { K ^ * _ { \\nu , \\frac { \\pi } { 2 } } } - \\lambda I ) = \\left \\{ \\begin{pmatrix} x \\\\ A _ { - } x \\end{pmatrix} , \\ ; \\ ; \\ ; x \\in \\mathbb { C } ^ 2 \\right \\} \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial x } - \\frac { \\partial v } { \\partial y } = 0 , ~ ~ \\frac { \\partial u } { \\partial y } + \\frac { \\partial v } { \\partial x } = 0 \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} b _ i = h ( g , r _ i ) : = g ^ { - 1 } r _ i , i = 1 , 2 , \\cdots , n . \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} e _ P x e _ P = E ( x ) e _ P \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} T _ S f ( z , w ) = \\int _ 0 ^ { \\infty } f ( t , w ) e ^ { i 2 \\pi z t } \\d t , z \\in S ( p ( w ) , \\infty ) \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} T = \\left ( \\tau ^ { ( 1 ) } , \\tau ^ { ( 2 ) } , \\ldots , \\tau ^ { ( \\ell - 1 ) } , \\tau ^ { ( \\ell ) } \\right ) , \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} X _ s : = \\{ x : c ( g _ s ^ { - 1 } , x ) = h ^ { - 1 } \\} , X _ k : = \\{ x : c ( g _ k ^ { - 1 } , x ) = \\bar { h } ^ { - 1 } \\} . \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} \\hat h ^ 2 ( t ) - & \\hat h ^ 1 ( t ) = \\int _ 0 ^ t P ( { t - r } ) \\left ( ( \\nabla \\hat \\gamma ^ 2 ( r ) - \\nabla \\hat \\gamma ^ 1 ( r ) ) ^ * \\hat b ( r ) \\right ) \\mathrm d r \\\\ & + \\int _ 0 ^ t P ( { t - r } ) \\left ( \\hat f ( r , \\hat \\gamma ^ 2 ( r ) , \\nabla \\hat \\gamma ^ 2 ( r ) - \\hat f ( r , \\hat \\gamma ^ 1 ( r ) , \\nabla \\hat \\gamma ^ 1 ( r ) ) \\right ) \\mathrm d r . \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} \\eta _ { t } \\ , : = \\ , \\int _ { \\mathbb R } g ( x ) { \\mathrm d } \\mathrm m _ { t } ( x ) - \\int ^ { t } _ { 0 } [ \\mathcal A _ { s } ( \\mathrm M _ { s } ) g ] { \\mathrm d } s \\ , \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu = 1 \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} v _ { x + y } ( t ) & = \\int _ t ^ 1 \\frac { ( s - t ) ^ { y - 1 } } { \\Gamma ( y ) } \\frac { v _ x ( s ) } { ( 1 - s ) ^ \\alpha } ( 1 - s ) ^ \\alpha d s \\\\ & \\le C \\int _ t ^ 1 \\frac { ( s - t ) ^ { y - 1 } } { \\Gamma ( y ) } \\frac { v _ x ( ( 1 - \\lambda ) + \\lambda s ) } { \\lambda ^ \\alpha ( 1 - s ) ^ \\alpha } ( 1 - s ) ^ \\alpha d s \\\\ & = C \\lambda ^ { - ( \\alpha + y ) } \\int _ { t ' } ^ 1 \\frac { ( u - t ' ) ^ { y - 1 } } { \\Gamma ( y ) } { v _ x ( u ) } d u \\\\ & = C \\lambda ^ { - ( \\alpha + y ) } v _ { x + y } ( t ' ) . \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\sup _ { a , b > 0 } \\frac { 4 ( 1 + a b ) ( 1 - e ^ { - 2 t } ) - 3 + 2 ( 2 e ^ { - 2 t } - 1 ) ( a + b ) } { a ^ 2 b ^ 2 } = 1 . \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} \\int _ G f ( x ) \\rho ( x ) d x & = \\int _ G f ( x ) \\rho ( x ) Q ( g ) ( \\ddot { x } ) d x \\\\ & = \\int _ K \\int _ H \\int _ G f ( x ) \\rho ( x ) g ( k ^ { - 1 } x h ) d x d h d k \\\\ & = \\int _ G \\int _ K \\int _ H f ( k x h ^ { - 1 } ) \\Delta _ H ( h ^ { - 1 } ) \\Delta _ K ( k ) \\rho ( x ) g ( x ) d h d k d x \\\\ & = \\int _ G g ( x ) \\rho ( x ) \\Big ( \\int _ K \\int _ H f ( k ^ { - 1 } x h ) d h d k \\Big ) d x . \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} { \\rm { i n v } } \\circ { \\rm { r e s } } \\stackrel { ? } { = } { \\rm { i n v } } \\circ [ k ' : k ] , \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} \\mbox { V a r } _ { \\mu _ z } ( f ) : = \\int _ { \\Sigma _ z } f ^ 2 \\ , d \\mu _ z - \\Big ( \\int _ { \\Sigma _ z } f \\ , d \\mu _ z \\Big ) ^ 2 \\le \\frac { 1 } { \\rho } \\mathcal { E } _ z ( f , f ) \\ , , \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} V ( a , b ) = \\{ z \\in \\C \\ ; | \\ ; a < \\arg \\ , z < b \\} . \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} p _ { L , N + 1 } ^ { \\ell _ 2 } ( x ) & = \\sum \\limits _ { \\ell = 0 } ^ N \\dfrac { \\sum _ { j = 0 } ^ N \\omega _ j \\tilde { \\Phi } _ { \\ell } ( x _ j ) f ( x _ j ) } { 1 + \\lambda \\mu _ { \\ell } ^ 2 } \\tilde { \\Phi } _ { \\ell } ( x ) \\\\ & = \\sum _ { j = 0 } ^ N \\omega _ j f ( x _ j ) \\sum _ { \\ell = 0 } ^ N \\dfrac { \\tilde { \\Phi } _ { \\ell } ( x _ j ) \\tilde { \\Phi } _ { \\ell } ( x ) } { 1 + \\lambda \\mu _ { \\ell } ^ 2 } . \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} B ( t ) = \\begin{cases} 2 ^ { - 1 } | t | ^ { s + 1 } | b | ^ { - 2 s - 1 } ( 2 ^ { 2 s + 2 } - 1 ) ( 1 - 2 ^ { - 2 s - 1 } ) ^ { - 1 } & ( | b | = 1 ) , \\\\ | t | ^ { s + 1 } | b | ^ { - 2 s - 1 } \\{ ( 2 ^ { 2 s + 1 } - 2 ^ { - 1 } ) ( 1 - 2 ^ { - 2 s - 1 } ) ^ { - 1 } + \\delta ( | t | = | 2 b | ) ( 2 ^ s - 2 ^ { 2 s + 1 } ) \\} & ( | b | > 1 ) . \\end{cases} \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} e ^ { ( 1 + o ( 1 ) ) k \\log k } \\le l \\le e ^ { ( 1 + o ( 1 ) ) n \\log n } . \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} \\mathcal G _ { \\mathcal X } f ( x ) & = f ( p x ) - f ( x ) + \\lambda \\big ( 1 - x - h ( 1 - x ) \\big ) f ' ( x ) \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} \\varphi \\big ( n , q ( x ) \\big ) : = K n x H . \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} \\mathcal { R } _ { s , p , q } : = R ( s - p - 1 , s ) \\cup L ^ p _ { s - q } , \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} \\Lambda = \\bigcup _ { j = 2 } ^ { \\infty } \\bigcup _ { n = 1 } ^ { \\infty } \\Lambda _ { j , n } . \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} \\theta ( F ) ( t ) ( m \\otimes \\sigma ) : = ( - 1 ) ^ { | \\sigma | | m | } F ( E Z ( t \\otimes \\sigma ) ) ( m ) \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} K _ { i \\bar j } = - \\frac 1 2 \\mu _ { i k } ^ r \\mu _ { \\bar j \\bar k } ^ { \\bar r } + \\frac 1 4 \\mu _ { \\bar k \\bar r } ^ { \\bar i } \\mu _ { k r } ^ j . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} N = \\sum _ { 1 \\leq b \\leq n } \\sigma _ { j _ b } \\omega _ b ( \\xi ) y _ b + \\sum _ { j \\in \\mathbb { Z } _ * } \\sigma _ { j } \\Omega _ j ( \\xi ) z _ j \\bar { z } _ j \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} \\begin{pmatrix} \\alpha & - \\overline { \\beta } \\\\ \\beta & \\overline { \\alpha } \\end{pmatrix} \\in G \\stackrel { \\gamma } { \\longrightarrow } \\begin{pmatrix} \\alpha \\\\ \\beta \\end{pmatrix} \\in S ^ 3 . \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} X = \\bigg \\{ \\psi ( t ) \\bigg | \\ & \\psi ( 0 ) = \\psi _ 0 , \\ \\| \\psi ( t ) \\| _ 2 \\leq 2 C _ 1 ( 1 + t ) ^ { - \\frac { d } { 4 } } \\| \\psi _ 0 \\| _ 1 \\ \\ { \\rm a n d } \\ \\\\ & \\sqrt { Q ( \\psi ( t ) ) } \\leq 2 C _ 1 C _ Q ( 1 + t ) ^ { - \\frac { d } { 4 } - \\frac { \\eta } { 2 } } \\| \\psi _ 0 \\| _ 1 , \\ \\forall \\ t \\geq 0 \\bigg \\} \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { D } ( p _ { _ 1 } | | p _ { _ 0 } ) } { \\partial P _ y } = \\frac { 1 } { 2 } \\left ( \\frac { 1 } { \\sigma _ w ^ 2 } - \\frac { 1 } { P _ y } \\right ) , \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} \\begin{cases} w _ 0 ^ + = b _ 0 + 2 \\log \\mu _ \\lambda - \\log h _ \\gamma , \\\\ \\partial _ n w _ 0 ^ + = - a _ 0 \\lambda \\mu _ \\lambda , \\end{cases} \\ \\hbox { a n d } \\begin{cases} w _ 0 ^ - = b _ 0 + 2 \\log \\mu _ \\lambda - \\log h _ \\gamma , \\\\ \\partial _ n w _ 0 ^ - = a _ 0 \\lambda \\mu _ \\lambda . \\end{cases} \\end{align*}"}
-{"id": "9950.png", "formula": "\\begin{align*} \\begin{aligned} \\tau _ Y & = ( \\tau ^ { n - 1 , - } - \\tau ^ { n - 1 , + } ) | _ Y \\\\ & = ( 0 , f | _ Y \\ , ( - 1 ) ^ { n - 1 } ( n - 2 ) ! \\ , \\check { \\chi } _ { H _ { n } \\cap Y } \\ , \\bar { \\partial } ( \\varphi _ 1 | _ Y ) \\wedge \\dots \\wedge \\bar { \\partial } ( \\varphi _ { n - 2 } | _ Y ) ) . \\end{aligned} \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 0 & 1 & 2 u ^ { 3 } \\\\ 1 & - 2 u ^ { 3 } & u ^ { 2 } \\\\ 2 u ^ { 3 } & u ^ { 2 } & - 4 u ^ { 1 } \\end{pmatrix} , \\\\ w _ { 1 2 } = 0 , w _ { 3 1 } = - \\frac { 1 } { \\sqrt { u ^ { 1 } + u ^ { 2 } u ^ { 3 } + 2 ( u ^ { 3 } ) ^ { 3 } } } , w _ { 2 3 } = \\frac { u ^ { 3 } } { \\sqrt { u ^ { 1 } + u ^ { 2 } u ^ { 3 } + 2 ( u ^ { 3 } ) ^ { 3 } } } . \\end{gather*}"}
-{"id": "9191.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial _ t y - D \\Delta y = R y & & \\Omega \\times ( 0 , \\infty ) , \\\\ & \\partial _ \\nu y = 0 & & \\partial \\Omega \\times ( 0 , \\infty ) , \\\\ & y ( 0 ) = y _ 0 & & \\quad \\Omega , \\end{aligned} \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\forall r , ~ \\sum _ { c } a _ { r , c } = 1 . \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} | A _ i | = ( i - 1 ) \\left ( q ^ { m ( d - 1 ) } - 1 \\right ) \\mbox { f o r a l l } 1 \\le i \\le M . \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} S _ { \\mu \\nu \\lambda } = a _ { \\mu } b _ { \\nu } c _ { \\lambda } + b _ { \\mu } c _ { \\nu } a _ { \\lambda } + c _ { \\mu } a _ { \\nu } b _ { \\lambda } , \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} \\begin{aligned} c ( x ' , y , t ) = { } & \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } \\bigg \\{ c ( x , y , t ) \\\\ & + ( 1 - 2 x ^ { 2 } ) h ( x ) \\left [ \\frac { 1 } { 2 } \\frac { \\partial } { \\partial y } \\beta _ { 1 } ( y , t ) - \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) \\right ] \\bigg \\} \\end{aligned} \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} P _ { \\mu _ t } = A _ t P _ { g _ t } A _ t ^ { - 1 } \\ , , g _ t ( P _ { g _ t } \\cdot , \\cdot ) = P ( g _ t ) ( \\cdot , \\cdot ) \\ , . \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\nu _ { F _ l } \\left ( \\bigg \\{ \\sum _ { i = 1 } ^ { k _ l } \\left ( G _ { l , i } - \\mathbb E _ { F _ l } ( G _ { l , i } ) \\right ) \\leqslant t \\hat { s } _ { l } \\bigg \\} \\right ) = \\mathcal { N } ( t ) , \\forall t \\in \\mathbb R . \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\sum _ { h + n = L _ 0 - k _ j } \\frac { \\alpha _ j ^ n } { n ! } b _ h = \\sum _ { h = 0 } ^ { \\min \\{ L _ 0 - k _ j , L \\} } \\frac { \\alpha _ j ^ { L _ 0 - k _ j - h } } { ( L _ 0 - k _ j - h ) ! } b _ h = 0 \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} U H _ \\mu ( \\xi ) U ^ * = H _ \\mu ( \\xi , A ) \\oplus \\bigoplus _ { n = 1 } ^ \\infty H _ { n , \\mu } ( \\xi , A ) \\mid _ { \\mathcal { F } ( \\mathcal { H } _ A ) \\otimes \\mathcal { H } _ { A ^ c } ^ { \\otimes _ s n } } : = G _ A ( \\xi ) \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} R ( \\lambda + i 0 ; x , y ) & = \\tfrac { i } { 2 \\sqrt { \\lambda } } \\bigl [ e ^ { i | x - y | \\sqrt { \\lambda } } - \\tfrac { q } { q - i \\sqrt { \\lambda } } e ^ { i ( | x | + | y | ) \\sqrt { \\lambda } } \\bigr ] , \\\\ R ( \\lambda - i 0 ; x , y ) & = \\tfrac { i } { 2 \\sqrt { \\lambda } } \\bigl [ e ^ { - i | x - y | \\sqrt { \\lambda } } - \\tfrac { q } { q + i \\sqrt { \\lambda } } e ^ { - i ( | x | + | y | ) \\sqrt { \\lambda } } \\bigr ] \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} \\max _ { t \\in [ 0 , T ] } \\left | ^ { W } D ^ 1 f ( t ) - \\tilde { f } ^ { ( 1 ) } ( t ) \\right | = O \\big ( 1 / N ^ { 1 + \\mu - 1 } \\big ) = O \\big ( 1 / N ^ { \\mu } \\big ) . \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} A ^ { I J } = \\partial _ x ^ { } \\Big ( G ^ { I J } ( u ) \\partial _ x ^ { } + C ^ { I J } _ K ( u ) u ^ K _ x \\Big ) \\partial _ x ^ { } , \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} \\left | \\frac { \\varphi ( u ) } { u } - \\sum _ { k = 1 } ^ { p - 1 } ( - 1 ) ^ { k r } b _ k \\frac { 1 } { u ^ { k r + 1 } } \\right | = \\frac { 1 } { | u | } \\left | \\psi ( u ^ { - r } ) - \\sum _ { k = 0 } ^ { p - 1 } ( - 1 ) ^ { k r } b _ k ( u ^ { - r } ) ^ k \\right | \\le \\frac { C A ^ p M _ p } { | u | ^ { p r + 1 } } \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{gather*} v : \\mathbb { R } \\times S ^ 1 \\rightarrow M \\\\ \\phi _ L ^ t ( v ( s , t ) ) = u ( s , t ) \\end{gather*}"}
-{"id": "767.png", "formula": "\\begin{align*} \\dim \\widehat { M } ^ { + } = n + g - 1 , \\ \\dim \\widehat { M } ^ { - } = - n + g - 1 \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\prod _ { 1 \\leq i \\leq j \\leq n } \\frac { 2 r + i + j - 1 } { i + j - 1 } \\cdot \\frac { i + j } { i + j } = 2 ^ n \\prod _ { 1 \\leq i \\leq j \\leq n } \\frac { 2 r + j + i - 1 } { j + i } , \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\rho ( m _ t ) R _ j & = \\epsilon ( [ g _ { \\alpha } m _ t g _ j ^ { - 1 } ] ) & & [ \\textrm { D e f i n i t i o n } ( \\ref { L i f t R e p } ) ] \\\\ & = \\epsilon ( [ g _ { \\alpha } g _ t ^ { - 1 } m g _ t g _ j ^ { - 1 } ] ) & & [ m _ t = g _ t ^ { - 1 } m g _ t ] \\\\ & = \\epsilon ( [ g _ { \\alpha } g _ t ^ { - 1 } g _ t g _ j ^ { - 1 } ] ) - \\epsilon ( [ g _ { \\alpha } g _ t ^ { - 1 } ] ) \\epsilon ( [ g _ t g _ j ^ { - 1 } ] ) & & [ \\textrm { S k e i n r e l a t i o n } ] \\\\ & = R _ j - \\epsilon ( [ g _ t g _ j ^ { - 1 } ] ) R _ t , \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} z _ B \\triangleright \\psi & = \\mathcal { D } ^ { - 1 } ( z ) \\big ( ? b _ i b _ j S ^ 2 ( a _ j a _ k ) g \\big ) \\psi ( ? a _ i b _ k ) \\\\ & = \\mathcal { D } ^ { - 1 } ( z ) \\big ( ? v ^ { - 1 } b _ i a _ k \\big ) \\psi ( ? a _ i b _ k ) = \\mathcal { D } ^ { - 1 } ( z ) \\big ( ? \\big ( v ^ { - 1 } \\big ) ' \\big ) \\psi \\big ( ? v \\big ( v ^ { - 1 } \\big ) '' \\big ) , \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} \\int _ { U \\times Y } ( \\phi ( y ) - \\phi ( y _ 0 ) ) \\gamma _ l ( d y , d u ) = \\frac { 1 } { \\lambda _ l } \\int _ { U \\times Y } \\nabla \\phi ( y ) ^ T f ( u , y ) \\gamma _ l ( d y , d u ) . \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} \\Xi _ r = \\{ \\xi \\in \\mathbb { R } ^ n _ + : | \\xi | \\leq r ^ { 3 / 2 } \\} . \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} q ( x ) = x ^ 3 - ( 7 i + 2 ) x ^ 2 - ( 7 i + 3 ) x + 1 2 i ^ 3 + 1 8 i ^ 2 + 6 i \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta H _ \\gamma & = 0 , \\mbox { i n } \\ \\Omega \\setminus \\gamma , \\\\ H _ \\gamma & = 0 , \\mbox { o n } \\ \\partial \\Omega , \\\\ H _ \\gamma & = 1 , \\quad \\mbox { o n } \\ \\gamma . \\end{aligned} \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\chi ( S ) = \\chi ( S ' ) \\ , . \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} ( \\lambda ) _ { \\upsilon } : = \\frac { \\Gamma ( \\lambda + \\upsilon ) } { \\Gamma ( \\lambda ) } = \\begin{cases} 1 , \\quad ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( \\upsilon = 0 ~ ; ~ \\lambda \\in \\mathbb { C } \\backslash \\{ 0 \\} ) \\\\ \\lambda ( \\lambda + 1 ) . . . ( \\lambda + n - 1 ) , ( \\upsilon = n \\in \\mathbb { N } ~ ; ~ \\lambda \\in \\mathbb { C } ) . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} \\norm { f } _ { \\mathcal { X } _ p } ^ 2 : = \\int _ { \\R } \\int _ { \\mathbb { B } _ p } \\abs { f ( t , \\zeta ) } ^ 2 \\lambda ( p ( \\zeta ) , t ) \\d V ( \\zeta ) \\d t < \\infty , \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} f _ u \\rho = d \\iota ( u ) \\rho . \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 g \\big ( \\dot x , \\tfrac { \\mathrm D V _ x } { \\mathrm d s } \\big ) \\ , \\mathrm d s = \\int _ 0 ^ 1 g \\big ( \\dot x , \\tfrac { \\mathrm D } { d s } \\mathcal R V _ x \\big ) \\ , \\mathrm d s \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} n I _ n = H _ 1 ^ \\top H _ 1 + ( H _ 2 ' ) ^ \\top H _ 2 ' + J _ n . \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} v _ { \\varepsilon ' } ( w ) = \\int _ { \\partial S _ { \\varepsilon } ^ + } e ^ { z w } u _ { \\varepsilon ' } ( z ) d z \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} E _ 2 = H ^ r ( \\Gamma _ 0 ^ { \\prime } , H ^ s ( \\Gamma _ { 0 , Z } , V ^ { 0 , q } _ { \\Psi } ) ) \\Rightarrow H ^ { r + s } ( \\Gamma _ 0 , V ^ { 0 , q } _ { \\Psi } ) \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\lambda ^ 2 } { \\rho _ \\lambda } \\int _ \\Omega e ^ { \\ , u } \\ , d x & = 2 c _ \\Omega \\int _ \\gamma | \\partial _ n K ^ \\pm _ \\gamma | + c _ \\Omega \\int _ \\alpha \\partial _ n K _ \\alpha \\\\ & = c _ \\Omega | \\Omega ^ + | + c _ \\Omega | \\Omega ^ - | \\\\ & = 1 , \\end{aligned} \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} f & : = e _ { 1 2 } + e _ { 1 3 } + e _ { 1 4 } + e _ { 2 3 } + e _ { 2 4 } + e _ { 2 5 } + e _ { 3 4 } + e _ { 3 5 } + e _ { 3 6 } \\\\ e & : = e _ { 0 1 } + e _ { 0 2 } + e _ { 0 3 } + e _ { 0 4 } + e _ { 2 3 } + e _ { 2 4 } + e _ { 2 5 } + e _ { 3 4 } + e _ { 3 5 } + e _ { 3 6 } \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} \\chi ( \\bar { X } _ { \\Gamma _ { 0 , s s } } , \\mathcal { O } _ { \\bar { X } _ { \\Gamma _ { 0 , s s } } } ) = \\mbox { v o l } ( X _ { \\Gamma _ { 0 , s s } } ) + L _ { c u s p } \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} B _ { T } = \\{ ( v , a ) : \\| v \\| _ { ( L _ t ^ \\infty H _ x ^ 1 \\cap L _ t ^ 4 L _ x ^ \\infty ) ( [ 0 , T ] \\times \\R ) } \\leq 2 C M , \\| a \\| _ { L _ t ^ \\infty ( [ 0 , T ] ) } \\leq 2 C M \\} , \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} h _ t ( x - y ) = \\frac { 1 } { ( 4 \\pi t ) ^ { n / 2 } } \\ , \\exp \\bigg ( - \\frac { | x - y | ^ 2 } { 4 t } \\bigg ) \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} \\zeta _ { f , g } ( s ) = \\prod _ { \\varpi } \\left ( 1 - \\exp \\left [ - s \\int _ 0 ^ { \\mathrm { p e r } ( \\varpi ) } g \\left ( f ^ t x ( \\varpi ) \\right ) \\ , d t \\right ] \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} h ( \\ell , \\ell + 1 ) = \\frac { f ( \\ell ) } { \\ell + 1 } h ( \\ell - 1 , \\ell ) = f ( \\ell ) \\prod _ { i = 1 } ^ \\ell \\frac { f ( i - 1 ) } { i + 1 } . \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} \\int e ^ { - \\kappa [ 1 + \\beta ( \\nabla \\phi ) ^ 2 ] } f ( \\kappa ) d \\kappa = e ^ { - V ( \\nabla \\phi ) } , \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} \\omega = \\delta _ z ( 1 / \\abs { z } ) = \\left ( \\frac { z _ 1 } { \\abs { z } ^ { M / 2 m _ 1 } } , \\cdots , \\frac { z _ n } { \\abs { z } ^ { M / 2 m _ n } } \\right ) . \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\tilde { w } ( v , v ' ) = \\left \\{ \\begin{array} { c l } H ' ( \\bar { x } _ { v ' } - \\bar { x } _ { v } ) & : v ' \\in N ( v ) , \\ v ' \\neq v \\\\ 1 + M - \\sum _ { v '' } H ' ( \\bar { x } _ { v '' } - \\bar { x } _ { v } ) & : v ' = v \\\\ 0 & : v ' \\notin N ( v ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} \\frac { w _ { \\lambda } ^ \\prime ( 1 ) } { w _ { \\lambda } ( 1 ) } = \\frac { h _ { 1 , \\lambda } ^ \\prime ( 1 ) } { h _ { 1 , \\lambda } ( 1 ) } + \\frac { n - 1 } { 2 } \\sqrt { K _ 0 } . \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} H ^ 0 ( M , E ) ^ \\Lambda = H ^ 0 ( X , E _ X ) , \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} A = C _ 0 \\otimes I + C _ 1 \\otimes T _ 1 + \\ldots + C _ r \\otimes T _ r \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} X _ V = p . \\partial _ q - \\partial _ q V ( q ) . \\partial _ p ~ , \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} \\alpha = s ^ { - 1 } d s \\wedge i _ X \\alpha = d s \\wedge ( s ^ { - 1 } i _ X \\alpha ) . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} y _ { i , j } = \\sum _ { k = 0 } ^ { M } p _ { k } ( i , j ) n ^ { - 1 + 2 \\beta - k } + o ( n ^ { - 1 + 2 \\beta - M } ) , \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} \\Psi ( x ) = \\textbf { T r } \\left ( \\psi ( x ) \\right ) , \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\langle \\pi \\left ( \\frac { ( t , \\tau ) } { ( s , \\sigma ) } \\right ) \\xi , \\eta \\rangle = \\langle \\theta ( \\sigma ) \\Phi ( s ) \\xi , \\theta ( \\tau ) \\Phi ( t ) \\eta \\rangle . \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} \\frac { \\delta F } { \\delta \\psi } & = D \\frac { \\partial \\epsilon } { \\partial \\psi } - \\left ( D \\frac { \\partial \\epsilon } { \\partial \\nabla \\psi } \\right ) \\\\ & = 2 D \\widehat { H } _ e \\psi + \\frac { i \\hbar } { M } D \\boldsymbol { A } \\cdot \\nabla \\psi + \\frac { i \\hbar } { M } \\operatorname { d i v } \\ ! \\big ( D ( i \\hbar \\nabla + \\boldsymbol { A } ) \\psi \\big ) , \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} s _ { \\ 1 } ^ R ( M _ { l , p } ^ * ) = \\zeta ^ { p } \\frac { \\zeta ^ { l } - 1 } { \\zeta - 1 } = s _ { M _ { d - 1 , 0 } } ^ R ( M _ { l , p } ) \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} f = \\left ( \\int _ { \\omega \\widehat { A _ X } } \\check f ( \\chi ) d \\chi \\right ) d a , \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\frac 1 { \\hat { s } _ { l } ^ 2 } \\sum _ { i = 1 } ^ { k _ l } L _ { F _ l } ( G _ { l , i } , \\eta \\hat s _ l ) = 0 , \\forall \\eta > 0 \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} d X _ t = u ( X _ t ) d t + \\beta _ j ( X _ t ) d \\Z ^ j _ t , \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} \\widetilde { \\Delta } _ h \\varphi _ { \\nu , \\ell } = \\lambda _ { \\nu } \\varphi _ { \\nu , \\ell } , \\langle \\varphi _ { \\nu , \\ell } , \\varphi _ { \\nu , \\ell ' } \\rangle _ { L ^ 2 ( Y ) } = \\delta _ { \\ell , \\ell ' } = \\begin{cases} 1 , \\ell = \\ell ' \\\\ 0 , \\ell \\neq \\ell ' . \\end{cases} \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{gather*} \\mathcal { N } ( z _ - , z _ + , [ 0 , 1 ] , \\rho , H ^ { ( k ) } , J ) \\\\ = \\Bigg \\{ ( \\tau , v ) \\in [ 0 , 1 ] \\times C ^ { \\infty } ( \\mathbb { R } \\times S ^ 1 \\rightarrow M ) \\ \\Bigg | \\ \\begin{matrix} v \\textrm { \\ s a t i s f i e s \\ } ( \\textrm { B } ) \\\\ \\lim _ { s \\rightarrow \\pm \\infty } v ( s , t ) = x _ { \\pm } ( t ) \\\\ [ u _ - \\sharp v , x _ - ] = [ u _ + , x _ + ] \\end{matrix} \\Bigg \\} \\end{gather*}"}
-{"id": "9492.png", "formula": "\\begin{align*} R _ 1 ( \\lambda ; x , y ) & = \\tfrac { i } { 2 \\sqrt { \\lambda } } [ e ^ { i | x - y | \\sqrt { \\lambda } } - e ^ { i ( | x | + | y | ) \\sqrt { \\lambda } } ] , \\\\ R _ 2 ( \\lambda ; x , y ) & = \\tfrac { 1 } { 2 ( q - i \\sqrt { \\lambda } ) } e ^ { i ( | x | + | y | ) \\sqrt { \\lambda } } . \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 1 , 2 ) = \\int _ { 0 } ^ { \\infty } \\frac { x \\cos ( 2 \\pi x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x = \\frac { 1 } { 6 4 } \\left ( \\frac { 1 } { 2 } - \\frac { 3 } { \\pi } + \\frac { 5 } { \\pi ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ \\tau \\phi _ j = - ( - \\Delta ) ^ s _ y \\phi _ j + p U ^ { p - 1 } ( y ) \\phi _ j + H _ j [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ( y , t ( \\tau ) ) , ~ y \\in \\mathbb { R } ^ n , ~ \\tau \\geq \\tau _ 0 , \\\\ & \\phi _ j ( y , \\tau _ 0 ) = e _ { 0 j } Z _ 0 ( y ) , ~ y \\in \\mathbb { R } ^ n , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} S ^ { \\phi } _ k = \\sum _ { a \\in A _ { + , k } } \\frac { \\mu ( a ) C _ { a } ( X _ 1 ) \\dots C _ { a } ( X _ n ) } { a } \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} f ( v + \\lambda ) \\rho ( \\lambda ) = f ( v ) \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} u _ { j } = \\sum _ { k = 1 } ^ { m } U _ { k , j } \\phi _ { k } , \\mbox { f o r $ j = 1 , \\ldots , s $ , } \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} \\textbf { J } _ { C } ( \\upsilon , b , c , \\lambda , y ) = \\int _ { 0 } ^ { \\infty } x ^ { \\upsilon - 1 } e ^ { - b \\lambda \\sqrt { x } } { } _ { r } \\Psi _ { s } \\left [ \\ \\begin{array} { l l l } ( \\alpha _ { 1 } , A _ { 1 } ) , . . . , ( \\alpha _ { r } , A _ { r } ) ; \\\\ ( \\beta _ { 1 } , B _ { 1 } ) , . . . , ( \\beta _ { s } , B _ { s } ) ; ~ \\end{array} e ^ { - c \\sqrt { x } } \\right ] \\cos ( x y ) d x , \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} ( \\gamma _ { k + 1 } ) ^ { - t } \\geq ( d _ k \\gamma _ k ) ^ { - t } \\big ( 1 - e _ k \\gamma _ k ^ { t } \\big ) ^ { - t } . \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} \\min ( u _ 0 ) = \\max \\{ \\min ( u ) \\ ; : u \\in G ( I ) \\} u _ 0 = x _ { i _ 1 } x _ { i _ 1 + t } \\cdots x _ { i _ 1 + t ( d - 1 ) } \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} \\mathrm { g r } ( G ) = \\liminf _ { n \\to \\infty } M _ n ^ { 1 / n } . \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} C = I _ { n - 2 } + \\frac { 2 \\bar { a } \\bar { a } ^ T } { 1 + \\sqrt { 1 + 2 \\| \\bar { a } \\| ^ 2 } } . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} \\chi _ \\mu ( e ( - v / u ) ; \\lambda ) : = \\chi _ \\mu ( \\lambda ) , \\chi _ { \\mu ' } ( e ( - u / v ) ; \\lambda ' ) : = \\chi ' _ { \\mu ' } ( \\lambda ' ) = \\sum _ { w \\in W } \\epsilon ( w ) e ^ { - 2 \\pi i \\frac { u } { v } \\left ( \\lambda ' + \\rho , w ( \\mu ' + \\rho ) \\right ) } \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} \\limsup _ { s \\to \\infty } \\mathbb P ( Z _ t \\xrightarrow { t \\to \\infty } 0 \\mid \\sigma ( Z _ r ; r \\leq s ) ) \\geq \\tfrac 1 { 1 + \\lambda } \\limsup _ { s \\to \\infty } ( 1 - p ) ^ { Z _ s } = \\tfrac 1 { 1 + \\lambda } ( 1 - p ) ^ { \\liminf \\limits _ { s \\to \\infty } Z _ s } . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} S ( p k , k ) \\equiv \\prod _ i \\binom { a _ i + a _ { i + 1 } } { a _ i } \\mod p . \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} [ n ] \\setminus \\big ( \\underbrace { \\bigcup _ { i \\in [ k ] } V ( E _ { P _ i } ) } _ { \\supseteq I \\cup J } \\big ) = ( J \\cup I ) ^ c \\setminus \\big ( \\bigcup _ { i \\in [ k ] } V ( E _ { P _ i } ) \\big ) = J ^ c \\cap I ^ c \\setminus \\big ( \\bigcup _ { i \\in [ k ] } { V ( E _ { P _ i } ) } \\big ) \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} g _ c = d r ^ 2 \\oplus f ^ 2 ( r ) \\cdot h ( r ) \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| v ^ { j _ 0 } _ n \\| _ { L ^ 2 } = 0 . \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} ( M ' ( t ) ) ^ 2 & = - ( M ' ( 0 ) ) ^ 2 + 2 M ' ( t ) M ' ( 0 ) + \\left ( \\int _ 0 ^ t \\int _ { \\Omega } u u _ t d x d \\tau \\right ) ^ 2 \\\\ & = - \\frac 1 4 | | u _ 0 | | ^ 2 + M ' ( t ) | | u _ 0 | | ^ 2 + \\left ( \\int _ 0 ^ t \\int _ { \\Omega } u u _ t d x d \\tau \\right ) ^ 2 \\\\ & \\leq M ' ( t ) | | u _ 0 | | ^ 2 + \\left ( \\int _ 0 ^ t \\int _ { \\Omega } u u _ t d x d \\tau \\right ) ^ 2 . \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} X : = ( 1 , x _ 1 , x _ 2 , \\dots , x _ n , x _ 1 ^ 2 , x _ 1 x _ 2 , \\dots , x _ 1 x _ n , \\dots , x _ 1 ^ d , x _ 1 ^ { d - 1 } x _ 2 , \\dots , x _ n ^ d ) ^ T . \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} \\rho _ { N + 1 } = \\bigl ( Q - \\rho _ N \\bigr ) ^ { - 1 } , \\rho _ 1 = 1 / Q . \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} \\hat H _ { \\partial M } & = t r _ { \\partial M } ( \\hat \\nabla \\mathbf { \\hat n } ) \\\\ & = t r _ { \\partial M } [ \\nabla d \\Phi _ f ( \\mathbf { \\hat n } ) ] \\\\ & = t r _ { \\partial M } [ \\nabla ( b \\mathbf { N } + a \\mathbf { n } ) ] \\\\ & = b t r _ { \\partial M } ( \\nabla \\mathbf { N } ) + a t r _ { \\partial M } ( \\nabla \\mathbf { n } ) \\\\ & = b t r _ { \\partial M } K + a H _ { \\partial M } . \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} \\frac { Z ( W , s , \\mu _ p ) } { L ( s , \\mu _ p \\tilde { \\pi } _ { p } ) } \\epsilon ( s , \\mu _ p \\tilde { \\pi } _ { p } ) = \\frac { Z ( \\tilde { \\pi } _ p ( w ) W , 1 - s , \\mu _ p ^ { - 1 } \\omega _ { \\tilde { \\pi } , p } ^ { - 1 } ) } { L ( 1 - s , \\mu _ p ^ { - 1 } \\pi _ { p } ) } . \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} q _ 0 ( y ) : = \\frac { p U ( y ) ^ { p - 1 } c _ 2 b _ j ^ { 2 s } } { ( n - 4 s ) c _ { n , s } ^ { n - 4 s } c _ 1 } + \\frac { b _ j ^ { 2 s } } { ( n - 4 s ) c _ { n , s } ^ { n - 4 s } } \\left ( Z _ { n + 1 } ( y ) + \\frac { n - 2 s } { 2 } \\alpha _ { n , s } \\frac { 1 } { \\left ( 1 + | y | ^ 2 \\right ) ^ { \\frac { n - 2 s } { 2 } } } \\right ) \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} ( f _ \\delta , ( - \\mathcal { L } _ X ^ \\omega ) ^ { - 1 } f _ \\delta ) = \\int _ { 0 } ^ { \\infty } d t \\delta ^ { - 2 } ( f _ \\delta , e ^ { t \\delta ^ { - 2 } \\mathcal { L } _ X ^ \\omega } f _ \\delta ) . \\end{align*}"}
-{"id": "9784.png", "formula": "\\begin{align*} T ^ * _ { \\pm } \\kappa ^ * ( - t ) ( i \\sigma ) \\kappa ( - t ) T _ { \\pm } = e ^ { \\mp t } \\Big [ & \\pm \\sin ( \\alpha ) \\Big ( C ^ 2 ( t ) + ( 1 - ( 2 z ) ^ 2 ) S ^ 2 ( t ) \\Big ) - 2 \\cos ( \\alpha ) C ( t ) S ( t ) \\\\ & + i \\cos ( \\alpha ) \\Big ( C ^ 2 ( t ) + ( 1 - ( 2 z ) ^ 2 ) S ^ 2 ( t ) \\Big ) \\widetilde { I } \\mp 2 i \\sin ( \\alpha ) C ( t ) S ( t ) \\widetilde { I } \\\\ & + 4 z i \\cos ( \\alpha ) S ^ 2 ( t ) \\widetilde { J } \\mp 4 z \\sin ( \\alpha ) S ^ 2 ( t ) \\widetilde { K } \\Big ] { ~ . } \\\\ & \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} R _ 4 ^ { \\frac 1 2 } = & ( \\sigma ( t _ { \\frac 1 2 } ) - \\sigma ^ { \\frac 1 2 } ) ) - \\bigtriangleup ( u ( t _ { \\frac 1 2 } ) - u ^ { \\frac 1 2 } ) = O ( \\Delta t ^ 2 ) . \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} v \\sim - w + a - a a - \\sum _ { i = 1 } ^ { 4 } a b ^ i a - \\sum _ { i = 1 } ^ { 4 } a b ^ i a ^ { - 1 } \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} T _ 2 ( \\omega ) = 2 \\mathrm { o r d } ( \\omega ) n \\sum _ { k ' = 0 } ^ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } \\sum _ { j = 0 } ^ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } - 1 } \\binom { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } { k ' } ^ 2 k ' \\binom { \\frac { n } { \\mathrm { o r d } ( \\omega ) } - 1 } { j } \\binom { k ' - 1 } { j } \\binom { k ' + j } { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } \\sum _ { i = 1 } ^ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } - 1 } \\frac { \\omega ^ i } { ( \\omega ^ i - 1 ) ^ 2 } . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} f ( t , x , \\omega ) ~ = ~ F \\bigl ( v ( t , x ) , \\omega \\bigr ) , \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} ( a \\tfrac { 1 } { c \\overline { c } } y ^ 2 + \\tfrac { 1 } { d \\overline { d } } z ^ 2 - ( a { + } 1 ) x ^ 2 - a ^ 2 - b ^ 2 - a ) ^ 2 \\ + \\ b ^ 2 ( x ^ 2 - \\tfrac { 1 } { c \\overline { c } } y ^ 2 + 1 ) ^ 2 \\ - \\ 4 b ^ 2 x ^ 2 \\ = \\ 0 \\ , . \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} R ( x ) = \\sum \\langle a _ i , x \\rangle b _ i \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} [ X , Y ] = [ X , Y ] _ 0 + \\frac { 1 } { | \\Gamma | } \\sum _ { q \\in \\pi ^ { - 1 } ( p ) } { \\rm R e s } _ { q } \\langle d X , Y \\rangle C , \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} | k _ b j _ b + \\frac { i - j } { n - \\frac 1 2 } | < \\frac { 1 } { 5 0 n \\sum _ { b = 1 } ^ n | j _ b | } | i - j | , b = 1 , \\cdots , n . \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} d V _ t & = d K _ t + X _ t d S _ t + S _ t d X _ t \\\\ & = \\Big [ - c _ { \\alpha } ( \\alpha _ t ) + \\gamma X _ t \\int _ { \\mathbb { R } } a d \\theta _ t ( a ) \\Big ] d t + \\sigma S _ t d W _ t + \\sigma _ 0 X _ t d W _ t ^ 0 . \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} H _ n ( q , t ) = \\sum _ { p = 0 } ^ { \\infty } \\frac { \\binom { h ( n , p ) + 1 } { p } _ q \\cdot q ^ { p ( p - 1 ) } t ^ p } { ( 1 - q ^ { n - h ( n , p ) } t ) \\cdots ( 1 - q ^ { n - 1 } t ) } , \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} f ( y ) = 2 \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } ( 4 y ^ { 2 } - 1 ) e ^ { - 2 y ^ { 2 } } - c o n s t \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{gather*} { } _ 4 F _ 3 \\biggl ( \\begin{matrix} \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 \\\\ 1 , \\ , 1 , \\ , 1 \\end{matrix} \\biggm | z \\biggr ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( \\frac 1 2 ) _ k ^ 4 } { k ! ^ 4 } z ^ k . \\end{gather*}"}
-{"id": "1296.png", "formula": "\\begin{align*} \\mathcal { B } _ v ( \\lambda ) \\otimes \\mathcal { B } _ w ( \\mu ) & = \\bigsqcup _ { \\pi \\in \\mathcal { B } _ w ( \\mu ) ^ \\lambda } \\left ( \\bigcup _ { a _ 1 , \\ldots , a _ l \\geq 0 } \\tilde { f } _ { i _ 1 } ^ { a _ 1 } \\cdots \\tilde { f } _ { i _ l } ^ { a _ l } ( C ( \\pi , e ) ) \\setminus \\{ 0 \\} \\right ) . \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} M \\cong \\bigoplus _ { j = 0 } ^ { n - 1 } R / ( d _ j ( X ) ) . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} \\mathbf { E } [ L _ { i } ] = \\sum _ { m = 1 } ^ { i } \\left [ K ^ { i - m } ( K - 1 ) \\prod _ { j = m } ^ { i } p _ j \\right ] . \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} \\Gamma ( f , h ) = \\frac { 1 } { \\beta } ( P \\nabla f ) \\cdot ( P \\nabla h ) \\ , . \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} u _ R ( x _ 1 ) & \\le c _ R ( x _ 1 , \\overline x _ 2 , \\dots , \\overline x _ N ) - \\frac { N - 1 } N c _ R ( \\overline x _ 1 , \\ldots , \\overline x _ N ) \\\\ & \\le \\frac { N ( N - 1 ) } { 2 } f ( \\frac { \\alpha } 2 ) - \\frac { N - 1 } N c _ R ( \\overline x _ 1 , \\ldots , \\overline x _ N ) = : M , \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} c _ { \\mu _ n } = I _ { \\mu _ n } ( u _ n ) \\to I ( u ) , \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} \\mathcal { F } _ { D , 0 } ( z , s ) : = 2 ^ { s - 1 } \\pi ^ { - \\frac { s + 1 } { 2 } } | D | ^ { \\frac { s } { 2 } } L _ D ( s ) B ( s ) \\Gamma \\biggl ( \\frac { s + 1 } { 2 } \\biggr ) P _ { \\frac { 1 } { 2 } , 0 } \\biggl ( z , \\frac { s } { 2 } + \\frac { 1 } { 4 } \\biggr ) . \\end{align*}"}
-{"id": "9693.png", "formula": "\\begin{align*} [ \\tilde { \\phi } ( \\tilde { A } / f \\tilde { A } ) ] _ { \\tilde { A } } = f + c ( f ) p _ 1 z ^ { m d } + c ( f ) p _ 2 z ^ { 2 m d } + \\dots + c ( f ) z ^ { r _ 0 m d } . \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} P u = 0 , u | _ { \\partial \\Omega } = \\partial _ { \\nu } u | _ { \\Omega } = 0 \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} \\phi ( t , \\zeta ) = R _ { \\mathcal { X } _ p } \\left ( e _ { ( z , w ) } \\circ T _ V \\right ) ( t , \\zeta ) - \\overline { X _ p ( t ; w , \\zeta ) } \\frac { \\overline { z } ^ { - 2 \\pi i t } } { \\overline { z } ^ { 1 + 1 / 2 \\mu } } . \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} \\mathbb { E ^ { * } } = - Q \\mathbb { I } - \\mathbb { E } , \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} \\dot { \\lambda } _ j + \\frac { 1 } { t } \\left ( P ^ T d i a g \\left ( \\frac { \\frac { n - 2 s } { 2 s } \\bar { \\sigma } _ r b _ r ^ { 2 - 2 s } + 1 } { n - 4 s } \\right ) P \\lambda \\right ) _ j = h ( t ) _ j \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\tilde \\rho ( r , t ) = \\sum _ { a = 1 } ^ N \\varrho _ a ( t ) \\delta ( r - q _ a ( t ) ) \\ , , \\varrho _ a ( t ) : = U ( q _ a ^ { ( 0 ) } , t ) \\varrho _ a ^ { ( 0 ) } U ^ \\dagger ( q _ a ^ { ( 0 ) } , t ) \\ , . \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} u ^ { \\zeta } _ { 0 , x _ { i } } ( 0 ) & = u _ { 0 , x _ { i } } ( 0 ) \\\\ - \\epsilon u ^ { \\zeta } _ { 0 , x _ { i } x _ { i } } ( 0 ) + H _ { i } ( 0 , 0 , u ^ { \\zeta } _ { 0 , x _ { i } } ( 0 ) ) & = - \\epsilon u ^ { \\zeta } _ { 0 , x _ { 1 } x _ { 1 } } ( 0 ) + H _ { 1 } ( 0 , 0 , u ^ { \\zeta } _ { 0 , x _ { 1 } } ( 0 ) ) \\\\ \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} q = { \\min } _ { \\prec } \\{ 1 , 2 , \\dots , k + n + 1 \\} \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} \\| S x _ k - f _ k \\| _ E < \\delta _ k \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\| f _ k \\| _ E \\ge 1 - \\delta _ k , \\ ; \\ ; k = 1 , 2 , \\dots \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} C = \\left \\{ \\sum _ { i = 1 } ^ n \\eta _ i \\varpi _ i \\in P ^ + \\ \\middle \\vert \\ \\sum _ { i = 1 } ^ n \\eta _ i \\leq d - ( n + 1 ) \\right \\} \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} w - \\lim _ { k \\rightarrow 0 , k \\in \\widetilde { S } _ \\varepsilon ( \\xi ) } Q ( k ) - P _ 0 Q ( k ) P _ 0 = 0 . \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} \\Phi ( x ) = \\sum _ { i = 0 } ^ n a _ i \\left ( \\frac { 3 i + 7 } { i + 2 } \\right ) \\| x \\| _ 2 ^ { i + 2 } . \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} | k _ b j _ b + \\frac { i + j } { n - \\frac 1 2 } | < \\frac { | i | + | j | } { 5 0 n \\sum _ { b = 1 } ^ n | j _ b | } , b = 1 , \\cdots , n . \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} \\mathbb { K } _ { n } ( x , y ) = \\min \\left \\{ 1 - \\rho _ { n , x , y } ^ { 2 } , \\frac { \\Delta } { \\left ( 1 - \\rho _ { n , x , y } ^ { 2 } \\right ) ^ { 3 / 2 } } \\right \\} , \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} \\textrm { s u p p } \\ o s c ( \\psi _ j , B _ \\epsilon ( \\cdot ) ) \\subset X _ \\epsilon , \\ j = 1 , 2 , \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} \\dot { v } _ t & = \\nabla _ x f ( x ^ * _ t , \\theta ^ * _ t ) ^ T v _ t , t \\in ( \\tau , T ] , \\\\ v _ \\tau & = f ( x ^ * _ \\tau , \\omega ) - f ( x ^ * _ \\tau , \\theta ^ * _ \\tau ) . \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} F ^ { ( u ) } _ { t } ( \\cdot ) \\ , : = \\ , u \\cdot \\delta _ { \\widetilde { X } ^ { ( u ) } _ { t } } ( \\cdot ) + ( 1 - u ) \\cdot \\mathcal L _ { X ^ { ( u ) } _ { t } } ( \\cdot ) \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} \\alpha ( h ) \\alpha ( \\varphi ( h ^ \\prime ) ) = \\alpha ( h ^ \\prime ) \\alpha ( \\varphi ( h ) ) . \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} & \\sup _ { R > 0 } \\int _ { R < | x | < 2 R } | \\mathbf { K } ( x ) | d x \\lesssim c _ 1 , \\\\ & \\sup _ { y \\not = 0 } \\int _ { | x | \\geq 2 | y | } \\left | \\mathbf { K } ( x - y ) - \\mathbf { K } ( x ) \\right | d x \\lesssim c _ 1 . \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} \\left ( w _ j / w _ k ; q \\right ) _ \\infty \\left ( w _ j / w _ k ; q \\right ) _ \\infty = \\left ( q ^ { n _ j - n _ k } ; q \\right ) _ \\infty \\left ( q ^ { n _ k - n _ j } ; q \\right ) _ \\infty = 0 . \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} \\norm { f } _ { \\mathcal { S } _ p ( 1 ) } & = \\norm { D _ t f } _ { \\mathcal { S } _ p ( t ) } . \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } A ^ { W , W } _ \\cdot ( l _ n ) & = \\lim _ { n \\to \\infty } \\left ( \\phi _ n ( \\cdot , W ) - \\phi _ n ( 0 , W _ 0 ) - \\int _ 0 ^ \\cdot \\nabla \\phi _ n ^ * ( r , W _ r ) \\mathrm d W _ r \\right ) \\\\ & = 0 . \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} | k _ b j _ b | < \\frac { | j | } { n - \\frac 1 2 } + \\frac { | j | } { 5 0 n \\sum _ { b = 1 } ^ n | j _ b | } < \\frac { | j | } { n - \\frac 5 9 } , b = 1 , \\cdots , n , \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} N ( x ^ k , r ) = { q \\choose r } q ^ { k - r } \\left ( \\sum _ { j = 0 } ^ { k - r } ( - 1 ) ^ j { q - r \\choose j } q ^ { - j } \\right ) . \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} B _ { i j } \\ ; = \\ ; - B _ { i i } \\sum _ { k } ^ { ( i ) } ( B ^ { - 1 } ) _ { i k } B _ { k j } ^ { ( i ) } \\ ; = \\ ; - B _ { j j } \\sum _ k ^ { ( j ) } B ^ { ( j ) } _ { i k } ( B ^ { - 1 } ) _ { k j } \\ , . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\| A _ j u _ { j + m N } - y _ j \\| _ { p } = 0 ; \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 1 } ^ r [ \\mathcal { F } _ { D , 0 } ( z , s ) ] ^ { \\mathrm { h o l } } & = \\mathrm { L C } _ { s = 1 } ^ r \\biggl [ \\frac { | D | ^ { \\frac { s } { 2 } } L _ D ( s ) } { 2 ^ { 2 s - 1 } \\pi ^ { \\frac { s - 1 } { 2 } } } \\frac { \\Gamma ( s - 1 / 2 ) } { \\Gamma ( s / 2 ) } \\frac { \\zeta ( 2 s - 1 ) } { \\zeta ( 2 s ) } \\biggr ] + \\sum _ { 0 < d \\equiv 0 , 1 ( 4 ) } \\frac { 1 } { \\sqrt { d } } \\mathrm { T r } _ { d , D } ( F _ { 0 , 0 , r } ) q ^ d . \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} \\pi ( k , \\mathbf { m } ) = \\sum _ { b = 1 } ^ n k _ b j _ b + \\mathbf { m } , \\ \\mathbf { m } \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} 2 e _ i ( \\frac { Y ^ { \\perp } } { u } ) \\nabla _ { e _ i } \\partial _ t = 2 e _ i ( \\frac { Y ^ { \\perp } } { u } ) [ - \\frac { 1 } { 2 } u ^ 2 d \\theta ( e _ i ) + u ^ { - 1 } e _ i ( u ) \\cdot \\partial _ t ] . \\end{align*}"}
-{"id": "9623.png", "formula": "\\begin{align*} \\{ f , g \\} _ { D B } = \\{ f , g \\} _ { P B } - \\left [ \\{ f , \\phi \\} _ { P B } \\{ \\eta , g \\} _ { P B } - \\{ f , \\eta \\} _ { P B } \\{ \\phi , g \\} _ { P B } \\right ] \\ ; . \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} Q = \\begin{pmatrix} q _ 1 ( 1 ) & 0 & 0 \\\\ 1 + A & 1 - A & A \\\\ B & 1 - B & B \\end{pmatrix} . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} [ \\mathfrak { M } ^ { - 1 } \\tilde { f } ] ( y ) = \\frac { \\log ( p ) } { 2 \\pi } \\sum _ { \\mu _ p \\in { } _ p \\mathfrak { X } } \\mu _ p ( y ) ^ { - 1 } \\int _ { - \\frac { \\pi } { \\log ( p ) } } ^ { \\frac { \\pi } { \\log ( p ) } } \\tilde { f } ( \\mu _ p , i t ) \\abs { y } _ { p } ^ { - i t } d t . \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} R T ^ { ( \\lambda ) } _ \\varphi R ^ * f = f * \\nu _ \\varphi \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} 0 & = \\Delta _ N ( y ) \\int _ N f ( x ) \\cdot 1 _ { A \\cap q ( N ) } \\big ( q ( x y ) \\big ) d \\omega ( x ) \\\\ & = \\int _ N f ( x y ^ { - 1 } ) 1 _ { q ^ { - 1 } ( A ) \\cap N } ( x ) d \\omega ( x ) . \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} \\frac { L ' } { L } ( s , \\pi ) = - \\sum _ { n = 2 } ^ \\infty \\frac { \\Lambda _ \\pi ( n ) } { n ^ s } \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} S _ 1 ( t , q ) & = - \\frac { 1 } { q ^ 2 } \\sum _ { j = 0 } ^ { n - 1 } j ( j - 1 ) \\sum _ { r \\ge 1 } r ( - t q ^ j ) ^ { r } \\\\ & = - \\frac { 1 } { q ^ 2 } \\sum _ { r \\ge 1 } r ( - t ) ^ r \\sum _ { j = 0 } ^ { n - 1 } j ( j - 1 ) q ^ { j r } \\implies \\\\ \\omega _ n ^ { 2 i } [ t ^ r ] S _ 1 ( t , \\omega _ n ^ i ) & = - r ( - 1 ) ^ r \\sum _ { j = 0 } ^ { n - 1 } j ( j - 1 ) = - 2 \\binom { n } { 3 } r ( - 1 ) ^ r . \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} L ^ t _ k : = \\frac { 1 } { m k } [ - t ^ { m k + 1 } \\partial _ t , L _ 0 ^ t ] = \\frac { 1 } { m k ( c + \\check { h } ) } \\left ( \\sum _ { n > 0 } n \\sum _ { a \\in A _ { - \\underline { n } } } \\left ( u _ { a } [ t ^ { - n + m k } ] u ^ a [ t ^ n ] - u _ a [ t ^ { - n } ] u ^ a [ t ^ { n + m k } ] \\right ) \\right ) , \\ , \\ , \\ , , \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} f _ { \\varepsilon , \\varepsilon ' } ^ * ( w ) = \\sup _ { z \\in S _ { \\varepsilon } } \\Re \\langle z , ( w - \\varepsilon ' \\xi _ 0 ) \\rangle = h _ { S _ { \\varepsilon } } ( w - \\varepsilon ' \\xi _ 0 ) , \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} Y = \\mathrm { S p e c } _ X \\mathrm { T } \\mathcal { K } _ X ^ { - 1 } \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} x _ { k _ { i n } + 1 } = \\Pi _ { \\mathcal { X } } [ x _ { k _ { i n } } - \\frac { 1 } { L _ p } ( ( H + \\rho A ^ T A ) x _ { k _ { i n } } + A ^ T \\lambda _ k - \\rho A ^ T b ) ] \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( - \\partial _ t - \\partial _ x ^ 2 - x ^ 2 \\partial _ y ^ 2 ) \\varphi ( t , x , y ) = g ( t , x , y ) & t \\in [ 0 , T ] , ( x , y ) \\in \\Omega _ L , \\\\ \\varphi ( t , x , y ) = 0 & t \\in [ 0 , T ] , ( x , y ) \\in \\partial \\Omega _ L , \\\\ \\varphi ( T , x , y ) = 0 & ( x , y ) \\in \\Omega _ L . \\end{array} \\right . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} \\| \\sigma ( s ) \\xi \\| ^ 2 = \\psi ( s ^ * s ) = \\| s \\| ^ 2 . \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} w _ { \\sigma } ^ 3 = s _ { \\rho _ { 1 , \\sigma } + \\rho _ { 2 , \\sigma } } . \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} \\N = D Q ( \\dot z + i E z ) + F _ 1 + F _ 2 + F _ 3 , \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} D _ { n } [ \\sigma ] = \\mathbf { G } [ \\tau ] ^ { n + 1 } n ^ { \\sum ( \\alpha _ { r } ^ { 2 } - \\beta _ { r } ^ { 2 } ) } E [ \\tau , \\alpha _ { 1 } , \\ldots , \\alpha _ { R } , \\beta _ { 1 } , \\ldots , \\beta _ { R } ] \\left ( 1 + o ( 1 ) \\right ) , \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{gather*} L ( z , s ) = \\prod _ { p } \\big ( 1 - a ( p ) p ^ { - s } + p ^ { 1 - 2 s } \\big ) ^ { - 1 } = \\sum _ { n = 1 } ^ \\infty \\frac { a ( n ) } { n ^ s } , \\end{gather*}"}
-{"id": "8104.png", "formula": "\\begin{align*} m _ B [ ( \\Omega , g , K ) ] = \\{ m _ { A D M } [ ( M , g , K ) ] \\} , \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} \\phi _ i ( s ) = \\sqrt { \\frac { 2 i + 1 } { T - t } } P _ i \\left ( \\left ( s - t - \\frac { T - t } { 2 } \\right ) \\frac { 2 } { T - t } \\right ) , \\ \\ \\ i = 0 , 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} \\psi _ 1 ( \\textstyle { \\frac { 1 } { 2 } } , \\beta ) = I ' ( \\textstyle { \\frac { 1 } { 2 } } u ^ + + \\beta u ^ - ) [ \\textstyle { \\frac { 1 } { 2 } } u ^ + ] > 0 , \\beta \\in [ \\textstyle { \\frac { 1 } { 2 } } , \\textstyle { \\frac { 3 } { 2 } } ] . \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} - \\frac 1 t \\log u _ 2 ( t ) \\xrightarrow { t \\to \\infty } \\ & 1 - p ^ k - \\hat \\alpha k + p ^ k \\Big ( 1 - \\frac 1 { p ^ k \\lambda } + \\frac 1 { p ^ k \\lambda } \\log \\frac 1 { p ^ k \\lambda } \\Big ) \\\\ [ 1 e m ] = \\ & 1 - \\hat \\alpha k - \\frac 1 \\lambda \\Big ( 1 - \\log \\frac 1 \\lambda - k \\log \\frac 1 p \\Big ) = A _ { \\lambda } . \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } ( n + j ) ( n + j - 1 ) \\cdots ( n + 1 ) A ^ n = j ! ( I - A ) ^ { - j - 1 } . \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} \\ddot { z } + \\eta ( t ) \\dot { z } + \\omega ^ 2 ( t ) z = 0 . \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} a ^ 2 + b c & = 1 & a b + b d & = 0 \\\\ c a + d c & = 0 & c b + d ^ 2 & = 1 \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} \\nabla _ { \\partial _ t } \\partial _ t = u \\nabla u . \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\lambda _ { n + 1 } \\left ( a _ { n + 1 } p _ n + p _ { n - 1 } \\right ) & = a _ { n + 1 } \\hat { q } _ { n - 1 } + \\hat { q } _ { n - 2 } \\\\ \\lambda _ { n + 1 } \\left ( a _ { n + 1 } q _ n + q _ { n - 1 } \\right ) & = a _ 1 \\left ( a _ { n + 1 } \\hat { q } _ { n - 1 } + \\hat { q } _ { n - 2 } \\right ) + \\left ( a _ { n + 1 } \\hat { p } _ { n - 1 } + \\hat { p } _ { n - 2 } \\right ) \\end{array} \\right . \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} \\nabla F _ i ( y ) ^ T f ( y , u ) = 0 , \\ i = 1 , . . . , k , \\ \\ \\ \\ \\forall \\ ( y , u ) \\in \\hat Y \\times U \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} M _ 1 ( \\kappa p + 1 ) H ( L ) c _ \\Omega ^ p & = \\frac { \\kappa p + 1 } { 4 ( \\kappa + 1 ) ^ p } , M _ 2 K ( G ) c _ { \\partial \\Omega } ^ p = \\frac { \\kappa p + 1 } { 4 ( \\kappa + 1 ) ^ p } . \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} \\theta ^ \\alpha _ r ( \\lambda ) \\le \\pi _ r ^ \\delta ( \\lambda ) + \\frac { c _ 1 } { ( \\delta ^ 2 \\alpha ) ^ d } \\ , \\max _ { \\substack { u , v \\ge \\delta \\\\ u + v = 1 - \\alpha } } \\theta ^ \\alpha _ { u r } ( \\lambda ) \\ , \\theta ^ \\alpha _ { v r } ( \\lambda ) . \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} I ( s u ^ + + t u ^ - ) & < \\left ( I ( u ^ + ) - \\frac { 1 } { 4 } I ' ( u ^ + ) [ u ^ + ] \\right ) + \\left ( I ( u ^ - ) - \\frac { 1 } { 4 } I ' ( u ^ - ) [ u ^ - ] \\right ) \\\\ & = \\left ( I ( u ^ + ) - \\frac { 1 } { 4 } I ' ( u ) [ u ^ + ] \\right ) + \\left ( I ( u ^ - ) - \\frac { 1 } { 4 } I ' ( u ) [ u ^ - ] \\right ) + \\frac { \\lambda } { 2 } \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } ( u ^ - ) ^ 2 d x \\\\ & = I ( u ^ + ) + I ( u ^ - ) + \\frac { \\lambda } { 2 } \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } ( u ^ - ) ^ 2 d x \\\\ & = I ( u ) , \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} c ( x _ 1 , \\ldots , x _ N ) = \\sum _ { 1 \\le i < j \\le N } f ( d ( x _ i , x _ j ) ) , \\ , ( x _ 1 , \\ldots , x _ N ) \\in X ^ N , \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} F & = P H P - P W \\overline { P } ( \\overline { F } _ { \\eta } ( v , \\omega ) - \\lambda ) ^ { - 1 } \\overline { P } W P \\\\ & = ( \\eta - \\lambda ) P + \\langle v , ( \\overline { F } _ { \\eta } ( v , \\omega ) - \\lambda ) ^ { - 1 } v \\rangle P . \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} g \\C e _ v = \\C e _ { z } , g \\in N \\cap ( 1 + \\j ) , v \\in C , \\end{align*}"}
-{"id": "9886.png", "formula": "\\begin{align*} \\widehat { w } ( \\xi ) = \\widehat { v } ( \\xi ) \\left ( ( 2 \\pi i \\xi ) ^ { - s } + ( - 2 \\pi i \\xi ) ^ s \\right ) = \\dfrac { 1 } { \\xi } \\widehat { \\psi } ( \\dfrac { \\xi } { \\epsilon } ) \\left ( ( 2 \\pi i \\xi ) ^ { - s } + ( - 2 \\pi i \\xi ) ^ s \\right ) . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} & \\sum _ { j = k } ^ { k + 1 } ( - 1 ) ^ { j - k } ( j ) _ { k } ( n - j ) _ { n - k - 1 } f _ j ( x ) y ^ { k - j + 1 } \\\\ = & \\ k ! ( n - k ) ! f _ k ( x ) y - ( k + 1 ) ! ( n - k - 1 ) ! f _ { k + 1 } ( x ) \\\\ = & \\ ( k + 1 ) ! ( n - k - 1 ) ! \\left ( \\frac { n - k } { k + 1 } f _ k ( x ) y - f _ { k + 1 } ( x ) \\right ) . \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} d f _ { p } = ( d f ^ 1 _ { p ^ 1 } , \\dots , d f ^ m _ { p ^ m } ) \\qquad \\forall p \\in G . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} C ( \\rho ^ 0 , \\rho ^ 1 ) : = \\inf _ { v _ t } ~ \\int _ 0 ^ 1 \\mathbb { E } L ( X _ t ( \\omega ) , v ( t , X _ t ( \\omega ) ) ) d t , \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} f _ { 3 / 2 , m } ( z ) = \\frac { \\sqrt { \\pi } } { 2 } \\biggl ( G _ { 3 / 2 , - m , 0 } ( z ) - \\frac { 4 } { \\sqrt { \\pi } } G _ { 3 / 2 , 0 , 0 } ( z ) \\biggr ) . \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} \\sum _ { \\tilde { \\alpha } \\in \\N _ 0 ^ 4 , | \\tilde { \\alpha } | = m } \\| \\partial ^ { \\tilde { \\alpha } } u _ n - \\partial ^ { \\tilde { \\alpha } } u \\| _ { G _ 0 ( J ' \\times \\R ^ 3 _ + ) } ^ 2 \\leq a _ { n } e ^ { C _ m T ' } \\longrightarrow 0 \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} - \\Delta Q - c | x | ^ { - 2 } Q + Q = | Q | ^ { \\frac { 4 } { d } } Q . \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{gather*} ( - 1 ) ^ { m - 1 } x _ i ^ + T _ { w ( i , m ) } \\big ( x _ { i - 1 } ^ + \\big ) = \\begin{cases} S _ { N , 1 } ( 1 ) & \\\\ S _ { i , i + 1 } ( 0 ) & \\end{cases} \\end{gather*}"}
-{"id": "6657.png", "formula": "\\begin{align*} \\left | \\sum _ { t = n _ 0 } ^ n f ( t ) \\frac { \\cos 4 \\theta ( t , E ) } { t - v } \\right | \\leq D ( E , A , \\varepsilon ) + { \\varepsilon } \\ln \\frac { n - v } { n _ 0 - v } , \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ { M _ 1 } X _ { j , 1 } , \\ldots , \\sum _ { j = 1 } ^ { M _ m } X _ { j , m } \\right ) \\le _ { s m } \\left ( \\sum _ { j = 1 } ^ { N _ 1 } X _ { j , 1 } , \\ldots , \\sum _ { j = 1 } ^ { N _ m } X _ { j , m } \\right ) . \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} E _ { 3 / 2 } ( z ) = \\sum _ { d \\leq 0 } H ( | d | ) q ^ { - d } + \\frac { 1 } { 1 6 \\pi \\sqrt { y } } \\sum _ { n \\in \\mathbb { Z } } \\beta ( 4 \\pi n ^ 2 y ) q ^ { - n ^ 2 } . \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} I ( t _ 1 u _ 1 + t _ 2 u _ 2 + t _ 3 u _ 3 ) & < \\sum _ { i = 1 } ^ 3 \\left ( I ( u _ i ) - \\frac { 1 } { 4 } I ' ( u _ i ) [ u _ i ] \\right ) \\\\ & = \\sum _ { i = 1 } ^ 3 \\left ( I ( u _ i ) - \\frac { 1 } { 4 } I ' ( u ) [ u _ i ] \\right ) + \\frac { \\lambda } { 2 } \\sum _ { i < j } \\int _ { \\mathbb R ^ 3 } \\phi _ j u _ i ^ 2 d x \\\\ & = \\sum _ { i = 1 } ^ 3 I ( u _ i ) + \\frac { \\lambda } { 2 } \\sum _ { i < j } \\int _ { \\mathbb R ^ 3 } \\phi _ j u _ i ^ 2 d x \\\\ & = I ( u _ 1 + u _ 2 + u _ 3 ) \\\\ & = I ( u ) , \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} & C _ 2 ( \\nu ) \\| e ^ { \\nu t \\Delta } u _ 0 \\| _ { \\widetilde { L } ^ 1 _ { T ' } ( \\dot { B } ^ { s + 1 } _ { 2 , 2 } ) } + T ' \\big ( M _ 0 ^ { \\frac { 1 } 2 } + C _ 2 ( \\nu ) M _ 0 \\big ) \\\\ & \\quad + C _ 2 ( \\nu ) ( T ' ) ^ { \\frac { r - 2 } { 2 r } } ( 2 C _ 1 ( \\nu ) M _ 0 + 3 2 C _ 2 M ^ 4 _ 0 ) < \\frac { 1 } { 2 C _ 3 } . \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} \\sum _ { \\substack { n > x \\\\ ( n , a _ 2 ) = 1 } } \\frac { 1 } { \\ell _ u ( n ) } = \\sum _ { \\substack { n > x \\\\ P ( n ) > y \\\\ ( n , a _ 2 ) = 1 } } \\frac { 1 } { \\ell _ u ( n ) } + \\sum _ { \\substack { n > x \\\\ P ( n ) \\leq y \\\\ ( n , a _ 2 ) = 1 } } \\frac { 1 } { \\ell _ u ( n ) } . \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} T _ S f ( z , w ) = \\int _ 0 ^ { \\infty } f ( t , w ) e ^ { i 2 \\pi z t } \\d t , ( z , w ) \\in \\mathcal { U } _ p \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} \\lvert T ^ { - n } [ f ] \\rvert _ S = & \\lvert \\sum _ { v \\in I } \\alpha ( v ) T ^ { - n } [ \\phi v ] \\rvert _ S \\\\ \\leq & \\sup _ { v \\in I } \\lvert T ^ { - n } [ \\phi v ] \\rvert _ S \\\\ = & O ( n ) \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} \\lambda x + Q s = 2 a ^ T Q s Q a + 2 a ^ T Q s \\bar { P } a - 2 ( s _ 1 - s _ 2 ) \\bar { P } a . \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} \\mathcal { L } ( e _ 1 , e _ 2 , e _ 3 , e _ 4 ) = \\alpha ( e _ 1 ) ^ 2 + \\beta \\ , e _ 2 \\ , , \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\Delta _ { n } ( x ) \\stackrel { \\mathrm { d e f } } { = } \\frac { \\left \\Vert K ^ { 3 } \\right \\Vert } { \\left \\Vert K ^ { 2 } \\right \\Vert ^ { 3 / 2 } \\sqrt { n \\ , h _ { n } \\ , f ( x ) } } . \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} Q f : = \\sum ^ d _ { i , j = 1 } q _ { i j } \\frac { \\partial ^ 2 } { \\partial _ i \\partial _ j } f . \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} w _ { \\sigma } ^ 3 = s _ { 2 \\rho _ { 1 , \\sigma } + \\rho _ { 2 , \\sigma } } . \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} \\begin{cases} \\delta _ { g _ 0 } Y = 0 \\\\ \\delta _ { g _ 0 } h + \\frac { 1 } { 2 } d ( t r _ { g _ 0 } h + 2 v ) = 0 , \\end{cases} \\quad M . \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { X _ { n , i } } \\underset { n \\to \\infty } { \\overset { \\mathrm { w e a k } } { \\longrightarrow } } \\nu _ \\beta \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} E ( V ^ j ( \\cdot - x ^ j _ n ) ) = \\frac { E ( \\tilde { V } ^ j _ n ) } { \\lambda _ j ^ 2 } + \\frac { \\lambda _ j ^ \\alpha - 1 } { \\alpha + 2 } \\| V ^ j \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } . \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} \\rho ( A , B ) = \\min \\{ \\rho ( a , b ) : a \\in A , b \\in B \\} . \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} \\det \\mathbb { F } _ \\pm = 0 . \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } D _ { t } U + F _ { i } ( D ^ { + } U , D ^ { - } U ) = 0 & \\ , \\ , ( J _ { i } \\setminus \\{ 0 \\} ) \\times \\{ 1 , \\dots , N \\} \\\\ U ( 0 , \\cdot ) = \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } U ( 1 _ { i } , \\cdot ) - B & \\ , \\ , \\{ 1 , 2 , \\dots , N \\} \\\\ U ( \\cdot , 0 ) = U _ { 0 } & \\ , \\ , J _ { i } \\times \\{ 0 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} \\begin{bmatrix} I - V U & O \\\\ O & I - U V \\end{bmatrix} \\begin{bmatrix} \\frac { \\phi } { \\sigma _ { - } } \\\\ \\frac { \\psi } { \\sigma _ { + } } \\end{bmatrix} = \\begin{bmatrix} - P _ { - } \\left ( z ^ { n } v P ^ { - } \\left ( \\frac { q } { z ^ { n } \\sigma _ { + } } \\right ) \\right ) \\\\ - P _ { + } \\left ( z ^ { - n } u P ^ { + } \\left ( \\frac { q } { \\sigma _ { - } } \\right ) \\right ) \\end{bmatrix} . \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} \\hat I ' ( t ) = \\hat U \\hat I \\hat U ^ \\dag = \\frac { 1 } { 2 } \\left [ \\rho ^ 2 ( \\hat p _ 1 ^ 2 + \\hat p _ 2 ^ 2 ) + \\frac { \\kappa ^ 2 } { \\rho ^ 2 } ( \\hat x _ 1 ^ 2 + \\hat x _ 2 ^ 2 ) \\right ] , \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} \\displaystyle \\| X _ 0 u \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } & = \\frac { 1 } { L ^ 2 } \\| \\varphi _ L - \\varphi \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\frac { 1 } { L ^ 2 } \\Big ( \\underbrace { \\| \\varphi _ L \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } } _ { = 1 } + \\underbrace { \\| \\varphi \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } } _ { = 1 } - 2 \\int _ { \\mathbb { R } ^ 2 } \\varphi _ L \\varphi \\ ; d q d p \\Big ) \\\\ & = \\frac { 2 } { L ^ 2 } \\Big ( 1 - \\int _ { \\mathbb { R } ^ 2 } \\varphi _ L \\varphi \\ ; d q d p \\Big ) { ~ . } \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} 0 = \\int _ { K \\backslash G / H } 1 _ A ( \\ddot { x } ) d \\mu _ 1 ( \\ddot { x } ) = \\int _ { K \\backslash G / H } 1 _ A ( \\ddot { x } ) \\varphi ( \\ddot { x } ) d \\mu _ 2 ( \\ddot { x } ) . \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} \\mathcal { L } u ( x ) = \\frac D 2 \\Delta u ( x ) + r ( x ) \\big ( u ( 0 ) - u ( x ) \\big ) . \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\frac { { \\rm { D } } } { \\partial \\overline { z } } w = 0 \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} & \\{ \\mathbf { q } _ i ^ { [ k + 1 ] } \\} _ { i = 1 } ^ N = \\underset { \\{ \\mathbf { q } _ i \\} _ { i = 1 } ^ N } { } \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ N \\| \\mathbf { q } _ i - ( \\mathbf { p } _ i ^ { [ k ] } + \\frac { \\boldsymbol { \\lambda } ^ { [ k ] } _ i } { \\rho } ) \\| ^ 2 \\\\ & \\underline { \\mathbf { p } } _ i ^ { [ k + 1 ] } \\leq \\mathbf { q } _ i \\leq \\overline { \\mathbf { p } } _ i ^ { [ k + 1 ] } \\\\ & \\sum _ { i = 1 } ^ N \\mathbf { q } _ i = \\mathbf { P } ^ { [ k + 1 ] } \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} \\partial _ n H ^ - _ \\gamma = | \\partial _ n \\psi | \\partial _ r H ^ - _ { C _ R } \\circ \\psi = - | \\partial _ n \\psi | \\partial _ r H ^ + _ { C _ R } \\circ \\psi = - \\partial _ n H ^ + _ \\gamma . \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} \\| p ' ( v ) - p ' ( x ) \\| _ { \\infty } = | v _ 1 - x _ 1 | = | v _ 2 - x _ 2 | . \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( r ) _ k ( 1 - r ) _ k ( t ) _ k ( 1 - t ) _ k } { k ! ^ 4 } \\end{gather*}"}
-{"id": "3106.png", "formula": "\\begin{align*} x \\mapsto x y ^ { 1 1 } , y \\mapsto y ^ 3 \\Longrightarrow \\mathcal { I } _ { 1 , 2 , \\beta _ 1 , \\beta _ 2 } = 1 2 \\int _ { \\mathcal { D } } \\check { f } _ { t , s } ( x , y ) ^ s x ^ { \\beta _ 1 } y ^ { { 4 ( 5 \\beta _ 1 + 3 \\beta _ 2 + 2 4 s ) } } \\frac { d x } { x } \\frac { d y } { y } , \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} d ( F ) : = \\frac { m _ K n + 1 } { n \\left ( \\sqrt { 2 } \\right ) ^ { 3 n - 1 } } d ( F | S ) : = \\frac { m _ K n + 1 } { \\left ( \\sqrt { 2 } \\right ) ^ { 3 n - 1 } } . \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} * \\mathbb { F } = \\pm i \\ , \\mathbb { F } \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} Q _ a Q _ b Q _ a - Q _ { Q _ a b } & = 4 ( Q _ a Q _ { a b } + Q _ { a b } Q _ a ) + 2 ( Q _ a Q _ { a ^ 2 , b ^ 2 } + Q _ { a ^ 2 , b ^ 2 } Q _ a ) + Q _ { a , b } Q _ a Q _ { a , b } \\\\ & - ( 2 Q _ { a ^ 2 b } + 8 Q _ { a ( a b ) } + 2 Q _ { a ^ 3 , b ^ 2 a } + 4 Q _ { a ^ 3 , b ( a b ) } ) . \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{gather*} \\check { f } _ { t , s } ( x , y ) : = t x ^ { 6 } y ^ { 3 6 } + s x y ^ { 8 } + x ^ { 7 } y ^ { 4 4 } + x ^ { 3 } + 1 . \\end{gather*}"}
-{"id": "4587.png", "formula": "\\begin{align*} \\frac { 1 } { H } \\sum _ { h = 1 } ^ { H } v a r \\left ( \\gamma ^ { \\tau } m ( y ) \\big | a _ { h - 1 } \\leq y < a _ { h } \\right ) \\leq \\frac { \\mathfrak { a } _ { 3 } } { H ^ { \\vartheta } } v a r \\left ( \\gamma ^ { \\tau } m ( y ) \\right ) . \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} G ^ { \\epsilon , N } ( z ) = G ^ { \\epsilon , N } _ y ( x ) = \\left ( \\psi _ N \\phi _ { \\epsilon } \\ast F _ y \\right ) ( x ) = \\int _ { \\R } F _ y ( x - t ) \\psi _ N ( t ) \\phi _ { \\epsilon } ( t ) \\d t = \\int _ { \\R } F ( z - t ) \\psi _ N ( t ) \\phi _ { \\epsilon } ( t ) \\d t . \\end{align*}"}
-{"id": "9960.png", "formula": "\\begin{align*} ( \\tau _ 1 , \\tau _ { 0 1 } ) - ( \\theta \\tau _ 1 + \\bar { \\partial } _ z \\theta \\wedge \\tau _ { 0 1 } , \\ , \\theta \\tau _ { 0 1 } ) = \\bar { \\vartheta } ( ( 1 - \\theta ) \\tau _ { 0 1 } , 0 ) . \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u + u ^ { \\frac { n + 2 } { n - 2 } } & \\Omega \\times ( 0 , \\infty ) , \\\\ u = 0 & \\partial \\Omega \\times ( 0 , \\infty ) , \\\\ u ( \\cdot , 0 ) = u _ 0 & \\Omega . \\end{cases} \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} \\beta ( y , t ) = \\int d x \\ b ( x , y , t ) \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} \\varphi ( \\sqrt L ) f ( x ) = \\langle f , \\varphi ( \\sqrt L ) ( x , \\cdot ) \\rangle , x \\in M . \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} \\langle f , H P _ c g \\rangle = \\langle f , - \\tfrac 1 2 \\partial _ x ^ 2 g \\rangle + B ( f , g ) , \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 2 \\backslash B _ { R _ 0 } ( 0 ) } A ( x ) ^ r ( e ^ { r \\alpha v ^ 2 } - 1 ) = \\sum _ { m = 1 } ^ { \\infty } \\dfrac { ( r \\alpha ) ^ m } { m ! } \\int _ { \\mathbb { R } ^ 2 \\backslash B _ { R _ 0 } ( 0 ) } A ( x ) ^ r v ^ { 2 m } . \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} ( Y _ { t } \\ , \\vert \\ , X _ { t } ) \\ , = \\ , ( \\widetilde { X } _ { t } \\ , \\vert \\ , X _ { t } ) \\ , ; t \\ge 0 \\ , . \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} & ( p _ 1 , \\dots , p _ r , w ) \\cdot ( b _ 1 , \\dots , b _ r ) \\\\ & \\qquad = ( \\Theta ( ( p _ 1 , \\dots , p _ r ) , ( b _ 1 , \\dots , b _ r ) ) , e ^ { \\lambda _ 1 } ( b _ 1 ) \\cdots e ^ { \\lambda _ r } ( b _ r ) w ) . \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} \\cot ( \\eta ( n + 1 , E ) + \\gamma ( n , E ) ) = \\cot ( \\eta ( n , E ) + \\gamma ( n , E ) ) - \\frac { 2 } { \\omega } b ' _ { n + 1 } \\vert \\varphi ( n , E ) \\vert ^ 2 \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} \\alpha _ 1 : = \\min \\left \\{ \\frac { 1 } { 2 } \\lambda _ { \\min } ^ { \\bar { Q } } , \\frac { 1 } { 2 } d \\right \\} s _ 1 , \\alpha _ 2 : = \\max \\left \\{ \\frac { 1 } { 2 } \\lambda _ { \\max } ^ { \\bar { Q } } , \\frac { 1 } { 2 } d \\right \\} s _ 2 . \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} r = n / ( p - 1 ) \\in \\mathbb { N } p \\nmid \\binom { l - n - 1 } { r } . \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} \\begin{bmatrix} v _ 1 \\\\ v _ 2 \\\\ \\vdots \\\\ v _ d \\\\ \\end{bmatrix} = Q \\begin{bmatrix} 1 \\otimes 1 \\\\ \\bar { \\theta } \\otimes 1 \\\\ \\vdots \\\\ \\bar { \\theta } ^ { d - 1 } \\otimes 1 \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} D _ { n } [ \\sigma ] = \\mathbf { G } [ \\sigma ] ^ { n + 1 } E [ \\sigma ] \\left ( 1 + o ( 1 ) \\right ) , \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} E [ \\tau ] = \\exp \\left ( \\sum _ { k = 1 } ^ { \\infty } k \\cdot \\widehat { \\log \\tau } ( k ) \\widehat { \\log \\tau } ( - k ) \\right ) , \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} \\mu P ( \\epsilon ; 0 , t ) r = \\tilde { \\pi } r - \\epsilon \\tilde { \\pi } \\tilde { Q } ^ { ( 1 ) } ( \\tilde { \\Pi } - \\tilde { Q } ) ^ { - 2 } r + O ( \\epsilon ^ 2 s ^ 3 ) \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} S ^ { \\delta } _ R ( L ) f = \\int _ 0 ^ { R ^ 2 } \\left ( 1 - \\frac { \\lambda } { R ^ 2 } \\right ) ^ { \\delta } _ + d E _ L ( \\lambda ) f \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} \\rho _ { n + 1 } ( x ' , y ' ) = \\sum _ { ( x , y ) \\in f ^ { - 1 } ( x ' , y ' ) } \\frac { \\rho _ { n } ( x , y ) } { \\lambda \\vert T ' ( x ) \\vert } \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} \\prod _ U R _ i : = \\varinjlim _ { I \\in U } \\prod _ { i \\in I } R _ i . \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} A _ { i , j } \\ , = \\ , \\begin{cases} 1 & : \\ , x _ { i , j } \\ge 1 / 2 \\ , ; \\\\ 0 & \\ , , \\end{cases} \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} | | \\Phi ' ( u ) - \\Phi ' ( v ) | | _ { H ^ { - 1 } ( \\Omega ) } & = | | f ( u ) - f ( v ) | | _ { H ^ { - 1 } ( \\Omega ) } \\\\ & \\leq C ( n , p , \\Omega ) | | f ( u ) - f ( v ) | | _ q \\\\ & \\leq C ( n , p , \\Omega ) \\left ( | | u - v | | _ { p q } + | | u - v | | _ { p q } ^ { p - 1 } \\right ) . \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } E _ { N } ^ { \\rm Q } = \\inf _ { u \\in H ^ { \\frac { 1 } { 2 } } ( \\mathbb R ^ 3 ) , \\| u \\| _ { L ^ 2 } = 1 } \\mathcal { E } _ { a } ^ { \\rm H } ( u ) = : E _ { a } ^ { \\rm H } \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} \\partial _ i K ( \\xi & - d \\Gamma ( m ) ) \\frac { 1 } { H _ \\mu ( \\xi ) - \\Sigma ( \\xi ) + \\omega ( k ) + 1 } - \\partial _ i K ( \\xi - d \\Gamma ( m ) ) \\frac { 1 } { H _ \\mu ( \\xi ) - \\Sigma ( \\xi ) + 1 } \\\\ & = \\partial _ i K ( \\xi - d \\Gamma ( m ) ) \\frac { 1 } { H _ \\mu ( \\xi ) - \\Sigma ( \\xi ) + 1 } \\frac { \\omega ( k ) } { H _ \\mu ( \\xi ) - \\Sigma ( \\xi ) + 1 + \\omega ( k ) } \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} \\langle \\psi , x \\rangle _ { X ' , X } = \\psi ( x ) x \\in X , \\ , \\psi \\in X ' . \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} \\begin{pmatrix} v \\\\ f \\end{pmatrix} \\mapsto \\begin{pmatrix} \\| \\xi \\| _ g ^ 2 \\ , v - \\langle \\xi , v \\rangle _ g \\ , \\xi + i f \\xi \\\\ - i \\ , \\langle \\xi , v \\rangle _ g \\end{pmatrix} . \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} e ^ { 1 / 2 } x e ^ { 1 / 2 } = E ( x ) e , \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} \\int _ { U } f ( k ) d \\lambda _ \\nu = \\nu \\lambda _ \\nu ( U \\cap B _ 1 ( 0 ) ) \\int _ { 0 } ^ { \\infty } f ( k e _ 1 ) k ^ { \\nu - 1 } d \\lambda _ 1 \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} v ( x , y , 0 ) = { } & \\delta ( y - x ) , \\\\ v ( x , x , t ) = { } & \\frac { e ^ { - \\eta t } } { \\sqrt { \\pi t } } , t > 0 , \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{gather*} \\big [ \\square \\big ( x _ i ^ + \\big ) , \\Omega _ + \\big ] = - x _ i ^ + \\otimes h _ i , \\big [ \\square \\big ( x _ i ^ - \\big ) , \\Omega _ + \\big ] = h _ i \\otimes x _ i ^ - \\end{gather*}"}
-{"id": "2405.png", "formula": "\\begin{align*} L = \\sum _ { s = ( a , b , k ) \\in S ^ 0 } \\sum _ { m \\in \\Z \\cap \\Z _ p ^ { \\times } [ a , b , k ] } \\lambda _ { \\pi } ( m ) F \\left ( \\frac { m } { M } \\right ) . \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty b _ s d s & \\leq \\int _ 0 ^ \\infty m e ^ { - L _ s } b _ s d s = m I < \\infty . \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} \\phi = p _ \\tau + \\frac { f ( t _ \\tau ) } { 2 m } ( p _ { 1 , \\tau } ^ 2 + p _ { 2 , \\tau } ^ 2 ) + \\frac { m \\omega ^ 2 ( t _ \\tau ) f ^ { - 1 } ( t _ \\tau ) } { 2 } ( x _ { 1 , \\tau } ^ 2 + x _ { 2 , \\tau } ^ 2 ) \\ ; \\sim 0 . \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} [ \\eta _ 1 , \\eta _ 2 ] = [ \\omega _ 1 , \\omega _ 2 ] \\gamma _ j . \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} T ^ \\mathrm { v e c } _ { \\alpha \\beta \\gamma } = \\frac 1 3 ( g _ { \\alpha \\beta } T ^ \\mu { } _ { \\mu \\gamma } - g _ { \\alpha \\gamma } T ^ \\mu { } _ { \\mu \\beta } ) , \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} \\nu ( a ) < \\nu ( b ) , \\nu ( a + b ) = \\nu ( a ) \\epsilon ( a + b ) \\equiv \\epsilon ( a ) \\mod p . \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} \\dot { B } \\frac { \\partial F } { \\partial P _ T } = 1 ; \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} f _ { K _ { \\varepsilon } } ^ * ( w ) = \\sup _ { z \\in K _ { \\varepsilon } } \\Re \\langle z , w \\rangle = h _ { K _ { \\varepsilon } } ( w ) = h _ { K } ( w ) + h _ { \\overline { D } ( 0 , \\varepsilon ) } ( w ) = h _ { K } ( w ) + \\varepsilon | | w | | , \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} P ^ { ( \\alpha , \\beta ) } _ { n } \\Big ( 1 - \\frac { 2 x } { \\beta } \\Big ) & = \\frac { ( \\alpha + 1 ) _ { n } } { n ! } \\ , _ 2 F _ 1 \\left ( - n , \\alpha + \\beta + n + 1 ; \\alpha + 1 ; \\frac { x } { \\beta } \\right ) \\\\ & = \\sum _ { k = 0 } ^ { n } \\frac { R _ { n , k } ^ { ( \\alpha ) } ( x ) } { \\beta ^ { k } } , \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} ( A _ 3 - A _ 4 ) ^ 2 = n ( f - 1 ) ( \\ell A _ 0 + a A _ 1 + b A _ 2 ) + n ( f - 2 ) ( A _ 3 - A _ 4 ) . \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 2 , 2 ) = \\int _ { 0 } ^ { \\infty } \\frac { x ^ { 2 } \\cos ( 2 \\pi x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x = \\frac { 1 } { 2 5 6 } \\left ( 1 - \\frac { 5 } { \\pi } + \\frac { 5 } { \\pi ^ { 2 } } \\right ) . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} \\big | \\partial _ { \\dot { \\xi } } \\Psi [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] [ \\dot { \\bar { \\xi } } ] ( x , t ) \\big | \\lesssim \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\| \\dot { \\bar { \\xi } } ( t ) \\| _ { n - 4 s + 1 + \\sigma } \\left ( \\sum _ { j = 1 } ^ k \\frac { \\mu _ 0 ^ { - \\frac { n - 6 s } { 2 } - 1 + \\sigma } ( t ) } { 1 + | y _ j | ^ { a - 2 s } } \\right ) , \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} e ( x _ 1 , \\ldots , x _ N ) : = g ( \\sum _ { j = 1 } ^ { m _ 1 } ( - 1 ) ^ { \\sigma ( j ) } \\ , x _ j , \\ldots , \\sum _ { j = m _ { n - 1 } + 1 } ^ { N } ( - 1 ) ^ { \\sigma ( j ) } \\ , x _ j ) , \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} d x ( s ) = - a ( x ( s ) ) \\nabla V ( x ( s ) ) \\ , d s + \\frac { 1 } { \\beta } ( \\nabla \\cdot a ) ( x ( s ) ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } \\sigma ( x ( s ) ) \\ , d w ( s ) \\ , , s \\ge 0 \\ , , \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} M ( \\lambda ) : = U _ q ( \\mathfrak { g } ) \\left / \\left ( \\sum _ { i \\in I } U _ q ( \\mathfrak { g } ) \\left ( K _ i - q ^ { ( \\lambda , \\alpha _ i ) } \\right ) + \\sum _ { i \\in I } U _ q ( \\mathfrak { g } ) E _ i \\right ) \\right . . \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} \\forall n \\geq 0 , ~ U _ { n + 1 } = U _ n X _ { n + 1 } , \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} \\frac { d } { d x } P ^ { ( \\alpha , \\beta ) } _ { n } ( x ) = \\frac { 1 } { 2 } ( n + \\alpha + \\beta + 1 ) P ^ { ( \\alpha + 1 , \\beta + 1 ) } _ { n - 1 } ( x ) \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} \\| \\pi ( s ) \\| \\geq \\| \\sigma ( s ) \\xi \\| = \\| s \\| \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{gather*} { } _ 4 F _ 3 \\biggl ( \\begin{matrix} r , \\ , 1 - r , \\ , t , \\ , 1 - t \\\\ 1 , \\ , 1 , \\ , 1 \\end{matrix} \\biggm | 1 \\biggr ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( r ) _ k ( 1 - r ) _ k ( t ) _ k ( 1 - t ) _ k } { k ! ^ 4 } . \\end{gather*}"}
-{"id": "613.png", "formula": "\\begin{align*} C \\left ( z \\right ) \\frac { \\partial w } { \\partial \\overline { z } } + w A \\left ( z \\right ) + B \\left ( z \\right ) \\overline { w } = 0 \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} \\int _ \\Omega S ( x ) \\d x = 0 . \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} q _ 1 & : = | f ( i ) | ^ 2 + | g ( i ) | ^ 2 = ( a _ 0 - a _ 2 ) ^ 2 + ( a _ 1 - a _ 3 ) ^ 2 + ( b _ 0 - b _ 2 ) ^ 2 + ( b _ 1 - b _ 3 ) ^ 2 , \\\\ q _ 2 & : = | f ( i ) | ^ 2 - | g ( i ) | ^ 2 = ( a _ 0 - a _ 2 ) ^ 2 + ( a _ 1 - a _ 3 ) ^ 2 - ( b _ 0 - b _ 2 ) ^ 2 - ( b _ 1 - b _ 3 ) ^ 2 , \\\\ q _ 3 & : = | f ( i ) + g ( i ) | ^ 2 = ( ( a _ 0 - a _ 2 ) + ( b _ 0 - b _ 2 ) ) ^ 2 + ( ( a _ 1 - a _ 3 ) + ( b _ 1 - b _ 3 ) ) ^ 2 , \\\\ q _ 4 & : = | f ( i ) - g ( i ) | ^ 2 = ( ( a _ 0 - a _ 2 ) - ( b _ 0 - b _ 2 ) ) ^ 2 + ( ( a _ 1 - a _ 3 ) - ( b _ 1 - b _ 3 ) ) ^ 2 . \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} v = ( \\lambda I - P ) u . \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{align*} \\exp _ { \\psi } ( L ( \\psi , \\tilde { \\mathbb { A } } ) ) = \\exp _ { \\psi } \\bigg ( \\sum _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) t ^ { \\deg _ { \\theta } ( a ) } } { a } \\bigg ) \\in A [ z _ 1 , \\dots , z _ n , t ] . \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\left | S ^ { ( 2 ) } _ { \\mu , \\xi } ( x , t ) \\right | \\lesssim \\mu _ 0 ^ { - \\frac { n + 2 s } { 2 } } \\frac { \\mu _ 0 ^ { n - 2 s + 2 } } { 1 + | y _ j | ^ { 4 s - 2 } } \\lesssim \\mu _ 0 ^ { 2 s - ( a - 2 s ) - \\sigma } ( t _ 0 ) \\sum _ { j = 1 } ^ k \\frac { \\mu _ j ^ { - 2 s } \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } + \\sigma } } { 1 + | y _ j | ^ { a } } . \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} Y ^ + _ n & = \\frac 1 { F ( \\tau _ n ) } \\sum _ { k \\leq h ( n ) } f ( \\tau _ k ) + \\frac 1 { F ( \\tau _ n ) } \\sum _ { h ( n ) < k \\leq n } \\frac 1 { f ( \\tau _ k ) } . \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} E _ { a _ { N } ' } ^ { \\rm H } - E _ { a _ { N } } ^ { \\rm H } & = \\big ( ( a _ { * } - a _ { N } ' ) ^ { \\frac { q } { q + 1 } } - ( a _ { * } - a _ { N } ) ^ { \\frac { q } { q + 1 } } \\big ) \\Big ( \\frac { q + 1 } { q } \\cdot \\frac { \\Lambda } { a _ { * } } + o ( 1 ) _ { N \\to \\infty } \\Big ) \\\\ & \\geq - ( a _ { N } ' - a _ { N } ) ^ { \\frac { q } { q + 1 } } \\Big ( \\frac { q + 1 } { q } \\cdot \\frac { \\Lambda } { a _ { * } } + o ( 1 ) _ { N \\to \\infty } \\Big ) . \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} 1 + a _ { n } ^ 2 + a _ { n + 1 } ^ 2 & = a _ { n + 1 } ^ 2 + a _ { n + 1 } a _ { n - 1 } \\\\ & = a _ { n + 1 } ( a _ { n + 1 } + a _ { n - 1 } ) \\\\ & = 3 a _ n a _ { n + 1 } \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { B ^ r _ { p , \\infty } } : = \\sup _ { m \\in \\mathbb { N } } 2 ^ { m r } \\Vert \\sum _ { 2 ^ m \\leq \\langle \\xi \\rangle < 2 ^ { m + 1 } } d _ { \\xi } [ \\xi ( x ) \\widehat { f } ( \\xi ) ] \\Vert _ { L ^ p ( G ) } < \\infty . \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} S \\mu & = \\mu \\tau _ { A , A } ( S \\otimes S ) , & S \\eta & = \\eta , \\\\ \\Delta S & = ( S \\otimes S ) \\tau _ { A , A } \\Delta , & \\varepsilon S & = \\varepsilon . \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} Q ( f ) ( K x H ) = \\int _ K \\int _ H f ( k ^ { - 1 } x h ) d h d k . \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} \\gamma _ n ( T _ m ) = \\begin{cases} E ^ { ( n ) } _ { 2 m - 1 , 2 m } & n = 2 m , \\\\ \\frac { 1 } { 2 } E ^ { ( n ) } _ { 2 m - 1 , 2 m } & n > 2 m , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} \\varphi ( d ^ { \\mathrm { c l o u d } } ) = e ^ { - \\beta _ c d ^ { \\mathrm { c l o u d } } } , \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} \\Delta \\ = \\ - 4 b ^ 2 ( x - y - 1 ) ( x - y + 1 ) ( x + y - 1 ) ( x + y + 1 ) \\ , . \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} D ^ \\alpha { } _ \\beta = \\begin{pmatrix} 1 & 0 & 0 & - 2 m a \\sin ( 2 m x ^ 4 ) \\\\ 0 & 1 & 0 & 2 m a \\cos ( 2 m x ^ 4 ) \\\\ 0 & 0 & 1 & 2 m b \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} \\left ( \\sum _ { n = 1 } ^ N \\lambda _ n t ^ { 1 / p ^ n } \\right ) \\frac { d w } { w } = f ^ * \\omega \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} \\frac { P _ n ( t ) } { ( 1 - t ) ^ { n + 1 } } = \\sum _ { k \\geq 0 } ( 1 + t ) ^ { k - 1 } \\frac { k ^ n } { 2 ^ { k - 1 } } . \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} \\theta _ u ^ n = e ^ { \\lambda u } f \\omega ^ n . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} c \\sum _ { i = 0 } ^ { \\lfloor ( t - 4 ) / 2 \\rfloor } F _ { t - 4 - 2 i } ( x ) + F _ { t - 2 } ( x ) = ( c - 1 ) { G } _ { t - 4 } ( x ) + { G } _ { t - 2 } ( x ) . \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\phi ' ( s ) = \\gamma ( s ) \\exp \\left \\{ - \\int _ 0 ^ s \\beta ( \\theta ) \\gamma ( \\theta ) d \\theta \\right \\} \\left \\{ \\psi ( s ) - \\beta ( s ) \\int _ 0 ^ s \\gamma ( \\theta ) \\psi ( \\theta ) d \\theta \\right \\} , \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\begin{cases} \\frac { b ( t , x ) } { 1 + | x | \\log ^ + | x | } \\in L ^ 1 ( 0 , T ; L ^ \\infty ( \\R ) ) ; \\\\ \\mathrm { d i v } _ \\mu b ( t , x ) \\in L ^ 1 ( 0 , T ; \\mathrm { E x p } _ \\mu ( \\frac { L } { \\log L } ) ) . \\end{cases} \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { B ( \\tau _ k ) } { \\tau _ k } \\cdot \\frac { t _ k } { B ( k t _ k ) } = \\infty , \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} \\begin{aligned} A _ { S } : = ( - S , - 1 / S ] \\times [ 0 , 1 ] , \\end{aligned} \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} \\beta ( a ) ( \\bar { \\nu } ) = a ^ { t } \\cdot \\bar { \\nu } , a \\in U ( \\mathfrak { g } ^ { \\Gamma _ q } ) \\bar { \\nu } \\in V ( \\mu ) . \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} \\epsilon ( s ( n , a p ^ h ) ) \\equiv ( - 1 ) ^ { a h + r - r _ h } \\frac { \\epsilon ( ( n - 1 ) ! ) } { ( a - 1 ) ! } \\binom { a - 1 } { r _ h } \\mod p , \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} 0 \\leftarrow R _ n / I _ n \\xleftarrow { } R _ n = F ( 0 , n ) \\xleftarrow { } F ( 1 , n ) \\xleftarrow { } F ( 2 , n ) \\xleftarrow { } F ( 3 , n ) \\cdots \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} \\| \\epsilon _ { o u t } ^ k \\| = L \\| f i ( f i ( M _ 3 ) \\tilde { x } _ k ) - M _ 3 \\tilde { x } _ k + V _ 2 - f i ( V _ 2 ) \\| \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} d ( x y ) = d ( x ) y + x d ( y ) + \\lambda d ( x ) d ( y ) , x , y \\in A . \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} P _ j : = P _ { j _ 0 } - \\frac { Q _ { j _ 0 j } L _ { j _ 0 j } } { C _ { j _ 0 j } } . \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} \\dot B _ { \\dot \\gamma } + ( B _ { \\dot \\gamma } ) ^ 2 + R ^ \\bot _ { \\dot \\gamma } = 0 , \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} X _ { \\perp } u = ( X _ { \\perp } u ) _ 0 + V a , a = \\sum _ { k \\neq 0 } \\frac { 1 } { i k } ( X _ { \\perp } u ) _ k . \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 1 } ^ r [ \\mathcal { G } _ { D , m } ( z , s ) ] ^ { \\mathrm { h o l } } & = - \\sum _ { n | m } \\biggl ( \\frac { D } { m / n } \\biggr ) n \\sqrt { D } \\mathrm { L C } _ { s = 1 } ^ r [ 1 ] q ^ { - n ^ 2 D } + \\sum _ { 0 > d \\equiv 0 , 1 ( 4 ) } \\mathrm { T r } _ { d , D } ( F _ { 0 , m , r } ) q ^ { - d } . \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} ( A ( q ^ { n - 1 } , t ) A ( q ^ { n - 2 } , t ) \\cdots A ( 1 , t ) ) _ { i , j } = \\sum _ { k = 0 } ^ { n } t ^ k \\sum _ { \\substack { w \\in S _ { n + 1 , k } \\\\ w _ 1 = i , w _ { n + 1 } = j } } q ^ { W _ 3 ( w ) } . \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} S _ { a + b t } ( \\rho \\| \\sigma ) = a \\sigma ( 1 ) + b \\rho ( 1 ) . \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda - \\mathcal { L } [ u ^ q ] + | D u ^ q | ^ m \\geq \\lambda & + \\frac 1 2 C _ 0 ^ { m ( q - 1 ) } \\Psi ^ { m ( q - 1 ) } f + C _ 1 ^ { q - 1 } \\Psi ^ { q - 1 } ( f - \\lambda ) \\\\ & + ( C _ 0 ^ { q - 1 } - C _ 1 ^ { q - 1 } ) \\Psi ^ { q - 1 } u . \\end{aligned} \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\sum _ { Q \\in \\mathcal { Q } _ { d D } / \\mathrm { S L } _ 2 ( \\mathbb { Z } ) } \\chi _ D ( Q ) \\int _ { \\mathrm { S L } _ 2 ( \\mathbb { Z } ) _ Q \\backslash S _ Q } - \\mathrm { l o g } ( y | \\eta ( z ) | ^ 4 ) \\frac { \\sqrt { d D } d z } { Q ( z , 1 ) } = 2 \\sqrt { D } L _ D ( 1 ) h ( d ) \\mathrm { l o g } \\varepsilon _ d . \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} c ( x _ 1 , \\ldots , x _ N ) = \\sum _ { 1 \\le i < j \\le N } f ( d ( x _ i , x _ j ) ) , \\ , ( x _ 1 , \\ldots , x _ N ) \\in X ^ N , \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} u _ { m + k } ( t , 1 ) = M _ { k } \\Big ( u _ { k + 1 } \\big ( t , x _ { k + 1 } ( - \\tau _ { m + k } , 0 , 0 ) \\big ) , \\dots , u _ { k + m - 1 } \\big ( t , x _ { k + m - 1 } ( - \\tau _ { m + k } , 0 , 0 ) \\big ) \\Big ) , \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} \\dot { \\psi } = \\mathcal { F } ( \\psi ) , \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} H _ n ^ { ( m ) } = \\sum _ { k = 1 } ^ n \\frac { 1 } { k ^ m } , \\ \\ H _ n ^ { ( 1 ) } = H _ n , \\ \\ H _ 0 ^ { ( m ) } = 0 . \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} K ( z , w ; Z , W ) & = \\left \\langle R _ { \\mathcal { H } _ p } \\left ( e _ { ( Z , W ) } \\circ T _ S \\right ) , R _ { \\mathcal { H } _ p } \\left ( e _ { ( z , w ) } \\circ T _ S \\right ) \\right \\rangle _ { \\mathcal { H } _ p } = 4 \\pi \\int _ 0 ^ { \\infty } t H _ p ( t ; w , W ) e ^ { i 2 \\pi ( z - \\overline { Z } ) t } \\d t . \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} n _ k = 2 { 3 k \\choose k } + 3 k \\leq 2 . 5 \\frac { ( 3 k ) ! } { k ! ( 2 k ) ! } < 3 \\frac { \\bigl ( \\frac { 3 k } { e } \\bigl ) ^ { 3 k } \\sqrt { 6 k \\pi } } { \\bigl ( \\frac { k } { e } \\bigl ) ^ { k } \\sqrt { 2 k \\pi } \\bigl ( \\frac { 2 k } { e } \\bigl ) ^ { 2 k } \\sqrt { 4 k \\pi } } = \\frac { 3 ^ { 3 / 2 } } { 2 } \\biggl ( \\frac { 3 ^ { 3 } } { 2 ^ 2 } \\biggl ) ^ k \\frac { 1 } { \\sqrt { k \\pi } } . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} \\begin{aligned} l = { } & h ( x ' ) \\beta _ { 0 } ( y , t ) + x ' h ( x ' ) \\beta _ { 1 } ( y , t ) \\\\ & = h ( T ( x ) ) \\left [ \\beta _ { 0 } ( y , t ) + ( 1 - 2 x ^ { 2 } ) \\beta _ { 1 } ( y , t ) \\right ] \\end{aligned} \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} \\big ( ( q + _ o r ( a ) ) / \\alpha , \\ , ( q + _ o r ( a ) ) - r ( q + _ o r ( a ) ) \\big ) & = \\big ( r ( a ) / \\alpha , \\ , ( q + _ o r ( a ) ) - _ o r ( a ) \\big ) . \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} a R _ { 1 1 } + b R _ { 2 1 } & = 0 \\\\ a R _ { 1 2 } + b R _ { 2 2 } & = [ b , a ] ( \\Delta ' ) ^ { - 1 } \\\\ c R _ { 1 1 } + d R _ { 2 1 } & = [ c , d ] ( \\Delta ' ) ^ { - 1 } \\\\ c R _ { 1 2 } + d R _ { 2 2 } & = ( [ d , a ] - [ c , b ] ) ( \\Delta ' ) ^ { - 1 } . \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} \\underset { \\mathbf { x } \\in \\mathbb { R } ^ { n p } } { } & g ( \\mathbf { x } ) \\triangleq f ( \\mathbf { x } ) + \\frac { \\rho ( 1 - \\eta ) } { 4 } \\| \\mathbf { E } _ { } \\mathbf { x } \\| ^ 2 , \\\\ & \\sqrt { \\eta } \\mathbf { E } _ { } \\mathbf { x } = \\mathbf { 0 } , \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} \\gamma ( t ) = ( 1 - t ) g ' + t g '' + ( g ^ M - g ^ P ) \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} \\psi _ \\eta ^ { \\ell } ( u ) = N ( \\rho , \\alpha ) \\frac { ( 1 - | \\eta | ^ 2 ) ^ { \\ell + 1 } } { \\sqrt { \\Gamma ( \\ell + 1 ) } } u ^ { \\frac { \\ell } { 2 } } e ^ { - \\frac { \\varpi } { 2 } u } \\frac { e ^ { \\frac { u \\eta } { \\eta - 1 } } } { ( 1 - \\eta ) ^ { 1 + \\ell } } e ^ { i \\theta _ { n , \\ell } ( t ) } . \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} \\left < \\exp ( \\sum - t _ { i _ k j _ k } ) \\right > _ { \\alpha , \\Lambda , \\epsilon } \\le \\prod _ { k = 1 } ^ { n } \\left < \\exp ( - n t _ { i _ k j _ k } ) \\right > _ { \\alpha , \\Lambda , \\epsilon } ^ { 1 / n } . \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} d \\Gamma _ A ( m ) = ( d \\Gamma _ A ( m _ 1 ) , \\dots , d \\Gamma _ A ( m _ { \\nu } ) ) \\\\ d \\Gamma _ A ^ { ( n ) } ( m ) = ( d \\Gamma _ A ^ { ( n ) } ( m _ 1 ) , \\dots , d \\Gamma _ A ^ { ( n ) } ( m _ { \\nu } ) ) \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} L ( x , y , \\mu , \\gamma , \\alpha ) = F ( x , y ) + C ( x , y , \\mu , \\gamma ) ^ T \\alpha . \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } x b ^ { m _ k } & \\sim \\phi b ^ { m _ 0 } x b ^ { m _ k - 1 } - \\sum _ { s \\in S _ b } \\phi b ^ { m _ 0 } x b ^ { m _ k - 1 } s \\\\ & \\sim . . . \\\\ & \\sim \\phi b ^ { m _ 0 } x b ^ { m _ k - ( m _ k - n _ k ) } - \\sum _ { i = 1 } ^ { m _ k - n _ k } \\sum _ { s \\in S _ b } \\phi b ^ { m _ 0 } x b ^ { m _ k - i } s \\\\ & = \\phi b ^ { m _ 0 } x b ^ { n _ k } - \\sum _ { i = n _ k } ^ { m _ k - 1 } \\sum _ { s \\in S _ b } \\phi b ^ { m _ 0 } x b ^ { i } s . \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\mu _ \\lambda = \\mathcal { O } \\left ( \\frac { \\log \\frac { 1 } { \\lambda } } { \\lambda } \\right ) . \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} 1 + z \\frac { T _ g '' ( z ) } { T _ g ' ( z ) } - \\alpha = z \\frac { f ' ( z ) } { f ( z ) } - \\alpha + z \\frac { g '' ( z ) } { g ' ( z ) } + 1 . \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} u _ { m + k - 2 } ( s , 0 ) = & F _ { m + k - 2 } ( s ) + \\int _ { 0 } ^ s { \\cal G } _ { m + k - 2 , m + k } ( \\xi ) u _ { m + k } ( \\xi , 0 ) \\ , d \\xi \\\\ [ 6 p t ] & + \\int _ { 0 } ^ s { \\cal G } _ { m + k - 2 , m + k - 1 } ( \\xi ) u _ { m + k - 1 } ( \\xi , 0 ) \\ , d \\xi \\mbox { f o r } 0 \\le s \\le t _ { k - 2 } , \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} I ( u ) : = \\dfrac { 1 } { 2 } M ( \\left \\| u \\right \\| ^ 2 ) - \\int _ { \\mathbb { R } ^ 2 } A ( x ) F ( u ) , \\ \\ u \\in H , \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} a ( q ) & = a ( q ^ 4 ) + 6 q \\dfrac { f _ 4 ^ 2 f _ { 1 2 } ^ 2 } { f _ 2 f _ 6 } , \\\\ a ( q ) & + 2 a ( q ^ 2 ) = 3 \\dfrac { f _ 2 f _ { 3 } ^ 6 } { f _ 1 ^ 2 f _ 6 ^ 3 } , \\\\ a ( q ) & + a ( q ^ 2 ) = 2 \\dfrac { f _ 2 ^ 6 f _ 3 } { f _ 1 ^ 3 f _ 6 ^ 2 } . \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} I _ { j } ( x ) = & \\int e ^ { i r ^ a } \\chi _ { j } ( r ) r ^ { \\frac { 2 a } { q } - 1 } ( r | x | ) ^ { - \\frac { d - 2 } { 2 } } J _ { \\frac { d - 2 } { 2 } } ( r | x | ) d r , d \\geq 2 , \\\\ I _ { j } ( x ) = & \\int e ^ { i | \\xi | ^ a } e ^ { i x \\xi } \\chi _ { j } ( \\xi ) | \\xi | ^ { \\frac { 2 a } { q } - d } d \\xi , d = 1 . \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} \\lim _ { m \\to + \\infty } \\deg ( p ) / q ^ m = 0 , \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} \\dim ( A + B ) + \\dim ( A \\cap B ) = \\dim A + \\dim B . \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} p ( z ) = \\sum _ { i = 0 } ^ { n } \\hat { p } ( i ) z ^ { i } q ( z ) = \\sum _ { i = 0 } ^ { n } \\hat { q } ( i ) z ^ { i } . \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} \\bar d ^ * ( y _ 0 , \\epsilon ) = d ^ * ( y _ 0 , \\epsilon ) . \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} & \\dot { \\Delta } _ j u = \\mathcal { F } ^ { - 1 } ( \\varphi _ j \\mathcal { F } u ) . \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} ( C ) = n ^ { r _ { p } - 2 } ( ( r _ { p } - 2 ) ( n - 1 ) - 4 ) . \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} - \\Delta w _ k + \\lambda w _ k \\leq - \\Delta w _ k + ( \\lambda + \\rho ( x _ 0 + \\epsilon _ k x ) \\phi _ { u _ k } ( x _ 0 + \\epsilon _ k x ) ) w _ k = w _ k ^ p \\leq \\frac { \\lambda } { 2 } w _ k . \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} g ^ { M _ 0 } - g ^ { P _ 0 } = ( g ^ M - g ^ P ) + g ' \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} G _ 1 ( L ) = \\{ ( \\left ( \\begin{array} { c c c c } \\rho & & & \\\\ & a _ { \\sigma } & & b _ { \\sigma } \\\\ & & 1 & \\\\ & c _ { \\sigma } & & d _ { \\sigma } \\end{array} \\right ) ) _ { \\sigma \\in I _ F } \\ \\vert a _ { \\sigma } , b _ { \\sigma } , c _ { \\sigma } , d _ { \\sigma } \\in L , \\rho \\in L ^ { \\times } , \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} \\begin{array} { l l } \\underset { \\sigma ^ 2 \\rightarrow 0 } { \\lim } \\mathbb { P } ( \\mathcal { A } _ 2 ) & = \\underset { \\sigma ^ 2 \\rightarrow 0 } { \\lim } \\mathbb { P } ( \\{ R R ( k ) > \\Gamma _ { R R T } ^ { \\alpha } ( k ) , \\ \\forall k \\} ) \\\\ & \\leq \\underset { \\sigma ^ 2 \\rightarrow 0 } { \\lim } \\mathbb { P } ( \\{ R R ( k _ 0 ) > \\Gamma _ { R R T } ^ { \\alpha } ( { k _ 0 } ) \\} ) = 0 . \\end{array} \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} \\mathcal { H } _ { i } = \\begin{bmatrix} D i a g ( H _ { 1 , a _ { i , 1 } } , \\dots , H _ { 1 , a _ { i , N } } ) \\\\ \\sigma _ { i } \\cdot D i a g ( H _ { 2 , a _ { i , 1 } } , \\dots , H _ { 2 , a _ { i , N } } ) \\\\ \\vdots \\\\ \\sigma _ { i } ^ { r - 1 } \\cdot D i a g ( H _ { r , a _ { i , 1 } } , \\dots , H _ { r , a _ { i , N } } ) \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} \\mathcal D _ 3 = \\{ \\Phi _ { ( \\psi , f ) } \\in \\mathcal D _ 4 : ~ f \\equiv 0 M \\} . \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} f ( x ) & = 2 ( x ^ 2 + 1 ) - h ( - x ) + m \\ : h ( x ) , \\\\ g ( x ) & = ( x ^ 3 + 1 ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\gamma _ { i , j ; n } = \\frac { 1 + \\rho _ { i , j } - \\rho _ { i , n } - \\rho _ { j , n } } { 2 \\sqrt { ( 1 - \\rho _ { i , n } ) ( 1 - \\rho _ { j , n } ) } } , \\ ; i \\neq n , j \\neq n \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} \\alpha = - 6 \\left ( \\sum _ { i = 1 } ^ { n + 1 } ( f ( i ) ^ 2 + d f ( i ) ) - \\sum _ { i = 1 } ^ { n } ( k _ { i - 1 } ( f ) ^ 2 + d k _ { i - 1 } ( f ) ) \\right ) . \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} \\delta _ { j } = \\frac { \\beta _ { j } - \\alpha _ { j } } { 2 } ( 1 + t ) , j = 1 , \\ldots , d , \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} v '' _ { \\lambda } ( z ) = z ^ { \\frac { 1 - 2 s } { s } } A _ { \\lambda } v _ { \\lambda } ( z ) + z ^ { \\frac { 1 - 2 s } { s } } \\ , \\delta \\ , v _ { \\lambda } ( z ) \\qquad \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} \\Delta ^ { * } = \\left ( \\ \\sum _ { j = 1 } ^ { q } B _ { j } - \\sum _ { i = 1 } ^ { p } A _ { i } \\right ) , \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} \\phi ( f ) : = \\int d x ~ f ( x ) \\phi _ { \\lfloor x \\rfloor } , \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\int _ { G } k _ l ( x ) d x & = \\int _ { G } \\sum _ { 2 ^ l \\leq \\langle \\xi \\rangle ^ 2 < 2 ^ { l + 1 } } \\sum _ { i , j = 1 } ^ { d _ \\xi } d _ \\xi \\xi _ { i j } ( x ) d x \\\\ & = \\sum _ { 2 ^ l \\leq \\langle \\xi \\rangle ^ 2 < 2 ^ { l + 1 } } \\sum _ { i , j = 1 } ^ { d _ \\xi } d _ \\xi \\int _ { G } \\xi _ { i j } ( x ) d x \\\\ & = \\sum _ { 2 ^ l \\leq \\langle \\xi \\rangle ^ 2 < 2 ^ { l + 1 } } \\sum _ { i , j = 1 } ^ { d _ \\xi } d _ \\xi \\int _ { G } \\xi _ { i j } ( x ) \\xi _ { 0 } ( x ) d x = 0 , \\\\ \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} { \\rm { d e t } } ( \\mathbf { M } ) = \\left ( \\prod _ { i = 1 } ^ n \\lambda _ i \\right ) ^ { f ( m + 1 ) } \\left ( \\prod _ { 1 \\leq i < j \\leq n } ( \\lambda _ j - \\lambda _ i ) \\right ) ^ { ( m + 1 ) ^ 2 } . \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} \\partial _ t \\xi = \\nu \\Delta \\xi + ( u \\nabla ) \\xi + \\beta _ j \\nabla \\xi \\dot { Z } _ t ^ j \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} ( s _ 1 , \\dots , s _ r ) \\cdot [ p _ 1 , \\dots , p _ r ] & = [ s _ 1 p _ 1 , s _ 1 ^ { - 1 } s _ 2 p _ 2 , \\dots , s _ { r - 1 } ^ { - 1 } s _ r p _ r ] \\\\ & = [ s _ 1 p _ 1 s _ 1 ^ { - 1 } , \\dots , s _ r p _ r s _ r ^ { - 1 } ] . \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} v _ n ( x ) = \\sum _ { j = 1 } ^ l V ^ j ( x - x ^ j _ n ) + v ^ l _ n ( x ) , \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} \\mathrm { R i c } = 0 . \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} b _ { k , m } ( n , s ) = ( - 1 ) ^ { \\lfloor \\frac { \\lambda _ k + 1 } { 2 } \\rfloor } b _ { 1 / 2 , ( - 1 ) ^ { \\lambda _ k } m } ( ( - 1 ) ^ { \\lambda _ k } n , s ) \\times \\left \\{ \\begin{array} { l l } | m n | ^ { \\frac { 1 - 2 k } { 4 } } & m \\neq 0 , n \\neq 0 , \\\\ 2 ^ { k - \\frac { 1 } { 2 } } \\pi ^ { \\frac { 2 k - 1 } { 4 } } | m + n | ^ { - \\frac { 2 k - 1 } { 4 } } & m n = 0 , m + n \\neq 0 , \\\\ 2 ^ { 2 k - 1 } \\pi ^ { k - \\frac { 1 } { 2 } } & m = n = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} \\ , ( { { \\mathrm d } { \\mathbb P } _ { 0 } } / { { \\mathrm d } \\mathbb P } ) \\vert _ { \\mathcal F _ { T } } \\ , : = \\ , Z _ { T } \\ , \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} s ( n ) = \\begin{dcases*} 0 & \\\\ - 2 ^ { - k } & \\end{dcases*} \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( \\ker \\beta ) = V + \\ker B , \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} D _ 2 \\cos _ k w & = 2 \\cos _ k \\left ( \\frac { w _ 1 + w _ 2 } 2 \\right ) \\cos _ k \\left ( \\frac { w _ 1 - w _ 2 } 2 \\right ) - 2 \\cos _ k \\left ( \\frac { w _ { - 1 } + w _ { - 2 } } 2 \\right ) \\cos _ k \\left ( \\frac { w _ { - 1 } - w _ { - 2 } } 2 \\right ) \\\\ & = 2 \\cos _ k \\left ( \\frac { w _ 1 + w _ 2 } 2 \\right ) \\left [ \\cos _ k \\left ( \\frac { w _ 1 - w _ 2 } 2 \\right ) - ( - 1 ) ^ j \\cos _ k \\left ( \\frac { w _ { - 1 } - w _ { - 2 } } 2 \\right ) \\right ] . \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} \\omega _ u = \\tilde { \\omega } _ u + \\omega ^ h _ u \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\mathbb { E } \\left [ \\frac { M ( t ) } { t } \\right ] = \\dot \\kappa ( 0 ) . \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} T _ V g ( z , w ) & = \\int _ { \\R } g \\left ( t , \\widehat { \\rho } _ { 1 / z } ( w ) \\right ) \\frac { z ^ { i 2 \\pi t } } { z ^ { 1 + 1 / 2 \\mu } } \\d t = \\int _ { \\R } g \\left ( t , \\frac { w _ 1 } { z ^ { 1 / 2 m _ 1 } } , \\cdots , \\frac { w _ n } { z ^ { 1 / 2 m _ n } } \\right ) \\frac { z ^ { i 2 \\pi t } } { z ^ { 1 + 1 / 2 \\mu } } \\d t \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} A \\xi _ i = \\mu _ i \\eta _ i , A ^ * \\eta _ i = \\mu _ i \\xi _ i , \\forall \\mu _ i > 0 . \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} 2 \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 6 n + 3 ) q ^ n & \\equiv \\left ( \\dfrac { 1 } { f _ 2 ^ 2 } - \\dfrac { a ( q ) } { f _ 2 ^ 2 } \\right ) \\dfrac { f _ { 6 } ^ 3 } { f _ 2 } \\\\ & \\equiv \\dfrac { f _ { 6 } ^ 3 } { f _ 2 ^ 3 } - \\dfrac { f _ { 6 } ^ 3 } { f _ 2 ^ 3 } \\left ( a ( q ^ 4 ) + 6 q \\dfrac { f _ 4 ^ 2 f _ { 1 2 } ^ 2 } { f _ 2 f _ 6 } \\right ) ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} \\mathcal { D } \\big ( \\mu ^ l \\big ( g ^ { - 1 } v ^ { - 1 } ? \\big ) \\big ) & = \\big \\langle \\mu ^ l \\big ( g ^ { - 1 } v ^ { - 1 } ? \\big ) \\otimes \\mathrm { i d } , ( g \\otimes 1 ) R R ' \\big \\rangle \\\\ & = \\big \\langle \\mu ^ l \\big ( g ^ { - 1 } v ^ { - 1 } ? \\big ) \\otimes \\mathrm { i d } , g v \\big ( v ^ { - 1 } \\big ) '' \\otimes v \\big ( v ^ { - 1 } \\big ) ' \\big \\rangle \\\\ & = \\mu ^ l \\big ( \\big ( v ^ { - 1 } \\big ) '' \\big ) v \\big ( v ^ { - 1 } \\big ) ' = \\mu ^ l \\big ( v ^ { - 1 } \\big ) v . \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} a = \\oplus _ { n = 1 } ^ \\infty E _ n a E _ n = \\lim _ { N \\to \\infty } \\oplus _ { n = 1 } ^ N E _ n a E _ n \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} \\tilde { \\Omega } _ j ( \\theta ) & = a _ j ^ 1 ( \\theta ) + a _ j ^ 2 ( \\theta ) \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} \\frac { \\lambda _ n } { | \\lambda _ n | } ( w _ n ) _ { i j } - \\delta ^ { k l } \\frac { \\partial ^ 2 ( w _ n ) _ { i j } } { \\partial y ^ k \\partial y ^ l } + \\widetilde { Z } ( w _ n ) _ { i j } = [ \\mathcal { E } _ 1 ] _ { i j } . \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} & T ^ { * } _ { W } : \\mathcal { K } _ { 2 , W } \\rightarrow H \\\\ & T ^ { * } _ { W } ( v _ i ( C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f ) = \\sum _ { i \\in I } v _ { i } ^ { 2 } C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } f . \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} \\mathcal { D T } ( \\pi ) & = \\{ \\pi _ i : \\pi _ i > \\pi _ { i + 1 } \\} \\mbox { a n d } \\\\ [ 5 p t ] \\mathcal { A T } ( \\pi ) & = \\{ \\pi _ { i + 1 } : \\pi _ i < \\pi _ { i + 1 } \\} \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t , i } \\ , = \\ , \\frac { \\ , 1 \\ , } { \\ , n \\ , } \\sum _ { \\stackrel { j = 1 } { j \\neq i } } ^ { n } h \\big ( X _ { t , i } , X _ { t , j } \\big ) { \\mathrm d } t + { \\mathrm d } W _ { t , i } \\ , ; t \\ge 0 \\ , , i \\ , = \\ , 1 , \\ldots , n \\ , . \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { x ^ { 2 n } } { n ! \\Gamma ( n + \\mu + 1 ) } = \\frac { I _ \\mu ( 2 x ) } { x ^ \\mu } . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } x b ^ { m _ k } & \\sim \\phi b ^ { m _ 0 - 1 } x b ^ { m _ k } - \\sum _ { s \\in S _ b } \\phi s b ^ { m _ 0 - 1 } x b ^ { m _ k } \\\\ & \\sim . . . \\\\ & \\sim \\phi b ^ { m _ 0 - ( m _ 0 - n _ 0 ) } x b ^ { m _ k } - \\sum _ { i = 1 } ^ { m _ 0 - n _ 0 } \\sum _ { s \\in S _ b } \\phi s b ^ { m _ 0 - i } x b ^ { m _ k } \\\\ & = \\phi b ^ { n _ 0 } x b ^ { m _ k } - \\sum _ { i = n _ 0 } ^ { m _ 0 - 1 } \\sum _ { s \\in S _ b } \\phi s b ^ i x b ^ { m _ k } . \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} \\mathbb { S } _ + ^ 1 = \\{ \\theta \\in \\R ^ d ; \\ ; | \\theta | = 1 , \\theta _ 1 \\geq 0 \\} . \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} E _ { i , j } ^ { ( r ) } : = [ E _ { i , j - 1 } ^ { ( r - s _ { j - 1 , j } ) } , E _ { j - 1 } ^ { ( s _ { j - 1 , j } + 1 ) } ] & 1 \\leq i < j \\leq n r > s _ { i , j } , \\\\ \\smallskip F _ { i , j } ^ { ( r ) } : = [ F _ { j - 1 } ^ { ( s _ { j , j - 1 } + 1 ) } , F _ { i , j - 1 } ^ { ( r - s _ { j , j - 1 } ) } ] & 1 \\leq i < j \\leq n r > s _ { j , i } . \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} \\lvert u _ n \\rvert _ S & = \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - ( n - i ) \\\\ & = \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - n + i \\\\ & \\geq \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - n + ( n - 2 \\lvert I \\rvert + 1 ) \\\\ & = \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - 2 \\lvert I \\rvert + 1 \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} ( q ^ 2 ; q ^ 2 ) _ { \\infty } = ( q ; q ) _ { \\infty } ( - q ; q ) _ { \\infty } , ( q ; q ) _ { \\infty } = ( q ^ 2 ; q ^ 2 ) _ { \\infty } ( q ; q ^ 2 ) _ { \\infty } , \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} I = & \\int \\log p ( y ) d y \\int p _ { \\beta } ( y | x ) p ( x ) d x = \\int p _ { 0 } ( y ) \\log p ( y ) d y \\leq \\int p _ { 0 } ( y ) \\log p _ { 0 } ( y ) \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} \\theta _ v ( t ) = \\Omega t + \\bar { \\theta } _ v , \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} p ' ( w ) = \\begin{cases} p ( w ) + \\epsilon ( 0 , - 1 ) & w \\sim y , \\\\ p ( w ) & \\end{cases} \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} S _ { ( \\lambda , \\mu ) , ( \\lambda ' , \\mu ' ) } = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho + \\lambda , ( w \\bullet ( \\lambda ' , \\mu ' ) ) _ 2 \\rangle + \\langle \\mu , ( w \\bullet ( \\lambda ' , \\mu ' ) ) _ 1 + 2 \\rho \\rangle } } { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho , w \\bullet 0 \\rangle } } , \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} \\int _ { V _ { \\zeta } } \\abs { F ( \\gamma , \\zeta ) } ^ 2 \\d V ( \\gamma ) = \\int _ { \\R } \\abs { f ( t , \\zeta ) } ^ 2 \\lambda ( p ( \\zeta ) , t ) \\d t . \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} x = \\cos \\left ( \\pi u _ { 0 } + \\frac { 2 \\pi } { N } \\cdot j \\right ) \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} c ( \\theta _ 0 , \\theta _ 1 ) = \\inf \\Big \\{ & \\int _ 0 ^ 1 \\mathbb { E } _ { z \\sim \\mu } L ( g ( \\theta _ t , z ) , \\frac { d } { d t } g ( \\theta _ t , z ) ) d t \\colon \\\\ & \\frac { d } { d t } g ( \\theta _ t , z ) = D _ p H ( g ( \\theta _ t , z ) , \\nabla _ x \\Phi ( t , g ( \\theta _ t , z ) ) ) , ~ \\theta ( 0 ) = \\theta _ 0 , ~ \\theta ( 1 ) = \\theta _ 1 \\Big \\} , \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} E ( \\bold { x } ) = \\dfrac { 1 } { 4 \\pi \\epsilon _ 0 } \\int _ { \\R ^ 3 } \\dfrac { \\bold { x } - s } { \\vert \\bold { x } - s \\vert ^ 3 } \\rho ( s ) \\ , \\d s , \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\theta _ { X \\otimes Y } = ( \\theta _ X \\otimes \\theta _ Y ) \\circ c _ { Y , X } \\circ c _ { X , Y } . \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} K _ 1 ( \\delta ) : = \\sup _ { | x | \\leq 2 \\delta } | H '' ( x ) | < \\infty . \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} \\mathcal { F } _ R : = \\left \\{ u \\in L ^ 1 _ \\rho ( X ) ~ \\Big | ~ u ( x _ 1 ) + \\cdots + u ( x _ N ) \\le c _ R ( x _ 1 , \\ldots , x _ N ) \\rho ^ { \\otimes ( N ) } ( x _ 1 , \\ldots , x _ N ) \\right \\} . \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} | | X _ K | | _ { \\tilde { r } , p - 1 , \\mathbf { a } } = O ( \\tilde { r } ^ 3 ) . \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} \\delta ( G _ k ) = \\min \\bigl ( \\frac { 1 } { 2 } n _ k + \\frac { k } { 2 } - 1 , \\frac { 2 } { 3 } n _ k + k - 1 \\bigl ) = \\frac { 1 } { 2 } n _ k + \\frac { k } { 2 } - 1 . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} \\mathrm { G L } ( 2 , \\mathbb { R } ) & = \\left \\{ A \\in \\mathrm { M a t } _ { 2 \\times 2 } ( \\mathbb { R } ) : \\det ( A ) = \\pm 1 \\right \\} \\\\ \\mathrm { S L } ( 2 , \\mathbb { R } ) & = \\left \\{ A \\in \\mathrm { M a t } _ { 2 \\times 2 } ( \\mathbb { R } ) : \\det ( A ) = 1 \\right \\} \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} \\dot { x } _ { 1 , \\tau } = \\{ x _ { 1 , \\tau } , { H _ { \\tau } } _ T \\} _ { P B } = \\lambda \\frac { f ( t _ \\tau ) } { m } p _ { 1 , \\tau } ; \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} \\dot { \\theta } = \\omega _ v + \\sum _ { v ' \\in N ( v ) } H ( \\theta _ { v ' } - \\theta _ v , v , v ' ) , \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} U ( o , \\ , o \\ , | \\ , z ) = \\sum _ { i = 1 } ^ r \\frac { - ( \\phi _ i ( z ) - m U ( o , \\ , o \\ , | \\ , z ) ) } { 2 m } + \\sum _ { i = 1 } ^ r \\frac { [ ( \\phi _ i ( z ) - m U ( o , \\ , o \\ , | \\ , z ) ) ^ 2 + 4 \\lambda m _ i z ^ 2 ] ^ { 1 / 2 } } { 2 m } , \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} [ ( x _ 1 , \\ldots , x _ { n - t ( d - 1 ) } ) ^ d ] ^ { \\sigma ^ t } = I _ { n , d , t } . \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} \\chi ^ 2 = m , \\mu \\chi = - \\chi \\mu . \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} p _ { y } ( \\textbf { z } ) & = P \\{ \\delta = 1 \\vert \\textbf { Z } = \\textbf { z } , Y = y \\} = E ( \\delta \\vert \\textbf { Z } = \\textbf { z } , Y = y ) , \\\\ [ 5 p t ] q _ { y } ( \\textbf { x } ) & = P \\{ \\delta = 1 \\vert \\textbf { X } = \\textbf { x } , Y = y \\} = E ( \\delta \\vert \\textbf { X } = \\textbf { x } , Y = y ) . \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} \\lambda _ { k + 1 } = \\Pi _ { D } [ \\lambda _ k + f i ( f i ( M _ 3 ) \\tilde { x } _ k ) - f i ( V _ 2 ) ] \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} y = C \\left ( - \\log _ { q } \\frac { | x | } { R } \\right ) ^ { - \\alpha } \\frac { x } { | x | } = C \\left ( \\frac { | x | } { R } \\right ) ^ { \\alpha ( q - 1 ) } \\left ( \\log _ { q } \\frac { R } { | x | } \\right ) ^ { - \\alpha } \\frac { x } { | x | } . \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} \\begin{cases} ( u _ { b } ) _ { x } & = p _ { b } \\\\ ( p _ { b } ) _ { x } & = - \\frac { b _ * u _ { b } + c _ * p _ { b } } { a _ * } \\end{cases} \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} \\ln R ( n + 1 , E ) - \\ln R ( n , E ) = - \\frac { b _ { n + 1 } ' } { \\omega } \\sin ( 2 \\eta ( n , E ) + 2 \\gamma ( n , E ) ) \\vert \\varphi ( n , E ) \\vert ^ 2 + O ( | b _ { n + 1 } ^ \\prime | ^ 2 ) . \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} \\tilde { \\lambda } ^ 2 W \\tilde { \\Pi } e = 0 , \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} I ( \\mathbf { x } ; \\mathbf { z } ) = N \\times I ( \\mathbf { x } [ i ] ; \\mathbf { z } [ i ] ) . \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\| \\nabla V ^ { j _ 0 } \\| ^ 2 _ { L ^ 2 } - \\frac { 1 } { \\alpha + 2 } \\| V ^ { j _ 0 } \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } - \\frac { c } { 2 } \\limsup _ { n \\rightarrow \\infty } \\int | x | ^ { - 2 } | V ^ { j _ 0 } ( x - x ^ { j _ 0 } _ n ) | ^ 2 d x = d _ M . \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} \\Re Z _ t ( A ) = - r - n + r ( g - 1 ) t ^ 2 \\ge 0 . \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} i \\partial _ t P _ c v = H P _ c v + P _ c \\ , \\N , \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} D ^ \\nu { } _ \\beta = \\begin{pmatrix} 9 & 0 \\\\ 0 & - 9 9 \\end{pmatrix} \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} v _ \\tau \\coloneqq \\lim _ { \\epsilon \\downarrow 0 } \\frac { 1 } { \\epsilon } ( x ^ { \\tau , \\epsilon } _ \\tau - x ^ * _ \\tau ) = f ( x ^ { * } _ \\tau , \\omega ) - f ( x ^ * _ \\tau , \\theta ^ * _ \\tau ) . \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} \\textup { C a p } _ \\chi ( \\cup E _ j ) = \\lim _ { j } \\textup { C a p } _ \\chi ( E _ j ) . \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } ( f ( z _ 1 ) \\dots f ( z _ n ) ) ^ i \\mu ( f ^ i ) x ^ i = D _ f ^ { \\phi } ( f ( z _ 1 ) \\dots f ( z _ n ) x ) ^ { - 1 } = D _ f ^ { \\varphi } ( x ) ^ { - 1 } . \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} { \\rm S } ( { \\rm a d } _ { H } ) ( X , Y ) = \\frac 1 2 \\big ( g ( \\mu ( H , X ) , Y ) + g ( \\mu ( H , Y ) , X ) \\big ) \\ , . \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} \\mathcal { A } g = h - \\mu ( h ) , \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t , i } ^ { ( u ) } \\ , = \\ , b \\big ( t , X _ { t , i } ^ { ( u ) } , \\widehat { F } _ { t , i } ^ { ( u ) } \\big ) { \\mathrm d } t + { \\mathrm d } W _ { t , i } \\ , ; t \\ge 0 \\ , , i \\ , = \\ , 1 , \\ldots , n - 1 \\ , , \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} x ( t ) = \\begin{bmatrix} 3 8 . 9 2 7 8 1 7 7 \\\\ 2 . 2 7 6 9 8 9 1 3 \\end{bmatrix} , x ( t + 1 ) = \\begin{bmatrix} 3 2 . 2 8 0 7 4 8 7 3 \\\\ - 5 . 8 3 8 7 4 0 1 2 \\end{bmatrix} , \\epsilon = 0 . 9 9 9 1 4 1 9 8 . \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} f '' ( h _ 0 , \\ldots , h _ { n - 1 } ) ( \\nu ) : = \\sum _ { i _ 0 , \\ldots , i _ { n - 1 } } \\langle f ' ( g _ { i _ 0 } , \\ldots , g _ { i _ { n - 1 } } ) , \\nu | _ { [ i _ 0 , \\ldots , i _ { n - 1 } ] } \\circ L \\rangle . \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} \\dim ^ - ( L ( \\lambda , \\mu ) ) = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } q ^ { - \\langle 2 \\rho , ( w \\bullet ( \\lambda , \\mu ) ) _ 2 \\rangle } } { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } q ^ { - \\langle 2 \\rho , w \\bullet ( 0 ) \\rangle } } , \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} h I _ D \\le \\sum _ { i = 1 } ^ a I _ { A _ i } - \\sum _ { j = 1 } ^ b I _ { B _ j } = \\sum _ { k _ 1 = 0 } ^ { t _ 1 - 1 } \\sum _ { k _ 2 = 0 } ^ { t _ 2 - 1 } d _ { k _ 1 , k _ 2 } I _ { R _ { k _ 1 , t _ 1 } \\times R _ { k _ 2 , t _ 2 } } \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} R ( n ) = \\vert Z ( n ) \\vert , \\eta ( n ) = \\mathrm { A r g } ( Z ( n ) ) . \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} x - v = y + w . \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} \\frac { S _ { \\mathcal { C } } } { \\sqrt { 2 0 } } = - S \\text T _ { \\mathcal { C } } = T . \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} u ( y + g _ k ) = ( 1 + o ( 1 ) ) u ( y ) , y \\in K _ { n , \\varepsilon } , | y | \\le \\sqrt { n } \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} ( \\mathrm { d } p ) _ \\alpha ( x ) = \\psi _ \\alpha { } ^ \\beta ( x ) \\ \\frac { \\partial ( p \\circ \\varphi ) } { \\partial x ^ \\beta } ( x ) \\ , , \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} \\frac { \\beta _ { \\l } } { \\alpha _ { \\l } } = \\frac { ( \\l - 1 ) ( \\l + d - 2 ) } { \\l ( \\l + d - 1 ) } . \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\frac { B ( \\tau _ t ) } { \\tau _ t } \\cdot \\frac { t } { B ( K t ) } = \\infty \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} T _ { \\pm } ^ * \\kappa _ 0 ( \\delta ) ( i \\sigma ) \\kappa _ 0 ( \\delta ) T _ { \\pm } = e ^ { \\pm \\delta ( t ) e ^ { 2 i \\alpha } } \\Big [ \\pm \\sin ( \\alpha ) c ( t ) - s ( t ) \\cos ( \\alpha ) + i \\Big ( \\cos ( \\alpha ) c ( t ) \\mp \\sin ( \\alpha ) s ( t ) \\Big ) \\widetilde { I } \\Big ] ~ , \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} \\| u \\| _ { [ L ^ 2 , E ] _ { \\frac { 2 } { 3 } } } = { \\sum _ { i = 1 } ^ { d } \\Big ( \\| \\ ; | D _ { q _ i } | ^ { \\frac { 2 } { 3 } } u \\| _ { L ^ 2 ( \\mathbb { R } ^ { 2 d } ) } ^ 2 + \\| \\ ; | \\partial _ { q _ i } V ( q _ i ) | ^ { \\frac { 2 } { 3 } } u \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ { 2 d } ) } \\Big ) + \\| u \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ { 2 d } ) } ~ . } \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} x = \\gamma ( s ) + t n ( s ) . \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} B ( t ) \\begin{cases} t ^ { q _ 0 } \\ , \\ell ( t ) ^ { \\alpha _ 0 } & , \\\\ t ^ { q _ \\infty } \\ , \\ell ( t ) ^ { \\alpha _ \\infty } & . \\end{cases} \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} | V _ { \\mu , \\xi } | \\lesssim \\sum _ { j = 1 } ^ k \\mu _ j ^ { - 2 s } \\frac { R ^ { - 2 s } } { 1 + | y _ j | ^ { 2 s } } . \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} T _ { y * } ( X ) : = T _ { y * } ( [ \\mathrm { i d } _ X ] ) , \\widehat { T } _ { y * } ( X ) : = \\widehat { T } _ { y * } ( [ \\mathrm { i d } _ X ] ) ; \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} x _ { k _ { i n } + 1 } = \\Pi _ { \\mathcal { X } } [ x _ { k _ { i n } } - f i ( f i ( M _ 1 ) x _ { k _ { i n } } ) - f i ( f i ( M _ 2 ) \\lambda _ k ) + f i ( V _ 1 ) ] \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} \\mathbb { E } e ^ { - \\lambda H _ { t } } & = e ^ { - t ( ( \\eta + \\lambda ) ^ { \\alpha } - \\eta ^ { \\alpha } ) } = e ^ { - t \\int _ { 0 } ^ { \\infty } ( 1 - e ^ { - \\lambda y } ) \\varPi ( \\mathrm { d } y ) } , \\quad \\lambda > 0 . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 u '' - r ( x ) u = 0 , \\ - \\infty < x < a ; \\\\ & u ( a ) = 1 ; \\\\ & 0 \\le u \\le 1 \\ \\ u \\ . \\end{aligned} \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} T = S + i B . \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} \\frac { { \\rm { D } } } { \\partial z } = \\frac { 1 } { 2 } \\left ( { { D } _ { x } } - \\sqrt { - 1 } { { D } _ { y } } \\right ) , ~ ~ ~ \\frac { { \\rm { D } } } { \\partial \\overline { z } } = \\frac { 1 } { 2 } \\left ( { { D } _ { x } } + \\sqrt { - 1 } { { D } _ { y } } \\right ) \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} [ \\wedge ^ { i + 1 } E ] \\otimes [ \\alpha _ * \\omega _ \\alpha ^ 0 ] & = [ \\wedge ^ { i + 1 } E ] \\otimes [ \\O + E ^ \\vee ] \\\\ & = [ K _ { i + 1 } ( i + 1 ) ] + [ \\wedge ^ { i + 1 } E ] \\otimes [ E ^ \\vee ] , \\ \\\\ [ \\wedge ^ i E ] \\otimes [ \\alpha _ * \\omega _ \\alpha ] & = [ \\wedge ^ i E ] \\otimes [ \\O + E ] \\\\ & = K _ { i + 1 } ( i ) + [ \\wedge ^ i E ] . \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} L ^ { \\chi _ g } = \\{ x \\in L : \\forall \\sigma \\in G , \\sigma ( x ) = \\chi _ g ( \\sigma ) x \\} . \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} ( \\delta _ \\alpha f ) ( a _ 1 , a _ 2 , \\ldots , a _ { n + 1 } ) = ~ & \\mu \\big ( \\alpha ^ { n - 1 } ( a _ 1 ) , f ( a _ 2 , \\ldots , a _ { n + 1 } ) \\big ) \\\\ ~ & + \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ i f \\big ( \\alpha ( a _ 1 ) , \\ldots , \\alpha ( a _ { i - 1 } ) , \\mu ( a _ i , a _ { i + 1 } ) , \\alpha ( a _ { i + 2 } ) , \\ldots , \\alpha ( a _ { n + 1 } ) \\big ) \\\\ ~ & + ( - 1 ) ^ { n + 1 } \\mu ( f ( a _ 1 , \\ldots , a _ n ) , \\alpha ^ { n - 1 } ( a _ { n + 1 } ) ) . \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} U _ + ^ { - 1 } ( w ) = \\frac { \\theta _ + w - \\theta _ - c } { w - c } U _ - ^ { - 1 } ( w ) = \\frac { \\theta _ - w - \\theta _ + c } { w - c } \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} ( ( \\sigma - \\Sigma _ h ) ^ { n - \\theta } , w _ h ) + ( \\nabla ( u - U _ h ) ^ { n - \\theta } , \\nabla w _ h ) = 0 . \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} j _ \\lambda ( h k , z _ 0 ) = j _ \\lambda ( h , z _ 0 ) \\chi _ \\lambda ( k ) ^ { - 1 } , \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ { m } \\alpha _ i ^ { l _ 0 l _ i } \\right ) \\prod _ { 1 \\le i < j \\le m } ( \\alpha _ i - \\alpha _ j ) ^ { l _ i l _ j } \\underset { \\mathbb { Q } [ \\alpha _ 1 , \\ldots , \\alpha _ m ] } { \\bigg | } \\mathcal { U } [ 0 ] . \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} L _ 0 ( t ; z ) = t _ 0 , L _ 1 ( t ; z ) = z _ { 0 1 } t _ 0 + z _ { 1 1 } t _ 1 + \\cdots + z _ { k 1 } t _ k . \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} F ( Q + v ) - F ( Q ) = \\tfrac { p + 2 } { 2 } \\mu v \\int _ 0 ^ 1 | Q + \\theta v | ^ p \\ , d \\theta + \\tfrac { p } { 2 } \\mu \\bar v \\int _ 0 ^ 1 | Q + \\theta v | ^ { p - 2 } ( Q + \\theta v ) ^ 2 \\ , d \\theta . \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} L ( x _ 1 , x _ 2 , \\dot { x } _ 1 , \\dot { x } _ 2 , t ) = f ^ { - 1 } ( t ) \\left [ \\frac { m } { 2 } ( \\dot { x } _ 1 ^ 2 + \\dot { x } _ 2 ^ 2 ) - \\frac { m \\omega ^ 2 ( t ) } { 2 } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) \\right ] , \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} G _ { n } ( x ) \\stackrel { \\mathrm { d e f } } { = } \\sup _ { | z - x | \\leq h _ { n } } f ( z ) \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} \\det \\left ( 1 - \\mathcal { L } _ { s , \\lambda } ^ { N } \\right ) & = \\exp \\left ( - \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n } \\mathrm { T r } \\ , \\left ( \\mathcal { L } _ { s , \\lambda } ^ { n N } \\right ) \\right ) . \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} \\begin{aligned} x _ { i , j } & = \\widehat { p _ { j } } ( n - p + i + 1 ) \\\\ & = \\sum _ { k = 1 } ^ { p - 1 } \\widehat { \\sigma _ { - } ^ { - 1 } } ( i - k ) \\widehat { \\sigma _ { - } p _ { j } } ( n - p + k + 1 ) \\end{aligned} \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} & M \\left ( ( 1 + x + x ^ 2 + x ^ 4 ) + t \\ : h ( x ) + y t \\ ; h ( x ) \\right ) = 2 ^ 6 ( 1 + 3 t ) , \\\\ & M \\left ( ( 1 + x ^ 2 ) + t \\ : h ( x ) + y \\left ( - ( 1 + x ^ 3 ) + t \\ : h ( x ) \\right ) \\right ) = 2 ^ 6 3 ^ 3 t . \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } p _ { n - 1 } & p _ { n } \\\\ q _ { n - 1 } & q _ { n } \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ 1 \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ 2 \\end{array} \\right ] \\cdots \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n } \\end{array} \\right ] \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} B ( t + 1 ) = \\left [ B ( t ) + X ( t ) \\right ] ^ { + } . \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} \\sum _ { \\substack { h , \\bar { h } , h \\bar { h } = h _ i \\\\ h \\in c ( g , X ) , \\bar { h } \\in c ( \\bar { g } , X ) } } \\pi ( \\xi ) | _ { [ t _ 1 , \\ldots , t _ { i - 1 } , g , \\bar { g } , t _ { i + 1 } , \\ldots , t _ { n - 1 } ] } = \\pi ( \\xi ) | _ { \\mathcal { J } } . \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} \\int _ { G } k _ l ( x ) d x = 0 , \\ , \\ , l \\geq 1 . \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\Phi _ G ( m _ x ) = \\imath \\ , \\begin{pmatrix} \\lambda ( m _ x ) & 0 \\\\ 0 & - \\lambda ( m _ x ) \\end{pmatrix} . \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} H _ N ( s _ 1 , \\ldots , s _ k ) : = \\sum _ { N \\geq n _ 1 > \\ldots > n _ k \\geq 1 } \\frac { 1 } { n _ 1 ^ { s _ 1 } \\ldots n _ k ^ { s _ k } } \\in \\Q \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} \\gamma _ n ( t _ { j k } ) = \\begin{cases} - \\gamma _ n ( a _ { j k } ) & n < M , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} B S _ { G , m , p } ( t _ 0 , A _ 0 , \\ldots , A _ 3 , D , f , u _ 0 ) = \\partial _ t ^ p g ( t _ 0 ) \\partial G \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} K _ { F , 1 } ( \\tau ) & = \\frac { 2 \\tau } { 3 \\pi } \\left ( e + \\frac { 1 } { e } \\right ) \\left ( 1 + \\left | \\log { \\frac { 4 q \\tau } { \\pi } } \\right | \\right ) \\\\ & + \\frac { 4 } { 2 7 } \\left ( 1 + \\frac { 1 } { e ^ 2 } \\right ) \\left ( \\frac { 0 . 6 3 4 } { 3 } \\log { 2 } + 0 . 3 1 7 \\log { ( 2 e ^ 4 q ^ 2 \\tau ^ 2 ) } + 1 2 . 8 0 2 \\right ) . \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} \\begin{cases} L _ { Y ^ T } g ^ { ( 4 ) } ( \\partial _ t , \\partial _ t ) = - 2 u Y ^ T ( u ) \\\\ [ L _ { Y ^ T } g ^ { ( 4 ) } ( \\partial _ t ) ] ^ T = - u ^ 2 d \\theta ( Y ^ T ) \\\\ [ L _ { Y ^ T } g ^ { ( 4 ) } ] ^ T = L _ { Y ^ T } g _ S . \\end{cases} \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\in \\Lambda } \\rho _ \\infty ( e _ \\lambda ) h = h \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} d \\widetilde { w } ( s ) = \\big ( A ^ { - 1 } \\nabla \\xi \\sigma \\big ) ( x ( s ) ) \\ , d w ( s ) = \\Big [ ( \\nabla \\xi a \\nabla \\xi ^ T ) ^ { - \\frac { 1 } { 2 } } \\nabla \\xi \\sigma \\Big ] ( x ( s ) ) \\ , d w ( s ) \\ , , s \\ge 0 \\ , . \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} \\int _ { x _ 0 } ^ x \\frac { \\cos ( 2 \\pi \\ell y - ( 2 \\tilde { \\theta } ( y , { E } ) - 2 \\tilde { \\theta } ( y , \\hat { E } ) ) ) } { 1 + y - b } d y = \\frac { 1 } { \\ell } \\frac { O ( 1 ) } { x _ 0 - b { + 1 } } . \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} N ( t ) = \\sum _ { i = 1 } ^ m \\int _ \\Omega u _ i ( t , x ) d x . \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} E ( G ^ { \\prime } ) = 5 i + 2 i + 2 + \\sum _ { i = 1 } ^ 3 | \\lambda ' _ i | \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} \\Upsilon ^ 1 { } _ { 3 2 } = \\pm 1 , \\Upsilon ^ 2 { } _ { 3 1 } = \\mp 1 , \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} U ^ { \\bar { i } \\bar { j } } ( x ) - V ^ { \\bar { i } \\bar { j } } ( x ) = U ^ { k \\bar { j } } ( x ) - V ^ { k \\bar { j } } ( x ) . \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\R ^ 3 } | \\nabla u | ^ 2 + \\frac { 3 b } { 2 } \\int _ { \\R ^ 3 } u ^ 2 + \\frac { 5 c } { 4 } \\int _ { \\R ^ 3 } \\rho \\phi _ u u ^ 2 + \\frac { c } { 2 } \\int _ { \\R ^ 3 } \\phi _ u u ^ 2 ( x , \\nabla \\rho ) - \\frac { 3 d } { p + 1 } \\int _ { \\R ^ 3 } | u | ^ { p + 1 } = 0 . \\\\ \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t , i } \\ , & = \\ , h ( X _ { t , i } , X _ { t , i + 1 } ) { \\mathrm d } t + { \\mathrm d } W _ { t , i } \\ , ; t \\ge 0 \\ , , i \\ , = \\ , 1 , \\ldots , n - 1 \\ , , \\\\ { \\mathrm d } X _ { t , n } \\ , & = \\ , h ( X _ { t , n } , X _ { t , 1 } ) { \\mathrm d } t + { \\mathrm d } W _ { t , n } \\ , . \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} [ T _ b , T _ { a ^ 2 } ] + 2 [ T _ a , T _ { a b } ] = 0 . \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} 0 \\leq \\eta _ n ( \\varphi _ m ) = \\eta _ n ( K _ m ) \\leq \\eta _ n ( K _ n ) = 0 , \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\log | \\Delta _ n | = \\frac { 1 } { n } \\log | q _ n \\theta - p _ n | \\leq - \\frac { 1 } { n } \\log q _ { n + 1 } \\leq \\frac { 1 } { n } \\log | \\Delta _ { n - 1 } | \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} \\partial _ t \\hat { C } = c _ 0 ^ 2 \\hat { C } | \\theta \\cdot \\nabla p | ^ 2 - \\hat { C } ^ \\gamma \\qquad \\mbox { o n s u p p } ( \\eta ) . \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\| \\nabla f \\| ^ 2 = b \\circ f , \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} | | \\Omega _ n | | _ { W ^ { \\alpha _ 0 , 1 } ( B _ 2 \\backslash B _ 1 ) } \\leq 2 c _ 1 , ~ ~ \\lim \\limits _ { n \\to \\infty } | | \\Omega _ n - \\Omega | | _ { W ^ { \\alpha _ 0 , 1 } ( B _ 2 \\backslash B _ 1 ) } = 0 , \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} - \\frac { n + m - 1 } { 2 } + D _ { n , m } + m - 1 & = - \\frac { n + m - 1 } { 2 } + \\frac { n - m + 1 } { 2 } + m - 1 \\\\ & = - m + 1 + m - 1 = 0 , \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} \\omega ( X ) ^ T C + C \\omega ( X ) = \\nabla ^ h _ X C . \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} \\left . \\frac { \\partial L } { \\partial e _ 2 } \\right | _ { e _ 2 = e _ 3 = e _ 4 = 0 } = - 1 , \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} L _ { n , \\beta } ^ t = \\left \\{ \\begin{array} { l l } P _ { n , \\beta } , & t \\gg 0 , \\\\ P _ { - n , \\beta } , & t \\ll 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} c = \\left \\{ \\begin{array} { l l } L & \\mbox { i f } , L \\leq 3 , \\\\ 8 L ^ 2 & \\mbox { i f } , L > 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} T \\Vert f ' _ h - f ' \\Vert _ 1 & = \\int _ 0 ^ T | g ' ( t ) | \\ , \\mathrm { d } t = \\sum _ { k = 1 } ^ M \\int _ { t _ { k - 1 } } ^ { t _ { k } } | g ' ( t ) | \\ , \\mathrm { d } t \\\\ & = \\sum _ { k = 1 } ^ M \\big | g ( t _ k ) - g ( t _ { k - 1 } ) \\big | , \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} \\phi ( t _ 0 , x _ 0 ) = \\inf _ { u \\in \\mathcal { U } } J [ t _ 0 , x _ 0 ; u ] = J [ t _ 0 , x _ 0 ; \\hat { u } ] ; \\qquad { \\forall ( t _ 0 , x _ 0 ) \\in [ 0 , T ] \\times S } . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} & R i c _ { g ^ { ( 4 ) } } = 0 \\quad M , \\\\ & \\begin{cases} g _ { \\partial M } = \\gamma \\\\ H _ { \\partial M } = H \\\\ t r _ { \\partial M } K = k \\\\ \\omega _ { \\mathbf n _ { \\partial M } } = \\tau \\end{cases} \\quad \\partial M , \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} \\rho ( \\lambda ) = \\frac { m - 2 + [ ( m _ 1 - m _ 2 ) ^ 2 + 4 \\lambda ( \\sqrt { m _ 1 } + \\sqrt { m _ 2 } ) ^ 2 ] ^ { 1 / 2 } } { 2 ( m + \\lambda - 1 ) } . \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} \\begin{cases} h _ { \\partial M } = 0 \\\\ ( H _ { \\partial M } ) ' _ { h ^ { ( 4 ) } } = 0 \\\\ ( t r _ { \\partial M } K ) ' _ { h ^ { ( 4 ) } } + 2 G = 0 \\\\ ( \\omega _ { \\mathbf n } ) ' _ { h ^ { ( 4 ) } } + \\nabla _ { \\partial M } G = 0 . \\end{cases} \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} ( y _ 1 \\otimes x _ 1 ) ( y _ 2 \\otimes x _ 2 ) = \\sum _ { ( x _ 1 ) ( y _ 2 ) } ( x _ 1 ' , y _ 2 ' ) ( x _ 1 ''' , S ( y _ 2 ''' ) ) y _ 1 y _ 2 '' \\otimes x _ 1 '' x _ 2 . \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} \\phi _ * ( Z ) = \\phi _ * ( X ) + i \\phi _ * ( Y ) , \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} P + Q = P ^ { \\infty } + [ P ^ { d - 1 } + Q ^ { d - 1 } ] + \\cdots + [ P ^ { 1 } + Q ^ { 1 } ] + [ P ^ { 0 } + Q ^ { 0 } ] . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ d L _ k L _ k ^ * \\leq I . \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} \\partial _ { \\omega } F ^ { z \\bar { z } } _ { i j } + \\mathbf { i } \\ \\Gamma _ K ( ( \\Omega _ i - \\Omega _ j ) F ^ { z \\bar { z } } _ { i j } ) = \\Gamma _ K R ^ { z \\bar { z } } _ { i j } , \\Gamma _ K F ^ { z \\bar { z } } _ { i j } = F ^ { z \\bar { z } } _ { i j } ; \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ \\infty ( 4 k + 1 ) \\frac { ( \\frac 1 2 ) _ k ^ 6 } { k ! ^ 6 } = \\frac { 3 2 } { \\pi ^ 2 } L ( f , 1 ) \\end{gather*}"}
-{"id": "6990.png", "formula": "\\begin{align*} a _ + ( v ) ( \\psi ^ { ( n ) } ) & = ( a _ { n } ( v ) \\psi ^ { ( n + 1 ) } ) \\\\ a ^ { \\dagger } _ + ( v ) ( \\psi ^ { ( n ) } ) & = ( 0 , a ^ \\dagger _ 0 ( v ) \\psi ^ { ( 0 ) } , a _ 1 ^ \\dagger ( v ) \\psi ^ { ( 1 ) } , \\dots ) \\\\ \\varphi _ + ( v ) & = a _ + ( v ) + a _ + ^ \\dagger ( v ) \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\cdot \\wedge \\tau _ M } g ( s , W _ s ) \\mathrm d s = A ^ { W , W } _ { \\cdot \\wedge \\tau _ M } ( g ) . \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} A ( t ) = \\int _ { 0 } ^ { t } a ( s ) \\ , \\d s \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} \\alpha _ { Q } = Q \\left ( \\frac { 2 \\pi ^ { ( k + \\ell ) / 2 } \\Gamma ( ( Q - \\ell ) / 2 ) } { 4 ^ { \\ell } \\Gamma ( k / 2 ) \\Gamma ( Q / 2 ) } \\right ) ^ { Q ' - 1 } , \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} | B _ i | \\le \\frac { i } { d + i - k } r ! \\prod _ { j = 1 } ^ h p _ j . \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} \\Xi \\colon D _ G ^ b ( Y ) \\to D _ { G \\times \\mathbb { C } ^ { \\ast } } ( Y , \\widetilde { \\chi } , w = 0 ) \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} \\ker B _ + = H _ 0 \\subseteq \\ker B _ { m + 1 } . \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} \\mathcal T ( f ) = r _ ! ( f \\times d n ) , \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} \\Phi ^ + ( B _ \\ell ) & = \\{ \\epsilon _ i = \\sum _ { i \\le k \\le \\ell } \\alpha _ k \\ , ( 1 \\le i \\le \\ell ) , \\epsilon _ i - \\epsilon _ j = \\sum _ { i \\le k < j } \\alpha _ k \\ , ( 1 \\le i < j \\le \\ell ) , \\\\ & \\epsilon _ i + \\epsilon _ j = \\sum _ { i \\le k < j } \\alpha _ k + 2 \\sum _ { j \\le k \\le \\ell } \\alpha _ k \\ , ( 1 \\le i < j \\le \\ell ) \\} . \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} k = \\frac { | \\Gamma | } { | \\Gamma _ { q _ i ( b ) } | } , \\ , \\ , \\ , . \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} 1 \\longrightarrow \\mathbf { G } _ m \\longrightarrow E \\longrightarrow \\mathbf { G } _ a \\longrightarrow 1 \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} \\lvert ( i , j ) \\rvert _ { W _ n ( w ) } = \\lvert l _ i \\rvert _ S + \\lvert ( 0 , j ) \\rvert _ { W _ n ( w _ k ) } = \\lvert ( i , 0 ) \\rvert _ { W _ n ( w _ 0 ) } + \\lvert r _ j \\rvert _ S . \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} ( [ n ] ^ i _ q ) ^ { ( j ) } = \\sum _ { a _ 1 + \\ldots + a _ i = j } \\binom { j } { a _ 1 , \\ldots , a _ i } \\prod _ { 1 \\le k \\le i } [ n ] _ q ^ { ( a _ k ) } . \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} \\beta = r [ F ] + k [ l ] , \\ r , k \\in \\mathbb { Z } _ { \\ge 0 } . \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} \\mu \\prod _ { k = 1 } ^ { n } P _ k ( \\epsilon ) r = \\tilde { \\pi } r - \\epsilon \\tilde { \\pi } \\tilde { P } ^ { ( 1 ) } ( I - \\tilde { P } + \\tilde { \\Pi } ) ^ { - 2 } \\tilde { P } r + O ( \\epsilon ^ 2 m ^ 3 ) \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} A _ { j k } = \\Im \\langle v , D _ j D _ k Q \\rangle + \\Im \\langle D _ j Q , D _ k Q \\rangle \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } ( y \\alpha ( y , t ) ) + \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) - \\frac { \\partial } { \\partial t } \\alpha ( y , t ) = - \\frac { 1 } { 8 } \\frac { \\partial ^ { 3 } } { \\partial y ^ { 3 } } p _ { 0 } ( y , t ) \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} ( \\l + 1 ) ^ { 2 \\tau } = \\left ( 1 + \\frac { 1 } { \\l } \\right ) ^ { 2 \\tau } \\l ^ { 2 \\tau } \\leq e \\l ^ { 2 \\tau } , \\l \\geq m . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} K _ X = \\mathcal { O } _ X ( ( 2 4 - 1 - 1 - 8 - 1 2 ) e ) = \\mathcal { O } _ X ( 2 e ) . \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} w \\left ( z \\right ) \\to \\widetilde { w } \\left ( z \\right ) = w \\left ( z \\right ) K \\left ( z \\right ) , ~ ~ z \\in U \\subset { { \\mathbb { C } } ^ { n } } \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} \\int _ { G } f ( y ) \\rho ( y ) d y & = \\int _ { K \\backslash G / H } Q ( f ) ( \\ddot { y } ) d \\mu _ { \\rho } ( \\ddot { y } ) \\\\ & = \\int _ { K \\backslash G / H } \\int _ { K } \\int _ { H } f ( k ^ { - 1 } y h ) d h d k d \\mu _ \\rho ( \\ddot { y } ) . \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} e ^ { - t \\omega } F _ 2 \\psi = F _ 2 F _ 2 ^ * e ^ { - t \\omega } F _ 2 \\psi . \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} D _ { \\infty } ^ { \\beta } f ( t ) = \\int _ 0 ^ { \\infty } ( f ( t - r ) - f ( t ) ) \\ , \\frac { \\Gamma ( - \\beta ) ^ { - 1 } d r } { r ^ { 1 + \\beta } } , t \\in \\mathbb R . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} { \\mathrm d } X ^ { ( u ) } _ { t } \\ , = \\ , b ( t , X _ { t } ^ { ( u ) } , \\ , F ^ { ( u ) } _ { t } ) \\ , { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c } m _ + & 0 \\\\ 0 & m _ - \\end{array} \\right ) , \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} p _ { } = P ( L _ { n - 1 } \\ge 1 ) & = 1 - P ( L _ { n - 1 } = 0 ) \\\\ & = 1 - G _ { L _ { n - 1 } } ( 0 ) , \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} \\beta ^ { u _ 1 } \\alpha _ { i _ 1 } ^ { u _ 1 + t } \\in K t = 0 , 1 , \\dots , p _ { i _ 1 } - 2 . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} U _ { i + 1 , j + 3 } & = \\left \\{ g + 3 ( i + 1 ) + 5 , \\frac { 1 } { 2 } ( 2 g - 3 ( j + 3 ) + 6 ( i _ 1 + 1 ) + 1 0 ) , \\right . \\\\ & \\phantom { \\ ; \\ ; \\ ; \\ ; \\ ; } \\left . \\frac { 1 } { 2 } ( 2 g + 3 ( j + 3 ) - 1 2 ( i _ 1 + 1 ) - 2 0 ) \\right \\} _ { g } \\\\ & = \\left \\{ g + 3 i + 8 , \\frac { 1 } { 2 } ( 2 g - 3 j + 6 i + 7 ) , \\frac { 1 } { 2 } ( 2 g + 3 j - 1 2 i - 2 3 ) \\right \\} _ { g } . \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} { a p \\brack b p } _ q \\equiv \\binom { a } { b } + \\frac { \\binom { a } { b } b ( a - b ) p } { 2 } ( q ^ p - 1 ) + \\frac { \\binom { a } { b } b ( a - b ) } { 2 } \\left ( \\frac { b ( a - b ) } { 4 } p ^ 2 + \\frac { a p ^ 2 - 5 } { 1 2 } - \\frac { p - 1 } { 2 } \\right ) ( q ^ p - 1 ) ^ 2 \\bmod [ p ] _ q ^ 3 . \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} \\begin{array} { r c l } x + _ o y & : = & d ( x , o , y ) \\\\ x - _ o y & : = & d ( x , y , o ) x , y \\in A . \\end{array} \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} U = \\{ d w = 0 \\} , \\ w | _ { U ^ { \\rm { r e d } } } = 0 . \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} Q _ { V _ 0 } ( \\xi ) : = \\int _ \\Omega | \\nabla \\xi | ^ 2 + V _ 0 \\xi ^ 2 = \\int _ \\Omega | \\nabla \\xi - \\vec { { \\bf f } } \\xi | ^ 2 \\ ; \\ ; \\forall \\xi \\in C _ c ^ 1 ( \\Omega ) . \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{gather*} \\varphi _ v = \\mu ^ l \\big ( v ^ { - 1 } \\big ) ^ { - 1 } \\mu ^ r \\big ( g v ^ { - 1 } ? \\big ) . \\end{gather*}"}
-{"id": "8142.png", "formula": "\\begin{align*} \\begin{cases} ( \\delta ^ * _ { \\tilde g ^ { ( 4 ) } } Y _ 2 ) ^ T = \\delta ^ * _ { g _ S } Y _ 2 ^ T \\\\ \\delta ^ * _ { \\tilde g ^ { ( 4 ) } } Y _ 2 ( \\partial _ t , \\partial _ t ) = - u Y _ 2 ^ T ( u ) \\\\ [ \\delta ^ * _ { \\tilde g ^ { ( 4 ) } } Y _ 2 ( \\partial _ t ) ] ^ T = - \\frac { 1 } { 2 } u ^ 2 d \\theta ( Y _ 2 ^ T ) + \\frac { 1 } { 2 } u ^ 2 d ( \\frac { Y _ 2 ^ { \\perp } } { u } ) . \\end{cases} \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} n _ b ( W ^ n ( w ) ) = - \\infty \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} ( z _ S ) _ i = \\begin{cases} 0 , & i \\in S \\\\ 1 , & i \\notin S . \\end{cases} \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} | u _ { x _ i x _ j } | & = \\left | \\frac { 1 } { { \\rm V o l } ( B ( x , r ) ) } \\int _ { \\partial B ( x , r ) } u _ { z _ i } ( z ) ( \\nu ( z ) , e _ j ) d z \\right | \\\\ & \\le \\frac { C ( d ) } { { \\rm V o l } ( B ( x , r ) ) { \\rm d i s t } ( x , \\partial K ) } \\int _ { \\partial B ( x , r ) } u ( z ) | ( \\nu ( z ) , e _ j ) | d z \\le C ( d ) ^ 2 \\frac { u ( x ) } { { \\rm d i s t } ( x , \\partial K ) ^ 2 } \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 , x _ 6 ) & = ( x _ 1 , x _ 2 , - x _ 3 , - x _ 4 , - x _ 5 , - x _ 6 ) , \\\\ g ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 , x _ 6 ) & = f ( x _ 1 + \\frac 1 2 , x _ 2 , x _ 3 , x _ 4 , - x _ 5 , - x _ 6 ) . \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} C = \\left \\{ \\lambda _ 1 \\varpi _ 1 + \\lambda _ 2 \\varpi _ 2 + \\lambda _ 3 \\varpi _ 3 \\in P ^ + \\ \\middle \\vert \\ \\lambda _ 1 + 2 \\lambda _ 2 + \\lambda _ 3 < 3 \\right \\} = \\{ 0 , \\varpi _ 1 , \\varpi _ 2 , \\varpi _ 3 , 2 \\varpi _ 1 , 2 \\varpi _ 3 , \\varpi _ 1 + \\varpi _ 3 \\} . \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} 1 \\longrightarrow T \\longrightarrow \\R _ { k ' / k } ( T ' ) \\longrightarrow S \\longrightarrow 1 \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } e ^ { | k | ^ { 4 } s - \\alpha s | k | } \\ d s = \\frac { e ^ { | k | ^ { 4 } t - \\alpha t | k | } - 1 } { | k | ^ { 4 } - \\alpha | k | } . \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} J _ i ( \\pi ) = \\int _ { \\mathbb { K } ^ \\infty } \\sum _ { n = 1 } ^ \\infty \\beta ^ { n - 1 } r ^ + _ i ( s _ n , a _ n ) d P _ { \\mu } ^ \\pi + \\int _ { \\mathbb { K } ^ \\infty } \\sum _ { n = 1 } ^ \\infty \\beta ^ { n - 1 } r ^ - _ i ( s _ n , a _ n ) d P _ { \\mu } ^ \\pi . \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} \\frac { 2 N - 1 } { 2 N } \\ \\frac { 2 N - 3 } { 2 N - 2 } \\ , \\cdots \\ , \\frac { 1 } { 2 } = \\frac { ( 2 N ) ! } { 2 ^ { 2 N } ( N ! ) ^ 2 } . \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} h ( t ) \\sim \\begin{cases} \\frac { t } { ( \\alpha - 1 ) \\log t } & \\textrm { f o r ( P ) } , \\\\ [ 5 p t ] t ^ \\frac { 1 } { \\alpha + 1 } \\ell ( t ) & \\textrm { f o r ( W ) } , \\end{cases} \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} E ( \\psi ( f ) ^ 2 ) = ( f , - \\Delta _ \\mathbb { R } ^ { - 1 } f ) . \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} \\int _ { B ( x _ { 0 } , r ) } \\frac { 1 } { | x | ^ { \\beta } } & \\left ( \\exp ( \\alpha | f ( x ) | ^ { p ' } ) - \\sum _ { 0 \\leq k < p - 1 , \\ ; k \\in \\mathbb { N } } \\frac { 1 } { k ! } ( \\alpha | f ( x ) | ^ { p ' } ) ^ { k } \\right ) d x \\\\ & \\leq \\sum _ { p ' k \\geq p , \\ ; k \\in \\mathbb { N } } \\frac { ( \\alpha p ' k C _ { 4 } ^ { p ' } ) ^ { k } } { k ! } . \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} \\frac { \\rho _ 1 ( k n h ) } { \\rho _ 2 ( k n h ) } = \\frac { \\frac { \\Delta _ H ( h ) \\Delta _ K ( k ) } { \\Delta _ G ( h ) } \\cdot \\rho _ 1 ( n ) } { \\frac { \\Delta _ H ( h ) \\Delta _ K ( k ) } { \\Delta _ G ( h ) } \\cdot \\rho _ 2 ( n ) } = \\frac { \\rho _ 1 ( n ) } { \\rho _ 2 ( n ) } \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} \\frac { m ^ 2 t _ h ' } { t _ h ^ 2 } + ( h - t _ h ) ^ { 2 b } ( 1 - t _ h ' ) = t _ h ' \\left ( \\frac { m ^ 2 } { t _ h ^ 2 } - ( h - t _ h ) ^ { 2 b } \\right ) + h - t _ h = h - t _ h . \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} \\frac { u ( 1 , E _ j ) } { u ( 0 , E _ j ) } = \\tan \\theta _ j , \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} f ( z _ 1 ) = \\prod _ { i = 1 } ^ d t _ { i } . \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} \\Pr \\Big ( Y > k \\Big ) = \\Pr \\Big ( e ^ Y > e ^ k \\Big ) \\le \\exp \\left ( \\frac { n } { L ! } - k \\right ) < \\exp \\left ( - \\frac { n } { L ! } \\right ) < e ^ { - \\sqrt n } . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} ( M i A e ^ { i A t } f ) ( \\tau , y ) = \\partial _ { t } e ^ { i t \\tau } ( M f ) ( \\tau , y ) = i \\tau e ^ { i t \\tau } ( M f ) ( \\tau , y ) , \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} \\frac { Y ^ + _ n } { Y ^ - _ n } & = \\frac { F ( \\tau _ { n + 1 } ) } { F ( \\tau _ n ) } \\xrightarrow { n \\to \\infty } 1 \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} \\begin{cases} \\dot { y } = u ( y ) , \\\\ \\left . y \\right | _ { \\tau = 0 } = x , \\end{cases} \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\frac { h ( x ) } { | T _ { N } ' ( x ) | } = \\frac { 1 } { N \\pi \\sqrt { 1 - \\cos ^ { 2 } \\left ( N \\pi u _ { 0 } \\right ) } } = \\frac { 1 } { N \\pi \\sqrt { 1 - x '^ { 2 } } } \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X _ { s } ^ { i } = & b ( s , X _ { s } ^ { i } , Y _ { s } ^ { i } , Z _ { s } ^ { i } ) d s + \\sigma ( s , X _ { s } ^ { i } , Y _ { s } ^ { i } , Z _ { s } ^ { i } ) d B _ { s } , \\\\ d Y _ { s } ^ { i } = & - g ( s , X _ { s } ^ { i } , Y _ { s } ^ { i } , Z _ { s } ^ { i } ) d s + Z _ { s } ^ { i } d B _ { s } , \\\\ X _ { t } ^ { i } = & \\xi , \\ Y _ { t + \\delta } ^ { i } = \\phi _ { i } ( X _ { t + \\delta } ^ { i } ) , i = 1 , 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} & \\psi ^ { \\prime } ( x ) : = \\psi ^ { \\prime } ( \\lambda _ 1 ( x ) ) v _ 1 + \\psi ^ { \\prime } ( \\lambda _ 2 ( x ) ) v _ 2 + \\psi ^ { \\prime } ( \\lambda _ 4 ( x ) ) v _ 4 \\\\ & \\psi ^ { \\prime \\prime } ( x ) : = \\psi ^ { \\prime \\prime } ( \\lambda _ 1 ( x ) ) v _ 1 + \\psi ^ { \\prime \\prime } ( \\lambda _ 2 ( x ) ) v _ 2 + \\psi ^ { \\prime \\prime } ( \\lambda _ 4 ( x ) ) v _ 4 \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} I ( u _ n ) & \\geq \\int _ { \\mathbb R ^ 3 } \\left [ \\frac { 1 } { 4 } f ( u _ n ) u _ n - F ( u _ n ) \\right ] d x \\\\ & \\geq \\int _ { B _ r ( y _ n ) } \\left [ \\frac { 1 } { 4 } f ( u _ n ) u _ n - F ( u _ n ) \\right ] d x \\\\ & = \\int _ { B _ r } \\left [ \\frac { 1 } { 4 } f ( u _ n ( x + y _ n ) ) u _ n ( x + y _ n ) - F ( u _ n ( x + y _ n ) ) \\right ] d x \\\\ & \\geq \\int _ { \\Lambda } \\left [ \\frac { 1 } { 4 } f ( u _ n ( x + y _ n ) ) u _ n ( x + y _ n ) - F ( u _ n ( x + y _ n ) ) \\right ] d x , \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} B ( x ) = \\begin{pmatrix} 0 & \\nu _ 3 ( x ) & - \\nu _ 2 ( x ) & 0 & 0 & 0 \\\\ - \\nu _ 3 ( x ) & 0 & \\nu _ 1 ( x ) & 0 & 0 & 0 \\\\ \\nu _ 2 ( x ) & - \\nu _ 1 ( x ) & 0 & 0 & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} \\frac { \\Delta \\mu _ \\varphi } { \\mu _ { \\boldsymbol \\varphi } } ( \\boldsymbol \\varphi ( x ) ) = \\frac { \\Delta \\rho _ \\varphi } { \\rho _ { \\boldsymbol \\varphi } } ( x ) \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} L _ 0 [ \\psi ] : = - ( - \\Delta ) ^ s _ y \\psi + p U ( y ) ^ { p - 1 } \\psi = h ( y ) \\mathbb { R } ^ n , \\ , \\ , \\psi ( y ) \\to 0 | y | \\to \\infty . \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} W _ t f ( n ) = \\sum _ { m = 0 } ^ { \\infty } f ( m ) K _ t ( m , n ) \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} \\lim _ { s \\to \\frac { 3 } { 4 } } \\biggl ( P _ { 1 / 2 , 0 } ( z , s ) - \\frac { \\frac { 3 } { 4 \\pi } \\theta ( z ) } { s - 3 / 4 } \\biggr ) = 3 \\widehat { \\mathbf { Z } } _ + ( z ) = F _ { 1 / 2 , 0 , 0 } ( z ) \\in H _ { 1 / 2 } ^ { 3 / 2 } , \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\varphi = A _ 0 + A _ 1 x + \\dots + A _ { d / 2 } x ^ { d / 2 } - \\frac { y } { ( \\alpha _ 1 - x ) ( \\alpha _ 3 - x ) \\dots ( \\alpha _ { d - 1 } - x ) } . \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\mathbf { H } \\times \\boldsymbol { \\nu } + \\kappa \\cdot ( \\mathbf { E } \\times \\boldsymbol { \\nu } ) \\times \\boldsymbol { \\nu } = \\mathbf { 0 } ( 0 , \\infty ) \\times \\Gamma . \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} w _ { \\sigma } ^ 1 = s _ { \\rho _ { 1 , \\sigma } } , \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} E ( V ) = \\frac { E ( \\tilde { V } ) } { \\lambda ^ 2 } + \\frac { \\lambda ^ \\alpha - 1 } { \\alpha + 2 } \\| V \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} & ( \\Delta ' ) ^ { - 1 } [ d , a ] = L _ { 1 1 } a + L _ { 1 2 } c \\\\ & ( \\Delta ' ) ^ { - 1 } [ d , b ] = L _ { 1 1 } b + L _ { 1 2 } d \\\\ & ( \\Delta ' ) ^ { - 1 } [ a , c ] = L _ { 2 1 } a + L _ { 2 2 } c \\\\ & ( \\Delta ' ) ^ { - 1 } [ b , c ] = L _ { 2 1 } b + L _ { 2 2 } d . \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} \\bigg \\{ { \\bf x } = ( x _ { k , l } ) _ { 2 \\le k \\le r , 1 \\le l \\le N } & \\in \\widehat { \\Delta } _ { { \\bf i } , \\lambda _ 1 , \\ldots , \\lambda _ r } \\cap \\mathbb { Z } ^ { ( r - 1 ) N } ~ \\bigg | \\\\ & \\lambda _ 1 + \\cdots + \\lambda _ r - \\sum _ { 2 \\le k \\le r , 1 \\le l \\le N } x _ { k , l } \\alpha _ { i _ { k , l } } = \\nu \\bigg \\} , \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{gather*} \\operatorname { c h a r } \\big ( p ^ ! ( A ) \\big ) = p ^ * ( \\operatorname { c h a r } ( A ) ) , \\end{gather*}"}
-{"id": "847.png", "formula": "\\begin{align*} & 2 \\Re \\iint _ { \\mathbb R ^ 3 \\times \\mathbb R ^ 3 } | x - y | ^ { - 1 } f ( x , y ) \\nabla _ { x } | x - y | ^ { - 1 } \\nabla _ { x } \\overline { f ( x , y ) } { \\rm d } x { \\rm d } y \\\\ & = - \\iint _ { \\mathbb R ^ 3 \\times \\mathbb R ^ 3 } | \\nabla _ { x } | x - y | ^ { - 1 } | ^ 2 | f ( x , y ) | ^ 2 { \\rm d } x { \\rm d } y . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} w \\bullet ( \\lambda , \\mu ) = \\left ( w \\bullet \\frac { \\lambda + \\mu } { 2 } + \\frac { \\lambda - \\mu } { 2 } , w \\bullet \\frac { \\lambda + \\mu } { 2 } - \\frac { \\lambda - \\mu } { 2 } \\right ) , \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\frac { 1 } { R _ { e } } \\sum _ { T ' \\in S ( G ' ) \\atop e ' \\in T ' } \\Pi ( T ' ) = \\frac { \\Pi ( G ' ) } { R _ { e } } \\sum _ { T \\in S ( G ) \\atop e \\notin T } \\Pi ( T ) . \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} & \\Delta r = ( n - 1 ) S ( r ) , \\\\ & K _ { { \\rm r a d } } = K ( r ) . \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} \\frac { \\rho } { \\int _ M V e ^ { \\ , u } \\ , d \\mu } = \\lambda ^ 2 \\longrightarrow 0 , \\rho \\to \\infty . \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} \\delta ^ + _ { D \\times F } ( ( d , f ) ) = \\delta _ D ^ + ( d ) \\delta _ { F } ^ + ( f ) \\ ; \\ ; { \\rm a n d } \\ ; \\ ; \\delta ^ - _ { D \\times F } ( ( d , f ) ) = \\delta _ D ^ - ( d ) \\delta _ { F } ^ - ( f ) . \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} \\begin{cases} \\dfrac { \\partial u ( n , t ) } { \\partial t } = \\mathcal { J } u ( n , t ) , \\\\ [ 4 p t ] u ( n , 0 ) = f ( n ) . \\end{cases} \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} f ( p ^ 1 , \\dots , p ^ m ) = ( f ^ 1 ( p ^ 1 ) , \\dots , f ^ m ( p ^ m ) ) \\qquad \\forall p \\in G . \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta u + u + \\rho ( x ) \\phi u = | u | ^ { p - 1 } u , & x \\in \\R ^ 3 , \\\\ \\ , \\ , \\ , - \\Delta \\phi = \\rho ( x ) u ^ 2 , \\ & x \\in \\R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} n \\leq \\frac { q } { r } - ( 1 + 2 \\beta ) & \\Longleftrightarrow q - ( r n + r + \\beta r ) \\geq \\beta r \\\\ & \\Longrightarrow ( q - ( r n + r + \\beta r ) ) ( q - 1 ) - \\beta r \\geq 0 \\\\ & \\Longleftrightarrow q ^ 2 - q ( r n + r + \\beta r + 1 ) + r ( n + 1 ) \\geq 0 \\\\ & \\Longleftrightarrow - 1 + \\frac { n + 1 } { q } + \\frac { q - 1 } { r } \\geq \\beta + n , \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} W ^ { n } _ { 0 , 1 , \\ldots , \\frac { \\l } { 2 } - 1 } = \\begin{cases} W ^ { n } _ { 0 , 1 } \\circ W ^ n _ { 2 , 3 } \\circ \\dots \\circ W ^ n _ { \\frac { \\l } { 2 } - 2 , \\frac { \\l } { 2 } - 1 } \\mbox { i f } \\l \\mbox { i s e v e n } \\\\ W ^ n _ { 0 , 1 } \\circ W ^ n _ { 2 , 3 } \\circ \\dots \\circ W ^ n _ { \\frac { \\l } { 2 } - 5 , \\frac { \\l } { 2 } - 4 } \\circ W ^ n _ { \\frac { \\l } { 2 } - 3 , \\frac { \\l } { 2 } - 2 , \\frac { \\l } { 2 } - 1 } \\mbox { i f } \\l \\mbox { i s o d d } \\\\ \\end{cases} \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} Q _ { 1 2 } = 1 , Q _ { 3 2 } = 2 , \\end{align*}"}
-{"id": "9947.png", "formula": "\\begin{align*} \\kappa ^ { n - 2 } _ { 0 1 } = - ( n - 2 ) ! \\ , \\check { \\chi } _ { H _ { n } \\cup H _ - } \\ , \\displaystyle \\sum _ { j = 1 } ^ { n - 1 } ( - 1 ) ^ { j } \\varphi _ { j } \\ , \\bar { \\partial } \\varphi _ 1 \\wedge \\dots \\wedge \\widehat { \\bar { \\partial } \\varphi _ j } \\wedge \\cdots \\wedge \\bar { \\partial } \\varphi _ { n - 1 } . \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { m i n i m i z e } & \\sum _ { t = 0 } ^ { N - 1 } \\| u ( t ) \\| _ 1 \\\\ \\mbox { s u b j e c t t o } & x ( t + 1 ) = A x ( t ) + B u ( t ) , t = 0 , \\ldots , N - 1 \\\\ & 0 . 2 / T ^ \\leq \\| u ( t ) \\| \\leq 1 / T ^ , t = 0 , \\ldots , N \\\\ & x ( 0 ) = ( 0 , 0 ) , x ( N ) = ( 1 , 0 ) . \\end{array} \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} \\lim _ { t \\uparrow T } \\| \\nabla u ( t ) \\| _ { L ^ 2 } = \\infty . \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} \\tilde { \\phi } _ \\tau = - ( - \\Delta ) ^ s \\tilde { \\phi } \\quad \\mathbb { R } ^ n \\setminus \\{ \\hat { e } \\} \\times ( - \\infty , 0 ] \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} \\sum _ { j \\in N ( i ) \\cap \\widetilde { \\mathcal { I } } } \\widetilde Q _ { i j } & = \\widetilde { Q } _ { i k } + \\widetilde { Q } _ { i \\ell } = Q _ { i k } + Q _ { i \\ell } , \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( \\ker \\beta ) = V + \\ker B . \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} 4 \\omega ( T _ { \\theta } ) & < \\max \\left \\{ u ^ { \\theta } ( 0 , t ) - v _ { \\theta } ( 0 , t ) - ( 2 D T _ { \\theta } + \\delta ) t \\ , \\mid \\ , t \\in [ T _ { \\theta } , T ] \\right \\} \\\\ & = u ^ { \\theta } ( 0 , T _ { \\theta } ) - v _ { \\theta } ( 0 , T _ { \\theta } ) - ( 2 D T _ { \\theta } + \\delta ) T _ { \\theta } \\\\ & < 4 \\omega ( T _ { \\theta } ) , \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} \\| \\tilde { B } ^ { l } \\| = \\limsup _ { \\epsilon \\downarrow 0 } \\| B _ 1 ( \\epsilon ) \\cdots B _ l ( \\epsilon ) \\| \\le \\beta , \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} a _ * ( x ) & : = a ( x , u _ * , p _ * ) \\\\ b _ * ( x ) & : = a _ u ( x , u _ * , p _ * ) . ( u _ * ) _ { x x } + D _ u f ( x , u _ * , p _ * ) \\\\ c _ * ( x ) & : = a _ p ( x , u _ * , p _ * ) . ( u _ * ) _ { x x } + D _ p f ( x , u _ * , p _ * ) . \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} \\mathcal G _ { \\mathcal W } f ( x ) & = f ( p x ) - f ( x ) + w ( x ) f ' ( x ) , \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} \\epsilon : = | | X _ P | | _ { s , r , p - 1 , \\mathbf { a } ; \\mathcal { O } _ r } + \\frac { \\alpha } { M } | | X _ P | | ^ { l i p } _ { s , r , p - 1 , \\mathbf { a } ; \\mathcal { O } _ r } = O ( r ^ { \\frac 7 4 } ) \\leq ( \\alpha \\gamma ) ^ { 1 + \\beta } , \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} \\Q _ { \\lambda } ( x ) = \\frac { x _ 1 ^ 2 } { a _ 1 - \\lambda } + \\dots + \\frac { x _ d ^ 2 } { a _ d - \\lambda } = 1 . \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} F ( X _ i ) = \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma _ i } e ^ { - \\frac { ( X _ i - \\mu _ i ) ^ 2 } { 2 \\sigma _ i ^ 2 } } . \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} \\mathbb { V } = \\{ ( V _ i , u _ e ) : \\ i \\in V ( Q ) , \\ e \\in E ( Q ) , \\ u _ e \\colon V _ { s ( e ) } \\to V _ { t ( e ) } \\} \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} \\partial _ t C _ t = C _ t r _ t \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} f | _ k W _ Q ( \\tau ) = Q ^ { k / 2 } \\left ( f | _ k h _ Q \\right ) ( Q \\tau ) . \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} \\delta _ { \\zeta } ( \\mu ) = \\left ( \\mu ^ { M / 2 m _ 1 } \\zeta _ 1 , \\cdots , \\mu ^ { M / 2 m _ n } \\zeta _ n \\right ) , \\ ; \\mu \\in \\C . \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} \\Pi _ { k \\ , \\boldsymbol { \\nu } } ( x , x ) _ j = \\Pi _ { k \\ , \\boldsymbol { \\nu } } ( x , x ) _ { j 1 } + \\Pi _ { k \\ , \\boldsymbol { \\nu } } ( x , x ) _ { j 2 } , \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } u + H _ { i } ( x , u _ { x _ { i } } ) = 0 & \\ , \\ , I _ { i } \\\\ \\sum _ { i = 1 } ^ { K } u _ { x _ { i } } = B & \\ , \\ , \\{ 0 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} d \\bar { x } _ i ( s ) = - ( \\Pi ^ T a ) _ { i j } \\frac { \\partial V } { \\partial x _ j } \\ , d s + \\frac { 1 } { \\beta } \\frac { \\partial ( \\Pi ^ T a ) _ { i j } } { \\partial x _ j } \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } ( \\Pi ^ T \\sigma ) _ { i j } \\ , d w _ j ( s ) \\ , , 1 \\le i \\le n \\ , . \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} & \\ \\bigcup _ { n \\geq 0 } \\tilde { f } _ { i } ^ { n } ( \\mathcal { B } _ { v } ( \\lambda ) \\otimes \\mathcal { B } _ { w } ( \\mu ) ) \\setminus \\{ 0 \\} \\\\ = & \\ ( \\mathcal { B } _ { s _ i v } ( \\lambda ) \\otimes \\mathcal { B } _ { w } ( \\mu ) ) \\sqcup ( S _ { i } ( \\tilde { e } _ i ^ { \\max } ( \\mathcal { B } _ v ( \\lambda ) ) ) \\otimes ( \\mathcal { B } _ { s _ i w } ( \\mu ) \\setminus \\mathcal { B } _ { w } ( \\mu ) ) ) . \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | \\sum _ { j = 1 } ^ k \\left \\{ \\big [ - ( - \\Delta ) ^ { \\frac { s } { 2 } } \\eta _ { j , R } , - ( - \\Delta ) ^ { \\frac { s } { 2 } } \\tilde { \\phi } _ j \\big ] + \\tilde { \\phi } _ j \\big ( - ( - \\Delta ) ^ { s } - \\partial _ t \\big ) \\eta _ { j , R } \\right \\} \\right | \\\\ & \\quad \\quad \\lesssim \\frac { 1 } { R ^ { a - 2 s } } \\| \\phi \\| _ { n - 2 s + \\sigma , a } \\sum _ { j = 1 } ^ k \\frac { \\mu _ j ^ { - 2 s } \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } + \\sigma } ( t ) } { 1 + | y _ j | ^ { a } } . \\end{aligned} \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} F _ 2 = F _ 2 ^ 1 + F _ 2 ^ 2 , \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} \\phi ' ( r ) = R \\left ( \\exp _ { q } \\left [ \\lambda \\log _ { q } \\frac { r } { R } \\right ] \\right ) ^ { q } \\lambda \\left ( \\frac { r } { R } \\right ) ^ { - q } \\frac { 1 } { R } = \\lambda \\left ( \\frac { | z | } { | x | } \\right ) ^ { q } . \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} Y _ t = \\xi + \\int _ t ^ T Z _ r b ( r , W _ r ) \\mathrm d r + \\int _ t ^ T f ( r , W _ r , Y _ r , Z _ r ) \\mathrm d r - \\int _ t ^ T Z _ r \\mathrm d W _ r . \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} d _ { C E } ( \\alpha _ { 1 } \\wedge \\dots \\wedge \\alpha _ { k } ) = \\sum _ { i < j } \\pm \\langle \\alpha _ { i } , \\alpha _ { j } \\rangle \\cdot \\alpha _ { 1 } \\wedge \\dots \\widehat { \\alpha } _ { i } \\dots \\widehat { \\alpha } _ { j } \\dots \\wedge \\alpha _ { k } . \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} J ( u ) : = \\dfrac { 1 } { 2 } \\left \\| u \\right \\| ^ 2 - \\int _ { \\mathbb { R } ^ 2 } A ( x ) F ( u ) , \\ \\ u \\in H . \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} \\ , { \\ , m _ { s } ( y _ { 1 } ) \\ , } \\widehat { \\mathrm M } _ { s } ( { \\mathrm d } y _ { 1 } { \\mathrm d } y _ { 2 } ) \\ , = \\ , { \\ , m _ { s } ( y _ { 2 } ) \\ , } \\mathrm M _ { s } ( { \\mathrm d } y _ { 1 } { \\mathrm d } y _ { 2 } ) \\ , ; 0 \\le s \\le T \\ , , \\ , \\ , ( y _ { 1 } , y _ { 2 } ) \\in \\mathbb R ^ { 2 } \\ , . \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} J _ { n , 1 } = \\begin{pmatrix} - I _ n & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{gather*} { } _ 2 F _ 1 \\biggl ( \\begin{matrix} r , \\ , 1 - r \\\\ 1 \\end{matrix} \\biggm | z \\biggr ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( r ) _ k ( 1 - r ) _ k } { k ! ^ 2 } z ^ k , r \\in \\big \\{ \\tfrac 1 3 , \\tfrac 1 4 , \\tfrac 1 6 \\big \\} . \\end{gather*}"}
-{"id": "139.png", "formula": "\\begin{align*} e ^ { i t \\sqrt { \\mathcal { L } _ V } } u _ 0 & = \\sum _ { \\nu \\in \\chi _ \\infty } \\sum _ { \\ell = 1 } ^ { d ( \\nu ) } \\varphi _ { \\nu , \\ell } ( y ) \\int _ 0 ^ \\infty ( r \\rho ) ^ { - \\frac { n - 2 } 2 } J _ { \\nu } ( r \\rho ) e ^ { i t \\rho } b _ { \\nu , \\ell } ( \\rho ) \\rho ^ { n - 1 } d \\rho \\\\ & = \\sum _ { \\nu \\in \\chi _ \\infty } \\sum _ { \\ell = 1 } ^ { d ( \\nu ) } \\varphi _ { \\nu , \\ell } ( y ) \\mathcal { H } _ { \\nu } \\big [ e ^ { i t \\rho } b _ { \\nu , \\ell } ( \\rho ) \\big ] ( r ) . \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} D ( t ) = \\inf \\left \\{ d \\ge { 0 } : A ( t - d ) \\le A ^ \\ast ( t ) \\right \\} , \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } \\big [ P ( T ( t ) \\ge i ) \\big ] ^ { 1 / k } & = \\sum _ { i = 1 } ^ { \\infty } \\big [ i ^ { \\ , p - 1 } P ( T ( t ) \\ge i ) \\big ] ^ { 1 / k } { \\strut i } ^ { \\ , - ( p - 1 ) / k } \\\\ & \\le \\bigg [ \\sum _ { i = 1 } ^ { \\infty } i ^ { \\ , p - 1 } P ( T ( t ) \\ge i ) \\bigg ] ^ { \\tfrac { 1 } { k } } \\ \\bigg [ \\sum _ { i = 1 } ^ { \\infty } { \\strut } i ^ { \\ , - \\tfrac { p - 1 } { k - 1 } } \\bigg ] ^ { \\tfrac { k - 1 } { k } } \\\\ & \\le \\big ( { \\| T ( t ) \\| } _ p \\big ) ^ { p / k } . \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} ( q ^ { a _ i - a _ j + n _ i - n _ j } ; q ) _ { n _ j + 1 } ( q ^ { a _ i - a _ j - n _ j } ; q ) _ { n _ i } = ( q ^ { a _ i - a _ j - n _ j } ; q ) _ { n _ i + n _ j } = ( q ^ { a _ i - a _ j - n _ j } ; q ) _ { n _ j + 1 } ( q ^ { a _ i - a _ j + 1 } ; q ) _ { n _ i } . \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} A _ t ^ { W , W } ( l _ n ) & = \\int _ 0 ^ t l _ n ( r , W _ r ) \\mathrm d r \\\\ & = \\int _ 0 ^ t \\left ( \\partial _ t \\phi _ n ( r , W _ r ) + \\frac 1 2 \\Delta \\phi _ n ( r , W _ r ) \\right ) \\mathrm d r \\\\ & = \\phi _ n ( t , W _ t ) - \\phi _ n ( 0 , W _ 0 ) - \\int _ 0 ^ t \\nabla \\phi _ n ^ * ( r , W _ r ) \\mathrm d W _ r , \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} \\theta ( r ) : = \\int _ 0 ^ r \\vartheta ( s ) d s . \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} \\Lambda = \\sum _ { 1 \\leq b \\leq n } \\sigma _ { j _ b } j _ b ^ 2 ( \\zeta _ b + y _ b ) + \\sum _ { j \\in \\mathbb { Z } _ * } \\sigma _ j j ^ 2 | z _ j | ^ 2 , \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} e ^ { - 2 \\pi i ( \\mu + \\rho , \\lambda + \\rho ) } = e ^ { - 2 \\pi i ( w ( \\mu + \\rho ) , \\lambda + \\rho ) } \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} u _ { t } + f \\left ( t , x , u \\right ) _ { x } = 0 . \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} [ f ; G _ { \\Lambda , \\epsilon } ( t ) f ] = \\sup \\nolimits _ \\varphi ( 2 [ f ; \\varphi ] - [ \\varphi ; D _ { \\Lambda , \\epsilon } ( t ) \\varphi ] ) . \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} \\mathbb { E } _ x : = \\{ ( v _ 1 , \\ldots , v _ s , e ) \\in \\mathbb { R } ^ s \\times \\mathbb { E } ' \\ , \\ , | \\ , \\ , v _ i = 0 x _ i = 0 \\} \\subset \\mathbb { E } \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{gather*} v = \\sum _ { s = 0 } ^ p v _ { \\mathcal { X } ^ + ( s ) } e _ s + \\hat q \\sum _ { s = 1 } ^ { p - 1 } v _ { \\mathcal { X } ^ + ( s ) } \\left ( \\frac { p - s } { [ s ] } w ^ + _ s - \\frac { s } { [ s ] } w ^ - _ s \\right ) \\end{gather*}"}
-{"id": "8581.png", "formula": "\\begin{align*} ( g \\otimes g ' ) \\circ ( f \\otimes f ' ) = ( - 1 ) ^ { | g ' | | f | } ( g \\circ f ) \\otimes ( g ' \\circ f ' ) . \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} | F ^ n | = ( 2 n + 1 ) ^ d \\ \\ \\ ( n = 0 , 1 , 2 , \\ldots ) . \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} s _ { \\mu , \\nu } = \\rho _ \\infty ( a _ { \\mu , \\nu } ) + \\sum _ { j = 1 } ^ { r _ { \\mu , \\nu } } t _ \\infty ( x ^ { ( \\mu , \\nu ) } _ { j , 1 } ) \\cdots t _ \\infty ( x ^ { ( \\mu , \\nu ) } _ { j , N _ j } ) . \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 0 } e ^ { - t ( 1 - x ) } ( 1 - x ) ^ { - 1 / 2 } \\ , d x & = \\int _ { 0 } ^ { 1 } e ^ { - t ( 1 + x ) } ( 1 + x ) ^ { - 1 / 2 } \\ , d x \\\\ & \\le \\int _ { 0 } ^ { 1 } e ^ { - t ( 1 - x ) } ( 1 - x ) ^ { - 1 / 2 } \\ , d x , \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} \\max \\limits _ { x \\in B _ R } \\max _ { i , j } ( U ^ { i j } - V ^ { i j } ) ( x ) = ( U ^ { \\bar { i } \\bar { j } } - V ^ { \\bar { i } \\bar { j } } ) ( \\bar { x } ) = \\eta > 0 . \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} p ( y ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( 1 + \\tau \\left ( - y ^ { 4 } + \\frac { 7 y ^ { 2 } } { 2 } - \\frac { 1 1 } { 1 6 } \\right ) \\right ) e ^ { - 2 y ^ { 2 } } + O ( \\tau ^ { 3 / 2 } ) \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} & ~ f \\circ \\delta _ \\alpha g - \\delta _ \\alpha ( f \\circ g ) + ( - 1 ) ^ { n - 1 } \\delta _ \\alpha f \\circ g \\\\ = & ~ ( - 1 ) ^ { n - 1 } f \\circ ( \\mu \\circ g ) - f \\circ ( g \\circ \\mu ) - ( - 1 ) ^ { m + n - 2 } \\mu \\circ ( f \\circ g ) + ( f \\circ g ) \\circ \\mu \\\\ & ~ + ( - 1 ) ^ { n - 1 } ( - 1 ) ^ { m - 1 } ( \\mu \\circ f ) \\circ g - ( - 1 ) ^ { n - 1 } ( f \\circ \\mu ) \\circ g . \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} \\lvert ( i _ 1 , j _ 1 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i _ 2 , j _ 2 ) \\rvert _ { W _ n ( w ) } = i _ 1 - i _ 2 + j _ 1 - j _ 2 . \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} \\psi _ { \\alpha } ^ { 0 } = \\bold { 1 } ( { t _ { 0 } > u _ { \\alpha } } ) . \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} \\overline { V } ^ { \\ ; i j } ( x ) = \\inf \\limits _ { \\beta \\in { \\Delta ^ j } } \\sup \\limits _ { \\delta \\in \\mathcal { A } ^ i } J ( x , \\delta , \\beta ( \\delta ) ) \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} Q _ k ^ l = \\frac { P _ l } { P _ k P _ { l - k } } , \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} \\mu _ { i r } ^ { r } = 0 \\ , , i = 1 , \\dots , n \\ , . \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} ( i ^ { \\mu _ 1 } ) ^ * \\omega = ( \\pi _ 1 ^ { \\mu _ 1 } ) ^ * \\omega _ { M _ { \\mu _ 1 } } . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { ( - 1 ) ^ { n s + j + s } \\binom k j \\binom { k - j } s \\frac { { G _ 0 ^ j G _ { n + r } ^ s } } { { G _ r ^ { j + s } } } H _ { m - ( n + r ) k + r j + n s } } } = ( - 1 ) ^ { ( n + 1 ) k } \\left ( { \\frac { { G _ n } } { { G _ r } } } \\right ) ^ k H _ m , G _ r \\ne 0 \\ , , \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} \\partial _ t g _ t = - { \\rm M } ( g _ t ) \\ , , g _ { | t = 0 } = g _ 0 \\ , , \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ { g ^ { ( 4 ) } } \\delta _ { g ^ { ( 4 ) } } ^ * Y = 0 \\quad M \\\\ Y = 0 \\quad \\quad \\partial M \\end{cases} \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} \\mathbf { B } ( x ) = \\left ( - \\operatorname { s i g n } ( x _ 2 ) \\left [ \\frac { x _ 1 } { | x _ 2 | ^ 2 } \\mathbf { 1 } _ { | x _ 1 | \\leq | x _ 2 | } + \\mathbf { 1 } _ { | x _ 1 | > | x _ 2 | } \\right ] , - \\left [ \\frac { 1 } { | x _ 2 | } \\mathbf { 1 } _ { | x _ 1 | \\leq | x _ 2 | } + \\mathbf { 1 } _ { | x _ 1 | > | x _ 2 | } \\right ] \\right ) \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} \\frac { a ( \\beta ) } { a ( \\alpha ) } \\cdot \\frac { b ( \\beta ) } { b ( \\alpha ) } = ( 1 + A ) ( 1 - B ) = 1 + B \\big ( A ( \\frac { 1 } { B } - 1 ) - 1 \\big ) . \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} \\sqcap _ p ^ { \\epsilon , i } [ n ] ( \\square ^ m ) = \\{ f \\in \\hom _ { \\square _ p } ( \\square ^ m , \\square ^ n ) | f = \\delta _ { n - 1 } ^ { \\epsilon ' , i ' } g ( \\epsilon ' , i ' ) \\neq ( \\epsilon , i ) \\} \\ . \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} B _ { \\overline { \\nu } , 0 } ( t , \\overline \\alpha ) e ^ { \\alpha _ j t } - B _ { \\overline { \\nu } , j } ( t , \\overline \\alpha ) = S _ { \\overline { \\nu } , j } ( t , \\overline \\alpha ) , j = 1 , \\ldots , m , \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} ( \\tau _ 1 + \\tau _ 2 + \\tau _ 3 ) ^ 2 = 0 \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} \\binom { y } { j } = \\binom { y _ m } { j } \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} v ( w ) = \\int _ { \\partial S _ { \\varepsilon } ^ + } e ^ { z w } u _ { \\varepsilon ' } ( z ) d z . \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} X ^ j _ { t _ { i + 1 } } ( \\uparrow \\downarrow ) = X _ { t _ { i } } ^ j + h \\ B ( t _ i , X ^ j _ { t _ i } , Y ^ j _ { t _ i } , Z ^ j _ { t _ i } , [ X ^ j _ { t _ i } , Y ^ j _ { t _ i } , Z ^ j _ { t _ i } ] ) \\pm \\sigma \\sqrt { h } . \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} [ \\Theta _ L ( \\eta ) * \\iota _ \\epsilon ] ( u ) = ( 2 \\epsilon L ) ^ { - d } \\sum _ { i \\in \\mathbb T _ L ^ d } \\mathbf 1 _ { [ \\frac { 2 i - 1 } { 2 L } , \\frac { 2 i + 1 } { 2 L } ) ^ d } ( u ) \\sum _ { j : | i - j | \\le \\lfloor \\epsilon L \\rfloor } \\eta ( j ) , \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} P ( 0 , 0 ) & = 2 , \\\\ P ( 0 , 1 ) & = \\dfrac { x ^ { 2 } } { y } - \\dfrac { y } { x ^ { 2 } } = \\dfrac { 4 q } { K } , \\\\ P ( 1 , 0 ) & = x y ^ { 2 } - \\dfrac { q ^ { 2 } } { x y ^ { 2 } } = K , \\\\ P ( 1 , - 1 ) & = \\dfrac { y ^ { 3 } } { x } + \\dfrac { q ^ { 2 } x } { y ^ { 3 } } = K - 2 q + \\dfrac { 4 q ^ { 2 } } { K } . \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\log r - \\frac { 1 } { 4 } r = - \\frac { 1 } { 4 } . \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} \\Delta K ( \\varphi , p ) = - \\int \\left [ \\frac { \\partial p } { \\partial y ^ \\alpha } ( \\varphi ( x ) ) \\right ] \\bigl [ \\Delta \\varphi ^ \\alpha ( x ) \\bigr ] \\ , \\bigl [ \\rho ( x ) \\bigr ] \\ , d x \\ , . \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} \\phi _ n ^ { \\ell } ( u ) = N ( \\rho , \\alpha ) \\sqrt { \\frac { n ! } { \\Gamma ( n + \\ell + 1 ) } } u ^ { \\frac { \\ell } { 2 } } e ^ { - \\frac { \\varpi } { 2 } u } L _ n ^ { \\ell } ( u ) , \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} \\mathcal { V } ( x ) = \\{ y \\in \\R ^ n : \\| y - x \\| \\le \\| y - x ' \\| , \\forall x ' \\in \\Lambda \\} . \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} P = \\sum _ { j = 1 } ^ N X _ j ^ * f _ j X _ j + X _ { N + 1 } + a _ 0 , \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} \\tilde { f } _ { i _ 1 } ^ { a _ 1 } \\cdots \\tilde { f } _ { i _ l } ^ { a _ l } \\left ( b _ \\lambda \\otimes b \\right ) = \\tilde { f } _ { i _ 1 } ^ { c _ 1 } \\cdots \\tilde { f } _ { i _ l } ^ { c _ l } ( b _ \\lambda ) \\otimes \\tilde { f } _ { i _ 1 } ^ { d _ 1 } \\cdots \\tilde { f } _ { i _ l } ^ { d _ l } ( b ) , \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} J _ 2 & = - 3 \\frac { \\chi _ { 1 2 } } { \\chi _ { 1 0 } } , \\\\ J _ 4 & = \\frac 1 { 2 4 } ( J _ 2 ^ 2 - \\psi _ 4 ) , \\\\ J _ 6 & = \\frac 1 { 2 1 6 } ( - J _ 2 ^ 3 + 3 6 J _ 2 J _ 4 + \\psi _ 6 ) , \\\\ J _ { 1 0 } & = - 4 \\chi _ { 1 0 } . \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} C _ { w } = C ( E _ 1 , E _ 2 , \\cdots , E _ { N ( w ) } ) . \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} | K ( r , r ' , y , y ' ) | \\lesssim \\begin{cases} r ^ { - \\alpha } r '^ { - \\beta } , & r \\leq r ' \\\\ 0 , & r > r ' , \\end{cases} \\end{align*}"}
-{"id": "9663.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - \\alpha } \\sum _ { j = 2 } ^ { \\infty } \\| F _ { j } \\| _ { \\mathcal { B } _ { \\alpha } ^ { 0 } } \\leq r _ { 1 } . \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{gather*} J ' _ s ( t , x ) = \\begin{cases} J ( t , x ) & s < - R \\\\ J ^ { + \\frac { 1 } { k } } ( t , x ) = J ( t + \\frac { 1 } { k } , x ) & s > R \\end{cases} \\end{gather*}"}
-{"id": "7333.png", "formula": "\\begin{align*} Q ( L _ n f ) ( K x H ) & = \\int _ K \\int _ H f ( n k ^ { - 1 } x h ) d h d k \\\\ & = \\int _ H \\int _ K f ( n k ^ { - 1 } n ^ { - 1 } n x h ) d k d h \\\\ & = \\int _ H \\int _ K f ( k ^ { - 1 } n x h ) d k d h \\\\ & = L _ n Q ( f ) ( K x H ) . \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} d \\mu _ z = \\frac { 1 } { Z } e ^ { - \\beta V _ 0 } \\Big [ \\mbox { d e t } ( \\nabla \\xi \\nabla \\xi ^ T ) \\Big ] ^ { - \\frac { 1 } { 2 } } d \\bar { \\nu } \\ , , \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} \\Upsilon ( n ) = \\frac { 1 } { n } \\sqrt { \\left ( \\frac { 2 } { n } \\right ) } ~ \\Phi \\left ( \\frac { 1 } { n } \\right ) + \\Phi ( n ) , \\end{align*}"}
-{"id": "9780.png", "formula": "\\begin{align*} n _ 1 = \\sqrt { N ( I + 2 z J ) } = \\sqrt { 1 + 4 z ^ 2 } \\neq 0 \\ ; \\ ; z \\neq \\pm \\frac { i } { 2 } \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} B = \\rm { d i a g } \\Big ( \\begin{bmatrix} 0 & b _ 1 \\\\ c _ 1 & 0 \\end{bmatrix} , \\dots , \\begin{bmatrix} 0 & b _ p \\\\ c _ p & 0 \\end{bmatrix} , \\underbrace { 0 , \\dots , 0 } _ m , b _ { p + 1 } , \\dots , b _ { p + r } \\Big ) . \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} \\| v \\| _ { X } & : = \\| v \\| _ { L _ t ^ \\infty H _ x ^ 1 \\cap L _ t ^ 4 L _ x ^ \\infty } + \\| \\langle x \\rangle ^ { - \\frac 3 2 } v \\| _ { L _ x ^ \\infty L _ t ^ 2 } + \\| \\partial _ x v \\| _ { L _ x ^ \\infty L _ t ^ 2 } , \\\\ \\| z \\| _ { Y } & : = \\| \\dot z + i E z \\| _ { L _ t ^ 1 \\cap L _ t ^ 2 } . \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} I _ { p _ 1 , q _ 1 } \\otimes u _ { h _ 1 } \\oplus \\cdots \\oplus I _ { p _ m , q _ m } \\otimes u _ { h _ m } \\oplus \\begin{bmatrix} 0 & I _ { k _ { m + 1 } } \\\\ I _ { k _ { m + 1 } } & 0 \\end{bmatrix} \\otimes 1 \\oplus \\cdots \\oplus \\begin{bmatrix} 0 & I _ { k _ l } \\\\ I _ { k _ l } & 0 \\end{bmatrix} \\otimes 1 \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} \\theta ^ \\prime ( x , E , a , b , \\theta _ 0 ) = \\gamma ^ \\prime ( x , E ) + \\frac { C } { \\gamma ^ \\prime ( x , E ) ( 1 + x - b ) } \\sin 2 \\theta \\sin ^ 2 \\theta , \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} \\theta _ { s , c } = \\begin{cases} - \\frac { l ^ { \\frac { n _ l } { 2 } - 2 c } } { a b } & \\frac { A M } { B Z ^ 2 l ^ { \\frac { n _ l } { 2 } } } \\geq 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} \\deg ( D | _ { S \\backslash B } ) = \\deg ( D | _ S ) - D ( p ) \\ , . \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} { } & ( \\mathrm { T 1 } ) & & T ^ * = T , & & T ^ 2 = Q \\ , T , Q > 0 , & & \\\\ { } & ( \\mathrm { T 2 } ) & & T _ { 1 2 } \\ , T _ { 2 3 } \\ , T _ { 1 2 } \\ , = T _ { 1 2 } , & & T _ { 2 3 } \\ , T _ { 1 2 } \\ , T _ { 2 3 } \\ , = T _ { 2 3 } , & & \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} f ( a _ 3 ) = ( 1 - t ) \\cdot f ( a _ 1 ) + t \\cdot f ( a _ 2 ) . \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} q ( x ) = \\frac { ( n - 1 ) ^ 2 } { 4 } K _ 0 + C ( E , A , K _ 0 ) \\frac { \\sin ( 2 \\sqrt { \\lambda } x + \\phi + \\phi ^ \\prime ) } { x } + V _ 1 ( x ) , \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} S : = L - \\mathrm { I d } \\end{align*}"}
-{"id": "9838.png", "formula": "\\begin{align*} L _ { u } ( u ^ 2 ) = & u \\left ( u ^ 2 \\right ) ^ { q ^ 2 } + ( u + 1 ) \\left ( u ^ 2 \\right ) ^ q + ( u ^ 2 + u ) u ^ 2 \\\\ = & u ^ 2 + ( u + 1 ) u ^ 4 + u ^ 4 + u ^ 3 \\\\ = & u ^ 2 \\left ( u ^ 3 + u + 1 \\right ) \\\\ = & 0 . \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - \\lambda x } f _ \\alpha ( x ) d x = e ^ { - \\lambda ^ \\alpha } . \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} \\nu _ 2 ( S ( 2 ^ h + 1 , k + 1 ) ) = \\sigma _ 2 ( k ) - 1 . \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} \\mu _ \\tau = \\sin ^ 2 ( \\mu ) + \\frac { b _ * u + c _ * p - \\lambda u } { a _ * } \\cos ^ 2 ( \\mu ) \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} \\frac { \\mathrm d } { \\mathrm d t } \\int _ \\Lambda \\rho _ t \\ ; \\ ! G \\ ; \\ ! \\mathrm d u = \\langle \\dot \\rho _ t , G \\rangle = \\int _ \\Lambda \\chi ( \\rho _ t ) v _ t \\cdot \\nabla G \\ ; \\ ! \\mathrm d u . \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} P ( s , x , D ) \\cdot f ( x ) ^ { s + 1 } = B ( s ) \\cdot f ( x ) ^ s . \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} \\langle \\psi _ n ^ { \\ell } | \\psi _ z ^ \\ell \\rangle = z ^ n \\sqrt { \\frac { \\Gamma ( 1 + \\ell ) } { n ! \\Gamma ( n + \\ell + 1 ) } } \\langle \\psi _ 0 ^ { \\ell } | \\psi _ z ^ \\ell \\rangle . \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} r _ - ( x ) : = \\frac D 2 \\frac { \\phi _ { 3 , - } '' ( x ) } { \\phi _ { 3 , - } ( x ) } \\le \\frac { D \\lambda _ 1 } { \\gamma _ 1 + x ^ 2 } , \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} c ( s ) = \\begin{cases} s , & s \\in [ 0 , \\frac 1 2 ] ; \\\\ [ . 2 c m ] 1 - s , & s \\in [ \\frac 1 2 , 1 ] . \\end{cases} \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} K _ g = \\lambda I + D \\ , , D ^ t = D \\ , . \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} D ^ \\alpha { } _ \\beta = \\begin{pmatrix} 1 & 0 & - a \\sin ( x ^ 3 + x ^ 4 ) & - a \\sin ( x ^ 3 + x ^ 4 ) \\\\ 0 & 1 & \\pm a \\cos ( x ^ 3 + x ^ 4 ) & \\pm a \\cos ( x ^ 3 + x ^ 4 ) \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} V ^ { ( n ) } = \\gamma _ n \\sum _ { k = 0 } ^ { n - 1 } z ^ { k } E ^ { ( n ) } _ { k + 1 , n - k } , Q _ n ( z ) = \\sum _ { k = 0 } ^ { n - 1 } | z | ^ { 2 k + 1 - n } , \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} f ( \\delta ) = \\Big ( \\frac { 3 \\delta - 1 } { 1 + \\delta } \\Big ) ^ 2 \\sqrt { 2 \\delta ( 1 + \\delta ) } > 0 . 1 5 ( \\pi + 1 ) . \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} \\kappa ^ \\star ( \\hat { x } , x ^ { \\star } , \\lambda ) = z \\left [ l _ 1 ^ { - 1 } + l _ 2 ^ { - 1 } - 1 \\right ] + b \\left ( 1 + \\lambda \\right ) \\left [ l _ 1 ^ { - 1 } - l _ 2 ^ { - 1 } + \\ln { \\hat { x } } - \\ln { x ^ \\star } \\right ] . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} { \\rm R i c } = { \\rm M } - \\tfrac 1 2 \\ , { \\rm B } - { \\rm S } ( { \\rm a d } _ { H } ) \\ , , \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} P = \\sum _ \\alpha a _ \\alpha ( z ) \\partial _ z ^ \\alpha . \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} \\nabla _ { e _ i } \\nabla _ { e _ i } ( \\frac { Y ^ { \\perp } } { u } \\partial _ t ) & = \\nabla _ { e _ i } [ e _ i ( \\frac { Y ^ { \\perp } } { u } ) \\partial _ t + \\frac { Y ^ { \\perp } } { u } \\nabla _ { e _ i } \\partial _ t ] \\\\ & = e _ i ( e _ i ( \\frac { Y ^ { \\perp } } { u } ) ) \\partial _ t + 2 e _ i ( \\frac { Y ^ { \\perp } } { u } ) \\nabla _ { e _ i } \\partial _ t + \\frac { Y ^ { \\perp } } { u } \\nabla _ { e _ i } \\nabla _ { e _ i } \\partial _ t . \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} & \\kappa _ 2 : ( \\kappa _ 2 + 1 ) p = ( \\kappa _ 1 + 1 ) p ^ * , \\\\ & \\kappa _ 3 : ( \\kappa _ 3 + 1 ) p = ( \\kappa _ 2 + 1 ) p ^ * , \\\\ & \\vdots \\vdots \\ , . \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} \\mathbb { H } [ ( \\mathfrak { Q } _ h u - U _ { h } ) ^ { 1 } ] \\leq & C ( \\| ( \\mathfrak { Q } _ h u - U _ { h } ) ^ { 1 } \\| ^ 2 + \\| ( \\mathfrak { Q } _ h u - U _ { h } ) ^ { 0 } \\| ^ 2 ) . \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} \\alpha _ n ( t , q ) = \\Big { ( } & ( n - 2 ) t q + ( n - 1 ) t + 2 q + 1 \\Big { ) } \\alpha _ { n - 1 } ( t , q ) \\\\ & + ( t - t ^ 2 ) ( q + 1 ) \\frac { \\partial } { \\partial t } \\alpha _ { n - 1 } ( t , q ) \\\\ & + ( 1 - t ) ( q ^ 2 + q ) \\frac { \\partial } { \\partial q } \\alpha _ { n - 1 } ( t , q ) \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} A = C _ 0 \\otimes I + C _ 1 \\otimes T _ 1 + \\ldots + C _ r \\otimes T _ r \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} f ( t ) = a + b ( t - 1 ) + c ( t - 1 ) ^ 2 + d \\ , { ( t - 1 ) ^ 2 \\over t } + \\int _ { ( 0 , + \\infty ) } { ( t - 1 ) ^ 2 \\over t + s } \\ , d \\mu ( s ) , t \\in ( 0 , + \\infty ) . \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{gather*} { } _ 3 F _ 2 \\biggl ( \\begin{matrix} \\frac 1 2 , \\ , r , \\ , 1 - r \\\\ 1 , \\ , 1 \\end{matrix} \\biggm | z \\biggr ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( \\frac 1 2 ) _ k ( r ) _ k ( 1 - r ) _ k } { k ! ^ 3 } z ^ k . \\end{gather*}"}
-{"id": "826.png", "formula": "\\begin{align*} H _ { N } = \\sum _ { i = 1 } ^ { N } \\big ( \\sqrt { - \\Delta _ { x _ i } + m ^ 2 } + V ( x _ i ) \\big ) - \\frac { a _ { N } } { N - 1 } \\sum _ { 1 \\leq i < j \\leq N } | x _ i - x _ j | ^ { - 1 } \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} Z ( n ) = \\frac { 2 } { \\omega } W _ { 0 , a ' } ( \\overline \\varphi , u ) ( n - 1 ) . \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} f ( x , y ) = \\arctan ( 2 5 0 ( \\cos ( t ) x + \\sin ( t ) y ) ) \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} ( \\Gamma _ { | k | \\geq K } X ) ( v ) = \\sum _ { \\mathbf { v } \\in V } \\sum _ { | k | \\geq K , i , \\alpha , \\beta } X ^ { ( \\mathbf { v } ) } _ { k , i , \\alpha , \\beta } e ^ { \\mathbf { i } k \\cdot x } y ^ { i } z ^ { \\alpha } \\bar { z } ^ { \\beta } \\partial _ { \\mathbf { v } } , \\Gamma _ { | k | < K } : = \\mbox { I d } - \\Gamma _ { | k | \\geq K } \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} \\frac { m ^ 2 } { t _ h } - \\frac { ( h - t _ h ) ^ { 2 b + 1 } } { 2 b + 1 } = \\frac { m ^ 2 } { \\tau _ h } - \\frac { ( h - \\tau _ h ) ^ { 2 b + 1 } } { 2 b + 1 } ; \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} \\Gamma ( f , h ) = & \\frac { 1 } { 2 } \\Big [ \\mathcal { L } ^ N ( f h ) - f \\mathcal { L } ^ N h - h \\mathcal { L } ^ N f \\Big ] = \\frac { 1 } { \\beta } \\ , \\nabla ^ N f \\cdot \\nabla ^ N h \\ , , \\\\ \\Gamma _ 2 ( f , h ) = & \\frac { 1 } { 2 } \\Big [ \\mathcal { L } ^ N \\Gamma ( f , h ) - \\Gamma ( f , \\mathcal { L } ^ N h ) - \\Gamma ( \\mathcal { L } ^ N f , h ) \\Big ] \\ , . \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} R _ { \\theta , n } = n ^ { - \\sigma } \\left \\{ \\cos \\left ( \\frac { \\theta } { 2 } t \\right ) \\Omega _ { n - 1 } ^ { - 1 } \\cos \\left ( \\varphi _ n - \\varphi _ 1 \\right ) + \\sin \\left ( \\frac { \\theta } { 2 } t \\right ) \\Omega _ { n + 1 } ^ { - 1 } \\sin \\left ( \\varphi _ { n + 1 } + \\varphi _ 1 \\right ) \\right \\} , \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} | z _ 1 | ^ 2 + | z _ 2 | ^ 2 + | z _ 3 | ^ 2 + | z _ 4 | ^ 2 = 1 , ( | z _ 1 | + | z _ 3 | ) ( | z _ 2 | + | z _ 4 | ) \\neq 0 . \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} { c c c c } q _ 1 ( 1 ) & 0 & 0 & 0 \\\\ q _ { 0 0 } ( 1 ) & q _ { 0 0 } ( 0 ) & 0 & 0 \\\\ B & 1 - B & 1 - B & B \\\\ 1 - A & A & A & 1 - A \\end{array} \\right ) . \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} \\Phi _ \\rho ( \\lambda ^ \\star ) & - \\Phi _ \\rho ( \\lambda _ { k + 1 } ) \\\\ & \\leq \\frac { L } { 2 } ( \\| \\lambda _ k - \\lambda ^ \\star \\| ^ 2 - \\| \\lambda _ { k + 1 } - \\lambda ^ \\star \\| ^ 2 ) + \\delta . \\\\ \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} \\mathcal F _ \\alpha ^ V ( \\rho ) = \\int _ \\Lambda \\Bigl [ f ( \\rho ( u ) ) - f ( \\bar \\rho _ { \\alpha , V } ( u ) ) - f ' ( \\bar \\rho _ { \\alpha , V } ( u ) ) \\bigl ( \\rho ( u ) - \\bar \\rho _ { \\alpha , V } ( u ) \\bigr ) \\Bigr ] \\ ; \\ ! \\mathrm d u , \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} m _ n ^ { - 1 } \\sum _ { i = 1 } ^ { n - 1 } m _ i + m _ n \\sum _ { i = n + 1 } ^ { \\infty } m _ i ^ { - 1 } < 2 ^ { - n - 1 } , \\ ; \\ ; n = 1 , 2 , \\dots \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} \\frac { \\partial \\rho ( t , x ) } { \\partial t } + \\nabla \\cdot ( \\rho ( t , x ) v ( t , x ) ) = 0 , \\rho ( 0 , x ) = \\rho ^ 0 ( x ) , \\rho ( 1 , x ) = \\rho ^ 1 ( x ) . \\end{align*}"}
-{"id": "9908.png", "formula": "\\begin{align*} \\psi _ { 1 } = \\frac 1 2 \\frac t { | t | } . \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} v ( x , y , 0 ) = { } & \\delta ( y - x ) , \\\\ v ( x , x , t ) = { } & \\frac { e ^ { - \\eta t } } { \\sqrt { \\pi t } } , t > 0 , \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t \\psi _ 0 = - ( - \\Delta ) ^ { s } \\psi _ 0 & \\Omega \\times ( t _ 0 , \\infty ) , \\\\ \\psi _ 0 = g & ( \\mathbb { R } ^ n \\setminus \\Omega ) \\times ( t _ 0 , \\infty ) , \\\\ \\psi _ 0 ( \\cdot , t _ 0 ) = h & \\mathbb { R } ^ n . \\end{array} \\right . \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} h _ k ( a _ k ) \\neq 0 \\mbox { a n d } h _ k ( a _ j ) = 0 , \\mbox { f o r } 1 \\leq j \\leq k - 1 . \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} S _ { n , 2 } ( \\omega ) = \\mathrm { o r d } ( \\omega ) ^ 4 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , ( \\frac { z ^ 2 } { 2 } - \\frac { 1 } { 6 } ) ( x - z ) ( y - z ) } + \\mathrm { o r d } ( \\omega ) ^ 2 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , \\frac { ( x - z ) ( y - z ) } { 6 } } . \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} f ( g K ) = K g ^ { - 1 } \\subseteq g ^ { - 1 } H \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} \\frac { m ^ 2 t _ h ' } { t _ h ^ 2 } + ( h - t _ h ) ^ { 2 b } ( 1 - t _ h ' ) = \\frac { m ^ 2 \\tau _ h ' } { \\tau _ h ^ 2 } + ( h - \\tau _ h ) ^ { 2 b } ( 1 - \\tau _ h ' ) . \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} \\Psi _ { 1 , 0 } ( z _ { v ^ { - 1 } A B ^ { - 1 } } ) & = \\sum _ I \\mathrm { t r } \\big ( \\Lambda _ I \\overset { I } { S ^ 2 ( a _ i ) } S ( \\overset { I } { T } ) \\overset { I } { S ( a _ j ) } b _ i b _ j \\big ) = \\mathcal { D } ^ { - 1 } ( z ) \\big ( S ^ 2 ( a _ i ) S ( ? ) S ( a _ j ) \\big ) b _ i b _ j \\\\ & = S \\big ( \\mathcal { D } ^ { - 1 } ( z ) \\big ) ( S ( a _ i ) a _ j ? ) b _ i b _ j = S \\big ( \\mathcal { D } ^ { - 1 } ( z ) \\big ) \\in \\mathcal { O } ( H ) \\otimes 1 . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\Sigma _ { z } = \\Big \\{ \\ , x \\in \\mathbb { R } ^ n ~ \\Big | ~ \\xi ( x ) = z \\Big \\} \\ , . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} \\frac { 1 } { \\widehat { \\alpha } p ' e } = A ^ { p ' } = B ^ { p ' } , \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} ( 1 + ( s ^ 2 / 2 + s ) ) t - t + U = s t + U . \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} 2 \\mathsf { R e } ( P ^ * u , i B u ) \\geq \\gamma _ 0 \\sum _ { j = 0 } ^ N | \\ ! | X _ j u | \\ ! | _ 0 ^ 2 + \\frac 3 2 | \\ ! | X _ 0 u | \\ ! | _ 0 ^ 2 - C | \\ ! | u | \\ ! | _ 0 ^ 2 , \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} \\zeta _ { f , g } ( z ) = \\prod _ { \\varpi \\ \\mbox { \\tiny p r i m e } } \\left ( 1 - z ^ { \\mathrm { p e r } ( \\varpi ) } \\prod _ { m = 0 , \\ldots , \\mathrm { p e r } ( \\varpi ) - 1 } g ( f ^ m x ( \\varpi ) ) \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} \\mathcal H _ r ^ * J _ { \\Pi } = \\gamma ( r ) J _ { \\Pi } . \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} \\gamma _ { k + 1 } \\leq - K _ 8 \\gamma _ k ^ { \\frac { 2 } { 1 + \\epsilon } } + \\alpha _ k \\gamma _ k + K _ { 1 1 } \\beta _ k \\leq - K _ 8 \\gamma _ k ^ { \\frac { 2 } { 1 + \\epsilon } } + d _ k \\gamma _ k \\\\ = d _ k \\gamma _ k \\bigg ( 1 - e _ k \\gamma _ k ^ { \\frac { 1 - \\epsilon } { 1 + \\epsilon } } \\bigg ) , \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { \\mathcal { E } _ { a _ { k } } ^ { \\rm H } ( u _ { k } ) } { E _ { a _ { k } } ^ { \\rm H } } = 1 . \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} ( 2 7 ) \\Leftrightarrow \\frac { \\partial } { \\partial y } \\left ( y p _ { 0 } ( y , t ) \\right ) + \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } p _ { 0 } ( y , t ) - \\frac { \\partial } { \\partial t } p _ { 0 } ( y , t ) = 0 \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} \\dim M ^ { \\pm } = \\pm n + g - 1 . \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} D ^ \\alpha { } _ \\beta = \\delta ^ \\alpha { } _ \\beta + \\operatorname { R e } \\left [ i u ^ \\alpha p _ \\beta \\ , e ^ { i p _ \\gamma x ^ \\gamma } \\right ] + v ^ \\alpha p _ \\beta . \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} [ \\hat f , \\hat g ] = i \\hbar \\{ f , g \\} _ { D B } \\ ; , \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} \\sum _ { j l } \\Big ( \\partial _ l \\big ( A _ \\lambda ^ { j k } ( \\nabla \\Q _ \\lambda ) \\big ) \\partial _ l u _ { \\lambda , j } \\Big ) & = \\sum _ l G ^ { k l } _ \\lambda ( \\nabla \\Q _ \\lambda ) \\partial _ l \\big ( \\phi _ \\lambda ( \\abs { \\nabla \\Q _ \\lambda } ) \\big ) . \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} [ L ^ m \\varphi ( \\delta \\sqrt { L } ) ] ( x , y ) = L ^ m [ \\varphi ( \\delta \\sqrt { L } ) ( \\cdot , y ) ] ( x ) = L ^ m [ \\varphi ( \\delta \\sqrt { L } ) ( x , \\cdot ) ] ( y ) . \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{gather*} y ^ 2 = x ^ 3 - 3 ( 9 - 8 z ) x + 2 \\big ( 2 7 - 3 6 z + 8 z ^ 2 \\big ) , \\\\ y ^ 2 = x ^ 3 - 2 7 ( 1 + 3 z ) x + 5 4 ( 1 - 9 z ) \\qquad y ^ 2 = x ^ 3 - 2 7 x + 5 4 ( 1 - 2 z ) . \\end{gather*}"}
-{"id": "4410.png", "formula": "\\begin{align*} p ^ { n , r } _ t = X ^ { n , r } _ { \\kappa ( n , t ) } - X ^ { n , r } _ { t ^ - } , q ^ { n , r } _ t = Y ^ { n , r } _ { \\kappa ( n , t ) } - Y ^ { n , r } _ { t ^ - } , t \\in [ - \\tau , T ] \\ , . \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} \\frac { { { \\rm { D } } ^ { 2 } } } { \\partial { { z } ^ { i } } \\partial \\overline { { { z } ^ { j } } } } = \\frac { { { \\partial } ^ { 2 } } } { \\partial { { z } ^ { i } } \\partial \\overline { { { z } ^ { j } } } } + \\frac { \\partial K } { \\partial \\overline { { { z } ^ { j } } } } \\frac { \\partial } { \\partial { { z } ^ { i } } } + \\frac { \\partial K } { \\partial { { z } ^ { i } } } \\frac { \\partial } { \\partial \\overline { { { z } ^ { j } } } } + { { \\psi } _ { i \\overline { j } } } \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} \\forall f \\exists e \\forall n \\exists p [ T _ 1 ( e , n , p ) \\wedge U ( p ) = f ( n ) ] \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} \\limsup _ { n , m \\to \\infty } \\mathbb { P } \\left \\lbrace \\sup _ { t \\in [ 0 , T ] } \\left \\lvert X ^ { n , r } _ t - X ^ { m , r } _ t \\right \\rvert \\ge \\varepsilon \\right \\rbrace \\le \\lim _ { R \\to \\infty } \\limsup _ { n , m \\to \\infty } \\left [ \\mathbb { P } \\left \\lbrace T > \\tau ^ { n , r } _ R \\right \\rbrace + \\mathbb { P } \\left \\lbrace T > \\tau ^ { m , r } _ R \\right \\rbrace + \\frac { C ( T , { \\omega ^ \\prime } ) I ^ { n , m } _ { R , T } } { \\varepsilon ^ 2 } \\right ] = 0 \\ , . \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} E _ { N } ^ { \\rm Q } = ( a _ { * } - a _ { N } ) ^ { \\frac { q } { q + 1 } } \\Big ( \\frac { q + 1 } { q } \\cdot \\frac { \\Lambda } { a _ { * } } + o ( 1 ) _ { N \\to \\infty } \\Big ) \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } ( X _ { t } ^ { ( u ) } ) \\ , = \\ , \\int ^ { \\infty } _ { 0 } e ^ { - 2 v } I _ { 0 } ( 2 u v ) { \\mathrm d } v \\ , = \\ , \\frac { \\ , 1 \\ , } { \\ , 2 \\sqrt { 1 - u ^ { 2 } } } \\ , < \\infty \\ , . \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} \\dfrac { d } { d t } g ( t ) + \\dfrac { b - 1 } { 2 } g ^ 2 ( t ) = - \\left ( p * ( \\frac { b } { 2 } u ^ 2 + \\frac { 3 - b } { 2 } u _ x ^ 2 ) \\right ) ( 0 ) \\leq 0 . \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( \\partial _ t - \\partial _ x ^ 2 - x ^ 2 \\partial _ y ^ 2 ) f _ { } ( t , x , y ) = \\mathbf 1 _ { \\omega _ { } } u _ { } ( t , x , y ) & t \\in [ 0 , T ] , ( x , y ) \\in \\Omega \\\\ f _ { } ( t , x , y ) = 0 & t \\in [ 0 , T ] , ( x , y ) \\in \\partial \\Omega \\\\ f _ { } ( 0 , x , y ) = f _ 0 ( x , y ) , ~ f _ { } ( T , x , y ) = 0 & ( x , y ) \\in \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} \\boldsymbol { B } ( k , 3 , 1 ) = \\begin{pmatrix} 0 & k & 0 \\\\ 1 & 0 & k - 1 \\\\ 0 & k & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ N c _ { \\ell } \\tilde { \\Phi } _ { \\ell } ( x ) = \\sum _ { j = 0 } ^ N \\left ( \\sum _ { \\ell = 0 } ^ N c _ { \\ell } \\tilde { \\Phi } _ { \\ell } ( x _ j ) \\right ) \\ell _ j ( x ) . \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} = \\frac { d ( d + 1 ) \\cdots ( d + i - 2 ) } { ( i - 1 ) ! } \\cdot \\frac { ( d + i ) ( d + i + 1 ) \\cdots ( d + p - 1 ) } { ( p - i ) ! } = \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} ( d _ i - \\lambda ) v _ i = \\sum _ { j \\sim i } v _ j , ~ ~ ~ \\forall i \\in \\{ 1 , \\dots , N \\} . \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t u ^ { \\frac 1 2 } , v \\Big { ) } - \\gamma ( \\nabla \\sigma ^ { \\frac 1 2 } , \\nabla v ) + ( \\nabla u ^ { \\frac 1 2 } , \\nabla v ) + ( f ^ { \\frac 1 2 } ( u ) , v ) = & ( g ^ { \\frac 1 2 } , v ) , ~ \\forall v \\in H _ 0 ^ 1 , \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} A ( \\underline { X } ) _ { ( n + i ) j } = \\begin{cases} - X _ { q } , & A ( \\underline { X } ) _ { i j } = X _ { n + q } , \\\\ 0 , & A ( \\underline { X } ) _ { i j } = 0 , \\end{cases} \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} i _ { c } \\colon S \\times \\{ c \\} \\hookrightarrow S \\times C = X . \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} \\underline { u } = u ^ { ( 0 ) } \\leq u ^ { ( n ) } \\leq u ^ { ( n + 1 ) } \\leq \\bar { u } \\ ; n \\geq 1 . \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} a ( \\zeta ) = \\sin ^ { - 1 } p ( \\zeta ) b ( \\zeta ) = \\pi - \\sin ^ { - 1 } p ( \\zeta ) . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} H _ { \\widetilde { \\mathcal S } } ( t ) = Q _ i ^ j Q _ { k - i } ^ { l - i } = \\frac { P _ j } { P _ i P _ { j - i } } \\cdot \\frac { P _ { l - i } } { P _ { k - i } P _ { l - k } } . \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} \\begin{bmatrix} x _ 1 ( 0 ) \\\\ x _ 2 ( 0 ) \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} , \\begin{bmatrix} x _ 1 ( N ) \\\\ x _ 2 ( N ) \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{gather*} Z ( S ^ { \\vee } ) = \\int _ { S } \\Omega , \\end{gather*}"}
