diff --git "a/process_3/tokenized_finally.jsonl" "b/process_3/tokenized_finally.jsonl"
new file mode 100644--- /dev/null
+++ "b/process_3/tokenized_finally.jsonl"
@@ -0,0 +1,9238 @@
+{"id": "4712.png", "formula": "\\begin{align*} Z ( T ) \\subseteq \\oplus _ { i = 0 } ^ d Z ( E _ i ^ * T E _ i ^ * ) . \\end{align*}"}
+{"id": "3440.png", "formula": "\\begin{align*} \\nu _ { t } ( [ a ] ) = \\exp \\left ( - a t - P _ G ( t \\phi ) \\right ) \\nu _ { t } ( [ a - 1 ] ) , \\end{align*}"}
+{"id": "3455.png", "formula": "\\begin{align*} q = \\begin{cases} \\ell / ( 2 \\ell - 1 ) & \\mbox { i n c a s e \\rm ( \\texttt { a } ) } , \\\\ 2 / ( 4 - \\beta ) & \\mbox { i n c a s e \\rm ( \\texttt { b } ) } . \\end{cases} \\end{align*}"}
+{"id": "89.png", "formula": "\\begin{align*} \\mathfrak { g } _ { \\mathbf { k } } = \\sum _ { \\iota = 0 } ^ { s - 1 } g ^ { ( \\iota ) } _ { \\mathbf { k } } \\end{align*}"}
+{"id": "4173.png", "formula": "\\begin{align*} \\int _ M | \\nabla ^ i S | ^ 2 d V _ g \\leq C ( \\int _ M | \\nabla ^ n S | ^ 2 d V _ g ) ^ { \\frac { i } { n } } ( \\int _ M | S | ^ 2 d V _ g ) ^ { 1 - \\frac { i } { n } } . \\end{align*}"}
+{"id": "4251.png", "formula": "\\begin{align*} ( I ) & \\geq ( 1 - 2 \\delta ) \\| \\nabla f \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } \\left ( 2 V ( x ) - \\frac { \\Omega ^ 2 } { 2 \\delta } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) \\right ) | f ( x ) | ^ 2 d x \\\\ & = ( 1 - 2 \\delta ) \\| \\nabla f \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } \\Big [ \\Big ( \\gamma ^ 2 - \\frac { \\Omega ^ 2 } { 2 \\delta } \\Big ) x _ 1 ^ 2 + \\Big ( \\gamma ^ 2 - \\frac { \\Omega ^ 2 } { 2 \\delta } \\Big ) x _ 2 ^ 2 + \\sum _ { j = 3 } ^ N \\gamma _ j ^ 2 x _ j ^ 2 \\Big ] | f ( x ) | ^ 2 d x . \\end{align*}"}
+{"id": "3269.png", "formula": "\\begin{align*} f _ { \\lambda } \\left ( x \\right ) \\equiv f \\left ( x \\right ) - \\lambda g \\left ( x \\right ) = y , f o r \\ y \\in Y , \\end{align*}"}
+{"id": "2611.png", "formula": "\\begin{align*} k \\geq r - 2 . \\end{align*}"}
+{"id": "6532.png", "formula": "\\begin{align*} G _ 2 ( Q ) & = \\int \\phi _ m ( x ) p _ 0 ^ m ( x _ 1 ) \\int \\phi _ m ( x ) \\ , p _ 0 ^ m ( x _ 2 ) Q ( x _ 1 , x _ 2 ) d x _ 2 d x _ 1 \\\\ & \\leq \\bar { G } \\int \\phi _ m ( x ) p _ 0 ^ m ( x _ 1 ) \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 ( x ^ j _ 2 ) p _ 0 ^ m ( x _ 2 ) Q ( x _ 1 , x _ 2 ) d x _ 2 d x _ 1 \\\\ & \\leq \\bar { G } ^ 2 \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 ( x ^ j _ 2 ) p _ 0 ^ m ( x _ 2 ) \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 ( x ^ j _ 1 ) p _ 0 ^ m ( x _ 1 ) Q ( x _ 1 , x _ 2 ) d x _ 1 d x _ 2 . \\end{align*}"}
+{"id": "5054.png", "formula": "\\begin{align*} F = \\sum _ { \\vec b = ( b _ 1 , \\ldots , b _ t ) \\in S } \\frac { P _ { \\vec b } } { Q _ 1 ^ { b _ 1 } \\cdots Q _ t ^ { b _ t } } , \\end{align*}"}
+{"id": "1442.png", "formula": "\\begin{align*} \\mathbb { P } ( | \\mathcal { C } ( V _ n ) | > k ) = \\mathbb { P } ( | \\mathcal { A } _ t | > 0 \\forall t \\leq k ) & = \\mathbb { P } ( 1 + \\sum _ { i = 1 } ^ { t } ( \\eta _ i - 1 ) > 0 \\forall t \\leq k ) \\\\ & \\leq \\mathbb { P } ( 1 + \\sum _ { i = 1 } ^ { t } ( X _ i - 1 ) > 0 \\forall t \\leq k ) . \\end{align*}"}
+{"id": "3997.png", "formula": "\\begin{align*} 0 \\geq L v = \\kappa L \\phi + \\tau L ( | D u | ^ 2 / 2 ) + L ( \\log ( w _ { 1 1 } ) ) . \\end{align*}"}
+{"id": "6650.png", "formula": "\\begin{align*} g : = ( a _ y - a ( t ' , x ' ) ) \\nabla \\delta _ y u . \\end{align*}"}
+{"id": "4120.png", "formula": "\\begin{align*} \\alpha _ { i j k } = \\langle [ e _ i , e _ j ] , e _ k \\rangle . \\end{align*}"}
+{"id": "8802.png", "formula": "\\begin{align*} \\left ( \\begin{array} { l l l } r _ { 1 } & r _ { 2 } & r _ { 3 } \\\\ r _ { 4 } & r _ { 5 } & r _ { 6 } \\\\ r _ { 7 } & r _ { 8 } & r _ { 9 } \\end{array} \\right ) M \\left ( \\begin{array} { l l l } r _ { 1 } & r _ { 2 } & r _ { 3 } \\\\ r _ { 4 } & r _ { 5 } & r _ { 6 } \\\\ r _ { 7 } & r _ { 8 } & r _ { 9 } \\end{array} \\right ) ^ { T } = [ A ] _ { \\mathbf { c b } _ i } . \\end{align*}"}
+{"id": "3732.png", "formula": "\\begin{align*} \\frac { q ^ { A ( \\ell - 2 ) } } { ( 1 - q ^ { \\ell - 2 } ) } f ( 0 ) + \\sum _ { j = - B } ^ { A - 1 } q ^ { j ( \\ell - 2 ) } f ( 0 , 0 , \\varpi ^ j ) = I ( f ) ( 0 ) . \\end{align*}"}
+{"id": "5736.png", "formula": "\\begin{align*} \\theta ^ { A A ^ { \\prime } } \\rightarrow \\theta _ { 1 } ^ { A A ^ { \\prime } } = ( L ^ { - 1 } ) _ { B ^ { \\prime } } ^ { A ^ { \\prime } } ( L ^ { - 1 } ) _ { B } ^ { A } \\theta ^ { B B ^ { \\prime } } , \\end{align*}"}
+{"id": "1949.png", "formula": "\\begin{align*} \\begin{gathered} \\varrho _ { E } ^ { ( N ) } [ u , X ] \\leq { \\varrho } _ { K } ^ { ( N , J ) } [ u , X ] \\ ; \\ ; u \\in U , \\\\ \\vartheta _ { E } ^ { ( N ) } \\leq \\vartheta _ { K } ^ { ( N , J ) } \\end{gathered} \\end{align*}"}
+{"id": "3700.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\ell \\left ( c _ i ( d _ { \\ell , i } ( \\mathcal { F } _ 2 ( f ) ) ) + \\sum _ { \\xi \\in X ^ { \\circ } _ i ( F ) } I _ i ( d _ { \\ell , i } ( \\mathcal { F } _ 2 ( f ) ) ) ( \\xi ) \\right ) + \\kappa d _ { \\ell , 0 } ( \\mathcal { F } _ 2 ( f ) ) ( 0 _ { V _ 0 } , 0 , 0 ) . \\end{align*}"}
+{"id": "3817.png", "formula": "\\begin{align*} P _ { ( - \\infty , \\lambda ] } ( H ) = \\sum _ { \\mu \\leq \\lambda } P _ { \\{ \\mu \\} } ( H _ 1 ) \\otimes P _ { ( - \\infty , \\lambda - \\mu ] } ( H _ 2 ) . \\end{align*}"}
+{"id": "1659.png", "formula": "\\begin{align*} D ( \\underline { k } ) = \\bigotimes _ { \\sigma \\in \\Sigma _ B } D ( k _ \\sigma ) \\otimes \\bigotimes _ { \\sigma \\in \\Sigma _ F \\setminus \\Sigma _ B } V ( k _ \\sigma - 2 ) , \\underline { k } = ( k _ \\sigma ) , \\end{align*}"}
+{"id": "2780.png", "formula": "\\begin{align*} & ( u _ 1 , u _ 2 , \\cdots \\ ! , u _ n ) \\\\ & = ( 0 , 0 , 1 , 1 , 1 , 2 , 1 , 2 , 1 , 2 , 1 , 1 , 1 , 1 , 1 , 0 ) . \\end{align*}"}
+{"id": "360.png", "formula": "\\begin{align*} T _ * : = \\lambda T \\textrm { a n d } { L } _ * = | v _ 0 | ^ { - 1 } \\lambda L \\ , . \\end{align*}"}
+{"id": "8201.png", "formula": "\\begin{align*} & \\Phi _ 1 ( t , - \\lambda ) : = { E } _ { ( \\beta , \\beta _ 1 , \\ldots , \\beta _ m ) , 1 } \\left ( - \\lambda t ^ \\beta , - \\lambda t ^ { \\beta _ 1 } , \\ldots , - \\lambda t ^ { \\beta _ m } \\right ) , \\\\ & \\Phi _ 2 ( t , - \\lambda ) : = 1 - \\lambda t ^ { \\beta } E _ { ( \\beta , \\beta - \\beta _ 1 , \\ldots , \\beta - \\beta _ m ) , \\beta + 1 } \\left ( - \\lambda t ^ \\beta , - \\lambda t ^ { \\beta _ 1 } , \\ldots , - \\lambda t ^ { \\beta _ m } \\right ) . \\end{align*}"}
+{"id": "1633.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ^ { 2 } _ { \\mathbb { R } ^ d } ) u = - u | u | ^ { \\frac { 8 } { d - 4 } } , u ( 0 , x ) = u _ 0 ( x ) \\in H ^ 2 ( \\mathbb { R } ^ d ) , \\end{align*}"}
+{"id": "8729.png", "formula": "\\begin{align*} G _ \\lambda ( x _ 1 , \\dots , x _ n , \\beta ) = \\det \\left [ \\sum _ { m = 0 } ^ \\infty { { i - l } \\choose m } \\beta ^ m G _ { \\lambda _ i - i + j + m } , \\right ] \\end{align*}"}
+{"id": "1225.png", "formula": "\\begin{align*} W _ 2 ( \\mu _ 0 , \\mu _ 1 ) : = \\sqrt { \\inf _ { \\gamma \\in \\Pi ( \\mu _ 0 , \\mu _ 1 ) } \\int _ { X \\times X } | x _ 0 - x _ 1 | ^ 2 d \\pi ( x _ 0 , x _ 1 ) } \\end{align*}"}
+{"id": "5089.png", "formula": "\\begin{align*} Q = ( 1 - c _ 1 u _ 1 ) \\cdots ( 1 - c _ s u _ s ) . \\end{align*}"}
+{"id": "4714.png", "formula": "\\begin{align*} B _ i : = ( r _ { i j } ^ k ) , 1 \\leq i \\leq s . \\end{align*}"}
+{"id": "907.png", "formula": "\\begin{gather*} f _ 0 = 1 , f _ { i + 1 } = z f _ i + z g _ i , i \\ge 0 , \\\\ g _ i = z f _ { i + 1 } + z g _ { i + 1 } + z h _ { i + 1 } , i \\ge 1 , \\\\ g _ 0 = z g _ { i + 1 } + z h _ { i + 1 } , i \\ge 1 , \\\\ h _ i = z h _ { i + 1 } + z g _ { i + 1 } , i \\ge 0 . \\end{gather*}"}
+{"id": "1131.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - k } ( - 1 ) ^ i \\binom { n } { i + k } \\binom { i + k } { i } & = \\sum _ { i = 0 } ^ { n - k } ( - 1 ) ^ i \\binom { n } { k } \\binom { n - k } { i } \\\\ & = \\binom { n } { k } \\sum _ { i = 0 } ^ { n - k } ( - 1 ) ^ i \\binom { n - k } { i } \\\\ & = \\left \\{ \\begin{array} { l l } \\binom { n } { k } \\cdot 1 & \\mbox { i f \\ } n - k = 0 \\\\ \\binom { n } { k } \\cdot 0 & \\mbox { i f \\ } n - k > 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "9030.png", "formula": "\\begin{align*} \\underline { V } _ t = \\underline { \\Psi } ( \\underline { V } ) _ t . \\end{align*}"}
+{"id": "5140.png", "formula": "\\begin{align*} \\det \\left ( M _ { n \\mathbf { m } } ( \\lambda , b , \\Omega _ { \\mathbf { m } } ^ { \\pm } ( \\lambda , b ) ) \\right ) \\underset { n \\rightarrow \\infty } { = } d _ { \\infty } ( \\lambda , b , \\mathbf { m } ) + \\frac { \\widetilde { d } _ { \\infty } ( \\lambda , b , \\mathbf { m } ) } { n } + O _ { \\lambda , b , \\mathbf { m } } \\left ( \\frac { 1 } { n ^ { 3 } } \\right ) , \\end{align*}"}
+{"id": "8336.png", "formula": "\\begin{align*} \\left | \\overline { Q _ l } ( s , y ) \\right | & = \\left | Q \\left ( I ^ { - 1 } \\frac { y _ 1 - l _ 1 s } { \\sqrt { 1 - l _ 1 ^ 2 } } , I ^ { - 1 } y _ 2 , I ^ { - 1 } y _ 3 \\right ) \\right | \\\\ & \\le K ( Q ) I \\left ( \\left ( \\frac { y _ 1 - l _ 1 s } { \\sqrt { 1 - l _ 1 ^ 2 } } \\right ) ^ 2 + y _ 2 ^ 2 + y _ 3 ^ 2 \\right ) ^ { - \\frac 1 2 } \\\\ & \\lesssim K ( Q ) I | y | ^ { - 1 } . \\end{align*}"}
+{"id": "692.png", "formula": "\\begin{align*} - d * d G ^ \\omega = \\omega = \\sum _ { k = 1 } ^ n \\Gamma _ k \\delta _ { w _ k } - \\frac { \\Gamma } { V } { \\rm v o l } . \\end{align*}"}
+{"id": "6962.png", "formula": "\\begin{gather*} \\big \\{ Y ^ { ( \\alpha ) } _ { j , m , n } \\big \\} _ { j = 1 } ^ n = \\big \\{ x _ { j , m } ^ { ( - \\alpha - 1 ) } \\big \\} _ { j = 1 } ^ m \\cup \\big \\{ x _ { j - m , n - m } ^ { ( \\alpha + 1 ) } \\big \\} _ { j = m + 1 } ^ { n } . \\end{gather*}"}
+{"id": "5106.png", "formula": "\\begin{align*} \\partial _ { t } \\mathbf { \\Phi } _ { t } ( z ) = \\mathbf { v } ( t , \\mathbf { \\Phi } _ { t } ( z ) ) \\quad \\mbox { a n d } \\quad \\mathbf { \\Phi } _ { 0 } = \\textnormal { I d } _ { \\mathbb { R } ^ { 2 } } . \\end{align*}"}
+{"id": "584.png", "formula": "\\begin{align*} \\mathbb { E } ^ \\dagger [ \\mathbb { X } ^ \\dagger _ { t _ n , t _ { n + 1 } } ] = \\mathcal { O } ( \\Delta t ^ 2 ) , \\end{align*}"}
+{"id": "6880.png", "formula": "\\begin{align*} X ^ { s _ i ( \\cdot ) } _ { p _ i ( \\cdot ) } : = \\left \\{ u \\colon \\R ^ N \\to \\R \\bigg | \\ | u | _ { { X } _ { p _ i } } < \\infty \\right \\} . \\end{align*}"}
+{"id": "8757.png", "formula": "\\begin{align*} & d _ \\infty ( ( x _ 1 , v _ 1 , w _ 1 ) , ( x _ 2 , v _ 2 , w _ 2 ) ) : = | \\mathcal { H } ( x _ 1 , v _ 1 , w _ 1 ) - \\mathcal { H } ( x _ 2 , v _ 2 , w _ 2 ) | _ { S _ \\infty } \\end{align*}"}
+{"id": "2543.png", "formula": "\\begin{align*} S _ 1 = \\bar S _ 1 + E _ 2 \\ , , \\end{align*}"}
+{"id": "2223.png", "formula": "\\begin{align*} h ( i , 4 ) = 1 + \\left \\lfloor \\dfrac { 5 i - 5 } { 6 } \\right \\rfloor \\qquad \\textrm { a n d } h ( i , 5 ) = 2 + \\left \\lfloor \\dfrac { 5 i - 6 } { 6 } \\right \\rfloor , \\end{align*}"}
+{"id": "7351.png", "formula": "\\begin{align*} M ( \\gamma ) = O \\left ( \\int _ { 1 } ^ { \\gamma } \\frac { K ( x ) ^ { \\alpha } } { x } d x \\right ) ( \\gamma \\to \\infty ) . \\end{align*}"}
+{"id": "3307.png", "formula": "\\begin{align*} ( \\tilde { \\mathcal A } + \\varepsilon ) ^ { - \\mu } = \\dfrac { \\sin ( \\pi \\mu ) } { \\pi } \\int _ { 0 } ^ { \\infty } \\dfrac { 1 } { ( \\lambda + \\varepsilon ) ^ { \\mu } } ( \\tilde { \\mathcal A } + \\lambda + \\varepsilon ) ^ { - 1 } { \\rm d } \\lambda . \\end{align*}"}
+{"id": "3469.png", "formula": "\\begin{align*} \\sum _ { k _ 1 + k _ 2 + k _ 3 = n } \\binom { n } { k _ 1 , k _ 2 , k _ 3 } ^ 2 \\le \\left ( \\sum _ { k _ 1 + k _ 2 + k _ 3 = n } \\binom { n } { k _ 1 , k _ 2 , k _ 3 } \\right ) ^ 2 = 9 ^ n \\end{align*}"}
+{"id": "8800.png", "formula": "\\begin{align*} r _ 1 ^ 2 + r _ 4 ^ 2 + r _ 7 ^ 2 = 1 , r _ 2 ^ 2 + r _ 5 ^ 2 + r _ 8 ^ 2 = 1 , r _ 3 ^ 2 + r _ 6 ^ 2 + r _ 9 ^ 2 = 1 , \\end{align*}"}
+{"id": "5730.png", "formula": "\\begin{align*} \\mathbf { D e } ^ { A A ^ { \\prime } } \\equiv d \\mathbf { e } ^ { A A ^ { \\prime } } + \\mathbf { A } _ { B } ^ { A } \\mathbf { e } ^ { B A ^ { \\prime } } + \\mathbf { A } _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\mathbf { e } ^ { A B ^ { \\prime } } = 0 , \\end{align*}"}
+{"id": "8709.png", "formula": "\\begin{align*} \\tilde \\psi ^ + _ { k - 1 / 2 } = \\sum _ { i } A _ { k , i } \\psi ^ + _ { i - 1 / 2 } , \\tilde \\psi ^ - _ { k - 1 / 2 } = \\sum _ { i } A _ { k - 1 , i - 1 } \\psi ^ - _ { i - 1 / 2 } \\end{align*}"}
+{"id": "7582.png", "formula": "\\begin{align*} u ( t , 0 ) = u ( t , 1 ) = 0 , \\ ; t \\geq 0 , \\end{align*}"}
+{"id": "8678.png", "formula": "\\begin{align*} \\mathcal { F } ( u _ 1 , \\dots , u _ l ; t ) & = \\prod _ { 1 \\le i < j \\le l } i _ { u _ i , t u _ j } \\left ( \\frac { u _ i - u _ j } { u _ i - t u _ j } \\right ) \\prod _ { i = 1 } ^ { l } E ( - t u _ i ) H ( u _ i ) , \\end{align*}"}
+{"id": "7046.png", "formula": "\\begin{align*} ( u + u b ' _ 1 x + u b ' _ 2 x ^ 2 + \\cdots ) ( 1 + d _ 1 x + d _ 2 x ^ 2 + \\cdots ) = 1 . \\end{align*}"}
+{"id": "9215.png", "formula": "\\begin{align*} v ( x , h ) = \\int _ { 0 } ^ { \\ell ( x , h ) } \\zeta ( s / \\tau ) e ^ { - s ^ { 2 } / 2 h } d s , \\end{align*}"}
+{"id": "8624.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } Z ^ \\lambda _ { i , k } ( x ) = e ^ { - \\lambda } \\frac { \\lambda ^ i } { i ! } . \\end{align*}"}
+{"id": "2602.png", "formula": "\\begin{align*} V _ { a , b } = \\left < v _ { a , b } ^ t \\ \\middle | \\ t = 1 , \\dots , d \\right > \\end{align*}"}
+{"id": "1256.png", "formula": "\\begin{align*} & \\dfrac { 1 6 \\alpha ( 1 - \\beta ) ^ 2 } { ( 1 + \\alpha ) ^ 2 ( 1 + 1 6 \\alpha ( 1 - \\beta ) ^ 2 ) } - \\dfrac { 1 - x ^ 2 } { ( 1 + \\alpha ) ^ 2 } \\\\ & { } - \\dfrac { 1 } { ( 1 - \\alpha ) ^ 2 } \\left ( x - \\dfrac { 2 \\sqrt { \\alpha } } { ( 1 + \\alpha ) \\sqrt { 1 + 1 6 \\alpha ( 1 - \\beta ) ^ 2 } } \\right ) ^ 2 = 0 , \\end{align*}"}
+{"id": "2653.png", "formula": "\\begin{align*} K ( a , t ; A _ 0 , A _ 1 ) : = \\inf _ { a _ 1 \\in A _ 1 } \\{ \\| a - a _ 1 \\| _ { A _ 0 } + t \\| a _ 1 \\| _ { A _ 1 } \\} \\ ; , t > 0 \\ ; . \\end{align*}"}
+{"id": "1042.png", "formula": "\\begin{align*} \\mathcal { R } _ { n , \\alpha } ( \\varepsilon ) = \\mathcal { R } _ { n , \\alpha } ( \\theta ( \\mathcal { P } _ k ) , ( \\cdot ) ^ 2 , \\varepsilon ) = \\inf _ { Q \\in \\mathcal { Q } _ \\alpha } \\inf _ { \\hat { \\mu } } \\sup _ { P _ { k , \\varepsilon } \\in \\mathcal { P } _ \\varepsilon ( \\mathcal { P } _ k ) } \\mathbb { E } _ { P _ { k , \\varepsilon } , Q } \\{ ( \\hat { \\mu } - \\mu ) ^ 2 \\} , \\end{align*}"}
+{"id": "1955.png", "formula": "\\begin{align*} w _ 1 ^ \\eta \\Big ( & \\int w _ 2 ^ x \\big ( h _ N \\| z _ 2 \\| ) K ( z _ 2 ) d z _ 2 \\Big ) + \\int w _ 1 ^ x ( h _ N \\| z _ 1 \\| ) K ( z _ 1 ) d z _ 1 \\\\ & \\ ; = w _ 1 ^ \\eta \\left ( \\ell h _ N ^ { \\alpha _ 2 } \\int \\| z _ 2 \\| ^ { \\alpha _ 2 } K ( z _ 2 ) d z _ 2 \\right ) + \\ell h _ N ^ { \\alpha _ 1 } \\int \\| z _ 1 \\| ^ { \\alpha _ 1 } K ( z _ 1 ) d z _ 1 \\\\ & \\ ; = \\ell \\left ( \\ell h _ N ^ { \\alpha _ 2 } m _ { \\alpha _ 2 } ( K ) \\right ) ^ { \\alpha _ 3 } + \\ell h _ N ^ { \\alpha _ 1 } m _ { \\alpha _ 1 } ( K ) \\end{align*}"}
+{"id": "9167.png", "formula": "\\begin{align*} Z _ 0 ( \\varphi ) = e ^ { \\frac { s _ 0 } { 2 } ( \\varphi , - \\Delta \\varphi ) + \\sum _ { x \\in \\Lambda _ N } \\tilde U ( \\varphi _ x ) } , \\varphi \\in \\R ^ { \\Lambda _ N } , \\end{align*}"}
+{"id": "1874.png", "formula": "\\begin{align*} \\alpha _ { z _ i } ( \\sigma ) = \\left \\{ \\begin{array} { l l } \\frac { \\pi } { 2 } & \\mbox { i f $ \\dot { z } _ i \\le z _ i - \\psi _ k $ } \\\\ \\sin ^ { - 1 } \\left ( \\frac { z _ i - \\psi _ k } { \\dot { z } _ i } \\right ) & \\mbox { i f $ \\dot { z } _ i \\ge z _ i - \\psi _ k $ } \\end{array} \\right . \\end{align*}"}
+{"id": "4367.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } \\frac { 1 } { a _ i } = \\theta . \\end{align*}"}
+{"id": "6140.png", "formula": "\\begin{align*} t = 0 : \\phi _ s = \\phi _ { 0 s } , \\phi _ s ' = \\phi _ { 1 s } , ( 1 \\leqslant s \\leqslant d ) , \\end{align*}"}
+{"id": "2107.png", "formula": "\\begin{align*} C \\ ! = \\ ! \\big \\{ ( c _ 1 \\ ! + \\ ! c _ { 1 2 } , \\ ; c _ 2 \\ ! + \\ ! c _ { 1 2 } g ) \\ , \\big | \\ , c _ 1 \\ ! \\in \\ ! C _ 1 , c _ 2 \\ ! \\in \\ ! C _ 2 , c _ { 1 2 } \\ ! \\in \\ ! C _ { 1 2 } \\big \\} \\ ! = \\ ! ( C _ 1 \\times C _ 2 ) \\oplus \\widehat C _ { 1 2 } . \\end{align*}"}
+{"id": "5302.png", "formula": "\\begin{align*} g ( U , U ) = e ^ { 2 x _ 2 } . \\end{align*}"}
+{"id": "2599.png", "formula": "\\begin{align*} \\left < v _ { 1 , 2 } ^ t , v _ { 1 , 3 } ^ t \\ \\middle | \\ t = 1 , \\dots , d \\right > \\cap \\left < v _ { 1 , 4 } ^ t \\ \\middle | \\ t = 1 , \\dots , d \\right > = \\{ 0 \\} \\ \\ \\left < v _ { 1 , 2 } ^ t \\ \\middle | \\ t = 1 , \\dots , d \\right > \\cap \\left < v _ { 1 , 3 } ^ t \\ \\middle | \\ t = 1 , \\dots , d \\right > = \\{ 0 \\} . \\end{align*}"}
+{"id": "7660.png", "formula": "\\begin{align*} g _ { I , J } = \\sum _ { \\textbf { p } \\in \\mathbb { Z } ^ { n } } c _ { \\textbf { p } } x _ { 1 } ^ { p _ { 1 } } \\cdots x _ { i } ^ { p _ { n } } = \\sum _ { p \\in \\mathbb { Z } ^ { n } } g _ { I , j , p } \\end{align*}"}
+{"id": "5093.png", "formula": "\\begin{align*} 1 / Q = \\frac { 1 } { ( 1 - u _ { 1 } ) \\cdots ( 1 - u _ { l } ) } \\ , \\odot \\ , \\frac { 1 } { ( 1 - \\alpha _ { 1 } ^ { - 1 } x _ 1 ) \\cdots ( 1 - \\alpha _ { d } ^ { - 1 } x _ d ) } . \\end{align*}"}
+{"id": "2600.png", "formula": "\\begin{align*} V _ { a , b } = \\left < v _ { a , b } ^ t \\ \\middle | \\ t = 1 , \\dots , d \\right > \\end{align*}"}
+{"id": "3121.png", "formula": "\\begin{align*} - \\Delta w = r a - \\int _ Y r a \\quad Y , w , \\int _ { Y } w = 0 , \\end{align*}"}
+{"id": "3719.png", "formula": "\\begin{align*} \\sum _ { s _ i \\in \\left \\{ \\frac { \\dim V _ i } { 2 } - 1 , \\frac { \\dim V _ i } { 2 } - 2 , 0 \\right \\} } \\mathrm { R e s } _ { s = s _ i } \\frac { e ^ { T s } Z _ { r _ i } ( \\mathcal { F } _ 2 ( f ) , s + 2 - \\tfrac { \\dim V _ i } { 2 } ) } { s } + o _ f ( 1 ) . \\end{align*}"}
+{"id": "5244.png", "formula": "\\begin{align*} d i v ( X ) = \\sum _ { k = 1 } ^ { m } g ( \\nabla _ { e _ k } X , e _ k ) , ~ \\forall X \\in \\Gamma ( T M ) , \\end{align*}"}
+{"id": "7418.png", "formula": "\\begin{align*} q _ { \\alpha } = 1 , q _ { \\alpha ^ * } = 1 . \\end{align*}"}
+{"id": "6287.png", "formula": "\\begin{align*} \\int _ { | t _ 0 , t _ 1 | } \\left | \\left ( \\frac { d } { d \\tau } \\big | _ { \\tau = s } f \\circ \\Phi ^ { p _ 0 } _ { \\tau , t _ 0 } \\circ ( \\Phi ^ { p } _ { s , t _ 0 } ) ^ { - 1 } - \\frac { d } { d \\tau } \\big | _ { \\tau = s } f \\circ \\Phi ^ { p _ 0 } _ { \\tau , t _ 0 } \\circ ( \\Phi ^ { p _ 0 } _ { s , t _ 0 } ) ^ { - 1 } \\right ) ( x ' ) \\right | \\d s < \\epsilon . \\end{align*}"}
+{"id": "4965.png", "formula": "\\begin{align*} g & : = x _ 1 ^ { - 1 } ( f ^ { l t + 1 } + r ^ t ) = x _ 1 ^ { - 1 } ( f ^ { l t } ( x _ 1 x _ 3 - x _ 2 ^ t ) + ( f ^ l x _ 2 + x _ 1 ^ m ) ^ t ) \\\\ & = f ^ { l t } x _ 3 + g ^ * + x _ 1 ^ { m t - 1 } , g ^ * : = x _ 1 ^ { - 1 } ( ( f ^ l x _ 2 + x _ 1 ^ m ) ^ t - f ^ { l t } x _ 2 ^ t - x _ 1 ^ { m t } ) . \\end{align*}"}
+{"id": "9041.png", "formula": "\\begin{align*} \\begin{cases} \\overline { V } _ { \\cdot } ^ { ( m ) , i , n } : = \\overline { \\Psi } _ { \\cdot } ^ { i } ( \\overline { \\textbf { V } } ^ { ( m - 1 ) , n } ) , \\ , \\ , \\overline { V } _ { \\cdot } ^ { ( 0 ) , i , n } = V _ { \\cdot } ^ { ( 0 ) , i , n } , \\\\ \\underline { V } _ { \\cdot } ^ { ( m ) , i , n } : = \\underline { \\Psi } _ { \\cdot } ( \\underline { \\textbf { V } } ^ { ( m - 1 ) , n } ) , \\ , \\ , \\underline { V } _ { \\cdot } ^ { ( 0 ) , i , n } = V _ { \\cdot } ^ { ( 0 ) , i , n } . \\end{cases} \\end{align*}"}
+{"id": "4912.png", "formula": "\\begin{align*} \\frac { \\delta \\mathcal { H } } { \\delta z ^ * } = \\frac { \\partial H } { \\partial z ^ * } - \\frac { \\mathrm d } { \\mathrm d x } \\frac { \\partial H } { \\partial z _ x ^ * } . \\end{align*}"}
+{"id": "6883.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\iint _ { \\R ^ { 2 N } \\setminus ( \\mathcal { C } \\Omega ) ^ 2 } \\frac { | u ( x ) - u ( y ) | ^ { p _ i ( x , y ) - 2 } ( u ( x ) - u ( y ) ) ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { N + s _ i ( x , y ) p _ i ( x , y ) } } \\ , d x \\ , d y \\\\ & = \\int _ { \\Omega } \\varphi ( - \\Delta ) ^ { s _ i ( \\cdot ) } _ { p _ i ( \\cdot ) } u \\ , d x + \\int _ { \\mathcal { C } \\Omega } \\varphi \\mathcal { N } ^ { s _ i ( \\cdot ) } _ { p _ i ( \\cdot ) } u \\ , d x , \\end{align*}"}
+{"id": "3302.png", "formula": "\\begin{align*} \\| ( \\tilde { \\mathcal A } + \\lambda ) ^ { \\frac 1 2 } W \\| ^ 2 _ { L ^ 2 [ B _ 0 ] } = \\int _ { \\mathbb R ^ 2 } | \\nabla W | ^ 2 + \\lambda \\langle W , W \\rangle \\forall \\ , W \\in \\mathcal { D } ( \\tilde { \\mathcal A } ^ { \\frac 1 2 x } ) . \\end{align*}"}
+{"id": "7288.png", "formula": "\\begin{align*} \\lim _ { x \\to + 0 } \\frac { ( - 1 ) ^ { k } G ^ { k } _ m ( x ) } { x } = \\infty . \\end{align*}"}
+{"id": "9175.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\epsilon ^ { - 2 } \\lambda ( \\epsilon p ) = | p | ^ 2 , \\lim _ { \\epsilon \\to 0 } \\epsilon ^ { - 2 } \\lambda _ { J } ( \\epsilon p ) = v _ { J } ^ 2 | p | ^ 2 , \\end{align*}"}
+{"id": "4661.png", "formula": "\\begin{align*} \\binom { m } { \\ell _ i } = \\sum _ { k = 0 } ^ { \\ell _ i } \\binom { \\ell _ j } { k } \\binom { m - \\ell _ j } { \\ell _ i - k } . \\end{align*}"}
+{"id": "4957.png", "formula": "\\begin{align*} Q _ { n , m } = \\sum _ { 0 \\le s \\le ( n - m ) / 3 } \\frac { n - m - 3 s + 2 } { n - s + 1 } \\ , \\binom { m + s - 2 } { s } \\binom { n + m - s - 1 } { m - 1 } . \\end{align*}"}
+{"id": "1991.png", "formula": "\\begin{align*} \\left [ \\begin{matrix} P U & \\\\ & I _ i \\end{matrix} \\right ] \\left [ \\begin{matrix} C & E \\\\ & D \\end{matrix} \\right ] \\left [ \\begin{matrix} V & \\\\ & I _ j \\end{matrix} \\right ] = \\left [ \\begin{matrix} P & P U E \\\\ & D \\end{matrix} \\right ] , \\end{align*}"}
+{"id": "341.png", "formula": "\\begin{align*} \\bar { G } _ { \\phi } ( p ^ 2 d _ { \\phi } N ) = c _ 1 G _ { \\phi ' } ( p ^ 2 d _ { \\phi } N ) + c _ 2 \\bar { G } _ { \\phi ' } ( p ^ 2 d _ { \\phi } N ) \\end{align*}"}
+{"id": "3941.png", "formula": "\\begin{align*} c ( q , p ) & = c ( 0 , 0 ) + c _ { q _ i } ( 0 , 0 ) q _ i + c _ { p _ j } ( 0 , 0 ) p _ j \\\\ & \\quad + \\frac { 1 } { 2 } c _ { q _ i q _ j } ( t q , t p ) q _ i q _ j + c _ { q _ i , p _ j } ( t q , t p ) q _ i p _ j + \\frac { 1 } { 2 } c _ { p _ i p _ j } ( t q , t p ) p _ i p _ j . \\end{align*}"}
+{"id": "9210.png", "formula": "\\begin{align*} \\varphi _ { \\Gamma } ( x ) = d _ { \\rm A g } ( x , \\Gamma ) : = \\inf _ { y \\in \\Gamma } d _ { \\rm A g } ( x , y ) . \\end{align*}"}
+{"id": "8503.png", "formula": "\\begin{align*} \\partial _ t C = C _ { \\xi ^ 2 } = A + f ( \\xi ) C _ { \\xi } \\end{align*}"}
+{"id": "1762.png", "formula": "\\begin{align*} - \\Delta v + \\lambda ^ { \\frac { p - 2 } { 4 ( 3 - p ) } } v + ( I _ 2 * | v | ^ 2 ) v = | v | ^ { p - 2 } v \\quad . \\end{align*}"}
+{"id": "3957.png", "formula": "\\begin{align*} g ( - a e _ 1 , t e _ 1 , 0 ) = - a t + O ( t ^ 2 ) < 0 . \\end{align*}"}
+{"id": "4753.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { b ' } x ' _ i ( t ) \\geq M \\forall t \\in \\{ 0 , \\ldots , T \\} . \\end{align*}"}
+{"id": "2912.png", "formula": "\\begin{align*} F ( \\zeta ) = \\sum _ { n = 1 } ^ \\infty c _ n \\zeta ^ { \\alpha ( n ) } , \\end{align*}"}
+{"id": "9255.png", "formula": "\\begin{align*} \\begin{aligned} 0 & \\neq d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots v _ { w ^ { - 1 } ( k ) } , \\ldots , v _ { h ( w ^ { - 1 } ( k ) ) } , e _ { k - 1 } , \\ldots \\widehat { v _ { r } } , \\ldots v _ n ) } \\\\ \\end{aligned} \\end{align*}"}
+{"id": "8885.png", "formula": "\\begin{align*} \\frac { 2 ^ { 1 - n - 2 \\nu } } { ( n + 2 \\nu - 1 ) ! } \\left ( - \\frac { d } { \\sin \\psi d \\psi } \\right ) ^ { n - 1 + 2 \\nu } \\left ( \\sum \\limits _ { m = 0 } ^ { + \\infty } \\frac { ( 2 m + 2 \\nu + n ) \\sin ( 2 m + 2 \\nu + n ) \\psi } { \\sin \\psi } e ^ { - 4 t ( m + \\nu + \\frac { n } { 2 } ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "7799.png", "formula": "\\begin{align*} & \\ ( \\Psi _ { c , b _ 2 } [ n _ 2 ] \\Psi _ { c , b _ 1 } [ n _ 1 ] ) ( y ^ { n _ 2 } ) \\\\ & = \\Psi _ { c , b _ 2 } [ n _ 2 ] ( y ^ { n _ 2 } ( 1 + q ^ { - c + b _ 1 } y ^ { n _ 1 } ) ^ { - 1 } ) \\\\ & = y ^ { n _ 2 } ( 1 + q ^ { - c + b _ 1 } y ^ { n _ 1 } ( 1 + q ^ { c + b _ 2 } y ^ { n _ 2 } ) ) ^ { - 1 } \\\\ & = y ^ { n _ 2 } ( 1 + q ^ { - c + b _ 1 } y ^ { n _ 1 } + q ^ { b _ 1 + b _ 2 } y ^ { n _ 1 } y ^ { n _ 2 } ) ^ { - 1 } , \\end{align*}"}
+{"id": "1894.png", "formula": "\\begin{align*} Z ( t ) = \\int _ 0 ^ 1 \\cos ( \\pi x ) z ( x , t ) d x , \\end{align*}"}
+{"id": "5262.png", "formula": "\\begin{align*} c g ( T _ U U , X ) = - c g ( U , U ) g ( X , \\nabla f ) - c ^ 2 \\frac { \\lambda ^ 2 } { 2 } g ( X , X ) g \\left ( U , \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } \\right ) . \\end{align*}"}
+{"id": "6696.png", "formula": "\\begin{align*} \\sum _ { p \\leq x } \\mu ( p ) \\mu ( p + a ) = - \\sum _ { p \\leq x } \\mu ( p + a ) , \\end{align*}"}
+{"id": "5705.png", "formula": "\\begin{align*} \\mathfrak { f } _ { a b c d } & = \\mathfrak { f } _ { [ a b ] c d } = \\mathfrak { f } _ { a b [ c d ] } , \\\\ \\mathfrak { f } _ { [ b c d ] } ^ { a } & = 0 . \\end{align*}"}
+{"id": "9142.png", "formula": "\\begin{align*} N _ n ( m ) = \\sum _ { k = 0 } ^ n { n \\choose k } N _ { n - k } ( m + k - 1 ) \\end{align*}"}
+{"id": "4934.png", "formula": "\\begin{align*} \\alpha \\delta _ n ^ + \\mathbf { z } _ n = \\widetilde { \\mathcal { J } } \\int _ 0 ^ 1 \\nabla \\widetilde { \\mathcal { H } } ( \\xi \\mathbf { z } _ { n + 1 } + ( 1 - \\xi ) \\mathbf { z } _ n ) \\ , \\mathrm { d } \\xi , \\end{align*}"}
+{"id": "2373.png", "formula": "\\begin{align*} E _ 2 = P E _ 1 Q , A _ 2 = P A _ 1 Q - P E _ 1 \\dot Q \\end{align*}"}
+{"id": "842.png", "formula": "\\begin{align*} \\mathbb { F } ^ { + } L _ d ( q _ k , q _ { k + 1 } ) \\mapsto ( q _ { k + 1 } , p _ { k + 1 } ) = ( q _ k , D _ 2 L _ d ( q _ k , q _ { k + 1 } ) ) \\end{align*}"}
+{"id": "8662.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { v } { u } \\right ) E ^ \\perp ( u ) E ( v ) = E ( v ) E ^ \\perp ( u ) , \\\\ \\left ( 1 - \\frac { v } { u } \\right ) H ^ \\perp ( u ) H ( v ) = H ( v ) H ^ \\perp ( u ) , \\\\ H ^ \\perp ( u ) E ( v ) = \\left ( 1 + \\frac { v } { u } \\right ) E ( v ) H ^ \\perp ( u ) , \\\\ E ^ \\perp ( u ) H ( v ) = \\left ( 1 + \\frac { v } { u } \\right ) H ( v ) E ^ \\perp ( u ) . \\end{align*}"}
+{"id": "2070.png", "formula": "\\begin{align*} \\min \\{ \\rho _ 1 , \\rho _ 2 \\} = 1 . \\end{align*}"}
+{"id": "3794.png", "formula": "\\begin{align*} \\Delta _ L h _ { i k } : = \\Delta h _ { i k } + 2 R _ { i j k l } h ^ { j l } - _ i ^ l h _ { k l } - _ k ^ l h _ { i l } . \\end{align*}"}
+{"id": "4098.png", "formula": "\\begin{align*} \\nabla _ i R & = \\frac { 1 } { 2 } ( \\nabla _ i H _ { j k l } - \\nabla _ k H _ { j i l } + \\nabla _ l H _ { j i k } ) H _ { j k l } \\\\ & = \\frac { 1 } { 4 } \\nabla _ i | H | ^ 2 - \\nabla _ k H _ { j i l } H _ { j k l } \\\\ & = \\frac { 1 } { 4 } \\nabla _ i | H | ^ 2 - 2 \\nabla _ i R \\end{align*}"}
+{"id": "4113.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda = \\int _ M \\langle \\mathcal { N } ( h , K , v _ { h , K } ) , ( h , K , v _ { h , K } ) \\rangle e ^ { - f } d V _ g = \\Big ( \\mathcal { N } ( h , K , v _ { h , K } ) , ( h , K , v _ { h , K } ) \\Big ) _ { f } \\end{align*}"}
+{"id": "32.png", "formula": "\\begin{align*} C ^ { k _ i + 1 } ( n _ i ) & = \\frac { ( 2 ( 2 a + 1 ) + 1 ) \\cdot 3 ^ { k _ i } - 1 } { 2 } = \\frac { 4 a \\cdot 3 ^ { k _ i } + 3 \\cdot 3 ^ { k _ i } - 1 } { 2 } \\\\ \\quad & = 2 a \\cdot 3 ^ { k _ i } + \\frac { 3 ^ { k _ i + 1 } - 1 } { 2 } \\end{align*}"}
+{"id": "6844.png", "formula": "\\begin{align*} w ( Y _ i , Y _ j ) = w ( R _ { \\alpha } Y _ i , R _ { \\alpha } Y _ j ) = w ( \\tilde Y _ i , \\tilde Y _ j ) = \\tilde X _ i ^ T ( \\tilde Q _ j - \\tilde Q _ i ) \\tilde X _ j , \\ ; \\tilde Q _ k = \\tilde U _ k \\tilde X _ k ^ { - 1 } , \\ , k = i , j . \\end{align*}"}
+{"id": "8068.png", "formula": "\\begin{align*} H _ { m , n } ^ - ( x ) = \\frac { 4 } { \\pi } \\int _ { - \\infty } ^ \\infty K _ { 2 i t } ( x ) \\sinh ( \\pi t ) k ( t ) V ( m ^ 2 n , t ) t \\ , d t . \\end{align*}"}
+{"id": "2156.png", "formula": "\\begin{align*} \\psi ' ( r ) & = - \\dfrac { 2 ( 1 - \\alpha ) ( 1 - r ) ^ 2 + \\beta ( 1 + r ) ^ 2 } { ( 1 - r ^ 2 ) ^ 2 } < 0 , \\end{align*}"}
+{"id": "6756.png", "formula": "\\begin{align*} \\sqrt { \\frac { 1 6 } { \\pi } } \\phi \\left ( \\left [ \\frac { 1 } { 2 } - \\sigma \\right ] + j \\omega \\right ) = \\frac { z \\left ( \\left [ \\frac { 1 } { 2 } - \\sigma \\right ] + j \\omega \\right ) } { \\left ( \\left [ \\frac { 1 } { 2 } - \\sigma \\right ] + j \\omega \\right ) } + \\frac { z \\left ( \\left [ \\frac { 1 } { 2 } + \\sigma \\right ] - j \\omega \\right ) } { \\left ( \\left [ \\frac { 1 } { 2 } + \\sigma \\right ] - j \\omega \\right ) } . \\end{align*}"}
+{"id": "448.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j \\leq n } a _ j ^ 2 \\leq \\frac { 1 } { d } \\sum _ { j = 1 } ^ { n } a _ j ^ 2 . \\end{align*}"}
+{"id": "4195.png", "formula": "\\begin{align*} \\begin{cases} ( \\phi , \\tilde { \\Lambda } ) | _ { \\{ t = 0 \\} } = \\varepsilon ( \\phi _ 0 , \\tilde { \\Lambda } _ 0 ) , ( \\phi _ 0 , \\tilde { \\Lambda } _ 0 ) \\in C _ c ^ { \\infty } ( \\mathbb { R } \\times \\mathbb { R } ) , \\\\ ( \\partial _ t \\phi , \\partial _ t \\tilde { \\Lambda } ) | _ { \\{ t = 0 \\} } = \\varepsilon ( \\phi _ 1 , \\tilde { \\Lambda } _ 1 ) , ( \\phi _ 1 , \\tilde { \\Lambda } _ 1 ) \\in C _ c ^ { \\infty } ( \\mathbb { R } \\times \\mathbb { R } ) . \\end{cases} \\end{align*}"}
+{"id": "7959.png", "formula": "\\begin{align*} \\pi : M \\to M ' = M / L \\end{align*}"}
+{"id": "2635.png", "formula": "\\begin{align*} V \\left ( \\xi \\right ) = 1 + O ( \\xi ^ { 1 / 2 - \\epsilon } ) \\ ; \\mbox { f o r } \\ ; 0 < \\xi < 1 \\mbox { a n d } V ^ { ( j ) } \\left ( \\xi \\right ) = O ( e ^ { - \\xi } ) \\ ; \\mbox { f o r } \\ ; \\xi > 0 , \\ ; j \\geq 0 . \\end{align*}"}
+{"id": "7120.png", "formula": "\\begin{align*} d \\cap x ^ { V } = 0 . \\end{align*}"}
+{"id": "7681.png", "formula": "\\begin{align*} \\omega = \\sum _ { k } \\lambda _ { k } \\frac { d f _ { k } } { f _ { k } } . \\end{align*}"}
+{"id": "8615.png", "formula": "\\begin{align*} T x = \\begin{cases} 1 , & x = 1 ; \\\\ - x , & . \\end{cases} \\end{align*}"}
+{"id": "2015.png", "formula": "\\begin{align*} \\alpha ( G ) \\leq s \\lambda _ n = n \\frac { - \\lambda _ n } { \\lambda _ 1 - \\lambda _ n } , \\end{align*}"}
+{"id": "2113.png", "formula": "\\begin{align*} P = \\begin{pmatrix} 0 & 1 & 0 & \\cdots & 0 \\\\ 0 & 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\ddots & 1 \\\\ 1 & 0 & 0 & \\cdots & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "5336.png", "formula": "\\begin{align*} \\int _ \\Omega u \\cdot \\operatorname { c u r l } \\varphi \\ , d x = \\int _ \\Omega \\operatorname { c u r l } u \\cdot \\varphi \\ , d x \\forall \\varphi \\in C ^ \\infty _ c ( \\Omega ) ^ 3 . \\end{align*}"}
+{"id": "8727.png", "formula": "\\begin{align*} g _ { \\lambda _ 1 ( u ) } \\dots g _ { \\lambda _ l ( u ) } = \\frac { u _ 1 ^ { \\lambda _ 1 } \\dots u _ l ^ { \\lambda _ l } } { ( 1 - u _ 1 ) ^ { \\lambda _ 1 } \\dots ( 1 - u _ l ) ^ { \\lambda _ l } } \\end{align*}"}
+{"id": "826.png", "formula": "\\begin{align*} \\pi _ { k p } ( ( z _ 1 , \\ldots , z _ k ) ) = \\widetilde { \\P } _ { p } ( [ z _ 1 , \\ldots , z _ k ] ) . \\end{align*}"}
+{"id": "3649.png", "formula": "\\begin{align*} \\Phi ( x , t ) : = \\int \\limits _ { \\mathbb { H } ^ { n } } \\mathcal { K } ( x , y , T - t ) u ( y , t ) ~ { \\rm d } v _ { \\mathbb { H } ^ { n } } ( y ) \\end{align*}"}
+{"id": "1065.png", "formula": "\\begin{align*} \\mathrm { T V } ( \\widetilde { R } _ 0 , \\widetilde { R } _ 1 ) = 0 . \\end{align*}"}
+{"id": "442.png", "formula": "\\begin{align*} \\| f _ j \\| = \\| \\tau _ j \\| = | f _ j ( \\tau _ j ) | = 1 , \\forall 1 \\leq j \\leq n , \\end{align*}"}
+{"id": "4209.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { t \\in [ 0 , T ] } \\norm { \\left ( \\Psi ^ { ( i ) } , \\partial _ t \\Psi ^ { ( i ) } \\right ) } _ { \\mathcal { H } } \\leq C ( 1 + T ) \\left ( \\norm { \\left ( \\Psi _ 0 , \\Psi _ 1 \\right ) } _ { \\mathcal { H } } + 2 B T \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "171.png", "formula": "\\begin{align*} \\alpha _ p = \\sum _ { j \\leq \\log ( 2 k ^ 2 + l ) / \\log p } \\# \\{ 1 \\leq k \\leq n : p ^ j \\mid 2 k ^ 2 + l \\} . \\end{align*}"}
+{"id": "582.png", "formula": "\\begin{align*} X ^ \\dagger _ { t _ n , t _ { n + 1 } } : = X _ { t _ { n + 1 } } ^ \\dagger - X _ { t _ n } ^ \\dagger \\end{align*}"}
+{"id": "2058.png", "formula": "\\begin{align*} \\Bigl ( \\frac { 1 } { h _ 2 } y _ 1 ' \\Bigr ) ' + \\biggl [ z \\Bigl ( \\frac { h _ 3 } { h _ 2 } \\Bigr ) ' + z ^ 2 \\Bigl ( h _ 1 - \\frac { h _ 3 ^ 2 } { h _ 2 } \\Bigr ) \\biggr ] y _ 1 = 0 \\end{align*}"}
+{"id": "5695.png", "formula": "\\begin{align*} ( \\beta _ { b } ^ { a } + \\overline { \\beta } _ { b } ^ { a } ) & = \\digamma _ { b } ^ { a } - \\mathfrak { f } _ { b } ^ { a } \\\\ \\mathfrak { f } _ { b } ^ { a } & = \\frac { 1 } { 2 } \\mathfrak { f } _ { b c d } ^ { a } \\theta ^ { c } \\theta ^ { d } , \\end{align*}"}
+{"id": "5611.png", "formula": "\\begin{align*} \\nabla m _ { 2 } = \\theta ( k l + l k + m _ { 3 } m _ { 3 } ) = \\theta h , \\end{align*}"}
+{"id": "2673.png", "formula": "\\begin{align*} H ' ( x ) = \\lambda q \\big ( 1 - H ( x ) \\big ) ^ { 1 / q } . \\end{align*}"}
+{"id": "2355.png", "formula": "\\begin{align*} A _ 0 ( 0 ) = V _ 0 ( 0 ) ^ T J _ n V _ 0 ( 0 ) , A _ 1 ( 0 ) = V _ 1 ( 0 ) ^ T J _ n V _ 1 ( 0 ) . \\end{align*}"}
+{"id": "7417.png", "formula": "\\begin{align*} \\mu ( s \\widetilde { \\alpha } , \\sigma ) = \\gamma ( G / P ) ^ 2 q ^ { n ( \\sigma ) + n ( \\omega ) + n ( \\sigma \\otimes \\omega ) } \\end{align*}"}
+{"id": "2986.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\ell } \\sum _ { i = 0 } ^ { \\ell - 1 } \\overrightarrow { W } ^ \\ell _ { j + 1 } Z _ { i , j } ( W _ { j + i - 1 } - W _ { j + i } ) \\psi _ i \\\\ & = \\frac { 1 } { \\ell } \\overrightarrow { W } ^ \\ell _ { j + 1 } ( W ^ 2 _ { j - 1 } - E _ { \\nu ^ n _ \\rho } [ W ^ 2 _ { j - 1 } ] - \\overrightarrow { ( W ^ 2 ) } ^ \\ell _ j ) - \\frac { \\ell - 1 } { \\ell ^ 2 } W _ j ( W _ j - W _ { j + 1 } ) - \\frac { 1 } { \\ell } W _ { j - 1 } ( W _ { j - 1 } - W _ j ) . \\end{aligned} \\end{align*}"}
+{"id": "6868.png", "formula": "\\begin{align*} ( u _ h ^ { \\bot } - u , v ) _ h = 0 \\forall v \\in X _ h . \\end{align*}"}
+{"id": "4104.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } g _ t = h , g _ 0 = g , \\\\ & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } H _ t = d K , H _ 0 = H \\end{align*}"}
+{"id": "1882.png", "formula": "\\begin{align*} K _ { n } = \\frac { ( S ^ { 1 } ) ^ { n } } { ( z _ { 1 } , \\dots , z _ { n - 1 } , z _ { n } ) \\sim ( \\bar { z } _ { 1 } , \\dots , \\bar { z } _ { n - 1 } , - z _ { n } ) } . \\end{align*}"}
+{"id": "4927.png", "formula": "\\begin{align*} u ( t + \\Delta t ) - u ( t ) = \\tfrac { \\Delta t } 2 ( u ' ( t + \\Delta t ) + u ' ( t ) ) + \\mathcal { R } ( \\Delta t ^ 2 ) . \\end{align*}"}
+{"id": "4094.png", "formula": "\\begin{align*} \\mathfrak { g d i f f } _ { H _ 0 } = \\{ ( X , B ) \\in \\Gamma ( T M ) \\times \\Omega ^ 2 : d ( i _ X H _ 0 + B ) = 0 \\} \\end{align*}"}
+{"id": "1263.png", "formula": "\\begin{align*} c _ f ( r ) & = a _ f ( r ) - 1 + \\dfrac { 1 } { 1 - \\alpha } , \\end{align*}"}
+{"id": "2784.png", "formula": "\\begin{align*} u _ i \\ ! = \\ ! \\left \\{ \\ ! \\begin{aligned} & 0 , ~ ~ ~ ~ ~ ~ ~ ~ \\ ! ~ ~ ~ ~ ~ i \\ ! \\in \\ ! [ 1 , d _ 1 ] ; \\\\ & x _ { i } - x ' _ { d _ 2 } , ~ ~ i \\in [ d _ 1 + 1 , d _ 2 ] ; \\\\ & 0 , ~ ~ ~ ~ ~ ~ ~ ~ i \\in [ d _ 2 + 1 , \\lambda _ 1 ] ; \\\\ & x _ { \\lambda _ 2 } - x ' _ { \\lambda _ 2 } , ~ ~ ~ ~ i \\in [ \\lambda _ 1 + 1 , \\lambda _ 2 ] ; \\\\ & 0 , ~ ~ ~ ~ ~ ~ ~ ~ \\ ! ~ ~ ~ ~ ~ i \\in [ \\lambda _ 2 + 1 , n ] . \\end{aligned} \\right . \\end{align*}"}
+{"id": "1969.png", "formula": "\\begin{align*} W _ { 6 4 } ( a ) = 1 + a y ^ 8 + ( 2 9 7 6 + 2 0 a ) y ^ { 1 2 } + ( 4 5 4 9 5 6 + 2 a ) y ^ { 1 6 } + \\cdots , \\end{align*}"}
+{"id": "9256.png", "formula": "\\begin{align*} \\begin{aligned} & 0 = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots v _ { w ^ { - 1 } ( k ) } , \\ldots , v _ { h ( w ^ { - 1 } ( k ) ) } , c e _ { j - 1 } , \\ldots \\widehat { v _ { r } } , \\ldots v _ n ) } . \\end{aligned} \\end{align*}"}
+{"id": "5605.png", "formula": "\\begin{align*} 0 = & \\langle \\nabla \\nabla S \\vert k \\vert m _ { 3 } k m _ { 2 } m _ 2 \\rangle _ { 3 } = - \\nabla _ { 3 } \\nabla _ { 3 } S _ { 2 2 } + \\nabla _ { 1 } \\nabla _ { 0 } S _ { 2 2 } = - \\nabla _ { 1 } S _ { 2 2 } \\nabla _ { 3 } k _ 3 , \\\\ 0 = & \\langle \\nabla \\nabla S \\vert k \\vert m _ { 2 } k m _ { 2 } m _ { 2 } \\rangle _ { 3 } = - \\nabla _ { 2 } \\nabla _ { 3 } S _ { 2 2 } = - \\nabla _ { 1 } S _ { 2 2 } \\nabla _ { 2 } k _ { 3 } , \\\\ 0 = & \\nabla _ { 0 } \\nabla _ { 3 } S _ { 2 2 } = \\nabla _ { 1 } S _ { 2 2 } \\nabla _ { 0 } k _ { 3 } . \\end{align*}"}
+{"id": "356.png", "formula": "\\begin{align*} U _ i ( f g ) = \\sum _ { i _ 1 + i _ 2 = i } U _ { i _ 1 } ( f ) U _ { i _ 2 } ( g ) , \\end{align*}"}
+{"id": "3082.png", "formula": "\\begin{align*} v ^ { k l } = v ^ { k l } _ B + \\bar { b } _ { k l } w , k , l \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "6397.png", "formula": "\\begin{align*} \\partial _ 2 ^ \\ast : = \\{ ( X , Y ) \\in U _ X \\times \\partial U _ Y : \\ X \\cdot N _ Y \\leq 0 \\} \\times [ 0 , T ) . \\end{align*}"}
+{"id": "3795.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } V ( t ) = \\Delta _ { g ( t ) } V ( t ) + _ { g ( t ) } ( V ( t ) ) , \\end{align*}"}
+{"id": "3818.png", "formula": "\\begin{align*} f = P _ { ( - \\infty , \\lambda ] } ( H ) f = \\sum _ { k \\colon \\mu _ k \\leq \\lambda } \\phi _ k \\otimes \\psi _ k \\end{align*}"}
+{"id": "4947.png", "formula": "\\begin{align*} \\mathrm { i } \\alpha \\delta _ n ^ + z _ { m , n } = \\delta _ m ^ { ( 2 ) } \\mu _ n z _ { m , n } + \\mu _ n ( | z _ { n , m } | ^ 2 ) \\mu _ n z _ { n , m } , \\end{align*}"}
+{"id": "128.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\lambda ^ p \\mathcal L ^ { 2 n } ( E _ { \\lambda , K } ) = \\frac { 2 } { n } \\int _ { \\mathbb R ^ n } | | \\nabla f ( x ) | | _ { Z _ p ^ * K } ^ p d x . \\end{align*}"}
+{"id": "8038.png", "formula": "\\begin{align*} L ' \\Bigl ( \\frac { 1 } { 2 } , f \\times u _ j \\Bigr ) = c L ' \\Bigl ( \\frac { 1 } { 2 } , g \\times u _ j \\Bigr ) \\end{align*}"}
+{"id": "4650.png", "formula": "\\begin{align*} \\mathsf { E } \\left [ \\prod _ { i = 1 } ^ N X _ i \\right ] = \\left \\{ \\begin{array} { l l } 0 & N , \\\\ \\sum _ { P \\in \\wp _ N ^ 2 } \\prod _ { ( i , j ) \\in P } C _ { i j } & N , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "7528.png", "formula": "\\begin{align*} Z _ g ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) = \\dfrac { G _ 0 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "2533.png", "formula": "\\begin{align*} \\bar T ^ \\varphi _ N ( x _ 1 + \\xi , y _ 1 + \\xi ) = \\frac 1 d \\sum _ { j = 1 } ^ N p _ { 1 , j } | w _ j - \\tilde u _ N ^ \\varphi ( y _ 1 + \\xi ) | ^ 2 \\ , . \\end{align*}"}
+{"id": "396.png", "formula": "\\begin{align*} h ( x ) : = \\begin{cases} f ( x , r - t _ { n - 1 } ( x ) ) \\ , , & x \\in \\R _ * ^ { n - 1 } t _ { n - 1 } ( x ) < r \\ , , \\\\ 0 \\ , , & \\ , . \\end{cases} \\end{align*}"}
+{"id": "1401.png", "formula": "\\begin{align*} \\pi _ 2 ^ { ( - , - , + ) } = L ( \\Delta [ 0 , - 3 ] , \\Delta [ 0 , - 1 ] ; \\pi ( 0 ^ - , 1 ^ - , 2 ^ + ) ) . \\end{align*}"}
+{"id": "7595.png", "formula": "\\begin{align*} N = \\bigg \\| \\int _ { 0 } ^ { s } \\int _ { 0 } ^ { 1 } \\bigg ( \\frac { \\partial G } { \\partial y } ( t - r , \\cdot , y ) - \\frac { \\partial G } { \\partial y } ( s - r , \\cdot , y ) \\bigg ) v ^ { \\delta + 1 } ( r , \\cdot ) \\d y \\d r \\bigg \\| _ { \\L ^ p } . \\end{align*}"}
+{"id": "8822.png", "formula": "\\begin{align*} P _ 2 ^ * A _ { 2 1 } P _ 1 = P _ 2 ^ * P _ 2 A _ { 2 1 } = A _ { 2 1 } . \\end{align*}"}
+{"id": "6502.png", "formula": "\\begin{align*} \\int e ^ { \\frac { n ^ 2 } { m ^ 2 } ( \\sqrt { \\Gamma } f ) ^ \\top \\sum _ { j = 1 } ^ { m } \\Xi ^ j _ u ( \\sqrt { \\Gamma } g ) } d N ( 0 , \\epsilon ^ 2 I _ { 2 d } ) ( f , g ) \\leq e ^ { C \\frac { n ^ 4 \\epsilon ^ 4 } { m ^ 4 } \\left ( ( \\sqrt { \\bar { \\Gamma } } ^ \\top \\Xi _ u \\sqrt { \\bar { \\Gamma } } ) ^ 2 \\right ) } , \\end{align*}"}
+{"id": "5168.png", "formula": "\\begin{align*} \\mathcal { I } ( C _ { 7 } ) = \\mathcal { I } ( D L G ( 1 , 2 ) ) \\geq \\frac { 5 } { 8 1 7 } . \\end{align*}"}
+{"id": "5429.png", "formula": "\\begin{align*} \\varphi _ t ^ * \\alpha = \\alpha ( \\varphi _ t ^ * \\omega ) \\end{align*}"}
+{"id": "6689.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { y } ^ { k + 1 } - \\bar { g } ^ { k + 1 } & = ( 1 - \\alpha ^ k ) ( \\bar { y } ^ k - \\bar { g } ^ k ) + \\gamma _ 2 ^ k \\bar { \\xi } _ w ^ k \\end{aligned} \\end{align*}"}
+{"id": "3005.png", "formula": "\\begin{align*} \\theta = g ^ { - 1 } \\mathbf { A } _ a ( x ) g e ^ a + g ^ { - 1 } \\hbox { d } g . \\end{align*}"}
+{"id": "3055.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\bar { A } : D ^ 2 z & = - \\sum _ { j , k , l = 1 } ^ n c _ j ^ { k l } ( A ) \\ , \\partial _ { j k l } ^ 3 u & & \\Omega , \\\\ z & = 0 & & \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4923.png", "formula": "\\begin{align*} \\mathcal { B } _ \\Theta = \\{ 1 , \\cos ( \\omega t ) , \\sin ( \\omega t ) \\} . \\end{align*}"}
+{"id": "7989.png", "formula": "\\begin{align*} j ' ( c ) \\in j ' ( a ) + j ' ( b ) = i ' ( x ) + i ' ( y ) = i ' ( x + y ) . \\end{align*}"}
+{"id": "4033.png", "formula": "\\begin{align*} \\det D Y ( \\cdot , v , D v ) & = \\frac { f _ \\delta ( \\cdot ) } { f _ \\epsilon ^ * ( Y ( \\cdot , v , D v ) ) } \\Omega _ \\delta \\\\ Y v ( \\Omega _ \\delta ) & = \\Omega ^ * _ { \\delta , \\epsilon } \\\\ v ( x _ 0 ) & = u _ \\epsilon ( x _ 0 ) \\end{align*}"}
+{"id": "8323.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ \\rho = \\frac 1 2 ( \\partial _ r + \\partial _ t ) , \\ \\partial _ { \\sigma } = \\frac 1 2 ( \\partial _ r - \\partial _ t ) , \\\\ \\partial _ { \\bar { \\rho } } = \\frac 1 2 ( \\partial _ { \\bar { r } } + \\partial _ s ) , \\ \\partial _ { \\sigma } = \\frac 1 2 ( \\partial _ { \\bar { r } } - \\partial _ s ) . \\end{cases} \\end{align*}"}
+{"id": "5708.png", "formula": "\\begin{align*} \\mathbf { A } _ { b } ^ { a } = ( \\mathbf { L } ^ { - 1 } ) _ { c } ^ { a } d \\mathbf { L } _ { b } ^ { c } \\mathbf { , } \\end{align*}"}
+{"id": "1023.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s / 2 } f ( x ) = - c _ { s } \\lim _ { t \\rightarrow { 0 ^ + } } t ^ { 1 - s } \\frac { \\partial u } { \\partial t } ( x , t ) . \\end{align*}"}
+{"id": "1260.png", "formula": "\\begin{align*} R _ { \\mathcal { B S } ( \\alpha ) } ( \\mathcal { K } ) & = \\begin{dcases} \\dfrac { 1 } { 2 - \\alpha } , & 0 \\leq \\alpha \\leq \\frac { 1 } { 5 } , \\\\ \\dfrac { 2 \\sqrt { \\alpha } } { ( 1 + \\alpha ) \\sqrt { 1 + 4 \\alpha } } , & \\frac { 1 } { 5 } \\leq \\alpha \\leq 1 . \\end{dcases} \\end{align*}"}
+{"id": "3137.png", "formula": "\\begin{align*} r ( y _ 1 , y _ 2 ) = r _ 1 ( y _ 1 + y _ 2 ) + r _ 2 ( y _ 1 - y _ 2 ) \\end{align*}"}
+{"id": "5315.png", "formula": "\\begin{align*} \\widetilde { \\nabla } _ { X } \\xi = ( \\xi _ { ( \\rho _ { 0 } , s _ { 0 } ) } \\varphi ) \\cdot X , \\mathrm { a n d } \\widetilde { \\nabla } _ { X } \\eta = ( \\eta _ { ( \\rho _ { 0 } , s _ { 0 } ) } \\varphi ) \\cdot X . \\end{align*}"}
+{"id": "3926.png", "formula": "\\begin{align*} \\det D Y w _ m & = \\frac { ( 1 + \\epsilon ) f ( \\cdot ) } { f ^ * ( Y w _ m ( \\cdot ) ) } B _ r \\\\ w _ m & = u _ m \\quad \\quad \\quad \\quad \\quad \\partial B _ r . \\end{align*}"}
+{"id": "4541.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { a _ i } \\leq \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < \\theta ? \\end{align*}"}
+{"id": "1453.png", "formula": "\\begin{align*} \\lambda = \\frac { k ( k - 1 ) } { m n ( m n - 1 ) } \\cdot b . \\end{align*}"}
+{"id": "9140.png", "formula": "\\begin{align*} \\int _ { \\R ^ { d n } } \\left ( \\prod _ { ( x , y ) \\in E ( f ) } \\nu ( x - y ) \\right ) d y _ 1 \\cdots d y _ n \\ ; = \\ ; ( C _ \\nu ) ^ n , \\ ; \\ ; C _ \\nu = \\int _ { \\R ^ d } \\nu ( y ) d y \\ ; = \\ ; C ( \\beta ) , \\end{align*}"}
+{"id": "3652.png", "formula": "\\begin{align*} \\Phi ( x , t ) = \\Phi ( x , 0 ) + \\int \\limits _ { 0 } ^ { t } \\int \\limits _ { \\mathbb { H } ^ { n } } \\mathcal { K } ( x , y , T - s ) e ^ { \\mu s } u ( y , s ) \\big ( e ^ { \\beta u ^ { p } ( y , s ) } - 1 \\big ) ~ { \\rm d } v _ { \\mathbb { H } ^ { n } } ( y ) \\ , { \\rm d } s . \\end{align*}"}
+{"id": "1063.png", "formula": "\\begin{align*} \\hat { f } _ { \\mathrm { L a p } } = \\sum _ { j = - 1 } ^ J \\sum _ { k \\in \\mathcal { N } _ j } \\hat { \\beta } _ { j k } \\psi _ { j k } . \\end{align*}"}
+{"id": "9027.png", "formula": "\\begin{align*} \\begin{cases} ( i ) Y _ t = \\xi + \\int _ t ^ T g ( s ) d s + ( K ^ 1 _ T - K ^ 1 _ t ) - ( K ^ 2 _ T - K ^ 2 _ t ) - \\int _ t ^ T Z _ s d B _ s , t \\in [ 0 , T ] , \\\\ ( i i ) \\zeta _ t \\geq Y _ { t } \\geq \\chi _ t , \\ , \\ , t \\in [ 0 , T ] , \\\\ ( i i i ) \\int _ 0 ^ T ( Y _ { t } - \\zeta _ t ) d K ^ 1 _ t = 0 , \\int _ 0 ^ T ( Y _ { t } - \\chi _ t ) d K ^ 2 _ t = 0 , \\end{cases} \\end{align*}"}
+{"id": "3490.png", "formula": "\\begin{align*} { \\bf { f } } _ l ^ { { \\rm { Z F } } } \\to { \\bf { f } } _ { l } ^ { { \\rm { M R T } } } = \\frac { { \\sqrt P { { \\bf { h } } _ l } } } { { \\sqrt { \\sum \\nolimits _ { l = 1 } ^ L { { { \\left \\| { { { \\bf { h } } _ l } } \\right \\| } ^ 2 } } } } } , \\ \\forall l , \\end{align*}"}
+{"id": "5902.png", "formula": "\\begin{align*} w = ( E , \\widetilde C _ { p - 1 } U ) , { \\cal L } \\theta = - ( E , \\widetilde C _ { p - 1 } A U ) , { \\cal R } \\theta = - ( E , \\widetilde C _ { p - 1 } B U ) . \\end{align*}"}
+{"id": "971.png", "formula": "\\begin{align*} \\widehat { h } : = \\psi - 2 r \\cdot \\varphi \\in H ^ 2 ( \\widehat { X } ) , \\end{align*}"}
+{"id": "613.png", "formula": "\\begin{align*} { \\rm d } \\mu _ t = \\frac { \\sigma _ t } { \\gamma } \\left ( ( A Z _ t ^ \\dagger ) ^ { \\rm T } { \\rm d } X _ t ^ \\dagger - \\mu _ t ( A ^ { \\rm T } A ) : \\tilde C \\ , { \\rm d } t \\right ) \\end{align*}"}
+{"id": "5629.png", "formula": "\\begin{align*} V _ { 0 } ( t ) = \\int ^ { S ( t ) } _ { S _ 0 } \\dfrac { F _ { 1 } ( x , 0 ) - F _ { 1 } ( S _ 0 , 0 ) } { F _ { 1 } ( x , 0 ) } d x + E ( t ) + \\dfrac { m _ { 1 } } { \\sigma ^ { \\alpha } } I . \\end{align*}"}
+{"id": "8488.png", "formula": "\\begin{align*} C _ { p t } = \\ ; & ( W C _ { \\xi } + U C _ { \\xi ^ 2 } ) _ p \\\\ = \\ ; & g ( W _ { \\xi } C _ { \\xi } + W C _ { \\xi ^ 2 } + U _ { \\xi } C _ { \\xi ^ 2 } + U C _ { \\xi ^ 3 } ) \\\\ = \\ ; & g \\Big ( W _ { \\xi } - U \\lambda \\Big ) C _ { \\xi } + g \\Big ( W + U _ { \\xi } - U \\varphi \\Big ) C _ { \\xi ^ 2 } , \\end{align*}"}
+{"id": "36.png", "formula": "\\begin{align*} T ( n _ { i + 1 } ) + T ( n _ { i + 2 } ) & < \\frac { 1 } { n _ { i + 1 } } \\cdot \\frac { 3 5 } { 9 } < \\frac { 3 5 } { 9 } \\cdot \\frac { 1 6 } { 2 7 } \\cdot \\frac { 1 } { n _ i } \\\\ & = \\frac { 5 6 0 } { 2 4 3 } \\cdot \\frac { 1 } { n _ i } , \\\\ T ( n _ i ) + T ( n _ { i + 1 } ) + T ( n _ { i + 2 } ) & < \\left ( \\frac { 1 9 } { 9 } + \\frac { 5 6 0 } { 2 4 3 } \\right ) \\cdot \\frac { 1 } { n _ i } = \\frac { 1 0 7 3 } { 2 4 3 } \\cdot \\frac { 1 } { n _ i } \\\\ & < 3 \\cdot \\frac { 9 7 } { 5 4 } \\cdot \\frac { 1 } { X _ 0 } . \\end{align*}"}
+{"id": "5931.png", "formula": "\\begin{align*} t \\geq T : \\widetilde C _ 1 U = 0 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "8941.png", "formula": "\\begin{align*} u _ 0 ^ 2 = 1 , u _ 1 ^ 2 = 0 , u _ i ^ 2 = \\frac { ( - 1 ) ^ { i } B _ { 2 i } } { i ( i - 2 ) ! } , \\ i \\geq 2 \\end{align*}"}
+{"id": "1513.png", "formula": "\\begin{align*} F ( x ) = \\frac { \\C _ r ( x + 1 ) } { \\C _ 1 ( x ) \\C _ 2 ( x ) ^ { \\binom { r - 1 } { 1 } } \\C _ 3 ( x ) ^ { \\binom { r - 1 } { 2 } } \\cdots \\C _ r ( x ) ^ { \\binom { r - 1 } { r - 1 } } } . \\end{align*}"}
+{"id": "3457.png", "formula": "\\begin{align*} F = f ( W ( \\varphi _ 1 ) , \\ldots , W ( \\varphi _ n ) ) , \\end{align*}"}
+{"id": "1769.png", "formula": "\\begin{align*} \\forall v \\in V ( G _ k ) \\pi ' _ k ( v ) = u ' \\in V ( G ' _ k ) , \\ ; \\ ; \\hbox { i f } Y _ { u ' } = - X _ v , \\end{align*}"}
+{"id": "1331.png", "formula": "\\begin{align*} ( ^ m ( \\mathcal { V } ) ) = { d + m - 1 \\choose m } , \\forall m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "7977.png", "formula": "\\begin{align*} \\textrm { I f } x _ i \\equiv y _ i \\textrm { f o r $ i = 1 , 2 $ , t h e n , $ ( x _ 1 + x _ 2 ) \\equiv ( y _ 1 + y _ 2 ) $ a n d $ ( r x _ i ) \\equiv ( r y _ i ) $ , } \\end{align*}"}
+{"id": "5376.png", "formula": "\\begin{align*} D F = \\left [ \\begin{array} { c c c c c } 0 & 0 & 0 & f _ { 1 4 } & f _ { 1 5 } \\\\ 0 & 0 & 0 & f _ { 2 4 } & f _ { 2 5 } \\\\ 0 & 0 & 0 & f _ { 3 4 } & f _ { 3 5 } \\\\ f _ { 4 1 } & f _ { 4 2 } & f _ { 4 3 } & 0 & 0 \\\\ f _ { 5 1 } & f _ { 5 2 } & f _ { 4 3 } & 0 & 0 \\\\ \\end{array} \\right ] \\end{align*}"}
+{"id": "2806.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n c _ i = \\sum _ { i = 1 } ^ n ( a _ i - b _ i ) = \\sum _ { i = 1 } ^ n a _ i - \\sum _ { i = 1 } ^ n b _ i = 0 . \\end{align*}"}
+{"id": "2477.png", "formula": "\\begin{align*} \\begin{aligned} J _ v ^ 1 ( E / L ) : = \\underset { w \\mid v } \\prod \\frac { H ^ 1 _ { f l } ( L _ w , E _ { p ^ \\infty } ) } { i m ( \\kappa _ w ) } \\ K _ v ^ 1 ( E / L ) : = \\underset { w \\mid v } \\prod H ^ 1 _ { f l } ( L _ w , E _ { p ^ \\infty } ) . \\end{aligned} \\end{align*}"}
+{"id": "3927.png", "formula": "\\begin{align*} v = u _ m + \\frac { k } { 2 } ( | x | ^ 2 - r ^ 2 ) , \\end{align*}"}
+{"id": "5136.png", "formula": "\\begin{align*} h _ { 1 } ( w ) = \\sum _ { n = 1 } ^ { \\infty } a _ { n } \\overline { w } ^ { n \\mathbf { m } - 1 } \\quad \\mbox { a n d } h _ { 2 } ( w ) = \\sum _ { n = 1 } ^ { \\infty } b _ { n } \\overline { w } ^ { n \\mathbf { m } - 1 } . \\end{align*}"}
+{"id": "7860.png", "formula": "\\begin{align*} x ^ 2 g '' ( x ) + ( x + 1 ) g ' ( x ) + \\alpha g ( x ) = 0 \\end{align*}"}
+{"id": "375.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 \\leq \\frac { \\sqrt { 2 } C v ^ 2 _ 0 L ^ { - 2 } T ^ 2 } { 3 } \\frac { \\widetilde { C } \\ln ( \\lambda T + 3 ) } { ( \\lambda T ) ^ { 3 / 2 } } = 3 ^ { - 1 } C _ 2 { L } ^ { - 2 } _ * { T } ^ { 1 / 2 } _ * \\ln ( { T } _ * + 3 ) \\ , , \\end{align*}"}
+{"id": "5193.png", "formula": "\\begin{align*} \\varphi ( z , t ) = A ( t ) v ( z ) \\ , , \\end{align*}"}
+{"id": "7318.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } c ^ d _ { m _ n } = 0 . \\end{align*}"}
+{"id": "4108.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } g _ t = h , g _ 0 = g , \\\\ & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } H _ t = d K , H _ 0 = H \\end{align*}"}
+{"id": "6627.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\norm { x ^ m ( t ) \\psi } ^ 2 & \\leq C _ { m , d } \\sum _ { j = 1 } ^ d \\norm { x _ j ^ m ( t ) \\psi } ^ { 2 - \\frac { 1 } { m } } \\norm { D _ j ^ m ( t ) \\psi } ^ { 1 / m } \\\\ & \\leq C _ { m , d , V } \\norm { x ^ m ( t ) \\psi } ^ { 2 - \\frac { 1 } { m } } \\sum _ { k = 0 } ^ m \\norm { H ^ k \\psi } ^ { 1 / m } . \\end{align*}"}
+{"id": "726.png", "formula": "\\begin{align*} \\Big | \\frac { d w } { d t } \\Big | = \\frac { C } { \\lambda ( w ) } . \\end{align*}"}
+{"id": "8213.png", "formula": "\\begin{align*} P \\{ N ( t ) = 2 k + 1 \\} = ( \\lambda _ 1 t ) ^ { k + 1 } ( \\lambda _ 2 t ) ^ k e ^ { - \\lambda _ 1 t } \\ , E ^ { k + 1 } _ { 1 , 2 k + 2 } \\Bigl ( t ( \\lambda _ 1 - \\lambda _ 2 ) \\Bigr ) , \\end{align*}"}
+{"id": "3653.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb { H } ^ { n } } \\mathcal { K } ( x , y , T - s ) ~ { \\rm d } v _ { \\mathbb { H } ^ { n } } ( y ) = 1 , \\end{align*}"}
+{"id": "8384.png", "formula": "\\begin{align*} w _ { X } ( \\mathcal { B } ) = \\aleph _ 0 + \\inf \\{ | \\mathcal { B } _ 0 | : \\mathcal { B } _ 0 \\mathcal { B } \\} . \\end{align*}"}
+{"id": "3284.png", "formula": "\\begin{align*} \\lambda _ { 1 } \\left ( p , p \\right ) = \\left ( p - 1 \\right ) \\inf \\ \\left \\{ \\frac { \\left \\Vert \\Delta u \\right \\Vert _ { p } } { \\left \\Vert \\nabla u \\right \\Vert _ { p } } \\left \\vert \\ u \\in W ^ { 2 , p } \\left ( \\Omega \\right ) \\cap W _ { 0 } ^ { 1 , p } \\left ( \\Omega \\right ) \\right . \\right \\} . \\end{align*}"}
+{"id": "1326.png", "formula": "\\begin{align*} | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | & \\leq \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ k ( \\tau _ j ) | \\\\ & = \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | = \\left ( \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | \\right ) ^ 2 . \\end{align*}"}
+{"id": "9152.png", "formula": "\\begin{align*} \\frac { e ^ { - q ( n \\pm 1 ) } ( q ( n \\pm 1 ) ) ^ { \\kappa - 1 } q } { e ^ { - q n } ( q n ) ^ { \\kappa - 1 } q } = e ^ { \\mp q } \\left ( 1 \\pm \\frac { 1 } { n } \\right ) ^ { \\kappa - 1 } , \\end{align*}"}
+{"id": "3570.png", "formula": "\\begin{align*} v _ A : = \\prod _ { k = 2 , \\dots , n } ^ \\rightarrow \\left ( y _ { k 1 } ^ { ( \\gamma _ A ) _ { k 1 } } \\cdots y _ { k , k - 1 } ^ { ( \\gamma _ A ) _ { k , k - 1 } } \\right ) ( z _ n ^ + ) ^ { \\lambda _ n } \\cdots ( z _ 1 ^ + ) ^ { \\lambda _ 1 } v _ 0 , \\end{align*}"}
+{"id": "4007.png", "formula": "\\begin{align*} w ^ { i i } D _ { p _ k p _ l } A _ { i i } D _ k \\eta D _ l \\eta & = \\sum _ { k , l \\neq i } D _ { p _ k p _ l } A _ { i i } D _ k \\eta D _ l \\eta + 2 \\sum _ { l \\neq i } D _ { p _ i p _ l } A _ { i i } D _ i \\eta D _ l \\eta \\\\ & \\quad \\quad + D _ { p _ i p _ i } A _ { i i } D _ i \\eta D _ i \\eta \\end{align*}"}
+{"id": "7086.png", "formula": "\\begin{align*} F _ n A & = F _ n M U ^ { C _ 2 } _ * \\cdot F _ 0 A \\\\ F _ n D & = F _ n M U ^ { C _ 2 } _ * \\cdot F _ 0 D . \\end{align*}"}
+{"id": "6751.png", "formula": "\\begin{align*} \\beta _ n ( s , \\rho ) = n \\sqrt { 2 \\pi } \\left ( 1 - \\rho \\right ) ^ { \\frac { 1 } { 2 } } \\left ( 1 - \\rho \\right ) ^ { \\frac { 1 } { 2 ( 1 - s ) } } \\end{align*}"}
+{"id": "3877.png", "formula": "\\begin{align*} g _ { j } + g _ { , k } D _ j Y ^ k + g _ z D _ j Z & = D _ j u \\\\ g _ { i j } + g _ { i , k } D _ j Y ^ k + g _ { i , z } D _ j Z & = D _ { i j } u . \\end{align*}"}
+{"id": "5331.png", "formula": "\\begin{align*} \\int _ { \\Omega } u \\cdot v \\ , d x = \\int _ { \\Omega } ( u _ 1 v _ 1 + u _ 2 v _ 2 + u _ 3 v _ 3 ) \\ , d x \\forall u , v \\in L ^ 2 ( \\Omega ) ^ 3 . \\end{align*}"}
+{"id": "1818.png", "formula": "\\begin{align*} \\begin{cases} u _ 1 ( t ) = u _ 1 ( - t ) , \\\\ u _ 2 ( t ) = u _ 3 ( - t ) , \\\\ u _ 3 ( t ) = u _ 2 ( - t ) . \\end{cases} \\end{align*}"}
+{"id": "9205.png", "formula": "\\begin{align*} E ( m _ { i , j } ) : = E _ { i , j } , \\end{align*}"}
+{"id": "8165.png", "formula": "\\begin{align*} \\mathcal { O ^ + } = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } \\sum _ { c > 0 } \\frac { S ( n , p ; c ) } { c } H _ { m , n } ^ + \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) , \\end{align*}"}
+{"id": "7146.png", "formula": "\\begin{align*} \\hat h \\circ \\hat g & = \\sum _ { i = 0 } ^ m \\sum _ { j = i } ^ { n + m } h _ i \\star \\delta _ 0 \\varphi ^ { j - i , m - i } _ 0 \\ x ^ { m + n - j } = \\sum _ { i = 0 } ^ m \\sum _ { s = 0 } ^ { n + m - i } h _ i \\star \\delta _ 0 \\varphi ^ { s , m - i } _ 0 \\ x ^ { m + n - i - s } . \\end{align*}"}
+{"id": "8388.png", "formula": "\\begin{align*} & w ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { s w } ) = w _ X ( \\mathcal { B } ) \\cdot i b ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { s w } ) = w _ X ( \\mathcal { B } ) \\cdot c ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { s w } ) \\\\ & = w _ X ( \\mathcal { B } ) \\cdot L ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { s w } ) = w _ X ( \\mathcal { B } ) \\cdot d ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { s w } ) = w _ X ( \\mathcal { B } ) \\cdot n w ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { s w } ) . \\end{align*}"}
+{"id": "3800.png", "formula": "\\begin{align*} 0 & \\leq - 2 \\ , | | ^ 2 + 4 n ( 1 2 + N _ 0 + N _ 1 r ) ( N _ 0 + N _ 1 \\sqrt { \\rho } ) r ^ 2 ( r ^ 2 - \\rho ) ^ { - 3 } \\\\ & + 4 8 n ( 1 2 + N _ 0 + N _ 1 r ) \\rho r ^ 2 ( r ^ 2 - \\rho ) ^ { - 4 } \\\\ & \\leq - 8 n ( 1 2 + N _ 0 + N _ 1 r ) ^ 2 r ^ 4 ( r ^ 2 - \\rho ) ^ { - 4 } \\\\ & + 4 n ( 1 2 + N _ 0 + N _ 1 r ) ( N _ 0 + N _ 1 r ) r ^ 4 ( r ^ 2 - \\rho ) ^ { - 4 } \\\\ & + 4 8 n ( 1 2 + N _ 0 + N _ 1 r ) r ^ 4 ( r ^ 2 - \\rho ) ^ { - 4 } \\\\ & = - 4 n ( 1 2 + N _ 0 + N _ 1 r ) ^ 2 r ^ 4 ( r ^ 2 - \\rho ) ^ { - 4 } \\end{align*}"}
+{"id": "6518.png", "formula": "\\begin{align*} \\P ^ U \\left ( \\frac { m - 1 } { 2 \\sqrt { b } } \\underset { i = 1 } { \\overset { b } { \\sum } } \\left ( ( p _ { f , U } ) _ i - \\frac { 1 } { 2 } \\right ) ^ 2 \\leq N _ \\alpha \\right ) & \\leq \\P ^ U \\left ( \\frac { m - 1 } { 2 4 \\sqrt { b } } \\underset { i = 1 } { \\overset { b } { \\sum } } \\min \\left \\{ \\frac { n } { m } ( U f ) _ i ^ 2 , 1 \\right \\} \\leq N _ \\alpha \\right ) . \\end{align*}"}
+{"id": "8511.png", "formula": "\\begin{align*} C _ { p } = \\bar { g } C _ { \\sigma } , C _ { p ^ 2 } = \\bar { g } \\bar { g } _ { \\sigma } C _ { \\sigma } + \\bar { g } ^ 2 C _ { \\sigma ^ 2 } , \\end{align*}"}
+{"id": "4322.png", "formula": "\\begin{align*} d _ p ( a _ 0 , a _ 1 , \\dots , a _ p ) = ( a _ p + a _ 0 , a _ 1 , \\dots , a _ { p - 1 } ) . \\end{align*}"}
+{"id": "4667.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { \\min \\{ 2 p , 2 m \\} } A _ { m , \\ell } = \\sum _ { \\ell = 0 } ^ { 2 p } \\sum _ { m = \\lceil \\ell / 2 \\rceil } ^ { p } A _ { m , \\ell } + \\sum _ { \\ell = 0 } ^ { 2 p } \\sum _ { m = p + 1 } ^ { \\infty } A _ { m , \\ell } = \\sum _ { \\ell = 0 } ^ { 2 p } \\sum _ { m = \\lceil \\ell / 2 \\rceil } ^ { \\infty } A _ { m , \\ell } \\end{align*}"}
+{"id": "3597.png", "formula": "\\begin{align*} F _ j ( \\lambda _ 1 , \\dots , \\lambda _ n ) = ( - 1 ) ^ { \\lambda _ { j + 1 } + \\cdots + \\lambda _ n } ( \\lambda _ j + n + 1 - j + [ \\lambda _ j + 1 ] _ 2 ( p - n ) ) ^ { \\frac { 1 } { 2 } } \\bigg ( \\prod _ { \\substack { \\ell = 1 , \\\\ \\ell \\neq i } } ^ n \\frac { \\lambda _ j - \\lambda _ \\ell - j + \\ell } { \\lambda _ j - \\lambda _ \\ell - j + \\ell + [ \\lambda _ j - \\lambda _ \\ell ] _ 2 } \\bigg ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "8451.png", "formula": "\\begin{align*} \\xi _ A ( X ) = 1 , \\quad \\| \\xi _ A \\| ^ 2 = w ( A ) . \\end{align*}"}
+{"id": "5948.png", "formula": "\\begin{align*} 0 = ( A \\widehat { U } , E _ r ) = ( \\widehat { U } , A ^ T E _ r ) = ( \\widehat { U } , C _ p ^ T P _ r ) = ( C _ p \\widehat { U } , P _ r ) \\end{align*}"}
+{"id": "4879.png", "formula": "\\begin{align*} \\int _ { t _ 2 } ^ 1 y ( t _ 1 ) ( t _ 2 - t _ 1 ) d t _ 1 & = t _ 2 \\int _ { t _ 2 } ^ 1 y ( t _ 1 ) d t _ 1 - \\int _ { t _ 2 } ^ 1 t _ 1 y ( t _ 1 ) d t _ 1 \\\\ & = t _ 2 \\int _ { t _ 2 } ^ 1 y ( t _ 1 ) d t _ 1 - \\left [ s \\int _ { s } ^ 1 y ( t _ 1 ) d t _ 1 \\right ] _ { s = t _ 2 } + \\int _ { t _ 2 } ^ 1 \\int _ { s } ^ 1 y ( t _ 1 ) d t _ 1 d s \\\\ & = \\int _ { t _ 2 } ^ 1 \\int _ { s } ^ 1 y ( t _ 1 ) d t _ 1 d s . \\end{align*}"}
+{"id": "8174.png", "formula": "\\begin{align*} \\sum \\limits _ { p + q = k } h ^ { p , q } _ { ( 2 ) } ( X _ j ) = \\sum \\limits _ { p + q = k } { \\rm d i m } _ \\mathbf { G _ j } ( \\mathcal { H } ^ { p , q } _ { ( 2 ) } ( X _ j ) ) \\rightarrow \\sum \\limits _ { p + q = k } { \\rm d i m } _ \\mathbf { G } ( \\mathcal { H } ^ { p , q } _ { ( 2 ) } ( \\widetilde { X } ) ) = \\sum \\limits _ { p + q = k } h ^ { p , q } _ { ( 2 ) } ( \\widetilde { X } ) , \\end{align*}"}
+{"id": "206.png", "formula": "\\begin{align*} \\int \\Big { | } \\frac { 1 } { L } \\sum \\limits _ { m = 1 } ^ { L } \\chi _ { B } \\circ T ^ { - m t _ n } - \\mu ( B ) \\Big { | } ^ { 2 } d \\mu & = \\frac { 1 } { L ^ { 2 } } \\sum \\limits _ { r , m = 1 } ^ { L } \\mu ( T ^ { ( m - r ) t _ { n } } ( B ) \\cap B ) - \\mu ( B ) \\mu ( B ) \\\\ & \\leq \\frac { 1 } { L } + \\frac { 1 } { L } \\sum \\limits _ { \\ell = 1 } ^ { L - 1 } \\frac { L - \\ell } { L } \\mu ( T ^ { \\ell t _ { n } } ( B ) \\cap B ) - \\mu ( B ) \\mu ( B ) < 2 \\epsilon ^ { 2 } / 2 = \\epsilon ^ { 2 } \\end{align*}"}
+{"id": "1072.png", "formula": "\\begin{align*} \\mathbb { E } _ { P , Q } \\Big [ \\frac { \\hat { N } _ 0 } { n } \\Big ] = \\frac { e ^ \\alpha } { e ^ \\alpha + 1 } \\mathbb { P } ( Y _ 1 = 0 ) + \\frac { 1 } { e ^ \\alpha + 1 } \\mathbb { P } ( Y _ 1 = 1 ) = \\frac { 1 } { e ^ \\alpha + 1 } \\Big \\{ 1 + ( e ^ \\alpha - 1 ) P ( A ) \\Big \\} . \\end{align*}"}
+{"id": "8333.png", "formula": "\\begin{align*} \\begin{cases} \\square _ { s , y } \\tilde { v } = \\sum _ { j = 1 } ^ 5 G _ j ^ { ( 3 ) } \\tilde { v } ^ j , \\\\ \\tilde { v } | _ { s = 0 } = v _ 0 , \\partial _ s \\tilde { v } | _ { s = 0 } = v _ 1 , \\end{cases} \\end{align*}"}
+{"id": "5046.png", "formula": "\\begin{align*} F _ 1 \\odot F _ 2 = \\sum _ { g \\in G } a _ g b _ g { \\vec x } ^ g . \\end{align*}"}
+{"id": "4649.png", "formula": "\\begin{align*} \\tilde { k } _ { i i } = 2 k _ { i i } , \\ \\ \\ \\ \\tilde { k } _ { i j } = k _ { i j } \\ \\ \\ \\ i \\neq j . \\end{align*}"}
+{"id": "8455.png", "formula": "\\begin{align*} \\| \\gamma _ A - \\gamma _ H \\| ^ 2 = \\| \\gamma _ A \\| ^ 2 - \\| \\gamma _ H \\| ^ 2 . \\end{align*}"}
+{"id": "6542.png", "formula": "\\begin{align*} { F _ X } \\left ( x \\right ) = \\frac { { { m ^ m } } } { { \\Gamma \\left ( m \\right ) } } \\sum \\limits _ { j = 0 } ^ M { \\frac { { { K ^ j } { \\alpha _ j } } } { { j ! } } } \\frac { 1 } { { \\Gamma \\left ( { j + 1 } \\right ) } } \\gamma \\left ( { j + 1 , \\frac { x } { { 2 { \\sigma ^ 2 } } } } \\right ) , \\end{align*}"}
+{"id": "3967.png", "formula": "\\begin{align*} \\det D Y ( \\cdot , v _ h , D v _ h ) = c / 2 B , \\\\ v _ h = u _ h + \\epsilon \\partial B , \\end{align*}"}
+{"id": "2549.png", "formula": "\\begin{align*} \\begin{aligned} Z ( \\sigma ^ { b , 1 } _ { r } ( v ) ; ( \\alpha , \\beta ) ) = & \\sum _ { \\begin{subarray} { c } c ^ { ' } _ { 1 } r _ 1 + \\sum _ { i = 2 } ^ { q } r _ i = r \\\\ c ^ { ' } _ { 1 } r _ 1 , r _ i \\ge 0 \\end{subarray} } Z ^ { * } _ { ( c ^ { ' } _ 1 r _ 1 , \\{ r _ i \\} _ { i = 2 } ^ { q } ) } ( \\tau ( v ) ; ( \\beta , \\alpha ) ) \\end{aligned} \\end{align*}"}
+{"id": "704.png", "formula": "\\begin{align*} a = \\oint _ \\alpha \\nu , b = \\oint _ \\beta \\nu . \\end{align*}"}
+{"id": "3902.png", "formula": "\\begin{align*} D ^ 2 \\tilde { u } ( x ) - A _ { i j } ( x , \\tilde { u } , D \\tilde { u } ) & = D ^ 2 u - A _ { i j } ( \\cdot , u , D u ) \\\\ & \\quad \\quad + \\epsilon [ I - A _ { i j , u } | x - x _ 0 | ^ 2 + A _ { i j , p _ k } \\cdot ( x - x _ 0 ) ] . \\end{align*}"}
+{"id": "381.png", "formula": "\\begin{align*} \\pi ^ * \\left ( A _ { \\eta _ * ( n ) } ~ \\big | K = n \\right ) \\leq C ^ { ( 2 ) } _ { \\textsf { K M T } } e ^ { - \\vartheta \\ln ( n ) } = \\frac { C ^ { ( 2 ) } _ { \\textsf { K M T } } } { n ^ \\vartheta } \\ , , \\end{align*}"}
+{"id": "5639.png", "formula": "\\begin{align*} \\zeta ( F ) = \\int _ { \\Delta _ F } \\prod _ { n = 0 } ^ { N _ F } \\prod _ { s _ v \\in S _ n ( F ) } d \\omega _ { s _ v } \\end{align*}"}
+{"id": "2019.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ \\infty \\theta ^ \\mathrm { m a x } _ t d t = \\frac { R ^ 2 } { 2 } . \\end{align*}"}
+{"id": "3742.png", "formula": "\\begin{align*} f _ s ( \\xi ) : & = \\int _ { F ^ \\times } | a | ^ { s + 1 } f ( a \\xi ) d ^ \\times a , \\xi \\in X _ { P _ 2 } ^ \\circ ( F ) \\end{align*}"}
+{"id": "2336.png", "formula": "\\begin{align*} & \\omega _ { N - 1 } ^ { 1 - \\frac { q } { N } } \\int _ 0 ^ { \\infty } \\frac { | v ( t ) | ^ q } { t ^ { 1 + q ( \\frac { N - 1 } { N } ) } } d t = \\omega _ { N - 1 } \\int _ 0 ^ 1 \\frac { | u ( r ) | ^ q } { r \\ ( \\log \\frac { 1 } { r } \\ ) ^ { 1 + q ( \\frac { N - 1 } { N } ) } } \\ , d r = \\int _ { B _ 1 ^ N } \\frac { | u ( x ) | ^ q } { | x | ^ N \\ ( \\log \\frac { 1 } { | x | } \\ ) ^ { 1 + q ( \\frac { N - 1 } { N } ) } } d x , \\\\ & \\int _ 0 ^ { \\infty } | v ' ( t ) | ^ N d t = \\int _ { B _ 1 ^ N } | \\nabla u | ^ N d x . \\end{align*}"}
+{"id": "3324.png", "formula": "\\begin{align*} \\theta = a _ i \\dd q ^ i + b _ i \\dd p _ i + c \\kappa , a _ i , b _ i \\in \\R , ~ i = 1 , \\dots , n , c \\neq 0 , \\end{align*}"}
+{"id": "4534.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { x _ 2 } \\leq \\frac { 1 } { 3 } + \\frac { 1 } { 4 } = \\frac { 7 } { 1 2 } < \\frac { 1 } { 2 } + \\frac { 1 } { x _ 2 - 1 } \\end{align*}"}
+{"id": "2514.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\alpha _ n ^ d } { n ^ d } = \\lim _ { n \\to \\infty } \\frac { n ^ d - ( 2 ^ d - 1 ) n ^ { d - 1 } + \\mathcal { O } ( n ^ { d - 2 } ) } { n ^ d } = 1 , \\end{align*}"}
+{"id": "8736.png", "formula": "\\begin{align*} \\left ( 1 + { \\beta } { u } \\right ) H ^ { \\perp } ( - 1 / \\beta ) H ( u ) = H ( u ) H ^ { \\perp } ( - 1 / \\beta ) , \\end{align*}"}
+{"id": "4005.png", "formula": "\\begin{align*} 0 & \\geq \\kappa ( w ^ { i i } - C ) + \\tau [ w _ { i i } - C ( 1 + w ^ { i i } ) ] + \\frac { 1 } { 2 w _ { 1 1 } ^ 2 } \\sum _ { i = 2 } ^ n w ^ { i i } w _ { 1 1 , i } ^ 2 \\\\ & \\quad - C ( 1 + w _ { i i } + w ^ { i i } ) + \\beta L ( \\log \\eta ) . \\end{align*}"}
+{"id": "1744.png", "formula": "\\begin{align*} I ( \\chi , \\underline { m } ) : = \\int _ { T ( \\C ) } \\chi ( t ) \\langle \\imath ( t ) \\delta s ( \\mu _ { \\underline { m } } ) , \\delta s ( \\mu _ { - \\underline { m } } ) \\rangle t ^ { \\underline { m } } d ^ \\times t , \\end{align*}"}
+{"id": "5091.png", "formula": "\\begin{align*} C / Q = \\alpha + \\sum _ { i = 1 } ^ s \\frac { \\beta _ i } { 1 - \\lambda _ i x _ d } , \\end{align*}"}
+{"id": "3156.png", "formula": "\\begin{align*} c ^ { 1 1 } _ j ( A ) + c ^ { 2 2 } _ j ( A ) = \\int _ Y r A e _ j \\cdot \\nabla ( v ^ { 1 1 } + v ^ { 2 2 } ) = 0 \\forall j \\in \\{ 1 , 2 \\} . \\end{align*}"}
+{"id": "4692.png", "formula": "\\begin{align*} \\mathfrak { R } ^ G = \\lbrace f \\in \\mathbb { C } [ x , y ] \\ \\vert \\ f ^ g = f , \\ \\forall g \\in G _ X \\rbrace . \\end{align*}"}
+{"id": "1678.png", "formula": "\\begin{align*} f _ n \\left ( u \\left ( \\begin{array} { c c } y & x \\\\ & y ^ { - 1 } \\end{array} \\right ) \\kappa ( \\theta ) \\right ) = y ^ k \\cdot e ^ { 2 i n \\theta } . \\end{align*}"}
+{"id": "4713.png", "formula": "\\begin{align*} e _ i e _ j = \\sum r _ { i j } ^ k e _ k . \\end{align*}"}
+{"id": "3287.png", "formula": "\\begin{align*} v | _ { t = t _ 0 } = v _ 0 \\textnormal { f o r } x \\in \\mathcal { F } _ 0 , \\ell ( t _ 0 ) = \\ell _ 0 , \\omega ( t _ 0 ) = \\omega _ 0 . \\end{align*}"}
+{"id": "8100.png", "formula": "\\begin{align*} k ^ * ( u ) = e ^ { - u ^ 2 } V ( m ^ 2 n , M u + T ) \\end{align*}"}
+{"id": "6038.png", "formula": "\\begin{align*} f ( z ) = \\int \\frac { \\dd \\mu ( x ) } { z - x } \\end{align*}"}
+{"id": "6375.png", "formula": "\\begin{align*} \\frac { d } { d r } \\ : \\frac { - m '' _ \\lambda ( r ) } { m _ \\lambda ( r ) } = \\frac { P _ \\lambda ( u ( r ) ) } { l _ \\lambda ^ 2 \\cdot ( x _ \\lambda ' ( u ( r ) ) ^ 2 + y ' ( u ( r ) ) ^ 2 ) ^ 3 } \\cdot \\frac { d u } { d r } , \\end{align*}"}
+{"id": "3060.png", "formula": "\\begin{align*} A ( y ) : = a ( y ) B ( y ) , B ( y ) : = \\mathrm { d i a g } ( 1 + b ( y ) , 1 - b ( y ) ) \\end{align*}"}
+{"id": "1619.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( \\hat S ^ { ( t ) } ) \\leq \\frac { 1 } { C } \\sum _ { c = 1 } ^ C M \\psi ( \\tau ^ { ( t ) } _ c , \\theta _ c , M ) \\end{align*}"}
+{"id": "8426.png", "formula": "\\begin{align*} \\Gamma _ { A , \\mu } ^ + : = \\bigl \\{ \\nu \\in \\mathcal E ^ + : \\ \\kappa \\nu \\geqslant \\kappa \\mu \\bigr \\} . \\end{align*}"}
+{"id": "3644.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } a ^ { \\prime } ( t ) = h ( t ) \\ , a ( t ) \\ , ( e ^ { \\beta ( C a ( t ) ) ^ p } - 1 ) \\\\ a ( 0 ) = 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "6654.png", "formula": "\\begin{align*} b ( t ' , x ' ) = \\nabla u ( t ' , x ' ) - \\nabla v _ { a ( t ' , x ' ) } ( t ' , x ' ) . \\end{align*}"}
+{"id": "2285.png", "formula": "\\begin{align*} u _ t = b ( u ) _ { x x } , \\end{align*}"}
+{"id": "7770.png", "formula": "\\begin{align*} \\mu \\ , \\rho ( x ) & = \\lim _ { n \\to \\infty } n \\ , x \\ , f ( x | \\tfrac { \\mu } { n } ) \\\\ { \\rm a n d } \\mu \\ , r _ \\alpha ( x ) & = \\lim _ { n \\to \\infty } n \\ , x \\ , m _ \\alpha ( x | \\tfrac { \\mu } { n } ) \\\\ & = \\phantom { \\mu \\ , } x \\int _ 0 ^ \\infty f _ \\alpha ( x | y ) \\lim _ { n \\to \\infty } n \\ , f ( y | \\tfrac { \\mu } { n } ) d y \\\\ & = \\mu \\ , x \\int _ 0 ^ \\infty f _ \\alpha ( x | y ) \\ , y ^ { - 1 } \\rho ( y ) d y \\end{align*}"}
+{"id": "4382.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n - 1 } \\frac { 1 } { a _ i } + \\frac { 1 } { a _ n - 1 } & = \\sum _ { i = 1 } ^ { n - 2 } \\frac { 1 } { a _ i } + \\frac { 1 } { a _ { n - 1 } } + \\frac { 1 } { a _ n - 1 } \\leq \\sum _ { i = 1 } ^ { n - 2 } \\frac { 1 } { a _ i } + \\frac { 1 } { a _ { n - 1 } - 1 } \\\\ & \\leq \\cdots \\leq \\sum _ { i = 1 } ^ { k - 1 } \\frac { 1 } { a _ i } + \\frac { 1 } { a _ k - 1 } \\end{align*}"}
+{"id": "5966.png", "formula": "\\begin{align*} & ( E _ r , A U ) = ( A ^ T E _ r , U ) \\\\ = & ( \\sum _ { s = 1 } ^ p \\alpha _ { r s } E _ s + A ^ T E _ r - \\sum _ { s = 1 } ^ p \\alpha _ { r s } E _ s , U ) \\\\ = & \\sum _ { s = 1 } ^ p \\alpha _ { r s } ( E _ s , U ) + ( A ^ T E _ r - \\sum _ { s = 1 } ^ p \\alpha _ { r s } E _ s , U ) \\\\ = & \\sum _ { s = 1 } ^ p \\alpha _ { r s } \\psi _ s + ( A ^ T E _ r - \\sum _ { s = 1 } ^ p \\alpha _ { r s } E _ s , U ) . \\end{align*}"}
+{"id": "4112.png", "formula": "\\begin{align*} \\Big ( ( h _ 1 , K _ 1 , v _ { h _ 1 , K _ 1 } ) , ( h _ 2 , K _ 2 , v _ { h _ 2 , K _ 2 } ) \\Big ) _ { f } = \\int _ M \\Big ( \\langle h _ 1 , h _ 2 \\rangle + \\langle K _ 1 , K _ 2 \\rangle + v _ { h _ 1 , K _ 1 } v _ { h _ 2 , K _ 2 } \\Big ) e ^ { - f } d V _ g \\end{align*}"}
+{"id": "8601.png", "formula": "\\begin{align*} f _ i = f _ { i _ s } + R ^ { i _ s } _ 1 + \\ldots + R ^ { i _ s } _ { i - i _ s } , \\end{align*}"}
+{"id": "4360.png", "formula": "\\begin{align*} L _ \\nu : = I \\tilde L _ \\nu , \\end{align*}"}
+{"id": "2809.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { n + 1 } u _ i & = u _ { n + 1 } \\prod _ { i = 1 } ^ n u _ i = \\min ( v _ n , u _ n ) \\left ( \\frac { \\prod _ { i = 1 } ^ n v _ i } { \\prod _ { i = 1 } ^ n u _ i } \\right ) \\prod _ { i = 1 } ^ n u _ i \\\\ & = v _ { n + 1 } \\prod _ { i = 1 } ^ n v _ i = \\prod _ { i = 1 } ^ { n + 1 } v _ i . \\end{align*}"}
+{"id": "2248.png", "formula": "\\begin{align*} ( U ^ 1 _ D ) _ { \\nu } : = \\big \\{ g \\in U ^ 1 _ D ~ | ~ \\o ( g ) \\geq \\nu \\big \\} , \\ \\ ( U ^ 1 _ D ) _ { \\nu + } : = \\big \\{ g \\in U ^ 1 _ D ~ | ~ \\o ( g ) > \\nu \\big \\} . \\end{align*}"}
+{"id": "1111.png", "formula": "\\begin{align*} R _ 0 : \\ , \\mathbb { P } ( X = r ) = \\frac { 1 + \\delta } { 2 } = 1 - \\mathbb { P } ( X = - r ) \\end{align*}"}
+{"id": "2291.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { g } ^ 0 _ { i \\pm 1 / 2 } & = g _ { i \\pm 1 / 2 } + \\frac { 1 } { 1 2 } h _ i ^ { ( 4 ) } \\Delta x ^ 4 + O ( \\Delta x ^ 5 ) , \\\\ \\hat { g } ^ 1 _ { i \\pm 1 / 2 } & = g _ { i \\pm 1 / 2 } - \\frac { 1 } { 9 0 } h _ i ^ { ( 5 ) } \\Delta x ^ 5 + O ( \\Delta x ^ 6 ) , \\\\ \\hat { g } ^ 2 _ { i \\pm 1 / 2 } & = g _ { i \\pm 1 / 2 } - \\frac { 1 } { 1 2 } h _ i ^ { ( 4 ) } \\Delta x ^ 4 + O ( \\Delta x ^ 5 ) . \\end{aligned} \\end{align*}"}
+{"id": "8427.png", "formula": "\\begin{align*} \\| \\mu ^ A \\| ^ 2 = \\min _ { \\nu \\in \\Gamma _ { A , \\mu } ^ + } \\ , \\| \\nu \\| ^ 2 . \\end{align*}"}
+{"id": "7509.png", "formula": "\\begin{align*} S _ { k , r } ( j ) = & \\sum \\limits _ { i = k - j } ^ k ( - 1 ) ^ { k - i } \\dbinom { k + r } { i + r } \\dbinom { i + r } { i + j - k } \\\\ = & \\sum \\limits _ { i = k - j } ^ k ( - 1 ) ^ { k - i } \\dbinom { k + r } { k - i } \\dbinom { i + r } { j - ( k - i ) } \\\\ = & \\sum \\limits _ { i = 0 } ^ j ( - 1 ) ^ i \\dbinom { k + r } { i } \\dbinom { k - i + r } { j - i } . \\end{align*}"}
+{"id": "7703.png", "formula": "\\begin{align*} \\Sigma _ { \\beta , a } = \\left \\{ z \\in \\mathbb { C } : \\left \\vert \\arg ( z - a ) \\right \\vert \\leq \\beta \\pi , \\ ; a \\geq 0 , \\ ; \\beta < \\frac { 1 } { 2 } \\right \\} , \\end{align*}"}
+{"id": "3101.png", "formula": "\\begin{align*} \\int _ Y ( \\partial _ 1 w _ A ) ( \\partial _ { 2 2 } ^ 2 w _ B ) = \\int _ Y ( \\partial _ 2 w _ A ) ( \\partial _ { 1 1 } ^ 2 w _ B ) = 0 . \\end{align*}"}
+{"id": "9056.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( \\int _ 0 ^ T \\int _ { \\R ^ * } | U _ s ( e ) | ^ 2 \\nu ( d e ) d s \\right ) ^ { \\frac { p } { 2 } } \\right ] \\leq l _ p \\mathbb { E } \\left [ \\left ( \\int _ 0 ^ T \\int _ { \\R ^ * } \\sum _ { j = 1 } ^ n | U ^ { j } _ s ( e ) | ^ 2 N ^ j ( d e , d s ) \\right ) ^ { \\frac { p } { 2 } } \\right ] . \\end{align*}"}
+{"id": "6632.png", "formula": "\\begin{align*} \\partial _ { x _ j } ^ p \\frac { x _ j ^ { 2 r } } { ( 1 + \\epsilon x ^ { 2 m } ) ^ 2 } & \\leq \\sum _ { \\ell = 0 } ^ p \\binom { p } { \\ell } ( 2 r ) \\cdots ( 2 r - \\ell + 1 ) x _ j ^ { 2 r - \\ell } \\sum _ { ( m _ 1 , \\dots , m _ { p - \\ell } ) } C _ { p - \\ell , m _ i } \\frac { x _ j ^ { \\ell - p } } { ( 1 + \\epsilon x ^ { 2 m } ) ^ 2 } \\\\ & = \\sum _ { \\ell = 0 } ^ p c _ { p , \\ell , r } \\frac { x _ j ^ { 2 r - p } } { ( 1 + \\epsilon x ^ { 2 m } ) ^ 2 } . \\end{align*}"}
+{"id": "5506.png", "formula": "\\begin{align*} \\log \\Gamma ( s ) & = \\left ( s - \\frac 1 2 \\right ) \\log s - s + O ( 1 ) \\\\ & = \\left ( s - \\frac 1 2 \\right ) \\log ( i t ) + \\left ( s - \\frac 1 2 \\right ) \\log \\left ( 1 + { \\sigma \\over i t } \\right ) - s + O ( 1 ) \\\\ & = \\left ( s - \\frac 1 2 \\right ) \\log ( i t ) + i t \\log \\left ( 1 + { \\sigma \\over i t } \\right ) - s + O ( 1 ) \\\\ & = \\left ( s - \\frac 1 2 \\right ) \\log ( i t ) - i t + O ( 1 ) \\end{align*}"}
+{"id": "4024.png", "formula": "\\begin{align*} ( G _ p \\cdot \\gamma ) ^ 2 = ( w ^ { m n } \\gamma _ m \\gamma _ n ) ( \\phi ^ * _ k \\phi ^ * _ l Y ^ l _ { p _ j } Y ^ k _ { p _ i } w _ { i j } ) , \\end{align*}"}
+{"id": "2555.png", "formula": "\\begin{align*} \\begin{aligned} v ^ { ' } _ { ( \\{ l _ i \\} _ { i = 1 } ^ { q - 1 } ) } & : = x _ { 1 } x _ { - 1 } ^ { k ^ { ' } _ 1 - 1 } \\left \\{ \\prod _ { i = 2 } ^ { q } x _ { c ^ { ' } _ { i - 1 } } x _ { 1 } ^ { l _ { i - 1 } } x _ { - 1 } ^ { k ^ { ' } _ { i } - 1 } \\right \\} \\\\ & = z _ { 1 } ( k ^ { ' } _ 1 ) \\left \\{ \\prod _ { i = 2 } ^ { q } z _ { c ^ { ' } _ { i - 1 } } ( 1 ) z _ { 1 } ( 1 ) ^ { l _ { i - 1 } - 1 } z _ { 1 } ( k ^ { ' } _ { i } ) \\right \\} , \\end{aligned} \\end{align*}"}
+{"id": "1237.png", "formula": "\\begin{align*} X ^ c _ \\geq ( y , p ) : = \\{ x \\in X : D _ y c ( x , y ) \\geq p \\} . \\end{align*}"}
+{"id": "4797.png", "formula": "\\begin{align*} \\max _ { j < \\beta N } \\left \\{ \\Pr _ { v \\sim \\mu _ t } [ | v H | = j ] \\cdot 2 ^ { 2 \\epsilon N \\log ( 1 - \\frac { 2 j } { N } ) } \\right \\} \\leq \\max _ { \\delta < \\beta } \\left \\{ 2 ^ { - ( 1 - h ( \\delta ) ) ( 1 - \\eta ) N } \\cdot 2 ^ { 2 \\epsilon N \\log ( 1 - 2 \\delta ) } \\right \\} . \\end{align*}"}
+{"id": "9095.png", "formula": "\\begin{align*} \\mathfrak { c } _ j ( \\mathfrak { n } ) = \\bigoplus _ { j + 1 \\leq l \\leq k } \\mathfrak { n } _ l j \\geq 0 . \\end{align*}"}
+{"id": "8132.png", "formula": "\\begin{align*} \\mathcal { R } ^ - = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { c > 0 } \\frac { S ( - n , p ; c ) } { c } H _ { m , n } ^ - \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) \\end{align*}"}
+{"id": "3144.png", "formula": "\\begin{align*} \\partial _ { 1 1 } ^ 2 w _ B = \\partial _ { 2 2 } ^ 2 w _ B = \\frac { 1 } { 2 } \\Delta w _ B = - \\frac { 1 } { c } \\left ( a - \\int _ Y a \\right ) . \\end{align*}"}
+{"id": "6229.png", "formula": "\\begin{align*} N _ p = \\inf _ { D \\in \\mathbb D _ p } \\hbox { r a n k } ( \\mathcal R _ D ) . \\end{align*}"}
+{"id": "5457.png", "formula": "\\begin{align*} & V ( X ^ n _ t , \\alpha ^ n _ t ) - V ( X ^ n _ 0 , \\alpha ^ n _ 0 ) \\\\ = & \\int _ { 0 } ^ { t } L ^ n V ( X ^ n _ s , \\alpha ^ n _ s ) d s + \\int _ { 0 } ^ { t } \\nabla V \\sigma ^ n ( s , X ^ n _ { s } , \\mathcal { L } _ { X ^ n _ { s } } , \\alpha ^ n _ s ) d W _ s \\\\ & + \\int _ { 0 } ^ { t } \\int _ { \\mathbb R } V ( X ^ n _ s , \\alpha _ 0 + h ( X ^ n _ s , \\alpha ^ n _ s , z ) ) - V ( X ^ n _ s , \\alpha ^ n _ s ) \\mu ( d s , d z ) , \\end{align*}"}
+{"id": "6524.png", "formula": "\\begin{align*} \\Psi _ L \\tilde { f } ^ L = \\underset { i = 0 } { \\overset { 2 ^ L - 1 } { \\sum } } \\tilde { f } _ { i } \\psi _ { L i } , \\end{align*}"}
+{"id": "408.png", "formula": "\\begin{align*} \\| \\tau _ { j _ 1 } + \\cdots + \\tau _ { j _ n } \\| = n ^ \\frac { 1 } { p } , \\forall j _ 1 , \\dots , j _ n . \\end{align*}"}
+{"id": "7546.png", "formula": "\\begin{align*} Z _ f ( s , \\chi ) = { \\left \\{ \\begin{array} { r l } \\dfrac { 1 - q ^ { - 1 } } { 1 - q ^ { - 1 - p s } } , \\ \\ \\ \\ & { \\rm i f } \\ \\chi ^ p = \\chi _ { { \\rm t r i v } } , \\\\ 0 , \\ \\ \\ \\ & { \\rm i f } \\ \\chi ^ p \\ne \\chi _ { { \\rm t r i v } } . \\end{array} \\right . } \\end{align*}"}
+{"id": "897.png", "formula": "\\begin{align*} H \\circledast X _ { j } = Y _ { j } , ~ ~ ~ ~ j = 1 , 2 , \\cdots , 2 7 , \\end{align*}"}
+{"id": "3928.png", "formula": "\\begin{align*} \\det [ D ^ 2 v - A ( \\cdot , v , D v ) ] & = \\det \\big [ D ^ 2 u _ m - A ( \\cdot , u _ m , D u _ m ) \\\\ & \\quad + k ( I - A _ { i j , u } ( | x | ^ 2 - r ^ 2 ) - A _ { i j , p _ k } \\cdot x ) \\big ] , \\\\ B ( \\cdot , u , D u ) & = B ( \\cdot , u _ m , D u _ m ) + \\frac { k } { 2 } B _ u ( | x | ^ 2 - r ^ 2 ) + k B _ { p _ k } \\cdot x . \\end{align*}"}
+{"id": "6859.png", "formula": "\\begin{gather*} J ( u ; F ) = \\Big ( Q u - F , Q u - F \\Big ) _ Y = \\\\ ( Q u , Q u ) _ Y - 2 ( Q u , F ) _ Y + ( F , F ) _ Y = \\\\ \\norm { Q u } ^ 2 _ Y + \\norm { F } ^ 2 _ Y - 2 ( Q u , F ) _ Y \\end{gather*}"}
+{"id": "6074.png", "formula": "\\begin{align*} \\ell _ k ( x ) - \\sum _ { i = 1 } ^ g \\ell _ i ( x ) \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac { \\ell _ k ( y ) \\dd y } { ( w \\tilde w ) ( y ) } \\equiv 0 \\end{align*}"}
+{"id": "3655.png", "formula": "\\begin{align*} H ( t ) = \\int _ 0 ^ t e ^ { ( \\mu - p \\lambda _ * ( M ) ) \\tau } { \\rm d } \\tau , \\end{align*}"}
+{"id": "4132.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda = \\int _ M - | \\nabla u | ^ 2 - 1 2 u ^ 2 - 1 0 \\langle \\nabla u , \\nabla \\omega \\rangle - ( \\triangle \\omega ) ^ 2 - 2 | \\nabla \\omega | ^ 2 d V _ g . \\end{align*}"}
+{"id": "2511.png", "formula": "\\begin{align*} \\Delta _ g f - a f = - c f ^ { q - 1 } , \\end{align*}"}
+{"id": "3870.png", "formula": "\\begin{align*} g ( x , Y ( x , u , p ) , Z ( x , u , p ) ) & = u , \\\\ g _ x ( x , Y ( x , u , p ) , Z ( x , u , p ) ) & = p . \\end{align*}"}
+{"id": "1387.png", "formula": "\\begin{align*} [ x , y ] _ \\rho = \\{ \\rho ^ { x } , \\rho ^ { x - 1 } , \\dots , \\rho | \\cdot | ^ y \\} , \\end{align*}"}
+{"id": "8196.png", "formula": "\\begin{align*} u ( t , x ) = u _ 0 ( x ) - \\int _ 0 ^ t k ( t , s ) \\left ( - \\frac 1 2 \\Delta - w \\nabla - c \\right ) ^ \\gamma u ( s , x ) d s . \\end{align*}"}
+{"id": "1053.png", "formula": "\\begin{align*} \\varphi _ 1 ( x ) = 1 , \\varphi _ { 2 j } ( x ) = \\sqrt { 2 } \\cos ( 2 \\pi j x ) \\mbox { a n d } \\varphi _ { 2 j + 1 } ( x ) = \\sqrt { 2 } \\sin ( 2 \\pi j x ) , j \\in \\mathbb { N } _ + . \\end{align*}"}
+{"id": "1429.png", "formula": "\\begin{align*} \\mathbb { P } ( \\tau _ h \\wedge \\tau _ 1 = N _ 2 ) \\leq \\frac { \\mathbb { E } [ \\tau _ h \\wedge \\tau _ 1 ] } { N _ 2 } \\leq \\frac { 3 ( \\sigma ^ 2 - 1 ) ^ { - 1 } } { N _ 2 } ( 2 h ^ 2 - z ^ 2 ) . \\end{align*}"}
+{"id": "1228.png", "formula": "\\begin{align*} & \\int _ { X \\times X \\times X } [ \\epsilon | x _ 0 - x _ 1 | ^ 2 + | x _ 0 - y | ^ 2 + | x _ 1 - y | ^ 2 ] d \\gamma _ \\epsilon ( y , x _ 0 , x _ 1 ) \\\\ & \\leq \\int _ { X \\times X \\times X } [ \\epsilon | x _ 0 - x _ 1 | ^ 2 + | x _ 0 - y | ^ 2 + | x _ 1 - y | ^ 2 ] d \\gamma ( y , x _ 0 , x _ 1 ) . \\end{align*}"}
+{"id": "391.png", "formula": "\\begin{align*} \\mathcal { I } _ { 0 } ( z ) = \\sum \\limits _ { k = 0 } ^ \\infty \\frac { ( z ^ 2 / 4 ) ^ k } { ( k ! ) ^ 2 } \\textrm { a n d } \\mathcal { I } _ { 1 } ( z ) = \\frac { z } { 2 } \\sum \\limits _ { k = 0 } ^ \\infty \\frac { ( z ^ 2 / 4 ) ^ k } { k ! ( k + 1 ) ! } \\ , \\textrm { f o r a n y } z \\in \\mathbb { R } \\ , . \\end{align*}"}
+{"id": "1195.png", "formula": "\\begin{align*} \\Upsilon ^ { * } J _ { 0 } - J _ { 0 } = O ( r _ { i } ^ { - 1 / c _ i } ) \\ ; \\ , \\textrm { w i t h t w o $ g _ { 0 , i } $ - d e r i v a t i v e s } . \\end{align*}"}
+{"id": "3903.png", "formula": "\\begin{align*} g _ x ( x , \\cdot , g ^ * ( x , \\cdot , u _ 0 ) ) ( Y u ( x ) ) & = g _ x ( x , \\cdot , g ^ * ( x , \\cdot , u _ 0 ) ) ( Y ( x , u _ 0 , \\partial u ( x ) ) ) \\\\ & = \\partial u ( x ) . \\end{align*}"}
+{"id": "7566.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 5 ) = \\dfrac { F _ 5 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "5879.png", "formula": "\\begin{align*} t \\geq T : U = \\sum _ { r = 1 } ^ p u _ r e _ r . \\end{align*}"}
+{"id": "5873.png", "formula": "\\begin{align*} w = ( e , U ) , { \\cal L } \\theta = - ( e , A U ) , { \\cal R } \\theta = - ( e , B U ) | _ { \\Sigma } , \\end{align*}"}
+{"id": "371.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n _ 1 + 1 } u _ k + \\sum \\limits _ { k = n _ 1 + 2 } ^ { n _ 2 } u _ k \\leq \\frac { t } { T r _ 2 } < \\sum \\limits _ { k = 1 } ^ { n _ 1 + 1 } u _ k + \\sum \\limits _ { k = n _ 1 + 2 } ^ { n _ 2 } u _ k + u _ { n _ 2 + 1 } \\ , . \\end{align*}"}
+{"id": "7281.png", "formula": "\\begin{align*} d ( m ) = \\inf \\left \\{ k \\geq 1 \\ ; \\middle | \\ ; \\int _ { 0 } ^ { 1 } ( - 1 ) ^ k G ^ k _ m ( x ) d m ( x ) < \\infty \\right \\} , \\end{align*}"}
+{"id": "2752.png", "formula": "\\begin{align*} \\varphi _ { R } & = ( \\varphi _ { R } ) ^ { [ + ] } + ( \\varphi _ { R } ) ^ { [ - ] } , \\\\ \\varphi _ { L } & = ( \\varphi _ { L } ) ^ { [ + ] } + ( \\varphi _ { L } ) ^ { [ - ] } , \\end{align*}"}
+{"id": "4009.png", "formula": "\\begin{align*} \\det [ D ^ 2 u - A ( \\cdot , u , D u ) ] & = B ( \\cdot , u , D u ) \\Omega \\\\ u & = \\phi \\partial \\Omega . \\end{align*}"}
+{"id": "1788.png", "formula": "\\begin{align*} f ^ { - 1 } ( U ^ \\diamondsuit ) = V ^ \\diamondsuit \\end{align*}"}
+{"id": "665.png", "formula": "\\begin{align*} d * \\nu = 0 . \\end{align*}"}
+{"id": "5525.png", "formula": "\\begin{align*} P _ { k } ( x ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n ^ k } \\exp \\left ( { - \\frac { x } { n ^ 2 } } \\right ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n ^ k } \\sum _ { m = 0 } ^ \\infty \\frac { ( - 1 ) ^ m x ^ m } { m ! n ^ { 2 m } } = \\sum _ { m = 0 } ^ \\infty \\frac { ( - 1 ) ^ m x ^ m } { m ! \\zeta ( k + 2 m ) } . \\end{align*}"}
+{"id": "8628.png", "formula": "\\begin{align*} d _ { T V } ( X , Y ) = d _ { T V } ( \\mu _ X , \\mu _ Y ) . \\end{align*}"}
+{"id": "771.png", "formula": "\\begin{align*} \\langle \\xi , g \\rangle = \\sup \\{ \\liminf _ { n \\to \\infty } \\langle \\xi ' _ n , g \\rangle : \\xi ' _ n \\to \\xi \\} \\end{align*}"}
+{"id": "763.png", "formula": "\\begin{align*} { \\mathbb { W } _ { \\kappa } } : = \\begin{cases} \\widetilde { \\mathbb { W } } _ { \\kappa } \\oplus \\left ( \\mathbb { W } ^ { \\ell , \\left ( \\frac { - } { \\ell } \\right ) } \\otimes _ { \\Lambda } \\mathcal { R } _ { \\mathcal { P } _ { \\kappa } } \\right ) [ f _ \\kappa ] & \\textrm { i f $ r = 1 $ a n d $ \\psi _ \\ell $ i s t h e t r i v i a l c h a r a c t e r , } \\\\ \\widetilde { \\mathbb { W } } _ { \\kappa } & \\textrm { o t h e r w i s e , } \\end{cases} \\end{align*}"}
+{"id": "1284.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { k - 1 } \\sum _ { j = 0 } ^ { \\ell - 1 } m _ { t , j } ( x ) q _ j ^ { ( t ) * } ( x ) \\eta _ j ^ * ( y ) \\zeta _ t ^ * ( z ) = a ( x , y , z ) ( x ^ s - \\alpha ) + b ( x , y , z ) ( y ^ \\ell - \\beta ) + c ( x , y , z ) ( z ^ k - \\gamma ) . \\end{align*}"}
+{"id": "2822.png", "formula": "\\begin{align*} [ f ] ( x _ 1 , x _ 2 , x _ 3 ) = 2 x _ 1 + 2 x _ 2 + 2 x _ 3 . \\end{align*}"}
+{"id": "6383.png", "formula": "\\begin{align*} \\delta = 1 - 2 \\sum _ { p \\mid P } \\frac 1 p , \\quad \\varepsilon = \\sum _ { p \\mid L } \\frac 1 p , \\quad \\theta = \\frac { \\varphi ( k ) } { k } , \\quad C _ q = \\begin{cases} 2 & 2 \\mid q , \\\\ 3 & 2 \\nmid q . \\end{cases} \\end{align*}"}
+{"id": "5273.png", "formula": "\\begin{align*} \\tilde { T } _ U V = T _ U V - g ( U , V ) \\nabla ( \\log \\lambda ) . \\end{align*}"}
+{"id": "406.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n = 1 } ^ \\infty a _ n \\tau _ n \\right \\| ^ p = \\sum _ { n = 1 } ^ \\infty | a _ n | ^ p . \\end{align*}"}
+{"id": "8823.png", "formula": "\\begin{align*} A = \\begin{bmatrix} A _ { 1 1 } & 0 \\\\ 0 & A _ { 2 2 } \\end{bmatrix} \\ , , \\ , B = \\begin{bmatrix} B _ { 1 1 } & 0 \\\\ 0 & B _ { 2 2 } \\end{bmatrix} . \\end{align*}"}
+{"id": "7370.png", "formula": "\\begin{align*} \\| g _ j \\| _ { \\dot { W } ^ { 1 , 1 } ( \\R ) } = 1 , \\end{align*}"}
+{"id": "6401.png", "formula": "\\begin{align*} w _ { j _ l } ( X _ { j _ l } , Y _ { j _ l } , t _ { j _ l } , \\tilde X , \\tilde Y , \\tilde t ) & = u ( X _ { j _ l } , Y _ { j _ l } , t _ { j _ l } ) - ( \\phi ( \\tilde X , \\tilde Y , \\tilde t ) - \\varphi _ { j _ l } ( \\tilde X , \\tilde Y , \\tilde t ) ) , \\end{align*}"}
+{"id": "2838.png", "formula": "\\begin{align*} b _ k = \\rho + \\sigma , b _ { \\ell } = \\rho - \\sigma . \\end{align*}"}
+{"id": "2271.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ \\Omega \\nabla _ x \\Phi \\cdot \\nabla _ x \\Psi \\ , d x d t = \\int _ 0 ^ T \\int _ \\Omega ( n - c ( x ) ) \\Psi \\ , d x d t , \\end{align*}"}
+{"id": "8456.png", "formula": "\\begin{align*} \\kappa ( \\gamma _ A , \\gamma _ H ) = \\lim _ { K \\uparrow H } \\ , \\kappa ( \\gamma _ A , \\gamma _ K ) = \\lim _ { K \\uparrow H } \\ , \\int \\kappa \\gamma _ A \\ , d \\gamma _ K = \\lim _ { K \\uparrow H } \\ , \\gamma _ K ( X ) = \\lim _ { K \\uparrow H } \\ , \\| \\gamma _ K \\| ^ 2 = \\| \\gamma _ H \\| ^ 2 , \\end{align*}"}
+{"id": "6231.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } \\widehat U '' - { \\Delta } \\widehat U + A \\widehat U = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\widehat U = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu \\widehat U + B \\widehat U = D \\widehat H & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 , \\\\ t = 0 : \\widehat U = \\widehat U ' = 0 & \\hbox { i n } \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "8059.png", "formula": "\\begin{align*} \\tilde g ( s ) = \\int _ 0 ^ \\infty x ^ { s - 1 } g ( x ) \\ , d x \\end{align*}"}
+{"id": "3105.png", "formula": "\\begin{align*} \\bar { \\gamma } : = \\left ( \\int _ Y \\frac { r } { \\gamma } \\right ) ^ { - 1 } = \\frac { 1 } { C : \\bar { A } } > 0 , \\end{align*}"}
+{"id": "7961.png", "formula": "\\begin{align*} L / K : = \\{ x \\in L \\mid x \\geq 1 _ K \\} . \\end{align*}"}
+{"id": "4820.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\epsilon } f ( \\epsilon , \\frac { 2 } { \\ln 2 } ) = - \\frac { 1 } { ( 1 - \\epsilon ) \\ln 2 } + 2 \\log ( \\ln 2 ) + \\frac { 2 } { \\ln 2 } , \\end{align*}"}
+{"id": "7107.png", "formula": "\\begin{align*} \\sum _ { | m | \\leq k } \\phi _ k ( \\alpha _ m ) = 0 \\in M U _ * / 2 . \\end{align*}"}
+{"id": "1677.png", "formula": "\\begin{align*} \\tilde I ( \\chi ) : = \\bigoplus _ { n \\in \\Z } I ( \\chi , n ) \\subseteq I ( \\chi ) . \\end{align*}"}
+{"id": "4939.png", "formula": "\\begin{align*} F _ 1 = 2 u v _ x - 2 u _ x v , G _ 1 = u ^ 2 + v ^ 2 , \\end{align*}"}
+{"id": "8331.png", "formula": "\\begin{align*} \\overline { Q _ l } ( s , y ) = Q _ l \\left ( \\frac { s } { | y | ^ 2 - s ^ 2 } , \\frac { y } { | y | ^ 2 - s ^ 2 } \\right ) , \\end{align*}"}
+{"id": "8117.png", "formula": "\\begin{align*} u _ 1 ( y ) = \\frac { 2 \\sqrt { y p } } { c } + 3 x ^ { \\frac { 1 } { 3 } } y ^ { \\frac { 1 } { 3 } } , u _ 2 ( y ) = \\frac { 2 \\sqrt { y p } } { c } - 3 x ^ { \\frac { 1 } { 3 } } y ^ { \\frac { 1 } { 3 } } , \\end{align*}"}
+{"id": "6928.png", "formula": "\\begin{align*} \\sum _ { n \\leq x \\atop P ^ - ( n ) > \\mathcal { L } } \\frac { \\abs { \\lambda _ f ( n ) } ^ 4 } { n } & = O \\big ( \\log x \\big ) + O \\big ( \\log _ 2 x \\big ) + O \\bigg ( \\int _ { L _ 2 } ^ x \\frac { u \\log u } { u ^ 2 } \\ d u \\bigg ) \\\\ & \\ll \\bigg ( \\int _ { L _ 2 } ^ x \\frac { \\log u } { u } \\ d u \\bigg ) + \\log x \\\\ & \\ll \\big ( \\log ^ 2 x \\big ) . \\end{align*}"}
+{"id": "1334.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | ^ m & \\geq \\sqrt { \\frac { \\frac { 1 } { { d + m - 1 \\choose m } } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) ^ m \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ { 2 m } } { n ^ 2 - n } } \\\\ & = \\sqrt { \\frac { \\frac { 1 } { ( ^ m ( \\mathcal { X } ) ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) ^ m \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ { 2 m } } { n ^ 2 - n } } . \\end{align*}"}
+{"id": "6171.png", "formula": "\\begin{align*} U _ n = \\begin{pmatrix} C _ 1 \\\\ E _ 1 ^ T \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} C _ 1 U _ n \\\\ E _ 1 ^ T U _ n \\end{pmatrix} \\rightarrow u \\begin{pmatrix} C _ 1 \\\\ E _ 1 ^ T \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} = u e _ 1 \\end{align*}"}
+{"id": "327.png", "formula": "\\begin{align*} R ( l ) = \\frac { 1 } { 2 l } \\sum _ { n = 1 } ^ \\infty \\frac { A ( n , 1 ) } { n ^ \\frac { 3 } { 2 } } ( \\frac { 1 6 } { l n } ) \\sum _ { \\substack { \\alpha \\leq Y \\\\ ( \\alpha , 2 l n ) = 1 } } \\frac { \\mu ( \\alpha ) } { \\alpha ^ 2 } \\sum _ { \\substack { k = - \\infty \\\\ k \\neq 0 } } ^ { \\infty } ( - 1 ) ^ k G _ k ( l n ) \\widetilde \\Phi _ { n } ( \\frac { k X } { 2 \\alpha ^ 2 l n } ) . \\end{align*}"}
+{"id": "2101.png", "formula": "\\begin{align*} F H = F H e _ 0 \\oplus F H e _ 1 \\oplus \\cdots \\oplus F H e _ s . \\end{align*}"}
+{"id": "3866.png", "formula": "\\begin{align*} \\det D Y ( \\cdot , u , D u ) = \\frac { f ( \\cdot ) } { f ^ * ( Y ( \\cdot , u , D u ) ) } \\Omega , \\end{align*}"}
+{"id": "5709.png", "formula": "\\begin{align*} \\mathbf { L } _ { b } ^ { a } \\mathbf { = } \\delta _ { b } ^ { a } - \\alpha _ { b } ^ { a } \\mathbf { m . } \\end{align*}"}
+{"id": "6106.png", "formula": "\\begin{align*} d _ 2 & = c _ 1 \\\\ d _ 3 & = c _ 2 \\\\ d _ 4 & = c _ 3 + c _ 4 \\hat { G } _ 1 \\\\ d _ 5 & = c _ 5 + c _ 6 \\hat { G } _ 1 \\\\ d _ 6 & = c _ 7 + c _ 8 \\hat { G } _ 1 + c _ 9 \\hat { G } _ 2 + c _ { 1 0 } \\hat { G } _ 3 \\\\ d _ 7 & = c _ { 1 1 } + c _ { 1 2 } \\hat { G } _ 1 ^ 2 + c _ { 1 3 } \\hat { G } _ 1 + c _ { 1 4 } \\hat { G } _ 2 + c _ { 1 5 } \\hat { G } _ 3 + c _ { 1 6 } \\hat { G } _ 4 . \\end{align*}"}
+{"id": "8854.png", "formula": "\\begin{align*} \\mathcal { A } _ { m } ^ { \\nu } : = \\{ F \\in L ^ { \\infty } ( \\mathbb { C } ^ { n } ) , \\Delta _ { \\nu } F = \\Lambda _ { n , \\nu } \\left ( \\lambda \\right ) F \\} \\subset L ^ { 2 } ( \\mathbb { C } ^ { n } , d \\mu _ { n } ) , \\end{align*}"}
+{"id": "6471.png", "formula": "\\begin{align*} | \\eta | ^ 2 & = O _ 0 ^ 4 ( \\eta ) + \\nu ^ 2 + \\rho ^ 2 , \\\\ ( w _ * \\cdot \\delta E ( \\eta ) ) ^ 2 & = O ^ 3 _ 3 ( \\eta ) + O _ 4 ^ 4 ( \\eta ) + \\left ( 2 \\partial _ { x x } \\theta _ * \\nu + 2 \\partial _ x \\theta _ * \\partial _ x \\nu \\right ) ^ 2 \\end{align*}"}
+{"id": "3312.png", "formula": "\\begin{align*} f \\in \\mathbb { Z } _ { n } ^ { \\mathbb { Z } _ { n } } \\left | f ^ { ( n - 1 ) } \\left ( \\mathbb { Z } _ { n } \\right ) \\right | = 1 , \\end{align*}"}
+{"id": "8009.png", "formula": "\\begin{align*} V _ H & = \\{ F _ 1 , F _ 2 , F _ 3 \\} , \\\\ E _ H & = \\left \\{ ( F _ 2 , F _ 1 ) , ( F _ 3 , F _ 1 ) , ( F _ 3 , F _ 2 ) \\right \\} . \\end{align*}"}
+{"id": "7708.png", "formula": "\\begin{align*} I ^ { ( 1 ) } ( \\lambda ) : = \\int _ { - 1 } ^ 1 \\frac { 1 } { 4 \\tau + \\lambda ( t + 1 ) ^ 2 } d t , I ^ { ( 2 ) } ( \\lambda ) : = \\int _ { - 1 } ^ 1 \\frac { 1 } { \\tau ( t + 1 ) ^ 2 + 4 \\lambda } d t . \\end{align*}"}
+{"id": "3282.png", "formula": "\\begin{align*} - \\left \\vert \\Delta u \\right \\vert ^ { p - 2 } \\Delta u = \\lambda \\left \\vert \\nabla u \\right \\vert ^ { \\mu - 2 } u , x \\in \\Omega , u \\left \\vert ~ _ { \\partial \\Omega } \\right . = 0 , \\end{align*}"}
+{"id": "4754.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ b x _ i ( t ) \\leq \\sum _ { i = 1 } ^ b \\hat { y } _ i = \\sum _ { i = 1 } ^ { b ' } \\tilde { y } _ i \\forall t \\leq T . \\end{align*}"}
+{"id": "1615.png", "formula": "\\begin{align*} V _ { \\sigma ^ 2 } = X + \\sigma W \\end{align*}"}
+{"id": "7267.png", "formula": "\\begin{align*} d ( m ) = \\inf \\left \\{ k \\geq 1 \\ ; \\middle | \\ ; \\int _ { 0 } ^ { 1 } ( - 1 ) ^ k G ^ k _ m ( x ) d m ( x ) < \\infty \\right \\} , \\end{align*}"}
+{"id": "3667.png", "formula": "\\begin{align*} \\mathcal { A } ^ k : = \\{ i : \\lambda ^ k ( x _ i ) + c ( u ^ k ( x _ i ) - g ( x _ i ) ) > 0 \\} . \\end{align*}"}
+{"id": "4203.png", "formula": "\\begin{align*} \\begin{cases*} \\partial _ t ^ 2 \\psi - \\Delta \\psi = f ( t , x ) , ( t , x ) \\in I \\times \\mathbb { R } , \\\\ ( \\psi , \\partial _ t \\psi ) | _ { \\{ t = 0 \\} } = ( \\psi _ 0 , \\psi _ 1 ) \\in H ^ { k } ( \\mathbb { R } ) \\times H ^ { k - 1 } ( \\mathbb { R } ) , \\end{cases*} \\end{align*}"}
+{"id": "7829.png", "formula": "\\begin{align*} A _ k ( x ) : = & \\sum _ { i = 1 } ^ 3 \\sum _ { t = 1 } ^ { 3 k - 2 } \\sum _ { s = 1 } ^ t x ^ { - 2 k i + 2 t + 2 s + 6 k - 4 } \\\\ = & \\ \\frac { ( 1 - x ^ { 6 k - 4 } ) ( 1 - x ^ { 6 k - 2 } ) ( 1 - x ^ { 6 k } ) } { ( 1 - x ^ 2 ) ( 1 - x ^ 4 ) ( 1 - x ^ { 2 k } ) } \\end{align*}"}
+{"id": "8586.png", "formula": "\\begin{align*} A : = a _ { n - 1 } \\ , 2 ^ { n - 1 } + a _ { n - 2 } \\ , 2 ^ { n - 2 } + \\ldots + a _ 1 \\ , 2 + a _ 0 . \\end{align*}"}
+{"id": "1426.png", "formula": "\\begin{align*} \\mathbb { P } ( \\tau _ h = N _ 2 ) + \\mathbb { P } ( t _ i - t _ { i - 1 } \\leq N _ 1 \\forall i , R _ { \\tau _ h } \\geq h ) . \\end{align*}"}
+{"id": "9252.png", "formula": "\\begin{align*} B _ { w ^ { - 1 } ( i + 1 ) , r } = \\left \\lbrace v _ 1 , \\ldots v _ { h ( w ^ { - 1 } ( i + 1 ) ) } , N v _ { w ^ { - 1 } ( i + 1 ) } , \\ldots \\widehat { v _ { r } } , \\ldots v _ n \\right \\rbrace \\end{align*}"}
+{"id": "1754.png", "formula": "\\begin{align*} F ( \\infty ) = - k _ M ( - 1 ) ^ { \\frac { k _ { \\rm i d } - 2 } { 2 } - m _ { \\rm i d } } \\langle \\mu _ { \\underline { m } } , \\mu _ { - \\underline { m } } \\rangle _ V \\sum _ { i \\geq \\frac { k _ c } { 2 } - m _ c } \\binom { k _ c - 1 } { i } \\binom { k _ { \\rm i d } - 1 } { \\lambda - \\bar m + i } . \\end{align*}"}
+{"id": "8750.png", "formula": "\\begin{align*} s _ { d _ 0 } ( \\hat { \\mathcal { O } } ^ + ) & \\geq s _ { d _ 0 } ( \\bar { \\mathcal { O } } ^ + ) - \\left ( \\frac { 2 } { \\sqrt { 2 } - 1 } \\right ) ^ { 1 / 2 } \\frac { | \\hat { \\mathcal { H } } _ { \\check { d } _ { \\xi } } - H g _ 0 ^ * | _ { S _ 2 } } { s ^ { 1 / 2 } _ { d _ 0 } ( H g _ 0 ^ * ) } \\\\ & \\geq s _ { d _ 0 } ( \\bar { \\mathcal { O } } ^ + ) - \\frac { s _ { d _ 0 } ( \\bar { \\mathcal { O } } ^ + ) } { 2 } = \\frac { s _ { d _ 0 } ( \\bar { \\mathcal { O } } ^ + ) } { 2 } , \\end{align*}"}
+{"id": "5796.png", "formula": "\\begin{align*} D K e r ( C _ p ) = \\{ 0 \\} . \\end{align*}"}
+{"id": "5962.png", "formula": "\\begin{align*} B ^ T E _ r = \\sum _ { s = 1 } ^ p \\beta _ { r s } E _ s , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "157.png", "formula": "\\begin{align*} L ^ { ' } _ { F } & = \\max \\bigg \\{ \\max _ { 1 \\leq j \\leq n } \\bigg ( \\bigg ( K L _ { \\nu _ { j } } + K L _ { q } \\kappa _ { b } \\gamma b \\bigg ) \\bigg ( 1 + \\frac { M ^ { 2 } K ^ { 2 } b } { \\delta ^ { j } } \\bigg ) \\bigg ) , \\\\ & \\qquad \\qquad \\bigg ( 1 + \\frac { M ^ { 2 } K ^ { 2 } b } { \\delta ^ { 0 } } \\bigg ) K L _ { q } \\kappa _ { b } \\gamma b , \\ \\max _ { 1 \\leq j \\leq n } L _ { \\nu _ { j } } \\bigg \\} . \\end{align*}"}
+{"id": "3564.png", "formula": "\\begin{align*} \\gamma ^ - : = \\gamma - e _ { \\ell j } + \\sum _ { u = 2 } ^ s e _ { v _ u i _ { u - 1 } } - e _ { v _ u i _ u } . \\end{align*}"}
+{"id": "6595.png", "formula": "\\begin{align*} I & \\leq A _ { \\beta , N } \\iint ( 1 + | \\xi - \\eta | ^ 2 ) ^ { - N } e ^ { - r | x - y | ^ { 1 / s } } ( 1 + | \\eta | ^ 2 ) ^ { - N _ 1 } e ^ { - r _ 1 | y | ^ { 1 / s } } d y d \\eta \\\\ & = A _ { \\beta , N } I _ 1 I _ 2 \\end{align*}"}
+{"id": "9170.png", "formula": "\\begin{align*} A = ( 1 + s \\gamma \\Delta ) ^ { - 1 } \\tilde C ( s , m ^ 2 ) = \\gamma + C ( s , m ^ 2 ) ( 1 + s \\gamma \\Delta ) . \\end{align*}"}
+{"id": "1207.png", "formula": "\\begin{align*} \\nabla ^ \\alpha \\Phi _ * : = \\nabla ^ { \\alpha _ 1 } \\Phi _ * \\otimes \\cdots \\otimes \\nabla ^ { \\alpha _ i } \\Phi _ * : T M ^ { \\otimes ( k + 1 ) } \\to \\Phi ^ * T M ^ { \\otimes i } . \\end{align*}"}
+{"id": "3138.png", "formula": "\\begin{align*} A ( y ) = \\frac { 1 } { r ( y ) } \\ , \\mathrm { d i a g } ( a ( y ) , c - a ( y ) ) \\end{align*}"}
+{"id": "994.png", "formula": "\\begin{align*} L ^ 2 _ \\rho ( \\Sigma ) : = \\left \\{ \\phi : \\rho \\phi \\in L ^ 2 ( \\Sigma ) \\right \\} , \\left < \\phi _ 1 , \\phi _ 2 \\right > _ { L ^ 2 _ \\rho ( \\Sigma ) } : = \\left < \\rho \\phi _ 1 , \\rho \\phi _ 2 \\right > _ { L ^ 2 ( \\Sigma ) } \\end{align*}"}
+{"id": "2908.png", "formula": "\\begin{align*} \\sup _ { k \\in \\mathbb { N } } \\sup _ { \\sigma > 0 } \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | \\mathcal { B } _ k f _ { \\sigma } | ) d m _ \\infty = \\sup _ { \\sigma > 0 } \\| \\mathcal { B } f _ \\sigma \\| _ 0 = \\sup _ { \\sigma > 0 } \\| f _ \\sigma \\| _ 0 = \\| f \\| _ 0 < \\infty . \\end{align*}"}
+{"id": "5492.png", "formula": "\\begin{align*} \\dot y ( t ) = J \\big ( \\nabla H ( t , y ( t ) ) - 2 \\pi \\lambda y ( t ) \\big ) . \\end{align*}"}
+{"id": "2111.png", "formula": "\\begin{align*} y \\big ( c ( x ) + c ' ( x ) y \\big ) = \\big ( \\overline c _ 2 ( x ) + \\overline c _ { 1 2 } ( x ) \\overline g ( x ) \\big ) + \\big ( \\overline c _ 1 ( x ) + \\overline c _ { 1 2 } ( x ) \\big ) \\ , y . \\end{align*}"}
+{"id": "6815.png", "formula": "\\begin{align*} z ' = \\mu \\frac { \\partial f } { \\partial y } z , \\end{align*}"}
+{"id": "7895.png", "formula": "\\begin{align*} f ( z ) ^ n + a _ { n - 1 } f ( z ) ^ { n - 1 } + \\cdots + a _ 1 f ( z ) + q ( z ) e ^ { Q ( z ) } f ^ { ( k ) } ( z + c ) = P ( z ) , \\end{align*}"}
+{"id": "815.png", "formula": "\\begin{align*} \\widehat { F } _ { n , t _ 1 , \\ldots , t _ d } ( x ) = \\widehat { \\nu } \\left ( y \\in Y : \\widehat { S } _ n y ( t _ 1 , \\ldots , t _ d ) \\in \\prod _ { i = 1 } ^ { d - 1 } ( - \\infty , x _ i ] \\right ) \\end{align*}"}
+{"id": "1496.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\frac \\pi 2 } \\theta ^ { r - 2 } \\log \\left ( \\cos \\frac \\theta 2 \\right ) d \\theta = - \\frac { \\pi ^ { r - 1 } } { r - 1 } \\left ( \\frac 1 { 2 ^ r } \\log 2 + 2 ^ { r - 1 } \\log \\C _ r \\left ( \\frac 1 4 \\right ) \\right ) . \\end{align*}"}
+{"id": "1430.png", "formula": "\\begin{align*} M ^ 2 _ t - M ^ 2 _ { t - 1 } = ( h - R _ { \\tau _ h + t } ) ^ 2 - ( h - R _ { \\tau _ h + t - 1 } ) ^ 2 & = ( \\eta _ { \\tau _ h + t } - 1 ) ^ 2 + 2 ( h - R _ { \\tau _ h + t - 1 } ) ( 1 - \\eta _ { \\tau _ h + t } ) \\\\ & = ( \\eta _ { \\tau _ h + t } - 1 ) ^ 2 + 2 M _ { t - 1 } ( 1 - \\eta _ { \\tau _ h + t } ) \\end{align*}"}
+{"id": "6504.png", "formula": "\\begin{align*} ( \\Xi ^ 2 _ u ) = \\| \\Xi _ u \\| \\ , ( \\Xi _ u ) \\leq \\frac { m ^ 2 } { n } ( \\Xi _ u ) \\lesssim \\frac { ( b \\wedge d ) m ^ 4 } { n ^ 2 } , \\end{align*}"}
+{"id": "7552.png", "formula": "\\begin{align*} \\int _ { \\mathcal { O } _ K ^ { \\times } } \\chi ( a c ( w ^ { k + r + l } ) ) | d w | = { \\left \\{ \\begin{array} { r l } 1 - q ^ { - 1 } , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } = \\chi _ { \\rm t r i v } , \\\\ 0 , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } \\neq \\chi _ { \\rm t r i v } . \\end{array} \\right . } \\end{align*}"}
+{"id": "1209.png", "formula": "\\begin{align*} \\frac { \\nabla V } { d t } = \\frac { \\nabla \\gimel _ { s _ 1 } } { \\partial t } = \\frac { \\nabla \\gimel _ t } { \\partial s _ 1 } = \\nabla X | _ { \\Phi _ t ( p ) } ( \\Phi ^ t _ * v _ 1 ) , \\end{align*}"}
+{"id": "5194.png", "formula": "\\begin{align*} \\alpha = \\sum _ { 1 \\le i < j < k \\le m } p _ { i j k } g _ { i j k } \\ . \\end{align*}"}
+{"id": "3763.png", "formula": "\\begin{align*} \\bar \\varepsilon _ i ( \\bar \\varepsilon _ j ( \\Gamma ) ) = \\bar \\varepsilon _ j ( \\bar \\varepsilon _ i ( \\Gamma ) ) \\quad \\bar \\varepsilon _ i ( \\bar \\varepsilon _ i ( \\Gamma ) ) = \\Gamma . \\end{align*}"}
+{"id": "5272.png", "formula": "\\begin{align*} \\tilde { T } _ U V = \\tilde { g } ( U , V ) \\tilde { H } = - g ( U , V ) \\nabla ( f + \\log \\lambda ) , \\end{align*}"}
+{"id": "6857.png", "formula": "\\begin{align*} J ( u ; F ) = \\norm { Q u - F } ^ 2 _ Y , u = \\underset { v \\in X } { \\arg \\min } \\ ; J ( v ; F ) . \\end{align*}"}
+{"id": "9254.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots v _ { w ^ { - 1 } ( k ) } , \\ldots , v _ { h ( w ^ { - 1 } ( k ) ) } , e _ { k - 1 } , \\ldots \\widehat { v _ { r } } , \\ldots v _ n ) } \\\\ & = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots v _ { w ^ { - 1 } ( k ) } , \\ldots , v _ { h ( w ^ { - 1 } ( k ) ) } , c e _ { j - 1 } , \\ldots \\widehat { v _ { r } } , \\ldots v _ n ) } . \\end{aligned} \\end{align*}"}
+{"id": "8699.png", "formula": "\\begin{align*} \\mathcal Q ( u _ 1 , . . , u _ { 2 l } ) = \\ , [ \\mathcal M _ { i , j } ] . \\end{align*}"}
+{"id": "3079.png", "formula": "\\begin{align*} A ( y ) : = a ( y ) B ( y ) \\quad y \\in \\R ^ n . \\end{align*}"}
+{"id": "5184.png", "formula": "\\begin{align*} A _ { \\overline { L } _ q ^ { 7 } ( r ; \\underline { m } ) } = \\begin{psmallmatrix} 1 & \\frac { r } { K } & y _ 0 & x _ 0 \\\\ 0 & 1 & r & \\frac { r ( r + 1 ) } { 2 } \\\\ 0 & 0 & 1 & r \\\\ 0 & 0 & 0 & 1 \\end{psmallmatrix} \\end{align*}"}
+{"id": "1394.png", "formula": "\\begin{align*} \\psi = ( S _ 2 \\boxtimes S _ 1 ) ^ { \\oplus 3 } + S _ 3 \\boxtimes S _ 2 . \\end{align*}"}
+{"id": "6450.png", "formula": "\\begin{align*} ( \\nabla _ { X , Y , t } f , \\nabla _ { X , Y , t } ^ 2 f ) ( \\tilde X ^ k _ j , \\tilde Y ^ k _ j , \\tilde t ^ k _ j ) & = ( - \\xi _ k , - E _ k ) , \\\\ ( \\nabla _ { X , Y , t } g , \\nabla _ { X , Y , t } ^ 2 g ) ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) & = ( \\eta _ k , H _ k ) . \\end{align*}"}
+{"id": "3311.png", "formula": "\\begin{align*} \\forall \\ , i \\in \\mathbb { Z } _ { n } , \\ ; f ^ { ( 0 ) } \\left ( i \\right ) \\ , : = i , \\mbox { a n d } \\forall \\ , k \\ge 0 , \\ ; f ^ { ( k + 1 ) } \\left ( i \\right ) = f ^ { ( k ) } \\left ( f \\left ( i \\right ) \\right ) = f \\left ( f ^ { ( k ) } \\left ( i \\right ) \\right ) . \\end{align*}"}
+{"id": "5655.png", "formula": "\\begin{align*} \\Lambda \\sum _ { m \\geq 0 } ( - \\Lambda ) ^ m & = \\Lambda ( 1 - \\Lambda + \\Lambda ^ 2 \\ldots ) = \\frac { \\Lambda } { 1 + \\Lambda } , \\\\ \\Lambda ^ 2 \\sum _ { m \\geq 0 } ( m + 1 ) ( - \\Lambda ) ^ m & = \\Lambda ^ 2 ( 1 - 2 \\Lambda + 3 \\Lambda ^ 2 \\ldots ) = \\Lambda ^ 2 \\frac { d } { d \\Lambda } \\frac { \\Lambda } { 1 + \\Lambda } = \\frac { \\Lambda ^ 2 } { ( 1 + \\Lambda ) ^ 2 } . \\end{align*}"}
+{"id": "8376.png", "formula": "\\begin{align*} { E } ( z ) : = Z ^ \\vartheta e ^ Z \\sum _ { j = 0 } ^ \\infty A _ j Z ^ { - j } , Z = \\kappa ( h z ) ^ { 1 / \\kappa } , \\end{align*}"}
+{"id": "782.png", "formula": "\\begin{align*} \\P _ x \\left \\{ ( x _ i ) : \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\xi ( x _ k , X ( x _ { k - 1 } ) \\ldots X ( x _ 0 ) c ) \\in [ \\ell ^ - _ \\xi - \\epsilon , \\ell ^ + _ \\xi + \\epsilon ] \\right \\} \\geq 1 - C _ 0 e ^ { - n \\alpha _ 0 } , \\end{align*}"}
+{"id": "4184.png", "formula": "\\begin{align*} D = \\left ( \\begin{array} { c c } \\alpha e ^ { \\Lambda } & 0 \\\\ 0 & \\alpha e ^ { - \\Lambda } \\end{array} \\right ) , R = \\left ( \\begin{array} { c c } \\cos \\phi & - \\sin \\phi \\\\ \\sin \\phi & \\cos \\phi \\end{array} \\right ) . \\end{align*}"}
+{"id": "694.png", "formula": "\\begin{align*} G ( z , w ) = \\frac { 1 } { 2 \\pi } ( - \\log | z - w | + H ( z , w ) ) , \\end{align*}"}
+{"id": "4866.png", "formula": "\\begin{align*} h ( 1 ) & = \\int _ { B _ { n - j + 1 } ( 1 ) } u ^ { j } ( s _ { j } ) \\dots u ^ n ( s _ n ) d \\L ^ { n - j + 1 } = I _ { n - j + 1 } ( u ) , \\\\ g ( 1 ) & = \\int _ { A _ { j - 1 } ( 1 ) } u ^ { j - 1 } ( s _ { j - 1 } ) \\dots u ^ { 1 } ( s _ { 1 } ) d \\L ^ { j - 1 } = I ^ \\flat _ { j - 1 } ( u ) . \\end{align*}"}
+{"id": "7982.png", "formula": "\\begin{align*} i ' ( [ x ] + [ y ] ) = i ' ( [ x + y ] ) = [ i ( x + y ) ] = [ i ( x ) + i ( y ) ] = [ i ( x ) ] + [ i ( y ) ] . \\end{align*}"}
+{"id": "6625.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\norm { x ( t ) ^ m \\psi } ^ 2 \\leq C _ m \\sum _ { j = 1 } ^ d \\norm { x _ j ^ m ( t ) \\psi } \\cdot \\norm { x _ j ^ m ( t ) \\psi } ^ { 1 - \\frac { 1 } { 2 ^ { \\ell - 1 } } } \\sum _ { r = 0 } ^ { 2 ^ { \\ell - 1 } } c _ { r , m } \\norm { x _ j ^ r ( t ) D _ j ^ r ( t ) \\psi } ^ { \\frac { 1 } { 2 ^ { \\ell - 1 } } } . \\end{align*}"}
+{"id": "471.png", "formula": "\\begin{align*} I ( G _ 1 \\cup G _ 2 ; x ) = I ( G _ 1 ; x ) \\cdot I ( G _ 2 ; x ) \\end{align*}"}
+{"id": "5711.png", "formula": "\\begin{align*} \\mathbf { x } ^ { a } = \\xi ^ { a } + ( d \\xi ^ { a } - \\theta ^ { a } ) \\mathbf { m } \\end{align*}"}
+{"id": "2742.png", "formula": "\\begin{align*} \\eqref { e q : G 2 - 1 } & = K _ i \\big [ E _ i , [ F _ i , F _ j ] _ { q ^ 3 } \\big ] = K _ i [ \\frac { K _ i - K _ i ^ { - 1 } } { q - q ^ { - 1 } } , F _ j ] _ { q ^ 3 } = q ^ 3 [ 3 ] K _ i K _ i ' F _ j = \\eqref { e q : G 2 - 1 } . \\end{align*}"}
+{"id": "4989.png", "formula": "\\begin{align*} V _ h : = V - h ^ 2 \\left ( | \\nabla \\chi _ < | ^ 2 + | \\nabla \\chi _ > | ^ 2 \\right ) . \\end{align*}"}
+{"id": "1379.png", "formula": "\\begin{align*} \\Omega _ m ^ R ( R x _ 1 ) = \\Omega _ { m - 1 } ^ R ( \\langle x _ 2 , \\ldots , x _ n \\rangle ) = \\Omega _ { m - 1 } ^ R ( R x _ 2 ) \\oplus \\cdots \\oplus \\Omega _ { m - 1 } ^ R ( R x _ n ) \\end{align*}"}
+{"id": "5595.png", "formula": "\\begin{align*} X ^ { a } ( \\nabla _ { a } W _ { b c d e } ) Q ^ { b c d e } = 0 \\ , \\{ = \\alpha \\pi ( X ) ^ { a } \\langle W \\vert k \\vert Q \\rangle _ { a } \\} , \\end{align*}"}
+{"id": "217.png", "formula": "\\begin{align*} \\mathbb { K } _ { N , m } ^ { \\mathrm { s p } } : = \\{ g \\in \\mathrm { G S p i n } _ { L _ { N } } ( \\widehat { \\Z } ) \\mid g = 1 \\textup { i n } C ^ { + } ( L _ { N , \\widehat { \\Z } / m \\widehat { \\Z } } ) \\} , \\end{align*}"}
+{"id": "4238.png", "formula": "\\begin{align*} I _ \\Omega ( c ) : = \\inf \\left \\{ E _ \\Omega ( f ) \\ : \\ f \\in S ( c ) \\right \\} , \\end{align*}"}
+{"id": "4851.png", "formula": "\\begin{align*} W _ \\beta ( v ) = W _ { \\beta _ 1 } ( v ) \\odot \\ . \\odot W _ { \\beta _ k } ( v ) . \\end{align*}"}
+{"id": "4049.png", "formula": "\\begin{align*} D _ { q _ i } A ^ * _ { k l } ( y , z , q ) & = \\frac { \\partial X ^ r } { \\partial q _ i } D _ { x _ r , y _ l } Q _ k + \\frac { \\partial X ^ r } { \\partial q _ i } D _ { x _ r , z } Q _ k Q _ l + \\delta _ { i l } D _ z ( Q _ k ) . \\end{align*}"}
+{"id": "7764.png", "formula": "\\begin{align*} - \\frac { \\widetilde h _ { \\alpha , \\theta } ^ \\prime ( s | \\mu ) } { \\widetilde h _ { \\alpha , \\theta } ( s | \\mu ) } & = \\mu \\dfrac { s ^ { \\alpha - 1 } } { ( 1 + s ) ^ { \\alpha - \\theta } } \\equiv \\mu \\ , \\rho _ { \\alpha , \\theta } ( s ) \\\\ \\implies \\widetilde h _ { \\alpha , \\theta } ( s | \\mu ) & = \\exp \\left \\{ - \\mu \\int _ 0 ^ s \\rho _ { \\alpha , \\theta } ( t ) d t \\right \\} \\end{align*}"}
+{"id": "4955.png", "formula": "\\begin{align*} C _ n : = \\frac { 1 } { n + 1 } \\binom { 2 n } { n } , \\end{align*}"}
+{"id": "1157.png", "formula": "\\begin{align*} & E _ \\mathcal { X } ( x _ 0 , x _ 1 ) \\iff x _ 0 + x _ 1 = b ; \\\\ & E _ \\mathcal { X } ( x _ 1 , x _ 2 ) \\iff x _ 1 + x _ 2 = b . \\end{align*}"}
+{"id": "2648.png", "formula": "\\begin{align*} H ( \\mu ) = \\left \\{ h = ( h _ j ) _ { j \\in \\N } \\in X : \\ , h _ j \\in H ( \\mu _ j ) , \\| h \\| ^ 2 _ { H ( \\mu ) } = \\sum _ { j \\in \\N } \\| h _ j \\| ^ 2 _ { H ( \\mu _ j ) } \\right \\} . \\end{align*}"}
+{"id": "869.png", "formula": "\\begin{align*} \\mathcal { A } \\ast \\mathcal { X } = \\mathcal { B } , \\end{align*}"}
+{"id": "2760.png", "formula": "\\begin{align*} \\hat { n } _ { A } = x ^ { A } \\frac { \\partial } { \\partial x ^ { A } } \\end{align*}"}
+{"id": "2251.png", "formula": "\\begin{align*} [ e , f ] = h , ~ [ h , e ] = [ h , f ] = 0 . \\end{align*}"}
+{"id": "2595.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { d } a _ h \\{ v ^ { t } _ { 1 , i } \\} _ { i = 2 , \\dots , r } = 0 , \\end{align*}"}
+{"id": "3267.png", "formula": "\\begin{align*} \\tilde { Y } _ { \\mathbf { d } , \\mathcal { A } } ( a b c ) = \\frac { ( n - 1 ) ^ 2 d _ a d _ b ( d _ b - 1 ) d _ c } { d ^ 2 n _ a n _ b ( n _ b - 1 ) n _ c } \\left ( 1 + \\mathcal { O } \\left ( \\frac { t ^ 2 } { n d } \\right ) \\right ) , \\end{align*}"}
+{"id": "8775.png", "formula": "\\begin{align*} \\dot { \\phi } _ i = & \\omega _ i + \\frac { 1 } { N } \\sum _ { j = 1 } ^ N W _ { i j } ( t ) g ( \\phi _ j - \\phi _ i ) , \\\\ \\dot { W } _ { i j } = & - \\varepsilon ( \\sum _ { k = 1 } ^ N f ( W _ { i k } , W _ { j k } ) + \\sum _ { k = 1 } ^ N ( h _ 1 ( \\phi _ k - \\phi _ i ) + h _ 2 ( \\phi _ k - \\phi _ j ) ) ) . \\end{align*}"}
+{"id": "3960.png", "formula": "\\begin{align*} \\overline { y } = y + O ( | y | ^ 2 ) . \\end{align*}"}
+{"id": "7028.png", "formula": "\\begin{align*} ( A ) / ( t - 0 ) = _ \\bullet A = \\bigoplus _ { n \\geq 0 } F _ n A / F _ { n - 1 } A . \\end{align*}"}
+{"id": "5299.png", "formula": "\\begin{align*} g _ { i j } = \\begin{pmatrix} e ^ { 2 x _ 2 } & 0 \\\\ 0 & e ^ { 2 x _ 2 } \\end{pmatrix} , g ^ { i j } = \\begin{pmatrix} e ^ { - 2 x _ 2 } & 0 \\\\ 0 & e ^ { - 2 x _ 2 } \\end{pmatrix} . \\end{align*}"}
+{"id": "2181.png", "formula": "\\begin{align*} \\phi ( r ) : = ( 4 - 6 \\alpha - 3 \\beta ) r ^ 2 - 3 ( 2 - 2 \\alpha + \\beta ) r + 2 . \\end{align*}"}
+{"id": "3062.png", "formula": "\\begin{align*} \\int _ Y ( \\partial _ 1 w _ A ) ( \\partial _ { 2 2 } ^ 2 w _ B ) = \\int _ Y ( \\partial _ 2 w _ A ) ( \\partial _ { 1 1 } ^ 2 w _ B ) = 0 . \\end{align*}"}
+{"id": "5975.png", "formula": "\\begin{align*} { h } ^ { - 1 } ( h ( \\mu ) ) \\cap f ^ { - 1 } ( z ) & = ( h \\times f ) ^ { - 1 } \\left ( h ( \\mu ) , z \\right ) \\\\ & = ( h \\times f ) ^ { - 1 } ( h \\times f ) ( \\mu ) . \\end{align*}"}
+{"id": "5955.png", "formula": "\\begin{align*} A e _ r = \\sum _ { s = 1 } ^ p \\alpha _ { s r } e _ s , B e _ r = \\sum _ { s = 1 } ^ p \\beta _ { s r } e _ s , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "3272.png", "formula": "\\begin{align*} y - \\lambda G \\left ( F ^ { - 1 } \\left ( y \\right ) \\right ) = 0 , y = F \\left ( x \\right ) , \\ x \\in X , \\end{align*}"}
+{"id": "1768.png", "formula": "\\begin{align*} & x _ { ( n , j ) } = - \\sum _ { k = j } ^ { n - 1 } x _ { ( k , n + j - k ) } , \\\\ & x _ { ( n - 2 , j ) } = - \\sum _ { k = 1 } ^ { j - 1 } x _ { ( n - 2 - j + k , k ) } - x _ { ( n - 1 , j + 1 ) } - x _ { ( n , j + 2 ) } , \\\\ & x _ { ( n - 1 , j ) } = - \\sum _ { i = 1 } ^ { n - 2 } x _ { ( i , j ) } - x _ { ( n , j ) } , \\end{align*}"}
+{"id": "2432.png", "formula": "\\begin{align*} \\bar \\partial _ \\tau ^ \\alpha w ^ n ( q ) + A ( q ) w ^ n ( q ) = 0 ~ ~ ~ ~ 1 \\le n \\le N w ^ 0 ( q ) = f + \\Delta v - q v . \\end{align*}"}
+{"id": "6782.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\frac { 1 } { P } } y ^ { \\frac { s } { 2 } - 1 } \\tilde { \\Psi } _ { \\chi } ( y ) d y = P ^ { - \\frac { s } { 2 } } \\int _ { 0 } ^ { 1 } y ^ { \\frac { s } { 2 } - 1 } \\tilde { \\Psi } _ { \\chi } \\left ( \\frac { y } { P } \\right ) d y . \\end{align*}"}
+{"id": "1984.png", "formula": "\\begin{align*} { \\textbf { Y } } _ { } = { \\textbf { G } } ^ { \\mathsf { H } } { \\textbf { S } } + { \\textbf { N } } . \\end{align*}"}
+{"id": "6811.png", "formula": "\\begin{align*} \\nabla \\alpha ( q _ m ) . ( q _ { m + 1 } - q _ { m - 1 } ) = \\alpha ( q _ { m + 1 } ) - \\alpha ( q _ { m - 1 } ) + 2 h \\nabla H ( q _ m ) . \\end{align*}"}
+{"id": "83.png", "formula": "\\begin{align*} \\mathcal { B } ^ { D } _ { n } ( t ) = \\big ( ( \\Sigma _ d ( w _ { 1 , n } ) ) ( t ) , \\cdots , ( \\Sigma _ d ( w _ { D , n } ) ) ( t ) \\big ) \\in \\mathbf { R } ^ D , \\ , \\ t \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "243.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho ^ 2 - \\beta \\rho \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "8366.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\int _ { | x | > t } \\left ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 \\right ) \\mathrm { d } x = 0 . \\end{align*}"}
+{"id": "3619.png", "formula": "\\begin{align*} l a m b d a & = 2 . 4 3 5 2 5 8 5 0 9 1 5 8 3 2 4 \\\\ C _ 0 & = 7 . 7 5 2 7 6 8 1 7 5 9 4 4 2 1 1 \\\\ R _ * & = 3 . 7 2 8 3 2 1 2 4 \\\\ L _ * & = 0 . 3 2 1 1 9 3 3 3 6 2 5 4 5 9 9 5 \\end{align*}"}
+{"id": "8449.png", "formula": "\\begin{align*} \\xi _ A : = \\gamma _ A / c _ * ( A ) \\in \\mathcal E ^ + . \\end{align*}"}
+{"id": "4935.png", "formula": "\\begin{align*} \\mathrm { i } z _ t + z _ { x x } + | z | ^ 2 z = 0 , ( x , t ) \\in ( a , b ) \\times ( 0 , T ) , \\end{align*}"}
+{"id": "7382.png", "formula": "\\begin{align*} X _ { \\alpha } ( \\chi ) : = \\chi ( h _ { \\alpha } ) , \\end{align*}"}
+{"id": "5421.png", "formula": "\\begin{align*} { \\mathcal { R } _ { m i } } ( A \\ast B ) \\simeq \\bigoplus _ { t = a } ^ { b } \\mathbf { 1 } _ { p t } ^ { f _ { m , t } ( v ) } \\boxtimes ( { \\mathcal { R } _ { t i } } ( A ) \\ast { \\mathcal { R } _ { ( m - t ) i } } ( B ) ) [ - P ' _ t ] ( - \\frac { P ' _ t } { 2 } ) , \\end{align*}"}
+{"id": "8241.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { N } } \\left ( 1 + 2 w ^ 2 \\right ) \\nabla w \\cdot \\nabla h \\ , d x + \\int _ { \\mathbb { R } ^ { N } } 2 w | \\nabla w | ^ 2 \\ , h \\ , d x + \\int _ { \\mathbb { R } ^ { N } } V \\left ( \\left | x \\right | \\right ) w h \\ , d x = \\int _ { \\mathbb { R } ^ { N } } K ( | x | ) g ( w ) h \\ , d x \\end{align*}"}
+{"id": "2745.png", "formula": "\\begin{align*} \\varphi \\circ D ( a b ) = & \\varphi \\circ i d _ { A } ( a ) \\varphi \\circ D b + \\varphi \\circ i d _ { A } ( b ) \\varphi \\circ D a \\\\ = & \\varphi ( a ) \\varphi \\circ D b + \\varphi ( b ) \\varphi \\circ D a \\\\ = & \\varphi ( a . D b + b . D a ) \\end{align*}"}
+{"id": "5188.png", "formula": "\\begin{align*} \\frac { r } { K } \\left ( \\frac { K ( K - 1 ) } { 2 } \\right ) m _ 2 ( a _ 1 - a _ 1 ' ) = r \\left ( \\frac { K - 1 } { 2 } \\right ) m _ 2 ( a _ 1 - a _ 1 ' ) \\equiv 0 \\mod r , \\end{align*}"}
+{"id": "7619.png", "formula": "\\begin{align*} \\psi _ k ( x ) : = \\chi _ { \\{ x \\geq 0 \\} } \\int _ { 0 } ^ { x } \\int _ { 0 } ^ { y } \\rho _ k ( z ) \\d z \\d y , \\end{align*}"}
+{"id": "1525.png", "formula": "\\begin{align*} \\sigma _ i = \\frac 1 4 p _ i ^ { a b } \\gamma _ a \\gamma _ b = \\frac 1 8 p _ i ^ { a b } [ \\gamma _ a , \\gamma _ b ] \\end{align*}"}
+{"id": "5776.png", "formula": "\\begin{align*} \\begin{cases} u '' + L u = \\mathcal R \\theta , \\\\ t = 0 : u = \\theta , \\ u ' = 0 \\end{cases} \\end{align*}"}
+{"id": "4695.png", "formula": "\\begin{align*} \\mathcal { I } ( G _ { I I } ) = \\frac { 1 } { ( 1 - t ^ 8 ) ( 1 - t ^ { 2 4 } ) } , \\end{align*}"}
+{"id": "6139.png", "formula": "\\begin{align*} \\begin{cases} \\phi _ s '' - \\Delta \\phi _ s + \\sum _ { r = 1 } ^ d \\alpha _ { r s } \\phi _ r = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\phi _ s = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu \\phi _ s + \\sum _ { r = 1 } ^ d \\beta _ { r s } \\phi _ r = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 . \\end{cases} \\end{align*}"}
+{"id": "2462.png", "formula": "\\begin{align*} | \\eta ^ { k + 1 } G _ k ( s ) | = \\eta \\Big | \\sum _ { p \\in [ N _ 0 , N _ 1 ] } \\frac { \\Lambda _ { \\pi } ( p ) + \\Lambda _ { \\pi \\otimes \\chi } ( p ) p ^ { \\beta _ { \\chi } - 1 } } { p ^ { 1 + \\eta + i \\tau } } \\frac { ( \\eta \\log p ) ^ k } { k ! } \\Big | + O \\Big ( \\frac { m ^ 2 \\eta \\log ( q Q T ) } { ( 1 1 0 ) ^ k } \\Big ) . \\end{align*}"}
+{"id": "5474.png", "formula": "\\begin{align*} E \\tilde V ( X _ t - Y _ t ) & = E \\tilde V ( X _ 0 - Y _ 0 ) + E \\int _ 0 ^ t \\tilde L \\tilde V ( X _ s - Y _ s ) d s \\\\ & \\leq E \\tilde V ( X _ 0 - Y _ 0 ) - \\gamma E \\int _ 0 ^ t \\tilde V ( X _ s - Y _ s ) d s . \\end{align*}"}
+{"id": "2047.png", "formula": "\\begin{align*} t _ { k + 1 } : = \\left \\{ \\begin{array} { l l l } \\inf \\{ n > t _ k \\mid \\xi _ { t _ k + 1 } + \\ldots + \\xi _ n > 0 \\} & { \\rm i f } & X _ { t _ k } ^ { ( 0 ) } \\leq - 1 , \\\\ \\\\ \\inf \\{ n > t _ k \\mid \\xi _ { t _ k + 1 } ' + \\ldots + \\xi _ n ' < 0 \\} & { \\rm i f } & X _ { t _ k } ^ { ( 0 ) } \\geq 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "3918.png", "formula": "\\begin{align*} \\int _ { E ^ * } f ^ * = \\int _ { Y u ^ { - 1 } ( E ^ * ) } f , \\end{align*}"}
+{"id": "7376.png", "formula": "\\begin{align*} { } ^ g K _ { M ^ 0 } = K _ { \\dot M ^ 0 } \\quad { } ^ g ( \\rho _ { M ^ 0 } \\otimes \\phi ) \\simeq \\rho _ { \\dot M ^ 0 } \\otimes \\dot \\phi , \\end{align*}"}
+{"id": "4685.png", "formula": "\\begin{align*} G _ I = \\left \\langle \\frac { 1 } { \\sqrt { 2 } } \\begin{pmatrix} 1 & 1 \\\\ 1 & - 1 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} \\right \\rangle , \\end{align*}"}
+{"id": "1147.png", "formula": "\\begin{align*} T _ { \\ell } ( f _ 1 ) & = \\lambda ( \\ell ) f _ 1 , \\mbox { \\ \\ a n d } \\\\ T _ { \\ell } ( f _ 0 ) + p ^ { j - 1 - \\nu _ p ( n ) } \\left ( T _ { \\ell } ( f _ 1 E _ { p - 1 } ^ n ) - T _ { \\ell } ( f _ 1 ) E _ { p - 1 } ^ n \\right ) & = \\lambda ( \\ell ) f _ 0 . \\end{align*}"}
+{"id": "6145.png", "formula": "\\begin{align*} V = \\hbox { K e r } \\begin{pmatrix} A ^ T - \\alpha I \\\\ B ^ T - \\beta I \\end{pmatrix} \\end{align*}"}
+{"id": "6838.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , Y _ 3 ) = \\mu _ c ^ { \\pm } ( Y _ 3 , Y _ 1 , Y _ 2 ) = \\mu _ c ^ { \\pm } ( Y _ 2 , Y _ 3 , Y _ 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "54.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m T ( n _ i ) & < K \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } + \\varepsilon \\\\ \\sum _ { i = 1 } ^ m T ( n _ i ) & \\leq K \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } \\end{align*}"}
+{"id": "7262.png", "formula": "\\begin{align*} \\frac { d } { d m } \\frac { d ^ + } { d x } u = \\lambda u , u ( 0 ) = 0 , u ^ + ( 0 ) = 1 \\end{align*}"}
+{"id": "3984.png", "formula": "\\begin{align*} y _ \\theta = Y ( x _ b , \\overline { u } ( x _ b ) , D \\overline { u } ( x _ b ) + \\theta s \\gamma ) . \\end{align*}"}
+{"id": "5803.png", "formula": "\\begin{align*} t \\geqslant 0 : \\| C _ p ( U ( t ) , U ' ( t ) ) \\| _ { ( V \\times H ) ^ { N - p } } \\leqslant M e ^ { - \\omega t } \\| C _ p ( U _ 0 , U _ 1 ) \\| _ { ( V \\times H ) ^ { N - p } } = 0 . \\end{align*}"}
+{"id": "1330.png", "formula": "\\begin{align*} \\lambda _ k = a , a > 0 , \\forall 1 \\leq k \\leq n \\end{align*}"}
+{"id": "4669.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { \\min \\{ 2 p + 1 , 2 m \\} } A _ { m , \\ell } = \\sum _ { \\ell = 0 } ^ { 2 p } \\sum _ { m = \\lceil \\ell / 2 \\rceil } ^ { \\infty } A _ { m , \\ell } + \\sum _ { m = p + 1 } ^ { \\infty } A _ { m , 2 p + 1 } . \\end{align*}"}
+{"id": "669.png", "formula": "\\begin{align*} ( \\frac { \\partial } { \\partial t } + \\mathcal { L } _ { \\bf v } ) ( { \\bf \\nu } ) = { \\rm e x a c t } = d \\phi . \\end{align*}"}
+{"id": "98.png", "formula": "\\begin{align*} z \\ast ( y \\ast x ) & = u ( z ( u ( y x ) ) ) - u ( z ( ( u y ) x ) ) - u ( z ( y ( u x ) ) ) \\\\ & - ( u z ) ( u ( y x ) ) + ( u z ) ( ( u y ) x ) + ( u z ) ( y ( u x ) ) \\\\ & - z ( u ( u ( y x ) ) ) + z ( u ( ( u y ) x ) ) + z ( u ( y ( u x ) ) ) \\end{align*}"}
+{"id": "7857.png", "formula": "\\begin{align*} - A _ 0 ( z ) = \\frac { f ^ { ( n ) } } { f } + A _ { n - 1 } ( z ) \\frac { f ^ { ( n - 1 ) } } { f } + \\cdots + A _ 1 ( z ) \\frac { f ' } { f } . \\end{align*}"}
+{"id": "7139.png", "formula": "\\begin{align*} u _ 1 ( a ' ( a e _ i ) ) & = a ' u _ 1 ( a e _ i ) - u _ 2 ( f _ P ( a ' \\otimes a e _ i ) ) + f _ Q ( a ' \\otimes u _ 0 ( a e _ i ) ) \\\\ & = a ' u _ 1 ( a e _ i ) - u _ 2 ( f ( a ' \\otimes a ) e _ i ) + f _ Q ( a ' \\otimes a b e _ j ) \\\\ & = a ' u _ 1 ( a e _ i ) - f ( a ' \\otimes a ) b + f ( a ' \\otimes a b ) e _ j \\\\ & = a ' u _ 1 ( a e _ i ) - a ' f ( a \\otimes b ) e _ j + f ( a ' a \\otimes b ) e _ j , \\end{align*}"}
+{"id": "2949.png", "formula": "\\begin{align*} E _ { \\nu ^ n _ \\rho } [ f ( \\eta ) ( W _ j - W _ { j + 1 } ) ] = - E _ { \\nu ^ n _ \\rho } [ \\nabla _ { j , j + 1 } f ( \\eta ) W _ j ] . \\end{align*}"}
+{"id": "5264.png", "formula": "\\begin{align*} \\frac { \\lambda ^ 2 } { 2 } g ( X , X ) g \\left ( U , \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } \\right ) = 0 , \\end{align*}"}
+{"id": "4262.png", "formula": "\\begin{align*} \\| g _ n \\| ^ { p + 1 } _ { L ^ { p + 1 } } = \\| \\phi \\| ^ { p + 1 } _ { L ^ { p + 1 } } + \\| g _ n - \\phi \\| ^ { p + 1 } _ { L ^ { p + 1 } } + o _ n ( 1 ) , \\end{align*}"}
+{"id": "3954.png", "formula": "\\begin{align*} \\sup _ { x \\in D } \\langle x , \\nu \\rangle = \\epsilon d , \\end{align*}"}
+{"id": "2102.png", "formula": "\\begin{align*} \\textstyle 1 = e _ 0 + e _ 1 + \\cdots + e _ s ; e _ i e _ j = \\begin{cases} e _ i , & i = j ; \\\\ 0 , & i \\ne j . \\end{cases} \\end{align*}"}
+{"id": "1200.png", "formula": "\\begin{align*} r ^ { 1 - \\epsilon } \\leq \\tilde { r } ' \\leq r ^ { 1 + \\epsilon } , ( 1 - \\epsilon ) r ^ { - \\epsilon } g _ C \\leq \\tilde { g } _ C ' \\leq ( 1 + \\epsilon ) r ^ \\epsilon g _ C , \\sum _ { k = 1 } ^ K r ^ k | \\nabla ^ k _ { g _ C } \\tilde { g } _ C ' | _ { g _ C } \\leq \\epsilon r ^ { \\epsilon } . \\end{align*}"}
+{"id": "6783.png", "formula": "\\begin{align*} \\phi _ { \\chi } ^ o ( s ) = \\frac { \\sqrt { \\pi } } { 4 } \\left [ P ^ { - \\frac { s } { 2 } } \\frac { z ( s ) } { 1 + s } - j \\frac { P ^ { - \\frac { 1 - s } { 2 } } } { G \\left ( \\overline { \\chi } , P \\right ) } \\frac { z ( 1 - s ) } { 2 - s } \\right ] ; \\mathfrak { R e } ( s ) \\in ( 0 , 1 ) \\end{align*}"}
+{"id": "5573.png", "formula": "\\begin{align*} v ( x , y ) = \\left ( \\sup _ { ( x ' , y ' ) \\in B _ { r \\sqrt { s } } ^ n ( 0 , 0 ) } \\{ u ( x ' , y ' ) \\} \\right ) \\max \\left \\{ 1 - \\frac { | ( x , y ) | ^ { 2 - n } - r ^ { 2 - n } } { ( s r ) ^ { 2 - n } - r ^ { 2 - n } } , 0 \\right \\} . \\end{align*}"}
+{"id": "6286.png", "formula": "\\begin{align*} \\pi _ k \\circ \\iota \\circ \\phi ' = \\psi \\circ \\iota ' . \\end{align*}"}
+{"id": "7732.png", "formula": "\\begin{align*} \\bar { \\tau } = \\frac { D _ { \\beta } ^ 2 } { 4 C _ { \\beta } ^ 4 e ^ 4 ( n - 1 ) ^ 4 } \\exp { \\left ( 4 W \\left ( H _ { \\beta } n ( n - 1 ) \\right ) \\right ) } , \\end{align*}"}