-{"id": "9849.png", "formula": "\\begin{align*} P ( \\alpha , \\beta + 1 ) & = \\dfrac { 4 q } { K } P ( \\alpha , \\beta ) + P ( \\alpha , \\beta - 1 ) , \\\\ P ( \\alpha + 2 , 0 ) & = K P ( \\alpha + 1 , 0 ) + q ^ 2 P ( \\alpha , 0 ) , \\\\ P ( \\alpha + 2 , - 1 ) & = \\left ( K - 2 q + \\dfrac { 4 q ^ { 2 } } { K } \\right ) P ( \\alpha + 1 , 0 ) - q ^ 2 P ( \\alpha , 1 ) . \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} & ( 2 \\ell - 3 k ^ 2 - 2 s _ 1 k + s _ 2 + 2 - \\alpha ) ^ 2 \\\\ & \\qquad \\qquad \\qquad = 4 \\ell ^ 2 - 4 \\ell ( 3 k ^ 2 + 2 s _ 1 k - s _ 2 - 2 ) - 1 2 k ^ 2 - 8 s _ 1 k + 4 s _ 2 + 4 \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} d s \\wedge i _ X \\alpha + i _ X ( d s \\wedge \\alpha ) = s \\alpha . \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 9 6 n + 1 2 ) q ^ n \\equiv \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 2 8 8 n + 3 6 ) q ^ n \\equiv 2 f _ 1 ^ 3 \\equiv 2 \\left ( f _ 3 a ( q ^ 3 ) - 3 q f _ 9 ^ 3 \\right ) ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} u _ { k + 1 } ( t , 0 ) = 0 \\mbox { f o r } t \\ge \\tau _ { k + 1 } , \\dots , u _ { k + m - 1 } ( t , 0 ) = 0 \\mbox { f o r } t \\ge \\tau _ { k + m - 1 } , \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} ( \\ln 2 + o ( 1 ) ) n = \\tau ' ( \\mathcal { P M } , n ) \\leq \\tau ( \\mathcal { P M } , n ) \\leq ( 1 + 2 / e + o ( 1 ) ) n , \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} \\begin{cases} - a ^ 2 + b ^ 2 = - 1 , \\\\ - c ^ 2 + d ^ 2 = 1 , \\\\ - a c + b d = 0 . \\end{cases} \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} \\chi _ { k \\boldsymbol { \\nu } } ( g ) = \\sum _ { j = 0 } ^ { k \\ , \\nu - 1 } e ^ { \\imath \\ , ( k \\nu - 1 - 2 j ) \\ , \\vartheta _ G ( g ) } . \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} E _ i ^ { l _ i ' } = 0 , F _ i ^ { l _ i ' } = 0 , K _ i ^ { 2 l _ i ' } = 1 , L _ i ^ { 2 l _ i ' } = 1 , K _ i ^ { l _ i ' } = L _ i ^ { l _ i ' } . \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} \\mathfrak F \\Phi ( v ^ * ) = \\int _ V \\Phi ( v ) \\psi ( \\omega ( v , v ^ * ) ) | \\omega | ( v ) . \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} - \\Delta Q - c | x | ^ { - 2 } Q + \\omega Q = | Q | ^ { \\frac { 4 } { d } } Q . \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} q _ 1 & = a _ 1 & q _ 2 & = a _ 2 q _ 1 + q _ 0 = a _ 2 a _ 1 + 1 \\\\ p _ 1 & = 1 & p _ 2 & = a _ 2 p _ 1 + p _ 0 = a _ 2 \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} P _ e ( o p t ) \\le \\frac { 1 } { 2 } \\Theta _ { \\Lambda } \\left ( q = \\exp ( - \\frac { 1 } { 8 \\sigma ^ 2 } ) \\right ) - \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} \\psi ( h ) = \\psi ( h _ { 1 , 2 } + h _ { 2 , 3 } + \\cdots + h _ { n - 1 , n } ) . \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{gather*} T _ { N - 1 } T _ { N - 2 } \\cdots T _ 3 \\big ( x _ 2 ^ + \\big ) = ( - 1 ) ^ { N - 3 } E _ { 2 , N } \\end{gather*}"}
-{"id": "5610.png", "formula": "\\begin{align*} I = ( x _ 1 x _ 3 , x _ 1 x _ 4 , x _ 1 x _ 5 , x _ 1 x _ 6 , x _ 2 x _ 4 , x _ 2 x _ 5 , x _ 2 x _ 6 , x _ 3 x _ 5 , x _ 3 x _ 6 ) . \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} \\widehat { V } ^ * b = V _ 0 ( H _ 1 \\oplus . . . \\oplus H _ d \\oplus W _ 1 ) ^ * V ^ * b = V _ 0 ( H _ 1 ^ * \\oplus . . . \\oplus H _ d ^ * \\oplus W _ 1 ^ * ) V ^ * b , \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} S ( \\xi ) = \\exp ( \\eta K _ + ) \\exp ( \\zeta K _ 0 ) \\exp ( - \\eta ^ * K _ - ) , \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} P _ { _ 0 } ( y ^ 2 ) & = \\frac { 1 } { \\Gamma ( 1 / 2 ) } \\gamma \\left ( \\frac { 1 } { 2 } , \\frac { y ^ 2 } { 2 \\sigma _ w ^ 2 } \\right ) , \\\\ P _ { _ 1 } ( y ^ 2 ) & = \\frac { 1 } { \\Gamma ( 1 / 2 ) } \\gamma \\left ( \\frac { 1 } { 2 } , \\frac { y ^ 2 } { 2 ( P _ x + \\sigma _ w ^ 2 ) } \\right ) , \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} \\widetilde { \\mathrm { T r } } _ { d , D } ( G _ m ( z , s ) ) : = \\left \\{ \\begin{array} { l l } \\mathrm { T r } _ { d , D } ( G _ m ( z , s ) ) , & d D < 0 , \\\\ B ( s ) ^ { - 1 } \\mathrm { T r } _ { d , D } ( G _ m ( z , s ) ) , & d D > 0 , d D \\neq \\square , \\end{array} \\right . \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} \\frac { b ( \\beta ) } { b ( \\alpha ) } = 1 - \\frac { Y ( X _ \\alpha - X _ \\beta ) } { ( d - 1 ) X _ \\alpha X _ \\beta + Y X _ \\alpha } . \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} D ( \\rho \\| \\sigma ) : = \\begin{cases} \\tau ( d _ \\rho ( \\log d _ \\rho - \\log d _ \\sigma ) ) & , \\\\ + \\infty & , \\end{cases} \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} \\partial \\square _ p [ n ] ( \\square ^ m ) = \\{ f \\in \\hom _ { \\square _ p } ( \\square ^ m , \\square ^ n ) | f = \\delta _ { n - 1 } ^ { \\epsilon , i } g \\} \\ . \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} \\mathcal { E } _ { p } ( \\gamma , \\zeta _ { p } ) = 0 . \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} \\Re \\sum _ { n = 1 } ^ { \\infty } { \\widehat { z } _ n } ^ { \\widehat { j } _ 0 } \\geq \\left ( \\frac { \\textit { \\v { c } } - 1 2 } { 4 \\textit { \\v { c } } } \\right ) | \\widehat { z } _ 1 | ^ { \\widehat { j } _ 0 } . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} & \\rho \\int _ { \\partial B _ { \\rho } } e _ L ( \\Q ) + \\int _ { B _ { \\rho } } \\left ( \\phi ^ \\prime ( | \\nabla \\Q | ) | \\nabla \\Q | - p \\phi ( | \\nabla \\Q | ) \\right ) \\\\ & = \\rho \\int _ { \\partial B _ { \\rho } } \\frac { \\phi ^ \\prime ( | \\nabla \\Q | ) } { | \\nabla \\Q | } | \\partial _ { \\boldsymbol { \\nu } } \\Q | ^ 2 + ( d - p ) \\int _ { B _ { \\rho } } e _ L ( \\Q ) + \\frac { p } { L } \\int _ { B _ { \\rho } } f _ B ( \\Q ) \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 1 & [ 3 ] & [ 3 ] & 1 \\\\ [ 3 ] & - 1 & - 1 & [ 3 ] \\\\ [ 3 ] & - 1 & - 1 & [ 3 ] \\\\ 1 & [ 3 ] & [ 3 ] & 1 \\end{pmatrix} T = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & - i & 0 & 0 \\\\ 0 & 0 & i & 0 \\\\ 0 & 0 & 0 & - 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} \\div \\mathcal { K } = \\varrho \\nabla \\Bigl ( \\kappa ( \\varrho ) \\Delta \\varrho + \\frac 1 2 \\kappa ' ( \\varrho ) | \\nabla \\varrho | ^ 2 \\Bigr ) \\cdotp \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} I _ { \\nu } ( x ) \\ , : = \\ , \\sum _ { k = 0 } ^ { \\infty } \\frac { ( x / 2 ) ^ { 2 k + \\nu } } { \\ , \\Gamma ( k + 1 ) \\cdot \\Gamma ( \\nu + k + 1 ) \\ , } \\ , ; x > 0 \\ , , \\nu \\ge - 1 \\ , . \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} \\tilde { f } _ { \\mathbf { t } } ( x , y ) = y ^ 2 ( 3 - 3 y + y ^ 2 ) ^ 2 + x ^ { 2 } ( 1 - y ) ^ 2 + t _ 1 x ^ { 4 } ( 1 - y ) + t _ 2 x ^ { 3 } . \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} \\begin{cases} ( R i c ) ' _ { h ^ { ( 4 ) } } = 0 \\\\ \\Delta G = 0 . \\end{cases} \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} \\widetilde E [ C , Q ] = \\sum _ { \\substack { ( i , j ) \\in \\mathcal { I } \\\\ i < j } } \\left ( \\frac { Q _ { i j } ^ 2 } { C _ { i j } } + \\frac { \\nu } { \\gamma } C _ { i j } ^ \\gamma \\right ) L _ { i j } \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} F ( X , Y + Z ) + F ( Y , Z ) - F ( X + Y , Z ) - F ( X , Y ) = 0 . \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} w = w ^ { ( 0 ) } ( \\vec { u } ) + \\sum _ { i , j } x _ i y _ j w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) + \\sum _ { i , i ' , j , j ' } x _ i x _ { i ' } y _ j y _ { j ' } w _ { i i ' j j ' } ^ { ( 2 ) } ( \\vec { u } ) + \\cdots \\end{align*}"}
-{"id": "9665.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - \\alpha } \\sum _ { j = 2 } ^ { \\infty } \\frac { ( r _ { 0 } + r _ { 1 } ) ^ { j } } { j ! } \\leq r _ { 1 } , \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} \\alpha ^ * \\Omega _ i = \\zeta _ 3 \\Omega _ i , \\beta ^ * \\Omega _ i = \\zeta _ { 1 1 } ^ { i + 1 } \\Omega _ i . \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} ( \\mathbb { R } ^ n \\times \\mathbb { E } ) _ x \\cap ( \\{ 0 \\} \\times \\mathbb { E } ) = ( \\{ 0 \\} \\times \\mathbb { E } ) _ x , \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} \\lambda _ { \\mu _ p \\pi _ p } ( p ^ m ) = \\begin{cases} \\delta _ { m = 0 } & \\mu _ p \\vert _ { \\Z _ p ^ { \\times } } \\neq \\chi _ i ^ { - 1 } \\vert _ { \\Z _ p ^ { \\times } } , \\\\ \\chi _ i ( p ^ m ) \\delta _ { m \\geq 0 } & \\mu _ p \\vert _ { \\Z _ p ^ { \\times } } = \\chi _ i ^ { - 1 } \\vert _ { \\Z _ p ^ { \\times } } \\end{cases} \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} [ \\mathfrak { M } f ] ( \\mu _ p \\abs { \\cdot } _ { p } ^ s ) = [ \\mathfrak { M } f ] ( \\mu _ p , s ) = \\int _ { \\Q ^ { \\times } _ p } f ( y ) \\mu _ p ( y ) \\abs { y } _ { p } ^ s d ^ { \\times } y . \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial \\overline { z } } + A w + B \\overline { w } = 0 \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} | k _ b j _ b + \\sum _ { j \\in \\mathbb { Z } _ * } \\frac { l _ j j } { n - \\frac 1 2 } | < \\frac { 1 } { 5 0 n \\sum _ { b = 1 } ^ n | j _ b | } \\sum _ { j \\in \\mathbb { Z } _ * } | j l _ j | , b = 1 , \\cdots , n . \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} - \\Delta u + u + \\rho _ { \\infty } \\bar { \\phi } _ u u = | u | ^ { p - 1 } u , \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} ( \\sum _ { j = 0 } ^ N X _ j ^ * X _ j u , u ) = \\sum _ { j = 0 } ^ N | \\ ! | X _ j u | \\ ! | _ 0 ^ 2 \\geq c _ K | \\ ! | u | \\ ! | _ { 1 / 2 } ^ 2 - C _ K | \\ ! | u | \\ ! | _ 0 ^ 2 , \\ , \\ , \\ , \\ , \\forall u \\in C _ 0 ^ \\infty ( K ) , \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} \\phi _ { n } ( T , R ) & = \\bigg ( \\iint _ { I } + \\iint _ { I I } \\bigg ) \\frac { r ^ { n - 2 } } { t ^ { n / 2 } } \\exp \\Big ( - \\frac { r ^ { 2 } } { 4 t } \\Big ) \\ , d r \\ , d t \\\\ & \\triangleq g _ { 1 } ( T , R ) + g _ { 2 } ( T , R ) . \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} \\mathbf { B } ^ i = \\sum _ { j = 1 } ^ { m } \\left ( \\frac { \\Omega _ { j R } ^ i ( \\cdot ) } { | \\cdot | ^ { d } } \\right ) \\star b _ { j R } ~ ~ ~ B _ { 2 R } . \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} q _ { \\rm S V C } ( \\mathbf p ) = \\sum _ { n \\in \\mathcal N } a _ { n } \\sum _ { \\ell \\in \\mathcal L } b _ { n , \\ell } \\sum _ { k \\in \\mathcal K _ { { \\rm S V C } , n , \\ell } } \\Pr [ K _ { { \\rm S V C } , n , \\ell } = k ] \\Pr [ { \\rm S I R } _ { { \\rm S V C } , n , \\ell } \\geq \\tau _ k ] , \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} \\mathcal { L } = \\mathcal { L } _ 0 + \\mathcal { L } _ 1 \\ , , \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} & \\mathbf { x } ^ { k + 1 } : = \\underset { \\mathbf { x } } { } f ( \\mathbf { x } ) + ( \\sqrt { \\eta } \\boldsymbol { \\nu } ^ k ) ^ T \\mathbf { E } _ { } \\mathbf { x } + \\frac { \\rho } { 4 } \\| \\mathbf { E } _ { } \\mathbf { x } \\| ^ 2 , \\\\ & \\boldsymbol { \\nu } ^ { k + 1 } : = \\boldsymbol { \\nu } ^ k + \\sqrt { \\eta } \\frac { \\rho } { 2 } \\mathbf { E } _ { } \\mathbf { x } ^ { k + 1 } . \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} Z _ t = f ( t , X _ t ) = f ( t , 0 ) - \\frac { X _ t ^ { 1 - m / 2 } } { ( 2 t ) ^ { ( n - m ) / 2 } } \\psi \\left ( \\frac { X _ t } { 2 t } \\right ) , t > 0 . \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} \\phi ( m \\otimes \\{ [ c _ 1 | . . . | c _ k ] \\} ) = \\left \\{ \\begin{array} { l l l } c _ 1 , & k = 1 , \\\\ m \\cdot [ c _ 1 | . . . | c _ { k - 1 } ] \\otimes c _ k , & k > 1 ; \\end{array} \\right . \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} \\int _ { ( \\partial S _ { \\varepsilon / 2 } \\cap \\overline { D } ( 0 , R ) ) ^ - } e ^ { z w } \\psi _ { \\varepsilon } u ( z ) d z = 0 \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} \\eta ( x ) = 1 \\textrm { f o r } x \\in B _ R ( x _ 0 ) , \\eta ( x ) = 0 \\textrm { f o r } x \\in B _ { 2 R } ( x _ 0 ) , | \\nabla \\eta | \\leq \\frac { C } { R } \\textrm { f o r } x \\in M , \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} \\omega ^ 2 a _ n '' ( \\omega ) = \\mathrm { o r d } ( \\omega ) ^ 4 b _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } + \\mathrm { o r d } ( \\omega ) ^ 2 c _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } . \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} \\hat \\rho _ i ( \\mu ) : = \\sum _ { \\eta \\in \\Omega _ L } \\mu ( \\eta ) \\eta ( i ) \\qquad \\textrm { a n d } \\hat \\jmath ^ V _ { i , i ' } ( \\mu ) : = \\sum _ { \\eta \\in \\Omega _ L } \\mu ( \\eta ) \\bigl ( \\hat r ^ V _ { \\eta , \\eta ^ { i , i ' } } - \\hat r ^ V _ { \\eta , \\eta ^ { i ' , i } } \\bigr ) . \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} \\int _ K \\int _ H \\frac { \\Delta _ G ( h ) } { \\Delta _ H ( h ) \\Delta _ K ( k ^ { - 1 } ) } f ( k ^ { - 1 } x h ) d h d k & = \\int _ H \\int _ K \\frac { \\Delta _ G ( h ) } { \\Delta _ H ( h ) \\Delta _ K ( k ^ { - 1 } ) } f ( k ^ { - 1 } x h ) d k d h \\\\ & = \\int _ { K \\times H } \\frac { \\Delta _ G ( h ) } { \\Delta _ H ( h ) \\Delta _ K ( k ^ { - 1 } ) } f ( k ^ { - 1 } x h ) d ( k \\times h ) . \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} \\tilde { f } _ i ^ a ( \\pi _ 1 \\otimes \\pi _ 2 ) & = \\tilde { f } _ i ^ M ( \\pi _ 1 ) \\otimes \\tilde { f } _ i ^ { a - M } ( \\pi _ 2 ) \\\\ & = \\tilde { f } _ i ^ M ( \\pi _ 1 ) \\otimes 0 \\\\ & = 0 \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} \\textup { P D } _ \\textup { t } ( 2 4 ( 3 n + 1 ) + 1 2 ) = \\textup { P D } _ \\textup { t } ( 3 ( 2 4 n + 1 2 ) ) \\equiv \\textup { P D } _ \\textup { t } ( 2 4 n + 1 2 ) ~ ( \\textup { m o d } ~ 4 ) . \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ m } u ^ 2 d V _ g & = \\int _ { \\mathbb { R } ^ m } u ^ 2 e ^ { - \\frac { m | x | ^ 2 } { m - 2 } } d V _ { g _ 0 } \\\\ & \\le \\frac { 2 ( m - 2 ) } { m } \\int _ { \\mathbb { R } ^ m } | \\nabla u | _ { g _ 0 } ^ 2 e ^ { - \\frac { m | x | ^ 2 } { m - 2 } } d V _ { g _ 0 } \\\\ & \\le \\frac { 2 ( m - 2 ) } { m } \\int _ { \\mathbb { R } ^ m } | \\nabla u | _ { g _ 0 } ^ 2 e ^ { \\frac { | x | ^ 2 } { 2 ( m - 2 ) } } e ^ { - \\frac { m | x | ^ 2 } { m - 2 } } d V _ { g _ 0 } \\\\ & = \\frac { 2 ( m - 2 ) } { m } \\int _ { \\mathbb { R } ^ m } | \\nabla u | _ g ^ 2 d V _ g . \\\\ \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\bigsqcup _ { \\pi \\in \\mathcal { B } _ w ( \\mu ) ^ \\lambda } E ( \\pi , v , i ) = S _ { i } ( \\tilde { e } _ i ^ { \\max } ( \\mathcal { B } _ v ( \\lambda ) ) ) \\otimes ( \\mathcal { B } _ { s _ i w } ( \\mu ) \\setminus \\mathcal { B } _ { w } ( \\mu ) ) . \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} S _ { N , r } ^ { ( 4 , 0 ) } = \\sum _ { k = 0 } ^ { r } N _ { 4 k } = \\frac { 1 } { 3 } \\left ( N _ { 4 ( r + 1 ) } - N _ { 4 r } + N _ { 4 ( r - 1 ) } - 1 \\right ) . \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} X _ { d _ { 2 } f } = \\frac { 1 } { 2 } \\sum _ { ( f ) } ( X _ { f _ { ( 1 ) } } X _ { f _ { ( 2 ) } } - ( - 1 ) ^ { | f _ { ( 1 ) } | \\cdot | f _ { ( 2 ) } | } X _ { f _ { ( 2 ) } } X _ { f _ { ( 1 ) } } ) , \\end{align*}"}
-{"id": "9902.png", "formula": "\\begin{align*} \\widehat { H } ^ \\mu ( \\mathbb { R } ) = \\left \\{ w \\in L ^ 2 ( \\mathbb { R } ) : \\int _ { \\mathbb { R } } ( 1 + | 2 \\pi \\xi | ^ { 2 \\mu } ) | \\widehat { w } ( \\xi ) | ^ 2 \\ , { \\rm d } \\xi < \\infty \\right \\} , \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} h ^ n _ i : t \\longmapsto \\sqrt { { n \\over T } } \\int _ 0 ^ t \\mathbf 1 _ { \\left [ t ^ n _ { i - 1 } , t ^ n _ i \\right ) } ( s ) \\ , d s , i = 1 , . . . , n . \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} \\begin{array} { c } m a x \\Big \\{ m i n \\Big [ r V ^ { i j } ( x ) - \\mathcal { A } V ^ { i j } ( x ) - f ^ { i j } ( x ) ; \\qquad \\qquad \\qquad \\qquad \\\\ V ^ { i j } ( x ) - M ^ { i j } [ V ] ( x ) \\Big ] ; V ^ { i j } ( x ) - N ^ { i j } [ V ] ( x ) \\Big \\} = 0 , \\end{array} \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} [ a ] _ { \\kappa } ^ { \\bar { n } } + [ b ] _ { \\kappa } ^ { \\bar { n } } = [ c ] _ { \\kappa } ^ { \\bar { n } } , \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} \\rho ( I ) : = \\max \\{ \\rho ( I _ j ) : \\j = 1 , \\ldots , n \\} . \\end{align*}"}
-{"id": "9901.png", "formula": "\\begin{align*} \\widehat { f } ( \\xi ) = \\left ( ( 2 \\pi i \\xi ) ^ { - s } + ( - 2 \\pi i \\xi ) ^ s \\right ) \\widehat { u } ( \\xi ) . \\end{align*}"}
-{"id": "9949.png", "formula": "\\begin{align*} \\kappa ^ { n - 2 } _ { 0 1 } | _ Y = ( - 1 ) ^ n ( n - 2 ) ! \\ , \\check { \\chi } _ { H _ { n } \\cap Y } \\ , \\bar { \\partial } ( \\varphi _ 1 | _ Y ) \\wedge \\dots \\wedge \\bar { \\partial } ( \\varphi _ { n - 2 } | _ Y ) , \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} \\| ( \\sqrt { A } + K _ { V } ) u \\| ^ 2 _ { L ^ 2 } = \\| ( \\sqrt { A } + O _ p ) u \\| ^ 2 _ { L ^ 2 } + \\| X _ V u \\| ^ 2 _ { L ^ 2 } + 2 \\mathbb { R } \\langle [ O _ p , X _ V ] u , u \\rangle { ~ , } \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} \\mathcal S _ { \\lambda } : = \\left \\{ { \\norm { \\phi _ \\lambda } } _ { D } ^ { - 1 / 2 } u _ { \\lambda } \\in H ^ 1 ( \\mathbb R ^ 3 ) \\setminus \\{ 0 \\} : u _ { \\lambda } \\eqref { o u r p r o b l e m } \\right \\} \\neq \\emptyset . \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} B _ j [ \\phi _ j ] : = \\mu _ { 0 j } ^ { 2 s - 1 } \\dot { \\mu } _ { 0 j } \\left ( \\frac { n - 2 s } { 2 } \\phi _ j + y \\cdot \\nabla _ y \\phi _ j \\right ) + \\mu _ { 0 j } ^ { 2 s - 1 } \\nabla \\phi _ j \\cdot \\dot { \\xi } _ j \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\beta & = 2 \\log \\frac { 1 } { a _ 0 \\lambda } + b _ 0 , a _ 0 = 2 , b _ 0 = \\log 2 , \\\\ \\mu _ \\lambda & = - \\frac { ( \\beta + 2 \\log \\beta ) } { a _ 0 \\lambda } \\partial _ n H _ \\gamma ^ + = \\frac { ( \\beta + 2 \\log \\beta ) } { a _ 0 \\lambda } \\partial _ n H _ \\gamma ^ - > 0 . \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} ( H ^ { s } ( W ) ) ^ { \\ast } = \\widetilde { H } ^ { - s } ( W ) , \\ ( \\widetilde { H } ^ { - s } ( W ) ) ^ { \\ast } = ( H ^ { s } ( W ) ) . \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} \\dim ( S / I ) = n - \\max \\{ \\min ( u ) \\ ; : u \\in G ( I ) \\} . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} \\mathcal { W } _ p ( W , Z ) = \\left ( \\inf \\mathbb { E } \\left [ | X _ 1 - Y _ 1 | ^ p \\right ] \\right ) ^ { 1 / p } \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} c _ k ( g ) = \\frac { 1 } { 1 + | k | ^ { 2 + \\mu } } \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} K = \\left \\lbrace x \\in \\R ^ n : g _ i ( x ) \\leq 0 , i \\in I , \\ ; h _ j ( x ) = 0 , j \\in J \\right \\rbrace . \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} - \\mathcal { L } [ w ] + c ( x ) | D w | + \\alpha w = 0 , \\alpha \\geq 0 . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\mathcal E = \\{ g ^ { ( 4 ) } \\in \\mathcal S : ~ R i c _ { g ^ { ( 4 ) } } = 0 \\} . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} R ( J ^ k ) = \\frac { J ^ { k + 1 } } { k + 1 } \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} \\sup _ { 1 \\le m \\le ( \\log ( 1 / \\epsilon ) ) ^ { 1 + \\delta } } \\left \\| B _ 1 ( \\epsilon ) \\cdots B _ m ( \\epsilon ) - \\left ( \\tilde { B } ^ m + \\epsilon \\sum _ { i = 1 } ^ { m - 1 } i \\tilde { B } ^ i \\tilde { B } ^ { ( 1 ) } \\tilde { B } ^ { m - i - 1 } \\right ) \\right \\| = O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ v ) \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} \\ \\int _ { U \\times Y } \\left ( \\nabla \\phi ( y ) ^ T f ( u , y ) + \\lambda ( \\phi ( y _ 0 ) - \\phi ( y ) \\right ) \\gamma ( d y , d u ) = 0 \\ \\ \\forall \\phi ( \\cdot ) \\in C ^ 1 \\} ; \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} & \\| u u _ h ^ { \\kappa } \\| _ { 1 , p } ^ p \\leq M _ { 2 6 } \\left ( ( \\kappa + 1 ) ^ p \\right ) ^ { M _ { 2 7 } } \\left [ \\| u u _ h ^ { \\kappa } \\| _ { \\tilde { q } _ 1 } ^ p + 1 \\right ] . \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} T ^ \\alpha { } _ { \\beta \\gamma } : = \\Upsilon ^ \\alpha { } _ { \\beta \\gamma } - \\Upsilon ^ \\alpha { } _ { \\gamma \\beta } \\ , . \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} \\widetilde { u } = { { k } _ { 1 } } u - v { { k } _ { 2 } } , ~ ~ ~ \\widetilde { v } = v { { k } _ { 1 } } + u { { k } _ { 2 } } \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} B ( x ) = \\begin{pmatrix} 0 & \\nu _ 3 ( x ) & - \\nu _ 2 ( x ) & 0 & 0 & 0 \\\\ - \\nu _ 3 ( x ) & 0 & \\nu _ 1 ( x ) & 0 & 0 & 0 \\\\ \\nu _ 2 ( x ) & - \\nu _ 1 ( x ) & 0 & 0 & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "10005.png", "formula": "\\begin{align*} A ( \\tau ) = \\begin{cases} 0 & \\tau \\in [ 0 , 1 ) \\\\ \\tau & \\tau \\ge 1 \\\\ \\end{cases} , \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} b _ n ( u ) = b * \\phi _ n ( u ) , \\ f _ n ( u ) = f * \\phi _ n ( u ) \\ \\mbox { a n d } \\ \\sigma _ n ( u ) = \\sigma * \\phi _ n ( u ) , \\ u \\in \\R , \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} R _ c ( f ) = R _ r ( f ) = f u \\ \\ \\ \\ \\ ( f \\in L ^ 2 ( G ) ) \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} L ^ { ( 2 ) } ( A ) = - 2 \\ , ( \\nabla _ \\alpha A ^ \\alpha ) ^ 2 + \\frac 1 2 \\ , ( \\nabla _ \\alpha A _ \\beta + \\nabla _ \\beta A _ \\alpha ) ( \\nabla ^ \\alpha A ^ \\beta + \\nabla ^ \\beta A ^ \\alpha ) \\ , , \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} u ( \\alpha ( i _ 1 + 1 , i _ 2 + 1 ) ) = 3 + u ( \\alpha ( i _ 1 , i _ 2 ) ) \\ , . \\end{align*}"}
-{"id": "9999.png", "formula": "\\begin{align*} A = F _ 0 * \\sum _ { i = 0 } ^ \\infty P ^ { * ( i ) } , \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} b _ i = a _ i + \\max \\{ l \\ ; : \\ ; i _ l < a _ i \\} \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} \\sup \\limits _ { z \\in \\mathbb { C } ^ 2 } | z _ q | ^ 2 e ^ { - \\frac { 1 } { 2 } \\Big ( | e ^ { t M } z | ^ 2 - | z | ^ 2 \\Big ) } = \\sup \\limits _ { s \\in \\mathbb { R } _ { + } } | l ( e ' _ q ) | ^ 2 e ^ { - \\frac { s Q _ t ( e ' _ q ) } { 2 } } = \\frac { 2 | l ( e ' _ q ) | ^ 2 } { Q _ t ( e ' _ q ) } \\sup \\limits _ { \\sigma \\in \\mathbb { R } _ { + } } \\sigma e ^ { - \\sigma } = c _ 0 \\frac { 2 | l ( e ' _ q ) | ^ 2 } { Q _ t ( e ' _ q ) } = c _ 0 \\frac { 2 } { Q _ t ( e ' _ q ) } \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} - \\mathbf { i } \\partial _ { \\omega } u + \\lambda u + \\mu ( x ) u = p ( x ) , x \\in \\mathbb { T } ^ n \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} \\nabla \\phi ( \\rho _ t ) = \\phi ' ( \\rho _ t ) \\nabla \\rho _ t = \\chi ( \\rho _ t ) f '' ( \\rho _ t ) \\nabla \\rho _ t . \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{gather*} \\widetilde { \\mathcal { M } } ( z _ - , z _ + , K ^ { ( k ) } , J _ 0 ) \\longrightarrow \\widetilde { \\mathcal { M } } ( \\widetilde { f } ( z _ - ) , \\widetilde { f } ( z _ + ) , H ^ { ( k ) } , J _ 1 ) \\\\ v ( s , t ) \\longrightarrow u ( s , t ) = \\phi _ L ^ t ( v ( s , t ) ) \\end{gather*}"}
-{"id": "1476.png", "formula": "\\begin{align*} \\binom { m + s } { m } b _ { m + s } = b _ m b _ s + \\sum _ { 0 < i , j < N } a _ { i j } b _ { m - i } b _ { s - j } , \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} d \\bar { \\nu } = \\frac { 1 } { Z } e ^ { - \\frac { \\beta } { \\epsilon } V _ 1 } d \\nu _ z \\ , , \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} \\mathrm { u s p } _ S ( X , [ a f ] ) = \\limsup _ n \\frac { \\lvert X ^ { n } [ a f ] \\rvert _ S } { n } = \\limsup _ n \\frac { \\lvert a \\rvert _ 0 \\lvert X ^ { n } [ f ] \\rvert _ S } { n } = \\lvert a \\rvert _ 0 \\mathrm { u s p } _ S ( X , [ f ] ) . \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} S ^ { \\delta } _ R ( H _ V ) f : = \\int _ 0 ^ { R ^ 2 } \\left ( 1 - \\frac { \\lambda } { R ^ 2 } \\right ) ^ { \\delta } d E _ { H _ V } ( \\lambda ) f , \\ \\ \\ \\ f \\in L ^ 2 . \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\log | \\Delta _ n | = \\frac { 1 } { n } \\left [ \\log | \\Delta _ 0 | + \\sum _ { m = 1 } ^ n \\log \\mathrm { T } ^ m ( \\theta ) \\right ] \\longrightarrow \\int _ { [ 0 , 1 ] } \\log x \\ d \\mu = \\frac { \\pi ^ 2 } { 1 2 \\log 2 } \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} 1 ^ 3 + 4 ^ 2 & = 2 ^ 3 + 3 ^ 2 , & 1 ^ 3 + 8 ^ 2 & = 4 ^ 3 + 1 ^ 2 , & 2 ^ 3 + 9 ^ 2 & = 4 ^ 3 + 5 ^ 2 \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} \\nabla f ( \\mathbf { x } ^ { k + 1 } ) + \\frac { ( 1 - \\eta ) \\rho } { 2 } \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } \\mathbf { x } ^ { k + 1 } + \\sqrt { \\eta } \\mathbf { E } _ { } ^ T \\boldsymbol { \\nu } ^ { k + 1 } = \\\\ + \\frac { \\rho } { 2 } ( 2 \\mathbf { D } + 2 \\epsilon \\mathbf { P } - \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } ) ( \\mathbf { x } ^ k - \\mathbf { x } ^ { k + 1 } ) \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} ( M U ( t ) f ) ( \\tau , y ) & = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { \\mathbb { R } } e ^ { - i s \\tau } ( F f ) ( s + t , y ) d s \\\\ & = \\frac { e ^ { i t \\tau } } { \\sqrt { 2 \\pi } } \\int _ { \\mathbb { R } } e ^ { - i s \\tau } ( F f ) ( s , y ) d s \\\\ & = e ^ { i t \\tau } ( M f ) ( \\tau , y ) . \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} \\sum _ { \\l _ N = - \\infty } ^ { \\infty } \\frac { 1 } { ( \\zeta q ^ { \\l _ N } ; q ) _ { \\infty } } P _ t \\left ( \\l _ N \\right ) , \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} \\tau ^ { n , r } _ R : = \\inf \\left \\lbrace t \\ge 0 : \\left \\lvert X ^ { n , r } _ t \\right \\rvert > \\frac { R } { 3 } \\right \\rbrace \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { \\binom k j \\binom j s \\left ( { \\frac { { f _ 3 } } { { f _ 1 } } } \\right ) ^ j \\left ( { \\frac { { f _ 2 } } { { f _ 3 } } } \\right ) ^ s X _ { m - a k + ( a - c ) j + ( c - b ) s } } } = \\frac { { X _ m } } { { f _ 1 { } ^ k } } \\ , , \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} H _ { n , \\mu } ( \\xi , A ) = \\int _ { ( A ^ c ) ^ n } ^ { \\oplus } H ^ { ( n ) } _ { \\mu , A } ( \\xi , k _ 1 , \\dots , k _ n ) d \\lambda _ { \\nu } ^ { \\otimes n } ( k _ 1 , \\dots , k _ n ) \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\mu ( Z _ 1 , Z _ 2 ) = Z _ 2 \\ , . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} x _ { k _ { i n } + 1 } = \\Pi _ { \\mathcal { X } } [ x _ { k _ { i n } } - \\frac { 1 } { L _ p } ( ( H + \\rho A ^ T A ) x _ { k _ { i n } } + A ^ T \\lambda _ k - \\rho A ^ T b + \\epsilon _ { g p } ) ] \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} w _ { k + 1 } ( t , 1 ) = W _ { k + 1 } ( t ) , \\dots , w _ { k + m } ( t , 1 ) = W _ { k + m } ( t ) \\mbox { f o r } t \\ge 0 , \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{gather*} ( \\tau _ a ) _ * ( a ) = a , ( \\tau _ a ) _ * ( b ) = b a ( \\tau _ b ) _ * ( a ) = b ^ { - 1 } a , ( \\tau _ b ) _ * ( b ) = b . \\end{gather*}"}
-{"id": "2004.png", "formula": "\\begin{align*} \\frac { P _ T \\left ( 1 - \\rho ^ * \\right ) \\sigma ^ { - 2 } + \\sum _ { k = 1 } ^ { r _ s } [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ { - 2 } } { r _ s \\sum _ { k = 1 } ^ { r _ s } \\left ( \\nu ^ * - \\rho ^ * [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 \\right ) ^ { - 1 } } = 2 ^ { \\frac { { R } } { r _ s } } \\left ( \\textstyle \\prod \\limits _ { k = 1 } ^ { r _ s } \\frac { [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } { \\nu ^ * - \\rho ^ * [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } \\right ) ^ { - \\frac { 1 } { r _ s } } , \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} \\Delta _ g u + \\rho \\left ( \\frac { V ( x ) e ^ { \\ , u } } { \\int _ M V e ^ { \\ , u } \\ , d \\mu } - c _ M \\right ) = 0 , c _ M = \\frac { 1 } { \\mathop { v o l } ( M ) } , \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} G ( s ) = \\frac { 1 } { h _ 1 + h _ 2 } \\left ( \\begin{array} { c c } h _ 2 e ^ { - s l } - h _ 2 e ^ { - ( h _ 1 + h _ 2 + s ) l } & h _ 2 e ^ { - ( h _ 1 + h _ 2 + s ) l } + h _ 1 e ^ { - s l } \\\\ h _ 2 e ^ { - s l } + h _ 1 e ^ { - ( h _ 1 + h _ 2 + s ) l } & - h _ 1 e ^ { - ( h _ 1 + h _ 2 + s ) l } + h _ 1 e ^ { - s l } \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} \\left < \\phi _ 0 ^ 2 \\right > _ n \\le C \\sum _ { j k \\in E } ( ( G _ d v ) _ j - ( G _ d v ) _ k ) ^ 2 = C [ v , G _ d v ] = C G _ d ( 0 , 0 ) . \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} \\limsup _ { t \\to 0 + } \\frac { B _ 1 ( t ) } { B ( \\lambda t ) } = \\infty \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} H ( L ) & : = \\left ( \\int _ { \\left \\{ x \\in \\Omega : \\ , a ( x ) > L \\right \\} } a ^ { \\frac { p ^ * } { p ^ * - p } } d x \\right ) ^ { \\frac { p ^ * - p } { p ^ * } } \\to 0 \\quad L \\to \\infty , \\\\ K ( G ) & : = \\left ( \\int _ { \\left \\{ x \\in \\partial \\Omega : \\ , b ( x ) > G \\right \\} } b ^ { \\frac { p _ * } { p _ * - p } } d \\sigma \\right ) ^ { \\frac { p _ * - p } { p _ * } } \\to 0 \\quad G \\to \\infty . \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} \\mathcal { R } _ { k l } ^ { \\nu } ( \\alpha _ { 1 , \\nu } ) = \\{ \\xi \\in \\mathcal { O } _ { \\nu } : | \\langle k , \\omega _ { \\nu } ( \\xi ) \\rangle + \\langle l , \\bar { \\Omega } _ { \\nu } ( \\xi ) \\rangle | < \\alpha _ { 1 , \\nu } \\frac { \\langle l \\rangle _ { \\infty } } { \\langle k \\rangle ^ { \\tau } } \\} , \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} [ T _ a , T _ { b c } ] + [ T _ b , T _ { a c } ] + [ T _ c , T _ { a b } ] = 0 \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} E _ { j } = ( q ^ { j } ; q ^ { j } ) _ { \\infty } . \\end{align*}"}
-{"id": "9976.png", "formula": "\\begin{align*} N ( \\tau ) = \\sup \\left \\{ i \\ge 0 : \\sum _ { j = 0 } ^ i X _ i \\le \\tau \\right \\} . \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} ( b _ 3 - b _ 4 ) a _ 1 + ( b _ 1 - b _ 2 ) a _ 4 = ( b _ 3 - b _ 4 ) a _ 2 + ( b _ 1 - b _ 2 ) a _ 3 + O ( \\delta ) . \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\mathcal { B } _ { a } ( x ) : = \\frac { 1 } { \\Gamma \\left ( \\frac { a } { \\nu } \\right ) } \\int _ { 0 } ^ { \\infty } t ^ { \\frac { a } { \\nu } - 1 } e ^ { - t } h _ { t } ( x ) d t \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} \\inf \\bigg \\{ \\int _ \\Omega \\varphi ( \\nabla u ) \\ , : \\ , u \\in B V ( \\Omega ) , \\ , u _ { | \\partial \\Omega } = g \\bigg \\} , \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} C _ * = \\overline { C } _ * \\delta _ { \\theta _ * } , Q _ * = q _ * \\cdot \\theta _ * \\delta _ { \\theta _ * } , \\theta _ * = \\frac { q _ * } { | q _ * | } \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} T _ { \\psi } A t = & \\sum _ { \\xi \\in \\mathcal { X } } ( A t ) _ { \\xi } \\psi _ { \\xi } = \\sum _ { \\xi \\in \\mathcal { X } } \\sum _ { \\eta \\in \\mathcal { X } } a _ { \\xi \\eta } t _ { \\eta } \\psi _ { \\xi } \\\\ & = \\sum _ { \\eta \\in \\mathcal { X } } \\Big ( \\sum _ { \\xi \\in \\mathcal { X } } a _ { \\xi \\eta } \\psi _ { \\xi } \\Big ) t _ { \\eta } = \\sum _ { \\eta \\in \\mathcal { X } } m _ { \\eta } t _ { \\eta } = : f . \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\mathcal D _ 4 = \\{ \\Phi _ { ( \\psi , f ) } | ~ & \\psi \\in D ^ { m + 1 , \\alpha } _ { \\delta } ( M ) \\psi | _ { \\partial M } = I d _ { \\partial M } ; \\\\ & ~ f \\in C ^ { m + 1 , \\alpha } _ { \\delta } ( M ) f | _ { \\partial M } = 0 ; \\\\ & \\Phi _ { ( \\psi , f ) } : V ^ { ( 4 ) } \\rightarrow V ^ { ( 4 ) } , \\\\ & \\Phi _ { ( \\psi , f ) } [ t , p ] = [ t + f , \\psi ( p ) ] , \\quad \\forall t \\in \\mathbb R , ~ p \\in M . ~ \\} , \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} K ( z , w ; Z , W ) & = \\left \\langle R _ { \\mathcal { X } _ p } \\left ( e _ { ( Z , W ) } \\circ T _ V \\right ) , R _ { \\mathcal { X } _ p } \\left ( e _ { ( z , w ) } \\circ T _ V \\right ) \\right \\rangle _ { \\mathcal { X } _ p } = \\int _ { \\R } X _ p ( t ; w , W ) \\frac { z ^ { 2 \\pi i t } \\overline { Z } ^ { - 2 \\pi i t } } { ( z \\overline { Z } ) ^ { 1 + 1 / 2 \\mu } } \\d t , \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} \\Theta = \\sum _ { \\substack { \\mu \\in Q \\\\ \\mu \\geq 0 } } \\Theta _ \\mu . \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} K _ i K _ j & = K _ j K _ i , & K _ i K _ i ^ { - 1 } & = 1 = K _ i ^ { - 1 } K _ i , \\\\ L _ i L _ j & = L _ j L _ i , & L _ i L _ i ^ { - 1 } & = 1 = L _ i ^ { - 1 } L _ i , \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} \\| \\sigma ( S ) \\xi \\| ^ 2 = \\psi ( S ^ * S ) = \\| S \\| ^ 2 . \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} & [ W , Y ] _ t \\\\ & = \\left [ W , Y _ 0 - \\int _ 0 ^ \\cdot Z _ r b ( r , W _ r ) \\mathrm d r - \\int _ 0 ^ \\cdot f ( r , W _ r , Y _ r , Z _ r ) \\mathrm d r + \\int _ 0 ^ \\cdot Z _ r \\mathrm d W _ r \\right ] _ t , \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} \\mathcal { V } ( 0 ) = \\{ y \\in \\mathbb { R } ^ { n } : y . v \\leq \\frac { 1 } { 2 } v . v \\} , \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} H ( | d | ) : = \\sum _ { Q \\in \\mathcal { Q } _ d / \\mathrm { S L } _ 2 ( \\mathbb { Z } ) } \\frac { 1 } { w _ Q ^ { } } d < 0 , \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} s _ i ( \\lambda , \\mu ) = ( \\lambda , \\mu ) - \\frac { 1 } { 2 } \\langle \\lambda + \\mu , \\alpha _ i ^ \\vee \\rangle ( \\alpha _ i , \\alpha _ i ) , \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} v = u \\cdots { \\rm ~ ~ a n d ~ } w = \\cdots [ u ] \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} { \\cal M } _ + ( Y \\times U ) \\times { \\cal M } _ + ( Y \\times U ) = \\{ \\lambda ( \\gamma , \\xi ) : \\ ( \\gamma , \\xi ) \\in \\mathcal { L } , \\ \\lambda \\geq 0 \\} , \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} a _ { n m } ( \\omega _ n ^ i ) = a _ { m ( n , i ) } ( 1 ) . \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} G ( z + 1 ) = ( 2 \\pi ) ^ { z / 2 } e ^ { - [ z ^ { 2 } ( \\gamma + 1 ) + z ] / 2 } \\prod _ { n = 1 } ^ { \\infty } \\left [ \\left ( 1 + \\frac { z } { n } \\right ) ^ { n } e ^ { z ^ { 2 } / ( 2 n ) - z } \\right ] , \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} g ( [ ( p _ { k , l } ) _ { k , l } , w ] ) \\coloneqq \\left ( [ ( p _ { k , l } ) _ { k , l } ] , \\left [ \\prod _ { l = 1 } ^ { N _ 1 } p _ { 1 , l } , \\prod _ { l = 1 } ^ { N _ 2 } p _ { 2 , l } , \\dots , \\prod _ { l = 1 } ^ { N _ r } p _ { r , l } , w \\right ] \\right ) . \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\sigma p = q + \\phi + z ^ { n } \\psi \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} R ( \\chi , \\sigma , m , r , \\kappa ) = \\max \\Big \\{ \\sqrt { 6 \\ , C _ { m , 0 } ( \\chi , \\sigma , r , \\kappa ) } \\ , r , \\ , C _ { \\ref { L e m m a C o r r e s p o n d e n c e L i n e a r N o n l i n e a r I n Z e r o } \\ref { I t e m N o n E m p t y F i x e d P o i n t S p a c e } } ( \\chi , \\sigma , m , r , \\mathcal { U } _ \\kappa ) ( m + 1 ) r + 1 \\Big \\} . \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} \\ , = \\ , \\Big \\lvert \\mathbb E ^ { 1 , 2 } \\big [ \\widetilde { b } ( v , x , \\omega _ { 1 } ) - \\widetilde { b } ( v , x , \\omega _ { 2 } ) \\big ] \\vert _ { \\{ x = X _ { v } ^ { \\mathrm m _ { 2 } } \\} } \\Big \\rvert \\ , \\le \\ , C _ { T } \\mathbb E ^ { 1 , 2 } \\big [ \\lvert \\omega _ { 1 , v } - \\omega _ { 2 , v } \\rvert \\wedge 1 \\big ] \\ , , \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} v _ { g , \\Lambda } ( k ) = g \\frac { \\chi _ \\Lambda ( \\omega ( k ) ) } { \\sqrt { \\omega ( k ) } } \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{gather*} T ^ { i j k } _ { P , Q } ( \\lambda , \\mu ) : = \\{ u ^ i _ \\lambda \\{ u ^ j _ \\mu u ^ k \\} _ P \\} _ Q , \\\\ J ^ { i j k } ( A , B ) : = T ^ { i j k } _ { A , B } ( \\lambda , \\mu ) - T ^ { j i k } _ { A , B } ( \\mu , \\lambda ) + T ^ { k i j } _ { A , B } ( - \\lambda - \\mu - \\partial , \\lambda ) , \\end{gather*}"}
-{"id": "8287.png", "formula": "\\begin{align*} p _ { _ 1 } ( y ) = p _ { _ 0 } ( y ) e ^ { - \\rho _ 1 y ^ 2 - \\rho _ 0 - 1 } . \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} N _ { D \\boxtimes F } ^ + [ ( d , f ) ] = N _ { D } ^ + [ d ] \\times N _ { F } ^ + [ f ] \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} f ( x ) & = - 1 - 2 x ^ 3 - m \\ : h ( x ) , \\\\ g ( x ) & = - 1 + ( x ^ 2 + x + 1 ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} D ( H ) = \\{ u \\in H ^ 1 ( \\R ) \\cap H ^ 2 ( \\R \\backslash \\{ 0 \\} ) : \\partial _ x u ( 0 + ) - \\partial _ x u ( 0 - ) = 2 q u ( 0 ) \\} \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} i \\partial _ t \\psi ( x , t ) = ( D _ m + V ( x ) ) \\psi ( x , t ) , \\psi ( x , 0 ) = \\psi _ 0 ( x ) . \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} f _ { k , m } ( z ) = q ^ { - m } + \\sum _ { \\substack { n > A _ k \\\\ ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) } } a _ k ( m , n ) q ^ n . \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} \\frac { \\partial ( G ^ { - 1 } ) _ i } { \\partial \\xi _ l } = ( \\Phi ^ { - 1 } ) _ { l l ' } ( a \\nabla \\xi _ { l ' } ) _ i \\ , , \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} \\Psi ( K _ i \\otimes 1 ) & = K _ i \\otimes 1 , & \\Psi ( 1 \\otimes K _ i ) & = 1 \\otimes K _ i , \\\\ \\Psi ( L _ i \\otimes 1 ) & = L _ i \\otimes 1 , & \\Psi ( 1 \\otimes L _ i ) & = 1 \\otimes L _ i , \\\\ \\Psi ( E _ i \\otimes 1 ) & = E _ i \\otimes L _ i ^ { - 1 } , & \\Psi ( 1 \\otimes E _ i ) & = K _ i ^ { - 1 } \\otimes E _ i , \\\\ \\Psi ( F _ i \\otimes 1 ) & = F _ i \\otimes L _ i , & \\Psi ( 1 \\otimes F _ i ) & = K _ i \\otimes F _ i . \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} S ( n , k ) = \\binom { n } { k } B _ { n - k } ^ { ( - k ) } s ( n , k ) = \\binom { n - 1 } { k - 1 } B _ { n - k } ^ { ( n ) } \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} ( a ^ { \\dagger } _ { \\oplus } ( v ) f ) ( k ) & = a ^ { \\dagger } _ + ( v ) f ( k ) \\\\ ( a _ { \\oplus } ( v ) f ) ( k ) & = a _ + ( v ) f ( k ) \\\\ ( \\varphi _ { \\oplus } ( v ) f ) ( k ) & = \\varphi _ + ( v ) f ( k ) \\\\ ( g ( d \\Gamma _ { \\oplus } ( \\omega ) + m ) f ) ( k ) & = g ( d \\Gamma _ + ( \\omega ) + m ( k ) ) f ( k ) . \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} \\mathrm { T r } _ { d , D } ( f ) : = \\sum _ { Q \\in \\mathcal { Q } ^ + _ { d D } / \\mathrm { S L } _ 2 ( \\mathbb { Z } ) } \\frac { \\chi _ D ^ { } ( Q ) } { w _ Q ^ { } } f ( \\alpha _ Q ) , \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { \\binom k j \\binom { k - j } s ( - 1 ) ^ { j + s } f _ 3 ^ j f _ 2 ^ s X _ { m + a k - c j - b s } } } = f _ 1 ^ k X _ m \\ , , \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} \\frac { d } { d t } I ( X _ 0 ; X _ t ) & = - J _ \\nu ( X _ 0 ; X _ t ) , \\\\ \\frac { d ^ 2 } { d t ^ 2 } I ( X _ 0 ; X _ t ) & = 2 K _ \\nu ( X _ 0 ; X _ t ) + 2 \\alpha J _ \\nu ( X _ 0 ; X _ t ) . \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} \\O ( I ) \\cap \\O ( K _ 2 ) & = \\O ( K _ 2 ) \\cap \\O ( K _ 1 ) = \\O ( K _ 1 ) \\cap \\O ( J ) \\\\ & = \\O ( J ) \\cap \\O ( K _ 3 ) = \\O ( K _ 3 ) \\cap \\O ( I ) = \\emptyset \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} U _ { i , j } = \\begin{cases} \\left \\{ \\frac { 1 } { 2 } ( 2 g - 3 j + 1 2 i + 2 5 ) , \\frac { 1 } { 2 } ( 2 g + 3 j - 6 i - 1 1 ) , g - 3 i - 7 \\right \\} _ { g } & \\mbox { i f $ j \\equiv 1 \\pmod { 2 } $ , } \\\\ \\left \\{ \\frac { 1 } { 2 } ( 2 g - 3 j + 1 2 i + 2 2 ) , \\frac { 1 } { 2 } ( 2 g + 3 j - 6 i - 1 4 ) , g - 3 k - 7 \\right \\} _ { g - 1 } & \\mbox { i f $ j \\equiv 0 \\pmod { 2 } $ , } \\\\ \\end{cases} \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} \\Gamma _ q ( z ) = \\dfrac { ( q ; q ) _ \\infty } { ( q ^ { z } ; q ) _ \\infty } ( 1 - q ) ^ { 1 - z } ( | q | < 1 ) . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ N \\omega _ j p ( x _ j ) = \\int _ { - 1 } ^ 1 w ( x ) p ( x ) d x \\quad \\forall p \\in \\mathbb { P } _ { 2 N + 1 } . \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} \\psi ( v ) = \\left ( \\psi ( v ^ 1 ) , \\psi ( v ^ 2 ) , . . . , \\psi ( v ^ N ) \\right ) , \\Psi ( v ) = \\sum \\limits _ { j = 1 } ^ { N } \\Psi ( v ^ j ) \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} \\lvert ( i _ 1 , j _ 1 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i _ 2 , j _ 2 ) \\rvert _ { W _ n ( w ) } = & \\lvert ( i _ 1 , j _ 1 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i _ 2 , j _ 1 ) \\rvert _ { W _ n ( w ) } \\\\ & + \\lvert ( i _ 2 , j _ 1 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i _ 2 , j _ 2 ) \\rvert _ { W _ n ( w ) } \\\\ = & i _ 1 - i _ 2 + j _ 1 - j _ 2 \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 1 - e ^ { \\pm t } ( 2 + 4 S ^ 2 - e ^ { \\pm t } ) \\mp e ^ { \\pm t } ( 1 - e ^ { \\pm 2 \\delta ( t ) e ^ { 2 i \\alpha } } ) \\Big ( 2 C S \\mp ( 1 + 2 S ^ 2 - e ^ { \\pm t } ) \\Big ) ~ . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} \\xi ' ( t ) = t ( 1 - t ^ 2 ) \\left ( \\abs { \\nabla u } ^ 2 + u ^ 2 \\right ) + t ^ 3 u ^ 4 \\left [ \\frac { f ( u ) } { u ^ 3 } - \\frac { f ( t u ) } { t ^ 3 u ^ 3 } \\right ] , \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} B _ t : = - \\int _ 0 ^ t u ( s , X _ s ) d s + \\tilde { B } _ t \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} \\gamma _ u = ( \\omega \\wedge \\omega ) _ 0 ^ u + u ( \\omega \\wedge \\omega ) _ 0 ' ( u ) . \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} 1 ^ 3 + 3 2 ^ 2 = 4 ^ 3 + 3 1 ^ 2 = 5 ^ 3 + 3 0 ^ 2 = 1 0 ^ 3 + 5 ^ 2 \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow \\infty } I ( u _ n ) \\geq \\int _ { \\Lambda } \\liminf _ { n \\rightarrow \\infty } \\left [ \\frac { 1 } { 4 } f ( u _ n ) u _ n - F ( u _ n ) \\right ] d x = \\infty , \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t , k } ^ { ( u ) } \\ , = \\ , - ( X _ { t , k } ^ { ( u ) } - u X _ { t , k + 1 } ^ { ( u ) } ) { \\mathrm d } t + { \\mathrm d } W _ { t , k } \\ , ; t \\ge 0 \\ , , k \\in \\mathbb N _ { 0 } \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} \\tilde \\rho ( n , m ) = \\tilde \\rho _ \\ell ( n , m ) = \\min _ { 1 \\leq i , j \\leq \\ell } | q _ i ( n ) - q _ j ( m ) | . \\end{align*}"}
-{"id": "9920.png", "formula": "\\begin{align*} \\psi _ { n } = C _ n \\dfrac { \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } y _ i d y _ 1 \\wedge \\dots \\wedge \\widehat { d y _ i } \\wedge \\dots \\wedge d y _ n } { \\Vert y \\Vert ^ n } \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} \\frac { Z ( n + 1 ) } { Z ( n ) } \\\\ = 1 - \\frac { i } { \\omega } b _ { n + 1 } ' \\vert \\varphi ( n ) \\vert ^ 2 ( e ^ { - 2 i ( \\eta ( n ) + \\gamma ( n ) ) } - 1 ) \\\\ \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } d ( y , A u _ k ) & = d ( y , A u ^ \\dag ) , \\\\ \\lim _ { k \\rightarrow \\infty } d ( u _ { k + 1 } , u _ k ) & = 0 . \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} C ( \\gamma ) = \\lim _ { R \\to \\infty } C ( \\gamma _ { R / 2 } ^ P ) = \\lim _ { R \\to \\infty } D _ R ( u _ R ) = D ( u ) . \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} \\lambda ^ { ( k s ) } _ { m a x } ( A ) = \\max _ { | S | = k s } \\lambda _ { m a x } ( A _ { S } ) . \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} A _ j & \\leq \\sum _ { i = - \\infty } ^ { 1 } ( 2 ^ { j } \\rho ) ^ d \\int _ { S ^ { d - 1 } } \\sup _ { r \\in [ 2 ^ { j } \\rho , 2 ^ { j + 1 } \\rho ] } | \\mathbf { K } _ { i , n } ( r \\theta ) | + ( 2 ^ { j } \\rho ) ^ d \\int _ { S ^ { d - 1 } } \\sup _ { r \\in [ 2 ^ { j } \\rho , 2 ^ { j + 1 } \\rho ] } | \\varphi _ { \\rho } ( r \\theta ) \\mathbf { K } _ { n } ( r \\theta ) | \\\\ & \\lesssim 2 ^ { j d } + \\sum _ { i = - \\infty } ^ { 1 } ( 2 ^ { j } \\rho ) ^ d \\int _ { S ^ { d - 1 } } \\sup _ { r \\in [ 2 ^ { j } \\rho , 2 ^ { j + 1 } \\rho ] } | \\mathbf { K } _ { i , n } ( r \\theta ) | , \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} \\rho ( x ) = \\rho ( \\varphi ( x ) ) \\left | \\det \\left ( \\frac { \\partial \\varphi ^ \\alpha } { \\partial x ^ \\beta } \\right ) \\right | . \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} W _ t & = \\frac { p ^ { P _ t } e ^ { \\delta t } } { X _ 0 ^ { - 1 } + \\vartheta \\int _ 0 ^ t p ^ { P _ s } e ^ { \\delta s } d s } . \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} f _ 2 \\colon \\mathcal { L } _ { \\mathbf { i } , \\lambda _ 1 , \\dots , \\lambda _ r } & \\stackrel { } { \\longrightarrow } \\mathcal { L } _ { \\mathbf { i } , \\mathbf { a } } \\\\ [ ( p _ { k , l } ) _ { k , l } , w ] & \\mapsto [ ( p _ { k , l } ) _ { k , l } , C ' w ] , \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} s _ { h } ( u , v ) = \\sum \\limits _ { D \\in \\mathcal { T } _ h } h _ D ^ { - 1 } \\langle u _ 0 - u _ b , v _ 0 - v _ b \\rangle _ { \\partial D } . \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} T _ 1 ( q ) & = 2 \\sum _ { 0 \\le k \\le n , \\ , \\mathrm { o r d } ( \\omega ) \\mid k } q ^ 2 ( { n \\brack k } _ q ' ) ^ 2 { n + k \\brack k } _ { \\omega } ^ 2 q ^ { f ( n , k ) } , \\\\ T _ 2 ( q ) & = 2 \\sum _ { 0 \\le k \\le n , \\ , \\mathrm { o r d } ( \\omega ) \\nmid k } q ^ 2 ( { n \\brack k } _ q ' ) ^ 2 { n + k \\brack k } _ { \\omega } ^ 2 q ^ { f ( n , k ) } . \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} ( x , F _ i y ) & = ( E _ i , F _ i ) ( r _ i ( x ) , y ) , & ( x , y F _ i ) & = ( E _ i , F _ i ) ( r _ i ' ( x ) , y ) , \\\\ ( E _ i x , y ) & = ( E _ i , F _ i ) ( x , \\rho _ i ' ( y ) ) , & ( x E _ i , y ) & = ( E _ i , F _ i ) ( x , \\rho _ i ( y ) ) . \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} X _ { P _ + } = X _ { \\hat { R } } + \\int _ 0 ^ 1 X _ { \\{ R ( t ) , F \\} \\circ \\Phi _ F ^ t } d t + X _ { ( P - R ) \\circ \\Phi _ F ^ 1 } . \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} \\bigcup _ { j = 0 } ^ { \\lceil N _ k / q _ k \\rceil } B \\Big ( ( i + j q _ k ) \\theta , { 1 \\over N _ k ^ \\tau } \\Big ) , \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } t ^ { k - 3 / 2 } ( 1 - x t ) ^ { - 1 } \\left ( \\tfrac { 1 - t } { 1 - t x } \\right ) ^ { k - 3 / 2 } \\d t \\leq ( 1 - x ) ^ { - 1 } \\int _ { 0 } ^ { 1 } t ^ { k - 3 / 2 } \\d t = ( 1 - x ) ^ { - 1 } \\tfrac { 1 } { k - 1 / 2 } . \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} & \\left \\| R ( \\epsilon ; t - s u _ { ( n ) } ) \\cdots R ( \\epsilon ; t - s u _ { ( 1 ) } ) - \\left ( \\tilde { R } ^ n - \\epsilon \\sum _ { i = 0 } ^ { n - 1 } s u _ { ( i + 1 ) } \\tilde { R } ^ { n - i - 1 } \\tilde { R } ^ { ( 1 ) } \\tilde { R } ^ { i } \\right ) \\right \\| \\\\ & \\le \\epsilon ^ 2 d '' s ^ 3 ( 1 + \\epsilon c ' s ) ^ n \\le \\epsilon ^ 2 d '' s ^ 3 ( 1 + \\epsilon c ' s ) ^ { \\frac { 5 } { 4 } \\tilde { \\lambda } s } \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} \\frac { q _ i ^ { - r } ( q _ i ^ { t - r + 1 } - q _ i ^ { - t + r - 1 } ) } { q _ i ^ { t + 1 } - q _ i ^ { - t - 1 } } + \\frac { q _ i ^ { t + 1 - r } ( q _ i ^ { r } - q _ i ^ { - r } ) } { q _ i ^ { t + 1 } - q _ i ^ { - t - 1 } } = \\frac { q _ i ^ { t - 2 r + 1 } - q _ i ^ { - t - 1 } + q _ i ^ { t + 1 } - q _ i ^ { t - 2 r + 1 } } { q _ i ^ { t + 1 } - q _ i ^ { - t - 1 } } = 1 , \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} h ( p ) - I ( X _ 0 ; X _ t ) & \\le 3 \\sum _ { i = 1 } ^ k p _ i \\sum _ { j \\neq i } \\| p \\| _ \\infty e ^ { - 0 . 0 8 5 \\frac { \\alpha m ^ 2 } { e ^ { 2 \\alpha t } - 1 } } \\\\ & = 3 ( k - 1 ) \\| p \\| _ \\infty e ^ { - 0 . 0 8 5 \\frac { \\alpha m ^ 2 } { e ^ { 2 \\alpha t } - 1 } } . \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\mathbf { p } _ i ( k + 1 ) : = & \\min _ { \\mathbf { p } _ c } f _ i ( \\mathbf { p } _ i , k + 1 ) + ( \\boldsymbol { \\lambda } ^ { [ k ] } - \\boldsymbol { \\lambda } ^ { [ k - 1 ] } ) ^ T \\mathbf { p } _ i + \\\\ & + \\frac { \\rho } { 2 } \\| \\mathbf { p } _ i - \\mathbf { p } _ i ^ { [ k ] } \\| ^ 2 \\\\ & \\underline { \\mathbf { p } } _ i ^ { [ k + 1 ] } \\leq \\mathbf { p } _ i \\leq \\overline { \\mathbf { p } } _ i ^ { [ k + 1 ] } \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} w ' ( e ) = \\begin{cases} ( n + m ) / 2 & \\\\ - ( n + m - 1 ) / 2 & \\end{cases} \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty n c ( n ) < \\infty . \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\alpha ( y ) = e ^ { - 2 y ^ { 2 } } \\left [ C - 2 \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\int _ { 1 } ( 4 y ^ { 2 } - 1 ) + c o n s t \\int _ { 1 } e ^ { 2 y ^ { 2 } } \\right ] \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} \\| e ^ { - \\Delta ^ { 2 } t } h _ { 0 } \\| _ { \\mathcal { B } _ { \\alpha } ^ { 2 } } = r _ { 0 } . \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( X \\cdot ( h \\otimes \\beta ( a ) ) ) = 0 . \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} \\mathbb { Z } ( \\mathcal { T } _ \\xi ) \\rtimes \\mathcal { S } = \\bigoplus _ { \\nu \\in P / ( ( l ' Q ^ \\vee ) \\cap Q ) } \\mathbb { Z } ( \\mathcal { T } _ \\xi ) _ \\nu . \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\varepsilon \\Delta \\rho - \\mathcal { L } _ R ^ { 0 } [ \\rho ] = F , \\quad & x \\in B _ R , \\\\ \\rho = 0 & x \\in \\partial B _ R , \\end{array} \\right . \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} { } _ { p } F _ { q } \\left ( \\ \\begin{array} { l l l } \\alpha _ { 1 } , . . . , \\alpha _ { p } ~ ; ~ \\\\ \\beta _ { 1 } , . . . , \\beta _ { q } ~ ; ~ \\end{array} z \\right ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\alpha _ { 1 } ) _ { n } . . . ( \\alpha _ { p } ) _ { n } } { ( \\beta _ { 1 } ) _ { n } . . . ( \\beta _ { q } ) _ { n } } \\frac { z ^ { n } } { n ! } ~ , \\newline \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} S _ { o u t } = S ^ { ( 2 ) } _ { \\mu , \\xi } + \\sum _ { j = 1 } ^ k ( 1 - \\eta _ { j , R } ) S _ { \\mu , \\xi , j } \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} G _ { i } ( p , p ) = H _ { i } ( p ) \\ , \\ , p \\in [ - L _ { c } , L _ { c } ] , \\ , \\ , i \\in \\{ 1 , 2 , \\dots , K \\} . \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} \\mathcal T ^ * I _ \\pi = J _ \\pi . \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} ( U _ { s } , W _ { t } ) : = ( \\bigcup _ { t \\notin u T s } V _ u , \\bigcup _ { s \\not \\in v T t } V _ v ) . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} \\dfrac { d } { d x } ( [ \\theta ( x , { E } ) - \\eta ( x , E ) ] - [ \\theta ( x , { \\hat E } ) - \\eta ( x , \\hat { E } ) ] = ( k ( E ) - k ( \\hat { E } ) ) + \\frac { O ( 1 ) } { 1 + x - b } . \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\liminf _ { \\lambda \\uparrow \\lambda _ 0 } \\theta ( \\lambda ) = \\liminf _ { \\lambda \\uparrow \\lambda _ 0 } \\sum _ { n = 1 } ^ { \\infty } f ^ { ( n ) } _ \\lambda ( o , \\ , o ) \\ge \\sum _ { n = 1 } ^ { \\infty } \\liminf _ { \\lambda \\uparrow \\lambda _ 0 } f ^ { ( n ) } _ \\lambda ( o , \\ , o ) = \\sum _ { n = 1 } ^ { \\infty } f ^ { ( n ) } _ { \\lambda _ 0 } ( o , \\ , o ) = \\theta ( \\lambda _ 0 ) , \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} \\alpha : = \\inf _ { r \\geq 0 } \\{ \\max _ { 1 \\leq i \\leq n } | c _ { i r } | \\} , \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} h = \\varphi ^ 3 + \\psi ^ 2 , \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} g \\big ( \\dot z ( 1 ) , V ( 1 ) \\big ) - g \\big ( \\dot z ( 0 ) , V ( 0 ) \\big ) = 0 . \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} A _ k = - \\frac { p _ k } { 2 } , \\ \\ \\ \\ a _ k = - \\frac { \\xi _ k } { 2 } . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{gather*} T _ 2 T _ 3 \\cdots T _ { N - 1 } \\big ( x _ { N - 1 } ^ - \\big ) = - T _ 2 T _ 3 \\cdots T _ { N - 2 } \\big ( x _ { N - 1 } ^ + \\big ) = - E _ { 2 , N } \\end{gather*}"}
-{"id": "6723.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { ( - 1 ) ^ { n ( j + s ) + j } \\binom k j \\binom j s \\frac { { G _ 0 ^ { j - s } G _ { n + r } ^ s } } { { G _ n ^ j } } H _ { m + ( n + r ) k - n j + ( n - r ) s } } } = ( - 1 ) ^ { ( n + 1 ) k } \\left ( { \\frac { { G _ r } } { { G _ n } } } \\right ) ^ k H _ m , G _ n \\ne 0 \\ , , \\end{align*}"}
-{"id": "10000.png", "formula": "\\begin{align*} \\mu * \\nu ( E ) = \\int _ { \\mathbb R ^ + } \\int _ { \\mathbb R ^ + } \\mathbf 1 _ { E } ( x + y ) d \\mu ( x ) d \\nu ( y ) . \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} \\begin{pmatrix} ( a _ n + a _ n ' ) u ( n ) \\\\ u ( n - 1 ) \\end{pmatrix} = & \\frac { 1 } { 2 i } \\left ( Z ( n ) \\begin{pmatrix} a _ n \\varphi ( n ) \\\\ \\varphi ( n - 1 ) \\end{pmatrix} - \\overline { Z ( n ) } \\begin{pmatrix} a _ n \\overline { \\varphi ( n ) } \\\\ \\overline { \\varphi ( n - 1 ) } \\end{pmatrix} \\right ) \\\\ = & \\mathrm { I m } \\left [ Z ( n ) \\begin{pmatrix} a _ n \\varphi ( n ) \\\\ \\varphi ( n - 1 ) \\end{pmatrix} \\right ] . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} A \\cdot \\mu ( \\cdot , \\cdot ) = A \\ , \\mu ( A ^ { - 1 } \\cdot , A ^ { - 1 } \\cdot ) \\ , , \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D O } _ \\textup { t } ( 8 n + 7 ) q ^ n & = 8 \\dfrac { f _ 2 ^ { 1 4 } f _ 3 f _ { 6 } ^ 4 f _ { 8 } ^ 2 } { f _ 1 ^ { 1 4 } f _ 4 ^ 3 f _ { 1 2 } ^ 2 } + 1 6 \\dfrac { f _ 2 ^ 9 f _ 3 ^ 2 f _ 4 ^ 5 f _ { 6 } } { f _ 1 ^ { 1 3 } f _ { 8 } ^ 2 } + 3 2 q \\dfrac { f _ 2 ^ { 8 } f _ 3 ^ 3 f _ 4 f _ { 8 } ^ 2 f _ { 1 2 } ^ 2 } { f _ 1 ^ { 1 2 } f _ { 6 } ^ 2 } , \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\mathrm { d } _ { \\left ( \\widetilde { \\mu } _ { g ^ { - 1 } } ( x _ { 1 } ) , x _ { 2 } \\right ) } \\psi = \\left ( \\alpha _ { \\widetilde { \\mu } _ { g ^ { - 1 } } } ( x _ { 1 } ) , - \\alpha _ { x _ { 2 } } \\right ) + O ( \\delta ) . \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} A _ { 1 , s } = a _ { 0 } b _ { 0 } c _ { 0 } , \\ \\ \\ A _ { 1 , s - 1 } = \\sum _ { r = 1 } ^ { n } \\frac { S _ { 0 0 r } } { r } , \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} K _ { 2 } \\left ( \\inf _ { | v | = 1 } | D f ( x ) ( v ) | \\right ) ^ d \\geq J _ { f } ( x ) \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} [ X , Y ] ( v ) : = d X ( v ) [ Y ( v ) ] - d Y ( v ) [ X ( v ) ] , \\quad \\forall \\ v \\in D ( s , r ) . \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} & ( 2 a _ { 1 } ^ 2 + 2 a _ { 2 } ^ 2 - 1 ) x _ 1 + 4 a _ 1 a _ 2 x _ 2 + 2 ( a _ 1 + a _ 2 ) \\bar { a } ^ T \\bar { x } = \\frac { s _ 1 } { \\lambda } , \\\\ & ( 2 a _ { 1 } ^ 2 + 2 a _ { 2 } ^ 2 - 1 ) x _ 2 + 4 a _ 1 a _ 2 x _ 1 + 2 ( a _ 1 + a _ 2 ) \\bar { a } ^ T \\bar { x } = \\frac { s _ 2 } { \\lambda } , \\\\ & 2 ( a _ 1 + a _ 2 ) x _ 1 \\bar { a } + 2 ( a _ 1 + a _ 2 ) x _ 2 \\bar { a } + \\bar { x } + 4 \\bar { a } ^ T \\bar { x } \\bar { a } = \\frac { \\bar { s } } { \\lambda } . \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} w _ { \\sigma } ^ 0 = \\mbox { i d } , \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} W ( x , y ) = \\sum _ { k = 1 } ^ { \\infty } \\zeta _ { 1 } ( n + \\{ y \\} + k ) \\zeta _ { 2 } ( n - \\{ x \\} - k ) \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} d _ M : = \\inf \\left \\{ E ( v ) \\ : \\ v \\in H ^ 1 , \\| v \\| ^ 2 _ { L ^ 2 } = M \\right \\} ; \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} t _ h ' & = - \\frac { b t _ h } { h - ( b + 1 ) t _ h } < 0 , \\\\ T _ h ' & = - \\frac { b T _ h } { h - ( b + 1 ) T _ h } > 0 . \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} \\frac { I } { 2 \\pi p } & = \\frac { \\alpha ^ 2 - 1 } { 2 - p + 2 \\alpha } + \\frac { 2 } { 2 - p + \\alpha } - \\frac { 1 } { 2 - p } \\\\ & = \\frac { \\alpha ^ 2 ( \\alpha ( 2 - p ) + p ^ 2 - 4 p + 2 ) } { ( 2 - p + 2 \\alpha ) ( 2 - p + \\alpha ) ( 2 - p ) } \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} c : = c _ i = c _ { i + 1 } . \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} ( \\d T ) _ x = \\left ( \\begin{matrix} 1 & 0 \\\\ 0 & * \\end{matrix} \\right ) \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} S _ f ( \\rho _ 1 + \\rho _ 2 \\| \\sigma _ 1 + \\sigma _ 2 ) = S _ f ( \\rho _ 1 \\| \\sigma _ 1 ) + S _ f ( \\rho _ 2 \\| \\sigma _ 2 ) . \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} ( - 1 ) ^ k & \\Big ( n \\frac { ( 3 d ^ 2 + 1 ) n ^ 2 - 6 d ^ 2 n + 2 d ^ 2 } { 1 2 d } - 2 \\binom { n } { 3 } \\Big ) \\sum _ { 0 \\le s \\le \\frac { k d } { n } - 1 } \\binom { d } { s } ( - 1 ) ^ { s } ( k - \\frac { n } { d } s ) \\\\ & - ( - 1 ) ^ k \\binom { n } { 2 } \\sum _ { 0 \\le s \\le \\frac { k d } { n } - 1 } \\binom { d } { s } ( k - \\frac { n } { d } s - 1 ) ( - 1 ) ^ { s } \\\\ & - ( - 1 ) ^ k \\binom { n } { 2 } ^ 2 \\sum _ { 0 \\le s \\le \\frac { k d } { n } - 1 } \\binom { d } { s } ( - 1 ) ^ { s } . \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} \\widehat { u } \\ , : = \\ , \\Big ( \\int ^ { T } _ { 0 } \\mathbb E \\big [ \\widetilde { X } _ { t } ^ { 2 } \\vert \\mathcal F _ { T } ^ { X } \\big ] { \\mathrm d } t \\Big ) ^ { - 1 } \\cdot \\mathbb E \\Big [ \\int ^ { T } _ { 0 } X _ { t } \\widetilde { X } _ { t } { \\mathrm d } t + \\int ^ { T } _ { 0 } \\widetilde { X } _ { t } { \\mathrm d } X _ { t } \\ , \\Big \\vert \\ , \\mathcal F _ { T } ^ { X } \\Big ] \\ , \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} \\begin{cases} v _ \\delta ( y ) = v ( y ) & y \\notin B _ { \\delta ^ { 1 / 2 } } ( \\bar x ) , \\\\ v ( y ) \\leq v _ \\delta ( y ) \\leq v ( y ) + \\delta & y \\in B _ { \\delta ^ { 1 / 2 } } ( \\bar x ) . \\end{cases} \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} \\begin{gathered} B ^ { ( m ) } _ { n - m } = A ^ { ( m - 1 ) } _ { n - m + 1 } A ^ { ( m ) } _ { n - m } = B ^ { ( m - 1 ) } _ { n - m + 1 } + a _ { n - m + 1 } A ^ { ( m - 1 ) } _ { n - m + 1 } \\\\ B ^ { ( 0 ) } _ { n } = 0 B ^ { ( 1 ) } _ { n - 1 } = 1 = A ^ { ( 0 ) } _ { n } \\end{gathered} \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\psi = - ( - \\Delta ) ^ { s } \\psi + V _ { \\mu , \\xi } \\psi + f ( x , t ) \\quad & \\quad \\Omega \\times ( t _ 0 , \\infty ) , \\\\ \\psi = g \\quad & ( \\mathbb { R } ^ n \\setminus \\Omega ) \\times ( t _ 0 , \\infty ) , \\\\ \\psi ( \\cdot , t _ 0 ) = h \\quad & \\quad \\mathbb { R } ^ n , \\end{cases} \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\sigma ( d \\Gamma ^ { ( n ) } ( \\omega ) ) & = \\overline { \\{ \\lambda _ 1 + \\cdots + \\lambda _ n \\mid \\lambda _ i \\in \\sigma ( \\omega ) \\} } , \\\\ \\inf ( \\sigma ( d \\Gamma ^ { ( n ) } ( \\omega ) ) ) & = n m . \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} \\mu _ { \\left \\{ x \\right \\} } ( d \\phi _ x ) = \\frac { 1 } { Z } \\exp \\left \\{ - \\frac { 1 } { 2 } \\phi _ x ^ 2 \\sum \\limits _ { y : | y - x | = 1 } \\omega _ { x y } + \\phi _ x \\sum \\limits _ { y : | y - x | = 1 } \\omega _ { x y } \\phi _ y \\right \\} \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} \\varphi _ { k , m } ( z , s ) : = \\left \\{ \\begin{array} { l l } \\Gamma ( 2 s ) ^ { - 1 } ( 4 \\pi | m | y ) ^ { - k / 2 } M _ { \\mathrm { s g n } ( m ) \\frac { k } { 2 } , s - 1 / 2 } ( 4 \\pi | m | y ) e ^ { 2 \\pi i m x } & m \\neq 0 , \\\\ y ^ { s - k / 2 } & m = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} \\begin{pmatrix} ( g _ 1 , h _ { 1 } ) _ { \\Omega } & \\dots & ( g _ m , h _ { 1 } ) _ { \\Omega } \\\\ \\vdots & \\vdots & \\vdots \\\\ ( g _ 1 , h _ { m } ) _ { \\Omega } & \\cdots & ( g _ m , h _ { m } ) _ { \\Omega } \\end{pmatrix} \\begin{pmatrix} a _ 1 \\\\ \\vdots \\\\ a _ m \\end{pmatrix} = v _ 1 ( \\epsilon ) + v _ { 2 } ( \\epsilon ) , \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} C _ \\alpha = \\frac { \\alpha ^ 2 ( 1 - \\alpha ) } { 2 } W + ( 1 - \\alpha ^ 2 ) P + \\frac { \\alpha ^ 2 ( 1 + \\alpha ) } { 2 } M , \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} U _ 1 ( q ) & = 2 \\sum _ { k , \\ell , \\ , \\mathrm { o r d } ( \\omega ) \\mid k } ( q { n \\brack k } ' _ q ) ^ 2 q ^ { f ( n , k , \\ell ) } { n \\brack \\ell } _ q { k \\brack \\ell } _ q { k + \\ell \\brack n } _ q , \\\\ U _ 2 ( q ) & = 2 \\sum _ { k , \\ell , \\ , \\mathrm { o r d } ( \\omega ) \\nmid k } ( q { n \\brack k } ' _ q ) ^ 2 q ^ { f ( n , k , \\ell ) } { n \\brack \\ell } _ q { k \\brack \\ell } _ q { k + \\ell \\brack n } _ q . \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} \\begin{cases} \\delta ^ 0 ( * ) = 0 \\ , \\\\ \\delta ^ 1 ( * ) = 1 \\ , \\\\ \\sigma ( i ) = * \\ \\forall \\ i \\in \\square ^ 1 \\ , \\\\ \\gamma ( i , j ) = \\mathrm { m a x } ( i , j ) \\ \\forall \\ i , j \\in \\square ^ 1 \\ . \\end{cases} \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} s _ + = \\frac { B + \\sqrt { B ^ 2 + 2 4 A C } } { 4 C } = \\frac { 3 + \\sqrt { 9 + 8 t } } { 1 2 C / B } \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} \\prod _ { 1 \\leq i \\leq j \\leq n } \\dfrac { r + i + j - 1 } { i + j - 1 } & = \\prod _ { j = 1 } ^ n \\dfrac { ( r + 2 j - 1 ) \\cdots ( r + j ) } { ( 2 j - 1 ) \\cdots j } = \\prod _ { j = 1 } ^ n \\dfrac { ( r + 2 j - 1 ) ! ( j - 1 ) ! } { ( r + j - 1 ) ! ( 2 j - 1 ) ! } \\\\ & = \\dfrac { \\Phi ( r + 2 n - 1 ) F ( r - 1 ) F ( n - 1 ) } { \\Phi ( r - 1 ) F ( r + n - 1 ) \\Phi ( 2 n - 1 ) } \\\\ & = \\dfrac { \\Phi ( r + 2 n - 1 ) \\Phi ( r - 2 ) F ( n - 1 ) } { F ( r + n - 1 ) \\Phi ( 2 n - 1 ) } , \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\| \\gamma ^ { ( d ) } _ { n } ( A ) \\| = \\max _ { 1 \\leq k \\leq r } \\| \\Gamma ' _ k \\| . \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} | | X _ { Q _ 2 } | | _ { s , r , p - 1 , \\mathbf { a } ; \\Xi _ r } = O ( r ^ { \\frac 7 4 } ) . \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} f _ { \\epsilon , t } ( x - 1 , x ) & = 1 + \\frac { \\epsilon } { M ( u t ) } \\sum _ { y \\in ( - \\epsilon t , x ) } G ^ \\omega _ { ( - \\epsilon t , x ) } ( x - 1 , y ) \\mu ( y ) f _ { \\epsilon , t } ( y , x ) \\\\ & \\le 1 + \\frac { \\epsilon } { M ( u t ) } \\sum _ { y \\in ( - \\epsilon t , x ) } G ^ \\omega _ { ( - \\epsilon t , x ) } ( x - 1 , y ) \\mu ( y ) ( 1 + \\eta ( \\epsilon ) ) ^ { x - y } . \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { s - 1 } \\beta ^ t \\alpha _ i ^ { u + p _ i - 1 } = \\Big ( \\sum _ { t = 0 } ^ { s - 1 } \\beta ^ t \\alpha _ i ^ { p _ i - 1 } \\Big ) \\alpha _ i ^ u \\in S _ i ^ { ( 2 ) } \\alpha _ i ^ u \\subseteq K . \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} \\dfrac { x ^ { 2 } } { y } - \\dfrac { y } { x ^ { 2 } } & = \\dfrac { 4 q } { K } , \\\\ x y ^ { 2 } - \\dfrac { q ^ { 2 } } { x y ^ { 2 } } & = K , \\\\ \\dfrac { y ^ { 3 } } { x } + \\dfrac { q ^ { 2 } x } { y ^ { 3 } } & = K - 2 q + \\dfrac { 4 q ^ { 2 } } { K } . \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} \\psi _ 1 ( \\textstyle { \\frac { 3 } { 2 } } , \\beta ) = I ' ( \\textstyle { \\frac { 3 } { 2 } } u ^ + + \\beta u ^ - ) [ \\textstyle { \\frac { 3 } { 2 } } u ^ + ] < 0 , \\beta \\in [ \\textstyle { \\frac { 1 } { 2 } } , \\textstyle { \\frac { 3 } { 2 } } ] . \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} | Q _ { - j , m } | ^ { - \\frac { 1 } { p } } \\Big ( \\sum _ { k : Q _ { 0 , k } \\subset Q _ { - j , m } } \\ ! \\ ! \\ ! | \\lambda _ k | ^ p \\Big ) ^ \\frac { 1 } { p } \\leq \\ | Q _ { - j , m } | ^ { - \\frac { 1 } { p } } \\| \\lambda | \\ell _ \\infty \\| \\ | Q _ { - j , m } | ^ { \\frac 1 p } = \\| \\lambda | \\ell _ \\infty \\| , \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\partial _ t v + ( v . \\nabla ) v = \\nu \\Delta v - \\nabla p , \\hbox { d i v } ~ v ( t , \\cdot ) = 0 , \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} \\ell ( G , \\mathcal M ^ 2 _ F ) = u ( G , \\mathcal M ^ 2 _ F ) = u ( G , \\mathcal M ^ 2 _ S ) = u ( G , \\mathcal M ^ 2 _ R ) = 6 / \\pi ^ 2 \\ , \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} \\langle \\tilde { J } _ \\mu ^ { ( \\alpha ) } ( X ) , s _ \\mu \\rangle & = \\sum _ { k = 0 } ^ n ( \\ell \\cdot \\ell ! h _ k ( B ( c _ 1 , \\ldots , c _ n ) ) + \\ell ! h _ k ( B ( d _ 1 , \\ldots , d _ n ) ) ) { \\alpha + k \\choose n } \\\\ & = \\sum _ { k = 0 } ^ n ( \\ell \\cdot \\ell ! r _ k ( B ( c _ 1 , \\ldots , c _ n ) ) + \\ell ! r _ k ( B ( d _ 1 , \\ldots , d _ n ) ) ) { \\alpha \\choose k } k ! \\\\ \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} A + B + C = ( q - \\ell + 1 ) \\prod _ { i = 2 } ^ { s - 1 } \\left ( q - p _ { i - 1 } + 1 \\right ) , \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} \\tilde { n } _ { \\kappa } ^ { [ m - 1 ] } ( a _ i ) = \\deg \\mathrm { r \\tilde { a } d } _ { \\kappa } ^ { [ m - 1 ] } ( a _ i ) = \\sum _ { w \\in \\C } \\Bigl ( \\mathrm { o r d } _ { w } ( a _ i ) - \\min _ { 0 \\leq j \\leq m - 1 } \\bigl \\{ \\mathrm { o r d } _ { w + j \\kappa } ( a _ i ) \\bigr \\} \\Bigr ) \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} G _ 2 ( L ) = \\{ ( \\left ( \\begin{array} { c c } \\beta \\cdot I _ 2 & \\\\ & I _ 2 \\end{array} \\right ) ) _ { \\sigma \\in I _ F } \\ \\vert \\beta \\in L ^ { \\times } \\} \\hookrightarrow T _ 1 ( L ) \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} C ( t ) = f _ { t } \\left ( X _ t ^ 1 , X _ t ^ 2 , \\ldots , X _ t ^ n \\right ) . \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} \\xi ( x ) = ( x _ 1 , x _ 2 , \\cdots , x _ m ) ^ T \\ , , \\forall \\ , x \\in \\mathbb { R } ^ n \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} \\mathcal { D } ( p _ { _ 1 } | | p _ { _ 0 } ) = \\int _ { \\mathcal { Y } } p _ { _ 1 } ( y ) \\log \\frac { p _ { _ 1 } ( y ) } { p _ { _ 0 } ( y ) } d { y } , \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} f ( x , u , \\bar { u } ) = \\mu | u | ^ 2 u + f _ { \\geq 4 } ( x , u , \\bar { u } ) , 0 \\neq \\mu \\in \\mathbb { R } , \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} J ^ \\alpha _ c ( u ) : = \\frac { \\| u \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } } { \\| u \\| ^ { \\frac { 4 - ( d - 2 ) \\alpha } { 2 } } _ { L ^ 2 } \\| u \\| ^ { \\frac { d \\alpha } { 2 } } _ { \\dot { H } ^ 1 _ c } } . \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} E ( v _ n ) = E ( \\tilde { V } ^ { j _ 0 } ) + E ( \\tilde { v } ^ { j _ 0 } _ n ) + o _ n ( 1 ) , \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } i \\partial _ t u + \\Delta u + c | x | ^ { - 2 } u & = & - | u | ^ \\alpha u , ( t , x ) \\in \\R ^ + \\times \\R ^ d , \\\\ u ( 0 ) & = & u _ 0 \\in H ^ 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} \\partial _ n w ^ - _ 0 = a _ 0 \\lambda \\nu _ \\lambda \\Longrightarrow \\nu _ \\lambda = \\frac { \\rho _ \\lambda c _ \\Omega \\partial _ n K _ \\gamma ^ - } { a _ 0 \\lambda } . \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} G _ i ( x ) = ( k - 1 ) ^ { i / 2 } U _ i ( \\frac { x } { 2 \\sqrt { k - 1 } } ) , \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} \\mathbf { M } ^ { \\varepsilon } ( \\mu ) ( x ) \\leq C \\mathbf { M } ^ 1 ( | D f ^ { \\nu } _ { x _ \\nu } | , \\tilde { H } _ { \\nu } ) ( \\langle x , \\nu \\rangle \\nu ) , ~ x _ \\nu : = x - \\langle x , \\nu \\rangle \\nu , \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} I _ { j k } ( x ; z ) = \\int _ { - \\infty } ^ { + \\infty } \\mathrm { d } \\theta \\ , \\int _ 0 ^ { + \\infty } \\mathrm { d } u \\left [ e ^ { \\imath \\ , \\sqrt { k } \\ , \\Psi _ x ( u , \\theta ) } \\cdot \\mathcal { A } _ { j k } ( x ; u , \\theta , z ) \\right ] , \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} \\left | \\sum _ { t = n _ 0 } ^ n f ( t ) \\frac { \\sin 2 \\theta ( t , E _ j ) \\sin 2 \\theta ( t , E ) } { t - v } \\right | \\leq D ( E , A , \\varepsilon ) + { \\varepsilon } \\ln \\frac { n - v } { n _ 0 - v } , \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} ( - 1 ) ^ r F _ n G _ m = F _ { n + r } G _ { m + r } - F _ r G _ { m + n + r } \\ , . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { L } ( t ) & = \\mathcal { T } ( t ) - \\mathcal { V } ( t ) \\\\ & = \\frac { 1 } { 2 } \\int _ { - \\infty } ^ \\infty u _ t ^ 2 - u _ x ^ 2 - ( u ^ 2 - 1 ) ^ 2 - \\frac { 1 } { 2 } \\Omega ^ 2 x ^ 2 u ^ 2 \\ , d x . \\end{aligned} \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\Gamma ( N , M ) \\coloneqq \\left \\{ \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in S L _ 2 ( \\mathbb { Z } ) \\colon a \\equiv d \\equiv 1 \\pmod { N } , \\ , c \\equiv 0 \\pmod { N M } \\right \\} . \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } u ^ { \\theta } _ { t } ( 0 , t _ { 0 } ) + H _ { i } ( t _ { 0 } , x , u ^ { \\theta } _ { x _ { i } } ( \\cdot , t _ { 0 } ) ) - ( D T _ { \\theta } + \\zeta ) = 0 & \\ , \\ , I _ { i } ^ { \\nu } \\\\ \\sum _ { i = 1 } ^ { K } u _ { x _ { i } } ^ { \\theta } ( \\cdot , t _ { 0 } ) = 0 & \\ , \\ , \\{ 0 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} h _ { t , s , a } \\cdot ( z ' , z _ n ) = \\left ( \\frac { t z ' } { ( \\sinh s ) z _ n + \\cosh s } , \\frac { ( \\cosh s ) z _ n + \\sinh s } { ( \\sinh s ) z _ n + \\cosh s } \\right ) . \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} T _ { k ' } = \\begin{cases} [ U _ { 2 k ' - 2 , 6 k ' - 3 } , U _ { 2 k ' - 1 , 6 k ' } , U _ { 2 k ' - 1 , 6 k ' + 1 } , U _ { 2 k ' , 6 k ' + 4 } ] & \\mbox { i f $ n \\equiv 1 \\pmod { 6 } $ , } \\\\ [ U _ { 2 k ' - 1 , 6 k ' + 1 } , U _ { 2 k ' , 6 k ' + 4 } , U _ { 2 k ' , 6 k ' + 5 } , U _ { 2 k ' + 1 , 6 k ' + 8 } ] & \\mbox { i f $ n \\equiv 5 \\pmod { 6 } $ . } \\\\ \\end{cases} \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} \\partial ^ 2 _ { i _ 1 i _ 2 } \\phi ( x ) = - ( R R ^ \\ast ) _ { i _ 1 i _ 2 } ( R R ^ \\ast ) _ { i _ 2 i _ 1 } = - ( R R ^ \\ast ) ^ 2 _ { i _ 1 i _ 2 } , \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} - \\Delta _ { g } \\bigl ( h ( r ) \\bigr ) = - \\left \\{ \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + ( n - 1 ) S ( r ) \\frac { \\partial } { \\partial r } \\right \\} h ( r ) , \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } g ( x ) \\mathrm m _ { t } ( { \\mathrm d } x ) \\ , = \\ , \\int _ { \\mathbb R } g ( x ) \\mathrm m _ { 0 } ( { \\mathrm d } x ) + \\int ^ { t } _ { 0 } [ \\mathcal A _ { s } ( \\mathrm M ) g ] \\ , { \\mathrm d } s \\ , ; 0 \\le t \\le T \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} d X = - \\nabla f ( X ) \\ , d t + \\sqrt { 2 } \\ , d W \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} e _ { n , k } ( \\omega ^ { g ' } _ n ) = | C _ { n , k } ^ g | . \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } \\int _ 0 ^ { \\pi } | \\cos n x | d x = \\frac { 2 } { \\pi } , \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} U \\phi ( x _ 1 , x _ 2 , t ) = \\phi ' ( x _ 1 , x _ 2 , t ) , \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} ( X + A + \\Phi ) u = - f , u | _ { \\partial _ - ( S M ) } = 0 . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} \\dot { \\psi } _ { i , j } = \\sum _ { i ' , j ' } \\sin ( \\psi _ { i ' , j ' } - \\psi _ { i , j } ) \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} H _ { i j } ( d x ) = \\mathbb { P } ( Y _ 1 \\in { d x } | J _ 0 = i , J _ 1 = j ) = \\frac { F _ { i j } ( d x ) } { p _ { i j } } . \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} c _ { i , j } : = \\sum _ { i ( \\ell ) = i } \\epsilon _ \\ell ^ j x _ \\ell . \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} \\bigcup _ { 0 \\leq j < N _ k } B \\big ( j \\theta , { 1 \\over N _ k ^ \\tau } \\big ) \\subset \\bigcup _ { i = 0 } ^ { q _ k - 1 } \\bigcup _ { j = 0 } ^ { \\lceil N _ k / q _ k \\rceil } B \\Big ( ( i + j q _ k ) \\theta , { 1 \\over N _ k ^ \\tau } \\Big ) . \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} \\check \\Phi ( \\chi ) ( x ) = \\int _ { A _ X } a \\cdot \\Phi ( x ) \\chi ^ { - 1 } ( a ) d a \\in C ^ \\infty ( A _ X , \\chi \\backslash X _ \\emptyset ) \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} \\nu ( B ) = \\mu \\big ( ( \\nabla \\varphi ) ^ { - 1 } ( B ) \\big ) \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} c _ { M , L _ \\xi ( \\lambda , \\mu ) } \\circ c _ { L _ \\xi ( \\lambda , \\mu ) , M } ( v _ { \\lambda , \\mu } \\otimes m ) = \\xi ^ { \\langle \\lambda , \\mu ' \\rangle } c _ { M , L _ \\xi ( \\lambda , \\mu ) } ( m \\otimes v _ { \\lambda , \\mu } ) \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} | { } _ 2 F _ 1 \\left ( k - \\tfrac { 1 } { 2 } , k + \\tfrac { z - 1 } { 2 } ; 2 k - 1 ; x \\right ) | & = \\left | \\tfrac { \\Gamma ( 2 k - 1 ) } { \\Gamma ( k - 1 / 2 ) ^ 2 } \\int _ { 0 } ^ { 1 } t ^ { k - 3 / 2 } ( 1 - t ) ^ { k - 3 / 2 } ( 1 - x t ) ^ { - k - ( z - 1 ) / 2 } \\d t \\right | \\\\ & \\leq \\tfrac { \\Gamma ( 2 k - 1 ) } { \\Gamma ( k - 1 / 2 ) ^ 2 } \\int _ { 0 } ^ { 1 } t ^ { k - 3 / 2 } ( 1 - x t ) ^ { - 1 } \\left ( \\tfrac { 1 - t } { 1 - t x } \\right ) ^ { k - 3 / 2 } \\d t . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} \\delta _ \\alpha f = - ( f \\circ \\mu - ( - 1 ) ^ { n - 1 } \\mu \\circ f ) = - [ f , \\mu ] _ \\alpha = ( - 1 ) ^ { n - 1 } [ \\mu , f ] _ \\alpha , ~ f \\in C ^ n _ \\alpha ( A , A ) . \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} 0 < \\beta _ { n } \\stackrel { \\mathrm { d e f } } { = } \\inf _ { y \\in E _ { n } } f ( y ) \\leq f ( x ) \\leq D _ { n } \\stackrel { \\mathrm { d e f } } { = } \\sup _ { y \\in E _ { n } } f ( y ) < \\infty , \\mbox { { f o r } } x \\in E _ { n } , \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} \\sum _ { i ' , j ' } \\sin ( \\bar { \\theta } _ { i ' , j ' } - \\bar { \\theta } _ { i , j } ) = 0 \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} \\mathbb { A ' } ^ { - 1 } _ L & = ( \\Delta ' ) ^ { - 1 } \\begin{pmatrix} \\Delta ' ( a - b d ^ { - 1 } c ) ^ { - 1 } & - b - ( [ d , b ] - [ d , a ] a ^ { - 1 } b ) ( d - c a ^ { - 1 } b ) ^ { - 1 } \\\\ - \\Delta ' ( a c ^ { - 1 } d - b ) ^ { - 1 } & a - ( [ b , c ] - [ a , c ] a ^ { - 1 } b ) ( d - c a ^ { - 1 } b ) ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} A ( \\dot z + i E z ) = b , \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} u ( t , x ) & = \\frac { u _ 0 ( \\overline { x } ) } { J X ( t , \\overline { x } ) } \\exp \\left ( \\int _ { 0 } ^ { t } G ( s , X ( s , \\overline { x } ) ) d s \\right ) \\\\ & + \\frac { 1 } { J X ( t , \\overline { x } ) } \\int _ { 0 } ^ { t } f ( \\tau , X ( \\tau , \\overline { x } ) ) \\exp \\left ( \\int _ { \\tau } ^ { t } G ( s , X ( s , \\overline { x } ) ) d s \\right ) J X ( \\tau , \\overline { x } ) d \\tau , \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} \\mathfrak F \\circ \\mathfrak J = \\mathfrak J . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} K \\cdot R ^ { ( 2 ) } ( D _ { f _ 2 } ) \\le I _ { \\rho } + N \\sum \\limits _ { n = 1 } ^ N N ^ { - 1 } I ( X _ { 2 n } ; Y _ { 1 n } | X _ { 1 n } ) \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} m _ j ^ + = m _ j + \\# a ( s _ j ) - \\# a ^ { - 1 } ( s _ { j + 1 } ) . \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} a _ 4 ( n ) & = \\frac { 2 ( n - 1 ) ^ 2 ( n - 2 ) } { n ^ 2 } ( 2 n - 3 ) ( 3 n - 5 ) ( n ^ 2 + 6 n - 1 2 ) ( n ^ 3 - 5 n ^ 2 + 6 n - 4 ) \\\\ & > 0 , \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 1 + \\lambda ( u ^ 2 ) ^ 2 & - \\lambda u ^ 1 u ^ 2 & 0 \\\\ - \\lambda u ^ 1 u ^ 2 & 4 + \\lambda ( u ^ 1 ) ^ 2 & 0 \\\\ 0 & 0 & \\lambda \\end{pmatrix} , \\\\ w _ { 1 2 } = \\frac { 2 \\lambda } { \\sqrt { \\det g } } , w _ { 3 1 } = w _ { 2 3 } = 0 , \\end{gather*}"}
-{"id": "9706.png", "formula": "\\begin{align*} b _ { i , j } : = \\left \\{ \\begin{array} { l l } a _ { i j } t _ i t _ { i + 1 } \\dots t _ { j - 1 + d } & i > j \\\\ a _ { i j } \\prod _ { k = i } ^ { j - 1 } t _ k & i < j \\\\ a _ { i j } & i = j \\\\ \\end{array} . \\right . \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} L ' _ { i j , i j } = \\frac { \\Pi ( G ' ) } { R _ { e } } \\Bigl ( \\sum _ { T \\in S ( G ) } \\Pi ( T ) - \\sum _ { T \\in S ( G ) \\atop e \\in T } \\Pi ( T ) \\Bigr ) . \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} & \\underset { \\{ \\Delta q \\} _ { i j } } { } & \\sum _ { i , j } \\| \\Delta q _ { i j } + \\mathbf { m } _ { i } ( j ) \\| ^ 2 \\\\ & & \\underline { x } _ { i j } \\leq \\Delta q _ { i j } \\leq \\overline { x } _ { i j } \\\\ & & \\sum _ { i = 1 } ^ N \\Delta q _ { i j } = \\mathbf { d } ( j ) . \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} \\gamma _ F ( a ) \\delta _ F ( x ) = \\delta _ F ( a \\cdot x ) , \\delta _ F ( x ) \\gamma _ F ( a ) = \\delta _ F ( x \\cdot a ) \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} \\begin{aligned} i \\partial _ t v & = H v + \\N , \\\\ \\N & : = F ( Q + v ) - F ( Q ) - i D Q ( \\dot z + i E z ) , \\end{aligned} \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} E _ { \\delta , k } : = \\left \\{ y \\in [ 0 , 1 ] : \\forall j = - N _ k + 1 , . . . , N _ k - 1 , \\ \\| j \\theta - y \\| \\geq { 1 \\over N _ k ^ { { \\tau } } } \\right \\} . \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} E _ 0 = \\{ r : T ( r + | c | , f ) \\geq C T ( r , f ) \\} . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} h _ { a , 0 } ( l ) : = h _ { a } ( l ) \\big | ^ { \\gamma \\equiv 0 } = \\frac { 1 } { 2 } \\sigma ^ 2 l ^ 2 + c \\delta b ^ { - 1 } l - \\delta . \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} X _ i & : = \\{ x : c ( g _ i ^ { - 1 } , x ) = h _ 0 ^ { - 1 } \\} , X _ j : = \\{ x : c ( g _ j ^ { - 1 } , x ) = h _ 1 ^ { - 1 } \\} , \\\\ X _ k & : = \\{ x : c ( g _ k ^ { - 1 } , x ) = h _ 2 ^ { - 1 } \\} , X _ s : = \\{ x : c ( g _ s ^ { - 1 } , x ) = h ^ { - 1 } \\} , \\\\ X _ t & : = \\{ x : c ( g _ t ^ { - 1 } , x ) = \\bar { h } ^ { - 1 } \\} . \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { H ^ { s , p } ( G ) } : = \\Vert \\Lambda _ s f \\Vert _ { L ^ p ( G ) } < \\infty . \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} I _ { n , d , t } ^ { \\sigma } = I _ { n + d - 1 , d , t + 1 } I _ { n , d , t } ^ { \\tau } = I _ { n - d + 1 , d , t - 1 } t \\geq 1 . \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} \\Pi : = - \\sum _ { j = 0 } ^ { k - 1 } a _ i A _ i , & & a _ i = [ ( 2 \\pi ) ^ { k } i ! ( 2 k - 2 i - 1 ) { ! } { ! } ] ^ { - 1 } , & & A _ i = ( \\pi ^ * \\mathcal { R } ) ^ i \\wedge I \\wedge ( D I ) ^ { 2 k - 2 i - 1 } . \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} G ( y ) = \\sum _ { q = m ( G ) } ^ { \\infty } \\frac { J ( q ) } { q ! } H _ q ( y ) \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} \\lambda = e ^ { - \\tau } = 1 - \\tau + \\frac { 1 } { 2 } \\tau ^ { 2 } \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} K _ { u , v } ^ w = t _ { \\lambda , \\mu } ^ \\nu , \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} Y = Y ^ T - u ^ { - 1 } Y ^ { \\perp } \\partial _ t . \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} L ( \\varphi , \\mathbb { A } ) = \\prod _ { f } \\frac { \\big [ \\mathbb { A } / f \\mathbb { A } \\big ] _ { \\mathbb { A } } } { \\big [ \\varphi ( \\mathbb { A } / f \\mathbb { A } ) \\big ] _ { \\mathbb { A } } } , \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} \\mathbb { B } _ L = \\Delta ^ { - 1 } \\left ( \\begin{matrix} [ d , a ] - [ b , c ] & [ d , b ] \\\\ [ a , c ] & 0 \\end{matrix} \\right ) , \\ ; \\ ; \\ ; \\ ; \\mathbb { B } _ R = \\left ( \\begin{matrix} [ c , b ] & [ b , a ] \\\\ [ c , d ] & [ d , a ] \\end{matrix} \\right ) \\Delta ^ { - 1 } . \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} \\int d \\mu ( z , \\ell ) | \\psi _ z ^ \\ell \\rangle \\langle \\psi _ z ^ \\ell | = \\sum _ { n = 0 } ^ \\infty | \\psi _ n ^ { \\ell } \\rangle \\langle \\psi _ n ^ { \\ell } | = \\mathbf { I } , \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} T ( r + s , f ) = T ( r , f ) + o \\left ( \\frac { T ( r , f ) } { ( \\log r ) ^ { 1 + \\varepsilon } } \\right ) , \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\eta _ k \\| x ^ k - y ^ k \\| ^ 2 = 0 . \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} \\lim _ { \\beta \\rightarrow \\infty } \\frac { \\beta \\big ( 1 - \\widehat { x } ^ { ( \\alpha , \\beta ) } _ { n , k } \\big ) } { 2 } = \\widehat { x } ^ { ( \\alpha ) } _ { n , n + 1 - k } . \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} j _ { \\lambda } ( h k , z _ 0 ) = j _ \\lambda ( h , k \\cdot z _ 0 ) j _ \\lambda ( k , z _ 0 ) = j _ \\lambda ( h , z _ 0 ) \\chi _ { \\lambda } ( k ) ^ { - 1 } . \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} \\tau ^ * = T \\wedge \\tau ^ { n , r } _ R \\wedge \\inf \\left \\lbrace t \\ge 0 : \\left \\lvert X ^ { n , r } _ t \\right \\rvert \\ge a \\right \\rbrace \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} \\lim _ { ( \\rho , \\delta ) \\to ( 0 , 0 ) } \\lambda ( \\rho , \\delta ) = 0 . \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} R & \\le E \\Bigg [ E \\bigg ( \\prod _ { i = 1 } ^ k e ^ { - \\alpha t _ i } \\phi _ 0 ( t _ i ) \\biggm | \\xi _ 0 \\bigg ) \\\\ & \\quad + \\varrho _ k \\sum _ { \\emptyset \\ne H \\subset \\{ 1 , \\dots , k \\} } E \\bigg ( \\prod _ { i \\notin H } e ^ { - \\alpha t _ i } \\phi _ 0 ( t _ i ) \\biggm | \\xi _ 0 \\bigg ) \\sum _ { \\substack { 1 \\le j _ i \\le \\xi _ 0 ( t _ i ) , \\\\ \\forall i \\in H } } \\ \\prod _ { i \\in H } e ^ { - \\alpha \\sigma _ { j _ i } } \\Bigg ] \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} v _ 1 \\wedge v _ 2 = \\tau _ 2 ( v _ 1 ) \\wedge \\tau _ 1 ( v _ 2 ) \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } \\log _ q r = \\log r , \\lim _ { q \\to 1 } \\exp _ q r = e ^ r \\ r > 0 . \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\Delta \\mu _ \\varphi ( y ) = \\frac { \\partial \\bigl ( \\bigl [ \\mu _ { \\boldsymbol \\varphi } ( y ) \\bigr ] \\bigl [ \\Delta \\varphi ^ \\alpha ( \\boldsymbol \\varphi ^ { - 1 } ( y ) ) \\bigr ] \\bigr ) } { \\partial y ^ \\alpha } \\ , . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} [ u ] _ { \\alpha ; B _ i } : = \\sup _ { x \\neq x ' , x , x ' \\in \\phi _ i ( B _ i ) } \\frac { | ( \\phi _ i ^ { - 1 } ) ^ * u ( x ) - ( \\phi _ i ^ { - 1 } ) ^ * u ( x ' ) | _ { \\delta } } { | x - x ' | ^ { \\alpha } } . \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { P T } { z _ P g _ P ( x ) } x \\in G . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} \\left | \\int _ { \\delta _ 2 } e ^ { t / u } \\left ( \\varphi ( u ) - \\sum _ { k = 0 } ^ { p - 1 } b _ k u ^ { k r } \\right ) \\ , \\frac { d u } { u } \\right | \\le \\pi \\beta _ 1 e ^ p C A ^ p M _ p \\left ( \\frac { | t | } { p } \\right ) ^ { p r } . \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} G _ x \\setminus Z _ x = \\left \\{ g _ 1 , \\ldots , g _ { a _ x } , g _ { a _ x + 1 } = g _ 1 ^ { - 1 } , \\ldots , g _ { b _ x } = g _ { a _ x } ^ { - 1 } \\right \\} \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} S _ i = \\bigcup _ { q _ { i + 1 } = 0 } ^ { p _ { i + 1 } - 1 } \\bigcup _ { q _ { i + 2 } = 0 } ^ { p _ { i + 2 } - 1 } \\dots \\bigcup _ { q _ h = 0 } ^ { p _ h - 1 } T _ i \\prod _ { i < j \\le h } \\alpha _ j ^ { q _ j } . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} \\beta _ { \\tilde g ^ { ( 4 ) } } g ^ { ( 4 ) } = 0 \\quad \\partial M . \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} P = \\left \\{ x \\in V \\mid \\langle x , u _ { F } \\rangle + \\lambda _ { F } \\geq 0 \\ \\right \\} . \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} \\varphi ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { | T ' ( x ) | } \\varphi ( x , y , t ) \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ M | \\beta _ j | & \\leq \\frac { 1 2 } { \\tilde { \\varepsilon } \\big | \\bigcup _ i B _ { \\ell ( i ) } \\big | } \\sum _ { j \\leq M } \\Big ( \\int f _ j - C \\alpha _ j \\Big ) \\leq \\frac { 1 2 } { \\tilde { \\varepsilon } \\big | \\bigcup _ i B _ { \\ell ( i ) } \\big | } ( 3 \\varepsilon _ 1 M ) , \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} \\frac { \\nu } { 2 } \\left ( \\| u _ { t } \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } ) } ^ { 2 } + \\| D _ { q } u _ { t } \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } ) } ^ { 2 } + \\| q u _ { t } \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } ) } ^ { 2 } \\right ) = \\| \\sqrt { \\nu } ( \\frac { - \\partial _ q + q } { \\sqrt { 2 } } ) e ^ { - t ( K _ { \\nu , \\frac { \\pi } { 2 } } + \\nu ^ { \\frac { 1 } { 3 } } ) } u \\| _ { L ^ 2 ( \\mathbb { R } ^ { 2 } ) } ^ { 2 } \\le \\frac { c } { t ^ { 3 } } \\| u \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } ) } ^ { 2 } \\ , . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ m } | u | _ g ^ 2 d V _ g & = \\int _ { \\mathbb { R } ^ m } { } | u | _ { g _ 0 } ^ 2 e ^ { - \\frac { m } { 4 ( m - 2 ) } | x | ^ 2 } d V _ { g _ 0 } \\leq \\liminf \\int _ { \\mathbb { R } ^ m } { } | u _ k | _ { g _ 0 } ^ 2 e ^ { - \\frac { m } { 4 ( m - 2 ) } | x | ^ 2 } d V _ { g _ 0 } \\\\ & = \\liminf \\int _ { \\mathbb { R } ^ m } | u _ k | _ g ^ 2 d V _ g . \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} \\dim F ^ + = g + a - 2 , \\ \\dim F ^ - = g + b - 2 , \\ \\dim B = g + a + b - 3 \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} \\inf _ { \\mathcal { T } , \\left ( y _ { \\mathcal { T } } ( \\cdot ) , u _ { \\mathcal { T } } ( \\cdot ) \\right ) } \\left \\{ \\frac { 1 } { \\mathcal { T } } \\int _ 0 ^ { \\mathcal { T } } k ( y _ { \\mathcal { T } } ( t ) , u _ { \\mathcal { T } } ( t ) ) d t \\right \\} : = v _ { p e r } ( y _ 0 ) , \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} \\Phi ( \\varphi ( a ^ { t } ) \\cdot h ) = 0 . \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} A ^ { - 1 } = J _ { n , 1 } A ^ * J _ { n , 1 } = \\begin{pmatrix} a ^ * & - \\overline { w } \\\\ - \\overline { v } ^ t & \\overline { d } \\end{pmatrix} . \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} P \\bigl ( | B ( t ) + \\mu t | + x < y \\bigr ) = { } & P \\bigl ( x - y - \\mu t < B ( t ) < y - x - \\mu t \\bigr ) \\\\ = { } & \\int _ { x - y - \\mu t } ^ { y - x - \\mu t } \\frac { e ^ { - \\frac { w ^ { 2 } } { 4 t } } } { \\sqrt { 4 \\pi t } } \\ , \\mathrm { d } w \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} f ( 1 ) = - \\frac { 1 } { 4 } , f ' ( r ) > 0 , r \\in ( 1 , 2 ) , f ' ( r ) < 0 , r > 2 \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\min & F ( x , y ) \\\\ & C ( x , y , \\mu , \\gamma ) = 0 \\\\ & s = ( x , y , \\mu , \\gamma ) \\in \\Omega \\times \\Delta \\end{array} \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} \\omega _ \\lambda ( t ) & : = \\frac { \\phi _ \\lambda '' ( t ) \\ , t - \\phi _ \\lambda ' ( t ) } { \\phi _ \\lambda ' ( t ) } . \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} Z \\stackrel { d e f } { = } ( I - P ) Y ( I - P ) - ( I - P ) D ^ * ( D P ) ^ \\dag D ( I - P ) \\ge 0 . \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } a _ { n + 1 } \\left ( \\lambda _ { n + 1 } p _ n - \\lambda _ n p _ n \\right ) + \\left ( \\lambda _ { n + 1 } p _ { n - 1 } - \\lambda _ { n - 1 } p _ { n - 1 } \\right ) & = 0 \\\\ a _ { n + 1 } \\left ( \\lambda _ { n + 1 } q _ n - \\lambda _ n q _ n \\right ) + \\left ( \\lambda _ { n + 1 } q _ { n - 1 } - \\lambda _ { n - 1 } q _ { n - 1 } \\right ) & = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} \\begin{cases} \\lambda _ { I } ( r ) = 0 & r < \\mathcal { R _ I } \\\\ \\lambda _ { I } ( r ) = \\lambda _ { b } \\mathbb { P } \\left [ \\mathbb { A _ { U E } } \\right ] & r \\ge \\mathcal { R _ I } , \\end{cases} \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} ( ( a ( n - 1 ) + \\ell ) b - ( - a + \\ell - n ) ) ( ( a ( n - 1 ) + \\ell - n ) b - ( a - \\ell ) ( \\ell - n ) ) = 0 . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} F _ t ( m ) & = | \\{ 1 \\le i \\le t \\ : \\ Z _ i = m \\} | , \\\\ A _ t ( m ) & = | \\{ 1 \\le i \\le t \\ : \\ Z ^ 0 _ i = m \\} | , \\\\ B _ t ( m ) & = | \\{ 1 \\le i \\le t \\ : \\ Z ^ 1 _ i = m \\} | . \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\rho _ { a , u , v } ( x ) = x + a h ( u , x ) v - \\varepsilon a ^ * h ( v , x ) u , x \\in V . \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} u _ { m + k } ( s , 0 ) = F _ { m + k } ( s ) \\mbox { f o r } 0 \\le s \\le t _ k , \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} \\psi _ 1 ( y ) = \\mu ( B ( y , r ) ) = \\int _ { B ( y , r ) } h ( z ) d z , \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } p _ n ^ { 1 / n } = e ^ { - \\pi ^ 2 / 1 2 \\log 2 } \\mbox { f o r L e b e s g u e - a l m o s t a l l } \\ \\theta \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} e ^ \\alpha ( \\lambda ( t ) ) = t ^ q \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} | A | \\ge q ^ { K } - \\binom { N } { 1 } q ^ { K - 1 } + \\binom { N } { 2 } q ^ { K - 2 } - \\dots + ( - 1 ) ^ { K - g } \\binom { N } { K - g } = q ^ { K } \\left ( 1 - \\dfrac { 1 } { q } \\right ) ^ { N } - \\sum _ { p = K - g + 1 } ^ { N } ( - 1 ) ^ { p } \\binom { N } { p } q ^ { K - p } . \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\Gamma _ 3 = \\left ( \\begin{array} { c c c } 2 & 1 & 0 \\\\ 1 & 2 & 1 \\\\ 0 & 1 & 2 \\end{array} \\right ) , \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} \\mathcal { B } = \\Big \\{ \\{ x \\in \\R ^ m \\colon x _ k = 0 \\} \\colon k = m ^ + + 1 , \\ldots , m \\Big \\} . \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} ( \\sigma ^ { n - \\theta } , w ) + ( \\nabla u ^ { n - \\theta } , \\nabla w ) = 0 , ~ \\forall w \\in H _ 0 ^ 1 . \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} & \\mathcal { E } ( \\omega _ { \\eta } , \\omega _ { \\eta } ) \\le \\beta \\int _ { \\mathbb { R } ^ n } \\Psi \\cdot ( a ^ { - 1 } \\Psi ) \\ , d \\mu \\ , , \\\\ & \\lim _ { \\eta \\rightarrow 0 } \\bigg ( \\eta ^ 2 \\int _ { \\mathbb { R } ^ n } \\omega _ { \\eta } ^ 2 \\ , d \\mu \\bigg ) = 0 \\ , . \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} C ( \\Sigma ( u _ \\pm ) ) = C ( \\Sigma ^ 0 ( u _ \\pm ) ) = C ( \\Sigma ^ \\tau ( u _ \\pm ) ) = C ( \\Sigma ^ 1 ( u _ \\pm ) ) = C ( \\Sigma _ \\mu ( w _ \\pm ) ) . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} a - _ o b = a + ( - b ) d ( a , b , c ) = a + ( - b ) + c . \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} R _ { \\mathcal { H } _ p } \\left ( e _ { ( z , w ) } \\circ T _ S \\right ) ( t , \\zeta ) = 4 \\pi t \\overline { H _ p ( t ; w , \\zeta ) } e ^ { - i 2 \\pi \\overline { z } t } . \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} \\sum _ { k = j } ^ { n - 1 } \\binom { k } { j } \\frac { 1 } { n - k } & = \\sum _ { k = j } ^ { n - 1 } [ z ^ { k } ] \\frac { z ^ { j } } { ( 1 - z ) ^ { j + 1 } } [ z ^ { n - k } ] \\log \\Bigl ( \\frac { 1 } { 1 - z } \\Bigr ) \\\\ & = [ z ^ { n } ] \\frac { z ^ { j } } { ( 1 - z ) ^ { j + 1 } } \\log \\Bigl ( \\frac { 1 } { 1 - z } \\Bigr ) \\\\ & = [ z ^ { n - j } ] \\frac { 1 } { ( 1 - z ) ^ { j + 1 } } \\log \\Bigl ( \\frac { 1 } { 1 - z } \\Bigr ) = \\binom { n } { j } ( H _ n - H _ j ) \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} \\widehat { V } V _ 0 = V ( H _ 1 \\oplus . . . \\oplus H _ d \\oplus W _ 1 ) U _ 0 \\widehat { U } = ( H _ 1 ^ * \\oplus . . . \\oplus H _ d ^ * \\oplus W _ 2 ^ * ) U \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} \\xi _ { 1 / 2 } f ( z ) & = \\frac { 1 } { 2 } c _ 0 ^ - u _ { 3 / 2 , 0 } ^ { [ 0 ] , + } ( y ) - \\sum _ { 0 < n \\equiv 0 , 3 ( 4 ) } c _ { - n } ^ - u _ { 3 / 2 , n } ^ { [ 0 ] , - } ( y ) e ^ { 2 \\pi i n x } \\\\ & = \\frac { 1 } { 2 } c _ 0 ^ - - \\sum _ { 0 < n \\equiv 0 , 3 ( 4 ) } c _ { - n } ^ - ( 4 \\pi n ) ^ { 3 / 4 } q ^ n , \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } x ^ { s - 1 } \\cos ( b x ) d x = \\frac { \\Gamma ( s ) \\cos ( \\frac { \\pi s } { 2 } ) } { b ^ { s } } . \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & L ( \\chi ( u ) , A _ 1 , A _ 2 , A _ 3 , \\sigma ( u ) ) v = f _ { \\alpha } , & & x \\in \\R ^ 3 _ + , & & t \\in ( 0 , T ) ; \\\\ & v ( 0 ) = \\partial ^ { ( 0 , \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 3 ) } S _ { \\chi , \\sigma , \\R ^ 3 _ + , m , \\alpha _ 0 } ( 0 , f , u _ 0 ) , & & x \\in \\R ^ 3 _ + ; \\end{aligned} \\right . \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} R ( u , J u , v , J v ) = R ( u , v , u , v ) + R ( u , J v , u , J v ) \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} \\lim _ { d \\rightarrow \\infty } \\frac { 1 } { d } { \\log \\mathbb { P } ( D > d ) } = - \\kappa ^ { A } ( \\theta ) , \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} R _ { a b } = \\log \\left ( 1 + \\frac { P _ x } { \\sigma _ b ^ 2 } \\right ) . \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} | c - 1 | | U | & = \\big | | c | U | - | U | \\big | \\\\ & = \\big | \\lambda _ n ( U ) - | U | \\big | = \\big | | n U n ^ { - 1 } | - | U | \\big | \\\\ & \\leq \\big | | U | - | U | \\big | = 0 . \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} \\mu \\prod _ { i = 1 } ^ { j } P _ i ( \\epsilon ) = \\mu \\tilde { P } ^ j r + \\epsilon \\mu \\sum _ { i = 1 } ^ { j - 1 } i \\tilde { P } ^ i \\tilde { P } ^ { ( 1 ) } \\tilde { P } ^ { j - 1 - i } r + O ( \\epsilon ^ 2 j ^ 3 ) \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} \\begin{aligned} g ( x ) = - & \\sum _ { i = 1 } ^ N \\min _ { l _ i \\leq y _ i \\leq u _ i } \\Big \\{ \\beta y ^ 2 _ i + \\Big { ( } \\beta \\sigma _ { ( - i ) } ( x ) - \\alpha \\Big { ) } y _ i + h _ i ( y _ i ) \\Big \\} \\\\ & + \\beta \\big { ( } \\sum _ { i = 1 } ^ N x _ i \\big { ) } ^ 2 - \\alpha ^ T x + \\sum _ { i = 1 } ^ N h _ i ( x _ i ) , \\end{aligned} \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 3 / 4 } ^ 0 \\bigg [ \\frac { b _ { 1 / 2 , 0 } ( d , s ) } { d ^ { 1 / 2 } \\Gamma ( s + 1 / 4 ) } \\bigg ] & = \\frac { \\pi } { \\sqrt { | d D | } } \\frac { 1 } { B ( 1 ) } \\frac { \\mathrm { T r } _ { d , D } ( ( - 3 / \\pi ) \\log ( y | \\eta ( z ) | ^ 4 ) ) } { L _ D ( 1 ) } \\\\ & = \\frac { 3 } { \\sqrt { d D } L _ D ( 1 ) } \\mathrm { T r } _ { d , D } ( - \\log ( y | \\eta ( z ) | ^ 4 ) ) . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} \\frac { \\rm { D } } { \\partial z } = \\frac { \\partial } { \\partial z } + \\frac { \\partial K } { \\partial z } , ~ ~ \\frac { \\rm { D } } { \\partial \\overline { z } } = \\frac { \\partial } { \\partial \\overline { z } } + \\frac { \\partial K } { \\partial \\overline { z } } \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} x \\vee y & = x \\vee ( a \\vee y ) = b \\vee y \\notin M , \\\\ x \\wedge y & = ( x \\wedge b ) \\wedge y = x \\wedge a \\notin M . \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} [ u ] _ { \\alpha ; B _ i } : = \\sup _ { x \\neq x ' , x , x ' \\in \\phi ( B _ i ) } \\frac { | u ( \\phi _ i ^ { - 1 } ( x ) ) - u ( \\phi _ i ^ { - 1 } ( x ' ) ) | } { | x - x ' | ^ { \\alpha } } . \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} X ( t ) = ( P ( t ) + Q ( t ) ) ^ { - 1 / 2 } P ( t ) = \\frac 1 2 \\left ( \\begin{matrix} \\sqrt { 1 + c _ t } + \\sqrt { 1 - c _ t } & 0 \\\\ - \\sqrt { 1 + c _ t } + \\sqrt { 1 - c _ t } & 0 \\end{matrix} \\right ) , { \\rm ~ f o r ~ a l l ~ } t \\in ( 0 , 1 ] . \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} d ( R | S ) : = \\lim _ { N \\to \\infty } \\frac { \\# R _ N } { \\# S _ N } \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} \\begin{pmatrix} p & 2 s D & - b _ 2 p / 2 - b _ 3 \\\\ 2 s D & 4 t D & 2 b _ 1 - b _ 2 s D \\\\ - b _ 2 p / 2 - b _ 3 & 2 b _ 1 - b _ 2 s D & b _ 2 ^ 2 / 4 \\end{pmatrix} . \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} \\tau ( T ) = ( T - I ) ( T + I ) ^ { - 1 } , \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} R _ { 1 2 } ( \\lambda ) R _ { 2 3 } ( \\lambda + \\mu ) R _ { 1 2 } ( \\mu ) = R _ { 2 3 } ( \\mu ) R _ { 1 2 } ( \\lambda + \\mu ) R _ { 2 3 } ( \\lambda ) . \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} 2 \\mathsf { R e } ( X _ j ^ * f X _ j u , i B u ) = 2 \\mathsf { R e } ( f X _ j u , i [ X _ j , B ] u ) + 2 \\mathsf { R e } ( f X _ j u , i B X _ j u ) \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} ( 1 - ( - t ) ^ { n / ( n , d ) } ) ^ { ( n , d ) } = \\sum _ { k = 0 } ^ { n } { n \\brack k } _ { \\omega _ n ^ d } t ^ k \\omega _ n ^ { d \\binom { k } { 2 } } . \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} ( 4 \\pi n ^ 2 | D | y ) ^ { - 1 / 4 } e ^ { - \\frac { \\pi i } { 4 } } \\mathcal { M } _ { - \\frac { 1 } { 4 } , \\frac { 1 } { 4 } } ^ + ( 4 \\pi n ^ 2 | D | y ) e ^ { 2 \\pi i n ^ 2 D x } & = ( 4 \\pi n ^ 2 | D | ) ^ { - 1 / 4 } e ^ { - \\frac { \\pi i } { 4 } } ( 4 \\pi n ^ 2 | D | e ^ { \\pi i } ) ^ { 1 / 4 } q ^ { n ^ 2 D } \\\\ & = q ^ { n ^ 2 D } . \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} \\chi _ R ( x ) \\mathcal { R } _ j ^ 2 ( \\chi _ { | x | / 8 } \\nu ) ( x ) , \\chi _ R ( x ) \\mathcal { R } _ j ^ 2 ( ( \\chi _ { 8 R } - \\chi _ { | x | } ) \\nu ) ( x ) \\in ( L ^ \\infty \\cap L ^ 1 ) ( \\mathbb { R } ^ 2 ) ~ j = 1 , 2 . \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} \\sum _ { d \\in \\mathbb { Z } } \\mathrm { T r } _ { d , D } ^ { \\mathrm { a d d } } ( F _ { 0 , m , r } ) q ^ d : = \\mathrm { L C } _ { s = 1 } ^ r [ ( \\ref { f i n p a r t } ) ] ^ { \\mathrm { h o l } } = \\sum _ { n | m } \\biggl ( \\frac { D } { m / n } \\biggr ) [ ( \\ref { f i n h o l } ) + ( \\ref { f i n h o l 2 } ) ] . \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\mathcal { L } ( y , p _ { _ 1 } ( y ) ) \\ ! = \\ ! p _ { _ 1 } ( y ) \\log \\frac { p _ { _ 1 } ( y ) } { p _ { _ 0 } ( y ) } \\ ! + \\ ! \\rho _ 0 p _ { _ 1 } ( y ) \\ ! + \\ ! \\rho _ 1 y ^ 2 p _ { _ 1 } ( y ) , \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty a ^ { k } \\cos ( b ^ k \\pi t ) , \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} \\phi \\left ( \\left ( \\begin{matrix} 1 & \\zeta \\\\ 0 & 1 \\end{matrix} \\right ) \\right ) = [ \\imath \\phi ] \\left ( \\left ( \\begin{matrix} 1 & 0 \\\\ - \\zeta & 1 \\end{matrix} \\right ) \\right ) . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} \\frac { 1 } { \\rho _ { \\zeta _ f } } = \\exp ( h _ \\mathrm { t o p } ( f ) ) \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} H _ \\tau = \\left ( \\frac { f ( t _ \\tau ) } { 2 m } ( p _ { 1 , \\tau } ^ 2 + p _ { 2 , \\tau } ^ 2 ) + p _ \\tau + \\frac { m \\omega ^ 2 ( t _ \\tau ) f ^ { - 1 } ( t _ \\tau ) } { 2 } ( x _ { 1 , \\tau } ^ 2 + x _ { 2 , \\tau } ^ 2 ) \\right ) \\dot { t } _ \\tau . \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} \\mathbf { x } ^ { k + 1 } = \\mathbf { W } _ 1 \\mathbf { x } ^ { k } - \\frac { 2 } { \\rho } ( 2 \\mathbf { D } + \\frac { 2 } { \\rho } \\mathbf { P } ) ^ { - 1 } ( \\nabla f ( \\mathbf { x } ^ { k + 1 } ) + \\mathbf { E } _ { } ^ T \\boldsymbol { \\alpha } ^ { k } ) , \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} & \\rho : G \\to ( 0 , + \\infty ) \\\\ & \\rho ( x ) = \\begin{cases} \\rho _ 1 ( x ) & x \\in N \\\\ 0 & x \\not \\in N \\end{cases} , \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} \\dot { R } ^ s \\wedge ( R ^ s ) ^ { k - 1 } = d ^ { \\nabla ^ s } \\left ( \\dot \\theta \\wedge ( R ^ s ) ^ { k - 1 } \\right ) . \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} U ( t ) f ( x ) : = e ^ { t Q / 2 } f ( e ^ { t } x ) . \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} U _ \\lambda ( o , o | z ) & = \\frac { - [ m + \\lambda - 1 - ( d - 1 ) z ] + \\sqrt { [ m + \\lambda - 1 - ( d - 1 ) z ] ^ 2 + 4 \\lambda ( d - m ) z ^ 2 } } { 2 ( d - m ) } ; \\\\ \\mathbb { G } _ \\lambda ( o , o | z ) & = \\frac { 1 } { 1 - U _ \\lambda ( o , o | z ) } \\\\ & = \\frac { 2 ( m - d ) } { 2 ( m - d ) - [ m + \\lambda - 1 - ( d - 1 ) z ] + \\sqrt { [ m + \\lambda - 1 - ( d - 1 ) z ] ^ 2 + 4 \\lambda ( d - m ) z ^ 2 } } . \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 0 , 6 ) = \\Phi ( 6 ) = \\int _ { 0 } ^ { \\infty } \\frac { \\cos ( 6 \\pi x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x = \\frac { 1 3 - 4 \\sqrt { 3 } } { 1 4 4 } , \\end{align*}"}
-{"id": "9859.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\tilde { c } ( 5 n + 4 ) q ^ { n } \\equiv 1 0 \\dfrac { E _ { 5 } ^ { 2 } E _ { 1 0 } ^ { 2 } } { E _ { 1 } ^ { 5 } E _ { 2 } } & \\equiv 1 0 \\dfrac { E _ { 5 } E _ { 1 0 } ^ { 2 } } { E _ { 2 } } \\pmod { 2 5 } , \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} A : L ^ 1 ( \\Omega ) \\rightarrow L ^ 1 ( \\Sigma ) , ( A u ) ( s ) = \\int _ { { \\Omega } } a ( s , t ) u ( t ) \\ , d t \\ , , \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} N _ r ^ { \\alpha , 1 } ( d , m ) = \\frac { 1 } { r ! } N _ 2 ^ * . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} \\begin{aligned} & w _ t = \\mathcal { L } w = \\frac D 2 w '' + r ( x ) \\big ( w ( 0 , t ) - w ( x , t ) \\big ) , \\ x \\in ( - L _ 1 , a ) ; \\\\ & w ( x , 0 ) = 1 , \\ x \\in ( - L _ 1 , a ) ; \\\\ & w ( a , t ) = 0 , \\ w ( - L _ 1 , t ) = w ( 0 , t ) , \\ t > 0 . \\end{aligned} \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} \\ddot \\rho + \\eta \\dot \\rho + \\omega ^ 2 \\rho = \\frac { \\nu ^ 2 f ^ 2 } { m ^ 2 \\rho ^ 3 } \\ ; . \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} \\Lambda ( Q _ 1 ) - \\Lambda ( Q _ 2 ) \\ , = \\ , ( A - Q _ 1 S ) ( Q _ 1 - Q _ 2 ) + ( Q _ 1 - Q _ 2 ) ( A - Q _ 2 S ) ^ { \\prime } \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} F ^ { - 1 } = P H ^ { - 1 } P = \\langle \\Omega , ( F _ { \\eta } ( v , \\omega ) - \\lambda ) ^ { - 1 } \\Omega \\rangle P . \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} \\mathbb { B } _ L & = ( \\Delta ' ) ^ { - 1 } \\left ( \\begin{matrix} [ d , a ] & [ d , b ] \\\\ [ a , c ] & [ b , c ] \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} \\mathcal { C } = 2 \\pi \\lambda _ { b } \\int _ { 0 } ^ { \\infty } x _ { 1 1 } \\mathrm { e } ^ { - \\left ( \\pi \\lambda _ { b } x _ { 1 1 } ^ 2 + \\frac { x _ { 1 1 } ^ { \\alpha } \\Theta N } { P } + \\pi \\lambda _ { b } x _ { 1 1 } ^ 2 \\Theta ^ { \\frac { 2 } { \\alpha } } \\int \\displaylimits _ { \\Theta ^ { \\frac { - 2 } { \\alpha } } } ^ { \\infty } \\frac { 1 } { 1 + z ^ { \\alpha / 2 } } \\mathrm { d } z \\right ) } \\mathrm { d } x _ { 1 1 } , \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{gather*} \\biggl ( \\frac { - 4 } p \\biggr ) \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { k ! ^ 2 } z ^ k \\equiv \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { k ! ^ 2 } ( 1 - z ) ^ k \\pmod p . \\end{gather*}"}
-{"id": "3254.png", "formula": "\\begin{align*} \\| u \\| _ { ( \\kappa + 1 ) p ^ * } & \\leq M _ { 2 9 } ^ { \\frac { 1 } { \\kappa + 1 } } M _ { 3 0 } ^ { \\frac { 1 } { \\sqrt { \\kappa + 1 } } } \\left [ \\| u ^ { \\kappa + 1 } \\| _ { \\tilde { q } _ 1 } ^ p + 1 \\right ] ^ { \\frac { 1 } { ( \\kappa + 1 ) p } } . \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} \\exists C = C _ { \\alpha , N , \\ell } > 0 \\ ; \\ ; \\forall t \\in ( 0 , 1 ] \\ ; \\ ; \\sup _ { | x | = 1 } | \\partial _ { t } ^ { \\ell } X ^ { \\alpha } h _ { t } ( x ) | \\leq C _ { \\alpha , N } t ^ { N } \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{gather*} w ( i , m ) = t _ { - \\alpha _ { i + 1 } } ^ { m - 1 } s _ { i + 1 } s _ { i + 2 } \\cdots s _ { i - 3 } s _ { i - 2 } . \\end{gather*}"}
-{"id": "5327.png", "formula": "\\begin{align*} \\Upsilon ( n ) = \\frac { 1 } { 2 \\pi n } + \\int _ { 0 } ^ { \\infty } \\frac { \\sin ( \\pi n x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x , \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} Z _ p ( A ; t ) \\coloneqq \\sum _ { m = 0 } ^ \\infty M _ p ( A ; m ) t ^ m . \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} \\pi = \\bigotimes _ { p \\leq \\infty } \\pi _ p , \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} { \\rm R i c } ^ { 1 , 1 } = { \\rm M } - { \\rm S } ( { \\rm a d } _ { H } ) \\ , , { \\rm R i c } ^ { 2 , 0 + 0 , 2 } = - \\frac 1 2 { \\rm B } . \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} F ( 1 , \\ , 0 ) & = \\frac { 1 } { 2 m } \\sum _ { i = 1 } ^ r \\left \\{ - ( m - m _ i + \\lambda ) + [ ( m - m _ i + \\lambda ) ^ 2 + 4 \\lambda m _ i ] ^ { 1 / 2 } \\right \\} > 0 , \\\\ F ( 1 , \\ , 1 ) & = \\frac { 1 } { 2 m } \\sum _ { i = 1 } ^ r \\left \\{ m _ i - \\lambda + [ ( m _ i - \\lambda ) ^ 2 + 4 \\lambda m _ i ] ^ { 1 / 2 } \\right \\} = \\frac { 1 } { 2 m } \\sum _ { i = 1 } ^ r \\{ m _ i - \\lambda + m _ i + \\lambda \\} = 1 . \\end{align*}"}
-{"id": "9893.png", "formula": "\\begin{align*} ( \\boldsymbol { D } ^ { - s } u , \\boldsymbol { D } ^ { 2 s * } \\phi ) = ( u , \\boldsymbol { D } ^ { - s * } \\boldsymbol { D } ^ { 2 s * } \\phi ) = ( u , \\boldsymbol { D } ^ { s * } \\phi ) = ( \\boldsymbol { D } ^ { s } u , \\phi ) \\ , , \\forall \\phi \\in C _ 0 ^ \\infty ( \\mathbb { R } ) . \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { \\binom k j \\binom j s \\left ( { \\frac { { f _ 1 } } { { f _ 3 } } } \\right ) ^ j \\left ( { - \\frac { { 1 } } { { f _ 1 } } } \\right ) ^ s X _ { m - ( c - b ) k + ( c - a ) j + a s } } } = \\left ( - \\frac { f _ 2 } { f _ 3 } \\right ) ^ k X _ m \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } Q ( f ) ( \\ddot { y } ) d \\mu _ { n } ( \\ddot { y } ) & = \\int _ { K \\backslash G / H } Q ( L _ n f ) ( \\ddot { y } ) d \\mu ( y ) \\\\ & = \\int _ { G } L _ n f ( y ) . \\rho ( y ) d y \\\\ & = \\int _ { K \\backslash G / H } Q \\big ( f . \\lambda ( n , . ) \\big ) ( \\ddot { y } ) d \\mu _ \\rho ( \\ddot { y } ) . \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} \\psi _ h ( h / ( b + 1 ) ) = - m ^ 2 + \\frac { b ^ { 2 b } } { ( b + 1 ) ^ { 2 b + 2 } } h ^ { 2 b + 2 } \\le 0 \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} \\| u u _ h ^ { \\kappa } \\| _ { p ^ * } & \\leq c _ \\Omega \\| u u _ h ^ { \\kappa } \\| _ { 1 , p } \\leq M _ { 2 7 } \\left ( ( \\kappa + 1 ) ^ { M _ { 2 8 } } \\right ) \\left [ \\| u u _ h ^ { \\kappa } \\| _ { \\tilde { q } _ 1 } ^ p + 1 \\right ] ^ { \\frac { 1 } { p } } \\\\ & \\leq M _ { 2 9 } \\left ( ( \\kappa + 1 ) ^ { M _ { 2 8 } } \\right ) \\left [ \\| u ^ { \\kappa + 1 } \\| _ { \\tilde { q } _ 1 } ^ p + 1 \\right ] ^ { \\frac { 1 } { p } } < \\infty . \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} g ( z ) = \\prod _ { j = 2 } ^ p \\sum _ { i = 0 } ^ { \\gamma _ j - 1 } z ^ { i e _ j } = \\prod _ { j = 1 } ^ p \\left ( 1 + z ^ { e _ j } + z ^ { 2 e _ j } + \\cdots z ^ { ( \\gamma _ j - 1 ) e _ j } \\right ) . \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } ( p _ n ) ^ r ( q _ n ) ^ 0 \\\\ ( p _ n ) ^ { r - 1 } ( q _ n ) ^ 1 \\\\ \\vdots \\\\ ( p _ n ) ^ 1 ( q _ n ) ^ { r - 1 } \\\\ ( p _ n ) ^ 0 ( q _ n ) ^ r \\end{array} \\right ] = E ( a _ 1 ; r ) \\left [ \\begin{array} { c } ( \\hat { p } _ { n - 1 } ) ^ r ( \\hat { q } _ { n - 1 } ) ^ 0 \\\\ ( \\hat { p } _ { n - 1 } ) ^ { r - 1 } ( \\hat { q } _ { n - 1 } ) ^ 1 \\\\ \\vdots \\\\ ( \\hat { p } _ { n - 1 } ) ^ 1 ( \\hat { q } _ { n - 1 } ) ^ { r - 1 } \\\\ ( \\hat { p } _ n ) ^ 0 ( \\hat { q } _ { n - 1 } ) ^ r \\end{array} \\right ] \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} g ( \\tau ) & = \\sum _ { n = 1 } ^ \\infty b ( n ) q ^ n = q + 2 0 q ^ 3 - 7 4 q ^ 5 - 2 4 q ^ 7 + 1 5 7 q ^ 9 + 1 2 4 q ^ { 1 1 } + \\dotsb \\\\ & = \\eta ( 2 \\tau ) ^ { 1 2 } + 3 2 \\eta ( 2 \\tau ) ^ 4 \\eta ( 8 \\tau ) ^ 8 \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} [ \\nabla ^ { \\oplus } _ u , \\tau _ u ( \\omega _ u ) ] = \\tau _ u ( \\nabla ^ { \\oplus } _ u \\omega _ u ) . \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} \\Gamma ( \\phi ) ^ * H _ \\mu ( \\xi ) \\Gamma ( \\phi ) = K ( \\xi - d \\Gamma ( m ) ) + d \\Gamma ( \\omega ) + \\mu \\varphi ( \\lvert v \\lvert ) \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ { M _ 1 } X _ { j , 1 } , \\ldots , \\sum _ { j = 1 } ^ { M _ m } X _ { j , m } \\right ) \\le _ { s m } \\left ( \\sum _ { j = 1 } ^ { N _ 1 } Y _ { j , 1 } , \\ldots , \\sum _ { j = 1 } ^ { N _ m } Y _ { j , m } \\right ) . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} T _ { y * } ( X ) = T _ y ^ * ( T _ X ) \\cap [ X ] , \\widehat { T } _ { y * } ( X ) = \\widehat { T } _ y ^ * ( T _ X ) \\cap [ X ] . \\end{align*}"}
-{"id": "9992.png", "formula": "\\begin{align*} \\bar G ( q ) = \\int _ 0 ^ \\infty e ^ { - q x } G ( d x ) , q \\in \\mathbb C ^ + . \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} T _ + & = T _ + ( m , t _ 0 , f , g , u _ 0 ) = T _ + ( k , t _ 0 , f , g , u _ 0 ) , \\\\ T _ { - } & = T _ { - } ( m , t _ 0 , f , g , u _ 0 ) = T _ { - } ( k , t _ 0 , f , g , u _ 0 ) \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} \\tau ( u ) = \\prod _ { j = 1 } ^ { d } x _ { i _ { j - ( j - 1 ) } } . \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} I H _ { \\mathcal S } ( t ) = \\frac { P _ { j } } { P _ { k - 1 } P _ { j - k + 1 } } \\cdot \\frac { P _ { l - k + 1 } } { P _ { l - k } } - \\left ( t ^ { 2 ( l - j ) } + t ^ { 2 ( l - j + 1 ) } + \\dots + t ^ { 2 ( k - 1 ) } \\right ) \\cdot \\frac { P _ { j } } { P _ { k } P _ { j - k } } , \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} ( 1 - x ) ^ { \\alpha } ( 1 + x ) ^ { \\beta } P _ n ^ { ( \\alpha , \\beta ) } ( x ) = \\frac { ( - 1 ) ^ n } { 2 ^ n \\ , n ! } \\frac { d ^ n } { d x ^ n } \\left ( ( 1 - x ) ^ { \\alpha + n } ( 1 + x ) ^ { \\beta + n } \\right ) . \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} F d t = d f + \\sum _ { i = 1 } ^ r ( g _ i ^ p h _ i ^ { p - 1 } d h _ i - g _ i d h _ i ) . \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} - \\Delta u + u + \\rho _ { \\infty } \\bar { \\phi } _ u u = \\mu | u | ^ { p - 1 } u , \\mu \\in \\left [ \\frac { 1 } { 2 } , 1 \\right ] , \\\\ \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} \\| u ( t , z ) \\| _ { L ^ 2 ( \\R ; L ^ { \\frac { 2 ( n - 1 ) } { n - 3 } } ( X ) ) } \\lesssim \\| u _ 0 \\| _ { \\dot H ^ { \\frac { n + 1 } { 2 ( n - 1 ) } } ( X ) } + \\| u _ 1 \\| _ { \\dot H ^ { \\frac { n + 1 } { 2 ( n - 1 ) } - 1 } ( X ) } . \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} \\min _ { x \\in \\R ^ d } ~ F ( x ) : = f ( x ) + r ( x ) . \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 2 4 n + 6 ) q ^ n & \\equiv \\dfrac { f _ 1 f _ 2 f _ { 6 } } { f _ 3 } \\cdot \\dfrac { f _ 2 ^ 6 f _ 3 ^ 3 } { f _ 1 ^ 9 f _ { 6 } ^ 2 } \\equiv f _ 2 ^ 3 ~ ( \\textup { m o d } ~ 2 ) , \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} \\psi _ { 3 } ( \\tau _ { n } '' ) = \\bold { 1 } ( t > \\tau _ { n } '' ) \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} S ( n , k ) = \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} [ b _ 0 , b _ 1 , \\ldots , b _ L ] = \\left [ \\mathcal { U } [ 0 ] : - \\mathcal { U } [ 1 ] : \\ldots : ( - 1 ) ^ L \\mathcal { U } [ L ] \\right ] , \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} - \\lambda \\mu _ { \\ell } \\leq 2 \\sum \\limits _ { j = 0 } ^ N \\omega _ j \\tilde { \\Phi } _ { \\ell } ( x _ j ) f ( x _ j ) \\leq \\lambda \\mu _ { \\ell } , \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\dim ^ R ( \\bar { \\ 1 } ) \\dim ^ R ( X ^ * ) = \\dim ^ R ( \\bar { \\ 1 } ) s _ { \\bar { \\ 1 } } ^ R ( X ) = S ^ { R , R } _ { \\bar { \\ 1 } , X } = \\dim ^ R ( X ) s _ X ^ R ( \\bar { \\ 1 } ) . \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} \\mathcal { F } _ o : = \\{ ( \\hat { g } , \\hat { b } _ a ) \\in S E ( 3 ) \\times \\mathbb { R } ^ 6 : \\mathcal { U } ( \\tilde { g } ) - \\min _ { g _ q \\in \\mathbb { Q } } \\mathcal { U } ( \\tilde { g } g _ q ) \\leq \\delta \\} , \\\\ \\mathcal { J } _ o : = \\{ ( \\hat { g } , \\hat { b } _ a ) \\in S E ( 3 ) \\times \\mathbb { R } ^ 6 : \\mathcal { U } ( \\tilde { g } ) - \\min _ { g _ q \\in \\mathbb { Q } } \\mathcal { U } ( \\tilde { g } g _ q ) \\geq \\delta \\} , \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} \\nu _ p ( B _ n ^ { ( l ) } ) = - \\sigma ( n ) / ( p - 1 ) , \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} \\sum _ { k = s } ^ { \\ell } C _ k = \\sum _ { n = 0 } ^ { s - 2 } \\left ( p _ { n + 1 } ^ { ( + ) } - p _ { n } ^ { ( + ) } \\right ) \\prod _ { i = 1 } ^ { n } \\left ( q - p _ i \\right ) \\prod _ { i = n + 1 } ^ { s - 2 } \\left ( q - p _ i + 1 \\right ) . \\end{align*}"}
-{"id": "9870.png", "formula": "\\begin{align*} ( \\boldsymbol { D } ^ { - \\mu } u , v ) = ( _ a D _ x ^ { - \\mu } u , v ) _ { L ^ 2 ( a , b ) } = ( u , { _ x D _ b ^ { - \\mu } } v ) _ { L ^ 2 ( a , b ) } = ( u , \\boldsymbol { D } ^ { - \\mu * } v ) . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} - \\Delta Q - c | x | ^ { - 2 } Q + Q = | Q | ^ { \\frac { 4 } { d } } Q . \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} u ( p _ k ) = \\omega ^ k u ( p _ 0 ) = \\omega _ 1 ^ { k _ 1 } \\cdots \\omega _ d ^ { k _ d } u ( p _ 0 ) \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} | k _ b j _ b + \\frac { i - j } { n - \\frac 1 2 } | < \\frac { 1 } { 5 0 n \\sum _ { b = 1 } ^ n | j _ b | } | i + j | , b = 1 , \\cdots , n . \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} g = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in \\Gamma _ 0 ( q ) \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} \\begin{aligned} | T _ { N } ' ( x ) | = & | \\sin \\left ( N \\arccos ( x ) \\right ) | \\cdot N \\cdot \\frac { 1 } { \\sqrt { 1 - x ^ { 2 } } } = \\frac { N \\cdot | \\sin \\left ( N \\pi u _ { 0 } \\right ) | } { \\sqrt { 1 - \\cos ^ { 2 } \\left ( \\pi u _ { 0 } \\right ) } } \\\\ = & \\frac { N \\cdot | \\sin \\left ( N \\pi u _ { 0 } \\right ) | } { | \\sin \\left ( \\pi u _ { 0 } \\right ) | } \\end{aligned} \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} \\begin{array} { l l } \\underset { s } { \\min } & \\dfrac { 1 } { 2 } | | z - v - s | | ^ 2 \\\\ s . a . & C ' _ z ( s - z ) = 0 . \\end{array} \\end{align*}"}
-{"id": "9921.png", "formula": "\\begin{align*} \\psi _ { n } ^ { ( 0 , n - 1 ) } = ( \\sqrt { - 1 } ) ^ n C _ n \\dfrac { \\sum _ { i = 1 } ^ n ( - 1 ) ^ i ( z _ i - \\bar { z } _ i ) d \\bar { z } _ 1 \\wedge \\dots \\wedge \\widehat { d \\bar { z } _ i } \\wedge \\dots \\wedge d \\bar { z } _ n } { \\Vert z - \\bar { z } \\Vert ^ n } . \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} a _ 3 ( n ) & = \\frac { 2 ( n - 2 ) } n ( 1 8 n ^ 8 - 1 3 5 n ^ 7 + 4 8 5 n ^ 6 - 1 0 8 7 n ^ 5 + 1 6 7 7 n ^ 4 - 1 9 2 6 n ^ 3 + 1 6 5 6 n ^ 2 - 8 8 8 n + 1 9 2 ) \\\\ & > 0 , \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ N S _ i + N \\right ) ^ s \\lesssim ( H _ N + N ) ^ s \\lesssim \\left ( \\sum _ { i = 1 } ^ N S _ i + N \\right ) ^ s . \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} \\max _ { u _ q \\in \\mathbb { U } } \\Delta ( u _ q , v ) = 2 \\lambda _ { 1 2 } ^ Q , v = v _ 3 . \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} X & = \\{ ( i , j ) u , ( i + 1 , j ) u , \\ldots , ( r , j ) u , ( r + 1 , j ) v , \\ldots \\} , \\\\ Y & = \\{ \\ldots , ( i - 1 , j + 1 ) u , ( i , j + 1 ) u \\} , \\\\ X ^ \\star & = \\{ ( i , j ) u , ( i + 1 , j ) u , \\ldots , ( r , j ) u \\} . \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} I ' ( u ) [ v ] = \\int _ { \\mathbb R ^ 3 } \\left ( \\nabla u \\nabla v + u v \\right ) d x + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ u u v d x - \\int _ { \\mathbb R ^ 3 } f ( u ) v d x , v \\in H ^ 1 _ { r a d } ( \\mathbb R ^ 3 ) \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} f \\circ g ( [ ( p _ { k , l } ) _ { k , l } , w ] ) & = f \\left ( [ ( p _ { k , l } ) _ { k , l } ] , \\left [ \\prod _ { l = 1 } ^ { N _ 1 } p _ { 1 , l } , \\prod _ { l = 1 } ^ { N _ 2 } p _ { 2 , l } , \\dots , \\prod _ { l = 1 } ^ { N _ r } p _ { r , l } , w \\right ] \\right ) ( ~ \\eqref { e q _ i s o m o r _ p r o p 2 . 8 _ 1 1 } ) \\\\ & = ( [ ( p _ { k , l } ) _ { k , l } , w ] ) . \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} P ^ { ( \\alpha , \\beta ) } _ t f ( n ) = \\frac { 1 } { \\sqrt { \\pi } } \\int _ { 0 } ^ { \\infty } \\frac { e ^ { - u } } { \\sqrt { u } } W _ { t ^ 2 / ( 4 u ) } ^ { ( \\alpha , \\beta ) } f ( n ) \\ , d u , t \\ge 0 . \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} \\big ( 1 - \\frac { \\varepsilon } { 3 } \\big ) | I \\cap V | \\leq \\sum _ { \\ell \\in \\Gamma } | A _ { \\ell } \\cap V | = \\Big | \\bigcup _ { \\ell \\in \\Gamma } A _ { \\ell } \\cap V \\Big | \\leq \\Big | \\bigcup _ { i } B _ { \\ell ( i ) } \\cap V \\Big | \\leq \\Big | \\bigcup _ { i } B _ { \\ell ( i ) } \\Big | . \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{gather*} \\langle s _ { i } , s _ { j o } \\rangle = \\langle s _ { i j } , s _ { i } \\rangle = \\langle s _ { j o } , s _ { i j } \\rangle = \\langle s _ { j } , s _ { k o } \\rangle ^ \\perp = ^ \\perp \\langle s _ { k } , s _ { i o } \\rangle . \\end{gather*}"}
-{"id": "1269.png", "formula": "\\begin{align*} A : = \\mathbb { Q } [ q ] _ { ( q ) } = \\left \\{ \\left . \\frac { f } { g } \\ \\right | \\ f , g \\in \\mathbb { Q } [ q ] , \\ g ( 0 ) \\not = 0 \\right \\} . \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} = \\left [ \\binom { k } { j } \\lambda _ i ^ { k - j } \\right ] ^ T _ { f \\leq k < f + ( m + 1 ) n } . \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} \\lceil \\langle v + \\varepsilon p , n _ j \\rangle \\rceil = \\lceil \\langle w + \\varepsilon p , n _ j \\rangle \\rceil \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\epsilon ( [ m _ t ^ { - 1 } \\gamma ] ) = \\epsilon ( [ g _ t ^ { - 1 } m ^ { - 1 } g _ t \\gamma ] ) & = \\epsilon ( [ g _ t ^ { - 1 } m m ^ { - 1 } g _ t \\gamma ] ) + \\epsilon ( [ g _ t ^ { - 1 } ] ) \\epsilon ( [ m ^ { - 1 } g _ t \\gamma ] ) \\\\ & = \\epsilon ( [ \\gamma ] ) + \\epsilon ( [ g _ t ^ { - 1 } ] ) \\epsilon ( [ m ^ { - 1 } g _ t \\gamma ] ) = 0 . \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} e _ 3 & = \\frac 1 6 \\left [ ( \\operatorname { t r } S ) ^ 3 - 3 ( \\operatorname { t r } S ) \\operatorname { t r } ( S ^ 2 ) + 2 \\operatorname { t r } ( S ^ 3 ) \\right ] , \\\\ e _ 4 & = \\frac 1 { 2 4 } \\left [ ( \\operatorname { t r } S ) ^ 4 - 6 ( \\operatorname { t r } S ) ^ 2 \\operatorname { t r } ( S ^ 2 ) + 3 ( \\operatorname { t r } ( S ^ 2 ) ) ^ 2 + 8 ( \\operatorname { t r } S ) \\operatorname { t r } ( S ^ 3 ) - 6 \\operatorname { t r } ( S ^ 4 ) \\right ] . \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } P f _ n ( x _ n ) - P f ( x ) \\leq \\lim _ { n \\to + \\infty } | P f _ n ( x _ n ) - P f ( x _ n ) | + \\lim _ { n \\to + \\infty } | P f ( x _ n ) - P f ( x ) | = 0 , \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} \\lvert u \\rvert _ S = \\lvert ( \\phi w ) _ n \\rvert _ S . \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} { \\mathbb P } _ { 0 } \\Big ( \\int ^ { t } _ { 0 } \\big \\lvert { \\mathbb E } _ { 0 } \\big [ \\lvert b ( s , X _ { s } , F _ { s } ) \\rvert \\ , \\big \\vert \\ , \\mathcal F _ { T } ^ { X } \\big ] \\big \\rvert ^ { 2 } { \\mathrm d } s < \\infty \\Big ) \\ , = \\ , 1 \\ , ; 0 \\le t \\le T \\ , . \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} J [ \\psi ^ { ( 1 ) } ] _ { T , t } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { p _ 1 \\to \\infty } } $ \\cr } } } \\sum _ { j _ 1 = 0 } ^ { p _ 1 } C _ { j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } , \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} \\cos ~ z = { } _ { 0 } F _ { 1 } \\left ( \\ \\begin{array} { l l l } \\overline { ~ ~ ~ ~ ~ } ; ~ \\\\ ~ ~ ~ \\frac { 1 } { 2 } ; ~ \\end{array} \\frac { - z ^ { 2 } } { 4 } \\right ) , \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} W ( t , x ) = \\underset { v \\in \\mathcal { U } ^ { t } [ t , T ] } { \\inf } Y _ { t } ^ { t , x ; v } . \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} & | \\nabla _ g u | ^ 2 _ g = g ^ { i j } \\frac { \\partial u ^ { \\alpha } } { \\partial x ^ i } \\frac { \\partial u ^ { \\alpha } } { \\partial x ^ j } = e ^ { \\frac { | x | ^ 2 } { 2 ( m - 2 ) } } | \\nabla _ { g _ 0 } u | ^ 2 _ { g _ 0 } ; \\\\ & d V _ g = \\sqrt { d e t g } d V _ { g _ 0 } = e ^ { - \\frac { m } { 4 ( m - 2 ) } | x | ^ 2 } d V _ { g _ 0 } . \\\\ \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} T ( S ^ { - 1 } ( F ) ) = \\tilde { T } ( \\pi ( \\pi ^ { - 1 } ( \\tilde { S } ^ { - 1 } ( F ) ) ) ) = \\tilde { T } ( \\tilde { S } ^ { - 1 } ( F ) ) = \\tilde { T } ( G ) \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} \\dfrac { \\log ( 1 \\pm x ) } { x } = \\pm H _ 1 ( \\mp x ) . \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} \\pi \\ast \\pi ^ \\prime ( t ) : = \\begin{cases} \\pi ( 2 t ) & ( 0 \\leq t \\leq 1 / 2 ) , \\\\ \\pi ( 1 ) + \\pi ^ \\prime ( 2 t - 1 ) & ( 1 / 2 \\leq t \\leq 1 ) . \\end{cases} \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} \\sigma : = \\sup _ { \\{ 0 \\} \\times \\beta _ { m + 1 } ( G ) } f _ { m + 1 } ^ { - 1 } < \\infty . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} p ^ * ( Y ) = \\inf _ { ( Y , - W ) \\in \\Omega ( A , B ) } h ^ * ( W ) . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} X _ { n + 1 } = \\bigg \\{ & \\psi ( t ) , \\ t \\in [ 0 , n + 1 ] \\bigg | \\ \\psi ( t ) = \\psi _ n ^ * ( t ) \\ t \\in [ 0 , n ] , \\ \\| \\psi ( t ) \\| _ 2 \\leq 2 C _ 1 ( 1 + t ) ^ { - \\frac { d } { 4 } } \\| \\psi _ 0 \\| _ 1 \\\\ & \\ { \\rm a n d } \\ \\sqrt { Q ( \\psi ( t ) ) } \\leq 2 C _ 1 C _ Q ( 1 + t ) ^ { - \\frac { d } { 4 } - \\frac { \\eta } { 2 } } \\| \\psi _ 0 \\| _ 1 , \\ n \\leq t \\leq n + 1 \\bigg \\} . \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} K ^ * _ { \\nu , \\frac { \\pi } { 2 } } = \\frac { 1 } { 2 } ( - \\partial _ p ^ 2 + p ^ 2 ) - \\sqrt { \\nu } \\Big ( p \\partial _ q - q \\partial _ p \\Big ) = O _ p - \\sqrt { \\nu } X _ { \\frac { \\pi } { 2 } } \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} \\textbf { I } _ { C } ( \\upsilon , b , \\lambda , y ) = \\int _ { 0 } ^ { \\infty } ~ x ^ { \\upsilon - 1 } e ^ { - ( \\lambda b ) \\sqrt { x } } \\left \\{ \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\lambda ) _ { k } } { k ! } e ^ { - ( b k ) \\sqrt { x } } \\right \\} \\cos ( x y ) d x . \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} \\mathbb { P } [ \\mu ] ( x ) = \\bar { C } \\ , \\mathcal { t } ( x ) \\otimes \\mathcal { t } ( x ) \\ , \\delta _ \\Gamma ( x ) . \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} b \\wedge r : = \\begin{bmatrix} b _ v \\times r _ v \\\\ b _ s r _ v - r _ s b _ v \\end{bmatrix} \\in \\mathbb { R } ^ 6 , \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} h = c - V V ^ { * } c , p = \\| h \\| _ { M } , Q = \\begin{bmatrix} \\Sigma & \\delta _ { s + 1 } ^ { 1 / 2 } V ^ { * } c \\\\ 0 & \\delta _ { s + 1 } ^ { 1 / 2 } p \\end{bmatrix} , \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} { R } _ k ^ i [ n _ 0 ] ( \\mathbf { p } , \\mathbf { s } , \\mathbf { r } ) = s _ { k } ^ i [ n _ 0 ] \\mathcal { B } \\log _ 2 \\Big ( 1 + \\frac { H _ k ^ i [ n _ 0 ] p _ k ^ i [ n _ 0 ] } { \\norm { \\mathbf { r } [ n _ 0 ] - \\mathbf { r } _ k } ^ 2 } \\Big ) . \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} \\frac { d } { d t } u ( t ) & = - e ^ { t ( - A ) } A u _ { i n } - \\int _ 0 ^ t e ^ { ( t - \\tau ) ( - A ) } \\lambda ( S , N ( u ) , F ) A u \\ , d \\tau + \\lambda ( S ( t ) , N ( u ( t ) ) , F ( t ) u ( t ) , \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} D ^ * _ { w } ( y ) : = \\{ p \\in \\R ^ m \\ | \\ p = \\lim _ { l \\rightarrow \\infty } \\nabla w ( y _ l ) \\ \\ { \\rm f o r \\ s o m e } \\ \\ y _ l \\rightarrow y \\} . \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} U _ { n i } & = u _ n + c _ i \\Delta t \\varphi _ { 1 } ( c _ i \\Delta t J _ n ) F ( u _ n ) + \\Delta t \\sum _ { j = 2 } ^ { i - 1 } a _ { i j } ( \\Delta t J _ n ) D _ { n j } , \\\\ u _ { n + 1 } & = u _ n + h \\varphi _ { 1 } ( \\Delta t J _ n ) F ( u _ n ) + \\Delta t \\sum _ { i = 2 } ^ { s } b _ { i } ( \\Delta t J _ n ) D _ { n i } \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} L ( s , \\pi ) = \\prod _ { p < \\infty } L ( s , \\pi _ p ) = \\sum _ { n \\in \\N } \\lambda _ { \\pi } ( n ) n ^ { - s } \\Re ( s ) \\gg 1 . \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} f ( k ^ { - 1 } k _ 0 ^ { - 1 } x h _ 0 h ) = f ( k ^ { - 1 } k _ 0 ^ { - 1 } y h _ 0 h ) = 0 . \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} \\dim V ( r \\varpi _ n ) = \\prod _ { i = 1 } ^ r \\dfrac { i } { n + i } \\binom { 2 n + 2 r } { n + r - i } \\binom { 2 n + 2 r } { r - i } ^ { - 1 } \\prod _ { 1 \\leq i < j \\leq r } \\frac { i + j } { 2 n + i + j } \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} t ^ { ( m ) _ h } = t ( t - h ) ( t - 2 h ) \\dots ( t - ( m - 1 ) h ) , \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} f ( g _ { t _ 1 } , \\ldots , g _ { t _ { n - 1 } } ) & = \\sum _ { g _ 0 \\in G } g _ 0 ^ { - 1 } \\theta ( g _ 0 , g _ { t _ 1 } , \\ldots , g _ { t _ { n - 1 } } ) \\\\ & \\quad + \\sum _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ i \\sum _ { \\substack { g , \\bar { g } \\in G \\\\ g \\bar { g } = g _ { t _ i } } } \\theta ( g _ { t _ 1 } , \\ldots , g _ { t _ { i - 1 } } , g , \\bar { g } , g _ { t _ { i + 1 } } , \\ldots , g _ { t _ { n - 1 } } ) \\\\ & \\quad + ( - 1 ) ^ n \\sum _ { g _ n \\in G } \\theta ( g _ { t _ 1 } , \\ldots , g _ { t _ { n - 1 } } , g _ n ) . \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} I _ { \\mu } ^ { \\infty } ( v _ 1 ) = c _ { \\mu } ^ { \\infty } = c _ { \\mu } . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} ( f | g ) ^ \\sigma = f ^ \\sigma | g _ \\lambda = f | g _ \\lambda g ^ { - 1 } g = \\chi ( A D - \\lambda ^ { - 1 } B C ) f | g = \\chi _ g ( \\sigma | _ L ) f | g . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} \\begin{aligned} H _ j [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] : = & \\Bigg \\{ \\mu _ { 0 j } ^ { \\frac { n + 2 s } { 2 } } S _ { \\mu , \\xi , j } ( \\xi _ j + \\mu _ { 0 j } y , t ) + B _ j [ \\phi _ j ] + B _ j ^ 0 [ \\phi _ j ] \\\\ & + p \\mu _ { 0 j } ^ { \\frac { n - 2 s } { 2 } } \\frac { \\mu _ { 0 j } ^ { 2 s } } { \\mu _ j ^ { 2 s } } U ^ { p - 1 } \\left ( \\frac { \\mu _ { 0 j } } { \\mu _ j } y \\right ) \\psi ( \\xi _ j + \\mu _ { 0 j } y , t ) \\Bigg \\} \\chi _ { B _ { 2 R } ( 0 ) } ( y ) \\end{aligned} \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} \\log \\bigg | \\dfrac { N ^ { ( 3 / 2 + i t ) / 2 } } { N ^ { ( \\sigma + i t ) / 2 } } \\bigg | = \\bigg ( \\dfrac { 3 } { 4 } - \\dfrac { \\sigma } { 2 } \\bigg ) \\log N . \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} E ( G ) = \\sum _ { i = 1 } ^ n | \\lambda _ i | \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} \\psi _ \\gamma ( \\sigma + i \\tau ) = \\tau \\dot { \\gamma } ( \\sigma ) \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} \\frac { \\rm { D } } { \\partial \\overline { z } } w = 0 \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} 0 & = \\sum _ { m \\in \\mathcal { I } } \\overline { \\alpha _ m } G _ { k , m , 0 } ( z ) = \\sum _ { m \\in \\mathcal { I } } \\xi _ { 2 - k } \\biggl ( \\alpha _ m \\frac { ( 4 \\pi m ) ^ { 1 - k } } { k - 1 } F _ { 2 - k , - m , 0 } ( z ) \\biggr ) \\\\ & = \\frac { ( 4 \\pi ) ^ { 1 - k } } { k - 1 } \\xi _ { 2 - k } \\bigg ( \\sum _ { m \\in \\mathcal { I } } \\frac { \\alpha _ m } { m ^ { k - 1 } } F _ { 2 - k , - m , 0 } ( z ) \\bigg ) . \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} E ( X ) = E \\left [ E ( X | Y ) \\right ] , \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\rho ( m _ t ) R _ 1 = R _ 1 - \\epsilon ( [ g _ t ] ) R _ t = ( 1 - c ) R _ 1 - \\sum _ { \\ast = 1 } ^ k c _ \\ast \\epsilon ( [ g _ t ] ) R _ { j _ \\ast } . \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} u _ { X } \\otimes u _ { Y } = u _ { X \\otimes Y } \\circ c _ { Y , X } \\circ c _ { X , Y } . \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} { { f } _ { u } } = { { g } _ { v } } , { { f } _ { v } } = - { { g } _ { u } } \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} 0 \\le Q x = P Q x \\le P x = 0 , \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} \\int \\limits _ t ^ T \\int \\limits _ t ^ s d { \\bf w } _ { \\tau } ^ { ( i _ 1 ) } d { \\bf w } _ { s } ^ { ( i _ 2 ) } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { q \\to \\infty } } $ \\cr } } } \\int \\limits _ t ^ T \\left ( { \\bf w } _ s ^ { ( i _ 1 ) } - { \\bf w } _ t ^ { ( i _ 1 ) } \\right ) ^ { ( q ) } d { \\bf w } _ s ^ { ( i _ 2 ) } . \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} \\beta _ { \\tilde g _ 0 ^ { ( 4 ) } } h ^ { ( 4 ) } = 0 \\quad M . \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} \\Phi _ n = \\Phi ^ + _ n - \\Phi ^ - _ n , \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} \\| S _ { k , \\theta } ^ { \\frac 1 2 } ( P ^ { ( k , \\alpha ) } _ t - P ^ { ( k ) } _ t ) S _ { k , \\theta } ^ { \\frac 1 2 } \\| _ { H S } & = \\| \\bar { f } \\otimes f - \\bar { g } \\otimes g \\| _ 2 \\\\ & \\leq \\| ( \\bar { f } - \\bar { g } ) \\otimes f \\| _ 2 + \\| \\bar { g } \\otimes ( f - g ) \\| _ 2 \\\\ & = \\| f - g \\| _ 2 ( \\| f \\| _ 2 + \\| g \\| _ 2 ) . \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 1 2 n + 9 ) q ^ n & \\equiv \\dfrac { f _ 6 ^ 4 } { f _ 3 ^ 2 } ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t } ^ { ( u ) } \\ , = \\ , - ( X _ { t } ^ { ( u ) } - u \\widetilde { X } ^ { ( u ) } _ { t } ) { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , ; t \\ge 0 \\ , . \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} - \\mathbf { i } \\partial _ { \\omega } u + \\lambda u + \\Gamma _ K ( \\mu u ) = \\Gamma _ K p , x \\in \\mathbb { T } ^ n \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} \\| u _ T ( 0 ) \\| _ { L ^ 2 } = \\| Q \\| _ { L ^ 2 } = M _ { } . \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} H ^ 3 ( \\mathbb P , \\mathcal T _ { \\mathbb P } \\otimes \\mathcal O _ { \\mathbb P } ( K _ { \\mathbb P } + \\widetilde X ) ) = 0 . \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} P _ { t } ^ { n } f = Q ^ { n } _ { t } ( f \\circ T _ { n } ) \\circ T _ { n } ^ { - 1 } . \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} K _ { \\mathcal I } = K _ { I _ 1 } \\times \\cdots \\times K _ { I _ r } \\hookrightarrow \\mathbf P _ { \\mathcal I } = P _ { I _ 1 } \\times \\cdots \\times P _ { I _ r } \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} s _ { m , t } ( n ) \\ ! = \\ ! \\frac { 1 } { N } \\sum _ { k = 0 } ^ { N - 1 } \\sqrt { P _ { m , k } } S _ { m , k } ( n ) e ^ { j 2 \\pi \\frac { k t } { N } } , \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} \\mathcal { U } _ 2 ( \\underline { \\tilde { g } } ) - \\min _ { g _ q \\in \\mathbb { Q } } \\mathcal { U } _ 2 ( g _ c ^ { - 1 } { \\tilde { g } } g _ q g _ c ) & = \\mathcal { U } _ 2 ( \\underline { \\tilde { g } } ) - \\min _ { g _ q \\in \\mathbb { Q } } \\mathcal { U } _ 2 ( \\underline { \\tilde { g } } g _ c ^ { - 1 } g _ q g _ c ) \\\\ & = \\mathcal { U } _ { 1 } ( \\tilde { g } ) - \\min _ { g _ q \\in \\mathbb { Q } } \\mathcal { U } _ { 1 } ( \\tilde { g } g _ q ) > \\delta , \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} ( \\tilde { e } _ i ( \\pi ) ) ( t ) : = \\begin{cases} \\pi ( t ) & ( 0 \\leq t \\leq t _ 0 ) , \\\\ s _ i ( \\pi ( t ) - \\pi ( t _ 0 ) ) + \\pi ( t _ 0 ) & ( t _ 0 \\leq t \\leq t _ 1 ) , \\\\ \\pi ( t ) + \\alpha _ i & ( t _ 1 \\leq t \\leq 1 ) . \\end{cases} \\end{align*}"}
-{"id": "9969.png", "formula": "\\begin{align*} \\Gamma = \\underset { 1 \\le k \\le n } { \\bigcap } H _ { k } \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} \\mathbf { G } \\triangleq \\begin{pmatrix} \\frac { 1 } { \\rho \\eta } \\mathbf { I } _ { m p } & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\rho \\mathbf { I } _ { n p } \\end{pmatrix} \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} \\exp _ { \\phi } ( \\mathcal { E } _ { \\phi } ( Y _ 1 , \\dots , Y _ n , W ) ) = \\exp _ { \\psi } ( L ( \\psi , \\tilde { \\mathbb { A } } ) ) \\cdot ( Y _ 1 \\dots Y _ n W ) . \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{gather*} T _ { t _ { - \\alpha _ 2 } } \\big ( E _ { k , 1 } ( 1 ) \\big ) = T _ { t _ { - \\alpha _ 2 } } T _ { k } \\big ( E _ { k + 1 , 1 } ( 1 ) \\big ) = T _ { k } T _ { t _ { - \\alpha _ 2 } } \\big ( E _ { k + 1 , 1 } ( 1 ) \\big ) = T _ k \\big ( E _ { k + 1 , 1 } ( 1 ) \\big ) = E _ { k , 1 } ( 1 ) . \\end{gather*}"}
-{"id": "5665.png", "formula": "\\begin{align*} \\beta ( t ) = \\rho ^ 2 ( t ) . \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} i \\hbar | \\chi | ^ 2 \\partial _ t { \\psi } + \\frac { i } { 2 } \\operatorname { I m } \\ ! \\bigg ( \\chi ^ * \\frac { \\delta h } { \\delta \\chi } \\bigg ) \\psi = & \\ \\frac { 1 } { 2 } \\frac { \\delta h } { \\delta \\psi } - \\lambda \\psi . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} \\gamma _ { 1 } ( y , t ) = \\frac { 1 } { 2 } \\frac { \\partial } { \\partial y } \\beta _ { 1 } ( y , t ) - \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} w _ i = x _ 0 ^ { q ^ i } + \\pi x _ 1 ^ { q ^ { i - 1 } } + \\cdots + \\pi ^ i x _ i \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} \\chi ( \\alpha ) : = \\hat \\chi _ { i , i + e _ k } ^ 0 ( \\nu _ \\alpha ) = \\frac 1 2 \\hat a _ { i , i + e _ k } ( \\nu _ \\alpha ) , \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} R _ { \\mathcal { X } _ p } \\left ( e _ { ( z , w ) } \\circ T _ V \\right ) ( t , \\zeta ) = \\overline { X _ p ( t ; w , \\zeta ) } \\frac { \\overline { z } ^ { - 2 \\pi i t } } { \\overline { z } ^ { 1 + 1 / 2 \\mu } } . \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} \\div \\Big ( \\frac { \\phi ' ( | \\nabla \\P | ) } { | \\nabla \\P | } \\nabla \\P \\Big ) = 0 \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} \\kappa ( \\epsilon ) = \\mu \\nu + \\epsilon e ^ { - 2 \\alpha } \\mu \\tilde { P } \\nu _ 2 + O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ w ) \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n r _ j M _ j = \\sum _ { j < i } r _ j I _ j + r _ i M _ i + \\sum _ { j > i } r _ j M _ j \\geq \\sum _ { j < i } r _ j I _ j + r _ i M _ i \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} \\widehat { P } _ { n } ^ { ( \\alpha , \\beta ) } ( x ) = - \\frac { 1 } { 2 } ( x - b ) P _ { n - 1 } ^ { ( \\alpha , \\beta ) } ( x ) + \\frac { b P _ { n - 1 } ^ { ( \\alpha , \\beta ) } ( x ) - P _ { n - 2 } ^ { ( \\alpha , \\beta ) } ( x ) } { 2 n - 2 + \\alpha + \\beta } \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} \\Delta _ h f ( t ) = f ( t + h ) - f ( t ) = f _ h ( t ) - f ( t ) . \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} \\begin{array} { l l } L _ 1 : = 2 \\xi - e _ 0 + e _ 1 ; & L _ 1 ' : = 4 e _ 0 - 2 e _ 1 - e _ 2 - e _ 3 - e _ 4 - e _ 5 - e _ 6 ; \\\\ L _ 2 : = 2 \\xi - 2 e _ 0 + e _ 1 + e _ 2 + e _ 3 + e _ 4 & L _ 2 ' : = 5 e _ 0 - 2 e _ 1 - 2 e _ 2 - 2 e _ 3 - 2 e _ 4 - e _ 5 - e _ 6 \\\\ L _ 3 : = 2 \\xi - 3 e _ 0 + 2 e _ 1 + e _ 2 + e _ 3 + e _ 4 + e _ 5 + e _ 6 & L _ 3 ' : = 6 e _ 0 - 3 e _ 1 - 2 e _ 2 - 2 e _ 3 - 2 e _ 4 - 2 e _ 5 - 2 e _ 6 . \\end{array} \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } e ^ { x t } P _ { k } ^ { ( \\alpha , \\beta ) } ( x ) \\ , d \\mu _ { \\alpha , \\beta } ( x ) & = \\frac { ( - 1 ) ^ { k } } { 2 ^ { k } k ! } \\int _ { - 1 } ^ { 1 } e ^ { x t } \\frac { d ^ { k } } { d x ^ { k } } \\left ( ( 1 - x ) ^ { \\alpha + k } ( 1 + x ) ^ { \\beta + k } \\right ) \\ , d x \\\\ & = \\frac { t ^ { k } } { 2 ^ { k } k ! } \\int _ { - 1 } ^ { 1 } e ^ { x t } ( 1 - x ) ^ { \\alpha + k } ( 1 + x ) ^ { \\beta + k } \\ , d x , \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} w = \\sum _ { i , j } x _ i y _ j w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) . \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} \\boldsymbol { \\Pi } = \\frac { 1 } { \\xi } ( \\mathbf { I } _ n - \\xi \\rho \\mathbf { D } _ { \\mathcal { G } } ) \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} \\Re \\sum _ { n = 1 } ^ { \\infty } z _ n ^ { j } \\leq & \\frac { 1 } { ( \\sigma _ 0 - 1 ) ^ { 2 j } } - \\frac { 1 } { ( \\sigma _ 0 - \\omega _ 0 ) ^ { 2 j } } + \\Re \\left [ \\frac { 1 } { \\{ ( \\sigma _ 0 - 1 ) + i t _ 0 \\} ^ { 2 j } } - \\frac { 1 } { \\{ ( \\sigma _ 0 - \\omega _ 0 ) + i t _ 0 \\} ^ { 2 j } } \\right ] . \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} \\nu _ 2 ( S ( 2 ^ h + 1 , k + 1 ) ) = \\sigma _ 2 ( k ) - 1 1 \\le k \\le 2 ^ h . \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} \\frac { h _ { 2 , \\lambda } ^ \\prime ( 1 ) } { h _ { 2 , \\lambda } ( 1 ) } = \\frac { h _ { 1 , \\lambda } ^ \\prime ( 1 ) } { h _ { 1 , \\lambda } ( 1 ) } . \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} f ( t , w ) = T ^ { - 1 } _ S F ( \\cdot , w ) = \\int _ { \\R } F ( x + i c , w ) e ^ { i 2 \\pi c t } e ^ { - i 2 \\pi x t } \\d x , \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} \\frac { d E } { d t } : = - \\int _ { 0 } ^ \\pi \\frac { L _ { p p } ( x , u , p ) } { a ( x , u , p ) } ( u _ t ) ^ 2 d x \\leq 0 \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } E ( u _ n ( t ) ) = 1 6 E ( u _ { 0 , n } ) < 0 , \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} \\beta _ { \\tilde g ^ { ( 4 ) } } ( \\alpha ^ 2 ) & = \\delta _ { \\tilde g ^ { ( 4 ) } } ( \\alpha ^ 2 ) + \\frac { 1 } { 2 } d ( t r _ { \\tilde g ^ { ( 4 ) } } \\alpha ^ 2 ) = \\delta _ { \\tilde g ^ { ( 4 ) } } ( \\alpha ^ 2 ) + u ^ { - 3 } d u . \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} \\big | \\det ( 1 + A ) \\big | = \\prod _ { j = 1 } ^ \\infty \\big ( 1 + \\lambda _ j ( A ) \\big ) \\leq \\prod _ { j = 1 } ^ \\infty \\big ( 1 + \\mu _ j ( A ) \\big ) = \\det \\big ( 1 + | A | \\big ) . \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} ( M , \\nu ) = \\prod _ { i = 1 } ^ p ( \\P ^ { N _ i } , \\theta _ i ) ^ { r _ i } \\times ( M ' , \\nu ' ) \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} L _ i L _ j u - L _ j L _ i u = \\bigg [ ( a \\nabla \\xi _ i ) _ l \\frac { \\partial ( a \\nabla \\xi _ j ) _ { l ' } } { \\partial x _ l } - ( a \\nabla \\xi _ j ) _ l \\frac { \\partial ( a \\nabla \\xi _ i ) _ { l ' } } { \\partial x _ l } \\bigg ] \\frac { \\partial u } { \\partial x _ { l ' } } = B _ { i j , l ' } \\frac { \\partial u } { \\partial x _ { l ' } } \\ , , \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} M = \\bigoplus _ { ( \\lambda , \\mu ) \\in P \\times P } M _ { ( \\lambda , \\mu ) } , \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} P _ { k , m } ( z , s ) = \\left \\{ \\begin{array} { l l } \\sum _ { r \\in \\mathbb { Z } } F _ { k , m , r } ( z ) \\bigl ( s + \\frac { k } { 2 } - 1 \\bigr ) ^ r & k \\leq 1 / 2 , \\\\ \\ \\\\ \\sum _ { r \\in \\mathbb { Z } } G _ { k , m , r } ( z ) \\bigl ( s - \\frac { k } { 2 } \\bigr ) ^ r & k \\geq 3 / 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{gather*} \\overset { I J } { R } _ { 1 2 } \\overset { I } { T _ 1 } \\overset { J } { T _ 2 } = \\overset { J } { T _ 2 } \\overset { I } { T _ 1 } \\overset { I J } { R } _ { 1 2 } . \\end{gather*}"}
-{"id": "5596.png", "formula": "\\begin{align*} - \\mathcal { L } [ v ] + | D v | ^ m + \\sigma v = f + \\sigma \\Theta . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\frac { h ( x ) } { | T _ { N } ' ( x ) | } = c o n s t _ { j } \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} D _ t f ( \\zeta ) = t ^ { 1 / 2 \\mu } f \\left ( \\widehat { \\rho } _ t ( \\zeta ) \\right ) \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} \\partial _ t f - \\nu \\Delta f = g , \\ ; f | _ { t = 0 } = f _ 0 , \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} \\Psi ( u , \\ , v ) : = 2 ( m + 1 - \\lambda ) - ( m - 2 ) u - \\sum _ { i = 1 } ^ 2 \\left [ \\left ( \\phi _ i ( u ) - \\frac { m v } { v - 1 } \\right ) ^ 2 + 4 \\lambda m _ i u ^ 2 \\right ] ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} \\tilde { U ' } & : = \\tilde { \\psi } ( \\tilde { U } ) , \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} \\langle \\eta , a ( k ) \\psi ( k ) \\rangle = \\mu \\frac { v ( k ) } { \\omega ( k ) } \\langle \\eta , Q ( k ) \\psi \\rangle . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} W ( t ) = \\begin{cases} \\{ e \\} & ( 0 \\leq t < 1 ) , \\\\ \\{ e , s _ 2 \\} & ( t = 1 ) . \\end{cases} \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{gather*} 0 = x _ { t - 1 } ^ 2 = R ( x _ t ) R ( x _ t ) = 2 R \\big ( x _ t x _ { t - 1 } - x _ t ^ 2 \\big ) , \\\\ R ( x _ { t - 1 } + x _ t R ( 1 ) - x _ t ) = R ( 1 ) R ( x _ t ) = R ( 1 ) x _ { t - 1 } = ( \\lambda _ { i _ 1 } + t - 2 ) x _ { t - 1 } + u , \\end{gather*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\langle L _ 0 , \\ldots , L _ n \\rangle = \\langle L _ 1 | _ { Z } , \\ldots , L _ n | _ { Z } \\rangle . \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} T = \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & \\zeta _ 7 ^ 3 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & \\zeta _ 7 ^ 5 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & \\zeta _ 7 ^ 6 \\end{pmatrix} , \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} \\gamma : = \\max \\left \\{ \\big | { } \\ln \\tfrac { m _ 1 } { M } \\big | \\ , , \\ , \\big | { } \\ln \\tfrac { M _ 1 } { m } \\big | \\right \\} , \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} H _ { i , j } = \\begin{cases} \\mathbb { E } _ i ( \\tau _ j ) H ( X ) , & i \\neq j , \\\\ \\mathbb { E } _ i ( \\tau _ i ^ + ) H ( X ) , & i = j . \\end{cases} \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} { \\rm R e } \\left \\{ 1 + z \\frac { T _ g '' ( z ) } { T _ g ' ( z ) } \\right \\} & = { \\rm R e } \\left \\{ z \\frac { f ' ( z ) } { f ( z ) } + z \\frac { g '' ( z ) } { g ' ( z ) } + 1 \\right \\} \\\\ & > \\frac { 1 + \\alpha - 2 \\gamma r + ( 1 - \\alpha ) r ^ 2 } { 1 - r ^ 2 } > 0 , \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} \\mu _ t : = A _ t \\cdot \\mu _ 0 \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} \\frac 1 { B _ { i i } } \\ ; = \\ ; ( B ^ { - 1 } ) _ { i i } - \\sum _ { k , l } ^ { ( i ) } ( B ^ { - 1 } ) _ { i k } B ^ { ( i ) } _ { k l } ( B ^ { - 1 } ) _ { l i } . \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} \\dot X = - \\lambda ( X - h ( X ) ) \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} & P _ { o u t } ( R _ { E } , \\phi _ { 2 } ) \\ ! \\ ! = \\ ! \\ ! \\\\ & ( 1 \\ ! - \\ ! e ^ { \\ ! - \\ ! \\frac { \\rho _ 1 | h _ 1 | ^ { 2 } } { \\rho _ e } } ) P _ { o u t , 1 } ( R _ { E } , \\phi _ { 2 } ) + e ^ { - \\frac { \\rho _ 1 | h _ 1 | ^ { 2 } } { \\rho _ e } } P _ { o u t , 2 } ( R _ { E } , \\phi _ { 2 } ) , \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} \\mathbb { L } _ { n } = O \\left ( h _ { n } \\left ( h _ { n } ^ { \\gamma } w _ { n } ^ { - v } + h _ { n } w _ { n } ^ { - v - 1 } + \\left ( n h _ { n } \\right ) ^ { - ( d + 1 ) / 5 d } \\right ) \\right ) . \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} B _ { \\overline { l } , 0 } ( t , \\overline \\alpha ) e ^ { \\alpha _ j t } - B _ { \\overline { l } , j } ( t , \\overline \\alpha ) = S _ { \\overline { l } , j } ( t , \\overline \\alpha ) , j = 1 , \\ldots , m , \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} I ( \\mathbf { x } , \\mathbf { z } ) & = h ( \\mathbf { z } ) - h ( \\mathbf { n } _ b ) \\\\ & = - \\int _ { - \\infty } ^ { \\infty } p ( z ) \\log p ( z ) d z - \\frac { 1 } { 2 } \\log ( 2 \\pi e \\sigma _ b ^ 2 ) , \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} H = \\begin{pmatrix} H _ { 1 } \\\\ H _ { 2 } \\\\ H _ { 3 } \\end{pmatrix} = \\begin{pmatrix} 1 & 1 & 1 & 1 \\\\ 1 & 1 & - 1 & - 1 \\\\ 1 & - 1 & 1 & - 1 \\\\ 1 & - 1 & - 1 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} \\| U ( t ) \\| _ { L ^ 2 \\rightarrow L ^ 2 } & \\leq C , t \\in \\mathbb { R } , \\\\ \\| U ( t ) U ( s ) ^ * f \\| _ { L ^ \\infty } & \\leq C h ^ { - \\alpha } ( h + | t - s | ) ^ { - \\sigma } \\| f \\| _ { L ^ 1 } . \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} \\lVert f + g \\rVert _ S & = \\lVert \\sum _ { v \\in I \\cup J } ( \\alpha ( v ) + \\beta ( v ) ) \\# v \\rVert _ S \\\\ & = \\max \\{ \\lvert v \\rvert _ S \\mid v \\in I \\cup J , \\alpha ( v ) + \\beta ( v ) \\neq 0 \\} \\\\ & \\leq \\max \\{ \\lvert v \\rvert _ S \\mid v \\in I \\cup J , \\alpha ( v ) \\neq 0 o r \\beta ( v ) \\neq 0 \\} \\\\ & = \\max ( \\lVert f \\rVert _ S , \\lVert g \\rVert _ S ) . \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} \\norm { f } _ { p , A } = \\left ( \\frac { 1 } { | A | } \\sum _ { x \\in A } | f ( x ) | ^ p \\right ) ^ { 1 / p } \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} E = E _ 1 \\oplus E _ 2 , \\ E _ 1 \\not \\cong E _ 2 \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} \\partial _ t ^ p \\Phi ( \\hat { u } ) ( 0 ) = S _ { G , m , p } ( 0 , \\chi ( \\hat { u } ) , A _ 1 ^ { \\operatorname { c o } } , A _ 2 ^ { \\operatorname { c o } } , A _ 3 ^ { \\operatorname { c o } } , \\sigma ( \\hat { u } ) , f , u _ 0 ) = S _ { \\chi , \\sigma , G , m , p } ( 0 , f , u _ 0 ) \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\pi ( t ) : = \\sum _ { i = 1 } ^ { k - 1 } ( a _ i - a _ { i - 1 } ) \\nu _ i + ( t - a _ { k - 1 } ) \\nu _ k \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} f ( x ) & = 2 ( x ^ 2 + 1 ) - ( x ^ 4 + x ^ 2 + 1 ) + m \\ : h ( x ) , \\\\ g ( x ) & = ( x ^ 3 + 1 ) + m \\ ; h ( x ) . \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} \\int _ { a } ^ x \\frac { 1 } { 1 + y } \\cos 4 \\theta ( y ) d y = O ( 1 ) . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} & \\phi _ { 2 } ^ { \\ast } \\ ! = \\ ! \\min \\ ! \\bigg ( \\ ! \\underbrace { \\frac { 1 } { 2 ^ { Q _ 1 } } \\ ! + \\ ! \\frac { \\sigma _ 1 ^ 2 } { P | h _ { 1 } | ^ { 2 } 2 ^ { Q _ { 1 } } } \\ ! - \\ ! \\frac { \\sigma _ 1 ^ 2 } { P | h _ { 1 } | ^ { 2 } } } _ { \\phi _ 2 ^ { \\dag } } , ~ ~ \\phi _ 2 ^ { \\ddag } \\ ! \\bigg ) , \\ ! \\\\ & \\phi _ { 1 } ^ { \\ast } = 1 - \\phi _ { 2 } ^ { * } , \\\\ & R _ { E } ^ { \\ast } = R _ E ^ { \\dag } ( \\phi _ 2 ^ { \\ast } ) , \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} z ( x ) \\equiv ( - 2 ) ^ t + ( - 6 ) \\left ( \\sum _ { k = 1 } ^ { t - 1 } ( - 2 ) ^ { k - 1 } y _ k \\right ) \\pmod { n } \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} | \\Phi \\rangle = \\int d \\mu ( z , \\ell ) \\frac { ( { z ^ * } ) ^ { \\frac { \\ell } { 2 } } } { \\sqrt { I _ { \\ell } ( 2 | z | ) } } f ( z ) | \\psi _ z ^ \\ell \\rangle , \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} x \\in K , \\ F ( x ) - \\lambda = 0 , \\ \\langle \\lambda , x \\rangle = 0 , \\ \\langle \\lambda , y \\rangle \\geq 0 \\ \\forall y \\in K . \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} \\Omega _ j = j ^ 2 + c j , j \\in \\bar { \\mathbb { Z } } \\setminus J , \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} T ^ R M : = \\{ v \\in T M \\mid g ( v , v ) < R ^ 2 \\} , \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} H ^ 2 ( E , j ^ * ( \\mathcal O _ { \\mathbb P } ( ( d _ i + x - 6 ) H + E ) ) ) = H ^ 3 ( E , j ^ * ( \\mathcal O _ { \\mathbb P } ( ( x - 6 ) H + 2 E ) ) ) = 0 , \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} u ( t , x ) = \\int _ { \\mathbb { R } } G ^ { \\varepsilon } ( t , x - y ) \\ , u _ 0 ( y ) \\ , d y - \\int _ 0 ^ t \\int _ { \\mathbb { R } } G _ { x } ^ { \\varepsilon } ( t - s , \\ , x - y ) f \\bigl ( s , y , u ( s , y ) \\bigr ) \\ , d y \\ , d s , \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\mathcal { U } [ h ] = ( - 1 ) ^ { h } ( L _ 0 - h ) ! \\sigma _ { L _ 0 - h } \\ ! \\left ( \\overline l , \\overline { \\alpha } \\right ) \\cdot ( - 1 ) ^ { L } F _ { m + 1 } \\cdot \\alpha _ 1 ^ { l _ 0 l _ 1 } \\cdots \\alpha _ m ^ { l _ 0 l _ m } \\prod _ { 1 \\le i < j \\le m } ( \\alpha _ i - \\alpha _ j ) ^ { l _ i l _ j } \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} v _ s : = v + s \\varphi , \\lambda _ s : = \\frac { \\| v \\| _ { L ^ 2 } } { \\| v _ s \\| _ { L ^ 2 } } w _ s : = \\lambda _ s v _ s . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} \\varphi ( \\beta ) = \\arg m i n _ { a \\in \\mathcal { A } } | \\beta - a | ^ { 2 } \\in \\mathcal { A } . \\\\ \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} \\phi _ \\sigma ( \\psi _ R ( a , b ) ) = \\phi _ \\sigma ( \\zeta _ { N , R } ^ b , a ) = ( \\zeta _ { N , R ^ \\sigma } ^ { \\lambda ^ { - 1 } b } , a ) = \\psi _ { R ^ \\sigma } ( a , \\lambda ^ { - 1 } b ) \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} \\bigcap _ { c \\in I ( C ; v ) } N [ c ] = \\{ v \\} \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} { \\rm \\Theta } _ L : = \\frac 1 2 \\int _ { - 1 } ^ 1 | T _ L ( x ) | d x = \\frac { L + 1 } { 2 } \\int _ { - 1 } ^ 1 \\left | \\frac { P _ L ( x ) - P _ { L + 1 } ( x ) } { 1 - x } \\right | d x , \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} T _ 2 : = \\tau _ k + \\tau _ { k + 1 } . \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} L ( e _ 2 , e _ 3 , e _ 4 ) = \\alpha ( e _ 2 + e _ 3 + e _ 4 ) ^ 2 + \\beta \\ , e _ 2 \\ , . \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} A ^ \\flat ( P ) : = - W ' ( P , \\varphi ( P ) ) . \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} \\underset { \\bar { \\mathbf { x } } \\in \\mathbb { R } ^ { p } } { } \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\| \\mathbf { h } _ i ^ T \\bar { \\mathbf { x } } - y _ i \\| , \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} \\norm { f } _ \\mathcal { H } : = [ ( f , f ) + ( f , - \\Delta _ \\mathbb { R } ^ { - 1 } f ) ] ^ { 1 / 2 } \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\Big | g _ 2 ( \\log C ( t , \\pi ) ) - g _ 2 ( \\log C ( t - \\nu , \\pi ) ) \\Big | & \\ll \\frac { \\mu _ { d , \\sigma } } { ( 1 - \\sigma ) } \\dfrac { | \\log C ( t , \\pi ) - \\log C ( t - \\nu , \\pi ) | } { ( \\log \\log C ( t , \\pi ) ) ^ 2 } \\\\ & \\ll \\dfrac { d \\ , \\mu _ { d , \\sigma } \\ , ( \\log C ( t , \\pi ) ) ^ { 2 - 2 \\sigma } } { ( 1 - \\sigma ) ^ 2 ( \\log \\log C ( t , \\pi ) ) ^ { 3 } } . \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} \\| u ^ k - q \\| = \\| T _ { \\rho _ k } ( x ^ k ) - T _ { \\rho _ k } ( q ) \\leq \\| x ^ k - q \\| . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} z ( x ) & = - 2 ( - 2 ( \\cdots ( - 2 ( - 2 ( - 2 x + 3 y _ { t - 1 } ) + 3 y _ { t - 2 } ) + 3 y _ { t - 3 } ) \\cdots ) + 3 y _ 1 ) + 3 y _ 0 \\\\ & = ( - 2 ) ^ t + 3 ( - 2 ) ^ { t - 1 } y _ { t - 1 } + 3 ( - 2 ) ^ { t - 2 } y _ { t - 2 } + \\cdots + 3 ( - 2 ) y _ 1 + 3 y _ 0 \\\\ & = ( - 2 ) ^ t x - 6 \\left ( \\sum _ { k = 1 } ^ { t - 1 } ( - 2 ) ^ { k - 1 } y _ k \\right ) + 3 y _ 0 \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} P ' ( V ) & : = S ^ d ( U \\oplus V ) = \\bigoplus _ { i = 0 } ^ d S ^ { d - i } U \\otimes S ^ i V , \\\\ X ' ( V ) & : = X ( U \\oplus V ) \\subseteq P ' ( V ) , \\\\ Z ' ( V ) & : = \\left \\{ q \\in X ' ( V ) ~ \\middle | ~ h ( ( S ^ d ( 1 _ U \\oplus 0 _ { V \\to U } ) ) q ) \\neq 0 \\right \\} ; \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} \\frac { 1 } { B } - 1 & > \\frac { 3 \\sqrt { 3 } \\big ( ( d - 1 ) ( k - 1 ) ^ 2 \\sin ^ 2 2 v \\sin ^ 2 2 w + ( k - c ) ( k + c - 2 ) \\sin ^ 2 2 w \\big ) } { 4 ( k - 1 ) ^ 2 ( \\cos 2 v - \\cos 2 w ) } \\\\ & > \\frac { 3 \\sqrt { 3 } ( d - 1 ) \\sin ^ 2 2 v \\sin ^ 2 2 w } { 4 ( \\cos 2 v - \\cos 2 w ) } , \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} k ( r ) / | r | : = \\int _ { [ 0 , t ] } l ( r / g ( s ) ) \\lambda ( d s ) / | r | \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} \\omega = \\left ( \\ \\sum _ { j = 1 } ^ { q } \\beta _ { j } - \\sum _ { i = 1 } ^ { p } \\alpha _ { i } \\right ) , \\newline \\end{align*}"}
-{"id": "9957.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( \\int _ { \\tilde f } \\varphi \\tau _ 1 + \\bar { \\partial } _ X \\varphi \\wedge \\tau _ { 0 1 } , \\ , \\ , \\int _ { \\tilde f } \\varphi \\tau _ { 0 1 } \\right ) & = \\left ( \\int _ { D _ 1 } \\varphi \\tau _ 1 + \\bar { \\partial } _ X \\varphi \\wedge \\tau _ { 0 1 } , \\ , \\ , \\int _ { D _ 1 } \\varphi \\tau _ { 0 1 } \\right ) \\\\ & = \\left ( \\int _ { D _ 1 } \\tau _ 1 - \\bar { \\partial } _ X ( ( 1 - \\varphi ) \\tau _ { 0 1 } ) , \\ , \\ , \\int _ { D _ 1 } \\varphi \\tau _ { 0 1 } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} J ( u ) = \\frac { 1 } { 2 p } I ( u ) + \\left ( \\frac 1 2 - \\frac { 1 } { 2 p } \\right ) | | \\nabla u | | ^ 2 , \\end{align*}"}
-{"id": "9882.png", "formula": "\\begin{align*} \\| u \\| _ { \\star } : = ( \\| u \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } + | u | _ \\star ^ 2 ) ^ { 1 / 2 } , ~ ~ \\star = L , R . \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} \\widetilde N ^ { [ q ] } _ \\kappa \\left ( r , \\frac { 1 } { [ f ] _ \\kappa ^ { \\overline { n } } } \\right ) \\leq \\sum _ { i = 0 } ^ { q - 1 } N \\left ( r , \\frac { 1 } { f ( z + i \\kappa ) } \\right ) , \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} \\sum _ { h _ 0 , i } \\theta ( g _ i , g _ j ) ( g _ i ( \\pi ( \\xi ) | _ { X _ j \\cap g _ i ^ { - 1 } X _ i } ) ) & = \\sum _ { g } \\sum _ { h _ 0 \\in c ( g , X ) } \\theta ( g , g _ j ) ( g ( \\pi ( \\xi ) | _ { X _ j \\cap Z _ { g , h _ 0 } } ) ) \\\\ & = \\sum _ { g } \\theta ( g , g _ j ) ( g ( \\pi ( \\xi ) | _ { X _ j } ) ) . \\end{align*}"}
-{"id": "9832.png", "formula": "\\begin{align*} L _ { u } ( \\beta ) : = u \\beta ^ { q ^ 2 } + ( u + 1 ) \\beta ^ q + ( u ^ 2 + u ) \\beta . \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} \\overline S _ { \\rho , \\sigma } = J \\Delta _ { \\rho , \\sigma } ^ { 1 / 2 } . \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{gather*} \\theta \\left ( c G _ - ^ 0 \\right ) = d G _ + ^ 0 \\theta \\left ( c G _ - ^ 1 \\right ) = d G _ + ^ 1 \\theta \\left ( a F _ + ^ 0 \\right ) = a F _ - ^ 0 \\theta \\left ( a F _ + ^ 1 \\right ) = a F _ - ^ 1 \\\\ \\zeta ^ 2 \\left ( c G _ - ^ 1 \\right ) = c G _ - ^ 0 \\zeta ^ 2 \\left ( a F _ + ^ 1 \\right ) = a F _ + ^ 0 S \\circ \\zeta \\left ( c G _ - ^ 0 \\right ) = a F _ + ^ { - 2 } = \\zeta ^ 4 \\left ( a F _ + ^ 0 \\right ) \\end{gather*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\left | X ( t ) - \\Pi _ n X ( t ) \\right | & = \\left | X ( t ) - X \\left ( t ^ n _ { i - 1 } \\right ) - { n \\over T } \\left ( t - t ^ n _ { i - 1 } \\right ) \\left ( X \\left ( t ^ n _ { i } \\right ) - X \\left ( t ^ n _ { i - 1 } \\right ) \\right ) \\right | \\\\ & \\leq 2 \\sup _ { t \\in \\left [ t ^ n _ { i - 1 } , t ^ n _ { i } \\right ] } \\left | X ( t ) - X \\left ( t ^ n _ { i - 1 } \\right ) \\right | , \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} U _ { i _ 1 , j } & = \\left \\{ g + 3 i _ 1 + 5 , \\frac { 1 } { 2 } ( 2 g - 3 j + 6 i _ 1 + 7 ) , \\frac { 1 } { 2 } ( 2 g + 3 j - 1 2 i _ 1 - 2 3 ) \\right \\} _ { g - 1 } = : \\{ \\alpha _ 1 , \\beta _ 1 , \\gamma _ 1 \\} , \\\\ U _ { i _ 2 , j } & = \\left \\{ g + 3 i _ 2 + 5 , \\frac { 1 } { 2 } ( 2 g - 3 j + 6 i _ 2 + 7 ) , \\frac { 1 } { 2 } ( 2 g + 3 j - 1 2 i _ 2 - 2 3 ) \\right \\} _ { g } = : \\{ \\alpha _ 2 , \\beta _ 2 , \\gamma _ 2 \\} . \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\left ( \\sum _ { j = 1 } ^ { k _ n ^ 1 } ( \\tau _ j ^ 1 - \\sigma _ j ^ 1 ) + \\sum _ { j = 1 } ^ { k _ n ^ 2 } ( \\tau _ j ^ 2 - \\sigma _ j ^ 2 ) \\right ) = 1 \\hbox { \\rm a . s . } \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} u _ { i , j } ( s ) \\cdot e _ { k , l } = e _ { k , l } + s \\delta _ { j , k } e _ { i , l } - s \\delta _ { l , i } e _ { k , j } - s ^ 2 \\delta _ { j , k } \\delta _ { l , i } e _ { i , j } . \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} \\min & \\ ; F ( d , X ) = \\eta _ 1 F _ 1 ( R X ) + \\eta _ 2 F _ 2 ( d ) \\\\ s . a . & \\ ; \\ ; T ( X ) + M ^ T \\alpha - \\beta = 0 \\\\ & \\ ; \\ ; \\Gamma d - M X = 0 \\\\ & \\ ; \\ ; \\beta ^ T X = 0 \\\\ & \\ ; \\ ; \\beta \\geq 0 , X \\geq 0 , d \\geq 0 \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} m + x + y = m x y \\mbox { \\ : a n d \\ : } y x + m x + y m = 1 . \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} \\omega ( T ) = \\sup _ { t \\in ( t _ 0 , T ) } \\| u ( t ) \\| _ { W ^ { 1 , \\infty } ( G ) } \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} \\P \\left ( \\sum _ { k = 1 } ^ n \\nu _ k > x \\right ) \\le \\sum _ { k = 1 } ^ n \\P ( \\nu _ k > y ) + \\left ( \\frac { e ^ 2 A _ t ^ + } { x y ^ { t - 1 } } \\right ) ^ { x / 2 y } . \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial } { \\partial y } \\int d x \\ x c ( x , y , t ) = & \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\biggl [ \\left ( 4 y ^ { 2 } - 1 \\right ) e ^ { - 2 y ^ { 2 } } - 4 y \\left ( \\frac { 4 } { 3 } y ^ { 3 } - y \\right ) e ^ { - 2 y ^ { 2 } } \\biggr ] \\\\ = & \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( - \\frac { 1 6 } { 3 } y ^ { 4 } + 8 y ^ { 2 } - 1 \\right ) e ^ { - 2 y ^ { 2 } } \\end{aligned} \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} \\begin{aligned} u ^ { i \\bar j } ( y ) ( u _ { i \\bar j } ( y ) - u _ { i \\bar j } ( x ) ) \\le \\log \\det u _ { i \\bar j } ( y ) - \\log \\det u _ { i \\bar j } ( x ) \\le | h | _ { C ^ { 0 , 1 } } | y - x | . \\end{aligned} \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} f ( \\xi ) = \\inf \\{ \\tau \\in I : \\ , \\xi \\in K _ \\tau \\} = \\min \\{ \\tau \\in I : \\ , \\xi \\in K _ \\tau \\} . \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} u _ 0 ( p _ a ) = ( \\cos \\pi / 6 , - \\sin \\pi / 6 ) , \\ , \\ , \\ , \\ , \\ , u _ 0 ( p _ b ) = ( \\cos \\pi / 6 , \\sin \\pi / 6 ) . \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} \\tilde { x } = \\exp ( a V ) \\exp ( v X ) g \\tilde { y } \\exp ( - r U ) \\tilde { x } = \\exp ( a ' V ) \\exp ( b ' X ) g ' \\tilde { y } \\gamma \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} \\frac { d C } { d t } ( x _ { i j } , \\theta _ { i j } ) = \\bigl ( c _ 0 ^ 2 | \\theta _ { i j } \\cdot \\nabla p ( x _ { i j } ) | ^ 2 - C ( x _ { i j } , \\theta _ { i j } ) ^ { \\gamma - 1 } \\bigr ) C ( x _ { i j } , \\theta _ { i j } ) , \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} ( e _ { ( z , w ) } \\circ T ) a = T a ( z , w ) & = \\sum _ { k = 0 } ^ { \\infty } a _ k ( w ) z ^ k = \\sum _ { k = 0 } ^ { \\infty } \\left \\langle a _ k , \\overline { Y _ p ( k ; w , \\cdot ) } \\right \\rangle _ { \\mathcal { W } _ p ( k ) } \\cdot z ^ k \\\\ & = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\pi } { k + 1 } \\left ( \\int _ { \\mathbb { B } _ p } a _ k ( \\zeta ) Y _ p ( k ; w , \\zeta ) \\frac { ( k + 1 ) z ^ { k } } { \\pi } ( 1 - p ( \\zeta ) ) ^ { k + 1 } \\d V ( \\zeta ) \\right ) , \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} \\mu \\prod _ { j = 1 } ^ { n } P _ j ( \\epsilon ) r = \\tilde { \\pi } r + \\epsilon n \\tilde { \\pi } \\tilde { { P } } ^ { ( 1 ) } ( I - \\tilde { P } + \\tilde { \\Pi } ) ^ { - 1 } r - \\epsilon \\tilde { \\pi } \\tilde { P } ^ { ( 1 ) } ( I - \\tilde { P } + \\tilde { \\Pi } ) ^ { - 2 } r + O ( \\epsilon ^ { 2 } n ^ 3 ) \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} \\sigma ^ { * } = ( 1 + A _ { 1 } + . . . + A _ { p } ) - ( B _ { 1 } + . . . + B _ { q } ) = 1 - \\Delta ^ { * } . \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} \\sum _ { \\lambda + \\nu = \\mu } \\Theta _ \\lambda \\Gamma _ \\nu = \\delta _ { \\mu , 0 } . \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} M = O \\Big ( ( d + \\Delta ) ^ 4 ( n + \\Delta ^ 3 ) + ( d + \\Delta ) ^ 8 d ^ 2 \\Delta ^ 2 / n + ( d + \\Delta ) ^ { 1 0 } d ^ 2 \\Delta ^ 3 / n ^ 2 \\Big ) . \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} h _ a ( l _ 1 ) = h _ a ( l _ 2 ) = 0 . \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} \\omega = \\frac { i } { 2 } \\sum _ { p = 1 } ^ n d z _ p \\wedge d \\bar { z } _ p + O ( | z | ^ 2 ) . \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} q _ { a } - q _ { a + 1 } & = - p _ { - a } + p _ { - ( a + 1 ) } \\\\ & = - ( p _ { - a - 1 } + p _ { - a - 3 } ) + p _ { - ( a + 1 ) } \\\\ & = - p _ { - ( a + 3 ) } = q _ { a + 3 } . \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{gather*} f _ { \\mathbf { t } } ( x , y ) : = f ( x , y ) + t _ 1 x ^ 7 y ^ 3 + t _ 2 x y ^ 9 + t _ 3 x ^ 9 y ^ 2 + t _ 4 x ^ 8 y ^ 3 + t _ 5 x ^ { 1 1 } y \\\\ + t _ 6 x ^ { 1 0 } y ^ 2 + t _ 7 x ^ 9 y ^ 3 + t _ 8 x ^ { 1 1 } y ^ 2 + t _ 9 x ^ { 1 0 } y ^ 3 + t _ { 1 0 } x ^ { 1 1 } y ^ 3 . \\end{gather*}"}
-{"id": "5159.png", "formula": "\\begin{align*} \\log \\frac { \\ , { \\mathrm d } \\mathbb P ^ { ( u ) } \\ , } { \\ , { \\mathrm d } \\mathbb P _ { 0 } \\ , } \\Big \\vert _ { \\mathcal F _ { T } } \\ , = \\ , \\int ^ { T } _ { 0 } ( X _ { t } - u \\widetilde { X } _ { t } ) { \\mathrm d } X _ { t } + \\frac { \\ , 1 \\ , } { \\ , 2 \\ , } \\int ^ { T } _ { 0 } ( X _ { t } - u \\widetilde { X } _ { t } ) ^ { 2 } { \\mathrm d } t \\ , . \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} \\begin{gathered} \\big | \\partial _ \\lambda ^ \\alpha c ( \\lambda , z , z ' ) \\big | \\leq C _ { \\alpha , X } \\lambda ^ { - \\alpha } . \\end{gathered} \\end{align*}"}
-{"id": "9968.png", "formula": "\\begin{align*} V = \\{ \\ , ( z = x + \\sqrt { - 1 } y ) \\in \\C ^ n \\mid \\ , x \\in M , \\ , | y | < \\epsilon \\ , \\} \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} \\frac { \\# B \\cap I _ N } { \\# I _ N } = \\frac { \\# B ' \\cap I ' _ N } { \\# I ' _ N } \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ { \\lambda } ( x ) : = \\lambda ^ { - \\frac { \\theta - 1 } { \\theta } } u ( x _ { \\lambda } ) , \\\\ & x _ { \\lambda } : = \\left [ \\lambda | x | ^ { - \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } } + ( 1 - \\lambda ) R ^ { - \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } } \\right ] ^ { - \\frac { p } { n - p } \\frac { \\theta - 1 } { \\theta } } \\frac { x } { | x | } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} \\gamma ( P ) = \\gamma ( v _ 0 v _ { 1 } ) + \\ldots + \\gamma ( v _ { \\ell - 1 } v _ \\ell ) . \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} x _ 0 = ( x ^ 0 _ { i j } ) \\in X _ 0 ^ { ( 1 ) } , x ^ 0 _ { \\imath \\jmath } = 0 \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{gather*} \\mu ( x ) = \\left \\{ \\begin{array} { c c } ( - 1 ) ^ { \\omega ( x ) } & \\mbox { i f } \\ \\omega ( x ) = \\Omega ( x ) \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\end{gather*}"}
-{"id": "4933.png", "formula": "\\begin{align*} \\begin{aligned} f = & \\partial _ { \\lambda _ 1 } S _ { o u t } [ \\bar { \\lambda } _ 1 ] + \\left ( \\partial _ { \\lambda _ 1 } V _ { \\mu , \\xi } \\right ) [ \\bar { \\lambda } _ 1 ] \\psi + p \\left [ ( u ^ * _ { \\mu , \\xi } + \\psi + \\phi ^ { i n } ) ^ { p - 1 } - ( u ^ * _ { \\mu , \\xi } ) ^ { p - 1 } \\right ] Z _ 1 \\\\ & + p ( p - 1 ) ( u ^ * _ { \\mu , \\xi } ) ^ { p - 2 } ( \\psi + \\phi ^ { i n } ) \\partial _ { \\lambda _ 1 } u ^ * _ { \\mu , \\xi } [ \\bar { \\lambda } _ 1 ] . \\end{aligned} \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\limsup _ { n , m \\to \\infty } I ^ { n , m , \\omega ^ \\prime } _ { R , T } = 0 \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} \\sum _ { l = N + 1 } ^ { n ^ { 1 - \\varepsilon } } \\mathbf { P } ( \\tau _ x > l - 1 , \\nu _ n > l - 1 ) & \\le \\sum _ { l = N + 1 } ^ { n ^ { 1 - \\varepsilon } } \\mathbf { P } ( \\tau _ x > l - 1 , \\nu _ l > l - 1 ) \\\\ & \\le \\sum _ { l = N + 1 } ^ { \\infty } e ^ { - C l ^ \\varepsilon } . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} ( g _ 1 , \\dots , g _ r ) \\cdot ( a _ 1 , \\dots , a _ r ) = ( g _ 1 a _ 1 , a _ 1 ^ { - 1 } g _ 2 a _ 2 , \\dots , a _ { r - 1 } ^ { - 1 } g _ r a _ r ) . \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} s _ h = u ^ { 1 - p _ n } \\big ( a _ n \\Delta _ g u + s _ g u \\big ) = u ^ { 1 - p _ n } L _ g ( u ) . \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} r = 0 \\leftrightarrow \\neg A \\wedge \\neg \\neg A \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} \\int _ \\zeta \\int _ u \\varphi ( u ) { ^ u \\Phi } ( \\zeta ) \\psi ( u \\frac { t } { \\zeta } ) d u d \\zeta = \\int _ u \\varphi ( u ) \\int _ \\zeta ^ * { ^ u \\Phi } ( \\zeta ) \\psi ( u \\frac { t } { \\zeta } ) d u d \\zeta . \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\| u _ h ^ { n } \\| ^ 2 + \\Delta t \\sum _ { k = 2 } ^ n \\| \\nabla u _ h ^ { k - \\theta } \\| ^ 2 + \\gamma \\Delta t \\sum _ { k = 2 } ^ n \\| \\sigma _ h ^ { k - \\theta } \\| ^ 2 \\leq & C ( \\| u _ h ^ { 0 } \\| ^ 2 + \\Delta t \\sum _ { k = 1 } ^ n \\| g ^ { k } \\| ^ 2 ) . \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } H _ \\nu ( \\rho ) = 2 K _ \\nu ( \\rho ) + 2 G _ \\nu ^ f ( \\rho ) \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} F : U \\subset \\mathcal { L } _ { + } ^ { \\infty } ( \\Omega ) \\to \\mathbb { L } ( H ^ { 1 / 2 } ( \\partial \\Omega ) , H ^ { - 1 / 2 } ( \\partial \\Omega ) ) : F ( \\gamma ) = \\Lambda _ { \\gamma } , \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } P _ { i } ( m , s , U ) = 0 & \\ , \\ , ( m , s ) \\in ( J _ { i } \\setminus \\{ 0 \\} ) \\times ( S \\setminus \\{ N \\} ) \\\\ U ( 0 , s + 1 ) = \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } U ( 1 _ { i } , s + 1 ) & \\ , \\ , s \\in S \\setminus \\{ N \\} \\\\ U ( m , 0 ) = u _ { 0 } ( - m \\Delta x ) & \\ , \\ , m \\in J _ { i } \\end{array} \\right . \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} \\mathbf { F } = b _ { 0 + } ^ { - 1 } ( \\tilde { F } _ 0 \\times \\tilde { F } _ 1 \\times \\cdots \\times \\tilde { F } _ { m ^ + } ) . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} A = A ( a , v , w , d ) = : \\begin{pmatrix} a & v \\\\ w ^ t & d \\end{pmatrix} , \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} M : = \\sup \\left \\{ | H _ { i } ( t , x , 0 ) | \\ , \\mid \\ , ( x , t ) \\in \\overline { I _ { i } } \\times [ 0 , T ] , \\ , \\ , i \\in \\{ 1 , 2 , \\dots , K \\} \\right \\} < \\infty . \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 u '' - r ( x ) u = 0 , \\ x \\in ( a , L _ 2 ) ; \\\\ & u ( a ) = 1 ; \\\\ & u ( L _ 2 ) = u ( 0 ) . \\end{aligned} \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { \\| u _ k - f _ k \\| _ E } { \\| f _ k \\| _ E } \\le \\sum _ { k = 1 } ^ \\infty \\frac { \\delta _ k } { 1 - \\delta _ k } . \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} \\int _ I \\left | \\log w - ( \\log w ) _ I \\right | \\ , d x & = \\int _ I \\left [ \\log w - ( \\log w ) _ I \\right ] _ + \\ , d x + \\int _ I \\left [ \\log w - ( \\log w ) _ I \\right ] _ - \\ , d x \\\\ & = 2 \\int _ I \\left [ \\log w - ( \\log w ) _ I \\right ] _ + \\ , d x , \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} \\pi ( f ) \\cup y = f \\cup \\widehat { \\pi } ( y ) = f \\cup d x = d f \\cup x - d ( f \\cup x ) . \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} ( s - 1 ) ( s - p - 1 ) + s ( s - p - 2 ) + 2 ( s - q ) - 1 = \\lfloor 2 ( s ^ 2 - s p - q ) - 2 \\sqrt { s ^ 2 - s p - q } \\rfloor , \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} ( a _ 0 , a _ 1 , a _ 2 , a _ 3 ) & = \\left ( m + \\frac { a } { 2 } , m + \\frac { ( b - 1 ) } { 2 } , m - \\frac { a } { 2 } , m - \\frac { ( b + 1 ) } { 2 } \\right ) , \\\\ ( b _ 0 , b _ 1 , b _ 2 , b _ 3 ) & = \\left ( m + \\frac { ( c - 1 ) } { 2 } , m + \\frac { ( d - 1 ) } { 2 } , m - \\frac { ( c + 1 ) } { 2 } , m - \\frac { ( d + 1 ) } { 2 } \\right ) , \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} w ( b ) = b - \\sum _ { i = 1 } ^ n \\left \\lfloor \\frac { b q _ i } { 1 + \\sum _ { j = 1 } ^ n q _ j } \\right \\rfloor \\ , . \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} \\mathcal { B } _ { , w , \\lambda , \\mu } : = \\bigcup _ { a _ 1 , \\ldots , a _ l \\geq 0 } \\tilde { f } _ { i _ 1 } ^ { a _ 1 } \\cdots \\tilde { f } _ { i _ l } ^ { a _ l } ( \\mathcal { B } _ e ( \\lambda ) \\otimes \\mathcal { B } _ w ( \\mu ) ) \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} \\int _ \\Omega \\frac { \\phi ^ \\prime ( | \\nabla \\Q | ) } { | \\nabla \\Q | } \\partial _ k Q _ { i j } \\ , \\partial _ p Q _ { i j } \\ , \\partial _ k X _ p ~ d V = \\int _ \\Omega e _ L ( \\Q ) ( \\div \\mathbf { X } ) ~ d V \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} N _ L \\leq C \\int _ { \\mathbb R ^ 3 } [ V ( x ) - L ] _ { - } ^ { 3 } { \\rm d } x = C \\int _ { | x | \\leq L ^ { \\frac { 1 } { p } } } ( L - | x | ^ p ) ^ { 3 } { \\rm d } x = C L ^ { 3 + \\frac { 3 } { p } } . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} R _ k ^ i [ n ] ( \\mathbf { p } , \\mathbf { s } , \\mathbf { r } ) = s _ { k } ^ i [ n ] \\mathcal { B } \\log _ 2 \\Big ( 1 + \\frac { H _ k ^ i [ n ] p _ k ^ i [ n ] } { \\norm { \\mathbf { r } [ n ] - \\mathbf { r } _ k } ^ 2 } \\Big ) , \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} b ^ { \\mathrm { a b s } } _ f \\cdot b ^ { \\mathrm { a b s } } _ g = \\sum _ { k \\in \\Psi ^ { \\# } ( Y , \\pi ) } \\left \\lvert N _ { f , g } ^ k \\right \\rvert b ^ { \\mathrm { a b s } } _ k . \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} A _ { k + 1 } & = A _ k + r - k , \\\\ B _ { k + 1 } & = B _ k + A _ k , \\\\ C _ { k + 1 } & = C _ k - ( r - k - 1 ) A _ k , \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} 1 & 0 & u ^ { 3 } \\\\ 0 & 1 & 0 \\\\ u ^ { 3 } & 0 & - 2 u ^ { 1 } \\end{pmatrix} , \\\\ w _ { 1 2 } = w _ { 2 3 } = 0 , w _ { 3 1 } = \\frac { 1 } { \\sqrt { 2 u ^ 1 + ( u ^ 3 ) ^ 2 } } . \\end{gather*}"}
-{"id": "1955.png", "formula": "\\begin{align*} \\prod _ { j , i _ j } A _ { j , i _ j } & < \\left ( \\prod _ { j , i _ j } \\left ( | a _ j | + 1 \\right ) ^ L \\right ) \\prod _ { j , i _ j } \\left ( 1 + \\frac { 1 } { | a _ j | } \\right ) ^ { i _ j } \\\\ & = \\left ( \\prod _ j \\left ( | a _ j | + 1 \\right ) ^ { \\nu _ j L } \\right ) \\prod _ j \\left ( 1 + \\frac { 1 } { | a _ j | } \\right ) ^ { \\frac { \\nu _ j ( \\nu _ j + 1 ) } { 2 } } \\leq f ^ { M L } g ^ { \\frac { M ^ 2 } { 2 } } . \\end{align*}"}
-{"id": "9698.png", "formula": "\\begin{align*} \\log _ { \\varphi } ( g ) = \\sum _ { j } \\ell _ j ( z _ 1 ) \\dots \\ell _ j ( z _ n ) \\gamma _ j \\tau ^ j ( g ) \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} \\tilde { f } _ i ( x ) & : = \\sum _ { j \\geq 0 , \\ j \\geq - \\langle \\lambda , \\alpha _ i ^ \\vee \\rangle } \\frac { 1 } { [ j + 1 ] ! _ i } F _ i ^ { j + 1 } x _ j , \\\\ \\tilde { e } _ i ( x ) & : = \\sum _ { j > 0 , \\ j \\geq - \\langle \\lambda , \\alpha _ i ^ \\vee \\rangle } \\frac { 1 } { [ j - 1 ] ! _ i } F _ i ^ { j - 1 } x _ j \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n + 1 } \\log | D _ { n } [ \\sigma - \\lambda ] | = \\log \\mathbf { G } \\left [ | \\sigma - \\lambda | \\right ] , \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} 2 v _ 2 X ^ 3 - 6 v _ 1 Y ^ 3 + \\frac { 6 p q } { v _ 1 ^ 2 + 3 v _ 2 ^ 2 } Z ^ 3 + 6 v _ 1 X ^ 2 Y - 1 8 v _ 2 X Y ^ 2 = 0 . \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} K _ { 2 } ( V , W ) = ( \\beta ^ { 2 } - 2 \\beta ) \\Phi ( V , W ) - \\alpha ( 1 - \\beta ) g ( V , W ) , \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} \\theta ^ { ( 2 ) } _ { 1 } ( l ) = h _ { 1 2 } ( l ) + e ^ { ( 2 ) } _ { 1 } ( l ) \\mbox { a n d } \\theta ^ { ( 2 ) } _ { 3 } ( l ) = h _ { 2 3 } ( l ) + e ^ { ( 2 ) } _ { 3 } ( l ) , \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} u _ t + u u _ x = - \\partial _ x \\Lambda ^ { - 2 } ( \\frac { b } { 2 } u ^ 2 + \\frac { 3 - b } { 2 } u _ x ^ 2 ) = - \\partial _ x p * ( \\frac { b } { 2 } u ^ 2 + \\frac { 3 - b } { 2 } u _ x ^ 2 ) ; \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} \\frac { \\lambda ^ 2 } { \\rho _ \\lambda } \\int _ \\Omega e ^ { \\ , u } \\ , d x = 1 . \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } C ( \\rho _ n ) = C ( \\rho ) . \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} Q _ 0 ( k , \\xi ) & = \\omega ( k ) ( H _ \\mu ( \\xi ) - \\Sigma ( \\xi ) + \\omega ( k ) ) ^ { - 1 } \\\\ P _ 0 ( \\xi ) & = 1 _ { \\{ \\Sigma ( \\xi ) \\} } ( H _ \\mu ( \\xi ) ) \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} E ( u _ { 0 , n } ) & = \\frac { 1 } { 2 } \\| u _ { 0 , n } \\| ^ 2 _ { \\dot { H } ^ 1 _ c } - \\frac { d } { 2 d + 4 } \\| u _ { 0 , n } \\| ^ { \\frac { 4 } { d } + 2 } _ { L ^ { \\frac { 4 } { d } + 2 } } \\\\ & = \\frac { 1 } { 2 } \\mu _ n ^ 2 \\lambda ^ 2 _ n \\| Q \\| ^ 2 _ { \\dot { H } ^ 1 _ c } - \\frac { d } { 2 d + 4 } \\mu _ n ^ { \\frac { 4 } { d } + 2 } \\lambda _ n ^ 2 \\| Q \\| ^ { \\frac { 4 } { d } + 2 } _ { L ^ { \\frac { 4 } { d } + 2 } } \\\\ & = \\frac { 1 } { 2 } \\left ( 1 - \\mu _ n ^ { \\frac { 4 } { d } } \\right ) \\mu _ n ^ 2 \\lambda _ n ^ 2 \\| Q \\| ^ 2 _ { \\dot { H } ^ 1 _ c } . \\end{align*}"}
-{"id": "9926.png", "formula": "\\begin{align*} \\varphi _ { \\Omega _ + } ( z ) = \\begin{cases} 1 & z \\in \\Omega _ + , \\\\ 0 & z \\in \\Omega _ - \\\\ \\end{cases} , \\varphi _ { \\Omega _ - } ( z ) = \\begin{cases} 0 & z \\in \\Omega _ + , \\\\ 1 & z \\in \\Omega _ - \\\\ \\end{cases} , \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ r \\prod _ { s = 1 } ^ n e ^ { d _ { j , s } \\varpi _ s } ( b _ { j , N _ j } ) = \\prod _ { j = 1 } ^ r e ^ { \\lambda _ j } ( b _ { j , N _ j } ) . \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} \\left \\langle \\frac { 1 } { ( \\zeta q ^ { \\l _ N } ; q ) _ { \\infty } } \\right \\rangle = \\det ( 1 + C D ) _ { L ^ 2 ( \\R ) } = \\det ( 1 + D C ) _ { L ^ 2 ( C _ a ) } . \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{gather*} r ( E _ 1 ) ^ 2 + r ( E _ 2 ) ^ 2 + r ( E _ 3 ) ^ 2 = 3 r ( E _ 1 ) r ( E _ 2 ) r ( E _ 3 ) \\end{gather*}"}
-{"id": "8885.png", "formula": "\\begin{align*} - c _ 4 x _ 4 p _ { 4 } + \\sum _ { v > 2 , v \\neq 4 } b _ v p _ { v } + \\sum _ { \\ell > 2 } p _ { \\ell } \\frac { \\partial } { \\partial x _ \\ell } r _ { 2 3 } = 0 \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} & \\ \\bigcup _ { n \\geq 0 } \\tilde { f } _ { i } ^ { n } ( \\mathcal { B } _ { v } ( \\lambda ) \\otimes \\mathcal { B } _ { w } ( \\mu ) ) \\setminus \\{ 0 \\} \\\\ = & \\ ( \\mathcal { B } _ { s _ i v } ( \\lambda ) \\otimes \\mathcal { B } _ { w } ( \\mu ) ) \\sqcup ( S _ { i } ( \\tilde { e } _ i ^ { \\max } ( \\mathcal { B } _ v ( \\lambda ) ) ) \\otimes ( \\mathcal { B } _ { s _ i w } ( \\mu ) \\setminus \\mathcal { B } _ { w } ( \\mu ) ) ) . \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} w _ { 0 , n } = \\partial ^ { ( 0 , \\alpha _ 1 , \\alpha _ 2 , 0 ) } S _ { \\chi , \\sigma , \\R ^ 3 _ + , m , \\alpha _ 0 } ( 0 , f _ { n } , u _ { 0 , n } ) , \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} \\tilde { \\mathbf { A } } ( p ) = \\int _ { [ 0 , \\infty [ } ( p + r ) ^ { - 1 } \\ , \\mathbf { G } ( r ) \\ , \\mu ( \\dd r ) \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} \\partial _ s ( C ^ r I C ^ r ) & = C ^ r \\partial _ s ( I ) C ^ r = \\partial _ t ( Y ^ { \\vee } ) \\\\ & = \\partial _ t ( C ^ { r + 1 } I C ^ { r - 1 } ) = C \\partial _ t ( C ^ r I C ^ { r - 1 } ) . \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} Z ^ { \\phi } ( t ) = \\sum _ { i = 0 } ^ { T ( t ) - 1 } \\phi _ i ( t - \\tau _ i ) . \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} u _ { 0 , n } ( x ) : = \\mu _ n Q ( x ) , \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} \\mathrm { S G } _ m \\ni \\eta : \\begin{pmatrix} x ^ 1 \\\\ x ^ 2 \\\\ x ^ 3 \\\\ x ^ 4 \\\\ \\end{pmatrix} \\mapsto \\begin{pmatrix} x ^ 1 \\cos ( 2 m q ^ 4 ) - x ^ 2 \\sin ( 2 m q ^ 4 ) ) \\\\ x ^ 1 \\sin ( 2 m q ^ 4 ) + x ^ 2 \\cos ( 2 m q ^ 4 ) \\\\ x ^ 3 \\\\ x ^ 4 \\end{pmatrix} + \\begin{pmatrix} q ^ 1 \\\\ q ^ 2 \\\\ q ^ 3 - 2 m b q ^ 4 \\\\ q ^ 4 \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} u ( 0 , x _ 0 ) = & \\mathbb E \\bigg [ \\Phi ( W _ T + x _ 0 ) \\\\ & + \\int _ 0 ^ T f \\left ( r , W _ r + x _ 0 , u ( r , W _ r + x _ 0 ) , \\nabla u ( r , W _ r + x _ 0 ) { \\mathrm d r } \\right ) \\mathrm d r \\\\ & + A _ T ^ { W , W } ( ( \\nabla u ^ * b ) ( x _ 0 + \\cdot ) ) \\bigg ] , \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} \\lambda _ { i j } ^ { k } & = \\lambda _ { ( n + i ) ( n + j ) } ^ { k } = 0 , \\\\ \\lambda _ { i ( n + j ) } ^ { k } & = 1 \\omega ( i , j ) = k . \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} h ( z ) : = h _ { m _ x } \\ , E \\big ( A ( z ) \\big ) , \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } \\int _ K \\int _ H f ( k ^ { - 1 } x h ) d h d k d \\mu ( \\ddot { x } ) = \\int _ G f ( x ) \\rho ( x ) d x \\ f \\in C _ c ( G ) \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} \\| u \\| _ { ( \\kappa _ n + 1 ) p ^ * } & \\leq M _ { 3 1 } ^ { \\sum \\limits _ { i = 1 } ^ n \\frac { 1 } { \\kappa _ i + 1 } } M _ { 3 0 } ^ { \\sum \\limits _ { i = 1 } ^ n \\frac { 1 } { \\sqrt { \\kappa _ i + 1 } } } \\| u \\| _ { ( \\kappa _ 0 + 1 ) p ^ * } \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ 2 } { d x ^ 2 } + q ( x ) - \\frac { ( n - 1 ) ^ 2 } { 4 } K _ 0 \\right ) w _ { \\lambda _ { t + 1 } } ( x ) = \\lambda _ { t + 1 } w _ { \\lambda _ { t + 1 } } ( x ) , \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} \\partial _ t g _ t = - S ( g _ t ) - \\frac 1 2 Q ^ 2 ( g _ t ) \\ , , g _ { | t = 0 } = g _ 0 . \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} a - a \\leq - K _ 7 \\underset { k \\to \\infty } { \\lim } \\| F ( u _ k ) - v \\| ^ { p } + 0 \\leq - \\frac { K _ 7 } { ( C _ F ) ^ { \\frac { 2 p } { 1 + \\epsilon } } } \\underset { k \\to \\infty } { \\lim } \\gamma _ k ^ { \\frac { 2 } { 1 + \\epsilon } } = - K _ 8 \\underset { k \\to \\infty } { \\lim } \\gamma _ k ^ { \\frac { 2 } { 1 + \\epsilon } } , \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} \\left \\| \\prod _ { 0 \\le k \\le m } P _ { n - k } ( \\epsilon ) - \\tilde { \\Pi } + \\epsilon \\tilde { \\Pi } \\tilde { P } ^ { ( 1 ) } ( I - \\tilde { P } + \\tilde { \\Pi } ) ^ { - 2 } \\tilde { P } \\right \\| = O ( \\epsilon ^ 2 m ^ 3 ) \\end{align*}"}
-{"id": "9729.png", "formula": "\\begin{align*} \\mathcal { L } _ { d , n } ( x , y ) ( z _ 1 , \\dots , z _ n ) : = x ^ { - d } \\sum \\limits _ { a \\in A _ { + , d } } \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) \\langle a \\rangle ^ y . \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} \\Theta _ t ( ( p _ 1 , \\dots , p _ r ) , ( b _ 1 , \\dots , b _ r ) ) = ( p _ 1 b _ 1 , \\Upsilon _ t ( b _ 1 ) ^ { - 1 } p _ 2 b _ 2 , \\dots , \\Upsilon _ t ( b _ { r - 1 } ) ^ { - 1 } p _ r b _ r ) \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} \\zeta ( s ) = \\frac { \\zeta ( 2 ) } { \\Delta } \\sum _ { \\nu = 1 } ^ { s - 2 } A _ { \\nu , 2 } \\Delta _ { \\nu s } + \\frac { 1 } { \\Delta } \\sum _ { \\nu = 1 } ^ { s - 2 } A _ { \\nu } \\Delta _ { \\nu s } + \\frac { 1 } { \\Delta } \\sum _ { \\nu = 1 } ^ { s - 2 } \\theta _ { s + 1 - \\nu } \\Delta _ { \\nu s } . \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( n ) q ^ n & = \\dfrac { 1 } { 2 } \\left ( \\dfrac { f _ 3 ^ 5 } { f _ 1 ^ 3 f _ 6 ^ 2 } - \\dfrac { f _ 6 } { f _ 1 f _ 2 f _ 3 } \\right ) \\\\ \\intertext { a n d } \\sum _ { n = 0 } ^ { \\infty } \\textup { P D O } _ \\textup { t } ( n ) q ^ n & = \\dfrac { q f _ 2 f _ 3 ^ 2 f _ { 1 2 } ^ 2 } { f _ 1 ^ 2 f _ 6 } , \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} G _ { 3 / 2 , m , 0 } ( z ) - \\frac { 4 } { \\sqrt { \\pi } } G _ { 3 / 2 , 0 , 0 } ( z ) = \\frac { 2 } { \\sqrt { \\pi } } \\biggl ( q ^ m - 2 + O ( q ) \\biggr ) . \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{gather*} \\rho ( E _ { i , i } ) = \\rho ( h _ i + E _ { i + 1 , i + 1 } ) = h _ { i - 1 } + E _ { i , i } = E _ { i - 1 , i - 1 } \\end{gather*}"}
-{"id": "4171.png", "formula": "\\begin{align*} 1 & = \\sum _ { i = 1 } ^ r \\frac { - ( ( \\lambda - 1 ) - ( m _ i - 1 ) R _ \\mathbb { G } ) + [ ( ( \\lambda - 1 ) - ( m _ i - 1 ) R _ \\mathbb { G } ) ^ 2 + 4 \\lambda m _ i R _ \\mathbb { G } ^ 2 ] ^ { 1 / 2 } } { 2 m } \\\\ & = \\sum _ { i = 1 } ^ r \\frac { ( 1 - \\lambda ) + ( m _ i - 1 ) R _ \\mathbb { G } + [ ( ( 1 - \\lambda ) + ( m _ i - 1 ) R _ \\mathbb { G } ) ^ 2 + 4 \\lambda m _ i R _ \\mathbb { G } ^ 2 ] ^ { 1 / 2 } } { 2 m } . \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} d ( R | S ) = \\frac { 1 + m _ K n } { 2 ^ n } = \\frac { \\# \\{ \\alpha \\in \\mathbf { M } _ 4 : \\star ( \\alpha ) = 1 \\} } { 2 ^ n } . \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} p _ 1 b _ 1 = \\prod _ { l = 1 } ^ { N _ 1 } p _ { 1 , l } , b _ 1 ^ { - 1 } p _ 2 b _ 2 = \\prod _ { l = 1 } ^ { N _ 2 } p _ { 2 , l } , \\dots , b _ { r - 1 } ^ { - 1 } p _ r b _ r = \\prod _ { l = 1 } ^ { N _ r } p _ { r , l } . \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} m _ { u , v } = \\left ( \\begin{array} { c c } - R _ { u } \\circ R _ { \\overline v } & 0 \\\\ 0 & - R _ { \\overline u } \\circ R _ v \\end{array} \\right ) . \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} | \\mathcal { O } _ r \\setminus \\mathcal { O } _ r ^ { - } | = O ( ( r ^ { \\frac 3 2 } ) ^ { n - 1 } \\alpha ) = O ( r ^ { \\frac 3 2 n + \\frac { 1 } { 1 0 } } ) , \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} \\dim V = \\sum _ { k = 1 } ^ m c _ k \\dim B _ k V . \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} D _ I : = \\{ x ^ T : = ( x _ 1 , \\ldots , x _ N ) : ( x _ 1 , \\ldots , x _ n ) \\in I , ( x _ { n + 1 } , \\ldots , x _ N ) \\in J ^ 0 \\} \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} \\phi ' _ { \\lambda } ( t ) = \\frac { \\phi ( \\lambda + t ) } { \\lambda + t } t \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} \\dot \\rho _ t = \\Delta \\phi ( \\rho _ t ) + \\nabla \\cdot ( \\chi ( \\rho _ t ) \\nabla ( V + \\tilde H _ t ) ) . \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\theta '' _ t = \\theta '' _ { t ' } = \\theta _ { | R } . \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} | \\rho ' _ t | = \\limsup _ { h \\to 0 } \\biggl ( \\frac { W _ 2 ( \\rho _ { t } , \\rho _ { t + h } ) } { h } \\biggr ) \\le \\sqrt { C _ { \\rm L i p } } \\| v _ t \\| _ { \\chi ( \\rho _ t ) } . \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} [ u ] _ { i , \\alpha } & = \\sup \\left \\{ \\frac { u ( x , t ) - u ( y , s ) } { ( | x - y | + | t - s | ^ { \\frac { 1 } { 2 } } ) ^ { \\alpha } } \\ , \\mid \\ , ( x , t ) , ( y , s ) \\in \\overline { I _ { i } } \\times [ 0 , T ] , \\ , \\ , \\right . \\\\ & \\left . ( x , t ) \\neq ( y , s ) \\right \\} , \\\\ [ u ] _ { i , 0 } & = \\sup \\left \\{ | u ( x , t ) | \\ , \\mid \\ , ( x , t ) \\in \\overline { I _ { i } } \\times [ 0 , T ] \\right \\} . \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} ( e ^ { - t \\mathbb { E } ^ { * } \\mathbb { E } } f ) ( x ) & = \\frac { e ^ { - t Q ^ { 2 } / 4 } } { \\sqrt { 4 \\pi t } } r ^ { - Q / 2 } \\int ^ { \\infty } _ { 0 } e ^ { - \\frac { ( \\ln r - \\ln s ) ^ { 2 } } { 4 t } } s ^ { - Q / 2 } f ( s y ) s ^ { Q - 1 } d s \\\\ & = \\frac { e ^ { - t Q ^ { 2 } / 4 } } { \\sqrt { 4 \\pi t } } | x | ^ { - Q / 2 } \\int ^ { \\infty } _ { 0 } e ^ { - \\frac { ( \\ln | x | - \\ln s ) ^ { 2 } } { 4 t } } s ^ { - Q / 2 } f ( s y ) s ^ { Q - 1 } d s , \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} M ' _ { k , \\ell } & = \\lambda _ { q + 1 } ^ { k - ( m + 1 ) q - r _ \\ell + f } \\sum _ { 0 \\leq s _ 1 , \\dots , s _ q \\leq m + 1 } \\left ( \\prod _ { i = 1 } ^ q \\binom { m + 1 } { s _ i } \\right ) \\left ( \\prod _ { i = 1 } ^ q ( - \\lambda _ i ) ^ { s _ i } \\right ) \\\\ & \\times \\lambda _ { q + 1 } ^ { ( m + 1 ) q - \\sum _ { i = 1 } ^ q s _ i } \\sum _ { s _ { q + 1 } = 0 } ^ r ( - 1 ) ^ { s _ { q + 1 } } \\binom { r } { s _ { q + 1 } } \\binom { ( k - \\sum _ { i = 1 } ^ q s _ i + f ) - s _ { q + 1 } } { r _ \\ell } . \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} g ( W _ { \\lambda } ( z ) ( 0 ) , \\nu ( z ( 0 ) ) = g ( W _ { \\lambda } ( z ) ( 1 ) , \\nu ( z ( 1 ) ) = 0 , \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} N ( x ) = \\exp ( - a _ { Q } u _ { Q } ( x ) ) \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} v ( E ) = ( k , H , k + n ) \\in H ^ { 2 \\ast } ( S , \\mathbb { Z } ) \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} d X = - \\alpha X \\ , d t + \\sqrt { 2 } \\ , d W \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} L ( \\tilde { C } , \\mathbb { A } ) = \\sum _ { a \\in A _ { + } } \\frac { a ( z _ 1 ) \\dots a ( z _ n ) } { a } = \\frac { \\log _ { C } ( \\omega _ n ) } { \\omega _ { n } } . \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} \\sum _ { j = 1 , 2 } \\mathcal { R } _ j ^ 2 ( \\chi _ { 8 R } \\nu ) ( x ) & = \\sum _ { j = 1 , 2 } \\chi _ R ( x ) \\mathcal { R } _ j ^ 2 ( \\chi _ { | x | / 8 } \\nu ) ( x ) + \\sum _ { j = 1 , 2 } \\chi _ R ( x ) \\mathcal { R } _ j ^ 2 ( ( \\chi _ { 8 R } - \\chi _ { | x | } ) \\nu ) ( x ) \\\\ & + \\sum _ { j = 1 , 2 } \\mathcal { R } _ j ^ 2 ( ( \\chi _ { | x | } - \\chi _ { | x | / 2 } ) \\nu ) ( x ) ~ ~ \\mathcal { D } ' ( B _ { R } ) , \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} k _ { i - 1 } ( g ) = \\left ( \\sum _ { j = 1 } ^ { i } \\left \\langle \\frac { \\lambda - \\mu } { 2 } + \\rho , \\alpha _ j \\right \\rangle \\right ) ^ { } . \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} C _ { m , 0 } ( \\chi , \\sigma , r , \\kappa ) & = C _ { \\ref { T h e o r e m E x i s t e n c e A n d U n i q u e n e s s O n D o m a i n } , m , 0 } ( \\eta ( \\chi ) , r _ 0 ( \\chi , \\sigma , m , r , \\kappa ) ) , \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} { V } ( x , E , a , b , \\theta _ 0 ) = - \\frac { C } { 1 + x - b } \\sin 2 \\theta ( x ) . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} j _ k ( \\gamma , z ) : = \\left \\{ \\begin{array} { l l } \\sqrt { c z + d } & k \\in \\mathbb { Z } , \\\\ \\bigl ( \\frac { c } { d } \\bigr ) \\epsilon _ d ^ { - 1 } \\sqrt { c z + d } & k \\in \\mathbb { Z } + 1 / 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} \\sum _ { ( i , j ) } ^ { } ( - 1 ) ^ { ( m - 1 ) ( i - 1 ) + ( n - 1 ) ( j - 1 ) } ~ \\big [ h _ { i , j } + ( - 1 ) ^ { m - 1 } h ' _ { i + 1 , j } + ( - 1 ) ^ { ( m - 1 ) + ( n - 1 ) } h '' _ { i + 1 , j + 1 } \\big ] = 0 , \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} f _ n = 2 x _ 0 x _ { n - 1 } + S ( f _ { n - 2 } ) . \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} \\begin{cases} x ^ 0 \\in C , \\\\ y ^ k = \\alpha _ k x ^ k + ( 1 - \\alpha _ k ) T x ^ k , \\\\ x ^ { k + 1 } = \\beta _ k x ^ k + ( 1 - \\beta _ k ) T y ^ k . \\end{cases} \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} D ( \\rho \\| \\sigma ) = \\lim _ \\alpha D ( E _ \\alpha \\rho E _ \\alpha \\| E _ \\alpha \\sigma E _ \\alpha ) , \\end{align*}"}
-{"id": "9868.png", "formula": "\\begin{align*} ( \\boldsymbol { D } ^ { - \\mu } u , v ) = ( u , \\boldsymbol { D } ^ { - \\mu * } v ) . \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} & { { k } _ { 1 } } \\left ( { { u } _ { y } } + { { v } _ { x } } \\right ) + { { k } _ { 2 } } \\left ( { { u } _ { x } } - { { v } _ { y } } \\right ) + u \\left ( { { k } _ { 1 y } } + { { k } _ { 2 x } } \\right ) + v \\left ( { { k } _ { 1 x } } - { { k } _ { 2 y } } \\right ) = 0 \\\\ & { { k } _ { 1 } } \\left ( { { u } _ { x } } - { { v } _ { y } } \\right ) - { { k } _ { 2 } } \\left ( { { u } _ { y } } + { { v } _ { x } } \\right ) + u \\left ( { { k } _ { 1 x } } - { { k } _ { 2 y } } \\right ) - v \\left ( { { k } _ { 1 y } } + { { k } _ { 2 x } } \\right ) = 0 \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} f | W _ Q = \\lambda _ Q ( f ) \\tilde { f } . \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} m ( g _ 1 K , g _ 2 H ) = g _ 1 K g _ 2 H \\subseteq g _ 1 g _ 2 H , \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} T = \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & \\zeta _ 5 ^ 3 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & \\zeta _ 5 ^ 2 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & - 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} \\eta ( x , \\theta ) : = \\delta _ \\Gamma ( x ) \\otimes \\delta ( \\theta - \\mathcal { t } ( x ) ) , \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} i \\bigl ( \\sigma ( a _ 1 , a _ 2 ) \\bigr ) = s ( a _ 1 ) s ( a _ 2 ) s ( a _ 1 a _ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\Lambda ^ { - \\frac { 3 } { 2 } } ( a _ { * } - a _ { k } ) ^ { \\frac { 3 } { 2 ( p + 1 ) } } u _ { k } ( \\Lambda ^ { - 1 } ( a _ { * } - a _ { k } ) ^ { \\frac { 1 } { p + 1 } } x ) = Q ( x ) . \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} J ( \\rho ) : = - \\nabla \\phi ( \\rho ) - \\chi ( \\rho ) \\nabla V , \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} \\partial _ z ^ \\alpha = \\left ( \\frac { \\partial } { \\partial z _ 1 } \\right ) ^ { \\alpha _ 1 } \\left ( \\frac { \\partial } { \\partial z _ 2 } \\right ) ^ { \\alpha _ 2 } \\cdots \\left ( \\frac { \\partial } { \\partial z _ n } \\right ) ^ { \\alpha _ n } \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} H _ p ( t ; w , W ) = \\left \\langle R _ { \\mathcal { S } _ p ( 1 ) } ( e _ W \\circ D _ t ) , R _ { \\mathcal { S } _ p ( 1 ) } ( e _ w \\circ D _ t ) \\right \\rangle _ { \\mathcal { S } _ p ( 1 ) } & = \\left \\langle t ^ { 1 / 2 \\mu } R _ { \\mathcal { S } _ p ( 1 ) } ( e _ { \\widehat { \\rho } _ t ( W ) } ) , t ^ { 1 / 2 \\mu } R _ { \\mathcal { S } _ p ( 1 ) } ( e _ { \\widehat { \\rho } _ t ( w ) } ) \\right \\rangle _ { \\mathcal { S } _ p ( 1 ) } \\\\ & = t ^ { 1 / \\mu } H _ p \\left ( 1 ; \\widehat { \\rho } _ t ( w ) , \\widehat { \\rho } _ t ( W ) \\right ) , \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} \\bigcap _ { c \\in I ( C ; u ) } N [ c ] = \\{ u \\} \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} \\sum _ { T \\in S ( G ) } \\Pi ( T ) = \\sum _ { T \\in S ( G ) \\atop e \\in T } \\Pi ( T ) + \\sum _ { T \\in S ( G ) \\atop e \\notin T } \\Pi ( T ) . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} E [ \\tau ] = \\exp \\left ( \\frac { 1 } { 2 \\pi } \\sum _ { k = 1 } ^ { \\infty } k \\cdot \\widehat { \\log \\tau } ( k ) \\widehat { \\log \\tau } ( - k ) \\right ) \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} A _ i B ' _ { i j } + B ' _ { i j } A _ j = 0 , \\forall 1 \\leq i , j \\leq r + s . \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{gather*} A B ^ m = \\langle a ^ m , b ^ { m + 1 } \\rangle C D ^ m = \\langle c ^ m , d ^ m \\rangle \\\\ F _ \\pm = \\langle F _ + , F _ - \\rangle G _ \\pm = \\langle G _ + , G _ - \\rangle F G _ + = \\langle F _ + , G _ + \\rangle F G _ - = \\langle F _ - , G _ - \\rangle . \\end{gather*}"}
-{"id": "5826.png", "formula": "\\begin{align*} \\tilde l _ N ( t ) : = \\sum _ { j , m } \\mu _ { j , m } ( t ) \\tilde h _ { j , m } , \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\dim V ( r \\varpi _ n ) & = F ( r ) \\dfrac { F ( n + r ) } { F ( n + 2 r ) } \\dfrac { F ( 2 n + 2 r ) } { F ( 2 n + r ) } \\dfrac { F ( n ) } { F ( n + r ) } \\times \\dfrac { F ( 2 n + r ) } { F ( 2 n + 1 ) } \\dfrac { \\Phi ( 2 n + 1 ) } { \\Phi ( 2 n + 2 r - 1 ) } \\dfrac { \\Phi ( 2 r - 1 ) } { F ( r ) } \\\\ & = \\dfrac { \\Phi ( 2 r - 1 ) \\Phi ( 2 n + 2 r ) F ( n ) } { \\Phi ( 2 n ) F ( n + 2 r ) } . \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} X ( v ) = ( X ^ { ( \\tilde { x } ) } ( v ) , X ^ { ( \\tilde { y } ) } ( v ) , X ^ { ( \\tilde { z } ) } ( v ) , X ^ { ( \\bar { { \\tilde z } } ) } ( v ) ) \\in \\mathcal { P } ^ { a , q } , \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{gather*} \\rho ( h _ i ( s ) ) = \\begin{cases} h _ { - \\theta } ( s ) & , \\\\ h _ { i - 1 } ( s ) & \\end{cases} \\end{gather*}"}
-{"id": "2464.png", "formula": "\\begin{align*} p \\nmid \\binom { n } { m } n _ i \\ge m _ i p . \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} \\begin{aligned} & D \\lambda _ 2 c _ 1 + D \\lambda _ 2 c _ 2 x ^ 2 + D \\lambda _ 2 | x | ^ M \\le \\\\ & D c _ 2 \\gamma _ 2 + D c _ 2 x ^ 2 + D ( \\lambda _ 2 + \\delta _ 2 ) \\gamma _ 2 | x | ^ { M - 2 } + D ( \\lambda _ 2 + \\delta _ 2 ) | x | ^ M . \\end{aligned} \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\nu _ j : = c _ j \\sum _ { \\alpha } \\rho _ { \\alpha } \\cdot ( \\tau _ { \\alpha } ) _ { * } ( \\mu _ { \\alpha } \\star \\chi _ { j } ) , \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} & M _ G \\left ( 1 + m \\frac { x ^ 4 - 1 } { x - 1 } + y m \\frac { x ^ 4 - 1 } { x - 1 } \\right ) = 8 m + 1 , \\\\ & M _ G \\left ( 2 + k \\frac { x ^ 4 - 1 } { x - 1 } + y k \\frac { x ^ 4 - 1 } { x - 1 } \\right ) = 2 ^ 8 ( 4 k + 1 ) , \\\\ & M _ G \\left ( ( x ^ 2 + 1 ) ( x - 1 ) + k \\frac { x ^ 4 - 1 } { x - 1 } + y \\left ( ( x + 1 ) + k \\frac { x ^ 4 - 1 } { x - 1 } \\right ) \\right ) = - 2 ^ 8 ( 4 k + 1 ) , \\\\ & M _ G \\left ( ( x ^ 2 + 1 ) + k \\frac { x ^ 4 - 1 } { x - 1 } + y \\left ( - ( x + 1 ) + k \\frac { x ^ 4 - 1 } { x - 1 } \\right ) \\right ) = 2 ^ 8 ( 2 k ) . \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} \\forall v ( \\forall k ( f ( j ( v , k ) ) = 0 ) \\leftrightarrow \\forall k A ( z , j ( v , k ) , 0 ) ) \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} ( b _ 2 - b _ 3 ) x ^ 2 \\ + \\ \\frac { b _ 3 } { a _ 2 ^ 2 } y ^ 2 \\ - \\ \\frac { b _ 2 } { a _ 3 ^ 2 } z ^ 2 \\ + \\ b _ 2 b _ 3 ( b _ 2 - b _ 3 ) \\ = \\ 0 \\ , , \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} \\mathfrak { s } = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} \\mathfrak { t } = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} b = a _ 2 ' ( b _ 2 + y b _ 3 ) . \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} B _ { i j } \\ ; = \\ ; B _ { i j } ^ { ( k ) } + \\frac { B _ { i k } B _ { k j } } { B _ { k k } } \\ , , \\frac 1 { B _ { i i } } \\ ; = \\ ; \\frac 1 { B _ { i i } ^ { ( k ) } } - \\frac { B _ { i k } B _ { k i } } { B _ { i i } ^ { ( k ) } B _ { i i } B _ { k k } } \\ , , \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} \\rho ( m _ t \\gamma ) R _ 1 & = \\epsilon ( [ g _ \\alpha g _ t ^ { - 1 } m g _ t \\gamma ] ) \\\\ & = \\epsilon ( [ g _ \\alpha \\gamma ] ) - \\epsilon ( [ g _ \\alpha g _ t ^ { - 1 } ] ) \\epsilon ( [ g _ t \\gamma ] ) \\\\ & = \\rho ( \\gamma ) R _ 1 - \\epsilon ( [ g _ t \\gamma ] ) R _ t . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 1 2 n + 9 ) q ^ n & \\equiv \\dfrac { f _ 2 ^ 2 f _ 3 ^ 2 f _ { 6 } ^ 2 } { f _ 1 ^ 4 } \\equiv f _ 6 ^ 3 ~ ( \\textup { m o d } ~ 2 ) . \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{gather*} \\overset { \\mathcal { X } ^ { \\epsilon } ( s ) } { W } = \\mathcal { D } ( \\chi ^ { \\epsilon } _ s ) , H ^ { s ' } = \\mathcal { D } ( G _ { s ' } ) \\end{gather*}"}
-{"id": "7124.png", "formula": "\\begin{align*} \\mathbf { Z } ( t , s , \\cdot ) = \\mathbf { E } ( t - \\tau s , \\cdot ) \\times \\boldsymbol { \\nu } s \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} | S _ { o u t } ( x , t ) | \\lesssim \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\sum _ { j = 1 } ^ k \\frac { \\mu _ j ^ { - 2 s } \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } + \\sigma } ( t ) } { 1 + | y _ j | ^ { a } } . \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} \\dot x ( t ) ~ \\in ~ \\Big [ g ' \\bigl ( v ( t , x ( t ) + ) ) \\bigr ) , \\ , g ' \\bigl ( v ( t , x ( t ) - ) ) \\bigr ) \\Big ] , x ( t _ 1 ) ~ = ~ y _ j \\ , . \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} | z ( t ) | = \\biggl | z ( t ) \\exp \\biggl \\{ i \\int _ 0 ^ t E [ z ( s ) ] \\ , d s \\biggr \\} \\biggr | , \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} \\gamma _ - ( s u ^ + + u ^ - ) & = I ' ( u ^ - ) [ u ^ - ] + s ^ 2 \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } ( u ^ - ) ^ 2 d x \\\\ & \\leq I ' ( u ^ - ) [ u ^ - ] + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } ( u ^ - ) ^ 2 d x = \\gamma _ - ( u ) \\leq 0 . \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} & \\quad \\mbox { T h e t o p e x p r e s s i o n i n ( b ) } \\\\ & = \\sum _ { t _ 1 , \\ldots , t _ { n - 1 } } \\sum _ { \\substack { g , \\bar { g } \\in G \\\\ g \\bar { g } = g _ { t _ i } } } \\big \\langle \\theta ( g _ { t _ 1 } , \\ldots , g _ { t _ { i - 1 } } , g , \\bar { g } , g _ { t _ { i + 1 } } , \\ldots , g _ { t _ { n - 1 } } ) , \\pi ( \\xi ) | _ { [ t _ 1 , \\ldots , t _ { n - 1 } ] } \\big \\rangle \\\\ & = \\mbox { T h e b o t t o m e x p r e s s i o n i n ( b ) } . \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} \\tilde { L } _ 2 : = \\frac { 1 } { ( 4 w _ { m a x } + 1 ) } L _ 2 \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} J _ p ( u _ { k + 1 } ^ { \\delta } - u _ 0 ) = ( 1 - \\beta _ k ) J _ p ( u _ k ^ { \\delta } - u _ 0 ) - \\mu F ' ( u _ k ^ { \\delta } ) ^ * j _ p ( F ( u _ k ^ { \\delta } ) - v ^ { \\delta } ) , \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} \\| \\underline { v } _ m \\| _ { 1 } : = \\sum _ { n = 1 } ^ { N } | v _ { m n } | \\in \\mathbb { Z } _ { \\ge 1 } , m = 1 , \\ldots , M , \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} \\underline { P } ^ { \\mathrm { s o l a r } } \\big ( z [ n ] \\big ) = \\frac { C _ 1 } { 1 + e ^ { - k _ { c } ( z [ n ] - \\alpha ) } } + C _ 2 , \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\| l ( t ) - P _ N l ( t ) \\| _ { H ^ { s } _ { r } ( \\mathbb R ) } = 0 . \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} A _ 0 \\partial _ t u + \\sum _ { j = 1 } ^ 3 A _ j \\partial _ j u + D u & = f , & & x \\in \\R ^ 3 _ + , & t \\in J ; \\\\ u ( 0 ) & = u _ 0 , & & x \\in \\R ^ 3 _ + . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} \\int x _ i x _ j \\ , \\widetilde \\mu ( \\d x ) = \\frac 1 { n ( n - 1 ) } \\int ( x _ 1 + \\dotsb + x _ n ) ^ 2 - ( x _ 1 ^ 2 + \\dotsb + x _ n ^ 2 ) \\ , \\widetilde \\mu ( \\d x ) = - \\frac \\beta { 2 n } . \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} L _ c ^ { \\Delta [ 1 ] } ( W _ n ) \\simeq N i ( W _ n ) \\simeq N F [ n ] = \\mathfrak { C } [ n ] . \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} T \\psi ( t ) = P _ t \\psi _ 0 + \\int _ 0 ^ t P _ { t - s } \\mathcal { G } ( \\psi ( s ) ) d s , \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} Q _ y ( x ) : = \\frac { x ( 1 + y e ^ { - x } ) } { 1 - e ^ { - x } } , \\widehat { Q } _ y ( x ) : = \\frac { x ( 1 + y e ^ { - x ( 1 + y ) } ) } { 1 - e ^ { - x ( 1 + y ) } } = 1 + \\frac { 1 - y } { 2 } x + \\cdots \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma _ { + } p & = \\frac { q } { \\sigma _ { - } } + \\frac { \\phi } { \\sigma _ { - } } + \\frac { z ^ { n } \\psi } { \\sigma { - } } \\\\ z ^ { - n } \\sigma _ { - } p & = \\frac { q } { z ^ { n } \\sigma _ { + } } + \\frac { \\phi } { z ^ { n } \\sigma _ { + } } + \\frac { \\psi } { \\sigma _ { + } } . \\end{aligned} \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\mathrm { T } ( \\theta ) = \\left \\{ \\begin{array} { l l } \\left \\{ \\frac { 1 } { \\theta } \\right \\} & \\theta \\in ( 0 , 1 ] \\\\ 0 & \\theta = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} \\begin{aligned} \\sin ( \\bar { \\theta } _ { i + 1 , j } - \\bar { \\theta } _ { i , j } ) + \\sin ( \\bar { \\theta } _ { i - 1 , j } - \\bar { \\theta } _ { i , j } ) = \\sin \\bigg ( \\frac { 2 \\pi } { N _ 1 } \\bigg ) + \\sin \\bigg ( - \\frac { 2 \\pi } { N _ 1 } \\bigg ) = 0 , \\end{aligned} \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} ( A _ 1 - A _ 2 ) ^ 2 & = I _ f \\otimes I _ { \\ell + 1 } \\otimes S ^ 2 \\\\ & = I _ f \\otimes I _ { \\ell + 1 } \\otimes ( \\frac { n - 2 \\ell } { a } S + \\frac { \\ell ( n - \\ell ) } { a ^ 2 } I _ n ) \\\\ & = \\frac { \\ell ( n - \\ell ) } { a ^ 2 } A _ 0 + \\frac { n - 2 \\ell } { a } ( A _ 1 - A _ 2 ) , \\\\ ( A _ 1 + A _ 2 ) ^ 2 & = I _ f \\otimes I _ { \\ell + 1 } \\otimes ( n J _ n - I _ n ) \\\\ & = ( n - 1 ) A _ 0 + ( n - 2 ) ( A _ 1 + A _ 2 ) . \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} ^ { W } D ^ z f ( t ) = \\sum _ { k = - \\infty } ^ \\infty \\left ( \\frac { i 2 \\pi k } { T } \\right ) ^ { z } c _ k ( f ) e ^ { i 2 \\pi k t / T } , \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} f \\cup _ \\alpha ( g \\cup _ \\alpha h ) = ( f \\cup _ \\alpha g ) \\cup _ \\alpha h , \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} a = \\frac { 1 + \\langle X , \\mathbf n \\rangle F } { \\sqrt { ( 1 + \\langle X , \\mathbf n \\rangle F ) ^ 2 - N ^ 2 F ^ 2 } } , ~ ~ b = \\sqrt { a ^ 2 - 1 } , \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} \\| v _ n \\| ^ 2 _ { L ^ 2 } = \\| V ^ { j _ 0 } \\| ^ 2 _ { L ^ 2 } + \\| v ^ { j _ 0 } _ n \\| ^ 2 _ { L ^ 2 } + o _ n ( 1 ) , \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} \\nu \\cdot ( \\lambda , \\nu ) = ( \\lambda + \\nu , \\mu - \\nu ) \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} \\Delta K _ 1 ( \\varphi , p ) = \\int \\left [ \\frac { \\partial p } { \\partial y ^ \\alpha } ( \\boldsymbol \\varphi ( x ) ) \\right ] \\bigl [ \\Delta \\varphi ^ \\alpha ( x ) \\bigr ] \\ , \\bigl [ \\rho _ { \\boldsymbol \\varphi } ( x ) - \\rho ( x ) \\bigr ] \\ , d x \\ , . \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} f _ C ( \\varphi _ C ( x ) ) = 9 0 0 \\int _ { N _ W ^ A ( x ) } \\deg _ W ^ F ( z ) \\dd z - \\deg _ W ^ B ( x ) \\mbox { a n d } \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} \\mathfrak S _ w h \\cdot \\varphi = h '^ \\vee \\cdot \\mathfrak R \\varphi , \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} \\hat K _ { i j } ( x , 0 ) = \\sum _ { ( r , s ) \\not \\in { \\cal J } } c _ { i j r s } ( B ) \\hat K _ { r s } ( x , 0 ) + \\sum _ { ( r , s ) \\not \\in { \\cal J } } \\frac { \\partial c _ { i j r s } ( B ) } { \\partial B _ { p q } } K _ { r s } ( x , 0 ) \\mbox { f o r } x \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} S ^ \\perp : = \\{ x \\in X : \\ , x \\perp y y \\in S \\} \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\frac { 1 + p } { c } \\int _ 0 ^ 1 ( 1 - u ) ^ { \\tfrac { \\alpha + 1 + b } { c } - 1 } \\exp \\Big ( \\frac { u ( 2 - p u ) } { 2 c } \\Big ) d u = 1 . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} t = \\tau _ 0 < \\ldots < \\tau _ N = T , \\ \\ \\ \\ \\Delta _ N = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm m a x } \\cr $ \\stackrel { } { { } _ { 0 \\le j \\le N - 1 } } $ \\cr } } } \\Delta \\tau _ j \\to 0 \\ \\ \\hbox { i f } \\ \\ N \\to \\infty , \\ \\ \\ \\ \\Delta \\tau _ j = \\tau _ { j + 1 } - \\tau _ j . \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} \\phi ( s , x ) = e ^ { - \\delta s } [ a _ 1 x ^ { l _ 1 } + a _ 2 x ^ { l _ 2 } + b \\ln { x } + c ] , \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ { N - 1 } i h _ i - N h _ { N - 1 } + c + N \\rho ( E _ { N , N } ) , \\end{gather*}"}
-{"id": "6684.png", "formula": "\\begin{align*} F _ { n + r } \\sum _ { j = 0 } ^ k { ( - 1 ) ^ { ( r + 1 ) j } \\left ( { \\frac { { F _ r } } { { F _ n } } } \\right ) ^ j G _ { m + r + ( n + r ) j } } = ( - 1 ) ^ r F _ n G _ m + ( - 1 ) ^ { ( r + 1 ) k } F _ n \\left ( { \\frac { { F _ r } } { { F _ n } } } \\right ) ^ { k + 1 } G _ { m + ( k + 1 ) ( n + r ) } , n \\ne 0 \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} E ( V ^ { j _ 0 } ) + \\frac { c } { 2 } \\int | x | ^ { - 2 } | V ^ { j _ 0 } ( x ) | ^ 2 d x = d _ M . \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} & W ^ j \\left ( x ^ j + \\alpha \\Delta x ^ j \\right ) = \\sqrt { \\mu } \\left ( v ^ j + \\alpha d _ x ^ j \\right ) , \\\\ & ( W ^ j ) ^ { - 1 } \\left ( s ^ j + \\alpha \\Delta s ^ j \\right ) = \\sqrt { \\mu } \\left ( v ^ j + \\alpha d _ s ^ j \\right ) . \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} \\lim _ { b \\rightarrow \\infty } \\frac { 1 } { b } { \\log \\mathbb { P } ( B > b ) } = - \\theta , \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} K ( z , z '' ) = & \\Big ( \\int _ { r ' < \\frac { r } { 2 } } + \\int _ { \\frac { r } { 2 } \\leq r ' \\leq 2 r } + \\int _ { r ' > 2 r } \\Big ) G ( z , z ' ) Q ( z ' , z '' ) \\ ; d \\mu ( z ' ) \\\\ = & K _ { 3 , 1 } ( z , z '' ) + K _ { 3 , 2 } ( z , z '' ) + K _ { 3 , 3 } ( z , z '' ) . \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial _ t y - D \\Delta y = - \\tilde { R } y & & \\Omega \\times ( 0 , \\infty ) , \\\\ & \\partial _ \\nu y = 0 & & \\partial \\Omega \\times ( 0 , \\infty ) , \\\\ & y ( 0 ) = y _ 0 & & \\quad \\Omega . \\end{aligned} \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} S _ + = \\{ x \\in S \\colon d \\theta ( x ) \\textup { i s p o s i t i v e s e m i - d e f i n i t e } \\} \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} \\| p ( u ) - p ( v ) \\| _ { \\infty } = | ( p ( u ) - p ( v ) ) \\cdot e _ { j _ k } | \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} v _ { \\theta } ( x , t ) = \\inf \\left \\{ v ( x , s ) + \\frac { ( t - s ) ^ { 2 } } { 2 \\theta } \\ , \\mid \\ , s \\in [ 0 , T ] \\right \\} . \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} f _ { K } ^ { \\leq 2 } : = \\Pi _ { | k | \\leq K } f ^ { \\leq 2 } , f ^ { \\leq 2 } : = f ^ { ( 0 ) } + f ^ { ( 1 ) } + f ^ { ( 2 ) } . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} \\left | N ( f ( x ) , r ) - \\mu _ { k + m - r } { q \\choose r } q ^ { k - r } \\right | \\leq \\sum _ { j = k + 1 } ^ { k + m } { j \\choose r } { \\frac { q } p + m \\sqrt { q } + j \\choose j } { m - 1 \\choose k + m - j } \\sqrt { q } ^ { k + m - j } . \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{gather*} \\left \\{ \\alpha \\in X _ n ^ k : \\alpha ( 0 ) = 0 , \\ P ( \\alpha ) = d \\right \\} = \\left \\{ \\alpha \\in X _ n ^ k : \\alpha \\ \\mbox { i s $ d $ - a d d i t i v e } \\right \\} \\setminus \\bigcup _ { p \\in P D _ d } \\left \\{ \\alpha \\in X _ n ^ k : \\alpha \\ \\mbox { i s $ \\frac { d } { p } $ - a d d i t i v e } \\right \\} \\end{gather*}"}
-{"id": "778.png", "formula": "\\begin{align*} & \\{ d \\widehat { w } ^ + = 0 \\} = \\left \\{ ( \\vec { x } , \\vec { u } ) \\in \\mathbb { P } ( V ^ { + } ) _ { \\widehat { U } _ p } : \\sum _ { i = 1 } ^ a x _ i w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) = 0 \\mbox { f o r a l l } 1 \\le j \\le b \\right \\} , \\\\ & \\{ d \\widehat { w } ^ - = 0 \\} = \\left \\{ ( \\vec { y } , \\vec { u } ) \\in \\mathbb { P } ( V ^ { - } ) _ { \\widehat { U } _ p } : \\sum _ { j = 1 } ^ b y _ j w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) = 0 \\mbox { f o r a l l } 1 \\le i \\le a \\right \\} . \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} \\beta ^ { u _ 1 ' } \\alpha _ { i _ 1 } ^ { u _ 1 + p _ { i _ 1 } - 1 } \\in K , \\ ; u _ 1 ' = 0 , 1 , \\dots , u _ 1 - 1 . \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} \\left | \\frac { \\partial ^ k f _ h ( x ) } { \\prod _ { j = 1 } ^ k \\partial x _ { i _ j } } \\right | \\le \\frac { 1 } { k } \\left | \\frac { \\partial ^ k h ( x ) } { \\prod _ { j = 1 } ^ k \\partial x _ { i _ j } } \\right | , \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} W ^ { \\star } _ { n + 1 } ( w ) = \\bigcup _ { u \\in W ^ { \\star } _ { n } ( w ) } W ^ { \\star } _ { 1 } ( u ) \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t E _ u ^ { n - \\theta } , v \\Big { ) } - \\gamma ( \\nabla E _ \\sigma ^ { n - \\theta } , \\nabla v ) + ( \\nabla E _ u ^ { n - \\theta } , \\nabla v ) + ( f ( u ( t _ { n - \\theta } ) ) - f ^ { n - \\theta } ( u ) , v ) = & ( R _ 1 ^ { n - \\theta } , v ) , ~ \\forall v \\in H _ 0 ^ 1 , \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { Q } \\ ; ^ { C } _ { t } D _ { T } ^ { q } \\Psi ( x , t ) = A ^ { * } \\mathcal { Q } \\Psi ( x , t ) + \\left \\langle \\mathcal { Q } \\varphi ( t ) , f \\right \\rangle _ { L ^ { 2 } ( D ) } \\chi _ { _ { D } } f ( x ) & \\hbox { i n } Q _ { T } \\\\ \\Psi ( \\xi , t ) = 0 & \\hbox { o n } \\Sigma _ { T } \\\\ \\Psi ( x , T ) = 0 & \\hbox { i n } \\Omega . \\end{cases} \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} f _ \\gamma ( s ) : = ( \\gamma + 1 ) | s | ^ \\frac { 2 \\gamma } { \\gamma + 1 } , \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\mbox { a n a l y t i c d - c r i t i c a l f l o p } & \\mbox { i f } h ^ 1 ( F ) > 1 \\\\ \\mbox { i s o m o r p h i s m s } & \\mbox { i f } h ^ 1 ( F ) = 1 \\\\ \\mbox { e m p t y s e t s } & \\mbox { i f } h ^ 1 ( F ) = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} A ( x [ f ] ) \\cdot ( h \\otimes v ) - h \\otimes ( x \\odot v ) = ( A ( x [ f ] ) \\cdot h ) \\otimes v , \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} \\left ( I - V U \\right ) \\left ( \\frac { \\phi } { \\sigma _ { - } } \\right ) = - P _ { - } \\left ( z ^ { n } v P ^ { - } \\left ( \\frac { q } { z ^ { n } \\sigma _ { + } } \\right ) \\right ) . \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} \\begin{aligned} & \\alpha ( q - 1 ) \\theta \\frac { n - p } { p } + \\theta = \\frac { n \\theta } { p } , & & \\alpha \\theta \\frac { n - p } { p } + 1 = \\theta , \\\\ & \\alpha ^ { - \\alpha \\theta \\frac { n - p } { p } } R ^ { - \\alpha ( q - 1 ) \\frac { n - p } { p } \\theta } C ^ { \\frac { n - p } { p } \\theta } = 1 , & & \\frac { 1 } { \\alpha } = q - 1 . \\end{aligned} \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} 4 m ^ 2 ( a ^ 2 + b ^ 2 ) = c \\ , . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} s _ 2 p _ { i _ 1 } p _ { i _ 2 } & + s _ 1 p _ { i _ 1 } p _ { i _ 2 } - p _ { i _ 1 } p _ { i _ 2 } \\\\ & = 2 s _ 1 p _ { i _ 1 } p _ { i _ 2 } \\\\ & = \\frac 2 { s _ 2 } s p _ { i _ 1 } p _ { i _ 2 } \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} \\bar { F } _ t ( \\bar { \\Q } ) : = \\int _ { B _ { \\delta _ t } } \\left \\{ \\frac { \\alpha } { p } \\abs { \\nabla \\bar { \\Q } } ^ p + \\bar { T } ( t ) \\left ( 1 - | \\bar { \\Q } | ^ 2 \\right ) ^ 2 + \\bar { H } ( t ) g ( \\bar { \\Q } ) \\right \\} \\d V \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} \\begin{aligned} \\| u ^ { k } - q \\| ^ 2 & = \\| T _ { \\rho _ k } ( x ^ k ) - T _ { \\rho _ k } ( q ) \\| ^ 2 \\leq \\langle T _ { \\rho _ k } ( x ^ k ) - T _ { \\rho _ k } ( q ) , x ^ { k } - q \\rangle \\\\ & = \\langle u ^ k - q , x ^ k - q \\rangle = \\frac { 1 } { 2 } \\big [ \\| u ^ k - q \\| ^ 2 + \\| x ^ { k } - q \\| ^ 2 - \\| u ^ k - x ^ k \\| ^ 2 \\big ] . \\end{aligned} \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} d \\theta ( \\tilde Y ^ T ) = d ( \\frac { \\tilde Y ^ { \\perp } } { u } ) . \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} \\max ( u ) - t ( \\deg ( u ) - 1 ) \\leq i _ 1 = \\min ( u _ 0 ) . \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} \\mathfrak { r } ^ + ( w ) : = \\{ \\alpha \\in \\mathfrak { r } ^ + \\vert w ^ { - 1 } \\alpha \\notin \\mathfrak { r } ^ + \\} , \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} ( - \\alpha , - \\alpha ^ \\sigma ) _ 2 = - 1 . \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} d x ( t ) = f ( t , x ( t ) ; \\alpha ) d t + \\sum \\limits _ { i = 1 } ^ { m } g _ i ( t , x ( t ) ; \\alpha ) d w ^ i ( t ) , t \\geq t _ 0 \\in \\Re . \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} | C _ m | < \\max _ { \\substack { 1 \\leq i \\leq t \\\\ 0 \\leq j \\leq m } } \\{ | c _ { i , j } | \\} = M . \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} \\lvert ( i _ 1 , j _ 1 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i _ 2 , j _ 2 ) \\rvert _ { W _ n ( w ) } = \\lvert l _ { i _ 1 } \\rvert _ S - \\lvert l _ { i _ 2 } \\rvert _ S + \\lvert r _ { j _ 1 } \\rvert _ S - \\lvert r _ { j _ 2 } \\rvert _ S . \\end{align*}"}
-{"id": "9703.png", "formula": "\\begin{align*} t _ { k , i } = z _ k - m _ i . \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} & h ( k + 1 , l + 1 ) - h ( k , l + 1 ) \\\\ & = \\left ( \\frac { \\ell } { \\ell + 1 } - \\frac { f ( k + 1 ) } { \\ell + 1 } \\right ) h ( k + 1 , \\ell ) + \\frac { f ( k + 1 ) } { \\ell + 1 } h ( k , \\ell ) \\\\ & - \\left ( \\frac { \\ell } { \\ell + 1 } - \\frac { f ( k ) } { \\ell + 1 } \\right ) h ( k , \\ell ) - \\frac { f ( k ) } { \\ell + 1 } h ( k - 1 , \\ell ) \\\\ & = \\left ( \\frac { \\ell } { \\ell + 1 } - \\frac { f ( k + 1 ) } { \\ell + 1 } \\right ) \\left ( h ( k + 1 , \\ell ) - h ( k , \\ell ) \\right ) + \\frac { f ( k ) } { \\ell + 1 } \\left ( h ( k , \\ell ) - h ( k - 1 , \\ell ) \\right ) , \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} \\int _ { Y \\times U } ( \\phi ( y ) - \\phi ( y _ 0 ) ) \\gamma ( d u , d y ) \\ = \\ \\int _ { Y \\times U } \\nabla \\phi ( y ) ^ T f ( u , y ) \\xi ( d u , d y ) \\ \\forall \\ \\phi ( \\cdot ) \\in C ^ 1 \\} . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\mathfrak { s e } ( 3 ) & : = \\left \\{ X \\in \\mathbb { R } ^ { 4 \\times 4 } | X = \\begin{bmatrix} \\Omega & v \\\\ 0 & 0 \\end{bmatrix} , \\Omega \\in \\mathfrak { s o } ( 3 ) , v \\in \\mathbb { R } ^ 3 \\right \\} . \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} D e & = - z ^ { - 2 } D z \\langle \\phi _ 0 , F ( z \\phi _ 0 + h ) \\rangle + z ^ { - 1 } \\langle \\phi _ 0 , D [ F ( z \\phi _ 0 + h ) ] \\rangle , \\\\ D h & = ( H + q ^ 2 / 2 ) ^ { - 1 } \\{ - P _ c D [ F ( z \\phi _ 0 + h ) ] + [ D e ] h + e [ D h ] \\} . \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} \\mathcal E \\bigl ( ( \\rho _ t ) _ { t \\in [ 0 , T ] } \\bigr ) = \\frac 1 2 \\int _ 0 ^ T \\| v _ t \\| _ { \\chi ( \\rho _ t ) } ^ 2 \\ ; \\ ! \\mathrm d t , \\mathcal E ^ \\star \\bigl ( ( \\rho _ t ) _ { t \\in [ 0 , T ] } \\bigr ) = \\frac 1 2 \\int _ 0 ^ T \\| w _ t \\| _ { \\chi ( \\rho _ t ) } ^ 2 \\ ; \\ ! \\mathrm d t . \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} h = g \\xi ( \\pi ( g ) ) ^ { - 1 } = g _ 1 g _ 2 \\xi ( \\pi ( g _ 1 ) \\pi ( g _ 2 ) ) ^ { - 1 } = h _ 1 h _ 2 \\sigma ( \\pi ( g _ 1 ) , \\pi ( g _ 2 ) ) . \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} z ( t ) = \\varphi ( b , T - t ) , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} \\mathcal { D } ( p _ { _ 0 } | | p _ { _ 1 } ) = \\int _ { \\mathcal { Y } } p _ { _ 0 } ( y ) \\log \\frac { p _ { _ 0 } ( y ) } { p _ { _ 1 } ( y ) } d { y } . \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} | S _ { o u t } ( x , t ) | \\lesssim \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } } ( \\mu _ 0 ^ { 2 s } + \\mu _ 0 ^ { n - 4 s } ) \\lesssim \\mu _ 0 ^ { \\min ( n - 4 s , 2 s ) - ( a - 2 s ) - \\sigma } ( t _ 0 ) \\sum _ { j = 1 } ^ k \\frac { \\mu _ j ^ { - 2 s } \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } + \\sigma } } { 1 + | y _ j | ^ { a } } . \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} f ( X + Y ) = f ( X ) f ( Y ) . \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} { \\sf M } \\left \\{ \\left ( I _ { T , t } ^ { ( i _ 1 i _ 2 ) } - I _ { T , t } ^ { ( i _ 1 i _ 2 ) q } \\right ) ^ { 2 n } \\right \\} \\le C _ { n , 2 } \\left ( \\frac { ( T - t ) ^ 2 } { 2 } \\left ( \\frac { 1 } { 2 } - \\sum _ { i = 1 } ^ q \\frac { 1 } { 4 i ^ 2 - 1 } \\right ) \\right ) ^ n \\ \\to 0 \\ \\ \\ \\hbox { i f } \\ q \\to \\infty , \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} J [ \\psi ^ { ( k ) } ] _ { T , t } = \\int \\limits _ t ^ T \\psi _ k ( t _ k ) \\ldots \\int \\limits _ t ^ { t _ { 2 } } \\psi _ 1 ( t _ 1 ) d { \\bf w } _ { t _ 1 } ^ { ( i _ 1 ) } \\ldots d { \\bf w } _ { t _ k } ^ { ( i _ k ) } \\ \\ \\ ( i _ 1 , \\ldots , i _ k = 0 , 1 , \\ldots , m ) , \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} \\Psi ( x \\otimes 1 ) & = x \\otimes L _ { \\mu } ^ { - 1 } , & \\Psi ( 1 \\otimes x ) & = K _ \\mu ^ { - 1 } \\otimes x , \\\\ \\Psi ( y \\otimes 1 ) & = y \\otimes L _ { \\mu } , & \\Psi ( 1 \\otimes y ) & = K _ \\mu \\otimes y . \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\eta \\int _ { \\mathbb { R } ^ n } \\omega _ { \\eta } ^ 2 \\ , d \\mu + \\mathcal { E } ( \\omega _ { \\eta } , \\omega _ { \\eta } ) = - \\int _ { \\mathbb { R } ^ n } \\Psi \\cdot \\nabla \\omega _ { \\eta } \\ , d \\mu \\ , , \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} f ( x ) & = - 1 - ( A - B x ) ( x ^ 3 - 1 ) ( 1 + x ) + h ( x ) \\\\ g ( x ) & = ( x ^ 2 + x + 1 ) - ( C - 1 - D x ) ( x ^ 3 - 1 ) ( 1 + x ) . \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} g _ { ( 6 , 1 0 , 1 5 ) } ^ { ( 5 c - 1 , 3 ( 3 c ) - 1 , 2 ( 7 c - 1 ) + 1 ) } ( z ) = ( 1 + z ^ { 7 c - 1 } ) ( 1 + z ^ { 3 c } + z ^ { 6 c } ) ( 1 + z + z ^ c + z ^ { 2 c } + z ^ { 3 c } + z ^ { 4 c } ) \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} \\varepsilon : = \\min \\bigg \\{ \\varepsilon _ 0 , \\frac { \\delta } { 2 C _ 1 } \\bigg \\} > 0 , \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} \\mathbb { T } _ L & = \\Delta ^ { - 1 } \\left ( \\begin{matrix} ( [ b , d ] d ^ { - 1 } c + [ d , a ] + [ c , b ] ) ( a - b d ^ { - 1 } c ) ^ { - 1 } & \\left ( [ d , a ] + [ c , b ] + [ b , d ] b ^ { - 1 } a \\right ) ( c - d b ^ { - 1 } a ) ^ { - 1 } \\\\ [ a , c ] ( a - b d ^ { - 1 } c ) ^ { - 1 } & [ a , c ] ( c - d b ^ { - 1 } a ) ^ { - 1 } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } G ^ { \\varepsilon } ( t - t _ 0 , \\delta _ 0 + y ) \\ , d y = \\int _ { \\delta _ 0 / \\sqrt { ( t - t _ 0 ) \\varepsilon } } ^ { + \\infty } G ( 1 , x ) \\ , d x , \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} W _ n ( i , j ) = \\square ^ { j - i - 1 } ; \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} ( a ; q ) _ \\infty & : = \\prod _ { n = 1 } ^ \\infty ( 1 - a q ^ { n - 1 } ) \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} \\widehat q : = q + \\left ( Q - P \\right ) ( z _ q ) , \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} x _ 1 - L T ~ = ~ y _ 0 ~ < ~ y _ 1 < ~ \\cdots ~ < ~ y _ N ~ < ~ y _ { N + 1 } ~ = ~ x _ 2 + L T \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\Lambda _ { 0 } = \\left [ \\begin{array} { c } 1 \\\\ 0 \\end{array} \\right ] \\mathbb { Z } \\oplus \\left [ \\begin{array} { c } \\Re \\tilde \\tau \\\\ \\Im \\tilde \\tau \\end{array} \\right ] \\mathbb { Z } \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} \\sum _ { n = 0 } \\frac { \\zeta ^ n } { ( q ; q ) _ n } \\left \\langle q ^ { n ( x _ N ( t ) + N ) } \\right \\rangle = \\left \\langle \\frac { 1 } { ( \\zeta q ^ { x _ N ( t ) + N } ; q ) _ { \\infty } } \\right \\rangle . \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} u _ { j - 1 } : = \\begin{cases} s _ { i _ { j - 1 } } u _ j & \\ell ( s _ { i _ { j - 1 } } u _ j ) > \\ell ( u _ j ) , \\\\ u _ j & \\ell ( s _ { i _ { j - 1 } } u _ j ) < \\ell ( u _ j ) . \\end{cases} \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} \\sum _ { \\eta ' \\in C } \\tilde { s } _ { \\eta , \\eta ' } \\tilde { s } _ { \\eta ' , 0 } \\xi ^ { 2 \\langle \\rho + \\eta ' , \\gamma \\rangle } = \\kappa ( - 1 ) ^ { l ( \\tilde { w } ) + l ( w _ 0 ) } \\delta _ { \\eta , - w _ 0 ( \\tilde { \\gamma } ) } . \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} \\psi _ { l } \\left ( \\frac { a \\overline { b } } { l ^ { \\frac { n _ l } { 2 } } } m \\right ) e \\left ( \\frac { a \\overline { b } } { l ^ { \\frac { n _ l } { 2 } } } m \\right ) = \\psi \\left ( \\frac { a \\overline { b } } { l ^ { \\frac { n _ l } { 2 } } } m \\right ) = 1 . \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} p _ { 1 , \\tau } = \\frac { \\partial L _ \\tau } { \\partial \\dot { x } _ { 1 , \\tau } } = \\frac { f ^ { - 1 } ( t _ \\tau ) m \\dot { x } _ { 1 , \\tau } } { \\dot { t } _ \\tau } ; \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} \\sum _ { i \\in K } [ [ x , y ] , t _ i ] \\otimes a b v _ i + \\lambda [ x , y ] \\otimes D ( a b ) + \\sum _ { i \\in I } \\Big ( [ y , \\varphi _ i ( x ) ] \\otimes a D ( b ) u _ i - [ x , \\varphi _ i ( y ) ] \\otimes D ( a ) b u _ i \\Big ) = 0 \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} \\Phi _ j ( y , t ) = \\gamma _ j \\mu _ 0 ^ { n - 2 s } p _ 0 ( y ) . \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} u ( 0 ) + \\min _ { i } \\min _ { \\tilde { \\theta } \\in [ 0 , 1 ] } H _ { i } \\left ( 0 , u _ { x _ { i } } ( 0 ) + \\tilde { \\theta } \\left ( \\sum _ { j = 1 } ^ { K } u _ { x _ { j } } ( 0 ) \\right ) ^ { - } \\right ) \\leq 0 \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} \\left | N ( f ( x ) , r ) - \\mu _ { k + m - r } { p \\choose r } p ^ { k - r } \\right | & \\leq \\sum _ { j = r } ^ { k + m } { j \\choose r } { m \\sqrt { p } + 1 + j \\choose m \\sqrt { p } + 1 } { m - 1 \\choose k + m - j } p ^ { \\frac { k + m - j } 2 } \\\\ & \\leq m \\cdot \\max _ { r \\leq j \\leq k + m } E _ j , \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} W _ t ^ { ( \\alpha , \\beta ) } f ( n ) = \\sum _ { m = 0 } ^ \\infty f ( m ) K ^ { ( \\alpha , \\beta ) } _ t ( m , n ) , \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} | T ^ { ( 2 ) } _ { a } ( x ) | \\leq C _ { 2 } \\begin{cases} | x | ^ { a - Q } , \\ ; x \\in \\mathbb { G } \\backslash \\{ 0 \\} , \\\\ | x | ^ { - Q } , \\ ; x \\in \\mathbb { G } \\ ; \\ ; | x | \\geq 1 , \\end{cases} \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} \\varphi ( u \\wedge v ) \\big ( ( \\vec z _ 1 , \\vec z _ 2 ) , \\dots , ( \\vec z _ { 2 n - 1 } , \\vec z _ { 2 n } ) \\big ) = \\big ( m _ { u , v } ( \\vec z _ 1 , \\vec z _ 2 ) , \\dots , m _ { u , v } ( \\vec z _ { 2 n - 1 } , \\vec z _ { 2 n } ) \\big ) , \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} Q _ 3 ^ k & = E \\bigg [ \\sup _ { i \\le 0 } e ^ { - \\beta \\tau _ i } Y _ i - \\max _ { i \\le n } e ^ { - \\beta \\tau _ i } Y _ i \\bigg ] ^ { \\ ! k } \\\\ & \\le E \\bigg [ \\sum _ { i > n } e ^ { - k \\beta \\tau _ i } Y _ i ^ k \\bigg ] \\\\ & = E ( Y _ 1 ^ k ) \\ , E \\bigg [ \\sum _ { i > n } e ^ { - k \\beta \\tau _ i } \\bigg ] \\\\ & \\le \\sup _ { t \\ge 0 } E \\big ( \\zeta _ i ^ k ( t ) \\big ) \\ , E \\bigg [ \\sum _ { i > n } e ^ { - k \\beta \\tau _ i } \\bigg ] , \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\eta \\omega _ { \\eta } - \\mathcal { L } \\omega _ { \\eta } = - \\nabla ^ * \\Psi \\ , . \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} \\chi _ { ( \\lambda , \\mu ) } = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } e ^ { ( w ( \\lambda + \\rho , \\mu + \\rho ) ) } } { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } e ^ { ( w ( \\rho , \\rho ) ) } } . \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} { a p \\brack b p } _ q \\equiv { a \\brack b } _ { q ^ { p ^ 2 } } - \\binom { a } { b } b ( a - b ) \\frac { p ^ 2 - 1 } { 2 4 } ( q ^ p - 1 ) ^ 2 \\bmod { [ p ] ^ 3 _ q } , \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} [ a , b , c , d ] : = \\left ( \\frac { a \\sqrt { 3 } } { 3 6 } + \\frac { b \\sqrt { 1 1 } } { 3 6 } , \\frac { c } { 3 6 } + \\frac { d \\sqrt { 3 } \\sqrt { 1 1 } } { 3 6 } \\right ) . \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} B ' = \\bigcap _ { B ' \\subset X \\subset A } X , \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} u _ 1 & > 0 E _ 1 , u _ 1 = 0 E _ 2 \\cup E _ 3 , \\\\ u _ 2 & < 0 E _ 2 , u _ 2 = 0 E _ 1 \\cup E _ 3 , \\\\ u _ 3 & \\neq 0 E _ 3 , u _ 3 = 0 E _ 1 \\cup E _ 2 . \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} u ( t ) = Q [ z ( t ) ] + v ( t ) . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} R _ 2 ^ { n - \\theta } = & ( \\sigma ( t _ { n - \\theta } ) - \\sigma ^ { n - \\theta } ) ) - \\bigtriangleup ( u ( t _ { n - \\theta } ) - u ^ { n - \\theta } ) = O ( \\Delta t ^ 2 ) , \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} R _ \\lambda ( h , k ) & = \\phi _ \\lambda ( h ) ( 1 - \\langle h \\cdot z _ 0 , k \\cdot z _ 0 \\rangle ) ^ { - \\lambda } \\overline { \\phi _ \\lambda ( k ) } \\\\ & = e ^ { - 2 \\pi i \\lambda ( x - y ) } \\left ( 1 - \\frac { 1 } { 2 n } \\sum _ { i = 1 } ^ n t _ i \\overline { s _ i } \\right ) ^ { - \\lambda } \\\\ & = e ^ { - 2 \\pi i \\lambda ( x - y ) } \\left ( 1 - \\frac { 1 } { 2 n } \\sum _ { i = 1 } ^ n t _ i ( s _ i ) ^ { - 1 } \\right ) ^ { - \\lambda } \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} | | X _ R | | _ { \\frac { \\tilde { r } } { 2 } , p - 1 , \\mathbf { a } } = O ( \\tilde { r } ^ 3 ) . \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} \\lvert l _ { i _ 1 } \\rvert _ S - \\lvert l _ { i _ 2 } \\rvert _ S = i _ 1 ' - i _ 2 ' = i _ 1 - i _ 2 . \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} \\Delta _ { 1 / 2 } P _ { 1 / 2 , 0 } ( z , s ) = \\biggl ( s - \\frac { 1 } { 4 } \\biggr ) \\biggl ( \\frac { 3 } { 4 } - s \\biggr ) P _ { 1 / 2 , 0 } ( z , s ) , \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} m _ { \\Phi , 1 } ( f \\otimes g ) ( t ) = \\int _ G f ( s ) g ( s ^ { - 1 } t ) L ( s , s ^ { - 1 } t ) d t \\ \\ ( s \\in G ) , \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { k = 1 } ^ { \\infty } \\dfrac { k + 1 } { \\big ( x ^ { \\sigma - \\frac { 1 } { 2 } } \\big ) ^ k } & \\Bigg | \\displaystyle \\sum _ { n \\leq x } \\Lambda ( n ) \\Bigg ( \\dfrac { 1 } { n ^ { \\sigma } ( k \\log x + \\log n ) ^ { 2 m + 2 } } - \\dfrac { 1 } { x ^ { 2 \\sigma - 1 } \\ , n ^ { 1 - \\sigma } ( ( k + 2 ) \\log x - \\log n ) ^ { 2 m + 2 } } \\Bigg ) \\Bigg | \\\\ & \\ \\ \\ \\ll _ { c } \\dfrac { x ^ { 1 - \\sigma } } { ( 1 - \\sigma ) ^ { 2 } ( \\log x ) ^ { 2 m + 3 } } . \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} d ( x , y ) = \\| x - y \\| + \\left | \\frac { 1 } { \\| x \\| } - \\frac { 1 } { \\| y \\| } \\right | , ( x , y ) \\in E \\times E , \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} \\widehat { P } _ { \\gamma , \\varepsilon } = \\widehat { P } _ { \\gamma , \\varepsilon } ( x , D ) : = \\sum _ { j = 0 } ^ N \\Bigr ( X _ j ^ * X _ j - \\varepsilon [ X _ j , X _ 0 ] ^ * [ X _ j , X _ 0 ] \\Bigr ) + \\frac { 1 } { \\gamma } Y , \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} G _ { m + n } + ( - 1 ) ^ n G _ { m - n } = L _ n G _ m \\ , , \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} \\theta = \\frac { p _ { n - 1 } \\mathrm { T } ^ n ( \\theta ) + p _ n } { q _ { n - 1 } \\mathrm { T } ^ n ( \\theta ) + q _ n } \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( \\frac { P _ x + \\sigma _ w ^ 2 } { \\sigma _ w ^ 2 } - 1 + \\log \\frac { \\sigma _ w ^ 2 } { P _ x + \\sigma _ w ^ 2 } \\right ) = 2 \\epsilon ^ 2 . \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} v _ i ( \\tau , \\xi ) = & \\int _ t ^ \\tau \\sum _ { j = 1 } ^ n \\Big ( C _ { i j } \\big ( x _ i ( s , \\tau , \\xi ) \\big ) v _ j \\big ( s , x _ i ( s , \\tau , \\xi ) \\big ) + D _ { i j } \\big ( x _ i ( s , \\tau , \\xi ) \\big ) v _ j ( s , 0 ) + f _ i \\big ( s , x _ i ( s , \\tau , \\xi ) \\big ) \\Big ) \\ , d s \\\\ [ 6 p t ] & + \\sum _ { r = 1 } ^ R \\sum _ { j = 1 } ^ n A _ { r , i j } ( t ) v _ j \\big ( t , x _ r \\big ) + \\int _ { 0 } ^ 1 \\sum _ { j = 1 } ^ n M _ { i j } ( t , x ) v _ j ( t , x ) \\ , d x + h ( t ) , \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} K _ { 1 } ( V , W ) = ( \\alpha \\beta - \\alpha ) \\Phi ( V , W ) + \\alpha ^ { 2 } g ( V , W ) + ( \\alpha ^ { 2 } + \\beta - \\beta ^ { 2 } ) \\eta ( V ) \\eta ( W ) , \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} - ( \\alpha + 1 ) = \\widehat { x } _ { 1 , 1 } ^ { ( \\alpha ) } < \\ldots < \\widehat { x } _ { n - 1 , 1 } ^ { ( \\alpha ) } < \\widehat { x } _ { n , 1 } ^ { ( \\alpha ) } < \\ldots < - \\alpha . \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\| P _ { N _ k } \\xi \\| & \\geq \\left \\| P _ { N _ k } \\left ( \\sum _ { j = 1 } ^ k c _ j \\xi _ j \\right ) \\right \\| - \\left \\| P _ { N _ k } \\left ( \\sum _ { j = k + 1 } ^ \\infty c _ j \\xi _ j \\right ) \\right \\| \\\\ & \\geq c _ k \\| P _ { N _ k } \\xi _ k \\| - \\sum _ { j = k + 1 } ^ \\infty c _ j \\| P _ { N _ k } \\xi _ j \\| \\\\ & \\geq c _ k \\| P _ { N _ k } \\xi _ k \\| \\left ( 1 - \\sum _ { j = k + 1 } ^ \\infty \\frac { 1 } { 2 ^ j } \\right ) \\geq \\frac { c _ k } { 2 } \\| P _ { N _ k } \\xi _ k \\| > 0 . \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} x F _ i - F _ i x = \\frac { K _ i r _ i ( x ) - r _ i ' ( x ) L _ i ^ { - 1 } } { q _ i - q _ i ^ { - 1 } } \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} \\sup _ { 1 \\le m \\le ( \\log ( 1 / \\epsilon ) ) ^ { 1 + \\delta } } \\| r _ m ( \\epsilon ) - ( \\tilde { r } + \\epsilon ( m - 1 ) \\tilde { r } ^ { ( 1 ) } ) \\| = O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ s ) \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} s = \\{ ( x , y ) \\in C _ { ( b , i ) } \\ | \\ y = - \\dfrac { n - 1 } { u } x + \\dfrac { n - 1 } { u } , \\ 0 < x < 1 \\} . \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} E [ \\tau , \\beta ] = G ( 1 + \\beta ) G ( 1 - \\beta ) E [ \\tau ] , \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} x _ { 0 , l } = 0 \\mbox { a n d } x _ { i , l } = x _ l ( t _ 0 , t _ i , 0 ) , \\mbox { f o r } 1 \\le i \\le l . \\end{align*}"}
-{"id": "9903.png", "formula": "\\begin{align*} | u | _ { \\widehat { H } ^ \\mu ( \\mathbb { R } ) } : = \\| | 2 \\pi \\xi | ^ \\mu \\widehat { u } \\| _ { L ^ 2 ( \\mathbb { R } ) } , \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} \\sum _ { i = j - 2 } ^ { j + 2 } ( 2 ^ j \\rho ) ^ d \\int _ { S ^ { d - 1 } } \\sup _ { r \\in [ 2 ^ { j } \\rho , 2 ^ { j + 1 } \\rho ] } | \\mathbf { K } _ { i , n , l } ( r \\theta ) | \\lesssim 2 ^ { j \\alpha } ~ ~ \\forall ~ ~ l \\geq - 4 . \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} \\Phi + \\Psi = ( \\Phi ^ + + \\Psi ^ + , \\Phi ^ - + \\Psi ^ - ) , \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} \\widetilde { u } = { { k } _ { 1 } } u - v { { k } _ { 2 } } , ~ ~ \\widetilde { v } = v { { k } _ { 1 } } + u { { k } _ { 2 } } \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} \\chi ^ A ( X ) = X ^ n + \\chi ^ A _ { n - 1 } X ^ { n - 1 } + \\cdots + \\chi ^ A _ 1 X + \\chi ^ A _ 0 \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\begin{aligned} F ( u ) = \\sup _ { \\varepsilon > 0 } \\bigg ( \\inf _ { \\substack { v \\in \\mathcal R \\\\ d ( u , v ) \\leq \\varepsilon } } F ( v ) \\bigg ) , \\ \\ u \\in \\overline { \\mathcal R } . \\end{aligned} \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} S _ { M , } : = \\left \\{ v \\in H ^ 1 _ { } \\ : \\ v ( \\ref { v a r i a t i o n a l p r o b l e m 2 } ) \\right \\} . \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} ( U ^ { i _ 0 j _ 0 } - V ^ { i _ 0 j _ 0 } ) ( \\bar { x } ) = \\eta > 0 . \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} s = \\frac { n } { p } - \\frac { n } { q } , ~ q > \\frac { n } { \\min \\{ \\alpha , n \\} } . \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\begin{aligned} K _ g = \\frac 1 2 \\left ( \\begin{matrix} - | a | ^ 2 - 2 | s | ^ 2 & 2 a \\bar s & 0 \\\\ 2 \\bar a s & - | a | ^ 2 & 0 \\\\ 0 & 0 & | a | ^ 2 \\end{matrix} \\right ) \\ , , \\end{aligned} \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} \\P ( X _ { n + 1 } = j + 1 | X _ { n } = j ) & = \\frac { f ( j ) } { n } \\times \\frac { n } { n + 1 } = \\frac { f ( j ) } { n + 1 } , \\\\ \\P ( X _ { n + 1 } = j | X _ { n } = j ) & = ( 1 - \\frac { f ( j ) } { n } ) \\times \\frac { n } { n + 1 } = \\frac { n - f ( j ) } { n + 1 } , \\\\ \\P ( X _ { n + 1 } = 0 | X _ { n } = j ) & = \\frac { 1 } { n + 1 } . \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} [ E _ i , F _ j ] = \\delta _ { i , j } \\frac { K _ i - L _ i ^ { - 1 } } { q _ i - q _ i ^ { - 1 } } , \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} \\det X = \\frac { ( - 1 ) ^ { ( n + 1 ) p } \\det \\tilde { T } } { D _ { n } [ \\sigma ] } \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} \\epsilon _ { 1 , \\sigma } = \\left \\{ \\begin{array} { c c } - k _ { 2 , \\sigma } - 3 \\ & \\mbox { i f } \\ \\sigma \\in I _ F ^ 2 \\\\ - k _ { 1 , \\sigma } - 4 \\ & \\mbox { i f } \\ \\sigma \\in I _ F ^ 3 \\end{array} \\right . , \\ \\epsilon _ { 2 , \\sigma } = \\left \\{ \\begin{array} { c c } k _ { 1 , \\sigma } + 1 \\ & \\mbox { i f } \\ \\sigma \\in I _ F ^ 2 \\\\ k _ { 2 , \\sigma } \\ & \\mbox { i f } \\ \\sigma \\in I _ F ^ 3 \\end{array} \\right . \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} \\mathcal { C } _ q = \\bigoplus _ { \\nu \\in P } \\mathcal { C } _ { q , \\nu } , \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} ( \\omega \\wedge \\omega ) _ 0 ^ u : = g ^ V _ u \\left ( \\tau _ 0 ^ u ( \\omega _ u ( a _ 2 ) ) \\xi _ 1 , \\tau _ 0 ^ u ( \\omega _ u ( a _ 1 ) ) \\xi _ 2 \\right ) - g ^ V _ u \\left ( \\tau _ 0 ^ u ( \\omega _ u ( a _ 1 ) ) \\xi _ 1 , \\tau _ 0 ^ u ( \\omega _ u ( a _ 2 ) ) \\xi _ 2 \\right ) . \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} C _ { \\hat \\chi } : = \\limsup _ { L \\to \\infty } \\frac 1 { L ^ d } \\sum _ { i \\in \\mathbb T _ L ^ d } \\sum _ { k = 1 } ^ d \\hat \\chi _ { i , i + e _ k } ( \\mu ^ L ) < \\infty , \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} = \\{ ( A _ { \\sigma } ) _ { \\sigma \\in I _ F } \\in \\prod _ { \\sigma \\in I _ F } \\mbox { G L } _ { 2 , L } ( L ) \\ \\mbox { s u c h t h a t } \\ \\mbox { d e t } ( A _ { \\sigma } ) = \\mbox { d e t } ( A _ { \\hat { \\sigma } } ) \\ \\forall \\ \\sigma , \\hat { \\sigma } \\in I _ F \\} . \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} K = { \\rm R i c } ^ { 1 , 1 } + \\frac 1 2 Q ^ 3 \\ , . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} \\inf _ { ( \\gamma , \\xi ) \\in \\Omega ( y _ 0 ) } \\int _ { Y \\times U } k ( y , u ) \\gamma ( d y , d u ) : = k ^ * ( y _ 0 ) , \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} A _ i = J _ { n _ i } ( \\vec { \\alpha } _ i ) , \\ , \\ , \\forall 1 \\leq i \\leq r ; A _ j = J _ { m _ { j - r } } ( \\vec { \\beta } _ j ) , \\ , \\ , \\forall r + 1 \\leq j \\leq r + s , \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t u = \\Delta u + A ( u ) ( \\nabla u , \\nabla u ) , \\quad \\mbox { o n $ \\mathbb { R } ^ n \\times \\mathbb { R } _ + $ } , & \\\\ & u | _ { t = 0 } = u _ 0 , & \\end{aligned} \\right . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } P _ { \\boldsymbol { \\theta } } } { \\mathrm { d } P _ { \\hat { \\boldsymbol { \\theta } } \\left ( \\boldsymbol { x } \\right ) } } \\left ( \\boldsymbol { x } \\right ) = \\exp \\left ( - D _ { A } \\left ( \\boldsymbol { \\theta } , \\hat { \\boldsymbol { \\theta } } \\left ( \\boldsymbol { x } \\right ) \\right ) \\right ) . \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} \\beta ^ { u _ 1 + t } \\alpha _ { i _ 1 } ^ { u _ 1 + p _ { i _ 1 } - 1 } = \\beta ^ { u _ 1 + t } \\alpha _ { i _ 1 } ^ { u _ 1 + t + \\lfloor \\frac { p _ { i _ 1 } - 1 - t } { s _ 1 } \\rfloor s _ 1 } \\alpha _ { i _ 1 } ^ { p _ { i _ 1 } - 1 - t - \\lfloor \\frac { p _ { i _ 1 } - 1 - t } { s _ 1 } \\rfloor s _ 1 } \\in W _ { i _ 1 } \\alpha _ { i _ 1 } ^ { p _ { i _ 1 } - 1 - t - \\lfloor \\frac { p _ { i _ 1 } - 1 - t } { s _ 1 } \\rfloor s _ 1 } \\subseteq K \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} \\hat I ( t ) = \\alpha ( t ) \\hat J _ + + \\beta ( t ) \\hat J _ - + \\delta ( t ) \\hat J _ 0 , \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} & \\stackrel { ( a ) } { \\geq } \\sqrt { \\frac { a _ { n } D _ 2 ( \\tau _ C ) } { v ^ * ( D _ 1 ( \\tau _ C ) ) ^ 2 } } \\left ( \\sqrt { \\frac { b _ { n , 1 } } { s _ 1 + \\frac { \\eta _ n ^ * } { v ^ * } } } - \\sqrt { \\frac { b _ { n , { 2 } } } { s _ { 2 } } } \\right ) = T _ { { \\rm S V C } , n , 1 } ^ * , \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} | \\partial _ z ^ \\alpha f ( z ) | \\le \\frac { \\alpha ! } { r ^ { | \\alpha | } } \\sup _ { | \\zeta _ j | = r } | f ( z + \\zeta ) | \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} & \\left | N ( f , r ) - \\sum _ { j = r } ^ { d } { j \\choose r } { q \\choose j } ( - 1 ) ^ { j - r } q ^ { k - j } \\right | \\\\ & \\leq \\sum _ { j = k + 1 } ^ d { j \\choose r } { \\frac { q } p + m \\sqrt { q } + j \\choose j } { m - 1 \\choose d - j } \\sqrt { q } ^ { d - j } . \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} \\beta : = \\{ ( a , b ) \\in A \\mid \\big ( \\psi _ 1 ( a ) , \\psi _ 1 ( b ) \\big ) \\in \\alpha _ i / \\alpha \\} , \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} \\exists \\ , C > 0 : \\| \\phi _ { u } \\| _ { D } ^ { 2 } = \\int _ { \\mathbb R ^ 3 } \\phi _ u u ^ 2 d x \\leq C \\norm { u } ^ 4 . \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} z ( h ( v , v ) - h ( w , w ) ) \\in R \\cap S = ( 0 ) . \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} \\begin{bmatrix} H _ { 1 , a _ { 1 , 1 } } & H _ { 1 , a _ { 2 , 1 } } & \\dots & H _ { 1 , a _ { M , 1 } } \\\\ \\sigma _ { 1 } H _ { 2 , a _ { 1 , 1 } } & \\sigma _ { 2 } H _ { 2 , a _ { 2 , 1 } } & \\dots & \\sigma _ { M } H _ { 2 , a _ { M , 1 } } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ \\sigma _ { 1 } ^ { r - 1 } H _ { r , a _ { 1 , 1 } } & \\sigma _ { 2 } ^ { r - 1 } H _ { r , a _ { 2 , 1 } } & \\dots & \\sigma _ { M } ^ { r - 1 } H _ { r , a _ { M , 1 } } \\end{bmatrix} . \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} U _ + ( z ) = c \\frac { z - \\theta _ - } { z - \\theta _ + } U _ - ( z ) = c \\frac { z - \\theta _ + } { z - \\theta _ - } \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} W _ 1 = \\hat { g _ 1 } ^ { - 1 } ( \\Gamma ( \\hat { g _ 1 } ) - \\Gamma ( g _ 0 ) ) \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} X _ { n } \\ , = \\ , \\sum _ { 0 \\le \\ell \\le k \\le n - 1 } { k \\choose \\ell } u ^ { \\ell } ( 1 - a ) ^ { \\ell } a ^ { k - \\ell } \\varepsilon _ { n - k , \\ell } \\ , , \\widetilde { X } _ { n } \\ , = \\ , \\sum _ { 0 \\le \\ell \\le k \\le n - 1 } { k \\choose \\ell } u ^ { \\ell } ( 1 - a ) ^ { \\ell } a ^ { k - \\ell } \\varepsilon _ { n - k , \\ell + 1 } \\ , , \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\| h \\| _ { C ( [ 0 , T ] ; B ) } ^ { ( \\rho ) } : = \\sup _ { 0 \\le t \\le T } e ^ { - \\rho t } \\| h ( t ) \\| _ { B } . \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} \\sum _ i u _ i e u _ i ^ * = r , r e ^ { 1 / 2 } = e ^ { 3 / 2 } = e ^ { 1 / 2 } r , \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} D = \\sup \\left \\{ \\frac { \\det ( Q + \\sum _ { k = 1 } ^ m c _ k B _ k ^ \\ast A _ k B _ k ) } { \\prod _ { k = 1 } ^ m ( \\det A _ k ) ^ { c _ k } } \\colon ( A _ 1 , \\ldots , A _ m ) \\in \\Lambda \\right \\} , \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} H ( \\tau , x ) ( s ) = x \\big ( ( 1 - \\tau ) s + \\tfrac \\tau 2 \\big ) ; \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} \\left . \\frac { \\partial L } { \\partial e _ 2 } \\right | _ { e _ 2 = e _ 3 = e _ 4 = 0 } \\ne 0 , \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq i \\leq t \\\\ 0 \\leq j \\leq m } } \\overline { c } _ { i , j } \\binom { s } { j } \\overline { \\mu } _ i ^ { s - j } \\neq 0 . \\end{align*}"}
-{"id": "9910.png", "formula": "\\begin{align*} \\int _ { X } [ \\o ] \\otimes a = n ( a ) \\int _ { X } [ \\o ] . \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} \\dfrac { 1 } { E _ { 1 } } & = \\dfrac { E _ { 2 5 } ^ { 5 } } { E _ { 5 } ^ { 6 } } \\Bigg ( \\dfrac { 1 } { R ( q ^ { 5 } ) ^ { 4 } } + \\dfrac { q } { R ( q ^ { 5 } ) ^ { 3 } } + \\dfrac { 2 q ^ { 2 } } { R ( q ^ { 5 } ) ^ { 2 } } + \\dfrac { 3 q ^ { 3 } } { R ( q ^ { 5 } ) } + 5 q ^ { 4 } \\\\ & \\quad \\quad \\quad \\quad - 3 q ^ { 5 } R ( q ^ { 5 } ) + 2 q ^ { 6 } R ( q ^ { 5 } ) ^ { 2 } - q ^ { 7 } R ( q ^ { 5 } ) ^ { 3 } + q ^ { 8 } R ( q ^ { 5 } ) ^ { 4 } \\Bigg ) . \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} \\frac { \\partial \\xi } { \\partial \\zeta } = \\frac { 1 } { \\pi } \\begin{pmatrix} \\frac 1 2 & 1 & \\cdots & 1 \\\\ 1 & \\frac 1 2 & \\cdots & 1 \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ 1 & 1 & \\cdots & \\frac 1 2 \\end{pmatrix} \\mbox { d i a g } ( j _ b : 1 \\leq b \\leq n ) \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} \\mathbb E _ { 0 } [ Z _ { t } ^ { - 1 } \\vert \\mathcal F ^ { X } _ { T } ] \\ , = \\ , \\rho _ { t } ( { \\bf 1 } ) \\ , = \\ , \\exp \\Big ( \\int ^ { t } _ { 0 } \\pi _ { s , 2 } ( b ) { \\mathrm d } X _ { s } - \\frac { \\ , 1 \\ , } { \\ , 2 \\ , } \\int ^ { t } _ { 0 } \\lvert \\pi _ { s , 2 } ( b ) \\rvert ^ { 2 } { \\mathrm d } s \\Big ) \\ , ; 0 \\le t \\le T \\ , . \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} \\min & \\frac { 1 } { 2 } x ^ T H x + \\langle \\lambda _ k , A x - b \\rangle + \\frac { \\rho } { 2 } \\| A x - b \\| ^ 2 \\\\ & x \\in \\mathcal { X } \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} P _ n ( t ) = 2 ^ { n - 1 } A _ n \\left ( \\frac { t + 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} F _ { k , - m , 0 } ( z ) & = P _ { k , - m } ( z , 1 - k / 2 ) \\\\ & = \\varphi _ { k , - m } ( z , 1 - k / 2 ) + \\sum _ { ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) } b _ { k , - m } ( n , 1 - k / 2 ) \\mathcal { W } _ { k , n } ( y , 1 - k / 2 ) e ^ { 2 \\pi i n x } \\\\ & = q ^ { - m } - \\frac { \\Gamma ( 1 - k , 4 \\pi m y ) } { \\Gamma ( 1 - k ) } q ^ { - m } + \\sum _ { ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) } b _ { k , - m } ( n , 1 - k / 2 ) \\mathcal { W } _ { k , n } ( y , 1 - k / 2 ) e ^ { 2 \\pi i n x } . \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} & p \\cdot [ p _ 1 , \\ldots , p _ r ] \\coloneqq [ p p _ 1 , p _ 2 , \\ldots , p _ r ] , \\\\ & p \\cdot [ p _ 1 , \\ldots , p _ r , v ] \\coloneqq [ p p _ 1 , p _ 2 , \\ldots , p _ r , v ] \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} \\sum _ { x \\in T _ { N } ^ { - 1 } ( x ' ) } T _ { 2 } ( x ) = 0 \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} R _ { \\mathcal { Y } _ p } \\left ( e _ { ( z , w ) } \\circ T \\right ) ( k , \\zeta ) = \\overline { Y _ p ( k ; w , \\zeta ) } \\frac { ( k + 1 ) \\overline { z } ^ k } { \\pi } , ( k , \\zeta ) \\in \\N \\times \\mathbb { B } _ p . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} & \\frac { \\prod _ { 1 \\le i < j \\le N } ( q ^ { a _ i } - q ^ { a _ j } ) ( q ^ { s _ j } - q ^ { s _ i } ) } { \\prod _ { i , j = 1 } ^ N ( q ^ { s _ i } - q ^ { a _ j } ) } = \\det \\left ( \\frac { 1 } { q ^ { s _ i } - q ^ { a _ j } } \\right ) _ { i , j = 1 } ^ N , \\\\ & \\frac { \\prod _ { 1 \\le i < j \\le N } \\sin \\pi ( a _ i - a _ j ) \\sin \\pi ( s _ j - a _ i ) } { \\prod _ { i , j = 1 } ^ N \\sin \\pi ( s _ i - a _ j ) } = \\det \\left ( \\frac { 1 } { \\sin \\pi ( s _ i - a _ j ) } \\right ) _ { i , j = 1 } ^ N . \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} \\ , \\widetilde { F } _ { t } ^ { ( u ) } \\ , = \\ , u \\cdot \\delta _ { \\widehat { X } _ { t } ^ { ( u ) } } + ( 1 - u ) \\cdot \\mathcal L _ { \\widetilde { X } _ { t } ^ { ( u ) } } \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} \\left | \\sum _ { \\ell = 1 } ^ n \\mu _ \\ell ^ { r _ e } x _ \\ell \\right | = M = \\max _ { \\substack { 1 \\leq i \\leq t \\\\ j \\geq 0 } } \\{ | c _ { i , j } | \\} \\neq 0 . \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} \\left ( \\prod _ { 1 \\le j \\le m } \\alpha _ j ^ { \\binom { \\nu _ j } { 2 } } \\right ) \\prod _ { 1 \\le i < j \\le m } ( \\alpha _ i - \\alpha _ j ) ^ { \\min \\left \\{ \\nu _ i ^ 2 , \\ , \\nu _ j ^ 2 \\right \\} } \\underset { \\mathbb { Z } [ \\alpha _ 1 , \\ldots , \\alpha _ m ] } { \\bigg | } \\widehat { D } ( \\overline { \\alpha } ) . \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} K _ { i j } ( x , 0 ) = 0 \\mbox { f o r } 1 \\le i \\le j \\le k \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} N ( k , t , c _ 2 ) - M ( k , t + 1 , c _ 1 ) & = \\left ( 1 + \\frac { k } { c _ 2 } - \\frac { 2 k } { c _ 1 } \\right ) ( k - 1 ) ^ { t - 2 } \\\\ & = \\left ( 1 - \\frac { k \\mathcal { G } _ { t - 2 } ( \\theta ) } { F _ { t - 1 } ( \\theta ) } + \\frac { 2 k G _ { t - 3 } ( \\theta ) } { F _ { t - 1 } ( \\theta ) } \\right ) ( k - 1 ) ^ { t - 2 } \\\\ & = \\left ( 1 - \\frac { k G _ { t - 2 } ( \\theta ) - k G _ { t - 3 } ( \\theta ) } { G _ { t - 1 } ( \\theta ) - G _ { t - 3 } ( \\theta ) } \\right ) ( k - 1 ) ^ { t - 2 } . \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} \\pi _ { 1 } ( ( X ^ q ) ^ { } ) = \\pi _ { 0 } ( ( X ^ q ) ^ { } ) = 0 , \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} E ( a ) = \\frac { 1 } { | G | } \\sum _ { g \\in G } \\alpha _ g ( a ) . \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} F ( \\tilde { \\nabla } ) ^ k = k \\ ; d s \\wedge \\dot { \\nabla } ^ s \\wedge F ( \\nabla ^ s ) ^ { k - 1 } + F ( \\nabla ^ s ) ^ k . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} c ( \\theta _ 0 , \\theta _ 1 ) : = \\inf _ { v _ t } ~ \\int _ 0 ^ 1 \\int _ \\Omega L ( x , v ( t , x ) ) \\rho ( \\theta _ t , x ) d x d t , \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} \\frac { 1 } { \\epsilon } ( x ^ { \\tau , \\epsilon } _ \\tau - x ^ * _ \\tau ) & = \\frac { 1 } { \\epsilon } \\int _ { \\tau - \\epsilon } ^ { \\tau } f ( x ^ { \\tau , \\epsilon } _ s , \\omega ) - f ( x ^ * _ s , \\theta ^ * _ s ) d s . \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} = ( 1 + e ^ { - 4 \\pi } ) ^ 2 a ^ 4 2 ^ { - 4 9 / 1 6 } e ^ { 1 1 \\pi / 1 6 } . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} a ( \\xi ) = \\xi _ 1 ^ 2 + \\xi _ 2 ^ 2 + \\xi _ 3 ^ 2 - \\frac { 1 } { N ^ 2 } ( X _ i \\xi ^ i ) ^ 2 . \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} \\phi b ^ { n _ 0 } x b ^ { m _ k } & \\sim \\phi b ^ { n _ 0 + 1 } x b ^ { m _ k } - \\sum _ { s \\in S _ b } \\phi s b ^ { n _ 0 + 1 } x b ^ { m _ k } \\\\ & \\sim . . . \\\\ & \\sim \\phi b ^ { n _ 0 + ( m _ 0 - n _ 0 ) } x b ^ { m _ k } - \\sum _ { i = 1 } ^ { m _ 0 - n _ 0 } \\sum _ { s \\in S _ b } \\phi s b ^ { n _ 0 + i } x b ^ { m _ k } \\\\ & = \\phi b ^ { m _ 0 } x b ^ { m _ k } - \\sum _ { i = n _ 0 + 1 } ^ { m _ 0 } \\sum _ { s \\in S _ b } \\phi s b ^ i x b ^ { m _ k } . \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} f ( z ) = \\prod _ { j = 1 } ^ p \\sum _ { i = 0 } ^ { \\gamma _ j - 1 } z ^ { i e _ j } = \\prod _ { j = 1 } ^ p \\left ( 1 + z ^ { e _ j } + z ^ { 2 e _ j } + \\cdots z ^ { ( \\gamma _ j - 1 ) e _ j } \\right ) . \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} u ^ { \\varepsilon , \\nu } _ t + f ^ { \\nu } ( t , x , u ^ { \\varepsilon , \\nu } ) _ x ~ = ~ \\varepsilon u ^ { \\varepsilon , \\nu } _ { x x } \\ , . \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} \\sum _ { W \\in J } \\chi _ 1 ( W ) \\chi _ 2 ( W ^ * ) = 0 . \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} \\bigl \\langle { \\bf p } _ i ^ { j + 2 } - { \\bf p } _ i ^ 1 , { \\bf p } _ i ^ { j + 1 } - { \\bf p } _ i ^ 1 \\bigr \\rangle = & \\bigl ( x _ i ^ { j + 2 } - x _ i ^ 1 \\bigr ) \\bigl ( x _ i ^ { j + 1 } - x _ i ^ 1 \\bigr ) + \\bigl ( y _ i ^ { j + 2 } - y _ i ^ 1 \\bigr ) \\bigl ( y _ i ^ { j + 1 } - y _ i ^ 1 \\bigr ) \\\\ & + \\bigl ( z _ i ^ { j + 2 } - z _ i ^ 1 \\bigr ) \\bigl ( z _ i ^ { j + 1 } - z _ i ^ 1 \\bigr ) . \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} 0 & = \\int _ { K \\backslash G / H } F ( \\ddot { x } ) d \\mu ( \\ddot { x } ) = \\int _ { K \\backslash G / H } Q ( f ) ( \\ddot { x } ) d \\mu ( \\ddot { x } ) \\\\ & = \\int _ { K \\backslash G / H } \\int _ K \\int _ H f ( k ^ { - 1 } x h ) d h d k d \\mu ( \\ddot { x } ) \\\\ & = \\int _ G f ( x ) \\rho ( x ) d x > 0 \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} \\frac { \\dim ( \\mathbb { Z } ( \\mathcal { T } _ \\xi ) \\rtimes \\mathcal { S } ) } { n + 1 } \\dim ^ { + } ( \\overline { \\mathbf { 1 } } ) = d ^ { 2 n } ( - 1 ) ^ { \\lvert \\Phi ^ + \\rvert + n } \\left ( \\sum _ { w \\in \\mathfrak { S } _ { n + 1 } } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho , w \\bullet 0 \\rangle } \\right ) ^ { - 2 } , \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} \\left < h , g \\right > : = E _ \\nu ( h ( \\omega ) g ( \\omega ) ) . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} J = \\Big ( \\{ 0 , 1 , \\dots , k \\} - \\{ i _ 1 , \\dots , i _ p \\} \\Big ) \\cup \\{ k + j _ 1 , \\dots , k + j _ p \\} , \\\\ 1 \\leq i _ 1 < \\cdots < i _ p \\leq k , \\ 1 \\leq j _ 1 < \\cdots < j _ p \\leq n , \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} u _ { k + 1 } ( s , 0 ) = F _ { k + 1 } ( s ) + \\int _ { 0 } ^ s \\sum _ { j = k + 2 } ^ { k + m } { \\cal G } _ { k + 1 , j } ( \\xi ) u _ j ( \\xi , 0 ) \\ , d \\xi \\mbox { f o r } 0 \\le s \\le t _ 1 , \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} - 1 - x + x ^ 2 + 2 x y - x ^ 2 y & = 0 , \\\\ - 1 - y + 2 x y + y ^ 2 - x y ^ 2 & = 0 , \\end{align*}"}
-{"id": "9934.png", "formula": "\\begin{align*} \\sigma ( \\alpha ) = \\dfrac { k ( k - 1 ) } { 2 } + \\sum _ { j = 1 } ^ k \\sum _ { i = 1 } ^ { \\alpha _ j - 1 } ( n _ i + p _ i ) . \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} ( 1 - z ) ^ { - \\alpha } = { } _ { 1 } F _ { 0 } \\left ( \\begin{array} { l l l } \\alpha ~ ; \\\\ \\overline { ~ ~ ~ } ; \\end{array} z \\right ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\alpha ) _ { n } } { n ! } z ^ { n } ~ , \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} u _ l : = \\begin{cases} s _ { i _ l } w ( \\pi ) & \\ell ( s _ { i _ l } w ( \\pi ) ) > \\ell ( w ( \\pi ) ) , \\\\ w ( \\pi ) & \\ell ( s _ { i _ l } w ( \\pi ) ) < \\ell ( w ( \\pi ) ) . \\end{cases} \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} \\int _ { B _ R } | x | ^ { \\frac { n \\theta } { p } } | \\nabla _ r u _ { \\lambda } ( x ) | ^ { \\theta } \\frac { d x } { | x | ^ n } = \\int _ { B _ R } | x | ^ { \\frac { n \\theta } { p } } | \\nabla _ r u ( x ) | ^ { \\theta } \\frac { d x } { | x | ^ n } \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} \\bar { I } _ { m o d } [ \\mathbf { Q } ] & = \\int _ { \\bar { \\Omega } } D ^ 3 \\phi \\left ( \\frac { | \\bar { \\nabla } \\mathbf { Q } | } { D } \\right ) + \\frac { D ^ 3 } { L } f _ B \\left ( \\mathbf { Q } \\right ) \\bar { d V } \\\\ & = D \\int _ { \\bar { \\Omega } } \\bar { \\psi } \\left ( | \\bar { \\nabla } \\mathbf { Q } | ^ 2 \\right ) + \\frac { 1 } { \\bar { L } } \\bar { f _ B } \\left ( \\mathbf { Q } \\right ) \\bar { d V } \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} T ( x _ 1 , \\dotsc , x _ n ) = ( x _ 1 + a , S ( x _ 2 , \\dotsc , x _ n ) ) \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} \\{ v ^ i _ \\lambda v ^ j \\} _ L = g ^ { j i } \\lambda + c ^ { j i } _ k v ^ k _ { 2 x } , \\{ v ^ i _ \\lambda v ^ j \\} _ N = w ^ j _ l v ^ l _ { 2 x } ( \\lambda + \\partial ) ^ { - 1 } w ^ i _ m v ^ m _ { 2 x } , \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} & 4 i ( \\Delta n - 2 i ) d ^ 2 ( d n - 6 ) - 2 i d ^ 3 \\Delta ^ 2 n - 1 6 i d ^ 4 \\Delta n \\\\ & \\geq 4 i \\Delta d ^ 3 n ^ 2 - 8 i ^ 2 d ^ 3 n - 2 4 i \\Delta d ^ 2 n - 2 i d ^ 3 \\Delta ^ 2 n - 1 6 i d ^ 4 \\Delta n \\\\ & = 4 i \\Delta d ^ 3 n ^ 2 - 2 i ( d n ) ^ 3 \\ , \\left ( 4 i / n ^ 2 + 1 2 \\Delta / ( d n ^ 2 ) + \\Delta ^ 2 / n ^ 2 + 8 d \\Delta / n ^ 2 \\right ) \\\\ & \\geq 4 i \\Delta d ^ 3 n ^ 2 - 2 i ( d n ) ^ 3 \\ , \\left ( 1 1 d \\Delta / n ^ 2 + \\Delta ^ 2 / n ^ 2 + 1 2 / n \\right ) \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} H _ N ^ { ( \\alpha ) } \\geq \\sum _ { i = 1 } ^ { N } S _ i + \\frac { N \\mu \\lambda } { 2 \\alpha } \\geq \\frac { N \\mu \\lambda } { 2 \\alpha } . \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} \\langle \\vec { m } ^ { \\star } , \\vec { 0 } \\rangle = 1 - \\sum _ { i \\in V ( Q ) } b _ i m _ i , \\ \\langle \\vec { 0 } , \\vec { m } ^ { \\star } \\rangle = 1 - \\sum _ { i \\in V ( Q ) } a _ i m _ i . \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{gather*} r _ { 0 0 } = - 3 , r _ { 0 1 } = \\sqrt { d _ 1 } , r _ { 1 0 } = - \\frac { 1 } { \\sqrt { d _ 1 } } , r _ { j j } = - 1 , \\ j = 1 , \\ldots , 2 n - 1 , \\\\ r _ { i \\ , i + 1 } = \\frac { d _ { i + 1 } } { d _ i } \\sqrt { - \\frac { d _ i } { d _ { i + 1 } } } , r _ { i + 1 \\ , i } = - \\sqrt { - \\frac { d _ i } { d _ { i + 1 } } } , \\ i = 2 , \\ldots , 2 n - 2 . \\end{gather*}"}
-{"id": "6073.png", "formula": "\\begin{align*} t \\cdot g : = \\lim _ { q \\rightarrow t } q g ( t \\in \\mathbb { R } _ { + } , q \\in \\mathbb { Q } _ { + } , g , \\in G ) . \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} \\bar { F } _ \\infty ( \\Q ; \\ , B _ R ) : = \\int _ { B _ R } \\left \\{ \\frac { \\alpha } { p } \\abs { \\nabla \\Q } ^ p + \\gamma \\left ( 1 - | \\Q | ^ 2 \\right ) ^ 2 \\right \\} \\d V \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} \\Lambda ( z , w _ 1 , \\dots , w _ n ) & = \\left ( \\frac { 1 + i z / 4 } { 1 - i z / 4 } , \\frac { w _ 1 } { ( 1 - i z / 4 ) ^ { 1 / m _ 1 } } , \\cdots , \\frac { w _ n } { ( 1 - i z / 4 ) ^ { 1 / m _ n } } \\right ) \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\Xi \\colon D ^ b ( Y ) \\stackrel { \\sim } { \\to } D _ { \\mathbb { C } ^ { \\ast } } ( Y , 0 ) . \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} B ( t ) = \\{ 2 ^ { - 1 } | t | ^ { s + 1 } [ | t | ^ { 1 / 2 } ] ^ { - 2 s - 1 } 2 ^ { - 2 s - 1 } ( 1 - 2 ^ { - 2 s - 1 } ) ^ { - 1 } + [ | t | ^ { 1 / 2 } ] \\} . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} { \\hat { \\mathfrak { g } } } ^ + _ { p ' } \\cdot \\Phi = 0 . \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t u ^ { n - \\theta } , v \\Big { ) } - ( \\nabla \\sigma ^ { n - \\theta } , \\nabla v ) + ( f ^ { n - \\theta } ( u ) , v ) = & ( g ^ { n - \\theta } , v ) , ~ \\forall v \\in H _ 0 ^ 1 , \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} P _ n ( x ) = a _ 0 + a _ 1 x + \\dots + a _ n x ^ n , \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} z _ { \\sf A f f } ( { \\cal U } ) & \\leq 2 \\cdot z _ { \\sf A f f } ( \\tilde { \\cal V } ) \\\\ & = 2 \\cdot O \\left ( \\frac { \\log n } { \\log \\log n } \\right ) \\cdot z _ { \\sf A R } ( \\tilde { \\cal V } ) \\\\ & \\leq 2 \\cdot O \\left ( \\frac { \\log n } { \\log \\log n } \\right ) \\cdot 4 \\log L \\cdot z _ { \\sf A R } ( { \\cal U } ) = O \\left ( \\frac { \\log n } { \\log \\log n } \\ \\cdot \\log L \\right ) \\cdot z _ { \\sf A R } ( { \\cal U } ) , \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} G = n \\ell A _ 0 + n ( a A _ 1 + b A _ 2 ) + n ( A _ 3 - A _ 4 ) . \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} \\beta = \\frac { 1 } { k + h ^ \\vee } + r ^ \\vee N \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} f _ \\delta ( x ) : = \\delta ^ { d / 2 + 1 } f ( \\delta x ) . \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} \\Omega ( 1 , 0 , n , \\alpha ( n ) ) = 0 . \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} [ L _ { \\bar { \\theta } } x ] _ v = \\sum _ { v ' \\in N ( v ) } H ' ( \\bar { \\theta } _ { v ' } - \\bar { \\theta } _ { v } ) ( x _ { v ' } - x _ v ) , \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{gather*} T _ i ( h _ { j , 1 } ) = T _ i \\big ( \\big [ x _ { j } ^ + , x _ { j , 1 } ^ - \\big ] \\big ) = \\big [ \\big [ x _ i ^ + , x _ { j } ^ + \\big ] , - \\big [ x _ i ^ - , x _ { j , 1 } ^ - \\big ] \\big ] \\\\ \\hphantom { T _ i ( h _ { j , 1 } ) } { } = - \\big [ \\big [ \\big [ x _ i ^ + , x _ j ^ + \\big ] , x _ i ^ - \\big ] , x _ { j , 1 } ^ - \\big ] - \\big [ x _ i ^ - , \\big [ \\big [ x _ i ^ + , x _ j ^ + \\big ] , x _ { j , 1 } ^ - \\big ] \\big ] . \\end{gather*}"}
-{"id": "1239.png", "formula": "\\begin{align*} n _ b ( W ^ n ( w ) ) = k _ w + \\lVert ( \\phi w ' ) _ n \\rVert _ S \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} \\beta ^ { u + t } \\alpha _ i ^ { u + p _ i - 1 } = \\beta ^ { u + t } \\alpha _ i ^ { u + t + \\lfloor \\frac { p _ i - 1 - t } { s } \\rfloor s } \\alpha _ i ^ { p _ i - 1 - t - \\lfloor \\frac { p _ i - 1 - t } { s } \\rfloor s } \\in S _ i \\alpha _ i ^ { p _ i - 1 - t - \\lfloor \\frac { p _ i - 1 - t } { s } \\rfloor s } \\subseteq K \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} \\mu ^ p _ { \\Lambda , \\epsilon } : = \\frac { 1 } { Z _ { \\Lambda , \\epsilon } } e ^ { - H ^ p _ { \\Lambda , \\epsilon } ( \\phi ) } \\prod _ { j \\in \\Lambda } d \\phi _ j . \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} \\tau _ 1 \\cdot \\tau _ 4 & = \\tau _ 1 \\cdot ( \\sigma _ 4 - \\sigma _ 0 - 4 e ) \\\\ & = \\tau _ 1 \\cdot \\sigma _ 4 \\\\ & = ( \\sigma _ 1 - \\sigma _ 0 - 4 e ) \\cdot \\sigma _ 4 \\\\ & = 1 2 - 0 - 4 \\\\ & = 8 . \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} - \\mathbf { i } \\partial _ { \\omega } F _ { ( - j ) j } + \\bar { \\Omega } _ { ( - j ) j } F _ { ( - j ) j } + \\tilde { \\Omega } _ { ( - j ) j } F _ { ( - j ) j } = - \\mathbf { i } R _ { ( - j ) j } , \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 5 7 6 n + 7 2 ) q ^ n & \\equiv 2 f _ 1 ^ 3 + 3 \\dfrac { f _ 1 ^ 3 f _ { 4 } ^ 4 f _ { 6 } ^ 6 } { f _ 2 ^ 8 f _ { 1 2 } ^ 3 } \\\\ & \\equiv \\left ( 2 + 3 \\dfrac { f _ 6 ^ 2 } { f _ { 1 2 } } \\right ) \\cdot f _ 1 ^ 3 \\\\ & \\equiv \\left ( 2 + 3 \\dfrac { f _ 6 ^ 2 } { f _ { 1 2 } } \\right ) \\left ( f _ 3 a ( q ^ 3 ) - 3 q f _ 9 ^ 3 \\right ) ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} R & = \\{ ( i , k ) \\bar { t } : j \\le k \\le \\lambda _ { i } \\} \\\\ S & = \\{ ( i + 1 , k ) \\bar { t } : \\mu _ { i + 1 } < k \\le j \\} . \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} \\sigma ^ \\alpha { } _ { \\dot a b } \\ , \\sigma ^ { \\beta \\dot c b } + \\sigma ^ \\beta { } _ { \\dot a b } \\ , \\sigma ^ { \\alpha \\dot c b } = - 2 g ^ { \\alpha \\beta } \\delta _ { \\dot a } { } ^ { \\dot c } \\ , . \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} I _ { p _ 1 , q _ 1 } \\oplus \\cdots \\oplus I _ { p _ m , q _ m } \\oplus \\begin{bmatrix} 0 & I _ { k _ { m + 1 } } \\\\ I _ { k _ { m + 1 } } & 0 \\end{bmatrix} \\oplus \\cdots \\oplus \\begin{bmatrix} 0 & I _ { k _ l } \\\\ I _ { k _ l } & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} \\sup _ { 0 \\le k \\le m } \\| P _ { n - k } ( \\epsilon ) - ( \\tilde { P } - \\epsilon k \\tilde { P } ^ { ( 1 ) } ) \\| = O ( \\epsilon ^ 2 m ^ 2 ) \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} ( \\tau _ 1 + \\tau _ 4 ) ^ 2 & = \\tau _ 1 ^ 2 + 2 \\tau _ 1 \\cdot \\tau _ 4 + \\tau _ 4 ^ 2 \\\\ & = - 8 + 1 6 - 8 \\\\ & = 0 . \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} D _ { G / T } : = 2 \\pi / V _ 3 . \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} b ^ p ( M ^ { 2 n } ) ( t ) & = b ^ p ( M ^ { 2 n } ) ( 1 ) = \\dim \\mathcal { H } ^ p ( M ^ { 2 n } , g _ J ) \\\\ & = b ^ p ( M ^ { 2 n } ) \\quad ( M ^ { 2 n } ) , \\\\ & t \\in [ 0 , 1 ) , 0 \\leq p \\leq 2 n . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\mathrm { T r } _ { d , D } ( - \\mathrm { l o g } ( y | \\eta ( z ) | ^ 4 ) ) & = \\sqrt { | D | } L _ D ( 1 ) \\mathrm { T r } _ { d , 1 } ( 1 ) , \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} { \\min } _ { \\prec } \\{ 1 , 2 , \\dots , k + n + 1 \\} = { \\min } _ { \\prec } ( J ^ { \\circ } ) = j _ l , \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\Theta _ { X / T } = ( \\C ^ k \\times \\Theta _ M ) / \\Lambda = \\C ^ k \\times ^ \\Lambda \\Theta _ M \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} \\det g _ { \\alpha \\beta } ( x ) = \\det h _ { \\mu \\nu } ( x ) . \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} f _ k ^ { ( r ) } ( x , y ) = f _ k ( x , y / r ) \\textup { f o r $ k = 1 , \\ldots , m $ } . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} M = \\max _ { \\substack { 1 \\leq i \\leq t \\\\ j \\geq 0 } } \\{ | c _ { i , j } | \\} \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} c _ 0 ( V ) = c _ 0 ( F ) \\oplus c _ 0 ( V \\setminus F ) . \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} d _ { C _ 1 \\times C _ 2 } ( ( r _ 1 \\times r _ 2 ) ( x _ 1 , x _ 2 ) ) & = d _ { C _ 1 \\times C _ 2 } ( r _ 1 ( x _ 1 ) , r _ 2 ( x _ 2 ) ) \\\\ & = d _ { C _ 1 } ( r _ 1 ( x _ 1 ) ) + d _ { C _ 2 } ( r _ 2 ( x _ 2 ) ) \\\\ & = d _ { C _ 1 } ( x _ 1 ) + d _ { C _ 2 } ( x _ 2 ) \\\\ & = d _ { C _ 1 \\times C _ 2 } ( x _ 1 , x _ 2 ) , \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} S _ p : = \\inf _ { u \\in H \\backslash \\{ 0 \\} } \\dfrac { \\left \\| u \\right \\| _ H } { \\left \\| u \\right \\| _ { L ^ p ( \\mathbb { R } ^ 2 ) } } , \\ \\ p \\geq 2 , \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} \\psi ( \\overline { ( H ( X , Y + Z ) + H ( Y , Z ) - H ( X + Y , Z ) - H ( X , Y ) ) d t } ) = 0 . \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} \\ , \\Theta \\ , : = \\ , ( X _ { 0 } ^ { ( u ) } , \\widetilde { X } _ { 0 } ^ { ( u ) } ) \\ , = \\ , ( X _ { 0 } ^ { ( u ) } ) \\otimes ( \\widetilde { X } _ { 0 } ^ { ( u ) } ) \\ , = \\ , \\theta ^ { \\otimes 2 } \\ , , \\theta \\ , : = \\ , ( X _ { 0 } ^ { ( u ) } ) \\ , \\equiv \\ , ( \\widetilde { X } _ { 0 } ^ { ( u ) } ) \\ , . \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} e _ 1 e _ 2 \\ldots e _ n , \\ e _ i \\in E ( Q ) , \\ t ( e _ i ) = s ( e _ { i + 1 } ) . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} w _ k ( x ) \\leq \\tilde w ( x ) : = C ( 1 + | x | ) ^ { - 1 } e ^ { - \\frac { \\sqrt { \\lambda } } { 2 } | x | } . \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} \\begin{aligned} & \\alpha _ t e ^ { \\beta _ t } = a _ V e ^ { a _ X } , \\\\ & e ^ { - \\beta _ t } = e ^ { - a _ X } - a _ V e ^ { a _ X } t . \\end{aligned} \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} ( \\sigma _ H ^ { n - \\theta } , w _ h ) + ( \\nabla u _ H ^ { n - \\theta } , \\nabla w _ H ) = 0 , ~ \\forall w _ H \\in L _ H , \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\mathbb { E } _ \\mu ( e ^ { i \\phi _ \\delta ( f ) } ) = \\exp \\left \\{ \\frac { 1 } { 2 } \\int d x f ( x ) ( Q ^ { - 1 } f ) ( x ) \\right \\} , \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} S _ N = \\sum _ { n = 1 } ^ { N } \\big ( F ( \\xi _ { q _ 1 ( n ) } , \\xi _ { q _ 2 ( n ) } , . . . , \\xi _ { q _ \\ell ( n ) } ) - \\bar F \\big ) \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} \\widetilde Q _ { 1 2 } : = 1 + q , \\widetilde Q _ { 3 2 } = 1 - q , \\widetilde Q _ { 3 1 } = q , \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} \\dfrac { \\partial \\varphi ^ \\alpha } { \\partial x ^ \\beta } = \\delta ^ \\alpha { } _ \\beta + \\dfrac { \\partial A ^ \\alpha } { \\partial x ^ \\beta } + O ( | A ^ 2 | ) . \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} N _ i ( \\delta ) = \\lambda \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} \\alpha _ n ( t , q ) = t ^ { n - 1 } \\alpha _ n \\left ( \\frac { 1 } { t } , \\frac { q } { t } \\right ) \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} \\bar { d } ( A ) = \\limsup _ { n \\rightarrow \\infty } \\frac { | A \\cap \\{ 1 , \\dots , n \\} | } n \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} ( - \\alpha , - \\alpha ^ \\sigma ) _ 2 & = ( - \\alpha , - 1 ) _ 2 ( - \\alpha , \\alpha ^ \\sigma ) _ 2 \\\\ & = ( - 1 , - 1 ) _ 2 ( \\alpha , - 1 ) _ 2 ( - 1 , \\alpha ^ \\sigma ) _ 2 ( \\alpha , \\alpha ^ \\sigma ) _ 2 . \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} D _ j ( | Q | ^ p Q ) = \\tfrac { p } { 2 } | Q | ^ { p - 2 } Q ^ 2 D _ j \\bar Q + \\tfrac { p + 2 } { 2 } | Q | ^ p D _ j Q , \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} d _ n ^ i ( ( x _ 0 , \\dots , x _ n ) ) = \\begin{cases} ( x _ 0 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots x _ n ) , & d ( x _ { i - 1 } , x _ i ) + d ( x _ i , x _ { i + 1 } ) = d ( x _ { i - 1 } , x _ { i + 1 } ) \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } y ^ 2 p _ { _ 1 } ( y ) d y & = \\frac { 2 } { \\sqrt { 2 \\pi } \\sigma _ w } e ^ { - \\rho _ 0 - 1 } \\int _ { 0 } ^ { \\infty } y ^ 2 e ^ { - \\left ( \\frac { 1 } { 2 \\sigma _ w ^ 2 } + \\rho _ 1 \\right ) y ^ 2 } d y \\\\ & = \\frac { e ^ { - \\rho _ 0 - 1 } \\sigma _ w ^ 2 } { ( 1 + 2 \\rho _ 1 \\sigma _ w ^ 2 ) ^ { 3 / 2 } } , \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} C = \\begin{pmatrix} 3 & 1 \\\\ 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} \\| v \\| : = \\sup _ { 1 \\le i \\le n } \\mbox { e s s s u p } _ { ( \\tau , \\xi ) \\in ( 0 , T ) \\times ( 0 , 1 ) } e ^ { - L _ 1 \\tau - L _ 2 \\xi } | v _ i ( \\tau , \\xi ) | . \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} \\chi _ \\mu ( e ( - v / u ) ; \\lambda ) : = \\sum _ { w \\in W } \\epsilon ( w ) e ^ { - 2 \\pi i \\frac { v } { u } \\left ( \\lambda + \\rho , w ( \\mu + \\rho ) \\right ) } , \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} y ' ( t ) = f ( y ( t ) , u ( t ) ) , \\ \\ \\ \\ \\ u ( t ) \\in U , \\ \\ \\ \\ \\ t \\in [ 0 , \\infty ) \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} \\gamma & 0 \\\\ 0 & 1 \\end{matrix} \\right ) w \\left ( \\begin{matrix} 1 & \\zeta _ { p } \\\\ 0 & 1 \\end{matrix} \\right ) = \\left ( \\begin{matrix} 1 & - \\gamma \\zeta _ { p } ^ { - 1 } \\\\ 0 & 1 \\end{matrix} \\right ) \\left ( \\begin{matrix} \\gamma \\zeta _ { p } ^ { - 1 } & 0 \\\\ 0 & 1 \\end{matrix} \\right ) \\left ( \\begin{matrix} - 1 & 0 \\\\ - 1 & - \\zeta _ { p } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} c _ 0 ^ 2 \\bar { C } | \\nabla p | ^ 2 - \\bar { C } ^ \\gamma & = 0 \\\\ - \\nabla \\cdot ( \\bar { C } \\nabla p ) & = S , \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} & \\hat { \\delta } ( f ) ( m \\otimes \\{ a _ 1 | . . . | a _ n \\} ) : = d _ N f ( m \\otimes \\{ a _ 1 | . . . | a _ n \\} ) - ( - 1 ) ^ { | f | } f ( b _ M ( m \\otimes \\{ a _ 1 | . . . | a _ n \\} ) ) \\\\ & + ( - 1 ) ^ { | m | + | a _ 1 | + . . . + | a _ { n - 1 } | - n + 1 } f ( m \\otimes \\{ a _ 1 | . . . | a _ { n - 1 } \\} ) \\cdot a _ n . \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} \\mathcal E \\bigl ( ( \\rho _ t ) _ { t \\in [ 0 , T ] } \\bigr ) = \\frac 1 2 \\int _ 0 ^ T \\| \\dot \\rho _ t \\| _ { - 1 , \\chi ( \\rho _ t ) } ^ 2 \\ ; \\ ! \\mathrm d t . \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} u _ { k } ( T _ 2 - \\delta , x ) = 0 \\mbox { f o r } x \\in [ x ^ * , 1 ] , \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} | [ P _ t \\psi ] _ v - [ P _ t \\psi ] _ { v ' } | & \\leq \\sum _ { v '' \\in V } | p _ t ( v , v '' ) - p _ t ( v ' , v '' ) | | \\psi _ { v '' } | \\\\ & \\leq C ( 1 + t ) ^ { - \\frac { \\eta } { 2 } } \\sum _ { v '' \\in V } p _ { 2 t } ( v , v '' ) | \\psi _ { v '' } | \\\\ & = C ( 1 + t ) ^ { - \\frac { \\eta } { 2 } } [ P _ { 2 t } | \\psi | ] _ v . \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} E _ \\alpha = T _ { i _ 1 } T _ { i _ 2 } \\cdots T _ { i _ { k } } ( E _ { i _ { k + 1 } } ) \\quad \\mathrm { a n d } F _ \\alpha = T _ { i _ 1 } T _ { i _ 2 } \\cdots T _ { i _ { k } } ( F _ { i _ { k + 1 } } ) . \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} \\| y _ j - y _ j ^ { \\delta } \\| _ { 1 } \\le \\delta _ j , j = 0 , \\dots , N - 1 , \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} \\int _ { V ^ { ( 4 ) } } \\langle \\beta _ { \\tilde g ^ { ( 4 ) } } \\delta ^ * _ { g ' } \\tilde Y , \\tilde Y \\rangle d v o l _ { \\tilde g ^ { ( 4 ) } } & = - \\int _ { V ^ { ( 4 ) } } \\langle \\beta _ { \\tilde g ^ { ( 4 ) } } \\delta ^ * _ { \\tilde g ^ { ( 4 ) } } Y ' , \\tilde Y \\rangle d v o l _ { \\tilde g ^ { ( 4 ) } } \\\\ & = - \\int _ { V ^ { ( 4 ) } } \\langle Y ' , \\beta _ { \\tilde g ^ { ( 4 ) } } \\delta ^ * _ { \\tilde g ^ { ( 4 ) } } \\tilde Y \\rangle d v o l _ { \\tilde g ^ { ( 4 ) } } \\\\ & = 0 . \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} \\begin{aligned} ( 2 8 ) \\Leftrightarrow & \\frac { \\partial } { \\partial y } \\int d x \\ x \\ b ( x , y , t ) \\\\ = & \\frac { \\partial } { \\partial y } \\left ( y \\alpha ( y , t ) \\right ) + \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) - \\frac { \\partial } { \\partial t } \\alpha ( y , t ) \\end{aligned} \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} [ \\iota _ t ( \\delta ) , f \\partial _ t ] = \\iota _ t ( \\delta ) ( f ) \\partial _ t , \\ , \\ , [ \\iota _ t ( \\delta _ 1 ) , \\iota _ t ( \\delta _ 2 ) ] = \\iota _ t [ \\delta _ 1 , \\delta _ 2 ] , \\ , \\ , \\ , \\ , [ \\iota _ t ( \\delta ) , r \\bar { C } ] = \\delta ( r ) \\bar { C } \\ , \\ , . \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} L ^ r { } _ s = \\begin{pmatrix} \\sqrt b \\\\ - \\sqrt { - c } \\end{pmatrix} \\begin{pmatrix} \\sqrt b & - \\sqrt { - c } \\end{pmatrix} \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} \\widehat { O } \\left ( \\begin{matrix} u \\\\ v \\\\ \\end{matrix} \\right ) = 0 \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} \\beta = ( q ^ 2 - q ) \\frac { m \\theta _ { r - 2 , q } } { q + 1 } + \\theta _ { r - 2 , q } . \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} = q ^ { \\frac { ( n - 1 ) ( n + 1 ) } { 1 2 } } \\frac { ( q ; q ^ 2 ) _ { \\infty } ^ 2 } { ( q ^ n ; q ^ { 2 n } ) _ { \\infty } ^ { 2 n } } \\frac { \\theta _ 1 ^ n \\big ( \\frac { \\pi } { 2 } \\mid \\frac { \\tau ' } { n } \\big ) } { \\theta _ 1 \\big ( \\frac { \\pi } { 2 } \\mid \\tau ' ) } \\theta _ 1 ( n z \\mid \\tau ' ) . \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} \\binom { k + r - 1 } { r } \\Delta ^ { r + 1 } ( ( k - 1 ) ^ n ) = \\sum _ { i = 0 } ^ { k - 1 } \\binom { n + k - i } { n - 1 } K ( n , i , r ) . \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} ( a ^ + ) ^ { q - 1 } = H , \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{gather*} \\overset { I } { L } { ^ { ( \\pm ) } } = \\big ( \\overset { I } { T } \\otimes \\mathrm { i d } \\big ) \\big ( R ^ { ( \\pm ) } \\big ) = \\bigl ( \\overset { I } { a _ i ^ { ( \\pm ) } } \\bigr ) b _ i ^ { ( \\pm ) } \\in \\mathrm { M a t } _ { \\dim ( I ) } ( H ) \\end{gather*}"}
-{"id": "7126.png", "formula": "\\begin{align*} E ( T ) \\leq \\gamma E ( 0 ) \\gamma = \\frac { \\tilde { c } } { \\tilde { c } + T / 2 } < 1 . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} D \\prod _ k \\lambda _ k ^ { c _ k } \\ge \\det \\left ( \\Big ( \\sum _ { k = 0 } ^ { m + 1 } c _ k \\lambda _ k \\ , u _ k \\otimes u _ k \\Big ) _ + \\right ) , \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 v '' - r ( x ) v = - 1 , \\ x \\in ( a , L _ 2 ) ; \\\\ & v ( a ) = 0 ; \\\\ & v ( L _ 2 ) = v ( 0 ) . \\end{aligned} \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} \\ker D _ 0 { \\tilde { r } } = \\ker D _ 0 r . \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} W _ { p , n } ( 0 , n ) = \\square _ p [ n - 1 ] \\ , \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} - \\eta _ 0 \\frac { p _ { _ 0 } ( y ) } { p _ { _ 1 } ( y ) } + \\eta _ 1 ^ b + \\eta _ 2 ^ b y ^ 2 = 0 , \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} \\sup _ { 0 \\le u \\le s } \\left \\| Q ( \\epsilon ; t - u ) - \\left ( \\tilde { Q } - \\epsilon \\tilde { Q } ^ { ( 1 ) } u + \\frac { \\epsilon ^ 2 } { 2 } \\tilde { Q } ^ { ( 2 ) } u ^ 2 \\right ) \\right \\| = O ( \\epsilon ^ 3 s ^ 3 ) \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} \\beta ' _ \\tau ( \\zeta _ N ^ m , n ) = \\beta _ \\tau ( \\zeta _ N ^ { m d + n b } , m c + n a ) . \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\pm \\infty } \\frac { e ^ { - \\pi | x | } } { 2 i \\sin ( \\pi i x ) } = 1 , \\end{align*}"}
-{"id": "9819.png", "formula": "\\begin{align*} & f : B _ { Q } \\to \\mathbb { M } _ 2 ( \\mathbb { C } ) \\\\ & a + b \\textbf { i } + c \\textbf { j } + d \\textbf { k } \\mapsto M = \\begin{pmatrix} a + b \\textbf { i } & c + d \\textbf { i } \\\\ - c + d \\textbf { i } & a - b \\textbf { i } \\end{pmatrix} { ~ . } \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { H ^ s } = \\left ( \\sum _ { k = - \\infty } ^ \\infty \\left ( 1 + | k | ^ 2 \\right ) ^ { s } | c _ k ( f ) | ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } < \\infty . \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u = H u + \\mu | u | ^ p u , \\\\ u ( 0 ) = u _ 0 . \\end{cases} \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} & \\| U _ h ^ { n } \\| ^ 2 + \\Delta t \\sum _ { k = 2 } ^ n \\| \\nabla U _ h ^ { k - \\theta } \\| ^ 2 + \\gamma \\Delta t \\sum _ { k = 2 } ^ n \\| \\Sigma _ h ^ { k - \\theta } \\| ^ 2 \\\\ \\leq & C ( \\| U _ h ^ { 0 } \\| ^ 2 + \\Delta t \\sum _ { k = 1 } ^ n \\| g ^ { k } \\| ^ 2 ) + \\Delta t \\sum _ { k = 1 } ^ n \\| U _ h ^ { n } \\| ^ 2 + \\Delta t \\sum _ { k = 1 } ^ n \\| u _ H ^ { n } \\| ^ 2 . \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\gamma u \\times H _ { e f f } + L _ 1 \\frac { 1 } { | u | ^ 2 } ( u \\cdot H _ { e f f } ) u - L _ 2 \\frac { 1 } { | u | ^ 2 } u \\times ( u \\times H _ { e f f } ) \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} \\frac { \\partial \\overline { \\mathcal { L } } ( y , p _ { _ 1 } ( y ) ) } { \\partial p _ { _ 1 } ( y ) } \\ ! = \\ ! \\ ! - \\ ! \\log p _ { _ 1 } ( y ) \\ ! - \\ ! 1 \\ ! - \\ ! \\eta _ 0 \\frac { p _ { _ 0 } ( y ) } { p _ { _ 1 } ( y ) } \\ ! + \\ ! \\eta _ 1 \\ ! + \\ ! \\eta _ 2 y ^ 2 . \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} u u ^ { \\dagger } x = x , \\ \\ x \\in R _ { u } \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} \\langle \\alpha , \\beta \\rangle _ g = \\det \\bigl ( \\langle \\alpha _ i , \\beta _ j \\rangle _ g \\bigr ) , \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} \\| \\mathrm { d } A ^ \\flat \\| _ g ^ 2 = - c \\ , , \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} \\beta _ { g ^ { ( 4 ) } } \\delta ^ * _ { g ^ { ( 4 ) } } [ \\beta _ { \\tilde g ^ { ( 4 ) } } g ^ { ( 4 ) } ] = 0 \\quad M . \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} ( - q ; - q ) _ \\infty = \\dfrac { f _ 2 ^ 2 } { f _ 1 f _ 4 } , \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} \\mu _ { j , m } : = 2 ^ { j ( s - \\frac 1 r + 1 ) } \\int _ { \\mathbb R } h ( x ) h _ { j , m } ( x ) \\mathrm d x , \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} \\d X _ t = \\sqrt 2 \\ , \\d B _ t - \\nabla U ( X _ t ) \\ , \\d t . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} \\mathbb E [ X _ { t } ^ { ( u ) } ] \\ , = \\ , \\mathbb E [ \\widetilde { X } _ { t } ^ { ( u ) } ] \\ , = \\ , \\mathbb E [ X _ { t } ^ { \\bullet } ] \\ , = \\ , \\mathbb E [ X _ { t } ^ { \\dagger } ] \\ , = \\ , 0 \\ , , t \\ge 0 \\ , , \\ , \\ , u \\in [ 0 , 1 ] \\ , , \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} ( \\beta , 1 ) = ( \\beta _ 1 , n _ 1 ) + ( \\beta _ 2 , n _ 2 ) , \\ \\beta _ i = r _ i [ F ] + k _ i [ l ] \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ j ( - 1 ) ^ { r j } \\binom k j \\left ( { \\frac { { F _ { n + r } } } { { F _ n } } } \\right ) ^ j G _ { m - ( n + r ) k + r j } } = ( - 1 ) ^ k ( - 1 ) ^ { r k } \\left ( { \\frac { { F _ r } } { { F _ n } } } \\right ) ^ k G _ m , n \\ne 0 \\ , . \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} \\Delta u - ( \\nabla h , \\nabla u ) = - \\lambda u e ^ { - \\frac { | x | ^ 2 } { 2 ( m - 2 ) } } . \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} \\mathcal { L } = \\mathcal { L } _ 0 + \\mathcal { L } _ 1 \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} - \\frac { f ^ { \\prime } } { f } & \\le r ^ { 1 - m } e ^ { \\frac { r ^ 2 } { 4 } } \\int ^ { + \\infty } _ { r } r ^ { m - 3 } e ^ { - \\frac { r ^ 2 } { 4 } } \\lambda _ k d r \\\\ & = \\lambda _ k \\frac { \\int ^ { + \\infty } _ { r } r ^ { m - 3 } e ^ { - \\frac { r ^ 2 } { 4 } } } { r ^ { m - 1 } e ^ { - \\frac { r ^ 2 } { 4 } } } \\\\ & = \\lambda _ k \\frac { 1 } { r ^ 3 } ( 2 + o ( 1 ) ) . \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { k = 0 } ^ { \\infty } f _ k ( z ) , z \\in \\mathbb { B } _ p \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} \\sum _ { \\lambda + \\nu = \\mu } \\Gamma _ \\lambda \\Theta _ \\nu = \\delta _ { \\mu , 0 } , \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} \\mbox { P r o b } \\left ( \\theta ^ { ( 2 ) } _ { 1 } ( l ) = b _ { k } ~ | ~ \\theta ( l ) = c _ { j } \\right ) = \\mbox { P r o b } \\left ( \\theta ^ { ( 3 ) } _ { 1 } ( l ) = \\Theta ^ { - 1 } ( \\phi ( c _ { j } ) \\ominus \\Theta ( b _ { k } ) ) \\right ) , \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} U \\epsilon ( f ) = U _ 2 U _ 1 \\epsilon ( f _ A , f _ { A ^ c } ) = U _ 2 \\epsilon ( f _ A ) \\otimes \\epsilon ( f _ { A ^ c } ) = \\epsilon ( f _ A ) \\oplus \\bigoplus _ { n = 1 } ^ \\infty \\epsilon ( f _ A ) \\otimes \\frac { 1 } { \\sqrt { n ! } } f _ { A ^ c } ^ { \\otimes n } \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { \\binom k j \\binom { k - j } s \\left ( { \\frac { { f _ 2 } } { { f _ 1 } } } \\right ) ^ j \\left ( { \\frac { { f _ 3 } } { { f _ 1 } } } \\right ) ^ s X _ { m - a k - ( b - a ) j - ( c - a ) s } } } = \\frac { { X _ m } } { { f _ 1 ^ k } } \\ , , \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} S ^ { R , R } _ { ( l , p ) , ( l ' , p ' ) } = \\frac { \\zeta } { 1 - \\zeta } \\zeta ^ { - l l ' - l p ' - p l ' - 2 p p ' } ( 1 - \\zeta ^ { l l ' } ) . \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\abs { c _ { p } ( \\pi _ { p } , l , t , \\mu _ p ) } \\leq 5 p ^ { \\frac { 1 } { 2 } } t \\max _ { i = 1 , 2 } ( \\abs { \\alpha _ i } ^ t ) , \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( \\frac { P _ x ^ { \\epsilon } } { \\sigma _ w ^ 2 } + \\log \\frac { \\sigma _ w ^ 2 } { P _ x ^ { \\epsilon } + \\sigma _ w ^ 2 } \\right ) = 2 \\epsilon ^ 2 . \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} \\partial _ { t } F ( t , x ) + \\inf \\limits _ { u \\in U } H ( t , x , F ( t , x ) , D F ( t , x ) , D ^ { 2 } F ( t , x ) , u ) = 0 , \\ F ( T , x ) = \\phi ( x ) , \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} \\begin{aligned} & r _ - ( x ) - \\hat r _ - ( x ) = \\frac D 2 \\lambda ( l + 1 ) ( \\gamma + x ^ 2 ) ^ { \\frac { l - 3 } 2 } \\times \\\\ & \\Big [ \\frac { 2 c _ 1 } { D \\lambda ( l + 1 ) } ( \\gamma + x ^ 2 ) ^ { \\frac { 3 - l } 2 } ( \\gamma _ 1 + x ^ 2 ) ^ l - ( l + 1 ) \\lambda x ^ 2 ( \\gamma + x ^ 2 ) ^ { \\frac { l + 1 } 2 } - \\gamma - l x ^ 2 \\Big ] . \\end{aligned} \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } p _ { n - 1 } & p _ { n } \\\\ q _ { n - 1 } & q _ { n } \\end{array} \\right ] = \\left [ \\begin{array} { c c } p _ { n - 2 } & p _ { n - 1 } \\\\ q _ { n - 2 } & q _ { n - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n } \\end{array} \\right ] , \\left [ \\begin{array} { c c } p _ { 0 } & p _ { 1 } \\\\ q _ { 0 } & q _ { 1 } \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} D _ K = \\frac { n - 1 } { 2 n } + \\left ( \\frac { 1 } { n } \\right ) d ( R | S ) \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} \\alpha : = \\left ( \\frac { 3 } { 2 } \\right ) ^ { 1 - p / 2 } \\bar { L } D , \\bar { L } : = \\frac { 2 7 C L } { 2 D ^ 2 B } \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} \\forall _ { 1 \\leq i , j \\leq l } \\exists _ { g \\in G } \\forall _ { c \\in C } \\varphi _ i ( c ) = \\varphi _ j ( g c g ^ { - 1 } ) . \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} z ^ { - n } \\sigma _ { - } p _ { i } = - z ^ { - n } u ( I - V U ) ^ { - 1 } \\left ( P ^ { + } \\left ( \\frac { z ^ { i } } { \\sigma _ { - } } \\right ) \\right ) , \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 1 , 1 / 2 ) = \\int _ { 0 } ^ { \\infty } \\frac { x \\cos ( \\frac { \\pi x } { 2 } ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x = \\frac { 1 3 - 4 \\pi } { 8 \\pi ^ { 2 } } , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} Y _ { t } ^ { t , x } = \\bar { Y } _ { t } ^ { t , x } . \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} \\phi ^ { S _ 1 } w ( v ) = \\phi ^ { X S _ 1 } _ { X w } ( X v ) = \\phi ^ { S _ 2 } _ { X w } ( X v ) . \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} \\gamma _ { u ( \\cdot ) , T } ( Q ) = \\frac { 1 } { T } \\int _ 0 ^ T 1 _ Q ( y ( t ) , u ( t ) ) d t , \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} Y _ t & : = \\frac 1 { F ( t ) } \\sum _ { k = 1 } ^ { D _ t } f ( \\tau _ k ) \\xrightarrow { t \\to \\infty } 1 \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} f ( x , y , t ) = \\frac { 1 } { \\sqrt { 2 } } \\sqrt { x ^ 2 + y ^ 2 + \\sqrt { ( x ^ 2 + y ^ 2 ) ^ 2 + 4 t ^ 2 } } . \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} \\left | \\Q - \\Pi ( \\Q ) \\right | ^ 2 = \\left ( \\lambda _ 1 + \\frac { s _ + } { 3 } \\right ) ^ 2 + \\left ( \\lambda _ 2 + \\frac { s _ + } { 3 } \\right ) ^ 2 + \\left ( \\lambda _ 1 + \\lambda _ 2 + \\frac { 2 s _ + } { 3 } \\right ) ^ 2 \\leq \\delta _ 0 ^ 2 . \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} d \\widetilde { \\mu } ( z ) = Q ( z ) \\ , d z \\ , . \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} \\mathrm { T r } _ d ( j - 7 4 4 ) : = \\sum _ { Q \\in \\mathcal { Q } _ d / \\mathrm { S L } _ 2 ( \\mathbb { Z } ) } \\frac { j ( \\alpha _ Q ) - 7 4 4 } { w _ Q ^ { } } d < 0 , \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} S _ { i _ 1 } = \\bigcup _ { q _ 2 = 0 } ^ { p _ { i _ 2 } - 1 } T _ { i _ 1 } \\alpha _ { i _ 2 } ^ { q _ 2 } . \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} \\mathcal { J } ^ { ( \\alpha , \\beta ) } f ( n ) = ( J ^ { ( \\alpha , \\beta ) } - I ) f ( n ) , \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t u _ h ^ { \\frac 1 2 } , v _ h \\Big { ) } - \\gamma ( \\nabla \\sigma _ h ^ { \\frac 1 2 } , \\nabla v _ h ) + ( \\nabla u _ h ^ { \\frac 1 2 } , \\nabla v _ h ) + ( f ^ { \\frac 1 2 } ( u _ h ) , v _ h ) = & ( g ^ { \\frac 1 2 } , v _ h ) , ~ \\forall v _ h \\in L _ h , \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} \\beta ( y ) = \\alpha _ { 0 } e ^ { - 2 y ^ { 2 } } + \\alpha _ { 1 } y ^ { 2 } e ^ { - 2 y ^ { 2 } } + \\alpha _ { 2 } y ^ { 4 } e ^ { - 2 y ^ { 2 } } \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} { \\tilde U } ( t , \\rho ) : = \\inf _ { m , \\rho } ~ \\Big \\{ \\sum _ { n = 1 } ^ N \\Delta t \\ , \\mathcal { L } ( m ^ n , \\rho ^ n ) - \\sum _ { n = 1 } ^ N \\Delta t \\ , F ( \\rho ^ n ) + G ( \\rho ^ 0 ) \\Big \\} \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} S _ f ( \\rho \\| \\sigma ) = S _ f ( e \\rho e \\| e \\sigma e ) . \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} S _ { 2 m + 1 , \\sigma } ( t , \\pi ) \\leq \\dfrac { 1 } { 2 \\pi ( 2 m + 2 ) ! } \\ , \\bigg ( \\dfrac { 3 } { 2 } - \\sigma \\bigg ) ^ { 2 m + 2 } \\log C ( t , \\pi ) - \\dfrac { 1 } { \\pi ( 2 m ) ! } \\ , \\displaystyle \\sum _ { \\gamma } g ^ { - } _ { 2 m + 1 , \\sigma , \\Delta } ( t - \\gamma ) + O _ m ( d ) . \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} \\frac { n } { \\omega _ n ^ { i k } - 1 } ( - 1 ) ^ { k + 1 } \\sum _ { 0 \\le s \\le ( k - 1 ) d / n } \\binom { d } { s } ( - 1 ) ^ s = \\omega _ n ^ { i \\binom { k } { 2 } + i } { n \\brack k } ^ { ' } _ { \\omega _ n ^ i } . \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} ( z , w ) \\mapsto \\left ( z , \\sum _ { p = 1 } ^ n \\left ( w _ p + \\frac { i } { 2 } \\bar { z } _ p \\right ) d z _ p \\right ) . \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| u _ { 0 , n } - v \\| _ { H ^ 1 } = 0 . \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} K _ + : = & \\left \\{ x \\in K : \\gamma > 0 R > 0 \\right . \\\\ & \\left . n \\mathbf { P } ( x + S ( n ) \\in D _ { R , \\gamma } , \\tau _ x > n ) > 0 \\right \\} , \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\phi _ r ( \\lambda ) : = \\mathbb P _ \\lambda [ \\exists ( z , R ) \\in \\eta \\mathsf B _ R ^ z ] , \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} \\nabla \\psi _ { \\epsilon } ( y ) ^ T f ( y , u ) \\geq - \\epsilon M \\ \\ \\forall ( y , u ) \\in Y \\times U , \\ \\ \\ { \\rm w h e r e } \\ \\ M : = \\max _ { ( y , u ) \\in Y \\times U } | | f ( y , u ) | | . \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} T _ g f ( z ) = \\int _ { 0 } ^ { z } f ( s ) g ' ( s ) { \\rm d } s = \\int _ { 0 } ^ { 1 } f ( t z ) z g ' ( t z ) { \\rm d } t = \\int _ { 0 } ^ { z } f { \\rm d } g . \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} \\big | f ( z ) - \\sum _ { n = 0 } ^ { p - 1 } a _ n z ^ n \\big | \\leq C _ p | z | ^ p , \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{gather*} \\overset { \\widetilde { I } \\widetilde { J } } { R } _ { 1 2 } \\overset { I } { B } _ 1 \\overset { \\widetilde { I } \\widetilde { J } } { ( R ' ) } _ { 1 2 } \\overset { J } { A } _ 2 = \\overset { J } { A } _ 2 \\overset { \\widetilde { I } \\widetilde { J } } { R } _ { 1 2 } \\overset { I } { B } _ 1 \\overset { \\widetilde { I } \\widetilde { J } } { ( R ^ { - 1 } ) } _ { 1 2 } , \\end{gather*}"}
-{"id": "352.png", "formula": "\\begin{align*} \\gamma ^ { ( d ) } _ { n } ( A ) = \\Gamma _ 1 ' \\oplus \\Gamma ' _ 2 \\oplus \\ldots \\oplus \\Gamma ' _ { p - 1 } \\oplus \\Gamma _ p . \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} A _ { f ^ { \\iota } } ( m _ { 1 } , \\ldots , m _ { n - 1 } ) = \\chi ( m _ { 1 } \\cdots m _ { n - 1 } ) A _ { f } ( m _ { n - 1 } , \\ldots , m _ { 1 } ) . \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} \\tilde { e } _ { i _ l } ^ { } \\cdots \\tilde { e } _ { i _ 1 } ^ { } \\left ( b \\otimes b ^ \\prime \\right ) & = b _ \\lambda \\otimes \\tilde { e } _ { i _ l } ^ { r _ l } \\cdots \\tilde { e } _ { i _ 1 } ^ { r _ 1 } ( b ^ \\prime ) . \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} ( t , s , x ) \\cdot ( t ' , s ' , x ' ) = ( t t ' , s + s ' , x + x ' ) . \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} \\sqrt { - \\Delta } Q + Q - a _ { * } ( \\left | \\cdot \\right | ^ { - 1 } \\star | Q | ^ { 2 } ) Q = 0 \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} e ^ { 2 \\pi c t } \\widehat { G ^ { \\epsilon } _ c } ( t ) - e ^ { 2 \\pi y t } \\widehat { G ^ { \\epsilon } _ y } ( t ) = 0 . \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{gather*} \\partial _ s f ( s , t ) + i \\partial _ t f ( s , t ) = 0 \\\\ f ^ { - 1 } ( \\partial B ( r ) ) \\neq \\emptyset , \\ \\ f ^ { - 1 } ( \\partial B ( 2 r ) ) \\neq \\emptyset \\end{gather*}"}
-{"id": "2312.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } ( | \\nabla \\bar { u } | ^ 2 + \\bar { u } ^ 2 ) + \\int _ { \\R ^ 3 } \\rho \\phi _ { \\bar { u } } \\bar { u } ^ 2 - \\mu \\int _ { \\R ^ 3 } \\bar { u } ^ { p + 1 } = 0 . \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} P ( a _ - + a _ 0 + a _ + ) = R _ 0 ( a _ 0 ) - \\lambda a _ + , a _ { \\pm } \\in A _ { \\pm } , \\ a _ 0 \\in A _ 0 , \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} \\int _ { A _ { r _ 2 , r _ 3 } } \\left ( \\mathfrak u ^ 2 - \\lambda \\phi _ { \\mathfrak u } - \\frac { 1 } { 2 C _ 2 } \\right ) \\mathfrak u ^ 2 \\ , d x > \\delta , \\int _ { A _ { r _ i , r _ { i + 1 } } } \\left ( \\mathfrak u ^ 2 + \\lambda \\phi _ { \\mathfrak u } + \\frac { 1 } { 2 C _ 2 } \\right ) \\mathfrak u ^ 2 \\ , d x < \\frac { \\delta } { 4 } , \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} Q _ h v | _ D : = ( Q _ { k , D } ^ 0 v _ 0 , Q _ { k , D } ^ b v _ b ) , \\ \\forall v \\in W ( D ) . \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} ( \\sigma ^ { \\frac 1 2 } , w ) + ( \\nabla u ^ { \\frac 1 2 } , \\nabla w ) = 0 , ~ \\forall w \\in H _ 0 ^ 1 . \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} \\mathbb { R } ^ 4 \\rtimes \\mathrm { O } ( 3 , 1 ) \\ni ( v , \\Lambda ) \\mapsto \\begin{pmatrix} \\Lambda & v \\\\ \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\end{array} & 1 \\end{pmatrix} \\in \\mathrm { S L } ( 5 , \\mathbb { R } ) . \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} ( A \\psi ) ( k ) = & ( H _ \\mu ( \\xi - k ) + \\omega ( k ) - \\Sigma ( \\xi ) ) ^ { - 1 } A ( ( H _ \\mu ( \\xi ) - \\Sigma ( \\xi ) ) \\psi ) ( k ) \\\\ & - \\mu v ( k ) ( H _ \\mu ( \\xi - k ) + \\omega ( k ) - \\Sigma ( \\xi ) ) ^ { - 1 } \\psi \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} y _ i = \\begin{cases} 0 & \\mbox { i f $ \\alpha $ i s a p p l i e d , o r } \\\\ 1 & \\mbox { i f $ \\beta $ i s a p p l i e d . } \\end{cases} \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} ( f , ( - \\mathcal { L } ^ \\omega _ X ) ^ { - 1 } f ) = \\int _ { 0 } ^ { \\infty } d t ( f , e ^ { t \\mathcal { L } ^ \\omega _ X } f ) . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} | N _ t ( e ) | & \\le d + d ( d - 1 ) + d ( d - 1 ) ^ 2 + \\dots + d ( d - 1 ) ^ { ( t - 1 ) / 2 } - 1 = \\tau ' _ t ( d ) - 1 . \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j > ( \\log ( 1 / \\epsilon ) ) ^ { 1 + \\delta } } j \\mu \\tilde { B } ^ j \\tilde { r } ^ { ( 1 ) } \\ , \\right \\| = O ( \\epsilon ^ 2 ) \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{gather*} T _ 2 T _ 3 \\cdots T _ 0 T _ 1 T _ 0 \\cdots T _ 3 \\big ( x _ 0 ^ + \\big ) = T _ 2 T _ 3 \\cdots T _ 0 T _ 1 \\big ( x _ { N - 1 } ^ + \\big ) = T _ 2 T _ 3 \\cdots T _ { N - 2 } \\big ( x _ 0 ^ + \\big ) = x _ 0 ^ + . \\end{gather*}"}
-{"id": "3500.png", "formula": "\\begin{align*} \\xi : \\begin{pmatrix} x ^ 1 \\\\ x ^ 2 \\\\ x ^ 3 \\\\ x ^ 4 \\\\ \\end{pmatrix} \\mapsto \\begin{pmatrix} x ^ 1 \\cos ( q ^ 3 + q ^ 4 ) \\mp x ^ 2 \\sin ( q ^ 3 + q ^ 4 ) \\\\ \\pm x ^ 1 \\sin ( q ^ 3 + q ^ 4 ) + x ^ 2 \\cos ( q ^ 3 + q ^ 4 ) \\\\ x ^ 3 \\\\ x ^ 4 \\end{pmatrix} + \\begin{pmatrix} q ^ 1 \\\\ q ^ 2 \\\\ q ^ 3 \\\\ q ^ 4 \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\mathcal { W } ( z ) : = \\left \\{ \\gamma \\in W \\ : \\ s u p p ( \\gamma ) \\in Y _ z \\times U \\right \\} \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} X = D + ( I - U _ A ^ * U _ A ) Y , \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{gather*} \\mathcal { N } ( z _ - , z _ + , F _ s , J ' _ s ) \\longrightarrow \\mathcal { N } ( \\widetilde { f } ( z _ - ) , \\widetilde { f } ( z _ + ) , G _ s , J '' _ s ) \\\\ v ( s , t ) \\longrightarrow u ( s , t ) = \\phi _ L ^ t ( v ( s , t ) ) \\end{gather*}"}
-{"id": "3558.png", "formula": "\\begin{align*} ( * T ^ \\mathrm { a x } _ \\pm ) _ \\alpha = \\mp \\ , \\frac 2 3 \\ , ( \\ , 0 \\ , , \\ , 0 \\ , , \\ , 1 \\ , , \\ , 1 \\ , ) \\ , , \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} \\theta _ v ( t ) = \\Omega t + \\bar { \\theta } _ v + \\psi _ v ( t ) , \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{gather*} = 2 \\binom { n + 2 } { 4 } + 2 \\sum _ { i = 2 } ^ { n + 1 } \\binom { i } { 2 } = 2 \\left ( \\binom { n + 2 } { 4 } + \\sum _ { i = 2 } ^ { n + 1 } \\binom { i } { 2 } \\right ) = 2 \\left ( \\binom { n + 2 } { 4 } + \\binom { n + 2 } { 3 } \\right ) = 2 \\binom { n + 3 } { 4 } . \\end{gather*}"}
-{"id": "6586.png", "formula": "\\begin{align*} \\sum _ { e \\in \\mathbf { E } } | { \\rm d } f ( e ) | \\le C ^ { 1 / 2 } \\sum _ { n = 0 } ^ \\infty \\lambda ^ { n / 2 } \\sum _ { e \\in \\mathbf { E } , \\ ; | e | = n } 1 < \\infty . \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} f ( x ) & = x ( x ^ 2 + x + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( 1 + x ) + m \\ : h ( x ) , \\\\ g ( x ) & = x ( x + 1 ) - ( x ^ 4 + x ^ 2 + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( 1 + x ) - m \\ : h ( x ) , \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} q ( 0 ) = 0 , q ' ( 0 ) = \\left < t _ { j k } \\right > , q '' ( \\lambda ) = \\left < ( t _ { j k } - \\left < t _ { j k } \\right > _ \\lambda ) ^ 2 \\right > _ \\lambda . \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} r _ i \\begin{cases} \\equiv 0 \\pmod { p } , & i \\neq i _ 0 , \\\\ \\not \\equiv 0 \\pmod { p } , & i = i _ 0 . \\end{cases} \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} y _ { i , j } = \\left ( z ^ { - n } u ( I - V U ) ^ { - 1 } z ^ { j } \\right ) \\mathbf { \\hat { } } \\ , ( - i ) . \\newline \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} \\Psi _ { S S S a } ( \\tau _ { n } , \\tau _ { n } ' , \\tau _ { n } '' ) = \\max \\{ \\Psi _ { S S S } ( \\tau _ { n } , \\tau _ { n } ' ) , \\psi _ { 3 } ( \\tau _ { n } '' ) \\} \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} \\sum _ { b = 0 } ^ { q _ 1 + q _ 2 + \\cdots + q _ n } z ^ { w ( b ) } \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} \\begin{pmatrix} a & v \\\\ w ^ t & d \\end{pmatrix} \\cdot z = \\frac { a z + v } { w ^ t z + d } . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} - \\Delta u _ n + u _ n + \\rho ( x ) \\phi _ { u _ n } u _ n = \\mu _ n u _ n ^ { p } , \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} ( X + \\Phi ) u = - f _ { 0 } \\in \\Omega _ 0 , u | _ { \\partial ( S M ) } = 0 \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} \\tilde { L } = ( C _ { L _ { i , j } } ) _ { i , j \\in S } . \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} \\begin{aligned} c _ { 0 } & = \\frac { \\Gamma ( 1 - 2 \\beta ) \\sin \\pi \\beta } { \\pi } \\cdot \\frac { \\tau _ { - } ( 1 ) } { \\tau _ { + } ( 1 ) } \\\\ c _ { 0 } ' & = \\frac { \\Gamma ( 1 + 2 \\beta ) \\sin \\pi \\beta } { \\pi } \\cdot \\frac { \\tau _ { + } ( 1 ) } { \\tau _ { - } ( 1 ) } \\end{aligned} \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} X ^ { \\dagger \\ ! \\dagger } _ { t } \\ , = \\ , \\int ^ { t } _ { 0 } e ^ { t - s } \\widetilde { X } ^ { \\dagger \\ ! \\dagger } _ { s } { \\mathrm d } s + \\int ^ { t } _ { 0 } e ^ { t - s } { \\mathrm d } B _ { s } \\ , , \\widetilde { X } ^ { \\dagger \\ ! \\dagger } _ { t } \\ , = \\ , \\int ^ { t } _ { 0 } \\sum _ { k = 0 } ^ { \\infty } e ^ { t - s } \\cdot \\frac { \\ , ( - 1 ) ^ { k } ( t - s ) ^ { k } \\ , } { \\ , k ! \\ , } \\ , { \\mathrm d } W _ { s , k } \\ , ; t \\ge 0 \\ , \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} \\left ( \\chi ^ + ( x ) - \\chi ^ - ( x ) \\right ) \\frac { 1 } { c _ 0 } \\alpha ( x ) \\hat { \\theta } ( x ) = w ( x ) , \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathbf { x } ] & = \\mu + \\omega \\delta \\sqrt { \\frac { 2 } { \\pi } } , \\\\ \\mathbb { E } [ | \\mathbf { x } - \\mathbb { E } [ \\mathbf { x } ] | ^ 2 ] & = \\omega ^ 2 \\left ( 1 - \\frac { 2 \\delta ^ 2 } { \\pi } \\right ) , \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} \\# \\{ \\alpha \\in \\mathbf { M } _ 4 : \\star ( \\alpha ) = 1 \\} = m _ K n + 1 . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} D ( t ) = \\int _ { 0 } ^ { 2 t } \\frac { D _ 1 ( s ) } { s } \\ , \\d s , t \\ge 0 . \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} \\lambda _ k ^ 0 = \\begin{cases} 2 ^ { - j \\frac { d } { u ' } } , & \\ Q _ { 0 , k } \\subset Q _ { - j , m _ 0 } , \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} \\Psi ( a ^ * t b ) = \\Psi ( a ) ^ * \\Psi ( t ) \\Psi ( b ) \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} \\hat { k } _ { R R T } = \\max \\{ k : R R ( k ) \\leq \\Gamma _ { R R T } ^ { \\alpha } ( k ) \\} \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} [ n ] ! _ i : = [ n ] _ i [ n - 1 ] _ i \\cdots [ 1 ] _ i \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\frac { x } { | x | } \\cdot D \\Phi = \\frac { x } { | x | } \\cdot \\frac { x } { | x | } \\Big ( f ^ { 1 / m } + \\frac { 1 } { m } f ^ { 1 / m - 1 } x \\cdot D f \\Big ) \\geq f ^ { 1 / m } \\Big ( 1 - \\frac 1 { m } \\Big ) , \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\mathrm { T } y ) ( 0 ) = \\sum _ { i = m } ^ { M } k _ i \\mu _ i ( 0 ) y ^ { ( i ) } _ 0 < 0 m < I _ 0 \\le M , \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} L _ s = \\sum _ { c \\geq - n _ l + 2 q + 2 \\abs { r } + 2 k } L _ { s , c } , \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} \\epsilon : = | | X _ P | | _ { s , r , q , \\mathbf { a } ; \\mathcal { O } } + \\frac { \\alpha } { M } | | X _ P | | _ { s , r , q , \\mathbf { a } ; \\mathcal { O } } ^ { l i p } \\leq ( \\alpha \\gamma ) ^ { 1 + \\beta } \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} \\mu _ X ( \\chi ) = \\gamma ( \\chi , - \\check \\alpha , 0 , \\psi ^ { - 1 } ) \\gamma ( \\chi , \\check \\alpha , 0 , \\psi ) . \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} \\begin{cases} L _ { \\frac { Y ^ { \\perp } } { u } \\partial _ t } g ^ { ( 4 ) } ( \\partial _ t , \\partial _ t ) = 0 \\\\ [ L _ { \\frac { Y ^ { \\perp } } { u } \\partial _ t } g ^ { ( 4 ) } ( \\partial _ t ) ] ^ T = - u ^ 2 d ( \\frac { Y ^ { \\perp } } { u } ) \\\\ [ L _ { \\frac { Y ^ { \\perp } } { u } \\partial _ t } g ^ { ( 4 ) } ] ^ T = 0 . \\end{cases} \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} \\dim ^ R ( \\delta _ k ) = \\xi ^ k \\dim ^ L ( \\delta _ k ) = \\xi ^ { - k } . \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} \\rho _ { t } ( \\varphi ) \\ , = \\ , \\pi _ { 0 } ( \\varphi ) + \\int ^ { t } _ { 0 } \\rho _ { s , 2 } ( \\varphi b ) { \\mathrm d } X _ { s } + \\int ^ { t } _ { 0 } \\rho _ { s , 3 } ( \\widetilde { \\mathcal A } _ { s } \\varphi ) { \\mathrm d } s \\ , , 0 \\le t \\le T \\ , , \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} \\partial _ i f _ i = \\partial _ i g _ + - \\partial _ i g _ - \\ ; \\ ; \\ ; \\ ; g _ { \\pm } \\in L ^ 2 ( E ) , \\ ; \\partial _ i g _ { \\pm } \\in L ^ { 1 } ( E ) \\ ; \\ ; \\ ; \\ ; \\partial _ i g _ { \\pm } \\geq 0 . \\end{align*}"}
-{"id": "9701.png", "formula": "\\begin{align*} L ( \\varphi , \\mathbb { A } ) = \\prod _ { f } \\frac { \\big [ \\mathbb { A } / f \\mathbb { A } \\big ] _ { \\mathbb { A } } } { \\big [ \\varphi ( \\mathbb { A } / f \\mathbb { A } ) \\big ] _ { \\mathbb { A } } } , \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{gather*} \\nabla ^ { \\mathbb { V } } = \\overline { \\nabla } ^ { \\mathrm { b a s } } | _ { \\mathbb { V } } = \\overline { \\nabla } ^ { \\mathrm { b a s } , \\overline { g } } | _ { \\mathbb { V } } , \\end{gather*}"}
-{"id": "7489.png", "formula": "\\begin{align*} \\norm { a } _ { \\mathcal { Y } _ p } ^ 2 : = \\pi \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { k + 1 } \\norm { a _ k } _ { \\mathcal { W } _ p ( k ) } ^ 2 = \\pi \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { k + 1 } \\int _ { \\mathbb { B } _ p } \\abs { a _ k ( w ) } ^ 2 ( 1 - p ( w ) ) ^ { k + 1 } d V ( w ) < \\infty . \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} F = - \\frac 1 2 \\operatorname { R e } \\left [ i ( p \\wedge u ^ \\flat ) e ^ { i p _ \\gamma x ^ \\gamma } \\ , \\right ] = - \\frac 1 2 \\mathrm { d } A ^ \\flat . \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t u _ H ^ { \\frac 1 2 } , v _ H \\Big { ) } - \\gamma ( \\nabla \\sigma _ H ^ { \\frac 1 2 } , \\nabla v _ H ) + ( \\nabla u _ H ^ { \\frac 1 2 } , \\nabla v _ H ) + ( f ^ { \\frac 1 2 } ( u _ H ) , v _ H ) = & ( g ^ { \\frac 1 2 } , v _ H ) , ~ \\forall v _ H \\in L _ H , \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { R } \\sum _ { j < j _ R } | C _ { j , R , m } ( x , y ) \\cap W _ R ( x , y ) \\cap h ^ { - 1 } ( W _ R ( x , y ) ) | & \\leq \\\\ \\frac { 1 } { R } \\sum _ { \\log f ( m ) \\leq j \\leq j _ R - 1 } \\frac { 2 c ( d ) R } { 2 ^ { d ^ { - 1 } \\delta j } } & \\leq \\frac { 1 } { 1 0 0 0 } , \\end{aligned} \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} u ( t ) = \\sum ^ { s } _ { j = 1 } u _ { j } \\ , \\chi _ j ( t ) , \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\frac { d ^ 2 w } { d z ^ 2 } + \\biggl ( - \\frac { 1 } { 4 } + \\frac { \\mu } { z } + \\frac { 1 - 4 \\nu ^ 2 } { 4 z ^ 2 } \\biggr ) w = 0 . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} q = { \\min } _ { \\prec } \\{ 1 , 2 , \\dots , k + n + 1 \\} \\precneqq { \\min } _ { \\prec } ( J ^ { \\circ } ) = j _ l . \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} - \\Delta u = f ( x ) u + h ( x ) u ^ { \\frac { N + 2 } { N - 2 } } , \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} S _ i ( k ) : = \\sum _ { a \\in A _ { + , i } } a ^ { k } . \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} \\mathbf { H } ( t , \\cdot ) \\times \\boldsymbol { \\nu } + \\gamma _ { 1 } \\mathbf { g } \\big ( \\mathbf { E } ( t , \\cdot ) \\times \\boldsymbol { \\nu } ) \\times \\boldsymbol { \\nu } + \\gamma _ { 2 } \\mathbf { g } \\big ( \\mathbf { E } ( t - \\tau , \\cdot ) \\times \\boldsymbol { \\nu } ) \\times \\boldsymbol { \\nu } = \\mathbf { 0 } ( 0 , \\infty ) \\times \\Gamma \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} [ n ] _ q ^ { ( i ) } & = ( ( q ^ n - 1 ) \\frac { 1 } { q - 1 } ) ^ { ( i ) } \\\\ & = \\sum _ { j = 0 } ^ { i } \\binom { i } { j } ( q ^ n - 1 ) ^ { ( i - j ) } ( \\frac { 1 } { q - 1 } ) ^ { ( j ) } \\\\ & = \\sum _ { j = 0 } ^ { i - 1 } \\binom { i } { j } n ( n - 1 ) \\cdots ( n - ( i - j - 1 ) ) q ^ { n - ( i - j ) } \\frac { j ! ( - 1 ) ^ j } { ( q - 1 ) ^ { j + 1 } } + \\frac { ( q ^ n - 1 ) i ! ( - 1 ) ^ i } { ( q - 1 ) ^ { i + 1 } } . \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} \\begin{pmatrix} x \\\\ 1 \\end{pmatrix} \\mapsto \\begin{pmatrix} \\Lambda & v \\\\ \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\end{array} & 1 \\end{pmatrix} \\begin{pmatrix} x \\\\ 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} \\int e ^ { - \\mathcal Q } \\prod _ { k = 1 } ^ m f _ k ^ { t c _ k + ( 1 - t ) d _ k } \\circ B _ k & \\le \\left ( \\int e ^ { - \\mathcal Q } \\prod _ { k = 1 } ^ m f _ k ^ { c _ k } \\circ B _ k \\right ) ^ t \\left ( \\int e ^ { - \\mathcal Q } \\prod _ { k = 1 } ^ m f _ k ^ { d _ k } \\circ B _ k \\right ) ^ { 1 - t } \\\\ & \\le ( \\sup J _ { \\mathcal Q , B , c } ) ^ t ( \\sup J _ { \\mathcal Q , B , d } ) ^ { 1 - t } \\prod _ { k = 1 } ^ m \\left ( \\int _ { H _ k } f _ k \\right ) ^ { t c _ k + ( 1 - t ) d _ k } . \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} \\prod _ { k = 0 } ^ { d ' - 1 } \\left ( 1 + \\zeta ^ { 2 k } X \\right ) = \\zeta ^ { d ' ( d ' - 1 ) } \\prod _ { k = 0 } ^ { d ' - 1 } \\left ( X + \\zeta ^ { - 2 k } \\right ) \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} - \\nabla \\cdot [ ( r I + m \\otimes m ) \\nabla p ] & = S , \\\\ \\frac { \\partial m } { \\partial t } - D ^ 2 \\Delta m - c _ 0 ^ 2 ( m \\cdot \\nabla p ) \\nabla p + | m | ^ { 2 ( \\gamma - 1 ) } m & = 0 , \\end{align*}"}
-{"id": "9974.png", "formula": "\\begin{align*} e ( G ) & \\leq \\binom { n - 2 k } { 2 } + 2 ( n - 2 k + 1 ) + 2 ( k - 1 ) \\cdot \\frac { n - k } { 2 } \\\\ [ 5 p t ] & = \\binom { n - k + 1 } { 2 } + \\frac { k ^ 2 - 3 k + 4 } { 2 } . \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} u ^ { * } ( x ) = \\limsup _ { \\delta \\to 0 ^ { + } } \\sup \\left \\{ U ^ { \\Delta x } ( m ) \\ , \\mid \\ , d ( - m \\Delta x , x ) + \\Delta x \\leq \\delta \\right \\} , \\\\ u _ { * } ( x ) = \\liminf _ { \\delta \\to 0 ^ { + } } \\inf \\left \\{ U ^ { \\Delta x } ( m ) \\ , \\mid \\ , d ( - m \\Delta x , x ) + \\Delta x \\leq \\delta \\right \\} . \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} \\inf _ { ( \\gamma , \\xi ) \\in \\Omega ( y _ 0 ) } \\langle ( k + 2 \\epsilon , M \\epsilon ) , ( \\gamma , \\xi ) \\rangle \\ = k ^ * ( y _ 0 , \\epsilon ) , \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} \\textbf { I } ^ { * } _ { C } ( \\upsilon , b , c , \\lambda , y ) = \\sum _ { k = 0 } ^ { \\infty } \\left [ \\frac { \\Theta ( k ) } { k ! } \\int _ { 0 } ^ { \\infty } ~ x ^ { \\upsilon - 1 } e ^ { - ( \\lambda b + c k ) \\sqrt { x } } \\cos ( x y ) d x \\right ] , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} 2 \\vert \\varphi ( n ) \\vert \\cdot \\vert \\varphi ( n + 1 ) \\vert a _ { n + 1 } \\sin ( \\gamma ( n + 1 ) - \\gamma ( n ) ) = \\omega . \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} { \\prod _ v } ' { \\rm { H } } ^ 2 ( k _ v , \\R _ { k ' / k } ( \\Z ) ) = { \\prod _ { v ' } } ' { \\rm { H } } ^ 2 ( k ' _ { v ' } , \\Z ) , \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} d X _ t = b ( t , X _ t ) d t + B _ t , \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\phi ( \\Pi _ { 1 , u } ) = & \\sum _ { n \\geq 1 } K ( n , 0 , u ) \\frac { x ^ n } { 2 ^ { n - 1 } n ! } \\\\ = & ~ 2 \\sum _ { n \\geq 1 } S ( n , u + 1 ) ( u + 1 ) ! \\frac { ( x / 2 ) ^ n } { n ! } \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} q ( a _ 1 , \\ldots , a _ N ) = t ( a _ 1 , \\ldots , a _ N , b _ 1 , \\ldots , b _ M ) \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} \\Pi _ { k \\ , \\boldsymbol { \\nu } } ( x , x ) _ { Z _ x } : = \\sum _ { g _ j \\in Z _ x } \\Pi _ { k \\ , \\boldsymbol { \\nu } } ( x , x ) _ j , \\Pi _ { k \\ , \\boldsymbol { \\nu } } ( x , x ) _ { G _ x \\setminus Z _ x } : = \\sum _ { g _ j \\not \\in Z _ x } \\Pi _ { k \\ , \\boldsymbol { \\nu } } ( x , x ) _ j . \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} I ( v _ n ) = \\frac { 1 } { 2 } \\norm { v _ n } ^ 2 + \\frac { \\lambda } { 4 } \\int _ { \\mathbb R ^ 3 } \\phi _ { v _ n } v _ n ^ { \\ , 2 } \\ , d x - \\int _ { \\mathbb R ^ 3 } F ( v _ n ) d x = 2 l + \\frac { \\lambda } { 4 } \\int _ { \\mathbb R ^ 3 } \\phi _ { v _ n } v _ n ^ 2 d x + o _ n ( 1 ) , \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} 0 = ( s \\circ \\alpha ) ( a ) = ( \\alpha _ E \\circ s ) ( a ) = \\alpha _ E ( x + \\nu y ) = y \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} \\phi ' ( \\cdot , \\hat { x } ) & = \\frac { 1 } { 1 + \\lambda } \\\\ \\phi ' ( \\cdot , x ^ { \\star } ) & = \\frac { 1 } { 1 + \\lambda } \\\\ \\phi ( \\cdot , x ^ { \\star } ) - \\phi ( \\cdot , \\hat { x } ) & = \\frac { x ^ { \\star } - \\hat { x } - \\kappa } { 1 + \\lambda } . \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} \\gamma : [ 0 , 1 ] \\to M , \\gamma ( 0 ) = \\varphi ( P ) , \\gamma ( 1 ) = P , \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} \\Delta = \\left ( \\begin{array} { c c } \\{ \\phi , \\phi \\} & \\{ \\phi , \\eta \\} \\\\ \\{ \\eta , \\phi \\} & \\{ \\eta , \\eta \\} \\end{array} \\right ) = \\left ( \\begin{array} { c c } 0 & - 1 \\\\ 1 & 0 \\end{array} \\right ) \\ ; , \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} \\theta _ r ^ \\alpha ( \\lambda ) \\le e ^ { - c ' r } + c _ 5 \\max _ { \\substack { u , v \\ge \\alpha \\\\ u + v = 1 - \\alpha } } \\theta _ { u r } ^ \\alpha ( \\lambda ) \\theta _ { v r } ^ \\alpha ( \\lambda ) . \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} 0 = \\varepsilon ( x ) = \\sum _ { \\substack { \\lambda + \\nu = \\mu \\\\ i , j } } \\left ( x , v _ i ^ { \\lambda } v _ j ^ { \\nu } \\right ) S ( u _ i ^ \\lambda ) K _ \\lambda u _ j ^ { \\nu } \\end{align*}"}
-{"id": "9965.png", "formula": "\\begin{align*} \\varphi _ { \\epsilon } ( z ) = \\dfrac { 1 } { ( \\sqrt { \\pi \\epsilon } ) ^ n } \\int _ { \\R ^ n } \\varphi ( t ) e ^ { - ( z - t ) ^ 2 / \\epsilon } \\ , d t . \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} \\Xi _ { ( 2 n - i + 1 , i ) } ( D ) = \\begin{array} { | c | c | c | c | c | } \\hline w ( N ^ { ( 1 ) } ; D ) & w ( N ^ { ( 2 ) } ; D ) & \\cdots & w ( N ^ { ( i - 1 ) } ; D ) & w ( N ^ { ( i ) } ; D ) \\\\ \\hline \\end{array} ^ { \\ , T } . \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 0 + } \\rho ( \\lambda ) = \\frac { ( m _ 1 \\vee m _ 2 ) - 1 } { m - 1 } . \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} \\int _ 0 ^ t g ( s , W _ s ) \\mathrm d s = \\phi ( t , W _ t ) - \\phi ( 0 , W _ 0 ) - \\int _ 0 ^ t \\nabla \\phi ^ * ( s , W _ s ) \\mathrm d W _ r , \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} \\mathcal { W } _ p ( k ) = A ^ 2 \\left ( \\mathbb { B } _ p , ( 1 - p ) ^ { k + 1 } \\right ) = \\left \\{ f \\in \\mathcal { O } ( \\mathbb { B } _ p ) | \\int _ { \\mathbb { B } _ p } \\abs { f } ^ 2 ( 1 - p ) ^ { k + 1 } d V < \\infty \\right \\} . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} \\hat { p } _ n ^ 2 ( s ) - \\hat { \\mathcal { P } } _ { 2 d } ( s ) \\hat { q } _ { n - d } ^ 2 ( s ) = 1 . \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} F ^ e ( \\Delta _ * \\mathrm { T } \\mathcal { K } _ X ^ { - 1 } ) = \\mathrm { T } ( \\Theta _ A [ d ] ) = \\Pi _ { d + 1 } ( A ) , \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} g ( x ) : = - \\min \\{ \\Phi ( x , y ) : y \\in D \\} . \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} | \\tilde { f } ( y _ 2 ) - \\tilde { f } ( \\theta _ 2 ) - \\tilde { f } ' ( \\theta _ 2 ) ( y _ 2 - \\theta _ 2 ) | \\lesssim | y _ 2 - \\theta _ 2 | ^ 2 , \\tilde { f } ( y _ 2 ) = y _ 2 ^ { - 1 } \\left [ \\chi \\left ( \\sqrt { 2 } y _ 2 \\right ) - \\chi \\left ( 8 \\sqrt { 2 } y _ 2 \\right ) \\right ] , \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} C ( \\Sigma ( u _ \\pm ) ) = C ( \\Sigma _ \\mu ( w _ \\pm ) ) = C ( \\Sigma _ \\mu ( v _ \\pm ) ) = [ 0 ] . \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} D _ { n } [ \\sigma ] = \\mathbf { G } [ \\tau ] ^ { n + 1 } n ^ { - \\beta ^ { 2 } } E [ \\tau , \\beta ] \\left ( 1 + o ( 1 ) \\right ) \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} \\lambda _ F ( n , \\tau ) = \\lim _ { t \\to \\infty } \\sum \\limits _ { \\substack { \\rho \\\\ | \\Im ( \\rho ) | \\le t } } \\left ( 1 - \\left ( \\frac { \\rho } { \\rho - \\tau } \\right ) ^ n \\right ) , \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} \\tilde L ^ { ( 2 ) } ( A ) = \\| \\mathrm { d } A ^ \\flat \\| _ g ^ 2 - 2 \\ , \\mathrm { R i c } ( A , A ) , \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} \\gamma _ + ( u ) < \\liminf _ { n \\rightarrow \\infty } \\left [ \\norm { u _ n ^ + } ^ 2 + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u _ n ^ + } u _ n ^ 2 d x - \\int _ { \\mathbb R ^ 3 } f ( u _ n ^ + ) u _ n ^ + d x \\right ] = \\liminf _ { n \\rightarrow \\infty } \\gamma _ + ( u _ n ) = 0 . \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { n - 1 } ( 1 + t q ^ j ) \\cdot & \\left ( \\sum _ { j = 0 } ^ { n - 1 } \\frac { j q ^ { j - 2 } t ( j - 1 - q ^ j t ) } { ( 1 + t q ^ j ) ^ 2 } + \\left ( \\sum _ { j = 0 } ^ { n - 1 } \\frac { j q ^ { j - 1 } t } { 1 + t q ^ j } \\right ) ^ 2 \\right ) \\\\ & = \\sum _ { r = 0 } ^ { n } q ^ { \\binom { r } { 2 } - 2 } t ^ r \\left ( \\left ( \\binom { r } { 2 } - 1 \\right ) \\binom { r } { 2 } { n \\brack r } _ q + q r ( r - 1 ) { n \\brack r } ^ { ' } _ { q } + q ^ 2 { n \\brack r } ^ { '' } _ q \\right ) . \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} \\widehat { b } ( i , n ) = \\widehat { b } ( i , n - 1 ) + q ^ { n - 1 } t ^ 2 \\widehat { b } ( i - 1 , n - 3 ) ( q , q t ) + q ^ { n - 1 } t ^ 3 \\widehat { b } ( i - 2 , n - 3 ) ( q , q t ) . \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} T U ^ { ( b ) } | _ { p } = \\left \\{ ( u _ 1 , \\ldots , u _ g ) \\in \\mathbb { C } ^ g : \\sum _ { k = 1 } ^ g a _ { i j k } u _ k = 0 : 1 \\le i \\le a , 1 \\le j \\le b \\right \\} . \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{gather*} \\big [ x _ { i } ^ + , T _ { w ( i , m ) } \\big ( x _ { i - 1 } ^ + \\big ) \\big ] = ( - 1 ) ^ { m - 1 } \\times \\begin{cases} h _ { - \\theta } ( m ) & , \\\\ h _ i ( m ) & \\end{cases} \\end{gather*}"}
-{"id": "2989.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| \\tilde { v } ^ { j _ 0 } _ n \\| ^ 2 _ { \\dot { H } ^ 1 _ c } = 0 . \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} e ^ { - \\beta t } M ^ { \\phi } ( t ) \\to \\sup _ { i \\ge 0 } e ^ { - \\beta \\tau _ i } Y _ i = : \\Delta \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} \\Psi _ { S S S } = \\max \\{ \\psi _ { 1 } ( \\tau _ { n } ) , \\psi _ { 2 } ( \\tau _ { n } ' ) \\} . \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} \\lvert ( n , 0 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i , 0 ) \\rvert _ { W _ n ( w ) } = n - i > 0 . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} D _ { \\varphi , F } ( q ) : = \\sum _ { m \\in M \\cap q F } \\varphi ( m ) \\ , . \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} \\| F \\| _ { L ^ 2 ( X , d \\mu ) } = \\| f \\| _ { \\ell ^ 2 ( \\mathbb { N } ) } . \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} L _ n v & = f _ { \\alpha , n } , & & x \\in \\R ^ 3 _ + , & & t \\in J ; \\\\ v ( 0 ) & = \\partial ^ { ( 0 , \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 3 ) } S _ { \\chi , \\sigma , \\R ^ 3 _ + , m , \\alpha _ 0 } ( 0 , f _ { n } , u _ { 0 , n } ) , & & x \\in \\R ^ 3 _ + ; \\end{aligned} \\right . \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} \\chi \\left ( \\Gamma \\backslash S _ 1 \\right ) + \\chi \\left ( \\Gamma \\backslash S _ 2 \\right ) = \\chi \\left ( \\Gamma \\backslash ( S _ 1 \\cap S _ 2 ) \\right ) + \\chi \\left ( \\Gamma \\backslash ( S _ 1 \\cup S _ 2 ) \\right ) \\ , . \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} h ( X , \\phi P Y ) + \\beta \\eta ( Y ) \\phi Q X + \\nabla _ { X } ^ { \\bot } \\phi Q Y = C h ( X , Y ) - \\alpha \\eta ( Y ) \\phi Q X + \\phi Q \\overline { \\nabla } ^ { ' } _ { X } Y , \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} \\xi _ a \\circ \\cdots \\circ \\xi _ k \\circ \\xi _ { 2 - k } \\circ \\xi _ k f ( z ) = 0 , , \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\MoveEqLeft \\norm { y - P ( x ) } = \\norm { \\frac { P ( x ) } { \\norm { P ( x ) } } - P ( x ) } = \\left | \\frac { 1 } { \\norm { P ( x ) } } - 1 \\right | \\norm { P ( x ) } \\\\ & = \\left | \\frac { 1 - \\norm { P ( x ) } } { \\norm { P ( x ) } } \\right | \\norm { P ( x ) } = 1 - \\norm { P ( x ) } < \\frac { \\delta } { 2 } . \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} A ^ { W , W } _ { t \\wedge \\tau _ M } ( g _ M ) = \\int _ 0 ^ { t \\wedge \\tau _ M } g _ M ( s , W _ s ) \\mathrm d s . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} P ( L _ 1 = k ) & = { K - 1 \\choose k } p _ 1 ^ k q _ 1 ^ { K - 1 - k } . \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} \\eta = \\frac { \\mu ^ { 2 } } { 4 } . \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} \\varphi _ k = 1 - q \\varphi _ { k + 1 } ~ . \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} \\Theta ^ 2 _ { \\alpha } = \\bigcup _ { \\nu \\geq 0 } \\bigcup _ { k \\in \\mathbb { Z } ^ n , \\pm j \\in \\mathbb { Z } _ * \\atop { | j | \\leq \\Pi _ { \\nu } } } \\mathcal { R } _ { k ( - j ) j } ^ { \\nu } ( \\alpha _ { 2 , \\nu } ) , \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} \\begin{aligned} & { \\mathfrak C } = \\big \\{ \\gamma ( A , B ) \\ , : \\ , A , B \\in \\partial \\Omega \\big \\} , \\\\ & { \\mathfrak C _ 0 } = \\big \\{ \\gamma ( A , A ) \\ , : \\ , A \\in \\partial \\Omega \\big \\} . \\end{aligned} \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ 2 } { d x ^ 2 } + q ( x ) - \\frac { ( n - 1 ) ^ 2 } { 4 } K _ 0 \\right ) w _ { \\lambda _ j } ( x ) = \\lambda _ j w _ { \\lambda _ j } ( x ) , \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} \\| E _ { \\rho , \\sigma } ( ( 0 , t ) ) \\xi _ \\sigma \\| ^ 2 & = \\| J E _ { \\sigma , \\rho } ( ( t ^ { - 1 } , + \\infty ) ) J J \\Delta _ { \\sigma , \\rho } ^ { 1 / 2 } \\xi _ \\rho \\| ^ 2 = \\| E _ { \\sigma , \\rho } ( ( t ^ { - 1 } , + \\infty ) ) \\Delta _ { \\sigma , \\rho } ^ { 1 / 2 } \\xi _ \\rho \\| ^ 2 \\\\ & = \\int _ { ( t ^ { - 1 } , + \\infty ) } s \\ , d \\| E _ { \\sigma , \\rho } ( s ) \\xi _ \\rho \\| ^ 2 = \\int _ { ( 0 , t ) } s ^ { - 1 } \\ , d \\| E _ { \\sigma , \\rho } ( s ^ { - 1 } ) \\xi _ \\rho \\| ^ 2 , \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} \\left | ( 1 - \\chi ( 4 r / r ' ) - \\chi ( 4 r ' / r ) ) ( \\mathcal { L } _ V + 1 ) ^ { - 1 } ( z , z ' ) \\right | \\lesssim \\begin{cases} d ( z , z ' ) ^ { 2 - n } , & d ( z , z ' ) \\leq 1 ; \\\\ d ( z , z ' ) ^ { - N } , & d ( z , z ' ) \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} ( \\tau _ 1 + \\tau _ 2 + \\tau _ 3 ) ^ 2 & = \\tau _ 1 ^ 2 + \\tau _ 2 ^ 2 + \\tau _ 3 ^ 2 + 2 ( \\tau _ 1 \\cdot \\tau _ 2 + \\tau _ 2 \\cdot \\tau _ 3 + \\tau _ 3 \\cdot \\tau _ 1 ) \\\\ & = - 8 - 8 - 8 + 2 ( 4 + 4 + 4 ) \\\\ & = 0 . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} A _ t ^ { W , Y } ( l ) = & \\left ( \\int _ 0 ^ t l ^ * ( r , W _ r ) \\mathrm d [ W , Y ] _ r \\right ) ^ * \\\\ = & \\left ( \\int _ 0 ^ t l ^ * ( r , W _ r ) \\nabla \\gamma ( r , W _ r ) \\mathrm d r \\right ) ^ * \\\\ = & \\int _ 0 ^ t \\nabla \\gamma ^ * ( r , W _ r ) l ( r , W _ r ) \\mathrm d r \\\\ = & A _ t ^ { W , W } ( \\nabla \\gamma ^ * \\ , l ) . \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} \\theta ( q , x ) = 1 + q x \\theta ( q , q x ) ~ . \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} \\norm { f } _ { L ^ 2 ( \\mathbb { R } , \\omega _ { a , b } ) } ^ 2 : = \\int _ { \\mathbb { R } } | f ( t ) | ^ 2 \\omega _ { a , b } ( t ) \\d t < \\infty . \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} \\| - \\nabla _ x \\log \\rho _ { 0 | t } & ( x \\ , | \\ , y ) \\| ^ 2 = \\| - \\nabla _ x \\log \\rho _ 0 ( x ) \\| ^ 2 \\\\ & + \\frac { \\alpha ^ 2 e ^ { - 2 \\alpha t } } { ( 1 - e ^ { - 2 \\alpha t } ) ^ 2 } \\| e ^ { - \\alpha t } x - y \\| ^ 2 \\\\ & + \\frac { 2 \\alpha e ^ { - \\alpha t } } { 1 - e ^ { - 2 \\alpha t } } \\langle - \\nabla _ x \\log \\rho _ 0 ( x ) , \\ , e ^ { - \\alpha t } x - y \\rangle . \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } ( y p _ { 0 } ( y , t ) ) + \\frac { 1 } { 2 } \\langle x ^ { 2 } \\rangle \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } p _ { 0 } ( y , t ) - \\frac { \\partial } { \\partial t } p _ { 0 } ( y , t ) = \\frac { \\partial } { \\partial y } \\int d x \\ x a ( x , y , t ) \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{gather*} F _ 0 ( z ) = { } _ 4 F _ 3 \\biggl ( \\begin{matrix} r , \\ , 1 - r , \\ , t , \\ , 1 - t \\\\ 1 , \\ , 1 , \\ , 1 \\end{matrix} \\biggm | z \\biggr ) \\end{gather*}"}
-{"id": "4075.png", "formula": "\\begin{align*} \\begin{array} { l l l } p _ { n + 1 } = a _ { n + 1 } p _ n + p _ { n - 1 } & \\hat { p } _ n = a _ { n + 1 } \\hat { p } _ { n - 1 } + \\hat { p } _ { n - 2 } \\\\ q _ { n + 1 } = a _ { n + 1 } q _ n + q _ { n - 1 } & \\hat { q } _ n = a _ { n + 1 } \\hat { q } _ { n - 1 } + \\hat { q } _ { n - 2 } \\end{array} \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} \\phi _ { j , \\lambda } ( x ) = \\lambda ^ { \\dim H _ j } \\prod _ { l = 1 } ^ { \\dim H _ j } \\phi ( \\lambda x _ l ) . \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} d _ s : = s ^ 2 - s \\lfloor s ^ { 1 / 2 } \\rfloor - \\lfloor s / 4 \\rfloor , s = 2 , 3 , \\ldots . \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} \\exp _ { \\phi } ( \\jmath ^ { \\phi } ) = \\sum _ { k \\geq 0 } Z ^ { \\phi } _ k w ^ { q ^ k } . \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} \\dfrac { d X ^ \\nu { } _ \\beta ( s ) } { d s } + X ^ \\alpha { } _ \\beta ( s ) \\ , \\Gamma ^ \\nu { } _ { \\alpha \\mu } ( \\gamma ( s ) ) \\dfrac { d \\gamma ^ \\mu ( s ) } { d s } = 0 \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} \\psi _ { \\alpha } ^ { 1 } = \\bold { 1 } ( { t _ { 1 } > u _ { \\alpha } } ) . \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} \\nu _ { \\alpha , 1 } ( n ) : = \\frac { \\mathrm e ^ { f ' ( \\alpha ) n } } { Z _ 1 ( f ' ( \\alpha ) ) } \\nu _ { * , 1 } ( n ) \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow \\infty } \\frac { \\left \\| \\frac { f } { | x | ^ { \\frac { \\beta } { q } } } \\right \\| _ { L ^ { q } ( B ( x _ { 0 } , r ) ) } } { ( \\Gamma ( q / p ' + 2 ) ) ^ { \\frac { 1 } { q } } \\| f \\| _ { L ^ { p } _ { Q / p } ( B ( x _ { 0 } , r ) ) } } \\leq \\lim _ { q \\rightarrow \\infty } C _ { \\varepsilon } ^ { \\frac { 1 } { q } } ( \\widehat { \\alpha } - \\varepsilon ) ^ { - \\frac { 1 } { p ' } } = ( \\widehat { \\alpha } - \\varepsilon ) ^ { - \\frac { 1 } { p ' } } . \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} W ( t ) = \\begin{cases} \\{ e \\} & ( 0 \\leq t < 1 ) , \\\\ \\{ e , s _ 1 \\} & ( t = 1 ) . \\end{cases} \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} \\sum _ { i \\in I _ 1 } \\sum _ { j \\in I _ l } \\lambda _ { i j } = & ~ \\lambda _ { 1 , ( l - 1 ) ( p + q ) - p + 2 } + 2 \\lambda _ { 1 , ( l - 1 ) ( p + q ) - p + 3 } + \\cdots + ( p - 1 ) \\lambda _ { 1 , ( l - 1 ) ( p + q ) } \\\\ & + p ~ \\lambda _ { 1 , ( l - 1 ) ( p + q ) + 1 } + ( p - 1 ) \\lambda _ { 1 , ( l - 1 ) ( p + q ) + 2 } + \\cdots + \\lambda _ { 1 , ( l - 1 ) ( p + q ) + p } . \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} g h : = \\lim _ { j \\to \\infty } S ^ j g S ^ j h , \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{align*} B = \\begin{pmatrix} 0 & \\nu _ 3 & - \\nu _ 2 & 0 & 0 & 0 \\\\ - \\nu _ 3 & 0 & \\nu _ 1 & 0 & 0 & 0 \\\\ \\nu _ 2 & - \\nu _ 1 & 0 & 0 & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} J ( g _ { A _ 1 } , \\ldots , g _ { A _ m } ) = \\begin{cases} \\left ( \\frac { \\det ( Q + \\sum _ { k = 1 } ^ m c _ k B _ k ^ \\ast A _ k B _ k ) } { \\prod _ { k = 1 } ^ m ( \\det A _ k ) ^ { c _ k } } \\right ) ^ { - 1 / 2 } & \\\\ \\infty & \\end{cases} \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} f '' ( h _ 0 , h _ 1 ) ( \\nu ) & = \\sum _ { i , j } \\sum _ { s , t } \\langle f ( h _ s , h _ t ) , ( \\nu | _ { h _ 0 Y _ j \\cap Y _ i } L ) | _ { g _ i X _ { g _ j , h _ t } \\cap X _ { g _ i , h _ s } } \\pi \\rangle \\\\ & = \\sum _ { i , j } \\sum _ { s , t } \\langle f ( h _ s , h _ t ) , \\nu | _ { h _ 0 Y _ j \\cap Y _ i \\cap \\phi ( g _ i X _ { g _ j , h _ t } \\cap X _ { g _ i , h _ s } ) } \\rangle . \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} R = t _ 0 ^ { \\rho } 0 < \\rho \\ll 1 . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} | G ( v , Q ) | = \\mathcal { O } ( v ^ 2 Q ^ { p - 1 } + v ^ { p + 1 } ) . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} \\begin{aligned} K ( x , y ) & = \\int _ { 0 } ^ { \\infty } \\zeta _ { 1 } ( n + y + z ) \\zeta _ { 2 } ( - n - x - z ) d z \\\\ K _ { e } ( x , y ) & = W ( x , y ) - K ( x , y ) \\\\ g _ { k } & = K ( x , - k ) \\\\ g _ { k , e } & = K _ { e } ( x , - k ) \\end{aligned} \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} \\int _ { \\Omega \\setminus \\omega } \\phi ( | \\nabla \\Q _ { 0 } | ) ~ d V & \\stackrel { \\eqref { e q : s _ p h i } } { = } \\lim _ { j \\to + \\infty } \\int _ { \\Omega } \\phi ( | \\nabla \\Q _ { L _ { k _ j } } | ) ~ d V - \\int _ \\omega \\phi ( | \\nabla \\Q _ { 0 } | ) ~ d V \\\\ & > \\lim _ { j \\to + \\infty } \\int _ { \\Omega \\setminus \\omega } \\phi ( | \\nabla \\Q _ { L _ { k _ j } } | ) ~ d V , \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} \\prod \\limits _ { j = 1 } ^ { k } x _ j ^ \\downarrow \\le \\prod \\limits _ { j = 1 } ^ { k } y _ j ^ \\downarrow . \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} u \\nu ( v \\otimes u ' ) = \\mu ( u \\otimes v ) u ' v \\mu ( u \\otimes v ' ) = \\nu ( v \\otimes u ) v ' . \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} G _ i ( X , Y ) = H _ i ( X + Y ) - H _ i ( X ) - H _ i ( Y ) \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} x _ 0 ^ { a _ 0 - 1 } y _ 0 u _ 1 x _ 0 ^ { - ( a _ 0 - 1 ) } & = x _ 0 ^ { - 1 } y _ 0 v _ 1 x _ 0 = x _ 0 ^ { - 1 } y _ 0 x _ 0 v _ 2 = x _ 0 ^ { - 1 } y _ 0 x _ 0 x _ 1 x _ 1 ^ { - 1 } v _ 2 \\\\ & = x _ 0 ^ { - 1 } y _ 0 ^ 3 x _ 1 ^ { - 1 } v _ 2 = x _ 1 y _ 0 v _ 3 x _ 1 ^ { - 1 } = y _ 0 x _ 2 v _ 3 x _ 1 ^ { - 1 } \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} \\sum _ { j \\in N ( i ) } \\widetilde Q _ { i j } = \\sum _ { j \\in N ( i ) \\cap \\widetilde { \\mathcal { I } } } \\widetilde Q _ { i j } + \\sum _ { j \\in N ( i ) \\setminus \\widetilde { \\mathcal { I } } } Q _ { i j } . \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} q + a = q + _ o a = a + _ o q . \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} S _ m & : = S \\cap E _ m \\subset E _ m m \\geq 0 , \\\\ S _ { \\infty } & : = S \\cap E _ { \\infty } \\subset E _ { \\infty } . \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} T _ \\theta x = x + \\theta . \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} I _ z = \\bigcup _ { i \\in J } \\left ] a _ i , b _ i \\right [ , \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } ( | \\nabla u _ n | ^ 2 + u _ n ^ 2 ) + \\int _ { \\R ^ 3 } \\rho \\phi _ { u _ n } u _ n ^ 2 - \\mu _ n \\int _ { \\R ^ 3 } u _ n ^ { p + 1 } = 0 . \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} [ \\dot { \\Delta } _ j , \\ , u \\cdot \\nabla ] v = & [ \\dot { \\Delta } _ j , \\ , \\dot { T } _ { u ^ i } ] \\partial _ i v + \\dot { \\Delta } _ j \\big ( \\dot { T } _ { \\partial _ i v } u ^ i \\big ) + \\dot { \\Delta } _ j \\big ( \\dot { R } ( u ^ i , \\ , \\partial _ i v ) \\big ) - \\dot { T } _ { \\partial _ i \\dot { \\Delta } _ j v } u ^ i - \\dot { R } \\big ( u ^ i , \\ , \\partial _ i \\dot { \\Delta } _ j v \\big ) \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} S ( \\phi U , \\phi V ) = S ( U , V ) + ( n - 1 ) \\eta ( U ) \\eta ( V ) \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} \\mathrm { s p } _ S ( T ^ { - 1 } , [ \\phi w ' ] ) < a ( I ) = \\mathrm { s p } _ S ( T ^ { - 1 } , [ \\phi v ] ) . \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} | D u ( x ) - D u ( y ) | & \\lesssim \\sup _ { y _ 0 \\in \\Omega _ { R / 4 } ( x _ 0 ) } | D u ( y _ 0 ) - \\Theta _ { y _ 0 , \\rho } | + \\Phi ( x , \\rho ) + \\Phi ( y , \\rho ) \\\\ & \\lesssim \\sup _ { y _ 0 \\in \\Omega _ { R / 4 } ( x _ 0 ) } \\Bigg ( \\sum _ { j = 0 } ^ \\infty \\Phi ( y _ 0 , \\kappa ^ j \\rho ) + \\Phi ( y _ 0 , \\rho ) \\Bigg ) \\\\ & \\lesssim \\sup _ { y _ 0 \\in \\Omega _ { R / 4 } ( x _ 0 ) } \\sum _ { j = 0 } ^ \\infty \\Phi ( y _ 0 , \\kappa ^ j \\rho ) \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} \\left | ( 1 - \\chi ( 4 r / r ' ) - \\chi ( 4 r ' / r ) ) ( \\mathcal { L } _ V + 1 ) ^ { - 1 } ( \\lambda z , \\lambda z ' ) \\right | \\lesssim \\begin{cases} \\lambda ^ { 2 - n } d ( z , z ' ) ^ { 2 - n } , & d ( z , z ' ) \\leq 1 / \\lambda ; \\\\ \\lambda ^ { - N } d ( z , z ' ) ^ { - N } , & d ( z , z ' ) \\geq 1 / \\lambda . \\end{cases} \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} S _ 0 = \\left \\{ \\mathbf { Q } \\in M ^ { 3 \\times 3 } \\colon Q _ { i j } = Q _ { j i } ; \\ Q _ { i i } = 0 \\right \\} \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} G _ { k , - m , 0 } ( z ) & = P _ { k , - m } ( z , k / 2 ) \\\\ & = \\varphi _ { k , - m } ( z , k / 2 ) + \\sum _ { ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) } b _ { k , - m } ( n , k / 2 ) \\mathcal { W } _ { k , n } ( y , k / 2 ) e ^ { 2 \\pi i n x } \\\\ & = \\frac { 1 } { \\Gamma ( k ) } \\biggl \\{ q ^ { - m } + \\sum _ { ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) , n > 0 } n ^ { k - 1 } b _ { k , - m } ( n , k / 2 ) q ^ n \\biggr \\} . \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n + 1 } f ( i ) \\equiv \\sum _ { i = 0 } ^ { n - 1 } k _ i ( f ) \\mod d . \\end{align*}"}
-{"id": "9645.png", "formula": "\\begin{align*} r _ 0 - r _ 1 \\ge \\sum _ { l = 0 } ^ m \\dim R _ { \\lambda ( l ) ^ T } & = \\sum _ { l = 0 } ^ m \\frac { 2 m - 2 l + 1 } { 2 m + 1 } \\binom { 2 m + 1 } { l } \\\\ & = \\sum _ { l = 0 } ^ m \\binom { 2 m + 1 } { l } - 2 \\sum _ { l = 0 } ^ m \\binom { 2 m } { l - 1 } . \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} Q _ { \\nu , \\ell } ( R , M ) = \\int _ { R } ^ { 2 R } \\int _ { 0 } ^ \\infty \\big | ( r \\rho ) ^ { - \\frac { n - 2 } 2 } J _ { \\nu } ( r \\rho ) b _ { \\nu , \\ell } ( M \\rho ) \\chi ( \\rho ) \\big | ^ 2 d \\rho d r . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} h ( \\boldsymbol \\mu , D ) = \\int \\left ( \\frac { 1 } { 2 m } \\frac { | \\boldsymbol \\mu | ^ 2 } { D } + \\frac { \\hbar ^ 2 } { 8 m } \\frac { | \\nabla D | ^ 2 } { D } + D V ( \\boldsymbol { x } ) \\right ) \\ , ^ 3 x \\ , . \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} \\tau _ u ( \\omega _ 1 \\wedge \\omega _ 2 ) ( \\xi ) = \\omega _ 2 ( \\xi ) \\omega _ 1 ^ { \\sharp _ u } - \\omega _ 1 ( \\xi ) \\omega _ 2 ^ { \\sharp _ u } . \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} A _ { N } ^ { \\ast } ( w _ { 1 , r } ) & = A _ { N } ^ { \\ast } ( a _ { 1 } , \\{ 1 \\} _ { b _ { 1 } - 1 } , w _ { 2 , r } ) \\\\ & = \\sum _ { i _ 1 = 1 } ^ { N } \\frac { 1 } { i _ 1 } \\dots \\sum _ { i _ { k - 1 } = 1 } ^ { i _ { k - 2 } } \\frac 1 { i _ { k - 1 } } \\sum _ { i _ { k } = 1 } ^ { i _ { k - 1 } } \\frac 1 { i _ { k } ^ { b _ 1 + 1 } } A _ { i _ k } ^ { \\ast } ( a _ 2 , \\{ 1 \\} _ { b _ 2 - 1 } , w _ { 3 , r } ) . \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} I ( \\mathbf { x } ; \\mathbf { z } ) = \\int _ { \\mathcal { Z } } \\int _ { \\mathcal { X } } p ( x , z ) \\log \\frac { p ( x , z ) } { p ( x ) p ( z ) } d x d z , \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} \\begin{cases} u _ t = - ( - \\Delta ) ^ { s } u + u ^ { \\frac { n + 2 s } { n - 2 s } } & \\Omega \\times ( t _ 0 , \\infty ) , \\\\ u = 0 & ( \\mathbb { R } ^ n \\setminus \\Omega ) \\times ( t _ 0 , \\infty ) , \\end{cases} \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} \\min _ { u _ V \\in ` R _ + ^ V } \\varphi ( u _ V ) & = \\min _ { u _ V \\in ` R _ + ^ V } \\max _ { e \\in E } \\frac { u ( ` x ( e ) ) } { u ( V ) } \\\\ & = \\min _ { u _ V \\in ` R _ + ^ V : u ( \\xi ( e ) ) \\leq 1 , \\forall e \\in E } \\frac { 1 } { u ( V ) } \\\\ & = \\frac { 1 } { ` t } \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} m U ( z _ 0 ) = m + \\lambda - 1 - ( \\sqrt { m _ 1 m _ 2 } - 1 ) z _ 0 ; \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} \\frac { B ( \\tau ) } { \\tau } = \\frac { D ( t ) } { t } , 0 < t < 1 . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} e ^ { 2 \\pi c t } \\widehat { G ^ { \\epsilon _ j } _ c } ( t ) = e ^ { 2 \\pi y t } \\widehat { G ^ { \\epsilon _ j } _ y } ( t ) , \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} F _ A ^ * ( \\overline { \\omega } ) = \\overline { b X ^ { \\frac { m + 1 - p } { p } } d X } \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} \\varphi ' ( x ) = - 4 ( 1 - \\lambda ) x ( x - 1 ) \\Big ( x - \\frac { \\lambda + 2 } { 2 ( \\lambda - 1 ) } \\Big ) , \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} \\overline { \\widetilde { S } ^ { R , R } _ { X , Y } } = \\frac { \\overline { \\widetilde { S } ^ { R , R } _ { 1 , Y } } } { \\widetilde { S } ^ { R , R } _ { 1 , Y } } \\widetilde { S } ^ { R , R } _ { X ^ * , Y } = \\frac { \\overline { \\widetilde { S } ^ { R , R } _ { 1 , Y } } } { \\widetilde { S } ^ { R , R } _ { 1 , \\bar { Y } } } \\widetilde { S } ^ { R , R } _ { X , \\bar { Y } } , \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} \\Psi ^ { \\# } ( Y , \\pi ) = \\left \\{ f \\in \\Psi ( Y , \\pi ) \\ \\middle \\vert \\ \\sum _ { y \\in Y } f ( y ) \\equiv n \\binom { d } { 2 } \\mod d \\right \\} . \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} L _ \\mathcal E ( h ) = \\int _ 0 ^ T \\int _ \\Lambda \\chi ( \\rho _ t ) v _ t \\cdot h _ t \\ ; \\ ! \\mathrm d u \\ ; \\ ! \\mathrm d t , L _ { \\mathcal E ^ \\star } ( h ) = \\int _ 0 ^ T \\int _ \\Lambda \\chi ( \\rho _ t ) w _ t \\cdot h _ t \\ ; \\ ! \\mathrm d u \\ ; \\ ! \\mathrm d t . \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} \\Psi ( x ) = \\textbf { T r } ( \\psi ( x ) ) = 2 ( \\psi ( x ) ) _ 1 = \\lambda _ 1 ( \\psi ( x ) ) + \\frac { 1 } { 2 } [ \\lambda _ 2 ( \\psi ( x ) ) + \\lambda _ 4 ( \\psi ( x ) ) ] \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} \\epsilon _ 0 \\leq ( \\frac { \\alpha _ 0 \\gamma _ 0 } { 8 0 } ) ^ { \\frac { 1 } { 1 - 2 \\beta ' } } \\prod _ { \\mu = 0 } ^ { \\infty } ( 2 ^ { \\mu } B _ { \\mu } ) ^ { - \\frac { 1 } { 3 \\kappa ^ { \\mu + 1 } } } , \\quad \\alpha _ 0 \\leq \\min \\{ \\frac { m _ 0 } { 1 0 } , \\frac { M _ { 3 , 0 } } { 2 } \\} . \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} \\mathfrak { u } = \\mathfrak { k } \\oplus \\mathfrak { m } \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} \\kappa _ n ( \\gamma ) = \\kappa ^ A _ n ( \\gamma ) + \\kappa ^ { - S } _ n ( \\gamma ) . \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ r s _ i \\ll \\det \\L \\le \\prod _ { i = 1 } ^ r s _ i . \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} c _ a ( x _ 1 , \\dots , x _ N ) = \\sum ^ N _ { i = 1 } \\sum ^ N _ { j = i + 1 } \\vert x _ j - x _ i \\vert ^ 2 , c _ w ( x _ 1 , \\dots , x _ N ) = - \\sum ^ N _ { i = 1 } \\sum ^ N _ { j = i + 1 } \\vert x _ j - x _ i \\vert ^ 2 . \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} \\kappa _ { 0 1 0 } = \\sum _ { p \\geq 1 } \\prod _ { k = 1 } ^ { p - 1 } q _ { 0 ^ k 1 0 } ( 0 ) { \\rm ~ ~ a n d ~ ~ } \\kappa _ { 1 1 0 } = \\sum _ { p \\geq 1 } \\prod _ { j = 0 } ^ { p - 1 } q _ { 0 ^ j 1 ^ q 0 } ( 0 ) \\prod _ { k = 2 } ^ { q - 1 } q _ { 0 ^ k 1 ^ q 0 } ( 1 ) . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} \\epsilon ( [ m _ t ^ { - 1 } ] ) & = \\epsilon ( [ g _ t ^ { - 1 } m ^ { - 1 } g _ t ] ) \\\\ & = \\epsilon ( [ g _ t ^ { - 1 } m m ^ { - 1 } g _ t ] ) + \\epsilon ( [ g _ t ^ { - 1 } ] ) \\epsilon ( [ m ^ { - 1 } g _ t ] ) \\qquad [ \\textrm { S k e i n r e l a t i o n } ] \\\\ & = \\epsilon ( [ e ] ) + \\epsilon ( [ g _ t ^ { - 1 } ] ) \\epsilon ( [ m ^ { - 1 } g _ t ] ) = 0 . \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} \\{ ( x , y , 0 ) \\in \\R ^ 3 \\ , : \\ , x ^ 2 + y ^ 2 \\le r ^ 2 \\} \\cup \\{ ( x , y , z ) \\in \\R ^ 3 \\ , : \\ , x ^ 2 + y ^ 2 = r ^ 2 \\ \\ z \\ge 0 \\} \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} R ^ t ( x ) = - \\frac { 1 } { \\chi } \\sum \\limits _ { l = 1 } ^ { t - 1 } \\sum \\limits _ { p _ 1 + \\ldots + p _ s = l } \\beta _ { l , p _ 1 , \\ldots , p _ s } R ^ { t - l } ( \\ldots ( ( x y ^ { p _ 1 } ) y ^ { p _ 2 } ) \\ldots y ^ { p _ s } ) , y = R ( 1 ) , \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\sum _ { \\substack { { r _ 1 , r _ 2 = 1 } \\\\ r _ 1 \\ne r _ 2 } } ^ { n } \\frac { S _ { r _ 1 r _ 1 r _ 2 } } { ( r _ 1 + k + 1 ) ^ 2 ( r _ 2 - r _ 1 ) } = \\sum _ { \\substack { { r _ 1 , r _ 2 = 1 } \\\\ r _ 1 \\ne r _ 2 } } ^ { n } \\frac { S _ { r _ 1 r _ 1 r _ 2 } } { r _ 2 - r _ 1 } \\left ( \\zeta ( 2 ) - H ^ { ( 2 ) } _ { r _ 1 } \\right ) . \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} ( m _ { n - 1 } ( a _ 1 , \\ldots , a _ { n - 1 } ) , a _ { n } ) = ( m _ { n - 1 } ( a _ 2 , \\ldots , a _ { n } ) , a _ 1 ) . \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} h ( X , \\phi Y ) = C h ( X , Y ) + \\phi Q \\overline { \\nabla } ^ { ' } _ { X } Y . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ r e ^ { d _ { j , s } \\varpi _ s } ( b _ { k ( j , s ) , m ( j , s ) } ) = \\prod _ { u = 1 } ^ x e ^ { ( d _ { j _ u , s } + \\dots + d _ { j _ { u + 1 } - 1 , s } ) \\varpi _ s } ( b _ { j _ u , m ( j _ u , s ) } ) . \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} b _ { 0 } ^ { ( \\alpha , \\beta ) } = \\frac { \\beta - \\alpha } { \\alpha + \\beta + 2 } . \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{gather*} \\varphi _ { v ^ { - 1 } } \\varphi _ { v ^ { - 1 } } ^ { v ^ { - 2 } } = \\frac { \\mu ^ l \\big ( v ^ { - 1 } \\big ) } { \\mu ^ l ( v ) } \\varepsilon . \\end{gather*}"}
-{"id": "1779.png", "formula": "\\begin{align*} \\Psi _ { b } ( x _ { 1 } , x _ 2 ; u , g ) : = u \\cdot \\psi \\left ( \\widetilde { \\mu } _ { g ^ { - 1 } } ( x _ { 1 } ) , x _ { 2 } \\right ) + 2 \\ , b \\cdot \\theta _ G ( g ) . \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} l = [ \\mathbb { K } : \\mathbb { F } _ p ] = s \\prod _ { i = 1 } ^ n p _ i . \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} j K ( n , i , j ) = ( n - i - j ) K ( n , i , j - 1 ) + ( i + 1 ) K ( n , i + 1 , j - 1 ) . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} \\Delta K _ 2 ( \\varphi , p ) = - \\int \\left [ \\frac { \\partial p } { \\partial y ^ \\alpha } ( \\boldsymbol \\varphi ( x ) ) \\right ] \\bigl [ \\Delta \\varphi ^ \\alpha ( x ) \\bigr ] \\ , \\bigl [ \\rho _ { \\boldsymbol \\varphi } ( x ) \\bigr ] \\ , d x \\ , . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} \\| u ( t _ n ) - u ^ n \\| + \\gamma ^ { \\frac 1 2 } \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\sigma ( t _ { k - \\theta } ) - \\sigma ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } + \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\nabla u ( t _ { k - \\theta } ) - u ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } \\leq C \\Delta t ^ 2 . \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} u ^ { \\theta } ( x , t ) = \\sup \\left \\{ u ( x , s ) - \\frac { ( t - s ) ^ { 2 } } { 2 \\theta } \\ , \\mid \\ , s \\in [ 0 , T ] \\right \\} . \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} k ( j , s ) & \\coloneqq \\max \\left ( \\{ k \\mid 1 \\leq k \\leq j , \\ i _ { k , l } = s 1 \\leq l \\leq N _ k \\} \\cup \\{ 0 \\} \\right ) , \\\\ m ( j , s ) & \\coloneqq \\max \\{ q \\mid i _ { k ( j , s ) , q } = s \\} \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} W _ { \\psi , \\varphi } = C _ U W _ { \\widetilde { \\psi } , \\widetilde { \\varphi } } C _ V W _ { \\widetilde { \\psi } , \\widetilde { \\varphi } } = C _ { U ^ * } W _ { \\psi , \\varphi } C _ { V ^ * } . \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} e ( u , v ) = \\sum _ { P \\colon u \\to v } \\prod _ { e \\in P } w ( e ) , \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} \\sum _ { p } \\big ( s _ { p j } ^ { k l } s _ { i p } ^ { s t } - s _ { i j } ^ { k p } s _ { p l } ^ { s t } + s _ { k l } ^ { s p } s _ { i j } ^ { p t } \\big ) = 0 . \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 v '' _ { n , + , a } - r ( x ) v _ { n , + , a } = - 1 , \\ x \\in ( - n , a ) ; \\\\ & v _ { n , + , a } ( a ) = v _ { n , + , a } ( - n ) = 0 , \\end{aligned} \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ t \\sum _ { j = 0 } ^ m c _ { i , j } \\binom { f } { j } \\mu _ i ^ { f - j } \\right | = \\max _ { \\substack { 1 \\leq i \\leq t \\\\ 0 \\leq j \\leq m } } \\{ | c _ { i , j } | \\} \\neq 0 . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} g = 1 + \\frac { 1 } { 2 } ( \\beta ^ 2 + K _ S \\cdot \\beta ) . \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\begin{bmatrix} A & b \\\\ c & d \\end{bmatrix} \\right ) : = \\begin{bmatrix} \\mathbb { P } _ a ( A ) & b \\\\ 0 _ { 1 \\times 3 } & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| x ^ k - q \\| = \\tau . \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} Z \\lambda ( a ) Z ^ { - 1 } & = X a X ^ { - 1 } \\oplus \\Phi ( a ) = \\rho ( \\Phi ( a ) ) \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} \\frac { p _ n } { q _ n } = \\frac { 1 } { a _ 1 + [ a _ 2 , \\ldots , a _ n ] } = \\frac { 1 } { a _ 1 + \\frac { \\hat { p } _ { n - 1 } } { \\hat { q } _ { n - 1 } } } = \\frac { \\hat { q } _ { n - 1 } } { a _ 1 \\hat { q } _ { n - 1 } + \\hat { p } _ { n - 1 } } \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} \\begin{aligned} T _ a T _ { a ^ 2 } & \\rightarrow T _ { a ^ 2 } T _ a \\\\ T _ b T _ { b ^ 2 } & \\rightarrow T _ { b ^ 2 } T _ b \\end{aligned} \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} \\nu ( 1 + \\varpi ^ l x + \\varpi ^ l y \\tau ) \\nu ' ( 1 + \\varpi ^ l x + \\varpi ^ l y \\sigma ( \\tau ) ) = \\psi ( \\varpi ^ l { \\tau } ( x + y { \\tau } ) + \\varpi ^ l \\sigma ( { \\tau } ) ( x + y \\sigma ( { \\tau } ) ) ) , \\forall x , y \\in { \\mathcal O } _ r . \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} \\bigcup _ { k \\geq k _ 0 } E _ { \\delta k } ^ c = \\bigcup _ { k \\geq k _ 0 } \\bigcup _ { - N _ k < j < N _ k } B \\big ( j \\theta , { 1 \\over N _ k ^ \\tau } \\big ) \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} \\mathbb E [ X _ { n } ^ { 2 } ] \\ , = \\ , \\sum _ { k = 0 } ^ { n - 1 } \\sum _ { \\ell = 0 } ^ { k } { k \\choose \\ell } ^ { 2 } u ^ { 2 \\ell } ( 1 - a ) ^ { 2 \\ell } a ^ { 2 ( k - \\ell ) } \\ , = \\ , \\sum _ { k = 0 } ^ { n - 1 } u ^ { k } ( 1 - a ) ^ { k } \\ , _ { 2 } F _ { 1 } \\Big ( - k , - k , 1 ; \\ , \\frac { a ^ { 2 } } { \\ , ( 1 - a ) ^ { 2 } \\ , } \\Big ) \\ , , \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} H ( \\sigma , \\sigma ' ) : = R ( u , J u , v , J v ) , \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} z ( v _ + - v _ - ) = n \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} D _ \\alpha : = \\left \\{ ( x _ 1 , \\ldots , x _ N ) \\in X ^ N ~ | ~ i , j d ( x _ i , x _ j ) < \\alpha \\right \\} . \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} Q = \\begin{bmatrix} J _ { c _ 1 } ( \\lambda _ 1 ) \\\\ & \\ddots \\\\ & & J _ { c _ p } ( \\lambda _ p ) \\\\ & & & \\check { I } _ { d _ 1 - 1 } \\\\ & & & & \\ddots \\\\ & & & & & \\check { I } _ { d _ q - 1 } \\\\ & & 0 _ { ( n - m - q ) \\times m } \\end{bmatrix} , P = \\begin{bmatrix} I _ { c _ 1 } \\\\ & \\ddots \\\\ & & I _ { c _ p } \\\\ & & & \\hat { I } _ { d _ 1 - 1 } \\\\ & & & & \\ddots \\\\ & & & & & \\hat { I } _ { d _ q - 1 } \\\\ & & & 0 _ { ( n - m - q ) \\times m } \\end{bmatrix} . \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{gather*} { f } ^ * _ { t , s } ( x , y ) : = t x y + s x ^ { 1 0 } y ^ { 7 } + x ^ { 1 1 } y ^ { 8 } + y ^ { 3 } + 1 . \\end{gather*}"}
-{"id": "4010.png", "formula": "\\begin{align*} \\Omega : = \\{ ( s , p , q ) : | s | \\leq s _ 0 , \\ ; p _ 0 \\leq p \\leq p _ 1 , \\ ; q _ 0 \\leq q < \\infty \\} . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} \\mu ^ { * } = \\sum _ { j = 1 } ^ { q } \\beta _ { j } - \\sum _ { i = 1 } ^ { p } \\alpha _ { i } + \\left ( \\frac { p - q } { 2 } \\right ) , \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} L ' _ { i j } = L _ { i j } \\Pi ( G ' ) \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} \\mathbf { x } ^ { k + 1 } = ( \\mathbf { W } \\otimes \\mathbf { I } _ p ) \\mathbf { x } ^ k - \\xi \\nabla f ( \\mathbf { x } ^ { k + 1 } ) + \\sum _ { t = 0 } ^ { k } ( \\mathbf { W } - \\tilde { \\mathbf { W } } ) \\otimes \\mathbf { I } _ p \\mathbf { x } ^ { t } , \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} f _ { j , n } \\ge \\varepsilon \\phi _ { j , 1 } \\ast \\phi _ { j , n } = \\varepsilon \\phi _ { j , \\frac { n } { n + 1 } } \\ge \\varepsilon 2 ^ { - \\dim H _ j } \\phi _ { j , 1 } \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\beta _ { i , i + j } ( I ) = \\sum _ { u \\in G ( I ) _ j } \\binom { \\max ( \\tau ^ t ( u ) ) - 1 } { i } . \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} N ( t ) \\geq N _ { i n } \\exp ( - T \\| g \\| _ { C ( [ 0 , T ] ) } ) = : \\delta ( T ) > 0 . \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} \\sigma ^ { \\alpha \\dot b a } \\ , \\partial _ { x ^ \\alpha } \\eta _ { \\dot b } = 0 . \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} H _ n ( q , t ) = \\frac { H _ { n - 2 } ( q , q t ) + t H _ { n - 3 } ( q , q ^ 2 t ) } { 1 - q ^ { n - 1 } t } \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} \\sum _ { j = 1 , 2 , 3 } | | \\mathbf { T } ^ j | | _ { \\mathcal { M } _ b \\to L ^ { 1 , \\infty } } \\lesssim c _ 1 + c _ 2 , \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} c _ \\lambda ( \\{ i - 1 , i \\} ) & = \\lambda ^ { - \\left ( \\vert i \\vert \\wedge \\vert i - 1 \\vert \\right ) } , i \\in [ - ( n - 1 ) , \\ , n ] \\cap \\Z ^ 1 , \\\\ c _ \\lambda ( \\{ z _ n , n \\} ) & = c _ \\lambda ( \\{ z _ n , - n \\} ) = \\lambda ^ { - n } . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} X = \\mathrm { T o t } _ C ( L _ 1 \\oplus L _ 2 ) \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} ( i _ k ) _ { k = 0 } ^ { \\ell - 1 } = ( \\underbrace { n , \\dotsc , n } _ { c _ n } , \\dotsc , \\underbrace { 2 , \\dotsc , 2 } _ { c _ 2 } , \\underbrace { 1 , \\dotsc , 1 } _ { c _ 1 } ) , \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} \\mathbf { H } = \\begin{pmatrix} \\mathbf { H } _ { 1 } & \\mathbf { H } _ { 2 } & \\dots & \\mathbf { H } _ { n } \\end{pmatrix} \\in \\mathbb { B } ^ { ( n - k ) \\ell \\times n \\ell } . \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} \\int _ { E _ { n } } \\Delta _ { n } ^ { 2 ( d + 1 ) / 5 d } ( x ) \\ , f ( x ) \\ , d x = O \\left ( \\left ( n h _ { n } \\right ) ^ { - ( d + 1 ) / 5 d } \\right ) . \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} [ p _ 1 ] _ { \\kappa } ^ { \\bar { n } } + [ p _ 2 ] _ { \\kappa } ^ { \\bar { n } } + \\cdots + [ p _ { m } ] _ { \\kappa } ^ { \\bar { n } } = 1 \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} \\bar { V } : = \\frac { \\hbar ^ 2 } { 2 M } K * \\left ( \\frac { \\| \\nabla \\bar \\rho \\| ^ 2 } { 2 \\bar { D } ^ 2 } - \\frac { \\nabla ^ 2 \\sqrt { \\bar { D } } } { \\sqrt { \\bar { D } } } \\right ) \\ , , \\widehat { H } _ \\textit { e f f } : = \\widehat { H } _ e - \\frac { \\hbar ^ 2 } { 2 M } \\left [ K * \\left ( \\bar { D } ^ { - 1 } \\nabla \\bar { \\rho } \\right ) \\right ] \\ , . \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} \\alpha & = - 6 \\left ( \\left ( \\sum _ { i = 1 } ^ n ( k _ { i - 1 } ( f ) - f ( i ) ) \\right ) ^ 2 + \\sum _ { i = 1 } ^ n ( k _ { i - 1 } ( f ) ^ 2 - f ( i ) ^ 2 ) \\right ) \\\\ & = 1 2 \\left ( \\sum _ { i = 1 } ^ n f ( i ) ( k _ { i - 1 } ( f ) - f ( i ) ) - 2 \\sum _ { 1 \\leq j < i \\leq n } ( k _ { i - 1 } ( f ) - f ( i ) ) ( k _ { j - 1 } ( f ) - f ( j ) ) \\right ) \\\\ & = 1 2 \\sum _ { i = 1 } ^ n ( k _ { i - 1 } ( f ) - f ( i ) ) \\left ( \\sum _ { j = 1 } ^ { i } f ( j ) - \\sum _ { j = 1 } ^ { i - 1 } k _ { j - 1 } ( f ) \\right ) \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} h \\sum _ { n = 0 } ^ { ( N - 1 ) h } f ( a + n h ) = \\left . S ( t ) \\right | _ a ^ { a + N h } , \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} \\| \\kappa ^ { ( d ) } ( A ) \\| = \\limsup _ { n \\to \\infty } \\| \\gamma _ n ^ { ( d ) } ( A ) \\| \\end{align*}"}
-{"id": "9936.png", "formula": "\\begin{align*} \\kappa _ 1 = \\tau _ { 1 , \\ , 1 } \\ , { \\wedge } \\ , \\tau _ { 2 , \\ , 1 } \\ , { \\wedge } \\ , \\cdots \\ , { \\wedge } \\ , \\tau _ { \\ell , \\ , 1 } . \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} M ^ + = M _ k ^ { \\star } , \\ M ^ - = M _ { k + 1 } ^ { \\star } , \\ U = U _ k ^ { \\star } , \\ \\pi ^ { \\pm } = \\pi _ k ^ { \\star \\pm } . \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} \\varphi : = \\sum _ { n = 0 } ^ \\infty \\phi ( n ) < \\infty . \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} \\begin{aligned} g ( x ) = - & \\sum _ { i = 1 } ^ N \\min _ { l _ i \\leq y _ i \\leq u _ i } \\Big \\{ ( \\tilde { B _ 1 } x - \\alpha ) _ i ( y _ i - x _ i ) + \\beta y ^ 2 _ i + h _ i ( y _ i ) \\Big \\} \\\\ & + \\beta \\big { ( } \\sum _ { i = 1 } ^ N x _ i \\big { ) } ^ 2 + \\sum _ { i = 1 } ^ N h _ i ( x _ i ) . \\end{aligned} \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} A ( x ) = \\big ( \\nabla \\xi a \\nabla \\xi ^ T \\big ) ^ { \\frac { 1 } { 2 } } ( x ) = \\Phi ^ { \\frac { 1 } { 2 } } ( x ) \\ , , \\forall ~ x \\in \\mathbb { R } ^ n \\ , , \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} \\sigma _ I ^ 2 = 1 . \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} c _ { \\mu } = I _ { \\mu } ( v _ 0 ) + \\sum _ { j = 1 } ^ { l } I _ { \\mu } ^ { \\infty } ( v _ j ) , \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} t ( n ) = r ( n ) + s ( n ) \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} \\gamma _ { - } ( u ) = I ' ( u ^ { - } ) [ u ^ { - } ] + \\lambda \\int _ { \\mathbb R ^ { 3 } } \\phi _ { u ^ { + } } ( u ^ { - } ) ^ { 2 } d x = \\gamma ( u ^ { - } ) + \\lambda \\int _ { \\mathbb R ^ { 3 } } \\phi _ { u ^ { + } } ( u ^ { - } ) ^ { 2 } d x . \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} s = \\sum _ { j = 1 } ^ r c ^ { ( j ) } _ 1 c ^ { ( j ) } _ 2 \\cdots c ^ { ( j ) } _ { N _ j } \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} \\det h _ { \\alpha \\beta } = \\det g _ { \\alpha \\beta } \\det ( \\operatorname { I d } + S ) , \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} \\Vert \\mathrm { H e s s } ( f ) \\Vert _ { \\mathrm { H S } } ^ 2 + \\langle \\left ( \\mathrm { H e s s } ( U ) - \\rho I _ n \\right ) \\nabla f , \\nabla f \\rangle = 0 . \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} = y ^ { - \\upsilon } \\sum _ { k = 0 } ^ { \\infty } \\bigg [ \\frac { \\Theta ( k ) } { k ! } \\sum _ { \\ell = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ { \\ell } ( \\lambda b + c k ) ^ { \\ell } \\Gamma \\left ( \\upsilon + \\frac { \\ell } { 2 } \\right ) } { y ^ { \\frac { \\ell } { 2 } } ~ \\ell ! } \\cos \\left ( \\frac { \\upsilon \\pi } { 2 } + \\frac { \\ell \\pi } { 4 } \\right ) \\bigg ] , \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} \\textstyle \\bar { v } _ i ^ \\mathcal { I } : = p _ i ^ { \\mathcal { I } } - p _ c ^ \\mathcal { I } , p ^ { \\mathcal { I } } _ c : = \\sum _ { i = 1 } ^ { n _ 1 } \\alpha _ i p ^ { \\mathcal { I } } _ i \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} | \\Omega ^ + | = \\int _ { \\Omega ^ + } \\Delta K ^ + = - \\int _ \\gamma \\partial _ n K ^ + = \\int _ \\gamma \\partial _ n K ^ - = \\int _ { \\Omega ^ + } \\Delta K ^ + = | \\Omega ^ - | , \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} \\nu ( k + 1 ) + \\nu ( S ( n , k + 1 ) ) & \\ge \\lceil ( \\nu ( k + 1 ) ( p - 1 ) + \\sigma ( k + 1 ) - \\sigma ( n ) ) / ( p - 1 ) \\rceil \\\\ & = \\lceil ( 1 + \\sigma ( k ) - \\sigma ( n ) ) / ( p - 1 ) \\rceil . \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} C & = K + K \\left [ \\sum _ { i = 1 } ^ { n - 2 } L _ i + n - 2 \\right ] , \\\\ \\mathbf { E } [ C ] & = K + K \\left [ \\sum _ { i = 1 } ^ { n - 2 } \\mathbf { E } [ L _ i ] + n - 2 \\right ] . \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\rho _ { X _ 1 } ( \\psi _ n ( t ) , \\psi ( t ) ) = 0 . \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} [ L ^ { p _ 0 } , L ^ { p _ 1 } ] _ { \\theta , r } = [ L ^ { p _ 0 , r _ 0 } , L ^ { p _ 1 , r _ 1 } ] _ { \\theta , r } = L ^ { p , r } , \\frac 1 p = \\frac { 1 - \\theta } { p _ 0 } + \\frac { \\theta } { p _ 1 } , ~ ~ 1 \\leq r \\leq \\infty ; \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} h _ Z ( j ) = h _ { Z \\setminus \\{ P \\} } ( j ) \\mbox { a n d } D h _ Z ( j ) = D h _ { Z \\setminus \\{ P \\} } ( j ) \\forall j \\leq i . \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} { \\mathrm d } \\overline { X } _ { t , i } \\ , = \\ , b ( t , \\overline { X } _ { t , i } , \\ , F _ { t , i } ) \\ , { \\mathrm d } t + { \\mathrm d } B _ { t , i } \\ , ; i \\ , = \\ , 1 , \\ldots , n \\ , , \\ , \\ , t \\ge 0 \\ , , \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} P _ { n , \\beta } - P _ { - n , \\beta } = ( - 1 ) ^ { n - 1 } n N _ { n , \\beta } . \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { j } \\log ( 1 - 2 t \\sigma ^ { 2 } _ { j } ) + t \\sum _ { j } \\sigma _ { j } ^ { 2 } + n t \\alpha \\geq n t \\alpha - 2 t ^ { 2 } \\sum _ { j } \\sigma _ { j } ^ { 4 } = n \\alpha ^ { 2 } ( \\kappa - 2 \\kappa ^ { 2 } \\sum _ { j } \\sigma _ { j } ^ { 4 } / n ) . \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} K ( z , z '' ) = & \\Big ( \\int _ { r ' < \\frac { r } { 2 } } + \\int _ { \\frac { r } { 2 } \\leq r ' \\leq 2 r } + \\int _ { r ' > 2 r } \\Big ) G ( z , z ' ) Q ( z ' , z '' ) \\ ; d \\mu ( z ' ) \\\\ = & K _ { 1 , 1 } ( z , z '' ) + K _ { 1 , 2 } ( z , z '' ) + K _ { 1 , 3 } ( z , z '' ) . \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} z _ { t _ k } = x ( t _ k ) + e _ { t _ k } \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} T _ 2 ( \\omega ) = - \\frac { \\mathrm { o r d } ( \\omega ) ^ 2 } { 6 } ( \\mathrm { o r d } ( \\omega ) ^ 2 - 1 ) a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , ( x - z ) ( y - z ) } . \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} \\begin{aligned} & \\cos \\left ( 2 \\pi u _ { 0 } + \\frac { 4 \\pi } { N } j \\right ) + \\cos \\left ( 2 \\pi u _ { 0 } - \\frac { 4 \\pi } { N } j \\right ) \\\\ & = 2 \\cos \\left ( 2 \\pi u _ { 0 } \\right ) \\cos \\left ( \\frac { 4 \\pi } { N } j \\right ) \\end{aligned} \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} f \\left ( t , x , \\omega \\right ) ~ \\doteq ~ \\begin{cases} f _ { 1 } \\left ( t , x , \\omega \\right ) & x < \\beta \\\\ f _ { 2 } \\left ( t , x , \\omega \\right ) & x > \\alpha \\end{cases} \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} \\lim _ { \\Lambda \\rightarrow \\infty } \\chi _ \\Lambda ( k ) = 1 \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} G = \\frac { 1 } { 4 \\pi } \\sum _ { j , k , l , m \\neq 0 \\atop { j - k + l - m = 0 } } \\gamma _ j \\gamma _ k \\gamma _ l \\gamma _ m q _ j \\bar { q } _ k q _ l \\bar { q } _ m , \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} \\left | \\frac { \\partial ^ 2 f _ h } { \\partial x _ i \\partial x _ j } ( x ) - \\frac { \\partial ^ 2 f _ h } { \\partial x _ i \\partial x _ j } ( y ) \\right | = \\mathcal O \\left ( | x - y | ^ \\alpha \\log | x - y | \\right ) , \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} h = f ^ 3 + g ^ 2 . \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} T _ { k } ( F , G ) = \\iint _ { s < t } \\langle U _ { k } ( t ) U _ { k } ^ * ( \\tau ) F ( \\tau ) , G ( t ) \\rangle _ { L ^ 2 } ~ d \\tau d t \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} I ' ( u ) [ u _ 1 ] & = I ' ( u ) [ v ^ + ] = I ' ( v ) [ v ^ + ] + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ 3 ( v ^ + ) ^ 2 d x = 0 , \\\\ I ' ( u ) [ u _ 2 ] & = I ' ( u ) [ v ^ - ] = I ' ( v ) [ v ^ - ] + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ 3 ( v ^ - ) ^ 2 d x = 0 , \\\\ I ' ( u ) [ u _ 3 ] & = 0 , \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} \\mathcal { A } ( u ) = ( A + \\Phi ) u \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} : \\ \\ & \\mathcal { D } = \\{ 1 \\} , Y = 1 ; \\ \\mathcal { D } = \\{ 2 \\} , Y = 1 ; \\ \\mathcal { D } = \\{ 1 , 2 \\} , Y = 1 ; \\ \\mathcal { D } = \\{ 1 \\} , Y ( \\eta ) = \\eta , \\\\ : \\ \\ & \\mathcal { D } = \\{ 1 \\} , Y = 1 ; \\ \\mathcal { D } = \\{ 2 \\} , Y = 1 , \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } \\sim \\phi b - \\sum _ { s \\in S _ b } \\sum _ { i = 1 } ^ { m _ 0 - 1 } \\phi b ^ i s . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} \\lambda _ j : = \\frac { \\sqrt { M } } { \\| V ^ j \\| _ { L ^ 2 } } \\geq 1 , \\lambda _ n ^ l : = \\frac { \\sqrt { M } } { \\| v ^ l _ n \\| _ { L ^ 2 } } \\geq 1 . \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} K ( \\varphi , p ) : = \\int \\bigl [ p ( \\varphi ( x ) ) \\bigr ] \\ , \\bigl [ \\rho _ \\varphi ( x ) - \\rho ( x ) \\bigr ] \\ , d x \\ , , \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} ( M , \\tilde { \\nu } ) = \\prod _ { i = 1 } ^ p ( \\P ^ { N _ i } , \\tilde { \\theta } _ i ) ^ { r _ i } \\times ( M ' , \\nu ' ) . \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} & \\Lambda ^ n : = \\left \\{ ( x _ 1 , x _ 2 , . . . , x _ n ) \\in \\R ^ n : x _ 1 ^ 2 \\geq \\sum \\limits _ { i = 2 } ^ { n } x _ i ^ 2 , x _ 1 \\geq 0 \\right \\} , & n \\in \\mathbb { Z } ^ + . \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} T _ { w + 1 } = T _ w C _ { w + 1 } \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} \\lambda ^ { ( k s ) } _ { m a x } ( A ) = \\max _ { | S | = k s } \\lambda _ { m a x } ( A _ { S } ) . \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\sharp E _ { 0 , i } = a _ i , \\ \\sharp E _ { i , 0 } = b _ i , \\ i \\in V ( Q ) , \\ \\sharp E _ { 0 , 0 } = c . \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} \\varepsilon _ k \\left ( \\begin{pmatrix} z _ 1 & 0 & 0 \\\\ 0 & z _ 2 & 0 \\\\ 0 & 0 & z _ 3 \\end{pmatrix} \\right ) : = z _ k . \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} B ( t ) = \\left ( | t | ^ { s + 1 } | b | ^ { - 2 s - 1 } \\frac { 1 - 2 ^ { - 2 s - 2 } } { 1 - 2 ^ { - 2 s - 1 } } - \\frac { 1 } { 2 } | t | ^ { s + 1 } | b | ^ { - 2 s - 1 } + \\frac { 1 } { 2 } | b | \\right ) . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} ( M A f ) ( \\tau , y ) = \\tau ( M f ) ( \\tau , y ) \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} \\| \\xi \\| \\leq \\sum _ { j = 1 } ^ \\infty \\frac { \\| \\xi _ j \\| } { 2 ^ j } = 1 \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} d f _ p ( X ) = \\lim _ { s \\to 0 + } \\log { } \\circ \\delta _ s ^ { - 1 } \\circ L _ { f ( p ) } ^ { - 1 } \\circ f \\circ L _ { p } \\circ \\delta _ t \\circ \\exp ( X ) \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} a ( t ) = A _ 0 \\cdot t + a _ 0 \\ , , b ( t ) = B _ 0 \\cdot t + b _ 0 \\ , , c ( t ) = C _ 0 \\cdot t + c _ 0 \\ , . \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - q ^ 2 x ^ 2 } d x = \\frac { \\sqrt { \\pi } } { 2 q } \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} d \\nu _ n ( s ) : = c ( 1 + n ) \\delta _ n + d ( 1 + n ) \\delta _ { 1 / n } + 1 _ { [ 1 / n , n ] } ( s ) { 1 + s \\over s } \\ , d \\mu ( s ) \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} \\kappa = \\sum _ { j = 0 } ^ { \\infty } e ^ { - \\alpha j } \\mu \\left ( \\prod _ { k = 1 } ^ { j } P _ k \\right ) r . \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t = \\Delta u + | u | ^ { p - 1 } u , & x \\in \\Omega , t > 0 , \\\\ u ( x , t ) = 0 , & x \\in \\partial \\Omega , t > 0 , \\\\ u ( x , 0 ) = u _ 0 ( x ) , & x \\in \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "9704.png", "formula": "\\begin{align*} \\big [ \\varphi ( \\mathbb { A } / f \\mathbb { A } ) \\big ] _ { \\mathbb { A } } = f + c ( f ) p _ 1 \\prod _ { k = 1 } ^ n f ( z _ k ) + c ( f ) p _ 2 \\prod _ { k = 1 } ^ n f ( z _ k ) ^ 2 + \\dots + c ( f ) \\prod _ { k = 1 } ^ n f ( z _ k ) ^ { r _ 0 } . \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} K _ t ^ { ( \\alpha , \\beta ) } ( m , n ) = \\int _ { - 1 } ^ 1 e ^ { - ( 1 - x ) t } p _ m ^ { ( \\alpha , \\beta ) } ( x ) p _ n ^ { ( \\alpha , \\beta ) } ( x ) \\ , d \\mu _ { \\alpha , \\beta } ( x ) . \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{gather*} \\overset { I } { M } \\cdot h = \\overset { I } { h ' } \\overset { I } { M } \\overset { I } { S ( h '' ) } \\end{gather*}"}
-{"id": "4643.png", "formula": "\\begin{align*} G _ i : = \\{ x / \\alpha _ { i - 1 } \\mid x \\in A , ( x , o ) \\in \\alpha _ i \\} . \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { \\binom k j \\binom { k - j } s \\left ( { \\frac { { f _ 1 } } { { f _ 2 } } } \\right ) ^ j \\left ( { \\frac { { f _ 3 } } { { f _ 2 } } } \\right ) ^ s X _ { m - b k - ( a - b ) j - ( c - b ) s } } } = \\frac { { X _ m } } { { f _ 2 ^ k } } \\ , , \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} J _ \\nu ( \\rho ) = \\int _ { \\R ^ n } \\rho ( x ) \\left \\| \\nabla \\log \\frac { \\rho ( x ) } { \\nu ( x ) } \\right \\| ^ 2 \\ , d x \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ { 2 ^ n } H _ { I _ j } ( f ) . \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} \\Big \\| { u \\choose \\dot { u } } - V _ 0 ( t ) { u _ 0 ^ + \\choose u _ 1 ^ + } \\Big \\| _ { \\dot H ^ 1 \\times L ^ 2 } & = \\Big \\| \\int _ t ^ \\infty V _ 0 ( t - s ) { 0 \\choose F ( u ( s ) ) } d s \\Big \\| _ { \\dot H ^ 1 \\times L ^ 2 } \\\\ & \\lesssim \\| ( | u | ^ { \\frac { 4 } { n - 2 } } ) u \\| _ { L _ t ^ 1 L _ z ^ 2 ( ( t , \\infty ) \\times X ) } \\\\ & \\lesssim \\| u \\| _ { L _ t ^ { ( n + 2 ) / ( n - 2 ) } L _ z ^ { 2 ( n + 2 ) / ( n - 2 ) } ( ( t , \\infty ) \\times X ) } ^ { \\frac { 4 } { n - 2 } } \\\\ & \\rightarrow 0 , t \\rightarrow \\infty . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} \\int _ M u ^ { p _ { m + n } - 2 } v _ i v _ j d v _ g = \\delta _ { i j } , \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} k ( 1 , 1 ) = 1 , k ( 1 , 2 ) = 1 , k ( 1 , 3 ) = 0 , k ( 2 , 1 ) = 1 , k ( 2 , 2 ) = 1 , k ( 2 , 3 ) = 2 . \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} P u = 0 , u | _ { \\partial ( S M ) } = 0 \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} - 2 \\ < x _ { n + 1 } - x _ n , x _ n - x _ { n - 1 } \\ > = \\| x _ { n + 1 } + x _ { n - 1 } - 2 x _ n \\| ^ 2 - \\| x _ { n + 1 } - x _ n \\| ^ 2 - \\| x _ n - x _ { n - 1 } \\| ^ 2 \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} a _ { n m } ( \\omega ) = a _ { \\frac { n m } { \\mathrm { o r d } ( \\omega ) } } ( 1 ) . \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} 0 = S _ { i j } = & 2 \\langle d \\phi ( e _ i ) , d \\phi ( e _ j ) \\rangle - g _ { i j } | d \\phi | ^ 2 + \\frac { 1 } { 2 } \\langle \\psi , e _ i \\cdot \\tilde \\nabla _ { e _ j } \\psi + e _ j \\cdot \\tilde \\nabla _ { e _ i } \\psi \\rangle - \\frac { 1 } { 6 } g _ { i j } \\langle R ^ N ( \\psi , \\psi ) \\psi , \\psi \\rangle . \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} t r ( g \\vert _ { \\Pi _ { \\mu , \\mu ' } } ) = \\sum _ { t \\in X } ( \\mu \\boxtimes \\mu ' ) ^ 0 ( t g t ^ { - 1 } ) = S _ { e } ( g ) + S _ f ( g ) , \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} \\big [ - ( - \\Delta ) ^ { \\frac { s } { 2 } } \\eta _ { j , R } , - ( - \\Delta ) ^ { \\frac { s } { 2 } } \\tilde { \\phi } _ j \\big ] : = C _ { n , s } \\int _ { \\mathbb { R } ^ n } \\frac { [ \\eta _ { j , R } ( y ) - \\eta _ { j , R } ( x ) ] [ \\tilde { \\phi } _ j ( x ) - \\tilde { \\phi } _ j ( y ) ] } { | x - y | ^ { n + 2 s } } d y . \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} \\frac { r _ e } { R _ e } + \\frac { r _ { e ' } } { R _ { e ' } } = \\frac { L _ { i j , i j } } { L _ { i i } R _ { e } } + R _ { e } \\frac { L ' _ { i j , i j } } { L ' _ { i i } } \\ , . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} A _ i = \\sum _ { j = 0 } ^ d P _ { j , i } E _ j , E _ j = \\frac { 1 } { | X | } \\sum _ { i = 0 } ^ { d } Q _ { i , j } A _ i . \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} \\sum _ { y \\in ( - \\epsilon t , u t ) } c ^ \\omega _ { - \\epsilon t } ( y ) ( \\mu ( y ) - 1 ) = o ( t ) \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} \\inf _ { \\gamma \\in \\Gamma _ T ( y _ 0 ) } \\int _ { Y \\times U } k ( y , u ) \\gamma ( d y , d u ) : = v _ T ( y _ 0 ) \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} ( \\tau _ z \\omega ) ( x , y ) : = \\omega ( x + z , y + z ) , ~ ~ \\forall \\{ x , y \\} \\in E . \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} \\widehat { g } ^ { \\pm } _ { \\sigma , \\Delta } ( 0 ) = 2 \\pi \\bigg ( \\dfrac { 3 } { 2 } - \\sigma \\bigg ) - \\dfrac { 2 } { \\Delta } \\log \\bigg ( \\dfrac { 1 \\mp e ^ { - ( 2 \\sigma - 1 ) \\pi \\Delta } } { 1 \\mp e ^ { - 2 \\pi \\Delta } } \\bigg ) . \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} x ^ * _ i > 0 ; & \\qquad \\frac { x ^ * _ { i } } { x ^ * _ { i - 1 } } = O ( i / d \\Delta ) \\ , \\ , \\ , ; \\\\ 0 \\le \\rho ^ * _ { \\tau } ( i ) \\le 1 \\ , \\ , ; & \\qquad \\sum _ { \\tau \\in \\Gamma } \\rho ^ * _ { \\tau } ( i ) \\le 1 . \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} R ( e _ { 1 i + 1 } ) R ^ 2 ( 1 ) = \\frac { 1 } { 2 } R ( e _ { 1 i + 1 } ) J ^ 2 = \\frac { 1 } { 2 } ( a _ { 1 2 } J ^ 3 + \\ldots + a _ { 1 i + 1 } J ^ { i + 2 } + B ' J ^ 2 ) . \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} & \\begin{aligned} Q _ 1 ( x , y , q , k ) : = & \\left ( l _ 1 x ^ { l _ 1 } + l _ 2 x ^ { l _ 2 } \\right ) \\left ( y - x - q + b ( 1 + k ) [ \\ln { x } - \\ln { y } ] \\right ) \\\\ & - ( x - b ( 1 + k ) ) ( y ^ { l _ 1 } - x ^ { l _ 1 } + y ^ { l _ 2 } - x ^ { l _ 2 } ) , \\end{aligned} \\\\ & \\begin{aligned} Q _ 2 ( x , y , q , k ) : = & \\left ( l _ 1 y ^ { l _ 1 } + l _ 2 y ^ { l _ 2 } \\right ) \\left ( y - x - q + b ( 1 + k ) [ \\ln { x } - \\ln { y } ] \\right ) \\\\ & - ( y - b ( 1 + k ) ) ( y ^ { l _ 1 } - x ^ { l _ 1 } + y ^ { l _ 2 } - x ^ { l _ 2 } ) . \\end{aligned} \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} \\Omega _ n : = \\{ x \\in \\Omega : d ( x , \\partial \\Omega ) > 1 / n \\} ; \\ ; \\phi _ { i , n } : = u _ i \\star \\rho _ n \\ ; \\ ; \\Omega _ n , \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} \\begin{aligned} S _ { \\lambda } ( T _ { \\alpha , q } v ) ( x ) & = T _ { \\alpha } ( D _ { \\lambda ^ { - \\alpha } } v ) ( x ) x \\in B _ R , \\\\ T _ { \\alpha } ^ { - 1 } ( S _ { \\lambda } u ) ( y ) & = D _ { \\lambda ^ { - \\alpha } } ( T ^ { - 1 } _ { \\alpha , q } u ) ( y ) y \\in \\mathbb { R } ^ n . \\end{aligned} \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} \\theta ( T ) = \\underset { k \\to + \\infty } { \\lim } \\ , \\Big ( \\inf \\big \\{ \\big \\| T ^ k - Q \\big \\| _ { \\mathcal { B } ( E ) } , \\ Q \\ \\mathrm { c o m p a c t } \\big \\} \\Big ) ^ { \\frac { 1 } { k } } = \\underset { k \\geq 1 } { \\inf } \\ , \\Big ( \\inf \\big \\{ \\big \\| T ^ k - Q \\big \\| _ { \\mathcal { B } ( E ) } , \\ Q \\ \\mathrm { c o m p a c t } \\big \\} \\Big ) ^ { \\frac { 1 } { k } } . \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 ^ + } \\| A u ^ { \\delta } _ { k _ * ( \\delta ) } - y \\| _ p = 0 , \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} \\begin{pmatrix} 8 & 1 2 & 0 & 1 & 1 \\\\ 1 & 1 & 1 & 0 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} \\dot X _ t ( \\omega ) = v ( t , X _ t ( \\omega ) ) , X _ 0 \\sim \\rho ^ 0 , X _ 1 \\sim \\rho ^ 1 . \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} \\mathbf P \\left [ N ^ t \\left ( \\tau _ 0 ( t ) \\right ) > \\varepsilon \\right ] \\ge \\mathbf P \\left [ \\right ] = \\int _ { t + \\varepsilon } ^ \\infty \\frac { \\nu ( d r ) } { \\lambda } , \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} \\frac { ( - 1 ) ^ e } { p ^ e } \\binom { n } { m } \\equiv \\prod \\frac { n _ i ! } { m _ i ! r _ i ! } \\mod p \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( r ) _ k ( 1 - r ) _ k } { k ! ^ 2 } z ^ k \\end{gather*}"}
-{"id": "1659.png", "formula": "\\begin{align*} \\Big | \\mu ( \\mathcal { C } ) - \\mu ( \\widehat { \\mathcal { C } } _ { N } ) \\Big | & = \\Big | \\mu ( \\mathcal { C } ) - \\mu ( \\widehat { \\mathcal { C } } _ { N } ) \\Big | I \\{ N > 0 \\} + \\Big | \\mu ( \\mathcal { C } ) - \\kappa _ { 0 } \\Big | I \\{ N = 0 \\} , \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} x \\mapsto x ^ { 1 1 } , \\ y \\mapsto x ^ 3 y \\Longrightarrow \\mathcal { I } _ { 1 , 1 , \\beta _ 1 , \\beta _ 2 } = \\ ! 4 4 \\ ! \\ ! \\int _ { \\mathcal { D } } \\ ! \\hat { f } _ { t , s } ( x , y ) ^ s x ^ { 4 ( 5 \\beta _ 1 + 3 \\beta _ 2 + 2 4 s ) } y ^ { 3 \\beta _ 1 + 4 \\beta _ 2 + 2 1 s } \\frac { d x } { x } \\ ! \\frac { d y } { y } , \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} P _ n ( x ) = \\frac { 1 } { n ! } \\left ( \\frac { d } { d x } \\right ) ^ n \\left ( x ^ n ( 1 - x ) ^ n \\right ) = a _ 0 + a _ 1 x + \\dots + a _ n x ^ n , \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} f ( X + Y ) = f ( X ) f ( Y ) h ( X , Y ) \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} p ^ r B _ n ^ { ( l ) } / n ! \\equiv ( - 1 ) ^ n \\binom { l - n - 1 } { r } \\equiv ( - 1 ) ^ r \\binom { n - l + r } { r } \\mod p . \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ { \\tilde g ^ { ( 4 ) } } \\delta ^ * _ { g _ { \\epsilon } ^ { ( 4 ) } } Y = 0 \\quad S \\\\ Y = 0 \\quad \\partial S \\end{cases} \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} \\left < \\phi _ 0 ^ 2 \\right > _ n = \\left < [ v , G _ { d , \\epsilon } ( t ) v ] \\right > _ { \\hat { n } } \\le \\sum _ { j k \\in E } ( ( G _ d v ) _ j - ( G _ d v ) _ k ) ^ 2 \\left < e ^ { - t _ { j k } } \\right > _ { \\hat { n } } . \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} \\eta _ n ( \\varphi ) = \\frac { 1 } { | K _ n | } \\int _ { K _ n } \\varphi ( x ) \\ , d x = 0 , \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} \\begin{cases} x ^ 0 \\in C , \\\\ u ^ k \\in C f ( u ^ k , y ) + \\frac { 1 } { r _ k } \\langle y - u ^ k , u ^ k - x ^ k \\rangle \\geq 0 , \\forall y \\in C , \\\\ x ^ { k + 1 } = \\alpha _ k x ^ k + ( 1 - \\alpha _ k ) T u ^ k . \\end{cases} \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} \\mathcal { L } _ { d , n } ( x , y _ m ) = \\frac { 1 } { x ^ d \\theta ^ { d y _ m } } H _ { d , n + \\ell _ q ( y _ m ) } ( z _ 1 , \\dots , z _ n , \\theta , \\dots , \\theta , \\dots , \\theta ^ { q ^ m } , \\dots , \\theta ^ { q ^ m } ) , \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} P _ { o u t } ( R _ { E } , \\phi _ { 2 } ) & \\ ! = \\ ! P r ( \\rho _ e | h _ { e } | ^ { 2 } \\ ! < \\ ! \\rho _ 1 | h _ { 1 } | ^ { 2 } ) P _ { o u t , 1 } ( R _ { E } , \\phi _ { 2 } ) \\\\ & \\ ! + \\ ! P r ( \\rho _ e | h _ { e } | ^ { 2 } \\ ! \\geq \\ ! \\rho _ 1 | h _ { 1 } | ^ { 2 } ) P _ { o u t , 2 } ( R _ { E } , \\phi _ { 2 } ) , \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} D _ { t } ( \\Phi ( \\mathrm m _ { 1 } ) , \\Phi ( \\mathrm m _ { 2 } ) ) \\ , & \\le \\ , C _ { T } e ^ { C _ { T } T } \\int ^ { t } _ { 0 } ( u D _ { v } ( \\mathrm m _ { 1 } , \\mathrm m _ { 2 } ) + ( 1 - u ) D _ { v } ( \\mathrm m _ { 1 } , \\mathrm m _ { 2 } ) ) { \\mathrm d } v \\\\ & \\ , = \\ , C _ { T } e ^ { C _ { T } T } \\int ^ { t } _ { 0 } D _ { v } ( \\mathrm m _ { 1 } , \\mathrm m _ { 2 } ) { \\mathrm d } v \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } u ( t , x ) \\ , d x ~ = ~ \\int _ { \\mathbb { R } } u _ 0 ( x ) \\ , d x \\qquad ~ t \\ge 0 . \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\gamma ( \\pi , r , s , \\psi ) L ( \\pi , r , s ) = \\epsilon ( \\pi , r , s , \\psi ) L ( \\pi , r ^ \\vee , 1 - s ) , \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} \\{ x _ { 1 , \\tau } , p _ { 1 , \\tau } \\} _ { D B } = 1 ; \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} \\frac { \\partial u ( n , t ) } { \\partial t } = \\mathcal { J } ^ { ( \\alpha , \\beta ) } u ( n , t ) \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} \\{ ( x , y , u _ 1 , \\dots , u _ n , v _ 1 , \\dots , v _ n ) ) \\in \\R ^ { 2 + 2 n } : v _ j = q _ j ( x , y , u _ 1 , \\dots , u _ n ) , j = 1 , \\dots , n \\} , \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\frac { K _ i - K _ i ^ { - 1 } } { q _ i - q _ i ^ { - 1 } } \\otimes K _ i + K _ i ^ { - 1 } \\otimes \\frac { K _ i - K _ i ^ { - 1 } } { q _ i - q _ i ^ { - 1 } } = \\frac { K _ i \\otimes K _ i - K _ i ^ { - 1 } \\otimes K _ i ^ { - 1 } } { q _ i - q _ i ^ { - 1 } } . \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} K \\left ( z \\right ) \\frac { \\partial w } { \\partial \\overline { z } } + w \\frac { \\partial K } { \\partial \\overline { z } } = 0 \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} \\rho _ C ( c ) = \\{ [ c ] \\} + \\sum _ { ( c ) } \\{ [ c ' ] | [ c '' ] \\} + \\sum _ { ( c ) } \\{ [ c ' ] | [ c '' ] | [ c ''' ] \\} + . . . \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} \\bar { Q } & = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n _ 1 } k _ i ( ( \\bar { v } _ i ^ { \\mathcal { I } } ) ^ \\times ) ^ 2 + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { n _ 2 } k _ { j + n _ 1 } ( ( v _ j ^ { \\mathcal { I } } ) ^ \\times ) ^ 2 , \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} \\begin{aligned} S _ { n , p } \\| u \\| _ { L ^ { p ^ * } ( \\Omega ) } = S _ { n , p } \\| u ^ { * } \\| _ { L ^ { p ^ * } ( B _ R ) } & < S _ { n , p } \\Bigg ( \\int _ { B _ { R } } \\frac { | u ^ { * } ( x ) | ^ { p ^ { * } } } { \\left [ \\frac { n - p } { p - 1 } \\log _ { { \\frac { n - 1 } { p - 1 } } } \\frac { R } { | x | } \\right ] ^ { \\frac { p ( n - 1 ) } { n - p } } } d x \\Bigg ) ^ { \\frac { 1 } { p ^ { * } } } \\\\ & \\le \\| \\nabla u ^ { * } \\| _ { L ^ { p } ( B _ R ) } \\le \\| \\nabla u \\| _ { L ^ { p } ( \\Omega ) } \\end{aligned} \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} m \\left ( r , \\frac { f ( z + c ) } { f ( z ) } \\right ) = O ( r ^ { \\sigma - 1 + \\varepsilon } ) , \\end{align*}"}
-{"id": "10003.png", "formula": "\\begin{align*} \\bar A ^ n ( q ) = \\frac { \\bar F ^ n _ 0 ( q ) } { 1 - \\bar P ^ n ( q ) } \\rightarrow \\frac { \\bar F _ 0 ( q ) } { 1 - \\bar P ( q ) } = \\bar A ( q ) . \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{gather*} \\overset { I } { M } \\ ! _ 1 \\overset { J } { M } \\ ! _ 2 = \\overset { J } { S ^ { - 1 } ( a _ i ) } _ 2 \\overset { I \\otimes J } { M } \\ ! \\ ! _ { 1 2 } \\big ( \\overset { I J } { R ' } \\big ) _ { 1 2 } \\big ( \\overset { I } { b _ i } \\big ) _ 1 \\end{gather*}"}