+{"id": "4358.png", "formula": "\\begin{align*} L _ \\nu u _ \\nu = f _ \\nu , \\end{align*}"}
+{"id": "1900.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } p _ { x x } ( x , t ) - c p ( x , t ) = 0 , \\ x \\in ( 0 , 1 ) , \\cr p _ x ( 0 , t ) = 0 , \\ p _ x ( 1 , t ) = \\widetilde { w } ( t ) . \\end{array} \\right . \\end{align*}"}
+{"id": "8995.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 _ + } | \\nabla v _ k ( t ) | ^ 2 d z = \\int _ { \\Omega _ k } | \\nabla u _ k ( t ) | ^ 2 d z \\le 2 E ( u _ 0 ) , \\end{align*}"}
+{"id": "7664.png", "formula": "\\begin{align*} \\phi : \\bigoplus _ { \\substack { I \\subseteq [ n ] \\\\ | I | = t } } \\Gamma ( D ( x _ { I } ) , \\Omega _ { X } ^ { j } ( \\log f ) ) \\to \\bigoplus _ { \\substack { I \\subseteq [ n ] \\\\ | I | = t } } \\Gamma ( D ( x _ { I } ) , \\Omega _ { X } ^ { j } ( \\log f ) ) \\end{align*}"}
+{"id": "7483.png", "formula": "\\begin{align*} V = \\bigcup _ { i = 1 } ^ { n } V _ i \\ , . \\end{align*}"}
+{"id": "7381.png", "formula": "\\begin{align*} { } ^ { w ^ \\vee } ( \\chi ^ \\vee \\cdot ( \\varphi _ \\sigma , \\varrho _ \\sigma ) ) = { } ^ { w ^ \\vee } ( \\varphi _ { \\chi \\otimes \\sigma } , \\varrho _ { \\chi \\otimes \\sigma } ) = ( \\varphi _ { { } ^ w ( \\chi \\otimes \\sigma ) } , \\varrho _ { { } ^ w ( \\chi \\otimes \\sigma ) } = ( \\varphi _ { \\chi \\otimes \\sigma } , \\varrho _ { \\chi \\otimes \\sigma } ) . \\end{align*}"}
+{"id": "8275.png", "formula": "\\begin{align*} \\omega _ { \\eta } ( x ) : = \\inf _ { y > 1 } \\left ( \\eta ( y ) \\log x + \\log y \\right ) . \\end{align*}"}
+{"id": "6586.png", "formula": "\\begin{align*} \\theta ( k , i ) & = \\beta _ { k , M _ - ' } \\circ \\beta _ { k , M _ + ' } ^ { - 1 } ( k , i ) \\\\ & = \\beta _ { k , M _ - ' } ( j ) \\\\ & = \\beta _ { k , M _ - } ( j + 1 ) \\\\ & = \\beta _ { k , M _ - } \\circ \\beta _ { k , M _ + } ^ { - 1 } \\circ \\beta _ { k , M _ + ' } ( j ) \\\\ & = \\beta _ { k , M _ - } \\circ \\beta _ { k , M _ + } ^ { - 1 } ( k , i ) \\\\ & = \\eta ( k , i ) \\end{align*}"}
+{"id": "2609.png", "formula": "\\begin{align*} \\left | \\bigcup _ { i = 1 } ^ r L _ i \\right | = \\sum _ { 1 \\leq i < j \\leq r } a _ { i , j } = \\sum _ { j \\in [ r ] \\setminus \\{ 1 \\} } a _ { 1 , j } + \\sum _ { l , t \\in [ r ] \\setminus \\{ 1 \\} , l < t } a _ { l , t } = ( k + 1 ) + \\frac { ( r - 2 ) ( k + 1 ) } { 2 } = \\frac { r ( k + 1 ) } { 2 } . \\end{align*}"}
+{"id": "8372.png", "formula": "\\begin{align*} \\Gamma ' _ { a , \\delta } : = \\{ \\ , ( | x | + \\delta ) ^ 2 - ( t - R + 2 \\delta ) ^ 2 = a ^ 2 , t > 0 \\} . \\end{align*}"}
+{"id": "484.png", "formula": "\\begin{align*} F [ u ] \\leq \\liminf \\limits _ { j \\to \\infty } F [ u _ j ] = \\inf _ { w \\in \\mathfrak { X } } F [ w ] . \\end{align*}"}
+{"id": "1603.png", "formula": "\\begin{align*} d _ p ^ p ( \\delta _ { x _ { n + 1 } } , \\mu ) = d _ p ^ p ( \\delta _ { x _ { n + 1 } } , \\mu ^ * ) - \\mu ( x _ { n + 1 } ) . \\end{align*}"}
+{"id": "7296.png", "formula": "\\begin{align*} X _ { m , j } ( t ) = \\left \\{ \\begin{aligned} & p [ u ] ( t - \\eta _ { m , j } ( u - ) ) & & \\ u \\in D ( p ) \\ \\ \\eta _ { m , j } ( u - ) \\leq t < \\eta _ { m , j } ( u ) , \\\\ & 0 & & . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4884.png", "formula": "\\begin{align*} C _ { s _ 0 } C _ { s _ 2 x _ l } = C _ { s _ 0 s _ 2 x _ l } + \\Box \\in C _ { s _ 0 s _ 2 x _ l } + H ^ { < 1 3 } . \\end{align*}"}
+{"id": "5224.png", "formula": "\\begin{align*} & p _ { 1 2 3 } = 0 , p _ { 3 5 6 } = 0 , p _ { 2 3 5 } = 1 , p _ { 1 3 5 } - p _ { 2 3 6 } = 0 , \\\\ & p _ { 1 2 4 } = 0 , p _ { 4 5 6 } = 0 , p _ { 2 4 5 } = 0 , p _ { 1 4 5 } - p _ { 2 4 6 } = 1 . \\end{align*}"}
+{"id": "2770.png", "formula": "\\begin{align*} \\Delta ( \\Lambda ) = \\Lambda \\otimes \\Lambda , S ( \\Lambda ) = \\Lambda ^ { - 1 } , \\qquad \\varepsilon ( \\Lambda ) = 1 . \\end{align*}"}
+{"id": "689.png", "formula": "\\begin{align*} \\eta = \\nu + * d G ^ \\omega \\end{align*}"}
+{"id": "4659.png", "formula": "\\begin{align*} m ^ { \\underline { \\ell _ i } } m ^ { \\underline { \\ell _ j } } = \\sum _ { k = 0 } ^ { \\min \\{ \\ell _ i , \\ell _ j \\} } G _ { \\ell _ i , \\ell _ j , k } m ^ { \\underline { \\ell _ i + \\ell _ j - k } } . \\end{align*}"}
+{"id": "5265.png", "formula": "\\begin{align*} g ( T _ W W , X ) + g ( T _ V V , X ) + 2 g ( T _ W V , X ) + g ( W , W ) g ( X , \\nabla f ) + g ( V , V ) g ( X , \\nabla f ) = 0 \\end{align*}"}
+{"id": "265.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\rho } - \\beta \\rho ^ 2 \\leqslant 0 , \\end{align*}"}
+{"id": "2094.png", "formula": "\\begin{align*} \\dim _ H ( \\mathcal { S } _ { m } ^ { ( b ) \\ast } ( w _ 1 ) ) & \\geq m - \\dim _ P ( \\mathcal { S } _ m ^ { ( b ) \\ast } ) \\\\ & \\geq m - \\dim _ P ( \\mathcal { S } _ m ) = m - ( m - 1 + \\frac { 1 } { m + 1 } ) = 1 - \\frac { 1 } { m + 1 } , \\end{align*}"}
+{"id": "8482.png", "formula": "\\begin{align*} ( \\alpha - \\beta ) \\varphi _ 1 ^ 2 + \\alpha \\frac { d \\varphi _ 1 } { d \\xi } + ( \\alpha - \\gamma ) \\lambda = \\epsilon , \\end{align*}"}
+{"id": "472.png", "formula": "\\begin{align*} D ( f ^ { n } ( y ) , f ^ { n } ( z ) ) \\leq D ( f ^ { n } ( y ) , f ^ { n } ( x ) ) + D ( f ^ { n } ( x ) , f ^ { n } ( z ) ) \\leq \\delta / 2 + \\delta / 2 = \\delta . \\end{align*}"}
+{"id": "3914.png", "formula": "\\begin{align*} | \\{ x ; f ( x ) > 0 Y u ( x ) \\setminus \\overline { \\Omega ^ * } \\} | = 0 . \\end{align*}"}
+{"id": "2300.png", "formula": "\\begin{align*} \\hat { g } ^ \\C _ { i + 1 / 2 } = g _ { i + 1 / 2 } + \\frac { 2 3 } { 3 6 0 } h _ i ^ { ( 5 ) } \\Delta x ^ 5 + O ( \\Delta x ^ 6 ) . \\end{align*}"}
+{"id": "4907.png", "formula": "\\begin{align*} \\partial ^ 2 _ s \\rho _ v = 1 - e ^ { - \\rho _ v } - b _ v . \\end{align*}"}
+{"id": "8883.png", "formula": "\\begin{align*} P _ { m } ^ { ( n - 1 , 2 \\nu ) } ( \\cos 2 \\rho ) = \\frac { 2 \\Gamma ( n + 2 \\nu ) \\Gamma ( m + 2 \\nu + 1 ) } { \\Gamma ( 2 \\nu + \\frac { 1 } { 2 } ) \\Gamma ( m + 2 \\nu + n ) } \\frac { 1 } { \\cos ^ { 4 \\nu } \\rho } \\int _ { \\rho } ^ { \\frac { \\pi } { 2 } } \\frac { \\sin \\psi } { \\left ( \\cos ^ { 2 } \\rho - \\cos ^ { 2 } \\psi \\right ) ^ { \\frac { 1 } { 2 } - 2 \\nu } } C _ { 2 m } ^ { n + 2 \\nu } ( \\cos \\psi ) d \\psi . \\end{align*}"}
+{"id": "8107.png", "formula": "\\begin{align*} C _ 2 = \\frac { \\sqrt { N p } } { T ^ { 1 - \\varepsilon } M } \\end{align*}"}
+{"id": "1221.png", "formula": "\\begin{align*} ( \\mathfrak { g } _ { 1 } ) _ { \\beta \\alpha } ( \\gamma p ) = \\sum _ { i = 1 } ^ { n - 1 } \\frac { \\partial F ^ { i } _ { \\beta \\alpha } } { \\partial z _ { \\beta } ^ { n } } \\biggr | _ { z _ \\beta ( \\gamma p ) } \\frac { \\partial } { \\partial z ^ { i } _ { \\alpha } } \\biggr | _ { z _ \\alpha ( \\gamma p ) } \\otimes d z _ { \\beta } ^ { n } | _ { z _ \\beta ( \\gamma p ) } . \\end{align*}"}
+{"id": "3370.png", "formula": "\\begin{align*} \\det ( x ^ 1 , \\dots , x ^ { i - 1 } , \\tilde { y } , x ^ { i + 1 } , \\dots , x ^ { n + 1 } ) = \\sum _ { \\sigma \\in S _ { n + 1 } } \\textnormal { s i g n } ( \\sigma ) \\tilde { y } _ { \\sigma ( i ) } \\prod _ { \\substack { j = 1 \\\\ j \\neq i } } ^ { n + 1 } x ^ j _ { \\sigma ( j ) } \\ , , \\end{align*}"}
+{"id": "6189.png", "formula": "\\begin{align*} \\hbox { K e r } ( C _ p ) = \\hbox { S p a n } \\{ x _ l ^ { ( k ) } : \\ 1 \\leqslant k \\leqslant \\overline d , \\ 1 \\leqslant l \\leqslant \\overline r _ k \\} . \\end{align*}"}
+{"id": "9178.png", "formula": "\\begin{align*} | \\hat { f } _ { \\epsilon } ( \\epsilon p ) | = \\Big | \\sum _ { y \\in \\epsilon \\Z ^ 2 } f _ { \\epsilon } ( y / \\epsilon ) ( e ^ { - i y \\cdot p } - 1 ) \\Big | \\leq \\| f _ \\epsilon \\| _ { \\ell ^ \\infty } \\sum _ { y \\in \\epsilon \\Z ^ 2 : | y | \\leq C _ f } | y \\cdot p | \\leq C C _ f ^ 4 | p | , \\end{align*}"}
+{"id": "2707.png", "formula": "\\begin{gather*} - \\Delta W ^ - = \\left ( W ^ - \\right ) ^ 2 , r > R _ - \\\\ \\lim _ { r \\to R _ - } W ^ - ( r ) = - \\infty . \\end{gather*}"}
+{"id": "5129.png", "formula": "\\begin{align*} \\Delta _ { n } ( \\lambda , b ) & : = B _ { n } ^ { 2 } ( \\lambda , b ) - 4 b C _ { n } ( \\lambda , b ) \\\\ & = \\Big ( b \\big [ \\Omega _ { n } ( \\lambda ) + \\Omega _ { n } ( \\lambda b ) \\big ] - ( 1 + b ^ { 2 } ) \\Lambda _ { 1 } ( \\lambda , b ) \\Big ) ^ { 2 } - 4 b ^ { 2 } \\Lambda _ { n } ^ { 2 } ( \\lambda , b ) . \\end{align*}"}
+{"id": "5075.png", "formula": "\\begin{align*} \\bigg \\{ \\ , { x \\choose l } = \\frac { x ( x - 1 ) \\cdots ( x - l + 1 ) } { l ! } \\ , : \\ , l \\ge 0 \\ , \\bigg \\} \\end{align*}"}
+{"id": "2840.png", "formula": "\\begin{align*} 0 < 1 - \\lambda = \\frac { \\tau - \\sigma } { 2 \\tau } < 1 . \\end{align*}"}
+{"id": "2586.png", "formula": "\\begin{align*} & \\{ g : \\mathbb { R } \\rightarrow G \\mid g \\in H ^ 1 , \\ g ( t + 2 \\pi ) = \\sigma ( g ( t ) ) \\ \\} \\\\ \\cong \\ & \\{ g : [ 0 , 2 \\pi ] \\rightarrow G \\mid g \\in H ^ 1 , \\ g ( 2 \\pi ) = \\sigma ( g ( 0 ) ) \\} . \\end{align*}"}
+{"id": "932.png", "formula": "\\begin{align*} m _ { \\alpha } = \\frac { \\theta _ { C } ^ { \\perp } \\cdot ( e _ { 1 } + \\alpha e _ { 2 } ) } { \\theta _ { C } \\cdot ( e _ { 1 } + \\alpha e _ { 2 } ) } , \\end{align*}"}
+{"id": "2108.png", "formula": "\\begin{align*} C = F H \\ ! \\cdot \\ ! ( g _ 1 + g _ { 1 2 } , \\ ; g _ { 1 2 } g ) + F H \\ ! \\cdot \\ ! ( 0 , g _ 2 ) . \\end{align*}"}
+{"id": "3999.png", "formula": "\\begin{align*} L ( | D u | ^ 2 / 2 ) & = w ^ { i i } ( u _ { k i } u _ { k i } - u _ k u _ { k i i } - D _ { p _ l } A _ { i i } u _ { k } u _ { l k } ) - B _ { p _ l } u _ { k } u _ { k l } \\\\ & = w ^ { i i } u _ { k i } u _ { k i } + u _ k [ w ^ { i i } ( u _ { k i i } - D _ { p _ l } A _ { i i } u _ { l k } ) - B _ { p _ l } u _ { k l } ] . \\end{align*}"}
+{"id": "6691.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } & \\left [ \\| \\bar { y } ^ { k + 1 } - \\bar { g } ^ { k + 1 } \\| ^ 2 | \\mathcal { F } ^ k \\right ] \\\\ & = ( 1 - 2 \\alpha ^ k + ( \\alpha ^ k ) ^ 2 ) \\| \\bar { y } ^ k - \\bar { g } ^ k \\| ^ 2 + ( \\gamma _ 2 ^ k ) ^ 2 \\mathbb { E } \\left [ \\| \\bar { \\xi } _ w ^ k \\| ^ 2 \\right ] \\end{aligned} \\end{align*}"}
+{"id": "2721.png", "formula": "\\begin{align*} \\lambda _ j \\beta _ j ' ( t ) = \\lambda _ j ' \\int \\left ( \\Lambda _ 0 \\Lambda W \\right ) _ { [ \\lambda _ j ] } \\partial _ t U - \\lambda _ j \\int \\left ( \\Lambda W \\right ) _ { [ \\lambda _ j ] } \\partial _ t ^ 2 U \\end{align*}"}
+{"id": "2651.png", "formula": "\\begin{align*} B _ t = W _ t - \\frac { t } { T } W _ T = \\frac { T - t } { \\sqrt { T } } W _ { t / ( T - t ) } . \\end{align*}"}
+{"id": "7119.png", "formula": "\\begin{align*} \\langle d , x ^ { V } \\rangle = \\langle \\Delta d , x ^ { \\rho _ 1 } \\otimes \\cdots \\otimes x ^ { \\rho _ k } \\rangle . \\end{align*}"}
+{"id": "8986.png", "formula": "\\begin{align*} w _ t = - ( \\varepsilon + d \\pi _ N ( v _ 1 ) ) w _ r - ( d \\pi _ N ( v _ 1 ) - d \\pi _ N ( v _ 2 ) ) u _ { 2 , r } \\hbox { o n } \\partial B = S ^ 1 . \\end{align*}"}
+{"id": "4338.png", "formula": "\\begin{align*} \\omega ( k , i , j ) ^ { - 1 } c ( i + j , k ) \\omega ( i , j , k ) ^ { - 1 } = c ( i , k ) \\omega ( i , k , j ) ^ { - 1 } c ( j , k ) . \\end{align*}"}
+{"id": "7478.png", "formula": "\\begin{align*} \\tilde { F } _ { A } \\left ( s \\right ) = \\frac { 1 } { \\mu _ A - s } \\cdot \\sum _ { ( u , v ) \\in E _ A } \\mu _ { u v } \\cdot \\tilde { F } _ { A \\cup \\{ u \\} } \\left ( s \\right ) . \\end{align*}"}
+{"id": "2835.png", "formula": "\\begin{align*} 1 = \\sum _ { i = 1 } ^ n p _ { \\tau ( i ) , j } < \\sum _ { i = 1 } ^ k p _ { \\tau ( i ) , j } \\leq \\sum _ { i = 1 } ^ k \\delta _ { i , j } \\leq 1 \\end{align*}"}
+{"id": "1959.png", "formula": "\\begin{align*} c _ g ( a H ) = a H g ^ { - 1 } = a g ^ { - 1 } g H g ^ { - 1 } = a g ^ { - 1 } H ' . \\end{align*}"}
+{"id": "4873.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ { n - 1 } } w _ { z _ n ; k _ n } ( t _ n ) \\ . w _ { z _ 2 ; k _ 2 } ( t _ 2 ) d \\L ^ { n - 1 } = \\ ; g ^ 1 _ { z _ 2 , k _ 2 } \\int _ { \\Sigma _ { n - 2 } } w _ { z _ n ; k _ n } ( t _ n ) \\ . w _ { z _ 3 ; k _ 3 } ( t _ 3 ) & d \\L ^ { n - 2 } \\\\ - \\frac { 1 } { 2 k _ 2 \\pi } \\int _ { \\Sigma _ { n - 2 } } w _ { z _ n ; k _ n } ( t _ n ) \\ . w _ { z _ 3 ; k _ 3 } ( t _ 3 ) w _ { z _ 2 ; k _ 2 } ( t _ 3 ) & d \\L ^ { n - 2 } \\end{align*}"}
+{"id": "7197.png", "formula": "\\begin{align*} \\overline q \\ , = \\ , x _ 0 - x _ 1 i - x _ 2 j - x _ 3 k , \\qquad | q | \\ , = \\ , \\sqrt { q \\overline q \\ , } \\ , = \\ , \\sqrt { x _ 0 ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 \\ , } . \\end{align*}"}
+{"id": "8576.png", "formula": "\\begin{align*} \\mathbf { s } = ( s _ 0 , s _ 1 , \\ldots , s _ { N - 1 } ) \\mapsto L ( \\mathbf { s } ) = ( s _ 1 , \\ldots , s _ { N - 1 } , s _ 0 ) . \\end{align*}"}
+{"id": "1913.png", "formula": "\\begin{align*} \\sum _ { p \\in \\P _ j } \\frac { 1 } { p } = \\sum _ { p \\leq \\lambda _ j : p \\nmid d } \\frac { 1 } { p } - \\sum _ { p \\leq \\lambda _ { j - 1 } : p \\nmid d } \\frac { 1 } { p } \\geq j - \\frac { 1 } { \\lambda _ j } - ( j - 1 ) = 1 - \\frac { 1 } { \\lambda _ j } . \\end{align*}"}
+{"id": "9044.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ | \\Delta V ^ { i , n } _ \\tau | ^ p \\right ] = \\mathbb { E } \\left [ | \\Delta V ^ { i , n } _ \\tau - \\Delta V ^ { ( m ) , i , n } _ \\tau | ^ p \\right ] \\leq 2 ^ p \\sup _ { \\tau \\in \\mathcal { T } _ 0 } \\mathbb { E } \\left [ | V ^ { i , n } _ \\tau - V ^ { ( m ) , i , n } _ \\tau | ^ p \\right ] , \\end{align*}"}
+{"id": "1715.png", "formula": "\\begin{align*} w \\delta s ( \\mu _ m ) = \\epsilon ( - 1 ) ^ { m } \\delta s ( \\mu _ m ) , w = \\left ( \\begin{array} { c c } 1 & \\\\ & - 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "9159.png", "formula": "\\begin{align*} \\Omega ^ { \\Lambda _ N } = \\{ \\sigma \\in ( 2 \\pi \\Z ) ^ { \\Lambda _ N } : \\sigma _ { x = 0 } = 0 \\} . \\end{align*}"}
+{"id": "5524.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } x ^ { - s - 1 } P _ k ( x ) { \\rm d } x = \\frac { \\Gamma ( - s ) } { \\zeta ( 2 s + k ) } . \\end{align*}"}
+{"id": "634.png", "formula": "\\begin{align*} I _ { D _ 3 } = \\ & ( x _ 1 x _ 5 , x _ 1 x _ 7 , x _ 1 x _ 9 , x _ 5 x _ 7 , x _ 5 x _ 8 , x _ 5 x _ 9 , x _ 6 x _ 8 , x _ 6 x _ 9 , x _ 7 x _ 9 , x _ 1 x _ 2 x _ 3 , x _ 1 x _ 4 x _ 6 , x _ 1 x _ 4 x _ 8 , \\\\ & \\ x _ 2 x _ 3 x _ 4 , x _ 2 x _ 3 x _ 6 , x _ 2 x _ 3 x _ 8 , x _ 2 x _ 4 x _ 6 , x _ 2 x _ 4 x _ 7 , x _ 2 x _ 4 x _ 8 , x _ 2 x _ 4 x _ 9 , x _ 2 x _ 5 x _ 6 , x _ 2 x _ 7 x _ 8 , \\\\ & \\ x _ 3 x _ 4 x _ 5 , x _ 3 x _ 4 x _ 6 , x _ 3 x _ 4 x _ 7 , x _ 3 x _ 4 x _ 8 , x _ 3 x _ 6 x _ 7 , x _ 3 x _ 8 x _ 9 ) \\subseteq k [ x _ 1 , \\dots , x _ 9 ] . \\end{align*}"}
+{"id": "8224.png", "formula": "\\begin{align*} A _ \\lambda = \\left [ \\begin{array} { c c } \\lambda B & C \\\\ C ^ * & D \\\\ \\end{array} \\right ] . \\end{align*}"}
+{"id": "1136.png", "formula": "\\begin{align*} a \\cdot m & = a m \\mbox { , \\ t h e \\ a s s u m e d \\ l e f t \\ $ A $ - a c t i o n \\ o n \\ $ M $ , \\ a n d } \\\\ m \\cdot a & = \\lambda ( a ) m . \\end{align*}"}
+{"id": "9.png", "formula": "\\begin{align*} \\begin{aligned} & \\quad + \\mathbb { E } \\bigg ( \\int _ 0 ^ \\infty e ^ { - \\beta _ 5 k t } \\int _ { \\mathcal { E } } \\bigg [ \\bigg ( \\int _ 0 ^ 1 l _ x d \\theta - \\bar { l } _ x \\bigg ) x _ { ( 1 , t ) } + \\bigg ( \\int _ 0 ^ 1 l _ u d \\theta - \\bar { l } _ u \\bigg ) v _ t \\bigg ] ^ { 2 k } \\nu ( d e ) d t \\bigg ) \\bigg ] \\\\ & : = \\mathbf { A } _ 1 + \\mathbf { A } _ 2 + \\mathbf { A } _ 3 + \\mathbf { A } _ 4 + \\mathbf { A } _ 5 . \\end{aligned} \\end{align*}"}
+{"id": "1666.png", "formula": "\\begin{align*} \\frac { \\langle \\Psi , \\Psi \\rangle } { \\langle \\Phi _ 1 , \\Phi _ 2 \\rangle } \\frac { \\langle \\Phi _ 1 ( \\underline { \\delta s } ( \\mu _ { \\underline m } ) ) , \\Phi _ 2 ( \\underline { \\delta s } ( \\mu _ { - \\underline m } ) ) \\rangle } { \\langle \\mathfrak { f } , \\mathfrak { f } \\rangle } \\prod _ { \\sigma \\mid \\infty } \\beta _ \\sigma \\alpha _ { \\sigma } = \\prod _ { \\sigma \\mid \\infty } \\frac { C _ \\sigma } { L ( 1 / 2 , \\Pi _ \\sigma , \\chi _ \\sigma ) } , \\end{align*}"}
+{"id": "2896.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb { T } ^ \\infty } \\log ^ { + } | G _ n ^ { * } | d m _ \\infty = \\int _ { \\mathbb { T } ^ \\infty } \\log ^ { + } | G ^ { * } | d m _ \\infty . \\end{align*}"}
+{"id": "3980.png", "formula": "\\begin{align*} w ^ { i j } D _ { p _ k p _ l } A _ { i j } \\eta _ k \\eta _ l & = \\sum _ { s } \\lambda _ s D _ { p _ k p _ l } A _ { i j } \\phi ^ s _ i \\phi ^ s _ j \\eta _ k \\eta _ l \\\\ & \\geq - C \\sum _ { s } \\lambda _ s | \\phi ^ s | | \\eta | ( \\phi ^ s \\cdot \\eta ) . \\end{align*}"}
+{"id": "3123.png", "formula": "\\begin{align*} r = 1 + \\partial _ { 1 1 } ^ 2 w - \\partial _ { 2 2 } ^ 2 w , v ^ { 1 1 } = - v ^ { 2 2 } = w , v ^ { 1 2 } = v ^ { 2 1 } \\equiv 0 . \\end{align*}"}
+{"id": "3769.png", "formula": "\\begin{align*} A _ \\Gamma = \\langle V \\mid ( s , t ; m ( e ) ) = ( t , s ; m ( e ) ) e = \\{ s , t \\} \\in E \\rangle , \\end{align*}"}
+{"id": "8557.png", "formula": "\\begin{align*} \\frac { \\lambda ( v - 1 ) ! } { ( v - t ) ! } = \\sum _ { j = 0 } ^ { v - 1 } d _ { w } ( j ) \\binom { j } { i } \\binom { v - 1 - j } { t - 1 - i } . \\end{align*}"}
+{"id": "8350.png", "formula": "\\begin{align*} \\Sigma _ { 1 , M , B } & = \\{ M + \\frac { B - R } { 2 } \\le t \\le M , \\ , | x | + t = 2 M + B \\} , \\\\ \\Sigma _ { 2 , M , B } & = \\{ t = M , \\ , M + B < | x | < R + M \\} , \\\\ \\Sigma _ { 3 , M , B } & = \\{ M + \\frac { B - R } { 2 } \\le t \\le M , \\ , | x | - t = R \\} . \\end{align*}"}
+{"id": "7188.png", "formula": "\\begin{align*} A _ { \\beta \\alpha } ( { \\Sigma _ 0 } ) y _ { \\alpha } = C _ \\beta ( { \\Sigma _ 0 } ) , \\end{align*}"}
+{"id": "4744.png", "formula": "\\begin{align*} x u _ i & = \\alpha _ i + y ^ { e _ i } ( r + y b ) = \\alpha _ i + y ^ { e _ i } r + y ^ { e _ i + 1 } b \\in U _ { e _ i + 1 } \\end{align*}"}
+{"id": "6536.png", "formula": "\\begin{align*} & \\bar { G } \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( x ^ j \\right ) \\int _ { A ^ c } \\phi _ m ( w ) \\ , Q ( x , w ) d P _ 0 ^ m ( w ) d P _ 0 ^ m ( x ) \\\\ & \\qquad \\geq \\int _ { A ^ c } \\phi _ m ( w ) \\int \\phi _ m ( x ) \\ , Q ( x , w ) d P _ 0 ^ m ( x ) d P _ 0 ^ m ( w ) . \\end{align*}"}
+{"id": "693.png", "formula": "\\begin{align*} G ^ { \\omega } ( z ) = \\sum _ { k = 1 } ^ n \\Gamma _ k { G ( z , w _ k ) } . \\end{align*}"}
+{"id": "7570.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 6 ) = \\dfrac { F _ 6 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "4910.png", "formula": "\\begin{align*} \\frac { \\partial z } { \\partial t } = \\mathcal { J } \\frac { \\delta \\mathcal H } { \\delta z ^ * } , \\end{align*}"}
+{"id": "1355.png", "formula": "\\begin{align*} \\mathcal { M } ( \\{ f _ j \\} _ { j = 1 } ^ n , \\{ \\tau _ j \\} _ { j = 1 } ^ n ) \\coloneqq \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | . \\end{align*}"}
+{"id": "831.png", "formula": "\\begin{align*} p _ i = \\frac { e _ i ^ T A ^ { \\lfloor c \\log n \\rfloor p } 1 } { \\lambda ^ { \\lfloor c \\log n \\rfloor p } } + O \\left ( n ^ { - 1 / 2 } \\right ) \\ \\ u _ i = \\frac { e _ \\ast ^ T A ^ { \\lfloor c \\log n \\rfloor p } e _ i } { \\lambda ^ { \\lfloor c \\log n \\rfloor p } } + O \\left ( n ^ { - 1 / 2 } \\right ) \\end{align*}"}
+{"id": "1352.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | \\geq I _ { } ( \\{ f _ j \\} _ { j = 1 } ^ n , \\{ \\tau _ j \\} _ { j = 1 } ^ n ) \\geq \\left ( \\frac { n - d } { d ( n - 1 ) } \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "1339.png", "formula": "\\begin{align*} \\operatorname { T r a } ( S ^ r _ { f , \\tau } ) \\leq \\frac { \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ r } { d ^ { r - 1 } } , \\forall r \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "649.png", "formula": "\\begin{align*} \\zeta : = \\min \\left \\{ \\frac { \\bar { \\epsilon } } { 2 0 N } , \\frac { \\bar { \\epsilon } _ { 0 } \\kappa } { 4 N } , \\frac { \\bar { \\alpha } - \\alpha } { 2 N ( N + 1 ) } , \\frac { \\beta - \\bar { \\beta } } { 2 N } , \\frac { \\epsilon _ { 0 } ( \\gamma - \\bar { \\gamma } ) } { 2 N } \\right \\} . \\end{align*}"}
+{"id": "1764.png", "formula": "\\begin{align*} u _ t ( x ) : = t ^ 2 u ( t x ) . \\end{align*}"}
+{"id": "810.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\nu \\left ( \\xi \\in \\partial \\Gamma : \\left | \\frac { d ( g \\cdot o , o ) } { n } - \\Lambda \\right | > \\epsilon \\right ) < 0 \\end{align*}"}
+{"id": "1571.png", "formula": "\\begin{align*} { \\rm R i c } + \\frac { 1 } { 2 } \\mathcal { L } _ X g - \\rho R g = \\mu g , \\end{align*}"}
+{"id": "6180.png", "formula": "\\begin{align*} S _ r = \\begin{pmatrix} 1 & - 1 & 0 & \\cdots & 0 \\\\ 0 & 1 & - 1 & \\cdots & 0 \\cr \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 & - 1 \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "4245.png", "formula": "\\begin{align*} A ( x ) = \\gamma ( - x _ 2 , x _ 1 , 0 , \\cdots , 0 ) , V _ \\gamma ( x ) : = \\frac { 1 } { 2 } \\sum _ { j = 3 } ^ N \\gamma _ j ^ 2 x _ j ^ 2 . \\end{align*}"}
+{"id": "5243.png", "formula": "\\begin{align*} g ( g r a d f , X ) = X ( f ) , ~ \\forall X \\in \\Gamma ( T M ) . \\end{align*}"}
+{"id": "6824.png", "formula": "\\begin{align*} \\mu ^ * ( Y , \\hat Y ) = \\mu _ 1 ( Y , \\hat Y ) + \\mu _ 2 ^ * ( Y , \\hat Y ) , \\ , \\mu _ 2 ^ * ( Y , \\hat Y ) = { \\rm { i n d } } ( - \\mathcal P ) , \\end{align*}"}
+{"id": "6406.png", "formula": "\\begin{align*} ( - \\nabla _ { \\tilde X , \\tilde Y , \\tilde t } \\Psi ( P _ j ) , H ) = ( \\nabla _ { X , Y , t } \\Psi ( P _ j ) , H ) & \\in \\overline { J } ^ { 2 , + } ( - u ) ( X _ j , Y _ j , t _ j ) = - \\overline { J } ^ { 2 , - } u ( X _ j , Y _ j , t _ j ) , \\\\ ( \\nabla _ { \\tilde X , \\tilde Y , \\tilde t } \\Psi ( P _ j ) , E ) & \\in \\overline { J } ^ { 2 , + } \\phi ( \\tilde X _ j , \\tilde Y _ j , \\tilde t _ j ) , \\end{align*}"}
+{"id": "3907.png", "formula": "\\begin{align*} \\mu ( E ) = \\inf _ { \\substack { B \\supset E \\\\ B } } \\int _ { Y u ( B \\cap \\Omega ) } f ^ * ( y ) \\ d y . \\end{align*}"}
+{"id": "3937.png", "formula": "\\begin{align*} \\overline { g } ( q , \\overline { Y } ( q , U , P ) , \\overline { Z } ( q , U , P ) ) = U , \\\\ \\overline { g } _ q ( q , \\overline { Y } ( q , U , P ) , \\overline { Z } ( q , U , P ) ) = P . \\end{align*}"}
+{"id": "4954.png", "formula": "\\begin{align*} z ( x , t ) & = \\left ( \\frac { 2 \\beta ^ 2 \\cosh \\theta + 2 \\mathrm { i } \\beta \\sqrt { 2 - \\beta ^ 2 } \\sinh \\theta } { 2 \\cosh \\theta - \\sqrt { 4 - 2 \\beta ^ 2 } \\cos ( \\sqrt \\omega \\beta x ) } - 1 \\right ) \\sqrt \\omega \\mathrm { e } ^ { \\mathrm { i } \\omega t } , \\theta = \\omega \\beta \\sqrt { 2 - \\beta ^ 2 } t , \\quad \\beta < \\sqrt { 2 } , \\\\ u ( x , t ) & = \\operatorname { R e } ( z ) , v ( x , t ) = \\operatorname { I m } ( z ) . \\end{align*}"}
+{"id": "1843.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\rho _ 0 ( t ) = \\frac { 1 } { 6 \\pi G t ^ 2 } , \\\\ & H ( t ) = \\frac { 2 } { 3 t } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2992.png", "formula": "\\begin{align*} \\int _ { \\textsf { f } _ { x _ 2 } } \\theta - \\int _ { \\textsf { f } _ { x _ 1 } } \\theta = \\int _ S \\hbox { d } \\theta = 0 \\end{align*}"}
+{"id": "4546.png", "formula": "\\begin{align*} \\frac { 1 } { s _ 1 } = \\frac { 1 } { 2 } = 1 - \\frac { 1 } { 2 } = 1 - \\frac { 1 } { s _ 1 } \\end{align*}"}
+{"id": "5475.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\sup _ { s \\geq 0 } W _ { \\tilde V } \\left ( P _ t ^ * \\nu _ 0 , P _ { t + s } ^ * \\nu _ 0 \\right ) = 0 . \\end{align*}"}
+{"id": "7253.png", "formula": "\\begin{align*} A ( t ) : = \\int _ { 0 } ^ { t } 1 _ { ( 0 , \\infty ) } ( X _ s ) d s . \\end{align*}"}
+{"id": "2309.png", "formula": "\\begin{align*} \\nu ( u ) = \\begin{cases} 4 u ( 1 - u ) , & 0 \\leqslant u \\leqslant 1 , \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "4075.png", "formula": "\\begin{align*} & \\langle F u , F v \\rangle = f _ { * } \\langle u , v \\rangle . \\\\ & [ F u , F v ] = F [ u , v ] . \\\\ & \\pi _ { T M } \\circ F = f _ { * } \\circ \\pi _ { T M } . \\end{align*}"}
+{"id": "7339.png", "formula": "\\begin{align*} \\tilde { m } ( x ) = 2 \\int _ { 1 } ^ { x } W ( y ) d y , \\tilde { s } ( x ) = \\int _ { 0 } ^ { x } \\frac { d y } { W ( y ) } , \\end{align*}"}
+{"id": "3751.png", "formula": "\\begin{align*} I ( f ) ( 0 , 0 , 0 , \\varpi ^ n ) & = n \\tilde { c } _ 2 ( I ( f ) ) + a _ 0 ( f ) ( ( 0 : 0 : 0 : 1 ) ) \\\\ & = n \\tilde { c } _ 2 ( I ( f ) ) + \\tilde { a } _ 2 ( I ( f ) ) , \\end{align*}"}
+{"id": "9228.png", "formula": "\\begin{align*} \\lll \\overline { u } = r + \\widehat { r } ( y , z ) , \\end{align*}"}
+{"id": "2984.png", "formula": "\\begin{align*} \\mathbb { E } _ n \\bigg [ \\sup _ { 0 \\le t \\le T } \\bigg | \\frac { 1 } { \\sqrt { n } } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\sum _ { i = 0 } ^ { \\ell - 1 } ( \\nabla _ { j + i - 1 , j + i } ( \\overrightarrow { W } _ { j + 1 } ^ \\ell ( s ) ) ^ 2 ) W _ { j + i - 1 } ( s ) \\psi _ i \\nabla ^ n \\varphi ^ n _ j ( s ) d s \\bigg | ^ 2 \\bigg ] . \\end{align*}"}
+{"id": "91.png", "formula": "\\begin{align*} \\deg ( \\mathfrak { h } _ { \\mathbf { k } } ) = \\deg ( \\mathfrak { g } _ { \\mathbf { k } } ) \\cdot \\deg ( [ s ] ) = \\deg ( \\mathfrak { g } _ { \\mathbf { k } } ) \\cdot s ^ 2 \\leq 3 s ^ 3 . \\end{align*}"}
+{"id": "6151.png", "formula": "\\begin{align*} \\begin{cases} u '' - \\Delta u = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\cr v '' - \\Delta v + u = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ u = v = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu u + \\alpha u = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 , \\\\ \\partial _ \\nu v + \\beta v = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{cases} \\end{align*}"}
+{"id": "6775.png", "formula": "\\begin{align*} \\frac { \\left | z \\left ( \\left ( \\frac { 1 } { 2 } - \\sigma \\right ) + j \\omega \\right ) \\right | } { \\left | z \\left ( \\left ( \\frac { 1 } { 2 } + \\sigma \\right ) - j \\omega \\right ) \\right | } = \\frac { \\left | P ^ { \\frac { 1 } { 2 } } \\right | } { \\left | G \\left ( \\overline { \\chi } , P \\right ) \\right | } \\frac { \\left | \\left ( \\frac { 1 } { 2 } - \\sigma \\right ) + j \\omega \\right | } { \\left | \\left ( \\frac { 1 } { 2 } + \\sigma \\right ) - j \\omega \\right | } \\end{align*}"}
+{"id": "6808.png", "formula": "\\begin{align*} D _ 1 L _ d ( q _ m , q _ { m + 1 } ) + D _ 2 L _ d ( q _ { m - 1 } , q _ { m } ) = 0 . \\end{align*}"}
+{"id": "6832.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) = \\mu _ c ^ { \\pm } ( Y _ m , Y _ 1 , Y _ 2 \\dots , Y _ { m - 1 } ) & = \\dots = \\mu _ c ^ { \\pm } ( Y _ 2 , Y _ 3 , \\dots , Y _ m , Y _ 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "1240.png", "formula": "\\begin{align*} & \\int _ X U ( \\bar \\mu _ t ( x ) ) d x = \\\\ & \\int _ Y \\frac { 1 } { \\bar \\nu ( y ) } \\int _ { T _ 0 ^ { - 1 } ( y ) } U \\Big ( \\frac { \\bar \\mu _ 0 ^ y ( x _ 0 ) \\bar \\nu ( y ) J T ^ y _ t ( \\nabla \\phi ^ y _ t ( x _ 0 ) ) } { \\det ( D ^ 2 \\phi ^ y _ t ( x _ 0 ) ) } \\Big ) \\frac { 1 } { J T ^ y _ t ( \\nabla \\phi ^ y _ t ( x _ 0 ) ) } \\det ( D ^ 2 \\phi _ t ^ y ( x _ 0 ) ) d \\mathcal H ^ { m - n } ( x _ 0 ) d y \\end{align*}"}
+{"id": "4407.png", "formula": "\\begin{align*} 0 < \\frac { 1 } { q \\prod _ { i = 1 } ^ n x _ i } \\leq \\frac { r } { q \\prod _ { i = 1 } ^ n x _ i } = \\frac { p } { q } - \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } \\leq \\frac { 1 } { q \\prod _ { i = 1 } ^ n a _ i } . \\end{align*}"}
+{"id": "3598.png", "formula": "\\begin{align*} \\mu : = \\sum _ { \\ell = 1 } ^ { j - 1 } ( \\lambda _ \\ell - \\lambda _ j ) \\epsilon _ \\ell . \\end{align*}"}
+{"id": "7135.png", "formula": "\\begin{align*} \\mu ( a _ 0 + a _ 1 t , b _ 0 + b _ 1 t ) = a _ 0 b _ 0 + ( a _ 0 b _ 1 + a _ 1 b _ 0 + f ( a _ 0 , b _ 0 ) ) t . \\end{align*}"}
+{"id": "493.png", "formula": "\\begin{align*} \\partial h ( x ) : = \\Big \\{ v \\in \\mathbb { X } \\ | \\ \\exists \\ , x ^ k \\to x \\ { \\rm w i t h } \\ h ( x ^ k ) \\to h ( x ) \\ { \\rm a n d } \\ v ^ k \\in \\widehat { \\partial } h ( x ^ k ) \\ { \\rm s u c h \\ t h a t } \\ v ^ k \\to v \\Big \\} . \\end{align*}"}
+{"id": "1323.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | \\geq \\sqrt { \\frac { n - d } { d ( n - 1 ) } } = \\sqrt { \\frac { n - ( \\mathcal { X } ) } { ( \\mathcal { X } ) ( n - 1 ) } } . \\end{align*}"}
+{"id": "1391.png", "formula": "\\begin{align*} k _ { \\rho , z } & \\coloneqq \\# \\{ i \\in \\{ 1 , \\dots , r \\} \\ ; | \\ ; \\} + m _ \\phi ( \\rho \\boxtimes S _ { 2 z + 1 } ) , \\end{align*}"}
+{"id": "6673.png", "formula": "\\begin{align*} \\begin{aligned} & \\lambda ^ k \\left \\langle g _ i ^ k - \\nabla f _ i ( \\bar { x } ^ k ) , \\bar x ^ k - \\theta ^ * \\right \\rangle \\geq - L \\lambda ^ k \\| x _ i ^ k - \\bar { x } ^ k \\| \\| \\bar x ^ k - \\theta ^ * \\| \\\\ & \\geq - \\frac { \\gamma ^ k } { 2 } \\| x _ i ^ k - \\bar { x } ^ k \\| ^ 2 - \\frac { L ^ 2 ( \\lambda ^ k ) ^ 2 } { 2 \\gamma ^ k } \\| \\bar x ^ k - \\theta ^ * \\| ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "3278.png", "formula": "\\begin{align*} \\lambda _ { p _ { 0 } , p _ { 1 } } = \\inf \\left \\{ \\left \\Vert \\nabla u \\right \\Vert _ { p } \\left [ \\underset { \\Omega } { \\int } \\left \\vert u \\right \\vert ^ { p _ { 0 } } \\left \\vert \\nabla u \\right \\vert ^ { p _ { 1 } } d x \\right ] ^ { - \\frac { 1 } { p } } \\left \\vert \\ u \\in S _ { 1 } ^ { W _ { 0 } ^ { 1 . p } \\left ( \\Omega \\right ) } \\left ( 0 \\right ) \\right . \\right \\} . \\end{align*}"}
+{"id": "7284.png", "formula": "\\begin{align*} u ( 0 ) = 1 , \\lim _ { x \\to + 0 } \\left ( u ^ + ( x ) - \\left ( \\lambda m ( x ) + \\sum _ { k = 1 } ^ { d - 1 } \\lambda ^ k \\int _ { x } ^ { 1 } G ^ k _ m ( y ) d m ( y ) \\right ) \\right ) = 0 . \\end{align*}"}
+{"id": "5146.png", "formula": "\\begin{align*} \\mathcal { Z } _ { \\nu } ' ( z ) = \\mathcal { Z } _ { \\nu - 1 } ( z ) - \\frac { \\nu } { z } \\mathcal { Z } _ { \\nu } ( z ) = \\mathcal { Z } _ { \\nu + 1 } ( z ) + \\frac { \\nu } { z } \\mathcal { Z } _ { \\nu } ( z ) . \\end{align*}"}
+{"id": "7879.png", "formula": "\\begin{align*} f '' ( z ) + ( \\log 1 2 ) f ' ( z ) + ( \\log 3 ) ( \\log 4 ) f ( z ) = ( \\log 3 ) ( \\log 4 ) + ( \\log 2 ) \\left ( \\log \\frac 3 2 \\right ) \\frac { 1 } { 2 ^ z } . \\end{align*}"}
+{"id": "1829.png", "formula": "\\begin{align*} \\mu _ i = \\frac { D \\cdot C _ i } { - ( K _ X + \\Delta ) \\cdot C _ i } = \\frac { k D \\cdot C _ i } { - k ( K _ X + \\Delta ) \\cdot C _ i } \\ > \\frac { ( k D ) \\cdot C _ i } { 2 n k } \\mbox { f o r a l l } i \\in I . \\end{align*}"}
+{"id": "8179.png", "formula": "\\begin{align*} u ( t ) & = u _ 0 + \\int _ 0 ^ t k ( t , s ) L u ( s ) d s , t > 0 , \\end{align*}"}
+{"id": "8280.png", "formula": "\\begin{align*} \\begin{alignedat} { 2 } \\dot { x } _ E & = u _ { x _ E } , \\dot { y } _ E = u _ { y _ E } , \\dot { z } _ E = u _ { z _ E } \\\\ \\dot { x } _ i & = u _ { x _ i } , \\ \\dot { y } _ i = u _ { y _ i } , \\ \\ \\dot { z } _ i = u _ { z _ i } \\end{alignedat} \\end{align*}"}
+{"id": "2315.png", "formula": "\\begin{align*} | \\Delta _ p U _ p | & = O \\ ( ( | x | ^ 2 + ( y - 1 ) ^ 2 ) ^ { ( - \\frac { N - p } { p - 1 } - 1 ) ( \\frac { p - 4 } { 2 } ) - \\frac { N - p } { 2 ( p - 1 ) } - \\frac { N - p } { 2 ( p - 1 ) } - 1 } \\ ) \\\\ & = O \\ ( \\ ( \\sqrt { | x | ^ 2 + ( y - 1 ) ^ 2 } \\ ) ^ { ( - \\frac { N - p } { p - 1 } - 1 ) ( p - 4 ) - \\frac { N - p } { ( p - 1 ) } - \\frac { N - p } { ( p - 1 ) } - 2 } \\ ) \\\\ & = O \\ ( \\ ( \\sqrt { | x | ^ 2 + ( y - 1 ) ^ 2 } \\ ) ^ { - \\frac { ( N - 1 ) ( p - 2 ) } { p - 1 } } \\ ) \\end{align*}"}
+{"id": "4810.png", "formula": "\\begin{align*} \\Pr _ { \\rho \\sim P _ \\epsilon } [ d ( c + \\rho ) \\neq c ] & \\leq 2 e ^ { - \\frac { \\sqrt { N } } { 3 \\epsilon } } + N \\mathop { \\max } _ { \\substack { S \\subseteq \\{ \\epsilon N \\pm N ^ { 3 / 4 } \\} \\\\ 1 \\leq | S | \\leq 2 } } \\Big \\{ \\frac { 1 } { \\binom { N } { S } } \\sum _ { j = 0 } ^ N \\Pr _ { c \\sim C ^ \\perp } \\big [ | c | = j \\big ] K _ S ( j ) ^ 2 - 1 \\Big \\} , \\end{align*}"}
+{"id": "1986.png", "formula": "\\begin{align*} { \\mathcal { R } } _ { } = \\frac { N } { L } \\max _ { { \\mathsf { t r } } \\left ( { \\textbf { S } } { \\textbf { S } } ^ { \\mathsf { H } } \\right ) \\leq p _ { } } \\log _ 2 \\det \\left ( { \\textbf { I } } _ L + { \\textbf { S } } ^ { \\mathsf { H } } { \\textbf { R } } _ { } { \\textbf { S } } \\right ) . \\end{align*}"}
+{"id": "6147.png", "formula": "\\begin{align*} \\Phi = 0 \\quad \\hbox { o n } ( 0 , T ) \\times \\Gamma _ 1 . \\end{align*}"}
+{"id": "9072.png", "formula": "\\begin{align*} f ( m a n g h ) = v ^ { * } \\big ( \\pi ( m a n g h ) v _ { H } \\big ) = a ^ { \\nu } v ^ { * } \\big ( \\pi ( g ) v _ { H } \\big ) = a ^ { \\nu } f ( g ) . \\end{align*}"}
+{"id": "6623.png", "formula": "\\begin{align*} \\norm { D ^ m \\phi } ^ 2 \\leq \\norm { \\phi } \\cdot \\norm { H ^ m \\phi } + c _ { V , m } ' \\sum _ { p = 0 } ^ { m - 1 } \\sum _ { r = 0 } ^ p \\sum _ { q = 0 } ^ p \\sum _ { s = 0 } ^ q \\norm { H ^ r \\phi } \\cdot \\norm { H ^ { m - 1 - p + s } \\phi } . \\end{align*}"}
+{"id": "3097.png", "formula": "\\begin{align*} c _ j ^ { 1 1 } ( A ) = \\bar { a } ( 1 + \\bar { b } ) \\int _ Y r _ B B e _ j \\cdot \\nabla w _ A , c _ j ^ { 2 2 } ( A ) = \\bar { a } ( 1 - \\bar { b } ) \\int _ Y r _ B B e _ j \\cdot \\nabla w _ A \\end{align*}"}
+{"id": "5458.png", "formula": "\\begin{align*} A ^ n ( s , x , \\mu , \\alpha ) & : = \\sigma ^ n ( s , x , \\mu , \\alpha ) \\sigma ^ n ( s , x , \\mu , \\alpha ) ^ \\top , \\\\ A ( s , x , \\mu , \\alpha ) & : = \\sigma ( s , x , \\mu , \\alpha ) \\sigma ( s , x , \\mu , \\alpha ) ^ \\top . \\end{align*}"}
+{"id": "273.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\tau } - \\beta \\rho \\tau \\leqslant - \\frac { \\beta ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\beta } { 2 } , \\end{align*}"}
+{"id": "5557.png", "formula": "\\begin{align*} | S _ 3 ( x ^ 2 ) | \\ll _ { \\epsilon } \\ell ^ { \\frac { 1 } { 2 } - k + \\epsilon } \\sum _ { n = \\ell } ^ { \\infty } \\frac { 2 x ^ 2 } { y _ n ^ 3 } e ^ { - \\frac { x ^ 2 } { y _ n ^ 2 } } . \\end{align*}"}
+{"id": "7550.png", "formula": "\\begin{align*} & Z _ f ( s , \\chi , A _ 1 ) \\\\ = & q ^ { - ( \\omega + 1 ) } \\\\ \\times & \\int _ { B _ 1 } \\chi ( a c ( \\pi ^ { k + r + l } x ^ p + \\pi ^ l y ^ r z ^ l \\mathbb { H } _ r ^ k ( y , t \\pi z - y ) | \\pi ^ { k + r + l } x ^ p + \\pi ^ l y ^ r z ^ l \\mathbb { H } _ r ^ k ( y , t \\pi z - y ) | ^ s | d x d y d z | \\\\ = & q ^ { - ( \\omega + 1 ) - l s } \\\\ \\times & \\int _ { B _ 1 } \\chi ( a c ( \\pi ^ { k + r } x ^ p + y ^ r z ^ l \\mathbb { H } _ r ^ k ( y , t \\pi z - y ) | \\pi ^ { k + r } x ^ p + y ^ r z ^ l \\mathbb { H } _ r ^ k ( y , t \\pi z - y ) | ^ s | d x d y d z | , \\end{align*}"}
+{"id": "2582.png", "formula": "\\begin{align*} & [ u , v ] = [ u , v ] + \\omega _ \\lambda ( u , v ) c , \\\\ & [ d , u ] = u ' , \\\\ & [ c , x ] = 0 , \\end{align*}"}
+{"id": "880.png", "formula": "\\begin{align*} i ^ { t } = \\mathop { \\arg \\max } _ { i = 1 , \\cdots , q } f _ { i } ( \\mathcal { X } ^ { t } ) = \\mathop { \\arg \\max } _ { i = 1 , \\cdots , q } \\| \\mathcal { A } * \\mathcal { X } ^ { t } - \\mathcal { B } \\| _ { F ( \\mathcal { G } _ { i } ) } ^ { 2 } , \\end{align*}"}
+{"id": "7042.png", "formula": "\\begin{align*} \\Omega _ { C _ 2 } = M U _ { C _ 2 } ^ { \\mathbf { C } ^ { \\infty } } \\subset M U _ { C _ 2 } ^ { \\mathbf { C } ^ { \\infty + \\sigma } } \\subset M U _ { C _ 2 } ^ { \\mathbf { C } ^ { \\infty + 2 \\sigma } } \\subset \\cdots \\subset M U _ { C _ 2 } . ^ { \\mathbf { C } ^ { \\infty + \\infty \\sigma } } = M U _ { C _ 2 } \\end{align*}"}
+{"id": "5482.png", "formula": "\\begin{align*} W _ d ( \\mu _ t , \\nu _ t ) & \\leq E d ( ( X _ t , \\alpha _ t ) , ( Y _ t , \\tilde \\alpha _ t ) ) = E \\sqrt { \\hat V ( X _ t - Y _ t ) } \\\\ & \\leq \\sqrt { E \\hat V ( X _ t - Y _ t ) } \\leq \\sqrt { E \\hat V ( X _ 0 - Y _ 0 ) e ^ { - \\theta t } } \\\\ & \\le e ^ { - \\frac { \\theta t } { 2 } } \\sqrt { K E \\hat V ( X _ 0 ) + K E \\hat V ( Y _ 0 ) } . \\end{align*}"}
+{"id": "3322.png", "formula": "\\begin{gather*} { \\rm F C } \\colon \\ z = \\eta - w \\bar { \\eta } , { \\rm F C } ^ { - 1 } \\colon \\ \\eta = \\frac { z + \\bar { z } w } { P } , \\end{gather*}"}
+{"id": "6681.png", "formula": "\\begin{align*} \\begin{aligned} & \\hat { W } _ k \\left ( { \\bf 1 } \\otimes \\bar { x } ^ k \\right ) = \\left ( \\left ( I + \\gamma ^ k W - \\frac { { \\bf 1 } { \\bf 1 } ^ T } { m } \\right ) { \\bf \\times 1 } \\right ) \\otimes \\left ( I _ d \\times \\bar { x } ^ k \\right ) = 0 \\end{aligned} \\end{align*}"}
+{"id": "1861.png", "formula": "\\begin{align*} \\mu _ { k + 1 } : = \\mu ( \\sigma _ k , { \\alpha } _ k ) = \\frac { 1 } { p } \\left [ a _ u ( \\alpha _ k ) \\sigma _ k + b _ u ( \\alpha _ k ) \\right ] , \\end{align*}"}
+{"id": "797.png", "formula": "\\begin{align*} \\mu _ { ( z , m ) } = \\frac { 1 } { m } \\sum _ { k = 0 } ^ { m - 1 } \\delta _ { \\sigma ^ k z } . \\end{align*}"}
+{"id": "8099.png", "formula": "\\begin{align*} H _ { m , n } ^ { + , 1 } ( x ) = \\frac { 4 M T } { \\pi } \\int _ { u = - \\infty } ^ \\infty \\int _ { \\zeta = - \\infty } ^ \\infty e ^ { - u ^ 2 } V ( m ^ 2 n , M u + T ) \\cos ( x \\cosh \\zeta ) e \\Bigl ( \\frac { u M \\zeta } { \\pi } \\Bigr ) e \\Bigl ( \\frac { T \\zeta } { \\pi } \\Bigr ) \\ , d u \\ , d \\zeta , \\end{align*}"}
+{"id": "7983.png", "formula": "\\begin{align*} \\beta ( n ) = \\beta ( i ( n ) ) = \\alpha ( \\pi ( n ) ) = 0 . \\end{align*}"}
+{"id": "8899.png", "formula": "\\begin{align*} \\sum \\limits _ { p = 0 } ^ { n - 1 } \\gamma _ { p } ^ { ( \\nu , n ) } \\mu ^ { 2 p } = 0 \\end{align*}"}
+{"id": "827.png", "formula": "\\begin{align*} \\mu _ r ( g ) = \\frac { e _ i ^ T A _ \\infty 1 } { e _ \\ast ^ T A ^ r A _ \\infty 1 } = \\lim _ { n \\to \\infty } \\frac { e _ i ^ T A ^ { n p } 1 } { e _ \\ast ^ T A ^ r A ^ { n p } 1 } \\end{align*}"}
+{"id": "3998.png", "formula": "\\begin{align*} D _ i ( | D u | ^ 2 / 2 ) & = u _ { k } u _ { k i } \\\\ D _ { i i } ( | D u | ^ 2 / 2 ) & = u _ { k i } u _ { k i } + u _ k u _ { k i i } . \\end{align*}"}
+{"id": "4377.png", "formula": "\\begin{align*} a _ 3 = G \\left ( \\frac { p } { q } - \\frac { 1 } { a _ 1 } - \\frac { 1 } { a _ 2 } \\right ) = q a _ 1 a _ 2 + 1 . \\end{align*}"}
+{"id": "1302.png", "formula": "\\begin{align*} H ^ i ( G _ n B / B , E ) = 0 i > 0 , \\end{align*}"}
+{"id": "2660.png", "formula": "\\begin{align*} R : = \\min \\left \\{ { \\alpha } , \\frac { { \\alpha } ( 2 p _ 1 ^ { - 1 } - 5 / 2 ) } { { \\alpha } + 2 p _ 1 ^ { - 1 } - 2 p _ 2 ^ { - 1 } } \\right \\} . \\end{align*}"}
+{"id": "1537.png", "formula": "\\begin{align*} j ^ { i j } _ A = p ^ { i j } _ A \\end{align*}"}
+{"id": "1411.png", "formula": "\\begin{align*} k _ { \\rho , z } & \\coloneqq \\# \\{ i \\in \\{ 1 , \\dots , r \\} \\ ; | \\ ; \\} + m _ \\phi ( \\rho \\boxtimes S _ { 2 z + 1 } ) \\end{align*}"}
+{"id": "8697.png", "formula": "\\begin{align*} \\mathcal Q ( u _ 1 , . . , u _ { 2 l - 1 } ) = - \\mathcal Q ( u _ 1 , . . , u _ { 2 l - 1 } , 0 ) , \\end{align*}"}
+{"id": "4144.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\lambda ( h , 0 ) & \\leq \\int _ M \\langle \\frac { 1 } { 2 } \\triangle h + ^ * h , h \\rangle - \\langle \\mathring { R } h , h \\rangle - R _ { i j } h _ { i k } h _ { j k } d V _ g \\\\ & = - \\frac { 1 } { 2 } \\| D h \\| _ { L ^ 2 } ^ 2 + 2 \\| h \\| _ { L ^ 2 } ^ 2 - 2 \\int _ M R _ { i j } h _ { i k } h _ { j k } d V _ g \\leq 0 \\quad \\end{align*}"}
+{"id": "7215.png", "formula": "\\begin{align*} p ( t ) \\ , = \\ , \\rho \\Big ( \\cos \\vartheta \\ , e ^ { i \\phi } \\ , + \\ , \\sin \\vartheta \\ , e ^ { i \\psi } \\ , j \\ , \\Big ) , \\qquad \\mbox { w h e r e } t \\in I = [ a , \\ , b ] \\end{align*}"}
+{"id": "7434.png", "formula": "\\begin{align*} K ( Z _ i , Z _ j ) ( P _ { i j } ) = \\phi _ { i j } ( P _ { i j } ) . \\end{align*}"}
+{"id": "2676.png", "formula": "\\begin{align*} G ( z ) & = z + { Q \\big ( 1 - p _ t + p _ t z \\big ) - ( 1 - p _ t ) - z p _ t \\over p _ t \\big ( 1 - Q ' ( 1 - p _ t ) \\big ) } \\\\ & = z + p _ t ^ { - 1 / q } \\left ( Q \\big ( z + ( 1 - z ) ( 1 - p _ t ) \\big ) - ( 1 - p _ t ) - z p _ t \\right ) \\\\ & = z + p _ t ^ { - 1 / q } q p _ t ^ { 1 / q } ( 1 - z ) ^ { 1 / q } ~ = Q ( z ) . \\end{align*}"}
+{"id": "2588.png", "formula": "\\begin{align*} & \\{ u : \\mathbb { R } \\rightarrow \\mathfrak { g } \\mid u \\in H ^ 0 , \\ u ( t + 2 \\pi ) = \\sigma ( u ( t ) ) , \\ \\rho ( u ( - t ) ) = - u ( t ) \\} \\\\ \\cong \\ & \\{ u : [ 0 , 2 \\pi ] \\rightarrow \\mathfrak { g } \\mid u \\in H ^ 0 , \\ \\rho ( \\sigma ^ { - 1 } u ( 2 \\pi - t ) ) = - u ( t ) \\} \\\\ \\cong \\ & \\{ u : [ 0 , \\pi ] \\rightarrow \\mathfrak { g } \\mid u \\in H ^ 0 \\} . \\end{align*}"}
+{"id": "8700.png", "formula": "\\begin{align*} Q _ \\alpha = \\ , [ q _ { \\alpha _ i , \\alpha _ j } ] _ { i , j = 1 , \\dots , 2 l } , \\end{align*}"}
+{"id": "599.png", "formula": "\\begin{align*} J ^ \\dagger _ { t _ n , t _ { n + 1 } } = \\lim _ { \\epsilon \\to 0 } J ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } - \\frac { \\Delta t \\ , \\gamma } { 2 } A ^ { \\rm T } : M \\ , . \\end{align*}"}
+{"id": "9104.png", "formula": "\\begin{align*} R _ { k , 0 } : = \\mathsf { C } ( \\mathsf { P } ) - \\sqrt { \\frac { \\mathsf { V } _ { \\mathrm { G } } ( \\mathsf { P } ) } { n } } , k = 1 , 2 \\end{align*}"}
+{"id": "6352.png", "formula": "\\begin{align*} F ' ( x ) = ( a ^ 2 - f ( x ) ^ 2 ) \\left ( f '' ( x ) / f ( x ) \\right ) ' , \\end{align*}"}
+{"id": "8916.png", "formula": "\\begin{align*} \\Omega _ { p } ^ { \\left ( \\nu \\right ) } \\left ( k \\right ) : = \\frac { ( - 1 ) ^ { k + 1 } B _ { 2 ( k + p + 1 ) } \\left ( \\nu + \\frac { 1 } { 2 } \\right ) } { ( k + p + 1 ) k ! } , k = 0 , 1 , 2 , . . . , \\end{align*}"}
+{"id": "8707.png", "formula": "\\begin{align*} \\tilde F _ \\lambda ( x _ 1 , \\dots , x _ n ; t ) = \\frac { ( 1 - t ) ^ n } { \\prod _ { i = 1 } ^ { n - l } ( 1 - t ^ i ) } \\sum _ { \\sigma \\in S _ n } \\sigma \\left ( ( 1 - x _ { 1 } ^ { \\lambda _ 1 } ) \\dots ( 1 - x _ { l } ^ { \\lambda _ n } ) \\prod _ { i = 1 } ^ { n } \\frac { x _ i } { 1 - x _ i } \\prod _ { i < j } \\frac { x _ { i } - t x _ { j } } { x _ { i } - x _ { j } } \\right ) , \\end{align*}"}
+{"id": "6915.png", "formula": "\\begin{align*} \\lambda ( t ) = \\frac { \\lambda _ 1 } { 1 + \\exp ( - \\lambda _ 2 ( t - \\lambda _ 3 ) ) } , \\end{align*}"}
+{"id": "5759.png", "formula": "\\begin{align*} T ^ * V = T ^ * U \\times T ^ * \\C ^ r = T ^ * U \\times \\C ^ r _ { \\underline { t } } \\times \\C ^ r _ { \\underline { s } } . \\end{align*}"}
+{"id": "6879.png", "formula": "\\begin{align*} | u | _ { { X { _ { p _ i } } } } : = [ u ] _ { s _ i ( \\cdot ) , p _ i ( \\cdot ) , \\R ^ { 2 N } \\setminus ( \\mathcal { C } \\Omega ) ^ 2 } + \\| u \\| _ { L ^ { \\overline { p } _ i ( \\cdot ) } ( \\Omega ) } + \\left \\| \\beta ^ { \\frac { 1 } { \\overline { p } _ i ( \\cdot ) } } u \\right \\| _ { L ^ { \\overline { p } _ i ( \\cdot ) } ( \\mathcal { C } \\Omega ) } , \\end{align*}"}
+{"id": "5918.png", "formula": "\\begin{align*} u = ( u _ 1 , u _ 2 ) ^ T . \\end{align*}"}
+{"id": "6951.png", "formula": "\\begin{align*} x y _ n ' ( x ) + \\alpha y _ n ( x ) = { } & L _ { n - m } ^ { ( \\alpha - 1 ) } ( x ) \\big ( x L _ { m - 1 } ^ { ( \\alpha + 1 ) } ( - x ) + \\alpha L _ m ^ { ( \\alpha ) } ( - x ) \\big ) \\\\ & + L _ m ^ { ( \\alpha - 1 ) } ( - x ) \\big ( \\alpha L _ { n - m - 1 } ^ { ( \\alpha ) } ( x ) - x L _ { n - m - 1 } ^ { ( \\alpha + 1 ) } ( x ) \\big ) . \\end{align*}"}
+{"id": "3777.png", "formula": "\\begin{align*} ( \\Phi _ \\# \\phi ) ( v ) = x _ e ^ { - 1 } \\phi ( v ) x _ e = \\phi ( x _ e ^ { - 1 } v x _ e ) = ( \\phi \\Phi _ \\# ) ( v ) , \\end{align*}"}
+{"id": "4302.png", "formula": "\\begin{align*} \\| \\nabla u ( t , \\cdot ) \\| ^ 2 _ { L ^ 2 } \\leq \\nu ^ { - 1 } _ 0 \\mathcal { H } ( u ; t ) = \\nu ^ { - 1 } _ 0 \\mathcal { H } ( u ; 0 ) = \\Lambda \\end{align*}"}
+{"id": "85.png", "formula": "\\begin{align*} \\xi = \\sum _ { i = 1 } ^ { \\infty } \\xi ( i ) p ^ { - i } , \\ , \\ \\xi ( i ) \\in \\{ 0 , 1 , \\cdots , p - 1 \\} \\end{align*}"}
+{"id": "8804.png", "formula": "\\begin{align*} \\min _ { r \\in [ 0 , t ] } f ^ n ( s - r ) = f ^ n ( s ) - \\max _ { r \\in [ 0 , t ] } ( M ^ n _ { 1 - s + r } - M ^ n _ { 1 - s } ) \\ge f ( s ) - Y _ { [ - n , n ] , 1 - s , t } . \\end{align*}"}
+{"id": "7049.png", "formula": "\\begin{align*} \\Omega _ { C _ 2 } = M U _ { C _ 2 } ^ { \\mathbf { C } ^ \\infty } \\to M U _ { C _ 2 } ^ { \\mathbf { C } ^ { \\infty , \\infty } } = M U _ { C _ 2 } . \\end{align*}"}
+{"id": "6024.png", "formula": "\\begin{align*} P _ { l _ 1 } ( h _ 2 , w _ { k , p } ) = 0 , \\end{align*}"}
+{"id": "5577.png", "formula": "\\begin{align*} ( r , \\mathcal { O } _ { ( 0 , 0 ) , \\rho } , R ) : = & \\sup \\{ | y | : ( x , y ) \\in \\left ( \\partial B ^ k _ { \\rho } ( 0 ) \\times B ^ { n - k } _ { R } ( 0 ) \\right ) \\cap \\mathcal { O } _ { ( 0 , 0 ) , \\rho } \\} \\\\ M ( r , \\mathcal { O } _ { ( 0 , 0 ) , \\rho } , R ) : = & \\sup \\{ u _ { ( 0 , 0 ) , \\rho } ( x , y ) : ( x , y ) \\in \\left ( \\partial B ^ k _ { \\rho } ( 0 ) \\times B ^ { n - k } _ { R } ( 0 ) \\right ) \\cap \\mathcal { O } _ { ( 0 , 0 ) , \\rho } \\} . \\end{align*}"}
+{"id": "5981.png", "formula": "\\begin{align*} ( \\Pi \\times e v ) _ { | \\mathcal { W } _ g } ^ { - 1 } ( m , x ) & = \\left ( M _ { m } \\times _ { X ^ r } \\{ x \\} \\right ) \\times _ { X ^ r } ( g \\prod _ i X ( u _ i ) ) \\\\ & = M _ m \\times _ { X ^ r } \\left ( \\{ x \\} \\times _ { X ^ r } ( g \\prod _ i X ( u _ i ) ) \\right ) \\\\ & = M _ m \\times _ { X ^ r } \\{ x \\} \\\\ & = ( \\Pi \\times e v ) ^ { - 1 } ( m , x ) , \\end{align*}"}
+{"id": "3160.png", "formula": "\\begin{align*} A ( y _ 1 , y _ 2 ) : = \\begin{pmatrix} 5 + \\sin ( 2 \\pi y _ 1 ) & 1 + \\cos ( 2 \\pi y _ 1 ) \\\\ 1 + \\cos ( 2 \\pi y _ 1 ) & 5 - \\sin ( 2 \\pi y _ 1 ) \\end{pmatrix} \\quad ( y _ 1 , y _ 2 ) \\in \\R ^ 2 \\end{align*}"}
+{"id": "8724.png", "formula": "\\begin{align*} f _ k ( x ) = x ( x - 1 ) ^ { k - 1 } = \\sum _ { j > 0 } ( - 1 ) ^ { k - j } { k - 1 \\choose j - 1 } x ^ j , f _ { - k } = \\frac { 1 } { ( x - 1 ) ^ { k } } = \\sum _ { j = 1 } ^ { \\infty } { { j - 1 } \\choose { k - 1 } } \\frac { 1 } { x ^ j } , f _ { 0 } = 0 . \\end{align*}"}
+{"id": "6256.png", "formula": "\\begin{align*} h _ { 0 } = j _ { \\mathcal { W } _ { 0 } } , h _ { 1 } ( \\mathcal { W } _ { 0 } ) = \\mathcal { W } _ { 1 } , h _ { t } \\big | _ { \\mathcal { S } } = j _ { \\mathcal { S } } \\end{align*}"}
+{"id": "1539.png", "formula": "\\begin{align*} \\Phi ^ { \\alpha j } = - \\frac i 2 \\kappa ^ { \\alpha j } \\end{align*}"}
+{"id": "1853.png", "formula": "\\begin{align*} B ^ 0 \\partial _ { \\tau } \\mathbf { U } + \\tau ^ { \\gamma - \\frac { 7 } { 3 } } B ^ i \\partial _ { i } \\mathbf { U } = \\frac { 1 } { \\tau } \\mathcal { B } \\mathbb { P } \\mathbf { U } + \\frac { 1 } { \\tau } H , \\end{align*}"}
+{"id": "6619.png", "formula": "\\begin{align*} \\norm { x _ k ^ j ( t ) \\psi } & \\leq \\norm { x _ k ^ m ( t ) \\psi } ^ { j / m } \\norm { \\psi } ^ { ( m - j ) / m } , \\\\ \\norm { D _ k ^ j ( t ) \\psi } & \\leq \\norm { D _ k ^ m ( t ) \\psi } ^ { j / m } \\norm { \\psi } ^ { ( m - j ) / m } . \\end{align*}"}
+{"id": "2981.png", "formula": "\\begin{align*} F ^ \\ell _ { i , j } = ( \\overrightarrow { W } _ { j + 1 } ^ \\ell ) ^ 2 ( W _ { j + i - 1 } - W _ { j + i } ) + ( \\nabla _ { j + i - 1 , j + i } ( \\overrightarrow { W } _ { j + 1 } ^ \\ell ) ^ 2 ) W _ { j + i - 1 } . \\end{align*}"}
+{"id": "302.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { H } / \\mathcal { Z } \\times \\mathcal { H } / \\mathcal { Z } & \\rightarrow \\mathcal { Z } \\\\ ( h _ 1 \\mathcal { Z } , h _ 2 \\mathcal { Z } ) & \\mapsto \\langle h _ 1 \\mathcal { Z } , h _ 2 \\mathcal { Z } \\rangle : = h _ { 1 } h _ { 2 } h _ { 1 } ^ { - 1 } h _ { 2 } ^ { - 1 } . \\end{aligned} \\end{align*}"}
+{"id": "5943.png", "formula": "\\begin{align*} \\widehat { U } = F ' ( 0 ) \\widehat H . \\end{align*}"}
+{"id": "1826.png", "formula": "\\begin{align*} K _ X = f ^ * K _ Y + R , \\end{align*}"}
+{"id": "6469.png", "formula": "\\begin{align*} \\delta E ( \\eta ) = - \\partial _ { x x } \\eta - 2 \\gamma e _ 1 \\wedge \\partial _ x \\eta + \\eta - \\eta _ 1 e _ 1 = O _ 2 ^ 1 ( \\eta ) . \\end{align*}"}
+{"id": "5104.png", "formula": "\\begin{align*} G = p _ 2 G ^ 2 + p _ 3 q G ^ 3 + \\cdots + p _ m q ^ { m - 2 } G ^ m . \\end{align*}"}
+{"id": "5836.png", "formula": "\\begin{align*} { \\mathcal H } _ 0 = \\Big \\{ u : ~ u \\in L ^ 2 ( \\Omega ) , \\int _ { \\Omega } u d x = 0 \\Big \\} , { \\mathcal H } _ 1 = H ^ 1 ( \\Omega ) \\cap { \\mathcal H } _ 0 . \\end{align*}"}
+{"id": "3709.png", "formula": "\\begin{align*} I ( \\mathcal { F } _ { X _ i } ( \\sigma _ i ( h ) f ) ) ( \\xi ) = | \\lambda ( h ) | ^ { - 1 } I ( \\sigma _ i ( \\lambda ( h ) I _ { V _ { i + 1 } } ) \\mathcal { F } _ { X _ i } ( f ) ) ( \\lambda ( h ) h ^ { - 1 } \\xi ) . \\end{align*}"}
+{"id": "6365.png", "formula": "\\begin{align*} x _ \\lambda ( u ) : = \\cos u + \\lambda ( \\cos u - ( \\cos ^ 4 u ) / 1 2 ) \\quad \\mbox { a n d } y ( u ) : = \\alpha \\sin u , \\end{align*}"}
+{"id": "9029.png", "formula": "\\begin{align*} X _ t = \\xi + \\int _ t ^ T f ( s , X _ s , \\pi _ s , \\psi _ s ) d s - \\int _ t ^ T \\pi _ s d W _ s - \\int _ t ^ T \\int _ { \\R ^ \\star } \\pi _ s ( d e ) d \\tilde { N } ( d s , d e ) t \\in [ 0 , T ] { \\rm \\ , \\ , a . s . } \\end{align*}"}
+{"id": "5172.png", "formula": "\\begin{align*} \\begin{aligned} & z _ j z _ i = q z _ i z _ j , \\ ; \\ ; \\ ; \\ ; i < j , \\ \\ z _ i z _ j ^ * = q z _ j ^ * z _ i , \\ ; \\ ; \\ ; \\ ; i \\neq j , \\\\ & z _ i ^ * z _ i = z _ i z _ i ^ * + ( 1 - q ^ 2 ) \\sum _ { j = i + 1 } ^ n z _ j z _ j ^ * , \\ ; \\ i = 0 , . . . , n , \\\\ & \\sum _ { j = 0 } ^ n z _ j z _ j ^ * = 1 , \\end{aligned} \\end{align*}"}
+{"id": "5119.png", "formula": "\\begin{align*} D G ( \\lambda , b , \\Omega , 0 , 0 ) ( h _ { 1 } , h _ { 2 } ) = \\left ( \\begin{array} { c } D _ { f _ { 1 } } G _ { 1 } ( \\lambda , b , \\Omega , 0 , 0 ) h _ { 1 } + D _ { f _ { 2 } } G _ { 1 } ( \\lambda , b , \\Omega , 0 , 0 ) h _ { 2 } \\\\ D _ { f _ { 1 } } G _ { 2 } ( \\lambda , b , \\Omega , 0 , 0 ) h _ { 1 } + D _ { f _ { 2 } } G _ { 2 } ( \\lambda , b , \\Omega , 0 , 0 ) h _ { 2 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "1106.png", "formula": "\\begin{align*} \\mathbb { P } \\{ W _ 1 = 1 \\} = 1 - \\mathbb { P } \\{ W _ 1 = - 1 \\} = \\frac { e ^ { \\alpha } } { e ^ { \\alpha } + 1 } . \\end{align*}"}
+{"id": "5810.png", "formula": "\\begin{align*} D e _ s = 0 , s = 1 , \\cdots , p . \\end{align*}"}
+{"id": "821.png", "formula": "\\begin{align*} \\mathcal { A } _ n ( x ) : = \\left \\{ \\xi \\in \\partial \\Gamma : \\frac { \\log \\| \\rho ( \\xi _ n ) \\| - \\Lambda n } { \\sqrt { n } } \\le x \\right \\} \\end{align*}"}
+{"id": "257.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = \\alpha ( u / v ) + \\beta ( w / v ) \\psi _ 2 ( u , v , w ) = \\alpha ( u / v ) + \\beta ( w / u ) . \\end{align*}"}
+{"id": "8808.png", "formula": "\\begin{align*} \\| T ^ n h \\| = \\| h \\| = \\| T ^ { * n } h \\| ( n = 1 , 2 , \\dots ) . \\end{align*}"}
+{"id": "2961.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\ell - 1 } ( W ^ 2 _ { j + i } - W ^ 2 _ { j + i - 1 } ) \\psi _ i = ( \\overrightarrow { W ^ 2 } ) ^ \\ell _ j - ( W ^ 2 _ j - E _ { \\nu _ \\rho } [ W ^ 2 _ j ] ) \\end{align*}"}
+{"id": "6747.png", "formula": "\\begin{align*} \\alpha _ n ( s , \\rho ) = n \\sqrt { 2 \\pi } \\left ( 1 - \\rho \\right ) ^ { \\frac { 1 } { 2 } } \\left ( 1 - \\rho \\right ) ^ { \\frac { 1 } { 2 s } } \\end{align*}"}
+{"id": "5837.png", "formula": "\\begin{align*} \\begin{cases} U '' - { \\Delta } U + A U = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\partial _ \\nu U = D H & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma , \\end{cases} \\end{align*}"}
+{"id": "8052.png", "formula": "\\begin{align*} \\omega ( t ) = \\frac { 4 \\pi \\abs [ \\Big ] { \\phi \\Bigl ( 1 , \\dfrac { 1 } { 2 } + i t \\Bigr ) } ^ 2 } { \\cosh ( \\pi t ) } . \\end{align*}"}
+{"id": "3856.png", "formula": "\\begin{align*} A _ { 2 n + j } - \\delta _ { j , 0 } = \\sum _ { k = \\delta _ { j , 0 } } ^ n A _ { 2 k + j - 1 } + \\sum _ { k = 1 } ^ n \\sum _ { i = 0 } ^ { 2 k + j - 2 } ( \\mu _ { 2 k + j - i } + \\delta _ { 3 , 2 k + j - i } ) A _ i . \\end{align*}"}
+{"id": "4402.png", "formula": "\\begin{align*} u _ n ( \\theta ) = \\sum _ { i = 1 } ^ n \\frac { 1 } { b _ i } < \\theta . \\end{align*}"}
+{"id": "6022.png", "formula": "\\begin{align*} \\O _ { h _ i } \\star \\O _ { i _ 2 , j _ 2 } = \\O _ { h _ i } \\cdot \\O _ v + Q _ 1 P _ { l _ 1 } ( i , v ) + Q _ 2 P _ { l _ 2 } ( i , v ) + Q _ 1 Q _ 2 P _ { l _ 1 + l _ 2 } ( i , v ) . \\end{align*}"}
+{"id": "8811.png", "formula": "\\begin{align*} \\sigma ( A B ) = \\{ \\lambda _ 1 \\lambda _ 2 \\ , : \\ , ( \\lambda _ 1 , \\lambda _ 2 ) \\in \\sigma _ T ( A , B ) \\} . \\end{align*}"}
+{"id": "1397.png", "formula": "\\begin{align*} \\pi _ 1 ^ { ( + , + , + ) } = L ( \\Delta [ 0 , - 2 ] , \\Delta [ 1 , - 2 ] ; \\pi ( 0 ^ + , 1 ^ + , 1 ^ + ) ) . \\end{align*}"}
+{"id": "3438.png", "formula": "\\begin{align*} p _ { a a } ^ { t } = \\sum _ { n = 0 } ^ { \\infty } p _ { a a } ^ { t } ( n ) , \\end{align*}"}
+{"id": "1467.png", "formula": "\\begin{align*} | \\mathcal { T } | - N = ( k - 2 ) \\left ( Y - \\frac { k ( k - 1 ) ( m - 2 ) } { ( m + 1 ) ( m ^ 2 - 2 ) } - \\frac { 2 k ( k - 1 ) ( m - 1 ) } { ( m + 1 ) ( m ^ 2 - 2 ) } \\right ) , \\end{align*}"}
+{"id": "710.png", "formula": "\\begin{align*} 2 \\mathcal { H } ( w _ 1 , \\dots , w _ n ; a _ 1 , \\dots , a _ { \\texttt { g } } , b _ 1 , \\dots , b _ { \\texttt { g } } ) = 2 ( \\nu , \\nu ) _ { 1 , { \\rm r e n o r m } } = \\end{align*}"}
+{"id": "270.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\rho } - \\beta \\rho \\tau \\leqslant - \\frac { \\beta ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\beta } { 2 } , \\end{align*}"}
+{"id": "2905.png", "formula": "\\begin{align*} \\| f \\| _ 0 = \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | \\mathcal { B } f | ) d m _ \\infty = \\| \\mathcal { B } f \\| _ 0 . \\end{align*}"}
+{"id": "6956.png", "formula": "\\begin{align*} { \\rm e } ^ { - x } L _ { m , n } ^ { I I , ( \\alpha ) } ( x ) = ( \\alpha + n + 1 - 2 m ) \\int _ { c } ^ { x } { \\rm e } ^ { - t } L _ m ^ { ( - \\alpha - 1 ) } ( t ) L _ { n - m } ^ { ( \\alpha + 1 ) } ( t ) \\ , { \\rm d } t \\end{align*}"}
+{"id": "857.png", "formula": "\\begin{align*} f ( w ) = [ e ^ { \\widehat { w } } J _ d - J _ d ( e ^ { \\widehat { w } } ) ^ T - \\widehat { h \\Pi _ k } ] ^ \\vee \\end{align*}"}
+{"id": "4505.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n v _ i \\leq \\sum _ { i = 1 } ^ n u _ i . \\end{align*}"}
+{"id": "8483.png", "formula": "\\begin{align*} \\lambda = \\omega \\frac { ( \\alpha - \\beta ) \\varphi _ 1 ^ 2 + \\alpha \\frac { d \\varphi _ 1 } { d \\xi } - \\epsilon } { 3 n ( n + 1 ) } . \\end{align*}"}
+{"id": "2848.png", "formula": "\\begin{align*} t _ { k , k } = t _ { \\ell , \\ell } = \\frac { \\tau + \\sigma } { 2 \\tau } \\end{align*}"}
+{"id": "1681.png", "formula": "\\begin{align*} \\mu ( P ) = \\rho ( s ( \\mu ) ) ( P ) = \\int _ { S ^ 1 } s ( \\mu ) ( \\kappa ( \\theta ) ) P ( - \\sin \\theta , \\cos \\theta ) d \\theta = \\sum _ { \\frac { 2 - k } { 2 } \\leq n \\leq \\frac { k - 2 } { 2 } } a _ n ( \\mu ) \\int _ { S ^ 1 } e ^ { 2 n i \\theta } P ( - \\sin \\theta , \\cos \\theta ) d \\theta . \\end{align*}"}
+{"id": "5000.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u - \\chi \\nabla \\cdot ( \\frac { u } { v } \\nabla v ) + g ( u ) , \\\\ v _ t = \\Delta v - u v , \\end{cases} \\end{align*}"}
+{"id": "2690.png", "formula": "\\begin{align*} \\mu _ 0 ( Q ( b _ i , 1 / ( 4 n ) ) ) = r ^ s = \\ell ^ { - 1 } , \\ \\ \\ i = 1 , \\ldots , \\ell . \\end{align*}"}
+{"id": "883.png", "formula": "\\begin{align*} p _ { i } ^ { t } = \\left \\{ \\begin{array} { l c l } \\frac { f _ { i } ( \\mathcal { X } ^ { t } ) } { \\sum _ { i \\in \\mathfrak { W } _ { t } } f _ { i } ( \\mathcal { X } ^ { t } ) } & & i \\in \\mathfrak { W } _ { t } \\\\ 0 & & i \\notin \\mathfrak { W } _ { t } . \\end{array} \\right . \\end{align*}"}
+{"id": "6998.png", "formula": "\\begin{align*} R _ c ( R _ p ) = \\inf _ { W : R _ { X | W } ( \\Delta ) + R _ { Y | W } ( \\Delta ) \\leq R _ p } I ( X , Y ; W ) . \\end{align*}"}
+{"id": "8414.png", "formula": "\\begin{align*} \\int \\partial _ { \\alpha } [ P _ { \\alpha } ^ k ] ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\hat \\Psi _ { \\alpha } \\ : d m = \\int \\sum _ { i = 1 } ^ k P _ { \\alpha } ^ { k - i } Q _ { \\alpha } P _ { \\alpha } ^ { i - 1 } ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\hat \\Psi _ { \\alpha } \\ : d m \\end{align*}"}
+{"id": "435.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\right \\| \\leq b \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "5425.png", "formula": "\\begin{align*} f ( \\bar { \\alpha } \\alpha ) = \\sum _ { a = 1 } ^ n 1 - \\sqrt { 1 - \\lambda _ a } + \\log \\frac { 1 + \\sqrt { 1 - \\lambda _ a } } { 2 } . \\end{align*}"}
+{"id": "3569.png", "formula": "\\begin{align*} \\prod _ { k = 2 , \\dots , n } ^ \\rightarrow \\left ( y _ { k 1 } ^ { ( \\gamma _ A ) _ { k 1 } } \\cdots y _ { k , k - 1 } ^ { ( \\gamma _ A ) _ { k , k - 1 } } \\right ) \\xi _ \\lambda , \\end{align*}"}
+{"id": "5320.png", "formula": "\\begin{align*} \\Phi _ 1 : = \\lambda \\mathrm { A d } _ { T _ 1 ^ { - 1 } } \\circ \\Psi \\qquad \\mbox { a n d } \\Phi _ 2 : = \\mathrm { A d } _ { T _ 2 ^ { - 1 } } \\circ ( \\Phi - \\lambda \\Psi ) \\end{align*}"}
+{"id": "2254.png", "formula": "\\begin{align*} g _ { t } : = 1 + p [ t ] \\in U _ D ^ 1 / Z _ D ^ 1 . \\end{align*}"}
+{"id": "2242.png", "formula": "\\begin{align*} P ( t ) = - Q ( - t ) . \\end{align*}"}
+{"id": "8994.png", "formula": "\\begin{align*} \\int _ { t _ 0 } ^ 0 & \\int _ { \\partial \\Omega _ k } | \\partial _ t u _ k | ^ 2 d s \\ ; d t = \\int _ { t _ 0 } ^ 0 \\int _ { \\partial \\Omega _ k } | d \\pi _ N ( u _ k ) \\partial _ { \\nu _ k } u _ k | ^ 2 d s \\ ; d t \\\\ & = \\int _ { t _ k + r _ k t _ 0 } ^ { t _ k } \\int _ { \\partial B } | u _ t | ^ 2 d \\phi \\ ; d t \\le \\int _ { t _ k + r _ k t _ 0 } ^ { T _ 0 } \\int _ { \\partial B } | u _ t | ^ 2 d \\phi \\ ; d t \\to 0 \\end{align*}"}
+{"id": "6866.png", "formula": "\\begin{gather*} J ^ h ( u ^ h , F ) = \\frac { 1 } { 2 } \\Bigg ( ( u _ h , u _ h ) _ h + ( u , u ) _ h + \\epsilon ( u , u ) \\Bigg ) - ( u , u _ h ) _ h \\\\ - e ( u , u _ h ) \\forall u \\in X , \\ ; \\forall u _ h \\in X _ h , \\end{gather*}"}
+{"id": "5941.png", "formula": "\\begin{align*} t \\geq T : \\phi _ r = ( E _ r , U ) = \\sum _ { s = 1 } ^ p ( E _ r , e _ s ) u _ s = \\sum _ { s = 1 } ^ p \\delta _ { r s } u _ s = u _ r . \\end{align*}"}
+{"id": "8175.png", "formula": "\\begin{align*} d s ^ 2 _ X ( V , V ) & = \\frac { \\left ( | V s _ 0 ( x ) | ^ 2 + | V s _ 1 ( x ) | ^ 2 \\right ) | s _ 0 ( x ) | ^ 2 - \\left ( s _ 0 ( x ) \\overline { V s _ 0 ( x ) } \\right ) \\left ( V s _ 0 ( x ) \\overline { s _ 0 ( x ) } \\right ) } { | s _ 0 ( x ) | ^ 4 } \\\\ & = \\frac { | V s _ 1 ( x ) | ^ 2 } { | s _ 0 ( x ) | ^ 2 } . \\end{align*}"}
+{"id": "3451.png", "formula": "\\begin{align*} x _ 0 f _ 4 = x _ 1 f _ 4 = x _ 2 f _ 4 = x _ 3 f _ 4 = 0 . \\end{align*}"}
+{"id": "3696.png", "formula": "\\begin{align*} J _ i = \\begin{pmatrix} J _ 0 & & & \\\\ & J & & \\\\ & & \\ddots & \\\\ & & & J \\end{pmatrix} \\end{align*}"}
+{"id": "976.png", "formula": "\\begin{align*} \\sigma _ 3 ( s ) = & \\bigg ( \\sum _ { j = 1 } ^ { 2 } R _ j + ( 1 - e ^ { - i s } ) R _ 3 , 2 \\big ( \\frac { \\pi } { 2 } - 1 \\big ) \\big ( \\sum _ { j = 1 } ^ { 2 } \\| R _ j \\| ^ 2 \\big ) + 2 ( s - \\sin s ) \\| R _ 3 \\| ^ 2 \\\\ & - 2 \\langle R _ 1 , R _ 2 \\rangle - 2 \\sin s \\langle R _ 1 + R _ { 2 } , R _ 3 \\rangle \\bigg ) \\end{align*}"}
+{"id": "3394.png", "formula": "\\begin{align*} T ( f ) = ( \\int _ 0 ^ 1 f ( x ) s i n x d x , \\int _ 0 ^ 1 f ( x ) s i n 2 x d x , \\ldots ) \\end{align*}"}
+{"id": "2895.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\mathbb { T } ^ \\infty } \\varphi ( \\log ^ { + } | F _ { [ r ] } | ) d m _ \\infty & = \\int _ { E _ r } \\varphi ( 0 ) d m _ \\infty + \\int _ { E _ r ^ c } \\varphi ( \\log | F _ { [ r ] } | ) d m _ \\infty \\\\ & \\leq \\varphi ( 0 ) + \\int _ { \\mathbb { T } ^ \\infty } \\varphi ( \\log | F _ { [ r ] } | ) d m _ \\infty \\\\ & \\leq \\varphi ( 0 ) + \\int _ { \\mathbb { T } ^ \\infty } \\varphi ( \\log | F ^ { * } | ) d m _ \\infty . \\end{aligned} \\end{align*}"}
+{"id": "9005.png", "formula": "\\begin{align*} \\theta \\ ! \\left ( \\partial _ K \\ ! \\left ( \\bigcup \\nolimits _ { i \\in F } A _ i \\right ) \\right ) \\ ! \\ , & \\leq \\ , \\theta \\ ! \\left ( \\bigcup \\nolimits _ { i \\in F } \\partial _ K A _ i \\right ) \\\\ & \\leq \\ , \\sum \\nolimits _ { i \\in F } \\theta ( \\partial _ K A _ i ) \\ , = \\ , \\sum \\nolimits _ { i \\in F } \\theta ( A _ i ) \\frac { \\theta ( \\partial _ K A _ i ) } { \\theta ( A _ i ) } \\ , \\leq \\ , M \\sum \\nolimits _ { i \\in F } \\theta ( A _ i ) . \\end{align*}"}
+{"id": "8000.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty t ^ k \\left ( \\sum _ { m + n = k } \\dfrac { ( - 1 ) ^ m } { m ! n ! } \\phi ( z a ^ n , a ^ m ) \\right ) = 0 \\end{align*}"}
+{"id": "7477.png", "formula": "\\begin{align*} \\mu _ A : = \\sum _ { ( u , v ) \\in E _ A } \\mu _ { u v } \\ , . \\end{align*}"}
+{"id": "7386.png", "formula": "\\begin{align*} \\lambda ( \\alpha ) : = \\log ( q _ \\alpha q _ { \\alpha ^ * } ) / \\log ( q _ F ) \\quad \\lambda ^ * ( \\alpha ) : = \\log ( q _ \\alpha q _ { \\alpha ^ * } ^ { - 1 } ) / \\log ( q _ F ) . \\end{align*}"}
+{"id": "5649.png", "formula": "\\begin{align*} a _ { i j } = \\begin{cases} & 1 \\quad i \\preceq j \\\\ & 0 \\quad . \\end{cases} \\end{align*}"}
+{"id": "2026.png", "formula": "\\begin{align*} C _ { k + 1 } : = \\inf \\{ n > C _ k : X _ { C _ k } ^ { ( 0 ) } + ( \\xi _ { C _ k + 1 } + \\xi _ { C _ k + 2 } + \\dotsi + \\xi _ { n } ) \\geq 0 X _ { C _ k } ^ { ( 0 ) } \\leq - 1 \\\\ X _ { C _ k } ^ { ( 0 ) } + ( \\xi ' _ { C _ k + 1 } + \\xi ' _ { C _ k + 2 } + \\dotsi + \\xi ' _ { n } ) < 0 X _ { C _ k } ^ { ( 0 ) } \\geq 0 \\} \\end{align*}"}
+{"id": "1249.png", "formula": "\\begin{align*} k _ { \\beta } ( z ) = \\dfrac { z } { ( 1 - z ) ^ { 2 - 2 \\beta } } . \\end{align*}"}
+{"id": "5055.png", "formula": "\\begin{align*} F = \\frac { P } { ( 1 - c _ 1 u _ 1 ) \\cdots ( 1 - c _ l u _ l ) } \\end{align*}"}
+{"id": "963.png", "formula": "\\begin{align*} \\mathcal R ( V , K ) : = \\left \\{ a \\in V ^ \\vee : \\mbox { t h e r e e x i s t s } b \\in V ^ \\vee \\mbox { s u c h t h a t } a \\wedge b \\in K ^ \\perp \\setminus \\{ 0 \\} \\right \\} \\cup \\{ 0 \\} \\end{align*}"}
+{"id": "6220.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p \\beta _ r u _ r ( T ) \\equiv \\sum _ { r = 1 } ^ p \\beta _ r u _ r ' ( T ) \\equiv 0 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "776.png", "formula": "\\begin{align*} S _ { n } ( t ) = \\frac { 1 } { ( n \\sigma _ 0 ^ 2 ) ^ { 1 / 2 } } \\left ( \\log \\| M _ { \\lfloor t n \\rfloor } \\| - n t \\lambda _ 1 + ( n t - \\lfloor n t \\rfloor ) ( \\log \\| M _ { \\lfloor t n \\rfloor + 1 } \\| - \\log \\| M _ { \\lfloor t n \\rfloor } \\| ) \\right ) \\end{align*}"}
+{"id": "7199.png", "formula": "\\begin{align*} q = z _ 0 + z _ 1 j , \\qquad \\mbox { w h e r e } z _ 0 = x _ 0 + x _ 1 i \\qquad \\mbox { a n d } z _ 1 = x _ 2 + x _ 3 i . \\end{align*}"}
+{"id": "4964.png", "formula": "\\begin{align*} \\phi ( x ^ p - a ^ { p - 1 } x ) = ( x ^ p + a ^ p ) - a ^ { p - 1 } ( x + a ) = x ^ p - a ^ { p - 1 } x . \\end{align*}"}
+{"id": "1889.png", "formula": "\\begin{align*} z ( x , t ) = y ( x , t ) - \\int _ 0 ^ x k ( x , \\zeta ) y ( \\zeta , t ) d \\zeta \\triangleq \\Lambda y ( x , t ) , \\end{align*}"}
+{"id": "8160.png", "formula": "\\begin{align*} H ^ - ( x ) = \\frac { 4 } { \\pi } \\int _ { - \\infty } ^ \\infty K _ { 2 i t } ( x ) \\sinh ( \\pi t ) h ( t ) t \\ , d t . \\end{align*}"}
+{"id": "1016.png", "formula": "\\begin{align*} \\lVert \\varphi _ k \\rVert _ { \\infty , \\alpha , \\beta } = \\sup _ { x \\in \\R } | x | ^ \\alpha | \\mathrm { D } ^ \\beta \\varphi _ k ( x ) | \\leq C \\lVert v _ k \\rVert _ { \\infty , \\alpha + 2 } \\end{align*}"}
+{"id": "2330.png", "formula": "\\begin{align*} V _ p ( x , y ) & = \\ ( \\frac { 4 ( p - 1 ) } { N - p } \\ ) ^ 2 ( 1 - X ) ^ { - 2 } + o \\ ( ( 1 - X ) ^ { - 2 } \\ ) \\\\ & = \\ ( \\frac { p - 1 } { N - p } \\ ) ^ 2 y ^ { - 2 } + o \\ ( y ^ { - 2 } \\ ) \\end{align*}"}
+{"id": "4388.png", "formula": "\\begin{align*} U _ n ( \\theta ) = \\left \\{ ( x _ i ) _ { i = 1 } ^ n \\in E _ n : \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < \\theta \\right \\} . \\end{align*}"}
+{"id": "2186.png", "formula": "\\begin{align*} R _ { \\mathcal { S } ^ { * } _ { S G } } ( F ) & = \\begin{dcases} \\sigma _ { 0 } & \\ 2 \\alpha + \\beta - 2 \\geqslant 0 \\\\ \\tilde { \\sigma _ { 0 } } & \\ 2 \\alpha + \\beta - 2 < 0 , \\end{dcases} \\end{align*}"}
+{"id": "358.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial t ^ 2 } f ( x , t ) + 2 \\lambda \\frac { \\partial } { \\partial t } f ( x , t ) = v _ 0 ^ 2 \\frac { \\partial ^ 2 } { \\partial x ^ 2 } f ( x , t ) \\ , . \\end{align*}"}
+{"id": "132.png", "formula": "\\begin{align*} \\mathcal L ^ { 2 n } ( \\widetilde E _ { \\lambda , K } ) & \\leq 2 \\int _ { \\mathbb R ^ n } \\int _ { \\mathbb R ^ n } \\mathbf { 1 } _ { \\left \\{ y : \\ | | y - x | | _ K \\leq \\left ( \\frac { 2 | f ( x ) | } { \\lambda } \\right ) ^ { \\frac p n } \\right \\} } d y d x \\\\ & = 2 \\int _ { \\mathbb R ^ n } \\left ( \\frac { 2 | f ( x ) | } { \\lambda } \\right ) ^ { p } | K | d x \\\\ & = \\frac { 2 ^ { p + 1 } | K | } { \\lambda ^ p } | | f | | _ { L ^ p ( \\mathbb R ^ n ) } ^ p . \\end{align*}"}
+{"id": "2545.png", "formula": "\\begin{align*} S _ 2 = \\bar S _ 2 + E _ 3 \\ , , \\end{align*}"}
+{"id": "5212.png", "formula": "\\begin{align*} f ^ 1 & = g ^ 1 ( z _ 1 , z _ 2 ) q ( z _ 3 ) \\\\ f ^ 2 & = g ^ 2 ( z _ 1 , z _ 2 ) \\\\ f ^ 3 & = g ^ 3 ( z _ 1 , z _ 2 ) q ( z _ 3 ) \\ . \\end{align*}"}
+{"id": "6677.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ^ k - x ^ { \\ast } \\| ^ 2 & \\leq \\| x ^ k - { \\bf 1 } \\otimes \\bar x ^ k + { \\bf 1 } \\otimes \\bar x ^ k - x ^ { \\ast } \\| ^ 2 \\\\ & \\leq 2 \\| x ^ k - { \\bf 1 } \\otimes \\bar x ^ k \\| ^ 2 + 2 \\| { \\bf 1 } \\otimes \\bar x ^ k - x ^ { \\ast } \\| ^ 2 \\\\ & \\leq 2 \\sum _ { i = 1 } ^ m \\| x _ i ^ k - \\bar { x } ^ k \\| ^ 2 + 2 m \\| \\bar x ^ k - \\theta ^ { \\ast } \\| ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "6438.png", "formula": "\\begin{align*} u _ \\epsilon ( X , Y , t ) \\leq \\tilde u _ \\epsilon ( X , Y , t ) = u _ \\epsilon ( \\hat X , \\hat Y , \\hat t ) + \\eta . \\end{align*}"}
+{"id": "586.png", "formula": "\\begin{align*} [ W _ { t _ n } ^ \\dagger , W _ { t _ n , t _ { n + 1 } } ^ \\dagger ] : = W ^ \\dagger _ { t _ n } \\otimes W ^ \\dagger _ { t _ n , t _ { n + 1 } } - W ^ \\dagger _ { t _ n , t _ { n + 1 } } \\otimes W _ { t _ n } ^ \\dagger . \\end{align*}"}
+{"id": "5558.png", "formula": "\\begin{align*} | S _ 3 ( x ^ 2 ) | \\ll _ { \\epsilon } \\ell ^ { \\frac { 1 } { 2 } - k + \\epsilon } \\left ( \\sum _ { n = \\ell } ^ { [ x ] } + \\sum _ { n = [ x ] + 1 } ^ \\infty \\right ) \\frac { x ^ 2 } { y _ n ^ 3 } e ^ { - \\frac { x ^ 2 } { y _ n ^ 2 } } . \\end{align*}"}
+{"id": "4537.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 3 0 } = \\frac { 1 } { 3 } + \\frac { 1 } { 5 } = \\frac { 8 } { 1 5 } < \\theta \\leq \\frac { 1 } { 2 } + \\frac { 1 } { 2 9 } = \\frac { 3 1 } { 5 8 } . \\end{align*}"}
+{"id": "4674.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell _ 1 = 0 } ^ { \\min \\{ n _ 1 , m \\} } \\sum _ { \\ell _ 2 = 0 } ^ { \\min \\{ n _ 2 , m \\} } A _ { m , \\ell _ 1 , \\ell _ 2 } = \\sum _ { \\ell _ 1 = 0 } ^ { n _ 1 } \\sum _ { \\ell _ 2 = 0 } ^ { n _ 2 } \\sum _ { m = \\max \\{ \\ell _ 1 , \\ell _ 2 \\} } ^ { \\infty } A _ { m , \\ell _ 1 , \\ell _ 2 } . \\end{align*}"}
+{"id": "4749.png", "formula": "\\begin{align*} v _ j = a + y ^ j t = ( a + y ^ j c ) + y ^ j ( t - c ) , \\end{align*}"}
+{"id": "646.png", "formula": "\\begin{align*} H _ { n + 1 } : = ( H _ { n } + c \\cdot N B _ { c } ) _ { \\delta } \\subset \\delta \\cdot \\Z , \\end{align*}"}
+{"id": "2247.png", "formula": "\\begin{align*} & \\sum _ { l = 1 } ^ { 2 k - 2 } \\biggl | \\tilde { a } ^ { \\infty } _ { l } - \\sum _ { j = 0 } ^ { l } \\frac { \\alpha _ { l - j } } { j ! } \\tau ^ i \\biggr | + \\biggl | \\tilde { a } ^ { \\infty } _ { 2 k - 1 } - \\sum _ { j = 1 } ^ { 2 k - 1 } \\frac { \\alpha _ { 2 k - 1 - j } } { j ! } \\tau ^ j \\biggr | \\\\ & + \\sum _ { j = 1 } ^ { 2 k - 2 } \\biggl | \\tilde { b } _ { j } ^ { + , \\infty } - \\sum _ { l = 0 } ^ { 2 k - 2 } \\frac { \\alpha _ { 2 k - 2 - l } } { ( l + j + 1 ) ! } \\tau ^ { l + j + 1 } \\biggr | = 0 . \\end{align*}"}
+{"id": "92.png", "formula": "\\begin{align*} d ' \\leq \\begin{cases} \\delta + 2 & r > 2 , \\\\ 2 \\delta + 1 & r = 2 . \\end{cases} \\end{align*}"}
+{"id": "5413.png", "formula": "\\begin{align*} \\delta ' ( q _ { i , ( i + k ' ) \\bmod p , k ' } , \\ell _ { k ' } ^ { ( i ) } ) = q _ { i , ( i + k ' + 1 ) \\bmod p , k ' + 1 } . \\end{align*}"}
+{"id": "708.png", "formula": "\\begin{align*} d B = d b + d \\oint _ \\beta * d G ^ \\omega = d b + \\sum _ { k = 1 } ^ n \\Gamma _ k d U _ \\beta ( w _ k ) . \\end{align*}"}
+{"id": "1409.png", "formula": "\\begin{align*} \\bigsqcup _ { j = 1 } ^ t \\{ \\underbrace { [ x + j - 1 , x ] _ \\rho , \\dots , [ x + j - 1 , x ] _ \\rho } _ { k _ { j - 1 } - k _ j } \\} \\end{align*}"}
+{"id": "2759.png", "formula": "\\begin{align*} \\overline { \\alpha \\ , u + \\beta \\ , v } = \\overline { \\alpha } \\ , \\overline { u } + \\overline { \\beta } \\ , \\overline { v } , \\quad \\overline { \\overline { u } } = u , \\quad \\overline { u \\ , v } = \\overline { v } \\ , \\overline { u } . \\end{align*}"}
+{"id": "9165.png", "formula": "\\begin{align*} \\| f \\| _ { C ^ 2 _ { j _ f } } : = \\max _ { n = 0 , 1 , 2 } L ^ { n j _ f } \\| \\nabla ^ n f \\| _ { L ^ \\infty ( \\Z ^ 2 ) } \\leq M L ^ { - 2 j _ f } , \\end{align*}"}
+{"id": "1932.png", "formula": "\\begin{align*} | k _ \\Sigma + A | ^ 2 & = \\tfrac { 1 } { n - 1 } h ^ 2 - \\tfrac { 2 } { n - 1 } h H + | A | ^ 2 \\\\ & = - \\tfrac { 1 } { n - 1 } h ^ 2 + | A | ^ 2 . \\end{align*}"}
+{"id": "6503.png", "formula": "\\begin{align*} ( \\Xi _ u ) = \\underset { j = 1 } { \\overset { m } { \\sum } } ( \\Xi ^ j _ u ) \\leq \\min \\{ 2 \\log 2 \\cdot \\frac { b } { d } , 1 \\} \\frac { m ^ 2 d } { n } . \\end{align*}"}
+{"id": "8444.png", "formula": "\\begin{align*} \\kappa \\gamma _ A = 1 \\end{align*}"}
+{"id": "7801.png", "formula": "\\begin{align*} \\epsilon _ { \\lambda } = \\begin{cases} 1 & \\langle n _ { \\lambda } , \\gamma ' \\rangle < 0 , \\\\ - 1 & \\langle n _ { \\lambda } , \\gamma ' \\rangle > 0 , \\end{cases} \\end{align*}"}
+{"id": "3383.png", "formula": "\\begin{align*} \\allowdisplaybreaks \\begin{aligned} \\bigvee _ { j = 1 } ^ { i - 1 } ( - u _ i + x _ j ) \\vee ( - u _ { i + 1 } + 1 + x _ i ) & \\leq \\bigvee _ { j = i + 1 } ^ { n - 1 } ( - u _ j + x _ j ) \\vee x _ n \\vee ( - u _ n ) & & \\enspace i = 1 , \\dots , n - 1 \\\\ \\bigvee _ { j = 1 } ^ n x _ j & \\leq 0 \\\\ - \\infty \\leq x _ 1 \\leq x _ 2 & \\leq \\dots \\leq x _ { n - 1 } \\leq u _ n + x _ n \\ , . \\end{aligned} \\end{align*}"}
+{"id": "5529.png", "formula": "\\begin{align*} & G _ { p , q } ^ { \\ , m , n } \\ ! \\left ( \\ , \\begin{matrix} a _ 1 , \\cdots , a _ p \\\\ b _ 1 , \\cdots , b _ q \\end{matrix} \\ ; \\Big | z \\right ) \\\\ & = \\sum _ { k = 1 } ^ { m } A _ { p , q , k } ^ { m , n } ( z ) { } _ p F _ { q - 1 } \\left ( \\begin{matrix} 1 + b _ k - a _ 1 , \\cdots , 1 + b _ k - a _ p \\\\ 1 + b _ k - b _ 1 , \\cdots , * , \\cdots , 1 + b _ k - b _ q \\end{matrix} \\Big | ( - 1 ) ^ { p - m - n } z \\right ) , \\end{align*}"}
+{"id": "9164.png", "formula": "\\begin{align*} C ( s , m ^ 2 ) = \\Gamma _ 1 ( s , m ^ 2 ) + \\dots + \\Gamma _ { N - 1 } ( s , m ^ 2 ) + \\Gamma _ N ^ { \\Lambda _ N } ( s , m ^ 2 ) + t _ N ( s , m ^ 2 ) Q _ N . \\end{align*}"}
+{"id": "264.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = & \\alpha ( w / u ) + u ( 1 + \\beta u ) , \\psi _ 2 ( u , v , w ) = \\alpha ( w / v ) + u ( 1 + \\beta u ) , \\\\ \\psi _ 3 ( u , v , w ) = & \\alpha ( w / v ) + v ( 1 + \\beta v ) \\psi _ 4 ( v , w ) = \\alpha ( w / u ) + v ( 1 + \\beta v ) . \\end{align*}"}
+{"id": "2055.png", "formula": "\\begin{align*} M _ 1 ( t ) = M _ 2 ( \\varphi ( t ) ) , t \\in [ 0 , L _ 1 ) . \\end{align*}"}
+{"id": "4027.png", "formula": "\\begin{align*} w ^ { i j } \\gamma _ i D _ j v & = \\chi ^ { - 1 } \\phi ^ * _ k Y ^ k _ { p _ j } D _ j v \\\\ & = \\chi ^ { - 1 } \\phi ^ * _ k Y ^ k _ { p _ j } D _ j ( G _ p \\cdot \\gamma ) - K \\chi ^ { - 1 } \\phi ^ * _ k Y ^ k _ { p _ j } D _ j G . \\end{align*}"}
+{"id": "2627.png", "formula": "\\begin{align*} V _ { [ r ] \\setminus \\{ l \\} } = \\left < v _ { a , b } ^ t \\ \\middle | \\ a , b \\in [ r ] \\setminus \\{ l \\} , t = 1 , \\dots , d \\right > \\end{align*}"}
+{"id": "7772.png", "formula": "\\begin{align*} f _ { 1 / 4 } ( x | \\mu ) & = \\int _ 0 ^ \\infty f _ { 1 / 2 } ( x | y ) f _ { 1 / 2 } ( y | \\mu ) d y \\\\ & = \\int _ 0 ^ \\infty \\frac { y } { 2 \\sqrt { \\pi x ^ 3 } } \\ , e ^ { - y ^ 2 / 4 x } \\ , \\frac { \\mu } { 2 \\sqrt { \\pi y ^ 3 } } \\ , e ^ { - \\mu ^ 2 / 4 y } \\ , d y \\\\ & = \\frac { \\mu } { 4 \\pi \\sqrt { x ^ 3 } } \\int _ 0 ^ \\infty \\frac { 1 } { \\sqrt { y } } \\ , e ^ { - \\mu ^ 2 / 4 y - y ^ 2 / 4 x } \\ , d y \\end{align*}"}
+{"id": "1862.png", "formula": "\\begin{align*} \\beta _ k = \\frac { \\min \\{ \\underline { s } _ k , \\underline { z } _ k \\} } { \\nu _ k } \\ge 0 , \\end{align*}"}
+{"id": "656.png", "formula": "\\begin{align*} | H _ { n } + \\bar { c } B ' | _ { \\delta } = | \\bar { A } + \\bar { c } B ' | _ { \\delta } \\geq \\delta ^ { - \\bar { \\epsilon } } | \\bar { A } | = \\delta ^ { - \\bar { \\epsilon } } | H _ { n } | . \\end{align*}"}
+{"id": "2203.png", "formula": "\\begin{align*} \\gamma = E ( q ) ^ 2 E ( q ^ { 1 0 } ) , \\qquad \\delta = E ( q ^ 2 ) E ( q ^ 5 ) ^ 2 , \\qquad \\xi = q \\dfrac { E ( q ^ 2 ) E ( q ^ { 1 0 } ) ^ 3 } { E ( q ) ^ 3 E ( q ^ 5 ) } , \\end{align*}"}
+{"id": "8006.png", "formula": "\\begin{align*} \\begin{aligned} \\dot x _ 1 & = f _ 1 , \\\\ \\dot x _ 1 & = x _ 2 + f _ 2 , \\\\ 0 & = x _ 1 + x _ 2 + \\Delta _ { - \\tau } x _ 3 + f _ 3 . \\end{aligned} \\end{align*}"}
+{"id": "1082.png", "formula": "\\begin{align*} I ( Z _ 1 ^ n ; J ) = \\frac { 1 } { M ( \\delta ) } \\sum _ { i = 1 } ^ { M ( \\delta ) } \\mathrm { K L } ( Q P ^ n _ i , \\overline { Q } ) \\leq 4 n ( e ^ \\alpha - 1 ) ^ 2 \\frac { 1 } { \\{ M ( \\delta ) \\} ^ 2 } \\sum _ { i , i ' = 1 } ^ { M ( \\delta ) } \\mathrm { T V } ( P _ i , P _ { i ' } ) \\leq 4 n ( e ^ \\alpha - 1 ) ^ 2 , \\end{align*}"}
+{"id": "8739.png", "formula": "\\begin{align*} \\begin{array} { l l l } g _ 0 : = [ C _ 0 B _ 0 , C _ 0 A _ 0 B _ 0 , \\dots , C _ 0 A _ 0 ^ { 2 T - 2 } B _ 0 ] \\in \\mathcal { M } _ { p \\times ( 2 T - 1 ) r } ( \\mathbb { R } ) \\\\ \\bar { g } _ 0 : = [ C _ 0 A _ 0 ^ { 2 T - 1 } B _ 0 , C _ 0 A _ 0 ^ { 2 T } B _ 0 , \\dots , C _ 0 A _ 0 ^ { N - 1 } B _ 0 ] \\in \\mathcal { M } _ { p \\times \\bar { N } r } ( \\mathbb { R } ) \\\\ h : = [ C _ 0 , C _ 0 A _ 0 , \\dots , C _ 0 A _ 0 ^ { N - 1 } ] \\in \\mathcal { M } _ { p \\times N d _ 0 } ( \\mathbb { R } ) . \\end{array} \\end{align*}"}
+{"id": "1264.png", "formula": "\\begin{align*} R _ { \\mathcal { B S } ( \\alpha ) } ( \\mathcal { S ^ { * } } [ A , B ] ) & = \\begin{dcases} \\min \\left \\{ 1 , \\dfrac { 2 \\sqrt { \\alpha } } { ( 1 + \\alpha ) \\sqrt { 4 \\alpha ( A - B ) ^ 2 + B ^ 2 } } \\right \\} , & 4 A \\alpha \\geq ( 5 \\alpha - 1 ) B , \\\\ \\min \\left \\{ 1 , \\dfrac { 1 } { ( 1 - \\alpha ) ( A - B ) - B } \\right \\} , & 4 A \\alpha \\leq ( 5 \\alpha - 1 ) B . \\end{dcases} \\end{align*}"}
+{"id": "487.png", "formula": "\\begin{align*} \\frac { 1 } { s } & \\int _ 0 ^ T \\iint \\limits _ { \\mathbb { R } ^ N \\times \\mathbb { R } ^ N } \\frac { H ( x , y , ( u + s \\psi ) ( x , t ) - ( u + s \\psi ) ( y , t ) ) - H ( x , y , u ( x , t ) - u ( y , t ) ) } { | x - y | ^ N } \\ , d x \\ , d y \\ , d t \\\\ & = \\int _ 0 ^ T \\iint \\limits _ { \\mathbb { R } ^ N \\times \\mathbb { R } ^ N } \\int _ 0 ^ 1 \\frac { D _ \\xi H ( x , y , ( u + s \\sigma \\psi ) ( x , t ) - ( u + s \\sigma \\psi ) ( y , t ) ) \\cdot ( \\psi ( x , t ) - \\psi ( y , t ) ) } { | x - y | ^ N } \\ , d \\sigma \\ , d x \\ , d y \\ , d t \\end{align*}"}
+{"id": "4073.png", "formula": "\\begin{align*} F ( X + \\xi ) = \\sigma X + \\frac { 1 } { 2 } \\pi ^ * \\xi X \\in T M , \\xi \\in T ^ * M . \\end{align*}"}
+{"id": "6068.png", "formula": "\\begin{align*} ( w _ { \\Delta _ 2 } \\tilde w _ { \\Delta _ 2 } ) ( t ) = \\prod _ { k = 1 } ^ { g + 1 } \\frac { ( 1 - x _ 0 ^ 2 ) ^ 2 } { ( 1 - a _ k x _ 0 ) ( 1 - b _ k x _ 0 ) } \\frac { ( w _ { \\Delta _ 1 } \\tilde w _ { \\Delta _ 1 } ) ( x ) } { ( 1 - x x _ 0 ) ^ { 2 g + 2 } } . \\end{align*}"}
+{"id": "7754.png", "formula": "\\begin{align*} f _ \\alpha ( x | \\mu = 1 ) \\equiv f _ \\alpha ( x ) & = \\frac { 1 } { \\pi } { \\rm I m } \\int _ 0 ^ \\infty e ^ { - x t } \\ , e ^ { - ( t e ^ { - i \\pi } ) ^ \\alpha } \\ , d t \\\\ & = \\frac { 1 } { \\pi } \\int _ 0 ^ \\infty e ^ { - x t } \\ , e ^ { - t ^ \\alpha \\cos \\pi \\alpha } \\sin ( t ^ \\alpha \\sin \\pi \\alpha ) \\ , d t \\end{align*}"}
+{"id": "4435.png", "formula": "\\begin{align*} J ( 2 , 6 ) & = \\left ( \\frac { 2 } { 3 } , \\frac { 7 } { 1 0 } \\right ] , J ( 2 , 1 2 ) = \\left ( \\frac { 7 } { 1 2 } , \\frac { 1 3 } { 2 2 } \\right ] , J ( 2 , 3 0 ) = \\left ( \\frac { 8 } { 1 5 } , \\frac { 3 1 } { 5 8 } \\right ] . \\end{align*}"}
+{"id": "519.png", "formula": "\\begin{align*} \\liminf _ { k \\to \\infty } H _ { \\delta } ( z ^ { \\ell ( k ) - i } ) \\ge \\lim _ { k \\to \\infty } H _ { \\delta } ( z ^ { \\ell ( k ) } ) \\ \\ { \\rm a n d } \\ \\ \\lim _ { k \\rightarrow \\infty } \\| z ^ { \\ell ( k ) - i } - z ^ { \\ell ( k ) - i - 1 } \\| = 0 . \\end{align*}"}
+{"id": "2076.png", "formula": "\\begin{align*} | \\sin ( \\eta _ 1 + \\eta _ 2 ) | & = | \\sin ( \\eta _ 1 ) \\cos ( \\eta _ 2 ) + \\sin ( \\eta _ 2 ) \\cos ( \\eta _ 1 ) | \\\\ & \\leq | \\sin ( \\eta _ 1 ) | + | \\sin ( \\eta _ 2 ) | , \\end{align*}"}
+{"id": "6113.png", "formula": "\\begin{align*} j = p n . \\end{align*}"}
+{"id": "1930.png", "formula": "\\begin{align*} 2 \\mu & = R _ g + \\tfrac { n } { n - 1 } h ^ 2 \\\\ J ^ i & = \\nabla ^ i h . \\end{align*}"}
+{"id": "5906.png", "formula": "\\begin{align*} \\alpha _ { s r } = 0 , 1 \\leq r , s \\leq p , \\end{align*}"}
+{"id": "8787.png", "formula": "\\begin{align*} k _ { \\lambda } ( z ) = \\frac { 1 - \\overline { \\theta ( \\lambda ) } \\theta ( z ) } { 1 - \\bar { \\lambda } z } , \\end{align*}"}
+{"id": "6549.png", "formula": "\\begin{align*} & { v _ { i } } \\left ( { k \\ ! + \\ ! 1 } \\right ) \\ ! = \\ ! \\omega { v _ { i } } \\ ! \\left ( k \\right ) \\ ! + \\ ! { c _ 1 } { R _ 1 } \\ ! \\left ( { { p _ { i } } \\ ! \\left ( k \\right ) \\ ! - \\ ! { x _ { i } } \\left ( k \\right ) } \\right ) \\ ! + \\ ! { c _ 2 } { R _ 2 } \\left ( { { p _ { g } } \\ ! \\left ( k \\right ) \\ ! - \\ ! { x _ { i } } \\left ( k \\right ) } \\right ) , \\\\ & { x _ { i } } \\left ( { k + 1 } \\right ) = { x _ { i } } \\left ( k \\right ) + { v _ { i } } \\left ( { k + 1 } \\right ) , \\end{align*}"}
+{"id": "7808.png", "formula": "\\begin{align*} p _ 1 ^ * ( n ) = ( p ^ * ( n ) , n ) \\in \\tilde M ^ { \\circ } . \\end{align*}"}
+{"id": "4415.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 1 7 } = \\frac { 2 0 } { 5 1 } < \\frac { 1 9 } { 4 8 } = \\frac { 1 } { 3 } + \\frac { 1 } { 1 6 } \\end{align*}"}
+{"id": "7937.png", "formula": "\\begin{align*} x \\sim y \\iff x + n = y + n ' \\textrm { f o r s o m e } n , n ' \\in N . \\end{align*}"}
+{"id": "46.png", "formula": "\\begin{align*} T ( n _ i ) + T ( n _ { i + 1 } ) & < \\left ( \\frac { 5 } { 3 } + 3 \\cdot \\frac { 9 } { 8 } \\right ) \\cdot \\frac { 1 } { X _ 0 } = \\frac { 1 2 1 } { 2 4 } \\cdot \\frac { 1 } { X _ 0 } \\\\ & < 7 \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } \\leq ( k _ i + k _ { i + 1 } ) \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } . \\end{align*}"}
+{"id": "7715.png", "formula": "\\begin{align*} E _ n ( \\lambda ) = \\frac { 4 \\sqrt { \\tau } } { \\pi } \\left ( e _ n ^ { ( 1 ) } ( \\lambda ) + e _ n ^ { ( 2 ) } ( \\lambda ) \\right ) , \\end{align*}"}
+{"id": "2021.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\{ D ' , \\ldots , D - 1 \\} & { \\rm i f } & \\alpha = 0 , \\\\ \\{ D ' , \\ldots , D \\} & { \\rm i f } & 0 < \\alpha < 1 , \\\\ \\{ D ' + 1 , \\ldots , D \\} & { \\rm i f } & \\alpha = 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "7894.png", "formula": "\\begin{align*} A ( z ) = H _ 0 ( z ) + H _ 1 ( z ) e ^ { \\zeta _ 1 z ^ n } + \\cdots + H _ m ( z ) e ^ { \\zeta _ m z ^ n } , m \\geq 2 . \\end{align*}"}
+{"id": "4951.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { H } } _ n = \\ , \\Delta x \\sum _ m \\left \\{ \\tfrac { 1 } 2 \\left [ ( \\delta _ m ^ + u _ { m , n } ) ^ 2 + ( \\delta _ m ^ - u _ { m , n } ) ^ 2 + ( \\delta _ m ^ + v _ { m , n } ) ^ 2 + ( \\delta _ m ^ - v _ { m , n } ) ^ 2 \\right ] - \\tfrac { 1 } 2 ( u _ { m , n } ^ 2 + v _ { m , n } ^ 2 ) ^ 2 \\right \\} . \\end{align*}"}
+{"id": "3256.png", "formula": "\\begin{align*} \\chi _ T ( f ) = \\sum \\limits _ { S \\subseteq T } ( - 1 ) ^ { | T | + | S | } f ( S ) . \\end{align*}"}
+{"id": "6926.png", "formula": "\\begin{align*} \\tilde { u } ( t ) = \\frac { \\tilde { x } _ 1 ( t ) \\left ( \\tilde { \\psi } _ 1 ( t ) - \\tilde { \\psi } _ 8 ( t ) \\right ) } { 2 w _ 2 } . \\end{align*}"}
+{"id": "5901.png", "formula": "\\begin{align*} D ^ T \\widetilde C _ { p - 1 } ^ T E = 0 . \\end{align*}"}
+{"id": "2341.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , \\cdots , n \\} , \\phi _ i : = \\frac { 1 } { \\sqrt { 2 } } ( \\psi _ i + S \\psi _ i ) , \\phi _ { n + i } = \\frac { 1 } { \\sqrt { 2 } } ( \\psi _ i - S \\psi _ i ) . \\end{align*}"}
+{"id": "5133.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Big ( \\big [ \\Omega _ { n + 1 } ( \\lambda ) - \\Omega _ { n } ( \\lambda ) \\big ] - \\big [ \\Omega _ { n + 1 } ( \\lambda b ) - \\Omega _ { n } ( \\lambda b ) \\big ] \\Big ) \\underset { n \\rightarrow \\infty } { = } O _ { \\lambda , b } \\left ( \\frac { 1 } { n ^ { 4 } } \\right ) . \\end{align*}"}
+{"id": "2999.png", "formula": "\\begin{align*} c ^ i _ { i j } = c ^ i _ { j i } = 0 \\end{align*}"}
+{"id": "5223.png", "formula": "\\begin{align*} & g _ { 1 2 5 } + g _ { 3 4 5 } = g _ { 1 2 6 } + g _ { 3 4 6 } = 0 \\\\ & g _ { 1 3 5 } - g _ { 2 3 6 } + g _ { 2 4 5 } + g _ { 1 4 6 } = 0 \\\\ & g _ { 1 3 6 } + g _ { 2 4 6 } - g _ { 1 4 5 } + g _ { 2 3 5 } = 0 \\ , . \\end{align*}"}
+{"id": "6012.png", "formula": "\\begin{align*} X ( w _ { i , j } ) & = \\left \\{ x = ( [ L _ x ] , [ H _ x ] ) \\in X \\mid L _ x \\subset < e _ 1 , \\cdots , e _ i > , \\ : H _ x \\subset < e _ j , \\cdots , e _ { n } > \\right \\} \\\\ & = ( L _ i \\times w _ 0 L _ { n - j + 1 } ) \\cap X . \\end{align*}"}
+{"id": "6473.png", "formula": "\\begin{align*} \\int | \\delta E ( m ) | ^ 2 d x - \\int | m \\cdot \\delta E ( m ) | ^ 2 d x & = \\int \\left ( | \\delta E ( w _ * + \\eta ) | ^ 2 - | \\delta E ( w _ * ) | ^ 2 \\right ) d x \\\\ & - \\int \\left ( | ( w _ * + \\eta ) \\cdot \\delta E ( w _ * + \\eta ) | ^ 2 - | w _ * \\cdot \\delta E ( w _ * ) | ^ 2 \\right ) d x \\end{align*}"}
+{"id": "5012.png", "formula": "\\begin{align*} c B ( [ X , Y ] , Z ) = \\xi ( [ X , Y ] , Z ) . \\end{align*}"}
+{"id": "8061.png", "formula": "\\begin{align*} \\Psi ^ \\pm ( x ) = x \\int _ 0 ^ \\infty \\psi ( y ) \\sum _ { j = 1 } ^ K \\frac { c _ j ^ \\pm e ( 3 x ^ \\frac { 1 } { 3 } y ^ { \\frac { 1 } { 3 } } ) + d _ j ^ \\pm e ( - 3 x ^ { \\frac { 1 } { 3 } } y ^ { \\frac { 1 } { 3 } } ) } { ( x y ) ^ { \\frac { j } { 3 } } } \\ , d y + O \\bigl ( ( x X ) ^ { \\frac { - K + 2 } { 3 } } \\bigr ) \\end{align*}"}
+{"id": "4739.png", "formula": "\\begin{align*} \\mathfrak { m } B = \\mathfrak { p } _ 1 \\cap \\mathfrak { p } _ 2 \\cap \\cdots \\cap \\mathfrak { p } _ n . \\end{align*}"}
+{"id": "1999.png", "formula": "\\begin{align*} \\zeta ( k _ 1 , k _ 2 ) \\zeta ( \\ell ) = & \\sum _ { \\substack { i _ 1 \\ge 1 , i _ 2 \\ge 2 \\\\ i _ 1 + i _ 2 = k _ 2 + \\ell } } \\binom { i _ 2 - 1 } { \\ell - 1 } \\zeta ( k _ 1 , i _ 1 , i _ 2 ) \\\\ & + \\sum _ { \\substack { i _ 1 , i _ 2 \\ge 1 , i _ 3 \\ge 2 \\\\ i _ 1 + i _ 2 + i _ 3 = k _ 1 + k _ 2 + \\ell } } \\binom { i _ 3 - 1 } { k _ 2 - 1 } \\left [ \\binom { i _ 2 - 1 } { k _ 1 - i _ 1 } + \\binom { i _ 2 - 1 } { k _ 1 - 1 } \\right ] \\zeta ( i _ 1 , i _ 2 , i _ 3 ) . \\end{align*}"}
+{"id": "4032.png", "formula": "\\begin{align*} v & : = w _ { 1 1 } - | \\tau | ^ 2 w _ { 1 1 } ( 0 ) - b ^ 2 ( 1 + M ) ^ { \\frac { n - 2 } { n - 1 } } . \\end{align*}"}
+{"id": "7136.png", "formula": "\\begin{align*} a f _ M ( b \\otimes m _ 0 ) - f _ M ( a b \\otimes m _ 0 ) + f _ M ( a \\otimes b m _ 0 ) - f ( a \\otimes b ) T _ M ( m _ 0 ) = 0 \\end{align*}"}
+{"id": "6407.png", "formula": "\\begin{align*} \\begin{pmatrix} H & 0 \\\\ 0 & E \\end{pmatrix} \\leq A + \\varepsilon A ^ 2 , \\end{align*}"}
+{"id": "8955.png", "formula": "\\begin{align*} { } _ { 2 } F { } _ { 1 } \\left ( \\begin{array} { c } - k , \\ , b \\\\ c \\end{array} \\left \\vert \\xi \\right . \\right ) = \\frac { k ! } { ( c ) _ { k } } P _ { k } ^ { ( c - 1 , b - c - k ) } ( 1 - 2 \\xi ) . \\end{align*}"}
+{"id": "1780.png", "formula": "\\begin{align*} \\rho ( \\beta ' ) = \\frac { \\max \\mathsf { L } ( \\beta ' ) } { \\min \\mathsf { L } ( \\beta ' ) } = \\frac { 1 + \\max \\mathsf { L } ( \\beta ) } { 1 + \\min \\mathsf { L } ( \\beta ) } = \\frac { L + 1 } { \\ell + 1 } . \\end{align*}"}
+{"id": "1188.png", "formula": "\\begin{align*} r _ i ^ k | \\nabla ^ { k } _ { g _ { 0 , i } } \\left ( \\Psi _ i ^ * g _ { 0 , i } - g _ { 0 , i } \\right ) | _ { g _ { 0 , i } } = O _ { i } ( r _ { i } ^ { - 1 / c _ i } ) \\ ; \\ , \\ ; \\ , r _ i \\to \\infty . \\end{align*}"}
+{"id": "6.png", "formula": "\\begin{align*} \\mathcal { U } _ { a d } [ 0 , \\infty ] = \\left \\{ u \\in U \\Big | u \\in \\mathcal { F } _ t ^ \\xi , \\bar { \\mathbb { E } } \\int _ 0 ^ \\infty e ^ { - \\beta _ 1 k t } | u _ t | ^ { 2 k } d t < \\infty , \\ \\beta _ 1 \\geq 0 , \\ k \\geq 1 \\right \\} . \\end{align*}"}
+{"id": "1299.png", "formula": "\\begin{align*} | q ( \\vec { z } ) | \\leq M \\ , \\frac { 1 + \\sum _ { j = 1 } ^ n | z _ j | ^ 2 } { \\left ( \\sum _ { j = 1 } ^ n y _ j ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } } , \\end{align*}"}
+{"id": "3867.png", "formula": "\\begin{align*} Y ( \\cdot , u , D u ) ( \\Omega ) = \\Omega ^ * , \\end{align*}"}
+{"id": "2642.png", "formula": "\\begin{align*} M _ 2 & = \\frac { 1 } { \\# h _ { ( 3 6 ) } } \\sum _ { \\substack { r _ 1 , r _ 2 \\geq 0 \\\\ a } } \\frac { 1 } { 2 ^ { r _ 1 } 3 ^ { r _ 2 / 2 } N ( a ) ^ { 1 / 2 } } V \\left ( \\frac { \\pi 2 ^ { 2 r _ 1 + 1 } 3 ^ { r _ 2 } N ( a ) } { z } \\right ) \\sum _ { \\psi \\bmod { 3 6 } } H ( a , \\psi , y ) , \\end{align*}"}
+{"id": "3107.png", "formula": "\\begin{align*} - \\tilde { A } : D ^ 2 w = - \\bar { \\gamma } \\ , \\gamma \\sum _ { i , j = 1 } ^ n c _ { i j } ( - A : D ^ 2 v ^ { i j } ) = - \\bar { \\gamma } \\ , \\gamma ( C : A - C : \\bar { A } ) = \\gamma - \\bar { \\gamma } , \\end{align*}"}
+{"id": "5517.png", "formula": "\\begin{align*} P _ 1 ( x ) : = \\sum _ { n = 1 } ^ \\infty \\frac { \\mu ( n ) } { n } \\exp \\left ( { - \\frac { x } { n ^ 2 } } \\right ) = O _ { \\epsilon } \\left ( x ^ { - \\frac { 1 } { 4 } + \\epsilon } \\right ) , \\mathrm { a s } \\ , \\ , x \\rightarrow \\infty , \\end{align*}"}
+{"id": "5526.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ k } P _ k \\left ( \\frac { x } { n ^ 2 } \\right ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ k } \\sum _ { m = 0 } ^ \\infty \\frac { ( - 1 ) ^ m x ^ m } { m ! n ^ { 2 m } \\zeta ( k + 2 m ) } = \\sum _ { m = 0 } ^ \\infty \\frac { ( - 1 ) ^ m x ^ m } { m ! } = e ^ { - x } . \\end{align*}"}
+{"id": "5715.png", "formula": "\\begin{align*} \\theta ^ { a } = ( \\mathbf { L } ^ { - 1 } ) _ { b } ^ { a } \\mathbf { d x } ^ { b } , \\end{align*}"}
+{"id": "2983.png", "formula": "\\begin{align*} \\mathbb { E } _ n \\bigg [ \\sup _ { 0 \\le t \\le T } \\bigg | \\frac { 1 } { \\sqrt { n } } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\sum _ { i = 0 } ^ { \\ell - 1 } F ^ \\ell _ { i , j } ( s ) \\psi _ i \\nabla ^ n \\varphi ^ n _ j ( s ) d s \\bigg | ^ 2 \\bigg ] \\le C \\frac { T } { n ^ 2 } \\| \\partial _ x \\varphi \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } . \\end{align*}"}
+{"id": "2114.png", "formula": "\\begin{align*} P ^ { - 1 } = \\begin{pmatrix} 0 & 0 & \\cdots & 0 & 1 \\\\ 1 & 0 & \\cdots & 0 & 0 \\\\ 0 & 1 & \\ddots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 & 0 \\end{pmatrix} = P ^ T . \\end{align*}"}
+{"id": "7118.png", "formula": "\\begin{align*} \\langle d , x ^ \\rho \\rangle = \\langle \\rho ^ { - 1 } d , x \\rangle , \\end{align*}"}
+{"id": "8349.png", "formula": "\\begin{align*} \\partial _ t \\left ( \\frac 1 2 | \\partial _ t u | ^ 2 + \\frac 1 2 | \\nabla u | ^ 2 \\right ) - \\div ( \\partial _ t u \\nabla u ) = \\partial _ t \\left ( \\frac { d - 2 } { 2 d } | u | ^ { \\frac { 2 d } { d - 2 } } \\right ) . \\end{align*}"}
+{"id": "6011.png", "formula": "\\begin{align*} w _ 0 L _ i & = \\{ [ 0 : \\dots ; 0 : x _ { n - i + 1 } : \\dots : x _ n ] \\} \\subset \\P ^ { n - 1 } \\\\ L _ i ^ \\bot & = \\{ [ 0 : \\dots ; 0 : x _ { i + 1 } : \\dots : x _ n ] \\} = w _ 0 L _ { n - i } . \\end{align*}"}
+{"id": "3389.png", "formula": "\\begin{align*} ( L \\otimes _ { \\mathfrak { p } } U _ { \\overline { T } } ) ( P _ j ' ) = L \\otimes _ { \\mathfrak { p } } U _ { \\overline { T } } ( P _ j ' , - ) . \\end{align*}"}
+{"id": "3421.png", "formula": "\\begin{align*} L _ { \\phi } f ( x ) : = \\sum _ { \\sigma ( y ) = x } \\exp \\left ( { \\phi ( y ) } \\right ) f ( y ) . \\end{align*}"}
+{"id": "9262.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , v _ 2 , \\ldots , v _ { h ( 1 ) } , N v _ 1 , \\ldots v _ n ) } . \\end{aligned} \\end{align*}"}
+{"id": "8198.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - \\sigma t } \\mathbb { E } \\left [ e ^ { - \\lambda A ( t ) } \\right ] d t = \\frac { 1 } { \\sigma } \\frac { 1 } { 1 + \\lambda ( \\mathcal { L } \\mathfrak { K } ) ( \\sigma ) } \\end{align*}"}
+{"id": "5956.png", "formula": "\\begin{align*} E _ r = P ^ { T } P e _ r , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "4590.png", "formula": "\\begin{align*} \\prod _ { i = m + k + 1 } ^ n a _ i \\leq \\left ( \\frac { \\prod _ { i = m + 1 } ^ { m + k } b _ i } { \\prod _ { i = m + 1 } ^ { m + k } a _ i } \\right ) \\prod _ { i = m + k + 1 } ^ n b _ i \\leq \\prod _ { i = m + k + 1 } ^ n b _ i \\end{align*}"}
+{"id": "8499.png", "formula": "\\begin{align*} \\frac { d } { d t } \\oint _ C \\sqrt { \\mu } d \\sigma & = \\ ; \\oint _ C \\left ( \\frac { \\mu _ t } { 2 \\sqrt { \\mu } } + \\sqrt { \\mu } \\frac { \\bar { g } _ t } { \\bar { g } } \\right ) d \\sigma \\\\ & = \\ ; \\oint _ C \\frac { 1 } { 6 \\sqrt { \\mu } } \\mu _ { \\sigma ^ 2 } d \\sigma \\\\ & = \\ ; \\frac { 1 } { 1 2 } \\oint _ C \\mu ^ { - 3 / 2 } \\mu _ { \\sigma } ^ 2 d \\sigma > 0 , \\end{align*}"}
+{"id": "5769.png", "formula": "\\begin{align*} f _ 2 = ( x _ 2 + x _ 3 + x _ 4 + x _ 5 ) ^ 3 - L \\cdot ( x _ 1 + x _ 2 + x _ 3 + x _ 4 ) ^ 2 \\end{align*}"}
+{"id": "4765.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\left ( \\lim _ { n \\rightarrow \\infty } \\int _ { \\Omega } | h ( x , | \\nabla u _ { n } | ) | \\ | \\nabla u _ { n } | | u _ { n } | \\ | \\nabla v | d x \\right ) = 0 . \\end{align*}"}
+{"id": "674.png", "formula": "\\begin{align*} \\int _ M G ^ \\omega \\ , { \\rm v o l } = 0 . \\end{align*}"}
+{"id": "7791.png", "formula": "\\begin{align*} \\Psi _ { a , b } [ n ] ( y ^ { n ' } ) & = \\exp \\Biggl ( \\ , \\sum _ { j = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { j + 1 } } { j [ j a ] _ q } q ^ { j b } X _ { j n } \\Biggr ) ( y ^ { n ' } ) \\\\ & = y ^ { n ' } \\exp \\Biggl ( \\sum _ { j = 1 } ^ { \\infty } \\frac { q ^ { 2 j \\{ n , n ' \\} } - 1 } { q ^ { 2 j a } - 1 } \\frac { ( - 1 ) ^ { j + 1 } } { j } q ^ { j a } q ^ { j b } y ^ { j n } \\Biggr ) . \\end{align*}"}
+{"id": "6202.png", "formula": "\\begin{align*} \\overline { \\mathcal R } = C _ p \\mathcal R . \\end{align*}"}
+{"id": "6981.png", "formula": "\\begin{gather*} \\frac { \\rm d } { { \\rm d } x } H _ { 2 , n } ^ { ( \\{ 1 , 1 \\} ) } ( x ) = 8 n ( n - 1 ) ( n - 2 ) H _ { \\{ 1 , 1 \\} } ( x ) H _ { n - 3 } ( x ) , \\end{gather*}"}
+{"id": "6017.png", "formula": "\\begin{align*} \\dim e v ^ { - 1 } ( x , y ) & = \\dim \\overline { \\mathcal { M } _ { 0 , 2 } } ( X , l _ 1 + l _ 2 ) - 2 \\dim X \\\\ & = \\dim X + \\int _ { l _ 1 + l _ 2 } c _ 1 ( T _ X ) - 1 - 2 \\dim X \\\\ & = \\int _ { l _ 1 + l _ 2 } ( ( n - 1 ) [ h _ 1 ] + ( n - 1 ) [ h _ 2 ] ) - \\dim X - 1 \\\\ & = 0 . \\end{align*}"}
+{"id": "7646.png", "formula": "\\begin{align*} \\omega = \\sum _ { k } \\lambda _ { k } \\frac { d f _ { k } } { f _ { k } } = \\sum _ { k } \\lambda _ { k } d \\log f _ { k } . \\end{align*}"}
+{"id": "9193.png", "formula": "\\begin{align*} D ( - \\Delta ^ { \\Lambda _ N } ) = \\{ f \\in \\ell ^ 2 ( \\Z ^ d ) : f ( 0 ) = 0 , \\ ; f ( x ) = f ( x + L ^ N y ) \\ ; \\} . \\end{align*}"}
+{"id": "8534.png", "formula": "\\begin{align*} & \\frac { n 2 ^ { Q ( n , n ) } } { \\sum _ { i = 1 } ^ { n } \\sum _ { p = 0 } ^ { \\lfloor ( n - i ) / 2 \\rfloor } \\binom { n } { i } \\binom { n - i } { 2 } ^ p ( 2 ^ i - 1 ) ^ n 2 ^ { Q ( n - i + 1 , n - i + 1 - 2 p ) } } \\\\ = & \\frac { 1 } { ( \\sum _ { i = 1 } ^ { n } \\sum _ { p = 0 } ^ { \\lfloor ( n - i ) / 2 \\rfloor } \\binom { n } { i } \\binom { n - i } { 2 } ^ p ( 2 ^ i - 1 ) ^ n 2 ^ { Q ( n - i + 1 , n - i + 1 - 2 p ) - Q ( n , n ) } ) / n } \\end{align*}"}
+{"id": "2878.png", "formula": "\\begin{align*} | L _ U | = N _ 1 + \\ldots + N _ k , \\end{align*}"}
+{"id": "7313.png", "formula": "\\begin{align*} P [ \\mathrm { e } ^ { - \\lambda X _ n } ] , P [ \\mathrm { e } ^ { - \\lambda X } ] < \\infty \\lim _ { n \\to \\infty } P [ \\mathrm { e } ^ { - \\lambda X _ n } ] = P [ \\mathrm { e } ^ { - \\lambda X } ] ( \\lambda > 0 ) . \\end{align*}"}
+{"id": "3753.png", "formula": "\\begin{align*} \\tilde { c } _ { 2 , v } = \\epsilon _ v ( 1 , \\psi _ v ) \\tilde { c } _ { 1 , v } \\circ \\tilde { d } _ { 2 , v } , \\end{align*}"}
+{"id": "2711.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u _ - - \\Delta u _ - = \\frac { 1 } { 2 } \\left ( F ( u ( t ) ) - F ( u ( - t ) ) \\right ) , \\vec { u } _ - ( 0 ) = ( 0 , \\partial _ t u ( 0 ) ) , \\end{align*}"}
+{"id": "8825.png", "formula": "\\begin{align*} P D _ Q ^ 2 = P ( I - Q ^ * Q ) = ( I - Q ^ * Q ) P = D _ Q ^ 2 P \\end{align*}"}
+{"id": "3225.png", "formula": "\\begin{align*} f ( u ) + f ( w ) \\geq ( 4 k _ 3 + 3 k _ 2 + 2 k _ 1 + d - 1 ) - k _ 3 - k _ 2 = 3 k _ 3 + 2 k _ 2 + 2 k _ 1 + d - 1 . \\end{align*}"}
+{"id": "8312.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { C } _ { x y } = \\{ x , y , z \\ | \\ z > 0 , \\ ( x , y ) \\in \\mathcal { H } _ { x y } \\} , \\end{array} \\right . \\end{align*}"}
+{"id": "4501.png", "formula": "\\begin{align*} \\frac { 1 1 } { 2 4 } - \\left ( \\frac { 1 } { 4 } + \\frac { 1 } { 5 } \\right ) = \\frac { 1 } { 1 2 0 } . \\end{align*}"}
+{"id": "2236.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m - 1 } n + 5 ^ { 2 m - 1 } \\right ) } q ^ n & \\equiv x _ { 2 m - 1 , 1 } E ( q ) ^ 2 E ( q ^ { 1 0 } ) \\pmod { 5 ^ { 2 m - 1 } } . \\end{align*}"}
+{"id": "8895.png", "formula": "\\begin{align*} \\mathcal { \\wp } = \\sum \\limits _ { p = 0 } ^ { n - 1 } \\gamma _ { p } ^ { ( \\nu , n ) } r ^ { 2 p } . \\end{align*}"}
+{"id": "6678.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { m ^ 2 } \\left \\| \\sum _ { i = 1 } ^ m g _ i ^ k \\right \\| ^ 2 \\leq \\frac { 2 L ^ 2 } { m } \\sum _ { i = 1 } ^ m \\| x _ i ^ k - \\bar { x } ^ k \\| ^ 2 + 2 L ^ 2 \\| \\bar x ^ k - \\theta ^ { \\ast } \\| ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "3212.png", "formula": "\\begin{align*} f ( x _ 2 ) = \\begin{cases} g ( \\phi ^ { - 1 } ( x _ 2 ) ) , \\mbox { i f } x _ 2 \\in \\phi ( X _ 1 ) , \\\\ \\infty , \\ ; \\ ; \\mbox { i f } x _ 2 \\in X _ 2 \\setminus \\phi ( X _ 1 ) , \\end{cases} \\end{align*}"}
+{"id": "5648.png", "formula": "\\begin{align*} \\zeta ( C , \\vec s ) : = \\sum _ { \\vec { n } \\in ( \\N ^ * ) ^ n } \\frac { 1 } { l _ 1 ( \\vec { n } ) ^ { s _ 1 } \\cdots l _ n ( \\vec { n } ) ^ { s _ n } } \\end{align*}"}
+{"id": "5618.png", "formula": "\\begin{align*} T = \\sum _ { b = - r } ^ r ( T ) _ b , ( T ) _ b \\in { \\cal T } _ { r , b } . \\end{align*}"}
+{"id": "5576.png", "formula": "\\begin{align*} \\max \\{ | y | : ( x , y ) \\in \\left ( \\partial B ^ k _ { N _ 0 } ( 0 ) \\times [ - 2 , 2 ] ^ { n - k } \\right ) \\cap \\partial \\mathcal { O } _ { ( 0 , 0 ) , \\rho } \\} & = 1 . \\end{align*}"}
+{"id": "2608.png", "formula": "\\begin{align*} \\sum _ { l , t \\in [ r ] \\setminus \\{ 1 \\} , l < t } a _ { l , t } = \\frac { ( r - 2 ) ( k + 1 ) } { 2 } . \\end{align*}"}
+{"id": "28.png", "formula": "\\begin{align*} \\left [ 1 - \\left ( u ^ 2 / 2 i _ 0 \\right ) \\right ] ^ { - 6 / 2 5 } = 1 + \\frac { 6 } { 2 5 } \\cdot \\frac { u ^ 2 } { 2 i _ 0 } + \\frac { 6 } { 2 5 } \\cdot \\frac { u ^ 4 } { 2 ( 2 i _ 0 ) ^ 2 } + \\frac { 1 } { 2 } \\cdot \\frac { 6 } { 2 5 } \\cdot \\frac { u ^ 4 } { ( 2 i _ 0 ) ^ 2 } + o ( u ^ 6 ) . \\end{align*}"}
+{"id": "3349.png", "formula": "\\begin{align*} \\Omega = \\dd q ^ i \\wedge \\dd p _ i , \\end{align*}"}
+{"id": "8675.png", "formula": "\\begin{align*} \\sum _ { k \\ge 0 } g _ k ( x ^ { - 1 } ) \\ , f _ k ( y ) = \\sum _ { k \\ge 0 } \\frac { y ^ { k } } { x ^ { k } } = i _ { x , y } \\left ( \\frac { x } { x - y } \\right ) , \\\\ \\sum _ { k < 0 } g _ k ( x ^ { - 1 } ) \\ , f _ k ( y ) = \\sum _ { k \\ge 0 } \\frac { x ^ k } { y ^ { k } } = i _ { y , x } \\left ( \\frac { x } { y - x } \\right ) , \\end{align*}"}
+{"id": "3973.png", "formula": "\\begin{align*} a ^ { i j } & = [ D ^ 2 u - A ( \\cdot , u , D u ) ] ^ { i j } , \\\\ b ^ i & = - a ^ { i j } ( A _ { i j } ) _ { p _ k } - \\tilde { B } _ { p _ k } , \\\\ c & = - a ^ { i j } ( A _ { i j } ) _ u - \\tilde { B } _ u , \\end{align*}"}
+{"id": "6020.png", "formula": "\\begin{align*} \\sum _ { r = r _ 1 + 1 } ^ { \\delta - \\delta _ 0 + r _ 1 } S ( r _ 1 , r ) & = \\sum _ { r = r _ 1 + 1 } ^ { \\delta - \\delta _ 0 + r _ 1 } ( - 1 ) ^ r r \\frac { ( \\delta - \\delta _ 0 + r _ 1 - 1 ) ! } { r _ 1 ! ( r - r _ 1 ) ! ( \\delta - \\delta _ 0 + r _ 1 - r ) ! } \\binom { d - d _ 0 - 1 } { r _ 1 - 1 } ; \\\\ & = ( - 1 ) ^ { r _ 1 + 1 } \\binom { \\delta - \\delta _ 0 + r _ 1 - 1 } { r _ 1 - 1 } \\binom { d - d _ 0 - 1 } { r _ 1 - 1 } . \\end{align*}"}
+{"id": "121.png", "formula": "\\begin{align*} \\int _ { ( n , 1 ) } h ( L ) d L = \\frac { 1 } { 2 } \\int _ { S ^ { n - 1 } } \\int _ { \\omega ^ { \\perp } } h ( \\hat { x } + L _ { \\omega } ) d \\mathcal H ^ { n - 1 } ( \\hat { x } ) d \\mathcal H ^ { n - 1 } ( \\omega ) . \\end{align*}"}
+{"id": "3995.png", "formula": "\\begin{align*} \\log \\det [ D ^ 2 u - A ( \\cdot , u , D u ) ] = B ( \\cdot , u , D u ) \\Omega , \\end{align*}"}
+{"id": "4620.png", "formula": "\\begin{align*} \\frac { 1 } { x _ 1 } + \\frac { 1 } { x _ 2 } = \\frac { 1 } { 6 } + \\frac { 1 } { 9 } = \\frac { 5 } { 1 8 } \\end{align*}"}
+{"id": "8646.png", "formula": "\\begin{align*} \\Psi ( x _ 1 , \\ldots , x _ N ) : = \\prod _ { i = 2 } ^ N F _ i ( x _ 1 , \\ldots , x _ i ) , \\end{align*}"}
+{"id": "2528.png", "formula": "\\begin{align*} \\Phi _ { x , y } ( \\tilde x , \\tilde y ) = \\varphi _ x ( \\tilde x ) \\delta ( \\tilde x - x - \\tilde y + y ) \\ , , \\end{align*}"}
+{"id": "7679.png", "formula": "\\begin{align*} H _ { \\mathfrak { m } } ^ { t } ( \\Omega _ { R } ^ { j } ( \\log f ) ) _ { - \\sum \\lambda _ { k } } = 0 . \\end{align*}"}
+{"id": "163.png", "formula": "\\begin{align*} B ( u ( \\theta ) ) ( \\hslash ) = u ( \\theta ) ( \\hslash ) = { \\mu } ( \\theta , \\hslash ) , \\ \\theta \\in J , \\ \\hslash \\in [ 0 , \\pi ] . \\end{align*}"}
+{"id": "1007.png", "formula": "\\begin{align*} \\S ( \\Z ) = \\bigcap _ { \\alpha \\geq 0 } \\ell _ { \\infty , \\alpha } ( \\Z ) . \\end{align*}"}
+{"id": "2918.png", "formula": "\\begin{align*} \\partial _ t Z = \\nu \\partial _ x ^ 2 Z + \\frac { \\lambda \\sqrt { D } } { \\nu } Z \\dot { W } ( t , x ) . \\end{align*}"}
+{"id": "938.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } | \\tau _ { E , \\alpha } ^ { \\prime } ( h ) | ^ { s } \\mathrm { d } h & \\le \\sum _ { k = 1 } ^ { M } \\int _ { I ( h _ { k } ) } C _ { h _ { k } } ^ { s } ( 1 + | h - h _ { 0 } | ^ { - \\frac { 1 } { \\sigma ^ { \\prime } } } ) ^ { s } \\mathrm { d } h \\le 2 ^ { s } \\overline { C } ^ { s } M \\int _ { - 1 } ^ { 1 } ( 1 + | h | ^ { \\frac { - s } { \\sigma ^ { \\prime } } } ) \\mathrm { d } h , \\end{align*}"}
+{"id": "7321.png", "formula": "\\begin{align*} m _ \\gamma ( x ) = \\frac { \\gamma ^ { \\alpha / 2 - 1 / 2 } } { u ( \\gamma ^ { \\alpha / 2 } ) } m ( \\gamma ^ { \\alpha / 2 } x ) , j _ \\gamma ( d x ) = \\frac { \\gamma } { v ( \\gamma ^ { \\alpha / 2 } ) } j ( d ( \\gamma ^ { \\alpha / 2 } x ) ) . \\end{align*}"}
+{"id": "3366.png", "formula": "\\begin{align*} N ^ 1 : = [ \\Phi , \\Phi ] + 2 \\dd \\eta \\otimes \\xi . \\end{align*}"}
+{"id": "7834.png", "formula": "\\begin{align*} r e ^ { i \\theta } = r \\cos \\left ( c \\frac { \\log r } { r } \\right ) \\pm i r \\sin \\left ( c \\frac { \\log r } { r } \\right ) \\sim r \\pm i c \\log r , r \\to \\infty . \\end{align*}"}
+{"id": "2396.png", "formula": "\\begin{align*} \\dot \\Phi _ 2 = J ^ { - 1 } C ( t ) \\Phi _ 2 , \\Phi _ 2 ( t _ 0 ) = I \\end{align*}"}
+{"id": "8526.png", "formula": "\\begin{align*} \\sum _ { x \\in X } a _ { x , y } = \\lambda \\mbox { f o r e a c h ~ $ y $ } . \\end{align*}"}
+{"id": "1839.png", "formula": "\\begin{align*} { \\mathrm { T r \\ , } } & \\left \\{ g ( X ^ * A X + Y ^ * B Y ) + g ( Y ^ * A Y + X ^ * B X ) \\right \\} \\\\ & \\le { \\mathrm { T r \\ , } } \\left \\{ X ^ * g ( A ) X + Y ^ * g ( B ) Y ) + Y ^ * g ( A ) Y + X ^ * g ( B ) X \\right \\} \\\\ & = { \\mathrm { T r \\ , } } \\left \\{ ( g ( A ) + g ( B ) ) ( X X ^ * + Y Y ^ * ) \\right \\} = { \\mathrm { T r \\ , } } \\left \\{ g ( A ) + g ( B ) \\right \\} \\end{align*}"}
+{"id": "4111.png", "formula": "\\begin{align*} 0 = \\int _ M v _ { h , K } \\triangle _ f v _ { h , K } e ^ { - f } d V _ g = \\int _ M - | \\nabla v _ { h , K } | ^ 2 e ^ { - f } d V _ g \\end{align*}"}
+{"id": "8480.png", "formula": "\\begin{align*} x _ { p ^ n } = x _ { r ^ n } \\left ( \\frac { d r } { d p } \\right ) ^ n + x _ { r ^ { n - 1 } } \\binom { n } { 2 } \\left ( \\frac { d r } { d p } \\right ) ^ { n - 2 } \\frac { d ^ 2 r } { d p ^ 2 } \\mod ( x _ { r ^ { n - 2 } } , \\ ; \\cdots , \\ ; x _ r ) . \\end{align*}"}
+{"id": "7629.png", "formula": "\\begin{align*} u _ n ( t ) = J _ 0 ( t ) + \\beta J _ 1 ^ n ( t ) + \\frac { \\alpha } { \\delta + 1 } J _ 2 ^ n ( t ) + J _ 3 ^ n ( t ) , \\ ; t \\in [ 0 , T ] \\end{align*}"}
+{"id": "7032.png", "formula": "\\begin{align*} k ^ { C _ 2 } _ \\diamond = \\dfrac { R ( C _ 2 ) [ v , \\mu , \\tau ] } { \\begin{matrix} \\tau ( \\sigma - 1 ) = v \\mu \\\\ \\mu ( \\sigma + 1 ) = 0 \\end{matrix} } \\end{align*}"}
+{"id": "5617.png", "formula": "\\begin{align*} 0 = \\langle R m \\vert k \\vert k l k m _ 3 \\rangle _ { 3 } = - R _ { 3 1 0 3 } + R _ { 0 1 0 1 } , \\end{align*}"}
+{"id": "3558.png", "formula": "\\begin{align*} z _ j ^ - \\Omega _ \\lambda = \\begin{cases} d _ { j } ^ - ( \\lambda ) \\prod _ { \\ell = j + 1 } ^ { n } ( \\lambda _ j - \\lambda _ \\ell - j + \\ell ) \\Omega _ { \\lambda - \\epsilon _ j } , & \\lambda - \\epsilon _ j \\in \\mathbb { P } \\ell ( \\lambda - \\epsilon _ j ) \\leq p , \\\\ 0 , . \\end{cases} \\end{align*}"}
+{"id": "3951.png", "formula": "\\begin{align*} g _ x ( \\hat { x } , \\hat { y } , h ) = \\beta e _ n . \\end{align*}"}
+{"id": "7316.png", "formula": "\\begin{align*} 1 - g _ { m _ n } + \\lambda G _ { m _ n } = 1 - ( \\varphi ^ d _ { m _ n } - c ^ d _ { m _ n } \\psi _ { m _ n } ) + \\lambda G _ { m _ n } = - \\Phi ^ d _ { m _ n } + c ^ d _ { m _ n } \\psi _ { m _ n } - \\sum _ { k = 1 } ^ { d } \\lambda ^ k G ^ k _ { m _ n } , \\end{align*}"}
+{"id": "5440.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d X _ t & = b ( t , X _ t , \\mathcal { L } _ { X _ t } , \\alpha _ t ) d t + \\sigma ( t , X _ t , \\mathcal { L } _ { X _ t } , \\alpha _ t ) d W _ t \\\\ X _ 0 & = \\xi , \\alpha _ 0 = \\zeta , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4332.png", "formula": "\\begin{align*} M = \\sum \\limits _ { w = 0 } ^ { n _ 1 } \\binom { n _ 1 } { w } M _ w . \\end{align*}"}
+{"id": "2080.png", "formula": "\\begin{align*} \\frac { \\partial \\Phi } { \\partial \\nu } ( \\tfrac { 1 } { 2 } , \\kappa , \\nu ) & = ( 1 - \\kappa ) \\ln ( \\tfrac { 1 } { 2 } ) ( \\tfrac { 1 } { 2 } ) ^ { 1 + \\nu } - \\ln ( \\tfrac { 1 + \\kappa } { 2 } ) ( \\tfrac { 1 + \\kappa } { 2 } ) ^ { 1 + \\nu } \\\\ & \\geq ( \\tfrac { 1 } { 2 } ) ^ 2 ( ( 1 - \\kappa ) \\ln ( \\tfrac { 1 } { 2 } ) - \\ln ( \\tfrac { 1 + \\kappa } { 2 } ) ( 1 + \\kappa ) ^ { 1 + \\nu } ) , \\end{align*}"}
+{"id": "6378.png", "formula": "\\begin{align*} & s _ { p , q } ^ 2 = 1 , & \\\\ & s _ { p , q } s _ { m , r } = s _ { m , r } s _ { p , q } , & [ p , q ] \\cap [ m , r ] = \\varnothing \\\\ & s _ { p , q } s _ { m , r } = s _ { p + q - r , p + q - m } s _ { p , q } , & [ p , q ] \\supset [ m , r ] \\end{align*}"}
+{"id": "8893.png", "formula": "\\begin{align*} \\mathcal { \\wp } = \\prod \\limits _ { j = 1 } ^ { n - 1 } \\left ( r - \\frac { n } { 2 } - \\nu + j \\right ) \\left ( r - \\frac { n } { 2 } + \\nu + j \\right ) . \\end{align*}"}
+{"id": "9032.png", "formula": "\\begin{align*} \\begin{cases} \\overline { V } ^ { ( n ) } : = \\overline { \\Psi } ( \\overline { V } ^ { n - 1 } ) , \\ , \\ , \\overline { V } ^ { 0 } = V ^ { ( 0 ) } , \\\\ \\underline { V } ^ { ( n ) } : = \\underline { \\Psi } ( \\underline { V } ^ { n - 1 } ) , \\ , \\ , \\underline { V } ^ { 0 } = V ^ { ( 0 ) } . \\end{cases} \\end{align*}"}
+{"id": "6252.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ { \\ell } } \\lesssim \\| u \\| _ { H ^ { \\ell _ 1 } } ^ { \\theta } \\| u \\| _ { H ^ { \\ell _ 2 } } ^ { 1 - \\theta } , \\quad \\quad \\ell = \\theta _ 1 \\ell _ 1 + ( 1 - \\theta ) \\ell _ 2 , \\theta _ 1 , \\theta _ 2 \\in ( 0 , 1 ) , \\end{align*}"}
+{"id": "8655.png", "formula": "\\begin{align*} m _ \\lambda = \\sum _ { ( i _ 1 , \\dots , i _ l ) \\in \\mathbb N ^ l } x ^ { \\lambda _ 1 } _ { i _ 1 } \\dots x ^ { \\lambda _ l } _ { i _ l } . \\end{align*}"}
+{"id": "5123.png", "formula": "\\begin{align*} D _ { f _ { 2 } } \\mathcal { I } _ { 2 } ( \\lambda , b , 0 , 0 ) ( h _ { 2 } ) ( w ) = \\sum _ { n = 0 } ^ { \\infty } ( n + 1 ) I _ { 1 } ( \\lambda b ) K _ { 1 } ( \\lambda ) b _ { n } e _ { n + 1 } ( w ) . \\end{align*}"}
+{"id": "7014.png", "formula": "\\begin{align*} W = \\alpha ( X + Y ) + N , \\end{align*}"}
+{"id": "7363.png", "formula": "\\begin{align*} \\zeta : = \\min \\left \\{ \\frac { 1 } { ( 2 d + 1 ) ^ 2 } , \\frac { \\varepsilon \\epsilon } { 4 d ^ 2 + \\varepsilon } \\right \\} > 0 . \\end{align*}"}
+{"id": "5846.png", "formula": "\\begin{align*} \\int _ \\Omega ( \\Phi '' , \\widehat { \\Phi } ) d x + \\int _ \\Omega \\langle \\nabla \\Phi , \\nabla \\widehat { \\Phi } \\rangle d x + \\int _ { \\Gamma } ( \\Phi , B \\widehat { \\Phi } ) d \\Gamma + \\int _ \\Omega ( \\Phi , A \\widehat { \\Phi } ) d x = 0 \\end{align*}"}
+{"id": "3635.png", "formula": "\\begin{align*} { \\rm d } s ^ 2 = { \\rm d } r ^ 2 + \\psi ^ 2 ( r ) \\ , { \\rm d } \\omega ^ 2 , \\end{align*}"}
+{"id": "6431.png", "formula": "\\begin{align*} u _ \\epsilon ( X _ k , Y _ { k } , t _ { k } ) & = u _ \\epsilon ( X _ k , Y _ { k - 1 } + { \\epsilon ^ 2 } X _ { k } / 2 , t _ { k } ) \\\\ & \\leq \\inf _ { \\tilde X \\in { B _ \\epsilon ( X _ { k - 1 } ) } } u _ \\epsilon ( \\tilde X , Y _ { k - 1 } + { \\epsilon ^ 2 } \\tilde X / 2 , t _ k ) + \\eta 2 ^ { - k } , \\end{align*}"}
+{"id": "6046.png", "formula": "\\begin{align*} f ( z ) - r _ n ( z ) = \\big ( 2 G _ { \\dot \\mu } + o ( 1 ) \\big ) \\frac { D _ { \\dot \\mu } ^ 2 ( z ) } { w ( z ) } \\left ( \\frac \\rho { \\varphi ( z ) } \\right ) ^ { 2 n } \\end{align*}"}
+{"id": "5124.png", "formula": "\\begin{align*} D _ { f _ { 2 } } \\mathcal { I } _ { 1 } ( \\lambda , b , 0 , 0 ) ( h _ { 2 } ) ( w ) = \\sum _ { n = 0 } ^ { \\infty } b ( n + 1 ) I _ { n + 1 } ( \\lambda b ) K _ { n + 1 } ( \\lambda ) b _ { n } e _ { n + 1 } ( w ) . \\end{align*}"}
+{"id": "5201.png", "formula": "\\begin{align*} p _ { 1 2 3 } = & b _ { 1 1 } b _ { 2 2 } b _ { 3 3 } = e _ { 1 2 3 } \\\\ p _ { 1 2 4 } = & b _ { 1 1 } b _ { 2 2 } b _ { 3 4 } = e _ { 1 2 4 } \\\\ p _ { 1 3 4 } = & b _ { 1 1 } \\left ( b _ { 2 3 } b _ { 3 4 } - b _ { 2 4 } b _ { 3 3 } \\right ) = e _ { 1 3 4 } \\end{align*}"}
+{"id": "261.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\rho } \\leqslant 0 , \\end{align*}"}
+{"id": "4259.png", "formula": "\\begin{align*} 1 = \\| f \\| ^ 2 _ { L ^ 2 ( \\R ^ N ) } & = \\sum _ { k \\geq 0 } \\Big ( \\int _ { \\R ^ 2 } | f _ k ( x _ \\perp ) | ^ 2 d x _ \\perp \\Big ) \\times \\Big ( \\int _ { \\R ^ { N - 2 } } | \\Phi _ k ( x _ \\intercal ) | ^ 2 d x _ \\intercal \\Big ) \\\\ & = \\sum _ { k \\geq 0 } \\| f _ k \\| ^ 2 _ { L ^ 2 ( \\R ^ 2 ) } . \\end{align*}"}
+{"id": "6553.png", "formula": "\\begin{align*} { I _ 1 } \\ ! = \\ ! \\ ! \\int _ 0 ^ \\infty \\ ! \\ ! \\ ! \\ ! { \\frac { { { x ^ { j + 1 } } { { \\left ( { x - { \\kappa ^ 2 } } \\right ) } ^ { { m _ { f k } } - 1 } } \\exp \\left ( { - u x { { \\left ( { 2 \\sigma _ k ^ 2 } \\right ) } ^ { - 1 } } } \\right ) } { \\rm d } x } { { { { \\left ( \\ ! { { m _ { f k } } x { C _ { 2 k } ^ { - 1 } } \\ ! - \\ ! { m _ { f k } } { \\kappa ^ 2 } { C _ { 2 k } ^ { - 1 } } \\ ! + \\ ! \\left ( { { m _ { s k } } \\ ! - \\ ! 1 } \\right ) { { \\bar z } _ k } } \\ ! \\right ) } ^ { { m _ { f k } } + { m _ { s k } } } } } } } . \\end{align*}"}
+{"id": "5549.png", "formula": "\\begin{align*} V ( x , k ) = \\frac { \\Gamma \\left ( \\frac { k } { 2 } \\right ) } { \\pi ^ k } \\sum _ { n = 1 } ^ \\infty \\frac { \\mu ( n ) n ^ { k - 1 } } { X _ n ^ { \\frac { k } { 2 } } } \\left \\{ { } _ 1 F _ { 1 } \\left ( \\frac { k } { 2 } ; \\frac { 1 } { 2 } ; - \\frac { 1 } { X _ n } \\right ) - 1 \\right \\} . \\end{align*}"}
+{"id": "3388.png", "formula": "\\begin{align*} { \\rm G e n } ( T ) : = \\{ M \\in \\mathcal { G } \\ | \\ \\exists \\textrm { a s e t } I \\textrm { a n d a n e p i m o r p h i s m } \\oplus _ { i \\in I } T \\to M \\to 0 \\} . \\end{align*}"}
+{"id": "5446.png", "formula": "\\begin{align*} W _ { \\rho } ( \\mu , \\nu ) : = \\inf _ { \\pi \\in \\mathcal C ( \\mu , \\nu ) } \\int _ { ( \\mathbb R ^ d \\times \\mathcal M ) \\times ( \\mathbb R ^ d \\times \\mathcal M ) } \\rho ( ( x , i ) , ( y , j ) ) \\pi ( d x \\times \\{ i \\} , d y \\times \\{ j \\} ) = \\inf E \\rho ( X , Y ) , \\end{align*}"}
+{"id": "6784.png", "formula": "\\begin{align*} \\frac { \\left | z \\left ( \\left ( \\frac { 1 } { 2 } - \\sigma \\right ) + j \\omega \\right ) \\right | } { \\left | z \\left ( \\left ( \\frac { 1 } { 2 } + \\sigma \\right ) - j \\omega \\right ) \\right | } = \\frac { \\left | j P ^ { \\frac { 1 } { 2 } } \\right | } { \\left | G \\left ( \\overline { \\chi } , P \\right ) \\right | } \\frac { \\left | \\left ( \\frac { 3 } { 2 } - \\sigma \\right ) + j \\omega \\right | } { \\left | \\left ( \\frac { 3 } { 2 } + \\sigma \\right ) - j \\omega \\right | } \\end{align*}"}
+{"id": "7790.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow 1 } \\Psi _ { a , b } [ n ] = \\Psi [ n ] ^ { 1 / a } . \\end{align*}"}
+{"id": "321.png", "formula": "\\begin{align*} L ( \\frac { 1 } { 2 } , \\phi \\otimes \\chi _ { 8 d } ) = \\kappa L ( \\frac { 1 } { 2 } , \\phi ' \\otimes \\chi _ { 8 d } ) \\end{align*}"}
+{"id": "4647.png", "formula": "\\begin{align*} k ^ { \\max } _ { i i } = \\lfloor \\ell _ { i i } / 2 \\rfloor , \\ \\ \\ \\ k ^ { \\max } _ { i j } = \\min \\{ \\ell _ { i j } , \\ell _ { j i } \\} \\ \\ \\ \\ i \\neq j , \\end{align*}"}
+{"id": "7491.png", "formula": "\\begin{align*} Z _ f ( s , \\chi ) = { \\left \\{ \\begin{array} { r l } \\dfrac { 1 - q ^ { - 1 } } { 1 - q ^ { - 1 - p s } } , \\ \\ \\ \\ & { \\rm i f } \\ \\chi ^ p = \\chi _ { { \\rm t r i v } } , \\\\ 0 , \\ \\ \\ \\ & { \\rm o t h e r w i s e } . \\end{array} \\right . } \\end{align*}"}
+{"id": "502.png", "formula": "\\begin{align*} { \\textstyle \\sum _ { k = \\widehat { k } _ 1 } ^ { \\nu } } \\sqrt { \\Phi ( x ^ { \\ell ( k ) } ) - \\Phi ( x ^ k ) } \\le \\sqrt { b ^ 2 / 2 } \\ , { \\textstyle \\sum _ { k = \\widehat { k } _ 1 } ^ { \\nu } } \\Xi _ k + \\sqrt { ( \\rho \\ ! + \\ ! 1 ) / 2 } \\ , { \\textstyle \\sum _ { k = \\widehat { k } _ 1 } ^ { \\nu } \\sum _ { j = k - \\widehat { k } _ 1 } ^ { k - 1 } } \\big \\| x ^ { j + 1 } \\ ! - \\ ! x ^ { j } \\big \\| . \\end{align*}"}
+{"id": "5634.png", "formula": "\\begin{align*} M : = \\prod _ { n \\ , = \\ , 0 } ^ \\infty F ( \\delta ^ { p ^ n } ) < \\infty \\end{align*}"}
+{"id": "5814.png", "formula": "\\begin{align*} ( A e _ r , e _ s ) = \\sum _ { q = 1 } ^ p \\beta _ { r q } { \\| e _ r \\| \\over \\| e _ q \\| } ( \\ ! ( e _ q , e _ s ) \\ ! ) = \\beta _ { r s } \\| e _ r \\| \\| e _ s \\| \\end{align*}"}
+{"id": "9121.png", "formula": "\\begin{align*} \\Phi _ { d } \\left ( x ^ { p } \\right ) = \\begin{cases} \\Phi _ { p d } ( x ) & p \\mid d \\\\ \\Phi _ { p d } ( x ) \\Phi _ { d } ( x ) & p \\nmid d . \\end{cases} \\end{align*}"}
+{"id": "3363.png", "formula": "\\begin{align*} \\eta = g \\xi , \\Phi ^ t g \\Phi = g - \\eta ^ t \\otimes \\eta . \\end{align*}"}
+{"id": "5401.png", "formula": "\\begin{align*} x = \\sum _ { n \\in \\N } \\frac { a _ n } { \\prod _ { i = 0 } ^ n \\beta _ i } . \\end{align*}"}
+{"id": "7950.png", "formula": "\\begin{align*} \\pi ( m _ 1 ) + \\pi ( d _ 1 ) = \\pi ( m _ 2 ) + \\pi ( d _ 2 ) , \\end{align*}"}
+{"id": "4607.png", "formula": "\\begin{align*} \\frac { 1 } { a _ n } \\leq \\frac { 1 } { b _ n } < \\frac { 1 } { q } - \\sum _ { i = 1 } ^ { n - 1 } \\frac { 1 } { a _ i } = \\frac { 1 } { a _ n - 1 } \\end{align*}"}
+{"id": "969.png", "formula": "\\begin{align*} W ( X , E ) : = W ( V , K ) , \\mbox { w h e r e } V : = H ^ 0 ( X , E ) ^ \\vee \\mbox { a n d } K = \\ker ( d ) ^ \\perp = \\mathrm { I m } ( d ) ^ \\vee \\subseteq \\bigwedge ^ 2 V . \\end{align*}"}
+{"id": "3840.png", "formula": "\\begin{align*} I _ 1 ( v ) \\sim & ( 1 - \\theta ^ 2 ) w _ 1 ^ 2 \\int _ 0 ^ 1 \\frac { x v ^ 2 } { 2 \\pi ( w _ 1 ^ 2 + ( 1 - \\theta ^ 2 ) v ^ 2 x ^ 2 ) ^ { 3 / 2 } } \\ , d x \\\\ = & w _ 1 \\int _ 0 ^ { v ( 1 - \\theta ^ 2 ) ^ { 1 / 2 } / w _ 1 } \\frac { x } { 2 \\pi ( x ^ 2 + 1 ) ^ { 3 / 2 } } \\ , d x \\to w _ 1 \\int _ 0 ^ \\infty \\frac { x } { 2 \\pi ( x ^ 2 + 1 ) ^ { 3 / 2 } } \\ , d x = w _ 1 / ( 2 \\pi ) . \\end{align*}"}
+{"id": "1233.png", "formula": "\\begin{align*} X _ 1 ( y , p ) : = \\{ x \\in X : - x \\cdot D f ( y ) = p \\} . \\end{align*}"}
+{"id": "511.png", "formula": "\\begin{align*} { \\textstyle \\sum _ { K _ 2 \\cup K _ { 3 1 } \\ni j = k } ^ { \\nu } } \\ , \\| x ^ { j + 1 } \\ ! - \\ ! x ^ { j } \\| \\le \\sqrt { 2 { a } ^ { - 1 } } { \\textstyle \\sum _ { K _ 2 \\cup K _ { 3 1 } \\ni j = k } ^ { \\nu } } \\sqrt { \\Phi ( x ^ { \\ell ( j + 1 ) } ) - \\Phi ( x ^ { j + 1 } ) } . \\end{align*}"}
+{"id": "5204.png", "formula": "\\begin{align*} c _ { 1 5 } & = c _ { 1 2 } \\cos ( d ) \\tanh ( 2 s ) \\\\ c _ { 1 4 } & = - c _ { 1 2 } \\sin ( d ) \\tanh ( 2 s ) \\ , . \\end{align*}"}
+{"id": "788.png", "formula": "\\begin{align*} \\mu = \\sum _ { j = 1 } ^ m \\alpha _ j \\mu _ { j } , \\end{align*}"}
+{"id": "1258.png", "formula": "\\begin{align*} x _ 0 & = \\dfrac { 1 + \\alpha } { 2 \\sqrt { \\alpha ( 1 + 1 6 \\alpha ( 1 - \\beta ) ^ 2 ) } } \\end{align*}"}
+{"id": "7639.png", "formula": "\\begin{align*} x \\in \\mathbb T , \\ \\ \\partial _ x U ( x ) = 0 \\ \\Rightarrow \\ \\alpha ( x ) = 0 , \\end{align*}"}
+{"id": "5856.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } \\psi '' - \\Delta \\psi + b ( \\psi ) = - e ^ { \\lambda h } ( e , A U ) & \\hbox { i n } ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu \\psi = e ^ { \\lambda h } ( e , D H ) & \\hbox { o n } ( 0 , T ) \\times \\Gamma , \\\\ t = 0 : \\psi = 0 , \\psi ' = 0 & \\hbox { i n } \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "4224.png", "formula": "\\begin{align*} \\mathcal { I } ( t ) : = - \\int \\rho \\left ( \\frac { x - v t } { \\lambda ( t ) } \\right ) \\left ( \\partial _ x \\Lambda \\partial _ t \\Lambda + 4 \\partial _ x \\phi \\partial _ t \\phi \\sinh ^ 2 ( \\Lambda ) \\right ) d x , v \\in ( - 1 , 1 ) . \\end{align*}"}
+{"id": "8115.png", "formula": "\\begin{align*} \\psi ( y ) = y ^ { - \\frac { 3 } { 4 } } g \\Bigl ( \\frac { m ^ 2 y } { N } \\Bigr ) e \\Bigl ( \\frac { 2 \\sqrt { y p } } { c } - \\frac { T ^ 2 c } { 4 \\pi ^ 2 \\sqrt { y p } } \\Bigr ) \\widehat { k ^ * } \\Bigl ( \\frac { M T c } { 2 \\pi ^ 2 \\sqrt { y p } } \\Bigr ) . \\end{align*}"}
+{"id": "3640.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { \\ln \\mathcal { K } _ M ( x , y , t ) } { t } = - \\lambda _ 1 ( M ) , \\end{align*}"}
+{"id": "7015.png", "formula": "\\begin{align*} Z _ X & = \\sqrt { \\alpha } V + N _ X , \\\\ Z _ Y & = \\sqrt { \\alpha } V + N _ Y , \\end{align*}"}
+{"id": "3917.png", "formula": "\\begin{align*} v ( y ) : = \\sup _ { x \\in \\Omega } g ^ * ( x , y , u ( x ) ) , \\\\ v ^ * ( x ) : = \\sup _ { y \\in V } g ( x , y , v ( y ) ) . \\end{align*}"}
+{"id": "194.png", "formula": "\\begin{align*} \\frac { m _ { n } } { c _ { n + 1 } } \\leq \\frac { 3 h _ { n } } { h _ { n + 1 } / ( n + 1 ) ^ { 1 + \\epsilon } } \\leq \\frac { 3 h _ { n } ( n + 1 ) ^ { 1 + \\epsilon } } { r _ { n } h _ { n } } = \\frac { 3 ( n + 1 ) ^ { 1 + \\epsilon } } { r _ { n } } \\to 0 , \\end{align*}"}
+{"id": "4220.png", "formula": "\\begin{align*} \\begin{aligned} & ~ { } p ( t , x ) : = \\partial _ x \\Lambda \\partial _ t \\Lambda + 4 \\sinh ^ { 2 } ( \\Lambda ) \\partial _ x \\phi \\partial _ t \\phi , \\\\ & ~ { } e ( t , x ) : = \\dfrac { 1 } { 2 } ( ( \\partial _ x \\Lambda ) ^ { 2 } + ( \\partial _ t \\Lambda ) ^ { 2 } ) + 2 \\sinh ^ { 2 } ( \\Lambda ) ( ( \\partial _ x \\phi ) ^ { 2 } + ( \\partial _ t \\phi ) ^ { 2 } ) . \\end{aligned} \\end{align*}"}
+{"id": "5712.png", "formula": "\\begin{align*} \\mathbf { \\mathbf { e } } ^ { a } & = ( \\mathbf { L } ^ { - 1 } ) _ { b } ^ { a } \\mathbf { d x } ^ { b } , \\\\ \\mathbf { A } _ { b } ^ { a } & = ( \\mathbf { L } ^ { - 1 } ) _ { c } ^ { a } d \\mathbf { L } _ { b } ^ { c } , \\end{align*}"}
+{"id": "1205.png", "formula": "\\begin{align*} A \\sum _ { \\substack { d , m _ 1 , \\ldots , m _ d \\in \\N _ 0 \\\\ m _ 1 + 2 m _ 2 + \\cdots + d m _ d = k } } \\left ( \\sum _ { \\substack { \\ell _ 1 , \\ell _ 2 \\in \\N _ 0 \\\\ \\ell _ 1 + \\ell _ 2 = m _ 1 + \\cdots + m _ d } } \\left | \\frac { \\nabla _ { g _ C } ^ { \\ell _ 1 } } { d t ^ { \\ell _ 1 } } ( \\nabla _ { g _ C } ^ { \\ell _ 2 } \\Phi ^ t _ * ) \\Bigg | _ { t = \\log r } \\right | _ { g _ C } \\right ) \\prod _ { j = 1 } ^ d \\ ; \\biggl | \\nabla ^ j _ { g _ C } \\biggl ( \\log r , { \\rm I d } + \\frac { d r } { r } X \\biggr ) \\biggr | _ { g _ C } ^ { m _ j } , \\end{align*}"}
+{"id": "3431.png", "formula": "\\begin{align*} { \\mu _ { t } ( \\underline { \\gamma } _ i ) } = { \\mu _ { t } ( [ a ] ) } \\exp \\left ( ( t \\phi - P _ G ( t \\phi ) ) ( \\underline { \\gamma } _ i ) \\right ) , \\end{align*}"}
+{"id": "4990.png", "formula": "\\begin{align*} - h ^ 2 \\Delta + V & = \\chi _ < \\left ( - h ^ 2 \\Delta _ \\Omega + V _ h \\right ) \\chi _ < + \\chi _ > \\left ( - h ^ 2 \\Delta _ \\Omega + V _ h \\right ) \\chi _ > \\ , . \\end{align*}"}
+{"id": "4944.png", "formula": "\\begin{align*} \\widetilde { A } ^ { D V D } : = \\left ( \\begin{array} { c } \\delta _ n ^ + u _ { m , n } + \\delta _ m ^ { ( 2 ) } \\mu _ n v _ { m , n } + \\mu _ n ( u _ { m , n } ^ 2 + v _ { m , n } ^ 2 ) \\mu _ n v _ { m , n } \\\\ - \\delta _ n ^ + v _ { m , n } + \\delta _ m ^ { ( 2 ) } \\mu _ n u _ { m , n } + \\mu _ n ( u _ { m , n } ^ 2 + v _ { m , n } ^ 2 ) \\mu _ n u _ { m , n } \\end{array} \\right ) = \\mathbf { 0 } . \\end{align*}"}
+{"id": "6005.png", "formula": "\\begin{align*} \\Phi _ { | D ( v ) } \\simeq \\{ ( t ; [ x _ 1 , \\dots , x _ h : 0 : \\dots : 0 ] ; [ 0 : \\dots : 0 : t y _ { n - p + r + 1 } : \\dots : t y _ { n - p + r + h - 1 } : y _ { h } : \\dots : y _ { n - p + r - h - 1 } : 0 \\dots : 0 ] \\\\ \\in \\mathbb { C } \\times \\mathbb { P } ^ { n - 1 } \\times \\mathbb { P } ^ { n - 1 } \\left | t \\sum _ { a = r + 1 } ^ { h - 1 } x _ a y _ { n - p + a } + x _ h y _ h = 0 \\right \\} . \\end{align*}"}
+{"id": "570.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } \\sigma _ t = - \\frac { \\sigma _ t ^ 2 } { \\gamma } \\ , ( A ^ { \\rm T } A ) : C \\end{align*}"}
+{"id": "4254.png", "formula": "\\begin{align*} \\omega ^ 0 _ \\gamma = \\omega ^ 0 = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ N \\gamma _ j . \\end{align*}"}
+{"id": "5029.png", "formula": "\\begin{align*} g _ o = 8 \\ , e ^ 1 \\odot e ^ 1 + \\left ( e ^ 2 \\odot e ^ 2 + e ^ 3 \\odot e ^ 3 \\right ) + \\left ( e ^ 4 \\odot e ^ 4 + e ^ 5 \\odot e ^ 5 \\right ) , H _ o = 2 e ^ { 1 2 3 } + 2 e ^ { 1 4 5 } . \\end{align*}"}
+{"id": "3047.png", "formula": "\\begin{align*} \\pi _ { \\R } ( x , a ) = \\pi _ { \\R } ( x ' , a ' ) \\Leftrightarrow \\pi _ { \\R } ( x , 0 ) = \\pi _ { \\R } ( x ' , a ' - a ) . \\end{align*}"}
+{"id": "3533.png", "formula": "\\begin{align*} \\mathcal { I } _ { i j } ( s ) : = \\big \\{ I = ( i _ 1 , \\dots , i _ s ) : i = i _ 1 < i _ 2 < \\dots < i _ s = j \\big \\} , \\end{align*}"}
+{"id": "2273.png", "formula": "\\begin{align*} \\int ^ { T } _ { 0 } \\int _ { \\Omega } \\big ( & \\rho u \\cdot \\partial _ t \\phi + ( \\rho u \\otimes u ) : \\nabla _ x \\phi + \\rho ^ \\gamma { \\rm { d i v } } _ x \\phi - \\mathbb { S } ( \\nabla _ x u ) : \\nabla _ x \\phi \\\\ & + ( j - n u ) \\cdot \\phi \\big ) \\ , d x d t + \\int _ { \\Omega } \\rho _ 0 u _ 0 \\cdot \\phi ( 0 , x ) \\ , d x = 0 , \\end{align*}"}
+{"id": "2670.png", "formula": "\\begin{align*} \\varphi ( T ) = \\textsc { h e i g h t } ( T ) , \\end{align*}"}
+{"id": "4843.png", "formula": "\\begin{align*} I _ n ( t , s ; v ) & = \\int _ { \\Sigma _ n ( t , s ) } v ( t _ n ) \\otimes \\dots \\otimes v ( t _ 1 ) d \\L ^ n , \\\\ I _ n ^ \\flat ( t , s ; v ) & = \\int _ { \\Sigma _ n ^ \\flat ( t , s ) } v ( t _ n ) \\otimes \\dots \\otimes v ( t _ 1 ) d \\L ^ n . \\end{align*}"}
+{"id": "4321.png", "formula": "\\begin{align*} d _ i ( a _ 0 , a _ 1 , \\dots , a _ p ) = ( a _ 0 , \\dots , a _ i + a _ { i + 1 } , \\dots , a _ p ) , \\end{align*}"}
+{"id": "6273.png", "formula": "\\begin{align*} \\mathcal { M } = \\{ ( x , y , z ) \\in \\mathbb { R } ^ 3 : x ^ { 2 } + z ^ { 2 } = r ( y ) \\} , \\end{align*}"}
+{"id": "8199.png", "formula": "\\begin{align*} \\mathfrak { K } _ 1 ( t ) : = \\frac { t ^ { \\beta - 1 } } { \\Gamma ( \\beta ) } + \\sum _ { j = 1 } ^ m b _ j \\frac { t ^ { \\beta _ j - 1 } } { \\Gamma ( \\beta _ j ) } \\end{align*}"}
+{"id": "967.png", "formula": "\\begin{align*} [ \\mathcal { D } _ \\mathfrak { K o s z } ] = { 2 n - 4 \\choose n - 1 } [ \\mathcal { L } ] . \\end{align*}"}
+{"id": "6069.png", "formula": "\\begin{align*} \\int _ { t ( c ) } ^ { t ( d ) } \\frac { \\ell ( t ) \\dd t } { ( w _ { \\Delta _ 2 } \\tilde w _ { \\Delta _ 2 } ) ( t ) } = \\int _ c ^ d \\frac { \\ell _ * ( x ) \\dd x } { ( w _ { \\Delta _ 1 } \\tilde w _ { \\Delta _ 1 } ) ( x ) } , \\end{align*}"}
+{"id": "5604.png", "formula": "\\begin{align*} 0 = & \\langle \\nabla \\nabla S \\vert k \\vert m _ { 3 } m _ { 2 } k m _ { 2 } \\rangle _ { 3 } = - \\nabla _ { 3 } \\nabla _ { 2 } S _ { 2 3 } + \\nabla _ { 1 } \\nabla _ { 2 } S _ { 0 2 } = - \\nabla _ { 2 } S _ { 1 2 } \\nabla _ { 3 } k _ 3 , \\\\ 0 = & \\langle \\nabla \\nabla S \\vert k \\vert m _ { 2 } m _ { 2 } k m _ { 2 } \\rangle _ { 3 } = - \\nabla _ { 2 } \\nabla _ { 2 } S _ { 2 3 } = - \\nabla _ { 2 } S _ { 1 2 } \\nabla _ { 2 } k _ { 3 } , \\\\ 0 = & \\nabla _ { 0 } \\nabla _ { 2 } S _ { 2 3 } = \\nabla _ { 2 } S _ { 1 2 } \\nabla _ { 0 } k _ { 3 } , \\end{align*}"}