-{"id": "1856.png", "formula": "\\begin{align*} Q ( x , y ) = \\sqrt { ( h - y ) _ + } h > 0 , \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} [ \\psi ( x , t ) ] _ { \\eta , B _ { \\mu _ j R } ( \\xi _ j ) } \\lesssim \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\sum _ { j = 1 } ^ k \\frac { \\mu _ j ^ { - \\eta } \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } + \\sigma } ( t ) } { 1 + | y _ j | ^ { a - 2 s + \\eta } } | y _ j | \\leq 2 R , \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} J _ { \\nu } ( r ) = \\frac { r ^ \\nu } { 2 ^ \\nu \\Gamma ( \\nu + 1 ) } + S _ \\nu ( r ) \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} b _ j ^ { n - 2 s - 1 } H ( q _ j , q _ j ) - \\sum _ { i \\neq j } b _ j ^ { \\frac { n - 2 s } { 2 } - 1 } b _ i ^ { \\frac { n - 2 s } { 2 } } G ( q _ j , q _ i ) = \\frac { 2 s b _ j ^ { 2 s - 1 } } { n - 2 s } j = 1 , \\cdots , k . \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} B _ j ^ 0 [ \\phi _ j ] : = p \\left [ U ^ { p - 1 } \\left ( \\frac { \\mu _ { 0 j } } { \\mu _ j } y \\right ) - U ^ { p - 1 } ( y ) \\right ] \\phi _ j + p \\left [ \\mu _ { 0 j } ^ { 2 s } ( u ^ { * } _ { \\mu , \\xi } ) ^ { p - 1 } - U ^ { p - 1 } \\left ( \\frac { \\mu _ { 0 j } } { \\mu _ j } y \\right ) \\right ] \\phi _ j . \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} O ( \\epsilon ^ 2 m ^ 3 ) & = \\left \\| \\prod _ { m < k < n } P _ { n - k } ( \\epsilon ) \\left ( \\prod _ { 0 \\le k \\le m } P _ { n - k } ( \\epsilon ) - \\tilde { \\Pi } + \\epsilon \\tilde { \\Pi } \\tilde { P } ^ { ( 1 ) } ( I - \\tilde { P } + \\tilde { \\Pi } ) ^ { - 2 } \\tilde { P } \\right ) \\right \\| \\\\ & = \\left \\| \\prod _ { k = 1 } ^ { n } P _ k ( \\epsilon ) - \\tilde { \\Pi } + \\epsilon \\tilde { \\Pi } \\tilde { P } ^ { ( 1 ) } ( I - \\tilde { P } + \\tilde { \\Pi } ) ^ { - 2 } \\tilde { P } \\right \\| , \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} \\mathcal { U } [ 0 ] = F _ m \\cdot \\alpha _ 1 ^ { l _ 0 l _ 1 } \\cdots \\alpha _ m ^ { l _ 0 l _ m } \\prod _ { 1 \\le i < j \\le m } ( \\alpha _ i - \\alpha _ j ) ^ { l _ i l _ j } \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} h ( z ) = \\int _ { - \\infty } ^ { \\infty } p ( z ) \\log \\frac { 1 } { p ( z ) } d z = \\int _ { - \\infty } ^ { \\infty } p _ { _ 1 } ( y ) \\log \\frac { 1 } { p _ { _ 1 } ( y ) } d y \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} & f _ 1 ^ { \\prime } ( \\alpha ) = \\frac { 1 } { 2 } \\textbf { T r } [ \\psi ^ { \\prime } ( v + \\alpha d _ x ) \\diamond d _ x + \\psi ^ { \\prime } ( v + \\alpha d _ s ) \\diamond d _ s ] , \\\\ & f _ 1 ^ { \\prime \\prime } ( \\alpha ) = \\frac { 1 } { 2 } \\textbf { T r } [ ( d _ x \\circ d _ x ) \\diamond \\psi ^ { \\prime \\prime } ( v + \\alpha d _ x ) + ( d _ s \\circ d _ s ) \\diamond \\psi ^ { \\prime \\prime } ( v + \\alpha d _ s ) ] . \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} \\sup _ { ( a , b ) \\in K _ q } \\{ a x + b y \\} = f _ p ( x , y ) : = \\begin{cases} \\frac { 1 } { p } \\frac { | y | ^ p } { x ^ { p - 1 } } & x > 0 \\\\ 0 \\quad & x = 0 , \\ , y = 0 \\\\ + \\infty & x = 0 , \\ , y \\neq 0 , x < 0 . \\end{cases} \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{gather*} \\rho ( t ) < \\epsilon \\\\ - \\pi < \\rho ' ( 0 ) < 0 \\\\ \\textrm { s u p p } ( \\rho ) \\subset [ 0 , ( 3 r ) ^ 2 ) \\\\ \\rho ' ( t ) \\le 0 \\\\ \\rho '' ( t ) < 0 \\ \\textrm { o n } \\ t < r ^ 2 \\\\ \\rho ' | _ { t \\in [ r ^ 2 , ( 2 r ) ^ 2 ] } \\equiv - \\frac { 4 } { 3 } \\pi \\ \\ \\Big ( \\Longrightarrow \\min _ { t \\in [ 0 , r ^ 2 ] } \\rho ' ( t ) = - \\frac { 4 } { 3 } \\pi \\Big ) \\end{gather*}"}
-{"id": "1932.png", "formula": "\\begin{align*} T \\ ! \\left ( \\overline { l } , \\overline { \\alpha } \\right ) : = \\alpha _ 1 ^ { \\binom { l _ 1 } { 2 } } \\cdots \\alpha _ m ^ { \\binom { l _ m } { 2 } } \\prod _ { 1 \\le i < j \\le m } ( \\alpha _ i - \\alpha _ j ) ^ { \\min \\left \\{ l _ i ^ 2 , \\ , l _ j ^ 2 \\right \\} } \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} W _ { 2 } ( r ) : = \\int _ { \\wp } r ^ { Q - 1 } \\phi _ { 3 } ( r y ) d \\sigma ( y ) \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} M ( t ) = \\begin{cases} g ^ { - 1 } ( ( 1 - \\epsilon ) \\log t ) , & \\textrm { o r } \\\\ g ^ { - 1 } ( \\log t ) , & \\textrm { w h e n \\eqref { e q : s t a b i l i t y } h o l d s . } \\end{cases} \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} | | X _ { \\tilde { Q } } | | _ { s , r , p - 1 , \\mathbf { a } ; \\Xi _ r } = O ( r ^ 2 ) . \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} \\alpha = ( d t + \\theta ) , \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{align*} \\sum _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) } { a } = \\log _ { \\varphi } ( P ( z _ 1 , \\dots , z _ n ) ) = \\frac { \\log _ { \\phi } ( \\omega _ { n } P _ { \\phi } ( z _ 1 , \\dots , z _ n ) ) } { \\omega _ { n } } . \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} n ^ { k } = \\sum _ { d \\mid n } J _ { k } ( d ) , \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} \\gamma _ + ( t u ) & = I ' ( t u ) [ t u ^ + ] \\\\ & = I ' ( t u ^ + ) [ t u ^ + ] + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { t u ^ + } ( t u ^ - ) ^ 2 d x \\\\ & = t ^ 2 \\norm { u ^ + } ^ 2 + \\lambda t ^ 4 \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } u ^ 2 d x - \\int _ { \\mathbb R ^ 3 } f ( t u ^ + ) t u ^ + d x . \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | \\mathcal { N } _ { F } ( T ) - A _ F T \\log { T } - B _ F T \\right | \\\\ & \\quad < C _ { F , 1 } ( T _ 0 ) \\log { T } + C _ { F , 2 } ( T _ 0 ) + \\frac { C _ { F , 3 } ( T _ 0 ) } { T } , \\end{aligned} \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} v _ i ( t , x ) = & \\int _ 0 ^ 1 g ( t , x , y ) h _ i ( y ) d y + \\int _ 0 ^ t \\int _ 0 ^ 1 g ( t - s , x , y ) b ( u _ i ( s , y ; h _ i ) ) d s d y \\\\ & + \\int _ 0 ^ t \\int _ 0 ^ 1 g ( t - s , x , y ) \\sigma ( u _ i ( s , y ; h _ i ) ) W ( d s d y ) . \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} S : V ^ { L } _ { \\mathbb { H } } \\longrightarrow V ^ { L } _ { \\mathbb { H } } , ~ S \\psi = T T ^ { \\dagger } \\psi = \\displaystyle \\sum _ { k \\in I } \\left \\langle \\psi | \\phi _ { k } \\right \\rangle \\phi _ { k } . \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} \\int _ \\Omega \\int _ { \\mathbb { S } _ + ^ 1 } \\left ( c _ 0 ^ 2 | \\theta \\cdot \\nabla p | ^ 2 \\right ) ^ { \\frac { 1 } { \\gamma - 1 } } ( \\theta \\cdot \\nabla p ) ( \\theta \\cdot \\nabla \\phi ) \\d \\ , \\theta \\d x = \\int _ \\Omega S \\phi \\d x \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} W _ 2 ( \\{ b ^ { - 2 } a , b ^ { - 1 } a , a b ^ { - 1 } a \\} ) = & W _ 2 ( b ^ { - 2 } a ) \\cup W _ 2 ( b ^ { - 1 } a ) \\cup W _ 2 ( a b ^ { - 1 } a ) \\\\ = & \\{ b ^ { - 2 } a , a b ^ { - 2 } a , a b ^ { - 3 } a \\} \\cup \\{ b ^ { - 1 } a , a b ^ { - 1 } a , a b ^ { - 2 } a \\} \\cup \\{ a b ^ { - 3 } a \\} \\\\ = & \\{ b ^ { - 2 } a , a b ^ { - 2 } a , a b ^ { - 3 } a , b ^ { - 1 } a , a b ^ { - 1 } a \\} . \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} u _ { m + k } ( t , 0 ) = u _ { k + m } ( 0 , \\lambda _ { k + m } t ) , \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} J ( \\tau ) = D \\cap ( A \\cup C ) \\cap ( B \\cup C ) = ( D \\cap B ) \\cap ( A \\cup C ) , \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} T _ { a , j , * } ^ \\nu ( h ) = \\chi _ j ( r ) r ^ { - \\frac { d - 2 } { 2 } } \\int _ 0 ^ \\infty \\frac { e ^ { i t \\rho ^ a } ( r \\rho / 2 ) ^ \\nu } { \\Gamma ( \\nu + 1 / 2 ) \\pi ^ { 1 / 2 } } & \\int _ { - 1 } ^ 1 \\sum _ { n = 0 } ^ \\infty \\frac { ( i \\theta r \\rho ) ^ n } { n ! } ( 1 - \\theta ^ 2 ) ^ { \\nu - 1 / 2 } d \\theta \\\\ & \\times \\rho ^ { \\frac { - d + 1 + a } { 2 } } \\chi _ { \\leq - j - 5 } ( \\rho ) h ( \\rho ) d \\rho . \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} \\tilde { \\mathbf { S } } _ { ( l , p ) , ( l ' , p ' ) } = \\frac { \\zeta ^ { - l l ' - l p ' - p l ' - 2 p p ' } ( \\zeta ^ { l l ' } - 1 ) } { d } . \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} W _ { \\pi , p } ( g _ { t , l , v } ) = \\sum _ { \\mu _ p \\in { } _ p \\mathfrak { X } _ l } c _ { t , l } ( \\mu _ p ) \\mu _ p ( v ) . \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\mathcal { R } ( C ) \\subseteq \\mathcal { R } ( A ) \\ \\mbox { a n d } \\ C A ^ * \\ \\mbox { i s H e r m i t i a n } . \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} \\rho ( x ) = \\ln n _ 1 + \\frac { 2 x } { e \\sqrt { n _ 1 } } , \\ x \\in [ n _ 1 , 2 n _ 1 ] . \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} \\ < D ^ { ( 2 ) } _ { n } f ( w ) , \\ , ( h _ { 1 } , h _ { 2 } ) \\ > _ { H _ { n } ^ { \\otimes ( 2 ) } } = \\left . \\frac { \\partial ^ { 2 } } { \\partial \\varepsilon _ { 1 } \\partial \\varepsilon _ { 2 } } f ( w + \\varepsilon _ { 1 } h _ { 1 } + \\varepsilon _ { 2 } h _ { 2 } ) \\right | _ { \\varepsilon _ { 1 } = \\varepsilon _ { 2 } = 0 } . \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\Pi ( x , y ) = \\int _ 0 ^ { + \\infty } e ^ { i u \\psi ( x , y ) } \\ , s ( x , y , u ) \\ , \\mathrm { d } u , \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} \\pi _ { t , k } ( \\varphi ) \\ , = \\ , \\pi _ { 0 , k } ( \\varphi ) + \\int ^ { t } _ { 0 } ( \\pi _ { s , k } ( \\varphi b ) - \\pi _ { s , k } ( b ) \\pi _ { s , k } ( \\varphi ) ) \\big ( { \\mathrm d } X _ { s } - \\pi _ { s , k } ( b ) { \\mathrm d } s \\big ) + \\int ^ { t } _ { 0 } \\pi _ { s , k + 1 } ( \\widetilde { \\mathcal A } _ { s } \\varphi ) { \\mathrm d } s \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} R ( x ) R ( y ) = R ( R ( x ) y + x R ( y ) + \\lambda x y ) \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} ( u \\cdot \\varphi ) ( m ) = \\varphi ( S ( u ) \\cdot m ) ( u \\cdot \\varphi ) ( m ) = \\varphi ( S ^ { - 1 } ( u ) \\cdot m ) . \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} V ^ 0 _ h : = \\{ v : v \\in V _ h \\ { \\rm a n d } \\ v | _ { \\partial \\Omega } = 0 \\} . \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} \\Pi _ { k \\boldsymbol { \\nu } } \\big ( x _ { 1 } , x _ { 2 } \\big ) = \\Pi _ { k \\boldsymbol { \\nu } } \\left ( \\widetilde { \\mu } _ g ( x _ { 1 } ) , \\widetilde { \\mu } _ g ( x _ { 2 } ) \\right ) \\forall \\ , g \\in G , \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} & x \\in C \\\\ & f ( x , y ) \\geq 0 y \\in C 0 \\in M _ i ( x ) i = 1 , \\ldots , m , \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} \\| u \\| _ { ( \\kappa _ 1 + 1 ) p _ * , \\partial \\Omega } = \\| u ^ { \\kappa _ 1 + 1 } \\| _ { p _ * , \\partial \\Omega } ^ { \\frac { 1 } { \\kappa _ 1 + 1 } } & \\leq M _ { 1 5 } ( \\kappa _ 1 , u ) \\left [ \\| u \\| _ { p _ * , \\partial \\Omega } ^ { p _ * } + 1 \\right ] ^ { \\frac { 1 } { ( \\kappa _ 1 + 1 ) p } } < \\infty . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} \\mathbb { E } [ f ( X - Z ) ] = \\mathbb { E } [ g ( Z ) ] \\le \\mathbb { E } [ h ( Z ) ] = \\mathbb { E } [ f ( Y - Z ) ] . \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\Delta _ { \\rho , \\sigma } = L _ { D _ \\rho } R _ { D _ \\sigma ^ { - 1 } } = \\sum _ { a > 0 , \\ , b > 0 } a b ^ { - 1 } L _ { P _ a } R _ { Q _ b } , \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} \\begin{aligned} & \\left ( \\frac { 5 \\tau ^ 2 A _ F } { 4 } + 1 \\right ) \\frac { \\log { N } } { N } + \\frac { \\log { \\log { N } } } { N } + \\frac { \\log { ( 2 0 K _ { F , 2 } ( \\tau ) ) } } { 2 N } \\\\ & \\le \\frac { 1 } { 3 } \\log { R } + \\frac { 1 } { 8 } \\log { R } + \\frac { 1 } { 2 4 } \\log { R } . \\end{aligned} \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} U + { \\cal K } U = G , \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} \\int ^ * \\left < r _ ! ( R _ x f _ 1 ) , \\Theta _ \\Pi \\right > \\psi ^ { - 1 } ( x ) d x = \\left < \\mathcal T f , \\Theta _ \\Pi \\right > , \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} b _ \\phi ( e _ 6 , e _ 6 ) = - \\phi _ { 2 6 7 } \\left ( \\phi _ { 1 4 7 } \\phi _ { 3 5 7 } - \\phi _ { 1 5 7 } \\phi _ { 3 4 7 } \\right ) e ^ { 1 2 3 4 5 6 7 } = - b _ \\phi ( e _ 7 , e _ 7 ) , \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} \\mathbf { E } [ C ] = K \\left [ n - 1 + \\sum _ { i = 1 } ^ { n - 2 } \\sum _ { m = 1 } ^ { i } \\left [ K ^ { i - m } ( K - 1 ) \\prod _ { j = m } ^ { i } p _ j \\right ] \\right ] . \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} \\frac { \\lambda ^ 2 } { \\rho _ \\lambda } \\int _ \\Omega V ( x ) e ^ { \\ , u } = 1 , \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} f _ { \\epsilon } = \\begin{cases} 0 , & t \\le \\epsilon , \\\\ \\epsilon ^ { - 1 } ( t - \\epsilon ) , & \\epsilon \\le t \\le 2 \\epsilon \\\\ 1 & t \\ge 2 \\epsilon . \\end{cases} \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} f ( q ' ) = 1 \\ , \\ , \\ , \\ , \\ , f _ { | \\Gamma \\cdot q '' \\cup ( \\Gamma \\cdot q ' \\setminus \\{ q ' \\} ) } = 0 . \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{gather*} y _ 1 = R ( y _ 2 ) R ( y _ 2 ) = 2 R ( y _ 1 y _ 2 ) - R \\big ( y _ 2 ^ 2 \\big ) , \\\\ \\lambda _ { i _ 1 } y _ 1 = R ( y _ 2 ) R ( 1 ) = R ( y _ 2 R ( 1 ) ) - y _ 1 , \\end{gather*}"}
-{"id": "9697.png", "formula": "\\begin{align*} \\exp _ { \\varphi } ( g ) = \\sum _ { j } \\ell _ j ( z _ 1 ) \\dots \\ell _ j ( z _ n ) \\alpha _ j \\tau ^ j ( g ) , \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{gather*} \\overline { \\nabla } ^ { \\mathrm { b a s } } = \\nabla ^ { \\mathbb { V } } \\oplus p ^ ! \\big ( \\nabla ^ { \\mathrm { b a s } } \\big ) . \\end{gather*}"}
-{"id": "1157.png", "formula": "\\begin{align*} \\phi b ^ { m } \\sim \\phi b - \\sum _ { i = 1 } ^ { m - 1 } \\sum _ { s \\in S _ b } \\phi b ^ i s . \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} ( D + 1 ) + r ( D - d + 1 ) & = ( D + 1 ) ( r + 1 ) - d \\cdot r \\\\ & = ( s - r + 1 ) ( 1 - \\alpha ) q ( r + 1 ) - d r \\\\ & > ( s - r + 1 ) \\left ( \\frac { \\ell } { r + 1 } + \\frac { r } { r + 1 } \\frac { d } { ( s - r + 1 ) q } \\right ) q ( r + 1 ) - d r \\\\ & = ( s - r + 1 ) \\ell q + d r - d r \\\\ & = ( s - r + 1 ) \\ell q . \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} L ( \\sigma ) : = \\inf \\left \\{ \\frac { | Z ( E ) | } { \\norm { [ E ] } } \\ , \\middle | \\ , [ E ] \\in \\widehat { \\ss } ( \\sigma ) \\ \\right \\} , \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} \\widehat { L } ^ { ( \\alpha ) } _ { n } ( x ) = - ( x + \\alpha + 1 ) L _ { n - 1 } ^ { ( \\alpha ) } ( x ) + L _ { n - 2 } ^ { ( \\alpha ) } ( x ) , \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} \\varphi _ { - 2 s + 3 / 2 } = q ^ { - ( 2 s - 1 ) ^ 2 / 2 } ( - \\phi ( q ) + q ^ { 2 s ^ 2 } \\varphi _ { ( 4 s + 1 ) / 2 } ( q ) ) ~ . \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} B _ { i } ^ { + } ( k , s ) & = \\frac { F _ { i } ( D ^ { + } U ( k , s + 1 ) , D ^ { - } U ( k , s + 1 ) ) - F _ { i } ( D ^ { + } U ( k , s ) , D ^ { - } U ( k , s + 1 ) ) } { D ^ { + } U ( k , s + 1 ) - D ^ { + } U ( k , s ) } , \\\\ B _ { i } ^ { - } ( k , s ) & = \\frac { F _ { i } ( D ^ { + } U ( k , s ) , D ^ { - } U ( k , s + 1 ) ) - F _ { i } ( D ^ { + } U ( k , s ) , D ^ { - } U ( k , s ) ) } { D ^ { - } U ( k , s + 1 ) - D ^ { - } U ( k , s ) } , \\\\ \\Gamma ( k , s ) & = f _ { i } ( s \\Delta t , - k \\Delta x ) . \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\theta _ { x _ 0 , \\kappa ^ i r } = p ( x _ 0 ) , \\lim _ { i \\to \\infty } \\Theta _ { x _ 0 , \\kappa ^ i r } = D u ( x _ 0 ) . \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} G _ { ( - \\infty , \\infty ) } ^ \\omega ( x , y ) = P ^ { \\omega } _ x ( H ( y ) < \\infty ) \\ , G _ { ( - \\infty , \\infty ) } ^ \\omega ( y , y ) . \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} \\tilde { Q } _ 0 ( F ) : = \\{ \\left ( \\begin{array} { c c } \\alpha A & A M \\\\ & ^ { t } A ^ { - 1 } \\end{array} \\right ) \\vert \\alpha \\in \\mathbb { G } _ { m , F } ( F ) , A \\in \\mbox { G L } _ { 2 , F } ( F ) , ^ { t } M = M \\} . \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} \\det \\left ( \\frac { \\partial x } { \\partial y } \\right ) = \\frac { | y | } { | x | } \\phi ' ( | y | ) \\frac { | x | ^ n } { | y | ^ n } , \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} a ( k - 1 ) = \\sum _ { p \\in { \\rm S i n g } ( \\mathcal { C } ) \\cap C _ { j } } ( r _ { p } - 1 ) , \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} ( v , \\pi ) = ( v _ 1 , \\pi _ 1 ) + ( v _ 2 , \\pi _ 2 ) , \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { ( - 1 ) ^ { n ( j + s ) } \\binom k j \\binom { k - j } s F _ n ^ { j + s } G _ { m - 2 n k + ( n - 1 ) j + ( n + 1 ) s } } } = ( - 1 ) ^ { n k } G _ m \\ , , \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} u ^ { \\prime \\prime } + \\frac { n - 1 } { r } u ^ { \\prime } + ( K _ 0 ( n - 1 ) ^ 2 / 4 + \\lambda ) u = 0 . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{gather*} \\tau _ { n } = \\frac { t } { 2 } ( 1 - \\sigma 2 ^ { - n } ) \\ , , h _ { n } = k ( 1 - \\sigma 2 ^ { - n } ) \\ , , \\\\ \\bar h _ { n } = \\frac { h _ { n } + h _ { n + 1 } } { 2 } = k ( 1 - 3 \\sigma 2 ^ { - n - 2 } ) \\ , , n \\ge 0 \\ , , \\end{gather*}"}
-{"id": "8662.png", "formula": "\\begin{align*} d ( B \\cap I | I ) = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "9925.png", "formula": "\\begin{align*} \\delta ( x ) = b _ { \\Omega _ + } \\left ( \\frac { - 1 } { 2 \\pi \\sqrt { - 1 } } \\frac 1 z \\right ) - b _ { \\Omega _ - } \\Big ( \\frac { - 1 } { 2 \\pi \\sqrt { - 1 } } \\frac 1 z \\Big ) . \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | \\mathcal { N } _ { F } ( T ) - \\frac { T } { \\pi } \\log { T } - \\frac { T } { \\pi } \\log { \\frac { q } { 2 \\pi e } } \\right | < 0 . 3 1 7 \\log { T } + 0 . 3 1 7 \\log { q } + 6 . 4 0 1 . \\end{aligned} \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} w _ \\infty = \\partial ^ \\alpha u _ \\infty = \\partial ^ \\alpha u . \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} A ( \\dot z + i E z ) = b , \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{gather*} m = \\frac { l + 1 } { 3 } \\ \\mbox { i s a M a r k o v n u m b e r . } \\end{gather*}"}
-{"id": "6103.png", "formula": "\\begin{align*} u ( t , x ) = \\sum _ { j \\in \\bar { \\mathbb { Z } } } \\gamma _ j q _ j ( t ) \\phi _ j ( x ) , \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\min & F ( x , y ) \\\\ s . t . & x \\in X \\\\ & y = \\underset { y } { a r g m i n } \\ ; f ( x , y ) \\\\ & \\begin{array} { l l } s . t . & h ( x , y ) = 0 \\\\ & y \\geq 0 \\end{array} \\end{array} , \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} A ^ \\top P ' = ( D ^ { - 1 } ) ^ \\top ( D ' ) ^ \\top P ' = ( D ^ { - 1 } ) ^ \\top \\begin{bmatrix} ( Q ' ) ^ \\top P ' \\\\ ( X ' ) ^ \\top P ' \\end{bmatrix} \\stackrel { \\eqref { i n i t i a l } } { = } ( D ^ { - 1 } ) ^ \\top \\begin{bmatrix} Q ^ \\top P \\\\ X ^ \\top P \\end{bmatrix} { = } ( D ^ { - 1 } ) ^ \\top D ^ \\top P = P . \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} L H S = & \\int p _ { \\beta } ( y , x ) \\log p _ { \\beta } ( y , x ) d y d x - \\int p _ { \\beta } ( y , x ) \\log p ( x ) d y d x \\\\ & + \\int p _ { \\beta } ( y , x ) \\log p ( y ) d y d x \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} h _ i ^ { - 1 } ( \\tau | _ { ( g _ 0 g _ 1 ) X _ l \\cap g _ 0 X _ j \\cap X _ i } \\pi ) = ( g _ 0 ^ { - 1 } \\tau ) | _ { g _ 1 X _ l \\cap X _ j \\cap g _ 0 ^ { - 1 } X _ i } \\pi . \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} \\sigma _ v \\left ( \\frac { \\chi _ \\mu ( e ( - 1 / u ) ; \\lambda ) } { \\chi _ \\mu ( e ( - 1 / u ) ; 0 ) } \\right ) = \\frac { \\chi _ \\mu ( e ( - v / u ) ; \\lambda ) } { \\chi _ \\mu ( e ( - v / u ) ; 0 ) } . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } Y _ { t } ^ { 1 , v } = \\mathbb { E } \\left [ \\int _ { t } ^ { t + \\delta } F _ { 1 } ( s , X _ { s } ^ { v } , Y _ { s } ^ { 1 , v } , Z _ { s } ^ { 1 , v } , v _ { s } ) d s \\right ] , \\\\ Y _ { t } ^ { 2 , v } = \\mathbb { E } \\left [ \\int _ { t } ^ { t + \\delta } F _ { 1 } ( s , x _ { 0 } , 0 , 0 , v _ { s } ) d s \\right ] , \\end{array} \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} Z ' _ { \\Lambda , \\epsilon } ( \\alpha ) = \\int \\det [ D _ { \\Lambda , \\epsilon } ( t ) ] ^ { - 1 / 2 } \\prod _ { j k \\in E ( \\mathbb { T } _ N ) } \\Big ( \\exp ( - e ^ { t _ { j k } } + t _ { j k } ) f _ \\alpha ( e ^ { t _ { j k } } ) d t _ { j k } \\Big ) . \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} \\phi = ( \\phi _ 1 , \\cdots , \\phi _ k ) = \\mathcal { A } ( \\phi ) : = ( \\mathcal { T } ( H _ 1 [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ) , \\cdots , \\mathcal { T } ( H _ k [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ) ) . \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t u _ h ^ { n - \\theta } , u _ h ^ { n - \\theta } \\Big { ) } + \\| \\nabla u _ h ^ { n - \\theta } \\| ^ 2 = & - \\gamma ( \\nabla \\sigma _ h ^ { n - \\theta } , \\nabla u _ h ^ { n - \\theta } ) - ( f ^ { n - \\theta } ( u _ h ) , u _ h ^ { n - \\theta } ) + ( g ^ { n - \\theta } , v _ h ) \\\\ = & - \\gamma \\| \\sigma _ h ^ { n - \\theta } \\| ^ 2 - ( f ^ { n - \\theta } ( u _ h ) , u _ h ^ { n - \\theta } ) + ( g ^ { n - \\theta } , u _ h ^ { n - \\theta } ) . \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} & \\kappa _ 1 : ( \\kappa _ 1 + 1 ) \\tilde { q } _ 1 = ( \\kappa _ 0 + 1 ) p ^ * , \\\\ & \\kappa _ 2 : ( \\kappa _ 2 + 1 ) \\tilde { q } _ 1 = ( \\kappa _ 1 + 1 ) p ^ * , \\\\ & \\kappa _ 3 : ( \\kappa _ 3 + 1 ) \\tilde { q } _ 1 = ( \\kappa _ 2 + 1 ) p ^ * , \\\\ & \\vdots \\vdots \\ , . \\end{align*}"}
-{"id": "9959.png", "formula": "\\begin{align*} \\sum _ { | I | = p , \\ , | J | = r } f ^ { I , J } ( z , t ) d z _ I d t _ J , \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} { F ( \\alpha ) - F ( 1 ) \\over \\alpha - 1 } = \\int { t ^ \\alpha - t \\over \\alpha - 1 } \\ , d \\mu ( t ) \\ \\nearrow \\ \\int t \\log t \\ , d \\mu ( t ) = S _ { t \\log t } ( \\rho \\| \\sigma ) = D ( \\rho \\| \\sigma ) . \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} G _ { L _ { n - 1 } } ( z ) & = \\mathbf { E } [ z ^ { L _ { n - 1 } } ] \\\\ & = \\mathbf { E } [ \\mathbf { E } [ z ^ { L _ { n - 1 } } \\lvert L _ { n - 2 } ] ] \\\\ & = \\mathbf { E } [ \\left ( q _ { n - 1 } + p _ { n - 1 } z \\right ) ^ { ( L _ { n - 2 } + 1 ) K - 1 } ] \\\\ & = \\left ( q _ { n - 1 } + p _ { n - 1 } z \\right ) ^ { K - 1 } G _ { L _ { n - 2 } } ( \\left ( q _ { n - 1 } + p _ { n - 1 } z \\right ) ^ { K } ) , \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\varepsilon \\Delta \\phi - \\mathcal { L } _ R ^ \\psi [ \\phi ] = M , \\quad & x \\in B _ R , \\\\ \\phi = g , & x \\in \\partial B _ R . \\end{array} \\right . \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} \\chi _ { h _ Q } ( \\sigma ) = \\chi _ Q ( - \\frac { N } { Q } z y \\lambda ( \\sigma ) ^ { - 1 } ) \\chi _ { N / Q } ( Q w x ) = \\chi _ Q ( \\lambda ( \\sigma ) ^ { - 1 } ) = \\overline { \\chi _ Q ( \\sigma ) } . \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} \\frac { R ( n + 1 ) } { R ( n ) } = \\left \\vert \\frac { Z ( n + 1 ) } { Z ( n ) } \\right \\vert . \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l } H ( x ) + C ^ T \\lambda = 0 , \\\\ \\lambda ^ T ( C x ) = 0 , \\ ; \\lambda \\geq 0 , \\ ; C x \\leq 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} | \\psi ( x , t ) | \\lesssim t ^ { - \\beta } \\| f \\| _ { * , \\beta , 2 s + \\alpha } \\sum _ { j = 1 } ^ k \\frac { 1 } { 1 + | y _ j | ^ \\alpha } . \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} N _ { 4 ( r + 2 ) } - & N _ { 4 ( r + 1 ) } + N _ { 4 r } - 1 \\\\ & = \\left ( 5 N _ { 4 ( r + 1 ) } - 2 N _ { 4 r } + N _ { 4 ( r - 1 ) } \\right ) - N _ { 4 ( r + 1 ) } + N _ { 4 r } - 1 \\\\ & = 4 N _ { 4 ( r + 1 ) } - N _ { 4 r } + N _ { 4 ( r - 1 ) } - 1 \\\\ & = 3 N _ { 4 ( r + 1 ) } + N _ { 4 ( r + 1 ) } - N _ { 4 r } + N _ { 4 ( r - 1 ) } - 1 \\\\ & = 3 N _ { 4 ( r + 1 ) } + 3 \\sum _ { k = 0 } ^ { r } N _ { 4 k } = 3 \\sum _ { k = 0 } ^ { r + 1 } N _ { 4 k } . \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{gather*} v , \\Delta ( v ) = ( R ' R ) ^ { - 1 } v \\otimes v , \\\\ S ( v ) = v , \\varepsilon ( v ) = 1 , v ^ 2 = u S ( u ) . \\end{gather*}"}
-{"id": "1348.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 B _ { n , c } '' - r ( x ) B _ { n , c } ( x ) = - 1 - c r ( x ) , \\ x < a ; \\\\ & B _ { n , c } ( a ) = B _ { n , c } ( - n ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{gather*} \\mbox { I f } \\ l \\geq 2 \\ \\mbox { a n d $ Q $ i s a f f i n e a c i c l i c q u i v e r } \\ \\Rightarrow \\abs { C _ { l } ^ { \\{ \\rm I d \\} } ( D ^ b ( Q ) ) } = 0 , \\\\ \\mbox { I f } \\ l \\geq 1 \\ \\mbox { a n d $ Q $ i s D y n k i n q u i v e r } \\ \\Rightarrow \\abs { C _ { l } ^ { \\{ \\rm I d \\} } ( D ^ b ( Q ) ) } = 0 . \\end{gather*}"}
-{"id": "1620.png", "formula": "\\begin{align*} - \\nabla \\cdot \\left ( D _ p ^ 2 L ( \\nabla u , \\cdot ) \\nabla z \\right ) = \\nabla \\cdot \\left ( \\int _ { 0 } ^ 1 \\left ( D _ p ^ 2 L ( \\nabla u { + } t \\nabla z , \\cdot ) { - } D _ p ^ 2 L ( \\nabla u , \\cdot ) \\right ) \\ , d t \\ , \\nabla z \\right ) . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t u _ H ^ { n - \\theta } , v _ H \\Big { ) } - \\gamma ( \\nabla \\sigma _ H ^ { n - \\theta } , \\nabla v _ H ) + ( \\nabla u _ H ^ { n - \\theta } , \\nabla v _ H ) + ( f ^ { n - \\theta } ( u _ H ) , v _ H ) = & ( g ^ { n - \\theta } , v _ H ) , ~ \\forall v _ H \\in L _ H , \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} \\lim _ { \\eta \\to 0 } \\int _ { - \\infty } ^ { + \\infty } e ^ { i \\gamma x } \\ , \\frac { \\partial ^ { \\alpha , \\eta } f } { \\partial | x | ^ { \\alpha } } \\ , \\mathrm { d } x = 2 C _ { \\alpha , 0 } \\ , | \\gamma | ^ { \\alpha } \\cos \\biggl ( \\frac { \\pi \\alpha } { 2 } \\biggr ) \\ , \\hat { F } ( \\gamma ) \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{gather*} ( \\Delta \\otimes \\mathrm { i d } ) ( R ) = R _ { 1 3 } R _ { 2 3 } , ( \\mathrm { i d } \\otimes \\Delta ) ( R ) = R _ { 1 3 } R _ { 1 2 } , \\\\ ( S \\otimes \\mathrm { i d } ) ( R ) = \\big ( \\mathrm { i d } \\otimes S ^ { - 1 } \\big ) ( R ) = R ^ { - 1 } , ( S \\otimes S ) ( R ) = R , \\\\ R _ { 1 2 } R _ { 1 3 } R _ { 2 3 } = R _ { 2 3 } R _ { 1 3 } R _ { 1 2 } \\end{gather*}"}
-{"id": "8935.png", "formula": "\\begin{align*} \\| \\varphi _ { _ { 0 } } \\| _ { \\mathcal { G } } ^ { 2 } = \\displaystyle \\int _ { 0 } ^ { T } \\displaystyle \\left \\langle \\varphi ( T - t ) , f \\right \\rangle _ { L ^ { 2 } ( D ) } ^ { 2 } d t \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} u _ N ( x , t ) = \\sum _ { j = 1 } ^ N u _ { N , j } ( t ) w _ j ( x ) \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} \\epsilon u _ { 0 , x _ { i } x _ { i } } ( 0 ) - H _ { i } ( 0 , 0 , u _ { 0 , x _ { i } } ( 0 ) ) & = \\epsilon u _ { 0 , x _ { 1 } x _ { 1 } } ( 0 ) - H _ { 1 } ( 0 , 0 , u _ { 0 , x _ { 1 } } ( 0 ) ) \\\\ \\sum _ { i = 1 } ^ { K } u _ { 0 , x _ { i } } ( 0 ) & = 0 \\\\ [ u _ { 0 } ] _ { 1 } + [ u _ { 0 } ] _ { 2 } + [ u _ { 0 } ] _ { 3 } & < \\infty \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} \\partial _ t P _ s ^ \\beta g ( t ) & = \\partial _ t \\left ( \\int _ 0 ^ t g ( t - r ) p ^ \\beta _ s ( r ) \\ , d r + g ( 0 ) \\int _ { - \\infty } ^ { - t } p ^ \\beta _ s ( - r ) \\ , d r \\right ) \\\\ & = \\int _ 0 ^ t g ' ( t - r ) p ^ \\beta _ s ( r ) \\ , d r \\pm g ( 0 ) p ^ \\beta _ s ( t ) . \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} ( f \\cup _ \\alpha g ) ( a _ 1 , \\ldots , a _ { m + n } ) = & ~ \\mu ( f ( \\alpha ^ { n - 1 } a _ 1 , \\ldots , \\alpha ^ { n - 1 } a _ m ) , g ( \\alpha ^ { m - 1 } a _ { m + 1 } , \\ldots , \\alpha ^ { m - 1 } a _ { m + n } ) ) \\\\ = & ~ \\mu ( \\alpha ^ { n - 1 } f ( a _ 1 , \\ldots , a _ m ) , \\alpha ^ { m - 1 } g ( a _ { m + 1 } , \\ldots , a _ { m + n } ) ) . \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} \\psi ( R ^ \\gamma _ { X , Y } \\sigma ) = R ^ N _ { X , Y } ( \\psi ( \\sigma ) ) = [ R ^ N _ { X , Y } , \\psi ( \\sigma ) ] \\ . \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} \\overline { p } _ i = \\begin{cases} p _ { i - 1 } & \\mbox { i f $ i \\equiv 1 , 3 \\pmod { 4 } $ } \\\\ p _ { i + 1 } & \\mbox { i f $ i \\equiv 0 , 2 \\pmod { 4 } $ . } \\\\ \\end{cases} \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} ( \\mathbf { I } - \\mathbf { K } - \\mathbf { K _ { e } } ) ^ { - 1 } ( g _ { k } + g _ { k , e } ) ( x ) \\sim \\sum _ { i = 0 } ^ { \\infty } ( n - k ) ^ { - 1 - i } h _ { i } \\left ( \\frac { j + k } { n - j } \\right ) \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} r V ^ { a _ s b _ t } ( X _ t ) - \\mathcal { A } V ^ { a _ t b _ t } ( X _ t ) = f ^ { a _ t b _ t } \\end{align*}"}
-{"id": "9985.png", "formula": "\\begin{align*} u ( x , \\tau ) = u _ 0 ( x + \\tau ) + \\int _ 0 ^ \\tau p ( x + \\tau - s ) u ( 0 , s ) d s . \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} \\mathbb E [ X _ { s } ^ { \\dagger } X _ { t } ^ { \\dagger } ] \\ , = \\ , \\mathbb E [ \\widetilde { X } _ { s } ^ { \\dagger } \\widetilde { X } _ { t } ^ { \\dagger } ] \\ , = \\ , e ^ { - ( t - s ) } \\int ^ { s } _ { 0 } e ^ { - 2 v } I _ { 0 } ( 2 \\sqrt { ( t - s + v ) v } ) { \\mathrm d } v \\ , ; 0 \\le s \\le t \\ , . \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} H _ { 1 , j } C ^ { I } _ { 1 } + \\sum _ { i = 2 } ^ { i = n } H _ { i , j } C ^ { I } _ { i } = 0 j \\in [ r ] . \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} \\prod _ { \\substack { 1 \\leq k \\leq r , 1 \\leq l \\leq N _ k , \\\\ i _ { k , l } = s } } e ^ { \\mathbf { a } _ { k } ( l ) \\varpi _ s } ( b _ { k , l } ) = \\prod _ { j = 1 } ^ r e ^ { d _ { j , s } \\varpi _ s } ( b _ { k ( j , s ) , m ( j , s ) } ) \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\phi _ t ( Q ) \\ , = \\ , E _ { t } ( Q ) \\ , Q \\ , E _ { t } ' ( Q ) + \\int _ { 0 } ^ t \\ , E _ { s , t } ( Q ) \\left [ R + \\phi _ { s } ( Q ) \\ , S \\ , \\phi _ { s } ( Q ) \\right ] E _ { s , t } ( Q ) ' \\ , d s \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} \\langle u s _ 0 \\rangle = N _ { A / K } ( u ) \\langle s _ 0 \\rangle \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} \\chi ( u ) \\partial _ t u + \\sum _ { j = 1 } ^ 3 A _ j ^ { \\operatorname { c o } } \\partial _ j u + \\sigma ( u ) u = f . \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} \\tilde { \\mathbf { W } } = \\mathbf { I } _ n + \\omega \\frac { \\xi \\rho } { 2 } \\mathbf { L } _ { \\mathcal { G } } , \\ , \\omega \\in \\left ( 0 . 5 , 1 \\right ) \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} p ( z ) = \\gamma \\prod _ { i = 1 } ^ m \\prod _ { j = 0 } ^ { l _ i } ( z - \\beta _ i + j \\kappa ) , \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} f _ 0 = e ^ { - \\mathcal Q _ + } f _ { m + 1 } = e ^ { - \\mathcal Q _ - } . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} L _ { 1 2 } & = - \\Delta ^ { - 1 } \\left ( b c - b d b ^ { - 1 } a + \\Delta \\right ) ( c - d b ^ { - 1 } a ) ^ { - 1 } \\\\ & = - \\Delta ^ { - 1 } b - ( c - d b ^ { - 1 } a ) ^ { - 1 } \\\\ & = - \\Delta ^ { - 1 } \\left ( b + \\Delta ( c - d b ^ { - 1 } a ) ^ { - 1 } \\right ) , \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} T ( \\alpha ( r + | c | ) , f ) = T \\left ( r + | c | + \\frac { r + | c | } { \\xi ( T ( r + | c | , f ) ) } , f \\right ) \\leq C T ( r + | c | , f ) \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D O } _ \\textup { t } ( n ) q ^ n & = q \\dfrac { f _ 2 f _ { 1 2 } ^ 2 } { f _ 6 } \\cdot \\dfrac { f _ 3 ^ 2 } { f _ 1 ^ 2 } \\\\ & = q \\dfrac { f _ 2 f _ { 1 2 } ^ 2 } { f _ 6 } \\left ( \\dfrac { f _ 4 ^ 4 f _ 6 f _ { 1 2 } ^ 2 } { f _ 2 ^ 5 f _ 8 f _ { 2 4 } } + 2 q \\dfrac { f _ 4 f _ 6 ^ 2 f _ 8 f _ { 2 4 } } { f _ 2 ^ 4 f _ { 1 2 } } \\right ) , \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} \\mu _ { i \\bar k } ^ { \\bar r } \\mu _ { \\bar r j } ^ l = \\mu _ { j \\bar k } ^ { \\bar r } \\mu _ { \\bar r i } ^ l \\ \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} \\mathrm { j } _ R ( E ) = - \\frac { 1 } { 2 } E ^ \\top \\mathbb { J } E . \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} \\mathrm { H e s s } ( f ) = 0 \\langle \\left ( \\mathrm { H e s s } ( U ) - \\rho I _ n \\right ) \\nabla f , \\nabla f \\rangle = 0 . \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} \\log _ p ( x ) = \\sum _ { n \\geq 1 } ( - 1 ) ^ { n + 1 } \\frac { ( x - 1 ) ^ n } { n } , \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} P _ { k , m } ( z , s ) = \\varphi _ { k , m } ( z , s ) + \\sum _ { n \\in \\mathbb { Z } } c _ { k , m } ( n , s ) \\mathcal { W } _ { k , n } ( y , s ) e ^ { 2 \\pi i n x } , \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} \\zeta _ { E _ 2 } ( 1 ) \\int _ { \\substack { ( x , y ) \\in \\Z _ 2 ^ 2 \\\\ x - y \\in \\Z _ 2 ^ \\times } } \\d x \\ , \\d y = ( 1 - 2 ^ { - 2 } ) ^ { - 1 } \\{ 1 \\times 1 - ( \\tfrac { 1 } { 2 } ) ^ 2 - ( \\tfrac { 1 } { 2 } ) ^ 2 \\} = \\tfrac { 2 } { 3 } \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{gather*} a _ { i j } = \\begin{cases} \\hphantom { - } 2 & i = j , \\\\ - 1 & i = j \\pm 1 , \\\\ \\hphantom { - } 0 & \\end{cases} m _ { i j } = \\begin{cases} \\hphantom { - } 1 & j = i - 1 , \\\\ - 1 & j = i + 1 , \\\\ \\hphantom { - } 0 & \\end{cases} \\end{gather*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\partial _ \\tau \\tilde { \\psi } = - ( - \\Delta ) ^ s \\tilde { \\psi } + a ( z , t ) \\cdot \\nabla _ z \\tilde { \\psi } + b ( z , t ) \\tilde { \\psi } + \\tilde { f } ( z , \\tau ) \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} | N [ c _ 1 ] \\cap N [ c _ 2 ] | = \\left \\{ \\begin{array} { l l } q , & \\hbox { i f $ d ( c _ 1 , c _ 2 ) = 1 $ ; } \\\\ 2 , & \\hbox { i f $ d ( c _ 1 , c _ 2 ) = 2 $ ; } \\\\ 0 , & \\hbox { i f $ d ( c _ 1 , c _ 2 ) > 2 $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} f _ 1 ( r ) = \\exp ( \\int _ 1 ^ r ( \\sqrt { K _ 0 } + f ( x ) ) d x ) , \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} \\langle \\Phi _ G \\left ( \\mu _ { g ^ { - 1 } } ( m _ { x _ { 1 } } ) \\right ) , \\beta \\rangle = \\langle \\Phi _ G \\left ( m _ { x _ 2 } \\right ) , \\beta \\rangle + O ( \\delta ) . \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} \\lVert ( \\widetilde { F } _ { \\eta } & ( v , \\omega _ k ) - \\xi ) ^ { - 1 } - ( \\widetilde { F } _ { \\eta } ( v , \\omega ) - \\xi ) ^ { - 1 } \\lVert \\\\ & = \\lVert ( \\widetilde { F } _ { \\eta } ( U v , f _ k ( U \\omega U ^ * ) ) - \\xi ) ^ { - 1 } - ( \\widetilde { F } _ { \\eta } ( U v , U \\omega U ^ * ) - \\xi ) ^ { - 1 } \\lVert \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} u _ { k , n } ^ { [ j ] , - } ( y ) & : = y ^ { - \\frac { k } { 2 } } \\frac { \\partial ^ j } { \\partial s ^ j } W _ { \\mathrm { s g n } ( n ) \\frac { k } { 2 } , s - \\frac { 1 } { 2 } } ( 4 \\pi | n | y ) \\bigg | _ { s = \\frac { k } { 2 } } , \\\\ u _ { k , n } ^ { [ j ] , + } ( y ) & : = y ^ { - \\frac { k } { 2 } } \\frac { \\partial ^ j } { \\partial s ^ j } \\mathcal { M } ^ + _ { \\mathrm { s g n } ( n ) \\frac { k } { 2 } , s - \\frac { 1 } { 2 } } ( 4 \\pi | n | y ) \\bigg | _ { s = \\frac { k } { 2 } } , \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} w _ 0 ^ \\pm = ( \\beta + 2 \\log \\beta ) H _ \\gamma ^ \\pm + \\tilde H ^ \\pm , \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} \\beta ( y ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( \\frac { 3 2 } { 9 } y ^ { 6 } - \\frac { 3 1 } { 3 } y ^ { 4 } + \\frac { 1 5 } { 2 } y ^ { 2 } - \\frac { 3 7 } { 4 8 } \\right ) e ^ { - 2 y ^ { 2 } } \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} \\le \\ , 2 C _ { T } \\int ^ { T } _ { 0 } \\sum _ { i = 1 } ^ { n } \\sup _ { 0 \\le t \\le s } \\lvert X _ { t , i } ^ { ( u ) } - \\overline { X } _ { t , i } \\rvert { \\mathrm d } s + 2 C _ { T } u \\int ^ { T } _ { 0 } \\lvert \\overline { X } _ { s , n + 1 } - \\overline { X } _ { s , 1 } \\rvert { \\mathrm d } s \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} Q _ 1 = \\frac { 1 } { 4 \\pi } \\sum _ { ( j , k , l , m ) \\in \\Delta _ 1 } \\gamma _ j \\gamma _ k \\gamma _ l \\gamma _ m q _ j \\bar { q } _ k q _ l \\bar { q } _ m , \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} C ( t ) = \\max \\limits _ { \\mathbb { P } ( x ) } I ( X ; Y | h ( t ) ) , \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\frac 1 { L ^ d } \\int _ 0 ^ T \\Psi ^ \\star _ L \\bigl ( \\mu ^ L _ t , F ^ V ( \\mu ^ L _ t ) \\bigr ) \\ ; \\ ! \\mathrm d t \\\\ = \\frac 1 2 \\int _ 0 ^ T \\| \\Delta \\phi ( \\rho _ t ) + \\nabla \\cdot ( \\chi ( \\rho _ t ) \\nabla V ) \\| _ { - 1 , \\chi ( \\rho _ t ) } ^ 2 \\ ; \\ ! \\mathrm d t . \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} u ( r ) = \\sum _ { j = 0 } ^ { \\infty } c _ j r ^ { j } . \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} \\mathcal { C } _ i ( p ) & : = \\{ ( p _ l , g ) \\in \\{ p _ 1 , \\dots , p _ m \\} \\times H ^ { - s } ( W ) \\subset \\widetilde { H } ^ s ( W ) \\times H ^ { - s } ( W ) , \\ ( p _ l , g ) \\in \\mathcal { C } _ i \\} \\subset \\mathcal { C } _ i . \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } \\mu \\prod _ { i = 1 } ^ { j } P _ i ( \\epsilon ) r = \\mu \\sum _ { j = 0 } ^ { n - 1 } \\tilde { P } ^ j r + \\epsilon \\mu \\sum _ { i = 1 } ^ { n - 2 } i \\tilde { P } ^ i \\tilde { P } ^ { ( 1 ) } \\sum _ { j = i + 1 } ^ { n - 1 } \\tilde { P } ^ { j - i - 1 } r + O ( \\epsilon ^ 2 n ^ 4 ) \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} C = e ^ { \\lambda _ 1 } \\left ( p _ 1 ^ { - 1 } p _ { 1 , 1 } p _ { 1 , 2 } p _ { 1 , 3 } \\right ) e ^ { \\lambda _ 2 } \\left ( p _ 2 ^ { - 1 } p _ 1 ^ { - 1 } p _ { 1 , 1 } p _ { 1 , 2 } p _ { 1 , 3 } p _ { 2 , 1 } \\right ) . \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} \\sup _ { \\mu , \\psi ( \\cdot ) , \\eta ( \\cdot ) \\in \\R ^ 1 \\times C ^ 1 \\times C ^ 1 } \\mu = d ( y _ 0 , \\epsilon ) \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} \\sum _ k \\sum _ { p \\in \\mathcal { P } _ { j , k } } \\frac { 1 } { p z _ u ( p ) } = \\sum _ { k > j / 2 } \\sum _ { p \\in \\mathcal { P } _ { j , k } } \\frac { 1 } { p z _ u ( p ) } + \\sum _ { k \\le j / 2 } \\sum _ { p \\in \\mathcal { P } _ { j , k } } \\frac { 1 } { p z _ u ( p ) } \\ll 2 ^ { - j / 2 } = y ^ { - a _ j / 2 } . \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} s p ( w ) = \\lvert A ^ + \\rvert + \\lvert A ^ - \\rvert + \\begin{cases} 0 , & \\\\ 1 , & \\\\ 2 , & \\end{cases} . \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\partial _ { t } W ( t , x ) + \\inf \\limits _ { u \\in U } H ( t , x , W ( t , x ) , D W ( t , x ) , D ^ { 2 } W \\left ( t , x \\right ) , u ) = 0 , \\\\ W ( T , x ) = \\phi ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} \\| { \\bf r } ^ k \\| _ 2 ^ 2 / \\sigma ^ 2 = \\chi ^ 2 _ { n - k } \\left ( \\dfrac { \\| ( { \\bf I } _ n - { \\bf P } _ k ) { \\bf X } _ { [ k _ 0 ] / [ k ] } \\boldsymbol { \\beta } _ { [ k _ 0 ] / [ k ] } \\| _ 2 ^ 2 } { \\sigma ^ 2 } \\right ) k < k _ 0 \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} \\det \\left [ \\binom { 2 ( n + i + j ) + 1 } { n + i + j } \\right ] _ { i , j = 0 } ^ { r - 1 } = \\mu _ 0 ^ r b _ 1 ^ { r - 1 } b _ 2 ^ { r - 2 } \\dotsm b _ { r - 2 } ^ 2 b _ { r - 1 } , \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} \\mathsf { I m } \\ , ( e ^ { \\lambda g } X _ j ^ * f _ j X _ j \\varphi , e ^ { \\lambda g } \\varphi ) = \\mathsf { I m } \\ , ( f _ j X _ j \\varphi , X _ j ( e ^ { 2 \\lambda g } \\varphi ) ) \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} | E _ 3 ( k ) | \\ll k ^ { - 1 / 2 } \\sum _ { \\substack { d _ 1 , d _ 2 \\in \\N \\\\ n = d _ 1 d _ 2 \\\\ d _ 1 \\neq d _ 2 } } | { \\bf B } ( \\nu ; ( d _ 1 - d _ 2 ) ^ { 2 } ) | \\ , C _ { \\nu _ \\infty , d _ 1 , d _ 2 } \\left ( \\tfrac { 4 \\sqrt { d _ 1 d _ 2 } } { ( \\sqrt { d _ 1 } + \\sqrt { d _ 2 } ) ^ 2 } \\right ) ^ { k } , \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} D ^ \\alpha { } _ \\beta = \\delta ^ \\alpha { } _ \\beta + \\partial A ^ \\alpha / \\partial x ^ \\beta \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} K ( \\pi ) : = R ( u , v , u , v ) , \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} \\Delta ( L _ i ) & = L _ i \\otimes L _ i & \\varepsilon ( L _ i ) & = 1 & S ( L _ i ) & = L _ i ^ { - 1 } , \\\\ \\Delta ( F _ i ) & = L _ i ^ { - 1 } \\otimes F _ i + F _ i \\otimes 1 & \\varepsilon ( F _ i ) & = 0 & S ( F _ i ) & = - L _ i F _ i . \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} \\begin{aligned} j = & - \\frac { N } { 2 } + 1 , - \\frac { N } { 2 } + 2 , \\dots , \\frac { N } { 2 } \\ \\ \\ ( N \\ e v e n ) \\\\ j = & - \\frac { N } { 2 } + \\frac { 1 } { 2 } , - \\frac { N } { 2 } + \\frac { 3 } { 2 } , \\dots , \\frac { N } { 2 } - \\frac { 1 } { 2 } \\ \\ \\ ( N \\ o d d ) \\end{aligned} \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} W _ j : = \\begin{pmatrix} \\binom { L + 1 } { e _ 1 } \\alpha _ j ^ { L - e _ 1 } & \\binom { L + 1 } { e _ 2 } \\alpha _ j ^ { L - e _ 2 } & \\cdots & \\binom { L + 1 } { e _ M } \\alpha _ j ^ { L - e _ M } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\binom { L + \\nu _ j } { e _ 1 } \\alpha _ j ^ { L - e _ 1 } & \\binom { L + \\nu _ j } { e _ 2 } \\alpha _ j ^ { L - e _ 2 } & \\cdots & \\binom { L + \\nu _ j } { e _ M } \\alpha _ j ^ { L - e _ M } \\end{pmatrix} _ { \\ ! \\nu _ j \\times M } , j = 1 , \\ldots , m , \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} \\exp _ { \\psi } ( g ) = \\sum _ { j } \\ell _ j ( z _ 1 ) \\dots \\ell _ j ( z _ n ) t ^ j \\alpha _ j \\tau ^ j ( g ) , \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} \\partial _ n w _ 0 ^ \\pm \\pm a _ 0 \\lambda \\mu _ \\lambda = \\partial _ n \\tilde H ^ \\pm = \\mathcal O ( 1 ) \\ \\hbox { o n } \\ \\gamma , \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} \\tilde { b } : = \\tilde { \\Psi } \\circ ( \\tilde { h } - \\tilde { \\mathfrak { s } } ) \\circ \\tilde { \\psi } ^ { - 1 } : \\tilde { U ' } \\rightarrow \\tilde { U ' } \\lhd ( \\mathbb { R } ^ { k ' } \\times \\mathbb { R } ^ n ) \\times \\tilde { \\mathbb { W } } \\end{align*}"}
-{"id": "9680.png", "formula": "\\begin{align*} L ( \\varphi , \\mathbb { A } ) = \\sum \\limits _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) } { a } = \\frac { \\log _ { \\phi } ( \\omega _ n ) } { \\omega _ n } . \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} \\int _ { x _ 0 } ^ x \\sin ( 2 \\pi \\ell y ) \\frac { \\sin ( 2 \\tilde { \\theta } ( y , { E } ) - 2 \\tilde { \\theta } ( y , \\hat { E } ) ) } { ( 1 + y - b ) } d y = \\frac { 1 } { \\ell } O \\left ( \\frac { 1 } { x _ 0 - b } \\right ) , \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} y = ( x _ { 1 } + . . . + x _ { k } ) - ( x _ { 1 } + . . . + x _ { k } ) ^ { 3 } / 3 k + \\epsilon . \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} | \\ ! | P ^ * u | \\ ! | _ 0 ^ 2 \\geq \\gamma _ 0 \\sum _ { j = 0 } ^ N | \\ ! | X _ j u | \\ ! | _ 0 ^ 2 + C | \\ ! | u | \\ ! | _ 0 ^ 2 \\geq C ' | \\ ! | u | \\ ! | _ { 1 / r } ^ 2 , \\ , \\ , \\forall u \\in C _ 0 ^ \\infty ( K ) , \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} n _ { s , b } \\cdot z _ 0 = \\frac { 1 } { \\frac { 1 } { 2 } ( - i s + | b | ^ 2 ) + 1 } \\left ( - i b _ 1 , \\ldots , - i b _ { n - 1 } , \\frac { 1 } { 2 } ( i s - | b | ^ 2 ) \\right ) \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} X ( t ) = X _ 0 + \\int _ { 0 } ^ { t } b ( X ( s ) ) d s + \\sigma B ( t ) , \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} \\Psi ^ \\star _ L ( \\mu , F ) : = \\sum _ { \\eta , \\eta ' \\in \\Omega _ L } a _ { \\eta , \\eta ' } ( \\mu ) \\Bigl [ \\cosh \\bigl ( \\tfrac 1 2 F _ { \\eta , \\eta ' } \\bigr ) - 1 \\Bigr ] \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ L ^ { A - S } _ { d , t } \\right ] = \\mathbb { E } \\left [ L ^ { A } _ { t - d } \\circ \\theta _ { d } \\right ] \\cdot \\mathbb { E } \\left [ L ^ { - S } _ { t } \\right ] = 1 . \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} F _ t : = \\{ ( s , t ) \\colon s t \\in E ( T ) \\} . \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} & \\mathcal { D } ( p _ { _ 1 } | | p _ { _ 0 } ) + \\rho _ 0 \\left [ \\int _ { - \\infty } ^ { \\infty } p _ { _ 1 } ( y ) d y - 1 \\right ] \\\\ & + \\rho _ 1 \\left [ \\int _ { - \\infty } ^ { \\infty } y ^ 2 p _ { _ 1 } ( y ) d y - P _ y \\right ] = \\int _ { - \\infty } ^ { \\infty } \\mathcal { L } ( y , p _ { _ 1 } ( y ) ) d y - \\tau , \\end{align*}"}
-{"id": "9661.png", "formula": "\\begin{align*} \\mathcal { T } h = e ^ { - \\Delta ^ { 2 } t } h _ { 0 } + \\sum _ { j = 2 } ^ { \\infty } I ^ { + } F _ { j } , \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} h t ( { \\mathcal F } ) & = \\sum _ i h t ( { \\mathcal T } _ i ) , \\\\ h t ( \\mathcal T ) & = \\sum _ { x \\in \\mathcal T } h t ( x ) \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} \\alpha = \\frac { p } { n - p } \\frac { \\theta - 1 } { \\theta } , q = 1 + \\frac { n - p } { p } \\frac { \\theta } { \\theta - 1 } , C = R \\alpha ^ { \\alpha } \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} u \\circ f _ { M , M ' } = f _ { M , M ' } \\circ \\Psi ( u ) . \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } I _ { \\{ X _ i = o \\} } = 0 \\hbox { \\rm a . s . } \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} \\alpha = [ E ] = \\sum _ { i } ^ m [ A _ i ] . \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} C _ { \\ref { E q u a t i o n L i p s c h i t z E s t i m a t e F o r F l o w O n F i r s t T i m e I n t e r v a l } } = C _ { \\ref { E q u a t i o n L i p s c h i t z E s t i m a t e F o r F l o w O n F i r s t T i m e I n t e r v a l } } ( \\chi , \\sigma , m , r , \\tilde { T } , \\kappa ) = C _ { \\ref { E q u a t i o n D i f f e r e n c e O f N o n l i n e a r S o l u t i o n s I n G m m i n u s 1 } } ( \\chi , \\sigma , m , r , R ( \\chi , \\sigma , m , r , \\kappa ) , \\tilde { U } _ \\kappa , \\tilde { T } ) \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} q _ { \\rm S V C } ^ * \\triangleq \\max _ { \\mathbf p } & q _ { { \\rm S V C } , \\infty } ( \\mathbf T ) \\\\ s . t . \\ & \\eqref { e q n : c o n s t r a i n t _ 0 _ 1 } , \\eqref { e q n : c o n s t r a i n t _ s u m } . \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} \\sup \\limits _ { \\| f _ 1 \\| _ { \\widetilde { H } ^ { s } ( W ) } = 1 = \\| f _ 2 \\| _ { \\widetilde { H } ^ s ( W ) } } \\int \\limits _ { W } ( ( \\Lambda _ { q _ 1 } - \\Lambda _ { q _ 2 } ) f _ 1 , f _ 2 ) d x & \\leq \\| \\Lambda _ { q _ 1 } - \\Lambda _ { q _ 2 } \\| _ { L ^ 2 ( W ) \\rightarrow L ^ 2 ( W ) } \\| f _ 1 \\| _ { L ^ 2 ( W ) } \\| f _ 2 \\| _ { L ^ 2 ( W ) } \\\\ & \\leq C _ p \\| \\Lambda _ { q _ 1 } - \\Lambda _ { q _ 2 } \\| _ { L ^ 2 ( W ) \\rightarrow L ^ 2 ( W ) } , \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} b _ j = - \\Im i \\biggl [ \\langle F ( Q + v ) - F ( Q ) , D _ j Q \\rangle - \\mu \\langle v , D _ j ( | Q | ^ p Q ) \\rangle \\biggr ] \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} h ( \\phi X , Y ) = C h ( \\phi X , \\phi Y ) + \\phi Q \\overline { \\nabla } ^ { ' } _ { \\phi X } \\phi Y . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } K _ { t _ 1 } ( m , n ) K _ { t _ 2 } ( m , j ) = K _ { t _ 1 + t _ 2 } ( n , j ) , n , j \\ge 0 . \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} \\tfrac { d } { d t } g _ t = - { \\rm M } ( g _ t ) \\ , , g _ { | t = 0 } = g _ 0 \\ , , \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} x _ s = \\frac { 2 } { \\Omega } \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} \\sum _ { d \\mid n } \\mu ( d ) a _ { n m / d } = \\sum _ { d \\in S _ n } \\mu ( d ) a _ { n m / d } = \\sum _ { d \\in T _ n } \\mu ( d ) ( a _ { n m / d } - a _ { n m / d p _ i } ) . \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} \\psi _ { \\alpha } ( x ) = \\alpha \\int _ { K _ 1 } G ( x , y ) d y - \\int _ { K _ 2 } G ( x , y ) d y . \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} \\begin{cases} m ' _ { F ( x ) , F ( y ) , F ( z ) } \\left ( F _ { x , y } \\otimes F _ { y , z } \\right ) & = F _ { x , z } m _ { x , y , z } \\ , \\\\ u _ { F ( x ) } = F _ { x , x } u _ x \\ , \\end{cases} \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} | | u ( t ) | | _ 2 ^ { \\theta ' / ( 1 - \\alpha ) } & \\leq C | | u ( t ) | | _ { 1 , p } ^ { \\theta ' / ( 1 - \\alpha ) } = C ( | | u ( t ) | | _ { 1 , p } ^ p ) ^ { \\theta ' / p ( 1 - \\alpha ) } \\\\ & \\leq C ( G ( t ) + | | u ( t ) | | _ { 1 , p } ^ p ) \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} \\dim ( \\mathcal { C } ) ^ 2 \\dim ^ R ( \\bar { \\ 1 } ) ^ 2 T = \\tau ^ { - } ( \\mathcal { C } ) S ^ { R , R } T ^ { - 1 } S ^ { R , R } T ^ { - 1 } S ^ { R , R } , \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathbb { E } _ { z \\sim \\mu } f ( g ( \\theta _ t , z ) ) = & \\frac { d } { d t } \\int _ { Z } f ( g ( \\theta _ t , z ) ) \\mu ( z ) d z \\\\ = & \\frac { d } { d t } \\int _ \\Omega f ( x ) \\rho ( \\theta _ t , x ) d x \\\\ = & \\int _ \\Omega f ( x ) \\frac { \\partial } { \\partial t } \\rho ( \\theta _ t , x ) d x , \\end{align*}"}
-{"id": "9871.png", "formula": "\\begin{align*} ( u , \\boldsymbol { D } ^ { - \\mu } v ) = ( \\boldsymbol { D } ^ { - \\mu * } u , v ) . \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} \\min & \\sum _ { t = 0 } ^ { N - 1 } [ \\frac { 1 } { 2 } x _ t ^ T Q _ t x _ t + \\frac { 1 } { 2 } u _ t ^ T R _ t u _ t ] + \\frac { 1 } { 2 } x _ N ^ T Q _ N x _ N \\\\ & x _ { t + 1 } = A x _ t + B u _ t , t = 0 , 1 , \\cdots , N - 1 \\\\ & l _ x \\leq x _ t \\leq u _ x , t = 0 , 1 , \\cdots , N \\\\ & l _ u \\leq u _ t \\leq u _ u , t = 0 , 1 , \\cdots , N - 1 \\\\ & x _ 0 = b _ 0 \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\sup _ { \\pi \\in \\Pi } E ^ \\pi _ \\mu \\left ( \\sum _ { k = n } ^ \\infty \\beta ^ { k - 1 } w ( s _ { k } ) \\right ) = 0 , \\end{align*}"}
-{"id": "9797.png", "formula": "\\begin{align*} N = ( 1 - e ^ { - t } ) + 2 S ( t ) \\Big ( S ( t ) + C ( t ) \\Big ) & = 2 t + ( 1 - \\frac { r _ 1 ^ 2 } { 6 } ) t ^ 3 - \\frac { 1 + r _ 1 ^ 2 } { 2 4 } t ^ 4 + \\mathcal { O } ( r _ 1 ^ 4 t ^ 5 ) \\\\ & = t \\Big ( 2 + ( 1 - \\frac { r _ 1 ^ 2 } { 6 } ) t ^ 2 - \\frac { 1 + r _ 1 ^ 2 } { 2 4 } t ^ 3 + \\mathcal { O } ( ( r _ 1 t ) ^ 4 ) \\Big ) ~ . \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} \\mathfrak { S } ( M ) = \\left \\{ N \\in \\mathrm { P G L } ( 2 , \\mathbb { R } ) : N ( \\gamma _ M ) = \\gamma _ M \\right \\} \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} f ' _ x ( t , x ) = - \\left ( 1 - \\frac { m } { 2 } \\right ) \\frac { x ^ { - m / 2 } } { ( 2 t ) ^ { ( n - m ) / 2 } } \\psi \\left ( \\frac { x } { 2 t } \\right ) - \\frac { x ^ { 1 - m / 2 } } { ( 2 t ) ^ { ( n - m ) / 2 } } \\psi ' \\left ( \\frac { x } { 2 t } \\right ) , t > 0 , x > 0 , \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} ( \\beta , 1 ) = v _ 1 + v _ 2 , \\ v _ i = ( \\beta _ i , n _ i ) \\in \\Gamma _ { \\le 1 } . \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} A _ { 1 , s - 2 } = \\sum _ { r = 1 } ^ { n } \\frac { S _ { 0 r r } - S _ { 0 0 r } } { r ^ 2 } - \\sum _ { r = 2 } ^ { n } \\sum _ { l = 1 } ^ { r - 1 } \\frac { S _ { 0 l r } + S _ { 0 r l } } { r - l } \\left ( \\frac { 1 } { r } - \\frac { 1 } { l } \\right ) , \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} \\o _ { \\tilde \\nu _ D } ( u , b ^ * J _ { \\nu _ D } u ) & = \\o _ { \\tilde \\nu _ D } ( u , ( b _ * ) ^ { - 1 } J _ { \\nu _ D } b _ * u ) \\\\ & = \\o _ { \\tilde \\nu _ D } ( u , J _ { \\tilde F } u ) = \\O _ { \\tilde F } ( u , J _ { \\tilde F } u ) > 0 \\ , . \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\lambda _ n p _ n = \\hat { q } _ { n - 1 } & \\lambda _ { n + 1 } p _ { n + 1 } = \\hat { q } _ n \\\\ \\lambda _ n q _ n = a _ 1 \\hat { q } _ { n - 1 } + \\hat { p } _ { n - 1 } & \\lambda _ { n + 1 } q _ { n + 1 } = a _ 1 \\hat { q } _ n + \\hat { p } _ n \\end{array} \\right . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} L ^ { p } = \\mathcal { M } _ { p } ^ { p } \\subset \\mathcal { M } _ { q _ { 1 } } ^ { p } \\subset \\mathcal { M } _ { q _ { 2 } } ^ { p } \\quad \\ , \\ , p \\geq q _ { 1 } \\geq q _ { 2 } > 0 . \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{gather*} x _ { w B ^ { m _ 1 } A ^ { n _ 1 } \\dots B ^ { m _ k } A ^ { n _ k } } = j _ { w B ^ { m _ 1 } A ^ { n _ 1 } \\dots B ^ { m _ k } A ^ { n _ k } } ( x ) . \\end{gather*}"}
-{"id": "4446.png", "formula": "\\begin{align*} [ c , b ] \\Delta ^ { - 1 } & = a R _ { 1 1 } + b R _ { 2 1 } \\\\ [ b , a ] \\Delta ^ { - 1 } & = a R _ { 1 2 } + b R _ { 2 2 } \\\\ [ c , d ] \\Delta ^ { - 1 } & = c R _ { 1 1 } + d R _ { 2 1 } \\\\ [ d , a ] \\Delta ^ { - 1 } & = c R _ { 1 2 } + d R _ { 2 2 } . \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ \\infty k ^ { 2 ( m + 1 ) } | c _ k ( f ) | ^ 2 \\leq \\sum _ { \\substack { k = - \\infty \\\\ k \\neq 0 } } ^ \\infty k ^ { 2 m + 2 } \\frac { C } { | k | ^ { 2 + 2 m + 2 \\mu } } = \\sum _ { \\substack { k = - \\infty \\\\ k \\neq 0 } } ^ \\infty \\frac { C } { | k | ^ { 2 \\mu } } < \\infty . \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} [ L ^ 2 , E ] _ { \\frac { 2 } { 3 } } = ( L ^ 2 , E ) _ { \\frac { 2 } { 3 } , 2 } = { \\lbrace u \\in L ^ 2 , \\ ; | D _ { q _ i } | ^ { \\frac { 2 } { 3 } } u \\in L ^ 2 , \\ ; | \\partial _ { q _ i } V ( q _ i ) | ^ { \\frac { 2 } { 3 } } u \\in L ^ 2 \\ ; \\ ; \\ ; 1 \\le i \\le d \\rbrace ~ , } \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} f _ 1 = x _ 0 ^ 2 , f _ 2 = 2 x _ 0 x _ 1 , \\ldots , f _ { n } = \\sum _ { i = 0 } ^ { n - 1 } x _ { i } x _ { n - 1 - i } . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} A _ 0 = I _ { ( \\ell + 1 ) n } , A _ 1 = I _ { \\ell + 1 } \\otimes A , A _ 2 = I _ { \\ell + 1 } \\otimes ( J _ n - A - I _ n ) , \\bar { L } = A _ 3 - A _ 4 . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} n ( r ; \\sigma + i t ) \\leq \\sum _ { i = 1 } ^ { 1 + [ r ] } n _ L ( t _ i ) & \\leq \\alpha _ 8 \\sum _ { i = 1 } ^ { 1 + [ r ] } \\{ \\log d _ L + n _ L \\log ( | t _ i | + 2 ) \\} \\\\ & \\leq \\alpha _ 8 ( 1 + r ) \\{ \\log d _ L + n _ L \\log ( | t | + r + 2 ) \\} . \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} l ( w ) : = \\vert \\mathfrak { r } ^ + ( w ) \\vert , \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} ( 1 - ( - t ) ^ { n / d } ) ^ d \\cdot \\sum _ { j = 0 } ^ { n - 1 } \\frac { j \\omega _ n ^ { i ( j - 1 ) } t } { 1 + t \\omega _ n ^ { i j } } = \\sum _ { r = 0 } ^ { n } \\omega _ n ^ { i ( \\binom { r } { 2 } - 1 ) } t ^ r \\Big ( 1 _ { n \\mid d r } \\cdot \\binom { r } { 2 } \\binom { d } { d r / n } + \\omega _ n ^ i { n \\brack r } ^ { ' } _ { \\omega _ n ^ i } \\Big ) . \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} p _ { n } & = a _ { n } p _ { n - 1 } + p _ { n - 2 } ; & p _ 0 & = 0 & p _ 1 & = 1 \\\\ q _ { n } & = a _ { n } q _ { n - 1 } + q _ { n - 2 } ; & q _ 0 & = 1 & q _ 1 & = a _ 1 \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| \\nabla v _ n \\| _ { L ^ 2 } = \\| \\nabla V \\| _ { L ^ 2 } , \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} h _ 1 ^ k ( x ) & : = \\lambda _ k f _ 1 ( x ^ k , x ) + \\frac { 1 } { 2 } \\| x - x ^ k \\| ^ 2 , \\\\ h _ 2 ^ k ( x ) & : = \\lambda _ k f _ 2 ( x ^ k , x ) + \\frac { 1 } { 2 } \\| x - y ^ k \\| ^ 2 . \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} A ( t ) \\begin{cases} \\exp ( - t ^ { \\beta _ 0 } ) & , \\\\ \\exp ( t ^ { \\beta _ \\infty } ) & , \\end{cases} \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} u _ { k + 1 } ( t , 1 ) = \\dots = u _ { m } ( t , 1 ) = 0 . \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} \\varphi ^ \\alpha ( x ) = x ^ \\alpha + A ^ \\alpha ( x ) + O ( | A | ^ 2 ) \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | \\mathcal { N } _ { F } ( T , 2 T ) - A _ F T \\log { T } - ( A _ F \\log { 4 } + B _ F ) T \\right | \\\\ & < c _ { F , 1 } ( T _ 0 ) \\log { T } + c _ { F , 2 } ( T _ 0 ) + \\frac { c _ { F , 3 } ( T _ 0 ) } { T } , \\end{aligned} \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} V ( x ) = V _ 0 ( x ) + \\frac { 1 } { \\epsilon } V _ 1 ( x ) \\ , , \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} \\tilde u ^ i = \\frac { l ^ i ( { \\ u } ) } { l ( { \\ u } ) } , ~ ~ ~ \\tilde g = \\frac { g } { l ^ 4 ( { \\ u } ) } , ~ ~ ~ \\tilde w = \\frac { w } { l ^ 2 ( { \\ u } ) } , \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( \\partial _ t - \\partial _ x ^ 2 - x ^ 2 \\partial _ y ^ 2 ) f ( t , x , y ) = 0 & t \\in [ 0 , T ] , ( x , y ) \\in \\Omega _ L , \\\\ f ( t , x , y ) = 0 & t \\in [ 0 , T ] , ( x , y ) \\in \\partial \\Omega _ L \\backslash \\Gamma _ L , \\\\ f ( t , x , y ) = v ( t , y ) & t \\in [ 0 , T ] , ( x , y ) \\in \\Gamma _ L , \\\\ f ( 0 , x , y ) = f _ 0 ( x , y ) & ( x , y ) \\in \\Omega _ L , \\end{array} \\right . \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} g _ { P } + ( x - 1 ) f _ { P } = ( a _ { m } - a _ { m + 1 } ) x ^ { m + 1 } + ( a _ { m + 1 } - a _ { m + 2 } ) x ^ { m + 2 } + \\dotsb \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} \\Delta _ { g _ S } u = \\frac { 1 } { 2 } u ^ 3 | d \\theta | _ { g _ S } ^ 2 . \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} \\begin{cases} X _ 1 q _ j & = - p _ { j } \\\\ X _ 2 q _ j & = 0 \\end{cases} \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} \\sum _ { j = 1 , 2 , 3 } | | \\mathbf { T } ^ j | | _ { L ^ p \\to L ^ p } + | | \\mathbf { T } ^ j | | _ { L ^ 1 \\to L ^ { 1 , \\infty } } \\lesssim c _ 1 + c _ 2 . \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} G : = \\mathbb { G } _ m \\times _ { \\mbox { R e s } _ { F | \\mathbb { Q } } \\mathbb { G } _ { m , F } } \\tilde { G } , \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} \\frac { Q _ { i j } ^ 2 } { C _ { i j } ^ 2 } = \\left ( \\frac { P _ j - P _ i } { L _ { i j } } \\right ) ^ 2 \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} \\Pi ( x ) = x _ k x \\in [ x - \\Delta x / 2 , x + \\Delta x / 2 ) \\Pi ( x ) = x _ 0 x < x _ 0 \\ , , \\ ; \\Pi ( x ) = x _ { N _ x } x \\ge x _ { N _ x } \\ , . \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 A _ n '' ( x ) - r ( x ) A _ n ( x ) = - 1 - r ( x ) A _ n ( 0 ) , \\ x \\in ( - n , a ) ; \\\\ & A _ n ( a ) = A _ n ( - n ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 N ^ 2 } \\Big ( \\sum _ { j = 1 } ^ { N } h _ { j } \\Big ) H _ { N } + \\frac { 1 } { 2 N ^ 2 } H _ { N } \\Big ( \\sum _ { j = 1 } ^ { N } h _ { j } \\Big ) \\\\ & = \\frac { 1 } { N ^ 2 } \\Big ( \\sum _ { j = 1 } ^ { N } h _ { j } \\Big ) ^ 2 - \\frac { a _ { N } } { 2 N ^ 2 ( N - 1 ) } \\sum _ { i = 1 } ^ { N } \\sum _ { 1 \\leq j < k \\leq N } \\big ( h _ { i } | x _ j - x _ k | ^ { - 1 } + | x _ j - x _ k | ^ { - 1 } h _ { i } \\big ) . \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} G ( x ) = ( \\xi ( x ) , \\phi ( x ) ) \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} \\alpha + \\omega ^ { l + 1 } \\cdot ( 1 + \\mu _ 1 ) + \\rho ~ & = ~ \\alpha _ i + \\omega ^ { l + 1 } \\cdot \\mu _ 0 + \\sigma + \\omega ^ { l + 1 } + \\omega ^ { l + 1 } \\cdot \\mu _ 1 + \\rho \\\\ & = ~ \\alpha _ i + \\omega ^ { l + 1 } \\cdot ( \\mu _ 0 + 1 + \\mu _ 1 ) + \\rho ~ = ~ \\alpha _ { i + 1 } , \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} \\frac { K _ n ' ( t ) } { \\sinh ( t ) ^ { n - 1 } } & = 1 - \\frac { ( ( n + 1 ) s + n ) ( ( n - 3 ) s + n + 2 ) + 2 ( n - 3 ) s ( s + 1 ) } { ( n - 1 ) ( n - 3 ) s ^ 2 + 2 n ( n - 1 ) s + n ( n + 2 ) } \\\\ & + 4 ( n - 1 ) \\frac { s ( s + 1 ) ( ( n - 3 ) s + n + 2 ) ( ( n - 3 ) s + n ) } { ( ( n - 1 ) ( n - 3 ) s ^ 2 + 2 n ( n - 1 ) s + n ( n + 2 ) ) ^ 2 } \\\\ & = \\frac { 2 4 s ^ 2 } { ( ( n - 1 ) ( n - 3 ) s ^ 2 + 2 n ( n - 1 ) s + n ( n + 2 ) ) ^ 2 } \\\\ & \\geq 0 . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} v _ { + , a } ( x ) = 2 \\phi _ 3 ( x ) \\frac { \\int _ { - \\infty } ^ x d y \\thinspace \\phi ^ { - 2 } _ 3 ( y ) \\int _ x ^ a d t \\phi ^ { - 2 } _ 3 ( t ) \\int _ y ^ t \\phi _ 3 ( z ) d z } { \\int _ { - \\infty } ^ a \\phi ^ { - 2 } _ 3 ( y ) d y } , \\ x \\le a ; \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} \\prod _ { k = 0 } ^ M A _ k ^ + - \\prod _ { k = 0 } ^ M A _ k ^ - = \\sum _ { \\ell = 0 } ^ M \\bigg ( \\prod _ { k = 0 } ^ { \\ell - 1 } A _ k ^ - \\bigg ) \\big ( A _ \\ell ^ + - A _ \\ell ^ - \\big ) \\bigg ( \\prod _ { k = \\ell + 1 } ^ M A _ k ^ + \\bigg ) , \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{gather*} V ( G \\sqcup H ) = V ( G ) \\sqcup V ( H ) \\\\ E ( G \\sqcup H ) = E ( G ) \\sqcup E ( H ) . \\end{gather*}"}
-{"id": "9445.png", "formula": "\\begin{align*} \\hat { k } ( \\xi ) = c _ { 0 } c _ { 0 } ' \\frac { \\pi ^ { 2 } \\csc \\left ( \\pi \\left ( \\frac { 1 } { 2 } - \\beta + i \\xi \\right ) \\right ) \\csc \\left ( \\pi \\left ( \\frac { 1 } { 2 } - \\beta - i \\xi \\right ) \\right ) } { \\Gamma ( 1 + 2 \\beta ) \\Gamma ( 1 - 2 \\beta ) } . \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } _ t = f ( \\xi _ t , \\theta _ t ^ * ) , \\\\ \\dot { p } _ t = - \\nabla _ x f ( x _ t , \\theta ^ * _ t ) p _ t - \\nabla _ x L ( x _ t , \\theta ^ * _ t ) . \\end{cases} \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} - D _ { \\infty } ^ { \\beta } \\tilde u ( t , x ) = & \\ \\int _ 0 ^ { \\infty } ( \\tilde u ( t - r , x ) - \\tilde u ( t , x ) ) \\ , \\nu ( r ) d r \\\\ = & \\ \\int _ 0 ^ { t } ( \\tilde u ( t - r , x ) - \\tilde u ( t , x ) ) \\ , \\nu ( r ) d r + \\int _ t ^ { \\infty } \\phi ( t - r , x ) \\ , \\nu ( r ) d r \\\\ & \\quad - \\tilde u ( t , x ) ) \\int _ t ^ \\infty \\nu ( r ) d r \\pm \\phi ( 0 , x ) \\int _ t ^ \\infty \\ , \\nu ( r ) d r \\\\ = & \\ - D _ { 0 } ^ { \\beta } \\tilde u ( t , x ) + f _ \\phi ( t , x ) . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ n \\mu _ i ^ { r _ j } x _ i \\right | = \\sup _ { r \\geq g } \\left \\{ \\left | \\sum _ { i = 1 } ^ n \\mu _ i ^ { r } x _ i \\right | \\right \\} > 0 \\mbox { f o r a l l } j \\geq 0 . \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} j ( n , k ) = \\frac { { ( n + k ) } ^ 2 + n + 3 k } { 2 } \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} g _ { P } + ( x - 1 ) f _ { P } = x ^ { n + 1 } g _ { P } ( \\tfrac 1 x ) \\ , . \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} \\phi b ^ i s \\sim \\phi b s - \\sum _ { j = 1 } ^ { i - 1 } \\sum _ { s ' \\in S _ b } \\phi s ' b ^ j s . \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} E _ t ( Q ) \\ , = \\ , e ^ { t B } \\ , ( P _ { \\infty } - P _ { \\infty } ^ - ) \\ , \\left [ e ^ { 2 t B } \\ , ( P _ { \\infty } - P _ { \\infty } ^ - ) + ( Q - P _ { \\infty } ^ - ) \\left ( I - e ^ { 2 t B } \\right ) \\right ] ^ { - 1 } \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} a _ { \\eta , \\eta ' } ( \\mu ) : = 2 \\bigl [ \\mu ( \\eta ) r ^ V _ { \\eta , \\eta ' } \\mu ( \\eta ' ) r ^ V _ { \\eta ' , \\eta } \\bigr ] ^ { 1 / 2 } \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} \\left ( \\int _ { \\mathbb R ^ { n } } | f ( x ) | ^ { 2 } d x \\right ) ^ { 2 } \\leq \\left ( \\frac { 2 } { n } \\right ) ^ { 2 } \\int _ { \\mathbb R ^ { n } } \\left | \\nabla f ( x ) \\right | ^ { 2 } d x \\int _ { \\mathbb R ^ { n } } \\| x \\| ^ { 2 } | f ( x ) | ^ { 2 } d x , n \\geq 2 , \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} \\varphi ( u \\wedge v ) \\big ( ( \\vec x _ 1 , \\vec x _ 2 ) , \\dots , ( \\vec x _ { 2 n - 1 } , \\vec x _ { 2 n } ) \\big ) = \\big ( m _ { u , v } ( \\vec x _ 1 , \\vec x _ 2 ) , \\dots , m _ { u , v } ( \\vec x _ { 2 n - 1 } , \\vec x _ { 2 n } ) \\big ) , \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} \\nu ^ w _ { l } ( U _ l ( t ) ) & = \\frac 1 { | \\mathcal P _ l | } \\sum _ { q \\in U _ l ( t ) } \\sum _ { \\{ p \\colon q \\in Q _ l ( p ) \\} } \\frac { 1 } { | Q _ l ( p ) | } \\\\ & \\leqslant \\frac { 1 } { | \\mathcal P _ { l } | } { \\sum _ { p \\in V ^ + _ l ( t ) } } \\sum _ { q \\in Q _ l ( p ) } \\frac { 1 } { | Q _ l ( p ) | } = \\nu _ { \\mathcal P _ { l } } ( V ^ + _ l ( t ) ) . \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} \\big [ \\tilde { \\phi } ( \\tilde { A } / f \\tilde { A } ) \\big ] _ { \\tilde { A } } = \\det _ { \\mathbb { F } _ q ( z ) [ X ] } ( X - \\tilde { \\phi } _ { \\theta } \\mid \\tilde { A } / f \\tilde { A } \\otimes _ { \\mathbb { F } _ q ( z ) } \\mathbb { F } _ q ( z ) [ X ] ) _ { | X = \\theta } . \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } g ( x ) { \\mathrm m } _ { t } ( { \\mathrm d } x ) \\ , = \\ , \\int _ { \\mathbb R } g ( x ) { \\mathrm m } _ { 0 } ( { \\mathrm d } x ) + \\int ^ { t } _ { 0 } [ \\widehat { \\mathcal A } _ { s } ( \\widehat { \\mathrm M } ) g ] \\ , { \\mathrm d } s \\ , ; 0 \\le t \\le T \\ , , \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} \\dot { x } _ t & = f ( x _ t , \\theta _ t ) \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\begin{aligned} & \\overline { \\textbf { d e t } } ( v \\diamond v ) = \\overline { \\textbf { d e t } } ( x ) \\overline { \\textbf { d e t } } ( s ) , \\\\ & \\textbf { T r } ( v \\diamond v ) = \\textbf { T r } ( x \\diamond s ) , \\\\ & \\lambda ^ 2 _ 1 ( v ) = \\lambda _ 1 \\left ( x \\diamond s \\right ) = \\lambda _ 1 ( x ) \\lambda _ 1 ( s ) . \\end{aligned} \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} F = \\{ j _ 1 , j _ 1 + 1 , \\ldots , j _ 1 + ( t - 1 ) , j _ 2 , j _ 2 + 1 , \\ldots , j _ 2 + ( t - 1 ) , \\ldots , j _ { d - 1 } , j _ { d - 1 } + 1 , \\ldots , j _ { d - 1 } + ( t - 1 ) \\} \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} ( ( K ( \\xi - d \\Gamma _ A ( m ) ) & + a \\cdot \\nabla K ( \\xi - d \\Gamma _ A ( m ) ) + E _ { \\xi , A } ( a ) ) \\psi ) ^ { ( n ) } \\\\ & = K ( \\xi + a - m ^ { ( n ) } ) \\psi ^ { ( n ) } \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} \\vec { x } = ( x _ 1 , \\ldots , x _ a ) , \\ \\vec { y } = ( y _ 1 , \\ldots , y _ b ) \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} \\int _ G f ( x ) d \\tilde { \\mu } & = \\int _ G f ( x ) Q \\big ( g ( q ( x ) \\big ) d \\tilde { \\mu } ( x ) \\\\ & = \\int _ G f ( x ) \\int _ K \\int _ H g ( k ^ { - 1 } x h ) d h d k d \\tilde { \\mu } ( x ) \\\\ & = \\int _ K \\int _ H \\int _ G f ( k x h ^ { - 1 } ) g ( x ) \\Delta _ K ( k ) \\Delta _ H ( h ) d \\tilde { \\mu } ( x ) d h d k \\\\ & = \\int _ G g ( x ) \\int _ H \\int _ K f ( k ^ { - 1 } x h ) d k d h d \\tilde { \\mu } ( x ) \\\\ & = \\int _ G g ( x ) Q ( f ) \\big ( q ( x ) \\big ) d \\tilde { \\mu } ( x ) = 0 . \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} [ v ( \\cdot , S ^ 1 ) , S ^ 2 ] _ t = \\int _ 0 ^ t \\nabla v ^ * ( r , S ^ 1 _ r ) \\mathrm d [ M ^ 1 , M ^ 2 ] _ r . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} \\mathfrak { s } _ p : = \\mathfrak { g } [ { \\Sigma } ^ o \\backslash \\pi ^ { - 1 } ( p ) ] ^ \\Gamma \\oplus \\mathbb { C } C , \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} \\omega _ k \\psi = \\omega _ k P _ { c , k } \\psi = ( c + 1 ) 2 ^ { - k } P \\psi _ { c , g , k } = ( c + 1 ) 2 ^ { - k } \\psi . \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} h _ { 0 , j } = & ~ ( f \\cup _ \\alpha ( h \\circ _ { j - m } g ) ) ( a _ 1 , \\ldots , a _ { m + n + p - 1 } ) j = m , m + 1 , \\ldots , m + p - 1 \\\\ h ' _ { i , m + i - 1 } = & ~ ( - 1 ) ^ { m + i - 1 } ( h \\circ _ { i - 1 } ( f \\cup _ \\alpha g ) ) ( a _ 1 , \\ldots , a _ { m + n + p - 1 } ) i = 1 , 2 , \\ldots , p \\\\ h '' _ { i , m + p } = & ~ ( - 1 ) ^ { m + n + p - 1 } ( ( h \\circ _ { i - 1 } f ) \\cup _ \\alpha g ) ( a _ 1 , \\ldots , a _ { m + n + p - 1 } ) i = 1 , 2 , \\ldots , p . \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} { C N ( X ) \\otimes _ { \\boldsymbol { [ 0 , \\infty ) } } A _ { [ 0 , \\epsilon ] } } _ n = \\Z \\left [ \\left \\{ ( x _ 0 , \\dots , x _ n ) \\mid \\sum _ { i = 0 } ^ n d ( x _ i , x _ { i + 1 } ) \\leq \\epsilon \\right \\} \\right ] \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} \\sum _ { i \\in I } v _ { i } ^ { 2 } \\langle \\pi _ { Z _ { i } } C ^ { \\prime } f , \\pi _ { Z _ { i } } C f \\rangle = \\Vert v _ { i } ( C ^ { * } \\pi _ { { Z } _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f \\Vert ^ { 2 } _ { 2 } & \\leq ( \\dfrac { ( 1 + \\lambda _ { 1 } ) \\sqrt { B } + \\Vert \\beta \\Vert _ { 2 } } { 1 - \\lambda _ { 2 } } \\Vert f \\Vert ) ^ { 2 } . \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} \\{ f , g \\} ( \\psi ) = \\frac 1 { 2 \\hbar } \\operatorname { I m } \\left \\langle \\frac { \\delta f } { \\delta \\psi } \\bigg | \\frac { \\delta g } { \\delta \\psi } \\right \\rangle = \\left \\langle \\frac { \\delta f } { \\delta \\psi } , - \\frac { i } { 2 \\hbar } \\frac { \\delta g } { \\delta \\psi } \\right \\rangle . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} X _ i = [ [ \\dots [ [ X _ 2 , X _ { i _ 1 } ] , X _ { i _ 2 } ] , \\dots , X _ { i _ { m - 1 } } ] , X _ { i _ m } ] , \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} U H _ \\mu ( \\xi ) U ^ * = H _ \\mu ( \\xi , A ) \\oplus \\bigoplus _ { n = 1 } ^ \\infty H _ { n , \\mu } ( \\xi , A ) \\mid _ { \\mathcal { F } ( \\mathcal { H } _ A ) \\otimes \\mathcal { H } _ { A ^ c } ^ { \\otimes _ s n } } \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} f _ { j , n } = ( f _ j + \\varepsilon \\phi _ { j , 1 } ) \\ast \\phi _ { j , n } . \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} Y _ i & : = \\{ y \\in Y ; c ' ( h _ 0 ^ { - 1 } , y ) = g _ i ^ { - 1 } \\} , \\\\ Y _ j & : = \\{ y \\in Y ; c ' ( h _ 1 ^ { - 1 } , y ) = g _ j ^ { - 1 } \\} . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} M _ k ( t , \\mathbf { h } ) = E \\Bigg [ \\sum _ { j = 1 } ^ { \\xi _ 0 ( t ) } e ^ { - k \\alpha \\sigma _ j } M _ k ( t - \\sigma _ j , \\mathbf { h } ) \\Bigg ] + R , \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} F _ { \\max } : = \\max _ { t \\in [ 0 , T ] } \\{ F ( t ) \\} . \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { f ( t ) } { g ( t ) } = 1 , \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mbox { m e a s } ( \\Omega _ { b , K } \\cap B _ r ( x _ n ) ) = 0 . \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} [ v , - \\Delta _ d v ] = \\sum _ { x y \\in E } ( v _ y - v _ x ) ^ 2 . \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} P f : = \\sum _ \\alpha a _ \\alpha ( z ) \\partial _ z ^ \\alpha f \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} ( V U ) _ { i , j } = \\sum _ { k = 1 } ^ { \\infty } \\zeta _ { 1 } ( n - j + k ) \\zeta _ { 2 } ( - n + i - k ) + o ( n ^ { - 1 - M } ) . \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} n ( x ) \\cdot \\mathbb { P } [ \\mu ] \\nabla p ( x ) = 0 \\quad \\mbox { f o r } x \\in \\partial \\Omega , \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} \\eta _ { j , R } \\partial _ t \\tilde { \\phi } _ j = \\eta _ { j , R } \\left [ - ( - \\Delta ) ^ s \\tilde { \\phi } _ j + p U _ j ^ { p - 1 } \\tilde { \\phi } _ j + p U _ j ^ { p - 1 } \\psi + S _ { \\mu , \\xi , j } \\right ] \\mathbb { R } ^ n \\times ( t _ 0 , \\infty ) , \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} & 2 \\langle \\nabla _ w v , \\partial _ t \\rangle = d \\xi ( v , w ) = - u ^ 2 d \\theta ( v , w ) \\\\ & \\xi ( [ v , w ] ) = \\langle \\nabla _ v w - \\nabla _ w v , \\partial _ t \\rangle = 2 \\langle \\nabla _ v w , \\partial _ t \\rangle = u ^ 2 d \\theta ( v , w ) . \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\left | ( 1 - \\eta _ { j , R } ) S _ { \\mu , \\xi , j } \\right | \\lesssim \\left ( \\frac { 1 } { R ^ { n - 2 s - a } } + \\frac { 1 } { R ^ { 4 s - a } } \\right ) \\frac { 1 } { R ^ { a - 2 s } } \\sum _ { j = 1 } ^ k \\frac { \\mu _ j ^ { - 2 s } \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } + \\sigma } } { 1 + | y _ j | ^ { a } } . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} \\dim Z _ t ( \\phi ) = \\frac { \\sqrt { | U ^ \\perp / U | } } { | N \\cap ( 1 + \\j ) / N \\cap ( 1 + \\i ) | } . \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} u ( x , t ) = u _ { \\Omega } ( x ) + \\epsilon e ^ { \\lambda t } \\chi ( x ) . \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\mathcal { D } ( p _ { _ 0 } | | p _ { _ 1 } ) = \\frac { 1 } { 2 } \\left ( \\frac { \\sigma _ w ^ 2 } { P _ y } - 1 + \\log \\frac { P _ y } { \\sigma _ w ^ 2 } \\right ) . \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\zeta _ { f , g } ( s ) = \\prod _ { \\varpi } \\left ( 1 - \\exp \\left [ - s \\cdot \\mathrm { p e r } ( \\varpi ) \\right ] \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} \\mathbf { M } _ { \\mathbf { m } , \\Phi } \\psi = \\displaystyle \\sum m _ { k } \\left \\langle \\psi | \\phi _ { k } \\right \\rangle \\phi _ { k } , \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} \\varepsilon _ 0 : = \\min \\bigg \\{ \\frac { 1 } { 2 C _ 1 } , \\frac { 1 } { 4 C _ 1 ^ 3 C _ Q ^ 2 C _ { \\frac { d } { 4 } , \\frac { d } { 2 } + \\eta } K } , \\frac { 1 } { 4 C _ 1 ^ 3 C _ Q ^ 2 C _ { \\frac { d } { 4 } , \\frac { d } { 2 } + \\eta } K } \\bigg \\} \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} \\tilde { L } _ { \\bar { \\theta } } : = \\frac { 1 } { M + 1 } L _ { \\bar { \\theta } } . \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} \\| L _ G f \\| _ \\infty & = \\sup _ { v \\in V } | L f ( v ) | \\\\ & \\leq \\sup _ { v \\in V } | f ( v ) | + \\sum _ { v \\sim v ' } | f ( v ' ) | \\\\ & \\leq ( D + 1 ) \\| f \\| _ \\infty . \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} s \\rho _ \\infty ( e _ \\lambda ) = s \\rho _ \\infty ( \\gamma _ F ( e _ \\lambda ) ) \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} g _ i = \\varphi \\left ( \\frac { \\partial } { \\partial x _ i } x _ 1 ^ { k _ 1 } \\dots x _ n ^ { k _ n } \\right ) , i = 1 , \\dots , n . \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} u _ { 1 } ( T _ { 2 } - \\delta , x ) = \\dots = u _ { k - 1 } ( T _ { 2 } - \\delta , x ) = 0 \\mbox { f o r } x \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( m , n ) = \\int _ { 0 } ^ { \\infty } x ^ { m } \\frac { \\cos ( \\pi n x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} \\varphi ( x ) + \\varphi _ - ( x ) = \\varphi _ 0 ( B _ 0 x ) + \\varphi _ + ( x ) - \\varphi _ { m + 1 } ( B _ { m + 1 } x ) \\ge \\varepsilon ' | x | ^ 2 - C _ 1 - C _ 2 , \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| x ^ k - z ^ k \\| = 0 . \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} g ^ { V } : = g ^ { V } ( 0 ) , & & g ^ N : = g ^ { V } \\oplus \\pi ^ * g ^ B . \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\begin{array} { c } m i n \\Big \\{ m a x \\Big [ r V ^ { i j } ( x ) - \\mathcal { A } V ^ { i j } ( x ) - f ( x , i , j ) ; \\qquad \\qquad \\qquad \\qquad \\\\ V ^ { i j } ( x ) - N ^ { i j } [ V ] ( x ) \\Big ] ; V ^ { i j } ( x ) - M ^ { i j } [ V ] ( x ) \\Big \\} = 0 . \\end{array} \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} s = \\sum _ { j = 1 } ^ r c ^ { ( j ) } _ 1 c ^ { ( j ) } _ 2 \\cdots c ^ { ( j ) } _ { N _ j } \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} f ( t ) - \\sum _ { k = 1 } ^ { p - 1 } ( - 1 ) ^ { k r } b _ k \\frac { t ^ { k r } } { ( k r ) ! } = \\frac { 1 } { 2 \\pi i } \\int _ { 1 - \\infty \\ , i } ^ { 1 + \\infty \\ , i } e ^ { t u } \\left ( \\frac { \\varphi ( u ) } { u } - \\sum _ { k = 1 } ^ { p - 1 } \\frac { ( - 1 ) ^ { k r } b _ k } { u ^ { k r + 1 } } \\right ) \\ , d u . \\end{align*}"}
-{"id": "9933.png", "formula": "\\begin{align*} \\omega _ j = \\left \\{ \\begin{array} { l l } \\tau _ { j , \\ , { 0 1 } } \\qquad & , \\\\ \\tau _ { j , \\ , { 1 } } \\qquad & , \\end{array} \\right . \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} F _ X ( u ) = u \\frac { Z _ X ' ( u ) } { Z _ X ( u ) } . \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha _ t = a _ V e ^ { a _ X } ( e ^ { - a _ X } - a _ V e ^ { a _ X } t ) , \\\\ \\beta _ t = - \\log ( e ^ { - a _ X } - a _ V e ^ { a _ X } t ) . \\end{aligned} \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} E \\Bigg [ \\sum _ { j = 1 } ^ { \\xi _ 0 ( t ) } e ^ { - k \\alpha \\sigma _ j } M _ k ( t - \\sigma _ j , \\mathbf { h } ) \\Bigg ] & = E \\Bigg [ \\int _ 0 ^ t e ^ { - k \\alpha s } M _ k ( t - s , \\mathbf { h } ) \\ , \\xi ( \\d s ) \\Bigg ] \\\\ & = \\int _ 0 ^ t e ^ { - k \\alpha s } M _ k ( t - s , \\mathbf { h } ) \\ , \\mu ( \\d s ) . \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} \\psi ( \\theta ( F ) ) ( s ) = \\sum _ { ( s _ p ) } F ( [ s _ { 1 , p - 1 } | E Z ( [ s _ { p } ' ] \\otimes s _ p '' ) | s _ { p + 1 , r } ] ) . \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} F _ t [ \\Q ] : = \\int _ { \\Omega } \\left \\{ \\frac { \\alpha } { p } \\abs { \\nabla \\Q } ^ p + T ( t ) \\left ( 1 - | \\Q | ^ 2 \\right ) ^ 2 + H ( t ) g ( \\Q ) \\right \\} \\d V , \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} M _ { i j } = \\left < \\phi _ i \\phi _ j \\right > _ F - \\left < \\phi _ i \\right > _ F \\left < \\phi _ j \\right > _ F . \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} \\mathbf { W } = \\mathbf { I } _ n - \\frac { \\xi \\rho } { 2 } \\mathbf { L } _ { \\mathcal { G } } \\\\ \\tilde { \\mathbf { W } } = \\mathbf { I } _ n - \\frac { \\xi \\rho } { 2 } ( 1 - \\eta ) \\mathbf { L } _ { \\mathcal { G } } . \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} T _ i ^ { 1 } | _ \\mathcal { P } & = \\mathsf { A d d D o t s } _ { \\gamma _ { i + 1 } } ( T _ i ^ { 0 } ) | _ \\mathcal { P } = \\mathsf { A d d D o t s } _ { \\theta _ { i + 1 } } ( T _ i ^ { 0 } ) | _ \\mathcal { P } \\\\ & = \\mathsf { A d d D o t s } _ { \\theta _ { i + 1 } } ( T _ i ^ { 0 } | _ \\mathcal { P } ) = \\mathsf { A d d D o t s } _ { \\theta _ { i + 1 } } ( C _ i ^ { 0 } | _ \\mathcal { P } ) \\\\ & = C _ i ^ 1 | _ \\mathcal { P } , \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} \\Pi _ { \\Gamma } ( t ) : = \\# \\{ \\overline { \\gamma } \\in \\overline { \\Gamma } : \\ell ( \\gamma ) \\leq t \\} \\leq C e ^ { \\delta t } . \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} g ( \\omega , x + b ) - g ( \\omega , x ) = g ( \\tau _ x \\omega , b ) , \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 1 } ^ r [ \\mathcal { F } _ { D , m } ( z , s ) ] ^ { \\mathrm { h o l } } = ( \\ref { a d d t r } ) + ( \\ref { s q t r } ) + ( \\ref { t r } ) . \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} \\sum _ { t _ 1 , \\ldots , t _ n } \\langle f ( h _ { t _ 1 } , \\ldots , h _ { t _ n } ) , ( g _ 0 ^ { - 1 } \\tau ) | _ { [ t _ 1 , \\ldots , t _ n ] } \\pi \\rangle = \\sum _ { t _ 0 , \\ldots , t _ n } \\langle h _ { t _ 0 } f ( h _ { t _ 1 } , \\ldots , h _ { t _ n } ) , \\tau | _ { [ t _ 0 , \\ldots , t _ n ] } \\pi \\rangle . \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} \\# \\bigr ( \\mathbb X \\cap { \\mathbb B } \\bigl ) \\ , : = \\ , \\sum _ { z \\in { \\mathbb B } } \\mathbb X ( z ) \\quad \\# { \\mathbb X } \\ , : = \\ , \\sum _ { z \\in [ 0 , 1 ] ^ d } \\mathbb X ( z ) \\ , , \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} \\begin{aligned} \\min \\ & \\sum _ { P \\in \\mathcal { P } } c _ P x _ P \\\\ & \\sum _ { P \\in \\mathcal { P } } \\alpha _ { v , P } \\cdot x _ P = 1 & & v \\in V \\\\ & x _ P \\in \\{ 0 , 1 \\} & & P \\in \\mathcal { P } \\\\ \\end{aligned} \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} A = \\partial _ x ^ { } \\Big ( g ^ { i j } \\partial _ x ^ { } + c ^ { i j } _ k u ^ k _ x + \\sum _ { \\alpha = 1 } ^ N w ^ i _ { \\alpha k } u ^ k _ x \\partial _ x ^ { - 1 } w ^ j _ { \\alpha l } u ^ l _ x \\Big ) \\partial _ x ^ { } , \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} - \\Delta _ { g } \\bigl ( h _ { 1 , \\lambda } ( r ) \\bigr ) = - h _ { 1 , \\lambda } ^ { \\prime \\prime } ( r ) - \\frac { n - 1 } { r } h _ { 1 , \\lambda } ^ { \\prime } ( r ) = ( K _ 0 ( n - 1 ) ^ 2 / 4 + \\lambda ) h _ { 1 , \\lambda } ( r ) \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} H = - \\tfrac 1 2 \\partial _ x ^ 2 + q \\delta ( x ) , q < 0 . \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} \\phi _ j ( s ) = \\frac { 1 } { \\sqrt { T - t } } \\begin{cases} 1 , \\ & \\ { \\rm i f } \\ j = 0 \\\\ ~ \\\\ \\sqrt { 2 } { \\rm s i n } ( 2 \\pi r ( s - t ) / ( T - t ) ) , \\ & \\ { \\rm i f } \\ j = 2 r - 1 \\\\ ~ \\\\ \\sqrt { 2 } { \\rm c o s } ( 2 \\pi r ( s - t ) / ( T - t ) ) , \\ & \\ { \\rm i f } \\ j = 2 r \\end{cases} , \\ \\ \\ r = 1 , \\ 2 , \\ldots \\end{align*}"}
-{"id": "9762.png", "formula": "\\begin{align*} V ( q ) = \\sum _ { i = 1 } ^ d \\frac { \\nu _ i } { 2 } q ^ 2 _ i { \\ , . } \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} c ( S ) = \\psi ( S ) - p _ a ( S ) + 1 \\ , . \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} u _ { 1 2 } u _ { 2 3 } - u _ { 1 3 } u _ { 2 2 } & = u _ { 1 2 } ( u _ { 2 3 } - u _ { 1 3 } u _ { 2 1 } ) - u _ { 1 3 } ( u _ { 2 2 } - u _ { 1 2 } u _ { 2 1 } ) \\\\ & = u _ { 1 2 } ( ( 1 + a _ { 1 1 } ) u _ { 2 3 } - u _ { 1 3 } u _ { 2 1 } ) - u _ { 1 3 } ( ( 1 + a _ { 1 1 } ) u _ { 2 2 } - u _ { 1 2 } u _ { 2 1 } ) \\\\ & - a _ { 1 1 } ( u _ { 1 2 } u _ { 2 3 } - u _ { 1 3 } u _ { 2 2 } ) , \\\\ \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| v _ n \\| ^ 2 _ { L ^ 2 } = M \\lim _ { n \\rightarrow \\infty } E ( v _ n ) = d _ { M , } . \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & a & b \\\\ 0 & 1 & c \\\\ 0 & 0 & 1 \\end{pmatrix} , & \\intertext { a n d w e h a v e t h a t } \\begin{pmatrix} 1 & a & b \\\\ 0 & 1 & c \\\\ 0 & 0 & 1 \\end{pmatrix} ^ 3 & = \\begin{pmatrix} 1 & 3 a & 3 b + 3 a c \\\\ 0 & 1 & 3 c \\\\ 0 & 0 & 1 \\end{pmatrix} \\equiv \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} \\begin{aligned} & \\mu ^ 2 \\overline { \\textbf { d e t } } ( v \\diamond v ) = \\overline { \\textbf { d e t } } ( x ) \\overline { \\textbf { d e t } } ( s ) , \\\\ & \\mu \\textbf { T r } ( v \\diamond v ) = \\textbf { T r } ( x ) \\textbf { T r } ( s ) , \\\\ & \\mu \\lambda _ 1 ( v \\diamond v ) = \\lambda _ 1 ( x ) \\lambda _ 1 ( s ) . \\end{aligned} \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} C _ K = D _ K \\left ( \\frac { 1 } { 2 } \\right ) ^ { \\left ( \\frac { n - 1 } { 2 } \\right ) } . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} S _ f ( \\rho \\| \\sigma ) = \\lim _ \\alpha \\biggl [ S _ f ( e _ \\alpha \\rho e _ \\alpha \\| e _ \\alpha \\sigma e _ \\alpha ) + \\sigma ( 1 - e _ \\alpha ) f \\biggl ( { \\rho ( 1 - e _ \\alpha ) \\over \\sigma ( 1 - e _ \\alpha ) } \\biggr ) \\biggr ] , \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} h _ { \\beta } ( x ) : = f _ { - 1 , \\sigma } ( x ) = \\dfrac { \\beta } { \\beta ^ 2 + x ^ 2 } . \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} V _ 1 = \\frac { 1 } { \\sqrt { 2 } } \\bigl ( E ^ { ( 2 ) } _ { 1 1 } + E ^ { ( 2 ) } _ { 2 2 } \\bigr ) , V _ 2 = \\frac { 1 } { \\sqrt { 2 } } \\bigl ( \\sqrt { - 1 } \\ , E ^ { ( 2 ) } _ { 1 2 } + E ^ { ( 2 ) } _ { 2 1 } \\bigr ) . \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} ( \\Sigma _ h ^ { n - \\theta } , w _ h ) + ( \\nabla U _ h ^ { n - \\theta } , \\nabla w _ h ) = 0 , ~ \\forall w _ h \\in L _ h , \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta u + b u + c \\rho ( x ) \\phi u = d | u | ^ { p - 1 } u , & x \\in \\R ^ 3 , \\\\ \\ , \\ , \\ , - \\Delta \\phi = \\rho ( x ) u ^ 2 , & x \\in \\R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "9781.png", "formula": "\\begin{align*} \\kappa _ 0 ( i \\sigma ) \\kappa _ 0 - ( \\kappa ^ * ) ^ { - 1 } ( i \\sigma ) ( \\kappa ) ^ { - 1 } = \\kappa _ 0 ( \\delta ) ( i \\sigma ) \\kappa _ 0 ( \\delta ) - \\kappa ^ * ( - t ) ( i \\sigma ) \\kappa ( - t ) \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} T _ { 1 } ^ { N } = \\nabla P _ { 0 } - \\frac { \\tau } { N } D ^ { 2 } P _ { 0 } u _ { 0 } ^ { N } + \\frac { \\tau } { N } J ( \\nabla P _ { 0 } - \\mathrm { i d } _ { \\Omega } ) , \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} f g = T _ f g + T _ g f + R ( f , g ) \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} R ( \\pi _ \\lambda ( h _ 1 ) f ) ( h ) & = j _ \\lambda ( h , z _ 0 ) ( \\pi _ \\lambda ( h _ 1 ) f ) ( h \\cdot z _ 0 ) \\\\ & = j _ \\lambda ( h , z _ 0 ) j _ \\lambda ( h _ 1 ^ { - 1 } , h \\cdot z _ 0 ) f ( h _ 1 ^ { - 1 } h \\cdot z _ 0 ) \\\\ & = j _ \\lambda ( h _ 1 ^ { - 1 } h , z _ 0 ) f ( h _ 1 ^ { - 1 } h \\cdot z _ 0 ) \\\\ & = ( \\rho _ \\lambda ( h _ 1 ) R f ) ( h ) \\ , . \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} - \\sum _ { m = 1 } ^ { N } e ^ { - r \\tau _ m } C ( \\xi _ { m - 1 } , \\xi _ m ) \\leq \\max _ { k \\in \\mathcal { I } } ( - C ( i , k ) ) \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} f ( x ) & = ( x ^ 2 + x + 1 ) + m \\ : h ( x ) , \\\\ g ( x ) & = ( x ^ 3 + 1 ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} q '' ( \\lambda ) = \\left < ( t _ { j k } - \\left < t _ { j k } \\right > _ \\lambda ) ^ 2 \\right > _ \\lambda \\le h ( \\alpha ) ^ { - 1 } / 2 . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} ( \\mathrm { d } A ) _ { \\alpha \\beta } = \\partial _ { x ^ \\alpha } A _ \\beta - \\partial _ { x ^ \\beta } A _ \\alpha \\ , . \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} \\tilde { F } _ k = F _ k + \\rho _ k ( - \\mathbf { G } ) \\subseteq V _ k , \\textup { f o r $ k = 0 , 1 , \\ldots , m ^ + $ } . \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} u _ { k + 1 } = S \\left ( \\mathcal { N } ( u _ k ) \\ , A ^ * \\frac { y } { A u _ k } \\right ) , k = 0 , \\dots , \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} g _ * ' = - 1 + ( 2 - ( 2 k + 1 ) \\ln 2 ) / 2 ^ { k + 1 } + ( 1 . 6 8 \\times ( 3 k + 3 ) ( \\ln 2 ) - 2 . 5 2 ) / 2 ^ { 2 k + 3 } ~ . \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} ( x ^ { k + 1 } , y ^ { k + 1 } ) g ( x ^ k , y ^ k ) + \\nabla g ( x ^ k , y ^ k ) \\begin{pmatrix} x ^ { k + 1 } - x ^ k \\\\ y ^ { k + 1 } - y ^ k \\end{pmatrix} + G ( x ^ { k + 1 } , y ^ { k + 1 } ) \\ni 0 . \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} \\psi ( \\zeta m ) = \\psi _ { } ( \\zeta _ { } m ) \\psi _ { \\infty } ( 0 ) = \\psi _ { } ( \\zeta _ { } m ) = \\psi _ { \\infty } \\left ( - \\zeta _ 0 m \\right ) . \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} \\left ( 1 - \\kappa _ 1 \\rho _ { } ( i ) \\right ) x _ { i } = \\kappa _ 2 , \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} \\iint _ { X \\times X } U ( x , y ) \\frac { d \\mu ( x ) d \\mu ( y ) } { d ^ \\gamma ( x , y ) } & = \\int _ { d ( x , x _ 0 ) < R } \\int _ { d ( y , x _ 0 ) \\geq 2 R } U ( x , y ) \\frac { d \\mu ( x ) d \\mu ( y ) } { d ^ \\gamma ( x , y ) } \\\\ & + \\int _ { d ( x , x _ 0 ) < 2 R } \\int _ { d ( y , x _ 0 ) < 2 R } U ( x , y ) \\frac { d \\mu ( x ) d \\mu ( y ) } { d ^ \\gamma ( x , y ) } \\\\ & + \\int _ { d ( x , x _ 0 ) \\geq 2 R } \\int _ { d ( y , x _ 0 ) < R } U ( x , y ) \\frac { d \\mu ( x ) d \\mu ( y ) } { d ^ \\gamma ( x , y ) } \\\\ & = I + I I + I I I . \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} J _ { \\mu } ^ { ( \\alpha ) } ( X ) = \\lim _ { t \\to 1 } J _ { \\mu } ( X ; t ^ { \\alpha } , t ) / ( 1 - t ) ^ n , \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} \\mathcal { I } _ { a } ( x ) : = \\frac { 1 } { \\Gamma \\left ( \\frac { a } { \\nu } \\right ) } \\int _ { 0 } ^ { \\infty } t ^ { \\frac { a } { \\nu } - 1 } h _ { t } ( x ) d t \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} \\Psi ^ * ( \\widetilde { T } _ V f ) ( z , w ) & = \\widetilde { T } _ V f ( \\Psi ( z , w ) ) \\cdot \\det \\ ; \\Psi ' ( z , w ) \\\\ & = \\left ( \\int _ { \\R } f \\left ( t , \\frac { w _ 1 } { z ^ { 1 / 2 m _ 1 } } , \\dots , \\frac { w _ n } { z ^ { 1 / 2 m _ n } } \\right ) \\frac { z ^ { 2 \\pi i t } } { z } \\d t \\right ) \\frac { 1 } { z ^ { 1 / 2 \\mu } } \\\\ & = T _ V f ( z , w ) , \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} \\lambda \\mathcal { L } ^ d \\left ( \\left \\{ \\mathbf { T } _ { \\varepsilon } ^ { 1 , n } ( \\mu ) > \\lambda \\right \\} \\cap B _ R \\right ) \\leq 9 \\sum _ { i = 1 } ^ { 9 } A _ { i } ( \\lambda / 9 , \\varepsilon ) . \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} T _ i ( E _ j ) & = \\begin{dcases} - F _ i L _ i & \\ i = j , \\\\ \\sum _ { k = 0 } ^ { r } ( - 1 ) ^ k q _ i ^ { - k } E _ i ^ { ( r - k ) } E _ j E _ i ^ { ( k ) } & , \\end{dcases} \\\\ T _ i ( F _ j ) & = \\begin{dcases} - K _ i ^ { - 1 } E _ i & \\ i = j , \\\\ \\sum _ { k = 0 } ^ { r } ( - 1 ) ^ k q _ i ^ k F _ i ^ { ( k ) } F _ j F _ i ^ { ( r - k ) } & , \\end{dcases} \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} Q _ { i j } = C _ { i j } \\frac { P _ i - P _ j } { L _ { i j } } . \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} \\rho = \\frac { 1 } { 2 i } \\sum _ { p = 1 } ^ n \\left ( \\bar { z } _ p + t _ p ( z ) + O ( | z | ^ 3 ) \\right ) d z _ p , \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} \\tau = - 2 \\ , e ^ { 4 5 } + 2 \\ , e ^ { 6 7 } - 2 \\ , e ^ { 4 6 } - 2 \\ , e ^ { 5 7 } + 2 \\ , e ^ { 4 7 } - 2 \\ , e ^ { 5 6 } . \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} a _ { k m } ( \\omega _ k ^ i ) + \\sum _ { d \\mid k , \\ , d \\neq 1 } \\mu ( d ) a _ { ( k / d , i ) } ( 1 ) = 0 . \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} 0 = \\mu _ j ^ { \\frac { n + 2 s } { 2 } } S ( u _ { \\mu , \\xi } + \\tilde { \\phi } ) = - ( - \\Delta ) ^ s _ y \\phi + p U ( y ) ^ { p - 1 } \\phi + \\mu _ j ^ { \\frac { n + 2 s } { 2 } } S ( u _ { \\mu , \\xi } ) + A [ \\phi ] \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } q _ n ^ { 1 / n } = e ^ { - \\pi ^ 2 / 1 2 \\log 2 } \\mbox { f o r L e b e s g u e - a l m o s t a l l } \\ \\theta \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} \\Delta ( X , f ^ { \\ast } \\mathcal L , v , f ^ { \\ast } \\tau ) = \\Delta ( Y , \\mathcal L , v , \\tau ) \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} x ' = T _ { N } ( x ) = \\cos \\left ( N \\arccos ( x ) \\right ) = \\cos \\left ( N \\pi u _ { 0 } \\right ) \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} \\mathcal { B } ( \\lambda ) \\otimes \\mathcal { B } ( \\mu ) = \\bigsqcup _ { \\pi \\in \\mathcal { B } ( \\mu ) ^ \\lambda } C ( \\pi , \\lambda , \\mu ) . \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\mathcal { E } \\sup _ { \\substack { t \\in [ - \\tau , T ] \\\\ r \\in \\mathcal { A } _ N } } \\mathbb { E } \\left \\lvert X ^ { r , \\mathcal { A } _ N } _ t - \\bar { X } ^ r _ t \\right \\rvert ^ 2 = 0 \\ , . \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 1 & \\delta t \\\\ 0 & 1 \\end{bmatrix} , B = \\begin{bmatrix} \\frac { \\delta t ^ 2 } { 2 m } \\\\ \\frac { \\delta t } { m } \\end{bmatrix} , \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} M ( X ) = ( - 1 ) ^ { n - 1 } \\sum _ { x \\in \\Sigma } \\mu _ { x } [ x ] \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} f ( \\langle \\mathrm m _ { t , n } , g \\rangle ) \\ , - \\ , f ( \\langle \\mathrm m _ { 0 , n } , g \\rangle ) - \\int ^ { t } _ { 0 } f ^ { \\prime } ( \\langle \\mathrm m _ { s , n } , g \\rangle ) \\Big ( \\frac { \\ , 1 \\ , } { \\ , n \\ , } \\sum _ { i = 1 } ^ { n } g ^ { \\prime } ( X _ { s , i } ^ { ( u ) } ) b ( s , X _ { s , i } ^ { ( u ) } , \\widehat { F } _ { s , i } ^ { ( u ) } ) + \\frac { \\ , 1 \\ , } { \\ , 2 \\ , } \\langle \\mathrm m _ { s , n } , g ^ { \\prime \\prime } \\rangle \\Big ) { \\mathrm d } s \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} & R ( n + 1 ) \\exp ( i \\eta ( n + 1 ) + i \\gamma ( n ) ) \\\\ = & R ( n ) \\exp ( i \\eta ( n ) + i \\gamma ( n ) ) - \\frac { i } { \\omega } b ' _ { n + 1 } \\vert \\varphi ( n ) \\vert ^ 2 ( \\exp ( - i \\eta ( n ) \\\\ & - i \\gamma ( n ) ) - \\exp ( i \\eta ( n ) + i \\gamma ( n ) ) ) R ( n ) . \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} f _ { X 1 1 } \\left ( x _ { 1 1 } | \\mathbb { A _ { U E } } \\right ) = \\frac { \\mathrm { d } } { \\mathrm { d } x _ { 1 1 } } \\frac { \\mathbb { P } \\left [ X _ { 1 1 } \\leq x _ { 1 1 } , \\mathbb { A _ { U E } } \\right ] } { \\mathbb { P \\left [ A _ { U E } \\right ] } } . \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} \\delta ^ { - 2 } ( f _ \\delta , e ^ { t \\delta ^ { - 2 } \\mathcal { L } _ X ^ \\omega } f _ \\delta ) \\le \\delta ^ { - 2 } C \\norm { f _ \\delta } ^ 2 _ \\infty \\lambda ( ~ f _ \\delta ) ^ 2 \\frac { 1 } { t ^ { d / 2 } } = C \\norm { f } ^ 2 _ \\infty \\lambda ( ~ f ) ^ 2 \\frac { 1 } { t ^ { d / 2 } } . \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} \\mathfrak { S } _ \\mathbb { R } = \\left \\{ N \\in \\mathrm { P S L } ( 2 , \\mathbb { R } ) : N ( \\gamma _ M ^ \\pm ) = \\gamma _ M ^ \\pm , \\ N ( \\theta _ \\pm ) = \\theta _ \\pm \\right \\} \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} \\textstyle { \\frac { P _ T } { \\sigma ^ 2 } } \\left ( \\prod \\limits _ { k = 1 } ^ { r } \\frac { \\beta \\ , [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } { \\beta + [ \\boldsymbol { \\Lambda } ] _ { 1 , 1 } ^ 2 - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } \\right ) ^ { \\frac { 1 } { r } } & \\ , = 2 ^ { \\frac { R } { r } } \\ , { \\textstyle \\sum \\limits _ { k = 1 } ^ { r } \\left ( \\frac { \\beta } { \\beta + [ \\boldsymbol { \\Lambda } ] _ { 1 , 1 } ^ 2 - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } \\right ) } . \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} c _ g : & = k ( 0 + ) + k ( 0 - ) \\\\ & = \\lim _ { r \\to 0 + } \\int _ { [ 0 , t ] } l ( r / g ( s ) ) \\lambda ( d s ) + \\lim _ { r \\to 0 - } \\int _ { [ 0 , t ] } l ( r / g ( s ) ) g ( s ) \\lambda ( d s ) \\\\ & = \\int _ { [ 0 , t ] } l ( 0 + ) \\lambda ( d s ) + \\int _ { [ 0 , t ] } l ( 0 - ) \\lambda ( d s ) \\\\ & = ( l ( 0 + ) + l ( 0 - ) ) t , \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} \\| T f \\| _ { L ^ q ( X ) } \\leq C \\| f \\| _ { L ^ p ( X ) } , s = \\frac { n } { p } - \\frac { n } { q } , ~ \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} \\frac { w _ { \\lambda _ j } ^ \\prime ( 1 ) } { w _ { \\lambda _ j } ( 1 ) } = \\frac { h _ { 1 , \\lambda _ j } ^ \\prime ( 1 ) } { h _ { 1 , \\lambda _ j } ( 1 ) } + \\frac { n - 1 } { 2 } \\sqrt { K _ 0 } , \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} \\displaystyle { x ^ { \\ell } - c = \\prod _ { j = 0 } ^ { \\ell - 1 } ( x - a b ^ j ) } , \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} \\frac { { \\rm { D } } w } { \\partial z } = w ' \\left ( z \\right ) + w \\left ( z \\right ) \\kappa ' \\left ( z \\right ) \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} e ^ { \\Re ( z _ R ( w - \\varepsilon ' \\xi _ 0 ) ) } = e ^ { | z _ R | | w - \\varepsilon ' \\xi _ 0 | \\cos ( \\theta _ R ) } = e ^ { R | w - \\varepsilon ' \\xi _ 0 | \\cos ( \\theta _ R ) } , \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} \\begin{gathered} \\big | \\partial _ \\lambda ^ \\alpha a _ \\pm ( \\lambda , z , z ' ) \\big | \\leq C _ \\alpha \\lambda ^ { - \\alpha } ( 1 + \\lambda d ( z , z ' ) ) ^ { - \\frac { n - 1 } 2 } , \\end{gathered} \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} \\lambda _ { m } : = m ( m + d - 2 ) . \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} & \\phi ( | \\nabla \\Q ^ t | ) = \\phi ( | \\nabla \\Q | ) \\circ \\mathbf { X } ^ t \\\\ & + t \\frac { \\phi ^ \\prime ( | \\nabla \\Q | ) \\circ \\mathbf { X } ^ t } { { | \\nabla \\Q | } \\circ \\mathbf { X } ^ t } ( \\partial _ k Q _ { i j } \\circ \\mathbf { X } ^ t ) ( \\partial _ p Q _ { i j } \\circ \\mathbf { X } ^ t ) \\partial _ k X _ p + \\mathrm { o } ( t ) \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} j ( n ) = \\inf _ { m \\geq 1 } ( q ( m + n ) - q ( m ) ) . \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} E = \\bigoplus _ { i = 1 } ^ k V _ i \\otimes E _ i \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} { } ( \\mathcal S ) \\cong \\begin{cases} \\mathbb G _ { k - j } ( \\mathbb C ^ { l - j } ) \\quad { } \\\\ \\mathbb G _ k ( \\mathbb C ^ j ) \\quad \\ , \\ , \\ , \\ , \\ , \\quad { } . \\end{cases} \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} k _ { i - 1 } ( f ) = \\left ( \\sum _ { j = 1 } ^ { i } \\left \\langle \\frac { \\lambda - \\mu } { 2 } + \\rho , \\alpha _ j \\right \\rangle \\right ) ^ { } . \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\nabla _ \\alpha A _ \\beta - \\nabla _ \\beta A _ \\alpha = \\partial _ \\alpha A _ \\beta - \\partial _ \\beta A _ \\alpha = ( \\mathrm { d } A ^ \\flat ) _ { \\alpha \\beta } \\ , , \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} P _ c v ( t ) = e ^ { - i t H } P _ c v ( 0 ) - i \\int _ 0 ^ t e ^ { - i ( t - s ) H } P _ c \\ , \\N \\ , d s . \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} ( \\frac { 2 ^ { \\nu } B _ { \\nu } } { \\alpha _ { 0 } } ) ^ { \\frac { 1 } { 1 - 2 \\beta ' } } = ( \\frac { 2 ^ { \\nu } B _ { \\nu } } { \\alpha _ { 0 } } ) ^ { \\frac { 1 } { 3 ( \\kappa - 1 ) } } = ( \\prod _ { \\mu = \\nu } ^ { \\infty } ( \\frac { 2 ^ { \\nu } B _ { \\nu } } { \\alpha _ { 0 } } ) ^ { \\frac { 1 } { 3 \\kappa ^ { \\mu + 1 } } } ) ^ { \\kappa ^ { \\nu } } \\leq ( \\prod _ { \\mu = \\nu } ^ { \\infty } ( \\frac { 2 ^ { \\mu } B _ { \\mu } } { \\alpha _ { 0 } } ) ^ { \\frac { 1 } { 3 \\kappa ^ { \\mu + 1 } } } ) ^ { \\kappa ^ { \\nu } } . \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} \\alpha _ 1 a + \\alpha _ 2 a ^ 2 + \\ldots + \\alpha _ k a ^ k = 0 \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} [ u ] _ { i , 1 + \\alpha } & = [ u _ { x _ { i } } ] _ { i , \\alpha } , \\\\ [ u ] _ { 1 + \\alpha } & = \\max _ { i } [ u ] _ { i , 1 + \\alpha } , \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} \\frac { N } { 2 L } = p _ { c } ( \\mathcal { Q } _ { b } ) . \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} \\chi ( M ^ { \\circ } ) = \\chi ( \\hat { M } ) - \\chi ( B ) . \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { K } f ( x ) & = \\int _ { 0 } ^ { \\infty } K ( x , y ) f ( y ) d y \\\\ \\mathbf { K _ { e } } f ( x ) & = \\int _ { 0 } ^ { \\infty } [ M ( x , y ) - K ( x , y ) ] f ( y ) d y . \\end{aligned} \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } H \\cdot l & D \\cdot l \\\\ H \\cdot F & D \\cdot F \\end{array} \\right ) = \\left ( \\begin{array} { c c } 1 & - 3 \\\\ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\sigma _ p ( x _ 1 , x _ 2 ) = p \\int _ 0 ^ 1 \\frac { \\big ( \\| x _ 1 + t x _ 2 \\| \\vee \\| x _ 1 \\| ) ^ p } { t } \\phi _ p \\bigg ( \\frac { t \\| x _ 2 \\| } { \\| x _ 1 + t x _ 2 \\| \\vee \\| x _ 1 \\| } \\bigg ) \\ , d t , \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{gather*} \\nabla _ { f a } s = f \\nabla _ a s , f \\in C ^ { \\ 8 } ( M ) , a \\in \\Gamma ( A ) , s \\in \\Gamma ( D ) , \\end{gather*}"}
-{"id": "4354.png", "formula": "\\begin{align*} \\sup _ { 0 \\le u \\le s } \\| R ( \\epsilon ; t - u ) - ( \\tilde { R } - \\epsilon \\tilde { R } ^ { ( 1 ) } u ) \\| = O ( \\epsilon ^ 2 s ^ 2 ) \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} & \\int _ { ( 0 , + \\infty ) } h _ n ( t ) \\ , d \\| E _ { \\rho , \\sigma } ( t ) \\xi _ \\sigma \\| ^ 2 \\\\ & \\quad = \\inf _ { x ( \\cdot ) } \\int _ { [ 1 / n , n ] } \\bigl \\{ \\sigma ( ( 1 - x ( s ) ) ^ * ( 1 - x ( s ) ) ) + s ^ { - 1 } \\rho ( x ( s ) x ( s ) ^ * ) \\bigr \\} ( 1 + s ) \\ , d \\nu _ n ( s ) , \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} & - \\rho \\int _ { \\partial B _ { \\rho } } \\frac { \\phi ^ \\prime ( | \\nabla \\Q | ) } { | \\nabla \\Q | } | \\partial _ { \\boldsymbol { \\nu } } \\Q | ^ 2 + \\int _ { B _ { \\rho } } \\phi ^ \\prime ( | \\nabla \\Q | ) | \\nabla \\Q | \\\\ & \\qquad = - \\rho \\int _ { \\partial B _ { \\rho } } \\left ( \\phi ( | \\nabla \\Q | ) + \\frac { 1 } { L } f _ B ( \\Q ) \\right ) + d \\int _ { B _ { \\rho } } \\left ( \\phi ( | \\nabla \\Q | ) + \\frac { 1 } { L } f _ B ( \\Q ) \\right ) \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} \\mathbf E \\left [ \\tau _ 0 ( t ) \\right ] = \\frac { t ^ \\beta } { \\Gamma ( \\beta + 1 ) } , \\mathbf E \\left [ e ^ { - \\lambda \\tau _ 0 ( t ) } \\right ] = E _ \\beta ( - \\lambda t ^ \\beta ) , t , \\lambda \\ge 0 , \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} \\left | f ( x , t ) \\right | \\lesssim \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\| \\bar { \\lambda } _ 1 \\| _ { 1 + \\sigma } \\sum _ { j = 1 } ^ k \\frac { \\mu _ j ^ { - 2 s } \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } - 1 + \\sigma } } { 1 + | y _ j | ^ { a } } . \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} \\phi h ( X , \\phi Y ) = \\phi C h ( X , Y ) + \\phi B h ( X , \\phi Y ) . \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} [ \\omega _ 1 , \\omega _ 2 ] \\Psi _ { D } ( \\alpha _ j ) = [ \\omega _ 1 , \\omega _ 2 ] \\gamma _ j \\ , k _ { j } \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\gamma _ - ( u ) \\leq \\liminf _ { n \\rightarrow \\infty } \\left [ \\norm { u _ n ^ - } ^ 2 + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u _ n ^ - } u _ n ^ 2 d x - \\int _ { \\mathbb R ^ 3 } f ( u _ n ^ - ) u _ n ^ - d x \\right ] = \\liminf _ { n \\rightarrow \\infty } \\gamma _ - ( u _ n ) = 0 . \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} \\overline { \\nabla } _ { X } N = - A _ { N } X + \\nabla _ { X } ^ { \\bot } N + \\alpha \\eta ( N ) X + \\beta \\eta ( N ) \\phi X - \\beta g ( \\phi X , N ) \\xi . \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} u ^ { q ^ 2 + q + 1 } = & u u ^ { ( q + 1 ) q } \\\\ = & u ( u + 1 ) ^ q \\\\ = & u ^ { q + 1 } + u \\\\ = & 1 . \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} \\mathrm { S G } _ m : = \\left \\{ { \\left . \\begin{pmatrix} \\cos ( 2 m q ^ 4 ) & - \\sin ( 2 m q ^ 4 ) & 0 & 0 & q ^ 1 \\\\ \\sin ( 2 m q ^ 4 ) & \\cos ( 2 m q ^ 4 ) & 0 & 0 & q ^ 2 \\\\ 0 & 0 & 1 & 0 & q ^ 3 \\\\ 0 & 0 & 0 & 1 & q ^ 4 \\\\ 0 & 0 & 0 & 0 & 1 \\end{pmatrix} \\right | q \\in \\mathbb { R } ^ 4 } \\right \\} . \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} p _ I ( s _ { \\pi ( 1 ) } , \\ldots , s _ { \\pi ( N ) } ) & = H _ I ( e ) \\ , ( s _ { \\pi ( \\pi ^ { - 1 } ( 1 ) ) } , \\ldots , s _ { \\pi ( \\pi ^ { - 1 } ( N ) ) } ) \\\\ & = H _ I ( e ) \\ , ( s _ 1 , \\ldots , s _ N ) . \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} \\Phi = I _ { m \\times m } \\ , , \\Pi = \\begin{pmatrix} 0 & 0 \\\\ 0 & I _ { ( n - m ) \\times ( n - m ) } \\end{pmatrix} \\ , , \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\hat { \\omega } : = [ R ^ y ] , \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} \\theta _ { n _ 1 , n _ 2 } ( t ) = \\int _ 0 ^ t \\langle \\phi _ { n _ 1 , n _ 2 } ( t ' ) | i \\frac { \\partial } { \\partial t ' } - \\hat H ( t ' ) | \\phi _ { n _ 1 , n _ 2 } ( t ' ) \\rangle d t ' . \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} T ( r , f ( z + c ) ) = ( 1 + o ( 1 ) ) T ( r , f ( z ) ) , \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} M _ t : = Y _ t - Y _ 0 + A ^ { W , Y } _ t ( b ) + \\int _ 0 ^ t f \\left ( r , W _ r , Y _ r , \\frac { \\mathrm d [ Y , W ] _ r } { \\mathrm d r } \\right ) \\mathrm d r \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\Big ( ( y z ) \\alpha _ i ( x ) - ( z x ) \\alpha _ i ( y ) \\Big ) \\otimes \\Big ( ( b c ) \\beta _ i ( a ) - ( c a ) \\beta _ i ( b ) \\Big ) = 0 . \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\dim ( I ) ^ 2 = 1 . \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} ( \\delta F ) _ \\alpha = \\nabla ^ \\beta F _ { \\alpha \\beta } . \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} K K ^ \\top & = K ^ 2 = \\frac { 1 } { 4 a ^ 2 } ( H _ 1 ^ \\top H _ 1 - H _ 2 ^ \\top H _ 2 ) ^ 2 = \\frac { 1 } { 4 a ^ 2 } ( H _ 1 ^ \\top ( H _ 1 H _ 1 ^ \\top ) H _ 1 + H _ 2 ^ \\top ( H _ 2 H _ 2 ^ \\top ) H _ 2 ) \\\\ & = \\frac { n } { 4 a ^ 2 } ( H _ 1 ^ \\top H _ 1 + H _ 2 ^ \\top H _ 2 ) = \\frac { n ^ 2 } { 4 a ^ 2 } I _ n . \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\varepsilon \\dot { S } & = F - N c = { G ( c , N , F ) - l ( c , N , F ) S } & & \\ [ 0 , T ] , \\\\ S ( 0 ) & = S _ { i n } \\ge 0 , \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} & = \\sum _ { m = 2 } ^ { 2 t - 1 } \\sum _ { i = m + 1 } ^ { 2 t } ( - 1 ) ^ { i + m - 1 } { G } _ { ( 1 , 2 ) , ( m , i ) } \\widehat { G } _ { \\{ 1 , 2 \\} , \\{ m , i \\} } . \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} \\| f \\| _ { \\mathcal { B } _ { \\alpha } ^ { j } } = \\sum _ { k \\in \\mathbb { Z } ^ { n } } | k | ^ { j } \\sup _ { t \\in [ 0 , \\infty ) } e ^ { \\alpha t | k | } | \\hat { f } ( t , k ) | . \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{gather*} v _ B ^ { - 1 } \\triangleright \\chi ^ { \\epsilon } _ { s + 1 } = - \\epsilon q ^ { - s - \\frac { 1 } { 2 } } \\overset { \\mathcal { X } ^ + ( 2 ) } { W } \\ ! \\ ! \\ ! \\ ! \\ ! _ B \\triangleright \\big ( v _ B ^ { - 1 } \\triangleright \\chi ^ { \\epsilon } _ { s } \\big ) - q ^ { - 2 s } v _ B ^ { - 1 } \\triangleright \\chi ^ { \\epsilon } _ { s - 1 } . \\end{gather*}"}
-{"id": "2187.png", "formula": "\\begin{align*} & \\mathcal { G } _ 0 = \\{ u _ 0 \\in H _ 0 ^ 1 ( \\Omega ) : T _ { \\max } ( u _ 0 ) = \\infty \\mathrm { a n d } u ( t ) \\to 0 , \\textrm { a s } \\ t \\to \\infty \\} , \\\\ & \\mathcal { B } = \\{ u _ 0 \\in H _ 0 ^ 1 ( \\Omega ) : T _ { \\max } ( u _ 0 ) < \\infty \\} , \\\\ & \\mathcal { N } : = \\{ u \\in H _ 0 ^ 1 ( \\Omega ) \\backslash \\{ 0 \\} : I ( u ) = 0 \\} . \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} n _ { i - 1 } ^ + = & n _ i + \\# a ( r _ i ) - \\# a ^ { - 1 } ( s _ { 1 } ) \\\\ n _ { i + k } ^ + = & n _ { i + k } + \\# a ( s _ k ) - \\# a ^ { - 1 } ( r _ { i + k + 1 } ) . \\end{align*}"}
-{"id": "9953.png", "formula": "\\begin{align*} o r _ { \\tilde f , x } = H ^ { 2 d } _ { \\{ x \\} } ( \\tilde f ^ { - 1 } ( \\tilde f ( x ) ) ; \\ , \\Z _ { \\tilde f ^ { - 1 } ( \\tilde f ( x ) ) } ) . \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\tilde { a } ( x , w , w _ x ) : = & \\frac { x _ \\xi ^ 2 } { l ( x ) } \\cdot a ( x , L ^ { - 1 } ( w ) , \\partial _ x L ^ { - 1 } ( w ) ) \\\\ \\tilde { f } ( x , w , w _ x ) : = & g _ n ( \\mu , x , L ^ { - 1 } ( w ) , \\partial _ x L ^ { - 1 } ( w ) ) + ( w _ x \\cdot x _ { \\xi \\xi } - \\partial _ \\xi ^ 2 u _ - ) - \\frac { l _ { \\xi \\xi } \\cdot ( w - u _ - ) _ \\xi } { l } \\\\ & - \\lambda a ( x , L ^ { - 1 } ( w ) , \\partial _ x L ^ { - 1 } ( w ) ) \\cdot L ^ { - 1 } ( w ) . \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} M ( D ) _ { c a } = \\begin{cases} 1 - t _ { \\kappa ( a _ 2 ) } & a = a _ 1 \\\\ t _ { \\kappa ( a _ 1 ) } & a = a _ 2 \\\\ - 1 & a = a _ 3 \\\\ 0 & a \\notin \\{ a _ 1 , a _ 2 , a _ 3 \\} \\end{cases} \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} K = \\left \\lbrace x \\in \\R ^ n : C x \\leq 0 \\right \\rbrace , \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} \\| \\pi ^ { ( d ) } ( A ) \\| \\geq \\| \\tau ^ { ( d ) } ( A ) U \\xi _ \\psi \\| = \\| U \\sigma _ \\psi ( A ) \\xi _ \\psi \\| = \\| A \\| . \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} p _ i = \\# \\{ \\epsilon _ i - \\mu _ { i , j } \\epsilon _ { j } \\in I \\mid \\mu _ { i , j } \\in \\{ \\pm 1 \\} \\} . \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} \\varphi ( x , y , t ) = h ( x ) p _ { 0 } ( y , t ) \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} C = \\left \\{ \\lambda _ 1 \\varpi _ 1 + \\lambda _ 2 \\varpi _ 2 \\in P ^ + \\ \\middle \\vert \\ \\lambda _ 1 + \\lambda _ 2 < 3 \\right \\} = \\{ 0 , \\varpi _ 1 , \\varpi _ 2 , \\varpi _ 1 + \\varpi _ 2 , 2 \\varpi _ 1 , 2 \\varpi _ 2 \\} . \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} \\ddot y + R ( t ) y = 0 \\quad ( 0 \\le t \\le \\pi / { \\sqrt k } ) , \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} s = \\sum _ { k = 1 } ^ { t - 1 } ( - 2 ) ^ { k - 1 } y _ { k - 1 } . \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} \\min f ( x ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; A x = b , \\ ; \\ ; \\ ; x \\in \\mathcal { X } \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} \\inf _ { \\gamma \\in \\Theta ^ { \\lambda } ( y _ 0 ) } \\int _ { Y \\times U } k ( y , u ) \\gamma ( d y , d u ) : = h ^ { \\lambda } ( y _ 0 ) \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} \\theta _ X \\theta _ Y \\mathbf { S } _ { X , Y } = \\sum _ { Z \\in J } s N _ { X , Y } ^ Z \\dim ^ R ( Z ) \\theta _ Z \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} u _ R ( x _ 1 ) = \\inf \\left \\{ \\sum _ { i = 1 } ^ N c _ R ( x _ 1 , x _ 2 , \\ldots , x _ N ) - \\sum _ { j = 2 } ^ N u _ R ( x _ j ) ~ \\bigg | ~ ( x _ 2 , \\ldots , x _ N ) \\in X ^ { N - 1 } \\right \\} . \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} b f ( 0 / b ) : = f ( 0 ^ + ) b \\quad \\mbox { f o r $ b \\ge 0 $ } , 0 f ( a / 0 ) : = \\lim _ { t \\searrow 0 } t f ( a / t ) = f ' ( + \\infty ) a \\quad \\mbox { f o r $ a > 0 $ } , \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} \\partial _ n ^ { ( 0 ) } ( f ) ( g _ 1 , \\dots , g _ { n - 1 } ) & = \\sum _ { g _ 0 \\in G } g _ 0 ^ { - 1 } f ( g _ 0 , g _ 1 , \\dots , g _ { n - 1 } ) , \\\\ \\partial _ n ^ { ( i ) } ( f ) ( g _ 1 , \\dots , g _ { n - 1 } ) & = \\sum _ { \\substack { g , \\bar { g } \\in G \\\\ g \\bar { g } = g _ i } } f ( g _ 1 , \\dots , g _ { i - 1 } , g , \\bar { g } , g _ { i + 1 } , g _ { n - 1 } ) ~ ~ 1 \\leq i \\leq n - 1 , \\\\ \\partial _ n ^ { ( n ) } ( f ) ( g _ 1 , \\dots , g _ { n - 1 } ) & = \\sum _ { g _ n \\in G } f ( g _ 1 , \\dots , g _ { n - 1 } , g _ n ) . \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } \\phi _ { 1 } ( x ) \\left | \\widetilde { f } ( | x | ) \\right | ^ { p } d x = \\int _ { \\mathbb { G } } \\phi _ { 1 } ( x ) \\left | f ( x ) \\right | ^ { p } d x \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} a \\equiv \\begin{pmatrix} I _ { m \\times m } & 0 \\\\ 0 & \\frac { 1 } { \\delta } I _ { ( n - m ) \\times ( n - m ) } \\end{pmatrix} \\ , , \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} I ( u _ n ) & = I _ { \\mu _ n } ( u _ n ) + \\frac { \\mu _ n - 1 } { p + 1 } \\int _ { \\R ^ 3 } u _ n ^ { p + 1 } \\\\ & = c _ { \\mu _ n } + \\frac { \\mu _ n - 1 } { p + 1 } \\int _ { \\R ^ 3 } u _ n ^ { p + 1 } , \\\\ \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t = \\Delta u + \\left ( \\frac { 1 } { | x | ^ { n - 2 } } * | u | ^ p \\right ) | u | ^ { p - 2 } u , & x \\in \\Omega , t > 0 \\\\ u ( x , t ) = 0 , & x \\in \\partial \\Omega , t > 0 \\\\ u ( x , 0 ) = u _ 0 ( x ) , & x \\in \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} \\rho _ { n } ( x , y ) = \\rho ( x , y , t ) \\ \\ \\ \\ ( t = n \\tau ) \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} m = & ( \\ell + 1 ) ( \\ell ' + 1 ) - \\# V = ( \\ell + 1 ) ( \\ell ' + 1 ) - ( \\ell + 1 ) \\left [ k ^ 2 \\ , + \\ , 2 k ( k + r - 1 ) \\right ] \\\\ = & ( \\ell + 1 ) ( \\ell ' + 1 - ( k ^ 2 + 2 k ( k + r - 1 ) ) \\\\ = & ( \\ell + 1 ) ( \\ell - 3 k ^ 2 - 2 s _ 1 k + s _ 2 + 1 ) \\\\ = & \\ell ^ 2 - \\ell ( 3 k ^ 2 + 2 s _ 1 k - s _ 2 - 2 ) - 3 k ^ 2 - 2 s _ 1 k + s _ 2 + 1 \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} \\phi ' ( \\alpha ) = f '' ( \\alpha ) \\chi ( \\alpha ) , \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} r ( G _ { \\lambda } ) \\geq r ( G _ { \\lambda ' } ) & \\geq \\frac { 1 } { 2 } ( d _ { l + 1 } / 2 - 1 + \\sum _ { j \\neq l + 1 } d _ j - 1 ) \\\\ & = \\frac { 1 } { 2 } ( \\dim ( V _ { \\lambda } ) - k - d _ { l + 1 } / 2 ) . \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} \\left ( \\sup _ { | v | = 1 } | D f ( x ) ( v ) | \\right ) ^ d \\leq K _ { 1 } J _ { f } ( x ) , \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} \\mathcal { F } _ { 2 d } ( l ) = \\{ v \\in ( l ^ { \\perp } / l ) \\otimes \\mathbb { R } \\mid ( v , v ) > 0 \\} / \\sim . \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} \\varphi ^ { G } = \\sum _ { 1 \\leq i \\leq m } \\eta _ { \\chi _ { i } } \\chi _ { i } , ~ { \\rm ~ w h e r e } ~ \\eta _ { \\chi _ { i } } \\geq 1 ~ { \\rm a r e ~ i n t e g e r s } . \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} \\begin{aligned} C ( \\{ s , v \\} , v ) & = c _ { s v } ( 0 ) & & \\forall v \\in V , v \\neq s \\\\ C ( S , v ) & = \\min _ { \\substack { u \\in S \\\\ u \\neq s , v } } C ( S \\setminus \\{ v \\} , u ) + c _ { u v } ( C ( S \\setminus \\{ v \\} , u ) ) & & \\forall S \\subseteq V , v \\in S . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} T _ { a ( b c ) } = T _ a T _ { b c } + T _ b T _ { a c } + T _ c T _ { a b } - T _ b T _ a T _ c - T _ c T _ a T _ b . \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} R [ J ' ( u ( t _ n ) ) ] = u ( t _ n ) - R [ f ( u ( t _ n ) ) ] \\to 0 \\ \\ \\mathrm { i n } \\ \\ , H _ 0 ^ 1 ( \\Omega ) . \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} B _ { q } ( u , w ) - \\lambda ( u , w ) = F ( w ) \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} L ^ { ( 2 ) } ( A ) = \\frac 1 2 \\ , ( \\nabla _ \\alpha A _ \\beta - \\nabla _ \\beta A _ \\alpha ) ( \\nabla ^ \\alpha A ^ \\beta - \\nabla ^ \\beta A ^ \\alpha ) - 2 \\ , \\mathrm { R i c } _ { \\mu \\nu } \\ , A ^ \\mu A ^ \\nu + \\nabla _ \\kappa B ^ \\kappa \\ , , \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\max _ { 1 \\leq j \\leq r } | \\alpha _ j | \\leq \\limsup _ { n \\to \\infty } \\left \\| \\gamma _ n \\left ( \\sum _ { j = 1 } ^ r \\alpha _ j T _ j \\right ) \\right \\| = \\left \\| \\sum _ { j = 1 } ^ r \\alpha _ j \\kappa ( T _ j ) \\right \\| = 0 \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\Phi _ \\delta ^ { \\gamma , \\varepsilon } ( t ) = \\frac { 1 } { 2 } \\int _ { D } \\log \\left ( 1 + \\frac { | X _ { 1 t } ( x ) - X _ { 2 t } ( x ) | ^ 2 + \\gamma \\langle \\eta _ t ^ \\varepsilon ( X _ { 1 t } ( x ) ) , X _ { 1 t } ( x ) - X _ { 2 t } ( x ) \\rangle ^ 2 } { \\delta ^ 2 } \\right ) d x . \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} d _ \\omega ( x , y ) : = \\inf \\limits _ \\gamma \\Big \\{ \\sum _ { i = 0 } ^ { l _ \\gamma - 1 } 1 \\land \\omega ( z _ i , z _ { i + 1 } ) ^ { - 1 / 2 } \\Big \\} \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} \\overline { S } ( V , \\xi ) = \\{ ( n - 1 ) ( 1 - \\beta + \\beta ^ { 2 } ) + \\alpha ( \\beta - 1 ) t r a c e \\Phi \\} \\eta ( V ) , \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} Z _ 1 ( \\theta ) : = \\sum _ { n \\in \\mathbb N _ 0 } \\mathrm e ^ { \\theta n } \\nu _ { * , 1 } ( n ) < \\infty . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} R ( x ) = \\begin{bmatrix} x _ { 1 } & x _ { 2 } & x _ { 3 : n } ^ T \\\\ x _ { 2 } & x _ { 1 } & x _ { 3 : n } ^ T \\\\ x _ { 3 : n } & x _ { 3 : n } & ( x _ 1 + x _ 2 ) I _ { n - 2 } \\end{bmatrix} . \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} K _ 1 = G ^ { ( 1 ) } \\rightarrow G ^ { ( 2 ) } \\rightarrow \\cdots \\rightarrow G ^ { ( n ) } = G \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} z ( L ( v ( \\xi ) - \\tilde { v } ( \\xi ) ) ) = z ( l ( x ) [ v ( \\xi ( x ) ) - \\tilde { v } ( \\xi ( x ) ) ] ) = z ( v ( x ) - \\tilde { v } ( x ) ) \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} \\eta _ { \\bf i , \\mathcal { I } } ^ { \\ast } \\mathcal { L } _ { \\mathcal I , \\lambda _ 1 , \\dots , \\lambda _ r } = \\{ ( p , q ) \\in Z _ { \\bf i } \\times \\mathcal { L } _ { \\mathcal I , \\lambda _ 1 , \\dots , \\lambda _ r } \\mid \\eta _ { { \\bf i } , \\mathcal I } ( p ) = \\pi _ { \\mathcal I , \\lambda _ 1 , \\dots , \\lambda _ r } ( q ) \\} , \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} \\left \\| G ^ { \\varepsilon } ( t , \\cdot ) \\right \\| _ { \\L ^ 1 } = 1 , \\left \\| G _ { x } ^ { \\varepsilon } ( t , \\cdot ) \\right \\| _ { \\L ^ 1 } = 2 G ^ { \\varepsilon } ( t , 0 ) = \\frac { 1 } { \\sqrt { \\pi \\varepsilon t } } , \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} A ( n ) & = \\sum _ { k = 0 } ^ \\infty { \\binom n k } ^ 2 { \\binom { n + k } k } ^ 2 = { } _ 4 F _ 3 \\biggl ( \\begin{matrix} - n , \\ , - n , \\ , n + 1 , \\ , n + 1 \\\\ 1 , \\ , 1 , \\ , 1 \\end{matrix} \\biggm | 1 \\biggr ) \\\\ & = \\sum _ { k = 0 } ^ n { \\binom n k } ^ 2 { \\binom { n + k } k } ^ 2 \\qquad n = 0 , 1 , 2 , \\dots \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} \\Omega ( K ) = \\omega ( K ) = r . \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { | X _ n | } { n } \\le \\mathcal { S } _ { \\mathbb { T } _ d } ( \\lambda ) \\hbox { \\rm a . s . } \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} B _ 0 ( U ) = b _ 0 ( U ) = b _ 0 ( V ) = B _ 0 ( V ) = B _ 0 ( H ) \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} \\dot { \\xi } _ j = \\mu _ 0 ^ { n - 4 s + 2 } c \\left [ b _ j ^ { n - 2 s } \\nabla H ( q _ j , q _ j ) - \\sum _ { i \\neq j } b _ j ^ { \\frac { n - 2 s } { 2 } } b _ i ^ { \\frac { n - 2 s } { 2 } } \\nabla G ( q _ j , q _ i ) \\right ] + h ( t ) \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} \\Big ( L ^ 2 , D \\Big ( ( \\sqrt { A } + K _ { V } ) ^ 3 \\Big ) \\Big ) _ { \\frac { 1 } { 3 } , 2 } = D ( \\sqrt { A } + K _ { V } ) \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} w = \\left ( \\begin{matrix} 0 & 1 \\\\ - 1 & 0 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} f _ { k , m } ( z ) = F _ { k , - m , 0 } ( z ) + \\sum _ { \\substack { A _ k < n < 0 \\\\ ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) } } a _ k ( m , n ) F _ { k , n , 0 } ( z ) . \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} - \\Delta u + W u = f \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} \\norm { F } _ { A ^ 2 ( S ( a , b ) ) } ^ 2 = \\int _ a ^ b \\norm { F _ y } _ { L ^ 2 ( \\R ) } ^ 2 \\d y < \\infty , \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} \\omega _ { a , b } ( t ) = \\frac { e ^ { - 4 \\pi a t } - e ^ { - 4 \\pi b t } } { 4 \\pi t } , t \\in \\R . \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} \\int _ { z } ^ { \\infty } \\frac { \\psi ( t , y ) } { t ^ { 2 } } \\ll \\int _ { z } ^ { \\infty } t ^ { - 1 - 1 / ( 2 \\log y ) } d t \\ll ( \\log y ) z ^ { - 1 / ( 2 \\log y ) } = ( \\log y ) \\exp \\left ( - \\frac { \\sqrt { y } } { 2 \\log y } \\right ) . \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} L _ G f ( v ) = \\frac { 1 } { m ( v ) } \\sum _ { v \\sim v ' } w ( v , v ' ) ( f ( v ' ) - f ( v ) ) . \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { \\binom k j \\binom j s \\left ( { \\frac { { f _ 2 } } { { f _ 3 } } } \\right ) ^ j \\left ( { \\frac { { f _ 1 } } { { f _ 2 } } } \\right ) ^ s X _ { m - c k + ( c - b ) j + ( b - a ) s } } } = \\frac { { X _ m } } { { f _ 3 { } ^ k } } \\ , , \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} u ( s ) = e ^ { [ \\rho ( 2 \\tau ( s ) ) - 2 \\rho ( \\tau ( s ) ) ] } \\ \\ \\ ( s \\in G ) . \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} \\o _ { \\tilde \\nu _ D } ( u , b ^ * J _ { \\nu _ D } u ) = K \\tilde \\pi ^ * \\o _ D ( u , b ^ * J _ { \\nu _ D } u ) + \\eta ( u , b ^ * J _ { \\nu _ D } u ) + d \\mu ( u , b ^ * J _ { \\nu _ D } u ) \\ , . \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} B _ i = \\bigcup _ { u _ { i + 1 } = 0 } ^ { s _ { i + 1 } - 1 } \\bigcup _ { u _ { i + 2 } = 0 } ^ { s _ { i + 2 } - 1 } \\dots \\bigcup _ { u _ { h + 1 } = 0 } ^ { s _ { h + 1 } - 1 } \\bigcup _ { q _ { i + 1 } = 0 } ^ { p _ { i + 1 } - 1 } \\bigcup _ { q _ { i + 2 } = 0 } ^ { p _ { i + 2 } - 1 } \\dots \\bigcup _ { q _ h = 0 } ^ { p _ h - 1 } \\Big ( G _ i \\beta ^ { \\sum _ { j = i + 1 } ^ { h + 1 } u _ j t _ j } \\prod _ { i < j \\le h } \\alpha _ j ^ { q _ j } \\Big ) . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} \\| g ^ n _ M - g _ M \\| _ { C ( [ 0 , T ] ; H ^ s _ r ( \\mathbb R ^ d ) ) } & = \\| g ^ n \\chi _ M - g \\chi _ M \\| _ { C ( [ 0 , T ] ; H ^ s _ r ( \\mathbb R ^ d ) ) } \\\\ & = \\sup _ { 0 \\leq t \\leq T } \\| ( g ^ n ( t , \\cdot ) - g ( t , \\cdot ) ) \\chi _ M \\| _ { H ^ s _ r ( \\mathbb R ^ d ) } \\\\ & \\leq c ( M ) \\sup _ { 0 \\leq t \\leq T } \\| g ^ n ( t , \\cdot ) - g ( t , \\cdot ) \\| _ { H ^ s _ r ( \\mathbb R ^ d ) } \\\\ & = c ( M ) \\| g ^ n - g \\| _ { C ( [ 0 , T ] ; H ^ s _ r ( \\mathbb R ^ d ) ) } , \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} X \\cdot Y \\cdot \\psi + Y \\cdot X \\cdot \\psi = - 2 g ( X , Y ) \\psi \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} ( \\lambda ) _ { m n } = m ^ { m n } \\prod _ { j = 1 } ^ { m } \\left ( \\frac { \\lambda + j - 1 } { m } \\right ) _ { n } ~ ~ ~ ~ ~ ~ ~ ~ ; m \\in \\mathbb { N } , ~ n \\in \\mathbb { N } _ { 0 } \\newline . \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} X u + \\mathcal { A } ( u ) = - f , u | _ { \\partial ( S M ) } = 0 , \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} \\sum _ { v ' \\in V } \\frac { w ( v , v ' ) } { M + 1 } = \\sum _ { v ' \\in N ( v ) } \\frac { H ' ( \\bar { \\theta } _ { v ' } - \\bar { \\theta } _ v ) } { M + 1 } \\leq \\frac { M } { M + 1 } < 1 . \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} U \\Delta w _ { i } = \\sigma _ { i } v _ { i } . \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} \\mathbb { S } _ { f , g } = i ^ { - \\lvert \\Phi ^ + \\rvert - n } D \\tilde { S } _ { \\iota ( f ) , \\iota ( g ) } D ^ { - 1 } \\mathbb { T } _ { f } = \\theta _ { L _ { \\xi } ( \\iota ( f ) ) } . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} \\gamma _ + ( u ^ + + t u ^ - ) & = I ' ( u ^ + ) [ u ^ + ] + t ^ 2 \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } ( u ^ - ) ^ 2 d x \\\\ & \\leq I ' ( u ^ + ) [ u ^ + ] + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } ( u ^ - ) ^ 2 d x = \\gamma _ + ( u ) < 0 , \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} \\frac { \\pi } { 2 } > \\bar { \\theta } _ { 2 , 0 } - \\bar { \\theta } _ { 2 , 1 } = \\frac { \\pi } { 2 } - 2 \\bar { \\theta } _ { 2 , 1 } \\geq \\frac { \\pi } { 2 } - 2 \\bar { \\theta } _ { i , 1 } = \\bar { \\theta } _ { i , 0 } - \\bar { \\theta } _ { i , 1 } \\geq 0 , \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\int d x \\ x \\ \\varphi ( x , y , t ) = \\langle x \\rangle p _ { 0 } ( y , t ) \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { \\ell ( s ) } \\left ( 1 + c _ 1 e ^ { c _ 1 | s | - c _ 2 k } \\right ) & \\leq \\left ( 1 + c _ 1 e ^ { c _ 1 | s | } \\right ) ^ { \\ell ( s ) } \\\\ & \\leq \\left ( c _ 3 e ^ { c _ 1 | s | } \\right ) ^ { \\ell ( s ) } \\\\ & \\leq \\exp \\left ( c _ 3 + c _ 4 | s | + c _ 5 | s | ^ 2 \\right ) \\\\ & \\leq \\exp \\left ( c _ 6 + c _ 7 | s | ^ 2 \\right ) \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} J _ { \\eta , \\eta ' } ( \\mu ) : = \\mu ( \\eta ) r ^ V _ { \\eta , \\eta ' } - \\mu ( \\eta ' ) r ^ V _ { \\eta ' , \\eta } . \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} \\gamma _ m = \\gamma _ m ( \\chi , \\sigma , T , r , \\kappa ) & : = \\gamma _ { \\ref { T h e o r e m E x i s t e n c e A n d U n i q u e n e s s O n D o m a i n } , m } ( \\eta ( \\chi ) , R _ 1 ( \\chi , \\sigma , m , R ( \\chi , \\sigma , m , r , \\kappa ) , \\kappa ) , T ) , \\\\ C _ m = C _ m ( \\chi , \\sigma , T , r ) & : = C _ { \\ref { T h e o r e m E x i s t e n c e A n d U n i q u e n e s s O n D o m a i n } , m } ( \\eta ( \\chi ) , R _ 1 ( \\chi , \\sigma , m , R ( \\chi , \\sigma , m , r , \\kappa ) , \\kappa ) , T ) , \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} \\dot { \\theta } _ { i , j } = \\omega + \\sin ( \\theta _ { i + 1 , j } - \\theta _ { i , j } ) + \\sin ( \\theta _ { i - 1 , j } - \\theta _ { i , j } ) + \\sin ( \\theta _ { i , j + 1 } - \\theta _ { i , j } ) + \\sin ( \\theta _ { i , j - 1 } - \\theta _ { i , j } ) , \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} 2 \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 2 4 n ) q ^ n & \\equiv \\dfrac { f _ 3 ^ 2 } { f _ 6 } \\cdot \\dfrac { f _ 2 } { f _ 1 ^ 2 } - \\dfrac { f _ 6 ^ 2 } { f _ 3 ^ 4 } + 2 \\\\ & \\equiv \\dfrac { f _ 3 ^ 2 } { f _ 6 } \\left ( \\dfrac { f _ 6 ^ 4 f _ 9 ^ 6 } { f _ 3 ^ 8 f _ { 1 8 } ^ 3 } + 2 q \\dfrac { f _ 6 ^ 3 f _ 9 ^ 3 } { f _ 3 ^ 7 } + 4 q ^ 2 \\dfrac { f _ 6 ^ 2 f _ { 1 8 } ^ 3 } { f _ 3 ^ 6 } \\right ) - \\dfrac { f _ 6 ^ 2 } { f _ 3 ^ 4 } + 2 ~ ( \\textup { m o d } ~ 8 ) , \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} \\pi ( [ X _ { h _ 1 } , Y _ { h _ 2 } ] ) = h _ 1 + h _ 2 = \\pi ( X _ { h _ 1 } ) + \\pi ( Y _ { h _ 2 } ) . \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} T _ y ^ * ( T _ X ) = \\prod \\limits _ { j = 1 } ^ n Q ( x _ j ) , \\widehat { T } _ y ^ * ( T _ X ) = \\prod \\limits _ { j = 1 } ^ n \\widehat { Q } ( x _ j ) \\in H ^ * ( X ) \\otimes \\mathbb { Q } [ y ] , \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ 2 } { d x ^ 2 } + q ( x ) - \\frac { ( n - 1 ) ^ 2 } { 4 } K _ 0 \\right ) w _ { \\lambda } ( x ) = \\lambda w _ { \\lambda } ( x ) , \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} 0 & = x + y + m + ( x + y ) m \\\\ & = x + y + m + 1 - x - m - 2 y + y - m - 1 = - m , \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} ( 2 6 ) \\Leftrightarrow \\langle x \\rangle = 0 \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} n \\left ( r , \\frac { 1 } { G } \\right ) \\leq \\sum _ { j = 1 } ^ { m + 1 } \\tilde n _ \\kappa ^ { [ m - 1 ] } \\left ( r , \\frac 1 { g _ j } \\right ) . \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} \\xi _ b = \\frac { 1 } { \\pi } ( \\frac 1 2 | j _ b | \\zeta _ b + \\sum _ { b ' \\neq b } | j _ { b ' } | \\zeta _ { b ' } ) , 1 \\leq b \\leq n . \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} \\lambda ^ \\star ( \\hat { x } , x ^ { \\star } ) & = \\left ( \\frac { z } { b } \\right ) \\frac { l _ 2 ^ { - 1 } z ^ { - l _ 2 } - l _ 1 ^ { - 1 } z ^ { - l _ 1 } } { l _ 1 ^ { - 1 } z ^ { - l _ 1 } + l _ 2 ^ { - 1 } z ^ { - l _ 2 } } - 1 \\\\ \\kappa ^ \\star ( \\hat { x } , x ^ { \\star } ) & = z \\left [ l _ 1 ^ { - 1 } + l _ 2 ^ { - 1 } - 1 \\right ] - z \\frac { l _ 1 ^ { - 1 } z ^ { - l _ 1 } - l _ 2 ^ { - 1 } z ^ { - l _ 2 } } { l _ 1 ^ { - 1 } z ^ { - l _ 1 } + l _ 2 ^ { - 1 } z ^ { - l _ 2 } } \\left [ l _ 1 ^ { - 1 } - l _ 2 ^ { - 1 } + \\ln { \\hat { x } } - \\ln { x ^ \\star } \\right ] , \\end{align*}"}
-{"id": "10004.png", "formula": "\\begin{align*} \\bar A ( q ) = \\frac { q + 1 } { q } e ^ { - q } = e ^ { - q } + \\frac { e ^ { - q } } q . \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} g ( x , y , t ) = \\frac { e ^ { - \\frac { ( y - x ) ^ { 2 } } { 4 t } } } { \\sqrt { 4 \\pi t } } , y > x > 0 . \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} - \\nabla \\cdot \\left ( D ^ 2 _ p L ( \\nabla u ( x ) , x ) \\nabla w ( x ) \\right ) = 0 . \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} a + b = c \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} \\mathcal F _ \\alpha ^ V ( \\rho _ { t _ 2 } ) - \\mathcal F _ \\alpha ^ V ( \\rho _ { t _ 1 } ) = \\int _ { t _ 1 } ^ { t _ 2 } \\bigl \\langle \\dot \\rho _ t , \\frac { \\delta \\mathcal F _ \\alpha ^ V } { \\delta \\rho _ t } \\bigr \\rangle \\ ; \\ ! \\mathrm d t = - \\int _ { t _ 1 } ^ { t _ 2 } \\langle \\dot \\rho _ t , \\Delta \\phi ( \\rho _ t ) + \\nabla \\cdot ( \\chi ( \\rho _ t ) \\nabla V ) \\rangle _ { - 1 , \\chi ( \\rho _ t ) } \\ ; \\ ! \\mathrm d t . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} f ( x ) & = ( x + 1 ) + ( x ^ 4 + x ^ 2 + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( x + 1 ) + ( m - 1 ) \\ : h ( x ) , \\\\ g ( x ) & = x - ( A - B x ) ( x ^ 3 - 1 ) ( x + 1 ) + m \\ : h ( x ) . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} \\big | \\big ( { \\cal F } ( v ) - { \\cal F } ( \\hat v ) \\big ) _ i ( \\tau , \\xi ) | & \\le { \\cal N } \\left ( \\int _ t ^ \\tau \\big ( | V ( s , x _ i ( s , \\tau , \\xi ) ) | + | V ( s , 0 ) | \\big ) \\ , d s + \\sum _ { j = k + 1 } ^ { k + m } | V _ { j } ( t , 0 ) | \\right ) \\\\ [ 6 p t ] & \\le 2 \\sqrt { n } { \\cal N } \\Big ( L _ 1 ^ { - 1 } e ^ { L _ 1 \\tau + L _ 2 } \\| V \\| + \\sum _ { j = k + 1 } ^ { k + m } | V _ { j } ( t , 0 ) | \\Big ) . \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} \\Vert \\pi _ { W _ i } C u ^ * f \\Vert = \\Vert \\pi _ { W _ i } u ^ * C f \\Vert = \\Vert \\pi _ { W _ i } u ^ * \\pi _ { u W _ i } C ^ * f \\Vert \\leq \\Vert u \\Vert \\Vert \\pi _ { u W _ i } C ^ * f \\Vert . \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} P _ { \\pm n , \\beta } = ( - 1 ) ^ { n + d - 1 } e ( P _ { \\pm n } ( X , \\beta ) ) , \\ N _ { n , \\beta } = ( - 1 ) ^ d e ( U _ n ( X , \\beta ) ) \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\left [ \\begin{array} { l l } B ^ { ( m ) } _ { n - m - 1 } & B ^ { ( m + 1 ) } _ { n - m - 1 } \\\\ A ^ { ( m ) } _ { n - m - 1 } & A ^ { ( m + 1 ) } _ { n - m - 1 } \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n - m } \\end{array} \\right ] \\left [ \\begin{array} { l l } B ^ { ( m - 1 ) } _ { n - m } & B ^ { ( m ) } _ { n - m } \\\\ A ^ { ( m - 1 ) } _ { n - m } & A ^ { ( m ) } _ { n - m } \\end{array} \\right ] \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } { \\| x ^ { { k } _ i } - y ^ { { k } _ i } \\| } = 0 . \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} \\square A = 0 , \\frac { \\partial A ^ \\alpha } { \\partial x ^ \\alpha } = 0 , \\left ( p ^ \\alpha \\frac { \\partial } { \\partial x ^ \\alpha } \\right ) A = 0 , \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} S _ 1 = - 1 , S _ 2 = 2 , S _ 3 = - 1 . \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} [ \\psi ( \\cdot , t ) ] _ { \\eta , B _ { \\mu _ j } ( \\xi _ j ) } \\lesssim \\| f \\| _ { * , \\beta , 2 s + \\alpha } \\left ( \\sum _ { j = 1 } ^ k \\frac { \\mu _ j ^ { - \\eta } t ^ { - \\beta } } { 1 + | y _ j | ^ { \\alpha + \\eta } } \\right ) \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} u ( \\phi ( \\alpha ( i _ 1 , i _ 2 ) ) ) = c ( i _ 2 + i _ 1 + 1 ) - ( - 1 ) = c ( i _ 1 + i _ 2 ) + c + 1 = u ( \\alpha ( i _ 1 , i _ 2 ) ) + c + 1 \\ , . \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} R _ { 2 2 } & = ( d - c a ^ { - 1 } b ) ^ { - 1 } ( [ d , a ] - c a ^ { - 1 } [ b , a ] ) \\Delta ^ { - 1 } \\\\ & = ( a - ( d - c a ^ { - 1 } b ) ^ { - 1 } \\Delta ) \\Delta ^ { - 1 } , \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} K _ { \\nu , \\alpha } & = \\frac { 1 } { 2 } ( - \\partial _ p ^ 2 + p ^ 2 ) + \\Big ( e ^ { i \\alpha } \\sqrt { \\nu } \\Big ) e ^ { - i \\alpha } \\Big ( p \\partial _ q + e ^ { 2 i \\alpha } q \\partial _ p \\Big ) \\\\ & = O _ p + z X _ { \\alpha } \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} A ^ { W , Y } _ t ( l ) : = \\left ( \\int _ 0 ^ t l ^ * ( r , W _ r ) \\mathrm d [ W , Y ] _ r \\right ) ^ * \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} \\tilde \\theta _ \\ell ( x ) = \\tilde { \\ell } x + O ( 1 ) \\ln ( 1 + x - b ) \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} T = T ^ { + } \\sqcup \\left [ \\begin{smallmatrix} 1 & 1 \\\\ 5 & 1 \\end{smallmatrix} \\right ] T ^ { + } \\sqcup \\left [ \\begin{smallmatrix} 3 & 1 \\\\ 5 & 3 \\end{smallmatrix} \\right ] T ^ { + } . \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} \\inf _ { v \\in \\overline { \\mathcal R } } F ( v ) = \\inf _ { v \\in \\mathcal R } F ( v ) . \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} \\mathcal { P } _ { 1 } = - \\frac { 8 } { 3 } y ^ { 3 } + 2 y \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} u _ \\gamma = \\begin{cases} W ^ + _ \\gamma , \\mbox { i n } \\Omega ^ + , \\\\ 2 - W ^ - _ \\gamma , \\mbox { i n } \\Omega ^ - . \\end{cases} \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} q _ t ( v , v ' ) : = \\frac { p _ t ( v , v ' ) } { m ( v ' ) } \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} \\mathbf { y } = \\sum _ { i = 1 } ^ { N } \\lambda _ i \\alpha _ i \\mathbf { s } _ i + \\mathbf { n } , \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} A _ { i } \\leq C _ { i } \\leq ( p ' ) ^ { \\frac { 1 } { p ' } } p ^ { \\frac { 1 } { q } } A _ { i } , \\ ; \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} F _ A ^ * ( b X ^ m d X ) = b ^ p X ^ m d X = b X ^ { \\frac { m + 1 - p } { p } } d X + ( C ^ { - 1 } - i ) ( b X ^ { \\frac { m + 1 - p } { p } } d X ) , \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} a _ { h } ( u , v ) = \\sum \\limits _ { D \\in \\mathcal { T } _ h } \\int _ { D } \\nabla _ { w , k - 1 , D } u \\cdot \\nabla _ { w , k - 1 , D } v \\ { \\rm d } x . \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} t r _ { \\partial M } \\hat K = a t r _ { \\partial M } K + b H _ { \\partial M } . \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} \\hat \\omega _ { \\mathbf { \\hat n } } = a ^ 2 d _ { \\partial M } ( b / a ) + \\omega _ { \\mathbf n } , \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} \\mathbf { w } ( \\theta ) = \\left ( \\cos ( j \\theta ) , \\ , \\sin ( j \\theta ) , 0 , \\ , \\ldots , \\ , 0 \\right ) \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} c _ w ( x _ 1 , \\dots , x _ N ) = \\sum ^ N _ { i = 1 } \\sum ^ N _ { j = i + 1 } - \\vert x _ i - x _ j \\vert ^ 2 , \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} C = R _ { j , k } = \\{ x : x \\equiv j \\mod k \\} \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} { R } _ k ^ i [ n _ 0 ] ( \\mathbf { p } , \\mathbf { s } , \\mathbf { r } ) = s _ { k } ^ i [ n _ 0 ] \\mathcal { B } \\log _ 2 \\Big ( 1 + \\frac { H _ k ^ i [ n _ 0 ] p _ k ^ i [ n _ 0 ] } { \\norm { \\mathbf { r } [ n _ 0 ] - \\mathbf { r } _ k } ^ 2 } \\Big ) . \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} \\left < ( \\phi \\cdot v ) ^ { 2 n } \\right > _ { \\alpha , \\Lambda , \\epsilon } = \\left < ( 2 n - 1 ) ! ! ( [ v ; G _ { \\Lambda , \\epsilon } ( t ) v ] ^ { n } ) \\right > _ { \\alpha , \\Lambda , \\epsilon } . \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} p _ \\tau = \\frac { \\partial L _ \\tau } { \\partial \\dot { t } _ \\tau } = - \\frac { f ( t _ \\tau ) } { 2 m } ( p _ { 1 , \\tau } ^ 2 + p _ { 2 , \\tau } ^ 2 ) - \\frac { m \\omega ^ 2 ( t _ \\tau ) f ^ { - 1 } ( t _ \\tau ) } { 2 } ( x _ { 1 , \\tau } ^ 2 + x _ { 2 , \\tau } ^ 2 ) . \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\tilde { V } _ k = \\frac { 1 } { \\sqrt { 1 + | z | ^ 2 } } \\ , V _ k \\otimes ( E ^ { ( 2 ) } _ { 1 2 } + z E ^ { ( 2 ) } _ { 2 1 } ) , Q ( z ) = | z | + 1 / | z | . \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} \\| V ^ { j _ 0 } \\| ^ 2 _ { L ^ 2 } = \\sup _ { j \\geq 1 } \\| V ^ j \\| ^ 2 _ { L ^ 2 } . \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} E ^ { p , q } _ { 2 } = H ^ { p } ( X , \\mathcal { H } ^ { q } ( \\Z ( r ) ) ) \\Rightarrow N ^ { \\bullet } H ^ { p + q } ( X , \\Z ( r ) ) , \\end{align*}"}
-{"id": "9696.png", "formula": "\\begin{align*} \\tau \\cdot v _ i = v _ { i - 1 } . \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} \\overline { P } ( f \\vert B ) = \\sup \\{ P ( f \\vert B ) \\colon P \\in \\mathbb { P } ^ * \\} = \\max \\{ P ( f \\vert B ) \\colon P \\in \\mathbb { P } ^ * \\} ~ ~ \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} z ( t ) = e ^ { - \\frac { 1 } { 2 } \\gamma t } \\left [ B _ 1 J _ 1 \\left ( \\frac { 2 \\omega _ 0 } { \\gamma } e ^ { - \\frac { 1 } { 2 } \\gamma t } \\right ) + B _ 2 Y _ 1 \\left ( \\frac { 2 \\omega _ 0 } { \\gamma } e ^ { - \\frac { 1 } { 2 } \\gamma t } \\right ) \\right ] , \\end{align*}"}
-{"id": "9863.png", "formula": "\\begin{align*} \\boldsymbol { D } ^ { - \\sigma } u ( x ) & : = \\dfrac { 1 } { \\Gamma ( \\sigma ) } \\int _ { - \\infty } ^ { x } ( x - s ) ^ { \\sigma - 1 } u ( s ) \\ , { \\rm d } s , \\\\ \\boldsymbol { D } ^ { - \\sigma * } u ( x ) & : = \\dfrac { 1 } { \\Gamma ( \\sigma ) } \\int _ { x } ^ { \\infty } ( s - x ) ^ { \\sigma - 1 } u ( s ) \\ , { \\rm d } s , \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} & \\mathbf { E } \\Big ( \\sup _ { 0 \\le t ' \\le t } \\big | \\xi ( x ( t ' ) ) - z ( t ' ) \\big | ^ 2 \\Big ) \\\\ \\le & \\Big ( \\frac { 3 \\kappa _ 1 ^ 2 \\ , t ^ 2 } { \\beta \\rho } + \\frac { 9 6 \\kappa _ 2 ^ 2 t } { \\beta ^ 2 \\rho } \\Big ) + \\Big ( 3 L _ b ^ 2 \\ , t + \\frac { 4 8 L _ \\sigma ^ 2 } { \\beta } \\Big ) \\ , \\int _ 0 ^ t \\Big ( \\mathbf { E } \\sup _ { 0 \\le t ' \\le s } \\big | \\xi ( x ( t ' ) ) - z ( t ' ) \\big | ^ 2 \\Big ) \\ , d s \\ , , \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} { \\mathrm d } \\widetilde { X } ^ { ( u ) } _ { t } \\ , = \\ , b ( t , \\widetilde { X } _ { t } ^ { ( u ) } , \\widetilde { F } _ { t } ^ { ( u ) } ) { \\mathrm d } t + { \\mathrm d } \\widetilde { B } _ { t } \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} \\Delta _ h u = - \\lambda u . \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} ( - \\Delta _ d ) G _ d ( x , y ) = \\delta ( x - y ) \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} \\frac { 1 } { c _ 1 } = - \\frac { G _ { t - 4 } ( \\theta ) } { F _ { t - 2 } ( \\theta ) } = - \\frac { G _ { t - 2 } ( \\theta ) } { F _ { t - 2 } ( \\theta ) } + 1 = - \\frac { \\lambda G _ { t - 3 } ( \\theta ) - ( k - 1 ) G _ { t - 4 } ( \\theta ) } { F _ { t - 2 } ( \\theta ) } + 1 = - \\frac { \\lambda G _ { t - 3 } ( \\theta ) } { F _ { t - 2 } ( \\theta ) } - \\frac { k - 1 } { c _ 1 } + 1 , \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\sum _ { t _ 0 , \\ldots , t _ { n - 1 } } \\langle f ( h _ { t _ 0 } , \\ldots , h _ { t _ { n - 1 } } ) , \\tau | _ { [ t _ 0 , \\ldots , t _ { n - 1 } ] } \\pi \\rangle = \\sum _ { t _ 0 , \\ldots , t _ n } \\langle ( - 1 ) ^ { n + 1 } f ( h _ { t _ 0 } , \\ldots , h _ { t _ { n - 1 } } ) , \\tau | _ { [ t _ 0 , \\ldots , t _ n ] } \\pi \\rangle . \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} 0 = \\frac { ( L + i _ j ) ! } { L ! } \\frac { 1 } { a _ j ^ { i _ j } } r _ { L + i _ j , j } = \\sum _ { \\substack { h + n = L + i _ j \\\\ 0 \\leq h \\leq L } } \\frac { ( L + i _ j ) ! } { h ! n ! } a _ j ^ { n - i _ j } c _ h = \\sum _ { h = 0 } ^ L \\binom { L + i _ j } { h } a _ j ^ { L - h } c _ h \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} C = \\{ v \\in M _ { \\mathbb R } \\ , | \\ , \\langle v , w \\rangle \\geq 0 \\ , \\ , \\ , { } w \\in \\sigma \\} , \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} F _ { m , \\ , j } ( X ) : = \\sum _ { s = 0 } ^ { m } ( - 1 ) ^ s \\binom { m } { s } \\binom { X - s } { j } \\in \\Q [ X ] . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} w = r _ 1 + B _ 1 w . \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} f ( x ) & = x ^ 3 + 1 + ( C + D x ) ( 1 - x ) ( x ^ 3 + 1 ) + m \\ : h ( x ) , \\\\ g ( x ) & = x + x ( C + D x ) ( 1 - x ) ( x ^ 3 + 1 ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} r ( q ) = \\sum _ { i = 0 } ^ { r - 1 } f _ i ( q ) [ n ] _ q ^ i \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{gather*} R _ { \\nabla } \\in \\Omega ^ 2 _ { \\mathrm { n l } } ( A ; \\operatorname { E n d } ( D ) ) , R _ { \\nabla } ( a , b ) = [ \\nabla _ a , \\nabla _ b ] - \\nabla _ { [ a , b ] } , a , b \\in \\Gamma ( A ) , \\end{gather*}"}
-{"id": "6687.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\binom k j G _ { s + j } } = G _ { s + 2 k } \\ , , \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} h ( m \\otimes \\{ [ c _ 1 | . . . | c _ k ] | a _ 2 | . . . | a _ n \\} ) = \\left \\{ \\begin{array} { l l l } 0 , & k = 1 , \\\\ \\{ [ c _ 1 ] | [ c _ 2 | . . . | c _ k ] | a _ 2 | . . . | a _ n \\} & k > 1 ; \\end{array} \\right . \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} \\int d x \\ x c ( x , y , t ) = \\langle x \\rangle \\gamma _ { 0 } ( y , t ) + \\langle x ^ { 2 } \\rangle \\gamma _ { 1 } ( y , t ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( \\frac { 4 } { 3 } y ^ { 3 } - y \\right ) e ^ { - 2 y ^ { 2 } } \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} P = \\sum _ \\alpha a _ \\alpha ( z ) \\partial _ z ^ \\alpha , \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} \\bigcap _ { i = 0 } ^ { m ^ + } ( \\pi ^ { - 1 } ( U ) + V + \\ker B _ i ) = \\pi ^ { - 1 } ( U ) , \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} K ^ + _ \\gamma ( r ) = a ^ + + \\frac { 1 } { 4 } r ^ 2 , K ^ - _ \\gamma ( r ) = a ^ - + b ^ - \\log r + \\frac { 1 } { 4 } r ^ 2 . \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} a ( i , j ) & = 2 \\ell - i - j + 2 n + 1 - \\lambda ' _ { \\ell + 1 - j } , \\\\ b ( i , j ) & = j - i + \\lambda ' _ { \\ell + 1 - j } . \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} \\max _ { 1 \\leq n \\leq m } \\| \\gamma ^ { ( d ) } _ n ( A ) \\| = \\| A \\| \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} t ^ { ( \\gamma ) _ h } = \\frac { h ^ \\gamma \\Gamma \\left ( \\frac { t } { h } + 1 \\right ) } { \\Gamma \\left ( \\frac { t } { h } - \\gamma + 1 \\right ) } . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} \\ell ^ p ( \\mathbb { N } , w ) = \\left \\{ f = \\{ f ( n ) \\} _ { n \\ge 0 } : \\| f \\| _ { \\ell ^ { p } ( \\mathbb { N } , w ) } : = \\Bigg ( \\sum _ { m = 0 } ^ { \\infty } | f ( m ) | ^ p w ( m ) \\Bigg ) ^ { 1 / p } < \\infty \\right \\} , \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} \\det \\mathcal { \\widehat { W } } = \\left ( \\left ( \\prod _ { j = 1 } ^ m \\prod _ { i _ j = 1 } ^ { \\nu _ j } \\frac { \\alpha _ j ^ { i _ j } } { ( L + i _ j ) ! } \\right ) \\prod _ { k = 1 } ^ M e _ k ! \\right ) \\det \\mathcal { W } , \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} \\Psi ^ \\star _ L ( \\mu , \\nabla \\tilde G _ L ) \\le \\Psi _ L ( \\mu , j ^ G ) \\le \\frac 1 2 \\sum _ { i \\in \\mathbb T _ L ^ d } \\sum _ { k = 1 } ^ d \\hat \\chi ^ 0 _ { i , i + e _ k } ( \\mu ) \\bigl [ 2 L \\sinh \\bigl ( \\tfrac 1 2 \\nabla ^ { i , i + e _ k } G ( \\cdot / L ) \\bigr ) \\bigr ] ^ 2 . \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} - \\div \\Big ( \\frac { \\phi ' _ { \\lambda } ( | \\nabla \\Q _ \\lambda | ) } { | \\nabla \\Q _ \\lambda | } \\nabla \\Q _ { \\lambda , i j } \\Big ) = - \\frac { 1 } { L } \\left ( \\frac { \\partial f _ B } { \\partial Q _ { i j } } ( \\bar \\Q ) + \\frac { B } { 3 } | \\bar \\Q | ^ 2 \\delta _ { i j } \\right ) \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} d _ { M , } : = \\inf \\left \\{ E ( v ) \\ : \\ v \\in H ^ 1 _ { } , \\| v \\| ^ 2 _ { L ^ 2 } = M \\right \\} , \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} u ' ( t ) = F ( u ( t ) ) = J _ n u ( t ) + N _ n ( u ( t ) ) , \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} \\tau _ i : = \\int _ { 0 } ^ 1 \\frac { 1 } { \\lambda _ i ( \\xi ) } \\ , d \\xi \\mbox { f o r } 1 \\le i \\le n \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} \\mu \\prod _ { k = 1 } ^ { n } P _ k ( \\epsilon ) r = \\tilde { \\pi } r - \\epsilon \\tilde { \\pi } \\tilde { P } ^ { ( 1 ) } ( I - \\tilde { P } + \\tilde { \\Pi } ) ^ { - 2 } \\tilde { P } r + O ( \\epsilon ^ 2 m ^ 3 ) \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{gather*} S \\big ( \\overset { I } { T } \\big ) = { } ^ t \\overset { \\ : I ^ * } { T } , \\end{gather*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\textstyle \\mu _ { _ { \\rm S M } } = \\frac { \\mathrm { t r } \\left ( \\mathbf { H } \\mathbf { S _ { _ \\mathrm { S M } } } \\mathbf { H } ^ { \\rm H } \\right ) } { \\mathrm { t r } \\left ( \\frac { \\mathbf { H } \\mathbf { S _ { _ \\mathrm { S M } } } \\mathbf { H } ^ { \\rm H } } { \\sigma ^ 2 \\ , \\ln 2 } \\left ( \\mathbf { I } _ { N _ R } + \\left ( 1 - \\rho _ { _ { \\rm S M } } \\right ) \\sigma ^ { - 2 } \\mathbf { H } \\mathbf { S _ { _ \\mathrm { S M } } } \\mathbf { H } ^ { \\rm H } \\right ) ^ { - 1 } \\right ) } . \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} B _ t ( f , g ) = \\sum _ { ( \\alpha , \\beta , \\gamma ) \\in ( \\{ - 1 , 1 \\} ^ d ) ^ 3 } K _ { \\alpha } ( Z _ { t , \\alpha , \\beta } f Z _ { t , \\alpha , \\gamma } g ) . \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } x b ^ { m _ k } \\sim \\phi b ^ { m _ 0 } x b ^ { n _ k } + \\sum _ { i = n _ k + 1 } ^ { m _ k } \\sum _ { s \\in S _ b } \\phi b ^ { m _ 0 } x b ^ { i } s . \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\pi _ { \\chi _ g } ( x ) = \\frac { 1 } { | G | } \\sum _ { \\tau \\in G } \\chi _ g ( \\tau ^ { - 1 } ) \\tau ( x ) ( x \\in L ) . \\end{align*}"}
-{"id": "9988.png", "formula": "\\begin{align*} \\bar u ( q , \\tau ) = e ^ { q \\tau } \\bar u _ 0 ( q ) + ( \\bar p ( q ) - 1 ) \\int _ 0 ^ \\tau e ^ { q ( \\tau - s ) } u ( 0 , s ) d s . \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} \\frac { 1 } { 4 \\sqrt { \\l } } \\leq \\prod _ { j = 0 } ^ { [ \\frac { \\l - 2 } { 2 } ] } \\frac { \\l - 1 - 2 j } { \\l - 2 j } \\leq \\frac { 4 } { \\sqrt { \\l } } . \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} { \\nu } _ x = \\sin ^ 2 ( { \\nu } ) + \\frac { b _ * u _ { b } + c _ * p _ { b } } { a _ * } \\cos ^ 2 ( { \\nu } ) \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\lambda _ c ( G ) = \\mathrm { g r } ( G ) = \\frac { 1 } { z _ * } . \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} 0 - f ^ { \\prime } r ^ { m - 1 } e ^ { - \\frac { r ^ 2 } { 4 } } & = \\int ^ { + \\infty } _ { r } r ^ { m - 1 } e ^ { - \\frac { r ^ 2 } { 4 } } ( - \\lambda e ^ { - \\frac { r ^ 2 } { 2 ( m - 2 ) } } + r ^ { - 2 } \\lambda _ k ) f d r \\\\ & \\le \\int ^ { + \\infty } _ { r } r ^ { m - 3 } e ^ { - \\frac { r ^ 2 } { 4 } } \\lambda _ k f d r \\\\ & \\le \\int ^ { + \\infty } _ { r } r ^ { m - 3 } e ^ { - \\frac { r ^ 2 } { 4 } } \\lambda _ k d r f . \\\\ \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} z _ { \\sf A f f } ( { \\cal U } ) & \\leq z _ { \\sf A f f } ( { \\cal V } ) \\\\ & = O \\left ( \\frac { \\log n } { \\log \\log n } \\right ) \\cdot z _ { \\sf A R } ( { \\cal V } ) \\\\ & \\leq O \\left ( \\frac { \\log n } { \\log \\log n } \\right ) \\cdot L \\cdot z _ { \\sf A R } ( { \\cal U } ) , \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} Y _ t = \\xi - \\int _ t ^ T Z _ r \\mathrm d X _ r . \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} \\Pi _ { k \\boldsymbol { \\nu } } \\big ( x _ { 1 } , x _ { 2 } \\big ) _ 2 = \\Pi _ { k \\boldsymbol { \\nu } } \\big ( x _ { 1 } , x _ { 2 } \\big ) _ 2 ' + \\Pi _ { k \\boldsymbol { \\nu } } \\big ( x _ { 1 } , x _ { 2 } \\big ) _ 2 '' , \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} L ^ t _ { \\theta } ( a \\cdot v ) = \\theta ( a ) \\cdot v + a \\cdot L ^ t _ { \\theta } ( v ) . \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} h _ { 1 , \\sigma } = k _ { 1 , \\sigma } \\ \\forall \\sigma \\in I _ F , \\ h _ { 2 , \\sigma } = \\left \\{ \\begin{array} { c c } - k _ { 2 , \\sigma } \\ & \\mbox { i f } \\ \\sigma = \\hat { \\sigma } \\\\ k _ { 2 , \\sigma } \\ & \\mbox { o t h e r w i s e } \\end{array} \\right . . \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} \\int ( x _ 1 + \\cdots + x _ n ) ^ 2 \\ , \\mu ( \\d x ) = 1 . \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { \\nu \\in \\chi _ \\infty } \\pi _ { \\nu } f = \\sum _ { \\nu \\in \\chi _ \\infty } \\sum _ { \\ell = 1 } ^ { d ( \\nu ) } a _ { \\nu , \\ell } ( r ) \\varphi _ { \\nu , \\ell } ( y ) \\end{align*}"}
-{"id": "9993.png", "formula": "\\begin{align*} \\frac d { d \\tau } \\bar U - q U = ( 1 - \\bar P ( q ) ) \\alpha ( \\tau ) , \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} \\det Y = \\sum _ { k _ { 0 } = 0 } ^ { M } \\cdots \\sum _ { k _ { p - 1 } = 0 } ^ { M } \\det \\left [ \\left ( p _ { k _ { i } } ( i , j ) \\right ) _ { 0 \\le i , j < p } \\right ] n ^ { - p + 2 \\beta p - k _ { 0 } - \\cdots - k _ { p - 1 } } . \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} \\int d x \\ x b ( x , y , t ) = \\langle x ^ { 2 } \\rangle \\beta _ { 1 } ( y , t ) = \\frac { 1 } { 2 } \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } ( 1 - 4 y ^ { 2 } ) e ^ { - 2 y ^ { 2 } } \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} H _ 0 = \\ker B _ + = \\bigcap _ { i = 1 } ^ { m ^ + } \\ker B _ i . \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} { \\bf b } & = Q \\circ \\pi _ { \\mathbb { R } ^ { k ' _ 1 } \\times \\mathbb { W } _ 1 \\times \\mathbb { R } ^ { k ' _ 2 } \\times \\mathbb { W } _ 2 } \\circ ( \\Psi _ 1 \\times \\Psi _ 2 ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\circ ( ( h _ 1 \\times h _ 2 ) - ( \\mathfrak { s } _ 1 \\times \\mathfrak { s } _ 2 ) ) \\circ ( \\psi _ 1 \\times \\psi _ 2 ) ^ { - 1 } \\circ q ^ { - 1 } \\\\ & = Q \\circ ( { \\bf b } _ 1 \\times { \\bf b } _ 2 ) \\circ q ^ { - 1 } . \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} \\prod _ { i \\in \\{ i _ 1 , \\dots , i _ j \\} } b _ { \\sigma ( i ) , i } = f ( z _ 1 ) ^ { k _ { \\sigma } } \\prod _ { i \\in \\{ i _ 1 , \\dots , i _ j \\} } a _ { \\sigma ( i ) , i } . \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} C _ p ( \\nu _ 2 ) & \\leq C _ p ( \\nu _ 1 ) - \\frac { C _ p ( \\nu _ 1 ) } { 4 } \\frac { ( C _ p ( \\nu _ 2 ) - 1 ) ^ 2 } { ( C _ p ( \\nu _ 2 ) - 1 ) ^ 2 + C _ p ( \\nu _ 2 ) } \\\\ & = C _ p ( \\nu _ 1 ) \\left ( 1 - \\frac { 1 } { 4 } \\frac { ( C _ p ( \\nu _ 2 ) - 1 ) ^ 2 } { ( C _ p ( \\nu _ 2 ) - 1 ) ^ 2 + C _ p ( \\nu _ 2 ) } \\right ) \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } \\phi ( x ) d x = \\int _ { \\mathbb { R } ^ d } \\phi ( X ( s , t , x ) ) J X ( s , t , x ) d x ~ ~ \\forall ~ ~ \\phi \\in L ^ 1 ( \\mathbb { R } ^ d ) \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} \\left \\| \\rho _ { t ^ { - 1 } } ( f ) - \\frac { M ' } { r } \\left ( \\sum _ { i = 1 } ^ r { \\rho _ { t ^ { - 1 } } ( \\gamma _ i ) } \\right ) \\right \\| _ { L _ 2 ( m _ { B ^ - } ) } \\leq \\delta \\sqrt { K } M ' t \\in T . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} { \\rm D e g } ( v ) = \\# \\{ v ' \\in V | \\ v \\sim v ' \\} , \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} \\alpha \\circ \\beta = \\cdot w , \\ \\beta \\circ \\alpha = \\cdot w . \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} n _ { \\lambda , i } = \\lambda ( [ x _ i , y _ i ] ) + \\langle x _ i , y _ i \\rangle \\frac { s _ i c } { m } . \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} S _ { 1 \\bar 1 } = \\mu _ { k \\bar r } ^ { \\bar 1 } \\mu _ { \\bar k r } ^ 1 , \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} k = \\frac { ( a - 1 ) \\ell ( a + \\ell ) } { 2 a ( \\ell - a ^ 2 ) } , \\lambda = \\frac { ( a + \\ell ) ( 3 a ^ 2 + a \\ell - a - 3 \\ell ) } { 4 a ( \\ell - a ^ 2 ) } , \\mu = \\frac { ( a - 1 ) \\left ( \\ell ^ 2 - a ^ 2 \\right ) } { 4 a ( \\ell - a ^ 2 ) } . \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} A _ { 2 } : = \\sup _ { R > 0 } \\left ( \\int _ { \\{ | x | \\leq R \\} } \\phi _ { 2 } ( x ) d x \\right ) ^ { \\frac { 1 } { q } } \\left ( \\int _ { \\{ | x | \\geq R \\} } ( \\psi _ { 2 } ( x ) ) ^ { - ( p ' - 1 ) } d x \\right ) ^ { \\frac { 1 } { p ' } } < \\infty . \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} \\bigcap _ { i = 0 } ^ { m ^ + } ( U + \\ker \\beta _ i ) = U . \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} \\begin{cases} \\hom _ { \\square _ c ^ + } ( \\emptyset , \\emptyset ) = \\ast , \\\\ \\hom _ { \\square _ c ^ + } ( \\emptyset , \\square ^ n ) = \\ast , \\\\ \\hom _ { \\square _ c ^ + } ( \\square ^ n , \\emptyset ) = \\emptyset . \\end{cases} \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} \\xi _ k \\circ \\Delta _ k ^ { r - 1 } \\tilde { G } _ { k , m , r } ( z ) & = ( 4 \\pi ) ^ { 1 - k } ( 1 - k ) ^ { r - 1 } \\biggl ( F _ { 2 - k , - m , 0 } ( z ) + \\sum _ { \\substack { A _ { 2 - k } < n < 0 \\\\ ( - 1 ) ^ { \\lambda _ { 2 - k } } n \\equiv 0 , 1 ( 4 ) } } a _ { 2 - k } ( m , n ) F _ { 2 - k , n , 0 } ( z ) \\biggr ) \\\\ & = ( 4 \\pi ) ^ { 1 - k } ( 1 - k ) ^ { r - 1 } f _ { 2 - k , m } ( z ) . \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} \\zeta ( j , s ) ' & = \\left ( \\prod _ { m = m ( j , s ) + 1 } ^ { N _ { k ( j , s ) } } b _ { k ( j , s ) , m - 1 } ^ { - 1 } p _ { k ( j , s ) , m } b _ { k ( j , s ) , m } \\right ) \\left ( \\prod _ { k = k ( j , s ) + 1 } ^ { j } \\prod _ { l = 1 } ^ { N _ k } b _ { k , l - 1 } ^ { - 1 } p _ { k , l } b _ { k , l } \\right ) \\\\ & = b _ { k ( j , s ) , m ( j , s ) } ^ { - 1 } \\zeta ( j , s ) b _ { j , N _ j } . \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} x ^ { \\ell } - c = ( x - a ) \\prod _ { j = 1 } ^ { ( \\ell - 1 ) / 2 } ( x ^ 2 - a ( b ^ j + b ^ { q j } ) x + a ^ 2 ) . \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} e ( t ) G _ { X , Y } = 0 . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} ( F - N c ) _ + = G ( c , N , F ) . \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} 0 \\geq \\Phi _ \\rho ( \\lambda ) - ( \\Phi _ { \\delta , L } ( \\lambda _ k ) + & \\langle s _ { \\delta , L } ( \\lambda _ k ) , \\lambda - \\lambda _ k \\rangle ) \\\\ & \\geq - \\frac { L } { 2 } \\| \\lambda - \\lambda _ k \\| ^ 2 - \\delta , \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} f _ { \\lambda u } = \\lambda f _ u + d \\lambda ( u ) , \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} G ^ i \\oplus D ^ { i - 1 } \\to & Z ^ i ( D ^ { \\bullet } ) \\\\ ( a , b ) \\ \\mapsto & \\ d ( b ) = ( \\alpha _ i \\beta _ i ( a ) + d ( b ) ) - \\alpha _ i \\beta _ i ( a ) . \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} t _ h & : = \\frac { 2 h } { 3 } \\cos \\frac { \\theta + 4 \\pi } { 3 } + \\frac { h } { 3 } \\in \\left ( 0 , \\frac { 2 h } { 3 } \\right ) , \\\\ T _ h & : = \\frac { 2 h } { 3 } \\cos \\frac { \\theta } { 3 } + \\frac { h } { 3 } \\in \\left ( \\frac { 2 h } { 3 } , h \\right ) . \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} \\overline { \\varphi } ( \\cdot | s ) : = \\sum _ { j = 1 } ^ N \\gamma _ j ( s ) \\delta _ { f _ j ( s ) } ( \\cdot ) . \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} h ( 0 , \\cdot ) = h _ { 0 } . \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| u ^ { k } - x ^ k \\| = 0 . \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} ( F f ) ( s , y ) : = e ^ { s Q / 2 } f ( e ^ { s } y ) \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} \\Tilde T ~ = ~ \\frac { \\pi \\varepsilon } { 1 6 L ^ { 2 } } \\ , , \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} E ( u _ { 0 , n } ) & = \\frac { \\mu _ n ^ 2 } { 2 } \\| Q \\| ^ 2 _ { \\dot { H } ^ 1 _ c } - \\frac { d \\mu _ n ^ { \\frac { 4 } { d } + 2 } } { 2 d + 4 } \\| Q \\| ^ { \\frac { 4 } { d } + 2 } _ { L ^ { \\frac { 4 } { d } + 2 } } \\\\ & = \\mu _ n ^ 2 E ( Q ) + \\frac { d } { 2 d + 4 } \\mu _ n ^ 2 \\left ( 1 - \\mu _ n ^ { \\frac { 4 } { d } } \\right ) \\| Q \\| ^ { \\frac { 4 } { d } + 2 } _ { L ^ { \\frac { 4 } { d } + 2 } } . \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} \\tilde { k } ^ * ( z ) = \\min _ { \\gamma \\in \\mathcal { W } ( z ) } \\int _ { Y _ z \\times U } k ( y , u ) \\gamma ( d y , d u ) , \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} \\phi \\left ( \\mathbb { E } \\left [ \\sum _ { i = 1 } ^ { n } X _ i \\right ] \\right ) \\le \\mathbb { E } \\left [ \\phi \\left ( \\sum _ { i = 1 } ^ { n } X _ i \\right ) \\right ] , \\ \\forall \\phi , \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} T ( t ) : = \\frac { t } { 8 } s _ + ^ { 2 - p } , H ( t ) : = \\frac { 3 + \\sqrt { 9 + 8 t } } { 3 2 } s _ + ^ { 2 - p } \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} \\int d x \\ x b ( x , y , t ) = \\langle x \\rangle \\beta ( y , t ) = 0 \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} h ^ 1 _ 1 ( F _ 1 ) = \\{ \\gamma _ 1 ( 0 ) \\} . \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} \\begin{aligned} K ^ \\pm _ \\gamma & = 1 , \\mbox { o n } \\gamma , \\\\ \\partial _ n K ^ + _ \\gamma + \\partial _ n K ^ - _ \\gamma & = 0 , \\mbox { o n } \\gamma . \\end{aligned} \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} & Q ^ 1 _ { i \\bar i } = T _ { i k \\bar m } T _ { \\bar i \\bar k m } \\ , , Q ^ 2 _ { i \\bar i } = T _ { \\bar k \\bar m i } T _ { k m \\bar i } \\ , , \\\\ & Q ^ 3 _ { i \\bar i } = T _ { i k \\bar k } T _ { \\bar i \\bar m m } \\ , , Q ^ 4 _ { i \\bar j } = \\frac 1 2 ( T _ { m k \\bar k } T _ { \\bar m \\bar i i } + T _ { \\bar m \\bar k k } T _ { m i \\bar i } ) \\ , , \\\\ & q ^ 1 = q ^ 2 = \\| T \\| ^ 2 \\ , , q ^ 3 = q ^ 4 = \\| w \\| ^ 2 \\ , , \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} E ( m T ) \\leq e ^ { - \\lambda m T } E ( 0 ) , \\ m = 1 , 2 , \\ldots . \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} C _ { \\lambda } : \\frac { x ^ 2 } { a - \\lambda } + \\frac { y ^ 2 } { b - \\lambda } = 1 , a > b > 0 . \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta K ^ - _ { \\gamma } & = 1 , \\mbox { i n } \\Omega , \\\\ K ^ - _ \\gamma & = 1 , \\mbox { o n } \\partial \\Omega = \\alpha , \\\\ K ^ - _ \\gamma & = 1 , \\mbox { o n } \\gamma . \\end{aligned} \\end{align*}"}
-{"id": "9916.png", "formula": "\\begin{align*} \\varPhi ( z ) = d z _ { 1 } \\wedge \\cdots \\wedge d z _ { n } \\quad \\varPhi _ i ( z ) = ( - 1 ) ^ { i - 1 } z _ i \\ , d z _ 1 \\wedge \\cdots \\wedge \\widehat { d z _ i } \\wedge \\cdots \\wedge d z _ n . \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} [ \\ln R ( x ) ] ^ \\prime = \\frac { V ( x ) } { 2 \\gamma ^ \\prime ( x ) } \\sin 2 \\theta ( x ) \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} x _ 1 + x _ 2 + x _ 3 = 0 . \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} z ( t ) = C y ( t ) , t \\in [ 0 , T ] \\ , , \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} H = \\ker B _ 0 + \\ker B _ i = \\ker B _ 0 + V , \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} \\left | \\sum _ { \\substack { 1 \\leq i \\leq t \\\\ 0 \\leq j \\leq m } } c _ { i , j } \\binom { s } { j } \\mu _ i ^ { s - j } \\right | = \\max _ { \\substack { 1 \\leq i \\leq t \\\\ 0 \\leq j \\leq m } } \\{ | c _ { i , j } | \\} . \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} = \\Delta _ p ^ { u _ 0 } ( u _ { k } ^ { \\delta } , u _ { k + 1 } ^ { \\delta } ) - \\mu \\langle j _ p ( F ( u _ k ^ { \\delta } ) - v ^ { \\delta } ) , F ' ( u _ k ^ { \\delta } ) ( u _ k ^ { \\delta } - u ^ { \\dagger } ) \\rangle + \\beta _ k \\langle J _ p ( u ^ { \\dagger } - u _ 0 ) , u ^ { \\dagger } - u _ k ^ { \\delta } \\rangle \\\\ - \\beta _ k \\langle J _ p ( u ^ { \\dagger } - u _ 0 ) - J _ p ( u _ k ^ { \\delta } - u _ 0 ) , u ^ { \\dagger } - u _ k ^ { \\delta } \\rangle . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} h _ { I _ F } = \\mbox { v o l } ( X _ { \\Gamma _ { 0 , s s } } ) ( 2 \\kappa + 3 ) ^ d + L _ { c u s p } , \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} \\lim \\limits _ { L \\rightarrow \\infty } \\frac { { \\rm \\Theta } _ L } { \\sqrt { L } } = 2 \\sqrt { \\frac { 2 } { \\pi } } . \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\tfrac 1 t \\log \\mathbb P ( P _ t \\geq x t ) & = - ( 1 - x + x \\log x ) x > 1 , \\\\ \\lim _ { t \\to \\infty } \\tfrac 1 t \\log \\mathbb P ( P _ t \\leq x t ) & = - ( 1 - x + x \\log x ) x \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ { j } } g ( x ) { \\mathrm M } _ { t } ^ { ( j ) } ( { \\mathrm d } x ) \\ , = \\ , \\int _ { \\mathbb R ^ { j } } g ( x ) { \\mathrm M } _ { 0 } ^ { ( j ) } ( { \\mathrm d } x ) + \\int ^ { t } _ { 0 } [ \\mathcal A _ { s } ^ { ( j ) } ( { \\mathrm M } ^ { ( j + 1 ) } ) g ] { \\mathrm d } s \\ , ; 0 \\le t \\le T \\ , , \\ , \\ , j \\ , = \\ , 2 , \\ldots , k - 1 \\ , \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} \\inf _ B u _ i > 0 , \\ ; i = 1 , 2 , \\ ; \\ ; B \\Subset \\Omega . \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} z _ 0 = \\frac { 1 } { 2 ( \\sqrt { m _ 1 } + \\sqrt { m _ 2 } ) ^ 2 } \\ , \\left [ x + \\frac { 4 ( m - 1 + \\sqrt { m _ 1 m _ 2 } ) ^ 2 } { x } - 2 ( m - 2 ) \\right ] , \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} R ( e _ { i j } ) = \\begin{cases} e _ { i , j + 1 } + e _ { i + 1 , j + 2 } + \\ldots + e _ { n - j + i - 1 , n } , & i \\leq j \\leq n - 1 , \\\\ - ( e _ { i - 1 , j } + e _ { i - 2 , j - 1 } + \\ldots + e _ { i - j + 1 , 1 } ) , & i > j , \\\\ 0 , & j = n . \\end{cases} \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{gather*} \\begin{binom} n { k - 1 } \\end{binom} \\left ( \\prod _ { l = k - 1 } ^ { n - 1 } ( m - l ) \\right ) + \\begin{binom} n { k } \\end{binom} \\left ( \\prod _ { l = k } ^ { n - 1 } ( m - l ) \\right ) ( m - 2 k ) \\\\ { } - \\begin{binom} n { k + 1 } \\end{binom} \\left ( \\prod _ { l = k + 1 } ^ { n - 1 } ( m - l ) \\right ) ( k + 1 ) ( m - k ) . \\end{gather*}"}
-{"id": "1437.png", "formula": "\\begin{gather*} \\operatorname { c h a r } \\big ( p ^ ! ( A ) \\big ) = \\mathfrak { u } \\big ( p ^ ! \\big ( \\nabla ^ { \\mathrm { b a s } } \\big ) \\big ) , \\end{gather*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\begin{aligned} & \\textbf { T r } \\left ( x \\diamond s \\right ) = 2 x ^ T s , \\\\ & \\textbf { T r } \\left ( x \\diamond x \\right ) = 2 x ^ T x = 2 | | x | | ^ 2 . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} \\mathcal F ^ a = \\big \\{ x \\in \\mathfrak M : \\mathcal F ( x ) \\leq a \\big \\} . \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} W _ 2 ( y _ 0 ) = \\mathcal { W } ( z ) \\ \\ \\ \\ \\ \\ \\forall y _ 0 \\in Y _ z . \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} \\alpha = \\min \\{ \\lambda _ { \\min } ( \\boldsymbol { \\varepsilon } ) , \\lambda _ { \\min } ( \\boldsymbol { \\mu } ) \\} . \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} C = \\max \\left ( 1 6 , \\ \\left ( \\frac { 8 c _ 1 } { c _ 3 } \\right ) ^ 2 , \\ \\frac { 5 1 2 } { 3 c _ 1 ^ 2 } \\right ) . \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} ( ( v , w ) ) _ D = ( q _ v , w ) , \\forall w \\in H ^ 2 _ 0 ( D ) , \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} \\mathcal { U } _ p = \\{ ( z , w ) \\in \\C \\times \\C ^ { n } \\ ; | \\ ; \\textrm { I m } \\ ; z > p ( w ) \\} . \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} N _ n ( u ( t _ n + \\tau ) ) = N _ n ( u ( t _ n ) ) + r _ { N } ( \\tau ) \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} { \\mathrm d } X ^ { \\bullet } _ { t } \\ , = \\ , b ( t , X _ { t } ^ { \\bullet } , \\ , \\mathcal L _ { X _ { t } ^ { \\bullet } } ) \\ , { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k - 1 } x _ { v _ { j } , v _ { j + 1 } } + x _ { v _ k , v _ 1 } + 2 x ( [ \\{ v _ 1 \\} : \\{ v _ 3 , \\ldots , v _ k \\} ] ) & \\\\ + \\sum _ { j = 4 } ^ { k } x ( [ \\{ v _ j \\} : \\{ v _ 3 , \\ldots , v _ { j - 1 } \\} ] ) \\leq k - 1 . \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} a _ 1 - a _ 2 \\ & = \\ \\sqrt { a _ 1 ^ 2 + a _ 2 ^ 2 - 2 a _ 1 a _ 2 } \\\\ & = \\sqrt { 2 ( a _ { 1 } ^ 2 + a _ { 1 } ^ 2 - \\| \\bar { a } \\| ^ 2 ) - ( ( a _ 1 + a _ { 2 } ) ^ 2 - 2 \\| \\bar { a } \\| ^ 2 ) } = 1 . \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} B + \\epsilon _ 1 B ^ 2 \\leq \\frac { 2 } { \\epsilon } \\begin{pmatrix} I & - I \\\\ - I & I \\end{pmatrix} . \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} \\widehat { \\Lambda } : = \\left \\{ \\left ( \\lambda , - \\tfrac { i } { 2 } \\bar { \\lambda } \\right ) \\mid \\lambda \\in \\Lambda \\right \\} . \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\tilde { \\mu } } ( u ( \\cdot , x ) ) = \\mathbb { E } _ { \\tilde { \\mu } } ( \\phi _ x - \\phi _ 0 ) = 0 . \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq b \\leq n } k _ b j _ b + \\sum _ { j \\in \\bar { Z } \\setminus J } ( l _ j - \\bar { l } _ j ) j = 0 . \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} \\alpha + \\beta + \\gamma = n + \\frac { n } { t } - \\frac { n } { q } , \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} \\| W _ t f \\| _ { \\ell ^ 2 ( \\mathbb { N } ) } & = \\| c _ n ( e ^ { ( \\cdot - s ^ { + } ) t } F ) \\| _ { \\ell ^ 2 ( \\mathbb { N } ) } = \\| e ^ { ( \\cdot - s ^ { + } ) t } F \\| _ { L ^ 2 ( X , d \\mu ) } \\\\ & \\le \\| F \\| _ { L ^ 2 ( X , d \\mu ) } = \\| f \\| _ { \\ell ^ 2 ( \\mathbb { N } ) } . \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} \\phi _ { \\theta } = \\theta + \\phi _ { \\theta , 1 } \\tau + \\dots + \\phi _ { \\theta , r } \\tau ^ r \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} u _ { m + i } ( t , 1 ) = 0 \\mbox { f o r } 0 \\le t < T _ { o p t } - \\tau _ i - \\tau _ { m + i } . \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} \\forall m \\exists p T _ 1 ( e , m , p ) \\rightarrow \\neg \\forall n \\left [ \\begin{gathered} \\neg \\exists m \\exists p ( T _ 1 ( e , m , p ) \\wedge U ( p ) = n ) \\rightarrow n \\not \\in X _ 0 \\\\ \\wedge \\\\ \\exists m \\exists p ( T _ 1 ( e , m , p ) \\wedge U ( p ) = n ) \\rightarrow n \\in X _ 0 \\end{gathered} \\right ] \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} | C | = \\frac { 1 } { 3 } \\sum _ { j = 1 } ^ 3 \\sum _ { i = 1 } ^ q | C ^ j _ i | = \\frac { 1 } { 3 } \\sum _ { j = 1 } ^ 3 \\sum _ { i = 1 } ^ q ( q - a ^ j _ i ) = q ^ 2 - \\frac { 1 } { 3 } \\sum _ { j = 1 } ^ 3 \\sum _ { i = 1 } ^ q a ^ j _ i . \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} \\Psi _ z ( y ) : = \\sum _ { i = 1 } ^ k ( F _ i ( y ) - z _ i ) ^ 2 . \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} \\lim _ { | x | \\to + \\infty } \\rho ( x ) = 0 , \\ , \\ , \\lim _ { | x | \\to + \\infty } V ( x ) = V _ { \\infty } > 0 , \\ , \\ , \\lim _ { | x | \\to + \\infty } K ( x ) = K _ { \\infty } > 0 , \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} \\infty = \\int _ { B _ R ( 0 ) } \\frac { \\lvert v ( k ) \\lvert ^ 2 } { \\omega ( k ) ^ 2 } d \\lambda _ \\nu = \\lambda _ \\nu ( B _ 1 ( 0 ) ) \\int _ { 0 } ^ \\infty 1 _ { B _ R ( 0 ) } ( k e _ 1 ) \\frac { \\lvert v ( k e _ 1 ) \\lvert ^ 2 } { \\omega ( k e _ 1 ) ^ 2 } k ^ { \\nu - 1 } d \\lambda _ 1 . \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} W ( D ) : = \\left \\{ v = ( v _ 0 , v _ b ) : v _ 0 \\in L _ 2 ( D _ 0 ) , v _ b \\in L _ 2 ( \\partial D ) \\right \\} . \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{gather*} { } _ 2 F _ 1 \\biggl ( \\begin{matrix} r , \\ , 1 - r \\\\ 1 \\end{matrix} \\biggm | z \\biggr ) ^ 2 = { } _ 3 F _ 2 \\biggl ( \\begin{matrix} \\frac 1 2 , \\ , r , \\ , 1 - r \\\\ 1 , \\ , 1 \\end{matrix} \\biggm | 4 z ( 1 - z ) \\biggr ) \\end{gather*}"}
-{"id": "6804.png", "formula": "\\begin{align*} c ( x , y , t ) = h ( x ) \\gamma _ { 0 } ( y , t ) + x h ( x ) \\gamma _ { 1 } ( y , t ) \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\mathbf { E } [ W ^ { ( 3 ) } _ k ( x ) ] \\to \\mathbf { E } \\left [ \\sum _ { l = 0 } ^ { \\infty } f ( x + g _ l + S ( l ) ) \\mathbb { I } \\{ \\tau > l \\} \\right ] . \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} \\dot \\rho _ t = - \\nabla \\cdot J ( \\rho _ t ) = \\Delta \\phi ( \\rho _ t ) + \\nabla \\cdot ( \\chi ( \\rho _ t ) \\nabla V ) . \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - f _ 2 ) ^ j \\binom k j X _ { m + a k - b j } } = f _ 1 { } ^ k X _ m \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} W _ { \\phi } = \\prod _ { p \\leq \\infty } W _ { \\phi , p } \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} \\int _ D v ( { \\Delta } ^ 2 q ) \\ { \\rm d } x = \\int _ { \\partial D } \\left ( Q _ { n } ( { { \\bf D } ^ 2 q } ) + \\frac { \\partial U _ { n t } ( { { \\bf D } ^ 2 q } ) } { \\partial t } \\right ) v \\ { \\rm d } s - \\int _ { \\partial D } U _ { n n } ( { { \\bf D } ^ 2 q } ) \\frac { \\partial v } { \\partial n } \\ { \\rm d } s . \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} G ( y ) = 2 y ^ { 2 } \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } B ^ { ( m ) } _ { n - m - 1 } & B ^ { ( m + 1 ) } _ { n - m - 1 } \\\\ A ^ { ( m ) } _ { n - m - 1 } & A ^ { ( m + 1 ) } _ { n - m - 1 } \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n - m } \\end{array} \\right ] \\cdots \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n } \\end{array} \\right ] \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} C ( \\gamma ) & \\le \\frac { N ( N - 1 ) } { 2 } f \\left ( f ^ { - 1 } \\left ( \\frac { N ^ 2 ( N - 1 ) } { 2 } f ( \\beta ) \\right ) \\right ) \\\\ & = \\frac { N ( N - 1 ) } { 2 } \\cdot \\frac { N ^ 2 ( N - 1 ) } { 2 } f ( \\beta ) = \\frac { N ^ 3 ( N - 1 ) ^ 2 } { 4 } f ( \\beta ) , \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} u _ 1 ( T , x ) = 0 \\mbox { f o r } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} f _ 1 ( r ) = r . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} c l W _ 1 ( y _ 0 ) = W _ 2 ( y _ 0 ) = \\mathcal { W } ( z ) \\ \\ \\ \\ \\ \\ \\forall y _ 0 \\in Y _ z , \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} P _ I / B = ( L _ I U _ I ) / B = L _ I / ( B \\cap L _ I ) = L _ I / B _ I , \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} \\lim _ { R \\to 1 } ( 1 - R ^ n ) = 0 , \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} 2 n - 1 \\nmid \\sum _ { b = 1 } ^ n j _ b \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\ell & = \\left ( \\frac { 1 } { k _ 1 } + \\frac { 1 } { k _ 2 } \\right ) N _ 1 ( \\delta ) , \\intertext { a n d h e n c e f o r a n y $ \\delta $ } N _ 1 ( \\delta ) & = \\frac { \\ell k _ 1 k _ 2 } { k _ 1 + k _ 2 } , \\\\ & = \\frac { k _ 1 k _ 2 } { n - 1 } \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} ( L \\circ \\widetilde { \\Phi } ) _ { \\ast } \\omega _ { \\mathbf i } ^ N = ( \\Phi \\circ \\eta ) _ { \\ast } ( \\eta ^ { \\ast } \\omega _ { \\mathcal I } ) ^ N = \\Phi _ { \\ast } \\omega _ { \\mathcal I } ^ N . \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} ( t _ 0 , t _ 1 , \\dots , t _ k ) z = ( L _ 0 , L _ 1 , \\dots , L _ { k + n + 1 } ) . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} \\mu _ 2 ( \\rho _ 2 ) & \\ge \\nu \\bigl ( \\partial ( \\rho _ 2 B ) \\cap \\left ( B + { \\bf v } _ 1 \\right ) \\bigr ) \\\\ & = 2 \\pi \\sqrt { 1 + { { 4 \\sqrt { 8 3 0 } } \\over { 2 1 } } } \\left ( \\sqrt { 1 + { { 4 \\sqrt { 8 3 0 } } \\over { 2 1 } } } - { { \\sqrt { 8 3 0 } } \\over { 2 1 } } - 1 \\right ) . \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} \\psi ( k , v ) & \\leq c _ 1 \\exp ( \\phi _ v ( k ) ) \\leq c _ 1 \\exp ( \\phi _ v ( \\varepsilon _ 0 v ) ) \\\\ & \\leq c _ 1 \\exp \\left ( - \\frac { c _ 2 ( \\varepsilon v - \\theta _ \\lambda v ) ^ 2 } { v } \\right ) = c _ 1 \\exp \\left ( - c _ 2 ( \\varepsilon _ 0 - \\theta _ \\lambda ) ^ 2 v \\right ) \\leq c _ 3 k ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} P C S = 1 - \\mathbb { P } ( \\hat { k } _ { R R T } \\neq k _ 0 ) = 1 - \\mathbb { P } _ { \\mathcal { U } } - \\mathbb { P } _ { \\mathcal { O } } . \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} N _ 0 = \\prod _ { \\substack { p \\mid N , \\\\ - v _ { p } ( \\zeta _ { p } ) \\leq 0 } } p ^ { n _ { p } } , N _ 1 = \\prod _ { \\substack { p \\mid N , \\\\ 0 < - v _ { p } ( \\zeta _ { p } ) < n _ { p } } } p ^ { n _ { p } } N _ 2 = \\prod _ { \\substack { p \\mid N , \\\\ n _ { p } \\leq - v _ { p } ( \\zeta _ { p } ) } } p ^ { n _ { p } } . \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} E ( t ) \\leq E ( m T ) \\leq e ^ { - \\lambda m T } E ( 0 ) = \\frac { 1 } { \\gamma } e ^ { - \\lambda ( m + 1 ) T } E ( 0 ) \\leq \\frac { 1 } { \\gamma } e ^ { - \\lambda t } E ( 0 ) \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} x _ { N , s } ( \\omega ) = \\mathbf { 1 } _ { A _ N ( s ) } ( \\omega ) \\tilde { x } _ N ( \\omega ) + \\mathbf { 1 } _ { A _ N ( s ) ^ c } ( \\omega ) x ^ * . \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} \\Big | \\psi ( z ) - \\sum _ { k = 0 } ^ { p - 1 } b _ k z ^ k \\Big | \\le C A ^ p M _ { p } | z | ^ p , z \\in S ( 0 , r \\beta _ 2 , R ^ r ) . \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} A ^ { ( n ) } _ { \\mathbb { Z } \\underline { l } } & : = A ^ { ( n ) } _ { l } / \\langle e ^ { ( n ) } _ { \\underline { l } } \\rangle . \\\\ A ^ { ( n ) } _ { \\underline { l } } & : = A _ { \\mathbb { Z } \\underline { l } } / \\tau ^ k _ n \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} \\mod ( t ) = 2 \\inf \\limits _ { n \\in \\mathbb { N } } \\left ( \\sum \\limits _ { k = n + 1 } ^ \\infty | x _ k ^ \\dagger | + t \\ , \\sum \\limits _ { k = 1 } ^ n \\| f ^ { ( k ) } \\| _ Y \\right ) , t > 0 . \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} \\mathbf { E \\ , } h _ { n } ^ { - 1 } \\ , H \\left ( \\frac { x - X } { h _ { n } } \\right ) = h _ { n } ^ { - 1 } \\int _ { \\mathbf { R } } f ( z ) \\ , H \\left ( \\frac { x - z } { h _ { n } } \\right ) \\ , d z = \\int _ { \\mathbf { R } } f ( x - h _ { n } y ) \\ , H ( y ) \\ , d y \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} G _ { [ i ] } & : = \\bigcup _ { u _ i = 0 } ^ { s _ i - 1 } \\bigcup _ { q _ i = 0 } ^ { p _ i - 1 } G _ { i - 1 } \\beta ^ { u _ i t _ i } \\alpha _ i ^ { q _ i } , \\\\ W _ { [ i ] } & : = \\bigcup _ { u _ 1 = 0 } ^ { s _ 1 - 1 } \\dots \\bigcup _ { u _ { i - 1 } = 0 } ^ { s _ { i - 1 } - 1 } \\bigcup _ { q _ 1 = 0 } ^ { p _ 1 - 1 } \\dots \\bigcup _ { q _ { i - 1 } = 0 } ^ { p _ { i - 1 } - 1 } \\Big ( W _ i \\beta ^ { \\sum _ { j = 1 } ^ { i - 1 } u _ j t _ j } \\prod _ { j = 1 } ^ { i - 1 } \\alpha _ j ^ { q _ j } \\Big ) . \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\alpha ( y ) + 4 y \\alpha ( y ) = 2 \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } ( 1 - 4 y ^ { 2 } ) e ^ { - 2 y ^ { 2 } } + c o n s t \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| x ^ k - y ^ k \\| = 0 . \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| v ^ { j _ 0 } _ n \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } = 0 . \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{gather*} k _ { i } = k [ 1 - \\sigma _ { 2 } + 2 ^ { - i } ( \\sigma _ { 2 } - \\sigma _ { 1 } ) ] \\ , , i = 0 \\ , , 1 \\ , , 2 \\ , , \\dots \\\\ t _ { i } = \\frac { t } { 2 } [ 1 - \\sigma _ { 2 } + 2 ^ { - i } ( \\sigma _ { 2 } - \\sigma _ { 1 } ) ] \\ , , i = 0 \\ , , 1 \\ , , 2 \\ , , \\dots \\end{gather*}"}
-{"id": "1537.png", "formula": "\\begin{align*} \\check { w } _ v = j ( \\check { y } _ v ) + d ( u _ v ) \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} f _ 1 ( x ) = \\begin{cases} \\frac { f ( x ) \\cdot F ( q ( x ) ) } { Q ( f ) ( q ( x ) ) } & Q ( f ) \\big ( q ( x ) \\big ) \\neq 0 \\\\ 0 & Q ( f ) \\big ( q ( x ) \\big ) = 0 \\end{cases} . \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\varrho _ 0 ^ \\sharp ( \\kappa ^ j r ) \\lesssim _ { \\gamma , \\varrho _ 0 , \\kappa } \\int _ 0 ^ r \\frac { \\varrho _ 0 ^ \\sharp ( t ) } { t } \\ , d t < \\infty , 0 < r \\le R _ 1 . \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} \\Sigma ( x ) = \\mbox { d i a g } \\big ( - \\lambda _ 1 ( x ) , \\cdots , - \\lambda _ { k } ( x ) , \\lambda _ { k + 1 } ( x ) , \\cdots , \\lambda _ { n } ( x ) \\big ) , \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} L _ s = \\sum _ { 1 < c \\leq \\frac { n _ l } { 2 } - q - \\abs { r } - k } L _ { s , - 2 c } + E , \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} \\| Q \\| ^ 2 _ { \\dot { H } ^ 1 _ c } = \\frac { d } { d + 2 } \\| Q \\| ^ { \\frac { 4 } { d } + 2 } _ { L ^ { \\frac { 4 } { d } + 2 } } = \\frac { d } { 2 } \\| Q \\| ^ 2 _ { L ^ 2 } . \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} { \\cal C } ( x ) : = C ( x ) - K ( x , x ) \\Sigma ( x ) + \\Sigma ( x ) K ( x , x ) = 0 \\mbox { f o r } x \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} A ^ { ( m ) } _ { n - m + \\ell - 1 } = A ^ { ( m ) } _ { n - m - 1 } , B ^ { ( m ) } _ { n - m + \\ell - 1 } = B ^ { ( m ) } _ { n - m - 1 } \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} = \\left \\| \\frac { f } { | x | ^ { \\frac { \\beta ( 1 - \\delta ) - \\gamma } { \\delta } } } \\right \\| ^ { \\delta } _ { L ^ { \\frac { \\delta p Q } { \\delta Q + p ( \\beta ( 1 - \\delta ) - \\gamma - a \\delta ) } } ( \\mathbb { G } ) } \\left \\| \\frac { f } { | x | ^ { - \\beta } } \\right \\| ^ { 1 - \\delta } _ { L ^ { q } ( \\mathbb { G } ) } . \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} | c _ h | \\le \\left ( f ^ { M L } g ^ { \\frac { M ^ 2 } { 2 } } \\right ) ^ { \\frac { 1 } { L + 1 - M } } , h = 0 , 1 , \\ldots , L , \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ n \\lambda _ i ^ { r _ m } c _ { i j _ 0 } \\right | \\geq \\beta \\mbox { f o r a l l } m \\geq 0 . \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} - \\nabla \\cdot ( \\bar { C } ( x ) \\nabla p ) = S . \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} X _ 0 = D + ( I - U _ A ^ * U _ A ) D ^ * + Z _ 0 , \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} Y = Y _ 1 > _ K Y _ 2 > _ K \\cdots > _ K Y _ { N - 1 } > _ K Y _ N . \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} | k _ b j _ b | < \\frac { | i - j | } { n - \\frac 1 2 } + \\frac { 1 } { 5 0 n \\sum _ { b = 1 } ^ n | j _ b | } | i + j | , b = 1 , \\cdots , n , \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} \\mathcal { E } _ p = \\{ ( z , w ) \\in \\C \\times \\C ^ n \\ ; | \\ ; \\abs { z } ^ 2 + p ( w ) < 1 \\} . \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\Pi _ { \\boldsymbol { \\mu } } \\big ( x ' , x '' \\big ) = \\mu \\cdot \\int _ G \\ , \\mathrm { d } V _ G ( g ) \\left [ \\overline { \\chi _ { \\boldsymbol { \\mu } } ( g ) } \\ , \\Pi \\left ( \\widetilde { \\mu } _ { g ^ { - 1 } } ( x ' ) , x '' \\right ) \\right ] . \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} ( \\delta _ \\alpha H ) ( a _ 1 , \\ldots , a _ { m + n + p - 1 } ) & ~ + \\sum _ { j = m } ^ { m + p - 1 } ( - 1 ) ^ { - ( m - 1 ) + ( n - 1 ) ( j - 1 ) } h _ { 0 , j } \\\\ & ~ + \\sum _ { i = 0 } ^ { p - 1 } ( - 1 ) ^ { ( m - 1 ) ( i - 1 ) + ( n - 1 ) ( m + i - 1 ) + ( m - 1 ) } h ' _ { i + 1 , m + i } \\\\ & ~ + \\sum _ { i = 0 } ^ { p - 1 } ( - 1 ) ^ { ( m - 1 ) ( i - 1 ) + ( n - 1 ) ( m + p - 2 ) + ( m - 1 ) + ( n - 1 ) } h '' _ { i + 1 , m + p } = 0 . \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} H _ 5 : = \\{ p \\mid p \\hbox { i s a p r i m e n u m b e r a n d $ p $ d i v i d e s $ l ^ 2 - 1 $ f o r s o m e } l \\in S \\} . \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} { \\rm { d e t } } ( \\mathbf { M } ) = { \\rm { d e t } } ( \\mathbf { M } ' ) = \\prod _ { 0 \\leq k < ( m + 1 ) n } M ' _ { k , k } = \\prod _ { 0 \\leq k < ( m + 1 ) n } \\lambda _ { q + 1 } ^ f \\prod _ { 1 \\leq i < q + 1 } ( \\lambda _ { q + 1 } - \\lambda _ i ) ^ { m + 1 } . \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} [ x [ g ] , y [ h ] ] = [ x , y ] [ g h ] + \\frac { 1 } { m } { \\rm R e s } _ { t = 0 } \\left ( ( d g ) h \\right ) \\langle x , y \\rangle C . \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } x b ^ { m _ k } \\sim \\phi b ^ { n _ 0 } x b ^ { m _ k } + \\sum _ { i = n _ 0 + 1 } ^ { m _ 0 } \\sum _ { s \\in S _ b } \\phi s b ^ i x b ^ { m _ k } . \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} E _ 0 ^ { ( r ; [ - L _ 1 , L _ 2 ] ) } T _ a = \\begin{cases} \\frac 1 r \\Big [ \\frac { \\sinh \\sqrt { \\frac { 2 r } D } ( a + L _ 1 ) - \\sinh \\sqrt { \\frac { 2 r } D } a - \\sinh \\sqrt { \\frac { 2 r } D } L _ 1 } { \\sinh \\sqrt { \\frac { 2 r } D } L _ 1 } \\Big ] , \\ 0 < a \\le L _ 2 \\\\ \\frac 1 r \\Big [ \\frac { \\sinh \\sqrt { \\frac { 2 r } D } ( | a | + L _ 2 ) - \\sinh \\sqrt { \\frac { 2 r } D } | a | - \\sinh \\sqrt { \\frac { 2 r } D } L _ 2 } { \\sinh \\sqrt { \\frac { 2 r } D } L _ 2 } \\Big ] , \\ - L _ 1 \\le a < 0 . \\end{cases} \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} \\lambda - \\mathcal { L } _ R ^ { \\psi _ n } [ w _ { R , n } ] + | D w _ { R , n } | ^ m = f _ n + \\mu _ { R , n } , x \\in B _ R , \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} E _ { a _ { k } } ^ { \\rm H } = ( a _ { * } - a _ { k } ) ^ { \\frac { q } { q + 1 } } \\Big ( \\frac { q + 1 } { q } \\cdot \\frac { \\Lambda } { a _ { * } } + o ( 1 ) _ { k \\to \\infty } \\Big ) \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} m _ K : = \\# \\{ X \\in \\mathbb { M } _ { 4 , G } : \\# X = n \\star ( X ) = 1 \\} . \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} v ^ { \\zeta } ( x , t ) - D t \\leq \\sup \\left \\{ v ^ { \\zeta } ( x , 0 ) \\ , \\mid \\ , x \\in \\bigcup _ { i = 1 } ^ { K } \\overline { I _ { i } ^ { a } } \\right \\} . \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} k ( x ) = c _ { 0 } c _ { 0 } ' e ^ { \\left ( \\frac { 1 } { 2 } + \\beta \\right ) x } \\int _ { 0 } ^ { \\infty } ( z + 1 ) ^ { - 1 + 2 \\beta } ( z + e ^ { x } ) ^ { - 1 - 2 \\beta } d z . \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} \\tilde { j } ( v _ 2 ( 0 ) - \\varphi ) - \\tilde { j } ( v _ 1 ( 0 ) - \\varphi ) = ( v _ 1 ' ( 0 ) , v _ 2 ( 0 ) - v _ 1 ( 0 ) ) . \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } e ^ { - N f ( x ) } g ( x ) d x = e ^ { - N f _ 0 } \\int _ { - \\pi } ^ { \\pi } e ^ { - N ( f ( x ) - f _ 0 ) } g ( x ) d x . \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} X ( t ) = X _ 0 - \\lambda \\int _ { 0 } ^ { t } X ( s ) d s + \\sigma B ( t ) , \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} N _ { 2 } = N _ { 2 1 } + N _ { 2 2 } , \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} \\ell _ { \\sigma } = \\left \\{ \\begin{array} { c c } - h _ { \\sigma } \\ & \\mbox { i f } \\ \\sigma = \\hat { \\sigma } \\\\ h _ { \\sigma } \\ & \\mbox { o t h e r w i s e } \\end{array} \\right . , \\ f = g - h _ { \\hat { \\sigma } } . \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} F ^ V _ { \\eta , \\eta ' } ( \\mu ) : = - \\nabla ^ { \\eta , \\eta ' } \\log \\Bigl ( \\frac { \\mu } { \\nu _ \\alpha ^ V } \\Bigr ) , \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} A _ p : = \\lim _ { \\xrightarrow [ \\tau > 0 ] { } } A _ { p , \\tau } ( \\mbox { r e s p . } \\ A _ { p , 0 } : = \\lim _ { \\xleftarrow [ \\tau > 0 ] { } } A _ { p , \\tau } ) . \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} H _ { + } ( \\xi ) & = K ( \\xi - d \\Gamma _ + ( m ) ) + d \\Gamma _ + ( \\omega ) + \\mu \\varphi _ + ( v ) \\\\ H _ { \\oplus } ( \\xi ) & = K ( \\xi - m - d \\Gamma _ { \\oplus } ( m ) ) + d \\Gamma _ { \\oplus } ( \\omega ) + \\omega + \\mu \\varphi _ { \\oplus } \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} [ E _ 1 ^ p \\dots E _ n ^ p ] = [ E _ 1 \\dots E _ n ] A ^ { - 1 } , \\end{align*}"}
-{"id": "9977.png", "formula": "\\begin{align*} R ( \\tau ) = \\sum _ { i = 1 } ^ { N ( \\tau ) + 1 } X _ i - \\tau . \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} \\Phi ( \\varphi ( a ^ { t } ) \\cdot h ) = 0 . \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} \\left | \\int _ { \\delta _ j } e ^ { t / u } \\left ( \\varphi ( u ) - \\sum _ { k = 0 } ^ { p - 1 } b _ k u ^ { k r } \\right ) \\ , \\frac { d u } { u } \\right | & \\le C A ^ p M _ p \\int _ 0 ^ { | t | / p } s ^ { p r } | e ^ { t / ( s e ^ { \\pm i \\pi \\beta _ 1 / 2 } ) } | \\ , \\frac { d s } { s } \\\\ & \\le C C _ 1 A ^ p M _ p \\left ( \\frac { | t | } { p } \\right ) ^ { p r } , \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} M ^ r v = Q \\cdot { \\rm { d i a g } } ( \\lambda _ 1 ^ r , \\dots , \\lambda _ n ^ r ) \\cdot Q ^ { - 1 } v = \\left [ \\sum _ { i = 1 } ^ n \\lambda _ i ^ r c _ { i j } \\right ] _ { 1 \\leq j \\leq n } ^ T \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} p _ { a + 1 } N _ { m - ( a + 1 ) } & + q _ { a + 1 } N _ { m - 2 ( a + 1 ) } + N _ { m - 3 ( a + 1 ) } \\\\ & = ( p _ { a } + p _ { a - 2 } ) N _ { m - ( a + 1 ) } + ( q _ { a - 2 } - q _ { a - 1 } ) N _ { m - 2 ( a + 1 ) } + N _ { m - 3 ( a + 1 ) } \\\\ & = ( p _ { a } + p _ { a - 2 } ) ( N _ { m - ( a - 2 ) } - N _ { m - ( a - 1 ) } ) \\\\ & \\ \\ + ( q _ { a - 2 } - q _ { a - 1 } ) ( N _ { m - 2 ( a - 2 ) } - N _ { m - 2 ( a - 1 ) } - 2 N _ { m - 2 a } ) \\\\ & \\ \\ + ( N _ { m - 3 ( a - 2 ) } - 4 N _ { m - 3 ( a - 1 ) } + 3 N _ { m - 3 a } ) \\\\ & = N _ { m } . \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} Z ^ N Y ^ { n _ { \\mu , i } + N + 1 } \\cdot v _ { + } & = \\left ( \\sum _ { \\gamma \\in \\Gamma / \\Gamma _ q } \\ , \\left ( \\gamma \\cdot ( x _ i [ f ] ) \\right ) _ q \\right ) ^ N Y ^ { n _ { \\mu , i } + N + 1 } \\cdot v _ { + } \\\\ & = ( x _ i [ f _ q ] ) ^ N Y ^ { n _ { \\mu , i } + N + 1 } \\cdot v _ { + } . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} \\frac { \\log ( 0 . 0 0 3 6 3 5 ) - \\log ( 2 . 0 7 3 \\cdot 1 0 ^ { - 8 } ) } { \\log ( 9 ) - \\log ( 9 9 9 9 ) } = - 1 . 7 2 1 \\ldots \\approx - 1 . 7 \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} \\lambda _ { k } = \\begin{cases} 2 ^ { - j \\frac { d } { u } } & Q _ { 0 , k } \\subset Q _ { - j , m _ j } \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} \\partial _ t ^ l \\hat { u } _ 1 ( 0 ) = S _ { \\chi , \\sigma , G , m , l } ( 0 , f , u _ 0 ) = \\partial _ t ^ l \\hat { u } _ 2 ( 0 ) \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} e _ { n - i } ( \\nu ) & = \\begin{cases} ( \\nu \\setminus \\{ i + 1 \\} ) \\cup \\{ i \\} & , \\\\ 0 & , \\end{cases} \\\\ f _ { n - i } ( \\nu ) & = \\begin{cases} ( \\nu \\setminus \\{ i \\} ) \\cup \\{ i + 1 \\} & , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} \\sum _ { \\lambda = j } ^ { j + n - 1 } = & ~ ( - 1 ) ^ j ~ h ( a ^ { m + n - 1 } _ { 1 , i - 1 } , ~ f ^ n , ~ a ^ { m + n - 1 } _ { i + m , j - 1 } , ~ a _ j ^ { m + n - 2 } \\cdot g ^ { m - 1 } , ~ a _ { j + n + 1 , m + n + p - 1 } ^ { m + n - 1 } ) \\\\ & ~ + ( - 1 ) ^ { j + n - 1 } ~ h ( a ^ { m + n - 1 } _ { 1 , i - 1 } , ~ f ^ n , ~ a ^ { m + n - 1 } _ { i + m , j - 1 } , ~ g ^ { m - 1 } \\cdot a _ { j + n } ^ { m + n - 2 } , ~ a _ { j + n + 1 , m + n + p - 1 } ^ { m + n - 1 } ) . \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} \\left \\{ \\gamma \\in \\mathcal { P } ( Y _ z \\times U ) \\ : \\ \\int _ { U \\times Y _ z } \\nabla \\phi ( y ) ^ T f ( u , y ) \\gamma ( d u , d y ) = 0 \\ \\ \\forall \\phi ( \\cdot ) \\in C ^ 1 \\right \\} , \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} E _ i y - y E _ i = \\frac { \\rho _ i ( y ) K _ i - L _ i ^ { - 1 } \\rho _ i ' ( y ) } { q _ i - q _ i ^ { - 1 } } . \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} \\dim H = \\sum _ { k = 1 } ^ m c _ k \\dim H _ k \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} \\lvert ( i _ 1 , j _ 1 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i _ 2 , j _ 2 ) \\rvert _ { W _ n ( w ) } = & ( \\lvert l _ { i _ 1 } \\rvert _ S + \\lvert m _ n \\rvert _ S + \\lvert r _ { j _ 1 } \\rvert _ S ) \\\\ & - ( \\lvert l _ { i _ 2 } \\rvert _ S + \\lvert m _ n \\rvert _ S + \\lvert r _ { j _ 2 } \\rvert _ S ) \\\\ = & \\lvert l _ { i _ 1 } \\rvert _ S - \\lvert l _ { i _ 2 } \\rvert _ S + \\lvert r _ { j _ 1 } \\rvert _ S - \\lvert r _ { j _ 2 } \\rvert _ S . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} E ( e ^ { i X } ) = e ^ { i E ( X ) - 1 / 2 ( X ) } , \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\left ( y \\beta ( y , t ) \\right ) + \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\beta ( y , t ) - \\frac { \\partial } { \\partial t } \\beta ( y , t ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( - \\frac { 4 } { 3 } y ^ { 4 } - 2 y ^ { 2 } + \\frac { 3 } { 4 } \\right ) e ^ { - 2 y ^ { 2 } } \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} & \\| R _ { \\alpha } \\tilde { v } \\| _ { H ^ s _ { \\overline { W } } } \\leq \\frac { 1 } { \\alpha } \\| \\overline { v } \\| _ { L ^ 2 ( \\Omega ) } , \\\\ & \\| A R _ { \\alpha } ( \\tilde { v } ) + \\sum \\limits _ { l = 1 } ^ { m } \\tilde { \\beta } _ l z _ l - \\overline { v } \\| _ { L ^ 2 ( \\Omega ) } = \\| A R _ { \\alpha } ( \\tilde { v } ) - \\tilde { v } \\| _ { L ^ 2 ( \\Omega ) } \\leq \\| \\overline { v } \\| _ { H ^ { s } _ { \\overline { \\Omega } } } \\left ( \\log \\left ( 1 / \\alpha \\right ) \\right ) ^ { - \\mu } . \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} A ^ { W , W } _ t ( l ) = \\phi ( t , W _ t ) - \\phi ( 0 , W _ 0 ) - \\int _ 0 ^ t \\nabla \\phi ^ * ( r , W _ r ) \\mathrm d W _ r . \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} - \\Delta Q _ { \\omega , } - c | x | ^ { - 2 } Q _ { \\omega , } + \\omega Q _ { \\omega , } = | Q _ { \\omega , } | ^ { \\frac { 4 } { d } } Q _ { \\omega , } . \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| \\nabla u _ { 0 , n } \\| _ { L ^ 2 } = \\lim _ { n \\rightarrow \\infty } \\mu _ n \\lambda _ n \\| \\nabla Q \\| _ { L ^ 2 } = \\| \\nabla Q \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} { \\bf x } _ t = { \\bf x } _ 0 + \\int \\limits _ 0 ^ t { \\bf a } ( { \\bf x } _ { \\tau } , \\tau ) d \\tau + \\int \\limits _ 0 ^ t B ( { \\bf x } _ { \\tau } , \\tau ) d { \\bf w } _ { \\tau } , \\ \\ \\ { \\bf x } _ 0 = { \\bf x } ( 0 , \\omega ) , \\ \\ \\ \\omega \\in \\Omega . \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} \\ln R ( n + 1 , E ) - \\ln R ( n , E ) = - \\vert \\varphi ( n , E ) \\vert ^ 2 \\frac { C } { n - v } + O ( 1 ) \\vert \\varphi ( n , E ) \\vert ^ 2 \\frac { \\cos 4 \\theta ( n , E ) } { n - v } + \\frac { O ( 1 ) } { ( n - v ) ^ 2 } . \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} V ( s ) = - \\dfrac { 1 } { 2 \\pi \\epsilon _ 0 } \\log \\dfrac { s } { s _ 0 } , s _ 0 > 0 . \\bigskip \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} \\| u \\| _ { \\widetilde { L } ^ p _ { T } ( \\dot { B } _ { 2 , 1 } ^ s ) } \\lesssim \\| u \\| ^ \\theta _ { \\widetilde { L } ^ { p _ 1 } _ { T } ( \\dot { B } _ { 2 , 1 } ^ { s _ 1 } ) } \\| u \\| ^ { 1 - \\theta } _ { \\widetilde { L } ^ { p _ 2 } _ { T } ( \\dot { B } _ { 2 , 1 } ^ { s _ 2 } ) } , \\ , \\frac { 1 } { p } = \\frac { \\theta } { p _ 1 } + \\frac { 1 - \\theta } { p _ 2 } , \\ , s = \\theta s _ 1 + ( 1 - \\theta ) s _ 2 \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\tilde { V } _ k = \\frac { 1 } { \\sqrt { Q _ 1 + Q _ 2 } } ( \\sqrt { Q _ 1 } V ^ { ( 1 ) } _ k ) \\oplus ( \\sqrt { Q _ 2 } V ^ { ( 2 ) } _ k ) . \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} \\mu _ { k } = s _ { \\beta _ k } ( \\mu _ { k - 1 } ) , \\ \\langle \\mu _ { k - 1 } , \\beta _ k ^ \\vee \\rangle < 0 . \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} \\rho ( \\omega ) = \\frac { 1 } { 2 \\beta } f _ \\alpha ( \\frac { \\omega } { 2 \\beta } ) e ^ { - \\frac { \\omega } { 2 \\beta } } , \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} \\delta _ { i } = \\int _ { a ^ - _ i } ^ { a ^ + _ i } \\int _ { a ^ - _ { i + 1 } } ^ { a ^ + _ { i + 1 } } P ( y _ { B } , y _ { C } ) d y _ B d y _ C + \\int _ { a ^ - _ { i + 1 } } ^ { a ^ + _ { i + 1 } } \\int _ { a ^ - _ i } ^ { a ^ + _ i } P ( y _ { B } , y _ { C } ) d y _ B d y _ C \\leq \\frac { \\eta } { 2 ^ { b } - 1 } \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} & \\| h \\| _ { C ^ { 0 + \\alpha } ( \\mathbb R ^ d ) } : = \\| h \\| _ { L ^ \\infty ( \\mathbb R ^ d ) } + \\sup _ { x \\neq y \\in \\mathbb R ^ d } \\frac { | h ( x ) - h ( y ) | } { | x - y | ^ \\alpha } \\\\ & \\| h \\| _ { C ^ { 1 + \\alpha } ( \\mathbb R ^ d ) } : = \\| h \\| _ { L ^ \\infty ( \\mathbb R ^ d ) } + \\| \\nabla h \\| _ { L ^ \\infty ( \\mathbb R ^ d ) } + \\sup _ { x \\neq y \\in \\mathbb R ^ d } \\frac { | \\nabla h ( x ) - \\nabla h ( y ) | } { | x - y | ^ \\alpha } , \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} u ( \\alpha ( 0 , i _ 2 ) ) = 2 i _ 2 - \\left \\lfloor i _ 2 \\left ( \\frac { 1 + \\sqrt { 5 } } { 2 } \\right ) \\right \\rfloor \\ , . \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} \\xi = \\alpha + \\beta , \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} = \\sum _ { \\substack { { r _ 1 , r _ 2 = 1 } \\\\ r _ 1 \\ne r _ 2 } } ^ { n } \\frac { S _ { 0 r _ 1 r _ 2 } } { r _ 2 - r _ 1 } \\left ( \\frac { H _ { r _ 2 } } { r _ 2 } - \\frac { H _ { r _ 1 } } { r _ 1 } \\right ) ; \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ z } f ^ 2 \\ , d \\mu _ z - \\Big ( \\int _ { \\Sigma _ z } f \\ , d \\mu _ z \\Big ) ^ 2 \\le \\frac { 2 \\epsilon C } { K } \\int _ { \\Sigma _ z } \\Gamma ( f , f ) \\ , d \\mu _ z = \\frac { 2 \\epsilon C } { \\beta K } \\int _ { \\Sigma _ z } \\ , | P \\nabla f | ^ 2 d \\mu _ z \\ , , \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} T _ { i + 1 } ^ 0 | _ \\mathcal { P } & = \\mathsf { R e m o v e D o t s } ( T _ { i } ^ { M + 1 } ) | _ \\mathcal { P } = \\mathsf { R e m o v e D o t s } ( T _ { i } ^ { M + 1 } | _ \\mathcal { P } ) \\\\ & = \\mathsf { R e m o v e D o t s } ( C _ { i } ^ { M + 1 } | _ \\mathcal { P } ) = \\mathsf { R e m o v e D o t s } ( C _ { i } ^ { M + 1 } ) | _ \\mathcal { P } \\\\ & = C _ { i + 1 } ^ 0 | _ \\mathcal { P } , \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} s ( n , k ) = \\binom { n - 1 } { k - 1 } B _ { n - k } ^ { ( n ) } \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} \\int _ { c _ { 2 j + 1 } } ^ { c _ { 2 j } } \\frac { \\eta _ { d - 1 } ( s ) } { \\sqrt { \\hat { \\mathcal { P } } _ { 2 d } ( s ) } } \\ , d s = 0 , \\ , j = d - 1 , d - 2 , . . . , 1 , \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} h ^ { - 1 } \\left \\{ \\sum ^ a _ { i = 1 } \\mu ( A _ i ) - \\sum ^ b _ { j = 1 } \\mu ( B _ j ) \\right \\} \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} Y _ z : = \\{ y \\in \\R ^ m \\ : \\ F ( y ) = z \\} . \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} \\begin{cases} \\i \\partial _ t u _ t \\ ; & = \\ ; S u _ t + \\frac { \\mu } { | \\cdot | ^ \\gamma } * | u _ t | ^ 2 u _ t \\\\ u _ t \\big | _ { t = 0 } \\ ; & = \\ ; u _ 0 . \\end{cases} \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} \\Delta = g _ 8 ^ 3 - 2 7 g _ { 1 2 } ^ 2 , \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} b _ r = ( - 1 ) ^ r \\frac { n ! } { r ! ( n - r ) ! } . \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\min ~ ~ & \\bar { c } ^ T z \\\\ ~ ~ & \\hat { A } z = \\hat { b } , \\\\ ~ ~ & z \\in \\R ^ n _ { + } \\times \\R _ { + } \\times \\Lambda ^ { n - 1 } \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} - \\sum _ { r = 0 } ^ p ( - 1 ) ^ { p + r } \\sigma ( i _ 0 , \\dots , \\widehat { i _ r } , \\dots , i _ p , j _ 0 , \\dots , j _ q ) - \\sum _ { r = 0 } ^ q ( - 1 ) ^ { r + 1 } \\sigma ( i _ 0 , \\dots , i _ p , j _ 0 , \\dots , \\widehat { j _ r } , \\dots , j _ q ) \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} \\mathcal { R } : = \\frac { d } { d r } . \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X _ { s } ^ { t , x } = & b ( s , X _ { s } ^ { t , x } , Y _ { s } ^ { t , x } , Z _ { s } ^ { t , x } ) d s + \\sigma ( s , X _ { s } ^ { t , x } , Y _ { s } ^ { t , x } ) d B _ { s } , \\\\ d Y _ { s } ^ { t , x } = & - g ( s , X _ { s } ^ { t , x } , Y _ { s } ^ { t , x } , Z _ { s } ^ { t , x } ) d s + Z _ { s } ^ { t , x } d B _ { s } , \\ ; s \\in \\lbrack t , T ] , \\\\ X _ { t } ^ { t , x } = & x , \\ Y _ { T } ^ { t , x } = \\phi ( X _ { T } ^ { t , x } ) . \\end{array} \\right . \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} d _ s = s ^ 2 - s \\lfloor s ^ { 1 / 2 } \\rfloor - \\lfloor s / 4 \\rfloor < s ^ 2 \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} \\| P _ N g ( t _ k , \\cdot ) - P _ N g ( t , \\cdot ) \\| _ { H ^ s _ r ( \\mathbb R ^ d ) } & = \\| P _ N ( g ( t _ k , \\cdot ) - g ( t , \\cdot ) ) \\| _ { H ^ s _ r ( \\mathbb R ^ d ) } \\\\ & \\leq \\| g ( t _ k , \\cdot ) - g ( t , \\cdot ) \\| _ { H ^ s _ r ( \\mathbb R ^ d ) } , \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} \\partial _ t \\mu _ t + \\partial _ C ( C \\mu _ t ( - \\partial _ C \\mathcal { E } ' [ \\mu _ t ] ) ) = 0 , \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial h _ { \\tau } } { \\partial \\tau } ( x _ { 0 } ) \\right ) _ { | \\tau = 0 } = \\frac { \\partial } { \\partial \\tau } \\varphi ^ { \\tau } _ { \\ell ( \\gamma _ { 0 } ) } ( x _ { 0 } ) _ { | \\tau = 0 } + \\partial _ { \\tau } \\ell ( \\gamma _ { \\tau } ) _ { | \\tau = 0 } X _ { 0 } ( x _ { 0 } ) + d \\varphi ^ { 0 } _ { \\ell ( \\gamma _ { 0 } ) } ( x _ { 0 } ) \\cdot \\left ( \\frac { \\partial h _ { \\tau } } { \\partial \\tau } ( x _ { 0 } ) \\right ) _ { | \\tau = 0 } . \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } \\left [ \\left ( \\frac { \\pi } { k \\ , \\nu } \\right ) ^ d \\cdot H ( X ) _ { k \\boldsymbol { \\nu } } \\right ] = \\int _ M \\ , \\mathrm { d } V _ M ( m ) \\left [ \\left ( \\frac { 1 } { 2 \\ , \\lambda ( m ) } \\right ) ^ { d + 1 } \\right ] . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t u ^ { n - \\theta } , v \\Big { ) } - \\gamma ( \\nabla \\sigma ^ { n - \\theta } , \\nabla v ) + ( \\nabla u ^ { n - \\theta } , \\nabla v ) + ( f ^ { n - \\theta } ( u ) , v ) = & ( g ^ { n - \\theta } , v ) , ~ \\forall v \\in H _ 0 ^ 1 , \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} V _ A ( f ) = ( P _ A f , P _ { A ^ c } f ) . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} P _ + = \\hat { R } + \\int _ { 0 } ^ 1 \\{ ( 1 - t ) ( \\hat { N } + \\hat { R } ) + t R , F \\} \\circ X _ F ^ t d t + ( P - R ) \\circ X _ F ^ 1 , \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} d \\Gamma _ A ( \\omega ) = \\bigoplus _ { n = 0 } ^ { \\infty } d \\Gamma _ A ^ { ( n ) } ( \\omega ) , \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} A ( u ) J = J A ( u ) = - A ( J u ) , A ( u ) ^ * J = J A ( u ) ^ * = A ( J u ) ^ * . \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ { \\int } _ { t } ^ { t + \\delta } | Z _ { s } ^ { 1 , v } | ^ { 2 } d s \\right ] & = | Y _ { t } ^ { 1 , v } | ^ { 2 } + \\mathbb { E } \\left [ \\left ( { \\int } _ { t } ^ { t + \\delta } F _ { 1 } ( s , X _ { s } ^ { v } , Y _ { s } ^ { 1 , v } , Z _ { s } ^ { 1 , v } , v _ { s } ) d s \\right ) ^ { 2 } \\right ] \\\\ & \\leq 2 \\mathbb { E } \\left [ \\left ( { \\int } _ { t } ^ { t + \\delta } | F _ { 1 } ( s , X _ { s } ^ { v } , Y _ { s } ^ { 1 , v } , Z _ { s } ^ { 1 , v } , v _ { s } ) | d s \\right ) ^ { 2 } \\right ] . \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\mathbf { E } [ L _ { i } ] & = \\sum _ { m = 1 } ^ { i } \\left ( K ^ { i - m } ( K - 1 ) \\textstyle \\prod _ { j = m } ^ { i } p _ j \\right ) \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| v _ n - v \\| _ { H ^ 1 } = 0 . \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} \\Re \\sum _ { n = 1 } ^ { \\infty } { \\widehat { z } _ n } ^ { j } \\leq & \\frac { 1 } { ( \\sigma _ 0 - 1 ) ^ { 2 j } } - \\frac { 1 } { ( \\sigma _ 0 - \\omega _ 0 ) ^ { 2 j } } + \\Re \\left [ \\frac { 1 } { \\{ ( \\sigma _ 0 - 1 ) + i t _ 0 \\} ^ { 2 j } } - \\frac { 1 } { \\{ ( \\sigma _ 0 - \\omega _ 0 ) + i t _ 0 \\} ^ { 2 j } } \\right ] . \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} \\int _ { x _ 0 } ^ x \\sin ( 2 \\pi \\ell y ) \\frac { \\sin ( 2 \\tilde { \\theta } ( y , { E } ) - 2 \\tilde { \\theta } ( y , \\hat { E } ) ) } { ( 1 + y - b ) } d y = & \\int _ { x _ 0 } ^ x \\frac { \\cos ( 2 \\pi \\ell y - ( 2 \\tilde { \\theta } ( y , { E } ) - 2 \\tilde { \\theta } ( y , \\hat { E } ) ) ) } { 2 ( 1 + y - b ) } d y \\\\ & - \\frac { \\cos ( 2 \\pi \\ell y + ( 2 \\tilde { \\theta } ( y , { E } ) - 2 \\tilde { \\theta } ( y , \\hat { E } ) ) ) } { 2 ( 1 + y - b ) } d y . \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} x \\star y = \\frac { x + y + \\hat { s } x y } { 1 - \\hat { \\Delta } x y } . \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { n } x _ i / ( v _ { i _ 1 , t } \\cdots v _ { i _ { d - 1 } , t } ) i _ { j + 1 } - i _ j \\geq t , 1 \\leq j \\leq d - 2 , \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} F : = \\mathbb { F } _ p ( \\{ \\alpha _ j : j \\in [ n ] \\setminus \\{ i _ 1 , i _ 2 \\} \\} ) . \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} A _ k : = \\left \\{ \\begin{array} { l l } 2 \\ell _ k - ( - 1 ) ^ { \\lambda _ k } & \\ell _ k , \\\\ 2 \\ell _ k & \\ell _ k . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} ( a b ) \\varphi ( c ) = 0 . \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} \\phi _ f ( G ) = \\inf _ d \\frac { \\phi \\bigl ( \\overline { \\overline { G } \\boxtimes K _ d } \\bigr ) } { d } . \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} \\varinjlim _ L { \\rm { H } } ^ j ( L ^ { \\otimes _ k n } , G ) = 0 , \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} I ( u ) : = \\frac { 1 } { 2 } \\int _ { \\mathbb R ^ { 3 } } \\left ( | \\nabla u | ^ { 2 } + u ^ { 2 } \\right ) d x + \\frac \\lambda 4 \\int _ { \\mathbb R ^ { 3 } } \\phi _ { u } u ^ { 2 } d x - \\int _ { \\mathbb R ^ { 3 } } F ( u ) d x , \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} \\int d x \\ x \\ a ( x , y , t ) = \\langle x \\rangle \\alpha ( y , t ) \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 4 } \\Delta e _ j \\ , d x \\ , = 0 \\ , , j = 2 , 3 , 4 . \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} Z _ 0 = ( I - U _ A ^ * U _ A ) Y _ 0 ^ * ( I - U _ A ^ * U _ A ) . \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} \\varphi : = \\sum _ { n = 0 } ^ \\infty \\phi ( n ) < \\infty . \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} | \\ ! | P ^ * u | \\ ! | _ 0 ^ 2 \\geq \\gamma _ 0 \\sum _ { j = 0 } ^ N | \\ ! | X _ j u | \\ ! | _ 0 ^ 2 + C | \\ ! | u | \\ ! | _ 0 ^ 2 , \\ , \\ , \\forall u \\in C _ 0 ^ \\infty ( K ) , \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} v _ n ( x ) = \\sum _ { j = 1 } ^ l V ^ j ( x - x ^ j _ n ) + v ^ l _ n ( x ) , \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} I _ { T , t } ^ { ( i _ 1 i _ 2 ) } = \\frac { T - t } { 2 } \\left ( \\zeta _ 0 ^ { ( i _ 1 ) } \\zeta _ 0 ^ { ( i _ 2 ) } + \\sum _ { i = 1 } ^ { \\infty } \\frac { 1 } { \\sqrt { 4 i ^ 2 - 1 } } \\left ( \\zeta _ { i - 1 } ^ { ( i _ 1 ) } \\zeta _ { i } ^ { ( i _ 2 ) } - \\zeta _ i ^ { ( i _ 1 ) } \\zeta _ { i - 1 } ^ { ( i _ 2 ) } \\right ) \\right ) , \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} \\left | N ( f ( x ) , r ) - \\mu _ { k + m - r } { q \\choose r } q ^ { k - r } \\right | \\leq \\sum _ { j = k + 1 } ^ { k + m } { j \\choose r } { \\frac { q } p + m \\sqrt { q } + j \\choose j } { m - 1 \\choose k + m - j } \\sqrt { q } ^ { k + m - j } . \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} \\sigma ( e ^ { i \\theta } ) = \\tau ( e ^ { i \\theta } ) e ^ { i \\beta ( \\theta - \\theta _ { 1 } ) } \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} \\binom { a _ 0 + a _ 1 p + a _ 2 p ^ 2 + \\dots + a _ m p ^ m } { b _ 0 + b _ 1 p + b _ 2 p ^ 2 + \\dots + b _ m p ^ m } \\equiv \\binom { a _ 0 } { b _ 0 } \\binom { a _ 1 } { b _ 1 } \\binom { a _ 2 } { b _ 2 } \\dots \\binom { a _ m } { b _ m } \\pmod { p } , \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} \\ o s c ( \\psi _ 1 , B _ \\epsilon ( x ) ) \\leq \\int _ { B ( x , r + \\epsilon ) } h ( z ) d z - \\int _ { B ( x , r - \\epsilon ) } h ( z ) d z = \\int _ { D } h ( z ) d z \\leq \\| h \\| _ { \\infty } L e b ( D ) , \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} g ^ { ( 4 ) } _ { \\epsilon } = \\tilde g ^ { ( 4 ) } + \\epsilon ( d t + \\theta ) ^ 2 . \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} { \\frac { \\partial w } { \\partial { \\bar { z } } } } = \\varphi ( z , { \\bar { z } } ) \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} m \\left ( r , \\frac { f ( z + c ) } { f ( z ) } \\right ) = o \\left ( \\frac { T ( r , f ) } { ( \\log r ) ^ { \\nu - \\varepsilon } } \\right ) , \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} a ^ 2 \\le r _ 1 ^ 2 = | \\omega _ 0 ( a u , a \\mu v + \\sqrt { 1 - \\mu ^ 2 } \\ , w ) | = a ^ 2 \\mu ^ 2 + a ( 1 - \\mu ^ 2 ) \\le a . \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} \\| D Q \\| _ W : = \\| \\langle x \\rangle D Q \\| _ { L _ x ^ 1 L _ t ^ \\infty } + \\| \\langle x \\rangle D Q \\| _ { L _ t ^ \\infty L _ x ^ 2 } + \\| \\partial _ x D Q \\| _ { L _ t ^ \\infty L _ x ^ 2 } , \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 1 - w _ 1 & w _ 1 & 0 & \\cdots \\\\ 0 & 1 - w _ 2 & w _ 2 & \\vdots \\\\ \\vdots & \\ddots & \\ddots & \\vdots \\\\ w _ n & \\cdots & 0 & 1 - w _ n \\end{pmatrix} \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} \\| ( \\tilde { P } - \\tilde { \\Pi } ) ^ { \\frac { n } { 2 } } \\| = O ( \\epsilon ^ k ) \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} \\log | N ( s ) | _ { N ( \\phi ) } ( x ) = \\int _ { X } \\log | s | _ { \\phi } \\delta _ { [ f ^ { - 1 } ( x ) ] } . \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} [ d _ { n } , d _ { k } ] = ( n - k ) d _ { n + k } + \\delta _ { n , - k } \\dfrac { n ^ { 3 } - n } { 1 2 } \\bar { C } ; \\ , \\ , \\ , [ d _ { n } , \\bar { C } ] = 0 . \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} Q ( z , z ' ) \\lesssim \\begin{cases} r '^ { - n } r ^ { s } ( r / r ' ) ^ { 1 - \\frac n 2 + \\nu _ 0 - s } , r < \\frac { r ' } 2 ; \\\\ d ( z , z ' ) ^ { - ( n - s ) } , r \\sim r ' ; \\\\ r ^ { - n + s } ( r ' / r ) ^ { 1 - \\frac n 2 + \\nu _ 0 } , r > 2 r ' ; \\end{cases} \\end{align*}"}
-{"id": "9851.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } c ( n ) q ^ { n } = \\dfrac { E _ { 2 } } { E _ { 1 } ^ { 3 } } . \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} P \\left ( \\beta \\in \\left [ 0 , \\frac { F ^ { - 1 } \\left ( 1 - \\tilde { \\alpha } \\right ) } { \\bar { X } } \\right ] \\right ) & = P \\left ( \\bar { X } \\in \\left [ 0 , \\frac { F ^ { - 1 } \\left ( 1 - \\tilde { \\alpha } \\right ) } { \\beta } \\right ] \\right ) \\\\ & = 1 - \\tilde { \\alpha } \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} h _ { 2 , \\lambda } : = f _ 1 ^ { - \\frac { n - 1 } { 2 } } w _ { \\lambda } . \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} \\langle U , \\mathrm { d } V \\rangle = \\langle \\delta U , V \\rangle . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} S _ N = \\sum _ { n = 1 } ^ N F ( \\xi _ n , \\xi _ { 2 n } , . . . , \\xi _ { \\ell n } ) \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { \\binom k j \\binom { k - j } s ( - 1 ) ^ { j + s } f _ 3 ^ j f _ 1 ^ s X _ { m + b k - c j - a s } } } = f _ 2 ^ k X _ m \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\mathbb E ( H _ q ( Z ( u ) ) H _ p ( Z ( v ) ) ) = q ! R ( v - u ) ^ q \\delta _ { p , q } \\textrm { $ ; $ } \\forall u , v \\in \\mathbb R _ + \\textrm { , } \\forall p , q \\in \\mathbb N . \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} x _ { k + 1 } = \\rho x _ k + \\sigma G _ k , \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} P = \\sum _ { j = 1 } ^ N X _ j ^ * f X _ j + X _ { N + 1 } + i X _ 0 + a _ 0 , \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} E ( w _ s ) = \\frac { \\lambda _ s ^ 2 } { 2 } \\| \\nabla v _ s \\| ^ 2 _ { L ^ 2 } - \\frac { c \\lambda _ s ^ 2 } { 2 } \\| | x | ^ { - 1 } v _ s \\| ^ 2 _ { L ^ 2 } - \\frac { \\lambda _ s ^ { \\alpha + 2 } } { \\alpha + 2 } \\| v _ s \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } , \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} \\Lambda _ 2 ( S _ 2 \\left [ f ( w _ 1 ) \\right ] , \\widehat { u } ) & = \\Lambda _ 2 ( f ( w _ 1 + w _ 2 - w _ { - 2 } ) - f ( w _ 1 ) , \\widehat { u } ) \\\\ & = \\Lambda _ 2 ( f ( w _ { - 1 } ) - f ( w _ 1 ) , \\widehat { u } ) \\\\ & = - \\frac 1 2 \\Lambda _ 2 ( f ( w _ 1 ) + f ( w _ 2 ) - f ( w _ { - 1 } ) - f ( w _ { - 2 } ) , \\widehat { u } ) . \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} \\theta = [ a _ 1 , a _ 2 , \\ldots , a _ n + \\mathrm { T } ^ n ( \\theta ) ] \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} T ^ { * } & : H \\longmapsto \\ell ^ { 2 } ( I ) \\\\ & T ^ { * } f = \\lbrace \\langle f , f _ { i } \\rangle \\rbrace _ { i \\in I } \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{gather*} F _ 0 ( z ) = { } _ 6 F _ 5 \\biggl ( \\begin{matrix} \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 \\\\ 1 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 1 \\end{matrix} \\biggm | z \\biggr ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( \\frac 1 2 ) _ k ^ 6 } { k ! ^ 6 } z ^ k \\end{gather*}"}
-{"id": "6232.png", "formula": "\\begin{align*} x _ i ( 0 ) = - i , ~ i = 1 , 2 , \\cdots , N , \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} - \\frac { 1 } { L } \\int \\eta \\ , \\partial _ j ( ( \\nabla _ { \\Q } f _ B ) ( \\bar { \\Q } ) ) \\cdot \\partial _ j \\bar { \\Q } \\ , \\d x = - \\frac { 1 } { L } \\int \\eta \\ , \\frac { \\partial ^ 2 f _ B ( \\bar { \\Q } ) } { \\partial Q _ { i \\ell } \\partial Q _ { h k } } \\partial _ j \\bar { Q } _ { i \\ell } \\partial _ j \\bar { Q } _ { h k } \\ , \\d x \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} w _ { 0 } ^ { ( \\alpha , \\beta ) } = \\frac { 1 } { \\| P _ { 0 } ^ { ( \\alpha , \\beta ) } \\| _ { L ^ 2 ( [ - 1 , 1 ] , d \\mu _ { \\alpha , \\beta } ) } } = \\sqrt { \\frac { \\Gamma ( \\alpha + \\beta + 2 ) } { 2 ^ { \\alpha + \\beta + 1 } \\Gamma ( \\alpha + 1 ) \\Gamma ( \\beta + 1 ) } } , \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} S _ { f _ n } ( \\rho \\| \\sigma ) & = \\sup _ { x ( \\cdot ) } \\biggl [ f _ n ( 0 ^ + ) \\sigma ( 1 ) + f _ n ' ( + \\infty ) \\rho ( 1 ) \\\\ & \\qquad - \\int _ { [ 1 / n , n ] } \\bigl \\{ \\sigma ( ( 1 - x ( s ) ) ^ * ( 1 - x ( s ) ) ) + s ^ { - 1 } \\rho ( x ( s ) x ( s ) ^ * ) \\bigr \\} ( 1 + s ) \\ , d \\nu _ n ( s ) \\biggr ] . \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} j = 4 ^ k l _ 0 + s , \\ , \\ , \\ , \\ , \\ , 0 \\leq s \\leq 2 u . \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} & \\Delta ( t _ i ) = t _ i \\otimes t _ i , & & \\varepsilon ( t _ i ) = 1 , & & S ( t _ i ) = t _ i ^ { - 1 } , \\\\ & \\Delta ( \\theta _ i ) = t _ i \\otimes \\theta _ i + \\theta _ i \\otimes 1 , & & \\varepsilon ( \\theta _ i ) = 0 , & & S ( \\theta _ i ) = - \\theta _ i t _ i ^ { - 1 } . \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} J ( \\varphi ) = \\int f \\left ( x ^ \\alpha , \\varphi ^ \\beta , \\frac { \\partial \\varphi ^ \\gamma } { \\partial x ^ \\kappa } \\right ) \\rho ( x ) \\ , d x \\ , , \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ { n } f ( B _ { u _ { i } } ) \\supset \\overline { U } . \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\log \\tfrac 1 { p _ s } d D _ s ( \\omega ) & = - \\sum _ { k \\geq 1 } \\log p _ { \\tau _ k ( \\omega ) } \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} f ( x ) & = x ^ 2 ( 1 - x ^ 3 ) - ( A - B x ) ( x ^ 3 - 1 ) ( 1 + x ) + m \\ : h ( x ) , \\\\ g ( x ) & = x ( x + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( 1 + x ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} \\sum _ { d ' \\mid \\frac { n } { g f ' } } \\mu ( g d ' ) = \\mu ( g ) \\sum _ { d ' \\mid \\tilde { ( \\frac { n } { g f ' } ) } } \\mu ( d ' ) = \\mu ( g ) 1 _ { \\tilde { ( \\frac { n } { g f ' } ) } = 1 } . \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} \\left \\| \\gamma _ n \\left ( \\sum _ { j = 1 } ^ r \\alpha _ j T _ j \\right ) \\right \\| \\geq \\frac { 1 } { 2 } \\max _ { 1 \\leq j \\leq r } | \\alpha _ j | . \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} \\gamma _ n ( T _ 1 ^ * T _ 1 - T _ 1 T _ 1 ^ * ) = \\frac { 1 } { 4 } ( E _ { 2 2 } ^ { ( n ) } - E _ { 1 1 } ^ { ( n ) } ) . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} \\| L _ G f \\| _ 1 & = \\sum _ { v \\in V } | L f ( v ) | \\\\ & \\leq \\sum _ { v \\in V } | f ( v ) | + \\sum _ { v \\in V } \\sum _ { v \\sim v ' } | f ( v ' ) | \\\\ & \\leq \\| f \\| _ 1 + D \\| f \\| _ 1 \\\\ & = ( D + 1 ) \\| f \\| _ 1 . \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| u _ { 0 , n } \\| ^ 2 _ { L ^ 2 } = \\| v \\| _ { L ^ 2 } ^ 2 = M , \\lim _ { n \\rightarrow \\infty } E ( u _ { 0 , n } ) = E ( v ) = d _ M . \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{gather*} G _ { s , t } ( x ) = H ^ { ( k ) } _ { t + \\rho ( s ) } \\\\ J ' _ s ( t , x ) = J _ { t + \\rho ( s ) } ( x ) \\end{gather*}"}
-{"id": "9577.png", "formula": "\\begin{align*} - \\real \\frac { L ' _ \\chi } { L _ \\chi } ( s ) = \\frac { 1 } { 2 } \\log ( \\Delta N ( \\mathfrak f _ { \\chi } ) ) + \\real \\left ( \\delta ( \\chi ) \\left ( \\frac { 1 } { s } + \\frac { 1 } { s - 1 } \\right ) - \\sum _ { \\rho \\in R _ { \\chi } } \\frac { 1 } { s - \\rho } + \\psi _ \\chi ( s ) \\right ) . \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} { N } ( t ) = N _ { i n } \\exp \\left ( \\int _ 0 ^ t g ( s ) \\ , d s \\right ) . \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} \\P \\left ( \\sum _ { k = 1 } ^ n \\nu _ k > \\delta n \\right ) \\le n \\P \\left ( \\mu _ 1 - 1 \\ge \\frac { \\epsilon n } { \\kappa } \\right ) + \\left ( \\frac { e ^ 2 c \\kappa ^ t } { \\delta \\epsilon ^ { t - 1 } n ^ { t - 1 } } \\right ) ^ { \\delta / 2 \\epsilon } \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} p _ { t , y , a } \\cdot ( z ' , z _ n ) = \\left ( \\frac { 2 } { - i y z _ n + 2 - i y } t z ' , \\frac { ( 2 + i y ) z _ n + i y } { - i y z _ n + 2 - i y } \\right ) . \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\sum _ { r _ 1 = 1 } ^ { n } \\frac { S _ { 0 0 r _ 1 } } { ( k + 1 ) ^ { 2 } ( r _ 1 + k + 1 ) } = \\zeta ( 2 ) \\sum _ { r _ 1 = 1 } ^ n \\frac { S _ { 0 0 r _ 1 } } { r _ 1 } - \\sum _ { r _ 1 = 1 } ^ n \\frac { S _ { 0 0 r _ 1 } H _ { r _ 1 } } { r _ 1 ^ 2 } ; \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & - 2 \\\\ - 1 & 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} \\langle D A ^ * u , A ^ * v \\rangle & = \\langle A ^ * u , D ^ * A ^ * v \\rangle = \\langle A ^ * u , C ^ * v \\rangle = \\langle C A ^ * u , v \\rangle \\\\ & = \\langle A C ^ * u , v \\rangle = \\langle C ^ * u , A ^ * v \\rangle = \\langle D ^ * A ^ * u , A ^ * v \\rangle , \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i , i ' , j , j ' \\leq n } = & \\sum _ { | i - i ' | > g , | j - j ' | > g } + \\sum _ { | i - i ' | > g , | j - j ' | \\leq g } + \\sum _ { | i - i ' | \\leq g , | j - j ' | > g } + \\sum _ { | i - i ' | \\leq g , | j - j ' | \\leq g } \\\\ = : & I + I I + I I I + I V . \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} \\begin{array} { l l } \\qquad \\quad \\mathcal { I } = \\{ 1 , 2 \\} \\\\ \\\\ f ^ { 1 1 } = 5 x , f ^ { 1 2 } = x , \\\\ \\\\ f ^ { 2 1 } = - x , f ^ { 2 2 } = - 4 x , \\\\ \\\\ r = 0 . 1 5 , \\sigma = 0 . 2 , b = 0 . 0 1 , \\end{array} \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow \\infty } I ( u _ n ) \\geq \\int _ { \\Lambda } \\liminf _ { n \\rightarrow \\infty } \\left [ \\frac { 1 } { 4 } f ( u _ n ( x + y _ n ) ) u _ n ( x + y _ n ) - F ( u _ n ( x + y _ n ) ) \\right ] d x = \\infty , \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} R _ 0 \\bigl ( \\check { X } ( j ) \\bigr ) & = \\bigl ( \\check { X } ( j + 1 ) - \\check { X } ( j ) \\bigr ) \\bigl ( \\check { X } ( j ) - \\check { X } ( j - 1 ) \\bigr ) \\ \\ ( j = 0 , 1 , \\ldots , N ) , \\\\ R _ 1 \\bigl ( \\check { X } ( j ) \\bigr ) & = \\check { X } ( j + 1 ) - 2 \\check { X } ( j ) + \\check { X } ( j - 1 ) \\ \\ ( j = 0 , 1 , \\ldots , N ) , \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} \\phi ( s ) = \\exp \\left \\{ - \\int _ 0 ^ s \\beta ( \\theta ) \\gamma ( \\theta ) d \\theta \\right \\} \\int _ 0 ^ s \\gamma ( \\theta ) \\psi ( \\theta ) d \\theta , \\ s \\geq 0 . \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} Q = ( V ( Q ) , E ( Q ) , s , t ) \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} Q _ { B _ n } P _ { B _ n } ( f ) & = V _ n ^ * ( 0 , P _ { B _ n } ( f ) ) = 1 _ { B _ n } P _ { B _ n } ( f ) = 1 _ { B _ n } f \\\\ Q _ { B _ n ^ c } P _ { B _ n ^ c } ( f ) & = V _ n ^ * ( P _ { B _ n ^ c } ( f ) , 0 ) = 1 _ { B _ n ^ c } P _ { B _ n ^ c } ( f ) = 1 _ { B _ n ^ c } f \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} \\phi _ 0 ( x ) : = | q | ^ { \\frac 1 2 } e ^ { q | x | } . \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} L _ 1 + L _ 2 + L _ 3 = 0 , \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { ( - 1 ) ^ { n ( j + s ) } \\binom k j \\binom { k - j } s F _ n ^ { j + s } G _ { m - 2 n k + ( n + 1 ) j + ( n - 1 ) s } } } = ( - 1 ) ^ { n k } G _ m \\ , , \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} g = \\exp ( a _ V ( g ) V ) \\exp ( a _ X ( g ) X ) \\exp \\left ( a _ U ( g ) U + \\sum a _ { i j } ( g ) X _ i ^ j \\right ) \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} p _ k ^ { ( \\rm i d , a ) } = \\frac { \\mu _ { 2 a } ^ * } { \\left ( \\nu _ { 2 a } ^ * - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 \\right ) \\ln 2 } . \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} f ( z ) = f ( 0 ) + \\sum _ { k } \\left [ f _ k \\left ( \\frac { 1 } { z - p _ k } \\right ) - f _ k \\left ( \\frac { 1 } { - p _ k } \\right ) \\right ] , \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} | S _ { m } ( \\omega ) | = | S _ { m } ( p _ k , q _ k , \\xi _ k , \\theta _ k ) | \\geq T _ m - c _ 3 \\sqrt { q _ k } , \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} s < r \\vee 0 < s \\rightarrow s = 0 \\vee s \\neq 0 \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} \\mu _ X ( \\chi ) = \\gamma ( \\chi , \\check \\alpha , 0 , \\psi ) ; \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} \\mathbb P ( Z _ t \\xrightarrow { t \\to \\infty } 0 ) + \\mathbb P ( Z _ t \\xrightarrow { t \\to \\infty } \\infty ) = 1 . \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} H _ { 2 m + 1 - \\epsilon } ( z ) = ( c - 1 ) P _ { m - 1 , \\epsilon } ( z ) + ( k - 1 ) P _ { m , \\epsilon } ( z ) , \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} K = \\int _ { \\mathbb { T } ^ n } F _ { \\geq 5 } ( x , \\sum _ { j \\in \\bar { \\mathbb { Z } } } \\gamma _ j q _ j \\phi _ j , \\sum _ { j \\in \\bar { \\mathbb { Z } } } \\gamma _ j \\bar { q } _ j \\phi _ { - j } ) d x . \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} | z _ q | ^ 2 e ^ { \\frac { - Q _ t ( z ) } { 2 } } = | \\alpha | ^ 2 | l ( e ' _ q ) | ^ 2 e ^ { \\frac { - | \\alpha | ^ 2 Q _ t ( e ' _ q ) - | \\beta | ^ 2 Q _ t ( e _ p ) } { 2 } } . \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} \\textbf { K } _ { C } ( \\upsilon , b , c , \\lambda , y ) = \\int _ { 0 } ^ { \\infty } x ^ { \\upsilon - 1 } e ^ { - b \\lambda \\sqrt { x } } { } _ { r } F _ { s } \\left ( \\ \\begin{array} { l l l } \\alpha _ { 1 } , . . . , \\alpha _ { r } ; \\\\ \\beta _ { 1 } , . . . , \\beta _ { s } ; ~ \\end{array} e ^ { - c \\sqrt { x } } \\right ) \\cos ( x y ) d x , \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} c = \\left \\{ \\begin{array} { l l } L & \\mbox { i f } L \\leq 3 , \\\\ 8 L ^ 2 & \\mbox { i f } L > 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} x _ i = c _ i s _ i + \\rho _ i . \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} C = \\left ( \\begin{array} { c c } 0 & 1 \\\\ - 1 & 0 \\end{array} \\right ) \\ ; . \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} \\widehat { F } _ { t , i } ^ { ( u ) } ( \\cdot ) \\ , : = \\ , u \\cdot \\delta _ { X _ { t , i + 1 } ^ { ( u ) } } ( \\cdot ) + ( 1 - u ) \\cdot \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } \\delta _ { X _ { t , j } ^ { ( u ) } } ( \\cdot ) \\ , , i \\ , = \\ , 1 , \\ldots , n - 1 \\ , \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} L ( \\tilde { u } - u ) = \\tilde { f } + ( \\chi ( u ) - \\chi ( \\tilde { u } ) ) \\partial _ t \\tilde { u } + ( \\sigma ( u ) - \\sigma ( \\tilde { u } ) ) \\tilde { u } - f = : F . \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} ( x ^ { k + 1 } , y ^ { k + 1 } ) g ( x ^ k , y ^ k ) + \\nabla g ( x ^ k , y ^ k ) \\begin{pmatrix} x ^ { k + 1 } - x ^ k \\\\ y ^ { k + 1 } - y ^ k \\end{pmatrix} = 0 . \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} \\sum _ { n \\in \\mathbb { Z } } I _ { n , \\beta } q ^ n = \\prod _ { n \\ge 1 } ( 1 - q ^ n ) ^ { - n \\cdot e ( X ) } \\sum _ { n \\in \\mathbb { Z } } P _ { n , \\beta } q ^ n . \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} \\mathsf { M } _ R : = \\inf _ { \\phi \\in \\mathcal { L } _ 1 } \\frac 1 R \\left \\| u - \\phi - ( u - \\phi ) _ { B _ R } \\right \\| _ { \\underline { L } ^ 2 ( B _ { R } ) } , \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} \\phi _ { i , n } > 0 \\ ; \\ ; \\phi _ { i , n } \\to u _ i \\ ; \\ ; W ^ { 1 , 1 } _ { l o c } ( \\Omega ) . \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} S ( g , p ) & = \\frac { 1 } { 2 } E \\int _ 0 ^ T \\int | D _ t g _ t ( x ) | ^ 2 d t d x + E \\int _ 0 ^ T \\int p ( t , g _ t ( x ) ) ( \\det \\nabla g _ t ( x ) - 1 ) d t d x \\\\ & : = S ^ 1 ( g , p ) + S ^ 2 ( g , p ) \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} \\Delta \\rho _ \\varphi ( x ) = \\frac 1 2 \\rho _ { \\boldsymbol \\varphi } ( x ) \\ , h ^ { \\alpha \\beta } ( x ) \\ , \\Delta h _ { \\alpha \\beta } ( x ) . \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} p _ { N } ^ { \\ell _ 2 - } ( x ) = \\dfrac { \\sum \\limits _ { j = 0 } ^ N \\dfrac { \\Omega _ j } { x - x _ j } f ( x _ j ) } { ( 1 + \\lambda \\mu _ { 0 } ^ 2 ) \\sum \\limits _ { j = 0 } ^ N \\dfrac { \\Omega _ j } { x - x _ j } } , \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} \\beta ^ u \\alpha _ i ^ { u + t } \\in K , \\ ; t = 0 , 1 , \\dots , p _ i - 2 . \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} f \\left ( v \\right ) : = \\exp \\left ( - \\sum _ { l = 1 } ^ r t _ k v _ k \\right ) \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} B _ { i j } = ( \\Phi ^ { - 1 } ) _ { k k ' } \\frac { \\partial \\xi _ { k ' } } { \\partial x _ { l ' } } B _ { i j , l ' } ( a \\nabla \\xi _ k ) \\ , , 1 \\le i , j \\le m \\ , , \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} \\Theta _ L ( \\eta ) : = \\frac 1 { L ^ d } \\sum _ { i \\in \\mathbb T _ L ^ d } \\eta ( i ) \\ ; \\ ! \\delta _ { i / L } . \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} K _ i E _ j = q ^ { \\langle \\alpha _ i , \\alpha _ j \\rangle } E _ j K _ i , \\forall 1 \\leq i , j \\leq n , \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} c ^ { ( \\ast ) } _ \\ell ( \\pi ) = \\oplus _ { \\epsilon } c ^ { ( \\ast ) } _ \\ell ( \\pi ) ^ { \\epsilon } . \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\rho : = \\frac { d - 2 } { 2 } - \\sqrt { \\left ( \\frac { d - 2 } { 2 } \\right ) ^ 2 - c } . \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\sup _ { \\eta \\in { \\cal M } } \\int _ Y | v ( y ) | 1 _ { \\{ | v ( y ) | \\geq l \\} } \\eta ( d y ) = 0 . \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} \\sup _ { ( \\mu , \\psi ( \\cdot ) , \\eta ( \\cdot ) ) \\in \\mathcal { D } ( \\epsilon ) } \\mu : = d ^ * ( y _ 0 , \\epsilon ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\mathbb { P } \\left \\lbrace \\left \\lvert p ^ { n , \\omega ^ \\prime } _ t \\right \\rvert \\ge \\varepsilon , - \\tau < t \\le \\tau ^ { n , \\omega ^ \\prime } _ R \\right \\rbrace = 0 \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} \\min _ { x } ~ F ( x ) : = f ( x ) + r ( x ) \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} \\| ( - \\Delta ) ^ { \\frac { 1 } { 4 } } Q \\| _ { L ^ { 2 } } ^ { 2 } = \\| Q \\| _ { L ^ { 2 } } ^ { 2 } = \\frac { a _ { * } } { 2 } \\iint _ { \\mathbb { R } ^ { 3 } \\times \\mathbb { R } ^ { 3 } } \\frac { | Q ( x ) | ^ { 2 } | Q ( y ) | ^ { 2 } } { | x - y | } { \\rm d } x { \\rm d } y = 1 . \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} - \\Delta v - c | x | ^ { - 2 } v + \\omega v = | v | ^ \\alpha v . \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} \\partial _ t F _ i ( t ) = - W _ i \\circ F ( t ) , \\ ; \\ ; \\ ; F _ i ( 0 ) = I d , \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} \\frac { \\delta _ { i } ^ { ( j ) } } { 2 ^ { R ^ { t h } } - 1 } - \\sum _ { m = 1 } ^ { i - 1 } \\delta _ { m } ^ { ( j ) } > 0 ~ ~ \\Leftrightarrow ~ ~ \\frac { \\delta _ { i } ^ { ( j ) } } { \\sum _ { m = 1 } ^ { i - 1 } \\delta _ { m } ^ { ( j ) } } > 2 ^ { R ^ { t h } } - 1 , i = 2 , \\cdots , \\ell \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { \\# \\left \\{ ( a , b ) \\in [ n ] ^ 2 , ~ { \\rm g c d } ( a , b ) = 1 \\right \\} } { n ^ 2 } = \\frac { 6 } { \\pi ^ 2 } \\ , . \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} K _ 1 ( \\varphi , p ) : = \\int \\bigl [ p ( \\varphi ( x ) ) \\bigr ] \\ , \\bigl [ \\rho _ { \\boldsymbol \\varphi } ( x ) - \\rho ( x ) \\bigr ] \\ , d x \\ , , \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} E [ C ] = \\sum _ { \\substack { ( i , j ) \\in \\mathcal { I } \\\\ i < j } } \\left ( \\frac { Q _ { i j } [ C ] ^ 2 } { C _ { i j } } + \\frac { \\nu } { \\gamma } C _ { i j } ^ \\gamma \\right ) L _ { i j } , \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} \\| u _ { 0 , n } \\| _ { L ^ 2 } = \\mu _ n \\| Q \\| _ { L ^ 2 } , \\| \\nabla u _ { 0 , n } \\| _ { L ^ 2 } = \\mu _ n \\lambda _ n \\| \\nabla Q \\| _ { L ^ 2 } , \\| u _ { 0 , n } \\| _ { \\dot { H } ^ 1 _ c } = \\mu _ n \\lambda _ n \\| Q \\| _ { \\dot { H } ^ 1 _ c } , \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} X _ { t } ^ { ( u ) } \\ , & = \\ , \\int ^ { t } _ { 0 } e ^ { - ( t - s ) } u \\widetilde { X } ^ { ( u ) } _ { s } { \\mathrm d } s + \\int ^ { t } _ { 0 } e ^ { - ( t - s ) } { \\mathrm d } B _ { s } \\ , , \\\\ \\widetilde { X } _ { t } ^ { ( u ) } \\ , & = \\ , \\int ^ { t } _ { 0 } \\sum _ { k = 0 } ^ { \\infty } \\mathfrak p _ { 0 , k } ( t - s ; u ) \\ , { \\mathrm d } W _ { s , k } \\ , , \\mathfrak p _ { 0 , k } ( t - s ; u ) \\ , : = \\ , \\frac { u ^ { k } ( t - s ) ^ { k } } { k ! } e ^ { - ( t - s ) } \\ , , \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} \\Gamma ( 2 s ) = \\frac { 2 ^ { 2 s - 1 } \\Gamma ( s ) \\Gamma ( s + 1 / 2 ) } { \\pi ^ { 1 / 2 } } , \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} q = r _ 1 s _ 1 + \\cdots + r _ k s _ k \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} L ( \\psi , \\tilde { \\mathbb { A } } ) = \\sum _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) t ^ { \\deg _ { \\theta } ( a ) } } { a } . \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} \\frac { 1 } { L } \\int _ { B _ R } \\abs { \\Q _ L - \\Pi ( \\Q _ L ) } ~ d V \\leq M \\left ( R ^ d \\phi ( R ^ { - 1 } ) + \\int _ { B _ R } \\phi ( | \\nabla \\Q _ L | ) d V \\right ) \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\frac { 1 } { m ^ k } \\binom { m + k - i - 1 } { k - 1 } \\frac { P _ { X / T } ( i ) } { m ^ { 2 n - k - 1 } } . \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} y _ { i , j } = \\hat { u } ( n - i - j ) + \\sum _ { k = 0 } ^ { \\infty } \\hat { u } ( n - i + k ) \\left [ ( I - V U ) ^ { - 1 } V U z ^ { j } \\right ] \\mathbf { \\hat { } } \\ , ( - k ) . \\newline \\end{align*}"}
-{"id": "9923.png", "formula": "\\begin{align*} \\nu _ { 0 1 } = ( - 1 ) ^ n ( n - 1 ) ! \\ , \\check { \\chi } _ { H _ { { n + 1 } } } \\ , d \\varphi _ 1 \\wedge \\dots \\wedge d \\varphi _ { n - 1 } , \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} \\bar { \\tau } : = \\tau ^ { n , m , r } _ R \\wedge \\inf \\left \\lbrace t \\ge 0 : \\left \\lvert X ^ { n , r } _ t - X ^ { m , r } _ t \\right \\rvert \\ge \\varepsilon \\right \\rbrace , \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} ( - 1 ) ^ k H ( e _ { k , k } \\otimes \\beta _ { i + 1 , i } ) = H ( e _ { k + 1 , k } \\otimes \\beta _ { i , i } ) & = \\begin{cases} \\beta _ { i + k + 1 , i + k } , & i + k + 1 \\leq r \\\\ 0 , & , \\end{cases} \\\\ H ( e _ { k , k } \\otimes \\beta _ { i , i } ) & = \\begin{cases} \\beta _ { i + k , i + k } , & i + k \\leq r - 1 \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} \\begin{cases} f ^ { \\sharp } ( t , x , \\omega ) \\le f ( t , x , \\omega ) + \\eta ( t ) & ~ ( t , x , \\omega ) \\in \\ , ] 0 , T [ \\times \\R \\times [ a , b ] , \\\\ U ( 0 , x ) \\le U ^ { \\sharp } ( 0 , x ) + \\bar \\eta \\quad \\ & ~ x \\in \\R . \\end{cases} \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} \\delta _ { i } ^ { ( j ) } = ( 2 ^ { R ^ { t h } } - 1 + C ) ( \\delta _ { 1 } ^ { ( j ) } + \\cdots + \\delta _ { i - 1 } ^ { ( j ) } ) , \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} E _ { \\bullet } = ( E _ 1 , E _ 2 , \\ldots , E _ k ) , \\ E _ i \\in D ^ b ( X ) . \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} \\lvert ( 0 , n ) \\rvert _ { W _ n ( w ) } - \\lvert ( 0 , j ) \\rvert _ { W _ n ( w ) } = n - j > 0 . \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta _ g u = f ; \\\\ & u \\in H ^ 1 _ 0 ( \\bar { M } ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} D ( \\rho \\| \\sigma ) : = \\begin{cases} - \\ < \\xi _ \\rho , ( \\log \\Delta _ { \\sigma , \\rho } ) \\xi _ \\rho \\ > = \\ < \\xi _ \\sigma , ( \\Delta _ { \\rho , \\sigma } \\log \\Delta _ { \\rho , \\sigma } ) \\xi _ \\sigma \\ > & , \\\\ + \\infty & , \\end{cases} \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} v _ p \\left ( \\frac { N } { \\gcd ( C ( u C + D ) , N ) } \\right ) & = v _ p ( N ) - \\min ( v _ p ( u C + D ) , v _ p ( N ) ) , \\\\ v _ p \\left ( \\frac { m } { \\gcd ( C ( u A + B ) , m ) } \\right ) & = v _ p ( m ) - \\min ( v _ p ( A ( u C + D ) - 1 ) , v _ p ( m ) ) . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} \\begin{aligned} \\Pi _ 1 [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ( t ) = & \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\mu _ 0 ^ { n + 1 - 4 s + \\sigma } ( t ) f ( t ) \\\\ & + \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\Theta \\left [ \\dot { \\lambda } , \\dot { \\xi } , \\mu _ 0 ^ { n - 4 s } ( t ) \\lambda , \\mu _ 0 ^ { n - 4 s } ( \\xi - q ) , \\mu _ 0 ^ { n + 1 - 4 s + \\sigma } \\phi \\right ] ( t ) \\end{aligned} \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} \\alpha ' _ \\tau ( a , b ) = \\alpha _ \\tau ( ( a , b ) g ) . \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n u ( \\tau _ { x _ j } \\omega , x _ { j + 1 } - x _ j ) = 0 \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} T _ { i _ 1 } : = \\bigcup _ { u _ 2 = 0 } ^ { s _ 2 - 1 } \\Big ( W _ { i _ 1 } \\beta ^ { u _ 2 s _ 1 } \\Big ) . \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} \\mathbb { A ' } ^ { - 1 } _ L = \\ , _ { c } \\mathbb { A } ^ { - 1 } _ L - \\mathbb { T } _ L . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} u _ { k + m } ( t , 0 ) = \\int _ { \\tau _ { k + 1 } - \\delta } ^ { \\tau _ { k + 1 } } K ( t , s ) u _ { k + m } ( s , 0 ) \\ , d s + f ( t ) , \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} d \\mu _ z ( x ) = \\frac { 1 } { Q ( z ) } \\frac { e ^ { - \\beta V ( x ) } } { Z } \\Big [ \\det ( \\nabla \\xi \\nabla \\xi ^ T ) ( x ) \\Big ] ^ { - \\frac { 1 } { 2 } } \\ , d \\nu _ z ( x ) \\ , , \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} \\sum _ { \\ell = - L } ^ { - 1 } \\frac { 1 } { ( \\zeta x ^ { - \\ell } ; q ) _ { \\infty } } P _ t \\left ( \\ell \\right ) . \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} \\delta ( \\epsilon ) = \\sum _ { j = 0 } ^ { \\infty } \\mu \\left ( \\prod _ { k = 1 } ^ { j } { B } _ k ( \\epsilon ) \\right ) r _ { j + 1 } ( \\epsilon ) , \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} m _ n ( x , y ) = \\min _ { - n \\leq j \\leq n } \\| ( x - y ) + j \\theta \\| . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} \\eta = \\eta ( \\theta ) : = \\sup \\{ \\beta \\geq 1 : \\liminf _ { j \\to \\infty } j ^ \\beta \\| j \\theta \\| = 0 \\} \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} \\Delta _ { g } f = a \\circ f . \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} \\mathrm H ( \\nu \\mid \\mu ) = \\int _ { \\R ^ n } \\log f \\ , d \\nu . \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} S ( n , k ) \\equiv ( - 1 ) ^ r \\epsilon ( n ! / k ! ) \\binom { n + r } { r } \\mod p . \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} f ( x ) & = x ^ 2 ( x ^ 2 + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( 1 + x ) + m \\ : h ( x ) , \\\\ g ( x ) & = x ( x ^ 3 - 1 ) + ( C - D x ) x ( x ^ 3 - 1 ) ( 1 + x ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} A = A ( \\mathbf { v } _ 1 , \\mathbf { v } _ 2 , \\vartheta , g \\ , T ) : = \\vartheta \\cdot \\Big \\langle \\Phi _ G ( m _ x ) , \\mathrm { A d } _ g ( \\beta ) \\Big \\rangle , \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} \\phi _ { t } ^ \\ell ( S ) & = | \\{ m \\in S \\ : \\ F _ t ( m ) \\ge \\ell \\} | \\\\ \\alpha _ { t } ^ \\ell ( S ) & = | \\{ m \\in S \\ : \\ A _ t ( m ) \\ge \\ell \\} | \\\\ \\beta _ { t } ^ \\ell ( S ) & = | \\{ m \\in S \\ : \\ B _ t ( m ) \\ge \\ell \\} | , \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} E [ \\sigma ] = \\exp \\left ( \\sum _ { k = 1 } ^ { \\infty } k \\cdot \\widehat { \\log \\sigma } ( k ) \\widehat { \\log \\sigma } ( - k ) \\right ) . \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\mathcal { D } } = - \\sqrt { B _ { \\mathcal { D } } ( x ) } \\ , e ^ { \\partial } \\sqrt { D _ { \\mathcal { D } } ( x ) } - \\sqrt { D _ { \\mathcal { D } } ( x ) } \\ , e ^ { - \\partial } \\sqrt { B _ { \\mathcal { D } } ( x ) } + B _ { \\mathcal { D } } ( x ) + D _ { \\mathcal { D } } ( x ) , \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} A = \\partial _ x ^ { } \\Big ( g ^ { i j } \\partial _ x ^ { } + c ^ { i j } _ k u ^ k _ x + w ^ i _ k u ^ k _ x \\partial _ x ^ { - 1 } w ^ j _ l u ^ l _ x \\Big ) \\partial _ x ^ { } , \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} L ^ { s } : = \\mathbb { Z } ^ 3 \\cap ( \\{ r + 1 \\} , [ 1 , s - q ] , \\{ s + 1 \\} ) , q : = \\lfloor s / 4 \\rfloor . \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} P _ n ( x ) = \\frac { 1 } { n ! } \\left ( \\frac { d } { d x } \\right ) ^ n \\left ( x ^ n ( 1 - x ) ^ n \\right ) , \\ \\ \\ Q _ n ( x ) = ( 1 - x ) ^ n . \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} p _ ! \\varphi ( a ) = \\delta ^ \\frac { 1 } { 2 } ( a ) O _ a ( \\Phi ) d ^ \\frac { 1 } { 2 } a . \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} b ( x , y , t ) = h ( x ) \\beta _ { 0 } ( y , t ) + x h ( x ) \\beta _ { 1 } ( y , t ) \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} \\bigoplus _ { j = 0 } ^ i U ^ { \\otimes m - j } \\otimes V ^ { \\otimes j } = U ^ { \\otimes m } \\oplus U ^ { \\otimes m - 1 } \\otimes V \\oplus \\cdots \\oplus U ^ { \\otimes m - i } \\otimes V ^ { \\otimes i } \\subseteq V ^ { \\otimes m } \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} H H _ 1 ^ \\top H _ 1 H ^ \\top = ( \\underbrace { n ^ 2 , \\ldots , n ^ 2 } _ { \\ell } , \\underbrace { 0 , \\ldots , 0 } _ { n - \\ell } ) . \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} ( d _ x f _ 1 ) | _ { \\{ 0 \\} \\times E _ 1 } = ( D _ x f ) | _ { \\{ 0 \\} \\times E _ 1 } : E _ 1 \\rightarrow \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} \\partial _ { \\tau } \\ell ( \\gamma _ { \\tau } ) _ { | \\tau = 0 } = - \\beta _ { x _ { 0 } } \\left ( \\frac { \\partial } { \\partial \\tau } \\varphi ^ { \\tau } _ { \\ell ( \\gamma _ { 0 } ) } ( x _ { 0 } ) _ { | \\tau = 0 } \\right ) . \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} \\pi ^ { ( d ) } ( S ) U \\xi = \\tau ^ { ( d ) } ( S ) U \\xi \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} Q \\overline { \\nabla } ^ { ' } _ { X } \\phi Y = \\{ ( 1 - \\beta ) g ( X , Y ) - \\alpha g ( X , \\phi Y ) \\} Q \\xi + B h ( X , Y ) \\end{align*}"}
-{"id": "9691.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { \\theta } = \\sum _ { i = 0 } ^ r \\tilde { \\phi } _ { \\theta , i } \\tau ^ i = \\sum _ { i = 0 } ^ r z ^ { m i } \\phi _ { \\theta , i } \\tau ^ i . \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} ( \\sigma ^ { n - \\theta } , w ) + \\gamma ( \\nabla u ^ { n - \\theta } , \\nabla w ) + ( u ^ { n - \\theta } , w ) = 0 , ~ \\forall w \\in H _ 0 ^ 1 . \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} R _ 1 \\bigl ( \\check { X } ( j ) \\bigr ) ^ 2 + 4 R _ 0 \\bigl ( \\check { X } ( j ) \\bigr ) = \\bigl ( \\check { X } ( j + 1 ) - \\check { X } ( j - 1 ) \\bigr ) ^ 2 \\ \\ ( j = 0 , 1 , \\ldots , N ) . \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} \\rho ^ { - 1 } ( g _ j ) = \\left \\{ ( h _ { m _ x } \\ , T , t _ j ) , \\ , ( k _ { m _ x } \\ , T , t _ j ^ { - 1 } ) \\right \\} , k _ { m _ x } : = h _ { m _ x } \\ , \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} \\bar { \\alpha } \\geq \\frac { 1 } { \\psi ^ { \\prime \\prime } \\left ( \\rho ( 2 \\sqrt { 2 } \\delta ( v ) ) \\right ) } = \\hat { \\alpha } \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} \\log | L ( \\sigma + i t , \\pi ) | = \\bigg ( \\dfrac { 3 } { 4 } - \\dfrac { \\sigma } { 2 } \\bigg ) \\log C ( t , \\pi ) - \\dfrac { 1 } { 2 } \\displaystyle \\sum _ { \\gamma } f _ { \\sigma } ( t - \\gamma ) + O ( d ) , \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} [ X _ j , X _ l ] = [ Y _ j , Y _ l ] = 0 \\quad [ X _ j , Y _ l ] = \\delta _ { j l } U \\qquad \\forall j , l = 1 , \\dots , m . \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} \\det ( A _ { m _ 1 } ( X _ 1 ) \\ , A _ { m _ 2 } ( X _ 2 ) \\dots A _ { m _ k } ( X _ k ) ) = \\left ( \\prod _ { i = 1 } ^ { k } \\prod _ { j = 1 } ^ { m _ i - 1 } j ! \\right ) \\prod _ { 1 \\le i < j \\le k } ( X _ j - X _ i ) ^ { m _ i m _ j } . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} q _ { \\rm D A S H } ( \\mathbf p ) = \\sum _ { n \\in \\mathcal N } a _ { n } \\sum _ { \\ell \\in \\mathcal L } b _ { n , \\ell } \\sum _ { k \\in \\mathcal K _ { { \\rm D A S H } , n , \\ell } } \\Pr [ K _ { { \\rm D A S H } , n , \\ell } = k ] \\Pr [ { \\rm S I R } _ { { \\rm D A S H } , n , \\ell } \\geq \\tau _ k ] , \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} { \\rm R e } \\left \\{ 1 + z \\frac { T _ g '' ( z ) } { T _ g ' ( z ) } \\right \\} & = { \\rm R e } \\left \\{ z \\frac { f ' ( z ) } { f ( z ) } + z \\frac { g '' ( z ) } { g ' ( z ) } + 1 \\right \\} \\\\ & > \\frac { 1 + \\alpha - 4 r + ( 1 - \\alpha ) r ^ 2 } { 1 - r ^ 2 } > 0 , \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} E _ 2 ^ { i , j } = { \\rm { H } } ^ i ( X , \\R ^ j f _ * \\Q ) \\Longrightarrow { \\rm { H } } ^ { i + j } ( \\eta , \\Q ) . \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} \\dim ( G _ { \\lambda ' } ) = \\frac { 1 } { 2 } ( ( d ' _ 1 ) ^ 2 - 1 + ( d ' _ k ) ^ 2 - 1 ) + \\frac { 1 } { 2 } \\sum _ { j \\neq 1 , k } d _ j ^ 2 - 1 . \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} v _ 0 \\circ X ^ { - 1 } _ \\gamma ( s , t ) = - a _ 0 \\lambda \\mu _ \\lambda | t | + b _ 0 + 2 \\log \\mu _ \\lambda - \\log h _ \\gamma + \\mathcal O \\left ( e ^ { - a _ 0 \\lambda \\mu _ \\lambda | t | } \\right ) . \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} z \\in \\mathcal { C } : = \\left \\{ u \\in \\mathrm { C } ^ { 2 , \\beta } ( \\overline { \\Omega } ) : \\int _ \\Omega u ( x ) d x = { N ( u ) } = 0 \\right \\} . \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 ( Z ) = T \\left ( \\partial _ { \\dot { \\lambda } _ 1 } S _ { o u t } [ \\dot { \\bar { \\lambda } } _ 1 ] + p \\left [ ( u ^ * _ { \\mu , \\xi } + \\psi + \\phi ^ { i n } ) ^ { p - 1 } - ( u ^ * _ { \\mu , \\xi } ) ^ { p - 1 } \\right ] Z , 0 , 0 \\right ) . \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\sup _ { s \\in [ t , T ] } \\int _ K | X _ n ( s , x ) - X ( s , x ) | \\wedge 1 d x = 0 . \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} N _ { i , j } ^ k = \\sum _ { l \\in I } \\frac { S _ { i , l } S _ { j , l } \\overline { S _ { k , l } } } { S _ { i _ 0 , l } } \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{gather*} I _ { a , b , d , ( K _ s , J ' _ s ) } : C F ^ { [ a , b ) } ( H ^ { ( k ) } , J ) \\rightarrow C F ^ { [ a + d , b + d ) } ( H ^ { ( k ) } , J ^ { + \\frac { 1 } { k } } ) \\\\ I _ { a , b , d , ( K _ s , J ' _ s ) } ( z _ - ) = \\sum _ { z _ + \\in \\widetilde { P } ( H ^ { ( k ) } ) } \\sharp \\mathcal { N } ( z _ - , z _ + , K _ s , J ' _ s ) \\cdot z _ + \\end{gather*}"}
-{"id": "1596.png", "formula": "\\begin{align*} u ' ( t ) = F ( u ( t ) ) , \\ ; \\ ; u ( t _ 0 ) = u _ 0 . \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} ( E _ \\sigma ^ { n - \\theta } , w ) + ( \\nabla E _ u ^ { n - \\theta } , \\nabla w ) = ( R _ 2 ^ { n - \\theta } , w ) , ~ \\forall w \\in H _ 0 ^ 1 , \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\int _ \\Omega \\mathbb { P } [ \\mu ] \\nabla p \\cdot \\nabla \\phi \\ , \\d x & = \\sum _ { i \\in \\mathcal { I } } G _ i \\int _ 0 ^ 1 \\partial _ { s _ i } \\phi \\ , \\d s _ i \\\\ & = \\sum _ { i \\in \\mathcal { I } } G _ i ( \\phi ( x _ i ^ - ) - \\phi ( x _ i ^ + ) ) = \\sum _ { j \\in \\mathcal { V } } S _ j \\phi ( x _ j ) , \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} \\begin{aligned} M ( x , y ) & = c _ { 0 } c _ { 0 } ' \\int _ { 0 } ^ { \\infty } ( n + \\{ x \\} + z ) ^ { - 1 - 2 \\beta } ( n + \\{ y \\} + z ) ^ { - 1 + 2 \\beta } d z \\\\ & + o \\left ( ( n + \\{ x \\} ) ^ { - \\frac { 1 } { 2 } - \\beta - \\delta } ( n + \\{ y \\} ) ^ { - \\frac { 1 } { 2 } + \\beta - \\delta } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} \\dot { \\theta } _ i = \\omega + \\sum _ { j = i - n } ^ { i + n } \\sin ( \\theta _ j - \\theta _ i ) , \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} W _ { n } ( 0 , n ) = \\square _ c [ n - 1 ] \\ . \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} I ( 0 ) & : = I _ 0 , \\\\ I ( t ) & : = \\begin{cases} \\min \\{ k \\in \\mathbb { N } : \\ \\mu _ k ( t ) \\le 0 \\} , & \\mu _ 0 ( t ) \\ge 0 , \\\\ - \\infty , & \\mu _ 0 ( t ) < 0 , \\\\ \\end{cases} \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} S _ { - 1 , \\sigma } ( t , \\pi ) & \\geq - \\frac { \\log C ( t , \\pi ) } { \\pi } \\left ( \\frac { e ^ { - 2 \\pi \\beta \\Delta } } { 1 + e ^ { - 2 \\pi \\beta \\Delta } } \\right ) - \\dfrac { 2 \\ , d \\ , \\beta \\ , e ^ { ( 1 - 2 \\beta ) \\pi \\Delta } } { \\pi ( \\frac { 1 } { 4 } - \\beta ^ 2 ) \\big ( 1 + e ^ { - 2 \\pi \\beta \\Delta } \\big ) ^ 2 } + O _ c \\left ( \\dfrac { d \\ , \\beta \\ , e ^ { ( 1 - 2 \\beta ) \\pi \\Delta } } { ( \\frac { 1 } { 2 } - \\beta ) ^ 2 \\Delta } \\right ) + O ( d ) . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} - \\Delta _ N u + b ( x ) | u | ^ { N - 2 } u = c ( x ) | u | ^ { N - 2 } u + g ( x ) f ( u ) + \\varepsilon h ( x ) , \\ \\ x \\in \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} \\gamma ( \\chi , 1 - s , \\psi ) = \\frac { \\epsilon ( \\chi , 1 - s , \\psi ) L ( \\chi ^ { - 1 } , s ) } { L ( \\chi , 1 - s ) } , \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} w \\left ( \\begin{matrix} 1 & \\zeta _ { p } \\\\ 0 & 1 \\end{matrix} \\right ) = \\left ( \\begin{matrix} 1 & 0 \\\\ - \\zeta _ { p } & 1 \\end{matrix} \\right ) w = \\left ( \\begin{matrix} 1 & - \\zeta _ { p } ^ { - 1 } \\\\ 0 & 1 \\end{matrix} \\right ) \\left ( \\begin{matrix} \\zeta _ { p } ^ { - 1 } & 0 \\\\ 0 & - \\zeta _ { p } \\end{matrix} \\right ) \\left ( \\begin{matrix} 0 & 1 \\\\ 1 & - \\zeta _ { p } ^ { - 1 } \\end{matrix} \\right ) w \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} L ^ t _ { \\theta '' } ( u _ 1 [ f _ 1 ] \\cdots u _ n [ f _ n ] \\cdot v ) = \\sum _ { i = 1 } ^ n u _ 1 [ f _ 1 ] \\cdots u _ i [ \\iota _ t ( \\theta '' ) ( f _ i ) ] \\cdots u _ n [ f _ n ] \\cdot v , \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} S _ n \\stackrel { \\mathrm { d } } { = } c _ n X + \\gamma _ n \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} \\inf _ { \\alpha \\in \\mathbb { A } } J ^ { \\mu } ( \\alpha ) = \\mathbb { E } \\left [ \\int _ 0 ^ T f ( t , X ^ { \\alpha } _ t , \\mu _ t , \\alpha _ t ) d t + g ( X ^ { \\alpha } _ t , \\mu _ T ) \\right ] , \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} \\begin{cases} u _ { \\tau } & = p \\\\ p _ { \\tau } & = - \\frac { f ( x , u , p ) } { a ( x , u , p ) } \\\\ x _ \\tau & = 1 \\end{cases} \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} { \\bf X } i _ { X } u = i _ { X } { \\bf X } u , { \\bf X } ( \\alpha \\wedge u ) = \\alpha \\wedge { \\bf X } u , { \\bf X } ( u \\wedge d \\alpha ) = ( { \\bf X } u ) \\wedge d \\alpha . \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\bar { \\phi } ( x , y ) : = \\langle \\tilde { B _ 1 } x - a , y - x \\rangle + y ^ T B _ 1 y - x ^ T B _ 1 x + \\bar { h } ( y ) - \\bar { h } ( x ) \\} , \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{gather*} \\nabla ^ { \\mathrm { R } , \\mathrm { b a s } } = \\nabla ^ { \\mathrm { R } } = \\nabla ^ { \\mathrm { R } , g _ { \\Sigma } } . \\end{gather*}"}
-{"id": "7655.png", "formula": "\\begin{align*} f '' ( h _ 0 , h _ 1 ) ( \\nu ) = \\sum _ { i , j } \\langle f ( h _ 0 , h _ 1 ) , \\nu | _ { h _ 0 Y _ j \\cap Y _ i } \\rangle = f ( h _ 0 , h _ 1 ) ( \\nu ) . \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} G _ i ( x ) = \\frac { \\sqrt { ( k - 1 ) ^ i } } { t ^ i ( t ^ 2 - 1 ) } ( t ^ { 2 i + 2 } - 1 ) . \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} & f \\left ( u , v \\right ) = v \\left ( { { \\beta } _ { y } } - { { \\alpha } _ { x } } \\right ) - u \\left ( { { \\beta } _ { x } } + { { \\alpha } _ { y } } \\right ) \\\\ & g \\left ( u , v \\right ) = v \\left ( { { \\beta } _ { x } } + { { \\alpha } _ { y } } \\right ) - u \\left ( { { \\alpha } _ { x } } - { { \\beta } _ { y } } \\right ) \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} \\mathbf a & = ( \\mathbf a _ 1 ( 1 ) , \\mathbf a _ 1 ( 2 ) , \\mathbf a _ 1 ( 3 ) , \\mathbf a _ 2 ( 1 ) ) \\\\ & = ( 0 , \\langle \\lambda _ 1 , \\alpha _ 2 ^ { \\vee } \\rangle + \\langle \\lambda _ 2 , \\alpha _ 2 ^ { \\vee } \\rangle , \\langle \\lambda _ 1 , \\alpha _ 1 ^ { \\vee } \\rangle + \\langle \\lambda _ 2 , \\alpha _ 1 ^ { \\vee } \\rangle , \\langle \\lambda _ 2 , \\alpha _ { 3 } ^ { \\vee } \\rangle ) . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { A } [ \\overline { u } ] ( x ) & = \\lambda - \\mathcal { L } [ u ] ( a x ) + a ^ { ( N + 1 ) m } | D u | ^ m ( a x ) - f ( a x ) \\\\ & \\qquad + \\mathcal { L } [ u ] ( a x ) - \\mathcal { L } [ \\overline { u } ] ( x ) + f ( a x ) - f ( x ) \\\\ & \\geq f ( a x ) - f ( x ) + \\mathcal { L } [ u ] ( a x ) - \\mathcal { L } [ \\overline { u } ] ( x ) . \\end{aligned} \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} \\sin _ q ( z ) = \\frac { \\theta _ 1 ( z \\mid \\tau ' ) } { \\theta _ 1 \\left ( \\frac { \\pi } { 2 } \\bigm | \\tau ' \\right ) } \\cos _ q ( z ) = \\frac { \\theta _ 1 \\left ( z + \\frac { \\pi } { 2 } \\bigm | \\tau ' \\right ) } { \\theta _ 1 \\left ( \\frac { \\pi } { 2 } \\bigm | \\tau ' \\right ) } ( \\tau ' = \\frac { - 1 } { \\tau } ) \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 0 } ^ { 2 p } A ^ { \\alpha } \\otimes \\mathbb Q _ { \\Delta } [ - \\alpha ] \\stackrel { \\theta } { \\cong } R \\rho _ * \\mathbb Q _ { \\widetilde \\Delta } . \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} { n ; b ; \\omega _ n ^ { g ' } \\brack k } = \\sum _ { \\substack { 0 \\le i \\le \\lfloor \\frac { n - k } { 2 } \\rfloor \\\\ n / ( g ' , n ) \\mid i } } \\binom { ( g ' , n ) } { i / ( n / ( g ' , n ) ) } { n - i \\brack i + k } _ { \\omega _ { n - i } ^ { g ' ( n - i ) / n } } . \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} \\mathbf { \\tilde { K } } f ( x ) = \\int _ { 0 } ^ { \\infty } k ( x - y ) f ( y ) d y , \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} - \\frac { 1 } { 4 } ( x - b ) ^ { 2 } P _ { n } ^ { ( \\alpha , \\beta ) } ( x ) = f _ { n + 1 } \\widehat { P } _ { n + 2 } ^ { ( \\alpha , \\beta ) } ( x ) - 2 b \\ g _ { n } \\widehat { P } _ { n + 1 } ^ { ( \\alpha , \\beta ) } ( x ) + h _ { n } \\widehat { P } _ { n } ^ { ( \\alpha , \\beta ) } ( x ) , \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ \\tau \\phi = - ( - \\Delta ) ^ s _ y \\phi + p U ^ { p - 1 } ( y ) \\phi + H [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ( y , t ( \\tau ) ) , & y \\in \\mathbb { R } ^ n , \\tau \\geq \\tau _ 0 , \\\\ \\phi ( y , \\tau _ 0 ) = e _ { 0 } Z _ 0 ( y ) , & y \\in \\mathbb { R } ^ n , \\end{cases} \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ 2 } { d x ^ 2 } + V _ 0 ( x ) + V ( x ) \\right ) u ( x , E _ j ) = E _ j u ( x , E _ j ) , \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} K _ i L _ j = L _ j K _ i , \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { L } ( y , p _ { _ 1 } ( y ) ) } { \\partial p _ { _ 1 } ( y ) } = \\log \\frac { p _ { _ 1 } ( y ) } { p _ { _ 0 } ( y ) } + 1 + \\rho _ 0 + \\rho _ 1 y ^ 2 . \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} \\int _ { \\widetilde { S } _ \\varepsilon ( \\xi ) \\cap B _ R ( 0 ) } \\frac { \\lvert v ( k ) \\lvert ^ 2 } { \\omega ( k ) ^ 2 } d \\lambda _ \\nu & = \\nu \\lambda _ \\nu ( \\widetilde { S } _ \\varepsilon ( \\xi ) \\cap B _ 1 ( 0 ) ) \\int _ { 0 } ^ \\infty 1 _ { B _ R ( 0 ) } ( x e _ 1 ) \\frac { \\lvert v ( k e _ 1 ) \\lvert ^ 2 } { \\omega ( k e _ 1 ) ^ 2 } k ^ { \\nu - 1 } d \\lambda _ 1 \\\\ & = \\infty \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} r _ { 1 3 } r _ { 1 2 } - r _ { 1 2 } r _ { 2 3 } + r _ { 2 3 } r _ { 1 3 } = 0 , \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} \\overline { \\mu } ( ( m , a ) , ( n , b ) ) = ~ ( \\mu ( m , b ) + \\mu ( a , n ) , ~ \\mu ( a , b ) ) ~ ~ ~ ~ ~ ~ \\overline { \\alpha } ( ( m , a ) ) = ( \\alpha _ M ( a ) , \\alpha ( a ) ) . \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} ( t _ 1 , t _ 2 ) \\cdot ( \\vec { x } , \\vec { y } ) = ( t _ 1 t _ 2 ^ { - 1 } \\cdot \\vec { x } , t _ 1 ^ { - 1 } t _ 2 \\cdot \\vec { y } ) . \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\sin \\pi ( s _ { j _ 2 } - s _ { j _ 1 } ) = ( - 1 ) ^ { n _ { j _ 2 } - n _ { j _ 1 } } \\sin \\left ( \\frac { 2 \\pi i ( \\ell _ { j _ 2 } - \\ell _ { j _ 1 } ) } { \\log q } \\right ) \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} \\gamma ( u _ 0 P v ) + \\gamma ( v P w ) = \\gamma ( u _ 0 P w ) = \\gamma _ w = \\gamma _ v = \\gamma ( u _ 0 P v ) , \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} E _ { 1 } ( x ) & = \\sum _ { i = 1 } ^ { n _ m } \\int _ { \\widetilde { K } _ { n , \\varepsilon } } \\mathbf P \\{ \\widetilde { \\nu } _ n = i , \\tau _ x > i , x + S ( i ) \\in d y \\} \\mathbf { E } [ u ( y + S ( n - i ) ) ; \\tau _ y > n - i ] . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} \\sigma _ q ( \\tilde { t } _ U ) = \\epsilon _ q ^ j \\tilde { t } _ U . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} \\textstyle p _ k ^ * = \\left ( \\frac { \\mu ^ * } { \\ln 2 \\left ( \\nu _ { _ { \\rm S M } } - \\rho _ { _ { \\rm S M } } [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 \\right ) } - \\frac { \\sigma ^ 2 } { \\left ( 1 - \\rho _ { _ { \\rm S M } } \\right ) [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } \\right ) ^ + , \\quad \\forall \\ , k = 1 , 2 , \\ldots , r . \\end{align*}"}
-{"id": "9739.png", "formula": "\\begin{align*} L ( \\tilde { C } , \\mathbb { A } ) = \\sum _ { a \\in A _ { + } } \\frac { a ( z _ 1 ) } { a } = \\log _ { \\tilde { C } } ( 1 ) = \\frac { \\log _ { C } ( \\omega _ 1 ) } { \\omega _ 1 } = - \\frac { \\tilde { \\pi } } { ( z _ 1 - \\theta ) \\omega _ 1 } , \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\mu _ { 0 } E _ { 0 } ( y , t ) Z _ { i } ( y ) d y = 0 , i = 1 , \\cdots , n + 1 \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} h ( x ) = q ^ { x } \\bigg ( \\frac { x } { r } - \\frac { \\beta + n } { q - 1 } \\bigg ) + \\frac { n } { q - 1 } & \\geq q \\bigg ( \\frac { 1 } { r } - \\frac { \\beta + n } { q - 1 } \\bigg ) + \\frac { n } { q - 1 } \\\\ & \\geq q \\bigg ( \\frac { 1 } { r } + \\frac { 1 } { q - 1 } - \\frac { n + 1 } { q ( q - 1 ) } - \\frac { 1 } { r } \\bigg ) + \\frac { n } { q - 1 } \\\\ & = 1 . \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} \\underset { n \\rightarrow \\infty } { \\lim } \\mathbb { P } \\left ( \\frac { \\tilde { Z } _ 2 } { n } > \\frac { M _ 1 } { 2 } \\right ) = 1 \\ \\ \\ \\underset { n \\rightarrow \\infty } { \\lim } \\mathbb { P } \\left ( \\frac { Z _ 2 } { n \\sigma ^ 2 } > \\frac { M _ 1 } { 2 } \\right ) = 1 . \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} F _ i ( x ) = ( x ^ 2 - 2 k + 2 ) F _ { i - 2 } ( x ) - ( k - 1 ) ^ 2 F _ { i - 4 } ( x ) \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} \\forall q , p \\geq 0 , ~ c _ { q , p } = S ^ { - 1 } \\left ( R _ p \\right ) ^ { \\frac 1 { q + 1 } } . \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} g _ { t , l , v } \\left ( \\begin{matrix} 0 & 1 \\\\ p ^ { n } & 0 \\end{matrix} \\right ) = n ( - v ^ { - 1 } p ^ { t + l } ) z ( v p ^ { n - l } ) g _ { t - n + 2 l , 0 , v ^ 2 } \\left ( \\begin{matrix} 1 & 1 + v ^ { - 1 } p ^ { l - n } \\\\ 0 & - v ^ { - 2 } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} \\textmd { I m } \\ ; z > p ( w ) = p \\left ( \\widehat { \\rho } _ { z } ( \\widehat { \\rho } _ { 1 / z } ( w ) ) \\right ) = p \\left ( \\widehat { \\rho } _ { 1 / z } ( w ) \\right ) \\abs { z } . \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} h ^ 1 ( G _ { \\Q , S \\cup \\{ p \\} } , \\overline { W } ) - h ^ 2 ( G _ { \\Q , S \\cup \\{ p \\} } , \\overline { W } ) = h ^ 0 ( G _ { \\Q , S \\cup \\{ p \\} } , \\overline { W } ) + \\dim ( \\overline { W } ) - h ^ 0 ( G _ { \\R } , \\overline { W } ) , \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} \\hat H ( t ) = \\frac { f ( t ) } { m } \\hat J _ - + f ^ { - 1 } ( t ) m \\omega ^ 2 ( t ) \\hat J _ + . \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} n 2 ^ { - m - 4 } \\ , & \\geq m 2 ^ { m } \\log ( 2 ^ { m + 3 } d ) + \\log \\Big ( \\frac { 2 n ^ k } { ( k - 1 ) ! } \\Big ) \\\\ \\ , & = \\ , m 2 ^ { m } \\log ( 2 ^ { m + 3 } d ) + k \\ , \\log \\Big ( c _ k \\frac { n } { k } \\Big ) , \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} P ( \\alpha , \\beta ) : = x ^ { \\alpha + 2 \\beta } y ^ { 2 \\alpha - \\beta } + \\dfrac { ( - 1 ) ^ { \\alpha + \\beta } q ^ { 2 \\alpha } } { x ^ { \\alpha + 2 \\beta } y ^ { 2 \\alpha - \\beta } } . \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} X _ { t _ i } & : = \\{ x : c ( g _ { t _ i } ^ { - 1 } , x ) = h _ i ^ { - 1 } \\} , \\\\ [ t _ 0 , \\ldots , t _ { n - 1 } ] & : = X _ { t _ 0 } \\cap g _ { t _ 0 } X _ { t _ 1 } \\cap \\dots \\cap ( g _ { t _ 0 } \\cdots g _ { t _ { n - 2 } } ) X _ { t _ { n - 1 } } . \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} \\mathcal { H } ^ t ( F _ C ) & \\leq \\mathcal { H } ^ t ( F _ \\emptyset ) + \\mathcal { H } ^ t ( \\mathcal { O } ) = \\mathcal { H } ^ t ( \\mathcal { O } ) . \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\widetilde { T } _ V f ( \\gamma , \\zeta ) = \\int _ { \\R } f ( t , \\zeta ) \\frac { \\gamma ^ { 2 \\pi i t } } { \\gamma } \\d t , ( \\gamma , \\zeta ) \\in \\mathcal { C } _ p , \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} \\int _ { B _ R } | x | ^ { \\frac { n } { p } \\theta } | \\nabla u _ { \\lambda } ( x ) | ^ { \\theta } \\frac { d x } { | x | ^ n } = \\int _ { B _ R } | x | ^ { \\frac { n } { p } \\theta } | \\nabla u ( x ) | ^ { \\theta } \\frac { d x } { | x | ^ n } . \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} = | \\wp | \\int _ { 0 } ^ { \\infty } \\frac { 1 } { | \\wp | } \\int _ { \\wp } | f ( r y ) | ^ { p } d \\sigma ( y ) \\phi _ { 1 } ( r ) r ^ { Q - 1 } d r = \\int _ { \\mathbb { G } } | f ( x ) | ^ { p } \\phi _ { 1 } ( x ) d x , \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} K _ 2 ( \\varphi , p ) : = \\int \\bigl [ p ( \\boldsymbol \\varphi ( x ) ) \\bigr ] \\ , \\bigl [ \\rho _ \\varphi ( x ) \\bigr ] \\ , d x \\ , , \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} \\mathrm { S G } _ 0 ^ \\pm : = \\left \\{ { \\left . \\begin{pmatrix} \\cos ( q ^ 3 + q ^ 4 ) & \\mp \\sin ( q ^ 3 + q ^ 4 ) & 0 & 0 & q ^ 1 \\\\ \\pm \\sin ( q ^ 3 + q ^ 4 ) & \\cos ( q ^ 3 + q ^ 4 ) & 0 & 0 & q ^ 2 \\\\ 0 & 0 & 1 & 0 & q ^ 3 \\\\ 0 & 0 & 0 & 1 & q ^ 4 \\\\ 0 & 0 & 0 & 0 & 1 \\end{pmatrix} \\right | q \\in \\mathbb { R } ^ 4 } \\right \\} . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} A ( s ) & = \\frac { 6 ( n - 1 ) ^ 6 ( n - 2 ) ^ 2 ( n ^ 2 + 6 n - 1 2 ) } { n ^ 3 } s ^ 4 \\\\ & + \\frac { ( n - 1 ) ^ 2 } { n ^ 2 } ( 2 4 n ^ 7 - 1 1 6 n ^ 6 - 8 8 n ^ 5 + 1 7 3 2 n ^ 4 - 4 9 2 0 n ^ 3 + 6 7 4 4 n ^ 2 - 4 8 0 0 n + 1 4 4 0 ) s ^ 3 \\\\ & + \\frac { n - 2 } { n ^ 2 } ( 3 6 n ^ 8 - 2 5 2 n ^ 7 + 5 0 6 n ^ 6 + 3 9 2 n ^ 5 - 3 3 8 6 n ^ 4 + 6 5 0 4 n ^ 3 - 6 4 8 0 n ^ 2 + 3 4 5 6 n - 7 6 8 ) s ^ 2 \\\\ & + ( n - 2 ) ^ 2 ( 2 4 n ^ 5 - 1 5 6 n ^ 4 + 3 1 6 n ^ 3 - 2 2 4 n ^ 2 + 3 2 n ) s \\\\ & + 2 n ^ 2 ( n - 2 ) ^ 3 ( 3 n - 4 ) ( n - 4 ) , \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} \\tau _ { \\ast } : = \\frac { 1 } { 1 + 2 c _ { \\ast } } \\log \\left ( 1 + \\frac { \\lambda _ { 0 } } { 6 C _ { M } ( \\kappa + \\| \\nabla P _ { 0 } \\| _ { W ^ { 3 , p } ( \\Omega , \\mathbb { R } ^ { 3 } ) } ) } \\right ) , \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} A : = \\sup _ { R > 0 } R ^ { Q } \\int _ { | x | \\geq R } \\frac { \\phi ( x ) \\exp \\left ( \\mathcal { M } \\log \\frac { 1 } { \\psi } \\right ) ( x ) } { | x | ^ { 2 Q } } d x < \\infty , \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { \\infty } \\Omega ( \\ell ) = \\sum _ { j = 0 } ^ { N - 1 } \\left \\{ \\sum _ { \\ell = 0 } ^ { \\infty } \\Omega ( N \\ell + j ) \\right \\} , \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} L ^ \\Phi ( G ) = \\left \\{ f : G \\to \\C : \\int _ G \\Phi ( \\alpha | f ( s ) | ) \\ , d s < \\infty \\mbox { f o r s o m e } \\alpha > 0 \\right \\} , \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} \\partial _ i A ( x ) & = e ^ { x _ i } U ^ \\ast e _ i e _ i ^ \\ast U \\textup { f o r $ i \\le n , $ } \\\\ \\partial _ j A ( x ) & = - e ^ { x _ j } V ^ \\ast f _ { j - n } f _ { j - n } ^ \\ast V \\textup { f o r $ j > n , $ } \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Delta _ f U + A ( U ) ( \\nabla U , \\nabla U ) = 0 , \\quad \\mbox { o n $ \\mathbb { R } ^ n $ } , & \\\\ & \\lim _ { | x | \\rightarrow + \\infty } U ( x ) = u _ 0 ( x / | x | ) , & \\end{aligned} \\right . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} A ^ { W , W } _ \\cdot ( l ) = \\lim _ { n \\to \\infty } A _ \\cdot ^ { W , W } ( l _ n ) \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} \\Big | \\mathrm { E } \\left ( e ^ { i t \\sum _ { j = 1 } ^ k \\eta _ j } \\right ) - \\prod _ { j = 1 } ^ { k } \\mathrm { E } \\left ( e ^ { i t \\eta _ j } \\right ) \\Big | \\leq & ~ C t ^ 2 \\frac { \\phi ( h _ n ) b _ n ^ 2 } { n h _ n ^ 2 \\left [ \\mathrm { E } \\Delta _ 1 ( x ) \\right ] ^ 2 } \\Big [ ( k - 1 ) \\sum _ { i \\in I _ 1 } \\sum _ { j \\in I _ 2 } \\lambda _ { i j } \\\\ & + ( k - 2 ) \\sum _ { i \\in I _ 1 } \\sum _ { j \\in I _ 3 } \\lambda _ { i j } + \\cdots + \\sum _ { i \\in I _ 1 } \\sum _ { j \\in I _ k } \\lambda _ { i j } \\Big ] . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} \\Tilde { \\boldsymbol { \\Lambda } } = \\frac { \\boldsymbol { \\Lambda } } { \\sqrt { \\sigma ^ { 2 } } } \\left ( \\frac { \\ln 2 } { \\mu _ 2 } \\left ( \\nu _ 2 \\mathbf { I } _ { r } - \\boldsymbol { \\Lambda } ^ { \\rm H } \\boldsymbol { \\Lambda } \\right ) \\right ) ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ { t } f + v \\cdot \\nabla _ { x } f + \\mathcal { L } ^ { 0 } f = \\Gamma ^ { 0 } ( f , f ) , ~ ~ t > 0 , x \\in \\mathbb { T } ^ { 3 } , v \\in \\R ^ 3 ; \\\\ & f | _ { t = 0 } = f _ { 0 } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} \\frac 1 { ( n + k ) n ^ { r _ 2 } } = \\sum _ { m = 2 } ^ { r _ 2 } \\frac { ( - 1 ) ^ { r _ 2 - m } } { n ^ m k ^ { r _ 2 + 1 - m } } + \\frac { ( - 1 ) ^ { r _ 2 + 1 } } { k ^ { r _ 2 } } \\Big ( \\frac { 1 } { n } - \\frac { 1 } { n + k } \\Big ) . \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t u ^ { n - \\theta } , v \\Big { ) } - \\gamma ( \\nabla \\sigma ^ { n - \\theta } , \\nabla v ) + ( \\sigma ^ { n - \\theta } , v ) + ( f ^ { n - \\theta } ( u ) , v ) = & ( g ^ { n - \\theta } , v ) , ~ \\forall v \\in H _ 0 ^ 1 , \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} \\tilde f \\left ( t , x , \\omega \\right ) \\ , = \\ , \\begin{cases} f _ { i } \\left ( t , x , \\omega \\right ) & \\hbox { i f } ~ t \\in [ a _ { i } , b _ { i } ] , \\ i = 1 , \\ldots , N , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} | Q _ { - \\nu , m } | ^ { \\frac { 1 } { u } - \\frac { 1 } { p } } \\left ( \\sum _ { k : Q _ { 0 , k } \\subset Q _ { - \\nu , m } } \\ ! \\ ! \\ ! | f ( e ^ { ( k ) } ) | ^ p \\right ) ^ { \\frac { 1 } { p } } & = \\| \\{ 2 ^ { \\nu d ( \\frac { 1 } { p } - \\frac { 1 } { u } ) } f ( e ^ { ( k ) } ) \\} _ k | \\ell ^ { 2 ^ { d \\nu } } _ p ( Q _ { - \\nu , m } ) \\| \\\\ & \\le \\| f \\| \\ , . \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} V _ 0 : = \\textrm { S p a n } _ k \\{ v _ I \\} . \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} \\inf _ { \\mathcal { C G } } J = \\begin{cases} ( 2 \\pi ) ^ { - 1 } \\sqrt { 1 - e ^ { - 2 t } } & e ^ { - 2 t } \\in \\big ( \\frac 1 4 , \\frac 1 2 \\big ] , \\\\ 0 & e ^ { - 2 t } \\in \\big ( \\frac 1 2 , 1 \\big ) . \\end{cases} \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} \\# w \\circ T = \\sum _ { u \\in W _ { 1 } ( w ) } \\# u . \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 4 8 n + 2 4 ) q ^ n & \\equiv \\dfrac { f _ 2 f _ 3 f _ 4 f _ { 1 2 } } { f _ 1 ^ 3 f _ 6 } - 2 q \\dfrac { f _ 6 ^ 2 f _ { 1 2 } ^ 4 } { f _ 3 ^ 8 } \\\\ & \\equiv \\dfrac { f _ 2 f _ 4 f _ { 1 2 } } { f _ 6 } \\cdot \\dfrac { f _ 3 } { f _ 1 ^ 3 } - 2 q f _ { 1 2 } ^ 3 \\\\ & \\equiv \\dfrac { f _ 2 f _ 4 f _ { 1 2 } } { f _ 6 } \\left ( \\dfrac { f _ 4 ^ 6 f _ 6 ^ 3 } { f _ 2 ^ 9 f _ { 1 2 } ^ 2 } + 3 q \\dfrac { f _ 4 ^ 2 f _ 6 f _ { 1 2 } ^ 2 } { f _ 2 ^ 7 } \\right ) - 2 q f _ { 1 2 } ^ 3 ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} Q ( k , \\xi ) = \\omega ( k ) ( H _ \\mu ( \\xi - k ) - \\Sigma ( \\xi ) + \\omega ( k ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} V _ 0 ^ T ( g ) & = \\sum _ { k = 1 } ^ M V _ { t _ { k - 1 } } ^ { t _ k } ( g ) = \\sum _ { k = 1 } ^ M \\big | g ( t _ k ) - g ( t _ { k - 1 } ) \\big | \\\\ & = \\sum _ { k = 1 } ^ M \\big | f ( t _ k + h ) - f ( t _ k ) - f ( t _ { k - 1 } + h ) + f ( t _ { k - 1 } ) \\big | \\\\ & \\leq \\sum _ { k = 1 } ^ M \\Big ( \\big | f ( t _ k + h ) - f ( t _ k ) \\big | + \\big | f ( t _ { k - 1 } + h ) - f ( t _ { k - 1 } ) \\big | \\Big ) \\\\ & \\leq 2 M C | h | ^ \\mu \\leq 2 L C | h | ^ \\mu , \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} \\sum _ { j = m } ^ { \\infty } E ( r _ { j } ) \\leq C r _ { m } ^ { - \\beta } . \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\theta ^ { \\ddagger } ( q , y ) = \\theta ( q ^ 4 , - y ^ 2 / q ) + i q y \\theta ( q ^ 4 , - q y ^ 2 ) = \\theta ( \\rho ^ 4 , y ^ 2 / \\rho ) - i \\rho y \\theta ( \\rho ^ 4 , \\rho y ^ 2 ) ~ . \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} X _ 0 u = \\frac { 1 } { L } \\int _ 0 ^ L X _ 0 e ^ { s X _ 0 } \\varphi d s = \\frac { 1 } { L } \\int _ 0 ^ L \\frac { d } { d s } \\varphi _ s d s = \\frac { 1 } { L } ( \\varphi _ L - \\varphi ) ~ . \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} f _ \\phi ( t , x ) : = \\int _ t ^ \\infty ( \\phi ( t - r , x ) - \\phi ( t , x ) ) \\frac { - \\Gamma ( - \\beta ) ^ { - 1 } d r } { r ^ { 1 + \\beta } } , \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} ( \\alpha \\cdot 1 + a ) ( \\beta \\cdot 1 + b ) = ( \\alpha \\beta + f ( a , b ) ) \\cdot 1 + ( \\alpha b + \\beta a ) , \\alpha , \\beta \\in F , \\ a , b \\in V , \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} u ( t , x ) = \\mathbf E \\left [ \\phi _ 0 ( Y ^ { x } ( t ) ) \\right ] , \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\rho ( \\theta _ t , x ) + \\nabla \\cdot ( \\rho ( \\theta _ t , x ) v ( t , x ) ) = 0 , \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} \\tilde { X } _ { s t } ^ { \\natural } = X _ { s t } ^ { \\natural } + \\nabla \\beta _ j ( X _ s ) e _ i \\left ( \\int _ s ^ t \\delta \\tilde { B } _ { s r } ^ i d Z _ r ^ j - \\int _ s ^ t \\delta B _ { s r } ^ i d Z _ r ^ j \\right ) . \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} \\xi ^ { \\pm } _ i = Z ^ { \\pm } _ { B \\pm \\varepsilon _ B , \\omega \\pm \\varepsilon _ { \\omega } } ( E _ i ) = Z _ { B , \\omega } ( E _ i ) \\pm ( \\varepsilon _ B + i \\varepsilon _ { \\omega } ) \\cdot [ E _ i ] . \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} c i _ 1 + ( ( k a - 1 ) c - k ) i _ 2 - \\left \\lfloor \\frac { i _ 1 - i _ 2 } { a } \\right \\rfloor = ( a c - 1 ) j _ 1 + c j _ 2 . \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} M _ t & = Y _ t - Y _ 0 + A ^ { W , W } _ t ( \\nabla u ^ * \\ , b ) + A ^ { W , W } _ t ( \\tilde f ) \\\\ & = Y _ t - Y _ 0 + A ^ { W , W } _ t ( \\nabla u ^ * \\ , b + \\tilde f ) \\\\ & = Y _ t - Y _ 0 - A ^ { W , W } _ t ( - \\nabla u ^ * \\ , b - \\tilde f ) . \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} \\begin{cases} \\dfrac { \\partial u ( n , t ) } { \\partial t } = \\mathcal { J } ^ { ( \\alpha , \\beta ) } u ( n , t ) , \\\\ [ 4 p t ] u ( n , 0 ) = f ( n ) . \\end{cases} \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} & { \\prod _ j } ^ { ( 1 ) } = \\prod _ { j = 1 } ^ { N _ 1 } , ~ ~ { \\prod _ { i , j } } ^ { ( 1 ) } = \\prod _ { i , j = 1 } ^ { N _ 1 } , ~ ~ { \\prod _ { i < j } } ^ { ( 1 ) } = \\prod _ { 1 \\le i < j \\le N _ 1 } , \\\\ & { \\prod _ j } ^ { ( 2 ) } = \\prod _ { j = N _ 1 + 1 } ^ { N } , ~ ~ { \\prod _ { i , j } } ^ { ( 2 ) } = \\prod _ { i , j = N _ 1 + 1 } ^ { N } , ~ ~ { \\prod _ { i < j } } ^ { ( 2 ) } = \\prod _ { N _ 1 + 1 \\le i < j \\le N } , ~ ~ { \\prod _ { i , j } } ^ { ( 3 ) } = \\prod _ { i = 1 } ^ { N _ 1 } \\prod _ { j = N _ 1 + 1 } ^ N . \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} L ^ t _ { \\theta } ( u _ 1 [ f _ 1 ] \\cdots u _ n [ f _ n ] \\cdot v ) = u _ 1 [ f _ 1 ] \\cdots u _ n [ f _ n ] \\cdot L ^ t _ \\theta ( v ) + \\sum _ { i = 1 } ^ n \\left ( u _ 1 [ f _ 1 ] \\cdots u _ i [ \\theta ( f _ i ) ] \\cdots u _ n [ f _ n ] \\cdot v \\right ) \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} V ^ { p , q } : = \\bigoplus _ { \\substack { \\Psi \\in \\mathcal { P } _ q ^ { ( \\lambda , 1 ) } } } H ^ p ( \\Gamma _ 1 , V ^ { 1 , q } _ { \\Psi } ) . \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} \\alpha \\beta + \\beta \\alpha = \\begin{cases} - b / 4 , & a , c \\equiv 0 \\mod 4 , \\\\ \\beta - b / 4 , & a \\equiv 3 \\mod 4 c \\equiv 0 \\mod 4 , \\\\ \\alpha - b / 4 , & a \\equiv 0 \\mod 4 c \\equiv 3 \\mod 4 , \\\\ \\alpha + \\beta - ( 2 + b ) / 4 , & a , c \\equiv 3 \\mod 4 . \\end{cases} \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} H ^ 2 ( E , F \\otimes j ^ * ( \\mathcal O _ { \\mathbb P } ( K _ { \\mathbb P } + \\widetilde X ) ) ) = 0 . \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} \\overline { \\tilde { \\mathbf { S } } _ { X , Y } } = \\frac { \\dim ^ R ( Y ^ * \\otimes \\bar { \\ 1 } ) } { \\dim ^ R ( \\bar { Y } ) } \\tilde { \\mathbf { S } } _ { X , \\bar { Y } } , \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{gather*} f : \\overline { S } \\longrightarrow B ( 3 r ) \\\\ f ( \\partial _ 1 S ) \\subset \\partial B ( r ) , \\ \\ \\ f ( \\partial _ 2 S ) \\subset \\partial B ( 2 r ) \\\\ \\partial _ s f ( s , t ) + i \\partial _ t f ( s , t ) = 0 \\ \\ \\ \\textrm { o n } \\ \\ ( s , t ) \\in S \\end{gather*}"}
-{"id": "5049.png", "formula": "\\begin{align*} \\Phi _ { 2 } ( s ) = \\left \\lfloor \\mathrm { l o g } _ 2 \\left ( \\frac { 2 ^ N - 1 } { \\mathrm { g c d } \\left ( 2 ^ N - 1 , S ( 2 ) \\right ) } + 1 \\right ) \\right \\rfloor , \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { L } } { \\partial \\rho } = - \\mu \\ ; \\mathrm { t r } \\left ( \\frac { \\mathbf { H S H } ^ { \\rm H } } { \\sigma ^ 2 \\ , \\ln 2 } \\left ( \\mathbf { I } _ { N _ R } + \\left ( 1 - \\rho \\right ) \\sigma ^ { - 2 } \\mathbf { H } \\mathbf { S } \\mathbf { H } ^ { \\rm H } \\right ) ^ { - 1 } \\right ) + \\mathrm { t r } \\left ( \\mathbf { H S H } ^ { \\rm H } \\right ) = 0 , \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{align*} \\big [ \\varphi ( \\mathbb { A } / f \\mathbb { A } ) \\big ] _ { \\mathbb { A } } = \\det _ { \\mathbb { F } _ q ( z _ 1 , \\dots , z _ n ) [ X ] } ( X - \\varphi _ { \\theta } \\mid \\mathbb { A } / f \\mathbb { A } \\otimes _ { \\mathbb { F } _ q ( z _ 1 , \\dots , z _ n ) } \\mathbb { F } _ q ( z _ 1 , \\dots , z _ n ) [ X ] ) _ { | X = \\theta } . \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} \\forall s , t > 0 \\ ; \\ ; h _ { t } \\ast h _ { s } = h _ { t + s } , \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} \\int _ { Y \\times U } ( \\phi ( y ) - \\phi ( y _ 0 ) ) \\gamma ( d u , d y ) \\ = \\ \\lim _ { l \\rightarrow \\infty } \\int _ { Y \\times U } \\nabla \\phi ( y ) ^ T f ( u , y ) \\xi _ l ( d u , d y ) \\ \\forall \\ \\phi ( \\cdot ) \\in C ^ 1 \\} . \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} \\mathcal { K } _ { 2 , W } : = \\lbrace v _ i ( C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f \\ : \\ f \\in H \\rbrace \\subset ( \\bigoplus _ { i \\in I } H ) _ { l ^ { 2 } } . \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} T _ { i j } = \\sum _ { k } \\left [ \\left ( 1 - S | M | ^ 2 \\right ) ^ { - 1 } S \\right ] _ { i k } \\left ( | G _ { k j } | ^ 2 - | M _ k | ^ 2 T _ { k j } \\right ) . \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} \\mathrm { T r } ( B ^ j _ a ( \\omega _ a ^ b ) ) = \\mathrm { T r } ( B ^ j _ 1 ( 1 ) ) \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} \\lambda ( n , \\ddot { y } ) = \\dfrac { \\rho ( n y ) } { \\rho ( y ) } . \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} O ( \\epsilon ^ k ) & = \\left \\| \\sum _ { 1 \\le j \\le m / 2 } j \\tilde { P } ^ { m - j } \\tilde { P } ^ { ( 1 ) } \\tilde { P } ^ j - \\sum _ { 1 \\le j \\le m / 2 } j \\tilde { \\Pi } \\tilde { P } ^ { ( 1 ) } \\tilde { P } ^ j \\right \\| \\\\ & = \\left \\| \\sum _ { 1 \\le j \\le m / 2 } j \\tilde { \\Pi } \\tilde { P } ^ { ( 1 ) } \\tilde { P } ^ j - \\sum _ { 1 \\le j \\le m / 2 } j \\tilde { \\Pi } \\tilde { P } ^ { ( 1 ) } ( \\tilde { P } - \\tilde { \\Pi } ) ^ j \\right \\| \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} K = \\sup \\{ k : \\ \\exists w _ 1 , \\ldots , w _ k , \\ \\alpha ( w _ 1 , \\cdots , w _ k ) > 0 \\} . \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} | g ( x , u ) | & = | ( - b u - c \\rho ( x ) \\phi u + d | u | ^ { p - 1 } u | \\\\ & \\leq C ( 1 + | \\rho \\phi _ u | + | u | ^ { p - 1 } ) ( 1 + | u | ) \\\\ & \\coloneqq h ( x ) ( 1 + | u | ) . \\\\ \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\Big \\{ \\tilde { f } _ { i _ 1 } ^ { x _ 1 } ( b _ { a _ 1 \\varpi _ { i _ 1 } } \\otimes \\tilde { f } _ { i _ 2 } ^ { x _ 2 } ( b _ { a _ 2 \\varpi _ { i _ 2 } } \\otimes \\cdots \\otimes \\tilde { f } _ { i _ { r - 1 } } ^ { x _ { r - 1 } } ( b _ { a _ { r - 1 } \\varpi _ { i _ { r - 1 } } } \\otimes \\tilde { f } _ { i _ r } ^ { x _ r } ( b _ { a _ r \\varpi _ { i _ r } } ) ) \\cdots ) ) ~ \\Big | ~ \\\\ { x _ 1 , \\dots , x _ r \\in \\mathbb { Z } _ { \\ge 0 } } \\Big \\} \\setminus \\{ 0 \\} ; \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\psi ( a _ 1 a _ 2 ) = \\lambda ( \\sigma ( \\pi ( g _ 1 ) , \\pi ( g _ 2 ) ) ^ { - 1 } \\psi ( a _ 1 ) \\psi ( a _ 2 ) , \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} P ' _ i = S _ { p _ { 2 i - 1 } } p _ { 2 i - 1 } P _ { 2 i - 1 } p ' _ { 2 i - 1 } R p ' _ { 2 i } P _ { 2 i } p _ { 2 i } S _ { p _ { 2 i } } \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} \\norm { \\widetilde { T } _ V f ( \\cdot , \\zeta ) } _ { A ^ 2 ( V _ { \\zeta } ) } ^ 2 = \\int _ { V _ { \\zeta } } \\abs { \\widetilde { T } _ V f ( \\gamma , \\zeta ) } ^ 2 \\d V ( z ) & = \\int _ { \\R } \\abs { f ( t , \\zeta ) } ^ 2 \\omega _ { a ( \\zeta ) , b ( \\zeta ) } ( t ) \\d t = \\int _ { \\R } \\abs { f ( t , \\zeta ) } ^ 2 \\lambda ( p ( \\zeta ) , t ) \\d t , \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} \\psi _ \\chi ( s ) = \\frac { r _ 1 + r _ 2 - \\beta _ \\chi } { 2 } \\psi \\left ( \\frac { s } { 2 } \\right ) + \\frac { r _ 2 + \\beta _ \\chi } { 2 } \\psi \\left ( \\frac { s + 1 } { 2 } \\right ) - \\frac { n \\log \\pi } { 2 } . \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{align*} Q P Q ^ { - 1 } = \\begin{bmatrix} \\eta & 0 & 0 & 0 & 0 \\\\ 0 & \\eta ^ q & 0 & \\ddots & 0 \\\\ 0 & 0 & \\ddots & \\ddots & \\vdots \\\\ \\vdots & \\vdots & \\ddots & \\ddots & 0 \\\\ 0 & 0 & 0 & 0 & \\eta ^ { q ^ { d - 1 } } \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} F _ i ^ { ( k ) } ( x ) = x F _ { i - 1 } ^ { ( k ) } ( x ) - ( k - 1 ) F _ { i - 2 } ^ { ( k ) } ( x ) \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} | \\omega | _ { \\mathcal { O } } ^ { l i p } = \\sup _ { \\xi , \\zeta \\in \\mathcal { O } \\atop { \\xi \\neq \\zeta } } \\frac { | \\Delta _ { \\xi \\zeta } \\omega | } { | \\xi - \\zeta | } , | \\Omega | _ { - \\delta , \\mathcal { O } } ^ { l i p } = \\sup _ { \\xi , \\zeta \\in \\mathcal { O } \\atop { \\xi \\neq \\zeta } } \\sup _ { j \\in \\mathbb { Z } _ * } | j | ^ { - \\delta } \\frac { | \\Delta _ { \\xi \\zeta } \\Omega _ j | } { | \\xi - \\zeta | } \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\alpha } ( f , g ) ( x ) = \\sup _ { d > 0 } \\frac { 1 } { ( 2 d ) ^ { n - \\alpha } } \\int _ { | y | _ { \\infty } \\leq d } | f ( x - y ) g ( x + y ) | d y . \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} \\Re Z _ t ( G [ - 1 ] ) = - r + 1 - n + ( r - 1 ) ( g - 1 ) t ^ 2 \\le 0 . \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\chi ( u ) \\partial _ t u + \\sum _ { j = 1 } ^ 3 A _ j ^ { \\operatorname { c o } } \\partial _ j u + \\sigma ( u ) u & = f , & & x \\in G , & & t \\in J ; \\\\ B u & = g , & & x \\in \\partial G , & & t \\in J ; \\\\ u ( t _ 0 ) & = u _ 0 , & & x \\in G , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } p _ { n - 1 } \\\\ q _ { n - 1 } \\end{array} \\right ] & = \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ 1 \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ 2 \\end{array} \\right ] \\cdots \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c } 0 \\\\ 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} R ^ k ( 1 ) R ^ l ( 1 ) = \\frac { 1 } { k ! } ( R ( 1 ) ) ^ k \\frac { 1 } { l ! } ( R ( 1 ) ) ^ l = \\frac { 1 } { k ! l ! } ( R ( 1 ) ) ^ { k + l } = \\frac { ( k + l ) ! } { k ! l ! } R ^ { k + l } ( 1 ) \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} = y - \\tau ^ { 1 / 2 } x + \\tau y - \\tau ^ { 3 / 2 } x + \\frac { 1 } { 2 } \\tau ^ { 2 } y \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} 1 \\ge \\mathbb E [ Z _ { t } ] \\ge \\mathbb E \\Big [ \\frac { Z _ { t } } { \\ , 1 + \\varepsilon Z _ { t } \\ , } \\Big ] \\ , & = \\ , \\frac { 1 } { \\ , 1 + \\varepsilon \\ , } - \\mathbb E \\Big [ \\int ^ { t } _ { 0 } \\frac { \\varepsilon Z _ { s } ^ { 2 } \\lvert b ( s , X _ { s } , F _ { s } ) \\rvert ^ { 2 } } { \\ , ( 1 + \\varepsilon Z _ { s } ) ^ { 3 } \\ , } { \\mathrm d } s \\Big ] \\ , \\\\ \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} ( c _ { i j , k } ^ { p } - c _ { i j } ^ { q } c _ { q k } ^ { p } ) w _ { p s } = ( c _ { i j , s } ^ { p } - c _ { i j } ^ { q } c _ { q s } ^ { p } ) w _ { p k } , \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} \\begin{aligned} & ( 1 + | y | ) | \\nabla _ y \\phi ( y , \\tau ) | \\chi _ { B _ { 2 R } ( 0 ) } ( y ) + | \\phi ( y , \\tau ) | \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\lesssim \\tau ^ { - \\nu } ( 1 + | y | ) ^ { - a } \\| h \\| _ { 2 s + a , \\nu , \\eta } , \\tau \\in ( \\tau _ 0 , + \\infty ) , y \\in \\mathbb { R } ^ n , \\end{aligned} \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} X _ { k + 1 } = \\begin{cases} X _ k - D _ k & ~ ~ X _ k - D _ k \\ge s \\\\ S & ~ ~ X _ k - D _ k < S , \\end{cases} \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} ( n ' ! / k ' ! ) / ( n ! / k ! ) = \\frac { n ' ! } { n ! b ! } \\bigg / \\frac { k ' ! } { k ! b ! } = \\binom { n ' } { n } \\bigg / \\binom { k ' } { k } . \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} \\langle - \\Delta u ' ( t ) , u ( t ) \\rangle _ 2 = \\frac { 1 } { 2 } \\frac { d } { d t } | | u ( t ) | | _ { 1 , 2 } ^ 2 \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} \\langle R ^ N ( \\psi , \\psi ) \\psi , \\psi \\rangle = \\sum _ { \\alpha , \\beta , \\gamma , \\delta } R _ { \\alpha \\beta \\gamma \\delta } \\langle \\psi ^ \\alpha , \\psi ^ \\gamma \\rangle \\langle \\psi ^ \\beta , \\psi ^ \\delta \\rangle , \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + n \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} \\Gamma _ 4 = \\left ( \\begin{array} { c c c c } 2 & 1 & 1 & 1 \\\\ 1 & 2 & 1 & 1 \\\\ 1 & 1 & 2 & 0 \\\\ 1 & 1 & 0 & 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{gather*} T _ i ( h _ { i , 1 } ) = T _ i \\big ( \\big [ x _ { i } ^ + , x _ { i , 1 } ^ - \\big ] \\big ) = \\big [ { - } x _ { i } ^ - , - x _ { i , 1 } ^ + + \\tfrac { \\hbar } { 2 } \\big \\{ h _ i , x _ i ^ + \\big \\} \\big ] = - h _ { i , 1 } - \\hbar \\big \\{ x _ i ^ + , x _ i ^ - \\big \\} + \\hbar h _ i ^ 2 . \\end{gather*}"}
-{"id": "5529.png", "formula": "\\begin{align*} v ^ * ( y _ 0 ) = k ^ * ( y _ 0 ) \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} \\lvert ( i , 2 \\lvert m _ k \\rvert ) \\rvert _ { W _ n ( w ) } - \\lvert ( i , j ) \\rvert _ { W _ n ( w ) } = 2 \\lvert m _ k \\rvert - j > 0 . \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} \\frac { \\partial ^ { k } } { ( \\partial z ) ^ M } t ( z ) = \\sum _ { i = 1 } ^ \\infty \\beta _ i z ^ { M _ i \\setminus M } , \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} S _ { i _ 1 } : = \\bigcup _ { u _ 2 = 0 } ^ { s _ 2 - 1 } \\bigcup _ { q _ 2 = 0 } ^ { p _ { i _ 2 } - 1 } \\Big ( \\beta ^ { u _ 2 s _ 1 } \\alpha _ { i _ 2 } ^ { q _ 2 } W _ { i _ 1 } \\Big ) , S _ { i _ 2 } : = \\bigcup _ { u _ 1 = 0 } ^ { s _ 1 - 1 } \\bigcup _ { q _ 1 = 0 } ^ { p _ { i _ 1 } - 1 } \\Big ( \\beta ^ { u _ 1 } \\alpha _ { i _ 1 } ^ { q _ 1 } W _ { i _ 2 } \\Big ) , \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} \\Lambda _ { s } \\varphi : = - \\lim _ { t \\to 0 + } t ^ { 1 - 2 s } u ' ( t ) \\qquad \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} \\mathcal { F } \\big \\{ f ( t ) \\big \\} ( \\nu ) = \\widehat { f } ( \\nu ) = \\int _ { - \\infty } ^ { \\infty } f ( t ) e ^ { - i 2 \\pi \\nu t } \\ , \\mathrm { d } t \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} R _ { \\ell , S _ K } ( ^ \\lambda \\mathcal { V } ) = \\mu _ { \\ell } ^ K ( V _ { \\lambda } ) [ - w ( \\lambda ) ] \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} \\vert d e t D f \\vert = \\lambda \\vert T ' ( x ) \\vert \\end{align*}"}
-{"id": "9874.png", "formula": "\\begin{align*} \\boldsymbol { D } ^ { \\mu } u & : = \\dfrac { 1 } { \\Gamma ( \\sigma ) } \\dfrac { { \\rm d } ^ n } { { \\rm d } x ^ n } \\int _ { - \\infty } ^ { x } ( x - s ) ^ { \\sigma - 1 } u ( s ) \\ , { \\rm d } s , \\\\ \\boldsymbol { D } ^ { \\mu * } u & : = \\dfrac { ( - 1 ) ^ n } { \\Gamma ( \\sigma ) } \\dfrac { { \\rm d } ^ n } { { \\rm d } x ^ n } \\int _ { x } ^ { \\infty } ( s - x ) ^ { \\sigma - 1 } u ( s ) \\ , { \\rm d } s . \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\tau _ { \\ast } } \\int _ { \\Omega } \\nabla P ( x , t ) \\cdot \\partial _ { t } \\psi ( x , t ) + ( u ( x , t ) \\cdot \\nabla ) \\psi ( x , t ) \\cdot \\nabla P ( x , t ) + J ( \\nabla P ( x , t ) - x ) \\cdot \\psi ( x , t ) \\ , d x d t = 0 \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} [ L ^ { p , r _ 0 } , L ^ { p , r _ 1 } ] _ { \\theta , r } = L ^ { p , r } , \\frac 1 r = \\frac { 1 - \\theta } { r _ 0 } + \\frac { \\theta } { r _ 1 } . \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} A = V \\Sigma _ Q W ^ { * } , V = V _ u V _ Q , W = W _ u W _ Q , \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} x _ { n + 1 } = T ( x _ { n } ) \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} \\Delta g _ j ( k ) = g _ j ( k + 1 ) - g _ j ( k ) = \\begin{cases} - \\frac { \\mu _ j } { \\mu _ { k } } \\frac { 1 } { f ( k ) ( 1 + f ( k ) ) } , & j \\geq k + 1 , \\\\ \\frac { 1 } { 1 + f ( j ) } , & j = k , \\\\ 0 , & j \\leq k - 1 . \\end{cases} \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\tilde \\Pi ( \\tilde g ^ { ( 4 ) } _ 0 , 0 ) = ( g _ { S ^ 2 } , 2 , 0 , 0 ) , \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} D : = - \\lim _ { \\mathrm { S N R } \\rightarrow \\infty } \\frac { \\log P _ { o u t } } { \\log \\mathrm { S N R } } . \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} \\Delta ^ { - 1 } \\left ( [ d , a ] - [ b , c ] \\right ) & = \\Delta ^ { - 1 } [ d , b ] b ^ { - 1 } a + L _ { 1 2 } ( c - d b ^ { - 1 } a ) \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} I _ 1 \\xleftarrow { \\begin{pmatrix} f _ 1 \\end{pmatrix} } R _ 1 \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} \\left \\{ ( t _ 1 , \\ldots , t _ { i - 1 } , t _ { i + 1 } , \\ldots , t _ { n - 1 } , s , k ) ~ \\begin{tabular} { | l } $ h \\bar { h } = h _ i $ \\\\ $ g _ s \\in c ' ( h , Y ) , g _ k \\in c ' ( \\bar { h } , Y ) $ \\\\ $ [ t _ 1 , \\ldots , t _ { i - 1 } , s , k , t _ { i + 1 } , \\ldots , t _ { n - 1 } ] \\neq \\emptyset $ \\end{tabular} \\right \\} \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} S ( r ) = \\frac { f _ 1 ^ \\prime ( r ) } { f _ 1 ( r ) } . \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\frac { d } { d t } H _ \\nu ( \\rho ) = - J _ \\nu ( \\rho ) \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} - \\mathbf { i } \\partial _ { \\omega } F _ { i j } + \\bar { \\Omega } _ { i j } F _ { i j } + \\tilde { \\Omega } _ { i j } F _ { i j } = - \\mathbf { i } \\ R _ { i j } , \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} \\psi ( G ) ( s ) : = \\sum _ { ( s _ p ) } \\psi ( G ) ( s _ { p + 1 , r } ) \\circ G ( [ s _ p ' ] ) ( \\_ \\otimes s _ p '' ) \\circ \\psi ( G ) ( s _ { 1 , p - 1 } ) , \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} \\underset { \\mathbf { x } \\in \\mathbb { R } ^ { n p } , \\mathbf { z } \\in \\mathbb { R } ^ { m p } } { } \\ , \\ , f ( \\mathbf { x } ) \\ , \\mathbf { A x } + \\mathbf { B z } = \\mathbf { 0 } , \\end{align*}"}
-{"id": "9872.png", "formula": "\\begin{align*} \\mathcal { F } ( \\boldsymbol { D } ^ { - \\sigma } u ) = ( 2 \\pi i \\xi ) ^ { - \\sigma } \\mathcal { F } ( u ) \\mathcal { F } ( \\boldsymbol { D } ^ { - \\sigma * } u ) = ( - 2 \\pi i \\xi ) ^ { - \\sigma } \\mathcal { F } ( u ) , \\xi \\ne 0 , \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} y = \\xi ( x ) , \\varphi ( y ) = \\eta ( \\varphi ( x ) ) \\ , , \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} F _ n ( m ) = \\frac { 1 } { n + 1 } \\sum \\limits _ { j = 0 } ^ n ( - 1 ) ^ j \\binom { n + 1 } { j } B _ j m ^ { n + 1 - j } , \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} \\sum _ { i \\in I } v _ { i } ^ { 2 } \\Vert \\pi _ { u W _ { i } } C f \\Vert ^ { 2 } & \\leq \\Vert u ^ { - 1 } \\Vert ^ { 2 } \\sum _ { i \\in I } v _ { i } ^ { 2 } \\Vert \\pi _ { W _ { i } } u ^ { * } C f \\Vert ^ { 2 } \\\\ & = \\Vert u ^ { - 1 } \\Vert ^ 2 \\sum _ { i \\in I } v _ i ^ 2 \\Vert \\pi _ { W _ i } C u ^ { * } f \\Vert ^ 2 \\\\ & \\leq B \\Vert u ^ { - 1 } \\Vert ^ { 2 } \\Vert u \\Vert ^ { 2 } \\Vert f \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} d X _ t & = B ( t , X _ t , Y _ t , Z _ t , [ X _ t , Y _ t , Z _ t ] ) d t + \\sigma d W _ t \\\\ X _ 0 & = \\xi \\in L ^ 2 ( \\Omega , \\mathcal { F } _ 0 , \\mathbb { P } ; \\mathbb { R } ) , \\\\ d Y _ t & = - F ( t , X _ t , Y _ t , Z _ t , [ X _ t , Y _ t , Z _ t ] ) d t + Z _ t d W _ t \\\\ Y _ T & = G ( X _ T , [ X _ T ] ) . \\\\ \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\Phi ^ { \\pm } : { \\cal L } _ { V } ( \\Omega ) \\to L ^ 1 ( \\Omega ) , \\hbox { g i v e n b y } \\Phi ^ { \\pm } ( \\xi ) = V ^ { \\pm } \\varphi \\xi . \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} ( H _ \\mu ( \\xi + a , A ) - H _ \\mu ( \\xi , A ) ) \\psi = ( K ( \\xi + a - d \\Gamma _ A ( m ) ) - K ( \\xi - d \\Gamma _ A ( m ) ) ) \\psi \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} | P \\nabla f | ^ 2 \\le | \\Pi P \\nabla f | ^ 2 = | \\Pi \\nabla f | ^ 2 \\le \\frac { 1 } { c _ 1 } ( a \\Pi \\nabla f ) \\cdot ( \\Pi \\nabla f ) \\ , . \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} \\psi ( a + b ) & = \\big ( ( a / \\alpha ) \\oplus ( b / \\alpha ) , \\ , \\psi _ 2 ( a ) + _ o \\psi _ 2 ( b ) \\big ) \\\\ & = \\big ( \\psi _ 1 ( a ) \\oplus \\psi _ 1 ( b ) , \\ , \\psi _ 2 ( a ) + _ o \\psi _ 2 ( b ) \\big ) \\\\ & = \\big ( \\psi _ 1 ( a ) , \\psi _ 2 ( a ) \\big ) \\boxplus \\big ( \\psi _ 2 ( b ) , \\psi _ 2 ( b ) \\big ) \\\\ & = \\psi ( a ) \\boxplus \\psi ( b ) \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} Q \\overline { \\nabla } ^ { ' } _ { Y } X = \\{ ( 1 - \\beta ) g ( \\phi X , Y ) - \\alpha g ( X , Y ) \\} Q \\xi + B h ( \\phi X , Y ) . \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} A _ { 1 } : = \\sup _ { R > 0 } \\left ( \\int _ { \\{ | x | \\geq R \\} } \\phi _ { 1 } ( x ) d x \\right ) ^ { \\frac { 1 } { q } } \\left ( \\int _ { \\{ | x | \\leq R \\} } ( \\psi _ { 1 } ( x ) ) ^ { - ( p ' - 1 ) } d x \\right ) ^ { \\frac { 1 } { p ' } } < \\infty \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } U ( m ) + F _ { i } ( D ^ { + } U ( m ) , D ^ { - } U ( m ) ) = f _ { i } ( - m \\Delta x ) & \\ , \\ , m \\in J _ { i } \\setminus \\{ 0 \\} \\\\ U ( 0 ) = \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } U ( 1 _ { i } ) \\end{array} \\right . \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\int _ N f ( x ) \\cdot 1 _ { A \\cap q ( N ) } ( x \\cdot \\ddot { y } ) d \\omega ( x ) = 0 . \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} \\int _ K f ( k ) d \\lambda _ n ( k ) = c \\int _ K f ( k ) d k . \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} p _ k ^ { ( \\rm i d ) } = \\begin{cases} \\frac { \\mu _ 2 ^ * } { \\left ( \\nu _ 2 ^ * - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 \\right ) \\ln 2 } - \\frac { \\sigma ^ 2 } { [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } , & \\\\ 0 , & \\end{cases} . \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} \\Phi ( M _ 1 , \\ldots , M _ m ) = \\log \\det \\left ( Q + \\sum _ { k = 1 } ^ m c _ k B _ k ^ \\ast M _ k B _ k \\right ) - \\sum _ { k = 1 } ^ m c _ k \\log \\det M _ k . \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} q _ { 0 ^ i 1 0 ^ j 1 } ( 1 ) = \\frac { a _ { j i } } { 1 - \\sum _ { k = 0 } ^ { i - 1 } ( 1 - a _ { j k } ) } , \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ j ( - 1 ) ^ { n j } \\binom k j L _ n { } ^ j G _ { n j } } = ( - 1 ) ^ k ( - 1 ) ^ { n k } G _ { 2 n k } \\ , . \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} V \\in L ^ p _ { l o c } ( \\Omega ) \\hbox { w i t h } \\left \\{ \\begin{array} { l l } p > N / 2 \\ , & \\hbox { f o r } N \\geq 2 , \\\\ p = 1 \\ , & \\hbox { f o r } N = 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} L _ { u } ( u + 1 ) = & u \\left ( u + 1 \\right ) ^ { q ^ 2 } + ( u + 1 ) \\left ( u + 1 \\right ) ^ q + ( u ^ 2 + u ) ( u + 1 ) \\\\ = & u \\left ( u + 1 \\right ) ^ { 4 } + ( u + 1 ) \\left ( u + 1 \\right ) ^ 2 + ( u ^ 2 + u ) ( u + 1 ) \\\\ = & ( u + 1 ) ^ 2 \\left ( u ^ 3 + u + u + 1 + u \\right ) \\\\ = & 0 . \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} p ' ( w ) = \\begin{cases} p ( w ) + \\epsilon ( 0 , - 1 ) & w \\sim y , \\\\ p ( w ) & \\end{cases} \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} \\vec { x } = ( x _ 1 , \\ldots , x _ a ) , \\ \\vec { y } = ( y _ 1 , \\ldots , y _ { b } ) , \\ \\vec { z } = ( z _ 1 , \\ldots , z _ c ) \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} L _ A = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { x _ i ( A ) } \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{gather*} \\Re F _ 1 ( z ) = - \\frac \\pi { \\sin \\pi r } \\Re F _ 0 ( 1 - z ) \\end{gather*}"}
-{"id": "8203.png", "formula": "\\begin{align*} F ^ { e } ( \\Delta _ * \\mathcal { K } _ X ^ { - 1 } ) = \\Theta _ { A } [ d ] . \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} u _ { k + 1 } = S \\left ( u _ k \\ , A ^ * \\frac { y } { A u _ k } \\right ) , k = 0 , \\dots , \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} ( S _ { + + } ) _ { p q } ( x ) = 0 \\mbox { f o r } 1 \\le q \\le p \\le m . \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} U _ { ( n - 7 ) / 3 , j } & = \\left \\{ g + n - 2 , \\frac { 1 } { 2 } ( 2 g - 3 j + 2 n - 7 ) , \\frac { 1 } { 2 } ( 2 g + 3 j - 4 n + 5 ) \\right \\} _ { g - 1 } \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} t : = \\frac { 2 7 | A | C } { B ^ 2 } > 0 \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} F [ Q ] : = \\inf _ { C \\in \\R ^ N _ + } \\widetilde E [ C , Q ] , \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} { R } _ k ^ i [ n _ 0 ] ( \\mathbf { p } , \\mathbf { s } , \\mathbf { r } ) = s _ { k } ^ i [ n _ 0 ] \\mathcal { B } \\log _ 2 \\Big ( 1 + \\frac { H _ k ^ i [ n _ 0 ] p _ k ^ i [ n _ 0 ] } { \\norm { \\mathbf { r } [ n _ 0 ] - \\mathbf { r } _ k } ^ 2 } \\Big ) . \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\gamma / 2 } Q - \\frac { 1 } { | \\cdot | ^ \\gamma } * | Q | ^ 2 Q = 0 . \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{gather*} v _ B ^ { - 1 } \\triangleright \\chi ^ + _ 1 = \\big ( \\varphi _ { v ^ { - 1 } } \\big ( \\chi ^ + _ 1 \\big ) ^ v \\big ) ^ { v ^ { - 1 } } = \\mu ^ l ( v ) ^ { - 1 } \\mu ^ l \\big ( K ^ { p - 1 } v ? \\big ) ^ { v ^ { - 1 } } = \\mu ^ l ( v ) ^ { - 1 } \\mu ^ r \\big ( K ^ { p + 1 } ? \\big ) . \\end{gather*}"}
-{"id": "848.png", "formula": "\\begin{align*} \\left \\langle f , | x - y | ^ { - 1 } f \\right \\rangle & \\leq \\lim _ { n \\to \\infty } \\left \\langle f _ n , | x - y | ^ { - 1 } f _ n \\right \\rangle \\\\ & \\leq 2 \\lim _ { n \\to \\infty } \\langle f _ n , ( - \\Delta _ { x } ) ^ { \\frac { 1 } { 4 } } ( - \\Delta _ { y } ) ^ { \\frac { 1 } { 4 } } f _ n \\rangle = 2 \\langle f , ( - \\Delta _ { x } ) ^ { \\frac { 1 } { 4 } } ( - \\Delta _ { y } ) ^ { \\frac { 1 } { 4 } } f \\rangle . \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ E = \\sum _ { i = 1 } ^ d \\| | D _ { q _ i } | u \\| _ { L ^ 2 ( \\mathbb { R } ^ { 2 d } ) } ^ 2 + \\| | \\partial _ { q _ { i } } V ( q _ { i } ) | u \\| _ { L ^ 2 ( \\mathbb { R } ^ { 2 d } ) } ^ { 2 } + \\| u \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ { 2 d } ) } ~ . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} \\kappa _ { v \\lambda } \\kappa _ { w \\mu } = \\sum _ { \\nu \\in P } a _ { v , w , \\lambda , \\mu } ^ { \\nu } \\kappa _ { \\nu } . \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} \\mathbb { T ' } _ R & = \\begin{pmatrix} ( d b ^ { - 1 } a - c ) ^ { - 1 } [ d , c ] & ( d b ^ { - 1 } a - c ) ^ { - 1 } \\left ( [ a , d ] + [ c , b ] + d b ^ { - 1 } [ b , a ] \\right ) \\\\ ( d - c a ^ { - 1 } b ) ^ { - 1 } [ c , d ] & ( c a ^ { - 1 } b - d ) ^ { - 1 } \\left ( [ a , d ] + [ c , b ] + c a ^ { - 1 } [ b , a ] \\right ) \\end{pmatrix} ( \\Delta ' ) ^ { - 1 } . \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} f _ n ( t ) = f _ n ( 0 ^ + ) + f _ n ' ( + \\infty ) t - h _ n ( t ) , t \\in ( 0 , + \\infty ) . \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{gather*} \\frac { L ( E _ z , 1 ) } { \\Re F _ 1 ( z ) } = - \\frac { L ( E _ z , 1 ) } { \\pi \\ , \\Re F _ 0 ( 1 - z ) } \\in \\mathbb Q . \\end{gather*}"}
-{"id": "5867.png", "formula": "\\begin{align*} \\nu D _ 2 f _ n = 4 \\mu _ n D _ 2 \\cos . \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} \\zeta _ { } = \\left ( \\zeta _ 0 , \\zeta _ 0 , \\zeta _ 0 , \\cdots \\right ) , \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} \\int f ( x ) ( \\alpha \\cdot x ) d \\mu ( x ) - | \\alpha | ^ 2 \\sigma ^ 2 & = \\int ( f ( x ) - \\alpha \\cdot x ) ( \\alpha \\cdot x ) d \\mu ( x ) \\\\ & \\leq \\left ( \\int | f ( x ) - \\alpha \\cdot x | ^ 2 d \\mu ( x ) \\right ) ^ { 1 / 2 } \\left ( \\int | \\alpha \\cdot x | ^ 2 d \\mu ( x ) \\right ) ^ { 1 / 2 } \\\\ & \\leq | \\alpha | \\sigma \\left ( C _ p \\int | \\nabla f - \\alpha | ^ 2 d \\mu \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} \\chi ( L _ \\xi ( \\lambda , \\mu ) ) = \\frac { \\sum _ { w \\in \\mathfrak { S } _ { n + 1 } } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle \\lambda + 2 \\rho , ( w \\bullet ( - 2 \\rho , 2 \\rho ) ) _ 2 \\rangle + \\langle ( w \\bullet ( - 2 \\rho , 2 \\rho ) ) _ 1 + 2 \\rho , \\mu \\rangle } } { \\xi ^ { \\langle 2 \\rho , 2 \\rho \\rangle } \\sum _ { w \\in \\mathfrak { S } _ { n + 1 } } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho , w \\bullet 0 \\rangle } } . \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} D _ { n } [ ( - z ) ^ { \\beta } \\tau ] n ^ { - \\beta ^ { 2 } } = O ( 1 ) \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} ( a _ 1 , \\ldots , a _ k ; q ) _ n = ( a _ 1 ; q ) _ n \\cdots ( a _ k ; q ) _ n , ( a _ 1 , \\ldots , a _ k ; q ) _ { \\infty } = ( a _ 1 ; q ) _ { \\infty } \\cdots ( a _ k ; q ) _ { \\infty } . \\end{align*}"}
-{"id": "9905.png", "formula": "\\begin{align*} H ^ { q } _ { \\{ 0 \\} } ( \\R ^ { l } ; \\Z _ { \\R ^ { l } } ) \\simeq \\begin{cases} \\Z & q = l , \\\\ 0 & q \\ne l , \\end{cases} \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} \\Phi _ n ( t ) = n \\int _ 0 ^ t \\sinh ( s ) ^ { n - 1 } d s , t \\geq 0 . \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} U _ \\lambda ( o , \\ , o \\ , | \\ , z ) = \\frac { ( d - 1 + \\lambda ) - \\sqrt { ( d - 1 + \\lambda ) ^ 2 - 4 \\lambda ( d - 1 ) z ^ 2 } } { 2 ( d - 1 ) } . \\end{align*}"}
-{"id": "9998.png", "formula": "\\begin{align*} \\bar F _ t ( q ) = ( 1 - \\bar P ( q ) ) \\int _ t ^ \\infty e ^ { q ( \\tau ( t ) - \\tau ( s ) ) } d s . \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { ( - 1 ) ^ { n ( j + s ) + s } \\binom k j \\binom j s \\frac { { G _ { m - 2 k - ( n - 1 ) j + 2 n s } } } { { F _ n ^ j } } } } = ( - 1 ) ^ k G _ m , n \\ne 0 \\ , . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta \\tilde H ^ \\pm & = 0 , \\mbox { i n } \\ \\Omega ^ \\pm , \\\\ \\tilde H ^ \\pm & = 0 , \\mbox { o n } \\ \\partial \\Omega ^ \\pm \\cap \\partial \\Omega , \\\\ \\tilde H ^ \\pm & = 2 \\log | \\partial _ n H _ \\gamma ^ \\pm | - \\log h _ \\gamma , \\mbox { o n } \\ \\gamma . \\end{aligned} \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} & \\mu \\big \\{ z \\in X : | S _ { \\alpha } f ( z ) | > 2 \\tau \\big \\} \\\\ & \\leq \\mu \\big \\{ z \\in X : | S _ { \\alpha } ^ 1 f ( z ) | > \\tau \\big \\} + \\mu \\big \\{ z \\in X : | S _ { \\alpha } ^ 2 f ( z ) | > \\tau \\big \\} . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} A _ k ^ { - 1 } - B _ k A ^ { - 1 } B _ k ^ \\ast = 0 \\quad \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\rho ( m _ t ^ { - 1 } ) R _ j & = \\epsilon ( [ g _ { \\alpha } g _ t ^ { - 1 } m ^ { - 1 } g _ t g _ j ^ { - 1 } ] ) \\\\ & = \\epsilon ( [ g _ { \\alpha } g _ t ^ { - 1 } m m ^ { - 1 } g _ t g _ j ^ { - 1 } ] ) + \\epsilon ( [ g _ { \\alpha } g _ t ^ { - 1 } ] ) \\epsilon ( [ m ^ { - 1 } g _ t g _ j ^ { - 1 } ] ) \\\\ & = R _ j + \\epsilon ( [ m ^ { - 1 } g _ t g _ j ^ { - 1 } ] ) R _ t . \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} L ( s , \\pi _ p ) = L ( s , \\chi _ 1 ) L ( s , \\chi _ 2 ) \\epsilon ( \\frac { 1 } { 2 } , \\pi _ p ) = \\epsilon ( \\frac { 1 } { 2 } , \\chi _ 1 ) \\epsilon ( \\frac { 1 } { 2 } , \\chi _ 2 ) . \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 \\setminus B _ R } | u _ n | ^ 3 = \\int _ { \\R ^ 3 \\setminus B _ R } \\frac { \\rho } { \\rho } | u _ n | ^ 3 < \\epsilon \\int _ { \\R ^ 3 \\setminus B _ R } \\rho | u _ n | ^ 3 < \\epsilon C , \\\\ \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} ( X + a ) u = - f , u | _ { \\partial _ - ( S M ) } = 0 . \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} \\varphi _ k ( q ) : = \\theta ( q , - q ^ { k - 1 } ) = \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ j q ^ { A _ j } ~ , ~ ~ ~ A _ j : = k j + j ( j - 1 ) / 2 ~ , ~ ~ ~ k \\in \\mathbb { R } ~ . \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} L _ 1 & = f ( 1 ) = ( a _ 0 + a _ 4 ) + ( a _ 1 + a _ 5 ) + ( a _ 2 + a _ 6 ) + ( a _ 3 + a _ 7 ) , \\\\ L _ 2 & = f ( - 1 ) = ( a _ 0 + a _ 4 ) - ( a _ 1 + a _ 5 ) + ( a _ 2 + a _ 6 ) - ( a _ 3 + a _ 7 ) , \\\\ H & = | f ( i ) | ^ 2 = ( ( a _ 0 + a _ 4 ) - ( a _ 2 + a _ 6 ) ) ^ 2 + ( ( a _ 1 + a _ 5 ) - ( a _ 3 + a _ 7 ) ) ^ 2 , \\\\ K & = | f ( w _ 8 ) f ( - w _ 8 ) | ^ 2 = ( ( a _ 0 - a _ 4 ) ^ 2 - ( a _ 2 - a _ 6 ) ^ 2 + 2 ( a _ 1 - a _ 5 ) ( a _ 3 - a _ 7 ) ) ^ 2 \\\\ & \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; + ( ( a _ 3 - a _ 7 ) ^ 2 - ( a _ 1 - a _ 5 ) ^ 2 + 2 ( a _ 0 - a _ 4 ) ( a _ 2 - a _ 6 ) ) ^ 2 . \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} { \\bf w } _ t ^ { ( i ) } = { \\bf w } _ { \\Delta } ^ { ( i ) } \\frac { t } { \\Delta } + \\frac { 1 } { 2 } a _ { i , 0 } + \\sum _ { r = 1 } ^ { \\infty } \\left ( a _ { i , r } { \\rm c o s } \\frac { 2 \\pi r t } { \\Delta } + b _ { i , r } { \\rm s i n } \\frac { 2 \\pi r t } { \\Delta } \\right ) , \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} \\hat { z } ( x ) = \\frac { x - \\hat { x } - \\kappa } { ( 1 + \\lambda ) } . \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} { \\rm R e } \\left \\{ 1 + z \\frac { T _ g '' ( z ) } { T _ g ' ( z ) } \\right \\} & = { \\rm R e } \\left \\{ z \\frac { f ' ( z ) } { f ( z ) } + z \\frac { g '' ( z ) } { g ' ( z ) } + 1 \\right \\} \\\\ & > \\frac { ( 1 - \\alpha ) r ^ 2 - k r + 1 + \\alpha } { 1 - r ^ 2 } = \\frac { \\lambda ( r ) } { 1 - r ^ 2 } \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\R ^ 3 } ( | \\nabla u _ n | ^ 2 + u _ n ^ 2 ) + \\frac { 1 } { 4 } \\int _ { \\R ^ 3 } \\rho \\phi _ { u _ n } u _ n ^ 2 - \\frac { \\mu _ n } { p + 1 } \\int _ { \\R ^ 3 } u _ n ^ { p + 1 } = c _ { \\mu _ n } , \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} \\forall n \\neg \\neg \\exists k ( a ^ * _ n ( k ) = 1 ) \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} 0 > \\chi ( E _ i ) = \\chi ( E _ 0 , E _ i ) = - a _ i + b _ i . \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\exists \\delta > 0 \\forall w \\in H ^ 1 _ { r a d } ( \\mathbb R ^ 3 ) \\norm { w } \\leq \\delta : \\gamma _ { \\pm } ( w ) = I ' ( w ) [ w ^ { \\pm } ] \\geq \\frac { 1 } { 4 } \\norm { w ^ { \\pm } } ^ 2 . \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } x b ^ { m _ k } \\sim \\phi b ^ { m _ 0 } x b ^ { 1 } - \\sum _ { i = 1 } ^ { m _ k - 1 } \\sum _ { s \\in S _ b } \\phi b ^ { m _ 0 } x b ^ i s . \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} D _ t f ( \\zeta ) = t ^ { 1 / 2 \\mu } \\cdot f ( \\widehat { \\rho } _ t ( \\zeta ) ) = t ^ { 1 / 2 \\mu } \\cdot t ^ { - 1 / 2 \\mu } F \\left ( \\widehat { \\rho } _ { 1 / t } ( \\widehat { \\rho } _ t ( \\zeta ) ) \\right ) = F ( \\zeta ) . \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} | f ( x , t ) | \\leq M \\sum _ { j = 1 } ^ k \\frac { \\mu _ j ^ { - 2 s } t ^ { - \\beta } } { 1 + | y _ j | ^ { 2 s + \\alpha } } , y _ j = \\frac { | x - \\xi _ j | } { \\mu _ j } \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} \\widetilde { F } _ { \\eta , \\widetilde { m } } ( v _ g , \\omega ) : & = W ( \\omega ^ { - 1 } P _ { \\widetilde { m } } v _ g , 1 ) F _ { \\eta } ( v _ g , \\omega ) W ( \\omega ^ { - 1 } P _ { \\widetilde { m } } v _ g , 1 ) ^ * + \\lVert \\omega ^ { - 1 / 2 } P _ { \\widetilde { m } } v _ g \\lVert ^ 2 \\\\ & = \\eta W ( 2 \\omega ^ { - 1 } P _ { \\widetilde { m } } v _ g , - 1 ) + d \\Gamma ( \\omega ) + \\varphi ( \\overline { P } _ { \\widetilde { m } } v _ g ) \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} Y _ t = & \\hat \\Phi ( W _ T ) + \\int _ t ^ T \\hat f \\left ( r , W _ r , Y _ r , \\frac { [ Y , W ] _ r } { \\mathrm d r } \\right ) \\mathrm d r \\\\ & + A _ { T } ^ { W , Y } ( \\hat b ) - A _ { t } ^ { W , Y } ( \\hat b ) - ( M _ T - M _ t ) , \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} & \\frac { \\epsilon ^ 2 } { 2 } e ^ { - \\alpha } \\mu \\sum _ { k = 0 } ^ { \\infty } k ^ 2 ( e ^ { - \\alpha } \\tilde { P } ) ^ k \\tilde { P } ^ { ( 2 ) } ( I - e ^ { - \\alpha } \\tilde { P } ) ^ { - 1 } r \\\\ = \\ , & \\frac { \\epsilon ^ 2 } { 2 } e ^ { - 2 \\alpha } \\mu \\tilde P \\left [ 2 e ^ { - \\alpha } \\tilde P ( I - e ^ { - \\alpha } \\tilde P ) ^ { - 3 } + ( I - e ^ { - \\alpha } \\tilde P ) ^ { - 2 } \\right ] \\tilde P ^ { ( 2 ) } ( I - e ^ { - \\alpha } \\tilde P ) ^ { - 1 } r . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} \\xi = ( \\xi _ 1 , \\xi _ 2 , \\cdots , \\xi _ m ) ^ T \\ , , \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} \\ , { \\mathrm m } _ { t } \\ , = \\ , ( X _ { t } ) \\ , \\equiv \\ , ( \\widetilde { X } _ { t } ) \\ , \\equiv \\ , ( Y _ { t } ) \\ , = \\ , \\widehat { \\mathrm m } _ { t } \\ , ; t \\ge 0 \\ , . \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\phi ( | \\nabla \\Q _ { 0 } | ) ~ d V = \\lim _ { k \\to + \\infty } \\int _ { \\Omega } \\phi ( | \\nabla \\Q _ { L _ k } | ) ~ d V . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} V _ \\lambda ( \\mathcal R z ) = \\mathcal R V _ \\lambda ( z ) . \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} \\begin{cases} \\dfrac { \\partial u ( n , t ) } { \\partial t } = \\mathcal { J } ^ { ( \\alpha , \\beta ) } u ( n , t ) , \\\\ [ 4 p t ] u ( n , 0 ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\beta ( y , t ) = 0 \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} ( M f ) ( \\tau , y ) : = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { \\mathbb { R } } e ^ { - i s \\tau } ( F f ) ( s , y ) d s , \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} \\sum _ { y \\in ( - \\epsilon t , u t ) } c ^ \\omega _ { - \\epsilon t } ( y ) & \\le \\sum _ { x \\in ( - \\epsilon t , u t ] } \\sum _ { y \\in ( - \\infty , x ) } G _ { ( - \\infty , x ) } ^ { \\omega } ( x - 1 , y ) \\\\ & = E ^ \\omega _ { - \\epsilon t } [ H ( u t ) ] . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} & R i c _ { g ^ { ( 4 ) } } + \\delta ^ * _ { g ^ { ( 4 ) } } \\beta _ { \\tilde g ^ { ( 4 ) } } g ^ { ( 4 ) } = 0 \\quad M , \\\\ & \\begin{cases} g _ { \\partial M } = \\gamma \\\\ H _ { \\partial M } = H \\\\ t r _ { \\partial M } K = k \\\\ \\omega _ { \\mathbf n } = \\tau \\\\ \\beta _ { \\tilde g ^ { ( 4 ) } } g ^ { ( 4 ) } = 0 . \\end{cases} \\quad \\quad \\partial M \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} I _ { \\alpha } f ( x ) : = \\int _ { \\mathbb { R } ^ { n } } \\frac { f ( y ) } { | x - y | ^ { n - \\alpha } } d y . \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} v _ j : = P _ { \\theta } [ \\phi ] ( \\widehat { u } _ j ) : = \\left ( \\lim _ { C \\to + \\infty } P _ { \\theta } ( \\phi + C , \\widehat { u } _ j ) \\right ) ^ * . \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} f \\sim g _ 2 = \\sum _ { v \\in I \\setminus \\{ v _ 2 \\} } \\alpha ( v ) \\phi v + \\alpha ( v _ 2 ) \\beta ( w _ 2 ) \\phi w _ 2 + \\sum _ { y \\in J _ 2 } \\alpha ( v _ 2 ) \\beta _ 2 ( y ) \\phi y . \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} L _ { \\mathbb { S } ^ { d - 1 } } m _ j ( x ) & = - \\lambda _ j m _ j ( x ) , x \\in \\Sigma \\\\ m _ j ( x ) & = 0 , x \\in \\partial \\Sigma . \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} & L _ j = t _ j \\ ( 0 \\le j \\le k ) , L _ { k + j } = t _ 0 + t _ 1 x _ { 1 j } + \\cdots + t _ k x _ { k j } \\ ( 1 \\le j \\le n ) , \\\\ & L _ { k + n + 1 } = t _ 0 + t _ 1 + \\cdots + t _ k . \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} A = \\frac { \\cos 2 v - \\cos 2 w } { 1 - \\cos 2 v } \\cdot \\frac { ( k - 2 ) ^ 2 } { ( k - 2 ) ^ 2 + 2 ( k - 1 ) ( 1 - \\cos 2 w ) } \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} h = \\sum _ { j = - 1 } ^ \\infty \\sum _ { m \\in \\mathbb Z } \\mu _ { j , m } 2 ^ { - j ( s - \\frac 1 r ) } h _ { j , m } \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\mathcal { Q } ( x ) = c _ 0 \\mathcal { Q } _ + ( B _ 0 x ) + c _ { m + 1 } \\mathcal { Q } _ - ( B _ { m + 1 } x ) \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} F ( z , \\ , U ) : = \\frac { 1 } { 2 m } \\sum _ { i = 1 } ^ r \\left \\{ - ( \\phi _ i ( z ) - m U ) + [ ( \\phi _ i ( z ) - m U ) ^ 2 + 4 \\lambda m _ i z ^ 2 ] ^ { 1 / 2 } \\right \\} . \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} \\# \\{ \\gamma \\in \\Gamma \\colon d ( \\widetilde { x _ 0 } , \\gamma \\widetilde { x _ 0 } ) = 2 n \\} = \\left \\{ \\begin{array} { c l } \\displaystyle 0 & \\textrm { i f } N ( x _ n ) = N ( x _ { n - 1 } ) \\\\ \\displaystyle \\frac { q - 1 } { q } N ( x _ n ) | \\Gamma _ { x _ 0 } | & \\textrm { i f } N ( x _ n ) = q N ( x _ { n - 1 } ) \\end{array} \\right . . \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} P _ { 1 / 2 , 0 } ( z , s ) = y ^ { s - 1 / 4 } + \\sum _ { n \\equiv 0 , 1 ( 4 ) } b _ { 1 / 2 , 0 } ( n , s ) \\mathcal { W } _ { 1 / 2 , n } ( y , s ) e ^ { 2 \\pi i n x } . \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} \\lvert ( 0 , n ) \\rvert _ { W _ n ( w ) } = \\lvert l _ 0 \\rvert _ S + \\lvert ( 0 , n ) \\rvert _ { W _ n ( w _ k ) } = \\lvert m _ 0 \\rvert + \\lVert ( \\phi w _ k ) _ n \\rVert _ S . \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} X = \\partial _ x + \\sum _ { j = 1 } ^ { n } p _ { X , j } ( x , y , u _ 1 , \\dots , u _ n ) \\partial _ { u _ j } \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} \\det d \\theta ( x ) \\le D \\prod _ { k = 1 } ^ m \\big ( \\det d T _ k ( B _ k ) \\big ) ^ { c _ k } \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} X _ { k } \\ , = \\ , a X _ { k - 1 } + ( 1 - a ) ( u \\widetilde { X } _ { k - 1 } + ( 1 - u ) \\mathbb E [ X _ { k - 1 } ] ) + \\varepsilon _ { k } \\ , ; k \\ , = \\ , 1 , 2 , \\ldots \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} f ( y ) - g ( y ) < f ( x ) - g ( x ) = \\inf ( f - g ) \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} c _ 0 : = \\max _ { \\partial Q } I \\circ h < c . \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} j _ \\lambda \\left ( \\begin{pmatrix} a & v \\\\ w ^ t & d \\end{pmatrix} , z \\right ) = ( w ^ t z + d ) ^ { - \\lambda } . \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} p ( y ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( 1 + \\tau \\left ( \\frac { 1 } { 3 } y ^ { 4 } + \\frac { 3 } { 2 } y ^ { 2 } - \\frac { 7 } { 1 6 } \\right ) \\right ) e ^ { - 2 y ^ { 2 } } + O ( \\tau ^ { 2 } ) . \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} \\int _ D \\nabla \\Pi _ { k , D } ^ \\Delta \\xi \\ { \\rm d } x = \\int _ D \\nabla \\xi \\ { \\rm d } x , \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} \\iota _ { \\Delta } ( a + b \\sqrt { 4 ^ { - 1 } \\Delta } ) & = \\left [ \\begin{smallmatrix} a & b \\\\ 4 ^ { - 1 } \\Delta b & a \\end{smallmatrix} \\right ] , a , b \\in \\Q . \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} \\inf ( \\sigma ( A ) ) = - \\log ( \\lVert \\exp ( - A ) \\lVert ) = \\lim _ { n \\rightarrow \\infty } - \\log ( \\lVert \\exp ( - A _ n ) \\lVert ) = \\lim _ { n \\rightarrow \\infty } \\inf ( \\sigma ( A _ n ) ) \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} 4 Q _ { a , b } ^ 2 = 4 Q _ { a b } + 2 Q _ { a ^ 2 , b ^ 2 } - Q _ a Q _ b - Q _ b Q _ a \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ 2 } { d x ^ 2 } + q ( x ) - \\frac { ( n - 1 ) ^ 2 } { 4 } \\right ) w _ { \\lambda _ j } ( x ) = \\lambda w _ { \\lambda _ j } ( x ) , \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} M = \\bigoplus _ { ( \\lambda , \\mu ) \\in P \\times P } M _ { \\lambda , \\mu } . \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} P h ( x ) : = \\sum _ { y \\sim x } p ( x , \\ , y ) h ( y ) , x \\in V ( G ) . \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} & \\left \\| \\sum _ { j = 0 } ^ { \\infty } \\mu \\left ( \\prod _ { k = 1 } ^ { j } B _ k ( \\epsilon ) \\right ) r _ { j + 1 } ( \\epsilon ) - \\sum _ { j = 0 } ^ { \\infty } \\mu \\tilde { B } ^ j \\tilde { r } - \\epsilon \\sum _ { j = 0 } ^ { \\infty } \\sum _ { i = 1 } ^ { j - 1 } i \\mu \\tilde { B } ^ i \\tilde { B } ^ { ( 1 ) } \\tilde { B } ^ { j - i } \\tilde { r } - \\epsilon \\sum _ { j = 0 } ^ { \\infty } j \\mu \\tilde { B } ^ j \\tilde { r } ^ { ( 1 ) } \\right \\| \\\\ & \\quad = O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ w ) \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} \\dim _ { F _ i } S _ i = p _ i , S _ i + S _ i \\alpha _ i + \\dots + S _ i \\alpha _ i ^ { s - 1 } = \\mathbb { K } , \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} \\frac { \\tilde { w } _ { m i n } } { 4 } m _ 1 ( n ) = \\tilde { w } _ { m i n } \\leq m _ 2 ( n ) \\leq 5 \\tilde { w } _ { m a x } = \\frac { 5 \\tilde { w } _ { m a x } } { 4 } m _ 1 ( n ) . \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\rho ( \\theta _ t , x ) + \\nabla \\cdot ( \\rho ( \\theta _ t , x ) v ( t , x ) ) = 0 , \\quad \\textrm { $ \\theta _ 0 , \\theta _ 1 $ a r e f i x e d } . \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} u = \\frac { 1 } { \\sqrt { n } } ( 1 , \\dotsc , 1 ) \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} | y ^ { - 1 } x | \\geq | x | - | y | > | x | - \\frac { | x | } { 2 } = \\frac { | x | } { 2 } , \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} \\begin{cases} H _ { 0 } : & \\beta _ { 1 } = \\beta _ { 2 } = . . . = \\beta _ { p } = 0 \\\\ H _ { 1 } : & \\mbox { g S N R } \\geq r ^ { 2 } . \\end{cases} \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} C _ 1 : = 3 + \\lvert \\eta \\lvert + \\sup _ { g \\in ( 0 , \\infty ) } \\lVert \\omega ^ { - 1 / 2 } P _ { \\omega } ( ( 0 , \\widetilde { m } ] ) v _ g \\lVert ^ 2 \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} \\sup _ { 0 \\le k \\le m } \\| P _ { n - k } ( \\epsilon ) - ( \\tilde { P } - \\epsilon k \\tilde { P } ^ { ( 1 ) } ) \\| = O ( \\epsilon ^ 2 m ^ 2 ) \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} P ^ { \\omega } _ x ( H ( y ) < \\infty ) = \\prod _ { z = x } ^ { y - 1 } P _ z ^ \\omega ( H ( z - 1 ) < \\infty ) \\le ( 1 - \\eta ) ^ { y - x } \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} \\widehat { \\mathbf { Z } } _ + ( z ) = \\sum _ { d > 0 } \\frac { 1 } { \\sqrt { d } } \\mathrm { T r } _ { d , 1 } ( 1 ) q ^ d + , \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} K ( g ) = - \\frac { { - a ^ 2 + b ^ 2 + c ^ 2 } } { 2 b c } \\ , \\zeta ^ 1 \\odot \\bar \\zeta ^ 1 - \\frac { a ^ 2 - b ^ 2 + c ^ 2 } { 2 a c } \\ , \\zeta ^ 2 \\odot \\bar \\zeta ^ 2 - \\frac { a ^ 2 + b ^ 2 - c ^ 2 } { 2 a b } \\ , \\zeta ^ 3 \\odot \\bar \\zeta ^ 3 \\ , . \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} U _ { g , k } ^ * \\omega _ { 1 , k } \\oplus \\omega _ { 2 , k } U _ { g , k } = \\omega _ k . \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { n - 1 } ( 1 + t \\omega ^ { i j } _ n ) \\cdot \\sum _ { j = 0 } ^ { n - 1 } \\frac { j \\omega _ n ^ { i ( j - 1 ) } t } { 1 + t \\omega _ n ^ { i j } } = \\sum _ { r = 0 } ^ { n } \\omega _ n ^ { i ( \\binom { r } { 2 } - 1 ) } t ^ r \\Big ( \\binom { r } { 2 } { n \\brack r } _ { \\omega ^ i _ n } + \\omega _ n ^ i { n \\brack r } ^ { ' } _ { \\omega _ n ^ i } \\Big ) . \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} p _ 1 = \\xi _ 3 , p _ 2 = e ^ { - \\lambda } ( \\cos ( x _ 3 ) \\xi _ 1 + \\sin ( x _ 3 ) \\xi _ 2 + [ - \\partial _ { x _ 1 } \\lambda \\sin ( x _ 3 ) + \\partial _ { x _ 2 } \\lambda \\cos ( x _ 3 ) ] \\xi _ 3 ) . \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { L } ^ 2 ( y , p _ { _ 1 } ( y ) ) } { \\partial p _ { _ 1 } ^ 2 ( y ) } = \\frac { 1 } { p _ { _ 1 } ( y ) } . \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} e _ 2 a _ 1 + e _ 1 a _ 4 = e _ 2 a _ 2 + e _ 1 a _ 3 + O ( \\delta ) . \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} \\Phi ^ 0 _ j ( q _ j , t ) = - ( 1 + o ( 1 ) ) \\int _ { t _ 0 } ^ { t } \\frac { \\dot { \\mu } _ j ( \\tilde { s } ) } { \\mu _ j ( \\tilde { s } ) } \\mu _ j ^ { - ( n - 2 s ) } ( \\tilde { s } ) F \\left ( \\frac { ( t - \\tilde { s } ) ^ { \\frac { 1 } { 2 s } } } { \\mu _ j ( \\tilde { s } ) } \\right ) d \\tilde { s } = c ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} \\langle \\mathfrak { s } , \\mathfrak { t } \\mid \\mathfrak { s } ^ 4 = 1 , ( \\mathfrak { s } \\mathfrak { t } ) ^ 3 = \\mathfrak { s } ^ 2 \\rangle \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} \\ddot { z } + \\gamma \\dot { z } + \\omega _ 0 ^ 2 e ^ { - \\gamma t } z = 0 . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ j ( - 1 ) ^ { r j } \\binom k j \\left ( { \\frac { { F _ { n + r } } } { { F _ n } } } \\right ) ^ j G _ { r j } } = ( - 1 ) ^ k ( - 1 ) ^ { r k } \\left ( { \\frac { { F _ r } } { { F _ n } } } \\right ) ^ k G _ { ( n + r ) k } , n \\ne 0 \\ , , \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} ( - \\mathcal L _ { \\beta , \\Omega } ^ { } ) ^ { - 1 } ( \\mathcal L _ \\Omega \\phi _ 0 ) ( t , x ) = \\mathbf E \\left [ \\int _ 0 ^ { \\tau _ { t , x } } \\mathcal L _ { \\beta , \\Omega } \\phi _ 0 \\left ( X ^ { x , \\alpha } ( s ) \\right ) d s \\right ] = \\mathbf E \\left [ \\phi _ 0 ( X ^ { x , \\alpha } ( \\tau _ { t , x } ) ) \\right ] - \\phi _ 0 ( x ) . \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} B _ { \\overline { l } , 0 } ( t , \\overline \\alpha ) e ^ { \\alpha _ j t } - B _ { \\overline { l } , j } ( t , \\overline \\alpha ) = S _ { \\overline { l } , j } ( t , \\overline \\alpha ) , j = 1 , \\ldots , m , \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} \\eta _ k & : = \\max \\{ \\beta _ k , \\| g _ 1 ^ k \\| , \\| g _ 2 ^ k \\| \\} , \\ \\lambda _ k : = \\dfrac { \\beta _ k } { \\eta _ k } , \\\\ y ^ k & : = \\arg \\min \\{ \\lambda _ k f _ 1 ( x ^ k , y ) + \\dfrac { 1 } { 2 } \\| y - x ^ k \\| ^ 2 \\mid y \\in C \\} , \\\\ z ^ k & : = \\arg \\min \\{ \\lambda _ k f _ 2 ( x ^ k , y ) + \\dfrac { 1 } { 2 } \\| y - y ^ k \\| ^ 2 \\mid y \\in C \\} , \\\\ x ^ { k + 1 } & : = \\gamma z ^ k + ( 1 - \\gamma ) T ( x ^ k ) . \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} q _ L ( \\theta ) & = \\frac { \\rho _ { 0 , L } } { 2 } a _ 0 + \\sum _ { \\ell = 1 } ^ L \\rho _ { \\ell , L } a _ { \\ell } \\cos \\ell \\theta \\\\ & = \\frac { \\rho _ { 0 , L } } { 2 } a _ 0 + \\sum _ { \\ell = 1 } ^ L \\rho _ { \\ell , L } a _ { \\ell } T _ { \\ell } ( x ) \\\\ & = \\frac { \\sqrt { \\pi / 2 } \\rho _ { 0 , L } } { 2 } a _ 0 \\tilde { T } _ 0 + \\sum _ { \\ell = 1 } ^ L \\sqrt { \\pi } \\rho _ { \\ell , L } a _ { \\ell } \\tilde { T } _ { \\ell } ( x ) , \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} p ' ( w ) = \\begin{cases} p ( w ) + \\epsilon ( 0 , 1 ) & w \\sim z , \\\\ p ( w ) & \\end{cases} \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} A _ 0 \\partial _ t u + \\sum _ { j = 1 } ^ 3 A _ j ^ { \\operatorname { c o } } \\partial _ j u + D u & = f , & & x \\in G , & & t \\in J ; \\\\ B u & = g , & & x \\in \\partial G , & & t \\in J ; \\\\ u ( t _ 0 ) & = u _ 0 , & & x \\in G ; \\end{aligned} \\right . \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} p _ { 2 , \\tau } = \\frac { \\partial L _ \\tau } { \\partial \\dot { x } _ { 2 , \\tau } } = \\frac { f ^ { - 1 } ( t _ \\tau ) m \\dot { x } _ { 2 , \\tau } } { \\dot { t } _ \\tau } ; \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} \\int _ { Z } f ( g ( \\theta , z ) ) \\mu ( z ) d z = \\int _ { \\Omega } f ( x ) \\rho ( \\theta , x ) d x , \\quad \\textrm { f o r a n y $ f \\in C _ c ^ { \\infty } ( \\Omega ) $ . } \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} \\gamma _ { s y m } = \\dfrac { 1 } { N ! } \\sum _ { \\sigma \\in \\mathfrak { S } _ N } \\sigma _ { \\sharp } \\gamma , \\sigma \\in \\mathfrak { S } _ N , \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} r ( D ) = \\begin{cases} d - g & d > g , \\\\ 0 & 0 \\leq d \\leq g . \\end{cases} \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} L ( e _ 2 , e _ 3 , e _ 4 ) = \\mathcal { L } ( - e _ 2 - e _ 3 - e _ 4 , e _ 2 , e _ 3 , e _ 4 ) . \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} a _ h ( u _ h , v _ h ) = ( f , \\Xi _ h v _ h ) , \\forall v _ h \\in \\mathcal { Q } _ h ^ k , \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} \\mathcal { L } : = \\{ ( \\gamma , \\xi ) \\in { \\cal M } _ + ( Y \\times U ) \\times { \\cal M } _ + ( Y \\times U ) \\ : \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} \\frac { \\partial ( G ^ { - 1 } ) _ i } { \\partial \\phi _ l } \\frac { \\partial \\phi _ l } { \\partial x _ j } = \\delta _ { i j } - \\frac { \\partial ( G ^ { - 1 } ) _ i } { \\partial \\xi _ l } \\frac { \\partial \\xi _ l } { \\partial x _ j } = \\delta _ { i j } - ( \\Phi ^ { - 1 } ) _ { l l ' } ( a \\nabla \\xi _ { l ' } ) _ i \\frac { \\partial \\xi _ l } { \\partial x _ j } = \\Pi _ { j i } \\ , . \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} \\begin{aligned} g _ { D _ I } ( x ) = - & \\sum _ { i = 1 } ^ N \\min _ { l _ i \\leq y _ i \\leq u _ i } \\Big \\{ \\beta y ^ 2 _ i + \\Big { ( } \\beta \\sigma _ { ( - i ) } ( x ) - \\alpha \\Big { ) } y _ i + h _ i ( y _ i ) \\Big \\} \\\\ & + \\beta \\sum _ { i = 1 } ^ N x ^ 2 _ i - a ^ T x + \\sum _ { i = 1 } ^ N h _ i ( x _ i ) , \\end{aligned} \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} \\mathcal F _ { L , \\alpha } ^ V ( \\mu ) : = \\mathcal H \\bigl ( \\mu | \\nu _ \\alpha ^ V \\bigr ) = \\sum _ { \\eta \\in \\Omega _ L } \\mu ( \\eta ) \\log \\Bigl ( \\frac { \\mu ( \\eta ) } { \\nu _ \\alpha ^ V ( \\eta ) } \\Bigr ) , \\end{align*}"}
-{"id": "9880.png", "formula": "\\begin{align*} \\widetilde { W } ^ { s } _ R ( \\mathbb { R } ) = \\{ u \\in L ^ 2 ( \\mathbb { R } ) , \\boldsymbol { D } ^ { s * } u \\in L ^ 2 ( \\mathbb { R } ) \\} , \\end{align*}"}
-{"id": "9649.png", "formula": "\\begin{align*} h _ { t } = \\sum _ { j = 0 } ^ { \\infty } \\Delta \\left ( \\frac { ( - \\Delta h ) ^ { j } } { j ! } \\right ) . \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & - 2 & 0 \\\\ - 1 & 2 & - 1 \\\\ 0 & - 1 & 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} x ^ 2 \\ = \\ p ^ 2 + q ^ 2 \\ , , \\ \\ y ^ 2 \\ = \\ ( p - 1 ) ^ 2 + q ^ 2 \\ , , \\ \\quad \\mbox { a n d } \\ z ^ 2 \\ = \\ ( p + a ) ^ 2 + ( q + b ) ^ 2 \\ , . \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} A ^ { ( n ) } _ { \\mathbb { Z } l } & : = \\mathbf { k } Q ^ { ( n ) } _ { \\mathbb { Z } l } / I ^ { ( n ) } _ l . \\\\ A ^ { ( n ) } _ { l } & : = A _ { \\mathbb { Z } l } / \\tau ^ k _ n . \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| T u ^ k - u ^ k \\| = 0 . \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} E _ 0 ^ { ( r ) } T _ a \\wedge T _ { - n } = A _ n ( 0 ) = c _ n = \\frac { v _ { n , + , a } ( 0 ) } { u _ { n , + , a } ( 0 ) } . \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} g ( \\omega , x + b ) - g ( \\omega , x ) = g ( \\tau _ x \\omega , b ) , \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} h ( t , \\cdot ) = e ^ { - \\Delta ^ { 2 } t } h _ { 0 } + \\sum _ { j = 2 } ^ { \\infty } I ^ { + } F _ { j } . \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D O } _ \\textup { t } ( 2 n ) q ^ n & = 2 q { f _ 2 f _ 4 f _ 6 f _ { 1 2 } } \\cdot \\dfrac { f _ 3 } { f _ 1 ^ 3 } \\\\ & = 2 q { f _ 2 f _ 4 f _ 6 f _ { 1 2 } } \\left ( \\dfrac { f _ 4 ^ 6 f _ 6 ^ 3 } { f _ 2 ^ 9 f _ { 1 2 } ^ 2 } + 3 q \\dfrac { f _ 4 ^ 2 f _ 6 f _ { 1 2 } ^ 2 } { f _ 2 ^ 7 } \\right ) . \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} f ( \\Delta _ { \\rho , \\sigma } ) : = \\int _ { ( 0 , + \\infty ) } f ( t ) \\ , d E _ { \\rho , \\sigma } ( t ) , \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} ( \\{ \\overline { X } _ { t , i } , t \\ge 0 \\} ) \\ , = \\ , ( \\{ \\overline { X } _ { t , 1 } , t \\ge 0 \\} ) \\ , ; i \\ , = \\ , 1 , 2 , \\ldots , n + 1 \\ , , \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} \\Pr \\left ( \\max _ { m \\in S } ( X _ m ) < a \\right ) & \\le 2 \\ , \\Pr \\left ( \\max _ { m \\in S } ( Y _ m ) < a \\right ) = 2 \\Pr \\left ( Y _ 1 < a \\right ) ^ { | S | } \\le 2 \\left ( 1 - e ^ { - \\theta } \\frac { \\theta ^ a } { a ! } \\right ) ^ { | S | } \\\\ & \\le 2 \\left ( 1 - \\frac { \\theta ^ a } { e a ! } \\right ) ^ { | S | } \\le 2 \\exp \\left ( - \\frac { \\theta ^ a | S | } { e a ! } \\right ) \\ . \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} \\frac { \\big [ \\tilde { \\mathbb { A } } / f \\tilde { \\mathbb { A } } \\big ] _ { \\tilde { \\mathbb { A } } } } { \\big [ \\psi ( \\tilde { \\mathbb { A } } / f \\tilde { \\mathbb { A } } ) \\big ] _ { \\tilde { \\mathbb { A } } } } & = \\frac { f } { f + c ( f ) p _ 1 t ^ d f ( z _ 1 ) + c ( f ) p _ 2 t ^ { 2 d } f ( z _ 1 ) ^ 2 + \\dots + c ( f ) t ^ { d r _ 0 } f ( z _ 1 ) ^ { r _ 0 } } \\\\ & = \\frac { 1 } { 1 + c ( f ) p _ 1 t ^ d f ( z _ 1 ) f ^ { - 1 } + \\dots + c ( f ) t ^ { r _ 0 d } f ^ { r _ 0 - 1 } f ( z _ 1 ) ^ { r _ 0 } f ^ { - r _ 0 } } \\\\ & = D ^ { \\psi } _ f ( f ^ { - 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} 0 \\leq { \\rm t r } _ g K ^ x = s - x _ i q ^ i = s - ( x _ 1 + x _ 2 ) q ^ 1 - ( x _ 3 + x _ 4 ) q ^ 3 \\leq 0 , \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} W _ \\mathrm { n e w } ^ { * } = W _ { u } ^ { T } \\Delta _ \\mathrm { n e w } , \\Delta _ \\mathrm { n e w } = \\mathrm { d i a g } ( \\delta _ 1 , \\ldots , \\delta _ { s + 1 } ) . \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} q _ A ( i ) = \\frac { \\sigma _ { i + 1 } \\ , \\rho _ { } ( i + 1 ) } { \\overline { m } _ { } ( i + 1 ) } . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{gather*} [ e _ 1 , e _ 2 ] = 0 , [ e _ 1 , e _ 3 ] = a e _ 1 + b e _ 2 , [ e _ 2 , e _ 3 ] = c e _ 1 + d e _ 2 , Q = \\left ( \\begin{matrix} a & b \\\\ c & d \\end{matrix} \\right ) \\in \\mathrm { G L } _ 2 ( \\R ) , \\end{gather*}"}
-{"id": "5091.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t , n } ^ { ( u ) } \\ , = \\ , b \\Big ( t , X _ { t , n } ^ { ( u ) } , u \\cdot \\delta _ { X _ { t , 1 } ^ { ( u ) } } + ( 1 - u ) \\cdot \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } \\delta _ { X _ { t , j } ^ { ( u ) } } \\Big ) { \\mathrm d } t + { \\mathrm d } W _ { t , n } \\ , . \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} \\{ ( \\left ( \\begin{array} { c c c c } \\sigma ( t ) & & & \\\\ & \\sigma ( t ) & & \\\\ & & \\sigma ( t ) ^ { - 1 } & \\\\ & & & \\sigma ( t ) ^ { - 1 } \\end{array} \\right ) ) _ { \\sigma \\in I _ F } \\ \\vert t = \\gamma _ 1 ^ { p _ 1 } \\dots \\gamma _ { d - 1 } ^ { p _ { d - 1 } } , p _ 1 , \\dots , p _ { d - 1 } \\in \\mathbb { Z } \\} \\hookrightarrow Q _ 0 / W _ 0 ( L ) . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\int _ { [ t , 1 ] } ( s - t ) ^ { x - 1 } d \\mu ( s ) d t & = \\int _ { [ 0 , 1 ] } \\int _ 0 ^ s ( s - t ) ^ { x - 1 } d t d \\mu ( s ) \\\\ & = \\int _ { [ 0 , 1 ] } \\frac { s ^ { x } } { x } d \\mu ( s ) < \\infty . \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} v _ n ( x ) = V ( x ) + r _ n ( x ) , \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} V ^ { \\xi ^ * _ { m } b _ { \\theta ^ * _ { k - 1 } } } ( X _ { \\theta ^ * _ { k } } ) - V ^ { \\xi ^ * _ { m - 1 } b _ { \\theta ^ * _ { k - 1 } } } ( X _ { \\theta ^ * _ { k } } ) - C ( \\xi ^ * _ { m - 1 } , \\xi ^ * _ m ) = 0 \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} \\frac { 1 } { y } = { { \\dot \\kappa } ^ { A } ( \\theta ) + { \\dot \\kappa } ^ { - S } ( \\theta ) } , \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial U } ( z , \\ , U ) = 1 + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ 2 \\frac { - \\left ( \\phi _ i ( z ) - m U \\right ) } { [ ( \\phi _ i ( z ) - m U ) ^ 2 + 4 \\lambda m _ i z ^ 2 ] ^ { 1 / 2 } } , \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} G _ { L _ { n - 1 } } ( z ) & = \\prod _ { i = 0 } ^ { n - 2 } f _ { n - 1 - i } ^ { K - 1 } ( z ) , \\\\ ~ f _ k ( z ) & = \\begin{cases} q _ k + p _ k f _ { k + 1 } ^ { K } ( z ) , & ~ 1 \\le k \\le n - 1 \\\\ z ^ { \\frac { 1 } { K } } , & ~ k = n . \\end{cases} \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } \\int _ { \\mathcal { I } _ R } e ^ { z w } \\psi _ { \\varepsilon } u ( z ) d z = \\lim _ { R \\to + \\infty } \\int _ { \\mathcal { J } _ R } e ^ { z w } \\psi _ { \\varepsilon } u ( z ) d z = 0 . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} g = a \\ \\zeta ^ 1 \\odot \\bar \\zeta ^ 1 + b \\ \\zeta ^ 2 \\odot \\bar \\zeta ^ 2 + c \\ \\zeta ^ 3 \\odot \\bar \\zeta ^ 3 , a , b , c > 0 . \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} A _ { T , t } ^ { ( i _ 1 i _ 2 ) q } = \\frac { T - t } { 2 } \\sum _ { i = 1 } ^ { q } \\frac { 1 } { \\sqrt { 4 i ^ 2 - 1 } } \\left ( \\zeta _ { i - 1 } ^ { ( i _ 1 ) } \\zeta _ { i } ^ { ( i _ 2 ) } - \\zeta _ i ^ { ( i _ 1 ) } \\zeta _ { i - 1 } ^ { ( i _ 2 ) } \\right ) , \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} \\vartheta ^ * : = \\min _ { g = \\mathcal { T } ( R , p ) \\in \\Upsilon } \\left ( 1 - \\frac { 1 } { 8 } \\| I _ 3 - R \\| _ F ^ 2 \\cos ^ 2 \\langle u , \\bar { Q } u \\rangle \\right ) . \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} \\mathbb A \\bigl ( ( \\pi _ t ) _ { t \\in [ 0 , T ] } \\bigr ) = \\frac 1 4 \\int _ 0 ^ T \\bigl \\| \\dot \\rho _ t - \\Delta \\phi ( \\rho _ t ) - \\nabla \\cdot ( \\chi ( \\rho _ t ) \\nabla V ) \\bigr \\| _ { - 1 , \\chi ( \\rho _ t ) } ^ 2 \\ ; \\ ! \\mathrm d t . \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} \\begin{array} { r l l } \\mathcal { N } : \\mathcal { G } & \\longrightarrow & \\mathcal { G } ^ { * } \\\\ \\varphi _ { _ { 0 } } & \\longrightarrow & \\mathcal { N } \\varphi _ { _ { 0 } } = \\mathcal { P } ( { } _ { 0 } I _ { T } ^ { 1 - q } \\Psi ( 0 ) ) , \\end{array} \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} A ^ p ( \\phi , t ) = \\sum _ { j k \\in E ( \\mathbb { T } _ N ) } \\Big ( 1 + \\beta ( \\phi ( j ) - \\phi ( k ) ) ^ 2 \\Big ) e ^ { t _ { j k } } + \\sum _ { j \\in \\Lambda } \\epsilon \\phi _ j ^ 2 , \\quad \\epsilon \\ge 0 . \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} M _ { q } = \\max \\{ \\varrho _ { v , q } : \\ , v \\in V \\} \\ , \\ , \\ , \\ , M _ q ^ k = ( M _ q ) ^ k . \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} u _ { m + 1 } ( t , 1 ) = M _ { 1 } \\Big ( u _ { k + 1 } \\big ( t , x _ { k + 1 } ( - \\tau _ { m + 1 } , 0 , 0 ) \\big ) , \\dots , u _ { m } \\big ( t , x _ { m + 1 } ( - \\tau _ { m + 1 } , 0 , 0 ) \\big ) \\Big ) \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} B _ { \\alpha } ( f , g ) ( x ) : = \\int _ { \\mathbb { R } ^ { n } } \\frac { f ( x - y ) g ( x + y ) } { | y | ^ { n - \\alpha } } d y , 0 < \\alpha < n . \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\binom { 2 k + 2 n + 1 } { k + n } x ^ k & = \\frac { 1 } { \\sqrt { 1 - 4 x } } \\left ( \\frac { 1 - \\sqrt { 1 - 4 x } } { 2 x } \\right ) ^ { 2 n + 1 } \\\\ & = \\frac { \\mu _ 0 } { \\displaystyle 1 + a _ 0 x - \\frac { b _ 1 x ^ 2 } { \\displaystyle 1 + a _ 1 x - \\frac { b _ 2 x ^ 2 } { \\displaystyle 1 + a _ 2 x - \\cdots } } } \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} F = \\sum _ { I \\in \\N ^ n } C _ I X ^ I \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} R + \\frac { \\partial \\phi } { \\partial s } + \\mathcal { L } \\phi = 0 . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} & \\Phi ( t ) : = \\Phi _ \\lambda ( t ) = \\frac { 2 d - m + \\lambda - 1 - ( d - 1 ) t + \\eta ( t ) } { 2 ( d + \\lambda - 1 ) } , \\\\ & \\Psi ( u , v ) = \\Phi ( u v ) - v ; \\end{align*}"}
-{"id": "9918.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\frac { n ( n - 1 ) } 2 } \\o _ { n } ^ { 0 } , \\qquad \\o _ { n } ^ { 0 } = \\Big ( \\frac 1 { 2 \\pi \\sqrt { - 1 } } \\Big ) ^ { n } \\frac 1 { z _ { 1 } \\cdots z _ { n } } \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { m } & > \\bar { f } _ R ^ { P _ m } ( x , y ) \\\\ & \\geq 1 - \\frac { | ( ( W _ R ( x , y ) ) ^ c \\cup h ^ { - 1 } ( W _ R ( x , y ) ) ^ c ) \\cap [ 0 , R ] | } { R } - \\\\ & \\frac { 1 } { R } \\sum _ { j \\geq 0 } | C _ { j , R , m } ( x , y ) \\cap W _ R ( x , y ) \\cap h ^ { - 1 } ( W _ R ( x , y ) ) | \\\\ & \\geq \\frac { 9 } { 1 0 } - \\frac { 1 } { R } \\sum _ { j \\geq 0 } | C _ { j , R , m } ( x , y ) \\cap W _ R ( x , y ) \\cap h ^ { - 1 } ( W _ R ( x , y ) ) | \\end{aligned} \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} D _ { s } \\varphi ( s , x ) \\ , : = \\ , \\frac { \\ , \\partial \\varphi \\ , } { \\ , \\partial s \\ , } ( s , x ) \\ , , D _ { i } \\varphi ( s , x ) \\ , : = \\ , \\frac { \\partial \\varphi } { \\partial x _ { i } } ( s , x ) \\ , , D _ { i } ^ { 2 } \\varphi ( s , x ) \\ , : = \\ , \\frac { \\ , \\partial ^ { 2 } \\varphi \\ , } { \\ , \\partial x _ { i } ^ { 2 } \\ , } ( s , x ) \\ , . \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} A _ 5 = I _ f \\otimes ( J _ { \\ell } - I _ { \\ell } ) \\otimes J _ n . \\end{align*}"}
-{"id": "9889.png", "formula": "\\begin{align*} ( ( 2 \\pi i \\xi ) ^ { - s } \\widehat { \\psi } , \\overline { ( - 2 \\pi i \\xi ) ^ { s } \\widehat { \\psi } } ) = \\| \\psi \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } . \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} \\bold { X } ( z ) = \\sum _ { n = 0 } ^ { \\infty } \\chi ( \\bold { X } _ n ) z ^ n , \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\varepsilon \\Delta v - \\mathcal { L } _ R ^ { \\theta _ 0 } [ v ] + | D v | ^ m + \\sigma v = f _ n + \\sigma \\Theta = \\tilde { f } _ n , & x \\in B _ R , \\\\ v = \\theta _ 0 , & x \\in \\partial B _ R . \\end{array} \\right . \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} F ( X , Y ) = F ( Y , X ) \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} A ( t ) = \\lambda { t } , \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} A = \\partial _ x ^ { } \\Big ( g ^ { i j } \\partial _ x ^ { } + c ^ { i j } _ k u ^ k _ x \\Big ) \\partial _ x ^ { } \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\chi ( C ) = c ( S ) \\ , . \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} f _ m = O ( \\epsilon ^ { 2 } n ^ 3 ) \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} \\check f ( \\chi ) = \\int _ { X _ \\emptyset } \\mathfrak R _ { 1 , { ^ w \\chi } \\delta ^ { \\frac { 1 } { 2 } } } \\Phi ( v , v ) d v , \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} x \\cdot ( v , w ) = ( v , w - x v ) . \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} t ( a _ { 2 } ^ { ( 1 ) } , \\ldots , a _ { 2 } ^ { ( k ) } , a _ { 1 } ^ { ( k + 1 ) } ) = t ( a _ { 2 } ^ { ( 1 ) } , \\ldots , a _ { 2 } ^ { ( k ) } , a _ { 2 } ^ { ( k + 1 ) } ) . \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} \\lvert ( 2 \\lvert m _ 0 \\rvert , j ) \\rvert _ { W _ n ( w ) } - \\lvert ( i , j ) \\rvert _ { W _ n ( w ) } = 2 \\lvert m _ 0 \\rvert - i > 0 . \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} S _ f ( \\rho \\| \\sigma ) : = \\ < \\xi _ \\sigma , f ( \\Delta _ { \\rho , \\sigma } ) \\xi _ \\sigma \\ > + f ( 0 ^ + ) \\sigma ( 1 - s _ M ( \\rho ) ) + f ' ( + \\infty ) \\rho ( 1 - s _ M ( \\sigma ) ) . \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} - \\Biggl . { \\bf 1 } _ { \\{ i _ 1 = i _ 2 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 1 = j _ 2 \\} } \\zeta _ { j _ 3 } ^ { ( i _ 3 ) } - { \\bf 1 } _ { \\{ i _ 2 = i _ 3 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 2 = j _ 3 \\} } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } - { \\bf 1 } _ { \\{ i _ 1 = i _ 3 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 1 = j _ 3 \\} } \\zeta _ { j _ 2 } ^ { ( i _ 2 ) } \\Biggr ) , \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\nabla \\rho _ t ( u ) = ( \\phi ^ { - 1 } ) ' ( \\phi ( \\rho _ t ( u ) ) ) \\nabla \\phi ( \\rho _ t ( u ) ) = \\frac { \\nabla \\phi ( \\rho _ t ( u ) ) } { \\phi ' ( \\rho _ t ( u ) ) } , \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\partial ^ n f ( g _ 0 , . . . , g _ n ) = \\ & g _ 0 f ( g _ 1 , . . . , g _ n ) + ( - 1 ) ^ { n + 1 } f ( g _ 0 , . . . , g _ { n - 1 } ) \\\\ & - \\sum _ { j = 0 } ^ { n - 1 } ( - 1 ) ^ { j + 1 } f ( g _ 0 , . . . , g _ j g _ { j + 1 } , . . . , g _ n ) . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} \\langle \\vec { m } ^ { \\star } , \\vec { i } \\rangle = \\langle \\vec { m } , \\vec { i } \\rangle - a _ i , \\ \\langle \\vec { i } , \\vec { m } ^ { \\star } \\rangle = \\langle \\vec { i } , \\vec { m } \\rangle - b _ i . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} \\delta ( \\epsilon ) = ( I - \\tilde { B } ) ^ { - 1 } r + \\epsilon \\mu \\tilde { B } ( I - \\tilde { B } ) ^ { - 2 } \\tilde { B } ^ { ( 1 ) } ( I - \\tilde { B } ) ^ { - 1 } \\tilde { r } + \\epsilon \\mu \\tilde { B } ( I - \\tilde { B } ) ^ { - 2 } \\tilde { r } ^ { ( 1 ) } + O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ w ) \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} \\| K _ { \\nu , 0 } ^ { - 1 } \\| _ { \\mathcal { L } ( L ^ 2 ( \\mathbb { R } ^ 2 ) ) } = \\| \\displaystyle \\int _ 0 ^ { + \\infty } e ^ { - t K _ { \\nu , 0 } } d t \\| _ { \\mathcal { L } ( L ^ 2 ( \\mathbb { R } ^ 2 ) ) } \\le \\int _ 0 ^ { + \\infty } \\| e ^ { - t K _ { \\nu , 0 } } \\| _ { \\mathcal { L } ( L ^ 2 ( \\mathbb { R } ^ 2 ) ) } d t { ~ , } \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} \\chi _ { ( \\lambda , \\mu ) } = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } e ^ { \\left ( w \\left ( \\frac { \\lambda + \\mu } { 2 } + \\rho \\right ) , w \\left ( \\frac { \\lambda + \\mu } { 2 } + \\rho \\right ) \\right ) } } { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } e ^ { ( w ( \\rho ) , w ( \\rho ) ) } } e ^ { \\left ( \\frac { \\lambda - \\mu } { 2 } , - \\frac { \\lambda - \\mu } { 2 } \\right ) } , \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} \\begin{aligned} g _ 1 & = \\exp ( x + y ) , & g _ 2 & = \\dfrac { 1 } { ( ( x - 1 . 1 ) ^ 2 ) + ( y - 1 . 1 ) ^ 2 ) ^ 2 } , \\\\ g _ 3 & = \\cos ( 2 4 x - 3 2 y ) \\sin ( 2 1 x - 2 8 y ) , & g _ 4 & = \\arctan ( 3 ( x ^ 2 + y ) ) . \\end{aligned} \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} \\psi ( S _ 1 ) + \\psi ( S _ 2 ) = \\psi ( S _ 1 \\cap S _ 2 ) + \\psi ( S _ 1 \\cup S _ 2 ) \\ , . \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} \\Lambda h ( x _ n ) = ( Q ^ f h ) ( x _ n ) \\leq \\gamma _ n W ( x _ n ) , \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} E _ 0 ^ { ( r ) } T _ a = \\begin{cases} 2 \\phi _ 1 ( a ) \\int _ 0 ^ a d y \\thinspace \\phi _ 1 ^ { - 2 } ( y ) \\int _ { - \\infty } ^ y \\phi _ 1 ( z ) d z , \\ a > 0 ; \\\\ \\frac { 2 \\phi _ 1 ( a ) } { \\int _ 0 ^ { \\infty } \\phi _ 1 ^ { - 2 } ( x ) d x } \\int _ 0 ^ { \\infty } d x \\thinspace \\phi _ 1 ^ { - 2 } ( x ) \\int _ a ^ 0 d y \\thinspace \\phi _ 1 ^ { - 2 } ( y ) \\int _ y ^ x \\thinspace \\phi _ 1 ( z ) d z , \\ a < 0 . \\end{cases} \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} R _ { 2 1 } & = ( d - c a ^ { - 1 } b ) ^ { - 1 } ( [ c , d ] - c a ^ { - 1 } [ c , b ] ) \\Delta ^ { - 1 } \\\\ & = ( a c ^ { - 1 } d - b ) ^ { - 1 } ( a c ^ { - 1 } [ c , d ] - [ c , b ] ) \\Delta ^ { - 1 } \\\\ & = ( ( a c ^ { - 1 } d - b ) ^ { - 1 } \\Delta - c ) \\Delta ^ { - 1 } , \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} - \\Delta w _ k + V ^ { + } w _ k = H _ k , w _ k \\in H ^ 1 _ 0 ( \\Omega _ * ) , \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ { r j } \\binom k j \\left ( { \\frac { { F _ r } } { { F _ n } } } \\right ) ^ j G _ { ( n + r ) j } } = ( - 1 ) ^ { r k } \\left ( { \\frac { { F _ { n + r } } } { { F _ n } } } \\right ) ^ k G _ { r k } , n \\ne 0 \\ , , \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} G ^ { - 1 } \\left ( 1 - \\tilde { \\alpha } \\right ) & = \\frac { F ^ { - 1 } \\left ( 1 - \\tilde { \\alpha } \\right ) } { \\bar { x } } \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { k ! ^ 2 } z ^ k \\end{gather*}"}
-{"id": "9102.png", "formula": "\\begin{align*} \\langle d x _ i , d x _ i \\rangle _ g & = x _ i , & \\langle d x _ i , d x _ j \\rangle _ g = 0 i \\neq j . \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} \\omega _ { M _ { \\mu _ 1 } } = ( \\chi ) ^ * \\omega _ { \\mathcal { O } _ { \\mu _ 2 } } ^ + , \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } x ^ { \\mu } \\exp ( - a x ^ { \\xi } ) \\cos ( x y ) d x = - y ^ { - \\mu - 1 } \\sum _ { \\ell = 0 } ^ { \\infty } \\left ( - \\frac { a } { y ^ { \\xi } } \\right ) ^ { \\ell } \\frac { 1 } { \\ell ! } \\Gamma ( \\mu + 1 + \\xi \\ell ) ~ \\sin \\left \\{ \\frac { \\pi } { 2 } ( \\mu + \\xi \\ell ) \\right \\} . \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} \\omega ( K ) = r . \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} \\mu _ t ^ { \\star } ( B ) \\ge \\mu _ t ^ { \\star } ( G [ - 1 ] ) = \\mu _ t ^ { \\star } ( - \\mathbf { v } _ { k - 1 } ) > 0 = \\mu _ t ^ { \\star } ( A ) \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\chi \\left ( \\frac { q c _ { 2 } \\cdots c _ { n - 1 } } { m d _ { n - 1 } \\cdots d _ { 2 } } \\right ) ^ { - 1 } \\frac { A _ { f } ( d _ { n - 1 } , \\ldots , d _ { 2 } , m ) } { \\abs { m } ^ { \\frac { n - 1 } { 2 } } \\prod _ { i = 2 } ^ { n - 1 } d _ { i } ^ { \\frac { i ( n - i ) } { 2 } } } . \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} A _ n ( 0 ) = c _ n . \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} A _ k : = \\left \\{ \\begin{array} { l l } 2 \\ell _ k - ( - 1 ) ^ { \\lambda _ k } & \\ell _ k , \\\\ 2 \\ell _ k & \\ell _ k . \\end{array} \\right . \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} ( x - \\alpha _ 2 ) \\varphi = ( \\alpha _ 1 - x ) ( \\alpha _ 2 - x ) ( A _ 0 ' + A _ 1 ' x + A _ 2 ' x ^ 2 ) + A _ 3 ' y + A _ 4 ' x y , \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} T ( e _ i ) = \\sum _ j r _ { i j } e _ j \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} \\varphi ( e , \\dot { e } ) = \\left \\{ \\begin{array} { l l } \\alpha e , & e \\dot { e } > 0 , \\\\ 0 , & , \\end{array} \\right . \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} b ( i , n ) = b ( i , n - 1 ) + b ( i - 1 , n - 3 ) + b ( i - 2 , n - 3 ) . \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\| ( - \\Delta ) ^ { \\frac { 1 } { 4 } } \\mathcal { Q } \\| _ { L ^ { 2 } } ^ 2 = \\frac { a _ { * } } { 2 } \\iint _ { \\mathbb { R } ^ { 3 } \\times \\mathbb { R } ^ { 3 } } \\frac { | \\mathcal { Q } ( x ) | ^ 2 | \\mathcal { Q } ( y ) | ^ { 2 } } { | x - y | } { \\rm d } x { \\rm d } y . \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} \\mathcal { H } : = H ( \\operatorname { d i v } _ { \\boldsymbol { \\varepsilon } } 0 , G ) \\times H ( \\operatorname { d i v } _ { \\boldsymbol { \\mu } } 0 , G ) \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} | ( i + j ) - \\frac { 2 } { 2 n - 1 } \\sum _ { b = 1 } ^ n j _ b | < \\frac { 1 } { 2 5 n } . \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{gather*} \\{ \\alpha _ { j + ( i - 1 ) } = \\alpha _ j + \\alpha _ { i - 1 } \\} _ { j = 1 } ^ { k + 1 - i } \\ \\mbox { a n d } \\ \\delta _ { n } ^ k = \\alpha _ { k + 2 - i } + \\alpha _ { i - 1 } \\ \\mbox { a n d } \\ \\{ \\alpha _ { j + k + 2 - i } = \\alpha _ { j } + \\alpha _ { k + 2 - i } \\} _ { j = 1 } ^ { i - 2 } . \\end{gather*}"}
-{"id": "6378.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } v ' ( s ) ^ 2 \\frac { ( B ( s ) - 1 ) \\sinh ( F _ n ( s ) ) ^ { 4 ( n - 1 ) } } s = 0 , \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { \\binom k j \\binom j s \\left ( { \\frac { { f _ 1 } } { { f _ 2 } } } \\right ) ^ j \\left ( { - \\frac { { 1 } } { { f _ 1 } } } \\right ) ^ s X _ { m - ( b - c ) k + ( b - a ) j + a s } } } = \\left ( - \\frac { f _ 3 } { f _ 2 } \\right ) ^ k X _ m \\ , . \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} \\| f \\| _ p : = \\bigg ( \\sum _ { v \\in V } | f ( v ) | ^ p \\bigg ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} x b = a _ 2 b _ 2 + a _ 3 b _ 3 . \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} V ^ { \\bar { i } \\bar { j } } ( x ) - U ^ { \\bar { i } \\bar { j } } ( x ) = \\max _ { i , j \\in \\mathcal { I } } \\{ V ^ { i j } ( x ) - U ^ { i j } ( x ) \\} \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} \\partial _ t \\hat u ( t , x ) = \\hat \\Sigma ( x ) \\partial _ x \\hat u ( t , x ) + \\hat S ( x ) \\hat u ( t , 0 ) , \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} E _ { s , s + t } ( Q ) = E _ { t } ( \\phi _ s ( Q ) ) \\mbox { \\rm a n d } E _ { s , s + t } ( P _ { \\infty } ) = E _ { t } ( P _ { \\infty } ) = e ^ { t B } \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} \\binom { n } { 2 } ( - 1 ) ^ { k + 1 } \\sum _ { 0 \\le s \\le ( k - 1 ) d / n } \\binom { d } { s } ( - 1 ) ^ s = \\omega _ n ^ { i \\binom { k } { 2 } } \\Big ( \\binom { k } { 2 } \\binom { d } { d k / n } + \\omega _ n ^ i { n \\brack k } ^ { ' } _ { \\omega _ n ^ i } \\Big ) . \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} H \\left ( \\theta ^ { ( 2 ) } _ { 1 } ( l ) ~ | ~ \\theta ( l ) = c _ { j } \\right ) = - \\sum _ { k = 1 } ^ { 2 ^ { m } } \\mbox { P r o b } \\left ( \\theta ^ { ( 2 ) } _ { 1 } ( l ) = b _ { k } ~ | ~ \\theta ( l ) = c _ { j } \\right ) \\mbox { l o g } _ { 2 } \\left ( \\mbox { P r o b } \\left ( \\theta ^ { ( 2 ) } _ { 1 } ( l ) = b _ { k } ~ | ~ \\theta ( l ) = c _ { j } \\right ) \\right ) \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} ( \\tilde { f } _ i ( \\pi ) ) ( t ) : = \\begin{cases} \\pi ( t ) & ( 0 \\leq t \\leq t _ 0 ) , \\\\ s _ i ( \\pi ( t ) - \\pi ( t _ 0 ) ) + \\pi ( t _ 0 ) & ( t _ 0 \\leq t \\leq t _ 1 ) , \\\\ \\pi ( t ) - \\alpha _ i & ( t _ 1 \\leq t \\leq 1 ) . \\end{cases} \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\phi ( y , \\tau ) \\cdot Z _ j ( y ) d y = 0 \\tau \\in ( \\tau _ 0 , \\tau _ 1 ) , j = 0 , 1 , \\cdots , n + 1 . \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} \\displaystyle \\arg \\max _ { \\mathcal { Q } _ { b } } ~ ~ p _ { c } ( \\mathcal { Q } _ { b } ) \\\\ \\mbox { s u c h t h a t } \\\\ H ( \\mathcal { Q } _ { b } ( y _ { B } ) ~ | ~ \\mathcal { Q } _ { b } ( y _ { C } ) , \\mathcal { Q } _ { b } ( y _ { B } ) \\in \\mathcal { C } ) = b , \\\\ p _ { c , m } ( \\mathcal { Q } _ { b } ) \\leq \\eta , \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} { \\mathfrak { C } } _ t = f _ t \\left ( C _ t ^ 1 , \\ldots , { C } _ t ^ { { M } } \\right ) . \\end{align*}"}
-{"id": "9909.png", "formula": "\\begin{align*} \\begin{aligned} & ( \\xi \\smallsmile \\eta ) _ 0 = \\xi _ 0 \\wedge \\eta _ 0 , \\quad ( \\xi \\smallsmile \\eta ) _ 1 = \\xi _ 1 \\wedge \\eta _ 1 \\quad \\\\ & ( \\xi \\smallsmile \\eta ) _ { 0 1 } = ( - 1 ) ^ { p + q } \\xi _ 0 \\wedge \\eta _ { 0 1 } + \\xi _ { 0 1 } \\wedge \\eta _ 1 . \\end{aligned} \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} \\Delta K _ 2 ( \\varphi , p ) = - \\int \\left [ \\frac { \\partial p } { \\partial y ^ \\alpha } ( y ) \\right ] \\bigl [ \\Delta \\varphi ^ \\alpha ( \\boldsymbol \\varphi ^ { - 1 } ( y ) ) \\bigr ] \\ , \\bigl [ \\mu _ { \\boldsymbol \\varphi } ( y ) \\bigr ] \\ , d y ^ 1 d y ^ 2 d y ^ 3 d y ^ 4 . \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} m _ N ^ q ( z ) = \\frac { 1 } { ( 2 \\pi i ) ^ N N ! } \\prod _ { 1 \\le i < j \\le N } ( z _ i / z _ j ; q ) _ { \\infty } ( z _ j / z _ i ; q ) _ { \\infty } \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} & q _ { g } ( \\mathbf p ) = \\sum _ { n \\in \\mathcal N } a _ { n } \\sum _ { \\ell \\in \\mathcal L } b _ { n , \\ell } q _ { g , n , \\ell } ( \\mathbf p ) , g \\in \\{ { \\rm S V C } , { \\rm D A S H } \\} . \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} P ^ { ( k ) } ( x _ 0 ) = 0 \\ k \\in \\{ 0 , 1 , \\ldots , n - 1 \\} , \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} [ F ] = \\beta , \\ \\chi ( F ) = n \\end{align*}"}
-{"id": "9928.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( x ) & = \\left [ \\left ( 0 , \\ , \\dfrac { 1 } { 2 \\pi \\sqrt { - 1 } } \\frac { \\varphi _ { \\Omega _ + } + \\varphi _ { \\Omega _ - } } z \\right ) \\right ] \\otimes [ \\nu _ { \\Omega _ + } ] = \\left [ \\left ( 0 , \\ , \\frac { 1 } { 2 \\pi \\sqrt { - 1 } } \\frac 1 z \\right ) \\right ] \\otimes [ \\nu _ { \\Omega _ + } ] \\end{aligned} \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} \\varphi _ { \\epsilon } ( x , y ) = \\displaystyle \\frac { 1 } { 2 \\epsilon } | x - y | ^ { 2 } . \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} \\int _ { B _ R } | x | ^ { \\frac { n } { p } \\theta } | \\nabla u _ { \\lambda } ( x ) | ^ { \\theta } \\frac { d x } { | x | ^ n } = \\omega _ { n - 1 } \\int _ 0 ^ R \\lambda ^ \\theta r ^ { \\frac { n } { p } \\theta - 1 } \\left | \\frac { d } { d r } u _ { \\lambda } ( \\phi _ { \\lambda } ( r ) ) \\right | ^ { \\theta } d r . \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} \\boldsymbol { \\Phi } _ { \\mathbf { s } , c } ^ { \\mathbb { \\epsilon } } ( x ) = \\tilde { \\Phi } _ { \\frac { n _ l } { 2 } - c } ^ { \\epsilon _ 1 } \\left ( \\frac { x b _ 1 } { a ^ { ( 1 ) } b ^ { ( 1 ) } N _ 0 N _ 1 } \\right ) \\overline { \\tilde { \\Phi } _ { \\frac { n _ l } { 2 } - c } ^ { \\epsilon _ 2 } \\left ( \\frac { x b _ 1 } { a ^ { ( 2 ) } b ^ { ( 2 ) } N _ 0 N _ 1 } \\right ) } \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} f _ { \\alpha , \\beta } ( x ) = | x | ^ \\alpha \\sin ( 1 / | x | ^ \\beta ) = \\frac { 1 } { 2 i } | x | ^ \\alpha e ^ { i | x | ^ { - \\beta } } - \\frac { 1 } { 2 i } | x | ^ \\alpha e ^ { - i | x | ^ { - \\beta } } . \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} x _ { i } = \\frac { \\kappa _ 2 - 2 \\kappa _ 1 \\kappa _ 3 } { 1 - \\kappa _ 1 ( 1 - \\epsilon ) } , \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} a _ { k m / d } ( \\omega ^ { i d } _ { k } ) = a _ { k m / d } ( \\omega ^ i _ { k / d } ) = a _ { m ( k / d , i ) } ( 1 ) . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} \\ell ( x ; a , b ) = 2 \\frac { x - a } { b - a } - 1 \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} \\langle \\mathrm { J } _ L ( E ) , \\zeta \\rangle = \\frac { 1 } { 2 } \\Omega ( \\zeta E , E ) . \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} \\overline { S } ( \\overline { R } ( X , Y ) Z , U ) + \\overline { S } ( Z , \\overline { R } ( X , Y ) U ) = 0 . \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} C = \\begin{pmatrix} i \\\\ & \\ddots \\\\ & & i \\\\ & & & - i & i \\\\ & & & 1 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} e _ { n , k } ( \\omega _ n ^ { g ' } ) = [ t ^ k ] \\mathrm { T r } ( A ^ { ( g ' , n ) } _ { \\omega } ( t ) ) = [ t ^ k ] \\mathrm { T r } ( A ^ { ( g , n ) } _ { \\omega } ( t ) ) . \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} | \\sum _ { b = 1 } ^ n k _ b j _ b ^ 2 + i ^ 2 - j ^ 2 | < \\frac { 1 } { 5 0 n } | i - j | , \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} \\rho _ { _ \\mathrm { U B } } \\triangleq & \\ , \\bigg \\{ \\rho \\ ; \\bigg | \\det \\left ( \\mathbf { I } _ { N _ R } + \\sigma ^ { - 2 } { \\left ( 1 - \\rho \\right ) \\mathbf { H } \\mathbf { S _ { _ \\mathrm { W F } } } \\mathbf { H } ^ { \\rm H } } \\right ) = 2 ^ { { R } } \\ , \\ , { \\rm a n d } \\ , \\ , 0 \\le \\rho \\le 1 \\bigg \\} . \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} \\frac { w ( v , v ' ) } { m ( v ) } \\leq \\sum _ { v \\sim v ' } \\frac { w ( v , v ' ) } { m ( v ) } = 1 , \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} K _ { t _ { 1 } } ( k , n ) = c _ k ( e ^ { t _ { 1 } ( \\cdot - s ^ { + } ) } p _ n ) , \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} S _ { n , 2 } ( \\omega ) = \\mathrm { o r d } ( \\omega ) ^ 4 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , \\frac { y ^ 2 ( x - y ) ^ 2 } { 2 } - \\frac { ( x - y ) ^ 2 } { 6 } } + \\mathrm { o r d } ( \\omega ) ^ 2 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , \\frac { ( x - y ) ^ 2 } { 6 } } . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) = \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ g ( x _ i ^ { \\vee } \\cdot y _ j ^ { \\vee } \\cdot u _ k ^ { \\vee } ) u _ k + ( \\mbox { h i g h e r o r d e r t e r m s i n } \\vec { u } ) . \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} \\theta _ { 1 2 } \\theta _ { 1 3 } \\theta _ { 2 3 } = \\theta _ { X _ 1 \\otimes X _ 2 \\otimes X _ 3 } \\theta _ 1 \\theta _ 2 \\theta _ 3 . \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} W _ \\xi ( \\lambda , \\mu ) = L ^ { } ( \\lambda , \\mu ) \\otimes _ { \\mathcal { A } } \\mathbb { C } . \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\theta } \\left ( { \\dot \\kappa } ^ { A } ( \\theta ) + { \\dot \\kappa } ^ { - S } ( \\theta ) \\right ) = { \\ddot \\kappa } ^ { A } ( \\theta ) + { \\ddot \\kappa } ^ { - S } ( \\theta ) \\ge 0 , \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} \\mu _ i = \\sum _ { j = 1 } ^ { i - 1 } k _ { j - 1 } ( f ) - \\sum _ { j = 1 } ^ i f ( j ) + \\frac { ( n - i + 1 ) ( n - i + 2 ) } { 2 } \\eta _ i = f ( i + 1 ) - f ( i ) - 1 . \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} | I | ^ { - 1 } \\int _ I \\left | \\log w - ( \\log w ) _ I \\right | \\ , d x & = 2 | I | ^ { - 1 } \\int _ I \\left [ \\log w - ( \\log w ) _ I \\right ] _ + \\ , d x \\\\ & \\le 2 \\log \\left ( | I | ^ { - 1 } \\int _ I \\exp \\left [ \\log w - ( \\log w ) _ I \\right ] _ + \\ , d x \\right ) \\\\ & \\le 2 \\log \\left ( | I | ^ { - 1 } \\int _ I \\exp \\left [ \\log w - ( \\log w ) _ I \\right ] \\ , d x + 1 \\right ) \\\\ & \\le 2 \\log ( [ w ] _ { A _ \\infty ( \\mathbb R ) } + 1 ) , \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} S ( x ) = \\sum _ { q \\geq 0 } v _ q x ^ { \\frac 1 { q + 1 } } . \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi ( x _ 0 , \\kappa ^ j r ) & \\le \\kappa ^ { \\gamma j } \\Phi ( x _ 0 , r ) \\\\ & + C \\big ( \\| D u \\| _ { L ^ \\infty ( B _ r ( x _ 0 ) ) } \\tilde { \\omega } _ { A ^ { \\alpha \\beta } } ( \\kappa ^ j r ) + \\tilde { \\omega } _ { f _ \\alpha } ( \\kappa ^ j r ) + \\tilde { \\omega } _ g ( \\kappa ^ j r ) \\big ) , \\end{aligned} \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} \\mathcal { C } _ { n e t } = \\mathbb { P \\left [ A _ { U E } \\right ] } \\mathcal { C } , \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} \\pi \\sum _ { k = 0 } ^ { \\infty } \\frac { ( 1 - p ( w ) ) ^ { k + 1 } } { k + 1 } \\abs { a _ k ( w ) } ^ 2 < \\infty . \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} N _ { D \\circ F } ^ + [ ( d , f ) ] = N _ { D } ^ + ( d ) \\times V ( F ) \\cup \\{ d \\} \\times N _ { F } ^ + [ f ] . \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\beta ( u , \\epsilon , P ) & = & \\displaystyle \\liminf _ { R \\to \\infty } \\frac { \\log K _ R ( \\epsilon , P ) } { u ( t ) } ; \\\\ e ( u , P ) & = & \\displaystyle \\limsup _ { \\epsilon \\to 0 } \\beta ( u , \\epsilon , P ) ; \\\\ e ( ( T _ t ) , u ) & = & \\displaystyle \\sup _ { P } e ( u , P ) . \\end{array} \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} G _ i ( x ) = x G _ { i - 1 } ( x ) - ( k - 1 ) G _ { i - 2 } ( x ) \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} \\begin{aligned} c ( x ' , y , t ) = { } & \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } ( c ( x , y , t ) - x \\frac { \\partial } { \\partial y } b ( x , y , t ) \\\\ & + h ( x ) \\left [ \\frac { \\partial } { \\partial y } ( y \\alpha ( y , t ) ) + \\frac { 1 } { 2 } x ^ { 2 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) - \\frac { \\partial } { \\partial t } \\alpha ( y , t ) \\right ] ) \\end{aligned} \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} \\tilde { \\chi } _ j ( t ) = \\begin{cases} \\delta _ j ^ { - 1 / 2 } , & t _ { j - 1 } < t < t _ { j } , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} \\hat I ' ( t ) \\phi ' ( x _ 1 , x _ 2 , t ) = E \\phi ' ( x _ 1 , x _ 2 , t ) , \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} \\| \\sigma _ \\psi ( A ) \\xi _ \\psi \\| ^ 2 = \\langle \\sigma _ \\psi ( A ^ * A ) \\xi _ \\psi , \\xi _ \\psi \\rangle = \\psi ( A ^ * A ) = \\| A \\| ^ 2 . \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} g = \\begin{pmatrix} A ( g ) \\ , e ^ { \\imath \\theta _ G ( g ) } & - \\overline { \\gamma ( g ) } \\\\ \\gamma ( g ) & A ( g ) \\ , e ^ { - \\imath \\theta _ G ( g ) } \\end{pmatrix} , \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} F _ { \\eta } = \\left \\{ n \\geq n _ 0 : \\frac { T _ { n + s } - T _ n } { T _ n } \\cdot h ( n ) \\geq \\eta \\right \\} \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} D _ { n - p } [ ( - z ) ^ { - p } \\sigma ] = ( - 1 ) ^ { p } \\det X \\cdot D _ { n } [ \\sigma ] . \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{gather*} S ^ { \\delta _ n ^ k - \\alpha _ k + 1 } \\left ( \\alpha \\right ) = \\alpha , S ^ { \\delta _ n ^ k + 2 - \\alpha _ { k - 1 } } \\left ( \\alpha \\right ) = \\alpha , \\dots , S ^ { \\delta _ n ^ k + 1 + u - \\alpha _ { k - u } } \\left ( \\alpha \\right ) = \\alpha , \\dots , S ^ { \\delta _ n ^ k + k - \\alpha _ { 1 } } \\left ( \\alpha \\right ) = \\alpha \\end{gather*}"}
-{"id": "7745.png", "formula": "\\begin{align*} [ f ; D _ { d , \\epsilon } ( t ) f ] = \\sum _ { j k \\in E } ( f _ j - f _ k ) ^ 2 e ^ { t _ { j k } } + \\epsilon \\sum _ { j \\in \\mathbb { Z } ^ d } f _ j ^ 2 \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} - 1 - x _ 1 ( 0 ) = X _ 1 , ~ x _ { i - 1 } ( 0 ) - x _ { i } ( 0 ) - 1 = X _ i ~ i = 2 , \\cdots , N , \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} | | D \\Phi ^ { \\nu } | | _ { s _ 0 , r _ 0 , s _ { \\nu } , r _ { \\nu } ; D _ { \\nu } \\times \\mathcal { O } _ { \\nu } } \\leq \\prod _ { \\mu = 1 } ^ { \\nu } | | D \\Phi _ { \\mu } | | _ { s _ { \\mu } , r _ { \\mu } , s _ { \\mu } , r _ { \\mu } ; D _ { \\mu } \\times \\mathcal { O } _ { \\mu } } \\leq \\prod _ { \\mu = 1 } ^ { \\infty } ( 1 + \\frac { \\gamma _ 0 } { 2 ^ { \\mu + 6 } } ) \\leq 2 , \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} ( 0 , w ) \\C e _ v = \\C e _ { u } , w \\in U ^ \\perp , v \\in C , \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\mathbb { A _ { U E } } \\right ] = 2 \\left ( \\pi \\lambda _ { b } \\right ) ^ 2 \\int _ { 0 } ^ { \\infty } x _ { 2 1 } \\mathrm { e } ^ { - \\pi \\lambda _ { b } x _ { 2 1 } ^ 2 } { \\min ^ 2 } \\left ( x _ { 2 1 } , \\beta ^ { \\frac { \\mu - 2 } { 4 \\mu } } x _ { 2 1 } ^ { \\frac { 2 } { \\mu } } \\right ) \\mathrm { d } x _ { 2 1 } \\approx \\frac { 2 \\ , \\Gamma \\left ( \\frac { 2 } { u } \\right ) } { \\mu \\left ( \\pi \\lambda _ b \\right ) ^ { \\frac { 2 } { \\mu } - 1 } \\beta ^ { \\frac { 1 } { \\mu } - \\frac { 1 } { 2 } } } , \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} ( H _ 1 { \\bf 1 } ) ^ \\top ( H _ 1 { \\bf 1 } ) = { \\bf 1 } ^ \\top ( \\ell I _ n + a A + b ( J _ n - A - I _ n ) ) { \\bf 1 } = ( \\ell + a k + b ( n - 1 - k ) ) n = 0 , \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\gamma ( \\chi , s , \\psi ) Z ( \\varphi , \\chi , s ) = Z ( \\hat \\varphi , \\chi ^ { - 1 } , 1 - s ) . \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} \\deg ( a \\prod _ { i = 1 } ^ n x _ i ^ { \\alpha _ i } ) : = \\sum _ { i = 1 } ^ n \\alpha _ i \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} \\bar \\rho : = \\frac { 1 } { 2 } \\left ( \\sum _ { \\alpha \\in \\Phi ( + ) \\cup \\Phi ( 0 ) _ + } \\alpha \\right ) - \\delta \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } \\alpha ( \\ddot { x } ) d \\mu _ \\rho ( \\ddot { x } ) = \\int _ G \\alpha \\big ( q ( x ) \\big ) f ( x ) d x . \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\frac { { X _ { m + b - ( a - b ) j } } } { { ( - f _ 2 / f _ 1 ) ^ j } } } = f _ 2 X _ m + \\frac { f _ 1 } { { ( - f _ 2 / f _ 1 ) ^ k } } X _ { m - ( k + 1 ) ( a - b ) } \\ , , \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} \\rho ^ { \\prime } ( \\sqrt { 2 } \\delta ( v ) ) = \\frac { 2 } { - \\psi ^ { \\prime \\prime } \\left ( \\rho ( \\sqrt { 2 } \\delta ( v ) ) \\right ) } < 0 \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} \\begin{aligned} a ^ + + \\frac { 1 } { 4 } R ^ 2 & = 1 , \\\\ a ^ - + b ^ - \\log R + \\frac { 1 } { 4 } R ^ 2 & = 1 , \\\\ b ^ - + \\frac { 1 } { 2 } R _ 1 ^ 2 & = 0 , \\\\ R ^ 2 + b ^ - = 0 . \\end{aligned} \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} & \\| f i ( f i ( M _ 1 ) x _ { k _ { i n } } ) - M _ 1 x _ { k _ { i n } } \\| \\\\ = & \\| f i ( f i ( M _ 1 ) x _ { k _ { i n } } ) - f i ( M _ 1 ) x _ { k _ { i n } } + f i ( M _ 1 ) x _ { k _ { i n } } - M _ 1 x _ { k _ { i n } } \\| \\\\ \\leq & \\| f i ( f i ( M _ 1 ) x _ { k _ { i n } } ) - f i ( M _ 1 ) x _ { k _ { i n } } \\| + \\| ( f i ( M _ 1 ) - M _ 1 ) x _ { k _ { i n } } \\| \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} u _ { } ( x ) = \\max \\bigg \\{ 0 , \\sqrt { 1 - \\frac { 1 } { 4 } \\Omega ^ 2 x ^ 2 } \\bigg \\} \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} f ( x ) = V _ 0 ^ x ( f ) - \\big ( V _ 0 ^ x ( f ) - f ( x ) \\big ) \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} & \\dim \\left ( ( \\pi ^ + ) ^ { - 1 } ( U ^ { ( b ) } ) \\times _ { U ^ { ( b ) } } ( \\pi ^ - ) ^ { - 1 } ( U ^ { ( b ) } ) \\right ) \\\\ & \\le ( a - 1 ) + ( b - 1 ) + ( g - a - b + 1 ) \\\\ & = g - 1 . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} \\mathcal { D } _ t u ^ { n - \\theta } + \\gamma \\bigtriangleup \\sigma ^ { n - \\theta } - \\bigtriangleup u ^ { n - \\theta } + f ^ { n - \\theta } ( u ) = g ( \\textbf { z } , t _ { n - \\theta } ) , \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} \\geq \\sum _ { j < i } r _ j I _ j + r _ i ( I _ i + 1 ) > \\sum _ { j \\leq i } r _ j I _ j + \\sum _ { j > i } r _ j I _ j = \\sum _ { j = 1 } ^ n r _ j I _ j , \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} - \\dfrac { \\log C ( t , \\pi ) } { 2 \\pi } + \\dfrac { 1 } { \\pi } \\displaystyle \\sum _ { \\gamma } m ^ { - } _ { \\beta , \\Delta } ( t - \\gamma ) + O ( d ) \\leq S _ { - 1 , \\sigma } ( t , \\pi ) . \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} u _ 1 ^ { q ^ 2 + 1 } + u _ 1 ^ { q ^ 2 } + 1 = 0 \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d - 1 } y _ i \\int _ { \\bf { a _ i } } \\omega _ j = 2 y _ d \\int _ { c _ 1 } ^ \\infty \\omega _ j , \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} \\Z _ l ^ { \\times } = \\bigsqcup _ { ( a , b , k ) \\in S ^ 0 } \\Z _ { l } ^ { \\times } [ a , b , k ] \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} F ( X , Y , Z , 0 ) & = F ( 0 , Y , Z , 0 ) + X \\int _ { 0 } ^ 1 \\partial _ X F ( \\alpha X , Y , Z , 0 ) { \\rm d \\alpha } \\\\ & = X \\int _ { 0 } ^ 1 \\int _ { 0 } ^ 1 \\partial _ X F ( \\alpha X , 0 , Z , 0 ) + Y \\partial _ X \\partial _ Y F ( \\alpha X , \\beta Y , Z , 0 ) { \\rm d \\beta } { \\rm d \\alpha } \\\\ & = X Y \\int _ { 0 } ^ 1 \\int _ { 0 } ^ 1 \\partial _ X \\partial _ Y F ( \\alpha X , \\beta Y , Z , 0 ) { \\rm d \\beta } { \\rm d \\alpha } . \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} \\bar { Q } = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n _ 1 } k _ i ( ( \\bar { v } _ i ^ { \\mathcal { I } } ) ^ \\times ) ^ 2 + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { n _ 2 } k _ { j + n _ 1 } ( ( v _ j ^ { \\mathcal { I } } ) ^ \\times ) ^ 2 , \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} Z _ { \\mathcal I } = P _ { I _ 1 } \\times ^ B Z _ { ( I _ 2 , \\dots , I _ r ) } . \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} h ( x ) = \\frac { 1 } { \\pi \\sqrt { 1 - x ^ { 2 } } } = \\frac { 1 } { \\pi | \\sin \\left ( \\pi u _ { 0 } \\right ) | } \\Rightarrow \\frac { h ( x ) } { | T _ { N } ' ( x ) | } = \\frac { 1 } { N \\pi | \\sin \\left ( N \\pi u _ { 0 } \\right ) | } \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{gather*} \\overline { \\nabla } ^ { \\mathrm { b a s } , \\overline { g } } _ v \\mathrm { h o r } ( b ) = 0 , v \\in \\Gamma ( V ) , b \\in \\Gamma ( A ) , \\end{gather*}"}
-{"id": "1119.png", "formula": "\\begin{align*} I _ \\times ( \\lambda ) \\ & = \\ \\{ \\lambda _ 1 , \\lambda _ 2 - 1 , . . . , \\lambda _ n - n + 1 \\} \\\\ I _ \\circ ( \\lambda ) \\ & = \\ \\{ 1 - n - \\lambda _ { n + 1 } , 2 - n - \\lambda _ { n + 2 } , . . . , - \\lambda _ { 2 n } \\} . \\end{align*}"}
-{"id": "9983.png", "formula": "\\begin{align*} \\mathbb P ( A ) & = \\mathbb P ( R ( \\tau - { \\Delta x } ) \\in [ x + \\Delta x , x + 2 \\Delta x ] ) = u ( x , \\tau ) \\Delta x - \\partial _ \\tau u ( x , \\tau ) ( \\Delta x ) ^ 2 + \\frac 3 2 \\partial _ x u ( x , t ) ( \\Delta x ) ^ 2 + o ( ( \\Delta x ) ^ 2 ) . \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to \\infty } \\frac { \\widehat { x } _ { n , k } ^ { ( \\alpha ) } - \\alpha } { \\sqrt { 2 \\alpha } } = h _ { n - 1 , k - 1 } \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} ( _ F ( n , k , \\Omega ) ) ^ \\bot = _ F ( n , n - k , \\Omega , v ) \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} \\sum _ { d \\mid n } \\mu ( d ) a _ { n m / d } ( \\omega _ n ^ { i d } ) = \\sum _ { d \\mid n } \\mu ( d ) a _ { \\frac { n } { d } m } ( \\omega _ { n / d } ^ i ) = \\sum _ { d \\mid n } \\mu ( d ) a _ { ( \\frac { n } { d } , i ) m } ( 1 ) . \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} F _ { i } ( p _ { 1 } , p _ { 2 } ) = - \\frac { \\epsilon } { \\Delta x } \\left ( p _ { 1 } - p _ { 2 } \\right ) + G _ { i } \\left ( p _ { 1 } , p _ { 2 } \\right ) . \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} e ( s _ 1 , \\ldots , s _ N ) = g ( t _ 1 , \\ldots , t _ n ) . \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} & a _ { 1 } ^ 2 + a _ { 1 } ^ 2 - \\| \\bar { a } \\| ^ 2 = 1 , \\\\ & 2 a _ 1 a _ { 2 } - \\| \\bar { a } \\| ^ 2 = 0 , \\\\ & - ( a _ 1 + a _ 2 ) \\bar { a } + C \\bar { a } = \\textbf { 0 } , \\\\ & C ^ 2 - 2 \\bar { a } \\bar { a } ^ T = I _ n . \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} \\mathrm { T } ( \\theta ) = \\left \\{ \\begin{array} { l l } \\left \\{ \\frac { 1 } { \\theta } \\right \\} & \\theta \\in ( 0 , 1 ] \\\\ 0 & \\theta = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} \\mu ( \\alpha ( a ) , \\mu ( b , c ) ) = \\mu ( \\mu ( a , b ) , \\alpha ( c ) ) , ~ ~ a , b , c \\in A . \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} v _ { k , j } : = P _ { \\theta } ( \\min ( \\varphi _ k , . . . , \\varphi _ { k + j } ) ) . \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} e _ { n , k } ( \\omega _ n ^ { g ' } ) = [ t ^ k ] \\mathrm { T r } ( A ( t ^ { \\frac { n } { ( g , n ) } } , 1 ) ^ { ( g , n ) } ) . \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} k ( x ) = \\int _ { ( x , \\infty ) } ( \\psi ^ { - 1 } ) ' ( \\log ( r ) - \\log ( x ) ) \\nu ( d r ) = \\int _ { ( x , \\infty ) } \\frac { 1 } { \\psi ' ( \\psi ^ { - 1 } ( \\log ( r ) - \\log ( x ) ) ) } \\nu ( d r ) . \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} x ' = \\cos \\left ( N \\pi u _ { 0 } + 2 \\pi j \\right ) = \\cos \\left ( N \\pi u _ { 0 } \\right ) \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} c _ k \\theta _ { 1 3 } ( I ) & = c _ k \\theta _ { 1 3 } ( v _ k v _ k ^ * ) = \\sum _ { n = 1 } ^ m c _ n \\theta _ { 1 3 } ( v _ n v _ k ^ * ) \\\\ & = \\left ( \\sum _ { n = 1 } ^ m c _ n \\theta _ { 1 2 } ( v _ n ) \\right ) \\theta _ { 2 3 } ( v _ k ^ * ) = 0 \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} \\mathbb E [ \\overline { b } ( s , x , \\overline { X } _ { s , j } ) ] \\ , = \\ , \\int _ { \\mathbb R } \\widetilde { b } ( s , x , z ) \\mathrm m _ { s } ( { \\mathrm d } z ) - \\int _ { \\mathbb R } \\widetilde { b } ( s , x , y ) \\mathrm m _ { s } ( { \\mathrm d } y ) \\ , = \\ , 0 \\ , ; s \\ge 0 , \\ , \\ , x \\in \\mathbb R \\ , . \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} \\lVert \\partial ^ 1 f \\rVert _ \\infty = \\sup _ { g , h \\in G } \\lvert f ( g h ) - f ( g ) - f ( h ) \\rvert = D ( f ) < \\infty . \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} B + \\epsilon _ 1 B ^ 2 \\leq \\frac { \\epsilon + \\epsilon _ 1 } { \\epsilon ^ 2 } \\begin{pmatrix} I & - I \\\\ - I & I \\end{pmatrix} , \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} Q = \\frac { 1 } { 2 \\pi ( 1 - e ^ { - 2 t } ) } \\left ( \\begin{array} { c c } 1 - ( 1 - e ^ { - 2 t } ) c _ 1 & - e ^ { - t } \\\\ - e ^ { - t } & 1 - ( 1 - e ^ { - 2 t } ) c _ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} E _ a ^ \\dagger E _ b = ( E _ a ' ) ^ \\dagger E ' _ b a , b = 1 , \\ldots , m . \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { a \\in K } \\| P _ N a - a \\| = 0 \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} + \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { ( k + 1 ) ^ { s - 3 } } \\sum _ { r _ 1 = 1 } ^ { n } \\sum _ { r _ 2 = 1 } ^ { n } \\sum _ { r _ 3 = 1 } ^ { n } \\frac { a _ { r _ 1 } b _ { r _ 2 } c _ { r _ 3 } } { ( r _ 1 + k + 1 ) ( r _ 2 + k + 1 ) ( r _ 3 + k + 1 ) } . \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} Q _ h \\phi = \\Lambda ^ { - 1 } h ^ { - 1 } Q ^ f ( h \\phi ) . \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} S _ 2 ^ \\epsilon : = \\epsilon ^ { - \\alpha } \\int _ { X _ \\epsilon } o s c ( \\psi _ 2 , B _ \\epsilon ( x ) ) d x \\leq C _ 0 \\epsilon ^ { - \\alpha } ( r + \\epsilon ) ^ N \\leq 2 ^ N \\epsilon _ 0 ^ { N - \\alpha } . \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} c ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } \\left [ c ( x , y , t ) - \\frac { 1 } { 6 } x ^ { 3 } h ( x ) \\frac { \\partial ^ { 3 } } { \\partial y ^ { 3 } } p _ { 0 } ( y , t ) \\right ] \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { ( - 1 ) ^ { n ( j + s ) + s } \\binom k j \\binom j s \\frac { { G _ 0 ^ { j - s } G _ n ^ s } } { { G _ { n + r } ^ j } } H _ { m + n k + ( r - n ) j + n s } } } = ( - 1 ) ^ { n k } \\left ( { \\frac { { G _ r } } { { G _ { n + r } } } } \\right ) ^ k H _ m , G _ { n + r } \\ne 0 \\ , , \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} y ^ 2 = 4 x ^ 3 - g _ 8 ( z , w ) x - g _ { 1 2 } ( z , w ) \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} \\tilde { e } _ { i _ 1 } ^ { } \\left ( b \\otimes b ^ \\prime \\right ) & = \\tilde { e } _ { i _ 1 } ^ { } \\left ( \\tilde { f } _ { i _ 1 } ^ { a _ 1 } \\cdots \\tilde { f } _ { i _ l } ^ { a _ l } ( b _ \\lambda ) \\otimes b ^ \\prime \\right ) \\\\ & = \\tilde { f } _ { i _ 2 } ^ { a _ 2 } \\cdots \\tilde { f } _ { i _ l } ^ { a _ l } ( b _ \\lambda ) \\otimes \\tilde { e } _ { i _ 1 } ^ { r _ 1 } ( b ^ \\prime ) \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} D : = \\{ \\left ( \\mathcal { A } ( \\gamma , \\xi ) , \\ \\langle ( k + 2 \\epsilon , M \\epsilon ) , ( \\gamma , \\xi ) \\rangle \\ \\right ) \\ : \\ ( \\gamma , \\xi ) \\in { \\cal M } _ + ( Y \\times U ) \\times { \\cal M } _ + ( Y \\times U ) \\} \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 4 8 n + 6 ) q ^ n & \\equiv f _ 1 ^ 3 \\equiv f _ 3 a ( q ^ 3 ) - 3 q f _ 9 ^ 3 ~ ( \\textup { m o d } ~ 2 ) , \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { Z } } \\varphi _ j ( \\cdot ) = 1 \\ \\mbox { i n } \\ \\R ^ n \\setminus \\{ 0 \\} \\ \\mbox { f o r } \\ , \\ , \\varphi _ j ( \\cdot ) = \\varphi ( 2 ^ { - j } \\cdot ) . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} v ^ * = \\sum _ \\sigma f _ \\sigma \\ , \\alpha _ { \\sigma ( 1 ) } \\wedge \\ldots \\wedge \\alpha _ { \\sigma ( k ) } , \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\widetilde { \\Psi } _ { S S S } = \\max \\{ \\psi _ { 1 } ( \\tau _ { n } ) , { \\widetilde \\psi } _ { 2 } ( \\tau _ { n } ' ) \\} \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\gamma = \\gamma _ 0 \\gamma _ s , \\gamma _ s = \\frac { 1 } { 8 0 } ( \\prod _ { \\mu = 0 } ^ { \\infty } ( 2 ^ { \\mu } B _ { \\mu } ) ^ { - \\frac { 1 } { 3 \\kappa ^ { \\mu + 1 } } } ) ^ { 1 - 2 \\beta ' } , \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} \\begin{array} { c c c c c } A _ { 1 , s } \\zeta ( s ) + A _ { 1 , s - 1 } \\zeta ( s - 1 ) & + \\dots + A _ { 1 , 3 } \\zeta ( 3 ) & = & A _ { 1 , 2 } \\zeta ( 2 ) + A _ 1 + \\theta _ s \\\\ A _ { 2 , s - 1 } \\zeta ( s - 1 ) & + \\dots + A _ { 2 , 3 } \\zeta ( 3 ) & = & A _ { 2 , 2 } \\zeta ( 2 ) + A _ 2 + \\theta _ { s - 1 } \\\\ \\dots \\dots \\dots \\dots & \\dots \\dots \\dots \\dots & \\dots \\dots \\dots \\dots & \\dots \\dots \\dots \\dots \\\\ & A _ { s - 2 , 3 } \\zeta ( 3 ) & = & A _ { s - 2 , 2 } \\zeta ( 2 ) + A _ { s - 2 } + \\theta _ { 3 } \\\\ \\end{array} , \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} \\inf \\bigg \\{ | | w | | ( \\overline \\Omega ) \\ , : \\ , w \\in \\mathcal M ^ 2 ( \\overline \\Omega ) , \\ , \\nabla \\cdot w = f \\bigg \\} , \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\tilde { m } _ { \\xi } = L ^ { N } \\tilde { b } _ { \\xi } , \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} \\theta ( x ) ^ \\prime = \\gamma ^ \\prime ( x ) - \\frac { V ( x ) } { 2 \\gamma ^ \\prime ( x ) } \\sin ^ 2 \\theta ( x ) . \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} ( S _ \\varepsilon ( t ) - p _ { \\varepsilon } ( t ) ) ^ 2 & = ( S _ \\varepsilon ( 0 ) - p _ { \\varepsilon } ( 0 ) ) ^ 2 + 2 \\int _ 0 ^ t ( \\dot { S _ \\varepsilon } ( \\tau ) - \\dot { p } _ { \\varepsilon } ( \\tau ) ) ( S _ \\varepsilon ( \\tau ) - p _ { \\varepsilon } ( \\tau ) ) d \\tau , \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} = - K _ 8 \\gamma _ k ^ { \\frac { 2 } { 1 + \\epsilon } } + \\alpha _ k \\gamma _ k + K _ 9 \\beta _ k ^ q + K _ { 1 0 } \\beta _ k \\leq - K _ 8 \\gamma _ k ^ { \\frac { 2 } { 1 + \\epsilon } } + \\alpha _ k \\gamma _ k + K _ { 1 1 } \\beta _ k , \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} \\nabla f ( \\mathbf { x } ^ { k + 1 } ) - \\nabla f ( \\mathbf { x } ^ { \\star } ) + ( 1 - \\eta ) \\frac { \\rho } { 2 } \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } ( \\mathbf { x } ^ { k + 1 } - \\mathbf { x } ^ { \\star } ) = \\\\ \\mathbf { E } _ { } ^ T ( \\boldsymbol { \\alpha } ^ { \\star } - \\boldsymbol { \\alpha } ^ { k + 1 } ) + \\frac { \\rho } { 2 } \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } ( \\mathbf { z } ^ { k } - \\mathbf { z } ^ { k + 1 } ) , \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} \\beta ^ { u ' } \\alpha _ i ^ { u + p _ i - 1 } \\in K , \\ ; u ' = 0 , 1 , \\dots , u - 1 . \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} \\ < \\xi _ \\sigma , f ( \\Delta _ { \\rho , \\sigma } ) \\xi _ \\sigma \\ > : = \\int _ { ( 0 , + \\infty ) } f ( t ) \\ , d \\| E _ { \\rho , \\sigma } ( t ) \\xi _ \\sigma \\| ^ 2 . \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} v _ { p e r } ( y _ 0 ) = \\liminf _ { T \\rightarrow \\infty } v _ T ( y _ 0 ) , \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} \\begin{aligned} S _ { n , p } \\left ( \\frac { n - p } { p - 1 } \\right ) ^ { - \\frac { n - 1 } { n } } \\to \\frac { \\sqrt { \\pi } n ^ { \\frac 1 n } } { \\Gamma \\left ( 1 + \\frac { n } { 2 } \\right ) ^ { \\frac 1 n } } \\end{aligned} \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow \\infty } I ( v _ n ) \\geq \\liminf _ { n \\rightarrow \\infty } \\left [ \\frac { 1 } { 2 } \\norm { v _ n } ^ 2 - \\int _ { \\mathbb R ^ 3 } F ( v _ n ) d x \\right ] = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{gather*} \\mu ^ r = \\mu ^ l \\circ S ^ { - 1 } . \\end{gather*}"}