+{"id": "4790.png", "formula": "\\begin{align*} \\mathop { \\mathsf { H } } _ { Z \\sim \\mathcal { D } ( C ^ { ( \\alpha ) } ) } ( Z _ j ) & \\leq ( 1 - \\alpha ) \\log \\frac { 1 } { 1 - \\alpha } + ( q - 1 ) \\cdot \\frac { \\alpha } { q - 1 } \\log \\frac { q - 1 } { \\alpha } \\\\ & = h _ q ( \\alpha ) \\log ( q ) . \\end{align*}"}
+{"id": "2298.png", "formula": "\\begin{align*} \\frac { d u _ { i } ( t ) } { d t } = \\frac { \\hat { g } _ { i + 1 / 2 } - \\hat { g } _ { i - 1 / 2 } } { \\Delta x ^ 2 } . \\end{align*}"}
+{"id": "6529.png", "formula": "\\begin{align*} V ^ { j } _ L : = \\frac { n } { m } \\epsilon _ { s } ^ { 2 } \\frac { \\| \\sqrt { \\frac { n } { m } } \\tilde { X } ^ { j } _ L \\| _ 2 ^ 2 } { 2 ( 1 + \\frac { n } { m } \\epsilon _ { s } ^ { 2 } ) } - 2 ^ { L - 1 } \\log ( 1 + \\frac { n } { m } \\epsilon _ { s } ^ { 2 } ) \\leq \\frac { \\frac { n } { m } \\epsilon _ { s } ^ { 2 } } { 2 } \\left ( \\| \\sqrt { \\frac { n } { m } } \\tilde { X } ^ { j } _ L \\| _ 2 ^ 2 - 2 ^ L \\right ) \\end{align*}"}
+{"id": "508.png", "formula": "\\begin{align*} \\sum _ { K _ 2 \\cup K _ { 3 1 } \\ni j = k } ^ { \\infty } \\ ! \\sqrt { \\Phi ( x ^ { \\ell ( j + 1 ) } ) \\ ! - \\ ! \\Phi ( x ^ { j + 1 } ) } \\le \\left \\{ \\begin{array} { c l } \\widetilde { \\gamma } \\widetilde { \\tau } ^ k & { \\rm i f } \\ \\theta \\ ! \\in ( 0 , 1 / 2 ] , \\\\ \\ ! \\widetilde { \\gamma } k ^ { \\frac { 1 - \\theta } { 1 - 2 \\theta } } & { \\rm i f } \\ \\theta \\ ! \\in ( 1 / 2 , 1 ) , \\end{array} \\right . \\end{align*}"}
+{"id": "9101.png", "formula": "\\begin{align*} \\bar { \\mathsf { C } } _ 1 & : = p \\mathsf { C } ( g _ 1 \\mathsf { P } _ 1 ) + \\bar { p } \\mathsf { C } ( h _ 1 \\mathsf { P } _ 1 ) , \\\\ \\bar { \\mathsf { V } } _ 1 & : = p \\mathsf { V } ( g _ 1 \\mathsf { P } _ 1 ) + \\bar { p } \\mathsf { V } ( h _ 1 \\mathsf { P } _ 1 ) . \\end{align*}"}
+{"id": "5964.png", "formula": "\\begin{align*} & \\hbox { K e r } ( C _ p ) \\cap \\hbox { I m } ( D ) \\\\ = & \\hbox { K e r } ( C _ p ) \\cap \\{ \\hbox { K e r } ( D ^ T ) \\} ^ \\perp = \\hbox { K e r } ( C _ p ) \\cap V ^ \\perp = \\{ 0 \\} , \\end{align*}"}
+{"id": "3398.png", "formula": "\\begin{align*} \\omega _ { \\theta } = \\partial _ { z } v ^ r - \\partial _ { r } v ^ z . \\end{align*}"}
+{"id": "8747.png", "formula": "\\begin{align*} \\bar { \\mathcal { O } } = U _ 0 \\Sigma _ 0 ^ { 1 / 2 } \\bar { \\mathcal { C } } = \\Sigma _ 0 ^ { 1 / 2 } V _ 0 ^ * , \\end{align*}"}
+{"id": "723.png", "formula": "\\begin{align*} = \\frac { \\Gamma _ 1 ^ 2 } { 4 \\pi } \\Big ( r ( w ) d w + \\overline { r ( w ) } d \\bar { w } + \\frac { d u } { u } + \\frac { d \\bar { u } } { \\bar { u } } \\Big ) . \\end{align*}"}
+{"id": "4162.png", "formula": "\\begin{align*} \\lambda ( g , b ) & = \\inf \\left \\{ \\int _ M ( R - \\frac { 1 } { 1 2 } | H | ^ 2 + | \\nabla f | ^ 2 ) e ^ { - f } d V _ g : \\int _ M e ^ { - f } d V _ g = 1 \\right \\} \\\\ & = \\inf \\left \\{ \\int _ M ( R - \\frac { 1 } { 1 2 } | H | ^ 2 ) \\omega ^ 2 + 4 | \\nabla \\omega | ^ 2 d V _ g : \\| \\omega \\| _ { L ^ 2 } = 1 \\right \\} . \\end{align*}"}
+{"id": "317.png", "formula": "\\begin{align*} \\begin{aligned} \\psi _ { N ( F ) } ( \\prod _ { \\alpha \\in \\Phi ^ { + } } x _ { \\alpha } ( t _ { \\alpha } ) ) : = \\psi _ { F } ( \\sum _ { \\alpha \\in \\Delta } t _ { \\alpha } ) \\end{aligned} \\end{align*}"}
+{"id": "4730.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ { e + d } c _ j ( v _ { j } , w _ { j } ) + \\sum \\limits _ { k = 1 } ^ { e } d _ k ( x _ k , y _ k ) = 0 . \\end{align*}"}
+{"id": "4355.png", "formula": "\\begin{align*} ( s , U ; U \\cap V ) = ( t , V ; V \\cap U ) \\end{align*}"}
+{"id": "5875.png", "formula": "\\begin{align*} t \\geq T : u ^ { ( i ) } = u _ r ( t , x ) , n _ { r - 1 } + 1 \\leq i \\leq n _ r , ~ 1 \\leq r \\leq p , \\end{align*}"}
+{"id": "4741.png", "formula": "\\begin{align*} ( y ^ e s ) ^ k = ( x u - \\delta ) ^ k = \\sum _ { i = 0 } ^ k \\binom { k } { i } ( x u ) ^ i \\delta ^ { k - i } , \\end{align*}"}
+{"id": "2366.png", "formula": "\\begin{align*} E ( t ) \\dot { x } + E ( t ) K ( t ) x & = ( J ( t ) - R ( t ) ) x + ( G ( t ) - P ( t ) ) u , \\\\ y & = ( G ( t ) + P ( t ) ) ^ T x + ( S ( t ) - N ( t ) ) u , \\end{align*}"}
+{"id": "5832.png", "formula": "\\begin{align*} \\| C _ p ( U ( t ) , U ' ( t ) ) \\| _ { ( V \\times H ) ^ { N - p } } = O ( ( 1 + t ) ^ { - \\delta } ) , t \\geqslant 0 , \\end{align*}"}
+{"id": "2009.png", "formula": "\\begin{align*} \\sum _ { \\phi \\neq 1 } N _ { i j } ( \\phi ) = k - 1 . \\end{align*}"}
+{"id": "5950.png", "formula": "\\begin{align*} & \\| ( E _ r , D \\widehat H ) \\| _ { H ^ { 2 \\alpha - 1 } ( \\Sigma ) } \\\\ = & \\| ( E _ r , B \\widehat { U } ) \\| _ { H ^ { 2 \\alpha - 1 } ( \\Sigma ) } \\leqslant c \\| \\widehat H \\| _ { L ^ 2 ( 0 , T ; ( L ^ 2 ( \\Gamma ) ) ^ M ) } , \\end{align*}"}
+{"id": "557.png", "formula": "\\begin{align*} \\Theta _ { n + 1 } = \\Theta _ n + K _ n \\left \\{ ( X _ { t _ { n + 1 } } ^ \\dagger - X _ { t _ n } ^ \\dagger ) - \\frac { 1 } { 2 } \\left ( \\Theta _ n + \\pi _ n [ \\theta ] \\right ) A X _ { t _ n } ^ \\dagger \\Delta t \\right \\} \\end{align*}"}
+{"id": "7644.png", "formula": "\\begin{align*} , \\ \\ \\ \\Psi _ j = \\varphi _ j + O ( e ^ { - c / h } ) L ^ 2 ( \\mathbb T ) . \\end{align*}"}
+{"id": "7445.png", "formula": "\\begin{align*} \\Omega = \\{ Z ^ { ( 1 ) } , \\dots , Z ^ { ( N ) } \\} \\end{align*}"}
+{"id": "1134.png", "formula": "\\begin{align*} D _ n : = \\sum _ { i = 1 } ^ n \\binom { n } { i } e ^ { n - i } \\mu ^ { i - 1 } \\delta ^ i \\in \\mathcal { O } . \\end{align*}"}
+{"id": "3703.png", "formula": "\\begin{align*} \\langle v _ 1 , v _ 2 \\rangle _ i : = v _ 1 ^ t J _ i v _ 2 \\end{align*}"}
+{"id": "315.png", "formula": "\\begin{align*} \\epsilon _ { \\mathrm { K a l } } | _ { E _ { 1 } ^ { \\times } } \\cdot \\epsilon _ { \\mathrm { H M } } \\cdot \\omega _ { \\mathrm { P r a } } | _ { E _ { 1 } ^ { \\times } } = \\zeta _ { K ^ { \\times } } | _ { E _ { 1 } ^ { \\times } } = \\omega _ { K / E _ 1 } . \\end{align*}"}
+{"id": "2453.png", "formula": "\\begin{align*} C ( \\pi \\otimes \\chi , t ) = q _ { \\pi \\otimes \\chi } \\prod _ { j = 1 } ^ { m } ( 3 + | i t + \\mu _ { \\pi \\otimes \\chi } ( j ) | ) , C ( \\pi \\otimes \\chi ) = C ( \\pi \\otimes \\chi , 0 ) . \\end{align*}"}
+{"id": "2826.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) = x _ 1 ^ 5 x _ 2 ^ 7 . \\end{align*}"}
+{"id": "9212.png", "formula": "\\begin{align*} \\begin{gathered} \\forall b \\in \\Gamma \\cap B ( a , r _ { a } ) , \\ \\forall 0 < r < r _ { a } , X _ { \\sigma } \\cap B ( b , r ) \\\\ A _ { \\pm } ( b , r ) = \\varphi ^ { - 1 } \\big ( \\{ - y _ { - } ^ { 2 } + y _ { + } ^ { 2 } < 0 \\} \\cap \\{ \\pm y _ { - } > 0 \\} \\big ) \\cap B ( b , r ) . \\end{gathered} \\end{align*}"}
+{"id": "8493.png", "formula": "\\begin{align*} l '' ( t _ 0 ) & = - \\frac { 1 } { 2 } \\int ^ 1 _ 0 \\big ( \\frac { 9 } { 2 } U _ { \\xi ^ 4 } ^ 2 + P _ 3 U _ { \\xi ^ 3 } ^ 2 + P _ 2 U _ { \\xi ^ 2 } ^ 2 + P _ 1 U _ { \\xi } ^ 2 + P _ 0 U ^ 2 \\big ) g d p , \\end{align*}"}
+{"id": "5597.png", "formula": "\\begin{align*} 0 = \\langle S \\vert k \\vert k m _ 2 \\rangle _ { 3 } = - S _ { 2 3 } , \\end{align*}"}
+{"id": "1609.png", "formula": "\\begin{align*} Y ^ n = \\sum _ { u = 1 } ^ K X _ u ^ n + Z ^ n \\end{align*}"}
+{"id": "7467.png", "formula": "\\begin{align*} F _ { ( M , P ) } ( v _ 1 , \\dots , v _ n ) = F _ { \\widehat { ( M , P ) } } ( v _ { a _ 1 } , \\dots , v _ { a _ { 2 k } } ) . \\end{align*}"}
+{"id": "6945.png", "formula": "\\begin{gather*} P _ { m , n } ^ { ( \\alpha , \\beta ) } ( x ) = \\frac { ( - 1 ) ^ m } { \\alpha + 1 + n - m } \\bigg ( \\frac { ( x - 1 ) ( 1 + \\alpha + \\beta + n - m ) } { 2 } P _ m ^ { ( - \\alpha - 1 , \\beta - 1 ) } ( x ) P _ { n - m - 1 } ^ { ( \\alpha + 2 , \\beta ) } ( x ) \\\\ \\hphantom { P _ { m , n } ^ { ( \\alpha , \\beta ) } ( x ) = } { } + ( \\alpha + 1 - m ) P _ m ^ { ( - \\alpha - 2 , \\beta ) } ( x ) P _ { n - m } ^ { ( \\alpha + 1 , \\beta - 1 ) } ( x ) \\bigg ) , \\end{gather*}"}
+{"id": "1416.png", "formula": "\\begin{align*} m _ { \\rho , x } = k _ { \\rho , x } - k _ { \\rho , x + 1 } . \\end{align*}"}
+{"id": "5840.png", "formula": "\\begin{align*} \\mathcal H _ 0 = L ^ 2 ( \\Omega ) , \\mathcal H _ 1 = H ^ 1 ( \\Omega ) , \\end{align*}"}
+{"id": "8416.png", "formula": "\\begin{align*} A \\subseteq B ~ ~ ~ ~ ~ ~ A \\cap B = A . \\end{align*}"}
+{"id": "8236.png", "formula": "\\begin{align*} I ( u ) : = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { N } } | \\nabla u | ^ 2 d x + \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { N } } V ( | x | ) f ( u ) ^ 2 d x - \\int _ { \\mathbb { R } ^ { N } } K ( | x | ) G ( f ( u ) ) \\ , d x . \\end{align*}"}
+{"id": "8946.png", "formula": "\\begin{align*} \\binom { a } { k } = \\frac { a ( a - 1 ) \\dots ( a - k + 1 ) } { k ! } = \\frac { ( - 1 ) ^ { k } ( - a ) _ { k } } { k ! } . \\end{align*}"}
+{"id": "2921.png", "formula": "\\begin{align*} \\mathcal { A } ^ \\varepsilon _ { s , t } ( \\varphi ) = \\int _ s ^ t \\int _ { \\mathbb { R } } u _ r ( \\iota _ \\varepsilon ( x ; \\cdot ) ) ^ 2 \\partial _ x \\varphi ( x ) d x d r . \\end{align*}"}
+{"id": "9002.png", "formula": "\\begin{align*} \\forall g _ { 0 } , g _ { 1 } \\in K \\ , \\forall x \\in G \\colon \\quad \\gamma ( g _ { 0 } , x ) = \\gamma ( g _ { 1 } , x ) \\ \\Longrightarrow \\ g _ { 0 } = g _ { 1 } . \\end{align*}"}
+{"id": "8556.png", "formula": "\\begin{align*} g ( v , t ) \\geq \\frac { \\binom { v } { t / 2 } - \\binom { v } { t / 2 - 1 } } { t ! } . \\end{align*}"}
+{"id": "2887.png", "formula": "\\begin{align*} \\sup \\limits _ { 0 < r < 1 } \\int _ { \\mathbb { T } ^ \\infty } \\log ^ + | F _ { [ r ] } | d m _ \\infty = \\lim _ { r \\rightarrow 1 } \\int _ { \\mathbb { T } ^ \\infty } \\log ^ + | F _ { [ r ] } | d m _ \\infty . \\end{align*}"}
+{"id": "9137.png", "formula": "\\begin{align*} M _ 2 : = & \\sum _ { k = 0 } ^ n { n \\choose k } ( k - 1 ) ( m + n - 1 ) ^ { n - k - 1 } = \\\\ = & n \\sum _ { k = 1 } ^ n { n - 1 \\choose k - 1 } ( m + n - 1 ) ^ { n - k - 1 } - ( m + n - 1 ) ^ { - 1 } \\sum _ { k = 0 } ^ n { n \\choose k } ( m + n - 1 ) ^ { n - k } = \\\\ = & n ( m + n - 1 ) ^ { - 1 } ( m + n ) ^ { n - 1 } - ( m + n - 1 ) ^ { - 1 } ( m + n ) ^ { n } \\\\ = & - m ( m + n - 1 ) ^ { - 1 } ( m + n ) ^ { n - 1 } . \\end{align*}"}
+{"id": "8225.png", "formula": "\\begin{align*} W = \\left [ \\begin{array} { c c c c c } 4 - 2 i & 2 + 3 i & - 4 + 4 i & - 3 - 4 i & 1 \\\\ 2 + 4 i & 3 i & 2 + 4 i & 3 i & - 5 + 7 i \\\\ \\end{array} \\right ] . \\end{align*}"}
+{"id": "2378.png", "formula": "\\begin{align*} \\alpha _ 1 \\phi _ 1 ( t _ 0 ) + \\cdots + \\alpha _ { d } \\phi _ { d } ( t _ 0 ) = 0 \\end{align*}"}
+{"id": "3697.png", "formula": "\\begin{align*} X _ i ( R ) : = \\{ u \\in V _ i ( R ) : Q _ i ( u ) = 0 \\} \\end{align*}"}
+{"id": "4745.png", "formula": "\\begin{align*} u = u _ j - \\sum _ { i < j } y ^ { f _ i } \\rho _ i u _ i . \\end{align*}"}
+{"id": "6060.png", "formula": "\\begin{align*} \\theta \\left ( \\vec u + \\vec j + \\mathsf B \\vec m \\right ) = \\exp \\bigg \\{ - \\pi \\mathrm { i } \\vec m ^ \\mathsf { T } \\mathsf B \\vec m - 2 \\pi \\mathrm { i } \\vec m ^ \\mathsf { T } \\vec u \\bigg \\} \\theta \\big ( \\vec u \\big ) , \\vec j , \\vec m \\in \\Z ^ g . \\end{align*}"}
+{"id": "8420.png", "formula": "\\begin{align*} \\kappa \\mu ( x ) & : = \\int \\kappa ( x , y ) \\ , d \\mu ( y ) , x \\in X , \\\\ \\kappa ( \\mu , \\nu ) & : = \\int \\kappa ( x , y ) \\ , d ( \\mu \\otimes \\nu ) ( x , y ) , \\end{align*}"}
+{"id": "1544.png", "formula": "\\begin{align*} \\hat \\sigma _ { a b \\dots k } ^ A : = \\sigma ^ A _ { I _ 1 I _ 2 \\dots I _ k } u ^ { I _ 1 } _ a u ^ { I _ 2 } _ b \\dots u ^ { I _ k } _ k \\end{align*}"}
+{"id": "277.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) & = \\alpha ( v / u ) - \\beta v w , \\psi _ 2 ( u , v , w ) = \\alpha ( v / u ) - \\beta u w , \\\\ \\psi _ 3 ( u , v , w ) & = \\alpha ( u / v ) + \\beta v w \\psi _ 4 ( v , w ) = \\alpha ( u / v ) + \\beta u w . \\end{align*}"}
+{"id": "5869.png", "formula": "\\begin{align*} U ( T ) = U ' ( T ) = 0 . \\end{align*}"}
+{"id": "1841.png", "formula": "\\begin{align*} \\rho = \\rho _ { 0 } ( t ) , \\boldsymbol v = \\boldsymbol v _ { 0 } = H ( t ) \\boldsymbol x . \\end{align*}"}
+{"id": "4275.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } ( \\nabla - i A ) ^ 2 \\phi + V _ \\gamma \\phi - | \\phi | ^ { p - 1 } \\phi + \\omega \\phi = 0 \\end{align*}"}
+{"id": "5339.png", "formula": "\\begin{align*} S _ \\varepsilon : = \\iota _ \\varepsilon \\circ T _ \\varepsilon ^ { - 1 } \\circ J _ \\varepsilon : L ^ 2 _ \\varepsilon ( \\Omega ) \\to L ^ 2 _ \\varepsilon ( \\Omega ) , \\end{align*}"}
+{"id": "2665.png", "formula": "\\begin{align*} \\Delta ^ { { \\rm Q } } _ m : = \\ Q _ m - Q _ { m - 2 } , \\end{align*}"}
+{"id": "8521.png", "formula": "\\begin{align*} U = \\{ 1 2 4 , 1 3 4 , 2 3 4 \\} , \\end{align*}"}
+{"id": "5596.png", "formula": "\\begin{align*} 0 = \\langle S \\vert k \\vert k m _ 3 \\rangle _ { 3 } = - S _ { 3 3 } + S _ { 0 1 } , \\end{align*}"}
+{"id": "5073.png", "formula": "\\begin{align*} 0 = \\sum _ { \\vec a \\in [ 0 , k ] ^ d } Q _ { j , \\vec a } ( n _ j ) f ( \\vec n - \\vec a ) = \\sum _ { \\vec a \\in [ 0 , k ] ^ d } Q _ { j , \\vec a } ( \\pi ( \\vec n ) ) f _ { \\vec a } ( \\vec n ) , \\end{align*}"}
+{"id": "4953.png", "formula": "\\begin{align*} \\widetilde { A } ^ { A V F } _ \\alpha : = \\left ( \\begin{array} { c } \\alpha \\delta _ n ^ + u _ { m , n } + \\delta _ m ^ { ( 2 ) } \\mu _ n v _ { m , n } + \\mu _ n ( v _ { m , n } ^ 2 + \\frac { 2 } 3 u _ { m , n } ^ 2 ) \\mu _ n v _ { m , n } + \\frac { 1 } 3 \\mu _ n ( u _ { m , n } ^ 2 v _ { m , n } ) \\\\ - \\alpha \\delta _ n ^ + v _ { m , n } + \\delta _ m ^ { ( 2 ) } \\mu _ n u _ { m , n } + \\mu _ n ( u _ { m , n } ^ 2 + \\frac { 2 } 3 v _ { m , n } ^ 2 ) \\mu _ n u _ { m , n } + \\frac { 1 } 3 \\mu _ n ( v _ { m , n } ^ 2 u _ { m , n } ) \\end{array} \\right ) = \\mathbf { 0 } , \\end{align*}"}
+{"id": "1511.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\C _ r \\left ( \\frac x { 2 } \\right ) & = - \\frac { ( r - 1 ) ! } { ( 2 \\pi i ) ^ { r - 1 } } \\sum _ { k = 0 } ^ { r - 1 } \\frac { ( \\pi i ) ^ k } { k ! } { \\rm L i } _ { r - k } ( - e ^ { - \\pi i x } ) x ^ k - \\frac { \\pi i } { r 2 ^ r } x ^ { r } \\\\ & \\qquad \\qquad - \\frac { ( r - 1 ) ! } { ( 2 \\pi i ) ^ { r - 1 } } ( 1 - 2 ^ { 1 - r } ) \\zeta ( r ) . \\end{aligned} \\end{align*}"}
+{"id": "222.png", "formula": "\\begin{align*} D _ { i } ( m ) ' : = \\{ g \\in D _ { i } \\mid g = 1 m \\} , \\end{align*}"}
+{"id": "5903.png", "formula": "\\begin{align*} t = T : ~ ~ w ( T ) = 0 . \\end{align*}"}
+{"id": "9115.png", "formula": "\\begin{align*} \\frac { d } { d \\rho } h _ { 1 2 , \\mathsf { P } } ( \\rho ) = \\frac { - 2 \\sqrt { h _ 1 h _ 2 } \\rho ^ 2 + 2 ( h _ 1 + h _ 2 ) \\rho - 2 \\sqrt { h _ 1 h _ 2 } } { \\left ( 1 - \\rho ^ 2 + \\mathsf { P } ( h _ 1 + h _ 2 - 2 \\rho \\sqrt { h _ 1 h _ 2 } ) \\right ) ^ 2 } \\end{align*}"}
+{"id": "452.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { m } a _ k ^ p \\leq \\sum _ { k = 1 } ^ { m } \\lambda _ k \\sum _ { j = 1 } ^ { n } a _ j ^ p = \\sum _ { k = 1 } ^ { d } \\lambda _ k , \\end{align*}"}
+{"id": "6597.png", "formula": "\\begin{align*} I _ 2 & = \\int ( 1 + | \\xi - \\eta | ^ 2 ) ^ { - N } ( 1 + | \\eta | ^ 2 ) ^ { - N _ 1 } d \\eta \\\\ & \\leq C ( 1 + | \\xi | ^ 2 ) ^ { - N } \\int ( 1 + | \\eta | ^ 2 ) ^ { N - N _ 1 } d \\eta \\\\ & = B _ N ( 1 + | \\xi | ^ 2 ) ^ { - N } , \\end{align*}"}
+{"id": "8817.png", "formula": "\\begin{align*} A _ { 1 1 } P _ 1 & = P _ 1 A _ { 1 1 } & A _ { 1 2 } P _ 2 = P _ 1 A _ { 1 2 } \\ , , \\\\ A _ { 2 1 } P _ 1 & = P _ 2 A _ { 2 1 } & A _ { 2 2 } P _ 2 = P _ 2 A _ { 2 2 } \\ , . \\end{align*}"}
+{"id": "1070.png", "formula": "\\begin{align*} \\mathbb { E } _ { P , Q } [ \\tilde { \\phi } ] & = \\mathbb { P } \\left ( | \\tilde { N } _ 0 / n - P _ 0 ( A ) | > | \\tilde { N } _ 0 / n - P _ 1 ( A ) | \\right ) \\\\ & = \\mathbb { P } \\left ( 2 \\{ \\widetilde { N } _ 0 / n - P ( A ) \\} < P _ 0 ( A ) + P _ 1 ( A ) - 2 P ( A ) \\right ) \\\\ & \\leq \\mathbb { P } \\left ( 2 \\{ \\widetilde { N } _ 0 / n - P ( A ) \\} < - \\mathrm { T V } ( P _ 0 , P _ 1 ) + 2 \\varepsilon \\right ) , \\end{align*}"}
+{"id": "3790.png", "formula": "\\begin{align*} R _ 4 & = \\{ \\iota ^ 2 \\} \\cup \\{ \\iota \\tilde \\alpha _ 0 = \\tilde \\alpha _ 0 \\iota \\} \\cup \\{ \\tilde \\alpha _ i \\iota = \\varepsilon _ i ^ 2 \\iota \\tilde \\alpha _ i \\mid 1 \\le i \\le 4 \\} \\cup \\{ \\varepsilon _ i \\iota \\varepsilon _ i \\iota \\mid 1 \\le i \\le 4 \\} \\end{align*}"}
+{"id": "6285.png", "formula": "\\begin{align*} \\psi \\circ \\iota ' \\circ \\phi = \\pi _ k \\circ \\iota \\end{align*}"}
+{"id": "6084.png", "formula": "\\begin{align*} G _ { \\lambda _ n } | D _ { \\lambda _ n \\pm } ^ 2 ( x ) | = \\lambda _ n ( x ) = B _ n ^ 2 ( x ) \\dot \\mu ( x ) m ( x ) / m _ n ( x ) \\end{align*}"}
+{"id": "8379.png", "formula": "\\begin{align*} [ B , \\epsilon ] ^ w = \\{ ( f , g ) : | f ( x ) - g ( x ) | < \\epsilon ( x ) x \\in B \\} \\ \\ \\ ( B \\in \\mathcal { B } , \\epsilon \\in C ^ + ( X ) ) . \\end{align*}"}
+{"id": "5062.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n Q _ j ( \\pi ( \\omega ) ) f _ j ( \\omega ) = 0 . \\end{align*}"}
+{"id": "7101.png", "formula": "\\begin{align*} \\underline { k } _ { - 2 m + n \\sigma } = \\left ( \\underline { R } \\overset { ( \\sigma - 1 ) ^ n } { \\longrightarrow } \\underline { J } ^ m \\right ) = \\circ \\end{align*}"}
+{"id": "6243.png", "formula": "\\begin{align*} \\Delta _ \\eta = \\partial _ { x _ 1 } ^ 2 + \\partial _ { x _ 2 } ^ 2 + \\eta ^ 2 \\partial _ { x _ 3 } ^ 2 . \\end{align*}"}
+{"id": "774.png", "formula": "\\begin{align*} \\Sigma _ A = \\bigsqcup _ { k = 0 } ^ { i - 1 } \\Sigma _ { A } ( i ) , \\end{align*}"}
+{"id": "562.png", "formula": "\\begin{align*} { \\rm d } \\Theta _ t = \\frac { \\sigma _ t } { \\gamma } \\left \\{ ( A X ^ \\dagger _ t ) ^ { \\rm T } \\circ { \\rm d } I _ t - \\frac { \\gamma } { 2 } \\mbox { t r } \\ , ( A ) \\ , { \\rm d } t \\right \\} . \\end{align*}"}
+{"id": "6362.png", "formula": "\\begin{align*} ( m '' / m ) ( x ) = f _ n '' / ( 1 + f _ n ) ( x ) + 2 f _ n ' / ( 1 + f _ n ) ( x ) \\cdot ( \\hat m ' / \\hat m ) ( x ) + ( \\hat m '' / \\hat m ) ( x ) \\ : \\ : \\ : \\mbox { o n } \\ : \\ : I _ n . \\end{align*}"}
+{"id": "7607.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\langle \\partial _ t v ( t ) , \\varphi \\rangle & = - \\langle \\mathfrak { A } _ 0 ( t ) , \\varphi \\rangle , \\\\ ( v ( 0 ) , \\varphi ) & = ( u _ 0 , \\varphi ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "3195.png", "formula": "\\begin{align*} A ( y ) : = \\mathrm { d i a g } ( a _ 1 ( y ) , a _ 2 ( y ) ) \\quad y \\in \\R ^ 2 , \\end{align*}"}
+{"id": "6257.png", "formula": "\\begin{align*} \\mathcal { B E } _ { r } = \\{ v \\in \\mathcal { E } : g ( v , v ) \\leq r ^ { 2 } \\} . \\end{align*}"}
+{"id": "1708.png", "formula": "\\begin{align*} C ( n ) = \\int _ { S ^ 1 } s ( \\mu _ m ) ( \\kappa ( \\theta ) ) \\rho ^ \\vee \\left ( \\mu _ { n - \\frac { k - 2 } { 2 } } \\right ) ( \\kappa ( \\theta ) ) d \\theta = \\left \\langle \\mu _ m , \\mu _ { n - \\frac { k - 2 } { 2 } } \\right \\rangle _ V . \\end{align*}"}
+{"id": "9100.png", "formula": "\\begin{align*} \\mathcal { S } ^ { ( n ) } ( \\mathsf { P } ) : = \\left \\{ x ^ n : \\| x ^ n \\| ^ 2 = n \\mathsf { P } \\right \\} . \\end{align*}"}
+{"id": "7243.png", "formula": "\\begin{align*} \\phi _ f ( u , z ) : = s ( \\partial B _ n \\cap H _ z ( u ) ^ - ) \\end{align*}"}
+{"id": "988.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { \\infty , \\mathcal { A } } & = \\sup \\lbrace \\Vert f ( x ) \\Vert _ { \\mathcal { A } } : \\ , x \\in X \\rbrace \\\\ & = \\sup \\lbrace \\Vert \\overline { f } ( \\delta _ { x } ) \\Vert _ { \\mathcal { A } } : \\ , x \\in X \\rbrace \\\\ & \\leq \\Vert \\overline { f } \\Vert _ { \\infty , \\mathcal { A } } . \\end{align*}"}
+{"id": "6600.png", "formula": "\\begin{align*} ( T p _ { \\phi _ 1 , \\phi _ 2 } ( a ) f , g ) _ { L ^ 2 ( \\mathbb { R } ^ { 2 n } ) } = ( a , \\overline { V _ { \\phi _ 1 } f } \\cdot V _ { \\phi _ 2 } g ) _ { L ^ 2 ( \\mathbb { R } ^ { 2 n } ) } \\end{align*}"}
+{"id": "2575.png", "formula": "\\begin{align*} & \\mathfrak { g } _ { 0 , \\epsilon } = ( \\mathfrak { g } ( 0 , \\epsilon ) + \\mathfrak { g } ( 0 , \\epsilon ^ { - 1 } ) \\cap \\mathfrak { g } , \\\\ & \\mathfrak { g } _ { \\alpha , \\epsilon } = ( \\mathfrak { g } ( \\alpha , \\epsilon ) + \\mathfrak { g } ( - \\alpha , \\epsilon ^ { - 1 } ) ) \\cap \\mathfrak { g } . \\end{align*}"}
+{"id": "3571.png", "formula": "\\begin{align*} ( z _ n ^ + ) ^ { \\lambda _ n } \\cdots ( z _ 1 ^ + ) ^ { \\lambda _ 1 } v _ 0 & = \\left ( \\prod _ { j = 1 } ^ n \\frac { ( - 1 ) ^ { \\frac { \\lambda _ j ( \\lambda _ j + 1 ) } { 2 } } } { \\lambda _ j ! j ^ { \\lambda _ j } } \\prod _ { k = 0 } ^ { \\lambda _ j - 1 } \\prod _ { \\ell = 1 } ^ { j - 1 } ( k - \\lambda _ \\ell - j + \\ell + 1 - [ k - \\lambda _ \\ell ] _ 2 ) \\right ) \\Omega _ \\lambda . \\end{align*}"}
+{"id": "2105.png", "formula": "\\begin{align*} E = E ' \\cup E '' , E ' = \\{ \\epsilon \\in E \\ , | \\ , \\overline \\epsilon = \\epsilon \\} , E '' = \\{ \\epsilon \\in E \\ , | \\ , \\overline \\epsilon \\ne \\epsilon \\} . \\end{align*}"}
+{"id": "1815.png", "formula": "\\begin{align*} X _ k = \\{ u \\in H ^ 1 _ T ( [ 0 , T ] , \\R ^ 2 \\setminus \\{ 0 \\} ) : ( u , 0 ) = k \\} , \\end{align*}"}
+{"id": "7510.png", "formula": "\\begin{align*} \\dbinom { k + r } { i } \\dbinom { k - i + r } { j - i } = & \\dfrac { ( k + r ) ! } { i ! ( k - i + r ) ! } \\cdot \\dfrac { ( k - i + r ) ! } { ( j - i ) ! ( k + r - j ) ! } \\\\ = & \\dfrac { ( k + r ) ! } { ( k + r - j ) ! } \\cdot \\dfrac { 1 } { i ! ( j - i ) ! } \\\\ = & \\dbinom { k + r } { j } \\cdot \\dfrac { j ! } { i ! ( j - i ) ! } \\\\ = & \\dbinom { k + r } { j } \\dbinom { j } { i } . \\end{align*}"}
+{"id": "1832.png", "formula": "\\begin{align*} K _ { X ' } + D ' = f ^ * ( K _ X + D ) \\quad \\mbox { a n d } L ' = f ^ * L . \\end{align*}"}
+{"id": "1083.png", "formula": "\\begin{align*} \\mathcal { J } _ 0 = \\{ j \\in \\mathcal { J } : P _ { k , \\varepsilon } ( A _ j ) > \\varepsilon + ( 1 - \\varepsilon ) ( 6 / M ) ^ k \\} . \\end{align*}"}
+{"id": "6620.png", "formula": "\\begin{align*} [ \\C { A } , f _ { \\epsilon } ( \\abs { x } ) ] \\phi ( x ) = \\sum _ { y \\sim x } [ f _ \\epsilon ( | y | ) - f _ \\epsilon ( \\abs { x } ) ] \\phi ( y ) = : \\sum _ { y \\sim x } \\alpha _ { x , y } \\phi ( y ) . \\end{align*}"}
+{"id": "7255.png", "formula": "\\begin{align*} u ( \\gamma ) ^ 2 v ( \\gamma ) \\sim N ( \\gamma ) ( \\gamma \\to \\infty ) \\lim _ { \\gamma \\to \\infty } \\frac { K ( \\gamma ) } { u ( \\gamma ) } = \\lim _ { \\gamma \\to \\infty } \\frac { L ( \\gamma ) } { v ( \\gamma ) } = 0 . \\end{align*}"}
+{"id": "6961.png", "formula": "\\begin{gather*} E _ n ( x ) = \\sum _ { j = 0 } ^ n \\frac { x ^ j } { j ! } . \\end{gather*}"}
+{"id": "7700.png", "formula": "\\begin{align*} Q ( \\textbf { s } ) f ^ { \\textbf { s } + \\textbf { 1 } } = b ( \\textbf { s } ) f ^ { \\textbf { s } } \\end{align*}"}
+{"id": "7293.png", "formula": "\\begin{align*} & ( - 1 ) ^ { k + 1 } G ^ { k + 1 } _ m ( x ) \\\\ = & \\int _ { 0 } ^ { x } ( - 1 ) ^ { k } z G ^ { k } _ m ( z ) d m ( z ) + x \\int _ { x } ^ { y } ( - 1 ) ^ { k } G ^ { k } _ m ( z ) d m ( z ) \\\\ & + x \\int _ { y } ^ { 1 } ( - 1 ) ^ { k } G ^ { k } _ m ( z ) d m ( z ) . \\end{align*}"}
+{"id": "3136.png", "formula": "\\begin{align*} \\int _ Y ( \\partial _ { 1 2 2 } ^ 3 w ) ( \\partial _ 1 w ) = - \\int _ Y ( \\partial _ { 1 2 } ^ 2 w ) ^ 2 \\neq 0 \\end{align*}"}
+{"id": "1771.png", "formula": "\\begin{align*} \\Delta ^ 0 ( \\Sigma ) \\ : = \\ \\Sigma \\ , \\Delta ^ { n + 1 } ( \\Sigma ) \\ : = \\ \\Delta ( \\Delta ^ n ( \\Sigma ) ) \\ , , \\end{align*}"}
+{"id": "7410.png", "formula": "\\begin{align*} q _ { \\alpha } = 1 , q _ { \\alpha ^ * } = 1 \\end{align*}"}
+{"id": "2443.png", "formula": "\\begin{align*} \\langle \\Psi , \\Xi \\rangle = \\int _ { \\Gamma \\backslash \\mathbb { H } } \\Psi ( x + i y ) \\overline { \\Xi ( x + i y ) } d \\mu , \\Psi , \\Xi \\in L ^ 2 ( \\Gamma \\backslash \\mathbb { H } , \\mu ) , \\end{align*}"}
+{"id": "1507.png", "formula": "\\begin{align*} \\zeta ( 3 ) = \\frac { 4 \\pi ^ 2 } { 2 1 } \\log \\left ( \\frac { e ^ { \\frac { 4 G } { \\pi } } \\C _ 3 \\left ( \\frac 1 4 \\right ) ^ { 1 6 } } { \\sqrt 2 } \\right ) . \\end{align*}"}
+{"id": "4077.png", "formula": "\\begin{align*} \\overline { f } & = \\begin{pmatrix} f _ { \\star } & 0 \\\\ 0 & ( f ^ * ) ^ { - 1 } \\end{pmatrix} : X + \\alpha \\longmapsto f _ { * } X + ( f ^ * ) ^ { - 1 } ( \\alpha ) , \\\\ e ^ B & = \\begin{pmatrix} I d & 0 \\\\ B & I d \\end{pmatrix} : X + \\alpha \\longmapsto X + \\alpha + i _ X B . \\end{align*}"}
+{"id": "1693.png", "formula": "\\begin{align*} \\delta s ( \\mu _ m ) = \\delta ( s ( \\mu _ m ) ) + m s ( \\mu _ m ) = \\sum _ { \\frac { 2 - k } { 2 } \\leq n \\leq \\frac { k - 2 } { 2 } } \\mu _ m ( P _ { n + \\frac { k - 2 } { 2 } } ) \\left ( \\delta ( f _ n ) + m f _ n \\right ) \\end{align*}"}
+{"id": "2946.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } _ n \\bigg [ \\sup _ { 0 \\le t \\le T } \\bigg | \\tilde { \\mathcal { B } } ^ n _ t ( \\varphi ) - \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\tau _ j \\mathcal { Q } ^ n _ \\rho ( \\ell ; s ) \\nabla ^ n \\varphi ^ n _ j ( s ) d s \\bigg | ^ 2 \\bigg ] \\le C \\bigg ( \\frac { \\ell } { n ^ 2 } + \\frac { T } { \\ell ^ { 2 } } \\bigg ) \\int _ 0 ^ T \\sum _ { j \\in \\mathbb { Z } } ( \\nabla ^ n \\varphi ^ n _ j ( t ) ) ^ 2 d t . \\end{aligned} \\end{align*}"}
+{"id": "5016.png", "formula": "\\begin{align*} H = \\begin{pmatrix} \\frac { i } { 2 } & 0 \\\\ 0 & - \\frac { i } { 2 } \\end{pmatrix} , E = \\begin{pmatrix} 0 & \\frac 1 { 2 \\sqrt { 2 } } \\\\ - \\frac 1 { 2 \\sqrt { 2 } } & 0 \\end{pmatrix} , V = \\begin{pmatrix} 0 & \\frac i { 2 \\sqrt { 2 } } \\\\ \\frac i { 2 \\sqrt { 2 } } & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "7380.png", "formula": "\\begin{align*} { } ^ L \\natural _ { \\chi ^ \\vee } = \\natural _ { \\chi } \\quad , \\end{align*}"}
+{"id": "7069.png", "formula": "\\begin{align*} k ^ { C _ 2 } _ \\diamond \\cong \\dfrac { R ( C _ 2 ) [ v , \\mu , \\tau ] } { \\begin{matrix} \\tau ( \\sigma - 1 ) = v \\mu \\\\ \\mu ( \\sigma + 1 ) = 0 \\end{matrix} } \\end{align*}"}
+{"id": "7168.png", "formula": "\\begin{align*} \\displaystyle { H ^ { \\rm { t r i g } } _ k = { \\tilde H } ^ { \\rm { t r i g } } _ k \\Big | _ { \\rm e q } \\ , . } \\end{align*}"}
+{"id": "735.png", "formula": "\\begin{align*} q ( z ) = \\frac { \\partial r ( z ) } { \\partial z } - \\frac { 1 } { 2 } r ( z ) ^ 2 . \\end{align*}"}
+{"id": "4119.png", "formula": "\\begin{align*} \\nabla _ X Y & = \\frac { 1 } { 2 } [ X , Y ] , \\\\ R ( X , Y ) Z & = - \\frac { 1 } { 4 } [ [ X , Y ] , Z ] , \\\\ K ( X , Y ) & = \\frac { 1 } { 4 } \\langle [ X , Y ] , [ X , Y ] \\rangle \\end{align*}"}
+{"id": "2408.png", "formula": "\\begin{align*} \\Delta ^ T = \\Delta , \\quad \\Sigma _ { 1 1 } ^ T = - \\Sigma _ { 1 1 } \\ . - \\dot \\Delta , \\Sigma _ { 2 1 } ^ T = - \\Sigma _ { 1 2 } \\ . , \\quad \\Sigma _ { 2 2 } ^ T = - \\Sigma _ { 2 2 } \\ . , A _ { 4 1 } ^ T = - A _ { 1 4 } \\ . - \\dot E _ { 1 4 } . \\end{align*}"}
+{"id": "8896.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { C } } \\mathcal { A } _ { m } ^ { \\nu } = \\frac { 2 } { n ! ( n - 1 ) ! } \\sum \\limits _ { p = 0 } ^ { n - 1 } \\gamma _ { p } ^ { ( \\nu , n ) } \\left ( m + \\nu + \\frac { n } { 2 } \\right ) ^ { 2 p + 1 } \\end{align*}"}
+{"id": "3899.png", "formula": "\\begin{align*} E ^ { i , j } = D _ { p _ j } Y ^ i \\end{align*}"}
+{"id": "64.png", "formula": "\\begin{align*} f ( i ) = \\begin{cases} - 1 , & x _ i = y _ i = 0 \\\\ 1 , & x _ i = 1 , y _ i = 0 \\\\ 2 , & x _ i = 0 , y _ i = 1 . \\end{cases} \\end{align*}"}
+{"id": "6337.png", "formula": "\\begin{align*} \\theta ( \\alpha _ { \\nu _ 0 } ( t _ c ) ) - \\theta ( q ) = \\Phi _ { \\nu _ 0 } ( r ( q ) , r ( \\alpha _ { \\nu _ 0 } ( t _ c ) ) ) = \\Phi _ { \\nu _ 0 } ( r ( q ) , h ( r ( q ) ) ) \\end{align*}"}
+{"id": "5758.png", "formula": "\\begin{align*} q : U \\times _ V T ^ * V = T ^ * V | _ U \\rightarrow T ^ * U . \\end{align*}"}
+{"id": "8385.png", "formula": "\\begin{align*} L ( X , \\mathcal { B } ) = & \\aleph _ 0 + \\min \\{ \\mathfrak { m } : \\mathcal { B } X \\mathcal { B } \\\\ & X \\leq \\mathfrak { m } \\} . \\end{align*}"}
+{"id": "1952.png", "formula": "\\begin{align*} f _ { j - 1 } \\left ( u , \\tilde { d } _ { N , { j ^ { * } } } , X \\right ) \\geq f _ { j - 1 } \\left ( u , \\frac { 1 } { N } \\sum _ { i = 1 } ^ N f _ j ( u , \\eta _ j , X _ i ) , X \\right ) . \\end{align*}"}
+{"id": "4530.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 1 2 } = \\frac { 1 } { 3 } + \\frac { 1 } { 4 } = \\frac { 7 } { 1 2 } < \\theta . \\end{align*}"}
+{"id": "4239.png", "formula": "\\begin{align*} \\Sigma : = \\left \\{ f \\in H ^ 1 ( \\R ^ N ) \\ : \\ | x | f \\in L ^ 2 ( \\R ^ N ) \\right \\} , \\end{align*}"}
+{"id": "419.png", "formula": "\\begin{align*} V : \\mathcal { X } \\ni \\sum _ { n = 1 } ^ { \\infty } f _ n ( x ) \\tau _ n \\mapsto \\sum _ { n = 1 } ^ { \\infty } f _ n ( x ) e _ n \\in \\ell ^ p ( \\mathbb { N } ) \\end{align*}"}
+{"id": "2028.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\bf C } ^ + ( r ) : = \\{ y \\in \\mathcal { I } ^ + ( r ) : ( y - S _ { \\mu } ) \\cap \\mathcal { I } ^ - ( r ) \\neq \\emptyset \\} , \\end{align*}"}
+{"id": "2071.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { m _ 3 ( t ) } { \\sqrt { m _ 1 ( t ) m _ 2 ( t ) } \\ , } = 0 . \\end{align*}"}
+{"id": "6066.png", "formula": "\\begin{align*} \\int _ { b _ k } ^ { a _ { k + 1 } } \\frac { \\ell _ j ( x ) \\dd x } { ( w \\tilde w ) ( x ) } = \\delta _ { k j } , k , j \\in \\{ 1 , \\ldots , g \\} . \\end{align*}"}
+{"id": "949.png", "formula": "\\begin{align*} \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { T r } _ C ( N ) ) \\ , = \\ , \\textrm { G } _ C \\textrm { - d i m } _ R ( M ) . \\end{align*}"}
+{"id": "1751.png", "formula": "\\begin{align*} \\rho ^ \\vee ( \\mu _ { \\underline { m } ' } ) ( \\kappa ( \\alpha , \\beta ) ) = \\mu _ { \\underline { m } ' } \\left ( \\left | \\begin{array} { c c } X & Y \\\\ - \\bar \\beta & \\bar \\alpha \\end{array} \\right | ^ { \\underline { k } - 2 } \\right ) = ( - \\bar \\beta ) ^ { \\frac { k _ { \\rm i d } - 2 } { 2 } - m _ { \\rm i d } ' } \\bar \\alpha ^ { \\frac { k _ { \\rm i d } - 2 } { 2 } + m _ { \\rm i d } ' } ( - \\beta ) ^ { \\frac { k _ { c } - 2 } { 2 } - m _ { c } ' } \\alpha ^ { \\frac { k _ c - 2 } { 2 } + m _ { c } ' } . \\end{align*}"}
+{"id": "1650.png", "formula": "\\begin{align*} x ^ { \\underline { k } } : = \\prod _ { \\sigma \\in \\Sigma _ F } \\prod _ { \\tilde \\sigma \\mid \\sigma } \\tilde \\sigma ( x ) ^ { k _ { \\tilde \\sigma } } . \\end{align*}"}
+{"id": "4573.png", "formula": "\\begin{align*} C ( x ) = \\{ y \\in G : y = g x g ^ { - 1 } g \\in G \\} . \\end{align*}"}
+{"id": "7555.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 1 ) = & ( 1 - q ^ { - 1 } ) q ^ { - ( \\omega + 1 ) - l s } Z _ { f _ 1 } ( s , \\chi ) \\\\ = & \\dfrac { F _ 1 ( q ^ { - s } ) } { ( 1 - q ^ { - 1 - s } ) ( 1 - q ^ { - p - l - p l s } ) } , \\end{align*}"}
+{"id": "8094.png", "formula": "\\begin{align*} \\frac { J _ { 2 i t } ( x ) - J _ { - 2 i t } ( x ) } { \\cosh ( \\pi t ) } = - \\frac { 2 i } { \\pi } \\tanh ( \\pi t ) \\int _ { - \\infty } ^ \\infty \\cos ( x \\cosh \\zeta ) e \\Bigl ( \\frac { t \\zeta } { \\pi } \\Bigr ) \\ , d \\zeta , \\end{align*}"}
+{"id": "3239.png", "formula": "\\begin{align*} | Z | | L ^ { - 1 } ( c _ { k - p _ 1 } ) \\cap X | \\ge \\sum _ { i = k - p _ 1 } ^ t | L ^ { - 1 } ( c _ i ) \\cap X | = \\sum _ { v \\in X } | L ( v ) \\cap Z | \\ge | X | ( p _ 1 + 1 ) . \\end{align*}"}
+{"id": "4500.png", "formula": "\\begin{align*} \\frac { 1 1 } { 2 4 } - \\left ( \\frac { 1 } { 3 } + \\frac { 1 } { 9 } \\right ) = \\frac { 1 } { 7 2 } \\end{align*}"}
+{"id": "8843.png", "formula": "\\begin{align*} \\Delta _ { \\nu } : = - \\left ( \\nabla _ { \\nu } \\right ) ^ { \\ast } \\nabla _ { \\nu } \\end{align*}"}
+{"id": "6759.png", "formula": "\\begin{align*} \\alpha _ n ( t ) = \\alpha _ n \\left ( \\frac { 1 } { 2 } + j \\left ( \\omega + j \\sigma \\right ) , \\rho ( t ) \\right ) = n \\sqrt { 2 \\pi } \\exp \\left ( - \\frac { 1 } { 2 } \\left [ 1 + \\gamma _ { \\alpha } \\right ] t \\right ) \\exp \\left ( \\frac { j } { 2 } \\omega _ { \\alpha } t \\right ) . \\end{align*}"}
+{"id": "2993.png", "formula": "\\begin{align*} s = f ( x , y ) = \\int _ 0 ^ y \\theta _ 4 ( x , y ' ) \\hbox { d } y ' \\hbox { m o d } [ q ] \\end{align*}"}
+{"id": "5004.png", "formula": "\\begin{align*} \\frac { \\frac { 1 - \\alpha } { 2 } - \\frac { 1 - \\alpha } { 2 q } } { \\frac { 1 - \\alpha } { 2 } + \\frac { 1 } { n } - \\frac { 1 } { 2 } } = \\frac { ( 1 - \\alpha ) \\frac { q - 1 } { q } } { 1 - \\alpha + \\frac { 2 } { n } - 1 } = \\frac { 1 - \\alpha } { q } = \\frac { q - \\frac { 2 } { n } } { q } \\in ( 0 , 1 ) , \\end{align*}"}
+{"id": "3843.png", "formula": "\\begin{align*} c ( p ) : = \\inf _ { \\left ( s , t \\right ) \\in \\mathbb { R } ^ { 2 } \\setminus \\left \\{ \\left ( 0 , 0 \\right ) \\right \\} } \\frac { \\left [ t ^ { 2 } + s ^ { 2 } + 2 s + 1 \\right ] ^ { \\frac { p } { 2 } } - 1 - p s } { \\left [ t ^ { 2 } + s ^ { 2 } \\right ] ^ { \\frac { p } { 2 } } } \\in \\left ( 0 , 1 \\right ] \\end{align*}"}
+{"id": "2960.png", "formula": "\\begin{align*} \\mathcal { R } ^ n _ t ( \\varphi ) = \\frac { 1 } { \\ell } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\bigg [ - \\sum _ { i = 1 } ^ { \\ell - 1 } ( W ^ 2 _ { j + i } - W ^ 2 _ { j + i - 1 } ) \\psi _ i - W _ j ( W _ j - W _ { j - 1 } ) + \\sigma _ n ^ 2 ( \\rho ) \\bigg ] \\varphi ^ n _ j ( s ) d s . \\end{align*}"}
+{"id": "1001.png", "formula": "\\begin{align*} f _ 2 \\big ( \\tfrac 1 3 , y \\big ) = - f _ 2 \\big ( \\tfrac 2 3 , y \\big ) = \\sin ( 2 \\pi y ) . \\end{align*}"}
+{"id": "5316.png", "formula": "\\begin{align*} \\Phi = \\sum _ { i = 1 } ^ n \\mathrm { A d } _ { T _ i } \\circ \\Phi _ i , \\end{align*}"}
+{"id": "9162.png", "formula": "\\begin{align*} \\Delta f ( x ) = \\sum _ { \\mu \\in \\hat { e } } ( f ( x + \\mu ) - f ( x ) ) = \\sum _ { \\mu \\in \\hat { e } } \\nabla ^ \\mu f ( x ) = \\frac 1 2 \\sum _ { \\mu \\in \\hat { e } } \\nabla ^ \\mu \\nabla ^ { - \\mu } f ( x ) , \\end{align*}"}
+{"id": "4365.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { i + 1 } \\frac { 1 } { a _ j } < \\theta \\leq \\sum _ { j = 1 } ^ { i } \\frac { 1 } { a _ j } + \\frac { 1 } { a _ { i + 1 } - 1 } . \\end{align*}"}
+{"id": "3191.png", "formula": "\\begin{align*} r ( y _ 1 , y _ 2 ) = r _ 1 ( y _ 1 + y _ 2 ) + r _ 2 ( y _ 1 - y _ 2 ) \\end{align*}"}
+{"id": "6846.png", "formula": "\\begin{align*} \\mathcal M = \\mathcal L \\ , \\tilde { \\mathcal M } . \\end{align*}"}
+{"id": "3754.png", "formula": "\\begin{align*} & \\frac { d } { d s } \\left ( - \\zeta _ v ( s ) ^ { - 1 } Z _ { r _ 2 } ( f , s ) + \\epsilon _ v ( 1 , \\psi _ v ) \\zeta _ v ( s + 1 ) ^ { - 1 } Z _ { r _ 1 } ( d _ 2 ( f ) , s + 1 ) \\right ) \\Big | _ { s = 0 } \\\\ & = ( \\log q _ v ) \\bigg ( ( e ( \\psi _ v ) + 1 ) \\tilde { c } _ { 2 , v } ( I ( f ) ) - \\tilde { a } _ { 2 , v } ( I ( f ) ) \\bigg ) . \\end{align*}"}
+{"id": "7462.png", "formula": "\\begin{align*} a ( W , M ' ) = a ( W \\setminus \\pi , M ' \\setminus \\{ i , i + 1 \\} ) \\end{align*}"}
+{"id": "5620.png", "formula": "\\begin{align*} ^ { C } D _ { t } ^ { \\alpha } f ( t ) = \\dfrac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int ^ { t } _ { t _ { 0 } } \\dfrac { f ^ { ' } ( x ) } { ( t - x ) ^ { \\alpha } } d x . \\end{align*}"}
+{"id": "7747.png", "formula": "\\begin{align*} \\psi ( s ) \\equiv \\int _ 0 ^ \\infty \\left ( 1 - e ^ { - s x } \\right ) \\ell ( x ) d x & = \\int _ 0 ^ \\infty \\int _ 0 ^ s e ^ { - x t } d t \\ , x \\ell ( x ) d x \\\\ & = \\int _ 0 ^ s \\int _ 0 ^ \\infty e ^ { - x t } \\rho ( x ) d x \\ , d t \\\\ & = \\int _ 0 ^ s \\widetilde \\rho ( t ) d t \\end{align*}"}
+{"id": "5914.png", "formula": "\\begin{align*} w = ( E , \\widetilde C _ { p - 1 } U ) , { \\mathcal L } \\theta = - ( E , \\widetilde C _ { p - 1 } A U ) , { \\mathcal R } \\theta = - ( E , \\widetilde C _ { p - 1 } B U ) . \\end{align*}"}
+{"id": "7062.png", "formula": "\\begin{align*} E _ { C _ 2 } ^ * ( X ) = \\widetilde { E } _ { C _ 2 } ^ { * + 2 \\dim \\xi } ( X ^ \\xi ) . \\end{align*}"}
+{"id": "7788.png", "formula": "\\begin{align*} \\Psi _ { a } [ n ] : = \\Psi _ { a , 0 } [ n ] . \\end{align*}"}
+{"id": "1493.png", "formula": "\\begin{align*} \\zeta ( 2 ) = 1 + \\frac { 1 } { 4 } + \\frac { 1 } { 9 } + \\cdots = \\frac { \\pi ^ { 2 } } { 6 } , \\end{align*}"}
+{"id": "7620.png", "formula": "\\begin{align*} ( \\Psi ' _ k ( h ) , h _ 1 ) = \\int _ { 0 } ^ { 1 } \\psi ' _ k ( h ( x ) ) h _ 1 ( x ) \\d x \\ \\ \\Psi '' _ k ( h ) ( h _ 1 , h _ 2 ) = \\int _ { 0 } ^ { 1 } \\psi '' _ k ( h ( x ) ) h _ 1 ( x ) h _ 1 ( x ) \\d x , \\end{align*}"}
+{"id": "8897.png", "formula": "\\begin{align*} \\mathcal { S } : = \\sum \\limits _ { p = 0 } ^ { n - 1 } \\gamma _ { p } ^ { ( \\nu , n ) } \\left ( 2 \\sum \\limits _ { m = 0 } ^ { + \\infty } \\left ( m + \\nu + \\frac { n } { 2 } \\right ) ^ { 2 p + 1 } e ^ { - \\left ( m + \\frac { n } { 2 } + \\nu \\right ) ^ { 2 } t } \\right ) . \\end{align*}"}
+{"id": "1308.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\| \\langle \\tau _ j , \\tau _ k \\rangle \\| ^ { 2 } \\geq \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , \\tau _ k \\rangle \\langle \\tau _ k , \\tau _ j \\rangle \\geq \\frac { n ^ 2 } { { d } } . \\end{align*}"}
+{"id": "4300.png", "formula": "\\begin{align*} \\mathcal { H } ( u ; t ) = \\mathcal { H } ( u ; 0 ) . \\end{align*}"}
+{"id": "191.png", "formula": "\\begin{align*} r _ { n } = 2 ^ { n ^ { \\alpha } } - 1 c _ { n } = \\Big { \\lceil } \\frac { h _ { n } } { n ^ { 1 + \\epsilon } } \\Big { \\rceil } . \\end{align*}"}
+{"id": "9226.png", "formula": "\\begin{align*} \\lll = \\lll _ { 0 } + \\lll _ { \\rm r e m } , \\end{align*}"}
+{"id": "5682.png", "formula": "\\begin{align*} L _ { c } ^ { a } L _ { d } ^ { b } \\eta _ { a b } = \\eta _ { c d } \\end{align*}"}
+{"id": "8193.png", "formula": "\\begin{align*} q _ 1 : = \\frac { b } { a } , q _ 2 : = \\frac { \\nu } { a } + \\mu , q _ 3 : = 1 + \\frac { \\nu - a } { b } , \\end{align*}"}
+{"id": "5807.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ p g u _ { 1 s } C _ p D e _ s + \\sum _ { s = 1 } ^ p u _ { 0 s } C _ p A e _ s = 0 , \\end{align*}"}
+{"id": "4490.png", "formula": "\\begin{align*} \\frac { 1 } { d ' _ 1 } + \\cdots + \\frac { 1 } { d ' _ n } = r . \\end{align*}"}
+{"id": "8716.png", "formula": "\\begin{align*} f _ k ( x ) = g _ { - k } ( x ^ { - 1 } ) = x ( x | a ) _ { k - 1 } , f _ { - k } ( x ) = g _ k ( x ^ { - 1 } ) = 1 / ( x | a ) _ { k } , f _ 0 ( x ) = g _ 0 ( x ) = 1 . \\end{align*}"}
+{"id": "8666.png", "formula": "\\begin{align*} \\psi ^ + ( u ) & = u ^ { - 1 } R ( u ) H ( u ) E ^ { \\perp } ( - u ) , \\\\ \\psi ^ - ( u ) & = R ^ { - 1 } ( u ) E ( - u ) H ^ { \\perp } ( u ) , \\end{align*}"}
+{"id": "4343.png", "formula": "\\begin{align*} [ \\vec { 0 } = \\vec { 1 } ] _ { M } \\cong \\phi _ { M } ( \\mathbf { 0 } / \\mathsf { C g } ^ { \\mathbf { 0 } } ( \\vec { 0 } , \\vec { 1 } ) ) ) \\cong \\phi _ { M } ( \\mathbf { 1 } ) \\cong H ( \\mathbf { 1 } ) \\cong 0 , \\end{align*}"}
+{"id": "5120.png", "formula": "\\begin{align*} D _ { f _ { 1 } } \\mathcal { S } _ { 1 } ( \\lambda , b , \\Omega , 0 ) h _ { 1 } ( w ) = \\sum _ { n = 0 } ^ { \\infty } ( n + 1 ) \\left ( \\Omega _ { n + 1 } ( \\lambda ) - \\Omega \\right ) a _ { n } e _ { n + 1 } ( w ) , \\end{align*}"}
+{"id": "8242.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { N } } V ( | x | ) f ( u ) ^ 2 d x = \\liminf _ n \\int _ { \\mathbb { R } ^ { N } } V ( | x | ) f ( u _ n ) ^ 2 d x . \\end{align*}"}
+{"id": "4001.png", "formula": "\\begin{align*} w ^ { i j } [ u _ { i j k } - A _ { i j , p _ l } u _ { l k } ] - B _ { p _ l } u _ { l k } = w ^ { i j } ( A _ { i j , k } + A _ { i j , u } u _ k ) + B _ k + B _ u u _ k . \\end{align*}"}
+{"id": "3799.png", "formula": "\\begin{align*} ( D ^ 2 \\sqrt { \\rho } ) _ { \\bar { x } } ( w , w ) & = \\int _ 0 ^ l \\chi ' ( s ) ^ 2 \\ , \\langle W ( s ) , W ( s ) \\rangle \\ , d s - \\int _ 0 ^ l \\chi ' ( s ) ^ 2 \\ , \\langle \\gamma ' ( s ) , W ( s ) \\rangle ^ 2 \\ , d s \\\\ & - \\int _ 0 ^ l \\chi ( s ) ^ 2 \\ , R ( \\gamma ' ( s ) , W ( s ) , \\gamma ' ( s ) , W ( s ) ) \\ , d s \\end{align*}"}
+{"id": "2201.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega ( n ) q ^ n = \\dfrac { E ( q ^ 2 ) ^ 9 } { E ( q ) ^ 6 } . \\end{align*}"}
+{"id": "218.png", "formula": "\\begin{align*} V : = \\bigcup _ { i = 1 } ^ { r } \\widetilde { f } _ { i } ^ { - 1 } ( V ' ) . \\end{align*}"}
+{"id": "5610.png", "formula": "\\begin{align*} 0 = & \\nabla _ { 1 } S _ { 2 3 } = ( - S _ { 3 3 } + S _ { 2 2 } ) \\nabla _ { 1 } ( m _ { 2 } ) _ { 3 } , \\\\ 0 = & \\nabla _ { 3 } S _ { 1 2 } = ( S _ { 2 2 } - S _ { 1 0 } ) \\nabla _ { 3 } ( m _ 2 ) _ { 1 } , \\\\ 0 = & \\nabla _ { 1 } S _ { 1 2 } = ( S _ { 2 2 } - S _ { 0 1 } ) \\nabla _ { 1 } ( m _ { 2 } ) _ { 1 } , \\\\ 0 = & \\nabla _ { 2 } S _ { 1 2 } = ( S _ { 2 2 } - S _ { 0 1 } ) \\nabla _ { 2 } ( m _ { 2 } ) _ { 1 } , \\end{align*}"}
+{"id": "72.png", "formula": "\\begin{align*} f _ 2 ( \\mathbf { z } ) = \\sum _ { i = 1 } ^ { n } { \\left ( 1 - \\sum _ { j \\in N [ j ] } { z _ j } \\right ) } \\end{align*}"}
+{"id": "935.png", "formula": "\\begin{align*} \\left | \\theta _ { \\alpha } \\cdot ( x _ { \\pm } ( h ) - x _ { \\pm } ( \\tilde { h } ) \\right | ^ { 2 } & = \\left | \\theta _ { \\alpha } \\cdot \\left ( \\left ( h - \\tilde { h } \\right ) \\theta _ { C } + \\left ( \\psi ( x ) - \\psi ( \\tilde { x } ) \\right ) \\theta _ { C } ^ { \\perp } \\right ) \\right | ^ { 2 } \\\\ & \\le | h - \\tilde { h } | ^ { 2 } + | \\psi ( x ) - \\psi ( \\tilde { x } ) | ^ { 2 } \\end{align*}"}
+{"id": "7011.png", "formula": "\\begin{align*} \\frac { \\partial \\ell } { \\partial \\nu } = \\log { \\frac { \\nu ( 1 - \\rho ) } { ( 2 \\nu - 1 ) \\Delta e ^ { R _ p } } } , \\end{align*}"}
+{"id": "3397.png", "formula": "\\begin{align*} \\vec { e } _ r = \\Big ( \\frac { x _ 1 } { r } , \\frac { x _ 2 } { r } , 0 \\Big ) , \\vec { e } _ { \\theta } = \\Big ( - \\frac { x _ 2 } { r } , \\frac { x _ 1 } { r } , 0 \\Big ) , \\vec { e } _ { z } = ( 0 , 0 , 1 ) . \\end{align*}"}
+{"id": "3171.png", "formula": "\\begin{align*} r _ A ( y ) : = r _ B ( y _ 1 , y _ 2 ) \\quad y = ( y _ 1 , y _ 2 , y _ 3 ) \\in \\R ^ 3 \\end{align*}"}
+{"id": "477.png", "formula": "\\begin{align*} | | v | | _ { L ^ \\phi ( A ) } = \\sup \\left \\{ \\left | \\int _ A \\ , v w \\ , d x \\right | : \\int _ A \\phi ^ * ( w ) \\leq 1 \\right \\} . \\end{align*}"}
+{"id": "5612.png", "formula": "\\begin{align*} A ( k , k ) = 0 , A ( k , l ) = - \\theta m _ 2 , A ( k , m _ 3 ) = 0 , \\end{align*}"}
+{"id": "1076.png", "formula": "\\begin{align*} Z _ i = \\begin{cases} Y _ i , & U _ i \\leq p , \\\\ 1 - Y _ i , & \\mbox { o t h e r w i s e } . \\end{cases} , \\end{align*}"}
+{"id": "4589.png", "formula": "\\begin{align*} \\prod _ { i = m + 1 } ^ { m + k } b _ i \\leq \\prod _ { i = m + 1 } ^ { m + k } a _ i \\end{align*}"}
+{"id": "3526.png", "formula": "\\begin{align*} F ( \\mathfrak { t } ) : = \\bigoplus _ { \\mu \\in \\mathfrak { h } ^ * } F _ \\mu ( \\mathfrak { t } ) . \\end{align*}"}
+{"id": "5211.png", "formula": "\\begin{align*} & f ^ 2 _ { 0 0 1 } = f ^ 3 _ { 1 0 0 } + f ^ 1 _ { 0 1 0 } = 0 \\\\ & f ^ 1 _ { 1 0 0 } f ^ 3 _ { 0 1 0 } - f ^ 1 _ { 0 1 0 } f ^ 3 _ { 1 0 0 } = 0 \\\\ & f ^ 2 _ { 1 0 0 } f ^ 3 _ { 0 0 1 } - f ^ 2 _ { 0 1 0 } f ^ 1 _ { 0 0 1 } = 0 \\ , . \\end{align*}"}
+{"id": "3180.png", "formula": "\\begin{align*} r ( y ) : = r _ B ( y _ 1 , y _ 2 ) = 1 + \\frac { 1 } { 4 } ( \\cos ( 2 \\pi y _ 1 ) - 2 \\sin ( 2 \\pi y _ 1 ) ) \\sin ( 2 \\pi y _ 2 ) \\end{align*}"}
+{"id": "3482.png", "formula": "\\begin{align*} \\begin{aligned} & y \\left [ n \\right ] = \\Big ( { \\sum \\nolimits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ l } } } \\Big ) s \\left [ { n - { n _ { \\max } } } \\right ] + \\\\ & \\sum \\nolimits _ { l = 1 } ^ L { \\sum \\nolimits _ { l ' \\ne l } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ { l ' } } s \\left [ { n - { n _ { \\max } } + { n _ { l ' } } - { n _ l } } \\right ] } } + z \\left [ n \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "5572.png", "formula": "\\begin{align*} A _ j ( x ' ) : = \\left ( B ^ k _ { 2 j } ( x ' ) \\setminus B ^ k _ { 2 ( j - 1 ) } ( x ' ) \\right ) \\times B ^ { n - k } _ 1 ( 0 ) . \\end{align*}"}
+{"id": "3310.png", "formula": "\\begin{align*} \\tau _ { \\lambda } ^ { ( 0 ) } [ W ] \\omega & = \\int _ { \\mathcal F _ 0 } \\Delta v ^ { ( 0 ) } _ { \\lambda } \\cdot u + \\int _ { \\mathcal F _ 0 } \\nabla v _ { \\lambda } ^ { ( 0 ) } : \\nabla u \\\\ & = - \\int _ { \\mathcal F _ 0 } w \\cdot u + \\int _ { \\mathcal F _ 0 } \\lambda v ^ { ( 0 ) } _ { \\lambda } \\cdot u - \\int _ { \\mathcal F _ 0 } v ^ { ( 0 ) } _ { \\lambda } \\cdot \\Delta u \\\\ & = - \\int _ { \\mathcal F _ 0 } w \\cdot u . \\end{align*}"}
+{"id": "8963.png", "formula": "\\begin{align*} \\int _ B | \\nabla u | ^ 2 d z = \\int _ { S ^ 1 } u \\partial _ r u \\ , d \\phi = \\int _ { S ^ 1 } u ( - \\Delta ) ^ { 1 / 2 } u \\ , d \\phi = \\int _ { S ^ 1 } | ( - \\Delta ) ^ { 1 / 4 } u | ^ 2 d \\phi , \\end{align*}"}
+{"id": "7589.png", "formula": "\\begin{align*} J _ 1 ( v ) ( t , x ) & : = \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { 1 } G ( t - s , x , y ) \\big ( ( 1 + \\gamma ) v ^ { \\delta + 1 } - \\gamma v - v ^ { 2 \\delta + 1 } \\big ) ( s , y ) \\d y \\d s \\\\ & = : ( ( 1 + \\gamma ) J _ { 1 1 } - \\gamma J _ { 1 2 } - J _ { 1 3 } ) v ( t , x ) , \\\\ J _ 2 ( v ) ( t , x ) & : = \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { 1 } \\frac { \\partial G } { \\partial y } ( t - s , x , y ) v ^ { \\delta + 1 } ( s , y ) \\d y \\d s , \\end{align*}"}
+{"id": "4557.png", "formula": "\\begin{align*} \\frac { a ' } { b ' } = \\frac { a } { b } - \\frac { 1 } { n } . \\end{align*}"}
+{"id": "96.png", "formula": "\\begin{align*} \\frac { \\eta _ { 1 , \\delta } } { \\eta _ { 0 , \\delta } } = \\frac { \\eta _ { 2 , \\delta } } { \\eta _ { 1 , \\delta } } = \\frac { \\eta _ { 3 , \\delta } } { \\eta _ { 2 , \\delta } } = \\dots = \\frac { \\eta _ { m - 1 , \\delta } } { \\eta _ { m - 2 , \\delta } } = \\gamma ^ \\delta \\frac { \\eta _ { 0 , \\delta } } { \\eta _ { m - 1 , \\delta } } . \\end{align*}"}
+{"id": "3738.png", "formula": "\\begin{align*} ( \\tilde { d } _ 1 ' , \\tilde { d } _ 1 ) ( b _ 1 ) = ( \\zeta ( 2 ) , \\zeta ( 2 ) ) , \\tilde { c } _ 1 ( b _ 1 ) = 1 = \\frac { b _ 1 ( 0 ) } { \\zeta ( 1 ) } . \\end{align*}"}
+{"id": "6464.png", "formula": "\\begin{align*} \\MoveEqLeft m \\wedge ( m \\wedge H ( m ) ) \\cdot \\delta E ( m ) = m \\wedge ( m \\wedge ( - \\delta E ( m ) ) \\cdot \\delta E ( m ) + h m \\wedge ( m \\wedge e _ 1 ) \\cdot \\delta E ( m ) \\\\ & = | m \\wedge \\delta E ( m ) | ^ 2 - h ( m \\wedge e _ 1 ) \\cdot ( m \\wedge \\delta E ( m ) ) \\\\ & = ( | \\delta E ( m ) | ^ 2 - | m \\cdot \\delta E ( m ) | ^ 2 ) - h ( m \\wedge e _ 1 ) \\cdot ( m \\wedge \\delta E ( m ) ) . \\end{align*}"}
+{"id": "838.png", "formula": "\\begin{align*} \\mu _ { n p + r } ( E ( t ) ) & = \\sum _ { | g _ 0 | = r } \\mu _ r ( g _ 0 ) \\tau _ { n p } ^ { t ( g _ 0 ) } ( g _ 1 : g _ 0 g _ 1 \\in E ( t ) ) \\\\ & = \\sum _ { | g _ 0 | = r } \\mu _ r ( g _ 0 ) \\left ( N ( t , \\sigma ) + O \\left ( \\frac { \\log n } { \\sqrt { n } } \\right ) \\right ) \\\\ & = N ( t , \\sigma ) + O \\left ( \\frac { \\log n } { \\sqrt { n } } \\right ) \\end{align*}"}
+{"id": "1733.png", "formula": "\\begin{align*} h ( g ) = f ( g ) \\left ( \\prod _ { \\sigma \\in \\Sigma } P _ \\sigma ( \\sigma ( c ) , \\sigma ( d ) ) \\sigma ( \\det ( g ) ) ^ { \\frac { 2 - k _ \\sigma } { 2 } } \\right ) , g = \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) , \\end{align*}"}
+{"id": "5927.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } z '' - \\Delta z = 0 & \\hbox { i n } ( T , + \\infty ) \\times \\Omega , \\\\ \\partial _ \\nu z = z = 0 & \\hbox { o n } ( T , + \\infty ) \\times \\Gamma . \\end{array} \\right . \\end{align*}"}
+{"id": "4723.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c [ \\xi _ { i , 1 } \\cdot ( m _ i , n _ i ) + \\xi _ { i , 2 } \\cdot ( r _ i , s _ i ) ] + \\sum _ { j = 1 } ^ d \\eta _ j \\cdot ( 0 , 0 ) \\end{align*}"}
+{"id": "392.png", "formula": "\\begin{align*} C ^ 2 _ 1 \\leq \\frac { \\widetilde { C } T ^ { p - 1 } _ * } { p L ^ { 2 ( p - 1 ) } _ * } + \\frac { { C } T ^ { p - 2 } _ * } { ( p - 1 ) L ^ { 2 ( p - 1 ) } _ * } \\textrm { a n d } C ^ 2 _ 2 = \\frac { 2 ^ { p - 1 } \\Gamma ( \\frac { 2 p - 1 } { 2 } ) } { p \\sqrt { \\pi } } \\frac { T ^ { p - 1 } _ * } { L ^ { 2 ( p - 1 ) } _ * } \\ , , \\end{align*}"}
+{"id": "4425.png", "formula": "\\begin{align*} \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 } = \\frac { 1 } { x _ 1 } + \\frac { 1 } { x _ 2 } \\end{align*}"}
+{"id": "8203.png", "formula": "\\begin{align*} \\Phi ( t , - \\lambda ) = 1 - \\lambda \\int _ 0 ^ t k ( t , s ) \\Phi ( s , - \\lambda ) d s , t > 0 . \\end{align*}"}
+{"id": "6247.png", "formula": "\\begin{align*} \\sigma _ 1 \\neq \\sigma _ 2 \\quad \\sigma _ 2 = \\sigma _ 3 , \\end{align*}"}
+{"id": "6224.png", "formula": "\\begin{align*} U _ n \\rightarrow \\begin{pmatrix} C _ p \\\\ E _ 1 ^ T \\\\ \\cdot \\\\ E _ p ^ T \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} 0 \\\\ u _ 1 \\\\ \\cdot \\\\ u _ p \\end{pmatrix} = : U \\end{align*}"}
+{"id": "3063.png", "formula": "\\begin{align*} \\gamma ( y ) : = \\frac { 1 } { C : A ( y ) } \\quad y \\in \\R ^ n \\end{align*}"}
+{"id": "697.png", "formula": "\\begin{align*} d s = e ^ { - h _ 0 ( w ) } | d w | . \\end{align*}"}
+{"id": "175.png", "formula": "\\begin{align*} ( n - 1 ) \\sum _ { \\substack { p \\leq n \\\\ p \\notin \\mathcal { S } } } \\dfrac { \\log p } { p - 1 } \\leq \\log ( 2 n ^ 2 + l ) \\pi ( n ) + \\dfrac { 1 } { \\lambda } \\sum _ { i = 1 } ^ { r } e _ i \\log p _ i + n \\log 4 \\end{align*}"}
+{"id": "5344.png", "formula": "\\begin{align*} \\operatorname { d i v } ( \\varepsilon \\nabla p ) + \\frac { \\sigma } { \\tau } \\ , p = 0 . \\end{align*}"}
+{"id": "5463.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { m \\geq n } \\mu ^ { n } \\circ \\phi _ { m } ^ { - 1 } = \\mu \\hbox { ~ w e a k l y ~ i n ~ } \\mathcal P ( \\mathbb R ^ d ) . \\end{align*}"}
+{"id": "3761.png", "formula": "\\begin{align*} \\mathrm { E i s } ( g ; f _ { \\chi _ s } ) : = \\sum _ { x \\in P _ \\ell ( F ) \\backslash \\mathrm { S O } _ { V _ \\ell } ( F ) } f _ { \\chi _ s } ( x g ) \\end{align*}"}
+{"id": "9068.png", "formula": "\\begin{align*} \\langle \\phi , \\psi \\rangle _ { S , \\xi , \\lambda } = \\int _ { K } \\langle \\phi ( k ) , \\psi ( k ) \\rangle _ { \\xi } \\ , d k \\qquad \\big ( \\phi , \\psi \\in C ^ { \\infty } ( S : \\xi : \\lambda ) \\big ) , \\end{align*}"}
+{"id": "4967.png", "formula": "\\begin{align*} \\begin{aligned} q _ 1 & : = a ^ { - 1 } \\xi ^ * ( q ) = a ^ { - 1 } \\Bigl ( \\xi ^ * ( c ^ p ) + \\sum _ { i \\ge 1 } ( \\xi ^ * ) _ i ( c ^ p ) { \\cdot } ( - ( a b ) ^ { p - 1 } c ) ^ i \\Bigr ) \\\\ & \\in w + \\widehat { \\xi } ( c ) + a ^ { p - 2 } b ^ { p - 1 } c S [ a b , c ] \\subset B . \\end{aligned} \\end{align*}"}
+{"id": "2197.png", "formula": "\\begin{align*} \\dfrac { ( \\phi \\ast ( H _ { k } G _ { n } ) ) ( z ) } { ( \\phi \\ast G _ { n } ) ( z ) } = \\dfrac { 2 z ( \\phi \\ast f _ { k } ) ' ( z ) } { ( \\phi \\ast F ) _ { n } ( z ) + \\overline { ( \\phi \\ast F ) _ { n } ( \\overline { z } ) } } \\end{align*}"}
+{"id": "6298.png", "formula": "\\begin{align*} \\pi _ i ^ { T v W } ( T ) = S & \\iff \\\\ & \\iff \\\\ & \\iff \\pi _ i ^ { r e v B S } ( S ) = s _ i ( S ) = T . \\end{align*}"}
+{"id": "2796.png", "formula": "\\begin{align*} G \\tau = \\{ \\sigma \\tau : \\sigma \\in G \\} = G \\end{align*}"}
+{"id": "7099.png", "formula": "\\begin{align*} \\langle \\Pi _ { V \\oplus W } , x ^ W \\rangle = \\pi _ V . \\end{align*}"}
+{"id": "3940.png", "formula": "\\begin{align*} & c ( 0 , 0 ) = 0 & & c _ { q } ( 0 , 0 ) = 0 & & & c _ { p } ( 0 , 0 ) = 0 \\\\ & c _ { p _ i , q _ j } ( 0 , 0 ) = \\delta _ { i j } & & c _ { q q , p } ( q , 0 ) = 0 & & & c _ { q , p p } ( 0 , p ) = 0 . \\end{align*}"}
+{"id": "8317.png", "formula": "\\begin{align*} K = & \\Big \\{ \\left ( \\lambda ( t ) ^ { \\frac { d } { 2 } - 1 } u ( t , \\lambda ( t ) \\cdot + x ( t ) ) , \\lambda ( t ) ^ { \\frac { d } { 2 } } u _ t ( t , \\lambda ( t ) \\cdot + x ( t ) \\right ) , \\\\ & t \\in \\left ( T ^ - ( u ) , T ^ + ( u ) \\right ) \\Big \\} \\end{align*}"}
+{"id": "7976.png", "formula": "\\begin{align*} b _ 1 \\equiv b _ 2 \\iff b _ 1 + A = b _ 2 + A , \\end{align*}"}
+{"id": "1182.png", "formula": "\\begin{align*} f _ i ( \\zeta _ 1 , \\ldots , \\zeta _ N ) + \\tau g _ i ( \\tau , \\zeta _ 1 , \\ldots , \\zeta _ N ) = 0 \\ ; \\ , ( i = 1 , \\ldots , I ) . \\end{align*}"}
+{"id": "4430.png", "formula": "\\begin{align*} \\theta \\in \\left ( \\frac { 1 7 } { 7 0 } , \\frac { 2 8 } { 1 1 5 } \\right ] \\subseteq \\left ( \\frac { 2 9 } { 1 2 0 } , \\frac { 2 8 } { 1 1 5 } \\right ] = J ( 5 , 2 4 ) . \\end{align*}"}
+{"id": "6087.png", "formula": "\\begin{align*} \\varkappa _ n \\gamma _ n \\gamma _ n ^ * = ( 1 + o ( 1 ) ) | ( T _ n \\psi _ n ) ( \\tau ) | | ( m / m _ n ) ( \\tau ) | | S _ { \\dot \\mu } ^ 2 ( \\tau ) | \\end{align*}"}
+{"id": "7586.png", "formula": "\\begin{align*} b ( u , u , | u | ^ { p - 2 } u ) = ( u ^ { \\delta } \\partial _ { \\xi } u , | u | ^ { p - 2 } u ) = 0 , \\end{align*}"}
+{"id": "1285.png", "formula": "\\begin{align*} q _ j ^ { ( t ) * } ( x ) \\eta _ j ^ * ( y ) \\zeta _ t ^ * ( z ) = \\sum _ { u = 0 } ^ { k - 1 } \\sum _ { v = 0 } ^ { \\ell - 1 } h _ { u v } ( x ) p _ v ^ { ( u ) } ( x ) \\eta _ v ( y ) \\zeta _ u ( z ) . \\end{align*}"}
+{"id": "3624.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } u = \\Delta _ { \\mathbb { R } ^ { n } } u + | u | ^ { p - 1 } u & \\hbox { i n } ~ \\mathbb { R } ^ { n } \\times ( 0 , T ) , \\\\ \\\\ u = u _ { 0 } \\in C ( \\mathbb { R } ^ { n } ) \\cap L ^ { \\infty } _ + ( \\mathbb { R } ^ { n } ) & \\hbox { i n } ~ \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{array} \\right . \\end{align*}"}
+{"id": "4192.png", "formula": "\\begin{align*} \\Lambda ( t , x ) : = \\lambda + \\tilde { \\Lambda } ( t , x ) , \\lambda \\neq 0 . \\end{align*}"}
+{"id": "4600.png", "formula": "\\begin{align*} a _ { n + 1 } = a _ n ^ 2 - a _ n + 1 = q \\prod _ { i = 1 } ^ n a _ k + 1 \\end{align*}"}
+{"id": "5980.png", "formula": "\\begin{align*} \\dim \\mathcal { W } _ 0 ' - \\dim p _ 1 ( \\mathcal { W } _ 0 ' ) & = \\dim p _ 1 ^ { - 1 } ( p _ 1 ( \\mu ) ) \\cap \\mathcal { W } _ 0 ' \\\\ & = \\dim \\pi _ { | Y } ^ { - 1 } \\left ( g ^ { - 1 } \\cdot ( f ' \\circ p _ 1 ) ( \\mu ) \\right ) \\\\ & = \\dim Y - \\dim \\pi ( Y ) , \\end{align*}"}
+{"id": "7819.png", "formula": "\\begin{align*} b ( i , j ) = ( 2 d ' _ 1 d ' _ 2 c + d ' _ 1 + d ' _ 2 - 2 d ' _ 1 j - 2 d ' _ 2 i ) / d ( ( 1 , 1 ) ) . \\end{align*}"}
+{"id": "6030.png", "formula": "\\begin{align*} \\dim X ( n - j + 1 , n - i + 1 ) & = n - j + 1 + n - ( n - i + 1 ) - 1 + \\Gamma _ { n - i + 1 - ( n - j + 1 ) , 0 } \\\\ & = n + i - j - 1 + \\Gamma _ { j - i , 0 } \\\\ & = \\dim X ( i , j ) , \\end{align*}"}
+{"id": "2261.png", "formula": "\\begin{align*} \\pi ^ B ( \\overline { r } ) : = ( M ^ { B } _ { \\infty } / \\frak { m } _ { \\infty } ) ^ { \\vee } . \\end{align*}"}
+{"id": "1159.png", "formula": "\\begin{align*} S T A ( \\mathbb { B } , \\mathbb { A } ) \\iff | B | = | A | \\wedge \\forall i < l , B ( i ) \\rightarrow A ( i ) \\wedge S w N U ( \\Omega , B ) . \\end{align*}"}
+{"id": "5603.png", "formula": "\\begin{align*} \\nabla _ { 1 } S _ { 1 0 } - \\nabla _ { 1 } S _ { 3 3 } = l ( S _ { 1 0 } - S _ { 3 3 } ) - S _ { 1 0 } \\nabla _ { 1 } l _ { 0 } - S _ { 1 0 } \\nabla _ { 1 } k _ { 1 } + - 2 S _ { 3 3 } \\nabla _ { 1 } ( m _ { 3 } ) _ { 3 } = 0 . \\end{align*}"}
+{"id": "5665.png", "formula": "\\begin{align*} \\lambda _ 1 = \\frac { 4 \\pi D } { | \\Omega | } \\sum _ { j = 1 } ^ N \\ell _ j . \\end{align*}"}
+{"id": "2492.png", "formula": "\\begin{align*} \\begin{gathered} \\inf _ { n \\geq 0 } \\alpha _ { 2 n + 1 } > 0 , \\ \\ \\inf _ { n \\geq 0 } \\alpha ' _ { 2 n } > 0 , \\\\ \\inf _ { n \\geq 0 } ( 1 + c _ n ^ 2 ) ^ { 1 / 2 } \\alpha _ { 2 n } > 0 , \\ \\ \\inf _ { n \\geq 0 } ( 1 + c _ n ^ 2 ) ^ { 1 / 2 } \\alpha ' _ { 2 n + 1 } > 0 . \\end{gathered} \\end{align*}"}
+{"id": "6994.png", "formula": "\\begin{align*} \\hat { X } ^ n = f _ { \\mathcal { D } _ x } ( S _ c , S _ x ) , \\hat { Y } ^ n = f _ { \\mathcal { D } _ y } ( S _ c , S _ y ) , \\end{align*}"}
+{"id": "5663.png", "formula": "\\begin{align*} \\phi = \\phi _ 0 + \\epsilon \\phi _ 1 + \\epsilon ^ 2 \\phi _ 2 + \\ldots , \\end{align*}"}
+{"id": "5693.png", "formula": "\\begin{align*} D \\theta ^ { a } & = \\tau ^ { a } + \\overline { \\tau } ^ { a } \\mathbf { , } \\\\ D \\tau ^ { a } - \\zeta ^ { a } - \\beta _ { b } ^ { a } \\theta ^ { b } & = 0 , \\\\ D \\zeta ^ { a } + \\beta _ { b } ^ { a } \\overline { \\tau } ^ { b } - \\overline { \\beta } _ { b } ^ { a } \\tau ^ { b } + \\gamma _ { b } ^ { a } \\theta ^ { b } & = 0 , \\end{align*}"}
+{"id": "4316.png", "formula": "\\begin{align*} h ^ 0 ( C , V / N ^ t ) - h ^ 0 ( C , U / N ^ t ) = h ^ 0 ( C , V ) - h ^ 0 ( C , U ) = \\delta . \\end{align*}"}
+{"id": "7068.png", "formula": "\\begin{align*} J ^ n v ^ { - n } \\oplus \\cdots \\oplus J v ^ { - 1 } \\oplus R ( C _ 2 ) [ v ] \\subset R ( C _ 2 ) [ v ^ { \\pm 1 } ] = K ^ { C _ 2 } _ * . \\end{align*}"}
+{"id": "4585.png", "formula": "\\begin{align*} \\frac { 1 } { \\prod _ { i = 1 } ^ n a _ i } = 1 - \\sum _ { i = 1 } ^ n \\frac { 1 } { a _ i } \\geq 1 - \\sum _ { i = 1 } ^ n \\frac { 1 } { b _ i } \\geq \\frac { 1 } { \\prod _ { i = 1 } ^ n b _ i } . \\end{align*}"}
+{"id": "4978.png", "formula": "\\begin{align*} \\begin{aligned} q _ 2 : = b ^ { - 1 } ( \\widetilde { q } _ 1 - \\lambda ( q , q _ 1 ) ( d F ) ^ { p - 1 } ) & \\in w ^ p + b ^ { - 1 } v + b ^ { p - 2 } R [ b x , y ] \\\\ & \\subset x ^ p + R [ b x , y ] + R [ x , y ] \\cap b ^ { - 1 } R [ b x , y ] . \\end{aligned} \\end{align*}"}
+{"id": "1858.png", "formula": "\\begin{align*} \\Vert \\mathbf { U } \\Vert _ { H ^ s } ^ { 2 } = \\sum _ { \\lvert \\alpha \\rvert \\leq s } \\langle \\mathnormal { D ^ { \\alpha } } \\mathbf { U } , \\mathnormal { D ^ { \\alpha } } \\mathbf { U } \\rangle . \\end{align*}"}
+{"id": "5673.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } W _ n ^ 2 & = \\lim _ { n \\to \\infty } \\frac { 1 } { n } ( \\sqrt { s } Z _ 1 + c \\sqrt { n } ) ^ 2 + s \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { i = 2 } ^ n Z _ i ^ 2 \\\\ & = c ^ 2 + s . \\end{align*}"}
+{"id": "5822.png", "formula": "\\begin{align*} ( \\widetilde U _ 0 , e _ r ) = ( U _ 0 , e _ r ) , ( \\widetilde U _ 1 , e _ r ) = ( U _ 1 , e _ r ) , r = 1 , \\cdots , p , \\end{align*}"}
+{"id": "1528.png", "formula": "\\begin{align*} \\nabla _ \\mu \\psi = \\nabla _ \\mu ^ { L C } \\psi + \\frac 1 2 K _ { \\mu \\nu } ^ \\pi \\frac 1 2 \\sigma _ \\pi { } ^ { \\nu } \\psi = \\nabla _ \\mu ^ { L C } \\psi + \\frac 1 8 T _ { \\tau \\mu \\nu } \\sigma ^ { \\tau \\nu } \\psi \\end{align*}"}
+{"id": "8713.png", "formula": "\\begin{align*} ( x | a ) _ n = ( x - a _ 1 ) ( x - a _ 2 ) \\dots ( x - a _ { n } ) , \\end{align*}"}
+{"id": "3660.png", "formula": "\\begin{align*} A u - f & \\le 0 , \\\\ u - g & \\le 0 , \\\\ \\langle A u - f , u - g \\rangle & = 0 . \\end{align*}"}
+{"id": "7633.png", "formula": "\\begin{align*} H : = \\big \\{ \\vec { x } \\in \\R ^ d : \\vec { f } ^ { \\ : \\mathrm { T } } \\vec { x } = 0 \\big \\} \\end{align*}"}
+{"id": "1822.png", "formula": "\\begin{align*} L _ { \\dot { u } } ( R u , R v ) = R L _ { \\dot { u } } ( u , v ) , \\end{align*}"}
+{"id": "959.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( \\Omega ^ { n - 1 } M , C ) \\ , = \\ , \\textrm { r . g r a d e } _ R ( M , C ) - n + 1 . \\end{align*}"}
+{"id": "944.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( M ' , C ) + \\textrm { r . g r a d e } _ R ( \\textrm { T r } _ C ( M ' ) , C ) \\ , = \\ , \\textrm { g r a d e } _ R ( \\textrm { E x t } _ { R } ^ n ( M ' , \\ , C ) ) + 1 . \\end{align*}"}
+{"id": "624.png", "formula": "\\begin{align*} \\beta ^ k _ { i , m } ( I ) = \\dim \\widetilde { H } _ { i - 1 } ( ( 1 , m ) _ { L _ I } ; k ) \\end{align*}"}
+{"id": "2032.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\bf C } ^ + ( r ) = \\mathcal { I } ^ + ( r ) & \\Longleftrightarrow \\left \\{ \\begin{array} { l l l } a _ 1 + D ' + r \\leq r \\\\ a _ { k + 1 } + D ' + r \\leq a _ k + r , \\forall k \\geq 1 \\\\ \\end{array} \\right . & \\Longleftrightarrow \\left \\{ \\begin{array} { l l l } a _ 1 \\leq - D ' \\\\ a _ { k + 1 } - a _ k \\leq - D ' , \\forall k \\geq 1 \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "3535.png", "formula": "\\begin{align*} \\mathbf { p } B _ j ^ + & = \\sum _ { i = 1 } ^ { j } \\sum _ { s = 1 } ^ { j - i + 1 } \\sum _ { I \\in \\mathcal { I } _ { i j } ( s ) } ( - 1 ) ^ { s - 1 } E ^ { e _ I } B _ i ^ + \\frac { \\prod _ { \\ell \\in I ^ \\complement } ( h _ \\ell - h _ j - 1 ) } { \\prod _ { \\ell = i } ^ { j - 1 } ( h _ \\ell - h _ j ) } , \\end{align*}"}
+{"id": "5353.png", "formula": "\\begin{align*} \\hat { \\Lambda } _ { F , s } [ \\varepsilon ] : = \\sum _ { \\substack { j _ 1 , \\dots , j _ s \\in F \\\\ j _ 1 < \\dots < j _ s } } ( \\lambda _ { j _ 1 } [ \\varepsilon ] + 1 ) \\cdots ( \\lambda _ { j _ s } [ \\varepsilon ] + 1 ) , \\end{align*}"}
+{"id": "7939.png", "formula": "\\begin{align*} 1 \\cdot 1 = 1 , 1 \\cdot 0 = 0 , 0 \\cdot 0 = 0 , 0 + 0 = 0 1 + 0 = 1 , 1 + 1 = \\mathbb { K } . \\end{align*}"}
+{"id": "1216.png", "formula": "\\begin{align*} \\tilde { E } ( \\gamma ( \\tilde { u } , h ) ) = \\tilde { E } _ { h _ 0 } ( \\gamma \\tilde { u } , \\iota ( \\gamma ) h ) = e ^ { \\tilde { f } ( ( \\gamma \\tilde { u } , h _ 0 ) , h _ 0 ^ { - 1 } \\iota ( \\gamma ) h ) - C _ { h _ 0 } ( \\gamma \\tilde { u } ) } . \\end{align*}"}
+{"id": "975.png", "formula": "\\begin{align*} \\| R _ m \\| ^ 2 = \\frac { 1 } { 4 \\pi } \\big [ 2 \\| R _ { m - 1 } \\| ^ 2 ( \\pi + 2 ) + 2 \\langle R _ 1 + . . . + R _ { m - 2 } , R _ { m - 1 } \\rangle \\big ] . \\end{align*}"}
+{"id": "1611.png", "formula": "\\begin{align*} Y = A U + Z \\end{align*}"}
+{"id": "8467.png", "formula": "\\begin{align*} ( \\gamma _ Q ) ^ A = \\gamma _ A . \\end{align*}"}
+{"id": "7758.png", "formula": "\\begin{align*} \\{ f \\star g \\} ( x ) & = \\int _ 0 ^ x f ( x - y ) g ( y ) d y = x \\int _ 0 ^ 1 f ( x ( 1 - t ) ) g ( x t ) d t \\end{align*}"}
+{"id": "7891.png", "formula": "\\begin{align*} f '' + \\left ( e ^ { P ( z ) } + e ^ { \\rho P ( z ) } + Q ( z ) \\right ) f = 0 . \\end{align*}"}
+{"id": "3190.png", "formula": "\\begin{align*} r ( y _ 1 , y _ 2 ) & : = 1 + \\frac { 1 } { 3 } \\sin ( 2 \\pi ( y _ 1 + y _ 2 ) ) + \\frac { 1 } { 3 } \\cos ( 2 \\pi ( y _ 1 - y _ 2 ) ) , \\\\ a ( y _ 1 , y _ 2 ) & : = 1 + \\frac { 1 } { 2 } \\sin ( 4 \\pi ( y _ 1 + y _ 2 ) ) \\end{align*}"}
+{"id": "130.png", "formula": "\\begin{align*} \\widetilde E _ { \\lambda , K } = \\left \\{ ( x , y ) \\in \\mathbb R ^ n \\times \\mathbb R ^ n : \\ x \\neq y , \\frac { | f ( x ) - f ( y ) | } { | | x - y | | _ K ^ { \\frac { n } { p } } } \\geq \\lambda \\right \\} . \\end{align*}"}
+{"id": "2687.png", "formula": "\\begin{align*} \\mathcal { H } ^ s ( K ) ^ { - 1 } & = \\sup \\left \\{ \\frac { \\mu ( U ) } { | U | ^ s } : U \\subset \\R ^ d \\right \\} \\\\ & = \\sup \\left \\{ \\frac { \\mu ( U ) } { | U | ^ s } : U \\subset \\R ^ d | U | \\geq \\Delta \\right \\} . \\end{align*}"}
+{"id": "876.png", "formula": "\\begin{align*} \\mathcal { Q } ^ { \\frac { 1 } { 2 } } * ( \\mathcal { X } ^ { t + 1 } - \\mathcal { X } ^ { \\star } ) & = \\mathcal { Q } ^ { \\frac { 1 } { 2 } } * ( \\mathcal { I } - \\mathcal { Q } ^ { - 1 } * \\mathcal { W } ) * \\mathcal { Q } ^ { - \\frac { 1 } { 2 } } * \\mathcal { Q } ^ { \\frac { 1 } { 2 } } * ( \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } ) = ( \\mathcal { I } - \\mathcal { Z } ) * \\mathcal { Q } ^ { \\frac { 1 } { 2 } } * ( \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } ) . \\end{align*}"}
+{"id": "1859.png", "formula": "\\begin{align*} \\bar { z } _ i \\hat { d } _ i + \\bar { s } _ i \\hat { e } _ i = \\bar { z } \\hat { d } _ i + \\bar { s } _ i \\hat { c } _ i = 0 \\end{align*}"}
+{"id": "7422.png", "formula": "\\begin{align*} Z \\in \\Omega _ n , \\ , \\widetilde Z \\in \\Omega _ { \\widetilde n } , \\ , \\alpha \\in { \\mathbb C } ^ { n \\times \\widetilde n } , \\ , \\alpha \\widetilde Z = Z \\alpha \\Rightarrow \\alpha f ( \\widetilde Z ) = f ( Z ) \\alpha . \\end{align*}"}
+{"id": "9147.png", "formula": "\\begin{align*} T ( \\eta | \\gamma ) \\ ; = \\ ; \\sum _ { f \\in \\mathfrak { F } _ { \\eta , \\gamma } } G _ C ( f ) , \\end{align*}"}
+{"id": "7666.png", "formula": "\\begin{align*} x _ { I } ^ { m } \\phi ( \\eta _ { I } ) - x _ { I } ^ { m } \\eta _ { I , - \\iota _ { E } ( \\omega ) } = \\sum _ { p \\neq - \\iota _ { E } ( \\omega ) } x _ { I } ^ { m } \\eta _ { I , p } \\in \\frac { 1 } { f } \\Omega _ { U } ^ { j } . \\end{align*}"}
+{"id": "4963.png", "formula": "\\begin{align*} B ^ { \\phi } : = \\{ b \\in B \\mid \\phi ( b ) = b \\} . \\end{align*}"}
+{"id": "816.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n ^ { 1 / 4 } } \\sum _ { k = 0 } ^ { n ^ { 1 / 4 } } \\sigma _ \\ast ^ k \\widehat { \\nu } ( E _ { n , t _ 1 , \\ldots , t _ d } ( x ) ) & = \\lim _ { n \\to \\infty } \\mu \\left ( y \\in \\Sigma _ A : S _ n y ( t _ 1 , \\ldots , t _ d ) \\in \\prod _ { i = 1 } ^ { d } ( - \\infty , x _ i ] \\right ) \\\\ & = F _ { t _ 1 , \\ldots , t _ d } ( x ) . \\end{align*}"}
+{"id": "6919.png", "formula": "\\begin{align*} x ( 0 ) = ( S _ 0 , E _ 0 , I _ 0 , Q _ 0 , R _ 0 , D _ 0 , P _ 0 , W _ 0 ) , \\end{align*}"}
+{"id": "6167.png", "formula": "\\begin{align*} ( u ( T ) , u ' ( T ) ) = ( ( x , C _ 1 U _ n ( T ) ) , ( x , C _ 1 U _ n ' ( T ) ) ) \\rightarrow ( 0 , 0 ) \\end{align*}"}
+{"id": "8555.png", "formula": "\\begin{align*} d _ { w } ( i ) : = \\big \\vert \\{ \\pi \\in X : \\pi ( i ) = w \\} \\big \\vert . \\end{align*}"}
+{"id": "791.png", "formula": "\\begin{align*} E = \\left \\{ \\xi \\in \\partial \\Gamma : \\lim _ { n \\to \\infty } \\frac { \\varphi ( \\xi _ n ) } { n } = \\Lambda \\right \\} . \\end{align*}"}
+{"id": "4183.png", "formula": "\\begin{align*} \\partial _ { t } ^ 2 \\alpha - \\partial _ { x } ^ 2 \\alpha = 0 . \\end{align*}"}
+{"id": "4079.png", "formula": "\\begin{align*} & \\langle \\mathcal { G } a , \\mathcal { G } b \\rangle = \\langle a , b \\rangle . \\\\ & \\langle \\mathcal { G } a , b \\rangle = \\langle a , \\mathcal { G } b \\rangle . \\\\ & \\langle \\mathcal { G } a , b \\rangle \\end{align*}"}
+{"id": "7433.png", "formula": "\\begin{align*} \\alpha _ { i _ 0 k _ 0 } Z _ { k _ 0 } = Z _ { i _ 0 } \\alpha _ { i _ 0 k _ 0 } \\end{align*}"}
+{"id": "792.png", "formula": "\\begin{align*} \\sigma _ \\ast ^ k \\widehat { \\nu } | _ { [ v ] } \\left \\{ x \\in \\Sigma _ B : \\lim _ { n \\to \\infty } \\frac { f _ n ( x ) } { n } = \\Lambda \\right \\} > 0 . \\end{align*}"}
+{"id": "3939.png", "formula": "\\begin{align*} \\overline { g } ( q , p , z ) & = \\overline { g } ( q , p , 0 ) - z \\\\ & \\quad \\quad + z [ a _ { i j } ( q , p ) q _ i p _ j + b _ { i j } ( q , p ) q _ j q _ i + c _ { i j } ( q , p ) p _ j p _ i ] + d ( q , p , z ) z ^ 2 . \\end{align*}"}
+{"id": "1837.png", "formula": "\\begin{align*} \\begin{bmatrix} A & 0 \\\\ 0 & B \\end{bmatrix} = U _ 1 \\begin{bmatrix} ( 1 - x ) A + x B & 0 \\\\ 0 & 0 \\end{bmatrix} U _ 1 ^ * + V _ 1 \\begin{bmatrix} 0 & 0 \\\\ 0 & x A + ( 1 - x ) B \\end{bmatrix} V _ 1 ^ * \\end{align*}"}
+{"id": "4658.png", "formula": "\\begin{align*} ( 2 m ) ^ { \\underline { \\ell } } = \\sum _ { p = \\lceil \\ell / 2 \\rceil } ^ { \\min \\{ m , \\ell \\} } C ( \\ell , p ; 2 ) m ^ { \\underline { p } } \\end{align*}"}
+{"id": "1356.png", "formula": "\\begin{align*} \\| ( ( x _ 1 , \\dots , x _ n ) , ( \\phi _ 1 , \\dots , \\phi _ n ) ) \\| \\coloneqq \\sum _ { j = 1 } ^ { n } ( \\| x _ j \\| + \\| \\phi _ j \\| ) \\end{align*}"}
+{"id": "3651.png", "formula": "\\begin{align*} & \\int \\limits _ { \\mathbb { H } ^ { n } } \\mathcal { K } ( x , z , T - t ) u ( z , t ) ~ { \\rm d } v _ { \\mathbb { H } ^ { n } } ( z ) = ~ \\int \\limits _ { \\mathbb { H } ^ { n } } \\mathcal { K } ( x , y , T ) \\ , u _ { 0 } ( y ) ~ { \\rm d } v _ { \\mathbb { H } ^ { n } } ( y ) \\\\ & + \\int \\limits _ { 0 } ^ { t } \\int \\limits _ { \\mathbb { H } ^ { n } } \\mathcal { K } ( x , y , T - s ) e ^ { \\mu s } \\ , u ( y , s ) \\big ( e ^ { \\beta u ^ { p } ( y , s ) } - 1 \\big ) ~ { \\rm d } v _ { \\mathbb { H } ^ { n } } ( y ) \\ , { \\rm d } s . \\end{align*}"}
+{"id": "4688.png", "formula": "\\begin{align*} G _ { I V } = \\left \\langle \\frac { 1 } { 2 } \\begin{pmatrix} 1 & 3 \\\\ 1 & - 1 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} \\right \\rangle . \\end{align*}"}
+{"id": "8401.png", "formula": "\\begin{align*} \\| \\partial _ { \\alpha } ( F _ { \\alpha , n _ k } ^ { - 1 } \\circ \\dots \\circ F _ { \\alpha , n _ 0 } ^ { - 1 } ) \\| _ { \\infty } \\leq C \\sum _ { j = 0 } ^ k \\| \\partial _ { \\alpha } F _ { \\alpha , n _ j } ^ { - 1 } \\| _ { \\infty } \\end{align*}"}
+{"id": "1888.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } d y ( x , t ) = y _ { x x } ( x , t ) d t + a ( x ) y ( x , t ) d t + \\sigma y ( x , t ) d B ( t ) , \\cr y _ x ( 0 , t ) = 0 , \\ ; t \\geq 0 , \\cr y _ x ( 1 , t ) = u ( t ) + w ( t ) , \\ ; t \\geq 0 , \\cr y ( x , 0 ) = y _ 0 ( x ) , \\ ; 0 \\leq x \\leq 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "2628.png", "formula": "\\begin{align*} \\dim \\sum _ { l \\in I } V _ { [ 4 ] \\setminus \\{ l \\} } ^ \\perp \\begin{cases} \\geq s | I | , \\\\ = ( r - 1 ) s - 1 , \\end{cases} \\end{align*}"}
+{"id": "5539.png", "formula": "\\begin{align*} \\mathcal { R } ( x ) = \\sum _ { \\rho } \\frac { 1 } { ( n _ \\rho - 1 ) ! } & \\lim _ { s \\rightarrow \\frac { k - \\rho } { 2 } } \\frac { { \\rm d } ^ { n _ \\rho - 1 } } { { \\rm d } s ^ { n _ \\rho - 1 } } \\Bigg \\{ \\left ( s - \\frac { k - \\rho } { 2 } \\right ) ^ { n _ \\rho } \\frac { \\Gamma ( s ) } { \\zeta ( k - 2 s ) x ^ s } \\Bigg \\} . \\end{align*}"}
+{"id": "6410.png", "formula": "\\begin{align*} \\eta & \\leq - ( m + p ) ( \\partial _ t - \\tilde X _ j \\cdot \\nabla _ Y ) \\phi ( \\tilde X _ j , \\tilde Y _ j , \\tilde t _ j ) + ( p - 2 ) \\lambda ( E ^ X ) + \\textrm { t r } ( E ^ X ) \\\\ & = - ( m + p ) ( \\partial _ { \\tilde t } - \\tilde X _ j \\cdot \\nabla _ { \\tilde Y } ) \\Psi ( P _ j ) + ( p - 2 ) \\lambda ( E ^ X ) + \\textrm { t r } ( E ^ X ) . \\end{align*}"}
+{"id": "368.png", "formula": "\\begin{align*} t ^ { ( n ) } _ m : = \\sum \\limits _ { k = 1 } ^ { m } { \\delta ^ { ( n ) } _ k } \\textrm { f o r a l l } m \\in \\mathbb { N } \\textrm { a n d } t ^ { ( n ) } _ 0 = 0 \\ , , \\end{align*}"}
+{"id": "288.png", "formula": "\\begin{align*} \\widehat { j } ( r ^ { - 1 } ( v ) v _ { \\widehat { S } } ( r ^ { - 1 } ( w ) ) r ^ { - 1 } ( v w ) ^ { - 1 } ) = t ( v , w ) . \\end{align*}"}
+{"id": "2210.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m - 1 } n + 5 ^ { 2 m - 1 } \\right ) } q ^ { n - 2 } & = q ^ { - 2 } \\gamma \\sum _ { i = 1 } ^ \\infty x _ { 2 m - 1 , i } \\xi ^ { i - 1 } . \\end{align*}"}
+{"id": "5848.png", "formula": "\\begin{align*} L _ t ( \\widehat { \\Phi } _ 0 , \\widehat { \\Phi } _ 1 ) = \\langle \\ ! \\langle ( \\widehat { U } _ 1 , - \\widehat { U } _ 0 ) , ( \\widehat { \\Phi } _ 0 , \\widehat { \\Phi } _ 1 ) \\rangle \\ ! \\rangle + \\int _ 0 ^ t \\int _ { \\Gamma } ( D H ( \\tau ) , \\Phi ( \\tau ) ) d x d t . \\end{align*}"}
+{"id": "765.png", "formula": "\\begin{align*} \\mathbb { W } _ { \\infty } : = \\begin{cases} \\left ( \\left ( \\mathbb { W } \\oplus \\mathbb { W } ^ { \\ell , \\left ( \\frac { - } { \\ell } \\right ) } \\right ) \\otimes _ { \\Lambda } \\mathcal { R } \\right ) [ f _ \\infty ] & \\textrm { i f $ r = 1 $ a n d $ \\psi _ \\ell $ i s t h e t r i v i a l c h a r a c t e r , } \\\\ \\left ( \\mathbb { W } \\otimes _ { \\Lambda } \\mathcal { R } \\right ) [ f _ \\infty ] & \\textrm { o t h e r w i s e . } \\end{cases} \\end{align*}"}
+{"id": "1091.png", "formula": "\\begin{align*} \\mathrm { T V } ( P _ { f _ 0 } , P _ { f _ 1 } ) = \\frac { 1 } { 2 } \\int _ { 0 } ^ 1 | f _ 0 ( x ) - f _ 1 ( x ) | \\ , \\mathrm { d } x = \\frac { \\gamma } { 2 } \\int _ { 0 } ^ 1 | \\psi _ { j k } ( x ) | \\ , \\mathrm { d } x = \\frac { c } { 2 } \\| \\psi \\| _ 1 2 ^ { - j ( \\beta + 1 / 2 ) } . \\end{align*}"}
+{"id": "4215.png", "formula": "\\begin{align*} \\mathcal { E } ( 0 ) = C _ 1 \\varepsilon ^ 2 . \\end{align*}"}
+{"id": "6530.png", "formula": "\\begin{align*} F ( r ) : = \\frac { \\int \\phi _ m ( x ) \\ , r ( x ) d x } { \\int \\underset { j = 1 } { \\overset { m } { \\Pi } } \\phi _ 1 ( x ^ j ) \\ , r ( x ) d x } \\in [ 0 , \\infty ] , \\end{align*}"}
+{"id": "8631.png", "formula": "\\begin{align*} U _ i = \\bigcap _ { n _ i \\leq n < n _ { i + 1 } } V _ n . \\end{align*}"}
+{"id": "4163.png", "formula": "\\begin{align*} \\lambda ( g , b ) = \\int _ M ( R - \\frac { 1 } { 1 2 } | H | ^ 2 ) \\omega _ { ( g , b ) } ^ 2 + 4 | \\nabla \\omega _ { ( g , b ) } | ^ 2 d V _ g . \\end{align*}"}
+{"id": "1099.png", "formula": "\\begin{align*} \\mathbb { E } ( \\| \\hat { f } - f \\| _ 2 ^ 2 ) & \\leq \\sum _ { j = 1 } ^ k \\left \\{ 2 \\varepsilon ^ 2 ( B _ 0 ^ 2 + \\theta _ j ^ 2 ) + \\frac { B ^ 2 } { n } \\right \\} + \\frac { r ^ 2 } { k ^ { 2 \\beta } } \\\\ & \\lesssim k \\varepsilon ^ 2 B _ 0 ^ 2 + \\varepsilon ^ 2 r ^ 2 + \\frac { B _ 0 ^ 2 k ^ 2 } { n \\alpha ^ 2 } + \\frac { r ^ 2 } { k ^ { 2 \\beta } } \\asymp k \\varepsilon ^ 2 + \\frac { k ^ 2 } { n \\alpha ^ 2 } + \\frac { 1 } { k ^ { 2 \\beta } } , \\end{align*}"}
+{"id": "4190.png", "formula": "\\begin{align*} \\begin{cases*} \\partial ^ 2 _ { t } \\Lambda - \\partial _ x ^ 2 \\Lambda = - 2 \\sinh ( 2 \\Lambda ) ( ( \\partial _ x \\phi ) ^ 2 - ( \\partial _ t \\phi ) ^ 2 ) , \\\\ \\partial _ t ^ 2 \\phi - \\partial _ x ^ 2 \\phi = - \\dfrac { \\sinh ( 2 \\Lambda ) } { \\sinh ^ 2 ( \\Lambda ) } ( \\partial _ t \\phi \\partial _ t \\Lambda - \\partial _ x \\phi \\partial _ x \\Lambda ) . \\end{cases*} \\end{align*}"}
+{"id": "7630.png", "formula": "\\begin{align*} \\nu _ { N } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\delta _ { y _ { i } } , \\end{align*}"}
+{"id": "7283.png", "formula": "\\begin{align*} u ( x ) = \\lambda ^ { d } G ^ { d } _ m ( x ) + \\lambda \\int _ { 0 } ^ { x } ( x - y ) u ( y ) d m ( y ) . \\end{align*}"}
+{"id": "2597.png", "formula": "\\begin{align*} \\dim \\sum _ { l \\in I } V _ { [ 4 ] \\setminus \\{ l \\} } ^ \\perp \\begin{cases} \\geq 3 | I | & , \\\\ = 8 & , \\end{cases} \\end{align*}"}
+{"id": "5334.png", "formula": "\\begin{align*} ( c - \\delta / 2 ) | \\tilde { \\xi } | ^ 2 \\leq \\varepsilon \\ , \\tilde { \\xi } \\cdot \\tilde { \\xi } + ( \\varepsilon _ k - \\varepsilon ) \\ , \\tilde { \\xi } \\cdot \\tilde { \\xi } = \\varepsilon _ k \\ , \\tilde { \\xi } \\cdot \\tilde { \\xi } \\leq ( c - \\delta ) | \\tilde { \\xi } | ^ 2 , \\end{align*}"}
+{"id": "7988.png", "formula": "\\begin{align*} f ( z ) \\in f ( x ) + f ( y ) = a + b \\subseteq C \\end{align*}"}
+{"id": "1864.png", "formula": "\\begin{align*} ( s _ i ^ { k + 1 } , z _ i ^ { k + 1 } ) : = ( s _ i ( \\sigma _ k , \\alpha _ k ) , z _ i ( \\sigma _ k , \\alpha _ k ) ) \\ge ( \\phi _ k , \\psi _ k ) > 0 \\end{align*}"}
+{"id": "4284.png", "formula": "\\begin{align*} \\displaystyle \\int _ { \\Omega } \\textbf { \\textit { v } } \\cdot \\nabla { \\varphi } \\ , d \\textbf { x } + \\displaystyle \\int _ { \\Omega } { \\varphi } \\ , \\mathrm { d i v } \\ , \\textbf { \\textit { v } } \\ , d \\textbf { x } = \\left \\langle \\textbf { \\textit { v } } \\cdot \\textbf { \\textit { n } } , \\varphi \\right \\rangle _ { { W } ^ { - 1 / p , p } ( \\Gamma ) \\times { W } ^ { 1 / p , p ' } ( \\Gamma ) } . \\end{align*}"}
+{"id": "759.png", "formula": "\\begin{align*} \\gamma \\cdot f ( x , y ) = f \\left ( ( x , y ) \\gamma \\right ) , \\end{align*}"}
+{"id": "6043.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { e \\in E _ n } \\big ( 1 - | \\psi ( e ) | \\big ) = \\infty . \\end{align*}"}
+{"id": "6089.png", "formula": "\\begin{align*} \\oint _ C \\partial _ { \\boldsymbol n } h ( s ) | \\dd s | = 0 , \\end{align*}"}
+{"id": "3693.png", "formula": "\\begin{align*} \\frac { \\partial \\psi _ w } { \\partial w _ i } \\circ \\psi _ { w } ^ { - 1 } = X _ { h _ i ( \\bullet , w ) } , \\end{align*}"}
+{"id": "9233.png", "formula": "\\begin{align*} \\theta _ { m } \\Delta _ { f } v _ { \\Gamma } e ^ { - ( f - f ( m ) ) / h } = \\theta _ { m } ( w + r ) e ^ { - ( f + \\ell _ { \\Gamma } ^ { 2 } / 2 - f ( m ) ) / h } , \\end{align*}"}
+{"id": "4374.png", "formula": "\\begin{align*} \\frac { p } { q } - \\frac { 1 } { a _ 1 } = \\frac { p } { q } - \\frac { 1 } { t } = \\frac { 1 } { q t } = \\frac { 1 } { q a _ 1 } \\end{align*}"}
+{"id": "6135.png", "formula": "\\begin{align*} \\begin{cases} & \\langle ( U ' ( t ) , - U ( t ) ) , ( \\Phi ( t ) , \\Phi ' ( t ) ) \\rangle = \\langle ( U _ 1 , - U _ 0 ) , ( \\Phi _ 0 , \\Phi _ 1 ) \\rangle \\\\ & \\displaystyle + \\int _ 0 ^ t \\int _ { \\Gamma _ 1 } ( D H ( \\tau ) , \\Phi ( \\tau ) ) d \\Gamma d \\tau , \\forall t \\geqslant 0 \\end{cases} \\end{align*}"}
+{"id": "6727.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\mu ( n ) \\mu ( n + a ) = \\sum _ { n \\leq x } ( - 1 ) ^ { \\omega ( n ) } \\mu ( n ) ^ 2 \\mu ( n + a ) . \\end{align*}"}
+{"id": "6251.png", "formula": "\\begin{align*} d c _ k = \\pm \\frac 1 { \\sqrt { | \\eta ^ 2 - 1 | } } \\frac { \\omega _ k } { \\sqrt { | \\omega _ k ^ 2 - 1 | } } d \\omega _ k \\quad c _ k ^ 2 - c _ m ^ 2 = \\frac { \\omega _ k ^ 2 - \\omega _ m ^ 2 } { \\eta ^ 2 - 1 } , \\end{align*}"}
+{"id": "5999.png", "formula": "\\begin{align*} g \\cdot X _ { i , j } ^ { k , l } & = \\{ ( L , H ) \\in X | L \\subset < e _ { 1 } , \\dots , e _ h > , \\ : < e _ { 1 } , \\dots e _ { r } , e _ { n - p + 1 } , \\dots , e _ n > \\subset H \\} \\\\ & = Y _ { r , p } ^ h \\end{align*}"}
+{"id": "7151.png", "formula": "\\begin{align*} \\displaystyle { { \\rm e q } : \\ z _ k = x _ k : = \\frac { k } { N } \\ , , v _ k = 0 \\ , , k = 1 , . . . , N \\ , , } \\end{align*}"}
+{"id": "385.png", "formula": "\\begin{align*} J ^ { ( 1 , 1 ) } _ 2 & \\leq 3 2 \\max \\{ 1 , C ^ 2 _ { \\textsf { K M T } } \\} L ^ { - 2 } _ * \\sum \\limits _ { n = 2 } ^ { \\infty } p _ n ( T _ * ) \\ln ^ 2 ( n ) + 8 4 \\sqrt { C ^ { ( 2 ) } _ { \\textsf { K M T } } } L ^ { - 2 } _ * \\sum \\limits _ { n = 2 } ^ { \\infty } { p _ n ( T _ * ) } n ^ { - 1 } \\\\ & \\leq 3 2 \\max \\{ 1 , C ^ 2 _ { \\textsf { K M T } } \\} L ^ { - 2 } _ * \\sum \\limits _ { n = 2 } ^ { \\infty } p _ n ( T _ * ) \\ln ^ 2 ( n ) + 2 5 2 \\sqrt { C ^ { ( 2 ) } _ { \\textsf { K M T } } } L ^ { - 2 } _ * T ^ { - 1 } _ * \\ , . \\end{align*}"}
+{"id": "8083.png", "formula": "\\begin{align*} \\mathcal { R ^ + } = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { c > 0 } \\frac { S ( n , p ; c ) } { c } H _ { m , n } ^ + \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) \\end{align*}"}
+{"id": "8015.png", "formula": "\\begin{align*} e & = ( F _ 2 , F _ 1 ) , \\\\ T & = \\left ( \\{ F _ 3 , F _ 2 , F _ 1 \\} , \\{ ( F _ 3 , F _ 2 ) , ( F _ 2 , F _ 1 ) \\} \\right ) , \\\\ F & = [ \\emptyset ] . \\end{align*}"}
+{"id": "3648.png", "formula": "\\begin{align*} u ( x , t ) = & ~ \\int \\limits _ { \\mathbb { H } ^ { n } } \\mathcal { K } ( x , y , t ) \\ , u _ { 0 } ( y ) ~ { \\rm d } v _ { \\mathbb { H } ^ { n } } ( y ) \\\\ & + \\int \\limits _ { 0 } ^ { t } \\int \\limits _ { \\mathbb { H } ^ { n } } \\mathcal { K } ( x , y , t - s ) e ^ { \\mu s } u ( y , s ) \\big ( e ^ { \\beta u ^ { p } ( y , s ) } - 1 \\big ) ~ { \\rm d } v _ { \\mathbb { H } ^ { n } } ( y ) \\ , { \\rm d } s . \\end{align*}"}
+{"id": "8799.png", "formula": "\\begin{align*} U _ r M U _ r ^ { T } = [ A ] _ { \\mathbf { c b } _ i } . \\end{align*}"}
+{"id": "3078.png", "formula": "\\begin{align*} A ( y ) = \\mathrm { d i a g } ( a _ 1 ( y ) , a _ 2 ( y ) ) \\quad y \\in \\R ^ 2 \\end{align*}"}
+{"id": "252.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - ( 1 - \\alpha ) \\tau ^ 2 - \\beta \\tau \\eta \\leqslant - \\frac { \\alpha } { 2 } ( 1 + 3 \\rho ^ 2 ) \\leqslant - \\frac { \\alpha } { 2 } , \\end{align*}"}
+{"id": "8031.png", "formula": "\\begin{align*} v = ( x _ 1 , \\ldots , x _ 6 , y _ 1 , \\ldots , y _ 6 , a _ 0 , \\ldots , a _ 3 , b _ 0 , \\ldots , b _ 3 ) \\in \\mathbb { R } ^ { 2 0 } . \\end{align*}"}
+{"id": "6405.png", "formula": "\\begin{align*} ( \\partial _ t \\phi ) ( \\tilde X _ j , \\tilde Y _ j , \\tilde t _ j ) & = \\partial _ { \\tilde t } \\Psi ( P _ j ) = - \\partial _ t \\Psi ( P _ j ) , \\\\ ( \\nabla _ X \\phi ) ( \\tilde X _ j , \\tilde Y _ j , \\tilde t _ j ) & = \\nabla _ { \\tilde X } \\Psi ( P _ j ) = - \\nabla _ X \\Psi ( P _ j ) , \\\\ ( \\nabla _ Y \\phi ) ( \\tilde X _ j , \\tilde Y _ j , \\tilde t _ j ) & = \\nabla _ { \\tilde Y } \\Psi ( P _ j ) = - \\nabla _ Y \\Psi ( P _ j ) . \\end{align*}"}
+{"id": "2478.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + ( 1 / t ^ n ) x , \\end{align*}"}
+{"id": "2325.png", "formula": "\\begin{align*} h ( x , y ) = \\left [ \\ , \\ ( | x | ^ 2 + ( 1 - y ) ^ 2 \\ ) ^ { - \\frac { N - p } { 2 ( p - 1 ) } } - \\ ( | x | ^ 2 + ( 1 + y ) ^ 2 \\ ) ^ { - \\frac { N - p } { 2 ( p - 1 ) } } \\ , \\right ] ^ { \\frac { p - 1 } { p - N } } . \\end{align*}"}
+{"id": "3083.png", "formula": "\\begin{align*} c _ j ^ { k l } ( A ) = \\bar { a } \\left ( c _ j ^ { k l } ( B ) + \\bar { b } _ { k l } \\int _ Y r _ B B e _ j \\cdot \\nabla w \\right ) , j , k , l \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "5050.png", "formula": "\\begin{align*} \\alpha _ { \\nu , j } \\coloneqq \\prod _ { i = 1 } ^ s \\gamma _ { \\nu , i } ^ { q _ { i , j } } \\end{align*}"}
+{"id": "104.png", "formula": "\\begin{align*} 2 z [ x , y ] = [ z x , y ] + [ x , z y ] \\ , \\ , \\ , \\ , \\mbox { ( d u a l L e i b n i z r u l e ) } . \\end{align*}"}
+{"id": "8166.png", "formula": "\\begin{align*} \\mathcal { O } ^ - = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } \\sum _ { c > 0 } \\frac { S ( - n , p ; c ) } { c } H _ { m , n } ^ - \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) . \\end{align*}"}
+{"id": "4445.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { a _ 2 } = \\frac { 1 } { x _ 1 } + \\frac { 1 } { x _ 2 } \\end{align*}"}
+{"id": "3507.png", "formula": "\\begin{align*} A ^ \\tau ( k , l ) : = A ( k , \\tau _ k ( l ) ) , \\big ( ( k , l ) \\in \\lambda ) \\big ) . \\end{align*}"}
+{"id": "604.png", "formula": "\\begin{align*} \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ \\dagger ) ^ { \\rm T } \\diamond { \\rm d } X _ t ^ \\dagger = \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ \\dagger ) ^ { \\rm T } { \\rm d } X _ t ^ \\dagger + \\frac { \\Delta t \\gamma } { 2 } A ^ { \\rm T } : M . \\end{align*}"}
+{"id": "7072.png", "formula": "\\begin{align*} F _ n \\widehat { E } ^ { C _ 2 } _ * & = E ^ { C _ 2 } _ { * + | n \\sigma | - n \\sigma } \\\\ F _ n \\widehat { E } ^ { C _ 2 } _ * ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) & = E ^ { C _ 2 } _ { * + | n \\sigma | - n \\sigma } ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) . \\end{align*}"}
+{"id": "137.png", "formula": "\\begin{align*} \\mathcal { L } ^ { 2 n } ( H _ { \\lambda , K } ^ + ) = \\int _ { r K } \\mathcal { L } ^ n ( H _ { \\lambda , K , x } ^ + ) d x . \\end{align*}"}
+{"id": "248.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = - ( 1 - \\alpha ) \\tau ^ 2 - \\beta \\rho \\eta \\leqslant 0 , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = - ( 1 - \\alpha ) \\rho ^ 2 - \\beta \\rho \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "8022.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 1 & \\cdots & 1 \\\\ & \\ddots & \\vdots \\\\ & & 1 \\\\ 1 & \\cdots & 1 \\end{bmatrix} , \\end{align*}"}
+{"id": "7085.png", "formula": "\\begin{align*} \\mathcal { E } ^ 1 _ { p , q } = E ^ { C _ 2 } _ { p + | n \\sigma | - n \\sigma } ( S ^ { V _ q \\rho _ { q + 1 } ^ { - 1 } } ) \\Rightarrow E ^ { C _ 2 } _ { p + q + | n \\sigma | - n \\sigma } ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) . \\end{align*}"}
+{"id": "1461.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\binom { x _ i } { 2 } + \\sum _ { j = 1 } ^ n \\binom { y _ j } { 2 } = \\frac { k ( k - 1 ) } { m + 1 } . \\end{align*}"}
+{"id": "1494.png", "formula": "\\begin{align*} \\zeta ( 2 m ) = 1 + \\frac { 1 } { 2 ^ { 2 m } } + \\frac { 1 } { 3 ^ { 2 m } } + \\cdots = \\frac { ( - 1 ) ^ { m - 1 } B _ { 2 m } 2 ^ { 2 m } } { 2 ( 2 m ) ! } \\pi ^ { 2 m } , \\end{align*}"}
+{"id": "3819.png", "formula": "\\begin{align*} \\lim _ n \\langle B _ n R A \\xi , C \\xi \\rangle & = \\lim _ n \\langle R A \\xi , B _ n C \\xi \\rangle = \\lim _ n \\langle R A \\xi , C B _ n \\xi \\rangle \\\\ & = \\langle R A \\xi , C T \\xi \\rangle = \\langle R A \\xi , T C \\xi \\rangle = \\langle T R A \\xi , C \\xi \\rangle \\ , , \\end{align*}"}
+{"id": "872.png", "formula": "\\begin{align*} \\mathcal { X } ^ { t + 1 } = \\mathop { \\arg \\min } _ { \\mathcal { X } \\in { \\mathbb { K } ^ { n \\times p } _ { l } } } \\| \\mathcal { X } - \\mathcal { X } ^ { t } \\| ^ 2 _ { F ( \\mathcal { Q } ) } ~ ~ ~ \\mathcal { S } ^ { T } * \\mathcal { A } * \\mathcal { X } = \\mathcal { S } ^ { T } * \\mathcal { B } , \\end{align*}"}
+{"id": "2824.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 ) = x _ 1 ^ 5 x _ 2 ^ 7 . \\end{align*}"}
+{"id": "1809.png", "formula": "\\begin{align*} \\begin{cases} \\dot { Y } = A Y + B Z , \\\\ \\dot { Z } = C Y - A ^ T Z , \\end{cases} \\end{align*}"}
+{"id": "7594.png", "formula": "\\begin{align*} M = \\bigg \\| \\int _ { s } ^ { t } \\int _ { 0 } ^ { 1 } \\frac { \\partial G } { \\partial y } ( t - r , \\cdot , y ) v ^ { \\delta + 1 } ( r , \\cdot ) \\d y \\d r \\bigg \\| _ { \\L ^ p } , \\end{align*}"}
+{"id": "6978.png", "formula": "\\begin{gather*} \\vec { d } = \\begin{pmatrix} d _ { 0 , n } \\\\ d _ { 1 , n } \\\\ \\vdots \\\\ d _ { n , n } \\end{pmatrix} \\ ! , \\vec { b } = \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ P _ { m , n } ^ { ( \\alpha , \\beta ) } ( x ) \\end{pmatrix} \\ ! , \\end{gather*}"}
+{"id": "3883.png", "formula": "\\begin{align*} u ( x _ 0 ) & = g ( x _ 0 , y _ 0 , z _ 0 ) , \\\\ u ( x ) & \\geq g ( x , y _ 0 , z _ 0 ) x \\in \\Omega , \\end{align*}"}
+{"id": "3748.png", "formula": "\\begin{align*} Z _ { r _ 1 } ( d _ 2 ( f ) , s + 1 ) & = \\int _ { F ^ \\times } \\int _ { F } f ( 0 , 0 , 0 , y , 0 , 0 ) \\psi ( a y ) d y | a | ^ { s + 1 } d ^ \\times a \\\\ & = \\int _ { | a | \\le q ^ A } \\int _ { F } f ( 0 , 0 , 0 , y , 0 , 0 ) \\psi ( a y ) d y | a | ^ { s + 1 } d ^ \\times a \\\\ & = \\int _ { F } \\int _ { | a | \\le q ^ { A } } f ( 0 , 0 , 0 , y , 0 , 0 ) \\psi ( a y ) | a | ^ { s + 1 } d ^ \\times a d y . \\end{align*}"}
+{"id": "6080.png", "formula": "\\begin{align*} \\lambda _ n ( x ) : = \\rho ( x ) B _ n ^ 2 ( x ) / m _ n ( x ) , x \\in \\Delta , \\end{align*}"}
+{"id": "4491.png", "formula": "\\begin{align*} 1 = \\frac { 1 } { 2 } + \\frac { 1 } { 3 } + \\frac { 1 } { 7 } + \\frac { 1 } { 7 8 } + \\frac { 1 } { 9 1 } . \\end{align*}"}
+{"id": "4869.png", "formula": "\\begin{align*} \\int _ t ^ 1 w _ { z ; k } ( s ) d s = \\frac { 1 } { 2 k \\pi } \\big ( w _ { i z ; k } ( 1 ) - w _ { i z ; k } ( t ) \\big ) , k \\neq 0 , \\end{align*}"}
+{"id": "2723.png", "formula": "\\begin{align*} \\tilde { U } _ 0 ^ j = W \\Longrightarrow \\lim _ { n \\to \\infty } \\frac { \\lambda _ { j + 1 } ( \\tilde { t } _ n ) } { \\lambda _ { j } ( \\tilde { t } _ n ) } = 0 . \\end{align*}"}
+{"id": "8021.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 1 & 1 & & \\\\ 1 & \\ddots & \\ddots & \\\\ & \\ddots & \\ddots & 1 \\\\ & & 1 & 1 \\\\ 1 & \\cdots & \\cdots & 1 \\end{bmatrix} , \\end{align*}"}
+{"id": "5631.png", "formula": "\\begin{align*} \\dfrac { d H } { d I } & \\leq \\dfrac { F ( S , I ) - F ( S , I ^ { * } ) } { F ( S , I ) F ( S , I ^ { * } ) } F _ { 1 } ( S , I ) - \\dfrac { I - I ^ { * } } { I I ^ { * } } \\\\ & = \\dfrac { 1 } { F ( S , I ^ { * } ) } \\left [ F _ { 1 } ( S , I ) - F _ { 1 } ( S , I ^ { * } ) \\right ] \\leq 0 . \\end{align*}"}
+{"id": "7838.png", "formula": "\\begin{align*} n ( r , \\Lambda _ { j k } ) = \\frac { | w _ j - w _ k | } { 2 \\pi } r + O ( 1 ) . \\end{align*}"}
+{"id": "8908.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d t } \\right ) ^ { p } \\vartheta _ { 2 } ( t ) \\simeq \\frac { ( - 1 ) ^ { p } p ! } { t ^ { 1 + p } } + \\sum \\limits _ { s = 0 } ^ { + \\infty } \\frac { B _ { s + p } t ^ { s } } { s ! } \\end{align*}"}
+{"id": "4642.png", "formula": "\\begin{align*} T ( r , \\gamma \\circ \\pi _ Y , H ) = T ( r , \\tilde { \\gamma } , \\pi ^ * H ) \\leq T ( r , \\tilde { \\gamma } , L ) = O ( \\log r ) , \\end{align*}"}
+{"id": "3280.png", "formula": "\\begin{align*} f \\left ( L x \\right ) - \\lambda g \\left ( x \\right ) = h , h \\in Z , \\ \\end{align*}"}
+{"id": "7067.png", "formula": "\\begin{align*} k ^ { C _ 2 } _ \\diamond = \\dfrac { R ( C _ 2 ) [ v , \\mu , \\tau ] } { \\begin{matrix} \\tau ( \\sigma - 1 ) = v \\mu \\\\ \\mu ( \\sigma + 1 ) = 0 \\end{matrix} } \\end{align*}"}
+{"id": "7129.png", "formula": "\\begin{align*} a _ i & = \\left \\langle d - \\left \\langle d , x ^ { \\rho _ 1 + \\cdots + \\rho _ { n - 1 } } ) \\right \\rangle \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) , x ^ { \\rho _ 1 + \\cdots + \\rho _ { i - 1 } } \\right \\rangle \\\\ & = \\begin{cases} \\left \\langle d , x ^ { \\rho _ 1 + \\cdots + \\rho _ { i - 1 } } \\right \\rangle & i < n \\\\ 0 & i \\geq n . \\end{cases} \\end{align*}"}
+{"id": "1239.png", "formula": "\\begin{align*} W ^ 2 _ \\nu ( \\mu _ s , \\mu _ t ) = \\int _ X W _ 2 ^ 2 ( \\mu _ t ^ y , \\mu _ s ^ y ) d \\nu ( y ) = ( t - s ) ^ 2 \\int _ X W _ 2 ^ 2 ( \\mu _ 0 ^ y , \\mu _ 1 ^ y ) d \\nu ( y ) = ( t - s ) ^ 2 W ^ 2 _ \\nu ( \\mu _ 0 , \\mu _ 1 ) \\end{align*}"}
+{"id": "4029.png", "formula": "\\begin{align*} w _ { \\xi \\xi } = w _ { \\tau \\tau } + 2 b w _ { \\tau \\beta } + b ^ 2 w _ { \\beta \\beta } . \\end{align*}"}
+{"id": "3969.png", "formula": "\\begin{align*} u ( x ) & = g ( x , Y u ( x ) , Z u ( x ) ) \\\\ & \\leq g ( x , Y v ( \\xi ) , Z v ( \\xi ) ) \\\\ & \\leq v ( x ) , \\end{align*}"}
+{"id": "2632.png", "formula": "\\begin{align*} \\tilde { \\gamma } _ { \\chi ^ { ( 7 2 c ) } , j } = \\frac { \\gamma _ { \\chi ^ { ( 7 2 c ) } , j } } { 2 \\pi } \\log X \\end{align*}"}
+{"id": "4878.png", "formula": "\\begin{align*} \\gamma ( t ) = ( 0 , t , 0 ) , t \\in I . \\end{align*}"}
+{"id": "5297.png", "formula": "\\begin{align*} g ( U , V ) = ( \\phi \\circ F ) ^ 2 g _ { | { L } } ( \\pi _ * U , \\pi _ * V ) . \\end{align*}"}
+{"id": "7470.png", "formula": "\\begin{align*} B _ n = \\sum _ { i = 1 } ^ n T _ i \\ , . \\end{align*}"}
+{"id": "1975.png", "formula": "\\begin{align*} \\left \\vert \\frac { f ( z + c _ { j } ) } { f ( z + c _ { l } ) } \\right \\vert \\leq \\exp _ { p } \\left [ \\left \\{ \\log _ { q - 1 } \\left ( r \\right ) \\right \\} ^ { \\rho _ { f } \\left ( p , q \\right ) + \\varepsilon } \\right ] < \\exp _ { p } \\left [ \\left \\{ \\log _ { q - 1 } \\left ( r \\right ) \\right \\} ^ { \\rho - 2 \\delta } \\right ] , \\left ( j \\neq l , j = 1 , 2 , . . . . , k \\right ) \\end{align*}"}
+{"id": "8302.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mu _ E \\nu _ 1 - \\nu _ E \\mu _ 1 - ( y _ 2 - y _ 3 ) \\Lambda = 0 \\\\ \\mu _ E \\nu _ 2 - \\nu _ E \\mu _ 2 - ( y _ 3 - y _ 1 ) \\Lambda = 0 \\\\ \\mu _ E \\nu _ 3 - \\nu _ E \\mu _ 3 - ( y _ 3 - y _ 1 ) \\Lambda = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "8607.png", "formula": "\\begin{gather*} R _ 1 , ~ R _ 2 , ~ R _ 3 , ~ R _ 4 , ~ R _ 7 , ~ R _ 8 , \\\\ A = \\frac { 1 } { 2 } ( R _ 2 + R _ 3 + R _ 4 + R _ 5 ) , ~ B = \\frac { 1 } { 2 } ( R _ 1 + R _ 2 + R _ 3 + R _ 6 ) , ~ C = \\frac { 1 } { 2 } ( R _ 7 + R _ 8 + R _ 9 + R _ { 1 2 } ) , \\\\ \\frac { 1 } { 2 } ( R _ 1 + R _ 2 + R _ 5 + R _ 8 + R _ { 1 0 } ) + \\frac { 1 } { 4 } ( R _ 2 + R _ 3 + R _ 4 + R _ 5 ) . \\end{gather*}"}
+{"id": "8645.png", "formula": "\\begin{align*} \\lambda _ 1 ( X ) \\geq \\frac { 1 } { 4 } - \\left ( \\frac { 7 } { 6 4 } \\right ) ^ 2 = \\frac { 9 7 5 } { 4 0 9 6 } . \\end{align*}"}
+{"id": "3204.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { C ^ \\kappa ( [ 0 , T ] ; \\mathbb { U } ) } : = \\vert u ( 0 ) \\vert _ \\mathbb { U } + \\sup \\Bigl \\{ \\frac { \\vert u ( t ) - u ( s ) \\vert _ \\mathbb { U } } { \\vert t - s \\vert ^ \\kappa } : s < t \\in [ 0 , T ] \\Bigr \\} < \\infty . \\end{align*}"}
+{"id": "2962.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\sup _ { \\gamma \\le \\delta } \\sup _ { \\tau \\in \\mathcal { T } _ T } \\mathbb { P } _ n \\big ( | X ^ n _ { \\tau + \\gamma } - X ^ n _ \\tau | > \\varepsilon \\big ) = 0 \\end{align*}"}
+{"id": "357.png", "formula": "\\begin{align*} C ^ \\infty ( M ) _ { ( i ) } = \\{ f \\colon \\ f ^ { ( i - 1 ) } | _ Q = 0 \\} . \\end{align*}"}
+{"id": "6354.png", "formula": "\\begin{align*} Y ' ( x ) = \\frac { f '' ( x ) - f '' ( \\xi \\circ f ( x ) ) \\cdot Y ( x ) ^ 2 } { f ' ( \\xi \\circ f ( x ) ) } \\end{align*}"}
+{"id": "8683.png", "formula": "\\begin{align*} F _ \\lambda ( x _ 1 , \\dots , x _ n ; t ) = F _ \\lambda ( x _ 1 , \\dots , x _ n , 0 ; t ) . \\end{align*}"}
+{"id": "1282.png", "formula": "\\begin{align*} m _ { t , j } ( x ) p _ { j } ^ { ( t ) } ( x ) = a ( x , \\theta ^ { 1 + j r ' } , \\omega ^ { 1 + t r } ) ( x ^ s - \\alpha ) . \\end{align*}"}
+{"id": "1856.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { H ^ s ( \\mathbb { T } ^ n ) } : = \\biggl ( \\sum _ { \\lvert \\alpha \\rvert \\leq s } \\int _ { \\mathbb { T } ^ n } \\lvert D ^ { \\alpha } u \\rvert ^ 2 \\ d x \\biggr ) ^ { 1 / 2 } < \\infty , \\end{align*}"}
+{"id": "1224.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } { ^ \\beta } \\gamma _ { i n } \\frac { \\partial F ^ { n } _ { \\beta \\alpha } } { \\partial z _ { \\beta } ^ { i } } \\biggr | _ { z _ \\beta ( \\gamma p ) } = { ^ \\alpha } \\gamma _ { n n } \\frac { \\partial F ^ { n } _ { \\beta \\alpha } } { \\partial z _ { \\beta } ^ { n } } \\biggr | _ { z _ \\beta ( p ) } . \\end{align*}"}
+{"id": "7802.png", "formula": "\\begin{align*} B = \\begin{pmatrix} 0 & - d _ 1 \\\\ d _ 2 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "4062.png", "formula": "\\begin{align*} D _ t v ( y , t ) & = g ^ * _ x ( x , y , u ( x , t ) ) \\frac { \\partial x } { \\partial t } + g ^ * _ u ( x , y , u ( x , t ) ) [ D u \\frac { \\partial x } { \\partial t } + u _ t ] . \\end{align*}"}
+{"id": "2591.png", "formula": "\\begin{align*} \\langle ( u _ 1 , u _ 2 ) + \\alpha c + \\beta d , ( v _ 1 , v _ 2 ) + \\gamma c + \\delta d \\rangle & = \\frac { 1 } { 2 } ( \\langle u _ 1 , u _ 2 \\rangle _ { L ^ 2 } + \\langle v _ 1 , v _ 2 \\rangle _ { L ^ 2 } ) + \\alpha \\delta + \\beta \\gamma . \\end{align*}"}
+{"id": "2964.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { n \\to \\infty } \\mathcal { X } ^ n = \\mathcal { X } , \\lim _ { n \\to \\infty } \\mathcal { M } ^ n = \\mathcal { M } , \\lim _ { n \\to \\infty } \\mathcal { S } ^ n = \\mathcal { S } , \\lim _ { n \\to \\infty } \\tilde { \\mathcal { B } } ^ n = \\tilde { \\mathcal { B } } \\end{aligned} \\end{align*}"}
+{"id": "1454.png", "formula": "\\begin{align*} \\lambda = \\frac { k ( k - 1 ) ( k - 2 ) } { m n ( m n - 1 ) ( m n - 2 ) } \\cdot b . \\end{align*}"}
+{"id": "3886.png", "formula": "\\begin{align*} & g _ y ^ * = \\frac { - g _ y } { g _ z } , & & g ^ * _ x = \\frac { - g _ x } { g _ z } , & g ^ * _ u = \\frac { 1 } { g _ z } , \\end{align*}"}
+{"id": "6371.png", "formula": "\\begin{align*} l _ \\lambda \\sin u ( r _ 1 ) = \\alpha ^ { - 1 } m _ \\lambda ( r _ 1 ) = \\alpha ^ { - 1 } m _ \\lambda ( r _ 2 ) = l _ \\lambda \\sin u ( r _ 2 ) . \\end{align*}"}
+{"id": "52.png", "formula": "\\begin{align*} T ( n _ i ) + T ( n _ { i + 1 } ) + T ( n _ { i + 1 } ) & < \\left ( \\frac { 2 9 7 } { 8 1 } + \\frac { 1 2 8 } { 8 1 } \\right ) \\cdot \\frac { 1 } { X _ 0 } \\\\ & = \\frac { 4 2 5 } { 8 1 } \\cdot \\frac { 1 } { X _ 0 } < 7 \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } \\\\ & \\leq ( k _ i + k _ { i + 1 } + k _ { i + 2 } ) \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } , \\end{align*}"}
+{"id": "183.png", "formula": "\\begin{align*} T _ 1 ( Z ) & = 3 m - 1 , T _ 2 ( Z ) = 3 m , \\\\ T _ j ( Z ) & = \\left [ \\frac { 6 m + ( j - 2 ) m + j - 1 } j \\right ] = m + \\left [ \\frac { 4 m + j - 2 } j \\right ] ; j = 3 , \\ldots , n - 1 . \\end{align*}"}
+{"id": "4096.png", "formula": "\\begin{align*} \\nabla _ i R = 2 \\nabla _ j R _ { i j } = \\frac { 1 } { 2 } \\nabla _ j H ^ 2 _ { i j } = \\frac { 1 } { 2 } ( \\nabla _ j H _ { i k l } ) H _ { j k l } \\end{align*}"}
+{"id": "281.png", "formula": "\\begin{align*} \\begin{aligned} \\bigcup _ { \\alpha \\in H ^ 1 ( \\mathrm { G a l } ( E / F ) , G ( E ) ) } \\Pi _ { ( G _ { \\alpha } ( F ) , \\omega _ { G _ { \\alpha } ( F ) , E } ) } ( G ( E ) ) & \\xhookrightarrow { \\iota } \\Pi ( G ( E ) ) \\xrightarrow { \\mathrm { L L C } } \\Phi ( G ( E ) ) \\xleftarrow { \\mathrm { B C } } \\Phi ( G ^ { \\mathrm { o p } } ( F ) ) . \\\\ \\mathrm { I m } ( \\mathrm { L L C } \\circ \\iota ) & = \\mathrm { I m } ( \\mathrm { B C } ) . \\end{aligned} \\end{align*}"}
+{"id": "2956.png", "formula": "\\begin{align*} \\begin{aligned} & W _ { j + i - 1 } ( W ^ - _ { j + i - 1 } - W _ { j + i - 1 } + W ^ + _ { j + i } - W _ { j + i } ) \\\\ & = - ( W _ { j + i - 1 } W _ { j + i } - W _ { j + i - 1 } W ^ - _ { j + i - 1 } ) + ( W _ { j + i - 1 } W ^ + _ { j + i } - W _ { j + i } ^ 2 ) + ( W _ { j + i } ^ 2 - W _ { j + i - 1 } ^ 2 ) . \\end{aligned} \\end{align*}"}
+{"id": "8453.png", "formula": "\\begin{align*} & \\max _ { \\nu \\in \\mathcal E ^ + _ A } \\ , G ( \\nu ) = G ( \\gamma _ A ) \\quad \\bigl ( { } = c _ * ( A ) \\bigr ) , \\\\ & \\max _ { \\nu \\in \\widehat { \\mathcal E } ^ + _ A } \\ , \\nu ( X ) = \\gamma _ A ( X ) \\quad \\bigl ( { } = c _ * ( A ) \\bigr ) . \\end{align*}"}
+{"id": "6377.png", "formula": "\\begin{align*} Q _ \\lambda ' ( u ) / \\sin 2 u = & \\lambda \\cdot ( 1 + F _ \\lambda ' ( \\cos u ) ) \\left ( \\cos u - \\frac { \\sin u \\sin 2 u } { 4 } + \\cos u \\cos 2 u \\right ) \\\\ & + \\frac { \\lambda ^ 2 \\cos ^ 2 u \\sin ^ 2 2 u } { 8 } \\end{align*}"}
+{"id": "334.png", "formula": "\\begin{align*} \\Phi _ { y } ( t ) = \\Phi ( t ) V ( y ( \\frac { \\pi } { 8 X t } ) ^ \\frac { 3 } { 2 } ) . \\end{align*}"}
+{"id": "2452.png", "formula": "\\begin{align*} \\Lambda ( s , \\pi ) = e ^ { a _ { \\pi } + b _ { \\pi } s } \\prod _ { \\rho } \\Big ( 1 - \\frac { s } { \\rho } \\Big ) e ^ { s / \\rho } \\end{align*}"}
+{"id": "7170.png", "formula": "\\begin{align*} \\sigma ^ { \\left ( s \\right ) } = \\rho s _ { , t } + \\rho s _ { , j } v _ { j } + J _ { k , k } - \\rho ( { { r } } / { \\vartheta } ) , \\end{align*}"}
+{"id": "8878.png", "formula": "\\begin{align*} \\varphi ( t , z ) = \\sum \\limits _ { m = 0 } ^ { + \\infty } e ^ { \\beta _ m t } g _ m ( z ) \\end{align*}"}
+{"id": "8924.png", "formula": "\\begin{align*} \\vartheta _ { 3 } \\left ( t \\right ) : = 2 \\sum \\limits _ { l = 1 } ^ { + \\infty } l e ^ { - l ^ { 2 } t } \\end{align*}"}
+{"id": "868.png", "formula": "\\begin{align*} \\left | B ' \\right | = \\left ( { \\frac { 1 } { 3 } + \\frac { 1 } { 1 2 } \\sqrt \\delta } \\right ) v ( F ) . \\end{align*}"}
+{"id": "168.png", "formula": "\\begin{align*} ( \\nu ( \\varkappa ) ) ( \\theta ) ( \\hslash ) : = \\sum _ { j = 1 } ^ { n } \\alpha _ { j } z ( \\theta _ { j } , \\hslash ) , \\ \\theta _ { j } \\in J , \\ \\hslash \\in [ 0 , \\pi ] , \\end{align*}"}
+{"id": "3354.png", "formula": "\\begin{align*} X _ f \\lrcorner \\theta & = \\epsilon f , \\\\ X _ f \\lrcorner \\Omega & = \\dd f - ( R f ) \\theta . \\end{align*}"}
+{"id": "7985.png", "formula": "\\begin{align*} \\gamma ( r [ x ] ) = \\gamma ( [ r x ] ) = \\beta ( r x ) = r \\beta ( x ) = r \\gamma ( [ x ] ) . \\end{align*}"}
+{"id": "7045.png", "formula": "\\begin{align*} M U _ * ( B U ) = M U _ * [ b _ 1 ' , b _ 2 ' , \\dots ] \\end{align*}"}
+{"id": "8611.png", "formula": "\\begin{align*} S = ( \\langle 1 \\rangle , { \\bf e } _ { \\ell _ 1 } , { \\bf e } _ { \\ell _ 1 - 1 } , \\ldots , \\langle 1 \\rangle , { \\bf e } _ { \\ell _ 2 } , { \\bf e } _ { \\ell _ 2 - 1 } , \\ldots , \\langle 1 \\rangle ) . \\end{align*}"}
+{"id": "5728.png", "formula": "\\begin{align*} \\mathbf { D } \\theta ^ { A A ^ { \\prime } } \\equiv d \\theta ^ { A A ^ { \\prime } } + \\mathbf { A } _ { B } ^ { A } \\theta ^ { B A ^ { \\prime } } + \\mathbf { A } _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\theta ^ { A B ^ { \\prime } } = 0 . \\end{align*}"}
+{"id": "160.png", "formula": "\\begin{align*} R e ( < A v , v > ) = - \\frac { 1 } { 2 } | z ( 0 ) | ^ { 2 } \\le 0 , \\end{align*}"}
+{"id": "460.png", "formula": "\\begin{align*} | \\phi ( \\tau _ n ) | = | \\psi ( \\tau _ n ) | | f _ n ( x ) | = | f _ n ( y ) | , \\forall n \\in \\mathbb { N } , \\end{align*}"}
+{"id": "4871.png", "formula": "\\begin{align*} g ^ 1 _ { z _ 1 , k _ 1 } = \\frac { 1 } { 2 k _ 1 \\pi } w _ { z _ 1 ; k _ 1 } ( 1 ) . \\end{align*}"}
+{"id": "3502.png", "formula": "\\begin{align*} \\lambda = \\big \\{ ( k , l ) : k \\in \\{ 1 , \\dots \\ell ( \\lambda ) \\} , \\ l \\in \\{ 1 , \\dots , \\lambda _ k \\} \\big \\} . \\end{align*}"}
+{"id": "1223.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } { ^ \\beta } \\gamma _ { i n } \\frac { \\partial F ^ { j } _ { \\beta \\alpha } } { \\partial z _ { \\beta } ^ { i } } \\biggr | _ { z _ \\beta ( \\gamma p ) } = \\sum _ { i = 1 } ^ { n } { ^ \\alpha } \\gamma _ { j i } \\frac { \\partial F ^ { i } _ { \\beta \\alpha } } { \\partial z _ { \\beta } ^ { n } } \\biggr | _ { z _ \\beta ( p ) } \\ ; \\ , \\ ; \\ , 1 \\leq j \\leq n . \\end{align*}"}
+{"id": "4689.png", "formula": "\\begin{align*} w _ C ( x , y ) : = \\sum _ { \\mathbf { c } \\in C } x ^ { n - w t ( \\mathbf { c } ) } y ^ { w t ( \\mathbf { c } ) } . \\end{align*}"}
+{"id": "4243.png", "formula": "\\begin{align*} I ^ m _ \\Omega ( c ) : = \\inf \\left \\{ E _ \\Omega ( f ) \\ : f \\in S ( c ) \\cap B _ \\Omega ( m ) \\right \\} , \\end{align*}"}
+{"id": "1782.png", "formula": "\\begin{align*} z _ k : = \\sum _ { i = j } ^ { m _ k } n _ i q ^ i \\in \\mathsf { Z } ( a ^ k ) \\end{align*}"}
+{"id": "9018.png", "formula": "\\begin{align*} \\limsup _ { i \\in I } \\frac { H _ \\mu ( \\alpha _ { F _ i } ) } { \\theta ( A _ i ) } + \\mu ( f ) & = \\lim _ { i \\in I } \\frac { H _ \\mu ( \\alpha _ { F _ i } ) + \\mu ( f _ { A _ i } ) } { \\theta ( A _ i ) } \\\\ & \\leq \\limsup _ { i \\in I } \\frac { \\operatorname { P } _ { f _ { A _ i } } ( \\mathcal { U } _ { F _ i } ) } { \\theta ( A _ i ) } + \\limsup _ { i \\in I } \\frac { | F _ i | } { \\theta ( A _ i ) } \\theta ( V ) \\epsilon \\\\ & \\leq \\operatorname { p } _ f ^ { ( O W ) } ( \\pi ) + \\epsilon . \\end{align*}"}
+{"id": "6807.png", "formula": "\\begin{align*} \\delta A _ d [ q _ d ] & = \\sum _ { m = 1 } ^ { N - 1 } D _ 1 L _ d ( q _ m , q _ { m + 1 } ) . \\delta { q _ m } + \\sum _ { m = 0 } ^ { N - 2 } D _ 2 L _ d ( q _ m , q _ { m + 1 } ) . \\delta { q _ { m + 1 } } . \\\\ & = \\sum _ { m = 1 } ^ { N - 1 } [ D _ 1 L _ d ( q _ m , q _ { m + 1 } ) . \\delta { q _ m } + D _ 2 L _ d ( q _ { m - 1 } , q _ { m } ) ] . \\delta { q _ { m } } . \\end{align*}"}
+{"id": "2069.png", "formula": "\\begin{align*} \\begin{alignedat} { 2 } \\tilde \\rho _ i & = \\beta \\rho _ i , & & i = 1 , 2 , \\\\ \\tilde \\kappa _ i & = \\beta c ^ { \\rho _ i } \\kappa _ i , & & i = 1 , 2 , 3 . \\end{alignedat} \\end{align*}"}
+{"id": "7007.png", "formula": "\\begin{align*} R _ { X | W } ( \\Delta ) & \\geq R _ { X } ( \\Delta ) - I ( X ; W ) \\\\ & \\geq \\frac { 1 } { 2 } \\log { \\frac { N ( X ) } { \\Delta } } - I ( X ; W ) \\\\ & = \\frac { 1 } { 2 } \\log { \\frac { N ( X | W ) } { \\Delta } } \\end{align*}"}
+{"id": "2353.png", "formula": "\\begin{align*} Q _ s ( 0 ) : = V _ s ( 0 ) ^ T V _ s ( 0 ) . \\end{align*}"}
+{"id": "8820.png", "formula": "\\begin{align*} P _ 2 P _ 2 ^ * A _ { 2 1 } = P _ 2 B _ { 1 2 } ^ * = B _ { 1 2 } ^ * P _ 1 = A _ { 2 1 } \\end{align*}"}
+{"id": "1717.png", "formula": "\\begin{align*} I ( \\chi , m ) = \\left ( 1 + \\chi ( - 1 ) \\epsilon \\right ) \\int _ 0 ^ { \\infty } t \\frac { d F ( t ) } { d t } \\frac { d t } { t } = \\left ( 1 + \\chi ( - 1 ) \\epsilon \\right ) \\left ( F ( \\infty ) - F ( 0 ^ + ) \\right ) \\end{align*}"}
+{"id": "1651.png", "formula": "\\begin{align*} \\Sigma _ T = \\Sigma _ { T } ^ { \\R } \\cup \\Sigma _ T ^ \\C , \\Sigma _ { T } ^ { \\R } = \\{ \\sigma \\mid \\infty , \\ ; T ( F _ \\sigma ) = \\R ^ \\times \\} , \\quad \\Sigma _ { T } ^ { \\C } = \\{ \\sigma \\mid \\infty , \\ ; T ( F _ \\sigma ) = \\C ^ \\times \\} . \\end{align*}"}
+{"id": "7219.png", "formula": "\\begin{align*} \\big \\langle p j , \\ , p ' \\big \\rangle \\ , = \\ , R ^ 2 \\vartheta ' . \\end{align*}"}
+{"id": "7299.png", "formula": "\\begin{align*} \\chi _ { m , j } ( \\lambda ) & = \\int _ { 0 } ^ { \\infty } ( 1 - g _ m ( \\lambda ; x ) ) j ( d x ) ( \\lambda > 0 ) . \\end{align*}"}
+{"id": "2593.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ r L _ i = \\bigcup _ { i \\in I \\subset [ r ] , \\mid I \\mid = r - 1 } L _ i \\mbox { a n d } L _ i \\cap L _ j \\neq \\emptyset \\end{align*}"}
+{"id": "6675.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 2 \\lambda ^ k } { m } \\sum _ { i = 1 } ^ m \\left \\langle g _ i ^ k , \\bar x ^ k - \\theta ^ * \\right \\rangle \\geq - \\frac { \\gamma ^ k } { m } \\sum _ { i = 1 } ^ m \\| x _ i ^ k - \\bar { x } ^ k \\| ^ 2 \\\\ & \\qquad - \\frac { L ^ 2 ( \\lambda ^ k ) ^ 2 } { \\gamma ^ k } \\| \\bar x ^ k - \\theta ^ * \\| ^ 2 + 2 \\lambda ^ k ( F ( \\bar { x } ^ k ) - F ( \\theta ^ { \\ast } ) ) \\end{aligned} \\end{align*}"}
+{"id": "9109.png", "formula": "\\begin{align*} \\mathsf { V } _ { \\rho , 1 } : = & \\frac { \\log ^ 2 e } { 4 } \\left ( \\frac { p \\cdot 2 h _ { \\rho } ^ 2 \\mathsf { P } ^ 2 + 4 h _ { \\rho } \\mathsf { P } } { ( 1 + h _ { \\rho } \\mathsf { P } ) ^ 2 } + \\bar { p } \\frac { 2 h _ 1 ^ 2 \\mathsf { P } ^ 2 } { ( 1 + h _ 1 \\mathsf { P } ) ^ 2 } \\right ) \\end{align*}"}
+{"id": "2173.png", "formula": "\\begin{align*} R _ { \\mathcal { S } ^ { * } _ { R } } ( F ) & = \\begin{dcases} \\sigma _ { 0 } & \\ 2 \\alpha + \\beta - 2 \\geqslant 0 \\\\ \\sigma _ { 0 } & \\ 2 \\alpha + \\beta - 2 < 0 \\ \\ X ( \\alpha , \\beta ) \\leqslant 0 \\\\ \\tilde { \\sigma _ { 0 } } & \\ 2 \\alpha + \\beta - 2 < 0 \\ X ( \\alpha , \\beta ) > 0 , \\end{dcases} \\end{align*}"}
+{"id": "8060.png", "formula": "\\begin{align*} \\gamma ^ { \\pm } ( s ) = \\gamma _ 0 ( s ) \\mp \\gamma _ 1 ( s ) \\end{align*}"}
+{"id": "2157.png", "formula": "\\begin{align*} F ( z ) & = z ( 1 - z ) ^ { 2 \\alpha - 2 } ( 1 + z ) ^ { \\beta } , \\\\ \\intertext { w h i c h i m p l i e s } \\dfrac { z F ' ( z ) } { F ( z ) } & = 1 + \\dfrac { ( 2 - 2 \\alpha ) z } { 1 - z } + \\dfrac { \\beta z } { 1 + z } \\\\ & = \\dfrac { ( 1 - 2 \\alpha - \\beta ) z ^ 2 + ( 2 - 2 \\alpha + \\beta ) z + 1 } { 1 - z ^ 2 } \\\\ & = \\lambda + \\dfrac { ( 1 - 2 \\alpha - \\beta + \\lambda ) z ^ 2 + ( 2 - 2 \\alpha + \\beta ) z + 1 - \\lambda } { 1 - z ^ 2 } . \\end{align*}"}
+{"id": "7593.png", "formula": "\\begin{align*} Q & = \\int _ { 0 } ^ { t } \\bigg \\{ \\int _ { 0 } ^ { 1 } \\chi _ { \\{ x + z \\in [ 0 , 1 ] \\} } \\bigg | \\int _ { 0 } ^ { 1 } \\chi _ { \\{ | x - y | > | z | \\} } | G ( t - s , x + z , y ) - G ( t - s , x , y ) | \\\\ & \\times v ^ { 2 \\delta + 1 } ( s , y ) \\d y \\bigg | ^ p \\d x \\bigg \\} ^ { \\frac { 1 } { p } } \\d s , \\end{align*}"}
+{"id": "5702.png", "formula": "\\begin{align*} \\mathbf { \\mathbf { e } } ^ { a } & \\mathbf { = } \\theta ^ { a } - \\tau ^ { a } \\mathbf { m } \\\\ \\mathbf { A } _ { b } ^ { a } & = \\alpha _ { b } ^ { a } - \\beta _ { b } ^ { a } \\mathbf { m } \\end{align*}"}
+{"id": "4721.png", "formula": "\\begin{align*} \\mathcal { C } ^ { \\perp _ { \\chi , t } } = \\{ ( v ' , w ' ) \\in \\mathcal { R } ^ { 2 n } : \\chi ( \\langle ( v , w ) \\mid ( v ' , w ' ) \\rangle _ { \\mathrm { s } } ) \\in H _ t ( v , w ) \\in \\mathcal { C } \\} . \\end{align*}"}
+{"id": "8907.png", "formula": "\\begin{align*} \\mathcal { S } = \\sum \\limits _ { p = 0 } ^ { n - 1 } \\gamma _ { p } ^ { ( \\nu , n ) } \\left [ \\left ( - \\frac { d } { d t } \\right ) ^ { p } \\vartheta _ { 2 } ( t ) + \\sum \\limits _ { \\ell = 0 } ^ { + \\infty } \\frac { ( - 1 ) ^ { \\ell } } { ( p + \\ell + 1 ) \\ell ! } \\left [ B _ { 2 ( p + \\ell + 1 ) } \\left ( \\frac { 1 } { 2 } \\right ) - B _ { 2 ( p + \\ell + 1 ) } \\left ( \\nu + \\frac { 1 } { 2 } \\right ) \\right ] \\ t ^ { \\ell } \\right ] \\end{align*}"}
+{"id": "7741.png", "formula": "\\begin{align*} s ( \\alpha ) = \\begin{cases} \\sup ( i '' \\alpha ) , & , \\\\ i ( \\alpha ) , & . \\end{cases} \\end{align*}"}
+{"id": "8801.png", "formula": "\\begin{align*} r _ 1 r _ 2 + r _ 4 r _ 5 + r _ 7 r _ 8 = 0 , r _ 2 r _ 3 + r _ 5 r _ 6 + r _ 8 r _ 9 = 0 , r _ 1 r _ 3 + r _ 4 r _ 6 + r _ 7 r _ 9 = 0 , \\end{align*}"}
+{"id": "2493.png", "formula": "\\begin{align*} k _ { { \\mathbf u } , \\zeta } ( z ) = \\frac { 1 - \\overline { { \\mathbf u } ( \\zeta ) } { \\mathbf u } ( z ) } { 1 - \\overline \\zeta z } , \\ \\ \\ z \\in \\mathbb D , \\end{align*}"}
+{"id": "6858.png", "formula": "\\begin{align*} J ( x ^ * ) = \\underset { x \\in K } { \\inf } J ( x ) . \\end{align*}"}
+{"id": "8816.png", "formula": "\\begin{align*} B _ { 1 1 } & = A _ { 1 1 } ^ * P _ 1 & B _ { 1 2 } & = A _ { 2 1 } ^ * P _ 2 \\ , , \\\\ B _ { 2 1 } & = A _ { 1 2 } ^ * P _ 1 & B _ { 2 2 } - A _ { 2 2 } ^ * P _ 2 & = D _ { P _ 2 } F _ { 2 2 2 } D _ { P _ 2 } \\ , . \\end{align*}"}
+{"id": "7466.png", "formula": "\\begin{align*} [ W ] ( v _ 1 , \\dots , v _ n ) = [ \\hat { W } ] ( v _ { a _ 1 } , \\dots , v _ { a _ { 2 k } } ) \\end{align*}"}
+{"id": "8949.png", "formula": "\\begin{align*} \\mathcal { B } ( a , b ) = \\int \\limits _ { 0 } ^ { 1 } t ^ { a - 1 } ( 1 - t ) ^ { b - 1 } d t = \\frac { \\Gamma ( a ) \\Gamma ( b ) } { \\Gamma ( a + b ) } . \\end{align*}"}
+{"id": "8869.png", "formula": "\\begin{align*} K _ { 0 , m } ( z , w ) = \\pi ^ { - n } \\left ( 2 m + n \\right ) \\frac { ( m + n - 1 ) ! } { m ! } P _ { m } ^ { ( n - 1 , 0 ) } ( \\cos 2 d _ { F S } ( z , w ) ) , \\end{align*}"}
+{"id": "5397.png", "formula": "\\begin{align*} \\mathsf { E } _ A ( G ) = D _ A ( G ) + | G | - 1 \\end{align*}"}
+{"id": "1469.png", "formula": "\\begin{align*} b = \\frac { | G | } { | G _ \\Delta | } = \\frac { 2 m ! ^ 2 } { | G _ \\Delta | } . \\end{align*}"}
+{"id": "3198.png", "formula": "\\begin{align*} E _ { r + 2 } ( \\psi , \\tau ) = ( * ) + \\sum _ { n \\ge 1 } q ^ n \\sum _ { \\substack { n = d _ 1 d _ 2 \\\\ ( N _ 1 , d _ 1 ) = ( N _ 2 , d _ 2 ) = 1 } } \\psi ( d _ 1 ) \\tau ( d _ 2 ) d _ 2 ^ { r + 1 } , \\end{align*}"}
+{"id": "105.png", "formula": "\\begin{align*} ( x \\ast y ) \\ast z - ( y \\ast x ) \\ast z & = ( x \\circ ( u y ) ) \\circ ( u z ) - ( y \\circ ( u x ) ) \\circ ( u z ) \\\\ & = ( ( x \\circ y ) u ) \\circ ( u z ) - ( ( y \\circ x ) u ) \\circ ( u z ) \\\\ & = x \\circ ( y ( u \\circ ( u z ) ) ) - y \\circ ( x ( u \\circ ( u z ) ) ) \\\\ & = x \\circ ( ( u y ) \\circ ( u z ) ) ) - y \\circ ( ( u x ) \\circ ( u z ) ) ) \\\\ & = x \\circ ( u ( y \\circ ( u z ) ) ) - y \\circ ( u ( x \\circ ( u z ) ) ) \\\\ & = x \\ast ( y \\ast z ) - y \\ast ( x \\ast z ) , \\end{align*}"}
+{"id": "7674.png", "formula": "\\begin{align*} [ [ \\bigoplus _ { I } \\sum _ { p \\in \\mathbb { Z } } \\eta _ { I , p } ] ] = [ [ \\bigoplus _ { I } \\eta _ { I , - \\iota _ { E } ( \\omega ) } ] ] . \\end{align*}"}
+{"id": "4466.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n v _ i < \\sum _ { i = 1 } ^ n u ' _ i . \\end{align*}"}
+{"id": "1738.png", "formula": "\\begin{align*} \\mu ( X ^ { \\underline m } Y ^ { \\underline k - 2 - \\underline m } ) = \\rho \\circ s ( \\mu ) ( X ^ { \\underline m } Y ^ { \\underline k - 2 - \\underline m } ) = C _ n \\cdot \\rho \\circ \\varphi _ n ( ( x ^ { 2 n } ) ^ \\vee ) ( X ^ { \\underline m } Y ^ { \\underline k - 2 - \\underline m } ) = C _ n \\frac { ( - 1 ) ^ { m _ { \\rm i d } + m _ c } } { n + M + 1 } \\binom { n + M } { m _ { \\rm i d } } ^ { - 1 } \\end{align*}"}
+{"id": "7512.png", "formula": "\\begin{align*} Z _ g ( s , \\chi , D ) = { \\left \\{ \\begin{array} { r l } M _ 1 ( q ^ { - s } ) , \\ \\ & { \\rm o r d } ( \\alpha _ { i _ 0 } ) < { \\rm o r d } ( \\beta _ { j _ 0 } ) , \\\\ \\dfrac { M _ 2 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\ \\ & { \\it o t h e r w i s e } , \\end{array} \\right . } \\end{align*}"}
+{"id": "3852.png", "formula": "\\begin{align*} T _ { n } ^ 2 = \\delta _ { n , 2 } + T _ { n - 1 } ^ 2 + 3 T _ { n - 2 } ^ 2 + 9 T _ { n - 3 } ^ 2 + 4 \\sum _ { l = 4 } ^ { n - 2 } ( T _ l + T _ { l - 1 } ) T _ { n - l } ^ 2 . \\end{align*}"}
+{"id": "1069.png", "formula": "\\begin{align*} \\mathrm { T V } ( P _ 0 , P ) = \\sup _ { A } | P _ 0 ( A ) - P ( A ) | = \\varepsilon \\sup _ { A } | P ( A ) - G ( A ) | \\leq \\varepsilon , \\end{align*}"}
+{"id": "9134.png", "formula": "\\begin{align*} N ( m | n ) = \\sum _ { k = 0 } ^ n { n \\choose k } N ( m + k - 1 | n - k ) . \\end{align*}"}
+{"id": "3373.png", "formula": "\\begin{align*} \\allowdisplaybreaks - u _ { n - 1 } + z ' _ j & = - 2 u _ { n - 1 } + ( ( u _ n - 1 ) - \\lambda ) \\vee ( - 1 ) ) \\\\ & = - ( u _ n - 1 ) + ( ( u _ n - 1 ) - \\lambda ) \\vee ( - 1 ) ) \\\\ & \\leq ( - \\lambda ) \\vee ( - u _ n ) = z ' _ n \\vee ( - u _ n ) \\ , . \\end{align*}"}
+{"id": "3150.png", "formula": "\\begin{align*} A ( y ) = \\frac { 1 } { r ( y ) } \\ , \\mathrm { d i a g } \\left ( a ( y ) , 2 - a ( y ) \\right ) \\quad y \\in \\R ^ 2 , \\end{align*}"}
+{"id": "155.png", "formula": "\\begin{align*} \\| h _ { j } ( \\varkappa ) - h _ { j } ( y ) \\| _ { \\mathbb { X } } & = K L _ { \\nu _ { j } } \\| \\varkappa - y \\| _ { \\mathbb { X } } + K L _ { q } b \\kappa _ { b } \\| \\tilde { \\varkappa } _ { \\eta } - \\tilde { y } _ { \\eta } \\| _ { D } . \\end{align*}"}
+{"id": "2837.png", "formula": "\\begin{align*} a _ k = \\rho + \\tau , a _ { \\ell } = \\rho - \\tau , \\end{align*}"}
+{"id": "4596.png", "formula": "\\begin{align*} u _ n ( \\theta ) = \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } . \\end{align*}"}
+{"id": "2402.png", "formula": "\\begin{align*} ( \\widetilde E , \\widetilde A ) = ( Q ^ T E Q , Q ^ T A Q - Q ^ T E \\dot Q ) \\end{align*}"}
+{"id": "1779.png", "formula": "\\begin{align*} \\ell : = \\min \\mathsf { L ( \\beta ) } L : = \\max \\mathsf { L ( \\beta ) } . \\end{align*}"}
+{"id": "1624.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { \\tau } } = \\mathcal { Q } ^ { - 1 } \\left ( \\frac { M \\tilde \\epsilon ^ * } { 2 } \\right ) + \\mathcal { Q } ^ { - 1 } \\left ( \\frac { M \\tilde \\epsilon ^ * } { 2 ( M - 1 ) } \\right ) \\end{align*}"}
+{"id": "4362.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { x + i } < \\theta \\end{align*}"}
+{"id": "454.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { m } c _ k ^ r \\leq \\sum _ { k = 1 } ^ { m } \\lambda _ k \\sum _ { j = 1 } ^ { n } c _ j ^ r = \\sum _ { k = 1 } ^ { d } \\lambda _ k , \\end{align*}"}
+{"id": "4216.png", "formula": "\\begin{align*} \\begin{cases} \\sinh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) = \\sinh ( 2 \\lambda ) + 2 \\cosh ( 2 \\lambda ) \\tilde { \\Lambda } + 4 \\sinh ( 2 \\lambda + 2 \\xi _ 1 ) \\tilde { \\Lambda } ^ 2 , \\\\ \\cosh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) = \\cosh ( 2 \\lambda ) + 2 \\sinh ( 2 \\lambda ) \\tilde { \\Lambda } + 4 \\cosh ( 2 \\lambda + 2 \\xi _ 2 ) \\tilde { \\Lambda } ^ 2 , \\end{cases} \\end{align*}"}
+{"id": "3070.png", "formula": "\\begin{align*} - A : D ^ 2 \\eta ^ { k l } = m _ { k l } ( a - \\bar { a } ) = ( c _ { k l } + m _ { k l } a ) - \\int _ Y r ( c _ { k l } + m _ { k l } a ) = a _ { k l } - \\int _ Y r a _ { k l } . \\end{align*}"}
+{"id": "2674.png", "formula": "\\begin{align*} H ( x ) = 1 - \\big ( ( \\lambda x + C ) ( 1 - q ) \\big ) ^ { - { q \\over 1 - q } } , \\end{align*}"}
+{"id": "2764.png", "formula": "\\begin{align*} \\partial ^ { A } \\triangleright F = \\left ( \\partial _ { \\operatorname * { c l a } } ^ { A } + \\partial _ { \\operatorname * { c o r } } ^ { A } \\right ) \\triangleright F . \\end{align*}"}
+{"id": "1654.png", "formula": "\\begin{align*} T ( F _ \\infty ) _ + = T ( F _ { \\infty } ) _ 0 \\times S , \\end{align*}"}
+{"id": "3670.png", "formula": "\\begin{align*} M _ { \\rho } ( \\phi , \\varphi ) & : = \\phi - [ \\phi + c \\varphi ] _ \\rho . \\end{align*}"}
+{"id": "6924.png", "formula": "\\begin{align*} H ( t , \\tilde { x } ( t ) , \\tilde { \\psi } ( t ) , \\tilde { u } ( t ) ) = \\min \\limits _ { u \\in [ 0 , 1 ] } H ( t , \\tilde { x } ( t ) , \\tilde { \\psi } ( t ) , u ) \\end{align*}"}
+{"id": "7726.png", "formula": "\\begin{align*} G _ { \\beta } = \\sqrt { D _ { \\beta } - \\sqrt { A ^ - _ { \\beta } } } , \\end{align*}"}
+{"id": "4670.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { \\min \\{ 2 p + 1 , 2 m \\} } A _ { m , \\ell } = \\sum _ { \\ell = 0 } ^ { 2 p + 1 } \\sum _ { m = \\lceil \\ell / 2 \\rceil } ^ { \\infty } A _ { m , \\ell } . \\end{align*}"}
+{"id": "7584.png", "formula": "\\begin{align*} \\lambda _ k = k ^ 2 \\pi ^ 2 \\ \\ e _ k ( \\xi ) = \\sqrt { \\frac { 2 } { \\pi } } \\sin ( k \\pi \\xi ) , \\ k = 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "1922.png", "formula": "\\begin{align*} - a \\Delta _ g u _ \\sigma + V u _ \\sigma & = 0 \\quad M _ \\sigma \\\\ \\nu _ g ( u _ \\sigma ) & = 0 \\quad \\partial M _ \\sigma , \\\\ u _ \\sigma - 1 & \\in W ^ { 2 , p } _ { - q } ( M _ \\sigma ) \\end{align*}"}
+{"id": "769.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\nu \\left ( \\xi \\in \\partial \\Gamma : \\left | \\frac { \\log \\| \\rho ( \\xi _ n ) \\| } { n } - \\Lambda \\right | > \\epsilon \\right ) < 0 . \\end{align*}"}
+{"id": "7040.png", "formula": "\\begin{align*} \\pi _ { 1 + \\sigma } & = - q _ 2 \\\\ \\pi _ { 2 + \\sigma } & = d _ { 1 , 0 } - a _ { 1 , 1 } q _ 2 \\\\ \\pi _ { 2 + 2 \\sigma } & = 4 d _ { 1 , 1 } + 2 q _ 4 - 2 q _ 2 q _ 3 - q _ 2 ^ 3 + ( 6 b _ 1 ^ 3 - 1 8 b _ 1 b _ 2 + 6 b _ 3 ) q _ 1 . \\end{align*}"}
+{"id": "1235.png", "formula": "\\begin{align*} \\int _ { X } h ( x ) \\bar \\mu ( x ) d x = \\int _ Y \\bar \\nu ( y ) d y \\int _ { T ^ { - 1 } ( y ) \\cap X ( v ^ s ) } \\frac { h ( x ) \\bar \\mu ( x ) } { \\bar \\nu ( y ) J T ^ y ( x ) } d \\mathcal H ^ { m - n } ( x ) . \\end{align*}"}
+{"id": "5255.png", "formula": "\\begin{align*} \\frac { d } { d t } g ( U , U ) = - 2 g ( A _ X X + T _ U X , U ) . \\end{align*}"}
+{"id": "2135.png", "formula": "\\begin{align*} R _ i ( L ) : = \\int _ { \\vert x \\vert \\geq L } J ^ { * ( i ) } ( x ) d x \\end{align*}"}
+{"id": "177.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq n \\\\ p \\notin \\mathcal { S } } } \\dfrac { \\log p } { p - 1 } \\leq \\dfrac { \\log ( 2 n ^ 2 + l ) } { n - 1 } \\left ( 2 \\log 4 \\dfrac { n } { \\log n } + \\sqrt { n } \\right ) + \\dfrac { \\log n } { \\log ( 2 n ^ 2 + l ) } \\sum _ { i = 1 } ^ { r } \\dfrac { \\gamma _ { l , p _ i } ( n ) \\log p _ i } { n - 1 } + \\dfrac { n \\log 4 } { n - 1 } . \\end{align*}"}
+{"id": "8976.png", "formula": "\\begin{align*} \\sup _ { 0 < t < T } \\int _ { \\partial B } | u _ { \\phi } ( t ) | ^ 2 d \\phi & + \\int _ 0 ^ T \\int _ B | \\nabla u _ { \\phi } | ^ 2 d x \\ , d t \\\\ & \\le C \\int _ { \\partial B } | u _ { 0 , \\phi } | ^ 2 d \\phi + C T R ^ { - 2 } E ( u _ 0 ) . \\end{align*}"}
+{"id": "5658.png", "formula": "\\begin{align*} \\phi _ 0 = \\phi _ 0 ^ { ( 0 ) } + \\epsilon \\phi _ 0 ^ { ( 1 ) } + \\epsilon ^ 2 \\phi _ 0 ^ { ( 2 ) } + \\ldots , \\end{align*}"}
+{"id": "4761.png", "formula": "\\begin{align*} \\int _ { ( \\Omega _ { q } ) _ { \\delta / 2 } } F ( x , u ) d x & \\leq c _ { 2 } ( \\| u \\| _ { L ^ { q _ { 1 } ^ { - } } ( \\Omega _ { q } ) _ { \\delta / 2 } } ^ { q _ { 1 } ^ { - } } + \\| u \\| ^ { \\beta } + \\| u \\| _ { L ^ { ( p ^ { * } ) ^ { - } } ( \\Omega _ { q } ) _ { \\delta / 2 } } ^ { ( p ^ { * } ) ^ { - } } ) \\\\ & \\leq c _ { 3 } ( \\| u \\| ^ { q _ { 1 } ^ { - } } + \\| u \\| ^ { \\beta } + \\| u \\| ^ { ( p ^ { * } ) ^ { - } } ) \\end{align*}"}
+{"id": "851.png", "formula": "\\begin{align*} L ( R , \\dot R ) = K = \\frac { 1 } { 2 } t r ( R ^ T \\dot R J _ d \\dot R ^ T R ) \\end{align*}"}
+{"id": "8643.png", "formula": "\\begin{align*} \\gamma = \\sum _ n p _ n \\sc { \\cdot , \\phi _ n } \\phi _ n , \\sum _ n p _ n = 1 , \\end{align*}"}
+{"id": "2890.png", "formula": "\\begin{align*} u ( \\zeta ) = \\int _ { \\mathbb { T } ^ \\infty } \\mathbf { P } _ \\zeta u d m _ \\infty . \\end{align*}"}
+{"id": "6895.png", "formula": "\\begin{align*} \\chi ( V , W ) = \\mbox { h o m } ( V , W ) - \\mbox { e x t } ( V , W ) . \\end{align*}"}
+{"id": "921.png", "formula": "\\begin{align*} \\psi \\ge v ( x ) : = \\min _ { z \\in M } u ( z ) + h ( z , x ) . \\end{align*}"}
+{"id": "61.png", "formula": "\\begin{align*} x _ i = \\begin{cases} 1 , & f ( i ) = 1 \\\\ 0 , & o t h e r w i s e . \\end{cases} \\end{align*}"}
+{"id": "970.png", "formula": "\\begin{align*} g \\bigl ( W ^ 1 _ { r + 2 } ( C ) \\bigr ) = 1 + \\frac { r } { r + 1 } { 2 r + 2 \\choose r } . \\end{align*}"}
+{"id": "5907.png", "formula": "\\begin{align*} A e _ r = 0 , 1 \\leq r \\leq p . \\end{align*}"}
+{"id": "313.png", "formula": "\\begin{align*} \\zeta _ { T ( E ) } | _ { S ( F ) } ( t ) = \\prod \\limits _ { \\substack { \\alpha \\in \\Phi _ { \\mathrm { s y m , r a m } } / \\Gamma _ { E } , \\\\ \\alpha ^ { \\mathrm { o p } } \\in \\Phi _ { \\mathrm { s y m , r a m } } / \\Gamma _ F } } \\omega _ { E _ \\alpha / F _ \\alpha } ( \\iota _ { \\alpha } \\alpha ( t ) ) . \\end{align*}"}
+{"id": "650.png", "formula": "\\begin{align*} | B \\cap B ( x , r ) | \\leq r ^ { \\kappa } | B | , x \\in \\R , \\ , \\delta \\leq r \\leq \\delta ^ { \\epsilon _ { 0 } } = \\delta ^ { \\bar { \\epsilon } _ { 0 } } . \\end{align*}"}
+{"id": "6268.png", "formula": "\\begin{align*} \\frac { d } { d t } ( x ' _ { t } , z ' _ { t } ) = \\big ( - x ' _ { t } z ' _ { t } , { x ' _ { t } } ^ { 2 } \\big ) . \\end{align*}"}
+{"id": "1407.png", "formula": "\\begin{align*} \\pi _ 2 ^ { ( + , - , - ) } = L ( \\Delta [ 0 , - 2 ] ^ 2 ; \\pi ( 0 ^ + , 1 ^ - , 1 ^ - ) ) . \\end{align*}"}
+{"id": "7873.png", "formula": "\\begin{align*} f ' = a _ 0 ( z ) + a _ 1 ( z ) f + a _ 2 ( z ) f ^ 2 , a _ 2 ( z ) \\not \\equiv 0 . \\end{align*}"}
+{"id": "8812.png", "formula": "\\begin{align*} A = \\begin{bmatrix} A _ { 1 1 } & A _ { 1 2 } \\\\ A _ { 2 1 } & A _ { 2 2 } \\end{bmatrix} \\ , , \\ , B = \\begin{bmatrix} B _ { 1 1 } & B _ { 1 2 } \\\\ B _ { 2 1 } & B _ { 2 2 } \\end{bmatrix} P = \\begin{bmatrix} P _ 1 & 0 \\\\ 0 & P _ 2 \\end{bmatrix} \\end{align*}"}
+{"id": "1425.png", "formula": "\\begin{align*} \\mathbb { P } ( t _ i - t _ { i - 1 } < T \\forall i ) \\leq \\mathbb { P } ( \\tau _ h = T ' ) + \\mathbb { P } ( t _ i - t _ { i - 1 } < T \\forall i , R _ { \\tau _ h } \\geq h ) , \\end{align*}"}
+{"id": "516.png", "formula": "\\begin{align*} H _ { \\delta } ( z ) : = F ( x ) + ( { \\delta } / { 2 } ) \\| x - u \\| ^ 2 \\quad { \\rm f o r } \\ z : = ( x , u ) \\in \\mathbb { X } \\times \\mathbb { X } . \\end{align*}"}
+{"id": "691.png", "formula": "\\begin{align*} \\Gamma = \\sum _ { k = 1 } ^ n \\Gamma _ k \\end{align*}"}
+{"id": "6923.png", "formula": "\\begin{align*} \\dot { \\psi } = - \\frac { \\partial H } { \\partial x } = \\bar { A } ( t , x , u ) \\psi + \\bar { B } ( x ) , \\end{align*}"}
+{"id": "8621.png", "formula": "\\begin{align*} x _ { n + 1 } = \\alpha _ n x _ n + ( 1 - \\alpha _ n ) \\tilde { T } x _ n \\end{align*}"}
+{"id": "8122.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { u _ 2 '' ( y _ 0 ) } } = \\sqrt { 6 } x ^ { \\frac { 3 } { 2 } } c ^ 5 p ^ { - \\frac { 5 } { 2 } } = \\sqrt { 6 } n _ 2 ^ { \\frac { 3 } { 2 } } n _ 1 ^ { 3 } c ^ { \\frac { 1 } { 2 } } m ^ { - \\frac { 3 } { 2 } } p ^ { - \\frac { 5 } { 2 } } \\end{align*}"}
+{"id": "4237.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\Delta \\phi + V \\phi - | \\phi | ^ { p - 1 } \\phi - \\Omega L _ z \\phi + \\omega \\phi = 0 . \\end{align*}"}
+{"id": "5147.png", "formula": "\\begin{align*} K _ { n } ( z ) = & \\frac { 1 } { 2 } \\left ( \\frac { z } { 2 } \\right ) ^ { - n } \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( n - k - 1 ) ! } { k ! } \\left ( \\frac { - z } { 4 } \\right ) ^ { k } + ( - 1 ) ^ { n + 1 } \\ln \\left ( \\frac { z } { 2 } \\right ) I _ { n } ( z ) \\\\ & + \\frac { 1 } { 2 } \\left ( \\frac { - z } { 2 } \\right ) ^ { n } \\sum _ { k = 0 } ^ { \\infty } \\left ( \\psi ( k + 1 ) + \\psi ( n + k + 1 ) \\right ) \\frac { \\left ( \\frac { z ^ { 2 } } { 4 } \\right ) ^ { k } } { k ! ( n + k ) ! } , \\end{align*}"}
+{"id": "8954.png", "formula": "\\begin{align*} P _ { k } ^ { ( a , b ) } ( x ) = \\frac { \\Gamma ( b + k + 1 ) } { k ! \\Gamma ( b + 1 ) } \\left ( \\frac { x - 1 } { 2 } \\right ) ^ { k } { } _ 2 F _ 1 \\left ( \\begin{matrix} - k , - a - k \\\\ b + 1 \\end{matrix} \\bigg | \\frac { x + 1 } { x - 1 } \\right ) , \\end{align*}"}
+{"id": "3264.png", "formula": "\\begin{align*} | \\chi _ S ( \\mathbf { d } ( c ) - \\tilde { B } _ { \\mathbf { d } - \\mathbf { e } _ a - \\mathbf { e } _ b , \\mathcal { A } } ( a b ) ) | = | \\chi _ S ( \\tilde { B } _ { \\mathbf { d } - \\mathbf { e } _ a - \\mathbf { e } _ b , \\mathcal { A } } ( a b ) ) | \\leq C _ 5 \\frac { d ^ 2 } { n } ( 2 4 d ) ^ { - | S | } ( 4 | S | ) ! . \\end{align*}"}
+{"id": "7440.png", "formula": "\\begin{align*} \\widehat \\Phi \\colon P \\mapsto \\begin{bmatrix} \\Phi _ { 1 1 } ( P ) & \\phi ( P ) \\\\ \\phi ( P ) & \\Phi _ { 2 2 } ( P ) \\end{bmatrix} \\end{align*}"}
+{"id": "879.png", "formula": "\\begin{align*} \\mathbf { E } [ d ] & = \\mathbf { E } [ \\mathbf { T r } ( ( \\mathcal { Z } ) ) ] = \\mathbf { T r } ( \\mathbf { E } [ ( \\mathcal { Z } ) ] ) \\geq n l \\lambda _ { \\min } ( \\mathbf { E } [ ( \\mathcal { Z } ) ] ) , \\end{align*}"}
+{"id": "3003.png", "formula": "\\begin{align*} \\mathcal { A } [ \\theta , \\pi ] = \\int _ { \\mathcal { Y } ^ N } \\frac { 1 } { 2 } \\| \\pi \\| ^ 2 e ^ { ( 4 ) } \\wedge \\bar { \\theta } ^ { ( r ) } + \\pi _ i \\wedge \\Theta ^ i \\end{align*}"}
+{"id": "1024.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 \\pi ) ^ n } \\int _ { \\mathbb { R } ^ n } | \\xi | ^ { s } | \\widehat { f } ( \\xi ) | ^ 2 d \\xi = \\frac { 2 ^ { 1 - s } \\Gamma ( 1 - s / 2 ) } { \\Gamma ( { s } / { 2 } ) } \\int _ { \\mathbb { R } ^ { n + 1 } _ { + } } | \\nabla _ { ( x , t ) } u ( x , t ) | ^ 2 t ^ { 1 - s } d x d t , \\end{align*}"}
+{"id": "696.png", "formula": "\\begin{align*} h _ { 1 1 } ( w ) = \\frac { \\pi } { 2 V } \\lambda ( w ) ^ 2 . \\end{align*}"}
+{"id": "6343.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Phi _ { \\nu _ 0 } ( t , h ( t ) ) = \\nu _ 0 ^ 3 f _ \\nu ( t , \\nu _ 0 ) ( m ( h ( t ) ) ^ { - 2 } - m ( t ) ^ { - 2 } ) . \\end{align*}"}
+{"id": "7665.png", "formula": "\\begin{align*} \\phi ( \\eta _ { I } ) = \\phi ( \\sum _ { p } \\eta _ { I , p } ) = \\sum _ { p \\neq - \\iota _ { E } ( w ) } \\frac { \\eta _ { I , p } } { p + \\iota _ { E } ( \\omega ) } . \\end{align*}"}
+{"id": "5360.png", "formula": "\\begin{align*} M = \\mu [ \\eta ] \\ , I _ m , \\end{align*}"}
+{"id": "8062.png", "formula": "\\begin{align*} L \\Bigl ( \\frac { 1 } { 2 } , f \\times u _ j \\Bigr ) & = \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { \\lambda _ j ( n ) A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } V _ - ( m ^ 2 n , t _ j ) + { } \\\\ & \\qquad { } + \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { \\lambda _ j ( n ) A ( m , n ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } V _ + ( m ^ 2 n , t _ j ) \\end{align*}"}
+{"id": "8410.png", "formula": "\\begin{align*} \\int \\partial _ { \\alpha } \\left ( P _ { \\alpha } ^ k ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\hat \\Psi _ { \\alpha } \\ : d m \\right ) & = \\int P _ { \\alpha } ^ k ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\partial _ { \\alpha } \\hat \\Psi _ { \\alpha } \\ : d m + \\int \\partial _ { \\alpha } [ P _ { \\alpha } ^ k ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) ] \\hat \\Psi _ { \\alpha } \\ : d m \\\\ & = ( I ) + ( I I ) \\end{align*}"}
+{"id": "389.png", "formula": "\\begin{align*} J _ 2 & \\leq ( C ^ { ( 1 ) } _ 2 + 2 4 ) T _ * L ^ { - 2 } _ * T ^ { - 1 / 2 } _ * ( 1 + T ^ { - 3 / 2 } _ * ) + 3 T _ * L ^ { - 2 } _ * T ^ { - 1 / 2 } _ * ( 1 + T ^ { - 1 / 2 } _ * ) + 1 1 T _ * L ^ { - 2 } _ * T ^ { - 1 / 2 } _ * ( 1 + T ^ { - 3 / 4 } _ * ) \\\\ & \\leq K _ 1 T _ * L ^ { - 2 } _ * T ^ { - 1 / 2 } _ * ( 1 + T ^ { - 1 / 2 } _ * + T ^ { - 3 / 4 } _ * + T ^ { - 3 / 2 } _ * ) \\ , . \\end{align*}"}
+{"id": "7173.png", "formula": "\\begin{align*} \\Sigma _ t = \\left \\{ z _ { \\alpha } ( x _ { j } , \\ , t ) , \\ , z _ { \\alpha , j } ( x _ { j } , \\ , t ) \\right \\} . \\end{align*}"}
+{"id": "7126.png", "formula": "\\begin{align*} a = \\epsilon ( a \\beta ( \\rho _ 1 ) ) = \\epsilon ( d ) = \\left \\langle d , x ^ 0 \\right \\rangle , \\end{align*}"}
+{"id": "4709.png", "formula": "\\begin{align*} G _ { I V } : & \\ \\varphi _ 8 = \\frac { 1 } { 2 } \\left ( x ^ { 2 } + 3 y ^ { 2 } \\right ) , \\\\ & \\ \\varphi _ { 2 4 } = \\frac { 1 } { 3 2 } ( 1 1 x ^ { 6 } + 4 5 x ^ { 4 } y ^ { 2 } + 4 0 5 x ^ { 2 } y ^ { 4 } + 2 4 3 y ^ { 6 } ) . \\\\ \\end{align*}"}
+{"id": "153.png", "formula": "\\begin{align*} ( \\delta \\varkappa ) ( \\theta ) = z ( \\theta ) , \\end{align*}"}
+{"id": "2147.png", "formula": "\\begin{align*} q ( t ) : = q _ 0 e ^ { - \\mu t } + b g ( t ) = q _ 0 e ^ { - \\mu t } + b \\int _ { 0 } ^ { t } e ^ { - \\mu ( t - s ) } ( 1 - U ( c s + s _ 0 ) ) d s , t \\geq 0 . \\end{align*}"}
+{"id": "7075.png", "formula": "\\begin{align*} \\pi _ { m + n \\sigma } & = [ \\mathbf { C P } ( m + n \\sigma ) ] \\in \\Omega ^ { C _ 2 } _ * \\\\ \\Pi _ { m + n \\sigma } & = [ \\mathbf { C P } ( m + n \\sigma ) \\to \\mathbf { C P } ^ \\infty _ { C _ 2 } ] \\in \\Omega ^ { C _ 2 } _ * ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) . \\end{align*}"}
+{"id": "571.png", "formula": "\\begin{align*} m _ t : = \\mathbb { E } ^ \\dagger [ \\mu _ t ] . \\end{align*}"}
+{"id": "3036.png", "formula": "\\begin{align*} w = \\sum \\limits _ { j = 0 } ^ { s _ 0 } \\sum \\limits _ { s = s _ 0 - j - 6 } ^ { { \\infty } } w ^ { s , j } _ { k } \\ \\ \\mathrm { i n } \\ \\ \\Omega _ 1 . \\end{align*}"}
+{"id": "7033.png", "formula": "\\begin{align*} \\Pi _ { m + n \\sigma } & = [ \\mathbf { C P } ( m + n \\sigma ) \\longrightarrow \\mathbf { C P } ^ { \\infty } _ { C _ 2 } ] \\in M U ^ { C _ 2 } _ * ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) . \\end{align*}"}
+{"id": "6367.png", "formula": "\\begin{align*} m _ \\lambda ' ( r ) = \\frac { y ' ( u ( r ) ) } { \\sqrt { x _ \\lambda ' ( u ( r ) ) ^ 2 + y ' ( u ( r ) ) ^ 2 } } . \\end{align*}"}
+{"id": "2987.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { E } _ n \\bigg [ \\sup _ { 0 \\le t \\le T } \\bigg | \\frac { 1 } { \\sqrt { n } } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\overline { W } _ { j - 1 } ( s ) \\overline { W } _ { j } ( s ) \\overline { W } _ { j + 1 } ( s ) \\nabla ^ n \\varphi ^ n _ j ( s ) d s \\bigg | ^ 2 \\bigg ] \\\\ & \\le C T ^ 2 \\| \\partial _ x \\varphi \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } \\le C \\frac { T ^ { 3 / 2 } } { n } \\| \\partial _ x \\varphi \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } . \\end{aligned} \\end{align*}"}
+{"id": "2189.png", "formula": "\\begin{align*} ( ( 1 - 2 \\alpha - \\beta ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + 1 ) \\mathit { e } = ( - 3 + 2 \\alpha + \\beta ) r ^ 2 + ( 2 - 2 \\alpha + \\beta ) r + 1 . \\end{align*}"}
+{"id": "9259.png", "formula": "\\begin{align*} \\begin{aligned} ( s _ { j k } w ) ^ { - 1 } ( k ) & = ( w ^ { - 1 } s _ { j k } ) ( k ) = w ^ { - 1 } ( j ) \\leq h ( ( s _ { j k } w ) ^ { - 1 } ( k + 1 ) ) = h ( w ^ { - 1 } ( k + 1 ) ) . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "3592.png", "formula": "\\begin{align*} m _ { i j } = \\lambda _ i , ( 1 \\leq i \\leq j \\leq n ) , \\end{align*}"}
+{"id": "9227.png", "formula": "\\begin{align*} \\lll \\widetilde { u } = \\widetilde { r } . \\end{align*}"}
+{"id": "7024.png", "formula": "\\begin{align*} \\langle d , x ^ { m + n \\sigma } \\rangle = \\langle \\Delta d , \\underbrace { x \\otimes \\cdots \\otimes x } _ { m } \\otimes \\underbrace { x ^ { \\sigma } \\otimes \\cdots \\otimes x ^ \\sigma } _ { n } \\rangle . \\end{align*}"}
+{"id": "3111.png", "formula": "\\begin{align*} A ^ 1 : = A ^ 0 + \\delta P \\end{align*}"}
+{"id": "1280.png", "formula": "\\begin{align*} \\eta _ j ( \\theta ^ { 1 + j r ' } ) = 1 { \\rm ~ a n d ~ } \\eta _ j ( \\theta ^ { 1 + j ' r ' } ) = 0 { \\rm ~ f o r ~ } j \\neq j ' . \\end{align*}"}
+{"id": "2641.png", "formula": "\\begin{align*} M _ 2 & = \\sum _ { \\substack { r _ 1 , r _ 2 \\geq 0 \\\\ a } } \\frac { 1 } { 2 ^ { r _ 1 } 3 ^ { r _ 2 / 2 } N ( a ) ^ { 1 / 2 } } V \\left ( \\frac { \\pi 2 ^ { 2 r _ 1 + 1 } 3 ^ { r _ 2 } N ( a ) } { z } \\right ) \\sum _ { c \\equiv 1 \\bmod { 3 6 } } \\ \\frac { g _ 6 ( c ) \\overline { \\chi _ c } ( a ) } { N ( c ) ^ { 1 / 2 } } W \\left ( \\frac { N ( c ) } { y } \\right ) . \\end{align*}"}
+{"id": "3760.png", "formula": "\\begin{align*} & - \\lim _ { s \\to 0 } \\left ( \\frac { d } { d s } \\zeta ( s ) ^ { - 1 } Z _ { r _ 2 } ( f , s ) \\right ) + \\epsilon ( 1 ) \\lim _ { s \\to 0 } \\left ( \\frac { d } { d s } \\zeta ( 1 + s ) ^ { - 1 } Z _ { r _ 1 } ( d _ 2 ( f ) , s + 1 ) \\right ) \\\\ & = \\sum _ { v } ( \\log q _ v ) \\bigg ( ( e ( \\psi _ v ) + 1 ) \\tilde { c } _ { 2 , v } ( I ( f _ v ) ) - \\tilde { a } _ { 2 , v } ( I ( f _ v ) ) \\bigg ) \\prod _ { v ' \\neq v } \\tilde { c } _ { 2 , v ' } ( I ( f _ { v ' } ) ) . \\end{align*}"}
+{"id": "8819.png", "formula": "\\begin{align*} B _ { 1 2 } ^ * P _ 1 ^ 2 = A _ { 2 1 } P _ 1 = P _ 2 A _ { 2 1 } = P _ 2 B _ { 1 2 } ^ * P _ 1 . \\end{align*}"}
+{"id": "6682.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{aligned} \\| g ^ k \\| ^ 2 & = m ^ 2 \\| \\nabla f ( x ^ k ) - \\nabla f ( x ^ * ) + \\nabla f ( x ^ * ) \\| ^ 2 \\\\ & \\leq 2 m ^ 2 \\| \\nabla f ( x ^ k ) - \\nabla f ( x ^ * ) \\| ^ 2 + 2 m ^ 2 \\| \\nabla f ( x ^ * ) \\| ^ 2 \\\\ & \\le 2 L ^ 2 \\| x ^ k - x ^ * \\| ^ 2 + 2 m ^ 2 \\| \\nabla f ( x ^ * ) \\| ^ 2 \\end{aligned} \\end{aligned} \\end{align*}"}
+{"id": "5345.png", "formula": "\\begin{align*} \\sigma _ j [ \\varepsilon ] = \\min _ { \\substack { V _ j \\subset H ^ 1 _ { \\rm \\scriptscriptstyle N } ( \\Omega ) , \\\\ \\operatorname { d i m } V _ j = j } } \\max _ { \\substack { u \\in V _ j , \\\\ u \\neq 0 } } \\frac { \\int _ { \\Omega } \\abs { \\operatorname { c u r l } u } ^ 2 d x + \\int _ \\Omega \\abs { \\operatorname { d i v } ( \\varepsilon u ) } ^ 2 d x } { \\int _ { \\Omega } \\varepsilon u \\cdot u \\ , d x } , \\end{align*}"}
+{"id": "5722.png", "formula": "\\begin{align*} d s ^ { 2 } = \\epsilon _ { A B } \\epsilon _ { A ^ { \\prime } B ^ { \\prime } } \\theta ^ { A A ^ { \\prime } } \\otimes \\theta ^ { B B ^ { \\prime } } \\end{align*}"}
+{"id": "180.png", "formula": "\\begin{align*} & ( t + r + 1 ) \\cdots ( t + n ) \\left [ \\underset { i = 1 } { \\overset { s } { \\sum } } \\binom { m _ i + r - 1 } { r } + \\dim _ K ( I _ { \\alpha } ) _ t \\right ] \\\\ & = ( r + 1 ) \\cdots ( n - 1 ) n \\left [ \\underset { i = 1 } { \\overset { s } { \\sum } } \\binom { m _ i + n - 1 } { n } + \\dim _ K I _ t \\right ] . \\end{align*}"}
+{"id": "1139.png", "formula": "\\begin{align*} d _ 0 ( e _ s ) & = s \\otimes 1 - 1 \\otimes s \\\\ d _ 1 ( e _ { s _ 0 , s _ 1 } ) & = ( s _ 0 \\otimes 1 - 1 \\otimes s _ 0 ) e _ { s _ 1 } - ( s _ 1 \\otimes 1 - 1 \\otimes s _ 1 ) e _ { s _ 0 } . \\end{align*}"}
+{"id": "8832.png", "formula": "\\begin{align*} H ( x , y , t ) = \\sum \\limits _ { k = 0 } ^ { + \\infty } e ^ { - \\lambda _ k t } \\phi _ k ( x ) \\phi _ k ( y ) \\end{align*}"}
+{"id": "4995.png", "formula": "\\begin{align*} \\partial _ Q S ( q _ { j - 1 } , q _ j ) + \\partial _ q S ( q _ j , q _ { j + 1 } ) = 0 \\forall j \\in \\{ 1 , \\ldots , n - 1 \\} . \\end{align*}"}
+{"id": "5350.png", "formula": "\\begin{align*} \\Lambda _ { F , s } [ \\varepsilon ] : = \\sum _ { \\substack { j _ 1 , \\dots , j _ s \\in F \\\\ j _ 1 < \\dots < j _ s } } \\lambda _ { j _ 1 } [ \\varepsilon ] \\cdots \\lambda _ { j _ s } [ \\varepsilon ] . \\end{align*}"}
+{"id": "3165.png", "formula": "\\begin{align*} r ( y ) = \\prod _ { i = 1 } ^ n \\left ( \\int _ 0 ^ 1 \\frac { \\mathrm { d } t } { a _ i ( t ) } \\right ) ^ { - 1 } \\frac { 1 } { a _ i ( y _ i ) } \\quad y = ( y _ 1 , \\dots , y _ n ) \\in \\R ^ n . \\end{align*}"}
+{"id": "5059.png", "formula": "\\begin{align*} F = \\frac { P } { ( 1 - \\omega _ 1 u _ 1 ) \\cdots ( 1 - \\omega _ l u _ l ) } \\end{align*}"}
+{"id": "7742.png", "formula": "\\begin{align*} \\widetilde f _ \\alpha ( s ) & = \\exp ( - s ^ \\alpha ) \\end{align*}"}
+{"id": "4510.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n v _ i = \\sum _ { i = 1 } ^ n e ^ { b _ i } < \\sum _ { i = 1 } ^ n e ^ { a _ i } = \\sum _ { i = 1 } ^ n u _ i . \\end{align*}"}
+{"id": "755.png", "formula": "\\begin{align*} \\tilde { U } _ \\ell \\varphi ( \\tilde { b } ) = \\varphi \\left ( \\tilde { b } \\tilde { \\varpi } _ \\ell \\right ) , \\end{align*}"}
+{"id": "5644.png", "formula": "\\begin{align*} \\zeta ^ t ( F ) : = \\sum _ { \\substack { n _ v \\geq 1 \\\\ v \\in V ( F ) } } \\prod _ { v \\in V ( F ) } \\left ( \\sum _ { \\substack { v ' \\in V ( F ) \\\\ v ' \\succeq v } } n _ { v ' } \\right ) ^ { - \\alpha _ v } \\end{align*}"}
+{"id": "3627.png", "formula": "\\begin{align*} p _ { F } ^ { * } ( \\mu ) : = 1 + \\frac { \\mu } { \\lambda _ 1 } : = 1 + \\frac { 4 \\mu } { ( n - 1 ) ^ { 2 } } , \\end{align*}"}
+{"id": "6350.png", "formula": "\\begin{align*} \\psi _ f ( \\nu ) = - 2 \\int _ 0 ^ \\infty A _ f \\circ f ^ { - 1 } ( \\sqrt { u ( \\tau , \\nu ) } ) / ( a ^ 2 \\tau ^ 2 + 1 ) d \\tau , \\end{align*}"}
+{"id": "3001.png", "formula": "\\begin{align*} \\bar { \\theta } ^ { ( r - \\alpha ) } _ { i _ 1 \\cdots i _ \\alpha } = \\frac { 1 } { ( r - \\alpha ) ! } \\epsilon _ { i _ 1 \\cdots i _ \\alpha j _ { \\alpha + 1 } \\cdots j _ r } \\theta ^ { j _ { \\alpha + 1 } } \\wedge \\cdots \\wedge \\theta ^ { j _ r } \\end{align*}"}
+{"id": "6040.png", "formula": "\\begin{align*} f ( z ) - \\frac { p _ n ( z ) } { q _ n ( z ) } = \\frac { v _ n ( z ) } { q _ n ^ 2 ( z ) } \\int \\frac { q _ n ^ 2 ( x ) } { v _ n ( x ) } \\frac { \\dd \\mu ( x ) } { z - x } . \\end{align*}"}
+{"id": "5098.png", "formula": "\\begin{align*} a c ^ n - \\sum _ { b \\in B ' } b g _ b ( n ) = 0 \\end{align*}"}
+{"id": "1169.png", "formula": "\\begin{align*} \\mu _ 0 | \\xi | ^ 2 \\le \\sum _ { i , j = 1 } ^ n a _ { i j } ( x ) \\xi _ i \\xi _ j \\le \\mu _ 0 ^ { - 1 } | \\xi | ^ 2 . \\end{align*}"}
+{"id": "1881.png", "formula": "\\begin{align*} \\tilde { \\alpha } = \\min _ { \\alpha } b _ u ( \\alpha ) . \\end{align*}"}
+{"id": "6686.png", "formula": "\\begin{align*} \\begin{aligned} x ^ { k + 1 } ( \\ell ) & - [ \\bar x ^ { k + 1 } ] _ \\ell { \\bf 1 } = ( I + \\gamma _ 1 ^ k R ) \\left ( x ^ k ( \\ell ) - [ \\bar x ^ k ] _ \\ell { \\bf 1 } \\right ) \\\\ & \\quad + \\gamma _ 1 ^ k \\left ( \\zeta _ w ^ k ( \\ell ) - [ \\bar \\zeta _ w ^ k ] _ \\ell { \\bf 1 } \\right ) - \\lambda ^ k \\left ( I - \\frac { { \\bf 1 } u ^ T } { m } \\right ) y ^ k ( \\ell ) \\end{aligned} \\end{align*}"}
+{"id": "9157.png", "formula": "\\begin{align*} ( - \\Delta _ J ) ^ { - 1 } ( x , y ) \\sim - \\frac { 1 } { 2 \\pi v _ J ^ 2 } \\log | x - y | , v _ J ^ 2 = \\frac { 1 } { 2 | J | } \\sum _ { x \\in J } x _ 1 ^ 2 . \\end{align*}"}
+{"id": "4883.png", "formula": "\\begin{align*} C _ { s _ 2 } C _ { x _ l } = C _ { s _ 2 x _ l } + \\Box \\in C _ { s _ 2 x _ l } + H ^ { < 1 3 } . \\end{align*}"}
+{"id": "5839.png", "formula": "\\begin{align*} \\mathcal H _ 0 = L ^ 2 ( \\Omega ) , \\mathcal H _ 1 = H ^ 1 _ { \\Gamma _ 0 } ( \\Omega ) , \\end{align*}"}
+{"id": "5989.png", "formula": "\\begin{align*} \\chi _ H ( \\alpha \\boxtimes \\beta ) & = \\chi _ H ( p _ 1 ^ * \\alpha \\otimes p _ 2 ^ * \\beta ) \\\\ & = \\chi _ H ( \\alpha \\otimes ( p _ 1 ) _ * p _ 2 ^ * \\beta ) \\\\ & = ( \\pi _ 1 ) _ * \\left ( \\alpha \\otimes \\pi _ 1 ^ * ( ( \\pi _ 2 ) _ * \\beta ) \\right ) \\\\ & = \\chi _ H ( \\alpha ) \\chi _ H ( \\beta ) , \\end{align*}"}
+{"id": "3450.png", "formula": "\\begin{align*} 4 n = x ^ 2 + y ^ 2 + 1 0 z ^ 2 . \\end{align*}"}
+{"id": "6333.png", "formula": "\\begin{align*} \\varphi ' ( \\nu _ 0 ) + \\int _ 1 ^ { r ( q ) } f _ \\nu ( x , \\nu _ 0 ) d x - \\int _ { r ( q _ c ) } ^ 1 f _ \\nu ( x , \\nu _ 0 ) d x = 0 \\end{align*}"}
+{"id": "5245.png", "formula": "\\begin{align*} \\Delta f = d i v ( \\nabla f ) . \\end{align*}"}
+{"id": "8461.png", "formula": "\\begin{align*} G ( \\gamma ) = 2 \\gamma ( X ) - \\| \\gamma \\| ^ 2 \\geqslant \\| \\gamma \\| ^ 2 . \\end{align*}"}
+{"id": "2738.png", "formula": "\\begin{align*} T ' _ { i , e } ( v ) & = \\sum _ { \\substack { a , b , c \\geq 0 ; \\\\ a - b + c = m } } ( - 1 ) ^ b q _ i ^ { e ( b - a c ) } F _ i ^ { ( a ) } E _ i ^ { ( b ) } F _ i ^ { ( c ) } v , v \\in M _ m , \\\\ T '' _ { i , e } ( v ) & = \\sum _ { \\substack { a , b , c \\geq 0 ; \\\\ - a + b - c = m } } ( - 1 ) ^ b q _ i ^ { e ( b - a c ) } E _ i ^ { ( a ) } F _ i ^ { ( b ) } E _ i ^ { ( c ) } v , v \\in M _ m . \\end{align*}"}
+{"id": "8886.png", "formula": "\\begin{align*} \\frac { ( 2 m + 2 \\nu + n ) \\sin ( 2 m + 2 \\nu + n ) \\psi } { \\sin \\psi } = - \\frac { d } { \\sin \\psi d \\psi } ( \\cos ( 2 m + 2 \\nu + n ) \\psi ) . \\end{align*}"}
+{"id": "1468.png", "formula": "\\begin{align*} Y & = \\frac { k ( k - 1 ) ( m - 1 ) ( m - 2 ) } { ( m + 1 ) ( m ^ 2 - 2 ) } + \\frac { k ( k - 1 ) ( m - 2 ) } { ( m + 1 ) ( m ^ 2 - 2 ) } + \\frac { 2 k ( k - 1 ) ( m - 1 ) } { ( m + 1 ) ( m ^ 2 - 2 ) } \\\\ & = \\frac { k ( k - 1 ) } { ( m + 1 ) ( m ^ 2 - 2 ) } \\left ( ( m - 1 ) ( m - 2 ) + ( m - 2 ) + 2 ( m - 1 ) \\right ) \\\\ & = \\frac { k ( k - 1 ) } { ( m + 1 ) ( m ^ 2 - 2 ) } \\left ( m ^ 2 - 2 \\right ) = \\frac { k ( k - 1 ) } { m + 1 } . \\end{align*}"}
+{"id": "7625.png", "formula": "\\begin{align*} \\mathcal { A } ^ { ( 2 ) } & = - \\big ( \\c ( v _ n ) , \\psi _ k ' ( w _ n ) \\big ) \\leq \\big ( | \\c ( v _ n ) | , \\psi _ k ' ( w _ n ) \\big ) \\leq C ( \\gamma , \\delta , n ) \\int _ { 0 } ^ { 1 } 1 \\cdot | u _ n - v _ n | _ + \\d x \\\\ & \\leq C ( \\gamma , \\delta , n ) \\int _ { 0 } ^ { 1 } | w _ n ( x ) | _ + ^ 2 \\d x . \\end{align*}"}
+{"id": "393.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\gamma } [ ( B ( t ) ) ^ 2 ] = \\sigma ^ 2 t \\textrm { f o r a l l } t \\geq 0 \\ , \\end{align*}"}
+{"id": "3350.png", "formula": "\\begin{align*} R \\lrcorner \\Omega = 0 , R \\lrcorner \\theta = 1 . \\end{align*}"}
+{"id": "4373.png", "formula": "\\begin{align*} a _ 1 = G \\left ( \\frac { p } { q } \\right ) = t = \\frac { q + 1 } { p } . \\end{align*}"}
+{"id": "1746.png", "formula": "\\begin{align*} s ' \\delta s ( \\mu _ { - \\underline { m } } ) = - 2 \\binom { 2 M } { k _ { \\rm i d } - 2 } \\varphi _ { M + 1 } ( t _ M ( \\mu _ { - \\underline { m } } ) ^ * ) . \\end{align*}"}
+{"id": "3683.png", "formula": "\\begin{align*} z _ 1 m _ 1 + \\cdots + z _ i m _ i = z _ 1 ( a _ 1 m ' _ 1 + z _ t w _ { t , 1 } ) + \\cdots + z _ { t - 1 } ( a _ { t - 1 } m ' _ { t - 1 } + z _ t w _ { t , t - 1 } ) , \\end{align*}"}
+{"id": "8591.png", "formula": "\\begin{align*} \\Psi = ( 6 , 3 , 9 , 1 3 , 5 , 1 0 , 1 1 ) \\circ ( 4 , 7 , 1 , 2 ) \\circ ( 1 4 , 1 2 , 8 , 1 5 ) . \\end{align*}"}
+{"id": "4581.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } = \\frac { 1 } { a _ 1 } \\leq \\frac { 1 } { b _ 1 } < 1 \\end{align*}"}
+{"id": "6399.png", "formula": "\\begin{align*} w _ j ( X , Y , t , \\tilde X , \\tilde Y , \\tilde t ) & : = u ( X , Y , t ) - \\phi ( \\tilde X , \\tilde Y , \\tilde t ) \\\\ & + \\bigl ( \\frac { j ^ 4 } { 4 } | X - \\tilde X | ^ 4 + \\frac { j ^ 4 } { 4 } | Y - \\tilde Y | ^ 4 + \\frac j 2 | t - \\tilde t | ^ 2 \\bigr ) , \\end{align*}"}
+{"id": "7557.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 2 ) = { \\left \\{ \\begin{array} { r l } ( 1 - q ^ { - 1 } ) q ^ { - ( \\omega + 1 ) - r s } Z _ { f _ 2 } ( s , \\chi ) , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } = \\chi _ { \\rm t r i v } , \\\\ 0 , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } \\neq \\chi _ { \\rm t r i v } . \\end{array} \\right . } \\end{align*}"}
+{"id": "2130.png", "formula": "\\begin{align*} f ( u ) > 0 , \\ ; \\forall u \\in ( \\theta , 1 ) , \\quad \\begin{cases} f ( u ) < 0 , \\ ; \\forall u \\in ( 0 , \\theta ) , \\quad ( ) , \\\\ \\\\ f ( u ) = 0 , \\ ; \\forall u \\in ( 0 , \\theta ) , \\quad ( ) . \\end{cases} \\end{align*}"}
+{"id": "4568.png", "formula": "\\begin{align*} a _ { n + 1 } = \\left ( \\prod _ { i = 1 } ^ n a _ i \\right ) + 1 \\end{align*}"}
+{"id": "3768.png", "formula": "\\begin{align*} \\Phi \\coloneqq \\begin{cases} \\operatorname { I d } & * _ \\Gamma ; \\\\ \\mathcal { E } _ { * _ \\Gamma } & * _ \\Gamma ; \\\\ \\mathcal { E } _ { * _ \\Gamma } \\alpha _ { * _ \\Gamma } & * _ \\Gamma . \\end{cases} \\end{align*}"}
+{"id": "3011.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\hbox { d } ^ \\mathbf { A } p _ i & = & ( \\partial ^ { \\gamma , \\mathbf { A } } _ b p { _ i } ^ { a b } + \\partial _ k p { _ i } ^ { a k } ) \\ e ^ { ( 3 ) } _ a \\wedge \\bar { e } ^ { ( r ) } \\\\ & & + \\ ( \\partial ^ { \\gamma , \\mathbf { A } } _ b p { _ i } ^ { j b } + \\partial _ k p { _ i } ^ { j k } + \\frac { 1 } { 2 } \\mathbf { F } { ^ j } _ { a b } p { _ i } ^ { a b } + \\frac { 1 } { 2 } c ^ j _ { k \\ell } p { _ i } ^ { k \\ell } ) \\ e ^ { ( 4 ) } \\wedge \\bar { e } ^ { ( r - 1 ) } _ j \\end{array} \\end{align*}"}
+{"id": "1774.png", "formula": "\\begin{align*} B = j _ B ( A ) \\subseteq j _ B ( g ^ { - 1 } ( Y ) ) = g ^ { - 1 } ( j _ Z ( Y ) ) = g ^ { - 1 } ( Y ) \\end{align*}"}
+{"id": "8484.png", "formula": "\\begin{align*} \\lambda = \\frac { 2 \\varphi ^ 2 + 3 \\varphi _ { \\xi } + \\epsilon } { 9 } , \\end{align*}"}
+{"id": "347.png", "formula": "\\begin{align*} | N _ { G } ( v ) \\cap ( A \\cup B ) | = | N _ { G } ( v ) \\cap B | + | N _ { G } ( v ) \\cap A | \\leq 1 + p - 1 = p . \\end{align*}"}
+{"id": "8828.png", "formula": "\\begin{align*} \\mathcal { T } _ { 1 1 } & = \\{ T _ { 1 1 1 } , T _ { 2 1 1 } , T _ { 3 1 1 } , \\dots \\} = ( T _ { n 1 1 } ) _ { n = 1 } ^ { \\infty } \\\\ \\mathcal { T } _ { 1 2 } & = \\{ T _ { 1 1 2 } , T _ { 2 1 2 } , T _ { 3 1 2 } , \\dots \\} = ( T _ { n 1 2 } ) _ { n = 1 } ^ { \\infty } \\\\ \\mathcal { T } _ { 2 1 } & = \\{ T _ { 1 2 1 } , T _ { 2 2 1 } , T _ { 3 2 1 } , \\dots \\} = ( T _ { n 2 1 } ) _ { n = 1 } ^ { \\infty } \\\\ \\mathcal { T } _ { 2 2 } & = \\{ T _ { 1 2 2 } , T _ { 2 2 2 } , T _ { 3 2 2 } , \\dots \\} = ( T _ { n 2 2 } ) _ { n = 1 } ^ { \\infty } \\end{align*}"}
+{"id": "5338.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { c u r l } \\operatorname { c u r l } u = \\lambda \\ , \\varepsilon \\ , u & \\mbox { i n } \\Omega , \\\\ \\operatorname { d i v } \\varepsilon u = 0 & \\mbox { i n } \\Omega , \\\\ \\nu \\times u = 0 & \\mbox { o n } \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "6993.png", "formula": "\\begin{align*} ( S _ c , S _ x , S _ y ) = f _ { \\mathcal { E } } ( X ^ n , Y ^ n ) , \\end{align*}"}
+{"id": "1679.png", "formula": "\\begin{align*} \\rho ( f _ n ) ( P _ m ) = \\int _ { S ^ 1 } e ^ { ( k - 2 - 2 m + 2 n ) i \\theta } d \\theta = \\left \\{ \\begin{array} { l l } 1 , & n = m - \\frac { k - 2 } { 2 } , \\\\ 0 , & n \\neq m - \\frac { k - 2 } { 2 } , \\end{array} \\right . \\end{align*}"}
+{"id": "6295.png", "formula": "\\begin{align*} s _ i s _ { i + 1 } s _ i ( T ) = s _ { i + 1 } s _ i s _ { i + 1 } ( T ) , \\end{align*}"}
+{"id": "7614.png", "formula": "\\begin{align*} \\P ( \\tau ^ { \\infty } < T ) \\leq \\P ( \\tau ^ n \\ ; { < } \\ ; T ) = \\P \\left ( \\sup _ { t \\in [ 0 , T ] } \\| u ^ n ( t ) \\| _ { \\L ^ p } ^ p \\geq n ^ p \\right ) , \\end{align*}"}
+{"id": "4856.png", "formula": "\\begin{align*} p ( \\tau _ 1 , \\dots , \\tau _ { k - 1 } ) = - ( b _ 1 \\tau _ 1 + \\dots + b _ { k - 1 } \\tau _ { k - 1 } ) ^ n + \\tau _ 1 ^ n + \\dots + \\tau _ k ^ n = 1 . \\end{align*}"}
+{"id": "6186.png", "formula": "\\begin{align*} C _ p B = \\overline B _ p C _ p , \\end{align*}"}
+{"id": "3915.png", "formula": "\\begin{align*} \\det D X v = \\frac { f ^ * ( \\cdot ) } { f ( X v ( \\cdot ) ) } \\Omega ^ * . \\end{align*}"}
+{"id": "9232.png", "formula": "\\begin{align*} \\ < \\Delta _ { f } \\psi _ { m } , \\psi _ { m } \\ > = h ^ { 2 } \\int _ { \\R ^ { d } } \\vert \\nabla g _ { m } ( x ) \\vert ^ { 2 } e ^ { - 2 ( f ( x ) - f ( m ) ) / h } d x . \\end{align*}"}
+{"id": "1490.png", "formula": "\\begin{align*} \\begin{aligned} \\C _ 2 ( x ) & = \\prod _ { n = 1 , n } ^ \\infty \\left \\{ \\left ( \\frac { 1 - \\frac x { ( \\frac n 2 ) } } { 1 + \\frac x { ( \\frac n 2 ) } } \\right ) ^ { \\frac n 2 } e ^ { 2 x } \\right \\} , \\\\ \\C _ 3 ( x ) & = \\prod _ { n = 1 , n } ^ \\infty \\left \\{ \\left ( 1 - \\frac { x ^ 2 } { ( \\frac n 2 ) ^ 2 } \\right ) ^ { ( \\frac n 2 ) ^ 2 } e ^ { x ^ 2 } \\right \\} \\end{aligned} \\end{align*}"}
+{"id": "8360.png", "formula": "\\begin{align*} v = 0 , \\mathrm { f o r } \\ | y | ^ 2 - s ^ 2 < R ^ { - 2 } , s \\ge 0 . \\end{align*}"}
+{"id": "5252.png", "formula": "\\begin{align*} g ( X , X ) = a \\cos ^ 2 \\omega ( t ) . \\end{align*}"}
+{"id": "7624.png", "formula": "\\begin{align*} \\mathcal { A } ^ { ( 2 ) } & = \\int _ { 0 } ^ { 1 } \\left [ ( 1 + \\gamma ) ( u _ n ^ { \\delta + 1 } - v _ n ^ { \\delta + 1 } ) ( x ) - \\gamma ( u _ n - v _ n ) ( x ) - ( u _ n ^ { 2 \\delta + 1 } - v _ n ^ { 2 \\delta + 1 } ) ( x ) \\right ] \\psi _ k ' ( w _ n ( x ) ) \\d x . \\end{align*}"}
+{"id": "8625.png", "formula": "\\begin{align*} M _ k ^ { x } ( S ) ( \\omega ) = \\sum _ { j \\in \\N \\cap b ^ k S } I _ j ( x , \\omega ) , \\end{align*}"}
+{"id": "4438.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 6 } = \\frac { 1 } { 3 } + \\frac { 1 } { 3 } = \\frac { 2 } { 3 } . \\end{align*}"}
+{"id": "829.png", "formula": "\\begin{align*} \\| \\pi _ { p n - 2 p \\lfloor c \\log n \\rfloor } - \\widetilde { \\tau } _ { n p } ^ c \\| _ { T V } = O \\left ( \\frac { 1 } { \\sqrt { n } } \\right ) \\end{align*}"}
+{"id": "3524.png", "formula": "\\begin{align*} Z ( \\mathfrak { g } , \\mathfrak { t } ) : = S ( \\mathfrak { g } , \\mathfrak { t } ) \\otimes _ { U ( \\mathfrak { h } ) } R ( \\mathfrak { h } ) . \\end{align*}"}
+{"id": "3419.png", "formula": "\\begin{align*} P _ G ( \\phi ) = P _ { t o p } ( \\phi ) . \\end{align*}"}
+{"id": "8590.png", "formula": "\\begin{align*} \\mathcal { C } = ( 1 , \\ , \\Psi ( 1 ) , \\ , \\ldots , \\ , 2 ^ { n - 1 } , \\ , 2 ^ n - 1 ) . \\end{align*}"}
+{"id": "87.png", "formula": "\\begin{align*} f _ { \\mathbf { k } } = \\eta _ { \\mathbf { k } } \\cdot x + \\overline { \\eta } _ { \\mathbf { k } } \\cdot y \\end{align*}"}
+{"id": "5534.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { C } } \\frac { \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } { \\rm d } s = R _ { 0 } + \\mathcal { R } _ { T } ( x ) , \\end{align*}"}
+{"id": "7750.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n \\widetilde f \\ , ^ \\prime ( s | \\tfrac { \\mu } { n } ) & = - \\lim _ { n \\to \\infty } \\mu \\widetilde \\rho ( s ) \\widetilde f ( s | \\tfrac { \\mu } { n } ) = - \\mu \\widetilde \\rho ( s ) \\widetilde f ( s | 0 ) \\\\ \\implies \\mu \\ , \\widetilde \\rho ( s ) & = - \\lim _ { n \\to \\infty } n \\ , \\widetilde f \\ , ^ \\prime ( s | \\tfrac { \\mu } { n } ) / \\widetilde f ( s | 0 ) = - \\lim _ { n \\to \\infty } n \\ , \\widetilde f \\ , ^ \\prime ( s | \\tfrac { \\mu } { n } ) \\end{align*}"}
+{"id": "8014.png", "formula": "\\begin{align*} \\begin{aligned} e & = ( F _ 3 , F _ 2 ) , \\\\ T & = \\left ( \\{ F _ 3 , F _ 2 \\} , \\{ ( F _ 3 , F _ 2 ) \\} \\right ) , \\\\ F & = \\left [ ( F _ 2 , F _ 1 ) \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "3882.png", "formula": "\\begin{align*} u ( x _ 0 ) & = x _ 0 \\cdot y _ 0 - z _ 0 , \\\\ u ( x ) & \\geq x \\cdot y _ 0 - z _ 0 x \\in \\Omega . \\end{align*}"}
+{"id": "6626.png", "formula": "\\begin{align*} & \\frac { \\dd } { \\dd t } \\norm { x ^ m ( t ) \\psi } ^ 2 \\\\ & \\qquad \\leq C _ m \\sum _ { j = 1 } ^ d \\norm { x _ j ^ m ( t ) \\psi } ^ { 2 - \\frac { 1 } { 2 ^ { \\ell - 1 } } } \\sum _ { r = 0 } ^ { 2 ^ { \\ell - 1 } } c _ { r , m } \\sum _ { p = 0 } ^ r c _ { p , r } \\norm { x _ j ^ { 2 r - p } ( t ) \\psi } ^ { \\frac { 1 } { 2 ^ \\ell } } \\norm { D _ j ^ { 2 r - p } ( t ) \\psi } ^ { \\frac { 1 } { 2 ^ { \\ell } } } \\ , . \\end{align*}"}
+{"id": "7945.png", "formula": "\\begin{align*} x + a + n _ 1 + n _ 1 ' = x + n _ 2 + n _ 1 ' = y + b + n _ 1 + n _ 1 ' = y + n _ 1 + n _ 2 ' , \\end{align*}"}
+{"id": "7562.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 4 ) = & \\sum _ { a = 0 } ^ { \\omega - 1 } q ^ { - a - 2 - a p s } \\int _ { B _ 4 ^ a } \\chi ( a c ( x ^ p + \\pi ^ { k + r + l - a p } y ^ r z ^ l \\mathbb { H } _ r ^ k ( y , t z - y ) ) ) | d x d y d z | \\\\ : = & F _ 4 ( q ^ { - s } ) , \\end{align*}"}
+{"id": "4760.png", "formula": "\\begin{align*} \\int _ { \\Omega } F ( x , t ) d x = \\int _ { ( \\Omega _ { q } ) _ { \\delta / 2 } } F ( x , u ) d x + \\lambda \\int _ { \\Omega _ { N } \\backslash ( \\Omega _ { q } ) _ { \\delta / 2 } } F _ { 1 } ( x , u ) d x + \\int _ { \\Omega _ { p } \\backslash ( \\Omega _ { q } ) _ { \\delta / 2 } } \\frac { | u | ^ { p ^ { * } ( x ) } } { p ^ { * } ( x ) } d x \\end{align*}"}
+{"id": "5025.png", "formula": "\\begin{align*} d * _ g H = 2 \\ , \\frac { h _ 4 } { \\mu } \\left ( e ^ { 1 2 5 } - e ^ { 1 3 4 } \\right ) - 2 \\ , \\frac { h _ 3 \\left ( a ^ 2 b ^ 2 + c ^ 2 \\right ) - c h _ 1 \\left ( a ^ 2 + b ^ 2 \\right ) } { \\mu \\left ( a ^ 2 b ^ 2 - c ^ 2 \\right ) } \\left ( e ^ { 1 2 4 } + e ^ { 1 3 5 } \\right ) . \\end{align*}"}
+{"id": "4992.png", "formula": "\\begin{align*} L = \\Gamma _ + \\cap g H g ^ { - 1 } \\cup ( - \\sigma ( \\Gamma _ + \\cap g H g ^ { - 1 } ) ) \\end{align*}"}
+{"id": "1690.png", "formula": "\\begin{align*} \\frac { x ^ { \\frac { k - 2 } { 2 } - m } y ^ { \\frac { k - 2 } { 2 } + m } } { 2 ^ { 2 - k } } = ( z + \\bar z ) ^ { \\frac { k - 2 } { 2 } - m } ( z - \\bar z ) ^ { \\frac { k - 2 } { 2 } + m } i ^ { \\frac { 2 - k } { 2 } - m } = i ^ { \\frac { 2 - k } { 2 } - m } \\sum _ { \\frac { 2 - k } { 2 } \\leq n \\leq \\frac { k - 2 } { 2 } } C ( n ) \\cdot z ^ { n + \\frac { k - 2 } { 2 } } \\bar z ^ { \\frac { k - 2 } { 2 } - n } \\end{align*}"}
+{"id": "466.png", "formula": "\\begin{align*} L _ { g , r , d } = ( r + 1 ) ^ g . \\end{align*}"}
+{"id": "2200.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega ( n ) q ^ n = \\sum _ { n = 1 } ^ \\infty \\dfrac { q ^ n ( - q ^ { n + 1 } ; q ) _ n ( - q ^ { 2 n + 2 } ; q ^ 2 ) _ \\infty } { ( 1 - q ^ n ) ^ 2 ( q ^ { n + 1 } ; q ) _ n ( q ^ { 2 n + 2 } ; q ^ 2 ) _ \\infty } . \\end{align*}"}
+{"id": "7740.png", "formula": "\\begin{align*} \\forall X \\in N _ { \\alpha } \\cap V _ { \\kappa + 1 } ( k ( X ) = X ) . \\end{align*}"}
+{"id": "4605.png", "formula": "\\begin{align*} \\frac { 1 } { q } - \\sum _ { i = 1 } ^ k \\frac { 1 } { a _ i } = \\frac { 1 } { q \\prod _ { i = 1 } ^ k a _ i } \\end{align*}"}
+{"id": "4228.png", "formula": "\\begin{align*} - v \\cosh ( t ) - \\frac { 1 } { \\sqrt { c } } \\sinh ( t ) \\tanh ( \\gamma ) = \\cosh \\Lambda . \\end{align*}"}
+{"id": "1616.png", "formula": "\\begin{align*} \\hat U ^ { ( t ) } = ( \\tilde Q ^ t \\odot A ) ^ * R ^ { ( t ) } + U ^ { ( t ) } \\end{align*}"}
+{"id": "5530.png", "formula": "\\begin{align*} A _ { p , q , k } ^ { m , n } ( z ) : = \\frac { z ^ { b _ k } \\prod _ { j = 1 , j \\neq k } ^ { m } \\Gamma ( b _ j - b _ k ) \\prod _ { j = 1 } ^ n \\Gamma ( 1 + b _ k - a _ j ) } { \\prod _ { j = m + 1 } ^ { q } \\Gamma ( 1 + b _ k - b _ { j } ) \\prod _ { j = n + 1 } ^ { p } \\Gamma ( a _ { j } - b _ k ) } . \\end{align*}"}
+{"id": "7844.png", "formula": "\\begin{align*} \\theta _ k ^ * = ( \\theta ^ \\bot + 2 k \\pi ) / q , k = 0 , \\ldots , q - 1 . \\end{align*}"}
+{"id": "5967.png", "formula": "\\begin{align*} ( E _ r , e _ s ) = \\delta _ { r s } ( r , s = 1 , \\cdots , p ) . \\end{align*}"}
+{"id": "7836.png", "formula": "\\begin{align*} f ( z ) = 1 + H _ 1 e ^ { w _ 1 z } + \\cdots + H _ n e ^ { w _ n z } \\end{align*}"}
+{"id": "8553.png", "formula": "\\begin{align*} z \\left ( n ; \\beta \\right ) = \\left \\{ \\begin{array} { l l } x \\left ( n ; \\beta \\right ) & \\left ( n ; \\beta \\right ) \\in F _ { \\downarrow } \\\\ 0 & \\end{array} \\right . \\end{align*}"}
+{"id": "7297.png", "formula": "\\begin{align*} \\chi _ { m , j } ( \\lambda ) & : = - \\log P [ \\mathrm { e } ^ { - \\lambda \\eta _ { m , j } ( 1 ) } ] \\\\ & \\ = \\int _ { 0 } ^ { \\infty } ( 1 - \\mathrm { e } ^ { - \\lambda u } ) n _ { m , j } ( T _ 0 \\in d u ) \\\\ & \\ = \\int _ { 0 } ^ { \\infty } P ^ m _ x [ 1 - \\mathrm { e } ^ { - \\lambda T _ 0 } ] j ( d x ) . \\end{align*}"}
+{"id": "4495.png", "formula": "\\begin{align*} \\frac { 4 } { 5 } = \\frac { 1 } { 2 } + \\frac { 1 } { 5 } + \\frac { 1 } { 1 0 } = \\frac { 1 } { 2 } + \\frac { 1 } { 4 } + \\frac { 1 } { 2 0 } . \\end{align*}"}
+{"id": "8419.png", "formula": "\\begin{align*} \\| a \\| = \\max _ { i = 1 , \\ldots , n } | \\lambda _ i | = \\| f _ a \\| \\end{align*}"}
+{"id": "3125.png", "formula": "\\begin{align*} - A : D ^ 2 \\hat { w } = a - \\bar { a } \\quad Y , \\hat { w } Y , \\int _ Y \\hat { w } = 0 . \\end{align*}"}
+{"id": "5216.png", "formula": "\\begin{align*} & p _ { 1 2 3 } = e _ { 1 2 3 } \\ , \\ p _ { 1 2 4 } = e _ { 1 2 4 } , p _ { 3 5 6 } = e _ { 3 5 6 } , p _ { 4 5 6 } = e _ { 4 5 6 } , \\\\ & p _ { 1 3 5 } - p _ { 2 3 6 } = e _ 0 , p _ { 1 3 6 } + p _ { 2 3 5 } = e _ 1 , p _ { 1 4 5 } - p _ { 2 4 6 } = e _ 2 , p _ { 1 4 6 } + p _ { 2 4 5 } = e _ 3 \\ , . \\end{align*}"}
+{"id": "7916.png", "formula": "\\begin{align*} F ( t - s , x ) & = \\P ^ { ( 2 ) } \\left ( M _ { t - s } \\geq \\sqrt { 2 } ( t - s ) - \\tfrac { 1 } { 2 \\sqrt { 2 } } \\log ( t ) - A - ( x - \\sqrt { 2 } s ) \\right ) \\\\ & \\leq C \\frac { \\sqrt { t + 1 } } { ( t - s + 1 ) ^ { \\frac { 3 } { 2 } } } \\left ( 1 + \\frac { \\log ( t ) } { \\sqrt { 2 } } + ( - x ) _ + \\right ) e ^ { - \\sqrt { 2 } ( \\sqrt { 2 } s - x - A ) } . \\end{align*}"}
+{"id": "151.png", "formula": "\\begin{align*} \\alpha _ { 2 } & = \\max \\bigg \\{ \\bigg ( K \\| \\phi ( 0 ) \\| _ { \\mathbb { X } } + M K R b + K S b \\kappa _ { b } \\bigg ) , \\max _ { 1 \\leq j \\leq n } \\bigg ( K C _ { \\nu _ { j } } \\\\ & \\qquad \\qquad + M K R b + K S b \\kappa _ { b } \\bigg ) , \\ \\max _ { 1 \\leq j \\leq n } C _ { \\nu _ { j } } \\bigg \\} . \\end{align*}"}
+{"id": "9086.png", "formula": "\\begin{align*} \\mathfrak { d } _ 0 = \\mathfrak { n } , \\mathfrak { d } _ j = [ \\mathfrak { d } _ { j - 1 } , \\mathfrak { n } ] + J [ \\mathfrak { d } _ { j - 1 } , \\mathfrak { n } ] \\end{align*}"}
+{"id": "133.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow 0 ^ + } \\lambda ^ p \\mathcal L ^ { 2 n } ( \\widetilde E _ { \\lambda , K } ) = 2 | K | | | f | | _ { L ^ p } ^ p . \\end{align*}"}
+{"id": "2810.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ j \\lambda v _ i \\leq \\prod _ { i = 1 } ^ j \\lambda u _ i \\qquad \\prod _ { i = 1 } ^ j v _ i \\leq \\prod _ { i = 1 } ^ j u _ i \\end{align*}"}
+{"id": "2936.png", "formula": "\\begin{align*} \\begin{aligned} \\langle \\mathcal { M } ^ n ( \\varphi ) \\rangle _ t & = \\int _ 0 ^ t \\big ( L _ n ( \\mathcal { X } ^ n _ s ( \\varphi ) ) ^ 2 - 2 \\mathcal { X } ^ n _ s ( \\varphi ) L _ n \\mathcal { X } ^ n _ s ( \\varphi ) \\big ) d s \\\\ & = \\frac { 1 } { n } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } g _ n ( \\eta ^ n _ j ( s ) ) \\big [ n ( \\varphi ^ n _ { j + 1 } ( s ) - \\varphi ^ n _ j ( s ) ) \\big ] ^ 2 d s , \\end{aligned} \\end{align*}"}
+{"id": "8284.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { R } _ p : = \\big \\{ \\ \\textbf { x } \\ | \\ B ( \\textbf { x } ) > 0 \\big \\} , \\ \\ \\mathcal { R } _ e : = \\big \\{ \\ \\textbf { x } \\ | \\ B ( \\textbf { x } ) < 0 \\big \\} . \\end{array} \\right . \\end{align*}"}
+{"id": "2359.png", "formula": "\\begin{align*} J _ n \\overline { Q } ( k ) J _ n = - Q ( - k ) , Q ( k ) ^ T J _ n Q ( - k ) = J _ n . \\end{align*}"}
+{"id": "5704.png", "formula": "\\begin{align*} \\beta _ { b } ^ { a } & = \\digamma _ { b } ^ { a } - \\mathfrak { f } _ { b } ^ { a } \\\\ \\mathfrak { f } _ { b } ^ { a } & = \\frac { 1 } { 2 } \\mathfrak { f } _ { b c d } ^ { a } \\theta ^ { c } \\theta ^ { d } \\end{align*}"}
+{"id": "910.png", "formula": "\\begin{gather*} a _ 0 = 1 , a _ { i + 1 } = z a _ i + z b _ i + z c _ i , i \\ge 0 , \\\\ b _ 0 = z b _ 1 , b _ i = z a _ { i + 1 } + z b _ { i + 1 } , i \\ge 1 , \\\\ c _ { i + 1 } = z a _ { i } + z c _ { i } , i \\ge 0 . \\end{gather*}"}
+{"id": "2565.png", "formula": "\\begin{align*} ( b , c ) \\cdot a = b a c ^ { - 1 } . \\end{align*}"}
+{"id": "7337.png", "formula": "\\begin{align*} b ( x ) = \\frac { \\delta - 1 + \\epsilon ( x ) } { x } + \\eta ( x ) ( x > 0 ) \\end{align*}"}
+{"id": "4155.png", "formula": "\\begin{align*} u = \\sum a _ i \\chi _ i , \\omega = \\sum b _ j \\chi _ j \\end{align*}"}
+{"id": "2357.png", "formula": "\\begin{align*} \\alpha _ 0 ( k = \\tfrac 1 2 ) = \\widetilde { \\alpha _ { s = 0 } } ( \\tfrac 1 2 ) , \\widetilde { \\alpha _ { s = 1 } } ( \\tfrac 1 2 ) = \\alpha _ 1 ( k = \\tfrac 1 2 ) , \\alpha _ 1 ( k = 0 ) = \\widetilde { \\alpha _ { s = 1 } } ( 0 ) . \\end{align*}"}
+{"id": "8637.png", "formula": "\\begin{align*} x \\in \\bigcap _ { n \\geq 1 } ( \\Omega ^ { \\mathbb N } \\setminus W _ n ) = \\bigcap _ { n \\geq 1 } \\Big ( \\bigcup _ { m \\geq n } T _ m \\Big ) = \\bigcap _ { n \\geq 1 } T _ { n } . \\end{align*}"}
+{"id": "659.png", "formula": "\\begin{align*} \\rho = 1 . \\end{align*}"}
+{"id": "3715.png", "formula": "\\begin{align*} \\Psi _ f ( x ) : = \\int _ K r _ i ( k ) f ( 0 , 0 , x ) d k . \\end{align*}"}
+{"id": "3543.png", "formula": "\\begin{align*} E _ { - \\alpha _ 1 } ^ { s _ 1 } \\cdots E _ { - \\alpha _ m } ^ { s _ m } B _ i ^ + = E _ { i _ 2 i _ 1 } \\cdots E _ { i _ s i _ { s - 1 } } B _ i ^ + = E ^ { e _ I } B _ i ^ + . \\end{align*}"}
+{"id": "7716.png", "formula": "\\begin{align*} s _ 1 & = \\left \\vert \\chi ^ { ( 2 ) } ( 1 + \\rho e ^ { i \\beta \\pi } ) + \\sqrt { \\left ( \\chi ^ { ( 2 ) } ( 1 + \\rho e ^ { i \\beta \\pi } ) \\right ) ^ 2 - 1 } \\right \\vert , \\\\ s _ 2 & = \\left \\vert \\chi ^ { ( 2 ) } ( 1 + \\rho e ^ { - i \\beta \\pi } ) + \\sqrt { \\left ( \\chi ^ { ( 2 ) } ( 1 + \\rho e ^ { - i \\beta \\pi } ) \\right ) ^ 2 - 1 } \\right \\vert . \\end{align*}"}
+{"id": "1013.png", "formula": "\\begin{align*} \\langle X _ { W , \\varphi } , v _ k \\rangle = \\sum _ { n \\in \\Z } \\langle W , \\varphi ( \\cdot - n ) \\rangle v _ k [ n ] = \\left \\langle W , \\sum _ { n \\in \\Z } v _ k [ n ] \\varphi ( \\cdot - n ) \\right \\rangle . \\end{align*}"}
+{"id": "1897.png", "formula": "\\begin{align*} h ' ( 0 ) = h ' ( 1 ) = 0 , h ( 1 ) \\neq 0 , h '' ( x ) = \\nu h ( x ) , \\end{align*}"}
+{"id": "1929.png", "formula": "\\begin{align*} R _ { \\tilde g _ i } - R _ g & = \\chi _ { \\lambda _ i } R _ g u ^ \\frac { 4 } { n - 2 } _ { \\lambda _ i } - R _ g \\\\ & = \\chi _ { \\lambda _ i } ( u ^ \\frac { 4 } { n - 2 } _ { \\lambda _ i } - 1 ) R _ g + ( \\chi _ \\lambda - 1 ) R _ g . \\end{align*}"}
+{"id": "4542.png", "formula": "\\begin{align*} s _ 1 = 2 \\end{align*}"}
+{"id": "3676.png", "formula": "\\begin{align*} ( ( I - \\alpha _ \\rho ^ k ) A + \\alpha _ \\rho ^ k c ) \\delta u & = M ( f - A u ^ k , u ^ k - g ) . \\end{align*}"}
+{"id": "2346.png", "formula": "\\begin{align*} C = \\begin{pmatrix} 0 & K _ M \\\\ - K _ M & 0 \\end{pmatrix} = J _ { 2 M } \\ , K _ { 2 M } . \\end{align*}"}
+{"id": "569.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } \\sigma _ t = - \\frac { \\sigma _ t ^ 2 } { \\gamma } \\ , ( A ^ { \\rm T } A ) : ( X _ t ^ \\dagger \\otimes X _ t ^ \\dagger ) \\end{align*}"}
+{"id": "7932.png", "formula": "\\begin{align*} M ( f , s ) : = \\frac { ( 2 \\pi ) ^ s } { \\Gamma ( s ) } \\int \\limits _ { \\sqrt { N } ^ { - 1 } } ^ { \\infty } f ( i z ) z ^ { s - 1 } d z . \\end{align*}"}
+{"id": "9135.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { n \\choose k } ( m + k - 1 ) ( m + n - 1 ) ^ { n - k - 1 } = M _ 1 \\ ; + \\ ; M _ 2 , \\end{align*}"}
+{"id": "4645.png", "formula": "\\begin{align*} \\mathsf { E } \\left [ g ( X ) X ^ n \\right ] = \\sum _ { \\ell = 0 } ^ n \\binom { n } { \\ell } \\mu ^ { n - \\ell } \\sum _ { k = 0 } ^ { \\lfloor \\ell / 2 \\rfloor } H _ { \\ell , k } \\sigma ^ { 2 ( \\ell - k ) } \\mathsf { E } \\left [ g ^ { ( \\ell - 2 k ) } ( X ) \\right ] , \\ \\ n \\in \\mathbb { N } . \\end{align*}"}
+{"id": "8702.png", "formula": "\\begin{align*} \\tilde Q _ \\lambda ( x _ 1 , \\dots , x _ n ) = 2 ^ l \\sum _ { \\sigma \\in S _ n } \\sigma \\left ( f _ { \\lambda _ 1 } ( x _ { 1 } ) \\dots f _ { \\lambda _ n } ( x _ n ) \\prod _ { i = 1 } ^ { n } \\prod _ { i < j } \\frac { x _ { i } + x _ { j } } { x _ { i } - x _ { j } } \\right ) , \\end{align*}"}
+{"id": "2214.png", "formula": "\\begin{align*} w _ i = U ( q ^ { - 2 } \\delta \\xi ^ { i - 1 } ) = \\gamma \\sum _ { j = 1 } ^ { 5 i } \\beta _ { i , j } \\xi ^ { j - 1 } . \\end{align*}"}
+{"id": "6447.png", "formula": "\\begin{align*} \\sup w _ j \\geq w _ j ( X ^ * , Y ^ * , t ^ * , X ^ * , Y ^ * , t ^ * ) = \\tilde u ( X ^ * , Y ^ * , t ^ * ) - v ( X ^ * , Y ^ * , t ^ * ) > 0 . \\end{align*}"}
+{"id": "7345.png", "formula": "\\begin{align*} j ( d x ) : = \\frac { \\alpha } { 2 } \\left ( \\frac { 1 } { x ^ { a + 1 } } \\wedge \\frac { ( \\log x ) ^ { t - 1 } } { x ^ { 2 / \\alpha + 1 } } \\right ) d x \\end{align*}"}
+{"id": "7662.png", "formula": "\\begin{align*} \\eta _ { I } = \\sum _ { p \\in \\mathbb { Z } } \\eta _ { I , p } \\eta _ { I , p } = \\sum _ { \\substack { J \\subseteq [ n ] \\\\ | J | = j } } \\frac { g _ { I , J , p + \\deg ( f ) - | J | } d x _ { J } } { f } \\end{align*}"}
+{"id": "6236.png", "formula": "\\begin{align*} B ^ T E _ i = \\sum _ { j = 1 } ^ p \\beta _ { i j } E _ j + C _ p ^ T x _ i . \\end{align*}"}
+{"id": "6386.png", "formula": "\\begin{align*} \\xi _ k ^ { \\frac { q ^ 3 - 1 } { p } } = ( g ^ k \\xi _ 0 ) ^ { \\frac { q ^ 3 - 1 } { p } } = g ^ { k \\frac { q ^ 2 + q + 1 } { p } ( q - 1 ) } \\xi _ 0 ^ { \\frac { q ^ 3 - 1 } { p } } = \\xi _ 0 ^ { \\frac { q ^ 3 - 1 } { p } } \\ne 1 , \\end{align*}"}
+{"id": "5358.png", "formula": "\\begin{align*} \\tilde \\varepsilon _ { t , \\eta } : = \\tilde \\varepsilon + t \\eta \\forall t \\in \\mathbb { R } , \\end{align*}"}
+{"id": "7021.png", "formula": "\\begin{align*} F _ { M U } ( y , z ) = \\sum _ { i , j \\geq 0 } a _ { i , j } y ^ i z ^ j \\in M U ^ * [ [ y , z ] ] \\end{align*}"}
+{"id": "8314.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ^ 2 u - \\Delta u - | u | ^ { \\frac { 4 } { d - 2 } } u = 0 , \\\\ u | _ { t = 0 } = u _ 0 \\in \\dot { H } ^ 1 , \\partial _ t u | _ { t = 0 } = u _ 1 \\in L ^ 2 . \\end{cases} \\end{align*}"}
+{"id": "9263.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { g } ' _ 1 & = ( x _ 1 - x _ 2 - 1 ) \\cdots ( x _ 1 - x _ { h ( 1 ) } - 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "9129.png", "formula": "\\begin{align*} K _ \\nu ( x ; \\xi ) \\ ; : = \\ ; \\begin{cases} & \\prod _ { y \\in \\xi } \\nu ( x - y ) , \\\\ & 1 , \\ ; \\xi = \\emptyset . \\end{cases} \\end{align*}"}
+{"id": "7167.png", "formula": "\\begin{align*} \\displaystyle { w '^ { \\{ k \\} } _ m = 0 \\ , . } \\end{align*}"}
+{"id": "7259.png", "formula": "\\begin{align*} \\frac { d } { d m } \\frac { d ^ + } { d x } u = \\lambda u , u ( 0 ) = 1 , u . \\end{align*}"}
+{"id": "4393.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } \\leq \\frac { n } { x _ 1 } \\leq \\frac { 1 } { c ^ { ( 1 ) } _ 1 } < \\sum _ { i = 1 } ^ n \\frac { 1 } { c ^ { ( 1 ) } _ i } < \\theta . \\end{align*}"}
+{"id": "1399.png", "formula": "\\begin{align*} \\psi = S _ 1 + S _ 1 + S _ 2 \\boxtimes S _ 2 + S _ 3 \\boxtimes S _ 3 . \\end{align*}"}
+{"id": "8250.png", "formula": "\\begin{align*} ( X = Y ) \\enspace & \\simeq \\enspace ( A \\simeq B ) \\\\ & \\simeq \\enspace \\sum _ { f : A \\cong B } \\mathsf { i s C o h e r e n t } ( f ) \\\\ \\enspace & \\simeq \\enspace ( A \\cong B ) \\end{align*}"}
+{"id": "7077.png", "formula": "\\begin{align*} F _ n D = \\left \\{ \\sum a _ i \\Pi _ { \\rho _ 1 + \\cdots + \\rho _ i } \\in D : a _ i \\in F _ { n + \\ell _ i } A \\right \\} , \\end{align*}"}
+{"id": "1641.png", "formula": "\\begin{align*} \\Phi ^ \\varepsilon ( t , x ) : = \\int _ \\mathbb { R } u ^ s ( t , x ) \\chi _ \\varepsilon ( s - 1 ) d s - B \\varepsilon ( t + 1 ) \\end{align*}"}
+{"id": "8985.png", "formula": "\\begin{align*} \\sup _ { t \\ge 0 } & \\| \\nabla u ^ { ( m ) } ( t ) \\| ^ 2 _ { L ^ 2 ( B ) } + \\varepsilon \\| u ^ { ( m ) } _ r \\| ^ 2 _ { L ^ 2 ( [ 0 , \\infty [ \\times S ^ 1 ) } + \\| u ^ { ( m ) } _ t \\| ^ 2 _ { L ^ 2 ( [ 0 , \\infty [ \\times S ^ 1 ) } \\\\ & \\le 2 \\| \\nabla u ^ { ( m ) } ( 0 ) \\| ^ 2 _ { L ^ 2 ( B ) } \\le 2 \\| \\nabla u _ 0 \\| ^ 2 _ { L ^ 2 ( B ) } \\le 2 R _ 0 ^ 2 . \\end{align*}"}
+{"id": "2876.png", "formula": "\\begin{align*} \\abs { \\frac { L _ f ' } { L _ f } ( \\sigma _ 0 + i t ) } ^ 2 & \\ll \\abs { \\sum _ { m = 1 } ^ { \\infty } \\frac { \\Lambda _ f ( m ) e ^ { - \\frac { m } { Y } } } { m ^ { \\sigma _ 0 + i t } } } ^ 2 + \\left ( Y ^ { \\frac { 1 } { 2 } - \\sigma _ 0 } \\log T \\right ) ^ 2 . \\\\ \\end{align*}"}
+{"id": "7733.png", "formula": "\\begin{align*} \\hat { \\rho } = \\frac { D _ { \\beta } ^ 2 } { e ^ 4 } \\exp { \\left ( 4 W \\left ( H _ { \\beta } n ( n - 1 ) \\right ) \\right ) } . \\end{align*}"}
+{"id": "9102.png", "formula": "\\begin{align*} ( \\tilde { \\mathsf { R } } _ 1 , \\tilde { \\mathsf { R } } _ 2 ) = \\left ( \\tilde { \\mathsf { R } } _ { \\mathrm { F O } , 1 } - \\frac { \\tilde { \\mathsf { R } } _ { \\mathrm { S O } , 1 } } { \\sqrt { n } } , \\tilde { \\mathsf { R } } _ { \\mathrm { F O } , 2 } - \\frac { \\tilde { \\mathsf { R } } _ { \\mathrm { S O } , 2 } } { \\sqrt { n } } \\right ) \\end{align*}"}
+{"id": "2109.png", "formula": "\\begin{align*} F G = \\big \\{ a ( x ) + a ' ( x ) y \\ : \\big | \\ ; a ( x ) , a ' ( x ) \\in F H \\big \\} . \\end{align*}"}
+{"id": "4042.png", "formula": "\\begin{align*} Q _ j ( x , y , z ) = - \\frac { g _ { y _ j } } { g _ z } ( x , y , z ) , \\end{align*}"}
+{"id": "3084.png", "formula": "\\begin{align*} \\bar { A } = \\int _ Y r A = \\int _ Y \\left ( \\bar { a } \\frac { r _ B } { a } \\right ) a B = \\bar { a } \\int _ Y r _ B B = \\bar { a } \\bar { B } . \\end{align*}"}
+{"id": "6583.png", "formula": "\\begin{align*} [ T _ - , \\sigma ( g _ 1 , \\dots , g _ n ) , T _ + ] [ T _ + , \\tau ( h _ 1 , \\dots , h _ n ) , U _ + ] = [ T _ - , \\sigma ( g _ 1 , \\dots , g _ n ) \\tau ( h _ 1 , \\dots , h _ n ) , U _ + ] \\end{align*}"}
+{"id": "7751.png", "formula": "\\begin{align*} \\mu \\rho ( x ) & = \\lim _ { n \\to \\infty } n x f ( x | \\tfrac { \\mu } { n } ) / \\widetilde f ( s | 0 ) = \\lim _ { n \\to \\infty } n x f ( x | \\tfrac { \\mu } { n } ) ( \\widetilde f ( s | 0 ) = 1 ) \\end{align*}"}
+{"id": "7599.png", "formula": "\\begin{align*} \\pi _ n y = \\begin{cases} y , & \\| y \\| _ { \\L ^ p } \\leq n , \\\\ \\frac { n } { \\| y \\| _ { \\L ^ p } } y , & \\| y \\| _ { \\L ^ p } > n . \\end{cases} \\end{align*}"}
+{"id": "8187.png", "formula": "\\begin{align*} u ( t , x ) = u _ 0 ( x ) + \\int _ 0 ^ t k ( t , s ) \\left ( L _ 0 + V \\right ) ^ f u ( s , x ) d s . \\end{align*}"}
+{"id": "3057.png", "formula": "\\begin{align*} A ( y ) = \\mathrm { d i a g } ( a _ 1 ( y ) , a _ 2 ( y ) ) \\quad y \\in \\R ^ 2 , \\end{align*}"}
+{"id": "784.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { P } _ x \\left ( \\frac { \\log \\| M _ n v \\| - \\log \\| M _ \\ell v \\| + \\ell \\lambda ' - n \\lambda _ 1 + e ^ { - \\epsilon \\ell } } { \\sigma \\sqrt { n } } \\leq t \\right ) \\\\ & \\geq \\mathbb { E } _ x \\left ( \\inf _ { v \\in \\R ^ d , \\| v \\| = 1 } \\mathbb { P } _ { z _ \\ell } \\left ( \\frac { \\log \\| M _ { n - \\ell } v \\| - ( n - \\ell ) \\lambda _ 1 } { \\sigma \\sqrt { n - \\ell } } \\leq t _ n \\right ) \\right ) - e ^ { c ' ( n - \\ell ) } , \\end{aligned} \\end{align*}"}
+{"id": "5535.png", "formula": "\\begin{align*} H _ 1 ( x ; T ) : = \\frac { 1 } { 2 \\pi i } \\int _ { \\lambda + i T } ^ { c + i T } \\frac { \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } { \\rm d } s , { \\rm a n d } H _ 2 ( x ; T ) : = \\frac { 1 } { 2 \\pi i } \\int _ { c - i T } ^ { \\lambda - i T } \\frac { \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } { \\rm d } s \\end{align*}"}
+{"id": "7191.png", "formula": "\\begin{align*} A _ { \\beta \\alpha } ( { \\Sigma _ 0 } ) y ^ 2 _ { \\alpha } = C _ \\beta ( { \\Sigma _ 0 } ) . \\end{align*}"}
+{"id": "6198.png", "formula": "\\begin{align*} t = 0 : W = C _ p \\widehat U _ 0 , W ' = C _ p \\widehat U _ 1 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "7598.png", "formula": "\\begin{align*} \\lim _ { C \\to \\infty } \\sup _ { n } \\P ( \\xi _ n \\geq C ) = 0 . \\end{align*}"}
+{"id": "1552.png", "formula": "\\begin{align*} & ( p ^ { + } + \\gamma ^ { + } - 1 ) \\int _ { M } | D \\mathrm { u } ( z ) | ^ { p ( z ) } \\ , \\ , d v _ { g } ( z ) + ( q ^ { + } + \\gamma ^ { + } - 1 ) \\int _ { M } \\mu ( z ) \\ , | D \\mathrm { u } ( z ) | ^ { q ( z ) } \\ , \\ , d v _ { g } ( z ) \\\\ & = \\lambda ( r ^ { - } + \\gamma ^ { - } - 1 ) \\int _ { M } | \\mathrm { u } ( z ) | ^ { r ( z ) } \\ , \\ , d v _ { g } ( z ) , \\end{align*}"}
+{"id": "351.png", "formula": "\\begin{align*} \\nu ( M , N ) = F _ { - r } \\supset \\cdots \\supset F _ 0 = 0 . \\end{align*}"}
+{"id": "7845.png", "formula": "\\begin{align*} n ( r , \\Lambda _ p ( \\theta ^ * , c ) ) = \\frac { | w _ j - w _ k | } { 2 \\pi } r ^ q + O \\left ( r ^ { q - p } \\log ^ { 3 + \\varepsilon } r \\right ) . \\end{align*}"}
+{"id": "4511.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k v _ i \\leq \\prod _ { i = 1 } ^ k u _ i \\end{align*}"}
+{"id": "4021.png", "formula": "\\begin{align*} \\phi ^ * _ i D _ { p _ k } Y ^ i = \\chi w ^ { j k } \\gamma _ j . \\end{align*}"}
+{"id": "5135.png", "formula": "\\begin{align*} b \\Big ( \\Omega _ { n + 1 } ( \\lambda ) - \\Omega _ { n } ( \\lambda ) + \\Omega _ { n + 1 } ( \\lambda b ) - \\Omega _ { n } ( \\lambda b ) \\Big ) = b \\Big ( \\varphi _ { n } ( \\lambda ) + \\varphi _ { n } ( \\lambda b ) \\Big ) \\underset { n \\rightarrow \\infty } { \\sim } \\frac { b } { n ^ { 2 } } \\end{align*}"}
+{"id": "5913.png", "formula": "\\begin{align*} D ^ T \\widetilde C _ { p - 1 } ^ T E = 0 . \\end{align*}"}
+{"id": "2619.png", "formula": "\\begin{align*} ( a _ { 2 , 3 } , a _ { 2 , 4 } , a _ { 3 , 4 } , a _ { 1 , 2 , 3 } , a _ { 1 , 2 , 4 } , a _ { 1 , 3 , 4 } , a _ { 2 , 3 , 4 } ) = ( 2 , 2 , 2 , 1 , 0 , 0 , 1 ) . \\end{align*}"}
+{"id": "8734.png", "formula": "\\begin{align*} g _ \\lambda ( x _ 1 , \\dots x _ n ; \\beta ) = \\det \\left [ \\sum _ { m = 0 } ^ \\infty { { 1 - i } \\choose m } \\beta ^ m h _ { \\lambda _ i - i + j - m } , \\right ] \\end{align*}"}
+{"id": "6732.png", "formula": "\\begin{align*} T _ { 1 } ( x ) = \\sum _ { x _ 1 \\leq t \\leq x } \\mu ( t ) d ( t ) \\sum _ { d ^ 2 \\leq x _ 0 } \\mu ( d ) \\sum _ { \\substack { k \\leq x / t \\\\ d ^ 2 \\mid k } } \\mu ( k t + a ) = O \\left ( \\frac { x } { ( \\log x ) ^ { C - 2 } } \\right ) , \\end{align*}"}
+{"id": "3209.png", "formula": "\\begin{align*} \\frac 1 p + \\frac 1 2 = \\frac 1 q . \\end{align*}"}
+{"id": "8676.png", "formula": "\\begin{align*} ( A ^ \\vee ) _ { i j } = A _ { - j , - i } . \\end{align*}"}
+{"id": "4785.png", "formula": "\\begin{align*} \\frac { 1 } { \\binom { N } { S } } \\sum _ { j \\in B } \\Pr _ { v \\sim \\mu _ { t } } \\big [ | v H | = j \\big ] K _ S ( j ) ^ 2 & \\leq \\frac { 1 } { \\binom { N } { S } } \\max _ { j \\in B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } [ | v H | = j ] \\cdot \\frac { 1 } { \\binom { N } { j } } \\right \\} \\sum _ { j \\in B } \\binom { N } { j } \\cdot K _ S ( j ) ^ 2 \\\\ & \\leq 2 ^ { N } \\max _ { j \\in B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } [ | v H | = j ] \\cdot \\frac { 1 } { \\binom { N } { j } } \\right \\} . \\end{align*}"}
+{"id": "382.png", "formula": "\\begin{align*} \\eta _ * ( n ) : = \\frac { \\sqrt { 2 } } { L _ * } \\left ( { C _ { \\textsf { K M T } } \\ln ( \\widetilde { n } ) + \\ln ( n ) } \\right ) \\ , . \\end{align*}"}
+{"id": "8731.png", "formula": "\\begin{align*} \\mathcal G ( u _ 1 , \\dots , u _ l ) = \\prod _ { i = 1 } ^ { l } E ( \\beta ) \\left ( 1 + { \\beta } / { u _ i } \\right ) ^ { i - l - 1 } \\cdot \\mathcal S ( u _ 1 , \\dots , u _ l ) , \\end{align*}"}
+{"id": "847.png", "formula": "\\begin{align*} J = t r ( J _ d ) I _ { 3 \\times 3 } - J _ d \\end{align*}"}
+{"id": "7238.png", "formula": "\\begin{align*} \\mathbb { E } [ T _ { a , b } ^ { n , n - 1 } ( P _ { N , n } ^ \\beta ) ] = C _ { N , n } ^ { \\beta , b } \\int _ { - 1 } ^ 1 | h | ^ a ( 1 - h ^ 2 ) ^ { n \\beta - { n - 1 \\over 2 } ( n + b + 1 ) } F _ { 1 , \\beta } ( h ) ^ { N - n } \\ , d h , \\end{align*}"}
+{"id": "1027.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\int _ 0 ^ \\infty G _ s ( t ) ^ 2 t ^ { a } d t < \\infty , & a > - ( s + 1 ) ; \\\\ \\int _ 0 ^ \\infty G ' _ s ( t ) ^ 2 t ^ { a } d t < \\infty , & a > 1 - 2 s . \\end{aligned} \\right . \\end{align*}"}
+{"id": "8752.png", "formula": "\\begin{align*} \\bar { d } _ 0 = \\check { d } _ { \\xi } \\vee \\check { \\eta } \\check { \\eta } \\geq \\eta = \\frac { \\log \\left ( \\psi _ { \\bar { A } } | \\bar { C } | _ { S _ \\infty } ^ 2 / s _ { d _ 0 } ( H g _ { 0 , d _ 0 } ) \\right ) } { 2 \\log \\left ( 1 / \\rho ( \\bar { A } ) \\right ) } , \\end{align*}"}
+{"id": "4059.png", "formula": "\\begin{align*} v ( y ) = g ^ * \\big ( Y u ^ { - 1 } ( y ) , y , u ( Y u ^ { - 1 } ( y ) ) \\big ) . \\end{align*}"}
+{"id": "436.png", "formula": "\\begin{align*} \\| f _ j \\| = \\| \\tau _ j \\| = | f _ j ( \\tau _ j ) | = 1 , \\forall 1 \\leq j \\leq n \\end{align*}"}
+{"id": "4818.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial c ^ 2 } f ( \\epsilon , c ) & = \\frac { 2 \\epsilon } { c ^ 2 \\ln 2 } > 0 , \\end{align*}"}
+{"id": "467.png", "formula": "\\begin{align*} s _ { 1 ^ r } ^ g \\cdot \\left [ \\sum _ { \\alpha _ 0 + \\cdots + \\alpha _ r = ( r + 1 ) ( d - r ) - r g } \\left ( \\prod _ { i = 0 } ^ r s _ { \\alpha _ i } \\right ) \\right ] \\end{align*}"}
+{"id": "4634.png", "formula": "\\begin{align*} \\left ( E _ 0 = \\bigoplus _ { p + q = k } E ^ { p , q } _ 0 , \\theta _ 0 = \\bigoplus _ { p + q = k } \\theta ^ { p , q } _ 0 \\right ) \\end{align*}"}
+{"id": "9179.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\lim _ { N \\rightarrow \\infty } \\lim _ { m ^ 2 \\rightarrow 0 } ( f _ \\epsilon , \\tilde { C } ( s , m ^ 2 ) f _ \\epsilon ) = \\frac { 1 } { 4 \\pi ^ 2 } \\int _ { \\R ^ 2 } \\frac { 1 } { v _ { J } ^ 2 + s } | p | ^ { - 2 } | \\hat { f } ( p ) | ^ 2 \\ , d p = \\frac { 1 } { v _ { J } ^ 2 + s } ( f , ( - \\Delta _ { \\R ^ 2 } ) ^ { - 1 } f ) \\end{align*}"}
+{"id": "6237.png", "formula": "\\begin{align*} \\begin{cases} \\phi _ r '' - \\Delta \\phi _ r + \\sum _ { s = 1 } ^ p \\widehat \\alpha _ { r s } \\phi _ s = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\phi _ r = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu \\phi _ r + \\sum _ { s = 1 } ^ p \\widehat \\beta _ { r s } \\phi _ s = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 , \\end{cases} \\end{align*}"}
+{"id": "1529.png", "formula": "\\begin{align*} \\boxed { S _ { E C } [ \\alpha , \\omega ] = \\int \\Omega ^ i \\rho ^ a _ { i , c } \\eta ^ { c b } \\wedge ( \\omega \\oplus \\alpha ) ^ { ( 8 ) } _ { a b } } \\end{align*}"}
+{"id": "3135.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( \\tilde { A } ) = 2 s ( 1 + \\bar { a } ) \\int _ Y ( \\partial _ { 1 2 2 } ^ 3 w ) ( \\partial _ 1 w ) . \\end{align*}"}
+{"id": "9107.png", "formula": "\\begin{align*} \\mathsf { C } _ { \\rho , 2 } & : = \\frac { \\log e } { 2 } \\left ( \\frac { h _ 1 } { 1 + h _ 1 \\mathsf { P } } - \\frac { h _ { \\rho } } { 1 + h _ { \\rho } \\mathsf { P } } \\right ) . \\end{align*}"}
+{"id": "6974.png", "formula": "\\begin{gather*} a ^ { ( n ) } = \\begin{cases} a ( a + 1 ) ( a + 2 ) \\cdots ( a + n - 1 ) & \\ n > 0 , \\\\ 1 & \\ n = 0 . \\end{cases} \\end{gather*}"}
+{"id": "7326.png", "formula": "\\begin{align*} M ( \\gamma ) = O ( \\gamma ^ { ( N + 1 - \\alpha ) / \\alpha } K ( \\gamma ) ^ { N + 1 } ) ( \\gamma \\to \\infty ) . \\end{align*}"}
+{"id": "6718.png", "formula": "\\begin{align*} \\omega ^ { - t s } e ^ { \\frac { i 2 \\pi u s } { q } } = e ^ { \\frac { - i 2 \\pi t s } { p } } \\cdot e ^ { \\frac { i 2 \\pi u s } { q } } = e ^ { \\frac { i 2 \\pi s ( u p - t q ) } { p q } } = e ^ { i 2 \\pi \\alpha s } , \\end{align*}"}
+{"id": "2405.png", "formula": "\\begin{align*} E _ { 2 2 } ^ T = - E _ { 2 2 } , E _ { 3 3 } ^ T = - E _ { 3 3 } , 0 = A _ { 1 2 } , 0 = A _ { 1 3 } . \\end{align*}"}
+{"id": "5660.png", "formula": "\\begin{align*} \\lambda _ 0 ^ { ( 2 ) } = - \\frac { 4 \\pi D } { | \\Omega | } \\sum _ { j = 1 } ^ N \\ell _ j \\chi _ j . \\end{align*}"}
+{"id": "1035.png", "formula": "\\begin{align*} \\mathcal { R } _ { n , \\alpha } ( \\theta ( \\mathcal { P } ) , \\Phi \\circ \\rho ) = \\mathcal { R } _ { n , \\alpha } ( \\theta ( \\mathcal { P } ) , \\Phi \\circ \\rho , 0 ) , \\end{align*}"}
+{"id": "323.png", "formula": "\\begin{align*} \\widetilde F ( \\xi ) = \\int _ { - \\infty } ^ { \\infty } ( \\cos ( 2 \\pi \\xi x ) + \\sin ( 2 \\pi \\xi x ) ) F ( x ) \\dd x . \\end{align*}"}
+{"id": "4099.png", "formula": "\\begin{align*} \\nabla _ i ( 3 R - \\frac { 1 } { 4 } | H | ^ 2 ) = 0 \\Longrightarrow 3 R - \\frac { 1 } { 4 } | H | ^ 2 = \\end{align*}"}
+{"id": "5763.png", "formula": "\\begin{align*} \\Lambda ^ \\sharp = q ^ { - 1 } \\Lambda . \\end{align*}"}
+{"id": "1658.png", "formula": "\\begin{align*} d ^ \\times t _ \\sigma = \\left \\{ \\begin{array} { l l } r ^ { - 1 } d r , & T ( F _ \\sigma ) _ + = \\R _ + , r \\in ( 0 , \\infty ) ; \\\\ \\pi ^ { - 1 } 2 d \\theta \\cdot r ^ { - 1 } d r , & T ( F _ \\sigma ) _ + = \\C ^ \\times , r \\in ( 0 , \\infty ) , \\quad \\theta \\in [ 0 , 2 \\pi ] \\\\ \\pi ^ { - 1 } 2 d \\theta , & T ( F _ \\sigma ) _ + = \\C ^ \\times / \\R ^ \\times , \\quad \\theta \\in [ 0 , \\pi ] \\end{array} \\right . \\end{align*}"}
+{"id": "701.png", "formula": "\\begin{align*} \\frac { d a _ k } { d t } = \\frac { \\Gamma } { V } \\oint _ { \\alpha _ k } * \\nu = \\frac { \\Gamma } { V } \\oint _ { \\alpha _ k } * \\eta , \\end{align*}"}
+{"id": "6624.png", "formula": "\\begin{align*} \\norm { D ( t ) ^ r \\psi } = \\norm { D ^ r \\psi _ t } \\leq C _ { r , V } \\sum _ { k = 0 } ^ r \\norm { H ^ k \\psi } \\leq C _ { r , V } ' \\norm { \\psi } _ { H ^ { 2 r } } , \\end{align*}"}
+{"id": "7038.png", "formula": "\\begin{align*} ( m + n ) m _ { m + n - 1 } = _ { x ^ { m + n - 1 } } \\dfrac { 1 } { ( 1 + b _ 1 x + b _ 2 x ^ 2 + \\cdots ) ^ { m + n } } , \\end{align*}"}
+{"id": "2517.png", "formula": "\\begin{align*} M _ f ( x , v , t ) = \\frac 1 { ( 2 \\pi T ( x , t ) ) ^ { 3 / 2 } } \\exp \\left ( - \\frac { | v - u ( x , t ) | ^ 2 } { 2 T ( x , t ) } \\right ) \\ , , \\end{align*}"}
+{"id": "4724.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c [ \\xi _ { i , 1 } \\cdot ( v _ i , w _ i ) + \\xi _ { i , 2 } \\cdot ( x _ i , y _ i ) ] + \\sum _ { j = 1 } ^ d \\eta _ j \\cdot ( v _ { c + j } , w _ { c + j } ) \\ = \\ ( 0 , 0 ) . \\end{align*}"}
+{"id": "4164.png", "formula": "\\begin{align*} 4 \\| \\nabla \\omega _ { ( g , b ) } \\| ^ 2 _ { L ^ 2 } & = \\lambda ( g , b ) - \\int _ M ( R - \\frac { 1 } { 1 2 } | H | ^ 2 ) \\omega _ { ( g , b ) } ^ 2 d V _ g \\\\ & \\leq \\sup _ { M } ( R - \\frac { 1 } { 1 2 } | H | ^ 2 ) - \\inf _ { M } ( R - \\frac { 1 } { 1 2 } | H | ^ 2 ) \\leq C . \\end{align*}"}
+{"id": "1158.png", "formula": "\\begin{align*} & I D M _ m ( F , A ) \\iff O P _ m ( F , A ) \\wedge \\forall a \\in A \\ , \\ , F ( a , a , . . . , a ) = a . \\end{align*}"}
+{"id": "3745.png", "formula": "\\begin{align*} \\textrm { i f } f \\in \\mathcal { S } ( V _ 2 ( F ) ) | _ { X _ { P _ 2 } ^ \\circ ( F ) } \\textrm { t h e n } \\tilde { a } _ 2 ( f ) = f ( 0 ) . \\end{align*}"}
+{"id": "8595.png", "formula": "\\begin{align*} \\deg ( g _ u ( x ) ) = \\deg ( \\widetilde { g } _ { t } ( x ) ) + 1 \\leq 2 ^ n - n - 2 . \\end{align*}"}
+{"id": "4683.png", "formula": "\\begin{align*} \\mathfrak { R } ^ { G _ { I I I } } = \\mathbb { C } [ \\varphi _ 4 , \\varphi _ { 1 2 } ] \\end{align*}"}
+{"id": "1586.png", "formula": "\\begin{align*} \\alpha s \\geq d _ p ^ p ( \\mu , \\xi ) = \\sum _ { ( x , y ) \\in X \\times X } \\varrho ^ p ( x , y ) \\cdot \\pi ^ * ( x , y ) . \\end{align*}"}
+{"id": "4074.png", "formula": "\\begin{align*} H ( X , Y , Z ) = 2 \\langle [ \\sigma X , \\sigma Y ] , \\sigma Z \\rangle X , Y , Z \\in T M . \\end{align*}"}
+{"id": "3177.png", "formula": "\\begin{align*} a _ 1 ( y ) : = b _ 1 ( y _ 1 , y _ 2 ) , a _ 2 ( y ) : = b _ 2 ( y _ 1 , y _ 2 ) , a _ 3 ( y ) : = 1 0 - b _ 1 ( y _ 1 , y _ 2 ) - b _ 2 ( y _ 1 , y _ 2 ) \\end{align*}"}
+{"id": "2488.png", "formula": "\\begin{align*} \\mathcal Q _ n = x _ n \\otimes x ' _ n n \\geq 0 . \\end{align*}"}
+{"id": "2072.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } m _ { r , 1 } ( x ) = x ^ { \\rho _ 1 } \\end{align*}"}
+{"id": "3115.png", "formula": "\\begin{align*} - A : D ^ 2 w = \\gamma - \\bar { \\gamma } \\quad Y , w Y , \\int _ Y w = 0 , \\end{align*}"}
+{"id": "424.png", "formula": "\\begin{align*} g _ n ( U x ) & = g _ n ( \\theta _ \\omega \\theta _ f x ) = g _ n \\left ( \\theta _ \\omega \\left ( \\sum _ { m = 1 } ^ { \\infty } f _ m ( x ) e _ m \\right ) \\right ) \\\\ & = g _ n \\left ( \\sum _ { m = 1 } ^ { \\infty } f _ m ( x ) \\theta _ \\omega e _ m \\right ) = g _ n \\left ( \\sum _ { m = 1 } ^ { \\infty } f _ m ( x ) \\omega _ m \\right ) \\\\ & = \\sum _ { m = 1 } ^ { \\infty } f _ m ( x ) g _ n ( \\omega _ m ) = f _ n ( x ) , \\forall x \\in \\mathcal { X } , \\end{align*}"}
+{"id": "4352.png", "formula": "\\begin{align*} x z = ( x z ) y ( x z ) = x ( z y ) ( x z ) = x ( z y ) ( z y ) ( x z ) = ( x z ) y z ( y x z ) \\end{align*}"}
+{"id": "8353.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ { 1 , M , A } } | ( \\partial _ t - \\partial _ r ) w | ^ 2 \\mathrm { d } t \\mathrm { d } S = \\varepsilon ( M ) . \\end{align*}"}
+{"id": "2844.png", "formula": "\\begin{align*} b _ i = a _ i \\end{align*}"}
+{"id": "6507.png", "formula": "\\begin{align*} \\P _ { 0 , K } ^ Y \\varphi ( Y ) + \\P _ { \\pi , K } ^ Y ( 1 - \\varphi ( Y ) ) & = 1 - \\big ( \\P _ { 0 , K } ^ Y ( Y \\in A _ \\varphi ) - \\P _ { \\pi , K } ^ Y ( Y \\in A _ \\varphi ) \\big ) . \\end{align*}"}
+{"id": "596.png", "formula": "\\begin{align*} J ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } : = \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ { ( \\epsilon ) } ) ^ { \\rm T } { \\rm d } X _ t ^ { ( \\epsilon ) } \\end{align*}"}
+{"id": "5179.png", "formula": "\\begin{align*} \\frac { r ( r - K ) } { 2 K } + ( K - 1 ) \\frac { r } { K } = \\frac { r ^ 2 - r K + 2 K r - 2 r } { 2 K } = \\frac { r ( r + K - 2 ) } { 2 K } . \\end{align*}"}
+{"id": "8123.png", "formula": "\\begin{align*} S ( 0 , a ; c ) = \\sum _ { v \\bar v \\equiv 1 \\pmod * { c } } e \\Bigl ( \\frac { a v } { c } \\Bigr ) , \\end{align*}"}
+{"id": "1114.png", "formula": "\\begin{align*} \\mathbb { E } _ { X \\sim R _ 0 } ( | X - \\theta | ) = r - \\delta \\theta \\mathbb { E } _ { X \\sim R _ 1 } ( | X - \\theta | ) = r + \\delta \\theta . \\end{align*}"}
+{"id": "5169.png", "formula": "\\begin{align*} k ( \\alpha ; p ) = \\frac { R ( x , y , y , x ) } { g ( x , x ) g ( y , y ) - [ g ( x , y ) ] ^ 2 } . \\end{align*}"}
+{"id": "9025.png", "formula": "\\begin{align*} _ k ( H ^ 0 ( \\Q _ \\Sigma / \\Q _ { n } , \\Phi ) ) + _ k ( H ^ 2 ( \\Q _ \\Sigma / \\Q _ { n } , \\Phi ) ) - _ k ( H ^ 1 ( \\Q _ \\Sigma / \\Q _ { n } , \\Phi ) ) = - d ^ - ( \\Phi ) [ \\Q _ n : \\Q ] . \\end{align*}"}
+{"id": "5022.png", "formula": "\\begin{align*} H ^ 2 ( e _ 1 , e _ 1 ) & = 2 \\ , \\frac { a ^ 4 p ^ 2 + b ^ 4 q ^ 2 } { a ^ 4 b ^ 4 } , \\\\ H ^ 2 ( e _ 2 , e _ 2 ) & = 2 \\ , \\frac { q ^ 2 } { a ^ 2 \\mu ^ 2 } = H ^ 2 ( e _ 3 , e _ 3 ) , \\\\ H ^ 2 ( e _ 4 , e _ 4 ) & = 2 \\ , \\frac { p ^ 2 } { b ^ 2 \\mu ^ 2 } = H ^ 2 ( e _ 5 , e _ 5 ) . \\end{align*}"}
+{"id": "8398.png", "formula": "\\begin{align*} \\int \\hat \\Psi _ { \\alpha } \\cdot \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha } ^ k \\cdot h _ { \\alpha } \\ : d m & = \\int P _ { \\alpha } ^ k ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\cdot \\hat \\Psi _ { \\alpha } \\ : d m \\\\ & = \\int P _ { \\alpha } \\left ( P _ { \\alpha } ^ k ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\cdot \\hat \\Psi _ { \\alpha } \\right ) \\ : d m \\end{align*}"}
+{"id": "5181.png", "formula": "\\begin{align*} A _ { \\overline { L } _ q ^ { 7 } ( r ; \\underline { m } ) } = \\begin{psmallmatrix} 1 & r & \\frac { r ( r + 1 ) } { 2 } & x _ 0 & \\cdots & \\cdots & x _ { K - 1 } \\\\ 0 & 1 & r & y _ 0 & \\cdots & \\cdots & y _ { K - 1 } \\\\ 0 & 0 & 1 & r / K & \\cdots & \\cdots & r / K \\\\ 0 & 0 & 0 & & & & \\\\ \\vdots & \\vdots & \\vdots & & I _ { K } & & \\\\ 0 & 0 & 0 & & & & \\\\ 0 & 0 & 0 & & & & \\end{psmallmatrix} \\end{align*}"}
+{"id": "8212.png", "formula": "\\begin{align*} P \\{ N ( t ) = 2 k \\} = ( \\lambda _ 1 t ) ^ k ( \\lambda _ 2 t ) ^ k e ^ { - \\lambda _ 1 t } \\ , E ^ { k } _ { 1 , 2 k + 1 } \\Bigl ( t ( \\lambda _ 1 - \\lambda _ 2 ) \\Bigr ) , \\end{align*}"}
+{"id": "109.png", "formula": "\\begin{align*} \\lim \\limits _ { s \\rightarrow 1 ^ - } ( 1 - s ) | | f | | ^ p _ { W ^ { s , p } ( \\mathbb R ^ n ) } = \\frac { 1 } { p } k ( p , n ) | | \\nabla f | | ^ p _ { L ^ p ( \\mathbb R ^ n ) } , \\end{align*}"}
+{"id": "2525.png", "formula": "\\begin{align*} M _ { Z _ N } ^ \\varphi ( x , v ) = M _ { u _ N ^ \\varphi ( x ) , T _ N ^ \\varphi ( x ) } ( v ) \\ , . \\end{align*}"}
+{"id": "901.png", "formula": "\\begin{align*} \\left | \\sin \\left ( \\pi ^ 3 q _ n \\right ) \\right | = \\left | \\sin \\left ( \\pi ^ 3 q _ n - \\pi p _ n \\right ) \\right | = \\left | \\sin \\pi \\left ( \\pi ^ 2 q _ n - p _ n \\right ) \\right | . \\end{align*}"}
+{"id": "2423.png", "formula": "\\begin{align*} M _ { 2 2 } \\dot v _ 2 = J _ { 2 2 } ( t ) v _ 2 + f _ 2 ( t ) \\end{align*}"}
+{"id": "3390.png", "formula": "\\begin{align*} A : = \\bigoplus _ { i , j \\in I } { \\rm H o m } ( P _ i , P _ j ) , \\end{align*}"}
+{"id": "7958.png", "formula": "\\begin{align*} \\pi ' i ( m ) = i ' \\pi ( m ) = \\pi ' \\beta ( x ) . \\end{align*}"}
+{"id": "1538.png", "formula": "\\begin{align*} \\Phi ^ { \\alpha a } = - \\frac 1 2 ( \\gamma ^ a s ) ^ \\alpha \\end{align*}"}
+{"id": "5957.png", "formula": "\\begin{align*} B ^ T E _ r = P ^ { T } P B e _ r \\subseteq P ^ { T } P \\hbox { K e r } ( C _ p ) \\subseteq V r = 1 , \\cdots , p , \\end{align*}"}
+{"id": "3028.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\hbox { E i n } ( \\mathbf { g } ) { _ a } ^ b + \\Lambda \\delta { _ a } ^ b & = & \\frac { 1 } { 2 } ( \\mathbf { F } { ^ i } _ { a c } \\mathbf { F } { _ i } ^ { b c } - \\frac { 1 } { 2 } \\| \\mathbf { F } \\| ^ 2 \\delta { _ a } ^ b ) \\\\ \\partial ^ { \\gamma , \\mathbf { A } } _ b \\mathbf { F } { _ i } ^ { a b } & = & 0 \\end{array} \\end{align*}"}
+{"id": "5005.png", "formula": "\\begin{align*} \\begin{cases} - \\varphi _ t = \\Delta \\varphi + g & , \\\\ \\partial _ \\nu \\varphi = 0 & , \\\\ \\varphi ( \\cdot , T ) = 0 & \\end{cases} \\end{align*}"}
+{"id": "9036.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ | \\Delta V _ \\tau | ^ p \\right ] = \\mathbb { E } \\left [ | \\Delta V _ \\tau - \\Delta V ^ { ( m ) } _ \\tau | ^ p \\right ] \\leq 2 ^ p \\sup _ { \\tau \\in \\mathcal { T } _ 0 } \\mathbb { E } \\left [ | V _ \\tau - V ^ { ( m ) } _ \\tau | ^ p \\right ] , \\end{align*}"}
+{"id": "6323.png", "formula": "\\begin{align*} Y ^ * { : = } \\frac { \\ell } { \\rm L } Y , Y { : = } \\left ( - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } \\right ] ^ 3 . \\end{align*}"}
+{"id": "5365.png", "formula": "\\begin{align*} ( E ^ { ( i ) } _ h ) ^ 2 - ( E ^ { ( j ) } _ h ) ^ 2 = 0 \\forall i , j \\in \\{ 1 , \\ldots , m \\} , \\forall h = 1 , 2 , 3 . \\end{align*}"}
+{"id": "1319.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j , k \\leq n } f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) \\geq \\frac { 1 } { d } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 = \\frac { 1 } { ( \\mathcal { X } ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 . \\end{align*}"}
+{"id": "478.png", "formula": "\\begin{align*} H ( x , y , \\xi ) = \\left ( \\frac { | \\xi | } { | x - y | ^ s } \\right ) ^ p + a ( x , y ) \\left ( \\frac { | \\xi | } { | x - y | ^ r } \\right ) ^ q \\end{align*}"}
+{"id": "3964.png", "formula": "\\begin{align*} & \\overline { G } \\subset \\{ \\overline { x } ; \\overline { x } _ 1 \\leq 0 \\} , & & \\overline { G } \\cap \\{ \\overline { x } _ 1 = 0 \\} = 0 , \\\\ & - a e _ 1 \\in \\overline { G } , & & b e _ 1 \\in \\overline { \\Omega ^ * } . \\end{align*}"}
+{"id": "1817.png", "formula": "\\begin{align*} \\begin{cases} u _ 1 ( t + T / 6 ) = G u _ 2 ( t ) , \\\\ u _ 2 ( t + T / 6 ) = G u _ 3 ( t ) , \\\\ u _ 3 ( t + T / 6 ) = G u _ 1 ( t ) , \\end{cases} G = \\begin{pmatrix} - 1 & 0 \\\\ 0 & 1 \\\\ \\end{pmatrix} , \\end{align*}"}
+{"id": "1572.png", "formula": "\\begin{align*} R i c + \\nabla ^ 2 f - \\rho R g = \\mu g . \\end{align*}"}
+{"id": "4499.png", "formula": "\\begin{align*} \\frac { 1 1 } { 2 4 } = \\frac { 1 } { 3 } + \\frac { 1 } { 8 } \\end{align*}"}
+{"id": "2803.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n c _ i = 0 . \\end{align*}"}
+{"id": "6212.png", "formula": "\\begin{align*} \\hbox { r a n k } ( C _ p D ) = \\hbox { r a n k } ( D ) = N - p . \\end{align*}"}
+{"id": "3497.png", "formula": "\\begin{align*} { C _ { { \\rm { D A M } } } } = \\frac { { \\left ( { { n _ c } - 2 { { \\tilde n } _ { \\max } } } \\right ) } } { { { n _ c } } } { \\log _ 2 } \\left ( { 1 + { \\gamma _ a } } \\right ) , \\end{align*}"}
+{"id": "5221.png", "formula": "\\begin{align*} b _ { 1 5 } = & \\frac { c _ { 3 6 } \\sin \\ ! \\left ( \\theta \\right ) - c _ { 4 6 } ( 1 - \\cos \\ ! \\left ( \\theta \\right ) ) } { 2 \\gamma \\sin \\ ! \\left ( \\theta \\right ) } \\ , b _ { 2 2 } \\\\ b _ { 2 5 } = & \\frac { m _ 1 \\sin \\ ! \\left ( \\theta \\right ) + m _ 0 \\left ( 1 + \\cos \\ ! \\left ( \\theta \\right ) \\right ) } { \\gamma ^ 2 \\left ( 1 + \\cos \\ ! \\left ( \\theta \\right ) \\right ) } \\ , b _ { 2 2 } \\ , \\end{align*}"}
+{"id": "1655.png", "formula": "\\begin{align*} \\int _ c \\omega = e ( \\omega ) \\cap c , \\omega \\in H ^ 0 ( \\Lambda , \\Omega _ M ^ u ) = \\Omega _ { M / \\Lambda } ^ u . \\end{align*}"}
+{"id": "796.png", "formula": "\\begin{align*} F ( t , n ) = \\left \\{ x \\in \\Sigma _ { B _ j } : \\frac { \\log \\| \\rho ( \\lambda ( x _ 0 , x _ 1 ) \\ldots \\lambda ( x _ { n - 1 } , x _ n ) ) \\| - \\Lambda n } { \\sqrt { n } } < t \\right \\} . \\end{align*}"}
+{"id": "3750.png", "formula": "\\begin{align*} \\lambda ( f ) & : = ( \\log q ) ^ { - 1 } \\frac { d } { d s } \\left ( - \\zeta ( s ) ^ { - 1 } Z _ { r _ 2 } ( f , s ) + \\zeta ( s + 1 ) ^ { - 1 } Z _ { r _ 1 } ( d _ 2 ( f ) , s + 1 ) \\right ) \\Big | _ { s = 0 } \\\\ & = - \\sum _ { j = - A } ^ { A - 1 } f ( 0 , 0 , \\varpi ^ j ) + 2 A f ( 0 ) - \\sum _ { j = - A } ^ { A - 1 } f ( 0 , 0 , 0 , \\varpi ^ j , 0 , 0 ) \\\\ & = ( n + 1 ) \\tilde { c } _ 2 ( I ( f ) ) - I ( f ) ( 0 , 0 , 0 , \\varpi ^ n ) \\end{align*}"}
+{"id": "8940.png", "formula": "\\begin{align*} b _ { j } ^ { \\left ( \\nu , 2 \\right ) } = 8 \\pi ^ 2 \\left [ \\frac { \\left ( 1 + \\nu ^ { 2 } \\right ) ^ { j } } { j ! } - \\frac { \\nu ^ 2 \\left ( 1 + \\nu ^ { 2 } \\right ) ^ { j - 1 } } { \\left ( j - 1 \\right ) ! } + \\sum \\limits _ { i = 2 } ^ { j } \\frac { \\left ( 1 + \\nu ^ { 2 } \\right ) ^ { j - i } } { \\left ( j - i \\right ) ! } c _ { i } ^ { ( \\nu , 2 ) } \\right ] \\end{align*}"}
+{"id": "6408.png", "formula": "\\begin{align*} \\nabla ^ 2 \\Psi ( P _ j ) = \\begin{pmatrix} M _ 1 & 0 & 0 & - M _ 1 & 0 & 0 \\\\ 0 & M _ 2 & 0 & 0 & - M _ 2 & 0 \\\\ 0 & 0 & j & 0 & 0 & - j \\\\ - M _ 1 & 0 & 0 & M _ 1 & 0 & 0 \\\\ 0 & - M _ 2 & 0 & 0 & M _ 2 & 0 \\\\ 0 & 0 & - j & 0 & 0 & j \\end{pmatrix} , \\end{align*}"}
+{"id": "5888.png", "formula": "\\begin{align*} \\hbox { r a n k } ( C _ p D ) = N - p . \\end{align*}"}
+{"id": "5153.png", "formula": "\\begin{align*} \\mathcal { S } _ { j } ( \\lambda , b , \\Omega , f _ { j } ) ( w ) & : = \\mbox { I m } \\left \\lbrace \\left [ \\Omega \\Phi _ { j } ( w ) + ( - 1 ) ^ { j } S ( \\lambda , \\Phi _ { j } , \\Phi _ { j } ) ( w ) \\right ] \\overline { w } \\overline { \\Phi _ { j } ' ( w ) } \\right \\rbrace , \\\\ \\mathcal { I } _ { j } ( \\lambda , b , f _ { 1 } , f _ { 2 } ) & : = ( - 1 ) ^ { j - 1 } \\mbox { I m } \\left \\lbrace S ( \\lambda , \\Phi _ { i } , \\Phi _ { j } ) ( w ) \\overline { w } \\overline { \\Phi _ { j } ' ( w ) } \\right \\rbrace . \\end{align*}"}
+{"id": "7182.png", "formula": "\\begin{align*} E _ t = \\left \\{ z _ { \\alpha , j k } ( x _ { j } , \\ , t ) \\right \\} , \\end{align*}"}
+{"id": "3013.png", "formula": "\\begin{align*} \\chi = \\chi { _ i } ^ { b k } e _ b ^ { ( 3 ) } \\wedge \\bar { \\theta } ^ { ( r - 1 ) } _ k + \\frac { 1 } { 2 } \\chi { _ i } ^ { j k } e ^ { ( 4 ) } \\wedge \\bar { \\theta } ^ { ( r - 2 ) } _ { j k } \\end{align*}"}
+{"id": "6010.png", "formula": "\\begin{align*} ( e v _ { r + 1 } ) _ * [ \\O _ { \\mathcal { W } _ { g _ k } } ] = [ \\O _ { e v _ { r + 1 } ( \\mathcal { W } _ { g _ k } ) } ] , \\end{align*}"}
+{"id": "772.png", "formula": "\\begin{align*} O ( g , R ) = \\{ \\xi \\in \\partial \\Gamma : \\langle \\xi , g \\rangle > | g | _ S - R \\} \\end{align*}"}
+{"id": "5817.png", "formula": "\\begin{align*} D e _ r = 0 , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "7224.png", "formula": "\\begin{align*} \\big | F ( q ) - a _ n \\big ( q - q _ 0 \\big ) ^ n \\big | _ { C _ { q _ 0 } } & = \\left | \\sum _ { \\ell = 0 } ^ { n - 1 } a _ { n - \\ell } \\big ( q - q _ 0 \\big ) ^ \\ell \\right | _ { C _ { q _ 0 } } \\\\ & \\leq \\sum _ { \\ell = 0 } ^ { n - 1 } \\big | a _ { n - \\ell } \\big | \\left | \\big ( q - q _ 0 \\big ) ^ \\ell \\right | _ { C _ { q _ 0 } } \\\\ & < R ^ n = a ^ n \\big | a _ n ( q - q _ 0 ) ^ n \\big | _ { C _ { q _ 0 } } \\end{align*}"}
+{"id": "7184.png", "formula": "\\begin{align*} C _ \\beta \\equiv r _ \\beta - \\frac { \\partial U _ { \\beta } } { \\partial z _ \\alpha } z _ { \\alpha , j } v _ j - \\frac { \\partial \\Phi ^ \\beta _ j } { \\partial z _ \\alpha } z _ { \\alpha , j } , \\ , \\ , \\ , \\ , \\beta = 1 \\dots \\omega , \\end{align*}"}
+{"id": "5468.png", "formula": "\\begin{align*} E | X ( t \\wedge \\tau _ n ) - Y ( t \\wedge \\tau _ n ) | ^ 2 & \\leq e ^ { 2 ( L _ n T + C \\cdot L _ n ) t } 2 ( L _ n T + C \\cdot L _ n ) T K e ^ { - L _ n \\epsilon } \\\\ & = 2 ( L _ n T + L _ n ) T K e ^ { - L _ n ( \\epsilon - 2 ( T + 1 ) t ) } . \\end{align*}"}
+{"id": "4047.png", "formula": "\\begin{align*} D _ { p _ k } A _ { i j } & = E ^ { r k } \\left ( D _ { x _ j } E _ { i r } - D _ { x _ j } ( Q _ r ) g _ { i , z } \\right ) \\\\ & = E ^ { r k } D _ { x _ j } E _ { i r } + \\delta _ { j k } \\frac { g _ { i , z } } { g _ z } . \\end{align*}"}
+{"id": "2541.png", "formula": "\\begin{align*} \\dot I _ N ( t ) & \\le C \\bigg ( \\varphi _ 0 + \\frac { \\| \\nabla \\varphi \\| _ \\infty } { r ^ { 2 d } \\varphi _ 0 } + \\frac { \\| \\nabla \\varphi \\| _ \\infty ^ 2 } { r ^ { 5 d } \\varphi _ 0 ^ 2 } \\bigg ) I _ N ( t ) \\\\ & + \\bigg ( \\frac { 1 } { r ^ { 2 d } N } + \\frac { 1 } { r ^ { 4 d } \\varphi _ 0 N ^ 2 } + \\frac { \\varphi _ 0 } { ( r ^ { 3 d } N ) ^ { 1 / 2 } } + \\frac { \\| \\nabla \\varphi \\| _ \\infty ^ 2 } { r ^ { 5 d } \\varphi _ 0 ^ 2 N ^ { 1 / 2 } } \\bigg ) \\ , . \\end{align*}"}
+{"id": "2556.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { l = 0 } ^ { r } \\sum _ { \\begin{subarray} { c } \\sum _ { i = 1 } ^ { p - 1 } \\mathbf { y } _ { k _ i - c _ { i } } ^ { ( i ) } + \\mathbf { y } ^ { ( p ) } _ { k _ p - 2 } = l \\\\ y _ { j } ^ { ( i ) } \\ge 0 \\end{subarray} } Z ( \\sigma _ { r - l } ( v _ { \\mathbf { y } } ) ; \\alpha ) \\\\ = & \\sum _ { l = 0 } ^ { r } \\sum _ { \\begin{subarray} { c } l _ 1 + \\cdots + l _ { q - 1 } = l \\\\ l _ i \\ge 0 \\end{subarray} } Z ( \\sigma _ { r - l } ( v ^ { ' } _ { ( \\{ l _ i \\} _ { i = 1 } ^ { q - 1 } ) } ) ; \\alpha ) \\end{aligned} \\end{align*}"}
+{"id": "8506.png", "formula": "\\begin{gather*} C _ t = \\frac { 1 } { 9 \\kappa - 5 \\kappa ^ { - 3 } \\kappa ^ 2 _ s + 3 \\kappa ^ { - 2 } \\kappa _ { s s } } N , C ( \\cdot , ~ 0 ) = C _ 0 ( \\cdot ) . \\end{gather*}"}
+{"id": "1598.png", "formula": "\\begin{align*} \\mu _ * ( u ) = \\mu ^ * ( x _ { n + 1 } ) = 0 , \\qquad \\mu _ * ( x _ { n + 1 } ) = \\mu ^ * ( u ) = c \\end{align*}"}
+{"id": "585.png", "formula": "\\begin{align*} \\mathbb { E } ^ \\dagger [ \\mathbb { W } _ { t _ n , t _ { n + 1 } } ^ \\dagger ] = \\frac { 1 } { 2 } \\mathbb { E } ^ \\dagger [ W ^ \\dagger _ { t _ n , t _ { n + 1 } } \\otimes W ^ \\dagger _ { t _ n , t _ { n + 1 } } - [ W ^ \\dagger _ { t _ n } , W ^ \\dagger _ { t _ n , t _ { n + 1 } } ] ] - \\frac { \\Delta t } { 2 } I = 0 , \\end{align*}"}
+{"id": "4016.png", "formula": "\\begin{align*} G ( \\cdot , u , D u ) = 0 \\partial \\Omega . \\end{align*}"}
+{"id": "6580.png", "formula": "\\begin{align*} \\tilde { E } : = \\sup _ { \\tilde { x } \\in B _ { \\frac { 1 } { 4 } } , \\ , 0 < \\tilde { r } < \\frac { 1 } { 4 } , \\frac 1 2 < \\tilde { t } \\leq 1 } \\frac { \\tilde { \\mu } _ { \\tilde { t } } ( B _ { \\tilde { r } } ( \\tilde { x } ) ) } { \\tilde { r } ^ { n - 1 } } \\leq 2 ^ { n - 1 } l ^ { 1 - n } E _ 1 . \\end{align*}"}
+{"id": "9077.png", "formula": "\\begin{align*} \\big ( \\mu ^ { \\circ } ( \\xi : \\lambda ) ( \\iota _ { e } \\eta ) \\big ) ( \\phi ) = \\int _ { M _ { Q } } \\int _ { A } \\int _ { \\overline { N } _ { Q } } a ^ { - \\lambda - \\rho _ { Q } } \\Big ( \\xi ^ { \\vee } ( m ) \\eta , \\phi ( m a \\overline { n } ) \\Big ) \\ , d \\overline { n } \\ , d a \\ , d m . \\end{align*}"}
+{"id": "6764.png", "formula": "\\begin{align*} p ( \\Delta x ) = \\frac { 1 } { 2 \\sigma } \\exp \\left ( - \\frac { \\left ( \\Delta x - \\mu \\right ) ^ 2 } { 4 \\sigma ^ 2 } \\right ) \\end{align*}"}
+{"id": "4304.png", "formula": "\\begin{align*} \\alpha ( t , \\xi ) = \\nu ^ { - 1 } _ 0 | c _ * ( t , \\xi ) - c ( t ) | | \\xi | + \\frac { | \\partial _ { t } c _ * ( t , \\xi ) | } { c _ * ( t , \\xi ) } . \\end{align*}"}
+{"id": "6519.png", "formula": "\\begin{align*} \\Big ( \\underset { i = 1 } { \\overset { b } { \\sum } } Z _ i ^ 2 & \\leq \\frac { 2 4 { b } m k } { C _ \\alpha ( m - 1 ) } N _ \\alpha \\Big ) + \\left ( \\underset { 1 \\leq i \\leq b } { \\max } Z _ i ^ 2 \\geq C _ \\alpha ^ { - 1 } { m \\sqrt { b } k } \\right ) \\\\ & \\qquad \\leq \\left ( \\underset { i = 1 } { \\overset { b } { \\sum } } Z _ i ^ 2 \\leq \\frac { 2 4 { b } m k } { C _ \\alpha ( m - 1 ) } N _ \\alpha \\right ) + \\alpha / 8 . \\end{align*}"}
+{"id": "8809.png", "formula": "\\begin{align*} A - B ^ * P = D _ P F _ 1 D _ P \\ , , B - A ^ * P = D _ P F _ 2 D _ P . \\end{align*}"}
+{"id": "4638.png", "formula": "\\begin{align*} \\mathrm { R a m } ( p _ Y ) = R _ 1 + R _ 2 \\end{align*}"}
+{"id": "8951.png", "formula": "\\begin{align*} { } _ { 2 } F { } _ { 1 } \\left ( \\begin{array} { c } a , \\ , b \\\\ c \\end{array} \\left \\vert \\xi \\right . \\right ) = ( 1 - \\xi ) ^ { - a } { } _ { 2 } F { } _ { 1 } \\left ( \\begin{array} { c } a , \\ , c - b \\\\ c \\end{array} \\left \\vert \\frac { \\xi } { 1 - \\xi } \\right . \\right ) , | \\arg ( 1 - \\xi ) | < \\pi , \\end{align*}"}
+{"id": "3747.png", "formula": "\\begin{align*} \\zeta ( s ) ^ { - 1 } Z _ { r _ 2 } ( f , s ) & = q ^ { - A s } f ( 0 ) + ( 1 - q ^ { - s } ) \\sum _ { j = - A } ^ { A - 1 } q ^ { - j s } f ( 0 , 0 , \\varpi ^ j ) \\\\ & = f ( 0 ) - s \\log q \\big ( A f ( 0 ) - \\sum _ { j = - A } ^ { A - 1 } f ( 0 , 0 , \\varpi ^ j ) \\big ) + O _ { f } ( s ^ 2 ) . \\end{align*}"}
+{"id": "4892.png", "formula": "\\begin{align*} t _ { u _ k } t _ { u _ l } = \\sum _ { 0 \\le i \\le \\min \\{ 2 k , 2 l \\} } t _ { u _ { k + l - i } } . \\end{align*}"}
+{"id": "4273.png", "formula": "\\begin{align*} H _ \\gamma ( u ( t ) ) & = 2 E _ \\gamma ( u ( t ) ) + \\frac { 4 } { p + 1 } \\| u ( t ) \\| ^ { p + 1 } _ { L ^ { p + 1 } } \\\\ & \\leq 2 E _ \\gamma ( u ( t ) ) + B ( H _ \\gamma ( u ( t ) ) ) ^ { \\frac { N ( p - 1 ) } { 4 } } ( M ( u ( t ) ) ) ^ { \\frac { 4 - ( N - 2 ) ( p - 1 ) } { 4 } } \\\\ & \\leq 2 | E ( u _ 0 ) | + B ( H _ \\gamma ( u ( t ) ) ) ^ { \\frac { N ( p - 1 ) } { 4 } } ( M ( u _ 0 ) ) ^ { \\frac { 4 - ( N - 2 ) ( p - 1 ) } { 4 } } . \\end{align*}"}
+{"id": "6615.png", "formula": "\\begin{align*} H = \\C { A } + W \\end{align*}"}
+{"id": "4071.png", "formula": "\\begin{align*} \\Gamma ( S ^ 2 M ) \\times \\Omega ^ 2 = ( \\mathcal { V } + \\mathcal { V } _ 1 ) \\oplus ( \\mathcal { V } ^ \\perp \\cap \\mathcal { V } _ 1 ^ \\perp ) \\end{align*}"}
+{"id": "3606.png", "formula": "\\begin{align*} R ^ { ( 2 ) } ( t ) \\ = \\ \\chi _ { ( - 1 , 1 ) } ( t ) \\ \\sqrt { 1 - t ^ 2 } . \\end{align*}"}
+{"id": "7310.png", "formula": "\\begin{align*} \\left | \\varphi ^ d _ { m _ n } ( \\lambda ; x ) - 1 - \\sum _ { k = 1 } ^ { d } \\lambda ^ k G ^ k _ { m _ n } ( x ) \\right | \\leq ( - 1 ) ^ { d } \\lambda ^ { d + 1 } \\left ( \\int _ { 0 } ^ { x } G ^ { d } _ { m _ n } ( y ) d m _ n ( y ) \\right ) x \\mathrm { e } ^ { \\lambda ( m _ n \\bullet s ) ( x ) } , \\end{align*}"}
+{"id": "2006.png", "formula": "\\begin{align*} m _ { X , Y } = \\prod _ { r = 1 } ^ n a _ { x _ r , y _ r } = \\phi ^ { \\sum _ { r = 1 } ^ n x _ r y _ r } = \\phi ^ { 0 } = 1 . \\end{align*}"}
+{"id": "7203.png", "formula": "\\begin{align*} \\big \\langle p , \\ , q \\big \\rangle \\ , = \\ , 0 , \\end{align*}"}
+{"id": "1311.png", "formula": "\\begin{align*} S _ { f , \\tau } : \\mathcal { X } \\ni x \\mapsto S _ { f , \\tau } x \\coloneqq \\sum _ { j = 1 } ^ n f _ j ( x ) \\tau _ j \\in \\mathcal { X } . \\end{align*}"}
+{"id": "10.png", "formula": "\\begin{align*} - d Q _ t = \\Big ( \\bar { f } + e ^ { - \\frac { \\beta } { 2 } t } \\bar { h } \\tilde { M } _ t - \\beta Q _ t \\Big ) d t - \\tilde { M } _ t d \\xi _ t , \\end{align*}"}
+{"id": "4664.png", "formula": "\\begin{align*} G _ { \\ell _ i , \\ell _ j , k } = \\binom { \\ell _ i + \\ell _ j - k } { k , \\ell _ i - k , \\ell _ j - k } \\frac { \\ell _ i ! \\ell _ j ! } { ( \\ell _ i + \\ell _ j - k ) ! } = \\binom { \\ell _ i } { k } \\binom { \\ell _ j } { k } k ! . \\end{align*}"}
+{"id": "9110.png", "formula": "\\begin{align*} \\min _ { \\rho \\in [ 0 , 1 ) } h _ { \\rho } \\mathsf { P } = \\min _ { \\rho \\in [ 0 , 1 ) } \\frac { h _ 1 + h _ 2 - 2 \\rho \\sqrt { h _ 1 h _ 2 } } { 1 - \\rho ^ 2 } \\mathsf { P } . \\end{align*}"}
+{"id": "3797.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\varphi _ t = \\Delta _ { g ( t ) , g ( t ) } \\varphi _ t . \\end{align*}"}
+{"id": "1935.png", "formula": "\\begin{align*} \\tilde { F } _ { \\tilde { g } } = \\tilde { H } ^ { - 1 } \\circ { F } _ g \\circ \\tilde { H } , \\end{align*}"}
+{"id": "3634.png", "formula": "\\begin{align*} \\begin{cases} \\psi '' ( r ) - G ( r ) \\psi ( r ) = 0 r > 0 , \\\\ \\psi ( 0 ) = 0 , \\ \\psi ' ( 0 ) = 1 . \\end{cases} \\end{align*}"}
+{"id": "7260.png", "formula": "\\begin{align*} \\tilde { \\eta } _ { m , j , \\gamma } ( t ) : = f ( \\gamma ) \\left ( \\frac { \\eta _ { m , j } ( \\gamma t ) } { \\gamma } - b t \\right ) ( \\gamma > 0 ) \\end{align*}"}
+{"id": "4039.png", "formula": "\\begin{align*} 0 & \\geq e ^ { \\kappa \\phi } a ^ { i j } D _ { i j } w ( x _ 0 ) \\\\ & = a ^ { i j } D _ { i j } v - 2 \\kappa a ^ { i j } D _ i \\phi D _ j v - \\kappa v a ^ { i j } D _ { i j } \\phi + \\kappa ^ 2 v a ^ { i j } D _ i \\phi D _ j \\phi . \\end{align*}"}
+{"id": "8859.png", "formula": "\\begin{align*} \\Phi _ { j } ^ { p , q , m } ( z ) = \\gamma _ { p , q } ^ { n , \\nu , m } ( 1 + \\left \\langle z , z \\right \\rangle ) ^ { - m - \\nu } { } _ { 2 } F { } _ { 1 } \\left ( \\begin{array} { c } p - m , q - m - 2 \\nu \\\\ n + p + q \\end{array} \\big | - \\left \\langle z , z \\right \\rangle \\right ) h _ { p , q } ^ { j } ( z , \\bar { z } ) \\end{align*}"}
+{"id": "9081.png", "formula": "\\begin{align*} & e ^ { - t \\rho _ { Q } ( X ) } \\Big ( R ^ { \\vee } \\big ( \\exp ( t X ) \\big ) D \\mu - R ^ { \\vee } \\big ( \\exp ( t X ) \\big ) \\delta _ { \\emptyset } ( D ) \\mu _ { \\emptyset } \\Big ) \\\\ & \\qquad = e ^ { - t \\rho _ { Q } ( X ) - \\epsilon t } R ^ { \\vee } \\big ( \\exp ( t X ) \\big ) R ^ { \\vee } ( u ) \\mu \\\\ & \\qquad \\qquad + e ^ { - t \\rho _ { Q } ( X ) } \\delta _ { \\emptyset } ( D ) \\Big ( R ^ { \\vee } \\big ( \\exp ( t X ) \\big ) \\mu - R ^ { \\vee } \\big ( \\exp ( t X ) \\big ) \\mu _ { \\emptyset } \\Big ) . \\end{align*}"}
+{"id": "1927.png", "formula": "\\begin{align*} \\partial _ j ( \\partial _ i g _ { \\lambda \\ , i j } - \\partial _ j g _ { \\lambda \\ , i i } ) & = \\chi _ \\lambda \\partial _ j ( \\partial _ i g _ { i j } - \\partial _ j g _ { i i } ) + \\partial \\chi _ \\lambda \\partial g + \\partial ^ 2 \\chi _ \\lambda ( g - \\delta ) \\\\ & = \\chi _ \\lambda \\partial _ j ( \\partial _ i g _ { i j } - \\partial _ j g _ { i i } ) + O ( | x | ^ { - q - 2 } ) . \\end{align*}"}
+{"id": "1640.png", "formula": "\\begin{align*} \\lambda _ s : = \\frac { \\lambda _ s } { 1 - \\alpha _ s } \\geq \\varepsilon _ 1 > 0 . \\end{align*}"}
+{"id": "6403.png", "formula": "\\begin{align*} \\Psi ( X , Y , t , \\tilde X , \\tilde Y , \\tilde t ) = \\frac { j ^ 4 } { 4 } | X - \\tilde X | ^ 4 + \\frac { j ^ 4 } { 4 } | Y - \\tilde Y | ^ 4 + \\frac { j } { 2 } | t - \\tilde t | ^ 2 , \\end{align*}"}
+{"id": "1670.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial x } ( 0 , 0 ) = \\delta ^ T _ \\tau \\Phi ( \\underline { s } ( z ^ { - 1 } \\mu ) ) ( z , t ) = \\Phi ( \\delta ^ T _ \\tau \\underline { s } ( z ^ { - 1 } \\mu ) ) ( z , t ) , \\frac { \\partial F } { \\partial y } ( 0 , 0 ) = - \\Phi ( \\underline { s } ( z ^ { - 1 } \\delta ^ T _ \\tau \\mu ) ) ( z , t ) . \\end{align*}"}
+{"id": "8681.png", "formula": "\\begin{align*} \\Gamma ^ + ( u _ 1 ) \\dots \\Gamma ^ + ( u _ l ) \\ , ( 1 ) = \\mathcal { F } ( u _ 1 , \\dots , u _ l ; t ) . \\end{align*}"}
+{"id": "660.png", "formula": "\\begin{align*} \\nabla \\cdot { \\bf v } = 0 . \\end{align*}"}
+{"id": "7817.png", "formula": "\\begin{align*} & \\ \\Psi _ { 1 / s d ( n ) , b } [ n ] ( x ^ { \\tilde m } ) = \\begin{cases} \\displaystyle x ^ { \\tilde m } \\prod _ { p = 1 } ^ { \\alpha } ( 1 + q ^ { ( 2 p - 1 ) / s d ( n ) } q ^ b x ^ { p _ 1 ^ * ( n ) } ) & \\alpha > 0 , \\\\ x ^ { \\tilde m } & \\alpha = 0 , \\\\ \\displaystyle x ^ { \\tilde m } \\prod _ { p = 1 } ^ { - \\alpha } ( 1 + q ^ { - ( 2 p - 1 ) / s d ( n ) } q ^ b x ^ { p _ 1 ^ * ( n ) } ) ^ { - 1 } & \\alpha < 0 . \\end{cases} \\end{align*}"}
+{"id": "3608.png", "formula": "\\begin{align*} \\widehat { u } ( \\xi _ 1 , \\xi ' ) \\ = \\ \\frac 1 { ( 2 \\pi ) ^ { \\frac { n - 1 } 2 } } \\widehat A ( \\xi ) + \\xi ' \\cdot \\nabla _ { \\xi ' } \\widehat { u } ( \\xi _ 1 , 0 ) + \\frac { 1 } { 2 } \\xi ' \\cdot D ^ 2 _ { \\xi ' } \\widehat { u } ( \\xi _ 1 , \\tilde { \\xi ' } ) \\xi ' \\end{align*}"}
+{"id": "4628.png", "formula": "\\begin{align*} ( \\omega \\cdot g ) ( h ) = \\omega ( g h ) \\end{align*}"}
+{"id": "3061.png", "formula": "\\begin{align*} - A : D ^ 2 w _ A = a - \\int _ Y r a & \\quad Y , w _ A Y , \\int _ Y w _ A = 0 , \\\\ - \\Delta w _ B = r _ B \\ , b - \\int _ Y r _ B \\ , b & \\quad Y , w _ B Y , \\int _ Y w _ B = 0 . \\end{align*}"}
+{"id": "5520.png", "formula": "\\begin{align*} P _ { k } ( x ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n ^ k } \\exp \\left ( { - \\frac { x } { n ^ 2 } } \\right ) = O \\bigg ( x ^ { - \\frac { k } { 2 } + \\frac { 1 } { 4 } } \\bigg ) , \\end{align*}"}
+{"id": "7374.png", "formula": "\\begin{align*} ( X / \\ ! / \\Gamma ) _ \\natural : = \\widetilde { X } _ \\natural / \\Gamma . \\end{align*}"}
+{"id": "6644.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v = \\Delta v + \\xi \\\\ v | _ { t \\leq 0 } = 0 \\end{cases} \\end{align*}"}
+{"id": "3142.png", "formula": "\\begin{align*} - A : D ^ 2 w _ A = \\frac { c } { 2 } \\left ( \\frac { 1 } { r } - 1 \\right ) \\quad Y , w _ A Y , \\int _ Y w _ A = 0 , \\end{align*}"}
+{"id": "4102.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } g = h , \\frac { \\partial } { \\partial t } H = d K , \\frac { \\partial } { \\partial t } f = \\phi \\end{align*}"}
+{"id": "6726.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\mu ( n + a ) e ^ { i 2 \\pi n / q } = O \\left ( \\frac { x } { ( \\log x / q ) ^ { D } } \\right ) , \\end{align*}"}
+{"id": "2757.png", "formula": "\\begin{align*} ( i ^ { \\ast } ) _ { A } \\ , \\bar { \\triangleleft } \\ , \\partial _ { t } + ( T ^ { \\ast } ) _ { A } ^ { D } \\ , \\bar { \\triangleleft } \\ , \\partial _ { D } = 0 \\end{align*}"}
+{"id": "1902.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } [ \\mathcal { A } f ] ( x ) = f '' ( x ) - c f ( x ) , \\cr D ( \\mathcal { A } ) = \\{ f \\in H ^ 2 ( 0 , 1 ) | \\ ; f ' ( 0 ) = f ' ( 1 ) = 0 \\} . \\end{array} \\right . \\end{align*}"}
+{"id": "8629.png", "formula": "\\begin{align*} \\mu ^ k \\Big ( \\omega \\in \\Omega ^ k : \\omega ( j - i , k ] = \\omega [ 1 , k - ( j - i ) ] \\Big ) = b ^ { - k + ( j - i ) } \\end{align*}"}
+{"id": "6459.png", "formula": "\\begin{align*} E ( g . m ) + ( \\delta E ( g . m ) , g . h ) _ { H ^ { - 1 } , H ^ 1 } + o ( \\| h \\| _ { H ^ 1 } ) & = E ( m ) + ( \\delta E ( m ) , h ) _ { H ^ { - 1 } , H ^ 1 } + o ( \\| h \\| _ { H ^ 1 } ) . \\end{align*}"}
+{"id": "679.png", "formula": "\\begin{align*} \\mathcal { E } ( \\omega _ 1 , \\omega _ 2 ) = ( d G ^ { \\omega _ 1 } , d G ^ { \\omega _ 2 } ) _ 1 = \\int _ M G ^ { \\omega _ 1 } \\wedge \\omega _ 2 . \\end{align*}"}
+{"id": "3222.png", "formula": "\\begin{align*} f ' ( w ) = f ( w ) \\ge ( 2 k _ 3 + 2 k _ 2 + k _ 1 ) - k _ 3 - k _ 2 = k _ 3 ' + k ' _ 2 + k _ 1 ' . \\end{align*}"}
+{"id": "1585.png", "formula": "\\begin{align*} \\nu ( \\hat { x } ) & = \\sum _ { x \\in X } \\pi ( x , \\hat { x } ) = \\pi ( \\hat { x } , \\hat { x } ) + \\sum _ { \\substack { x \\in X \\\\ x \\neq \\hat { x } } } \\pi ( x , \\hat { x } ) \\\\ & \\leq \\sum _ { y \\in X } \\pi ( \\hat { x } , y ) + \\sum _ { \\substack { x \\in X \\\\ x \\neq \\hat { x } } } \\pi ( x , \\hat { x } ) = \\mu ( \\hat { x } ) + \\sum _ { \\substack { x \\in X \\\\ x \\neq \\hat { x } } } \\pi ( x , \\hat { x } ) . \\end{align*}"}
+{"id": "3197.png", "formula": "\\begin{align*} \\tilde { A } ( y ) : = \\frac { 1 } { a _ 1 ( y ) } A ( y ) = \\mathrm { d i a g } \\left ( 1 , \\frac { a _ 2 ( y ) } { a _ 1 ( y ) } \\right ) \\quad y \\in \\R ^ 2 , \\end{align*}"}
+{"id": "4414.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 3 0 } = \\frac { 1 } { 3 } + \\frac { 1 } { 5 } = \\frac { 8 } { 1 5 } \\end{align*}"}
+{"id": "7495.png", "formula": "\\begin{align*} \\sigma ( \\bar { f } , D , \\chi ) : = { \\left \\{ \\begin{array} { r l } q ^ { - n } \\cdot \\# \\{ \\bar { P } \\in \\bar { D } | \\bar { P } \\ { \\rm i s \\ a \\ n o n s i n g u l a r \\ p o i n t \\ o f } \\ V _ { \\bar { f } } ( \\mathbb { F } _ q ) \\} , \\ \\ & { \\rm i f } \\ \\chi = \\chi _ { { \\rm t r i v } , } \\\\ 0 , \\ \\ & { \\rm i f } \\ \\chi \\neq \\chi _ { { \\rm t r i v } } . \\end{array} \\right . } \\end{align*}"}
+{"id": "7702.png", "formula": "\\begin{align*} \\phi ( \\sum _ { \\mid I \\mid = j } \\frac { a _ { I } } { f } d x _ { I } ) = \\sum _ { \\mid I \\mid = j } a _ { i } \\partial _ { I ^ { \\complement } } \\end{align*}"}
+{"id": "5066.png", "formula": "\\begin{align*} \\sum _ { ( j , s ) \\in I _ { \\nu } } q _ { j , s } l ^ { \\varepsilon ( \\omega ) s } f _ j ( \\omega ) = 0 \\end{align*}"}
+{"id": "3853.png", "formula": "\\begin{align*} T _ { n + 4 } ^ 2 = 1 + \\sum _ { k = 0 } ^ n \\left \\{ 3 T _ { k + 2 } ^ 2 + 9 T _ { k + 1 } ^ 2 + 4 \\sum _ { i = 2 } ^ k ( T _ { k + 4 - i } + T _ { k + 3 - i } ) T _ i ^ 2 \\right \\} . \\end{align*}"}
+{"id": "7047.png", "formula": "\\begin{align*} d _ { i , j } - c _ { i , j } & = u d _ { i , j + 1 } \\\\ q _ j - p _ j & = u q _ { j + 1 } \\\\ q _ 0 & = 0 \\end{align*}"}
+{"id": "8795.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 \\mu _ i \\begin{pmatrix} v _ 1 ( t _ i ) \\begin{pmatrix} \\overline { v _ 1 ( t _ i ) } \\\\ \\overline { v _ 2 ( t _ i ) } \\\\ \\overline { v _ 3 ( t _ i ) } \\\\ \\end{pmatrix} & v _ 2 ( t _ i ) \\begin{pmatrix} 0 \\\\ \\overline { v _ 2 ( t _ i ) } \\\\ \\overline { v _ 3 ( t _ i ) } \\\\ \\end{pmatrix} & v _ 3 ( t _ i ) \\begin{pmatrix} 0 \\\\ 0 \\\\ \\overline { v _ 3 ( t _ i ) } \\\\ \\end{pmatrix} \\end{pmatrix} \\end{align*}"}
+{"id": "2017.png", "formula": "\\begin{align*} \\frac { d } { d t } S _ t ( x ) = V _ t ( S _ t ( x ) ) , \\ \\ \\ S _ 0 ( x ) = x . \\end{align*}"}
+{"id": "8327.png", "formula": "\\begin{align*} v _ 0 ( y ) = | y | ^ { - d + 1 } u ( 0 , \\frac { y } { | y | ^ 2 } ) , v _ 1 ( y ) = | y | ^ { - d - 1 } \\partial _ t u ( 0 , \\frac { y } { | y | ^ 2 } ) . \\end{align*}"}
+{"id": "2167.png", "formula": "\\begin{align*} \\sigma _ { 0 } : = & \\dfrac { 4 } { 3 ( 2 - 2 \\alpha + \\beta ) + \\sqrt { ( 6 - 6 \\alpha + 3 \\beta ) ^ 2 + 8 ( 6 \\alpha + 3 \\beta - 4 ) } } , \\intertext { a n d } \\tilde { \\sigma _ { 0 } } : = & \\dfrac { 4 } { ( 2 - 2 \\alpha + \\beta ) + \\sqrt { ( 2 - 2 \\alpha + \\beta ) ^ 2 - 8 ( 2 \\alpha + \\beta - 4 ) } } . \\end{align*}"}
+{"id": "678.png", "formula": "\\begin{align*} ( \\nu _ 1 , \\nu _ 2 ) _ 1 = \\int _ M \\nu _ 1 \\wedge * \\nu _ 2 . \\end{align*}"}
+{"id": "8126.png", "formula": "\\begin{align*} \\frac { \\pm \\bar u p - n _ 1 } { m c n _ 1 ^ { - 1 } p } = \\frac { \\bar u ' } { c ' } \\end{align*}"}
+{"id": "2823.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 ) = x _ 4 . \\end{align*}"}
+{"id": "6523.png", "formula": "\\begin{align*} \\frac { x } { \\sqrt { 2 \\pi } } \\Big ( \\underset { i = 0 } { \\overset { \\infty } { \\sum } } \\frac { ( - 1 ) ^ i ( x / \\sqrt { 2 } ) ^ { 2 i } } { ( 2 i + 1 ) i ! } \\Big ) \\geq \\frac { x } { \\sqrt { 2 \\pi } } \\left ( \\underset { i = 0 } { \\overset { \\infty } { \\sum } } \\frac { ( - 1 ) ^ i c ^ { 2 i } } { ( 2 i + 1 ) i ! } \\right ) & = \\frac { x } { \\sqrt { 2 } c } \\left ( \\Phi ( \\sqrt { 2 } c ) - \\frac { 1 } { 2 } \\right ) , \\end{align*}"}
+{"id": "1869.png", "formula": "\\begin{align*} \\beta = \\sin ^ { - 1 } \\left ( \\frac { - \\ddot { s } _ i } { \\sqrt { \\dot { s } _ i ^ 2 + \\ddot { s } _ i ^ 2 } } \\right ) = \\sin ^ { - 1 } \\left ( \\frac { - ( p _ { s _ i } \\sigma + q _ { s _ i } ) } { \\sqrt { \\dot { s } _ i ^ 2 + ( p _ { s _ i } \\sigma + q _ { s _ i } ) ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "4758.png", "formula": "\\begin{align*} \\begin{bmatrix} \\frac { 2 - q } { 3 } & 0 & \\frac { 1 } { 3 } & \\frac { q } { 3 } & 0 \\\\ 0 & \\frac { 2 } { 3 } & 0 & 0 & \\frac { 1 } { 3 } \\\\ \\frac { q } { 3 } & \\frac { 1 } { 3 } & \\frac { 1 - q } { 3 } & 0 & \\frac { 1 } { 3 } \\\\ \\frac { 1 } { 3 } & \\frac { q } { 3 } & 0 & \\frac { 2 - 2 q } { 3 } & \\frac { q } { 3 } \\\\ 0 & 0 & \\frac { q } { 3 } & \\frac { 1 } { 3 } & \\frac { 2 - q } { 3 } \\end{bmatrix} \\end{align*}"}
+{"id": "4178.png", "formula": "\\begin{align*} \\| ( g _ c , H _ c ) - ( \\widetilde { g } ^ s _ i ( T _ i ) , \\widetilde { H } _ i ^ s ( T _ i ) ) \\| _ { C ^ { k - 3 } } = \\| ( g _ c , H _ c ) - ( \\widetilde { g } _ i ( 1 ) , \\widetilde { H } _ i ( 1 ) ) \\| _ { C ^ { k - 3 } } \\leq \\| ( g _ c , 0 ) - ( \\widetilde { g } _ i ( 1 ) , \\widetilde { b } _ i ( 1 ) ) \\| _ { C ^ { k - 2 } } \\rightarrow 0 \\end{align*}"}
+{"id": "5555.png", "formula": "\\begin{align*} S _ 2 ( x ^ 2 ) = S _ 3 ( x ^ 2 ) + O _ { \\epsilon } \\left ( \\ell ^ { \\frac { 1 } { 2 } - k + \\epsilon } \\right ) , \\end{align*}"}
+{"id": "4480.png", "formula": "\\begin{align*} \\frac { 3 } { 8 5 } = \\frac { 1 } { 3 0 } + \\frac { 1 } { 5 1 0 } \\end{align*}"}
+{"id": "1336.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | ^ m & \\geq \\sqrt { \\frac { n - { d + m - 1 \\choose m } } { { d + m - 1 \\choose m } ( n - 1 ) } } = \\sqrt { \\frac { n - ( ^ m ( \\mathcal { X } ) ) } { ( ^ m ( \\mathcal { X } ) ) ( n - 1 ) } } \\\\ & = \\sqrt { \\frac { 1 } { n - 1 } \\left [ \\frac { n } { { d + m - 1 \\choose m } } - 1 \\right ] } . \\end{align*}"}
+{"id": "7389.png", "formula": "\\begin{align*} \\alpha ^ \\vee ( a ) = \\eta _ \\alpha ^ { - 1 } ( a , a ^ { - 1 } ) \\quad \\beta ^ \\vee ( a ) = \\eta _ \\alpha ^ { - 1 } ( 1 , a ) \\quad \\end{align*}"}
+{"id": "2845.png", "formula": "\\begin{align*} \\rho = \\frac { a _ k + a _ { \\ell } } { 2 } = \\frac { b _ k + b _ { \\ell } } { 2 } \\end{align*}"}
+{"id": "1378.png", "formula": "\\begin{align*} \\beta ( t ) = \\max \\Biggl \\{ \\frac { \\delta M ( t ) } { ( t - t _ 0 ) ^ { \\frac { 1 } { 1 + \\epsilon _ 0 } } \\xi ( t ) } , \\frac { M ( t ) } { ( t - t _ 1 ) ^ { \\frac { 1 } { 1 + \\epsilon } } } \\Biggr \\} . \\end{align*}"}
+{"id": "1662.png", "formula": "\\begin{align*} \\langle \\mathfrak { f } _ 1 , \\mathfrak { f } _ 2 \\rangle = 2 \\Lambda ( 1 , \\Pi , { \\rm a d } ) \\cdot \\Lambda _ F ( 2 ) ^ { - 1 } \\cdot \\prod _ v \\alpha _ v ( W _ { \\mathfrak { f } _ 1 , v } , W ^ - _ { \\mathfrak { f } _ 2 , v } ) , \\end{align*}"}
+{"id": "5908.png", "formula": "\\begin{align*} \\varepsilon _ i = ( 0 , \\cdots , \\overset { ( i ) } { 1 } , \\cdots , 0 ) ^ T , 1 \\leq i \\leq N \\end{align*}"}
+{"id": "9237.png", "formula": "\\begin{align*} ( 1 - \\Pi _ { h } ) \\varphi _ { j } = \\mathcal { O } ( h ^ { \\infty } \\sqrt { \\mu _ { j } } ) . \\end{align*}"}
+{"id": "4948.png", "formula": "\\begin{align*} \\widetilde { A } _ \\alpha ^ { D V D } : = \\left ( \\begin{array} { c } \\alpha \\delta _ n ^ + u _ { m , n } + \\delta _ m ^ { ( 2 ) } \\mu _ n v _ { m , n } + \\mu _ n ( u _ { m , n } ^ 2 + v _ { m , n } ^ 2 ) \\mu _ n v _ { m , n } \\\\ - \\alpha \\delta _ n ^ + v _ { m , n } + \\delta _ m ^ { ( 2 ) } \\mu _ n u _ { m , n } + \\mu _ n ( u _ { m , n } ^ 2 + v _ { m , n } ^ 2 ) \\mu _ n u _ { m , n } \\end{array} \\right ) = \\mathbf { 0 } . \\end{align*}"}
+{"id": "4880.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\left ( \\int _ t ^ 1 v _ 1 ^ 1 ( \\tau ) d \\tau \\right ) ^ n d t = \\frac { 6 } { 5 ( n + 1 ) 4 ^ { n } } \\big ( 3 ^ { n - 1 } - 2 ^ { n - 1 } \\big ) . \\end{align*}"}
+{"id": "2034.png", "formula": "\\begin{align*} \\rho ( n ) : = \\left \\{ \\begin{array} { l l l } \\displaystyle \\sum _ { k = 1 } ^ { + \\infty } \\mu ( k ) \\ , \\mu ' [ n - k + 1 , n ] & { \\rm i f } & n \\leq - 1 , \\\\ \\displaystyle \\sum _ { k = 1 } ^ { + \\infty } \\mu ' ( - k ) \\ , \\mu [ n + 1 , n + k ] & { \\rm i f } & n \\geq 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "6082.png", "formula": "\\begin{align*} ( f - r _ n ) ( z ) = ( 2 + o ( 1 ) ) ( T _ n \\psi _ n ) ( z ) \\frac { \\big ( m S _ { \\dot \\mu } ^ 2 \\big ) ( z ) } { w ( z ) } \\end{align*}"}
+{"id": "3921.png", "formula": "\\begin{align*} \\int _ { Y u ^ { - 1 } ( E ^ * ) } f = \\int _ { E ^ * } f ^ * , \\end{align*}"}
+{"id": "984.png", "formula": "\\begin{align*} \\hat f ( \\psi _ i ) & = \\psi _ i ( f ) = x _ i \\otimes \\varphi _ i ( f ) \\\\ & = \\varphi _ i ( f ( x _ i ) ) = \\varphi _ i ( a ) \\\\ & = \\hat a ( \\varphi _ i ) \\ , \\ , \\ , \\ , ( 1 \\leq i \\leq n ) \\end{align*}"}
+{"id": "7964.png", "formula": "\\begin{align*} \\pi ' i ( m ) = i ' \\pi ( m ) = \\pi ' \\beta ( x ) . \\end{align*}"}
+{"id": "6845.png", "formula": "\\begin{align*} { \\mathcal M } = F _ { w _ { m , 1 } } \\mathcal K = ( I - w _ { m , 1 } ^ { \\dag } w _ { m , 1 } ) ( w _ { 1 , 2 } \\ , w _ { 1 , 3 } \\ , \\dots \\ , w _ { 1 , m - 1 } ) . \\end{align*}"}
+{"id": "8774.png", "formula": "\\begin{align*} \\dot { \\phi } _ i = & h _ i ( t ) + \\frac { 1 } { N } \\sum _ { j = 1 } ^ N W _ { i j } ( t ) g ( t , \\phi _ j , \\phi _ i ) \\\\ \\dot { W } _ { i j } = & ( G ' ( W _ { i j } ) ) ^ { - 1 } + h ( t , \\phi _ i , \\phi _ j ) , \\end{align*}"}
+{"id": "4617.png", "formula": "\\begin{align*} \\prod _ { i = m } ^ { m + k - 1 } \\frac { 1 } { b _ i } < \\prod _ { i = m } ^ { m + k - 1 } \\frac { 1 } { a _ i } . \\end{align*}"}
+{"id": "1947.png", "formula": "\\begin{align*} \\phi ( x ) = \\begin{cases} 1 + x & - 1 \\le x < 0 , \\\\ 1 - x & 0 \\le x \\le 1 , \\\\ 0 & \\hbox { o t h e r w i s e . } \\\\ \\end{cases} \\end{align*}"}
+{"id": "5157.png", "formula": "\\begin{align*} \\mathcal { I } ( H ) = \\lim _ { n \\rightarrow \\infty } \\mathcal { I } ( H , n ) . \\end{align*}"}
+{"id": "1327.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | \\geq \\sqrt { \\frac { \\frac { 1 } { d } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ 2 } { n ^ 2 - n } } . \\end{align*}"}
+{"id": "4698.png", "formula": "\\begin{align*} \\varphi _ k ( \\bar { x } ) = \\varphi _ k ( x , y ) = \\frac { 1 } { | G | } \\sum _ { \\sigma \\in G } ( \\sigma _ 1 \\bar { x } ) ^ k \\end{align*}"}
+{"id": "8551.png", "formula": "\\begin{align*} y \\left ( n ; \\beta \\right ) : = \\left \\{ \\begin{array} { l l } x \\left ( n ; \\beta \\right ) & \\left ( n ; \\beta \\right ) \\in F _ { \\downarrow } \\\\ 1 _ { G } & \\end{array} \\right . \\end{align*}"}
+{"id": "1130.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - k } ( - 1 ) ^ i \\binom { n } { i + k } \\binom { i + k } { i } & = \\left \\{ \\begin{array} { l l } 0 & \\mbox { i f \\ } k < n \\\\ 1 & \\mbox { i f \\ } k = n . \\end{array} \\right . \\end{align*}"}
+{"id": "1675.png", "formula": "\\begin{align*} \\kappa ( \\theta ) : = \\left ( \\begin{array} { c c } \\cos \\theta & \\sin \\theta \\\\ - \\sin \\theta & \\cos \\theta \\end{array} \\right ) . \\end{align*}"}
+{"id": "5084.png", "formula": "\\begin{align*} T _ n = \\sum _ { i = 1 } ^ k \\sum _ { l = 1 } ^ { N _ 0 } { n \\choose l } \\beta _ i c _ i ^ { n - l } \\theta _ i ^ l + \\sum _ { i = 1 } ^ k \\sum _ { l = N _ 0 + 1 } ^ n { n \\choose l } \\beta _ i c _ i ^ { n - l } \\theta _ i ^ l . \\end{align*}"}
+{"id": "3217.png", "formula": "\\begin{align*} \\sum _ { A \\subseteq 2 ^ { \\mathbb { A } } \\setminus \\left \\{ \\emptyset \\right \\} } \\sum _ { A \\subseteq B \\subseteq \\mathbb { A } } \\mathbb { P } \\left [ W = B \\right ] \\cdot p ^ { \\left | A \\right | } \\cdot p ^ { \\left | B \\setminus A \\right | } \\end{align*}"}
+{"id": "2589.png", "formula": "\\begin{align*} \\hat { \\tau } ( c ) = - c , \\hat { \\tau } ( d ) = - d , \\hat { \\tau } ( u , v ) ( t ) = ( \\sigma ^ { - 1 } v ( - t ) , \\sigma u ( - t ) ) . \\end{align*}"}
+{"id": "4044.png", "formula": "\\begin{align*} D _ { p _ k } = D _ { p _ k } Y ^ r D _ { y _ r } + D _ { p _ k } Z D _ z = E ^ { r k } \\left ( D _ { y _ r } - \\frac { g _ { y _ r } } { g _ z } D _ z \\right ) . \\end{align*}"}
+{"id": "8347.png", "formula": "\\begin{align*} v ( s , y ) = M _ 0 ^ { 1 - \\frac { d } { 2 } } J ^ { \\frac { 1 - d } { 2 } } v ^ { ( M _ 0 ) } \\left ( \\frac { s - I M _ 0 } { M _ 0 J } , \\frac { y } { M _ 0 J } \\right ) , \\end{align*}"}
+{"id": "4406.png", "formula": "\\begin{align*} 0 < \\frac { p } { q } - \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } \\leq \\frac { p } { q } - \\sum _ { i = 1 } ^ n \\frac { 1 } { a _ i } = \\frac { 1 } { q \\prod _ { i = 1 } ^ n a _ i } . \\end{align*}"}
+{"id": "4325.png", "formula": "\\begin{align*} c ( m , y ^ { n _ 1 } ) = ( ( c _ { 1 } ( m ) , c _ { 2 } ( m , y ^ { n _ 1 } ) ) , \\end{align*}"}
+{"id": "4181.png", "formula": "\\begin{align*} d s ^ 2 = f ( t , x ) ( d x ^ 2 - d t ^ 2 ) + g _ { a b } ( t , x ) d x ^ a d x ^ b . \\end{align*}"}
+{"id": "5207.png", "formula": "\\begin{align*} A _ 6 = & \\ell _ 1 A _ 1 , A _ 4 = \\ell _ 2 A _ 2 , A _ 5 = 1 / ( \\ell _ 1 \\ell _ 2 ) A _ 3 \\quad \\mathrm { o r } \\\\ A _ 5 = & \\ell _ 1 A _ 1 , A _ 6 = \\ell _ 2 A _ 2 , A _ 4 = 1 / ( \\ell _ 1 \\ell _ 2 ) A _ 3 \\ , . \\end{align*}"}
+{"id": "9013.png", "formula": "\\begin{align*} H _ \\mu ( \\alpha _ { F _ i } ) \\ , & \\leq \\ , H _ \\mu ( \\beta ) + H _ \\mu ( \\alpha _ { F _ i } | \\beta ) \\ , \\leq \\ , H _ \\mu ( \\beta ) + H _ \\mu ( \\alpha _ g | \\beta ) \\\\ & = \\ , H _ \\mu ( \\beta ) + H _ \\mu ( \\alpha | \\beta _ { g ^ { - 1 } } ) \\ , \\leq \\ , H _ \\mu ( \\beta ) + | F _ i | \\epsilon . \\end{align*}"}
+{"id": "3161.png", "formula": "\\begin{align*} c _ 2 ^ { 1 2 } ( A ) = \\int _ 0 ^ 1 \\left ( \\frac { 1 } { 2 \\pi } - \\frac { \\sqrt { 6 } } { \\pi } \\frac { 1 + \\cos ( 2 \\pi t ) } { 5 + \\sin ( 2 \\pi t ) } \\right ) \\ln ( 5 + \\sin ( 2 \\pi t ) ) \\ , \\mathrm { d } t = 0 . 0 0 3 \\dots \\neq 0 , \\end{align*}"}
+{"id": "6454.png", "formula": "\\begin{gather*} ( g . m ) \\wedge ( g . \\tilde m ) = g . ( m \\wedge \\tilde m ) , ( g . m ) \\cdot ( g . \\tilde m ) = \\tau _ y ( m \\cdot \\tilde m ) , \\\\ \\partial _ y ( g . m ) = - g . \\partial _ x m , \\partial _ \\phi ( g . m ) = e _ 1 \\wedge g . m = g . ( e _ 1 \\wedge m ) . \\end{gather*}"}
+{"id": "1792.png", "formula": "\\begin{align*} A _ { \\overline { w } } = ( R g _ * \\Lambda [ \\langle 2 \\rho , \\mu \\rangle ] ) _ { \\overline { w } } , \\end{align*}"}
+{"id": "5883.png", "formula": "\\begin{align*} t = 0 : W = C _ p \\widehat { U } _ 0 , W ' = C _ p \\widehat { U } _ 1 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "7344.png", "formula": "\\begin{align*} K ( x ) = ( \\log x ) ^ { ( s - 1 ) / \\alpha } . \\end{align*}"}
+{"id": "3260.png", "formula": "\\begin{align*} B _ { \\mathbf { d } , \\mathcal { A } } ( a b ) = \\sum \\limits _ { c \\in \\mathcal { A } ( a ) \\setminus \\mathcal { A } ( b ) } P _ { \\mathbf { d } , \\mathcal { A } } ( a c ) + \\sum \\limits _ { c \\in \\mathcal { A } ( a ) \\cap \\mathcal { A } ( b ) } Y _ { \\mathbf { d } , \\mathcal { A } } ( a c b ) . \\end{align*}"}
+{"id": "6083.png", "formula": "\\begin{align*} H _ n ( z ) : = ( T _ n \\psi _ n ) ( z ) \\frac { m ( z ) } { m _ n ( z ) } \\frac { B _ n ^ 2 ( z ) S _ { \\dot \\mu } ^ 2 ( z ) } { G _ { \\lambda _ n } D _ { \\lambda _ n } ^ 2 ( z ) } \\left ( \\frac { \\varphi ( z ) } \\rho \\right ) ^ { 2 ( n - d _ n ) } . \\end{align*}"}
+{"id": "6166.png", "formula": "\\begin{align*} t = 0 : u = ( E _ 1 , \\widehat U _ 0 ) , u ' = ( E _ 1 , \\widehat U _ 1 ) \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "2204.png", "formula": "\\begin{align*} \\emph { \\textbf { x } } _ 1 = ( 1 8 , 7 2 0 , 7 6 2 5 , 3 2 5 0 0 , 5 0 0 0 0 , 0 , 0 , \\ldots ) , \\end{align*}"}
+{"id": "9201.png", "formula": "\\begin{align*} d X _ { t } = - 2 \\nabla f ( X _ { t } ) d t + \\sqrt { 2 h } d B _ { t } , \\end{align*}"}
+{"id": "4618.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\frac { 1 } { b _ i } < \\sum _ { i = 1 } ^ { n } \\frac { 1 } { a _ i } . \\end{align*}"}
+{"id": "4154.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda \\Big ( ( u g , K ) + ( h , 0 ) \\Big ) = \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda ( u g , K ) + \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda ( h , 0 ) = 0 . \\end{align*}"}
+{"id": "5532.png", "formula": "\\begin{align*} e ^ { - x } - 1 = \\frac { 1 } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } \\Gamma ( s ) x ^ { - s } { \\rm d } s . \\end{align*}"}
+{"id": "6096.png", "formula": "\\begin{align*} \\mathcal { F } _ k \\circ \\mathcal { F } _ k ( f ) = f . \\end{align*}"}
+{"id": "7918.png", "formula": "\\begin{align*} F _ { \\phi } ( r , x ) = \\gamma ( \\phi ) \\frac { \\sqrt { t } ( - x ) e ^ { \\sqrt { 2 } x } } { r ^ { \\frac { 3 } { 2 } } } e ^ { - x ^ 2 / 2 r } ( 1 + o ( 1 ) ) \\end{align*}"}
+{"id": "1756.png", "formula": "\\begin{align*} I ( \\chi , { \\underline m } ) = \\left \\{ \\begin{array} { l l } ( - 1 ) ^ { M + m } 8 ( 2 M + 1 ) ( 2 M + 2 ) \\binom { 2 M } { k _ { \\rm i d } - 2 } \\binom { k _ { \\rm i d } - 2 } { \\frac { k _ { \\rm i d } - 2 } { 2 } - m _ { \\rm i d } } ^ { - 1 } \\binom { k _ { c } - 2 } { \\frac { k _ { c } - 2 } { 2 } - m _ { c } } ^ { - 1 } , & \\chi = 1 , \\\\ 0 , & \\chi \\neq 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "1376.png", "formula": "\\begin{align*} \\mathcal { K } ( t ) = \\Big [ \\Big ( \\bar { B } ^ { - 1 } \\Big ) ^ { \\frac { 1 } { 1 + \\epsilon _ 0 } } \\Big ] ^ { - 1 } ( t ) , \\alpha ( t ) = \\frac { \\delta M ( t ) } { ( t - t _ 0 ) ^ { \\frac { 1 } { 1 + \\epsilon _ 0 } } \\xi ( t ) } . \\end{align*}"}
+{"id": "6715.png", "formula": "\\begin{align*} E _ 0 ( x ) = \\frac { 1 } { p } \\sum _ { n \\leq x , } \\sum _ { 1 \\leq s \\leq p - 1 } \\mu ( s + a ) e ^ { i 2 \\pi u s / q } \\ll \\frac { x } { p } \\frac { x } { ( \\log x ) ^ { D _ 0 } } \\ll \\frac { x } { ( \\log x ) ^ { D _ 0 } } , \\end{align*}"}
+{"id": "2579.png", "formula": "\\begin{align*} \\frac { 1 } { - \\pi + 2 m \\pi } ( e _ i \\pm e _ j ) = ( - 1 ) \\times \\frac { 1 } { - \\pi + 2 ( - m + 1 ) \\pi } ( e _ i \\pm e _ j ) , \\end{align*}"}
+{"id": "2613.png", "formula": "\\begin{align*} 3 = a _ { 2 , 3 } + a _ { 2 , 4 } + a _ { 3 , 4 } = 1 + 1 + 1 \\leftrightarrow ( a _ { 1 , 2 } , a _ { 1 , 3 } , a _ { 1 , 4 } , a _ { 2 , 3 } , a _ { 2 , 4 } , a _ { 3 , 4 } ) = ( 1 , 1 , 1 , 1 , 1 , 1 ) . \\end{align*}"}
+{"id": "5920.png", "formula": "\\begin{align*} L = ( P e _ i , P e _ j ) \\hbox { a n d } \\Lambda = ( \\widehat B P e _ i , P e _ j ) , 1 \\leqslant i , j \\leqslant 2 , \\end{align*}"}
+{"id": "6852.png", "formula": "\\begin{align*} \\begin{aligned} S _ { 1 , 2 , \\dots , N + 2 } ^ { [ 0 ] } ( i , j ) & = \\mathcal X ^ { [ i - 1 ] \\ , T } _ { j - 1 } = - \\mathcal X ^ { [ j - 1 ] } _ { i - 1 } , S _ { 1 , 2 , \\dots , N + 2 } ^ { [ 0 ] } ( j , i ) = S _ { 1 , 2 , \\dots , N + 2 } ^ { [ 0 ] \\ , T } ( i , j ) , \\ ; j \\ge i , i , j = 1 , 2 , \\dots , N + 2 , \\\\ \\end{aligned} \\end{align*}"}
+{"id": "8447.png", "formula": "\\begin{align*} \\gamma _ { A ' } = \\gamma _ H = \\gamma _ A . \\end{align*}"}
+{"id": "2154.png", "formula": "\\begin{align*} \\dfrac { z Q ' ( z ) } { Q ( z ) } = \\sum _ { k = 1 } ^ { n } \\dfrac { z } { z - z _ { k } } . \\end{align*}"}
+{"id": "4081.png", "formula": "\\begin{align*} \\| u \\| _ k = \\Bigl ( \\sum _ { i = 0 } ^ k \\int _ M \\langle \\nabla ^ i u , \\nabla ^ i u \\rangle _ h d V _ g \\Bigr ) ^ { 1 / 2 } \\end{align*}"}
+{"id": "6441.png", "formula": "\\begin{align*} W _ \\rho & : = B _ \\rho ( \\hat X ) \\times B _ \\rho ( \\hat Y ) \\times ( \\hat t - \\rho , \\hat t + \\rho ) , \\ K _ \\rho : = \\overline { W _ { 2 \\rho } } \\setminus W _ \\rho . \\end{align*}"}
+{"id": "8830.png", "formula": "\\begin{align*} \\Delta = - \\frac { 1 } { \\sqrt { \\det g } } \\sum \\limits _ { i , j } \\partial _ i \\left ( g ^ { i j } \\sqrt { \\det g } \\partial _ j \\right ) . \\end{align*}"}
+{"id": "4308.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\nu _ 0 \\leq c ( t ) \\leq M , & t \\in [ 0 , T _ { m } ] , \\\\ & \\left | c ^ \\prime ( t ) \\right | \\leq \\frac { K } { ( T _ m - t ) ^ { \\frac { s + 1 } { s } } } , & \\mathrm { a . e . } \\ , t \\in [ 0 , T _ m ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2.png", "formula": "\\begin{align*} y _ t = y _ T + \\int _ t ^ T g ( x _ s , y _ s , z _ s , \\tilde { z } _ s , \\gamma _ { ( s , e ) } ) d s - \\int _ t ^ T z _ s d W _ s - \\int _ t ^ T \\tilde { z } _ s d \\xi _ s - \\int _ t ^ T \\int _ { \\mathcal { E } } \\gamma _ { ( s , e ) } \\tilde { N } ( d e , d s ) . \\end{align*}"}
+{"id": "8466.png", "formula": "\\begin{align*} c _ * ( A ) = \\inf _ { \\nu \\in \\Gamma ^ + _ A } \\ , \\nu ( X ) . \\end{align*}"}
+{"id": "3090.png", "formula": "\\begin{align*} - A : D ^ 2 w _ A = a - \\bar { a } \\quad Y , w _ A Y , \\int _ Y w _ A = 0 , \\end{align*}"}
+{"id": "1706.png", "formula": "\\begin{align*} F ( t ) = ( - 1 ) ^ { \\frac { k } { 2 } } ( i t ) ^ { \\frac { 2 - k } { 2 } + m } ( k - 1 ) \\sum _ { 0 \\leq j \\equiv \\frac { k - 2 } { 2 } + m \\ ; ( { \\rm m o d } \\ ; 2 ) } \\binom { k } { j } \\sum _ { s \\geq 0 } C ( j + 2 s ) ( i t ) ^ { j + 2 s } \\end{align*}"}
+{"id": "2061.png", "formula": "\\begin{align*} u '' ( x ) + \\bigl ( c x ^ { \\gamma - 2 } - d ^ 2 x ^ { 2 \\gamma - 2 } \\bigr ) u ( x ) = 0 , x > 0 , \\end{align*}"}
+{"id": "2039.png", "formula": "\\begin{align*} \\ell _ + : = \\inf \\{ k > 0 : S _ k > 0 \\} \\ell _ - ' : = \\inf \\{ k > 0 : S _ k ' < 0 \\} \\end{align*}"}
+{"id": "8835.png", "formula": "\\begin{align*} \\left ( e ^ { - t \\Delta _ { 0 } } \\right ) \\sim \\frac { 1 } { ( 4 \\pi t ) ^ n } \\sum \\limits _ { k = 0 } ^ { + \\infty } a _ k ^ { ( n ) } t ^ k , t \\searrow 0 ^ + , \\end{align*}"}
+{"id": "1657.png", "formula": "\\begin{align*} d ^ \\times x _ \\sigma = \\frac { d x _ \\sigma } { | x _ \\sigma | } = \\pm \\frac { d r } { r } , F _ \\sigma = \\R \\ ; r = | x _ \\sigma | ; d ^ \\times x _ \\sigma = \\frac { 2 d s d t } { \\pi ( s ^ 2 + t ^ 2 ) } = \\frac { 2 d r d \\theta } { \\pi r } , F _ \\sigma = \\C , \\ ; x _ v = s + i t = r e ^ { i \\theta } . \\end{align*}"}
+{"id": "7104.png", "formula": "\\begin{align*} \\Omega ^ { C _ 2 } _ { + 2 n \\alpha } \\cong \\frac { M U _ { * - 1 } [ u ] } { \\begin{pmatrix} u ^ n \\ ; , \\ ; \\sum _ { \\ell = 0 } ^ { n - 1 } c _ { i , j + \\ell } u ^ \\ell \\ ; , \\ ; \\sum _ { \\ell = 0 } ^ { n - 1 } p _ { j + \\ell } u ^ \\ell \\end{pmatrix} } \\end{align*}"}
+{"id": "6814.png", "formula": "\\begin{align*} y _ { 2 } ^ { [ m ] } + ( - 1 ) ^ { n } z _ { m } & = h \\displaystyle \\sum _ { i = 1 } ^ { 2 } b _ { 2 i } F _ { i } - y _ { 2 } ^ { [ m - 1 ] } + ( - 1 ) ^ { n - 1 } h \\displaystyle \\sum _ { i = 1 } ^ { 2 } \\frac { \\partial f } { \\partial y } b _ { 2 i } u _ { i 2 } z _ { m - 1 } - ( - 1 ) ^ { n - 1 } z _ { m - 1 } , \\\\ \\Rightarrow z _ { m } & = \\big ( 1 - h \\displaystyle \\sum _ { i = 1 } ^ { 2 } \\frac { \\partial f } { \\partial y } b _ { 2 i } u _ { i 2 } \\big ) z _ { m - 1 } . \\end{align*}"}
+{"id": "9097.png", "formula": "\\begin{align*} [ J \\mathfrak { n } _ 2 , \\mathfrak { n } ] \\ni [ J Z _ 2 , J X ] = J [ J Z _ 2 , X ] \\in J [ J \\mathfrak { n } _ 2 , \\mathfrak { n } ] . \\end{align*}"}
+{"id": "1828.png", "formula": "\\begin{align*} ( D + \\lambda ( K _ X + \\Delta ) ) \\cdot R = 0 . \\end{align*}"}
+{"id": "653.png", "formula": "\\begin{align*} \\nu ^ { 0 } ( \\Omega _ { c _ { 1 } \\cdots c _ { N - 1 } c } ) : = \\mathbf { 1 } _ { \\Omega } ( c _ { 1 } , \\ldots , c _ { N - 1 } , c ) . \\end{align*}"}
+{"id": "7581.png", "formula": "\\begin{align*} \\nu ( Q _ r ( x ) ) = \\lim _ { k \\rightarrow \\infty } \\nu _ k ( Q _ r ( x ) ) \\end{align*}"}
+{"id": "1795.png", "formula": "\\begin{align*} u ^ n + a _ 1 ( t ) u ^ { n - 1 } + \\dots + a _ n ( t ) = 0 \\end{align*}"}
+{"id": "4115.png", "formula": "\\begin{align*} S ( g ) = \\int _ M R d V _ g . \\end{align*}"}
+{"id": "6526.png", "formula": "\\begin{align*} \\pi _ L \\left ( \\tilde { f } ^ L \\in \\R ^ { 2 ^ L } : c _ \\alpha \\rho ^ 2 \\leq \\| \\tilde { f } ^ L \\| _ 2 ^ 2 \\leq 2 ^ { - 2 L s } R ^ 2 \\right ) & \\geq \\left ( c _ \\alpha \\rho ^ 2 \\leq Z ^ \\top \\Gamma Z \\leq R ^ 2 \\rho ^ 2 \\right ) \\\\ & = \\left ( \\sqrt { c _ \\alpha } 2 ^ L \\leq Z ^ \\top \\bar { \\Gamma } Z \\leq \\frac { R ^ 2 } { \\sqrt { c _ \\alpha } } 2 ^ L \\right ) , \\end{align*}"}
+{"id": "296.png", "formula": "\\begin{align*} \\begin{aligned} H ^ 1 ( F , \\{ \\pm 1 \\} ) & \\rightarrow H ^ 1 ( W _ F , Z ( \\widehat { G } ) ) \\\\ [ E ] & \\mapsto ( \\omega _ { U _ { 1 , E ' / F } ( F ) , E } ) ^ { n - 1 } \\circ \\det = \\omega _ { E E ' / E ' } ^ { n - 1 } \\circ \\iota _ { E ' } \\circ \\det , \\end{aligned} \\end{align*}"}
+{"id": "2832.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ n } t _ { \\sigma } P _ { \\sigma } = S . \\end{align*}"}
+{"id": "7657.png", "formula": "\\begin{align*} L _ { \\omega } ( [ \\zeta ] ) = ( \\deg ( [ \\zeta ] ) + \\iota _ { E } ( \\omega ) ) [ \\zeta ] = ( \\deg ( [ \\zeta ] ) + \\sum \\lambda _ { k } ) [ \\zeta ] . \\end{align*}"}
+{"id": "3221.png", "formula": "\\begin{align*} f ' ( x ) \\geq f ( x ) - 1 \\geq ( 2 k _ 3 + k _ 2 + k _ 1 + i ) - 1 = 2 k _ 3 ' + k ' _ 2 + k _ 1 ' + i + 1 . \\end{align*}"}
+{"id": "6959.png", "formula": "\\begin{gather*} L _ { m , n } ^ { I , ( \\alpha ) } ( x ) = \\sum _ { j = 0 } ^ n a _ { j , n } x ^ j . \\end{gather*}"}
+{"id": "5087.png", "formula": "\\begin{align*} [ x ^ h ] P _ n = [ x ^ h ] T _ n = [ x ^ h ] F _ n - \\sum _ { i = 1 } ^ k [ x ^ h ] \\beta _ i c _ i ^ n - \\underbrace { [ x ^ h ] \\sum _ { i = k + 1 } ^ s \\beta _ i \\alpha _ i ^ n } _ { = 0 } . \\end{align*}"}
+{"id": "8935.png", "formula": "\\begin{align*} \\frac { \\omega _ { n } } { ( 4 \\pi t ) ^ { n } } e ^ { \\left ( \\frac { n ^ { 2 } } { 4 } + \\nu ^ { 2 } \\right ) t } \\sum \\limits _ { i = 0 } ^ { + \\infty } c _ { i } ^ { \\left ( \\nu , n \\right ) } t ^ { i } \\end{align*}"}
+{"id": "1032.png", "formula": "\\begin{align*} \\max _ { i = 1 , \\ldots , n } \\sup _ { A } \\sup _ { z _ 1 , \\dotsc , z _ { i - 1 } \\in \\mathcal { Z } } \\sup _ { x , x ' \\in \\mathcal { X } } \\frac { Q _ i ( A | x , z _ 1 , \\dotsc , z _ { i - 1 } ) } { Q _ i ( A | x ' , z _ 1 , \\dotsc , z _ { i - 1 } ) } \\leq e ^ \\alpha , \\end{align*}"}
+{"id": "6984.png", "formula": "\\begin{gather*} \\{ U _ { j , n } \\} _ { j = 1 } ^ { n - 1 } = \\bigg \\{ \\frac { \\rm i } { \\sqrt { 2 } } , \\frac { - { \\rm i } } { \\sqrt { 2 } } \\bigg \\} \\cup \\{ y _ { j - 2 , n - 3 } \\} _ { j = 3 } ^ { n - 1 } . \\end{gather*}"}
+{"id": "7656.png", "formula": "\\begin{align*} L _ { \\omega } ( \\eta ) = ( \\deg ( \\eta ) + \\iota _ { E } ( \\omega ) ) \\eta = ( \\deg ( \\eta ) + \\sum \\lambda _ { k } ) \\eta . \\end{align*}"}
+{"id": "6063.png", "formula": "\\begin{align*} \\mathsf W : = \\left [ \\int _ { b _ i } ^ { a _ { i + 1 } } \\big ( x + x ^ { - 1 } \\big ) ^ l \\frac { x ^ g \\dd x } { ( w \\tilde w ) ( x ) } \\right ] _ { i = 1 , l = 1 } ^ { g , g } , \\end{align*}"}
+{"id": "4792.png", "formula": "\\begin{align*} \\Pr _ { c \\sim \\mathcal { D } ( C ) } \\Big [ | c | = \\alpha N \\Big ] & = \\frac { \\left | C ^ { ( \\alpha ) } \\right | } { q ^ r } \\\\ & = 2 ^ { \\mathsf { H } ( Z ) } \\cdot q ^ { - r } \\\\ & \\leq q ^ { - ( 1 - h _ q ( \\alpha ) ) \\cdot r } . \\end{align*}"}
+{"id": "2332.png", "formula": "\\begin{align*} \\ ( ( v \\otimes v ) u \\ ) _ i = \\sum _ { j = 1 } ^ N ( v \\otimes v ) _ { i , j } u _ j = \\sum _ { j = 1 } ^ N v _ i v _ j u _ j = \\ ( \\sum _ { j = 1 } ^ N v _ j u _ j \\ ) v _ i = 0 . \\end{align*}"}
+{"id": "8649.png", "formula": "\\begin{align*} \\epsilon _ { i k } : = \\begin{cases} 1 & : i = k t _ i = d ( x _ i , x _ k ) , \\\\ 0 & : \\end{cases} \\end{align*}"}
+{"id": "5422.png", "formula": "\\begin{align*} { _ i r } ^ { m } ( x y ) = \\sum _ { t = 0 } ^ { m } v ^ { ( \\nu - t i , ( m - t ) i ) + t ( m - t ) } \\frac { [ m ] _ { v } ! } { [ t ] _ { v } ! [ m - t ] _ { v } ! } { _ i r } ^ { t } ( x ) { _ i r } ^ { m - t } ( y ) \\end{align*}"}
+{"id": "4266.png", "formula": "\\begin{align*} I _ \\gamma ( c ) = \\lim _ { n \\rightarrow \\infty } E _ \\gamma ( f _ n ) = \\lim _ { n \\rightarrow \\infty } E _ \\gamma ( g _ n ) & = E _ \\gamma ( \\phi ) + \\lim _ { n \\rightarrow \\infty } E _ \\gamma ( g _ n - \\phi ) \\\\ & > \\frac { \\| \\phi \\| ^ 2 _ { L ^ 2 } } { c } I _ \\gamma ( c ) + \\frac { c - \\| \\phi \\| ^ 2 _ { L ^ 2 } } { c } I _ \\gamma ( c ) = I _ \\gamma ( c ) \\end{align*}"}
+{"id": "2100.png", "formula": "\\begin{align*} \\dim _ P ( B ) = \\dim _ H ( B ) = \\dim ( C ^ m ) = \\frac { \\log 2 } { \\log 3 } m < m . \\end{align*}"}
+{"id": "775.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\log \\| \\wedge ^ k M _ n \\| \\underset { n \\to \\infty } { \\longrightarrow } \\sum _ { i = 1 } ^ k \\lambda _ i . \\end{align*}"}
+{"id": "5037.png", "formula": "\\begin{align*} 0 = - \\Delta _ { g _ t } H _ t = \\frac { \\partial } { \\partial t } H _ t = \\frac { d } { d t } \\lambda _ t H _ o , \\end{align*}"}
+{"id": "7846.png", "formula": "\\begin{align*} \\phi = \\sum e ^ { f _ \\nu } \\big / \\sum e ^ { g _ \\nu } , \\end{align*}"}
+{"id": "5602.png", "formula": "\\begin{align*} 0 \\equiv \\nabla _ { X } S _ { 3 3 } - \\nabla _ { X } S _ { 0 1 } = X ( S _ { 3 3 } - S _ { 0 1 } ) - 2 S _ { a 3 } \\nabla _ { X } m _ { 3 } ^ { a } + S _ { a 1 } \\nabla _ { X } k ^ { a } + S _ { 0 a } \\nabla _ { X } l ^ { a } = 3 S _ { 1 3 } \\nabla _ { X } k _ { 3 } , \\end{align*}"}
+{"id": "8029.png", "formula": "\\begin{align*} & = B _ n A _ { n + 1 } - A _ n B _ { n + 1 } \\\\ & = B _ n B _ { n + 1 } ( h ( A _ n / B _ n ) - A _ n / B _ n ) \\\\ & = B _ n B _ { n + 1 } ( p ( A _ n / B _ n ) - q ( A _ n / B _ n ) A _ n / B _ n ) q ( A _ n / B _ n ) ^ { - 1 } \\\\ & = C ' B _ n B _ { n + 1 } ( A _ n - \\gamma _ 1 B _ n ) ^ { c _ 1 } \\cdots ( A _ n - \\gamma _ u B _ n ) ^ { c _ u } B _ n ^ { - c _ 1 - \\cdots - c _ u } B _ { n + 1 } ^ { - 1 } B _ n ^ { { \\rm d e g } ( p ) } \\end{align*}"}
+{"id": "1711.png", "formula": "\\begin{align*} F ( \\infty ) = ( - 1 ) ^ { m } ( k - 1 ) \\binom { k - 2 } { \\frac { k - 2 } { 2 } - m } ^ { - 1 } \\sum _ { s \\geq 0 } \\binom { k } { \\frac { k - 2 } { 2 } - m + 2 s + 2 } . \\end{align*}"}
+{"id": "7575.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 3 ) = & \\dfrac { ( 1 - 5 ^ { - 1 } ) } { 1 - 5 ^ { - 1 } t } \\Big ( 5 ^ { - 4 } t + 5 ^ { - 2 } - 2 \\times 5 ^ { - 3 } \\Big ) , \\\\ Z _ f ( s , \\chi , A _ 4 ) = & ( 1 - 5 ^ { - 1 } ) \\Big ( 5 ^ { - 3 } t ^ 5 + 5 ^ { - 2 } \\Big ) , \\\\ Z _ f ( s , \\chi , A _ 5 ) = & ( 1 - 5 ^ { - 1 } ) ^ 2 \\Big ( 5 ^ { - 3 } t ^ 5 + ( 5 ^ { - 2 } - 5 ^ { - 3 } ) t ^ 3 + 5 ^ { - 1 } \\Big ) , \\\\ Z _ f ( s , \\chi , A _ 6 ) = & ( 1 - 5 ^ { - 1 } ) ^ 2 \\Big ( 5 ^ { - 4 } t ^ 5 + ( 5 ^ { - 3 } - 5 ^ { - 4 } ) t ^ 4 + ( 5 ^ { - 2 } - 5 ^ { - 3 } ) t ^ 2 + 5 ^ { - 1 } \\Big ) . \\end{align*}"}
+{"id": "455.png", "formula": "\\begin{align*} \\phi ( x ) = \\sum _ { n = 1 } ^ { \\infty } \\phi ( \\tau _ n ) f _ n ( x ) , \\forall x \\in \\mathcal { X } , \\forall \\phi \\in \\mathcal { X } ^ * \\end{align*}"}
+{"id": "7233.png", "formula": "\\begin{align*} A _ 5 ^ 2 & = k A _ 0 + \\frac { 1 } { 2 } ( k + s + t + s t ) A _ 2 + \\frac { 1 } { 2 } ( k + s t ) A _ 3 , \\\\ A _ 5 A _ 6 & = A _ 6 A _ 5 = k A _ 1 + \\frac { 1 } { 2 } ( k + s + t + s t ) A _ 2 + \\frac { 1 } { 2 } ( k + s t ) A _ 3 , \\\\ A _ 6 ^ 2 & = k A _ 0 + \\frac { 1 } { 2 } ( k + s + t + s t ) A _ 2 + \\frac { 1 } { 2 } ( k + s t ) A _ 3 . \\end{align*}"}
+{"id": "926.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\chi _ { E } ( Y ^ { x } _ { \\alpha } ( t ) ) d t - T \\lambda _ { 2 } ( E ) = o \\left ( \\log ( T ) ^ { \\frac { 1 } { 2 } + \\varepsilon } \\right ) \\forall \\varepsilon > 0 . \\end{align*}"}
+{"id": "6283.png", "formula": "\\begin{align*} p ^ { \\nu } _ { K , \\mathbb { S } , 1 } ( \\xi ) = \\int _ \\mathbb { S } p ^ { \\nu } _ { K } \\circ \\xi _ t \\d t , \\end{align*}"}
+{"id": "5802.png", "formula": "\\begin{align*} U _ 0 = \\sum _ { r = 1 } ^ p u _ { 0 r } e _ r , U _ 1 = \\sum _ { r = 1 } ^ p u _ { 1 r } e _ r , \\end{align*}"}
+{"id": "9014.png", "formula": "\\begin{align*} \\operatorname { p } _ f [ \\pi , \\eta ] \\leq \\sup _ { \\epsilon \\in \\mathbb { U } _ X } \\limsup _ { i \\in I } \\frac { P _ { f _ { A _ i } } ( \\epsilon _ { F _ i } ) } { \\theta ( K F _ i ) } \\leq \\sup _ { \\epsilon \\in \\mathbb { U } _ X } \\lim _ { i \\in I } \\frac { P _ { f _ { A _ i } } ( \\epsilon _ { K F _ i } ) } { \\theta ( K F _ i ) } = \\sup _ { \\epsilon \\in \\mathbb { U } _ X } \\operatorname { p } _ f [ \\pi , \\epsilon ] . \\end{align*}"}
+{"id": "75.png", "formula": "\\begin{align*} P _ n = [ ( e ^ n - 1 ) / ( e - 1 ) ] ( Q ) + [ e ^ n ] ( P _ 0 ) . \\end{align*}"}
+{"id": "610.png", "formula": "\\begin{align*} \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ { ( \\epsilon ) } ) ^ { \\rm T } { \\rm d } X _ t ^ { ( \\epsilon ) } = A : \\left ( X ^ { ( \\epsilon ) } _ { t _ { n + 1 / 2 } } \\otimes X ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } \\right ) , \\end{align*}"}
+{"id": "756.png", "formula": "\\begin{align*} & \\mathsf { H } ^ 1 _ \\infty ( A ) = \\varprojlim \\mathsf { H } ^ 1 _ n ( A ) , & \\mathsf { H } _ \\infty ( A ) = \\varprojlim \\mathsf { H } _ n ( A ) & & & & \\mathsf { h } _ \\infty ( A ) = \\varprojlim \\mathsf { h } _ n ( A ) , \\end{align*}"}
+{"id": "7813.png", "formula": "\\begin{align*} \\Lambda = \\begin{pmatrix} O & D \\\\ - D & D B \\end{pmatrix} , D = \\mathrm { d i a g } ( d _ 1 ^ { - 1 } , \\cdots , d _ r ^ { - 1 } ) . \\end{align*}"}
+{"id": "361.png", "formula": "\\begin{align*} \\mathbb { S } _ { [ 0 , T ] } : = ( S ( t ) : 0 \\leq t \\leq T ) \\ , . \\end{align*}"}
+{"id": "7661.png", "formula": "\\begin{align*} g _ { I , J , p } = \\sum _ { \\substack { \\textbf { p } \\in \\mathbb { Z } ^ { n } \\\\ | \\textbf { p } | = p } } c _ { \\textbf { p } } x ^ { \\textbf { p } } = \\sum _ { \\substack { \\textbf { p } \\in \\mathbb { Z } ^ { n } \\\\ | \\textbf { p } | = p } } c _ { \\textbf { p } } x _ { 1 } ^ { p _ { 1 } } \\cdots x _ { n } ^ { p _ { n } } \\end{align*}"}
+{"id": "8433.png", "formula": "\\begin{align*} c _ * ( A ) = \\sup _ { \\nu \\in \\widehat { \\mathcal E } ^ + _ A } \\ , \\nu ( X ) . \\end{align*}"}