-{"id": "3150.png", "formula": "\\begin{align*} | a - b | = \\sqrt { 2 ( 1 - ( a . b ) ) } \\leq \\sqrt { 2 ( 1 - ( a . b ) ^ 2 ) } = \\sqrt { 2 | a - ( a . b ) b | ^ 2 } \\leq \\varepsilon / \\sqrt { 2 } , \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} f ( ( p _ 1 , \\dots , p _ r ) \\cdot ( b _ 1 , \\dots , b _ r ) ) = e ^ { \\lambda _ 1 } ( b _ 1 ) \\cdots e ^ { \\lambda _ r } ( b _ r ) f ( p _ 1 , \\dots , p _ r ) \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} c \\int _ { K } V ^ + | \\xi | \\leq \\| \\xi \\| ^ { U } _ { V , \\Omega } : = \\sqrt { Q _ V ( \\xi ) } + \\int _ { U } | \\xi | \\ ; \\ ; \\ ; \\forall \\xi \\in W _ c ^ { 1 , \\infty } ( \\Omega ) . \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} I _ { T , t } ^ { ( i _ 1 i _ 2 ) } = \\int \\limits _ t ^ T \\int \\limits _ t ^ s d { \\bf w } _ { \\tau } ^ { ( i _ 1 ) } d { \\bf w } _ s ^ { ( i _ 2 ) } \\ \\ \\ ( i _ 1 , i _ 2 = 1 , \\ldots , m ) , \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} K _ { i j } ( 1 , y ) = 0 \\mbox { f o r } y \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} \\pi _ \\lambda ( g ) \\circ T ^ { ( \\lambda ) } _ \\varphi = T ^ { ( \\lambda ) } _ { \\pi ( g ) \\varphi } \\circ \\pi _ \\lambda ( g ) \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} 1 + \\frac { 1 } { ( 2 \\pi ) ^ k } \\sum _ { j = 0 } ^ { k - 1 } f ' ( 0 ) ^ { 2 k - 1 - 2 j } \\tilde { c } ( k - 1 - j ) \\int _ N P _ { j , 2 k - 1 } ( h ) . \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} \\nabla _ { \\partial _ t } \\nabla _ { \\partial _ t } Y - \\nabla _ { \\nabla _ { \\partial _ t } \\partial _ t } Y & = \\nabla _ { \\nabla _ { Y ^ T } \\partial _ t } \\partial _ t - u \\nabla _ { \\nabla u } Y ^ T + \\langle u \\nabla u , \\nabla \\frac { Y ^ { \\perp } } { u } \\rangle \\cdot \\partial _ t \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} z _ { t _ k } = x _ 2 ( t _ k ) - 0 . 0 0 1 x _ 2 ^ 3 ( t _ k ) + ( x _ 2 ( t _ k ) - 0 . 0 1 x _ 2 ^ 2 ( t _ k ) ) \\xi _ { t _ k } + e _ { t _ k } , \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} & \\nabla _ { \\nabla _ { Y ^ T } \\partial _ t } \\partial _ t = \\nabla _ { - \\frac { 1 } { 2 } u ^ 2 d \\theta ( Y ^ T ) + u ^ { - 1 } Y ^ T ( u ) \\cdot \\partial _ t } \\partial _ t \\\\ & = \\nabla _ { - \\frac { 1 } { 2 } u ^ 2 d \\theta ( Y ^ T ) } \\partial _ t + u ^ { - 1 } Y ^ T ( u ) \\nabla _ { \\partial _ t } \\partial _ t \\\\ & = \\frac { 1 } { 4 } u ^ 4 d \\theta ( d \\theta ( Y ^ T ) ) - \\frac { 1 } { 2 } u d \\theta ( Y ^ T , \\nabla u ) \\cdot \\partial _ t + Y ^ T ( u ) \\nabla u . \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} \\mathcal { M } ^ { ( k \\boldsymbol { \\nu } ) } _ { a , a } ( g ) = e ^ { - 2 \\imath a \\theta _ G ( g ) } \\cdot R _ { \\ell , a } \\left ( A ( g ) ^ 2 \\right ) , \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} \\mathcal F ( x ) = \\int _ 0 ^ 1 g ( \\dot x , \\dot x ) \\ , \\mathrm d s . \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} f ( x ) & = ( 1 - x ^ 2 ) - 2 ( x ^ 3 + 1 ) + ( m + 1 ) \\ : h ( x ) , \\\\ g ( x ) & = - x ^ 4 + x ^ 2 ( x ^ 3 + 1 ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { \\infty } \\Omega ( \\ell ) = \\sum _ { j = 0 } ^ { 3 } \\left \\{ \\sum _ { \\ell = 0 } ^ { \\infty } \\Omega ( 4 \\ell + j ) \\right \\} , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} w = \\beta _ 1 \\beta _ 2 \\dots \\beta _ { p _ w } \\alpha _ w s _ w , \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} \\left \\Vert D ^ 1 f - \\tilde { f } ^ { ( 1 ) } \\right \\Vert _ \\infty = \\max _ { t \\in [ 0 , T ] } \\left | D ^ 1 f ( t ) - \\tilde { f } ^ { ( 1 ) } ( t ) \\right | \\leq \\frac { E '' \\Vert f \\Vert _ { H ^ s } } { N ^ { s - 3 / 2 } } . \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} N ( r ) = \\sup _ { j \\in \\{ 1 , \\ldots , m + 1 \\} } N \\left ( r , \\frac { 1 } { g _ j } \\right ) , \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} \\big | \\xi ( x ( t ) ) - z ( t ) \\big | ^ 2 \\le & 3 \\bigg | \\int _ 0 ^ t \\varphi ( x ( s ) ) \\ , d s \\bigg | ^ 2 + 3 L _ b ^ 2 \\bigg ( \\int _ 0 ^ t \\big | \\xi ( x ( s ) ) - z ( s ) \\big | \\ , d s \\bigg ) ^ 2 + \\frac { 6 } { \\beta } \\big | M ( t ) \\big | ^ 2 \\ , . \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} \\xi _ k F _ { k , m , r } ( z ) & = ( 4 \\pi | m | ) ^ { 1 - k } \\biggl \\{ ( 1 - k ) G _ { 2 - k , - m , r } ( z ) + G _ { 2 - k , - m , r - 1 } ( z ) \\biggr \\} , \\\\ \\xi _ k G _ { k , m , r } ( z ) & = ( 4 \\pi | m | ) ^ { 1 - k } F _ { 2 - k , - m , r - 1 } ( z ) , \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\left | N ( f , 0 ) - \\sum _ { i = 0 } ^ { k + m } ( - 1 ) ^ i { q \\choose i } q ^ { k - i } \\right | & \\leq \\sum _ { j = k + 1 } ^ d { \\frac { q } p + m \\sqrt { q } + j \\choose j } { m - 1 \\choose d - j } \\sqrt { q } ^ { d - j } . \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} \\mathcal { L } : = \\left \\{ \\Lambda _ { q } : q \\in \\mathcal { Q } _ N , \\ \\| q \\| _ { L ^ { \\infty } ( \\Omega ) } \\leq \\frac { \\lambda _ 1 } { 2 } \\right \\} \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} J _ \\infty \\equiv \\bigl \\{ 2 \\cos \\bigl ( \\frac { \\pi } { k + 2 } \\bigl ) , \\ k = 1 , 2 , \\ldots \\bigr \\} . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} ( \\mathbb { A ' } ^ { - 1 } _ L ) _ { 1 2 } & = ( \\Delta ' ) ^ { - 1 } \\left ( a - ( [ b , c ] - [ a , c ] a ^ { - 1 } b ) ( d - c a ^ { - 1 } b ) ^ { - 1 } \\right ) \\\\ & = ( \\Delta ' ) ^ { - 1 } \\left ( a ( d - c a ^ { - 1 } b ) - ( [ b , c ] - [ a , c ] a ^ { - 1 } b ) \\right ) ( d - c a ^ { - 1 } b ) ^ { - 1 } \\\\ & = ( \\Delta ' ) ^ { - 1 } \\left ( a d - b c \\right ) ( d - c a ^ { - 1 } b ) ^ { - 1 } \\\\ & = ( d - c a ^ { - 1 } b ) ^ { - 1 } \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} \\tau = \\frac { \\sqrt { x _ 0 ^ 4 \\sin ^ 2 \\theta + t _ 0 ^ 2 } - x _ 0 ^ 2 \\sin \\theta } { t _ 0 } , - v = ( \\alpha , \\beta ) = ( x _ 0 ( \\tau \\cos \\theta - 1 ) , x _ 0 \\tau \\sin \\theta ) . \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} p _ 1 ^ { - 1 } ( 0 ) \\cap p _ 2 ^ { - 1 } ( 0 ) = \\{ ( x , \\xi ) \\in T ^ * ( S M ) \\ , ; \\ , \\xi _ 3 = 0 , \\ \\cos ( x _ 3 ) \\xi _ 1 + \\sin ( x _ 3 ) \\xi _ 2 = 0 \\} \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} H = - \\tfrac 1 2 \\partial _ x ^ 2 + q \\delta ( x ) , \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} F ( t ; x ' , x '' ) : = \\int _ { G / T } \\ , \\mathrm { d } V _ { G / T } ( g \\ , T ) \\ , \\left [ \\Pi \\left ( \\widetilde { \\mu } _ { g \\ , t ^ { - 1 } \\ , g ^ { - 1 } } ( x ' ) , x '' \\right ) \\right ] ( t \\in T ) . \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} \\varphi _ { 0 , m } ( z , s ) = \\Gamma ( s ) ^ { - 1 } \\phi _ m ( z , s ) m \\neq 0 , \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} w ( x ) = \\frac { \\sqrt { 4 q ^ 2 - x ^ 2 } } { k ^ 2 - x ^ 2 } \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} R ( U , V ) \\xi = \\eta ( V ) U - \\eta ( U ) V , \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} c _ m ( f ) = \\int _ { X } f ( t ) p _ m ( t ) \\ , d \\mu ( t ) , \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\Psi _ { 0 , 1 } ( M ) _ 1 R _ { 2 1 } \\Psi _ { 0 , 1 } ( M ) _ 2 R ^ { - 1 } _ { 2 1 } & = L ^ { ( + ) } _ 1 L ^ { ( - ) - 1 } _ 1 R _ { 2 1 } L ^ { ( + ) } _ 2 L ^ { ( - ) - 1 } _ 2 R _ { 2 1 } ^ { - 1 } \\\\ & = L ^ { ( + ) } _ 1 L ^ { ( + ) } _ 2 R _ { 2 1 } L ^ { ( - ) - 1 } _ 1 L ^ { ( - ) - 1 } _ 2 R _ { 2 1 } ^ { - 1 } = L ^ { ( + ) } _ 1 L ^ { ( + ) } _ 2 L ^ { ( - ) - 1 } _ 2 L ^ { ( - ) - 1 } _ 1 \\\\ & = L ^ { ( + ) } _ { 1 2 } L ^ { ( - ) - 1 } _ { 1 2 } = \\Psi _ { 0 , 1 } ( M ) _ { 1 2 } . \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\left \\langle \\frac { 1 } { ( \\zeta q ^ { \\l ^ { ( N ) } _ N } ; q ) _ { \\infty } } \\right \\rangle = \\det \\left ( \\delta _ { i , j } + \\int _ { - \\infty } ^ { \\infty } d x A ( i , x ) B ( x , j ) \\right ) _ { i , j = 1 } ^ N , \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} h ( \\boldsymbol { \\mu } , D , \\psi ) = \\int \\left ( \\frac { 1 } { 2 M } \\frac { | \\boldsymbol { \\mu } + D \\boldsymbol { A } | ^ 2 } { D } + \\frac { \\hbar ^ 2 } { 8 M } \\frac { ( \\nabla D ) ^ 2 } { D } + D \\ , \\epsilon ( \\psi , \\nabla \\psi ) \\right ) ^ 3 r \\ , . \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} ( i ^ { \\mu _ 1 } ) ^ * \\omega = ( \\pi _ 1 ^ { \\mu _ 1 } ) ^ * ( \\chi ) ^ * \\omega _ { \\mathcal { O } _ { \\mu _ 2 } } ^ + . \\end{align*}"}
-{"id": "9980.png", "formula": "\\begin{align*} \\alpha ( \\tau ) = u _ 0 ( \\tau ) * \\sum _ { i = 0 } ^ \\infty p ^ { * ( i ) } ( \\tau ) , \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} \\limsup _ { \\epsilon \\to 0 } \\limsup _ { L \\to \\infty } \\frac 1 { L ^ d } \\sum _ { i \\in \\mathbb T _ L ^ d } \\sum _ { \\eta \\in \\Omega _ L } \\mu ^ L _ { [ 0 , T ] } ( \\eta ) \\ ; \\ ! \\Bigl | \\hat \\phi _ i ^ { \\lfloor \\epsilon L \\rfloor } ( \\delta _ \\eta ) - \\hat \\phi _ i ( \\nu _ { \\eta ^ { \\lfloor \\epsilon L \\rfloor } ( i ) } ) \\Bigr | = 0 . \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} B ( x ) = \\begin{pmatrix} 0 & \\nu _ 3 ( x ) & - \\nu _ 2 ( x ) & 0 & 0 & 0 \\\\ - \\nu _ 3 ( x ) & 0 & \\nu _ 1 ( x ) & 0 & 0 & 0 \\\\ \\nu _ 2 ( x ) & - \\nu _ 1 ( x ) & 0 & 0 & 0 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} \\widetilde { u } = u + u \\alpha - v \\beta , ~ ~ \\widetilde { v } = v + \\alpha v + \\beta u \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} \\tau ' ( h _ 1 , h _ 2 ) = \\tilde \\tau ( h _ 1 , h _ 2 ) \\gamma ( h _ 1 ) \\gamma ( h _ 2 ) \\gamma ( h _ 1 h _ 2 ) ^ { - 1 } \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} \\tau ^ { 2 d - 2 } = \\frac { ( c - 1 ) \\tau ^ 2 + k - 1 } { ( k - 1 ) \\tau ^ 2 + c - 1 } . \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} [ f ; G _ { \\Lambda , \\epsilon } ( t ) f ] = \\sup \\nolimits _ \\varphi ( 2 [ f ; \\varphi ] - [ \\varphi ; D _ { \\Lambda , \\epsilon } ( t ) \\varphi ] ) \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} \\Theta _ { M , M ' } \\circ \\Delta ( u ) = ( \\Psi \\circ \\Delta ^ { \\mathrm { o p } } ) ( u ) \\circ \\Theta _ { M , M ' } , \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} \\begin{cases} u _ t + f ( t , x , u ) _ x ~ = ~ \\varepsilon u _ { x x } \\ , , \\\\ u ( 0 , x ) ~ = ~ u _ 0 ( x ) . \\end{cases} \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} \\Pr \\left ( | \\{ m \\in S : Y _ m > 0 \\} | \\le \\frac { | S | \\theta } { 2 e } \\right ) \\le \\exp \\left ( - 2 | S | \\left ( \\frac { \\theta } { e } - \\frac { \\theta } { 2 e } \\right ) ^ 2 \\right ) = \\exp \\left ( - \\frac { \\theta ^ { 2 } | S | } { 2 e ^ 2 } \\right ) . \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} \\mathbf { T } ^ { l , 1 } _ { \\varepsilon _ 1 , i , j } ( \\mu _ l ) ( x ) = \\sup _ { \\rho \\in ( 0 , 2 R ) , e \\in S ^ { d - 1 } } \\varepsilon _ 1 ^ { - d + 1 } \\left | \\left ( \\frac { 1 } { | . | ^ { d - 1 } } \\Theta ^ { \\varepsilon _ 1 , e } _ { l , \\rho } ( . ) \\right ) \\star \\mathbf { K } _ j ^ i \\star \\mu _ l ( x ) \\right | ~ ~ \\forall ~ ~ x \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } k _ { 2 , \\sigma } = \\theta & \\forall \\sigma \\in I _ F ^ 0 \\\\ - k _ { 2 , \\sigma } - 2 = \\theta & \\forall \\sigma \\in I _ F ^ 1 \\\\ - k _ { 1 , \\sigma } - 3 = \\theta & \\forall \\sigma \\in I _ F ^ 2 \\\\ - k _ { 1 , \\sigma } - 3 = \\theta & \\forall \\sigma \\in I _ F ^ 3 \\end{array} \\right . \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} 2 \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 1 2 n ) q ^ n & \\equiv \\dfrac { f _ 2 ^ 4 f _ 6 ^ 2 } { f _ 1 ^ 4 f _ 4 f _ { 1 2 } } - \\dfrac { f _ 1 } { f _ 3 } \\dfrac { f _ 2 ^ 4 f _ 3 f _ { 6 } ^ 2 } { f _ 1 ^ 5 f _ 4 f _ { 1 2 } } ~ a ( q ^ 2 ) \\\\ & \\equiv \\dfrac { f _ 2 ^ 2 f _ 6 ^ 2 } { f _ 4 f _ { 1 2 } } - \\dfrac { f _ 2 ^ 2 f _ { 6 } ^ 2 } { f _ 4 f _ { 1 2 } } ~ a ( q ^ 2 ) ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} \\Delta _ h f = - \\frac { ( n - 2 ) \\psi ' ( f ) } { 2 \\phi ^ 2 } \\| \\nabla _ g f \\| ^ 2 _ g + \\frac { 1 } { \\phi } \\Delta _ g ( f ) \\in S _ f , \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} f _ \\alpha ( x ) = \\frac { z ( x ) ^ { 1 / \\alpha } } { 2 \\pi ( 1 - \\alpha ) ^ { 1 / \\alpha } } \\int _ { - \\pi } ^ \\pi U _ \\alpha ( \\phi ) \\exp \\left \\{ - z ( x ) U _ \\alpha ( \\phi ) \\right \\} d \\phi . \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} P \\circ { \\bf b } ( a , z ) = z - B ( a , z ) , \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 0 + } v ' _ { \\lambda } ( z ) = v ' _ { \\delta } ( z ) \\qquad \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} f ( x ) = & ( x ^ 2 + 1 ) - B ( x ^ 4 + x ^ 2 + 1 ) ( x - 1 ) + m \\ : h ( x ) , \\\\ g ( x ) = & 1 - A ( x ^ 4 + x ^ 2 + 1 ) ( x - 1 ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} [ \\mathcal { O } _ { X _ u } ] \\cdot [ \\mathcal { O } _ { X _ v } ] = \\sum _ { w \\in W ^ P } K _ { u , v } ^ w [ \\mathcal { O } _ { X _ w } ] . \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} \\O ( J _ { _ U } ) \\cap \\bigcap \\mathcal { U } & = \\O ( J _ { _ U } ) \\cap \\bigg [ \\Big ( \\bigcup _ { U \\neq V \\in \\mathcal { U } } V \\Big ) \\cup \\Big ( \\bigcup _ { U ' \\neq V \\in \\mathcal { U } } V \\Big ) \\bigg ] \\\\ & = \\bigg [ \\O ( J _ { _ U } ) \\cap \\Big ( \\bigcup _ { U \\neq V \\in \\mathcal { U } } V \\Big ) \\bigg ] \\cup \\bigg [ \\O ( J _ { _ U } ) \\cap \\Big ( \\bigcup _ { U ' \\neq V \\in \\mathcal { U } } V \\Big ) \\bigg ] = \\emptyset . \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} & ( - 1 ) ^ n ( g a ( v ) ( T - \\mu ) ^ { - 1 } ) ^ { n } u \\\\ & = I _ n \\left ( ( - 1 ) ^ n g ^ n \\overline { v ( k _ 1 ) \\dots v ( k _ n ) } u ( k _ 1 , \\dots , k _ n ) \\prod _ { \\ell = 1 } ^ { n } ( T ^ { ( \\ell ) } ( k _ 1 , \\dots , k _ \\ell ) - \\lambda ) ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} [ \\mathfrak { M } W _ l ] ( \\mu _ p ) = \\sum _ { x \\in \\Z _ { l } ^ { \\times } / ( 1 + l ^ { \\kappa } \\Z _ l ) } W _ l ( x ) \\mu _ p ( x ) \\int _ { 1 + l ^ { \\kappa } \\Z _ l } \\mu _ p ( y ) d ^ { \\times } y = 0 . \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} P _ { 3 / 2 , 0 } ( z , \\frac { 3 } { 4 } ) = G _ { 3 / 2 , 0 , 0 } ( z ) = - 1 2 E _ { 3 / 2 } ( z ) , \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { k - 1 } | g _ { l } | ^ { p - 2 - \\delta } \\mathbf { P } ( \\tau _ x > l ) & \\le C \\sum _ { l = 1 } ^ { k - 1 } l ^ { p / 2 - 1 - \\gamma } \\mathbf { P } ( \\tau _ x > l ) \\\\ & \\le C \\mathbf { E } [ \\tau _ x ^ { p / 2 - \\gamma / 2 } ] \\le C ( 1 + | x | ^ { p - \\gamma } ) . \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} \\gamma ^ { \\lambda } _ { u ( \\cdot ) } ( Q ) = \\lambda \\int _ 0 ^ \\infty e ^ { - \\lambda t } 1 _ Q ( y ( t ) , u ( t ) ) d t , \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} \\sup _ { k \\in \\{ 0 , \\ldots , N \\} } \\tfrac { 1 } { M ^ { ( N - k ) } k ! } = \\frac { 1 } { M ^ { N } } \\sup _ { k \\in \\{ 0 , \\ldots , N \\} } \\tfrac { M ^ { k } } { k ! } \\leq \\frac { 1 } { M ^ { N } } \\sum _ { k = 0 } ^ \\infty \\tfrac { M ^ { k } } { k ! } = \\frac { e ^ { M } } { M ^ { N } } . \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\mu _ j ^ { \\frac { n + 2 s } { 2 } } S ( u _ { \\mu , \\xi } ) ( y , t ) Z _ { n + 1 } ( y ) d y = 0 . \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} \\frac { S _ { \\widehat { \\mathcal { C } } } } { i \\sqrt { 2 8 } } = S T _ { \\widehat { \\mathcal { C } } } = T , \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} \\mathbb B _ m ( s , p ) \\ , : = \\ , \\bigcap _ { \\mathbb B \\in \\Omega _ m ( s , p ) } \\mathbb B \\ , = \\ , \\prod _ { \\ell = 1 } ^ d \\Bigl [ p _ \\ell , p _ \\ell + \\frac { s _ \\ell - 1 } { 2 ^ { m } } \\Bigr ] . \\end{align*}"}
-{"id": "9955.png", "formula": "\\begin{align*} \\tau - \\hat { \\tau } = \\bar { \\vartheta } ( ( 1 - \\varphi ) \\tau _ { 0 1 } , \\ , 0 ) , \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} M _ { 1 i } ( \\textbf { a } ) & = ( 1 + a _ { 1 1 } ) a _ { 2 i } - a _ { 1 i } a _ { 2 1 } , i = 2 , 3 , \\\\ M _ { 2 3 } ( \\textbf { a } ) & = a _ { 1 2 } a _ { 2 3 } - a _ { 1 3 } a _ { 2 2 } . \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} i _ g : S _ { \\pi _ m ( K _ { m , g } ) } : = S _ { \\pi _ m ( K _ { m , g } ) } ( \\mathcal { G } _ m , \\mathfrak { H } _ m ) \\rightarrow S _ K ^ * \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} Z _ { 5 } ( t ) & = \\frac { 2 0 - 5 0 4 t ^ { 2 } + 3 5 1 6 t ^ { 4 } - 6 7 7 6 t ^ { 6 } + 2 3 0 4 t ^ { 8 } } { 1 - 3 0 t ^ { 2 } + 2 7 3 t ^ { 4 } - 8 2 0 t ^ { 6 } + 5 7 6 t ^ { 8 } } \\\\ & = 2 0 + 9 6 t ^ { 2 } + 9 3 6 t ^ { 4 } + \\cdots + 2 2 1 2 9 7 6 6 8 4 6 1 6 t ^ { 2 0 } + \\cdots . \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} L ( \\chi , t ) & = \\prod _ { i = 1 } ^ r ( 1 - \\rho _ i t ) \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} \\epsilon ( [ m _ t ] ) & = \\epsilon ( [ g _ t ^ { - 1 } m g _ t ] ) \\\\ & = \\epsilon ( [ g _ t ^ { - 1 } g _ t ] ) - \\epsilon ( [ g _ t ^ { - 1 } ] ) \\epsilon ( [ g _ t ] ) & & [ \\textrm { S k e i n r e l a t i o n } ] \\\\ & = \\epsilon ( [ e ] ) - \\epsilon ( [ g _ t ^ { - 1 } ] ) \\epsilon ( [ g _ t ] ) = 0 , \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} E ( v _ n ) = E ( V ) + E ( r _ n ) + o _ n ( 1 ) , \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} \\sup _ { n \\ge 1 } \\frac { \\ , 1 \\ , } { \\ , \\sqrt { n } \\ , } \\sum _ { i = 1 } ^ { n } \\mathbb E [ \\sup _ { 0 \\le t \\le T } \\lvert X _ { t , i } ^ { ( u ) } - \\overline { X } _ { t , i } \\rvert ] \\le 2 C _ { T } e ^ { 2 C _ { T } T } \\sup _ { n \\ge 1 } \\big ( \\frac { \\ , 2 T \\ , } { \\sqrt { n } } ( \\mathbb E [ \\lvert \\overline { X } _ { 0 , 1 } \\rvert ] + c ) e ^ { c T } + c \\big ) < \\infty \\ , . \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} q ( \\lambda ) = q ( 0 ) + q ' ( 0 ) \\lambda + q '' ( \\xi _ \\lambda ) \\lambda ^ 2 / 2 . \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} X ( x , D ) ^ * = \\bar { X } ( x , D ) + d _ { \\bar { X } } ( x ) , \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} \\theta _ { \\bar { X } } \\dim ^ R ( \\bar { X } ) = \\theta _ { X ^ * } \\theta _ { \\bar { \\mathbf { 1 } } } S ^ { R , R } _ { X ^ * , \\bar { \\mathbf { 1 } } } & = \\theta _ { X ^ * } \\theta _ { \\bar { \\mathbf { 1 } } } \\dim ^ R ( \\bar { \\mathbf { 1 } } ) s ^ R _ { \\bar { \\mathbf { 1 } } } ( X ^ * ) \\\\ & = \\theta _ { X ^ * } \\theta _ { \\bar { \\mathbf { 1 } } } \\dim ^ R ( \\bar { \\mathbf { 1 } } ) \\dim ^ R ( X ) . \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} r _ 0 & \\ge \\left \\lceil \\frac { 1 0 } { 3 } \\right \\rceil \\cdot 1 + \\left \\lceil \\frac { 8 } { 3 } \\right \\rceil \\cdot 2 + \\left \\lceil \\frac { 1 } { 3 } \\right \\rceil \\cdot 1 = 4 + 6 + 1 = 1 1 \\\\ \\intertext { a n d } r _ 2 & \\le \\left \\lfloor \\frac { 1 0 } { 3 } \\right \\rfloor \\cdot 1 + \\left \\lfloor \\frac { 8 } { 3 } \\right \\rfloor \\cdot 2 + \\left \\lfloor \\frac { 1 } { 1 } \\right \\rfloor \\cdot 1 = 3 + 4 + 0 = 7 . \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} \\sigma ^ { 1 \\dot a b } = \\begin{pmatrix} 0 & - 1 \\\\ - 1 & 0 \\end{pmatrix} , \\sigma ^ { 2 \\dot a b } = \\begin{pmatrix} 0 & - i \\\\ i & 0 \\end{pmatrix} , \\sigma ^ { 3 \\dot a b } = \\begin{pmatrix} - 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\sigma ^ { 4 \\dot a b } = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} T _ S f ( z ) = \\int _ { \\R } f ( t ) e ^ { i 2 \\pi z t } \\d t , z \\in S ( a , b ) \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} D _ k \\frac { \\lvert k \\lvert } { \\omega ( k ) } \\sum _ { i = 1 } ^ { n } \\widehat { k } _ i \\langle \\partial _ i K ( \\xi - d \\Gamma ( m ) ) P _ 0 \\psi , ( 1 - P _ 0 ) Q ( k ) P _ 0 \\phi \\rangle . \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} S _ { n , 2 } ( q ) = 2 \\sum _ { k = 0 } ^ { n } q ^ 2 \\Big ( { n \\brack k } _ q ' \\Big ) ^ 2 { n + k \\brack k } _ q ^ 2 q ^ { f ( n , k ) } . \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} \\frac { P _ { X / T } ( m ) } { m ^ { 2 n - k - 1 } } \\leq \\frac { P _ M ( m ) } { m ^ { 2 n - k - 1 } } = \\frac { P _ M ( m ) } { m ^ { 2 ( n - k ) - 1 } } \\cdot \\frac { 1 } { m ^ k } \\ ; \\overset { m \\to \\infty } { \\longrightarrow } \\ ; 0 \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} \\rho _ i ( y y ' ) = y \\rho _ i ( y ' ) + q ^ { - \\langle \\alpha _ i , \\mu ' \\rangle } \\rho _ i ( y ) y ' , \\rho _ i ' ( y y ' ) = q ^ { - \\langle \\alpha _ i , \\mu \\rangle } y \\rho _ i ' ( y ' ) + \\rho _ i ' ( y ) y ' . \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{gather*} \\alpha _ 1 + \\alpha _ k = \\delta _ n ^ k , \\{ \\alpha _ j = \\alpha _ 1 + \\alpha _ { j - 1 } \\} _ { j = 2 } ^ k \\\\ \\{ \\alpha _ { j + ( i - 1 ) } = \\alpha _ j + \\alpha _ { i - 1 } \\} _ { j = 1 } ^ { k + 1 - i } , \\ \\ \\delta _ { n } ^ k = \\alpha _ { k + 2 - i } + \\alpha _ { i - 1 } , \\ \\{ \\alpha _ { j + k + 2 - i } = \\alpha _ { j } + \\alpha _ { k + 2 - i } \\} _ { j = 1 } ^ { i - 2 } k \\geq i \\geq 2 . \\end{gather*}"}
-{"id": "9456.png", "formula": "\\begin{align*} \\hat { u } ( n ) - \\zeta _ { 1 } ( n ) = o ( n ^ { - 1 + 2 \\beta - M } ) \\hat { v } ( - n ) - \\zeta _ { 2 } ( - n ) = o ( n ^ { - 1 - 2 \\beta - M } ) \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | = k } 1 = \\binom { n + k - 1 } { k } \\le 2 ^ { n + k - 1 } . \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} \\frac { \\partial T } { \\partial t } + ( u \\cdot \\nabla ) T = J ( T - \\mathrm { i d } _ { \\Omega } ) , \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} ( X _ { \\cdot , i } ^ { ( u ) } , X _ { \\cdot , i + 1 } ^ { ( u ) } , \\ldots , X _ { \\cdot , i + k - 1 } ^ { ( u ) } ) \\ , = \\ , ( X _ { \\cdot , 1 } ^ { ( u ) } , X _ { \\cdot , 2 } ^ { ( u ) } , \\ldots , X _ { \\cdot , k } ^ { ( u ) } ) \\ , ; i = 1 , \\ldots , n - k + 1 \\ , . \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} \\mathbf { i } \\partial _ { \\omega } ( \\Delta F _ { i j } ) + \\bar { \\Omega } _ { i j } \\Delta F _ { i j } + \\Gamma _ K ( \\tilde { \\Omega } _ { i j } \\Delta F _ { i j } ) = - \\mathbf { i } \\partial _ { \\Delta { \\omega } } F _ { i j } - \\Gamma _ K ( ( \\Delta \\Omega _ { i j } ) F _ { i j } - \\mathbf { i } \\Delta R _ { i j } ) : = Q _ { i j } , \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} ( I _ { \\mu } ^ { \\infty } ) ' ( u _ n ^ 1 ( \\cdot + y _ n ^ 1 ) - v _ 1 ) = ( I _ { \\mu } ^ { \\infty } ) ' ( u _ n ^ 1 ( \\cdot + y _ n ^ 1 ) ) - ( I _ { \\mu } ^ { \\infty } ) ' ( v _ 1 ) + o ( 1 ) \\textrm { i n } \\ , \\ , H ^ { - 1 } ( \\R ^ 3 ) . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} H ' ( \\bar { \\theta } _ { v ' } - \\bar { \\theta } _ v ) = H ' ( \\bar { \\theta } _ v - \\bar { \\theta } _ { v ' } ) \\geq 0 \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} \\mathcal L _ { \\mathcal I , \\lambda _ 1 , \\dots , \\lambda _ r } = \\mathbf P _ { \\mathcal I } \\times _ { B ^ r } \\C _ { - \\lambda _ 1 , \\dots , - \\lambda _ r } , \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{gather*} \\hat { f } _ { \\mathbf { t } } ( x , y ) = y ^ 2 ( 3 - 3 x y + x ^ 2 y ^ 2 ) ^ 2 + ( 1 - x y ) ^ 2 + t _ 1 x ^ 2 ( 1 - x y ) + t _ 2 x , \\\\ \\tilde { f } _ { \\mathbf { t } } ( x , y ) = ( 3 - 3 y + y ^ 2 ) ^ 2 + x ^ { 2 } ( 1 - y ) ^ 2 + t _ 1 x ^ { 4 } y ^ 2 ( 1 - y ) + t _ 2 x ^ { 3 } y . \\end{gather*}"}
-{"id": "636.png", "formula": "\\begin{align*} A ^ 2 = \\frac { 1 } { ( a - b ) ^ 2 } \\big ( & ( a - b ) ( n - 2 \\ell + 2 b ) A \\\\ & + ( \\ell - b ) ( n - \\ell + b ) I _ n + b ( n - n b - 2 \\ell + 2 b ) J _ n - ( a - b ) b ( A J _ n + J _ n A ) \\big ) . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} h ' _ { x , y } = h _ { x , y } - \\sum _ { z = 0 } ^ { x - 1 } \\frac { h ' _ { z , x } h ' _ { z , y } } { \\sqrt { h ' _ { z , z } } } \\ \\ ( 0 \\leq x \\leq y \\leq x _ { } ) , \\end{align*}"}
-{"id": "10006.png", "formula": "\\begin{align*} \\tau ( t ) = A ^ { \\dagger } ( t ) = \\begin{cases} 1 & t \\in ( 0 , 1 ) \\\\ t & t > 1 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} \\theta _ n : = \\min _ { C _ n \\in \\mathcal { C } _ n } \\# \\Theta ( C _ n ) , \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} & \\delta _ { g _ 0 } ^ * Y = 0 \\quad M , \\\\ & t r _ { \\partial M } \\delta _ { g _ 0 } ^ * Y = 0 \\quad \\partial M . \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} \\frac { 2 C _ G T } { L ^ d } \\sum _ { i \\in \\mathbb T _ L ^ d } \\sum _ { k = 1 } ^ d \\bigl ( \\cosh \\bigl ( \\tfrac 1 2 \\nabla ^ { i , i + e _ k } V ( \\cdot / L ) \\bigr ) - 1 \\bigr ) \\hat \\chi _ { i , i + e _ k } ^ 0 \\biggl ( \\frac 1 T \\int _ 0 ^ T \\mu ^ L _ t \\ ; \\ ! \\mathrm d t \\biggr ) . \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} \\sup _ { 1 \\le i \\le n } \\| P _ i ( \\epsilon ) - ( \\tilde { P } + \\epsilon ( i - 1 ) \\tilde { P } ^ { ( 1 ) } ) \\| = O ( \\epsilon ^ 2 n ^ 2 ) \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} \\bigg | \\frac { \\eta \\bigl ( \\frac { - 1 + 2 \\sqrt { 2 } i } { 3 } \\bigr ) } { \\eta ( 2 \\sqrt { 2 } i ) } \\bigg | ^ 4 = 3 + 3 \\sqrt { 2 } . \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} \\frac { Z _ { t } } { \\ , 1 + \\varepsilon Z _ { t } \\ , } \\ , = \\ , \\frac { 1 } { \\ , 1 + \\varepsilon \\ , } + \\int ^ { t } _ { 0 } \\frac { Z _ { s } b ( s , X _ { s } , F _ { s } ) } { ( 1 + \\varepsilon Z _ { s } ) ^ { 2 } } { \\mathrm d } B _ { s } - \\int ^ { t } _ { 0 } \\frac { \\varepsilon Z _ { s } ^ { 2 } \\lvert b ( s , X _ { s } , F _ { s } ) \\rvert ^ { 2 } } { \\ , ( 1 + \\varepsilon Z _ { s } ) ^ { 3 } \\ , } { \\mathrm d } s \\ , ; t \\ge 0 \\ , , \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} M _ X ( \\sigma , T ) = O _ { \\sigma , \\varepsilon } ( T ^ { \\min \\{ 2 ( \\delta - \\sigma ) , \\delta \\} + \\varepsilon } ) , \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} b < \\ldots < \\widehat { x } _ { n , n } ^ { ( \\alpha , \\beta ) } < \\ldots < \\widehat { x } _ { 2 , 2 } ^ { ( \\alpha , \\beta ) } < \\widehat { x } _ { 1 , 1 } ^ { ( \\alpha , \\beta ) } = c . \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} { \\mathrm d } \\overline { X } _ { t , j } \\ , = \\ , b ( t , \\overline { X } _ { t , j } , u \\cdot \\delta _ { \\overline { X } _ { t , j + 1 } } + ( 1 - u ) \\cdot \\mathcal L _ { \\overline { X } _ { t , j } } ) { \\mathrm d } t + { \\mathrm d } W _ { t , j } \\ , ; t \\ge 0 \\ , \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} \\lvert [ f ] + \\mathrm { H o m } ( F _ n , \\mathbb { R } ) \\rvert _ S = \\begin{cases} \\lvert [ f ] \\rvert _ S , & \\lvert [ f ] \\rvert _ S \\geq 2 \\\\ 0 , & \\end{cases} . \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} \\widehat { \\phi _ { N ( k , n ) } } ( \\alpha , y , \\xi ) = \\frac { 2 ^ { \\lambda - ( n - k ) / 2 } \\sqrt { \\pi } } { \\Gamma ( \\lambda ) k ^ { | \\alpha | } } y ^ { \\lambda + | \\alpha | - 1 - ( n - k ) / 2 } e ^ { - 2 y } e ^ { - | \\xi | ^ 2 / 8 y } \\frac { 1 } { \\alpha _ 1 ! \\cdots \\alpha _ k ! } , \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} C _ 2 A _ 2 ' = 1 \\ ; \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} R R ^ * f ( s , b ) & = \\int _ { \\R \\times \\R ^ { n - 1 } } f ( s ' , b ' ) \\left ( \\frac { 1 } { 2 } ( - i ( s - s ' ) + | b - b ' | ^ 2 ) + 1 \\right ) ^ { - \\lambda } d s ' d b ' \\\\ & = ( f * \\phi _ { N ( n ) } ) ( t , y ) \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} Q _ \\alpha ( \\rho \\| \\sigma ) = \\begin{cases} S _ { f _ \\alpha } ( \\rho \\| \\sigma ) & , \\\\ - S _ { f _ \\alpha } ( \\rho \\| \\sigma ) & . \\end{cases} \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} D _ { A } ( A X ) = \\left [ \\begin{array} { c c c c } X & 0 _ { 1 \\times m } & \\cdots & 0 _ { 1 \\times m } \\\\ 0 _ { 1 \\times m } & X & \\cdots & 0 _ { 1 \\times m } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 _ { 1 \\times m } & 0 _ { 1 \\times m } & \\cdots & X \\\\ \\end{array} \\right ] . \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} H \\circ \\Phi = N _ + + P _ + , \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} \\sum _ { p \\geq 0 , q \\geq 1 } \\prod _ { j = 0 } ^ { p - 1 } q _ { 0 ^ j 1 ^ q 0 } ( 0 ) \\prod _ { k = 1 } ^ { q - 1 } q _ { 1 ^ k 0 } ( 1 ) . \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} \\frac { \\Gamma ( - \\lambda + 1 ) } { \\Gamma ( - \\lambda - k + 1 ) } ( - 1 ) ^ k = \\frac { \\Gamma ( \\lambda + k ) } { \\Gamma ( \\lambda ) } \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} \\lvert ( i _ 1 , j _ 1 ) \\rvert _ { W _ n ( w ) } - \\lvert ( i _ 2 , j _ 2 ) \\rvert _ { W _ n ( w ) } = \\lvert l _ { i _ 1 } \\rvert _ S - \\lvert l _ { i _ 2 } \\rvert _ S + \\lvert r _ { j _ 1 } \\rvert _ S - \\lvert r _ { j _ 2 } \\rvert _ S . \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} v ( k , \\theta ) \\leq N ( k , t , c ) = 1 + \\sum _ { i = 0 } ^ { t - 3 } k ( k - 1 ) ^ { i } + \\frac { k ( k - 1 ) ^ { t - 2 } } { c } , \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} e _ 1 = - c , e _ 2 = c , e _ 3 = e _ 4 = 0 . \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\frac 1 { L ^ d } \\int _ 0 ^ T \\Psi _ L \\bigl ( \\mu ^ L _ t , \\jmath ^ L _ t \\bigr ) \\ ; \\ ! \\mathrm d t \\\\ = \\frac 1 2 \\int _ 0 ^ T \\| \\dot \\rho _ t \\| _ { - 1 , \\chi ( \\rho _ t ) } ^ 2 \\ ; \\ ! \\mathrm d t \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} \\tilde s ( \\bar g _ 1 ) \\tilde s ( \\bar g _ 2 ) \\tilde s ( \\bar g _ 1 \\bar g _ 2 ) ^ { - 1 } = [ ( e , \\iota \\bigl ( \\sigma ( \\bar g _ 1 , \\bar g _ 2 ) \\bigr ) ] = [ f \\bigl ( \\sigma ( \\bar g _ 1 , \\bar g _ 2 ) \\bigr ) , e ] . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\xi } [ f _ { x } ( x , \\xi ) ] = f ( x ) \\textrm { a n d } \\mathbb { E } _ { \\xi } [ f _ { x } ( y , \\xi ) - f ( y ) ] \\leq \\frac { \\tau } { 2 } \\| y - x \\| ^ 2 _ 2 \\qquad \\forall x , y , \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\begin{aligned} & \\hat r ( x ) : = \\frac D 2 \\frac { \\phi _ 3 '' ( x ) } { \\phi _ 3 ( x ) } = \\frac D 2 \\big ( ( \\psi ' ( x ) ) ^ 2 + \\psi '' ( x ) \\big ) = \\\\ & \\frac D 2 \\lambda ( l + 1 ) ( \\gamma + x ^ 2 ) ^ { \\frac { l - 3 } 2 } \\big [ ( l + 1 ) \\lambda x ^ 2 ( \\gamma + x ^ 2 ) ^ { \\frac { l + 1 } 2 } + \\gamma + l x ^ 2 \\big ] . \\end{aligned} \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } \\int _ K \\int _ H f ( k ^ { - 1 } x h ) d h d k d \\mu ( \\ddot { x } ) = \\int _ G f ( x ) \\rho ( x ) d x \\ \\ f \\in C _ c ( N ) , \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\beta e _ { \\mathbb { Q } } ( \\varphi ) = \\sum _ { 1 \\leq i \\leq k } \\alpha _ { i } e _ { \\mathbb { Q } } ( \\chi _ { i } ) . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\theta } \\left ( \\frac { { \\dot \\kappa } ^ { - S } ( \\theta ) } { { \\dot \\kappa } ^ { A } ( \\theta ) } \\right ) = \\frac { { \\ddot \\kappa } ^ { - S } ( \\theta ) \\cdot { \\dot \\kappa } ^ { A } ( \\theta ) - { \\ddot \\kappa } ^ { A } ( \\theta ) \\cdot { \\dot \\kappa } ^ { - S } ( \\theta ) } { \\left [ { \\dot \\kappa } ^ { A } ( \\theta ) \\right ] ^ { 2 } } \\ge 0 , \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} 2 D T _ { \\theta } + \\delta & \\leq u _ { t } ^ { \\theta } ( 0 , t _ { 0 } ) - v _ { \\theta , t } ( 0 , t _ { 0 } ) \\\\ & = ( a - b ) + 2 D T _ { \\theta } + 2 \\zeta \\\\ & \\leq 2 D T _ { \\theta } + 2 \\zeta . \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} R ( \\lambda ) & \\otimes R ( \\mu ) = \\\\ & \\bigoplus _ { | \\nu | = | \\lambda | + | \\mu | , k ( \\nu ) \\leq n } ( c _ { \\lambda \\mu } ^ { \\nu } ) ^ 2 R ( \\nu ) \\oplus \\bigoplus _ { | \\nu | < | \\lambda | + | \\mu | , k ( \\nu ) \\leq n } d _ { \\lambda \\mu } ^ { \\nu } R ( \\nu ) \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} \\mathbf { P } \\left ( \\max _ { j \\le n } | X ( j ) | > a r , \\tau _ x > n \\right ) & \\le \\sum _ { j = 1 } ^ n \\mathbf { P } \\left ( | X ( j ) | > a r , \\tau _ x > n \\right ) \\\\ & \\le \\sum _ { j = 1 } ^ n \\mathbf { P } \\left ( | X ( j ) | > a r , \\tau _ x > j - 1 \\right ) \\\\ & = \\mathbf { E } [ \\tau _ x \\wedge n ] \\mathbf { P } ( | X | > a r ) . \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} u _ T ( t , x ) : = \\frac { e ^ { - i \\frac { | x | ^ 2 } { 4 ( T - t ) } } } { ( T - t ) ^ { \\frac { d } { 2 } } } u \\left ( \\frac { 1 } { T - t } , \\frac { x } { T - t } \\right ) . \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} = - \\frac { ( T - t ) ^ 2 } { 8 } { \\rm l n } \\left | 1 - \\frac { 2 } { 2 q + 1 } \\right | \\le C _ 1 \\frac { ( T - t ) ^ 2 } { q } , \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ d } | \\eta _ t - \\eta _ t ^ \\varepsilon | d | D ^ { s } b _ t | d t + \\lim _ { \\varepsilon \\to 0 } \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ d } | \\xi _ t - \\xi _ t ^ \\varepsilon | d | D ^ { s } b _ t | d t = 0 . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} \\Delta ( \\det A ) = ( \\det A ) \\operatorname { t r } \\left ( A ^ { - 1 } \\ , \\Delta A \\right ) . \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} \\frac { 1 } { H } \\sum _ { h = 1 } ^ { H } v a r \\left ( \\gamma ^ { \\tau } m ( y ) \\big | a _ { h - 1 } \\leq y < a _ { h } \\right ) \\leq \\frac { \\mathfrak { a } _ { 3 } } { H ^ { \\vartheta } } v a r \\left ( \\gamma ^ { \\tau } m ( y ) \\right ) . \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} ( \\ell I _ n + a A + b ( J _ n - A - I _ n ) ) ^ 2 & = n ( \\ell I _ n + a A + b ( J _ n - A - I _ n ) ) . \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\mbox { V a r } _ { \\mu _ z } ( f ) = \\int _ { \\Sigma _ z } f ^ 2 \\ , d \\mu _ z - \\Big ( \\int _ { \\Sigma _ z } f \\ , d \\mu _ z \\Big ) ^ 2 \\le \\frac { 1 } { \\beta \\rho _ 0 } \\int _ { \\Sigma _ z } | P \\nabla f | ^ 2 d \\mu _ z \\ , , \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} d ( B _ 1 ) = & \\frac { m _ K n + 1 } { n 2 ^ n } \\\\ d ( B _ 2 ) = & \\frac { 2 ^ { n - 1 } ( n - 1 ) + m _ K n + 1 } { n 2 ^ n } \\\\ d ( B _ 3 ) = & \\frac { 2 ^ n ( n - 1 ) + m _ K n + 1 } { n 2 ^ n } \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n + 1 } \\log D _ { n } [ \\sigma ] = \\log \\mathbf { G } [ \\sigma ] , \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} \\varphi ( d ^ { \\mathrm { c l o u d } } ) = e ^ { - \\beta _ c d ^ { \\mathrm { c l o u d } } } , \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} N _ J ( u , v ) = [ u , v ] + J [ J u , v ] + J [ u , J v ] - [ J u , J v ] \\end{align*}"}
-{"id": "9962.png", "formula": "\\begin{align*} u * \\bar \\vartheta _ z ( \\tau \\otimes a _ X ) & = u * ( \\bar \\partial _ z \\tau _ 1 \\otimes a _ X , \\ , ( \\tau _ 1 - \\bar \\partial _ z \\tau _ { 0 1 } ) \\otimes a _ X ) ) \\\\ & = ( \\bar \\partial ( u * ( \\tau _ 1 \\otimes a _ X ) ) , \\ , u * ( \\tau _ 1 \\otimes a _ X ) - \\bar \\partial ( u * ( \\tau _ { 0 1 } \\otimes a _ X ) ) \\\\ & = \\bar \\vartheta ( u * ( \\tau \\otimes a _ X ) ) . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} P \\overline { \\nabla } ^ { ' } _ { X } Y = P \\nabla ^ { ' } _ { X } Y + \\alpha \\eta ( Y ) P X - \\alpha g ( X , Y ) P \\xi + \\beta \\eta ( Y ) \\phi P X - \\beta g ( \\phi X , Y ) P \\xi , \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} \\int _ G f ( x ) \\rho ( x ) d x = \\int _ { \\frac { G } { H } } \\int _ H f ( x h ) d h d \\mu ( x H ) , \\end{align*}"}
-{"id": "9966.png", "formula": "\\begin{align*} L = \\bigcap _ { a \\in W } \\{ \\ , z = x + \\sqrt { y } \\in \\C ^ n \\mid \\ , | y | < | x - a | \\ , \\} . \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} \\hat { \\alpha } = \\frac { 1 } { \\psi ^ { \\prime \\prime } \\left ( \\rho ( 2 \\sqrt { 2 } \\delta ( v ) ) \\right ) } . \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} | \\sum _ { b = 1 } ^ 2 k _ b j _ b ^ 2 | \\leq \\max \\{ | k _ 1 j _ 1 ^ 2 | , | k _ 2 j _ 2 ^ 2 | \\} \\leq \\frac { 1 } { ( 2 - \\frac { 9 } { 1 7 } ) ^ 2 } ( i + j ) ^ 2 \\leq \\frac { 2 } { ( 2 - \\frac { 9 } { 1 7 } ) ^ 2 } ( i ^ 2 + j ^ 2 ) . \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} \\begin{aligned} x ( t + 1 ) & = A x ( t ) + B u ( t ) , \\\\ u ( t ) & = ( F x ( t ) ) , \\end{aligned} \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} T & : \\ell ^ { 2 } ( I ) \\longmapsto H \\\\ & T ( c _ { i } ) = \\sum _ { i \\in I } c _ { i } f _ { i } \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} e _ q ' = e _ q - \\frac { S _ t ( e _ p , e _ q ) } { S _ t ( e _ p , e _ p ) } e _ p = \\begin{pmatrix} 1 \\\\ \\\\ \\ \\frac { 4 \\sqrt { \\nu } \\ ; S ^ 2 ( t ) } { ( 1 - e ^ { - t } ) + 2 S ^ 2 ( t ) + 2 S ( t ) C ( t ) } \\end{pmatrix} \\ ; , \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} g _ { \\l + 1 } > c _ { \\l } c _ { \\l - 2 } \\cdots c _ 2 g _ 1 = \\frac { 1 } { l + 1 } g _ 1 . \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 4 } \\Delta e _ 3 \\ , d x \\ , = - \\int _ { \\mathbb { R } ^ 4 } \\left ( \\frac { \\partial B ^ \\beta { } _ \\alpha } { \\partial x ^ \\beta } \\right ) \\Delta A ^ \\alpha \\ , d x \\ , = - \\int _ { \\mathbb { R } ^ 4 } \\left ( \\frac { \\partial B ^ 4 { } _ \\alpha } { \\partial x ^ 4 } \\right ) \\Delta A ^ \\alpha \\ , d x \\ , = \\ , 0 \\ , . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { \\binom k j \\binom { k - j } s ( - 1 ) ^ { j + s } f _ 2 ^ j f _ 1 ^ s X _ { m + c k - b j - a s } } } = f _ 3 ^ k X _ m \\ , . \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} - \\mathrm d Y _ t = \\mathcal D _ x \\mathcal H ( t , X _ t , \\alpha _ t , Y _ t , Z _ t ) \\mathrm d t - Z _ t \\mathrm d W _ t , \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} \\tilde S _ N = \\sum _ { n = 1 } ^ N F ( \\xi _ { p _ 1 ( n ^ l ) } , \\xi _ { p _ 2 ( n ^ l ) } , . . . , \\xi _ { p _ \\ell ( n ^ l ) } ) \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} \\gamma _ n \\left ( \\sum _ { j = 1 } ^ r \\alpha _ j T _ j \\right ) = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ r \\alpha _ j E ^ { ( n ) } _ { 2 j - 1 , 2 j } \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} d ( ( v , a ) , ( \\tilde v , \\tilde a ) ) = \\| v - \\tilde v \\| _ { L _ t ^ \\infty L _ x ^ 2 ( [ 0 , T ] \\times \\R ) } + \\| a - \\tilde a \\| _ { L _ t ^ \\infty ( [ 0 , T ] ) } \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} E = \\left \\{ r : T \\left ( r + | c | + \\frac { r + | c | } { \\xi ( T ( r + | c | , f ) ) } , f \\right ) \\geq C T ( r + | c | , f ) \\right \\} \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} F ( \\tau ) - c ( S _ \\varepsilon ( \\tau ) , N _ \\varepsilon ( \\tau ) , F ( \\tau ) ) N _ \\varepsilon ( \\tau ) = ( f _ \\varepsilon ( \\tau ) - c ( S _ \\varepsilon ( \\tau ) , N _ \\varepsilon ( \\tau ) , F ( \\tau ) ) ) N _ \\varepsilon ( \\tau ) \\xrightarrow { \\varepsilon \\to 0 } 0 \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} u _ { t t } - \\sigma ( u _ x ) _ x - u _ { x x t } = f ( u ) + g ( x ) \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\sum _ { x \\in T _ { N } ^ { - 1 } ( x ' ) } T _ { 2 } ( x ) = \\cos \\left ( 2 \\pi u _ { 0 } \\right ) + 2 \\cos \\left ( 2 \\pi u _ { 0 } \\right ) \\left ( \\frac { - 1 } { 2 } \\right ) = 0 \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} X _ { t _ { i + 1 } } ^ { j } = X _ { t _ { i } } ^ { j } + h \\ B ( X _ { t _ i } ^ { j } , ( u ^ { j - 1 } _ i , v ^ { j - 1 } _ i ) ( X _ { t _ { i } } ^ { j } ) , \\mu ^ { j - 1 } _ i , ( u ^ { j - 1 } _ i , v ^ { j - 1 } _ i ) \\sharp \\mu ^ { j - 1 } _ i ) + \\sigma \\Delta W _ { i } . \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{gather*} \\frac { L ( z , 1 ) } { \\Re F _ 1 ( z ) } = - \\frac { L ( z , 1 ) } { \\Gamma ( r ) \\Gamma ( 1 - r ) \\Re F _ 0 ( 1 - z ) } \\in \\mathbb Q , \\end{gather*}"}
-{"id": "346.png", "formula": "\\begin{align*} \\| \\kappa ( T _ 1 ^ * T _ 1 - T _ 1 T _ 1 ^ * ) \\| = \\limsup _ { n \\to \\infty } \\| \\gamma _ n ( T _ 1 ^ * T _ 1 - T _ 1 T _ 1 ^ * ) \\| = \\frac { 1 } { 4 } \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } ( 2 ^ { j } \\rho ) ^ d \\int _ { S ^ { d - 1 } } \\sup _ { r \\in [ 2 ^ { j } \\rho , 2 ^ { j + 1 } \\rho ] } | \\mathbf { K } ^ { \\varepsilon , n } _ { e , \\rho } ( r \\theta ) | d \\mathcal { H } ^ { d - 1 } ( \\theta ) \\lesssim 1 . \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} 1 + n _ 2 ( G ) = 2 n _ 8 ( G ) & \\Longleftrightarrow 2 ^ { n + a + b } = 2 ^ { n + 2 a + 2 b - 1 } ( 2 ^ b - 1 ) \\\\ & \\Longleftrightarrow 1 = 2 ^ { a + b - 1 } ( 2 ^ b - 1 ) \\\\ & \\Longleftrightarrow a = 0 b = 1 . \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} & \\mathbf { E } \\Big ( \\sup _ { 0 \\le t ' \\le t } \\big | \\xi ( x ( t ' ) ) - z ( t ' ) \\big | ^ 2 \\Big ) \\\\ \\le & \\ , 3 \\mathbf { E } \\bigg [ \\sup _ { 0 \\le t ' \\le t } \\Big | \\int _ 0 ^ { t ' } \\varphi ( x ( s ) ) \\ , d s \\Big | ^ 2 \\bigg ] + 3 L _ b ^ 2 \\ , \\mathbf { E } \\Big ( \\int _ 0 ^ t \\big | \\xi ( x ( s ) ) - z ( s ) \\big | \\ , d s \\Big ) ^ 2 + \\frac { 6 } { \\beta } \\mathbf { E } \\sup _ { 0 \\le s \\le t } \\big | M ( s ) \\big | ^ 2 \\ , . \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} X \\cap U ' = X \\cap U _ s = \\emptyset , \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} \\Omega _ n ( x ) : = \\int _ { 0 } ^ { \\infty } \\tilde { \\Omega } \\star \\varrho _ { n } \\left ( \\frac { x } { | x | } r \\right ) r ^ { d - 1 } d r ~ ~ \\forall ~ x \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} ( I - P _ n ) h _ { n 1 } & = P _ n r - \\pi _ n r e , \\\\ ( I - P _ n ) h _ { n 2 } & = h _ { n 1 } - \\pi _ n h _ { n 1 } e , \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } p _ { _ 1 } ( y ) d y & = \\frac { 2 } { \\sqrt { 2 \\pi } \\sigma _ w } e ^ { \\rho _ 0 + 1 } \\int _ { 0 } ^ { \\infty } e ^ { - \\left ( \\frac { 1 } { 2 \\sigma _ w ^ 2 } + \\rho _ 1 \\right ) y ^ 2 } d y \\\\ & = \\frac { e ^ { - \\rho _ 0 - 1 } } { \\sqrt { 1 + 2 \\rho _ 1 \\sigma _ w ^ 2 } } , \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} \\mathbb { V } = \\{ p \\in L ^ 2 ( \\Omega ) : | p | _ { H _ m ^ { \\beta / 2 , \\lambda } ( \\mathbb { R } ^ n ) } < \\infty \\} , \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} | \\phi _ r - \\omega | _ { \\mathcal { O } _ r ^ - } + \\frac { \\alpha } { M } | \\phi _ r - \\omega | ^ { l i p } _ { \\mathcal { O } _ r ^ - } = O ( \\epsilon ) = O ( r ^ { 7 / 4 } ) . \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} f _ i ( q ) = \\frac { 1 } { i ! n ^ i } \\left ( ( q - 1 ) ^ { i } G _ { g _ { i } , n } ( q ) - \\sum _ { m _ 1 = 0 } ^ { i - 1 } \\sum _ { m _ 2 = m _ 1 } ^ { i } \\binom { i } { m _ 2 } f _ { m _ 1 } ^ { ( i - m _ 2 ) } ( q ) R _ { n , m _ 1 , m _ 2 } ( q ) ( q - 1 ) ^ { i - m _ 2 } q ^ { i - m _ 2 } \\right ) , \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} & a ^ \\prime = \\frac { - a ^ 2 + b ^ 2 + c ^ 2 } { 2 b c } \\ , , b ^ \\prime = \\frac { a ^ 2 - b ^ 2 + c ^ 2 } { 2 a c } \\ , , c ^ \\prime = \\frac { a ^ 2 + b ^ 2 - c ^ 2 } { 2 a b } \\ , , \\\\ & a ( 0 ) = a _ 0 \\ , , b ( 0 ) = b _ 0 \\ , , c ( 0 ) = c _ 0 \\ , , \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} \\mathcal U _ x ( \\rho ) = B ( x , \\rho ) \\bigcup B ( \\mathcal R x , \\rho ) , \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} g _ 0 = a _ 0 \\ \\zeta ^ 1 \\odot \\bar \\zeta ^ 1 + b _ 0 \\ \\zeta ^ 2 \\odot \\bar \\zeta ^ 2 + c _ 0 \\ \\zeta ^ 3 \\odot \\bar \\zeta ^ 3 \\ , . \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} e ^ { - s X _ 0 } a ^ w ( q , p , D _ q , D _ p ) e ^ { s X _ 0 } = a ^ w ( e ^ { - s M } ( q , p ) , e ^ { s M } ( D _ q , D _ p ) ) . \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} \\sum _ { \\substack { d \\leq x \\\\ ( d , a _ { 2 } ) = 1 } } J _ { k } ( d ) \\bigg \\lfloor \\frac { x } { \\ell _ u ( d ) } \\bigg \\rfloor \\leq x \\sum _ { \\substack { d \\leq x \\\\ ( d , a _ { 2 } ) = 1 } } \\frac { J _ { k } ( d ) } { \\ell _ { u } ( d ) } \\leq x \\sum _ { \\substack { d \\leq x \\\\ ( d , a _ { 2 } ) = 1 } } \\frac { d ^ { k } } { \\ell _ { u } ( d ) } . \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} \\langle u , ( H - \\lambda ) ^ { - 1 } w \\rangle & \\geq \\langle ( T - \\lambda ) ^ { - 1 } u , ( - g a ^ { \\dagger } ( v ) ( T - \\lambda ) ^ { - 1 } ) ^ { n _ 1 } ( - g a ( v ) ( T - \\lambda ) ^ { - 1 } ) ^ { n _ 2 } w \\rangle \\\\ & = \\langle ( T - \\lambda ) ^ { - 1 } ( - g a ( v ) ( T - \\lambda ) ^ { - 1 } ) ^ { n _ 1 } u , ( - g a ( v ) ( T - \\lambda ) ^ { - 1 } ) ^ { n _ 2 } w \\rangle \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} \\gamma ^ 1 ( t , W _ t ) + h ^ 1 ( t , W _ t ) = \\gamma ^ 2 ( t , W _ t ) + h ^ 2 ( t , W _ t ) \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} ( 1 - \\epsilon ( \\mu ) ) \\epsilon ( [ \\ell \\gamma ] ) = \\epsilon ( [ \\ell ] ) \\epsilon ( [ \\gamma ] ) . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} D = N \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} \\lambda _ { \\mu _ p \\pi _ p } ( p ^ m ) = \\begin{cases} \\delta _ { m = 0 } & \\mu _ p \\vert _ { \\Z _ p ^ { \\times } } \\neq \\chi _ 1 ^ { - 1 } \\vert _ { \\Z _ p ^ { \\times } } , \\\\ \\frac { \\chi _ 1 ( p ^ { m + 1 } ) - \\chi _ 2 ( p ^ { m + 1 } ) } { \\chi _ 1 ( p ) - \\chi _ 2 ( p ) } \\delta _ { m \\geq 0 } & \\mu _ p \\vert _ { \\Z _ p ^ { \\times } } = \\chi _ 1 ^ { - 1 } \\vert _ { \\Z _ p ^ { \\times } } \\end{cases} \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} c _ { t , l } ( \\mu _ p ) = c _ { p } ( \\pi _ { p } , l , t , \\mu _ p ) \\zeta _ { p } ( 1 ) p ^ { - \\frac { l + t + a ( \\mu _ p \\pi _ p ) } { 2 } } \\lambda _ { \\chi _ { \\mu _ p } \\pi } ( p ^ { t + a ( \\mu _ p \\pi _ { p } ) + \\delta _ { \\mu _ p \\pi _ { p } } } ) , \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} & d _ { \\Lambda } E Z ( t \\otimes \\sigma ) = E Z ( d _ { \\Lambda } t \\otimes \\sigma ) + ( - 1 ) ^ { | t | } E Z ( t \\otimes \\partial ' \\sigma ) \\\\ & - \\sum _ { ( \\sigma ) } [ ( i , \\sigma ' ) | E Z ( t \\otimes \\sigma '' ) ] + \\sum _ { ( \\sigma ) } ( - 1 ) ^ { | t | | \\sigma ' | } [ E Z ( t \\otimes \\sigma ' ) | ( j , \\sigma '' ) ] , \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} N _ i ( \\delta ) = \\lambda _ i \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} \\mathcal C : & = \\big \\{ c \\in ( 0 , + \\infty ) ^ { m ^ + } \\times ( - \\infty , 0 ] ^ { m - m ^ + } ; \\ ; ( H , B , c ) \\ , \\mathrm { s a t i s f i e s } \\ , \\mathrm { C o n d i t i o n \\ , ( C ) } \\big \\} \\\\ & = \\big \\{ c \\in D _ 1 ; \\ ; ( H , B , c ) \\ , \\mathrm { s a t i s f i e s } \\ , \\mathrm { C o n d i t i o n \\ , ( C ) } \\big \\} , \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} \\ln R ( n + 1 , E ) - \\ln R ( n , E ) = - \\vert \\varphi ( n , E ) \\vert ^ 2 \\frac { C } { n - v } \\sin ^ 2 ( 2 \\eta ( n ) + 2 \\gamma ( n ) ) + \\frac { \\vert O ( 1 ) \\vert } { ( n - v ) ^ 2 } . \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} K L ( p _ { \\mathbb { S } } | | p _ { 0 } ( y , x ) ) = \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} Z _ { \\Lambda } ( \\psi _ { \\Lambda ^ c } ) = \\int _ { \\mathbb { R } ^ \\Lambda } e ^ { - H _ { \\Lambda } ( \\phi ) } \\prod _ { j \\in \\Lambda } d \\phi _ j . \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} \\mathbf { H } _ { S } = \\begin{pmatrix} \\mathbf { H } _ { i _ 1 } & \\mathbf { H } _ { i _ 2 } & \\dots & \\mathbf { H } _ { i _ { | S | } } \\end{pmatrix} \\in \\mathbb { B } ^ { ( n - k ) \\ell \\times | S | \\ell } . \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} h _ { I _ F } = \\chi ( \\bar { X } _ { \\Gamma _ { 0 , s s } } , \\mathcal { O } _ { \\bar { X } _ { \\Gamma _ { 0 , s s } } } ) + \\epsilon \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} | | \\nabla u | | ^ 2 = \\int _ { \\Omega } z ( u ) | u | ^ { p - 2 } u u \\ , d x \\leq | | z ( u ) | u | ^ { p - 1 } u | | _ q \\cdot | | u | | _ { q ' } , \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} d ( \\mathcal { C } _ 1 ( p ) , \\mathcal { C } _ 2 ( p ) ) = \\max \\left \\{ \\sup \\limits _ { h \\in \\mathcal { C } _ 2 ( p ) , h \\neq 0 } \\inf \\limits _ { k \\in \\mathcal { C } _ 1 ( p ) } \\frac { \\| h - k \\| _ { H } } { \\| h \\| _ { H } } , \\sup \\limits _ { h \\in \\mathcal { C } _ 1 ( p ) , h \\neq 0 } \\inf \\limits _ { k \\in \\mathcal { C } _ 2 ( p ) } \\frac { \\| h - k \\| _ { H } } { \\| h \\| _ { H } } \\right \\} . \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} T _ I \\coloneqq \\{ s \\in T \\mid \\alpha _ i ( s ) = 1 i \\in I \\} ^ 0 \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} \\begin{aligned} K _ { F , 3 } ( T , \\tau ) & \\coloneqq \\frac { 0 . 4 3 2 \\tau ^ 2 } { T } \\left ( \\frac { A _ F } { 2 } \\log { ( 2 T ) } + \\frac { A _ F \\log { 4 } + B _ F } { 2 } \\right . \\\\ & \\left . - \\frac { c _ { F , 1 } ( T _ 0 ) } { 3 T } \\log ( 2 ^ \\frac { 1 } { 3 } T ) - \\frac { c _ { F , 2 } ( T _ 0 ) } { 3 T } - \\frac { 2 c _ { F , 3 } ( T _ 0 ) } { 7 T ^ 2 } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| v _ n \\| ^ 2 _ { L ^ 2 } = M \\lim _ { n \\rightarrow \\infty } E ( v _ n ) = d _ M . \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} [ d , a ] \\Delta ^ { - 1 } = c a ^ { - 1 } ( [ b , a ] \\Delta ^ { - 1 } - b R _ { 2 2 } ) + d R _ { 2 2 } \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } p _ n \\\\ q _ n \\end{array} \\right ] & = \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ 1 \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ 2 \\end{array} \\right ] \\cdots \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ { n - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c } 1 \\\\ a _ { n } \\end{array} \\right ] \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} { { D } _ { x } } = \\frac { \\partial } { \\partial x } + { { \\alpha } _ { x } } - { { \\beta } _ { y } } , ~ ~ { { D } _ { y } } = \\frac { \\partial } { \\partial y } + { { \\alpha } _ { y } } + { { \\beta } _ { x } } \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} M _ 1 = \\max _ { 1 \\leq b \\leq n } | j _ b | , M _ 2 = \\frac { n } { n - \\frac 1 2 } ; \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\varphi _ { 0 , m } ( z , s ) = \\Gamma ( 2 s ) ^ { - 1 } 2 ^ { 2 s - 1 } \\Gamma ( s + 1 / 2 ) \\pi ^ { - 1 / 2 } \\phi _ m ( z , s ) . \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} \\lvert m _ { n + 1 } \\rvert _ S - \\lvert m _ { n } \\rvert _ S = & ( k + \\sum _ { j \\in A ^ + } ( m _ j + n + 1 ) - \\sum _ { j \\in A ^ - } ( m _ j - n - 1 ) + \\sum _ { j \\in A ^ 0 } \\lvert m _ j \\rvert ) \\\\ & - ( k + \\sum _ { j \\in A ^ + } ( m _ j + n ) - \\sum _ { j \\in A ^ - } ( m _ j - n ) + \\sum _ { j \\in A ^ 0 } \\lvert m _ j \\rvert ) \\\\ = & \\sum _ { j \\in A ^ + } 1 + \\sum _ { j \\in A ^ - } 1 \\\\ = & \\lvert A ^ + \\rvert + \\lvert A ^ - \\rvert . \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\frac { \\partial m } { \\partial t } = \\lambda _ 1 m \\times H _ { e f f } - \\lambda _ 2 m \\times ( m \\times H _ { e f f } ) \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} { \\rm M } ( X , Y ) = - \\frac 1 2 \\ , g ( \\mu ( X , X _ k ) , \\mu ( Y , X _ k ) ) + \\frac 1 4 \\ , g ( \\mu ( X _ k , X _ j ) , X ) g ( \\mu ( X _ k , X _ j ) , Y ) \\ , , \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} \\alpha = n d ( 1 - d ^ 2 ) - 6 \\sum _ { y \\in Y } ( f ( y ) ^ 2 + d f ( y ) ) , \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} \\sqrt { \\sigma ^ { - 2 } } \\mathbf { H } \\mathbf { Q } ^ { - \\frac { 1 } { 2 } } = \\mathbf { U } \\frac { \\boldsymbol { \\Lambda } } { \\sqrt { \\sigma ^ { 2 } } } \\left ( \\frac { \\ln 2 } { \\mu _ 2 } \\left ( \\nu _ 2 \\mathbf { I } _ { r } - \\boldsymbol { \\Lambda } ^ { \\rm H } \\boldsymbol { \\Lambda } \\right ) \\right ) ^ { - \\frac { 1 } { 2 } } \\mathbf { V } ^ { \\rm H } . \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} M ( D ) _ { c a } = \\begin{cases} 1 - t _ { \\kappa ( a _ 2 ) } & a = a _ 1 \\\\ t _ { \\kappa ( a _ 1 ) } - 1 & a = a _ 2 = a _ 3 \\\\ 0 & a \\notin \\{ a _ 1 , a _ 2 \\} \\end{cases} \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} \\alpha ( y ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } e ^ { - 2 y ^ { 2 } } \\left ( 2 y - \\frac { 8 } { 3 } y ^ { 3 } \\right ) \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\langle \\eta _ t ^ \\varepsilon ( x _ 1 ) , A _ { 1 } ^ { } \\rangle = E ^ { } + E ^ { } + E ^ { } + E ^ { } + \\sum _ { i = 1 , 2 } \\tilde { \\Theta } ^ { \\varepsilon _ 1 , \\mathbf { e } _ i } _ { 1 , r } \\star \\left [ \\operatorname { d i v } ( \\mathbf { B } _ t ) \\right ] ( x _ i ) , \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} p _ { \\zeta } ' ( x ) & = ( 1 - \\varphi _ { \\zeta } ( x ) ) p ' ( x ) + \\varphi _ { \\zeta } ( x ) q ' ( x ) + \\varphi _ { \\zeta } ' ( x ) ( q ( x ) - p ( x ) ) , \\\\ p _ { \\zeta } '' ( x ) & = ( 1 - \\varphi _ { \\zeta } ( x ) ) p '' ( x ) + \\varphi _ { \\zeta } ( x ) q '' ( x ) + 2 \\varphi _ { \\zeta } ' ( x ) ( q ' ( x ) - p ' ( x ) ) \\\\ & \\qquad + \\varphi _ { \\zeta } '' ( x ) ( q ( x ) - p ( x ) ) . \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} V ^ { \\xi ^ * _ { m } \\eta ^ * _ { n } } ( X _ { \\theta ^ * _ { k } } ) - V ^ { \\xi ^ * _ { m - 1 } \\eta ^ * _ { n - 1 } } ( X _ { \\theta ^ * _ { k } } ) + \\chi ( \\eta ^ * _ { n - 1 } , \\eta ^ * _ n ) - C ( \\xi ^ * _ { m - 1 } , \\xi ^ * _ m ) = 0 . \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\mathrm { V a r } g _ n ( x ) = & ~ \\mathrm { E } \\left [ g _ n ( x ) - \\hat { g } _ n ( x ) - \\mathrm { E } \\left ( g _ n ( x ) - \\hat { g } _ n ( x ) \\right ) \\right ] ^ 2 + \\mathrm { V a r } \\hat { g } _ n ( x ) \\\\ & + 2 \\mathrm { E } \\left [ \\left ( g _ n ( x ) - \\hat { g } _ n ( x ) - \\mathrm { E } \\left ( g _ n ( x ) - \\hat { g } _ n ( x ) \\right ) \\right ) \\left ( \\hat { g } _ n ( x ) - \\mathrm { E } \\hat { g } _ n ( x ) \\right ) \\right ] . \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} | N _ t ( e ) | & \\le ( d - 2 ) \\cdot \\iota _ { s , d } - 1 + 2 ( t - 2 s + 1 ) ( ( d - 2 ) \\cdot \\iota _ { s , d } + 1 ) + 2 \\iota _ { s , d } \\\\ & = ( 2 t - 4 s + 3 ) ( d - 1 ) ^ s - 2 + 2 \\frac { ( d - 1 ) ^ s - 1 } { d - 2 } . \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} \\begin{cases} X q _ j & = - p _ { Y , j } ( x , y , u _ 1 , \\dots , u _ n ) \\\\ Y q _ j & = p _ { X , j } ( x , y , u _ 1 , \\dots , u _ n ) . \\end{cases} \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} \\beta _ i - f _ { 0 , \\ , i } ( \\epsilon _ i ) = f _ { 0 , \\ , i } ( \\gamma _ i - \\epsilon _ i ) , \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} S = \\{ ( a , b , k ) \\in \\Z \\times \\N \\times \\N _ 0 \\vert b \\leq l ^ { k + 2 r ^ - } , \\abs { a } \\leq l ^ { k + 2 r ^ + } , ( a , b ) = ( a , l ) = ( b , l ) = 1 \\} . \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} \\bold { X } = \\sum _ { n = 0 } ^ { \\infty } \\bold { X } _ n , \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} y ^ { n - k } \\sum _ { j = k } ^ { n } ( j ) _ { k } f _ j ( x ) ( - \\frac { 1 } { y } ) ^ { j - k } = \\sum _ { j = k } ^ { n } ( - 1 ) ^ { j - k } ( j ) _ { k } f _ j ( x ) y ^ { n - j } . \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} K _ - | \\psi _ z ^ \\ell \\rangle = z | \\psi _ z ^ \\ell \\rangle , \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} t & = \\sum _ { i = 1 } ^ { t ' } \\left ( 1 + \\left \\lfloor \\frac { d - d _ i } { q } \\right \\rfloor \\right ) \\\\ & \\leq t ' \\left ( 1 + \\frac { d } { q } \\right ) - \\frac { D } { q } . \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} | G _ i | \\le \\frac { i } { d + i - k } \\prod _ { j = 1 } ^ i ( s _ j p _ j ) . \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} w _ 0 = y ( t _ k ) , \\ w _ j = A w _ { j - 1 } + \\sum _ { \\ell = 0 } ^ { p - j } \\frac { t ^ { \\ell } _ k } { \\ell ! } v _ { j + \\ell } , \\ j = 1 , \\ldots , p . \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} \\mu ^ * = \\frac { P _ T + \\textstyle \\sum _ { k = 1 } ^ { r _ s } { \\sigma ^ 2 } \\left ( \\left ( 1 - \\rho ^ * \\right ) [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 \\right ) ^ { - 1 } } { r _ s \\textstyle \\sum _ { k = 1 } ^ { r _ s } \\left ( \\left ( \\nu ^ * - \\rho ^ * [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 \\right ) \\ln 2 \\right ) ^ { - 1 } } . \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} \\widehat { g } ^ { \\pm } _ { \\sigma , \\Delta } ( \\xi ) = \\sum _ { k = - \\infty } ^ { \\infty } ( \\pm 1 ) ^ k \\frac { ( k + 1 ) } { | \\xi + k \\Delta | } \\Big ( e ^ { - 2 \\pi | \\xi + k \\Delta | ( \\sigma - \\frac 1 2 ) } - e ^ { - 2 \\pi | \\xi + k \\Delta | } \\Big ) . \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} \\tau ^ { n , \\omega ^ \\prime } _ R : = \\mathbf { 1 } _ { \\left \\lbrace 3 \\left \\lvert z ^ { \\omega ^ \\prime } \\right \\rvert _ \\infty < R \\right \\rbrace } \\cdot \\inf \\left \\lbrace t \\ge 0 : \\left \\lvert X ^ { n , \\omega ^ \\prime } _ t \\right \\rvert > \\frac { R } { 3 } \\right \\rbrace \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} \\theta ^ { \\tau , \\epsilon } _ t = \\begin{cases} \\omega & t \\in [ \\tau - \\epsilon , \\tau ] , \\\\ \\theta ^ * _ t & . \\end{cases} \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} \\theta _ \\pm = \\frac { a - d \\pm \\sqrt { ( a + d ) ^ 2 + 4 ( a d - b c ) } } { 2 c } \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} \\beta ( \\theta ) = \\frac { 1 } { \\ell } \\log \\mathrm { r a d } ( N _ \\theta ) \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} { \\rm R e } \\left \\{ 1 + z \\frac { g '' ( z ) } { g ' ( z ) } \\right \\} \\geq \\frac { 1 - 2 \\delta r + r ^ 2 } { 1 - r ^ 2 } \\quad ( | z | = r < 1 ) . \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} \\rho _ q ( d , m _ d ) \\le \\rho _ q ( d , m _ d ) - \\rho _ q ( b + e - 1 , m _ d - a ) + \\sum _ { i = 1 } ^ b \\rho _ q ( e + i - 1 , m _ d - a - 1 ) . \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} & T _ { W } : H \\rightarrow \\mathcal { K } _ { 2 , W } \\\\ & T _ { W } ( f ) = ( v _ { i } ( C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f ) . \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} R : = \\min \\left \\{ \\frac { 1 } { 2 } , \\ , \\left ( \\frac { 1 } { 4 C _ 2 } \\right ) ^ { \\frac { 1 } { 2 ( 1 - d / q ) } } \\right \\} \\ ! , \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} E _ k ^ p = I _ k - \\sum _ { j _ 1 , \\ldots , j _ k = 0 } ^ { p } C _ { j _ k \\ldots j _ 1 } { \\sf M } \\left \\{ J [ \\psi ^ { ( k ) } ] _ { T , t } \\sum \\limits _ { ( j _ 1 , \\ldots , j _ k ) } \\int \\limits _ t ^ T \\phi _ { j _ k } ( t _ k ) \\ldots \\int \\limits _ t ^ { t _ { 2 } } \\phi _ { j _ { 1 } } ( t _ { 1 } ) d { \\bf f } _ { t _ 1 } ^ { ( i _ 1 ) } \\ldots d { \\bf f } _ { t _ k } ^ { ( i _ k ) } \\right \\} , \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} s = \\sum _ { k = 1 } ^ \\infty ( - 2 ) ^ { k - 1 } y _ k \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} p _ { j } ( x ) : = \\frac { ( - 1 ) ^ { d ( j ) } } { I ( j ) ! } x ^ { I ( j ) } , \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} y ' ( t ) = A y ( t ) + v _ 1 + t v _ 2 + \\cdots + \\frac { t ^ { p - 1 } } { ( p - 1 ) ! } v _ p , \\ y ( 0 ) = v _ 0 . \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} \\begin{cases} x + y + z + w & = \\ell , \\\\ x + y - z - w & = a , \\\\ x - y + z - w & = a , \\\\ x - y - z + w & = a . \\end{cases} \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} ( \\Gamma _ 2 - \\rho \\Gamma ) ( f ) = 0 . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} G ( X _ 1 , \\dots , X _ n ) = \\sum _ { i = 0 } ^ d G _ i ( X _ 1 , \\dots , X _ { n - 1 } ) X _ n ^ i . \\end{align*}"}
-{"id": "9869.png", "formula": "\\begin{align*} ( _ a D _ x ^ { - \\sigma } u , v ) _ { L ^ 2 ( a , b ) } = ( u , { _ x D _ b ^ { - \\sigma } } v ) _ { L ^ 2 ( a , b ) } . \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} { \\cal E } _ { x _ i } '' ( 0 ) = \\theta _ i ^ 2 - l ^ 2 \\int _ 0 ^ 1 K ^ { \\dot \\gamma } _ X ( x _ i ( t ) ) \\ , d t \\ge 0 , { \\cal E } _ { \\bar e _ j } '' ( 0 ) = - \\ / l ^ 2 \\int _ 0 ^ 1 K ^ { \\dot \\gamma } _ X ( \\bar e _ j ) \\ , d t \\ge 0 . \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\tilde { X } _ H m _ X ( x , \\xi ) = \\frac { 1 } { 2 T _ { \\alpha _ 0 } ' } \\left ( m _ 0 \\circ \\tilde { \\Phi } ^ X _ { T _ { \\alpha _ 0 } ' } ( x , \\xi ) - m _ 0 \\circ \\tilde { \\Phi } ^ X _ { - T _ { \\alpha _ 0 } ' } ( x , \\xi ) \\right ) , \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c r } \\ddot { x } _ 1 + \\eta ( t ) \\dot { x } _ 1 + \\omega ^ 2 ( t ) x _ 1 = 0 , \\\\ \\ddot { x } _ 2 + \\eta ( t ) \\dot { x } _ 2 + \\omega ^ 2 ( t ) x _ 2 = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} M ' _ { k , \\ell } = \\begin{cases} 0 , & \\mbox { i f } k > \\ell \\\\ \\displaystyle \\lambda _ { q + 1 } ^ f \\left ( \\prod _ { i = 1 } ^ q ( \\lambda _ { q + 1 } - \\lambda _ i ) \\right ) ^ { m + 1 } , & \\mbox { i f } k = \\ell . \\end{cases} \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} Y = \\bigcup _ { z \\in Z } Y _ z , \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} L ( \\phi ^ { \\vee } , z _ 1 , \\dots , z _ n , s ) : = L ( \\phi ^ { \\vee } , z _ 1 , \\dots , z _ n ; \\theta ^ { s } , - s ) = \\sum \\limits _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) } { a ^ s } . \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} g _ { t , l , v } = \\left ( \\begin{matrix} \\varpi _ { p } ^ t & 0 \\\\ 0 & 1 \\end{matrix} \\right ) w \\left ( \\begin{matrix} 1 & v \\varpi _ { p } ^ { - l } \\\\ 0 & 1 \\end{matrix} \\right ) = \\left ( \\begin{matrix} 0 & \\varpi _ { p } ^ t \\\\ - 1 & - v \\varpi _ { p } ^ { - l } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\left < - e ^ { t _ { j k } } [ 1 + \\frac { 1 } { 2 } ( \\phi _ j - \\phi _ k ) ^ 2 ] + \\frac { d } { d t _ { j k } } \\ln f _ \\alpha ( e ^ { t _ { j k } } ) + 1 \\right > = 0 . \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} a _ 1 + \\frac { e _ 1 } { e _ 2 } a _ 4 = a _ 2 + \\frac { e _ 1 } { e _ 2 } a _ 3 + O \\big ( \\delta | e _ 2 | ^ { - 1 } \\big ) , \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} & \\mathbf { a } _ { k } ( l ) = \\langle \\lambda _ k + \\cdots + \\lambda _ r , \\alpha _ { u _ { k , l } } ^ { \\vee } + \\cdots + \\alpha _ { u _ { k , m _ k } } ^ { \\vee } \\rangle 1 \\leq l \\leq m _ k , \\\\ & \\mathbf a _ { k } ( m _ k + 1 ) = 0 . \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { q + 2 } y _ i / x _ i = 0 . \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} \\overline { U } _ { ( n - 7 ) / 3 , j } & = \\left \\{ g - n + 1 , \\frac { 1 } { 2 } ( 2 g + 3 j - 2 n + 5 ) , \\frac { 1 } { 2 } ( 2 g - 3 j + 4 n - 7 ) \\right \\} _ { g } . \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} u _ { k + 1 } ^ { \\delta } = u _ 0 + J _ q ^ * ( J _ p ( u _ { k + 1 } ^ { \\delta } - u _ 0 ) ) , \\ 0 < \\beta _ k \\leq \\beta _ { \\max } < 1 , \\ k = 0 , 1 , 2 , \\cdots \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ { N _ 1 } X _ { j , 1 } , \\ldots , \\sum _ { j = 1 } ^ { N _ m } X _ { j , m } \\right ) \\le _ { s m } \\left ( \\sum _ { j = 1 } ^ { N _ 1 } Y _ { j , 1 } , \\ldots , \\sum _ { j = 1 } ^ { N _ m } Y _ { j , m } \\right ) . \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} v ( t ) & = e ^ { - i t H } P _ c u _ 0 - i \\int _ 0 ^ t e ^ { - i ( t - s ) H } P _ c F ( v ( s ) + a ( s ) \\phi _ 0 ) \\ , d s , \\\\ a ( t ) & = e ^ { i \\frac { 1 } { 2 } q ^ 2 t } a ( 0 ) - i \\int _ 0 ^ t e ^ { i \\frac 1 2 q ^ 2 ( t - s ) } \\langle \\phi _ 0 , F ( v ( s ) + a ( s ) \\phi _ 0 ) \\rangle \\ , d s . \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} 0 = \\frac { \\partial } { \\partial y } ( y \\beta ( y , t ) ) + \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\beta ( y , t ) + \\biggl [ \\frac { 1 } { 2 } y \\frac { \\partial } { \\partial y } + \\frac { 1 } { 2 } ( y ^ { 2 } + 1 ) \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } + \\frac { 1 } { 4 } y \\frac { \\partial ^ { 3 } } { \\partial y ^ { 3 } } + \\frac { 1 } { 6 4 } \\frac { \\partial ^ { 4 } } { \\partial y ^ { 4 } } - \\frac { 1 } { 2 } \\biggr ] p _ { 0 } ( y , t ) \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} \\big [ \\psi ( \\tilde { \\mathbb { A } } / f \\tilde { \\mathbb { A } } ) \\big ] _ { \\tilde { \\mathbb { A } } } = \\det _ { \\mathbb { F } _ q ( z _ 1 , \\dots , z _ n , t ) [ X ] } ( X - \\psi _ { \\theta } \\mid \\tilde { \\mathbb { A } } / f \\tilde { \\mathbb { A } } \\otimes _ { \\mathbb { F } _ q ( z _ 1 , \\dots , z _ n , t ) } \\mathbb { F } _ q ( z _ 1 , \\dots , z _ n , t ) [ X ] ) _ { | X = \\theta } . \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} \\mu \\big ( Q ( f ) \\big ) = \\int _ G f ( x ) d \\tilde { \\mu } . \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} \\begin{aligned} V _ 1 = \\frac { 1 } { \\sqrt { 3 } } \\bigl ( E ^ { ( 3 ) } _ { 1 3 } + E ^ { ( 3 ) } _ { 2 2 } + E ^ { ( 3 ) } _ { 3 1 } \\bigr ) , & V _ 2 = \\frac { 1 } { \\sqrt { 3 } } \\bigl ( q \\ , E ^ { ( 3 ) } _ { 1 2 } + E ^ { ( 3 ) } _ { 2 1 } + E ^ { ( 3 ) } _ { 3 3 } \\bigr ) , \\\\ V _ 3 = \\frac { 1 } { \\sqrt { 3 } } \\bigl ( E ^ { ( 3 ) } _ { 1 1 } & + \\bar { q } \\ , E ^ { ( 3 ) } _ { 2 3 } + E ^ { ( 3 ) } _ { 3 2 } \\bigr ) . \\end{aligned} \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} & \\quad \\sum _ { h _ 2 , k } \\theta ( g _ j , g _ k ) ( \\pi ( \\xi ) | _ { g _ j X _ k \\cap X _ j } ) \\\\ & = \\sum _ { h _ 2 , k } \\theta ( g _ j , g _ k ) ( \\pi ( \\xi ) | _ { g _ j X _ { g _ k , h _ 2 } \\cap X _ j } ) ~ ( \\mbox { a s $ X _ k = X _ { g _ k , h _ 2 } $ b y c o n v e n t i o n } ) \\\\ & = \\sum _ { g } \\sum _ { \\substack { h _ 2 \\\\ h _ 2 \\in c ( g , X ) } } \\theta ( g _ j , g ) ( \\pi ( \\xi ) | _ { g _ j X _ { g , h _ 2 } \\cap X _ j } ) \\\\ & = \\sum _ g \\theta ( g _ j , g ) ( \\pi ( \\xi ) | _ { X _ j } ) . \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} \\theta _ { u _ t } ^ n \\leq C t ^ { - 1 } \\sum _ { k = 1 } ^ n \\theta _ { u } ^ k \\wedge \\theta _ { \\phi } ^ { n - k } + \\theta _ { \\phi } ^ n , \\ \\ \\forall t > 1 , \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} \\{ u ^ i { } _ { \\lambda } u ^ j \\} = ( \\lambda + \\partial ) \\left ( g ^ { j i } \\lambda + c ^ { j i } _ l u ^ l _ x + w ^ j _ l u ^ l _ x ( \\lambda + \\partial ) ^ { - 1 } w ^ i _ m u ^ m _ x \\right ) \\lambda . \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} ( M e ^ { - t A ^ { 2 } } f ) ( \\tau , y ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( - t ) ^ { k } } { k ! } ( M A ^ { 2 k } f ) ( \\tau , y ) . \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} G ( g ) ( m \\otimes \\{ [ c _ 1 | . . . | c _ k ] \\} ) : = \\sum _ { i = 0 } ^ { k } g ( m \\cdot [ c _ 1 | . . . | c _ i ] \\otimes c _ { i + 1 } ) \\cdot [ c _ { i + 2 } | . . . | c _ k ] . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} \\nabla _ X ^ P Y = ( \\pi ^ * \\nabla ^ B ) _ X Y + \\frac { 1 } { 2 } \\Omega ( X , Y ) . \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} o s c ( \\psi _ 1 , B _ \\epsilon ( x ) ) = \\underset { y \\in B ( x , \\epsilon ) \\cap X } { \\textrm { e s s - s u p } } \\int _ { B ( y , r ) } h ( z ) d z - \\underset { \\tilde y \\in B ( x , \\epsilon ) \\cap X } { \\textrm { e s s - i n f } } \\int _ { B ( \\tilde y , r ) } h ( z ) d z . \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} J u ( n , E _ j ) = E _ j u ( n , E _ j ) , \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} f _ { \\mathrm { c o s p } } ( i ) = \\begin{cases} 0 & \\ i = 1 \\\\ d - n + i - 2 & \\end{cases} k _ i ( f _ { \\mathrm { c o s p } } ) = d - i - 1 . \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} B ( t ) = \\sup _ { 0 \\le { s } \\le { t } } ( { A } ( s , t ) - { S } ( s , t ) ) . \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} J \\big ( ( g _ { A _ k } ( \\cdot + b _ k ) ) \\big ) < + \\infty \\iff A \\textup { i s p o s i t i v e d e f i n i t e ( i . e . $ ( A _ k ) _ { k = 1 } ^ m \\in \\Lambda $ ) } \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} \\{ G \\in d g C a t _ { \\mathbf { k } } ( \\Lambda ( J ^ n ) , ^ { \\tau } _ { \\Omega C } ) : G ( x ) = \\theta ( P ) , G ( y ) = \\theta ( Q ) \\} , \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{gather*} ( e _ { s - 1 } ) _ A \\overset { \\mathcal { X } ^ + ( 2 ) } { W } \\ ! \\ ! \\ ! \\ ! _ { v B ^ { - 1 } A } \\triangleright \\big ( \\chi ^ + _ s + \\chi ^ - _ { p - s } \\big ) = \\chi ^ + _ { s - 1 } + \\chi ^ - _ { p - s + 1 } , \\\\ ( e _ { s + 1 } ) _ A \\overset { \\mathcal { X } ^ + ( 2 ) } { W } \\ ! \\ ! \\ ! \\ ! _ { v B ^ { - 1 } A } \\triangleright \\big ( \\chi ^ + _ s + \\chi ^ - _ { p - s } \\big ) = \\chi ^ + _ { s + 1 } + \\chi ^ - _ { p - s - 1 } . \\end{gather*}"}
-{"id": "2070.png", "formula": "\\begin{align*} \\mathrm G ( \\left \\vert \\cdot \\right \\vert ^ 2 ) ( x ) = 2 n - 2 n \\vert x \\vert ^ 2 + 2 \\beta \\sum _ { i \\neq j } \\frac { x _ i } { x _ i - x _ j } = 2 n - 2 n \\vert x \\vert ^ 2 + n ( n - 1 ) \\beta . \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\left ( e _ { ( z , w ) } \\circ T _ V \\right ) f & = \\left \\langle f , R _ { \\mathcal { X } _ p } \\left ( e _ { ( z , w ) } \\circ T _ V \\right ) \\right \\rangle _ { \\mathcal { X } _ p } \\\\ & = \\int _ { \\R } \\int _ { \\mathbb { B } _ p } f \\left ( t , \\frac { \\zeta _ 1 } { z ^ { 1 / 2 m _ 1 } } , \\dots , \\frac { \\zeta _ n } { z ^ { 1 / 2 m _ n } } \\right ) \\overline { R _ { \\mathcal { X } _ p } \\left ( e _ { ( z , w ) } \\circ T _ V \\right ) ( t , \\zeta ) } \\lambda ( p ( \\zeta ) , t ) \\d V ( \\zeta ) \\d t . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} \\Lambda = ( \\Lambda _ 1 ^ 1 \\oplus \\cdots \\oplus \\Lambda _ m ^ 1 ) \\oplus ( \\Lambda _ 1 ^ 2 \\oplus \\cdots \\oplus \\Lambda _ m ^ 2 ) \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} q ( - w ) = q ( ( j - n ) y - z _ j + c ) \\le q ( ( j - n ) y ) + q ( - z _ j + c ) = q ( - z _ j + c ) . \\end{align*}"}
-{"id": "9981.png", "formula": "\\begin{align*} A ( \\tau ) : = \\mathbb E [ N ( \\tau ) ] = \\int _ 0 ^ \\tau \\alpha ( s ) d s . \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} \\begin{gathered} \\mathfrak { S } ( M ) ^ + = \\mathfrak { S } ( M ) ^ + _ + \\sqcup \\mathfrak { S } ( M ) ^ + _ - \\mathfrak { S } ( M ) ^ - = \\mathfrak { S } ( M ) ^ - _ + \\sqcup \\mathfrak { S } ( M ) ^ - _ - \\\\ \\mathfrak { S } ( M ) _ + = \\mathfrak { S } ( M ) ^ + _ + \\sqcup \\mathfrak { S } ( M ) ^ - _ + \\mathfrak { S } ( M ) _ - = \\mathfrak { S } ( M ) ^ + _ - \\sqcup \\mathfrak { S } ( M ) ^ - _ - \\\\ \\end{gathered} \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} | A | & \\ge q ^ { u g } \\left ( 1 - \\dfrac { 1 } { q } \\right ) ^ { N } - \\left ( \\sqrt { q } - 1 \\right ) ^ { ( u - 1 ) g + 1 } q ^ { g } \\\\ & \\ge q ^ { u g } \\left ( 1 - \\dfrac { 1 } { q } \\right ) ^ { N } - \\sqrt { q } ^ { ( u - 1 ) g + 1 } q ^ { g } \\\\ & = q ^ { u g } \\left ( 1 - \\dfrac { 1 } { q } \\right ) ^ { N } - q ^ { \\frac { ( u + 1 ) g + 1 } { 2 } } . \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} R _ { 1 2 } & = ( d b ^ { - 1 } a - c ) ^ { - 1 } \\left ( a d - d a + c b - b c + d a - d b ^ { - 1 } a b \\right ) ( \\Delta ' ) ^ { - 1 } \\\\ & = ( d b ^ { - 1 } a - c ) ^ { - 1 } \\left ( a d - b c + c b - d b ^ { - 1 } a b \\right ) ( \\Delta ' ) ^ { - 1 } \\\\ & = ( d b ^ { - 1 } a - c ) ^ { - 1 } \\left ( \\Delta ' + ( c - d b ^ { - 1 } a ) b \\right ) ( \\Delta ' ) ^ { - 1 } \\\\ & = \\left ( - b + ( d b ^ { - 1 } a - c ) ^ { - 1 } \\Delta ' \\right ) ( \\Delta ' ) ^ { - 1 } \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} \\langle f , h \\rangle _ \\mu = \\int _ { \\mathbb { R } ^ n } f ( x ) \\ , h ( x ) \\ , d \\mu ( x ) \\ , , \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} E _ 2 ^ { i , j } = { \\rm { H } } ^ i ( k , \\R ^ j f _ * G _ { k ' } ) \\Longrightarrow { \\rm { H } } ^ { i + j } ( k ' , G _ { k ' } ) , \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } f ( x ) d x = \\int _ { 0 } ^ { \\infty } \\int _ { \\wp } f ( r y ) r ^ { Q - 1 } d \\sigma ( y ) d r . \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} g ( T ^ { ' } ( U , V ) , W ) = g ( T ( W , U ) , V ) . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} \\ker B _ 0 + \\ker B _ { m + 1 } = H \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} \\sigma _ s ( [ u , m ] ) = [ u , s ( m ) ] . \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} L e b \\left ( \\bigcup _ { 0 \\leq j < N _ k } B \\big ( j \\theta , { 1 \\over N _ k ^ \\tau } \\big ) \\right ) \\leq q _ k \\cdot { 5 \\over q _ k ^ { 1 + \\epsilon _ 2 } } = { 5 \\over q _ k ^ { \\epsilon _ 2 } } \\leq 5 \\beta ^ { - \\epsilon _ 2 k } , \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\lambda ( n _ 1 n _ 2 , p ) = \\lambda ( n _ 1 , n _ 2 p ) \\lambda ( n _ 2 , p ) . \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} \\mu _ t ( x , \\theta , C ) = \\delta ( C - \\hat { C } ( t , x ) ) \\otimes \\delta ( \\theta - \\hat { \\theta } ( t , x ) ) \\end{align*}"}
-{"id": "9970.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ S _ 1 , \\dots , S _ { n + 1 } , S _ { n + 2 } \\} , \\mathcal { S } ' = \\{ S _ 1 , \\dots , S _ { n + 1 } \\} , \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} \\sigma _ x = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\ , \\ , \\ , \\ , \\ , \\sigma _ y = \\begin{pmatrix} 0 & - i \\\\ i & 0 \\end{pmatrix} \\ , \\ , \\ , \\ , \\ , \\sigma _ z = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} R & \\leq | x - x _ 0 | + C _ 0 t \\leq | x - y | + C _ 0 ( t - s ) + | y - x _ 0 | + C _ 0 s \\\\ & \\leq \\epsilon + C _ 0 \\epsilon + R - C _ 0 \\Big ( 2 + \\frac { 1 } { C _ 0 } \\Big ) \\epsilon = R - C _ 0 \\epsilon , \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} a _ n ( \\omega ) = a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } } ( 1 ) \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} | \\nabla \\Q ^ t | ^ 2 = | \\nabla \\Q | ^ 2 \\circ \\mathbf { X } ^ t + 2 t ( \\partial _ k Q _ { i j } \\circ \\mathbf { X } ^ t ) ( \\partial _ p Q _ { i j } \\circ \\mathbf { X } ^ t ) \\partial _ k X _ p \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} ( Q ' ) ^ \\top P ' = Q ^ \\top P . \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} M ( u ( t ) ) & = \\int | u ( t , x ) | ^ 2 d x = M ( u _ 0 ) , \\\\ E ( u ( t ) ) & = \\frac { 1 } { 2 } \\int | \\nabla u ( t , x ) | ^ 2 d x - \\frac { c } { 2 } \\int | x | ^ { - 2 } | u ( t , x ) | ^ 2 d x - \\frac { 1 } { \\alpha + 2 } \\int | u ( t , x ) | ^ { \\alpha + 2 } d x , \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} - \\varphi _ t - ( - \\Delta ) ^ s \\varphi + \\alpha _ { n , s } ( n - 2 s ) \\mu ^ { - ( n - 2 s ) - 1 } _ j \\frac { | y _ j | ^ 2 } { \\left ( 1 + | y _ j | ^ 2 \\right ) ^ { \\frac { n - 2 s } { 2 } + 2 } } \\dot { \\xi } _ j \\cdot y _ j = 0 \\mathbb { R } ^ n \\times ( t _ 0 , + \\infty ) . \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} S _ { 0 , \\sigma } ( t , \\pi ) - S _ { 0 , \\sigma } ( t - h , \\pi ) = h \\ , S _ { - 1 , \\sigma } ( t _ h ^ * , \\pi ) & \\geq - h \\ , M _ { - 1 , \\sigma } ^ - ( t _ h ^ * ) \\ , \\ell _ { 0 , \\sigma } ( t _ h ^ * ) + h \\ , O _ c ( r _ { 1 , \\sigma } ( t _ h ^ * ) ) \\\\ & = - h \\ , M _ { - 1 , \\sigma } ^ - ( t _ h ^ * ) \\ , \\ell _ { 0 , \\sigma } ( t _ h ^ * ) + h \\ , O _ c ( r _ { 1 , \\sigma } ( t ) ) , \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} \\sin ~ z = z ~ { } _ { 0 } F _ { 1 } \\left ( \\ \\begin{array} { l l l } \\overline { ~ ~ ~ ~ ~ } ; ~ \\\\ ~ ~ ~ \\frac { 3 } { 2 } ; ~ \\end{array} \\frac { - z ^ { 2 } } { 4 } \\right ) . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} S _ { X ^ * , Y ^ * } ^ { R , R } & = S _ { X , Y } ^ { L , L } , \\\\ S _ { X ^ * , Y ^ * } ^ { L , R } & = S _ { Y , X } ^ { L , R } . \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} \\begin{aligned} \\det Y & = \\det \\left [ \\left ( w _ { i , j } \\right ) _ { 0 \\le i , j < p } + \\left ( \\epsilon _ { i , j } \\right ) _ { 0 \\le i , j < p } \\right ] \\\\ & = \\det \\left [ \\left ( w _ { i , j } \\right ) _ { 0 \\le i , j < p } \\right ] + \\epsilon \\end{aligned} \\end{align*}"}
-{"id": "9805.png", "formula": "\\begin{align*} u ( q , p ) = \\frac { 1 } { L } \\int _ { 0 } ^ L e ^ { s X _ 0 } \\varphi d s = \\frac { 1 } { L } \\int _ { 0 } ^ L \\varphi _ s ( q , p ) d s \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} \\| P _ N a - a \\| & \\leq \\| P _ N ( a - a _ j ) \\| + \\| P _ N a _ j - a _ j \\| + \\| a _ j - a \\| \\\\ & \\leq ( 1 + c ) \\| a - a _ j \\| + \\max _ { i = 1 , \\ldots , m } \\| P _ N a _ i - a _ i \\| \\\\ & \\leq ( 1 + c ) \\delta + \\max _ { i = 1 , \\ldots , m } \\| P _ N a _ i - a _ i \\| , \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} F _ n & \\leq 2 \\left ( \\epsilon ^ { \\frac { 8 } { 7 } } \\left ( \\phi _ { ( 2 u _ n - v _ 0 ) } ^ { 6 } + | u _ n | ^ 2 \\right ) + { \\epsilon ^ { - 8 } } \\phi _ { v _ 0 } ^ 6 + \\epsilon ^ 4 \\phi _ { ( u _ n - v _ 0 ) } ^ 6 + { \\epsilon ^ { - \\frac { 4 } { 3 } } } v _ 0 ^ 2 + \\left | \\phi _ { v _ 0 } v _ 0 \\right | ^ { \\frac { 3 } { 2 } } \\right ) = : G _ n , \\\\ \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} \\iota _ { k , d } : = \\sum _ { i = 0 } ^ { k - 1 } ( d - 1 ) ^ i = \\frac { 1 } { d - 2 } ( ( d - 1 ) ^ k - 1 ) . \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\equiv 0 , 1 ( 4 ) } c _ n ^ - u _ { 1 / 2 , n } ^ { [ 0 ] , - } ( y ) e ^ { 2 \\pi i n x } + c _ 0 ^ + . \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\bar { T } ( t ) = 2 ^ { p / 2 - 4 } \\cdot 3 ^ { p - 2 } \\left ( \\frac { C } { B } \\right ) ^ { p - 2 } = : \\gamma > 0 . \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n } \\left \\vert \\mathrm { T r } \\ , \\left ( \\mathcal { L } _ { s , \\lambda } ^ { n N } \\right ) \\right \\vert = N \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { N n } \\left \\vert \\mathrm { T r } \\ , \\left ( \\mathcal { L } _ { s , \\lambda } ^ { n N } \\right ) \\right \\vert \\leq N \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m } \\left \\vert \\mathrm { T r } \\ , \\left ( \\mathcal { L } _ { s , \\lambda } ^ { m } \\right ) \\right \\vert . \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} e ^ { i t \\sqrt { \\mathcal { L } _ V } } u _ 0 & = \\int _ 0 ^ \\infty ( s \\rho ) ^ { - \\frac { n - 2 } 2 } J _ { \\nu _ 0 } ( s \\rho ) e ^ { i t \\rho } ( \\mathcal { H } _ { \\nu _ 0 } u _ 0 ) ( \\rho ) \\rho ^ { n - 1 } d \\rho \\\\ & = \\int _ 0 ^ \\infty ( s \\rho ) ^ { - \\frac { n - 2 } 2 } J _ { \\nu _ 0 } ( s \\rho ) e ^ { i t \\rho } \\chi ( \\rho ) \\rho ^ { n - 1 } d \\rho . \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\tau ' _ t ( d ) & : = \\begin{cases} \\frac { 1 } { d - 2 } ( 2 ( d - 1 ) ^ { t / 2 + 1 } - d ) & 2 \\mid t \\\\ \\frac { 1 } { d - 2 } ( d ( d - 1 ) ^ { ( t + 1 ) / 2 } - d ) & 2 \\nmid t \\end{cases} \\ \\ \\\\ \\tau _ t ( d ) & : = \\begin{cases} \\frac { 1 } { d - 2 } ( d ( d - 1 ) ^ { t / 2 } - 2 ) & 2 \\mid t \\\\ \\frac { 1 } { d - 2 } ( 2 ( d - 1 ) ^ { ( t + 1 ) / 2 } - 2 ) & 2 \\nmid t \\end{cases} . \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} 0 \\le J _ i ^ + ( \\pi ) : = E _ { \\mu } ^ \\pi \\left ( \\sum _ { n = 1 } ^ \\infty \\beta ^ { n - 1 } r _ i ^ + ( s _ n , a _ n ) \\right ) \\le E _ { \\mu } ^ \\pi \\left ( \\sum _ { n = 1 } ^ \\infty \\beta ^ { n - 1 } w ( s _ n ) \\right ) < \\infty . \\end{align*}"}
-{"id": "9866.png", "formula": "\\begin{align*} { _ a D _ x ^ { - \\mu } } { _ a D _ x ^ { - \\sigma } } u ( x ) = { _ a D _ x ^ { - \\mu - \\sigma } } u ( x ) ~ { _ x D _ b ^ { - \\mu } } { _ x D _ b ^ { - \\sigma } } u ( x ) = { _ x D _ b ^ { - \\mu - \\sigma } } u ( x ) . \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} T a ( z , w ) = \\sum _ { k = 0 } ^ { \\infty } a _ k ( w ) z ^ k , ( z , w ) \\in \\mathcal { E } _ p \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} \\dot { S } & = 0 = G ( S , N , F ) - l ( S , N , F ) \\ , S = \\left ( f - c _ 1 ( S , N , F ) \\right ) N \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} W _ { \\infty } ( x ) = e \\left ( \\frac { a } { b l ^ { \\frac { n _ l } { 2 } } } x \\right ) F \\left ( \\frac { x } { M } \\right ) . \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} \\Xi _ h = \\begin{cases} \\Pi _ { k , h } ^ 0 & 2 \\leq k \\leq 3 , \\\\ \\Pi _ { k - 1 , h } ^ 0 & k \\geq 4 . \\end{cases} \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} \\frac { T _ { w + 1 } } { J _ w } = \\frac { T _ { w } C _ { w + 1 } } { J _ w } < M C _ { w + 1 } \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} \\lvert x \\rvert _ 0 = \\begin{cases} 0 , \\ x = 0 \\\\ 1 , \\end{cases} . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} \\varepsilon _ n = \\sup \\left \\{ \\tfrac { 1 } { \\sqrt { M ^ j } } \\left \\| U _ { n , M } ^ { 0 } - u \\right \\| _ { k } \\colon j , k \\in \\N _ 0 , j + n + k = N \\right \\} \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} \\phi _ { H ( n ) } ( t , s ) & = \\sum _ { k = 0 } ^ \\infty \\frac { \\Gamma ( - \\lambda + 1 ) } { \\Gamma ( - \\lambda - k + 1 ) \\Gamma ( k + 1 ) } ( \\cosh x ) ^ { - \\lambda - k } \\left ( - \\frac { 1 } { 2 ( n - 1 ) } \\sum _ { i = 1 } ^ { n - 1 } t _ i \\right ) ^ k \\\\ & = \\sum _ { k = 0 } ^ \\infty ( 2 ( n - 1 ) ) ^ { - k } \\frac { \\Gamma ( \\lambda + k ) } { \\Gamma ( \\lambda ) \\Gamma ( k + 1 ) } ( \\cosh x ) ^ { - \\lambda - k } \\left ( \\sum _ { i = 1 } ^ { n - 1 } t _ i \\right ) ^ k , \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\frac { n + m } { 2 } + D _ { n , m } = \\frac { n + m } { 2 } + \\frac { n - m } { 2 } = n . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} d f _ { \\tau } \\odot ( - u ^ 2 d t + X _ i d x ^ i ) = L _ Z g ^ { ( 4 ) } . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} \\ln f _ \\alpha ( e ^ { t } ) = \\int _ { 0 } ^ t g ( s ) d s + \\ln f _ \\alpha ( 0 ) > K t + \\ln f _ \\alpha ( 1 ) . \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} s - \\lim _ { k \\rightarrow 0 } \\widehat { k } \\cdot \\nabla K ( \\xi - d \\Gamma ( m ) ) Q _ 0 ( k , \\xi ) ( 1 - P _ 0 ( \\xi ) ) = 0 \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} \\varphi ( u _ V ) : = \\max _ { e \\in E } \\frac { u ( ` x ( e ) ) } { u ( V ) } , \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} \\begin{cases} ( \\gamma \\nabla v ) = 0 , \\ \\ \\ \\Omega \\\\ \\ \\ v = g , \\ \\ \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} 0 = ( P u , u ) = ( - \\Delta u , u ) + ( V u , u ) \\geq \\norm { \\nabla u } ^ 2 . \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} r _ i r _ i & = 1 \\\\ r _ i r _ j & = r _ j r _ i a ( 1 ) _ { i j } = 0 \\\\ r _ i r _ j r _ i & = r _ j r _ i r _ j \\ , a ( 1 ) _ { i j } = - 1 . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} \\mu ( Z _ 1 , Z _ 2 ) = Z _ 3 \\ , , \\mu ( Z _ 1 , Z _ 3 ) = - Z _ 2 \\ , , \\mu ( Z _ 2 , Z _ 3 ) = Z _ 1 \\ , . \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\sup _ { \\| f \\| _ { 1 , \\tau } \\leq 1 } \\int _ { \\mathbb { H } ^ { n } } \\frac { 1 } { ( \\rho ( \\xi ) ) ^ { \\beta } } \\left ( \\exp ( \\alpha | f ( \\xi ) | ^ { Q ' } ) - \\sum _ { k = 0 } ^ { Q - 2 } \\frac { \\alpha ^ { k } | f ( \\xi ) | ^ { k Q ' } } { k ! } \\right ) d \\xi < \\infty , \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} P _ { k , m } ( z , s ) = \\varphi _ { k , m } ( z , s ) + \\sum _ { ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) } b _ { k , m } ( n , s ) \\mathcal { W } _ { k , n } ( y , s ) e ^ { 2 \\pi i n x } , \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{gather*} P _ n = \\left \\{ \\exp \\left ( \\frac { 2 \\pi } { n + 2 } j \\right ) \\right \\} _ { j = 0 } ^ { n + 1 } P _ n ^ k = \\{ U \\subset P _ n : \\abs { U } = k + 1 \\} . \\end{gather*}"}
-{"id": "5779.png", "formula": "\\begin{align*} A ^ { W , W } _ t ( l ) = \\left ( \\int _ 0 ^ t l ^ * ( r , W _ r ) I _ d \\mathrm d r \\right ) ^ * = \\int _ 0 ^ t l ( r , W _ r ) \\mathrm d r , \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} \\frac { \\partial x _ i } { \\partial y _ j } = \\frac { | x | } { | y | } \\left [ \\delta _ { i j } + \\left ( \\frac { | y | } { | x | } \\phi ' ( | y | ) - 1 \\right ) \\frac { y _ i y _ j } { | y | ^ 2 } \\right ] , \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} 0 \\to \\pi ^ * N _ { d - 2 } ( - d ) \\to \\pi ^ * N _ { d - 3 } ( - d + 2 ) \\to \\pi ^ * N _ { d - 4 } ( - d + 3 ) \\to \\dots \\\\ \\dots \\to \\pi ^ * N _ 2 ( - 3 ) \\to \\pi ^ * N _ 1 ( - 2 ) \\to \\O _ { \\P E } \\to \\O _ X \\to 0 , \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} \\psi ( \\overline { ( H ( X , Y ) - H ( Y , X ) ) d t } ) = 0 , \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} F = \\frac { P _ T } { \\dot { B } } - \\frac { \\dot { A } _ 1 } { A _ 1 ' \\dot { B } } P _ 1 - \\frac { \\dot { A } _ 2 } { A _ 2 ' \\dot { B } } P _ 2 + \\frac { 1 } { \\dot { B } } \\left [ \\int \\left ( \\dot { D } _ 1 A _ 1 ' - \\dot { A } _ 1 D _ 1 ' \\right ) d Q _ 1 + \\int \\left ( \\dot { D } _ 1 A _ 2 ' - \\dot { A } _ 2 D _ 2 ' \\right ) d Q _ 2 \\right ] \\ ; , \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} F ( n ) = \\prod _ { i = 1 } ^ n i ! , \\qquad \\Phi ( n ) = n ! \\cdot ( n - 2 ) ! \\cdot ( n - 4 ) ! \\cdots . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} C _ { d , s , m } = \\sup _ { \\l \\geq m + 1 } \\frac { \\alpha _ { \\l } \\beta _ { \\l } ( \\l ^ 2 - 1 ) ^ { 2 s } } { \\l ^ { 2 s + 2 } ( \\alpha _ { \\l } ( \\l + 1 ) ^ { 2 s } - \\beta _ { \\l } ( \\l - 1 ) ^ { 2 s } ) } . \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} \\omega ( v , w ) = \\varepsilon _ { r s } \\ , v ^ r w ^ s , \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} Q _ 2 = \\frac { 1 } { 4 \\pi } \\sum _ { ( j , k , l , m ) \\in \\Delta \\setminus \\Delta _ 1 } \\gamma _ j \\gamma _ k \\gamma _ l \\gamma _ m q _ j \\bar { q } _ k q _ l \\bar { q } _ m . \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j > ( \\log ( 1 / \\epsilon ) ) ^ { 1 + \\delta } } \\mu \\left ( \\prod _ { k = 1 } ^ { j } B _ k ( \\epsilon ) \\right ) r _ { j + 1 } ( \\epsilon ) \\right \\| = O ( \\epsilon ^ 2 ) , \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} W _ { 0 , 0 } ( \\overline \\varphi , \\varphi ) ( n ) = 2 i a _ { n + 1 } \\mathrm { I m } ( \\overline { \\varphi ( n ) } \\varphi ( n + 1 ) ) = i \\omega , \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 3 } \\left ( \\abs { \\nabla u } ^ 2 + u ^ 2 \\right ) d x & < \\int _ { \\mathbb R ^ 3 } \\left ( \\abs { \\nabla u } ^ 2 + u ^ 2 \\right ) d x + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ u u ^ 2 d x \\\\ & = \\int _ { \\mathbb R ^ 3 } f ( u ) u d x \\\\ & \\leq \\frac { 1 } { 2 } \\int _ { \\mathbb R ^ 3 } u ^ 2 d x + C _ { 1 / 2 } \\int _ { \\mathbb R ^ 3 } \\abs { u } ^ q d x \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial U } ( 1 , \\ , U ) & = \\frac { r } { 2 } - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ r \\frac { m - m _ i + \\lambda - m U } { [ ( m - m _ i + \\lambda - m U ) ^ 2 + 4 \\lambda m _ i ] ^ { 1 / 2 } } > 0 , \\\\ \\frac { \\partial ^ 2 F } { \\partial U ^ 2 } ( 1 , \\ , U ) & = \\frac { m } { 2 } \\sum _ { i = 1 } ^ r \\frac { 4 \\lambda m _ i } { \\{ ( m - m _ i + \\lambda - m U ) ^ 2 + 4 \\lambda m _ i \\} ^ { 3 / 2 } } > 0 . \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} \\sum ^ { N } _ { i = 1 } u ( x _ 1 ) \\leq \\sum _ { 1 \\leq i < j \\leq N } \\dfrac { 1 } { \\vert x _ i - x _ j \\vert } , \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} \\lim _ { \\l \\rightarrow \\infty } \\frac { 1 } { T _ l } \\int _ 0 ^ { T _ l } k ( y ^ l ( t ) , u ^ l ( t ) ) d t = \\tilde { k } ^ * ( z ) . \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n v _ j \\alpha _ j ^ t h ( \\alpha _ j ) c _ j = 0 t = 0 , \\dots , s - 1 . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} 4 m ^ 2 \\left ( \\frac 1 2 u _ \\alpha \\bar u ^ \\alpha + v _ \\beta v ^ \\beta \\right ) = c \\ , , \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\frac { \\rho ( n _ 1 ) } { n _ 1 } = \\frac { \\ln n _ 1 + \\frac { 2 \\sqrt { n _ 1 } } { e } } { n _ 1 } . \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} ( h u ^ { i \\bar j } w _ i ) _ { \\bar j } \\ge \\eta ^ { \\b l } ( \\eta ^ k h _ k ) _ { \\b l } - g , g : = \\frac { | h _ \\eta | ^ 2 } { h } . \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} \\langle f , \\pi ( g ) K _ z \\rangle & = \\langle \\pi ( g ^ { - 1 } ) f , K _ z \\rangle \\\\ & = ( \\pi ( g ^ { - 1 } ) f ) ( z ) \\\\ & = j _ \\lambda ( g ^ { - 1 } , z ) f ( g ^ { - 1 } \\cdot z ) \\\\ & = \\langle f , \\overline { j _ \\lambda ( g ^ { - 1 } , z ) } K _ { g ^ { - 1 } \\cdot z } \\rangle . \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} \\varphi ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } \\varphi ( x , y , t ) \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} \\pi _ { \\mathbb { R } ^ n } \\circ r = \\pi _ { \\mathbb { R } ^ n } U , \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} \\lvert u _ n \\rvert _ S & = \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - ( n - j ) \\\\ & = \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - n + j \\\\ & \\geq \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - n + ( n - 2 \\lvert I \\rvert + 1 ) \\\\ & = \\lvert ( \\phi v _ 0 ) _ n \\rvert _ S - 2 \\lvert I \\rvert + 1 \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} \\epsilon _ 0 : = | | X _ { P _ 0 } | | _ { s _ 0 , r _ 0 , p - 1 , \\mathbf { a } _ 0 ; \\mathcal { O } _ 0 } ^ { \\lambda _ 0 } \\leq ( \\alpha \\gamma ) ^ { 1 + \\beta } \\leq ( \\alpha _ 0 \\gamma _ 0 \\gamma _ s ) ^ { \\frac { 1 } { 1 - 2 \\beta ' } } . \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{gather*} \\mathrm { u } ( \\nabla , h ) : = \\sum i ^ { q + 1 } \\mathrm { c s } ^ q \\big ( \\nabla , \\nabla ^ h \\big ) \\in \\Omega ^ { \\mathrm { o d d } } ( A ; \\C ) . \\end{gather*}"}
-{"id": "219.png", "formula": "\\begin{align*} \\sup _ { \\begin{smallmatrix} 0 \\le n \\le m \\\\ n , m \\in \\mathbb { N } \\end{smallmatrix} } \\frac { 1 } { ( m - n + 1 ) ^ p } \\Bigg ( \\sum _ { k = n } ^ m w ( k ) \\Bigg ) \\Bigg ( \\sum _ { k = n } ^ m w ( k ) ^ { - 1 / ( p - 1 ) } \\Bigg ) ^ { p - 1 } < \\infty , \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} \\det \\mathcal { S } = ( - 1 ) ^ { L } \\sum _ { h = 0 } ^ L ( - 1 ) ^ h \\mathcal { U } [ h ] \\frac { x ^ { L _ 0 - h } } { ( L _ 0 - h ) ! } . \\end{align*}"}
-{"id": "9964.png", "formula": "\\begin{align*} \\operatorname { S u p p } ( [ u * h ] ) = \\operatorname { S u p p } ( [ u * \\tau _ \\epsilon ] ) \\subset \\{ \\ , x \\in U \\mid \\operatorname { d i s t } ( x , \\operatorname { S u p p } ( u ) ) \\le \\epsilon \\ , \\} . \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} Q [ z ( t ) ] = z ( t ) \\phi _ 0 + h ( z ( t ) ) , \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} \\| \\chi _ { A _ n } \\| _ k \\le m _ k ^ { - 1 } \\phi _ { X _ 0 } ( t _ n ) = \\frac { m _ n } { m _ k } \\| \\chi _ { A _ n } \\| _ n . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} \\max \\{ \\max ( u ) - t ( \\deg ( u ) - 1 ) \\ ; : u \\in G ( I ) \\} = \\max \\{ \\min ( u ) \\ ; : u \\in G ( I ) \\} . \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} \\frac { | A | ^ 2 } { ( 2 k ) ^ { 2 l + 3 } } & \\leq \\sum _ { u , v } { 1 _ { w A ^ { \\sigma _ { s _ 1 } } } ( u ) 1 _ { A ^ { - \\sigma _ { s _ 2 } } y ^ { - 1 } } ( v ) 1 _ { A z ^ { - 1 } } ( u v ^ { - 1 } ) } \\\\ & = \\langle 1 _ { w A ^ { \\sigma _ { s _ 1 } } } , 1 _ { A z ^ { - 1 } } \\ast 1 _ { A ^ { - \\sigma _ { s _ 2 } } y ^ { - 1 } } \\rangle _ { \\ell _ 2 ( G ) } \\leq \\| 1 _ { w A ^ { \\sigma _ { s _ 1 } } } \\| _ { \\ell _ 2 ( G ) } \\| 1 _ { A z ^ { - 1 } } \\ast 1 _ { A ^ { - \\sigma _ { s _ 2 } } y ^ { - 1 } } \\| _ { \\ell _ 2 ( G ) } . \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{gather*} \\alpha _ k < \\delta _ n ^ k \\ \\ \\Rightarrow \\ \\ S \\left ( ( \\alpha _ 0 , \\dots , \\alpha _ { k - 1 } , \\alpha _ k ) \\right ) = ( \\alpha _ 0 + 1 , \\dots , \\alpha _ { k - 1 } + 1 , \\alpha _ k + 1 ) \\\\ S \\left ( ( \\alpha _ 0 , \\dots , \\alpha _ { k - 1 } , \\delta _ n ^ k ) \\right ) = ( 0 , \\alpha _ 0 , \\dots , \\alpha _ { k - 1 } ) . \\end{gather*}"}
-{"id": "6780.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\int d x \\ x c ( x , y , t ) = 0 \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} \\| \\tilde l _ N ( t ) - l _ N ( t ) \\| _ { H ^ { s } _ { r } ( \\mathbb R ) } & = \\| \\sum _ { j , m } \\mu _ { j , m } ( t ) ( h _ { j , m } - \\tilde h _ { j , m } ) \\| _ { H ^ { s } _ r ( \\mathbb R ) } \\\\ & \\leq \\max _ { t \\in [ 0 , T ] } \\sum _ { j , m } | \\mu _ { j , m } ( t ) | \\| ( h _ { j , m } - \\tilde h _ { j , m } ) \\| _ { H ^ { s } _ { r } ( \\mathbb R ) } \\\\ & < \\varepsilon . \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} v _ i ( \\tau , \\xi ) = & \\int _ t ^ \\tau \\sum _ { j = 1 } ^ n \\Big ( C _ { i j } ( x _ i ( s , \\tau , \\xi ) ) v _ j ( s , x _ i ( s , \\tau , \\xi ) ) + D _ { i j } ( x _ i ( s , \\tau , \\xi ) ) v _ j ( s , 0 ) \\\\ [ 6 p t ] & + f _ i \\big ( s , x _ i ( s , \\tau , \\xi ) \\big ) \\Big ) \\ , d s + \\sum _ { j = 1 } ^ { m } B _ { i j } v _ { j + k } ( t , 0 ) + g _ i ( t ) , \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} I ( t u ) & = \\left ( \\frac { t ^ 2 } { 2 } - \\frac { t ^ 4 } { 4 } \\right ) \\norm { u } ^ 2 + \\int _ { \\mathbb R ^ 3 } \\left [ \\frac { t ^ 4 } { 4 } f ( u ) u - F ( t u ) \\right ] d x \\\\ & < \\frac { 1 } { 4 } \\norm { u } ^ 2 + \\int _ { \\mathbb R ^ 3 } \\left [ \\frac { 1 } { 4 } f ( u ) u - F ( u ) \\right ] d x \\\\ & = I ( u ) . \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| v ^ k - T u ^ k \\| = 0 . \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} H _ { F } = \\left \\{ x \\mid \\langle x , u _ { F } \\rangle + \\lambda _ { F } = 0 \\right \\} , \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} W _ t f ( n ) = \\int _ X e ^ { ( x - s ^ { + } ) t } F ( x ) p _ n ( x ) \\ , d \\mu ( x ) = c _ n ( e ^ { ( \\cdot - s ^ { + } ) t } F ) \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} \\phi ^ { \\ast } = \\frac { ( P _ x + \\sigma _ w ^ 2 ) \\sigma _ w ^ 2 } { P _ x } \\ln \\left ( \\frac { P _ x + \\sigma _ w ^ 2 } { \\sigma _ w ^ 2 } \\right ) . \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} v ^ * ( y _ 0 ) = \\sup _ { w ( \\cdot ) \\in \\mathcal { H } } w ( y _ 0 ) \\ \\ \\ \\ \\ \\forall \\ y _ 0 \\in Y , \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} \\Pi _ n f ( t ) = { n \\over T } \\left ( f \\left ( t ^ n _ k \\right ) - f \\left ( t ^ n _ { k - 1 } \\right ) \\right ) \\left ( t - t ^ n _ { k - 1 } \\right ) + f \\left ( t ^ n _ { k - 1 } \\right ) . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\| a _ \\alpha \\| _ { p , \\sigma + \\tau } & \\le \\sum _ { \\beta \\le \\alpha } \\frac { \\| z ^ { \\alpha - \\beta } \\| _ { p , \\tau } } { ( \\alpha - \\beta ) ! \\beta ! } \\| F z ^ \\beta \\| _ { p , \\sigma } \\\\ & \\le C \\sum _ { \\beta \\le \\alpha } \\frac { 1 } { ( \\alpha - \\beta ) ! \\beta ! } \\left ( \\frac { | \\alpha - \\beta | } { e \\tau p } \\right ) ^ \\frac { | \\alpha - \\beta | } { p } \\left ( \\frac { | \\beta | } { e \\sigma ' p } \\right ) ^ \\frac { | \\beta | } { p } . \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} c _ d ( X _ 1 , \\dotsc , X _ d ) = \\begin{bmatrix} X _ 1 1 _ d \\\\ \\hline \\begin{array} { c | c } 0 _ { \\binom d 2 \\times 1 } & c _ { d - 1 } ( X _ 2 , \\dotsc , X _ d ) \\end{array} \\end{bmatrix} . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} W ( h , U ) \\varphi ( g ) W ( h , U ) ^ * & = \\varphi ( U g ) - 2 \\textup { R e } ( \\langle U g , h \\rangle ) \\\\ W ( h , U ) a ( g ) W ( h , U ) ^ * & = a ( U g ) - \\langle U g , h \\rangle \\\\ W ( h , U ) a ^ \\dagger ( g ) W ( h , U ) ^ * & = a ^ \\dagger ( U g ) - \\langle h , U g \\rangle . \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} P f ( x ) & = \\frac { d _ x ^ + g ( | x | + 1 ) + d _ x ^ 0 g ( | x | ) + \\lambda d _ x ^ - g ( | x | - 1 ) } { d _ x ^ + + d _ x ^ 0 + \\lambda d _ x ^ { - } } \\\\ \\ , \\\\ & \\ge \\frac { ( d _ x ^ + + d _ x ^ 0 ) g ( | x | + 1 ) + \\lambda d _ x ^ - g ( | x | - 1 ) } { d _ x ^ + + d _ x ^ 0 + \\lambda d _ x ^ { - } } . \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { m _ k } \\langle \\lambda _ k + \\cdots + \\lambda _ r , \\alpha _ { u _ { k , l } } ^ { \\vee } \\rangle \\varpi _ l = \\sum _ { l = 1 } ^ { m _ k + 1 } \\mathbf { a } _ k ( l ) \\varepsilon _ l , \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} \\xi ^ { \\ast } = 1 - \\mathcal { V } _ T ( p _ { _ 0 } , p _ { _ 1 } ) = 1 - \\frac { 1 } { 2 } \\| p _ { _ 0 } ( y ) - p _ { _ 1 } ( y ) \\| _ 1 , \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\Omega ^ { * } _ { a d R } ( \\R ^ { r } ) = \\bigl ( S ( y _ { 1 } , \\dots , y _ { r } ) \\otimes \\Lambda ( d y _ { 1 } , \\dots , d y _ { r } ) , d _ { d R } = \\sum _ { k } \\frac { \\partial } { \\partial y _ { k } } \\cdot d y _ { k } \\bigr ) . \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} g ( t , x , \\omega ) ~ = ~ f \\bigl ( t , x , \\omega + \\hat u ^ { \\varepsilon } ( t , x ) \\bigr ) - \\hat f \\bigl ( t , x , \\hat u ^ { \\varepsilon } ( t , x ) \\bigr ) . \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\mathcal { L } ^ N = - \\frac { 1 } { \\epsilon } \\nabla ^ N V _ 1 \\cdot \\nabla ^ N + \\frac { 1 } { \\beta } \\Delta ^ N \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} Z _ p ^ { C } ( t ) \\coloneqq \\sum _ { m = 0 } ^ \\infty M _ p ^ { C } ( m ) t ^ m \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , Z _ p ^ { H } ( t ) \\coloneqq \\sum _ { m = 0 } ^ \\infty M _ p ^ { H } ( m ) t ^ m . \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} M _ t = & Y _ t - Y _ 0 - A ^ { W , W } _ t ( - \\nabla u ^ * \\ , b - \\tilde f ) \\\\ = & u ( t , W _ t ) - u ( 0 , W _ 0 ) \\\\ & - u ( t , W _ t ) + u ( 0 , W _ 0 ) + \\int _ 0 ^ t \\nabla u ^ * ( r , W _ r ) \\mathrm d W _ r \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} a _ { n m } ( q ) \\equiv \\sum _ { i = 0 } ^ { \\min \\{ p - 2 , r - 1 \\} } f _ i ( q ) [ n ] _ q ^ i \\bmod [ n ] _ q ^ { 1 + \\min \\{ p - 2 , r - 1 \\} } . \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} l = [ \\mathbb { K } : \\mathbb { F } _ p ] = r ! \\prod _ { i = 1 } ^ n p _ i . \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} \\bar e ^ { \\AA } _ r ( d , m ) = \\sum _ { i = 1 } ^ m \\mu _ i q ^ { m - i } , \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} \\begin{aligned} C & = \\{ v \\in M _ { \\mathbb R } \\ , | \\ , \\langle v , n _ i \\rangle \\geq 0 \\ , \\ , \\ , { } i = 1 , \\dots , t \\} \\\\ & = \\bigcap _ { i = 1 } ^ t \\{ v \\in M _ { \\mathbb R } \\ , | \\ , \\langle v , n _ i \\rangle \\geq 0 \\} , \\end{aligned} \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} T _ { a , j , * } ^ \\nu ( h ) ( t , r ) = & \\chi _ j ( r ) r ^ { - \\frac { d - 2 } { 2 } } \\int _ 0 ^ \\infty e ^ { i t \\rho ^ a } J _ \\nu ( r \\rho ) \\rho ^ { \\frac { - d + 1 + a } { 2 } } \\chi _ { \\leq - j - 5 } ( \\rho ) h ( \\rho ) d \\rho , \\\\ T _ { a , j , j ' } ^ \\nu ( h ) ( t , r ) = & \\chi _ j ( r ) r ^ { - \\frac { d - 2 } { 2 } } \\int _ 0 ^ \\infty e ^ { i t \\rho ^ a } J _ \\nu ( r \\rho ) \\rho ^ { \\frac { - d + 1 + a } { 2 } } \\chi _ { j ' } ( \\rho ) h ( \\rho ) d \\rho . \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\underline { P } ^ { \\mathrm { s o l a r } } \\big ( z [ n ] \\big ) = \\frac { C _ 1 } { 1 + e ^ { - k _ { c } ( z [ n ] - \\alpha ) } } + C _ 2 , \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} \\overline { C } ( t ) = \\frac { S ( t ) } { t } . \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} \\overline { S } ( \\phi V , \\phi W ) = \\overline { S } ( V , W ) + \\{ ( n - 1 ) ( 1 - \\beta + \\beta ^ { 2 } ) + \\alpha ( \\beta - 1 ) t r a c e \\Phi \\} \\eta ( V ) \\eta ( W ) \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} H _ { j , i } = H _ { i } ^ { j - 1 } , j \\in [ r ] , i \\in [ n ] , \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} A = V \\begin{pmatrix} I _ j & 0 \\\\ 0 & G \\end{pmatrix} U , \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} \\| W _ t f - f \\| _ { \\ell ^ 2 ( \\mathbb { N } ) } ^ 2 = \\| c _ n ( ( e ^ { ( \\cdot - s ^ { + } ) t } - 1 ) F ) \\| _ { \\ell ^ 2 ( \\mathbb { N } ) } ^ 2 = \\int _ { X } ( e ^ { ( x - s ^ { + } ) t } - 1 ) ^ 2 F ^ 2 ( x ) \\ , d \\mu ( x ) \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} p _ k ^ { ( \\rm i d ) } = \\left ( \\frac { \\mu _ 2 ^ * } { \\ln 2 \\left ( \\nu _ 2 ^ * - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 \\right ) } - \\frac { \\sigma ^ 2 } { [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } \\right ) ^ + , \\forall \\ , k = 1 , 2 , \\ldots , r . \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} I \\phi ( \\cdot , x ) : = \\sum _ { j = 1 } ^ { l } \\int _ { \\mathbb { R } ^ p } \\{ \\phi ( \\cdot , x + \\gamma ^ { j } ( x , z _ j ) ) - \\phi ( \\cdot , x ) - \\nabla \\phi ( \\cdot , x ) \\gamma ^ { j } ( x , z _ { j } ) \\} \\nu _ { j } ( d z _ { j } ) , \\ ; { \\forall x \\in \\mathbb { R } ^ p . } \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} \\sum _ { k , l } G _ \\lambda ^ { k l } ( \\Q ) \\xi _ k \\xi _ l & \\leq \\abs { \\xi } ^ 2 + c _ 1 \\ , \\abs { \\xi } ^ 2 = ( c _ 1 + 1 ) \\ , \\abs { \\xi } ^ 2 , \\\\ \\sum _ { k , l } G _ \\lambda ^ { k l } ( \\Q ) \\xi _ k \\xi _ l & \\geq \\abs { \\xi } ^ 2 \\big ( 1 + \\min \\{ 0 , c _ 0 - 1 \\} \\big ) = \\min \\{ c _ 0 , 1 \\} , \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} C = [ 0 , \\infty ) ^ s \\times \\mathbb { E } ' \\subset \\mathbb { R } ^ s \\times \\mathbb { E } ' = \\mathbb { E } . \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} \\gamma ^ 1 ( t , x ) + h ^ 1 ( t , x ) = \\gamma ^ 2 ( t , x ) + h ^ 2 ( t , x ) , \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} T _ n ( x ) = c _ 0 + c _ 1 x + \\dots + c _ n x ^ n ; \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{gather*} C _ c ^ { \\infty } ( S ^ 1 \\times M ) = \\{ H : S ^ 1 \\times M \\longrightarrow \\mathbb { R } \\ | \\ \\textrm { s u p p } ( H ) \\subset S ^ 1 \\times M \\ \\textrm { i s \\ c o m p a c t } \\} \\end{gather*}"}
-{"id": "5616.png", "formula": "\\begin{align*} \\beta _ { i , i + j } ( I ) = \\sum _ { u \\in G ( I ) _ j } \\binom { \\max ( u ) - t ( j - 1 ) - 1 } { i } . \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} \\dim ^ + ( L ( \\lambda , \\mu ) ) = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } q ^ { \\langle 2 \\rho , ( w \\bullet ( \\lambda , \\mu ) ) _ 2 \\rangle } } { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } q ^ { \\langle 2 \\rho , w \\bullet ( 0 ) \\rangle } } \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} E _ 1 ^ { i , 0 } = E _ 1 ^ { i , 1 } = 0 \\mbox { f o r a l l } i . \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} 2 l + o _ n ( 1 ) < 2 l + \\frac { \\lambda } { 4 } \\int _ { \\mathbb R ^ 3 } \\phi _ { v _ n } v _ n ^ 2 d x + o _ n ( 1 ) = I ( v _ n ) \\leq I ( u _ n ) = l + o _ n ( 1 ) , \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} \\gamma _ + ( v ^ + + t v ^ - ) & = I ' ( v ^ + ) [ v ^ + ] + t ^ 2 \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { v ^ - } ( v ^ + ) ^ 2 d x \\\\ & \\leq I ' ( v ^ + ) [ v ^ + ] + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { v ^ - } ( v ^ + ) ^ 2 d x = \\gamma _ + ( v ) < 0 . \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 0 , 2 / 5 ) = \\Phi \\left ( \\frac { 2 } { 5 } \\right ) = \\int _ { 0 } ^ { \\infty } \\frac { \\cos \\left ( \\frac { 2 \\pi x } { 5 } \\right ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x = \\frac { 8 - 3 \\sqrt { 5 } } { 1 6 } . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} | k G _ { t - 2 } ( \\theta ) - k G _ { t - 3 } ( \\theta ) | - | G _ { t - 1 } ( \\theta ) - G _ { t - 3 } ( \\theta ) | & = ( k - 1 ) G _ { t - 3 } ( \\theta ) + G _ { t - 1 } ( \\theta ) - k G _ { t - 2 } ( \\theta ) \\\\ & = ( \\theta - k ) G _ { t - 2 } ( \\theta ) < 0 \\end{align*}"}
-{"id": "10008.png", "formula": "\\begin{align*} d Y _ t = - F ( t , Y _ t , Z _ t ) d t + ( Z _ t ) ^ { t r } d W _ t , t \\geq 0 \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} \\theta _ y = \\theta - y \\left ( \\kappa ^ { A } ( \\theta ) + \\kappa ^ { - S } ( \\theta ) \\right ) , \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} Q + \\sum _ { k = 1 } ^ m c _ k B _ k ^ * B _ k = \\mathrm { I d } _ H \\mathrm { a n d } \\dim H \\ge s ^ + ( Q ) + \\sum _ { k : \\ , c _ k > 0 } \\dim H _ k . \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} \\pi _ r ^ \\alpha ( \\lambda ) = g ( \\alpha , r ) r ^ { d - \\gamma } \\exp [ - c ( \\frac r 2 - { 2 \\alpha } r ) ^ \\gamma ] . \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ^ { k + 1 } - q \\| & = \\| \\beta _ k v ^ k + ( 1 - \\beta _ k ) T u ^ k - q \\| \\\\ & \\leq \\beta _ k \\| v ^ k - q \\| + ( 1 - \\beta _ k ) \\| T u ^ k - q \\| \\\\ & \\leq \\beta _ k \\| v ^ k - q \\| + ( 1 - \\beta _ k ) \\| u ^ k - q \\| . \\end{aligned} \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} M _ f = \\begin{pmatrix} a _ 1 & a _ 2 & 0 & b \\\\ a _ 3 & a _ 4 & - b & 0 \\\\ 0 & c & a _ 1 & a _ 3 \\\\ - c & 0 & a _ 2 & a _ 4 \\end{pmatrix} . \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} R _ { \\mathcal { S } _ p ( 1 ) } ( e _ { \\zeta } \\circ D _ t ) = t ^ { 1 / 2 \\mu } R _ { \\mathcal { S } _ p ( 1 ) } ( e _ { \\widehat { \\rho } _ t ( \\zeta ) } ) . \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} \\int _ { \\partial S _ { \\varepsilon } ^ + } e ^ { z w } ( u _ { \\varepsilon ' } ( z ) - u _ { \\varepsilon _ 1 ' } ( z ) ) d z = \\ ! \\ ! \\lim _ { R \\to + \\infty } \\int _ { \\partial ( S _ { \\varepsilon } \\cap \\overline { D } ( 0 , R ) ) ^ + } e ^ { z w } ( u _ { \\varepsilon ' } ( z ) - u _ { \\varepsilon _ 1 ' } ( z ) ) d z = 0 . \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} A ^ + & = \\{ j \\in \\{ 1 , . . . , l - 1 \\} \\mid s _ j \\neq a , s _ { j + 1 } = a ^ { - 1 } \\} , \\\\ A ^ - & = \\{ j \\in \\{ 1 , . . . , l - 1 \\} \\mid s _ j = a , s _ { j + 1 } \\neq a ^ { - 1 } \\} . \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} e _ n ( q , t ) = \\sum _ { k = 0 } ^ { n } t ^ k \\sum _ { w \\in S ^ { ' } _ { n + 1 , k } } q ^ { W _ 3 ( w ) } . \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} \\| u \\| _ { ( \\kappa _ n + 1 ) p ^ * } & \\leq M _ { 3 1 } ^ { \\frac { 1 } { \\kappa _ n + 1 } } M _ { 3 0 } ^ { \\frac { 1 } { \\sqrt { \\kappa _ n + 1 } } } \\left \\| u \\right \\| _ { ( \\kappa _ n + 1 ) \\tilde { q } _ 1 } = M _ { 3 1 } ^ { \\frac { 1 } { \\kappa _ n + 1 } } M _ { 3 0 } ^ { \\frac { 1 } { \\sqrt { \\kappa _ n + 1 } } } \\left \\| u \\right \\| _ { ( \\kappa _ { n - 1 } + 1 ) p ^ * } \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} V ( x ) = \\widetilde V \\left ( x , E _ { t + 1 } , B _ { w + 1 } \\backslash \\{ E _ { t + 1 } \\} , J _ k + t T _ { k + 1 } , J _ k + ( t + 1 ) T _ { k + 1 } , t T _ { k + 1 } , \\frac { u ^ \\prime ( J _ w + t T _ { w + 1 } , E _ { t + 1 } ) } { u ( J _ w + t T _ { w + 1 } , E _ { t + 1 } ) } \\right ) , \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} v _ t = a ( \\xi , v , v _ \\xi ) [ v _ { \\xi \\xi } + \\lambda _ n v ] + g _ n ( \\mu , \\xi , v , v _ \\xi ) \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} \\mu ^ { \\psi } _ { \\Lambda , \\epsilon } : = \\frac { 1 } { Z _ { \\Lambda , \\epsilon } ( \\psi _ { \\Lambda ^ c } ) } e ^ { - H _ { \\Lambda , \\epsilon } ( \\phi \\vee \\psi ) } \\prod _ { j \\in \\Lambda } d \\phi _ j , \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} \\mathcal { C } ( x ) : = \\sum _ { k = 0 } ^ { \\infty } c _ k x ^ k = \\frac { 1 - \\sqrt { 1 - 4 x } } { 2 x } , x \\in \\left [ - \\frac { 1 } { 4 } , \\ , \\frac { 1 } { 4 } \\right ] . \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} | \\tilde { f } | _ { \\alpha ; \\bar { \\Omega } } = | z ( u ) | u | ^ { p - 2 } u | _ { \\alpha ; \\bar { \\Omega } } \\leq | z ( u ) | _ { \\alpha ; \\bar { \\Omega } } | | u | ^ { p - 2 } u | _ { \\alpha ; \\bar { \\Omega } } . \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} \\frac { \\tilde { \\phi } ^ { \\prime \\prime } _ { K _ L } ( t ) t - \\tilde { \\phi } ^ \\prime _ { K _ L } ( t ) } { \\tilde { \\phi } ^ \\prime _ { K _ L } ( t ) } = \\frac { \\phi ^ { \\prime \\prime } ( K _ L t ) K _ L t - \\phi ^ \\prime ( K _ L t ) } { \\phi ^ \\prime ( K _ L t ) } \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} \\psi _ \\nu ( \\vect { x } ) = \\prod _ { j = 1 } ^ d \\psi _ 0 \\bigl ( \\ell ( x _ j ; \\bar { \\alpha } _ { j } , \\bar { \\beta } _ { j } ) \\bigr ) . \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} \\P ( \\mu ( 0 ) > r ) = \\exp \\{ - g ( r ) \\} \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\sup _ { 0 < R _ 1 < R _ 2 < 0 } \\left | \\int _ { R _ 1 < | x | < R _ 2 } \\left ( \\mathbf { K } _ n ( x ) - \\mathbf { K } ( x ) \\right ) d x \\right | = 0 . \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} J ' ( u ) v = \\int _ { \\mathbb { R } ^ 2 } ( \\nabla u \\cdot \\nabla v + b ( x ) u v ) - \\int _ { \\mathbb { R } ^ 2 } A ( x ) f ( u ) v , \\ \\ \\forall \\ u , v \\in H , \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{gather*} | \\rho ' ( \\tau ^ 2 ) | = \\pi \\end{gather*}"}
-{"id": "6127.png", "formula": "\\begin{align*} | k _ b j _ b + \\sum _ { j \\in \\mathbb { Z } _ * } \\frac { l _ j j } { n - \\frac 1 2 } | < \\frac { | k | } { 1 0 0 n \\sum _ { b = 1 } ^ n | j _ b | } , \\end{align*}"}
-{"id": "9865.png", "formula": "\\begin{align*} \\begin{aligned} { _ a D _ x ^ { - \\mu } } { _ a D _ x ^ { - \\sigma } } u ( x ) & = { _ a D _ x ^ { - \\mu - \\sigma } } u ( x ) , x > a \\\\ { _ x D _ b ^ { - \\mu } } { _ x D _ b ^ { - \\sigma } } u ( x ) & = { _ x D _ b ^ { - \\mu - \\sigma } } u ( x ) , b < x . \\end{aligned} \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} \\lambda ( p ( \\zeta ) , t ) = \\begin{cases} \\dfrac { 1 } { 4 \\pi t } \\left ( e ^ { - 4 \\pi t \\sin ^ { - 1 } p ( \\zeta ) } - e ^ { - 4 \\pi t ( \\pi - \\sin ^ { - 1 } p ( \\zeta ) ) } \\right ) \\ ; t \\neq 0 \\\\ \\pi - 2 \\sin ^ { - 1 } ( p ( \\zeta ) ) t = 0 . \\end{cases} \\end{align*}"}
-{"id": "9861.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\Delta _ { 2 } ( 5 n + 4 ) q ^ { n } & = 4 1 \\dfrac { E _ { 1 0 } ^ { 3 } } { E _ { 1 } E _ { 2 } ^ { 3 } E _ { 5 } } + 8 6 0 q \\dfrac { E _ { 1 0 } ^ { 6 } } { E _ { 1 } ^ { 4 } E _ { 2 } ^ { 2 } E _ { 5 } ^ { 2 } } + 6 8 0 0 q ^ { 2 } \\dfrac { E _ { 1 0 } ^ { 9 } } { E _ { 1 } ^ { 7 } E _ { 2 } E _ { 5 } ^ { 3 } } \\\\ & \\quad + 2 4 0 0 0 q ^ { 3 } \\dfrac { E _ { 1 0 } ^ { 1 2 } } { E _ { 1 } ^ { 1 2 } E _ { 5 } ^ { 4 } } + 3 2 0 0 0 q ^ { 4 } \\dfrac { E _ { 2 } E _ { 1 0 } ^ { 1 5 } } { E _ { 1 } ^ { 1 5 } E _ { 5 } ^ { 5 } } , \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} \\begin{aligned} | \\psi ( x , t ) | & \\lesssim \\| f \\| _ { * , \\beta , 2 s + \\alpha } \\left ( \\sum _ { j = 1 } ^ k \\frac { t ^ { - \\beta } } { 1 + | y _ j | ^ { \\alpha } } \\right ) \\\\ & + e ^ { - \\delta ( t - t _ 0 ) } \\| h \\| _ { L ^ \\infty ( \\mathbb { R } ^ n ) } + t ^ { - \\beta } \\| \\tau ^ \\beta g ( x , \\tau ) \\| _ { L ^ \\infty ( ( \\mathbb { R } ^ n \\setminus \\Omega ) \\times ( t _ 0 , \\infty ) ) } , \\end{aligned} \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} \\sum _ { i \\in N ( j ) } { C } _ i \\frac { p ( x _ i ^ - ) - p ( x _ i ^ + ) } { L _ i } & = S _ j j \\in \\mathcal { V } , \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} E _ X : = \\C ^ k \\times ^ \\Lambda E , \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} \\dot { \\theta } _ v ( t ) = \\omega _ v + \\sum _ { v ' \\in N ( v ) } H ( \\theta _ { v ' } ( t ) - \\theta _ v ( t ) ) , \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ { \\tilde g ^ { ( 4 ) } } \\delta _ { \\tilde g ^ { ( 4 ) } } ^ * Y ' + \\beta _ { \\tilde g ^ { ( 4 ) } } \\delta ^ * _ { g ' } \\tilde Y = 0 \\quad S \\\\ Y ' = 0 \\quad \\partial S , \\end{cases} \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} n \\cdot \\max _ { 1 \\leq i \\leq m + 1 } \\deg p _ { i } & \\leq \\sum _ { i = 1 } ^ { m + 1 } \\deg \\mathrm { r \\tilde { a } d } _ { \\kappa } ^ { [ m - 1 ] } ( [ p _ { i } ] _ { \\kappa } ^ { \\bar { n } } ) - \\frac { 1 } { 2 } m ( m - 1 ) . \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} g ( \\omega , 0 ) = 0 . \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} \\rho ( Y ) \\le 4 6 - \\dim V ^ { \\oplus 2 } = 6 , \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} \\varphi _ t : = \\lim _ { k \\to \\infty } \\varphi _ t ( k ) , \\ t \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 ( Z ) = T ( f + p \\left [ ( u ^ * _ { \\mu , \\xi } + \\psi + \\phi ^ { i n } ) ^ { p - 1 } - ( u ^ * _ { \\mu , \\xi } ) ^ { p - 1 } \\right ] Z , 0 , 0 ) , \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} T X = T \\left ( ( \\C ^ k \\times M ) / \\pi \\right ) \\cong T ( \\C ^ k \\times M ) / \\pi \\cong Z _ { \\C ^ k \\times M } / \\pi \\cong Z _ { ( \\C ^ k \\times M ) / \\pi } = Z _ X , \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} c _ 1 ( \\xi ) = \\sum _ { j = 1 } ^ k \\sum _ { l = 1 } ^ { m _ j + 1 } { \\mathbf { a } _ { j } ( l ) } x _ { j , l } . \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} \\phi _ { B } ( \\textbf { x } ) & = I \\left \\lbrace \\textbf { x } \\in \\mathcal { C } _ { 1 } \\cap \\mathcal { C } _ { 0 } , P \\{ Y = 1 \\vert \\textbf { X } = \\textbf { x } \\} > P \\{ Y = 0 \\vert \\textbf { X } = \\textbf { x } \\} \\right \\rbrace \\\\ & + I \\left \\lbrace \\textbf { x } \\in \\mathcal { C } _ { 1 } - \\mathcal { C } _ { 0 } \\right \\rbrace . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } x b ^ { m _ k } \\sim \\phi b ^ { 1 } x b ^ { m _ k } - \\sum _ { i = 1 } ^ { m _ 0 - 1 } \\sum _ { s \\in S _ b } \\phi s b ^ i x b ^ { m _ k } . \\end{align*}"}
-{"id": "9884.png", "formula": "\\begin{align*} f = \\boldsymbol { D } ^ { - s } u + \\boldsymbol { D } ^ { s } u . \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} \\mathbb { A } ^ \\alpha ( x ) = u ^ \\alpha \\ , e ^ { i p _ \\beta x ^ \\beta } . \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} \\Pr [ { \\rm S I R } _ { { \\rm S V C } , n , \\ell } \\geq \\tau | d _ 0 = x ] = \\mathbb E _ { I _ { n , \\ell } , \\overline { I } _ { n , \\ell } } \\left [ \\exp ( - s ( I _ { n , \\ell } + \\overline { I } _ { n , \\ell } ) ) \\right ] = \\mathcal L _ { I _ { n , \\ell } } ( s ) \\mathcal L _ { \\overline { I } _ { n , \\ell } } ( s ) , \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} \\mathcal { H } : ~ \\begin{cases} \\dot { x } ~ ~ \\in F ( x ) , & x \\in \\mathcal { F } \\\\ x ^ { + } \\in G ( x ) , & x \\in \\mathcal { J } \\end{cases} \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} a ( q ) : = \\sum _ { m , n = { - \\infty } } ^ \\infty q ^ { m ^ 2 + m n + n ^ 2 } = 1 + 6 \\sum _ { n = 0 } ^ \\infty \\left ( \\dfrac { q ^ { 3 n + 1 } } { 1 - q ^ { 3 n + 1 } } - \\dfrac { q ^ { 3 n + 2 } } { 1 - q ^ { 3 n + 2 } } \\right ) . \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} \\Psi ( \\textstyle { \\frac { 1 } { 2 } } , \\beta ) & = \\left ( I ' ( h ( \\textstyle { \\frac { 1 } { 2 } } , \\beta ) ) [ h ( \\textstyle { \\frac { 1 } { 2 } } , \\beta ) ^ + ] , I ' ( h ( \\textstyle { \\frac { 1 } { 2 } } , \\beta ) ) [ h ( \\textstyle { \\frac { 1 } { 2 } } , \\beta ) ^ - ] \\right ) \\\\ & = \\left ( I ' ( \\textstyle { \\frac { 1 } { 2 } } u ^ + + \\beta u ^ - ) [ \\textstyle { \\frac { 1 } { 2 } } u ^ + ] , I ' ( \\textstyle { \\frac { 1 } { 2 } } u ^ + + \\beta u ^ - ) [ \\beta u ^ - ] \\right ) . \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} \\tilde { \\mathcal { U } } _ \\kappa = \\mathcal { U } _ \\kappa + \\overline { B } ( 0 , \\kappa / 2 ) \\subseteq \\mathcal { U } \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} \\overline { \\nabla } _ { X } Y = \\overline { \\nabla } ^ { ' } _ { X } Y + h ( X , Y ) + \\beta \\eta ( Y ) \\phi Q X . \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} m _ j ^ - = m _ j - n \\# a ( s _ j ) + n \\# a ^ { - 1 } ( s _ { j + 1 } ) . \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} \\N = D Q ( \\dot z + i E z ) + F _ 1 + F _ 2 + F _ 3 \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } x h ( x ) \\frac { \\partial } { \\partial y } p _ { 0 } ( y , t ) = 0 \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} \\nu ( S ( n , k ) ) = \\frac { \\sigma ( k ) - \\sigma ( n ) } { p - 1 } = \\frac { \\sigma ( k ) - a } { p - 1 } . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} h _ { i , j } + h ' _ { i , j } + h '' _ { i , j } = \\delta _ \\alpha ( ( h \\circ _ { i - 1 } f ) \\circ _ { j - 1 } g ) ( a _ 1 , a _ 2 , \\ldots , a _ { m + n + p - 1 } ) . \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} m \\ll l ^ { 2 c } \\left ( \\frac { B ^ 2 Z ^ 2 } { M } + \\frac { A ^ 2 M } { l ^ { n _ l } } \\right ) l ^ { 3 \\epsilon n _ l } = \\mathcal { M } l ^ { 3 \\epsilon n _ l } . \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} H _ n ' ( t ) & = \\Bigg ( \\frac { ( n - 1 ) ^ 3 ( n ^ 2 + 6 n - 1 2 ) } { n ^ 2 } + \\frac { ( n - 2 ) ( 2 n ^ 2 - 7 n + 7 ) } { \\sinh ( t ) ^ 2 } + \\frac { ( n - 2 ) ^ 2 ( n - 4 ) } { \\sinh ( t ) ^ 4 } \\Bigg ) \\\\ & \\qquad \\times J _ n ( t ) \\ , \\sinh ( t ) ^ { n - 1 } , \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} \\big [ \\psi ( \\tilde { \\mathbb { A } } / f \\tilde { \\mathbb { A } } ) \\big ] _ { \\tilde { \\mathbb { A } } } & = \\det ( Q ) \\\\ & = f + c ( f ) p _ 1 t ^ d f ( z _ 1 ) + c ( f ) p _ 2 t ^ { 2 d } f ( z _ 1 ) ^ 2 + \\dots + c ( f ) t ^ { r _ 0 d } f ( z _ 1 ) ^ { r _ 0 } . \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} \\hat \\theta = [ a _ 2 , a _ 3 , a _ 4 , \\ldots ] \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} & Y _ { m , k } ( n ) = \\\\ & \\sqrt { P _ { m , k } } \\sqrt { \\alpha _ m } F _ { m , k } G _ { m , k } S _ { m , k } ( n ) X _ m ( n ) + W _ { m , k } ( n ) , \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} \\left ( \\left ( \\prod _ { j = 1 } ^ m \\prod _ { i _ j = 1 } ^ { \\nu _ j } ( L + i _ j ) ! \\right ) \\prod _ { k = 1 } ^ M \\frac { 1 } { e _ k ! } \\right ) \\det \\mathcal { \\widehat { W } } = \\left ( \\prod _ { j = 1 } ^ m \\prod _ { i _ j = 1 } ^ { \\nu _ j } \\alpha _ j ^ { i _ j } \\right ) \\det \\mathcal { W } \\in \\Z [ \\alpha _ 1 , \\ldots , \\alpha _ m ] . \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} Q = \\sum _ { k \\in \\{ 0 , m + 1 \\} } c _ k B ^ \\ast _ k A _ k B _ k , \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} { G } _ { i } ( x ) = \\frac { { F } _ { i + 2 } ( x ) - ( k - 1 ) ^ 2 { F } _ { i } ( x ) } { x ^ 2 - k ^ 2 } \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} s _ { \\mu , \\upsilon } ( z ) = \\frac { z ^ { \\mu + 1 } } { ( \\mu - \\upsilon + 1 ) ( \\mu + \\upsilon + 1 ) } { } _ { 1 } F _ { 2 } \\left ( \\ \\begin{array} { l l l } ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1 ; ~ \\\\ \\frac { \\mu - \\upsilon + 3 } { 2 } , \\frac { \\mu + \\upsilon + 3 } { 2 } ; ~ \\end{array} \\frac { - z ^ { 2 } } { 4 } \\right ) , \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} \\mathcal { B } _ { a } ( x ) & = \\frac { 1 } { \\Gamma \\left ( \\frac { a } { \\nu } \\right ) } \\int _ { 0 } ^ { \\infty } t ^ { \\frac { a } { \\nu } - 1 } e ^ { - t } h _ { t } ( x ) d t \\\\ & = \\frac { 1 } { \\Gamma \\left ( \\frac { a } { \\nu } \\right ) } \\int _ { 0 } ^ { | x | ^ { \\nu } } t ^ { \\frac { a } { \\nu } - 1 } e ^ { - t } h _ { t } ( x ) d t + \\frac { 1 } { \\Gamma \\left ( \\frac { a } { \\nu } \\right ) } \\int _ { | x | ^ { \\nu } } ^ { \\infty } t ^ { \\frac { a } { \\nu } - 1 } e ^ { - t } h _ { t } ( x ) d t \\\\ & = : J _ { 1 } + J _ { 2 } . \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} L ( \\phi , A ) = \\prod _ { f } \\frac { \\big [ A / f A \\big ] _ { A } } { \\big [ \\phi ( A / f A ) \\big ] _ { A } } , \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} \\Vert f ' _ h - f ' \\Vert _ 1 & \\leq \\big | f ( t _ 1 + h ) - f ( t _ 1 ) \\big | + \\big | f ( t _ { M - 1 } + h ) - f ( t _ { M - 1 } ) \\big | \\\\ & \\enspace \\enspace + \\sum _ { k = 2 } ^ { M - 1 } \\Big ( \\big | f ( t _ k + h ) - f ( t _ k ) \\big | + \\big | f ( t _ { k - 1 } + h ) - f ( t _ { k - 1 } ) \\big | \\Big ) \\\\ & \\leq 2 ( M - 1 ) C | h | ^ \\mu \\leq 2 ( L - 1 ) C | h | ^ \\mu , \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } | y | ^ { \\frac { n \\theta } { p } } | \\nabla _ r v ( y ) | ^ { { \\theta } } \\frac { d y } { | y | ^ { { n } } } = \\int _ { { B _ { R } } } | x | ^ { \\frac { n \\theta } { p } } | \\nabla _ r u ( x ) | ^ { \\theta } \\frac { d x } { | x | ^ { { n } } } . \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} \\begin{aligned} = & 2 \\cos \\left ( 2 \\pi u _ { 0 } \\right ) + 2 \\sum _ { j = 1 } ^ { \\frac { N } { 2 } - 1 } \\left ( \\cos \\left ( 2 \\pi u _ { 0 } \\right ) \\cos \\left ( \\frac { 4 \\pi } { N } j \\right ) \\right ) \\\\ = & 2 \\cos \\left ( 2 \\pi u _ { 0 } \\right ) \\left ( 1 + \\sum _ { j = 1 } ^ { \\frac { N } { 2 } - 1 } \\cos \\left ( \\frac { 4 \\pi } { N } j \\right ) \\right ) \\end{aligned} \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} \\mathbb { A ' } ^ { - 1 } _ R = \\mathbb { A } ^ { - 1 } _ R = \\mathbb { A } _ \\mathcal { G } ^ { - 1 } . \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} \\psi ( t ) = P _ t \\psi ( 0 ) + \\int _ 0 ^ t P _ { t - s } \\mathcal { G } ( \\psi ( s ) ) d s \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} \\alpha _ n ( t , q ) = ( 1 + q ) ^ { n - 1 } A _ n \\left ( \\frac { t + q } { 1 + q } \\right ) , \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} \\frac { \\partial ( G ^ { - 1 } ) _ i } { \\partial \\xi _ l } \\frac { \\partial \\xi _ l } { \\partial x _ j } + \\frac { \\partial ( G ^ { - 1 } ) _ i } { \\partial \\phi _ l } \\frac { \\partial \\phi _ l } { \\partial x _ j } = \\delta _ { i j } \\ , , \\forall ~ 1 \\le i , j \\le n \\ , . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} k ( E ) = \\theta ( n _ 0 + q , E ) - \\theta ( n _ 0 , E ) + O \\left ( \\frac { 1 } { n _ 0 - v } \\right ) \\mod \\Z . \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} v ( t ) = e ^ { t ( - A ) } u _ { i n } + \\int _ 0 ^ t e ^ { ( t - \\tau ) ( - A ) } \\lambda ( S , N ( A ^ { - 1 } x ) , F ( \\tau ) ) A ^ { - 1 } x d \\tau . \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} \\frac { \\ , 1 \\ , } { \\ , n \\ , } \\sum _ { i = 1 } ^ { n } \\mathbb E \\big [ 2 b ( s , X _ { s , i } ^ { ( u ) } , \\widehat { F } _ { t , i } ^ { ( u ) } ) ( X _ { s , i } - X _ { t , i } ) \\vert \\mathcal F _ { t } ] \\le c _ { 5 } \\int _ { \\mathbb R } \\lvert x \\rvert ^ { 2 } { \\mathrm d } m _ { t , n } ( x ) + c _ { 6 } \\ , . \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} \\widehat { \\phi _ { N ( n ) } } ( y , \\xi ) = \\left \\{ \\begin{array} { l l } \\sqrt { 2 \\pi } 2 ^ \\lambda y ^ { \\lambda - 1 } \\left ( \\frac { 1 } { 2 y } \\right ) ^ { \\frac { n - 1 } { 2 } } e ^ { - 2 y } e ^ { - | \\xi | ^ 2 / 8 y } , & y > 0 \\\\ 0 , & y < 0 \\end{array} \\right . \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} S _ V = \\Big \\{ x \\in \\R ^ m \\colon \\sum _ { k = 1 } ^ m x _ k \\dim B _ k V \\le \\dim V \\Big \\} . \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} g _ e = d r ^ 2 \\oplus r ^ 2 g ^ V ( r ) \\oplus \\pi ^ * g ^ B ( r ) \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} G ( z ) = T _ S ( \\widehat { F } _ c e ^ { i 2 \\pi c ( \\cdot ) } ) ( z ) = \\int _ { \\mathbb { R } } \\widehat { F } _ c ( t ) e ^ { i 2 \\pi ( z - i c ) t } \\ , d t . \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} \\mathcal { Q } _ d = \\{ Q ( X , Y ) = a X ^ 2 + b X Y + c Y ^ 2 \\ | \\ a , b , c \\in \\mathbb { Z } , b ^ 2 - 4 a c = d \\} . \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} \\mathbf { S } = \\begin{pmatrix} 1 & [ 3 ] \\\\ [ 3 ] & - 1 \\end{pmatrix} \\mathbf { T } = \\begin{pmatrix} 1 & 0 \\\\ 0 & - i \\end{pmatrix} \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } X _ n = v _ { \\mathbf { P } } > 0 . \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\gamma ( s , \\chi \\pi _ { v } , \\psi _ { v } ) = \\varepsilon ( s , \\chi \\pi _ { v } , \\psi _ { v } ) \\dfrac { L ( 1 - s , \\chi ^ { - 1 } \\tilde { \\pi } _ { v } ) } { L ( s , \\chi \\pi _ { v } ) } . \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} \\chi _ \\infty = \\Big \\{ \\nu : \\nu = \\sqrt { ( n - 2 ) ^ 2 / 4 + \\lambda } ; ~ \\lambda ~ ~ \\Delta _ h + V _ 0 ( y ) \\Big \\} . \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} \\forall x \\forall y ( f ( x ) = y = y \\leftrightarrow A ( z , x , y ) ) \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} p _ { L , N + 1 } ^ { \\ell _ 1 } ( x ) = \\frac { 1 } { 2 } \\sum \\limits _ { \\ell = 0 } ^ N { S _ { \\lambda \\mu _ { \\ell } } \\left ( 2 \\alpha _ { \\ell } \\right ) } \\tilde { \\Phi } _ { \\ell } ( x ) . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} \\begin{cases} d \\Phi _ f ( \\mathbf { \\hat N } ) = a \\mathbf N + b \\mathbf n , \\\\ d \\Phi _ f ( \\mathbf { \\hat n } ) = c \\mathbf N + d \\mathbf n . \\end{cases} \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} \\theta _ 1 ( z , q ) = \\theta _ 1 ( z \\mid \\tau ) = 2 \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n q ^ { ( 2 n + 1 ) ^ 2 / 4 } \\sin ( 2 n + 1 ) z . \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} n _ b ( W ^ n ( w ) ) = \\lvert ( 0 , 0 ) \\rvert _ { W _ n ( w ) } = \\lvert ( 0 , 0 ) \\rvert _ { W _ n ( w _ 0 ) } + \\lvert r _ 0 \\rvert _ S = \\lVert ( \\phi w _ k ) _ n \\rVert _ S + \\lvert m _ k \\rvert . \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} \\mathcal { H } _ 0 : = \\{ \\Delta _ \\mathbb { R } g : g \\in C ^ \\infty _ 0 ( \\mathbb { R } ^ d ) \\} . \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { N _ k } | S _ m ( \\omega ) | ^ 2 \\geq \\frac { 3 c _ 1 ^ 2 C } { 2 5 6 } N _ k ^ 2 > 2 N _ k ^ 2 , \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} + \\sum _ { \\substack { { r _ 1 , r _ 2 , r _ 3 = 1 } \\\\ r _ 1 \\ne r _ 2 \\ne r _ 3 } } ^ { n } \\frac { a _ { r _ 1 } b _ { r _ 2 } c _ { r _ 3 } } { ( r _ 3 - r _ 2 ) ( r _ 3 - r _ 1 ) } \\left ( \\frac { 1 } { r _ 3 + k + 1 } - \\frac { 1 } { r _ 1 + k + 1 } \\right ) . \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} { H _ { \\tau } } _ T = \\lambda \\phi \\ ; , \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} | X _ i | , | Y _ i | \\leq ( 1 + \\xi ) \\frac { n } { 2 \\ell } + 2 \\hat \\ell \\beta n = \\left ( 1 + \\xi + 4 \\ell \\hat \\ell \\beta \\right ) \\frac { n } { 2 \\ell } = ( 1 + 2 \\xi ) \\frac { n } { 2 \\ell } \\leq | S _ { \\min } | . \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} \\norm { v '' _ { \\lambda } } _ { L ^ { 2 } _ { \\overline { s } ^ { \\star } ( H ) } } ^ { 2 } & \\le - ( w _ { \\lambda } ( 0 ) , v ' _ { \\lambda } ( 0 ) ) _ { H } \\\\ & = - ( A _ { \\lambda } v _ { \\lambda } ( 0 ) + \\delta v _ { \\lambda } ( 0 ) , v ' _ { \\lambda } ( 0 ) ) _ { H } \\\\ & \\le - ( A _ { \\lambda } \\varphi + \\delta \\varphi , v ' _ { \\lambda } ( 0 ) ) _ { H } , \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( \\ker \\beta _ + ) = V + \\ker B _ + \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} g ^ { ( 4 ) } = - N ^ 2 d t ^ 2 + g _ { i j } ( d x ^ i + X ^ i d t ) ( d x ^ j + X ^ j d t ) , \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} d v + ( v . \\nabla ) v = \\sqrt { 2 \\nu } \\nabla v \\circ d W _ t - \\nabla p , \\div ~ v ( t , \\cdot ) = 0 . \\end{align*}"}
-{"id": "9799.png", "formula": "\\begin{align*} c \\sqrt { \\nu } e ^ { - \\nu ^ { \\frac { 1 } { 3 } } t } = \\frac { c } { t ^ { \\frac { 3 } { 2 } } } \\left ( \\nu ^ { \\frac { 1 } { 3 } } t \\right ) ^ { \\frac { 3 } { 2 } } e ^ { - \\nu ^ { \\frac { 1 } { 3 } } t } \\end{align*}"}
-{"id": "9855.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } b ( 5 n + 4 ) q ^ { n } & = 3 5 \\dfrac { E _ { 5 } ^ { 2 } E _ { 1 0 } ^ { 2 } } { E _ { 1 } ^ { 4 } E _ { 2 } ^ { 4 } } + 7 0 0 q \\dfrac { E _ { 5 } ^ { 4 } E _ { 1 0 } ^ { 4 } } { E _ { 1 } ^ { 6 } E _ { 2 } ^ { 6 } } + 6 8 7 5 q ^ 2 \\dfrac { E _ { 5 } ^ { 6 } E _ { 1 0 } ^ { 6 } } { E _ { 1 } ^ { 8 } E _ { 2 } ^ { 8 } } \\\\ & + 3 1 2 5 0 q ^ 3 \\dfrac { E _ { 5 } ^ { 8 } E _ { 1 0 } ^ { 8 } } { E _ { 1 } ^ { 1 0 } E _ { 2 } ^ { 1 0 } } + 7 8 1 2 5 q ^ 4 \\dfrac { E _ { 5 } ^ { 1 0 } E _ { 1 0 } ^ { 1 0 } } { E _ { 1 } ^ { 1 2 } E _ { 2 } ^ { 1 2 } } . \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} c _ 1 = - \\frac { G _ { t - 2 } ( \\theta ) - G _ { t - 4 } ( \\theta ) } { G _ { t - 4 } ( \\theta ) } = - \\frac { F _ { t - 2 } ( \\theta ) } { G _ { t - 4 } ( \\theta ) } . \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} \\gcd ( g ( n ) , p D ) = 1 n . \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} J [ \\psi ^ { ( 2 ) } ] _ { T , t } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { p _ 1 , p _ 2 \\to \\infty } } $ \\cr } } } \\sum _ { j _ 1 = 0 } ^ { p _ 1 } \\sum _ { j _ 2 = 0 } ^ { p _ 2 } C _ { j _ 2 j _ 1 } \\Biggl ( \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\zeta _ { j _ 2 } ^ { ( i _ 2 ) } - { \\bf 1 } _ { \\{ i _ 1 = i _ 2 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 1 = j _ 2 \\} } \\Biggr ) , \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} \\int _ { \\R } h ( t ) \\int _ { \\mathbb { B } _ p } q ( \\zeta ) \\overline { \\phi ( t , \\zeta ) } \\lambda ( p ( \\zeta ) , t ) \\d V ( \\zeta ) \\d t & = 0 . \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} U _ k ( t ) = \\exp i t k \\hat { H } _ k \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} \\| u \\| _ { ( \\kappa + 1 ) p ^ * } & \\leq M _ { 2 9 } ^ { \\frac { 1 } { \\kappa + 1 } } M _ { 3 0 } ^ { \\frac { 1 } { \\sqrt { \\kappa + 1 } } } \\left [ 2 \\| u ^ { \\kappa + 1 } \\| _ { \\tilde { q } _ 1 } ^ p \\right ] ^ { \\frac { 1 } { ( \\kappa + 1 ) p } } \\leq M _ { 3 1 } ^ { \\frac { 1 } { \\kappa + 1 } } M _ { 3 0 } ^ { \\frac { 1 } { \\sqrt { \\kappa + 1 } } } \\| u \\| _ { ( \\kappa + 1 ) \\tilde { q } _ 1 } \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} \\Delta _ { \\rho , \\sigma } ^ p ( x h _ \\sigma ^ { 1 / 2 } ) = h _ \\rho ^ p x h _ \\sigma ^ { 1 / 2 - p } , \\qquad 0 \\le p \\le 1 / 2 , \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} \\overline { \\mathcal { L } } ( y , p _ { _ 1 } ( y ) ) & = p _ { _ 1 } ( y ) \\log \\frac { 1 } { p _ { _ 1 } ( y ) } + \\eta _ 0 p _ { _ 0 } ( y ) \\log \\frac { p _ { _ 0 } ( y ) } { p _ { _ 1 } ( y ) } \\\\ & ~ ~ ~ ~ + \\eta _ 1 p _ { _ 1 } ( y ) + \\eta _ 2 y ^ 2 p _ { _ 1 } ( y ) , \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} \\Psi _ { 1 , 0 } \\big ( C ^ { ( + ) } _ 1 A _ 2 C ^ { ( + ) - 1 } _ 1 \\big ) & = L ^ { ( + ) } _ 1 \\widetilde { L } ^ { ( + ) } _ 1 L ^ { ( + ) } _ 2 L ^ { ( - ) - 1 } _ 2 \\widetilde { L } ^ { ( + ) - 1 } _ 1 L ^ { ( + ) - 1 } _ 1 = L ^ { ( + ) } _ 1 L ^ { ( + ) } _ 2 L ^ { ( - ) - 1 } _ 2 L ^ { ( + ) - 1 } _ 1 \\\\ & = R _ { 1 2 } ^ { - 1 } L ^ { ( + ) } _ 2 L ^ { ( + ) } _ 1 R _ { 1 2 } L ^ { ( - ) - 1 } _ 2 L ^ { ( + ) - 1 } _ 1 = R _ { 1 2 } ^ { - 1 } L ^ { ( + ) } _ 2 L ^ { ( - ) - 1 } _ 2 R _ { 1 2 } \\\\ & = \\Psi _ { 1 , 0 } \\big ( R _ { 1 2 } ^ { - 1 } A _ 2 R _ { 1 2 } \\big ) \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} [ \\tilde { L } _ 1 x _ { i , j } ] = \\sum _ { i ' , j ' } \\frac { 1 } { 4 } ( x _ { i ' , j ' } - x _ { i , j } ) , \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} \\mathcal { P } _ q : = \\{ \\mbox { d e c o m p o s i t i o n s } \\ I _ F = \\bigsqcup _ { \\substack { i = 0 , \\dots , 3 } } I _ F ^ i \\ \\vert \\sum \\limits _ { i = 0 } ^ 3 i \\vert I ^ i _ F \\vert = q \\} . \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} \\int _ { H } \\prod _ { k = 1 } ^ m f _ k ^ { c _ k } ( B _ k x ) \\ , d \\gamma _ H ( x ) \\ge \\prod _ { k = 1 } ^ m \\Big ( \\int _ { H _ k } f _ k \\ , d \\gamma _ { H _ k } \\Big ) ^ { c _ k } , \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} q _ n = a _ n q _ { n - 1 } + q _ { n - 2 } = a _ { n + \\ell } A ^ { ( n - 1 ) } _ \\ell + A ^ { ( n - 2 ) } _ \\ell = A ^ { ( n ) } _ \\ell \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} ( w _ i , w _ j ) _ \\Delta & = \\frac { 1 } { \\sigma _ j } \\big ( w _ i , ( U \\Delta ) ^ * v _ j ) _ \\Delta \\\\ & = \\frac { 1 } { \\sigma _ j } \\big ( ( U \\Delta ) w _ i , v _ j \\big ) _ { M } \\\\ & = \\frac { \\sigma _ { i } } { \\sigma _ j } ( v _ i , v _ j ) _ { M } \\\\ & = \\frac { \\sigma _ { i } } { \\sigma _ j } \\delta _ { i j } = \\delta _ { i j } . \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} f ( t , \\zeta ) = \\int _ { \\R } \\left ( \\Phi ^ * \\right ) ^ { - 1 } F ( x + i c , \\zeta ) e ^ { 2 \\pi c t } e ^ { - i 2 \\pi x t } \\d x : = \\widetilde { T } _ V ^ { - 1 } F ( t , \\zeta ) , \\ ; t \\in \\R , \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} \\lvert m _ { n + 1 } \\rvert _ S - \\lvert m _ { n } \\rvert _ S = \\lvert A ^ + \\rvert + \\lvert A ^ - \\rvert . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} 5 \\kappa ^ { ( 2 + \\delta ) } \\leq ( \\sqrt { 5 e } ( 1 + 2 L T ) ) ^ { ( 2 + \\delta ) } \\leq ( 4 ( 1 + 2 L T ) ) ^ { ( 2 + \\delta ) } = ( 4 + 8 L T ) ^ { ( 2 + \\delta ) } . \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} \\phi ( \\alpha ) = \\frac { \\alpha + 2 ^ { \\frac { m } { 2 } } - 1 + i ( 2 ^ { \\frac { m } { 2 } } - 1 ) } { 2 } . \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} d \\mu ( z , \\ell ) = \\frac { 2 } { \\pi } K _ { \\ell } ( 2 | z | ) I _ { \\ell } ( 2 | z | ) d ^ 2 z , \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} = \\sum _ { \\ell = 0 } ^ { \\infty } \\Omega ( 4 \\ell ) + \\sum _ { \\ell = 0 } ^ { \\infty } \\Omega ( 4 \\ell + 1 ) + \\sum _ { \\ell = 0 } ^ { \\infty } \\Omega ( 4 \\ell + 2 ) + \\sum _ { \\ell = 0 } ^ { \\infty } \\Omega ( 4 \\ell + 3 ) , \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} \\tau _ 1 ^ { - 1 } \\tau ' \\tau _ 1 = \\tau '^ { - 1 } . \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} d ^ * ( y _ 0 , \\epsilon ) = k ^ * ( y _ 0 , \\epsilon ) \\ \\ \\ \\forall \\ \\epsilon > 0 . \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} w - \\lim _ { k \\rightarrow 0 , k \\in \\widetilde { S } _ \\varepsilon ( \\xi ) } Q ( k , \\xi ) - ( 1 - C _ \\omega \\widehat { k } \\cdot u ( \\xi ) ) ^ { - 1 } P _ 0 ( \\xi ) = 0 . \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} h _ { \\alpha \\beta } ( x ) : = g _ { \\mu \\nu } ( \\varphi ( x ) ) \\frac { \\partial \\varphi ^ \\mu } { \\partial x ^ \\alpha } \\frac { \\partial \\varphi ^ \\nu } { \\partial x ^ \\beta } \\ , . \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\tau ( s t ) \\leq \\tau ( s ) + \\tau ( t ) \\ \\ \\ \\ \\tau ( s ) = \\tau ( s ^ { - 1 } ) \\ \\ ( s , t \\in G ) . \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} \\partial _ t ^ p \\Phi ( \\hat { u } ) ( 0 ) = S _ { G , m , p } ( 0 , \\chi ( \\hat { u } ) , A _ 1 ^ { \\operatorname { c o } } , A _ 2 ^ { \\operatorname { c o } } , A _ 3 ^ { \\operatorname { c o } } , \\sigma ( \\hat { u } ) , f , u _ 0 ) \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} W _ { \\phi , \\infty } \\left ( \\left ( \\begin{matrix} \\gamma & 0 \\\\ 0 & 1 \\end{matrix} \\right ) \\right ) = \\omega _ { \\tilde { \\pi } , \\infty } ( \\gamma ) \\abs { \\gamma } _ { \\infty } ^ { \\frac { 1 } { 2 } } W _ { \\infty } ( \\gamma ) , \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} \\phi ( t ) = P ( T - t ) \\Psi + \\int _ t ^ T P ( r - t ) l ( r ) \\mathrm d r , \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{gather*} S ^ { \\delta _ n ^ k + 1 + u - \\alpha _ { k - u } } \\left ( \\alpha \\right ) = \\alpha \\iff \\begin{array} { c } \\{ \\alpha _ { j + ( k - u ) } = \\alpha _ j + \\alpha _ { k - u } \\} _ { j = 1 } ^ { u } \\\\ \\mbox { a n d } \\ \\ \\delta _ { n } ^ k = \\alpha _ { 1 + u } + \\alpha _ { k - u } \\\\ \\mbox { a n d } \\ \\ \\{ \\alpha _ { j + 1 + u } = \\alpha _ { j } + \\alpha _ { 1 + u } \\} _ { j = 1 } ^ { k - u - 1 } . \\end{array} \\end{gather*}"}
-{"id": "4761.png", "formula": "\\begin{align*} \\Phi _ { i j } = \\nabla \\xi _ { i } \\cdot ( a \\nabla \\xi _ { j } ) \\ , , \\forall \\ , 1 \\le i , j \\le m \\ , . \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} S _ 1 = - 1 , S _ 2 = 3 , S _ 3 = - 2 . \\end{align*}"}
-{"id": "9764.png", "formula": "\\begin{align*} O _ p = \\frac { 1 } { 2 } ( D ^ 2 _ p + p ^ 2 ) \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Big ( \\sum _ { x \\in \\partial B _ G ( n ) } ( d _ x ^ { + } + d _ x ^ 0 ) \\Big ) ^ { 1 / n } = 1 . \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} A f ( x ) = \\sum _ { [ \\xi ] \\in \\widehat { G } } d _ { \\xi } [ \\xi ( x ) \\sigma _ A ( x , \\xi ) \\widehat { f } ( \\xi ) ] . \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} E = \\Big \\{ \\{ j , k \\} : j , k \\in \\mathbb { Z } ^ d , \\norm { j - k } _ 2 = 1 \\Big \\} \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} \\lambda _ { \\min } ( \\boldsymbol { \\varphi } ) : = \\min _ { x \\in \\bar { G } } \\min _ { | \\boldsymbol { \\xi } | = 1 } \\boldsymbol { \\xi } \\cdot \\big ( \\boldsymbol { \\varphi } ( \\mathbf { x } ) \\boldsymbol { \\xi } \\big ) \\boldsymbol { \\varphi } \\in C ^ { 0 } \\big ( \\bar { G } , \\mathbb { R } ^ { 3 \\times 3 } \\big ) . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} \\Psi ( a ) \\Psi ( a ) ^ * \\leq \\Psi ( a a ^ * ) \\leq \\Psi ( a ^ * a ) = \\Psi ( a ) ^ * \\Psi ( a ) = \\Psi ( a ) \\Psi ( a ) ^ * . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} \\left | \\sum _ { j = 0 } ^ { N - 1 } \\cos ( 4 j k ( E ) + \\phi ) \\right | \\leq N \\varepsilon . \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} \\left ( \\prod _ { j = 1 } ^ r \\prod _ { s = 1 } ^ n e ^ { d _ { j , s } \\varpi _ s } ( b _ { k ( j , s ) , m ( j , s ) } ) \\right ) \\left ( \\prod _ { s = 1 } ^ n \\prod _ { j = 1 } ^ r e ^ { d _ { j , s } \\varpi _ s } ( \\zeta ( j , s ) ) ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} b & = \\epsilon \\delta ^ { - 1 } , \\\\ c & = \\epsilon \\delta ^ { - 2 } \\left ( \\Gamma - \\frac { 1 } { 2 } \\sigma ^ 2 \\right ) . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( m , n ) = \\int _ { 0 } ^ { \\infty } x ^ { m } \\frac { \\cos ( \\pi n x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x , \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} H = \\Lambda + B + Q _ 1 + Q _ 2 + K . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} \\begin{cases} u _ { t } + \\hat f ( t , x , u ) _ { x } ~ = ~ \\varepsilon u _ { x x } , \\\\ u ( 0 , x ) = u _ 0 ( x ) . \\end{cases} \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} \\xi ( z ) = z ^ { \\beta } ( z + h ) ^ { - \\frac { 1 - 2 s } { s } } \\qquad \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} 0 & = \\int _ M \\langle \\delta _ { g _ 0 } \\delta _ { g _ 0 } ^ * Y , Y \\rangle _ { g _ 0 } d v o l _ { g _ 0 } \\\\ & = \\int _ M | \\delta _ { g _ 0 } ^ * Y | ^ 2 - \\int _ { \\partial M } \\delta _ { g _ 0 } ^ * Y ( \\mathbf n , Y ) - \\int _ { \\infty } \\delta ^ * _ { g _ 0 } Y ( \\mathbf n , Y ) \\\\ & = \\int _ M | \\delta _ { g _ 0 } ^ * Y | ^ 2 - \\int _ { \\partial M } \\delta _ { g _ 0 } ^ * Y ( \\mathbf n , Y ^ T ) - \\int _ { \\partial M } \\delta _ { g _ 0 } ^ * Y ( \\mathbf n , \\mathbf n ) Y ^ { \\perp } . \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} p ^ \\omega ( t , x , y ) : = P ^ { \\omega } _ x [ X _ t = y ] . \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} \\iota _ \\chi : M \\hookrightarrow \\P \\left ( H ^ 0 \\big ( M , \\O _ M ( 1 ) \\big ) ^ \\vee \\right ) = \\P ^ N , N = h ^ 0 \\big ( M , \\O _ M ( 1 ) \\big ) - 1 ; \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} ( ( \\sigma - \\Sigma _ h ) ^ { \\frac 1 2 } , w _ h ) + ( \\nabla ( u - U _ h ) ^ { \\frac 1 2 } , \\nabla w _ h ) = 0 . \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} \\chi _ { \\ell } ( ( K _ n \\vee C _ { 2 l + 1 } ) \\square K _ { 1 , s } ) = \\begin{cases} n + 3 & s < \\frac { 1 } { 3 } ( n + 3 ) ! ( 4 ^ l - 1 ) \\\\ n + 4 & s \\geq \\frac { 1 } { 3 } ( n + 3 ) ! ( 4 ^ l - 1 ) . \\end{cases} \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} \\sigma ^ { \\frac 1 2 } - \\bigtriangleup u ^ { \\frac 1 2 } = 0 . \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} \\mathcal { J } = J - s ^ { + } I , \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} \\rho ( \\gamma ) R _ j : = \\epsilon ( [ g _ \\alpha \\gamma g _ j ^ { - 1 } ] ) , \\textrm { f o r a n y } \\gamma \\in \\pi _ K . \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} \\bar { Y } _ { s } ^ { t , x } = \\tilde { W } \\left ( s , \\bar { X } _ { s } ^ { t , x } \\right ) s \\in \\lbrack t , T ] . \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} m ( l ) : = \\frac { 1 } { 2 } \\sigma ^ 2 l ( l - 1 ) + l \\Gamma - \\delta + \\int _ \\mathbb { R } \\Big \\{ ( 1 + \\gamma ( z ) ) ^ l - 1 - l \\gamma ( z ) \\Big \\} \\nu ( d z ) . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} h ( l ) : = \\frac { 1 } { 2 } \\sigma ^ 2 l ( l - 1 ) + l \\Gamma - \\delta + \\int _ \\mathbb { R } \\Big \\{ ( 1 + \\gamma ( z ) ) ^ l - 1 - l \\gamma ( z ) \\Big \\} \\nu ( d z ) . \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} \\mu + t \\zeta _ p : = \\begin{pmatrix} \\mu _ 1 \\\\ \\vdots \\\\ \\mu _ { \\bar k - 1 } \\\\ \\mu _ { \\bar k } \\end{pmatrix} + t \\begin{pmatrix} P _ 1 e _ p \\\\ \\vdots \\\\ P _ { \\bar k - 1 } e _ p \\\\ e _ p \\end{pmatrix} , \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} K ( z , Z ) = \\left \\langle R _ L ( e _ Z \\circ T ) , R _ L ( e _ z \\circ T ) \\right \\rangle _ L , \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} s _ i ( \\varepsilon _ k ) = \\begin{cases} \\varepsilon _ { i + 1 } & ( k = i ) , \\\\ \\varepsilon _ i & ( k = i + 1 ) , \\\\ \\varepsilon _ k & ( k \\not = i , i + 1 ) . \\end{cases} \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} \\mathbf { H } = \\begin{pmatrix} H _ { 1 , 1 } & H _ { 1 , 2 } & \\dots & H _ { 1 , n } \\\\ H _ { 2 , 1 } & H _ { 2 , 2 } & \\dots & H _ { 2 , n } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ H _ { r , 1 } & H _ { r , 2 } & \\dots & H _ { r , n } \\end{pmatrix} \\in \\mathbb { B } ^ { r \\ell \\times n \\ell } . \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} \\sum _ { j } \\big ( t _ { i a } ^ { b j } t _ { j c } ^ { d l } - t _ { j a } ^ { b c } t _ { i j } ^ { d l } - t _ { b c } ^ { d j } t _ { i a } ^ { j l } \\big ) = 0 . \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} \\ < \\xi _ \\sigma , f ( \\Delta _ { \\rho , \\sigma } ) \\xi _ \\sigma \\ > = \\ < \\xi _ \\rho , \\widetilde f ( \\Delta _ { \\sigma , \\rho } ) \\xi _ \\rho \\ > . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} \\zeta _ { j } ^ { ( i ) } = \\int \\limits _ t ^ T \\phi _ { j } ( s ) d { \\bf w } _ s ^ { ( i ) } \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} W ( u ) = \\begin{cases} - \\beta \\log u , & u > 0 \\\\ + \\infty & \\end{cases} \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} \\frac { d } { d t } H _ \\nu ( X _ t ) & = - J _ \\nu ( X _ t ) , \\\\ \\frac { d ^ 2 } { d t ^ 2 } H _ \\nu ( X _ t ) & = 2 K _ \\nu ( X _ t ) + 2 \\alpha J _ \\nu ( X _ t ) . \\end{align*}"}
-{"id": "9887.png", "formula": "\\begin{align*} ( \\boldsymbol { D } ^ { - s } \\psi , \\boldsymbol { D } ^ { s * } \\psi ) = \\| \\psi \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } , ( \\boldsymbol { D } ^ { - s } \\psi , \\boldsymbol { D } ^ { s } \\psi ) = \\cos ( s \\pi ) \\| \\psi \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } . \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} R _ 1 ( \\lambda ; x , y ) = \\tfrac { i } { 2 \\sqrt { \\lambda } } \\begin{cases} [ e ^ { - i x \\sqrt { \\lambda } } - e ^ { i x \\sqrt { \\lambda } } ] e ^ { i y \\sqrt { \\lambda } } & y \\geq x \\geq 0 , \\\\ 0 & y \\geq 0 \\geq x , \\\\ e ^ { - i x \\sqrt { \\lambda } } [ e ^ { i y \\sqrt { \\lambda } } - e ^ { - i y \\sqrt { \\lambda } } ] & 0 \\geq y \\geq x , \\end{cases} \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} \\nu ( s ( n - 1 , k - 1 ) ) & = \\nu ( s ( n , k ) ) \\\\ \\epsilon ( s ( n - 1 , k - 1 ) ) & \\equiv \\epsilon ( s ( n , k ) ) p . \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\mathbf { i } , \\lambda _ 1 , \\dots , \\lambda _ r } & \\coloneqq \\mathcal { L } _ { \\mathbf { i } , \\scriptsize { \\underbrace { 0 , \\dots , 0 , \\lambda _ 1 } _ { N _ 1 } , \\underbrace { 0 , \\dots , 0 , \\lambda _ 2 } _ { N _ 2 } } , \\dots , \\scriptsize { \\underbrace { 0 , \\dots , 0 , \\lambda _ r } _ { N _ r } } } \\\\ & = ( { \\bf P } _ { \\bf i } \\times \\C _ { 0 , \\dots , 0 , \\lambda _ 1 , 0 , \\dots , 0 , \\lambda _ 2 , \\dots , 0 , \\dots , 0 , \\lambda _ r } ) / B ^ { N _ 1 + \\dots + N _ r } . \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} \\mathcal { J } & : = \\bigsqcup _ { \\substack { h , \\bar { h } , h \\bar { h } = h _ i \\\\ h \\in c ( g , X ) , \\bar { h } \\in c ( \\bar { g } , X ) } } [ t _ 1 , \\ldots , t _ { i - 1 } , g , \\bar { g } , t _ { i + 1 } , \\ldots , t _ { n - 1 } ] \\\\ & = [ t _ 1 , \\ldots , t _ { n - 1 } ] . \\qquad ( \\mbox { b y F a c t ~ 2 ~ + ~ ( \\ref { e q f o r t h m 1 . 1 : f i n a l f o r m f o r [ t 1 , . . . , g , g b a r , . . . ] } ) } ) \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} g _ { ( 6 , 1 0 , 1 5 ) } ^ { ( 5 c - 1 , 3 c - 1 , 2 ( 4 c - 1 ) + 1 ) } ( z ) = ( 1 + z ^ { 4 c - 1 } ) ( 1 + z ^ c + z ^ { 2 c } ) ( 1 + z + z ^ c + z ^ { 2 c } + z ^ { 3 c } + z ^ { 4 c } ) \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} \\Gamma _ { b } ( x _ { 1 } , x _ 2 ; u , g ) : = u \\ , \\psi \\left ( \\widetilde { \\mu } _ { g ^ { - 1 } } ( x _ { 1 } ) , x _ { 2 } \\right ) - ( \\nu + b ) \\cdot \\vartheta _ G ( g ) . \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} & 2 \\ell - 3 k ^ 2 - 2 s _ 1 k + s _ 2 \\\\ & \\qquad \\qquad = 2 \\sqrt { \\ell ^ 2 - \\ell ( 3 k ^ 2 + 2 s _ 1 k - s _ 2 - 2 ) - 3 k ^ 2 - 2 s _ 1 k + s _ 2 + 1 } - 2 + \\alpha \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} \\sum ^ a _ { i = 1 } I _ { A _ i } \\leq \\sum ^ b _ { j = 1 } I _ { B _ j } \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} \\Pi _ 0 = \\Pi \\circ \\mathcal P _ 0 , \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} \\gamma _ { k + 1 } ^ { \\delta } - \\gamma _ k ^ { \\delta } \\leq \\mu \\delta \\| F ( u _ k ^ { \\delta } ) - v ^ { \\delta } \\| ^ { p - 1 } + \\bigg [ 2 ^ { p + q - 1 } \\beta _ k ^ q \\frac { G _ q } { q } \\frac { p } { C _ p } + \\beta _ k \\bigg ( \\frac { ( p - 1 ) \\epsilon _ 2 ^ { \\frac { p } { p - 1 } } } { C _ p } + \\frac { \\epsilon _ 2 ^ { - p } } { C _ p } - ( 1 + r _ k ) \\bigg ) \\bigg ] \\rho ^ 2 . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} \\frac { \\partial f _ i } { \\partial x _ j } = \\frac { \\partial f _ j } { \\partial x _ i } , i , j = 1 , \\dots , n . \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} f _ B ( v _ 0 ) = \\frac { 1 } { { \\rm V o l } ( v _ 0 , r ) } \\sum _ { v \\in B ( v _ 0 , r ) } m ( v ) f ( v ) . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} N _ { m } & = N _ { m - 1 } + N _ { m - 3 } \\\\ & = ( 1 0 N _ { m - 7 } - N _ { m - 1 3 } + N _ { m - 1 9 } ) + ( 1 0 N _ { m - 9 } - N _ { m - 1 5 } + N _ { m - 2 1 } ) \\\\ & = 1 0 ( N _ { m - 7 } + N _ { m - 9 } ) - ( N _ { m - 1 3 } + N _ { m - 1 5 } ) + ( N _ { m - 1 9 } + N _ { m - 2 1 } ) \\\\ & = 1 0 N _ { m - 6 } + 5 N _ { m - 1 2 } + N _ { m - 1 8 } , \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} \\| u \\| _ { \\tilde { L } ^ 1 _ T ( B ^ { s + 1 } _ { 2 , 2 } ) } \\leq & C _ 2 ( \\nu ) \\| e ^ { \\nu t \\Delta } u _ 0 \\| _ { \\widetilde { L } ^ 1 _ T ( \\dot { B } ^ { s + 1 } _ { 2 , 2 } ) } + T M _ 0 ^ { \\frac { 1 } 2 } + C _ 2 ( \\nu ) \\int ^ T _ 0 \\| b \\| ^ 2 _ { B ^ { s } _ { 2 , 2 } } \\ , d t \\\\ & + C _ 2 ( \\nu ) T ^ { \\frac { r - 2 } { 2 r } } \\| u \\| ^ { \\frac { r - 2 } { r } } _ { \\widetilde { L } ^ { \\infty } _ T ( B ^ { s - 1 } _ { 2 , 2 } ) } \\| u \\| ^ { 1 + \\frac { 2 } { r } } _ { \\widetilde { L } ^ 2 _ T ( B ^ { s } _ { 2 , 2 } ) } . \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} \\begin{aligned} C ( X ) \\ddot { X } + Q ( X ) \\ddot { A } = - & Q ' ( X ) \\dot { X } \\dot { A } - C ' ( X ) \\dot { X } ^ 2 + F ( X ) + K ( X ) A \\\\ 2 I ( X ) \\ddot { X } + 2 C ( X ) \\ddot { A } = - & I ' ( X ) \\dot { X } ^ 2 - U ' ( X ) + 2 F ' ( X ) A \\\\ & + K ' ( X ) A ^ 2 + Q ' ( X ) \\dot { A } ^ 2 \\end{aligned} \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } E ( v _ n ) = d _ M , \\lim _ { n \\rightarrow \\infty } \\| \\tilde { v } ^ { j _ 0 } _ n \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } = 0 , E ( \\tilde { V } ^ { j _ 0 } ) = d _ M . \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} N \\Delta _ { X } = N \\Delta _ { 0 } + \\cdots + N \\Delta _ { s } + N \\Delta ^ { s + 1 } \\in C H ^ { d } ( X \\times X ) . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} \\mathbf { E } [ W ^ { ( 3 ) } _ { \\nu _ n } ( x ) ; \\nu _ n \\le n ^ { 1 - \\varepsilon } ] \\to \\mathbf { E } \\left [ \\sum _ { l = 0 } ^ { \\infty } f ( x + g _ l + S ( l ) ) \\mathbb { I } \\{ \\tau > l \\} \\right ] . \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} \\min _ { Q } \\widetilde { \\mathcal { E } } [ C , Q ] \\eqref { l i n e a r C } , \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} w _ t = \\tilde { a } ( x , w , w _ x ) w _ { x x } + \\tilde { f } ( x , w , w _ x ) \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} \\Z _ { l } ^ { \\times } [ a , b , k ] = \\{ m \\in \\Z _ l ^ { \\times } \\vert b \\alpha / m - a \\in l ^ { q + \\abs { r } + k } \\Z _ l \\} . \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( \\partial _ t - \\partial _ x ^ 2 - x ^ 2 \\partial _ y ^ 2 ) f ( t , x , y ) = \\mathbf 1 _ \\omega u ( t , x , y ) & t \\in [ 0 , T ] , ( x , y ) \\in \\Omega , \\\\ f ( t , x , y ) = 0 & t \\in [ 0 , T ] , ( x , y ) \\in \\partial \\Omega , \\\\ f ( 0 , x , y ) = f _ 0 ( x , y ) & ( x , y ) \\in \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} \\phi _ 0 = g _ 0 \\sqrt { \\frac { 2 } { f _ 2 } } \\quad \\quad \\quad \\phi _ 2 = \\left ( \\frac { 2 g _ 2 } { f _ 2 } - \\frac { g _ 0 f _ 4 } { 2 f ^ 2 _ 2 } \\right ) \\sqrt { \\frac { 2 } { f _ 2 } } . \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{gather*} R _ { 1 2 } B _ 1 R _ { 2 1 } A _ 2 = A _ 2 R _ { 1 2 } B _ 1 R _ { 1 2 } ^ { - 1 } . \\end{gather*}"}
-{"id": "9915.png", "formula": "\\begin{align*} \\int _ { M } u \\otimes a = n ( a ) \\int _ { M } u \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} \\hat \\jmath _ { i , i ' } ^ { \\ ; \\ ! 0 } ( \\mu ) = \\hat \\phi _ i ( \\mu ) - \\hat \\phi _ { i ' } ( \\mu ) = - \\nabla ^ { i , i ' } \\hat \\phi ( \\mu ) , \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} \\theta ( r ) = \\left \\{ \\begin{array} { c l } r ^ 2 & 0 \\leq r \\leq 1 , \\\\ & r \\geq 2 , \\end{array} \\right . \\theta '' ( r ) \\leq 2 r \\geq 0 . \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} f ' ( u ; v ) = \\int _ { \\Omega } v ( t ) [ 1 + \\log u ( t ) ] \\ , d t , \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} & W _ { m , j , i } = \\begin{cases} ( 1 - w _ { m , i } ) \\prod _ { z = i + 1 } ^ j w _ { m , z } , & i \\in \\{ 1 , 2 , \\cdots , j \\} \\\\ \\prod _ { z = 1 } ^ j w _ { m , z } , & i = 0 \\end{cases} . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} b = a _ 2 ' b _ 2 + a _ 3 ' b _ 3 . \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} I H _ { \\mathcal S } ( t ) = Q _ i ^ j Q _ { k - i } ^ { l - i } = \\frac { P _ j } { P _ i P _ { j - i } } \\cdot \\frac { P _ { l - i } } { P _ { k - i } P _ { l - k } } . \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} \\Pi ( T ' ) = \\frac { \\Pi ( G ' ) } { \\Pi ( G ' - T ' ) } . \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} ( V ^ 2 ) ' = - 2 \\ , g ( R ( t ) y , \\widetilde { y ' } ) \\ , y ^ 2 = - 2 \\ , g ( \\widetilde { R ( t ) y } , \\widetilde { y ' } ) \\ , y ^ 2 , \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\theta ( n + 1 , E ) - \\theta ( n , E ) = \\gamma ( n + 1 , E ) - \\gamma ( n , E ) + O ( | b _ { n + 1 } ^ \\prime | ) . \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\nabla \\cdot \\left ( D _ p L \\left ( \\nabla u , x \\right ) \\right ) = 0 & \\mbox { i n } & \\ B _ r , \\\\ & - \\nabla \\cdot \\left ( D _ p \\overline { L } \\left ( \\nabla \\overline { u } \\right ) \\right ) = 0 & \\mbox { i n } & \\ B _ r , \\\\ & u , \\overline { u } \\in f + H ^ 1 _ 0 ( B _ r ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} \\hat { k } ( \\xi ) = - \\frac { \\sin ^ { 2 } \\pi \\beta } { \\cosh ^ { 2 } \\pi \\xi - \\sin ^ { 2 } \\pi \\beta } . \\newline \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} \\partial _ { s _ i } p = \\frac { p ( x _ i ^ - ) - p ( x _ i ^ + ) } { L _ i } \\qquad \\mbox { f o r } s _ i \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} \\lim \\limits _ { M \\to \\infty } \\sup \\limits _ { 0 < \\delta < 1 } \\int _ { M } ^ { \\infty } d t \\delta ^ { - 2 } ( f _ \\delta , e ^ { t \\delta ^ { - 2 } \\mathcal { L } _ X ^ \\omega } f _ \\delta ) = 0 . \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} | \\sum _ { b = 1 } ^ n k _ b j _ b ^ 2 + \\sum _ { j \\in \\mathbb { Z } _ * } l _ j j ^ 2 | \\geq \\frac { 1 } { 1 0 0 n } \\max \\{ | k | , \\sum _ { j \\in \\mathbb { Z } _ * } | j l _ j | \\} , \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} - ( - \\Delta ) ^ s _ y \\phi _ { 0 } + p U ( y ) ^ { p - 1 } \\phi _ { 0 } = - \\mu _ { 0 } E _ { 0 } \\mathbb { R } ^ n , ~ ~ \\phi _ 0 ( y , t ) \\to 0 | y | \\to \\infty . \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} H = N + P = \\sum _ { 1 \\leq b \\leq n } \\sigma _ { j _ b } \\omega _ b y _ b + \\sum _ { j \\in \\mathbb { Z } _ * } \\sigma _ j \\Omega _ j z _ j \\bar { z } _ j + \\tilde { Q } + Q _ 2 + R \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} \\Big \\| { u \\choose \\dot { u } } - V _ 0 ( t ) { u _ 0 ^ + \\choose u _ 1 ^ + } \\Big \\| _ { \\dot H ^ 1 \\times L ^ 2 } & = \\Big \\| \\int _ t ^ \\infty V _ 0 ( t - s ) { 0 \\choose F ( u ( s ) ) } d s \\Big \\| _ { \\dot H ^ 1 \\times L ^ 2 } \\\\ & \\lesssim \\| ( | u | ^ { \\frac { 4 } { n - 2 } } ) u \\| _ { L _ t ^ { 2 ( n - 2 ) / ( n + 2 ) } L _ z ^ { 2 n ( n - 2 ) / ( n - 3 ) ( n + 2 ) } } \\\\ & \\lesssim \\| u \\| _ { L _ t ^ { 2 } L _ z ^ { 2 n / ( n - 3 ) } ( ( t , \\infty ) \\times X ) } ^ { \\frac { 4 } { n - 2 } } \\\\ & \\rightarrow 0 , t \\rightarrow \\infty . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} A _ \\cdot ^ { W , W } ( l ) & = \\lim _ { n \\to \\infty } A _ \\cdot ^ { W , W } ( l _ n ) \\\\ & = \\lim _ { n \\to \\infty } \\left ( \\phi _ n ( \\cdot , W _ \\cdot ) - \\phi _ n ( 0 , W _ 0 ) - \\int _ 0 ^ \\cdot \\nabla \\phi _ n ^ * ( r , W _ r ) \\mathrm d W _ r \\right ) \\\\ & = \\phi ( \\cdot , W _ t ) - \\phi ( 0 , W _ 0 ) - \\int _ 0 ^ \\cdot \\nabla \\phi ^ * ( r , W _ r ) \\mathrm d W _ r \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} ( n - 1 ) \\ell = ( m - 1 ) T . \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} A \\varphi _ k = \\sigma _ k \\psi _ k , \\ A ^ { \\ast } \\psi _ k = \\sigma _ k \\varphi _ k . \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\tau ^ * = T \\wedge \\tau ^ { n , \\omega ^ \\prime } _ R \\wedge \\inf \\left \\lbrace t \\ge 0 : \\left \\lvert X ^ { n , \\omega ^ \\prime } _ t \\right \\rvert \\ge a \\right \\rbrace \\ , . \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\frac 1 { L ^ d } \\mathcal F _ { L , \\alpha } ^ V \\bigl ( \\mu ^ L _ t \\bigr ) = \\mathcal F _ \\alpha ^ V ( \\rho _ t ) . \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} \\| T f \\| _ { L ^ q ( X ) } = & \\Big ( \\int _ X \\Big | \\int _ X K ( z , z ' ) f ( z ' ) \\ ; d \\mu ( z ' ) \\Big | ^ q \\ ; d \\mu ( z ) \\Big ) ^ { 1 / q } \\\\ \\lesssim & \\Big ( \\int _ 0 ^ \\infty \\Big | \\int _ { r < r ' } \\tilde { K } ( r , r ' ) \\tilde { f } ( r ' ) r '^ { - 1 } \\ ; d r ' \\Big | ^ q \\ ; r ^ { - 1 } d r \\Big ) ^ { 1 / q } , \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} p ( \\delta _ { \\zeta } ( \\mu ) ) = \\abs { \\mu } ^ { M } p ( \\zeta ) . \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} \\delta _ 1 ^ { ( j ) } = \\frac { 1 } { ( 2 ^ { R ^ { t h } } + C ) ^ { \\ell - 1 } } , \\mbox { a n d } \\delta _ { i } ^ { ( j ) } = \\frac { 2 ^ { R ^ { t h } } - 1 + C } { ( 2 ^ { R ^ { t h } } + C ) ^ { \\ell - i + 1 } } , i = 2 , \\cdots , \\ell , \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\kappa _ { 1 0 ^ q 1 } = \\sum _ { p \\geq 0 } c _ { q , p } , ~ \\forall q \\geq 0 . \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} \\eta _ 2 \\star \\eta _ 1 ( t , x ) = \\begin{cases} \\eta _ 1 ( 2 t , x ) , & t \\in [ 0 , \\tfrac 1 2 ] , \\\\ \\eta _ 2 ( 2 t - 1 , \\eta _ 1 ( 1 , x ) ) , & t \\in [ \\tfrac 1 2 , 1 ] . \\end{cases} \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} [ | u | ^ p ] _ { \\alpha ; \\bar { \\Omega } } = \\sup _ { x \\neq y } \\frac { | | u ( x ) | ^ p - | u ( y ) | ^ p | } { | x - y | ^ { \\alpha } } & = \\sup _ { x \\neq y } p \\xi ^ { p - 1 } \\frac { | | u ( x ) | - | u ( y ) | | } { | x - y | ^ { \\alpha } } \\\\ & \\leq C ( \\Omega , p ) [ u ] _ { \\alpha ; \\bar { \\Omega } } \\leq C , \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} f ( x ) = \\langle u , x \\rangle + b . \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} H \\circ \\Psi = \\Lambda + B + Q _ 2 + R \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} G _ { ( a , b ) } ^ \\omega ( x , y ) = E _ x ^ \\omega \\left [ \\int _ 0 ^ { H ( a ) \\wedge H ( b ) } 1 _ { \\{ y \\} } ( S _ r ) \\dd r \\right ] . \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} { \\tilde J } _ { \\mu } ^ { ( \\alpha ) } ( X ) = \\lim _ { q \\to 1 } J _ { \\mu } ( X ; q , q ^ { \\alpha } ) / ( 1 - q ) ^ n , \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u + \\lambda ^ 2 V ( x ) e ^ { \\ , u } & = 0 , \\quad \\mbox { i n } \\ \\Omega \\subset \\R ^ 2 , \\\\ u & = 0 , \\quad \\mbox { o n } \\ , \\partial \\Omega , \\end{aligned} \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} z \\in \\mathbb { C } ^ n \\mapsto \\mathrm { e } ^ { - n \\sum _ { k = 1 } ^ n | z _ i | ^ 2 } \\prod _ { j < k } | z _ j - z _ k | ^ \\beta , \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} \\sum _ { i \\in I } v _ { i } ^ { 2 } \\langle \\pi _ { Z _ { i } } C ^ { \\prime } f , \\pi _ { Z _ { i } } C f \\rangle & = \\Vert v _ { i } ( C ^ { * } \\pi _ { { Z } _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f \\Vert ^ { 2 } _ { 2 } \\\\ & \\geq ( \\dfrac { ( 1 - \\lambda _ { 1 } ) \\sqrt { A } - \\Vert \\beta \\Vert _ { 2 } } { 1 + \\lambda _ { 2 } } \\Vert f \\Vert ) ^ { 2 } \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} \\| \\chi _ A \\| _ n = \\min \\{ m _ n ^ { - 1 } \\phi _ { X _ 0 } ( t ) , m _ n \\phi _ { X _ 1 } ( t ) \\} . \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} s = 0 \\vee s \\neq 0 \\rightarrow \\neg A \\vee \\neg \\neg A \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} \\varphi _ { _ { 0 } } \\longmapsto \\| \\varphi _ { _ { 0 } } \\| _ { \\mathcal { G } } ^ { 2 } = \\displaystyle \\int _ { 0 } ^ { T } \\left \\| C \\varphi ( T - t ) \\right \\| ^ { 2 } d t . \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} + \\sum _ { r = 2 } ^ { n } \\sum _ { l = 1 } ^ { r - 1 } \\sum _ { i = 0 } ^ { l - 1 } \\left ( S _ { i r l } + S _ { i l r } \\right ) \\left ( \\frac { H _ { i } } { ( i - r ) ( i - l ) } + \\frac { H _ { r } } { ( r - i ) ( r - l ) } + \\frac { H _ { l } } { ( l - i ) ( l - r ) } \\right ) , \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} [ P _ t x _ 0 ] _ v = \\sum _ { v ' \\in V } q _ t ( v , v ' ) x _ { v ' , 0 } m ( v ' ) , \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} \\begin{aligned} b ( x ' , y , t ) = { } & \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } ( b ( x , y , t ) + h ( x ) ( \\frac { \\partial } { \\partial y } ( y p _ { 0 } ( y , t ) ) \\\\ & + \\frac { 1 } { 2 } x ^ { 2 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } p _ { 0 } ( y , t ) - \\frac { \\partial } { \\partial t } p _ { 0 } ( y , t ) ) ) \\end{aligned} \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} \\begin{aligned} \\max \\varphi ( s , 1 ) < C ( 1 + 1 / s ) . \\end{aligned} \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} x \\coloneqq \\Re \\left ( \\frac { \\rho } { \\rho - \\tau } \\right ) = \\frac { | \\rho | ^ 2 - \\Re ( \\rho ) \\tau } { | \\rho - \\tau | ^ 2 } = 1 + \\frac { \\Re ( \\rho ) \\tau - \\tau ^ 2 } { | \\rho - \\tau | ^ 2 } \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} C ^ { - 1 } \\Big \\| \\sum _ { k = 1 } ^ \\infty b _ k e _ k ^ * \\Big \\| _ { E _ p ^ * } \\le \\Big \\| \\sum _ { k = 1 } ^ \\infty b _ k y _ k \\Big \\| _ { X _ p ^ * } \\le C \\Big \\| \\sum _ { k = 1 } ^ \\infty b _ k e _ k ^ * \\Big \\| _ { E _ p ^ * } , \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} [ p _ 1 ] _ { \\kappa } ^ { \\bar { n } } + [ p _ 2 ] _ { \\kappa } ^ { \\bar { n } } + \\cdots + [ p _ { m } ] _ { \\kappa } ^ { \\bar { n } } = [ p _ { m + 1 } ] _ { \\kappa } ^ { \\bar { n } } \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} & f '' ( h _ 0 , \\ldots , h _ { n - 1 } ) ( \\nu ) \\\\ & = \\sum _ { i _ 0 , \\ldots , i _ { n - 1 } } \\langle f ' ( g _ { i _ 0 } , \\ldots , g _ { i _ { n - 1 } } ) , \\nu | _ { [ i _ 0 , \\ldots , i _ { n - 1 } ] } \\circ L \\rangle \\\\ & = \\sum _ { i _ 0 , \\ldots , i _ { n - 1 } } \\sum _ { t _ 0 , \\ldots , t _ { n - 1 } } \\langle f ( h _ { t _ 0 } , \\ldots , h _ { t _ { n - 1 } } ) , ( \\nu | _ { [ i _ 0 , \\ldots , i _ { n - 1 } ] } \\circ L ) | _ { [ t _ 0 , \\ldots , t _ { n - 1 } ] } \\pi \\rangle . \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} g \\cdot u = \\{ g _ { t ( e ) } ^ { - 1 } \\circ u _ e \\circ g _ { s ( e ) } \\} _ { e \\in E ( Q ) } \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} \\theta _ u ^ n = f \\omega ^ n . \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} N _ m \\log N _ m = O ( m ^ n \\log m ) = o ( m ^ { n + 1 } ) , \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} d Y _ { s } ^ { 2 , u } = - F _ { 1 } ( s , x _ { 0 } , 0 , 0 , u _ { s } ) d s + Z _ { s } ^ { 2 , u } d B _ { s } , \\ Y _ { t + \\delta } ^ { 2 , u } = 0 . \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} \\Delta _ g u = - \\lambda u . \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} \\mathcal { L } \\phi ( \\cdot , x ) = \\sum _ { i = 1 } ^ p \\mu _ i ( x ) \\frac { \\partial \\phi } { \\partial x _ i } ( \\cdot , x ) + \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ p ( \\sigma \\sigma ^ T ) _ { i j } ( x ) \\frac { \\partial ^ 2 \\phi } { \\partial x _ i \\partial x _ { j } } + I \\phi ( \\cdot , x ) , \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} u _ { k + 1 } = u _ k \\int _ \\Sigma \\frac { a _ j ( s , \\cdot ) \\ , y _ j ( s ) } { ( A _ j u _ k ) ( s ) } \\ , d s , k = 0 , \\dots , \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} u _ 2 ( t , 0 ) = a u _ { 3 } ( t , 0 ) + b u _ { 4 } ( t , 0 ) \\mbox { f o r } t \\ge 0 . \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} m y = y - m - 1 . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\max _ { 1 \\leqslant i \\leqslant k _ l } \\nu _ { F _ l } ( \\{ | G _ { l , i } - \\mathbb E _ { F _ l } ( G _ { l , i } ) | \\geqslant \\eta \\hat s _ l \\} ) = 0 , \\forall \\eta > 0 \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} I ( t _ 1 u _ 1 + t _ 2 u _ 2 + t _ 3 u _ 3 ) & = \\sum _ { i = 1 } ^ 3 I ( t _ i u _ i ) + \\frac { \\lambda } { 2 } \\sum _ { i < j } t _ { i } ^ { 2 } t _ { j } ^ { 2 } \\int _ { \\mathbb R ^ 3 } \\phi _ i u _ j ^ 2 d x \\\\ I ' ( u ) [ u _ i ] & = I ' ( u _ i ) [ u _ i ] + \\lambda \\sum _ { j \\neq i } \\int _ { \\mathbb R ^ 3 } \\phi _ j u _ i ^ 2 d x , \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} [ \\ , U \\ , c \\ , ] \\Delta _ \\mathrm { n e w } = V _ \\mathrm { n e w } \\Sigma _ { Q } W _ \\mathrm { n e w } ^ { * } , \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} U _ { n 2 } & = u _ n + \\tfrac { 1 } { 2 } \\Delta t \\varphi _ { 1 } ( \\tfrac { 1 } { 2 } \\Delta t J _ n ) F ( u _ n ) , \\\\ U _ { n 3 } & = u _ n + \\Delta t \\varphi _ { 1 } ( \\Delta t J _ n ) F ( u _ n ) , \\\\ u _ { n + 1 } & = u _ n + \\Delta t \\varphi _ { 1 } ( \\Delta t J _ n ) F ( u _ n ) + \\Delta t \\varphi _ 3 ( \\Delta t J _ n ) ( 1 6 D _ { n 2 } - 2 D _ { n 3 } ) \\\\ & + \\Delta t \\varphi _ 4 ( \\Delta t J _ n ) ( - 4 8 D _ { n 2 } + 1 2 D _ { n 3 } ) . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{gather*} c G _ - ^ m = \\langle c ^ m , G _ - \\rangle a F _ + ^ m = \\langle a ^ m , F _ + \\rangle d G _ + ^ m = \\langle d ^ m , G _ + \\rangle a F _ - ^ m = \\langle a ^ m , F _ - \\rangle \\\\ c F _ - ^ m = \\langle c ^ m , F _ - \\rangle b G _ + ^ m = \\langle b ^ m , G _ + \\rangle d F _ + ^ m = \\langle d ^ m , F _ + \\rangle b G _ - ^ m = \\langle b ^ m , G _ - \\rangle \\end{gather*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\varepsilon = & p ( p - 1 ) + n + ( 2 n - p ) ( 2 n - p - 1 ) + n \\\\ = & 4 n ^ 2 - 4 n p + 2 p ^ 2 \\equiv 2 p \\mod 4 . \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\beta ( y ) = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( - y ^ { 4 } + \\frac { 7 y ^ { 2 } } { 2 } + C \\right ) e ^ { - 2 y ^ { 2 } } \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} \\Psi _ \\lambda : = c _ \\lambda \\Psi , c _ \\lambda = ( 2 + \\lambda ^ - ) \\lambda ^ - = \\max ( 0 , - \\lambda ) \\geq 0 . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} g _ { \\rm s t d } = \\zeta ^ 1 \\odot \\bar \\zeta ^ 1 + \\zeta ^ 2 \\odot \\bar \\zeta ^ 2 + \\zeta ^ 3 \\odot \\bar \\zeta ^ 3 \\ , , \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} e = \\sum _ { I \\subseteq \\{ 1 , \\ldots , N \\} } H _ I ( e ) . \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} d ( y _ j , A _ j u ) , j = 0 , \\dots , N - 1 . \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} a _ k ( m , n ) = - a _ { 2 - k } ( n , m ) . \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} \\lambda _ { k } = \\ , \\begin{cases} | Q _ { - j , m _ j } | ^ { - \\frac { 1 } { u _ 1 } } & Q _ { 0 , k } \\subset Q _ { - j , m _ j } \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} L ( \\phi ^ { \\vee } , z _ 1 , \\dots , z _ n ; x , y ) = \\sum _ { d \\geq 0 } x ^ { - d } \\sum \\limits _ { a \\in A _ { + , d } } \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) \\langle a \\rangle ^ y \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} \\vec { x } = ( x _ 1 , \\ldots , x _ a ) , \\ \\vec { y } = ( y _ 1 , \\ldots , y _ { b } ) \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = e _ 2 + \\eta \\ , e _ 3 , [ e _ 1 , e _ 3 ] = - \\eta \\ , e _ 2 + e _ 3 , \\ , \\ , \\eta \\in \\R . \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} ( A ) _ + = \\begin{cases} A & \\textup { i f $ A $ i s p o s i t i v e s e m i - d e f i n i t e , } \\\\ 0 & \\textup { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} { a m \\brack b m } _ q \\equiv { a \\brack b } _ { q ^ { m ^ 2 } } - \\binom { a } { b } b ( a - b ) \\frac { m ^ 2 - 1 } { 2 4 } ( q ^ m - 1 ) ^ 2 \\bmod { \\Phi _ m ( q ) ^ 3 } \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} \\sum _ { i \\in I } v _ { i } ^ { 2 } \\langle \\pi _ { Z _ { i } } C ^ { \\prime } f , \\pi _ { Z _ { i } } C f \\rangle = \\Vert v _ { i } ( C ^ { * } \\pi _ { Z _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f \\Vert ^ { 2 } \\geq ( A - \\epsilon ^ { 2 } ) \\Vert f \\Vert ^ { 2 } \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} Q = \\frac 1 2 Q ^ 1 - \\frac 1 4 Q ^ 2 - \\frac 1 2 Q ^ 3 + Q ^ 4 \\ , , \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} \\omega _ 0 : = \\sup _ { t \\in ( t _ 0 , T _ + ) } \\| u ( t ) \\| _ { W ^ { 1 , \\infty } ( G ) } < \\infty \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} \\| f \\| _ { L _ { Q / p } ^ { p } ( B ( x _ { 0 } , r ) ) } : = \\left ( \\int _ { B ( x _ { 0 } , r ) } ( | ( - \\L ) ^ { \\frac { Q } { 2 p } } f ( x ) | ^ { p } + | f ( x ) | ^ { p } ) d x \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} \\frac { { { \\rm { D } } ^ { 2 } } } { \\partial { { z } ^ { i } } \\partial \\overline { { { z } ^ { j } } } } d { { z } ^ { i } } \\wedge d \\overline { { { z } ^ { j } } } & = \\partial \\overline { \\partial } w + \\partial w \\overline { \\partial } K + \\partial K \\overline { \\partial } w + w \\psi \\\\ & = 0 \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} \\Lambda ^ n = \\Upsilon ^ n : = \\left \\{ \\mathbf { x } \\in \\R ^ n \\ | \\ x _ 1 ^ 2 \\geq \\sum \\limits _ { i = 3 } ^ { n } x _ i ^ 2 , \\ x _ 1 \\geq 0 \\right \\} . \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} \\theta _ y = - y \\kappa ^ { - S } ( \\theta ) - ( y - 1 ) \\kappa ^ { A } ( \\theta ) , \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} g ( z ) = \\langle g , T R _ L ( e _ z \\circ T ) \\rangle _ H , \\textrm { i . e . , } T R _ L ( e _ z \\circ T ) = R _ H ( e _ z ) \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} \\widehat { K _ h } ( \\tau ) = C \\int _ { \\R ^ d } \\frac { | \\xi | ^ a ( 1 + | \\xi | ^ 2 ) ^ { \\frac { a - d } { 2 } } } { ( \\tau - | \\xi | ^ a ) ^ 2 + | \\xi | ^ { 2 a } } \\hat h ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} \\phi _ P ( z ) : = H _ { P } ( z ) - \\frac { r } { 2 } \\log ( 1 + \\| z \\| ^ 2 ) . \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} C _ K : = \\frac { 2 ^ { n - 1 } ( n - 1 ) + m _ K n + 1 } { \\left ( \\sqrt { 2 } \\right ) ^ { 3 n - 1 } n } . \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} V \\phi = \\begin{cases} e _ { - \\textup { s i g n } ( \\eta ) } \\otimes \\psi & \\eta \\neq 0 \\\\ e _ { - 1 } \\otimes \\psi _ { - 1 } + e _ { 1 } \\otimes \\psi _ 1 & \\eta = 0 \\end{cases} \\end{align*}"}
-{"id": "9935.png", "formula": "\\begin{align*} \\kappa _ i = \\underset { \\beta \\in \\Lambda ^ { \\ell - 1 } } { \\sum } ( \\bar { \\partial } \\varphi ) _ \\beta \\wedge \\tau _ { \\beta \\ , i } + \\displaystyle \\sum _ { 0 \\le k \\le \\ell - 2 , \\ , \\beta \\in \\Lambda ^ { k } } ( \\bar { \\partial } \\varphi ) _ \\beta \\wedge \\left ( \\displaystyle \\sum _ { \\lambda \\in \\Lambda , \\ , 1 \\le j \\le k + 1 } ( - 1 ) ^ { j + 1 } \\varphi _ \\lambda \\ , \\ , \\tau _ { \\overset { \\overset { j } { \\vee } } { \\lambda \\ , \\beta \\ , i } } \\right ) , \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} \\dfrac { d } { d t } g ( t ) + \\dfrac { 1 } { 2 } g ^ 2 ( t ) = - \\left ( p * ( u ^ 2 + \\dfrac { u _ x ^ 2 } { 2 } ) \\right ) ( 0 ) \\leq 0 . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ { \\lambda } ( x ) : = \\lambda ^ { - \\frac { p - 1 } { p } } u ( x _ { \\lambda } ) , \\\\ & x _ { \\lambda } : = \\left [ \\lambda | x | ^ { - \\frac { n - p } { p - 1 } } + ( 1 - \\lambda ) R ^ { - \\frac { n - p } { p - 1 } } \\right ] ^ { - \\frac { p - 1 } { n - p } } \\frac { x } { | x | } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} v _ { - , a } ( x ) = 2 \\phi _ 1 ( x ) \\frac { \\int _ x ^ \\infty d y \\thinspace \\phi ^ { - 2 } _ 1 ( y ) \\int _ a ^ x d t \\phi ^ { - 2 } _ 1 ( t ) \\int _ t ^ y \\phi _ 1 ( z ) d z } { \\int _ a ^ \\infty \\phi ^ { - 2 } _ 1 ( y ) d y } , \\ x \\ge a . \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} F _ { k , m , r } ( z ) & = \\frac { 1 } { r ! } \\frac { \\partial ^ r } { \\partial s ^ r } P _ { k , m } ( z , s ) \\bigg | _ { s = 1 - k / 2 } \\\\ & = \\sum _ { j = 0 } ^ r c _ { m , j } ^ + u _ { k , m } ^ { [ j ] , + } ( y ) e ^ { 2 \\pi i m x } + \\sum _ { ( - 1 ) ^ { \\lambda _ k } n \\equiv 0 , 1 ( 4 ) } \\sum _ { j = 0 } ^ r c _ { n , j } ^ - u _ { k , n } ^ { [ j ] , - } ( y ) e ^ { 2 \\pi i n x } , \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} L _ Y \\alpha ^ 2 ( \\partial _ t , v ) & = - \\alpha ^ 2 ( [ Y , v ] , \\partial _ t ) \\\\ & = - \\alpha ( [ Y , v ] ) \\\\ & = u ^ { - 2 } \\xi ( [ Y , v ] ) . \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} \\begin{aligned} \\Im \\langle v & , D _ j D Q ( \\dot z + i E z ) \\rangle + \\Im \\langle D _ j Q , D Q ( \\dot z + i E z ) \\rangle \\\\ & = - \\Im i \\biggl [ \\langle F ( Q + v ) - F ( Q ) , D _ j Q \\rangle - \\langle v , D _ j ( | Q | ^ p Q ) \\rangle \\biggr ] . \\end{aligned} \\end{align*}"}
-{"id": "9978.png", "formula": "\\begin{align*} \\partial _ \\tau u ( x , \\tau ) - \\partial _ x u ( x , \\tau ) & = p ( x ) u ( 0 , \\tau ) , \\tau , x \\ge 0 , u ( x , 0 ) = u _ 0 ( x ) . \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} H _ \\mu ( \\xi , A ) = K ( \\xi - d \\Gamma _ A ( m ) ) + d \\Gamma _ A ( \\omega ) + \\mu \\varphi ( v _ A ) \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} \\| u ^ n - U _ h ^ { n } \\| + \\gamma ^ { \\frac 1 2 } \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| ( \\sigma - \\Sigma _ h ) ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } + \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\nabla ( u - U _ { h } ) ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } \\leq C [ h ^ { m + 1 } + H ^ { 2 m + 2 } ] . \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} \\lvert K ( \\xi - m ^ { ( n ) } ( k ) ) \\lvert = K ( \\xi ) + \\lvert m ^ { ( n ) } ( k ) \\lvert \\lVert \\nabla K ( \\xi ) \\lvert + \\lvert m ^ { ( n ) } ( k ) \\lvert ^ 2 C _ K \\leq \\widetilde { C } ( 1 + n ^ 2 R ^ 2 ) \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} N = \\{ g \\in G ; \\ g K = K g \\} . \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} { \\bf 1 } _ { \\{ \\tau < s \\} } = \\sum _ { j = 0 } ^ { \\infty } \\int \\limits _ t ^ T { \\bf 1 } _ { \\{ \\tau < s \\} } \\phi _ j ( \\tau ) d \\tau \\cdot \\phi _ j ( \\tau ) = \\sum _ { j = 0 } ^ { \\infty } \\int \\limits _ t ^ s \\phi _ j ( \\tau ) d \\tau \\cdot \\phi _ j ( \\tau ) , \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{gather*} g _ { i j } = \\begin{pmatrix} a _ 3 ( u ^ 2 ) ^ 2 + a _ 2 ( u ^ 3 ) ^ 2 & - a _ 3 u ^ 1 u ^ 2 + \\alpha u ^ 3 & - a _ 2 u ^ 1 u ^ 3 + \\alpha u ^ 2 \\\\ - a _ 3 u ^ 1 u ^ 2 + \\alpha u ^ 3 & a _ 2 + a _ 3 ( u ^ 1 ) ^ 2 & - 2 \\alpha u ^ 1 \\\\ - a _ 2 u ^ 1 u ^ 3 + \\alpha u ^ 2 & - 2 \\alpha u ^ 1 & a _ 3 + a _ 2 ( u ^ 1 ) ^ 2 \\end{pmatrix} , \\\\ w _ { 1 2 } = \\sqrt { \\frac { ( a _ 2 ^ 2 - \\alpha ^ 2 ) ( a _ 3 ^ 2 - \\alpha ^ 2 ) } { \\det g } } u ^ 3 , w _ { 2 3 } = 0 , w _ { 3 1 } = \\sqrt { \\frac { ( a _ 2 ^ 2 - \\alpha ^ 2 ) ( a _ 3 ^ 2 - \\alpha ^ 2 ) } { \\det g } } u ^ 2 , \\end{gather*}"}
-{"id": "4527.png", "formula": "\\begin{align*} C = \\left \\{ u \\in X : u ( x ) \\geq 0 x \\in \\Omega \\right \\} . \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{gather*} \\beta ( j + i u ) = \\beta ( j ) + i \\beta ( u ) = \\beta ( j ) + \\beta ( i u ) i \\geq 0 , j \\geq 0 , j + i u \\leq k , \\end{gather*}"}
-{"id": "6254.png", "formula": "\\begin{align*} \\det \\left ( \\delta _ { i , j } + \\int _ { - \\infty } ^ { \\infty } d x A ( i , x ) B ( x , j ) \\right ) _ { i , j = 1 } ^ N = \\det \\left ( 1 + B A \\right ) _ { L ^ 2 ( \\R ) } , \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} w _ { \\sigma } ^ 2 = s _ { \\rho _ { 1 , \\sigma } + \\rho _ { 2 , \\sigma } } s _ { \\rho _ { 2 , \\sigma } } , \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} H _ 0 ^ { \\perp _ { \\mathcal Q } } = \\{ x \\in H \\colon \\forall { y \\in H _ 0 } , \\ \\mathcal Q ( x , y ) = 0 \\} , \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} I : = \\sum _ { j = 1 } ^ { d _ k } f ( \\sqrt { k } ( \\mu _ { k , j } - E ) ) \\Pi _ { k , { \\mu _ { k , j } } } ( z _ k , z _ k ) = \\int _ { \\R } \\hat { f } ( t ) e ^ { - i E \\sqrt { k } t } { U } _ { k } ( t / \\sqrt { k } , z _ k , z _ k ) \\frac { d t } { 2 \\pi } \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} \\mathfrak R \\Phi ( 0 , 1 ) = \\int \\Phi ( 1 , x ) d x . \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} S _ { N , r + 1 } ^ { ( 4 , b ) } & = \\sum _ { k = 0 } ^ { r + 1 } N _ { 4 k + b } = S _ { N , r } ^ { ( 4 , b ) } + N _ { 4 ( r + 1 ) + b } \\\\ & = ( 5 S _ { N , r - 1 } ^ { ( 4 , b ) } - 2 S _ { N , r - 2 } ^ { ( 4 , b ) } + S _ { N , r - 3 } ^ { ( 4 , b ) } + 1 ) + ( 5 N _ { 4 r + b } - 2 N _ { 4 ( r - 1 ) + b } + N _ { 4 ( r - 2 ) + b } ) \\\\ & = 5 S _ { N , r } ^ { ( 4 , b ) } - 2 S _ { N , r - 1 } ^ { ( 4 , b ) } + S _ { N , r - 2 } ^ { ( 4 , b ) } + 1 , \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} ( s _ 1 , s _ 2 ) \\cdot ( \\vec { x } , \\vec { y } , t ) = ( s _ 1 \\vec { x } , s _ 2 ^ { - 1 } \\vec { y } , s _ 1 ^ { - 1 } s _ 2 t ) . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} L _ 0 [ \\psi ] : = - ( - \\Delta ) ^ s _ y \\psi + p U ( y ) ^ { p - 1 } \\psi = h ( y ) \\mathbb { R } ^ n , ~ ~ \\psi ( y ) \\to 0 | y | \\to \\infty . \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} \\mathrm { S I N R } = \\frac { h _ { \\mathrm { 1 1 } } X _ { \\mathrm { 1 1 } } ^ { - \\alpha } } { \\sum _ { i \\in \\Phi _ { b } ^ { ' } } h _ { i } D _ { i } ^ { - \\alpha } \\mathbf { 1 } \\left ( D _ { i } \\geq \\max ( X _ { 1 1 } , X _ { 1 1 } ^ { \\frac { \\mu } { 2 } } \\left ( \\frac { P } { N } \\right ) ^ { \\frac { 2 - \\mu } { 2 \\alpha } } \\left ( \\frac { 1 } { M } \\right ) ^ { \\frac { 1 } { 2 \\alpha } } ) \\right ) + \\frac { N } { P } } , \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} i \\hbar D \\xi & = \\frac { 1 } { 2 } \\left ( \\frac { \\delta F } { \\delta \\psi } \\psi ^ { \\dagger } + \\psi \\frac { \\delta F } { \\delta \\psi } ^ { \\dagger } \\right ) + \\alpha \\psi \\psi ^ { \\dagger } \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} \\Delta K ( \\varphi , p ) = \\Delta K _ 1 ( \\varphi , p ) + \\Delta K _ 2 ( \\varphi , p ) , \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} \\mu \\cdot ( z _ 1 , \\ldots , z _ n ) = \\left ( e ^ { 2 \\pi i m _ 1 } z _ 1 , \\ldots , e ^ { 2 \\pi i m _ n } z _ n \\right ) , \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} \\| A + \\delta K \\| & \\geq \\| \\gamma _ n ^ { ( d ) } ( A + \\delta K ) \\| = \\| \\gamma _ n ^ { ( d ) } ( A ) + \\delta R \\| \\\\ & \\geq \\| \\gamma _ n ^ { ( d ) } ( A ) \\xi + \\delta R \\xi \\| = ( 1 + \\delta ) \\| \\gamma _ n ^ { ( d ) } ( A ) \\xi \\| \\\\ & = ( 1 + \\delta ) \\| \\gamma _ n ^ { ( d ) } ( A ) \\| \\geq ( 1 + \\delta ) ( 1 - \\delta / 2 ) \\| A \\| \\\\ & > \\| A \\| . \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\lambda ^ 2 \\exp \\left ( ( \\beta + 2 \\log \\beta ) H _ \\gamma ^ \\pm + \\tilde H ^ \\pm \\right ) \\lesssim e ^ { - c _ 0 M \\log \\beta } = o ( \\beta ) , \\mathop { d i s t } ( x , \\gamma ) > \\delta _ \\lambda . \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} P = \\sum _ \\alpha a _ \\alpha ( z ) \\partial _ z ^ \\alpha , \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} L ( \\phi ^ { \\vee } , z _ 1 , \\dots , z _ n ; x , y ) : = \\sum _ { d \\geq 0 } \\mathcal { L } _ { d , n } ( x , y ) ( z _ 1 , \\dots , z _ n ) , \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} \\dot { A } _ 1 ( C _ 1 ' P _ 1 + D _ 1 ' ) + \\dot { B } \\frac { \\partial F } { \\partial Q _ 1 } = ( \\dot { C } _ 1 P _ 1 + \\dot { D } _ 1 ) A ' _ 1 ; \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} \\mathcal { F } [ p ] : = \\int _ \\Omega \\int _ { \\mathbb { S } _ + ^ 1 } \\frac { \\gamma - 1 } { 2 \\gamma - 1 } c _ 0 ^ { \\frac { 2 } { \\gamma - 1 } } | \\theta \\cdot \\nabla p | ^ { \\frac { 2 \\gamma - 1 } { \\gamma - 1 } } \\ , d \\theta \\ , d x - \\int _ \\Omega p S \\ , d x \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} \\dot { p } _ { 2 , \\tau } = \\{ p _ { 2 , \\tau } , { H _ { \\tau } } _ T \\} _ { P B } = - \\lambda m \\omega ^ 2 ( t _ \\tau ) f ^ { - 1 } ( t _ \\tau ) x _ { 2 , \\tau } ; \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t ( u - u _ h ) ^ { n - \\theta } , v _ h \\Big { ) } & - \\gamma ( \\nabla ( \\mathfrak { Q } _ h \\sigma - \\sigma _ h ) ^ { n - \\theta } , \\nabla v _ h ) \\\\ & + ( \\nabla ( \\mathfrak { Q } _ h u - u _ h ) ^ { n - \\theta } , \\nabla v _ h ) + ( f ^ { n - \\theta } ( u ) - f ^ { n - \\theta } ( u _ h ) , v _ h ) = 0 , \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\abs { \\mu } < \\frac { \\abs { z } } { ( p ( z ) ) ^ { 1 / M } } = \\frac { 1 } { ( p ( \\omega ) ) ^ { 1 / M } } : = R ( \\omega ) . \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} \\mathbf E \\left [ \\phi \\big ( 0 , X ^ { x , \\alpha } ( \\tau _ { t , x } ) \\big ) + \\int _ 0 ^ { \\tau _ { t , x } } f _ \\phi \\left ( - X ^ { t , \\beta } ( s ) , X ^ { x , \\alpha } ( s ) \\right ) d s \\right ] = \\mathbf E \\left [ \\phi \\left ( - X ^ { t , \\beta } ( \\tau _ { t , x } ) , X ^ { x , \\alpha } ( \\tau _ { t , x } ) \\right ) \\right ] . \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} E ( v _ n ) = \\sum _ { j = 1 } ^ l E ( V ^ j ( \\cdot - x ^ j _ n ) ) + E ( v ^ l _ n ) + o _ n ( 1 ) , \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} ( t , x ) \\cdot ( s , y ) = ( t s , x + y ) \\end{align*}"}
-{"id": "9858.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\tilde { c } ( 5 n + 4 ) q ^ { n } & = 3 5 \\dfrac { E _ { 5 } ^ { 2 } E _ { 1 0 } ^ { 2 } } { E _ { 1 } ^ { 5 } E _ { 2 } } + 7 0 0 q \\dfrac { E _ { 5 } ^ { 4 } E _ { 1 0 } ^ { 4 } } { E _ { 1 } ^ { 7 } E _ { 2 } ^ { 3 } } + 6 8 7 5 q ^ 2 \\dfrac { E _ { 5 } ^ { 6 } E _ { 1 0 } ^ { 6 } } { E _ { 1 } ^ { 9 } E _ { 2 } ^ { 5 } } \\\\ & + 3 1 2 5 0 q ^ 3 \\dfrac { E _ { 5 } ^ { 8 } E _ { 1 0 } ^ { 8 } } { E _ { 1 } ^ { 1 1 } E _ { 2 } ^ { 7 } } + 7 8 1 2 5 q ^ 4 \\dfrac { E _ { 5 } ^ { 1 0 } E _ { 1 0 } ^ { 1 0 } } { E _ { 1 } ^ { 1 3 } E _ { 2 } ^ { 9 } } . \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} [ v ( \\cdot , S ^ 1 ) , S ^ 2 ] _ t & = [ M ^ v , M ^ 2 ] _ t \\\\ & = \\left [ \\int _ 0 ^ \\cdot \\nabla v ^ * ( r , S ^ 1 _ r ) \\mathrm d M ^ 1 _ r , M ^ 2 \\right ] \\\\ & = \\int _ 0 ^ t \\nabla v ^ * ( r , S ^ 1 _ r ) \\mathrm d [ M ^ 1 , M ^ 2 ] _ r , \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} Z = \\{ x y + z ^ 2 = 0 \\} \\subset \\mathbb { C } ^ 3 \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{gather*} T _ { w ( 1 , m ) } ( E _ { k , 1 } ( s + 1 ) ) = \\begin{cases} E _ { 1 , 1 } ( s + 1 ) & , \\\\ ( - 1 ) ^ { ( m - 1 ) ( N + 1 ) } ( - E _ { 3 , 1 } ( s - m + 2 ) ) & , \\\\ - E _ { k + 1 , 1 } ( s + 1 ) & , \\\\ ( - 1 ) ^ { m - 1 } E _ { 2 , 1 } ( s + m ) & \\end{cases} \\end{gather*}"}
-{"id": "2615.png", "formula": "\\begin{align*} \\min \\bigg \\{ | | w | | ( \\overline \\Omega ) \\ , : \\ , w \\in \\mathcal M ^ d ( \\overline \\Omega ) , \\ , \\nabla \\cdot w = f \\bigg \\} . \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} \\norm { F } ^ 2 _ { A ^ 2 ( \\mathcal { C } _ p ) } = \\int _ { \\mathbb { B } _ p } \\int _ { V _ { \\zeta } } \\abs { F ( \\gamma , \\zeta ) } ^ 2 \\d V ( \\gamma ) \\d V ( \\zeta ) < \\infty , \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} \\frac { B ( \\tau _ { k _ j } ) } { \\tau _ { k _ j } } \\cdot \\frac { t _ { k _ j } } { B ( k _ j t _ { k _ j } ) } \\le \\frac { B ( 2 t _ { k _ j } ) } { 2 t _ { k _ j } } \\cdot \\frac { t _ { k _ j } } { B ( k _ j t _ { k _ j } ) } \\le \\frac { B ( 2 t _ { k _ j } ) } { B ( 2 t _ { k _ j } ) } \\cdot \\frac { 1 } { k _ j } = \\frac { 1 } { k _ j } \\to 0 \\quad k \\to \\infty , \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\sigma _ { 1 } ^ { j - 1 } \\sum _ { k = 0 } ^ { s - 1 } \\lambda _ { a _ { 1 , 1 } , k } ^ { j - 1 } c ^ { 1 } _ { 1 , b ( a _ { 1 , 1 } , k ) } + \\sum _ { q _ { i } \\in Q } \\sigma _ { q _ i } ^ { j - 1 } \\sum _ { k = 0 } ^ { s - 1 } \\lambda _ { a _ { 1 , 1 } , k } ^ { j - 1 } c ^ { q _ i } _ { 1 , b ( a _ { 1 , 1 } , k ) } + \\sum _ { i = 1 } ^ { | V | } \\sigma _ { v _ i } ^ { j - 1 } \\lambda _ { a _ { v _ { i , 1 } } , b _ { \\gamma _ { i } } } ^ { j - 1 } \\mu _ { v _ i , 1 , 1 } ^ { ( b ) } = 0 j \\in [ r ] . \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} w _ { i j , l } = c _ { i j } ^ { s } w _ { s l } , \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} \\mathrm M _ { t , n } \\ , : = \\ , \\frac { \\ , 1 \\ , } { \\ , n \\ , } \\sum _ { i = 1 } ^ { n } \\delta _ { ( X _ { t , i } ^ { ( u ) } , X _ { t , i + 1 } ^ { ( u ) } ) } \\ , , \\mathrm m _ { t , n } \\ , : = \\ , \\frac { 1 } { \\ , n \\ , } \\sum _ { i = 1 } ^ { n } \\delta _ { X _ { t , i } ^ { ( u ) } } \\ , , 0 \\le t \\le T \\ , , \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} x _ 1 ( t _ \\tau ( \\tau ) ) = x _ { 1 , \\tau } ( \\tau ) ; x _ 2 ( t _ \\tau ( \\tau ) ) = x _ { 2 , \\tau } ( \\tau ) , \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} \\Vert f _ h - f \\Vert _ p = \\left ( \\frac { 1 } { T } \\int _ 0 ^ T | f ( t + h ) - f ( t ) | ^ p \\mathrm { d } t \\right ) ^ { 1 / p } \\leq C h ^ \\mu , \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} \\lambda ( A ^ { 1 - t } B ^ t ) = \\lambda \\left ( A ^ { \\frac { 1 - t } { 2 } } B ^ t A ^ { \\frac { 1 - t } { 2 } } \\right ) \\prec _ { \\log } \\lambda \\left ( A ^ { \\frac { 1 - t } { 2 t } } B A ^ { \\frac { 1 - t } { 2 t } } \\right ) ^ t . \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} X \\cdot ( v _ { 1 } \\otimes \\cdots \\otimes v _ { a } ) = \\sum ^ { a } _ { k = 1 } v _ { 1 } \\otimes \\cdots \\otimes X | _ { p _ k } \\cdot v _ { k } \\otimes \\cdots \\otimes v _ { a } , \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} \\chi _ { 1 , R } = 2 - \\varphi '' _ R , \\chi _ { 2 , R } = 2 d - \\Delta \\varphi _ R . \\end{align*}"}
-{"id": "9899.png", "formula": "\\begin{align*} \\begin{aligned} \\widehat { \\boldsymbol { D } ^ { - s } u } & = \\lim _ { n \\rightarrow \\infty } \\widehat { \\boldsymbol { D } ^ { - s } { \\psi _ n } _ i } = \\lim _ { n \\rightarrow \\infty } ( 2 \\pi i \\xi ) ^ { - s } \\widehat { { \\psi _ n } _ i } \\quad , \\\\ \\widehat { \\boldsymbol { D } ^ { s * } u } & = \\lim _ { n \\rightarrow \\infty } \\widehat { \\boldsymbol { D } ^ { s * } { \\psi _ n } _ i } = \\lim _ { n \\rightarrow \\infty } ( - 2 \\pi i \\xi ) ^ { s } \\widehat { { \\psi _ n } _ i } \\quad . \\end{aligned} \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\Big ( ( x y ) \\alpha _ i ( z ) + ( z x ) \\alpha _ i ( y ) + ( y z ) \\alpha _ i ( x ) \\Big ) \\otimes \\Big ( ( a b ) \\beta _ i ( c ) + ( c a ) \\beta _ i ( b ) + ( b c ) \\beta _ i ( a ) \\Big ) = 0 . \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} a ( g ) ( f _ 1 \\otimes _ s \\cdots \\otimes _ s f _ n ) & = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } \\langle g , f _ i \\rangle f _ 1 \\otimes _ s \\cdots \\otimes _ s \\hat { f } _ i \\otimes _ s \\cdots \\otimes _ s f _ n \\\\ a ^ \\dagger ( g ) ( f _ 1 \\otimes _ s \\cdots \\otimes _ s f _ n ) & = \\sqrt { n + 1 } g \\otimes _ s f _ 1 \\otimes _ s \\cdots \\otimes _ s f _ n \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} \\tilde { L } _ 3 = \\frac { 1 } { 2 \\cos ( \\frac { 2 \\pi } { N _ 1 } ) + 2 \\cos ( \\frac { 2 \\pi } { N _ 2 } ) } L _ 3 . \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} \\varepsilon ( f ) = ( - 1 ) ^ { \\left \\lvert \\left \\{ ( y , y ' ) \\in Y \\times Y \\ \\middle \\vert \\ y < y ' f ( y ) < f ( y ' ) \\right \\} \\right \\rvert } . \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{gather*} \\Re \\int _ 1 ^ \\infty \\frac { \\d x } { \\sqrt { x ( x - 1 ) ( x - z ) } } = \\Re \\int _ 0 ^ 1 \\frac { \\d t } { \\sqrt { t ( 1 - t ) ( 1 - z t ) } } = \\pi \\ , \\Re { } _ 2 F _ 1 \\biggl ( \\begin{matrix} \\frac 1 2 , \\ , \\frac 1 2 \\\\ 1 \\end{matrix} \\biggm | z \\biggr ) , \\end{gather*}"}
-{"id": "5910.png", "formula": "\\begin{align*} K = \\sum _ { i \\le m ^ + } c _ i B _ i ^ * A _ i B _ i = A + \\sum _ { m ^ + < j \\le m } | c _ j | B _ j ^ * A _ j B _ j = \\sum _ { m ^ + < j \\le m + 1 } | c _ j | B _ j ^ * A _ j B _ j . \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} k \\cdot B = \\sum _ { i = 1 } ^ { \\nu } k _ i B _ i . \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} Y : = - \\frac 1 2 \\Bigl ( ( \\mathsf { I m } \\ , d _ { X _ 0 } ) X _ { N + 1 } + ( ( \\mathsf { I m } \\ , d _ { X _ 0 } ) X _ { N + 1 } ) ^ * \\Bigr ) = Y ^ * . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} y ( t _ { k + 1 } ) = \\tau ^ p _ k \\varphi _ p ( \\tau _ k A ) w _ p + \\sum _ { j = 0 } ^ { p - i } \\frac { \\tau ^ j _ k } { j ! } w _ j , \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} S = S ( v , \\tau ) = \\bigcup _ { k > 0 } \\left \\{ ( k , v ( \\sigma / \\tau ^ k ) ) \\mid \\sigma \\in H ^ 0 ( X , \\mathcal L ^ { \\otimes k } ) \\setminus \\{ 0 \\} \\right \\} \\subset \\mathbb { N } \\times \\mathbb { Z } ^ r , \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { B _ { R _ 0 } } | D w _ { R , n } | ^ m & \\leq \\sup _ { B _ { R _ 0 } } | f _ n | + | \\lambda | + \\sup _ { B _ { R _ 0 } } \\big | \\mathcal { L } _ R ^ { \\psi _ n } [ w _ { R , n } ] \\big | . \\end{aligned} \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} \\begin{array} { c c c c c c c } L & < & \\sum _ { r = 1 } ^ { \\infty } \\ln ( 1 - y ^ r ) & < & - \\sum _ { r = 1 } ^ { \\infty } y ^ r - ( 1 / 2 ) \\sum _ { r = 1 } ^ { \\infty } y ^ { 2 r } & & \\\\ \\\\ & = & - y / ( 1 - y ) - y ^ 2 / 2 ( 1 - y ^ 2 ) & = & - y ( 2 + 3 y ) / 2 ( 1 - y ^ 2 ) & = : & L _ 0 ~ . \\end{array} \\end{align*}"}
-{"id": "9919.png", "formula": "\\begin{align*} \\o _ { n } = \\o _ { n } ^ { 0 } \\ , \\varPhi ( z ) . \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} M _ { 2 1 } = M _ { 2 1 } \\left ( \\| u \\| _ { \\frac { p ^ * - p } { \\tilde { q } _ 1 - p } \\tilde { q } _ 1 } \\right ) \\quad M _ { 2 2 } = M _ { 2 2 } \\left ( \\| u \\| _ { \\frac { p _ * - p } { \\tilde { q } _ 2 - p } \\tilde { q } _ 2 } \\right ) . \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\Delta _ k : = - y ^ 2 \\biggl ( \\frac { \\partial ^ 2 } { \\partial x ^ 2 } + \\frac { \\partial ^ 2 } { \\partial y ^ 2 } \\biggr ) + i k y \\biggl ( \\frac { \\partial } { \\partial x } + i \\frac { \\partial } { \\partial y } \\biggr ) = - \\xi _ { 2 - k } \\circ \\xi _ k . \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} A + B + C _ { s - 1 } = \\prod _ { i = 1 } ^ { s - 1 } \\left ( q - p _ i \\right ) . \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\sqrt { 2 } B _ t - \\int _ 0 ^ t \\nabla U ( X _ s ) \\ , \\d s + \\Phi _ t , \\ t \\geq 0 , \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} = \\sum _ { r _ 1 = 1 } ^ { n } \\frac { a _ { r _ 1 } b _ { r _ 1 } } { ( r _ 1 + k + 1 ) ^ 2 } + \\sum _ { \\substack { { r _ 1 , r _ 2 = 1 } \\\\ r _ 1 \\ne r _ 2 } } ^ { n } \\frac { a _ { r _ 1 } b _ { r _ 2 } } { r _ 2 - r _ 1 } \\left ( \\frac { 1 } { r _ 1 + k + 1 } - \\frac { 1 } { r _ 2 + k + 1 } \\right ) . \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} D \\mathbf { B } = \\left ( { \\begin{array} { c c } D _ { x _ 1 } \\mathbf { B } _ 1 & D _ { x _ 1 } \\mathbf { B } _ 2 \\\\ D _ { x _ 2 } \\mathbf { B } _ 1 & D _ { x _ 2 } \\mathbf { B } _ 2 \\\\ \\end{array} } \\right ) = \\left ( { \\begin{array} { c c } \\mathbf { K } _ 1 \\star f _ 1 & \\mathbf { K } _ 2 \\star f _ 2 \\\\ \\mathbf { K } _ 0 \\star \\mu & \\mathbf { K } _ 3 \\star f _ 3 \\\\ \\end{array} } \\right ) ~ ~ ~ x = ( x _ 1 , x _ 2 ) , ~ ~ \\mathbf { B } = ( \\mathbf { B } _ 1 , \\mathbf { B } _ 2 ) , \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} \\gamma = \\sigma _ 2 \\sigma _ 1 ^ 2 \\sigma _ 2 \\sigma _ 3 \\sigma _ 4 \\sigma _ 5 w _ 1 \\sigma _ 2 \\sigma _ 3 \\sigma _ 4 \\sigma _ 3 ^ { - 1 } \\sigma _ 2 ^ { - 1 } w _ 2 \\sigma _ 2 \\sigma _ 3 \\sigma _ 2 ^ { - 1 } w _ 3 \\sigma _ 2 \\sigma _ 5 ^ { - 1 } \\sigma _ 4 ^ { - 1 } \\sigma _ 3 ^ { - 1 } \\sigma _ 1 ^ 2 . \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { L } } { \\partial \\mathbf { S } } = \\frac { \\mu \\ , \\left ( 1 - \\rho \\right ) } { \\sigma ^ 2 \\ , \\ln 2 } \\ ; \\mathbf { H } ^ { \\rm H } \\left ( \\mathbf { I } _ { N _ R } + \\left ( 1 - \\rho \\right ) \\sigma ^ { - 2 } \\mathbf { H } \\mathbf { S } \\mathbf { H } ^ { \\rm H } \\right ) ^ { - 1 } \\mathbf { H } + \\rho \\ , \\mathbf { H } ^ { \\rm H } \\mathbf { H } - \\nu \\mathbf { I } _ { N _ T } = 0 , \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} e ( t ) \\sigma = e ( t \\sigma ) . \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} \\widehat { X } ( w _ 1 , \\dots , w _ r ) & = \\overline { ( P _ { w _ 1 } \\cap G _ { w _ 1 } ) w _ 1 ( P ^ { w _ 1 } \\cap G _ { w _ 1 } ) } \\times ^ { ( P ^ { w _ 1 } \\cap G _ { w _ 1 } ) } \\widehat { X } ( w _ 2 , \\dots , w _ { r } ) \\\\ & = \\overline { B _ { I _ 1 } w _ 1 L _ { I _ 1 } } \\times ^ { B _ { I _ 1 } } \\widehat { X } ( w _ 2 , \\dots , w _ { r } ) \\\\ & = L _ { I _ 1 } \\times ^ { B _ { I _ 1 } } \\widehat { X } ( w _ 2 , \\dots , w _ { r } ) \\\\ & \\simeq P _ { I _ 1 } \\times ^ B \\widehat { X } ( w _ 2 , \\dots , w _ { r } ) . \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} ( M , s ) , \\ M = U ^ + \\cup U ^ - , \\ s | _ { U ^ { \\pm } } = s ^ { \\pm } . \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} \\langle q ^ { n ( x _ N ( t ) + N ) } \\rangle = \\frac { ( - 1 ) ^ n q ^ { \\frac 1 2 n ( n - 1 ) } } { ( 2 \\pi i ) ^ n } \\int \\prod _ { 1 \\leq j < k \\leq n } \\frac { z _ j - z _ k } { z _ j - q z _ k } \\prod _ { j = 1 } ^ n \\left ( \\prod _ { m = 1 } ^ N \\frac { a _ m } { a _ m - z _ j } \\right ) e ^ { ( q - 1 ) t z _ j } \\frac { d z _ j } { z _ j } , \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} = \\frac { 3 ( T - t ) ^ { 2 } } { 2 \\pi ^ 2 q } \\le C _ 2 \\frac { ( T - t ) ^ 2 } { q } , \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} \\begin{aligned} \\chi ( S ) + \\deg ( D | _ S ) = \\chi ( S \\cap U ) & + \\sum _ { i \\colon p _ i \\notin S } \\big ( \\chi ( S \\cap e _ i ) - \\chi ( S \\cap U \\cap e _ i ) \\big ) \\\\ & + \\sum _ { i \\colon p _ i \\in S } \\big ( \\chi ( S \\cap e _ i ) - \\chi ( S \\cap U \\cap e _ i ) + 1 \\big ) \\ , , \\end{aligned} \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} A ^ t ( T - A ) = A ^ t T - A ^ t A = & \\begin{bmatrix} 0 & n _ 1 - 1 & n _ 1 - 1 & \\ldots & n _ 1 - 1 \\\\ n _ 2 - 1 & 0 & n _ 2 - 1 & \\ldots & n _ 2 - 1 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ n _ d - 1 & n _ d - 1 & n _ d - 1 & \\dots & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} & \\widetilde \\Phi ( X \\ , | \\ , Y \\ ! = \\ ! y ) \\\\ & = \\int _ { \\R ^ n } \\rho _ { X | Y } ( x | y ) ( \\nabla _ y \\log \\rho _ { X | Y } ( x | y ) ) ( \\nabla _ y \\log \\rho _ { X | Y } ( x | y ) ) ^ \\top d x \\\\ & = - \\int _ { \\R ^ n } \\rho _ { X | Y } ( x \\ , | \\ , y ) \\ , \\nabla ^ 2 _ y \\log \\rho _ { X | Y } ( x \\ , | \\ , y ) \\ , d x \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} u _ { \\mu , \\xi } ( x , t ) = \\sum _ { j = 1 } ^ k u _ j ( x , t ) ~ \\mbox { w i t h } ~ u _ j ( x , t ) : = U _ { \\mu _ j , \\xi _ j } ( x ) + \\mu _ j ^ { \\frac { n - 2 s } { 2 } } \\Phi _ j ^ * ( x , t ) - \\mu _ j ^ { \\frac { n - 2 s } { 2 } } H ( x , q _ j ) . \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { i _ { k } } \\binom { i _ { k } } { n } ( - 1 ) ^ { n - 1 } \\zeta ^ { \\ast } _ { n } ( \\{ 1 \\} _ { b _ 1 - 1 } , w _ { 2 , r } ) = \\frac { 1 } { i _ k ^ { b _ 1 - 1 } } \\sum _ { j = 1 } ^ { i _ k } ( - 1 ) ^ { j - 1 } \\binom { i _ { k } } { j } \\zeta ^ { \\ast } _ { j } ( w _ { 2 , r } ) . \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} P ( \\log g ) = \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\left ( \\sum _ { x \\in \\mathrm { F i x } f ^ n } \\prod _ { m = 1 } ^ { n } g ( f ^ m x ) \\right ) \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} \\dot { g } = g \\xi ^ \\wedge , \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } A ^ { W , W } ( P _ N g ) = A ^ { W , W } ( g ) \\mathcal C . \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} V _ { \\mu , \\xi } = p \\sum _ { j = 1 } ^ k \\left ( ( u ^ * _ { \\mu , \\xi } ) ^ { p - 1 } - \\left [ \\mu _ j ^ { - \\frac { n - 2 s } { 2 } } U \\left ( \\frac { x - \\xi _ j } { \\mu _ j } \\right ) \\right ] ^ { p - 1 } \\right ) \\eta _ { j , R } + p ( 1 - \\sum _ { j = 1 } ^ k \\eta _ { j , R } ) ( u ^ * _ { \\mu , \\xi } ) ^ { p - 1 } . \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} & \\# \\{ ( h , k ) : \\} \\\\ & = \\sum _ { i = 1 } ^ d { n _ i \\choose 2 } = \\sum _ { i = 1 } ^ d { g _ i \\choose 2 } = \\sum _ { m \\geq 2 } { m \\choose 2 } t _ m = { d \\choose 2 } . \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} F _ m ( \\beta _ m ) = \\left ( \\frac { C _ m } { C _ m + g } \\right ) ^ { C _ m / g } \\frac { g } { C _ m + g } \\left \\{ \\begin{array} { c l } \\leq & \\frac { g } { C _ m + g } = \\frac { g } { a m ^ 2 + b m + c + g } \\\\ \\\\ = & ( e ^ { - 1 } g / a m ^ 2 ) ( 1 + o ( 1 ) ) \\end{array} \\right . \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} F ( \\mathcal { L } _ V ) g ( r , y ) = \\sum _ { \\nu \\in \\chi _ \\infty } \\sum _ { \\ell = 1 } ^ { d ( \\nu ) } \\varphi _ { \\nu , \\ell } ( y ) \\int _ 0 ^ \\infty F ( \\rho ^ 2 ) ( r \\rho ) ^ { - \\frac { n - 2 } 2 } J _ \\nu ( r \\rho ) b _ { \\nu , \\ell } ( \\rho ) \\rho ^ { n - 1 } d \\rho \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} J _ { S } ( f ^ n ) = J _ { S } ( f ^ { n q } ) = J _ { S } ( f ^ { q } ) . \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} g = ( g _ i ) _ { i \\in V ( Q ) } \\mapsto ( \\det g _ i ) ^ { \\Re ( \\xi ^ { \\pm } _ i ) } . \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} H _ \\mu ( \\xi , A ) : = H _ 0 ( \\xi , A ) + \\mu \\varphi ( v _ A ) \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} | G _ i | & \\le \\Big ( \\frac { i - 1 } { d + i - 1 - k } + \\frac { 1 } { d + i - k } \\Big ) \\prod _ { j = 1 } ^ i s _ j p _ j - \\Big ( \\frac { i - 1 } { d + i - 1 - k } - \\frac { i - 1 } { d + i - k } \\Big ) \\prod _ { j = 1 } ^ i s _ j p _ j \\\\ & = \\frac { i } { d + i - k } \\prod _ { j = 1 } ^ i s _ j p _ j . \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\mathbf { K _ { e } } f ( x ) \\| _ { 2 , - \\beta , n } & = \\| ( 1 + x ) ^ { \\beta } \\mathbf { K _ { e } } f ( n x ) \\| _ { 2 } \\\\ & \\le \\int _ { 0 } ^ { \\infty } ( 1 + x ) ^ { \\beta } ( n + n x ) ^ { - \\frac { 1 } { 2 } - \\beta - \\delta } d x \\cdot n ^ { \\frac { 1 } { 2 } + \\beta - \\delta } \\cdot \\| f \\| _ { 2 , - \\beta , n } \\\\ & = c ' n ^ { - 2 \\delta } \\| f \\| _ { 2 , - \\beta , n } . \\end{aligned} \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} m ( v ) = \\sum _ { v \\sim v ' } w ( v , v ' ) \\leq D w _ { m a x } . \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} h ( x ) = q ^ { x } \\bigg ( \\frac { x } { r } - \\frac { \\beta + n } { q - 1 } \\bigg ) + \\frac { n } { q - 1 } . \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} | X | ^ 2 : = \\dim ^ R ( X ) \\dim ^ L ( X ) = \\dim ^ R ( X ) \\dim ^ R ( X ^ * ) . \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} T ( r + s ) = T ( r ) + ( \\zeta + o ( 1 ) ) \\left ( \\frac { T ( r ) } { h ( r ) } \\right ) , \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} n \\not \\in K \\leftrightarrow \\exists m f ( n , m ) = 1 \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} \\omega _ \\eta ( x ( t ' ) ) - \\omega _ \\eta ( x ( 0 ) ) = \\int _ 0 ^ { t ' } ( \\mathcal { L } \\omega _ \\eta ) ( x ( s ) ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } \\int _ 0 ^ { t ' } \\big [ ( \\nabla \\omega _ \\eta ) ^ T \\sigma \\big ] ( x ( s ) ) \\ , d w ( s ) \\ , , \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} & \\| \\eta \\| = \\sup _ { B \\subset S } | \\eta ( B ) | , \\\\ & \\| A \\| = \\sup _ { x \\in S } \\| A ( x , \\cdot ) \\| , \\\\ & \\| f \\| = \\sup _ { x \\in S } | f ( x ) | , \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} W : = \\left \\{ \\gamma \\in \\mathcal { P } ( Y \\times U ) \\ : \\ \\int _ { Y \\times U } \\nabla \\phi ( y ) ^ T f ( u , y ) \\gamma ( d u , d y ) = 0 \\ \\ \\forall \\phi ( \\cdot ) \\in C ^ 1 \\right \\} , \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} \\Xi _ { a _ 1 , a _ 2 , b _ 1 , b _ 2 } = \\sum _ m \\kappa _ { a _ 1 , b _ 1 } \\overline { \\kappa _ { a _ 2 , b _ 2 } } , \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} T _ + ( m , t _ 0 , f , g , u _ 0 ) & = \\sup \\{ \\tau \\geq t _ 0 \\colon \\exists \\ , G _ m \\eqref { E q u a t i o n N o n l i n e a r I B V P } [ t _ 0 , \\tau ] \\} , \\\\ T _ { - } ( m , t _ 0 , f , g , u _ 0 ) & = \\inf \\{ \\tau \\leq t _ 0 \\colon \\exists \\ , G _ m \\eqref { E q u a t i o n N o n l i n e a r I B V P } [ \\tau , t _ 0 ] \\} . \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} \\bar { \\alpha } : = \\frac { \\rho ( \\sqrt { 2 } \\delta ( v ) ) - \\rho ( 2 \\sqrt { 2 } \\delta ( v ) ) } { 2 \\sqrt { 2 } \\delta ( v ) } . \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} x _ { i , j } = \\hat { p } ( n - p + 1 + 1 ) . \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{gather*} \\hat \\varPhi ( t ) = t \\hat \\varPhi ' ( t ) , \\end{gather*}"}
-{"id": "2014.png", "formula": "\\begin{align*} P ( a _ 1 + a _ 2 ) = - \\lambda a _ 2 , a _ 1 \\in A _ 1 , \\ a _ 2 \\in A _ 2 , \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} F _ { j } = \\frac { ( - 1 ) ^ { j } } { j ! } \\left ( ( \\Delta h ) ^ { j } \\right ) . \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} & \\widetilde f _ n ( 0 ^ + ) \\sigma ( 1 ) + \\widetilde f _ n ' ( + \\infty ) \\rho ( 1 ) \\\\ & \\quad - \\int _ { [ 1 / n , n ] } \\bigl \\{ \\sigma ( ( 1 - x ( s ) ) ^ * ( 1 - x ( s ) ) ) + s ^ { - 1 } \\rho ( x ( s ) x ( s ) ^ * ) \\bigr \\} ( 1 + s ) \\ , d \\widetilde \\nu _ n ( s ) \\\\ & = f _ n ( 0 ^ + ) \\rho ( 1 ) + f _ n ' ( + \\infty ) \\sigma ( 1 ) \\\\ & \\quad - \\int _ { [ 1 / n , n ] } \\bigl \\{ \\rho ( ( 1 - y ( s ) ) ^ * ( 1 - y ( s ) ) ) + s ^ { - 1 } \\sigma ( y ( s ) y ( s ) ^ * ) \\bigr \\} ( 1 + s ) \\ , d \\nu _ n ( s ) , \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} \\big | \\partial _ { \\phi } \\Psi [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] [ \\bar { \\phi } ] ( x , t ) \\big | \\lesssim \\frac { 1 } { R ^ { a - 2 s } } \\| \\bar { \\phi } ( t ) \\| _ { n - 2 s + \\sigma , a } \\left ( \\sum _ { j = 1 } ^ k \\frac { \\mu _ 0 ^ { \\frac { n - 2 s } { 2 } + \\sigma } ( t ) } { 1 + | y _ j | ^ { a - 2 s } } \\right ) . \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} ( u _ 1 , \\dots , u _ n ) : = \\exp ( u _ 1 U _ 1 + \\dots u _ n U _ n ) , \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} \\chi _ M ( x ) = \\begin{cases} 1 & | x | \\leq M \\\\ 0 & | x | \\geq M + 1 \\end{cases} \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} | z ( u _ k ) - z ( u ) | & = \\left | \\int _ { \\Omega } \\frac { | u _ k ( y ) | ^ p - | u ( y ) | ^ p } { | x - y | ^ { n - 2 } } \\ , d y \\right | \\\\ & \\leq C _ { \\Omega , p , q } \\left ( \\int _ { \\Omega } | | u _ k | ^ p - | u | ^ p | ^ q \\ , d y \\right ) ^ { 1 / q } \\\\ & \\leq C _ { \\Omega , p , q } \\left ( \\int _ { \\Omega } \\xi ^ { ( p - 1 ) q } | | u _ k | - | u | | ^ q \\ , d y \\right ) ^ { 1 / q } , \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} \\kappa ( \\epsilon ) = \\mu ( I - e ^ { - \\alpha } \\tilde { { P } } ) ^ { - 1 } r + \\epsilon e ^ { - \\alpha } \\mu \\sum _ { k = 0 } ^ { \\infty } a _ { k + 1 , 1 } e ^ { - \\alpha k } \\tilde { P } ^ k \\tilde { P } ^ { ( 1 ) } ( I - e ^ { - \\alpha } \\tilde { P } ) ^ { - 1 } r + O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ w ) \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} H = - \\mathbf { i } \\int _ { \\mathbb { T } } u _ x \\bar { u } d x + \\frac 1 2 \\int _ { \\mathbb { T } } | u | ^ 4 d x + \\int _ { \\mathbb { T } } F _ { \\geq 5 } ( x , u , \\bar { u } ) d x . \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} e _ n ( q ) = F _ { n + 1 } ( q ) + G _ { n - 1 } ( q ) \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} P _ { j } = \\int _ { a ^ { - } _ { j } } ^ { a ^ { + } _ { j } } P ( y _ { B } ) d y _ { B } \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} K ^ { - 1 } \\Big \\| \\sum _ { i = 1 } ^ { \\infty } c _ i f _ { k _ i } \\Big \\| _ E \\le \\Big \\| \\sum _ { i = 1 } ^ { \\infty } c _ i u _ { k _ i } \\Big \\| _ E \\le K \\Big \\| \\sum _ { i = 1 } ^ { \\infty } c _ i f _ { k _ i } \\Big \\| _ E , \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} | N _ t ( u ) | & \\le 1 + d + d ( d - 1 ) + d ( d - 1 ) ^ 2 + \\dots + d ( d - 1 ) ^ { t / 2 - 1 } - 1 \\\\ & = \\tau _ t ( d ) - 1 . \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\frac { \\alpha _ j ^ { L _ 0 - k _ j - h } } { ( L _ 0 - k _ j - h ) ! } , k _ j = 0 , \\ldots , l _ j - 1 , j = 1 , \\ldots , m , h = 0 , 1 , \\dots , L \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = - \\frac { d } { d x } \\frac { \\partial H } { \\partial \\bar { u } } \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} S _ i \\circ S _ j \\circ S _ i \\circ S _ j \\circ \\cdots = S _ j \\circ S _ i \\circ S _ j \\circ S _ i \\circ \\cdots , \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} u ( t , x ) - u ( 0 , x ) & = \\int _ 0 ^ \\infty e ^ { - \\lambda s } P ^ \\Omega _ s P ^ \\beta _ s ( g - g ( 0 ) ) ( t , x ) \\ , d s \\\\ & = \\int _ 0 ^ \\infty e ^ { - \\lambda s } P ^ \\Omega _ s P ^ { \\beta , } _ s ( g - g ( 0 ) ) ( t , x ) \\ , d s \\in ( \\mathcal L _ { \\beta , \\Omega } ^ { } ) , \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} I \\otimes L _ \\xi ( \\lambda , \\mu ) \\simeq L _ \\xi \\left ( \\lambda + \\sum _ { i = 1 } ^ n ( \\eta _ { i - 1 } - \\eta _ i ) \\varpi _ i + d \\varpi _ 1 , \\mu + \\sum _ { i = 1 } ^ n ( \\eta _ { i - 1 } - \\eta _ i ) \\varpi _ i - d \\varpi _ 1 \\right ) . \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} \\overline { C } { ( d , \\epsilon ) } = \\sup _ { P ( D ( t ) > d ) \\le \\epsilon , \\forall t } E \\left [ \\frac { A ( t ) } { t } \\right ] . \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} K _ V = p . \\partial _ q - \\partial _ q V ( q ) . \\partial _ p + \\frac { 1 } { 2 } ( - \\Delta _ p + p ^ 2 ) , ( q , p ) \\in \\mathbb { R } ^ { 2 d } \\ , , \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} \\nabla _ \\xi G ( x , \\xi ) = ( G _ 0 ) _ t ' ( x , | \\xi | ) \\frac { \\xi } { | \\xi | } = \\mathcal { A } _ 0 ( x , | \\xi | ) \\xi = \\mathcal { A } ( x , \\xi ) \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} \\phi b ^ { m _ 0 } x b ^ { n _ k } & \\sim \\phi b ^ { m _ 0 } x b ^ { n _ k + 1 } - \\sum _ { s \\in S _ b } \\phi b ^ { m _ 0 } x b ^ { n _ k + 1 } s \\\\ & \\sim . . . \\\\ & \\sim \\phi b ^ { m _ 0 } x b ^ { n _ k + ( m _ k - n _ k ) } - \\sum _ { i = 1 } ^ { m _ k - n _ k } \\sum _ { s \\in S _ b } \\phi b ^ { m _ 0 } x b ^ { n _ k + i } s \\\\ & = \\phi b ^ { m _ 0 } x b ^ { m _ k } - \\sum _ { i = n _ k + 1 } ^ { m _ k } \\sum _ { s \\in S _ b } \\phi b ^ { m _ 0 } x b ^ { i } s . \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} l = s \\biggl ( \\prod _ { \\substack { i = 1 \\\\ [ . 0 2 i n ] p _ i \\equiv 1 s } } ^ n p _ i \\biggr ) , \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} J _ w = \\sum _ { i } ^ w N ( i ) T _ i . \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} f ( X + Y ) = f ( X ) f ( Y ) h ( X , Y ) . \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} \\tau ( z ) = z + \\mu . \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} E ( G ) = 5 i + 1 + 2 i + 1 + \\sum _ { i = 1 } ^ 3 | \\lambda _ i | \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} L ( s , \\pi _ p ) = \\begin{cases} L ( s , \\abs { \\cdot } _ p ^ { \\frac { 1 } { 2 } } ) & \\chi = 1 , \\\\ 1 & \\chi \\neq 1 , \\end{cases} \\epsilon ( \\frac { 1 } { 2 } , \\pi _ p ) = \\begin{cases} - 1 & \\chi = 1 , \\\\ \\epsilon ( \\frac { 1 } { 2 } , \\chi ) ^ 2 & \\chi \\neq 1 . \\end{cases} \\end{align*}"}
-{"id": "9922.png", "formula": "\\begin{align*} \\psi _ { 1 } ^ { ( 0 , 0 ) } = \\psi _ { 1 } = \\frac 1 2 \\frac y { | y | } . \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} \\partial ^ n f ( g _ 0 , . . . , g _ n ) = \\ & g _ 0 f ( g _ 1 , . . . , g _ n ) + ( - 1 ) ^ { n + 1 } f ( g _ 0 , . . . , g _ { n - 1 } ) \\\\ & - \\sum _ { j = 0 } ^ { n - 1 } ( - 1 ) ^ { j + 1 } f ( g _ 0 , . . . , g _ j g _ { j + 1 } , . . . , g _ n ) . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} Q _ \\mu ^ \\pi ( Z ) : = E _ { \\mu } ^ \\pi \\left ( \\sum _ { n = 1 } ^ \\infty \\beta ^ { n - 1 } 1 _ { Z } ( s _ n , a _ n ) \\right ) , Z \\in { \\cal B } ( S \\times A ) . \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} \\Psi _ z ( y ) = 0 \\ \\forall \\ y \\in Y _ z , \\ \\ \\ \\ \\ \\ \\Psi _ z ( y ) > 0 \\ \\forall \\ y \\in Y / Y _ z \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{gather*} \\partial _ s u ( s , t ) + J ' _ { s , t } ( \\partial _ t u ( s , t ) - X _ { K _ { s } , t } ( u ( s , t ) ) ) = 0 \\\\ \\lim _ { s \\rightarrow \\pm \\infty } u ( s , t ) = x _ { \\pm } ( t ) , \\ \\ \\ \\ \\ [ v _ - \\sharp u , x _ + ] = [ v _ + , x _ + ] \\end{gather*}"}
-{"id": "3003.png", "formula": "\\begin{align*} - \\Delta Q _ \\omega - c | x | ^ { - 2 } Q _ \\omega + \\omega Q _ \\omega = | Q _ \\omega | ^ { \\frac { 4 } { d } } Q _ \\omega . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} B _ { \\pm } = S _ 1 - S _ 1 ( M ^ \\pm ( \\lambda ) + S _ 1 ) ^ { - 1 } S _ 1 \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} p ^ * = \\begin{cases} \\frac { N p } { N - p } & p < N , \\\\ q \\in ( 1 , \\infty ) & p \\geq N . \\end{cases} \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} \\hat r _ { \\eta , \\eta ^ { i , i ' } } ^ V = g _ 1 ( \\eta ( i ) ) g _ 2 ( \\eta ( i ' ) ) \\mathrm e ^ { - \\frac 1 2 ( V ( i ' / L ) - V ( i / L ) ) } . \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} { \\rm g c d } \\left [ p _ 1 ( a _ 1 r _ 1 + s _ 1 ) , a _ 2 r _ 2 + s _ 2 \\right ] = 1 \\ , . \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { n - 1 } ( 1 + t q ^ i ) = \\sum _ { k = 0 } ^ { n } { n \\brack k } _ { q } t ^ k q ^ { \\binom { k } { 2 } } . \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} 2 n ( n + & \\alpha + \\beta ) ( 2 n + \\alpha + \\beta - 2 ) P _ { n } ^ { ( \\alpha , \\beta ) } ( x ) \\\\ & = ( 2 n + \\alpha + \\beta - 1 ) \\left ( ( 2 n + \\alpha + \\beta ) ( 2 n + \\alpha + \\beta - 2 ) x + \\alpha ^ 2 - \\beta ^ 2 \\right ) P _ { n - 1 } ^ { ( \\alpha , \\beta ) } ( x ) \\\\ & \\phantom { = } - 2 ( n + \\alpha - 1 ) ( n + \\beta - 1 ) ( 2 n + \\alpha + \\beta ) P _ { n - 2 } ^ { ( \\alpha , \\beta ) } ( x ) \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{gather*} \\mathrm { c s } \\big ( \\overline { \\nabla } ^ { \\mathrm { b a s } } , \\overline { \\nabla } ^ { \\mathrm { b a s } , \\overline { g } } \\big ) = p ^ * \\mathrm { c s } \\big ( \\nabla ^ { \\mathrm { b a s } } , \\nabla ^ { \\mathrm { b a s } , g } \\big ) , \\end{gather*}"}
-{"id": "1025.png", "formula": "\\begin{align*} I ' ( \\alpha u ^ { + } + \\beta u ^ { - } ) [ \\alpha u ^ { + } ] = I ' ( \\alpha u ^ { + } ) [ \\alpha u ^ { + } ] + \\alpha ^ { 2 } \\beta ^ { 2 } \\lambda \\int _ { \\mathbb R ^ { 3 } } \\phi _ { u ^ { + } } ( u ^ { - } ) ^ { 2 } d x \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} \\int _ G f ( x ) d \\tilde { \\mu } ( x ) = \\int _ { K \\backslash G / H } Q ( f ) ( \\ddot { x } ) d \\mu ( \\ddot { x } ) , \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} = : \\sum _ { k \\geq p - 1 , \\ ; k \\in \\mathbb { N } } I _ { 1 , k } + \\sum _ { k \\geq p - 1 , \\ ; k \\in \\mathbb { N } } I _ { 2 , k } = : I _ { 1 } + I _ { 2 } . \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} \\varphi _ \\pm ^ { - 1 } : \\begin{pmatrix} x ^ 1 \\\\ x ^ 2 \\\\ x ^ 3 \\\\ x ^ 4 \\\\ \\end{pmatrix} \\mapsto \\begin{pmatrix} x ^ 1 \\\\ x ^ 2 \\\\ x ^ 3 \\\\ x ^ 4 \\\\ \\end{pmatrix} - a \\begin{pmatrix} \\cos ( x ^ 3 + x ^ 4 ) \\\\ \\pm \\sin ( x ^ 3 + x ^ 4 ) \\\\ 0 \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} P _ r ( x ) = \\sum a _ i x b _ i \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N a _ i ^ { s } \\ , \\leq \\ , N ^ { 1 - s } \\left ( \\sum _ { i = 1 } ^ N a _ i \\right ) ^ { s } \\ , \\leq \\ , N ^ { 1 - s } \\sum _ { i = 1 } ^ N a _ i ^ { s } . \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} \\sum _ { v ' \\in V } p _ t ( v , v ' ) = 1 \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} \\mathfrak { g } _ { p } [ { \\Sigma } ^ o ] ^ \\Gamma : = \\left ( \\mathfrak { g } \\otimes \\mathbb { C } _ { p } [ { \\Sigma } ^ o ] \\right ) ^ \\Gamma , \\ , \\ , \\ , \\ , \\ , \\mathfrak { g } _ { p '' } [ { \\Sigma } '^ o ] ^ \\Gamma : = \\left ( \\mathfrak { g } \\otimes \\mathbb { C } _ { p '' } [ { \\Sigma } '^ o ] \\right ) ^ \\Gamma . \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} ( \\frac { \\partial \\xi } { \\partial \\zeta } ) ^ { - 1 } = \\frac { 4 \\pi } { 2 n - 1 } \\mbox { d i a g } ( j _ b ^ { - 1 } : 1 \\leq b \\leq n ) \\begin{pmatrix} \\frac 3 2 - n & 1 & \\cdots & 1 \\\\ 1 & \\frac 3 2 - n & \\cdots & 1 \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ 1 & 1 & \\cdots & \\frac 3 2 - n \\end{pmatrix} . \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} \\Gamma ( m z ) = ( 2 \\pi ) ^ { \\frac { ( 1 - m ) } { 2 } } m ^ { m z - \\frac { 1 } { 2 } } \\prod _ { j = 1 } ^ { m } \\Gamma \\left ( z + \\frac { j - 1 } { m } \\right ) \\newline , ~ ~ ~ m z \\in \\mathbb { C } \\backslash \\mathbb { Z } _ { 0 } ^ { - } \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} \\mu = \\sum _ { i = 1 } ^ n \\mu _ i \\alpha _ i \\lambda = - \\mu + 2 \\sum _ { i = 1 } ^ n \\eta _ i \\varpi _ i \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} \\eqref { G f o u r i e r 5 } = \\int _ { x _ 0 } ^ x \\frac { c _ 0 } { 2 } \\frac { \\sin ( 2 \\tilde { \\theta } ( y , { E } ) - 2 \\tilde { \\theta } ( y , \\hat { E } ) ) } { ( 1 + y - b ) } d x + \\sum _ { k = 1 } ^ \\infty c _ k \\cos ( 2 \\pi k y ) \\frac { \\sin ( 2 \\tilde { \\theta } ( y , { E } ) - 2 \\tilde { \\theta } ( y , \\hat { E } ) ) } { ( 1 + y - b ) } d y \\\\ + \\sum _ { k = 1 } ^ \\infty d _ k \\sin ( 2 \\pi k y ) \\frac { \\sin ( 2 \\tilde { \\theta } ( y , { E } ) - 2 \\tilde { \\theta } ( y , \\hat { E } ) ) } { ( 1 + y - b ) } d y . \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} & \\mathbf { T } ^ l _ { \\varepsilon _ 1 , i , j } ( \\mu _ l ) ( x ) = \\sup _ { \\rho \\in ( 0 , 2 R ) , e \\in S ^ { d - 1 } } \\frac { \\varepsilon _ 1 ^ { - d + 1 } } { \\rho } \\left | \\left ( \\frac { 1 } { | . | ^ { d - 1 } } \\Theta ^ { \\varepsilon _ 1 , e } _ { l , \\rho } ( . ) \\right ) \\star \\mathbf { K } _ j ^ i \\star \\mu _ l ( x ) \\right | ~ ~ \\forall ~ ~ x \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} Z _ p ^ C ( t ) \\equiv \\frac { t ^ 2 } { 1 - t ^ 2 } \\equiv \\sum _ { m = 1 } ^ { \\infty } t ^ { 2 m } \\pmod { p } . \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} G ( s ) : = \\int _ { \\wp } s ^ { Q - 1 } h ( s y ) ( \\psi _ { 3 } ( s y ) ) ^ { 1 - p ' } d \\sigma ( y ) \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{gather*} T _ { t _ { - \\alpha _ 2 } } ( E _ { 2 , 1 } ( 1 ) ) = T _ 2 T _ 3 \\cdots T _ 0 T _ 1 T _ 0 ( E _ { 2 , 1 } ( 1 ) ) . \\end{gather*}"}
-{"id": "7400.png", "formula": "\\begin{align*} Y \\cdot \\Psi = 0 . \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{gather*} G _ s ( t , x ) = L \\sharp F _ s ( t , s ) \\\\ J '' _ s = ( \\phi _ L ^ t ) _ * ^ { - 1 } J ' _ s ( \\phi _ L ^ t ) _ * \\end{gather*}"}
-{"id": "198.png", "formula": "\\begin{align*} a _ { 0 } ^ { ( \\alpha , \\beta ) } = \\frac { 2 } { \\alpha + \\beta + 2 } \\sqrt { \\frac { ( \\alpha + 1 ) ( \\beta + 1 ) } { ( \\alpha + \\beta + 3 ) } } , \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} Y ^ { + } > _ K Y ^ - , \\ ( \\mbox { r e s p . } ~ Y ^ { + } = _ K Y ^ - ) . \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} \\left < g ( t _ { j k } ) \\right > = \\left < e ^ { t _ { j k } } [ 1 + \\frac { 1 } { 2 } ( \\phi _ j - \\phi _ k ) ^ 2 ] \\right > - 1 . \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} \\mathbb { P } [ x _ N ( t ) + N = \\l _ N ] = P _ t ( \\l _ N ) , \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} W ( \\tilde { \\lambda } ^ 2 \\tilde { \\Pi } + \\tilde { Q } ^ 2 ) = \\tilde { Q } ^ { ( 1 ) } . \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} \\left | z \\frac { f ' ( z ) } { f ( z ) } - \\frac { 1 - A \\overline { B } r ^ 2 } { 1 - | B | ^ 2 r ^ 2 } \\right | \\leq \\frac { | B - A | r } { 1 - | B | ^ 2 r ^ 2 } \\quad ( | z | = r < 1 ) . \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\partial _ { t } W ( t , x ) + \\inf \\limits _ { u \\in U } H ( t , x , W ( t , x ) , D W ( t , x ) , D ^ { 2 } W ( t , x ) , u ) = 0 , \\\\ W ( T , x ) = \\phi ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} g \\equiv \\sum _ { j = 1 } ^ { M } f _ j \\otimes ( x _ j - \\bar x ) + f _ { M + 1 } \\otimes \\sum _ { j = 1 } ^ M \\beta _ j ( x _ j - \\bar x ) , \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} \\mbox { $ \\lambda _ i $ i s L i p s c h i t z o n $ [ 0 , 1 ] $ f o r $ 1 \\le i \\le n \\ , ( = k + m ) $ . } \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} \\begin{aligned} & \\Re \\Bigg ( \\sum \\limits _ { \\substack { \\rho \\\\ | \\Im ( \\rho ) | \\le N } } \\left ( 1 - \\left ( \\frac { \\rho } { \\rho - \\tau } \\right ) ^ n \\right ) \\Bigg ) < \\sum \\limits _ { \\substack { \\rho \\\\ | \\Im ( \\rho ) | \\le N } } 1 - \\frac { 1 } { 2 0 } R ^ n \\end{aligned} \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} ( \\delta _ n * \\delta _ m ) ( C _ i ) = \\sum _ { \\underset { ( l _ 1 - l _ 0 ) 4 ^ k + 2 j = 2 u ( 4 ^ k ) } { 0 \\leq j \\leq l _ 0 4 ^ k + 2 u } } \\frac { n - m + 2 j + 1 } { ( n + 1 ) ( m + 1 ) } . \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} p _ ! f ( a ) = \\delta ( a ) O _ a ( \\Phi ) d a , \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } p _ n \\\\ q _ n \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & a _ 1 \\end{array} \\right ] \\left [ \\begin{array} { c } \\hat { p } _ { n - 1 } \\\\ \\hat { q } _ { n - 1 } \\end{array} \\right ] \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} \\mathbb { J } _ { s , n } ( \\alpha ) = \\alpha I _ { s n } + \\mathbb { J } _ { s , n } ( 0 ) . \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X _ { s } ^ { m } = & b ( s , X _ { s } ^ { m } , Y _ { s } ^ { m } , Z _ { s } ^ { m } , u _ { s } ^ { m } ) d s + \\sigma ( s , X _ { s } ^ { m } , u _ { s } ^ { m } ) d B _ { s } , \\\\ d \\tilde { Y } _ { s } ^ { m } = & - g ( s , X _ { s } ^ { m } , \\tilde { Y } _ { s } ^ { m } , \\tilde { Z } _ { s } ^ { m } , u _ { s } ^ { m } ) d s + \\tilde { Z } _ { s } ^ { m } d B _ { s } , s \\in \\lbrack t , T ] , \\\\ X _ { t } ^ { m } = & x , \\ \\tilde { Y } _ { T } ^ { m } = \\phi ( X _ { T } ^ { m } ) . \\end{array} \\right . \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\mathbb { A ' } ^ { - 1 } _ L = \\mathbb { A } ^ { - 1 } _ L = \\mathbb { A } _ \\mathcal { G } ^ { - 1 } . \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} S _ { j _ 0 } = 1 , S _ { j _ 1 } = - 1 , \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} \\mathfrak { C } _ { \\square _ c } ( K ) ( x , y ) : = \\underset { T \\to K \\in ( N e c \\downarrow K ) _ { x , y } } { } [ C _ { \\square _ c } ( T ) ( \\alpha _ T , \\omega _ T ) ] , \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} 0 < \\norm { u _ \\lambda } ^ 2 = \\int _ { \\mathbb R ^ 3 } \\left ( \\frac { f ( u _ \\lambda ) } { u _ \\lambda ^ 3 } - \\lambda \\frac { \\phi _ \\lambda } { u _ \\lambda ^ 2 } \\right ) u _ \\lambda ^ 4 \\ , d x < \\int _ { \\mathbb R ^ 3 } \\left ( 1 - \\lambda \\frac { \\phi _ \\lambda } { u _ \\lambda ^ 2 } \\right ) u _ \\lambda ^ 4 \\ , d x = \\int _ { \\mathbb R ^ 3 } \\left ( u _ \\lambda ^ 4 - \\lambda \\phi _ \\lambda u _ \\lambda ^ 2 \\right ) d x \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} [ u , \\varphi ] = ( \\nabla _ g u , \\nabla _ g \\varphi ) _ g ( \\forall u , \\varphi \\in H ^ 1 _ 0 ( \\bar { M } ) ) . \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} \\partial _ x u ( t , x ) = \\partial _ x w ( t , x ) - \\int _ 0 ^ x \\partial _ x K ( x , y ) w ( t , y ) \\ , d y - K ( x , x ) w ( t , x ) . \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} \\mathcal Q _ { | \\bigcap _ { i = 1 } ^ { m ^ + } \\ker B _ i } \\mbox { i s p o s i t i v e d e f i n i t e a n d } \\dim H \\ge s ^ + ( \\mathcal Q ) + \\dim H _ 1 + \\cdots + \\dim H _ { m ^ + } . \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} \\underline { P } ^ { \\mathrm { s o l a r } } \\big ( z [ n ] \\big ) = \\frac { C _ 1 } { 1 + e ^ { - k _ { c } ( z [ n ] - \\alpha ) } } + C _ 2 , \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } Q ( f ) ( \\ddot { n } ) d \\mu _ 1 ( \\ddot { n } ) & = \\int _ N f ( n ) \\rho _ 1 ( n ) d n \\\\ & = \\int _ N f ( n ) \\big ( \\frac { \\rho _ 1 ( n ) } { \\rho _ 2 ( n ) } \\big ) \\rho _ 2 ( n ) d n \\\\ & = \\int _ { K \\backslash G / H } Q ( f ) ( \\ddot { n } ) \\varphi ( \\ddot { n } ) d \\mu _ 2 ( \\ddot { n } ) . \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} \\lambda _ { \\mu _ p \\pi _ p } ( p ^ m ) = \\begin{cases} \\delta _ { m = 0 } & \\mu _ p \\neq \\omega _ { \\pi , p } ^ { - 1 } , \\\\ \\chi _ 2 ( p ^ m ) \\delta _ { m \\geq 0 } & \\mu _ p = \\omega _ { \\pi , p } ^ { - 1 } . \\end{cases} \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} A ^ { W , W } _ { \\cdot \\wedge \\tau _ M } ( g _ M ) = A ^ { W , W } _ { \\cdot \\wedge \\tau _ M } ( g ) . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} \\alpha \\Vert y \\Vert _ U = & \\Vert x - z ( \\alpha ) \\Vert _ U \\\\ \\leq & K _ \\rho \\Vert F ( x ) - F ( z ( \\alpha ) ) \\Vert _ V \\\\ \\leq & K _ \\rho ( \\alpha \\Vert D F ( x ) y \\Vert _ V + \\Vert F ( x + \\alpha y ) - F ( x ) - D F ( x ) \\alpha y \\Vert _ V ) . \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} a ^ \\tau & : = \\tau \\tilde { a } + ( 1 - \\tau ) a + \\sum _ { i = - + } \\chi _ { u _ i } \\mu _ { u _ i } ( \\tau ) [ u - { u _ i } ( x ) ] \\\\ f ^ \\tau ( x , u , u _ x ) & : = \\tau \\tilde { f } + ( 1 - \\tau ) f + \\sum _ { i = - + } \\chi _ { u _ i } \\mu _ { u _ i } ( \\tau ) [ u - { u _ i } ( x ) ] \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { ( - 1 ) ^ { n j } \\binom k j \\binom { k - j } s \\frac { { G _ { m - ( n + 1 ) k - ( n - 1 ) j + 2 s } } } { { F _ n ^ j } } } } = \\frac { { G _ m } } { { F _ n ^ k } } , n \\ne 0 \\ , , \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} \\sigma _ 1 \\cdot \\sigma _ 4 = \\deg g = 1 2 \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} d X _ t = b ( X _ t ) \\ , d t + \\sigma ( X _ t ) \\ , d B _ t , \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} g \\big ( \\nabla \\phi ( z ( s ) ) , \\dot z ( s ) \\big ) = 0 , s \\in ] 0 , 1 [ \\phi \\big ( z ( s ) \\big ) = 0 , \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} \\mathsf { r e c t s } | _ { \\mathcal { P } } ( T _ \\mathcal { S } ) = \\mathsf { r e c t s } | _ { \\mathcal { P } } ( T _ \\mathcal { C } ) . \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} x ^ 2 = p ^ 2 + q ^ 2 \\ , , \\ \\ y ^ 2 = a _ 2 ^ 2 ( p ^ 2 + q ^ 2 - 2 b _ 2 p + b _ 2 ^ 2 ) \\ , , \\ \\mbox { a n d } \\ z ^ 2 = a _ 3 ^ 2 ( p ^ 2 + q ^ 2 - 2 b _ 3 p + b _ 3 ^ 2 ) \\ , . \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} ( \\nabla ( y - \\mathfrak { Q } _ { \\mathfrak { h } } y ) , \\nabla y _ { \\mathfrak { h } } ) = 0 , ~ \\forall y _ { \\mathfrak { h } } \\in V _ { \\mathfrak { h } } , \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} I ( v _ n ) \\leq I ( u _ n ) = l + o _ n ( 1 ) . \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} & e _ 1 = d x _ { 1 } , e _ 2 = - d x _ { 2 } , e _ 3 = d x _ { 3 } - x _ 1 d x _ 2 - d x _ 4 - x _ 2 d x _ 1 = d ( x _ 3 - x _ 4 - x _ 1 x _ 2 ) , \\\\ & e _ 4 = d x _ 4 + x _ 2 d x _ 1 , e _ 5 = d x _ { 5 } + x _ { 1 } d x _ { 4 } , e _ 6 = d x _ { 6 } - x _ 2 d x _ { 3 } \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} 0 \\leq \\eta _ n \\leq 1 , | \\eta _ n ' ( s ) | \\leq \\frac { C } { n } , \\quad \\begin{cases} \\eta _ n ( s ) = 1 , & | s | \\leq n , \\\\ \\eta _ n ( s ) = 0 , & | s | > 2 n \\end{cases} \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} \\omega ( m , n ) = m ^ 5 + \\frac { n ^ 2 } { m } , \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} \\begin{cases} \\tau ( 0 ) = \\tau ( 1 ) = 1 \\\\ w ( i ) = ( i ) \\otimes ( i ) i \\in \\{ 0 , 1 \\} \\\\ w ( 0 1 ) = ( 0 ) \\otimes ( 0 1 ) + ( 0 1 ) \\otimes ( 1 ) \\ . \\end{cases} \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} ( x , y ) . a & : = ( x , x ( a \\theta ) + y ) & ( a \\in M , x \\in V , y \\in W ) \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { P } \\left \\lbrace \\sup _ { t \\in [ 0 , T ] } \\left \\lvert X ^ { n , r } _ t - X ^ { r } _ t \\right \\rvert \\ge \\varepsilon \\right \\rbrace = 0 , \\lim _ { n \\to \\infty } \\tilde { \\mathbb { P } } \\left \\lbrace \\sup _ { t \\in [ 0 , T ] } \\left \\lvert Y ^ { n , r } _ t - Y ^ { r } _ t \\right \\rvert \\ge \\varepsilon \\right \\rbrace = 0 . \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} \\sum _ { i = \\ell } ^ L T _ { { \\rm S V C } , n , i } ^ * = \\begin{cases} \\frac { 1 } { D _ 1 ( \\tau _ C ) } \\sqrt { \\frac { a _ { n } b _ { n , 1 } D _ 2 ( \\tau _ C ) } { v ^ * s _ 1 + \\eta _ n ^ * } } - \\frac { D _ 2 ( \\tau _ C ) } { D _ 1 ( \\tau _ C ) } , & \\ell = 1 \\\\ \\frac { 1 } { D _ 1 ( \\tau _ C ) } \\sqrt { \\frac { a _ { n } b _ { n , \\ell } D _ 2 ( \\tau _ C ) } { v ^ * s _ \\ell } } - \\frac { D _ 2 ( \\tau _ C ) } { D _ 1 ( \\tau _ C ) } , & \\ell \\in \\{ 2 , \\cdots , L \\} \\end{cases} . \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} x = \\sum _ { j \\geq 0 , \\ j \\geq - \\langle \\lambda , \\alpha _ i ^ \\vee \\rangle } \\frac { 1 } { [ j ] ! _ i } F _ i ^ j x _ j , \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\mathfrak v : = \\nu \\mathfrak u \\eta \\ \\ e _ t : = t \\mathfrak v , t > 1 . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ \\pi f _ { \\alpha , \\beta } ( x ) e ^ { - i k x } \\ , \\mathrm { d } x & = \\frac { 1 } { \\pi } \\int _ 0 ^ \\pi f _ { \\alpha , \\beta } ( x ) \\cos ( k x ) \\\\ & = \\frac { 1 } { i 4 \\pi } \\int _ 0 ^ \\pi x ^ \\alpha \\big ( e ^ { i x ^ { - \\beta } } - e ^ { - i x ^ { - \\beta } } \\big ) \\big ( e ^ { i k x } + e ^ { - i k x } \\big ) \\ , \\mathrm { d } x , \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| \\beta _ k ( v ^ k - q ) + ( 1 - \\beta _ k ) ( T u ^ k - q ) \\| = \\lim _ { k \\to \\infty } \\| x ^ { k + 1 } - q \\| = \\tau . \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} s \\cdot ( \\vec { x } , \\vec { y } ) = ( \\vec { x } , s \\vec { y } ) , \\ s \\cdot ( \\vec { x } , \\vec { y } ) = ( s \\vec { x } , \\vec { y } ) , \\ s \\cdot ( \\vec { x } , \\vec { y } , t ) = ( \\vec { x } , \\vec { y } , s t ) \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| \\nabla v _ n \\| _ { L ^ 2 } = \\| \\nabla \\tilde { V } ^ { j _ 0 } \\| _ { L ^ 2 } , \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi ^ 0 _ j ( q _ j , t ) & = - ( 1 + o ( 1 ) ) \\int _ { t _ 0 } ^ { t } \\frac { \\dot { \\mu } _ j ( \\tilde { s } ) } { \\mu _ j ( \\tilde { s } ) } \\mu _ j ^ { - ( n - 2 s ) } ( \\tilde { s } ) F \\left ( \\frac { ( t - \\tilde { s } ) ^ { \\frac { 1 } { 2 s } } } { \\mu _ j ( \\tilde { s } ) } \\right ) d \\tilde { s } \\\\ & = \\frac { 2 s b _ j ^ { 4 s - n } } { ( n - 4 s ) c _ { n , s } ^ { n - 4 s } } A + o ( 1 ) : = B b _ j ^ { 4 s - n } + o ( 1 ) \\end{aligned} \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} J _ \\nu ( r ) = \\frac { ( r / 2 ) ^ \\nu } { \\Gamma ( \\nu + 1 / 2 ) \\pi ^ { 1 / 2 } } \\int _ { - 1 } ^ 1 e ^ { i r \\theta } ( 1 - \\theta ^ 2 ) ^ { \\nu - 1 / 2 } d \\theta , \\ \\ \\nu > - 1 / 2 . \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} \\left \\| R ( \\epsilon ; t - s u _ { ( n ) } ) \\cdots R ( \\epsilon ; t - s u _ { ( 1 ) } ) - \\left ( \\tilde { R } ^ n - \\epsilon \\sum _ { i = 0 } ^ { n - 1 } s u _ { ( i + 1 ) } \\tilde { R } ^ { n - i - 1 } \\tilde { R } ^ { ( 1 ) } \\tilde { R } ^ { i } \\right ) \\right \\| = O ( \\epsilon ^ 2 s ^ 3 ) \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} ( \\alpha , \\alpha ^ \\sigma ) _ 2 = 1 \\implies ( \\beta , \\beta ^ \\sigma ) _ 2 = 1 . \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} \\begin{aligned} W _ i ^ { ( 1 ) } & : = \\Big \\{ \\beta ^ { u _ i t _ i } \\alpha _ i ^ { u _ i + q s _ i } : u _ i = 0 , 1 , \\dots , s _ i - 1 ; q = 0 , 1 , \\dots , \\frac { p _ i - 1 } { s _ i } - 1 \\Big \\} , \\\\ W _ i ^ { ( 2 ) } & : = \\Big \\{ \\sum _ { u _ i = 0 } ^ { s _ i - 1 } \\beta ^ { u _ i t _ i } \\alpha _ i ^ { p _ i - 1 } \\Big \\} , \\\\ W _ i & : = W _ i ^ { ( 1 ) } \\cup W _ i ^ { ( 2 ) } . \\end{aligned} \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} f ( z ) = 1 + z ^ { \\mu _ 1 } + z ^ { \\mu _ 2 } + \\cdots + z ^ { \\mu _ M } \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} A _ { r + 1 } & = \\tfrac 1 2 ( r + 1 ) ( r + 2 ) , \\\\ B _ { r + 1 } & = \\tfrac 1 6 ( r + 1 ) ( r + 2 ) ( 2 r + 3 ) , \\\\ C _ { r + 1 } & = - \\tfrac 1 8 ( r + 1 ) ( r + 2 ) ( r ^ 2 - r - 4 ) . \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} \\mu ^ { q - 1 } < \\frac { q } { 2 ^ q \\hat { L } ^ q G _ q } . \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { s - 1 } \\lambda _ { 1 , k } ^ { j - 1 } c ^ { I } _ { 1 , b ( 1 , k ) } + \\sum _ { i = 2 } ^ { i = n } \\lambda _ { i , b _ { i } } ^ { j - 1 } \\sum _ { k = 0 } ^ { s - 1 } c ^ { I } _ { i , b ( 1 , k ) } = 0 j \\in [ r ] . \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ n | S _ m ( \\omega ) | ^ 2 \\geq 2 n ^ 2 . \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} G ( e _ t ) & = t ^ 2 \\left ( \\norm { \\mathfrak v } ^ 2 + \\int _ { \\mathbb R ^ 3 } \\mathfrak v ^ 2 d x \\right ) + \\lambda t ^ 4 \\int _ { \\mathbb R ^ 3 } \\phi _ { \\mathfrak v } \\mathfrak v ^ 2 d x - \\int _ { \\mathbb R ^ 3 } f ( t \\mathfrak v ) t \\mathfrak v d x \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} \\int _ X F _ 1 ( x ) F _ 2 ( x ) \\ , d \\mu ( x ) = \\sum _ { m = 0 } ^ \\infty f _ 1 ( m ) f _ 2 ( m ) , \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} \\begin{cases} H _ { 0 } : & \\beta _ { 1 } = . . . = \\beta _ { p } = 0 , \\\\ H _ { 1 } : & \\mbox { a t l e a s t o n e } \\beta _ { j } \\neq 0 . \\end{cases} \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} \\sigma ^ 1 { } _ { \\dot a b } : = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , \\sigma ^ 2 { } _ { \\dot a b } : = \\begin{pmatrix} 0 & - i \\\\ i & 0 \\end{pmatrix} , \\sigma ^ 3 { } _ { \\dot a b } : = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\sigma ^ 4 { } _ { \\dot a b } : = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} f _ t = ( \\Pi _ t ) _ { \\# } ( | | x - y | | \\cdot \\gamma ) \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } \\det \\Big ( Q + c _ i B _ i ^ \\ast A ^ { ( t ) } _ i B _ i + \\sum _ { \\stackrel { 1 \\le k \\le m } { k \\neq i } } c _ k B _ k ^ \\ast A _ k B _ k \\Big ) = \\det \\Big ( Q + c _ i \\tilde { B } _ i ^ \\ast \\tilde { A } _ i \\tilde { B } _ i + \\sum _ { \\stackrel { 1 \\le k \\le m } { k \\neq i } } c _ k B _ k ^ \\ast A _ k B _ k \\Big ) > 0 \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} Y _ t - Y _ 0 + A ^ { W , Y } _ t ( b ) + \\int _ 0 ^ t f \\left ( r , W _ r , Y _ r , \\frac { \\mathrm d [ Y , W ] _ r } { \\mathrm d r } \\right ) \\mathrm d r = \\int _ 0 ^ t Z _ r \\mathrm d W _ r \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\mathbb { H } [ ( \\mathfrak { Q } _ h u - u _ { h } ) ^ { 1 } ] = & ( 3 - 2 \\theta ) { \\| ( \\mathfrak { Q } _ h u - u _ { h } ) ^ { 1 } \\| } ^ { 2 } - ( 1 - 2 \\theta ) { \\| ( \\mathfrak { Q } _ h u - u _ { h } ) ^ { 0 } \\| } ^ { 2 } \\\\ & + ( 2 - \\theta ) ( 1 - 2 \\theta ) { \\| ( \\mathfrak { Q } _ h u - u _ { h } ) ^ { 1 } - ( \\mathfrak { Q } _ h u - u _ { h } ) ^ { 0 } \\| } ^ { 2 } \\\\ \\leq & C ( \\| ( \\mathfrak { Q } _ h u - u _ { h } ) ^ { 1 } \\| ^ 2 + \\| ( \\mathfrak { Q } _ h u - u _ { h } ) ^ { 0 } \\| ^ 2 ) . \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { 1 } ( \\textbf { z } ) & = I \\left \\lbrace \\textbf { z } \\in \\mathcal { C } _ { 1 } \\cap \\mathcal { C } _ { 0 } , p ( \\textbf { z } , 1 ) \\dfrac { p } { \\mu { ( \\mathcal { C } _ { 1 } ) } } > p ( \\textbf { z } , 0 ) \\dfrac { 1 - p } { \\mu { ( \\mathcal { C } _ { 0 } ) } } \\right \\rbrace \\\\ & + I \\left \\lbrace \\textbf { z } \\in \\mathcal { C } _ { 1 } - \\mathcal { C } _ { 0 } , p ( \\textbf { z } , 1 ) > 0 \\right \\rbrace . \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} V ^ { ( n + 1 ) } ( t ) = \\int _ 0 ^ t f ( t - s ) V ^ { ( n + 1 ) } ( s ) + \\int _ t ^ T g ( t - s ) V ^ { ( n + 1 ) } ( s ) \\ , d s + \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { n - k + 1 } g ^ { ( n - k ) } ( T - t ) V ^ { ( k ) } ( T ) . \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} \\mathcal L _ { \\beta , \\Omega } u _ n = ( - D ^ \\beta _ 0 + \\mathcal L _ \\Omega ) u _ n , \\quad n \\in \\mathbb N . \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} \\frac { c _ 0 ^ 2 \\overline { Q } ^ 2 } { \\overline { C } } - \\overline { C } ^ \\gamma = 0 \\qquad \\mbox { o n s u p p } ( \\eta ) . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} \\Psi ( v ) & = \\psi ( \\lambda _ 1 ( v ) ) + \\frac { 1 } { 2 } \\left [ \\psi ( \\lambda _ 2 ( v ) ) + \\psi ( \\lambda _ 4 ( v ) ) \\right ] \\\\ & \\leq \\psi \\left ( \\sqrt { \\lambda _ 1 ( x ) \\lambda _ 1 ( s ) } \\right ) + \\frac { 1 } { 2 } \\psi \\left ( \\sqrt { \\lambda _ 2 ( x ) \\lambda _ 2 ( s ) } \\right ) + \\frac { 1 } { 2 } \\psi \\left ( \\sqrt { \\lambda _ 4 ( x ) \\lambda _ 4 ( s ) } \\right ) . \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} \\sup _ { ( \\mu , \\psi ( \\cdot ) , \\eta ( \\cdot ) ) \\in \\mathcal { D } } \\mu : = d ^ * ( y _ 0 ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} \\ker S + \\ker T = H . \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} \\langle \\epsilon ( g ) , \\Gamma ( \\widehat { O } ) ^ * H _ \\mu ( \\xi ) \\Gamma ( \\widehat { O } ) \\epsilon ( f ) \\rangle & = \\langle \\epsilon ( \\widehat { O } g ) , H _ \\mu ( \\xi ) \\epsilon ( \\widehat { O } f ) \\rangle \\\\ & = \\langle \\epsilon ( g ) , H _ \\mu ( O \\xi ) \\epsilon ( f ) \\rangle . \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} H _ { \\Lambda } ( \\phi ) : = \\sum _ { j k \\in E ( \\Lambda ) } V ( \\phi _ j - \\phi _ k ) = \\sum _ { j k \\in E ( \\Lambda ) } \\left ( \\left ( 1 + \\beta ( \\phi _ j - \\phi _ k \\right ) ^ 2 \\right ) ^ \\alpha . \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} & \\sum _ { l = l _ 0 } ^ { \\infty } | \\mathbf { K } _ { i , n , l } ( r \\theta ) | \\\\ & \\quad \\quad \\lesssim \\sum _ { l = l _ 0 } ^ { \\infty } \\frac { \\varepsilon ^ { - d + 1 } } { \\rho ^ \\alpha } \\int _ { \\mathbb { R } ^ d } \\mathbf { 1 } _ { | y | \\sim 2 ^ { i - l } \\rho } | \\mathbf { K } _ { n } ( y ) | | \\Theta ( r \\theta - y ) - \\Theta ( r \\theta ) | d y + \\frac { \\varepsilon ^ { - d + 1 } } { \\rho ^ \\alpha } | \\Theta ( r \\theta ) | , \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\nu _ { \\mathcal P _ { l } } \\left ( \\Big \\{ S ^ { k _ l ( n _ l + M _ l ) } h _ l - \\mathbb { E } _ { \\mathcal { P } _ { l } } ( S ^ { k _ l ( n _ l + M _ l ) } h _ l ) \\leqslant t s _ l \\Big \\} \\right ) = \\mathcal { N } ( t ) . \\end{align*}"}
-{"id": "9877.png", "formula": "\\begin{align*} ( u , \\boldsymbol { D } ^ { \\mu * } \\psi ) = ( w , \\psi ) ~ \\forall \\psi \\in C _ 0 ^ \\infty ( \\mathbb { R } ) . \\end{align*}"}
-{"id": "9712.png", "formula": "\\begin{align*} \\frac { \\big [ \\mathbb { A } / f \\mathbb { A } \\big ] _ { \\mathbb { A } } } { \\big [ \\varphi ( \\mathbb { A } / f \\mathbb { A } ) \\big ] _ { \\mathbb { A } } } & = \\frac { f } { f + c ( f ) p _ 1 \\prod _ { i = 1 } ^ n f ( z _ i ) + c ( f ) p _ 2 \\prod _ { i = 1 } ^ n f ( z _ i ) ^ 2 + \\dots + c ( f ) \\prod _ { i = 1 } ^ n f ( z _ i ) ^ { r _ 0 } } \\\\ & = \\frac { 1 } { 1 + c ( f ) p _ 1 f ^ { - 1 } \\prod _ { i = 1 } ^ n f ( z _ i ) + \\dots + c ( f ) f ^ { r _ 0 - 1 } f ^ { - r _ 0 } \\prod _ { i = 1 } ^ n f ( z _ i ) ^ { r _ 0 } } \\\\ & = D ^ { \\varphi } _ f ( f ^ { - 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} W _ { n , r } ^ { ( N ) } = \\sum _ { i = 1 } ^ \\ell W _ { i , n , r } ^ { ( N ) } , \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} \\int _ { H } \\prod _ { k = 1 } ^ m f _ k ^ { c _ k } ( B _ k x ) \\ , d x \\ge \\prod _ { k = 1 } ^ m \\Big ( \\int _ { H _ k } f _ k \\Big ) ^ { c _ k } . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} \\dot { q } _ j = - \\mathbf { i } \\sigma _ j \\frac { \\partial H } { \\partial \\bar { q } _ j } , \\sigma _ j = \\begin{cases} 1 , \\ & j \\geq 1 \\\\ - 1 , \\ & j \\leq - 1 \\end{cases} \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} \\exp \\big ( \\sum _ { n \\ge 1 } a _ n x ^ n / n \\big ) = \\sum _ { n \\ge 1 } \\left ( [ u ^ { n - 1 } ] f ^ n ( u ) / n \\right ) x ^ { n - 1 } . \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} M _ { \\sigma } ( v ) = M _ { p _ { S \\ast } \\sigma } ( v ) \\times \\mathbb { C } \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} \\| \\alpha _ k ( \\beta ) ( x ) \\| _ 2 & = \\sqrt { \\sum _ \\sigma \\left ( \\int _ 0 ^ 1 t ^ { k - 1 } f _ \\sigma ( t x ) d t \\right ) ^ 2 } \\\\ & \\le \\frac { 1 } { k } \\sqrt { \\sum _ \\sigma \\max _ { 0 \\le t \\le 1 } f _ \\sigma ^ 2 ( t x ) } \\\\ & \\le \\frac { 1 } { k } \\sqrt { m \\choose k } \\max _ \\sigma \\max _ { 0 \\le t \\le 1 } | f _ \\sigma ( t x ) | \\\\ & \\le \\frac { 1 } { k } \\sqrt { m \\choose k } \\max _ { 0 \\le t \\le 1 } \\| \\beta ( t x ) \\| _ 2 , \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} \\bar { x } ( s ) = G ^ { - 1 } ( z , \\bar { y } ( s ) ) \\ , , \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} u ' ( t ) = v ' ( z ) z ^ { - \\frac { 1 - 2 s } { 2 s } } . \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} G _ m ( z , s ) : = \\sum _ { \\gamma \\in \\mathrm { S L } _ 2 ( \\mathbb { Z } ) _ { \\infty } \\backslash \\mathrm { S L } _ 2 ( \\mathbb { Z } ) } ( \\phi _ m | _ 0 \\gamma ) ( z , s ) , \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} k & = \\frac { \\ell n ( n - \\ell - 1 ) } { n ( a ^ 2 + \\ell ) - ( a - \\ell ) ^ 2 } , \\\\ \\lambda & = \\frac { n ( n ^ 2 ( a ^ 3 + \\ell ^ 2 ) - 2 ( \\ell + 1 ) n ( a ^ 3 + \\ell ^ 2 ) + ( 2 a \\ell + a + \\ell ( \\ell + 2 ) ) ( a - \\ell ) ^ 2 ) } { ( ( a - \\ell ) ^ 2 - n ( a ^ 2 + \\ell ) ) ^ 2 } , \\\\ \\mu & = \\frac { \\ell n ( a - \\ell ) ( \\ell - n + 1 ) ( a - \\ell + n ) } { ( ( a - \\ell ) ^ 2 - n ( a ^ 2 + \\ell ) ) ^ 2 } . \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} [ T _ a , T _ { a ^ 2 } ] = 0 . \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - f _ 1 ) ^ j \\binom k j X _ { m + b k - a j } } = f _ 2 { } ^ k X _ m \\ , , \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} \\beta ^ 2 ( \\Q ) = 1 - 6 \\frac { \\left ( \\textrm { t r } \\ , \\Q ^ 3 \\right ) ^ 2 } { \\left ( \\textrm { t r } \\ , \\Q ^ 2 \\right ) ^ 3 } . \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} C _ \\infty ( x ) & = ( c _ 0 B ( x ) ) ^ { \\frac { 2 } { \\gamma + 1 } } . \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} \\| W _ { \\widetilde { \\psi } _ 0 , \\widetilde { \\varphi } _ 0 } f - W _ { \\psi _ { i , 1 , 0 } , \\varphi _ { i , 1 , 0 } } f \\| _ { n , q } = e ^ { \\frac { \\left | ( z ^ 0 ) ' _ { [ j ] } \\right | ^ 2 } { 2 } } \\| W _ { \\widetilde { \\psi } , \\widetilde { \\varphi } } f - W _ { \\psi _ { i , 1 } , \\varphi _ { i , 1 } } f \\| _ { n , q } , \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} X _ s ^ { t _ 0 , x _ 0 } = x _ 0 + \\int _ { t _ 0 } ^ { s \\wedge \\tau _ S } \\Gamma X _ r ^ { t _ 0 , x _ 0 } d r + \\int _ { t _ 0 } ^ { s \\wedge \\tau _ S } \\sigma X _ r ^ { t _ 0 , x _ 0 } d B _ r + \\int ^ { s \\wedge \\tau _ S } _ { t _ 0 } \\int X _ { r - } ^ { t _ 0 , x _ 0 } \\gamma ( r , z ) \\tilde { N } ( d r , d z ) , \\\\ \\mathbb { P } - { \\rm a . s } , & \\\\ X ^ { t _ 0 , x _ 0 } _ { t _ 0 } : = x _ 0 , & \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} 4 \\delta ( W ) + ( 5 a + 2 b ) k + t _ { 2 } = 3 t _ { 6 } . \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} W _ { 0 , a ' } ( f , g ) ( n ) - W _ { 0 , a ' } ( f , g ) ( n - 1 ) = & - b ' _ { n + 1 } f ( n ) g ( n ) \\\\ & - a ' _ n ( f ( n ) g ( n - 1 ) + f ( n - 1 ) g ( n ) ) . \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} Q ( B \\times D ) = \\int _ B \\varphi ( D | s ) q ( d s ) , \\quad \\mbox { f o r a l l } B \\in { \\cal B } ( S ) , \\ D \\in { \\cal B } ( A ) , \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} 1 + \\sum _ { i = 1 } ^ d x _ i r _ i + y M = ( \\ell + y ) M \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} m ( v ) = \\sum _ { v ' \\in N ( v ) } H ' ( \\bar { \\theta } _ { v ' } - \\bar { \\theta } _ v ) \\leq M \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} d z _ i ( s ) = & - \\frac { \\partial V _ 0 } { \\partial z _ i } \\big ( z ( s ) , y ( s ) \\big ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } d w _ i ( s ) \\ , , 1 \\le i \\le m \\ , , \\\\ d y _ j ( s ) = & - \\frac { \\partial V _ 0 } { \\partial y _ j } \\big ( z ( s ) , y ( s ) \\big ) \\ , d s - \\frac { 1 } { \\epsilon } \\frac { \\partial V _ 1 } { \\partial y _ j } \\big ( y ( s ) \\big ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } \\ , d w _ j ( s ) \\ , , m + 1 \\le j \\le n \\ , . \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} \\Gamma ( u ) = \\Phi ( b ^ { - 2 } u ) - \\Phi ( b u ) , \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} C _ { \\infty } \\leq C \\sum _ { k = 0 } ^ { \\infty } \\frac { | 2 ^ { k + 1 } Q _ { 0 } | ^ { \\frac { \\alpha t } { n } } } { | 2 ^ { k + 1 } Q _ { 0 } | ^ { t } | Q _ { 0 } | } \\left ( \\int _ { 2 ^ { k + 3 } Q _ { 0 } } f d x \\cdot \\int _ { 2 ^ { k + 3 } Q _ { 0 } } g d x \\right ) ^ { t } \\left ( \\int _ { Q _ { 0 } } v ^ { \\frac { t } { 1 - t } } d x \\right ) ^ { 1 - t } . \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} \\begin{aligned} b ( x ' , y , t ) = { } & \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } \\biggl [ b ( x , y , t ) - x \\frac { \\partial } { \\partial y } a ( x , y , t ) \\\\ & + \\frac { \\partial } { \\partial y } ( y \\varphi ( x , y , t ) ) + \\frac { 1 } { 2 } x ^ { 2 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\varphi ( x , y , t ) - \\frac { \\partial } { \\partial t } \\varphi ( x , y , t ) \\biggr ] \\end{aligned} \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} \\min ~ ~ & c ^ T x \\\\ ~ ~ & A x = b , \\\\ ~ ~ & x \\in \\Upsilon , \\end{align*}"}
-{"id": "9890.png", "formula": "\\begin{align*} \\begin{aligned} ( \\boldsymbol { D } ^ { - s } \\psi , \\boldsymbol { D } ^ { s } \\psi ) & = ( \\widehat { \\boldsymbol { D } ^ { - s } \\psi } , \\overline { \\widehat { \\boldsymbol { D } ^ { s } \\psi } } ) \\\\ & = ( \\mathcal { F } ( \\boldsymbol { D } ^ { - s } \\psi ) , \\overline { \\mathcal { F } ( \\boldsymbol { D } ^ { s } \\psi ) } ) \\\\ & = ( ( 2 \\pi i \\xi ) ^ { - s } \\widehat { \\psi } , \\overline { ( 2 \\pi i \\xi ) ^ { s } \\widehat { \\psi } } ) . \\end{aligned} \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} \\det B H _ { r , n } & = \\frac { 1 } { 2 ^ r } \\det \\left [ \\binom { 2 i + 2 j + 2 ( n + 1 ) } { i + j + ( n + 1 ) } \\right ] _ { i , j = 0 } ^ { r - 1 } \\allowdisplaybreaks \\\\ & = \\frac { 1 } { 2 ^ r } 2 ^ { r - 1 + ( n + 1 ) } \\prod _ { j = 0 } ^ n \\prod _ { i = 1 } ^ j \\frac { 2 r + j + i - 1 } { j + i } = 2 ^ n \\prod _ { j = 0 } ^ n \\prod _ { i = 1 } ^ j \\frac { 2 r + j + i - 1 } { j + i } \\allowdisplaybreaks \\\\ & = 2 ^ n \\prod _ { 1 \\leq i \\leq j \\leq n } \\frac { 2 r + j + i - 1 } { j + i } . \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} \\lim _ n \\frac { \\lvert T ^ { - n } [ \\phi w ] \\rvert _ S } { n } & = \\lim _ n \\frac { \\lvert ( \\phi w ) _ n \\rvert _ S } { n } \\\\ & = \\lim _ n \\frac { ( n - N ) s p ( w ) + \\lvert ( \\phi w ) _ N \\rvert _ S } { n } \\\\ & = s p ( w ) + \\lim _ n \\frac { - N s p ( w ) + \\lvert ( \\phi w ) _ N \\rvert _ S } { n } \\\\ & = s p ( w ) . \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} \\int _ { x } ^ { \\infty } \\frac { d \\psi ( t , y ) } { t } = \\frac { \\psi ( t , y ) } { t } \\bigg | _ { x } ^ { \\infty } + \\int _ { x } ^ { \\infty } \\frac { \\psi ( t , y ) } { t ^ { 2 } } d t . \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} s K g ^ { - 1 } = r _ { g ^ { - 1 } } ( s K ) \\subseteq s g ^ { - 1 } H \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} \\mathcal { E } _ z ( f , h ) = - \\int _ { \\Sigma _ z } ( \\mathcal { L } _ 0 f ) \\ , h \\ , d \\mu _ z \\ , , \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} \\Pi _ { k , I } ( z , w ) = \\sum _ { \\mu _ { k , j } \\in I } s _ { k , j } ( z ) \\overline { s _ { k , j } ( w ) } . \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} ( \\mathcal { L } ^ \\omega _ X f ) ( x ) = \\sum _ { y \\sim x } \\omega ( x , y ) ( f ( y ) - f ( x ) ) . \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} i \\partial _ t u = \\partial _ x ^ 2 u + \\nu | u | ^ 2 u , \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} \\dim U + \\sum _ { i = 1 } ^ { \\lfloor d / 2 \\rfloor } \\dim S ^ { d - i } U . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} \\mathbf { H } \\triangleq \\begin{pmatrix} ( 2 / \\rho \\eta ) \\mathbf { I } _ { m p } & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\mathbf { M } , \\end{pmatrix} \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{gather*} g _ { \\Sigma } \\big ( \\nabla ^ { \\Sigma } _ u v , w \\big ) = g _ { \\Sigma } \\big ( \\nabla ^ { \\mathrm { R } } _ u \\mathrm { P } _ H v , \\mathrm { P } _ H w \\big ) + g _ { \\Sigma } \\big ( \\nabla ^ { \\mathrm { R } } _ u \\mathrm { P } _ V v , \\mathrm { P } _ V w \\big ) \\end{gather*}"}
-{"id": "4374.png", "formula": "\\begin{align*} ( \\mathbf { \\sqrt { V } } \\otimes \\mathbf { I } _ p ) \\mathbf { x } ^ { \\star } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} \\alpha _ n ( x ) & = \\min \\{ \\alpha ( x ) , x \\alpha ( \\tfrac 1 n ) \\} , \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } j ( \\omega _ n ^ m ) ^ j = \\begin{cases} \\binom { n } { 2 } & \\mbox { i f $ n \\mid m $ , } \\\\ \\frac { n } { \\omega _ n ^ m - 1 } & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} \\| f \\| _ { E ^ \\prime } : = \\sup _ { g \\in E , \\ , \\| g \\| _ { E } \\leq 1 } \\int _ { I } | f ( x ) g ( x ) | \\ , d x < \\infty . \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} D _ { t } V ( k , s ) + B _ { i } ^ { + } ( k , s ) D ^ { + } V ( k , s ) + B _ { i } ^ { - } ( k , s ) D ^ { - } V ( k , s ) - D _ { t } \\Gamma ( k , s ) = 0 , \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} \\int x _ i \\ , \\widetilde \\mu ( \\d x ) = \\frac 1 n \\int ( x _ 1 + \\dotsb + x _ n ) \\ , \\mu ( \\d x ) = 0 , \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} \\lim _ { y \\rightarrow \\pm \\infty } \\frac { ( - \\zeta x ^ \\ell ) ^ { i y } } { e ^ { \\pi | y | } } = 0 \\Leftrightarrow \\frac { - \\pi - \\mu } { \\ell } < \\theta < \\frac { \\pi - \\mu } { \\ell } \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} \\frac { d \\hat I ( t ) } { d t } = \\frac { \\partial \\hat I ( t ) } { \\partial t } + \\frac { 1 } { i } [ \\hat I ( t ) , \\hat H ( t ) ] \\equiv 0 , \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} M ' ( \\theta _ - ) M ' ( \\theta _ + ) = \\frac { ( \\det M ) ^ 2 } { ( ( c \\theta _ - + d ) ( c \\theta _ + + d ) ) ^ 2 } = 1 \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} & ( I - e ^ { - \\alpha } \\tilde { P } ) \\nu = r , \\\\ & ( I - e ^ { - \\alpha } \\tilde { P } ) \\nu _ 1 = \\tilde { P } ^ { ( 1 ) } \\nu , \\\\ & ( I - e ^ { - \\alpha } \\tilde { P } ) \\nu _ 2 = \\nu _ 1 . \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} v ^ { k } & = \\alpha _ k x ^ k + ( 1 - \\alpha _ k ) T x ^ k , \\\\ x ^ { k + 1 } & = \\beta _ k v ^ k + ( 1 - \\beta _ k ) T u ^ k , \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} ( A \\stackrel { C ( \\mathbf { z } ) } { \\star } B ) ( \\mathbf { x } , \\mathbf { y } ) = \\int _ { 0 } ^ { \\mathbf { z } } A _ { , C } ( \\mathbf { x } , \\mathbf { r } ) \\cdot B _ { C , } ( \\mathbf { r } , \\mathbf { y } ) C ( d \\mathbf { r } ) , \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} M _ { k , \\ell } = \\binom { k + f } { r _ \\ell } \\lambda _ { q _ \\ell + 1 } ^ { k + f - r _ \\ell } . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} \\alpha _ M ( \\mu _ l ( a , m ) ) = \\mu _ l ( \\alpha ( a ) , \\alpha _ M ( m ) ) ~ ~ ~ \\mu _ l ( \\alpha ( a ) , \\mu _ l ( b , m ) ) = \\mu _ l ( \\mu ( a , b ) , \\alpha _ M ( m ) ) . \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\Lambda = \\sum _ { j \\neq 0 } \\sigma _ j j ^ 2 | q _ j | ^ 2 , \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} \\begin{aligned} & \\Delta x = \\sqrt { \\mu } W ^ { - 1 } d _ x , & \\Delta s = \\sqrt { \\mu } W d _ s . \\end{aligned} \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} \\tilde { m } _ { \\xi } = L ^ { N } \\tilde { b } _ { \\xi } , \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} E = \\left \\{ r : T \\left ( r + | c | + \\frac { r + | c | } { ( \\log T ( r + | c | , f ) ) ^ { 1 + \\varepsilon / 3 } } , f \\right ) \\geq C T ( r + | c | , f ) \\right \\} , \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} f ( x ) & = ( x ^ 2 + x + 1 ) + m \\ : h ( x ) , \\\\ g ( x ) & = ( x ^ 2 - x + 1 ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} \\beta : U ( \\mathfrak { g } ^ { \\Gamma _ q } ) \\to \\bigoplus _ { \\mu \\in D _ { c , p '' } } \\ , V ( \\mu ^ { * } ) \\otimes V ( \\mu ) , \\beta ( a ) = a \\odot \\sum \\limits _ { \\mu } I _ { \\mu } . \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} | F ( u _ N ) - F ( u ) | = | f ( \\xi ) | | u _ N - u | \\leq C ( | \\xi | ^ q + 1 ) | u _ N - u | \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} D _ t v ( t , g _ t ( x ) ) = D _ t D _ t g _ t ( x ) = \\big ( \\frac { \\partial } { \\partial t } v + ( v . \\nabla ) v + \\nu \\Delta v \\big ) ( t , g _ t ( x ) ) . \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} U f = \\bigoplus _ { j = 1 } ^ { 2 m } \\varrho ( \\gamma _ j ) f _ j \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ H \\mathcal { I } ( ( f _ i ) , ( f _ { j , n } ) ) = \\int _ H \\mathcal { I } ( ( f _ i ) , ( f _ j + \\varepsilon \\phi _ { j , 1 } ) ) \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} S ( x ) = \\sum _ { j \\in \\mathcal { V } } S _ j \\delta ( x - x _ j ) , \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } u _ { t } + H _ { i } ( t , x , u _ { x _ { i } } ) = 0 & \\ , \\ , I _ { i } \\times ( 0 , T ) \\\\ \\sum _ { i = 1 } ^ { K } u _ { x _ { i } } = B & \\ , \\ , \\{ 0 \\} \\times ( 0 , T ) \\\\ u = u _ { 0 } & \\ , \\ , \\mathcal { I } \\times \\{ 0 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} V ( m , s ) + F _ { i } ( D ^ { + } U ( m , s ) , D ^ { - } U ( m , s ) ) = f _ { i } ( s \\Delta t , - m \\Delta x ) \\ , \\ , J _ { i } . \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} - ( - \\Delta ) ^ s _ y \\phi _ { 0 j } + p U ( y ) ^ { p - 1 } \\phi _ { 0 j } = - \\mu _ j ^ { \\frac { n + 2 s } { 2 } } S ( u _ { \\mu , \\xi } ) \\mathbb { R } ^ n , \\ , \\ , \\phi _ { 0 j } ( y , t ) \\to 0 | y | \\to \\infty \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} | ( i ^ 2 - j ^ 2 ) - \\frac { i - j } { n - \\frac 1 2 } \\sum _ { b = 1 } ^ n j _ b | < \\frac { 1 } { 2 5 n } | i - j | , \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} L ^ { A - S } _ { d , t } = \\left ( L ^ { A } _ { t - d } \\circ \\theta _ { d } \\right ) \\cdot L ^ { - S } _ { t } \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} & G _ N ( \\omega ) ( x ) - G _ N ( \\omega ) ( y ) \\\\ = \\ , & { D F _ N ( \\omega ) ( x ^ * ) } ^ { - 1 } [ D F _ N ( \\omega ) ( x ^ * ) ( x - y ) - ( F _ N ( \\omega ) ( x ) - F _ N ( \\omega ) ( y ) ) ] \\\\ = \\ , & { D F _ N ( \\omega ) ( x ^ * ) } ^ { - 1 } [ D F _ N ( \\omega ) ( x ^ * ) - R _ N ( \\omega ) ( x , y ) ] ( x - y ) , \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} \\textbf { I } _ { C } ( \\upsilon , b , \\lambda , y ) = \\int _ { 0 } ^ { \\infty } x ^ { \\upsilon - 1 } \\frac { \\cos ( x y ) } { \\{ - 1 + \\exp { ( b \\sqrt { x } ) } \\} ^ { \\lambda } } d x , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} \\tfrac { d } { d t } \\ , g _ t = - { \\rm R i c } ^ { 1 , 1 } ( g _ t ) \\ , , \\end{align*}"}
-{"id": "9817.png", "formula": "\\begin{align*} b ^ 2 t ^ 3 \\exp \\Big ( u ( t ) b ^ 2 \\Big ) \\le \\frac { - t ^ 3 e ^ { u ( t ) } } { u ( t ) } = : F ( t ) \\ , . \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} \\overline { R } ( U , V ) W = { \\overline { \\nabla } } _ { U } { \\overline { \\nabla } } _ { V } W - { \\overline { \\nabla } } _ { V } { \\overline { \\nabla } } _ { U } W - { \\overline { \\nabla } } _ { [ U , V ] } W . \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} C _ { k } = 1 0 + 4 ^ { N ( k ) } + N ( k ) ^ { 1 0 0 } + K _ k + \\max _ { 1 \\leq j \\leq N ( k ) } C ( \\lambda _ j , A _ k \\backslash \\{ \\lambda _ j \\} , K _ 0 ) , \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} \\begin{aligned} & u _ { + , a , \\hat r _ \\pm } ( 0 ) \\sim \\frac { C _ \\pm } { \\phi _ { 3 , + } ( a ) } , \\ \\ a \\to \\pm \\infty , \\ \\ C _ \\pm > 0 ; \\\\ & \\lim _ { a \\to 0 ^ + } u _ { + , a , \\hat r _ \\pm } ( 0 ) = 1 . \\end{aligned} \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} \\varphi ( x ) = ( \\mathcal { L } \\xi ) ( x ) - \\widetilde { b } ( \\xi ( x ) ) \\ , , \\forall ~ x \\in \\mathbb { R } ^ n \\ , . \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} \\int _ { V ^ { ( 4 ) } } \\langle \\delta ^ * _ { g ' } \\tilde Y , \\delta ^ * _ { \\tilde g ^ { ( 4 ) } } \\tilde Y + \\frac { 1 } { 2 } ( \\delta \\tilde Y ) \\tilde g ^ { ( 4 ) } \\rangle d v o l _ { \\tilde g ^ { ( 4 ) } } = 0 . \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} I ( \\mathbf { x } , \\mathbf { z } ) = h ( \\mathbf { z } ) - h ( \\mathbf { n } _ b ) , \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} \\mathcal { D } _ t u ^ { \\frac 1 2 } + \\gamma \\bigtriangleup \\sigma ^ { \\frac 1 2 } - \\bigtriangleup u ^ { \\frac 1 2 } + f ^ { \\frac 1 2 } ( u ) = g ( \\textbf { z } , t _ { \\frac 1 2 } ) , \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} H _ 3 ( \\mathbb C ) = \\left \\{ \\left [ \\begin{smallmatrix} 1 & z _ 1 & z _ 3 \\\\ 0 & 1 & z _ 2 \\\\ 0 & 0 & 1 \\end{smallmatrix} \\right ] \\mid z _ 1 , z _ 2 , z _ 3 \\in \\mathbb C \\right \\} \\ , . \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} \\bar \\rho _ { \\alpha , V } ( u ) : = \\frac { E _ { \\nu _ { \\alpha , 1 } } \\bigl [ \\eta ( 0 ) \\mathrm e ^ { - V ( u ) \\eta ( 0 ) } \\bigr ] } { E _ { \\nu _ { \\alpha , 1 } } \\bigl [ \\mathrm e ^ { - V ( u ) \\eta ( 0 ) } \\bigr ] } < \\infty . \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} \\kappa ( \\epsilon ) = \\mu ( I - e ^ { - \\alpha } \\tilde { P } ) ^ { - 1 } r + \\epsilon e ^ { - \\alpha } \\mu ( I - e ^ { - \\alpha } \\tilde { P } ) ^ { - 1 } \\tilde { P } ^ { ( 1 ) } ( I - e ^ { - \\alpha } \\tilde { P } ) ^ { - 1 } r + O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ w ) \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} M _ { 0 , \\nu } ( z ) = 2 ^ { 2 \\nu } \\Gamma ( \\nu + 1 ) z ^ { 1 / 2 } I _ { \\nu } \\bigl ( \\frac { z } { 2 } \\bigr ) , \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} \\Theta \\Delta ( u ) = \\left ( \\Psi \\circ \\Delta ^ { \\mathrm { o p } } \\right ) ( u ) \\Theta \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} g _ { \\pm } = \\exp \\left ( \\frac { 1 } { 2 } \\left ( \\log g \\pm i \\widetilde { \\log g } \\right ) \\right ) , \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} \\Lambda ( P _ { \\infty } ) = 0 \\quad \\mbox { \\rm a n d s u c h t h a t } B : = A - P _ { \\infty } S \\quad \\mbox { \\rm i s a s t a b l e m a t r i x ( a . k . a . H u r w i t z m a t r i x ) . } \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} H ^ q ( W _ { 2 , L } , U _ { \\chi } ) \\simeq \\bigoplus \\limits _ { I \\subset I _ F \\ \\mbox { \\tiny { s . t . } } \\ \\vert I \\vert = q } U ^ q _ I , \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} A ^ \\psi ( \\phi , t ) = \\sum _ { j k \\in E ( \\Lambda ) } \\Big ( 1 + \\beta ( \\phi \\vee \\psi ( j ) - \\phi \\vee \\psi ( k ) ) ^ 2 \\Big ) e ^ { t _ { j k } } + \\sum _ { j \\in \\Lambda } \\epsilon \\phi _ j ^ 2 , \\quad \\epsilon \\ge 0 . \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{gather*} \\langle X \\rangle = \\langle X ' \\rangle \\ \\mbox { w h e n e v e r $ X , X ' $ a r e e x c e p t i o n a l p a i r s f r o m t h e s a m e r o w o f t a b l e \\eqref { 1 p a i r s } } \\end{gather*}"}
-{"id": "885.png", "formula": "\\begin{align*} \\xi ^ { \\pm } _ { i } = \\sqrt { - 1 } , \\ i \\in V ( Q ) , \\ \\Re ( \\xi ^ { + } _ 0 ) < 0 , \\ \\Re ( \\xi ^ { - } _ 0 ) > 0 . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} \\begin{aligned} & \\overline { A } d _ x = 0 \\\\ & \\overline { A } ^ T \\Delta y + d _ s = 0 \\\\ & d _ x + d _ s = \\psi ^ \\prime ( v ) . \\end{aligned} \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} - \\sum _ { t = 1 } ^ d \\rho _ { t } r _ { t } i _ { t } \\equiv i _ j \\pmod { s _ j } . \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} \\langle g X , g Y \\rangle _ g : = \\langle \\langle X , Y \\rangle \\rangle , g \\in S E ( 3 ) , X , Y \\in \\mathfrak { s e } ( 3 ) . \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} [ t _ 0 , \\ldots , t _ { n - 1 } ] : = X _ { t _ 0 } \\cap g _ 0 X _ { t _ 1 } \\cap g _ 0 g _ 1 X _ { t _ 2 } \\cap \\dots \\cap ( g _ 0 \\cdots g _ { n - 2 } ) X _ { t _ { n - 1 } } . \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} A _ 5 = I _ f \\otimes ( J _ { \\ell + 1 } - I _ { \\ell + 1 } ) \\otimes J _ n , A _ 6 = ( J _ f - I _ f ) \\otimes I _ { \\ell + 1 } \\otimes J _ n . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} L _ { G } ^ { ( i ) } & = \\sup \\left \\{ \\frac { | G _ { i } ( p _ { 1 } , p _ { 2 } ) - G _ { i } ( q _ { 1 } , q _ { 2 } ) | } { | p _ { 1 } - q _ { 1 } | + | p _ { 2 } - q _ { 2 } | } \\ , \\mid \\ , ( p _ { 1 } , p _ { 2 } ) , ( q _ { 1 } , q _ { 2 } ) \\in \\mathbb { R } ^ { 2 } \\right \\} , \\\\ L _ { G } & = \\max \\{ L _ { G } ^ { ( 1 ) } , \\dots , L _ { G } ^ { ( K ) } \\} . \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} B _ { i + 1 } + B _ { i + 1 } \\subset B _ i | B _ { i + 1 } | = \\Omega _ K ( | B _ i | ) i \\in \\N _ 0 \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} V : = \\{ \\tilde { x } _ 1 , \\cdots , \\tilde { x } _ n , \\tilde { y } _ 1 , \\cdots , \\tilde { y } _ n , \\cdots , \\tilde { z } _ j , \\cdots , \\bar { { \\tilde z } } _ j , \\cdots \\} , j \\in \\mathbb { Z } _ * . \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} b ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { | T ' ( x ) | } b ( x , y , t ) \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} S ^ \\alpha { } _ \\beta = \\begin{pmatrix} 0 & 0 & - a \\sin ( x ^ 3 + x ^ 4 ) & - a \\sin ( x ^ 3 + x ^ 4 ) \\\\ 0 & 0 & \\pm a \\cos ( x ^ 3 + x ^ 4 ) & \\pm a \\cos ( x ^ 3 + x ^ 4 ) \\\\ - a \\sin ( x ^ 3 + x ^ 4 ) & \\pm a \\cos ( x ^ 3 + x ^ 4 ) & a ^ 2 & a ^ 2 \\\\ a \\sin ( x ^ 3 + x ^ 4 ) & \\mp a \\cos ( x ^ 3 + x ^ 4 ) & - a ^ 2 & - a ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} L ( \\psi , \\tilde { \\mathbb { A } } ) = \\sum _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) t ^ { \\deg _ { \\theta } ( a ) } } { a } , \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} \\frac { p _ t ( v , v ' ) } { m ( v ' ) } = \\frac { p _ t ( v ' , v ) } { m ( v ) } . \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} 1 + 2 ( d - 1 ) \\cdot \\iota _ { t / 2 , d } = \\tau ' _ t ( d ) . \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} w \\left ( z \\right ) & = \\Phi \\left ( z \\right ) { { e } ^ { - \\kappa \\left ( z \\right ) } } = \\Phi \\left ( z \\right ) { { e } ^ { - \\alpha \\left ( z \\right ) } } { { e } ^ { - \\sqrt { - 1 } \\beta \\left ( z \\right ) } } \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} N = \\sum _ { i = 1 } ^ d \\rho _ q ( i , m _ i ) = \\sum _ { i = e } ^ d \\rho _ q ( i , n _ i ) \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} \\ell & = \\frac { 1 } { k _ 1 } N _ 1 ( \\delta ) + \\frac { 1 } { k _ 2 } N _ 2 ( \\delta ) , \\\\ & = \\frac { 1 } { k _ 1 } N _ 1 ( \\delta ) + \\frac { 1 } { k _ 2 } N _ 1 ( - \\delta ) . \\intertext { R e p l a c i n g $ \\delta $ b y $ - \\delta $ i n t h e a b o v e a r g u m e n t g i v e s } \\ell & = \\frac { 1 } { k _ 1 } N _ 1 ( - \\delta ) + \\frac { 1 } { k _ 2 } N _ 1 ( \\delta ) , \\intertext { s o } \\left ( \\frac { 1 } { k _ 1 } - \\frac { 1 } { k _ 2 } \\right ) N _ 1 ( \\delta ) & = \\left ( \\frac { 1 } { k _ 1 } - \\frac { 1 } { k _ 2 } \\right ) N _ 1 ( - \\delta ) \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} K ( n , i , j ) = ( & i + j + 1 ) \\Big { [ } K ( n - 1 , i , j ) + K ( n - 1 , i , j - 1 ) \\Big { ] } \\\\ & + ( n - i - j ) \\Big { [ } K ( n - 1 , i - 1 , j ) + K ( n - 1 , i - 1 , j - 1 ) \\Big { ] } . \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} g _ 2 '' ( \\omega ) = \\frac { - m ^ 2 ( m ^ 2 - 1 ) ( c _ n + a _ n ' ( 1 ) ) } { \\omega ^ 2 } . \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} g _ t ( 1 ) & = \\gamma _ + ( u ^ + + t u ^ - ) = I ' ( u ^ + ) [ u ^ + ] + t ^ 2 \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ - } ( u ^ + ) ^ 2 d x \\\\ & \\leq I ' ( u ^ + ) [ u ^ + ] + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ - } ( u ^ + ) ^ 2 d x = \\gamma _ + ( u ) < 0 \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} & ( d w = 0 ) \\cap ( V ^ { + \\ast } \\times V ^ - \\times U ) \\subset ( V ^ { + \\ast } \\times \\{ 0 \\} \\times U ) , \\\\ & ( d w = 0 ) \\cap ( V ^ + \\times V ^ { - \\ast } \\times U ) \\subset ( \\{ 0 \\} \\times V ^ { - \\ast } \\times U ) . \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} a _ s ( u _ h , v _ h ) = a _ h ( u _ h , v _ h ) + s _ h ( u _ h , v _ h ) = ( f , v _ 0 ) _ { \\Omega } , \\ \\forall v = ( v _ 0 , v _ b ) \\in V _ h ^ 0 . \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} B S _ { \\chi , \\sigma , G , m , p } ( t _ 0 , f , u _ 0 ) = \\partial _ t ^ p g ( t _ 0 ) \\partial G \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} \\lim _ { \\Delta x \\to 0 ^ { + } } \\sup \\left \\{ | U ^ { \\Delta x } ( m ) - u ( - m \\Delta x ) | \\ , \\mid \\ , d ( - m \\Delta x , 0 ) \\leq R \\right \\} = 0 . \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} a _ r = \\frac { ( - 1 ) ^ r ( n + r ) ! } { ( r ! ) ^ 2 ( n - r ) ! } , \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} \\hat \\jmath ^ V _ { i , i + e _ k } ( \\mu ) = \\hat \\jmath ^ { \\ ; \\ ! 0 } _ { i , i + e _ k } ( \\mu ) \\cosh \\bigl ( \\tfrac 1 2 \\nabla ^ { i , i + e _ k } V ( \\cdot / L ) \\bigr ) + \\hat \\chi _ { i , i + e _ k } ^ 0 ( \\mu ) 2 \\sinh \\bigl ( - \\tfrac 1 2 \\nabla ^ { i , i + e _ k } V ( \\cdot / L ) \\bigr ) . \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} \\sum _ { a \\in A _ { + , k } } \\mu ( a ) C _ a ( X _ 1 ) \\dots C _ a ( X _ { n - 1 } ) = \\sum _ { a \\in A _ { + , k } } \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ { n - 1 } ) \\cdot ( X _ 1 \\dots X _ { n - 1 } ) . \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} f _ { \\alpha ^ k } = \\partial _ k f _ \\alpha - \\partial _ k \\chi ( u ) \\partial _ t \\partial ^ { \\alpha } u - \\sum _ { j = 1 } ^ 2 \\partial _ k A _ j \\partial _ j \\partial ^ \\alpha u - \\partial _ k \\sigma ( u ) \\partial ^ { \\alpha } u . \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} \\left | \\frac { \\varphi ( w ) } { w } \\right | = \\frac { 1 } { | w | } | \\psi ( w ^ { - r } ) - b _ 0 | \\le \\frac { C A M _ { 1 } } { | w | ^ { r + 1 } } . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} \\widehat { \\phi _ { P ( n ) } } ( \\alpha , \\xi ) = \\left \\{ \\begin{array} { l l } \\frac { 2 ^ { \\lambda + 1 / 2 } \\sqrt { \\pi } } { \\Gamma ( \\lambda ) } e ^ { - 2 \\xi } \\frac { \\xi ^ { \\lambda + | \\alpha | - 1 } } { ( n - 1 ) ^ k } \\frac { 1 } { \\alpha _ 1 ! \\cdots \\alpha _ { n - 1 } ! } & \\xi > 0 , \\alpha _ 1 \\geq 0 , \\ldots , \\alpha _ { n - 1 } \\geq 0 \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} \\| I ^ { + } f \\| _ { \\mathcal { B } _ { \\alpha } ^ { 2 } } = \\sum _ { k \\in \\mathbb { Z } ^ { n } \\setminus \\{ 0 \\} } | k | ^ { 4 } \\sup _ { t \\in [ 0 , \\infty ) } e ^ { \\alpha t | k | } \\left | \\int _ { 0 } ^ { t } e ^ { - | k | ^ { 4 } ( t - s ) } \\hat { f } ( s , k ) \\ d s \\right | . \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} \\{ p _ { 2 , \\tau } , p _ \\tau \\} _ { D B } = m \\omega ^ 2 ( t _ \\tau ) f ^ { - 1 } ( t _ \\tau ) x _ { 2 , \\tau } . \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} & g \\in M e t ^ { m , \\alpha } _ { \\delta } ( M ) , \\\\ & N - 1 \\in C ^ { m , \\alpha } _ { \\delta } ( M ) , \\\\ & X \\in T ^ { m , \\alpha } _ { \\delta } ( M ) , \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} ( x , y , u _ 1 , \\dots u _ n ) : = \\exp ( x X + y Y + u _ 1 U _ 1 + \\dots + u _ n U _ n ) . \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} \\mathcal { G } : = \\left \\{ ( x , \\theta , C ) ; \\ ; x \\in \\Gamma _ i \\mbox { f o r s o m e } i \\in \\mathcal { I } , \\ , \\theta = \\mathcal { t } _ i ( x ) , \\ , C = C _ i \\right \\} , \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} h ( u , v ) = \\lambda g ( u , v ) , \\qquad \\forall v \\in V . \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} S _ { i \\bar j } = g ^ { k \\bar l } \\Omega _ { k \\bar l i \\bar j } \\ , . \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} h ^ { \\alpha \\beta } ( x ) = g ^ { \\mu \\nu } ( \\varphi ( x ) ) \\ , \\psi _ \\mu { } ^ \\alpha \\ , \\psi _ \\nu { } ^ \\beta , \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta H ^ \\pm _ { C _ R } & = 0 , \\mbox { i n } A ^ \\pm , \\\\ H ^ \\pm _ { C _ R } & = 0 , \\mbox { o n } \\partial A ^ \\pm \\cap \\partial A , \\\\ H ^ \\pm _ { C _ R } & = 1 , \\mbox { o n } C _ R , \\\\ \\partial _ r H ^ + _ { C _ R } + \\partial _ r H ^ - _ { C _ R } & = 0 , \\mbox { o n } C _ R . \\end{aligned} \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} u _ { t t } ( t ) + A u ( t ) + G u _ t ( t ) + B ( t ) u _ t ( t ) = f ( t ) \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} F f = \\sum _ \\mu \\frac { f _ \\mu } { \\mu ! } \\ , F z ^ \\mu . \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} C _ { d _ 1 , d _ 2 , \\frac { 2 p q } { d _ 1 d _ 2 } } : d _ 1 X ^ 3 + d _ 2 Y ^ 3 + \\frac { 2 p q } { d _ 1 d _ 2 } Z ^ 3 = 0 . \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} C _ 0 ( \\alpha , \\gamma _ 1 , \\ldots , \\gamma _ j ) = C ( ( 0 , \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 3 ) , ( 0 , \\gamma _ { 1 , 1 } , \\gamma _ { 1 , 2 } , \\gamma _ { 1 , 3 } ) , \\ldots , ( 0 , \\gamma _ { j , 1 } , \\gamma _ { j , 2 } , \\gamma _ { j , 3 } ) ) \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} \\left | \\int _ { \\R ^ 3 } \\rho \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) ^ 2 - \\int _ { \\R ^ 3 } \\rho _ { \\infty } \\bar \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) ^ 2 \\right | & \\leq \\int _ { \\R ^ 3 } \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) ^ 2 | \\rho ( x ) - \\rho _ { \\infty } | \\\\ & + \\int _ { \\R ^ 3 } \\bar \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) ^ 2 | \\rho ( x ) - \\rho _ { \\infty } | \\\\ \\\\ & = : I _ 1 + I _ 2 . \\\\ \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} { V ( u , x ) } = \\frac { 1 } { 2 } ( u ^ 2 - 1 ) ^ 2 - \\frac { 1 } { 2 } + \\frac { 1 } { 4 } \\Omega ^ 2 x ^ 2 u ^ 2 . \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} a \\cdot x = \\gamma _ F ( a ) \\cdot x , x \\cdot a = x \\cdot \\gamma _ F ( a ) \\end{align*}"}
-{"id": "9876.png", "formula": "\\begin{align*} \\begin{aligned} \\tau _ h ( \\boldsymbol { D } ^ { \\mu } u ) = \\boldsymbol { D } ^ { \\mu } ( \\tau _ h u ) & , \\tau _ h ( \\boldsymbol { D } ^ { \\mu * } u ) = \\boldsymbol { D } ^ { \\mu * } ( \\tau _ h u ) \\\\ \\Pi _ \\kappa ( \\boldsymbol { D } ^ { \\mu } u ) = \\kappa ^ { - \\mu } \\boldsymbol { D } ^ { \\mu } ( \\Pi _ \\kappa u ) & , \\Pi _ \\kappa ( \\boldsymbol { D } ^ { \\mu * } u ) = \\kappa ^ { - \\mu } \\boldsymbol { D } ^ { \\mu * } ( \\Pi _ \\kappa u ) . \\end{aligned} \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { x \\in T _ { N } ^ { - 1 } ( x ' ) } x = & \\cos \\left ( \\pi u _ { 0 } \\right ) + 2 \\cos \\left ( \\pi u _ { 0 } \\right ) \\sum _ { j = 1 } ^ { \\frac { N } { 2 } - \\frac { 1 } { 2 } } \\cos \\left ( \\frac { 2 \\pi } { N } j \\right ) \\\\ = & \\cos \\left ( \\pi u _ { 0 } \\right ) \\left ( 1 + 2 \\left ( \\frac { - 1 } { 2 } \\right ) \\right ) = 0 \\ \\ \\ \\square \\end{aligned} \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} c ' = \\frac { c - p _ 0 } { p _ 1 - p _ 0 } , v _ 0 ' = \\frac { f ^ { n _ 0 } ( p _ 0 ) - p _ 0 } { p _ 1 - p _ 0 } , v _ 1 ' = \\frac { f ^ { n _ 1 } ( p _ 1 ) - p _ 0 } { p _ 1 - p _ 0 } , \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} S _ { 2 m + 1 , \\sigma } ( t , \\pi ) = \\dfrac { ( - 1 ) ^ m } { 2 \\pi ( 2 m + 2 ) ! } \\ , \\bigg ( \\dfrac { 3 } { 2 } - \\sigma \\bigg ) ^ { 2 m + 2 } \\log C ( t , \\pi ) - \\dfrac { ( - 1 ) ^ m } { \\pi ( 2 m ) ! } \\ , \\displaystyle \\sum _ { \\gamma } f _ { 2 m + 1 , \\sigma } ( t - \\gamma ) + O _ m ( d ) , \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\frac { m _ i z _ * } { 1 + m _ i z _ * } = 1 . \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} \\begin{aligned} p ( y ) = { } & \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\biggl [ 1 + \\tau ^ { 1 / 2 } \\left ( \\frac { 8 } { 3 } y ^ { 3 } - 2 y \\right ) \\\\ & + \\tau \\left ( \\frac { 3 2 } { 9 } y ^ { 6 } - \\frac { 3 1 } { 3 } y ^ { 4 } + \\frac { 1 5 } { 2 } y ^ { 2 } - \\frac { 3 7 } { 4 8 } \\right ) \\biggr ] e ^ { - 2 y ^ { 2 } } + O ( \\tau ^ { 3 / 2 } ) \\end{aligned} \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} c = G ( 1 + \\beta ) G ( 1 - \\beta ) E [ \\tau ] \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} { n ; b ; \\omega _ n ^ { g ' } \\brack k } = | B _ { n , k } ^ { g } | . \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} \\left | \\varphi ( u ) \\right | = | \\psi ( u ^ { r } ) | \\le C M _ { 0 } . \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} \\phi b ^ { - 1 } x b ^ { m _ k } \\sim \\phi b ^ { n _ 0 } x b ^ { m _ k } + \\sum _ { i = n _ 0 + 1 } ^ { - 1 } \\sum _ { s \\in S _ b } \\phi s b ^ i x b ^ { m _ k } . \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} S _ M : = \\left \\{ v \\in H ^ 1 \\ : \\ v ( \\ref { v a r i a t i o n a l p r o b l e m 1 } ) \\right \\} ; \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} ( 3 d _ 1 + d _ 2 ) x - d _ 1 y = \\frac { r _ 1 d _ 1 + k _ 1 d _ 2 - n _ 1 ( r d _ 1 + k d _ 2 ) } { r k _ 1 - k r _ 1 } . \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N S _ i + N \\lesssim H _ N + N \\lesssim \\sum _ { i = 1 } ^ N S _ i + N \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} ( f _ 1 , f _ 2 ) ^ t \\cdot ( g _ 1 , g _ 2 ) ^ t = \\int f _ 1 g _ 1 + \\int f _ 2 g _ 2 . \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} \\overline { \\tilde { S } ^ { R , R } _ { X , Y } } = \\tilde { S } ^ { R , R } _ { X , \\bar { Y } } \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} S _ m ( \\omega ) : = \\sum _ { j = 1 } ^ m e [ \\omega ( j ^ 2 - j ) ] , \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} g ( \\bar x ) & \\leq \\lambda f ( x ) + r ( x ) + ( 1 - \\lambda ) ( f ( y ) + r ( y ) ) \\\\ & \\qquad + \\lambda \\tau \\Phi ( x ) + ( 1 - \\lambda ) \\tau \\Phi ( y ) + \\lambda \\rho \\Phi ( x ) + ( 1 - \\lambda ) \\rho \\Phi ( y ) \\\\ & = \\lambda [ f ( x ) + r ( x ) + ( \\tau + \\rho ) \\Phi ( x ) ] + ( 1 - \\lambda ) [ f ( y ) + r ( y ) + ( \\tau + \\rho ) \\Phi ( y ) ] \\\\ & \\leq \\lambda g ( x ) + ( 1 - \\lambda ) g ( y ) . \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} | | \\Phi | | ^ 2 = \\int d \\mu ( z , \\ell ) \\frac { | z | ^ { \\ell } } { I _ { \\ell } ( 2 | z | ) } | f ( z ) | ^ 2 < \\infty . \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} r _ { \\partial } & : = r | _ { U _ { \\partial } } : U _ { \\partial } \\rightarrow U _ { \\partial } \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} \\sum \\limits _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) t ^ { \\deg _ { \\theta } ( a ) } } { a } = \\log _ { \\psi } ( 1 ) . \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} \\nabla \\Psi _ z ( y ) ^ T f ( y , u ) = 2 \\sum _ { i = 1 } ^ k ( F _ i ( y ) - z _ i ) \\left ( \\nabla F _ i ( y ) ^ T f ( y , u ) \\right ) = 0 \\ \\ \\ \\ \\forall \\ ( y , u ) \\in Y \\times U . \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} \\bigg | \\int _ { \\R ^ 3 } \\big ( \\rho \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) - \\rho _ { \\infty } \\bar \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) \\big ) h \\bigg | & \\leq \\left | \\int _ { \\R ^ 3 } ( \\rho - \\rho _ { \\infty } ) \\phi _ { ( u _ n - v _ 0 ) } ( u _ n - v _ 0 ) h \\right | \\\\ & \\quad + \\left | \\int _ { \\R ^ 3 } \\rho _ { \\infty } ( \\phi _ { ( u _ n - v _ 0 ) } - \\bar \\phi _ { ( u _ n - v _ 0 ) } ) ( u _ n - v _ 0 ) h \\right | \\\\ & = : I _ 1 + I _ 2 . \\\\ \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} \\dot { \\xi } _ j = \\Pi _ { 2 , j } [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] ( t ) , \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} R = \\frac { R _ 1 } { \\sqrt { 2 } } , \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} h ^ * ( p _ 1 , p _ 2 ) & = n _ 1 p _ 1 + n _ 2 p _ 2 + n e ^ { - p _ 1 A _ { 2 1 } - p _ 2 A _ { 1 2 } } \\\\ & = \\frac { n _ 1 } { A _ { 2 1 } } ( p _ 1 A _ { 2 1 } + p _ 2 A _ { 1 2 } ) + n e ^ { - p _ 1 A _ { 2 1 } - p _ 2 A _ { 1 2 } } \\\\ & = \\frac { n _ 1 } { A _ { 2 1 } } \\ln \\frac { n A _ { 2 1 } } { n _ 1 } + n e ^ { - \\ln \\frac { n A _ { 2 1 } } { n _ 1 } } = \\frac { n _ 1 } { A _ { 2 1 } } ( 1 + \\ln \\frac { n A _ { 2 1 } } { n _ 1 } ) \\\\ \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} M _ t = \\int _ 0 ^ t \\nabla u ^ * ( r , W _ r ) \\mathrm d W _ r , \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} \\sup _ { a , b > 0 } \\frac { 4 ( 1 + a b ) ( 1 - e ^ { - 2 t } ) - 3 + 2 ( 2 e ^ { - 2 t } - 1 ) ( a + b ) } { a ^ 2 b ^ 2 } \\\\ [ 1 e x ] \\ge \\sup _ { a > 0 , b = 1 / a } 8 ( 1 - e ^ { - 2 t } ) - 3 + 2 ( 2 e ^ { - 2 t } - 1 ) ( a + 1 / a ) = + \\infty . \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } | D _ n ( x ) | d x = \\frac { 4 } { \\pi ^ 2 } \\log n + \\mathcal { O } ( 1 ) , \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} G = d s ^ 2 + g _ s . \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} J ( \\mu ) : = \\int _ { \\mathbb { R } } \\frac { | f ' ( x ) | ^ 2 } { f ( x ) } d x \\end{align*}"}
-{"id": "10002.png", "formula": "\\begin{align*} \\bar F _ t ^ n ( q ) = ( 1 - \\bar p ^ n ( q ) ) \\int _ t ^ \\infty e ^ { q ( \\tau ^ n ( t ) - \\tau ^ n ( s ) ) } d s . \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} \\epsilon ( S ( n , k ) ) \\equiv ( - 1 ) ^ r \\epsilon ( n ! / k ! ) \\binom { n + r } { r } \\mod p . \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} H = \\frac { 1 } { \\sqrt { 2 n } } \\begin{pmatrix} \\mathcal { N } ( 0 , 2 ) & \\chi _ { ( n - 1 ) \\beta } & & & \\\\ \\chi _ { ( n - 1 ) \\beta } & \\mathcal { N } ( 0 , 2 ) & \\chi _ { ( n - 2 ) \\beta } & & \\\\ & \\ddots & \\ddots & \\ddots & \\\\ & & \\chi _ { 2 \\beta } & \\mathcal { N } ( 0 , 2 ) & \\chi _ { \\beta } \\\\ & & & \\chi _ { \\beta } & \\mathcal { N } ( 0 , 2 ) \\end{pmatrix} \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} c _ { \\mu , h } = \\frac { 1 } { \\mu ( h ) \\mu _ h ( h ^ { - 1 } ) } . \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} S ^ \\alpha { } _ \\beta = \\begin{pmatrix} 0 & 0 & 0 & - 2 m a \\sin ( 2 m x ^ 4 ) \\\\ 0 & 0 & 0 & 2 m a \\cos ( 2 m x ^ 4 ) \\\\ 0 & 0 & 0 & 2 m b \\\\ 2 m a \\sin ( 2 m x ^ 4 ) & - 2 m a \\cos ( 2 m x ^ 4 ) & - 2 m b & - c \\end{pmatrix} . \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} r _ + ( x ) : = \\frac D 2 \\frac { \\phi _ { 3 , + } '' ( x ) } { \\phi _ { 3 , + } ( x ) } \\ge \\frac { D \\lambda _ 2 } { \\gamma _ 2 + x ^ 2 } . \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} e _ { \\delta } = \\frac { 1 } { m } \\left ( \\frac { 1 } { k _ 1 } N _ 1 ( \\delta ) + \\frac { 1 } { k _ 2 } N _ 2 ( \\delta ) + \\dotsb + \\frac { 1 } { k _ m } N _ m ( \\delta ) \\right ) . \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} E ( \\tilde { V } ) = \\frac { \\lambda ^ 2 } { 2 } \\| V \\| ^ 2 _ { \\dot { H } ^ 1 _ c } - \\frac { \\lambda ^ { \\alpha + 2 } } { \\alpha + 2 } \\| V \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } . \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} f ^ { ( R ) } _ k ( x , y ) = f _ k ( x / R , y ) \\textup { f o r $ k = 1 , \\ldots , m $ . } \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} & \\max _ { x , b } \\| \\nabla f ( x ) \\| ^ 2 \\\\ & A x = b \\\\ & x \\in \\mathcal { X } , b \\in \\mathcal { B } \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} \\big \\langle \\varphi _ { v ^ { - 1 } } \\varphi _ { v ^ { - 1 } } ^ { v ^ { - 1 } } , x \\big \\rangle & = \\mu ^ l ( v ) ^ { - 2 } \\mu ^ l \\big ( v g ^ { - 1 } x ' \\big ) \\mu ^ l \\big ( g ^ { - 1 } x '' \\big ) = \\mu ^ l ( v ) ^ { - 2 } \\big \\langle \\mu ^ l ( v ? ) \\mu ^ l , g ^ { - 1 } x \\big \\rangle \\\\ & = \\mu ^ l ( v ) ^ { - 1 } \\mu ^ l ( g ^ { - 1 } x ) = \\varphi _ { v ^ { - 1 } } ^ { v ^ { - 1 } } ( x ) . \\end{align*}"}
-{"id": "9853.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\tilde { c } ( n ) q ^ { n } = \\dfrac { E _ { 1 0 } ^ { 3 } } { E _ { 1 } ^ 2 E _ { 2 } ^ 2 E _ { 5 } } . \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} \\partial _ t E _ { s , t } ( Q ) = ( A - \\phi _ t ( Q ) S ) \\ , E _ { s , t } ( Q ) \\quad \\mbox { \\rm a n d } \\partial _ s E _ { s , t } ( Q ) = - E _ { s , t } ( Q ) \\ , ( A - \\phi _ s ( Q ) S ) \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} M ( z ) = \\frac { z ( \\theta _ + s - \\theta _ - ) - \\theta _ + \\theta _ - ( s - 1 ) } { z ( s - 1 ) - ( \\theta _ - s - \\theta _ + ) } \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} \\mathbf { T } _ { \\varepsilon } ( \\mu ) ( x ) : = \\mathbf { T } _ { \\varepsilon } ^ { \\mathbf { K } } ( \\mu ) ( x ) & = \\sup _ { \\rho \\in ( 0 , \\rho _ 0 ) , e \\in S ^ { d - 1 } } \\frac { \\varepsilon ^ { - d + 1 } } { \\rho ^ \\alpha } \\left | \\left ( \\frac { 1 } { | \\cdot | ^ { d - \\alpha } } \\phi ^ { e , \\varepsilon } _ \\rho ( \\cdot ) \\right ) \\star \\mathbf { K } \\star \\mu ( x ) \\right | ~ ~ \\forall ~ x \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} \\frac { A _ n ( t ) } { ( 1 - t ) ^ { n + 1 } } = \\sum _ { k = 0 } ^ { \\infty } ( k + 1 ) ^ n t ^ { k } , \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} H ( \\mathcal { Q } _ { b } ( y _ { B } ) ~ | ~ \\mathcal { Q } _ { b } ( y _ { C } ) , \\mathcal { Q } _ { b } ( y _ { B } ) \\in \\mathcal { C } ) = - \\sum _ { j = 1 } ^ { 2 ^ b } g _ j \\log _ { 2 } { g _ j } , \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} A _ t ^ { Y ^ i , W } ( b ) = A _ t ^ { W , W } ( ( \\nabla \\gamma ^ i ) ^ * \\ , b ) . \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{gather*} K ' _ k ( t , x ) = H ( t , x ) \\ \\ ( x \\in M \\backslash \\iota ( B ( 3 r ) ) ) \\\\ K ' _ k ( t , \\iota ( z ) ) = \\frac { 1 } { k } \\rho ( | \\iota ( z ) | ^ 2 ) \\ \\ ( z \\in B ( 3 r ) ) \\end{gather*}"}
-{"id": "1273.png", "formula": "\\begin{align*} \\bigcup _ { n \\geq 0 } \\tilde { f } _ k ^ n ( \\mathcal { B } _ w ( \\lambda ) ) \\setminus \\{ 0 \\} = \\mathcal { B } _ { s _ k w } ( \\lambda ) . \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{gather*} \\big [ \\big [ \\big [ x _ i ^ + , x _ j ^ + \\big ] , x _ i ^ - \\big ] , x _ { j , 1 } ^ - \\big ] = \\big [ \\big [ h _ i , x _ j ^ + \\big ] , x _ { j , 1 } ^ - \\big ] = \\big [ { - } x _ j ^ + , x _ { j , 1 } ^ - \\big ] = - h _ { j , 1 } \\end{gather*}"}
-{"id": "8278.png", "formula": "\\begin{align*} \\mathcal { D } ( p _ { _ 1 } | | p _ { _ 0 } ) & = N \\times \\mathcal { D } ( p _ { _ 1 } ( y [ i ] ) | | p _ { _ 0 } ( y [ i ] ) ) , \\\\ \\mathcal { D } ( p _ { _ 0 } | | p _ { _ 1 } ) & = N \\times \\mathcal { D } ( p _ { _ 0 } ( y [ i ] ) | | p _ { _ 1 } ( y [ i ] ) ) , \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} & \\left \\langle \\frac { 1 } { ( \\zeta q ^ { x _ N ( t ) + N } ; q ) _ { \\infty } } \\right \\rangle = \\det \\left ( 1 + K _ { \\zeta } \\right ) _ { L ^ 2 ( C _ a ) } , \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} \\eta _ { { \\bf i } , \\mathcal I } ( A ( t ) \\cdot x ) = t \\cdot \\eta _ { { \\bf i } , \\mathcal I } ( x ) \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} \\frac { 1 } { \\widetilde { \\alpha } p ' e } = D ^ { p ' } = F ^ { p ' } , \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} \\begin{cases} H _ { 0 } : & \\beta _ { 1 } = \\beta _ { 2 } = . . . = \\beta _ { p } = 0 , o r \\mbox { g S N R } = 0 , \\\\ H _ { a } : & \\mbox { g S N R } \\geq \\lambda _ 0 . \\end{cases} \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{gather*} \\alpha , S ^ { \\delta _ n ^ k + 1 - \\alpha _ k } \\left ( \\alpha \\right ) , \\dots , S ^ { \\delta _ n ^ k + 1 + u - \\alpha _ { k - u } } \\left ( \\alpha \\right ) , \\dots , S ^ { \\delta _ n ^ k + k - \\alpha _ { 1 } } \\left ( \\alpha \\right ) , S ^ { \\delta _ n ^ k + k + 1 } \\left ( \\alpha \\right ) = \\alpha . \\end{gather*}"}
-{"id": "7377.png", "formula": "\\begin{align*} \\hat { \\mathfrak { g } } _ p : = \\mathfrak { g } [ \\pi ^ { - 1 } ( \\mathbb { D } ^ \\times _ p ) ] ^ \\Gamma \\oplus \\mathbb { C } C \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} N ( k , t _ 2 , c _ 2 ) = 2 \\sum _ { i = 0 } ^ { t _ 2 - 3 } ( k - 1 ) ^ i + ( k - 1 ) ^ { t _ 2 - 2 } + \\frac { k ( k - 1 ) ^ { t _ 2 - 2 } } { c _ 2 } , \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} r _ i ( x x ' ) = q ^ { - \\langle \\alpha _ i , \\mu \\rangle } x r _ i ( x ' ) + r _ i ( x ) x ' , r _ i ' ( x x ' ) = x r _ i ' ( x ' ) + q ^ { - \\langle \\alpha _ i , \\mu ' \\rangle } r _ i ' ( x ) x ' . \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\mathbf { B } ^ i = \\sum _ { j = 1 } ^ { m } \\mathbf { K } _ { j R } ^ i \\star b _ { j R } ~ ~ ~ B _ { 2 R } . \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} 1 + \\sum _ { n \\geq 1 } \\frac { t \\alpha _ n ( t , q ) } { ( 1 - t ) ^ n } \\frac { x ^ n } { 2 ^ { n - 1 } n ! } = \\sum _ { v \\geq 0 } \\left ( \\frac { 2 t } { 1 - t } \\right ) ^ v \\left [ \\sum _ { u \\geq 0 } \\left ( \\frac { q - t } { 1 - t } \\right ) ^ u \\left ( e ^ { x / 2 } - 1 \\right ) ^ u \\right ] ^ v \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} & \\left \\langle \\frac { 1 } { ( \\zeta q ^ { \\l _ N } ; q ) _ { \\infty } } \\right \\rangle = \\det \\left ( 1 + K _ { \\zeta } \\right ) _ { L ^ 2 ( C _ a ) } , \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} \\mathbb { A } ^ \\alpha ( x ) = a \\begin{pmatrix} 1 \\\\ - i \\\\ 0 \\\\ 0 \\end{pmatrix} e ^ { 2 i m x ^ 4 } \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} c ( x ' , y , t ) = \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } \\left [ c ( x , y , t ) + \\frac { 1 } { 4 } h ( x ) ( 2 x ^ { 2 } - 1 ) \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) - \\frac { 1 } { 6 } x ^ { 3 } h ( x ) \\frac { \\partial ^ { 3 } } { \\partial y ^ { 3 } } p _ { 0 } ( y , t ) \\right ] \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} W _ { n } ( w ) = \\amalg _ { i = 1 } ^ n \\amalg _ { j = 0 } ^ n d L _ { i } ( w ) \\cdot M _ { n } ( w ) \\cdot d R _ { j } ( w ) \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} \\int _ { Y \\times U } q ( y , u ) \\gamma ^ { \\lambda } _ { u ( \\cdot ) } ( d y , d u ) = \\lambda \\int _ 0 ^ \\infty e ^ { - \\lambda t } q ( y ( t ) , u ( t ) ) d t \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { n - 1 } ( 1 + t q ^ j ) \\cdot \\sum _ { j = 0 } ^ { n - 1 } \\frac { j q ^ { j - 1 } t } { 1 + t q ^ j } = \\sum _ { r = 0 } ^ { n } q ^ { \\binom { r } { 2 } - 1 } t ^ r \\Big ( \\binom { r } { 2 } { n \\brack r } _ q + q { n \\brack r } ^ { ' } _ { q } \\Big ) . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} \\Psi ( \\alpha , \\beta ) : = ( \\psi _ 1 ( \\alpha , \\beta ) , \\psi _ 2 ( \\alpha , \\beta ) ) : = \\left ( I ' ( \\varphi ( \\alpha , \\beta ) ) [ \\varphi ( \\alpha , \\beta ) ^ + ] , I ' ( \\varphi ( \\alpha , \\beta ) ) [ \\varphi ( \\alpha , \\beta ) ^ - ] \\right ) . \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} A _ 0 \\partial _ t u + \\sum _ { j = 1 } ^ 3 A _ j \\partial _ j u + D u & = f , & & x \\in \\R ^ 3 _ + , & t \\in J ; \\\\ B u & = g , & & x \\in \\partial \\R ^ 3 _ + , & t \\in J ; \\\\ u ( t _ 0 ) & = u _ 0 , & & x \\in \\R ^ 3 _ + ; \\end{aligned} \\right . \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} \\P ( J _ { n + 1 } = J _ { n } | J _ { n } ) = \\frac { n } { n + 1 } \\mbox { a n d } \\P ( J _ { n + 1 } = n + 1 | J _ { n } ) = \\frac { 1 } { n + 1 } . \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} \\int e ^ { - 2 y ^ { 2 } } d y = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } , \\ \\ \\ \\ \\ \\int y ^ { 2 } e ^ { - 2 y ^ { 2 } } d y = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\frac { 1 } { 4 } , \\ \\ \\ \\ \\ \\ \\int y ^ { 4 } e ^ { - 2 y ^ { 2 } } d y = \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\frac { 3 } { 1 6 } , \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} P _ { X } ( h , \\lambda ) : = \\mathbf { A } _ h ( N _ 0 , X ) ( - h \\mathbf { X } - h \\lambda ) \\mathbf { A } _ h ( N _ 0 , X ) ^ { - 1 } , \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} L _ 1 \\widetilde { F } _ { \\eta } ( ( \\widetilde { v } _ { g , k } , 0 ) , \\omega _ { k , 1 } \\oplus \\omega _ { k , 2 } ) L _ 1 ^ * = & d \\Gamma ( \\omega _ 1 ) \\otimes 1 + 1 \\otimes d \\Gamma ( \\omega _ { k _ 2 } ) \\\\ & + \\eta W ( \\widetilde { v } _ { g , k } , - 1 ) \\otimes \\Gamma ( - 1 ) . \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { \\mu } \\frac { f _ \\mu } { \\mu ! } \\ , z ^ \\mu ( f _ { \\mu } = \\partial _ { z } ^ { \\mu } f ( 0 ) ) . \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} K _ t ( m , n ) = \\int _ X e ^ { ( x - s ^ { + } ) t } p _ m ( x ) p _ n ( x ) \\ , d \\mu ( x ) , \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t E _ u ^ { \\frac 1 2 } , v \\Big { ) } - \\gamma ( \\nabla E _ \\sigma ^ { \\frac 1 2 } , \\nabla v ) + ( \\nabla E _ u ^ { \\frac 1 2 } , \\nabla v ) + ( f ( u ( t _ { \\frac 1 2 } ) ) - f ^ { \\frac 1 2 } ( u ) , v ) = & ( R _ 3 ^ { \\frac 1 2 } , v ) , ~ \\forall v \\in H _ 0 ^ 1 , \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\mathbb { A } ^ { - 1 } _ L = \\mathbb { A } ^ { - 1 } _ R = \\mathbb { A } _ \\mathcal { G } ^ { - 1 } = & \\left ( \\begin{matrix} ( a - b d ^ { - 1 } c ) ^ { - 1 } & - ( d b ^ { - 1 } a - c ) ^ { - 1 } \\\\ - ( a c ^ { - 1 } d - b ) ^ { - 1 } & ( d - c a ^ { - 1 } b ) ^ { - 1 } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} \\phi _ k = \\psi _ k + x _ j ( v / x _ j - v ' ) \\quad v / x _ j - v ' \\in I _ H . \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} d _ { C _ 1 \\times C _ 2 } ( x _ 1 , x _ 2 ) = d _ { C _ 1 } ( x _ 1 ) + d _ { C _ 2 } ( x _ 2 ) ( x _ 1 , x _ 2 ) \\in C _ 1 \\times C _ 2 . \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} K ( z , w ; Z , W ) = 4 \\pi \\int _ { 0 } ^ { \\infty } t H _ p ( t ; w , W ) e ^ { i 2 \\pi ( z - \\overline { Z } ) t } \\d t \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\overline { \\gamma } \\in \\overline { \\Gamma } \\\\ \\ell ( \\gamma ) \\geq \\ell _ { 0 } ( \\widetilde { X } ) } } e ^ { - \\ell ( \\gamma ) \\mathrm { R e } ( s ) } = \\mathrm { R e } ( s ) \\int _ { \\ell _ { 0 } ( \\widetilde { X } ) } ^ { \\infty } e ^ { - \\mathrm { R e } ( s ) x } \\Pi _ { \\Gamma } ( x ) d x \\leq C \\frac { \\mathrm { R e } ( s ) e ^ { - ( \\mathrm { R e } ( s ) - \\delta ) \\ell _ { 0 } ( \\widetilde { X } ) } } { \\mathrm { R e } ( s ) - \\delta } . \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} A _ 0 \\partial _ t u _ \\tau & + \\sum _ { j = 1 } ^ 3 A _ j \\partial _ j u _ \\tau + D u _ \\tau = f _ \\tau - \\tau \\Big ( A _ 0 - \\sum _ { j = 1 } ^ 3 \\partial _ j \\psi A _ j \\Big ) u _ \\tau \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} X _ t \\stackrel { d } { = } e ^ { - \\alpha t } X _ 0 + \\sqrt { \\frac { 1 - e ^ { - 2 \\alpha t } } { \\alpha } } Z \\end{align*}"}
-{"id": "9854.png", "formula": "\\begin{align*} \\dfrac { 1 } { E _ 1 ^ 2 E _ 2 ^ 2 } = : \\sum _ { n = 0 } ^ \\infty b ( n ) q ^ n \\end{align*}"}
-{"id": "9883.png", "formula": "\\begin{align*} f = \\boldsymbol { D } ^ { - s } u + \\boldsymbol { D } ^ { s * } u . \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = e _ 2 , [ e _ 1 , e _ 3 ] = e _ 3 . \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} \\frac { T _ { j + 1 } - T _ { j } } { \\varepsilon } = - ( u _ { j } \\cdot \\nabla ) T _ { j } + J ( T _ { j } - \\mathrm { i d } _ { \\Omega } ) \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} \\ell _ 1 & : = F ( 1 , 1 ) = ( a _ 0 + a _ 2 ) + ( a _ 1 + a _ 3 ) + ( b _ 0 + b _ 2 ) + ( b _ 1 + b _ 3 ) , \\\\ \\ell _ 2 & : = F ( 1 , - 1 ) = ( a _ 0 + a _ 2 ) + ( a _ 1 + a _ 3 ) - ( b _ 0 + b _ 2 ) - ( b _ 1 + b _ 3 ) , \\\\ \\ell _ 3 & : = F ( - 1 , 1 ) = ( a _ 0 + a _ 2 ) - ( a _ 1 + a _ 3 ) + ( b _ 0 + b _ 2 ) - ( b _ 1 + b _ 3 ) , \\\\ \\ell _ 3 & : = F ( - 1 , - 1 ) = ( a _ 0 + a _ 2 ) - ( a _ 1 + a _ 3 ) - ( b _ 0 - b _ 2 ) + ( b _ 1 + b _ 3 ) . \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} H _ { \\Lambda , \\epsilon } ( \\phi ) = H _ { \\Lambda } ( \\phi ) + \\epsilon \\sum _ { j \\in \\Lambda } \\phi _ j ^ 2 , \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{gather*} \\forall \\ , x \\in H , \\big ( \\mathrm { i d } \\otimes \\mu ^ l \\big ) \\circ \\Delta ( x ) = \\mu ^ l ( x ) 1 \\big ( \\ \\big ( \\mu ^ r \\otimes \\mathrm { i d } \\big ) \\circ \\Delta ( x ) = \\mu ^ r ( x ) 1 \\big ) . \\end{gather*}"}
-{"id": "5227.png", "formula": "\\begin{align*} f _ i ( x ) = ( \\alpha - \\beta \\sum _ { j = 1 } ^ { N } x _ j ) x _ i - h _ i ( x _ i ) \\ ( i = 1 , \\ldots , N ) , \\end{align*}"}
-{"id": "9873.png", "formula": "\\begin{align*} \\begin{aligned} \\tau _ h ( \\boldsymbol { D } ^ { - \\mu } u ) = \\boldsymbol { D } ^ { - \\mu } ( \\tau _ h u ) & , \\tau _ h ( \\boldsymbol { D } ^ { - \\mu * } u ) = \\boldsymbol { D } ^ { - \\mu * } ( \\tau _ h u ) \\\\ \\Pi _ \\kappa ( \\boldsymbol { D } ^ { - \\mu } u ) = \\kappa ^ \\mu \\boldsymbol { D } ^ { - \\mu } ( \\Pi _ \\kappa u ) & , \\Pi _ \\kappa ( \\boldsymbol { D } ^ { - \\mu * } u ) = \\kappa ^ \\mu \\boldsymbol { D } ^ { - \\mu * } ( \\Pi _ \\kappa u ) . \\end{aligned} \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} \\| r ^ { - \\beta } u ( t , z ) \\| _ { L ^ 2 _ t ( \\R ; L ^ 2 ( X ) ) } \\leq C \\left ( \\| u _ 0 \\| _ { \\dot H ^ { \\beta - \\frac 1 2 } ( X ) } + \\| u _ 1 \\| _ { \\dot H ^ { \\beta - \\frac 3 2 } ( X ) } \\right ) , \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} P \\{ A _ { I , J } \\} = \\prod _ { i \\in I } p _ i ^ { - 1 } \\prod _ { i \\in J } ( 1 - p _ i ^ { - 1 } ) \\ , . \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} ( e ^ { - t \\mathbb { E } ^ { * } \\mathbb { E } } f ) ( x ) & = \\frac { e ^ { - t Q ^ { 2 } / 4 } } { \\sqrt { 4 \\pi t } } r ^ { - Q / 2 } \\int ^ { \\infty } _ { 0 } e ^ { - \\frac { ( \\ln r - \\ln s ) ^ { 2 } } { 4 t } } s ^ { - Q / 2 } f ( s y ) s ^ { Q - 1 } d s \\\\ & = \\frac { e ^ { - t Q ^ { 2 } / 4 } } { \\sqrt { 4 \\pi t } } | x | ^ { - Q / 2 } \\int ^ { \\infty } _ { 0 } e ^ { - \\frac { ( \\ln | x | - \\ln s ) ^ { 2 } } { 4 t } } s ^ { - Q / 2 } f ( s y ) s ^ { Q - 1 } d s . \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} f ( x ) & = ( x ^ 2 + 1 ) + ( A + B x ) ( x ^ 3 + 1 ) ( 1 - x ) + m \\ : h ( x ) , \\\\ g ( x ) & = x ( 1 - x ) + x ( A + B x ) ( x ^ 3 + 1 ) ( 1 - x ) + m \\ : h ( x ) , \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} 0 = \\frac { \\partial } { \\partial y } ( y \\beta ( y ) ) + \\frac { 1 } { 4 } \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\beta ( y ) + \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\left ( \\frac { 6 4 } { 3 } y ^ { 6 } - 6 8 y ^ { 4 } + 4 6 y ^ { 2 } - \\frac { 1 5 } { 4 } \\right ) e ^ { - 2 y ^ { 2 } } \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} p _ { 0 } ( y , t ) = \\int d x \\ \\varphi ( x , y , t ) \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\widehat { f } ( y ) = c _ p \\int _ { \\Q _ p } f ( x ) \\psi _ p ( x y ) d \\mu ( x ) , c _ p = \\begin{cases} 1 & p < \\infty , \\\\ \\frac { 1 } { \\sqrt { 2 \\pi } } & p = \\infty . \\end{cases} \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} \\pi _ { t , k } ( \\varphi ) \\ , : = \\ , \\mathbb E [ \\varphi ( t , \\overline { X } _ { t , 1 } , \\ldots , \\overline { X } _ { t , k } ) \\ , \\vert \\ , \\mathcal F _ { t } ^ { X } ] \\ , ; k \\ , = \\ , 2 , \\ldots , n , \\ , \\ , 0 \\le t \\le T \\ , \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} \\textbf { R } _ { C } ( 0 , 2 ) = \\Phi ( 2 ) = \\int _ { 0 } ^ { \\infty } \\frac { \\cos ( 2 \\pi x ) } { \\{ - 1 + \\exp { ( 2 \\pi \\sqrt { x } ) } \\} } d x = \\frac { 1 } { 1 6 } , \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} \\mathcal { L } _ 0 u = \\varphi \\ , , \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} A ^ \\alpha ( x ) = a \\begin{pmatrix} \\cos ( x ^ 3 + x ^ 4 ) \\\\ \\pm \\sin ( x ^ 3 + x ^ 4 ) \\\\ 0 \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} \\eta = \\lambda \\mu _ \\lambda t \\Longrightarrow t = \\frac { \\eta } { \\lambda \\mu _ \\lambda } \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} \\lambda ^ { - 1 } ( y - \\tau ^ { 1 / 2 } x ) = ( 1 + \\tau + \\frac { 1 } { 2 } \\tau ^ { 2 } ) ( y - \\tau ^ { 1 / 2 } x ) \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} \\overline { h } ( X , Y ) = h ( X , Y ) + \\beta \\eta ( Y ) \\phi Q X , \\end{align*}"}
-{"id": "9979.png", "formula": "\\begin{align*} u ( x , \\tau ) = u _ 0 ( x + \\tau ) + \\int _ 0 ^ \\tau p ( x + \\tau - s ) \\alpha ( s ) d s . \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} \\deg E & = ( g + d - 1 ) , \\\\ \\deg N _ i & = ( d - 2 - i ) ( g + d - 1 ) { { d - 2 } \\choose i - 1 } . \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} \\check { I } _ { d - 1 } \\hat { I } _ { d - 1 } ^ \\top = J _ d ( 0 ) . \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} \\mu _ t ( x , \\theta , C ) = \\eta ( x , \\theta ) \\delta ( C - \\hat { C } ( t , x , \\theta ) ) , \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} ( \\mathbb { F } _ + ) ^ { \\alpha \\beta } = i \\sigma ^ { \\alpha \\dot b a } \\ , \\theta _ { \\dot b } { } ^ { \\dot c } \\ , \\sigma ^ \\beta { } _ { \\dot c a } \\ , . \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} K ( n , i , j ) = \\sum _ { k = 0 } ^ { n - 1 } \\binom { k } { i } \\binom { n - 1 - k } { i + j - k } A ( n , k ) , \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\frac { d } { d t } x _ i ( t , s , \\xi ) = - \\lambda _ i \\big ( x _ i ( t , s , \\xi ) \\big ) \\mbox { a n d } x _ i ( s , s , \\xi ) = \\xi \\mbox { i f } k + 1 \\le i \\le k + m . \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} \\mathrm { v a r } _ \\mu ( f ) = \\int _ { \\R ^ n } f ^ 2 \\ , \\mathrm d \\mu - \\left ( \\int _ { \\R ^ n } f \\ , \\mathrm d \\mu \\right ) ^ 2 . \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} - \\Delta ( c + [ a , c ] ( a - b d ^ { - 1 } c ) ^ { - 1 } ) & = - \\Delta ( c ( a - b d ^ { - 1 } c ) + [ a , c ] ) ( a - b d ^ { - 1 } c ) ^ { - 1 } \\\\ & = - \\Delta \\ , ( - c b d ^ { - 1 } + a ) c ( a - b d ^ { - 1 } c ) ^ { - 1 } \\\\ & = - \\Delta \\ , ( - c b + a d ) d ^ { - 1 } c ( a - b d ^ { - 1 } c ) ^ { - 1 } \\\\ & = - d ^ { - 1 } c ( a - b d ^ { - 1 } c ) ^ { - 1 } \\\\ & = - ( a c ^ { - 1 } d - b ) ^ { - 1 } , \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} \\exp ( \\psi ( t ) U ) h \\exp ( - t U ) = \\exp ( \\alpha _ t V ) \\exp ( \\beta _ t X ) , \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} D _ K = \\frac { 2 ^ { n - 1 } ( n - 1 ) + m _ K n + 1 } { 2 ^ n n } . \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} \\sigma _ 1 \\cdot \\sigma _ 2 = \\sigma _ 2 \\cdot \\sigma _ 3 = \\sigma _ 1 \\cdot \\sigma _ 3 = \\deg f = 8 \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} f _ { d - 1 } ( \\theta ) = - \\frac { c ( k - c ) \\sqrt { ( k - 1 ) ^ { d - 2 } } } { \\tau ^ { d - 2 } ( ( k - 1 ) \\tau ^ 2 + c - 1 ) } , \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{gather*} \\overset { \\mathcal { X } ^ + ( 2 ) } { W } = - q a - q ^ { - 1 } d = - \\hat q ^ 2 F E - q K - q ^ { - 1 } K ^ { - 1 } = - \\hat q ^ 2 C , \\end{gather*}"}
-{"id": "7424.png", "formula": "\\begin{align*} \\dot { \\hat { \\rho } } \\circ \\dot { \\sigma } ( x ) ( v _ + ) = x \\cdot v _ + + \\lambda ( x ) c v _ + = \\lambda ( x ) c v _ + . \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} u ^ * _ { \\mu , \\xi } ( x , t ) = u _ { \\mu , \\xi } ( x , t ) + \\tilde { \\Phi } ( x , t ) \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} \\begin{cases} u _ { \\tau } & = p \\\\ p _ { \\tau } & = - \\lambda u [ 1 - u ^ 2 ] \\\\ x _ \\tau & = 1 . \\end{cases} \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} \\zeta _ { M } ( s ) = \\prod _ { p \\mid M } \\zeta _ p ( s ) , \\ \\abs { \\cdot } _ M = \\prod _ { p \\mid M } \\abs { \\cdot } _ p ( m , M ^ { \\infty } ) = \\prod _ { p \\mid M } p ^ { v _ p ( m ) } . \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { - i t H } P _ c & = \\int _ 0 ^ \\infty e ^ { - i t \\lambda } E ( \\lambda ) \\ , d \\lambda , \\\\ E ( \\lambda ) & : = \\tfrac { 1 } { 2 \\pi i } [ R ( \\lambda + i 0 ) - R ( \\lambda - i 0 ) ] . \\end{aligned} \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} \\Theta ( q , x ) = \\prod _ { j = 1 } ^ { \\infty } ( 1 - q ^ j ) ( 1 + x q ^ j ) ( 1 + q ^ { j - 1 } / x ) ~ . \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} ( \\rho ( m _ t \\gamma ) - \\rho ( \\gamma ) ) R _ 1 = - \\epsilon ( [ g _ t \\gamma ] ) R _ t . \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} \\nu _ j ( t ) = t ^ { - \\frac { 1 + \\frac { n - 2 s } { 2 s } \\bar { \\sigma } _ j b _ j ^ { 2 - 2 s } } { n - 4 s } } \\left [ d _ j + \\int _ { t _ 0 } ^ t \\tau ^ { \\frac { 1 + \\frac { n - 2 s } { 2 s } \\bar { \\sigma } _ j b _ j ^ { 2 - 2 s } } { n - 4 s } } ( P h ) _ j ( \\tau ) d \\tau \\right ] , \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} \\mathrm { E x t } _ A ^ { i } ( P , Q ) & = \\mathrm { E x t } _ A ^ { i } ( \\Omega ^ { m } ( I ) , \\Omega ^ n ( M ) ) \\\\ & = \\mathrm { E x t } ^ { i } _ A ( I , \\Omega ^ { n - m } ( M ) ) \\\\ & = 0 , \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} \\deg ( D ' | _ S ) = \\deg ( D ' | _ { S ' } ) \\geq p _ a ( S ' ) = p _ a ( S ) \\ , . \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} \\hat { h } _ 1 ( r ) = h _ 1 - 2 r \\mathrm { I I } \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} \\partial _ t P _ t ~ = ~ \\Lambda ( P _ t ) ~ : = & ~ A P _ t + P _ t A ^ { \\prime } + R - P _ t S P _ t \\\\ : = & ~ ( A - P _ t S ) P _ t + P _ t ( A - P _ t S ) ^ { \\prime } + R + P _ t S P _ t , ~ P _ 0 = Q \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} e \\left ( \\frac { a \\overline { b } } { l ^ { \\frac { n _ l } { 2 } } } m \\right ) = e \\left ( - \\frac { a \\overline { l ^ { \\frac { n _ l } { 2 } } } } { b } m \\right ) e \\left ( \\frac { a } { b l ^ { \\frac { n _ l } { 2 } } } m \\right ) \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} \\frac { 1 } { \\ , n \\ , } \\sum _ { j = 1 } ^ { n } \\sum _ { k = 1 } ^ { n } \\mathbb E [ \\overline { b } ( s , \\overline { X } _ { s , i } , \\overline { X } _ { s , j } ) \\overline { b } ( s , \\overline { X } _ { s , i } , \\overline { X } _ { s , k } ) ] \\le \\frac { 2 C } { n } \\sum _ { j = 1 } ^ { n } \\sum _ { k = j } ^ { n } \\Big [ \\frac { \\ , c ^ { k - j } \\ , } { \\ , ( k - j ) ! \\ , } \\Big ] ^ { 1 / 2 } \\le 2 C \\sum _ { k = 0 } ^ { \\infty } \\Big [ \\frac { \\ , c ^ { k } \\ , } { \\ , k ! \\ , } \\Big ] ^ { 1 / 2 } < + \\infty \\ , . \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} \\epsilon ( s ( n , k ) ) \\equiv \\epsilon ( ( n - 1 ) ! / ( k - 1 ) ! ) \\binom { k - 1 } { r } \\mod p . \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\eta _ { k _ i } \\| x ^ { k _ i } - y ^ { k _ i } \\| ^ 2 = 0 . \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} - v _ j '' + \\omega v _ j + \\left ( ( x \\pm \\tfrac { \\alpha } { 2 k - N } ) ^ 2 - 3 \\right ) v _ j = \\frac { - 1 } { \\varphi _ { k , j } ' } \\frac { d } { d x } \\left [ ( \\varphi _ { k , j } ' ) ^ 2 \\frac { d } { d x } \\left ( \\frac { v _ j } { \\varphi _ { k , j } ' } \\right ) \\right ] , x \\in \\mathbb { R } _ + \\setminus \\{ a _ k \\} . \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} u _ { t t } - \\Delta u - \\Delta u _ t = f \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{gather*} T _ { w ( 1 , m ) } ( E _ { N , k } ( - s ) ) = \\begin{cases} ( - 1 ) ^ { m - 1 } E _ { 2 , 1 } ( - s + m - 1 ) & , \\\\ ( - 1 ) ^ { ( m - 1 ) N } ( - E _ { 2 , 3 } ( - s + 2 m - 2 ) ) & , \\\\ ( - 1 ) ^ { m - 1 } ( - E _ { 2 , k + 1 } ( - s + m - 1 ) ) & , \\\\ E _ { 2 , 2 } ( - s ) + \\delta _ { s , 0 } ( m - 1 ) c & . \\end{cases} \\end{gather*}"}
-{"id": "1831.png", "formula": "\\begin{align*} f _ 1 ^ { \\prime } ( 0 ) = \\frac { 1 } { 2 } \\textbf { T r } \\left ( \\psi ^ { \\prime } ( v ) \\diamond ( d _ x + d _ s ) \\right ) = - \\frac { 1 } { 2 } \\textbf { T r } \\left ( \\psi ^ { \\prime } ( v ) \\diamond \\psi ^ { \\prime } ( v ) \\right ) = - 2 \\delta ( v ) ^ 2 \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} \\# E _ 5 ( x ) \\leq \\sum _ j \\sum _ k \\sum _ { p \\in \\mathcal { P } _ { j , k } } \\sum _ { \\substack { n \\leq x \\\\ P ( n ) | u _ n \\\\ P ( n ) = p } } 1 \\leq \\sum _ j \\sum _ k \\sum _ { p \\in \\mathcal { P } _ { j , k } } \\psi \\left ( \\frac { x } { p z _ u ( p ) } , p \\right ) \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} u _ t = a ( x , u , u _ x ) [ u _ { x x } + \\lambda u [ 1 - u ^ 2 ] ] \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} | D \\psi _ n | ^ { m } = \\lambda _ n ^ { - 1 } f + \\lambda _ n ^ { 1 / m - 1 } \\mathcal { L } [ \\psi _ n ] - 1 . \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} W _ { \\phi , p } ( g ) = W _ { \\tilde { \\pi } , p } ( g ) \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} \\phi w \\circ T ^ n = \\sum _ { u \\in W ^ { \\star } _ { n } ( w ) } \\phi u . \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ^ { k + 1 } - q \\| ^ 2 & = \\| \\beta _ k ( v ^ { k } - q ) + ( 1 - \\beta _ k ) ( T z ^ k - q \\| ^ 2 \\\\ & \\leq \\beta _ k \\| v ^ { k } - q \\| ^ 2 + ( 1 - \\beta _ k ) \\| z ^ k - q \\| ^ 2 . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t ( u - U _ h ) ^ { \\frac 1 2 } , v _ h \\Big { ) } & - \\gamma ( \\nabla ( \\sigma - \\Sigma _ h ) ^ { \\frac 1 2 } , \\nabla v _ h ) + ( \\nabla ( u - U _ h ) ^ { \\frac 1 2 } , \\nabla v _ h ) \\\\ & + ( f ^ { \\frac 1 2 } ( u ) - \\frac 1 2 \\mathfrak { F } ( U _ { h } ^ { 1 } , u _ { H } ^ { 1 } ) - \\frac 1 2 f ( U _ { h } ^ { 0 } ) , v _ h ) = 0 , \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} \\inf \\left ( e ^ { t } - \\frac { d ^ 2 } { d t ^ 2 } \\ln f _ \\alpha ( e ^ { t } ) \\right ) \\delta _ { j k , i l } = h ( \\alpha ) \\delta _ { j k , i l } . \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} \\det ( \\dot { \\sigma } ) & = 1 \\ , \\ , \\ , \\\\ & = - 1 \\ , \\ , \\ , \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} f _ { \\alpha , \\beta } ( x ) = x ^ \\alpha \\sin ( 1 / x ^ \\beta ) . \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} \\varphi \\mapsto \\int _ { \\mathbb R ^ d } \\mathcal A _ t ( \\Delta f ) ( x _ 0 ) \\varphi ( x _ 0 ) \\mathrm d x _ 0 = \\int _ 0 ^ t ( \\Delta f \\star \\varphi ) ( W _ s + x _ 0 ) \\mathrm d s . \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} g _ { ( a _ { n + 1 } , a _ { n } ) } ^ { ( a _ { n + 1 } , a _ { n } ) } ( z ) = \\left ( \\sum _ { i = 0 } ^ { a _ n - 1 } z ^ i \\right ) \\left ( \\sum _ { i = 0 } ^ { a _ { n + 1 } - 1 } z ^ i \\right ) \\ , . \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} A ^ { W , Y } ( l _ n ) = A ^ { W , W } ( \\nabla \\gamma ^ * \\ , l _ n ) . \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} \\bigl ( D ^ { \\frac { 1 } { 2 } } f \\bigr ) ( t ) = \\frac { 1 } { \\sqrt { \\pi } } \\int _ { 0 } ^ { t } f ' ( s ) ( t - s ) ^ { - \\frac { 1 } { 2 } } \\ , \\mathrm { d } s \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} C _ { \\hat \\phi } : = \\limsup _ { L \\to \\infty } \\frac 1 { L ^ d } \\sum _ { i \\in \\mathbb T _ L ^ d } \\hat \\phi _ i ( \\mu ^ L ) < \\infty . \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} S _ { n , 3 } ( \\omega ) = \\mathrm { o r d } ( \\omega ) ^ 4 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , \\frac { 1 } { 2 } y ^ 2 ( x - y ) ^ 2 - \\frac { ( x - y ) ^ 2 } { 6 } } + \\mathrm { o r d } ( \\omega ) ^ 2 a _ { \\frac { n } { \\mathrm { o r d } ( \\omega ) } , \\frac { 1 } { 6 } ( x - y ) ^ 2 } . \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} ( x ^ 0 _ { \\imath \\jmath } ) ^ { m _ { \\imath \\jmath } } = \\begin{cases} 1 & ( \\textrm { i f } m _ { \\imath \\jmath } = 0 ) , \\\\ 0 & ( \\textrm { i f } m _ { \\imath \\jmath } \\neq 0 ) . \\end{cases} \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} ( \\vec { m } ^ { \\star } ) _ i = \\dim V _ i , \\ 0 \\le i \\le k \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} I _ { m o d } [ \\mathbf { Q } ] & = \\int _ { \\Omega } \\phi \\left ( | \\nabla \\mathbf { Q } | \\right ) + \\frac { 1 } { L } f _ B \\left ( \\mathbf { Q } \\right ) d V \\\\ & = \\int _ { \\Omega } \\psi \\left ( | \\nabla \\mathbf { Q } | ^ 2 \\right ) + \\frac { 1 } { L } f _ B \\left ( \\mathbf { Q } \\right ) d V \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} { a n \\brack b n } _ { \\omega _ n ^ i } = \\binom { a ( n , i ) } { b ( n , i ) } . \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} V _ x = \\mathcal R V _ x . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\beta _ 1 , \\beta _ 2 } = \\int _ { [ 0 , 1 ] ^ 2 } f _ { t , s } ( x , y ) ^ s x ^ { \\beta _ 1 } y ^ { \\beta _ 2 } \\frac { d x } { x } \\frac { d y } { y } \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} \\Phi ( \\bar { \\Q } , B _ { \\kappa ^ j \\rho } ) \\leq \\frac 1 { 2 ^ j } \\Phi ( \\bar { \\Q } , B _ { \\rho } ) + \\sum _ { i = 0 } ^ { j - 1 } 2 ^ { - j } c _ { \\kappa , \\alpha , L } \\rho ^ \\alpha \\leq \\frac 1 { 2 ^ j } \\Phi ( \\bar { \\Q } , B _ { \\rho } ) + 2 c _ { \\kappa , \\alpha , L } \\rho ^ \\alpha . \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} \\sigma _ { 1 } ^ { j - 1 } \\lambda _ { a _ { 1 , 1 } , k } ^ { j - 1 } c ^ { 1 } _ { 1 , b ( a _ { 1 , 1 } , k ) } + \\sum _ { q _ { i } \\in Q } \\sigma _ { q _ i } ^ { j - 1 } \\lambda _ { a _ { 1 , 1 } , k } ^ { j - 1 } c ^ { q _ i } _ { 1 , b ( a _ { 1 , 1 } , k ) } + \\sum _ { i = 1 } ^ { | V | } \\sigma _ { v _ i } ^ { j - 1 } \\lambda _ { a _ { v _ { i , 1 } } , b _ { \\gamma _ { i } } } ^ { j - 1 } c ^ { v _ i } _ { 1 , b ( a _ { 1 , 1 } , k ) } = 0 \\textbf { \\space } \\forall j \\in [ r ] , k \\in [ 0 , s - 1 ] , \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} R _ 3 ^ { \\frac 1 2 } = & \\mathcal { D } _ t ( u ( t _ { \\frac 1 2 } ) - u ^ { \\frac 1 2 } ) + \\gamma \\bigtriangleup ( \\sigma ( t _ { \\frac 1 2 } ) - \\sigma ^ { \\frac 1 2 } ) - \\bigtriangleup ( u ( t _ { n - \\theta } ) - u ^ { n - \\theta } ) + ( f ( u ( t _ { \\frac 1 2 } ) ) - f ^ { \\frac 1 2 } ( u ) ) \\\\ = & O ( \\Delta t ^ 2 ) , \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} f _ { n } ( \\alpha , \\beta ) = 2 n ^ { 2 } + 2 n ( \\alpha + \\beta + 1 ) + ( \\alpha + 1 ) ( \\beta + 1 ) \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} f _ j & = \\frac { 1 } { ( L _ 0 - c _ { j , 1 } ) ! } \\cdot \\frac { 1 } { ( L _ 0 - c _ { j , 2 } ) ! } \\cdots \\frac { 1 } { ( L _ 0 - c _ { j , l _ j } ) ! } \\cdot \\prod _ { 1 \\le h < k \\le l _ j } ( ( L _ 0 - c _ { j , k } ) - ( L _ 0 - c _ { j , h } ) ) \\\\ & = \\frac { 1 } { ( L _ 0 - c _ { j , 1 } ) ! } \\cdot \\frac { 1 } { ( L _ 0 - c _ { j , 2 } ) ! } \\cdots \\frac { 1 } { ( L _ 0 - c _ { j , l _ j } ) ! } \\cdot \\prod _ { 1 \\le h < k \\le l _ j } ( h - k ) \\neq 0 \\\\ \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} \\int _ \\Omega \\int _ { \\mathbb { S } _ + ^ 1 } \\theta \\cdot \\nabla \\phi \\d Q ( x , \\theta ) = \\int _ 0 ^ 1 ( \\partial _ s \\phi ) \\overline { Q } \\d s & = - \\int _ 0 ^ 1 ( \\partial _ s \\overline { Q } ) \\phi \\d s \\\\ & = \\int _ \\Omega S \\phi \\ , \\d x . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\theta ^ \\prime ( x , E ) = \\gamma ^ \\prime ( x , E ) + \\frac { C } { \\gamma ^ \\prime ( x , E ) ( 1 + x - b ) } \\sin 2 \\theta ( x , E ) \\sin ^ 2 \\theta ( x , E ) , \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} f _ n ( t ) & : = a + b ( t - 1 ) + c \\ , { n ( t - 1 ) ^ 2 \\over t + n } + d \\ , { ( t - 1 ) ^ 2 \\over t + ( 1 / n ) } \\\\ & \\qquad + \\int _ { [ 1 / n , n ] } { ( t - 1 ) ^ 2 \\over t + s } \\ , d \\mu ( s ) , t \\in ( 0 , + \\infty ) . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } U + F _ { i } ( D ^ { + } U , D ^ { - } U ) = 0 & \\ , \\ , J _ { i } \\setminus \\{ 0 \\} \\\\ U ( 0 ) = \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } U ( 1 _ { i } ) - B \\end{array} \\right . \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} & F _ { m , \\ , j } ( X ) - F _ { m - 1 , \\ , j } ( X ) \\\\ & = ( - 1 ) ^ m \\binom { X - m } { j } + \\sum _ { s = 1 } ^ { m - 1 } ( - 1 ) ^ s \\binom { m - 1 } { s - 1 } \\binom { X - s } { j } \\\\ & = ( - 1 ) ^ m \\binom { X - m } { j } - \\sum _ { s = 0 } ^ { m - 2 } ( - 1 ) ^ s \\binom { m - 1 } { s } \\binom { ( X - 1 ) - s } { j } \\\\ & = ( - 1 ) ^ m \\binom { X - m } { j } + ( - 1 ) ^ { m - 1 } \\binom { X - m } { j } - F _ { m - 1 , \\ , j } ( X - 1 ) = - F _ { m - 1 , \\ , j } ( X - 1 ) . \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} \\left . \\frac { \\partial L } { \\partial e _ 2 } \\right | _ { e _ 2 = c , \\ e _ 3 = e _ 4 = 0 } = 0 \\quad c > 0 . \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} Z _ { \\Lambda , \\epsilon } ( \\alpha ) = \\int e ^ { - \\sum _ { j k } \\left ( 1 + ( \\phi _ j - \\phi _ k ) ^ 2 \\right ) e ^ { t _ { j k } } - \\sum _ { j \\in \\Lambda } \\epsilon \\phi _ j ^ 2 } \\prod _ { j k \\in E } \\Big ( f _ \\alpha ( e ^ { t _ { j k } } ) e ^ { t _ { j k } } d t _ { j k } \\Big ) \\prod _ { j \\in \\Lambda } d \\phi _ j . \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} ( ( \\sigma - \\mathfrak { Q } _ h \\sigma ) ^ { n - \\theta } , w _ h ) + ( \\nabla ( \\mathfrak { Q } _ h u - u _ h ) ^ { n - \\theta } , \\nabla w _ h ) = 0 . \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} c n ^ { - p + 2 \\beta p - ( p ^ { 2 } - p ) } = c n ^ { - p ^ { 2 } + 2 \\beta p } , \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} f \\sim g _ 3 = \\sum _ { v \\in I \\setminus \\{ v _ 3 \\} } \\alpha ( v ) \\phi v + \\alpha ( v _ 3 ) \\beta ( w _ 3 ) \\phi w _ 3 + \\sum _ { y \\in J _ 3 } \\alpha ( v _ 3 ) \\beta _ 3 ( y ) \\phi y . \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} \\widetilde { w } = t \\sum _ { i , j } x _ i y _ j w _ { i j } ( \\vec { u } ) \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\big ( V + ( \\ker B _ + ) ^ { \\perp _ { \\mathcal Q } } \\big ) \\cap \\bigcap _ { i = 1 } ^ { m ^ + } ( V + \\ker B _ i ) = V . \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} ( 1 ) & \\ ; Q \\ ; \\ ; Q _ { n _ i } \\subset Q \\ ; \\ ; i , \\\\ ( 2 ) & \\lim _ { i \\rightarrow \\infty } | Q _ { n _ i } | = \\infty , \\\\ ( 3 ) & \\sup _ i | Q _ { n _ i } | < \\infty Q _ { n _ i } \\cap Q _ { n _ j } = \\emptyset i \\not = j . \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} \\lambda _ { \\chi _ { \\mu _ p } \\pi _ p } ( p ^ m ) = \\begin{cases} \\delta _ { m = 0 } & \\mu _ p \\neq \\chi ^ { - 1 } , \\\\ q ^ { - \\frac { m } { 2 } } \\delta _ { m \\geq 0 } & \\mu _ p = \\chi ^ { - 1 } \\end{cases} \\end{align*}"}
-{"id": "9894.png", "formula": "\\begin{align*} ( \\boldsymbol { D } ^ t ( \\boldsymbol { D } ^ { s * } u ) , \\phi ) = ( \\boldsymbol { D } ^ { s * } u , \\boldsymbol { D } ^ t \\phi ) , \\forall \\phi \\in C _ 0 ^ \\infty ( \\mathbb { R } ) . \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} \\omega _ n ^ { 2 i } [ t ^ r ] S _ 3 ( t , \\omega _ n ^ i ) = ( - 1 ) ^ r \\Big ( - \\binom { n } { 2 } ^ 2 + r n \\frac { ( 3 d ^ 2 + 1 ) n ^ 2 - 6 d ^ 2 n + 2 d ^ 2 } { 1 2 d } \\Big ) . \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} h ( u , v ) = g ( L u , v ) , \\qquad \\forall u , v \\in V . \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} \\limsup _ { \\epsilon \\to 0 } \\limsup _ { L \\to \\infty } \\sup _ { \\mu } \\frac 1 { L ^ d } \\sum _ { i \\in \\mathbb T _ L ^ d } \\sum _ { k = 1 } ^ d \\sum _ { \\eta \\in \\Omega _ L } \\mu ( \\eta ) \\ ; \\ ! \\Bigl | \\hat \\chi _ { i , i + e _ k } ^ { \\lfloor \\epsilon L \\rfloor } ( \\delta _ \\eta ) - \\hat \\chi _ { i , i + e _ k } ( \\nu _ { \\eta ^ { \\lfloor \\epsilon L \\rfloor } ( i ) } ) \\Bigr | = 0 , \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} Q _ 1 ( x , y , q , k ) & : = \\left ( l _ 1 x ^ { l _ 1 } + l _ 2 x ^ { l _ 2 } \\right ) \\left ( y - x - q + b ( 1 + k ) [ \\ln { x } - \\ln { y } ] \\right ) \\\\ & \\qquad - ( x - b ( 1 + k ) ) ( y ^ { l _ 1 } - x ^ { l _ 1 } + y ^ { l _ 2 } - x ^ { l _ 2 } ) , \\\\ Q _ 2 ( x , y , q , k ) & : = \\left ( l _ 1 y ^ { l _ 1 } + l _ 2 y ^ { l _ 2 } \\right ) \\left ( y - x - q + b ( 1 + k ) [ \\ln { x } - \\ln { y } ] \\right ) \\\\ & \\qquad - ( y - b ( 1 + k ) ) ( y ^ { l _ 1 } - x ^ { l _ 1 } + y ^ { l _ 2 } - x ^ { l _ 2 } ) . \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} ( f + g ) ^ { p - 1 } d ( f + g ) - f ^ { p - 1 } d f - g ^ { p - 1 } d g = d \\left ( \\frac { ( f + g ) ^ p - f ^ p - g ^ p } { p } \\right ) , \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} \\sigma _ q \\cdot z _ q ^ { - 1 } = \\epsilon _ q z _ q ^ { - 1 } . \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} \\tilde { e } _ { i _ 2 } ^ { } \\tilde { e } _ { i _ 1 } ^ { } \\left ( b \\otimes b ^ \\prime \\right ) & = \\tilde { f } _ { i _ 3 } ^ { a _ 3 } \\cdots \\tilde { f } _ { i _ l } ^ { a _ l } ( b _ \\lambda ) \\otimes \\tilde { e } _ { i _ 2 } ^ { r _ 2 } \\tilde { e } _ { i _ 1 } ^ { r _ 1 } ( b ^ \\prime ) . \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} \\tilde { \\tau } ^ { n , r } _ R : = \\inf \\left \\lbrace t \\ge 0 : \\left \\lvert Y ^ { n , r } _ t \\right \\rvert > \\frac { R } { 3 } \\right \\rbrace , \\tilde { \\tau } ^ { n , m , r } _ R : = T \\wedge \\tilde { \\tau } ^ { n , r } _ R \\wedge \\tilde { \\tau } ^ { m , r } _ R . \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { 0 } ( \\textbf { X } ) = \\begin{cases} 1 ( 1 - q ( \\textbf { X } , 1 ) ) P \\{ Y = 1 \\vert \\textbf { X } \\} > ( 1 - q ( \\textbf { X } , 0 ) ) P \\{ Y = 0 \\vert \\textbf { X } \\} \\\\ 0 \\end{cases} \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} ( 1 - \\alpha ^ { 2 } ) \\overline { S } ( Y , U ) = ( 1 - \\alpha ^ { 2 } ) ( n - 1 - \\alpha \\ , t r a c e \\Phi ) g ( Y , U ) , \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ 2 } { d x ^ 2 } + q ( x ) - \\frac { ( n - 1 ) ^ 2 } { 4 } K _ 0 \\right ) w _ { \\lambda _ 1 } ( x ) = \\lambda _ 1 w _ { \\lambda _ 1 } ( x ) , \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } { \\log m _ n ( x , y ) \\over - \\log n } = { 1 \\over \\eta } , \\limsup _ { n \\to \\infty } { \\log m _ n ( x , y ) \\over - \\log n } = 1 . \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} B _ { \\pi _ { p } , \\kappa } ( y ) = \\frac { \\log ( p ) } { 2 \\pi } \\int _ { - \\frac { \\pi } { \\log ( p ) } } ^ { \\frac { \\pi } { \\log ( p ) } } \\frac { L ( 1 - \\kappa - i t , \\tilde { \\pi } _ { p } ) } { L ( \\kappa + i t , \\pi _ { p } ) } \\abs { y } _ { p } ^ { - i t } d t . \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } \\int _ K \\int _ H f ( k ^ { - 1 } x h ) d h d k d \\mu ( \\ddot { x } ) = \\int _ G f ( x ) \\rho ( x ) d x \\ f \\in C _ c ( G ) . \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} N _ { r + 1 } = N _ { r } + N _ { r - 2 } , \\ N _ { 0 } = 0 , \\ N _ { 1 } = N _ { 2 } = 1 \\ \\ ( r \\geq 2 ) . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} \\dim G \\cdot ( A , B ) = \\dim G - \\dim G _ { ( A , B ) } = n ^ 2 - n + p . \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\Psi \\big ( [ z _ 0 : z _ 1 ] \\big ) : = \\frac { \\imath } { | z _ 0 | ^ 2 + | z _ 1 | ^ 2 } \\ , \\begin{pmatrix} \\frac { 1 } { 2 } \\ , \\left ( | z _ 0 | ^ 2 - | z _ 1 | ^ 2 \\right ) & z _ 0 \\cdot \\overline { z } _ 1 \\\\ \\overline { z } _ 0 \\cdot z _ 1 & \\frac { 1 } { 2 } \\ , \\left ( | z _ 1 | ^ 2 - | z _ 0 | ^ 2 \\right ) \\end{pmatrix} . \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} \\begin{aligned} C ( \\{ s , v \\} , v ) & = c _ { s v } & & \\forall v \\in V , v \\neq s \\\\ C ( S , v ) & = \\min _ { \\substack { u \\in S \\\\ u \\neq s , v } } C ( S \\setminus \\{ v \\} , u ) + c _ { u v } & & \\forall S \\subseteq V , v \\in S . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} \\widehat { Q } _ y ( \\alpha ) = \\frac { \\alpha ( 1 + y e ^ { - \\alpha ( 1 + y ) } ) } { 1 - e ^ { - \\alpha ( 1 + y ) } } = \\frac { \\alpha ( 1 + y ) } { 1 - e ^ { - \\alpha ( 1 + y ) } } - \\alpha y \\in \\mathbb { Q } [ y ] [ [ \\alpha ] ] \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial x } - \\frac { \\partial v } { \\partial y } + a u + b v = 0 , ~ ~ \\frac { \\partial u } { \\partial y } + \\frac { \\partial v } { \\partial x } + c u + d v = 0 \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} \\frac { P _ n ( t ) } { ( 1 - t ) ^ { n + 1 } } = \\sum _ { k = 1 } ^ { \\infty } \\left ( 1 + t \\right ) ^ { k - 1 } \\frac { k ^ n } { 2 ^ { k + 1 } } \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} Q _ { z _ 1 , z _ 2 , z _ 3 , z _ 4 } = \\frac { 1 } { \\sqrt { ( | z _ 1 | ^ 2 + | z _ 3 | ^ 2 ) ( | z _ 2 | ^ 2 + | z _ 4 | ^ 2 ) } } \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} \\varphi _ 0 ( 1 _ 2 ) = \\begin{cases} | m | ^ { - 1 } ( 1 + 2 ^ { \\frac { z + 1 } { 2 } } ) & ( \\tau \\in \\{ \\pm 2 , \\pm 1 0 \\} ) , \\\\ | m | ^ { - 1 } ( 1 + 2 ^ { - \\frac { z + 1 } { 2 } } ) & ( \\tau \\in \\{ - 1 , - 5 \\} ) , \\\\ | m | ^ { - 1 } 3 ^ { - 1 } 2 ( 1 + 2 ^ { - z } ) & ( \\tau = 5 ) . \\end{cases} \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} 0 = E [ \\vert \\eta ( t , W _ t ) \\vert ] = \\int _ { [ 0 , T ] \\times \\mathbb R ^ d } \\vert \\eta ( t , x ) \\vert p _ t ( x ) \\mathrm d t \\mathrm d x \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} \\sum _ { x \\in T _ { N } ^ { - 1 } ( x ' ) } T _ { M } ( x ) = 0 \\end{align*}"}
-{"id": "9973.png", "formula": "\\begin{align*} \\C _ { \\overline { S _ { 1 } } } \\otimes _ { { } _ { \\C _ X } } \\C _ { \\overline { S _ { 2 } } } \\otimes _ { { } _ { \\C _ X } } \\cdots \\otimes _ { { } _ { \\C _ X } } \\C _ { \\overline { S _ { n + 1 } } } = \\C _ M \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} P _ G ( x ) & = ( - 1 ) ^ { \\sum a _ i } x ^ { m _ 0 ( G ) } ( x + 1 ) ^ { m _ { - 1 } ( G ) } \\{ x ^ 2 ( x + 1 ) ^ 2 - x ^ 2 ( x + 1 ) ( a _ 2 + a _ 4 ) \\\\ & - ( a _ 1 a _ 2 + a _ 1 a _ 4 + a _ 3 a _ 4 ) x ( x + 1 ) + ( a _ 2 a _ 3 a _ 4 ) x - a _ 1 a _ 2 a _ 3 a _ 4 \\} . \\\\ \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{gather*} ( u , x ) \\sim ( v , y ) \\Longleftrightarrow \\begin{cases} x = y \\\\ \\omega ( u \\sharp \\overline { v } ) = 0 \\\\ c _ 1 ( u \\sharp \\overline { v } ) = 0 \\end{cases} \\end{gather*}"}
-{"id": "7186.png", "formula": "\\begin{align*} Q = V _ { Q } \\ , \\Sigma _ { Q } \\ , W _ { Q } ^ { T } , \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} \\nabla a = \\frac { w - v } \\alpha \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} \\begin{cases} u _ x & = p \\\\ p _ x & = - \\frac { b _ * u + c _ * p - \\lambda u } { a _ * } \\end{cases} \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\| v _ n \\| ^ 2 _ { L ^ 2 } & = \\| V \\| ^ 2 _ { L ^ 2 } + \\| r _ n \\| ^ 2 _ { L ^ 2 } + o _ n ( 1 ) , \\\\ E ( v _ n ) & = E ( V ) + E ( r _ n ) + o _ n ( 1 ) , \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} A ( q ^ { n - 1 } , t ) A ( q ^ { n - 2 } , t ) \\cdots A ( 1 , t ) \\Big | _ { q = \\omega } = \\left ( A ( \\omega ^ { \\frac { n } { ( g ' , n ) } - 1 } , t ) A ( \\omega ^ { \\frac { n } { ( g ' , n ) } - 2 } , t ) \\cdots A ( 1 , t ) \\right ) ^ { ( g ' , n ) } . \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} \\mathrm { T r } _ { d , D } ( 1 ) = \\left \\{ \\begin{array} { l l } \\dfrac { \\sqrt { | d D | } L _ D ( 1 ) } { \\pi } , & d = \\square , \\\\ 0 , & d \\neq \\square . \\end{array} \\right . \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} t = \\frac { n ' } { \\omega \\cdot \\beta ' } \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} h ( p ) - I ( X _ 0 ; X _ t ) \\le 3 \\sum _ { i = 1 } ^ k p _ i \\sum _ { j \\neq i } b _ { i j } e ^ { - 0 . 0 8 5 c _ { i j } ^ 2 } . \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} Q _ h \\varphi ( x ) = \\Lambda ^ { - 1 } \\frac { Q ^ f ( h \\varphi ) ( x ) } { h ( x ) } \\geq \\frac { 1 } { \\Lambda \\sup _ { K _ n } h } \\alpha _ n \\eta _ n ( h \\varphi ) \\geq \\widetilde { \\alpha } _ n \\ , \\widetilde { \\eta } _ { n } ( \\varphi ) , \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} \\mu _ f ^ \\star ( \\varphi ) = \\frac { \\mu _ h \\left ( h ^ { - 1 } \\varphi \\right ) } { \\mu _ h \\left ( h ^ { - 1 } \\right ) } , \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} w ' ( N E P ) & = \\frac { ( n - 0 + ( m + 1 ) - 1 ) } { 2 } + w ' ( P ) = \\frac { n + m } { 2 } + v _ { n - 1 , m } \\\\ & = \\frac { n + m } { 2 } - \\sum _ { k = D _ { n - 1 , m } } ^ { D _ { n - 1 , m } + m - 1 } k = \\frac { n + m } { 2 } - \\sum _ { k = D _ { n , m + 1 } + 1 } ^ { D _ { n , m + 1 } + m } k \\\\ & = - \\sum _ { k = D _ { n , m + 1 } } ^ { D _ { n , m + 1 } + ( m + 1 ) - 1 } k = v _ { n , m + 1 } . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} P ^ { ( 1 ) } = Q ^ { ( 1 ) } , ~ \\sum _ { \\substack { \\ell _ { N _ 1 + 1 } , \\cdots , \\ell _ N = - \\infty \\\\ \\ell _ { N _ 1 + 1 } + \\cdots + \\ell _ N = \\ell } } ^ { \\infty } P ^ { ( 2 ) } = Q ^ { ( 2 ) } , ~ P ^ { ( 3 ) } = Q ^ { ( 3 ) } , \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} ( \\delta _ { [ x ] } * \\delta _ { [ y ] } ) ( E ) = ( \\delta _ x * \\delta _ y ) ( \\pi ^ { - 1 } ( E ) , \\ , \\ , , \\ , \\ , \\chi _ E ( [ x ] * [ y ] ) = \\chi _ { \\pi ^ { - 1 } ( E ) } ( x * y ) . \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} \\Delta _ h & = \\Delta _ { g _ 0 } - \\nabla _ { g _ 0 } h \\cdot \\nabla _ { g _ 0 } \\\\ & = \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + \\frac { m - 1 } { r } \\frac { \\partial } { \\partial r } + \\frac { 1 } { r ^ 2 } \\Delta _ \\theta - \\frac { r } { 2 } \\frac { \\partial } { \\partial r } . \\\\ \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( v ) & = \\frac { 1 } { \\sqrt { 2 } } \\sqrt { \\sum \\limits _ { i = 1 } ^ { N } \\| \\psi ^ { \\prime } ( v ^ j ) \\| ^ 2 } \\\\ & = \\frac { 1 } { 2 \\sqrt { 2 } } \\sqrt { \\sum _ { j = 1 } ^ { N } \\left [ 2 \\psi ^ { \\prime } ( \\lambda _ 1 ( v ^ j ) ) ^ 2 + \\psi ^ { \\prime } ( \\lambda _ 2 ( v ^ j ) ) ^ 2 + \\psi ^ { \\prime } ( \\lambda _ 4 ( v ^ j ) ) ^ 2 \\right ] } . \\end{aligned} \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} f _ 1 \\sum _ { j = 0 } ^ k { f _ 2 ^ j X _ { m - a - b j } } = X _ m - f _ 2 ^ { k + 1 } X _ { m - ( k + 1 ) b } \\ , , \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} N _ \\varepsilon \\ge \\min \\left \\{ N _ { i n } , \\frac { F _ { \\min } } { c _ { \\min } } \\right \\} = \\frac { F _ { \\min } } { c _ { \\min } } , \\qquad \\forall t \\in [ 0 , T ] , \\ \\forall \\varepsilon > 0 , \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} R _ { 1 2 } & = a ^ { - 1 } ( [ b , a ] - b ( a - ( d - c a ^ { - 1 } b ) ^ { - 1 } \\Delta ) ) \\Delta ^ { - 1 } \\\\ & = ( - b + ( d b ^ { - 1 } a - c ) ^ { - 1 } \\Delta ) \\Delta ^ { - 1 } . \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} Y = \\partial _ y + \\sum _ { j = 1 } ^ { n } p _ { Y , j } ( x , y , u _ 1 , \\dots , u _ n ) \\partial _ { u _ j } \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} H _ { \\{ 2 , 3 , 4 \\} } ( 5 x _ 2 ^ 2 x _ 3 + 7 x _ 2 ^ 2 x _ 3 x _ 4 ^ 5 + x _ 1 x _ 2 x _ 3 x _ 4 + 4 x _ 4 ^ 3 x _ 5 + 1 3 x _ 2 ^ 6 x _ 3 ^ 8 x _ 4 ^ 7 ) = 7 x _ 2 ^ 2 x _ 3 x _ 4 ^ 5 + 1 3 x _ 2 ^ 6 x _ 3 ^ 8 x _ 4 ^ 7 \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} \\lvert u \\rvert _ S = \\lvert l \\rvert _ S + \\lvert m \\rvert _ S + \\lvert r \\rvert _ S . \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} a \\Delta _ \\kappa a + a \\Delta _ \\kappa b = a \\Delta _ \\kappa c \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} \\chi _ { \\ell } ( M \\square K _ { 1 , s } ) = \\begin{cases} k & s < P _ { \\ell } ( M , k ) \\\\ k + 1 & s \\geq P _ { \\ell } ( M , k ) . \\end{cases} \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} p _ { L } ( x ) = \\sum _ { \\ell = 0 } ^ L \\beta _ { \\ell } \\tilde { \\Phi } _ { \\ell } ( x ) \\in \\mathbb { P } _ L , x \\in [ - 1 , 1 ] , \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\begin{cases} \\ - \\Delta u & = f , \\\\ \\ u | _ { \\partial \\Omega } & = 0 . \\end{cases} \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} \\lvert ( 2 \\lvert m _ 0 \\rvert , j ) \\rvert _ { W _ n ( w ) } - \\lvert ( i , j ) \\rvert _ { W _ n ( w ) } = \\lvert l _ { 2 \\lvert m _ 0 \\rvert } \\rvert _ S - \\lvert l _ i \\rvert _ S > 0 . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} \\varphi _ \\tau : \\begin{pmatrix} u & - \\overline { v } \\\\ v & \\overline { u } \\end{pmatrix} \\mapsto \\begin{pmatrix} u \\cdot e ^ { \\imath \\tau } & - \\overline { v } \\cdot e ^ { - \\imath \\tau } \\\\ v \\cdot e ^ { \\imath \\tau } & \\overline { u } \\cdot e ^ { - \\imath \\tau } \\end{pmatrix} . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} W _ Q : = \\left < r _ 1 , \\dots , r _ n \\right > \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\tilde { V } _ k = V _ k \\otimes V ^ { ( n ) } , \\tilde { Q } ( z ) = Q _ n ( z ) \\ , Q . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} \\theta _ 1 ( z \\mid \\tau ) = i q ^ { \\frac { 1 } { 4 } } e ^ { - i z } ( q ^ 2 e ^ { - 2 i z } , e ^ { 2 i z } , q ^ 2 ; q ^ 2 ) _ { \\infty } . \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} ( \\widetilde { M } v ) _ i & = ( 1 - \\widetilde { w } _ i ) v _ i + \\widetilde { w } _ i v _ { i + 1 } \\\\ & = \\left ( 1 - \\frac { v _ i } { v _ { i + 1 } - v _ i } \\right ) v _ i + \\frac { v _ i } { v _ { i + 1 } - v _ i } v _ { i + 1 } \\\\ & = v _ i - \\frac { v _ i } { v _ { i + 1 } - v _ i } v _ i + \\frac { v _ i } { v _ { i + 1 } - v _ i } v _ { i + 1 } \\\\ & = 2 v _ i . \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} a \\tau _ 1 + b \\tau _ 2 + c \\tau _ 3 + d ( \\tau _ 2 ^ 2 - \\tau _ 1 \\tau _ 3 ) + e = 0 \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} M ^ i _ t : = Y ^ i _ t - Y ^ i _ 0 + A _ t ^ { W , Y ^ i } ( b ) + \\int _ 0 ^ t f \\left ( r , W _ r , Y ^ i _ r , \\frac { \\mathrm d [ Y ^ i , W ] _ r } { \\mathrm d r } \\right ) \\mathrm d r , \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } v _ { \\theta , t } + H _ { i } ( t , x , v _ { \\theta , x _ { i } } ) + D T _ { \\theta } = 0 & \\ , \\ , I _ { i } \\times ( T _ { \\theta } , T ) \\\\ \\sum _ { i = 1 } ^ { K } v _ { \\theta , x _ { i } } = 0 & \\ , \\ , \\{ 0 \\} \\times ( T _ { \\theta } , T ) \\\\ v _ { \\theta } = u _ { 0 } - 2 \\omega ( T _ { \\theta } ) & \\ , \\ , \\mathcal { I } \\times \\{ T _ { \\theta } \\} ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} C ( \\Sigma _ \\mu ( v _ \\pm ) ) = C ( L ( \\Sigma _ \\mu ( v _ \\pm ) ) ) = C ( \\Sigma _ \\mu ( w _ \\pm ) ) . \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} R R ^ * f = \\chi _ { - \\lambda } \\cdot ( \\widetilde { f } * \\phi _ H ) , \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} K _ 2 ( \\varphi , p ) = \\int \\bigl [ p ( y ) \\bigr ] \\ , \\bigl [ \\mu _ \\varphi ( y ) \\bigr ] \\ , d y ^ 1 d y ^ 2 d y ^ 3 d y ^ 4 , \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} \\sum _ { y \\in Y } ( f ( y ) ^ 2 + d f ( y ) ) = \\sum _ { i = 1 } ^ { n + 1 } ( f ( i ) ^ 2 + d f ( i ) ) + n \\sum _ { i = 0 } ^ { d - 1 } ( i ^ 2 + d i ) - \\sum _ { i = 1 } ^ { n } ( k _ { i - 1 } ( f ) ^ 2 + d k _ { i - 1 } ( f ) ) . \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} \\| u u _ h ^ { \\kappa _ 1 } \\| _ { p ^ * } & \\leq M _ 5 ( \\kappa _ 1 , u ) \\left [ \\| u u _ h ^ { \\kappa _ 1 } \\| _ { p } ^ p + 1 \\right ] ^ { \\frac { 1 } { p } } \\leq M _ 6 ( \\kappa _ 1 , u ) \\left [ \\| u ^ { \\kappa _ 1 + 1 } \\| _ { p } ^ p + 1 \\right ] ^ { \\frac { 1 } { p } } \\\\ & = M _ 6 ( \\kappa _ 1 , u ) \\left [ \\| u \\| _ { p ^ * } ^ { p ^ * } + 1 \\right ] ^ { \\frac { 1 } { p } } < \\infty , \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} \\tilde { f } _ i ^ a ( \\pi _ 1 \\otimes \\pi _ 2 ) = \\tilde { f } _ i ^ M ( \\pi _ 1 ) \\otimes \\tilde { f } _ i ^ { a - M } ( \\pi _ 2 ) \\not \\in \\mathcal { B } _ { s _ i v } ( \\lambda ) \\otimes \\mathcal { B } _ w ( \\mu ) \\sqcup \\{ 0 \\} . \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} \\mu _ t ^ { \\star } ( A ) \\le \\mu _ t ^ { \\star } ( i _ { c \\ast } F [ - 1 ] ) = \\mu _ t ^ { \\star } ( - \\mathbf { v } _ k ) < 0 = \\mu _ t ^ { \\star } ( B ) \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} x ( t + 1 ) = \\left \\{ \\begin{array} { l l } A ^ - x ( t ) , & x _ 1 ( t ) \\leq 0 , \\\\ A ^ + x ( t ) , & , \\end{array} \\right . t = 0 , 1 , 2 \\ldots , \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} { F } _ { s , i } \\ , : = \\ , u \\cdot \\delta _ { \\overline { X } _ { s , i + 1 } } + ( 1 - u ) \\cdot \\mathcal L _ { \\overline { X } _ { s , i } } \\ , , i \\ , = \\ , 1 , \\ldots , n \\ , , \\ , \\ , s \\ge 0 \\ , \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} \\{ u _ h > 0 \\} = \\{ u _ d > 0 \\} . \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} \\mathbb { A } ^ { - 1 } _ L = \\ , _ { c } \\mathbb { A } ^ { - 1 } _ L - \\mathbb { T } _ L , \\ ; \\ ; \\ ; \\ ; \\mathbb { A } ^ { - 1 } _ R = \\ , _ { c } \\mathbb { A } ^ { - 1 } _ R - \\mathbb { T } _ R , \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} \\| v _ n \\| ^ 2 _ { L ^ 2 } = M n \\geq 1 \\lim _ { n \\rightarrow \\infty } E ( v _ n ) = d _ { M , } , \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} Q = \\sum _ { k \\in \\{ 0 , m + 1 \\} } c _ k B _ k ^ \\ast A _ k B _ k . \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} F ( d _ { \\Lambda } ( E Z ( t \\otimes \\sigma ) ) ) ( m ) = d _ { F _ i } ( F ( E Z ( t \\otimes \\sigma ) ) ( m ) ) \\pm F ( E Z ( t \\otimes \\sigma ) ) ( d _ { F _ i } m ) . \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} \\langle f , H P _ c g \\rangle = \\langle f , - \\tfrac 1 2 \\partial _ x ^ 2 g \\rangle + B ( f , g ) , \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} b _ j = - \\Im i \\langle G ( v , Q ) , D _ j Q \\rangle , \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} A ^ { W , Y } _ t ( b ) = A ^ { W , W } _ t ( \\nabla u ^ * \\ , b ) . \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} & d ^ { - 2 n } \\sum _ { \\substack { n _ 1 + \\cdots + n _ d = n \\\\ n _ i \\ge 1 , \\ ; 1 \\le i \\le d } } { 2 n \\choose 2 n _ 1 , \\ldots , 2 n _ d } \\prod _ { j = 1 } ^ d n _ j ^ { - 3 / 2 } \\\\ & \\le c _ 3 \\exp ( - c _ 4 n ) + \\sum _ { \\substack { n _ 1 + \\cdots + n _ d = n \\\\ n _ i \\in [ \\frac { n } { d } , \\ , \\frac { 3 n } { d } ] , \\ ; 1 \\le i \\le d } } \\P ( S _ i = 2 n _ i , \\ 1 \\le i \\le d ) \\prod _ { j = 1 } ^ d n _ j ^ { - 3 / 2 } \\ , . \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} \\nu _ p \\binom { n p } { m p } = \\nu _ p \\binom { n } { m } \\epsilon _ p \\binom { n p } { m p } \\equiv \\epsilon _ p \\binom { n } { m } \\mod p . \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{gather*} { F _ * } _ p ( R _ p ) = R _ { F ( p ) } \\end{gather*}"}
-{"id": "7535.png", "formula": "\\begin{align*} \\mathcal { C } _ p = \\left \\{ ( \\gamma , \\zeta ) \\in \\C \\times \\C ^ { n } \\ ; | \\ ; \\textmd { I m } \\ ; \\gamma > p ( \\zeta ) \\abs { \\gamma } \\right \\} . \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} = \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { ( r _ 1 + k + 1 ) ( r _ 2 + k + 1 ) ( r _ 3 + k + 1 ) ( k + 1 ) ^ { s - 3 } } . \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} \\theta ^ \\prime ( x , \\hat { E } ) = \\gamma ^ \\prime ( x , \\hat { E } ) + \\frac { C } { \\gamma ^ \\prime ( x , \\hat { E } ) ( 1 + x - b ) } \\sin 2 \\theta ( x , E ) \\sin ^ 2 \\theta ( x , \\hat { E } ) . \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} \\mathcal V ^ { - } _ \\delta ( x ) = \\big \\{ V \\in \\mathcal V ^ { - } ( x ) : g ( \\nabla \\phi ( x ( s ) ) , V ( s ) ) \\leq 0 \\ \\ \\phi ( x ( s ) ) \\in [ - \\delta , \\delta ] \\big \\} . \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} \\inf \\tilde { H } _ { N } = ( a _ { * } - a _ { N } ) ^ { \\frac { p } { p + 1 } } \\Big ( \\frac { \\tilde { \\Lambda } } { a _ { * } } \\cdot \\frac { p + 1 } { p } + o ( 1 ) _ { N \\to \\infty } \\Big ) \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} u _ R ( x _ 1 ) & = \\sum _ { 1 \\le i < j \\le N } f ( d ( x _ i , x _ j ) ) - \\sum _ { j = 2 } ^ N u _ R ( x _ j ) \\\\ & \\ge \\frac { N ( N - 1 ) } { 2 } f ( 2 L ) - ( N - 1 ) M . \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} L _ h ( v ) = u ^ { 1 - p _ n } L _ g ( u v ) . \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{gather*} A ( - 1 / 2 ) = { } _ 4 F _ 3 \\biggl ( \\begin{matrix} \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 , \\ , \\frac 1 2 \\\\ 1 , \\ , 1 , \\ , 1 \\end{matrix} \\biggm | 1 \\biggr ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( \\frac 1 2 ) _ k ^ 4 } { k ! ^ 4 } \\end{gather*}"}
-{"id": "7923.png", "formula": "\\begin{align*} \\mathbf { w } ^ { \\prime \\prime } = \\abs { \\mathbf { w } ^ \\prime } ^ 2 \\mathbf { w } . \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} f \\colon \\mathbb { R } ^ d / \\mathbb { Z } ^ d \\to \\mathbb { R } ^ d / \\mathbb { Z } ^ d f ( x + \\mathbb { Z } ^ d ) = M _ f x + \\mathbb { Z } ^ d \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} [ \\mathcal { M } W _ l ( s ; \\cdot ) ] ( \\mu _ l ) = \\frac { \\zeta _ l ( 1 ) } { l ^ { \\frac { n _ l } { 2 } - q - \\abs { r } - k } } \\hat { f } _ s ( \\xi ) , \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} r _ 1 ( x ) - \\frac { y } { \\prod _ { j = 1 } ^ { d - 1 } ( \\alpha _ j - x ) } \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} x ( x + \\alpha ) y '' - ( x - \\alpha ) ( x + \\alpha + 1 ) y ' + [ n x + ( n - 2 ) \\alpha ] y = 0 , \\ \\ \\ y ( x ) = \\widehat { L } ^ { ( \\alpha ) } _ { n } ( x ) . \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\{ g ( 1 ) , \\ldots , g ( n + 1 ) \\} = \\left \\{ \\left ( - \\langle \\mu , \\varpi _ 1 \\rangle + \\sum _ { j = 1 } ^ { i - 1 } \\left \\langle \\frac { \\lambda + \\mu } { 2 } + \\rho , \\alpha _ j \\right \\rangle \\right ) ^ { } \\ \\middle \\vert \\ 1 \\leq i \\leq n + 1 \\right \\} \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} \\vec x _ \\alpha = ( x _ { 1 , \\alpha } , x _ { 2 , \\alpha } , \\dots , x _ { 8 , \\alpha } ) = x _ { 1 , \\alpha } + i x _ { 2 , \\alpha } + \\dots + h x _ { 8 , \\alpha } , \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} \\Psi _ { s i r } = \\max \\{ \\psi _ { 1 } ( \\tau _ { n } ) , \\psi _ { 2 } ( \\tau _ { n } ' ) \\} \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} e _ 1 ( \\varphi ) & : = \\operatorname { t r } S , \\\\ e _ 2 ( \\varphi ) & : = \\frac 1 2 \\left [ ( \\operatorname { t r } S ) ^ 2 - \\operatorname { t r } ( S ^ 2 ) \\right ] , \\\\ e _ 3 ( \\varphi ) & : = \\operatorname { t r } \\operatorname { a d j } S , \\\\ e _ 4 ( \\varphi ) & : = \\det S . \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} \\tau ( s _ { \\mu , \\nu } ) \\xi _ m = \\rho ( a _ { \\mu , \\nu } ) \\xi _ m + \\sum _ { j = 1 } ^ { r _ { \\mu , \\nu } } t ( x ^ { ( \\mu , \\nu ) } _ { j , 1 } ) t ( x ^ { ( \\mu , \\nu ) } _ { j , 2 } ) \\cdots t ( x ^ { ( \\mu , \\nu ) } _ { j , N _ j } ) \\xi _ m \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{gather*} \\partial : C F ( H , J ) \\rightarrow C F ( H , J ) \\\\ \\partial ( z _ - ) = \\sum _ { z _ + \\in \\widetilde { P } ( H ) } \\sharp \\mathcal { M } ( z _ - , z _ + , H , J ) \\cdot z _ + \\end{gather*}"}
-{"id": "7903.png", "formula": "\\begin{align*} & \\frac { 1 } { L ^ { q + 1 } } \\int _ { B _ { \\delta _ { q + 1 } } } \\left ( \\frac { \\partial f _ B } { \\partial Q _ { i j } } ( \\Q ) + \\frac { B } { 3 } | \\Q | ^ 2 \\delta _ { i j } \\right ) \\nu _ { i j } ( \\Q ) \\left | \\Q - \\Pi ( \\Q ) \\right | ^ q \\\\ & \\geq \\frac { 1 } { L ^ { q + 1 } } \\int _ { B _ { \\delta _ { q + 1 } } } \\left | \\Q - \\Pi ( \\Q ) \\right | ^ { q + 1 } \\\\ \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} F [ Q ] = \\sum _ { \\substack { ( i , j ) \\in \\mathcal { I } \\\\ i < j } } f _ \\gamma ( Q _ { i j } ) L _ { i j } . \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} f \\left ( \\langle u _ { j } , C u _ { j } \\rangle \\right ) & = f \\left ( \\frac { \\langle u _ { j } , A u _ { j } \\rangle + \\langle u _ { j } , B u _ { j } \\rangle } { 2 } \\right ) \\\\ & \\leq \\frac { 1 } { 2 } \\left [ f ( \\langle u _ { j } , A u _ { j } \\rangle ) + f ( \\langle u _ { j } , B u _ { j } \\rangle ) \\right ] . \\end{align*}"}
-{"id": "9906.png", "formula": "\\begin{align*} o r _ { M / E } ( U ) = H ^ { l } _ { U } ( \\pi ^ { - 1 } ( U ) ; \\Z _ { E } ) . \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} \\mathbb { G } _ \\lambda ( o , \\ , o \\ , | \\ , z ) & = \\frac { 1 } { 1 - U _ \\lambda ( o , \\ , o \\ , | \\ , z ) } \\\\ & = \\frac { 2 ( d - 1 ) } { 2 ( d - 1 ) - ( d - 1 + \\lambda ) + \\sqrt { ( d - 1 + \\lambda ) ^ 2 - 4 \\lambda ( d - 1 ) z ^ 2 } } . \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\mathsf { I m } \\ , ( e ^ { \\lambda g } P ^ * \\varphi , e ^ { \\lambda g } \\varphi ) = \\sum _ { j = 1 } ^ N \\underset { ( \\ref { C P * } . 1 ) } { \\mathsf { I m } \\ , ( e ^ { \\lambda g } X _ j ^ * f _ j X _ j \\varphi , e ^ { \\lambda g } \\varphi ) } \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} \\mathcal L _ { \\beta , \\Omega } ^ { } \\bar u = \\mathcal L _ { \\beta , \\Omega } ( u - \\phi _ 0 ) = \\mathcal L _ { \\beta , \\Omega } u - \\mathcal L _ \\Omega \\phi _ 0 = - f - \\mathcal L _ \\Omega \\phi _ 0 , \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} \\frac { 1 } { y } = 1 + \\frac { { \\dot \\kappa } ^ { - S } ( \\theta ) } { { \\dot \\kappa } ^ { A } ( \\theta ) } , \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { \\binom k j \\binom j s \\left ( { \\frac { { f _ 3 } } { { f _ 2 } } } \\right ) ^ j \\left ( { \\frac { { f _ 1 } } { { f _ 3 } } } \\right ) ^ s X _ { m - b k + ( b - c ) j + ( c - a ) s } } } = \\frac { { X _ m } } { { f _ 2 { } ^ k } } \\ , , \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{gather*} \\langle s _ { 1 2 3 } , s _ k \\rangle = \\langle s _ { i j } , s _ { 1 2 3 } \\rangle = \\langle s _ { k } , s _ { i j } \\rangle = \\langle s _ i , s _ j \\rangle ^ { \\perp } = ^ { \\perp } \\langle s _ { k i } , s _ { k j } \\rangle \\end{gather*}"}
-{"id": "9375.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 2 } { 3 } \\log { R } + \\frac { 1 } { 4 } \\log { R } + \\frac { 1 } { 1 2 } \\log { R } \\\\ & \\quad \\ge \\frac { \\log { n } } { n } + \\frac { \\log { \\log { n } } } { n } + \\frac { \\log { \\left ( 4 0 \\left ( 0 . 5 + K _ { F , 1 } ( \\tau ) + K _ { F , 4 } ( \\tau ) \\right ) \\right ) } } { n } . \\end{aligned} \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\psi = - ( - \\Delta ) ^ { s } \\psi + V _ { \\mu , \\xi } \\psi + \\tilde { \\phi } ( - ( - \\Delta ) ^ { s } ) \\eta _ { R } + \\cdots , & \\quad \\Omega \\times ( t _ 0 , \\infty ) , \\\\ \\psi = - u ^ * _ { \\mu , \\xi } , & \\quad ( \\mathbb { R } ^ n \\setminus \\Omega ) \\times ( t _ 0 , \\infty ) \\end{cases} \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} \\ell ^ 2 + a ^ 2 k _ x + b ^ 2 ( n - 1 - k _ x ) = n \\ell . \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} \\dim ( \\mathbb { Z } ( \\mathcal { T } _ \\xi ) \\rtimes \\mathcal { S } ) = d ^ n N , \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} \\frac { x } { e ^ x - 1 } = \\sum _ { n \\geq 0 } B _ n \\frac { x ^ n } { n ! } . \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} \\| u \\| _ { S ^ s _ { q , r } ( I ) } \\ ; : = \\ ; \\| | \\nabla | ^ { - 3 ( 1 - \\sigma ) ( \\frac 1 2 - \\frac 1 r ) } u _ \\tau \\| _ { L ^ q _ t ( I ; W ^ { s , r } _ x ) } \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} G _ { L _ { n - 1 } } ( z ) & = \\textstyle \\prod _ { i = 0 } ^ { n - 2 } f _ { n - 1 - i } ^ { K - 1 } ( z ) \\\\ f _ k ( z ) & = \\begin{cases} q _ k + p _ k f _ { k + 1 } ^ { K } ( z ) , & ~ 1 \\le k \\le n - 1 \\\\ z ^ { \\frac { 1 } { K } } , & ~ k = n , \\end{cases} \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} \\sum _ l \\langle f ( h _ j ) , \\tau | _ { g _ 1 X _ l \\cap X _ j } \\pi \\rangle = \\langle f ( h _ j ) , \\tau | _ { \\sqcup _ l g _ 1 X _ l \\cap X _ j } \\pi \\rangle = \\langle f ( h _ j ) , \\tau | _ { X _ j } \\pi \\rangle = \\langle \\pi ^ { * * } ( f ( h _ j ) ) , \\tau _ j \\rangle . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 3 6 n + 1 8 ) q ^ n & \\equiv 2 q f _ { 1 2 } ^ 3 + 3 q f _ { 1 2 } ^ 3 \\cdot \\dfrac { f _ 2 ^ 2 } { f _ 4 } + \\dfrac { f _ 6 ^ 2 } { f _ { 1 2 } } \\cdot f _ 4 ^ 3 \\\\ & \\equiv 2 q f _ { 1 2 } ^ 3 + 3 q f _ { 1 2 } ^ 3 \\left ( \\dfrac { f _ { 1 8 } ^ 2 } { f _ { 3 6 } } - 2 q ^ 2 \\dfrac { f _ 6 f _ { 3 6 } ^ 2 } { f _ { 1 2 } f _ { 1 8 } } \\right ) \\\\ & \\quad + \\dfrac { f _ 6 ^ 2 } { f _ { 1 2 } } \\left ( f _ { 1 2 } a ( q ^ { 1 2 } ) - 3 q ^ 4 f _ { 3 6 } ^ 3 \\right ) ~ ( \\textup { m o d } ~ 4 ) . \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} [ \\omega _ { \\mathbf i } ] = [ \\omega _ { \\mathbf i } ' ] H ^ 2 ( Z _ { \\bf i } ; \\mathbb { R } ) , [ \\omega _ { \\mathcal I } ] = [ \\omega _ { \\mathcal I } ' ] H ^ 2 ( Z _ { \\mathcal I } ; \\mathbb { R } ) \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} \\theta _ { \\lambda , \\mu } = \\xi ^ { \\langle \\lambda + 2 \\rho , \\mu \\rangle } . \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{gather*} X ^ 2 Y + Y ^ 2 Z + Z ^ 2 X = z ^ { 1 / 3 } X Y Z , \\\\ X ^ 4 + Y ^ 2 + Z ^ 4 = z ^ { 1 / 4 } X Y Z \\qquad X ^ 3 + Y ^ 2 + Z ^ 6 = z ^ { 1 / 6 } X Y Z , \\end{gather*}"}
-{"id": "3212.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\gamma ( s ) \\psi ( s ) d s & = \\phi ( t ) \\exp \\left \\{ \\int _ 0 ^ t \\beta ( \\theta ) \\gamma ( \\theta ) d \\theta \\right \\} \\\\ & \\leq \\int _ 0 ^ t \\alpha ( s ) \\gamma ( s ) \\exp \\left \\{ \\int _ s ^ t \\beta ( \\theta ) \\gamma ( \\theta ) d \\theta \\right \\} d s . \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} \\phi _ { p , m , r } \\big ( g , ( a _ 1 , \\dots , a _ { p + m } ) , ( b _ 1 , c _ 1 , \\dots , b _ p , c _ p , b _ { p + 1 } , \\dots , b _ { p + r } ) \\big ) = ( g A g ^ { - 1 } , g B g ^ { - 1 } ) , \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} \\phi _ { B } ( \\textbf { x } ) = \\begin{cases} 1 P \\{ Y = 1 \\vert \\textbf { X } = \\textbf { x } \\} > P \\{ Y = 0 \\vert \\textbf { X } = \\textbf { x } \\} \\\\ 0 \\end{cases} \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} m _ K = \\# \\ker ( \\star ) - 1 . \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} \\begin{aligned} c ( x ' , y , t ) = { } & \\sum _ { x \\in T ^ { - 1 } ( x ' ) } \\frac { 1 } { \\vert T ' ( x ) \\vert } \\bigg \\{ c ( x , y , t ) - x \\frac { \\partial } { \\partial y } b ( x , y , t ) \\\\ & + h ( x ) \\left [ \\frac { \\partial } { \\partial y } \\int d x \\ x b ( x , y , t ) + \\frac { 1 } { 2 } ( x ^ { 2 } - \\langle x ^ { 2 } \\rangle ) \\frac { \\partial ^ { 2 } } { \\partial y ^ { 2 } } \\alpha ( y , t ) \\right ] \\bigg \\} \\end{aligned} \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} \\int _ { K \\backslash G / H } \\int _ N f ( x ) \\cdot 1 _ { A \\cap q ( N ) } ( x \\cdot \\ddot { y } ) d \\omega ( x ) d \\mu ( \\ddot { y } ) = \\int _ N \\int _ { K \\backslash G / H } f ( x ) \\cdot 1 _ { A \\cap q ( N ) } ( x \\cdot \\ddot { y } ) d \\mu ( \\ddot { y } ) d \\omega ( x ) . \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} 2 \\mathsf { R e } \\big ( ( X ^ * _ { N + 1 } - i X ^ * _ 0 ) u , i B u \\big ) = 2 \\mathsf { I m } ( X ^ * _ { N + 1 } u , B u ) - 2 \\mathsf { R e } ( i X ^ * _ 0 u , i B u ) \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} \\mathbb { P } [ \\mu ] = \\hat { C } \\hat { \\theta } \\otimes \\hat { \\theta } = m \\otimes m . \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} F ( z , w ) = \\int _ 0 ^ { \\infty } f ( t , w ) e ^ { i 2 \\pi z t } \\d t \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} \\lim \\limits _ { l \\to \\infty } \\sup _ { \\pi \\in \\Pi } E _ \\mu ^ \\pi \\left ( \\sum _ { k = 1 } ^ \\infty \\beta ^ { k - 1 } w ( s _ k ) 1 _ { \\{ w ( s _ k ) \\ge l \\} } \\right ) = 0 . \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } p _ { 0 } ( y , t ) - \\frac { 1 } { 2 } \\frac { \\partial ^ { 2 } } { \\partial t ^ { 2 } } p _ { 0 } ( y , t ) = 0 \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} \\theta = \\left [ a _ 1 , a _ 2 , \\ldots \\right ] = \\frac { 1 } { a _ 1 + } \\frac { 1 } { a _ 2 + } \\cdots \\frac { 1 } { a _ n + } \\cdots \\ , \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} D _ s : = | x | ^ s \\psi ( x ) d ^ \\times x \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} \\dot B ^ X _ { \\dot \\gamma } + ( B ^ X _ { \\dot \\gamma } ) ^ 2 + { 2 \\ , g ( X / n , \\dot \\gamma ) } \\ , B ^ X _ { \\dot \\gamma } + R ^ { \\bot } _ { X , \\dot \\gamma } = 0 . \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} \\lim \\limits _ { l \\to \\infty } \\sup _ { \\pi \\in \\Pi } E ^ \\pi _ \\mu \\left ( w ( s _ n ) 1 _ { \\{ w ( s _ n ) \\ge l \\} } \\right ) = 0 \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} \\sigma _ q ( x _ i ) = \\epsilon _ q ^ { s _ i } x _ i , \\ , \\sigma _ q ( y _ i ) = \\epsilon _ q ^ { - s _ i } y _ i . \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} A = U _ A ( A ^ * A ) ^ { \\frac { 1 } { 2 } } \\ \\mbox { a n d } \\ U _ A ^ * U _ A = P _ { \\overline { \\mathcal { R } ( A ^ * ) } } . \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} ( ( \\sigma - \\sigma _ h ) ^ { \\frac 1 2 } , w _ h ) + ( \\nabla ( u - u _ h ) ^ { \\frac 1 2 } , \\nabla w _ h ) = 0 . \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} \\mathrm { G L } ( 2 , \\mathbb { R } ) = \\mathrm { S L } ( 2 , \\mathbb { R } ) \\cup E \\cdot \\mathrm { S L } ( 2 , \\mathbb { R } ) \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} \\theta _ { \\sigma ( W ) } = \\theta _ { \\sigma ( V ' ) } \\theta _ { \\sigma ( \\coprod \\limits _ { i = 1 } ^ r b _ i ^ { \\beta _ i } ) } = \\theta _ { \\sigma ( V ' ) } p _ 1 ^ { \\beta _ 1 } \\cdots p _ r ^ { \\beta _ r } . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} \\frac { | x | } { | z | } \\phi ' ( | x | ) = \\lambda \\left ( \\frac { | x | } { | z | } \\right ) ^ { 1 - q } = 1 - \\left ( \\frac { R } { | z | } \\right ) ^ { 1 - q } + \\lambda \\left ( \\frac { R } { | z | } \\right ) ^ { 1 - q } = ( q - 1 ) \\log _ { q } \\frac { R } { | z | } + \\lambda \\left ( \\frac { R } { | z | } \\right ) ^ { 1 - q } \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} u _ R ( \\overline x _ i ) = \\frac 1 N c _ R ( \\overline x _ 1 , \\ldots , \\overline x _ N ) = \\frac 1 N c ( \\overline x _ 1 , \\ldots , \\overline x _ N ) i , \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} \\begin{aligned} & Q _ a ^ 2 Q _ b + Q _ a Q _ b Q _ a + Q _ b Q _ a ^ 2 + 4 Q _ { a , b } ^ 2 Q _ a + 4 Q _ a Q _ { a , b } ^ 2 + 4 Q _ { a , b } Q _ a Q _ { a , b } \\\\ = & Q _ { a ^ 2 b } + 4 Q _ { a ( a b ) } + 4 Q _ { a ^ 2 b , a ( a b ) } + 4 Q _ { a ^ 3 , b ( a b ) } + 2 Q _ { a ^ 3 , b ^ 2 a } \\end{aligned} \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } \\mathbb { E } \\left [ { \\int } _ { t } ^ { T } \\left ( | \\tilde { Y } _ { s } ^ { m } - Y _ { s } ^ { m } | ^ { 2 } + | \\tilde { Z } _ { s } ^ { m } - Z _ { s } ^ { m } | ^ { 2 } \\right ) d s \\right ] & \\leq C ^ { 2 } \\sum \\limits _ { i = 0 } ^ { m - 1 } \\mathbb { E } \\left [ \\left \\vert W ( t _ { i + 1 } ^ { m } , X _ { t _ { i + 1 } ^ { m } } ^ { m } ) - \\tilde { Y } _ { t _ { i + 1 } ^ { m } } ^ { m } \\right \\vert ^ { 2 } \\right ] \\\\ & \\leq \\frac { C ^ { 2 } } { m } . \\end{array} \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} f _ { P } ( x ) = x ^ { n } f _ { P } ( \\tfrac 1 x ) \\ , . \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} I = \\left ( \\begin{array} { c c c c } s _ { 0 } ( 0 ) & s _ { 1 } ( 0 ) & \\cdots & s _ { u - 1 } ( 0 ) \\\\ s _ { 0 } ( 1 ) & s _ { 1 } ( 1 ) & \\cdots & s _ { u - 1 } ( 1 ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ s _ { 0 } ( v - 1 ) & s _ { 1 } ( v - 1 ) & \\cdots & s _ { u - 1 } ( v - 1 ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} \\| T _ { \\alpha , q } ^ { - 1 } u \\| _ { X ( \\mathbb { R } ^ n ) } = \\| u \\| _ { \\tilde { X } ( B _ R ) } \\| \\nabla ( T _ { \\alpha , q } ^ { - 1 } u ) \\| _ { Y ( \\mathbb { R } ^ n ) } = \\| L _ { p , \\theta } u \\| _ { \\tilde { Y } ( B _ R ) } . \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} \\Big { ( } \\mathcal { D } _ t u _ h ^ { n - \\theta } , v _ h \\Big { ) } - \\gamma ( \\nabla \\sigma _ h ^ { n - \\theta } , \\nabla v _ h ) + ( \\nabla u _ h ^ { n - \\theta } , \\nabla v _ h ) + ( f ^ { n - \\theta } ( u _ h ) , v _ h ) = & ( g ^ { n - \\theta } , v _ h ) , ~ \\forall v _ h \\in L _ h , \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} ( \\alpha _ 1 - x ) ( \\alpha _ 2 - x ) p _ 2 ^ 2 ( x ) - ( a _ 1 - x ) ( a _ 2 - x ) ( a _ 3 - x ) p _ 1 ^ 2 ( x ) = x ^ 6 . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\frac { \\partial ^ { k } } { ( \\partial z ) ^ M } t ( z ) = \\sum _ { i = 1 } ^ \\infty \\alpha _ i \\frac { \\partial ^ { k } } { ( \\partial z ) ^ M } z ^ { M _ i } . \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} u _ { k , 0 } ^ { [ j ] , - } ( y ) & : = \\frac { \\partial ^ j } { \\partial s ^ j } y ^ { 1 - \\frac { k } { 2 } - s } \\bigg | _ { s = \\frac { k } { 2 } } = ( - 1 ) ^ j ( \\mathrm { l o g } \\ y ) ^ j y ^ { 1 - k } , \\\\ u _ { k , 0 } ^ { [ j ] , + } ( y ) & : = \\frac { \\partial ^ j } { \\partial s ^ j } y ^ { s - \\frac { k } { 2 } } \\bigg | _ { s = \\frac { k } { 2 } } = ( \\mathrm { l o g } \\ y ) ^ j . \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} L ( s , \\tilde { \\pi } ) = \\sum _ { n \\in \\N } \\overline { \\lambda _ { \\pi } ( n ) } n ^ { - s } . \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} | P \\nabla f | ^ 2 \\le \\frac { \\delta } { c _ 2 } ( a \\Pi P \\nabla f ) \\cdot ( P \\nabla f ) = \\frac { \\delta } { c _ 2 } ( \\Pi ^ T a \\Pi \\nabla f ) \\cdot ( P \\nabla f ) = \\frac { \\delta } { c _ 2 } ( a \\Pi \\nabla f ) \\cdot ( \\Pi \\nabla f ) \\ , . \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 2 4 n + 1 2 ) q ^ n & \\equiv 2 f _ 4 ^ 3 - 2 q f _ { 1 2 } ^ 3 \\\\ & \\equiv 2 f _ { 1 2 } a ( q ^ { 1 2 } ) - 6 q ^ 4 f _ { 3 6 } ^ 3 - 2 q f _ { 1 2 } ^ 3 ~ ( \\textup { m o d } ~ 4 ) , \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} \\max ( \\| X _ n \\| _ \\infty , \\| X _ { n , r } \\| _ \\infty ) \\leq 2 K ( 1 + \\ell ) = \\varrho _ { \\infty } \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} \\Lambda : = \\sup _ { B _ 1 } \\phi ( \\abs { \\nabla \\Q _ L } ) < + \\infty . \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} T _ { \\pm } = \\frac { 1 } { \\sqrt { 2 } } \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\\\ \\mp i e ^ { \\pm 2 i \\alpha } & 0 \\\\ 0 & \\pm i \\end{pmatrix} , T ^ * _ { \\pm } T _ { \\pm } = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ~ . \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} \\mathbf { E } [ L _ i ] & = \\mathbf { E } [ \\mathbf { E } [ L _ i \\lvert L _ { i - 1 } ] ] \\\\ & = \\mathbf { E } [ ( ( L _ { i - 1 } + 1 ) K - 1 ) p _ i ] \\\\ & = p _ i K \\mathbf { E } [ L _ { i - 1 } ] + p _ i ( K - 1 ) , \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} \\Big | f ( z ) - \\sum _ { n = 0 } ^ { p - 1 } a _ n z ^ n \\Big | \\le C A ^ p M _ { p } | z | ^ p , z \\in G . \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X _ { s } ^ { u } = & b ( s , X _ { s } ^ { u } , Y _ { s } ^ { u } , Z _ { s } ^ { u } , u _ { s } ) d s + \\sigma ( s , X _ { s } ^ { u } , Y _ { s } ^ { u } , Z _ { s } ^ { u } , u _ { s } ) d B _ { s } , \\\\ d Y _ { s } ^ { u } = & - g ( s , X _ { s } ^ { u } , Y _ { s } ^ { u } , Z _ { s } ^ { u } , u _ { s } ) d s + Z _ { s } ^ { u } d B _ { s } , \\\\ X _ { t } ^ { u } = & x _ { 0 } , \\ Y _ { t + \\delta } ^ { u } = \\varphi ( t + \\delta , X _ { t + \\delta } ^ { u } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} \\left ( { \\bf w } _ s ^ { ( i ) } - { \\bf w } _ t ^ { ( i ) } \\right ) ^ { ( q ) } = \\int \\limits _ t ^ T \\left ( \\sum _ { j = 0 } ^ { q } \\int \\limits _ t ^ s \\phi _ j ( \\tau ) d \\tau \\cdot \\phi _ j ( \\tau ) \\right ) d { \\bf w } _ { \\tau } ^ { ( i ) } = \\sum _ { j = 0 } ^ { q } \\int \\limits _ t ^ s \\phi _ j ( \\tau ) d \\tau \\cdot \\int \\limits _ t ^ T \\phi _ j ( \\tau ) d { \\bf w } _ { \\tau } ^ { ( i ) } . \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} \\| u ( t _ n ) - u _ h ^ { n } \\| + \\gamma ^ { \\frac 1 2 } \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\sigma ( t _ { k - \\theta } ) - \\sigma _ h ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } + \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\nabla ( u ( t _ { k - \\theta } ) - u _ { h } ^ { k - \\theta } ) \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } \\leq C [ \\Delta t ^ 2 + h ^ { m + 1 } ] , \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} \\alpha _ i ^ t \\in K , \\ ; t = 0 , 1 , \\dots , p _ i - 2 . \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} H ^ \\pm _ \\gamma = H _ \\gamma \\left | _ { \\Omega ^ \\pm } \\right . . \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} \\eta ( s ) = g \\big ( \\xi ( s ) , \\nu ( z ( s ) ) \\big ) ^ + \\cdot \\nu ( z ( s ) ) . \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} \\tilde { K } _ k ( m , n , c ) : = \\sum _ { \\substack { d ( c ) ^ * \\\\ a d \\equiv 1 ( c ) } } \\biggl ( \\frac { c } { d } \\biggr ) \\epsilon _ d ^ { 2 k } e \\biggl ( \\frac { a m + d n } { c } \\biggr ) . \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{gather*} \\Delta ( g ) = g \\otimes g \\forall \\ , x \\in H , S ^ 2 ( x ) = g x g ^ { - 1 } . \\end{gather*}"}
-{"id": "7081.png", "formula": "\\begin{align*} W ( h _ 1 , U _ 1 ) W ( h _ 2 , U _ 2 ) = e ^ { - i ( \\langle h _ 1 , U _ 1 h _ 2 \\rangle ) } W ( ( h _ 1 , U _ 1 ) ( h _ 2 , U _ 2 ) ) , \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} & \\underset { \\{ \\mathbf { p } _ i \\} _ { i = 1 } ^ N , \\ , \\{ \\mathbf { q } _ i \\} _ { i = 1 } ^ N } { \\min } & \\sum _ { i = 1 } ^ M C _ i ( \\mathbf { p } _ i , k ) - \\sum _ { i = M + 1 } ^ N U _ i ( \\mathbf { q } _ i , k ) \\\\ & & \\underline { \\mathbf { p } } _ i ^ { [ k ] } \\leq \\mathbf { q } _ i \\leq \\overline { \\mathbf { p } } _ i ^ { [ k ] } , \\ , i = 1 , \\hdots , N \\\\ & \\sum _ { i = 1 } ^ N \\mathbf { q } _ i = \\mathbf { P } ^ { [ k ] } & \\mathbf { q } _ i = \\mathbf { p } _ i , \\ , i = 1 , \\hdots , N . \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} \\Pi ^ T a = a \\Pi = \\mathcal { O } \\Big ( \\frac { 1 } { \\delta } \\Big ) \\ , , \\mbox { a n d } ( I - \\Pi ^ T ) a = a ( I - \\Pi ) = \\mathcal { O } ( 1 ) \\ , . \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { Q } \\ ; ^ { C } _ { t } D _ { T } ^ { q } \\Theta ( x , t ) = A ^ { * } \\mathcal { Q } \\Theta ( x , t ) + \\left \\langle \\mathcal { Q } z ( t ) , f \\right \\rangle _ { L ^ { 2 } ( D ) } \\chi _ { _ { D } } f ( x ) & \\hbox { i n } Q _ { T } \\\\ \\Theta ( \\xi , t ) = 0 & \\hbox { o n } \\Sigma _ { T } \\\\ \\Theta ( x , T ) = 0 & \\hbox { i n } \\Omega . \\end{cases} \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} a _ { F } ( \\gamma ) = \\chi _ { \\gamma } ( m _ { v } ^ { F } ) \\ , . \\end{align*}"}
-{"id": "9867.png", "formula": "\\begin{align*} \\boldsymbol { D } ^ { - \\mu } \\boldsymbol { D } ^ { - \\sigma } u = \\boldsymbol { D } ^ { - ( \\mu + \\sigma ) } u \\boldsymbol { D } ^ { - \\mu * } \\boldsymbol { D } ^ { - \\sigma * } u = \\boldsymbol { D } ^ { - ( \\mu + \\sigma ) * } u . \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} \\int _ \\R e ^ { - i ( t - s ) H } P _ c F ( s ) \\ , d s = \\int _ { - \\infty } ^ t e ^ { - i ( t - s ) H } P _ c F ( s ) \\ , d s - \\int _ t ^ \\infty e ^ { - i ( t - s ) H } P _ c F ( s ) \\ , d s , \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} \\mathbb { B } _ p = \\{ w \\in \\C ^ n \\ ; | \\ ; p ( w ) < 1 \\} . \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} \\tilde u ( t , x ) = & \\ \\mathbf E \\left [ \\phi \\left ( - X ^ { t , \\beta } ( \\tau _ 0 ( t ) ) , X ^ { x , \\alpha } ( \\tau _ 0 ( t ) ) \\right ) \\mathbf 1 _ { \\{ \\tau _ 0 ( t ) < \\tau _ \\Omega ( x ) \\} } \\right ] \\\\ & \\ + \\mathbf E \\left [ \\int _ 0 ^ { \\tau _ 0 ( t ) \\wedge \\tau _ \\Omega ( x ) } g \\left ( - X ^ { t , \\beta } ( s ) , X ^ { x , \\alpha } ( s ) \\right ) d s \\right ] , \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} x _ { L _ { 0 } - 1 } & = x _ { L _ { 0 } } ( 1 - x _ { L _ { 0 } } ) ^ { q - 1 } \\\\ & > \\frac { 1 } { 3 q } \\Big ( 1 - \\frac { 1 } { q } \\Big ) ^ { q - 1 } = \\frac { E _ { q } } { 3 } . \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{gather*} \\abs { \\left \\{ \\alpha \\in X _ n ^ k : \\alpha ( 0 ) = 0 , \\ P ( \\alpha ) = d \\right \\} } = \\sum _ { B \\subset P D _ d } ( - 1 ) ^ { \\abs { B } } \\left ( \\begin{array} { c } \\frac { d } { \\prod _ { p \\in B } p } \\frac { n + 2 } { k + 1 } - 1 \\\\ \\frac { d } { \\prod _ { p \\in B } p } - 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "137.png", "formula": "\\begin{align*} ( \\mathcal { H } _ { \\nu } f ) ( \\rho , y ) = \\int _ 0 ^ \\infty ( r \\rho ) ^ { - \\frac { n - 2 } 2 } J _ { \\nu } ( r \\rho ) f ( r , y ) r ^ { n - 1 } d r . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\Delta _ g u + \\lambda ^ 2 V ( x ) e ^ { \\ , u } = \\rho _ \\lambda c _ M . \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\tilde f ( r , W _ r ) \\mathrm d r = A ^ { W , W } _ t ( \\tilde f ) . \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} \\langle R ^ N ( \\psi , \\psi ) \\psi , \\psi \\rangle = \\langle R ^ N ( v , v ) v , v \\rangle | \\Psi | ^ 4 = 0 \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} f ( x ) & = ( x + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( x + 1 ) + ( m - 1 ) \\ : h ( x ) , \\\\ g ( x ) & = ( x ^ 2 + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( x + 1 ) - m \\ : h ( x ) , \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} r _ n ( x ) - r ( x ) = & ~ ( g _ n - r f _ n ) ( x ) - \\mathrm { E } ( ( g _ n - r f _ n ) ( x ) ) \\\\ & - ( r ( x ) - \\mathrm { E } g _ n ( x ) ) + ( r ( x ) - \\mathrm { E } g _ n ( x ) ) ( f _ n ( x ) - \\mathrm { E } f _ n ( x ) ) \\\\ & - ( g _ n ( x ) - \\mathrm { E } g _ n ( x ) ) ( f _ n ( x ) - \\mathrm { E } f _ n ( x ) ) + r _ n ( x ) ( f _ n ( x ) - \\mathrm { E } f _ n ( x ) ) ^ 2 . \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\mathfrak F _ { \\chi ^ { - 1 } } \\circ \\mathfrak J _ { \\chi } = \\mathfrak J _ { \\chi ^ { - 1 } } . \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} c _ 0 ^ 2 C | \\nabla p | ^ 2 - C ^ \\gamma = 0 \\qquad \\mbox { o n s u p p } ( \\lambda ) . \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} Q \\big ( f \\cdot \\frac { \\rho _ 1 } { \\rho _ 2 } \\big ) ( K n H ) & = \\int _ K \\int _ H f ( k ^ { - 1 } n h ) \\frac { \\rho _ 1 ( k ^ { - 1 } n h ) } { \\rho _ 2 ( k ^ { - 1 } n h ) } d h d k \\\\ & = \\frac { \\rho _ 1 ( n ) } { \\rho _ 2 ( n ) } \\int _ K \\int _ H f ( k ^ { - 1 } n h ) d h d k , \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} d \\bar { x } _ i ( s ) = - ( \\Pi ^ T a ) _ { i j } \\frac { \\partial V } { \\partial x _ j } \\ , d s + \\frac { 1 } { \\beta } \\frac { \\partial ( \\Pi ^ T a ) _ { i j } } { \\partial x _ j } \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } ( \\Pi ^ T \\sigma ) _ { i j } \\ , d w _ j ( s ) \\ , . \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} \\mathcal { W } _ { \\frac { 1 } { 2 } , 0 } \\biggl ( y , \\frac { s } { 2 } + \\frac { 1 } { 4 } \\biggr ) = \\frac { ( 4 \\pi ) ^ { 1 / 2 } y ^ { 1 / 2 - s / 2 } } { ( s - 1 / 2 ) \\Gamma ( s / 2 ) \\Gamma ( s / 2 + 1 / 2 ) } . \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} P _ n ( x ) = a _ 0 + a _ 1 x + \\dots + a _ n x ^ n , \\\\ Q _ n ( x ) = b _ 0 + b _ 1 x + \\dots + b _ n x ^ n , \\\\ T _ n ( x ) = c _ 0 + c _ 1 x + \\dots + c _ n x ^ n . \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} 0 = b _ { p ^ n } b _ { ( p - 1 ) p ^ n } \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} p ^ { ( n ) } ( o , \\ , x ) = \\frac { 1 } { ( 2 \\pi n ) ^ { d / 2 } ( \\det \\Sigma ) ^ { 1 / 2 } } \\exp \\left ( - \\frac { ( x - n \\mathbf { m } ) \\cdot \\Sigma ^ { - 1 } ( x - n \\mathbf { m } ) } { 2 n } \\right ) + o ( n ^ { - d / 2 } ) , \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\begin{aligned} & \\bigg | \\lim _ { t \\to \\infty } \\sum _ { T ( n ) < | \\Im ( \\rho ) | \\le t } \\Re \\left ( 1 - \\left ( \\frac { \\rho } { \\rho - \\tau } \\right ) ^ n \\right ) \\bigg | \\\\ & \\quad < \\sum _ { h = 0 } ^ { \\infty } \\sum _ { | \\Im ( \\rho ) | \\in ( 2 ^ h T ( n ) , 2 ^ { h + 1 } T ( n ) ] } \\frac { 3 n \\tau ^ 2 + ( n e \\tau ) ^ 2 } { 3 \\Im ( \\rho ) ^ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} & \\langle r ( \\tilde { x } _ k ) , \\epsilon _ { o u t } ^ k \\rangle \\leq \\langle s _ { \\delta , L } ( \\lambda _ k ) , \\lambda _ { \\delta k } - \\lambda _ k \\rangle \\\\ & \\leq \\Phi _ \\rho ( \\lambda _ { \\delta k } ) - \\Phi _ { \\delta , L } ( \\lambda _ k ) + \\frac { L } { 2 } B _ { o u t } ^ 2 + \\delta . \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} Q _ n ( x ) = b _ 0 + b _ 1 x + \\dots + b _ n x ^ n ; \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} L _ \\xi ( ( d - ( n + 1 ) ) \\varpi _ 1 , ( d - ( n + 1 ) ) \\varpi _ 1 ) \\otimes L _ \\xi ( \\eta , \\eta ) \\simeq L _ \\xi \\left ( \\sum _ { i = 1 } ^ n \\eta _ { i - 1 } \\varpi _ i , \\sum _ { i = 1 } ^ n \\eta _ { i - 1 } \\varpi _ i \\right ) , \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\lambda - \\varepsilon \\Delta v - \\mathcal { L } _ R ^ \\psi [ v ] + | D v | ^ m = f , \\quad & x \\in B _ R , \\\\ v = g , & x \\in \\partial B _ R . \\end{array} \\right . \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} \\mathcal { L } ( X , \\dot X ) = \\int _ { R } L ( u _ 1 ( x , X ( t ) ) ) \\ , d x = a _ 0 ( X ) \\dot X ^ 2 - a _ 1 ( X ) , \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} S ( n , n ) = 1 , S ( n , n - 1 ) = \\binom { n } { 2 } , S ( n , n - 2 ) = \\binom { n } { 3 } + 3 \\binom { n } { 4 } . \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} ( t , b , h ) \\cdot ( z ' , z '' , z _ n ) = ( t z ' , z '' + b , z _ n + 2 i \\langle z '' , b \\rangle + s + i | b | ^ 2 ) , \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} - \\varphi _ 0 \\partial _ 1 ( \\epsilon _ { f , i j } ) = d _ 1 ( - \\varphi _ 1 ( \\epsilon _ { f , i j } ) + d _ 2 ( \\sigma _ { f , i j } ) ) = d _ 1 ( - \\varphi _ 1 ( \\epsilon _ { f , i j } ) ) \\in ( x _ i , x _ j ) \\varphi _ 0 ( F _ 0 ) . \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} \\mathcal { P } ^ { a , p } : = \\mathbb { C } ^ n \\times \\mathbb { C } ^ n \\times \\ell ^ { a , p } _ J \\times \\ell ^ { a , p } _ J \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} \\varphi _ { \\theta } = \\sum _ { i = 0 } ^ { r } \\varphi _ { \\theta , i } \\tau ^ i = \\sum _ { i = 0 } ^ { r } \\ell _ i ( z _ 1 ) \\dots \\ell _ i ( z _ n ) \\phi _ { \\theta , i } \\tau ^ i . \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} \\Gamma _ { \\varphi } ( \\varphi , \\sigma , \\mu ) & = \\Lambda _ { \\varphi } ( \\varphi , \\sigma ) - \\theta _ { \\varphi } ( \\varphi , \\sigma ) \\mu , \\\\ \\Gamma _ { \\sigma } ( \\varphi , \\sigma , \\mu ) & = \\Lambda _ { \\sigma } ( \\varphi , \\sigma ) - \\theta _ { \\sigma } ( \\varphi , \\sigma ) \\mu , \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} \\rho ( \\gamma _ 1 ) \\rho ( m _ t ) R _ j & = \\rho ( \\gamma _ 1 ) \\big ( R _ j - \\epsilon ( [ g _ t g _ j ^ { - 1 } ] ) R _ t \\big ) & & [ \\textrm { E q u a t i o n } ( \\ref { g e n e r a t o r a c t i o n } ) ] \\\\ & = \\epsilon ( [ g _ \\alpha \\gamma _ 1 g _ j ^ { - 1 } ] ) - \\epsilon ( [ g _ t g _ j ^ { - 1 } ] ) \\epsilon ( [ g _ \\alpha \\gamma _ 1 g _ t ^ { - 1 } ] ) , & & [ \\textrm { D e f i n i t i o n } ( \\ref { L i f t R e p } ) ] \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} w _ a = \\frac { M _ a } M \\ , , \\qquad \\kappa _ a = \\frac { Z _ a } Z \\ , . \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} l = r ! \\ , e ^ { ( 1 + o ( 1 ) ) n \\log n } . \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} \\begin{aligned} p ( y ) = { } & \\left ( \\frac { 2 } { \\pi } \\right ) ^ { 1 / 2 } \\biggl [ 1 + \\tau ^ { 1 / 2 } \\left ( - \\frac { 8 } { 3 } y ^ { 3 } + 2 y \\right ) \\\\ & + \\tau \\left ( \\frac { 3 2 } { 9 } y ^ { 6 } - \\frac { 3 1 } { 3 } y ^ { 4 } + \\frac { 1 5 } { 2 } y ^ { 2 } - \\frac { 3 7 } { 4 8 } \\right ) \\biggr ] e ^ { - 2 y ^ { 2 } } + O ( \\tau ^ { 3 / 2 } ) \\end{aligned} \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} Q ^ { ( \\tau , Z ) } [ t _ 0 , x _ 0 ] = \\mathbb { E } \\left [ \\int _ { t _ 0 } ^ { \\tau _ S } W ( r , X ^ { t _ 0 , x _ 0 , ( \\tau , Z ) } _ r ) d r + \\sum _ { j \\geq 1 } e ^ { - \\delta _ p \\tau _ { j } } c _ P ( \\tau _ { j } ^ - , z _ { j } ) \\cdot 1 _ { \\{ \\tau _ { j } \\leq \\tau _ S \\} } \\right ] & , \\\\ \\forall ( t _ 0 , x _ 0 ) \\in [ 0 , T ] \\times S , \\ ; \\forall ( \\tau , Z ) & \\in U , \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} \\frac { \\partial w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) } { \\partial u _ k } = a _ { i j k } + O ( \\vec { u } ) . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} \\tilde { g } _ { j } = \\frac { \\alpha _ { j } + \\gamma _ { \\alpha _ { j } } } { ( \\sum _ { k = 1 } ^ { 2 ^ { b } } \\alpha _ { k } ) + \\gamma _ { \\alpha _ { j } } - \\gamma _ { \\alpha _ { j + 1 } } } . \\end{align*}"}
-{"id": "9850.png", "formula": "\\begin{align*} K - 3 q - \\dfrac { 4 q ^ { 2 } } { K } = \\dfrac { E _ { 1 } ^ { 2 } E _ { 2 } ^ { 2 } } { E _ { 5 } ^ { 2 } E _ { 1 0 } ^ { 2 } } . \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} b _ i : = \\Psi _ i \\circ ( h _ i - \\mathfrak { s } _ i ) \\circ \\psi _ i ^ { - 1 } : U ' _ i \\rightarrow U ' _ i \\lhd \\mathbb { R } ^ { k ' _ i } \\times \\mathbb { W } _ i \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} \\left ( n h _ { n } \\right ) ^ { - ( d + 1 ) / 5 d } \\leq \\left ( n h _ { n } \\right ) ^ { - 3 / 1 0 } = \\left ( h _ { n } ^ { 2 / 5 } \\right ) ^ { 3 / 4 } \\left ( \\left ( n h _ { n } ^ { 2 } \\right ) ^ { - 6 / 5 } \\right ) ^ { 1 / 4 } \\leq h _ { n } ^ { 2 / 5 } + \\left ( n h _ { n } ^ { 2 } \\right ) ^ { - 6 / 5 } . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} B _ n ( q ) = \\sum _ { r = 0 } ^ { n - 1 } S ( n , r + 1 ) ( r + 1 ) ! q ^ r \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} \\begin{aligned} & \\bar { \\theta } _ { i , i } = 0 , \\\\ & \\bar { \\theta } _ { i , 0 } = \\frac { \\pi } { 2 } - \\bar { \\theta } _ { i , 1 } , \\end{aligned} \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} S _ { ( \\lambda , \\mu ) , ( \\lambda ' , \\mu ' ) } = \\xi ^ { \\langle 2 \\rho , \\mu - \\eta + \\mu ' - \\eta ' \\rangle - 2 \\langle \\mu - \\eta , \\mu ' - \\eta ' \\rangle } \\tilde { s } _ { \\eta , \\eta ' } \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{gather*} \\Delta \\big ( \\big [ x _ i ^ + , x _ { j , 1 } ^ + \\big ] \\big ) = \\big [ \\square \\big ( x _ i ^ + \\big ) , \\square \\big ( x _ { j , 1 } ^ + \\big ) - \\hbar \\big [ 1 \\otimes x _ j ^ + , \\Omega _ + \\big ] \\big ] \\\\ \\hphantom { \\Delta \\big ( \\big [ x _ i ^ + , x _ { j , 1 } ^ + \\big ] \\big ) } { } = \\square \\ , T _ i \\big ( x _ { j , 1 } ^ - \\big ) - \\hbar \\big [ \\square \\big ( x _ i ^ + \\big ) , \\big [ 1 \\otimes x _ j ^ + , \\Omega _ + \\big ] \\big ] . \\end{gather*}"}