+{"id": "871.png", "formula": "\\begin{align*} ( \\mathcal { X } ^ { \\frac { 1 } { 2 } } ) & = ( ^ { - 1 } ( ( \\mathcal { X } ) ^ { \\frac { 1 } { 2 } } ) ) = ( \\mathcal { X } ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "8024.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\sum _ { i \\ne j } e _ i e _ j / | x _ 1 - x _ j | = \\frac { 1 } { 2 } \\int _ { \\mathbb R ^ 3 } E ^ 2 ( x ) \\mathrm d ^ 3 x - \\sum \\\\ & \\qquad \\ge - N \\varepsilon ( D ) , | x _ 1 - x _ j | \\ge D . \\end{align*}"}
+{"id": "3554.png", "formula": "\\begin{align*} d _ { i } ^ + ( \\lambda ) = \\frac { ( - 1 ) ^ { \\sum _ { \\alpha = i } ^ n ( \\alpha + 1 ) ( \\lambda _ \\alpha - \\lambda _ { \\alpha + 1 } + \\delta _ { \\alpha i } ) } } { ( \\lambda _ i + 1 ) i } \\bigg ( \\prod _ { \\ell = 1 } ^ { i - 1 } \\frac { \\lambda _ i - \\lambda _ \\ell - i + \\ell + 1 } { \\lambda _ i - \\lambda _ \\ell - i + \\ell + [ \\lambda _ i - \\lambda _ \\ell ] _ 2 } \\bigg ) \\end{align*}"}
+{"id": "3595.png", "formula": "\\begin{align*} c _ { j } ^ + ( \\lambda ) = \\bigg ( \\prod _ { \\ell = 1 } ^ { j - 1 } \\frac { \\lambda _ j - \\lambda _ \\ell - j + \\ell + 1 } { \\lambda _ j - \\lambda _ \\ell - j + \\ell } \\bigg ) ^ { \\frac { 1 } { 2 } } F _ j ( \\lambda _ 1 , \\dots , \\lambda _ n ) \\end{align*}"}
+{"id": "3274.png", "formula": "\\begin{align*} \\lambda = \\inf \\left \\{ \\frac { \\left \\langle F \\left ( x \\right ) , g \\left ( x \\right ) \\right \\rangle } { \\left \\langle G \\left ( x \\right ) , g \\left ( x \\right ) \\right \\rangle } \\left \\vert \\ x \\in \\right . B _ { r _ { 0 } } ^ { X } \\left ( 0 \\right ) \\backslash \\left \\{ 0 \\right \\} \\right \\} , r _ { 0 } > 0 . \\end{align*}"}
+{"id": "4247.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { \\Sigma _ \\gamma } : = \\| ( \\nabla - i A ) f \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } V _ \\gamma ( x ) | f ( x ) | ^ 2 d x + \\| f \\| ^ 2 _ { L ^ 2 } . \\end{align*}"}
+{"id": "1222.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n - 1 } { ^ \\alpha } \\gamma _ { j i } \\frac { \\partial F _ { \\beta \\alpha } ^ i } { \\partial z ^ n _ \\beta } \\biggr | _ { z _ \\beta ( p ) } = \\begin{cases} { ^ \\beta } \\gamma _ { n n } \\frac { \\partial F ^ j _ { \\beta \\alpha } } { \\partial z ^ n _ \\beta } \\Bigr | _ { z _ \\beta ( \\gamma p ) } & \\ ; \\ , 1 \\leq j \\leq n - 1 , \\\\ 0 & \\ ; \\ , j = n . \\end{cases} \\end{align*}"}
+{"id": "7005.png", "formula": "\\begin{align*} C ( X ; Y ) = \\log { \\frac { N ( X , Y ) } { 1 - \\rho } } , \\end{align*}"}
+{"id": "2786.png", "formula": "\\begin{align*} \\limsup \\limits _ { t \\to \\infty } \\frac { | B _ t | } { t ^ { \\frac { 1 } { 2 } + \\varepsilon } } = 0 , \\end{align*}"}
+{"id": "4533.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { x _ 2 } = \\frac { 2 } { 3 } < \\theta \\leq \\frac { 1 } { 2 } + \\frac { 1 } { 5 } = \\frac { 7 } { 1 0 } . \\end{align*}"}
+{"id": "2011.png", "formula": "\\begin{align*} | \\{ ( x , y ) \\in E \\times E : x \\cdot y = t \\} | = \\frac { | E | ^ 2 } { q } + R ( E ) , \\end{align*}"}
+{"id": "6100.png", "formula": "\\begin{align*} \\mathcal { F } _ { - 2 k } ( F ( z ) ) = - \\overline { c _ F ^ + ( 0 ) } - \\frac { 1 } { ( 2 k ) ! } \\sum _ { \\substack { n \\ll \\infty \\\\ n \\neq 0 } } \\overline { c _ F ^ + ( - n ) } \\Gamma ( 1 + 2 k , - 4 \\pi n v ) q ^ n . \\end{align*}"}
+{"id": "5783.png", "formula": "\\begin{align*} U '' + \\mathcal L U + D \\mathcal G U ' = 0 \\end{align*}"}
+{"id": "354.png", "formula": "\\begin{align*} V _ a ( x _ b ) | _ N = \\delta _ { a b } , \\ \\ a > k _ 0 . \\end{align*}"}
+{"id": "6033.png", "formula": "\\begin{align*} \\forall 1 \\leq i , j , k , p , s , t \\leq n , \\ : i \\neq j , \\ : k \\neq p , \\ : s \\neq t , \\ : \\ : \\ : \\ : & d _ 1 ( w _ { i , j } , w _ { k , p } , w _ { s , t } ) = 1 - \\lfloor \\frac { i + k - s } { n } \\rfloor ; \\\\ & d _ 2 ( w _ { i , j } , w _ { k , p } , w _ { s , t } ) = \\lfloor \\frac { j + p - t } { n } \\rfloor , \\end{align*}"}
+{"id": "5554.png", "formula": "\\begin{align*} P _ { k } ( x ^ 2 ) & = \\sum _ { n = 1 } ^ { \\ell - 1 } \\frac { \\mu ( n ) } { n ^ k } \\exp \\left ( { - \\frac { x ^ 2 } { n ^ 2 } } \\right ) + \\sum _ { n = \\ell } ^ { \\infty } \\frac { \\mu ( n ) } { n ^ k } \\exp \\left ( { - \\frac { x ^ 2 } { n ^ 2 } } \\right ) \\\\ & : = S _ 1 ( x ^ 2 ) + S _ 2 ( x ^ 2 ) , \\end{align*}"}
+{"id": "6085.png", "formula": "\\begin{align*} ( f - r _ n ) ( z ) = \\varkappa _ n \\big ( b _ n ^ { - 2 } q _ n R _ n \\big ) ( z ) . \\end{align*}"}
+{"id": "5877.png", "formula": "\\begin{align*} C _ p = \\left ( \\begin{array} { c c c c } S _ { 1 } & & & \\\\ & S _ { 2 } & & \\\\ & & \\ddots & \\\\ & & & S _ { p } \\end{array} \\right ) . \\end{align*}"}
+{"id": "3088.png", "formula": "\\begin{align*} A ( y ) : = a ( y ) B ( y ) \\quad y \\in \\R ^ 2 . \\end{align*}"}
+{"id": "2551.png", "formula": "\\begin{align*} \\begin{aligned} & Z ( x _ { 1 } x _ { e _ 1 } { \\cdots } x _ { e _ { n - 1 } } x _ { - 1 } ; ( \\alpha , \\beta ) ) \\\\ = & \\idotsint \\displaylimits _ { \\begin{subarray} { c } 0 < t _ 0 < \\cdots < t _ n < 1 \\end{subarray} } ( 1 - t _ 0 ) ^ { 1 - \\alpha } t _ 0 ^ { \\beta - 1 } \\omega _ 1 ( t _ 0 ) \\left \\{ \\prod _ { i = 1 } ^ { n - 1 } \\omega _ { e _ i } ( t _ i ) \\right \\} \\omega _ { - 1 } ( t _ n ) t _ n ^ { 1 - \\beta } ( 1 - t _ n ) ^ { \\alpha - 1 } \\mathrm { d } t _ { 0 } \\cdots \\mathrm { d } t _ { n } \\end{aligned} \\end{align*}"}
+{"id": "2985.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla _ { j + i - 1 , j + i } ( \\overrightarrow { W } ^ \\ell _ { j + 1 } ) ^ 2 = \\ell ^ { - 1 } \\overrightarrow { W } ^ \\ell _ { j + 1 } \\nabla _ { j + i - 1 , j + i } Z _ { i , j } - \\ell ^ { - 2 } ( \\nabla _ { j + i - 1 , j + i } Z _ { i , j } ) ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "4920.png", "formula": "\\begin{align*} \\rho _ 1 = & \\ , \\rho ( p _ \\ell ; f _ \\ell ( \\mathbf { a } _ m ) , f _ \\ell ( \\mathbf { b } _ m ) ) , \\\\ \\rho _ 2 = & \\ , \\rho ( q _ \\ell ^ + ; g _ \\ell ^ + ( \\delta ^ + _ m \\mathbf { a } _ m ) , g _ \\ell ^ + ( \\delta ^ + _ m \\mathbf { b } _ m ) ) , \\\\ \\rho _ 3 = & \\ , \\rho ( q _ \\ell ^ - ; g _ \\ell ^ - ( \\delta ^ - _ m \\mathbf { a } _ m ) , g _ \\ell ^ - ( \\delta ^ - _ m \\mathbf { b } _ m ) ) , \\end{align*}"}
+{"id": "2626.png", "formula": "\\begin{align*} \\left | \\bigcup _ { i = 1 } ^ r L _ i \\right | = r ( k + 1 ) - \\sum _ { l = 2 } ^ r ( - 1 ) ^ l \\binom { r } { l } ( r - l ) s = r s , \\left | \\bigcap _ { j = 1 } ^ { r - 1 } L _ { i _ j } \\right | = s \\mbox { a n d } \\bigcap _ { j = 1 } ^ { r - 1 } L _ { i _ j } \\cap \\bigcap _ { j = 1 } ^ { r - 1 } L _ { i ' _ { j } } = \\emptyset . \\end{align*}"}
+{"id": "114.png", "formula": "\\begin{align*} \\lim \\limits _ { s \\rightarrow 0 ^ + } s \\int _ { \\mathbb R ^ n } \\int _ { \\mathbb R ^ n } \\frac { | f ( x ) - f ( y ) | ^ p } { | | x - y | | ^ { n + s p } _ K } d x d y = \\frac { 2 n } { p } | K | | | f | | _ { L ^ p ( \\mathbb R ^ n ) } ^ p . \\end{align*}"}
+{"id": "5493.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x ( t ) = J \\big ( \\nabla H ( t , x ( t ) ) - 2 \\pi \\lambda x ( t ) \\big ) , \\\\ \\| x \\| _ 2 = 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "3202.png", "formula": "\\begin{align*} \\frac 1 p + \\frac 1 2 = \\frac { 1 } { p ^ \\ast } . \\end{align*}"}
+{"id": "7880.png", "formula": "\\begin{align*} g '' ( z ) + p ( z ) g ' ( z ) + q ( z ) g ( z ) = 0 , \\end{align*}"}
+{"id": "5105.png", "formula": "\\begin{align*} Z = \\lbrace \\mathbf { y } = ( y _ 1 , z _ 1 , y _ 2 , z _ 2 , . . . , z _ { k - 2 } , y _ { k - 1 } ) : y _ i , z _ i \\in \\mathbb { N } , \\\\ 1 \\leq y _ 1 , . . . , y _ { k - 1 } \\leq 2 ^ { k / 3 - \\frac { k } { 2 \\log k } } , 1 \\leq z _ i \\leq i + 1 \\rbrace . \\end{align*}"}
+{"id": "8329.png", "formula": "\\begin{align*} \\tilde { v } = v , \\mathrm { i n } \\ | y | > s \\ge 0 , \\end{align*}"}
+{"id": "7874.png", "formula": "\\begin{align*} u ' = A ( z ) + u ^ 2 \\end{align*}"}
+{"id": "3015.png", "formula": "\\begin{align*} \\theta ^ a \\wedge \\theta ^ b \\wedge \\pi _ c = \\theta ^ a \\wedge \\theta ^ b \\wedge \\pi _ i = 0 \\end{align*}"}
+{"id": "1486.png", "formula": "\\begin{align*} 1 + \\frac { 1 } { 3 ^ { 3 } } + \\frac { 1 } { 5 ^ 3 } + \\cdots = \\frac { \\pi ^ 2 } { \\log 2 } + 2 \\int _ 0 ^ { \\frac { \\pi } { 2 } } \\theta \\log ( \\sin \\theta ) d \\theta , \\end{align*}"}
+{"id": "4743.png", "formula": "\\begin{align*} x u _ i = \\alpha _ i + y ^ { e _ i } s _ i \\end{align*}"}
+{"id": "2284.png", "formula": "\\begin{align*} & f ( 0 , x , v ) = f _ { 0 } ( x , v ) , \\rho ( 0 , x ) = \\rho _ 0 ( x ) , u ( 0 , x ) = u _ 0 ( x ) , \\\\ & \\gamma ^ - f ( t , x , v ) \\big | _ { ( 0 , T ) \\times \\Sigma ^ - } = g ( t , x , v ) , \\Phi ( t , x ) \\big | _ { ( 0 , T ) \\times \\partial \\Omega } = 0 , \\\\ & \\rho ( t , x ) \\big | _ { ( 0 , T ) \\times \\Gamma _ { \\rm { i n } } } = \\rho _ B ( x ) , u ( t , x ) \\big | _ { ( 0 , T ) \\times \\partial \\Omega } = u _ B ( x ) , \\end{align*}"}
+{"id": "3066.png", "formula": "\\begin{align*} c _ j ^ { k l } ( A ) = \\int _ Y r A e _ j \\cdot \\nabla v ^ { k l } = 0 \\forall \\ , 1 \\leq j , k , l \\leq n . \\end{align*}"}
+{"id": "4318.png", "formula": "\\begin{align*} \\omega _ 3 = d z _ 3 - ( a . d x + b . d y ) & = 0 \\\\ \\omega _ 2 = d z _ 2 & = 0 \\\\ \\omega _ 1 = d z _ 1 & = 0 \\end{align*}"}
+{"id": "6904.png", "formula": "\\begin{align*} m a x \\{ b _ i , b _ { i ' } + b _ { j ' } - 1 \\} & \\leq - ( a + 1 ) \\\\ 0 & \\leq b _ i \\end{align*}"}
+{"id": "3010.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\partial ^ { \\gamma , \\mathbf { A } } _ b p { _ i } ^ { a b } & = & \\partial _ b p { _ i } ^ { a b } - \\mathbf { A } { ^ k } _ b c ^ \\ell _ { k i } p { _ \\ell } ^ { a b } + \\gamma { ^ a } _ { c b } p { _ i } ^ { c b } + \\gamma { ^ b } _ { c b } p { _ i } ^ { a c } \\\\ \\partial ^ { \\gamma , \\mathbf { A } } _ b p { _ i } ^ { j b } & = & \\partial _ b p { _ i } ^ { j b } - \\mathbf { A } { ^ k } _ b c ^ \\ell _ { k i } p { _ \\ell } ^ { j b } + \\mathbf { A } { ^ k } _ b c ^ j _ { k \\ell } p { _ i } ^ { \\ell b } + \\gamma { ^ b } _ { c b } p { _ i } ^ { j c } \\end{array} \\end{align*}"}
+{"id": "6704.png", "formula": "\\begin{align*} \\mu ( n ) = \\sum _ { d ^ 2 \\mid n } \\mu ( d ) \\lambda ( n / d ^ 2 ) . \\end{align*}"}
+{"id": "5628.png", "formula": "\\begin{align*} \\begin{cases} _ { 0 } D _ { t } ^ { \\alpha } S ( t ) = \\Lambda ^ { \\alpha } - d ^ { \\alpha } S ( t ) - F \\left ( S ( t ) , I ( t ) \\right ) , \\\\ [ 0 . 2 c m ] _ { 0 } D _ { t } ^ { \\alpha } E ( t ) = F \\left ( S ( t ) , I ( t ) \\right ) - m _ { 1 } E ( t ) , \\\\ [ 0 . 2 c m ] _ { 0 } D _ { t } ^ { \\alpha } I ( t ) = \\sigma ^ { \\alpha } E ( t ) - m _ { 2 } I ( t ) , \\end{cases} \\end{align*}"}
+{"id": "7353.png", "formula": "\\begin{align*} \\partial _ E S : = \\{ u v \\in E ( G ) : u \\in S v \\in V ( G ) \\setminus S \\} . \\end{align*}"}
+{"id": "5683.png", "formula": "\\begin{align*} \\theta ^ { a } & \\rightarrow \\theta _ { 1 } ^ { a } = ( L ^ { - 1 } ) _ { b } ^ { a } \\theta ^ { b } , \\\\ \\omega _ { b } ^ { a } & \\rightarrow \\omega _ { 1 b } ^ { a } = ( L ^ { - 1 } ) _ { c } ^ { a } d L _ { b } ^ { c } + ( L ^ { - 1 } ) _ { c } ^ { a } \\omega _ { d } ^ { c } L _ { b } ^ { d } , \\\\ \\Omega _ { b } ^ { a } & \\rightarrow \\Omega _ { 1 b } ^ { a } = ( L ^ { - 1 } ) _ { c } ^ { a } \\Omega _ { d } ^ { c } L _ { b } ^ { d } . \\end{align*}"}
+{"id": "1568.png", "formula": "\\begin{align*} g ( X ^ v ) = X ^ v + X ^ { v + r v ( q - 1 ) } + X ^ { v + s v ( q - 1 ) } . \\end{align*}"}
+{"id": "2421.png", "formula": "\\begin{align*} L \\dot I = - V _ 2 = 0 , \\end{align*}"}
+{"id": "6201.png", "formula": "\\begin{align*} \\overline A _ p ^ r \\overline B _ p ^ s \\overline D = \\overline A _ p ^ r \\overline B _ p ^ s C _ p D = C _ p A ^ r B ^ s D , \\end{align*}"}
+{"id": "4274.png", "formula": "\\begin{align*} I ^ m _ \\gamma ( c ) & \\leq \\inf \\left \\{ E _ \\gamma ( f ) \\ : \\ f \\in S ( c ) \\cap B _ \\gamma ( m / 4 ) \\right \\} \\\\ & < \\inf \\left \\{ E _ \\gamma ( f ) \\ : \\ f \\in S ( c ) \\cap \\left ( B _ \\gamma ( m ) \\backslash B _ \\gamma ( m / 2 ) \\right ) \\right \\} \\\\ & \\leq E _ \\gamma ( \\phi ) = I ^ m _ \\gamma ( c ) \\end{align*}"}
+{"id": "7124.png", "formula": "\\begin{align*} ( d - a \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) ) \\cap x ^ { \\rho _ 1 + \\cdots + \\rho _ { n - 1 } } & = a \\beta ( \\rho _ n ) - a \\beta ( \\rho _ n ) \\\\ & = 0 , \\end{align*}"}
+{"id": "8815.png", "formula": "\\begin{align*} A _ { 1 1 } & = B _ { 1 1 } ^ * P _ 1 & A _ { 1 2 } & = B _ { 2 1 } ^ * P _ 2 \\ , , \\\\ A _ { 2 1 } & = B _ { 1 2 } ^ * P _ 1 & A _ { 2 2 } - B _ { 2 2 } ^ * P _ 2 & = D _ { P _ 2 } F _ { 1 2 2 } D _ { P _ 2 } \\ , . \\end{align*}"}
+{"id": "7862.png", "formula": "\\begin{align*} x [ L _ n ^ { ( \\alpha ) } ] '' ( x ) + ( \\alpha + 1 - x ) [ L _ n ^ { ( \\alpha ) } ] ' ( x ) + n L _ n ^ { ( \\alpha ) } ( x ) = 0 , \\end{align*}"}
+{"id": "4746.png", "formula": "\\begin{align*} x u & = x u _ j - \\sum _ { i < j } y ^ { f _ i } \\rho _ i x u _ i \\\\ & = \\alpha _ j + y ^ { e _ j } s _ j - \\sum _ { i < j } y ^ { f _ i } \\rho _ i ( \\alpha _ i + y ^ { e _ i } s _ i ) \\\\ & = \\left ( \\alpha _ j - \\sum _ { i < j } y ^ { f _ i } \\rho _ i \\alpha _ i \\right ) + y ^ { e _ j } \\left ( s _ j - \\sum _ { i < j } \\rho _ i s _ i \\right ) \\end{align*}"}
+{"id": "5970.png", "formula": "\\begin{align*} A ^ T E _ r - \\sum _ { s = 1 } ^ p \\alpha _ { r s } E _ s = C _ p ^ T R _ r , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "2096.png", "formula": "\\begin{align*} m = \\dim _ H ( A + B ) \\leq \\dim _ H ( A ) + \\dim _ P ( B ) \\leq m k e w ^ { \\prime } + \\dim _ P ( B ) . \\end{align*}"}
+{"id": "2182.png", "formula": "\\begin{align*} \\psi ( r ) : = ( 6 \\alpha - 8 + 3 \\beta ) r ^ 2 - 3 ( 2 - 2 \\alpha + \\beta ) r + 2 . \\end{align*}"}
+{"id": "4970.png", "formula": "\\begin{align*} \\lambda : = q _ 1 ^ { - 1 } ( q ^ { t / p } - ( f ^ { l p } q _ 2 ) ^ { t / p } ) = q _ 1 ^ { - 1 } ( q ^ { t / p } - ( q - \\xi ) ^ { t / p } ) \\in k [ f , q _ 1 , q ] . \\end{align*}"}
+{"id": "889.png", "formula": "\\begin{align*} \\mathbf { V a r } _ { i \\sim \\mathbf { u } } [ p _ { i } ^ { t + 1 } ] & = \\frac { 1 } { q } \\sum _ { i = 1 } ^ { q } \\left ( p _ { i } ^ { t + 1 } - \\frac { 1 } { q } \\sum _ { s = 1 } ^ { q } p _ { s } ^ { t + 1 } \\right ) ^ { 2 } = \\frac { 1 } { q } \\sum _ { i = 1 } ^ { q } \\left ( p _ { i } ^ { t + 1 } - \\frac { 1 } { q } \\right ) ^ { 2 } \\geq \\frac { 1 } { q } \\left ( p _ { i ^ { t } } ^ { t + 1 } - \\frac { 1 } { q } \\right ) ^ { 2 } = \\frac { 1 } { q ^ { 3 } } . \\end{align*}"}
+{"id": "4945.png", "formula": "\\begin{align*} \\delta _ n ^ + \\widetilde { G } _ \\ell + \\delta _ m ^ + \\widetilde { F } _ \\ell , \\ell = 1 , 2 , \\end{align*}"}
+{"id": "6721.png", "formula": "\\begin{align*} \\sum _ { m \\leq x / q } \\mu ^ { + } ( q m + a ) \\leq \\sum _ { m \\leq x / q } \\mu ^ { + } ( m ) = \\frac { 3 } { \\pi ^ 2 } \\frac { x } { q } + O \\left ( \\frac { x } { q ( \\log x / q ) ^ D } \\right ) . \\end{align*}"}
+{"id": "1637.png", "formula": "\\begin{align*} \\dot { u } _ t ( x ) + F ( t , x , u _ t ( x ) ) & = - \\gamma ' ( t ) + F ( t , x , u _ t ( x ) ) \\\\ & \\leq - \\gamma ' ( t ) + F ( t , x , 0 ) - \\lambda _ F ( \\phi ( x ) - \\gamma ( t ) ) \\\\ & \\leq \\lambda _ F | \\phi ( x ) | - C + F ( t , x , 0 ) \\\\ & \\leq c _ - , \\end{align*}"}
+{"id": "3467.png", "formula": "\\begin{align*} & B _ 2 ( w , z , x ; t ) = 2 \\widetilde { f } _ 2 ( w , z , x ; t ) = f _ 2 ( w , z , x ; t ) + f _ 2 ( z , w , x ; t ) \\\\ & = \\int _ { 0 < \\theta < r < t } G _ { t - r } ( x - z ) G _ { r - \\theta } ( z - w ) d r d \\theta + \\int _ { 0 < r < \\theta < t } G _ { t - \\theta } ( x - w ) G _ { \\theta - r } ( w - z ) d r d \\theta . \\end{align*}"}
+{"id": "4597.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k v _ i \\leq \\prod _ { i = 1 } ^ k u _ i \\end{align*}"}
+{"id": "7786.png", "formula": "\\begin{align*} X _ n ( y ^ { n ' } ) : = & \\ [ \\{ n , n ' \\} ] _ q y ^ { n ' + n } \\\\ = & \\ \\frac { q ^ { 2 \\{ n , n ' \\} } - 1 } { q - q ^ { - 1 } } y ^ { n ' } y ^ { n } . \\end{align*}"}
+{"id": "7474.png", "formula": "\\begin{align*} \\tilde { F } _ { A } ( s ) = F _ { A } ( s ) \\end{align*}"}
+{"id": "7736.png", "formula": "\\begin{align*} \\rho _ N ^ { - \\frac { 1 } { 2 } } \\left [ 1 + C _ { \\beta } \\sqrt { 2 } \\left ( \\frac { \\tau } { \\rho _ N } \\right ) ^ { \\frac { 1 } { 4 } } \\right ] ^ { - 2 n } = \\frac { 1 } { D _ { \\beta } } \\left ( 1 + \\sqrt { 2 } G _ { \\beta } \\tau ^ { - \\frac { 1 } { 4 } } \\right ) ^ { - 2 n } . \\end{align*}"}
+{"id": "3424.png", "formula": "\\begin{align*} h ( \\mu _ { \\infty } ) = \\lim _ { t \\to \\infty } h ( \\mu _ t ) = \\sup _ { \\nu \\in \\mathcal { M } _ { m a x } ( \\phi ) } h ( \\nu ) , \\end{align*}"}
+{"id": "7196.png", "formula": "\\begin{align*} i ^ 2 \\ , = \\ , j ^ 2 \\ , = \\ , k ^ 2 \\ , = - 1 , \\qquad \\mbox { a n d } i j k = - 1 . \\end{align*}"}
+{"id": "6308.png", "formula": "\\begin{align*} B _ i \\cdot B _ j = \\sum _ { k } b _ { i , j } ^ k B _ k & \\Leftrightarrow \\Delta D _ k = \\sum _ { i , j } b _ { i , j } ^ k D _ i \\otimes D _ j \\\\ D _ i \\cdot D _ j = \\sum _ { k } d _ { i , j } ^ k D _ k & \\Leftrightarrow \\Delta B _ k = \\sum _ { i , j } d _ { i , j } ^ k B _ i \\otimes B _ j \\end{align*}"}
+{"id": "6612.png", "formula": "\\begin{align*} \\nabla _ { \\theta } ( P _ n H ( \\theta ) P _ n ) & = ( \\nabla _ { \\theta } P _ n ) H ( \\theta ) P _ n + P _ n ( \\nabla _ { \\theta } H ( \\theta ) ) P _ n + P _ n H ( \\theta ) \\nabla _ { \\theta } P _ n \\\\ & = P _ n ( \\nabla _ { \\theta } H ( \\theta ) ) P _ n + E _ n \\nabla _ { \\theta } ( P _ n P _ n ) = P _ n ( \\nabla _ { \\theta } H ( \\theta ) ) P _ n + E _ n \\nabla _ { \\theta } P _ n . \\end{align*}"}
+{"id": "6799.png", "formula": "\\begin{align*} & B _ \\eta \\left ( \\left ( \\overline { x } _ { n + \\i - 1 , \\widehat { n + \\eta - 1 } } \\right ) _ { \\i = 1 } ^ { \\infty } , \\left ( \\overline { w } \\right ) _ { \\i = 1 } ^ { \\infty } \\right ) \\\\ & = d \\left ( \\overline { x } _ { n + \\eta - 1 } , \\overline { w } \\right ) \\end{align*}"}
+{"id": "9189.png", "formula": "\\begin{align*} \\# ( X _ 0 , X _ 1 , Z ) : = | X _ 0 | _ { j + 1 } + | \\operatorname { C o m p } _ { j + 1 } ( X _ 1 ) | + | \\operatorname { C o m p } _ { j + 1 } ( Z ) | \\leq 1 \\end{align*}"}
+{"id": "1775.png", "formula": "\\begin{align*} \\mathsf { L } ( x ) : = \\{ | z | \\mid z \\in \\mathsf { Z } ( x ) \\} . \\end{align*}"}
+{"id": "4078.png", "formula": "\\begin{align*} ( f , F ) \\circ ( f ' , F ' ) = \\overline { f \\circ f ' } \\circ e ^ { B ' + f '^ * B } F = \\overline { f } \\circ e ^ B , F ' = \\overline { f ' } \\circ e ^ { B ' } . \\end{align*}"}
+{"id": "608.png", "formula": "\\begin{align*} \\mathbb { E } ^ \\dagger \\left [ X ^ { ( \\epsilon ) } _ { \\tau _ l , \\tau _ { l + 1 } } \\otimes X ^ { ( \\epsilon ) } _ { \\tau _ { l + 1 } , \\tau _ { l + 2 } } \\right ] = \\mathcal { O } ( \\epsilon ^ { - 1 } \\Delta \\tau ^ 2 ) \\end{align*}"}
+{"id": "5173.png", "formula": "\\begin{align*} C ( L _ q ^ { 2 n + 1 } ( r ; \\underline { m } ) ) \\cong C ( L _ { 2 n + 1 } ) ^ { \\varrho _ { \\underline { m } } ^ r } \\cong \\left ( \\sum _ { i = 0 } ^ n p _ { ( v _ i , 0 ) } \\right ) C ^ * ( L _ { 2 n + 1 } \\times _ { c } \\mathbb { Z } _ r ) \\left ( \\sum _ { i = 0 } ^ n p _ { ( v _ i , 0 ) } \\right ) . \\end{align*}"}
+{"id": "2438.png", "formula": "\\begin{align*} b ( 0 ) = b ( 1 0 ) = 1 , v = \\frac { 1 } { 5 0 } x ( 1 0 - x ) + 1 f = 1 0 . \\end{align*}"}
+{"id": "4110.png", "formula": "\\begin{align*} \\int _ M K _ { i j } \\delta ( i _ { \\nabla f } H ) _ { i j } e ^ { - f } d V _ g & = - \\int _ M K _ { i j } h _ { l k } H _ { k i j } \\nabla _ l f e ^ { - f } d V _ g + \\int _ M K _ { i j } [ \\nabla _ l \\phi H _ { l i j } + \\nabla _ l f ( d K ) _ { l i j } ] e ^ { - f } d V _ g \\\\ & = - \\int _ M K _ { i j } h _ { l k } H _ { k i j } \\nabla _ l f e ^ { - f } d V _ g - \\frac { 1 } { 3 } \\int _ M \\langle d K , H \\rangle \\phi e ^ { - f } d V _ g + \\int _ M ( d K ) _ { l i j } K _ { i j } \\nabla _ l f e ^ { - f } d V _ g \\end{align*}"}
+{"id": "5235.png", "formula": "\\begin{align*} \\nabla _ X U = A _ X U + \\nu \\nabla _ X U , \\end{align*}"}
+{"id": "5531.png", "formula": "\\begin{align*} e ^ { - x } = \\frac { 1 } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } \\Gamma ( s ) x ^ { - s } { \\rm d } s , \\end{align*}"}
+{"id": "8233.png", "formula": "\\begin{align*} \\int _ { B _ { R _ { 1 } } } K \\left ( \\left | x \\right | \\right ) \\left | u \\right | ^ { q _ { 1 } } d x = \\left \\| u \\right \\| ^ { q _ { 1 } } \\int _ { B _ { R _ { 1 } } } K \\left ( \\left | x \\right | \\right ) \\frac { \\left | u \\right | ^ { q _ { 1 } } } { \\left \\| u \\right \\| ^ { q _ { 1 } } } d x \\leq \\left \\| u \\right \\| ^ { q _ { 1 } } \\mathcal { S } _ { 0 } \\left ( q _ { 1 } , R _ { 1 } \\right ) \\end{align*}"}
+{"id": "41.png", "formula": "\\begin{align*} t \\cdot \\sum _ { i = 1 } ^ m T ( n _ i ) & = \\sum _ { i = 1 } ^ { t \\cdot m } T ( n _ i ) < \\frac { 9 7 ( t \\cdot m ) + 7 3 } { 5 4 } \\cdot \\frac { 1 } { X _ 0 } , \\\\ \\sum _ { i = 1 } ^ m T ( n _ i ) & < \\frac { 9 7 } { 5 4 } \\cdot m \\cdot \\frac { 1 } { X _ 0 } + \\frac { 1 } { t } \\cdot \\frac { 7 3 } { 5 4 } \\cdot \\frac { 1 } { X _ 0 } . \\end{align*}"}
+{"id": "1459.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m x _ i ( x _ i - 1 ) = \\sum _ { j = 1 } ^ m y _ j ( y _ j - 1 ) = \\frac { k ( k - 1 ) } { m + 1 } , \\end{align*}"}
+{"id": "182.png", "formula": "\\begin{align*} T ( Z , L ) & = \\max \\left \\{ \\left [ \\frac { \\sum _ { l = 1 } ^ q m _ { i _ l } + j - 2 } { j } \\right ] | \\ P _ { i _ 1 } , \\ldots , P _ { i _ q } \\in L \\right \\} \\\\ & = T _ j ( Z ) . \\end{align*}"}
+{"id": "5283.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 , j \\neq i } ^ { m - n } s e c ( U _ i , U _ j ) & = \\sum \\limits _ { j = 1 , j \\neq i } ^ { m - n } \\hat { s } e c ( U _ i , U _ j ) - \\lambda ^ 2 ( m - n - 1 ) \\| \\nabla ( f + \\log \\lambda ) \\| ^ 2 \\\\ & + \\frac { \\lambda ^ 4 } { 4 } ( m - n - 1 ) \\| g r a d _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\| ^ 2 - \\lambda ^ 2 ( m - n - 1 ) g \\left ( \\nabla f , \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\right ) , \\end{align*}"}
+{"id": "908.png", "formula": "\\begin{align*} \\sum _ { i \\ge 0 } u ^ i f _ { i + 1 } & = \\sum _ { i \\ge 0 } u ^ i z f _ i + \\sum _ { i \\ge 0 } u ^ i z g _ i , \\\\ \\sum _ { i \\ge 1 } u ^ i g _ i & = \\sum _ { i \\ge 1 } u ^ i z f _ { i + 1 } + \\sum _ { i \\ge 1 } u ^ i z g _ { i + 1 } + \\sum _ { i \\ge 1 } u ^ i z h _ { i + 1 } , \\\\ \\sum _ { i \\ge 0 } u ^ i h _ i & = \\sum _ { i \\ge 0 } u ^ i z h _ { i + 1 } + \\sum _ { i \\ge 0 } u ^ i z g _ { i + 1 } . \\end{align*}"}
+{"id": "6332.png", "formula": "\\begin{align*} \\varphi ( \\nu ) : = \\int _ { \\xi ( \\nu ) } ^ { 1 } \\frac { 2 \\nu } { m ( x ) \\sqrt { m ( x ) ^ 2 - \\nu ^ 2 } } d x \\quad \\mbox { a n d } \\psi ( \\nu ) : = \\int ^ { \\eta ( \\nu ) } _ { 1 } \\frac { 2 \\nu } { m ( x ) \\sqrt { m ( x ) ^ 2 - \\nu ^ 2 } } d x . \\end{align*}"}
+{"id": "9184.png", "formula": "\\begin{align*} G ^ * _ j ( X , \\varphi ) = \\begin{array} { l l } \\begin{cases} G _ j ( X , \\varphi ) & \\ ; \\ ; * = 0 \\\\ G _ j ^ { \\Psi } ( X , \\varphi ) & \\ ; \\ ; * = \\Psi . \\end{cases} \\end{array} \\end{align*}"}
+{"id": "1573.png", "formula": "\\begin{align*} f ( t ) - f ( 0 ) = f ' ( 0 ) t + \\frac { \\mu } { 2 } t ^ 2 - \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { s } R i c \\big ( \\gamma ' ( \\tau ) , \\gamma ' ( \\tau ) \\big ) d \\tau d s + \\rho \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { s } R d \\tau d s . \\end{align*}"}
+{"id": "6496.png", "formula": "\\begin{align*} S _ { } ( L ) = \\frac { 1 } { \\sqrt { b ' } m ' } \\underset { i = 1 } { \\overset { b ' } { \\sum } } \\left [ \\left ( \\underset { j \\in M _ L } { \\sum } \\left [ ( Y _ { } ^ { j } ( L ) ) _ { i } - 1 / 2 \\right ] \\right ) ^ 2 - \\frac { m ' } { 4 } \\right ] . \\end{align*}"}
+{"id": "186.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { n - 1 } ( r _ { j } + 1 ) \\leq h _ { n } \\leq \\frac { 1 } { \\mu ( C _ { 1 } ) } \\prod _ { j = 1 } ^ { n - 1 } ( r _ { j } + 1 ) \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\frac { 1 } { h _ { n } } \\prod \\limits _ { j = 1 } ^ { n - 1 } ( r _ { j } + 1 ) \\to \\mu ( C _ { 1 } ) . \\end{align*}"}
+{"id": "3006.png", "formula": "\\begin{align*} \\begin{array} { l l l l l l } p _ i & = & ( S ^ { - 1 } ) ^ j _ i \\pi _ j & p { _ i } ^ { a b } & = & ( S ^ { - 1 } ) ^ j _ i \\pi { _ j } ^ { a b } \\\\ p { _ i } ^ { a k } & = & ( S ^ { - 1 } ) ^ j _ i S ^ k _ \\ell \\pi { _ j } ^ { a \\ell } & p { _ i } ^ { j k } & = & ( S ^ { - 1 } ) ^ { \\ell } _ i S ^ j _ { m } S ^ k _ { n } \\pi { _ \\ell } ^ { m n } \\end{array} \\end{align*}"}
+{"id": "312.png", "formula": "\\begin{align*} \\epsilon _ { \\mathrm { H M } } ( t ) = 1 , \\end{align*}"}
+{"id": "2377.png", "formula": "\\begin{align*} E ( t ) \\dot x = A ( t ) x \\end{align*}"}
+{"id": "9191.png", "formula": "\\begin{align*} ( E ^ { \\Lambda _ N } _ { j + 1 } , U ^ { \\Lambda _ N } _ { j + 1 } , K _ { j + 1 } ^ { \\Lambda _ N } ( \\cdot ; 0 ) ) = \\Phi _ { j + 1 } ^ { \\Lambda _ N } ( E ^ { \\Lambda _ N } _ j , U ^ { \\Lambda _ N } _ j , K _ j ^ { \\Lambda _ N } ( \\cdot ; 0 ) ) , 0 \\leq j \\leq N - 1 . \\end{align*}"}
+{"id": "8570.png", "formula": "\\begin{align*} \\begin{aligned} \\langle & ( 0 , 1 , 2 0 , 4 , 2 ) ( 3 , 8 , 9 , 1 2 , 1 3 ) ( 5 , 1 6 , 1 0 , 1 1 , 1 8 ) ( 6 , 7 , 1 5 , 1 9 , 1 4 ) , \\\\ & ( 0 , 1 3 , 1 6 , 5 , 1 0 ) ( 1 , 1 4 , 1 9 , 4 , 2 ) ( 3 , 1 8 , 7 , 1 2 , 1 5 ) ( 9 , 2 1 , 1 1 , 2 0 , 1 7 ) \\rangle \\end{aligned} \\end{align*}"}
+{"id": "6320.png", "formula": "\\begin{align*} \\mu _ { i j k } { = } \\mathsf { A } ^ { i j k } _ { n \\ell p } \\frac { \\partial ^ 2 u _ n } { \\partial x _ \\ell \\partial x _ p } + \\mathsf { S } _ { n \\ell } ^ { i j k } \\frac { \\partial u _ n } { \\partial x _ \\ell } , \\end{align*}"}
+{"id": "4832.png", "formula": "\\begin{align*} \\phi ( s ) = \\sum _ { j = 1 } ^ n \\frac { \\phi ^ { ( j ) } ( 0 ) } { j ! } s ^ j + \\frac { \\phi ^ { ( n + 1 ) } ( \\bar s ) } { ( n + 1 ) ! } s ^ { n + 1 } , s \\in [ . 0 , 1 ] , \\end{align*}"}
+{"id": "7174.png", "formula": "\\begin{align*} \\mathcal { S } = \\bigcup _ { t \\in [ 0 , \\ , \\infty ] } \\{ t \\} \\times \\Sigma _ t , \\end{align*}"}
+{"id": "2352.png", "formula": "\\begin{align*} Q _ 0 ( 0 ) = V _ 0 ( 0 ) ^ T V _ 0 ( 0 ) , Q _ 1 ( 0 ) = V _ 1 ( 0 ) ^ T V _ 1 ( 0 ) . \\end{align*}"}
+{"id": "5442.png", "formula": "\\begin{align*} V ( X _ t , \\alpha _ t ) - V ( X _ 0 , \\alpha _ 0 ) = \\int _ 0 ^ t L V ( X _ s , \\alpha _ s ) d s + M _ 1 ( t ) + M _ 2 ( t ) , \\end{align*}"}
+{"id": "1524.png", "formula": "\\begin{align*} \\epsilon = \\tau \\oplus \\beta \\end{align*}"}
+{"id": "2162.png", "formula": "\\begin{align*} \\psi ( r ) : = ( - 1 + 2 \\alpha + \\beta - \\mathit { e } ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + ( \\mathit { e } - 1 ) \\end{align*}"}
+{"id": "6334.png", "formula": "\\begin{align*} \\left . \\frac { \\partial } { \\partial \\nu } \\right | _ { \\nu _ 0 } \\theta ( \\alpha _ \\nu ( t _ c ) ) = \\left . \\frac { \\partial } { \\partial \\nu } \\right | _ { \\nu _ 0 } r ( \\alpha _ \\nu ( t _ c ) ) = 0 . \\end{align*}"}
+{"id": "7052.png", "formula": "\\begin{align*} d _ { i , j + 1 } ( d _ { k , \\ell } - c _ { k , \\ell } ) & = ( d _ { i , j } - c _ { i , j } ) d _ { k , \\ell + 1 } \\\\ d _ { i , j + 1 } ( q _ \\ell - p _ \\ell ) & = ( d _ { i , j } - c _ { i , j } ) q _ { \\ell + 1 } \\\\ q _ { j + 1 } ( q _ \\ell - p _ \\ell ) & = ( q _ j - p _ j ) q _ { \\ell + 1 } \\\\ q _ 0 & = 0 \\end{align*}"}
+{"id": "1753.png", "formula": "\\begin{align*} F ( 0 ) = k _ M ( - 1 ) ^ { \\frac { k _ { \\rm i d } - 2 } { 2 } - m _ { \\rm i d } } \\langle \\mu _ { \\underline { m } } , \\mu _ { - \\underline { m } } \\rangle _ V \\sum _ { i \\leq \\frac { k _ c - 2 } { 2 } - m _ c } \\binom { k _ c - 1 } { i } \\binom { k _ { \\rm i d } - 1 } { \\lambda - \\bar m + i } . \\end{align*}"}
+{"id": "7156.png", "formula": "\\begin{align*} \\displaystyle { [ D _ k , D _ l ] = 0 \\ , , k , l = 1 , . . . , N \\ , . } \\end{align*}"}
+{"id": "2227.png", "formula": "\\begin{align*} \\nu ( x _ { 2 m , 1 } ) & = \\nu { \\left ( \\sum _ { k = 1 } ^ \\infty x _ { 2 m - 1 , k } \\alpha _ { k , 1 } \\right ) } \\\\ & = \\nu { \\left ( x _ { 2 m - 1 , 1 } \\alpha _ { 1 , 1 } + \\sum _ { k = 2 } ^ \\infty x _ { 2 m - 1 , k } \\alpha _ { k , 1 } \\right ) } . \\end{align*}"}
+{"id": "5551.png", "formula": "\\begin{align*} M ( \\ell ; n ) : = \\sum _ { m = \\ell } ^ { n } \\frac { \\mu ( m ) } { m ^ k } & = A ( n ) f ( n ) - A ( \\ell - 1 ) f ( \\ell - 1 ) - \\int _ { \\ell - 1 } ^ n A ( t ) f ' ( t ) { \\rm d } t . \\end{align*}"}
+{"id": "3725.png", "formula": "\\begin{align*} \\tilde { P } _ \\ell ^ { \\mathrm { d e r } } \\backslash \\mathrm { S p i n } _ { V _ \\ell } = P _ \\ell ^ { \\mathrm { d e r } } \\backslash \\mathrm { S O } _ { V _ \\ell } \\end{align*}"}
+{"id": "7446.png", "formula": "\\begin{align*} 0 = K ( Z , Z ) ( I ) & = \\Gamma _ 1 ( Z , Z ) ( Q ( Z , Z ) ( I ) \\succeq \\epsilon _ 0 \\Gamma _ 1 ( Z , Z ) ( I ) \\\\ 0 = K ( Z , Z ) ( I ) & = - \\Gamma _ 3 ( Z , Z ) ( Q ( Z , Z ) ( I ) \\preceq - \\epsilon _ 0 \\Gamma _ 3 ( Z , Z ) ( I ) \\end{align*}"}
+{"id": "9224.png", "formula": "\\begin{align*} \\lll u & = 2 \\nabla f \\cdot \\nabla u + 2 \\ell _ { 0 } \\nabla \\ell _ { 0 } \\cdot \\nabla u + \\vert \\nabla \\ell _ { 0 } \\vert ^ { 2 } u \\\\ & = 2 \\nabla \\phi _ { + } \\cdot \\nabla u + \\vert \\nabla \\ell _ { 0 } \\vert ^ { 2 } u , \\end{align*}"}
+{"id": "4475.png", "formula": "\\begin{align*} \\frac { 2 } { 3 } & = \\frac { 1 } { 2 } + \\frac { 1 } { 6 } \\\\ & = \\frac { 1 } { 2 } + \\frac { 1 } { 7 } + \\frac { 1 } { 4 2 } \\\\ & = \\frac { 1 } { 2 } + \\frac { 1 } { 7 } + \\frac { 1 } { 4 3 } + \\frac { 1 } { 1 8 0 6 } . \\end{align*}"}
+{"id": "9250.png", "formula": "\\begin{align*} \\tilde { d } = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots v _ { w ^ { - 1 } ( k ) } , \\ldots , v _ { h ( w ^ { - 1 } ( k ) ) } , X v _ { w ^ { - 1 } ( k ) } , \\ldots \\widehat { v _ { r } } , \\ldots v _ n ) } = 0 , \\end{align*}"}
+{"id": "5310.png", "formula": "\\begin{align*} ( \\nabla _ { X } A _ { \\xi } ) ( Y ) - ( \\nabla _ { Y } A _ { \\xi } ) ( X ) = \\omega ( X ) A _ { \\xi } Y - \\omega ( Y ) A _ { \\xi } X . \\end{align*}"}
+{"id": "1905.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathbb { E } | z ( \\cdot , t \\wedge \\tau _ { n } ) | ^ 2 _ { L ^ 2 ( 0 , 1 ) } = \\mathbb { E } | z ( \\cdot , 0 ) | ^ 2 _ { L ^ 2 ( 0 , 1 ) } + 2 \\mathbb { E } \\int _ 0 ^ { t \\wedge \\tau _ { n } } z ( 1 , s ) \\widetilde { w } ( s ) d s \\cr - 2 \\mathbb { E } \\int _ 0 ^ { t \\wedge \\tau _ { n } } | z _ x ( \\cdot , s ) | ^ 2 _ { L ^ 2 ( 0 , 1 ) } d s + \\mathbb { E } \\int _ 0 ^ { t \\wedge \\tau _ { n } } ( \\sigma ^ 2 - 2 c ) | z ( \\cdot , s ) | ^ 2 _ { L ^ 2 ( 0 , 1 ) } d s . \\end{array} \\end{align*}"}
+{"id": "724.png", "formula": "\\begin{align*} \\frac { d } { d t } \\log u + r ( w ) \\frac { d w } { d t } = 0 . \\end{align*}"}
+{"id": "4226.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { | v | } { \\lambda ( t ) } \\int \\rho ' \\left ( \\frac { x - v t } { \\lambda ( t ) } \\right ) \\left | \\partial _ x \\Lambda \\partial _ t \\Lambda + 4 \\partial _ x \\phi \\partial _ t \\phi \\sinh ^ 2 ( \\Lambda ) \\right | = & ~ { } \\frac { | v | } { \\lambda ( t ) } \\int \\rho ' \\left ( \\frac { x - v t } { \\lambda ( t ) } \\right ) | p ( t , x ) | \\\\ \\leq & ~ { } \\frac { | v | } { \\lambda ( t ) } \\int \\rho ' \\left ( \\frac { x - v t } { \\lambda ( t ) } \\right ) e ( t , x ) . \\end{aligned} \\end{align*}"}
+{"id": "6290.png", "formula": "\\begin{align*} \\psi ( F _ \\alpha ) = F _ { \\alpha ^ c } . \\end{align*}"}
+{"id": "303.png", "formula": "\\begin{align*} \\begin{aligned} \\epsilon _ { \\mathrm { H M } } ( t ) & = \\epsilon ^ { - } _ { j _ { \\pi } ( T ) ( E ) , [ g ] } ( t ) \\epsilon ^ { + } | _ { j _ { \\pi } ( T ) ( E ) , [ g ] } ( t ) \\\\ & : = \\det ( \\mathrm { A d } ( t ) _ { g G ^ { 0 } ( E ) _ { [ x ] } g ^ { - 1 } \\cap G ( F ) } ) \\prod _ { i = 0 } ^ { d - 1 } \\mathrm { s g n } _ { k _ { F } ^ { \\times } } ( \\det \\mathrm { A d } ( t ) | _ { \\mathfrak { W } _ i } ) . \\end{aligned} \\end{align*}"}
+{"id": "76.png", "formula": "\\begin{align*} P _ { n } = [ e _ 0 ] ( P _ { n - 1 } ) + \\cdots + [ e _ { \\mathfrak { L } } ] ( P _ { n - 1 - \\mathfrak { L } } ) + Q , \\ , \\ n \\geq \\mathfrak { L } + 1 . \\end{align*}"}
+{"id": "5998.png", "formula": "\\begin{align*} p = j + l - 2 - r & = j - 1 + n - ( n - l + 1 ) - r \\\\ & = \\# ( e _ 1 , \\dots e _ { j - 1 } , e _ { n - l + 2 } , \\dots , e _ n ) - r , \\end{align*}"}
+{"id": "1770.png", "formula": "\\begin{align*} \\Delta ( \\Sigma ) \\ : = \\ \\big \\{ \\ , \\Delta u \\ | \\ u \\in \\Sigma \\ , \\big \\} \\ , \\end{align*}"}
+{"id": "1535.png", "formula": "\\begin{align*} j ^ { b c } _ a = 0 \\end{align*}"}
+{"id": "7642.png", "formula": "\\begin{align*} - E _ { - + } ( \\lambda ) = \\Gamma _ h \\big ( F _ h ( \\lambda ) + \\mathsf R _ h ( \\lambda ) \\big ) F _ h ( \\lambda ) = \\left ( \\begin{array} { c c } - \\lambda & 0 \\\\ 0 & \\mu _ { 2 , h } ^ \\Delta / \\gamma _ { 2 , h } - \\lambda \\end{array} \\right ) , \\end{align*}"}
+{"id": "4942.png", "formula": "\\begin{align*} \\widetilde { H } ( \\mathbf U , \\mathbf V ) _ m = \\frac { 1 } 2 \\left \\{ ( \\delta _ m ^ + U _ { m } ) ^ 2 + ( \\delta _ m ^ + V _ { m } ) ^ 2 + ( \\delta _ m ^ - U _ { m } ) ^ 2 + ( \\delta _ m ^ - V _ { m } ) ^ 2 \\right \\} - \\frac { 1 } 2 ( U _ { m } ^ 2 + V _ { m } ^ 2 ) ^ 2 . \\end{align*}"}
+{"id": "3395.png", "formula": "\\begin{align*} \\limsup _ { t \\uparrow T ^ \\star } \\int _ { 0 } ^ { t } \\| \\omega ( \\tau ) \\| _ { L ^ \\infty } d \\tau = \\infty . \\end{align*}"}
+{"id": "1123.png", "formula": "\\begin{align*} \\Theta = \\left [ \\begin{smallmatrix} \\alpha _ 0 & \\alpha _ 1 \\\\ \\beta _ 0 & \\beta _ 1 \\\\ \\gamma _ 0 & \\gamma _ 1 \\end{smallmatrix} \\right ] \\end{align*}"}
+{"id": "440.png", "formula": "\\begin{align*} \\| f _ j \\| = \\| \\tau _ j \\| = | f _ j ( \\tau _ j ) | = 1 , \\forall 1 \\leq j \\leq n , \\end{align*}"}
+{"id": "3788.png", "formula": "\\begin{align*} \\alpha _ { 0 } & = \\operatorname { I d } _ V , & \\alpha _ { 1 } ( v ) & = \\begin{cases} f & v = c ; \\\\ e & v = d ; \\\\ d & v = e ; \\\\ c & v = f ; \\\\ v & v \\in \\{ a , b \\} , \\end{cases} & \\alpha _ { 2 } ( v ) & = \\begin{cases} e & v = d ; \\\\ d & v = e ; \\\\ v & v \\neq d , e , \\end{cases} & \\alpha _ { 3 } & = \\alpha _ { 1 } \\alpha _ { 2 } . \\end{align*}"}
+{"id": "1590.png", "formula": "\\begin{align*} \\xi ( u ) = \\eta ( u ) + ( 1 - s ) \\alpha \\delta _ u ( u ) \\qquad \\mbox { a n d } \\qquad \\xi ( v ) = \\eta ( v ) + s \\alpha \\delta _ v ( v ) \\end{align*}"}
+{"id": "1925.png", "formula": "\\begin{align*} - a \\Delta _ \\lambda v _ { \\lambda , \\sigma } + ( R _ \\lambda - \\chi _ \\lambda R _ g ) v _ { \\lambda , \\sigma } & = - ( R _ \\lambda - \\chi _ \\lambda R _ g ) \\quad M _ \\sigma , \\\\ \\nu _ g ( v _ { \\lambda , \\sigma } ) & = 0 \\quad \\partial M _ \\sigma , \\\\ v _ { \\lambda , \\sigma } & \\in W ^ { 2 , p } _ { - q ' } ( M _ \\sigma ) \\end{align*}"}
+{"id": "3489.png", "formula": "\\begin{align*} { \\gamma _ { { \\rm { Z F } } } } = \\bar P \\sum \\nolimits _ { l = 1 } ^ L { { { \\left \\| { { { \\bf { Q } } _ l } { { \\bf { h } } _ l } } \\right \\| } ^ 2 } } , \\end{align*}"}
+{"id": "5963.png", "formula": "\\begin{align*} ( D ^ T ) = V . \\end{align*}"}
+{"id": "3923.png", "formula": "\\begin{align*} X ^ { - 1 } _ { v _ { \\overline { \\mathbf { u } } } } ( x _ i ) \\setminus E _ v \\subset \\bigcup _ { j = 1 } ^ \\infty \\bigcap _ { k = j } ^ \\infty X ^ { - 1 } _ { v _ { \\mathbf { u } ^ k } } ( x _ i ) . \\end{align*}"}
+{"id": "2518.png", "formula": "\\begin{align*} f ( n \\tau ) & \\approx S _ 0 ( \\tau ) ( S _ 0 ( \\tau ) P ) ^ { n - 1 } f _ 0 + S _ 0 ( \\tau ) ( P - 1 ) ( S _ 0 ( \\tau ) P ) ^ { n - 1 } f _ 0 \\\\ & = \\cdots \\cdots \\cdots \\\\ & = S _ 0 ( n \\tau ) f _ 0 + \\sum _ { k = 1 } ^ n S _ 0 ( k \\tau ) ( P - 1 ) ( S _ 0 ( \\tau ) P ) ^ { n - k } f _ 0 \\\\ & \\approx S _ 0 ( n \\tau ) f _ 0 + \\sum _ { k = 1 } ^ n S _ 0 ( k \\tau ) ( P - 1 ) f [ ( n - k ) \\tau ] \\ , . \\end{align*}"}
+{"id": "6018.png", "formula": "\\begin{align*} [ C ' ] & = [ L \\times C ] \\cup [ X ] \\\\ & = ( l _ 1 \\boxtimes \\delta l _ 2 ) \\cup ( h _ 1 + h _ 2 ) \\\\ & = l _ 1 + \\delta l _ 2 , \\end{align*}"}
+{"id": "4244.png", "formula": "\\begin{align*} \\gamma _ 1 = \\gamma _ 2 = : \\gamma \\end{align*}"}
+{"id": "8489.png", "formula": "\\begin{align*} F \\ ; = & g ( g \\varphi - 3 g _ { \\xi } ) \\bigg ( - U _ { \\xi ^ 3 } + 2 \\varphi U _ { \\xi ^ 2 } - 3 W _ { \\xi ^ 2 } + \\varphi W _ { \\xi } + \\varphi _ { \\xi } W \\\\ & \\qquad \\qquad \\qquad + U _ { \\xi } ( 2 \\lambda - \\varphi ^ 2 + 3 \\varphi _ { \\xi } ) + U ( 2 \\lambda _ { \\xi } - 2 \\varphi \\varphi _ { \\xi } + \\varphi _ { \\xi ^ 2 } ) \\bigg ) . \\end{align*}"}
+{"id": "591.png", "formula": "\\begin{align*} \\sigma _ t ^ { ( \\epsilon ) } = \\pi _ t ^ { ( \\epsilon ) } \\left [ ( \\theta - \\pi _ t ^ { ( \\epsilon ) } [ \\theta ] ) ^ 2 \\right ] \\end{align*}"}
+{"id": "5021.png", "formula": "\\begin{align*} H = q \\ , e ^ { 1 2 3 } + p \\ , e ^ { 1 4 5 } . \\end{align*}"}
+{"id": "1418.png", "formula": "\\begin{align*} F = \\begin{cases} \\max _ { \\precsim } & \\\\ \\pi _ 1 \\ , \\ , \\pi _ 2 & \\end{cases} \\end{align*}"}
+{"id": "5234.png", "formula": "\\begin{align*} \\nabla _ U V = T _ U V + \\nu \\nabla _ U V , \\end{align*}"}
+{"id": "8424.png", "formula": "\\begin{align*} \\kappa \\mu = \\kappa \\nu \\iff \\mu = \\nu . \\end{align*}"}
+{"id": "798.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\int _ { \\Sigma _ { B _ j } } 1 _ { F ( t , n ) } \\ d \\mu _ { ( z , m ) } = \\mu _ j ( F ( t , n ) ) . \\end{align*}"}
+{"id": "2144.png", "formula": "\\begin{align*} \\partial _ t \\tilde { w } = J \\ast \\tilde { w } - \\tilde { w } + \\tilde { f } ( \\tilde { w } ) , \\end{align*}"}
+{"id": "2361.png", "formula": "\\begin{align*} A = W ^ T J _ { 2 m } W W \\in \\O ( 2 m ) . \\end{align*}"}
+{"id": "4583.png", "formula": "\\begin{align*} \\frac { 5 } { 6 } \\leq \\frac { 1 } { b _ 1 } + \\frac { 1 } { b _ 2 } \\leq \\frac { 1 } { 3 } + \\frac { 1 } { 3 } = \\frac { 2 } { 3 } \\end{align*}"}
+{"id": "5710.png", "formula": "\\begin{align*} \\mathbf { \\mathbf { e } } ^ { a } = ( \\mathbf { L } ^ { - 1 } ) _ { b } ^ { a } d \\mathbf { x } ^ { b } \\end{align*}"}
+{"id": "795.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\limsup _ { m \\to \\infty } \\frac { 1 } { m } \\# \\left \\{ 0 \\le k \\le m : \\frac { \\log \\| \\rho ( \\xi _ k ^ { - 1 } \\xi _ { k + n } ) \\| - \\Lambda n } { \\sqrt { n } } < t \\right \\} = \\frac { 1 } { \\sigma _ j \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ t e ^ { - \\frac { s ^ 2 } { 2 \\sigma _ j ^ 2 } } d s . \\end{align*}"}
+{"id": "1283.png", "formula": "\\begin{align*} \\zeta _ t ( z ) \\eta _ j ( y ) q _ j ^ { ( t ) } ( x ) \\zeta _ u ( z ) \\eta _ v ( y ) p _ v ^ { ( u ) } ( x ) = 0 \\end{align*}"}
+{"id": "2434.png", "formula": "\\begin{align*} \\theta _ h ^ n = F _ \\tau ^ h ( n ; q ) \\theta _ h ( 0 ) + \\tau \\sum _ { j = 1 } ^ { n } E _ \\tau ^ h ( j ; q ) A _ h ( q ) ( R _ h ( q ) - P _ h ) w ^ { n + 1 - j } ( q ) = : I + I I . \\end{align*}"}
+{"id": "2834.png", "formula": "\\begin{align*} \\prod _ { \\gamma \\in G } \\left ( \\prod _ { i = 1 } ^ n x _ { \\sigma ( i ) } ^ { a _ { \\gamma ^ { - 1 } ( i ) } } \\right ) ^ { t _ { \\gamma } } \\\\ \\leq \\sum _ { \\gamma \\in G } t _ { \\gamma } \\left ( \\prod _ { i = 1 } ^ n x _ { \\sigma ( i ) } ^ { a _ { \\gamma ^ { - 1 } ( i ) } } \\right ) . \\end{align*}"}
+{"id": "6716.png", "formula": "\\begin{align*} E _ 1 ( x ) = \\frac { 1 } { p } \\sum _ { n \\leq x , } \\sum _ { 1 \\leq s \\leq p - 1 } \\mu ( s + a ) e ^ { i 2 \\pi s / q } \\ll \\frac { x } { p } \\frac { x } { ( \\log x ) ^ { D _ 1 } } \\ll \\frac { x } { ( \\log x ) ^ { D _ 1 } } , \\end{align*}"}
+{"id": "1627.png", "formula": "\\begin{align*} u ( t , x ) = e ^ { i t \\Delta ^ 2 } u _ 0 + i \\int _ 0 ^ t e ^ { i ( t - s ) \\Delta ^ 2 } ( | u | ^ 2 u ) ( s ) \\ , d s : = u _ l + u _ { n l } . \\end{align*}"}
+{"id": "8978.png", "formula": "\\begin{align*} | \\nabla ^ { k + 1 } u | ^ 2 \\le C ( | \\nabla ^ k \\partial _ { \\phi } u | ^ 2 + | \\nabla ^ k u | ^ 2 ) \\le C \\sum _ { j = 0 } ^ k | \\nabla \\partial _ { \\phi } ^ j u | ^ 2 \\ \\hbox { i n } B _ { R _ 0 } ( z _ 0 ) \\end{align*}"}
+{"id": "3646.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } w = \\Delta _ { \\mathbb { H } ^ { n } } w + e ^ { \\mu t } w ^ { p + 1 } & \\hbox { i n } ~ \\mathbb { H } ^ { n } \\times ( 0 , + \\infty ) , \\\\ \\\\ w = u _ { 0 } \\in L ^ { \\infty } ( \\mathbb { H } ^ { n } ) & \\hbox { i n } ~ \\mathbb { H } ^ { n } \\times \\{ 0 \\} , \\end{array} \\right . \\end{align*}"}
+{"id": "7272.png", "formula": "\\begin{align*} K ( \\gamma ) : = \\int _ { 0 } ^ { \\gamma } m ( x , \\infty ) ^ 2 d x \\end{align*}"}
+{"id": "3458.png", "formula": "\\begin{align*} F = \\delta ( - D L ^ { - 1 } F ) . \\end{align*}"}
+{"id": "2748.png", "formula": "\\begin{align*} D ( f _ { \\circ } g _ { \\circ } ) = & D ( ( \\varphi \\circ f ) ( \\varphi \\circ g ) ) \\\\ = & D ( \\varphi \\circ f g ) \\\\ = & D _ { \\varphi } ( f g ) \\\\ = & \\varphi \\circ f ( x ) D _ { \\varphi } g + \\varphi \\circ g ( x ) D _ { \\varphi } f \\\\ = & \\varphi \\circ f ( x ) D ( \\varphi \\circ g ) + \\varphi \\circ g ( x ) D ( \\varphi \\circ f ) \\\\ = & f _ { \\circ } ( x ) D g _ { \\circ } + g _ { \\circ } ( x ) D f _ { \\circ } \\end{align*}"}
+{"id": "3182.png", "formula": "\\begin{align*} - A : D ^ 2 v ^ { 3 3 } = - \\left ( a _ 1 - \\int _ { [ 0 , 1 ] ^ 3 } r a _ 1 \\right ) - \\left ( a _ 2 - \\int _ { [ 0 , 1 ] ^ 3 } r a _ 2 \\right ) = a _ 3 - \\int _ { [ 0 , 1 ] ^ 3 } r a _ 3 , \\end{align*}"}
+{"id": "8575.png", "formula": "\\begin{align*} F ( x ) = \\prod _ { r ( x ) \\in \\mathcal { A } } r ( x ) = \\frac { x ^ { 2 ^ n } + x } { x \\ , ( x + 1 ) } . \\end{align*}"}
+{"id": "3360.png", "formula": "\\begin{align*} T M = \\dd \\eta \\oplus ~ \\eta , \\end{align*}"}
+{"id": "4825.png", "formula": "\\begin{align*} d _ 0 ^ n F ( v _ 1 , \\dots , v _ n ) = & \\frac { \\partial ^ n } { \\partial t _ 1 \\dots \\partial t _ n } F \\Big ( \\sum _ { h = 1 } ^ n t _ h v _ h \\Big ) \\bigg | _ { t _ 1 = \\dots = t _ n = 0 } \\\\ = & \\frac { 1 } { n ! } \\frac { \\partial ^ n } { \\partial t _ 1 \\dots \\partial t _ n } d _ 0 ^ n F \\Big ( \\sum _ { h = 1 } ^ n t _ h v _ h \\Big ) \\bigg | _ { t _ 1 = \\dots = t _ n = 0 } . \\end{align*}"}
+{"id": "528.png", "formula": "\\begin{align*} \\| z ^ k - \\widetilde { z } \\| \\le \\sum _ { j = k } ^ { \\infty } \\| z ^ { j + 1 } \\ ! - \\ ! z ^ j \\| \\le \\left \\{ \\begin{array} { c l } \\gamma \\varrho ^ { k } & { \\rm i f } \\ \\theta = 1 / 2 , \\\\ \\gamma k ^ { \\frac { 1 - \\theta } { 1 - 2 \\theta } } & { \\rm i f } \\ \\theta \\in ( 1 / 2 , 1 ) \\end{array} \\right . \\ \\ { \\rm f o r \\ a l l } \\ k \\ge \\overline { k } . \\end{align*}"}
+{"id": "5575.png", "formula": "\\begin{align*} H ( r , ( x , y ) , u ) : = \\int _ { \\partial B _ r ^ n ( x , y ) } u ^ 2 d \\sigma . \\end{align*}"}
+{"id": "263.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\rho } \\leqslant 0 , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\tau } \\leqslant 0 , \\end{align*}"}
+{"id": "1435.png", "formula": "\\begin{align*} \\eta _ t \\geq J _ t ( d - 1 ) - J _ t ( d - 1 ) \\mathbb { 1 } _ { \\{ v ( h _ t ) \\in \\cup _ { k = 1 } ^ { d - 1 } \\mathcal { V } ^ { ( k ) } _ { i - 1 } \\cup \\{ v ( e _ i ) \\} \\} } \\eqqcolon \\eta '' _ t . \\end{align*}"}
+{"id": "6933.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { h _ k ( n ) } { n ^ s } & = \\prod _ p ( 1 + p ^ { - s } + \\cdots + p ^ { - ( k - 1 ) s } ) \\\\ & = \\prod _ p \\frac { 1 - p ^ { - k s } } { 1 - p ^ { - s } } \\\\ & = \\frac { \\zeta ( s ) } { \\zeta ( k s ) } . \\\\ \\end{align*}"}
+{"id": "1009.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z } \\log \\left ( 1 - F ( C | n | ^ \\alpha ) \\right ) = - \\infty \\end{align*}"}
+{"id": "7719.png", "formula": "\\begin{align*} C _ { \\beta } = \\sqrt { 2 } \\cos \\left ( \\frac { \\pi } { 4 } ( \\beta + 1 ) \\right ) . \\end{align*}"}
+{"id": "3025.png", "formula": "\\begin{align*} \\hbox { E i n } ( \\mathbf { h } ) { _ a } ^ b + \\Lambda _ 0 \\delta { _ a } ^ b = \\hbox { E i n } ( \\mathbf { h } ) { _ i } ^ b = 0 \\end{align*}"}
+{"id": "4482.png", "formula": "\\begin{align*} 2 = 1 + \\frac { 1 } { 2 } + \\frac { 1 } { 3 } + \\frac { 1 } { 6 } \\end{align*}"}
+{"id": "3346.png", "formula": "\\begin{align*} 2 \\Phi _ { x x } & = - \\frac { 1 } { \\Phi _ { x q } } [ \\Phi _ { x y } ( \\Phi _ { y q } + \\Phi _ { y p } ) + \\Phi _ { x q } ( \\Phi _ { p q } + \\Phi _ { q p } ) ] , \\\\ 2 \\Phi _ { q q } & = \\frac { 1 } { \\Phi _ { x q } } [ \\Phi _ { x y } ( - \\Phi _ { y q } + \\Phi _ { y p } ) + \\Phi _ { x q } ( - \\Phi _ { p q } + \\Phi _ { q p } ) ] . \\end{align*}"}
+{"id": "3464.png", "formula": "\\begin{align*} g _ { k } ( t _ 1 , x _ 1 , \\ldots , t _ { k } , x _ { k } , r , z , t , x ) = G _ { t - t _ { k } } ( x - x _ { k } ) \\ldots G _ { t _ 1 - r } ( x _ 1 - z ) . \\end{align*}"}
+{"id": "3486.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( { { \\rm { P 1 } } } \\right ) \\ \\ \\mathop { \\max } \\limits _ { \\left \\{ { { { \\bf { f } } _ l } } \\right \\} _ { l = 1 } ^ L } & \\ \\ { \\Big | { \\sum \\nolimits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ l } } } \\Big | ^ 2 } \\\\ { \\rm { s . t . } } & \\ \\ { \\bf { h } } _ l ^ H { { \\bf { f } } _ { l ' } } = 0 , \\ \\forall l \\ne l ' , \\ \\ \\sum \\nolimits _ { l = 1 } ^ L { { { \\left \\| { { { \\bf { f } } _ l } } \\right \\| } ^ 2 } } \\le P . \\end{aligned} \\end{align*}"}
+{"id": "7411.png", "formula": "\\begin{align*} \\mu ( s \\widetilde { \\alpha } , \\sigma ) = \\begin{cases} \\gamma ( G / P ) ^ 2 q _ F ^ { n ( \\sigma ) + n ( \\sigma \\otimes \\omega ) } \\frac { ( 1 - \\omega ( \\varpi ) q _ F ^ { - 2 s } ) ( 1 - \\omega ( \\varpi ) ^ { - 1 } q _ F ^ { 2 s } ) } { ( 1 - \\omega ( \\varpi ) q ^ { - 1 - 2 s } ) ( 1 - \\omega ( \\varpi ) ^ { - 1 } q _ F ^ { - 1 + 2 s } ) } & \\\\ \\gamma ( G / P ) ^ 2 q _ F ^ { n ( \\sigma ) + n ( \\omega ) + n ( \\sigma \\otimes \\omega ) } & \\end{cases} \\end{align*}"}
+{"id": "3215.png", "formula": "\\begin{align*} Q \\cap \\psi _ { n } ^ { - 1 } \\circ \\psi \\left ( A _ { i } \\right ) = \\psi _ { n } ^ { - 1 } \\left ( \\psi _ { n } \\left ( Q \\right ) \\cap \\psi \\left ( A _ { i } \\right ) \\right ) \\underset { n \\to \\infty } { \\longrightarrow } Q \\cap \\psi \\left ( A \\right ) . \\end{align*}"}
+{"id": "5516.png", "formula": "\\begin{align*} P _ 2 ( x ) : = \\sum _ { n = 1 } ^ \\infty \\frac { \\mu ( n ) } { n ^ 2 } \\exp \\left ( - \\frac { x } { n ^ 2 } \\right ) = O _ { \\epsilon } \\left ( x ^ { - \\frac { 3 } { 4 } + \\epsilon } \\right ) , { \\rm a s } \\ , \\ , x \\rightarrow \\infty , \\end{align*}"}
+{"id": "221.png", "formula": "\\begin{align*} [ N ^ { \\vee } : N ] \\leq [ M ^ { \\vee } : M ] [ M : ( N + N _ { M ^ { \\vee } } ^ { \\perp } ) \\cap M ] = d [ M : ( N + N ^ { \\perp } ) ] \\end{align*}"}
+{"id": "6853.png", "formula": "\\begin{align*} \\begin{aligned} S _ { 1 , 2 , \\dots , N + 2 } ^ { [ 0 ] } ( i + 1 , i ) & = S _ { 1 , 2 , \\dots , N + 2 } ^ { [ 0 ] \\ ; T } ( i , i + 1 ) = \\mathcal B _ { i - 1 } , \\end{aligned} \\end{align*}"}
+{"id": "5240.png", "formula": "\\begin{align*} ( \\nabla F _ \\ast ) ( X , Y ) = { \\nabla } _ X ^ F F _ \\ast Y - F _ \\ast ( { \\nabla } _ X ^ M Y ) , ~ \\forall X , Y \\in \\Gamma ( T M ) , \\end{align*}"}
+{"id": "3851.png", "formula": "\\begin{align*} A _ { n } = \\delta _ { n , 0 } + A _ { n - 1 } + 3 A _ { n - 2 } + 9 A _ { n - 3 } + \\sum _ { l = 4 } ^ n \\mu _ l A _ { n - l } , \\end{align*}"}
+{"id": "6338.png", "formula": "\\begin{align*} \\Phi _ { \\nu _ 0 } ( u , v ) : = \\varphi ( \\nu _ 0 ) + \\int _ 1 ^ { u } f ( x , \\nu _ 0 ) d x - \\int ^ 1 _ { v } f ( x , \\nu _ 0 ) d x \\end{align*}"}
+{"id": "5961.png", "formula": "\\begin{align*} B e _ r = \\sum _ { s = 1 } ^ p \\beta _ { s r } e _ s , r = 1 , \\cdots , p , \\end{align*}"}
+{"id": "5586.png", "formula": "\\begin{align*} 0 = - ( \\rho ^ l { } _ { j } \\rho _ { i k } + \\rho ^ l { } _ { k } \\rho _ { i j } ) v _ l - \\rho ^ l { } _ { k } \\rho _ { l j } v _ i . \\end{align*}"}
+{"id": "2208.png", "formula": "\\begin{align*} U { \\left ( \\sum _ { n = n _ 0 } ^ \\infty a ( n ) q ^ n \\right ) } = \\sum _ { n = n _ 0 } ^ \\infty a ( 5 n ) q ^ n . \\end{align*}"}
+{"id": "6802.png", "formula": "\\begin{align*} L = \\left < \\alpha ( q ) , \\dot { q } \\right > - H ( q ) , \\end{align*}"}
+{"id": "3014.png", "formula": "\\begin{align*} \\hat { \\theta } ^ { ( N - \\alpha ) } _ { I _ 0 \\cdots I _ \\alpha } = \\frac { 1 } { ( N - \\alpha ) ! } \\epsilon _ { I _ 0 \\cdots I _ \\alpha J _ { \\alpha + 1 } \\cdots J _ N } \\theta ^ { J _ { \\alpha + 1 } } \\wedge \\cdots \\wedge \\theta ^ { J _ N } \\end{align*}"}
+{"id": "7925.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { P _ r H _ 1 ( x ) } { r } = \\lim _ { r \\to 0 } \\frac { P _ r H _ 2 ( y ) } { r } = 0 \\ , . \\end{align*}"}
+{"id": "912.png", "formula": "\\begin{align*} b _ n = ( 1 - \\frac { 1 } { 2 ! } ) ^ { ( \\frac { 1 } { 2 ! } - \\frac { 1 } { 3 ! } ) ^ { \\ldots ^ { ( \\frac { 1 } { n ! } - \\frac { 1 } { ( n + 1 ) ! } ) } } } \\end{align*}"}
+{"id": "5789.png", "formula": "\\begin{align*} e _ r = \\sum _ { i = n _ { r - 1 } + 1 } ^ { n _ r } \\epsilon _ i , 1 \\leqslant r \\leqslant p , \\end{align*}"}
+{"id": "300.png", "formula": "\\begin{align*} \\# \\{ \\Gamma _ F \\} = n - 1 ( \\mathrm { m o d } ~ 2 ) . \\end{align*}"}
+{"id": "7012.png", "formula": "\\begin{align*} R _ c ( R _ p ) & \\geq \\ell \\left ( \\frac { \\Delta e ^ { R _ p } } { 2 \\Delta e ^ { R _ p } - 1 + \\rho } \\right ) \\\\ & = \\frac { 1 } { 2 } \\log ^ + \\frac { N ^ 2 ( X , Y ) } { ( 1 - \\rho ) \\left ( 2 \\Delta e ^ { R _ p } - 1 + \\rho \\right ) } , \\end{align*}"}
+{"id": "386.png", "formula": "\\begin{align*} J ^ { ( 1 , 1 ) } _ 2 & \\leq 3 2 C ^ 2 ( \\gamma ) \\max \\{ 1 , C ^ 2 _ { \\textsf { K M T } } \\} T _ * L ^ { - 2 } _ * T ^ { 2 \\gamma - 1 } _ * + 2 5 2 \\sqrt { C ^ { ( 2 ) } _ { \\textsf { K M T } } } T _ * L ^ { - 2 } _ * T ^ { - 2 } _ * \\ , . \\end{align*}"}
+{"id": "363.png", "formula": "\\begin{align*} \\sigma ^ 2 : = L ^ { - 2 } \\frac { v _ 0 ^ 2 } { \\lambda } \\ , . \\end{align*}"}
+{"id": "24.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\mathbf { D } _ { 2 3 } = \\mathbb { E } \\int _ 0 ^ \\infty e ^ { - \\beta t } \\bar { \\mathcal { Z } } \\bar { f } _ z z _ 1 d t . \\end{align*}"}
+{"id": "9185.png", "formula": "\\begin{align*} F ( X , \\varphi + t ) = \\sum _ { q \\in \\Z } e ^ { i \\sqrt { \\beta } q t } \\hat { F } _ q ( X , \\varphi ) , t \\in \\R , \\end{align*}"}
+{"id": "2370.png", "formula": "\\begin{align*} \\begin{bmatrix} M & 0 \\\\ 0 & 0 \\end{bmatrix} \\begin{bmatrix} \\dot { v } \\\\ \\dot { p } \\end{bmatrix} = \\begin{bmatrix} A _ S ( t ) - A _ H ( t ) & - B \\\\ B ^ T & - C \\end{bmatrix} \\begin{bmatrix} v \\\\ p \\end{bmatrix} + \\begin{bmatrix} f ( t ) \\\\ 0 \\end{bmatrix} , \\end{align*}"}
+{"id": "7744.png", "formula": "\\begin{align*} f _ { \\tfrac { 1 } { 2 } } ( x ) & = \\frac { 1 } { 2 \\sqrt { \\pi } } \\ , x ^ { - 3 / 2 } \\ , e ^ { - 1 / 4 x } \\end{align*}"}
+{"id": "7271.png", "formula": "\\begin{align*} g _ m ( \\lambda ; x ) = \\varphi ^ d _ m ( \\lambda ; x ) - c ^ d _ m ( \\lambda ) \\psi _ m ( \\lambda ; x ) , \\end{align*}"}
+{"id": "6768.png", "formula": "\\begin{align*} L ( s , \\chi ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\chi ( n ) } { n ^ s } ; \\mathfrak { R e } ( s ) > 1 \\end{align*}"}
+{"id": "4961.png", "formula": "\\begin{align*} v _ { i + 1 } = c _ i v _ i - v _ { i - 1 } , \\end{align*}"}
+{"id": "1177.png", "formula": "\\begin{align*} \\max _ { i = 1 , \\ldots , p } \\sqrt { n / p } \\ , | S _ { i i } - 1 | \\le \\max _ { i = 1 , \\ldots , p } \\sqrt { n / p } \\ , | S _ i ( 1 ) - 1 | + \\max _ { i = 1 , \\ldots , p } \\sqrt { n / p } \\ , | S _ i ( 2 ) | \\ , . \\end{align*}"}
+{"id": "7811.png", "formula": "\\begin{align*} x ^ { \\tilde m } x ^ { \\tilde m ' } = q ^ { - \\{ \\tilde m , \\tilde m ' \\} _ 1 } x ^ { \\tilde m + \\tilde m ' } . \\end{align*}"}
+{"id": "6943.png", "formula": "\\begin{align*} L _ { m , n } ^ { I I I , ( \\alpha ) } ( x ) = x L _ { n - m - 2 } ^ { ( \\alpha + 2 ) } ( x ) L _ { m } ^ { ( - \\alpha - 1 ) } ( - x ) + ( m + 1 ) L _ { m + 1 } ^ { ( - \\alpha - 2 ) } ( - x ) L _ { n - m - 1 } ^ { ( \\alpha + 1 ) } ( x ) \\end{align*}"}
+{"id": "2776.png", "formula": "\\begin{align*} & ( u _ 1 , u _ 2 , \\cdots \\ ! , u _ n ) \\\\ & = ( 0 , 0 , - 1 , - 1 , - 1 , - 1 , 0 , 0 , 0 , 0 , 1 , 0 , 1 , 1 , 0 , 0 ) . \\end{align*}"}
+{"id": "3268.png", "formula": "\\begin{align*} f \\left ( x \\right ) = \\lambda g \\left ( x \\right ) , x \\in M \\subseteq X , \\end{align*}"}
+{"id": "2638.png", "formula": "\\begin{align*} h _ { a ^ * \\varpi } ( r ^ * \\varpi ^ { 4 - j } , s ; \\psi _ { \\varpi ^ { j + 1 } } \\chi ) & = \\sum _ { ( n , a ^ * \\varpi ) = 1 } \\frac { \\psi _ { \\varpi ^ { j + 1 } } ( n ) \\chi ( n ) g _ 6 ( a \\varpi ^ { 4 - j } , n ) } { N ( n ) ^ s } \\\\ & = \\sum _ { ( n , a ^ * ) = 1 } \\frac { \\psi _ { \\varpi ^ { j + 1 } } ( n ) \\chi ( n ) g _ 6 ( a \\varpi ^ { 4 - j } , n ) } { N ( n ) ^ s } - \\sum _ { \\substack { ( n , a ^ * ) = 1 \\\\ \\varpi | n } } \\frac { \\psi _ { \\varpi ^ { j + 1 } } ( n ) \\chi ( n ) g _ 6 ( a \\varpi ^ { 4 - j } , n ) } { N ( n ) ^ s } . \\end{align*}"}
+{"id": "7426.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ K b _ i ^ * K ( Z _ i , Z _ j ) ( a _ i ^ * a _ j ) b _ j \\succeq 0 . \\end{align*}"}
+{"id": "2207.png", "formula": "\\begin{align*} D & = 1 - \\big ( 2 0 5 y + 4 3 0 0 y ^ 2 + 3 4 0 0 0 y ^ 3 + 1 2 0 0 0 0 y ^ 4 + 1 6 0 0 0 0 y ^ 5 \\big ) x \\\\ & \\quad - \\big ( 2 1 5 y + 4 4 7 5 y ^ 2 + 3 5 0 0 0 y ^ 3 + 1 2 2 0 0 0 y ^ 4 + 1 6 0 0 0 0 y ^ 5 \\big ) x ^ 2 \\\\ & \\quad - \\big ( 8 5 y + 1 7 5 0 y ^ 2 + 1 3 5 2 5 y ^ 3 + 4 6 5 0 0 y ^ 4 + 6 0 0 0 0 y ^ 5 \\big ) x ^ 3 \\\\ & \\quad - \\big ( 1 5 y + 3 0 5 y ^ 2 + 2 3 2 5 y ^ 3 + 7 8 7 5 y ^ 4 + 1 0 0 0 0 y ^ 5 \\big ) x ^ 4 \\\\ & \\quad - \\big ( y + 2 0 y ^ 2 + 1 5 0 y ^ 3 + 5 0 0 y ^ 4 + 6 2 5 y ^ 5 \\big ) x ^ 5 . \\end{align*}"}
+{"id": "1599.png", "formula": "\\begin{align*} \\gamma : [ 0 , c ] \\to \\mathcal { W } _ p ( X ) ; \\qquad \\gamma ( t ) : = \\mu ^ * + t \\delta _ { x _ { n + 1 } } - t \\delta _ u . \\end{align*}"}
+{"id": "7300.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } m _ n ( x ) = 0 \\ \\ x > 0 . \\\\ & \\lim _ { n \\to \\infty } \\int _ { 0 } ^ { 1 } y d m _ n ( y ) = 0 . \\end{align*}"}
+{"id": "4462.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k v _ i \\leq \\prod _ { i = 1 } ^ k u _ i = \\prod _ { i = 1 } ^ k u ' _ i . \\end{align*}"}
+{"id": "5380.png", "formula": "\\begin{align*} \\mathbb { E } \\bigg [ \\exp \\bigg ( \\sum _ { j = 1 } ^ m 2 \\alpha _ j \\sum _ { x < k \\le y } \\frac { \\Re X ( k ) e ^ { i k t _ j } } { \\sqrt { k } e ^ { i k \\sigma } } \\bigg ) \\bigg ] = \\exp \\bigg ( \\sum _ { j _ 1 , j _ 2 = 1 } ^ m \\sum _ { x < k \\le y } \\alpha _ { j _ 1 } \\alpha _ { j _ 2 } \\frac { \\cos \\big ( k ( t _ { j _ 2 } - t _ { j _ 1 } ) \\big ) } { k e ^ { 2 i k \\sigma } } \\bigg ) . \\end{align*}"}
+{"id": "4211.png", "formula": "\\begin{align*} \\partial _ x Q _ 0 ( \\phi , \\tilde { \\Lambda } ) = Q _ 0 ( \\partial _ x \\phi , \\tilde { \\Lambda } ) + Q _ 0 ( \\phi , \\partial _ x \\tilde { \\Lambda } ) . \\end{align*}"}
+{"id": "6699.png", "formula": "\\begin{align*} \\mu _ { * } ( n ) = ( - 1 ) ^ { \\omega ( n ) } \\lambda ( n ) = ( - 1 ) ^ { \\Omega ( n ) } \\end{align*}"}
+{"id": "4124.png", "formula": "\\begin{align*} \\mathcal { V } _ 1 = \\begin{cases} \\{ ( u g , K ) : u \\in C ^ \\infty ( M ) , \\ , K \\in \\Omega ^ 2 \\} & n \\geq 4 \\\\ \\{ ( u g , - d ^ * ( \\omega d V _ g ) ) : u , \\omega \\in C ^ \\infty ( M ) \\} & n = 3 . \\end{cases} \\end{align*}"}
+{"id": "5417.png", "formula": "\\begin{align*} w ^ { ( m ) } = \\left ( \\prod _ { s = 1 } ^ { q - 1 } \\prod _ { k = 1 } ^ { \\alpha _ s } \\ell _ 0 ^ { ( i _ s ) } \\cdots \\ell _ { g - 1 } ^ { ( i _ s ) } ( \\lceil \\beta _ { i _ { s , k } + g } \\rceil - 1 ) \\cdots ( \\lceil \\beta _ { i _ { s , k + 1 } + p ( 2 N + 1 ) - 1 } \\rceil - 1 ) \\right ) \\ell _ 0 ^ { ( j ) } \\cdots \\ell _ { m - 1 } ^ { ( j ) } . \\end{align*}"}
+{"id": "7307.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\varphi ^ d _ { m _ n } ( \\lambda ; x ) = 1 , \\ \\lim _ { n \\to \\infty } \\psi _ { m _ n } ( \\lambda ; x ) = x , \\ \\lim _ { n \\to \\infty } g _ { m _ n } ( \\lambda ; x ) = 1 \\end{align*}"}
+{"id": "4561.png", "formula": "\\begin{align*} \\frac { 4 } { 1 7 } = \\frac { 1 } { 5 } + \\frac { 1 } { 3 0 } + \\frac { 1 } { 5 1 0 } \\end{align*}"}
+{"id": "2397.png", "formula": "\\begin{align*} E _ { 3 3 } ( t ) \\dot x _ 3 = A _ { 3 3 } ( t ) x _ 3 + f _ 3 ( t ) , \\end{align*}"}
+{"id": "6669.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { E } \\left [ \\| \\bar x ^ { k + 1 } - \\theta ^ * \\| ^ 2 | \\mathcal { F } ^ k \\right ] \\le ( 1 + a ^ k ) \\| \\bar x ^ k - \\theta ^ * \\| ^ 2 \\cr & + \\left ( \\frac { \\gamma ^ k } { m } + a ^ k \\right ) \\sum _ { i = 1 } ^ m \\| x _ i ^ k - \\bar x ^ k \\| ^ 2 + b ^ k - c ^ k ( F ( \\bar x ^ k ) - F ( \\theta ^ * ) ) \\end{aligned} \\end{align*}"}
+{"id": "4538.png", "formula": "\\begin{align*} \\frac { 1 } { 9 } < \\frac { 9 } { 2 0 } - \\frac { 1 } { 3 } < \\theta - \\frac { 1 } { 3 } \\leq \\frac { 1 1 } { 2 4 } - \\frac { 1 } { 3 } = \\frac { 1 } { 8 } \\end{align*}"}
+{"id": "1672.png", "formula": "\\begin{align*} \\chi ( t _ \\infty ) = t _ \\infty ^ { \\underline { m } } , \\underline { m } \\in \\Z ^ { \\Sigma _ T } , \\frac { 2 - \\underline { k } } { 2 } \\leq \\underline { m } \\leq \\frac { \\underline { k } - 2 } { 2 } . \\end{align*}"}
+{"id": "316.png", "formula": "\\begin{align*} \\epsilon _ { \\mathrm { K a l } } | _ { E _ { 1 } ^ { \\times } } \\cdot \\epsilon _ { \\mathrm { H M } } \\cdot \\omega _ { \\mathrm { P r a } } | _ { E _ { 1 } ^ { \\times } } = \\zeta _ { K ^ { \\times } } | _ { E _ { 1 } ^ { \\times } } = \\omega _ { K / E _ 1 } . \\end{align*}"}
+{"id": "4305.png", "formula": "\\begin{align*} E ( t , \\xi ) = \\left [ | \\partial _ { t } w ( t , \\xi ) | ^ 2 + c _ * ( t , \\xi ) | \\xi | ^ 2 | w ( t , \\xi ) | ^ 2 \\right ] | \\xi | ^ { 4 \\sigma } k ( t , \\xi ) , \\end{align*}"}
+{"id": "3588.png", "formula": "\\begin{align*} E ^ { \\gamma _ { A ( 1 ) } } \\Omega _ \\lambda & = - 2 v _ { A ( 1 ) } + 3 v _ { A ( 2 ) } - v _ { A ( 3 ) } \\\\ E ^ { \\gamma _ { A ( 2 ) } } \\Omega _ \\lambda & = - 3 v _ { A ( 2 ) } + 3 v _ { A ( 3 ) } \\\\ E ^ { \\gamma _ { A ( 3 ) } } \\Omega _ \\lambda & = - 1 2 v _ { A ( 3 ) } . \\end{align*}"}
+{"id": "8002.png", "formula": "\\begin{align*} \\phi ( z a , 1 ) = \\phi ( z , a ) \\end{align*}"}
+{"id": "780.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\sigma ( X ( x _ k ) , X ( x _ { k - 1 } ) \\ldots X ( x _ 0 ) c ) = \\sum _ { k = 1 } ^ n \\phi _ k ( ( x _ i ) ) + \\sum _ { k = 0 } ^ { n - 1 } \\xi ( y _ k ) , \\end{align*}"}
+{"id": "3406.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } v ( t , x ) = v ^ r ( t , r , z ) \\vec { e } _ { r } + v ^ z ( t , r , z ) \\vec { e } _ { z } \\\\ B ( t , x ) = B ^ \\theta ( t , r , z ) \\vec { e } _ { \\theta } , \\end{array} \\right . \\end{align*}"}
+{"id": "4138.png", "formula": "\\begin{align*} \\langle \\mathring { R } h , h \\rangle = R _ { i j k l } h _ { i l } h _ { j k } = - \\frac { 1 } { 4 } H _ { j l b } H _ { i k b } h _ { i l } h _ { j k } = \\frac { 1 } { 4 } H _ { l j b } H _ { i k b } h _ { i l } h _ { j k } . \\end{align*}"}
+{"id": "6254.png", "formula": "\\begin{align*} \\tau \\big | _ { \\bar { 0 } _ { \\mathcal { S } } } = \\iota _ { \\mathcal { S } } . \\end{align*}"}
+{"id": "8837.png", "formula": "\\begin{align*} \\Delta _ { \\nu } : = 4 ( 1 + \\langle z , z \\rangle ) \\left ( \\sum _ { i , j = 1 } ^ { n } ( \\delta _ { i j } + z _ { i } \\bar { z _ { j } } ) \\frac { \\partial ^ { 2 } } { \\partial z _ { i } \\partial \\bar { z _ { j } } } - \\nu \\sum _ { j = 1 } ^ { n } \\left ( z _ { j } \\frac { \\partial } { \\partial z _ { j } } - \\bar { z _ { j } } \\frac { \\partial } { \\partial \\bar { z _ { j } } } \\right ) - \\nu ^ { 2 } \\right ) + 4 \\nu ^ { 2 } \\end{align*}"}
+{"id": "5979.png", "formula": "\\begin{align*} \\mathrm { d i m } ( h \\times f ) ( \\mathcal { W } _ 0 ) & = \\mathrm { d i m } \\mathcal { W } _ 0 - \\mathrm { d i m } M + \\mathrm { d i m } ( h \\times f ) ( M ) \\\\ & = \\mathrm { d i m } \\mathcal { W } _ g - \\mathrm { d i m } M + \\mathrm { d i m } ( h \\times f ) ( M ) , \\end{align*}"}
+{"id": "2750.png", "formula": "\\begin{align*} \\overline { \\varphi _ { R } ^ { \\ast } } = \\varphi _ { L } ^ { \\ast } . \\end{align*}"}
+{"id": "3433.png", "formula": "\\begin{align*} p _ { i i } ^ { t } + p _ { i j } ^ { t } p _ { j i } ^ { t } \\sum _ { n \\geq 1 } ( p _ { j j } ^ { t } ) ^ n = 1 . \\end{align*}"}
+{"id": "2800.png", "formula": "\\begin{align*} b _ i = a _ i \\qquad . \\end{align*}"}
+{"id": "6592.png", "formula": "\\begin{align*} \\xi ( k , i ) = k + ( i - 1 ) n \\end{align*}"}
+{"id": "4198.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\int _ { v t - \\lambda ( t ) } ^ { v t + \\lambda ( t ) } \\left ( ( \\partial _ { t } \\Lambda ) ^ { 2 } + ( \\partial _ { x } \\Lambda ) ^ { 2 } + \\sinh ^ { 2 } { \\Lambda } ( ( \\partial _ { t } \\phi ) ^ { 2 } + ( \\partial _ { x } \\phi ) ^ { 2 } ) \\right ) ( t , x ) d x = 0 . \\end{align*}"}
+{"id": "6266.png", "formula": "\\begin{align*} Z _ { 0 } ( x , y , z ) = x \\cdot \\left ( x \\frac { \\partial } { \\partial z } - z \\frac { \\partial } { \\partial x } \\right ) \\Big | _ { \\mathcal { M } _ { 0 } } . \\end{align*}"}
+{"id": "3133.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( \\tilde { A } ) = s ( 1 + \\bar { a } ) \\int _ Y r ( 1 + a ) \\partial _ { 1 1 } ^ 2 w = - s ( 1 + \\bar { a } ) \\int _ Y ( \\partial _ 1 [ r ( 1 + a ) ] ) ( \\partial _ 1 w ) . \\end{align*}"}
+{"id": "5327.png", "formula": "\\begin{align*} \\mathrm { c u r l } E - { \\rm i } \\ , \\omega \\mu H = 0 , \\mathrm { c u r l } H + { \\rm i } \\ , \\omega \\varepsilon E = 0 \\mbox { i n } \\Omega . \\end{align*}"}
+{"id": "7383.png", "formula": "\\begin{align*} \\chi _ s ( m ) : = | \\det ( m ) | _ F ^ s \\end{align*}"}
+{"id": "1143.png", "formula": "\\begin{align*} \\tilde { T } _ { m n } & = \\tilde { T } _ m \\circ \\tilde { T } _ n , ( m , n ) = 1 \\ \\mbox { \\ \\ \\ a n d } \\\\ \\tilde { T } _ { p ^ { r + 2 } } ( x ) & = \\tilde { T } _ p \\left ( \\tilde { T } _ { p ^ { r + 1 } } ( x ) \\right ) - \\frac { 1 } { p } \\Psi ^ p \\tilde { T } _ { p ^ r } ( x ) p \\in P , \\ r \\geq 0 , \\ x \\in E ^ * ( X ) . \\end{align*}"}
+{"id": "1197.png", "formula": "\\begin{align*} \\Psi ( r , x ) = \\Psi ^ { \\log r } ( 1 , x ) . \\end{align*}"}
+{"id": "7212.png", "formula": "\\begin{align*} \\big \\langle ( p - p _ 0 ) \\omega , \\ , ( p - p _ 0 ) ' \\big \\rangle \\ , = \\ , \\rho ^ 2 \\theta ' \\end{align*}"}
+{"id": "6642.png", "formula": "\\begin{align*} f _ r : = f _ { r , 0 } : = f * \\psi _ r \\end{align*}"}
+{"id": "8421.png", "formula": "\\begin{align*} w ( A ) : = \\inf _ { \\mu \\in \\breve { \\mathcal E } ^ + _ A } \\ , \\kappa ( \\mu , \\mu ) \\in [ 0 , \\infty ] . \\end{align*}"}
+{"id": "9153.png", "formula": "\\begin{align*} \\varphi _ \\kappa \\ , \\le \\varphi _ \\beta ^ p \\ , \\varphi _ \\alpha ^ { 1 - p } = \\varphi _ \\beta ^ { \\frac { \\alpha - \\kappa } { \\alpha - \\beta } } \\ , \\varphi _ \\alpha ^ { \\frac { \\kappa - \\beta } { \\alpha - \\beta } } \\qquad \\varphi _ \\beta \\ge \\varphi _ \\kappa ^ { \\frac { 1 } { p } } \\varphi _ \\alpha ^ { 1 - \\frac { 1 } { p } } = \\varphi _ \\kappa ^ { 1 + \\frac { \\kappa - \\beta } { \\alpha - \\kappa } } \\varphi _ \\alpha ^ { - \\frac { \\kappa - \\beta } { \\alpha - \\kappa } } . \\end{align*}"}
+{"id": "6101.png", "formula": "\\begin{align*} E _ 2 ^ * ( z ) = 1 - 2 4 \\sum _ { n \\geq 1 } \\sigma _ { 1 } ( n ) e ^ { 2 \\pi i n z } - \\frac { 3 } { \\pi y } . \\end{align*}"}
+{"id": "8901.png", "formula": "\\begin{align*} \\mathcal { R } _ { p } ^ { \\left ( \\nu \\right ) } \\left ( t \\right ) = \\sum \\limits _ { \\mu = \\nu + \\frac { 1 } { 2 } } ^ { + \\infty } \\mu ^ { 2 p + 1 } e ^ { - \\mu ^ { 2 } t } \\end{align*}"}
+{"id": "8054.png", "formula": "\\begin{align*} H ^ + ( x ) = 2 i \\int _ { - \\infty } ^ \\infty J _ { 2 i t } ( x ) \\frac { h ( t ) t } { \\cosh ( \\pi t ) } \\ , d t , \\end{align*}"}
+{"id": "2460.png", "formula": "\\begin{align*} \\Big | \\sum _ { \\substack { L ( \\rho , \\pi ) = 0 \\\\ | s - \\rho | \\leq 2 0 0 \\eta } } \\frac { 1 } { ( s - \\rho ) ^ { k + 1 } } + \\sum _ { \\substack { L ( \\rho ' , \\pi \\otimes \\chi ) = 0 \\\\ | s - \\rho ' | \\leq 2 0 0 \\eta } } \\frac { 1 } { ( s + 1 - \\beta _ { \\chi } - \\rho ) ^ { k + 1 } } \\Big | \\geq \\Big ( \\frac { 1 } { 5 0 | s - \\rho _ 0 | } \\Big ) ^ { k + 1 } \\geq \\frac { 1 } { ( 1 0 0 \\eta ) ^ { k + 1 } } . \\end{align*}"}
+{"id": "923.png", "formula": "\\begin{align*} H ( d u ( x ) ) - c = 0 , \\end{align*}"}
+{"id": "53.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m _ 1 } T ( n _ i ) & < \\left ( \\sum _ { i = 1 } ^ { m _ 1 } k _ i \\right ) \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } + 2 \\cdot 3 \\cdot \\frac { 1 } { X _ 0 } \\\\ & = \\left ( \\sum _ { i = 1 } ^ { m _ 1 } k _ i + 8 \\right ) \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } . \\end{align*}"}
+{"id": "7567.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 6 ) & = \\sum _ { a = 0 } ^ { \\omega - 1 } { Z _ f ( s , \\chi , A _ 6 ^ a ) } \\\\ & = \\sum _ { a = 0 } ^ { \\omega - 1 } q ^ { - 1 - a } Z _ { f _ { 6 , a } } ( s , \\chi , B _ 6 ^ a ) , \\end{align*}"}
+{"id": "6804.png", "formula": "\\begin{align*} L _ d ( q _ { m } , q _ { m + 1 } ) & = \\frac { h } { 2 } [ L ( q _ { m } , \\frac { q _ { m + 1 } - q _ { m } } { h } ) + L ( q _ { m + 1 } , \\frac { q _ { m + 1 } - q _ { m } } { h } ) ] \\end{align*}"}
+{"id": "1359.png", "formula": "\\begin{align*} \\mathcal { M } ( \\{ f _ j \\} _ { j = 1 } ^ n , \\{ \\tau _ j \\} _ { j = 1 } ^ n ) \\geq \\sqrt { \\frac { n - d } { d ( n - 1 ) } } \\eqqcolon \\gamma . \\end{align*}"}
+{"id": "1915.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k e ^ { t _ j } = \\sum _ { t = 1 } ^ { M - 2 T } b _ t e ^ { T + t } \\leq \\sum _ { t = 1 } ^ { M - 2 T } T ( M - 2 T - t + 1 ) e ^ { T + t } = \\sum _ { s = 1 } ^ { M - 2 T } T s e ^ { M - T + 1 - s } < T e ^ { M - T + 1 } \\end{align*}"}
+{"id": "5819.png", "formula": "\\begin{align*} E ( t ) = \\| u ( t ) \\| _ { V ^ p } ^ 2 + ( B u ( t ) , u ( t ) ) _ { H ^ p } + \\| u ' ( t ) \\| ^ 2 _ { H ^ p } . \\end{align*}"}
+{"id": "1966.png", "formula": "\\begin{align*} r ' _ i = r _ i + \\langle r _ i , y \\rangle x - \\langle r _ i , x \\rangle y . \\end{align*}"}
+{"id": "6870.png", "formula": "\\begin{gather*} Q : X \\to Y : = A ( \\Omega ) \\times B ( \\partial \\Omega ) \\\\ u \\mapsto \\Big ( \\mathcal { L } u , \\mathcal { B } u \\Big ) , \\end{gather*}"}
+{"id": "6534.png", "formula": "\\begin{align*} & \\int _ A \\phi _ m ( w ) \\int \\phi _ m ( x ) \\ , Q ( x , w ) d P _ 0 ^ m ( x ) d P _ 0 ^ m ( w ) \\\\ & \\qquad < \\bar { G } \\int _ A \\phi _ m ( w ) \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( x ^ j \\right ) \\ , Q ( x , w ) d P _ 0 ^ m ( x ) d P _ 0 ^ m ( w ) . \\end{align*}"}
+{"id": "2769.png", "formula": "\\begin{align*} \\Lambda \\triangleright X ^ { A } = q ^ { 4 } X ^ { A } , \\qquad \\Lambda \\triangleright \\partial ^ { A } = q ^ { - 4 } \\partial ^ { A } . \\end{align*}"}
+{"id": "853.png", "formula": "\\begin{align*} A t t i t u d e \\ ; D y n a m i c s \\ ; \\begin{cases} \\widehat { h \\Pi } = F _ k J _ d - J _ d F _ k ^ T \\\\ R _ { k + 1 } = R _ k F _ k \\\\ \\Pi _ { k + 1 } = F _ k ^ T \\Pi _ k + h u _ k \\end{cases} \\end{align*}"}
+{"id": "6554.png", "formula": "\\begin{align*} { F _ { T _ w } } \\left ( t \\right ) & = \\Pr \\left ( { { T _ w } < t } \\right ) = \\Pr \\left ( { { \\kappa ^ 2 } + { C _ { 2 w } } Z _ w < t } \\right ) \\\\ & = \\Pr \\left ( { Z _ w < \\frac { { t - { \\kappa ^ 2 } } } { { { C _ { 2 w } } } } } \\right ) = { F _ { Z _ w } } \\left ( { \\frac { { t - { \\kappa ^ 2 } } } { { { C _ { 2 w } } } } } \\right ) , \\end{align*}"}
+{"id": "4045.png", "formula": "\\begin{align*} D _ { p _ k } A _ { i j } = D _ { p _ k } g _ { i j } = E ^ { r , k } \\left ( g _ { i j , r } - \\frac { g _ { , r } } { g _ z } g _ { i j , z } \\right ) . \\end{align*}"}
+{"id": "4694.png", "formula": "\\begin{align*} \\mathcal { I } ( G _ { I } ) = \\frac { 1 } { ( 1 - t ^ 2 ) ( 1 - t ^ 8 ) } , \\end{align*}"}
+{"id": "7886.png", "formula": "\\begin{align*} f ''' ( z ) - K f ' ( z ) + e ^ z f ( z ) = 0 , K \\in \\C . \\end{align*}"}
+{"id": "8694.png", "formula": "\\begin{align*} \\tilde s _ { \\lambda } = \\det \\left [ \\tilde h _ { \\lambda _ i ; i - j } \\right ] _ { i , j = 1 , \\dots l } , \\end{align*}"}
+{"id": "8742.png", "formula": "\\begin{align*} { H ^ { \\dagger } } ^ { * } h = \\begin{bmatrix} & h _ 1 & \\frac { 1 } { 2 } h _ 2 & \\frac { 1 } { 3 } g _ 3 & & \\frac { 1 } { T } h _ T \\\\ & \\frac { 1 } { 2 } h _ 2 & \\frac { 1 } { 3 } h _ 3 & & & \\frac { 1 } { T - 1 } h _ { T + 1 } \\\\ & \\frac { 1 } { 3 } h _ 3 & & & & \\frac { 1 } { T - 2 } h _ { T + 2 } \\\\ & & & & & \\\\ & \\frac { 1 } { T } h _ { T } & \\frac { 1 } { T - 1 } g _ { T + 1 } & \\frac { 1 } { T - 2 } h _ { T + 2 } & & h _ { 2 T - 1 } \\\\ \\end{bmatrix} . \\end{align*}"}
+{"id": "8692.png", "formula": "\\begin{align*} \\mathcal S ( u _ 1 , . . , u _ l ) = \\det [ u _ i ^ { - j + i } H ( u _ i ) ] . \\end{align*}"}
+{"id": "176.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq n \\\\ p \\notin \\mathcal { S } } } \\dfrac { \\log p } { p - 1 } \\leq \\dfrac { \\log ( 2 n ^ 2 + l ) } { n - 1 } \\left ( 2 \\log 4 \\dfrac { n } { \\log n } + \\sqrt { n } \\right ) + \\sum _ { i = 1 } ^ { r } \\dfrac { e _ i \\log p _ i } { \\lambda ( n - 1 ) } + \\dfrac { n \\log 4 } { n - 1 } . \\end{align*}"}
+{"id": "5363.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\xi \\left ( ( E ^ { ( i ) } _ h ) ^ 2 - ( E ^ { ( j ) } _ h ) ^ 2 \\right ) \\ , d x = 0 \\forall i , j \\in \\{ 1 , \\ldots , m \\} , \\forall h = 1 , 2 , 3 . \\end{align*}"}
+{"id": "4972.png", "formula": "\\begin{align*} \\begin{gathered} \\phi ( x _ 1 ) = x _ 1 + f ^ { 2 p } g ^ { p - 1 } , \\phi ( x _ 2 ) = x _ 2 + f g - \\left \\{ \\begin{array} { c c } f ^ 7 g ^ 2 & \\ p = 2 \\\\ f ^ { 2 p - 1 } g ^ { p - 1 } & \\ p \\ge 3 \\end{array} \\right . \\\\ \\phi ( x _ 3 ) = x _ 3 - f ^ { - p } \\bigl ( ( x _ 1 + f ^ { 2 p } g ^ { p - 1 } ) ^ { s } - x _ 1 ^ { s } \\bigr ) \\end{gathered} \\end{align*}"}
+{"id": "6645.png", "formula": "\\begin{align*} A ( \\nabla u ) = a ( t ' , x ' ) \\nabla u \\end{align*}"}
+{"id": "3487.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( { { \\rm { P 2 } } } \\right ) \\ \\ \\mathop { \\max } \\limits _ { \\left \\{ { { { \\bf { b } } _ l } } \\right \\} _ { l = 1 } ^ L } & \\ \\ { \\Big | { \\sum \\nolimits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { { \\bf { Q } } _ l } { { \\bf { b } } _ l } } } \\Big | ^ 2 } \\\\ { \\rm { s . t . } } & \\ \\ \\sum \\nolimits _ { l = 1 } ^ L { { { \\left \\| { { { \\bf { Q } } _ l } { { \\bf { b } } _ l } } \\right \\| } ^ 2 } } \\le P . \\end{aligned} \\end{align*}"}
+{"id": "952.png", "formula": "\\begin{align*} \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { T r } _ C ( N ) ) \\ , = \\ , \\textrm { r . g r a d e } _ R ( \\textrm { T r } _ C ( N ) , C ) . \\end{align*}"}
+{"id": "4728.png", "formula": "\\begin{align*} \\mathcal { H } ' = \\{ ( e _ i , 0 , f _ i , - \\beta _ { i } ) , ( g _ i , \\theta _ i \\gamma _ { i } , h _ i , 0 ) : 1 \\leq i \\leq m \\} \\cup \\{ ( e _ { k } , 0 , f _ { k } , 0 ) , ( g _ { \\ell } , 0 , h _ { \\ell } , 0 ) : m + 1 \\leq k \\leq t + s m + 1 \\leq \\ell \\leq t \\} . \\end{align*}"}
+{"id": "2288.png", "formula": "\\begin{align*} b ( u ( x ) ) = \\frac { 1 } { \\Delta x ^ 2 } \\int ^ { x + \\Delta x / 2 } _ { x - \\Delta x / 2 } \\left ( \\ , \\int ^ { \\eta + \\Delta x / 2 } _ { \\eta - \\Delta x / 2 } h ( \\xi ) d \\xi \\right ) d \\eta . \\end{align*}"}
+{"id": "7667.png", "formula": "\\begin{align*} \\phi ( \\eta _ { I } ) = \\sum _ { p \\neq - \\iota _ { E } ( \\omega ) } \\frac { \\eta _ { I , p } } { p + \\iota _ { E } ( \\omega ) } \\in \\frac { 1 } { f } \\Omega _ { U } ^ { j } [ x _ { I } ^ { - 1 } ] . \\end{align*}"}
+{"id": "5815.png", "formula": "\\begin{align*} ( \\ ! ( e _ r , A e _ s ) \\ ! ) = \\sum _ { q = 1 } ^ p \\beta _ { s q } { \\| e _ s \\| \\over \\| e _ q \\| } ( \\ ! ( e _ r , e _ q ) \\ ! ) = \\beta _ { s r } \\| e _ s \\| \\| e _ r \\| . \\end{align*}"}
+{"id": "3906.png", "formula": "\\begin{align*} \\mu _ { u , f ^ * } ( E ) : = \\int _ { Y u ( E ) } f ^ * ( y ) \\ d y . \\end{align*}"}
+{"id": "7923.png", "formula": "\\begin{align*} \\frac { \\partial \\mu _ t } { \\partial t } + \\div ( - \\nabla \\eta _ t \\mu _ t ) = 0 \\ , , \\quad \\ , . \\end{align*}"}
+{"id": "7306.png", "formula": "\\begin{align*} g _ { m _ n } ( \\lambda ; x ) = \\varphi ^ d _ { m _ n } ( \\lambda ; x ) - c ^ d _ { m _ n } ( \\lambda ) \\psi _ { m _ n } ( \\lambda ; x ) ( x \\in [ 0 , 1 ] ) , \\end{align*}"}
+{"id": "4888.png", "formula": "\\begin{align*} & C _ { s _ 1 s _ 2 u _ 0 } = ( C _ { s _ 1 } C _ { s _ 2 } - 1 ) C _ { s _ 0 s _ 1 s _ 3 } \\\\ & C _ { s _ 0 s _ 1 s _ 3 } C _ { u _ l } = \\xi ^ 3 C _ { u _ l } . \\end{align*}"}
+{"id": "3494.png", "formula": "\\begin{align*} { \\gamma _ { { \\rm { M R T } } } } \\to \\bar P { \\left \\| { { \\bf { \\bar h } } } \\right \\| ^ 2 } = \\bar P \\sum \\nolimits _ { l = 1 } ^ L { { { \\left \\| { { { \\bf { h } } _ l } } \\right \\| } ^ 2 } } , \\end{align*}"}
+{"id": "1174.png", "formula": "\\begin{align*} R _ { i j } = \\frac { S _ { i j } } { \\sqrt { S _ { i i } S _ { j j } } } \\ , , i , j = 1 , \\ldots , p \\ , , \\end{align*}"}
+{"id": "2829.png", "formula": "\\begin{align*} \\left ( P _ { \\sigma } \\right ) _ { i , j } = \\delta _ { i , \\sigma ( j ) } . \\end{align*}"}
+{"id": "529.png", "formula": "\\begin{align*} \\Psi ( x ^ k , y ^ k ) - \\omega ^ * \\le \\left \\{ \\begin{array} { c l } \\gamma ' \\widehat { \\varrho } ^ { \\lceil \\frac { k - 1 } { m + 1 } \\rceil } & { \\rm i f } \\ \\theta = 1 / 2 , \\\\ \\gamma ' { k } ^ { \\frac { 1 - \\theta } { 1 - 2 \\theta } } & { \\rm i f } \\ \\theta \\in ( 1 / 2 , 1 ) . \\end{array} \\right . \\end{align*}"}
+{"id": "2666.png", "formula": "\\begin{align*} \\Delta ^ { { \\rm I } * } _ m : = \\ I _ m - I _ { m - 2 } , \\end{align*}"}
+{"id": "8391.png", "formula": "\\begin{align*} f _ { \\alpha } ( x ) = \\begin{cases} x ( 1 + 2 ^ { \\alpha } x ^ { \\alpha } ) & 0 \\leq x \\leq 1 / 2 \\\\ 2 x - 1 & 1 / 2 < x \\leq 1 . \\end{cases} \\end{align*}"}
+{"id": "3544.png", "formula": "\\begin{align*} \\mathbf { p } B _ j ^ + = \\sum _ { i = 1 } ^ { j } \\sum _ { s = 1 } ^ { j - i + 1 } \\sum _ { I \\in \\mathcal { I } _ { i j } ( s ) } E ^ { e _ I } B _ i ^ + H _ I ^ + . \\end{align*}"}
+{"id": "4182.png", "formula": "\\begin{align*} \\partial _ t \\left ( \\alpha \\partial _ t g g ^ { - 1 } \\right ) - \\partial _ x \\left ( \\alpha \\partial _ x g g ^ { - 1 } \\right ) = 0 , \\det g = \\alpha ^ 2 . \\end{align*}"}
+{"id": "2199.png", "formula": "\\begin{align*} ( a ; q ) _ n : = \\prod _ { j = 1 } ^ n ( 1 - a q ^ { j - 1 } ) , n \\in \\mathbb { N } \\cup \\{ \\infty \\} . \\end{align*}"}
+{"id": "7273.png", "formula": "\\begin{align*} f ( \\gamma ) = \\sqrt { \\frac { \\gamma } { K ( \\gamma ) } } \\ b = m [ 0 , \\infty ) . \\end{align*}"}
+{"id": "9155.png", "formula": "\\begin{align*} \\frac { e ^ { - q ( n \\pm 1 ) } ( q ( n \\pm 1 ) ) ^ { \\kappa - 1 } q } { e ^ { - q n } ( q n ) ^ { \\kappa - 1 } q } = e ^ { \\mp q } \\left ( 1 \\pm \\frac { 1 } { n } \\right ) ^ { \\kappa - 1 } , \\end{align*}"}
+{"id": "3058.png", "formula": "\\begin{align*} A ( y ) : = C + a ( y ) M \\quad y \\in \\R ^ n \\end{align*}"}
+{"id": "6911.png", "formula": "\\begin{align*} ( x _ { d - i } u _ i ) \\sim ( x _ { d - i } \\tilde { u } _ i ) & \\exists ( w _ i ) _ i \\in \\hat { A } _ { \\sigma } ^ * , \\ ; \\frac { u _ i } { \\tilde { u } _ i } = \\frac { w _ i ^ q } { w _ { i + 1 } } \\\\ & \\exists w _ 1 \\in \\hat { A } _ { \\sigma } ^ * , \\ ; w _ 1 ^ N V ( ( u _ i ) _ i ) = V ( ( \\tilde { u } _ i ) _ i ) \\\\ & \\tilde { V } ( ( u _ i ) _ i ) = \\tilde { V } ( ( \\tilde { u } _ i ) _ i ) \\pmod { ( { A } _ { \\sigma } ^ * ) ^ N } \\end{align*}"}
+{"id": "539.png", "formula": "\\begin{align*} S _ 1 \\geq \\frac { X \\log X } { \\zeta ( 2 - 2 \\theta _ 3 - 0 . 0 1 ) ^ { 3 5 } } \\prod _ { j = 1 } ^ R \\Bigg ( ( 1 - 2 ^ { - \\frac { \\ell _ j } { 2 } } ) \\exp \\Big ( \\frac { k ^ 2 - 1 } { 2 } \\sum _ { p _ j \\in P _ j } ( \\frac { A ( p _ j , 1 ) ^ 2 } { p _ j } + O ( \\frac { | A ( p _ j , 1 ) | } { p _ j ^ { 2 - 2 \\theta _ 3 } } ) ) \\Big ) \\Bigg ) . \\end{align*}"}
+{"id": "7285.png", "formula": "\\begin{align*} g _ m ( \\lambda ; \\cdot ) = \\varphi ^ d _ m ( \\lambda ; \\cdot ) - c ^ d _ m ( \\lambda ) \\psi _ m ( \\lambda ; \\cdot ) \\end{align*}"}
+{"id": "3950.png", "formula": "\\begin{align*} \\partial K ( 0 ) & \\supset C _ n h R ^ * \\\\ | \\partial K ( 0 ) | & \\geq C _ n h ^ n \\prod _ { i = 1 } ^ n \\left ( \\frac { 1 } { b _ i } + \\frac { 1 } { a _ i } \\right ) . \\end{align*}"}
+{"id": "9083.png", "formula": "\\begin{align*} [ J _ e X , J _ e Y ] - [ X , Y ] - J _ e ( [ J _ e X , Y ] + [ X , J _ e Y ] ) = 0 , \\end{align*}"}
+{"id": "5882.png", "formula": "\\begin{align*} W = C _ p U , \\overline D _ p = C _ p D , \\end{align*}"}
+{"id": "7460.png", "formula": "\\begin{align*} \\# \\partial ( \\pi ) = \\# - \\# . \\end{align*}"}
+{"id": "700.png", "formula": "\\begin{align*} \\eta = - \\sum _ { j = 1 } ^ \\texttt { g } A _ j d U _ { \\beta _ j } + \\sum _ { j = 1 } ^ \\texttt { g } B _ j d U _ { \\alpha _ j } . \\end{align*}"}
+{"id": "3823.png", "formula": "\\begin{align*} a _ 2 ( f ) & = 2 g - d + 2 ( e - 1 ) \\\\ a _ { 1 , 1 } ( f ) & = d - 3 g - e ( e - 1 ) + \\frac { 3 } { 2 } ( e - 1 ) ( e - 2 ) \\end{align*}"}
+{"id": "3484.png", "formula": "\\begin{align*} y \\left [ n \\right ] = \\Big ( { \\sum \\nolimits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ l } } } \\Big ) s \\left [ { n - { n _ { \\max } } } \\right ] + z \\left [ n \\right ] . \\end{align*}"}
+{"id": "6863.png", "formula": "\\begin{gather*} j ' ( t ) = 2 t ( Q v , Q v ) _ Y + 2 ( Q v , Q u - F ) _ Y . \\end{gather*}"}
+{"id": "6900.png", "formula": "\\begin{align*} A = m a x & \\left \\{ { a + 2 \\choose 2 } - { d _ 1 + 1 \\choose 2 } , 0 \\right \\} \\\\ A = m a x & \\left \\{ { a + 2 \\choose 2 } - { d _ 1 + 1 \\choose 2 } - { d _ 2 + 1 \\choose 2 } + { d _ 1 + d _ 2 - a \\choose 2 } , 0 \\right \\} \\\\ A = m a x & \\Big \\{ { a + 2 \\choose 2 } - { d _ 1 + 1 \\choose 2 } - { d _ 2 + 1 \\choose 2 } - { d _ 3 + 1 \\choose 2 } + { d _ 1 + d _ 2 - a \\choose 2 } + \\\\ & + { d _ 2 + d _ 3 - a \\choose 2 } + { d _ 3 + d _ 1 - a \\choose 2 } - { d _ 1 + d _ 2 + d _ 3 - 2 a - 1 \\choose 2 } , 0 \\Big \\} \\end{align*}"}
+{"id": "6742.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } \\int _ { - \\infty } ^ { \\infty } p \\left [ x _ 1 , x _ 2 ; ( m _ 1 , m _ 2 , \\rho ) \\right ] d x _ 1 d x _ 2 = 1 \\end{align*}"}
+{"id": "9042.png", "formula": "\\begin{align*} \\overline { V } _ \\cdot ^ { ( 1 ) , i , n } = \\overline { \\Psi } _ \\cdot ( \\overline { \\textbf { V } } ^ { ( 0 ) , n } ) = \\underline { \\Psi } _ \\cdot ( \\overline { \\textbf { V } } ^ { ( 0 ) , n } ) = \\underline { \\Psi } _ \\cdot ( \\underline { \\textbf { V } } ^ { ( 0 ) , n } ) = \\underline { V } _ \\cdot ^ { ( 1 ) , i , n } . \\end{align*}"}
+{"id": "4421.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 1 ^ 2 - a _ 1 + 1 } , \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 1 ^ 2 - a _ 1 } \\right ] = \\left ( \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 1 ^ 2 - a _ 1 + 1 } , \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 1 - 1 } \\right ] \\end{align*}"}
+{"id": "8987.png", "formula": "\\begin{align*} \\frac 1 2 & \\frac { d } { d t } \\big ( \\| \\nabla \\partial ^ k _ { \\phi } u \\| ^ 2 _ { L ^ 2 ( B ) } \\big ) = ( - 1 ) ^ k \\int _ B \\nabla \\partial ^ { 2 k } _ { \\phi } u \\nabla u _ t \\ ; d x \\\\ & = ( - 1 ) ^ k ( \\partial ^ { 2 k } _ { \\phi } u _ r , u _ t ) _ { L ^ 2 ( S ^ 1 ) } = ( - 1 ) ^ { k + 1 } ( \\partial ^ { 2 k } _ { \\phi } u _ r , ( \\varepsilon + d \\pi _ N ( u ) ) u _ r ) _ { L ^ 2 ( S ^ 1 ) } \\\\ & \\le - \\| d \\pi _ N ( u ) \\partial ^ k _ { \\phi } u _ r \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } - I , \\end{align*}"}
+{"id": "5459.png", "formula": "\\begin{align*} E V ( X ^ n _ { t \\wedge \\tau ^ n } , \\alpha ^ n _ { t \\wedge \\tau ^ n } ) & \\leq e ^ { ( \\lambda _ 1 + \\lambda _ 2 ) t } E V ( X ^ n _ 0 , \\alpha ^ n _ 0 ) \\\\ & \\leq e ^ { ( \\lambda _ 1 + \\lambda _ 2 ) T } E V ( X ^ n _ 0 , \\alpha ^ n _ 0 ) = : \\delta . \\end{align*}"}
+{"id": "0.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { y } ^ n _ t = \\zeta _ t = \\mathbb { E } [ \\zeta | \\mathcal { F } _ t ] , \\hat { z } ^ n _ t = \\hat { \\tilde { z } } ^ n _ t = \\hat { \\gamma } ^ n _ { ( t , e ) } = g ( x , \\hat { y } ^ n , \\hat { z } ^ n , \\hat { \\tilde { z } } ^ n , \\hat { \\gamma } ^ n ) = 0 . \\end{aligned} \\end{align*}"}
+{"id": "4133.png", "formula": "\\begin{align*} \\triangle \\chi _ k = - \\mu _ k \\chi _ k , \\mu _ k = k ( k + 2 ) , k = 0 , 1 , 2 , . . . . \\end{align*}"}
+{"id": "6102.png", "formula": "\\begin{align*} E _ { 3 / 2 } ^ { * } ( \\tau ) = \\sum _ { D \\geq 0 } H ( D ) e ^ { 2 \\pi i D \\tau } + \\frac { 1 } { 1 6 \\pi } \\sum _ { n \\in \\Z } v ^ { - 1 / 2 } \\beta _ { 3 / 2 } ( 4 \\pi n ^ { 2 } v ) e ^ { - 2 \\pi i n ^ { 2 } \\tau } , \\end{align*}"}
+{"id": "6485.png", "formula": "\\begin{align*} \\mathrm { G a l } ( \\mathbb { Q } ( \\zeta _ { 4 n } ) / \\mathbb { Q } ( \\zeta _ n ) ^ { { \\eta ' } ^ { - 1 } ( \\mathcal { H } ' ) } ) = \\eta ^ { - 1 } ( \\mathcal { H } ) . \\end{align*}"}
+{"id": "1153.png", "formula": "\\begin{align*} c = a \\dot - b \\longleftrightarrow ( ( b + c = a ) \\vee ( a \\leq b \\wedge c = 0 ) ) , \\end{align*}"}
+{"id": "7841.png", "formula": "\\begin{align*} m ( r , 1 / f ) = O ( r ^ { q - p } + \\log r ) , \\end{align*}"}
+{"id": "1372.png", "formula": "\\begin{align*} J ( t ) = \\frac { 1 } { 2 } I ( t ) + \\frac { k } { 4 } \\left \\| u \\right \\| _ { 2 } ^ { 2 } \\end{align*}"}
+{"id": "5747.png", "formula": "\\begin{align*} D \\theta ^ { A A ^ { \\prime } } & = 0 , \\\\ \\Omega _ { B } ^ { A } \\theta ^ { B A ^ { \\prime } } - \\Omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\theta ^ { A B ^ { \\prime } } & = 0 , \\end{align*}"}
+{"id": "95.png", "formula": "\\begin{align*} \\theta _ { 0 , 1 } = \\theta _ { 1 , 1 } = \\theta _ { 2 , 1 } = \\dots = \\theta _ { m - 1 , 1 } = \\gamma ^ j , \\end{align*}"}
+{"id": "5600.png", "formula": "\\begin{align*} \\nabla _ { 2 } ( m _ { 2 } ) _ { 3 } = 0 , \\quad \\nabla _ { 1 } ( m _ 2 ) _ { 0 } = \\nabla _ { 0 } ( m _ 2 ) _ { 1 } = \\nabla _ { 3 } ( m _ 2 ) _ { 3 } . \\end{align*}"}
+{"id": "1710.png", "formula": "\\begin{align*} F ( t ) = - ( - 1 ) ^ { \\frac { k } { 2 } } ( i t ) ^ { \\frac { 2 - k } { 2 } + m } ( k - 1 ) \\sum _ { 0 \\leq j \\equiv \\frac { k - 2 } { 2 } + m \\ ; ( { \\rm m o d } \\ ; 2 ) } \\binom { k } { j } \\sum _ { s \\geq 0 } C ( j - 2 - 2 s ) ( i t ) ^ { j - 2 - 2 s } . \\end{align*}"}
+{"id": "9247.png", "formula": "\\begin{align*} { ( G _ { j k } ( c ) w ) } _ { i \\ell } = \\begin{cases} 1 & w ( \\ell ) = i , \\\\ c & i = j w ^ { - 1 } ( k ) = \\ell , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "6465.png", "formula": "\\begin{align*} w _ * = \\begin{pmatrix} \\cos \\theta _ * \\\\ \\sin \\theta _ * \\cos \\varphi _ * \\\\ \\sin \\theta _ * \\sin \\varphi _ * \\end{pmatrix} , n _ * = \\begin{pmatrix} - \\sin \\theta _ * \\\\ \\cos \\theta _ * \\cos \\varphi _ * \\\\ \\cos \\theta _ * \\sin \\varphi _ * \\end{pmatrix} , p _ * = \\begin{pmatrix} 0 \\\\ - \\sin \\varphi _ * \\\\ \\cos \\varphi _ * \\end{pmatrix} , \\end{align*}"}
+{"id": "5274.png", "formula": "\\begin{align*} \\tilde { s e c } ( U , V ) = \\frac { \\tilde { g } ( \\tilde { R } ( U , V ) V , U ) } { \\tilde { g } ( U , U ) \\tilde { g } ( V , V ) - \\tilde { g } ( U , V ) ^ 2 } = \\frac { 1 } { \\lambda ^ 2 } g ( \\tilde { R } ( U , V ) V , U ) . \\end{align*}"}
+{"id": "5450.png", "formula": "\\begin{align*} W _ { 2 , N } ^ 2 ( \\mu , \\nu ) : = \\inf _ { \\pi \\in \\mathcal C ( \\mu , \\nu ) } \\int _ { \\mathbb R ^ d \\times \\mathbb R ^ d } | \\phi _ N ( x ) - \\phi _ N ( y ) | ^ 2 \\pi ( d x , d y ) , \\phi _ N ( x ) : = \\frac { N x } { N \\vee | x | } . \\end{align*}"}
+{"id": "8358.png", "formula": "\\begin{align*} ( \\Box + V _ 3 ) v = 0 , V _ 3 = \\sum _ { j = 1 } ^ 3 G _ j ^ { ( 4 ) } v ^ { j - 1 } . \\end{align*}"}
+{"id": "2133.png", "formula": "\\begin{align*} \\partial _ t \\psi ( t , x ) = J \\ast \\psi ( t , \\cdot ) ( x ) - \\psi ( t , x ) + e ^ { - t } J ( x ) . \\end{align*}"}
+{"id": "7453.png", "formula": "\\begin{align*} Q ( \\widehat M _ \\chi ) ^ * = \\widehat M ^ * _ Q \\end{align*}"}
+{"id": "6878.png", "formula": "\\begin{align*} \\lim _ { \\mu \\to \\infty } \\| v _ \\mu \\| = 0 . \\end{align*}"}
+{"id": "4823.png", "formula": "\\begin{align*} D _ 0 ^ h F ( v _ 1 , \\dots , v _ h ) = \\frac { d ^ h } { d t ^ h } F \\Big ( \\sum _ { h = 1 } ^ n \\frac { t ^ h v _ h } { h ! } \\Big ) \\bigg | _ { t = 0 } . \\end{align*}"}
+{"id": "5591.png", "formula": "\\begin{align*} 0 = \\nabla _ 0 ( \\nabla W ) _ { 0 0 i 1 j } = 2 \\kappa _ i \\kappa ^ k \\Psi _ { k j } - \\kappa ^ k \\kappa _ k \\Psi _ { i j } . \\end{align*}"}
+{"id": "823.png", "formula": "\\begin{align*} \\nu ( \\mathcal { A } _ n ( x ) ) = \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma } \\int _ { - \\infty } ^ x e ^ { - t ^ 2 / 2 \\sigma ^ 2 } \\ d t + O \\left ( n ^ { - 1 / 4 } \\right ) \\end{align*}"}
+{"id": "2206.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty \\sum _ { j = 1 } ^ \\infty \\alpha _ { i , j } x ^ i y ^ j & = \\dfrac { N _ \\alpha } { D } , \\\\ \\sum _ { i = 1 } ^ \\infty \\sum _ { j = 1 } ^ \\infty \\beta _ { i , j } x ^ i y ^ j & = \\dfrac { N _ \\beta } { D } , \\end{align*}"}
+{"id": "3348.png", "formula": "\\begin{align*} \\flat ( X _ H ) = \\dd H , { } \\flat : T M \\rightarrow T ^ * M , \\flat ( X ) = X \\lrcorner \\Omega . \\end{align*}"}
+{"id": "2530.png", "formula": "\\begin{align*} A _ \\varphi = \\frac { A r ^ d \\varphi _ 0 } { 2 \\| \\nabla \\varphi \\| _ \\infty } \\end{align*}"}
+{"id": "581.png", "formula": "\\begin{align*} J ^ \\dagger _ { t _ n , t _ { n + 1 } } = A ^ { \\rm T } : ( X ^ \\dagger _ { t _ n } \\otimes X ^ \\dagger _ { t _ n , t _ { n + 1 } } ) + A ^ { \\rm T } : \\mathbb { X } _ { t _ n , t _ { n + 1 } } ^ \\dagger \\ , . \\end{align*}"}
+{"id": "7809.png", "formula": "\\begin{align*} - \\{ p _ 1 ^ * ( n ) , \\tilde m \\} _ 1 = \\langle n , \\tilde m \\rangle _ 1 . \\end{align*}"}
+{"id": "8011.png", "formula": "\\begin{align*} T & = ( \\{ F _ 3 \\} , \\emptyset ) , \\\\ F & = \\left [ ( F _ 3 , F _ 2 ) , ( F _ 3 , F _ 1 ) \\right ] , \\end{align*}"}
+{"id": "4061.png", "formula": "\\begin{align*} v ( y , t ) & : \\overline { \\Omega ^ * } \\times [ 0 , T ] \\rightarrow \\mathbf { R } \\\\ v ( y , t ) & = \\sup _ { x \\in \\Omega } g ^ * ( x , y , u ( x , t ) ) . \\end{align*}"}
+{"id": "6770.png", "formula": "\\begin{align*} \\phi _ { \\chi } ^ e ( s ) : = \\pi ^ { - \\frac { s } { 2 } } \\Gamma \\left ( \\frac { s } { 2 } \\right ) L ( s , \\chi ) = \\int _ { 0 } ^ { \\infty } y ^ { \\frac { s } { 2 } - 1 } \\Psi _ { \\chi } ( y ) d y \\end{align*}"}
+{"id": "8842.png", "formula": "\\begin{align*} \\nabla _ { \\nu } = d + \\nu \\left ( \\partial - \\overline { \\partial } \\right ) \\partial \\log \\left ( 1 + \\langle z , z \\rangle \\right ) . \\end{align*}"}
+{"id": "4412.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\frac { 1 } { a _ i } \\leq \\sum _ { i = 1 } ^ m \\frac { 1 } { x _ i } < \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < \\frac { p } { q } . \\end{align*}"}
+{"id": "3520.png", "formula": "\\begin{align*} \\sum _ { k = i } ^ { j } \\gamma _ { k i } \\geq \\sum _ { k = i + 1 } ^ { j + 1 } \\gamma _ { k , i + 1 } , \\end{align*}"}
+{"id": "3671.png", "formula": "\\begin{align*} - M _ \\rho ( f - A u , u - g ) = A u - f + [ f - A u + c ( u - g ) ] _ \\rho = 0 \\end{align*}"}
+{"id": "1825.png", "formula": "\\begin{align*} f ^ * ( K _ Y + B ) = K _ X + \\sum _ { i , j } ( 1 - ( 1 - b _ i ) e _ { i j } ) D _ { i j } . \\end{align*}"}
+{"id": "4257.png", "formula": "\\begin{align*} \\left ( - \\frac { 1 } { 2 } \\Delta _ { \\R ^ { N - 2 } } + V _ \\gamma \\right ) \\Phi _ k = \\lambda _ k \\Phi _ k , \\| \\Phi _ k \\| ^ 2 _ { L ^ 2 ( \\R ^ { N - 2 } ) } = 1 , \\lambda _ k \\leq \\lambda _ { k + 1 } , \\forall k \\geq 0 . \\end{align*}"}
+{"id": "2296.png", "formula": "\\begin{align*} \\omega _ k = \\sigma ^ + \\omega ^ + _ k - \\sigma ^ - \\omega ^ - _ k . \\end{align*}"}
+{"id": "7551.png", "formula": "\\begin{align*} & Z _ f ( s , A _ 1 ) \\\\ = & q ^ { - ( \\omega + 1 ) - l s } \\int _ { \\mathcal { O } _ K ^ 2 } \\chi ( a c ( \\pi ^ { k + r } u ^ p w ^ { k + r + l } + w ^ { r + l } v ^ l \\mathbb { H } _ r ^ k ( w , t \\pi v w - w ) \\\\ \\times & | \\pi ^ { k + r } u ^ p w ^ { k + r + l } + w ^ { r + l } v ^ l \\mathbb { H } _ r ^ k ( w , t \\pi v w - w ) | ^ s | - w ^ { \\frac { k + r + l } { p } + 1 } | | d u d v d w | \\\\ : = & q ^ { - ( \\omega + 1 ) - l s } Z _ { f _ 1 } ( s , \\chi ) \\int _ { \\mathcal { O } _ K ^ { \\times } } \\chi ( a c ( w ^ { k + r + l } ) ) | d w | , \\end{align*}"}
+{"id": "5982.png", "formula": "\\begin{align*} p _ 1 ^ { - 1 } ( m ) & = \\{ m \\} \\times _ { ( X ' ) ^ r } \\left ( g _ 1 X ( u _ 1 ) \\times \\dots g _ r X ( u _ r ) \\right ) \\\\ & \\simeq \\pi _ u ^ { - 1 } ( e v ( m ) ) \\\\ & \\simeq g \\cdot \\pi _ u ^ { - 1 } ( g ^ { - 1 } e v ( m ) ) , \\end{align*}"}
+{"id": "7265.png", "formula": "\\begin{align*} G ^ 1 _ m ( x ) = \\int _ { 0 } ^ { x } ( m ( y ) - m ( 1 ) ) d y \\end{align*}"}
+{"id": "3189.png", "formula": "\\begin{align*} A ( y ) : = \\frac { 1 } { r ( y ) } \\ , \\mathrm { d i a g } ( a ( y ) , 2 - a ( y ) ) \\quad y \\in \\R ^ 2 , \\end{align*}"}
+{"id": "7193.png", "formula": "\\begin{align*} \\sigma ^ 3 = \\lambda \\left [ B _ { \\alpha } ( { \\Sigma _ 0 } ) y ^ 1 _ { \\alpha } - D ( { \\Sigma _ 0 } ) \\right ] + ( 1 - \\lambda ) \\left [ B _ { \\alpha } ( { \\Sigma _ 0 } ) y ^ 2 _ { \\alpha } - D ( { \\Sigma _ 0 } ) \\right ] = \\end{align*}"}
+{"id": "82.png", "formula": "\\begin{align*} v _ { j , n } = ( v _ n ^ { ( 1 + ( j - 1 ) d ) } , \\cdots , v _ n ^ { ( d + ( j - 1 ) d ) } ) \\in \\mathbf { R } ^ d \\end{align*}"}
+{"id": "6982.png", "formula": "\\begin{gather*} H _ { 2 , n } ^ { ( \\{ 1 , 1 \\} ) } ( x ) = \\det \\begin{pmatrix} 2 x & 4 x ^ 2 - 2 & H _ n ( x ) \\\\ 2 & 8 x & H _ n ' ( x ) \\\\ 0 & 8 & H _ n '' ( x ) \\end{pmatrix} \\end{gather*}"}
+{"id": "6766.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } I _ n ( t ) = \\lim _ { t \\rightarrow \\infty } \\left | \\mathfrak { R e } ( \\alpha _ n ( t ) ) \\right | \\exp \\left [ \\mp \\frac { j } { 2 } e ^ { t } \\mathfrak { R e } ( \\alpha _ n ( t ) ) \\mathfrak { I m } ( \\alpha _ n ( t ) ) \\right ] = \\lim _ { t \\rightarrow \\infty } \\left | \\mathfrak { R e } ( \\alpha _ n ( t ) ) \\right | . \\end{align*}"}
+{"id": "3656.png", "formula": "\\begin{align*} \\theta e ^ { \\delta t + \\gamma t ^ { \\frac { q } { \\alpha } } - \\lambda _ * ( M ) t } \\phi _ M ( x ) < \\varepsilon , \\ \\ \\gamma = \\delta ^ { - \\frac { 1 } { \\alpha } } . \\end{align*}"}
+{"id": "6477.png", "formula": "\\begin{align*} G = \\mathbb { Z } _ { n _ 1 } \\otimes \\mathbb { Z } _ { n _ 2 } \\otimes \\cdots \\otimes \\mathbb { Z } _ { n _ r } , \\end{align*}"}
+{"id": "5236.png", "formula": "\\begin{align*} \\nabla _ X Y = A _ X Y + \\mathcal { H } \\nabla _ X Y . \\end{align*}"}
+{"id": "5195.png", "formula": "\\begin{align*} h = v ^ \\ast \\beta = \\sum _ { 1 \\leq i < j \\leq m } Q _ { i j } G _ { i j } = \\sum _ { 1 \\leq i < j \\leq m } Q _ { i j } d v ^ i \\wedge d v ^ j \\ . \\end{align*}"}
+{"id": "3498.png", "formula": "\\begin{align*} { C _ { { \\rm { O F D M } } } } { \\rm = } \\frac { { { n _ c } - { n _ { { \\rm { O F D M } } } } { { \\tilde n } _ { \\max } } } } { { { n _ c } } } \\frac { 1 } { K } \\sum \\nolimits _ { k = 1 } ^ K { { { \\log } _ 2 } \\Big ( { 1 + \\frac { { { p _ k } { { \\left \\| { { \\bf { h } } \\left [ k \\right ] } \\right \\| } ^ 2 } } } { { \\sigma ^ 2 } / K } } \\Big ) } , \\end{align*}"}
+{"id": "7456.png", "formula": "\\begin{align*} K _ \\mu ( Z , W ) ( P ) = K ( Z , W ) ( P ) P \\in { \\mathbb S } ^ { n _ Z \\times n _ W } . \\end{align*}"}
+{"id": "6500.png", "formula": "\\begin{align*} \\int x _ i q _ u ( x _ 1 , x _ 2 ) d ( P _ 0 \\times P _ 0 ) ( x _ 1 , x _ 2 ) = \\int x _ i d P _ 0 ( x _ i ) = 0 \\in \\R ^ { m d } \\end{align*}"}
+{"id": "4142.png", "formula": "\\begin{align*} \\int _ M \\nabla _ i h _ { j k } \\nabla _ j h _ { k i } d V _ g & = - \\int _ M h _ { j k } \\nabla _ i \\nabla _ j h _ { k i } d V _ g \\\\ & = - \\int _ M h _ { j k } ( \\nabla _ j \\nabla _ i h _ { k i } - \\mathring { R } ( h ) _ { j k } + R _ { j l } h _ { l k } ) d V _ g \\\\ & = \\int _ M | h | ^ 2 + \\langle \\mathring { R } h , h \\rangle - R _ { j l } h _ { j k } h _ { l k } d V _ g \\end{align*}"}
+{"id": "6392.png", "formula": "\\begin{align*} ( ( p - 2 ) \\Delta _ { \\infty , X } ^ N + \\Delta _ X ) u ( X , t ) & = | \\nabla u ( X , t ) | ^ { 2 - p } \\Delta _ { p , X } u ( X , t ) \\\\ & : = | \\nabla _ X u ( X , t ) | ^ { 2 - p } \\nabla _ X \\cdot ( | \\nabla _ X u ( X , t ) | ^ { p - 2 } \\nabla _ X u ( X , t ) ) , \\end{align*}"}
+{"id": "3731.png", "formula": "\\begin{align*} I ( f ) ( 0 ) & = \\sum _ { j = - B } ^ { A - 1 } q ^ { j ( \\ell - 2 ) } f ( 0 , 0 , \\varpi ^ j ) + \\frac { q ^ { A ( \\ell - 2 ) } } { 1 - q ^ { \\ell - 2 } } f ( 0 ) - \\frac { q ^ { ( n - A + 1 ) ( \\ell - 2 ) } } { 1 - q ^ { \\ell - 2 } } f ( 0 ) \\\\ & + q ^ { ( n - A + 1 ) ( \\ell - 2 ) } \\sum _ { j = 0 } ^ { B + A - 1 } q ^ { j ( \\ell - 2 ) } f ( \\varpi ^ { A - j - 1 } \\xi , 0 , 0 ) . \\end{align*}"}
+{"id": "4091.png", "formula": "\\begin{align*} \\langle \\nabla _ X ^ { \\pm } Y , Z \\rangle = \\langle \\nabla _ X Y , Z \\rangle \\pm \\frac { 1 } { 2 } H ( X , Y , Z ) \\end{align*}"}
+{"id": "7781.png", "formula": "\\begin{align*} m _ \\alpha ( x | \\mu ) & = \\mu \\int _ 0 ^ \\infty f _ \\alpha ( x | y ) \\ , e ^ { - y / 2 } \\ , I _ \\mu ( y / 2 ) \\frac { d y } { y } \\\\ \\rho _ \\alpha ( x ) & = x \\int _ 0 ^ \\infty f _ \\alpha ( x | y ) \\ , e ^ { - y / 2 } \\ , I _ 0 ( y / 2 ) \\frac { d y } { y } \\end{align*}"}
+{"id": "3762.png", "formula": "\\begin{align*} \\Gamma & = \\Gamma _ 1 \\cup \\Gamma _ 2 , & \\Gamma _ 0 & = \\Gamma _ 1 \\cap \\Gamma _ 2 . \\end{align*}"}
+{"id": "9200.png", "formula": "\\begin{align*} Z _ { j + 1 } ( \\varphi ' , 0 ; ( \\Psi _ k ) _ { k \\leq j } ) = e ^ { - E _ { j + 1 } | \\Lambda | + e _ { j + 1 } } \\sum _ { Z \\in \\mathcal { P } _ { j + 1 } } e ^ { U _ { j + 1 } ( \\Lambda \\backslash Z , \\varphi ' ) } K _ { j + 1 } ( Z , \\varphi ' ; ( \\Psi _ k ) _ { k \\leq j } ) \\end{align*}"}
+{"id": "3659.png", "formula": "\\begin{align*} \\min _ { v \\in V } J ( v ) & = \\min _ { v \\in V } \\frac { 1 } { 2 } \\langle A v , v \\rangle - \\langle f , v \\rangle \\\\ v & \\le g \\end{align*}"}
+{"id": "5713.png", "formula": "\\begin{align*} d \\theta ^ { a } + \\mathbf { A } _ { b } ^ { a } \\theta ^ { b } & = 0 , \\\\ \\mathbf { A } _ { b } ^ { a } & = \\omega _ { b } ^ { a } - \\Omega _ { b } ^ { a } \\mathbf { m , } \\end{align*}"}
+{"id": "7696.png", "formula": "\\begin{align*} M \\otimes _ { \\mathbb { C } [ S ] } \\mathbb { C } _ { ( \\tau ^ { \\sharp } ) ^ { - j } ( \\textbf { a } ) } = 0 . \\end{align*}"}
+{"id": "9171.png", "formula": "\\begin{align*} A f = \\sum _ { j \\leq N } u _ j + t _ N Q _ N ( f + s \\gamma \\Delta f ) = \\sum _ { j \\leq N } u _ j , \\end{align*}"}
+{"id": "3410.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } \\zeta _ { z } + v \\cdot \\nabla \\zeta _ { z } - \\mu \\Delta \\zeta _ { z } = \\zeta _ { r } \\partial _ { r } v _ { z } + \\zeta _ { z } \\partial _ { z } v _ { z } , \\\\ { \\zeta _ { z } } _ { \\vert t = 0 } = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "793.png", "formula": "\\begin{align*} E _ Y : = \\left \\{ x \\in Y : \\lim _ { n \\to \\infty } \\frac { f _ n ( x ) } { n } = \\Lambda \\right \\} \\ \\ \\ \\ \\widehat { \\nu } ( E _ Y ) > 0 . \\end{align*}"}
+{"id": "9138.png", "formula": "\\begin{align*} Q _ { h , \\nu } ( \\eta | \\gamma ) \\ ; = \\ ; \\sum _ { f \\in \\mathfrak { F } _ { \\eta , \\gamma } } G ( f ) , \\end{align*}"}
+{"id": "8650.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ N \\int \\abs \\Psi ^ 2 F ^ { - 2 } _ i f ' ( t _ i ) ^ 2 + \\sum _ { j < i } \\int \\abs \\Psi ^ 2 V ( d ( x _ i , x _ j ) ) \\\\ & \\leq 2 \\mu \\sum _ { j < i } \\int F _ 1 ^ 2 \\ldots F _ { j - 1 } ^ 2 F _ { j + 1 , j } ^ 2 \\cdots F _ { i - 1 , j } ^ 2 F _ { i + 1 , i j } ^ 2 \\cdots F _ { N , i j } ^ 2 \\d x _ { 1 , \\ldots , N , i j } \\\\ & \\quad \\times \\int \\left ( 2 \\mu f ' ( d ( x _ i , x _ j ) ) ^ 2 + f ( d ( x _ i , x _ j ) ) ^ 2 V ( d ( x _ i , x _ j ) ) \\right ) \\d x _ i \\d x _ j , \\end{align*}"}
+{"id": "6537.png", "formula": "\\begin{align*} & \\bar { G } ^ 2 \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( w ^ j \\right ) \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( x ^ j \\right ) \\ , Q ( x , w ) ( d P _ 0 ^ m \\times P _ 0 ^ m ) ( x , w ) \\\\ & \\qquad > \\int \\int \\phi _ m \\left ( x \\right ) \\phi _ m ( w ) \\ , Q ( x , w ) d P _ 0 ^ m ( w ) d P _ 0 ^ m ( x ) , \\end{align*}"}
+{"id": "2573.png", "formula": "\\begin{align*} \\mathfrak { g } _ 0 & = \\{ x \\in \\mathfrak { g } \\mid \\operatorname { a d } ( \\eta ) x = 0 \\ \\ \\eta \\in \\mathfrak { t } \\} , \\\\ \\mathfrak { g } _ \\alpha & = \\{ x \\in \\mathfrak { g } \\mid \\operatorname { a d } ( \\eta ) ^ 2 x = - \\langle \\alpha , \\eta \\rangle ^ 2 x \\ \\ \\eta \\in \\mathfrak { t } \\} . \\end{align*}"}
+{"id": "6947.png", "formula": "\\begin{gather*} \\mathbb { N } _ { \\lambda } = \\{ n \\geq | \\lambda | - m \\colon n \\neq | \\lambda | + \\lambda _ j - j , \\ \\ j = 1 , 2 , \\dots , m \\} . \\end{gather*}"}
+{"id": "943.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( K , C ) \\ , = \\ , \\textrm { r . g r a d e } _ R ( M , C ) - k + 1 . \\end{align*}"}
+{"id": "6211.png", "formula": "\\begin{align*} \\hbox { K e r } ( C _ p ) \\cap V ^ \\perp = \\hbox { K e r } ( C _ p ) \\cap \\hbox { I m } ( D ) = \\{ 0 \\} , \\end{align*}"}
+{"id": "504.png", "formula": "\\begin{align*} \\frac { 2 m \\ ! + \\ ! 1 } { 4 ( m \\ ! + \\ ! 1 ) } \\ ! \\sum _ { K _ 1 \\ni j = k } ^ { \\nu } \\ ! \\sqrt { \\Phi ( x ^ { \\ell ( j + 1 ) } ) - \\Phi ( x ^ { j + 1 } ) } & \\le \\ ! \\frac { \\sqrt { 2 } ( m \\ ! + \\ ! 1 ) ^ 2 b } { \\sqrt { a } } \\sum _ { j = k } ^ { \\nu } \\ ! \\Xi _ { j + 1 } + a _ 1 ( m \\ ! + \\ ! 1 ) \\sum _ { j = k - m } ^ { k - 1 } \\| x ^ { j + 1 } - x ^ j \\| \\\\ & \\quad \\ ! + \\ ! \\frac { a _ 1 ( m + 1 ) } { 2 } \\Big [ \\sum _ { j = k } ^ { \\nu } \\ , \\Xi _ j \\ ! + \\ ! 2 b a ^ { - 1 } \\varphi ( \\Phi ( x ^ { \\ell ( k ) } ) \\ ! - \\ ! \\Phi ^ * ) \\Big ] . \\end{align*}"}
+{"id": "6701.png", "formula": "\\begin{align*} \\mu ( n ) = ( - 1 ) ^ { \\omega ( n ) } \\mu ^ 2 ( n ) \\mu ( n ) = ( - 1 ) ^ { \\Omega ( n ) } \\mu ^ 2 ( n ) . \\end{align*}"}
+{"id": "8627.png", "formula": "\\begin{align*} d _ { T V } ( P , Q ) = \\sup _ { A \\in \\mathcal { F } } \\left | P ( A ) - Q ( A ) \\right | . \\end{align*}"}
+{"id": "228.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - \\beta \\tau \\eta \\leqslant - \\frac { \\alpha } { 2 } ( 1 + 3 \\rho ^ 2 ) \\leqslant - \\frac { \\alpha } { 2 } , \\end{align*}"}
+{"id": "2991.png", "formula": "\\begin{align*} \\begin{array} { r c c c c } \\hbox { d } \\theta ( e _ a , \\partial _ y ) / \\theta _ 4 = \\Theta _ a & = & 0 & \\hbox { ( a ) } \\\\ \\Theta _ { a b } + \\pi _ { a b } & = & 0 & \\hbox { ( b ) } \\\\ \\hbox { d } \\pi & = & \\frac { 1 } { 2 } \\| \\pi \\| ^ 2 e ^ { ( 4 ) } & \\hbox { ( c ) } \\end{array} \\end{align*}"}
+{"id": "4135.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda = \\sum _ i \\chi _ i ^ 2 \\Big ( \\int _ M - ( \\mu _ i + 1 2 ) a _ i ^ 2 - 1 0 a _ i b _ i \\mu _ i - ( \\mu _ i ^ 2 + 2 \\mu _ i ) b _ i ^ 2 d V _ g \\Big ) . \\end{align*}"}
+{"id": "563.png", "formula": "\\begin{align*} K _ { n + 1 / 2 } = \\sigma _ n ( A X _ { t _ { n + 1 / 2 } } ^ \\dagger ) ^ { \\rm T } \\left ( \\gamma + \\Delta t \\sigma _ n ( A X _ { t _ { n + 1 / 2 } } ^ \\dagger ) ^ { \\rm T } A X _ { t _ { n + 1 / 2 } } ^ \\dagger \\right ) ^ { - 1 } , \\end{align*}"}
+{"id": "3064.png", "formula": "\\begin{align*} A ( y ) : = C + a ( y ) M \\quad y \\in \\R ^ n \\end{align*}"}
+{"id": "3303.png", "formula": "\\begin{align*} \\log { ( 1 + \\tau + t ) } ^ 2 - \\log { ( 1 + \\tau ) } ^ 2 & = \\log { \\Bigr ( 1 + \\frac { t } { 1 + ( \\chi t ) ^ 2 } \\Bigl ) } \\Bigl ( \\log { ( 1 + t + ( \\chi t ) ^ 2 ) } + \\log { ( 1 + ( \\chi t ) ^ 2 ) } \\Bigr ) \\\\ & \\leq C \\frac { t \\log { ( 1 + ( \\chi t ) ^ 2 ) } } { 1 + ( \\chi t ) ^ 2 } \\leq \\frac { C } { \\chi } . \\end{align*}"}
+{"id": "7572.png", "formula": "\\begin{align*} Z _ { f _ { 7 , a } } ( s , \\chi , B _ 7 ^ a ) = \\dfrac { F _ { 7 , a } ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "8701.png", "formula": "\\begin{align*} \\tilde Q _ { \\lambda } = \\tilde \\Gamma ^ + _ { - \\lambda _ l } | _ { t = - 1 } \\dots \\tilde \\Gamma ^ + _ { - \\lambda _ 1 } | _ { t = - 1 } ( 1 ) . \\end{align*}"}
+{"id": "3638.png", "formula": "\\begin{align*} \\psi ( \\rho ) = e ^ { f ( \\rho ) } \\ a n d \\ \\psi ( \\rho ) = O ( \\rho ^ { 1 + \\frac { \\gamma } { 2 } } ) \\ a s \\ r \\rightarrow \\infty \\ \\ i f \\ \\gamma > 0 . \\end{align*}"}
+{"id": "6817.png", "formula": "\\begin{align*} \\mu _ i = ( \\xi _ i ) ^ { - 1 } w _ i ^ * B U w _ i , \\end{align*}"}
+{"id": "3238.png", "formula": "\\begin{align*} L ' ( v ) = \\begin{cases} L ( v ) \\cup \\{ c ' \\} & , \\cr L ( v ) & . \\end{cases} \\end{align*}"}
+{"id": "7774.png", "formula": "\\begin{align*} \\widetilde m _ { \\alpha , \\lambda } ( s | \\mu ) & = \\widetilde g _ { \\mu , \\lambda } ( s ^ \\alpha ) = \\frac { \\lambda ^ \\mu } { ( \\lambda + s ^ \\alpha ) ^ \\mu } \\end{align*}"}
+{"id": "5131.png", "formula": "\\begin{align*} \\Omega _ { n } ^ { \\pm } ( \\lambda , b ) & : = \\frac { B _ { n } ( \\lambda , b ) \\pm \\sqrt { \\Delta _ { n } ( \\lambda , b ) } } { 2 b } \\\\ & = \\frac { 1 - b ^ { 2 } } { 2 b } \\Lambda _ { 1 } ( \\lambda , b ) + \\frac { 1 } { 2 } \\Big ( \\Omega _ { n } ( \\lambda ) - \\Omega _ { n } ( \\lambda b ) \\Big ) \\\\ & \\quad \\pm \\frac { 1 } { 2 b } \\sqrt { \\Big ( b \\big [ \\Omega _ { n } ( \\lambda ) + \\Omega _ { n } ( \\lambda b ) \\big ] - ( 1 + b ^ { 2 } ) \\Lambda _ { 1 } ( \\lambda , b ) \\Big ) ^ { 2 } - 4 b ^ { 2 } \\Lambda _ { n } ^ { 2 } ( \\lambda , b ) } . \\end{align*}"}
+{"id": "5108.png", "formula": "\\begin{align*} A _ { b } : = \\big \\{ z \\in \\mathbb { C } \\quad \\textnormal { s . t . } b < | z | < 1 \\big \\} \\quad \\mbox { f o r } b \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "2407.png", "formula": "\\begin{align*} \\widetilde E ^ T = Q ^ T E ^ T Q = Q ^ T E Q = \\widetilde E \\end{align*}"}
+{"id": "3175.png", "formula": "\\begin{align*} b _ 1 ( y _ 1 , y _ 2 ) & : = \\frac { 1 - \\frac { 1 } { 2 } \\sin ( 2 \\pi y _ 1 ) \\sin ( 2 \\pi y _ 2 ) } { 1 + \\frac { 1 } { 4 } ( \\cos ( 2 \\pi y _ 1 ) - 2 \\sin ( 2 \\pi y _ 1 ) ) \\sin ( 2 \\pi y _ 2 ) } , \\\\ b _ 2 ( y _ 1 , y _ 2 ) & : = \\frac { 1 + \\frac { 1 } { 2 } \\sin ( 2 \\pi y _ 1 ) \\sin ( 2 \\pi y _ 2 ) } { 1 + \\frac { 1 } { 4 } ( \\cos ( 2 \\pi y _ 1 ) - 2 \\sin ( 2 \\pi y _ 1 ) ) \\sin ( 2 \\pi y _ 2 ) } \\end{align*}"}
+{"id": "1314.png", "formula": "\\begin{align*} S _ { f , \\tau } : \\mathcal { X } \\ni x \\mapsto S _ { f , \\tau } x \\coloneqq \\sum _ { j = 1 } ^ n f _ j ( x ) \\tau _ j \\in \\mathcal { X } . \\end{align*}"}
+{"id": "8833.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { k = 0 } ^ { + \\infty } e ^ { - t \\lambda _ k } \\sim \\frac { 1 } { ( 2 \\sqrt { \\pi t } ) ^ d } ( a _ 0 + a _ 1 t + a _ 2 t ^ 2 + \\cdots ) , t \\searrow 0 ^ { + } . \\end{align*}"}
+{"id": "1686.png", "formula": "\\begin{align*} \\delta \\mu _ m = \\frac { d } { d x } { \\rm e x p } ( x \\delta ) \\mu _ m \\mid _ { x = 0 } = \\frac { d } { d x } \\imath ( e ^ x ) \\mu _ m \\mid _ { x = 0 } = \\frac { d } { d x } e ^ { - m x } \\mid _ { x = 0 } \\mu _ m = - m \\mu _ m . \\end{align*}"}
+{"id": "4471.png", "formula": "\\begin{align*} n = 4 q + 3 . \\end{align*}"}
+{"id": "4987.png", "formula": "\\begin{align*} \\iint _ { \\R ^ d \\times \\R ^ d } V ( q ) \\left | ( G _ { p , q } , f ) \\right | ^ 2 \\ , \\frac { d p \\ , d q } { ( 2 \\pi ) ^ d } = \\int _ { \\R ^ d } ( V * | g | ^ 2 ) ( x ) | f ( x ) | ^ 2 \\ , d x \\ , . \\end{align*}"}
+{"id": "3662.png", "formula": "\\begin{align*} M ( \\phi , \\varphi ) : = \\phi - [ \\phi + c \\varphi ] _ + , \\end{align*}"}
+{"id": "5731.png", "formula": "\\begin{align*} \\mathbf { A } _ { B } ^ { A } & = \\alpha _ { B } ^ { A } - \\digamma _ { B } ^ { A } \\mathbf { m , } \\\\ \\digamma _ { B } ^ { A } & = d \\alpha _ { B } ^ { A } + \\alpha _ { C } ^ { A } \\alpha _ { B } ^ { C } \\end{align*}"}
+{"id": "6150.png", "formula": "\\begin{align*} \\begin{cases} u '' - \\Delta u = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ v '' - \\Delta v = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ u = v = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu u + \\alpha u = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 , \\\\ \\partial _ \\nu v + \\beta v = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{cases} \\end{align*}"}
+{"id": "865.png", "formula": "\\begin{align*} \\left | { { Q _ { { R _ i } } } } \\right | + 2 \\left | { { S _ { { R _ i } } } } \\right | = \\min \\left \\{ { \\left | { V ( { R _ i } ) } \\right | , \\left ( { \\frac { 1 } { 3 } - 3 \\sqrt \\delta } \\right ) v ( F ) - 1 } \\right \\} . \\end{align*}"}
+{"id": "5843.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n a _ { i j } ( x ) \\eta _ i \\eta _ j \\geq c \\sum _ { i = 1 } ^ n \\eta _ i ^ 2 \\end{align*}"}
+{"id": "6461.png", "formula": "\\begin{align*} m _ * \\wedge H ( m _ * ) & = g _ * ( t ) . ( w _ * \\wedge H ( w _ * ) ) = - h ( t ) e _ 1 \\wedge m _ * , \\\\ m _ * \\wedge ( m _ * \\wedge H ( m _ * ) ) & = g _ * ( t ) . ( w _ * \\wedge ( w _ * \\wedge H ( w _ * ) ) ) = - h ( t ) m _ * \\wedge ( e _ 1 \\wedge m _ * ) . \\end{align*}"}
+{"id": "8121.png", "formula": "\\begin{align*} u _ 2 '' ( y ) = \\frac { 2 } { 3 } x ^ { \\frac { 1 } { 3 } } y ^ { - \\frac { 5 } { 3 } } - \\frac { 1 } { 2 } p ^ { \\frac { 1 } { 2 } } c ^ { - 1 } y ^ { - \\frac { 3 } { 2 } } , \\end{align*}"}
+{"id": "2663.png", "formula": "\\begin{align*} \\vartheta : = \\frac { 1 } { q _ 1 } + \\frac { 1 } { 2 \\alpha } \\bigg ( \\frac { 1 } { q _ 1 } - \\frac { 1 } { q _ 2 } \\bigg ) . \\end{align*}"}
+{"id": "4485.png", "formula": "\\begin{align*} 2 \\cdot 6 & = 1 2 = 6 + 3 + 2 + 1 \\\\ 2 \\cdot 2 8 & = 5 6 = 2 8 + 1 4 + 7 + 4 + 2 + 1 . \\end{align*}"}
+{"id": "1038.png", "formula": "\\begin{align*} \\mathcal { R } _ { n , \\alpha } ( \\varepsilon ) \\geq \\inf _ { Q \\in \\mathcal { Q } _ \\alpha } \\inf _ { \\phi \\in \\Phi _ Q } \\left \\{ \\mathbb { E } _ { \\tilde { P } , Q } [ \\phi ] + \\mathbb { E } _ { \\tilde { P } , Q } [ 1 - \\phi ] \\right \\} = 1 . \\end{align*}"}
+{"id": "1665.png", "formula": "\\begin{align*} \\ell ( \\Phi _ 1 ( \\underline { \\delta s } ( \\mu _ { \\underline m } ) ) , \\chi ) \\cdot \\ell ( \\Phi _ 2 ( \\underline { \\delta s } ( \\mu _ { - \\underline m } ) , \\chi ^ { - 1 } ) = C ( \\underline k , \\underline m ) \\cdot L ( 1 / 2 , \\Pi , \\chi ) \\cdot \\frac { \\langle \\Phi _ 1 , \\Phi _ 2 \\rangle } { \\langle \\Psi , \\Psi \\rangle } \\cdot \\prod _ { v \\nmid \\infty } \\left ( \\beta _ v ( \\Phi _ { 1 , v } , \\Phi _ { 2 , v } ) \\cdot \\alpha _ { v } ( W _ { \\mathfrak { f } , v } , W ^ - _ { \\mathfrak { f } , v } ) \\right ) , \\end{align*}"}
+{"id": "2198.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty p ( n ) q ^ n = \\dfrac { 1 } { ( q ; q ) _ \\infty } , \\end{align*}"}
+{"id": "5352.png", "formula": "\\begin{align*} M _ { F , s } [ \\varepsilon ] : = \\sum _ { \\substack { j _ 1 , \\dots , j _ s \\in F \\\\ j _ 1 < \\dots < j _ s } } \\mu _ { j _ 1 } [ \\varepsilon ] \\cdots \\mu _ { j _ s } [ \\varepsilon ] \\end{align*}"}
+{"id": "6665.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| x _ i ^ k - \\bar x ^ k \\| = \\lim _ { k \\to \\infty } \\| y _ i ^ k - v _ i \\bar y ^ k \\| = 0 , \\ : \\forall i a . s . \\end{align*}"}
+{"id": "8640.png", "formula": "\\begin{align*} \\Delta = \\sinh ( r ) ^ { 1 - d } \\partial _ r \\left ( \\sinh ( t ) ^ { d - 1 } \\partial _ r \\right ) + \\sinh ( r ) ^ { - 2 } \\Delta _ \\Sigma , \\end{align*}"}
+{"id": "1392.png", "formula": "\\begin{align*} \\psi = \\bigoplus _ \\rho \\bigoplus _ { z \\in ( 1 / 2 ) \\Z _ { \\geq 0 } } \\left ( \\rho \\boxtimes S _ { z + 1 + \\delta _ z } \\boxtimes S _ { z + 1 - \\delta _ z } \\right ) ^ { \\oplus ( k _ { \\rho , z } - k _ { \\rho , z + 1 } ) } \\end{align*}"}
+{"id": "4375.png", "formula": "\\begin{align*} a _ 2 = G \\left ( \\frac { p } { q } - \\frac { 1 } { a _ 1 } \\right ) = q a _ 1 + 1 . \\end{align*}"}
+{"id": "5686.png", "formula": "\\begin{align*} \\overset { p } { \\mathbf { p } } = \\overset { p } { \\pi } + \\overset { p + 1 } { \\pi _ { 1 } } \\mathbf { m } + \\overset { p + 1 } { \\pi _ { 2 } } \\overline { \\mathbf { m } } + \\overset { p + 2 } { \\pi } \\mathbf { m } \\overline { \\mathbf { m } } . \\end{align*}"}
+{"id": "8598.png", "formula": "\\begin{align*} m _ { \\mathbf { s } } ( x ) = f ^ { * } ( x ) = f ( x ) = \\frac { F ( x ) } { d ( x ) } = x ^ { 1 2 } + x ^ 9 + x ^ 6 + x ^ 3 + 1 . \\end{align*}"}
+{"id": "2695.png", "formula": "\\begin{align*} \\frac { d M } { d t } & = f ( x ( t ) ) x ' ( t ) \\\\ \\frac { d P } { d t } & = f ' ( x ( t ) ) x ' ( t ) . \\end{align*}"}
+{"id": "4889.png", "formula": "\\begin{align*} C _ { s _ 1 s _ 2 u _ 0 } C _ { u _ l } = \\xi ^ 3 ( C _ { s _ 1 } C _ { s _ 2 } - 1 ) C _ { u _ l } . \\end{align*}"}
+{"id": "8362.png", "formula": "\\begin{align*} w = 0 , \\mathrm { f o r } \\ | x | > k | t | + R , \\ | t | < \\nu ( k ) . \\end{align*}"}
+{"id": "5560.png", "formula": "\\begin{align*} \\sum _ { n = [ x ] + 1 } ^ \\infty \\frac { x ^ 2 } { y _ n ^ 3 } e ^ { - \\frac { x ^ 2 } { y _ n ^ 2 } } \\ll \\sum _ { n > x } \\frac { x ^ 2 } { n ^ 3 } \\ll O ( 1 ) . \\end{align*}"}
+{"id": "3943.png", "formula": "\\begin{align*} c _ { q _ i , p _ j } ( t q , t p ) & = c _ { q _ i , p _ j } ( 0 , 0 ) + c _ { q _ i q _ k , p _ j } ( \\tau q , \\tau p ) q _ k + c _ { q _ i , p _ j p _ k } ( \\tau q , \\tau p ) p _ k \\\\ & = c _ { q _ i , p _ j } ( 0 , 0 ) + c _ { q _ i q _ k , p _ j } ( \\tau q , 0 ) q _ k + a ^ { ( 3 ) } _ { i j , k l } ( q , p ) q _ k p _ l \\\\ & \\quad \\quad + c _ { q _ i , p _ j p _ k } ( 0 , \\tau p ) p _ k + a ^ { ( 4 ) } _ { i j , k l } ( q , p ) q _ k p _ l . \\end{align*}"}
+{"id": "3493.png", "formula": "\\begin{align*} { \\gamma _ { { \\rm { M R T } } } } = \\frac { { P { { \\left \\| { { \\bf { \\bar h } } } \\right \\| } ^ 2 } } } { { P \\sum \\nolimits _ { i = - { n _ { { \\rm { s p a n } } } } , i \\ne 0 } ^ { { n _ { { \\rm { s p a n } } } } } { { { \\left | { { { { \\bf { \\bar g } } } ^ H } \\left [ i \\right ] { \\bf { \\bar h } } / \\left \\| { { \\bf { \\bar h } } } \\right \\| } \\right | } ^ 2 } } + { \\sigma ^ 2 } } } . \\end{align*}"}
+{"id": "1433.png", "formula": "\\begin{align*} \\mathbb { P } ( | \\mathcal { C } _ { \\max } ( \\mathbb { G } ( n , p ) ) | < n ^ { 2 / 3 } / A ) \\leq \\mathbb { P } ( | \\mathcal { C } _ { \\max } ( \\mathbb { G } ( n , p ) ) | < T ) = \\mathbb { P } ( t _ i - t _ { i - 1 } < T \\forall i ) . \\end{align*}"}
+{"id": "2862.png", "formula": "\\begin{align*} ( \\sigma f ) ( x _ 1 , x _ 2 , x _ 3 ) & = f ( x _ 2 , x _ 3 , x _ 1 ) = x _ 2 x _ 3 ^ 2 x _ 1 ^ 3 = x _ 1 ^ 3 x _ 2 x _ 3 ^ 2 \\\\ ( \\tau f ) ( x _ 1 , x _ 2 , x _ 3 ) & = f ( x _ 2 , x _ 1 , x _ 3 ) = x _ 2 x _ 1 ^ 2 x _ 3 ^ 3 = x _ 1 ^ 2 x _ 2 x _ 3 ^ 3 \\\\ ( ( \\sigma \\tau ) f ) ( x _ 1 , x _ 2 , x _ 3 ) & = f ( x _ 3 , x _ 2 , x _ 1 ) = x _ 3 x _ 2 ^ 2 x _ 1 ^ 3 = x _ 1 ^ 3 x _ 2 ^ 2 x _ 3 \\\\ ( \\tau \\sigma f ) ( x _ 1 , x _ 2 , x _ 3 ) & = f ( x _ 1 , x _ 3 , x _ 2 ) = x _ 1 x _ 3 ^ 2 x _ 2 ^ 3 = x _ 1 x _ 2 ^ 3 x _ 3 ^ 2 \\\\ \\end{align*}"}
+{"id": "8248.png", "formula": "\\begin{align*} x = y \\enspace \\to \\enspace \\forall P , \\left ( P ( x ) \\leftrightarrow P ( y ) \\right ) . \\end{align*}"}
+{"id": "1039.png", "formula": "\\begin{align*} Z _ i = \\begin{cases} Y _ i , & U _ i \\leq e ^ { \\alpha } / ( 1 + e ^ { \\alpha } ) , \\\\ 1 - Y _ i , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "3582.png", "formula": "\\begin{align*} - \\frac { ( 1 - \\delta _ { \\lambda _ 1 \\lambda _ 2 } ) d _ 3 ^ - ( \\lambda ) } { \\lambda _ 1 - \\lambda _ 3 + 2 } E _ { 3 1 } \\Omega _ { \\lambda - \\epsilon _ 3 } \\end{align*}"}
+{"id": "2730.png", "formula": "\\begin{align*} [ E _ i , F _ j ] = \\delta _ { i j } \\frac { K _ i - K _ i ' } { q - q ^ { - 1 } } , K _ i E _ j & = q _ i ^ { c _ { i j } } E _ j K _ i , K _ i F _ j = q _ i ^ { - c _ { i j } } F _ j K _ i , \\\\ K _ i ' E _ j = q _ i ^ { - c _ { i j } } E _ j K _ i ' , & K _ i ' F _ j = q _ i ^ { c _ { i j } } F _ j K _ i ' , \\end{align*}"}
+{"id": "3108.png", "formula": "\\begin{align*} c _ j ^ { k l } ( \\tilde { A } ) & = \\bar { \\gamma } \\left ( c _ j ^ { k l } ( A ) - \\bar { \\gamma } \\ , \\bar { a } _ { k l } \\sum _ { s , t = 1 } ^ n c _ { s t } \\int _ Y r A e _ j \\cdot \\nabla v ^ { s t } \\right ) \\\\ & = \\bar { \\gamma } \\left ( c _ j ^ { k l } ( A ) - \\bar { \\gamma } \\ , \\bar { a } _ { k l } \\sum _ { s , t = 1 } ^ n c _ { s t } \\ , c _ j ^ { s t } ( A ) \\right ) \\\\ & = 0 , \\end{align*}"}
+{"id": "1554.png", "formula": "\\begin{align*} \\tilde { \\zeta } _ { \\mathrm { u } } ( t ) = & \\ , t ^ { p ^ { + } - r ^ { - } } \\int _ { M } | D \\mathrm { u } ( z ) | ^ { p ( z ) } \\ , \\ , d v _ { g } ( z ) + t ^ { q ^ { + } - r ^ { - } } \\int _ { M } \\mu ( z ) \\ , | D \\mathrm { u } ( z ) | ^ { q ( z ) } \\ , \\ , d v _ { g } ( z ) \\\\ & - t ^ { - r ^ { - } - \\gamma ^ { - } + 1 } \\int _ { M } g ( z ) | \\mathrm { u } ( z ) | ^ { 1 - \\gamma ( z ) } \\ , \\ , d v _ { g } ( z ) \\ , \\ , \\ , \\mbox { f o r a l l } \\ , \\ , \\ , t > 0 . \\end{align*}"}
+{"id": "2900.png", "formula": "\\begin{align*} S ( K ) = \\int _ { \\mathbb { T } ^ \\infty } | 1 - \\varphi | ^ p K d m _ { \\infty } . \\end{align*}"}
+{"id": "1097.png", "formula": "\\begin{align*} \\big | \\mathbb { E } \\left ( \\overline { Z } _ { j } - \\theta _ j \\right ) \\big | = \\Big | \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\varepsilon ( \\mathbb { E } _ g [ \\varphi _ j ( X _ i ) ] - \\theta _ j ) \\Big | \\leq \\varepsilon ( B _ 0 + | \\theta _ j | ) . \\end{align*}"}
+{"id": "3233.png", "formula": "\\begin{align*} | V _ 1 | = m _ 1 = k , \\ \\ | V _ 2 | = 0 , \\ \\ | V _ 3 | = k + 2 . \\end{align*}"}
+{"id": "8359.png", "formula": "\\begin{align*} \\Gamma _ { a , \\delta } : = \\left \\{ - ( | y | - \\delta ) ^ 2 + ( s - 2 \\delta ) ^ 2 + a ^ 2 = 0 , | y | > \\delta , s > 0 \\right \\} , \\end{align*}"}
+{"id": "3471.png", "formula": "\\begin{align*} & \\| D _ { z , w } ^ 2 F _ R ( t ) \\| _ 4 = \\left \\| \\int _ { B _ R } D _ { z , w } ^ 2 u ( t , x ) d x \\right \\| _ 4 \\leq \\int _ { B _ R } \\| D _ { z , w } ^ 2 u ( t , x ) \\| _ 4 d x \\leq C \\int _ { B _ R } \\widetilde { f } _ 2 ( w , z , x ; t ) d x = \\\\ & C \\int _ { B _ R } \\left ( \\int _ { 0 < \\theta < r < t } G _ { t - r } ( x - z ) G _ { r - \\theta } ( z - w ) d r d \\theta + \\int _ { 0 < r < \\theta < t } G _ { t - \\theta } ( x - w ) G _ { \\theta - r } ( w - z ) d r d \\theta \\right ) d x . \\end{align*}"}
+{"id": "7962.png", "formula": "\\begin{align*} f _ 2 ( f _ 1 ( x ) ) = f _ 2 ( u ( x _ 1 \\vee x ) ) = v ( x _ 2 \\vee u ( x _ 1 \\vee x ) ) = v ( u ( u ^ { - 1 } ( x _ 2 ) \\vee x _ 1 \\vee x ) ) . \\end{align*}"}
+{"id": "2754.png", "formula": "\\begin{align*} \\overline { h _ { [ \\varepsilon ] } ^ { \\mathbf { p } } } = f _ { [ \\varepsilon ] } ^ { \\mathbf { p } } . \\end{align*}"}
+{"id": "6225.png", "formula": "\\begin{align*} t \\geqslant T : U = \\sum _ { r = 1 } ^ p ( E _ r , U ) e _ r = \\sum _ { r = 1 } ^ p u _ r e _ r \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "7301.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { y \\in [ 0 , x _ 0 ] } | G ^ k _ { m _ n } ( y ) | = 0 \\ \\ k \\geq 1 . \\end{align*}"}
+{"id": "7527.png", "formula": "\\begin{align*} Z _ g ( s , \\chi ) = Z _ g \\big ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) + \\sum _ { i = 1 } ^ { 5 } Z _ g \\Big ( s , \\chi , S ( \\Delta _ { \\gamma _ i } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) ) \\Big ) , \\end{align*}"}
+{"id": "8005.png", "formula": "\\begin{align*} V _ E & : = \\left \\{ F _ 1 , . . . , F _ n \\right \\} , \\\\ V _ V & : = \\Big \\{ \\Delta _ { p \\tau } x _ k ^ { ( q ) } \\ ; \\Big | \\ ; \\exists k , p , q \\in \\mathbb { N } _ 0 \\cup \\{ - 1 \\} \\\\ & \\qquad \\qquad \\qquad \\ ; \\ ; \\Delta _ { p \\tau } x _ k ^ { ( q ) } \\Big \\} , \\\\ E & : = \\left \\{ \\left . \\{ F _ i , v _ k \\} \\in V _ E \\times V _ V \\ ; \\right | \\ ; v _ k F _ i \\right \\} . \\end{align*}"}
+{"id": "6779.png", "formula": "\\begin{align*} \\phi _ { \\chi } ^ o ( s ) : = \\pi ^ { - \\frac { s + 1 } { 2 } } \\Gamma \\left ( \\frac { s + 1 } { 2 } \\right ) L ( s , \\chi ) = \\int _ { 0 } ^ { \\infty } y ^ { \\frac { s } { 2 } - 1 } \\tilde { \\Psi } _ { \\chi } ( y ) d y \\end{align*}"}
+{"id": "5032.png", "formula": "\\begin{align*} F _ { * x _ o } = \\left ( \\begin{array} { c c c c c } 1 2 8 \\sqrt { 2 } & 0 & 0 & 0 & - 1 2 8 \\\\ 0 & 2 5 6 & 0 & 0 & - 6 4 \\\\ 0 & 0 & 2 5 6 & 0 & - 6 4 \\\\ 0 & 0 & 0 & - 1 9 2 & 0 \\end{array} \\right ) , \\end{align*}"}
+{"id": "146.png", "formula": "\\begin{align*} ( \\xi \\varkappa ) ( \\theta ) = z ( \\theta ) , \\end{align*}"}
+{"id": "1622.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { \\tau } } = \\mathcal { Q } ^ { - 1 } \\left ( \\epsilon ^ * \\right ) + \\mathcal { Q } ^ { - 1 } \\left ( \\frac { \\epsilon ^ * } { M - 1 } \\right ) \\end{align*}"}
+{"id": "4787.png", "formula": "\\begin{align*} K _ { \\{ w , w ' \\} } ( y ) & = K _ { w } ( y ) + K _ { w ' } ( y ) \\\\ & \\leq 2 \\cdot \\max \\big \\{ K _ { w } ( y ) , K _ { w ' } ( y ) \\big \\} . \\end{align*}"}
+{"id": "5137.png", "formula": "\\begin{align*} \\forall w \\in \\mathbb { T } , D G ( \\lambda , b , \\Omega , 0 , 0 ) ( h _ { 1 } , h _ { 2 } ) ( w ) = \\sum _ { n = 1 } ^ { \\infty } n \\mathbf { m } M _ { n \\mathbf { m } } ( \\lambda , b , \\Omega ) \\left ( \\begin{array} { c } a _ { n } \\\\ b _ { n } \\end{array} \\right ) e _ { n \\mathbf { m } } ( w ) . \\end{align*}"}
+{"id": "6630.png", "formula": "\\begin{align*} \\partial _ { x _ j } ^ p \\frac { x _ j ^ { 2 r } } { ( 1 + \\epsilon x ^ { 2 m } ) ^ 2 } = \\sum _ { \\ell = 0 } ^ p \\binom { p } { \\ell } ( 2 r ) \\cdots ( 2 r - \\ell + 1 ) x _ j ^ { 2 r - \\ell } \\partial _ { x _ j } ^ { p - \\ell } \\frac { 1 } { ( 1 + \\epsilon x ^ { 2 m } ) ^ 2 } . \\end{align*}"}
+{"id": "7728.png", "formula": "\\begin{align*} A ^ - _ { \\beta } = \\frac { - 1 - \\rho _ 0 \\cos ( \\beta \\pi ) + \\sqrt { 1 + 2 \\rho _ 0 \\cos ( \\beta \\pi ) + \\rho _ 0 ^ 2 } } { 2 } . \\end{align*}"}
+{"id": "8776.png", "formula": "\\begin{align*} \\nu _ t ^ x = \\nu _ 0 ^ x \\circ ( \\mathcal { S } ^ x _ { t , 0 } ) ^ { - 1 } [ \\eta _ 0 , \\nu _ { \\cdot } , \\omega ] , x \\in X , \\end{align*}"}
+{"id": "5561.png", "formula": "\\begin{align*} | S _ 2 ( x ^ 2 ) | = O _ { \\epsilon } \\left ( x ^ { \\frac { 1 } { 2 } - k + \\epsilon } \\right ) . \\end{align*}"}
+{"id": "7364.png", "formula": "\\begin{align*} Q _ i : = \\left [ 0 , \\gamma _ i \\right ) \\times [ 0 , 1 ) ^ { k - 1 } , \\end{align*}"}
+{"id": "5167.png", "formula": "\\begin{align*} \\mathcal { I } ( G \\wedge G ' ) & \\geq \\dfrac { 6 ! \\cdot 4 ^ { 4 } \\cdot 2 ^ { 2 } } { 4 ! \\cdot 2 ! \\cdot 6 ^ { 6 } } \\cdot \\frac { 3 } { 8 } \\cdot 1 = \\frac { 1 0 } { 8 1 } \\end{align*}"}
+{"id": "126.png", "formula": "\\begin{align*} | f ( x ) - f ( y ) | & = \\left | \\tilde { f } ( s _ x ) - \\tilde { f } ( s _ y ) \\right | \\\\ & = \\left | \\int _ { s _ y } ^ { s _ x } \\tilde { f } ^ { \\prime } ( s ) d s \\right | \\\\ & \\leq \\left | \\int _ { s _ y } ^ { s _ x } \\left | \\nabla f ( \\hat { x } + s \\omega ) \\cdot \\omega \\right | d s \\right | \\\\ & \\leq | x - y | ^ { \\frac { p - 1 } { p } } \\left | \\int _ { s _ y } ^ { s _ x } \\left | \\nabla f ( \\hat { x } + s \\omega ) \\cdot \\omega \\right | ^ p d s \\right | ^ { \\frac 1 p } . \\end{align*}"}
+{"id": "6865.png", "formula": "\\begin{align*} J ^ h ( u ; F ) = \\norm { Q u - F } ^ 2 _ h , u _ h = \\underset { v \\in X _ h } { \\arg \\min } \\ ; J ^ h ( v ; F ) . \\end{align*}"}
+{"id": "5485.png", "formula": "\\begin{align*} b ( x , \\mu , 1 ) & = - x ^ 3 - x , ~ b ( x , \\mu , 2 ) = - \\frac { 1 } { 2 } x , \\\\ \\sigma ( x , \\mu , 1 ) & = \\int _ { \\mathbb R } x \\mu ( d x ) , ~ \\sigma ( x , \\mu , 2 ) = x + 2 \\int _ { \\mathbb R } x \\mu ( d x ) . \\end{align*}"}
+{"id": "4242.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\Delta Q + Q - | Q | ^ { \\frac { 4 } { N } } Q = 0 , \\end{align*}"}
+{"id": "7092.png", "formula": "\\begin{align*} \\pi _ { m + n \\sigma } = \\tilde { \\pi } _ { m + n \\sigma } + \\gamma q _ 1 \\end{align*}"}
+{"id": "9053.png", "formula": "\\begin{align*} W _ T ^ { i , n } = Y _ 0 ^ { i , n } - \\xi ^ { i , n } & - \\int _ 0 ^ T f ( s , Y _ s ^ { i , n } , Z _ s ^ { i , i , n } , U _ s ^ { i , i , n } , L _ { n } [ \\textbf { Y } _ s ^ n ] ) d s \\\\ & + \\sum _ { j = 1 } ^ n \\int _ 0 ^ T Z _ s ^ { i , j , n } d B ^ j _ s + \\int _ 0 ^ T \\int _ { \\R ^ * } \\sum _ { j = 1 } ^ n U _ s ^ { i , j , n } ( e ) \\Tilde { N } ^ j ( d s , d e ) \\end{align*}"}
+{"id": "1736.png", "formula": "\\begin{align*} s ( \\mu ) = \\sum _ { | \\lambda | \\leq n \\leq M } ( 2 n + 1 ) \\binom { 2 n } { n + \\lambda } \\cdot \\varphi _ n \\circ t _ n ( \\mu ) . \\end{align*}"}
+{"id": "1059.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { j \\geq - 1 } \\sum _ { k \\in \\mathbb { Z } } \\beta _ { j k } \\psi _ { j k } ( x ) = \\sum _ { j \\geq - 1 } \\sum _ { k \\in \\mathbb { Z } } \\left \\{ \\int _ { \\mathbb { R } } f ( x ' ) \\psi _ { j k } ( x ' ) \\ , \\mathrm { d } x ' \\right \\} \\psi _ { j k } ( x ) . \\end{align*}"}
+{"id": "4002.png", "formula": "\\begin{align*} w ^ { i i } A _ { i i , p _ 1 p _ 1 } w _ { 1 1 } ^ 2 & = w ^ { 1 1 } A _ { 1 1 , p _ 1 p _ 1 } w _ { 1 1 } ^ 2 + \\sum _ { i = 2 } ^ n w ^ { i i } A _ { i i , p _ 1 p _ 1 } w _ { 1 1 } ^ 2 \\\\ & \\geq - C w _ { 1 1 } . \\end{align*}"}
+{"id": "1417.png", "formula": "\\begin{align*} \\bigsqcup _ { j = 1 } ^ t \\{ \\underbrace { [ x + j - 1 , x ] _ \\rho , \\dots , [ x + j - 1 , x ] _ \\rho } _ { k _ { j - 1 } - k _ j } \\} \\end{align*}"}
+{"id": "5139.png", "formula": "\\begin{align*} \\| R _ { N } \\| _ { L ^ { \\infty } ( \\mathbb { T } ) } \\leqslant \\left ( \\sum _ { n = N + 1 } ^ { \\infty } \\frac { 1 } { n ^ { 2 } } \\right ) ^ { \\frac { 1 } { 2 } } \\| g _ { 2 } \\| _ { C ^ { \\alpha } ( \\mathbb { T } ) } \\underset { N \\rightarrow \\infty } { \\longrightarrow } 0 . \\end{align*}"}
+{"id": "3814.png", "formula": "\\begin{align*} \\bigcap _ { j = 0 } ^ { d - 1 } \\ker ( R B ^ j ) = \\ker ( R ) \\cap \\ker ( R B ) . \\end{align*}"}
+{"id": "8864.png", "formula": "\\begin{align*} ( 2 \\pi ^ { n } ) ^ { - 1 } \\Gamma ( n ) \\left ( \\gamma _ { p , q } ^ { n , \\nu , m } \\right ) ^ { 2 } = \\beta _ { \\nu , n , m } \\frac { ( - 1 ) ^ { p + q } ( - m ) _ { p } ( - m - 2 \\nu ) _ { q } ( n + m + 2 \\nu ) _ { p } ( n + m ) _ { q } } { ( n ) _ { p + q } ( n ) _ { p + q } } , \\end{align*}"}
+{"id": "4857.png", "formula": "\\begin{align*} Q _ \\delta = \\big \\{ b = ( b _ 1 , \\dots , b _ { k - 1 } ) \\in \\R ^ { k - 1 } \\mid \\min \\{ | b _ 1 | , \\dots , | b _ { k - 1 } | \\} \\geq \\delta , \\ , \\max \\{ | b _ 1 | , \\dots , | b _ { k - 1 } | \\} \\leq 1 \\big \\} . \\end{align*}"}
+{"id": "2037.png", "formula": "\\begin{align*} \\nu ( n ) : = \\mathbb { 1 } _ { ] - \\infty , 0 [ } ( n ) \\left ( \\displaystyle \\sum _ { k = 0 } ^ { + \\infty } \\mathcal { U } ( k ) \\ , \\rho ( n - k ) \\right ) + \\mathbb { 1 } _ { [ 0 , + \\infty [ } ( n ) \\left ( \\displaystyle \\sum _ { k = 0 } ^ { + \\infty } \\mathcal { U } ' ( - k ) \\ , \\rho ( n + k ) \\right ) . \\end{align*}"}
+{"id": "3434.png", "formula": "\\begin{align*} \\frac { \\mu _ { t } ( [ b ] ) } { \\mu _ { t } ( [ a ] ) } = \\sum _ { m = 1 } ^ { \\infty } m p ^ t _ { a b } ( p _ { b b } ^ t ) ^ { m - 1 } p ^ t _ { b a } = \\frac { p ^ t _ { a b } p ^ t _ { b a } } { ( 1 - p ^ t _ { b b } ) ^ 2 } . \\end{align*}"}
+{"id": "7534.png", "formula": "\\begin{align*} \\Delta _ { \\gamma _ 2 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) = \\bigcup \\limits _ { m = 0 } ^ { j _ 0 - 1 } \\big \\{ ( m + b j _ 0 , \\frac { m i _ 0 + n _ m } { j _ 0 } + b i _ 0 + a ) | a , b \\in \\mathbb { Z } ^ { + } \\} \\end{align*}"}
+{"id": "6203.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ d \\beta _ r E _ r = C _ p ^ T x . \\end{align*}"}
+{"id": "7017.png", "formula": "\\begin{align*} X & = W + N _ X \\\\ Y & = W + N _ Y , \\end{align*}"}
+{"id": "1161.png", "formula": "\\begin{align*} & E _ { \\mathcal { X } } ( j , k ) : \\sigma _ k ( a , b ) \\iff \\exists a ' , b ' \\in D _ j , \\ , \\sigma _ j ( a ' , b ' ) \\wedge E ^ { j k } _ { \\ddot { \\mathcal { A } } } ( a , a ' ) \\wedge E ^ { j k } _ { \\ddot { \\mathcal { A } } } ( b , b ' ) . \\end{align*}"}
+{"id": "8943.png", "formula": "\\begin{align*} b _ { j } ^ { \\left ( \\nu , 4 \\right ) } = \\frac { ( 4 \\pi ) ^ 4 } { 4 ! } \\sum \\limits _ { i = 0 } ^ { j } \\frac { \\left ( 1 + \\nu ^ { 2 } \\right ) ^ { j - i } } { \\left ( j - i \\right ) ! } c _ { i } ^ { \\left ( \\nu , 4 \\right ) } , \\end{align*}"}
+{"id": "2427.png", "formula": "\\begin{align*} \\begin{aligned} F ( t ; q ) = \\frac { 1 } { 2 \\pi \\mathrm { i } } \\int _ { \\Gamma _ { \\theta , \\kappa } } e ^ { z t } z ^ { \\alpha - 1 } ( z ^ \\alpha + A ( q ) ) ^ { - 1 } \\ , \\d z ~ ~ ~ ~ E ( t ; q ) = \\frac { 1 } { 2 \\pi \\mathrm { i } } \\int _ { \\Gamma _ { \\theta , \\kappa } } e ^ { z t } ( z ^ \\alpha + A ( q ) ) ^ { - 1 } \\ , \\d z , \\end{aligned} \\end{align*}"}
+{"id": "4288.png", "formula": "\\begin{align*} & \\varphi , \\ , \\psi \\in \\mathcal { C } ^ \\infty ( \\R ^ 3 ) , 0 \\le \\varphi , \\ , \\psi \\le 1 , \\varphi + \\psi = 1 \\quad \\quad \\R ^ 3 , \\\\ & \\varphi = 1 \\quad B _ R , \\quad \\varphi \\subset B _ { R + 1 } . \\end{align*}"}
+{"id": "1548.png", "formula": "\\begin{align*} T _ { \\tau \\mu \\nu } & : = g _ { \\tau \\pi } T ^ \\pi { } _ { \\mu \\nu } \\\\ T ^ { \\pi \\tau \\rho } & : = g ^ { \\tau \\mu } g ^ { \\rho \\nu } T ^ \\pi { } _ { \\mu \\nu } \\end{align*}"}
+{"id": "8436.png", "formula": "\\begin{align*} c ( K ) = \\gamma _ K ( X ) \\leqslant \\sup _ { \\nu \\in \\widehat { \\mathcal E } ^ + _ K } \\ , \\nu ( X ) \\leqslant \\sup _ { \\nu \\in \\widehat { \\mathcal E } ^ + _ A } \\ , \\nu ( X ) . \\end{align*}"}
+{"id": "2007.png", "formula": "\\begin{align*} ( N ( \\phi ) \\chi _ F , \\chi _ F ) = \\sum _ { i , j \\in F } N _ { i j } ( \\phi ) \\end{align*}"}
+{"id": "5669.png", "formula": "\\begin{align*} \\Lambda \\sum _ { m \\geq 0 } ( - \\Lambda ) ^ m & = \\Lambda ( 1 - \\Lambda + \\Lambda ^ 2 \\ldots ) = \\frac { \\Lambda } { 1 + \\Lambda } , \\\\ \\Lambda ^ 2 \\sum _ { m \\geq 0 } ( m + 1 ) ( - \\Lambda ) ^ m & = \\Lambda ^ 2 ( 1 - 2 \\Lambda + 3 \\Lambda ^ 2 \\ldots ) = \\Lambda ^ 2 \\frac { d } { d \\Lambda } \\frac { \\Lambda } { 1 + \\Lambda } = \\frac { \\Lambda ^ 2 } { ( 1 + \\Lambda ) ^ 2 } . \\end{align*}"}
+{"id": "5162.png", "formula": "\\begin{align*} \\dfrac { P ( D _ { 0 } ) + P ( D _ { 1 } ) + P ( D _ { 2 } ) + \\ldots + P ( D _ { k - 1 } ) } { k } \\geq \\left ( \\prod ^ { k - 1 } _ { j = 0 } P ( D _ { j } ) \\right ) ^ { \\frac { 1 } { k } } . \\end{align*}"}
+{"id": "662.png", "formula": "\\begin{align*} d s ^ 2 = \\lambda ( z ) ^ 2 | d z | ^ 2 = \\lambda ( z ) ^ 2 ( d x ^ 2 + d y ^ 2 ) \\end{align*}"}
+{"id": "1073.png", "formula": "\\begin{align*} & \\mathbb { E } ( \\widetilde { N } _ 0 / n ) - \\frac { 1 - \\varepsilon } { 2 } \\{ P _ 0 ( A ) + P _ 1 ( A ) \\} - \\frac { \\varepsilon } { 2 } = P ( A ) - \\frac { 1 - \\varepsilon } { 2 } \\{ P _ 0 ( A ) + P _ 1 ( A ) \\} - \\frac { \\varepsilon } { 2 } \\\\ \\geq & ( 1 - \\varepsilon ) P _ 0 ( A ) - \\frac { 1 - \\varepsilon } { 2 } \\{ P _ 0 ( A ) + P _ 1 ( A ) \\} - \\frac { \\varepsilon } { 2 } = \\frac { 1 - \\varepsilon } { 2 } \\mathrm { T V } ( P _ 0 , P _ 1 ) - \\frac { \\varepsilon } { 2 } . \\end{align*}"}
+{"id": "2969.png", "formula": "\\begin{align*} \\frac { g ^ { \\prime \\prime } ( 0 ) } { 2 } \\mathcal { A } ^ \\varepsilon _ { s , t } ( \\varphi ) = \\lim _ { n \\to \\infty } \\int _ s ^ t \\sum _ { j \\in \\mathbb { Z } } \\tau _ j \\mathcal { Q } ^ n _ \\rho ( \\varepsilon n ; r ) \\nabla ^ n \\varphi ^ n _ j ( r ) d r \\end{align*}"}
+{"id": "7415.png", "formula": "\\begin{align*} w ( X _ { \\alpha } ) = X _ { w ( \\alpha ) } . \\end{align*}"}
+{"id": "2658.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\Delta w \\| _ { L ^ 2 ( I , L ^ 2 ) } & \\leq C ' \\bigg ( \\frac { 1 + \\| a \\| _ { { W ^ { 1 } _ \\infty } } } { \\min ( { \\rho ( a ) } , 1 ) } \\bigg ) ^ 5 \\bigg ( \\frac { 1 + \\| b \\| _ { { W ^ { 1 } _ \\infty } } } { \\min ( { \\rho ( b ) } , 1 ) } \\bigg ) ^ 5 \\| a - b \\| _ { { W ^ { 1 } _ \\infty } } . \\end{aligned} \\end{align*}"}
+{"id": "8092.png", "formula": "\\begin{align*} H _ { m , n } ^ + ( x ) \\ll x T ^ { - 1 } ( m ^ 2 n ) ^ { - \\frac { 1 } { 2 } } T ^ { \\frac { 3 } { 2 } } T ^ { 1 + \\varepsilon } M = x T ^ { \\frac { 3 } { 2 } + \\varepsilon } ( m ^ 2 n ) ^ { - \\frac { 1 } { 2 } } M . \\end{align*}"}
+{"id": "884.png", "formula": "\\begin{align*} \\mathop { \\max } _ { i = 1 , \\cdots , q } f _ { i } ( \\mathcal { X } ^ { t } ) \\geq \\delta _ { \\infty } ^ { 2 } ( \\mathcal { Q } , \\boldsymbol { \\mathcal { S } } ) \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } , \\\\ \\mathbf { E } _ { i \\sim \\mathbf { p } } [ f _ { i } ( \\mathcal { X } ^ { t } ) ] \\geq \\delta _ { \\mathbf { p } } ^ { 2 } ( \\mathcal { Q } , \\boldsymbol { \\mathcal { S } } ) \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } . \\end{align*}"}
+{"id": "6079.png", "formula": "\\begin{align*} m _ n ( z ) : = ( z - x _ { n , 1 } ) \\cdots ( z - x _ { n , g } ) . \\end{align*}"}
+{"id": "7919.png", "formula": "\\begin{align*} & \\mathbb { E } \\left ( \\exp \\left ( - \\gamma ( \\phi ) \\int _ { \\epsilon t } ^ { ( 1 - \\epsilon ) t } \\frac { \\sqrt { t } \\tilde { Z } _ s ^ { \\epsilon } } { ( t - s ) ^ { \\frac { 3 } { 2 } } } d s \\right ) \\right ) = \\mathbb { E } \\left ( \\exp \\left ( - \\gamma ( \\phi ) \\int _ { \\epsilon } ^ { 1 - \\epsilon } \\frac { \\tilde { Z } ^ { \\epsilon } _ { \\lambda t } } { ( 1 - \\lambda ) ^ { \\frac { 3 } { 2 } } } d \\lambda \\right ) \\right ) . \\end{align*}"}
+{"id": "1180.png", "formula": "\\begin{align*} | \\nabla _ { g _ 0 } ^ j ( \\Phi ^ { * } g - g _ { 0 } ) | _ { g _ { 0 } } = O ( r ^ { \\lambda _ 1 - j } ) , \\\\ | \\nabla _ { g _ 0 } ^ j ( \\Phi ^ { * } J - J _ { 0 } ) | _ { g _ { 0 } } = O ( r ^ { \\lambda _ 2 - j } ) , \\\\ | \\nabla _ { g _ 0 } ^ j ( \\Phi ^ { * } \\Omega - \\Omega _ { 0 } ) | _ { g _ { 0 } } = O ( r ^ { \\lambda _ 2 - j } ) . \\end{align*}"}
+{"id": "8652.png", "formula": "\\begin{align*} I ( f _ R ) \\leq \\begin{cases} \\frac { 2 \\pi } { \\ln \\frac { \\tanh ( R / 2 ) } { \\tanh ( a / 2 ) } } ( R ^ 2 - a ^ 2 ) & : d = 2 , \\\\ \\frac { 4 \\pi \\tanh a } { \\tanh R - \\tanh a } \\tanh R ( R ^ 2 - a ^ 2 ) & : d = 3 . \\end{cases} \\end{align*}"}
+{"id": "3443.png", "formula": "\\begin{align*} \\mu _ { \\infty } ( [ 1 1 \\ldots 1 a ] ) = 0 , \\qquad \\mbox { f o r a l l } a \\geq 2 . \\end{align*}"}
+{"id": "9194.png", "formula": "\\begin{align*} | S ( f + g ) - S ( f ) | = \\lim _ { N \\to \\infty } | S _ N ( f + g ) - S _ N ( f ) | \\leq \\lim _ { N \\to \\infty } ( g , ( - \\Delta _ J ^ { \\Lambda _ N } ) ^ { - 1 } g ) = ( g , ( - \\Delta _ J ) ^ { - 1 } g ) , \\end{align*}"}
+{"id": "7356.png", "formula": "\\begin{align*} r _ { i } ' : = 1 0 0 ^ { 2 ^ { i + 1 } - 1 } \\end{align*}"}
+{"id": "6291.png", "formula": "\\begin{align*} { s _ i } ^ 2 & = 1 ; \\\\ s _ i s _ { i + 1 } s _ i & = s _ { i + 1 } s _ i s _ { i + 1 } ; \\\\ s _ i s _ j & = s _ j s _ i , \\ | i - j | \\ge 2 . \\end{align*}"}
+{"id": "6160.png", "formula": "\\begin{align*} E = \\sum _ { r = 1 } ^ d \\alpha _ r E _ r , \\end{align*}"}
+{"id": "7833.png", "formula": "\\begin{align*} \\Lambda ( \\theta ^ \\bot , c ) = \\left \\{ r e ^ { i \\theta } : r > 1 , \\ , | \\theta - \\theta ^ \\bot | < c \\frac { \\log r } { r } \\right \\} , \\end{align*}"}
+{"id": "9242.png", "formula": "\\begin{align*} { ( G _ { j k } ( c ) ) } _ { i l } = \\begin{cases} 1 & i = l , \\\\ c & i = j k = l , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "8279.png", "formula": "\\begin{align*} \\bar x = y , \\ ; \\bar y = z , \\ ; \\bar z = B x + C y + A z - y ^ 2 , \\end{align*}"}
+{"id": "4766.png", "formula": "\\begin{align*} \\omega : H _ 1 ( M , \\mathbb { Z } ) = \\left < [ m _ 1 ] , [ m _ 2 ] \\middle | \\left ( \\begin{matrix} m & - 1 \\\\ - 1 & n \\end{matrix} \\right ) \\left ( \\begin{matrix} [ m _ 1 ] \\\\ [ m _ 2 ] \\end{matrix} \\right ) = 0 \\right > \\to G , \\end{align*}"}
+{"id": "6618.png", "formula": "\\begin{align*} \\biggl ( \\sum _ { j = 1 } ^ d c _ j \\alpha _ j ( x ) \\biggr ) ^ m = \\sum _ { l _ 1 + \\dots + l _ r = m } \\frac { m ! } { l _ 1 ! \\cdots l _ r ! } c _ 1 ^ { l _ 1 } \\cdots c _ r ^ { l _ r } \\alpha _ 1 ( x ) ^ { l _ 1 } \\cdots \\alpha _ r ( x ) ^ { l _ r } . \\end{align*}"}
+{"id": "1312.png", "formula": "\\begin{align*} \\theta _ f : \\mathcal { X } \\ni x \\mapsto \\theta _ f x \\coloneqq ( f _ j ( x ) ) _ { j = 1 } ^ n \\in \\mathbb { K } ^ n . \\end{align*}"}
+{"id": "6548.png", "formula": "\\begin{align*} { \\xi _ k } = & { { \\mathbb P } _ { F A } } + { { \\mathbb P } _ { M D } } = \\Pr \\left ( { { \\kappa ^ 2 } + D _ { j w } ^ { - { \\alpha _ { j w } } } { P _ { j k } } h _ { j w } ^ 2 > { \\varepsilon _ k } } \\right ) \\\\ & + \\Pr \\left ( { D _ { a w } ^ { - { \\alpha _ { a w } } } { P _ { a k } } h _ { a w } ^ 2 + { \\kappa ^ 2 } + D _ { j w } ^ { - { \\alpha _ { j w } } } { P _ { j k } } h _ { j w } ^ 2 < { \\varepsilon _ k } } \\right ) , \\end{align*}"}
+{"id": "1382.png", "formula": "\\begin{align*} R _ { 1 2 } ( u - v ) K _ { 1 } ^ - ( u ) R _ { 2 1 } ( u + v ) K ^ - _ { 2 } ( v ) = K ^ { - } _ { 2 } ( v ) R _ { 1 2 } ( u + v ) K ^ { - } _ { 1 } ( u ) R _ { 2 1 } ( u - v ) , \\end{align*}"}
+{"id": "2468.png", "formula": "\\begin{align*} G _ { \\alpha } ( z _ { 0 } , R ) = G ( z _ { 0 } , R ) \\omega ( R ) ^ { \\alpha } \\leq G . \\end{align*}"}
+{"id": "839.png", "formula": "\\begin{align*} \\frac { \\# ( S _ n \\setminus ( T _ \\epsilon \\cap S _ n ) ) } { \\# S _ n } = O ( e ^ { - \\alpha n } ) \\end{align*}"}
+{"id": "3812.png", "formula": "\\begin{align*} A = \\Delta _ { x ^ { ( 2 ) } } - \\frac { 1 } { 4 } | x ^ { ( 2 ) } | ^ 2 - x ^ { ( 2 ) } \\cdot \\nabla _ { x ^ { ( 1 ) } } + \\nabla _ { x ^ { ( 1 ) } } V ( x ^ { ( 1 ) } ) \\cdot \\nabla _ { x ^ { ( 2 ) } } , \\end{align*}"}
+{"id": "1377.png", "formula": "\\begin{align*} W ( t ) = \\Big ( \\big ( \\bar { B } ^ { - 1 } \\big ) ^ { \\frac { 1 } { 1 + \\epsilon } } + \\big ( \\bar { H } ^ { - 1 } \\big ) ^ { \\frac { 1 } { 1 + \\epsilon } } \\Big ) ^ { - 1 } ( t ) , \\end{align*}"}
+{"id": "6004.png", "formula": "\\begin{align*} \\Phi _ { | D ( v ) } \\simeq \\{ ( t ; [ x _ 1 , \\dots , x _ h ] ; [ 0 : \\dots : 0 : t y _ { n - p + 1 } : \\dots : t y _ { n } : y _ { r + p + 1 } : \\dots : y _ { n } ] \\in \\mathbb { C } \\times \\mathbb { P } ^ { n - 1 } \\times \\mathbb { P } ^ { n - 1 } \\\\ | t \\sum _ { a = r + 1 } ^ { r + p } x _ a y _ { n - p + a - r } + \\sum _ { a = r + p + 1 } ^ h x _ a y _ a = 0 \\} . \\end{align*}"}
+{"id": "8931.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { m } k ^ q = \\frac { 1 } { q + 1 } \\left [ B _ { q + 1 } ( m + 1 ) - B _ { q + 1 } \\right ] , q = 1 , 2 , \\cdots , \\end{align*}"}
+{"id": "6049.png", "formula": "\\begin{align*} \\mathsf V : = \\left [ \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac { x ^ l \\dd x } { w ( x ) } \\right ] _ { i = 1 , l = 0 } ^ { g , g - 1 } , \\end{align*}"}
+{"id": "6910.png", "formula": "\\begin{align*} u _ i = \\frac { \\prod _ { \\tilde { b } \\in R _ { d - 1 - i } } P _ { \\tilde { b } } } { \\prod _ { \\tilde { b } \\in R _ { d - i } } P _ { \\tilde { b } } } \\in \\hat { A } _ { \\sigma } ^ * . \\end{align*}"}
+{"id": "5072.png", "formula": "\\begin{align*} \\alpha ^ { - n } [ x _ 1 ^ { n } \\vec u ] P = ( n + 1 ) [ \\vec u ] B _ 0 + [ \\vec u ] B _ 1 . \\end{align*}"}
+{"id": "1201.png", "formula": "\\begin{align*} \\Psi _ * | _ p v = \\Phi ^ { \\log r } _ * | _ p ( v + r ^ { - 1 } d r ( v ) X | _ p ) . \\end{align*}"}
+{"id": "862.png", "formula": "\\begin{align*} \\eta = \\frac { \\gamma } { 1 5 } , \\ ; \\ ; \\delta = \\min \\left \\{ 1 0 ^ { - 1 6 } , \\ ; \\frac { \\eta ^ { 2 } } { 1 0 0 } \\right \\} \\ ; \\ ; \\epsilon = \\min \\left \\{ \\frac { \\delta } { 1 2 9 6 } , \\epsilon _ 1 \\right \\} \\ ; \\ ; \\ ; \\ ; 0 < \\beta \\ll \\epsilon . \\end{align*}"}
+{"id": "1873.png", "formula": "\\begin{align*} \\alpha _ { z _ i } ( \\sigma ) = \\left \\{ \\begin{array} { l l } \\frac { \\pi } { 2 } & \\mbox { i f $ z _ i - \\psi _ k + p _ { z _ i } \\sigma + q _ { z _ i } \\ge 0 $ } \\\\ \\cos ^ { - 1 } \\left ( \\frac { z _ i - \\psi _ k + p _ { z _ i } \\sigma + q _ { z _ i } } { p _ { z _ i } \\sigma + q _ { z _ i } } \\right ) & \\mbox { i f $ z _ i - \\psi _ k + p _ { z _ i } \\sigma + q _ { z _ i } \\le 0 $ } . \\end{array} \\right . \\end{align*}"}
+{"id": "6050.png", "formula": "\\begin{align*} \\det ( \\mathsf V ) = \\int _ { b _ 1 } ^ { a _ 2 } \\cdots \\int _ { b _ g } ^ { a _ { g + 1 } } \\frac { V ( x _ 1 , \\ldots , x _ g ) } { w ( x _ 1 ) \\cdots w ( x _ g ) } \\dd x _ g \\cdots \\dd x _ 1 \\neq 0 , \\end{align*}"}
+{"id": "6942.png", "formula": "\\begin{align*} L _ { m , n } ^ { I I , ( \\alpha ) } ( x ) = x L _ m ^ { ( - \\alpha - 1 ) } ( x ) L _ { n - m - 1 } ^ { ( \\alpha + 2 ) } ( x ) + ( m - \\alpha - 1 ) L _ { m } ^ { ( - \\alpha - 2 ) } ( x ) L _ { n - m } ^ { ( \\alpha + 1 ) } ( x ) \\end{align*}"}
+{"id": "6486.png", "formula": "\\begin{align*} \\frac { \\varphi ( 4 n ) } { | \\mathcal { H } | } = \\frac { | \\mathbb { Z } _ { 4 n } ^ * | } { | \\mathcal { H } | } = | \\mathcal { O } _ S | = \\frac { | \\mathbb { Z } _ { n } ^ * | } { | \\mathcal { H } ' | } = \\frac { \\varphi ( n ) } { | \\mathcal { H } ' | } . \\end{align*}"}
+{"id": "8382.png", "formula": "\\begin{align*} d n _ { \\mathcal { B } } ( X ) = \\aleph _ 0 + \\min \\{ & | F _ { \\mathcal { B } } | : F _ { \\mathcal { B } } C ( X ) \\\\ & \\mathcal { B } \\} . \\end{align*}"}
+{"id": "5616.png", "formula": "\\begin{align*} \\begin{gathered} \\hat { R } _ { 3 3 } = 2 \\hat { R } _ { 0 3 1 3 } = 2 [ R _ { 0 3 1 3 } - g ( A ( k , l ) , A ( m _ 3 , m _ 3 ) ) + g ( A ( m _ { 3 } , l ) , A ( k , m _ 3 ) ) ] \\\\ = 2 R _ { 0 3 1 3 } - 2 \\theta ^ 2 . \\end{gathered} \\end{align*}"}
+{"id": "1587.png", "formula": "\\begin{align*} \\sum _ { y \\in X } \\varrho ^ p ( u , y ) \\cdot \\pi ^ * ( u , y ) + \\pi ^ * ( u , u ) \\geq \\sum _ { y \\in X } \\pi ^ * ( u , y ) = \\mu ( u ) . \\end{align*}"}
+{"id": "630.png", "formula": "\\begin{align*} I ( G ) = & \\ ( x _ 1 x _ 5 , x _ 1 x _ 6 , x _ 1 x _ 8 , x _ 1 x _ { 1 0 } , x _ 2 x _ 5 , x _ 2 x _ 6 , x _ 2 x _ 9 , x _ 2 x _ { 1 1 } , x _ 3 x _ 7 , x _ 3 x _ 8 , x _ 3 x _ 9 , x _ 3 x _ { 1 1 } , x _ 4 x _ 7 , \\\\ & \\ \\ x _ 4 x _ 8 , x _ 4 x _ { 1 0 } , x _ 4 x _ { 1 1 } , x _ 5 x _ 8 , x _ 5 x _ 9 , x _ 6 x _ { 1 0 } , x _ 6 x _ { 1 1 } , x _ 7 x _ 9 , x _ 7 x _ { 1 0 } , x _ 8 x _ { 1 1 } ) \\end{align*}"}
+{"id": "8632.png", "formula": "\\begin{align*} \\bigcup _ { k \\geq 1 } \\bigcap _ { n \\geq k } V _ n = \\bigcup _ { i \\geq 1 } \\bigcap _ { n \\geq n _ i } V _ n = \\bigcup _ { i \\geq 1 } \\bigcap _ { j \\geq i } U _ j . \\end{align*}"}
+{"id": "8923.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { C } } \\mathcal { A } _ { m } ^ { \\nu } = \\frac { 2 } { n ! ( n - 1 ) ! } \\sum \\limits _ { p = 0 } ^ { n - 1 } \\tau _ { p } ^ { ( \\nu , n ) } \\left ( m + \\nu + \\frac { n } { 2 } \\right ) ^ { 2 p + 1 } \\end{align*}"}
+{"id": "464.png", "formula": "\\begin{align*} | \\phi ( \\tau _ \\alpha ) | = | \\psi ( \\tau _ \\alpha ) | | f _ \\alpha ( x ) | = | f _ \\alpha ( y ) | , \\forall \\alpha \\in \\Omega , \\end{align*}"}
+{"id": "7319.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { 0 } ^ { 1 } \\sum _ { k = 1 } ^ { d } \\lambda ^ k G ^ k _ { m _ n } d j _ n = \\lim _ { n \\to \\infty } \\lambda ^ 2 \\int _ { 0 } ^ { 1 } G ^ 2 _ { m _ n } d j _ n = \\kappa \\lambda ^ 2 . \\end{align*}"}
+{"id": "40.png", "formula": "\\begin{align*} \\mu & = \\frac { 1 } { K } \\cdot \\sum _ { i = 1 } ^ m T ( n _ i ) , \\\\ \\delta & < \\frac { K + L } { K } < \\delta + \\frac { 1 } { K \\cdot 3 \\cdot \\log ( 2 ) } \\cdot \\sum _ { i = 1 } ^ m T ( n _ i ) . \\end{align*}"}
+{"id": "5512.png", "formula": "\\begin{align*} I & = x ^ { 1 - \\beta } \\Gamma ( 1 - \\beta ) \\lambda + f ( \\beta ) + x ^ { - \\beta } \\int _ { - \\beta - i \\infty } ^ { - \\beta + i \\infty } | \\Gamma ( s ) f ( s + \\beta ) | \\mathrm d s \\\\ & = x ^ { 1 - \\beta } \\Gamma ( 1 - \\beta ) \\lambda + f ( \\beta ) + O _ \\varepsilon \\left \\{ x ^ { - \\beta } ( q _ 1 q _ 2 ) ^ { 1 + \\varepsilon } \\int _ 0 ^ \\infty t ^ { 2 + \\varepsilon } | \\Gamma ( - \\beta + i t ) | \\mathrm d t \\right \\} \\end{align*}"}
+{"id": "8442.png", "formula": "\\begin{align*} \\Gamma _ A ^ + : = \\bigl \\{ \\nu \\in \\mathcal E ^ + : \\ \\kappa \\nu \\geqslant 1 \\bigr \\} . \\end{align*}"}
+{"id": "3359.png", "formula": "\\begin{align*} \\{ f , g \\} _ J = - \\dd \\eta ( \\sharp \\dd f , \\sharp \\dd g ) - R ( g ) f + g R ( f ) . \\end{align*}"}
+{"id": "8887.png", "formula": "\\begin{align*} \\Theta _ { n + 1 , \\nu } ( t , \\psi ) = \\sum \\limits _ { m = 0 } ^ { + \\infty } e ^ { - 4 t ( m + \\nu + \\frac { n } { 2 } ) ^ 2 } \\cos ( 2 m + 2 \\nu + n ) \\psi . \\end{align*}"}
+{"id": "2444.png", "formula": "\\begin{align*} \\lim _ { \\lambda _ { \\varphi } \\to \\infty } \\mu _ { \\varphi } ( \\Psi ) = \\frac { 3 } { \\pi } \\mu ( \\Psi ) . \\end{align*}"}
+{"id": "1784.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { s - t } \\sum _ { i = k _ j } ^ { m _ { k _ j } } n _ i ^ { ( j ) } q ^ i \\in \\mathsf { Z } ( x ) . \\end{align*}"}
+{"id": "1175.png", "formula": "\\begin{align*} \\nu _ i + 1 : = \\min \\{ 1 \\le \\ell \\le p \\ , : \\ , \\delta _ \\ell = \\alpha _ i \\} \\ , . \\end{align*}"}
+{"id": "2683.png", "formula": "\\begin{align*} \\big \\{ \\Delta _ { x , \\S ( \\varphi , T ) } \\ , : x \\in \\S ( \\varphi , T ) | _ { = s } \\big \\} & = \\big \\{ \\S ( \\varphi , \\Delta _ { x , T } ) \\ , : x \\in \\S ( \\varphi , T ) | _ { = s } \\big \\} \\\\ & = \\big \\{ \\S ( \\varphi , \\Delta _ { x , T } ) \\ , : x \\in T | _ { = s } \\ , \\ , \\S ( \\varphi , \\Delta _ { x , T } ) \\not = \\phi \\big \\} , \\end{align*}"}
+{"id": "6510.png", "formula": "\\begin{align*} \\left ( \\bigg | \\frac { 1 } { \\sqrt { k } l } \\underset { i = 1 } { \\overset { k } { \\sum } } \\left ( \\underset { j = 1 } { \\overset { l } { \\sum } } B ^ j _ i - l p _ i \\right ) ^ 2 - \\mu _ p + \\frac { l - 1 } { \\sqrt { k } } \\underset { i = 1 } { \\overset { k } { \\sum } } \\left ( p _ i - \\frac { 1 } { 2 } \\right ) ^ 2 + \\zeta \\bigg | \\leq c _ { \\alpha , n } \\right ) \\end{align*}"}
+{"id": "2481.png", "formula": "\\begin{align*} X _ n X _ n T | _ { \\mathcal H _ n } = U _ { 0 n } X _ n \\ n \\geq 0 . \\end{align*}"}
+{"id": "2420.png", "formula": "\\begin{align*} A _ { 3 3 } ^ T = - A _ { 3 3 } - \\dot E _ { 3 3 } , A _ { 4 3 } ^ T = - A _ { 3 4 } - \\dot E _ { 3 4 } , A _ { 4 4 } ^ T = - A _ { 4 4 } - \\dot E _ { 4 4 } . \\end{align*}"}
+{"id": "3393.png", "formula": "\\begin{align*} M = \\bigoplus _ { i \\in I } e _ i M . \\end{align*}"}
+{"id": "1478.png", "formula": "\\begin{align*} \\lambda & = \\frac { 2 k ( k - 1 ) ( m - 1 ) ! ( m - 2 ) ! } { ( m + 1 ) | G _ \\Delta | } \\\\ & = \\frac { 2 k ( k - 1 ) ( k - 2 ) ! ( k - 3 ) ! } { k | G _ \\Delta | } \\\\ & = \\frac { 2 ( k - 1 ) ! ( k - 3 ) ! } { | G _ \\Delta | } \\end{align*}"}
+{"id": "2297.png", "formula": "\\begin{align*} \\omega ^ { \\pm } _ k = \\frac { \\alpha ^ { \\pm } _ k } { \\sum ^ 2 _ { l = 0 } \\alpha ^ { \\pm } _ l } , ~ ~ \\alpha ^ { \\pm } _ k = \\gamma ^ { \\pm } _ k \\left ( 1 + \\left ( \\frac { \\tau } { \\beta _ k + \\epsilon } \\right ) ^ 2 \\right ) , ~ ~ k = 0 , 1 , 2 , \\end{align*}"}
+{"id": "7102.png", "formula": "\\begin{align*} \\underline { k } _ { - 2 m + n \\sigma } = \\left ( \\underline { R } \\overset { ( \\sigma - 1 ) ^ n } { \\longrightarrow } \\underline { J } ^ m \\right ) = \\underline { J } ^ m / \\underline { J } ^ n \\cong m / n . \\end{align*}"}
+{"id": "5165.png", "formula": "\\begin{align*} \\dfrac { \\dfrac { n ^ { 3 } ( n - 1 ) ^ { 3 } } { 8 } } { \\binom { 3 n } { 6 } } & = \\dfrac { n ^ { 3 } ( n - 1 ) ^ { 3 } } { 8 } \\cdot \\dfrac { 6 ! } { 3 n ( 3 n - 1 ) ( 3 n - 2 ) ( 3 n - 3 ) ( 3 n - 4 ) ( 3 n - 5 ) } . \\end{align*}"}
+{"id": "62.png", "formula": "\\begin{align*} y _ i = \\begin{cases} 1 , & f ( i ) = 2 \\\\ 0 , & o t h e r w i s e . \\end{cases} \\end{align*}"}
+{"id": "7128.png", "formula": "\\begin{align*} d - \\left \\langle d , x ^ { \\rho _ 1 + \\cdots + \\rho _ { n - 1 } } \\right \\rangle \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) = \\sum _ { i \\geq 1 } a _ i \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) \\end{align*}"}
+{"id": "9231.png", "formula": "\\begin{align*} \\psi _ { m } : = \\left \\{ \\begin{aligned} & \\theta _ { m } ( v _ { \\Gamma } + 1 ) e ^ { - ( f - f ( m ) ) / h } & & U _ { \\Gamma } \\Gamma \\in { \\bf j } ( m ) , \\\\ & 2 \\theta _ { m } e ^ { - ( f - f ( m ) ) / h } & & \\R ^ { d } \\setminus \\bigcup _ { \\Gamma \\in { \\bf j } ( m ) } U _ { \\Gamma } , \\end{aligned} \\right . \\end{align*}"}
+{"id": "147.png", "formula": "\\begin{align*} \\| u ( \\theta ) \\| _ \\mathbb { U } & \\leq Q , \\ \\mbox { f o r } \\ \\theta \\in J , \\ \\mbox { w h e r e } \\ Q = \\max _ { 0 \\leq j \\leq n } \\{ Q _ { j } \\} . \\end{align*}"}
+{"id": "6936.png", "formula": "\\begin{align*} P _ { n } ^ { ( \\alpha , \\beta ) } ( x ) = \\frac { \\Gamma ( \\alpha + n + 1 ) } { n ! \\Gamma ( \\alpha + \\beta + n + 1 ) } \\sum _ { m = 0 } ^ n \\binom { n } { m } \\frac { \\Gamma ( \\alpha + \\beta + n + m + 1 ) } { \\Gamma ( \\alpha + m + 1 ) } \\bigg ( \\frac { x - 1 } { 2 } \\bigg ) ^ m . \\end{align*}"}
+{"id": "964.png", "formula": "\\begin{align*} \\mathcal { F } ( a ) : = \\bigl \\{ b \\in V ^ \\vee : a \\wedge b \\in K ^ \\perp \\bigr \\} \\end{align*}"}
+{"id": "43.png", "formula": "\\begin{align*} \\frac { m _ 2 } { m } \\cdot K & \\leq \\sum _ { i = m - m _ 2 + 1 } ^ { m } k _ i = \\sum _ { \\ell = 0 } ^ { m _ 2 - 1 } k _ { ( m - m _ 2 + 1 ) + \\ell } \\\\ & \\leq \\frac { \\log ( n _ { m - m _ 2 + 1 } + 1 ) } { \\log ( 2 ) } \\cdot \\sum _ { \\ell = 0 } ^ { m _ 2 - 1 } \\delta ^ { \\ell } \\\\ & = \\frac { \\log ( n _ { m - m _ 2 + 1 } + 1 ) } { \\log ( 2 ) } \\cdot \\frac { \\delta ^ { m _ 2 } - 1 } { \\delta - 1 } . \\end{align*}"}
+{"id": "3352.png", "formula": "\\begin{align*} \\dd \\theta = 0 , \\dd \\Omega = 0 . \\end{align*}"}
+{"id": "7773.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n x f _ { 1 / 4 } ( x | \\tfrac { \\mu } { n } ) & = \\frac { \\mu } { 4 \\pi \\sqrt { x } } \\int _ 0 ^ \\infty y ^ { - 1 / 2 } \\ , e ^ { - y ^ 2 / 4 x } \\ , d y \\\\ & = \\frac { \\mu } { 8 \\pi \\sqrt { x } } \\int _ 0 ^ \\infty y ^ { - 3 / 4 } e ^ { - y / 4 x } \\ , d y ( y ^ 2 \\to y ) \\\\ & = \\frac { \\mu } { 8 \\pi \\sqrt { x } } \\ , \\Gamma ( \\tfrac { 1 } { 4 } ) \\ , ( 4 x ) ^ { 1 / 4 } \\\\ & = \\frac { \\mu } { 4 } \\ , \\frac { x ^ { - 1 / 4 } } { \\Gamma ( \\frac { 3 } { 4 } ) } \\end{align*}"}
+{"id": "1089.png", "formula": "\\begin{align*} \\mathbb { P } ( | X | > M ) ^ { 1 / k ' } = \\mathbb { P } ( | X - \\mu | > M - \\mu ) ^ { 1 / k ' } \\leq ( M - \\mu ) ^ { 1 - k } \\end{align*}"}
+{"id": "7944.png", "formula": "\\begin{align*} x \\leq y \\iff x + y = y . \\end{align*}"}
+{"id": "7134.png", "formula": "\\begin{align*} R \\cong \\prod _ { i = 1 } ^ \\infty A \\{ x ^ { \\rho _ 1 + \\cdots + \\rho _ { i - 1 } } \\} D \\cong \\bigoplus _ { i = 1 } ^ \\infty A \\{ \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) \\} , \\end{align*}"}
+{"id": "3686.png", "formula": "\\begin{align*} A + B \\coloneqq \\left \\{ ( x ; a + b ) \\ ; \\middle | \\ ; \\begin{aligned} & x \\in \\pi ( A ) \\cap \\pi ( B ) , \\\\ & a \\in A \\cap \\pi ^ { - 1 } ( x ) , b \\in B \\cap \\pi ^ { - 1 } ( x ) \\end{aligned} \\right \\} \\subset T ^ * X . \\end{align*}"}
+{"id": "4444.png", "formula": "\\begin{align*} \\theta \\in \\left ( \\frac { 1 1 } { 2 8 } , \\frac { 1 9 } { 4 8 } \\right ] \\subseteq \\left ( \\frac { 2 0 } { 5 1 } , \\frac { 1 9 } { 4 8 } \\right ] = J ( 3 , 1 7 ) . \\end{align*}"}
+{"id": "7869.png", "formula": "\\begin{align*} f ^ { ( n ) } + P _ { n - 1 } ( z ) f ^ { ( n - 1 ) } + \\cdots + P _ 0 ( z ) f = H ( z ) . \\end{align*}"}
+{"id": "5732.png", "formula": "\\begin{align*} \\mathbf { e } ^ { A A ^ { \\prime } } = \\theta ^ { A A ^ { \\prime } } \\end{align*}"}
+{"id": "2820.png", "formula": "\\begin{align*} [ f ] ( x _ 1 , x _ 2 ) = x _ 1 + x _ 2 . \\end{align*}"}
+{"id": "1074.png", "formula": "\\begin{align*} & \\mathbb { E } ( \\widetilde { N } _ 0 / n ) - \\frac { 1 - \\varepsilon } { 2 } \\{ P _ 0 ( A ) + P _ 1 ( A ) \\} - \\frac { \\varepsilon } { 2 } \\\\ \\leq & ( 1 - \\varepsilon ) P _ 1 ( A ) + \\varepsilon - \\frac { 1 - \\varepsilon } { 2 } \\{ P _ 0 ( A ) + P _ 1 ( A ) \\} - \\frac { \\varepsilon } { 2 } = \\frac { \\varepsilon } { 2 } - \\frac { 1 - \\varepsilon } { 2 } \\mathrm { T V } ( P _ 0 , P _ 1 ) . \\end{align*}"}
+{"id": "1272.png", "formula": "\\begin{align*} c ( x , y , z ) = \\displaystyle \\sum _ { i = 0 } ^ { s - 1 } \\sum _ { j = 0 } ^ { \\ell - 1 } \\sum _ { t = 0 } ^ { k - 1 } c _ { i , j , t } ~ x ^ i y ^ j z ^ t . \\end{align*}"}
+{"id": "3989.png", "formula": "\\begin{align*} [ D _ { i j } \\phi - D _ { p _ k } A _ { i j } & ( x , u , D u ) D _ k \\phi ] \\tau _ i \\tau _ j \\\\ & = [ D _ i \\gamma _ j - A _ { i j , p _ k } ( x , u , p ) \\gamma _ k ] \\tau _ i \\tau _ j \\geq c _ 0 | \\tau | ^ 2 , \\end{align*}"}
+{"id": "5469.png", "formula": "\\begin{align*} X _ s ( \\omega ) = Y _ s ( \\omega ) , \\alpha _ s ( \\omega ) = \\tilde \\alpha _ s ( \\omega ) , ~ \\forall ~ s \\leq \\tau ( \\omega ) < N . \\end{align*}"}
+{"id": "2731.png", "formula": "\\begin{align*} \\Delta ( E _ i ) = E _ i \\otimes 1 + K _ i \\otimes E _ i , & \\Delta ( F _ i ) = 1 \\otimes F _ i + F _ i \\otimes K _ { i } ' , \\\\ \\Delta ( K _ { i } ) = K _ { i } \\otimes K _ { i } , & \\Delta ( K _ { i } ' ) = K _ { i } ' \\otimes K _ { i } ' . \\end{align*}"}
+{"id": "5300.png", "formula": "\\begin{align*} T _ { U } U = - X . \\end{align*}"}
+{"id": "6514.png", "formula": "\\begin{align*} \\left ( \\bigg | \\frac { 1 } { l k } \\left ( \\underset { i = 1 } { \\overset { k } { \\sum } } \\underset { j = 1 } { \\overset { l } { \\sum } } ( B ^ j _ i - p _ i ) \\right ) ^ 2 - \\mu _ p ' + \\frac { l - 1 } { k } \\left ( \\underset { i = 1 } { \\overset { k } { \\sum } } ( p _ i - \\frac { 1 } { 2 } ) \\right ) ^ 2 + \\zeta \\bigg | \\leq c _ { \\alpha , n } \\right ) , \\end{align*}"}
+{"id": "3562.png", "formula": "\\begin{align*} E ^ { \\gamma - e _ J + \\sum _ { u = 2 } ^ s e _ { v _ u i _ { u - 1 } } - e _ { v _ u i _ u } } \\Omega _ { \\lambda + \\epsilon _ i } \\quad E ^ { \\gamma - e _ { \\ell j } + \\sum _ { u = 2 } ^ s e _ { v _ u i _ { u - 1 } } - e _ { v _ u i _ u } } \\Omega _ { \\lambda - \\epsilon _ i } . \\end{align*}"}
+{"id": "3911.png", "formula": "\\begin{align*} Y u ( K ) \\setminus \\mathcal { Z } \\subset \\bigcup _ { i = 1 } ^ \\infty \\bigcap _ { k = i } ^ \\infty Y u _ k ( U ) , \\end{align*}"}
+{"id": "6544.png", "formula": "\\begin{align*} { F _ Z } \\left ( z \\right ) = & \\frac { { { m _ f } ^ { { m _ f } - 1 } { z ^ { { m _ f } } } } } { { B \\left ( { { m _ f } , { m _ s } } \\right ) \\left ( { { m _ s } - 1 } \\right ) { ^ { { m _ { f } } } } { { \\bar z } ^ { { m _ f } } } } } \\\\ & \\times { } _ 2 { F _ 1 } \\left ( { { m _ f } , { m _ f } + { m _ s } , { m _ f } + 1 ; - \\frac { { { m _ f } z } } { { \\left ( { { m _ s } - 1 } \\right ) \\bar z } } } \\right ) . \\end{align*}"}
+{"id": "8856.png", "formula": "\\begin{align*} \\beta _ { m } : = \\Lambda _ { n , \\nu } \\left ( \\pm ( 2 ( m + \\nu ) + n ) \\right ) = - 4 ( m + \\nu ) ( m + \\nu + n ) + 4 \\nu ^ { 2 } \\end{align*}"}
+{"id": "3232.png", "formula": "\\begin{align*} | V _ { 1 } | = m _ { i _ 0 + 1 } , \\ S _ { f _ 1 , i _ 0 + 1 } = V _ 1 \\cap P _ { i _ 0 + 1 } = \\emptyset , \\ \\ P _ { { i _ 0 } + 1 } = R _ { f _ 1 , { i _ 0 } + 1 } . \\end{align*}"}
+{"id": "817.png", "formula": "\\begin{align*} C _ { n , t _ 1 , \\ldots , t _ d } ^ { \\pm } ( x ) = E _ { n , t _ 1 , \\ldots , t _ d } ( x \\pm C n ^ { - 1 / 4 } ( 1 , 1 , \\ldots , 1 ) ) \\end{align*}"}
+{"id": "143.png", "formula": "\\begin{align*} & \\max \\bigg \\{ \\max _ { 1 \\leq j \\leq n } \\bigg ( \\bigg ( 1 + \\frac { M ^ { 2 } K ^ { 2 } b } { \\delta ^ { j } } \\bigg ) \\bigg ( K K _ { 1 } \\gamma b + K L _ { \\nu _ { j } } \\bigg ) \\bigg ) , \\\\ & \\qquad \\qquad \\bigg ( \\bigg ( 1 + \\frac { M ^ { 2 } K ^ { 2 } b } { \\delta ^ { 0 } } \\bigg ) \\bigg ( K _ { 1 } K \\gamma b + K C _ { \\nu } \\bigg ) \\bigg ) , \\max _ { 1 \\leq j \\leq n } L _ { \\nu _ { j } } \\bigg \\} < 1 , \\mbox { w h e r e } \\ \\gamma = \\frac { b } { \\beta } . \\end{align*}"}
+{"id": "636.png", "formula": "\\begin{align*} \\beta _ i ^ { k } ( ( I + ( w ) ) ^ q ) = \\sum _ { \\ell = 1 } ^ q \\left [ \\beta _ i ^ k ( I ^ { \\ell } ) + \\beta _ { i - 1 } ^ k ( I ^ { \\ell } ) \\right ] . \\end{align*}"}
+{"id": "3871.png", "formula": "\\begin{align*} \\det D Y ( \\cdot , u , D u ) = \\psi ( \\cdot , u , D u ) , \\end{align*}"}
+{"id": "4877.png", "formula": "\\begin{align*} \\Delta _ n ^ \\perp ( q ) = \\big \\{ \\lambda \\in T _ q ^ * M \\mid \\lambda ( v ) = 0 \\textrm { f o r a l l $ v \\in \\Delta _ n ( q ) $ } \\big \\} . \\end{align*}"}
+{"id": "122.png", "formula": "\\begin{align*} E ( F , \\gamma ) : = \\left \\{ ( x , y ) \\in \\mathbb R \\times \\mathbb R : \\ x \\neq y , \\left | \\int _ y ^ x F \\right | \\geq | x - y | ^ { \\gamma + 1 } \\right \\} . \\end{align*}"}
+{"id": "7123.png", "formula": "\\begin{align*} a = \\epsilon ( a \\beta ( \\rho _ 1 ) ) = \\epsilon ( \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) - \\gamma ( \\rho _ 1 , \\dots , \\rho _ n ) ) & = \\epsilon ( \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) ) - \\epsilon ( \\gamma ( \\rho _ 1 , \\dots , \\rho _ n ) ) \\\\ & = 0 \\end{align*}"}
+{"id": "6585.png", "formula": "\\begin{align*} \\theta ( \\ell , i ) = \\beta _ { \\ell , M _ - ' } \\circ \\beta _ { \\ell , M _ + ' } ^ { - 1 } ( \\ell , i ) = \\beta _ { \\ell , M _ - } \\circ \\beta _ { \\ell , M _ + } ^ { - 1 } ( \\ell , i ) = \\eta ( \\ell , i ) \\end{align*}"}
+{"id": "3688.png", "formula": "\\begin{align*} \\phi ^ H _ s \\circ \\phi ^ { H ' } _ s = \\phi ^ { H \\sharp H ' } _ s , \\end{align*}"}
+{"id": "8370.png", "formula": "\\begin{align*} w = 0 , \\mathrm { f o r } \\ | x | > R , \\ 0 \\le t \\le R , \\end{align*}"}
+{"id": "799.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\int 1 _ { F ( t , n ) } \\ d \\mu _ { ( z , m ) } = \\mu _ j ( F ( t , n ) ) . \\end{align*}"}
+{"id": "8813.png", "formula": "\\begin{align*} \\| P _ 2 ^ n x \\| = \\| x \\| = \\| { P _ 2 ^ * } ^ n x \\| , n = 1 , 2 , \\hdots \\end{align*}"}
+{"id": "8074.png", "formula": "\\begin{align*} \\mathcal { D } & = \\frac { 1 } { 2 } \\sum _ { ( m , p ) = 1 } \\frac { A ( p , m ) } { m p ^ { \\frac { 1 } { 2 } } } H _ { m , p } + \\frac { 1 } { 2 } \\sum _ { ( m , p ) > 1 } \\frac { A ( p , m ) } { m p ^ { \\frac { 1 } { 2 } } } H _ { m , p } \\\\ & = \\frac { 1 } { 2 } \\sum _ { ( m , p ) = 1 } \\frac { A ( p , m ) } { m p ^ { \\frac { 1 } { 2 } } } H _ { m , p } + \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\frac { A ( p , m p ) } { ( m p ) p ^ { \\frac { 1 } { 2 } } } H _ { m p , p } . \\end{align*}"}
+{"id": "7971.png", "formula": "\\begin{align*} f ( w + q ' ) = f ( w ) + f ( q ' ) = x + f ( q ' ) = \\sum f ( a _ i ) = x + \\sum f ( b _ i ) . \\end{align*}"}
+{"id": "8527.png", "formula": "\\begin{align*} \\{ \\ell \\not \\in X : a _ { \\ell , y } = 0 \\mbox { f o r a l l } y \\not \\in Y \\} \\cap \\{ \\ell \\not \\in X : a _ { \\ell , y } + r _ y \\le \\lambda \\mbox { f o r a l l } y \\in Y \\} , \\end{align*}"}
+{"id": "8582.png", "formula": "\\begin{align*} L ^ k \\mathbf { a } : = \\left ( a _ k , a _ { k + 1 } , \\ldots , a _ { N - 1 } , a _ 0 , \\ldots , a _ { k - 1 } \\right ) . \\end{align*}"}
+{"id": "917.png", "formula": "\\begin{align*} T _ n = - \\dfrac { \\log T _ n } { \\log { ( n ( n + 1 ) ) } } \\end{align*}"}
+{"id": "593.png", "formula": "\\begin{align*} K _ n ^ { ( \\epsilon ) } = \\sigma _ n ^ { ( \\epsilon ) } ( A X _ { t _ n } ^ { ( \\epsilon ) } ) ^ { \\rm T } \\left ( \\gamma + \\Delta t \\sigma _ n ^ { ( \\epsilon ) } ( A X _ { t _ n } ^ { ( \\epsilon ) } ) ^ { \\rm T } A X _ { t _ n } ^ { ( \\epsilon ) } \\right ) ^ { - 1 } \\ , . \\end{align*}"}
+{"id": "8412.png", "formula": "\\begin{align*} & P _ { \\alpha } ( \\partial _ { \\alpha } [ \\hat \\Psi _ { \\alpha } h _ { \\alpha } ] ) \\\\ & = \\sum _ { n = 1 } ^ { \\infty } ( \\partial _ { \\alpha } h _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) ( \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) G _ { \\alpha , n } + ( h _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) ( \\partial _ { \\alpha } \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) G _ { \\alpha , n } \\end{align*}"}
+{"id": "6359.png", "formula": "\\begin{align*} m ( x ) = \\hat m ( x ) o n ( 0 , t _ 0 ] , \\end{align*}"}
+{"id": "1920.png", "formula": "\\begin{align*} R _ { \\tilde g } = ( R _ g - V ) u ^ { - \\frac { 4 } { n - 2 } } \\end{align*}"}
+{"id": "7292.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\frac { G ^ { k + 2 } _ m ( x ) } { G ^ 2 _ m ( x ) } = 0 . \\end{align*}"}
+{"id": "3601.png", "formula": "\\begin{align*} d _ I ^ + ( \\mu ) = d _ { i } ^ + ( \\mu ) \\frac { \\prod _ { \\ell \\in I ^ \\complement } ( \\mu _ i - \\mu _ \\ell - i + \\ell + 1 ) } { \\prod _ { \\ell = i + 1 } ^ { j } ( \\mu _ i - \\mu _ \\ell - i + \\ell ) } . \\end{align*}"}
+{"id": "4159.png", "formula": "\\begin{align*} \\mathcal { F } ( g , b , f ) = \\int _ M ( R - \\frac { 1 } { 1 2 } | H _ c + d b | ^ 2 + | \\nabla f | ^ 2 ) e ^ { - f } d V _ g \\end{align*}"}
+{"id": "2843.png", "formula": "\\begin{align*} a _ { \\ell } + a _ k = b _ { \\ell } + b _ k \\end{align*}"}
+{"id": "3076.png", "formula": "\\begin{align*} A ( y ) : = \\mathrm { d i a g } ( a ( y ) , c - a ( y ) ) \\quad y \\in \\R ^ 2 . \\end{align*}"}
+{"id": "4812.png", "formula": "\\begin{align*} K _ s ( x ) = \\sum _ { j = 0 } ^ s ( - 1 ) ^ j \\binom { x } { j } \\binom { N - x } { s - j } \\leq \\sum _ { j = 0 } ^ s \\binom { x } { j } \\binom { N - x } { s - j } = \\binom { N } { s } . \\end{align*}"}
+{"id": "7153.png", "formula": "\\begin{align*} \\displaystyle { u ^ { \\{ k \\} } _ m = u ^ { \\{ k \\} } _ l } \\end{align*}"}
+{"id": "8688.png", "formula": "\\begin{align*} \\tilde { \\tilde { F } } _ \\lambda ( x _ 1 , \\dots , x _ n ; t ) = \\tilde { \\tilde { F } } _ \\lambda ( x _ 1 , \\dots , x _ n , 0 ; t ) . \\end{align*}"}
+{"id": "1286.png", "formula": "\\begin{align*} p _ j ^ { ( t ) } ( x ) \\eta _ j ( y ) \\zeta _ t ( z ) = \\sum _ { u = 0 } ^ { k - 1 } \\sum _ { v = 0 } ^ { \\ell - 1 } h ' _ { u v } ( x ) q _ v ^ { ( u ) * } ( x ) \\eta _ v ^ * ( y ) \\zeta _ u ^ * ( z ) . \\end{align*}"}
+{"id": "144.png", "formula": "\\begin{align*} \\alpha _ { 1 } & = \\max \\bigg \\{ \\bigg ( M K Q b + K N b + K \\left [ \\| \\phi ( 0 ) \\| _ { \\mathbb { X } } + \\bar { M } \\right ] \\bigg ) , \\max _ { 1 \\leq j \\leq n } \\bigg ( M K Q b + K N b + K C _ { \\nu _ { j } } \\bigg ) , \\\\ & \\qquad \\max _ { 1 \\leq j \\leq n } C _ { \\nu _ { j } } \\bigg \\} . \\end{align*}"}
+{"id": "3230.png", "formula": "\\begin{align*} \\begin{aligned} | R _ { f _ 1 , { i _ 0 } + 1 } | & \\ge 2 , \\\\ | V _ 2 | & \\ge 2 { i _ 0 } , \\\\ | V _ 3 | & \\ge k - { i _ 0 } + 1 , \\\\ | V _ 1 | & = | V | - | V _ 2 | - | V _ 3 | \\le k - { i _ 0 } + 1 . \\end{aligned} \\end{align*}"}
+{"id": "7610.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\int _ 0 ^ T \\left ( 1 + \\| v ^ n ( t ) \\| _ { \\L ^ { 2 \\delta } } ^ { 4 \\delta } + \\| u ( t ) \\| _ { \\L ^ { 2 \\delta } } ^ { 4 \\delta } \\right ) \\| v ^ n ( t ) - u ( t ) \\| _ { \\L ^ 2 } ^ 2 \\d t \\\\ & = \\int _ 0 ^ T \\left ( 1 + \\| v ( t ) \\| _ { \\L ^ { 2 \\delta } } ^ { 4 \\delta } + \\| u ( t ) \\| _ { \\L ^ { 2 \\delta } } ^ { 4 \\delta } \\right ) \\| v ( t ) - u ( t ) \\| _ { \\L ^ 2 } ^ 2 \\d t . \\end{align*}"}
+{"id": "6977.png", "formula": "\\begin{gather*} P _ { m , n } ^ { ( \\alpha , \\beta ) } ( x ) = \\sum _ { j = 0 } ^ n d _ { j , n } x ^ j . \\end{gather*}"}
+{"id": "4234.png", "formula": "\\begin{align*} \\hat { \\Lambda } = \\ln ( \\cosh ( \\gamma ) ) + \\ln \\left ( | v | - \\frac { \\tanh ( \\gamma ) \\tanh ( \\beta \\gamma ) } { \\sqrt { c } } + \\sqrt { \\left ( | v | - \\frac { \\tanh ( \\gamma ) \\tanh ( \\beta \\gamma ) } { \\sqrt { c } } \\right ) ^ 2 - 1 } \\right ) . \\end{align*}"}
+{"id": "3227.png", "formula": "\\begin{align*} f ' ( u ) = | L ( u ) | \\geq ( 3 k _ 3 + 2 k _ 2 + k _ 1 + d ) - ( k _ 2 + k _ 3 ) = 2 k _ 3 + k _ 2 + k _ 1 + d = 2 k _ 3 ' + k ' _ 2 + k _ 1 ' + d ' . \\end{align*}"}
+{"id": "8090.png", "formula": "\\begin{align*} H _ { m , n } ^ + ( x ) = 2 i \\int _ { - \\infty } ^ \\infty J _ { 2 i y + 1 } ( x ) \\frac { k \\Bigl ( - \\dfrac { 1 } { 2 } i + y \\Bigr ) V \\Bigl ( m ^ 2 n , - \\dfrac { 1 } { 2 } i + y \\Bigr ) \\Bigl ( - \\dfrac { 1 } { 2 } i + y \\Bigr ) } { \\cosh \\Bigl ( \\pi \\Bigl ( - \\dfrac { 1 } { 2 } i + y \\Bigr ) \\Bigr ) } \\ , d y \\end{align*}"}
+{"id": "6314.png", "formula": "\\begin{align*} Y ^ { i f } _ t = 2 n c Z ^ { i f } _ { k - 1 } + ( 1 - Z ^ { i f } _ { k - 1 } ) ( Y ^ { i f } _ { k n c - 1 } + n c ) \\end{align*}"}
+{"id": "7181.png", "formula": "\\begin{align*} \\mathcal { W } = \\bigcup _ { t \\in [ 0 , \\ , \\infty ] } \\{ t \\} \\times W _ t . \\end{align*}"}
+{"id": "4271.png", "formula": "\\begin{align*} \\| ( \\nabla - i A ) \\phi \\| ^ 2 _ { L ^ 2 } = \\frac { 2 N } { N + 2 } \\| \\phi \\| ^ { 2 + \\frac { 4 } { N } } _ { L ^ { 2 + \\frac { 4 } { N } } } \\end{align*}"}
+{"id": "1933.png", "formula": "\\begin{align*} - a \\Delta _ g u + R _ g u = 0 \\end{align*}"}
+{"id": "232.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho ^ 2 - \\beta \\rho \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "367.png", "formula": "\\begin{align*} \\delta ^ { ( n ) } _ k : = \\frac { T } { \\beta _ n } u ^ { ( n ) } _ k \\textrm { f o r } k \\in \\N \\ , . \\end{align*}"}
+{"id": "7390.png", "formula": "\\begin{align*} \\alpha ^ \\vee ( a ) = \\eta _ { \\beta ^ \\vee } ^ { - 1 } ( 1 , a ) \\quad \\beta ^ \\vee ( a ) = \\eta _ { \\beta ^ \\vee } ^ { - 1 } ( a , a ^ { - 1 } ) \\quad \\end{align*}"}
+{"id": "299.png", "formula": "\\begin{align*} & \\# \\{ \\Gamma _ F \\} + 2 \\# \\{ \\Gamma _ F \\} \\\\ = & \\# \\{ \\Gamma _ F \\} = n - 1 , \\end{align*}"}
+{"id": "5898.png", "formula": "\\begin{align*} t \\geq T : c _ { p + 1 } U = \\sum _ { r = 1 } ^ p \\alpha _ r u _ r = 0 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "3990.png", "formula": "\\begin{align*} D _ { q _ k q _ l } X ^ \\alpha = \\frac { g _ { j , z } } { g _ z } D _ { q _ k } X ^ \\alpha D _ { q _ l } X ^ j + \\frac { g _ { i , z } } { g _ z } D _ { q _ k } X ^ i D _ { q _ l } X ^ \\alpha - D _ { p _ \\alpha } g _ { i j } D _ { q _ k } X ^ i D _ { q _ l } X ^ j . \\end{align*}"}
+{"id": "3858.png", "formula": "\\begin{align*} T _ n ^ 2 = T _ n + \\sum _ { k = 2 } ^ { n - 2 } \\sum _ { l = 2 } ^ k \\{ 4 ( T _ l + T _ { l - 1 } ) - 2 \\delta _ { l , 2 } \\} T _ { k - l + 2 } ^ 2 T _ { n - k } . \\end{align*}"}
+{"id": "9244.png", "formula": "\\begin{align*} X \\cdot V _ j ( g ) \\subseteq V _ { h ( j ) } ( g ) & \\Leftrightarrow \\left \\langle v _ 1 , \\ldots v _ { h ( j ) } , X v _ 1 , \\ldots , X v _ j \\right \\rangle = V _ { h ( j ) } ( g ) \\\\ & \\Leftrightarrow X v _ j \\in V _ { h ( j ) } ( g ) . \\end{align*}"}
+{"id": "3296.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 2 } | \\nabla V | ^ 2 = 2 \\int _ { \\mathcal F _ 0 } | D ( V ) | ^ 2 . \\end{align*}"}
+{"id": "6235.png", "formula": "\\begin{align*} ( E _ i , B \\widehat U ) = 0 \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 . \\end{align*}"}
+{"id": "470.png", "formula": "\\begin{align*} I ( G ; x ) = I ( G - v ; x ) + x \\cdot I ( G - N [ v ] ; x ) \\end{align*}"}
+{"id": "7549.png", "formula": "\\begin{align*} Z _ f ( s , \\chi ) = \\dfrac { 1 } { 1 - q ^ { - ( \\omega + 2 ) - ( k + r + l ) s } } \\sum _ { i = 1 } ^ { 7 } Z _ f ( s , \\chi , A _ i ) . \\end{align*}"}
+{"id": "597.png", "formula": "\\begin{align*} J ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } = A ^ { \\rm T } : ( X ^ { ( \\epsilon ) } _ { t _ n } \\otimes X ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } ) + A ^ { \\rm T } : \\mathbb { X } _ { t _ n , t _ { n + 1 } } ^ { ( \\epsilon ) } . \\end{align*}"}
+{"id": "7294.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } ( u ( x ) - u ( 0 ) ) j ( d x ) = 0 . \\end{align*}"}
+{"id": "2715.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} V ^ k _ 0 & = \\sum _ { j = j _ k } ^ { j _ { k + 1 } - 1 } W _ { ( \\nu _ j ) } + \\check { h } _ 0 ^ k \\\\ V ^ k _ 1 & = \\sum _ { j = j _ k } ^ { j _ { k + 1 } - 1 } \\check { \\alpha } _ j ( \\Lambda W ) _ { [ \\nu _ j ] } + \\check { g } _ 1 ^ k , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1183.png", "formula": "\\begin{align*} \\{ ( q , s ) \\in L \\times \\C ^ * : s \\cdot q = \\sigma ( \\pi ( q ) ) \\} , \\end{align*}"}
+{"id": "4862.png", "formula": "\\begin{align*} X _ n = \\bigcup _ { j = 1 } ^ n X _ { n j } . \\end{align*}"}
+{"id": "6451.png", "formula": "\\begin{align*} ( p - 2 ) \\sum ^ m _ { j = 1 } \\lambda _ j z _ j ^ 2 + \\sum ^ m _ { j = 1 } \\lambda _ j = ( p - 1 ) \\sum ^ m _ { j = 1 } \\lambda _ j z _ j ^ 2 + \\sum ^ m _ { j = 1 } \\lambda _ j ( 1 - z _ j ^ 2 ) \\leq 0 . \\end{align*}"}
+{"id": "1939.png", "formula": "\\begin{align*} \\varrho [ u , X ] = \\mathbb { E } \\left [ f _ 1 \\left ( u , \\mathbb { E } [ f _ 2 ( u , \\mathbb { E } [ \\ldots f _ k ( u , \\mathbb { E } [ f _ { k + 1 } ( u , X ) ] , X \\right ) \\right ] \\ldots , X ) ] , X ) ] , \\end{align*}"}
+{"id": "7966.png", "formula": "\\begin{align*} f ( \\vee x _ i ) = \\vee f ( x _ i ) \\in L . \\end{align*}"}
+{"id": "6855.png", "formula": "\\begin{align*} \\tilde { \\mathcal M _ d } = { \\mathcal M _ d } = 0 , S _ { 2 , \\dots , N + 1 } ^ { [ 0 ] } - \\mathcal D _ d - \\mathcal D _ d ^ T \\le 0 , \\end{align*}"}
+{"id": "3838.png", "formula": "\\begin{align*} { \\rm c o v } ( X _ i ( s ) , X _ j ( t ) ) = \\min ( s , t ) \\ , { \\rm c o v } ( X _ i ( 1 ) , X _ j ( 1 ) ) , \\ s , t \\geq 0 , \\ i , j = 1 , \\ldots , n . \\end{align*}"}
+{"id": "5312.png", "formula": "\\begin{align*} \\mathrm { R i c } ( X , Y ) = \\frac { n } { 2 } \\| \\mathbf { H } \\| ^ { 2 } \\ , g ( X , Y ) - ( n - 2 ) \\widetilde { g } ( \\mathbf { H } , \\xi ) g ( X , A _ { \\eta } Y ) . \\end{align*}"}
+{"id": "4693.png", "formula": "\\begin{align*} \\mathcal { I } ( G _ X ) = \\sum _ k ^ \\infty \\dim \\mathfrak { R } ^ { G _ X } _ k t ^ k . \\end{align*}"}
+{"id": "2547.png", "formula": "\\begin{align*} \\begin{aligned} & Z ( z _ 1 ( k _ 1 ) z _ { c _ 1 } ( k _ 2 ) { \\cdots } z _ { c _ { p - 1 } } ( k _ p ) ; ( \\alpha , \\beta ) ) \\\\ = & \\sum _ { \\begin{subarray} { c } 0 { \\le } m _ 1 < _ { c _ 1 } \\cdots < _ { c _ { p - 1 } } m _ p < \\infty \\end{subarray} } \\frac { ( \\alpha ) _ { m _ 1 } } { { m _ 1 } ! } \\frac { { m _ p } ! } { ( \\alpha ) _ { m _ p + 1 } } \\left \\{ \\prod _ { i = 1 } ^ { p - 1 } \\frac { 1 } { ( m _ i + \\beta ) ^ { k _ i } } \\right \\} \\frac { 1 } { ( m _ p + \\beta ) ^ { k _ p - 1 } } , \\end{aligned} \\end{align*}"}
+{"id": "3426.png", "formula": "\\begin{align*} \\mu ( [ x _ 0 x _ 1 \\ldots x _ n ] ) = \\pi ( x _ 0 ) p ( x _ 0 x _ 1 ) p ( x _ 1 x _ 2 ) \\ldots p ( x _ { n - 1 } x _ n ) , \\end{align*}"}
+{"id": "4855.png", "formula": "\\begin{align*} \\tau _ 1 ^ n + \\dots + \\tau _ k ^ n = 1 . \\end{align*}"}
+{"id": "129.png", "formula": "\\begin{align*} \\liminf _ { \\lambda \\rightarrow \\infty } \\lambda ^ p \\mathcal L ^ { 2 n } \\left ( E _ { \\lambda , K } \\right ) \\geq & \\frac { 1 } { n } \\int _ { \\mathbb R ^ n } \\left ( \\int _ { S ^ { n - 1 } } \\left | \\nabla f ( x ) \\cdot \\omega \\right | ^ p | | \\omega | | _ K ^ { - n - p } d \\omega \\right ) d x \\\\ = & \\frac { 2 } { n } \\int _ { \\mathbb R ^ n } | | \\nabla f ( x ) | | _ { Z _ p ^ * K } ^ p d x . \\end{align*}"}
+{"id": "2700.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u - \\Delta u = | u | ^ { \\frac { 4 } { N - 2 } } u \\end{align*}"}
+{"id": "8710.png", "formula": "\\begin{align*} \\sum _ { k > 0 } \\frac { ( 1 - x ^ k ) ( y - 1 ) } { y ^ { k + 1 } \\ , ( 1 - x ) } = \\frac { 1 } { x - y } . \\end{align*}"}
+{"id": "2004.png", "formula": "\\begin{align*} ( A \\chi _ I , \\chi _ I ) = \\sum c _ i ^ 2 \\lambda _ i \\leq \\left ( \\sum c _ i ^ 2 \\right ) \\lambda _ { m a x } = | I | \\cdot \\lambda _ { m a x } , \\end{align*}"}
+{"id": "7378.png", "formula": "\\begin{align*} g \\cdot ( \\varphi , \\varrho ) = ( g \\varphi g ^ { - 1 } , g \\cdot \\varrho ) . \\end{align*}"}
+{"id": "8050.png", "formula": "\\begin{align*} \\rho _ j ( \\pm n ) = \\rho _ j ( \\pm 1 ) \\lambda _ j ( n ) n ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "3712.png", "formula": "\\begin{align*} \\int _ { F ^ \\times \\times K } & | b | ^ { s } \\chi ( b ) | a | ^ { - 1 } \\left ( r _ i ( k ) f \\right ) ( 0 , 0 , a ^ { - 1 } b ) d k d ^ \\times b \\\\ = & | a | ^ { s - 1 } \\chi ( a ) \\int _ { F ^ \\times \\times K } \\chi ( b ) | b | ^ { s } \\left ( r _ i ( k ) f \\right ) ( 0 , 0 , b ) d ^ \\times b . \\end{align*}"}
+{"id": "117.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow 0 ^ + } \\lambda ^ p \\mathcal L ^ { 2 n } ( \\widetilde E _ { \\lambda } ) = 2 | B ^ n | | | f | | ^ p _ { L ^ p ( \\mathbb R ^ n ) } . \\end{align*}"}
+{"id": "5985.png", "formula": "\\begin{align*} V _ { p _ i } : & = \\left \\{ \\mu = ( P _ { i , j } ) \\in V \\mid \\langle ( F _ \\mu ) _ { p _ i } \\rangle \\subset \\pi _ k ( x _ i ) \\right \\} \\\\ & = \\left \\{ \\mu = ( P _ { i , j } ) \\in V \\mid \\langle P _ { n _ { k - 1 } + 1 } ( p _ i ) , \\dots , P _ { n _ k } ( p _ i ) \\rangle \\subset \\pi _ k ( x _ i ) \\right \\} . \\end{align*}"}
+{"id": "4779.png", "formula": "\\begin{align*} \\Pr _ { \\rho \\sim P _ \\epsilon } [ \\rho \\neq \\tilde { D } _ { 1 } ( H \\rho ^ \\intercal ) ] & \\leq 2 e ^ { - \\frac { l ^ 2 } { 3 \\epsilon N } } + ( 2 l + 1 ) \\mathop { \\max } _ { \\substack { w , w ' \\in \\{ \\epsilon N \\pm l \\} \\\\ w ' \\leq w } } \\left \\{ \\frac { \\Pr _ { \\rho \\sim \\lambda _ { w , w ' } } \\big [ | \\Omega _ \\rho ^ { \\{ w , w ' \\} } | > 1 \\big ] } { \\Pr _ { \\rho \\sim \\lambda _ { w , w ' } } \\big [ | \\rho | = w \\big ] } \\right \\} . \\end{align*}"}
+{"id": "8258.png", "formula": "\\begin{align*} \\zeta ( s ) \\ne 0 ~ ~ ~ s = \\sigma + i t , ~ ~ \\sigma > 1 - \\frac { c } { \\log t } , \\end{align*}"}
+{"id": "1002.png", "formula": "\\begin{align*} f _ 3 \\big ( \\tfrac 1 3 , y \\big ) = f _ 3 \\big ( \\tfrac 2 3 , y \\big ) = \\sin ( 3 \\pi y ) . \\end{align*}"}
+{"id": "1477.png", "formula": "\\begin{align*} \\lambda = \\frac { ( k - 1 ) ! ( k - 3 ) ! } { ( k / 2 - 2 ) ! ( k / 2 - 1 ) ! } . \\end{align*}"}
+{"id": "2245.png", "formula": "\\begin{align*} & \\sum _ { l = 1 } ^ { 2 k - 2 } \\biggl | \\tilde { a } ^ { n } _ { l } - \\sum _ { j = 0 } ^ { l } \\frac { \\alpha _ { l - j } } { j ! } \\tau ^ j \\biggr | + \\biggl | \\tilde { a } ^ { n } _ { 2 k - 1 } - \\sum _ { j = 1 } ^ { 2 k - 1 } \\frac { \\alpha _ { 2 k - 1 - j } } { j ! } \\tau ^ j \\biggr | \\\\ & + \\sum _ { j = 1 } ^ { 2 k - 2 } \\biggl | \\tilde { b } ^ { + , n } _ { j } - \\sum _ { l = 0 } ^ { 2 k - 2 } \\frac { \\alpha _ { 2 k - 2 - l } } { ( l + j + 1 ) ! } \\tau ^ { l + j + 1 } \\biggr | \\geq \\epsilon > 0 \\end{align*}"}
+{"id": "47.png", "formula": "\\begin{align*} T ( n _ i ) + T ( n _ { i + 1 } ) & < \\left ( \\frac { 5 } { 3 } + \\frac { 6 5 } { 2 7 } \\cdot \\frac { 9 } { 8 } \\right ) \\cdot \\frac { 1 } { X _ 0 } = \\frac { 1 0 5 } { 2 4 } \\cdot \\frac { 1 } { X _ 0 } \\\\ & < 6 \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } = ( k _ i + k _ { i + 1 } ) \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } . \\end{align*}"}
+{"id": "4633.png", "formula": "\\begin{align*} \\mathrm { R } _ { K / \\mathbb { Q } } ( G ) : = \\prod ^ d _ { i = 1 } G ^ { \\sigma _ i } , \\end{align*}"}
+{"id": "9151.png", "formula": "\\begin{align*} f ( \\lambda _ * ) = \\left ( 1 - \\frac { \\tau } { m + \\tau } \\right ) ^ { \\frac { m + \\tau } { \\tau } \\cdot \\tau } \\cdot \\left ( 1 - \\frac { \\tau } { m + \\tau } \\right ) ^ { - \\tau } \\cdot \\left ( \\frac { \\tau } { m + \\tau } \\right ) ^ \\tau . \\\\ \\end{align*}"}
+{"id": "3705.png", "formula": "\\begin{align*} \\chi ( a ) : = ( a , ( - 1 ) ^ { \\dim V _ i / 2 } \\det J _ i ) \\end{align*}"}
+{"id": "1277.png", "formula": "\\begin{align*} \\zeta _ t ( \\omega ^ { 1 + t r } ) = 1 { \\rm ~ a n d ~ } \\zeta _ t ( \\omega ^ { 1 + t ' r } ) = 0 { \\rm ~ f o r ~ } t \\neq t ' . \\end{align*}"}
+{"id": "6258.png", "formula": "\\begin{align*} \\phi ( \\mathcal { B E } _ { r } ) = \\mathcal { K } , \\phi \\big | _ { \\bar { 0 } _ { \\mathcal { S } } } = \\iota _ { \\mathcal { S } } . \\end{align*}"}
+{"id": "7709.png", "formula": "\\begin{align*} I _ n ^ { ( 1 ) } ( \\lambda ) : = \\sum _ { j = 1 } ^ n \\omega _ j \\left ( 4 \\tau + \\lambda ( t _ j + 1 ) ^ 2 \\right ) ^ { - 1 } , I _ n ^ { ( 2 ) } ( \\lambda ) : = \\sum _ { j = 1 } ^ n \\omega _ j \\left ( \\tau ( t _ j + 1 ) ^ 2 + 4 \\lambda \\right ) ^ { - 1 } , \\end{align*}"}
+{"id": "3289.png", "formula": "\\begin{align*} { \\partial _ t V + A V = 0 , } \\end{align*}"}
+{"id": "6194.png", "formula": "\\begin{align*} \\overline A _ p \\overline x _ l ^ { ( k ) } = C _ p A C _ p ^ T ( C _ p C _ p ^ T ) ^ { - 1 } C _ p \\widehat x _ l ^ { ( k ) } = C _ p A \\widehat x _ l ^ { ( k ) } . \\end{align*}"}
+{"id": "5382.png", "formula": "\\begin{align*} \\mathbb { E } [ Z _ t ( m ) ^ 2 ] = \\sigma _ m ^ 2 & : = \\sum _ { e ^ { W + C + m - 1 } < k \\le e ^ { W + C + m } } \\frac { 1 } { 2 k e ^ { 2 k \\sigma } } \\\\ \\mathbb { E } [ Z ( m ) Z _ t ( m ) ] = \\rho _ { m , t } \\sigma _ m ^ 2 & : = \\sum _ { e ^ { W + C + m - 1 } < k \\le e ^ { W + C + m } } \\frac { \\cos ( k t ) } { 2 k e ^ { 2 k \\sigma } } \\end{align*}"}
+{"id": "7357.png", "formula": "\\begin{align*} r _ i : = 1 0 0 ^ { 2 ^ { i + 1 } - 2 } . \\end{align*}"}
+{"id": "3129.png", "formula": "\\begin{align*} \\bar { \\gamma } : = \\left ( \\int _ Y \\frac { r } { \\gamma } \\right ) ^ { - 1 } = \\left ( \\int _ Y ( r + r A : D ^ 2 \\phi ) \\right ) ^ { - 1 } = 1 , \\end{align*}"}
+{"id": "4171.png", "formula": "\\begin{align*} L _ { \\nabla \\lambda } = ( A ( h ) + C ( K ) , B ( K ) + D ( h ) ) . \\end{align*}"}
+{"id": "1061.png", "formula": "\\begin{align*} \\sigma _ J = C 2 ^ { J / 2 } / \\alpha , \\mbox { w i t h } C = ( 8 \\lceil A \\rceil + 4 ) \\| \\psi \\| _ \\infty \\sqrt { 2 } ( \\sqrt { 2 } - 1 ) ^ { - 1 } , \\end{align*}"}
+{"id": "3357.png", "formula": "\\begin{align*} \\flat ( X _ H ) = \\dd H - ( R \\lrcorner H + H ) \\eta , \\end{align*}"}
+{"id": "5994.png", "formula": "\\begin{align*} X ^ { w _ 0 w _ { k , l } } = w _ 0 X ( w _ { k , l } ) = \\left \\{ ( L , H ) \\in X \\bigr \\rvert L \\subset < e _ { n - k + 1 } , \\cdots , e _ n > , \\ : < e _ { n - l + 2 } , \\cdots , e _ { n } > \\subset H \\right \\} . \\end{align*}"}
+{"id": "2351.png", "formula": "\\begin{align*} Q ( - k ) ^ T = Q ( k ) . \\end{align*}"}
+{"id": "942.png", "formula": "\\begin{align*} \\textrm { g r a d e } _ R ( \\textrm { E x t } _ { R } ^ i ( M , \\ , C ) ) \\ , = \\ , \\min \\{ \\textrm { d e p t h } ( R _ { \\mathfrak p } ) \\mid { \\mathfrak p } \\in \\textrm { S u p p } _ R ( \\textrm { E x t } _ { R } ^ i ( M , \\ , C ) ) \\} . \\end{align*}"}
+{"id": "8338.png", "formula": "\\begin{align*} \\Box _ { s , y } v & = 3 I ^ { - 2 } \\overline { Q _ l } ^ 2 v + 3 I ^ { - \\frac 1 2 } \\overline { Q _ l } v ^ 2 + I v ^ 3 \\\\ & = : \\sum _ { j = 1 } ^ 3 G _ j ^ { ( 4 ) } v ^ j . \\end{align*}"}
+{"id": "7431.png", "formula": "\\begin{align*} & \\phi ( \\alpha \\cdot P ) = \\bigg [ K ( Z _ i , Z _ j ) \\big ( \\sum _ { k = 1 } ^ N \\alpha _ { i k } P _ { k j } \\big ) \\bigg ] _ { 1 \\le i , j \\le N } , \\\\ & L _ \\alpha \\phi ( P ) = \\bigg [ \\sum _ { k = 1 } ^ N \\alpha _ { i k } K ( Z _ k , Z _ j ) ( P _ { k j } ) \\bigg ] _ { 1 \\le i , j \\le N } . \\end{align*}"}
+{"id": "4035.png", "formula": "\\begin{align*} \\int _ { \\Omega ^ * } f ^ * ( y ) \\ d y & = \\int _ { \\Omega } f ^ * ( Y ( \\cdot , v , D v ) ) \\det D Y ( \\cdot , v , D v ) \\\\ & > \\int _ { \\Omega } t f ( \\cdot ) + ( 1 - t ) f ^ * ( Y ( \\cdot , u _ 0 , D u _ 0 ) ) \\det D Y ( \\cdot , u _ 0 , D u _ 0 ) \\\\ & = \\int _ { \\Omega ^ * } f ^ * . \\end{align*}"}
+{"id": "8773.png", "formula": "\\begin{align*} \\dot { \\phi } _ i = & \\omega _ i + \\frac { 1 } { N } \\sum _ { j = 1 } ^ N W _ { i j } ( t ) g ( t , \\phi _ j , \\phi _ i ) , \\\\ \\dot { W } _ { i j } = & ( G ' ( W _ { i j } ) ) ^ { - 1 } h ( t , \\phi _ j , \\phi _ i ) , \\end{align*}"}
+{"id": "6800.png", "formula": "\\begin{align*} d \\left ( f ( \\overline { x } ) , g ( \\overline { x } ) \\right ) = \\sup _ { \\overline { x } \\in [ a , b ] } \\left | f ( \\overline { x } ) - g ( \\overline { x } ) \\right | e ^ { - m \\overline { x } } \\end{align*}"}
+{"id": "544.png", "formula": "\\begin{align*} { \\rm d } X _ t = f ( X _ t , \\theta ) { \\rm d } t + \\gamma ^ { 1 / 2 } { \\rm d } W _ t \\end{align*}"}
+{"id": "8604.png", "formula": "\\begin{align*} e _ 1 \\cdot ( x _ 1 , x _ 2 ) & = ( - x _ 1 , x _ 2 + b ) , \\\\ e _ 2 \\cdot ( x _ 1 , x _ 2 ) & = ( x _ 1 + a , - x _ 2 ) . \\end{align*}"}
+{"id": "5225.png", "formula": "\\begin{align*} A _ 1 & = \\ell _ 1 A _ 2 \\quad , A _ 4 = \\ell _ 2 A _ 5 \\\\ A _ 6 & = \\frac { 1 } { \\ell _ 2 } A _ 3 + \\ell _ 1 A _ 5 \\end{align*}"}
+{"id": "903.png", "formula": "\\begin{align*} \\pi ^ 2 = [ 9 ; 1 , 6 , 1 , 2 , 4 7 , 1 , 8 , 1 , 1 , 2 , 2 , 1 , 1 , 8 , 3 , 1 , 1 0 , 5 , 1 , 3 , 1 , 2 , 1 , 1 , 3 , 1 5 , \\ldots ] . \\end{align*}"}
+{"id": "3515.png", "formula": "\\begin{align*} ( A _ { ( k , l ) \\to i } ) ^ \\tau = ( A ^ \\tau ) _ { ( k , \\tau _ k ^ { - 1 } ( l ) ) \\to i } . \\end{align*}"}
+{"id": "6647.png", "formula": "\\begin{align*} \\partial _ t \\delta _ y u - \\nabla \\cdot a _ y \\nabla \\delta _ y u = \\partial _ t \\delta _ y v - \\nabla \\cdot a ( t ' , x ' ) \\nabla \\delta _ y v \\end{align*}"}
+{"id": "5253.png", "formula": "\\begin{align*} g ( U , U ) = a \\sin ^ 2 \\omega ( t ) . \\end{align*}"}
+{"id": "8036.png", "formula": "\\begin{align*} L ( s , f ) = \\sum _ { m = 1 } ^ \\infty \\frac { A ( m , 1 ) } { m ^ s } \\end{align*}"}
+{"id": "3581.png", "formula": "\\begin{align*} E _ { i + 1 , i } ^ { \\mu _ i - \\mu _ { i + 1 } + 1 } \\Omega _ \\mu = 0 , ( 1 \\leq i \\leq n - 1 ) . \\end{align*}"}
+{"id": "5113.png", "formula": "\\begin{align*} \\mbox { I m } \\left \\lbrace \\partial _ { t } \\gamma ( t , s ) \\overline { \\partial _ { s } \\gamma ( t , s ) } \\right \\rbrace = \\Omega \\mbox { R e } \\left \\lbrace \\gamma ( 0 , s ) \\overline { \\partial _ { s } \\gamma ( 0 , s ) } \\right \\rbrace . \\end{align*}"}
+{"id": "6512.png", "formula": "\\begin{align*} ( \\zeta ) = \\frac { 4 l } { k } \\underset { i = 1 } { \\overset { k } { \\sum } } ( p _ i - \\frac { 1 } { 2 } ) ^ 2 p _ i ( 1 - p _ i ) \\leq \\frac { l } { k } \\underset { i = 1 } { \\overset { k } { \\sum } } \\left ( p _ i - \\frac { 1 } { 2 } \\right ) ^ 2 . \\end{align*}"}
+{"id": "3521.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { j } \\gamma _ { k i } \\geq \\sum _ { k = 1 } ^ { j + 1 } \\gamma _ { k , i + 1 } . \\end{align*}"}
+{"id": "5250.png", "formula": "\\begin{align*} F _ \\ast ( \\nabla _ X X ) + ( \\nabla F _ \\ast ) ( X , X ) = 0 . \\end{align*}"}
+{"id": "4544.png", "formula": "\\begin{align*} s _ { i + 1 } = \\left ( \\prod _ { j = 1 } ^ i s _ i \\right ) + 1 \\end{align*}"}
+{"id": "3208.png", "formula": "\\begin{align*} \\frac 1 p + \\frac 1 r = \\frac 1 q . \\end{align*}"}
+{"id": "131.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ n } \\mathbf { 1 } _ { \\left \\{ y : \\ | | y - x | | _ K \\leq \\left ( \\frac { 2 | f ( x ) | } { \\lambda } \\right ) ^ { \\frac p n } \\right \\} } d y = \\int _ { \\mathbb R ^ n } \\mathbf { 1 } _ { \\left \\{ z + x : \\ | | z | | _ { \\left ( \\frac { 2 | f ( x ) | } { \\lambda } \\right ) ^ { \\frac p n } K } \\leq 1 \\right \\} } d z = \\left ( \\frac { 2 | f ( x ) | } { \\lambda } \\right ) ^ { p } | K | . \\end{align*}"}
+{"id": "8144.png", "formula": "\\begin{align*} \\phi ( \\xi ) = - \\frac { T \\xi } { M } \\pm \\frac { x } { 2 \\pi } \\sinh \\frac { \\xi \\pi } { M } \\end{align*}"}
+{"id": "280.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\rho } { \\tau } - \\beta \\tau \\eta \\leqslant \\frac { 2 \\alpha \\rho ^ 2 } { 1 + 3 \\rho ^ 2 } \\leqslant \\frac { 2 \\alpha } { 3 } , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\rho } { \\tau } - \\beta \\rho \\eta \\leqslant \\frac { 2 \\alpha \\rho ^ 2 } { 1 + 3 \\rho ^ 2 } \\leqslant \\frac { 2 \\alpha } { 3 } , \\end{align*}"}
+{"id": "1579.png", "formula": "\\begin{align*} d _ p ( \\mu , \\nu ) : = \\Big ( \\inf _ { \\pi \\in \\Pi ( \\mu , \\nu ) } \\sum _ { ( x , x ' ) \\in X \\times X } \\varrho ^ p ( x , x ' ) \\cdot \\pi ( x , x ' ) \\Big ) ^ { 1 / p } . \\end{align*}"}
+{"id": "3713.png", "formula": "\\begin{align*} \\Theta _ f ( g ) : = \\sum _ { \\xi \\in V _ { i + 1 } ( F ) } \\rho _ { i + 1 } ( g ) f ( \\xi ) . \\end{align*}"}
+{"id": "4903.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty e ^ { - \\rho } \\big ( b + ( \\partial _ s \\rho ) ^ 2 \\big ) d s + \\int _ { - \\infty } ^ \\infty \\big ( 1 - e ^ { - \\rho } \\big ) ^ 2 d s = | | b | | _ { L ^ 1 } . \\end{align*}"}
+{"id": "5424.png", "formula": "\\begin{align*} F ( J , \\alpha ) = \\int _ M f ( \\bar { \\alpha } \\alpha ) \\frac { \\omega ^ n } { n ! } \\end{align*}"}
+{"id": "8609.png", "formula": "\\begin{align*} \\wp _ A ( { \\bf { b } } ) = \\# \\{ { \\bf { x } } \\in \\mathbb { N } ^ d \\ ; : \\ ; A { \\bf { x } } = { \\bf { b } } . \\} \\end{align*}"}
+{"id": "5175.png", "formula": "\\begin{align*} p _ { ( v _ i , k ) } = \\sum _ { \\alpha \\in A } s _ { \\alpha } s _ { \\alpha } ^ * . \\end{align*}"}
+{"id": "6141.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } u _ r '' - \\Delta u _ r + \\sum _ { s = 1 } ^ d \\alpha _ { r s } u _ s = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ u _ r = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu u _ r + \\sum _ { s = 1 } ^ d \\beta _ { r s } u _ s = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "8661.png", "formula": "\\begin{align*} H ( u ) E ( - u ) = 1 , H ^ \\perp ( u ) E ^ { \\perp } ( - u ) = 1 , \\end{align*}"}
+{"id": "1094.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ \\infty \\theta _ j \\varphi _ j = \\sum _ { j = 1 } ^ \\infty \\left \\{ \\int _ { [ 0 , 1 ] } f ( x ) \\varphi _ j ( x ) \\ , \\mathrm { d } x \\right \\} \\varphi _ j . \\end{align*}"}
+{"id": "7408.png", "formula": "\\begin{align*} \\begin{matrix} X _ { \\alpha } ( s ) = \\omega ( \\varpi ) q _ F ^ { - 2 s } \\\\ q _ { \\alpha ^ * } = 1 , q _ { \\alpha } = q _ F \\end{matrix} \\end{align*}"}
+{"id": "2190.png", "formula": "\\begin{align*} \\psi ( r ) : = ( ( 2 - 2 \\alpha - \\beta ) ( \\mathit { e } + 1 ) + ( \\mathit { e } - 1 ) ) r ^ 2 + ( 2 - 2 \\alpha + \\beta ) ( \\mathit { e } + 1 ) r - ( \\mathit { e } - 1 ) . \\end{align*}"}
+{"id": "9078.png", "formula": "\\begin{align*} & \\int _ { M / ( M \\cap H ) } \\left | \\int _ { A / ( A \\cap H ) } a ^ { \\lambda - \\rho _ { Q } } f ( k m a H _ { \\emptyset } ) \\ , d a \\right | ^ { 2 } \\ , d m \\\\ & \\qquad = \\sum _ { [ \\sigma ] \\in \\widehat { M } } \\dim ( V _ { \\sigma } ) \\sum _ { \\eta \\in E _ { \\sigma } } \\left \\| \\int _ { M / ( M \\cap H ) } \\int _ { A / ( A \\cap H ) } a ^ { \\lambda - \\rho _ { Q } } f ( k m a H _ { \\emptyset } ) \\sigma ( m ) \\eta \\ , d a \\ , d m \\right \\| ^ { 2 } _ { \\sigma } . \\end{align*}"}
+{"id": "2718.png", "formula": "\\begin{align*} & V ^ k _ 0 = \\sum _ { j _ k } ^ { j _ { k + 1 } - 1 } W _ { ( \\nu _ j ) } + \\check h _ 0 ^ k , \\\\ & V ^ k _ 1 = \\sum _ { j _ k } ^ { j _ { k + 1 } - 1 } \\check \\alpha _ j ( \\Lambda W _ { ( \\nu _ j ) } ) _ { [ \\nu _ j ] } + \\check g _ 1 ^ k , \\end{align*}"}
+{"id": "101.png", "formula": "\\begin{align*} x \\ast y & = u ( x y ) - ( u x ) y - x ( u y ) = x ( u y ) - ( x u ) y - x ( u y ) = - ( x u ) y . \\end{align*}"}
+{"id": "7155.png", "formula": "\\begin{align*} \\displaystyle { \\frac { d ^ 2 z _ m } { d t ^ 2 _ k } \\Bigg | _ { \\rm e q } = 0 \\ , , k , m = 1 , . . . , N } \\end{align*}"}
+{"id": "4359.png", "formula": "\\begin{align*} L _ \\nu L _ \\nu ^ * \\tilde w ( \\nu , \\eta ) + \\eta \\tilde w ( \\nu , \\eta ) = f _ \\nu . \\end{align*}"}
+{"id": "6071.png", "formula": "\\begin{align*} \\varphi ( z ) : = \\exp \\left \\{ \\pi \\int _ 1 ^ z \\frac { u ( s ) \\dd s } { ( w \\tilde w ) ( s ) } \\right \\} , \\end{align*}"}
+{"id": "2082.png", "formula": "\\begin{align*} \\mathcal { S } _ m = \\mathcal { S } _ m ( 1 / m ) , \\mathcal { S } _ m ^ { \\ast } = \\mathcal { S } _ m ^ { \\ast } ( 1 / m ) , \\end{align*}"}
+{"id": "6493.png", "formula": "\\begin{align*} \\rho ^ 2 _ s \\asymp \\begin{cases} \\left ( \\frac { \\sqrt { b } n } { \\sqrt { \\log ( n ) } } \\right ) ^ { - \\frac { 2 s } { 2 s + 1 } } , & \\ ; m \\geq n ^ { \\frac { 1 } { 2 s + 1 } } , \\\\ \\left ( \\frac { \\sqrt { b } n } { \\sqrt { m \\log ( n ) } } \\right ) ^ { - \\frac { 2 s } { 2 s + 1 / 2 } } , & \\ ; m < n ^ { \\frac { 1 } { 2 s + 1 } } , \\end{cases} \\end{align*}"}
+{"id": "5177.png", "formula": "\\begin{align*} \\begin{aligned} & ( e _ { i i _ 1 } , m _ i + k ) ( e _ { i _ 1 i _ 2 } , m _ i + m _ { i _ 1 } + k ) ( e _ { i _ 2 i _ 3 } , m _ i + m _ { i _ 1 } + m _ { i _ 2 } + k ) \\\\ & \\cdots \\cdots ( e _ { i _ { s - 1 } i _ s } , m _ i + m _ { i _ 1 } + m _ { i _ 2 } + \\cdots + m _ { i _ { s - 1 } } + k ) \\end{aligned} \\end{align*}"}
+{"id": "88.png", "formula": "\\begin{align*} g _ { \\mathbf { k } } = f _ { \\mathbf { k } } \\circ \\tau _ { P _ 0 } \\end{align*}"}
+{"id": "6693.png", "formula": "\\begin{align*} \\frac { 1 } { \\log x } \\sum _ { n \\leq x } \\frac { \\lambda ( n ) \\lambda ( n + a ) } { n } = O \\left ( \\frac { 1 } { ( \\log \\log \\log x ) ^ c } \\right ) , \\end{align*}"}
+{"id": "1564.png", "formula": "\\begin{align*} \\rho \\circ X ^ { R - Q } \\circ \\rho & = \\frac { ( X + \\omega ) ^ { R - Q } + \\omega ( \\omega X + 1 ) ^ { R - Q } } { \\omega ( X + \\omega ) ^ { R - Q } + ( \\omega X + 1 ) ^ { R - Q } } \\\\ & = \\frac { ( X + \\omega ) ^ R ( \\omega X + 1 ) ^ Q + \\omega ( \\omega X + 1 ) ^ R ( X + \\omega ) ^ Q } { \\omega ( X + \\omega ) ^ R ) ( \\omega X + 1 ) ^ Q + ( \\omega X + 1 ) ^ R ( X + \\omega ) ^ Q } \\\\ & = X ^ { ( - 1 ) ^ \\ell } \\circ g ( X ) . \\end{align*}"}
+{"id": "5465.png", "formula": "\\begin{align*} \\tilde Q ( x _ 1 , x _ 2 ) f ( k , l ) & = \\sum _ { ( j , i ) \\in \\mathcal M \\times \\mathcal M } \\tilde q _ { ( k , l ) , ( j , i ) } ( x _ 1 , x _ 2 ) ( f ( j , i ) - f ( k , l ) ) \\\\ & = \\sum _ { j \\in \\mathcal M } ( q _ { k j } ( x _ 1 ) - q _ { l j } ( x _ 2 ) ) ^ + ( f ( j , l ) - f ( k , l ) ) \\\\ & + \\sum _ { j \\in \\mathcal M } ( q _ { l j } ( x _ 2 ) - q _ { k j } ( x _ 1 ) ) ^ + ( f ( k , j ) - f ( k , l ) ) \\\\ & + \\sum _ { j \\in \\mathcal M } ( q _ { k j } ( x _ 1 ) \\wedge q _ { l j } ( x _ 2 ) ) ( f ( j , j ) - f ( k , l ) ) \\end{align*}"}
+{"id": "6144.png", "formula": "\\begin{align*} \\hbox { K e r } ( \\mathcal R ^ T ) = \\{ 0 \\} . \\end{align*}"}
+{"id": "5170.png", "formula": "\\begin{align*} [ e _ 0 , e _ 1 ] = - e _ 2 , [ e _ 0 , e _ 2 ] = - e _ 1 , [ e _ 1 , e _ 2 ] = 0 . \\end{align*}"}
+{"id": "5472.png", "formula": "\\begin{align*} P ( \\left \\{ \\eta _ { \\alpha } > \\eta + \\delta \\right \\} \\cap \\left \\{ \\eta _ { \\tilde \\alpha } > \\eta + \\delta \\right \\} \\cap B ) \\geq 1 - \\frac { 1 } { 4 } P ( B ) - ( 1 - \\frac { 3 } { 4 } ) P ( B ) = \\frac { 1 } { 2 } P ( B ) > 0 . \\end{align*}"}
+{"id": "4105.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } g _ t = h , g _ 0 = g , \\\\ & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } H _ t = d K , H _ 0 = H , \\\\ & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } f _ t = \\phi , f _ 0 = f \\end{align*}"}
+{"id": "7825.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { k d _ 1 } \\prod _ { s = 1 } ^ { j - 1 } \\prod _ { i = 1 } ^ { d _ 1 - 1 } \\prod _ { t = 1 } ^ { i } \\begin{bmatrix} 2 \\\\ 2 \\end{bmatrix} _ { 1 , \\frac { - 2 j - 2 s + 2 k i + 2 k t + 2 } { k d _ 1 } - 1 } \\begin{bmatrix} 2 \\\\ 2 \\end{bmatrix} _ { 1 , \\frac { - 2 j - 2 s + 2 k i + 2 k t + 2 } { k d _ 1 } + 1 } , \\end{align*}"}
+{"id": "7999.png", "formula": "\\begin{align*} \\overline { \\lim _ { N \\rightarrow \\infty } } \\left | \\int _ { 0 } ^ { 1 } f _ N \\left ( x \\right ) Q _ N \\left ( x , \\varepsilon ^ 0 \\right ) d x \\right | = \\overline { \\lim _ { N \\rightarrow \\infty } } D _ N \\left ( \\varepsilon ^ 0 \\right ) = + \\infty . \\end{align*}"}
+{"id": "7563.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 5 ) = & \\sum _ { a = 0 } ^ { \\omega - 1 } { Z _ f ( s , \\chi , A _ 5 ^ a ) } \\\\ = & \\sum _ { a = 0 } ^ { \\omega - 1 } q ^ { - a - 1 } Z _ { f _ { 5 , a } } ( s , \\chi , B _ 5 ^ a ) , \\end{align*}"}
+{"id": "3023.png", "formula": "\\begin{align*} \\begin{array} { l } \\partial ^ { \\mathbf { A } } _ c p { _ a } ^ { j c } = \\partial _ c p { _ a } ^ { j c } + \\mathbf { A } { ^ k } _ c c ^ j _ { k \\ell } p { _ a } ^ { \\ell c } \\\\ \\partial ^ { \\mathbf { A } } _ c p { _ i } ^ { j c } = \\partial _ c p { _ i } ^ { j c } - \\mathbf { A } { ^ k } _ c c ^ \\ell _ { k i } p { _ \\ell } ^ { j c } + \\mathbf { A } { ^ k } _ c c ^ j _ { k \\ell } p { _ i } ^ { \\ell c } \\end{array} \\end{align*}"}
+{"id": "480.png", "formula": "\\begin{align*} \\Delta _ h w ( t ) : = \\frac { 1 } { h } ( w ( t + h ) - w ( t ) ) . \\end{align*}"}
+{"id": "5589.png", "formula": "\\begin{align*} & 0 = \\nabla _ 0 W _ { i j k l } = \\kappa _ i \\Psi _ { j k l } - \\kappa _ j \\Psi _ { i k l } + \\kappa _ k \\Psi _ { l i j } - \\kappa _ l \\Psi _ { k i j } \\ ; ; \\\\ & 0 = \\nabla _ 0 W _ { 0 1 i j } = \\kappa ^ m \\Psi _ { m i j } - \\kappa _ i \\Psi _ j + \\kappa _ j \\Psi _ i \\ ; ; \\\\ & 0 = \\nabla _ 0 W _ { 0 i 1 j } = - \\kappa ^ m \\Psi _ { j m i } - \\kappa _ i \\Psi _ j \\ ; . \\end{align*}"}
+{"id": "6195.png", "formula": "\\begin{align*} \\overline A _ p \\overline x _ l ^ { ( k ) } = C _ p A ( \\widehat x _ l ^ { ( k ) } + \\widetilde x _ l ^ { ( k ) } ) = C _ p A x _ l ^ { ( k ) } . \\end{align*}"}
+{"id": "8873.png", "formula": "\\begin{align*} N _ t ^ { ( n , \\nu ) } ( z , w ) : = \\pi ^ { - n } \\left ( \\frac { ( 1 + \\langle w , z \\rangle ) ^ 2 } { ( 1 + | z | ^ { 2 } ) ( 1 + | w | ^ { 2 } ) } \\right ) ^ { \\nu } e ^ { 4 t ( \\nu ^ 2 + \\frac { n ^ 2 } { 4 } ) } . \\end{align*}"}
+{"id": "3271.png", "formula": "\\begin{align*} f _ { \\lambda } \\left ( x \\right ) \\equiv F \\left ( x \\right ) - \\lambda G \\left ( x \\right ) = y , y \\in Y . \\end{align*}"}
+{"id": "687.png", "formula": "\\begin{align*} \\int _ b ^ a d U _ { \\alpha _ j } = \\oint _ { \\alpha _ j } * d G ^ { \\delta _ a - \\delta _ b } , \\end{align*}"}
+{"id": "9176.png", "formula": "\\begin{align*} \\hat { f } _ { \\epsilon } ( \\epsilon p ) = \\sum _ { y \\in \\epsilon \\Z ^ 2 } f _ { \\epsilon } ( y / \\epsilon ) e ^ { - i y \\cdot p } . \\end{align*}"}
+{"id": "6395.png", "formula": "\\begin{align*} u ( X , Y , t ) & = \\frac 1 2 \\biggl \\{ \\sup _ { \\tilde X \\in { { B _ \\epsilon ( X ) } } } u ( \\tilde X , Y + { \\epsilon ^ 2 } \\tilde X / 2 , t - { \\epsilon ^ 2 } / 2 ) + \\inf _ { \\tilde X \\in { { B _ \\epsilon ( X ) } } } u ( \\tilde X , Y + { \\epsilon ^ 2 } \\tilde X / 2 , t - { \\epsilon ^ 2 } / 2 ) \\biggr \\} \\\\ & + { o } ( \\epsilon ^ 2 ) , \\mbox { a s } \\epsilon \\to 0 . \\end{align*}"}
+{"id": "4319.png", "formula": "\\begin{align*} \\phi _ 1 ( D ^ 1 \\times ( - \\epsilon , \\epsilon ) = \\phi _ 2 ( D ^ 1 \\times ( - \\epsilon , \\epsilon ) ) . \\end{align*}"}
+{"id": "7218.png", "formula": "\\begin{align*} p _ 0 ( t ) = R \\Big ( e ^ { i \\phi } + e ^ { i \\psi } j \\Big ) , \\phi = \\psi , \\qquad \\mbox { a n d } p = q - p _ 0 \\end{align*}"}
+{"id": "5919.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } L u '' - L \\Delta u = 0 & \\hbox { i n } ( T , + \\infty ) \\times \\Omega , \\\\ L \\partial _ \\nu u + \\Lambda u = 0 & \\hbox { o n } ( T , + \\infty ) \\times \\Gamma , \\end{array} \\right . \\end{align*}"}
+{"id": "1724.png", "formula": "\\begin{align*} D = \\{ ( z _ 1 , z _ 2 ) \\in \\C ^ 2 , \\ ; | z _ 1 | ^ 2 + | z _ 2 | ^ 2 \\leq 1 \\} . \\end{align*}"}
+{"id": "2560.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta ( z _ 1 ( k _ 1 ) z _ { c _ 1 } ( k _ 2 ) { \\cdots } z _ { c _ { p - 1 } } ( k _ p ) ; \\alpha ) = \\sum _ { \\begin{subarray} { c } 0 { \\le } m _ 1 < _ { c _ 1 } \\cdots < _ { c _ { p - 1 } } m _ p < \\infty \\end{subarray} } \\prod _ { i = 1 } ^ { p } \\frac { 1 } { ( m _ i + \\alpha ) ^ { k _ i } } , \\end{aligned} \\end{align*}"}
+{"id": "5455.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\d X ^ n _ t & = b ( t , X ^ n _ { t \\wedge \\tau ^ n } , \\mathcal { L } _ { X ^ n _ { t \\wedge \\tau ^ n } } , \\alpha ^ n _ t ) d t + \\sigma ( t , X ^ n _ { t \\wedge \\tau ^ n } , \\mathcal { L } _ { X ^ n _ { t \\wedge \\tau ^ n } } , \\alpha ^ n _ t ) d W _ t \\\\ X ^ n _ 0 & = X _ 0 , \\alpha ^ n _ 0 = \\alpha \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "235.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = & \\beta u w + \\alpha ( 1 + v ) ^ 2 , \\psi _ 2 ( u , v , w ) = \\beta u w + \\alpha ( 1 + u ) ^ 2 , \\\\ \\psi _ 3 ( u , v , w ) = & \\beta v w + \\alpha \\left ( 1 + v \\right ) ^ 2 \\psi _ 4 ( v , w ) = \\beta v w + \\alpha ( 1 + u ) ^ 2 . \\end{align*}"}
+{"id": "7850.png", "formula": "\\begin{align*} T ( z ) = f ( z ) - a = \\frac { g ( z ) - a h ( z ) } { h ( z ) } \\end{align*}"}
+{"id": "2872.png", "formula": "\\begin{align*} \\sum _ { m \\leq Y } \\frac { c _ m } { m ^ l } & = \\int _ 1 ^ Y \\frac { d \\left ( \\sum _ { m \\leq u } c _ m \\right ) } { u ^ l } \\\\ & = \\frac { \\sum _ { m \\leq u } c _ m } { u ^ l } \\bigg | _ 1 ^ Y - \\int _ 1 ^ Y ( - l ) \\frac { \\sum _ { m \\leq u } c _ m } { u ^ { l + 1 } } d u . \\\\ \\end{align*}"}
+{"id": "6489.png", "formula": "\\begin{align*} N ^ { j } = \\sum _ { l = 1 } ^ { C _ { b , d } } \\underset { i = 1 } { \\overset { d } { \\sum } } B ^ { j } _ { l i } \\in \\{ 0 , 1 , \\dots , C _ { b , d } d \\} , \\end{align*}"}
+{"id": "5332.png", "formula": "\\begin{align*} \\mathcal { E } : = \\Big \\{ \\varepsilon \\in W ^ { 1 , \\infty } \\left ( \\Omega \\right ) \\cap \\ , \\ , & \\mathrm { S y m } _ 3 ( \\Omega ) : \\\\ & \\exists \\ , c > 0 \\varepsilon ( x ) \\ , \\xi \\cdot \\xi \\geq c \\ , \\abs { \\xi } ^ 2 x \\in \\Omega , \\xi \\in \\mathbb { R } ^ 3 \\Big \\} , \\end{align*}"}
+{"id": "915.png", "formula": "\\begin{align*} t _ { n + 1 } = - \\dfrac { \\log t _ { n } } { \\log { ( n ( n + 1 ) ) } } \\end{align*}"}
+{"id": "3330.png", "formula": "\\begin{align*} \\dd \\omega = 0 , \\theta \\wedge \\omega ^ 2 = 2 \\frac { k \\nu \\sqrt { \\delta } } { y ^ 2 } \\dd x \\wedge \\dd y \\wedge \\dd q \\wedge \\dd p \\wedge \\dd \\kappa , \\end{align*}"}
+{"id": "5462.png", "formula": "\\begin{align*} \\phi _ N ( X ^ { j } _ { t \\wedge \\tau _ N ^ { n , m } } ) = X ^ { j } _ { t \\wedge \\tau _ N ^ { n , m } } , ~ j \\in \\{ n , m \\} \\end{align*}"}
+{"id": "6418.png", "formula": "\\begin{align*} \\phi ( X _ 1 , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) & = \\min _ { \\tilde X \\in \\overline { { B _ \\epsilon ( X ) } } } \\phi ( \\tilde X , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) = \\inf _ { \\tilde X \\in { { B _ \\epsilon ( X ) } } } \\phi ( \\tilde X , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) . \\end{align*}"}
+{"id": "7694.png", "formula": "\\begin{align*} \\gamma = \\mathbb { C } [ s _ { 1 } , \\dots , s _ { d } ] \\xrightarrow [ ] { } T _ { j } ^ { - 1 } \\mathbb { C } [ S ] . \\end{align*}"}
+{"id": "2598.png", "formula": "\\begin{align*} \\dim \\sum _ { l = 2 } ^ 4 V _ { [ 4 ] \\setminus \\{ l \\} } ^ \\perp = 8 \\iff \\dim \\bigcap _ { l = 2 } ^ 4 V _ { [ 4 ] \\setminus \\{ l \\} } = 0 . \\end{align*}"}
+{"id": "7432.png", "formula": "\\begin{align*} & \\alpha _ { i k } Z _ k = Z _ i \\alpha _ { i k } k \\Rightarrow \\\\ & \\bigg [ K ( Z _ i , Z _ j ) \\big ( \\sum _ { k = 1 } ^ N \\alpha _ { i k } P _ { k j } \\big ) \\bigg ] _ { i j } = \\bigg [ \\sum _ { k = 1 } ^ N \\alpha _ { i k } K ( Z _ k , Z _ j ) ( P _ { k j } ) \\bigg ] _ { i j } . \\end{align*}"}
+{"id": "2240.png", "formula": "\\begin{align*} \\mathcal { H } _ { m + \\frac 1 2 } : = \\{ v : v \\widehat { \\mathbb { R } ^ { 2 } } , x \\cdot \\nabla v = ( m + \\frac 1 2 ) v \\} . \\end{align*}"}
+{"id": "4586.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n a _ i \\leq \\prod _ { i = 1 } ^ n b _ i . \\end{align*}"}
+{"id": "7710.png", "formula": "\\begin{align*} f ^ { ( 1 ) } ( t ) = \\frac { 1 } { 4 \\tau + \\lambda ( t + 1 ) ^ 2 } { \\rm a n d } f ^ { ( 2 ) } ( t ) = \\frac { 1 } { \\tau ( t + 1 ) ^ 2 + 4 \\lambda } . \\end{align*}"}
+{"id": "6387.png", "formula": "\\begin{align*} \\xi _ k ^ { \\frac { q ^ 3 - 1 } { p } } = \\xi _ k ^ { ( q ^ 2 + q + 1 ) \\frac { q - 1 } { p } } = g ^ { ( 3 k + d ) \\frac { q - 1 } { p } } \\ne 1 , \\end{align*}"}
+{"id": "9236.png", "formula": "\\begin{align*} \\ < \\varphi _ { j } , \\varphi _ { k } \\ > = \\delta _ { j , k } + \\mathcal { O } ( e ^ { - c / h } ) \\ < \\Delta _ { f } \\varphi _ { j } , \\varphi _ { k } \\ > = \\delta _ { j , k } \\mu _ { j } , \\end{align*}"}
+{"id": "3869.png", "formula": "\\begin{align*} \\det D ^ 2 u & = \\psi ( \\cdot , u , D u ) \\Omega , \\end{align*}"}
+{"id": "4593.png", "formula": "\\begin{align*} \\sum _ { i = m + 1 } ^ { n } \\frac { 1 } { b _ i } < \\sum _ { i = m + 1 } ^ { n } \\frac { 1 } { a _ i } . \\end{align*}"}
+{"id": "8435.png", "formula": "\\begin{align*} G ( \\nu ) = \\lim _ { K \\uparrow A } \\ , G ( \\nu | _ K ) . \\end{align*}"}
+{"id": "3323.png", "formula": "\\begin{align*} \\alpha : = k / 2 , \\quad \\gamma : = \\nu . \\end{align*}"}
+{"id": "4582.png", "formula": "\\begin{align*} \\frac { 5 } { 6 } = \\frac { 1 } { 2 } + \\frac { 1 } { 3 } \\leq \\frac { 1 } { b _ 1 } + \\frac { 1 } { b _ 2 } < 1 . \\end{align*}"}
+{"id": "2733.png", "formula": "\\begin{align*} { } ^ \\omega \\Delta ( E _ j ) = F _ j \\otimes 1 + K _ j ' \\otimes E _ j , & { } ^ \\omega \\Delta ( F _ j ) = 1 \\otimes F _ j + E _ j \\otimes K _ j ' , \\\\ { } ^ \\omega \\Delta ( K _ j ) = K _ j ' \\otimes K _ j , & { } ^ \\omega \\Delta ( K _ j ' ) = K _ j \\otimes K _ j ' . \\end{align*}"}
+{"id": "5064.png", "formula": "\\begin{align*} Q _ j ( y + b ) = \\sum _ { s = 0 } ^ m q _ { j , s } y ^ s \\qquad q _ { j , s } \\in K . \\end{align*}"}
+{"id": "9012.png", "formula": "\\begin{align*} \\sup _ { \\eta \\in \\mathbb { U } _ X } \\operatorname { E } _ \\mu ^ { ( O W ) } [ \\pi , \\eta ] \\ , & = \\ , \\sup _ { \\eta \\in \\mathbb { U } _ X } \\limsup _ { i \\in I } \\frac { H _ \\mu [ \\eta _ { A _ i } ] } { \\theta ( A _ i ) } \\\\ & = \\ , \\sup _ { \\eta \\in \\mathbb { U } _ X } \\limsup _ { i \\in I } \\frac { H _ \\mu [ \\eta _ { K F _ i } ] } { \\theta ( A _ i ) } \\ , = \\ , \\sup _ { \\eta \\in \\mathbb { U } _ X } \\limsup _ { i \\in I } \\frac { H _ \\mu [ \\eta _ { F _ i } ] } { \\theta ( A _ i ) } . \\end{align*}"}
+{"id": "1489.png", "formula": "\\begin{align*} \\begin{aligned} \\C _ r ( x ) & = \\prod _ { n = - \\infty , n } ^ \\infty P _ r \\left ( \\frac { x } { \\frac n 2 } \\right ) ^ { ( \\frac n 2 ) ^ { r - 1 } } \\\\ & = \\prod _ { n = 1 , n } ^ \\infty \\left \\{ P _ r \\left ( \\frac { x } { \\frac n 2 } \\right ) P _ r \\left ( - \\frac { x } { \\frac n 2 } \\right ) ^ { ( - 1 ) ^ { r - 1 } } \\right \\} ^ { ( \\frac n 2 ) ^ { r - 1 } } \\end{aligned} \\end{align*}"}
+{"id": "350.png", "formula": "\\begin{align*} \\nu ( M , N ) = F _ { - r } \\supset \\cdots \\supset F _ { - 1 } \\supset F _ 0 = 0 \\end{align*}"}
+{"id": "8906.png", "formula": "\\begin{align*} \\sigma _ { p } ^ { \\left ( \\nu \\right ) } \\left ( \\ell \\right ) = \\frac { ( - 1 ) ^ { \\ell } } { 2 ( p + \\ell + 1 ) \\ell ! } \\left [ B _ { 2 ( p + \\ell + 1 ) } \\left ( \\nu + \\frac { 1 } { 2 } \\right ) - B _ { 2 ( p + \\ell + 1 ) } \\left ( \\frac { 1 } { 2 } \\right ) \\right ] . \\end{align*}"}
+{"id": "9088.png", "formula": "\\begin{align*} \\mathfrak { Z } ( \\mathfrak { n } / \\mathfrak { d } ^ { j _ 0 - 1 } ) \\cap J \\mathfrak { Z } ( \\mathfrak { n } / \\mathfrak { d } ^ { j _ 0 - 1 } ) = \\mathfrak { n } / \\mathfrak { d } ^ { j _ 0 - 1 } . \\end{align*}"}
+{"id": "8721.png", "formula": "\\begin{align*} s ^ { F } _ \\lambda ( x _ 1 , \\dots , x _ n ) = \\frac { \\det [ ( x _ i | a ) _ { \\lambda _ j + n - j } ] _ { 1 \\le i , j \\le n } } { \\det [ x ^ { n - j } _ i ] _ { 1 \\le i , j \\le n } } . \\end{align*}"}
+{"id": "3577.png", "formula": "\\begin{align*} \\Omega _ A & = \\sum _ { \\tau \\in S _ \\lambda } \\omega _ { A ^ \\tau } \\\\ & = \\sum _ { \\tau \\in S _ { \\lambda } } [ B _ { A ^ \\tau ( 1 , 1 ) } ^ + , \\dots , B _ { A ^ \\tau ( \\lambda _ 1 ' , 1 ) } ^ + ] \\cdots [ B _ { A ^ \\tau ( 1 , \\ell ( \\lambda ' ) ) } ^ + , \\dots , B _ { A ^ \\tau ( \\lambda _ { \\ell ( \\lambda ' ) } ' , \\ell ( \\lambda ' ) ) } ^ + ] v _ 0 . \\end{align*}"}
+{"id": "3446.png", "formula": "\\begin{align*} \\begin{cases} d : = 4 a c - b ^ 2 > 0 ; \\\\ b ^ 2 \\leq a c \\leq \\textstyle \\frac { d } { 3 } ; \\\\ - a \\leq b \\leq a \\leq c . \\end{cases} \\end{align*}"}
+{"id": "9139.png", "formula": "\\begin{align*} G ( f ) \\ ; = \\ ; h ^ { | \\eta | + | \\gamma | } \\prod _ { ( x , y ) \\in E ( f ) } \\nu ( x - y ) , \\end{align*}"}
+{"id": "5878.png", "formula": "\\begin{align*} \\hbox { K e r } ( C _ p ) = \\hbox { S p a n } \\{ e _ 1 , \\cdots , e _ p \\} , \\end{align*}"}
+{"id": "9158.png", "formula": "\\begin{align*} \\avg { F } _ { J , \\beta } ^ { \\Lambda _ N } \\propto \\sum _ { \\sigma \\in \\Omega ^ { \\Lambda _ N } } e ^ { - \\frac { 1 } { 2 \\beta } ( \\sigma , - \\Delta _ J \\sigma ) } \\ , F ( \\sigma ) = \\sum _ { \\sigma \\in \\Omega ^ { \\Lambda _ N } } e ^ { - \\frac { 1 } { 4 \\beta | J | } \\sum _ { x - y \\in J } ( \\sigma _ x - \\sigma _ y ) ^ 2 } \\ , F ( \\sigma ) \\end{align*}"}
+{"id": "8448.png", "formula": "\\begin{align*} \\lim _ { s \\in S } \\ , \\| \\mu _ s \\| ^ 2 = w ( A ) . \\end{align*}"}
+{"id": "8044.png", "formula": "\\begin{align*} \\phi ( s ) = \\sqrt { \\pi } \\frac { \\Gamma \\Bigl ( s - \\dfrac { 1 } { 2 } \\Bigr ) } { \\Gamma ( s ) } \\frac { \\zeta ( 2 s - 1 ) } { \\zeta ( 2 s ) } , \\end{align*}"}
+{"id": "4050.png", "formula": "\\begin{align*} \\frac { \\partial X ^ r } { \\partial q _ i } & = - g _ z E ^ { i r } , \\end{align*}"}
+{"id": "2018.png", "formula": "\\begin{align*} \\| S ( x ) - S ( y ) \\| = \\lim \\limits _ { k \\to \\infty } \\| S _ { t _ k } ( x ) - S _ { t _ k } ( y ) \\| \\leq \\limsup _ { k \\to \\infty } L _ { t _ k } \\| x - y \\| \\leq L \\| x - y \\| . \\end{align*}"}
+{"id": "5052.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ l C _ j ( \\vec x ) \\vec \\gamma _ j , \\end{align*}"}
+{"id": "1946.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\int \\| z \\| \\ , d \\mu _ N ( z ) = \\int _ { \\| z \\| \\leq r } \\| z \\| \\ , d P ( z ) . \\end{align*}"}
+{"id": "3184.png", "formula": "\\begin{align*} c _ j ^ { 3 3 } ( A ) = \\int _ { [ 0 , 1 ] ^ 3 } r a _ j \\partial _ j v ^ { 3 3 } = - \\int _ { [ 0 , 1 ] ^ 3 } r a _ j \\partial _ j v ^ { 1 1 } - \\int _ { [ 0 , 1 ] ^ 3 } r a _ j \\partial _ j v ^ { 2 2 } = - c _ j ^ { 1 1 } ( A ) - c _ j ^ { 2 2 } ( A ) \\end{align*}"}
+{"id": "5700.png", "formula": "\\begin{align*} \\mathbf { A } _ { b } ^ { a } = ( \\mathbf { L } ^ { - 1 } ) _ { c } ^ { a } d \\mathbf { L } _ { b } ^ { c } \\mathbf { , } \\end{align*}"}
+{"id": "4886.png", "formula": "\\begin{align*} \\Pi ( t _ w t _ u ) = \\pi ( w ) \\cdot \\pi ( u ) , w , u \\in D . \\end{align*}"}
+{"id": "6914.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dot { S } ( t ) = - \\dfrac { \\beta S ( t ) I ( t ) } { N } - \\omega S ( t ) , \\\\ \\dot { E } ( t ) = \\dfrac { \\beta S ( t ) I ( t ) } { N } - \\gamma E ( t ) , \\\\ \\dot { I } ( t ) = \\gamma E ( t ) - \\delta I ( t ) , \\\\ \\dot { Q } ( t ) = \\delta I ( t ) - \\lambda ( t ) Q ( t ) - \\kappa ( t ) Q ( t ) , \\\\ \\dot { R } ( t ) = \\lambda ( t ) Q ( t ) , \\\\ \\dot { D } ( t ) = \\kappa ( t ) Q ( t ) , \\\\ \\dot { P } ( t ) = \\omega S ( t ) , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "639.png", "formula": "\\begin{align*} J _ G = ( x _ i y _ j - x _ j y _ i : \\{ i , j \\} \\in E ( G ) ) \\end{align*}"}
+{"id": "4084.png", "formula": "\\begin{align*} ( f , B ) \\cdot ( g , b ) \\longmapsto ( f ^ * g , f ^ * b - B ) \\end{align*}"}
+{"id": "3739.png", "formula": "\\begin{align*} \\tilde { c } _ 1 ( I ( f ) ) = \\frac { Z _ { r _ 1 } ( f , 1 ) } { \\zeta ( 1 ) } = \\frac { I ( f ) ( 0 ) } { \\zeta ( 1 ) } . \\end{align*}"}
+{"id": "894.png", "formula": "\\begin{align*} \\mathbf { E } \\left [ \\left \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\right \\| _ { F } ^ { 2 } | \\mathcal { X } ^ { 0 } \\right ] \\leq \\left ( 1 - \\mathop { \\min } _ { k = 1 , 2 , \\cdots , l } \\frac { \\lambda _ { \\min } \\left ( \\widehat { \\mathcal { A } } _ { ( k ) } \\widehat { \\mathcal { A } } ^ { H } _ { ( k ) } \\right ) } { \\| \\mathcal { A } \\| _ { F } ^ { 2 } } \\right ) ^ { t } \\left \\| \\mathcal { X } ^ { 0 } - \\mathcal { X } ^ { \\star } \\right \\| _ { F } ^ { 2 } \\end{align*}"}
+{"id": "2469.png", "formula": "\\begin{align*} T ( x ) = \\left \\{ T _ x \\in X ^ * : \\| T _ x \\| = 1 T _ x ( x ) = \\| x \\| \\right \\} . \\end{align*}"}
+{"id": "1428.png", "formula": "\\begin{align*} \\mathbb { E } [ R ^ 2 _ { t - 1 } | R _ { t - 1 } ] & = R ^ 2 _ { t - 1 } + 2 R _ { t - 1 } ( \\mathbb { E } [ \\eta _ t | R _ { t - 1 } ] - 1 ) + \\mathbb { E } [ \\eta ^ 2 _ t | R _ { t - 1 } ] - 2 \\mathbb { E } [ \\eta _ t | R _ { t - 1 } ] + 1 \\\\ & \\geq R ^ 2 _ { t - 1 } + \\sigma ^ 2 - 1 - 2 h \\varepsilon _ 1 - \\varepsilon _ 2 \\\\ & \\ge R ^ 2 _ { t - 1 } + 2 \\frac { \\sigma ^ 2 - 1 } { 3 } - \\varepsilon _ 2 \\\\ & \\geq R ^ 2 _ { t - 1 } + \\frac { \\sigma ^ 2 - 1 } { 3 } . \\end{align*}"}
+{"id": "4882.png", "formula": "\\begin{align*} C _ { x _ 1 } C _ { x _ l } = C _ { \\tau s _ 3 } C _ { s _ 0 } C _ { s _ 2 } C _ { s _ 1 } C _ { s _ 3 } C _ { x _ l } + \\Box = \\xi ^ 2 C _ { \\tau s _ 3 } C _ { s _ 0 } C _ { s _ 2 } C _ { x _ l } + \\Box \\in \\xi ^ 2 C _ { \\tau s _ 3 } C _ { s _ 0 } C _ { s _ 2 } C _ { x _ l } + H ^ { < 1 3 } . \\end{align*}"}
+{"id": "6508.png", "formula": "\\begin{align*} \\eta _ { p , l , k } : = \\frac { l - 1 } { 2 \\sqrt { k } } \\underset { i = 1 } { \\overset { k } { \\sum } } \\left ( p _ i - \\frac { 1 } { 2 } \\right ) ^ 2 \\geq c _ { \\alpha , n } , \\end{align*}"}
+{"id": "4040.png", "formula": "\\begin{align*} Y u ^ { - 1 } ( K ) \\supset \\bigcap _ { i = 1 } ^ \\infty \\bigcup _ { k = i } ^ \\infty Y u ^ { - 1 } _ k ( K ) , \\end{align*}"}
+{"id": "1740.png", "formula": "\\begin{align*} \\delta s ( \\underline { \\mu } ) = - 2 \\binom { 2 M } { k _ { \\rm i d } - 2 } \\varphi _ { M + 1 } ( t _ M ( \\underline { \\mu } ) ^ * ) , \\end{align*}"}
+{"id": "2996.png", "formula": "\\begin{align*} \\hbox { d } ^ \\gamma e ^ a : = \\hbox { d } e ^ a + \\gamma { ^ a } _ b \\wedge e ^ b = 0 . \\end{align*}"}
+{"id": "3555.png", "formula": "\\begin{align*} d _ { i } ^ - ( \\lambda ) = \\frac { ( \\lambda _ i + 1 ) i ( \\lambda _ i + n - i + [ \\lambda _ i ] _ 2 ( p - n ) ) } { ( - 1 ) ^ { \\sum _ { \\alpha = i } ^ n ( \\alpha + 1 ) ( \\lambda _ \\alpha - \\lambda _ { \\alpha + 1 } ) } } \\bigg ( \\prod _ { \\ell = i + 1 } ^ n \\frac { \\lambda _ i - \\lambda _ \\ell - i + \\ell - 1 } { \\lambda _ i - \\lambda _ \\ell - i + \\ell - [ \\lambda _ i - \\lambda _ \\ell ] _ 2 } \\bigg ) . \\end{align*}"}
+{"id": "5809.png", "formula": "\\begin{align*} ( \\ ! ( D e _ s , d ) \\ ! ) = ( \\ ! ( e _ s , D d ) \\ ! ) = 0 , d \\in K e r ( D ) , \\end{align*}"}
+{"id": "8704.png", "formula": "\\begin{align*} \\Omega ( \\tau \\otimes \\tau ) = \\tau \\otimes \\tau \\end{align*}"}
+{"id": "8920.png", "formula": "\\begin{align*} c _ { d } ^ { \\left ( \\nu , n \\right ) } = \\left \\{ \\begin{array} { c } \\frac { ( n - d - 1 ) ! } { ( n - 1 ) ! } \\gamma _ { n - d - 1 } ^ { ( \\nu , n ) } , 0 \\leq d \\leq n - 1 \\\\ \\sum \\limits _ { p = 0 } ^ { n - 1 } \\frac { c _ { n - p - 1 } ^ { \\left ( \\nu , n \\right ) } } { p ! } \\Omega _ { p } ^ { \\left ( \\nu \\right ) } \\left ( d - n \\right ) , d \\geq n . \\end{array} \\right . \\end{align*}"}
+{"id": "7246.png", "formula": "\\begin{align*} N ( \\gamma ) : = \\int _ { 0 } ^ { \\gamma } j ( d x ) \\int _ { 0 } ^ { x } d y \\int _ { y } ^ { \\gamma } d m ( z ) \\int _ { 0 } ^ { z } m ( w , \\infty ) d w . \\end{align*}"}
+{"id": "8880.png", "formula": "\\begin{align*} \\sum \\limits _ { m = 0 } ^ { + \\infty } U ( - 4 m ( m + n ) ) K _ { 0 , m } ( z , w ) . \\end{align*}"}
+{"id": "8942.png", "formula": "\\begin{align*} b _ { j } ^ { \\left ( \\nu , 3 \\right ) } = \\frac { ( 4 \\pi ) ^ 3 } { 3 ! } \\sum \\limits _ { i = 0 } ^ { j } \\frac { \\left ( \\frac { 9 } { 4 } + \\nu ^ { 2 } \\right ) ^ { j - i } } { \\left ( j - i \\right ) ! } c _ { i } ^ { \\left ( \\nu , 3 \\right ) } , \\end{align*}"}
+{"id": "459.png", "formula": "\\begin{align*} | \\phi ( \\tau _ n ) | = | \\psi ( \\tau _ n ) | | f _ n ( x ) | = | f _ n ( y ) | , \\forall n \\in \\mathbb { N } , \\end{align*}"}
+{"id": "4595.png", "formula": "\\begin{align*} u _ n ( \\theta ) = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ 1 , \\ldots , x _ n ) \\in \\N ^ n , x _ 1 \\geq \\cdots \\geq x _ n , \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < \\theta \\right \\} \\end{align*}"}
+{"id": "5159.png", "formula": "\\begin{align*} \\mathcal { P } ( K _ { \\rho , \\rho + 1 } ) = \\dfrac { \\binom { 2 \\rho + 1 } { \\rho } } { 2 ^ { 2 \\rho } } . \\end{align*}"}
+{"id": "1802.png", "formula": "\\begin{align*} \\frac { d } { d t } L _ { \\dot { u } } \\big ( t , u _ 0 ( t ) , \\dot { u } _ 0 ( t ) \\big ) = L _ u \\big ( t , u _ 0 ( t ) , \\dot { u } _ 0 ( t ) \\big ) . \\end{align*}"}
+{"id": "1931.png", "formula": "\\begin{align*} 2 ( \\mu + \\langle J , \\nu \\rangle ) & = R _ g + \\tfrac { n } { n - 1 } h ^ 2 + 2 \\langle \\nabla h , \\nu \\rangle , \\\\ 2 ( \\mu - | J | ) & = R _ g + \\tfrac { n } { n - 1 } h ^ 2 - 2 | \\nabla h | , \\end{align*}"}
+{"id": "3326.png", "formula": "\\begin{align*} X ^ { \\flat } _ H \\ ! = \\ ! [ - B _ i \\ ! + \\ ! a _ i ( A _ j a _ j \\ ! + \\ ! B _ j b _ j \\ ! + \\ ! C ) ] \\ ! \\dd \\ ! q ^ i \\ ! + \\ ! [ A _ i \\ ! + \\ ! b _ i ( A _ j a _ j \\ ! + \\ ! B _ j b _ j \\ ! + \\ ! C ) ] \\ ! \\dd \\ ! p _ i \\ ! + \\ ! ( A _ j a _ j \\ ! + \\ ! B _ j b _ j + \\ ! C ) c \\dd \\ ! \\kappa . \\end{align*}"}
+{"id": "5656.png", "formula": "\\begin{align*} \\frac { d } { d s } \\frac { \\Lambda ^ n } { ( 1 + \\Lambda ) ^ n } = - \\frac { 1 } { s } \\frac { n { \\Lambda } ^ n } { ( 1 + { \\Lambda } ) ^ { n + 1 } } \\rightarrow - \\frac { n | \\Omega | } { 4 \\pi \\epsilon D \\bar { \\ell } } \\mbox { a s } \\ { s \\rightarrow 0 } . \\end{align*}"}
+{"id": "5279.png", "formula": "\\begin{align*} \\hat { \\tilde { s e c } } ( U , V ) = \\hat { s } e c ( U , V ) . \\end{align*}"}
+{"id": "8016.png", "formula": "\\begin{align*} S = \\big \\{ & \\big ( \\{ F _ 3 , F _ 1 , F _ 2 \\} , \\{ ( F _ 3 , F _ 1 ) , ( F _ 3 , F _ 2 ) \\} \\big ) , \\\\ & \\big ( \\{ F _ 3 , F _ 2 , F _ 1 \\} , \\{ ( F _ 3 , F _ 2 ) , ( F _ 2 , F _ 1 ) \\} \\big ) \\big \\} . \\end{align*}"}
+{"id": "3126.png", "formula": "\\begin{align*} - \\Delta \\hat { w } = r a - \\bar { a } \\quad Y , \\hat { w } Y , \\int _ Y \\hat { w } = 0 , \\end{align*}"}
+{"id": "6988.png", "formula": "\\begin{gather*} \\frac { 1 } { n } \\ , \\det \\begin{pmatrix} 1 & U _ { 1 , n } & ( U _ { 1 , n } ) ^ 2 & \\cdots & ( U _ { 1 , n } ) ^ { n - 2 } \\\\ 1 & U _ { 2 , n } & ( U _ { 2 , n } ) ^ 2 & \\cdots & ( U _ { 2 , n } ) ^ { n - 2 } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 & U _ { n - 1 , n } & ( U _ { n - 1 , n } ) ^ 2 & \\cdots & ( U _ { n - 1 , n } ) ^ { n - 2 } \\end{pmatrix} = \\frac { 1 } { n } \\prod _ { 1 \\leq i < j \\leq n - 1 } ( U _ { j , n } - U _ { i , n } ) . \\end{gather*}"}
+{"id": "5568.png", "formula": "\\begin{align*} \\Sigma : = \\{ x \\in \\partial \\{ u > 0 \\} : \\Theta ^ n _ { \\{ u > 0 \\} } ( x ) = 0 \\} \\end{align*}"}
+{"id": "8463.png", "formula": "\\begin{align*} & \\max _ { \\nu \\in \\mathcal E ' _ A } \\ , G ( \\nu ) = G ( \\gamma _ A ) \\quad \\bigl ( { } = c _ * ( A ) \\bigr ) , \\\\ & \\max _ { \\nu \\in \\widehat { \\mathcal E } ' _ A } \\ , \\nu ( X ) = \\gamma _ A ( X ) \\quad \\bigl ( { } = c _ * ( A ) \\bigr ) . \\end{align*}"}
+{"id": "5249.png", "formula": "\\begin{align*} \\nabla ^ F _ X F _ \\ast X = 0 . \\end{align*}"}
+{"id": "4732.png", "formula": "\\begin{align*} \\alpha ^ s & = a _ 0 ^ { s - 4 } ( a _ 0 a _ 1 ^ { s n _ 1 } a _ 2 ^ { s n _ 2 } \\cdots a _ m ^ { s n _ m } ) a _ 0 ^ 3 a _ { m + 1 } ^ { s n _ { m + 1 } } \\in A \\intertext { f o r a l l $ s \\ge 4 $ b y t h e i n d u c t i v e h y p o t h e s i s , a n d } \\beta ^ s & = p ^ { s - 3 } ( p a _ 1 ^ { s n _ 1 } a _ 2 ^ { s n _ 2 } \\cdots a _ m ^ { s n _ m } ) p ^ 2 a _ { m + 1 } ^ { s n _ { m + 1 } } \\in A \\end{align*}"}
+{"id": "7704.png", "formula": "\\begin{align*} { \\mathcal { L } } ^ { - \\alpha } v = \\frac { \\sin ( \\alpha \\pi ) } { \\pi } \\int \\nolimits _ { 0 } ^ { \\infty } \\varsigma ^ { - \\alpha } ( \\varsigma { \\mathcal { I } } + \\mathcal { L } ) ^ { - 1 } v d \\varsigma , v \\in { \\mathcal { H } } , \\end{align*}"}
+{"id": "2324.png", "formula": "\\begin{align*} \\dot { W } _ { 0 , } ^ { 1 , N } \\hookrightarrow L ^ { \\infty , N } ( \\log L ) ^ { - 1 } \\hookrightarrow L ^ { \\infty , q } ( \\log L ) ^ { - 1 + \\frac { 1 } { N } - \\frac { 1 } { q } } \\hookrightarrow L ^ { \\infty , \\infty } ( \\log L ) ^ { - 1 + \\frac { 1 } { N } } = { \\rm E x p L } ^ { \\frac { N } { N - 1 } } . \\end{align*}"}
+{"id": "6162.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ d \\alpha _ r ( E _ r , \\widehat U _ 0 ) \\equiv \\sum _ { r = 1 } ^ d \\alpha _ r ( E _ r , \\widehat U _ 1 ) \\equiv 0 \\end{align*}"}
+{"id": "7554.png", "formula": "\\begin{align*} f _ 1 ( u , v ) = & \\pi ^ { k + r } u ^ p + v ^ l \\sum _ { i = 0 } ^ k ( - 1 ) ^ i \\dbinom { k + r } { k - i } \\dbinom { i + r - 1 } { i } ( \\pi t ) ^ { k - i } v ^ { k - i } \\\\ = & \\pi ^ { k + r } u ^ p + ( - 1 ) ^ k \\dbinom { k + r - 1 } { k } v ^ l + \\sum _ { i = 0 } ^ { k - 1 } ( - 1 ) ^ i \\dbinom { k + r } { k - i } \\dbinom { i + r - 1 } { i } ( \\pi t ) ^ { k - i } v ^ { k + l - i } . \\end{align*}"}
+{"id": "2483.png", "formula": "\\begin{align*} Z X _ T x = X x = \\alpha _ n X _ n x x \\in \\mathcal H _ n \\ n \\geq 0 . \\end{align*}"}
+{"id": "4232.png", "formula": "\\begin{align*} \\phi _ 0 ( x ) & = - \\dfrac { 1 } { 2 \\varepsilon } \\arctan \\left ( \\cosh ( \\beta ( \\lambda + \\varepsilon \\theta ( x ) ) ) \\cosh ( \\lambda + \\varepsilon \\theta ( x ) ) ( \\tanh ( \\beta ( \\lambda + \\varepsilon \\theta ( x ) ) ) + v \\sqrt { c } \\tanh ( \\lambda + \\varepsilon \\theta ( x ) ) ) \\right ) \\\\ & + \\dfrac { 1 } { 2 \\varepsilon } \\arctan \\left ( \\cosh ( \\beta \\lambda ) \\cosh ( \\lambda ) ( \\tanh ( \\beta \\lambda ) + v \\sqrt { c } \\tanh ( \\lambda ) ) \\right ) , \\end{align*}"}
+{"id": "6854.png", "formula": "\\begin{align*} { \\mathcal M _ d } = ( I - \\mathcal X _ { N + 1 } ^ { [ 0 ] \\ ; \\dag } \\mathcal X _ { N + 1 } ^ { [ 0 ] } ) \\mathcal K _ d . \\end{align*}"}
+{"id": "416.png", "formula": "\\begin{align*} g _ n ( \\omega _ m ) = f _ n ( V ^ { - 1 } V \\tau _ m ) = f _ n ( \\tau _ m ) = \\delta _ { n , m } , \\forall n , m \\in \\mathbb { N } , \\end{align*}"}
+{"id": "4726.png", "formula": "\\begin{align*} b _ { \\ell 1 } = \\sum \\limits _ { j \\in \\mathcal { J } } ^ { } u _ j b _ { j 1 } + p c _ { \\ell 1 } { b } _ { \\ell 2 } = \\sum \\limits _ { j \\in \\mathcal { J } } ^ { } v _ j b _ { j 1 } + p c _ { \\ell 2 } u _ j , v _ j \\in \\mathbb { Z } _ p c _ { \\ell 1 } , c _ { \\ell 2 } \\in \\mathbb { Z } _ { p ^ { a - 1 } } ^ { 2 n } \\end{align*}"}
+{"id": "864.png", "formula": "\\begin{align*} \\deg _ { F _ 1 \\cup F _ 2 } ( u ) & = { \\deg _ { { F _ 1 } } } ( u ) + { \\deg _ { { F _ 2 } } } ( u ) \\geq \\delta ( F ) - d _ { F _ 3 } ( u ) \\geq v ( F ) - \\sqrt \\delta t / 2 - \\left ( \\frac { 1 } { 2 } + \\sqrt { \\delta } \\right ) t \\\\ & \\overset { ( \\ref { F - d e g r e e } ) } { > } \\left ( { 1 - \\frac { { \\frac { 1 } { 2 } + \\frac { 3 } { 2 } \\sqrt \\delta } } { { 1 - \\frac { 3 } { 2 } \\sqrt \\delta } } } \\right ) v ( F ) > \\left ( \\frac { 1 } { 2 } - 3 \\sqrt { \\delta } \\right ) v ( F ) . \\end{align*}"}
+{"id": "4221.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ { t } p ( t , x ) = & ~ { } \\partial _ { x } e ( t , x ) , \\\\ \\partial _ { t } e ( t , x ) = & ~ { } \\partial _ { x } p ( t , x ) , \\end{aligned} \\end{align*}"}
+{"id": "7514.png", "formula": "\\begin{align*} Z _ g ( s , \\chi , D ) = & q ^ { - e _ 2 s } Z _ { g _ 1 } ( s , \\chi , D ) \\\\ = & q ^ { - e _ 2 s } \\Big ( v ( \\bar { g } _ 1 , D , \\chi ) + \\sigma ( \\bar { g } _ 1 , D , \\chi ) \\dfrac { ( 1 - q ^ { - 1 } ) q ^ { - s } } { 1 - q ^ { - 1 - s } } + Z _ { g _ 1 } ( s , D _ { S ( g _ 1 , D ) } ) \\Big ) \\\\ = & \\dfrac { M _ { 2 , 3 } ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } + q ^ { - e _ 2 s } Z _ { g _ 1 } ( s , \\chi , D _ { S ( g _ 1 , D ) } ) , \\end{align*}"}
+{"id": "8786.png", "formula": "\\begin{align*} \\mathcal { G } [ \\varepsilon , W ( \\tau ) + h ( \\tau ) ] = & [ W ' ( \\tau ) + h ' ( \\tau ) - K ( \\varepsilon , \\tau , W + h ) , W ( 0 ) + h ( 0 ) , W ( t _ 1 ) + h ( t _ 1 ) ] \\\\ = & \\mathcal { G } [ \\varepsilon , W ( \\tau ) + h ( \\tau ) ] + [ A _ { \\varepsilon } h , h ( 0 ) , h ( t _ 1 ) ] + o ( h ) , \\end{align*}"}
+{"id": "8961.png", "formula": "\\begin{align*} d \\pi _ N ( u ) \\big ( u _ t + \\partial _ r u \\big ) = u _ t + d \\pi _ N ( u ) \\partial _ r u = 0 \\ \\hbox { o n } S ^ 1 \\times [ 0 , \\infty [ , \\end{align*}"}
+{"id": "1799.png", "formula": "\\begin{align*} \\begin{cases} u ( 0 ) = u _ 0 , \\\\ u ( T ) = u _ T , \\end{cases} \\end{align*}"}
+{"id": "3699.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ \\ell \\left ( c _ i ( d _ { \\ell , i } ( f ) ) + \\sum _ { \\xi \\in X ^ { \\circ } _ i ( F ) } I ( d _ { \\ell , i } ( f ) ) ( \\xi ) \\right ) + \\kappa d _ { \\ell , 0 } ( f ) ( 0 _ { V _ 0 } , 0 , 0 ) \\\\ & = \\sum _ { i = 1 } ^ \\ell \\left ( c _ i ( d _ { \\ell , i } ( \\mathcal { F } _ { X _ { \\ell } } ( f ) ) ) + \\sum _ { \\xi \\in X ^ { \\circ } _ i ( F ) } I ( d _ { \\ell , i } ( \\mathcal { F } _ { X _ { \\ell } } ( f ) ) ) ( \\xi ) \\right ) + \\kappa d _ { \\ell , 0 } ( \\mathcal { F } _ { X _ { \\ell } } ( f ) ) ( 0 _ { V _ 0 } , 0 , 0 ) . \\end{align*}"}
+{"id": "9000.png", "formula": "\\begin{align*} v ^ 4 ( z ) & = | ( v ^ 2 \\varphi ) ( z ) | ^ 2 \\le \\int _ 0 ^ { 2 r } | \\partial _ x ( v ^ 2 \\varphi ) ( x + s , y ) | d s \\cdot \\int _ 0 ^ { 2 r } | \\partial _ y ( v ^ 2 \\varphi ) ( x , y + t ) | d t \\\\ & \\le \\int _ { \\{ s ; ( s , y ) \\in B \\} } | \\partial _ x ( v ^ 2 \\varphi ) ( s , y ) | d s \\cdot \\int _ { \\{ t ; ( x , t ) \\in B \\} } | \\partial _ y ( v ^ 2 \\varphi ) ( x , t ) | d t , \\end{align*}"}
+{"id": "4307.png", "formula": "\\begin{align*} K : = \\frac { 2 L \\max ( M , 1 ) } { \\min ( \\nu _ 0 , 1 ) } e ^ { 2 \\nu _ 0 ^ { - 1 } M } T _ { m } ^ { \\frac { s + 1 } { s } } \\left \\| \\left ( ( - \\Delta ) ^ { \\frac { 3 } { 4 } } u _ { 0 } , ( - \\Delta ) ^ { \\frac { 1 } { 4 } } u _ { 1 } \\right ) \\right \\| _ { \\gamma _ { \\eta , L ^ { 2 } } ^ { s } \\times \\gamma _ { \\eta , L ^ { 2 } } ^ { s } } ^ { 2 } . \\end{align*}"}
+{"id": "1602.png", "formula": "\\begin{align*} d _ p ^ p ( \\delta _ { x _ { n + 1 } } , \\Phi ( \\mu ) ) = d _ p ^ p ( \\delta _ { x _ { n + 1 } } , \\mu ^ * ) - \\mu ( x _ { n + 1 } ) + \\varrho ^ p ( x _ { n + 1 } , w ) \\mu ( x _ { n + 1 } ) . \\end{align*}"}
+{"id": "8845.png", "formula": "\\begin{align*} L _ { \\nu } = - ( 1 + \\left \\vert z \\right \\vert ^ { 2 } ) \\left ( ( 1 + \\left \\vert z \\right \\vert ^ { 2 } ) \\frac { \\partial ^ { 2 } } { \\partial z \\partial \\bar { z } } + \\nu \\left ( z _ { j } \\frac { \\partial } { \\partial z _ { j } } - \\bar { z _ { j } } \\frac { \\partial } { \\partial \\bar { z _ { j } } } \\right ) - \\nu ^ { 2 } \\right ) - \\nu ^ { 2 } \\equiv - \\frac { 1 } { 4 } \\Delta _ { \\nu } . \\end{align*}"}
+{"id": "577.png", "formula": "\\begin{align*} \\alpha _ t = \\min \\left \\{ \\bar \\alpha , \\frac { 1 } { ( A ^ { \\rm T } A ) : ( A ^ { \\rm T } + A ) ^ { - 1 } \\ , t } \\right \\} , \\end{align*}"}
+{"id": "7806.png", "formula": "\\begin{align*} \\langle ( n , m ) , ( m ' , n ' ) \\rangle _ 1 : = \\langle n , m ' \\rangle + \\langle n ' , m \\rangle . \\end{align*}"}
+{"id": "4502.png", "formula": "\\begin{align*} R _ n \\left ( \\frac { a } { b } \\right ) = R _ { n - 1 } \\left ( \\frac { a } { b } \\right ) + \\frac { 1 } { m } \\end{align*}"}
+{"id": "9209.png", "formula": "\\begin{align*} \\lambda ( m _ { 2 } , h ) = \\frac { 2 s _ { 2 } } { \\sqrt { \\pi } } F ^ { \\prime \\prime } ( 0 ) \\sqrt { \\vert F ^ { \\prime \\prime } ( s _ { 2 } ) \\vert } \\sqrt { h } e ^ { - \\frac { 2 S ( m _ { 2 } ) } { h } } \\alpha ( h ) , \\end{align*}"}
+{"id": "7573.png", "formula": "\\begin{align*} Z _ { f _ { 7 , 0 } } ( s , \\chi , B _ 7 ^ a ) = \\dfrac { F _ { 7 , 0 } ( q ^ { - s } ) } { ( 1 - q ^ { - 1 - s } ) ( 1 - q ^ { - p - ( k + 1 ) - p ( k + 1 ) s } ) } , \\end{align*}"}
+{"id": "8733.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { \\beta } { u } \\right ) E ^ { \\perp } ( - u ) E ( { \\beta } ) = E ( \\beta ) E ^ { \\perp } ( - u ) , \\end{align*}"}
+{"id": "3897.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d \\theta ^ 2 } = ( \\dot { x _ \\theta } ) _ i ( \\dot { x _ \\theta } ) _ j & D _ { x _ i x _ j } - g _ z ^ 2 q _ m q _ n E ^ { n , j } E ^ { m , a } D _ { x _ j } ( E _ { a b } ) E ^ { b , i } D _ { x _ i } \\\\ & + g _ z g _ { j , z } E ^ { n , j } E ^ { m , i } q _ m q _ n D _ { x _ i } . \\end{align*}"}
+{"id": "6653.png", "formula": "\\begin{align*} g = ( a _ y - a ( t ' , x ' ) ) \\nabla \\delta _ y u \\end{align*}"}
+{"id": "699.png", "formula": "\\begin{align*} + \\sum _ { j = 1 } ^ { \\texttt g } \\big ( \\oint _ { \\alpha _ j } \\eta \\oint _ { \\beta _ j } * \\eta - \\oint _ { \\alpha _ j } * \\eta \\oint _ { \\beta _ j } \\eta \\big ) . \\end{align*}"}
+{"id": "5206.png", "formula": "\\begin{align*} A _ 4 & = \\ell _ 1 A _ 2 + \\ell _ 2 A _ 3 \\\\ A _ 5 & = \\ell _ 3 A _ 3 \\quad , A _ 6 = ( 1 / \\ell _ 3 ) A _ 2 \\ , . \\end{align*}"}
+{"id": "4919.png", "formula": "\\begin{align*} \\left ( \\frac { \\delta \\widetilde H } { \\delta ( \\mathbf { a } , \\mathbf { b } ) } \\right ) _ m = \\left ( \\frac { \\partial \\widetilde H } { \\partial ( \\mathbf { a } , \\mathbf { b } ) } \\right ) _ m \\ ! \\ ! \\ ! \\ ! - \\delta _ m ^ - \\left ( \\frac { \\partial \\widetilde H } { \\partial \\delta ^ + ( \\mathbf { a } , \\mathbf { b } ) } \\right ) _ m \\ ! \\ ! \\ ! \\ ! - \\delta _ m ^ + \\left ( \\frac { \\partial \\widetilde H } { \\partial \\delta ^ - ( \\mathbf { a } , \\mathbf { b } ) } \\right ) _ m , \\end{align*}"}
+{"id": "335.png", "formula": "\\begin{align*} | H ( s ; \\phi , k , l , \\alpha ) | \\ll | k | ^ \\varepsilon \\alpha ^ \\varepsilon l ^ { \\frac { 1 } { 2 } + \\varepsilon } ( l , k _ 2 ^ 2 ) ^ { \\frac { 1 } { 2 } } \\sum _ { \\substack { h _ 0 \\mid \\frac { k \\cdot \\textrm { r a d } k } { ( l , k ) } \\\\ p \\mid h _ 0 \\Rightarrow p \\mid ( l , k _ 2 ) } } \\sum _ { \\substack { h \\mid k \\\\ ( h , l ) = 1 } } | A ( h _ 0 h , 1 ) | , \\end{align*}"}
+{"id": "7475.png", "formula": "\\begin{align*} F _ A ( s ) = \\tilde { F } _ A ( s ) = 1 \\ , . \\end{align*}"}
+{"id": "405.png", "formula": "\\begin{align*} \\| x \\| ^ p = \\sum _ { n = 1 } ^ \\infty | f _ n ( x ) | ^ p . \\end{align*}"}
+{"id": "4914.png", "formula": "\\begin{align*} x _ m = & \\ , a + ( m - 1 ) \\Delta x , m = 1 , \\ldots , M , \\Delta x = \\frac { b - a } { M - 1 } , \\\\ t _ n = & \\ , n \\Delta t , n = 0 , \\ldots , N , \\Delta t = \\frac { T } { N } , \\end{align*}"}
+{"id": "5518.png", "formula": "\\begin{align*} { } _ p F _ q \\left ( c _ 1 , \\cdots , c _ p ; \\ , d _ 1 , \\cdots , d _ q ; \\ , z \\right ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( c _ 1 ) _ n \\cdots ( c _ p ) _ n } { ( d _ 1 ) _ n \\cdots ( d _ q ) _ n } \\frac { z ^ n } { n ! } , \\end{align*}"}
+{"id": "329.png", "formula": "\\begin{align*} \\sum _ { h = 0 } ^ \\infty \\frac { A ( p ^ { 2 h + 1 } , 1 ) } { ( p + 1 ) p ^ { s h - 1 } } = \\frac { p } { p + 1 } \\frac { 1 + p ^ s A ( p , 1 ) } { p ^ s + A ( 1 , p ) } \\sum _ { h = 0 } ^ \\infty \\frac { A ( p ^ { 2 h } , 1 ) } { p ^ { s h } } , \\end{align*}"}
+{"id": "2486.png", "formula": "\\begin{align*} \\cap _ { n = 0 } ^ \\infty \\vee _ { k = n } ^ \\infty \\mathcal H _ k = \\{ 0 \\} . \\end{align*}"}
+{"id": "8844.png", "formula": "\\begin{align*} \\Delta _ { \\nu } = 4 ( 1 + \\langle z , z \\rangle ) \\left ( \\sum _ { i , j = 1 } ^ { n } ( \\delta _ { i j } + z _ { i } \\bar { z _ { j } } ) \\frac { \\partial ^ { 2 } } { \\partial z _ { i } \\partial \\bar { z _ { j } } } - \\nu \\sum _ { j = 1 } ^ { n } \\left ( z _ { j } \\frac { \\partial } { \\partial z _ { j } } - \\bar { z _ { j } } \\frac { \\partial } { \\partial \\bar { z _ { j } } } \\right ) - \\nu ^ { 2 } \\right ) + 4 \\nu ^ { 2 } \\end{align*}"}
+{"id": "1617.png", "formula": "\\begin{align*} \\hat S ^ { ( t ) } _ i ( \\theta _ { c ( i ) } ) = 1 [ \\hat U _ i ^ { ( t ) } > \\theta _ { c ( i ) } ] \\end{align*}"}
+{"id": "2616.png", "formula": "\\begin{align*} 9 & = a _ { 2 , 3 } + a _ { 2 , 4 } + a _ { 2 , 5 } + a _ { 3 , 4 } + a _ { 3 , 5 } + a _ { 4 , 5 } = 1 + 1 + 1 + 2 + 2 + 2 \\\\ & \\leftrightarrow ( a _ { 1 , 2 } , a _ { 1 , 3 } , a _ { 1 , 4 } , a _ { 1 , 5 } , a _ { 2 , 3 } , a _ { 2 , 4 } , a _ { 2 , 5 } , a _ { 3 , 4 } , a _ { 3 , 5 } , a _ { 4 , 5 } ) = ( 3 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 ) . \\end{align*}"}
+{"id": "2277.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta \\Phi - \\varepsilon \\Delta ^ { 2 m + 1 } \\Phi = \\int _ { \\mathbb { R } ^ 3 } f \\ , d v - c ( x ) , \\\\ & \\Phi | _ { ( 0 , T ) \\times \\partial \\Omega } = \\Delta \\Phi | _ { ( 0 , T ) \\times \\partial \\Omega } = \\cdot \\cdot \\cdot = \\Delta ^ { 2 m } \\Phi | _ { ( 0 , T ) \\times \\partial \\Omega } = 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "5865.png", "formula": "\\begin{align*} w ( T ) = w ' ( T ) = 0 . \\end{align*}"}
+{"id": "7855.png", "formula": "\\begin{align*} \\log | f ( r e ^ { i \\theta } ) | = ( h _ f ( \\theta ) + o ( 1 ) ) r ^ { \\rho } , r e ^ { i \\theta } \\not \\in \\bigcup _ k D ( z _ k , r _ k ) , \\end{align*}"}
+{"id": "7484.png", "formula": "\\begin{align*} \\kappa ( X ) = \\kappa ( F ) + \\dim Y . \\end{align*}"}
+{"id": "5351.png", "formula": "\\begin{align*} d \\rvert _ { \\varepsilon = \\tilde { \\varepsilon } } \\Lambda _ { F , s } [ \\eta ] = - \\sum _ { k = 1 } ^ n c _ k \\sum _ { l \\in F _ k } \\int _ \\Omega \\eta \\tilde { E } ^ { ( l ) } \\cdot \\tilde { E } ^ { ( l ) } \\ , d x , \\end{align*}"}
+{"id": "2562.png", "formula": "\\begin{align*} \\zeta ( \\sigma _ r ^ { b , 2 } ( v ) ; \\alpha ) = \\sum _ { \\begin{subarray} { c } r _ 1 + \\cdots + r _ { q } = r \\\\ r _ i \\ge 0 \\end{subarray} } H ^ { * } _ { ( \\{ r _ i \\} _ { i = 1 } ^ { q } ) } ( \\tau ( v ) ; \\alpha ) \\end{align*}"}
+{"id": "8636.png", "formula": "\\begin{align*} W _ n = \\bigcap _ { m \\geq n } ( \\Omega ^ { \\mathbb N } \\setminus T _ m ) , \\end{align*}"}
+{"id": "637.png", "formula": "\\begin{align*} \\beta _ i ^ { \\mathbb { Z } _ p } ( ( I + ( w ) ) ^ q ) - \\beta _ i ^ \\mathbb { Q } ( ( I + ( w ) ) ^ q ) & = \\sum _ { \\ell = 1 } ^ q \\left [ \\beta _ i ^ { \\mathbb { Z } _ p } ( I ^ { \\ell } ) + \\beta _ { i - 1 } ^ { \\mathbb { Z } _ p } ( I ^ { \\ell } ) - \\beta _ i ^ { \\mathbb { Q } } ( I ^ { \\ell } ) - \\beta _ { i - 1 } ^ { \\mathbb { Q } } ( I ^ { \\ell } ) \\right ] \\\\ & \\geq \\sum _ { \\ell = 1 } ^ h \\left [ \\beta _ { i - 1 } ^ { \\mathbb { Z } _ p } ( I ^ { \\ell } ) - \\beta _ { i - 1 } ^ { \\mathbb { Q } } ( I ^ { \\ell } ) \\right ] + | B | \\geq | B | , \\end{align*}"}
+{"id": "1569.png", "formula": "\\begin{align*} r v ( q - 1 ) & \\equiv r ( Q - R ) ( q - 1 ) \\pmod { q ^ 2 - 1 } \\\\ & = r ( T - 1 ) R ( q - 1 ) \\\\ & \\equiv T R ( q - 1 ) \\pmod { q ^ 2 - 1 } \\\\ & = Q ( q - 1 ) , \\end{align*}"}
+{"id": "9046.png", "formula": "\\begin{align*} W : = K ^ 1 - K ^ 2 , W ^ i : = K ^ { 1 , i } - K ^ { 2 , i } , W ^ { i , n } : = K ^ { 1 , i , n } - K ^ { 2 , i , n } , \\end{align*}"}
+{"id": "8148.png", "formula": "\\begin{align*} \\psi ( y ) = g \\Bigl ( \\frac { m ^ 2 y } { N } \\Bigr ) k ^ * \\Biggl ( \\frac { \\dfrac { 4 \\pi \\sqrt { y p } } { c } - 2 T } { 2 M } \\Biggr ) y ^ { - \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "4976.png", "formula": "\\begin{align*} ( y g _ 2 - g _ 1 ) h _ 2 - b _ 1 ^ e h _ 1 \\equiv _ { b _ 1 ^ { e + 1 } } t h _ 2 - b _ 1 ^ e h _ 1 = b _ 1 ^ e ( b _ 1 ^ { - e } t h _ 2 - h _ 1 ) \\equiv _ { b _ 1 ^ { e + 1 } } 0 . \\end{align*}"}
+{"id": "1512.png", "formula": "\\begin{align*} \\C _ r ( x + 1 ) = C \\prod _ { k = 1 } ^ r \\C _ k ( x ) ^ { \\binom { r - 1 } { k - 1 } } \\end{align*}"}
+{"id": "7986.png", "formula": "\\begin{align*} \\gamma ( [ z ] ) = \\beta ( z ) \\in \\beta ( x ' ) + \\beta ( y ' ) = \\beta ( x ) + \\beta ( y ) = \\gamma ( [ x ] ) + \\gamma ( [ y ] ) , \\end{align*}"}
+{"id": "8509.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } \\int ^ t _ 0 \\oint _ { C } \\varphi _ { \\xi } ^ 2 d \\xi d t = L ( t ) - L ( 0 ) . \\end{align*}"}
+{"id": "112.png", "formula": "\\begin{align*} | | x | | _ K = \\inf \\{ \\lambda \\geq 0 : \\ x \\in \\lambda K \\} \\end{align*}"}
+{"id": "1241.png", "formula": "\\begin{align*} X _ 1 ^ y ( y , D v ( y ) ) = \\partial ^ c v ( y ) : = \\{ x \\in X ^ y : - x \\cdot f ( y ) - v ( y ) \\leq - x \\cdot f ( \\tilde y ) - v ( \\tilde y ) \\ ; \\forall \\tilde y \\in Y \\} , \\end{align*}"}
+{"id": "5346.png", "formula": "\\begin{align*} \\operatorname { d i v } ( \\varepsilon ' u ) = \\operatorname { t r } ( \\varepsilon ' D u ) + ( \\operatorname { d i v } \\varepsilon ' ) \\cdot u . \\end{align*}"}
+{"id": "4101.png", "formula": "\\begin{align*} \\lambda ( g , H ) = \\inf \\Big \\{ \\mathcal { F } ( g , H , f ) \\big | \\ , f \\in C ^ \\infty ( M ) , \\ , \\int _ M e ^ { - f } d V _ g = 1 \\Big \\} . \\end{align*}"}
+{"id": "3283.png", "formula": "\\begin{align*} - \\left \\vert \\Delta u \\right \\vert ^ { p - 2 } \\Delta u = \\lambda \\left \\vert u \\right \\vert ^ { \\nu } u , x \\in \\Omega , u \\left \\vert ~ _ { \\partial \\Omega } \\right . = 0 , \\end{align*}"}
+{"id": "7785.png", "formula": "\\begin{align*} y ^ n y ^ { n ' } = q ^ { \\{ n , n ' \\} } y ^ { n + n ' } . \\end{align*}"}
+{"id": "2065.png", "formula": "\\begin{align*} a _ \\pm = \\frac { 1 } { 2 } \\biggl ( 1 \\pm \\frac { \\rho _ 2 } { \\rho _ 3 } - \\frac { i \\kappa _ 3 } { \\kappa } \\cdot \\frac { \\rho _ 1 - \\rho _ 2 } { \\rho _ 1 + \\rho _ 2 } \\biggr ) , b _ \\pm = 1 \\pm \\frac { \\rho _ 2 } { \\rho _ 3 } \\ , . \\end{align*}"}
+{"id": "8634.png", "formula": "\\begin{align*} U _ k = \\Omega ^ { \\N } \\setminus O _ k , \\end{align*}"}
+{"id": "237.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho ^ 2 - \\beta \\rho \\eta + \\alpha \\leqslant \\alpha , \\end{align*}"}
+{"id": "2178.png", "formula": "\\begin{align*} r _ a & = \\begin{dcases} a - \\frac { 1 } { 3 } & \\ \\frac { 1 } { 3 } < a \\leq 1 \\\\ \\frac { 5 } { 3 } - a & \\ 1 \\leq a < \\frac { 5 } { 3 } . \\end{dcases} \\end{align*}"}
+{"id": "4152.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda ( h , K ) = 0 \\Longleftrightarrow ( h , K ) \\in \\mathcal { V } + \\widetilde { \\mathcal { V } } \\end{align*}"}
+{"id": "3836.png", "formula": "\\begin{align*} v ^ 2 g _ V ( v ) = v ^ 2 \\left [ \\int _ 1 ^ \\infty + \\int _ 1 ^ \\infty \\right ] + v ^ 2 \\left [ \\int _ 0 ^ 1 + \\int _ 0 ^ 1 \\right ] = I _ 1 ( v ) + I _ 2 ( v ) . \\end{align*}"}
+{"id": "6474.png", "formula": "\\begin{align*} | \\delta E ( w _ * + \\eta ) | ^ 2 - | \\delta E ( w _ * ) | ^ 2 = 2 \\delta E ( w _ * ) \\cdot \\delta E ( \\eta ) + | \\delta E ( \\eta ) | ^ 2 = 2 \\beta _ * w _ * \\cdot \\delta E ( \\eta ) + | \\delta E ( \\eta ) | ^ 2 . \\end{align*}"}
+{"id": "5435.png", "formula": "\\begin{align*} - u ^ { a b } _ { , a b } = A _ t . \\end{align*}"}
+{"id": "4513.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n v _ i = \\prod _ { i = 1 } ^ n u _ i \\end{align*}"}
+{"id": "5042.png", "formula": "\\begin{align*} f ( \\vec a _ 0 + m _ 1 \\vec a _ 1 + \\cdots + m _ s \\vec a _ s ) = g _ { 0 } g _ { 1 } ^ { m _ 1 } \\cdots g _ { s } ^ { m _ s } \\qquad \\end{align*}"}
+{"id": "5651.png", "formula": "\\begin{align*} ( s - 1 ) A \\cup \\{ ( s - 1 ) a _ n + a _ 2 , \\cdots , ( s - 1 ) a _ n + a _ n \\} = s A . \\end{align*}"}
+{"id": "1976.png", "formula": "\\begin{align*} T ( r , A _ { j } ) \\leq \\exp _ { p - 1 } \\left [ \\left \\{ \\log _ { q - 1 } \\left ( r \\right ) \\right \\} ^ { \\rho - \\delta } \\right ] , j = 0 , 1 , . . . , k , j \\neq l \\end{align*}"}
+{"id": "4652.png", "formula": "\\begin{align*} \\mathcal { T } _ { i i } = \\exp \\left ( \\frac { \\sigma ^ 2 _ { i } } { 2 } \\partial ^ 2 _ i \\right ) , \\ \\ i = 1 , \\ldots , N , \\end{align*}"}
+{"id": "154.png", "formula": "\\begin{align*} \\| u ( \\theta ) \\| _ \\mathbb { U } & \\leq R , \\ \\mbox { f o r } \\ \\theta \\in J , \\ \\mbox { w h e r e } \\ R = \\max _ { 0 \\leq j \\leq n } \\{ R _ { j } \\} . \\end{align*}"}
+{"id": "4052.png", "formula": "\\begin{align*} 0 & \\leq a ^ { i j } D _ { i j } u + b ^ i D _ i u + c u B _ r ( 0 ) , \\\\ u & = 0 \\partial B _ r , \\end{align*}"}
+{"id": "2664.png", "formula": "\\begin{align*} \\beta : = \\left ( \\frac 1 { q _ 1 } - \\frac 1 2 \\right ) \\frac { \\alpha } { \\alpha + \\delta } , \\delta : = \\frac 1 { q _ 1 } - \\frac 1 { q _ 2 } . \\end{align*}"}
+{"id": "7398.png", "formula": "\\begin{align*} \\begin{matrix} \\lambda ( \\alpha ) = \\log ( q _ { \\alpha } q _ { \\alpha ^ * } ) / \\log ( q _ F ) = 1 + f ( L / F ) , \\\\ \\lambda ^ * ( \\alpha ) = | \\log ( q _ { \\alpha } q _ { \\alpha ^ * } ^ { - 1 } ) / \\log ( q _ F ) | = | 1 - f ( L / F ) | . \\end{matrix} \\end{align*}"}
+{"id": "832.png", "formula": "\\begin{align*} \\frac { u _ i p _ j } { p _ \\ast } \\cdot \\left ( \\frac { 1 } { u _ i } + O ( n ^ { - 1 / 2 } ) \\right ) \\cdot \\left ( \\frac { 1 } { p _ j } + O ( n ^ { - 1 / 2 } ) \\right ) \\cdot \\left ( p _ \\ast + O ( \\theta ^ n ) \\right ) = 1 + O ( n ^ { - 1 / 2 } ) \\end{align*}"}
+{"id": "2762.png", "formula": "\\begin{align*} \\overline { f \\circledast g } = \\overline { g } \\circledast \\overline { f } . \\end{align*}"}
+{"id": "948.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( \\textrm { T r } _ C ( N ) , C ) \\ , = \\ , \\textrm { r . g r a d e } _ R ( M , C ) . \\end{align*}"}
+{"id": "2316.png", "formula": "\\begin{align*} ( M ^ { \\# } f ) ( z ) = | \\det M ' ( z ) | ^ { \\frac { N - p } { N p } } f ( M ( z ) ) \\end{align*}"}
+{"id": "4552.png", "formula": "\\begin{align*} \\frac { 1 } { 7 } = \\frac { 1 } { 8 } + \\frac { 1 } { 5 6 } . \\end{align*}"}
+{"id": "2104.png", "formula": "\\begin{align*} & e _ C + e _ { C ' } - e _ C e _ { C ' } = ( f _ 1 + f _ 0 ) + ( f _ 2 + f _ 0 ) - ( f _ 1 + f _ 0 ) ( f _ 2 + f _ 0 ) \\\\ & \\textstyle = f _ 1 + f _ 2 + 2 f _ 0 - f _ 0 = f _ 1 + f _ 2 + f _ 0 = \\sum _ { \\epsilon \\in E _ C \\cup E _ { C ' } } \\epsilon . = e _ { C + C ' } . \\end{align*}"}
+{"id": "5786.png", "formula": "\\begin{align*} U = e ^ { i t \\sqrt { \\beta ^ 2 - \\lambda } } \\phi E \\end{align*}"}
+{"id": "4153.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { V } } = \\{ ( u g , - d ^ * ( \\omega d V _ g ) ) : u = a _ 1 \\chi _ 1 + a _ 2 \\chi _ 2 , \\ , \\omega = - a _ 1 \\chi _ 1 - \\frac { a _ 2 } { 2 } \\chi _ 2 , \\ , \\} , \\end{align*}"}
+{"id": "2087.png", "formula": "\\begin{align*} K = C _ { b , W _ 1 } \\times \\cdots \\times C _ { b , W _ m } \\end{align*}"}
+{"id": "4742.png", "formula": "\\begin{align*} ( x u ) ^ k = ( x ^ 2 ) \\cdot ( x ^ { k - 2 } u ^ k ) \\in \\mathfrak { q } B = \\mathfrak { q } \\end{align*}"}
+{"id": "4172.png", "formula": "\\begin{align*} \\widetilde { g } ( t ) = \\begin{cases} g ( t ) & t \\in [ 0 , 1 ] \\\\ \\psi ^ * _ t g ( t ) & t \\geq 1 \\end{cases} , \\widetilde { H } ( t ) = \\begin{cases} H ( t ) & t \\in [ 0 , 1 ] \\\\ \\psi ^ * _ t H ( t ) & t \\geq 1 \\end{cases} . \\end{align*}"}
+{"id": "2448.png", "formula": "\\begin{align*} L ( s , \\pi _ { p } ) = \\prod _ { j = 1 } ^ { m } \\Big ( 1 - \\frac { \\alpha _ { j , \\pi } ( p ) } { p ^ { s } } \\Big ) ^ { - 1 } = \\sum _ { j = 0 } ^ { \\infty } \\frac { \\lambda _ { \\pi } ( p ^ j ) } { p ^ { j s } } . \\end{align*}"}
+{"id": "2029.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\bf C } ^ - ( r ) : = \\{ y \\in \\mathcal { I } ^ - ( r ) : ( y - S _ { \\mu ' } ) \\cap \\mathcal { I } ^ + ( r ) \\neq \\emptyset \\} . \\end{align*}"}
+{"id": "6013.png", "formula": "\\begin{align*} p _ 2 \\left ( g _ 1 X ( w _ { i _ 1 , j _ 1 } ) \\times g _ 2 X ( w _ { i _ 2 , j _ 2 } ) \\right ) & = g _ 1 ^ \\bot \\cdot p _ 2 ( X ( w _ { i _ 1 , j _ 1 } ) ) \\times g _ 2 ^ \\bot \\cdot p _ 2 ( X ( w _ { i _ 2 , j _ 2 } ) ) \\subset \\P ^ { n - 1 } \\times \\P ^ { n - 1 } \\\\ & = ( g _ 1 ^ \\bot \\cdot w _ 0 L _ { n - j _ 1 + 1 } ) \\times ( g _ 2 ^ \\bot \\cdot w _ 0 L _ { n - j _ 2 + 1 } ) \\\\ & = L _ { g _ 1 } \\times L _ { g _ 2 } , \\end{align*}"}
+{"id": "7252.png", "formula": "\\begin{align*} n _ { m , j } [ T _ 0 > s ] = o ( s ^ { - 2 } U ^ { \\sharp } ( s ^ 2 ) ^ { - 1 } v ( s ^ { \\alpha } U ^ { \\sharp } ( s ^ 2 ) ^ { \\alpha / 2 } ) ) ( s \\to \\infty ) , \\end{align*}"}
+{"id": "8188.png", "formula": "\\begin{align*} v ( \\tau ) = u _ 0 + \\int _ 0 ^ \\tau \\kappa _ g ( \\tau , \\sigma ) L v ( \\sigma ) d \\sigma , \\qquad \\tau > 0 , \\end{align*}"}
+{"id": "5727.png", "formula": "\\begin{align*} \\mathbf { F } _ { B } ^ { A } \\mathbf { e } ^ { B A ^ { \\prime } } + \\mathbf { F } _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\mathbf { e } ^ { A B ^ { \\prime } } & = 0 , \\\\ \\mathbf { D F } _ { B } ^ { A } & = \\mathbf { D F } _ { B ^ { \\prime } } ^ { A ^ { \\prime } } = 0 , \\end{align*}"}
+{"id": "4442.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 3 0 } = \\frac { 1 } { 3 } + \\frac { 1 } { 5 } = \\frac { 8 } { 1 5 } . \\end{align*}"}
+{"id": "4763.png", "formula": "\\begin{align*} a _ { 0 } & \\int _ { \\Omega } h ( x , | \\nabla u _ { n } | ) \\ u _ { n } \\nabla u _ { n } \\ \\nabla v d x \\leq \\int _ { \\Omega } \\widetilde { \\psi } _ { q } ( x ) \\varphi ( x , u _ { n } ) v u _ { n } d x \\\\ & + \\int _ { \\Omega } \\psi _ { p } ( x ) | u _ { n } | ^ { p ^ { \\ast } ( x ) } v d x - a _ { 0 } \\int _ { \\Omega } h ( x , | \\nabla u _ { n } | ) \\ | \\nabla u _ { n } | ^ { 2 } \\ v d x + o _ { n } ( 1 ) . \\end{align*}"}
+{"id": "4929.png", "formula": "\\begin{align*} \\alpha = \\frac { \\omega \\Delta t ( 1 + \\cos ( \\omega \\Delta t ) ) } { 2 \\sin ( \\omega \\Delta t ) } . \\end{align*}"}
+{"id": "2911.png", "formula": "\\begin{align*} ( \\hat { \\mathcal { B } } f ) ( \\zeta ) = \\lim _ { k \\rightarrow \\infty } ( \\mathcal { B } _ k f ) ( \\zeta ) , \\quad \\zeta \\in \\mathbb { D } _ 1 ^ \\infty . \\end{align*}"}
+{"id": "7008.png", "formula": "\\begin{align*} \\hat { X } _ w = \\alpha ( X _ w + Z _ w ) , \\end{align*}"}
+{"id": "4592.png", "formula": "\\begin{align*} \\prod _ { i = m + 1 } ^ { m + k } \\frac { 1 } { b _ i } < \\prod _ { i = m + 1 } ^ { m + k } \\frac { 1 } { a _ i } \\end{align*}"}
+{"id": "2923.png", "formula": "\\begin{align*} \\eta ^ { j , j + 1 } _ k = \\begin{cases} \\begin{aligned} & \\eta _ j - 1 & & k = j , \\\\ & \\eta _ { j + 1 } + 1 & & k = j + 1 , \\\\ & \\eta _ k & & \\end{aligned} \\end{cases} \\end{align*}"}
+{"id": "858.png", "formula": "\\begin{align*} \\frac { \\partial F ( w ) } { \\partial w _ i } & = e ^ { \\widehat { w } } \\frac { \\partial \\widehat { w } } { \\partial w _ i } J _ d - J _ d ( e ^ { \\widehat { w } } \\frac { \\partial \\widehat { w } } { \\partial w _ i } ) ^ T \\\\ & = e ^ { \\widehat { w } } \\frac { \\partial \\widehat { w } } { \\partial w _ i } J _ d + J _ d \\frac { \\partial \\widehat { w } } { \\partial w _ i } ( e ^ { \\widehat { w } } ) ^ T \\end{align*}"}
+{"id": "6204.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } u _ r '' - \\Delta u _ r + \\sum _ { s = 1 } ^ d \\alpha _ { r s } u _ s = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ u _ r = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu u _ r + \\sum _ { s = 1 } ^ d \\beta _ { r s } u _ s = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "5152.png", "formula": "\\begin{align*} G _ { j } ( \\lambda , b , \\Omega , f _ { 1 } , f _ { 2 } ) = \\mathcal { S } _ { j } ( \\lambda , b , \\Omega , f _ { j } ) + \\mathcal { I } _ { j } ( \\lambda , b , f _ { 1 } , f _ { 2 } ) , \\end{align*}"}
+{"id": "139.png", "formula": "\\begin{align*} A _ g : = \\left \\{ ( x , y ) \\in \\mathbb R ^ n \\times \\mathbb R ^ n \\colon \\frac { | g _ r ( x ) - g _ r ( y ) | } { | | x - y | | _ K ^ { \\frac { n } { p } } } \\geq { \\lambda \\sigma } \\right \\} . \\end{align*}"}
+{"id": "3000.png", "formula": "\\begin{align*} \\begin{array} { c c l } \\theta & = & \\theta ^ i \\mathbf { t } _ i = ( \\theta { ^ i } _ \\mu ( z ) \\hbox { d } x ^ \\mu + \\theta { ^ i } _ j ( z ) \\hbox { d } y ^ j ) \\mathbf { t } _ i \\\\ \\pi & = & \\pi _ i \\mathbf { t } ^ i = \\frac { 1 } { ( N - 1 ) ! } \\pi _ { i I _ 1 \\cdots I _ { N - 1 } } ( z ) \\hbox { d } z ^ { I _ 1 } \\wedge \\cdots \\wedge \\hbox { d } z ^ { I _ { N - 1 } } \\mathbf { t } ^ i \\end{array} \\end{align*}"}
+{"id": "2283.png", "formula": "\\begin{align*} & \\partial _ t f + v \\cdot \\nabla _ x f + { \\rm { d i v } } _ v ( ( u - v ) f ) - \\nabla _ x \\Phi \\cdot \\nabla _ v f - \\Delta _ v f = 0 , \\\\ & - \\Delta \\Phi = \\int _ { \\mathbb { R } ^ 3 } f \\ , d v - c ( x ) , \\\\ & \\partial _ t \\rho + { \\rm { d i v } } _ x ( \\rho u ) = 0 , \\\\ & \\partial _ t ( \\rho u ) + { \\rm { d i v } } ( \\rho u \\otimes u ) + \\nabla \\rho ^ { \\gamma } + \\delta \\nabla \\rho ^ { \\beta } = { \\rm { d i v } } \\mathbb { S } ( \\nabla u ) - \\int _ { \\mathbb { R } ^ 3 } ( u - v ) f \\ , d v , \\end{align*}"}
+{"id": "3734.png", "formula": "\\begin{align*} \\sum _ { \\xi \\in X _ \\ell ^ \\circ ( E ) } I ( \\mathcal { F } _ { X _ \\ell } ( f _ { v } f _ { v _ 1 } f _ { v _ 2 } f ^ { v v _ 1 v _ 2 } ) ) ( \\xi ) = 0 . \\end{align*}"}
+{"id": "7688.png", "formula": "\\begin{align*} \\sum _ { q \\neq - \\iota _ { E } ( \\omega ) } \\frac { 1 } { q + \\iota _ { E } ( \\omega ) } \\cdot \\iota _ { E } ( \\eta _ { q } ) \\xmapsto { \\nabla _ { \\omega } } \\sum _ { q \\neq - \\iota _ { E } ( \\omega ) } \\eta _ { q } = \\eta - \\eta _ { - \\iota _ { E } ( \\omega ) } , \\end{align*}"}
+{"id": "7362.png", "formula": "\\begin{align*} \\| f _ m \\| _ \\infty \\leq \\| f _ 0 \\| _ \\infty + \\sum _ { k = 1 } ^ m \\| f _ k - f _ { k - 1 } \\| _ \\infty \\leq \\frac { c \\ , 2 ^ { 1 + \\varepsilon } } { 2 ^ d ( 2 ^ \\varepsilon - 1 ) } \\end{align*}"}
+{"id": "197.png", "formula": "\\begin{align*} \\liminf \\frac { r _ { n } + 1 } { \\log ( h _ { n } ) } \\geq \\liminf \\frac { ( n + 1 ) \\log ( n + 1 ) z _ { n } } { 9 n \\log ( n ) } = \\liminf \\frac { z _ { n } } { 9 } = \\infty . \\end{align*}"}
+{"id": "4550.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } = \\frac { 1 } { 4 } + \\frac { 1 } { 1 2 } \\end{align*}"}
+{"id": "444.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n \\in \\mathbb { M } } f _ n ( x ) \\tau _ n - \\sum _ { n = 1 } ^ { \\infty } r _ n f _ n ( x ) \\tau _ n \\right \\| \\leq c \\varepsilon ^ \\frac { 1 } { d } \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "8443.png", "formula": "\\begin{align*} & \\gamma _ A ( X ) = \\| \\gamma _ A \\| ^ 2 = c _ * ( A ) , \\\\ & \\kappa \\gamma _ A \\geqslant 1 , \\\\ & \\kappa \\gamma _ A \\leqslant 1 , \\\\ & \\kappa \\gamma _ A = 1 \\end{align*}"}
+{"id": "3779.png", "formula": "\\begin{align*} N & = \\langle S _ N \\mid R _ N \\rangle , & Q & = \\langle S _ Q \\mid R _ Q \\rangle , \\end{align*}"}
+{"id": "6008.png", "formula": "\\begin{align*} [ Y ( \\alpha _ 1 ) ] \\cup [ Y ( i , j ) ] & = [ Y ( { i + 1 , j } ) ] & \\ : \\mathrm { i f } \\ : j \\neq i + 1 , \\ : i \\neq n ; \\\\ [ Y ( \\alpha _ 1 ) ] \\cup [ Y ( n , j ) ] & = 0 ; \\\\ [ Y ( \\alpha _ 1 ) ] \\cup [ Y ( i , i + 1 ) ] & = [ Y ( i + 2 , i + 1 ) ] + [ Y ( i + 1 , i ) ] , \\end{align*}"}
+{"id": "5886.png", "formula": "\\begin{align*} C _ p B P P ^ T C _ p ^ T = \\overline B _ p C _ p P P ^ T C _ p ^ T . \\end{align*}"}
+{"id": "6179.png", "formula": "\\begin{align*} 0 = n _ 0 < n _ 1 < n _ 2 < \\cdots < n _ p = N . \\end{align*}"}
+{"id": "6750.png", "formula": "\\begin{align*} z ( s ) = \\lim _ { N \\rightarrow \\infty } z _ N ( s ) = - \\sqrt { \\frac { 1 6 } { \\pi } } s \\left [ \\int _ { 0 } ^ { 1 } y ^ { \\frac { s } { 2 } - 1 } \\Psi ( y ) d y + \\frac { 1 } { s } \\right ] . \\end{align*}"}
+{"id": "8057.png", "formula": "\\begin{align*} \\alpha = - \\nu _ 1 - 2 \\nu _ 2 + 1 , \\beta = - \\nu _ 1 + \\nu _ 2 , \\gamma = 2 \\nu _ 1 + \\nu _ 2 - 1 \\end{align*}"}
+{"id": "3616.png", "formula": "\\begin{align*} \\int \\frac { \\alpha } { \\beta } & = 2 \\frac { \\alpha } { \\beta } \\\\ \\beta \\int R ^ 2 & = \\frac { \\beta } { \\pi R _ * ^ 2 L _ * } \\\\ \\int R & = 2 \\frac { R ^ \\frac { 3 } { 2 } } { L } \\int R ^ 2 / \\sqrt { . . . } d R = 2 \\frac { R ^ { 3 / 2 } } { L } \\left ( \\frac { 4 \\alpha + 1 4 7 } { 8 \\beta ^ { 5 / 2 } } ( - \\frac \\pi 2 + \\arctan ( 7 / ( 2 \\sqrt { \\alpha } ) ) ) + \\frac { 2 1 \\sqrt { \\alpha / \\beta } } { 4 \\beta ^ 2 } \\right ) \\end{align*}"}
+{"id": "6293.png", "formula": "\\begin{align*} \\pi _ i \\pi _ { i + 1 } \\pi _ i ( T ) = \\pi _ { i + 1 } \\pi _ i \\pi _ { i + 1 } ( T ) . \\end{align*}"}
+{"id": "8561.png", "formula": "\\begin{align*} 0 \\equiv \\sum _ { j = 0 } ^ { v - 1 } j ^ { i } d _ { w } ( j ) \\equiv \\sum _ { j = 1 } ^ { p - 1 } j ^ { i } y _ { w } ( j ) \\mod p . \\end{align*}"}
+{"id": "8034.png", "formula": "\\begin{align*} ( p _ 1 , \\ldots , p _ 5 ) = \\frac { 1 } { 2 } ( \\det ( p _ 2 - p _ 1 , p _ 3 - p _ 1 ) + \\det ( p _ 3 - p _ 1 , p _ 4 - p _ 1 ) + \\det ( p _ 4 - p _ 1 , p _ 5 - p _ 1 ) ) . \\end{align*}"}
+{"id": "4611.png", "formula": "\\begin{align*} q r < \\prod _ { i = 1 } ^ n b _ i . \\end{align*}"}
+{"id": "5280.png", "formula": "\\begin{align*} \\hat { s } e c ( U , V ) - \\lambda ^ 2 \\mid \\nabla ( f + \\log \\lambda ) \\mid ^ 2 & = s e c ( U , V ) - \\frac { \\lambda ^ 4 } { 4 } \\mid \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\mid ^ 2 \\\\ & - \\frac { \\lambda ^ 2 } { 2 } \\left ( g ( T _ U U , \\nabla \\frac { 1 } { \\lambda ^ 2 } ) + g ( T _ V V , \\nabla \\frac { 1 } { \\lambda ^ 2 } ) \\right ) . \\end{align*}"}
+{"id": "2544.png", "formula": "\\begin{align*} x ( t ) & = x _ 0 + v _ 0 t _ 1 + v _ 1 ( t _ 2 - t _ 1 ) + v _ 2 ( t _ 3 - t _ 2 ) \\cdots \\ , , \\\\ y ( t ) & = x _ 0 + v _ 0 t _ 1 + \\xi _ 1 + w _ 1 ( t _ 2 - t _ 1 ) + \\xi _ 2 + w _ 2 ( t _ 3 - t _ 2 ) \\cdots \\end{align*}"}
+{"id": "4026.png", "formula": "\\begin{align*} - Y ^ { k } _ { p _ l p _ j } A _ { l i } - Y ^ k _ { p _ l } A _ { l i , p _ j } = Y ^ { k } _ { i p _ j } + Y ^ k _ { u p _ j } p _ i + Y ^ k _ z \\delta _ { i j } . \\end{align*}"}