-{"id": "6553.png", "formula": "\\begin{gather*} T _ 2 T _ 3 \\cdots T _ { N - 1 } ( E _ { 1 , 1 } ( s ) ) = E _ { 1 , 1 } ( s ) , T _ 2 T _ 3 \\cdots T _ { N - 1 } ( E _ { N , N } ( - s ) ) = E _ { 2 , 2 } ( - s ) \\end{gather*}"}
-{"id": "9825.png", "formula": "\\begin{align*} \\begin{aligned} \\bar x + g ( t ) & = \\bar x + f _ { M + 1 } ( t ) \\sum _ { j = 1 } ^ M \\beta _ j ( x _ j - \\bar x ) \\\\ & = \\big ( 1 - f _ { M + 1 } ( t ) \\big ) \\bar x + f _ { M + 1 } ( t ) \\big ( \\bar x + \\sum _ { j = 1 } ^ M \\beta _ j ( x _ j - \\bar x ) \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } \\int _ { ( \\partial S _ { \\varepsilon } \\cap \\overline { D } ( 0 , R ) ) ^ + } e ^ { z w } \\psi _ { \\varepsilon } u ( z ) d z = \\int _ { \\partial S _ { \\varepsilon } ^ + } e ^ { z w } u ( z ) d z . \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} x \\cdot _ \\chi y = \\chi \\Bigl ( s ( \\overline { g _ 1 } ) s ( \\overline { g _ 2 } ) s ( \\overline { g _ 1 } \\overline { g _ 2 } ) ^ { - 1 } \\Bigr ) x y \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} \\frac { d - 2 } { 2 } \\| Q \\| ^ 2 _ { \\dot { H } ^ 1 _ c } + \\frac { d } { 2 } \\| Q \\| ^ 2 _ { L ^ 2 } = \\frac { d ^ 2 } { 2 d + 4 } \\| Q \\| ^ { \\frac { 4 } { d } + 2 } _ { L ^ { \\frac { 4 } { d } + 2 } } . \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} U ( E _ c ) = \\sum _ { i = 1 } ^ k \\beta _ i U ( E _ { a _ i } ) = \\sum _ { i = 1 } ^ k \\beta _ i E _ { a _ i } ' = E _ c ' \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} s _ \\nu = \\sum _ { j = 1 } ^ { r _ \\nu } x ^ { ( \\nu ) } _ { j , 1 } \\cdots x ^ { ( \\nu ) } _ { j , N _ j } . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} i _ A = \\sup _ { 0 < t < 1 } \\frac { \\log \\widehat { h } _ A ( t ) } { \\log t } \\quad I _ A = \\inf _ { 1 < t < \\infty } \\frac { \\log \\widehat { h } _ A ( t ) } { \\log t } . \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} \\big [ \\mathbb { A } / f \\mathbb { A } \\big ] _ { \\mathbb { A } } = f , \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} ( A _ 1 + A _ 2 ) ( A _ 3 + A _ 4 ) & = ( n - 1 ) ( J _ f - I _ f ) \\otimes ( J _ { \\ell + 1 } - I _ { \\ell + 1 } ) \\otimes J _ n \\\\ & = ( n - 1 ) ( A _ 3 + A _ 4 ) , \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} G _ { m + n } = F _ { n - 1 } G _ m + F _ n G _ { m + 1 } \\ , , \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} \\int _ X | u - \\phi | \\theta _ u ^ n = \\int _ { 0 } ^ { + \\infty } \\theta _ u ^ n ( u < \\phi - t ) d t = 2 \\int _ 0 ^ { + \\infty } \\theta _ u ^ n ( u < \\phi - 2 t ) d t . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| u _ n ( t _ n ) \\| ^ 2 _ { L ^ 2 } = M , \\lim _ { n \\rightarrow \\infty } E ( u _ n ( t _ n ) ) = d _ M . \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} [ T _ { a b } , T _ { a ^ 2 } ] = 2 ( T _ a ^ 2 [ T _ a , T _ b ] + [ T _ a , T _ b ] T _ a ^ 2 - [ T _ a T _ { a ^ 2 } , T _ b ] ) \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} \\mathcal { R } ^ { \\nu } _ { k i j } ( \\alpha _ { 1 , \\nu } ) = \\{ \\xi \\in \\mathcal { O } _ { \\nu } : | \\langle k , \\omega _ { \\nu } ( \\xi ) \\rangle + \\bar { \\Omega } _ { \\nu , i } ( \\xi ) - \\bar { \\Omega } _ { \\nu , j } ( \\xi ) | < \\alpha _ { 1 , \\nu } \\frac { \\max \\{ | i | , | j | \\} } { \\langle k \\rangle ^ { \\tau } } \\} , i \\neq \\pm j . \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} J ( f , g ) ^ 2 = \\begin{cases} ( 2 \\pi ) ^ { - 2 } ( 1 - e ^ { - 2 t } ) \\frac { a ^ 2 b ^ 2 } { 4 ( 1 + a b ) ( 1 - e ^ { - 2 t } ) - 3 + 2 ( 2 e ^ { - 2 t } - 1 ) ( a + b ) } & \\\\ [ 1 e x ] + \\infty & \\end{cases} \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n r _ j M _ j > \\sum _ { j = 1 } ^ n r _ j I _ j . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} \\lvert \\phi v ( w ) - \\phi v ( x ) - \\phi v ( y ) \\rvert = & \\lvert \\phi v ( x _ 1 y _ 2 ) - \\phi v ( x _ 1 z ) - \\phi v ( z ^ { - 1 } y _ 2 ) \\rvert \\\\ = & \\lvert \\phi v ( x _ 1 y _ 2 ) - \\phi v ( x _ 1 ) - \\phi v ( y _ 2 ) \\\\ & - \\phi v ( x _ 1 z ) + \\phi v ( x _ 1 ) + \\phi v ( z ) \\\\ & - \\phi v ( z ^ { - 1 } y _ 2 ) - \\phi v ( z ) + \\phi v ( y _ 2 ) \\rvert \\\\ \\leq & 2 \\lvert v \\rvert _ S + 2 \\lvert v \\rvert _ S + 2 \\lvert v \\rvert _ S \\\\ < & \\infty . \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} \\phi ( q ^ 2 ) = \\prod _ { k = 1 } ^ { \\infty } ( ( 1 - q ^ k ) / ( 1 + q ^ k ) ) ~ . \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} \\lambda _ 1 ^ \\star & = \\left ( \\frac { z _ 2 } { b } \\right ) \\frac { l _ 2 ^ { - 1 } z _ 2 ^ { - l _ 2 } - l _ 1 ^ { - 1 } z _ 2 ^ { - l _ 1 } } { l _ 1 ^ { - 1 } z _ 2 ^ { - l _ 1 } + l _ 2 ^ { - 1 } z _ 2 ^ { - l _ 2 } } - 1 \\\\ \\kappa _ 1 ^ \\star & = z \\left [ l _ 1 ^ { - 1 } + l _ 2 ^ { - 1 } - 1 \\right ] - z _ 2 \\frac { l _ 1 ^ { - 1 } z _ 2 ^ { - l _ 1 } - l _ 2 ^ { - 1 } z _ 2 ^ { - l _ 2 } } { l _ 1 ^ { - 1 } z _ 2 ^ { - l _ 1 } + l _ 2 ^ { - 1 } z _ 2 ^ { - l _ 2 } } \\left [ l _ 1 ^ { - 1 } - l _ 2 ^ { - 1 } + \\ln { \\hat { x } _ 2 } - \\ln { x ^ \\star _ 2 } \\right ] , \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} S ( x ) = \\delta ( x - { x ^ + } ) - \\delta ( x - { x ^ - } ) . \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} \\mathrm { T } y : = \\underbrace { k _ m } _ { > 0 } \\langle y , \\phi _ m \\rangle + \\sum _ { i = m + 1 } ^ { M - 1 } \\underbrace { k _ i } _ { \\ge 0 } \\langle y , \\phi _ i \\rangle + \\underbrace { k _ M } _ { > 0 } \\langle y , \\phi _ M \\rangle , 1 \\le m < M , m , M \\in \\mathbb { N } . \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} \\lambda _ { i j } ^ { k } & = \\lambda _ { ( n + i ) ( n + j ) } ^ { k } = 0 , \\\\ \\lambda _ { i ( n + j ) } ^ { k } & = \\lambda _ { j ( n + i ) } ^ { k } \\omega ( i , j ) = k . \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} \\inf _ { ( \\gamma , \\xi ) \\in \\Omega ( y _ 0 ) } \\left \\{ \\int _ { Y \\times U } ( k ( y , u ) + 2 \\epsilon ) \\gamma ( d y , d u ) + M \\epsilon \\int _ { Y \\times U } \\xi ( d y , d u ) \\right \\} : = k ^ * ( y _ 0 , \\epsilon ) . \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} E _ { L _ { 1 } } = ( \\lambda _ { 1 , k } ^ { j - 1 } ) _ { j \\in [ r ] , k \\in [ 0 , s - 1 ] } , F _ { L _ 1 } = ( c ^ { 1 } _ { 1 , b ( 1 , k ) } ) _ { k \\in [ 0 , s - 1 ] } , E _ { V } = ( \\lambda _ { i , b _ { i } } ^ { j - 1 } ) _ { j \\in [ r ] , i \\in [ 2 , n ] } , F _ { V } = ( \\mu ^ { ( b ) } _ { i , 1 } ) _ { i \\in [ 2 , n ] } . \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { 0 } ( \\textbf { x } ) = I \\left \\lbrace \\textbf { x } \\in \\mathcal { C } ^ { ' } _ { 1 } \\cap \\mathcal { C } ^ { ' } _ { 0 } , ( 1 - q ( \\textbf { x } , 1 ) ) \\dfrac { p } { \\mu { ( \\mathcal { C } ^ { ' } _ { 1 } ) } } > ( 1 - q ( \\textbf { x } , 0 ) ) \\dfrac { 1 - p } { \\mu { ( \\mathcal { C } ^ { ' } _ { 0 } ) } } \\right \\rbrace + I \\left \\lbrace \\textbf { x } \\in \\mathcal { C } ^ { ' } _ { 1 } - \\mathcal { C } ^ { ' } _ { 0 } \\right \\rbrace . \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} f _ 1 ^ n + f _ 2 ^ n + \\cdots + f _ m ^ n = 1 \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} \\mathcal { O } \\setminus \\mathcal { O } _ { \\alpha } = \\Theta ^ 1 _ { \\alpha } \\cup \\Theta ^ 2 _ { \\alpha } , \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} ( \\mathbf { I } - \\mathbf { M } ) ^ { - 1 } \\mathbf { M } z ^ { k } = ( \\mathbf { I } - \\mathbf { K } - \\mathbf { K _ { e } } ) ^ { - 1 } ( g _ { k } + g _ { k , e } ) + o ( n ^ { - M } ) , \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} g t - t = r t + c ( ( 1 + r ) ^ { - 1 } - ( 1 + r ) ) u . \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} t _ 1 = 1 ; t _ i = \\prod _ { j = 1 } ^ { i - 1 } s _ j , \\ ; i = 2 , 3 , \\dots , h + 1 \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty p ^ \\beta _ s ( t - r ) \\ , d s = \\frac { ( t - r ) ^ { \\beta - 1 } } { \\Gamma ( \\beta ) } , t > r . \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} ( E _ i x ' , y ) = \\sum _ { 0 \\leq \\nu \\leq \\mu } ( x ' , L _ { \\nu } ^ { - 1 } y ' _ { - ( \\mu - \\nu ) } ) ( E _ i , y '' _ { - \\nu } ) = ( x ' , y ' _ { - ( \\mu - \\alpha _ i ) } ) ( E _ i , y '' _ { - \\alpha _ i } ) . \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} ( a ( z _ 1 ) \\dots a ( z _ n ) ) \\cdot \\tau ^ m ( Y _ 1 \\dots Y _ n ) = \\tau ^ m ( C _ a ( Y _ 1 ) \\dots C _ a ( Y _ n ) ) . \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} C _ 2 \\dot { A } _ 2 + \\dot { B } \\frac { \\partial F } { \\partial P _ 2 } = 0 ; \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} \\left | \\varphi ( u ) - \\sum _ { k = 0 } ^ { p - 1 } b _ k u ^ { k r } \\right | = \\left | \\psi ( u ^ { r } ) - \\sum _ { k = 0 } ^ { p - 1 } b _ k ( u ^ { r } ) ^ k \\right | \\le C A ^ p M _ p | u | ^ { p r } . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} \\hat { I } _ { d - 1 } ^ \\top \\check { I } _ { d - 1 } = J _ { d - 1 } ( 0 ) . \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} Y _ { t } ^ { t , x ; u ^ { m } } = \\sum _ { i = 1 } ^ { m } Y _ { t } ^ { t , x ; v ^ { i , m } } I _ { A _ { i } } \\geq \\underset { v \\in \\mathcal { U } ^ { t } [ t , T ] } { \\inf } Y _ { t } ^ { t , x ; v } P a . s . . \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { E } ( t ) = - \\sum _ { n = 0 } ^ { \\infty } ( \\delta u ( n , t ) ) ^ 2 \\leq 0 . \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} \\liminf _ { t \\to \\infty } e ^ { - \\beta t } M ^ { \\phi } ( t ) = \\Delta \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} R ^ { ( i ) } ( D ) = \\begin{cases} \\frac { 1 } { 2 } \\log \\frac { 1 } { D } & 0 < D \\le 1 , \\\\ 0 & D > 1 , \\end{cases} \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} \\mu _ \\lambda = \\frac { \\rho _ \\lambda c _ \\Omega | \\partial _ n K ^ \\pm _ \\gamma | } { a _ 0 \\lambda } . \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} t \\log t & = \\frac { n } { d } \\log \\frac { n } { d } + \\left ( 1 + \\log \\frac { n } { d } \\right ) \\left ( t - \\frac { n } { d } \\right ) + \\frac { 1 } { 2 } \\xi ^ { - 1 } \\left ( t - \\frac { n } { d } \\right ) ^ 2 \\\\ & \\leq \\frac { n } { d } \\log \\frac { n } { d } + \\left ( 1 + \\log \\frac { n } { d } \\right ) \\left ( t - \\frac { n } { d } \\right ) + \\frac { 1 } { 2 } \\frac { n } { n / d - \\sqrt { n } } \\\\ & \\leq \\frac { n } { d } \\log \\frac { n } { d } + \\left ( 1 + \\log \\frac { n } { d } \\right ) \\left ( t - \\frac { n } { d } \\right ) + d . \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} \\gamma _ + ( t u ) > t ^ 4 \\left [ \\norm { u ^ + } ^ 2 + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } u ^ 2 d x - \\int _ { \\mathbb R ^ 3 } \\frac { f ( t u ^ + ) } { t ^ 3 } u ^ + d x \\right ] > t ^ 4 \\gamma _ + ( u ) = 0 , \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} \\lvert ( \\phi v ) _ n \\rvert _ S = \\lvert T ^ { - n } [ v ] \\rvert _ S > n _ b ( W ^ n ( w ) ) + C . \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} T ( r , f ( z + c ) ) & = m ( r , f ( z + c ) ) \\leq m ( r , f ( z ) ) + m \\left ( r , \\frac { f ( z + c ) } { f ( z ) } \\right ) \\\\ & = T ( r , f ( z ) ) + m \\left ( r , \\frac { f ( z + c ) } { f ( z ) } \\right ) . \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} d _ a = \\begin{cases} 2 , a \\ne 1 ; \\\\ 3 , a = 1 . \\end{cases} \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} O _ { \\varpi } = \\bigl ( \\underbrace { \\emptyset , \\dotsc , \\emptyset } _ K , \\underbrace { ( 1 ) , \\dotsc , ( 1 ) } _ { c _ { n - 1 } } , \\dotsc , \\underbrace { ( n - 1 , \\dotsc , 1 ) , \\dotsc , ( n - 1 , \\dotsc , 1 ) } _ { n _ 1 } \\bigr ) , \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} D ^ b ( C ^ { [ n + g - 1 ] } ) = \\langle \\overbrace { D ^ b ( J _ C ) , \\ldots , D ^ b ( J _ C ) } ^ { n } \\rangle \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} \\int _ { [ 0 , T ] \\times \\mathbb R ^ d } p _ s ( y ) \\mathrm d s \\mathrm d y = T \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} { D } \\ { \\rm i s \\ s t a r \\ s h a p e d \\ w i t h \\ r e s p e c t \\ t o \\ a \\ d i s c \\ \\mathfrak { B } \\ w i t h \\ r a d i u s \\ = \\ \\rho } h _ D . \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} a \\Delta _ \\kappa a + b \\Delta _ \\kappa a = c \\Delta _ \\kappa a . \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} G _ { L _ { n - 1 } } ( z ) & = \\mathbf { E } [ z ^ { L _ { n - 1 } } ] \\\\ & = \\sum _ { k = 0 } ^ { K ^ { n - 1 } - 1 } P ( L _ { n - 1 } = k ) z ^ k . \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\Big ( ( x y ) \\alpha _ i ( z ) \\otimes ( a b ) \\beta _ i ( c ) + ( z x ) \\alpha _ i ( y ) \\otimes ( c a ) \\beta _ i ( b ) + ( y z ) \\alpha _ i ( x ) \\otimes ( b c ) \\beta _ i ( a ) \\Big ) = 0 \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} \\mathsf { r e c t s } | _ { \\mathcal { P } } ( T _ \\mathcal { C } ) = \\mathsf { r e c t s } | _ { \\mathcal { P } } ( T _ \\mathcal { S } ) = \\mathsf { r e c t s } | _ { \\mathcal { P } } ( T ) , \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} \\mathbf { B } ^ i _ { t } ( x _ 1 ) - \\mathbf { B } ^ i _ { t } ( x _ 2 ) = r A _ { i 1 } ^ { } + r A _ { i 1 } ^ { } + r A _ { i 1 } ^ { } + r A _ { i 1 } ^ { } + r \\varepsilon _ 1 A _ { i 2 } + r ( \\mathbf { e } _ 1 . \\eta _ t ^ \\varepsilon ( x _ 1 ) ) A _ { i 1 } ^ { } , \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} \\left | \\left \\langle \\left ( \\frac { 1 } { 2 k } \\sum _ { j = 1 } ^ k { \\left ( \\omega _ j \\pi ( x _ j ) + \\omega _ j ^ { - 1 } \\pi ( x _ j ) ^ * \\right ) } \\right ) ^ { 2 r + 1 } v , w \\right \\rangle \\right | & \\leq \\epsilon + M O \\left ( \\frac { r } { k } \\right ) ^ r . \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} W ( \\tilde { \\Pi } - \\tilde { Q } ) ^ 2 = \\tilde { Q } ^ { ( 1 ) } . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} R = \\frac { R _ 1 } { \\sqrt { r ^ * } } \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} 2 \\leq q , p \\leq \\infty , \\quad \\frac { 1 } { q } = ( { d - \\frac { d _ a } { 2 } + \\frac { 1 } { 2 } } ) ( \\frac { 1 } { 2 } - \\frac { 1 } { p } ) , ( q , p ) \\neq ( 2 , \\frac { 4 d - 2 } { 2 d - 3 } ) . \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} \\| I ^ { + } f \\| _ { \\mathcal { B } _ { \\alpha } ^ { 2 } } \\leq \\left ( \\sup _ { k \\in \\mathbb { Z } ^ { n } \\setminus \\{ 0 \\} } \\frac { 1 } { 1 - \\alpha / | k | ^ { 3 } } \\right ) \\| f \\| _ { \\mathcal { B } _ { \\alpha } ^ { 0 } } = \\frac { 1 } { 1 - \\alpha } \\| f \\| _ { \\mathcal { B } _ { \\alpha } ^ { 0 } } . \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} a _ 1 ( z ) + a _ 2 ( z ) + a _ 3 ( z ) & = \\frac { A ( 8 \\alpha ^ 2 - 4 \\alpha - 1 ) \\left ( 3 2 \\alpha ^ 3 - 8 \\alpha ^ 2 + 4 \\alpha - 1 \\right ) } { 4 ( 1 - 4 \\alpha ) ^ 2 } z \\\\ & + \\frac { A ( 8 \\alpha ^ 2 - 4 \\alpha - 1 ) \\left ( 3 2 \\alpha ^ 4 + 1 6 0 \\alpha ^ 3 - 8 \\alpha ^ 2 + 8 \\alpha - 3 \\right ) } { 1 6 ( 1 - 4 \\alpha ) ^ 2 } . \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} Z ( n ) = R ( n ) e ^ { i \\eta ( n ) } . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} \\tilde { \\theta } ' _ \\ell ( x ) = \\tilde { \\ell } + \\frac { O ( 1 ) } { 1 + x - b } \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} \\tilde { c } ( l ) = ( - 1 ) ^ { l } \\cdot ( 2 l - 1 ) ! ! . \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} \\mathbb { T ' } _ L & = ( \\Delta ' ) ^ { - 1 } \\begin{pmatrix} d - \\Delta ' ( a - b d ^ { - 1 } c ) ^ { - 1 } & ( [ d , b ] - [ d , a ] a ^ { - 1 } b ) ( d - c a ^ { - 1 } b ) ^ { - 1 } \\\\ - c + \\Delta ' ( a c ^ { - 1 } d - b ) ^ { - 1 } & ( [ b , c ] - [ a , c ] a ^ { - 1 } b ) ( d - c a ^ { - 1 } b ) ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} \\mathcal { F } ^ { - 1 } \\big \\{ \\widehat { f } ( \\nu ) \\big \\} ( t ) = \\int _ { - \\infty } ^ { \\infty } \\widehat { f } ( \\nu ) e ^ { i 2 \\pi \\nu t } \\ , \\mathrm { d } \\nu . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} w _ { \\sigma } ^ 0 = \\mbox { i d } , \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} h ( s , x , y , z , u ) { = z + { D \\varphi ( s , x ) } ^ { \\intercal } \\sigma } \\left ( s , x , y + \\varphi ( s , x \\right ) , h ( s , x , y , z , u ) , u ) . \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} ( u \\wedge v ) = ( \\iota u \\wedge v \\tau ) \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} A ( t ) \\begin{cases} t ^ { p _ 0 } \\ , \\ell ( t ) ^ { \\alpha _ 0 } & , \\\\ t ^ { p _ \\infty } \\ , \\ell ( t ) ^ { \\alpha _ \\infty } & . \\end{cases} \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} \\dot { x } _ { 2 , \\tau } = \\{ x _ { 2 , \\tau } , { H _ { \\tau } } _ T \\} _ { P B } = \\lambda \\frac { f ( t _ \\tau ) } { m } p _ { 2 , \\tau } ; \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ \\infty \\frac { M _ p ( m ) } { p - 1 } t ^ m & = \\sum _ { m = 0 } ^ \\infty \\frac { ( 2 m ) ! } { m ! ( m + 1 ) ! } p ^ { m + 1 } t ^ { 2 m } - \\sum _ { m = 0 } ^ \\infty \\sum _ { k = 1 } ^ m \\frac { ( 2 k + 1 ) ( 2 m ) ! } { ( m - k ) ! ( m + k + 1 ) ! } p ^ { m - k } t ^ { 2 m } \\\\ & = p C ( p t ^ 2 ) - \\sum _ { k = 1 } ^ \\infty C ( p t ^ 2 ) ^ { 2 k + 1 } t ^ { 2 k } \\\\ & = p C ( p t ^ 2 ) - C ( p t ^ 2 ) ^ 3 \\cdot \\frac { t ^ 2 } { 1 - t ^ 2 C ( p t ^ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} f ( n , m , k ) = ( 1 + o ( 1 ) ) \\frac { e ^ { - m / n } } { k ! } \\cdot \\frac { m ^ k } { n ^ { k - 1 } } . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} \\varphi _ { \\zeta } ( x ) = 0 \\ , \\ , x \\in ( - \\infty , - 2 \\zeta ] & , \\varphi _ { \\zeta } ( x ) = 1 \\ , \\ , x \\in [ - \\zeta , 0 ] , \\\\ \\max \\left \\{ \\zeta | \\varphi _ { \\zeta } ' ( x ) | , \\zeta ^ { 2 } | \\varphi _ { \\zeta } '' ( x ) | \\right \\} & \\leq C _ { 0 } \\ , \\ , x \\in ( - \\infty , 0 ] , \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} \\det ( D ^ \\alpha { } _ \\beta ) = - 1 . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} Q _ { n } \\stackrel { \\mathrm { d e f } } { = } \\sup _ { x \\in E _ { n } } \\frac { q _ { n } ( x ) } { f ( x ) } \\rightarrow 0 , \\mbox { a s } n \\rightarrow \\infty , \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} u ( n , t ) = W _ t f ( n ) , \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} u _ 1 ^ { ( i + 1 , l ) } : = e _ { i , l } ( e _ { i + 1 , l } ^ { p - 2 } ) \\prod e _ { j , k } ^ { p - 1 } . \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} & \\liminf _ { x \\to 0 + } ( \\psi ^ { - 1 } ) ' ( - \\log ( x ) ) \\nu ( ( x , 1 ) ) + \\liminf _ { x \\to 0 - } ( \\psi ^ { - 1 } ) ' ( - \\log | x | ) \\nu ( ( - 1 , x ) ) \\\\ = & \\liminf _ { x \\to 0 + } \\frac { \\nu ( ( x , 1 ) ) } { \\psi ' ( \\psi ^ { - 1 } ( - \\log ( x ) ) ) } + \\liminf _ { x \\to 0 - } \\frac { \\nu ( ( - 1 , x ) ) } { \\psi ' ( \\psi ^ { - 1 } ( - \\log | x | ) ) } > 1 . \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} [ n ] ^ { ( i ) } _ { \\omega } = \\frac { P _ { n , i } ( \\omega ) } { ( \\omega - 1 ) ^ i \\omega ^ i } \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} \\Theta ^ 1 _ { \\alpha } = \\bigcup _ { \\nu \\geq 0 } \\bigcup _ { k \\in \\mathbb { Z } ^ n \\setminus \\{ 0 \\} , | l | \\leq 2 \\atop { l \\neq e _ { - j } - e _ j } } \\mathcal { R } _ { k l } ^ { \\nu } ( \\alpha _ { 1 , \\nu } ) , \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} p ( w _ 1 , \\cdots , w _ { n } ) = \\sum _ { \\textrm { w t } _ { m } ( \\alpha ) = \\textrm { w t } _ { m } ( \\beta ) = 1 / 2 } C _ { \\alpha , \\beta } w ^ { \\alpha } \\overline { w } ^ { \\beta } . \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} | M '^ { r + 1 } - M ^ { r + 1 } | = | ( M + E ) ( M ^ r + E _ r ) - M ^ { r + 1 } | = | M E _ r + E M ^ r + E E _ r | < \\epsilon , \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} f _ { X 1 1 } \\left ( x _ { 1 1 } | \\mathbb { A _ { U E } } \\right ) = \\frac { 2 \\pi \\lambda _ { b } x _ { 1 1 } \\mathrm { e ^ { - \\pi \\lambda _ { b } \\max ^ 2 \\left ( x _ { 1 1 } , x _ { 1 1 } ^ { \\mu / 2 } \\left ( \\frac { P } { N } \\right ) ^ { \\frac { 2 - \\mu } { 2 \\alpha } } \\left ( \\frac { 1 } { M } \\right ) ^ { \\frac { 1 } { 2 \\alpha } } \\right ) } } } { \\mathbb { P \\left [ A _ { U E } \\right ] } } , \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} \\left | \\begin{array} { l l l l } C _ 3 & C _ 4 & \\dots & C _ { m + 1 } \\\\ C _ 4 & C _ 5 & \\dots & C _ { m + 2 } \\\\ & & \\dots \\\\ C _ { m + 1 } & C _ { m + 2 } & \\dots & C _ { 2 m - 1 } \\end{array} \\right | = 0 . \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ j { ( - 1 ) ^ { n j + s } \\binom k j \\binom j s \\frac { { G _ { n + r } ^ { j - s } G _ n ^ s } } { { G _ 0 ^ j } } H _ { m + r k + ( n - r ) j + r s } } } = \\left ( { \\frac { { G _ r } } { { G _ 0 } } } \\right ) ^ k H _ m , G _ 0 \\ne 0 \\ , , \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\sup _ { ( \\mu , \\psi ( \\cdot ) , \\eta ( \\cdot ) ) \\in \\R ^ 1 \\times C ^ 1 \\times C ^ 1 } \\mu = d ^ * ( y _ 0 ) \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} P ( E _ k ) = P ( E _ k \\lvert C _ k ) P ( C _ k ) + P ( E _ k \\lvert \\overline { C _ k } ) P ( \\overline { C _ k } ) . \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} [ u ^ { n - 1 } ] f ^ n ( u ) = \\frac { 1 } { n - 1 } [ u ^ { n - 2 } ] ( f ^ n ) ' ( u ) = \\frac { n } { n - 1 } [ u ^ { n - 2 } ] f ' ( u ) f ^ { n - 1 } ( u ) . \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} D ^ 2 I ( b ) = P ^ T ( \\bar { \\sigma } _ 1 , \\cdots , \\bar { \\sigma } _ k ) P \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} \\mathbb { E } _ i \\left [ e ^ { \\theta M ( t ) } h _ { J _ t } ^ { ( \\theta ) } \\right ] = h _ i ^ { ( \\theta ) } e ^ { t \\kappa ( \\theta ) } . \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} \\MoveEqLeft \\norm { \\lambda x + y } _ { X } \\geq \\norm { \\lambda x + P ( x ) } _ { X } - \\norm { y - P ( x ) } _ { X } > \\norm { \\lambda x + P ( x ) } _ { X } - \\frac { \\delta } { 2 } = \\norm { \\lambda x + P ( x ) } _ { X } \\norm { P } _ { X \\to X } - \\frac { \\delta } { 2 } \\\\ & \\geq \\norm { P ( \\lambda + P ) ( x ) } _ { X } = | \\lambda + 1 | \\norm { P ( x ) } _ { X } \\\\ & \\geq | \\lambda + 1 | \\bigg ( 1 - \\frac { \\delta } { 2 } \\bigg ) . \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} Q ( \\sigma ) : = \\left \\{ \\sum _ { s \\in \\sigma ( 1 ) } \\rho _ { s } s \\bigm | 0 \\leq \\rho _ { s } < 1 \\right \\} . \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} P = \\Big \\{ x \\in \\R ^ m \\colon \\sum _ { k = 1 } ^ m p _ k x _ k = b \\Big \\} \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} & [ \\partial _ n S _ n ( \\theta ) ] ( h _ 1 , \\ldots , h _ { n - 1 } ) \\\\ & = [ \\partial _ n \\theta ' ] ( h _ 1 , \\ldots , h _ { n - 1 } ) \\\\ & = \\sum _ { h _ 0 \\in H } h _ 0 ^ { - 1 } \\theta ' ( h _ 0 , \\ldots , h _ { n - 1 } ) + \\sum _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ i \\sum _ { \\substack { h , \\bar { h } \\in H \\\\ h \\bar { h } = h _ i } } \\theta ' ( h _ 1 , \\ldots , h _ { i - 1 } , h , \\bar { h } , h _ { i + 1 } , \\ldots , h _ { n - 1 } ) \\\\ & \\quad { } + ( - 1 ) ^ n \\sum _ { h _ n \\in H } \\theta ' ( h _ 1 , \\ldots , h _ n ) . \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\left ( { \\frac { { F _ r } } { { F _ { n + r } } } } \\right ) ^ j G _ { m - r + n j } } = ( - 1 ) ^ r \\frac { { F _ { n + r } } } { { F _ n } } G _ m - ( - 1 ) ^ r \\frac { { F _ r } } { { F _ n } } \\left ( { \\frac { { F _ r } } { { F _ { n + r } } } } \\right ) ^ k G _ { m + ( k + 1 ) n } , n + r \\ne 0 \\ , . \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} p ' ( w ) = \\begin{cases} p ( w ) + \\epsilon ( 0 , 1 ) & w \\sim z , \\\\ p ( w ) & \\end{cases} \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} \\mathrm { H e s s } ( U ) u = \\rho \\ , u \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow 0 , k \\in \\widetilde { S } _ \\varepsilon ( \\xi ) } \\langle \\eta , Q ( k ) \\psi \\rangle - ( 1 - C _ \\omega \\widehat { k } \\cdot v ( \\xi ) ) ^ { - 1 } \\langle \\eta , \\psi \\rangle = 0 . \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} \\dot { \\psi } _ v = ( \\omega _ v - \\Omega ) + \\sum _ { v ' \\in N ( v ) } H ( \\bar { \\theta } _ { v ' } + \\psi _ { v ' } - \\bar { \\theta } _ { v } - \\psi _ { v } ) , \\ \\ \\ \\ \\ v \\in V , \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{gather*} \\sum _ { s \\geq 0 } \\big ( E _ { 1 , 1 } ( s + 1 ) h _ 1 ( - s + m - 1 ) + h _ 1 ( s + m ) E _ { 2 , 2 } ( - s ) \\big ) \\\\ { } = \\sum _ { s \\geq 0 } \\big ( S _ { 1 , 1 } ( s + 1 ) - S _ { 2 , 2 } ( s + m ) \\big ) - \\sum _ { s = 0 } ^ { m - 2 } E _ { 1 , 1 } ( s + 1 ) E _ { 2 , 2 } ( - s + m - 1 ) , \\end{gather*}"}
-{"id": "6992.png", "formula": "\\begin{align*} \\langle \\psi , \\phi \\rangle _ + : = \\langle \\psi , Q _ n \\phi \\rangle = \\sum _ { i = 0 } ^ { K } \\langle \\psi ^ { ( i ) } , \\phi ^ { ( i ) } \\rangle , \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} \\mathcal { F } : = \\left \\{ u \\in L ^ 1 _ \\rho ( X ) ~ \\Big | ~ u ( x _ 1 ) + \\cdots + u ( x _ N ) \\le c ( x _ 1 , \\ldots , x _ N ) \\rho ^ { \\otimes ( N ) } ( x _ 1 , \\ldots , x _ N ) \\right \\} \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} L _ s = \\sum _ { m \\in \\Z \\cap \\Z _ p ^ { \\times } [ a , b , k ] } \\lambda _ { \\pi } ( m ) F \\left ( \\frac { m } { M } \\right ) . \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} ( I - U _ A ^ * U _ A ) Y _ 0 & = ( I - U _ A ^ * U _ A ) \\big [ D + ( I - U _ A ^ * U _ A ) Y _ 0 \\big ] \\\\ & = ( I - U _ A ^ * U _ A ) \\big [ D ^ * + Y _ 0 ^ * ( I - U _ A ^ * U _ A ) \\big ] \\\\ & = ( I - U _ A ^ * U _ A ) D ^ * + Z _ 0 . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} \\| f ( z ) \\| ^ 2 _ { L ^ 2 ( Y ) } = \\sum _ { \\nu \\in \\chi _ \\infty } \\sum _ { \\ell = 1 } ^ { d ( \\nu ) } | a _ { \\nu , \\ell } ( r ) | ^ 2 . \\end{align*}"}
-{"id": "9939.png", "formula": "\\begin{align*} \\bar { \\vartheta } \\tau _ Y = \\bar { \\vartheta } ( ( \\tau ^ { n - 1 , - } - \\tau ^ { n - 1 , + } ) | _ Y ) = ( \\bar { \\vartheta } \\tau ^ { n - 1 , - } - \\bar { \\vartheta } \\tau ^ { n - 1 , + } ) | _ Y = ( \\tau - \\tau ) | _ Y = 0 , \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} W _ { 3 } ( x ) : = \\phi ( x ) \\exp \\left ( \\mathcal { M } \\log \\frac { 1 } { \\psi } \\right ) ( x ) \\end{align*}"}
-{"id": "9932.png", "formula": "\\begin{align*} \\tau _ \\alpha = ( - 1 ) ^ { \\sigma ( \\alpha ) } \\omega _ 1 \\ , { \\wedge } \\ , \\omega _ 2 \\ , { \\wedge } \\ , \\cdots \\ , { \\wedge } \\ , \\omega _ \\ell , \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} t ^ \\alpha Y _ t & \\geq \\frac 1 { F ( t ) } \\int _ 0 ^ t s ^ \\alpha f ( s ) d D _ s , \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} \\mathbb { F } = F _ { i _ 1 } ( \\alpha _ { i _ 1 } ) , [ \\mathbb { F } : F _ { i _ 1 } ] = p _ { i _ 1 } . \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} \\lambda _ n ( w _ n ) _ { i j } - | \\lambda _ n | \\delta ^ { k l } \\frac { \\partial ^ 2 ( w _ n ) _ { i j } } { \\partial y ^ k \\partial y ^ l } - | \\lambda _ n | \\widetilde { Z } ( w _ n ) _ { i j } = \\frac { \\left ( f _ n \\left ( \\frac { 1 } { \\sqrt { | \\lambda _ n | } } y \\right ) \\right ) _ { i j } } { \\| R _ { \\lambda _ n } f _ n \\| _ { C ^ { 0 , \\alpha } } } . \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} x = m - 2 + [ ( m _ 1 - m _ 2 ) ^ 2 + 4 \\lambda ( \\sqrt { m _ 1 } + \\sqrt { m _ 2 } ) ^ 2 ] ^ { 1 / 2 } , \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} f ( x ) & = ( x + 1 ) - ( A - B x ) ( x ^ 3 - 1 ) ( x + 1 ) + m \\ : h ( x ) , \\\\ g ( x ) & = x - ( A - B x ) ( x ^ 3 - 1 ) ( x + 1 ) + m \\ : h ( x ) . \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} P _ { n , g } = ( - 1 ) ^ { n - 1 } \\sum _ { k = 0 } ^ { N } ( n + 2 k ) e ( U _ k ) . \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} Q ( k ) & = Q _ 0 ( k ) + \\frac { \\lvert k \\lvert } { \\omega ( k ) } Q _ 0 ( k ) ( \\widehat { k } \\cdot \\nabla K ( \\xi - d \\Gamma ( m ) ) ) Q ( k ) + o ( k ) \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} F ( D , \\psi ) : = \\int \\ ! D \\epsilon ( \\psi , \\nabla \\psi ) \\ , ^ 3 r \\ , , \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} \\gamma = ( \\operatorname { I d } , T , T ^ { ( 2 ) } , \\dots , T ^ { ( N - 1 ) } ) _ { \\sharp } \\rho , \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} { \\mathcal A } _ h = \\big \\{ x \\in \\mathcal D : h ( 1 , x ) \\in \\mathcal O _ { r _ * } \\big \\} . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D O } _ \\textup { t } ( 8 n + 6 ) q ^ n & = 1 6 \\dfrac { f _ 2 ^ { 1 6 } f _ { 6 } ^ { 1 0 } } { f _ 1 ^ { 1 7 } f _ 3 ^ 3 f _ { 1 2 } ^ 4 } - 8 q \\dfrac { f _ 2 ^ { 2 8 } f _ 3 f _ { 1 2 } ^ { 4 } } { f _ 1 ^ { 2 1 } f _ 4 ^ 8 f _ { 6 } ^ 2 } - 1 2 8 q ^ 2 \\dfrac { f _ 2 ^ { 4 } f _ 3 f _ 4 ^ 8 f _ { 1 2 } ^ { 4 } } { f _ 1 ^ { 1 3 } f _ { 6 } ^ 2 } , \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} | | ( u _ n ^ 1 ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } & = | | ( u _ n ^ 1 ( \\cdot + y _ n ^ 1 ) - v _ 1 ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } + | | ( v _ 1 ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } + o ( 1 ) \\\\ & = | | ( u _ n ^ 1 - v _ 1 ( \\cdot - y _ n ^ 1 ) ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } + | | ( v _ 1 ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } + o ( 1 ) \\\\ & = | | ( u _ n ^ 2 ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } + | | ( v _ 1 ) _ + | | _ { L ^ { p + 1 } } ^ { p + 1 } + o ( 1 ) , \\\\ \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} R _ k ^ i [ n ] ( \\mathbf { p } , \\mathbf { s } , \\mathbf { r } ) = s _ { k } ^ i [ n ] \\mathcal { B } \\log _ 2 \\Big ( 1 + \\frac { H _ k ^ i [ n ] p _ k ^ i [ n ] } { \\norm { \\mathbf { r } [ n ] - \\mathbf { r } _ k } ^ 2 } \\Big ) , \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} T _ V f ( z ) = \\int \\limits _ { \\mathbb { R } } f ( t ) \\dfrac { z ^ { i 2 \\pi t } } { z } \\d t , z \\in V ( a , b ) \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} \\int x _ i ^ 2 \\ , \\widetilde \\mu ( \\d x ) = \\frac 1 n \\int \\vert x \\vert ^ 2 \\ , \\mu ( \\d x ) = \\frac { \\beta } 2 + \\frac { 2 - \\beta } { 2 n } , \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} A ^ { W , W } _ { \\cdot \\wedge \\tau _ M } ( g ^ n _ M ) = A ^ { W , W } _ { \\cdot \\wedge \\tau _ M } ( g ^ n ) \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} \\begin{aligned} x _ { 1 , l } = \\max \\{ x \\in \\mathbb { Z } _ { \\ge 0 } \\mid \\tilde { e } _ { i _ { 1 , l } } ^ x \\tilde { e } _ { i _ { 1 , l - 1 } } ^ { x _ { 1 , l - 1 } } \\cdots \\tilde { e } _ { i _ { 1 , 1 } } ^ { x _ { 1 , 1 } } b \\neq 0 \\} \\end{aligned} \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n + 1 } \\sum _ { i = 0 } ^ { n } F ( \\lambda _ { i , n } ) = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } F ( \\sigma ( e ^ { i \\theta } ) ) d \\theta , \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} \\limsup _ { \\lambda \\to \\infty } A _ { 1 2 } ( \\lambda , \\varepsilon ) & \\leq C ( n , \\varepsilon , \\zeta ) \\sum _ { k = 0 } ^ { d - 2 } \\sum _ { y _ { \\tau } \\in S _ { \\tau } } \\int _ { \\mathbb { R } ^ d } \\mathbf { 1 } _ { B _ { 2 0 \\tau } ( y _ { \\tau } ) } ( x ) | \\langle \\eta _ { d - k } ^ { \\kappa } ( y _ { \\tau } ) , \\eta ( x ) \\rangle | d | \\mu | ^ { s } ( x ) . \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} & [ S _ { n - 1 } \\partial _ n \\theta ] ( h _ 1 , \\ldots , h _ { n - 1 } ) ( \\xi ) \\\\ & = [ S _ { n - 1 } f ] ( h _ 1 , \\ldots , h _ { n - 1 } ) ( \\xi ) \\\\ & = \\sum _ { t _ 1 , \\ldots , t _ { n - 1 } } f ( g _ { t _ 1 } , \\ldots , g _ { t _ { n - 1 } } ) ( \\pi ( \\xi ) | _ { [ t _ 1 , \\ldots , t _ { n - 1 } ] } ) . \\quad ( \\mbox { d e f . o f $ S _ { n - 1 } $ } ) \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} \\delta ^ { * } = \\left ( \\prod _ { i = 1 } ^ { p } | A _ { i } | ^ { - A _ { i } } \\right ) \\left ( \\prod _ { j = 1 } ^ { q } | B _ { j } | ^ { B _ { j } } \\right ) , \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} c _ { M , M ' } = \\tau \\circ f _ { M , M ' } \\circ \\Theta _ { M , M ' } . \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} - \\sum _ v \\mbox { i n v } _ v ( \\chi ( \\check { y } _ v ) ) = \\sum _ v \\mbox { i n v } _ v ( ( \\pi ( u _ v ) \\cup \\gamma _ v ) - h ) . \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 6 n + 4 ) q ^ n & = \\dfrac { f _ 3 ^ 3 } { f _ 1 ^ 3 } \\Bigg ( 2 \\dfrac { f _ 2 ^ 2 f _ 4 ^ 4 } { f _ 1 ^ { 1 0 } } \\left ( 4 \\dfrac { f _ 2 ^ 6 f _ 3 } { f _ 1 ^ 3 f _ { 6 } ^ 2 } a ( q ^ 2 ) - a ( q ^ 2 ) - a ^ 2 ( q ^ 2 ) + 3 6 q \\dfrac { f _ 2 ^ 4 f _ { 6 } ^ 4 } { f _ 1 ^ 2 f _ 3 ^ 2 } \\right ) \\\\ & \\quad + \\dfrac { f _ 2 ^ { 1 4 } } { f _ 1 ^ { 1 4 } f _ 4 ^ 4 } \\left ( 1 2 \\dfrac { f _ 2 ^ 8 } { f _ 1 ^ 4 } - 3 \\dfrac { f _ 2 ^ 2 f _ { 6 } ^ 2 } { f _ 1 f _ 3 } \\right ) \\Bigg ) . \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} \\mu \\prod _ { j = 1 } ^ { n } P _ j ( \\epsilon ) r = \\mu \\tilde { P } ^ n r + \\epsilon \\mu \\sum _ { i = 1 } ^ { n - 1 } i \\tilde { P } ^ { i } \\tilde { P } ^ { ( 1 ) } \\tilde { P } ^ { n - i - 1 } r + O ( \\epsilon ^ { 2 } n ^ 3 ) \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} d y = \\left ( \\sum _ { i = 1 } ^ { p - 1 } i e _ i ^ p a ^ i \\right ) \\frac { d a } { a } . \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} - \\alpha \\div \\left ( \\abs { \\nabla \\bar { \\Q } _ t } ^ { p - 2 } \\nabla \\bar { \\Q } _ t \\right ) = 4 \\bar { T } ( t ) ( 1 - | \\bar { \\Q } _ t | ^ 2 ) \\bar { \\Q } _ t + 1 2 \\bar { H } ( t ) ( \\sqrt { 6 } \\bar { \\Q } _ t ^ 2 - | \\bar { \\Q } _ t | ^ 2 \\bar { \\Q } _ t ) \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} \\begin{aligned} w _ 0 ^ + \\circ X ^ { - 1 } _ \\gamma ( s , 0 ) + \\frac { \\eta } { \\lambda \\mu _ \\lambda } \\left ( \\partial _ n w _ 0 ^ + \\circ X ^ { - 1 } _ \\gamma \\right ) ( s , 0 ) & = - a _ 0 \\eta + b _ 0 + 2 \\log \\mu _ \\lambda - \\log h _ \\gamma , \\\\ w _ 0 ^ - \\circ X ^ { - 1 } _ \\gamma ( s , 0 ) + \\frac { \\eta } { \\lambda \\mu _ \\lambda } \\left ( \\partial _ n w _ 0 ^ - \\circ X ^ { - 1 } _ \\gamma \\right ) ( s , 0 ) & = a _ 0 \\eta + b _ 0 + 2 \\log \\mu _ \\lambda - \\log h _ \\gamma . \\end{aligned} \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} L ( s , \\pi _ { \\infty } ) L ( s , \\pi ) = \\left ( \\prod _ { p \\leq \\infty } \\epsilon ( s , \\pi _ p ) \\right ) L ( 1 - s , \\tilde { \\pi } _ { \\infty } ) L ( 1 - s , \\tilde { \\pi } ) . \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} \\nabla f ( \\mathbf { x } ^ { \\star } ) + \\frac { \\rho ( 1 - \\eta ) } { 2 } \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } \\mathbf { x } ^ { \\star } + \\sqrt { \\eta } \\mathbf { E } _ { } ^ T \\boldsymbol { \\alpha } ^ { \\star } = \\mathbf { 0 } \\\\ \\mathbf { E } _ { } \\mathbf { x } ^ { \\star } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "9989.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - \\bar p ( q ) } \\bar u _ 0 ( q ) = \\int _ { \\mathbb { R ^ + } } e ^ { - q s } \\alpha ( s ) d s = [ \\bar u ( 0 , \\cdot ) ] ( q ) . \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } E ( v _ n ) = d _ { M , } , \\lim _ { n \\rightarrow \\infty } \\| r _ n \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } = 0 , E ( V ) = d _ { M , } . \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} r _ i = \\begin{bmatrix} p _ i ^ \\mathcal { I } \\\\ 1 \\end{bmatrix} , i = 1 , \\cdots , n _ 1 r _ { j + n _ 1 } = \\begin{bmatrix} v _ { j } ^ \\mathcal { I } \\\\ 0 \\end{bmatrix} , j = 1 , , \\cdots , n _ 2 \\end{align*}"}
-{"id": "9791.png", "formula": "\\begin{align*} ( e ^ { t M } ) ^ * e ^ { t M } & = e ^ { t } \\Big ( 1 + 2 S ^ 2 ( t ) - 2 C ( t ) S ( t ) \\sigma _ 3 - 4 \\sqrt { \\nu } S ^ 2 ( t ) \\sigma _ 1 \\Big ) \\\\ & = e ^ { t } ( a + v ) ~ , \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} A _ 2 \\gamma = A _ 2 \\gamma \\in \\mathbb { F } _ q ( \\{ \\alpha _ i : i \\in \\{ 1 , 2 , \\dots , k - 1 \\} \\setminus \\{ j \\} \\} ) . \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} ( * T ^ \\mathrm { a x } _ m ) _ \\alpha = - \\ , \\frac 4 3 \\ , ( \\ , 0 \\ , , \\ , 0 \\ , , \\ , m \\ , , 0 \\ , ) \\ , . \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} & = \\left ( 1 - q ^ { - n } + 2 q ^ { - 1 + ( n - 1 ) s _ 1 } - q ^ { n ^ 2 - n s _ 1 - s _ 2 } - q ^ { n ^ 2 - n - n s _ 1 - s _ 2 } \\right ) \\cdot \\\\ & \\frac { ( 1 - q ^ { - 1 } ) ( 1 - q ^ { - n } ) q ^ { n ^ 2 + 1 - ( 2 n - 1 ) s _ 1 - s _ 2 } } { ( 1 - q ^ { n ^ 2 + 1 - ( 2 n - 1 ) s _ 1 - s _ 2 } ) ( 1 - q ^ { n ^ 2 - n s _ 1 - s _ 2 } ) } , \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} \\dim V \\ge \\sum _ { k = 1 } ^ m c _ k \\dim B _ k V , \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} P _ 0 Q ( k ) P _ 0 - D _ k P _ 0 & = D _ k \\frac { \\lvert k \\lvert } { \\omega ( k ) } P _ 0 ( \\widehat { k } \\cdot \\nabla K ( \\xi - d \\Gamma ( m ) ) ) ( 1 - P _ 0 ) Q ( k ) P _ 0 \\\\ & + D _ k \\left ( \\frac { \\lvert k \\lvert } { \\omega ( k ) } - C _ \\omega \\right ) \\widehat { k } \\cdot u P _ 0 Q ( k ) P _ 0 + D _ k P _ 0 o _ 1 ( k ) P _ { 0 } . \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} | k _ b j _ b | < \\frac { | i + j | } { n - \\frac 1 2 } + \\frac { | i | + | j | } { 5 0 n \\sum _ { b = 1 } ^ n | j _ b | } < \\frac { | i | + | j | } { n - \\frac { 9 } { 1 7 } } , b = 1 , \\cdots , n . \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} m _ { \\xi } = L ^ { K } b _ { \\xi } , \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} \\widehat { \\mathcal A } _ { s } ( \\widehat { \\mathrm M } ) g \\ , : = \\ , u \\int _ { \\mathbb R ^ { 2 } } \\widetilde { b } ( s , y _ { 1 } , y _ { 2 } ) \\cdot \\frac { \\ , m _ { s } ( y _ { 1 } ) \\ , } { \\ , m _ { s } ( y _ { 2 } ) \\ , } g ^ { \\prime } ( y _ { 1 } ) \\widehat { \\mathrm M } _ { s } ( { \\mathrm d } y _ { 1 } { \\mathrm d } y _ { 2 } ) + ( 1 - u ) \\int _ { \\mathbb R ^ { 2 } } \\widetilde { b } ( s , y _ { 1 } , y _ { 2 } ) g ^ { \\prime } ( y _ { 1 } ) \\mathrm m _ { s } ( { \\mathrm d } y _ { 1 } ) \\mathrm m _ { s } ( { \\mathrm d } y _ { 2 } ) \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} G ( t , z ) = \\phi ( t , x - \\kappa - ( 1 + \\lambda ) z ) + z , \\forall \\ ; z \\in \\mathcal { Z } , \\forall ( t , x ) \\in [ 0 , T ] \\times S . \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} { } + ( 1 - u ) \\int ^ { T } _ { 0 } \\frac { 1 } { \\ , n \\ , } \\sum _ { i = 1 } ^ { n } \\Big \\lvert \\sum _ { j = 1 } ^ { n } \\overline { b } ( s , \\overline { X } _ { s , i } , \\overline { X } _ { s , j } ) \\Big \\rvert { \\mathrm d } s \\ , , \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} & F ( d _ { \\Lambda } ( E Z ( t \\otimes \\sigma ) ) ) ( m ) = F ( E Z ( d _ { \\Lambda } t \\otimes \\sigma ) ) ( m ) \\pm F ( E Z ( t \\otimes \\partial ' \\sigma ) ) ( m ) \\\\ & \\pm \\sum _ { ( \\sigma ) } F ( E Z ( t \\otimes \\sigma '' ) ) ( \\tau ( \\sigma ' ) \\cdot m ) \\pm \\sum _ { ( \\sigma ) } \\tau ( \\sigma '' ) \\cdot F ( E Z ( t \\otimes \\sigma ' ) ) ( m ) \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} \\sigma ( u ) = \\prod _ { j = 1 } ^ { d } x _ { i _ { j } + ( j - 1 ) } . \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} \\sum _ { i \\in I } v _ { i } ^ { 2 } \\langle \\pi _ { Z _ { i } } C ^ { \\prime } f , \\pi _ { Z _ { i } } C f \\rangle = \\Vert v _ { i } ( C ^ { * } \\pi _ { Z _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f \\Vert ^ { 2 } \\leq ( B + \\epsilon ^ { 2 } ) \\Vert f \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 0 . 7 0 0 5 & - 0 . 2 6 3 8 \\\\ - 0 . 2 2 7 8 & - 0 . 4 6 2 7 \\end{bmatrix} \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} K _ 1 = G ^ { ( 1 ) } \\rightarrow G ^ { ( 2 ) } \\rightarrow \\cdots \\rightarrow G ^ { ( n ) } = G \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} B ( s ) & = \\frac { 6 ( n - 1 ) ^ 5 ( n ^ 2 + 6 n - 1 2 ) ( n - 2 ) ^ 2 } { n ^ 2 } s ^ 3 \\\\ & + \\frac { 2 ( n - 1 ) ( n - 2 ) } { n ^ 2 } ( 9 n ^ 7 - 4 3 n ^ 6 - 2 4 n ^ 5 + 5 0 2 n ^ 4 - 1 2 4 2 n ^ 3 + 1 4 2 4 n ^ 2 - 8 1 6 n + 1 9 2 ) s ^ 2 \\\\ & + 2 ( n - 2 ) ^ 2 ( 9 n ^ 5 - 5 9 n ^ 4 + 1 3 1 n ^ 3 - 1 2 3 n ^ 2 + 4 6 n - 8 ) s \\\\ & + 2 n ^ 2 ( n - 2 ) ^ 3 ( n - 4 ) ( 3 n - 4 ) . \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} e & = z ^ { - 1 } \\langle \\phi _ 0 , F ( z \\phi _ 0 + h ) \\rangle , \\\\ h & = ( H + q ^ 2 / 2 ) ^ { - 1 } \\{ - P _ c F ( z \\phi _ 0 + h ) + e h \\} , \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} R _ \\mathbb { G } = R _ \\mathbb { G } ( \\lambda ) = \\frac { 1 } { \\limsup _ { n \\to \\infty } \\sqrt [ n ] { p ^ { ( n ) } ( x , y ) } } \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} \\tilde { X } _ H m _ X ( x , \\xi ) = \\frac { 1 } { 2 T _ { \\alpha _ 0 } ' } \\left ( m _ 0 \\circ \\tilde { \\Phi } ^ X _ { T _ { \\alpha _ 0 } ' } ( x , \\xi ) - m _ 0 \\circ \\tilde { \\Phi } ^ X _ { - T _ { \\alpha _ 0 } ' } ( x , \\xi ) \\right ) = \\frac { 1 } { 2 T _ { \\alpha _ 0 } ' } > 0 . \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\left ( \\frac { \\sqrt { \\frac { \\pi } { 2 } } \\rho ( y ) } { e ^ { - 2 y ^ { 2 } } } - 1 \\right ) \\frac { 1 } { \\tau } = \\frac { 1 } { 3 } y ^ { 4 } + \\frac { 3 } { 2 } y ^ { 2 } - \\frac { 7 } { 1 6 } + O ( \\tau ) \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} | \\ ! | P ^ * u | \\ ! | _ 0 ^ 2 \\geq \\gamma _ 0 \\sum _ { j = 0 } ^ N | \\ ! | X _ j u | \\ ! | _ 0 ^ 2 + C | \\ ! | u | \\ ! | _ 0 ^ 2 \\geq C ' | \\ ! | u | \\ ! | _ { 1 / 2 } ^ 2 , \\ , \\ , \\forall u \\in C _ 0 ^ \\infty ( K ) , \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} P ( X _ { k + 1 } = y \\mid X _ 0 , \\ldots , X _ k ) = P _ { k + 1 } ( X _ k , y ) a . s . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} A _ 2 h _ { 2 , i } = A _ 1 i \\in [ k ] \\setminus \\{ j \\} . \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} \\nu ( \\Q ) : = \\frac { \\Q - \\Pi \\left ( \\Q \\right ) } { \\left | \\Q - \\Pi \\left ( \\Q \\right ) \\right | } \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\| \\tilde { V } ^ j _ n \\| ^ 2 _ { L ^ 2 } = \\lambda _ j ^ 2 \\| V ^ j ( \\cdot - x ^ j _ n ) \\| ^ 2 _ { L ^ 2 } = M , \\| \\tilde { v } ^ l _ n \\| ^ 2 _ { L ^ 2 } = ( \\lambda ^ l _ n ) ^ 2 \\| v ^ l _ n \\| ^ 2 _ { L ^ 2 } = M . \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} g = d r ^ 2 \\oplus g ( r ) , & & g ( r ) = r ^ 2 g ^ { V } \\oplus \\pi ^ * g ^ B \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} [ L x ] _ { i , j } = ( x _ { i + 1 , j } - x _ { i , j } ) + ( x _ { i - 1 , j } - x _ { i , j } ) + ( x _ { i , j + 1 } - x _ { i , j } ) + ( x _ { i , j - 1 } - x _ { i , j } ) , \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} \\Delta ( x ) = \\sum _ { \\substack { \\lambda + \\nu = \\mu \\\\ i , j } } \\left ( x , v _ i ^ { \\lambda } v _ j ^ { \\nu } \\right ) u _ i ^ \\lambda \\otimes K _ \\lambda u _ j ^ { \\nu } , \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| \\nabla r _ n \\| _ { L ^ 2 } = 0 . \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} \\mathcal { O } _ { \\nu + 1 } = \\mathcal { O } _ { \\nu } \\setminus \\big ( \\bigcup _ { k \\in \\mathbb { Z } ^ n \\setminus \\{ 0 \\} , | l | \\leq 2 \\atop { l \\neq e _ { - j } - e _ j } } \\mathcal { R } _ { k l } ^ { \\nu } ( \\alpha _ { 1 , \\nu } ) \\cup \\bigcup _ { k \\in \\mathbb { Z } ^ n , \\pm j \\in \\mathbb { Z } _ * \\atop { | j | \\leq \\Pi _ { \\nu } } } \\mathcal { R } _ { k ( - j ) j } ^ { \\nu } ( \\alpha _ { 2 , \\nu } ) \\big ) , \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} H Q + \\mu | Q | ^ p Q = E Q , \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} \\frac { \\Gamma ( k - 1 / 2 ) } { \\Gamma \\left ( \\tfrac { k } { 2 } + \\tfrac { - z - 1 } { 4 } \\right ) \\Gamma \\left ( \\tfrac { k } { 2 } + \\tfrac { z - 1 } { 4 } \\right ) } = 2 ^ { k - 3 / 2 } \\sqrt { \\pi } ^ { - 1 } \\times \\frac { \\Gamma ( k / 2 - 1 / 4 ) } { \\Gamma \\left ( \\tfrac { k } { 2 } + \\tfrac { - z - 1 } { 4 } \\right ) } \\times \\frac { \\Gamma ( k / 2 + 1 / 4 ) } { \\Gamma \\left ( \\tfrac { k } { 2 } + \\tfrac { z - 1 } { 4 } \\right ) } \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} u _ { t t } = u _ { x x } - { \\frac { \\partial V } { \\partial u } } \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} \\langle \\vec { m } , \\vec { i } \\rangle = m _ i - \\sum _ { j \\in V ( Q ) , e \\in E _ { j , i } } m _ j > 0 \\end{align*}"}
-{"id": "9684.png", "formula": "\\begin{align*} L ( \\phi ^ { \\vee } , z _ 1 \\dots , z _ n , s ) : = L ( \\phi ^ { \\vee } , z _ 1 , \\dots , z _ n ; \\theta ^ s , - s ) = \\sum _ { a \\in A _ { + } } \\frac { \\mu ( a ) a ( z _ 1 ) \\dots a ( z _ n ) } { a ^ s } . \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} C ( \\Sigma _ \\mu ( v _ \\pm ) ) = C ( \\Sigma _ 0 ( v _ \\pm ) ) = [ 0 ] . \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} I _ { p _ 1 , q _ 1 } \\otimes \\delta _ 1 u _ { h _ 1 } \\oplus \\cdots \\oplus I _ { p _ m , q _ m } \\otimes \\delta _ m u _ { h _ m } \\oplus \\begin{bmatrix} 0 & I _ { k _ { m + 1 } } \\\\ I _ { k _ { m + 1 } } & 0 \\end{bmatrix} \\otimes 1 \\oplus \\cdots \\oplus \\begin{bmatrix} 0 & I _ { k _ l } \\\\ I _ { k _ l } & 0 \\end{bmatrix} \\otimes 1 \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} \\sigma ^ \\alpha { } _ { \\dot a b } \\ , \\partial _ { x ^ \\alpha } \\xi ^ b = 0 . \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} D ^ \\alpha { } _ \\beta = U ^ \\alpha { } _ \\gamma \\ , V ^ \\gamma { } _ \\beta \\ , , \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} G _ X ^ { N _ 0 , N _ 1 } ( x , \\xi ) : = m _ { X } ^ { N _ 0 , N _ 1 } ( x , \\xi ) \\ln ( 1 + f ( x , \\xi ) ) , \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{gather*} L ( z , s ) = \\prod _ { p } \\big ( 1 - a ( p ) p ^ { - s } + p ^ { 1 - 2 s } \\big ) ^ { - 1 } = \\sum _ { n = 1 } ^ \\infty \\frac { a ( n ) } { n ^ s } , \\end{gather*}"}
-{"id": "9300.png", "formula": "\\begin{align*} \\sigma \\cdot ( z _ 1 , \\ldots , z _ n ) = \\left ( \\prod _ { \\alpha = 1 } ^ n z _ \\alpha ^ { s _ { 1 \\alpha } ^ \\lambda } , \\ldots , \\prod _ { \\alpha = 1 } ^ n z _ \\alpha ^ { s _ { n \\alpha } ^ \\lambda } \\right ) . \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} v ( s ) = e ^ { [ \\tau ( s ) ^ 2 q ' ( \\tau ( s ) ) ] } \\ \\ \\ ( s \\in G ) \\end{align*}"}
-{"id": "9911.png", "formula": "\\begin{align*} \\psi = \\begin{cases} \\ \\ \\frac 1 2 \\\\ - \\frac 1 2 , \\end{cases} \\ \\psi _ { + } = \\begin{cases} 1 \\\\ 0 , \\end{cases} \\ \\psi _ { - } = \\begin{cases} \\ 0 \\\\ - 1 \\end{cases} \\ , \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} ( \\omega _ v - \\Omega ) + \\sum _ { v ' \\in N ( v ) } H ( \\bar { \\theta } _ { v ' } - \\bar { \\theta } _ v ) = 0 \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} \\pi ^ b = \\chi _ { \\omega _ { \\pi , b } ^ { - 1 } } \\pi , \\omega _ { \\pi , b } = \\prod _ { p \\mid b } \\omega _ { \\pi , p } . \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} & \\textbf { i } ^ 2 = \\textbf { j } ^ 2 = \\textbf { k } ^ 2 = \\textbf { i } \\textbf { j } \\textbf { k } = - 1 \\\\ & \\textbf { i } \\textbf { j } = - \\textbf { j } \\textbf { i } = \\textbf { k } \\\\ & \\textbf { j } \\textbf { k } = - \\textbf { k } \\textbf { j } = \\textbf { i } \\\\ & \\textbf { k } \\textbf { i } = - \\textbf { i } \\textbf { k } = \\textbf { j } ~ . \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} \\phi ( \\Pi _ { 1 , u } ) = 2 \\left ( e ^ { x / 2 } - 1 \\right ) ^ { u + 1 } \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} v _ B ^ { - 1 } \\overset { \\mathcal { X } ^ + ( 2 ) } { W } \\ ! \\ ! \\ ! \\ ! \\ ! _ A \\triangleright \\chi ^ { \\epsilon } _ s & = v _ B ^ { - 1 } \\triangleright \\chi ^ { \\epsilon } _ s \\big ( { - } \\hat q ^ 2 C ? \\big ) \\\\ & = \\sum _ { \\substack { \\ell = 1 \\\\ \\sigma \\in \\{ \\pm \\} } } ^ p - \\epsilon \\big ( q ^ s + q ^ { - s } \\big ) \\lambda ^ { \\sigma } _ { \\ell } ( \\epsilon , s ) \\chi ^ { \\sigma } _ { \\ell } + \\sum _ { j = 1 } ^ { p - 1 } - \\epsilon \\big ( q ^ s + q ^ { - s } \\big ) \\delta _ j ( \\epsilon , s ) G _ j . \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} \\mathbb { G } _ \\lambda ( o , \\ , o \\ , | \\ , z ) & = \\frac { 1 } { 1 - U _ \\lambda ( o , \\ , o \\ , | \\ , z ) } \\\\ & = \\frac { 2 ( d - 1 ) } { 2 ( d - 1 ) - ( d - 1 + \\lambda ) + \\sqrt { ( d - 1 + \\lambda ) ^ 2 - 4 \\lambda ( d - 1 ) z ^ 2 } } . \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} z _ { t _ k } = h _ 0 ( t _ k , x ( t _ k ) ) + e _ { t _ k } , \\ k = 0 , 1 , . . , N \\in \\mathbb { N } , \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} g _ 0 = d t ^ 2 + h ( 0 ) , \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} 1 \\ge \\liminf _ { k \\to \\infty } \\sum _ { n = { 1 } } ^ { \\infty } f _ { \\lambda _ { n _ k } } ^ { ( n ) } ( o , o ) z _ 0 ^ n \\ge \\sum _ { n = { 1 } } ^ { \\infty } \\liminf _ { k \\to \\infty } f _ { \\lambda _ { n _ k } } ^ { ( n ) } ( o , o ) z _ 0 ^ n = \\sum _ { n = { 1 } } ^ { \\infty } f _ { \\lambda _ { 0 } } ^ { ( n ) } ( o , o ) z _ 0 ^ n = U _ { \\lambda _ 0 } ( o , \\ , o \\ , | \\ , z _ 0 ) . \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} J ( f , g ) ^ 2 = \\begin{cases} \\frac { ( 2 \\pi ) ^ { 2 - ( c _ 1 + c _ 2 ) } a ^ { c _ 1 } b ^ { c _ 2 } ( 1 - e ^ { - 2 t } ) ^ 2 } { \\det \\left ( \\begin{array} { c c } 1 + ( 1 - e ^ { - 2 t } ) c _ 1 ( a - 1 ) & - e ^ { - t } \\\\ [ 1 e x ] - e ^ { - t } & 1 + ( 1 - e ^ { - 2 t } ) c _ 2 ( b - 1 ) \\end{array} \\right ) } & \\\\ + \\infty & \\end{cases} \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} L : = \\sum _ { m \\in \\Z } \\lambda _ { \\pi } ( m ) F ( \\frac { m } { M } ) , \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & a & 0 & d \\\\ - c & b & - d & 0 \\\\ 0 & e & 0 & - c \\\\ - e & 0 & a & b \\end{pmatrix} \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} \\eta = t _ \\tau - \\frac { ( t _ 2 - t _ 1 ) ( \\tau - \\tau _ 1 ) } { \\tau _ 2 - \\tau _ 1 } + t _ 1 \\sim 0 \\ ; , \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} D ( s , r ) : = \\mathbb { T } ^ n _ s \\times D ( r ) : = \\mathbb { T } ^ n _ s \\times B _ { r ^ 2 } \\times B _ r \\times B _ r \\subset \\mathcal { P } ^ { a , p } , \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} \\epsilon = 5 \\left ( \\frac { \\Delta + d } { n } \\right ) ^ 2 . \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} A f ( x ) = \\sum _ { [ \\xi ] \\in \\widehat { G } } d _ { \\xi } ( \\xi ( x ) \\sigma _ { A } ( x , \\xi ) \\widehat { f } ( \\xi ) ) , \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} p ( x ) = \\inf _ { v \\in \\R } \\varphi ( x , v ) + h ( v ) = \\inf _ { v > 0 } \\left \\{ \\frac { x ^ 2 } { 2 v } - v \\right \\} = - \\infty ( x \\in \\R ) . \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} L _ { p , \\theta } u : = \\nabla _ { \\mathbb { S } ^ { n - 1 } } u ( x ) \\slash \\left [ ( q - 1 ) \\log _ { q } \\frac { R } { | x | } \\right ] + \\nabla _ r u ( x ) . \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{gather*} T _ i ^ M ( X m ) = T _ i ( X ) T _ i ^ M ( m ) \\end{gather*}"}
-{"id": "1036.png", "formula": "\\begin{align*} \\gamma _ + ( t u ) < t ^ 4 \\left [ \\norm { u ^ + } ^ 2 + \\lambda \\int _ { \\mathbb R ^ 3 } \\phi _ { u ^ + } u ^ 2 d x - \\int _ { \\mathbb R ^ 3 } \\frac { f ( t u ^ + ) } { t ^ 3 } u ^ + d x \\right ] < t ^ 4 \\gamma _ + ( u ) = 0 . \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} \\mathrm { l s p } _ S ( X , [ f ] ) \\leq & \\mathrm { u s p } _ S ( X , [ f ] ) \\\\ \\leq & \\sup _ { v \\in I } \\mathrm { u s p } _ S ( X , [ \\phi v ] ) \\\\ \\leq & \\mathrm { l s p } _ S ( X , [ f ] ) . \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} d ( x _ 1 + y _ 1 , x _ 2 + y _ 2 , x _ 3 + y _ 3 ) = x _ 4 + y _ 4 . \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} \\sum _ { \\substack { n > x \\\\ P ( n ) \\leq y \\\\ ( n , a _ 2 ) = 1 } } \\frac { 1 } { \\ell _ u ( n ) } \\ll _ u ( \\log y ) e ^ { - \\sqrt { y } / 2 \\log y } + \\frac { \\log y } { \\log v } e ^ { - v \\log v } . \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} ( a \\delta _ g ) ( b \\delta _ h ) = \\begin{cases} a \\alpha _ g ( b 1 _ { g ^ { - 1 } } ) \\delta _ { g h } & \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} h _ x | x - y | ^ { - 2 } h _ x \\leq 4 h _ x ( - \\Delta _ { y } ) h _ x \\leq 4 h _ x h _ { y } ^ { 2 } h _ x = 4 h _ { x } ^ { 2 } h _ { y } ^ { 2 } . \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} \\mathcal { I } ( f , g , \\beta _ 1 , \\beta _ 2 , \\beta _ 3 ) ( s ) : = \\int _ { \\mathcal { D } } f ( x , y ) ^ s x ^ { \\beta _ 1 } y ^ { \\beta _ 2 } g ( x , y ) ^ { \\beta _ 3 } \\frac { d x } { x } \\frac { d y } { y } \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } [ f ( t ) t - 4 F ( t ) ] = + \\infty . \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} 0 & = \\int _ { K \\backslash G / H } f ( x ) \\cdot 1 _ { A \\cap q ( N ) } ( x \\cdot \\ddot { y } ) d \\mu ( \\ddot { y } ) \\\\ & = f ( x ) \\cdot \\mu \\big ( x ^ { - 1 } \\cdot A \\cap q ( N ) \\big ) . \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\int \\bar f g \\ , d x \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} B _ 0 = \\frac { d _ 2 } { r ( 3 d _ 1 + d _ 2 ) } [ D ] , \\ B _ { \\pm } = B _ 0 \\pm \\varepsilon [ D ] , \\ 0 < \\varepsilon \\ll 1 . \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} y ^ { d } \\cdot \\Phi = \\hat { y } ^ { d } \\cdot \\Phi . \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} \\sum _ { \\l = m } ^ { \\infty } e ^ { 2 \\tau \\log ( \\l ) } \\norm { u _ { \\l } } ^ 2 \\leq \\frac { C } { \\kappa \\tau } \\sum _ { \\l = m + 1 } ^ { \\infty } e ^ { 2 \\tau \\log ( \\l ) } \\norm { ( X u ) _ { \\l } } ^ 2 \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} H ( x _ 1 , x _ 2 , p _ 1 , p _ 2 , t ) = \\frac { f ( t ) } { 2 m } ( p _ 1 ^ 2 + p _ 2 ^ 2 ) + f ^ { - 1 } ( t ) \\frac { m \\omega ^ 2 ( t ) } { 2 } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) . \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} J ( \\varphi ) = \\int _ M L \\bigl ( e _ 2 ( \\varphi ) , e _ 3 ( \\varphi ) , e _ 4 ( \\varphi ) \\bigr ) \\ , \\rho ( x ) \\ , d x \\ , , \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} \\exp \\left ( \\sum _ k \\sum _ { l \\not = m _ k / 2 } t _ { k l } X _ l ^ k \\right ) S _ { i , \\gamma } \\cap \\exp \\left ( \\sum _ k \\sum _ { l \\not = m _ k / 2 } s _ { k l } X _ l ^ k \\right ) S _ { i , \\gamma } = \\emptyset , \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} \\begin{aligned} k _ { 0 } + \\cdots + k _ { p - 1 } & \\ge k _ { 1 , i _ { 0 } } , \\ldots , k _ { 1 , i _ { p - 1 } } + k _ { 2 , i _ { 0 } } , \\ldots , k _ { 2 , i _ { p - 1 } } \\\\ & \\ge 2 \\sum _ { i = 0 } ^ { p - 1 } i = p ^ { 2 } - p . \\end{aligned} \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ R \\lambda ^ \\theta r ^ { \\frac { n } { p } \\theta - 1 } \\left | \\frac { d } { d r } u _ { \\lambda } ( \\phi _ { \\lambda } ( r ) ) \\right | ^ { \\theta } d r & = \\int _ 0 ^ R \\lambda ^ { \\theta } r ^ { \\frac { n } { p } \\theta - 1 } \\left | s ' u ' ( s ) \\right | ^ { \\theta } \\frac { 1 } { s ' } d s \\\\ & = \\int _ 0 ^ R \\lambda ^ { \\theta } r ^ { \\frac { n } { p } \\theta - 1 } s '^ { \\theta - 1 } \\left | u ' ( s ) \\right | ^ { \\theta } d s . \\end{aligned} \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} \\limsup _ { \\lambda \\to \\infty } \\lambda \\mathcal { H } ^ k \\left ( \\left \\{ \\mathbf { M } ^ k ( \\mu , X ) > \\lambda \\right \\} \\right ) = 0 . \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} - \\Delta _ { g } \\bigl ( h _ { 1 , \\lambda } ( r ) \\bigr ) = - \\left \\{ \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + ( n - 1 ) \\frac { f _ 1 ' ( r ) } { f _ 1 ( r ) } \\frac { \\partial } { \\partial r } \\right \\} h _ { 1 , \\lambda } ( r ) = \\left ( \\frac { ( n - 1 ) ^ 2 } { 4 } { K _ 0 } + \\lambda \\right ) h _ { 1 , \\lambda } ( r ) . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} w ( \\lambda , \\mu ) = \\left ( w \\left ( \\frac { \\lambda + \\mu } { 2 } \\right ) + \\frac { \\lambda - \\mu } { 2 } , w \\left ( \\frac { \\lambda + \\mu } { 2 } \\right ) - \\frac { \\lambda - \\mu } { 2 } \\right ) . \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} f ^ \\nu = f _ { \\delta _ \\nu } , f _ \\delta ( t , x , u ) = F ( v _ \\delta ( t , x ) , u ) , v _ \\delta ( t , x ) = \\int _ { \\Omega } \\rho _ \\delta ( t - s ) \\rho _ \\delta ( x - y ) v ( s , y ) \\ ; d y \\ , d s . \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} \\min ~ ~ & c ^ T x \\\\ ~ ~ & A x = b , \\\\ ~ ~ & x \\in \\Lambda . \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} z _ j = \\begin{cases} \\lambda _ 1 ( v ^ j ) & 1 \\leq j \\leq 2 N , \\\\ \\lambda _ 2 ( v ^ j ) & 2 N + 1 \\leq j \\leq 3 N , \\\\ \\lambda _ 4 ( v ^ j ) & 3 N + 1 \\leq j \\leq 4 N . \\end{cases} \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} \\mathcal { Q } _ p ( t ) = A ^ 2 \\left ( \\mathbb { B } _ p , \\lambda ( p , t ) \\right ) . \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} \\Omega ^ n : = \\{ \\textbf { x } \\in \\R ^ n : [ \\sum _ { i = 1 } ^ { k } x _ i ] ^ 2 \\geq \\xi _ k \\sum \\limits _ { j = k + 1 } ^ { n } x _ j ^ 2 , g _ l ( x _ { 1 : k } ) \\geq 0 , \\ x _ r \\geq 0 , \\ r , l \\in ( 1 , . . . , k ) \\} . \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} E = \\bigoplus _ { i = 1 } ^ k V _ i \\otimes E _ i \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t \\phi + \\frac 1 2 \\Delta \\phi = l \\\\ \\phi ( T ) = \\Psi , \\end{array} \\right . \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} b _ { 1 / 2 , d } \\bigl ( m ^ 2 D , \\frac { s } { 2 } + \\frac { 1 } { 4 } \\bigr ) = \\left \\{ \\begin{array} { l l } \\sum _ { 0 < n | m } \\mu ( n ) \\bigl ( \\frac { D } { n } \\bigr ) \\widetilde { \\mathrm { T r } } _ { d , D } ( G _ { m / n } ( z , s ) ) , & m \\neq 0 , \\\\ 2 ^ { 1 - s } \\pi ^ { \\frac { s + 1 } { 2 } } | D | ^ { - \\frac { s } { 2 } } L _ D ( s ) ^ { - 1 } \\widetilde { \\mathrm { T r } } _ { d , D } ( G _ 0 ( z , s ) ) , & m = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} \\int _ { \\{ 2 R < | x | \\} } \\left | \\widetilde { B } _ { Q / p } \\left ( \\frac { x } { 2 } \\right ) \\right | ^ { p ^ { \\prime } } d x = \\int _ { \\{ 2 R < | x | < 2 \\} } \\left | \\widetilde { B } _ { Q / p } \\left ( \\frac { x } { 2 } \\right ) \\right | ^ { p ^ { \\prime } } d x + \\int _ { \\{ | x | \\geqslant 2 \\} } \\left | \\widetilde { B } _ { Q / p } \\left ( \\frac { x } { 2 } \\right ) \\right | ^ { p ^ { \\prime } } d x . \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} d X _ t ( \\omega ) = V _ j ( X _ t ( \\omega ) ) d \\tilde { \\Z } ( B ) _ t ^ j ( \\omega ) , X _ 0 = x \\in \\R ^ d \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} A ( t ) \\begin{cases} t ^ { p _ 0 } \\ , e ^ { - \\sqrt { \\log 1 / t } } & , \\\\ t ^ { p _ \\infty } \\ , e ^ { \\sqrt { \\log t } } & . \\end{cases} \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} n _ b ( W ^ n ( w ) ) = \\lVert ( \\phi w _ 0 ) _ n \\rVert _ S + \\lvert m _ k \\rvert . \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} c = I ( v _ 0 ) + \\sum _ { j = 1 } ^ { l } I ^ { \\infty } ( v _ j ) , \\\\ \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{gather*} \\hat { f } _ { t , s } ( x , y ) : = t x ^ { 3 6 } y + s x ^ { 8 } y ^ { 1 0 } + x ^ { 4 4 } + y ^ { 1 1 } + 1 , \\end{gather*}"}
-{"id": "5843.png", "formula": "\\begin{align*} u ( t , x ) = & \\ \\mathbf E \\left [ \\phi _ 0 \\left ( X ^ { x , \\alpha } ( \\tau _ 0 ( t ) ) \\right ) \\mathbf 1 _ { \\{ \\tau _ 0 ( t ) < \\tau _ \\Omega ( x ) \\} } \\right ] \\\\ & \\ + \\mathbf E \\left [ \\int _ 0 ^ { \\tau _ 0 ( t ) \\wedge \\tau _ \\Omega ( x ) } f \\left ( - X ^ { t , \\beta } ( s ) , X ^ { x , \\alpha } ( s ) \\right ) d s \\right ] , \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { - \\infty } \\phi _ 1 ^ { - 2 } ( x ) d x = \\infty , \\ \\ \\int ^ { + \\infty } \\phi _ 1 ^ { - 2 } ( x ) d x < \\infty , \\\\ & \\int _ { - \\infty } \\phi _ 2 ^ { - 2 } ( x ) d x < \\infty , \\ \\ \\int ^ { + \\infty } \\phi _ 2 ^ { - 2 } ( x ) d x = \\infty , \\\\ & \\int _ { - \\infty } ^ { + \\infty } \\phi _ 3 ^ { - 2 } ( x ) d x < \\infty . \\end{aligned} \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} \\sum _ v \\mbox { i n v } _ v ( ( \\check { w } \\cup \\zeta ) - h ) \\stackrel { ? } { = } 0 \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { ( - 1 ) ^ { n j } \\binom k j G _ { 2 n j } } = ( - 1 ) ^ { n k } L _ n { } ^ k G _ { n k } \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} \\min _ { l _ i \\leq y _ i \\leq u _ i } \\Big \\{ \\beta y ^ 2 _ i + \\Big { ( } \\beta \\sigma ^ { ( - i ) } ( x ) - \\alpha \\Big { ) } y _ i + h _ i ( y _ i ) \\Big \\} , \\ i = 1 , 2 , \\ldots , N . \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} t _ j : = \\begin{pmatrix} e ^ { \\imath \\vartheta _ j } & 0 \\\\ 0 & e ^ { - \\imath \\vartheta _ j } \\end{pmatrix} , g _ j : = h _ { m _ x } \\ , t _ j \\ , h _ { m _ x } ^ { - 1 } . \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} \\tilde { g } _ j = \\frac { \\alpha _ j } { \\alpha _ 1 + \\alpha _ 2 + . . . + \\alpha _ { 2 ^ b } } , \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} \\chi _ { \\ell } ( C _ { 2 l + 1 } \\square K _ { 1 , s } ) = \\begin{cases} 3 & s < 2 ^ { 2 l + 1 } - 2 \\\\ 4 & s \\geq 2 ^ { 2 l + 1 } - 2 . \\end{cases} \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} Q = ( 1 , 1 , - 1 , . . . , - 1 ) \\in \\R ^ { n \\times n } , \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d } d _ { T _ i ' } ( v ) = d _ { G ' } ( v ) \\leq 2 d - 1 . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} \\widetilde { S } _ \\varepsilon ( \\xi ) & = S _ \\varepsilon ( \\xi ) \\cap S _ \\varepsilon ( C _ \\omega u ( \\xi ) ) . \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} \\varphi ( y ) = \\left \\{ \\begin{array} { r l } u ( x ) + K L | x - y | , & \\ , \\ , y \\in I _ { j } \\\\ u ( x ) + K L | x | + L | y | , & \\ , \\ , y \\in I _ { i } , \\ , \\ , i \\neq j \\end{array} \\right . \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} R R ( k ) = \\dfrac { \\| { \\bf r } ^ k \\| _ 2 ^ 2 / \\sigma ^ 2 } { \\| { \\bf r } ^ { k - 1 } \\| _ 2 ^ 2 / \\sigma ^ 2 } \\sim \\dfrac { \\chi ^ 2 _ { n - k } } { \\chi ^ 2 _ { n - k } + \\chi ^ 2 _ 1 } \\sim \\mathbb { B } ( \\dfrac { n - k } { 2 } , \\dfrac { 1 } { 2 } ) \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} & \\bigsqcup _ { \\substack { h , \\bar { h } , h \\bar { h } = h _ i \\\\ h \\in c ( g , X ) , \\bar { h } \\in c ( \\bar { g } , X ) } } ( g _ { t _ 1 } \\cdots g _ { t _ { i - 1 } } ) X _ { g , h } \\cap ( g _ { t _ 1 } \\cdots g _ { t _ { i - 1 } } g ) X _ { \\bar { g } , \\bar { h } } \\cap ( g _ { t _ 1 } \\cdots g _ { t _ { i - 1 } } g \\bar { g } ) X _ { t _ { i + 1 } } \\\\ & \\quad { } = ( g _ { t _ 1 } \\cdots g _ { t _ { i - 1 } } ) X _ { t _ i } \\cap ( g _ { t _ 1 } \\cdots g _ { t _ i } ) X _ { t _ { i + 1 } } . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} a _ 1 ( z ) & = A ( z + \\alpha ) ( z + \\alpha + 1 ) ( z + \\alpha + 2 ) ^ 2 , \\\\ a _ 2 ( z ) & = B ( z + \\beta ) ( z + \\beta + 1 ) ( z + \\beta + 2 ) ^ 2 , \\\\ a _ 3 ( z ) & = - ( A + B ) z ( z + 1 ) ( z + 2 ) ( z + 3 ) . \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} a _ { n + 1 } \\le \\sum _ { k = 0 } ^ n a _ { n - k } b _ { k } . \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} I _ a f = u | _ { \\partial _ + ( S M ) } \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} \\begin{cases} v _ t + g ( v ) _ x = 0 , & t \\in [ 0 , T ] , ~ x \\in \\R , \\\\ [ 1 m m ] v ( 0 , x ) = v _ 0 ( x ) , & x \\in \\R . \\end{cases} \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} S _ { ( \\lambda , \\mu ) , ( \\lambda ' , \\mu ' ) } = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho + \\lambda , ( w \\bullet ( \\lambda ' , \\mu ' ) ) _ 2 \\rangle + \\langle \\mu , ( w \\bullet ( \\lambda ' , \\mu ' ) ) _ 1 + 2 \\rho \\rangle } } { \\sum _ { w \\in W } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho , w \\bullet 0 \\rangle } } . \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} a b \\alpha ( 1 ) + a \\alpha ( b ) + b \\alpha ( a ) + \\Big ( \\beta ( b ) 1 + \\beta ( 1 ) b \\Big ) D ( a ) + \\Big ( \\beta ( a ) 1 + \\beta ( 1 ) a \\Big ) D ( b ) = 0 . \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} \\sum _ { a \\in A _ { + , k } } \\mu ( a ) C _ a ( X _ 1 ) \\dots C _ a ( X _ { n - 1 } ) = 0 . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} \\begin{array} { l l } F ^ { - 1 } _ { a , b } ( z ) = \\rho ( n , 1 ) + \\dfrac { b - 1 } { a + 1 } \\rho ( n , 2 ) \\\\ + \\dfrac { ( b - 1 ) ( a ^ 2 + 3 a b - a + 5 b - 4 ) } { 2 ( a + 1 ) ^ 2 ( a + 2 ) } \\rho ( n , 3 ) + O ( z ^ { ( 4 / a ) } ) \\end{array} \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} \\begin{aligned} T ( t ) & = e ^ { - t ( \\sqrt { A } + K _ { V } ) } ~ , \\\\ L ^ 2 & = L ^ 2 ( \\mathbb { R } ^ { 2 d } ) ~ , \\\\ E & = \\lbrace u \\in L ^ 2 ( \\mathbb { R } ^ { 2 d } ) , q u , \\partial _ q u \\in L ^ 2 ( \\mathbb { R } ^ { 2 d } ) \\rbrace \\end{aligned} \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} \\mbox { V a r } _ { \\mu _ z } ( f ) = \\int _ { \\Sigma _ z } f ^ 2 \\ , d \\mu _ z - \\Big ( \\int _ { \\Sigma _ z } f \\ , d \\mu _ z \\Big ) ^ 2 \\le \\frac { C \\epsilon } { c _ 1 K } \\mathcal { E } _ z ( f , f ) \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} \\tilde { f } ( x , q ) = f \\left ( x , ( \\underline { u } ( x ) \\vee q ) \\wedge \\bar { u } ( x ) \\right ) + m ( \\underline { u } ( x ) \\vee q ) \\wedge \\bar { u } ( x ) . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} ( ( q _ 1 - q _ 2 ) u _ 1 , u _ 2 ) _ { \\Omega } = ( ( \\Lambda _ { q _ 1 } - \\Lambda _ { q _ 2 } ) f _ 1 , f _ 2 ) _ { W } . \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} \\dot { A } _ 2 ( C _ 2 ' P _ 2 + D _ 2 ' ) + \\dot { B } \\frac { \\partial F } { \\partial Q _ 2 } = ( \\dot { C } _ 2 P _ 2 + \\dot { D } _ 2 ) A ' _ 2 ; \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} \\phi _ { n } ( \\textbf { Z } , \\delta ) = \\delta \\phi _ { n , 1 } ( \\textbf { Z } ) + ( 1 - \\delta ) \\phi _ { n , 0 } ( \\textbf { X } ) \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} \\min \\left \\{ \\sum _ { i = 1 } ^ { K } \\tilde { p } _ { i } , u ( 0 ) + \\min _ { i } H _ { i } ( 0 , \\tilde { p } _ { i } ) \\right \\} \\leq 0 . \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} \\psi ( e ^ { - \\pi } ) & = a 2 ^ { - 5 / 8 } e ^ { \\pi / 8 } , \\\\ \\psi ( e ^ { - 2 \\pi } ) & = a 2 ^ { - 5 / 4 } e ^ { \\pi / 4 } , \\\\ \\psi ( e ^ { - \\pi / 2 } ) & = a 2 ^ { - 7 / 1 6 } ( \\sqrt { 2 } + 1 ) ^ { 1 / 4 } e ^ { \\pi / 1 6 } , \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} \\mathbb { S } _ { f , g } = \\frac { \\sum _ { w \\in \\mathfrak { S } _ { n + 1 } } ( - 1 ) ^ { l ( w ) } \\xi ^ { \\langle 2 \\rho , w \\bullet 0 \\rangle } } { d ^ n } \\varepsilon ( f ) \\varepsilon ( g ) ( - 1 ) ^ { \\sum _ { i = 1 } ^ { n } ( k _ { i - 1 } ( f ) + k _ { i - 1 } ( g ) ) } S _ { ( \\lambda _ f , \\mu _ f ) , ( \\lambda _ g , \\mu _ g ) } . \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} \\tau _ 1 \\cdot \\tau _ 2 & = 4 , \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\mu ^ { - n } N _ 1 ^ n u ( \\mu ) & = \\sum _ { n \\geq 0 } \\mu ^ { - n } \\left [ \\lambda ^ n P _ v u ( \\mu ) + O ( \\lambda _ 2 ^ n ) \\right ] \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} \\begin{array} { r c l } a + b & : = & \\varphi \\big ( ( a / \\alpha ) \\oplus ( b / \\alpha ) , \\ , \\psi _ 2 ( a ) + _ o \\psi _ 2 ( b ) \\big ) \\\\ - b & : = & \\varphi \\big ( \\ominus ( b / \\alpha ) , \\ , - _ o \\ , \\psi _ 2 ( b ) \\big ) . \\end{array} \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} F _ \\sharp \\gamma ( A \\times \\R \\times \\R ) & = F _ \\sharp \\gamma ( ( A \\cap C ) \\times \\R \\times \\R ) \\\\ & = \\gamma ( ( A \\cap C ) \\times \\R \\times \\R ) + \\gamma ( \\R \\times ( A \\cap C ) \\times \\R ) + \\gamma ( \\R \\times \\R \\times ( A \\cap C ) ) \\\\ & = 3 \\mu ( A \\cap C ) = \\mu _ C ( A ) . \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} \\mathbb F ( g ) = \\int _ M k \\ , d V _ g \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{gather*} T _ { w ( 1 , 1 ) } \\big ( E _ { N , k } ( - s ) \\big ) = - E _ { N , k + 1 } ( - s ) \\end{gather*}"}
-{"id": "2637.png", "formula": "\\begin{align*} { \\rm \\Lambda } _ L = \\frac { 4 \\log L / \\pi ^ 2 } { 1 + \\lambda \\tilde { \\mu } ^ 2 } + \\mathcal { O } ( 1 ) , \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} 1 + \\sum _ { n \\geq 1 } \\frac { t \\alpha _ n ( t , q ) } { ( 1 - t ) ^ n } \\frac { x ^ n } { 2 ^ { n - 1 } n ! } = \\frac { ( 1 - t ) - ( q - t ) ( e ^ { x / 2 } - 1 ) } { ( 1 - t ) - ( q + t ) ( e ^ { x / 2 } - 1 ) } . \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} \\mathfrak { s } _ p = \\mathfrak { s } _ p ^ { \\geq 0 } \\oplus V . \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\deg ( \\omega _ { M ^ + , s ^ + } | _ { C } ) = 2 K _ { Y ^ + } \\cdot i ^ + ( C ) < 0 \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} \\| f \\| _ { L _ { Q / p } ^ { p } ( B ( x _ { 0 } , r ) ) } : = \\left ( \\int _ { B ( x _ { 0 } , r ) } ( | \\R ^ { \\frac { Q } { \\nu p } } f ( x ) | ^ { p } + | f ( x ) | ^ { p } ) d x \\right ) ^ { 1 / p } , \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} \\left \\| T \\right \\| = \\sup _ { \\left \\| \\phi \\right \\| = 1 } \\left | \\left \\langle \\phi | T \\phi \\right \\rangle \\right | \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} \\frac { S ( 2 ) } { 2 ^ N - 1 } = \\frac { \\sum \\limits _ { i = 0 } ^ { N - 1 } s ( i ) 2 ^ i } { 2 ^ N - 1 } = \\frac { e } { f } , \\ 0 \\leq e \\leq f , \\ \\mathrm { g c d } ( e , f ) = 1 . \\end{align*}"}
-{"id": "9557.png", "formula": "\\begin{align*} g _ 0 : = - k + ( 2 k + 1 ) q ^ { k + 1 } - ( 3 k + 3 ) q ^ { 2 k + 3 } + ( 4 k + 6 ) q ^ { 3 k + 6 } < 0 ~ . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} P D = D \\ \\mbox { a n d t h u s } \\ D ^ * ( I - P ) = 0 , \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} \\beta ( \\theta ) = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log q _ n \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} \\sum _ { \\lambda = 1 } ^ { i - 1 } = \\sum _ { \\lambda = 1 } ^ { i - 1 } ( - 1 ) ^ \\lambda ~ h ( a ^ { m + n - 1 } _ { 1 , \\lambda - 1 } , ~ \\alpha ^ { m + n - 2 } ( a _ \\lambda \\cdot a _ { \\lambda + 1 } ) , ~ a ^ { m + n - 1 } _ { \\lambda + 2 , i } , ~ f ^ n , ~ a _ { i + m + 1 , j } ^ { m + n - 1 } , ~ g ^ m , ~ a ^ { m + n - 1 } _ { j + n + 1 , m + n + p - 1 } ) , \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} Z ^ { ( 0 1 ) } _ { \\R } = \\{ z \\in Z ^ { ( 0 1 ) } \\mid \\textrm { e a c h e n t r y o f $ z $ i s a r e a l n u m b e r } \\} . \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} I _ 3 \\xleftarrow { \\begin{pmatrix} f _ 1 & f _ 2 & f _ 3 \\end{pmatrix} } R _ 3 ^ 3 \\xleftarrow { \\begin{pmatrix} - 2 x _ 0 & - 4 x _ 2 \\\\ x _ 1 & - x _ 1 \\\\ 0 & 2 x _ 0 \\end{pmatrix} } R _ 3 ^ 2 . \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} T - \\sum _ { i = 1 } ^ { N } ( b _ i - a _ i ) ~ < ~ \\delta \\ , , \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} t _ { H C } = \\max _ { i , m _ { ( i ) } \\leq \\frac { 1 } { 2 } } \\frac { \\sqrt { p } \\left ( i / p - m _ { ( i ) } \\right ) } { \\sqrt { m _ { ( i ) } ( 1 - m _ { ( i ) } ) } } \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{gather*} T _ { t _ { - \\alpha _ 2 } } ( E _ { 2 , 1 } ( 1 ) ) = - E _ { 2 , 1 } ( 2 ) . \\end{gather*}"}
-{"id": "4396.png", "formula": "\\begin{align*} { \\cal I } ( x , x ) = \\int _ a ^ b \\big ( | \\dot x - g ( \\dot \\gamma , X ) \\ , x | ^ 2 - K ^ \\top _ X ( { \\dot \\gamma } , x ) \\ , | x | ^ 2 \\big ) \\ , d t + g ( \\dot \\gamma , X ) | x | ^ 2 \\ , | _ a ^ b . \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} \\sup _ { a , b > 0 } \\frac { \\lambda - 1 - \\lambda ( a + b ) + ( \\lambda + 2 ) a b } { a ^ 2 b ^ 2 } = \\sup _ { x = ( a b ) ^ { - 1 / 2 } > 0 } ( \\lambda - 1 ) x ^ 4 - 2 \\lambda x ^ 3 + ( \\lambda + 2 ) x ^ 2 = : \\sup _ { x > 0 } \\varphi ( x ) . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} P f ( x ) \\ge \\frac { ( d - 1 ) g ( | x | + 1 ) + \\lambda g ( | x | - 1 ) } { d - 1 + \\lambda } = \\rho _ { \\mathbb { T } _ d } ( \\lambda ) f ( x ) , x \\not = o . \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} & \\widehat { M } ^ { + } = \\left \\{ ( \\vec { x } , \\vec { u } ) \\in \\mathbb { P } ( V ^ { + } ) _ { \\widehat { U } } : \\sum _ { i = 1 } ^ a x _ i w _ { i j } ( \\vec { u } ) = 0 \\mbox { f o r a l l } 1 \\le j \\le b \\right \\} , \\\\ & \\widehat { M } ^ { - } = \\left \\{ ( \\vec { y } , \\vec { u } ) \\in \\mathbb { P } ( V ^ { - } ) _ { \\widehat { U } } : \\sum _ { j = 1 } ^ b y _ j w _ { i j } ( \\vec { u } ) = 0 \\mbox { f o r a l l } 1 \\le i \\le a \\right \\} . \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} u _ { 1 } ( T _ { o p t } , x ) = \\dots = u _ { k } ( T _ { o p t } , x ) = 0 \\mbox { i n } ( 0 , 1 ) . \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} \\sum _ { j \\in N ( i ) } Q _ { i j } = \\sum _ { j \\in N ( i ) } C _ { i j } \\frac { P _ i - P _ j } { L _ { i j } } = S _ i i \\in \\mathcal { V } , \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} \\theta \\left ( \\begin{bmatrix} 0 & 0 & I \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} \\right ) \\neq 0 . \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} g _ { \\delta } ( t ) = \\begin{cases} \\delta ^ { - 1 } ( \\delta - t ) & 0 \\le t \\le \\delta , \\\\ 0 & \\delta \\le t \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} - \\frac { ( q ^ { a _ j - a _ i - n _ i } ; q ) _ { n _ i } } { q ^ { a _ j n _ i } ( q ^ { a _ i - a _ j + 1 } ; q ) _ { n _ i } } = - \\frac { ( q ^ { a _ i - a _ j + 1 } ; q ) _ { n _ i } ( - q ^ { a _ j - a _ i - n _ i } ) ^ { n _ i } q ^ { \\frac { n _ i ( n _ i - 1 ) } { 2 } } } { q ^ { a _ j n _ i } ( q ^ { a _ i - a _ j + 1 } ; q ) _ { n _ i } } = ( - 1 ) ^ { n _ i + 1 } q ^ { - ( a _ i + n _ i ) n _ i } q ^ { \\frac { n _ i ( n _ i - 1 ) } { 2 } } . \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} \\big \\| e ^ { - \\beta t } M ^ { \\phi } ( t ) - \\Delta \\big \\| _ k & \\le \\Big \\| e ^ { - \\beta t } M ^ { \\phi } ( t ) - \\max _ { i \\le n } e ^ { - \\beta t } D _ i ^ { \\phi } ( t - \\tau _ i ) \\Big \\| _ k \\\\ & + \\Big \\| \\max _ { i \\le n } e ^ { - \\beta t } D _ i ^ { \\phi } ( t - \\tau _ i ) - \\max _ { i \\le n } e ^ { - \\beta \\tau _ i } Y _ i \\Big \\| _ k \\\\ & + \\Big \\| \\Delta - \\max _ { i \\le n } e ^ { - \\beta \\tau _ i } Y _ i \\Big \\| _ k \\\\ & = : Q _ 1 + Q _ 2 + Q _ 3 . \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} \\left | 1 - x ^ n \\right | \\le \\left | 1 - x \\right | \\sum \\limits _ { k = 0 } ^ { n - 1 } \\left | x \\right | ^ k \\le \\frac { \\left | - \\Re ( \\rho ) \\tau + \\tau ^ 2 \\right | } { | \\rho - \\tau | ^ 2 } n < \\frac { n \\tau ^ 2 } { \\Im ( \\rho ) ^ 2 } . \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac D 2 u _ { n , + , a } '' - r ( x ) u _ { n , + , a } = 0 , \\ x \\in ( - n , a ) ; \\\\ & u _ { n , + , a } ( a ) = u _ { n , + , a } ( - n ) = 1 . \\end{aligned} \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} h ( l ) = 0 \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} y _ { i } = \\mu + \\beta _ { i } + \\epsilon _ { i } , i = 1 , . . . , p , \\end{align*}"}
-{"id": "9878.png", "formula": "\\begin{align*} ( u , \\boldsymbol { D } ^ { \\mu } \\psi ) = ( w , \\psi ) ~ \\forall \\psi \\in C _ 0 ^ \\infty ( \\mathbb { R } ) . \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} \\Psi ( y ) = \\sup \\{ x y - \\Phi ( x ) : x \\ge 0 \\} \\quad ( y \\geq 0 ) . \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} \\frac { 1 } { u _ { 1 } / l } = \\frac { 1 } { p _ { 1 } / l } - \\frac { \\alpha } { n } . \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } G _ R ( \\mathbf { w } _ R ; \\ , B _ { \\rho _ 0 } \\setminus B _ { \\rho _ 0 ( 1 - \\mu _ R ) } ) = 0 . \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} \\begin{array} { r c l } I & : = & \\{ i \\in \\{ 1 , \\ldots , l \\} : p _ i x _ 1 , \\ldots , x _ N \\} , \\\\ J & : = & \\{ 1 , \\ldots , l \\} \\setminus I . \\end{array} \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} \\langle a \\rangle : = a \\theta ^ { - \\deg _ { \\theta } ( a ) } \\in 1 + \\frac { 1 } { \\theta } \\mathbb { F } _ q \\bigg [ \\frac { 1 } { \\theta } \\bigg ] . \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} - S \\leq - l S = \\left [ f - c _ 1 ( S , N , F ) \\right ] N \\leq 0 . \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} ( H - \\lambda ) ^ { - 1 } = \\sum _ { n = 0 } ^ { \\infty } ( T - \\lambda ) ^ { - 1 } ( - g \\varphi ( v ) ( T - \\lambda ) ^ { - 1 } ) ^ n . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} b : = \\Psi \\circ ( h - \\mathfrak { s } ) \\circ \\psi ^ { - 1 } : \\hat { U } ' \\rightarrow \\hat { U } ' \\lhd \\mathbb { R } ^ { k ' } \\times \\mathbb { W } \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} ( a \\cdot \\mu ) ( x ) = \\delta ^ { - \\frac { 1 } { 2 } } ( a ) \\mu ( a \\cdot x ) \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} ( \\beta u ) ^ q + ( \\beta u ) = \\beta + \\beta ^ { q ^ 2 } . \\end{align*}"}
-{"id": "9721.png", "formula": "\\begin{align*} D ^ { \\psi } _ f ( x ) : & = D ^ { \\phi } _ { f } ( t ^ d f ( z _ 1 ) x ) \\\\ & = 1 + c ( f ) p _ 1 t ^ d f ( z _ 1 ) x + c ( f ) p _ 2 t ^ { 2 d } f ( z _ 1 ) ^ 2 f x ^ 2 + \\dots + c ( f ) t ^ { d r _ 0 } f ( z _ 1 ) ^ { d r _ 0 } f ^ { r _ 0 - 1 } x ^ { r _ 0 } , \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} n _ s ( I \\cup J ) & = \\max \\{ \\lvert v \\rvert _ S \\mid v \\in I \\cup J , \\tau _ s ( v ) \\neq v \\} \\\\ & = \\max ( \\max \\{ \\lvert v \\rvert _ S \\mid v \\in I , \\tau _ s ( v ) \\neq v \\} , \\max \\{ \\lvert v \\rvert _ S \\mid v \\in J , \\tau _ s ( v ) \\neq v \\} ) \\\\ & = \\max ( n _ s ( I ) , n _ s ( J ) ) \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} \\langle x , y \\rangle = \\langle x | _ { E _ F } , y | _ { E _ F } \\rangle + \\langle x | _ { E \\setminus E _ F } , y | _ { E \\setminus E _ F } \\rangle . \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} i \\partial _ { t } u + D ^ a u = g , u ( 0 , x ) = f ( x ) \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} \\underset { r \\in [ 0 , t _ 0 ) } { \\sup } \\ \\frac { \\mu _ r ( W ) } { \\mu _ r ( h ) } = \\underset { r \\in [ 0 , t _ 0 ) } { \\sup } \\ \\frac { \\mu ( P _ r ^ f W ) } { \\mu ( P _ r ^ f h _ { t _ 0 } ) } , \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} A u = y \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { t \\in [ 0 , 1 ] } \\sqrt { Q ( \\psi _ n ( t ) - \\psi ( t ) ) } \\leq 2 D \\lim _ { n \\to \\infty } \\rho _ { X _ 1 } ( \\psi _ n ( t ) , \\psi ( t ) ) = 0 . \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} I ^ { \\sigma } ( X ) = I ^ { \\sigma ( j ) } I ^ { - j } ( X ) \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } ( f ( z _ 1 ) t ^ d ) ^ i \\mu ( f ^ i ) x ^ i = D _ f ^ { \\phi } ( f ( z _ 1 ) t ^ d x ) ^ { - 1 } = D _ f ^ { \\psi } ( x ) ^ { - 1 } . \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} Q ( z ' , z '' ) \\lesssim \\begin{cases} r ''^ { - n } r '^ { s } ( r ' / r '' ) ^ { 1 - \\frac n 2 + \\nu _ 0 - s } , r ' < \\frac { r '' } 2 ; \\\\ d ( z ' , z '' ) ^ { - ( n - s ) } , r ' \\sim r '' ; \\\\ r '^ { - n + s } ( r '' / r ' ) ^ { 1 - \\frac n 2 + \\nu _ 0 } , r ' > 2 r '' ; \\end{cases} \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} \\widehat { I ^ { + } f } ( t , k ) = - \\int _ { 0 } ^ { t } e ^ { - | k | ^ { 4 } ( t - s ) } | k | ^ { 2 } \\hat { h } ( s , k ) \\ d s . \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} z ( t ) = \\displaystyle \\int _ { D } y ( x , T - t ) f ( x ) d x . \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} \\biggl \\lVert \\frac { y ^ { q ^ j } } { \\tau ^ j ( \\alpha ) } - 1 \\biggr \\rVert = \\biggl \\lVert \\frac { \\tau ^ j ( y - \\alpha ) } { \\tau ^ j ( \\alpha ) } \\biggr \\rVert \\to 0 \\textup { a s } j \\to \\infty . \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} \\sum _ { 0 \\le s \\le ( k - 1 ) d / n } \\binom { d } { s } ( - 1 ) ^ { ( \\frac { n } { d } + 1 ) s } \\sum _ { j = 0 } ^ { n - 1 } ( - j ) ( - \\omega _ n ^ { i j } ) ^ { k - \\frac { n } { d } s } = \\omega _ n ^ { i \\binom { k } { 2 } } \\Big ( 1 _ { n \\mid d k } \\cdot \\binom { k } { 2 } \\binom { d } { d k / n } + \\omega _ n ^ i { n \\brack k } ^ { ' } _ { \\omega _ n ^ i } \\Big ) . \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} \\left ( \\int _ { \\mathbb R ^ { n } } | f | ^ { 2 } d x \\right ) ^ { 2 } \\leq \\left ( \\frac { 2 } { n } \\right ) ^ { 2 } \\int _ { \\mathbb R ^ { n } } \\left | \\nabla f \\right | ^ { 2 } d x \\int _ { \\mathbb R ^ { n } } \\| x \\| ^ { 2 } | f | ^ { 2 } d x , n \\geq 2 , \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} \\langle \\partial _ t - X + N ^ 2 \\nabla f , ( \\partial _ i f ) \\partial _ t + \\partial _ { x ^ i } \\rangle _ { g ^ { ( 4 ) } } = 0 , ~ \\forall i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} \\sum _ { j = 1 , 2 , 3 } \\limsup _ { \\lambda \\to \\infty } \\lambda \\mathcal { L } ^ d \\left ( \\{ | \\mathbf { T } ^ j ( \\mu ) | > \\lambda \\} \\right ) \\lesssim ( c _ 1 + c _ 2 ) | \\mu | ^ { s } ( \\mathbb { R } ^ d ) . \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} \\lim _ { | x | \\rightarrow \\infty } b ( x ) = \\infty , \\ \\textrm { o r } \\ 1 / b \\in L ^ 1 ( \\mathbb { R } ^ 2 ) , \\ \\textrm { o r } \\ \\mbox { m e a s } ( \\Omega _ { b , K } ) < \\infty \\ \\forall \\ , K > 0 , \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\sup _ { 0 \\le k \\le m } \\left \\| P _ { n - k } ( \\epsilon ) - \\left ( \\tilde { P } - \\epsilon k \\tilde { P } ^ { ( 1 ) } + \\frac { \\epsilon ^ 2 } { 2 } k ^ 2 \\tilde { P } ^ { ( 2 ) } \\right ) \\right \\| = O ( \\epsilon ^ 3 m ^ 3 ) \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} C ( \\{ u _ - , u _ + \\} ) & = \\left [ \\frac { N _ - \\cup N _ + } { \\partial _ e ( N _ - \\cup N _ + ) } \\right ] \\\\ & = \\left [ \\frac { N _ - } { \\partial _ e N _ - } \\vee \\frac { N _ + } { \\partial _ e N _ + } \\right ] = C ( u _ - ) \\vee C ( u _ + ) . \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} 2 \\beta \\langle - \\Delta _ 2 u ' ( t ) , u ( t ) \\rangle _ 2 & \\leq 4 \\beta | | u ' ( t ) | | _ { 1 , 2 } | | u ( t ) | | _ { 1 , 2 } \\\\ & \\leq \\frac { 4 \\beta } { 2 \\epsilon G ( t ) ^ \\alpha } | | u ' ( t ) | | _ { 1 , 2 } ^ 2 + \\frac { 4 \\beta \\epsilon G ( t ) ^ \\alpha } { 2 } | | u ( t ) | | _ { 1 , 2 } ^ 2 \\\\ & = 2 \\beta \\epsilon G ( t ) ^ { - \\alpha } G ' ( t ) + 2 \\beta \\epsilon G ( t ) ^ \\alpha | | u ( t ) | | _ { 1 , 2 } ^ 2 . \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} A _ { s - 2 , 3 } = \\sum _ { r = 1 } ^ { n } a _ { r } b _ { r } c _ { r } , \\ \\ \\ A _ { s - 2 , 2 } = \\sum _ { r = 1 } ^ { n } \\sum _ { l = 0 } ^ { r - 1 } \\frac { S _ { r r l } - S _ { l l r } } { r - l } \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} F e ^ { - t A ^ { 2 } } F ^ { - 1 } f ( r , y ) = \\frac { 1 } { \\sqrt { 4 \\pi t } } \\int _ { \\mathbb { R } } \\exp \\left ( - \\frac { ( r - s ) ^ { 2 } } { 4 t } \\right ) f ( s y ) d s . \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} i \\hbar \\dot { \\varrho } _ a = \\big [ \\widehat { H } _ e ( q _ a ) , { \\varrho } _ a \\big ] + \\frac { \\hbar ^ 2 } { 2 M } \\sum _ { b } \\left [ { \\varrho _ b } , { \\varrho } _ a \\right ] { \\int \\frac { K ' ( r - q _ a ) K ' ( r - q _ b ) } { \\sum _ { c } w _ c K ( r - q _ c ) } \\ , r } \\ , , \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} t ^ i \\ell _ { i } ( z _ j ) \\cdot ( Y _ 1 Y _ 2 \\dots Y _ { j - 1 } Y _ j Y _ { j + 1 } \\dots Y _ n W ) & = t ^ i \\ell _ { i } ( z _ j ) \\cdot ( Y _ j Y _ 1 \\dots Y _ { j - 1 } Y _ { j + 1 } \\dots Y _ n W ) \\\\ & = t ^ i \\cdot \\tau ^ i ( Y _ j ) Y _ 1 \\dots Y _ { j - 1 } Y _ { j + 1 } \\dots Y _ n W \\\\ & = \\tau ^ i ( Y _ j W ) Y _ 1 \\dots Y _ { j - 1 } Y _ { j + 1 } \\dots Y _ n . \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { F } _ q } ( B h _ i ) = t _ i = l / r . \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} | B _ { n , k } | & \\le k \\ , 2 ^ k \\sum _ { \\substack { n _ 1 + \\cdots + n _ k = n \\\\ n _ i \\ge 1 , \\ , 2 \\le i \\le k , \\ ; n _ 1 \\ge n / k } } C _ { n _ 1 - 1 } C _ { n _ 2 - 1 } \\cdots C _ { n _ k - 1 } \\\\ & = k \\ , 2 ^ k \\sum _ { \\substack { n _ 2 + \\cdots + n _ k \\le n - ( n / k ) \\\\ n _ i \\ge 1 , \\ , 2 \\le i \\le k } } C _ { n - n _ 2 - \\cdots - n _ k - 1 } C _ { n _ 2 - 1 } \\cdots C _ { n _ k - 1 } \\ , . \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\tilde { K } _ { 3 / 2 } ( m , n , c ) & = - i \\tilde { K } _ { 1 / 2 } ( - m , - n , c ) , \\\\ \\tilde { K } _ { k + 2 } ( m , n , c ) & = \\tilde { K } _ k ( m , n , c ) . \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} \\sigma ( e ^ { i \\theta } ) = \\tau ( e ^ { i \\theta } ) \\prod _ { r = 1 } ^ { R } \\left ( 2 - 2 \\cos ( \\theta - \\theta _ { r } ) \\right ) ^ { \\alpha _ { r } } e ^ { i \\beta _ { r } ( \\theta - \\theta _ { r } ) } , \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d + 1 } ( - 1 ) ^ i \\det ( \\widehat { n _ i } ) n _ i = 0 \\ . \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} & l _ 1 \\cap ( u _ 1 \\cup u _ 2 ) = \\{ 1 < 2 < 3 < 4 < 5 < 6 < 7 \\} , \\\\ & l _ 2 \\cap ( u _ 1 \\cup u _ 2 ) = \\{ a < b < c < d < e \\} , \\\\ & u _ 1 \\cap ( l _ 1 \\cup l _ 2 ) = \\{ 1 < 2 < 3 < 4 < a < 6 < c \\} , \\\\ & u _ 2 \\cap ( l _ 1 \\cup l _ 2 ) = \\{ b < 7 < d < e < 5 \\} . \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} L \\ll l ^ { n _ l \\epsilon } \\max _ { \\substack { 0 \\leq k \\leq q - \\abs { r } , \\\\ A \\leq \\frac { 1 } { 2 } l ^ { k + 2 r ^ + } , \\\\ B \\leq \\frac { 1 } { 2 } l ^ { k + 2 r ^ - } } } \\abs { L _ { A , B , k } } , L _ { A , B , k } = \\sum _ { \\substack { s = ( a , b , k ) \\in S ^ 0 , \\\\ A \\leq \\abs { a } < 2 A , \\\\ B \\leq b < 2 B } } L _ s . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} \\nu ( s ( n , a p ^ h ) ) = \\frac { a - 1 - \\sigma ( n - 1 ) } { p - 1 } + h . \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} A ^ { ( 1 ) } _ \\ell & = a _ 1 & A ^ { ( 2 ) } _ \\ell & = 1 + a _ 1 a _ 2 \\\\ A ^ { ( 0 ) } _ 1 & = 1 & A ^ { ( 1 ) } _ 1 & = a _ 2 \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( \\ker \\beta _ + ) \\subseteq V + \\ker B _ { m + 1 } = \\pi ^ { - 1 } ( \\ker \\beta _ { m + 1 } ) . \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} \\phi ( t ) = \\frac { k ^ { 1 - p / 2 } } { p } \\left ( t ^ 2 + k \\right ) ^ { p / 2 } - \\frac { k } { p } \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} \\beta x ^ 2 + \\beta ^ \\sigma y ^ 2 = z ^ 2 \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} \\begin{cases} \\ \\Delta ^ 2 u & = f , \\\\ \\ u | _ { \\partial \\Omega } & = 0 , \\\\ \\ \\left . \\frac { \\partial u } { \\partial n } \\right | _ { \\partial \\Omega } & = 0 . \\end{cases} \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} \\omega ( y ) = \\prod _ { p \\mid N } \\chi ( y _ p ) \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} \\Psi ( u ) : = \\Phi ( u ) - \\Phi ( b u ) . \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} \\begin{aligned} U _ R ( x ) = \\left [ a + b \\left \\{ \\frac { 1 } { | x | ^ { \\frac { n - p } { p - 1 } } } - \\frac { 1 } { R ^ { \\frac { n - p } { p - 1 } } } \\right \\} ^ { - \\frac { p } { n - p } } \\right ] ^ { 1 - \\frac { n } { p } } \\end{aligned} \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\nabla f ( \\mathbf { x } ^ { * } ) + \\mathbf { E } _ { } ^ T \\boldsymbol { \\alpha } ^ { * } = \\mathbf { 0 } \\\\ \\mathbf { E } _ { } \\mathbf { x } ^ * = \\mathbf { 0 } , \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} \\widetilde { H } : = ( d \\Gamma ( \\omega ) + \\varphi ( \\widetilde { v } ) ) \\oplus ( d \\Gamma ( \\omega ) + \\varphi ( \\widetilde { v } ) ) \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} N _ { 4 0 } & = p _ { 1 0 } N _ { 4 0 - 1 0 } + q _ { 1 0 } N _ { 4 0 - 2 0 } + N _ { 4 0 - 3 0 } \\\\ & = 4 6 \\cdot 3 9 8 6 5 - 1 3 \\cdot 8 7 2 + 1 9 \\\\ & = 1 8 2 2 4 7 3 , \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} M _ 1 ^ \\top M _ 1 & = \\begin{pmatrix} q J _ q & - J _ q & \\cdots & - J _ q \\\\ - J _ q & ( q + 1 ) I _ q - J _ q & \\cdots & ( q + 1 ) I _ q - J _ q \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ - J _ q & ( q + 1 ) I _ q - J _ q & \\cdots & ( q + 1 ) I _ q - J _ q \\end{pmatrix} . \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} \\int _ 0 ^ { T ^ \\ast } ( \\| \\nabla v \\| _ { \\dot { B } _ { 2 , 1 } ^ { \\frac { n } { 2 } } } + \\| d \\| ^ 2 _ { \\dot { B } _ { 2 , 1 } ^ { 1 + \\frac { n } { 2 } } } ) d t = \\infty . \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i , j } x _ i y _ j \\frac { \\partial w _ { i j } ^ { ( 1 ) } ( \\vec { u } ) } { \\partial u _ k } = 0 : 1 \\le k \\le g \\right \\} \\subset ( V ^ { + \\ast } \\times V ^ - ) _ { \\widehat { U } _ p } \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} u ^ { \\prime } ( x ) = R ( x ) | \\varphi ^ \\prime ( x ) | \\cos ( \\theta ( x ) + \\delta ( x ) - \\gamma ( x ) ) . \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} H ( \\sigma ) : = R ( u , J u , u , J u ) . \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{gather*} V ( \\overline { G } ) = V ( G ) \\\\ E ( \\overline { G } ) = \\bigl \\{ \\{ u , v \\} : \\{ u , v \\} \\not \\in E ( G ) , u \\neq v \\bigr \\} . \\end{gather*}"}
-{"id": "6976.png", "formula": "\\begin{align*} H _ { \\mu , A } ^ { ( n ) } ( \\xi , k _ 1 , \\dots , k _ n ) = H _ \\mu ( \\xi - k _ 1 - \\dots - k _ n , A ) + \\omega ( k _ 1 ) + \\dots + \\omega ( k _ n ) \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} \\dot { S } ( t ) = \\frac { 1 } { \\varepsilon } ( F ( t ) - N ( u ( t ) ) \\ , c ( S ( t ) , N ( u ( t ) ) , F ( t ) ) ) , \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} \\rho _ { X _ 1 } ( \\psi _ 1 ( t ) , \\psi _ 2 ( t ) ) : = \\sup _ { t \\in [ 0 , 1 ] } \\| \\psi _ 1 ( t ) - \\psi _ 2 ( t ) \\| _ 2 , \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\nabla ^ * \\Psi = \\beta \\nabla V \\cdot \\Psi - \\mbox { d i v } \\Psi \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{gather*} D ( A _ { K ^ { ( k ) } } ( z ) ) X = - \\int _ { S ^ 1 } \\omega ( X ( t ) , \\dot { x } ( t ) ) d t + \\int _ { S ^ 1 } d K _ t ^ { ( k ) } \\cdot X ( t ) d t \\end{gather*}"}
-{"id": "1450.png", "formula": "\\begin{align*} P _ { k } = \\begin{cases} [ U _ { k , 3 k - 1 } , U _ { k , 3 k } , U _ { k , 3 k + 1 } , U _ { k , 3 k + 2 } ] & \\mbox { i f $ k $ i s o d d , } \\\\ [ U _ { k , 3 k + 2 } , U _ { k , 3 k + 1 } , U _ { k , 3 k } , U _ { k , 3 k - 1 } ] & \\mbox { i f $ k $ i s e v e n . } \\\\ \\end{cases} \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} \\{ p _ { 1 , \\tau } , p _ \\tau \\} _ { D B } = m \\omega ^ 2 ( t _ \\tau ) f ^ { - 1 } ( t _ \\tau ) x _ { 1 , \\tau } ; \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| x ^ { k } - q \\| = \\tau . \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} V ( x ) = \\frac { o ( 1 ) } { 1 + x } \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} n \\in X \\leftrightarrow \\exists m ( f ( j ( n , m ) ) = n ) \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} & \\frac { \\kappa p + 1 } { ( \\kappa + 1 ) ^ p } \\| u u _ h ^ { \\kappa } \\| _ { 1 , p } ^ p \\leq M _ { 2 3 } \\frac { \\kappa p + 1 } { ( \\kappa + 1 ) ^ p } \\| u u _ h ^ { \\kappa } \\| _ { \\tilde { q } _ 1 } ^ p + M _ { 2 4 } \\| u u _ h ^ { \\kappa } \\| _ { \\tilde { q } _ 2 , \\partial \\Omega } ^ p + M _ { 2 5 } \\kappa . \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} f \\times d n = \\int R _ n f _ 1 \\cdot \\psi ^ { - 1 } ( n ) d n . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} \\ , \\mathrm m \\ , = \\ , ( \\widetilde { X } _ { t } ^ { \\mathrm m } , 0 \\le t \\le T ) \\ , \\ , ( X _ { 0 } ^ { \\mathrm m } ) \\ , = \\ , ( \\widetilde { X } _ { 0 } ^ { \\mathrm m } ) \\ , = \\ , \\theta \\ , . \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} \\mathbf { v } _ { i , j } : = \\left [ \\begin{array} { c c c c } \\binom { f } { j } \\lambda _ i ^ { f - j } & \\binom { f + 1 } { j } \\lambda _ i ^ { ( f + 1 ) - j } & \\dots & \\binom { f + ( m + 1 ) n - 1 } { j } \\lambda _ i ^ { ( f + ( m + 1 ) n - 1 ) - j } \\end{array} \\right ] ^ T \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} ( a ; q ) _ { \\infty } & : = \\prod _ { n = 0 } ^ { \\infty } ( 1 - a q ^ { n } ) . \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} F ( \\nabla ^ { h _ u } ) = F ( \\nabla ^ { \\oplus } _ u ) + [ \\nabla ^ { \\oplus } _ u , \\tau _ u ( \\omega _ u ) ] + \\tau _ u ( \\omega _ u ) \\wedge \\tau _ u ( \\omega _ u ) . \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} \\# \\Omega ( X _ q ) = O \\big ( [ \\Gamma : \\Gamma _ q ] ^ { 1 - \\alpha } \\big ) \\qquad \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} \\phi _ 0 ( x ) : = | q | ^ { \\frac 1 2 } e ^ { q | x | } . \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} \\chi _ \\lambda ( k k ' ) = j _ \\lambda ( k k ' , z _ 0 ) ^ { - 1 } = \\left ( j _ \\lambda ( k , k ' \\cdot z _ 0 ) j _ \\lambda ( k ' , z _ 0 ) \\right ) ^ { - 1 } = \\chi _ \\lambda ( k ) \\chi _ \\lambda ( k ' ) \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } G ( y ) = g ( y ) \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} ( - 1 ) ^ n p ^ r B _ n ^ { ( l ) } / n ! \\equiv ( - 1 ) ^ r \\binom { n + r - l } { r } \\mod p . \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\mu ^ { \\mu } \\left ( \\pi \\lambda _ b \\right ) ^ 4 \\beta ^ { 2 - \\mu } - \\left ( \\pi ^ 3 \\lambda _ b ^ 2 \\sqrt { \\Theta } \\ , \\Gamma \\left ( \\frac { 2 } { \\mu } \\right ) \\right ) ^ { \\mu } = 0 . \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} P _ 1 ( a _ 1 , . . . , a _ n ) & = ( a _ 2 , a _ 1 , . . . , a _ n ) , \\\\ P _ 2 ( a _ 1 , . . . , a _ n ) & = ( a _ 2 , . . . , a _ n , a _ 1 ) , \\\\ H ( a _ 1 , . . . , a _ n ) & = ( a _ 1 ^ { - 1 } , a _ 2 , . . . , a _ n ) , \\\\ T ( a _ 1 , . . . , a _ n ) & = ( a _ 1 a _ 2 , a _ 2 , . . . , a _ n ) . \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{gather*} g = u v ^ { - 1 } . \\end{gather*}"}
-{"id": "5647.png", "formula": "\\begin{align*} | B ( t ) + \\mu t | + x = | B ^ { \\mu } ( t ) | + x , x > 0 . \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} \\chi _ { \\mu , p } ( a p ^ k ) = \\begin{cases} \\mu ( a ) & p = l , \\\\ \\mu ( p ) ^ { - k } & p \\neq l . \\end{cases} \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} g = ( g _ i ) _ { i \\in V ( Q ) } \\mapsto \\prod _ { e \\in E ( Q ) } \\det g _ { s ( e ) } \\cdot ( \\det g _ { t ( e ) } ) ^ { - 1 } . \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\Vert v _ { i } ( C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } ) ^ { \\frac { 1 } { 2 } } f \\Vert ^ { 2 } = \\sum _ { i \\in I } v _ { i } ^ { 2 } \\langle \\pi _ { W _ { i } } C ^ { \\prime } f , \\pi _ { W _ { i } } C f \\rangle \\geq A \\Vert f \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "9967.png", "formula": "\\begin{align*} \\langle \\theta u , \\varphi _ \\epsilon | _ { U } \\rangle \\to \\langle \\theta u , \\varphi \\rangle = \\langle u , \\varphi \\rangle . \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} \\bigcap _ { k = 0 } ^ { m ^ + } ( V + \\ker B _ k ) = V , \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} \\int _ { \\mathbb { C P } ^ n } \\left ( r \\omega _ { F S } + i \\partial \\bar { \\partial } \\phi _ { P _ 1 } \\right ) \\wedge \\cdots \\wedge \\left ( r \\omega _ { F S } + i \\partial \\bar { \\partial } \\phi _ { P _ n } \\right ) = \\int _ { ( \\mathbb { C } ^ * ) ^ n } ( i \\partial \\bar { \\partial } H _ { P _ 1 } ) \\wedge \\cdots \\wedge ( i \\partial \\bar { \\partial } H _ { P _ n } ) . \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} & a = { { \\alpha } _ { x } } - { { \\beta } _ { y } } , ~ ~ b = - \\left ( { { \\alpha } _ { y } } + { { \\beta } _ { x } } \\right ) \\\\ & c = { { \\alpha } _ { y } } + { { \\beta } _ { x } } , ~ ~ d = { { \\alpha } _ { x } } - { { \\beta } _ { y } } \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} | \\partial \\mathcal F | ( \\rho ) = \\min \\{ \\| \\zeta \\| _ \\rho \\ ; \\ ! : \\ ; \\ ! \\zeta \\in \\partial \\mathcal F ( \\rho ) \\} . \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{gather*} \\{ \\alpha _ { j + ( k - u ) } = \\alpha _ j + \\alpha _ { k - u } \\} _ { j = 1 } ^ { u } \\ \\ \\mbox { a n d } \\ \\ \\delta _ { n } ^ k = \\alpha _ { 1 + u } + \\alpha _ { k - u } \\ \\ \\mbox { a n d } \\ \\ \\{ \\alpha _ { j + 1 + u } = \\alpha _ { j } + \\alpha _ { 1 + u } \\} _ { j = 1 } ^ { k - u - 1 } . \\end{gather*}"}
-{"id": "9963.png", "formula": "\\begin{align*} \\operatorname { S u p p } ( u * \\tau _ { 1 , \\epsilon } ) \\cup \\operatorname { S u p p } ( u * \\tau _ { 0 1 , \\epsilon } ) \\subset \\{ \\ , z = x + \\sqrt { - 1 } y \\in \\mathbb { C } ^ n \\mid \\operatorname { d i s t } ( x , \\operatorname { S u p p } ( u ) ) \\le \\epsilon \\ , \\} , \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} \\min _ { u \\in L ^ 1 _ + ( \\Omega ) } f ( u ) \\quad A u = y . \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} B '' = \\begin{bmatrix} B '' _ 1 & * & * & * \\\\ & B ^ { ( 1 ) } _ { 1 1 } & * & * \\\\ & & - B ^ { ( 1 ) } _ { 2 2 } & \\star \\\\ & & & B ^ { ( 1 ) } _ { 3 3 } \\end{bmatrix} , \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} \\overline { \\tilde { \\mathbf { S } } _ { X , Y } } = ( - 1 ) ^ { \\delta _ { Y ^ * \\otimes \\bar { \\ 1 } \\not \\in J } } \\tilde { \\mathbf { S } } _ { X , \\bar { Y } } \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} X _ { \\langle F ^ { z \\bar { z } } z , \\bar { z } \\rangle } = ( 0 , - ( \\sigma _ { j _ b } \\partial _ { x _ b } \\langle F ^ { z \\bar { z } } z , \\bar { z } \\rangle ) _ { 1 \\leq b \\leq n } , - \\mathbf { i } ( \\sigma _ j \\partial _ { \\bar { z } _ j } \\langle F ^ { z \\bar { z } } z , \\bar { z } \\rangle ) _ { j \\in \\mathbb { Z } _ * } , \\mathbf { i } ( \\sigma _ j \\partial _ { z _ j } \\langle F ^ { z \\bar { z } } z , \\bar { z } \\rangle ) _ { j \\in \\mathbb { Z } _ * } ) ^ T . \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} J = \\begin{pmatrix} b _ 0 & a _ 0 & 0 & 0 & \\cdots \\\\ a _ 0 & b _ 1 & a _ 1 & 0 & \\cdots \\\\ 0 & a _ 1 & b _ 2 & a _ 2 & \\cdots \\\\ 0 & 0 & a _ 2 & b _ 3 & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{pmatrix} , \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} \\varepsilon _ { \\lambda } = \\pm 1 & \\Longrightarrow \\varepsilon _ { \\mathcal L } = \\pm ( - 1 ) ^ { n / 2 } \\Longrightarrow \\varepsilon _ { { \\mathcal L } _ { n - 2 } } = \\pm ( - 1 ) ^ { n / 2 } \\\\ & \\Longrightarrow \\varepsilon _ { \\lambda _ { n - 2 } } = \\pm ( - 1 ) ^ { n / 2 } ( - 1 ) ^ { ( n - 2 ) / 2 } = \\mp 1 \\ . \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} W _ { 0 , 0 } ( f , g ) ( n ) = & a _ { n + 1 } f ( n ) g ( n + 1 ) - a _ { n + 1 } f ( n + 1 ) g ( n ) , \\\\ W _ { a ' , a ' } ( f , g ) ( n ) = & ( a _ { n + 1 } + a _ { n + 1 } ' ) f ( n ) g ( n + 1 ) - ( a _ { n + 1 } + a _ { n + 1 } ' ) f ( n + 1 ) g ( n ) , \\\\ W _ { 0 , a ' } ( f , g ) ( n ) = & ( a _ { n + 1 } + a _ { n + 1 } ' ) f ( n ) g ( n + 1 ) - a _ { n + 1 } f ( n + 1 ) g ( n ) . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} \\mu ( [ k , \\infty ) ) = \\prod _ { i = 0 } ^ { k - 1 } \\frac { f ( i ) } { 1 + f ( i ) } = f ( k - 1 ) \\mu _ { k - 1 } . \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} \\pi _ \\lambda ( k ) ( \\varphi \\ , K _ { k \\cdot z _ 0 } ) ( z ) & = j _ \\lambda ( k ^ { - 1 } , z ) \\varphi ( k ^ { - 1 } \\cdot z ) K _ { k \\cdot z _ 0 } ( k ^ { - 1 } \\cdot z ) \\\\ & = \\varphi ( k ^ { - 1 } \\cdot z ) \\pi _ \\lambda ( k ) ( K _ { k \\cdot z _ 0 } ) ( z ) \\\\ & = \\overline { j _ \\lambda ( k ^ { - 1 } , k \\cdot z _ 0 ) } \\varphi ( z ) K _ { z _ 0 } ( z ) , \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} \\Xi ' _ { ( i , 2 n - i + 1 ) } ( D ' ) = \\begin{array} { | c | c | c | c | c | } \\hline w ' ( E ^ { ( i ) } ; D ' ) & w ' ( E ^ { ( i - 1 ) } ; D ' ) & \\cdots & w ' ( E ^ { ( 2 ) } ; D ' ) & w ' ( E ^ { ( 1 ) } ; D ' ) \\\\ \\hline \\end{array} ^ { \\ , T } . \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} \\sum _ { d \\mid k } \\mu ( d ) a _ { k m / d } ( \\omega ^ { i d } _ k ) = 0 \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} f _ n ( \\omega ) : = 1 - \\cos ( \\omega ) \\sum _ { k = 0 } ^ { n - 1 } \\frac { C _ { 2 k } ^ k } { 4 ^ { k } } ( \\sin \\omega ) ^ { 2 k } , \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} \\sum _ { \\lambda + \\nu = \\mu } \\Gamma _ \\lambda \\Theta _ \\nu = \\sum _ { \\substack { \\lambda + \\nu = \\mu \\\\ i , j } } S ( u _ i ^ \\lambda ) K _ \\lambda u _ j ^ { \\nu } \\otimes v _ i ^ \\lambda v _ j ^ \\nu \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} \\widetilde { w } ' \\left ( { { z } _ { 0 } } \\right ) = \\underset { z \\to { { z } _ { 0 } } } { \\mathop { \\lim } } \\ , \\frac { \\widetilde { w } \\left ( z \\right ) - \\widetilde { w } \\left ( { { z } _ { 0 } } \\right ) } { z - { { z } _ { 0 } } } \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} d ( 1 - d ^ 2 ) - 6 \\sum _ { i = 0 } ^ { d - 1 } ( i ^ 2 + d i ) = 6 d ^ 2 ( 1 - d ) \\equiv 0 \\mod 1 2 d , \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} \\ln f _ \\alpha ( e ^ t ) = C + \\frac { 1 } { \\alpha - 1 } t + \\ln \\left ( \\int _ { - \\pi } ^ \\pi U _ \\alpha ( x ) \\exp \\left \\{ - z U _ \\alpha ( x ) \\right \\} d x \\right ) , \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} Q _ { z _ 1 , z _ 2 } = | z _ 1 | / | z _ 2 | + | z _ 2 | / | z _ 1 | \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} \\int _ { B _ { u _ p } ^ { \\ell } } f ( x ) \\lvert g _ { u _ p } ( x ) \\lvert ^ 2 d \\lambda _ \\nu ( x ) = \\frac { 1 } { \\ell ! } \\int _ { A _ { u _ p } } f ( x ) \\prod _ { i = 1 } ^ \\nu \\lvert g _ { u _ p } ^ { ( i ) } ( x _ i ) \\lvert ^ 2 d \\lambda _ \\nu ( x ) \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} E [ C ] = \\left ( \\frac { 1 } { \\bar C } + \\frac { \\nu } { \\gamma } \\bar C ^ \\gamma \\right ) \\sum _ { \\substack { ( i , j ) \\in \\bar { \\mathcal { J } } \\\\ i < j } } L _ { i j } . \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} P _ { k , m } ( z , s ) = \\left \\{ \\begin{array} { l l } \\sum _ { r \\in \\mathbb { Z } } F _ { k , m , r } ( z ) \\bigl ( s + \\frac { k } { 2 } - 1 \\bigr ) ^ r & k \\leq 1 / 2 , \\\\ \\ \\\\ \\sum _ { r \\in \\mathbb { Z } } G _ { k , m , r } ( z ) \\bigl ( s - \\frac { k } { 2 } \\bigr ) ^ r & k \\geq 3 / 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{gather*} T _ { w ( 1 , m ) } ( E _ { 1 , 1 } ( s ) ) = E _ { 1 , 1 } ( s ) , T _ { w ( 1 , m ) } ( E _ { N , N } ( - s ) ) = E _ { 2 , 2 } ( - s ) + \\delta _ { s , 0 } ( m - 1 ) c \\end{gather*}"}
-{"id": "9122.png", "formula": "\\begin{align*} P = \\begin{bmatrix} p _ { 0 , 0 } & p _ { 0 , 1 } & \\dots & p _ { 0 , r - 1 } \\\\ p _ { 1 , 0 } & p _ { 1 , 1 } & \\dots & p _ { 1 , r - 1 } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ p _ { r - s - 1 , 0 } & p _ { r - s - 1 , 1 } & \\dots & p _ { r - s - 1 , r - 1 } \\end{bmatrix} . \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) : = \\sum _ { \\substack { r \\geq 0 \\\\ i _ 1 , \\ldots , i _ r \\in I } } A \\tilde { f } _ { i _ 1 } \\cdots \\tilde { f } _ { i _ r } ( v _ \\lambda ) . \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} ( c + 1 ) \\cdot k _ 1 & \\le \\sum V o l ( B _ { 1 / 3 } ( x _ i ) ) + V o l ( B _ { 1 / 3 } ( p ) ) \\\\ & = V o l \\big ( \\bigcup B _ { 1 / 3 } ( x _ i ) \\cup B _ { 1 / 3 } ( p ) \\big ) \\le V o l ( B _ { 3 0 } ( p ) ) \\le k _ 2 . \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\widetilde { H } _ n ( q , t ) = \\widetilde { H } _ { n - 1 } ( q , t ) - q ^ { n - 1 } t ^ 2 ( 1 - t ^ 2 ) \\widetilde { H } _ { n - 3 } ( q , q t ) . \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} [ t _ 0 , \\ldots , t _ { i - 2 } , s _ { i - 1 } , t _ { i + 1 } , \\ldots , t _ n ] = \\sqcup _ { ( t _ { i - 1 } , t _ i ) \\in \\Delta _ { i - 1 } } [ t _ 0 , \\ldots , t _ n ] . \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{gather*} - \\hbar E _ { N , N } ( E _ { 1 , 1 } - c ) - \\tfrac { \\hbar } { 2 } h _ 0 ^ 2 = - \\tfrac { \\hbar } { 2 } \\big ( E _ { N , N } ^ 2 + ( E _ { 1 , 1 } - c ) ^ 2 \\big ) , \\\\ - \\hbar E _ { i , i } E _ { i + 1 , i + 1 } - \\tfrac { \\hbar } { 2 } h _ i ^ 2 = - \\tfrac { \\hbar } { 2 } \\big ( E _ { i , i } ^ 2 + E _ { i + 1 , i + 1 } ^ 2 \\big ) , i \\neq 0 . \\end{gather*}"}
-{"id": "30.png", "formula": "\\begin{align*} \\| u _ H ^ { n } \\| ^ 2 + \\Delta t \\sum _ { k = 2 } ^ n \\| \\nabla u _ H ^ { k - \\theta } \\| ^ 2 + \\gamma \\Delta t \\sum _ { k = 2 } ^ n \\| \\sigma _ H ^ { k - \\theta } \\| ^ 2 \\leq & C ( \\| u _ H ^ { 0 } \\| ^ 2 + \\Delta t \\sum _ { k = 1 } ^ n \\| g ^ { k } \\| ^ 2 ) . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} k = | U | + \\frac 1 2 ( | V ( A ) | - 1 ) . \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta u + V ( x ) u + \\lambda \\rho ( x ) \\phi u = K ( x ) | u | ^ { p - 1 } u , & x \\in \\R ^ 3 , \\\\ \\ , \\ , \\ , - \\Delta \\phi = \\rho ( x ) u ^ 2 , & x \\in \\R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\| u ( t _ n ) - u _ H ^ { n } \\| + \\gamma ^ { \\frac 1 2 } \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\sigma ( t _ { k - \\theta } ) - \\sigma _ H ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } + \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\nabla ( u ( t _ { k - \\theta } ) - u _ { H } ^ { k - \\theta } ) \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } \\leq C [ \\Delta t ^ 2 + H ^ { m + 1 } ] , \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} ( d z _ p , 0 ) 1 \\leq p \\leq n , \\left ( \\frac { i } { 2 } \\sum _ { p = 1 } ^ n \\bar { z } _ p \\ , d z _ p , 1 \\right ) . \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d h _ { s } = & \\left [ g ^ { 2 } ( s ) h _ { s } + b ^ { 2 } ( s ) m _ { s } + \\sigma ^ { 2 } ( s ) n _ { s } \\right ] d s + \\left [ g ^ { 3 } ( s ) h _ { s } + b ^ { 3 } ( s ) m _ { s } + \\sigma ^ { 3 } ( s ) n _ { s } \\right ] d B _ { s } , \\\\ d m _ { s } = & - \\left [ g ^ { 1 } ( s ) h _ { s } + b ^ { 1 } ( s ) m _ { s } + \\sigma ^ { 1 } ( s ) n _ { s } \\right ] d s + n _ { s } d B _ { s } , \\\\ h _ { t } = & 1 , \\ m _ { t + \\delta } = \\phi ^ { 1 } ( t + \\delta ) h _ { t + \\delta } . \\end{array} \\right . \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} T ^ { ' } ( U , V ) = \\alpha \\{ \\eta ( U ) V - g ( U , V ) \\xi \\} + \\beta \\{ \\eta ( U ) \\phi V - g ( \\phi U , V ) \\xi \\} . \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{gather*} v : \\mathbb { R } \\times S ^ 1 \\rightarrow M \\\\ \\partial _ s v + J ( \\partial _ t v - X _ { K ^ { ( k ) } } ( v ) ) = 0 \\end{gather*}"}
-{"id": "5658.png", "formula": "\\begin{align*} \\ddot { z } + \\gamma \\dot { z } + \\omega _ 0 ^ 2 z = 0 , \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} ( X + A + \\Phi ) u = - f , u | _ { \\partial _ - ( S M ) } = 0 . \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} \\tau _ a w _ s = \\sum _ { \\ell } a _ { \\ell } ( s ) w _ { \\ell } , \\tau _ b w _ s = \\sum _ { \\ell } b _ { \\ell } ( s ) w _ { \\ell } . \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} \\begin{array} { c c c c c } \\theta ^ { \\ddagger } ( q , y ) & = & \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ j q ^ { j ( 2 j + 1 ) } y ^ { 2 j } & + & i q y \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ j q ^ { j ( 2 j + 3 ) } y ^ { 2 j } \\\\ \\\\ & = & \\theta ( q ^ 4 , - y ^ 2 / q ) & + & i q y \\theta ( q ^ 4 , - q y ^ 2 ) ~ . \\end{array} \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\psi ( a ) = ( \\psi _ 1 ( a ) , \\psi _ 2 ( a ) ) : = ( a / \\alpha , \\ , a - _ o r ( a ) ) \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} \\chi ( X ) = \\sum _ { i = 0 } ^ \\infty ( - 1 ) ^ i \\beta _ i ( X ) \\ , , \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} & M _ G \\left ( 1 + x ^ 4 + m \\ : k ( x ) \\right ) = 2 ^ 4 ( 1 + 6 m ) , \\\\ & M _ G \\left ( ( 1 + x ^ 4 ) ^ 2 - \\ ; h ( - x ^ 2 ) + m \\ : k ( x ) \\right ) = 2 ^ 6 ( 1 + 3 m ) , \\\\ & M _ G \\left ( x - 1 + m \\ ; k ( x ) \\right ) = 2 ^ 4 3 ^ 2 m . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} \\lVert a f \\rVert _ S & = \\max \\{ \\lvert v \\rvert _ S \\mid v \\in I , 0 \\alpha ( v ) \\neq 0 \\} \\\\ & = 0 \\\\ & = \\lvert a \\rvert _ 0 \\lVert f \\rVert _ S . \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} \\| \\epsilon _ { g p } \\| & = L _ p \\| f i ( f i ( M _ 1 ) x _ { k _ { i n } } ) - M _ 1 x _ { k _ { i n } } + f i ( f i ( M _ 2 ) \\lambda _ k ) \\\\ & - M _ 2 \\lambda _ k + ( V _ 1 - f i ( V _ 1 ) ) \\| \\\\ & \\leq L _ p ( \\| f i ( f i ( M _ 1 ) x _ { k _ { i n } } ) - M _ 1 x _ { k _ { i n } } \\| \\\\ & + \\| f i ( f i ( M _ 2 ) \\lambda _ k ) - M _ 2 \\lambda _ k \\| + \\| ( V _ 1 - f i ( V _ 1 ) ) \\| ) \\\\ \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} ( a _ { n + 1 } + a _ { n + 1 } ' ) u ( n + 1 ) + ( b _ { n + 1 } + b _ { n + 1 } ' ) u ( n ) + ( a _ n + a _ n ' ) u ( n - 1 ) = E u ( n ) . \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} P _ A ( v ) = v \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} & \\mathcal { T } ( t ) = \\frac { 1 } { 2 } \\int _ { - \\infty } ^ \\infty u _ t ^ 2 \\ , d x \\\\ & \\mathcal { V } ( t ) = \\frac { 1 } { 2 } \\int _ { - \\infty } ^ \\infty u _ x ^ 2 + ( u ^ 2 - 1 ) ^ 2 + \\frac { 1 } { 2 } \\Omega ^ 2 x ^ 2 u ^ 2 \\ , d x . \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} \\psi ^ { \\prime } \\left ( \\varrho \\left ( 2 \\Psi ( v ) \\right ) \\right ) & = \\psi ^ { \\prime } \\left ( \\varrho \\left ( \\sum _ { j = 1 } ^ { 4 N } \\psi ( z _ j ) \\right ) \\right ) \\ \\leq \\ \\sqrt { \\sum _ { j = 1 } ^ { 4 N } \\psi ^ { \\prime } ( z _ j ) ^ 2 } = \\\\ & = \\sqrt { \\sum _ { j = 1 } ^ { N } \\left [ 2 \\psi ^ { \\prime } ( \\lambda _ 1 ( v ^ j ) ) ^ 2 + \\psi ^ { \\prime } ( \\lambda _ 2 ( v ^ j ) ) ^ 2 + \\psi ^ { \\prime } ( \\lambda _ 4 ( v ^ j ) ) ^ 2 \\right ] } = \\\\ & = 2 \\sqrt { 2 } \\delta ( v ) . \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } ( \\log g _ { u } ( n ) ) ^ { \\lambda } = M _ { u , \\lambda } x + E _ { u , \\lambda } ( x ) , \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} { \\cal X } : = L ^ 2 ( t _ 1 , t _ 0 ) \\times L ^ 2 ( t _ 2 , t _ 0 ) \\times \\cdots \\times L ^ 2 ( t _ k , t _ 0 ) \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} \\partial _ j \\phi ( x ) = - e ^ { x _ j } e _ { j - n } ^ \\ast V A ^ { - 1 } ( x ) V ^ \\ast e _ { j - n } . \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} \\left . \\frac { \\partial L } { \\partial e _ 2 } \\right | _ { e _ 2 = e _ 3 = e _ 4 = 0 } = \\beta , \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} L _ n v & = f _ { \\alpha , n } , & & x \\in \\R ^ 3 _ + , & & t \\in J ; \\\\ B v & = \\partial ^ \\alpha g _ { n } , & & x \\in \\partial \\R ^ 3 _ + , & & t \\in J ; \\\\ v ( 0 ) & = \\partial ^ { ( 0 , \\alpha _ 1 , \\alpha _ 2 , 0 ) } S _ { \\chi , \\sigma , \\R ^ 3 _ + , m , \\alpha _ 0 } ( 0 , f _ { n } , u _ { 0 , n } ) , & & x \\in \\R ^ 3 _ + ; \\end{aligned} \\right . \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} K _ { k } = 1 0 + \\max _ { 1 \\leq j \\leq N ( k ) } K ( \\lambda _ j , A _ k \\backslash \\{ \\lambda _ j \\} , K _ 0 ) , \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} \\tilde { v } ^ { j _ 0 } _ n \\rightharpoonup 0 H ^ 1 \\lim _ { n \\rightarrow \\infty } \\| \\tilde { v } ^ { j _ 0 } _ n \\| _ { L ^ 2 } = 0 . \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} B ^ { \\mu } ( t ) = B ( t ) + \\mu t + x , \\quad \\mu \\in \\mathbb { R } , \\ ; x \\in \\mathbb { R } . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} & \\int _ 0 ^ t f \\left ( r , W _ r , Y ^ i _ r , \\frac { \\mathrm d [ Y ^ i , W ] _ r } { \\mathrm d r } \\right ) \\mathrm d r \\\\ & = \\int _ 0 ^ t f \\left ( r , W _ r , \\gamma ^ i ( r , W _ r ) , \\nabla \\gamma ^ i ( r , W _ r ) \\right ) \\mathrm d r \\\\ & = \\int _ 0 ^ t \\tilde f ^ i \\left ( r , W _ r \\right ) \\mathrm d r , \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} \\rho ( \\gamma _ 1 m _ t ) R _ j & = \\epsilon ( [ g _ \\alpha \\gamma _ 1 m _ t g _ j ^ { - 1 } ] ) \\\\ & = \\epsilon ( [ g _ \\alpha \\gamma _ 1 g _ t ^ { - 1 } m g _ t g _ j ^ { - 1 } ] ) \\\\ & = \\epsilon ( [ g _ \\alpha \\gamma _ 1 g _ t ^ { - 1 } g _ { t } g _ j ^ { - 1 } ] ) - \\epsilon ( [ g _ \\alpha \\gamma _ 1 g _ t ^ { - 1 } ] ) \\epsilon ( [ g _ t g _ j ^ { - 1 } ] ) . \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} \\chi _ M = \\sum _ { ( \\lambda , \\mu ) \\in P \\times P } \\dim ( M _ { ( \\lambda , \\mu ) } ) e ^ { ( \\lambda , \\mu ) } . \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} \\Lambda _ { + } = \\underset { \\textrm { I m } \\ ; \\lambda > 0 } \\bigoplus \\ker ( H _ { K ^ * _ { \\nu , \\frac { \\pi } { 2 } } } - \\lambda I ) = \\left \\{ \\begin{pmatrix} x \\\\ A _ { + } x \\end{pmatrix} , \\ ; \\ ; \\ ; x \\in \\mathbb { C } ^ 2 \\right \\} , \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\langle \\psi _ { z _ 1 } ^ \\ell | \\psi _ { z _ 2 } ^ \\ell \\rangle = \\frac { I _ { \\ell } ( 2 \\sqrt { z _ 1 ^ * z _ 2 } ) } { \\sqrt { I _ { \\ell } ( 2 | z _ 1 | ) | I _ { \\ell } ( 2 | z _ 2 | ) } } . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} h _ { l } ^ { ( k ) } = u _ { l } ^ { ( k ) } + z _ { l } ^ { ( k ) } + r _ { l } ^ { ( k ) } \\mbox { i n } \\Omega , \\ u _ l ^ { ( k ) } = f _ l ^ { ( k ) } \\mbox { i n } \\Omega _ e , \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} \\times \\left ( I _ k - \\sum _ { j _ 1 = 0 } ^ { p _ 1 } \\ldots \\sum _ { j _ k = 0 } ^ { p _ k } C ^ 2 _ { j _ k \\ldots j _ 1 } \\right ) ^ n , \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} b _ { \\alpha } ( G ) = \\Delta ^ 2 d ^ 4 n ^ 2 \\left ( 1 + O \\left ( \\frac { d + \\Delta } { n } \\right ) \\right ) . \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} \\int d x ~ f ( x ) = 0 . \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\mathrm { L C } _ { s = 1 } ^ r [ \\mathcal { G } _ { D , 0 } ( z , s ) ] ^ { \\mathrm { h o l } } & = - \\mathrm { L C } _ { s = 1 } ^ r \\bigg [ 2 ^ { s - 2 } \\pi ^ { - \\frac { s } { 2 } - 1 } | D | ^ { \\frac { s } { 2 } } L _ D ( s ) \\Gamma \\biggl ( \\frac { s } { 2 } + 1 \\biggr ) \\bigg ] + \\sum _ { 0 > d \\equiv 0 , 1 ( 4 ) } \\mathrm { T r } _ { d , D } ( F _ { 0 , 0 , r } ) q ^ { - d } . \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} \\tilde { \\xi } _ n \\to 0 \\ ; L ^ 1 _ { l o c } ( \\Omega ) \\ ; \\ ; \\ ; \\ ; T _ { \\vec { { \\bf f } } } ( \\tilde { \\xi } _ n ) = \\varphi T _ { \\vec { { \\bf f } } } ( \\xi _ n ) + \\nabla \\varphi \\xi _ n : = \\vec { \\tilde { { \\bf g } } } _ n \\to \\varphi \\vec { { \\bf g } } \\ ; ( L ^ 1 ( \\Omega ) ) ^ N . \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} | F ( x ) | \\leq K [ 1 + \\sum _ { i = 1 } ^ \\ell | x _ i | ^ \\lambda ] \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} \\bar u _ { t } + \\bar f \\left ( t , x \\right ) _ { x } = 0 . \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} \\begin{pmatrix} A P _ 1 & 0 & 0 & \\cdots & - A \\\\ 0 & A P _ 2 & 0 & \\cdots & - A \\\\ \\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\ 0 & \\cdots & 0 & A P _ { \\bar k - 1 } & - A \\\\ 0 & \\cdots & \\cdots & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} w _ { i , j } = \\sum _ { k = 0 } ^ { p ^ { 2 } - p } p _ { k } ( i , j ) n ^ { - 1 + 2 \\beta - k } \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} v ( w ) = \\int _ { \\partial S _ { \\varepsilon } ^ + } e ^ { z w } u ( z ) d z , \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\{ f , h \\} ( \\rho \\ , ) = - i \\hbar ^ { - 1 } \\operatorname { T r } \\left ( \\rho \\left [ \\frac { \\delta f } { \\delta \\rho } \\ , , \\frac { \\delta h } { \\delta \\rho } \\right ] \\right ) \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} ( \\nabla \\xi a \\nabla \\phi ^ T ) \\equiv 0 , \\ , \\Longleftrightarrow ( a \\nabla \\xi _ i ) _ { l } \\frac { \\partial \\phi _ { j } } { \\partial x _ { l } } = 0 \\ , , \\forall \\ , ~ 1 \\le i \\le m \\ , , 1 \\le j \\le n - m \\ , . \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} f ( h h _ 1 ) = \\chi _ { \\lambda } ( h _ 1 ) ^ { - 1 } f ( h ) = j _ \\lambda ( h _ 1 , z _ 0 ) f ( h ) , \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} f = \\sum _ { n \\ge 0 , \\{ 1 , \\ldots , n + 1 \\} \\stackrel { \\psi } { \\to } V ( Q ) } \\sum _ { e _ i \\in E _ { \\psi ( i ) , \\psi ( i + 1 ) } } a _ { \\psi , e _ { \\bullet } } \\cdot e _ 1 e _ 2 \\ldots e _ { n } . \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} d y ( s ) = - a ( y ( s ) ) \\nabla V ( y ( s ) ) \\ , d s + \\frac { 1 } { \\beta } ( \\nabla \\cdot a ) ( y ( s ) ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } \\sigma ( y ( s ) ) \\ , d \\bar { w } ( s ) \\ , , s \\in [ 0 , t ] \\ , . \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} E _ a ^ \\top \\mathbb { J } E _ b = ( E ' _ a ) ^ \\top \\mathbb { J } E ' _ b \\qquad \\textrm { f o r } a , b = 1 , \\ldots , m . \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} S _ { W } f = T ^ { * } _ { W } T _ { W } ( f ) = \\sum _ { i \\in I } v _ { i } ^ { 2 } C ^ { * } \\pi _ { W _ { i } } C ^ { \\prime } f . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} ( \\mathbb { A } ^ { - 1 } _ R ) _ { 1 1 } & = ( d + ( d b ^ { - 1 } a - c ) ^ { - 1 } [ c , d ] ) ( \\Delta ' ) ^ { - 1 } \\\\ & = ( d b ^ { - 1 } a - c ) ^ { - 1 } ( ( d b ^ { - 1 } a - c ) d + [ c , d ] ) ( \\Delta ' ) ^ { - 1 } \\\\ & = ( d b ^ { - 1 } a - c ) ^ { - 1 } d \\underbrace { ( b ^ { - 1 } a d - c ) } _ { = b ^ { - 1 } \\Delta ' } ( \\Delta ' ) ^ { - 1 } \\\\ & = ( d b ^ { - 1 } a - c ) ^ { - 1 } d b ^ { - 1 } = ( a - b d ^ { - 1 } c ) ^ { - 1 } . \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\mathbf { E } _ \\mu | \\varphi | ^ 2 = \\int _ { \\mathbb { R } ^ n } | \\varphi | ^ 2 \\ , d \\mu \\le \\frac { \\kappa _ 1 ^ 2 } { \\beta \\rho } \\ , . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} \\sum _ { j = n _ 0 } ^ { n } \\frac { 1 } { j - v } \\leq O ( 1 ) \\ln \\left ( \\frac { n - v } { n _ 0 - v } \\right ) . \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } z ( t ) \\exp \\biggl \\{ i \\int _ 0 ^ t E [ z ( s ) ] \\ , d s \\biggr \\} = z _ + . \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} u _ { k + 1 } = u _ k \\ , A ^ * \\frac { y } { A u _ k } , k = 0 , \\dots , \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} \\tilde I _ { j } ( x ) : = & \\int _ \\R e ^ { i ( r ^ a - r | x | ) } \\chi _ { j } ( r ) | r | ^ { \\frac { 2 a } { q } - 1 } h ( r | x | ) d r , d \\geq 2 , \\\\ \\tilde I _ { j } ( x ) : = & \\int _ \\R e ^ { i | \\xi | ^ a } e ^ { i x \\xi } \\chi _ { j } ( \\xi ) | \\xi | ^ { \\frac { 2 a } { q } - 1 } d \\xi , d = 1 . \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} T ( n , k ) = ( n - k ) T ( n - 1 , k - 1 ) + ( n + 1 ) T ( n - 1 , k ) + ( k + 1 ) T ( n - 1 , k + 1 ) . \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} k ^ * : = \\min _ { \\gamma \\in W } \\int _ { Y \\times U } k ( y , u ) \\gamma ( d y , d u ) , \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} \\mu ^ L _ 0 \\biggl ( \\eta \\in \\Omega _ L \\ ; \\ ! \\Bigl | \\ ; \\ ! \\frac 1 { L ^ d } \\sum _ { i \\in \\mathbb T _ L ^ d } \\eta ( i ) \\le C _ { \\rm t o t } \\biggr ) = 1 . \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} \\tau _ 1 + \\tau _ 2 + \\tau _ 3 = 0 . \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} g = d t ^ 2 + h ( t ) , \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} q ( - j y + z _ j - ( - n y + c ) ) = q ( ( - j + n ) y + z _ j - c ) < r _ { n , c } . \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} \\alpha : = \\inf _ { r \\geq 0 } | M ^ r \\mathbf { v } | > 0 . \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} \\tilde { \\theta } _ \\ell ( x _ i ) = 2 \\pi i _ 0 + \\frac { 2 i - 1 } { 2 } \\pi \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { \\sum _ { s = 0 } ^ { k - j } { ( - 1 ) ^ { n ( j + s ) + j } \\binom k j \\binom { k - j } s \\frac { { G _ { n + r } ^ j G _ n ^ s } } { { G _ r ^ { j + s } } } H _ { m - r k + n j + ( n + r ) s } } } = \\left ( { \\frac { { G _ 0 } } { { G _ r } } } \\right ) ^ k H _ m , G _ r \\ne 0 \\ , . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} \\mu ( [ 0 , a ] ) = \\lambda _ \\nu ( a ( U \\cap B _ 1 ( 0 ) ) ) = \\nu \\lambda _ \\nu ( U \\cap B _ 1 ( 0 ) ) \\int _ { 0 } ^ { a } k ^ { \\nu - 1 } d \\lambda _ 1 \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} \\sum _ { \\lambda = j + n } ^ { m + n + p - 2 } = \\sum _ { \\lambda = j + n } ^ { m + n + p - 2 } ( - 1 ) ^ \\lambda ~ h ( a ^ { m + n - 1 } _ { 1 , i - 1 } , ~ f ^ n , ~ a ^ { m + n - 1 } _ { i + m , j - 1 } , ~ g ^ m , ~ a ^ { m + n - 1 } _ { j + n , \\lambda - 1 } , ~ \\alpha ^ { m + n - 2 } ( a _ \\lambda \\cdot a _ { \\lambda + 1 } ) , ~ a ^ { m + n - 1 } _ { \\lambda + 2 , m + n + p - 1 } ) . \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} z = - \\left ( \\frac { n + 1 } { n } \\right ) x \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} \\left ( A ^ { * } \\right ) ' \\left ( \\beta \\right ) & = - \\frac { \\kappa } { 2 \\beta ^ { 2 } } \\ , \\textrm { a n d } \\ , \\left ( A ^ { * } \\right ) '' \\left ( \\beta \\right ) = \\frac { \\kappa } { \\beta ^ { 3 } } . \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} g _ t ( \\cdot , \\cdot ) = g _ 0 ( A _ t \\cdot , A _ t \\cdot ) , \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} X _ { t _ { i + 1 } } ^ { j } & = \\Pi \\left ( X _ { t _ { i } } ^ { j } + h \\ B ( X _ { t _ i } ^ { j } , ( u ^ { j - 1 } _ i , v ^ { j - 1 } _ i ) ( X _ { t _ { i } } ^ { j } ) , \\mu ^ { j - 1 } _ i , ( u ^ { j - 1 } _ i , v ^ { j - 1 } _ i ) \\sharp \\mu ^ { j - 1 } _ i ) + \\sigma \\Delta W _ { i } \\right ) \\\\ [ X _ { t _ { i } } ^ { j } ] & = \\mu ^ j _ i ( \\cdot ) \\mu ^ j _ { i + 1 } ( \\cdot ) = [ X _ { t _ { i + 1 } } ^ { j } ] . \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} p _ L ^ * ( x ) = \\sum \\limits _ { \\ell = 0 } ^ L \\frac { \\tilde { \\Phi } _ { \\ell } ( x ) } { 1 + \\lambda \\mu _ { \\ell } ^ 2 } \\int _ { - 1 } ^ 1 w ( x ) \\tilde { \\Phi } _ { \\ell } ( x ) f ( x ) d x . \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} \\Vert f \\Vert ^ { 2 } \\leq \\Vert T ^ { \\dagger } _ { W } \\Vert ^ { 2 } . \\Vert T _ { w } f \\Vert ^ { 2 } = \\Vert T ^ { \\dagger } _ { W } \\Vert ^ { 2 } . \\sum _ { i \\in I } v _ { i } ^ { 2 } \\langle \\pi _ { W _ { i } } C ^ { \\prime } f , \\pi _ { W _ { i } } C f \\rangle ^ { 2 } \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} f ( x , u ) - f ( x , \\underline { u } ) + U ( x ) w = \\left ( \\frac { f ( x , \\underline { u } ( x ) ) - f ( x , u ( x ) ) } { \\underline { u } ( x ) - u ( x ) } \\right ) ^ + w \\geq 0 , \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} \\binom { \\bar { \\partial } } { - \\rho _ { K _ { \\varepsilon } } } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\begin{pmatrix} - \\rho _ { K _ { \\varepsilon } } & - \\bar { \\partial } \\end{pmatrix} . \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} \\mathcal { R } : = \\{ r _ { \\Omega } P _ q f : \\ f \\in H _ 1 \\} \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} \\mathcal { C } : = \\bigcup _ { j \\geq 0 } \\mathcal { C } _ { r _ j } , \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} \\int _ G f ( h x h ^ { - 1 } ) d \\tilde { \\mu } ( x ) = \\int _ G f ( x ) d \\tilde { \\mu } ( x ) \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} P _ { G ^ { \\prime } } ( x ) = x ^ { 4 i + 1 } ( x + 1 ) ^ { 5 i } ( x + 2 i + 2 ) ( x ^ 3 - ( 7 i + 2 ) x ^ 2 - ( 7 i + 3 ) x + 1 2 i ^ 3 + 1 8 i ^ 2 + 6 i ) \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} E ( Q ) = \\frac { 1 } { 2 } \\| Q \\| ^ 2 _ { \\dot { H } ^ 1 _ c } - \\frac { d } { 2 d + 4 } \\| Q \\| ^ { \\frac { 4 } { d } + 2 } _ { L ^ { \\frac { 4 } { d } + 2 } } = 0 . \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} T _ { { \\rm S V C } , n , \\ell + 1 } ^ * = \\sqrt { \\frac { a _ { n } D _ 2 ( \\tau _ C ) } { v ^ * ( D _ 1 ( \\tau _ C ) ) ^ 2 } } \\left ( \\sqrt { \\frac { b _ { n , \\ell + 1 } } { s _ { \\ell + 1 } } } - \\sqrt { \\frac { b _ { n , { \\ell + 2 } } } { s _ { \\ell + 2 } } } \\right ) & \\geq \\sqrt { \\frac { a _ { n } D _ 2 ( \\tau _ C ) } { v ^ * ( D _ 1 ( \\tau _ C ) ) ^ 2 } } \\left ( \\sqrt { \\frac { b _ { n , \\ell } } { s _ { \\ell } } } - \\sqrt { \\frac { b _ { n , { \\ell + 1 } } } { s _ { \\ell + 1 } } } \\right ) \\\\ & = T _ { { \\rm S V C } , n , 1 } ^ * . \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} P _ { 3 / 2 , 0 } ( z , s ) = y ^ { s - 3 / 4 } + \\sum _ { n \\equiv 0 , 3 ( 4 ) } b _ { 3 / 2 , 0 } ( n , s ) \\mathcal { W } _ { 3 / 2 , n } ( y , s ) e ^ { 2 \\pi i n x } . \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} W _ { \\rho ( n ( \\zeta ) \\xi ) \\varphi ' } ( a ( \\gamma ) ) = W _ { \\varphi ' } ( a ( \\gamma ) n ( \\zeta ) \\xi ) = \\psi ( \\gamma \\zeta ) W _ { \\varphi ' } ( a ( \\gamma ) \\xi ) . \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} \\det ( 1 + A ) = \\prod _ { j = 1 } ^ { E ( A ) } ( 1 + \\lambda _ j ( A ) ) . \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ n \\lambda _ i ^ r x _ i \\right | = \\left | \\lambda _ 1 ^ r \\sum _ { i = 1 } ^ n \\mu _ i ^ r x _ i \\right | . \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\frac { p + 1 } { p } \\cdot \\frac { \\Lambda } { a _ { * } } & \\geq \\frac { \\Lambda } { a _ { * } } \\| ( - \\Delta ) ^ { \\frac { 1 } { 4 } } W \\| _ { L ^ { 2 } } ^ 2 + \\frac { 1 } { \\Lambda ^ p } \\int _ { \\mathbb { R } ^ { 3 } } V ( x ) | W ( x ) | ^ { 2 } { \\rm d } x \\\\ & = \\frac { \\Lambda b } { a _ { * } } \\| ( - \\Delta ) ^ { \\frac { 1 } { 4 } } \\mathcal { Q } \\| _ { L ^ { 2 } } ^ 2 + \\frac { 1 } { \\Lambda ^ p b ^ p } \\int _ { \\mathbb { R } ^ { 3 } } V ( x - x _ 0 ) | \\mathcal { Q } ( x ) | ^ { 2 } { \\rm d } x . \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{gather*} = 2 \\binom { n + 2 } { 4 } + \\sum _ { i = 2 } ^ { n + 1 } \\frac { ( i - 1 ) i } { 2 } + \\sum _ { \\alpha = 0 } ^ { n - 1 } \\frac { ( n - \\alpha ) ( n - \\alpha + 1 ) } { 2 } = 2 \\binom { n + 2 } { 4 } + \\sum _ { i = 2 } ^ { n + 1 } \\frac { ( i - 1 ) i } { 2 } + \\sum _ { j = 1 } ^ { n } \\frac { j ( j + 1 ) } { 2 } \\end{gather*}"}
-{"id": "2301.png", "formula": "\\begin{align*} \\lim _ { | x | \\to + \\infty } \\rho ( x ) = \\rho _ { \\infty } > 0 , \\ , \\ , V ( x ) \\equiv 1 , \\ , \\ , \\lim _ { | x | \\to + \\infty } K ( x ) = K _ { \\infty } > 0 , \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} u ' ( t ) = A u ( t ) , \\ ; \\ ; u ( t _ 0 ) = u _ 0 , \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { f } ^ { ( z ) } ( t ) = & \\sum _ { 0 \\leq k < N / 2 } G _ k e ^ { i 2 \\pi k t / T } + \\sum _ { - N / 2 < k < 0 } G _ { N + k } e ^ { i 2 \\pi k t / T } \\\\ & + \\left ( \\frac { \\pi N } { T } \\right ) ^ { z } F _ { N / 2 } \\cos \\left ( \\frac { \\pi N t } { T } + z \\frac { \\pi } { 2 } \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} m _ { \\sigma } ( E ) : = \\sum _ { i = 1 } ^ m | Z ( A _ i ) | . \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} \\dfrac { \\partial ^ 2 u } { \\partial t ^ 2 } - \\left ( k _ 1 + k _ 2 \\int _ 0 ^ L \\left | \\dfrac { \\partial u } { \\partial x } \\right | ^ 2 \\textrm { d } x \\right ) \\dfrac { \\partial ^ 2 u } { \\partial x ^ 2 } = 0 , \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} \\binom { p ^ { n + 1 } } { p ^ n } b _ { p ^ { n + 1 } } = b _ { p ^ n } b _ { ( p - 1 ) p ^ n } + \\sum _ { 0 < i , j < N } a _ { i j } b _ { p ^ n - i } b _ { ( p - 1 ) p ^ n - j } . \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| \\nabla \\tilde { v } ^ { j _ 0 } _ n \\| _ { L ^ 2 } = 0 . \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} [ z ( u ) ] _ { \\alpha ; \\bar { \\Omega } } & = \\sup _ { x \\neq y } \\frac { | z ( u ) ( x ) - z ( u ) ( y ) | } { | x - y | ^ \\alpha } \\\\ & = \\sup _ { x \\neq y } \\left | \\int _ { \\Omega } \\frac { | u ( x - z ) | ^ p - | u ( y - z ) | ^ p } { ( | x - y | ^ { \\alpha } ) | z | ^ { n - 2 } } \\ , d z \\right | \\\\ & \\leq [ | u | ^ p ] _ { \\alpha ; \\bar { \\Omega } } \\int _ { \\Omega } \\frac { 1 } { | z | ^ { n - 2 } } \\ , d z \\\\ & = C ( \\Omega ) [ | u | ^ p ] _ { \\alpha ; \\bar { \\Omega } } \\leq C . \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} \\int _ \\Omega v _ { n _ j ( m ) } ( - \\Delta \\xi + V ^ + \\xi ) = \\int _ \\Omega ( V _ { n _ j ( m ) } ^ - + \\lambda _ { n _ j ( m ) , n _ j ( m ) } ) v _ { n _ j ( m ) } \\xi , \\ ; \\forall \\xi \\in C _ c ^ { \\infty } ( \\Omega _ m ) . \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{gather*} f _ 2 ( \\tau ) = \\sum _ { n = 1 } ^ \\infty a _ 2 ( n ) q ^ n = \\eta ( \\tau ) ^ 2 \\eta ( 2 \\tau ) \\eta ( 4 \\tau ) \\eta ( 8 \\tau ) ^ 2 , f _ 3 ( \\tau ) = \\sum _ { n = 1 } ^ \\infty a _ 3 ( n ) q ^ n = \\eta ( 2 \\tau ) ^ 3 \\eta ( 6 \\tau ) ^ 3 \\end{gather*}"}
-{"id": "6498.png", "formula": "\\begin{align*} \\liminf _ { r \\to 0 } \\frac { \\int _ { [ - r , r ] } x ^ 2 \\ , \\nu ( d x ) } { r ^ 2 \\log ( \\frac { 1 } { r } ) } : = C > \\frac { 1 } { 2 p } . \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} \\{ f _ { i , j } & ( \\beta ^ { r ^ t } ) \\} _ { j \\in [ l ] } = \\{ \\beta ^ a : a _ i = 0 \\} \\bigcup \\\\ & \\Big ( \\bigcup _ { u = 0 } ^ { r - 2 } \\{ \\beta ^ a : a _ i = 1 , a _ { i - 1 } = \\dots = a _ { t + 2 } = 0 , a _ { t + 1 } = u \\} \\Big ) . \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} | | \\partial _ x ^ k ( \\Phi _ r - \\Phi _ 0 ) | | _ { s , r , p ; \\mathbb { T } ^ n \\times \\mathcal { O } _ r ^ - } + \\frac { \\alpha } { M } | | \\partial _ x ^ k ( \\Phi _ r - \\Phi _ 0 ) | | ^ { l i p } _ { s , r , p ; \\mathbb { T } ^ n \\times \\mathcal { O } _ r ^ - } = O ( \\epsilon ^ { \\frac { 1 } { 1 + \\beta } } / \\alpha ) = O ( r ^ { 1 / 4 0 } ) , \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} Y = Y _ 1 \\dashrightarrow Y _ 2 \\dashrightarrow \\cdots \\dashrightarrow Y _ { N - 1 } \\dashrightarrow Y _ N \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{gather*} T _ { w ( 1 , m ) } ( E _ { k , 1 } ( s + 1 ) E _ { N , k } ( - s ) ) \\\\ \\qquad { } = ( - 1 ) ^ { m - 1 } \\times \\begin{cases} E _ { 1 , 1 } ( s + 1 ) E _ { 2 , 1 } ( - s + m - 1 ) & , \\\\ E _ { 3 , 1 } ( s - m + 2 ) E _ { 2 , 3 } ( - s + 2 m - 2 ) & , \\\\ E _ { k + 1 , 1 } ( s + 1 ) E _ { 2 , k + 1 } ( - s + m - 1 ) & , \\\\ E _ { 2 , 1 } ( s + m ) \\Big ( E _ { 2 , 2 } ( - s ) + \\delta _ { s , 0 } ( m - 1 ) c \\Big ) & . \\end{cases} \\end{gather*}"}
-{"id": "7646.png", "formula": "\\begin{align*} \\sum _ { j , l } \\langle h _ j f ( h _ l ) , \\tau | _ { g _ 1 X _ l \\cap X _ j } \\pi \\rangle & = \\sum _ l \\langle \\pi ^ { * * } ( f ( h _ l ) ) , ( g _ 1 ^ { - 1 } \\tau ) _ l \\rangle , \\\\ \\sum _ { j , l } \\langle f ( h _ j h _ l ) , \\tau | _ { g _ 1 X _ l \\cap X _ j } \\pi \\rangle & = \\sum _ s \\langle \\pi ^ { * * } ( f ( h _ s ) ) , \\tau _ s \\rangle , \\\\ \\sum _ { j , l } \\langle f ( h _ j ) , \\tau | _ { g _ 1 X _ l \\cap X _ j } \\pi \\rangle & = \\sum _ j \\langle \\pi ^ { * * } ( f ( h _ j ) ) , \\tau _ j \\rangle . \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} t ( a _ { f ( 1 ) } ^ { ( 1 ) } , \\ldots , a _ { f ( k ) } ^ { ( k ) } , a _ { 1 } ^ { ( k + 1 ) } ) = t ( a _ { f ( 1 ) } ^ { ( 1 ) } , \\ldots , a _ { f ( k ) } ^ { ( k ) } , a _ { 2 } ^ { ( k + 1 ) } ) , \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} d ^ * ( y _ 0 ) = \\min _ { \\gamma \\in W _ 2 ( y _ 0 ) } \\int _ { Y \\times U } k ( y , u ) \\gamma ( d y , d u ) . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} B _ 0 ( \\tilde { U } ) \\supseteq B _ 0 ( V ) = B _ 0 ( H ) , \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} \\Psi ^ \\star _ L ( \\mu , \\nabla \\tilde G _ L ) = 2 \\sum _ { i \\in \\mathbb T _ L ^ d } \\sum _ { k = 1 } ^ d \\hat a _ { i , i + e _ k } ( \\mu ) L ^ 2 \\Bigl [ \\cosh \\bigl ( \\tfrac 1 2 \\nabla ^ { i , i + e _ k } G ( \\cdot / L ) \\bigr ) - 1 \\Bigr ] \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} \\partial _ t C & = \\frac { c _ 0 ^ 2 B ^ 2 } { C } - C ^ \\gamma , \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} c ( x _ 1 , \\ldots , x _ N ) = \\sum _ { 1 \\le i < j \\le N } f ( d ( x _ i , x _ j ) ) , \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} d z ( s ) = & \\ , \\widetilde { b } ( z ( s ) ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } \\widetilde { \\sigma } ( z ( s ) ) \\ , d \\widetilde { w } ( s ) \\\\ = & \\ , \\widetilde { b } ( z ( s ) ) \\ , d s + \\sqrt { 2 \\beta ^ { - 1 } } \\widetilde { \\sigma } ( z ( s ) ) \\Big [ \\big ( \\nabla \\xi a \\nabla \\xi ^ T \\big ) ^ { - \\frac { 1 } { 2 } } \\nabla \\xi \\sigma \\Big ] ( x ( s ) ) \\ , d w ( s ) \\ , . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a _ { 1 1 } & a _ { 1 2 } & \\cdots & a _ { 1 m } \\\\ a _ { 2 1 } & a _ { 2 2 } & \\cdots & a _ { 2 m } \\\\ & & \\vdots & \\\\ a _ { n 1 } & a _ { n 2 } & \\cdots & a _ { n m } \\end{bmatrix} , \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} I & = \\frac { 1 } { 2 \\pi i } \\int _ { 2 - i \\infty } ^ { 2 + i \\infty } F _ C ( s ) k ( s ) \\ , d s \\\\ & = \\sum _ { \\mathfrak { p } } \\sum _ { m = 1 } ^ { \\infty } \\theta ( { \\mathfrak { p } } ^ m ) ( \\log N _ { K / { \\mathbb Q } } { \\mathfrak { p } } ) \\widehat { k } ( N _ { K / { \\mathbb Q } } { \\mathfrak { p } } ^ { m } ) , \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} | D u ( x _ 0 ) - \\Theta _ { x _ 0 , r } | + | p ( x _ 0 ) - \\theta _ { x _ 0 , r } | \\lesssim \\sum _ { j = 0 } ^ \\infty \\Phi ( x _ 0 , \\kappa ^ j r ) . \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} \\binom { a p ^ k } { b p ^ k } & \\equiv \\binom { a p ^ { k - 1 } } { b p ^ { k - 1 } } \\bmod p ^ { 3 k } , \\\\ g _ { n p ^ k } ( 1 ) & \\equiv g _ { n p ^ { k - 1 } } ( 1 ) \\bmod p ^ { 3 k } , \\\\ h _ { n p ^ k } ( 1 ) & \\equiv h _ { n p ^ { k - 1 } } ( 1 ) \\bmod p ^ { 3 k } . \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} P ( 0 ) = \\mathrm { T r } \\left ( \\begin{bmatrix} 1 & \\omega ^ { n - 1 } \\\\ 0 & 0 \\end{bmatrix} \\begin{bmatrix} 1 & \\omega ^ { n - 2 } \\\\ 0 & 0 \\end{bmatrix} \\cdots \\begin{bmatrix} 1 & 1 \\\\ 0 & 0 \\end{bmatrix} \\right ) = \\mathrm { T r } \\left ( \\begin{bmatrix} 1 & 1 \\\\ 0 & 0 \\end{bmatrix} \\right ) = 1 . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} \\mathbf { x } ^ { k + 1 } = ( \\mathbf { I } - \\frac { \\xi \\rho } { 2 } \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } ) \\mathbf { x } ^ k - \\xi \\nabla f ( \\mathbf { x } ^ { k + 1 } ) - \\sum _ { t = 0 } ^ k ( \\frac { \\rho \\eta \\xi } { 2 } ) \\mathbf { E } _ { } ^ T \\mathbf { E } _ { } \\mathbf { x } ^ t . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left ( ( 1 - x ) ^ { \\alpha } ( 1 + x ) ^ { \\beta } P ^ { ( \\alpha , \\beta ) } _ { n } ( x ) \\right ) = - 2 ( n + 1 ) ( 1 - x ) ^ { \\alpha - 1 } ( 1 + x ) ^ { \\beta - 1 } P ^ { ( \\alpha - 1 , \\beta - 1 ) } _ { n + 1 } ( x ) , \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} H ( x , p , \\theta ) = p \\cdot f ( x , \\theta ) - L ( x , \\theta ) . \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} \\lambda _ n ( w _ n ) _ { i j } - ( \\Delta _ L w _ n ) _ { i j } = \\frac { ( f _ n ( z ) ) _ { i j } } { \\| R _ { \\lambda _ n } f _ n \\| _ { C ^ { 0 , \\alpha } } } . \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} A ( P ) : = \\dot \\gamma ( 0 ) . \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ g = \\delta _ x & \\Omega , \\\\ G = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} B = B ( \\mathbf { v } _ 1 , \\vartheta , g \\ , T ) : = \\vartheta \\cdot \\omega _ { m _ x } \\big ( \\mathrm { A d } _ g ( \\beta ) _ M ( m ) , \\mathbf { v } _ 1 \\big ) . \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} L ^ t i D _ j Q = - \\tilde E z _ j [ z _ 2 ( i D _ 1 Q ) - z _ 1 ( i D _ 2 Q ) ] , \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} M & = \\max \\left \\{ 3 , \\frac { 9 L _ \\sigma ^ 2 } { \\sqrt { \\pi } } , \\frac { 8 L _ b ^ 2 } { L _ \\sigma ^ { 4 } } , \\frac { 1 4 4 L _ b ^ 2 } { L _ \\sigma ^ { 2 } \\sqrt { \\pi } } , \\frac { 8 6 4 L _ b ^ 2 } { \\sqrt { \\pi } } \\right \\} , \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} P : = \\exp \\left ( - i \\left ( t z + \\frac { t ^ 3 } { 6 } \\right ) \\right ) \\exp \\left ( \\frac { i t } { 2 } \\frac { \\partial ^ 2 } { \\partial z ^ 2 } + \\frac { t ^ 2 } { 2 } \\frac { \\partial } { \\partial z } \\right ) . \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} \\hat I ( t ) \\phi ( x _ 1 , x _ 2 , t ) = E \\phi ( x _ 1 , x _ 2 , t ) , \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} M ' _ { k , \\ell } = 0 \\mbox { i f } q > q _ \\ell . \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} N ( X ) ( \\epsilon ) _ n = \\left \\{ ( x _ 0 , \\dots , x _ n ) \\mid x _ i \\in X \\sum _ { i = 0 } ^ { n - 1 } d ( x _ i , x _ { i + 1 } ) \\leq \\epsilon ) \\right \\} \\ , \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} ( x ) _ n = \\sum _ { k = 1 } ^ \\infty s ( n , k ) x ^ k \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} \\widetilde V _ x = \\frac { V _ x + \\mathcal R V _ x } { \\Vert V _ x + \\mathcal R V _ x \\Vert _ * } . \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} \\sup _ { z \\in \\Z } E ^ \\omega _ { z - 1 } [ \\exp \\{ \\epsilon H ( z ) \\} ] & \\le E ^ { \\underline { \\omega } } _ { 0 } [ \\exp \\{ \\epsilon H ( 1 ) \\} ] = : 1 + \\eta ( \\epsilon ) . \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} N ( h ) = O ( h ^ { - \\delta } ) \\quad \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} \\left ( \\frac { t } { e ^ t - 1 } \\right ) ^ l e ^ { t x } = \\sum _ { n = 0 } ^ \\infty B _ n ^ { ( l ) } ( x ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} N _ { m } & = N _ { m - 1 } + N _ { m - 3 } \\\\ & = ( 6 N _ { m - 6 } + 5 N _ { m - 1 1 } + N _ { m - 1 6 } ) + ( 6 N _ { m - 8 } + 5 N _ { m - 1 3 } + N _ { m - 1 8 } ) \\\\ & = 6 ( N _ { m - 6 } + N _ { m - 8 } ) + 5 ( N _ { m - 1 1 } + N _ { m - 1 3 } ) + ( N _ { m - 1 6 } + N _ { m - 1 8 } ) \\\\ & = 6 N _ { m - 5 } + 5 N _ { m - 1 0 } + N _ { m - 1 5 } , \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} \\tau _ { r , u } t - t = r u ( \\tau _ { s , u } t - t = s u ) \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} 2 = | V ( G ' ) | - | E ( G ' ) | + | F ( G ' ) | \\le \\frac 2 3 | E ( G ' ) | - | E ( G ' ) | + \\frac 1 3 | E ( G ' ) | = 0 , \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} \\ell _ { n _ s } & = \\lfloor \\sqrt [ 3 ] { n _ s - d _ s + d _ s } \\rfloor = \\left \\lfloor \\sqrt [ 3 ] { n _ s - d _ s } \\ , \\sqrt [ 3 ] { 1 + \\frac { d _ s } { n _ s - d _ s } } \\right \\rfloor \\\\ & = \\left \\lfloor s \\ , \\sqrt [ 3 ] { 1 + \\frac { d _ s } { s ^ 3 } } \\right \\rfloor \\le \\left \\lfloor s \\ , \\Big ( 1 + \\frac { 1 } { 3 } \\frac { d _ s } s ^ 3 \\Big ) \\right \\rfloor = s \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} M ( X ) _ n = \\bigoplus _ { l \\in [ 0 , \\infty ) } \\Z \\left [ \\left \\{ ( x _ 0 , \\dots , x _ n ) \\mid \\sum _ { i = 0 } ^ { n } d ( x _ i , x _ { i + 1 } ) = l \\right \\} \\right ] \\ , . \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} X _ n = m ^ { - 1 } X _ { n - 1 } + \\varepsilon _ n , n \\geq 1 , \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } Y ^ + _ n & = \\liminf _ { n \\to \\infty } Y ^ - _ n = \\liminf _ { t \\to \\infty } Y _ t \\leq 1 \\leq \\limsup _ { t \\to \\infty } Y _ t = \\limsup _ { n \\to \\infty } Y ^ + _ n . \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\boldsymbol { \\lambda } _ i ^ { [ k + 1 ] } = \\boldsymbol { \\lambda } _ i ^ { [ k ] } + \\rho ( \\mathbf { p } _ i ^ { [ k + 1 ] } - \\mathbf { q } _ i ^ { [ k + 1 ] } ) \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} u ( \\rho \\ , ) = - \\frac { i } { 2 \\hbar } \\left [ \\{ \\widehat { u } ^ k , \\widehat { P } _ k \\} _ { + \\ , } , \\rho \\right ] \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow \\infty } E ( V ^ { j _ 0 } ( \\cdot - x ^ { j _ 0 } _ n ) ) = d _ M , \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} u _ { \\lambda } ( x ) : = \\lambda u ( \\phi _ { \\lambda } ( x ) ) , \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} f _ n ( t ) & = \\biggl ( a - b + c + n d + \\int _ { [ 1 / n , n ] } s ^ { - 1 } \\ , d \\mu ( s ) \\biggr ) \\\\ & \\qquad + \\biggl ( b + n c + d + \\int _ { [ 1 / n , n ] } d \\mu ( s ) \\biggr ) t \\\\ & \\qquad - c ( 1 + n ) { t ( 1 + n ) \\over t + n } - d ( 1 + n ) { t \\bigl ( 1 + { 1 \\over n } \\bigr ) \\over t + { 1 \\over n } } \\\\ & \\qquad - \\int _ { [ 1 / n , n ] } { 1 + s \\over s } \\cdot { t ( 1 + s ) \\over t + s } \\ , d \\mu ( s ) \\\\ & = f _ n ( 0 ^ + ) + f _ n ' ( + \\infty ) t - h _ n ( t ) \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} d g ( z ) = d \\mu ( z ) = r ^ { n - 1 } d r d h = r ^ { n - 1 } d r d \\mu _ Y ( y ) . \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} \\kappa ^ { A } ( \\theta ) + \\kappa ^ { - S } ( \\theta ) = 0 , \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} ( - 1 ) ^ { k + 1 } \\sum _ { 0 \\le s \\le ( k - 1 ) d / n } \\binom { d } { s } ( - 1 ) ^ s \\sum _ { j = 0 } ^ { n - 1 } j \\omega _ n ^ { i k j } = \\omega _ n ^ { i \\binom { k } { 2 } } \\Big ( 1 _ { n \\mid d k } \\cdot \\binom { k } { 2 } \\binom { d } { d k / n } + \\omega _ n ^ i { n \\brack k } ^ { ' } _ { \\omega _ n ^ i } \\Big ) . \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} \\sum _ { \\lambda + \\nu = \\mu } \\Theta _ \\lambda \\Gamma _ \\nu = \\sum _ { \\substack { \\lambda + \\nu = \\mu \\\\ i , j } } u _ i ^ { \\lambda } S ( u _ j ^ \\nu ) K _ \\nu \\otimes v _ i ^ \\lambda v _ j ^ \\nu = 0 . \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} \\hat \\gamma ( A , B ) ( s ) = \\Psi ^ { - 1 } \\big ( ( 1 - s ) \\Psi ( A ) + s \\Psi ( B ) \\big ) , \\ , A , B \\in \\overline \\Omega . \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{gather*} \\overset { I } { B } \\cdot h = \\overset { I } { h ' } \\overset { I } { B } \\overset { I } { S ( h '' ) } , \\overset { I } { A } \\cdot h = \\overset { I } { h ' } \\overset { I } { A } \\overset { I } { S ( h '' ) } . \\end{gather*}"}
-{"id": "7719.png", "formula": "\\begin{align*} [ \\phi ; - \\Delta ^ p _ \\Lambda \\phi ] = \\sum \\limits _ { j , k \\in \\Lambda , j k \\in E } ( \\phi _ j - \\phi _ k ) ^ 2 \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} b _ n ^ { ( \\alpha , \\beta ) } = \\frac { \\beta ^ 2 - \\alpha ^ 2 } { ( 2 n + \\alpha + \\beta ) ( 2 n + \\alpha + \\beta + 2 ) } , n \\geq 1 , \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\sigma ( x _ 1 , x _ 2 , \\cdot \\cdot \\cdot , x _ N ) = \\frac { \\sqrt { - V ( x ) } } { 4 \\sqrt { N } ( \\rho ( x _ 1 ) + 1 ) } \\Big ( V \\int _ 0 ^ { x _ 1 } \\rho ( t ) d t - \\rho ^ { \\ ; \\prime } ( x _ 1 ) \\Big ) . \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} w \\left ( z \\right ) \\to \\widetilde { w } \\left ( z \\right ) = w \\left ( z \\right ) K \\left ( z \\right ) \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} \\int _ 0 ^ t f \\left ( r , W _ r , Y _ r , \\frac { \\mathrm d [ Y , W ] _ r } { \\mathrm d r } \\right ) \\mathrm d r & = \\int _ 0 ^ t f \\left ( r , W _ r , u ( r , W _ r ) , \\nabla u ( r , W _ r ) \\right ) \\mathrm d r \\\\ & = \\int _ 0 ^ t \\tilde f ( r , W _ r ) \\mathrm d r , \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} \\begin{pmatrix} \\delta \\mathrm { d } - 2 \\ , \\mathrm { R i c } & \\mathrm { d } \\\\ \\delta & 0 \\end{pmatrix} \\begin{pmatrix} A ^ \\flat \\\\ \\tilde p \\end{pmatrix} = 0 . \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} { \\mathrm d } X _ { t } ^ { \\dagger } \\ , = \\ , - ( X _ { t } ^ { \\dagger } - \\widetilde { X } _ { t } ^ { \\dagger } ) { \\mathrm d } t + { \\mathrm d } B _ { t } \\ , , t \\ge 0 \\ , , \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} \\phi ( k , \\zeta ) = R _ { \\mathcal { Y } _ p } \\left ( e _ { ( z , w ) } \\circ T \\right ) ( k , \\zeta ) - \\overline { Y _ p ( k ; w , \\zeta ) } \\frac { ( k + 1 ) \\overline { z } ^ k } { \\pi } , ( k , \\zeta ) \\in \\N \\times \\mathbb { B } _ p . . \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} ( n - 1 ) x _ j - x _ { j + 1 } - x _ { j + 2 } - \\ldots - x _ { n + 1 } - x _ 1 - \\ldots - x _ { j - 1 } = n x _ j - ( x _ 1 + \\ldots + x _ n ) - x _ { n + 1 } = n ( x _ j - x _ { n + 1 } ) . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} s _ n s _ { n - 1 } \\cdots s _ 1 ( d \\varpi _ n + \\rho ) - \\rho \\equiv d \\varpi _ n - \\sum _ { i = 1 } ^ n s _ n s _ { n - 1 } \\cdots s _ { n + 2 - i } ( \\alpha _ { n + 1 - i } ) \\mod d Q ^ \\vee . \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} \\mbox { $ u _ { l } ( t , 1 ) = 0 $ f o r $ 0 \\le t \\le T _ { o p t } - \\tau _ l $ a n d $ k + 1 \\le l \\le m $ } . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} | v | ^ 2 _ { k - 1 , w } = \\sum \\limits _ { D \\in \\mathcal { T } _ h } \\int _ { D } \\nabla _ { w , k - 1 , D } v \\cdot \\nabla _ { w , k - 1 , D } v \\ { \\rm d } x + h _ D ^ { - 1 } \\langle v _ 0 - v _ b , v _ 0 - v _ b \\rangle _ { \\partial D } , \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} G ^ { - 1 } ( \\xi ( x ) , \\phi ( x ) ) = x \\ , , \\forall \\ , x \\in \\mathbb { R } ^ n \\ , , \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} \\textstyle \\mu _ 2 ^ * = \\left ( { P _ T + \\sum \\limits _ { k = 1 } ^ { r _ s } \\frac { \\sigma ^ 2 } { [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 } } \\right ) \\left ( { r _ s \\sum \\limits _ { k = 1 } ^ { r _ s } \\frac { 1 } { \\left ( \\nu _ 2 ^ * - [ \\boldsymbol { \\Lambda } ] _ { k , k } ^ 2 \\right ) \\ln 2 } } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} \\left ( \\int _ { \\{ 2 R < | x | \\} } \\left ( \\frac { | x | } { 2 } \\right ) ^ { ( a - Q ) q } \\frac { d x } { | x | ^ { b } } \\right ) ^ { \\frac { 1 } { q } } & \\left ( \\int _ { \\{ | x | < R \\} } d x \\right ) ^ { \\frac { 1 } { p ^ { \\prime } } } \\\\ & \\leq C R ^ { a - Q - \\frac { b } { q } + \\frac { Q } { q } } R ^ { Q / p ' } \\leq C \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} \\partial _ t g _ t = - P ( g _ t ) \\ , , g _ { | t = 0 } = g _ 0 . \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} \\overline { C } { ( d , \\epsilon ) } = \\sup _ { \\mathbb { P } ( D ( t ) > d ) \\le \\epsilon , \\forall t } \\lambda , \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} L _ \\tau ( x _ { 1 , \\tau } , x _ { 2 , \\tau } , t _ \\tau , \\dot { x } _ { 1 , \\tau } , \\dot { x } _ { 2 , \\tau } , \\dot { t } _ \\tau ) = \\frac { f ^ { - 1 } ( t _ \\tau ) m } { 2 \\dot { t } _ \\tau } \\left ( \\dot { x } _ { 1 , \\tau } ^ 2 + \\dot { x } _ { 2 , \\tau } ^ 2 \\right ) - \\frac { m \\omega ^ 2 ( t _ \\tau ) \\dot { t } _ \\tau } { 2 } f ^ { - 1 } ( t _ \\tau ) \\left ( x _ { 1 , \\tau } ^ 2 + x _ { 2 , \\tau } ^ 2 \\right ) . \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } P _ { \\boldsymbol { \\theta } } } { \\mathrm { d } \\lambda } \\left ( \\boldsymbol { x } \\right ) & = \\frac { \\exp \\left ( \\boldsymbol { \\theta } \\cdot \\boldsymbol { x } \\right ) } { Z \\left ( \\boldsymbol { \\theta } \\right ) } = \\exp \\left ( \\boldsymbol { \\theta } \\cdot \\boldsymbol { x } - A \\left ( \\boldsymbol { \\theta } \\right ) \\right ) . \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial \\overline { z } } + A w = 0 \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} \\| f _ i \\| _ E \\ge 1 - \\delta _ i , \\ ; \\ ; i = 1 , 2 , \\dots \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\partial ^ p _ 1 g ( \\cdot , \\tilde { x } ) = \\partial _ 1 f ( \\cdot , \\tilde { x } ) \\ ; \\ ; [ a _ 1 , b _ 1 ] . \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} | k _ b j _ b + \\sum _ { j \\in \\mathbb { Z } _ * } \\frac { l _ j j } { n - \\frac 1 2 } | \\geq \\frac { 1 } { 1 0 0 n \\sum _ { b = 1 } ^ n | j _ b | } \\max \\{ | k | , \\sum _ { j \\in \\mathbb { Z } _ * } | j l _ j | \\} , b = 1 , \\cdots , n . \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} I _ { A + \\Phi } f = u | _ { \\partial _ + ( S M ) } , \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} 0 < \\sigma < \\bar { \\sigma } \\bar { \\sigma } \\leq \\frac { n - 2 s } { 2 s } \\bar { \\sigma } _ j b _ j ^ { 2 - 2 s } , j = 1 , \\cdots , k , \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} \\frac { 1 } { s } = \\frac { 1 } { p } - \\frac { \\alpha } { n } \\quad \\quad \\frac { t } { s } = \\frac { q } { p } , \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} E ( t ) + \\int _ 0 ^ t | | u ' ( \\tau ) | | _ { 1 , 2 } ^ 2 \\ , d \\tau & \\leq \\liminf _ { N \\to \\infty } \\left ( E _ N ( t ) + \\int _ 0 ^ t | | u _ N ' ( \\tau ) | | _ { 1 , 2 } ^ 2 \\ , d \\tau \\right ) \\\\ & = \\liminf _ { N \\to \\infty } E _ N ( 0 ) \\\\ & = E ( 0 ) \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} 0 & \\le \\pi _ { \\psi } ( a e _ P ) e e \\pi _ { \\psi } ( a e _ P ) ^ * \\\\ & \\le \\pi _ { \\psi } ( a e _ P ) e ^ { 1 / 2 } e e ^ { 1 / 2 } \\pi _ { \\psi } ( a e _ P ) ^ * \\\\ & \\le \\pi _ { \\psi } ( a e _ P ) e ^ { 1 / 2 } h e ^ { 1 / 2 } \\pi _ { \\psi } ( a e _ P ) ^ * \\\\ & \\le \\pi _ { \\psi } ( a e _ P ) h e \\pi _ { \\psi } ( a e _ P ) ^ * = 0 ( ) \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} g ^ { i j } = g ^ { j i } , g ^ { i j } _ { , k } = c ^ { i j } _ k + c ^ { j i } _ k . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{gather*} \\operatorname { c h a r } ( \\mathfrak { g } ) = 0 \\Longleftrightarrow \\mathrm { m o d } ( \\mathfrak { g } ) = 0 \\Longleftrightarrow \\operatorname { T r } Q = 0 . \\end{gather*}"}
-{"id": "9169.png", "formula": "\\begin{align*} t ( z ) = \\sum _ { i = 1 } ^ \\infty \\alpha _ i z ^ { M _ i } \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} P _ G ( x ) = x ^ { 3 i - 1 } ( x + 1 ) ^ { 6 i + 1 } ( x + 2 i + 1 ) ( x ^ 4 - ( 8 i + 2 ) x ^ 3 - ( - 8 i ^ 2 + 4 i + 3 ) x ^ 2 - ( - 8 i ^ 3 - 2 0 i ^ 2 - 8 i ) x - 8 i ^ 4 - 1 2 i ^ 3 - 4 i ^ 2 ) \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} p _ m ^ { ( \\alpha , \\beta ) } ( x ) p _ n ^ { ( \\alpha , \\beta ) } ( x ) = \\sum _ { k = | m - n | } ^ { m + n } c ( k , n , m , \\alpha , \\beta ) p _ k ^ { ( \\alpha , \\beta ) } ( x ) , \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} X _ { \\tau } = \\left \\{ ( x _ 1 , \\dots , x _ k ) \\in X , x _ { i _ 1 } = \\cdots = x _ { i _ { a _ 1 } } , \\ldots , x _ { l _ 1 } = \\cdots = x _ { l _ { a _ s } } \\right \\} . \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} \\nabla u ( x ) = - \\sum ^ N _ { i = 1 } \\dfrac { x - T ^ { ( i ) } ( x ) } { \\vert x - T ^ { ( i ) } ( x ) \\vert ^ 3 } . \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} u _ { k + m } ( t , 0 ) = 0 \\mbox { f o r } ( t \\ge \\tau _ { k + m } \\mbox { a n d } t \\not \\in [ \\tau _ { k + 1 } - \\delta , \\tau _ { k + 1 } ] ) , \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} \\varphi _ R ( x ) = \\varphi _ R ( r ) : = R ^ 2 \\theta ( r / R ) , r = | x | . \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\frac { \\sigma ^ 2 _ { \\mathcal P _ { l } } ( S ^ { n _ l } _ { a _ { i } } h _ l ) } { \\sigma ^ 2 _ { \\mathcal P _ { l } } ( S ^ { n _ l } h _ l ) } = 1 , 1 \\leqslant i \\leqslant k _ l . \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\int d x \\ x \\varphi ( x , y , t ) = 0 \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} G _ 3 & = \\{ f _ 1 , f _ 2 , f _ 3 \\} \\\\ G _ 4 & = \\{ f _ 1 , f _ 2 , f _ 3 , f _ 4 , x _ 0 x _ 2 ^ 2 \\} \\\\ G _ 5 & = \\{ f _ 1 , f _ 2 , f _ 3 , f _ 4 , f _ 5 , x _ 0 x _ 2 x _ 3 \\} \\\\ G _ 6 & = \\{ f _ 1 , \\ldots , f _ 6 , x _ 0 x _ 3 ^ 2 + 2 x _ 0 x _ 2 x _ 4 , 2 x _ 1 x _ 3 ^ 2 + 3 x _ 0 x _ 3 x _ 4 - x _ 0 x _ 2 x _ 5 \\} . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} \\Big \\{ \\langle c , x \\rangle : x \\in \\lambda ^ { - 1 } ( Q ) \\Big \\} = [ \\delta , \\Delta ] . \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} A _ j : L ^ 1 ( \\Omega ) \\to L ^ 1 ( \\Sigma _ j ) , ( A _ j u ) ( s ) : = \\int _ \\Omega a _ j ( s , t ) \\ , u ( t ) \\ , d t , j = 0 , \\dots , N - 1 . \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} { C } _ i ^ { \\gamma + 1 } = c _ 0 ^ 2 { C } _ i ^ 2 | \\partial _ { s _ i } p | ^ 2 = c _ 0 ^ 2 G _ i ^ 2 \\qquad \\mbox { f o r } i \\in \\mathcal { I } . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} | T ^ { ( 2 ) } _ { a } ( x ) | \\leq C _ { 2 } \\begin{cases} | x | ^ { a - Q } , \\ ; x \\in \\mathbb { G } \\backslash \\{ 0 \\} , \\\\ | x | ^ { - Q } , \\ ; x \\in \\mathbb { G } \\ ; \\ ; | x | \\geq 1 , \\end{cases} \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} \\delta _ { \\tilde g ^ { ( 4 ) } } ( \\alpha ^ 2 ) & = - \\frac { 1 } { u ^ 2 } \\{ - \\nabla _ { \\partial _ t } [ \\alpha ^ 2 ( \\partial _ t ) ] + \\alpha ^ 2 ( \\nabla _ { \\partial _ t } \\partial _ t ) \\} \\\\ & = \\frac { 1 } { u ^ 2 } \\nabla _ { \\partial _ t } \\alpha = - \\frac { 1 } { u ^ 2 } \\nabla _ { \\partial _ t } ( \\frac { 1 } { u ^ 2 } \\xi ) \\\\ & = - \\frac { 1 } { u ^ 4 } \\nabla _ { \\partial _ t } \\xi = - u ^ { - 3 } d u . \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} | x | ^ 2 = \\langle x , u \\rangle ^ 2 + | \\pi ^ \\perp ( x ) | ^ 2 . \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} & \\Delta ^ { - 1 } \\left ( [ d , a ] - [ b , c ] \\right ) = L _ { 1 1 } a + L _ { 1 2 } c \\\\ & \\Delta ^ { - 1 } [ d , b ] = L _ { 1 1 } b + L _ { 1 2 } d \\\\ & \\Delta ^ { - 1 } [ a , c ] = L _ { 2 1 } a + L _ { 2 2 } c \\\\ & 0 = L _ { 2 1 } b + L _ { 2 2 } d . \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} E ( \\varphi ) - \\mathrm { d } p = 0 , \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} \\partial _ { ( 0 ) } ^ n ( f ) ( g _ 0 , \\dots , g _ n ) & = g _ 0 f ( g _ 1 , \\dots , g _ n ) , \\\\ \\partial _ { ( i ) } ^ n ( f ) ( g _ 0 , \\dots , g _ n ) & = f ( g _ 0 , \\cdots , g _ { i - 1 } g _ i , \\dots , g _ n ) ~ ~ 1 \\leq i \\leq n , \\\\ \\partial _ { ( n + 1 ) } ^ n ( f ) ( g _ 0 , \\dots , g _ n ) & = f ( g _ 0 , \\dots , g _ { n - 1 } ) . \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} P _ { ( i , j ) , \\ell } = 0 \\forall { \\ell \\neq i } . \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} r _ { t _ 1 , t _ 2 } ( D ) = & \\# \\{ ( k _ 1 , k _ 2 ) \\in ( [ t _ 1 ] - 1 ) \\times ( [ t _ 2 ] - 1 ) : ~ D \\cap ( R _ { k _ 1 , t _ 1 } \\times R _ { k _ 2 , t _ 2 } ) \\neq \\emptyset \\} \\ , . \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} \\log _ { \\psi } ( g ) = \\sum _ { j } \\ell _ j ( z _ 1 ) \\dots \\ell _ j ( z _ n ) t ^ j \\gamma _ j \\tau ^ j ( g ) \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} L _ P ( t ) : = | t P \\cap \\Z ^ n | \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{gather*} F _ 0 ( z ) = { } _ 2 F _ 1 \\biggl ( \\begin{matrix} r , \\ , 1 - r \\\\ 1 \\end{matrix} \\biggm | z \\biggr ) , \\end{gather*}"}
-{"id": "4953.png", "formula": "\\begin{align*} ( \\Lambda , \\Xi ) = \\mathcal { A } ( \\Lambda , \\Xi ) \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} v _ n ( x ) = \\tilde { V } ^ { j _ 0 } ( x ) + \\tilde { v } ^ { j _ 0 } _ n ( x ) , \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} R ^ g = R ^ \\tau + \\tau ^ 2 \\mbox { w i t h } ( \\tau ^ 2 ) _ { X , Y } Z = [ \\tau _ X , \\tau _ Y ] Z - 2 \\tau _ { \\tau _ X Y } Z \\ . \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left ( \\| f _ n \\| ^ p _ { L ^ p } - \\| f _ n - f \\| ^ p _ { L ^ p } \\right ) = \\| f \\| ^ p _ { L ^ p } . \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} n _ b ( W ^ n ( w ) ) = \\max ( \\lvert ( 0 , n ) \\rvert _ { W _ n ( w ) } , \\lvert ( n , 0 ) \\rvert _ { W _ n ( w ) } ) . \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} \\| Q \\| ^ 2 _ { \\dot { H } ^ 1 _ c } = \\frac { d } { d + 2 } \\| Q \\| ^ { \\frac { 4 } { d } + 2 } _ { L ^ { \\frac { 4 } { d } + 2 } } = \\frac { d \\omega } { 2 } \\| Q \\| ^ 2 _ { L ^ 2 } . \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} R ( x ) R ( y ) - R ( R ( x ) y + x R ( y ) ) = - x y . \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} x = - \\sum _ { n = 1 } ^ { \\infty } s _ n e _ { n } + \\sum _ { n = 1 } ^ { \\infty } t _ n n e _ { n } + \\sum _ { n = 1 } ^ { \\infty } r _ n ( n + 1 ) ( e _ { n + 1 } + e _ 1 ) , \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} S _ { \\nu } ( r ) = \\frac { ( r / 2 ) ^ { \\nu } } { \\Gamma \\left ( \\nu + \\frac 1 2 \\right ) \\Gamma ( 1 / 2 ) } \\int _ { - 1 } ^ { 1 } ( e ^ { i s r } - 1 ) ( 1 - s ^ 2 ) ^ { ( 2 \\nu - 1 ) / 2 } d s \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} u ^ { \\pm } \\in W ^ { 1 , p } ( \\Omega ) , | u | = u ^ + + u ^ - , u = u ^ + - u ^ - . \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} [ L _ 3 x ] _ { i , j } = \\cos \\bigg ( & \\frac { 2 \\pi } { N _ 1 } \\bigg ) \\bigg [ ( x _ { i + 1 , j } - x _ { i , j } ) + ( x _ { i - 1 , j } - x _ { i , j } ) \\bigg ] \\\\ & + \\cos \\bigg ( \\frac { 2 \\pi } { N _ 2 } \\bigg ) \\bigg [ ( x _ { i , j + 1 } - x _ { i , j } ) + ( x _ { i , j - 1 } - x _ { i , j } ) \\bigg ] . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } W _ t f ( n ) = f ( n ) , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} \\norm { P ( x _ k ) } _ { X } \\geq \\norm { \\lambda x _ k - P ( x _ k ) } _ { X } - \\norm { \\lambda x _ k } _ { X } = \\norm { \\lambda x _ k - P ( x _ k ) } _ { X } - 1 . \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} \\| \\Phi ^ * \\omega _ 0 - \\omega _ 0 \\| _ 2 ^ 2 = \\sum _ { j = 1 } ^ n ( \\lambda _ j ^ 2 - \\mu _ j ^ 2 ) ^ 2 + ( n - \\sum _ { j = 1 } ^ n \\mu _ j ^ 4 ) \\to 0 ^ + \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} & \\int \\limits _ { \\{ | v _ { j , n } | \\geq | v _ { j } | \\} } \\left ( 1 + | \\log | v _ { j , n } | ^ 2 | \\right ) \\left | | v _ { j , n } | ^ 2 - | v _ { j } | ^ 2 \\right | d x \\\\ & \\leq \\left ( \\int \\limits _ { \\{ | v _ { j , n } | \\geq | v _ { j } | \\} } \\left ( 1 + | \\log | v _ { j , n } | ^ 2 | \\right ) ^ 2 ( | v _ { j , n } | + | v _ { j } | ) ^ 2 d x \\right ) ^ { \\tfrac 1 { 2 } } \\left ( \\int \\limits _ { \\mathbb { R } _ + } ( | v _ { j , n } | - | v _ { j } | ) ^ 2 d x \\right ) ^ { \\tfrac 1 { 2 } } . \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{gather*} i ^ * \\operatorname { c h a r } ( A ) = z ^ * \\operatorname { c h a r } \\big ( p ^ ! i ^ ! ( A ) \\big ) = ( p z ) ^ * \\operatorname { c h a r } \\big ( i ^ ! ( A ) \\big ) = \\operatorname { c h a r } \\big ( i ^ ! ( A ) \\big ) , \\end{gather*}"}
-{"id": "9048.png", "formula": "\\begin{align*} p ' ( w ) = \\begin{cases} p ( w ) + \\epsilon ( 0 , - 1 ) & w \\sim y , \\\\ p ( w ) & \\end{cases} \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} \\kappa ( \\epsilon ) = \\ , & \\mu \\sum _ { j = 0 } ^ { \\infty } e ^ { - \\alpha j } \\tilde { P } ^ j r + \\epsilon \\mu \\sum _ { j = 1 } ^ { \\infty } e ^ { - \\alpha j } \\sum _ { k = 1 } ^ { j } a _ { k 1 } \\tilde { P } ^ { k - 1 } \\tilde { P } ^ { ( 1 ) } \\tilde { P } ^ { j - k } r + O ( \\epsilon ^ 2 ( \\log ( 1 / \\epsilon ) ) ^ w ) \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} u _ { \\lambda } ( x ) = \\lambda ^ { - \\frac { n - 1 } { n } } u \\left ( \\left ( \\frac { | x | } { R } \\right ) ^ { \\lambda - 1 } x \\right ) , \\lambda > 0 . \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} \\Delta J ( \\varphi ) = \\int E _ \\lambda \\left ( x ^ \\alpha , \\varphi ^ \\beta , \\frac { \\partial \\varphi ^ \\gamma } { \\partial x ^ \\kappa } \\ , , \\frac { \\partial ^ 2 \\varphi ^ \\sigma } { \\partial x ^ \\mu \\partial x ^ \\nu } \\right ) \\Delta \\varphi ^ \\lambda \\ , \\rho ( x ) \\ , \\ , d x \\ , . \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} \\ln ( ( 1 - \\beta / \\eta ) ^ D ) = - ( \\beta / 2 ) \\eta - \\beta - \\beta ^ 2 / 4 - K ~ , \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} \\sigma _ { i , i } = 1 - \\frac { \\rho _ { i j } - \\rho _ { i n } \\rho _ { j n } } { 1 - \\rho _ { j n } ^ 2 } \\rho _ { i j } - \\frac { \\rho _ { i n } - \\rho _ { i j } \\rho _ { j n } } { 1 - \\rho _ { j n } ^ 2 } \\rho _ { i n } . \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} ( \\d T ) _ x ( \\nabla f ( T x ) ) = \\nabla f ( T x ) , \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} \\dot { \\zeta } & \\leq - H _ 4 \\zeta , H _ 4 : = \\begin{bmatrix} \\bar { c } _ 3 & 0 \\\\ - \\frac { \\eta _ 3 } { 2 } & \\frac { \\eta _ 2 } { 2 } \\end{bmatrix} . \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} \\rho ^ 2 = \\hat { L } ^ { - p } ( L C _ F ^ 2 ) ^ { \\frac { - p } { \\epsilon } } \\bigg ( \\frac { C _ p } { p } \\bigg ) ^ { 1 + \\frac { 2 } { \\epsilon } } . \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} \\max _ { r / 2 \\leq s \\leq r } ( r - s ) ^ p \\max _ { B _ s } e _ { L } ( \\Q _ { L } ) = ( r - r _ 1 ^ L ) ^ p \\max _ { B _ { r _ 1 ^ L } } e _ L ( \\Q _ { L } ) \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} \\kappa ( \\epsilon ) = \\sum _ { j = 0 } ^ { \\infty } e ^ { - \\alpha j } \\mu \\left ( \\prod _ { k = 1 } ^ { j } P _ k ( \\epsilon ) \\right ) r , \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} \\nabla p : = \\frac { 1 } { c _ 0 } \\frac { w } { | w | } , \\hat \\theta : = \\left ( \\chi ^ + - \\chi ^ - \\right ) \\frac { w } { | w | } , \\alpha : = c _ 0 | w | \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} W _ t ( g ) = S ' ( f ( g t , t ) , g t - t ) W ' ( g | _ { U ^ \\perp } ) , g \\in G _ t . \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} B = \\limsup _ { q \\rightarrow \\infty } \\sup _ { f \\in L ^ { p } _ { Q / p } ( B ( x _ { 0 } , r ) ) \\backslash \\{ 0 \\} } \\frac { \\left \\| \\frac { f } { | x | ^ { \\frac { \\beta } { q } } } \\right \\| _ { L ^ { q } ( B ( x _ { 0 } , r ) ) } } { q ^ { 1 - 1 / p } \\| f \\| _ { L ^ { p } _ { Q / p } ( B ( x _ { 0 } , r ) ) } } . \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} ( t , y , x ) \\cdot ( t ' , y ' , x ' ) = ( t t ' , y + y ' , x + x ' ) . \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} J = J _ v + J _ r + J _ m , \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} 2 q \\partial \\theta / \\partial q = x ( \\partial ^ 2 / \\partial x ^ 2 ) ( x \\theta ) = x ^ 2 \\partial ^ 2 \\theta / \\partial x ^ 2 + 2 x \\partial \\theta / \\partial x \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} \\dot { \\psi } _ { i , j } = \\sum _ { i ' , j ' } \\sin ( \\bar { \\theta } _ { i ' , j ' } + \\psi _ { i ' , j ' } - \\bar { \\theta } _ { i , j } - \\psi _ { i , j } ) \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} \\frac { \\phi _ 1 ( z _ 0 ) - m U ( z _ 0 ) } { [ ( \\phi _ 1 ( z _ 0 ) - m U ( z _ 0 ) ) ^ 2 + 4 \\lambda m _ 1 z _ 0 ^ 2 ] ^ { 1 / 2 } } = \\frac { m U ( z _ 0 ) - \\phi _ 2 ( z _ 0 ) } { [ ( \\phi _ 2 ( z _ 0 ) - m U ( z _ 0 ) ) ^ 2 + 4 \\lambda m _ 2 z _ 0 ^ 2 ] ^ { 1 / 2 } } , \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} M _ { \\mu , \\nu } ( y ) = \\frac { \\Gamma ( 1 + 2 \\nu ) } { \\Gamma ( \\nu - \\mu + \\frac { 1 } { 2 } ) } e ^ { \\pi i \\mu } \\mathcal { M } ^ + _ { \\mu , \\nu } ( y ) + \\frac { \\Gamma ( 1 + 2 \\nu ) } { \\Gamma ( \\nu + \\mu + \\frac { 1 } { 2 } ) } e ^ { - \\pi i ( \\nu - \\mu + \\frac { 1 } { 2 } ) } W _ { \\mu , \\nu } ( y ) , 2 \\nu \\not \\in \\mathbb { Z } _ { < 0 } , \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} Q \\overline { \\nabla } ^ { ' } _ { \\phi X } \\phi Y = \\{ ( 1 - \\beta ) g ( \\phi X , Y ) - \\alpha g ( X , Y ) \\} Q \\xi + B h ( \\phi X , Y ) . \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} X _ G = X _ { H } \\oplus Y . \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} G ( z , z ' ) \\lesssim \\begin{cases} r '^ { - n } r ^ { - s } ( r / r ' ) ^ { s - \\frac n 2 + \\nu ' _ 0 } , r ' > 2 r ; \\\\ r ^ { - 1 } d ( z , z ' ) ^ { - ( n - 1 + s ) } , r \\sim r ' ; \\\\ r ^ { - n - s } ( r ' / r ) ^ { 1 - \\frac n 2 + \\nu ' _ 0 } , r ' < \\frac { r } { 2 } ; \\end{cases} \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} a = ( a _ k ) _ { k = 0 } ^ { \\infty } \\longmapsto \\sum _ { k = 0 } ^ { \\infty } a _ k z ^ k \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} Q _ n ( x ) = ( 1 - x ) ^ n = b _ 0 + b _ 1 x + \\dots + b _ n x ^ n ; \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} \\hat u ( T _ { o p t } , x ) = 0 \\mbox { f o r } x \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} w _ { i , j } & = \\int ^ { T } _ { 0 } \\delta ^ { - 1 } _ j \\chi _ j ( t ) f _ { i } ( t ) \\ , d t , \\\\ f _ i ( t ) & = \\sum _ { \\ell = 1 } ^ s w _ { i , \\ell } \\ , \\chi _ \\ell ( t ) . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\alpha + g ( y ) \\alpha + f ( y ) = 0 \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} j k ( E ) = \\theta ( n _ 0 + j q , E ) - \\theta ( n _ 0 , E ) + O \\left ( \\frac { 1 } { n _ 0 - v } \\right ) \\mod \\Z . \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} & \\Delta u = \\frac { 1 } { 2 } \\frac { \\partial } { \\partial u } \\left ( { { f } ^ { 2 } } + { { g } ^ { 2 } } \\right ) \\\\ & \\Delta v = \\frac { 1 } { 2 } \\frac { \\partial } { \\partial v } \\left ( { { f } ^ { 2 } } + { { g } ^ { 2 } } \\right ) \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} \\frac { F ( t _ { n + 1 } ) } { F ( t _ n ) } & = \\Big ( \\frac { t _ { n + 1 } } { t _ n } \\Big ) ^ { \\beta ' } \\cdot \\frac { A ( t _ { n + 1 } ) } { A ( t _ n ) } \\cdot \\exp \\Big ( \\int _ { t _ n } ^ { t _ { n + 1 } } \\frac { \\mathcal E ( y ) } y d y \\Big ) . \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathcal { I } _ R } e ^ { z w } \\psi _ { \\varepsilon } ( z ) u ( z ) d z \\right | & < 2 \\pi R \\sup _ { z \\in \\mathcal { I } _ R } | e ^ { z w } u ( z ) | \\\\ & = 2 \\pi R \\sup _ { z \\in \\mathcal { I } _ R } | e ^ { z ( w - \\varepsilon ' \\xi _ 0 ) } | \\sup _ { z \\in \\mathcal { I } _ R } | e ^ { z ( \\varepsilon ' \\xi _ 0 ) } u ( z ) | . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} Z _ 0 ^ { p , q } ( A ) : = A ^ { p , q } B _ 0 ^ { p , q } ( A ) : = 0 . \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} \\begin{cases} [ L _ { Y _ 1 } \\alpha ^ 2 ] ^ T = 0 \\\\ [ L _ { Y _ 1 } \\alpha ^ 2 ] ( \\partial t , \\partial t ) = 0 \\\\ \\{ [ L _ { Y _ 1 } \\alpha ^ 2 ] ( \\partial t ) \\} ^ T = d \\theta ( Y _ 1 ^ T ) - d ( \\frac { Y _ 1 ^ { \\perp } } { u } ) , \\end{cases} \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} X _ 1 = \\frac { \\partial } { \\partial { x _ 1 } } X _ 2 = \\frac { \\partial } { \\partial { x _ 2 } } + \\sum _ { j \\geq 3 } p _ { j } \\frac { \\partial } { \\partial { x _ j } } \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} u ( \\P _ { W _ t } ) = u ( \\P _ { W _ 0 } ) + \\int _ 0 ^ t \\langle \\partial _ { \\mu } u ( \\P _ { W _ s } ) ( . ) \\cdot \\bar { f } ( . ) , \\ , \\P _ { W _ s } \\rangle \\ , d s , \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} ( H _ { n - 1 } ( \\Sigma ) , \\mathbb { Z } _ p ) & \\cong ( \\mathbb { Z } _ { q _ { n - 1 , 1 } } \\oplus \\ldots \\oplus \\mathbb { Z } _ { q _ { n - 1 , l _ { n - 1 } } } , \\mathbb { Z } _ p ) \\\\ & \\cong ( \\mathbb { Z } _ { q _ { n - 1 , 1 } } , \\mathbb { Z } _ p ) \\oplus \\ldots \\oplus ( \\mathbb { Z } _ { q _ { n - 1 , l _ { n - 1 } } } , \\mathbb { Z } _ p ) = 0 . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} \\| u ( t _ n ) - U _ h ^ { n } \\| + \\gamma ^ { \\frac 1 2 } \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\sigma ( t _ { k - \\theta } ) - \\Sigma _ h ^ { k - \\theta } \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } + \\Big { ( } \\Delta t \\sum _ { k = 1 } ^ n \\| \\nabla ( u ( t _ { k - \\theta } ) - U _ { h } ^ { k - \\theta } ) \\| ^ 2 \\Big { ) } ^ { \\frac 1 2 } \\leq C [ \\Delta t ^ 2 + h ^ { m + 1 } + H ^ { 2 m + 2 } ] . \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} \\Psi ( a ^ * t b ) = \\Psi ( a ) ^ * \\Psi ( t ) \\Psi ( b ) \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} K ( z , Z ) = \\overline { R _ H ( e _ z ) ( Z ) } , \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} G = n \\ell A _ 0 + n ( a A _ 1 + b A _ 2 ) + n ( A _ 3 - A _ 4 ) . \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} { w _ 1 } N _ 1 ( \\delta ) + { w _ 2 } N _ 2 ( \\delta ) + \\dotsb + { w _ m } N _ m ( \\delta ) = \\ell \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} [ \\mathfrak { M } _ { } ^ { - 1 } \\tilde { f } ] ( \\mu _ p , y ) = \\frac { \\log ( p ) } { 2 \\pi } \\int _ { - \\frac { \\pi } { \\log ( p ) } } ^ { \\frac { \\pi } { \\log ( p ) } } \\tilde { f } ( \\mu _ p , i t ) \\abs { y } _ { p } ^ { - i t } d t . \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ { t } f + v \\cdot \\nabla _ { x } f + \\mathcal { L } ^ { \\epsilon } f = \\Gamma ^ { \\epsilon } ( f , f ) , ~ ~ t > 0 , x \\in \\mathbb { T } ^ { 3 } , v \\in \\R ^ 3 ; \\\\ & f | _ { t = 0 } = f _ { 0 } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} \\pi _ \\lambda ( g ) f ( z ) = j _ \\lambda ( g ^ { - 1 } , z ) f ( g ^ { - 1 } \\cdot z ) . \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} [ \\Tilde { \\boldsymbol { \\Lambda } } _ { \\rm o } ] _ { i , i } = \\left ( 1 - [ \\Tilde { \\boldsymbol { \\Lambda } } ] _ { i , i } ^ { - 2 } \\right ) ^ + , \\ , \\forall \\ , i = 1 , 2 , \\ldots r . \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} d v + ( v . \\nabla ) v = \\sqrt { 2 \\nu } \\nabla v . d W _ t + \\nu \\Delta v - \\nabla p , \\div ~ v ( t , \\cdot ) = 0 , \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} D _ k = N \\times ( K - k + 1 ) . \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { N - 1 } f ( n _ 0 + j q + p ) \\frac { \\cos 4 \\theta ( n _ 0 + j q + p , E ) } { n _ 0 + j q - p } \\leq N \\left ( \\frac { \\varepsilon } { n _ 0 - v } + \\frac { O ( 1 ) } { ( n _ 0 - v ) ^ 2 } \\right ) , \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} \\mathbf { B } ^ i = \\sum _ { j = 1 } ^ { m } \\mathbf { K } _ j ^ i \\star b _ j ~ ~ ~ B _ { 2 R } , ~ ~ b _ j \\in L ^ 1 ( [ 0 , T ] , B V ( \\mathbb { R } ^ d ) ) , \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} G _ \\nu ^ f ( \\rho ) = \\int _ { \\R ^ n } \\rho ( x ) \\left \\langle \\nabla \\log \\frac { \\rho ( x ) } { \\nu ( x ) } , \\big ( \\nabla ^ 2 f ( x ) \\big ) \\ , \\nabla \\log \\frac { \\rho ( x ) } { \\nu ( x ) } \\right \\rangle d x \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\mathcal { G } _ { D , m } ( z , s ) : = - \\Gamma \\biggl ( \\frac { s } { 2 } + 1 \\biggr ) \\sum _ { n | m } \\biggl ( \\frac { D } { n } \\biggr ) \\bigg | \\frac { m } { n } \\bigg | \\sqrt { D } P _ { \\frac { 3 } { 2 } , - \\frac { m ^ 2 D } { n ^ 2 } } \\biggl ( z , \\frac { s } { 2 } + \\frac { 1 } { 4 } \\biggr ) . \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} \\begin{cases} a ^ { \\varepsilon , \\nu } ~ \\doteq ~ - \\eta ' ( u ^ { \\varepsilon , \\nu } ) f _ { x } ^ { \\nu } ( t , x , u ^ { \\varepsilon , \\nu } ) + q _ { x } ^ { \\nu } ( t , x , u ^ { \\varepsilon , \\nu } ) , \\\\ b ^ { \\varepsilon , \\nu } ~ \\doteq ~ - \\varepsilon \\eta '' ( u ^ { \\varepsilon , \\nu } ) \\bigl ( u ^ { \\varepsilon , \\nu } _ { x } \\bigr ) ^ 2 , \\\\ c ^ { \\varepsilon , \\nu } ~ \\doteq ~ \\varepsilon \\eta ( u ^ { \\varepsilon , \\nu } ) _ { x x } . \\end{cases} \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} X _ P = ( ( \\sigma _ { j _ b } P _ { y _ b } ) _ { 1 \\leq b \\leq n } , - ( \\sigma _ { j _ b } P _ { x _ b } ) _ { 1 \\leq b \\leq n } , - \\mathbf { i } ( \\sigma _ j P _ { \\bar { z } _ j } ) _ { j \\in \\mathbb { Z } _ * } , \\mathbf { i } ( \\sigma _ j P _ { z _ j } ) _ { j \\in \\mathbb { Z } _ * } ) ^ T \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} \\max _ { 1 \\leq i \\leq m + 1 } \\{ \\deg a _ i \\} \\leq \\sum _ { i = 1 } ^ { m + 1 } \\tilde { n } _ { \\kappa } ^ { [ m - 1 ] } ( a _ i ) - \\frac { 1 } { 2 } m ( m - 1 ) , \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} - \\langle \\mu , \\varpi _ 1 \\rangle + \\sum _ { j = 1 } ^ { i - 1 } \\left \\langle \\frac { \\lambda + \\mu } { 2 } + \\rho , \\alpha _ j \\right \\rangle = f ( i ) - \\frac { n ( n + 1 ) } { 2 } \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} y = \\sum _ { i = 0 } ^ { p - 1 } e _ i ^ p a ^ i \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} \\sum _ { x \\in T _ { N } ^ { - 1 } ( x ' ) } x = 0 \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} t = \\frac { n _ i } { \\omega \\cdot \\beta _ i } \\le \\frac { m _ i } { \\omega \\cdot \\beta _ i } \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\tau = \\rho _ 0 + \\rho _ 1 P _ y . \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} \\psi ( t , y ) \\ll t e ^ { - \\log t / 2 \\log y } = t ^ { 1 - 1 / 2 \\log y } , \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} \\left \\| \\begin{bmatrix} C _ 0 & C _ k \\\\ 0 & C _ 0 \\end{bmatrix} \\right \\| ^ 2 & \\geq \\left \\| \\begin{bmatrix} C _ 0 & C _ k \\end{bmatrix} \\right \\| ^ 2 \\\\ & = \\| C _ 0 C _ 0 ^ * + C _ k C _ k ^ * \\| \\\\ & \\geq \\| C _ 0 \\| ^ 2 + \\delta _ k > \\| C _ 0 \\| ^ 2 \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow r _ { + } } y ( x ; \\tau ) = 0 \\ \\ \\mbox { a n d } \\ \\ \\lim _ { x \\rightarrow s _ { - } } y ( x ; \\tau ) = 0 \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} K ( x , y ) = c _ { 0 } c _ { 0 } ' \\int _ { 0 } ^ { \\infty } ( n + x + z ) ^ { - 1 - 2 \\beta } ( n + y + z ) ^ { - 1 + 2 \\beta } d z . \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} D _ m = - i \\alpha \\cdot \\nabla + m \\beta = - i \\sum _ { k = 1 } ^ { 2 } \\alpha _ k \\partial _ { k } + m \\beta \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\| \\gamma ^ { ( d ) } _ { n } ( A ) \\| & = \\max _ { 1 \\leq k \\leq r } \\| \\Gamma ' _ k \\| < \\max _ { 1 \\leq k \\leq r } \\| \\Gamma _ k \\| \\\\ & \\leq \\max _ { 1 \\leq p \\leq r } \\| \\gamma _ { 2 p } ^ { ( d ) } ( A ) \\| \\leq \\| A \\| . \\end{align*}"}
-{"id": "9927.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( x ) & = \\left [ \\left ( 0 , \\ , \\dfrac 1 { 2 \\pi \\sqrt { - 1 } } \\frac { \\varphi _ { \\Omega _ + } } z \\right ) \\right ] \\otimes [ \\nu _ { \\Omega _ + } ] - \\left [ \\left ( 0 , \\ , \\dfrac 1 { 2 \\pi \\sqrt { - 1 } } \\frac { \\varphi _ { \\Omega _ - } } z \\right ) \\right ] \\otimes [ \\nu _ { \\Omega _ - } ] \\\\ & \\\\ & = \\dfrac { - 1 } { 2 \\pi \\sqrt { - 1 } } \\left ( \\dfrac { 1 } { x + \\sqrt { - 1 } \\ , 0 } - \\dfrac { 1 } { x - \\sqrt { - 1 } \\ , 0 } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} \\beta = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) , \\alpha _ 1 = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ) , \\alpha _ 2 = \\left ( \\begin{array} { c c } 0 & - i \\\\ i & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\rho ( 0 + ) = \\frac { \\max _ { 1 \\le i \\le r } ( m _ i - 1 ) } { m - 1 } . \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\P ( Z _ m = W _ n ) = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\sum _ { x \\in \\Z ^ d } p ^ { ( m ) } ( o , \\ , x ) \\ , p ^ { ( n ) } ( o , \\ , x ) , \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} \\mathcal { C } _ { n e t } = 2 \\pi \\lambda _ { b } \\int _ { 0 } ^ { \\infty } x _ { 1 1 } \\mathrm { e } ^ { - \\pi \\lambda _ { b } \\max ^ { 2 } \\left ( x _ { 1 1 } , x _ { 1 1 } ^ { \\mu / 2 } \\left ( \\frac { P } { N } \\right ) ^ { \\frac { 2 - \\mu } { 2 \\alpha } } \\left ( \\frac { 1 } { M } \\right ) ^ { \\frac { 1 } { 2 \\alpha } } \\right ) } \\mathrm { e } ^ { \\left ( - \\frac { s N } { P } \\right ) } \\mathbb { \\mathcal { L _ { I } } } \\left ( s \\right ) \\mathrm { d } x _ { 1 1 } , \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} & & T f ( z ) & = ( e _ z \\circ T ) f = \\langle f , R _ L ( e _ z \\circ T ) \\rangle _ L = \\langle T f , T R _ L ( e _ z \\circ T ) \\rangle _ H , \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } u ^ { \\theta } _ { t } + H _ { i } ( t , x , u ^ { \\theta } _ { x _ { i } } ) - D T _ { \\theta } = 0 & \\ , \\ , I _ { i } \\times ( T _ { \\theta } , T ) \\\\ \\sum _ { i = 1 } ^ { K } u ^ { \\theta } _ { x _ { i } } = 0 & \\ , \\ , \\{ 0 \\} \\times ( T _ { \\theta } , T ) \\\\ u ^ { \\theta } = u _ { 0 } + 2 \\omega ( T _ { \\theta } ) & \\ , \\ , \\mathcal { I } \\times \\{ T _ { \\theta } \\} \\end{array} \\right . \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} n ^ 2 A = \\theta H _ 1 ^ \\top H _ 1 + \\tau ( H _ 2 ' ) ^ \\top H _ 2 ' + k J _ n . \\end{align*}"}
-{"id": "9994.png", "formula": "\\begin{align*} \\frac 1 { \\alpha ( \\tau ) } \\frac d { d \\tau } ( e ^ { - q \\tau } \\bar U _ \\tau ) = ( 1 - \\bar P ( q ) ) e ^ { - q \\tau } . \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} \\begin{aligned} & u '' + e ^ { \\ , u } = 0 , \\mbox { i n } \\ \\R , \\\\ & u ' ( 0 ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} v _ i ( \\tau , \\xi ) = & \\int _ 0 ^ \\tau \\sum _ { j = 1 } ^ n \\Big ( C _ { i j } ( x _ i ( s , \\tau , \\xi ) ) v _ j ( s , x _ i ( s , \\tau , \\xi ) ) + D _ { i j } ( x _ i ( s , \\tau , \\xi ) ) v _ j ( s , 0 ) \\\\ [ 6 p t ] & + f _ i \\big ( s , x _ i ( s , \\tau , \\xi ) \\big ) \\Big ) \\ , d s + v _ { 0 , i } \\big ( x _ i ( 0 , \\tau , \\xi ) \\big ) , \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} \\frac { d } { d t } x _ i ( t , s , \\xi ) = \\lambda _ i \\big ( x _ i ( t , s , \\xi ) \\big ) \\mbox { a n d } x _ i ( s , s , \\xi ) = \\xi \\mbox { i f } 1 \\le i \\le k , \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} - \\Delta _ { g } \\bigl ( h _ { \\lambda } ( r ) \\bigr ) = \\left ( \\frac { ( n - 1 ) ^ 2 } { 4 } { K _ 0 } + \\lambda \\right ) h _ { \\lambda } ( r ) . \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} \\left | \\Re \\left ( \\left ( \\frac { \\rho } { \\rho - \\tau } \\right ) ^ n \\right ) \\right | = \\left | \\Re \\left ( \\left ( x + y i \\right ) ^ n \\right ) \\right | \\le \\left ( 1 + \\frac { \\tau ^ 2 } { \\Im ( \\rho ) ^ 2 } \\right ) ^ \\frac { n } { 2 } . \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} \\big | \\partial _ { \\dot { \\lambda } } \\Psi [ \\lambda , \\xi , \\dot { \\lambda } , \\dot { \\xi } , \\phi ] [ \\dot { \\bar { \\lambda } } ] ( x , t ) \\big | \\lesssim \\frac { t _ 0 ^ { - \\varepsilon } } { R ^ { a - 2 s } } \\| \\dot { \\bar { \\lambda } } ( t ) \\| _ { n - 4 s + 1 + \\sigma } \\left ( \\sum _ { j = 1 } ^ k \\frac { \\mu _ 0 ^ { - \\frac { n - 6 s } { 2 } - 1 + \\sigma } ( t ) } { 1 + | y _ j | ^ { a - 2 s } } \\right ) , \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} \\Delta _ p & : = \\big | ( S ^ { n _ l } _ { a _ { i } } h ( p ) - S ^ { n _ l } h _ l ( q _ i ) ) - ( \\mathbb E _ { \\mathcal P _ { l } } ( S _ { a _ i } ^ { n _ l } h _ l ) - \\mathbb E _ { \\mathcal P _ { l } } ( S ^ { n _ l } h _ l ) ) \\big | \\\\ & \\leqslant \\omega ^ { n _ l } ( h _ l , 2 \\epsilon _ l , q _ i ) + \\int \\omega ^ { n _ l } ( h _ l , 2 \\epsilon _ l , p ) \\ , d \\nu _ { \\mathcal P _ { l } } ( p ) . \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} h _ F ( x ) : = | h _ M ( x ) | , h _ { - 1 , m } ( x ) : = \\sqrt 2 h _ F ( x - m ) , \\ ; m \\in \\mathbb Z , \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\textup { P D } _ \\textup { t } ( 1 0 8 n + 5 4 ) q ^ n & \\equiv \\left ( 2 + 3 \\dfrac { f _ 6 ^ 2 } { f _ { 1 2 } } \\right ) \\cdot f _ 4 ^ 3 + q f _ { 1 2 } ^ 3 \\cdot \\dfrac { f _ 2 ^ 2 } { f _ 4 } ~ ( \\textup { m o d } ~ 4 ) . \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} \\left \\{ ( t _ 1 , \\ldots , t _ { n - 1 } , g , \\bar { g } ) ~ \\begin{tabular} { | l } $ g \\bar { g } = g _ { t _ i } , h \\bar { h } = h _ i $ \\\\ $ h \\in c ( g , X ) , \\bar { h } \\in c ( \\bar { g } , X ) , g _ { t _ i } \\in c ' ( h _ i , Y ) $ \\\\ $ [ t _ 1 , \\ldots , t _ { i - 1 } , g , \\bar { g } , t _ { i + 1 } , \\ldots , t _ { n - 1 } ] \\neq \\emptyset $ \\end{tabular} \\right \\} . \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} \\tilde { \\phi } ( x , t ) = \\mu _ j ^ { - \\frac { n - 2 s } { 2 } } \\phi \\left ( \\frac { x - \\xi _ j } { \\mu _ j } , t \\right ) . \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} | X | \\geq \\sum _ { j = 1 } ^ 3 \\sum _ { i = 1 } ^ q ( a ^ j _ i + f ^ j _ i ) ^ 2 . \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} x _ { i } = \\sum _ { k = 1 } ^ { m } v _ { i , k } \\ , \\phi _ { k } , \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} \\begin{cases} R i c _ { g _ S } = \\frac { 1 } { u } D ^ 2 _ { g _ S } u + 2 u ^ { - 4 } ( \\omega ^ 2 - | \\omega | ^ 2 _ { g _ S } \\cdot g _ S ) \\\\ \\Delta _ { g _ S } u = 2 u ^ { - 3 } | \\omega | ^ 2 _ { g _ S } \\\\ \\delta _ { g _ S } \\omega + 3 u ^ { - 1 } \\langle d u , \\omega \\rangle _ { g _ S } = 0 \\\\ d \\omega = 0 \\end{cases} , \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} q _ i 1 + \\sum _ { j = 1 } ^ n q _ j , \\ , . \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} \\frac { 1 } { \\mathcal { T } _ l } \\int _ 0 ^ { \\mathcal { T } _ l } k ( \\tilde y ^ l ( t ) , \\tilde u ^ l ( t ) ) d t \\geq v _ { p e r } ( y _ 0 ) , \\ \\ l = 1 , 2 , . . . \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{gather*} T _ { t _ { - \\alpha _ 2 } } \\big ( E _ { 3 , 1 } ( s + 1 ) \\big ) = ( - 1 ) ^ { N + 1 } E _ { 3 , 1 } ( s ) , \\\\ T _ { t _ { - \\alpha _ 2 } } \\big ( E _ { k + 1 , 1 } ( s + 1 ) \\big ) = E _ { k + 1 , 1 } ( s + 1 ) . \\end{gather*}"}
-{"id": "4192.png", "formula": "\\begin{align*} \\mu ( x ) = \\mu _ \\lambda ( x ) : = \\lambda ^ { - | x | } \\{ d _ x + ( \\lambda - 1 ) d _ x ^ { - } \\} = \\lambda ^ { - | x | } \\{ d _ x ^ + + d _ x ^ 0 + \\lambda d _ x ^ - \\} , x \\in V ( G ) . \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} f ( a ) = \\sup _ { \\theta \\in \\mathbb R } \\ ; \\ ! \\bigl ( a \\ ; \\ ! \\theta - \\log Z _ 1 ( \\theta ) \\bigr ) = a \\ ; \\ ! f ' ( a ) - \\log Z _ 1 ( f ' ( a ) ) , \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} \\frac { \\lambda ^ 2 } { \\beta } \\int _ \\Omega V ( x ) e ^ { \\ , u } \\ , d x = \\frac { \\lambda ^ 2 } { \\beta } \\int _ { | t | < \\delta _ \\lambda } V ( x ) e ^ { \\ , u } \\ , d x + o ( 1 ) , \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} \\mathcal { B } _ v ( \\lambda ) \\otimes \\mathcal { B } _ w ( \\mu ) & = \\bigsqcup _ { \\pi \\in \\mathcal { B } _ w ( \\mu ) ^ \\lambda } \\left ( \\bigcup _ { a _ 1 , \\ldots , a _ l \\geq 0 } \\tilde { f } _ { i _ 1 } ^ { a _ 1 } \\cdots \\tilde { f } _ { i _ l } ^ { a _ l } ( C ( \\pi , e ) ) \\setminus \\{ 0 \\} \\right ) \\\\ & \\subsetneq \\bigsqcup _ { \\pi \\in \\mathcal { B } _ w ( \\mu ) ^ \\lambda } C ( \\pi , v ) \\\\ & = \\mathcal { B } _ v ( \\lambda ) \\otimes \\mathcal { B } _ w ( \\mu ) , \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} B ( v , r ) : = \\{ v ' \\ | \\ \\rho ( v , v ' ) \\leq r \\} . \\end{align*}"}