+{"id": "4717.png", "formula": "\\begin{align*} \\gamma = \\alpha \\beta , \\zeta = \\alpha \\beta ^ 2 , \\sigma = \\beta \\alpha , \\mu = \\alpha \\beta \\alpha , \\omega = \\beta \\alpha \\beta . \\end{align*}"}
+{"id": "8910.png", "formula": "\\begin{align*} B _ d \\left ( \\frac { 1 } { 2 } \\right ) = - ( 1 - 2 ^ { 1 - d } ) B _ d , \\end{align*}"}
+{"id": "3532.png", "formula": "\\begin{align*} M ' ( \\mathfrak { g } , \\mathfrak { t } ) = D \\oplus \\mathfrak { t } ^ - M ' ( \\mathfrak { g } , \\mathfrak { t } ) . \\end{align*}"}
+{"id": "684.png", "formula": "\\begin{align*} { \\lambda ^ 2 } ( \\dot { x } d x + \\dot { y } d y ) = \\frac { \\lambda ^ 2 } { 2 } ( \\dot { \\bar { z } } d z + \\dot { { z } } d \\bar { z } ) , \\end{align*}"}
+{"id": "2491.png", "formula": "\\begin{align*} \\Bigl \\| P _ { \\mathcal H _ 0 } T \\sum _ { n \\geq 0 } a _ n e _ { 2 n + 1 } \\Bigr \\| ^ 2 & = \\sum _ { n \\geq 0 } | a _ n c _ n ( \\lambda _ { 2 n + 1 } - \\lambda _ { 2 n } ) | ^ 2 \\\\ & \\leq ( \\sup _ { n \\geq 0 } c _ n | \\lambda _ { 2 n } - \\lambda _ { 2 n + 1 } | ) ^ 2 \\sum _ { n \\geq 0 } | a _ n | ^ 2 . \\end{align*}"}
+{"id": "6946.png", "formula": "\\begin{gather*} H _ { | \\lambda | , n } ^ { ( \\lambda ) } = \\operatorname { W r } [ H _ { \\lambda _ m } , H _ { \\lambda _ { m - 1 } + 1 } , \\dots , H _ { \\lambda _ 2 + m - 2 } , H _ { \\lambda _ 1 + m - 1 } , H _ { n - | \\lambda | + m } ] \\end{gather*}"}
+{"id": "5804.png", "formula": "\\begin{align*} U = \\sum _ { s = 1 } ^ p u _ s e _ s . \\end{align*}"}
+{"id": "7641.png", "formula": "\\begin{align*} \\Gamma _ h ^ { - 1 } = O ( h ^ { - 1 / 2 } ) . \\end{align*}"}
+{"id": "2722.png", "formula": "\\begin{align*} \\lambda _ j \\int \\left ( \\Lambda W \\right ) _ { [ \\lambda _ j ] } \\partial _ t ^ 2 U = & - \\int \\left ( \\Lambda W \\right ) _ { ( \\lambda _ j ) } L _ { W _ { ( \\lambda _ j ) } } h \\\\ & + 2 \\int \\sum _ { j \\neq k } ( \\Lambda W ) _ { ( \\lambda _ j ) } W _ { ( \\lambda _ k ) } h + \\int ( \\Lambda W ) _ { ( \\lambda _ j ) } h ^ 2 \\\\ & + \\int ( \\Lambda W ) _ { ( \\lambda _ j ) } ( 2 M v _ L + v _ L ^ 2 + 2 v _ L h ) \\\\ & + \\int ( \\Lambda W ) _ { ( \\lambda _ j ) } ( M ^ 2 + \\Delta M ) . \\end{align*}"}
+{"id": "2758.png", "formula": "\\begin{align*} X ^ { 0 } X ^ { A } = X ^ { A } X ^ { 0 } , A \\in \\{ + , 3 , - \\} . \\end{align*}"}
+{"id": "6672.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 2 \\lambda ^ k } { m } \\sum _ { i = 1 } ^ m \\left \\langle g _ i ^ k , \\bar x ^ k - \\theta ^ * \\right \\rangle = \\frac { 2 \\lambda ^ k } { m } \\sum _ { i = 1 } ^ m \\left \\langle g _ i ^ k - \\nabla f _ i ( \\bar { x } ^ k ) , \\bar x ^ k - \\theta ^ * \\right \\rangle \\\\ & \\qquad + \\frac { 2 \\lambda ^ k } { m } \\sum _ { i = 1 } ^ m \\left \\langle \\nabla f _ i ( \\bar { x } ^ k ) , \\bar x ^ k - \\theta ^ * \\right \\rangle \\end{aligned} \\end{align*}"}
+{"id": "2522.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla _ x f = \\varrho _ f M _ f - f \\ , , \\end{align*}"}
+{"id": "4170.png", "formula": "\\begin{align*} \\| \\frac { d } { d t } \\Big | _ { t = 0 } \\lambda ( g + t h , b + t K ) \\| _ { C ^ { 0 , \\alpha } } \\leq C ( \\| h \\| _ { C ^ { 2 , \\alpha } } + \\| K \\| _ { C ^ { 2 , \\alpha } } ) . \\end{align*}"}
+{"id": "5691.png", "formula": "\\begin{align*} \\mathbf { L } _ { b } ^ { a } & = ( \\delta _ { c } ^ { a } + \\mathbf { l } _ { c } ^ { a } \\mathbf { m } ^ { 2 } ) ( \\delta _ { d } ^ { c } + _ { 1 } \\lambda _ { d } ^ { c } \\mathbf { m } ^ { 1 } ) L _ { b } ^ { d } , \\\\ \\mathbf { l } _ { c } ^ { a } & = _ { 2 } \\lambda _ { c } ^ { a } + \\lambda _ { c } ^ { a } \\mathbf { m } ^ { 1 } , \\end{align*}"}
+{"id": "5076.png", "formula": "\\begin{align*} F _ n = \\sum _ { i = 1 } ^ s \\beta _ i \\alpha _ i ^ n \\in L . \\end{align*}"}
+{"id": "8882.png", "formula": "\\begin{align*} P _ { k } ^ { ( \\alpha , \\beta ) } ( 2 t ^ { 2 } - 1 ) = \\frac { 2 \\Gamma ( \\alpha + \\beta + 1 ) \\Gamma ( k + \\beta + 1 ) } { \\Gamma ( \\beta + \\frac { 1 } { 2 } ) \\Gamma ( k + \\alpha + \\beta + 1 ) } \\int _ { 0 } ^ { 1 } ( 1 - u ^ { 2 } ) ^ { \\beta - \\frac { 1 } { 2 } } C _ { 2 k } ^ { \\alpha + \\beta + 1 } ( t u ) d u , \\end{align*}"}
+{"id": "5590.png", "formula": "\\begin{align*} & 0 = 2 \\rho ^ q ( \\rho _ j \\Psi _ { i k q } - \\rho _ i \\Psi _ { j k q } - \\rho _ k \\Psi _ { q i j } ) + \\rho ^ q \\rho _ q \\Psi _ { k i j } + 2 \\rho _ k ( \\rho _ i \\Psi _ j - \\rho _ j \\Psi _ i ) \\ ; ; \\\\ & 0 = 2 \\rho ^ q \\rho ^ r \\Psi _ { q i r } - 4 \\rho _ i ( \\rho ^ q \\Psi _ q ) + \\rho ^ q \\rho _ q \\Psi _ i \\ ; . \\end{align*}"}
+{"id": "2741.png", "formula": "\\begin{align*} \\big [ K _ i E _ i , [ F _ i , F _ j ] _ { q ^ 3 } \\big ] _ q & = q ^ 3 [ 3 ] F _ j K _ i K _ i ' , \\\\ \\Big [ K _ i E _ i , \\big [ F _ i , [ F _ i , F _ j ] _ { q ^ 3 } \\big ] _ q \\Big ] _ { q ^ { - 1 } } & = q ( 1 + [ 3 ] ) [ B _ i ^ \\sigma , F _ j ] _ { q ^ 3 } K _ i K _ i ' . \\end{align*}"}
+{"id": "6361.png", "formula": "\\begin{align*} C _ n : = \\min \\big \\{ a _ n ( 2 | \\hat m | _ { 2 , n } ) ^ { - 1 } , ( 3 [ 1 + 4 | \\hat m ' / \\hat m | _ { 1 , n } + | \\hat m '' / \\hat m | _ { 1 , n } ] ) ^ { - 1 } , ( n ^ 2 | \\hat m | _ { 2 , n } ) ^ { - 1 } \\big \\} , \\end{align*}"}
+{"id": "6330.png", "formula": "\\begin{align*} I ( \\mathcal { C } _ 0 ) _ 2 = \\langle I ( S _ 1 ) _ 2 , I ( S _ 2 ) _ 2 , I ( S _ 3 ) _ 2 \\rangle . \\end{align*}"}
+{"id": "1276.png", "formula": "\\begin{align*} { \\rm I f ~ } r = 1 , ~ \\zeta _ { t } ( z ) = \\xi _ { t + 1 } ( z ) { \\rm ~ a n d ~ } \\zeta _ { k - 1 } ( z ) = \\xi _ 0 ( z ) \\end{align*}"}
+{"id": "2939.png", "formula": "\\begin{align*} \\mathcal { X } ^ n _ t ( \\varphi ) = \\mathcal { X } ^ n _ 0 ( \\varphi ) + \\mathcal { S } ^ n _ t ( \\varphi ) + \\mathcal { B } ^ n _ t ( \\varphi ) + \\mathcal { M } ^ n _ t ( \\varphi ) . \\end{align*}"}
+{"id": "2404.png", "formula": "\\begin{align*} [ \\ > E _ { 1 2 } \\ > \\ > E _ { 1 3 } \\ > ] V = [ \\ > I _ p \\ > \\ > 0 \\ > ] \\end{align*}"}
+{"id": "4121.png", "formula": "\\begin{align*} \\alpha _ { i j k } = \\langle [ e _ i , e _ j ] , e _ k \\rangle = \\langle e _ i , [ e _ j , e _ k ] \\rangle = \\alpha _ { j k i } = \\alpha _ { k i j } . \\end{align*}"}
+{"id": "8405.png", "formula": "\\begin{align*} ( \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) ' ( z ) & = \\left ( \\hat \\psi _ { \\alpha } \\left ( \\frac { z _ { n } + 1 } { 2 } \\right ) + \\sum _ { j = 0 } ^ { n - 1 } \\hat \\psi _ { \\alpha } ( f _ { \\alpha } ^ j ( z _ { n } ) ) \\right ) ' \\\\ & = \\psi ' \\left ( \\frac { z _ { n } + 1 } { 2 } \\right ) z _ { n } ' + \\sum _ { j = 1 } ^ { n } \\psi ' ( z _ { j } ) z _ j ' \\end{align*}"}
+{"id": "7419.png", "formula": "\\begin{align*} ( T _ { w _ i } - q _ F ) ( T _ { w _ i } + 1 ) = 0 . \\end{align*}"}
+{"id": "3894.png", "formula": "\\begin{align*} ( \\dot { x _ \\theta } ) _ i = g _ z E ^ { m , i } q _ m . \\end{align*}"}
+{"id": "5296.png", "formula": "\\begin{align*} - 2 g ( U , V ) X ( f ) & = 2 g ( H , X ) g ( U , V ) = g ( \\nabla _ U V , X ) + g ( \\nabla _ V U , X ) \\\\ & = - g ( \\nabla _ U X , V ) - g ( \\nabla _ V X , U ) \\\\ & = g ( [ X , U ] , V ) - g ( \\nabla _ X U , V ) + g ( [ X , V ] , U ) - g ( \\nabla _ X V , U ) \\\\ & = - g ( \\nabla _ X U , V ) - g ( \\nabla _ X V , U ) \\\\ & = - X ( g ( U , V ) ) . \\end{align*}"}
+{"id": "5696.png", "formula": "\\begin{align*} \\mathfrak { f } _ { a b c d } & = \\mathfrak { f } _ { [ a b ] c d } = \\mathfrak { f } _ { a b [ c d ] } , \\\\ \\mathfrak { f } _ { [ b c d ] } ^ { a } & = 0 . \\end{align*}"}
+{"id": "2360.png", "formula": "\\begin{align*} L ^ * L & = W ^ * A ^ * \\overline { W } W ^ T A W = W ^ * A ^ * A W = \\Lambda ^ 2 , \\\\ L L ^ * & = W ^ T A W W ^ * A ^ * \\overline { W } = W ^ T A \\overline { A } \\overline { W } = \\overline { W ^ * A ^ * A W } = \\overline { \\Lambda ^ 2 } = \\Lambda ^ 2 , \\end{align*}"}
+{"id": "1287.png", "formula": "\\begin{align*} q _ { \\ell - 1 } ^ { ( k - 1 ) * } ( x ) = m _ { k - 1 , \\ell - 1 } ( x ) p _ { \\ell - 1 } ^ { ( k - 1 ) } ( x ) , \\end{align*}"}
+{"id": "8220.png", "formula": "\\begin{align*} P \\lbrace \\mathcal { T } ( t ) = x + v _ 0 ( t - s ) \\ | \\ \\mathcal { T } ( s ) = x \\rbrace = \\frac { e ^ { - \\lambda ( t - s ) } } { 2 } \\Bigl ( 1 + \\frac { | v _ 1 s - x | } { ( a _ 1 - a _ 2 ) s } \\ , P \\lbrace N ( s ) \\ \\ | \\ \\mathcal { T } ( s ) = x \\rbrace \\Bigr ) . \\end{align*}"}
+{"id": "4085.png", "formula": "\\begin{align*} H ' = H _ 0 + d ( f ^ * b - B ) = H _ 0 + f ^ * ( d b ) - d B = f ^ * ( H _ 0 + d b ) = f ^ * H . \\end{align*}"}
+{"id": "295.png", "formula": "\\begin{align*} \\begin{aligned} \\widehat { 2 \\rho } : \\{ \\pm 1 \\} & \\rightarrow Z ( \\widehat { G } _ { \\mathrm { s c } } ) \\rightarrow Z ( \\widehat { G } ) \\\\ - 1 & \\mapsto \\prod _ { \\alpha \\in \\Phi ^ { + } } \\widehat { \\alpha } ( - 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "8181.png", "formula": "\\begin{align*} u ( t ) : = \\Phi ( t , L ) u _ 0 \\end{align*}"}
+{"id": "5795.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ N a _ { k p } = \\alpha , k = 1 , \\cdots , N . \\end{align*}"}
+{"id": "3452.png", "formula": "\\begin{align*} q _ 1 f _ 4 = \\ldots = q _ s f _ 4 = 0 \\end{align*}"}
+{"id": "1761.png", "formula": "\\begin{align*} v ( x ) = \\lambda ^ { \\frac { 1 } { 2 ( 3 - p ) } } u \\big ( \\lambda ^ { \\frac { p - 2 } { 4 ( 3 - p ) } } x \\big ) \\end{align*}"}
+{"id": "1783.png", "formula": "\\begin{align*} a ^ { k - j } b ^ { m _ k } = n _ j b ^ { m _ k - j } + \\sum _ { i = j + 1 } ^ { m _ k } n _ i a ^ { i - j } b ^ { m _ k - i } \\end{align*}"}
+{"id": "5677.png", "formula": "\\begin{align*} \\lim _ { h \\rightarrow 0 } f _ { Q _ j } ( y ; \\rho + h , \\rho ) = f _ { \\chi ^ 2 _ { n + 2 } ( \\frac { \\rho ^ 2 } { \\sigma _ j ^ 2 } ) } ( y ) , y > 0 . \\end{align*}"}
+{"id": "8348.png", "formula": "\\begin{align*} ( s , y ) \\mapsto ( s ' , y ' ) = \\left ( \\frac { s - I M _ 0 } { M _ 0 J } , \\frac { y } { M _ 0 J } \\right ) \\end{align*}"}
+{"id": "1420.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( | \\mathcal C _ { \\max } | < n ^ { 2 / 3 } / A \\right ) = O ( A ^ { - 3 / 5 } ) \\mathbb { P } \\left ( | \\mathcal C _ { \\max } | > A n ^ { 2 / 3 } \\right ) = O ( A ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "2258.png", "formula": "\\begin{align*} \\psi _ 1 = [ \\xi ] ^ { a + 2 + ( p + 1 ) b } , ~ \\psi _ { 2 } = [ \\xi ] ^ { a + 3 + ( p + 1 ) ( b - 1 ) } , ~ \\psi _ { 3 } = [ \\xi ] ^ { a + 1 + ( p + 1 ) b } . \\end{align*}"}
+{"id": "2390.png", "formula": "\\begin{align*} E _ 2 = Q ^ T E _ 1 Q , A _ 2 = Q ^ T A _ 1 Q - Q ^ T E _ 1 \\dot Q \\end{align*}"}
+{"id": "2686.png", "formula": "\\begin{align*} \\mathcal { H } ^ s ( K ) ^ { - 1 } & = \\max \\left \\{ \\frac { \\mu ( U ) } { | U | ^ s } : U \\subset \\R ^ d \\right \\} \\\\ & = \\max \\left \\{ \\frac { \\mu ( U ) } { | U | ^ s } : U \\subset \\R ^ d | U | \\geq \\Delta \\right \\} . \\end{align*}"}
+{"id": "1470.png", "formula": "\\begin{align*} \\lambda = \\frac { k ( k - 1 ) } { m ^ 2 ( m ^ 2 - 1 ) } \\cdot b = \\frac { k ( k - 1 ) } { m ^ 2 ( m + 1 ) ( m - 1 ) } \\cdot b . \\end{align*}"}
+{"id": "1385.png", "formula": "\\begin{align*} \\psi = \\psi _ 1 \\oplus \\psi _ 0 \\oplus \\psi _ 1 ^ \\vee , \\end{align*}"}
+{"id": "6591.png", "formula": "\\begin{align*} [ M \\setminus \\{ j \\} , g ' , M \\setminus \\{ j \\} ] = [ M , g , M ] \\end{align*}"}
+{"id": "8725.png", "formula": "\\begin{align*} \\tilde F _ \\lambda ( x _ 1 , \\dots , x _ n ; t ) = \\frac { ( 1 - t ) ^ n } { \\prod _ { i = 1 } ^ { n - l } ( 1 - t ^ i ) } \\sum _ { \\sigma \\in S _ n } \\sigma \\left ( x _ 1 ( x _ { 1 } - 1 ) ^ { \\lambda _ 1 - 1 } \\dots x _ n ( x _ { n } - 1 ) ^ { \\lambda _ n - 1 } \\prod _ { i < j } \\frac { x _ { i } - t x _ { j } } { x _ { i } - x _ { j } } \\right ) \\end{align*}"}
+{"id": "272.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\rho } - \\beta \\rho \\tau \\leqslant - \\frac { \\beta ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\beta } { 2 } \\end{align*}"}
+{"id": "4619.png", "formula": "\\begin{align*} J ( 5 , 1 3 ) = \\left ( \\frac { 1 } { 5 } + \\frac { 1 } { 1 3 } , \\frac { 1 } { 5 } + \\frac { 1 } { 1 2 } \\right ] = \\left ( \\frac { 1 8 } { 6 5 } , \\frac { 1 7 } { 6 0 } \\right ] . \\end{align*}"}
+{"id": "1450.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\binom { y _ j } { 3 } = \\frac { k ( k - 1 ) ( k - 2 ) ( m - 1 ) ( m - 2 ) } { 6 ( m n - 1 ) ( m n - 2 ) } ; \\end{align*}"}
+{"id": "2235.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m + 1 } ( 5 n + 2 ) + 5 ^ { 2 m + 1 } \\right ) } q ^ n \\equiv x _ { 2 m , 1 } E ( q ^ 2 ) E ( q ^ 5 ) ^ 2 \\pmod { 5 ^ { 2 m } } . \\end{align*}"}
+{"id": "2713.png", "formula": "\\begin{align*} \\partial _ t U ( t ) = \\sum _ { j = 1 } ^ J \\alpha _ j ( t ) \\Lambda W _ { [ \\lambda _ j ( t ) ] } + g _ 1 ( t ) , \\end{align*}"}
+{"id": "5454.png", "formula": "\\begin{align*} & { ( \\mathcal { L } _ { X ^ n _ t } ) \\circ \\phi _ n ^ { - 1 } } ( A ) = P ( X ^ n _ t \\in \\phi _ n ^ { - 1 } ( A ) ) \\\\ = & P ( \\phi _ n ( X ^ n _ t ) \\in A ) = \\mathcal { L } _ { \\phi _ n ( X ^ n _ t ) } ( A ) = \\mathcal { L } _ { X ^ n _ { t \\wedge \\tau ^ n } } ( A ) . \\end{align*}"}
+{"id": "5433.png", "formula": "\\begin{align*} \\alpha _ 0 ( \\omega _ 0 ) = \\gamma _ t ^ { - 1 } { } ^ * \\left ( \\alpha _ 0 ( \\gamma _ t ^ * \\omega _ 0 ) \\right ) , \\end{align*}"}
+{"id": "2263.png", "formula": "\\begin{align*} [ \\chi ] = [ \\xi ] ^ { a + 2 + ( p + 1 ) b } , \\end{align*}"}
+{"id": "5785.png", "formula": "\\begin{align*} L \\phi = ( \\beta ^ 2 - \\lambda ) \\phi \\hbox { a n d } g \\phi = 0 . \\end{align*}"}
+{"id": "1532.png", "formula": "\\begin{align*} \\omega _ L ( X ) _ { | g } & = \\d R _ { g ^ { - 1 } } X _ { | g } \\\\ \\omega _ R ( X ) _ { | g } & = \\d L _ { g ^ { - 1 } } X _ { | g } \\end{align*}"}
+{"id": "6999.png", "formula": "\\begin{align*} R _ c ( R _ p ) = \\frac { 1 } { 2 } \\log ^ { + } { \\frac { 1 - \\rho ^ 2 } { ( 1 - \\rho ) \\left ( 2 \\Delta e ^ { R _ p } + \\rho - 1 \\right ) } } \\end{align*}"}
+{"id": "4169.png", "formula": "\\begin{align*} & \\| \\frac { d } { d t } \\Big | _ { t = 0 } f _ { ( g + t h , b + t K ) } \\| _ { C ^ { 2 , \\alpha } } \\leq C ( \\| h \\| _ { C ^ { 2 , \\alpha } } + \\| K \\| _ { C ^ { 2 , \\alpha } } ) ( g , b ) \\in \\mathcal { U } , \\\\ & \\| \\frac { d } { d t } \\Big | _ { t = 0 } f _ { ( g + t h , b + t K ) } \\| _ { W ^ { 2 , 2 } } \\leq C ( \\| h \\| _ { W ^ { 2 , 2 } } + \\| K \\| _ { W ^ { 2 , 2 } } ) ( g , b ) \\in \\mathcal { U } . \\end{align*}"}
+{"id": "7533.png", "formula": "\\begin{align*} Z _ g \\big ( s , \\chi , S ( \\Delta _ { \\gamma _ 1 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) ) \\big ) = \\dfrac { G _ 1 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "8917.png", "formula": "\\begin{align*} = \\frac { \\omega _ { n } } { ( 4 \\pi t ) ^ { n } } \\left [ \\sum \\limits _ { d = 0 } ^ { + \\infty } c _ { d } ^ { \\left ( \\nu , n \\right ) } \\ t ^ { d } \\right ] \\left [ \\sum \\limits _ { i = 0 } ^ { + \\infty } \\frac { 1 } { i ! } \\left ( \\frac { n ^ { 2 } } { 4 } + \\nu ^ { 2 } \\right ) ^ { i } \\right ] \\end{align*}"}
+{"id": "6255.png", "formula": "\\begin{align*} h ( x , t ) = G ( f _ { 0 } ^ { - 1 } ( x ) , t ) \\end{align*}"}
+{"id": "6210.png", "formula": "\\begin{align*} \\hbox { I m } ( D ) = V ^ \\perp , \\end{align*}"}
+{"id": "8473.png", "formula": "\\begin{align*} c ^ * ( A ) = \\sup _ { \\nu \\in \\mathcal E ' _ A } \\ , G ( \\nu ) = \\sup _ { \\nu \\in \\widehat { \\mathcal E } ' _ A } \\ , \\nu ( X ) . \\end{align*}"}
+{"id": "8771.png", "formula": "\\begin{align*} \\mathcal { J } _ { i , j } = \\begin{cases} 1 / | \\bar { l } | ( U _ l \\varepsilon ) ^ * ( i , j ) \\in \\mathbb { N } ^ 2 , \\ \\ 1 \\leq | i - j | = l \\leq 2 T - 1 , \\\\ 0 . \\end{cases} \\end{align*}"}
+{"id": "9065.png", "formula": "\\begin{align*} ( N _ { P } \\cap w N _ { Q } w ^ { - 1 } ) \\cdot z = N _ { P } \\cdot z . \\end{align*}"}
+{"id": "4982.png", "formula": "\\begin{align*} \\langle V \\rangle ( q ) : = \\int _ { \\R ^ d } V ( x ) g ( x - q ) ^ 2 \\ , d x \\ , , q \\in \\R ^ d \\ , , \\end{align*}"}
+{"id": "5233.png", "formula": "\\begin{align*} T _ { X _ 1 } X _ 2 = \\mathcal { H } \\nabla _ { \\nu X _ 1 } \\nu X _ 2 + \\nu \\nabla _ { \\nu X _ 1 } \\mathcal { H } X _ 2 , \\end{align*}"}
+{"id": "4329.png", "formula": "\\begin{align*} & \\frac { 2 ^ { n _ 1 } ( 1 + o ( 1 ) ) } { \\binom { n _ 1 / 2 + \\sqrt { n _ 1 \\ln n _ 1 } + t } { t } } 2 ^ { n _ 2 } \\left ( 1 - \\sum \\limits _ { i = 1 } ^ t \\frac { \\binom { n _ 2 } { i } ( 1 + o ( 1 ) ) } { n _ 2 ^ { 2 i } } \\right ) \\\\ & = \\frac { t ! 2 ^ { n + t } } { n _ 1 ^ t } ( 1 + o ( 1 ) ) . \\end{align*}"}
+{"id": "4072.png", "formula": "\\begin{align*} & [ a , [ b , c ] ] = [ [ a , b ] , c ] + [ b , [ a , c ] ] . \\\\ & \\pi [ a , b ] = [ \\pi ( a ) , \\pi ( b ) ] . \\\\ & [ a , f b ] = f [ a , b ] + \\pi ( a ) f b . \\\\ & \\pi ( a ) \\langle b , c \\rangle = \\langle [ a , b ] , c \\rangle + \\langle a , [ b , c ] \\rangle . \\\\ & [ a , b ] + [ b , a ] = \\mathcal { D } \\langle a , b \\rangle \\end{align*}"}
+{"id": "5275.png", "formula": "\\begin{align*} \\tilde { s e c } ( U , V ) & = s e c ( U , V ) + \\frac { \\lambda ^ 2 } { 2 } \\left ( H e s s \\frac { 1 } { \\lambda ^ 2 } ( U , U ) + H e s s \\frac { 1 } { \\lambda ^ 2 } ( V , V ) \\right ) \\\\ & - \\frac { \\lambda ^ 4 } { 4 } \\mid g r a d \\frac { 1 } { \\lambda ^ 2 } \\mid ^ 2 - \\frac { \\lambda ^ 4 } { 4 } \\left ( \\left ( U \\frac { 1 } { \\lambda ^ 2 } \\right ) ^ 2 + \\left ( V \\frac { 1 } { \\lambda ^ 2 } \\right ) ^ 2 \\right ) , \\end{align*}"}
+{"id": "3172.png", "formula": "\\begin{align*} v ^ { 1 1 } _ A ( y ) : = v ^ { 1 1 } _ B ( y _ 1 , y _ 2 ) \\quad y = ( y _ 1 , y _ 2 , y _ 3 ) \\in \\R ^ 3 \\end{align*}"}
+{"id": "2751.png", "formula": "\\begin{align*} \\overline { A ^ { 0 } } = A _ { 0 } , \\qquad \\overline { A ^ { C } } = A _ { C } . \\end{align*}"}
+{"id": "346.png", "formula": "\\begin{align*} \\begin{array} { l l l } e ( G - X ) & = & e ( G ) - e ( X , B ) \\\\ & \\geq & \\left ( n - \\left \\lfloor k _ { \\ell } / 2 \\right \\rfloor + 1 \\right ) ( s _ \\ell - 1 ) + \\left ( m - s _ \\ell + 1 \\right ) \\left ( \\left \\lfloor k _ { \\ell } / 2 \\right \\rfloor - 1 \\right ) - s _ { \\ell - 1 } n \\\\ & = & \\left ( n + m - 2 s _ { \\ell } + 2 \\right ) ( \\lfloor k _ { \\ell } / 2 \\rfloor - 1 ) . \\end{array} \\end{align*}"}
+{"id": "7856.png", "formula": "\\begin{align*} f '' + A ( z ) f ' + B ( z ) f = 0 . \\end{align*}"}
+{"id": "4309.png", "formula": "\\begin{align*} \\eta ' = \\eta - ( K s + 2 M ) \\nu ^ { - 1 } _ { 0 } > 0 , \\end{align*}"}
+{"id": "1785.png", "formula": "\\begin{align*} \\ell _ x = \\sum _ { j = 1 } ^ { s - t } \\sum _ { i = k _ j } ^ { m _ { k _ j } } n _ i ^ { ( j ) } = \\ell _ { k _ 1 } + \\dots + \\ell _ { k _ { s - t } } , \\end{align*}"}
+{"id": "6728.png", "formula": "\\begin{align*} S _ 0 ( x ) = \\sum _ { t \\leq x } \\mu ( t ) d ( t ) \\sum _ { d ^ 2 \\leq x _ 0 } \\mu ( d ) \\sum _ { \\substack { k \\leq x / t \\\\ d ^ 2 \\mid k } } \\mu ( k t + a ) \\ll \\frac { x } { ( \\log x ) ^ { ( C - 3 ) / 2 } } , \\end{align*}"}
+{"id": "728.png", "formula": "\\begin{align*} S ( z ) = \\bar { z } \\partial \\Omega . \\end{align*}"}
+{"id": "5085.png", "formula": "\\begin{align*} P _ n \\coloneqq \\sum _ { i = 1 } ^ k \\sum _ { l = 1 } ^ { N _ 0 } { n \\choose l } \\beta _ i c _ i ^ { n - l } \\theta _ i ^ l . \\end{align*}"}
+{"id": "1506.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\C _ 5 \\left ( \\frac 1 4 \\right ) & = \\frac 1 { 2 ^ 9 } \\log 2 - \\frac { G } { 2 ^ 5 \\pi } - \\frac { 3 ^ 2 \\zeta ( 3 ) } { 2 ^ 9 \\pi ^ 2 } - \\frac { 3 ^ 2 \\cdot 5 \\cdot 3 1 \\cdot \\zeta ( 5 ) } { 2 ^ { 1 0 } \\pi ^ 4 } \\\\ & \\quad + \\frac 3 { 2 ^ { 1 0 } \\pi ^ 3 } \\left ( \\zeta \\left ( 4 , \\frac 1 4 \\right ) - \\zeta \\left ( 4 , \\frac 3 4 \\right ) \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "5202.png", "formula": "\\begin{align*} A = \\hat R \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & \\cos ( \\theta ) & - \\sin ( \\theta ) \\\\ 0 & \\sin ( \\theta ) & \\cos ( \\theta ) \\end{pmatrix} \\begin{pmatrix} b _ { 1 1 } & b _ { 1 2 } & b _ { 1 3 } & b _ { 1 4 } \\\\ 0 & b _ { 2 2 } & b _ { 2 3 } & 0 \\\\ 0 & 0 & 1 / ( b _ { 1 1 } b _ { 2 2 } ) & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "996.png", "formula": "\\begin{align*} a ( f , g ) = \\sum _ { j = 1 } ^ k \\int _ { \\Omega _ j } \\big ( \\nabla u _ j ^ f \\cdot \\nabla u _ j ^ g - \\lambda _ * u _ j ^ f u _ j ^ g \\big ) , \\end{align*}"}
+{"id": "1601.png", "formula": "\\begin{align*} \\mu ( v ) = \\mu ^ * ( v ) = \\mu ^ * ( u ) = \\mu ( u ) + \\mu ( x _ { n + 1 } ) , \\end{align*}"}
+{"id": "1908.png", "formula": "\\begin{align*} \\sum _ { \\frac { e a ^ 2 } { L ^ 2 } < x _ 1 < \\sqrt { a + d L } } 2 ^ { - \\omega ( x _ 1 ) } \\frac { ( \\log x _ 1 ) ^ { - \\frac 1 2 } } { x _ 1 } \\ll \\sum _ { k = \\lfloor \\log \\frac { e a ^ 2 } { L ^ 2 } \\rfloor } ^ { \\lfloor \\log \\sqrt { a + d L } \\rfloor } k ^ { - 1 } \\leq 2 \\log \\log ( a + d L ) . \\end{align*}"}
+{"id": "8265.png", "formula": "\\begin{align*} U _ 1 : = \\int _ 1 ^ q , ~ U _ 2 : = \\int _ q ^ Q , ~ U _ 3 : = \\int _ Q ^ { \\infty } . \\end{align*}"}
+{"id": "3822.png", "formula": "\\begin{align*} d ^ 2 - 2 n ( g - 1 ) = ( d ' ) ^ 2 - 2 n ' ( g - 1 ) \\hbox { a n d } d \\equiv d ' \\mod 2 g - 2 ; \\end{align*}"}
+{"id": "4720.png", "formula": "\\begin{align*} ( v , w ) = a _ 1 ( v _ 1 , w _ 1 ) + a _ 2 ( v _ 2 , w _ 2 ) + \\cdots + a _ c ( v _ c , w _ c ) + b _ 1 ( x _ 1 , y _ 1 ) + b _ 2 ( x _ 2 , y _ 2 ) + \\cdots + b _ c ( x _ c , y _ c ) + ( f , g ) \\end{align*}"}
+{"id": "5349.png", "formula": "\\begin{align*} d \\rvert _ { \\varepsilon = \\tilde { \\varepsilon } } J _ \\varepsilon [ \\eta ] [ \\tilde { u } ] [ \\tilde { v } ] = \\int _ \\Omega \\eta \\tilde { u } \\cdot \\tilde { v } \\ , d x \\end{align*}"}
+{"id": "1741.png", "formula": "\\begin{align*} \\varphi _ n ( \\mu ) \\left ( \\begin{array} { c c } \\ast & \\ast \\\\ r & s \\end{array} \\right ) : = \\Delta ^ { \\frac { \\underline { k } } { 2 } } \\frac { \\mu \\left ( P _ n \\right ) } { ( | r | ^ 2 + | s | ^ 2 ) ^ { M + n + 2 } } , P _ n ( r , s ) : = \\left | \\begin{array} { c c } \\bar s & - \\bar r \\\\ x & y \\end{array} \\right | ^ { n + \\lambda } \\left | \\begin{array} { c c } r & s \\\\ x & y \\end{array} \\right | ^ { n - \\lambda } . \\end{align*}"}
+{"id": "524.png", "formula": "\\begin{align*} E ( t ) : = \\min \\bigg \\{ \\frac { F _ { \\lambda , \\mu } ( x ^ k ) - F _ { \\lambda , \\mu } ^ { \\rm m i n } } { F _ { \\lambda , \\mu } ( x ^ 0 ) - F _ { \\lambda , \\mu } ^ { \\rm m i n } } \\ , | \\ , k \\in \\big \\{ i \\ ! : T ( i ) \\le t \\big \\} \\bigg \\} . \\end{align*}"}
+{"id": "2497.png", "formula": "\\begin{align*} Z ( { \\mathbf v } h + g ) = { \\mathbf u } \\psi h + \\frac { \\psi } { \\varphi _ 0 } g , \\ \\ \\ h \\in H ^ 2 , \\ g \\in \\mathcal K _ { \\mathbf v } . \\end{align*}"}
+{"id": "1625.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ^ { 2 } _ { \\mathbb { R } ^ d } ) u = \\mu u | u | ^ 2 , u ( 0 , x ) = u _ 0 ( x ) \\in H ^ { d + \\epsilon } ( \\mathbb { R } ^ d ) , \\end{align*}"}
+{"id": "5768.png", "formula": "\\begin{align*} d \\log l _ { { u } } = u _ 1 d \\log x _ 1 + \\cdots + u _ n d \\log x _ n + u _ 0 d \\log ( 1 - x _ 1 - \\cdots - x _ n ) \\end{align*}"}
+{"id": "7683.png", "formula": "\\begin{align*} \\omega ^ { \\prime } = \\sum _ { k } \\lambda _ { k } \\frac { d f ^ { \\prime } _ { k } } { f ^ { \\prime } _ { k } } . \\end{align*}"}
+{"id": "1357.png", "formula": "\\begin{align*} | f _ j ( \\tau _ k ) | ^ 2 = \\gamma , \\forall 1 \\leq j , k \\leq n , j \\neq k . \\end{align*}"}
+{"id": "2219.png", "formula": "\\begin{align*} \\nu ( x _ { 2 m - 1 , 1 } ) & = 0 , \\\\ \\nu ( x _ { 2 m , 1 } ) & = 0 . \\end{align*}"}
+{"id": "2400.png", "formula": "\\begin{align*} U ^ H \\Phi = \\left [ \\begin{array} { c } I _ { 2 p } \\\\ 0 \\end{array} \\right ] . \\end{align*}"}
+{"id": "8610.png", "formula": "\\begin{align*} g _ k ( n ) = g _ { k } ( n - 2 ) + g _ { k - 2 } ( n - 4 ) + \\binom { n - k - 1 } { k - 1 } . \\end{align*}"}
+{"id": "6856.png", "formula": "\\begin{align*} f i n d \\ ; u \\in X \\ ; s u c h \\ ; t h a t \\ ; Q u = F . \\end{align*}"}
+{"id": "6039.png", "formula": "\\begin{align*} \\int x ^ m q _ n ( x ) \\frac { \\dd \\mu ( x ) } { v _ n ( x ) } = 0 , m \\in \\{ 0 , \\ldots , n - 1 \\} , \\end{align*}"}
+{"id": "1292.png", "formula": "\\begin{align*} T _ m \\mu : = \\lim _ { \\ell \\to \\infty } T _ m \\mu _ \\ell , \\end{align*}"}
+{"id": "6174.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } W '' - \\Delta W + \\overline A _ 1 W = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ W = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu W + \\overline B _ 1 W = C _ 1 D H & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "7767.png", "formula": "\\begin{align*} \\widetilde m _ \\alpha ( s | \\mu ) & = \\int _ 0 ^ \\infty \\widetilde f _ \\alpha ( s | y ) f ( y | \\mu ) d y = \\int _ 0 ^ \\infty e ^ { - y s ^ \\alpha } f ( y | \\mu ) d y = \\widetilde f ( s ^ \\alpha | \\mu ) \\end{align*}"}
+{"id": "8971.png", "formula": "\\begin{align*} \\| \\partial _ r u \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } & \\le \\| f \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } + \\| \\partial _ r ( d i s t _ N ( u ) ) \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } \\\\ & \\le \\| f \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } + C \\delta \\| \\nabla u \\| ^ 2 _ { H ^ { 1 / 2 } ( B ) } . \\end{align*}"}
+{"id": "6791.png", "formula": "\\begin{align*} d ( \\mathcal { U } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) , \\mathcal { U } ( ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) ) \\leq ~ c F ^ { * } _ { \\eta } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } , ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) ) , \\end{align*}"}
+{"id": "3537.png", "formula": "\\begin{align*} \\mathbf { p } \\{ B _ i ^ + , B _ j ^ + \\} & = \\sum _ { k = 1 } ^ { i } \\sum _ { s = 1 } ^ { i - k + 1 } \\sum _ { I \\in \\mathcal { I } _ { k i } ( s ) } \\sum _ { l = 1 } ^ { j } \\sum _ { t = 1 } ^ { j - l + 1 } \\sum _ { J \\in \\mathcal { I } _ { l j } ( t ) } E ^ { e _ I + e _ J } \\{ B _ k ^ + , B _ l ^ + \\} H _ { I J } ^ + \\end{align*}"}
+{"id": "4464.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k v _ i \\leq \\prod _ { i = 1 } ^ k u _ i = \\left ( \\prod _ { i = 1 } ^ { \\ell - 2 } u _ i \\right ) \\left ( u ' _ { \\ell - 1 } u ' _ { \\ell } \\right ) \\left ( \\prod _ { i = \\ell + 1 } ^ k u _ i \\right ) = \\prod _ { i = 1 } ^ k u ' _ i . \\end{align*}"}
+{"id": "2558.png", "formula": "\\begin{align*} x _ { 1 } x _ { - 1 } ^ { k ^ { ' } _ 1 - 1 } \\left \\{ \\prod _ { i = 2 } ^ { q } x _ { c ^ { ' } _ { i - 1 } } x _ { 1 } ^ { l _ { i - 1 } } x _ { - 1 } ^ { k ^ { ' } _ { i } - 1 } \\right \\} = z _ { 1 } ( k ^ { ' } _ 1 ) \\left \\{ \\prod _ { i = 2 } ^ { q } z _ { c ^ { ' } _ { i - 1 } } ( 1 ) z _ { 1 } ( 1 ) ^ { l _ { i - 1 } - 1 } z _ { 1 } ( k ^ { ' } _ { i } ) \\right \\} \\end{align*}"}
+{"id": "1051.png", "formula": "\\begin{align*} \\mathcal { R } _ { n , \\alpha } ( \\mathcal { F } _ \\beta , \\Phi \\circ \\rho , \\varepsilon ) = \\inf _ { Q \\in \\mathcal { Q } _ \\alpha } \\inf _ { \\tilde { f } } \\sup _ { P _ { f _ \\varepsilon } \\in \\mathcal { P } _ \\varepsilon ( \\mathcal { F } _ \\beta ) } \\mathbb { E } _ { P _ { f _ \\varepsilon } , Q } \\Big \\{ \\Phi \\circ \\rho ( \\tilde { f } , f ) \\Big \\} , \\end{align*}"}
+{"id": "4847.png", "formula": "\\begin{align*} v ^ \\flat ( t ) = v ( 1 - t ) , t \\in I . \\end{align*}"}
+{"id": "3203.png", "formula": "\\begin{align*} B _ \\eta : = \\{ \\mathrm { v } \\in F : \\| \\mathrm { v } \\| _ { L ^ \\infty ( 0 , T ; L ^ { 3 / 2 } ) \\cap L ^ 1 ( 0 , T ; W ^ { 1 , 1 } ) } \\le \\eta \\} . \\end{align*}"}
+{"id": "8069.png", "formula": "\\begin{align*} \\mathcal { D } = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } \\delta ( n , p ) H _ { m , n } , \\end{align*}"}
+{"id": "7590.png", "formula": "\\begin{align*} M & = \\bigg \\| \\int _ { s } ^ { t } \\big ( G ( t - r , 0 , \\cdot ) * v ^ { 2 \\delta + 1 } ( r , \\cdot ) \\big ) \\d r \\bigg \\| _ { \\L ^ p } , \\end{align*}"}
+{"id": "8999.png", "formula": "\\begin{align*} \\int _ { t _ 0 } ^ 0 & \\int _ { \\partial \\Omega _ k } | \\partial _ t u _ k | ^ 2 d s \\ ; d t = \\int _ { t _ 0 } ^ 0 \\int _ { \\partial \\Omega _ k } | d \\pi _ N ( u _ k ) \\partial _ { \\nu _ k } u _ k | ^ 2 d s \\ ; d t \\\\ & = \\int _ { t _ k + r _ k t _ 0 } ^ { t _ k } \\int _ { \\partial B } | u _ t | ^ 2 d \\phi \\ ; d t \\le \\int _ { t _ k + r _ k t _ 0 } ^ { \\infty } \\int _ { \\partial B } | u _ t | ^ 2 d \\phi \\ ; d t \\to 0 \\end{align*}"}
+{"id": "1531.png", "formula": "\\begin{align*} S [ \\phi ] = \\int j \\phi ^ * \\Theta \\end{align*}"}
+{"id": "1844.png", "formula": "\\begin{align*} \\ddot { \\varrho } + 2 H \\dot { \\varrho } - \\frac { c _ s ^ 2 } { a ^ 2 } \\Delta \\varrho - 4 \\pi \\mathnormal { G } \\rho _ 0 \\varrho = 0 , \\end{align*}"}
+{"id": "1567.png", "formula": "\\begin{align*} g ( X ) : = X ^ r \\Bigl ( X ^ { ( S + T ) ( q - 1 ) } + X ^ { T ( q - 1 ) } + 1 \\Bigr ) \\end{align*}"}
+{"id": "7936.png", "formula": "\\begin{align*} a \\oplus b = \\max \\{ a , b \\} , a \\odot b = a + b , \\end{align*}"}
+{"id": "1125.png", "formula": "\\begin{align*} c _ { ( a ^ i ) ( b ^ j ) } ^ { \\rho } = c _ { ( a ^ i ) ( b ^ j ) } ^ { ( a + 1 , a ^ { i - 1 } , b ^ { j - 1 } , b - 1 ) } \\ge c _ { ( a ^ i ) ( b ) } ^ { ( a + 1 , a ^ { i - 1 } , b - 1 ) } \\ge c _ { ( a ) ( b ) } ^ { ( a + 1 , b - 1 ) } = 1 \\end{align*}"}
+{"id": "4515.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n v _ i \\leq \\sum _ { i = 1 } ^ n u _ i \\end{align*}"}
+{"id": "4031.png", "formula": "\\begin{align*} w _ { \\tau \\beta } = 0 . \\end{align*}"}
+{"id": "578.png", "formula": "\\begin{align*} \\sigma _ { n + 1 } = \\sigma _ n - \\frac { \\sigma _ n ^ 2 } { \\gamma } \\ , ( A ^ { \\rm T } A ) : C \\Delta t + \\mathcal { O } ( \\Delta t ^ 2 ) , \\end{align*}"}
+{"id": "6635.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v = \\Delta v + \\xi \\\\ v | _ { t \\leq 0 } = 0 , \\end{cases} \\end{align*}"}
+{"id": "5391.png", "formula": "\\begin{align*} \\sharp V + \\sharp W & = \\sharp ( V \\cap W ) + \\sharp ( V \\cup W ) \\\\ & \\le \\sharp ( F ( V ) \\cap F ( W ) ) + \\sharp ( F ( V ) \\cup F ( W ) ) \\\\ & = \\sharp F ( V ) + \\sharp F ( W ) \\\\ & = \\sharp V + \\sharp W \\end{align*}"}
+{"id": "4631.png", "formula": "\\begin{align*} m ( v , j , k ) = ( u , l ) \\end{align*}"}
+{"id": "896.png", "formula": "\\begin{align*} \\mathbf { E } \\left [ \\left \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\right \\| _ { F } ^ { 2 } | \\mathcal { X } ^ { 0 } \\right ] \\leq \\left ( 1 - \\mathop { \\min } _ { k = 1 , 2 , \\cdots , l } \\frac { \\lambda _ { \\min } \\left ( \\widehat { \\mathcal { A } } _ { ( k ) } \\widehat { \\mathcal { A } } _ { ( k ) } ^ { H } \\right ) } { \\| \\widehat { \\mathcal { A } } _ { ( k ) } \\| _ { F } ^ { 2 } } \\right ) ^ { t } \\| \\mathcal { X } ^ { 0 } - \\mathcal { X } ^ { \\star } \\| _ { F } ^ { 2 } . \\end{align*}"}
+{"id": "7915.png", "formula": "\\begin{align*} & \\lim _ { \\epsilon \\to 0 } \\limsup _ { t \\to \\infty } \\P ( \\exists u \\in \\mathcal { N } ^ 2 _ t : T ( u ) \\notin [ \\epsilon t , ( 1 - \\epsilon ) t ] , X _ u ( t ) \\geq m _ t - A ) = 0 \\end{align*}"}
+{"id": "6329.png", "formula": "\\begin{align*} N ^ { - m } \\sum _ { i = 1 } ^ { m } \\sum _ { s ' \\in \\mathcal { O } ( l _ { i } ) } W _ { s ' } \\prod _ { j \\neq i } W _ { l _ { j } } = N ^ { - m } \\sum _ { s ' \\in \\mathcal { O } ( s ) } W _ { s ' } \\end{align*}"}
+{"id": "8936.png", "formula": "\\begin{align*} \\gamma _ { p } ^ { ( \\nu , n ) } = \\frac { ( n - 1 ) ! } { p ! } c _ { n - p - 1 } ^ { \\left ( \\nu , n \\right ) } , 0 \\leq p \\leq n - 1 \\end{align*}"}
+{"id": "3931.png", "formula": "\\begin{align*} \\det D Y u _ \\epsilon & = \\det E ^ { - 1 } ( \\cdot , u _ \\epsilon , D u _ \\epsilon ) [ D ^ 2 u + \\epsilon I - A ( \\cdot , u _ \\epsilon , D u _ \\epsilon ) ] , \\end{align*}"}
+{"id": "4922.png", "formula": "\\begin{align*} \\mathbf { Z } ' ( t ) = - \\mathrm i \\mathcal { F } ( \\mathbf { Z } ( t ) , \\mathbf { Z } ( t ) ) , \\end{align*}"}
+{"id": "4231.png", "formula": "\\begin{align*} \\phi ( t , x ) = \\hat { \\phi } - \\epsilon \\mbox { w i t h } \\epsilon = C _ 1 ( \\lambda ) . \\end{align*}"}
+{"id": "1628.png", "formula": "\\begin{align*} u _ { n l } = F _ { 1 } + F _ { 2 } + F _ { 3 } \\end{align*}"}
+{"id": "8814.png", "formula": "\\begin{align*} A - B ^ * P = D _ P F _ 1 D _ P , B - A ^ * P = D _ P F _ 2 D _ P . \\end{align*}"}
+{"id": "9076.png", "formula": "\\begin{align*} N _ { P } \\cap w N _ { Q } w ^ { - 1 } = N _ { Q } \\cap w N _ { Q } w ^ { - 1 } . \\end{align*}"}
+{"id": "802.png", "formula": "\\begin{align*} A _ \\epsilon = \\left \\{ g \\in \\Gamma : \\left | \\frac { \\log \\| \\rho ( g ) \\| } { | g | _ S } - \\Lambda \\right | > \\epsilon \\right \\} . \\end{align*}"}
+{"id": "8205.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty g ( a ) \\nu _ t ( d a ) = \\int _ 0 ^ t k ( t , s ) \\int _ 0 ^ \\infty g ( a ) \\mathcal { P } _ { A ( s ) } ( d a ) d s , \\end{align*}"}
+{"id": "8355.png", "formula": "\\begin{align*} \\lim _ { t \\to \\pm \\infty } \\int _ { | x | > | t | } \\left ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 \\right ) \\mathrm { d } x = 0 . \\end{align*}"}
+{"id": "4255.png", "formula": "\\begin{align*} \\Phi ( x ) : = \\pi ^ { - \\frac { N } { 4 } } \\left ( \\prod _ { j = 1 } ^ N \\sqrt { \\gamma _ j } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { - \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ N \\gamma _ j x _ j ^ 2 } . \\end{align*}"}
+{"id": "7438.png", "formula": "\\begin{align*} { \\mathbb K } ( Z , W ) ( P ) = \\begin{bmatrix} { \\mathbb L } _ 1 ( Z , W ) ( P ) & K ( Z , W ) ( P ) \\\\ K ( Z , W ) ( P ) & { \\mathbb L } _ 2 ( Z , W ) ( P ) \\end{bmatrix} \\end{align*}"}
+{"id": "1121.png", "formula": "\\begin{align*} \\alpha : ( w , \\theta ) \\mapsto ( \\alpha ( w ) , \\alpha \\theta \\alpha ^ { - 1 } ) = ( 1 , \\alpha ) ( w , \\theta ) ( 1 , \\alpha ) ^ { - 1 } . \\end{align*}"}
+{"id": "5642.png", "formula": "\\begin{align*} & [ 0 , 1 ] ^ { | V ( F ) | - | | S _ 0 ( F ) | | } \\ni ( z _ { v _ 1 } , \\cdots , z _ { v _ p } ) \\in \\Delta _ F \\setminus S _ 0 ( F ) \\\\ : \\Longleftrightarrow & \\left ( \\{ v _ 1 , \\cdots , v _ p \\} = V ( F ) \\setminus \\{ v ' \\in s _ v | s _ v \\in S _ 0 ( F ) \\} ~ \\wedge ~ ( v _ i \\preceq v _ j \\Leftrightarrow z _ { v _ j } \\leq z _ { v _ i } ) \\right ) . \\end{align*}"}
+{"id": "3281.png", "formula": "\\begin{align*} \\lambda = \\inf \\left \\{ \\frac { \\left \\langle f \\left ( L x \\right ) , L x \\right \\rangle } { \\left \\langle g \\left ( x \\right ) , L x \\right \\rangle } \\left \\vert \\ x \\in X \\right . \\right \\} . \\end{align*}"}
+{"id": "2781.png", "formula": "\\begin{align*} x _ i = \\ ! \\left \\{ \\ ! \\begin{aligned} & x ' _ i , ~ ~ ~ ~ ~ i \\in [ 1 , d _ 1 - 1 ] , \\\\ & x ' _ { i - 1 } , ~ ~ i \\in [ d _ 1 + 1 , d _ 2 ] \\backslash \\{ \\lambda _ 1 \\} , \\\\ & x ' _ i , ~ ~ ~ ~ ~ i \\in [ d _ 2 + 1 , n ] \\backslash \\{ \\lambda _ 2 \\} . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2139.png", "formula": "\\begin{align*} \\begin{cases} J \\ast U - U - c U ' + f ( U ) = 0 , \\ , \\R , \\\\ U ( - \\infty ) = 0 , U ( + \\infty ) = 1 , \\end{cases} \\end{align*}"}
+{"id": "11.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d Q _ t & = \\Big ( \\bar { f } + e ^ { - \\frac { \\beta } { 2 } t } \\bar { h } \\tilde { M } _ t - \\beta Q _ t \\Big ) d t - \\tilde { M } _ t d \\xi _ t , \\\\ Q _ T & = 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "887.png", "formula": "\\begin{align*} \\| \\mathcal { X } ^ { t + 1 } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } & = \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } - \\mathop { \\max } _ { i ^ { t } = 1 , 2 , \\dots , q } f _ { i ^ { t } } ( \\mathcal { X } ^ { t } ) \\leq ( 1 - \\delta _ { \\infty } ^ { 2 } ( \\mathcal { Q } , \\boldsymbol { \\mathcal { S } } ) ) \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } . \\end{align*}"}
+{"id": "9264.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { g } _ 1 & = ( x _ 1 - 1 ) ( x _ 1 - x _ 2 - 1 ) \\cdots ( x _ 1 - x _ { h ( 1 ) } - 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "5513.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n } \\exp \\left ( { - \\frac { x } { n ^ 2 } } \\right ) = \\sqrt { \\frac { \\pi } { x } } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n } \\exp \\left ( - \\frac { \\pi ^ 2 } { n ^ 2 x } \\right ) , \\end{align*}"}
+{"id": "7634.png", "formula": "\\begin{align*} \\| A ^ n x _ n \\| _ { \\beta } \\leq \\beta ^ n \\| x _ n \\| _ { \\beta } \\leq K \\beta ^ n \\qquad A ^ n x _ n = \\big ( ( M _ \\varphi ^ { - 1 } ) ^ { \\mathrm { T } } \\big ) ^ n ( M ^ n _ \\varphi ) ^ { \\mathrm { T } } \\vec { f } = \\vec { f } n \\in \\N . \\end{align*}"}
+{"id": "173.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq n \\\\ p \\notin \\mathcal { S } } } \\beta _ p \\log p \\leq \\sum _ { \\substack { p \\leq n \\\\ p \\in \\mathcal { S } } } \\log ( 2 n ^ 2 + l ) + \\dfrac { 1 } { \\lambda } \\sum _ { i = 1 } ^ { r } e _ i \\log p _ i + \\sum _ { n < p < 2 n } \\dfrac { \\alpha _ p } { \\lambda } \\log p . \\end{align*}"}
+{"id": "79.png", "formula": "\\begin{align*} v _ n ^ { ( i ) } = \\Psi ^ { - 1 } \\big ( u _ n ^ { ( i ) } \\big ) . \\end{align*}"}
+{"id": "1203.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { K } r ^ k | \\nabla ^ k _ { g _ C } \\Psi _ * | _ { g _ C } \\leq A r ^ \\epsilon . \\end{align*}"}
+{"id": "1230.png", "formula": "\\begin{align*} \\mu _ i ( \\{ ( x _ i ^ 1 , x _ i ^ 2 , . . . x _ i ^ m ) : x _ i ^ 1 \\leq z _ i ( y ) \\} ) = \\nu ( - \\infty , y ) . \\end{align*}"}
+{"id": "6110.png", "formula": "\\begin{align*} m ( j _ i ) _ { n _ i } = \\left ( \\dfrac { n _ i ^ { j _ i } } { j _ i ! } \\right ) \\ln \\left ( \\sum _ { h \\geq 0 } \\dfrac { \\hat { a } _ h ( j _ i , \\{ \\epsilon _ t \\} ) } { n _ i ^ h } \\right ) . \\end{align*}"}
+{"id": "2306.png", "formula": "\\begin{align*} u ( x , 0 ) = \\begin{cases} 1 , & - 4 < x < - 1 , \\\\ 2 , & 0 < x < 3 , \\\\ 0 , & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
+{"id": "7347.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & 2 ( s - 1 ) / \\alpha + t \\geq 0 & ( \\alpha > 2 ) , \\\\ & t > 0 s + t > 0 & ( \\alpha = 2 ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3546.png", "formula": "\\begin{align*} B _ j ^ + = \\mathbf { p } B _ j ^ + + \\sum _ { i = 1 } ^ { j - 1 } \\sum _ { s = 2 } ^ { j - i + 1 } \\sum _ { I \\in \\mathcal { I } _ { i j } ( s ) } E ^ { e _ I } \\mathbf { p } B _ i ^ + \\frac { \\prod _ { \\ell \\in I ^ \\complement } ( h _ i - h _ \\ell + 1 ) } { \\prod _ { \\ell = i + 1 } ^ { j } ( h _ i - h _ \\ell ) } \\end{align*}"}
+{"id": "5183.png", "formula": "\\begin{align*} P _ \\ell = \\frac { r - ( m _ 2 ^ { - 1 } m _ 1 ( \\ell + 1 ) + r k _ \\ell - h ) } { K } - 1 \\end{align*}"}
+{"id": "5361.png", "formula": "\\begin{align*} \\eta _ h : = \\norm { \\xi } _ { W ^ { 1 , \\infty } ( \\Omega ) } ^ { - 1 } \\xi \\ , e _ { h h } \\end{align*}"}
+{"id": "3031.png", "formula": "\\begin{align*} \\mathcal { F } ^ { s , j } _ { Q _ k } = \\emptyset \\ \\ \\mathrm { f o r } \\ \\ j > s _ 0 \\ \\ \\mathrm { o r } \\ \\ s < s _ 0 - j - 6 \\end{align*}"}
+{"id": "5609.png", "formula": "\\begin{align*} \\nabla _ { [ a } S _ { b ] c } + \\frac { 1 } { 1 2 } \\nabla _ { [ a } R g _ { b ] c } = 0 , \\end{align*}"}
+{"id": "8270.png", "formula": "\\begin{align*} F ( x ) : = \\int _ 1 ^ x d f ( y ) \\textrm { w h e r e } f ' ( y ) : = \\frac { 1 - 1 / y } { \\log y } f ' _ 0 ( y ) , \\textrm { i . e . } d f ( y ) : = \\frac { 1 - 1 / y } { \\log y } d f _ 0 ( y ) \\end{align*}"}
+{"id": "1345.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | \\geq \\sqrt { \\frac { \\frac { 1 } { ( G _ { f , \\tau } ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ 2 } { n ^ 2 - n } } . \\end{align*}"}
+{"id": "721.png", "formula": "\\begin{align*} 2 \\pi \\left ( \\begin{array} { c c } R _ { \\rm r o b i n } ( w _ 1 ) & G ( w _ 1 , w _ 2 ) \\\\ G ( w _ 2 , w _ 1 ) & R _ { \\rm r o b i n } ( w _ 2 ) \\\\ \\end{array} \\right ) = \\end{align*}"}
+{"id": "6785.png", "formula": "\\begin{align*} \\frac { \\left | z \\left ( \\left ( \\frac { 1 } { 2 } - \\sigma \\right ) + j \\omega \\right ) \\right | } { \\left | z \\left ( \\left ( \\frac { 1 } { 2 } + \\sigma \\right ) - j \\omega \\right ) \\right | } = \\frac { \\left | \\left ( \\frac { 3 } { 2 } - \\sigma \\right ) + j \\omega \\right | } { \\left | \\left ( \\frac { 3 } { 2 } + \\sigma \\right ) - j \\omega \\right | } \\end{align*}"}
+{"id": "5793.png", "formula": "\\begin{align*} ( C _ p C _ p ^ T ) ^ { - 1 / 2 } C _ p D = \\overline D _ p ( C _ p C _ p ^ T ) ^ { - 1 / 2 } C _ p . \\end{align*}"}
+{"id": "6021.png", "formula": "\\begin{align*} \\O _ { h _ i } \\star \\O _ { i _ 2 , j _ 2 } = \\O _ { h _ i } \\cdot \\O _ v + Q _ 1 P _ { l _ 1 } ( i , v ) + Q _ 2 P _ { l _ 2 } ( i , v ) + Q _ 1 Q _ 2 P _ { l _ 1 + l _ 2 } ( i , v ) , \\end{align*}"}
+{"id": "1000.png", "formula": "\\begin{align*} f _ 1 \\big ( \\tfrac 1 3 , y \\big ) = f _ 1 \\big ( \\tfrac 2 3 , y \\big ) = \\sin ( 2 \\pi y ) . \\end{align*}"}
+{"id": "7991.png", "formula": "\\begin{align*} \\pi : A \\to A ' = A / C . \\end{align*}"}
+{"id": "4420.png", "formula": "\\begin{align*} J ( a _ 1 , a _ 2 ) \\subseteq \\left ( \\frac { 1 } { a _ 1 } , \\frac { 1 } { a _ 1 - 1 } \\right ] = I ( a _ 1 ) . \\end{align*}"}
+{"id": "3407.png", "formula": "\\begin{align*} \\Delta \\zeta _ { \\alpha } ( t , x ) = \\mathcal { R } _ { \\alpha } ^ { - 1 } ( \\Delta \\zeta ) ( t , \\mathcal { R } _ { \\alpha } x ) . \\end{align*}"}
+{"id": "8758.png", "formula": "\\begin{align*} & d _ 2 ( ( x _ 1 , v _ 1 , w _ 1 ) , ( x _ 2 , v _ 2 , w _ 2 ) ) : = | \\mathcal { H } ( x _ 1 , v _ 1 , w _ 1 ) - \\mathcal { H } ( x _ 2 , v _ 2 , w _ 2 ) | _ { S _ 2 } . \\end{align*}"}
+{"id": "7884.png", "formula": "\\begin{align*} g '' ( z ) - ( 2 \\gamma e ^ { z / 2 } + 1 ) g ' ( z ) + \\gamma e ^ { z / 2 } g ( z ) = 0 , \\end{align*}"}
+{"id": "8950.png", "formula": "\\begin{align*} { } _ p F _ q \\left ( \\begin{matrix} a _ { 1 } , \\cdots , a _ { p } \\\\ c _ { 1 } , \\cdots , c _ { q } \\end{matrix} \\bigg | \\xi \\right ) = \\sum _ { k = 0 } ^ { + \\infty } \\frac { ( a _ { 1 } ) _ { k } \\cdots ( a _ { p } ) _ { k } } { ( c _ { 1 } ) _ { k } \\cdots ( c _ { q } ) _ { k } } \\frac { \\xi ^ { k } } { k ! } , \\end{align*}"}
+{"id": "9146.png", "formula": "\\begin{align*} \\frac { N _ n ( | \\eta | ) } { n ! } = \\frac { m ( m + n ) ^ { n - 1 } } { n ! } < \\frac { m ( m + n ) ^ { n - 1 } } { n ^ n e ^ { - n } \\sqrt { 2 \\pi n } } = \\frac { m } { m + n } \\frac { 1 } { \\sqrt { 2 \\pi n } } e ^ n ( 1 + \\frac { m } { n } ) ^ n < e ^ { m + n } . \\end{align*}"}
+{"id": "985.png", "formula": "\\begin{align*} \\hat f ( \\psi _ i ) = \\psi _ i ( f ) & = x _ 0 \\otimes \\varphi _ i ( f ) \\\\ & = \\varphi _ i ( f ( x _ 0 ) ) = \\varphi _ i ( a ) \\\\ & = \\hat a ( \\varphi _ i ) \\ , \\ , \\ , \\ , ( 1 \\leq i \\leq n ) \\end{align*}"}
+{"id": "5298.png", "formula": "\\begin{align*} g ( X , Y ) = ( { \\tilde \\lambda } \\circ F ) ^ 2 g ' ( F _ * X , F _ * Y ) , \\end{align*}"}
+{"id": "555.png", "formula": "\\begin{align*} C = - \\gamma ( A + A ^ { \\rm T } ) ^ { - 1 } . \\end{align*}"}
+{"id": "170.png", "formula": "\\begin{align*} \\xi ( \\varkappa ) ( \\theta ) ( \\hslash ) : = 0 . \\end{align*}"}
+{"id": "2888.png", "formula": "\\begin{align*} z \\star \\zeta = ( z \\zeta _ 1 , \\ldots , z ^ n \\zeta _ n , \\ldots ) \\in \\overline { \\mathbb { D } } ^ \\infty . \\end{align*}"}
+{"id": "7901.png", "formula": "\\begin{align*} f ( z ) ^ 2 + a _ 1 f ( z ) + q ( z ) e ^ { Q ( z ) } f ' ( z ) = 0 , \\end{align*}"}
+{"id": "2075.png", "formula": "\\begin{align*} \\Pr _ \\theta [ D ( \\theta ) \\leq \\tfrac { \\lambda n } { 4 } ] & \\leq \\Pr _ \\theta \\Big [ ( | G | - 2 D ( \\theta ) ) ^ { 2 \\ell } \\geq ( \\tfrac { \\lambda n } { 2 } ) ^ { 2 \\ell } \\Big ] \\leq \\frac { 2 ^ { - 2 \\ell } r _ \\ell ( A ) } { ( \\lambda n / 2 ) ^ { 2 \\ell } } = \\frac { r _ \\ell ( A ) } { ( \\lambda n ) ^ { 2 \\ell } } . \\end{align*}"}
+{"id": "8772.png", "formula": "\\begin{align*} W _ { u , v , w , k } = \\begin{bmatrix} & & 0 & w _ { u , v , w , 2 T - 1 } ^ * & w _ { u , v , w , 2 T - 2 } ^ * & \\cdots & w _ { u , v , w , 1 } ^ * & & \\end{bmatrix} , \\end{align*}"}
+{"id": "7907.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } w _ { x x } + \\sqrt { 2 } w _ x + w ( w - 1 ) = 0 . \\end{align*}"}
+{"id": "3924.png", "formula": "\\begin{align*} \\det [ D ^ 2 u - A ( \\cdot , u , D u ) ] & = B ( \\cdot , u , D u ) \\Omega \\\\ u & = \\phi \\partial \\Omega . \\end{align*}"}
+{"id": "6026.png", "formula": "\\begin{align*} P _ { l _ 1 + l _ 2 } ( h _ 1 , w _ { k , p } ) & = Q _ 1 Q _ 2 \\left ( \\O _ { n , 1 } - \\O _ { n , 1 } - \\O _ { n , 1 } + \\O _ { n , 1 } + \\O _ { n , 1 } - \\O _ { n , 1 } \\right ) \\\\ & = 0 & \\mathrm { i f } \\ : k \\neq 1 . \\end{align*}"}
+{"id": "134.png", "formula": "\\begin{align*} \\mathcal L ^ { 2 n } ( \\widetilde E _ { \\lambda , K } ) = 2 \\mathcal L ^ { 2 n } ( H _ { \\lambda , K } ^ + ) . \\end{align*}"}
+{"id": "2232.png", "formula": "\\begin{align*} E ( q ) = E ( q ^ { 2 5 } ) { \\left ( \\dfrac { 1 } { R ( q ^ 5 ) } - q - q ^ 2 R ( q ^ 5 ) \\right ) } , \\end{align*}"}
+{"id": "1668.png", "formula": "\\begin{align*} C _ \\sigma = \\left \\{ \\begin{array} { l l } ( - 1 ) ^ { m _ { \\sigma _ 1 } + m _ { \\sigma _ 2 } } 1 6 ( 2 \\pi ) ^ { 1 - k _ { \\sigma _ 1 } - k _ { \\sigma _ 2 } } \\Gamma ( \\frac { k _ { \\sigma _ 1 } } { 2 } - m _ { \\sigma _ 1 } ) \\Gamma ( \\frac { k _ { \\sigma _ 1 } } { 2 } + m _ { \\sigma _ 1 } ) \\Gamma ( \\frac { k _ { \\sigma _ 2 } } { 2 } - m _ { \\sigma _ 2 } ) \\Gamma ( \\frac { k _ { \\sigma _ 2 } } { 2 } + m _ { \\sigma _ 2 } ) , & \\chi _ { 0 , \\sigma } = 1 ; \\\\ 0 , & \\chi _ { 0 , \\sigma } \\neq 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "3427.png", "formula": "\\begin{align*} p ( a , b ) = \\frac { h \\left ( b \\right ) } { h ( a ) } \\exp \\left ( \\phi ( a b ) - P _ { t o p } ( \\phi ) \\right ) . \\end{align*}"}
+{"id": "9063.png", "formula": "\\begin{align*} \\widehat { w } _ { 2 } ( x , \\tau ) = : \\sum _ { l = 0 } ^ { \\infty } \\sum _ { m = 1 } ^ { d _ { l } } \\varphi _ { l m } ( \\rho , \\tau ) Y _ { l m } ( \\omega ) \\ , , \\end{align*}"}
+{"id": "4916.png", "formula": "\\begin{align*} P _ \\ell ( Z _ m ) = | f _ \\ell ( Z _ m ) | ^ { p _ \\ell } , Q _ \\ell ^ + ( Z _ m ) = | g _ \\ell ^ + ( \\delta _ m ^ + Z _ m ) | ^ { q ^ + _ \\ell } , Q _ \\ell ^ - ( Z _ m ) = | g _ \\ell ^ - ( \\delta _ m ^ - Z _ m ) | ^ { q ^ - _ \\ell } , \\end{align*}"}
+{"id": "3981.png", "formula": "\\begin{align*} u _ 1 & : \\Omega _ \\delta \\rightarrow \\mathbf { R } , \\\\ u _ 1 ( x ) & = \\sup \\{ g ( \\cdot , y , z ) ; y \\in \\Omega ^ * , g ( \\cdot , y , z ) \\leq u \\Omega \\} . \\end{align*}"}
+{"id": "7338.png", "formula": "\\begin{align*} W ( x ) : = \\exp \\left ( \\int _ { x _ 0 } ^ { x } b ( y ) d y \\right ) = x ^ { \\delta - 1 } c ( x ) \\exp \\left ( \\int _ { x _ 0 } ^ { x } \\frac { \\epsilon ( y ) } { y } d y \\right ) \\end{align*}"}
+{"id": "2024.png", "formula": "\\begin{align*} \\overset { \\sim } { \\lambda } \\overset { \\sim } { P } ( C ' ) & = \\displaystyle \\sum _ { C } \\overset { \\sim } { \\lambda } ( C ) \\overset { \\sim } { P } ( C , C ' ) \\\\ & = \\displaystyle \\sum _ { C } \\displaystyle \\int _ { C } P ( x , C ' ) \\lambda ( d x ) \\\\ & = \\displaystyle \\int _ { X } P ( x , C ' ) \\lambda ( d x ) \\\\ & = \\overset { \\sim } { \\lambda } ( C ' ) . \\end{align*}"}
+{"id": "8715.png", "formula": "\\begin{align*} \\frac { 1 } { x ^ n } & = \\sum _ { k = 1 } ^ { \\infty } ( - 1 ) ^ { n - k } e _ { k - n } ( a _ 1 , \\dots , a _ { k - 1 } ) \\frac { 1 } { ( x | a ) _ { k } } \\\\ & = \\sum _ { k , r = 1 } ^ { \\infty } ( - 1 ) ^ { n - k } e _ { k - n } ( a _ 1 , \\dots , a _ { k - 1 } ) h _ { r - k } ( a _ 1 , \\dots , a _ { k } ) \\frac { 1 } { x ^ r } = \\sum _ { k , r = 1 } ^ { \\infty } A _ { - k , - n } A _ { r , k } \\frac { 1 } { x ^ r } . \\end{align*}"}
+{"id": "843.png", "formula": "\\begin{align*} \\mathbb { F } ^ { - } L _ d ( q _ k , q _ { k + 1 } ) \\mapsto ( q _ { k } , p _ { k } ) = ( q _ k , - D _ 1 L _ d ( q _ k , q _ { k + 1 } ) ) \\end{align*}"}
+{"id": "2410.png", "formula": "\\begin{align*} O ( p , q ) = \\{ \\Phi \\in { \\mathbb R } ^ { n , n } \\mid \\Phi ^ T S \\Phi = S \\} , \\end{align*}"}
+{"id": "7559.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 2 ) = & ( 1 - q ^ { - 1 } ) q ^ { - \\omega - 1 - r s } Z _ { f _ 2 } ( s , \\chi ) \\\\ = & \\dfrac { F _ 2 ( q ^ { - s } ) } { ( 1 - q ^ { - 1 - s } ) ( 1 - q ^ { - p - r - p r s } ) } \\end{align*}"}
+{"id": "3623.png", "formula": "\\begin{align*} f ( s , t ) : = e ^ { \\mu t } s \\big ( e ^ { \\beta s ^ { p } } - 1 \\big ) , \\end{align*}"}
+{"id": "6424.png", "formula": "\\begin{align*} \\Gamma _ X ^ \\epsilon & : = \\{ X \\in \\mathbb R ^ m \\setminus U _ X : \\ d ( X , \\partial U _ X ) \\leq \\epsilon \\} , \\end{align*}"}
+{"id": "5909.png", "formula": "\\begin{align*} V _ r = \\{ \\varepsilon _ { n _ { r - 1 } + 1 } , \\cdots , \\varepsilon _ { n _ { r } } \\} , 1 \\leq r \\leq p . \\end{align*}"}
+{"id": "1482.png", "formula": "\\begin{align*} \\sin x = x \\prod _ { n = 1 } ^ { \\infty } \\left ( 1 - \\frac { x ^ 2 } { n ^ 2 \\pi ^ { 2 } } \\right ) . \\end{align*}"}
+{"id": "517.png", "formula": "\\begin{align*} \\tau _ { k , 0 } = \\max \\Big \\{ \\min \\Big \\{ \\frac { \\| \\Delta z ^ k \\| ^ 2 } { \\langle \\Delta z ^ k , \\Delta \\zeta ^ k \\rangle } , \\frac { \\langle \\Delta z ^ k , \\Delta \\zeta ^ k \\rangle } { \\| \\Delta \\zeta ^ k \\| ^ 2 } , { \\tau } _ { \\rm m a x } \\Big \\} , { \\tau } _ { \\rm m i n } \\Big \\} , \\end{align*}"}
+{"id": "7931.png", "formula": "\\begin{align*} \\mathcal L ^ { 0 } _ { \\mathcal P , t _ { 0 } } : = \\langle \\{ \\textnormal { R e } ( M _ { 0 } ^ { n } ) : n \\in \\mathbb Z \\} \\rangle + \\langle \\{ \\textnormal { I m } ( M _ { 0 } ^ { n } ) i : n \\in \\mathbb Z \\} \\rangle \\subset \\mathcal L _ { \\mathcal P , t _ { 0 } } \\end{align*}"}
+{"id": "6745.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\left | x _ 1 - x _ 2 \\right | } { \\partial x _ 1 \\partial x _ 2 } = - 2 \\delta \\left ( x _ 1 - x _ 2 \\right ) \\end{align*}"}
+{"id": "8571.png", "formula": "\\begin{align*} \\begin{aligned} \\langle & ( 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , 1 7 , 1 8 , 1 9 , 2 0 , 2 1 , 2 2 ) , \\\\ & ( 0 , 2 3 ) ( 1 , 2 2 ) ( 2 , 1 1 ) ( 3 , 1 5 ) ( 4 , 1 7 ) ( 5 , 9 ) ( 6 , 1 9 ) ( 7 , 1 3 ) ( 8 , 2 0 ) ( 1 0 , 1 6 ) ( 1 2 , 2 1 ) ( 1 4 , 1 8 ) , \\\\ & ( 2 , 1 6 , 9 , 6 , 8 ) ( 3 , 1 2 , 1 3 , 1 8 , 4 ) ( 7 , 1 7 , 1 0 , 1 1 , 2 2 ) ( 1 4 , 1 9 , 2 1 , 2 0 , 1 5 ) \\rangle \\end{aligned} \\end{align*}"}
+{"id": "1855.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { H ^ s ( \\mathbb { T } ^ n ) } : = \\biggl ( \\sum _ { \\xi \\in \\mathbb { Z } ^ n } \\lvert \\hat { u } ( \\xi ) \\rvert ^ { 2 } ( 1 + \\lvert \\xi \\rvert ^ { 2 } ) ^ s \\biggr ) ^ { 1 / 2 } < \\infty , \\end{align*}"}
+{"id": "6497.png", "formula": "\\begin{align*} \\| \\tilde { f } ^ { L _ s } \\| _ 2 ^ 2 & \\geq \\| f \\| _ 2 ^ 2 / 2 - R ^ 2 2 ^ { - 2 L _ s s } \\gtrsim C _ 0 \\sqrt { \\log \\log ( n ) } \\rho _ s ^ 2 \\\\ & = \\frac { C _ 0 2 ^ { 3 L _ s / 2 } \\sqrt { \\log \\log n } } { 2 n ( \\frac { b } { \\log ( n ) } \\wedge 2 ^ { L _ s } ) } \\gtrsim \\frac { C _ 0 2 ^ { L _ s } \\sqrt { \\log \\log n } } { n b ' \\frac { m ' } { m } } , \\end{align*}"}
+{"id": "8944.png", "formula": "\\begin{align*} ( a ) _ { k } = a ( a + 1 ) \\dots ( a + k - 1 ) , k \\in \\mathbb { N } , \\end{align*}"}
+{"id": "4525.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n v _ i \\leq \\sum _ { i = 1 } ^ n u _ i - \\left ( v _ { n + 1 } - u _ { n + 1 } \\right ) \\leq \\sum _ { i = 1 } ^ n u _ i . \\end{align*}"}
+{"id": "4826.png", "formula": "\\begin{align*} D _ 0 ^ n F ( v _ 1 , \\dots , v _ n ) = \\frac { d ^ n } { d t ^ n } F \\Big ( \\sum _ { h = 1 } ^ n \\frac { t ^ h v _ h } { h ! } \\Big ) \\bigg | _ { t = 0 } , v _ 1 , \\dots , v _ n \\in X . \\end{align*}"}
+{"id": "4853.png", "formula": "\\begin{align*} d _ 0 G ( w _ { 2 ; t _ 0 , s } ) = - d _ 0 ^ 2 G ( v _ { t _ 0 , s } , v _ { t _ 0 , s } ) = O ( s ^ 3 ) , \\end{align*}"}
+{"id": "9217.png", "formula": "\\begin{align*} \\Lambda _ { \\pm } = \\{ ( x , \\nabla \\phi _ { \\pm } ( x ) ) ; \\ x \\in \\R ^ { d } \\} . \\end{align*}"}
+{"id": "449.png", "formula": "\\begin{align*} \\| f _ j \\| = a _ j , \\| \\tau _ j \\| = b _ j , | f _ j ( \\tau _ j ) | = c _ j , \\forall 1 \\leq j \\leq n \\end{align*}"}
+{"id": "7592.png", "formula": "\\begin{align*} P & = \\int _ { 0 } ^ { t } \\bigg \\{ \\int _ { 0 } ^ { 1 } \\chi _ { \\{ x + z \\in [ 0 , 1 ] \\} } \\bigg | \\int _ { 0 } ^ { 1 } \\chi _ { \\{ | x - y | \\leq | z | \\} } | G ( t - s , x + z , y ) - G ( t - s , x , y ) | \\\\ & \\times v ^ { 2 \\delta + 1 } ( s , y ) \\d y \\bigg | ^ p \\d x \\bigg \\} ^ { \\frac { 1 } { p } } \\d s , \\end{align*}"}
+{"id": "3527.png", "formula": "\\begin{align*} f u : = \\sum _ { s , t } E _ { - \\alpha _ 1 } ^ { s _ 1 } \\cdots E _ { - \\alpha _ m } ^ { s _ m } E _ { \\alpha _ m } ^ { t _ m } \\cdots E _ { \\alpha _ 1 } ^ { t _ 1 } q _ { s , t } u , \\end{align*}"}
+{"id": "5614.png", "formula": "\\begin{align*} g ( R ( X , Y ) Z , W ) = g ( \\hat { R } ( X , Y ) Z , W ) + g ( A ( X , Z ) , A ( Y , W ) ) - g ( A ( Y , Z ) , A ( X , W ) ) , \\end{align*}"}
+{"id": "5690.png", "formula": "\\begin{align*} \\mathbf { L } _ { b } ^ { a } \\eta _ { a c } \\mathbf { L } _ { d } ^ { c } = \\eta _ { b d } . \\end{align*}"}
+{"id": "8642.png", "formula": "\\begin{align*} L = C \\frac { 1 } { \\sqrt { \\rho a } } \\end{align*}"}
+{"id": "4901.png", "formula": "\\begin{align*} \\partial ^ 2 _ s \\rho = 1 - e ^ { - \\rho } - b . \\end{align*}"}
+{"id": "3584.png", "formula": "\\begin{align*} y _ { 1 2 } & = E _ { 1 2 } \\\\ y _ { 1 3 } & = E _ { 1 3 } \\\\ y _ { 2 3 } & = E _ { 2 3 } ( h _ 2 - h _ 1 ) + E _ { 2 1 } E _ { 1 3 } \\\\ y _ { 2 1 } & = E _ { 2 1 } \\\\ y _ { 3 1 } & = E _ { 3 1 } ( h _ 1 - h _ 2 ) + E _ { 2 1 } E _ { 3 2 } \\\\ y _ { 3 2 } & = E _ { 3 2 } . \\end{align*}"}
+{"id": "667.png", "formula": "\\begin{align*} \\omega = d \\nu = ( \\frac { \\partial \\nu _ y } { \\partial x } - \\frac { \\partial \\nu _ x } { \\partial y } ) \\ , d x \\wedge d y . \\end{align*}"}
+{"id": "6819.png", "formula": "\\begin{gather*} L ( q , \\dot { q } ) = \\begin{bmatrix} q ^ { [ 2 ] } & 0 \\end{bmatrix} \\begin{bmatrix} \\dot { q } ^ { [ 1 ] } \\\\ \\dot { q } ^ { [ 2 ] } \\end{bmatrix} + c o s ( q ^ { [ 1 ] } ) - \\frac { ( q ^ { [ 2 ] } ) ^ 2 } { 2 } . \\end{gather*}"}
+{"id": "102.png", "formula": "\\begin{align*} A s ( x , y , z ) _ \\ast & = ( x \\ast y ) \\ast z - x \\ast ( y \\ast z ) \\\\ & = ( - ( x u ) y ) \\ast z - x \\ast ( - ( y u ) z ) = ( ( ( x u ) y ) u ) z - ( x u ) ( ( y u ) z ) . \\\\ \\end{align*}"}
+{"id": "3154.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( A ) = c _ 1 ^ { 2 2 } ( A ) = - \\frac { 1 } { 1 2 8 \\pi } , c _ 2 ^ { 1 1 } ( A ) = c _ 2 ^ { 2 2 } ( A ) = 0 , \\end{align*}"}
+{"id": "6122.png", "formula": "\\begin{align*} \\mathcal R = ( \\mathcal R _ { ( p , q , \\cdots , r , s ) } , \\mathcal R _ { ( p ' , q ' , \\cdots , r ' , s ' ) } , \\cdots ) \\end{align*}"}
+{"id": "8834.png", "formula": "\\begin{align*} \\Delta _ { 0 } : = 4 ( 1 + \\langle z , z \\rangle ) \\sum _ { i , j = 1 } ^ { n } ( \\delta _ { i j } + z _ { i } \\bar { z } _ { j } ) \\frac { \\partial ^ { 2 } } { \\partial z _ { i } \\partial \\bar { z } _ { j } } , \\end{align*}"}
+{"id": "6188.png", "formula": "\\begin{align*} \\overline x _ l ^ { ( k ) } = C _ p x _ l ^ { ( k ) } , 1 \\leqslant k \\leqslant \\overline d , 1 \\leqslant l \\leqslant \\overline r _ k , \\end{align*}"}
+{"id": "6995.png", "formula": "\\begin{align*} P _ { e _ x } = \\mathbb { E } \\left [ \\frac { 1 } { n } \\sum _ { k = 1 } ^ n d _ x ( X _ k , \\hat { X } _ k ) \\right ] , P _ { e _ y } = \\mathbb { E } \\left [ \\frac { 1 } { n } \\sum _ { k = 1 } ^ n d _ y ( Y _ k , \\hat { Y } _ k ) \\right ] . \\end{align*}"}
+{"id": "1261.png", "formula": "\\begin{align*} R _ { \\mathcal { B S } ( \\alpha ) } ( M ( \\beta ) ) & = \\begin{dcases} \\dfrac { 1 } { 1 + 2 ( 1 - \\alpha ) ( \\beta - 1 ) } , & 1 < \\beta \\leq 1 + \\dfrac { 1 - \\alpha } { 8 \\alpha } , \\\\ \\dfrac { 2 \\sqrt { \\alpha } } { ( 1 + \\alpha ) \\sqrt { 1 + 1 6 \\alpha ( \\beta - 1 ) ^ 2 } } , & 1 + \\dfrac { 1 - \\alpha } { 8 \\alpha } \\leq \\beta < \\frac { 4 } { 3 } . \\end{dcases} \\end{align*}"}
+{"id": "2667.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { k _ n } 2 ^ k | \\Gamma ( \\Lambda _ k ( \\xi _ n ) ) | , \\end{align*}"}
+{"id": "5428.png", "formula": "\\begin{align*} f ' ( \\bar { \\alpha } _ t \\alpha _ t ) \\bar { \\alpha } _ t \\alpha _ t = g _ t ^ { \\frac { 1 } { 2 } } f ' ( H _ t ) H _ t g _ t ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "2484.png", "formula": "\\begin{align*} \\lim _ k \\| \\mathcal Q _ { n _ k } \\| \\frac { \\| X _ T x _ { n _ k } \\| } { \\| x _ { n _ k } \\| } = 0 . \\end{align*}"}
+{"id": "2033.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\bf C } ^ - ( r ) = \\mathcal { I } ^ - ( r ) & \\Longleftrightarrow \\left \\{ \\begin{array} { l l l } - b _ 1 + D + r - \\delta \\geq r - \\delta \\\\ - b _ \\ell + r \\leq - b _ { \\ell + 1 } + D + r , \\forall \\ell \\geq 1 \\\\ \\end{array} \\right . & \\Longleftrightarrow \\left \\{ \\begin{array} { l l l } b _ 1 \\leq D \\\\ b _ { \\ell + 1 } - b _ \\ell \\leq D , \\forall \\ell \\geq 1 . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "2269.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t f + v \\cdot \\nabla _ x f + { \\rm { d i v } } _ v ( ( u - v ) f ) - \\Delta _ v f = 0 , \\\\ & \\partial _ t \\rho + { \\rm { d i v } } _ x ( \\rho u ) = 0 , \\\\ & \\partial _ t ( \\rho u ) + { \\rm { d i v } } _ x ( \\rho u \\otimes u ) + \\nabla _ x \\rho ^ \\gamma - \\Delta _ x u = - \\int _ { \\mathbb { R } ^ 3 } ( u - v ) f \\ , d v . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5381.png", "formula": "\\begin{align*} & \\mathbb { P } \\bigg [ \\sum _ { m = 1 } ^ j G _ m \\le a + h ( j ) \\ , \\forall 1 \\le j \\le n \\bigg ] \\\\ = & \\mathbb { P } [ G _ 1 \\le - C - 1 - 3 \\log 2 ] \\mathbb { P } \\bigg [ \\sum _ { m = 1 } ^ j G _ m \\le a + h ( j ) \\ , \\forall 1 \\le j \\le n \\ , \\big \\vert \\ , G _ 1 \\le - C - 1 - 3 \\log 2 \\bigg ] . \\end{align*}"}
+{"id": "5156.png", "formula": "\\begin{align*} \\mathcal { I } ( H , n ) = \\dfrac { \\mathcal { P } ( H , n ) } { \\binom { n } { k } } . \\end{align*}"}
+{"id": "882.png", "formula": "\\begin{align*} \\mathfrak { W } _ { t } = \\left \\{ i | f _ { i } ( \\mathcal { X } ^ { t } ) \\geq \\theta \\mathop { \\max } _ { j = 1 , 2 , \\cdots , q } f _ { j } ( \\mathcal { X } ^ { t } ) + ( 1 - \\theta ) \\mathbf { E } _ { j \\sim \\mathbf { p } } [ f _ { j } ( \\mathcal { X } ^ { t } ) ] \\right \\} , \\end{align*}"}
+{"id": "1980.png", "formula": "\\begin{align*} { \\textbf { Y } } = \\sum \\nolimits _ { k = 1 } ^ { K } \\left ( \\sqrt { \\alpha _ k } { \\textbf { h } } _ k { \\textbf { x } } _ k ^ { \\mathsf { H } } + \\sqrt { \\alpha _ { k ' } } { \\textbf { h } } _ { k ' } { \\textbf { x } } _ { k ' } ^ { \\mathsf { H } } \\right ) + { \\textbf { G } } ^ { \\mathsf { H } } { \\textbf { S } } + { \\textbf { N } } , \\end{align*}"}
+{"id": "3603.png", "formula": "\\begin{align*} d _ { j } ^ + ( \\mu ) = \\frac { \\kappa ( \\mu ) } { \\kappa ( \\mu + \\epsilon _ j ) } c _ { j } ^ + ( \\mu ) = \\frac { c _ { j } ^ - ( \\mu + \\epsilon _ j ) } { d _ { j } ^ - ( \\mu + \\epsilon _ j ) } c _ { j } ^ + ( \\mu ) = \\frac { c _ { j } ^ + ( \\mu ) ^ 2 } { d _ { j } ^ - ( \\mu + \\epsilon _ j ) } \\end{align*}"}
+{"id": "3996.png", "formula": "\\begin{align*} L ( \\phi ) : = w ^ { i j } [ D _ { i j } \\phi - D _ { p _ k } A _ { i j } D _ k \\phi ] - D _ { p _ k } B D _ k \\phi \\geq w ^ { i i } - C , \\end{align*}"}
+{"id": "7690.png", "formula": "\\begin{align*} Z _ { r - 1 } ( B _ { F , 0 } ) = Z ( B _ { F , 0 } ) . \\end{align*}"}
+{"id": "1629.png", "formula": "\\begin{align*} \\begin{aligned} & F _ 1 ( t ) = i \\int _ 0 ^ M e ^ { i ( t - s ) \\Delta ^ 2 } ( | u | ^ 2 u ) ( s ) \\ , d s , \\\\ & F _ 2 ( t ) = i \\int _ M ^ { t - M } e ^ { i ( t - s ) \\Delta ^ 2 } ( | u | ^ 2 u ) ( s ) \\ , d s , \\\\ & F _ 3 ( t ) = i \\int _ { t - M } ^ t e ^ { i ( t - s ) \\Delta ^ 2 } ( | u | ^ 2 u ) ( s ) \\ , d s . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "2944.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\mathbb { E } _ n \\bigg [ \\sup _ { 0 \\le t \\le T } \\big | \\mathcal { B } ^ n _ t ( \\varphi ) - \\tilde { \\mathcal { B } } ^ n _ t ( \\varphi ) \\big | ^ 2 \\bigg ] = 0 . \\end{align*}"}
+{"id": "7882.png", "formula": "\\begin{align*} g ( z ) = f ( z ) \\exp \\left ( - \\frac { 1 } { 2 } \\int ^ z p ( t ) d t \\right ) A ( z ) = q ( z ) - \\frac { 1 } { 2 } p ' ( z ) - \\frac { 1 } { 4 } p ( z ) ^ 2 . \\end{align*}"}
+{"id": "2688.png", "formula": "\\begin{align*} ( 8 n + 4 ) r = ( 8 n + 4 ) \\ell ^ { - 1 / s } \\leq ( 8 n + 4 ) \\omega _ d ^ { - 1 / s } \\left ( n - \\frac { \\sqrt { d } } { 2 } \\right ) ^ { - d / s } < \\epsilon ^ { 1 / s } < 1 . \\end{align*}"}
+{"id": "6130.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } U '' - \\Delta U + A U = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ U = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu U + B U = D H & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "6561.png", "formula": "\\begin{align*} \\mathcal { A } _ q : = \\bigcup _ { \\substack { 1 \\leq p \\leq q , \\\\ \\gcd ( p , q ) = 1 } } \\left ( \\frac { p } { q } - \\frac { \\psi ( q ) } { q } , \\frac { p } { q } + \\frac { \\psi ( q ) } { q } \\right ) , q = 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "6827.png", "formula": "\\begin{align*} \\tau ( L _ 1 , L _ 2 , L _ 3 ) = \\mu _ c ^ { + } ( Y _ 1 , Y _ 2 , Y _ 3 ) - \\mu _ c ^ { - } ( Y _ 1 , Y _ 2 , Y _ 3 ) . \\end{align*}"}
+{"id": "5841.png", "formula": "\\begin{align*} \\begin{cases} U '' - { \\Delta } U + A U = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\partial _ \\nu U + B U = D H & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma \\end{cases} \\end{align*}"}
+{"id": "8129.png", "formula": "\\begin{align*} v _ 1 ' ( y ) = x ^ { \\frac { 1 } { 3 } } y ^ { - \\frac { 2 } { 3 } } + \\frac { T ^ 2 c m p } { 4 \\pi ^ 2 y ^ 2 n _ 1 ^ 2 } \\gg \\frac { T ^ 2 c m p } { y ^ 2 n _ 1 ^ 2 } \\end{align*}"}
+{"id": "6871.png", "formula": "\\begin{align*} B _ 1 ( \\mathbb { D } ) = \\overline { \\Big \\{ \\sum _ { j = 1 } ^ { n } a _ j d _ j \\ ; : \\ ; n \\in \\mathbb { N } , d _ j \\in \\mathbb { D } , \\ ; \\norm { \\{ a _ j \\} _ { j = 1 } ^ n } _ { \\ell _ 1 } \\leq 1 \\Big \\} } . \\end{align*}"}
+{"id": "298.png", "formula": "\\begin{align*} ( \\omega _ { S _ { \\alpha } ( F _ { \\pm \\alpha } ) , E } \\circ \\alpha ) | _ { S ( F ) } = ( \\omega _ { E F _ { \\alpha } / F _ { \\alpha } } \\circ \\iota _ { F _ \\alpha } \\circ \\alpha ) | _ { S ( F ) } . \\end{align*}"}
+{"id": "1078.png", "formula": "\\begin{align*} R _ 0 ( \\{ D - 1 \\} ) = 1 \\mbox { a n d } \\begin{cases} R _ 1 ( \\{ D - 1 \\} ) = 1 - \\eta , \\\\ R _ 1 \\left ( \\{ D - 1 + \\left ( 2 \\eta \\right ) ^ { - 1 / k } \\} \\right ) = \\eta , \\end{cases} \\ , \\eta \\in ( 0 , 1 ) , \\end{align*}"}
+{"id": "8399.png", "formula": "\\begin{align*} ( \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha , n _ 0 } ^ { - 1 } ) ( z ) & = \\hat \\psi _ { \\alpha } \\left ( \\frac { z _ { n _ 0 } + 1 } { 2 } \\right ) + \\sum _ { j = 0 } ^ { n _ 0 - 1 } \\hat \\psi _ { \\alpha } ( f _ { \\alpha } ^ j ( z _ { n _ 0 } ) ) \\\\ & = \\hat \\psi _ { \\alpha } \\left ( \\frac { z _ { n _ 0 } + 1 } { 2 } \\right ) + \\sum _ { j = 1 } ^ { n _ 0 } \\hat \\psi _ { \\alpha } ( z _ { j } ) \\end{align*}"}
+{"id": "140.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\| f _ r \\| _ { L ^ p ( \\mathbb R ^ n ) } = \\| f \\| _ { L ^ p ( \\mathbb R ^ n ) } \\lim _ { r \\rightarrow \\infty } \\| g _ r \\| _ { L ^ p ( \\mathbb R ^ n ) } = 0 \\end{align*}"}
+{"id": "2078.png", "formula": "\\begin{align*} \\frac { \\partial \\Phi } { \\partial p } & = 1 - \\xi ^ { 1 + \\nu } - ( 1 + \\nu ) ( p + ( 1 - p ) \\xi ) ^ { \\nu } ( 1 - \\xi ) \\\\ & \\geq 1 - \\xi ^ { 1 + \\nu } - ( 1 + \\nu ) ( 1 - \\xi ) \\\\ & \\geq \\xi ( 1 + \\nu - \\xi ^ { \\nu } ) > 0 , \\end{align*}"}
+{"id": "8019.png", "formula": "\\begin{align*} V _ H = \\{ & F _ 1 , F _ 2 , F _ 3 , F _ 4 \\} , \\\\ E _ H = \\big \\{ & ( F _ 1 , F _ 2 ) , ( F _ 1 , F _ 3 ) , ( F _ 2 , F _ 3 ) , ( F _ 3 , F _ 2 ) , ( F _ 4 , F _ 1 ) , ( F _ 4 , F _ 2 ) , \\\\ & ( F _ 4 , F _ 3 ) \\big \\} . \\end{align*}"}
+{"id": "1820.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } L ( u , \\dot { u } ) \\ , d t = M \\int _ { 0 } ^ { \\frac { T } { M } } L ( u , \\dot { u } ) \\ , d t . \\end{align*}"}
+{"id": "1553.png", "formula": "\\begin{align*} & ( p ^ { + } - r ^ { + } ) \\int _ { M } | D \\mathrm { u } ( z ) | ^ { p ( z ) } \\ , \\ , d v _ { g } ( z ) + ( q ^ { + } - r ^ { + } ) \\int _ { M } \\mu ( z ) | D \\mathrm { u } ( z ) | ^ { q ( z ) } \\ , \\ , d v _ { g } ( z ) \\\\ & = ( r ^ { - } + \\gamma ^ { - } - 1 ) \\int _ { M } g ( z ) | \\mathrm { u } ( z ) | ^ { 1 - \\gamma ( z ) } \\ , \\ , d v _ { g } ( z ) . \\end{align*}"}
+{"id": "4852.png", "formula": "\\begin{align*} g _ i ^ { s \\ 0 _ j + t _ 0 } = g _ i ^ { t _ 0 } + O ( s ) , \\end{align*}"}
+{"id": "1126.png", "formula": "\\begin{align*} c _ { ( a + b - 1 , 1 ) ( c ^ k ) } ^ { ( a + b + c - 1 , c ^ { k - 1 } , 1 ) } \\ge c _ { ( a + b - 1 , 1 ) ( c ) } ^ { ( a + b + c - 1 , 1 ) } \\ge c _ { ( a + b - 1 ) ( c ) } ^ { ( a + b + c - 1 ) } = 1 \\end{align*}"}
+{"id": "7113.png", "formula": "\\begin{align*} L T ( q _ t ( u x _ { i _ t } + p _ { i _ t } ) ) = L T ( q _ t ) u x _ { i _ t } \\end{align*}"}
+{"id": "3167.png", "formula": "\\begin{align*} A ( y ) = \\tilde { A } ( y _ 1 ) \\quad y = ( y _ 1 , \\dots , y _ n ) \\in \\R ^ n \\end{align*}"}
+{"id": "56.png", "formula": "\\begin{align*} s ^ { t o t } ( i ) = \\sum _ { j \\in N ( i ) } { f ( j ) } \\geq 1 , i \\in V , \\end{align*}"}
+{"id": "5538.png", "formula": "\\begin{align*} \\mathcal { R } ( x ) = \\sum _ { \\rho } \\lim _ { s \\rightarrow \\frac { k - \\rho } { 2 } } \\frac { \\left ( s - \\frac { k - \\rho } { 2 } \\right ) \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } = - \\frac { 1 } { 2 } \\sum _ { \\rho } \\frac { \\Gamma ( \\frac { k - \\rho } { 2 } ) } { \\zeta ' ( \\rho ) } x ^ { - \\frac { k - \\rho } { 2 } } , \\end{align*}"}
+{"id": "4227.png", "formula": "\\begin{align*} E _ [ \\Lambda , \\phi ] ( t ) = 0 . \\end{align*}"}
+{"id": "6608.png", "formula": "\\begin{align*} P _ n ( \\theta ) ( U \\psi ) _ { \\theta } = 0 n > N . \\end{align*}"}
+{"id": "6717.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq n \\leq x } \\omega ^ { n t } = \\sum _ { 1 \\leq n \\leq x } e ^ { i 2 \\pi n t / p } = \\frac { 1 - e ^ { \\frac { i 2 \\pi t ( x + 1 ) } { p } } } { 1 - e ^ { \\frac { i 2 \\pi t } { p } } } . \\end{align*}"}
+{"id": "9199.png", "formula": "\\begin{align*} \\mathcal { E } ^ { \\Psi } K _ j ( \\varphi ' ) ^ { [ Z ] } = \\sum _ { ( B _ { Z '' } ) _ { Z '' } } \\prod _ { Z '' } J _ j ^ { \\Psi } ( B , Z '' ) \\end{align*}"}
+{"id": "5389.png", "formula": "\\begin{align*} & \\ll N \\left ( C p ^ { 2 4 } \\log \\log N \\right ) ^ { 2 p } \\sum _ { \\sigma \\in S _ { 2 p } } \\sum _ { 0 \\le h _ 1 , \\dots , h _ { 2 p } \\le \\log K } \\bigg ( \\prod _ { \\substack { i = 1 \\\\ i \\ne i _ 0 } } ^ { 2 p } \\alpha ( h _ i ) ^ 2 e ^ { h _ i } \\bigg ) \\mathrm { v o l } ( I _ { \\sigma , h } ) . \\end{align*}"}
+{"id": "1730.png", "formula": "\\begin{align*} ( \\kappa ( a , b ) P ) ( x , y ) : = P ( ( x , y ) \\kappa ( a , b ) ) = P ( a x - \\bar b y , b x + \\bar a y ) , ( a , b ) \\in S ^ 3 . \\end{align*}"}
+{"id": "4429.png", "formula": "\\begin{align*} \\frac { 2 9 } { 1 2 0 } = \\frac { 1 } { 5 } + \\frac { 1 } { 2 4 } < \\frac { 1 } { 7 } + \\frac { 1 } { 1 0 } = \\frac { 1 7 } { 7 0 } < \\frac { 1 } { 5 } + \\frac { 1 } { 2 3 } = \\frac { 2 8 } { 1 1 5 } . \\end{align*}"}
+{"id": "7097.png", "formula": "\\begin{align*} \\mathbf { C P } ( V \\oplus W ) / \\mathbf { C P } ( W ) & \\to ( \\nu \\to \\mathbf { C P } ( V ) ) \\\\ [ \\vec { v } : \\lambda _ 1 : \\cdots : \\lambda _ k ] & \\mapsto \\begin{cases} \\infty & \\vec { v } = 0 \\\\ ( [ \\vec { v } ] , s ( \\lambda _ 1 ) \\vec { v } , \\cdots , s ( \\lambda _ k ) \\vec { v } ) & \\vec { v } \\neq 0 . \\end{cases} \\end{align*}"}
+{"id": "5887.png", "formula": "\\begin{align*} \\overline B _ p = C _ p P \\widehat { B } P ^ T C _ p ^ T ( C _ p P P ^ T C _ p ^ T ) ^ { - 1 } , \\end{align*}"}
+{"id": "441.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\right \\| \\leq b \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "6216.png", "formula": "\\begin{align*} u _ r = ( E _ r , U ) , \\end{align*}"}
+{"id": "5499.png", "formula": "\\begin{align*} \\jmath ^ * F _ { \\lambda _ 0 + 1 } = \\jmath ^ * ( ^ { - 1 } \\circ F _ { \\lambda _ 0 } \\circ ) . \\end{align*}"}
+{"id": "5335.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ \\varepsilon = J _ \\varepsilon [ u ] [ v ] : = \\int _ \\Omega \\varepsilon u \\cdot v \\ , d x \\forall u , v \\in L ^ 2 ( \\Omega ) ^ 3 . \\end{align*}"}
+{"id": "2761.png", "formula": "\\begin{align*} \\lbrack \\lbrack a ] ] _ { q } = \\frac { 1 - q ^ { a } } { 1 - q } , \\end{align*}"}
+{"id": "6360.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } m ' ( x ) = \\hat m ' ( \\infty ) , \\end{align*}"}
+{"id": "4107.png", "formula": "\\begin{align*} 0 = \\delta \\lambda = \\delta R - \\frac { 1 } { 1 2 } \\delta | H | ^ 2 + 2 \\delta \\triangle f - \\delta | \\nabla f | ^ 2 . \\end{align*}"}
+{"id": "6316.png", "formula": "\\begin{align*} g | _ \\Sigma = & ( h ^ { - 1 } - h u ^ 2 _ r ) d r ^ 2 + ( r ^ 2 - h u ^ 2 _ \\theta ) d \\theta ^ 2 + ( r ^ 2 \\sin ^ 2 \\theta - h u ^ 2 _ \\phi ) d \\phi ^ 2 \\\\ & - 2 h u _ r u _ \\theta d r d \\theta - 2 h u _ r u _ \\phi d r d \\phi - 2 h u _ \\theta u _ \\phi d \\theta d \\phi \\end{align*}"}
+{"id": "5431.png", "formula": "\\begin{align*} \\alpha _ t ( \\omega ' ) = \\gamma _ t ^ { - 1 } { } ^ * \\left ( \\alpha ( \\gamma _ t ^ * \\omega ' ) \\right ) . \\end{align*}"}
+{"id": "8102.png", "formula": "\\begin{align*} \\widehat { k ^ * } \\Bigl ( - \\frac { M \\zeta } { \\pi } \\Bigr ) = \\int _ { - \\infty } ^ \\infty k ^ * ( u ) e \\Bigl ( \\frac { u M \\zeta } { \\pi } \\Bigr ) \\ , d u . \\end{align*}"}
+{"id": "9011.png", "formula": "\\begin{align*} \\inf _ { F \\in \\mathcal { F } _ + ( G ) } \\frac { H _ \\mu ( \\alpha _ { F } ) } { | F | } + \\mu ( f ) \\ , & = \\ , \\inf _ { F \\in \\mathcal { F } _ + ( G ) } \\frac { H _ \\mu ( \\alpha _ { F } ) + \\mu ( \\sum _ { F } { f } ) } { | F | } \\\\ & \\leq \\ , \\inf _ { F \\in \\mathcal { F } _ + ( G ) } \\frac { \\operatorname { P } _ { \\sum _ { F } { f } } ( \\mathcal { U } _ { F } ) } { | F | } + \\epsilon \\\\ & \\leq \\ , \\operatorname { p } _ f ^ { ( n v ) } ( \\pi ) + \\epsilon . \\end{align*}"}
+{"id": "397.png", "formula": "\\begin{align*} \\mathbb { E } [ | 1 - Z | ^ 6 ] = \\frac { 1 5 m ^ 2 + 1 3 0 m + 1 2 0 } { m ^ 5 } \\ , . \\end{align*}"}
+{"id": "150.png", "formula": "\\begin{align*} & \\max \\bigg \\{ \\max _ { 1 \\leq j \\leq n } \\bigg ( \\bigg ( K L _ { \\nu _ { j } } + K L _ { q } \\kappa _ { b } \\gamma b \\bigg ) \\bigg ( 1 + \\frac { M ^ { 2 } K ^ { 2 } b } { \\delta ^ { j } } \\bigg ) \\bigg ) , \\\\ & \\qquad \\qquad \\bigg ( 1 + \\frac { M ^ { 2 } K ^ { 2 } b } { \\delta ^ { 0 } } \\bigg ) K L _ { q } \\kappa _ { b } \\gamma b , \\ \\max _ { 1 \\leq j \\leq n } L _ { \\nu _ { j } } \\bigg \\} < 1 , \\mbox { w h e r e } \\ \\gamma = \\frac { b } { \\beta } . \\end{align*}"}
+{"id": "8063.png", "formula": "\\begin{align*} F ( u ) = \\Bigl ( \\cos \\frac { \\pi u } { A } \\Bigr ) ^ { - 3 A } , \\end{align*}"}
+{"id": "3868.png", "formula": "\\begin{align*} \\det D Y ( \\cdot , u , D u ) = \\psi ( \\cdot , u , D u ) \\Omega , \\end{align*}"}
+{"id": "8151.png", "formula": "\\begin{align*} a ( y ) = \\psi ( y ) y ^ { - \\frac { 1 } { 3 } } = g \\Bigl ( \\frac { m ^ 2 y } { N } \\Bigr ) k ^ * \\Biggl ( \\frac { \\dfrac { 4 \\pi \\sqrt { y p } } { c } - 2 T } { 2 M } \\Biggr ) y ^ { - \\frac { 5 } { 6 } } . \\end{align*}"}
+{"id": "7143.png", "formula": "\\begin{align*} f ( a \\otimes b ) = \\begin{cases} a ' e b ' , & \\mbox { i f $ a b = a ' \\alpha ^ r b ' $ } , \\\\ 0 , & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
+{"id": "4315.png", "formula": "\\begin{align*} E _ \\alpha : = \\cap _ { j = 1 } ^ n \\ker ( E \\to E _ { x _ j } / E ^ { \\beta ( \\alpha , j ) } _ j ) . \\end{align*}"}
+{"id": "6611.png", "formula": "\\begin{align*} ( \\nabla _ { \\theta } E _ n ( \\theta ) ) P _ n ( \\theta ) = 2 P _ n ( \\theta ) ( D + \\theta ) P _ n ( \\theta ) . \\end{align*}"}
+{"id": "4018.png", "formula": "\\begin{align*} \\phi ^ * _ i D _ j Y ^ i = \\chi \\gamma _ j , \\end{align*}"}
+{"id": "3112.png", "formula": "\\begin{align*} \\int _ Y r ^ 1 A ^ 1 e _ 1 \\cdot \\nabla q = \\delta \\int _ Y r ^ 0 P e _ 1 \\cdot \\nabla q = \\delta \\int _ Y Z e _ 1 \\cdot \\nabla q \\neq 0 , \\end{align*}"}
+{"id": "6233.png", "formula": "\\begin{align*} ( D ^ T E _ i , \\widehat H ) = ( E _ i , B \\widehat U ) . \\end{align*}"}
+{"id": "5303.png", "formula": "\\begin{align*} \\nabla f = e ^ { - 2 x _ 2 } X , \\end{align*}"}
+{"id": "3559.png", "formula": "\\begin{align*} B _ \\ell ^ + E ^ \\gamma \\Omega _ \\lambda = \\sum _ { j = \\ell } ^ n \\sum _ { t = 1 } ^ { j - \\ell + 1 } \\sum _ { J \\in \\mathcal { I } _ { \\ell j } ( t ) } ( - 1 ) ^ { t - 1 } \\prod _ { u = 1 } ^ { t - 1 } \\gamma _ { j _ { u + 1 } j _ u } E ^ { \\gamma - e _ J } B _ j ^ + \\Omega _ \\lambda , \\end{align*}"}
+{"id": "2253.png", "formula": "\\begin{align*} \\alpha _ j ( x ) : = \\sigma _ j ( \\overline { x } ) ^ { q - 1 } . \\end{align*}"}
+{"id": "3474.png", "formula": "\\begin{align*} \\int _ { T _ n ( s ) } G _ { s - t _ n } ( x - x _ n ) \\prod _ { j = 1 } ^ { n - 1 } G _ { t _ { j + 1 } - t _ j } ( x _ { j + 1 } - x _ j ) d \\pmb { t } = \\int _ { T _ n ( t ) } G _ { s - t _ n } ( x - x _ n ) \\prod _ { j = 1 } ^ { n - 1 } G _ { t _ { j + 1 } - t _ j } ( x _ { j + 1 } - x _ j ) d \\pmb { t } . \\end{align*}"}
+{"id": "5148.png", "formula": "\\begin{align*} K _ { 0 } ( z ) = - \\log \\left ( \\frac { z } { 2 } \\right ) I _ { 0 } ( z ) + \\sum _ { m = 0 } ^ { \\infty } \\frac { \\left ( \\frac { z } { 2 } \\right ) ^ { 2 m } } { ( m ! ) ^ { 2 } } \\psi ( m + 1 ) , \\end{align*}"}
+{"id": "456.png", "formula": "\\begin{align*} x - x _ n & = x - x _ { n - 1 } - \\frac { 2 } { a + b } S _ { f , \\tau } ( x - x _ { n - 1 } ) \\\\ & = \\left ( I _ \\mathcal { X } - \\frac { 2 } { b + a } S _ { f , \\tau } \\right ) ( x - x _ { n - 1 } ) \\\\ & = \\cdots = \\left ( I _ \\mathcal { X } - \\frac { 2 } { b + a } S _ { f , \\tau } \\right ) ^ n x , \\quad \\forall x \\in \\mathcal { X } , \\forall n \\geq 1 . \\end{align*}"}
+{"id": "7329.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { 1 } f ( x ) G ^ 2 _ { m _ \\gamma } ( x ) j _ \\gamma ( d x ) \\\\ = & \\frac { - 1 } { u ( \\gamma ' ) ^ 2 v ( \\gamma ' ) } \\int _ { 0 } ^ { \\gamma ' } f ( \\gamma '^ { - 1 } x ) j ( d x ) \\int _ { 0 } ^ { x } d y \\int _ { y } ^ { \\gamma ' } G _ m ( z ) d m ( z ) \\\\ & + \\frac { m ( \\gamma ' ) } { u ( \\gamma ' ) ^ 2 v ( \\gamma ' ) } \\int _ { 0 } ^ { \\gamma ' } f ( \\gamma '^ { - 1 } x ) j ( d x ) \\int _ { 0 } ^ { x } d y \\int _ { y } ^ { \\gamma ' } z d m ( z ) . \\end{align*}"}
+{"id": "3813.png", "formula": "\\begin{align*} ( Q ^ { 1 / 2 } , B Q ^ { 1 / 2 } , \\dots , B ^ { d - 1 } Q ^ { 1 / 2 } ) = ( Q ^ { 1 / 2 } , B Q ^ { 1 / 2 } , Q ^ { 1 / 2 } , \\dots , B Q ^ { 1 / 2 } ) \\end{align*}"}
+{"id": "8409.png", "formula": "\\begin{align*} \\sum _ { k \\geq 1 } \\partial _ { \\alpha } \\left ( \\int \\hat \\Psi _ { \\alpha } \\cdot \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha } ^ k \\cdot h _ { \\alpha } \\ : d m \\right ) = \\sum _ { k \\geq 1 } \\int \\partial _ { \\alpha } \\left ( P _ { \\alpha } ^ k ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\hat \\Psi _ { \\alpha } \\ : d m \\right ) . \\end{align*}"}
+{"id": "1160.png", "formula": "\\begin{align*} P o l y n o m ( \\mathbb { A } ) = C l o n e ( \\Omega , a _ 1 , . . . , a _ { | A | } ) . \\end{align*}"}
+{"id": "7287.png", "formula": "\\begin{align*} ( - 1 ) ^ { k + 1 } G ^ { k + 1 } _ m ( x ) = \\int _ { 0 } ^ { x } d y \\int _ { y } ^ { 1 } ( - 1 ) ^ { k } G ^ { k } _ m ( z ) d m ( z ) \\end{align*}"}
+{"id": "7794.png", "formula": "\\begin{align*} \\frac { q ^ { 2 j \\{ n , n ' \\} } - 1 } { q ^ { 2 j a } - 1 } & = \\begin{cases} 1 & \\{ n , n ' \\} > 0 , \\\\ \\displaystyle - q ^ { - 2 j a } & \\{ n , n ' \\} < 0 . \\end{cases} \\end{align*}"}
+{"id": "359.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ e ^ { - z V ( t ) } \\big ] = \\cosh ( z ) \\textrm { f o r a n y } z \\in \\mathbb { R } \\textrm { a n d } t \\geq 0 \\ , , \\end{align*}"}
+{"id": "2970.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\bigg \\langle \\sqrt { n } \\sum _ { j \\in \\mathbb { Z } } ( \\overline { W } _ { j - 1 } - \\overline { W } _ j ) \\nabla ^ n \\varphi ^ n _ j , f \\bigg \\rangle _ { L ^ 2 ( \\nu ^ n _ \\rho ) } = - 2 \\sqrt { n } E _ { \\nu ^ n _ \\rho } \\bigg [ \\sum _ { j \\in \\mathbb { Z } } W _ { j - 1 } \\nabla ^ n \\varphi ^ n _ j \\nabla _ { j - 1 , j } f \\bigg ] . \\end{aligned} \\end{align*}"}
+{"id": "1644.png", "formula": "\\begin{align*} \\frac { \\ell ( f _ 1 , \\chi ) \\cdot \\ell ( f _ 2 , \\chi ^ { - 1 } ) } { \\langle f _ 1 , f _ 2 \\rangle } = \\frac { \\Lambda _ F ( 2 ) \\cdot \\Lambda ( 1 / 2 , \\Pi , \\chi ) } { 2 \\Lambda ( 1 , \\Pi , { \\rm a d } ) } \\cdot \\prod _ v \\beta _ v ( f _ { 1 , v } , f _ { 2 , v } ) , f _ 1 , f _ 2 \\in \\pi , \\end{align*}"}
+{"id": "8262.png", "formula": "\\begin{align*} \\log F _ p ( X ) = o ( \\log X ) ( X \\to \\infty ) , \\end{align*}"}
+{"id": "4503.png", "formula": "\\begin{align*} \\lambda _ 1 + \\cdots + \\lambda _ n = 1 . \\end{align*}"}
+{"id": "8970.png", "formula": "\\begin{align*} \\Delta ( d i s t ^ i _ N ( u ) ) = d i v ( \\nu _ i ( u ) \\cdot \\nabla u ) = \\nabla u \\cdot d \\nu _ i ( u ) \\nabla u \\ \\hbox { i n } B . \\end{align*}"}
+{"id": "3675.png", "formula": "\\begin{align*} ( - \\alpha _ \\rho ^ k c - ( I - \\alpha _ \\rho ^ k ) A ) \\delta u = - M _ \\rho ( f - A u ^ k , u ^ k - g ) . \\end{align*}"}
+{"id": "6527.png", "formula": "\\begin{align*} \\underset { f \\in H _ { c _ \\alpha \\rho _ { s _ L } } ^ { s _ L , R } } { \\sup } \\P _ f ^ Y ( T = 0 ) \\geq \\P _ { \\pi _ L } ^ Y ( T = 0 ) - \\pi _ L \\circ \\Psi ^ { - 1 } _ L \\left ( f \\notin H _ { c _ \\alpha \\rho _ { s _ L } } ^ { s _ L , R } \\right ) , \\end{align*}"}
+{"id": "1608.png", "formula": "\\begin{align*} d _ p ( \\nu , \\nu ' ) \\leq \\sqrt [ p ] { \\sum _ { 1 \\leq i , j \\leq L } \\varrho ^ p ( u _ i , u _ j ) \\cdot \\pi ( u _ i , u _ j ) } < \\sqrt [ p ] { L ^ 2 K ^ p \\frac { \\widetilde \\varepsilon ^ p } { K ^ p L } } = \\widetilde \\varepsilon \\leq \\varepsilon . \\end{align*}"}
+{"id": "8580.png", "formula": "\\begin{align*} \\deg ( g ( x ) ) < \\deg ( f ( x ) ) , ~ \\gcd ( g ( x ) , f ( x ) ) = 1 , ~ f ( 0 ) = 1 . \\end{align*}"}
+{"id": "3038.png", "formula": "\\begin{align*} \\begin{aligned} | | D ^ 2 w ^ { s , 0 } _ { k } | | _ { L ^ { p } ( Q _ k ) } & \\leq C \\left ( d _ k ^ { - 2 } | | w ^ { s , 0 } _ { k } | | _ { L ^ { p } ( \\widetilde Q _ k ) } + | | f \\chi _ { \\mathcal { F } ^ { s , 0 } _ { Q _ k } } | | _ { L ^ { p } ( \\widetilde Q _ k ) } \\right ) \\\\ \\\\ & \\leq C \\left ( 2 ^ { - \\frac { 2 m } { n p ' } } | | f | | _ { L ^ p ( \\mathcal { F } ^ { s , 0 } _ { Q _ k } ) } + | | f | | _ { L ^ p ( \\mathcal { F } ^ { s , 0 } _ { Q _ k } \\cap \\widetilde Q _ k ) } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "6468.png", "formula": "\\begin{align*} \\mu = - \\frac { 1 } { 2 } ( \\mu ^ 2 + \\nu ^ 2 + \\rho ^ 2 ) = - \\frac { 1 } { 2 } ( \\nu ^ 2 + \\rho ^ 2 ) + O ^ 4 _ 0 ( \\eta ) . \\end{align*}"}
+{"id": "5632.png", "formula": "\\begin{align*} \\dfrac { d H } { d I } & \\geq \\dfrac { F ( S , I ) - F ( S , I ^ { * } ) } { F ( S , I ) F ( S , I ^ { * } ) } F _ { 1 } ( S , I ) - \\dfrac { I - I ^ { * } } { I I ^ { * } } \\\\ & = \\dfrac { 1 } { F ( S , I ^ { * } ) } \\left [ F _ { 1 } ( S , I ) - F _ { 1 } ( S , I ^ { * } ) \\right ] \\geq 0 . \\end{align*}"}
+{"id": "4815.png", "formula": "\\begin{align*} | T | = 2 ^ { \\epsilon N \\log \\frac { 4 \\epsilon ( 1 - \\epsilon ) } { ( 1 - \\eta ) ^ { 4 \\ln 2 } } + o ( N ) } \\end{align*}"}
+{"id": "3318.png", "formula": "\\begin{align*} \\prod _ { i \\in \\mathbb { Z } _ { n } } \\left ( \\lambda - x _ { i } \\right ) & = \\sum _ { 0 \\le k \\le n } e _ { k } \\left ( \\mathbf { x } \\right ) \\lambda ^ { n - k } \\end{align*}"}
+{"id": "6538.png", "formula": "\\begin{align*} \\int \\phi _ m ( x ) q _ 2 ( x ) d P _ 0 ^ m ( x ) & = \\int \\int \\phi _ m ( x ) q \\left ( \\frac { x - w } { \\sqrt { 2 } } \\right ) q \\left ( \\frac { x + w } { \\sqrt { 2 } } \\right ) d P _ 0 ^ m ( x ) \\ , d P _ 0 ^ m ( w ) \\\\ & = \\bar { G } \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( x ^ j \\right ) q _ 2 ( x ) d P _ 0 ^ m ( x ) . \\end{align*}"}
+{"id": "6353.png", "formula": "\\begin{align*} ( \\xi \\circ f ) ' ( x ) = \\xi ' ( f ( x ) ) f ' ( x ) = \\frac { f ' ( x ) } { f ' ( \\xi \\circ f ( x ) ) } \\end{align*}"}
+{"id": "1649.png", "formula": "\\begin{align*} \\Z ^ { \\Sigma _ F } : = \\left \\{ \\underline { k } = ( k _ \\sigma ) _ { \\sigma \\in \\Sigma _ F } ; k _ \\sigma = ( k _ { \\tilde \\sigma } ) _ { \\tilde \\sigma \\mid \\sigma } \\in \\Z ^ { [ F _ \\sigma : \\R ] } \\right \\} \\simeq \\Z ^ { [ F : \\Q ] } . \\end{align*}"}
+{"id": "3721.png", "formula": "\\begin{align*} \\sum _ { s _ i \\in \\left \\{ \\frac { \\dim V _ i } { 2 } - 1 , \\frac { \\dim V _ i } { 2 } - 2 , 0 \\right \\} } \\mathrm { R e s } _ { s = s _ i } \\frac { e ^ { T s } Z _ { r _ i } ( f , s + 2 - \\tfrac { \\dim V _ i } { 2 } ) } { s } \\end{align*}"}
+{"id": "2429.png", "formula": "\\begin{align*} \\bar \\partial _ \\tau ^ \\alpha \\varphi ^ n = \\tau ^ { - \\alpha } \\sum _ { j = 0 } ^ n b _ j ^ { ( \\alpha ) } ( \\varphi ^ n - \\varphi ^ 0 ) , \\quad \\mbox { w i t h } ( 1 - \\xi ) ^ \\alpha = \\sum _ { j = 0 } ^ \\infty b _ j ^ { ( \\alpha ) } \\xi ^ j . \\end{align*}"}
+{"id": "6029.png", "formula": "\\begin{align*} P _ { l _ 1 + l _ 2 } ( h _ 2 , w _ { k , p } ) = \\left \\{ \\begin{array} { l r } 0 & \\mathrm { i f } \\ : ( k , p ) \\neq ( 1 , n ) \\\\ Q _ 1 Q _ 2 \\left ( [ \\O _ X ] - \\O _ { h _ 2 } \\right ) & \\mathrm { i f } \\ : ( k , p ) = ( 1 , n ) \\end{array} \\right . \\end{align*}"}
+{"id": "5214.png", "formula": "\\begin{align*} & g _ { 0 1 } + g \\ , g _ { 1 0 } = 0 \\quad , g ^ 1 _ { 0 1 } + g \\ , g ^ 1 _ { 1 0 } = 0 \\\\ & \\nabla g ^ 3 = g \\ , \\nabla g ^ 1 \\quad , g ^ 1 g ^ 2 _ { 0 1 } + g ^ 3 g ^ 2 _ { 1 0 } = 0 \\ , . \\end{align*}"}
+{"id": "951.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( N , C ) + \\textrm { r . g r a d e } _ R ( \\textrm { T r } _ C ( N ) , C ) \\ , = \\ , \\textrm { g r a d e } _ R ( \\textrm { E x t } _ { R } ^ t ( N , \\ , C ) ) + 1 . \\end{align*}"}
+{"id": "1231.png", "formula": "\\begin{align*} \\mu _ i ( \\{ ( x _ i ^ 1 , x _ i ^ 2 ) : x _ i ^ 1 \\leq z _ i ( y ) \\} ) = \\nu ( - \\infty , y ) . \\end{align*}"}
+{"id": "8402.png", "formula": "\\begin{align*} \\partial _ { \\alpha } ( \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha , n _ 0 } ^ { - 1 } ) ( z ) = \\psi ' \\left ( \\frac { z _ { n _ 0 } + 1 } { 2 } \\right ) \\partial _ { \\alpha } z _ { n _ 0 } + \\sum _ { j = 1 } ^ { n _ 0 } \\psi ' ( z _ { j } ) \\partial _ { \\alpha } z _ { n _ j } - ( n _ 0 + 1 ) \\partial _ { \\alpha } \\hat \\psi _ { \\alpha } \\end{align*}"}
+{"id": "7938.png", "formula": "\\begin{align*} A + B : = \\bigcup _ { a \\in A , b \\in B } ( a + b ) . \\end{align*}"}
+{"id": "6187.png", "formula": "\\begin{align*} A x _ l ^ { ( k ) } = \\lambda _ k x _ l ^ { ( k ) } + x _ { l + 1 } ^ { ( k ) } , 1 \\leqslant l \\leqslant r _ k \\hbox { w i t h } x _ { r _ k + 1 } ^ { ( k ) } = 0 . \\end{align*}"}
+{"id": "8075.png", "formula": "\\begin{align*} A ( p , m p ) & = \\sum _ { d \\mid ( p , p m ) } \\mu ( d ) A \\Big ( \\frac { p } { d } , 1 \\Big ) A \\Big ( 1 , \\frac { p m } { d } \\Big ) \\\\ & = A ( p , 1 ) A ( 1 , p m ) - A ( 1 , 1 ) A ( 1 , m ) \\\\ & = A ( p , 1 ) A ( 1 , p m ) - A ( 1 , m ) \\end{align*}"}
+{"id": "2506.png", "formula": "\\begin{align*} X ^ { - 1 } \\mathcal E = \\{ 0 \\} \\ \\mathcal E \\in \\operatorname { H l a t } U \\mathcal E \\neq \\mathcal K . \\end{align*}"}
+{"id": "877.png", "formula": "\\begin{align*} \\| \\Gamma ^ { t + 1 } \\| _ { F } ^ { 2 } = \\| ( \\mathcal { I } - \\mathcal { Z } ) * \\Gamma ^ { t } \\| ^ { 2 } _ { F } = \\| \\Gamma ^ { t } \\| ^ { 2 } _ { F } - \\| \\mathcal { Z } * \\Gamma ^ { t } \\| ^ { 2 } _ { F } , \\end{align*}"}
+{"id": "1004.png", "formula": "\\begin{align*} \\phi ( x , y ) = - \\frac { \\sin ( \\pi x ) \\sin ( 3 \\pi y ) } { \\nu \\cdot \\nabla ( \\sin ( 3 \\pi x ) \\sin ( \\pi y ) ) } \\bigg | _ { x = 1 / 3 , 2 / 3 } = \\frac { \\sqrt 3 } { 6 \\pi } \\frac { \\sin ( 3 \\pi y ) } { \\sin ( \\pi y ) } , \\end{align*}"}
+{"id": "8654.png", "formula": "\\begin{align*} K ( f _ R ) \\leq \\begin{cases} \\frac { 2 \\pi R } { \\ln \\frac { \\tanh ( R / 2 ) } { \\tanh ( a / 2 ) } } & : d = 2 , \\\\ \\frac { 4 \\pi \\tanh a R } { 1 - \\frac { \\tanh a } { \\tanh R } } & : d = 3 . \\end{cases} \\end{align*}"}
+{"id": "530.png", "formula": "\\begin{align*} M _ { i _ t , j _ t } = M _ { i _ t , j _ t } ^ * + \\sigma ( { \\xi _ { t } } / { \\| \\xi \\| } ) \\| M _ { \\Omega } ^ * \\| _ F , \\end{align*}"}
+{"id": "1763.png", "formula": "\\begin{align*} - \\Delta v + ( I _ 2 * | v | ^ 2 ) v = | v | ^ { p - 2 } v \\quad \\end{align*}"}
+{"id": "8798.png", "formula": "\\begin{align*} S = \\begin{pmatrix} a & 0 & 1 \\\\ 0 & b & 0 \\\\ 1 & 0 & c \\end{pmatrix} , S = \\begin{pmatrix} a & 1 & 0 \\\\ 1 & b & 0 \\\\ 0 & 0 & c \\end{pmatrix} S = \\begin{pmatrix} a & 0 & 0 \\\\ 0 & b & 1 \\\\ 0 & 1 & c \\end{pmatrix} \\end{align*}"}
+{"id": "3207.png", "formula": "\\begin{align*} \\vert g _ i ( x _ 2 ) - g _ i ( x _ 1 ) \\vert \\leq i \\vert x _ 2 - x _ 1 \\vert ^ \\alpha , \\ ; \\ ; x _ 1 , x _ 2 \\in \\mathbb { R } , \\ ; \\ ; i = 1 , 2 . \\end{align*}"}
+{"id": "2631.png", "formula": "\\begin{align*} \\dim \\left < v _ { 1 , i } ^ t \\ \\middle | \\ i = 2 , \\dots , r , t = 1 , \\dots , d \\right > = \\sum _ { i = 2 } ^ r \\dim \\left < v _ { 1 , i } ^ t \\ \\middle | \\ t = 1 , \\dots , d \\right > = \\sum _ { i = 2 } ^ r d _ { 1 , i } . \\end{align*}"}
+{"id": "3977.png", "formula": "\\begin{align*} \\overline { u } ( x ) & = g ( x , y , v _ \\rho ( y ) ) \\\\ & \\leq g ( x , y _ 0 , z _ 0 ) + | g _ y | | y - y _ 0 | + | g _ z | | v _ \\rho ( y ) - z _ 0 | \\\\ & \\leq g _ 0 ( x ) + 2 \\rho \\Vert g \\Vert _ { C ^ 1 ( \\Gamma ) } . \\end{align*}"}
+{"id": "1164.png", "formula": "\\begin{align*} c = a + _ { ( m o d \\ , p ) } b \\longleftrightarrow c < p \\wedge c \\equiv ( a + b ) \\ , ( m o d \\ , p ) . \\end{align*}"}
+{"id": "4578.png", "formula": "\\begin{align*} 1 = \\frac { 1 } { 2 } + \\frac { 1 } { 3 } + \\frac { 1 } { 6 } \\end{align*}"}
+{"id": "5314.png", "formula": "\\begin{align*} \\widetilde { g } = \\pi _ { B } ^ { * } ( g _ { B } ) + \\varphi ( \\rho , s ) ^ { 2 } \\ , \\pi _ { M } ^ { * } ( g ) , \\end{align*}"}
+{"id": "5317.png", "formula": "\\begin{align*} \\Phi = \\sum _ { i = 1 } ^ { m + 1 } \\mathrm { A d } _ { T _ i } \\circ \\Phi _ { i } = \\sum _ { i = 1 } ^ m \\mathrm { A d } _ { T _ i } \\circ \\Phi _ i + \\mathrm { A d } _ { T _ { m + 1 } } \\circ \\Phi _ { m + 1 } , \\end{align*}"}
+{"id": "6099.png", "formula": "\\begin{align*} F ( z ) = \\sum _ { n \\gg - \\infty } c _ F ^ + ( n ) q ^ n \\in M ^ ! _ { - 2 k } , \\end{align*}"}
+{"id": "2584.png", "formula": "\\begin{align*} & \\operatorname { A d } ( \\tilde { g } ) c = c , \\\\ & \\operatorname { A d } ( \\tilde { g } ) d = d - g ' g ^ { - 1 } - \\frac { 1 } { 2 } \\| g ' g ^ { - 1 } \\| ^ 2 c , \\\\ & \\operatorname { A d } ( \\tilde { g } ) u = g u g ^ { - 1 } + \\langle g ' g ^ { - 1 } , g u g ^ { - 1 } \\rangle c \\end{align*}"}
+{"id": "1680.png", "formula": "\\begin{align*} \\rho ^ \\vee ( \\mu ) \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) = \\mu \\left ( \\left | \\begin{array} { c c } X & Y \\\\ c & d \\end{array} \\right | ^ { k - 2 } \\right ) \\cdot ( a d - b c ) ^ { \\frac { 2 - k } { 2 } } . \\end{align*}"}
+{"id": "9145.png", "formula": "\\begin{align*} M _ 2 : = & \\sum _ { k = 0 } ^ n { n \\choose k } ( k - 1 ) ( m + n - 1 ) ^ { n - k - 1 } = \\\\ = & n \\sum _ { k = 1 } ^ n { n - 1 \\choose k - 1 } ( m + n - 1 ) ^ { n - k - 1 } - ( m + n - 1 ) ^ { - 1 } \\sum _ { k = 0 } ^ n { n \\choose k } ( m + n - 1 ) ^ { n - k } = \\\\ = & n ( m + n - 1 ) ^ { - 1 } ( m + n ) ^ { n - 1 } - ( m + n - 1 ) ^ { - 1 } ( m + n ) ^ { n } \\\\ = & - m ( m + n - 1 ) ^ { - 1 } ( m + n ) ^ { n - 1 } . \\end{align*}"}
+{"id": "6161.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ d \\alpha _ r u _ r ( T ) \\equiv \\sum _ { r = 1 } ^ d \\alpha _ r u _ r ' ( T ) \\equiv 0 . \\end{align*}"}
+{"id": "8251.png", "formula": "\\begin{align*} ( A = B ) \\xrightarrow { ~ ~ \\simeq ~ ~ } ( A \\cong B ) \\end{align*}"}
+{"id": "7425.png", "formula": "\\begin{align*} K ( Z , W ) ( P ) = K _ 1 ( Z , W ) ( P ) - K _ 2 ( Z , W ) ( P ) + i ( K _ 3 ( Z , W ) ( P ) - K _ 4 ( Z , W ) ( P ) ) \\end{align*}"}
+{"id": "8766.png", "formula": "\\begin{align*} x & = [ u ^ * _ 0 , \\dots , u ^ * _ { N - 2 } , u ^ * _ { N - 1 } ] ^ * \\in \\mathbb { R } ^ { r N } \\\\ y & = [ w _ 0 ^ * , \\dots , w ^ * _ { N - 1 } ] ^ * \\in \\mathbb { R } ^ { d _ 0 N } \\\\ z & = [ x ^ * , y ^ * ] ^ * \\in \\mathbb { R } ^ { ( d _ 0 + r ) N } \\\\ U _ l & = [ u _ { 2 T - l } , u _ { 2 T + 1 - l } , \\dots , u _ { N - l } ] \\in \\mathcal { M } _ { p \\times \\bar { N } } ( \\mathbb { R } ) . \\end{align*}"}
+{"id": "3775.png", "formula": "\\begin{align*} f & = \\mathcal { E } \\alpha , & \\mathcal { E } & = \\mathcal { E } ( \\eta ) , & g & = \\mathcal { E } ' \\alpha ' , & \\mathcal { E } ' & = \\mathcal { E } ( \\eta ' ) \\end{align*}"}
+{"id": "550.png", "formula": "\\begin{align*} \\Theta _ { n + 1 } = \\Theta _ n + K _ n \\left \\{ ( X _ { t _ { n + 1 } } ^ \\dagger - X _ { t _ n } ^ \\dagger ) - \\frac { 1 } { 2 } \\left ( f ( X _ { t _ n } ^ \\dagger , \\Theta _ n ) + \\pi _ n [ f ( X ^ \\dagger _ { t _ n } , \\theta ) ] \\right ) \\Delta t \\right \\} \\end{align*}"}
+{"id": "3912.png", "formula": "\\begin{align*} \\det D Y u ( \\cdot ) = \\frac { f ( \\cdot ) } { f ^ * ( Y u ( \\cdot ) ) } \\Omega , \\end{align*}"}
+{"id": "2755.png", "formula": "\\begin{align*} \\overline { i ^ { A } } = i _ { A } ^ { \\ast } . \\end{align*}"}
+{"id": "4497.png", "formula": "\\begin{align*} \\frac { 4 6 9 9 } { 7 3 2 0 } = \\end{align*}"}
+{"id": "3971.png", "formula": "\\begin{align*} Y v ( Y v ^ { - 1 } ( Y u ( \\Omega ' ) ) ) = Y v ( \\Omega ) \\cap Y u ( \\Omega ' ) = Y u ( \\Omega ' ) \\setminus \\mathcal { F } , \\end{align*}"}
+{"id": "6934.png", "formula": "\\begin{align*} L _ n ^ { ( \\alpha ) } ( x ) = \\sum _ { m = 0 } ^ n \\frac { ( - 1 ) ^ m } { m ! } \\binom { n + \\alpha } { n - m } x ^ m . \\end{align*}"}
+{"id": "4041.png", "formula": "\\begin{align*} Y u ^ { - 1 } ( V ) \\setminus E _ u \\subset \\bigcup _ { i = 1 } ^ \\infty \\bigcap _ { k = i } ^ \\infty Y u _ k ^ { - 1 } ( U ) . \\end{align*}"}
+{"id": "8663.png", "formula": "\\begin{align*} q _ { a , b } = - q _ { b , a } , q _ { a , a } = 0 . \\end{align*}"}
+{"id": "2637.png", "formula": "\\begin{align*} h _ { a ^ * \\varpi } ( a \\varpi ^ j , s ; \\chi ) & = \\sum _ { ( n , a ^ * \\varpi ) = 1 } \\frac { \\chi ( n ) g _ 6 ( a \\varpi ^ j , n ) } { N ( n ) ^ s } \\\\ & = \\sum _ { ( n , a ^ * ) = 1 } \\frac { \\chi ( n ) g _ 6 ( a \\varpi ^ j , n ) } { N ( n ) ^ s } - \\sum _ { \\substack { ( n , a ^ * ) = 1 \\\\ \\varpi | n } } \\frac { \\chi ( n ) g _ 6 ( a \\varpi ^ j , n ) } { N ( n ) ^ s } . \\end{align*}"}
+{"id": "3682.png", "formula": "\\begin{align*} \\textrm { d e p t h } _ R \\textrm { T r } \\Omega ^ { d + 1 } \\overline { R } \\geq \\min \\{ \\textrm { d e p t h } _ R F , \\textrm { d e p t h } _ R \\lambda \\Omega ^ { d + 1 } \\overline { R } - 1 \\} = d - 1 > 0 . \\end{align*}"}
+{"id": "855.png", "formula": "\\begin{align*} F ( \\widehat { w } ) = e ^ { \\widehat { w } } J _ d - J _ d ( e ^ { \\widehat { w } } ) ^ T - \\widehat { h \\Pi _ k } \\end{align*}"}
+{"id": "7280.png", "formula": "\\begin{align*} ( - 1 ) ^ { k + 1 } G ^ { k + 1 } _ m ( x ) & = ( - 1 ) ^ k \\int _ { 0 } ^ { x } d y \\int _ { y } ^ { 1 } G ^ k _ m ( z ) d m ( z ) \\\\ & = ( - 1 ) ^ { k + 1 } \\int _ { 0 } ^ { x } d y \\left ( G ^ k _ m ( y ) \\tilde { m } ( y ) + \\int _ { y } ^ { 1 } ( G ^ k _ m ) ^ + ( z ) \\tilde { m } ( z ) d z \\right ) \\\\ & \\leq ( - 1 ) ^ { k + 1 } G ^ k _ m ( 1 ) G ^ 1 _ m ( x ) + ( - 1 ) ^ { k + 1 } \\int _ { 0 } ^ { x } \\tilde { m } ( y ) d y \\int _ { y } ^ { 1 } ( G ^ k _ m ) ^ + ( z ) d z \\\\ & \\leq 2 ( - 1 ) ^ { k + 1 } G ^ k _ m ( 1 ) G ^ 1 _ m ( x ) < \\infty . \\end{align*}"}
+{"id": "5011.png", "formula": "\\begin{align*} c \\ , \\omega ( Z , X , Y ) = - d \\xi ( Z , X , Y ) = \\xi ( [ X , Y ] , Z ) . \\end{align*}"}
+{"id": "6379.png", "formula": "\\begin{align*} & \\gamma _ I ^ 2 = 1 , \\\\ & \\gamma _ I \\gamma _ J = \\gamma _ J \\gamma _ { w _ J ( I ) } , & \\mathrm { i f } \\ , \\ , I \\subset J \\ , \\ , \\mathrm { o r } \\ , \\ , W _ { I \\cup J } = W _ I \\times W _ J . \\end{align*}"}
+{"id": "6722.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\leq x \\\\ n \\equiv a \\bmod q } } \\mu ^ 2 ( n ) = \\frac { 6 } { \\pi ^ 2 } \\prod _ { p \\mid q } \\left ( 1 - \\frac { 1 } { p ^ 2 } \\right ) \\frac { x } { q } + O \\left ( ( x / q ) ^ { 1 / 4 + \\varepsilon } \\right ) , \\end{align*}"}
+{"id": "6119.png", "formula": "\\begin{align*} t = 0 : \\Phi = \\widehat \\Phi _ 0 , \\Phi ' = \\widehat \\Phi _ 1 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "7165.png", "formula": "\\begin{align*} \\displaystyle { w '^ { \\{ k \\} } _ m = - \\frac { d v _ m } { d t ' _ k } \\Big | _ { \\rm e q } \\ , . } \\end{align*}"}
+{"id": "6275.png", "formula": "\\begin{align*} h ( x , y , z ) = \\big ( \\sqrt { r ( y ) } \\cdot x , \\ , y , \\ , \\sqrt { r ( y ) } \\cdot z \\big ) \\end{align*}"}
+{"id": "9009.png", "formula": "\\begin{align*} \\alpha ( A _ { i _ M } C _ { M } , K ) & \\leq \\frac { 1 } { 1 - \\epsilon } \\max _ { g \\in C _ { M } } \\alpha ( A _ { i _ M } g , K ) \\\\ & = \\frac { \\alpha ( A _ { i _ M } , K ) } { 1 - \\epsilon } \\\\ & \\leq \\frac { \\epsilon ^ { 4 } } { 1 - \\epsilon } \\\\ & \\leq \\epsilon ^ 3 . \\end{align*}"}
+{"id": "3841.png", "formula": "\\begin{align*} \\Delta _ { A , p } u : = \\textrm { d i v } _ A ( | \\nabla _ A u | ^ { p - 2 } \\nabla _ A u ) , \\end{align*}"}
+{"id": "5958.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p a _ r e _ r \\in V ^ \\perp . \\end{align*}"}
+{"id": "8303.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\frac { \\partial V ( \\textbf { x } ) } { \\partial x _ E } \\ ! \\ ! \\ ! \\ ! & = \\frac { \\nu _ E } { \\Lambda } ( x _ E - x ^ * ) . \\end{array} \\right . \\end{align*}"}
+{"id": "2767.png", "formula": "\\begin{align*} \\overline { T ^ { + } } = q ^ { - 2 } T ^ { - } , \\qquad \\overline { T ^ { - } } = q ^ { 2 } T ^ { + } , \\overline { T ^ { 3 } } = T ^ { 3 } . \\end{align*}"}
+{"id": "1335.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | ^ m & \\geq \\frac { n - { d + m - 1 \\choose m } } { { d + m - 1 \\choose m } ( n - 1 ) } = \\frac { n - ( ^ m ( \\mathcal { X } ) ) } { ( ^ m ( \\mathcal { X } ) ) ( n - 1 ) } \\\\ & = \\frac { 1 } { n - 1 } \\left [ \\frac { n } { { d + m - 1 \\choose m } } - 1 \\right ] \\end{align*}"}
+{"id": "3052.png", "formula": "\\begin{align*} - A : D ^ 2 v ^ { k l } = a _ { k l } - \\bar { a } _ { k l } \\quad Y , v ^ { k l } Y , \\int _ Y v ^ { k l } = 0 \\end{align*}"}
+{"id": "1257.png", "formula": "\\begin{align*} 4 \\alpha \\left ( 1 + 1 6 \\alpha ( 1 - \\beta ) ^ 2 \\right ) x ^ 2 - 4 ( 1 + \\alpha ) \\sqrt { \\alpha ( 1 + 1 6 \\alpha ( 1 - \\beta ) ^ 2 ) } x + ( 1 + \\alpha ) ^ 2 = 0 . \\end{align*}"}
+{"id": "4150.png", "formula": "\\begin{align*} R _ { i j k l } h _ { i l } h _ { j k } & = 2 \\Big [ ( h _ { 1 1 } h _ { 2 2 } - h ^ 2 _ { 1 2 } ) + ( h _ { 1 1 } h _ { 3 3 } - h ^ 2 _ { 1 3 } ) + ( h _ { 2 2 } h _ { 3 3 } - h ^ 2 _ { 2 3 } ) \\Big ] , \\\\ R _ { j l } h _ { l k } h _ { j k } & = 2 ( h _ { 1 1 } ^ 2 + h _ { 2 2 } ^ 2 + h _ { 3 3 } ^ 2 + 2 h _ { 1 2 } ^ 2 + 2 h _ { 1 3 } ^ 2 + 2 h _ { 2 3 } ^ 2 ) . \\end{align*}"}
+{"id": "2475.png", "formula": "\\begin{align*} | r _ p ( C l ( K _ n ) ) - r _ p ( C l _ S ( K _ n ) ) | = O ( 1 ) . \\end{align*}"}
+{"id": "7922.png", "formula": "\\begin{align*} l ( 0 , x ) = 0 \\ , , l ' ( 0 , x ) = a ^ { - 1 } \\Delta u ( x ) \\end{align*}"}
+{"id": "2123.png", "formula": "\\begin{align*} \\forall \\kappa \\in \\S ^ 1 , F ( \\kappa ) : = \\sum _ { j \\in \\Z } a _ j \\kappa ^ j . \\end{align*}"}
+{"id": "8268.png", "formula": "\\begin{align*} | S _ 1 | + | S _ 3 | \\le 4 e ^ { - 2 m } m = \\frac 1 4 \\log q \\ge \\max \\left ( \\frac 1 4 \\log | \\rho _ 0 | , ( A + \\kappa ) , 4 \\right ) . \\end{align*}"}
+{"id": "7261.png", "formula": "\\begin{align*} \\tilde { \\chi } _ { m , j , \\gamma } ( \\lambda ) & = \\gamma \\int _ { 0 } ^ { \\infty } \\left ( 1 - g _ m \\left ( \\frac { f ( \\gamma ) } { \\gamma } \\lambda ; x \\right ) \\right ) j ( d x ) - b f ( \\gamma ) \\lambda \\\\ & = \\int _ { 0 } ^ { \\infty } ( 1 - g _ { m _ \\gamma } ( \\lambda ; x ) ) j _ \\gamma ( d x ) - b _ \\gamma \\lambda \\\\ & = \\chi _ { m _ { \\gamma } , j _ { \\gamma } } ( \\lambda ) - b _ \\gamma \\lambda \\end{align*}"}
+{"id": "3436.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { 1 - p _ { a a } ^ { t } } { 1 - p _ { b b } ^ { t } } = \\lim _ { t \\to \\infty } \\frac { 1 - q _ { a a } ^ { t } } { 1 - q _ { b b } ^ { t } } . \\end{align*}"}
+{"id": "4222.png", "formula": "\\begin{align*} \\begin{aligned} & ~ { } \\partial _ { t } \\left ( \\partial _ { x } \\Lambda \\partial _ { t } \\Lambda + 4 \\sinh ^ { 2 } { \\Lambda } \\partial _ { x } \\phi \\partial _ { t } \\phi \\right ) \\\\ & ~ { } \\quad - \\partial _ { x } \\left ( \\frac { 1 } { 2 } ( ( \\partial _ { x } \\Lambda ) ^ { 2 } + ( \\partial _ { t } \\Lambda ) ^ { 2 } ) + 2 \\sinh ^ { 2 } ( \\Lambda ) ( ( \\partial _ { x } \\phi ) ^ { 2 } + ( \\partial _ { t } \\phi ) ^ { 2 } ) \\right ) = 0 . \\end{aligned} \\end{align*}"}
+{"id": "3916.png", "formula": "\\begin{align*} X v ( \\Omega ^ * ) = \\Omega . \\end{align*}"}
+{"id": "6957.png", "formula": "\\begin{gather*} \\big \\{ X ^ { ( \\alpha ) } _ { j , m , n } \\big \\} _ { j = 1 } ^ n = \\big \\{ { - } x _ { j , m } ^ { ( \\alpha - 1 ) } \\big \\} _ { j = 1 } ^ m \\cup \\big \\{ x _ { j - m , n - m } ^ { ( \\alpha - 1 ) } \\big \\} _ { j = m + 1 } ^ { n } . \\end{gather*}"}
+{"id": "9202.png", "formula": "\\begin{align*} \\Delta _ { f } = - h ^ { 2 } \\Delta + \\vert \\nabla f \\vert ^ { 2 } - h \\Delta f , \\end{align*}"}
+{"id": "7213.png", "formula": "\\begin{align*} p ( t ) = q ( t ) - p _ 0 ( t ) \\qquad \\Rightarrow \\langle p \\omega , \\ , p ' \\rangle = R ^ 2 . \\end{align*}"}
+{"id": "4604.png", "formula": "\\begin{align*} a _ { k + 1 } = a _ k ^ 2 - a _ k + 1 = q \\prod _ { i = 1 } ^ k a _ i + 1 \\end{align*}"}
+{"id": "6430.png", "formula": "\\begin{align*} { \\mathcal { I } } _ k : = \\sigma ( ( X _ 0 , Y _ 0 ) , ( X _ 1 , Y _ 1 ) , \\ldots , ( X _ k , Y _ k ) ) \\mbox { f o r } k \\geq 1 . \\end{align*}"}
+{"id": "2149.png", "formula": "\\begin{align*} U ' ( \\zeta _ + ( t , x ) ) + U ' ( \\zeta _ - ( t , x ) ) \\geq U ' ( \\zeta _ - ( t , x ) ) \\geq \\min _ { U ^ { - 1 } ( \\delta ) \\leq z \\leq U ^ { - 1 } ( 1 - \\delta ) } U ' ( z ) : = \\vartheta > 0 . \\end{align*}"}
+{"id": "8745.png", "formula": "\\begin{align*} s _ { \\min } ( H g _ 0 ^ * ) & = | s _ { d _ 0 } ( H g _ 0 ^ * ) - s _ { d _ 0 } ( H \\hat { g } ^ * ) | \\leq | s _ 1 ( H \\Delta g ^ * ) | \\\\ & \\leq | H \\Delta g ^ * | _ { S _ 2 } \\leq \\frac { 5 \\sqrt { 2 } } { 3 \\sigma _ u ^ 2 } d _ 0 ^ { 1 / 2 } \\lambda T \\leq \\xi , \\end{align*}"}
+{"id": "6834.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ c ^ { - } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) & = \\sum \\limits _ { k = 1 } ^ { m - 1 } \\mu ( Y _ k , Y _ { k + 1 } ) + \\mu ( Y _ m , Y _ 1 ) = \\sum \\limits _ { k = 1 } ^ { m - 1 } \\mu ^ * ( Y _ { k + 1 } , Y _ { k } ) + \\mu ^ * ( Y _ 1 , Y _ m ) + s _ m = \\mu ^ { + } ( Y _ m , Y _ { m - 1 } , \\dots , Y _ 1 ) + s _ m , \\end{aligned} \\end{align*}"}
+{"id": "7933.png", "formula": "\\begin{align*} A \\cdot B : = \\sum _ { C \\in \\textrm { I s o } ( \\mathcal { A } ) } \\textbf { a } ^ C _ { A , B } C , \\end{align*}"}
+{"id": "2215.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m } ( 5 n + 2 ) + 5 ^ { 2 m } \\right ) } q ^ n & = \\sum _ { i = 1 } ^ \\infty x _ { 2 m , i } U ( q ^ { - 2 } \\delta \\xi ^ { i - 1 } ) \\\\ & = \\sum _ { i = 1 } ^ \\infty x _ { 2 m , i } \\gamma \\sum _ { j = 1 } ^ { 5 i } \\beta _ { i , j } \\xi ^ { j - 1 } \\\\ & = \\gamma \\sum _ { j = 1 } ^ \\infty { \\left ( \\sum _ { i = 1 } ^ \\infty x _ { 2 m , i } \\beta _ { i , j } \\right ) } \\xi ^ { j - 1 } \\\\ & = \\gamma \\sum _ { j = 1 } ^ \\infty x _ { 2 m + 1 , i } \\xi ^ { j - 1 } , \\end{align*}"}
+{"id": "3685.png", "formula": "\\begin{align*} X X + Y ^ T Y & = X X - X C ( - C ^ T X ) \\\\ & = X I _ { n - 1 } X + X C C ^ T X \\\\ & = X ( C C ^ T + I _ { n - 1 } ) X \\\\ & = X ( C C ^ T + I _ { n - 1 } ) ( C C ^ T + I _ { n - 1 } ) ^ { - 1 } ( J _ { n - 1 } - n I _ { n - 1 } ) \\\\ & = X ( J _ { n - 1 } - n I _ { n - 1 } ) \\\\ & = - J _ { n - 1 } - n X \\end{align*}"}
+{"id": "1645.png", "formula": "\\begin{align*} V ( \\underline { k } - 2 ) : = \\bigotimes _ { { \\tilde \\sigma } : F \\hookrightarrow \\C } { \\rm S y m } ^ { k _ { \\tilde \\sigma } - 2 } ( \\C ^ 2 ) . \\end{align*}"}
+{"id": "8865.png", "formula": "\\begin{align*} \\beta _ { \\nu , n , m } = \\frac { ( 2 m + 2 \\nu + n ) \\Gamma ( n + m ) \\Gamma ( n + m + 2 \\nu ) } { \\pi ^ { n } \\Gamma ( n ) m ! ( m + 2 \\nu ) ! } \\end{align*}"}
+{"id": "6432.png", "formula": "\\begin{align*} S ( X , Y ) : = S _ { \\hat Y } ( X ) . \\end{align*}"}
+{"id": "3590.png", "formula": "\\begin{align*} d _ { j } ^ + ( \\lambda ) = \\frac { ( - 1 ) ^ { \\sum _ { \\alpha = j } ^ n ( \\alpha + 1 ) ( \\lambda _ \\alpha - \\lambda _ { \\alpha + 1 } + \\delta _ { \\alpha j } ) } } { ( \\lambda _ j + 1 ) j } \\bigg ( \\prod _ { \\ell = 1 } ^ { j - 1 } \\frac { \\lambda _ j - \\lambda _ \\ell - j + \\ell + 1 } { \\lambda _ j - \\lambda _ \\ell - j + \\ell + [ \\lambda _ j - \\lambda _ \\ell ] _ 2 } \\bigg ) \\end{align*}"}
+{"id": "5546.png", "formula": "\\begin{align*} I ( X _ n , k ) : = \\frac { 1 } { 2 \\pi i } \\int _ { \\lambda - i \\infty } ^ { \\lambda + i \\infty } \\frac { \\Gamma ( s ) \\Gamma \\left ( \\frac { k } { 2 } - s \\right ) } { \\Gamma \\left ( \\frac { 1 - k } { 2 } + s \\right ) } X _ n ^ { - s } { \\rm d } s , \\end{align*}"}
+{"id": "2030.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\bf C } ^ + ( r ) = \\mathcal { I } ^ + ( r ) \\cap \\bigcup _ { k \\geq 1 } A _ k A _ k : = \\{ a _ k + D ' + r , \\ldots , a _ k + r - \\delta \\} \\end{align*}"}
+{"id": "3130.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( \\tilde { A } ) = ( 1 + \\bar { a } ) \\int _ Y r A e _ 1 \\cdot \\nabla w _ { \\gamma } = ( 1 + \\bar { a } ) \\int _ Y r ( 1 + a ) \\partial _ 1 w _ { \\gamma } . \\end{align*}"}
+{"id": "3400.png", "formula": "\\begin{align*} v ( t , x ) = v ^ r ( t , r , z ) \\vec { e } _ { r } + v ^ z ( t , r , z ) \\vec { e } _ { z } , B = B ^ \\theta \\vec { e } _ { \\theta } . \\end{align*}"}
+{"id": "4839.png", "formula": "\\begin{align*} | h _ 2 ( r , t ) | & \\leq L ( | \\tau | + | R _ 2 ( r , t ) | ) \\\\ & \\leq L ( | \\tau | + \\sigma ( | r | + | t | ) ) \\\\ & \\leq L ( \\lambda _ 0 \\delta + 2 \\sigma \\varepsilon ) . \\end{align*}"}
+{"id": "51.png", "formula": "\\begin{align*} T ( n _ i ) + T ( n _ { i + 1 } ) + T ( n _ { i + 1 } ) & < \\left ( \\frac { 2 9 7 } { 8 1 } + \\frac { 1 2 8 } { 2 4 3 } \\right ) \\cdot \\frac { 1 } { X _ 0 } \\\\ & = \\frac { 1 0 1 9 } { 2 4 3 } \\cdot \\frac { 1 } { X _ 0 } < 6 \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } \\\\ & = ( k _ i + k _ { i + 1 } + k _ { i + 2 } ) \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } . \\end{align*}"}
+{"id": "8496.png", "formula": "\\begin{align*} P _ 0 & = \\ ; \\frac { 8 1 \\Big ( 4 \\sqrt { 2 } \\xi ( { \\xi } ^ 2 - 8 1 ) - 6 { \\xi } ^ 2 + 2 4 7 5 \\Big ) } { 4 { \\xi } ^ 8 } , \\\\ P _ 1 & = \\ ; \\frac { 2 7 \\Big ( \\sqrt { 2 } \\xi ( 9 3 - 2 { \\xi } ^ 2 ) + 3 { \\xi } ^ 2 - 6 0 3 \\Big ) } { 2 { \\xi } ^ 6 } , \\\\ P _ 2 & = \\ ; \\frac { 9 \\Big ( 1 2 \\sqrt { 2 } \\xi ( 2 { \\xi } ^ 2 - 4 3 ) + 4 { \\xi } ^ 4 - 3 6 { \\xi } ^ 2 + 2 7 8 1 \\Big ) } { 3 2 { \\xi } ^ 4 } , \\\\ P _ 3 & = \\ ; \\frac { 9 ( 6 \\sqrt { 2 } \\xi + 2 { \\xi } ^ 2 - 2 7 ) } { 4 { \\xi } ^ 2 } . \\end{align*}"}
+{"id": "2168.png", "formula": "\\begin{align*} \\zeta ( \\sigma _ { 0 } ) = & \\dfrac { 1 } { 4 ( - 4 + 6 \\alpha + 3 \\beta ) ^ 2 } ( 2 ( 8 - 6 \\alpha - 3 \\beta ) ( 6 \\alpha - 6 - 3 \\beta ) ( 6 \\alpha - 6 - 3 \\beta \\\\ & \\quad { } + \\sqrt { ( 6 - 6 \\alpha + 3 \\beta ) ^ 2 + 8 ( 6 \\alpha + 3 \\beta - 4 ) } ) \\\\ & \\quad + 4 8 ( 6 \\alpha + 3 \\beta - 4 ) ( 2 - 2 \\alpha - \\beta ) ) , \\end{align*}"}
+{"id": "8235.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { N } } \\left ( 1 + 2 w ^ 2 \\right ) \\nabla w \\cdot \\nabla h \\ , d x + \\int _ { \\mathbb { R } ^ { N } } 2 w | \\nabla w | ^ 2 \\ , h \\ , d x + \\int _ { \\mathbb { R } ^ { N } } V \\left ( \\left | x \\right | \\right ) w h \\ , d x = \\int _ { \\mathbb { R } ^ { N } } K ( | x | ) g ( w ) h \\ , d x , \\end{align*}"}
+{"id": "1434.png", "formula": "\\begin{align*} \\eta _ t = \\mathbb { 1 } _ { \\{ h _ t \\in \\mathcal { U } _ { t - 1 } \\} } J _ t \\left | \\mathcal { S } ( h _ t ) \\cap \\mathcal { U } _ { t - 1 } \\setminus \\{ h _ t \\} \\right | - \\mathbb { 1 } _ { \\{ h _ t \\in \\mathcal { A } _ { t - 1 } \\} } . \\end{align*}"}
+{"id": "7565.png", "formula": "\\begin{align*} Z _ { \\tilde { f } _ { 5 , a } } ( s , \\chi , C _ 5 ^ a ) = { \\left \\{ \\begin{array} { r l } F _ { 5 , a } ( q ^ { - s } ) , \\ \\ & 0 \\le a < \\dfrac { l } { p } , \\\\ \\dfrac { \\tilde F _ { 5 , a } ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\ \\ & { \\dfrac { l } { p } \\le a < \\omega } , \\end{array} \\right . } \\end{align*}"}
+{"id": "6349.png", "formula": "\\begin{align*} \\psi _ { f } ( \\nu ) : = 2 \\int _ 1 ^ { ( f | _ { ( 1 , r _ 1 ) } ) ^ { - 1 } ( \\nu ) } \\frac { \\nu } { f ( x ) \\sqrt { f ( x ) ^ 2 - \\nu ^ 2 } } d x \\end{align*}"}
+{"id": "3314.png", "formula": "\\begin{align*} F \\left ( \\mathbf { x } \\right ) = \\prod _ { 0 \\le i < m } \\left ( P _ { i } \\left ( \\mathbf { x } \\right ) \\right ) ^ { \\alpha _ { i } } , G \\left ( \\mathbf { x } \\right ) = \\prod _ { 0 \\le i < m } \\left ( P _ { i } \\left ( \\mathbf { x } \\right ) \\right ) ^ { \\beta _ { i } } , \\end{align*}"}
+{"id": "2221.png", "formula": "\\begin{align*} \\nu ( x _ { 2 m - 1 , 3 } ) + \\nu ( \\alpha _ { 3 , i } ) & \\geq 2 m - 1 + \\left \\lfloor \\dfrac { 5 i - 4 } { 6 } \\right \\rfloor \\\\ & = 2 m + \\left \\lfloor \\dfrac { 5 i - 1 0 } { 6 } \\right \\rfloor . \\end{align*}"}
+{"id": "731.png", "formula": "\\begin{align*} \\{ S ( z ) , z \\} _ 1 = \\frac { S '' ( z ) } { S ' ( z ) } = - 2 \\frac { T ' ( z ) } { T ( z ) } . \\end{align*}"}
+{"id": "2089.png", "formula": "\\begin{align*} \\dim _ P ( \\mathcal { S } _ K ^ { \\ast } ( w ) ) \\geq \\dim _ P ( \\mathcal { S } _ K ^ { ( b ) \\ast } ( w ) ) \\geq ( 1 - w ) \\dim ( K ) = ( 1 - w ) \\frac { \\sum _ { i = 1 } ^ { m } \\log | W _ i | } { \\log b } > 0 . \\end{align*}"}
+{"id": "5986.png", "formula": "\\begin{align*} p ^ { - 1 } ( g ) = \\overline { { \\mathcal { M } } _ { 0 , r } } ( F l _ J , \\d ) \\times _ { ( F l _ I ) ^ r } \\left ( g \\cdot ( F l _ J ) ^ r \\right ) & \\simeq \\pi ^ { - 1 } ( g ) \\\\ & \\simeq g \\cdot \\left ( \\overline { { \\mathcal { M } } _ { 0 , r } } ( F l _ J , \\d ) \\times _ { ( F l _ I ) ^ r } ( F l _ J ) ^ r \\right ) \\\\ & \\simeq \\overline { { \\mathcal { M } } _ { 0 , r } } ( F l _ J , \\d ) \\times _ { ( F l _ I ) ^ r } ( F l _ J ) ^ r \\end{align*}"}
+{"id": "279.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\tau } { \\rho } + \\beta \\rho \\eta \\geqslant \\frac { \\alpha ( 1 + 3 \\rho ^ 2 ) } { 2 \\rho ^ 2 } \\geqslant \\frac { 3 \\alpha } { 2 } , \\end{align*}"}
+{"id": "1120.png", "formula": "\\begin{align*} \\frac { 1 } { \\sum _ { j = 1 } ^ n \\eta _ j } \\sum _ { i = 1 } ^ n \\eta _ i \\left [ \\mathbb { E } _ { X , Y } \\{ | X - \\theta _ i | \\} - \\mathbb { E } _ X \\{ | X - \\theta | \\} \\right ] \\leq \\frac { 2 r } { 1 - \\varepsilon } \\left ( \\frac { 1 + ( e + 1 ) ^ 2 } { \\sqrt { n \\alpha ^ 2 } } + \\varepsilon \\right ) , \\end{align*}"}
+{"id": "1310.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\| \\langle \\tau _ j , \\tau _ k \\rangle \\| ^ { 2 } \\geq \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , \\tau _ k \\rangle \\langle \\tau _ k , \\tau _ j \\rangle \\geq \\frac { n ^ 2 } { { d } } . \\end{align*}"}
+{"id": "6569.png", "formula": "\\begin{align*} \\sum _ { ( q , r ) \\in \\mathcal { E } ^ { 1 } } \\lambda ( \\mathcal { A } _ q \\cap \\mathcal { A } _ r ) = \\sum _ { q = 1 } ^ Q \\lambda ( \\mathcal { A } _ q ) = \\Psi ( Q ) . \\end{align*}"}
+{"id": "7348.png", "formula": "\\begin{align*} u ( \\gamma ) : = \\sqrt { c _ 1 } ( \\log \\gamma ) ^ { s / 2 } v ( \\gamma ) : = c _ 2 ( \\log \\gamma ) ^ t ( \\gamma \\geq 1 ) \\end{align*}"}
+{"id": "6860.png", "formula": "\\begin{gather*} j ( t ) = J ( t u + ( 1 - t ) v ; F ) = \\\\ t ^ 2 ( Q u - F , Q u - F ) _ Y + ( 1 - t ) ^ 2 ( Q v - F , Q v - F ) _ Y + \\\\ t ( 1 - t ) ( Q u - F , Q v - F ) _ Y . \\end{gather*}"}
+{"id": "3892.png", "formula": "\\begin{align*} \\frac { g _ y } { g _ z } ( x _ \\theta , y _ 0 , z _ 0 ) = \\theta q _ 1 + ( 1 - \\theta ) q _ 0 , \\end{align*}"}
+{"id": "3845.png", "formula": "\\begin{align*} \\{ a \\in \\mathcal { P } _ \\kappa \\lambda \\mid f _ q ( a ) \\in G \\} \\in U & \\Leftrightarrow [ f _ q ] _ { U _ \\alpha } = q \\in i ( G ) ( \\alpha ) \\\\ & \\Leftrightarrow k ( q ) = q \\in k \\circ i ( G ) = \\overline { G } . \\end{align*}"}
+{"id": "5916.png", "formula": "\\begin{align*} t \\geq T : U = e _ 1 u _ 1 + e _ 2 u _ 2 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "5372.png", "formula": "\\begin{align*} { \\rm m a x r a n k } ( { \\mathcal { F } } ) = \\max _ { F \\in \\mathcal { F } , x \\in U } { \\rm r a n k } ( D F ( x ) ) . \\end{align*}"}
+{"id": "4141.png", "formula": "\\begin{align*} \\| D h \\| _ { L ^ 2 } ^ 2 & = \\frac { 1 } { 3 } \\int _ M ( \\nabla _ i h _ { j k } + \\nabla _ j h _ { k i } + \\nabla _ k h _ { i j } ) ^ 2 d V _ g = \\| \\nabla h \\| _ { L ^ 2 } ^ 2 + 2 \\int _ M \\nabla _ i h _ { j k } \\nabla _ j h _ { k i } d V _ g . \\end{align*}"}
+{"id": "6136.png", "formula": "\\begin{align*} D ^ T E _ r = 0 , 1 \\leqslant r \\leqslant d \\end{align*}"}
+{"id": "2567.png", "formula": "\\begin{align*} \\Upsilon ( g ' g ^ { - 1 } ) ( t ) & = ( \\frac { 1 } { 2 } g ' ( t / 2 ) g ( t / 2 ) ^ { - 1 } , - \\frac { 1 } { 2 } g ' ( 1 - t / 2 ) g ( 1 - t / 2 ) ^ { - 1 } ) \\\\ & = ( \\tilde { g } _ 1 ' ( t ) \\tilde { g } _ 1 ( t ) ^ { - 1 } , \\tilde { g } _ 2 ' ( t ) \\tilde { g } _ 2 ( t ) ^ { - 1 } ) = ( \\tilde { g } _ 1 , \\tilde { g } _ 2 ) ' ( \\tilde { g } _ 1 , \\tilde { g } _ 2 ) ^ { - 1 } = \\Omega ( g ) ' \\Omega ( g ) ^ { - 1 } ( t ) . \\end{align*}"}
+{"id": "6517.png", "formula": "\\begin{align*} ( \\zeta ) = \\frac { 4 l } { k ^ 2 } \\left ( \\underset { i = 1 } { \\overset { k } { \\sum } } ( p _ i - \\frac { 1 } { 2 } ) \\right ) ^ 2 \\underset { i = 1 } { \\overset { k } { \\sum } } p _ i ( 1 - p _ i ) \\leq \\frac { l } { k ^ 2 } \\left ( \\underset { i = 1 } { \\overset { k } { \\sum } } ( p _ i - \\frac { 1 } { 2 } ) \\right ) ^ 2 . \\end{align*}"}
+{"id": "8110.png", "formula": "\\begin{align*} \\tilde H _ { m , n } ^ { + , 1 } ( x ) & = 4 T \\int _ { - \\infty } ^ \\infty \\widehat { k ^ * } ( \\xi ) e \\Bigl ( - \\frac { T \\xi } { M } - \\frac { x } { 2 \\pi } - \\frac { \\pi x \\xi ^ 2 } { 4 M ^ 2 } - \\frac { \\pi ^ 3 x \\xi ^ 4 } { 4 8 M ^ 4 } - \\frac { \\pi ^ 5 x \\xi ^ 6 } { 1 4 4 0 M ^ 6 } \\Bigr ) \\ , d \\xi + { } \\\\ & \\qquad \\qquad { } + O \\Bigl ( T \\int _ { - \\infty } ^ \\infty \\abs { \\widehat { k ^ * } ( \\xi ) } \\frac { \\abs { \\xi } ^ 8 \\abs { x } } { M ^ 8 } \\ , d \\xi \\Bigr ) . \\end{align*}"}
+{"id": "1718.png", "formula": "\\begin{align*} \\Upsilon = \\iota ^ { - 1 } \\left | \\begin{array} { c c } x _ 1 & y _ 1 \\\\ x _ 2 & y _ 2 \\end{array} \\right | ^ { k - 2 } \\in V ( k - 2 ) \\otimes V ( k - 2 ) , \\end{align*}"}
+{"id": "8211.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 p } { \\partial t ^ 2 } + ( a _ 1 + a _ 2 ) \\frac { \\partial ^ 2 p } { \\partial t \\partial x } + ( \\lambda _ 1 + \\lambda _ 2 ) \\frac { \\partial p } { \\partial t } = - a _ 1 a _ 2 \\frac { \\partial ^ 2 p } { \\partial x ^ 2 } + \\frac { ( a _ 1 + a _ 2 ) ( \\lambda _ 1 + \\lambda _ 2 ) + ( a _ 1 - a _ 2 ) ( \\lambda _ 1 - \\lambda _ 2 ) } { 2 } \\frac { \\partial p } { \\partial x } . \\end{align*}"}
+{"id": "8780.png", "formula": "\\begin{align*} \\mathbf { d } ( ( \\phi ( t , \\cdot ) , \\eta _ t ) , ( \\varphi ( t , \\cdot ) , \\zeta _ t ) ) = 0 . \\end{align*}"}
+{"id": "3574.png", "formula": "\\begin{align*} m _ { i j } = \\sum _ { k = i } ^ j ( \\gamma _ A ) _ { k i } , ( 1 \\leq i \\leq j \\leq n ) . \\end{align*}"}
+{"id": "6097.png", "formula": "\\begin{gather*} \\begin{aligned} \\mathcal { F } _ k ( f ( \\tau ) ) & = - \\overline { c _ f ^ - ( 0 ) } v ^ { 1 - k } - ( - k ) ! \\sum _ { \\substack { n \\gg - \\infty \\\\ n \\neq 0 } } \\overline { c _ f ^ - ( - n ) } q ^ n - \\overline { c _ f ^ + ( 0 ) } \\\\ & - \\frac { 1 } { ( - k ) ! } \\sum _ { \\substack { n \\ll \\infty \\\\ n \\neq 0 } } \\overline { c _ f ^ + ( - n ) } \\Gamma ( 1 - k , - 4 \\pi n v ) q ^ n . \\end{aligned} \\end{gather*}"}
+{"id": "7887.png", "formula": "\\begin{align*} g ''' ( z ) & + c _ j e ^ { z / 3 } g '' ( z ) + \\left ( \\frac { c _ j ^ 2 } { 3 } e ^ { 2 z / 3 } + \\frac { c _ j } { 3 } e ^ { z / 3 } - K \\right ) g ' ( z ) \\\\ & + \\frac { c _ j } { 3 } e ^ { z / 3 } \\left ( \\frac { c _ j } { 3 } e ^ { z / 3 } + \\frac { c _ j } { 9 } - K \\right ) g ( z ) = 0 . \\end{align*}"}
+{"id": "6169.png", "formula": "\\begin{align*} \\begin{pmatrix} C _ 1 \\\\ E _ 1 ^ T \\end{pmatrix} e _ 1 = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "1204.png", "formula": "\\begin{align*} \\frac { \\nabla _ { g _ C } } { d t } [ ( \\nabla ^ k _ { g _ C } \\Phi _ \\ast ^ t ) ( v _ 1 , \\ldots , v _ { k + 1 } ) ] = \\nabla _ { g _ C } X | _ { \\Phi ^ t ( p ) } [ ( \\nabla ^ k _ { g _ C } \\Phi _ \\ast ^ t ) ( v _ 1 , \\ldots , v _ { k + 1 } ) ] + \\Theta ^ { t } . \\end{align*}"}
+{"id": "6935.png", "formula": "\\begin{gather*} \\frac { \\rm d } { { \\rm d } x } L _ n ^ { ( \\alpha ) } ( x ) = - L _ { n - 1 } ^ { ( \\alpha + 1 ) } ( x ) , \\\\ [ . 5 e x ] L _ n ^ { ( \\alpha ) } ( x ) = L _ { n } ^ { ( \\alpha + 1 ) } ( x ) - L _ { n - 1 } ^ { ( \\alpha + 1 ) } ( x ) , \\\\ x L _ { n } ^ { ( \\alpha + 1 ) } ( x ) = ( n + \\alpha + 1 ) L _ n ^ { ( \\alpha ) } ( x ) - ( n + 1 ) L _ { n + 1 } ^ { ( \\alpha ) } ( x ) . \\end{gather*}"}
+{"id": "7821.png", "formula": "\\begin{align*} d ( ( 2 , 1 ) ) = k d _ 1 , d ( ( 1 , 2 ) ) = \\begin{cases} k d _ 1 / 2 & ( k : ) , \\\\ k d _ 1 & ( k : ) . \\end{cases} \\end{align*}"}
+{"id": "3337.png", "formula": "\\begin{align*} \\Phi _ { z \\kappa } = 0 , z = x , y , q , p , \\kappa . \\end{align*}"}
+{"id": "5470.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathbb R ^ d , i \\in \\mathcal M } P ( \\eta _ { \\alpha } > \\eta + \\delta : \\alpha _ { \\eta } = i , X _ { \\eta } = x ) \\geq 1 - \\frac { 1 } { 4 } P ( B ) , \\\\ \\inf _ { y \\in \\mathbb R ^ d , i \\in \\mathcal M } P ( \\eta _ { \\tilde \\alpha } > \\eta + \\delta : \\tilde \\alpha _ { \\eta } = i , Y _ { \\eta } = y ) \\geq 1 - \\frac { 1 } { 4 } P ( B ) . \\end{align*}"}
+{"id": "5833.png", "formula": "\\begin{align*} U '' + \\mathcal L U + A U + D _ 1 \\mathcal G _ 1 U ' + D _ 2 \\mathcal G _ 2 U ' = 0 , \\end{align*}"}
+{"id": "8264.png", "formula": "\\begin{align*} m \\geq \\log Y , & M : = 1 6 m , ~ ~ \\mu : = 1 2 m , q : = e ^ { M - \\mu } = e ^ { 4 m } , Q : = e ^ { M + \\mu } = e ^ { 2 8 m } , \\end{align*}"}
+{"id": "1307.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\| \\langle \\tau _ j , \\tau _ k \\rangle \\| ^ { 2 m } \\geq \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , \\tau _ k \\rangle ^ { m } \\langle \\tau _ k , \\tau _ j \\rangle ^ { m } \\geq \\frac { n ^ 2 } { { d + m - 1 \\choose m } } , \\forall m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "2267.png", "formula": "\\begin{align*} & \\rho ( t , x ) = \\rho _ B ( x ) , ( t , x ) \\in ( 0 , T ) \\times \\Gamma _ { \\rm { i n } } , \\\\ & u ( t , x ) = u _ B ( x ) , ( t , x ) \\in ( 0 , T ) \\times \\partial \\Omega , \\end{align*}"}
+{"id": "3722.png", "formula": "\\begin{align*} & - \\sum _ { j = 1 } ^ { i + 1 } \\sum _ { \\xi \\in X _ { j } ^ \\circ ( F ) } I ( d _ { i + 1 , j } ( f ) ) ( \\xi ) + \\sum _ { j = 1 } ^ { i + 1 } \\sum _ { \\xi \\in X _ { j } ^ \\circ ( F ) } I ( d _ { i + 1 , j } ( \\mathcal { F } _ { X _ { i + 1 } } ( f ) ) ) ( \\xi ) \\\\ & - \\kappa d _ { i + 1 , 0 } ( f ) ( 0 _ { V _ 0 } , 0 , 0 ) + \\kappa d _ { i + 1 , 0 } ( \\mathcal { F } _ { X _ { i + 1 } } ( f ) ) ( 0 _ { V _ 0 } , 0 , 0 ) \\\\ & - c _ { i + 1 } ( f ) + c _ { i + 1 } ( \\mathcal { F } _ { X _ { i + 1 } } ( f ) ) . \\end{align*}"}
+{"id": "4580.png", "formula": "\\begin{align*} 1 = \\frac { 1 } { n } + \\frac { 1 } { n } + \\frac { 1 } { n } + \\cdots + \\frac { 1 } { n } \\end{align*}"}
+{"id": "6862.png", "formula": "\\begin{align*} f i n d \\ ; u \\in X \\ ; s u c h \\ ; t h a t \\ ; a ( u , v ) = G ( v ) \\ ; \\forall v \\in X , \\end{align*}"}
+{"id": "5904.png", "formula": "\\begin{align*} A e _ r = \\sum _ { s = 1 } ^ p \\alpha _ { s r } e _ s 1 \\leqslant r \\leqslant p , \\end{align*}"}
+{"id": "2656.png", "formula": "\\begin{align*} u _ N ( y _ 1 , \\dots , y _ N ) = u _ M ( y _ 1 , \\dots , y _ N , 0 , \\dots , 0 ) \\qquad \\forall ( y _ j ) _ { j = 1 } ^ N \\in \\R ^ N , \\end{align*}"}
+{"id": "2505.png", "formula": "\\begin{align*} \\varphi _ 0 = \\frac { 1 - { \\mathbf v } } { 1 - { \\mathbf u } } . \\end{align*}"}
+{"id": "6031.png", "formula": "\\begin{align*} \\O _ u \\star \\O _ v & = \\O _ { h _ i } \\star \\mathcal { O } _ { u ' } \\star \\mathcal { O } _ v = \\sum _ { w \\in W ^ P , \\beta _ 1 \\in E ( X ) } N _ { u ' , v } ^ { w , \\beta } Q ^ { \\beta _ 1 } \\O _ { h _ i } \\star \\mathcal { O } _ w \\\\ & = \\sum _ { w , \\ : w ' \\in W ^ P , \\ : \\beta _ 1 , \\beta _ 2 \\in E ( X ) } N _ { u ' , v } ^ { w , \\beta _ 1 } Q ^ { \\beta _ 1 } N _ { h _ i , w } ^ { w ' , \\beta _ 2 } Q ^ { \\beta _ 2 } \\mathcal { O } _ { w ' } \\end{align*}"}
+{"id": "786.png", "formula": "\\begin{align*} \\nu _ n : = \\frac { \\sum _ { | g | _ S \\leq n } \\lambda ^ { - | g | _ S } \\delta _ g } { \\sum _ { | g | _ S \\leq n } \\lambda ^ { - | g | _ S } } . \\end{align*}"}
+{"id": "2365.png", "formula": "\\begin{align*} E ^ T = E , A ^ T = - A - \\dot E \\end{align*}"}
+{"id": "4264.png", "formula": "\\begin{align*} E _ \\gamma ( \\phi ) = \\frac { 1 } { \\lambda ^ 2 } E _ \\gamma ( \\lambda \\phi ) + \\frac { 2 ( \\lambda ^ { p - 1 } - 1 ) } { p + 1 } \\| \\phi \\| ^ { p + 1 } _ { L ^ { p + 1 } } . \\end{align*}"}
+{"id": "1972.png", "formula": "\\begin{align*} A _ { k } ( z ) f ( z + c _ { k } ) + A _ { k - 1 } ( z ) f ( z + c _ { k - 1 } ) + \\cdots + A _ { 1 } ( z ) f ( z + c _ { 1 } ) + A _ { 0 } ( z ) f ( z ) = F ( z ) \\end{align*}"}
+{"id": "90.png", "formula": "\\begin{align*} \\mathfrak { h } _ { \\mathbf { k } } = \\mathfrak { g } _ { \\mathbf { k } } \\circ [ s ] \\end{align*}"}
+{"id": "3147.png", "formula": "\\begin{align*} Q _ 1 & : = \\int _ 0 ^ 1 \\int _ 0 ^ 1 a ( y _ 1 , y _ 2 ) R _ 1 ' ( y _ 1 + y _ 2 ) \\mathrm { d } y _ 1 \\mathrm { d } y _ 2 , \\\\ Q _ 2 & : = \\int _ 0 ^ 1 \\int _ 0 ^ 1 a ( y _ 1 , y _ 2 ) R _ 2 ' ( y _ 1 - y _ 2 ) \\mathrm { d } y _ 1 \\mathrm { d } y _ 2 . \\end{align*}"}
+{"id": "3402.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t \\Omega + v \\cdot \\nabla \\Omega - \\mu \\big ( \\Delta + \\frac { 2 } { r } \\partial _ r \\big ) \\Omega = - \\partial _ z \\Sigma ^ 2 , & \\\\ \\partial _ { t } \\Sigma + v \\cdot \\nabla \\Sigma - \\big ( \\Delta + \\frac { 2 } { r } \\partial _ r \\big ) \\Sigma = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "9119.png", "formula": "\\begin{align*} \\tilde { i } ( X ^ n ; Y ^ n ) : = \\log \\frac { d P _ { Y ^ n | X ^ n } ( Y ^ n | X ^ n ) } { d Q _ { Y ^ n } ( Y ^ n ) } , \\end{align*}"}
+{"id": "5643.png", "formula": "\\begin{align*} & [ 0 , 1 ] ^ { | V ( F ) | - | | S ^ k ( F ) | | } \\ni ( z _ { v _ 1 } , \\cdots , z _ { v _ p } ) \\in \\Delta _ F \\setminus S ^ k ( F ) \\\\ : \\Longleftrightarrow & \\left ( \\{ v _ 1 , \\cdots , v _ p \\} = V ( F ) \\setminus \\{ v ' \\in s _ v | s _ v \\in S ^ k ( F ) \\} ~ \\wedge ~ ( v _ i \\preceq v _ j \\Leftrightarrow z _ { v _ j } \\leq z _ { v _ i } ) \\right ) \\end{align*}"}
+{"id": "6416.png", "formula": "\\begin{align*} \\alpha = 1 , \\beta = 0 , \\textrm { i f } p = \\infty , \\textrm { a n d } \\alpha = \\frac { p - 2 } { m + p } , \\beta = \\frac { m + 2 } { m + p } , \\textrm { i f } p < \\infty . \\end{align*}"}
+{"id": "4383.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } \\frac { 1 } { a _ i } \\leq \\sum _ { i = 1 } ^ { n } \\frac { 1 } { a _ i } < \\sum _ { i = 1 } ^ { n - 1 } \\frac { 1 } { a _ i } + \\frac { 1 } { a _ n - 1 } \\leq \\sum _ { i = 1 } ^ { k - 1 } \\frac { 1 } { a _ i } + \\frac { 1 } { a _ k - 1 } \\end{align*}"}
+{"id": "5891.png", "formula": "\\begin{align*} D ^ T C _ p ^ T E = 0 . \\end{align*}"}
+{"id": "4666.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { \\min \\{ 2 p , 2 m \\} } A _ { m , \\ell } = \\sum _ { m = 0 } ^ { p } \\sum _ { \\ell = 0 } ^ { 2 m } A _ { m , \\ell } + \\sum _ { m = p + 1 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { 2 p } A _ { m , \\ell } . \\end{align*}"}
+{"id": "7929.png", "formula": "\\begin{align*} \\Delta f _ t = \\frac { \\partial f _ t } { \\partial \\lambda } \\ , , \\quad \\ , . \\end{align*}"}
+{"id": "3764.png", "formula": "\\begin{align*} \\bar \\varepsilon ^ 2 ( \\Delta ) = ( \\bar \\varepsilon \\bar \\varepsilon ) ( \\Delta ) = \\Delta \\end{align*}"}
+{"id": "400.png", "formula": "\\begin{align*} T \\omega _ n = \\tau _ n , \\forall n \\in \\mathbb { N } , \\end{align*}"}
+{"id": "4529.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 6 } = \\frac { 1 } { 3 } + \\frac { 1 } { 3 } = \\frac { 2 } { 3 } < \\theta . \\end{align*}"}
+{"id": "2625.png", "formula": "\\begin{align*} ( r - 2 ) ( k + 1 ) = ( r - 2 ) ( r - 1 ) s \\end{align*}"}
+{"id": "2358.png", "formula": "\\begin{align*} W : = \\dfrac { 1 } { 2 \\pi } \\left [ \\widetilde { \\alpha _ 0 } ( 0 ) - \\alpha _ 0 ( 0 ) \\right ] \\in \\Z . \\end{align*}"}
+{"id": "7183.png", "formula": "\\begin{align*} \\mathcal { E } = \\bigcup _ { t \\in [ 0 , \\ , \\infty ] } \\{ t \\} \\times E _ t . \\end{align*}"}
+{"id": "1808.png", "formula": "\\begin{align*} \\begin{cases} \\dot { y } = A y + B z , \\\\ \\dot { z } = C y - A ^ T z , \\end{cases} \\end{align*}"}
+{"id": "4621.png", "formula": "\\begin{align*} \\frac { 1 } { 5 } + \\frac { 1 } { 1 3 } < \\frac { 1 } { 6 } + \\frac { 1 } { 9 } < \\frac { 1 } { 4 } + \\frac { 1 } { 1 9 } < \\theta = \\frac { 1 } { 4 } + \\frac { 1 } { 1 8 } < \\frac { 1 } { 3 } . \\end{align*}"}
+{"id": "7622.png", "formula": "\\begin{align*} \\mathtt { A } ( s ) & : = \\langle A ^ n ( s , u _ n ) - A ^ n ( s , v _ n ) , \\psi _ k ' ( w _ n ( s ) ) \\rangle , \\\\ \\mathtt { B } ( s ) & : = \\sum _ { i = 1 } ^ { n } \\big ( \\psi '' _ k ( w _ n ( s ) ) ( g ( u _ n ( s ) ) - g ( v _ n ( s ) ) ) \\phi _ i , ( g ( u _ n ( s ) ) - g ( v _ n ( s ) ) \\phi _ i ) \\big ) , \\end{align*}"}
+{"id": "7935.png", "formula": "\\begin{align*} 0 \\cdot 0 = 0 , 1 \\cdot 1 = 1 , 0 \\cdot 1 = 0 , 0 + 0 = 0 , 1 + 0 = 1 , 1 + 1 = 1 . \\end{align*}"}
+{"id": "898.png", "formula": "\\begin{align*} \\mathcal { A } * \\mathcal { X } = \\mathcal { B } , \\end{align*}"}
+{"id": "8657.png", "formula": "\\begin{align*} s _ \\lambda = \\det [ h _ { \\lambda _ i - i + j } ] _ { 1 \\le i , j \\le l } \\end{align*}"}
+{"id": "7521.png", "formula": "\\begin{align*} S _ { k , r } ( k + 1 ) = & \\dbinom { k + r } { k + 1 } \\Big ( \\sum \\limits _ { i = 0 } ^ k ( - 1 ) ^ i \\dbinom { k + 1 } { i } \\Big ) \\\\ = & \\dbinom { k + r } { k + 1 } \\Big ( \\sum \\limits _ { i = 0 } ^ { k + 1 } ( - 1 ) ^ i \\dbinom { k + 1 } { i } - ( - 1 ) ^ { k + 1 } \\Big ) \\\\ = & ( - 1 ) ^ k \\dbinom { k + r } { k + 1 } . \\end{align*}"}
+{"id": "825.png", "formula": "\\begin{align*} p _ i = \\lim _ { n \\to \\infty } \\frac { e _ i ^ T A ^ { n p } 1 } { \\lambda ^ { n p } } \\ \\ \\ \\ u _ i = \\lim _ { n \\to \\infty } \\frac { e _ \\ast ^ T A ^ { n p } e _ i } { \\lambda ^ { n p } } \\end{align*}"}
+{"id": "510.png", "formula": "\\begin{align*} { \\textstyle \\sum _ { \\overline { K } _ { \\ ! 1 } \\ni j = k } ^ { \\nu } } \\| x ^ { j + 1 } \\ ! - \\ ! x ^ j \\| \\le ( { 1 } / { 2 } ) { \\textstyle \\sum _ { j = k } ^ { \\nu } } \\Xi _ j + \\widehat { c } _ 1 ( \\theta ) \\Xi _ k ^ { \\frac { 1 - \\theta } { \\theta } } . \\end{align*}"}
+{"id": "2836.png", "formula": "\\begin{align*} \\gamma = \\frac { a _ k + a _ { \\ell } } { 2 } = \\frac { b _ k + b _ { \\ell } } { 2 } \\end{align*}"}
+{"id": "1523.png", "formula": "\\begin{align*} p ^ { b c } _ A = 2 \\delta ^ i _ A \\rho _ { i , d } ^ b \\eta ^ { d c } \\end{align*}"}
+{"id": "7787.png", "formula": "\\begin{align*} \\Psi _ { a , b } [ n ] : = \\exp \\Biggl ( \\sum _ { j = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { j + 1 } } { j [ j a ] _ q } q ^ { j b } X _ { j n } \\Biggr ) \\in G _ { n _ 0 } ^ { \\parallel } , \\end{align*}"}
+{"id": "2513.png", "formula": "\\begin{align*} \\alpha _ n ^ d = n ^ d - ( 2 ^ d - 1 ) n ^ { d - 1 } + \\mathcal { O } ( n ^ { d - 2 } ) , \\end{align*}"}
+{"id": "2652.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\xi _ i Z _ { x _ i } \\ ; \\ ; \\mbox { i s a c e n t e r e d G a u s s i a n R V } . \\end{align*}"}
+{"id": "1366.png", "formula": "\\begin{align*} | f _ \\alpha ( \\tau _ \\beta ) | = \\gamma , \\forall \\alpha , \\beta \\in \\Omega , \\alpha \\neq \\beta . \\end{align*}"}
+{"id": "717.png", "formula": "\\begin{align*} d s _ { \\rm m e t r i c } ^ 2 = \\lambda ( w ) ^ 2 | d w | ^ 2 = \\frac { 2 V } { \\pi } h _ { 1 1 } ( w ) | d w | ^ 2 , \\end{align*}"}
+{"id": "1129.png", "formula": "\\begin{align*} T _ { m n } & = T _ m T _ n ( m , n ) = 1 \\\\ T _ { p ^ { r + 2 } } & = T _ p T _ { p ^ { r + 1 } } - p ^ { k - 1 } T _ { p ^ r } p , r \\geq 0 . \\end{align*}"}
+{"id": "808.png", "formula": "\\begin{align*} \\widehat { \\nu } ( E _ n ( \\epsilon ) ) \\leq \\widetilde { \\nu } _ { \\epsilon ' n } ( E _ n ( \\epsilon ) ) + C _ 0 \\theta ^ n \\le \\sum _ { k = 1 } ^ { n \\epsilon ' } \\widehat { \\nu } ( E _ n ( \\epsilon ) \\cap A _ k ) + C _ 0 \\theta ^ n \\end{align*}"}
+{"id": "1296.png", "formula": "\\begin{align*} f ( { \\vec { \\zeta } } ) = \\sum _ { \\vec { k } \\in \\Z ^ m } \\mu _ { \\vec { k } } \\vec { \\zeta } ^ { \\vec { k } } , \\end{align*}"}
+{"id": "8183.png", "formula": "\\begin{align*} u ( t , x ) & = u _ 0 ( x ) + \\int _ 0 ^ t k ( t , s ) L _ 0 u ( s , x ) d s , t > 0 , x \\in Q , \\\\ \\lim _ { t \\searrow 0 } u ( t , x ) & = u _ 0 ( x ) , x \\in Q . \\end{align*}"}
+{"id": "2243.png", "formula": "\\begin{align*} P ( t ) = - Q ( - t ) . \\end{align*}"}
+{"id": "3020.png", "formula": "\\begin{align*} \\Theta ^ I = \\frac { 1 } { 2 } \\Theta { ^ I } _ { J K } \\theta ^ J \\wedge \\theta ^ K \\end{align*}"}
+{"id": "4349.png", "formula": "\\begin{align*} z = \\prod _ { j = 1 } ^ { k } a _ { i _ { p + j - 1 } , i _ { p + j } } . \\end{align*}"}
+{"id": "8130.png", "formula": "\\begin{align*} v _ 2 ' ( y ) = - x ^ { \\frac { 1 } { 3 } } y ^ { - \\frac { 2 } { 3 } } + \\frac { T ^ 2 c m p } { 4 \\pi ^ 2 y ^ 2 n _ 1 ^ 2 } , \\end{align*}"}
+{"id": "2164.png", "formula": "\\begin{align*} r _ a & = \\begin{dcases} \\frac { 3 a - 1 } { 3 } & \\ \\frac { 1 } { 3 } < a \\leq \\frac { 5 } { 3 } \\\\ 3 - a & \\ \\frac { 5 } { 3 } \\leq a < 3 . \\end{dcases} \\end{align*}"}
+{"id": "1390.png", "formula": "\\begin{align*} D _ { \\rho | \\cdot | ^ z } ( \\pi ) \\not = 0 \\implies z \\in \\{ 0 , 1 / 2 \\} . \\end{align*}"}
+{"id": "3528.png", "formula": "\\begin{align*} E _ \\alpha \\mathbf { p } = \\mathbf { p } E _ { - \\alpha } = 0 , ( \\alpha \\in \\Delta ^ + ) . \\end{align*}"}
+{"id": "5359.png", "formula": "\\begin{align*} M : = \\left ( - \\tilde \\lambda \\int _ \\Omega \\eta \\tilde { E } ^ { ( i ) } \\cdot \\tilde { E } ^ { ( j ) } \\ , d x \\right ) _ { i , j = 1 , \\ldots , m } \\end{align*}"}
+{"id": "7596.png", "formula": "\\begin{align*} P & = \\int _ { 0 } ^ { t } \\bigg \\{ \\int _ { 0 } ^ { 1 } \\chi _ { \\{ x + z \\in [ 0 , 1 ] \\} } \\bigg | \\int _ { 0 } ^ { 1 } \\chi _ { \\{ | x - y | \\leq | z | \\} } \\bigg ( \\frac { \\partial G } { \\partial y } ( t - s , x + z , y ) - \\frac { \\partial G } { \\partial y } ( t - s , x , y ) \\bigg ) \\\\ & \\times v ^ { \\delta + 1 } ( s , y ) \\d y \\bigg | ^ p \\d x \\bigg \\} ^ { \\frac { 1 } p } \\d s , \\end{align*}"}
+{"id": "7823.png", "formula": "\\begin{align*} d ( ( 3 , 1 ) ) & = k d _ 1 , \\\\ d ( ( 2 , 2 ) ) & = \\begin{cases} k d _ 1 / 2 & ( k d _ 1 : \\textrm { e v e n } ) , \\\\ k d _ 1 & ( k d _ 1 : \\textrm { o d d } ) , \\end{cases} \\\\ d ( ( 1 , 3 ) ) & = \\begin{cases} k d _ 1 / 3 & ( k \\equiv 0 \\mod 3 ) , \\\\ k d _ 1 & ( k \\not \\equiv 0 \\mod 3 ) , \\end{cases} \\end{align*}"}
+{"id": "322.png", "formula": "\\begin{align*} V ( y ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( u ) } \\prod _ { i = 1 } ^ 3 \\frac { \\Gamma ( \\frac { s + \\frac { 1 } { 2 } - \\gamma _ i } { 2 } ) } { \\Gamma ( \\frac { \\frac { 1 } { 2 } - \\gamma _ i } { 2 } ) } y ^ { - s } H ( s ) \\frac { \\dd s } { s } \\end{align*}"}
+{"id": "7009.png", "formula": "\\begin{align*} \\ell ( \\nu ) & : = \\frac { 1 } { 2 } \\log { ( 2 \\pi e ) ^ 2 N ^ 2 ( X , Y ) } - \\nu R _ p - \\nu \\log { ( 2 \\pi e \\Delta ) } \\\\ & + \\frac { \\nu } { 2 } \\log { \\frac { \\nu ^ 2 } { 2 \\nu - 1 } } - \\frac { 1 - \\nu } { 2 } \\log { ( 2 \\pi e ) ^ 2 \\frac { ( 1 - \\rho ) ^ 2 } { 2 \\nu - 1 } } , \\end{align*}"}
+{"id": "4194.png", "formula": "\\begin{align*} ( \\Psi , \\partial _ t \\Psi ) \\in \\mathcal { H } : = H ^ 1 ( \\mathbb { R } ) \\times H ^ { 1 } ( \\mathbb { R } ) \\times L ^ 2 ( \\mathbb { R } ) \\times L ^ 2 ( \\mathbb { R } ) . \\end{align*}"}
+{"id": "4750.png", "formula": "\\begin{align*} v _ { j + 1 } = v _ j - \\sum _ { i \\le h } y ^ { f _ i } \\rho _ i z _ i = \\biggl ( a - \\sum _ { i \\le h } y ^ { f _ i } \\rho _ i b _ i \\biggr ) + y ^ j \\biggl ( t - \\sum _ { i \\le h } \\rho _ i t _ i \\biggr ) , \\end{align*}"}
+{"id": "750.png", "formula": "\\begin{align*} \\widetilde { \\chi } ( b ) = \\prod _ { q \\mid N } \\widetilde { \\chi _ q } ( b ) . \\end{align*}"}
+{"id": "5047.png", "formula": "\\begin{align*} \\sum _ { m , n = 0 } ^ \\infty f ( m , n ) x ^ m y ^ n & = \\frac { 1 } { 1 - 2 x y } + \\frac { 1 } { 1 - 3 x y } + \\frac { y } { ( 1 - 3 x y ) ^ 2 ( 1 - 5 y ) } + \\frac { x } { ( 1 - x y ) ( 1 - x ) } \\\\ & = \\sum _ { k = 0 } ^ \\infty ( 2 ^ k + 3 ^ k ) x ^ k y ^ k + \\sum _ { k , l = 0 } ^ \\infty ( k + 1 ) 3 ^ k 5 ^ l x ^ k y ^ { k + l + 1 } + \\sum _ { k , l = 0 } ^ \\infty x ^ { k + l + 1 } y ^ { l } \\end{align*}"}
+{"id": "8184.png", "formula": "\\begin{align*} u ( t , x ) & = u _ 0 ( x ) + \\int _ 0 ^ t k ( t , s ) \\big ( L _ 0 u ( s , x ) + V ( x ) u ( s , x ) \\big ) d s , t > 0 , x \\in Q , \\\\ \\lim _ { t \\searrow 0 } u ( t , x ) & = u _ 0 ( x ) , x \\in Q . \\end{align*}"}
+{"id": "388.png", "formula": "\\begin{align*} \\hat { s } _ \\ell : = s _ { 2 \\ell } + s _ { 2 \\ell - 1 } \\textrm { a n d } \\omega _ \\ell : = \\frac { s _ { 2 \\ell } - s _ { 2 \\ell - 1 } } { \\hat { s } _ \\ell } \\ , \\end{align*}"}
+{"id": "6307.png", "formula": "\\begin{align*} s _ \\lambda ( X , Y ) = \\sum _ { \\mu \\subset \\lambda } s _ \\mu ( X ) s _ { \\lambda / \\mu } ( Y ) , \\end{align*}"}
+{"id": "9143.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { n \\choose k } ( m + k - 1 ) ( m + n - 1 ) ^ { n - k - 1 } = M _ 1 \\ ; + \\ ; M _ 2 , \\end{align*}"}
+{"id": "8171.png", "formula": "\\begin{align*} \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } H _ { m , p } & = \\frac { 3 L ( 1 , \\tilde f ) } { \\pi } \\int _ { - \\infty } ^ { \\infty } k ( t ) \\tanh ( \\pi t ) t \\log \\abs { t } \\ , d t + { } \\\\ & \\qquad { } + K \\int _ { - \\infty } ^ \\infty k ( t ) \\tanh ( \\pi t ) t \\ , d t + O ( T ^ { \\frac { 1 } { 7 } + \\varepsilon } M p ^ { \\frac { 1 } { 7 } - \\varepsilon } ) \\end{align*}"}
+{"id": "7073.png", "formula": "\\begin{align*} \\Delta \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) = \\sum _ { i = 1 } ^ n \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) \\otimes \\beta ( \\rho _ i , \\dots , \\rho _ n ) , \\end{align*}"}
+{"id": "2804.png", "formula": "\\begin{align*} 1 = \\left ( \\prod _ { \\sigma \\in S _ n } x _ { \\sigma ( 1 ) } ^ { c _ 1 } x _ { \\sigma ( 2 ) } ^ { c _ 2 } \\cdots x _ { \\sigma ( n ) } ^ { c _ n } \\right ) ^ { 1 / n ! } < \\frac { 1 } { n ! } \\sum _ { \\sigma \\in S _ n } x _ { \\sigma ( 1 ) } ^ { c _ 1 } x _ { \\sigma ( 2 ) } ^ { c _ 2 } \\cdots x _ { \\sigma ( n ) } ^ { c _ n } \\end{align*}"}
+{"id": "3959.png", "formula": "\\begin{align*} \\tilde { x } = \\frac { - g _ y } { g _ z } ( x , 0 , 0 ) . \\end{align*}"}
+{"id": "9225.png", "formula": "\\begin{align*} \\lll = ( d F ) 2 \\nabla _ { x } \\phi _ { + } \\circ F ^ { - 1 } \\cdot \\nabla _ { y , z } + \\vert \\nabla _ { x } \\ell _ { 0 } \\vert ^ { 2 } \\circ F ^ { - 1 } . \\end{align*}"}
+{"id": "1543.png", "formula": "\\begin{align*} p _ i ^ { b c } A _ { b c D } = 0 \\Leftrightarrow A _ { b c [ D ] } - A _ { c b [ D ] } = 0 \\end{align*}"}
+{"id": "6989.png", "formula": "\\begin{align*} w ^ * ( G ) \\ = \\{ g \\in G \\mid w ( g x _ 1 , x _ 2 , \\dots , x _ k ) & = w ( x _ 1 , \\dots , g x _ i , \\dots , x _ k ) \\\\ & = w ( x _ 1 , \\dots , g x _ k ) \\\\ & = w ( x _ 1 , \\dots , x _ k ) \\ \\forall x _ i \\in G \\} . \\end{align*}"}
+{"id": "8670.png", "formula": "\\begin{align*} \\alpha _ k = \\partial / \\partial p _ k , \\alpha _ { - k } = k p _ k , ( k = 1 , 2 , \\dots ) , \\alpha _ 0 = a _ 0 \\cdot 1 , 1 = 1 . \\end{align*}"}
+{"id": "5041.png", "formula": "\\begin{align*} f ( \\vec a _ 0 + m _ 1 \\vec a _ 1 + \\cdots + m _ s \\vec a _ s ) = \\sum _ { i = 1 } ^ l g _ { i , 0 } g _ { i , 1 } ^ { m _ 1 } \\cdots g _ { i , s } ^ { m _ s } \\qquad \\end{align*}"}
+{"id": "7249.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\infty } \\frac { K ( \\gamma ) ^ 2 L ( \\gamma ) } { N ( \\gamma ) } = 0 . \\end{align*}"}
+{"id": "3476.png", "formula": "\\begin{align*} { \\bf J } _ { n , R } ( t ) = I _ n ( g _ { n , R } ( \\cdot ; t ) ) \\mbox { a n d } g _ { n , R } ( \\cdot ; t ) = \\int _ { B _ R } f _ n ( \\cdot , x ; t ) d x . \\end{align*}"}
+{"id": "3405.png", "formula": "\\begin{align*} \\partial _ r v ^ r + \\frac { v ^ r } { r } + \\partial _ z v ^ z = 0 , \\partial _ r B ^ r + \\frac { B ^ r } { r } + \\partial _ z B ^ z = 0 . \\end{align*}"}
+{"id": "7998.png", "formula": "\\begin{align*} f _ N \\left ( x \\right ) = \\int _ { 0 } ^ { x } s i g n \\int _ { 0 } ^ { y } Q _ N \\left ( z , \\varepsilon ^ 0 \\right ) d z d y . \\end{align*}"}
+{"id": "8127.png", "formula": "\\begin{align*} v _ 1 ( y ) = 3 x ^ { \\frac { 1 } { 3 } } y ^ { \\frac { 1 } { 3 } } - \\frac { T ^ 2 c m p } { 4 \\pi ^ 2 y n _ 1 ^ 2 } v _ 2 ( y ) = - 3 x ^ { \\frac { 1 } { 3 } } y ^ { \\frac { 1 } { 3 } } - \\frac { T ^ 2 c m p } { 4 \\pi ^ 2 y n _ 1 ^ 2 } , \\end{align*}"}
+{"id": "4962.png", "formula": "\\begin{align*} A = \\begin{pmatrix} c _ 1 & - 1 \\\\ [ 4 p t ] 1 & \\phantom { - } 0 \\end{pmatrix} \\begin{pmatrix} c _ 2 & - 1 \\\\ [ 4 p t ] 1 & \\phantom { - } 0 \\end{pmatrix} \\cdots \\begin{pmatrix} c _ N & - 1 \\\\ [ 4 p t ] 1 & \\phantom { - } 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "552.png", "formula": "\\begin{align*} \\partial _ t u = - U \\partial _ y u + \\rho \\partial _ y ^ 2 u + \\dot { \\mathcal { W } } \\end{align*}"}
+{"id": "1487.png", "formula": "\\begin{align*} \\cos x = \\prod _ { n = 1 , n } ^ \\infty \\left ( 1 - \\frac { x ^ 2 } { ( \\frac { n \\pi } 2 ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "2106.png", "formula": "\\begin{align*} C = ( C _ 1 \\times C _ 2 ) \\oplus \\widehat C _ { 1 2 } , ~ ~ \\mbox { w h e r e } ~ ~ \\widehat C _ { 1 2 } = \\big \\{ \\big ( c _ { 1 2 } , \\ , c _ { 1 2 } g \\big ) \\ , \\big | \\ , c _ { 1 2 } \\in C _ { 1 2 } \\big \\} \\cong C _ { 1 2 } . \\end{align*}"}
+{"id": "4988.png", "formula": "\\begin{align*} \\iint _ { \\R ^ d \\times \\R ^ d } p ^ 2 \\left | ( G _ { p , q } , f ) \\right | ^ 2 \\ , \\frac { d p \\ , d q } { ( 2 \\pi ) ^ d } = \\| \\nabla f \\| ^ 2 + \\| \\nabla g \\| ^ 2 \\| f \\| ^ 2 \\ , . \\end{align*}"}
+{"id": "5229.png", "formula": "\\begin{align*} l _ { 1 } = & k _ 1 + k _ 0 t ^ 3 / k _ 4 & l _ { 2 } = & k _ 4 / t ^ 3 & b _ { 1 5 } = & k _ 6 t ^ 2 & b _ { 2 5 } = & k _ 5 t ^ 2 \\\\ b _ { 1 2 } = & k _ 3 / t & b _ { 1 3 } = & - k _ 0 k _ 6 t ^ 2 + k _ 7 / t & b _ { 2 3 } = & - k _ 0 k _ 5 t ^ 2 + k _ 8 / t \\end{align*}"}
+{"id": "645.png", "formula": "\\begin{align*} \\tau _ { n } : = N ^ { 4 N - 3 n } \\epsilon , 0 \\leq n \\leq N . \\end{align*}"}
+{"id": "7920.png", "formula": "\\begin{align*} \\liminf _ { s \\to 0 } \\frac { P _ s \\mathcal { Q } ^ t u ( x ) - \\mathcal { Q } ^ t u ( x ) } { s } \\ge \\liminf _ { s \\to 0 } \\frac { P _ s u ( x _ t ) - u ( x _ t ) } { s } = \\Delta u ( x _ t ) = 0 \\ , , \\end{align*}"}
+{"id": "2170.png", "formula": "\\begin{align*} 3 ( ( 1 - 2 \\alpha - \\beta ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + 1 ) = 1 - r ^ 2 . \\end{align*}"}
+{"id": "4675.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell _ 1 = 0 } ^ { \\min \\{ n _ 1 , m \\} } \\sum _ { \\ell _ 2 = 0 } ^ { \\min \\{ n _ 2 , m \\} } A _ { m , \\ell _ 1 , \\ell _ 2 } = \\sum _ { m = 0 } ^ { n _ 1 } \\sum _ { \\ell _ 1 = 0 } ^ m \\sum _ { \\ell _ 2 = 0 } ^ m A _ { m , \\ell _ 1 , \\ell _ 2 } & + \\sum _ { m = n _ 1 + 1 } ^ { n _ 2 } \\sum _ { \\ell _ 1 = 0 } ^ { n _ 1 } \\sum _ { \\ell _ 2 = 0 } ^ m A _ { m , \\ell _ 1 , \\ell _ 2 } + \\sum _ { m = n _ 2 + 1 } ^ { \\infty } \\sum _ { \\ell _ 1 = 0 } ^ { n _ 1 } \\sum _ { \\ell _ 2 = 0 } ^ { n _ 2 } A _ { m , \\ell _ 1 , \\ell _ 2 } . \\end{align*}"}
+{"id": "8851.png", "formula": "\\begin{align*} d ( n , p , q ) = \\delta ( n , p , q ) - \\delta ( n , p - 1 , q - 1 ) , p , q \\neq 0 , \\end{align*}"}
+{"id": "3226.png", "formula": "\\begin{align*} f ' ( x ) = \\norm { L ( u ) \\cap L ( w ) } \\geq k _ 3 + k _ 2 + k _ 1 = k _ 3 ' + k ' _ 2 + k _ 1 ' . \\end{align*}"}
+{"id": "2531.png", "formula": "\\begin{align*} \\varrho _ N ^ \\varphi ( x ) & \\ge \\tilde \\varrho _ N ^ \\varphi ( x ) - | \\varrho _ N ^ \\varphi ( x ) - \\tilde \\varrho _ N ^ \\varphi ( x ) | \\ge A r ^ d \\varphi _ 0 - \\frac { \\| \\nabla \\varphi \\| _ \\infty } N \\sum _ { j = 1 } ^ N | x _ j - y _ j | \\\\ & \\ge A r ^ d \\varphi _ 0 - \\| \\nabla \\varphi \\| _ \\infty A _ \\varphi = \\frac { A r ^ d \\varphi _ 0 } 2 \\ , , \\end{align*}"}
+{"id": "8319.png", "formula": "\\begin{align*} \\Box ( \\chi u ) = \\chi \\Box u + 2 \\partial _ t \\chi \\partial _ t u - 2 \\nabla _ x \\chi \\cdot \\nabla _ x u + u \\Box \\chi \\in L ^ \\frac { 2 ( d + 1 ) } { d + 3 } . \\end{align*}"}
+{"id": "6394.png", "formula": "\\begin{align*} ( \\tilde X , \\tilde Y , \\tilde t ) ^ { - 1 } \\circ ( X , Y , t ) = ( X - \\tilde X , Y - \\tilde Y - ( \\tilde t - t ) \\tilde X , t - \\tilde t ) . \\end{align*}"}
+{"id": "5239.png", "formula": "\\begin{align*} A _ X Y = \\frac { 1 } { 2 } \\left \\{ \\nu [ X , Y ] - \\lambda ^ 2 g ( X , Y ) ( \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } ) \\right \\} , ~ \\forall X , Y \\in \\Gamma ( k e r F _ \\ast ) ^ \\bot . \\end{align*}"}
+{"id": "6112.png", "formula": "\\begin{align*} \\lim _ { \\substack { n \\to \\infty \\\\ p } } L ( j , n ) = R ( p ) \\end{align*}"}
+{"id": "1519.png", "formula": "\\begin{align*} \\int _ 0 ^ x \\theta ^ { r - 2 } \\log \\left ( \\cos \\frac \\theta 2 \\right ) d \\theta = \\frac { x ^ { r - 1 } } { r - 1 } \\log \\left ( \\cos \\frac x 2 \\right ) - \\frac { ( 2 \\pi ) ^ { r - 1 } } { r - 1 } \\log \\C _ r \\left ( \\frac x { 2 \\pi } \\right ) . \\end{align*}"}
+{"id": "8827.png", "formula": "\\begin{align*} \\dfrac { P + P ^ * Q } { 2 } - \\left ( \\dfrac { P + P ^ * Q } { 2 } \\right ) ^ * Q = \\dfrac { 1 } { 2 } P D _ Q ^ 2 = \\dfrac { 1 } { 2 } D _ Q ^ 2 P = \\dfrac { 1 } { 2 } D _ Q P D _ Q [ P D _ Q = D _ Q P ] . \\end{align*}"}
+{"id": "8200.png", "formula": "\\begin{align*} \\mathfrak { K } _ 2 ( t ) : = t ^ { \\beta - 1 } { E } _ { ( \\beta - \\beta _ 1 , \\ldots , \\beta - \\beta _ m ) , \\beta } \\left ( - b _ 1 t ^ { \\beta - \\beta _ 1 } , \\ldots , - b _ m t ^ { \\beta - \\beta _ m } \\right ) \\end{align*}"}
+{"id": "8860.png", "formula": "\\begin{align*} K _ { \\nu , m } ( z , w ) : = \\frac { ( 2 m + 2 \\nu + n ) \\Gamma ( m + n + 2 \\nu ) } { \\pi ^ { n } \\Gamma ( m + 2 \\nu + 1 ) } \\left ( \\frac { \\left ( 1 + \\langle w , z \\rangle \\right ) ^ { 2 } } { ( 1 + | z | ^ { 2 } ) ( 1 + | w | ^ { 2 } ) } \\right ) ^ { \\nu } P _ { m } ^ { ( n - 1 , 2 \\nu ) } ( \\cos 2 d _ { F S } ( z , w ) ) , \\end{align*}"}
+{"id": "1301.png", "formula": "\\begin{align*} S t _ n = S t _ 1 \\otimes S t _ 1 ^ { ( 1 ) } \\otimes \\cdots \\otimes S t _ 1 ^ { ( n - 1 ) } . \\end{align*}"}
+{"id": "6292.png", "formula": "\\begin{align*} { \\pi _ i } ^ 2 & = \\pi _ i ; \\\\ \\pi _ i \\pi _ { i + 1 } \\pi _ i & = \\pi _ { i + 1 } \\pi _ i \\pi _ { i + 1 } ; \\\\ \\pi _ i \\pi _ j & = \\pi _ j \\pi _ i , \\ | i - j | \\ge 2 . \\end{align*}"}
+{"id": "3157.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( A ) = c _ 2 ^ { 2 2 } ( A ) = c _ 2 ^ { 1 1 } ( A ) + 2 \\ , c _ 1 ^ { 1 2 } ( A ) = c _ 1 ^ { 2 2 } ( A ) + 2 \\ , c _ 2 ^ { 1 2 } ( A ) = 0 , \\end{align*}"}
+{"id": "6617.png", "formula": "\\begin{align*} H = \\C { A } + Q , \\end{align*}"}
+{"id": "118.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\lambda ^ p \\mathcal L ^ { 2 n } ( E _ { \\lambda , K } ) = \\frac { 2 } { n } \\int _ { \\mathbb R ^ n } | | \\nabla f | | _ { Z _ p ^ * K } ^ p d x . \\end{align*}"}
+{"id": "8159.png", "formula": "\\begin{align*} H ^ + ( x ) = 2 i \\int _ { - \\infty } ^ \\infty J _ { 2 i t } ( x ) \\frac { h ( t ) t } { \\cosh ( \\pi t ) } \\ , d t , \\end{align*}"}
+{"id": "6806.png", "formula": "\\begin{align*} \\delta A _ d [ q _ d ] & = D _ 1 L _ d ( q _ 0 , q _ 1 ) \\cdot \\delta { q _ 0 } + \\sum _ { m = 1 } ^ { N - 1 } D _ 1 L _ d ( q _ m , q _ { m + 1 } ) \\cdot \\delta { q _ m } \\\\ & + \\sum _ { m = 0 } ^ { N - 2 } D _ 2 L _ d ( q _ m , q _ { m + 1 } ) \\cdot \\delta { q _ { m + 1 } } + D _ 2 L _ d ( q _ { N - 1 } , q _ { N } ) \\cdot \\delta { q _ { N - 1 } } . \\end{align*}"}
+{"id": "3054.png", "formula": "\\begin{align*} C _ { j k l } ( A ) : = c _ j ^ { k l } ( A ) + c _ k ^ { j l } ( A ) + c _ l ^ { j k } ( A ) = 0 \\qquad \\forall j , k , l \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "5652.png", "formula": "\\begin{align*} \\frac { \\Gamma _ 0 } { | \\Omega | } < \\Phi _ j \\mbox { f o r a l l } j = 1 , \\ldots , N . \\end{align*}"}
+{"id": "6966.png", "formula": "\\begin{gather*} \\big \\{ Z ^ { ( \\alpha ) } _ { j , m , n } \\big \\} _ { j = 1 } ^ { n - 1 } = \\big \\{ { - } x _ { j , m } ^ { ( - \\alpha - 1 ) } \\big \\} _ { j = 1 } ^ m \\cup \\big \\{ x _ { j - m , n - m - 1 } ^ { ( \\alpha + 1 ) } \\big \\} _ { j = m + 1 } ^ { n - 1 } . \\end{gather*}"}
+{"id": "3228.png", "formula": "\\begin{align*} f ' ( w ) = f ( w ) \\ge k _ 2 + k _ 3 = k ' _ 2 + k _ 3 ' + 1 = k ' _ 2 + k ' _ 3 + k ' _ 1 . \\end{align*}"}
+{"id": "4454.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { a _ i } < \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < \\theta ? \\end{align*}"}
+{"id": "5770.png", "formula": "\\begin{align*} \\mathcal L = L I \\quad \\hbox { a n d } \\quad \\mathcal G = g I . \\end{align*}"}
+{"id": "7779.png", "formula": "\\begin{align*} { \\rm a n d } r _ { \\alpha , \\lambda } ( x ) & = - \\frac { 1 } { \\pi } \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ k } { k } \\sin ( \\pi k \\alpha ) \\frac { \\Gamma ( k \\alpha + 1 ) } { \\lambda ^ k x ^ { k \\alpha } } = - \\alpha \\sum _ { k = 1 } ^ \\infty \\frac { ( - \\lambda x ^ \\alpha ) ^ { - k } } { \\Gamma ( 1 - k \\alpha ) } \\end{align*}"}
+{"id": "8309.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { B } _ s = \\big \\{ \\ \\textbf { x } \\ | \\ B ( \\textbf { x } ) = 0 \\big \\} \\end{array} \\right . \\end{align*}"}
+{"id": "3170.png", "formula": "\\begin{align*} a _ i ( y ) : = b _ i ( y _ 1 , y _ 2 ) \\quad y = ( y _ 1 , y _ 2 , y _ 3 ) \\in \\R ^ 3 \\end{align*}"}
+{"id": "5384.png", "formula": "\\begin{align*} \\exp \\bigg ( 2 \\sum _ { m = 1 } ^ { M _ 1 } Z _ m ^ { ( W ) } ( t ) \\bigg ) \\ll \\exp \\bigg ( 4 \\sum _ { m = 1 } ^ { M _ 1 } \\sigma _ { m , W } ^ 2 \\bigg ) \\frac { 1 } { M _ 1 ^ 4 } \\ll \\frac { e ^ { 2 M _ 1 } } { M _ 1 ^ 4 } . \\end{align*}"}
+{"id": "2238.png", "formula": "\\begin{align*} U _ { \\tau } ( x _ 1 , x _ 2 ) = ( x _ 1 + \\tau , x _ 2 ) , \\ U _ { \\tau } ( f ) ( x ) = f ( U _ { - \\tau } x ) . \\end{align*}"}
+{"id": "3641.png", "formula": "\\begin{align*} \\lambda _ * ( M ) = \\begin{cases} \\lambda _ 1 ( M ) , \\ \\ \\ \\ \\ \\mbox { i f } \\ \\gamma > 0 \\\\ \\frac { ( n - 1 ) ^ 2 } { 4 } \\kappa ^ 2 , \\ \\ \\ \\mbox { i f o n l y } \\ \\gamma \\geq 0 \\ \\mbox { i s a s s u m e d } , \\end{cases} \\end{align*}"}
+{"id": "5379.png", "formula": "\\begin{align*} A ( N ) = \\frac { 1 } { 2 \\pi i } \\int _ { | z | = 1 } F _ K ( z ) \\frac { d z } { z ^ { N + 1 } } . \\end{align*}"}
+{"id": "2614.png", "formula": "\\begin{align*} 4 = a _ { 2 , 3 } + a _ { 2 , 4 } + a _ { 3 , 4 } = 2 + 1 + 1 \\leftrightarrow ( a _ { 1 , 2 } , a _ { 1 , 3 } , a _ { 1 , 4 } , a _ { 2 , 3 } , a _ { 2 , 4 } , a _ { 3 , 4 } ) = ( 1 , 1 , 2 , 2 , 1 , 1 ) \\end{align*}"}
+{"id": "7780.png", "formula": "\\begin{align*} \\{ g \\star m \\} _ { \\alpha , \\lambda } ( x ) & \\equiv g _ { 1 - \\alpha , 0 } ( x ) \\star m _ { \\alpha , \\lambda } ( x ) / \\lambda \\\\ \\mathrm { a n d } \\phi _ { \\alpha , \\lambda } ( t ) & \\equiv \\frac { \\alpha } { \\pi } { \\rm I m } \\left \\{ \\frac { ( e ^ { - i \\pi } t ) ^ { \\alpha - 1 } } { \\lambda + e ^ { - i \\pi \\alpha } \\ , t ^ \\alpha } \\right \\} = \\frac { \\alpha } { \\pi } \\frac { \\lambda \\ , t ^ { \\alpha - 1 } \\sin \\pi \\alpha } { \\lambda ^ 2 + 2 \\lambda \\ , t ^ \\alpha \\cos \\pi \\alpha + t ^ { 2 \\alpha } } \\end{align*}"}
+{"id": "2171.png", "formula": "\\begin{align*} ( 1 - 2 \\alpha - \\beta ) r ^ 2 + ( 2 - 2 \\alpha + \\beta ) r + 1 = 3 ( 1 - r ^ 2 ) . \\end{align*}"}
+{"id": "4443.png", "formula": "\\begin{align*} \\theta \\in \\left ( \\frac { 9 } { 2 0 } , \\frac { 1 1 } { 2 4 } \\right ] \\subseteq \\left ( \\frac { 4 } { 9 } , \\frac { 1 1 } { 2 4 } \\right ] = J ( 3 , 9 ) . \\end{align*}"}
+{"id": "7832.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { j = 1 } ^ n \\alpha _ j e ^ { w _ j z } \\end{align*}"}
+{"id": "533.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ n ( X - e ^ { 2 i \\pi a _ j / d } ) ( X - \\overline { e ^ { 2 i \\pi a _ j / d } } ) = P ( g ) \\overline { P ( g ) } = \\prod _ { m = 1 } ^ k { \\Phi _ { u _ m } } ^ { \\alpha _ m } . \\end{align*}"}
+{"id": "2938.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { B } ^ n _ t ( \\varphi ) & = \\int _ 0 ^ t ( \\partial _ s + A _ n ) \\mathcal { X } ^ n _ s ( \\varphi ) d s \\\\ & = \\frac { 1 } { \\sqrt { n } } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\big [ n g _ n ( \\eta ^ n _ j ( s ) ) \\nabla ^ n \\varphi ^ n _ j ( s ) - \\frac { f _ n } { n } ( \\eta ^ n _ j ( s ) - \\rho ) \\partial _ x \\varphi ^ n _ j ( s ) \\big ] d s . \\end{aligned} \\end{align*}"}
+{"id": "6658.png", "formula": "\\begin{align*} \\begin{aligned} & x _ i ^ { k + 1 } - { x ' _ i } ^ { k + 1 } & = ( 1 + w _ { i i } \\gamma ^ k ) ( x _ i ^ k - { x ' _ i } ^ k ) - \\lambda ^ k ( g _ i ^ k - { g ' _ i } ^ k ) , \\end{aligned} \\end{align*}"}
+{"id": "7006.png", "formula": "\\begin{align*} ( R _ c , R _ p ) = \\left ( \\log { \\frac { N ( X , Y ) } { 1 - \\rho } } , \\log { \\frac { 1 - \\rho } { \\Delta } } \\right ) , \\end{align*}"}
+{"id": "925.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\chi _ { E } ( Y ^ { x } _ { \\alpha } ( t ) ) d t - T \\lambda _ { 2 } ( E ) = o \\left ( \\log ( T ) ^ { 1 + \\varepsilon } \\right ) . \\end{align*}"}
+{"id": "8619.png", "formula": "\\begin{align*} x _ { n + 1 } = \\alpha _ n u + ( 1 - \\alpha _ n ) [ \\beta _ n x _ n + ( 1 - \\beta _ n ) \\tilde { T } x _ n ] \\end{align*}"}
+{"id": "954.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( M ^ C , C ) \\ , = \\ , \\textrm { r . g r a d e } _ R ( \\textrm { T r } _ C ( M ) , C ) - 2 \\ , = \\ , t - 2 . \\end{align*}"}
+{"id": "7669.png", "formula": "\\begin{align*} ( x _ { I } ^ { m } \\eta - x _ { I } ^ { m } \\eta _ { I , - \\iota _ { E } ( \\omega ) } ) \\wedge \\frac { d f } { f } = \\sum _ { p \\neq - \\iota _ { E } ( \\omega ) } x _ { I } ^ { m } \\eta _ { I , p } \\wedge \\frac { d f } { f } \\in \\frac { 1 } { f } \\Omega _ { U } ^ { j + 1 } [ x _ { I } ^ { - 1 } ] . \\end{align*}"}
+{"id": "3122.png", "formula": "\\begin{align*} \\bar { a } : = \\int _ Y r a , \\end{align*}"}
+{"id": "4575.png", "formula": "\\begin{align*} | C ( x ) | = \\frac { | G | } { | H ( x ) | } \\end{align*}"}
+{"id": "4270.png", "formula": "\\begin{align*} 0 = I _ \\gamma ( M ( Q ) ) & = E _ \\gamma ( \\phi ) \\\\ & = \\frac { 1 } { 2 } \\| ( \\nabla - i A ) \\phi \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } V _ \\gamma ( x ) | \\phi ( x ) | ^ 2 d x - \\frac { N } { N + 2 } \\| \\phi \\| ^ { 2 + \\frac { 4 } { N } } _ { L ^ { 2 + \\frac { 4 } { N } } } \\\\ & \\geq \\int _ { \\R ^ N } V _ \\gamma ( x ) | \\phi ( x ) | ^ 2 d x \\geq 0 . \\end{align*}"}
+{"id": "7187.png", "formula": "\\begin{align*} z _ { \\alpha } ( x _ { j } , \\ , t _ 0 ) = z _ { \\alpha \\ , 0 } ( { x } _ { j } ) , \\ , \\ , \\ , \\ , \\forall { P } \\in C . \\end{align*}"}
+{"id": "4048.png", "formula": "\\begin{align*} g ^ * _ { y _ k y _ l } ( x , y , u ) = - D _ { y _ l } \\left ( \\frac { g _ { _ { , k } } } { g _ z } \\right ) \\Big | _ { ( x , y , g ^ * ( x , y , u ) ) } - g ^ * _ { , l } D _ { z } \\left ( \\frac { g _ { _ { , k } } } { g _ z } \\right ) \\Big | _ { ( x , y , g ^ * ( x , y , u ) ) } . \\end{align*}"}
+{"id": "2639.png", "formula": "\\begin{align*} L ( s , \\psi \\chi ^ { ( a ) } ) = \\sum _ { \\mathcal { A } \\neq 0 } \\frac { \\psi ( \\mathcal { A } ) \\chi ^ { ( a ) } ( \\mathcal { A } ) } { N ( \\mathcal { A } ) ^ s } . \\end{align*}"}
+{"id": "7734.png", "formula": "\\begin{align*} g ^ { ( 1 ) } ( \\tau , \\rho _ N ) = g ^ { ( 2 ) } ( \\tau , \\rho _ 0 ) . \\end{align*}"}
+{"id": "5540.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n ^ k } \\exp \\left ( { - \\frac { x } { n ^ 2 } } \\right ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\lambda - i \\infty } ^ { \\lambda + i \\infty } \\frac { \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } { \\rm d } s + \\frac { 1 } { 2 } \\sum _ { \\rho } \\frac { \\Gamma ( \\frac { k - \\rho } { 2 } ) } { \\zeta ' ( \\rho ) } x ^ { - \\frac { k - \\rho } { 2 } } . \\end{align*}"}
+{"id": "1178.png", "formula": "\\begin{align*} \\sum _ { I \\sqcup J = \\{ 1 , \\dots , N \\} } \\ 1 _ { i \\in I } \\ 1 _ { j \\in J } = 2 ^ { N - 2 } \\sum _ { I \\sqcup J = \\{ 1 , \\dots , N \\} ; \\ , \\ I , J \\neq \\emptyset } 1 = 2 ^ { N } - 2 \\ , , \\end{align*}"}
+{"id": "1759.png", "formula": "\\begin{align*} \\langle \\delta s ( \\mu _ { \\underline 0 } ) , \\delta s ( \\mu _ { \\underline 0 } ) \\rangle = 4 ( - 1 ) ^ { M } ( 2 M + 2 ) ^ 2 ( 2 M + 1 ) ^ 2 ( 2 M + 3 ) ^ { - 1 } \\binom { 2 M + 2 } { k _ { \\rm i d } - 1 } ^ { - 1 } \\binom { 2 M } { k _ { \\rm i d } - 2 } ^ 2 \\binom { 2 M + 2 } { M + 1 } ^ { - 1 } \\end{align*}"}
+{"id": "2127.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) = \\int _ \\R J ( y ) \\left [ u ( t , x - y ) - u ( t , x ) \\right ] d y + f \\left ( u ( t , x ) \\right ) , t > 0 , \\ , x \\in \\R , \\end{align*}"}
+{"id": "747.png", "formula": "\\begin{align*} \\tilde { h } _ { 1 1 } ( \\tilde { w } ) | d \\tilde { w } | ^ 2 = h _ { 1 1 } ( w ) | d w | ^ 2 . \\end{align*}"}
+{"id": "5778.png", "formula": "\\begin{align*} I m ( \\widetilde C _ q ^ T ) \\cap K e r ( D ) \\not = \\{ 0 \\} . \\end{align*}"}
+{"id": "8373.png", "formula": "\\begin{align*} w = 0 , \\mathrm { f o r } \\ | x | > | t - R | \\ \\mathrm { a n d } \\ t \\ge 0 . \\end{align*}"}
+{"id": "3475.png", "formula": "\\begin{align*} Q _ R ( t _ i ) = \\delta ( - { R ^ { - d / 2 } } D L ^ { - 1 } F _ R ( t _ i ) ) . \\end{align*}"}
+{"id": "6558.png", "formula": "\\begin{align*} I _ { C } = \\int _ 0 ^ \\infty { \\log _ 2 \\left ( { 1 + \\gamma } \\right ) { \\gamma ^ { { s _ 1 } - 1 } } { \\rm d } \\gamma } = \\frac { { - 1 } } { { \\ln 2 } } \\Gamma \\left ( { { s _ 1 } } \\right ) \\Gamma \\left ( { - { s _ 1 } } \\right ) . \\end{align*}"}
+{"id": "4540.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { a _ i } \\leq \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < \\theta ? \\end{align*}"}
+{"id": "7847.png", "formula": "\\begin{align*} S ( z ) = \\frac { 1 } { f ( z ) - a } - b = \\frac { h ( z ) - b g ( z ) + a b h ( z ) } { g ( z ) - a h ( z ) } = \\frac { \\sum _ { j = 0 } ^ n \\tilde G _ j ( z ) e ^ { w _ j z ^ q } } { \\sum _ { j = 0 } ^ n \\tilde H _ j ( z ) e ^ { w _ j z ^ q } } \\end{align*}"}
+{"id": "1875.png", "formula": "\\begin{align*} \\beta = \\sin ^ { - 1 } \\left ( \\frac { p _ { z _ i } \\sigma + q _ { z _ i } } { \\sqrt { \\dot { z } _ i ^ 2 + ( p _ { z _ i } \\sigma + q _ { z _ i } ) ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "148.png", "formula": "\\begin{align*} \\| p _ { j } ( \\varkappa ) - p _ { j } ( y ) \\| _ { \\mathbb { X } } \\le & K L _ { \\nu _ { j } } \\| \\varkappa - y \\| _ { \\mathbb { X } } + K K _ { 1 } b \\| \\tilde { \\varkappa } _ { \\tau } - \\tilde { y } _ { \\tau } \\| _ { D } \\\\ \\le & K L _ { \\nu _ { j } } \\| \\varkappa - y \\| _ { \\mathbb { X } } + K K _ { 1 } \\gamma b \\| \\varkappa - y \\| _ { \\mathbb { X } } , \\ \\mbox { f o r } \\ j = 1 , \\ldots , n . \\end{align*}"}
+{"id": "5090.png", "formula": "\\begin{align*} C / Q = C Q _ 0 ^ { - 1 } ( 1 - \\lambda _ 1 x _ d ) ^ { - 1 } \\cdots ( 1 - \\lambda _ s x _ d ) ^ { - 1 } , \\end{align*}"}
+{"id": "6767.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } e ^ { 2 \\left [ \\gamma _ { \\beta } - \\gamma _ { \\alpha } \\right ] t } \\frac { \\left | \\cos \\frac { 1 } { 2 } \\omega _ { \\alpha } t \\right | } { \\left | \\cos \\frac { 1 } { 2 } \\omega _ { \\beta } t \\right | } = \\frac { \\left ( \\frac { 1 } { 2 } - \\sigma \\right ) ^ 2 + \\omega ^ 2 } { \\left ( \\frac { 1 } { 2 } + \\sigma \\right ) ^ 2 + \\omega ^ 2 } . \\end{align*}"}
+{"id": "4676.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell _ 1 = 0 } ^ { \\min \\{ n _ 1 , m \\} } \\sum _ { \\ell _ 2 = 0 } ^ { \\min \\{ n _ 2 , m \\} } & A _ { m , \\ell _ 1 , \\ell _ 2 } = \\sum _ { \\ell _ 1 = 0 } ^ { n _ 1 } \\sum _ { \\ell _ 2 = 0 } ^ { n _ 2 } \\sum _ { m = \\max \\{ \\ell _ 1 , \\ell _ 2 \\} } ^ { n _ 1 } A _ { m , \\ell _ 1 , \\ell _ 2 } + \\sum _ { \\ell _ 1 = 0 } ^ { n _ 1 } \\sum _ { \\ell _ 2 = 0 } ^ { n _ 2 } \\sum _ { m = n _ 1 + 1 } ^ { n _ 2 } A _ { m , \\ell _ 1 , \\ell _ 2 } + \\sum _ { \\ell _ 1 = 0 } ^ { n _ 1 } \\sum _ { \\ell _ 2 = 0 } ^ { n _ 2 } \\sum _ { m = n _ 2 + 1 } ^ { \\infty } A _ { m , \\ell _ 1 , \\ell _ 2 } , \\end{align*}"}
+{"id": "2561.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { \\begin{subarray} { c } 0 { \\le } m _ 1 < _ { c _ 1 } \\cdots < _ { c _ { p - 1 } } m _ p < \\infty \\end{subarray} } z ^ { m _ p } \\frac { { m _ p } ! } { ( \\alpha ) _ { m _ p } } \\left \\{ \\prod _ { i = 1 } ^ { p } \\frac { 1 } { ( m _ i + \\alpha ) ^ { a _ i } ( m _ i + 1 ) ^ { b _ i } } \\right \\} , \\end{aligned} \\end{align*}"}
+{"id": "7137.png", "formula": "\\begin{align*} u _ 1 ( a m _ 0 ) & = a u _ 1 ( m _ 0 ) - u _ 2 ( f _ M ( a \\otimes m _ 0 ) ) + f _ N ( a \\otimes u _ 0 ( m _ 0 ) ) . \\end{align*}"}
+{"id": "3975.png", "formula": "\\begin{align*} & \\phi ^ * < 0 \\Omega ^ * & & \\phi ^ * = 0 \\partial \\Omega ^ * \\\\ & D _ { p } ^ 2 \\phi ^ * ( Y ( x , u , p ) ) \\geq 0 & & | D \\phi ^ * | \\neq 0 \\partial \\Omega ^ * . \\end{align*}"}
+{"id": "5965.png", "formula": "\\begin{align*} \\hbox { r a n k } ( C _ p D ) = \\hbox { r a n k } ( D ) = M = N - p . \\end{align*}"}
+{"id": "7082.png", "formula": "\\begin{align*} \\mathcal { E } ^ 2 _ { p , * } = \\mathcal { E } ^ \\infty _ { p , * } = \\begin{cases} E ^ { C _ 2 } _ * / _ e ^ { C _ 2 } & p = 0 \\\\ E _ * & p = 2 \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "6086.png", "formula": "\\begin{align*} | ( f - r _ n ) ( \\tau ) | = ( 2 + o ( 1 ) ) | \\varkappa _ n \\gamma _ n \\gamma _ n ^ * | | ( m _ n / w ) ( \\tau ) | \\end{align*}"}
+{"id": "8185.png", "formula": "\\begin{align*} f ( \\lambda ) = a + b \\lambda + \\int _ { ( 0 , \\infty ) } \\left ( 1 - e ^ { - \\lambda s } \\right ) \\nu ( d s ) , \\end{align*}"}
+{"id": "7235.png", "formula": "\\begin{align*} A _ 4 ^ 2 & = k A _ 0 + \\frac { ( k - m ) k } { 2 m ( n - 1 ) } A _ 2 + \\frac { k ^ 2 } { 2 m n } A _ 3 , \\\\ A _ 4 A _ 5 & = A _ 5 A _ 4 = k A _ 1 + \\frac { ( k - m ) k } { 2 m ( n - 1 ) } A _ 2 + \\frac { k ^ 2 } { 2 m n } A _ 3 , \\\\ A _ 5 ^ 2 & = k A _ 0 + \\frac { ( k - m ) k } { 2 m ( n - 1 ) } A _ 2 + \\frac { k ^ 2 } { 2 m n } A _ 3 . \\end{align*}"}
+{"id": "6986.png", "formula": "\\begin{gather*} H _ { 2 , n } ^ { ( \\{ 1 , 1 \\} ) } ( x ) = \\sum _ { j = 0 } ^ n t _ { j , n } x ^ j . \\end{gather*}"}
+{"id": "7987.png", "formula": "\\begin{align*} \\pi ' i ( a ) = i ' \\pi ( a ) = \\pi ' \\beta ( x ) . \\end{align*}"}
+{"id": "895.png", "formula": "\\begin{align*} \\mathcal { X } ^ { t + 1 } = \\mathcal { X } ^ { t } - \\mathcal { Q } ^ { - 1 } * \\underline { \\mathcal { W } } * ( \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } ) , \\end{align*}"}
+{"id": "4984.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } V ( x ) | G _ { p , q } ( x ) | ^ 2 \\ , d x = \\langle V \\rangle ( q ) \\ , . \\end{align*}"}
+{"id": "2865.png", "formula": "\\begin{align*} \\prod _ { \\sigma \\in S _ n } x _ { \\sigma ( 1 ) } ^ { c _ 1 } x _ { \\sigma ( 2 ) } ^ { c _ 2 } \\cdots x _ { \\sigma ( n ) } ^ { c _ n } = 1 . \\end{align*}"}
+{"id": "4204.png", "formula": "\\begin{align*} \\begin{aligned} & \\sup _ { t \\in [ 0 , T ] } \\norm { ( \\psi , \\partial _ t \\psi ) } _ { \\mathcal { H } } \\leq C ( 1 + T ) \\left ( \\norm { ( \\psi _ 0 , \\psi _ 1 ) } _ { H ^ { k } ( \\mathbb { R } ) \\times H ^ { k - 1 } ( \\mathbb { R } ) } + \\int _ { 0 } ^ { T } \\norm { f } _ { H ^ { k - 1 } ( \\mathbb { R } ) } ( t ) d t \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "1965.png", "formula": "\\begin{align*} \\rho ( C ) = \\{ ( \\rho ( c _ { 1 } ) , \\rho ( c _ { 2 } ) , \\ldots , \\rho ( c _ { n } ) ) \\ \\vert \\ ( c _ { 1 } , c _ 2 , \\ldots , c _ { n } ) \\in C \\} . \\end{align*}"}
+{"id": "8003.png", "formula": "\\begin{align*} F ( t , x ( t ) , \\dot x ( t ) , \\Delta _ { - \\tau } x ( t ) ) = 0 , \\end{align*}"}
+{"id": "8406.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { \\infty } | \\partial _ x [ ( h _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) ( \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) G _ { \\alpha , n } ] | \\leq C \\sum _ { n = 1 } ^ { \\infty } n ^ { - \\left ( 1 + \\frac { 1 } { \\alpha } \\right ) } ( n + 1 ) < \\infty \\end{align*}"}
+{"id": "8667.png", "formula": "\\begin{align*} \\psi ^ + ( u ) ( z ^ m f ) & = z ^ { m + 1 } u ^ { m } H ( u ) E ^ { \\perp } ( - u ) ( f ) , \\\\ \\psi ^ - ( u ) ( z ^ m f ) & = z ^ { m - 1 } { u ^ { - m } } E ( - u ) H ^ { \\perp } ( u ) ( f ) . \\end{align*}"}
+{"id": "3765.png", "formula": "\\begin{align*} \\alpha _ 0 ( j ) & = \\ell , & \\alpha _ 0 ( \\ell ) & = j , & \\alpha _ 0 ( k ) & = m , & \\alpha _ 0 ( m ) & = k , & \\end{align*}"}
+{"id": "7094.png", "formula": "\\begin{align*} \\pi _ { 2 + \\sigma } = d _ { 1 , 0 } - a _ { 1 , 1 } q _ 2 \\end{align*}"}
+{"id": "7303.png", "formula": "\\begin{align*} | G ^ { k + 1 } _ { m _ n } ( x ) | & = \\left | - x \\int _ { 1 } ^ { x } ( - 1 ) ^ k G ^ k _ { m _ n } ( y ) d m _ n ( y ) + \\int _ { 0 } ^ { x } ( - 1 ) ^ k y G ^ k _ { m _ n } ( y ) d m _ n ( y ) \\right | \\\\ & \\leq \\sup _ { z \\in [ 0 , x _ 0 ] } | G ^ k _ { m _ n } ( z ) | \\left ( x _ 0 | \\tilde { m } _ n ( x _ 0 ) | + \\int _ { 0 } ^ { x _ 0 } y d m _ n ( y ) \\right ) . \\end{align*}"}
+{"id": "7.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\mathcal { Z } _ { ( 1 , t ) } & = e ^ { - \\frac { \\beta } { 2 } t } \\Big [ \\bar { \\mathcal { Z } } \\big ( \\bar { h } _ x x _ { ( 1 , t ) } + \\bar { h } _ u v _ t \\big ) + \\bar { h } \\mathcal { Z } _ { ( 1 , t ) } \\Big ] d \\xi _ t , \\\\ \\mathcal { Z } _ { ( 1 , 0 ) } & = 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "3030.png", "formula": "\\begin{align*} \\mathcal { F } ^ { s } = \\bigcup _ { k } \\ \\{ Q _ k : 2 ^ { - s - 1 } < d _ k \\leq 2 ^ { - s } , \\ \\widetilde { Q } _ k \\subset \\Omega _ { 1 / 4 } \\} , \\ s = 2 , 3 , . . . . \\end{align*}"}
+{"id": "275.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 5 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\rho } - \\beta \\rho ^ 2 \\leqslant 0 \\end{align*}"}
+{"id": "2122.png", "formula": "\\begin{align*} a _ j ^ n - \\frac { 1 } { \\sqrt { 2 \\pi V n } } \\exp \\left ( - \\frac { X _ { n , j } ^ 2 } { 2 } \\right ) - \\sum _ { \\sigma = 2 } ^ s \\frac { q _ \\sigma ( X _ { n , j } ) } { n ^ \\frac { \\sigma } { 2 } } \\underset { n \\rightarrow + \\infty } = o \\left ( \\frac { 1 } { n ^ \\frac { s } { 2 } } \\right ) \\end{align*}"}
+{"id": "6382.png", "formula": "\\begin{align*} K ^ { s + 1 } = ( K ^ s _ { 1 } , \\cdots , K ^ s _ { t + 1 } , K ^ s _ { t } , \\cdots , K ^ s _ { n _ s } ) . \\end{align*}"}
+{"id": "602.png", "formula": "\\begin{align*} \\theta _ { n + 1 } ^ { ( \\epsilon ) } = \\theta _ n ^ { ( \\epsilon ) } + \\frac { \\alpha _ { t _ n } } { \\gamma } ( A X _ { t _ n } ^ { ( \\epsilon ) } ) ^ { \\rm T } \\left ( ( X _ { t _ { n + 1 } } ^ { ( \\epsilon ) } - X _ { t _ n } ^ { ( \\epsilon ) } ) - \\theta _ n ^ { ( \\epsilon ) } A X _ { t _ n } ^ { ( \\epsilon ) } \\Delta t \\right ) . \\end{align*}"}
+{"id": "3798.png", "formula": "\\begin{align*} \\Delta V + D _ X V = 0 , \\end{align*}"}
+{"id": "8257.png", "formula": "\\begin{align*} \\mathcal { I } _ n = n D _ n ( \\hat { \\theta } ) ' ( V _ n ( \\hat { \\theta } ) ) ^ { - 1 } D _ n ( \\hat { \\theta } ) \\end{align*}"}
+{"id": "3124.png", "formula": "\\begin{align*} r = 1 + M : D ^ 2 \\hat { w } , v ^ { k l } = m _ { k l } \\hat { w } \\end{align*}"}
+{"id": "7109.png", "formula": "\\begin{align*} I = ( u x _ i + p _ i \\ ; : \\ ; i \\geq 1 ) \\end{align*}"}
+{"id": "6643.png", "formula": "\\begin{align*} B _ { x ' } ( x ) : = B ( x - x ' ) + b \\end{align*}"}
+{"id": "8438.png", "formula": "\\begin{align*} \\lim _ { K \\uparrow A } \\ , \\nu ( K ) = \\nu _ * ( A ) = \\nu ( X ) = \\infty , \\end{align*}"}
+{"id": "7638.png", "formula": "\\begin{align*} x \\in \\mathbb T , \\ \\ \\partial _ x U ( x ) = 0 \\ \\Rightarrow \\ \\alpha ( x ) > 0 . \\end{align*}"}
+{"id": "4080.png", "formula": "\\begin{align*} H ( X , Y , Z ) & = 2 \\langle [ \\sigma X , \\sigma Y ] _ { H _ 0 } , \\sigma Z \\rangle \\\\ & = 2 \\langle [ X , Y ] + L _ { X } b ( Y ) - i _ { Y } ( d b ( X ) ) + i _ Y i _ X H _ 0 , Z + i _ Z b \\rangle \\\\ & = H _ 0 ( X , Y , Z ) + ( L _ X b ( Y ) - i _ Y ( d b ( X ) ) ) ( Z ) + b ( Z , [ X , Y ] ) \\\\ & = H _ 0 ( X , Y , Z ) + d b ( X , Y , Z ) , \\end{align*}"}
+{"id": "5061.png", "formula": "\\begin{align*} g ( ( n _ j ) _ { j \\in J } ) = f ( n _ 1 , \\ldots , n _ d ) \\quad n _ i = c _ i \\quad i \\in [ 1 , d ] \\setminus J . \\end{align*}"}
+{"id": "6065.png", "formula": "\\begin{align*} \\ell _ j ( x ) : = x ^ g \\sum _ { i = 1 } ^ g \\ell _ { i j } \\big ( x + x ^ { - 1 } \\big ) ^ i = h _ j \\prod _ { i = 1 } ^ g ( x - x _ { i j } ) ( 1 - x x _ { i j } ) , \\end{align*}"}
+{"id": "3437.png", "formula": "\\begin{align*} 1 - p _ { a a } ^ { t } = p _ { a b } ^ { t } p _ { b a } ^ { t } \\sum _ { n \\geq 1 } ( p _ { b b } ^ { t } ) ^ n \\geq \\exp \\left ( - t C - N P _ G ( t \\phi ) \\right ) . \\end{align*}"}
+{"id": "2644.png", "formula": "\\begin{align*} S _ R ( X , Y ; \\hat { \\phi } , \\Phi ) = o ( X \\log X ) , \\mbox { a s } X \\rightarrow \\infty . \\end{align*}"}
+{"id": "7623.png", "formula": "\\begin{align*} \\mathtt { A } ( s ) = - \\nu \\mathcal { A } ^ { ( 1 ) } ( s ) + \\beta \\mathcal { A } ^ { ( 2 ) } ( s ) + \\frac { \\alpha } { \\delta + 1 } \\mathcal { A } ^ { ( 3 ) } ( s ) , \\end{align*}"}
+{"id": "8925.png", "formula": "\\begin{align*} \\left ( - 1 \\right ) ^ { p } \\left ( \\frac { d } { d t } \\right ) ^ { p } \\left [ \\vartheta _ { 3 } ( t ) \\right ] = 2 \\sum \\limits _ { l = 1 } ^ { + \\infty } l ^ { 2 p + 1 } e ^ { - l ^ { 2 } t } \\end{align*}"}
+{"id": "5739.png", "formula": "\\begin{align*} _ { c } \\mathbf { L } _ { B } ^ { A } = [ \\delta _ { C } ^ { A } + ( d L _ { D } ^ { A } ) ( L ^ { - 1 } ) _ { C } ^ { D } \\mathbf { m } ] L _ { B } ^ { C } , \\end{align*}"}
+{"id": "6351.png", "formula": "\\begin{align*} A _ f ( x ) : = \\sqrt { a ^ 2 - f ( x ) ^ 2 } / f ' ( x ) \\end{align*}"}
+{"id": "7416.png", "formula": "\\begin{align*} w X _ { \\alpha } ( \\chi ) = w ( X _ { \\alpha } ( \\chi ) ) = w ( \\chi ( h _ { \\alpha } ) ) = \\chi ( w ( h _ { \\alpha } ) ) = \\chi ( h _ { w ( \\alpha ) } ) = \\chi ( h _ { \\alpha } ) = X _ { \\alpha } ( \\chi ) . \\end{align*}"}
+{"id": "4201.png", "formula": "\\begin{align*} g ^ { ( 1 ) } = \\left [ \\begin{array} { c c } \\dfrac { e ^ { t } Q _ c ( x - v t ) } { Q _ c ( x - v t - x _ 0 ) } & - \\dfrac { 1 } { c } Q _ c ( x - v t ) \\\\ - \\dfrac { 1 } { c } Q _ c ( x - v t ) & \\dfrac { e ^ { - t } Q _ c ( x - v t ) } { Q _ c ( x - v t + x _ 0 ) } \\end{array} \\right ] , \\end{align*}"}
+{"id": "4609.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\frac { 1 } { b _ i } = \\sum _ { i = 1 } ^ { n - 1 } \\frac { 1 } { b _ i } + \\frac { 1 } { b _ n } < \\sum _ { i = 1 } ^ { n - 1 } \\frac { 1 } { a _ i } + \\frac { 1 } { a _ n } = \\sum _ { i = 1 } ^ { n } \\frac { 1 } { a _ i } . \\end{align*}"}
+{"id": "8515.png", "formula": "\\begin{align*} \\begin{cases} z ' = w + z ( z ^ 2 + w ^ 2 + 1 ) \\\\ w ' = - z + w ( z ^ 2 + w ^ 2 + 1 ) \\end{cases} \\end{align*}"}
+{"id": "5471.png", "formula": "\\begin{align*} & P ( \\eta _ { \\alpha } > \\eta + \\delta ) \\\\ & = \\int _ { \\mathbb R ^ d \\times \\mathcal M } P ( \\eta _ { \\alpha } > \\eta + \\delta : \\alpha _ { \\eta } = i , X _ { \\eta } = x ) P ( ( X _ { \\eta } , \\alpha _ { \\eta } ) \\in ( d x , d i ) ) \\\\ & \\geq 1 - \\frac { 1 } { 4 } P ( B ) . \\end{align*}"}
+{"id": "2898.png", "formula": "\\begin{align*} \\int _ { E _ k } \\log ( 1 + | q _ { n _ k } - h | ) d m _ \\infty = \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + \\chi _ { E _ k } | q _ { n _ k } - h | ) d m _ \\infty \\rightarrow 0 \\quad ( k \\rightarrow \\infty ) . \\end{align*}"}
+{"id": "2320.png", "formula": "\\begin{align*} { \\bf B } ( z ) = R \\circ J \\circ T _ { e _ N } \\circ S _ 2 \\circ J \\circ T _ { - e _ N } ( z ) , \\ , \\ , R = \\begin{pmatrix} 1 & & & \\\\ & \\ddots & & \\\\ & & 1 & \\\\ & & & - 1 \\end{pmatrix} , \\ , e _ N = \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ 1 \\end{pmatrix} , \\end{align*}"}
+{"id": "5791.png", "formula": "\\begin{align*} D = C _ p ^ T R C _ p . \\end{align*}"}
+{"id": "5306.png", "formula": "\\begin{align*} \\mathbf { H } = - \\dfrac { 1 } { n } \\left ( \\mathrm { t r a c e } \\ , ( A _ { \\eta } ) \\xi + \\mathrm { t r a c e } \\ , ( A _ { \\xi } ) \\eta \\right ) . \\end{align*}"}
+{"id": "8328.png", "formula": "\\begin{align*} \\tilde { v } = 0 , \\mathrm { i n } \\ \\ | y | + | s | \\le R ^ { - 1 } . \\end{align*}"}
+{"id": "7967.png", "formula": "\\begin{align*} \\pi : M \\to M ' = M / L \\end{align*}"}
+{"id": "135.png", "formula": "\\begin{align*} H _ { \\lambda , K , x , r } ^ + : & = \\left \\{ y \\in \\mathbb R ^ n \\colon | | y | | _ K > r , \\ , \\frac { | f ( x ) - f ( y ) | } { | | x - y | | _ K ^ { \\frac { n } { p } } } \\geq \\lambda \\right \\} \\\\ & = \\left \\{ y \\in \\mathbb R ^ n \\colon | | y | | _ K > r , \\ , \\frac { | f ( x ) | } { | | x - y | | _ K ^ { \\frac { n } { p } } } \\geq \\lambda \\right \\} . \\end{align*}"}
+{"id": "5033.png", "formula": "\\begin{align*} c ^ 4 = \\frac 1 4 \\lambda h ^ 2 . \\end{align*}"}
+{"id": "6540.png", "formula": "\\begin{align*} \\| f \\| _ 2 ^ 2 = \\sum _ { j = j _ 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { 2 ^ { j } - 1 } f _ { j k } ^ 2 . \\end{align*}"}
+{"id": "1635.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\omega = - { \\rm R i c } ( \\omega ) , \\quad \\omega | _ { t = 0 } = \\omega _ 0 , \\end{align*}"}
+{"id": "3403.png", "formula": "\\begin{align*} \\forall { q } \\in \\mathbb { Z } \\quad \\dot { \\Delta } _ { q } = \\varphi ( 2 ^ { q } D ) u , \\dot { S } _ { q } = \\sum _ { j \\le q - 1 } \\dot { \\Delta } _ { j } v . \\end{align*}"}
+{"id": "3116.png", "formula": "\\begin{align*} \\bar { \\gamma } : = \\left ( \\int _ Y \\frac { r ^ 1 } { \\gamma } \\right ) ^ { - 1 } = \\left ( \\int _ Y ( r ^ 1 + r ^ 1 A ^ 1 : D ^ 2 \\phi ) \\right ) ^ { - 1 } = 1 , \\end{align*}"}
+{"id": "4474.png", "formula": "\\begin{align*} \\frac { 4 } { n } = \\frac { 1 } { u } + \\frac { 1 } { v } + \\frac { 1 } { w } . \\end{align*}"}
+{"id": "5408.png", "formula": "\\begin{align*} \\prod _ { k \\in \\N _ { \\ge 1 } } \\big ( 1 - \\tfrac { 1 } { ( \\alpha \\beta ) ^ { 2 ^ { k - 1 } } } \\big ) = f ( \\tfrac { 1 } { \\alpha \\beta } ) . \\end{align*}"}
+{"id": "7972.png", "formula": "\\begin{align*} y + x + f ( q ) + f ( q ' ) = y + \\sum f ( a _ i ) + f ( q ) = y + \\sum f ( a _ i ) + \\sum f ( y _ i ) = \\sum f ( x _ i ) + \\sum f ( b _ i ) . \\end{align*}"}
+{"id": "8232.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow 0 ^ { + } } \\mathcal { S } _ { 0 } \\left ( q _ { 1 } , R \\right ) = \\lim _ { R \\rightarrow + \\infty } \\mathcal { S } _ { \\infty } \\left ( q _ { 2 } , R \\right ) = 0 , \\tag * { $ \\left ( { \\cal S } _ { q _ { 1 } , q _ { 2 } } ^ { \\prime \\prime } \\right ) $ } \\end{align*}"}
+{"id": "5321.png", "formula": "\\begin{align*} \\mathrm { A d } _ { V _ 1 } \\circ \\pi = \\lambda \\mathrm { A d } _ W \\circ \\mathrm { A d } _ { T _ 1 ^ { - 1 } } \\circ \\mathrm { A d } _ { T ^ { \\frac { 1 } { 2 } } V } \\circ \\pi = \\lambda \\mathrm { A d } _ { W } \\circ \\mathrm { A d } _ { T _ 1 ^ { - 1 } } \\circ \\Psi = \\mathrm { A d } _ W \\circ \\Phi _ 1 = \\Phi \\end{align*}"}
+{"id": "3925.png", "formula": "\\begin{align*} \\det [ D ^ 2 u - A ( \\cdot , u , D u ) ] = B ( \\cdot , u , D u ) \\Omega , \\\\ u = g ( \\cdot , y , z ) \\partial \\Omega , \\end{align*}"}
+{"id": "179.png", "formula": "\\begin{align*} T _ 1 & = \\max \\{ m _ i + m _ j - 1 \\ | \\ i \\ne j ; i , j = 1 , \\ldots , s \\} , \\\\ T _ r & = \\left [ \\frac { m _ 1 + \\cdots + m _ { s } + r - 2 } { r } \\right ] . \\end{align*}"}
+{"id": "4427.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { 5 } , \\frac { 1 } { 4 } \\right ] = I ( 5 ) = \\bigcup _ { a _ 2 = 2 1 } ^ { \\infty } J ( 5 , a _ 2 ) = \\bigcup _ { a _ 2 = 2 1 } ^ { \\infty } \\left ( \\frac { 1 } { 5 } + \\frac { 1 } { a _ 2 } , \\frac { 1 } { 5 } + \\frac { 1 } { a _ 2 - 1 } \\right ] . \\end{align*}"}
+{"id": "1259.png", "formula": "\\begin{align*} R _ { \\mathcal { B S } ( \\alpha ) } ( \\mathcal { S } ^ { * } ) & = \\begin{dcases} \\dfrac { 1 } { 3 - 2 \\alpha } , & 0 \\leq \\alpha \\leq \\frac { 1 } { 9 } , \\\\ \\dfrac { 2 \\sqrt { \\alpha } } { ( 1 + \\alpha ) \\sqrt { 1 + 1 6 \\alpha } } , & \\frac { 1 } { 9 } \\leq \\alpha \\leq 1 . \\end{dcases} \\end{align*}"}
+{"id": "200.png", "formula": "\\begin{align*} \\tilde { B } _ { 2 } = \\prod _ { i = 0 } ^ { r _ { 1 } } \\tilde { B } _ { 1 } 1 ^ { e _ { 1 } + i } = \\Big { ( } \\prod _ { i = 0 } ^ { r _ { 1 } - 1 } \\tilde { B } _ { 1 } 1 ^ { e _ { 1 } + i } \\Big { ) } \\tilde { B } _ { 1 } 1 ^ { e _ { 1 } + r _ { 1 } } = \\Big { ( } \\prod _ { i = 0 } ^ { r _ { 1 } - 1 } B _ { 1 } 1 ^ { c _ { 1 } + i } \\Big { ) } B _ { 1 } 1 ^ { e _ { 1 } + r _ { 1 } } \\end{align*}"}
+{"id": "8112.png", "formula": "\\begin{align*} W _ { m , n } ^ + ( x ) = 4 T e \\Bigl ( - \\frac { x } { 2 \\pi } \\Bigr ) \\int _ { - \\infty } ^ \\infty k _ 0 ^ * ( \\xi ) e \\Bigl ( - \\frac { T \\xi } { M } - \\frac { \\pi x \\xi ^ 2 } { 4 M ^ 2 } \\Bigr ) \\ , d \\xi \\end{align*}"}
+{"id": "6616.png", "formula": "\\begin{align*} \\sum _ { u \\in \\C { N } _ v \\setminus \\{ w \\} } | \\Im \\zeta _ v ^ \\gamma ( u ) | = \\frac { | \\Im \\zeta _ w ^ \\gamma ( u ) | } { | \\zeta _ w ^ \\gamma ( v ) | ^ 2 } - \\Im \\gamma . \\end{align*}"}
+{"id": "4780.png", "formula": "\\begin{align*} \\Pr _ { \\rho \\sim P _ \\epsilon } [ \\rho \\notin D _ { k , l } ( H \\rho ^ \\intercal ) ] & \\leq \\Pr _ { \\rho \\sim P _ \\epsilon } [ \\rho \\in B ] + \\Pr _ { \\rho \\sim P _ \\epsilon } [ \\rho \\notin D _ { k , l } ( H \\rho ^ \\intercal ) \\big | \\rho \\notin B ] \\\\ & \\leq 2 e ^ { - \\frac { l ^ 2 } { 3 \\epsilon N } } + \\max _ { w \\in \\{ \\epsilon N \\pm l \\} } \\Pr _ { \\rho \\sim P _ \\epsilon } [ \\rho \\notin D _ { k , l } ( H \\rho ^ \\intercal ) \\big | | \\rho | = w ] . \\end{align*}"}
+{"id": "5761.png", "formula": "\\begin{align*} q \\left ( \\xi + \\sum _ { i = 1 } ^ r s _ i d \\log f _ i + \\sum _ { i = 1 } ^ { r } s _ i ( - d \\log t _ i ) \\right ) ( v ) = 0 \\end{align*}"}
+{"id": "7723.png", "formula": "\\begin{align*} \\sin \\varphi _ 0 & = \\sqrt { \\frac { 2 } { \\tau } } \\sqrt { \\sqrt { 1 + \\tau } - 1 } , \\\\ \\cos \\varphi _ 0 & = - \\sqrt { 1 - \\frac { 2 } { \\tau } ( \\sqrt { 1 + \\tau } - 1 ) } , \\end{align*}"}
+{"id": "5118.png", "formula": "\\begin{align*} \\left ( \\overline { f } \\right ) ' ( w ) = - \\frac { \\overline { f ' ( w ) } } { w ^ { 2 } } . \\end{align*}"}
+{"id": "2116.png", "formula": "\\begin{align*} e _ { C _ { 1 2 } } ( P ) e _ { C _ { 1 2 } } ( P ) ^ T + g ( P ) g ( P ) ^ T = \\big ( e _ { C _ { 1 2 } } \\ ! ( P ) \\ , ~ g ( P ) \\big ) \\big ( e _ { C _ { 1 2 } } \\ ! ( P ) \\ , ~ g ( P ) \\big ) ^ T = 0 . \\end{align*}"}
+{"id": "2074.png", "formula": "\\begin{align*} H \\left ( \\tfrac { 1 } { \\log ( 1 / \\kappa ) } \\right ) = \\tau + H \\left ( \\tau \\right ) = \\lambda , \\end{align*}"}
+{"id": "668.png", "formula": "\\begin{align*} ( \\frac { \\partial } { \\partial t } + \\mathcal { L } _ { \\bf v } ) ( { \\bf \\nu } ) = d ( \\frac { 1 } { 2 } | { \\bf v } | ^ 2 - { p } ) . \\end{align*}"}
+{"id": "4034.png", "formula": "\\begin{align*} f ^ * ( Y ( \\cdot , v , D v ) ) & \\det D Y ( \\cdot , v , D v ) = e ^ { [ ( 1 - t ) \\tau + \\eta _ a ( \\cdot ) ] ( v - u _ 0 ) } \\big [ t f ( \\cdot ) \\\\ & \\quad + ( 1 - t ) f ^ * ( Y ( \\cdot , u _ 0 , D u _ 0 ) ) \\det D Y ( \\cdot , u _ 0 , D u _ 0 ) \\big ] \\\\ Y u ( \\Omega ) & = \\Omega ^ * . \\end{align*}"}
+{"id": "3518.png", "formula": "\\begin{align*} E _ { i j } ^ m \\Omega _ { D ( \\gamma ) } = \\frac { \\gamma _ { j k } ! } { ( \\gamma _ { j k } - m ) ! } \\Omega _ { D ( \\gamma + m ( e _ { i k } - e _ { j k } ) ) } , \\end{align*}"}
+{"id": "545.png", "formula": "\\begin{align*} { \\rm d } X _ t ^ \\dagger = f ^ \\dagger ( X _ t ^ \\dagger ) { \\rm d } t + \\gamma ^ { 1 / 2 } { \\rm d } W ^ \\dagger _ t , \\end{align*}"}
+{"id": "5494.png", "formula": "\\begin{align*} F _ \\lambda ( x ) : = - J \\dot x - \\nabla H ( x ) + 2 \\pi \\lambda x , \\forall \\lambda \\in \\R , \\end{align*}"}
+{"id": "5269.png", "formula": "\\begin{align*} { \\tilde H } = - { \\tilde \\nabla } ( f + \\log \\phi ) = - { \\tilde \\nabla } \\log ( \\phi e ^ f ) . \\end{align*}"}
+{"id": "2183.png", "formula": "\\begin{align*} \\dfrac { z F ' ( z ) } { F ( z ) } & = \\dfrac { ( 1 - 2 \\alpha - \\beta ) z ^ 2 + ( 2 - 2 \\alpha + \\beta ) z + 1 } { 1 - z ^ 2 } . \\end{align*}"}
+{"id": "5464.png", "formula": "\\begin{align*} E \\sum _ { k = 0 } ^ { [ T / \\epsilon ] } \\int _ { k \\epsilon } ^ { ( k + 1 ) \\epsilon } \\mathcal \\mathbb 1 _ { [ 0 , \\tau _ N ^ { n , m } ] } ( r ) | b ( r , X ^ m _ { k \\epsilon \\wedge \\tau ^ m } , \\mu ^ m _ r , \\alpha ^ m _ r ) - b ( r , X ^ m _ { r \\wedge \\tau ^ m } , \\mu ^ m _ r , \\alpha ^ m _ r ) | ^ 2 d r \\leq C ( N , T ) \\epsilon . \\end{align*}"}
+{"id": "4800.png", "formula": "\\begin{align*} \\Pr _ { v \\sim \\mu _ t } \\big [ | v H | = j \\big ] \\leq 2 ^ { - ( 1 - h ( \\frac { j } { N } ) ) ( 1 - \\eta ) N } , \\end{align*}"}
+{"id": "1703.png", "formula": "\\begin{align*} s ' { \\delta s ( \\mu _ { - m } ) } = \\frac { ( k - 1 ) } { 2 } \\left ( ( - i ) ^ { \\frac { k - 2 } { 2 } - m } f _ { \\frac { k } { 2 } } + i ^ { \\frac { k - 2 } { 2 } - m } f _ { - \\frac { k } { 2 } } \\right ) , \\end{align*}"}
+{"id": "3553.png", "formula": "\\begin{align*} B _ j ^ - \\Omega _ \\lambda = \\sum _ { i = j } ^ { n } \\sum _ { s = 1 } ^ { i - j + 1 } \\sum _ { I \\in \\mathcal { I } _ { j i } ( s ) } d _ { i } ^ - ( \\lambda ) \\frac { 1 } { \\prod _ { \\ell \\in I , \\ell \\neq i } ( \\lambda _ i - \\lambda _ \\ell - i + \\ell ) } E ^ { e _ I } \\Omega _ { \\lambda - \\epsilon _ i } , \\end{align*}"}
+{"id": "4395.png", "formula": "\\begin{align*} u _ n ( \\theta ) & = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ( \\theta ) \\right \\} \\\\ & = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ( \\theta ) x _ 1 < x _ 1 ^ * \\right \\} \\\\ & = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ^ { ( 1 ) } ( \\theta ) \\right \\} . \\end{align*}"}
+{"id": "1515.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left ( \\frac 1 { \\pi x ^ { r - 1 } } \\frac { \\C _ r ' ( x ) } { \\C _ r ( x ) } \\right ) = - \\frac { r - 1 } { \\pi x ^ { r } } \\frac { \\C _ r ' ( x ) } { \\C _ r ( x ) } + \\frac 1 { \\pi x ^ { r - 1 } } \\left ( \\frac { \\C _ r '' ( x ) } { \\C _ r ( x ) } - \\frac { \\C _ r ' ( x ) ^ 2 \\C _ r ( x ) ^ { - 1 } } { \\C _ r ( x ) } \\right ) . \\end{align*}"}
+{"id": "7070.png", "formula": "\\begin{align*} d \\cap x ^ { m + n \\sigma } = 0 . \\end{align*}"}
+{"id": "3401.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t \\omega _ { \\theta } + v \\cdot \\nabla \\omega _ { \\theta } - \\mu \\big ( \\Delta - \\frac { 1 } { r ^ 2 } \\big ) \\omega _ { \\theta } = \\frac { v _ r } { r } \\omega _ { \\theta } - \\partial _ z \\frac { ( B _ { \\theta } ) ^ 2 } { r } , & \\\\ \\partial _ { t } B _ { \\theta } + v \\cdot \\nabla B _ { \\theta } - \\big ( \\Delta - \\frac { 1 } { r ^ 2 } \\big ) B _ { \\theta } = \\frac { v _ r } { r } B _ { \\theta } . \\end{array} \\right . \\end{align*}"}
+{"id": "7161.png", "formula": "\\begin{align*} \\displaystyle { H _ i = { \\tilde H } _ i \\Big | _ { \\rm e q } \\ , . } \\end{align*}"}
+{"id": "65.png", "formula": "\\begin{align*} x '' = x ' + y ' \\textnormal { a n d } y '' = y ' . \\end{align*}"}
+{"id": "8741.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\mathcal { H } _ \\infty ( \\hat { g } _ { } , g _ 0 ) \\lesssim \\begin{cases} \\frac { \\sigma _ z } { \\sigma _ u } \\sqrt { \\frac { r T ^ 2 } { N } } \\log ( T ) , & N \\geq d _ 0 ^ 2 \\wedge T , \\\\ \\frac { \\sigma _ z } { \\sigma _ u } \\sqrt { \\frac { d _ 0 r T ^ 2 } { N } } \\log ( T ) , & d _ 0 \\leq N \\leq d _ 0 ^ 2 \\wedge T . \\end{cases} \\end{align*}"}
+{"id": "789.png", "formula": "\\begin{align*} A _ \\epsilon = \\left \\{ g \\in \\Gamma : \\left | \\frac { \\varphi ( g ) } { | g | _ S } - \\Lambda \\right | > \\epsilon \\right \\} . \\end{align*}"}
+{"id": "6460.png", "formula": "\\begin{align*} w _ * \\wedge H ( w _ * ) & = w _ * \\wedge ( h ( t ) e _ 1 ) = - h ( t ) e _ 1 \\wedge w _ * , \\\\ w _ * \\wedge ( w _ * \\wedge H ( w _ * ) ) & = - h ( t ) w _ * \\wedge ( e _ 1 \\wedge w _ * ) . \\end{align*}"}
+{"id": "1202.png", "formula": "\\begin{align*} \\frac { \\nabla _ { g _ C } } { d t } ( \\Phi ^ t _ * | _ p w ) = \\nabla _ { g _ C } X | _ { \\Phi ^ t ( p ) } ( \\Phi ^ t _ * | _ p w ) \\end{align*}"}
+{"id": "6578.png", "formula": "\\begin{align*} K ( \\mu , s , s ' ) : = \\sup _ { x \\in B _ s , r \\leq s ' } \\frac { 1 } { r ^ { n - 1 } } \\mu ( B _ r ( x ) ) < \\infty . \\end{align*}"}
+{"id": "5519.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n ^ k } \\exp \\left ( { - \\frac { x } { n ^ 2 } } \\right ) = \\frac { \\Gamma ( \\frac { k } { 2 } ) } { x ^ \\frac { k } { 2 } } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n } { } _ 1 F _ 1 \\left ( \\frac { k } { 2 } ; \\frac { 1 } { 2 } ; - \\frac { \\pi ^ 2 } { n ^ 2 x } \\right ) + \\frac { 1 } { 2 } \\sum _ { \\rho } \\frac { \\Gamma ( \\frac { k - \\rho } { 2 } ) } { \\zeta ' ( \\rho ) } x ^ { - \\frac { ( k - \\rho ) } { 2 } } , \\end{align*}"}
+{"id": "1146.png", "formula": "\\begin{align*} \\overline { H } ( f _ n \\otimes v _ n ) = T _ { \\ell } ( f _ n ) \\otimes v _ n \\end{align*}"}
+{"id": "6118.png", "formula": "\\begin{align*} \\begin{cases} \\Phi '' - { \\Delta } \\Phi + A ^ T \\Phi = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\Phi = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\cr \\partial _ \\nu \\Phi + B ^ T \\Phi = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{cases} \\end{align*}"}
+{"id": "6588.png", "formula": "\\begin{align*} \\theta ( k , i ) = \\beta _ { k , M _ - } ( g _ k ( 1 ) ) = \\beta _ { k , M _ - } \\circ g _ k \\circ \\beta _ { k , M _ + } ^ { - 1 } ( k , i ) = \\eta ( k , i ) \\end{align*}"}
+{"id": "2507.png", "formula": "\\begin{align*} V : = \\begin{pmatrix} S _ \\infty & K \\\\ \\mathbb O & R _ 0 \\end{pmatrix} \\cong S _ \\infty . \\end{align*}"}
+{"id": "5544.png", "formula": "\\begin{align*} \\frac { 1 } { \\zeta ( 1 - k + 2 s ) } = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) n ^ { k - 1 } } { n ^ { 2 s } } . \\end{align*}"}
+{"id": "6223.png", "formula": "\\begin{align*} \\begin{pmatrix} C _ p \\\\ E _ 1 ^ T \\\\ \\cdot \\\\ E _ p ^ T \\end{pmatrix} U _ n = \\begin{pmatrix} C _ p U _ n \\\\ E _ 1 ^ T U _ n \\\\ \\cdot \\\\ E _ p ^ T U _ n \\end{pmatrix} \\rightarrow \\begin{pmatrix} 0 \\\\ u _ 1 \\\\ \\cdot \\\\ u _ p \\end{pmatrix} \\end{align*}"}
+{"id": "9257.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots v _ { w ^ { - 1 } ( k ) } , \\ldots , v _ { h ( w ^ { - 1 } ( k ) ) } , e _ { k - 1 } , \\ldots \\widehat { v _ { r } } , \\ldots v _ n ) } \\end{aligned} \\end{align*}"}
+{"id": "3884.png", "formula": "\\begin{align*} g ^ * ( x , y , g ( x , y , z ) ) = z . \\end{align*}"}
+{"id": "5808.png", "formula": "\\begin{align*} C _ p A e _ s = 0 , C _ p D e _ s = 0 , s = 1 , \\cdots , p . \\end{align*}"}
+{"id": "3556.png", "formula": "\\begin{align*} ( \\mathbf { p } B _ i ^ \\pm ) \\Omega _ \\lambda = \\begin{cases} d _ { i } ^ \\pm ( \\lambda ) \\Omega _ { \\lambda \\pm \\epsilon _ i } , & \\lambda \\pm \\epsilon _ i \\in \\mathbb { P } \\ell ( \\lambda \\pm \\epsilon _ i ) \\leq p , \\\\ 0 , , \\end{cases} \\end{align*}"}
+{"id": "7081.png", "formula": "\\begin{align*} \\mathcal { E } ^ 1 _ { p , q } = \\begin{cases} E ^ { C _ 2 } _ q & p = 0 \\\\ E _ { p + q } & p = 1 , 2 \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "5367.png", "formula": "\\begin{align*} | E ^ { ( i ) } | ^ 2 - | E ^ { ( j ) } | ^ 2 = 0 \\forall i , j \\in \\{ 1 , \\ldots , m \\} . \\end{align*}"}
+{"id": "4297.png", "formula": "\\begin{align*} w = e ^ { - j 2 \\pi / N _ b } . \\end{align*}"}
+{"id": "8672.png", "formula": "\\begin{align*} \\varphi ( u ) \\varphi ( v ) + \\varphi ( v ) \\varphi ( u ) = 2 v \\delta ( v , - u ) , \\end{align*}"}
+{"id": "1606.png", "formula": "\\begin{align*} C = \\prod _ { i = 1 } ^ n \\left [ a _ i - \\frac { \\widetilde \\varepsilon ^ p } { K ^ p L ^ 2 } , a _ i + \\frac { \\widetilde \\varepsilon ^ p } { K ^ p L ^ 2 } \\right ] \\subseteq \\mathbb { R } ^ L \\end{align*}"}
+{"id": "6822.png", "formula": "\\begin{align*} p _ m & = - D _ 1 L _ d ( q _ m , q _ { m + 1 } ) , \\\\ p _ { m + 1 } & = D _ 2 L _ d ( q _ m , q _ { m + 1 } ) . \\end{align*}"}
+{"id": "6684.png", "formula": "\\begin{align*} \\nabla F ( \\bar { x } ^ k ) - \\bar { y } ^ k = \\frac { 1 } { m } \\sum _ { i = 1 } ^ m ( \\nabla f _ i ( \\bar x ^ k ) - g _ i ^ k ) + \\bar { g } ^ k - \\bar { y } ^ k \\end{align*}"}
+{"id": "8498.png", "formula": "\\begin{align*} \\oint _ C \\mu d s & = \\ ; \\oint _ C \\left ( \\kappa ^ { 4 / 3 } - \\frac { 5 } { 9 } \\kappa ^ { - 8 / 3 } \\kappa ^ 2 _ s + \\frac { 1 } { 3 } \\kappa ^ { - 5 / 3 } \\kappa _ { s s } \\right ) d s \\\\ & = \\ ; \\oint _ C \\Big ( \\kappa ^ { 4 / 3 } - \\frac { 5 } { 9 } \\kappa ^ { - 8 / 3 } \\kappa ^ 2 _ s \\Big ) d s - \\oint _ C \\Big ( \\frac { 1 } { 3 } \\kappa _ { s } \\Big ) d \\left ( \\kappa ^ { - 5 / 3 } \\right ) \\\\ & = \\ ; \\oint _ C \\kappa ^ { 4 / 3 } d s > 0 . \\end{align*}"}
+{"id": "7746.png", "formula": "\\begin{align*} \\widetilde f ( s | \\mu ) = e ^ { - \\mu } \\sum _ { n = 0 } ^ \\infty \\frac { \\mu ^ n } { n ! } \\ , \\widetilde \\ell \\ , ^ { n } ( s ) & = \\exp \\{ - \\mu ( 1 - \\widetilde \\ell ( s ) ) \\} \\\\ & = \\exp \\left \\{ - \\mu \\int _ 0 ^ \\infty \\left ( 1 - e ^ { - s x } \\right ) \\ell ( x ) d x \\right \\} \\end{align*}"}
+{"id": "7192.png", "formula": "\\begin{align*} A _ { \\beta \\alpha } ( { \\Sigma _ 0 } ) y ^ 3 _ { \\alpha } = C _ \\beta ( { \\Sigma _ 0 } ) , \\end{align*}"}
+{"id": "6248.png", "formula": "\\begin{align*} { \\begin{cases} \\eta \\geq 2 - \\delta , & \\eta > 1 , \\\\ \\eta \\leq \\frac { \\delta + 1 } 2 , & \\eta < 1 . \\end{cases} } \\end{align*}"}
+{"id": "6413.png", "formula": "\\begin{align*} \\lambda ( ( p - 2 ) B ) + \\textrm { t r } B = ( p - 2 ) \\lambda _ m ( B ) + \\sum ^ m _ { i = 1 } \\lambda _ i ( B ) = ( p - 1 ) \\lambda _ m ( B ) + \\sum ^ { m - 1 } _ { i = 1 } \\lambda _ i ( B ) . \\end{align*}"}
+{"id": "762.png", "formula": "\\begin{align*} \\widetilde { \\mathbb { W } } _ { \\kappa } : = \\left ( \\mathbb { W } \\otimes _ { \\Lambda } \\mathcal { R } _ { \\mathcal { P } _ \\kappa } \\right ) [ f ^ { B } _ \\kappa ] \\end{align*}"}
+{"id": "3422.png", "formula": "\\begin{align*} \\nu \\left ( L _ { \\phi } f \\right ) = \\exp ( P _ G ( \\phi ) ) \\nu ( f ) , \\mbox { f o r e a c h } f \\in L ^ 1 ( \\nu ) . \\end{align*}"}
+{"id": "7078.png", "formula": "\\begin{align*} F _ n E ^ { C _ 2 } _ * = E ^ { C _ 2 } _ { * + | n \\sigma | - n \\sigma } \\end{align*}"}
+{"id": "4981.png", "formula": "\\begin{align*} G _ { p , q } ( x ) : = e ^ { i p \\cdot x } g ( x - q ) \\ , , x \\in \\R ^ d \\ , , \\end{align*}"}
+{"id": "7279.png", "formula": "\\begin{align*} \\frac { d } { d m } \\frac { d ^ + } { d x } g _ m = \\lambda g _ m \\end{align*}"}
+{"id": "311.png", "formula": "\\begin{align*} \\epsilon _ { \\mathrm { K a l } } | _ { S ( F ) } ( t ) = 1 , \\end{align*}"}
+{"id": "7000.png", "formula": "\\begin{align*} R _ c ( R _ p ) = \\frac { 1 } { 2 } \\log ^ { + } { \\frac { 1 - \\rho ^ 2 } { \\Delta ^ 2 e ^ { 2 R _ p } } } \\end{align*}"}
+{"id": "9249.png", "formula": "\\begin{align*} \\tilde { d } = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots v _ { w ^ { - 1 } ( k ) } , v _ { w ^ { - 1 } ( k ) + 1 } , X v _ { w ^ { - 1 } ( k ) } , \\ldots \\widehat { v _ { r } } , \\ldots v _ n ) } \\neq 0 , \\end{align*}"}
+{"id": "2498.png", "formula": "\\begin{align*} \\operatorname { c l o s } Y H ^ 2 = H ^ 2 \\end{align*}"}
+{"id": "4289.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\textbf { \\textit { u ' } } + \\nabla \\pi ' = \\boldsymbol { 0 } \\quad \\quad \\mathrm { d i v } \\ , \\textbf { \\textit { u ' } } = 0 \\quad \\quad \\Omega ' , \\\\ \\textbf { \\textit { u ' } } = \\textbf { \\textit { u } } \\quad \\quad \\Gamma . \\end{cases} \\end{align*}"}
+{"id": "7621.png", "formula": "\\begin{align*} \\Psi _ k ( w _ n ( t ) ) = \\Psi _ k ( w _ n ( 0 ) ) + \\int _ { 0 } ^ { t } \\mathtt { A } ( s ) \\d s + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } \\mathtt { B } ( s ) \\d s + M _ { n } ( t ) , \\end{align*}"}
+{"id": "7981.png", "formula": "\\begin{align*} ( r + s ) [ m ] = [ ( r + s ) m ] = [ r m + s m ] = [ r m ] + [ s m ] = r [ m ] + s [ m ] . \\end{align*}"}
+{"id": "2363.png", "formula": "\\begin{align*} E ( t ) \\dot x = A ( t ) x + f ( t ) , \\begin{array} { l } E , A \\in C ( { \\mathbb I } , { \\mathbb R } ^ { n , n } ) , \\ f \\in C ( { \\mathbb I } , { \\mathbb R } ^ { n } ) \\mbox { s u f f i c i e n t l y s m o o t h } , \\end{array} \\end{align*}"}
+{"id": "8425.png", "formula": "\\begin{align*} g = 1 , \\end{align*}"}
+{"id": "4258.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { k \\geq 0 } f _ k ( x _ \\perp ) \\Phi _ k ( x _ \\intercal ) . \\end{align*}"}
+{"id": "4465.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\ell - 1 } v _ i = v _ { \\ell - 1 } \\prod _ { i = 1 } ^ { \\ell - 2 } v _ i \\leq u ' _ { \\ell - 1 } \\prod _ { i = 1 } ^ { \\ell - 2 } u ' _ i = \\prod _ { i = 1 } ^ { \\ell - 1 } u ' _ i . \\end{align*}"}
+{"id": "4058.png", "formula": "\\begin{align*} f ( X v ( \\cdot ) ) \\det D X v = & f ^ * ( \\cdot ) \\overline { \\Omega ^ * } \\\\ X v ( \\Omega ^ * ) & = \\Omega . \\end{align*}"}
+{"id": "3879.png", "formula": "\\begin{align*} D Y ( x , u , D u ) = E ^ { - 1 } [ D ^ 2 u - g _ { x x } ( x , Y ( x , u , D u ) , Z ( x , u , D u ) ) ] , \\end{align*}"}
+{"id": "8028.png", "formula": "\\begin{align*} B _ { n + 1 } = B _ n ( A _ n - \\beta _ 1 B _ n ) ^ { b _ 1 } \\cdots ( A _ n - \\beta _ t B _ n ) ^ { b _ t } = q ( A _ n / B _ n ) \\cdot B _ n ^ { { \\rm d e g } ( p ( x ) ) } . \\end{align*}"}
+{"id": "6970.png", "formula": "\\begin{gather*} P _ { m , n } ^ { ( \\alpha , \\beta ) } ( x ) = ( - 1 ) ^ { m } \\frac { ( \\beta + n ) ( \\alpha + n - 2 m + 1 ) } { ( \\alpha + n - m + 1 ) ( x + 1 ) ^ { \\beta } } \\int _ { - 1 } ^ x ( t + 1 ) ^ { \\beta - 1 } P _ m ^ { ( - \\alpha - 1 , \\beta - 1 ) } ( t ) P _ { n - m } ^ { ( \\alpha + 1 , \\beta - 1 ) } ( t ) \\ , { \\rm d } t \\end{gather*}"}
+{"id": "6443.png", "formula": "\\begin{align*} \\phi _ j ( X , Y , t ) : = \\phi ( X , Y , t ) + u _ { \\epsilon _ j } ( X _ j , Y _ j , t _ j ) - \\phi ( X _ j , Y _ j , t _ j ) . \\end{align*}"}
+{"id": "113.png", "formula": "\\begin{align*} \\lim \\limits _ { s \\rightarrow 1 ^ - } ( 1 - s ) \\int _ { \\mathbb R ^ n } \\int _ { \\mathbb R ^ n } \\frac { | f ( x ) - f ( y ) | ^ p } { | | x - y | | ^ { n + s p } _ K } d x d y = \\frac { 2 } { p } \\int _ { \\mathbb R ^ n } | | \\nabla f ( x ) | | _ { Z _ p ^ * K } ^ p d x . \\end{align*}"}
+{"id": "3904.png", "formula": "\\begin{align*} \\det D Y u = \\frac { f ( \\cdot ) } { f ^ * ( Y u ) } \\Omega , \\end{align*}"}
+{"id": "6848.png", "formula": "\\begin{align*} \\tau ( L _ 1 , L _ 2 , \\dots , L _ m ) = \\mu _ c ^ { + } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) - \\mu _ c ^ { - } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) = \\nu _ c ^ { + } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) - \\nu _ c ^ { - } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) , \\end{align*}"}
+{"id": "4846.png", "formula": "\\begin{align*} I _ n ^ \\flat ( 0 , t ; v ) & = \\int _ { 0 } ^ t v ( t _ n ) \\otimes I _ { n - 1 } ^ \\flat ( 0 , t _ n ; v ) d t _ n \\\\ & = - \\int _ { t } ^ 1 v ( t _ n ) \\otimes I _ { n - 1 } ^ \\flat ( 0 , t _ n ; v ) d t _ n \\\\ & = ( - 1 ) ^ { n } \\int _ { t } ^ 1 v ( t _ n ) \\otimes I _ { n - 1 } ( t _ n , 1 - t _ n ; v ) d t _ n \\\\ & = ( - 1 ) ^ { n } I _ n ( t , 1 - t ; v ) . \\end{align*}"}
+{"id": "111.png", "formula": "\\begin{align*} \\lim \\limits _ { s \\rightarrow 0 ^ + } s | | f | | ^ p _ { W ^ { s , p } ( \\mathbb R ^ n ) } = \\frac { 2 n } { p } | B ^ n | | | f | | ^ p _ { L ^ p ( \\mathbb R ^ n ) } , \\end{align*}"}
+{"id": "3674.png", "formula": "\\begin{align*} \\begin{bmatrix} A & I \\\\ - \\alpha _ \\rho ^ k c & I - \\alpha _ \\rho ^ k \\end{bmatrix} \\begin{bmatrix} \\delta u \\\\ \\delta \\lambda \\end{bmatrix} & = - \\begin{bmatrix} A u ^ k + \\lambda ^ k - f \\\\ M _ \\rho ( \\lambda ^ k , u ^ k - g ) \\end{bmatrix} . \\end{align*}"}
+{"id": "6661.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { x } ^ { k + 1 } & = \\bar { x } ^ k + \\gamma _ 1 ^ k \\bar { \\zeta } _ w ^ k - \\lambda ^ k \\frac { ( u \\otimes { I _ d } ) ^ T } { m } y ^ k \\\\ \\bar { y } ^ { k + 1 } & = ( 1 - \\alpha ^ k ) \\bar { y } ^ k + \\gamma _ 2 ^ k \\bar { \\xi } _ w ^ k + \\bar { g } ^ { k + 1 } - ( 1 - \\alpha ^ k ) \\bar { g } ^ k \\end{aligned} \\end{align*}"}
+{"id": "305.png", "formula": "\\begin{align*} \\begin{aligned} \\mathfrak { W } _ i = \\bigoplus _ { \\mathcal { O } \\in ( \\Phi ^ { i + 1 } - \\Phi ^ { i } ) / \\Gamma _ { F } } ( \\bigoplus _ { \\alpha \\in \\mathcal { O } } \\mathfrak { g } _ { \\alpha } ) ( F ) _ { x , s _ i , s _ { i } ^ { + } } . \\end{aligned} \\end{align*}"}
+{"id": "2161.png", "formula": "\\begin{align*} \\phi ( r ) : = ( 1 - \\mathit { e } ( - 1 + 2 \\alpha + \\beta ) ) r ^ 2 - \\mathit { e } ( 2 - 2 \\alpha + \\beta ) r + ( \\mathit { e } - 1 ) \\end{align*}"}
+{"id": "1830.png", "formula": "\\begin{align*} ( D + \\mu ( K _ X + \\Delta ) ) \\cdot R = 0 . \\end{align*}"}
+{"id": "6348.png", "formula": "\\begin{align*} \\Psi _ { \\nu _ 0 } ( t , h ( t ) ) = \\psi ( \\nu _ 0 ) - \\int ^ t _ 1 f ( x , \\nu _ 0 ) d x + \\int ^ 1 _ { h ( t ) } f ( x , \\nu _ 0 ) d x \\end{align*}"}
+{"id": "1275.png", "formula": "\\begin{align*} \\mathcal { C } _ 3 = \\left \\{ \\big ( c ^ { ( z ^ 0 ) } | c ^ { ( z ^ 1 ) } | \\cdots | c ^ { ( z ^ { k - 1 } ) } \\big ) : c ( x , y , z ) \\in \\mathcal { C } \\right \\} . \\end{align*}"}
+{"id": "7556.png", "formula": "\\begin{align*} & Z _ f ( s , \\chi , A _ 2 ) \\\\ = & q ^ { - \\omega - 1 - r s } \\int _ { B _ 2 } \\chi ( a c ( \\pi ^ { k + l } w ^ { k + r + l } u ^ p + w ^ { r + l } v ^ r \\mathbb { H } _ r ^ k ( \\pi v w , t w - \\pi v w ) ) ) \\\\ \\times & | \\pi ^ { k + l } w ^ { k + r + l } u ^ p + w ^ { r + l } v ^ r \\mathbb { H } _ r ^ k ( \\pi v w , t w - \\pi v w ) | ^ s | w ^ { \\frac { k + r + l } { p } + 1 } | | d u d v d w | , \\\\ : = & q ^ { - ( \\omega + 1 ) - r s } Z _ { f _ 2 } ( s , \\chi ) \\int _ { \\mathcal { O } _ K ^ { \\times } } \\chi ( a c ( w ^ { k + r + l } ) ) | d w | , \\end{align*}"}
+{"id": "2375.png", "formula": "\\begin{align*} \\overline { \\bigcup _ { j \\in { \\mathbb N } } { \\mathbb I } _ j } = { \\mathbb I } \\end{align*}"}
+{"id": "7341.png", "formula": "\\begin{align*} \\ell ( x ) : = c ( x ) \\exp \\left ( \\int _ { x _ 0 } ^ { x } \\frac { \\epsilon ( y ) } { y } d y \\right ) . \\end{align*}"}
+{"id": "7532.png", "formula": "\\begin{align*} \\sum _ { a = d _ 1 + 1 } ^ { \\infty } q ^ { - a } Z _ { \\tilde { g } _ { 1 , a } } ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) = & q ^ { - e _ 1 s } \\sum _ { a = d _ 1 + 1 } ^ { \\infty } q ^ { - a } c _ 1 ( \\chi , a ) \\\\ : = & c _ 1 ( \\chi ) q ^ { - e _ 1 s } , \\end{align*}"}
+{"id": "5079.png", "formula": "\\begin{align*} T _ n \\coloneqq \\sum _ { i = 1 } ^ k \\beta _ i ( \\alpha _ i ^ n - c _ i ^ n ) = \\sum _ { \\nu = 1 } ^ p \\bigg [ \\sum _ { j = k _ \\nu } ^ { k _ { \\nu + 1 } - 1 } \\beta _ j \\alpha _ j ^ n - \\Big ( \\sum _ { j = k _ \\nu } ^ { k _ { \\nu + 1 } - 1 } \\beta _ j \\Big ) c _ { k _ \\nu } ^ n \\bigg ] . \\end{align*}"}
+{"id": "713.png", "formula": "\\begin{align*} d \\mathcal { H } = \\sum _ { k = 1 } ^ n \\frac { \\Gamma _ k ^ 2 } { 2 } \\big ( \\frac { \\partial R _ { \\rm r o b i n } ( w _ k ) } { \\partial w _ k } d w _ k + \\frac { \\partial R _ { \\rm r o b i n } ( w _ k ) } { \\partial \\bar { w } _ k } d \\bar { w } _ k \\big ) + \\end{align*}"}
+{"id": "2041.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { h = 1 } ^ { + \\infty } C ( h ) \\ , C ' ( - h ) = + \\infty . \\end{align*}"}
+{"id": "4100.png", "formula": "\\begin{align*} \\mathcal { F } ( g , H , f ) = \\int _ M ( R - \\frac { 1 } { 1 2 } | H | ^ 2 + | \\nabla f | ^ 2 ) e ^ { - f } d V _ g \\end{align*}"}
+{"id": "8475.png", "formula": "\\begin{align*} \\Gamma _ { A , \\mu } ^ + = \\Gamma _ { A , \\mu } ^ * , \\end{align*}"}
+{"id": "5343.png", "formula": "\\begin{align*} \\int _ \\Omega p \\left ( \\tau \\ , \\operatorname { d i v } ( \\varepsilon \\nabla \\psi ) + \\sigma \\psi \\right ) \\ , d x = 0 . \\end{align*}"}
+{"id": "1293.png", "formula": "\\begin{align*} T _ { m _ { \\vec { y } , k , j , \\delta } } \\mu = \\lim _ { \\varepsilon \\to 0 } T _ { m _ { \\vec { y } , k , j , \\delta } } \\mu _ \\varepsilon = 0 . \\end{align*}"}
+{"id": "7106.png", "formula": "\\begin{align*} \\phi ( m ) = \\phi _ 0 ( m ) + \\phi _ 1 ( m ) u + \\phi _ 2 ( m ) u ^ 2 + \\cdots . \\end{align*}"}
+{"id": "6668.png", "formula": "\\begin{align*} \\begin{aligned} \\| x _ i ^ { k + 1 } - { x ' _ i } ^ { k + 1 } \\| _ 1 & \\leq ( 1 - \\gamma ^ k _ 1 | R _ { i i } | ) \\| x _ i ^ k - { x ' _ i } ^ k \\| _ 1 \\\\ & + \\lambda ^ k \\| y _ i ^ k - { y ' _ i } ^ k \\| _ 1 . \\end{aligned} \\end{align*}"}
+{"id": "5987.png", "formula": "\\begin{align*} ( M \\times _ { S } S ' ) _ { ( m , s ) } & = M \\times _ { S } \\{ s \\} \\times _ { M ' } \\{ m \\} \\\\ & = ( M \\times _ { M ' } \\{ m \\} ) \\times _ { S } \\{ s \\} \\simeq M _ m , \\end{align*}"}
+{"id": "1054.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty j ^ { 2 \\beta } \\theta _ j ^ 2 = \\sum _ { j = 1 } ^ \\infty j ^ { 2 \\beta } \\left \\{ \\int _ { 0 } ^ 1 f ( x ) \\varphi _ j ( x ) \\ , \\mathrm { d } x \\right \\} ^ 2 \\leq r ^ 2 < \\infty , \\end{align*}"}
+{"id": "9085.png", "formula": "\\begin{align*} \\mathfrak { d } ^ j = \\{ X \\in \\mathfrak { n } : [ X , \\mathfrak { n } ] \\subseteq \\mathfrak { d } ^ { j - 1 } , [ J X , \\mathfrak { n } ] \\subseteq \\mathfrak { d } ^ { j - 1 } \\} \\end{align*}"}
+{"id": "3141.png", "formula": "\\begin{align*} \\int _ Y ( \\partial _ 1 w _ A ) ( \\partial _ { 2 2 } ^ 2 w _ B ) = \\int _ Y ( \\partial _ 2 w _ A ) ( \\partial _ { 1 1 } ^ 2 w _ B ) = 0 , \\end{align*}"}
+{"id": "7525.png", "formula": "\\begin{align*} & Z _ f ( s , \\chi , D ) \\\\ = & \\int _ { D } \\chi ( a c f ( x ) ) | f ( x ) | ^ s | d x | \\\\ = & \\int _ { D _ 1 } \\chi ( a c f ( \\pi ^ { t _ 1 } y _ 1 , \\cdots , \\pi ^ { t _ n } y _ n ) ) | f ( \\pi ^ { t _ 1 } y _ 1 , \\cdots , \\pi ^ { t _ n } y _ n ) | ^ s | J ( T _ { t _ 1 , \\cdots , t _ n } ) | | d y _ 1 \\cdots d y _ n | \\\\ = & q ^ { - ( t _ 1 + \\cdots + t _ n ) } \\int _ { D _ 1 } \\chi ( a c f ( \\pi ^ { t _ 1 } y _ 1 , \\cdots , \\pi ^ { t _ n } y _ n ) ) | f ( \\pi ^ { t _ 1 } y _ 1 , \\cdots , \\pi ^ { t _ n } y _ n ) | ^ s | d y | , \\end{align*}"}
+{"id": "7098.png", "formula": "\\begin{align*} F ( x , t ) = \\begin{cases} \\infty & x \\in \\mathbf { C P } ( W ) x \\in E ( \\nu ) | x | \\geq 1 / t \\\\ \\tan ( \\frac { \\pi } { 2 } t | x | ) x & x \\in E ( \\nu ) | x | < 1 / t . \\end{cases} \\end{align*}"}
+{"id": "2702.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u - \\partial _ r ^ 2 u - \\frac { 1 } { r } \\partial _ r u + \\frac { 2 u ( 1 - u ^ 2 ) } { r ^ 2 } = 0 , \\end{align*}"}
+{"id": "2265.png", "formula": "\\begin{align*} d X _ { M , i } ( t ) & = V _ { M , i } ( t ) d t , 1 \\leq i \\leq M , \\\\ d V _ { M , i } ( t ) & = \\Big ( - u ( X _ { M , i } ( t ) ) + V _ { M , i } ( t ) - \\nabla U * c ( X _ { M , i } ( t ) ) + \\frac { 1 } { M } \\sum _ { j \\neq i } ^ { M } \\nabla U ( X _ { M , i } ( t ) - X _ { M , j } ( t ) ) \\Big ) d t + d W _ t ^ i , \\end{align*}"}
+{"id": "5844.png", "formula": "\\begin{align*} y | _ { \\Sigma } \\in H ^ { 2 \\alpha - 1 } ( \\Sigma ) = L ^ 2 ( 0 , T ; H ^ { 2 \\alpha - 1 } ( \\Gamma ) ) \\cap H ^ { 2 \\alpha - 1 } ( 0 , T ; L ^ 2 ( \\Gamma ) ) , \\end{align*}"}
+{"id": "254.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = \\alpha ( v / u ) + \\beta ( w / u ) \\psi _ 2 ( u , v , w ) = \\alpha ( v / u ) + \\beta ( w / v ) . \\end{align*}"}
+{"id": "2079.png", "formula": "\\begin{align*} \\frac { \\partial \\Phi } { \\partial \\xi } & = ( 1 - p ) ( 1 + \\nu ) \\xi ^ { \\nu } - ( 1 + \\nu ) ( p + ( 1 - p ) \\xi ) ^ { \\nu } ( 1 - p ) \\\\ & = ( 1 - p ) ( 1 + \\nu ) ( \\xi ^ \\nu - ( p + ( 1 - p ) \\xi ) ^ \\nu ) < 0 , \\end{align*}"}
+{"id": "2854.png", "formula": "\\begin{align*} c _ { \\ell - 1 } = a _ { \\ell - 1 } = b _ { \\ell - 1 } \\geq b _ { \\ell } \\geq c _ { \\ell } = \\rho - \\sigma > \\rho - \\tau = a _ { \\ell } \\geq a _ { \\ell + 1 } = c _ { \\ell + 1 } . \\end{align*}"}
+{"id": "2146.png", "formula": "\\begin{align*} g ( t _ 0 ) = \\max _ { t \\geq 0 } g ( t ) = \\frac { 1 } { \\mu } ( 1 - U ( c t _ 0 + s _ 0 ) ) \\le \\frac { 1 } { \\mu } ( 1 - U ( s _ 0 ) ) . \\end{align*}"}
+{"id": "3796.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } h ( t ) = \\Delta _ { L , g ( t ) } h ( t ) . \\end{align*}"}
+{"id": "1575.png", "formula": "\\begin{align*} f ( \\bar x ) \\geq f ( x ^ k ) + ( c ^ k ) ^ \\top ( \\bar x - x ^ k ) = c _ 0 + ( c ^ k ) ^ \\top \\bar x \\geq c _ 0 + \\min _ { x \\in P } \\ ; ( c ^ k ) ^ \\top x = c _ 0 + ( c ^ k ) ^ \\top \\hat x ^ k \\end{align*}"}
+{"id": "4246.png", "formula": "\\begin{align*} \\Sigma _ \\gamma : = \\left \\{ f \\in H ^ 1 _ A ( \\R ^ N ) \\ : \\ \\int _ { \\R ^ N } V _ \\gamma ( x ) | f ( x ) | ^ 2 d x < \\infty \\right \\} \\end{align*}"}
+{"id": "5082.png", "formula": "\\begin{align*} w _ i = \\begin{cases} \\beta _ i & \\\\ - \\sum _ { j = k _ \\nu } ^ { k _ { \\nu + 1 } - 1 } \\beta _ j & \\end{cases} \\end{align*}"}
+{"id": "93.png", "formula": "\\begin{align*} \\beta _ 0 ^ \\delta - 1 & = ( 1 + \\beta _ 0 + \\dots + \\beta _ 0 ^ { \\delta - 1 } ) ( \\beta _ 0 - 1 ) , \\\\ \\beta _ i ^ \\delta - 1 & = ( 1 + \\beta _ i + \\dots + \\beta _ i ^ { \\delta - 1 } ) ( \\beta _ i - 1 ) . \\\\ \\end{align*}"}
+{"id": "4569.png", "formula": "\\begin{align*} a _ { n + 2 , 2 } = 2 ^ { 2 ^ n } + 1 \\qquad \\end{align*}"}
+{"id": "4651.png", "formula": "\\begin{align*} A = A _ 0 + \\sum _ { j = 1 } ^ N A _ j , \\end{align*}"}
+{"id": "6975.png", "formula": "\\begin{gather*} \\big \\{ W ^ { ( \\alpha , \\beta ) } _ { j , m , n } \\big \\} _ { j = 1 } ^ n = \\big \\{ z _ { j , m } ^ { ( - \\alpha - 1 , \\beta - 1 ) } \\big \\} _ { j = 1 } ^ m \\cup \\big \\{ z _ { j - m , n - m } ^ { ( \\alpha + 1 , \\beta - 1 ) } \\big \\} _ { j = m + 1 } ^ { n } . \\end{gather*}"}
+{"id": "7225.png", "formula": "\\begin{align*} U = \\begin{pmatrix} 0 & 1 & 0 & \\cdots & 0 \\\\ 0 & 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & 1 \\\\ \\omega & 0 & 0 & \\cdots & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "4276.png", "formula": "\\begin{align*} \\omega M ( \\phi ) & = - \\frac { 1 } { 2 } \\| ( \\nabla - i A ) \\phi \\| ^ 2 _ { L ^ 2 } - \\int _ { \\R ^ N } V _ \\gamma ( x ) | \\phi ( x ) | ^ 2 d x + \\| \\phi \\| ^ { p + 1 } _ { L ^ { p + 1 } } \\\\ & = - E _ \\gamma ( \\phi ) + \\frac { p - 1 } { p + 1 } \\| \\phi \\| ^ { p + 1 } _ { L ^ { p + 1 } } > - E _ \\gamma ( \\phi ) . \\end{align*}"}
+{"id": "407.png", "formula": "\\begin{align*} \\| \\tau _ n \\| = 1 = \\| f _ n \\| , \\forall n \\in \\mathbb { N } , \\end{align*}"}
+{"id": "8017.png", "formula": "\\begin{align*} P = \\big \\{ & \\big \\{ ( F _ 3 , \\{ x _ 1 , \\dot x _ 1 \\} , F _ 1 ) , ( F _ 3 , \\{ x _ 2 \\} , F _ 2 ) \\big \\} , \\\\ & \\big \\{ ( F _ 3 , \\{ x _ 2 \\} , F _ 2 ) , ( F _ 2 , \\{ x _ 1 , \\dot x _ 1 \\} , F _ 1 ) \\big \\} \\big \\} . \\end{align*}"}
+{"id": "7910.png", "formula": "\\begin{align*} Z _ \\infty : = \\lim _ { t \\to \\infty } \\sum _ { u \\in \\mathcal { N } ^ 1 _ t } ( \\sqrt { 2 } t - X _ u ( t ) ) e ^ { \\sqrt { 2 } X _ u ( t ) - 2 t } . \\end{align*}"}
+{"id": "6971.png", "formula": "\\begin{gather*} ( 1 + x ) P _ { m , n } ^ { ( \\alpha , \\beta ) } ( x ) ' + \\beta P _ { m , n } ^ { ( \\alpha , \\beta ) } ( x ) \\\\ { } = ( - 1 ) ^ { m } \\frac { ( \\beta + n ) ( \\alpha + n - 2 m + 1 ) } { ( \\alpha + n - m + 1 ) } P _ { n - m } ^ { ( \\alpha + 1 , \\beta - 1 ) } ( x ) P ^ { ( - \\alpha - 1 , \\beta - 1 ) } _ m ( x ) . \\end{gather*}"}
+{"id": "6143.png", "formula": "\\begin{align*} \\mu = \\sup _ { \\alpha , \\beta \\in \\mathbb C } \\hbox { d i m K e r } \\begin{pmatrix} A ^ T - \\alpha I \\\\ B ^ T - \\beta I \\end{pmatrix} . \\end{align*}"}
+{"id": "9133.png", "formula": "\\begin{align*} N ( m | n ) \\ ; = \\ ; m ( n + m ) ^ { n - 1 } . \\end{align*}"}
+{"id": "1049.png", "formula": "\\begin{align*} J = \\max \\widehat { \\mathcal { J } } - 1 \\mbox { a n d } L = \\begin{cases} 0 , & J \\equiv 0 \\ , ( 3 ) , \\\\ 1 , & J \\equiv 1 \\ , ( 3 ) , \\\\ 2 , & J \\equiv 2 \\ , ( 3 ) . \\end{cases} \\end{align*}"}
+{"id": "8371.png", "formula": "\\begin{align*} w = 0 , \\mathrm { f o r } \\ | x | ^ 2 - ( t - R ) ^ 2 > R ^ 2 , \\ t > R . \\end{align*}"}
+{"id": "4630.png", "formula": "\\begin{align*} P ^ { - * } g = \\bigcup _ { n = 0 } ^ { \\infty } P ^ { - n } g \\end{align*}"}
+{"id": "1404.png", "formula": "\\begin{align*} \\pi _ 2 ^ { ( - , + , - ) } = L ( \\Delta [ 0 , - 3 ] , \\Delta [ 0 , - 1 ] ; \\pi ( 0 ^ - , 1 ^ + , 2 ^ - ) ) . \\end{align*}"}
+{"id": "4461.png", "formula": "\\begin{align*} u _ { \\ell - 1 } > \\frac { u _ { \\ell - 1 } } { t } = u ' _ { \\ell - 1 } \\geq u ' _ { \\ell } = t u _ { \\ell } > u _ { \\ell } \\end{align*}"}
+{"id": "2138.png", "formula": "\\begin{align*} r ^ + : = \\sup _ { u \\in ( \\theta , 1 ] } \\frac { f ( u ) } { u - \\theta } > 0 . \\end{align*}"}
+{"id": "494.png", "formula": "\\begin{align*} \\limsup _ { j \\to \\infty } \\Phi ( x ^ { k _ j } ) \\le \\Phi ( \\widehat { x } ) \\quad { \\rm f o r \\ e a c h } \\ \\{ x ^ { k _ j } \\} _ { j \\in \\mathbb { N } } \\ { \\rm w i t h } \\ \\lim _ { j \\to \\infty } x ^ { k _ j } = \\widehat { x } . \\end{align*}"}
+{"id": "4751.png", "formula": "\\begin{align*} v _ { j + 1 } - v _ j = \\sum _ { i \\leq h } y ^ { f _ i } \\rho _ i z _ i \\in y ^ { j - e _ g } B . \\end{align*}"}
+{"id": "6938.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } x } H _ n ( x ) = 2 n H _ { n - 1 } ( x ) , n \\geq 1 . \\end{align*}"}
+{"id": "6687.png", "formula": "\\begin{align*} \\begin{aligned} y ^ { k + 1 } ( \\ell ) - v [ \\bar y ^ { k + 1 } ] _ \\ell = & \\bar C ^ k ( y ^ k ( \\ell ) - v [ \\bar y ^ k ] _ \\ell ) + \\gamma _ 2 ^ k \\Pi _ v \\xi _ w ^ { k } ( \\ell ) \\\\ & + \\Pi _ v \\left ( g ^ { k + 1 } ( \\ell ) - ( 1 - \\alpha ^ k ) g ^ { k } ( \\ell ) \\right ) \\end{aligned} \\end{align*}"}
+{"id": "5757.png", "formula": "\\begin{align*} i ^ * ( [ \\Lambda ] ) \\vert _ { t \\mapsto 1 } = c _ 0 ^ E ( \\Lambda ) + c _ 1 ^ E ( \\Lambda ) \\ldots + c _ r ^ E ( \\Lambda ) . \\end{align*}"}
+{"id": "813.png", "formula": "\\begin{align*} F _ { n , t _ 1 , \\ldots , t _ d } ( x ) = \\nu \\left ( \\xi \\in \\partial \\Gamma : S _ n \\xi ( t _ 1 , \\ldots , t _ d ) \\in \\prod _ { i = 1 } ^ { d } ( - \\infty , x _ i ] \\right ) , \\end{align*}"}
+{"id": "4526.png", "formula": "\\begin{align*} \\frac { 2 } { 3 } = \\frac { 1 } { 2 } + \\frac { 1 } { 6 } = \\frac { 1 } { 3 } + \\frac { 1 } { 3 } \\end{align*}"}
+{"id": "5110.png", "formula": "\\begin{align*} \\Omega _ { \\mathbf { m } } ( \\lambda ) = I _ { 1 } ( \\lambda ) K _ { 1 } ( \\lambda ) - I _ { \\mathbf { m } } ( \\lambda ) K _ { \\mathbf { m } } ( \\lambda ) \\end{align*}"}
+{"id": "1015.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z } \\frac { ( 1 + | x | ) ^ \\alpha } { ( 1 + | n | ) ^ { \\alpha + 2 } ( 1 + | x - n | ) ^ \\alpha } = \\sum _ { n \\in \\Z } \\frac { F ( n , x - n ) } { ( 1 + | n | ) ^ { 2 } } \\leq \\sum _ { n \\in \\Z } \\frac { 1 } { ( 1 + | n | ) ^ { 2 } } = \\frac { \\pi ^ 2 } { 3 } < \\infty . \\end{align*}"}
+{"id": "7105.png", "formula": "\\begin{align*} \\phi ( d _ { i , j } - c _ { i , j } ) & = c _ { i , j + 1 } u + c _ { i , j + 2 } u ^ 2 + \\cdots , \\\\ \\phi ( q _ j - p _ j ) & = p _ { j + 1 } u + p _ { j + 2 } u ^ 2 + \\cdots . \\end{align*}"}
+{"id": "7803.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} _ { 1 , 0 } \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} _ { 1 , 0 } = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} _ { 1 , 0 } \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} _ { 1 , 0 } \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} _ { 1 , 0 } . \\end{align*}"}
+{"id": "3320.png", "formula": "\\begin{align*} \\prod _ { i \\in \\mathbb { Z } _ { n } } \\left ( \\lambda - x _ { i } \\right ) & = \\sum _ { 0 \\le k \\le n } e _ { k } \\left ( \\mathbf { x } \\right ) \\ , \\lambda ^ { n - k } \\\\ & = \\lambda ^ { n } - 1 \\end{align*}"}
+{"id": "6964.png", "formula": "\\begin{gather*} L _ { m , n } ^ { I I , ( \\alpha ) } ( x ) = \\sum _ { j = 0 } ^ n c _ { j , n } x ^ j . \\end{gather*}"}
+{"id": "7402.png", "formula": "\\begin{align*} X _ { \\alpha } ( \\psi _ s ) = ( \\chi ^ 2 \\chi '^ { - 1 } ) ( \\varpi _ L ) q _ L ^ { - s } \\end{align*}"}
+{"id": "18.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d p _ t & = ( \\bar { H } _ y + \\beta p _ t ) d t + \\bar { H } _ z d W _ t + \\bar { H } _ { \\tilde { z } } d \\xi _ t + \\int _ { \\mathcal { E } } \\bar { H } _ \\gamma \\tilde { N } ( d e , d t ) , \\\\ - d Q _ t & = ( \\bar { H } _ { \\mathcal { Z } } - \\beta Q _ t ) d t - \\tilde { M } _ t d \\xi _ t , \\\\ - d q _ t & = ( \\bar { H } _ x - \\beta q _ t ) d t - m _ t d W _ t - \\tilde { m } _ t d \\xi _ t - \\int _ { \\mathcal { E } } n _ { ( t , e ) } \\tilde { N } ( d e , d t ) , \\\\ p _ 0 & = \\phi _ y ( \\bar { y } _ 0 ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1278.png", "formula": "\\begin{align*} \\zeta _ { t } ^ \\ast ( z ) = \\left \\{ \\begin{array} { l l l } b _ { t } ~ \\zeta _ { k - 2 - t } ( z ) & { \\rm i f } & \\gamma = 1 \\\\ b _ { t } ~ \\zeta _ { k - 1 - t } ( z ) & { \\rm i f } & \\gamma = - 1 \\end{array} \\right . \\end{align*}"}
+{"id": "8981.png", "formula": "\\begin{align*} u _ t = - ( \\varepsilon + d \\pi _ N ( v ) ) u _ r \\hbox { o n } \\partial B \\times [ 0 , T _ 2 [ , \\end{align*}"}
+{"id": "1509.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\C _ r \\left ( \\frac 1 4 \\right ) & = \\left ( \\frac 1 4 \\right ) ^ { r - 1 } \\left ( - \\frac 1 2 \\log 2 + ( r - 1 ) \\sum _ { m = 1 } ^ \\infty \\frac { ( 1 - 2 ^ { - 2 m } ) \\zeta ( 2 m ) } { m ( 2 m + r - 1 ) 2 ^ { 2 m } } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "3199.png", "formula": "\\begin{align*} \\Sigma _ K ^ { ( 1 ) } & = \\{ \\xi \\in \\Sigma _ K ( a , b ) , a \\leq 0 , b \\geq 1 \\} , \\\\ \\Sigma _ K ^ { ( 2 ) } & = \\{ \\xi \\in \\Sigma _ K ( a , b ) , a \\geq 1 , b \\leq 0 \\} . \\end{align*}"}
+{"id": "8150.png", "formula": "\\begin{align*} \\phi ( y ) = 3 x ^ { \\frac { 1 } { 3 } } y ^ { \\frac { 1 } { 3 } } , \\end{align*}"}
+{"id": "1484.png", "formula": "\\begin{align*} \\mathcal S _ r ( x ) = e ^ { \\frac { x ^ { r - 1 } } { r - 1 } } \\prod _ { n = 1 } ^ \\infty \\left \\{ P _ r \\left ( \\frac x n \\right ) P _ r \\left ( - \\frac x n \\right ) ^ { ( - 1 ) ^ { r - 1 } } \\right \\} ^ { n ^ { r - 1 } } , \\end{align*}"}
+{"id": "250.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = - ( 1 - \\alpha ) \\rho ^ 2 - \\beta \\tau \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "8597.png", "formula": "\\begin{align*} c _ { \\mathcal { H } } ( x ) : = d ( x ) \\ , \\widetilde { c } ( x ) \\mbox { a n d } F ( x ) = d ( x ) \\ , f ( x ) . \\end{align*}"}
+{"id": "5545.png", "formula": "\\begin{align*} V ( x , k ) : = \\pi ^ { \\frac { 1 } { 2 } - k } \\sum _ { n = 1 } ^ \\infty \\mu ( n ) n ^ { k - 1 } I ( X _ n , k ) , \\end{align*}"}
+{"id": "6491.png", "formula": "\\begin{align*} \\rho ^ 2 \\asymp \\begin{cases} n ^ { - \\frac { 2 s } { 2 s + 1 / 2 } } , & b \\geq n ^ { \\frac { 1 } { 2 s + 1 / 2 } } , \\\\ \\left ( { \\sqrt { b } n } \\right ) ^ { - \\frac { 2 s } { 2 s + 1 } } , & n ^ { \\frac { 1 } { 2 s + 1 / 2 } } / m ^ { \\frac { 2 s + 1 } { 2 s + 1 / 2 } } \\leq b < n ^ { \\frac { 1 } { 2 s + 1 / 2 } } , \\\\ ( n / \\sqrt { m } ) ^ { - \\frac { 2 s } { 2 s + 1 / 2 } } , & b < n ^ { \\frac { 1 } { 2 s + 1 / 2 } } / m ^ { \\frac { 2 s + 1 } { 2 s + 1 / 2 } } . \\end{cases} \\end{align*}"}
+{"id": "6228.png", "formula": "\\begin{align*} C _ p A e _ r = 0 \\quad \\hbox { a n d } C _ p B e _ r = 0 \\hbox { f o r } 1 \\leqslant r \\leqslant p . \\end{align*}"}
+{"id": "3557.png", "formula": "\\begin{align*} z _ j ^ + \\Omega _ \\lambda = \\begin{cases} d _ { j } ^ + ( \\lambda ) \\prod _ { \\ell = 1 } ^ { j - 1 } ( \\lambda _ \\ell - \\lambda _ j - \\ell + j ) \\Omega _ { \\lambda + \\epsilon _ j } , & \\lambda + \\epsilon _ j \\in \\mathbb { P } \\ell ( \\lambda + \\epsilon _ j ) \\leq p , \\\\ 0 , , \\end{cases} \\end{align*}"}
+{"id": "737.png", "formula": "\\begin{align*} \\frac { d } { d t } \\log \\frac { d z } { d t } + r ( z ) \\frac { d z } { d t } = 0 . \\end{align*}"}
+{"id": "651.png", "formula": "\\begin{align*} \\Omega : = \\{ ( c _ { 1 } , \\ldots , c _ { N } ) \\in C ^ { N } : | ( A _ { c _ { 1 } } \\cap \\ldots \\cap A _ { c _ { N } } ) \\times ( B _ { c _ { 1 } } \\cap \\ldots \\cap B _ { c _ { N } } ) | \\geq \\tfrac { 1 } { 2 } \\delta ^ { N \\zeta } | A | | B | \\} \\end{align*}"}
+{"id": "1255.png", "formula": "\\begin{align*} ( ( u - 1 ) ^ 2 + v ^ 2 ) ^ 2 & = \\dfrac { 1 6 ( { \\rho } _ 0 ) ^ 4 ( 1 - \\beta ) ^ 4 } { ( 1 + ( { \\rho } _ 0 ) ^ 2 - 2 { \\rho } _ 0 \\cos t ) ^ 2 } , \\intertext { a n d } \\left ( \\dfrac { u - 1 } { 1 - \\alpha } \\right ) ^ 2 + \\left ( \\dfrac { v } { 1 + \\alpha } \\right ) ^ 2 & = \\dfrac { 4 ( { \\rho } _ 0 ) ^ 2 ( 1 - \\beta ) ^ 2 \\left [ ( { \\rho } _ 0 - \\cos t ) ^ 2 ( 1 + \\alpha ) ^ 2 + ( \\sin t ) ^ 2 ( 1 + \\alpha ) ^ 2 \\right ] } { ( 1 + ( { \\rho } _ 0 ) ^ 2 - 2 { \\rho } _ 0 \\cos t ) ^ 2 ( 1 - \\alpha ) ^ 2 ( 1 + \\alpha ) ^ 2 } . \\end{align*}"}
+{"id": "6714.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\mu ( n + a ) e ^ { i 2 \\pi u n / q } = \\frac { 1 } { p } \\sum _ { n \\leq x , } \\sum _ { 0 \\leq t \\leq p - 1 , } \\sum _ { 1 \\leq s \\leq p - 1 } \\mu ( s + a ) \\omega ^ { t ( n - s ) } e ^ { i 2 \\pi u s / q } . \\end{align*}"}
+{"id": "9067.png", "formula": "\\begin{align*} \\big ( \\theta ^ { S } _ { \\xi , \\lambda } \\eta \\big ) ( \\phi ) = \\eta \\Big ( x \\mapsto \\int _ { M _ { S } } \\int _ { A _ { S } } \\int _ { N _ { S } } a ^ { - \\lambda + \\rho _ { S } } \\xi ( m ^ { - 1 } ) \\phi ( m a n x ) \\ , d n \\ , d a \\ , d m \\Big ) , \\end{align*}"}
+{"id": "1745.png", "formula": "\\begin{align*} \\delta \\mu _ { \\underline { m } } = \\frac { d } { d r } ( \\imath ( e ^ r ) \\mu _ { \\underline { m } } ) \\mid _ { r = 0 } = \\frac { d } { d r } ( e ^ { - m r } \\mu _ { \\underline { m } } ) \\mid _ { x = 0 } = - m \\cdot \\mu _ { \\underline { m } } , \\end{align*}"}
+{"id": "269.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\rho } - \\beta \\rho \\tau \\leqslant - \\frac { \\beta ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\beta } { 2 } , \\end{align*}"}
+{"id": "3110.png", "formula": "\\begin{align*} Z ( y ) = \\zeta ( y _ 1 + y _ 2 ) \\ , \\mathrm { d i a g } ( 1 , - 1 , 0 , \\dots , 0 ) \\quad y = ( y _ 1 , \\dots , y _ n ) \\in \\R ^ n \\end{align*}"}
+{"id": "2947.png", "formula": "\\begin{align*} \\| F \\| ^ 2 _ { 1 , n } = \\frac { n ^ 2 } { 2 } \\sum _ { j \\in \\mathbb { Z } , \\ , | j - j ^ \\prime | = 1 } E _ { \\nu ^ n _ \\rho } [ g _ n ( \\eta _ j ) ( \\nabla _ { j , j ^ \\prime } F ( \\eta ) ) ^ 2 ] . \\end{align*}"}
+{"id": "2456.png", "formula": "\\begin{align*} \\# \\{ \\rho \\colon | \\rho - ( 1 + i t ) | \\leq \\eta , ~ L ( \\rho , \\pi \\otimes \\chi ) = 0 \\} \\leq 5 m \\eta \\log ( q C ( \\pi ) ) + O ( m ^ 2 \\eta + 1 ) \\end{align*}"}
+{"id": "1048.png", "formula": "\\begin{align*} \\widehat { \\mathcal { J } } = \\biggl \\{ j \\in \\mathcal { J } : \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Z _ { i j } \\geq \\tau \\biggr \\} . \\end{align*}"}
+{"id": "3565.png", "formula": "\\begin{align*} \\bar { \\gamma } ^ \\pm _ { k l } : = \\begin{cases} \\gamma ^ \\pm _ { k l } , & k > l , \\\\ \\lambda _ l \\pm \\delta _ { i l } - \\sum _ { m = 1 } ^ { l - 1 } \\gamma ^ \\pm _ { m l } , & k = l , \\\\ 0 & k < l , \\end{cases} \\end{align*}"}
+{"id": "1626.png", "formula": "\\begin{align*} A ( \\tau ) : = \\sup _ { 0 \\leq s \\leq \\tau } s ^ { \\frac { d } { 4 } } \\| u ( s ) \\| _ { L _ { x } ^ { \\infty } ( \\R ^ d ) } . \\end{align*}"}
+{"id": "3561.png", "formula": "\\begin{align*} E ^ \\gamma E ^ { e _ I } = \\sum _ { v _ 2 = i _ 2 } ^ n \\cdots \\sum _ { v _ t = i _ s } ^ n \\prod _ { u = 2 } ^ s \\gamma _ { v _ u i _ u } ^ { ( 1 - \\delta _ { v _ u i _ u } ) } E ^ { \\gamma + \\sum _ { u = 2 } ^ s e _ { v _ u i _ { u - 1 } } - e _ { v _ u i _ u } } . \\end{align*}"}
+{"id": "580.png", "formula": "\\begin{align*} J ^ \\dagger _ { t _ n , t _ { n + 1 } } : = \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ \\dagger ) ^ { \\rm T } { \\rm d } X _ t ^ \\dagger , \\end{align*}"}
+{"id": "2647.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ m \\sqrt { k _ i ! } H _ { k _ i } ( x _ i ) = \\frac { \\partial ^ { k _ 1 + \\ldots + k _ m } } { \\partial t _ 1 ^ { k _ 1 } \\ldots \\partial t _ m ^ { k _ m } } \\exp \\left ( \\sum _ { i = 1 } ^ m t _ i x _ i - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ m t _ i ^ 2 \\right ) \\mid _ { t _ 1 = \\ldots = t _ m = 0 } . \\end{align*}"}
+{"id": "1244.png", "formula": "\\begin{align*} \\mathcal H ^ { m - 1 } ( X _ = ( y , k ) ) \\geq C \\mu ( X _ \\leq ( y , k ) ) \\end{align*}"}
+{"id": "5983.png", "formula": "\\begin{align*} a _ { k , j } = a _ { k - 1 , j } \\ : \\mathrm { f o r } \\ : 1 \\leq j \\leq r _ k . \\end{align*}"}
+{"id": "4522.png", "formula": "\\begin{align*} u _ { n + 1 } = v _ { n + 1 } \\left ( \\frac { \\prod _ { i = 1 } ^ { n } v _ i } { \\prod _ { i = 1 } ^ n u _ i } \\right ) . \\end{align*}"}
+{"id": "676.png", "formula": "\\begin{align*} - d * d G ^ \\omega = \\omega \\end{align*}"}
+{"id": "1396.png", "formula": "\\begin{align*} \\psi = S _ 3 \\boxtimes S _ 2 + S _ 4 \\boxtimes S _ 3 . \\end{align*}"}
+{"id": "3229.png", "formula": "\\begin{align*} f ' ( x ) = f ( x ) - 1 \\ge k _ 2 + i + 1 - 1 = k ' _ 2 + i . \\end{align*}"}
+{"id": "3026.png", "formula": "\\begin{align*} \\gamma { ^ a } _ { b c } = \\frac { 1 } { 2 } ( \\Theta { ^ a } _ { b c } - \\eta ^ { a d } \\eta _ { b e } \\Theta { ^ e } _ { d c } - \\eta ^ { a d } \\eta _ { c e } \\Theta { ^ e } _ { d b } ) \\end{align*}"}
+{"id": "7465.png", "formula": "\\begin{align*} a _ 1 , \\dots , a _ { 1 8 } = 1 , 2 , 3 , 3 , 4 , 5 , 5 , 7 , 8 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 4 , 1 4 , 1 5 , 1 6 , 1 8 . \\end{align*}"}
+{"id": "6606.png", "formula": "\\begin{align*} H ( \\theta ) = \\sum _ { n = 1 } ^ { \\infty } E _ n ( \\theta ) P _ n ( \\theta ) , \\end{align*}"}
+{"id": "1381.png", "formula": "\\begin{align*} t ( u ) = t r _ 0 \\{ K _ 0 ^ { + } ( u ) T _ 0 ( u ) K ^ { - } _ 0 ( u ) \\hat { T } _ 0 ( u ) \\} , \\end{align*}"}
+{"id": "3878.png", "formula": "\\begin{align*} g _ { , k } D _ j Y ^ k + g _ z D _ j Z = 0 . \\end{align*}"}
+{"id": "286.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ 0 : W _ F & \\rightarrow N _ { \\widehat { G } } ( \\widehat { S } _ 0 ) \\\\ w & \\mapsto N ( w ) , \\end{aligned} \\end{align*}"}
+{"id": "7043.png", "formula": "\\begin{align*} \\Omega _ { C _ 2 } [ 1 / \\tau ] & = \\left ( \\Omega _ { C _ 2 } \\to \\Sigma ^ { \\sigma - | \\sigma | } \\Omega _ { C _ 2 } \\to \\Sigma ^ { 2 \\sigma - | 2 \\sigma | } \\Omega _ { C _ 2 } \\to \\dots \\right ) \\\\ & \\simeq \\left ( M U _ { C _ 2 } ^ { \\mathbf { C } ^ \\infty } \\to M U _ { C _ 2 } ^ { \\mathbf { C } ^ { \\infty , 1 } } \\to M _ { C _ 2 } U ^ { \\mathbf { C } ^ { \\infty , 2 } } \\to \\dots \\right ) \\\\ & = M U _ { C _ 2 } \\end{align*}"}
+{"id": "5662.png", "formula": "\\begin{align*} \\lambda = \\epsilon \\lambda _ 1 + \\epsilon ^ 2 \\lambda _ 2 + \\ldots \\end{align*}"}
+{"id": "3508.png", "formula": "\\begin{align*} A _ \\sigma ( k , l ) : = A ( \\sigma _ l ( k ) , l ) , \\big ( ( k , l ) \\in \\lambda ) \\big ) . \\end{align*}"}
+{"id": "3536.png", "formula": "\\begin{align*} \\mathbf { p } B _ j ^ - & = \\sum _ { i = j } ^ { n } \\sum _ { s = 1 } ^ { i - j + 1 } \\sum _ { I \\in \\mathcal { I } _ { j i } ( s ) } E ^ { e _ I } B _ i ^ - \\frac { 1 } { \\prod _ { \\ell \\in I , \\ell \\neq j } ( h _ j - h _ \\ell ) } , \\end{align*}"}
+{"id": "1127.png", "formula": "\\begin{align*} c _ { ( a + 1 , 1 ^ { b - 1 } ) ( c ^ k ) } ^ { ( a + c , c ^ { k - 1 } , 1 ^ b ) } \\ge c _ { ( a + 1 , 1 ^ { b - 1 } ) ( c ) } ^ { ( a + c , 1 ^ b ) } \\ge c _ { ( a + 1 ) ( c ) } ^ { ( a + c , 1 ) } \\ge c _ { ( 1 ) ( c ) } ^ { ( c , 1 ) } = 1 \\end{align*}"}
+{"id": "7222.png", "formula": "\\begin{align*} v _ \\alpha ( t ) = \\frac { q ( \\alpha , \\ , t ) - p ( \\alpha , \\ , t ) } { \\big | q ( \\alpha , \\ , t ) - p ( \\alpha , \\ , t ) \\big | } \\end{align*}"}
+{"id": "4798.png", "formula": "\\begin{align*} \\max _ { j < \\beta N } \\left \\{ \\Pr _ { v \\sim \\mu _ t } [ | v H | = j ] \\cdot 2 ^ { 2 \\epsilon N \\log ( 1 - \\frac { 2 j } { N } ) } \\right \\} \\leq \\max _ { \\alpha > 1 } \\left \\{ 2 ^ { - \\frac { 2 \\alpha \\tilde { \\epsilon } ( 1 - \\tilde { \\epsilon } ) } { \\ln 2 } ( 1 - \\eta ) N } \\cdot 2 ^ { \\epsilon N \\log ( 4 \\alpha \\tilde { \\epsilon } ( 1 - \\tilde { \\epsilon } ) ) } \\right \\} . \\end{align*}"}
+{"id": "7830.png", "formula": "\\begin{align*} B _ k ( x ) : = \\frac { A _ k ( x ) } { 1 + x ^ 2 + x ^ 4 } = \\frac { ( 1 - x ^ { 6 k - 4 } ) ( 1 - x ^ { 6 k - 2 } ) ( 1 - x ^ { 6 k } ) } { ( 1 - x ^ 4 ) ( 1 - x ^ 6 ) ( 1 - x ^ { 2 k } ) } . \\end{align*}"}
+{"id": "7405.png", "formula": "\\begin{align*} w ( X _ { \\alpha } ) = X _ { w ( \\alpha ) } \\end{align*}"}
+{"id": "5174.png", "formula": "\\begin{align*} \\psi : C ^ * ( \\overline { L } _ { 2 n + 1 } ^ { r ; \\underline { m } } ) \\to \\left ( \\sum _ { \\substack { ( v _ i , k ) \\\\ i = 0 , . . . , n , \\ k \\in S _ i } } p _ { ( v _ i , k ) } \\right ) C ^ * ( L _ { 2 n + 1 } \\times _ c \\Z _ r ) \\left ( \\sum _ { \\substack { ( v _ i , k ) \\\\ i = 0 , . . . , n , \\ k \\in S _ i } } p _ { ( v _ i , k ) } \\right ) \\end{align*}"}
+{"id": "5552.png", "formula": "\\begin{align*} M ( \\ell ; n ) = O _ { \\epsilon } ( \\ell ^ { \\frac { 1 } { 2 } - k + \\epsilon } ) , \\end{align*}"}
+{"id": "8037.png", "formula": "\\begin{align*} L ( s , f \\times u _ j ) = \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { \\lambda _ j ( n ) A ( n , m ) } { ( m ^ 2 n ) ^ s } \\end{align*}"}
+{"id": "672.png", "formula": "\\begin{align*} ( \\omega _ 1 , \\omega _ 2 ) _ 2 = \\int _ M \\omega _ 1 \\wedge * \\omega _ 2 . \\end{align*}"}
+{"id": "7166.png", "formula": "\\begin{align*} \\displaystyle { u '^ { \\{ k \\} } _ m = u '^ { \\{ k \\} } _ l } \\end{align*}"}
+{"id": "3181.png", "formula": "\\begin{align*} v ^ { 1 1 } ( y ) : = v ^ { 1 1 } _ B ( y _ 1 , y _ 2 ) , v ^ { 2 2 } ( y ) : = v ^ { 2 2 } _ B ( y _ 1 , y _ 2 ) , v ^ { 3 3 } ( y ) : = - v ^ { 1 1 } _ B ( y _ 1 , y _ 2 ) - v ^ { 2 2 } _ B ( y _ 1 , y _ 2 ) \\end{align*}"}
+{"id": "3664.png", "formula": "\\begin{align*} M ( f - A u , u - g ) & = 0 , \\end{align*}"}
+{"id": "290.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta _ { S ( F ) , \\chi , \\chi ' } : S ( F ) & \\rightarrow \\mathbb { C } ^ { \\times } \\\\ t & \\mapsto \\prod _ { \\alpha \\in \\Phi _ { \\mathrm { s y m } } / \\Gamma _ F } \\zeta _ { \\alpha } ( \\iota _ { \\alpha } \\alpha ( t ) ) \\prod _ { \\alpha \\in \\Phi _ { \\mathrm { a s y } } / \\Gamma _ F \\times \\{ \\pm 1 \\} } \\zeta _ { \\alpha } ( \\alpha ( t ) ) , \\end{aligned} \\end{align*}"}
+{"id": "8781.png", "formula": "\\begin{align*} \\omega ^ m ( t , x ) = \\omega ^ m _ i ( t ) . \\end{align*}"}
+{"id": "8737.png", "formula": "\\begin{align*} y _ n & = \\sum \\limits _ { i = 0 } ^ { n - 1 } C _ 0 A _ 0 ^ { n - 1 - i } B _ 0 u _ i + \\sum \\limits _ { i = 1 0 } ^ { n - 1 } C _ 0 A _ 0 ^ { n - 1 - i } w _ i + v _ { n } \\\\ & = \\sum \\limits _ { l = t - 2 T + 1 } ^ { t - 1 } C _ 0 A _ 0 ^ { t - 1 - l } B u _ l + \\sum \\limits _ { l = 0 } ^ { t - 2 T } C _ 0 A _ 0 ^ { t - 1 - l } B _ 0 u _ l + \\sum \\limits _ { l = 0 } ^ { t - 1 } C _ 0 A _ 0 ^ { t - 1 - l } w _ l + v _ { t } \\\\ & : = g _ 0 X _ t + \\bar { g } _ 0 \\bar { X } _ t + h W _ t + v _ t , \\end{align*}"}
+{"id": "4677.png", "formula": "\\begin{align*} \\sum _ { m = \\max \\{ \\ell _ 1 , \\ell _ 2 \\} } ^ { \\infty } \\sum _ { k = \\max \\{ 0 , \\ell _ 1 + \\ell _ 2 - m \\} } ^ { \\min \\{ \\ell _ 1 , \\ell _ 2 \\} } A _ { m , k } = \\sum _ { k = 0 } ^ { \\min \\{ \\ell _ 1 , \\ell _ 2 \\} } \\sum _ { m = \\ell _ 1 + \\ell _ 2 - k } ^ { \\infty } A _ { m , k } . \\end{align*}"}
+{"id": "6828.png", "formula": "\\begin{align*} \\tau ( L _ 1 , L _ 2 , \\dots , L _ m ) = \\mu _ c ^ { + } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) - \\mu _ c ^ { - } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) , \\end{align*}"}
+{"id": "2928.png", "formula": "\\begin{align*} \\begin{aligned} & b _ 2 = g ^ { \\prime } ( 0 ) , b _ 1 = \\frac { 1 } { 2 } g ^ { \\prime \\prime } ( 0 ) ( 1 + 2 \\Phi _ n ( \\rho ) ) , \\\\ & b _ 0 = \\frac { g ^ { ( 3 ) } ( 0 ) } { 6 g ^ \\prime ( 0 ) } ( 1 + 6 \\Phi _ n ( \\rho ) + 3 \\Phi _ n ( \\rho ) ^ 2 ) - \\frac { g ^ { \\prime \\prime } ( 0 ) ^ 2 } { 4 g ^ \\prime ( 0 ) ^ 2 } ( 1 + 1 0 \\Phi _ n ( \\rho ) + 9 \\Phi _ n ( \\rho ) ^ 2 ) . \\end{aligned} \\end{align*}"}
+{"id": "8389.png", "formula": "\\begin{align*} & w ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { w } ) = w _ X ( \\mathcal { B } ) \\cdot i b ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { w } ) = w _ X ( \\mathcal { B } ) \\cdot c ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { w } ) \\\\ & = w _ X ( \\mathcal { B } ) \\cdot L ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { w } ) = w _ X ( \\mathcal { B } ) \\cdot d ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { w } ) = w _ X ( \\mathcal { B } ) \\cdot n w ( C ( X ) , \\tau _ { \\mathcal { B } } ^ { w } ) . \\end{align*}"}
+{"id": "3173.png", "formula": "\\begin{align*} [ - A : D ^ 2 v ^ { 1 1 } _ A ] ( y ) = [ - B : D ^ 2 v ^ { 1 1 } _ B ] ( y _ 1 , y _ 2 ) = b _ 1 ( y _ 1 , y _ 2 ) - \\bar { b } _ 1 = a _ 1 ( y ) - \\bar { a } _ 1 , \\end{align*}"}
+{"id": "7960.png", "formula": "\\begin{align*} f ( \\underset { i \\in I } { \\vee } ~ y _ i ) = \\underset { i \\in I } { \\vee } ~ f ( y _ i ) . \\end{align*}"}
+{"id": "641.png", "formula": "\\begin{align*} \\Big | B \\ , \\setminus \\ , \\bigcup _ { j = 1 } ^ { n } B _ { j } \\Big | < \\delta ^ { 2 \\epsilon } | B | \\Big | B \\ , \\setminus \\ , \\bigcup _ { j = 1 } ^ { n } B _ { j } \\Big | \\geq \\delta ^ { 2 \\epsilon } | B | . \\end{align*}"}
+{"id": "7542.png", "formula": "\\begin{align*} Z _ g \\big ( s , \\chi , S ( \\Delta _ { \\gamma _ 5 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) ) \\big ) = \\dfrac { G _ 5 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "1941.png", "formula": "\\begin{align*} & f _ 1 ( u , \\eta , x ) = ( 1 - \\kappa ) \\eta _ 1 + \\kappa \\big ( u _ 0 + \\frac { 1 } { \\alpha } \\eta _ 2 ^ { 1 / q } \\big ) , \\\\ & f _ 2 ( u , x ) = \\begin{pmatrix} - \\langle x , u \\rangle \\\\ [ \\max ( 0 , - \\langle x , u \\rangle - u _ 0 ) ] ^ q \\end{pmatrix} , \\end{align*}"}
+{"id": "4456.png", "formula": "\\begin{align*} u _ n ( \\theta ) = u _ { n _ 0 } ( \\theta ) + u _ { n - n _ 0 } \\left ( \\theta - u _ { n _ 0 } ( \\theta ) \\right ) \\end{align*}"}
+{"id": "6434.png", "formula": "\\begin{align*} d ( ( X , Y , t ) , ( \\hat X , \\hat Y , \\hat t ) ) : = | X - \\hat X | + | Y - \\hat Y - ( \\hat t - t ) \\hat X | ^ { 1 / 3 } + | \\hat t - t | ^ { 1 / 2 } . \\end{align*}"}
+{"id": "5439.png", "formula": "\\begin{align*} Q ( x ) V ( x , \\cdot ) ( i ) = \\sum _ { j \\in \\mathcal M } q _ { i j } ( x ) V ( x , j ) = \\sum _ { j \\in \\mathcal M , j \\neq i } q _ { i j } ( x ) ( V ( x , j ) - V ( x , i ) ) , ~ x \\in \\mathbb R ^ d , ~ i \\in \\mathcal M . \\end{align*}"}
+{"id": "8344.png", "formula": "\\begin{align*} Q _ l ^ { ( M _ 0 ) } ( t , x ) = Q ^ { ( M _ 0 ) } \\left ( \\frac { x _ 1 - l _ 1 t } { \\sqrt { 1 - l _ 1 ^ 2 } } , x _ 2 , x _ 3 \\right ) , \\end{align*}"}
+{"id": "4356.png", "formula": "\\begin{align*} s _ 1 | ^ { U _ 1 } _ W = s _ 2 | ^ { U _ 2 } _ W \\end{align*}"}
+{"id": "6241.png", "formula": "\\begin{align*} \\quad \\omega _ { k m n } ^ { \\vec \\sigma } : = \\sigma _ 1 \\omega _ k + \\sigma _ 2 \\omega _ m + \\sigma _ 3 \\omega _ n , \\quad \\quad \\vec \\sigma = ( \\sigma _ 1 , \\sigma _ 2 , \\sigma _ 3 ) \\in \\{ 0 , \\pm \\} ^ { 3 } . \\end{align*}"}
+{"id": "2437.png", "formula": "\\begin{align*} q _ { n + 1 } = K _ { h , \\tau } q _ n , \\forall ~ ~ n = 0 , 1 , \\ldots , \\end{align*}"}
+{"id": "9080.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - t \\rho _ { Q } ( X ) - \\epsilon t } R ^ { \\vee } \\big ( \\exp ( t X ) \\big ) R ^ { \\vee } ( u ) \\mu = 0 . \\end{align*}"}
+{"id": "658.png", "formula": "\\begin{align*} \\frac { \\partial { \\bf v } } { \\partial t } + ( { \\bf { v } } \\cdot \\nabla ) { \\bf v } = - \\frac { 1 } { \\rho } \\nabla p . \\end{align*}"}
+{"id": "2001.png", "formula": "\\begin{align*} ( A \\chi _ I , \\chi _ I ) = \\sum c _ i ^ 2 \\lambda _ i \\leq \\left ( \\sum c _ i ^ 2 \\right ) \\lambda _ { m a x } = | I | \\cdot \\lambda _ { m a x } , \\end{align*}"}
+{"id": "73.png", "formula": "\\begin{align*} f _ 3 ( \\mathbf { z } ) = \\frac { o b j _ f ( \\mathbf { z } ) + n } { 3 n } . \\end{align*}"}
+{"id": "8853.png", "formula": "\\begin{align*} \\Delta _ { \\nu } F ( z ) = \\Lambda _ { n , \\nu } \\left ( \\lambda \\right ) F ( z ) , \\end{align*}"}
+{"id": "2046.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { n = 0 } ^ { + \\infty } H ( n ) \\ , H ' ( - n ) & = \\displaystyle \\sum _ { n = 0 } ^ { + \\infty } H ^ { q } ( n ) H '^ { p } ( - n ) \\left [ H ^ { p } ( n ) H '^ { q } ( - n ) \\right ] \\\\ & \\leq \\mathbb { E } \\left [ \\xi _ 1 ^ p \\right ] ^ p \\ , \\mathbb { E } [ ( - { \\xi _ 1 ' } ) ^ q ] ^ q \\displaystyle \\sum _ { n = 0 } ^ { + \\infty } \\left [ ( n + 1 ) ^ { - q ^ 2 } H ^ { q } ( n ) \\ , ( n + 1 ) ^ { - p ^ 2 } H '^ { p } ( - n ) \\right ] . \\end{align*}"}
+{"id": "5613.png", "formula": "\\begin{align*} A ( l , l ) = 0 , A ( l , m _ 3 ) = 0 , A ( m _ 3 , m _ 3 ) = - \\theta m _ 2 . \\end{align*}"}
+{"id": "364.png", "formula": "\\begin{align*} C \\sqrt { { T _ * } { L ^ { - 2 } _ * } } T ^ { - 1 / 4 } _ * \\big ( \\sqrt { \\ln ( T _ * + 3 ) } + T ^ { - 3 / 4 } _ * \\big ) + C L ^ { - 1 } _ * = \\mathcal { O } \\big ( \\lambda ^ { - 1 / 4 } \\sqrt { \\ln ( \\lambda ) } \\big ) \\ , . \\end{align*}"}
+{"id": "6890.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\rho ( v _ n - v _ { \\mu } ) = \\lim _ { n \\rightarrow \\infty } \\int _ { \\Omega } b ( x ) | v _ n - v _ { \\mu } | ^ { \\alpha ( x ) } d x . \\end{align*}"}
+{"id": "39.png", "formula": "\\begin{align*} \\mu = \\frac { 1 } { K } \\cdot \\sum _ { n \\in \\Omega _ o } \\frac { 1 } { n } = \\frac { 1 } { K } \\cdot \\sum _ { i = 1 } ^ m \\sum _ { t = 0 } ^ { k _ i - 1 } \\frac { 1 } { C ^ t ( n _ i ) } , \\end{align*}"}
+{"id": "5200.png", "formula": "\\begin{align*} b _ { 1 2 } ' = & \\frac { b _ { 1 2 } b _ { 1 1 } ' - w _ 3 b _ { 1 1 } b _ { 2 2 } + c _ { 1 2 } } { b _ { 1 1 } } \\\\ [ 1 m m ] b _ { 1 3 } ' = & \\frac { b _ { 1 3 } b _ { 1 1 } ' + w _ 2 b _ { 1 1 } b _ { 3 3 } - w _ 3 b _ { 1 1 } b _ { 2 3 } + c _ { 1 3 } } { b _ { 1 1 } } \\\\ [ 1 m m ] b _ { 2 3 } ' = & \\frac { b _ { 1 1 } b _ { 2 3 } b _ { 2 2 } ' - w _ 1 + b _ { 1 1 } c _ { 2 3 } - b _ { 1 2 } c _ { 1 3 } + b _ { 1 3 } c _ { 1 2 } } { b _ { 2 2 } b _ { 1 1 } } \\ . \\end{align*}"}
+{"id": "8219.png", "formula": "\\begin{align*} x ( y - x ) f ( s , t - s ; y - x ) = \\frac { x ( y - x ) \\ , \\lambda e ^ { - \\lambda ( t - s ) } } { 2 c ^ 2 s \\sqrt { c ^ 2 ( t - s ) ^ 2 - ( y - x ) ^ 2 } } \\ , I _ 1 \\Bigl ( \\frac { \\lambda } { c } \\sqrt { c ^ 2 ( t - s ) ^ 2 - ( y - x ) ^ 2 } \\Bigr ) \\end{align*}"}
+{"id": "3035.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } \\Delta w ^ { s , j } _ { k } = f \\chi _ { { F } ^ { s , j } _ { Q _ k } } \\ \\ \\ \\mathrm { i n } \\ \\ \\ \\ \\Omega _ 1 , \\\\ \\ \\ \\ w ^ { s , j } _ { k } = 0 \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathrm { o n } \\ \\ \\ \\partial \\Omega _ 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "8294.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\Lambda \\ ! \\ ! \\ ! \\ ! & = x _ E ( y _ 3 z _ 1 \\ ! - \\ ! y _ 1 z _ 3 ) \\ ! + \\ ! y _ E ( x _ 1 z _ 3 \\ ! - \\ ! x _ 3 z _ 1 ) \\ ! + \\ ! z _ E ( x _ 3 y _ 1 \\ ! - \\ ! x _ 1 y _ 3 ) \\\\ & ~ ~ - x _ 2 [ y _ 3 ( z _ 1 \\ ! - \\ ! z _ E ) + y _ 1 ( z _ E \\ ! - \\ ! z _ 3 ) + y _ E ( z _ 3 \\ ! - \\ ! z _ 1 ) ] \\\\ & ~ ~ - y _ 2 [ x _ 3 ( z _ E \\ ! - \\ ! z _ 1 ) + x _ 1 ( z _ 3 \\ ! - \\ ! z _ E ) + x _ E ( z _ 1 \\ ! - \\ ! z _ 3 ) ] \\\\ & ~ ~ - z _ 2 [ x _ 3 ( y _ 1 \\ ! - \\ ! y _ E ) + x _ 1 ( y _ E \\ ! - \\ ! y _ 3 ) + x _ E ( y _ 3 \\ ! - \\ ! y _ 1 ) ] . \\end{array} \\right . \\end{align*}"}
+{"id": "4166.png", "formula": "\\begin{align*} d \\mathcal { L } ( 0 , 0 , \\phi ) | _ { ( g _ c , 0 , f _ c ) } = ( 2 \\triangle \\phi - 2 \\langle \\nabla \\phi , \\nabla f _ c \\rangle , - \\int _ M \\phi e ^ { - f _ c } d V _ { g _ c } ) \\in C ^ { 0 , \\alpha } _ { g _ c } ( M ) \\times \\mathbb { R } \\end{align*}"}
+{"id": "1923.png", "formula": "\\begin{align*} m _ \\mathrm { A D M } ( \\mathcal E , \\tilde g _ { \\sigma } ) - m _ \\mathrm { A D M } ( \\mathcal E , g ) & = \\lim _ { r \\to \\infty } \\frac { - 2 } { ( n - 2 ) \\omega _ { n - 1 } } \\int _ { | x | = r } \\nu _ g ( u _ { \\sigma } ) \\ , d \\mu _ { S _ r , g } \\\\ & = \\frac { - 2 } { ( n - 2 ) \\omega _ { n - 1 } } \\int _ { M _ \\sigma } \\Delta _ g u _ { \\sigma } \\ , d \\mu _ { g } \\\\ & = \\frac { - 1 } { 2 ( n - 1 ) \\omega _ { n - 1 } } \\int _ M V u _ { \\sigma } \\ , d \\mu _ g , \\end{align*}"}
+{"id": "7435.png", "formula": "\\begin{align*} \\phi _ K ( [ P _ { i j } ] ) & = H ( Z ^ { ( 0 ) } ) ( { \\rm i d } _ { N \\times N } \\otimes \\pi ) ( [ P _ { i j } ] ) H ( Z ^ { ( 0 ) } ) ^ * \\\\ & = V \\Pi ( [ P _ { i j } ] ) V ^ * \\end{align*}"}
+{"id": "2326.png", "formula": "\\begin{align*} \\nabla h ( x , y ) = h ( x , y ) ^ { \\frac { N - 1 } { p - 1 } } \\left [ \\ , \\ ( | x | ^ 2 + ( 1 - y ) ^ 2 \\ ) ^ { \\frac { p - N } { 2 ( p - 1 ) } - 1 } \\begin{pmatrix} x \\\\ y - 1 \\end{pmatrix} - \\ ( | x | ^ 2 + ( 1 + y ) ^ 2 \\ ) ^ { \\frac { p - N } { 2 ( p - 1 ) } - 1 } \\begin{pmatrix} x \\\\ y + 1 \\end{pmatrix} \\ , \\right ] \\end{align*}"}
+{"id": "4523.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { n + 1 } v _ i = \\prod _ { i = 1 } ^ { n + 1 } u _ i . \\end{align*}"}
+{"id": "7743.png", "formula": "\\begin{align*} f _ \\alpha ( x ) & = - \\frac { 1 } { \\pi } \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { k ! } \\sin ( \\pi k \\alpha ) \\frac { \\Gamma ( k \\alpha + 1 ) } { x ^ { k \\alpha + 1 } } \\end{align*}"}
+{"id": "166.png", "formula": "\\begin{align*} \\| \\nu _ { j } ( \\theta , \\varkappa ) - \\nu _ { j } ( \\theta , y ) \\| _ { \\mathbb { X } } & = \\| \\theta \\varkappa - \\theta y \\| _ { \\mathbb { X } } = | \\theta | \\| \\varkappa - y \\| _ { \\mathbb { X } } \\\\ & \\le b \\| \\varkappa - y \\| _ { \\mathbb { X } } . \\end{align*}"}
+{"id": "7065.png", "formula": "\\begin{align*} H \\underline { R } ^ { C _ 2 } _ \\diamond = R [ \\mu , \\tau ] / ( 2 \\mu ) \\end{align*}"}
+{"id": "2293.png", "formula": "\\begin{align*} d _ k = \\sigma ^ + \\gamma _ k ^ + - \\sigma ^ - \\gamma _ k ^ - . \\end{align*}"}
+{"id": "383.png", "formula": "\\begin{align*} J ^ { ( 1 , 1 ) } _ 2 \\leq 8 \\sum \\limits _ { n = 2 } ^ { \\infty } p _ n ( T _ * ) \\eta ^ 2 _ * ( n ) + 8 \\sum \\limits _ { n = 2 } ^ { \\infty } \\frac { p _ n ( T _ * ) } { \\widetilde { n } } \\left ( \\frac { C ^ { ( 2 ) } _ { \\textsf { K M T } } } { n ^ \\vartheta } \\right ) ^ { 1 / 2 } \\sum _ { j = 1 } ^ { \\widetilde { n } } a _ j \\ , , \\end{align*}"}
+{"id": "7027.png", "formula": "\\begin{align*} ( A ) / ( t - 1 ) = A \\end{align*}"}
+{"id": "8693.png", "formula": "\\begin{align*} \\tilde s _ { \\lambda } = \\tilde \\Gamma ^ + _ { - \\lambda _ 1 } | _ { t = 0 } \\dots \\tilde \\Gamma ^ + _ { - \\lambda _ l } | _ { t = 0 } \\ , ( 1 ) . \\end{align*}"}
+{"id": "968.png", "formula": "\\begin{align*} [ \\mathcal { D } _ \\mathfrak { K o s z } ] = ( n - 2 ) [ \\mathcal { D } _ \\mathfrak { R e s } ] . \\end{align*}"}
+{"id": "3353.png", "formula": "\\begin{align*} \\theta = \\dd \\kappa + \\epsilon p _ i \\dd q ^ i , i = 1 , \\dots , n , ~ ~ \\epsilon \\in \\R . \\end{align*}"}
+{"id": "8300.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\eta _ E = x _ 1 ( z _ 3 \\ ! - \\ ! z _ 2 ) + x _ 2 ( z _ 1 \\ ! - \\ ! z _ 3 ) + x _ 3 ( z _ 2 \\ ! - \\ ! z _ 1 ) \\\\ \\eta _ 1 = x _ 2 ( z _ E \\ ! - \\ ! z _ 3 ) + x _ 3 ( z _ 2 \\ ! - \\ ! z _ E ) + x _ E ( z _ 3 \\ ! - \\ ! z _ 2 ) \\\\ \\eta _ 2 = x _ 3 ( z _ E \\ ! - \\ ! z _ 1 ) + x _ 1 ( z _ 3 \\ ! - \\ ! z _ E ) + x _ E ( z _ 1 \\ ! - \\ ! z _ 3 ) \\\\ \\eta _ 3 = x _ 1 ( z _ E \\ ! - \\ ! z _ 2 ) + x _ 2 ( z _ 1 \\ ! - \\ ! z _ E ) + x _ E ( z _ 2 \\ ! - \\ ! z _ 1 ) . \\end{array} \\right . \\end{align*}"}
+{"id": "4286.png", "formula": "\\begin{align*} \\forall ( \\boldsymbol { \\lambda } , \\mu ) \\in N _ { [ 1 - 3 / p ' + k ] } , \\langle \\textbf { \\textit { f } } , \\boldsymbol { \\lambda } \\rangle _ { W ^ { - 1 , p } _ { k } ( \\R ^ 3 ) \\times W ^ { 1 , p ' } _ { - k } ( \\R ^ 3 ) } - \\langle \\chi , \\mu \\rangle _ { W ^ { 0 , p } _ { k } ( \\R ^ 3 ) \\times W ^ { 0 , p ' } _ { - k } ( \\R ^ 3 ) } = 0 , \\end{align*}"}
+{"id": "5186.png", "formula": "\\begin{align*} \\begin{aligned} ( x _ 0 ' - x _ t ' ) - ( x _ 0 - x _ t ) & \\equiv u _ { 1 2 } ( y _ 0 - y _ t ) + ( v _ { 3 4 } - v _ { 3 , t + 4 } ) \\frac { r ( r + 1 ) } { 2 } \\pmod { r } \\\\ & \\equiv - u _ { 1 2 } a _ 2 t \\frac { r } { K } \\pmod { r } , \\end{aligned} \\end{align*}"}
+{"id": "6436.png", "formula": "\\begin{align*} d ( ( Z , W , \\tau ) , ( \\hat Z , \\hat W , \\hat \\tau ) \\circ ( Z , W , \\tau ) ) = \\| ( \\hat Z , \\hat W + ( \\hat \\tau - \\tau ) \\hat Z + \\hat \\tau Z , \\hat \\tau ) \\| < \\rho _ 1 , \\end{align*}"}
+{"id": "4707.png", "formula": "\\begin{align*} \\varphi _ 4 & = \\frac { 1 } { 3 } \\left ( x ^ { 4 } + 8 x y ^ { 3 } \\right ) , \\\\ \\varphi _ { 1 2 } & = 2 4 3 \\left ( 6 1 x ^ { 1 2 } + 4 4 0 x ^ { 9 } y ^ { 3 } + 1 4 7 8 4 x ^ { 6 } y ^ { 6 } + 2 8 1 6 0 x ^ { 3 } y ^ { 9 } + 1 0 2 4 y ^ { 1 2 } \\right ) . \\end{align*}"}
+{"id": "5598.png", "formula": "\\begin{align*} k \\mapsto k ^ { \\prime } = k , l \\mapsto l ^ { \\prime } = l - \\frac { 1 } { 2 } z ^ { 2 } k + z m _ { 2 } , m _ { 2 } \\mapsto m _ { 2 } ^ { \\prime } = m _ { 2 } - z k , m _ 3 \\mapsto m _ { 3 } ^ { \\prime } = m _ { 3 } , \\end{align*}"}
+{"id": "4762.png", "formula": "\\begin{align*} A \\left ( m ( u _ { n } ) \\right ) - \\int _ { \\Omega } F ( x , u _ { n } ) \\ d x = c _ { M } + \\delta _ { n } , \\end{align*}"}
+{"id": "673.png", "formula": "\\begin{align*} \\omega = - d * d G ^ \\omega + { \\rm c o n s t a n t } \\cdot { \\rm v o l } , \\end{align*}"}
+{"id": "5889.png", "formula": "\\begin{align*} M = \\hbox { r a n k } ( D ) \\geq N - p . \\end{align*}"}
+{"id": "4993.png", "formula": "\\begin{align*} f _ 0 ( q , p ) : = ( q + \\nabla \\ell _ 0 ( p ) , p ) . \\end{align*}"}
+{"id": "434.png", "formula": "\\begin{align*} \\| f _ j \\| = \\| \\tau _ j \\| = | f _ j ( \\tau _ j ) | = 1 , \\forall 1 \\leq j \\leq n \\end{align*}"}
+{"id": "2881.png", "formula": "\\begin{align*} B \\left ( \\begin{matrix} t _ 1 \\\\ \\vdots \\\\ t _ r \\end{matrix} \\right ) = \\left ( \\begin{matrix} a _ 1 \\\\ \\vdots \\\\ a _ r \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "4565.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } < 1 = \\frac { 1 } { 1 } \\end{align*}"}
+{"id": "8876.png", "formula": "\\begin{align*} g ( z ) = \\sum \\limits _ { m = 0 } ^ { + \\infty } g _ m ( z ) , g _ m \\in \\mathcal { A } _ { m } ^ { \\nu } . \\end{align*}"}
+{"id": "8215.png", "formula": "\\begin{align*} P \\{ \\mathcal { T } ( t ) = x + v _ k ( t - s ) \\ | \\ \\mathcal { T } ( s ) = x , N ( s ) = k , V ( 0 ) = v _ 0 \\} = P \\{ N ( t - s ) = 0 \\ | \\ V ( 0 ) = v _ k \\} , \\end{align*}"}
+{"id": "8403.png", "formula": "\\begin{align*} \\partial _ { \\alpha } ( G _ { \\alpha , \\underline { n } } \\circ F _ { \\alpha , n _ 0 } ^ { - 1 } ) = \\sum _ { j = 1 } ^ k & \\partial _ { \\alpha } [ G _ { \\alpha , n _ j } \\circ F _ { \\alpha , n _ { j - 1 } } ^ { - 1 } \\circ \\dots \\circ F _ { \\alpha , n _ { 0 } } ^ { - 1 } ] \\\\ & \\times \\Pi _ { i \\neq j } G _ { \\alpha , n _ i } \\circ F _ { \\alpha , n _ { i - 1 } } ^ { - 1 } \\circ \\dots \\circ F _ { \\alpha , n _ { 0 } } ^ { - 1 } \\end{align*}"}
+{"id": "3679.png", "formula": "\\begin{align*} \\vert U _ 2 ( E , F _ Y ) \\vert \\leq \\frac { Y } { 2 } + \\frac { 4 g + 2 } { ( \\sqrt { q } - 1 ) ( 1 - q ^ { - 1 } ) } , \\end{align*}"}
+{"id": "2103.png", "formula": "\\begin{align*} C + C ' & \\textstyle = \\Big ( \\bigoplus _ { \\epsilon \\in E _ 1 } F H \\epsilon \\Big ) + \\Big ( \\bigoplus _ { \\epsilon \\in E _ 0 } F H \\epsilon \\Big ) + \\Big ( \\bigoplus _ { \\epsilon \\in E _ 2 } F H \\epsilon \\Big ) + \\Big ( \\bigoplus _ { \\epsilon \\in E _ 0 } F H \\epsilon \\Big ) \\\\ & \\textstyle = \\bigoplus _ { \\epsilon \\in E _ 0 \\cup E _ 1 \\cup E _ 2 } F H \\epsilon = \\bigoplus _ { \\epsilon \\in E _ C \\cup E _ { C ' } } F H \\epsilon . \\end{align*}"}
+{"id": "7650.png", "formula": "\\begin{align*} C ( q _ { 1 } , \\dots , q _ { p } ; M ) = 0 \\to M \\to \\iffalse \\bigoplus _ { 1 \\leq \\ell \\leq p } M [ q _ { \\ell } ^ { - 1 } ] \\to \\fi \\cdots \\to \\bigoplus _ { \\substack { I \\subseteq [ p ] \\\\ \\mid I \\mid = t } } M [ q _ { I } ^ { - 1 } ] \\to \\cdots \\to M [ q _ { [ p ] } ^ { - 1 } ] \\to 0 \\end{align*}"}
+{"id": "8635.png", "formula": "\\begin{align*} T _ m = \\bigcup _ { k \\geq m + k _ 0 } O _ { k } . \\end{align*}"}
+{"id": "4010.png", "formula": "\\begin{align*} \\tilde { A } _ { i j } ( \\cdot , \\tilde { u } , D \\tilde { u } ) & = D _ i \\Upsilon ^ k D _ j \\Upsilon ^ l A _ { k l } ( \\cdot , \\tilde { u } , [ D \\Upsilon ] ^ { - 1 } D \\tilde { u } ) + [ D \\Upsilon ] ^ { \\alpha k } D _ \\alpha \\tilde { u } D _ { i j } \\Upsilon ^ k , \\\\ \\tilde { B } ( \\cdot , \\tilde { u } , D \\tilde { u } ) & = [ \\det D \\Upsilon ] ^ 2 B ( \\cdot , \\tilde { u } , [ D \\Upsilon ] ^ { - 1 } D \\tilde { u } ) . \\end{align*}"}
+{"id": "2145.png", "formula": "\\begin{align*} - ( J \\ast U ) ( x ) + U ( x ) + c U ' ( x ) = f ( U ( x ) ) \\ge C _ 1 ( 1 - U ( x ) ) > 0 , \\forall x \\geq R . \\end{align*}"}
+{"id": "2095.png", "formula": "\\begin{align*} w ^ { \\prime } = \\frac { \\nu _ 1 - 1 } { \\nu _ 0 \\nu _ 1 } . \\end{align*}"}
+{"id": "8644.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} z : = \\frac { a z + b } { c z + d } , \\end{align*}"}
+{"id": "230.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - \\beta \\rho \\eta \\leqslant - \\frac { \\alpha } { 2 } ( 1 + 3 \\rho ^ 2 ) \\leqslant - \\frac { \\alpha } { 2 } , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - \\beta \\tau \\eta \\leqslant - \\frac { \\alpha } { 2 } ( 1 + 3 \\rho ^ 2 ) \\leqslant - \\frac { \\alpha } { 2 } , \\end{align*}"}
+{"id": "3240.png", "formula": "\\begin{align*} ( 2 k + 2 ) k = \\sum _ { v \\in V } | L ( v ) | = \\sum _ { c \\in C } | L ^ { - 1 } ( c ) | \\le | C - F | ( k + 1 - b ) + | F | ( k + p _ 1 + 2 ) . \\end{align*}"}
+{"id": "8368.png", "formula": "\\begin{align*} \\liminf _ { t \\to + \\infty } \\int _ { | x | > - R + t } \\left ( | \\nabla u | ^ 2 + | \\partial _ t u | ^ 2 \\right ) = 0 . \\end{align*}"}
+{"id": "6249.png", "formula": "\\begin{align*} k = | n | \\lambda _ k ( \\sin \\theta _ k \\cos \\phi _ k , \\sin \\theta _ k \\sin \\phi _ k , c _ k ) , \\quad k \\in \\Z ^ 3 . \\end{align*}"}
+{"id": "2282.png", "formula": "\\begin{align*} \\int ^ { T } _ { 0 } & \\int _ { \\Omega } \\big ( \\rho u \\cdot \\partial _ t \\phi + ( \\rho u \\otimes u ) : \\nabla \\phi + \\rho ^ \\gamma { \\rm { d i v } } \\phi + \\delta \\rho ^ \\beta { \\rm { d i v } } \\phi - \\varepsilon \\nabla \\rho \\cdot \\nabla u \\cdot \\phi \\\\ & - Z _ \\varepsilon : \\nabla \\phi - \\mathbb { S } ( \\nabla u ) : \\nabla \\phi + ( j - n u ) \\cdot \\phi \\big ) \\ , d x d t + \\int _ { \\Omega } \\rho _ 0 u _ 0 \\cdot \\phi ( 0 , x ) \\ , d x = 0 , \\end{align*}"}
+{"id": "7943.png", "formula": "\\begin{align*} [ a ] \\cdot [ b ] = [ a b ] , [ a ] + [ b ] = \\{ [ c ] \\mid c = g _ 1 a + g _ 2 b \\textrm { f o r s o m e } g _ 1 , g _ 2 \\in G \\} . \\end{align*}"}
+{"id": "5328.png", "formula": "\\begin{align*} \\mathrm { d i v } \\ , \\varepsilon E = 0 \\mbox { i n } \\Omega . \\end{align*}"}
+{"id": "3067.png", "formula": "\\begin{align*} - A : D ^ 2 w = a - \\bar { a } \\quad Y , w Y , \\int _ Y w = 0 . \\end{align*}"}
+{"id": "8452.png", "formula": "\\begin{align*} \\lambda _ A = \\xi _ A = \\gamma _ A / c _ * ( A ) . \\end{align*}"}
+{"id": "6903.png", "formula": "\\begin{align*} - ( a + 1 ) & \\leq m i n \\{ b _ i , b _ { i ' } + b _ { j ' } \\} \\\\ b _ i & \\leq 1 \\end{align*}"}
+{"id": "3778.png", "formula": "\\begin{align*} R _ \\Phi ( \\Gamma ) & = \\begin{cases} \\varnothing & * _ \\Gamma ; \\\\ \\{ \\mathcal { E } _ { * _ \\Gamma } \\} & * _ \\Gamma ; \\\\ \\{ \\tilde \\alpha _ { * _ \\Gamma } \\} & * _ \\Gamma . \\end{cases} \\end{align*}"}
+{"id": "8568.png", "formula": "\\begin{align*} X = \\bigcup _ { i = 0 } ^ { n - 1 } T f ^ { i } . \\end{align*}"}
+{"id": "3214.png", "formula": "\\begin{align*} Q ^ { \\circ } \\cap \\left ( \\bigcup _ { j = 1 } ^ { n } \\psi _ { j } A _ { j } \\right ) \\subseteq M \\subseteq Q \\cap \\left ( \\bigcup _ { j = 1 } ^ { n } \\psi _ { j } A _ { j } \\right ) . \\end{align*}"}
+{"id": "8049.png", "formula": "\\begin{align*} u _ j ( z ) = \\sum _ { n \\neq 0 } \\rho _ j ( n ) W _ { s _ j } ( n z ) . \\end{align*}"}
+{"id": "7858.png", "formula": "\\begin{align*} f '' + e ^ { - z } f ' + \\alpha f = 0 , \\alpha \\in \\C \\setminus \\{ 0 \\} , \\end{align*}"}
+{"id": "8612.png", "formula": "\\begin{align*} \\norm { x - x _ 0 } = \\min \\{ \\norm { x - y } : y \\in D \\} . \\end{align*}"}
+{"id": "3929.png", "formula": "\\begin{align*} & \\det [ D ^ 2 w - A ( \\cdot , w , D w ) ] = t B ( \\cdot , w , D w ) \\\\ & \\quad \\quad + ( 1 - t ) \\det [ D ^ 2 v - A ( \\cdot , v , D v ) ] B _ r , \\\\ & w = u _ m \\partial B _ r . \\end{align*}"}
+{"id": "1903.png", "formula": "\\begin{align*} \\begin{array} { l l } & d ( | z ( \\cdot , t ) | ^ 2 _ { L ^ 2 ( 0 , 1 ) } ) = ( \\sigma ^ 2 - 2 c ) | z ( \\cdot , t ) | ^ 2 _ { L ^ 2 ( 0 , 1 ) } d t - 2 | z _ x ( \\cdot , t ) | ^ 2 _ { L ^ 2 ( 0 , 1 ) } d t \\cr & \\ \\ \\ + 2 z ( 1 , t ) \\widetilde { w } ( t ) d t + 2 \\sigma | z ( \\cdot , t ) | _ { L ^ 2 ( 0 , 1 ) } ^ 2 d B ( t ) . \\end{array} \\end{align*}"}
+{"id": "8754.png", "formula": "\\begin{align*} L _ k = \\begin{bmatrix} 0 & \\cdots & 0 & u _ { 2 T - 1 } & u _ { 2 T } & \\cdots & u _ { \\bar { N } } & 0 & \\cdots & 0 \\end{bmatrix} ^ * , \\end{align*}"}
+{"id": "9148.png", "formula": "\\begin{align*} K _ { n + p } = \\sum _ { i = 1 } ^ n e _ { n - i + 1 } ( X ) \\cdot K _ { i + p - 1 } . \\end{align*}"}
+{"id": "1045.png", "formula": "\\begin{align*} R _ i ^ { ( \\ell ) } = \\min \\bigl \\{ X _ i - ( j - 1 ) M / 3 : j \\in \\mathcal { J } , j \\equiv \\ell \\ , ( 3 ) , X _ i \\geq ( j - 1 ) M / 3 \\bigr \\} \\end{align*}"}
+{"id": "3361.png", "formula": "\\begin{align*} X = ( X \\lrcorner \\eta ) R + ( X - ( X \\lrcorner \\eta ) R ) . \\end{align*}"}
+{"id": "1342.png", "formula": "\\begin{align*} G _ { f , \\tau } \\coloneqq [ f _ j ( \\tau _ k ) ] _ { 1 \\leq j , k \\leq n } = \\begin{pmatrix} f _ 1 ( \\tau _ 1 ) & f _ 1 ( \\tau _ 2 ) & \\cdots & f _ 1 ( \\tau _ n ) \\\\ f _ 2 ( \\tau _ 1 ) & f _ 2 ( \\tau _ 2 ) & \\cdots & f _ 2 ( \\tau _ n ) \\\\ \\vdots & \\vdots & & \\vdots \\\\ f _ n ( \\tau _ 1 ) & f _ n ( \\tau _ 2 ) & \\cdots & f _ n ( \\tau _ n ) \\\\ \\end{pmatrix} _ { n \\times n } \\in \\mathbb { M } _ n ( \\mathbb { K } ) . \\end{align*}"}
+{"id": "8546.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Bigg ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sum _ { p = 1 } ^ { \\lfloor ( n - i ) / 2 \\rfloor } \\binom { n } { i } \\binom { n - i } { 2 } ^ p ( 2 ^ i - 1 ) ^ n 2 ^ { Q ( n - i + 1 , n - i + 1 - 2 p ) - Q ( n , n ) } \\Bigg ) \\le 1 , \\end{align*}"}
+{"id": "2603.png", "formula": "\\begin{align*} \\dim \\sum _ { a , b \\in I } V _ { a , b } ^ \\perp \\begin{cases} \\geq | I | & \\mbox { f o r a n y $ I \\subset [ 5 ] $ , $ 1 \\leq | I | \\leq 2 $ , a n d } \\\\ = 3 & \\mbox { f o r a n y $ I \\subset [ 5 ] $ , $ | I | = 3 $ } , \\end{cases} \\end{align*}"}
+{"id": "9062.png", "formula": "\\begin{align*} D ^ \\alpha _ x \\partial _ t ^ k w _ 2 ( x , t ) = \\int _ { - \\infty } ^ \\infty ( i \\tau ) ^ k D _ x ^ \\alpha \\widehat { w } _ 2 ( x , \\tau ) e ^ { i \\tau t } d \\tau \\end{align*}"}
+{"id": "1338.png", "formula": "\\begin{align*} \\operatorname { T r a } ( S ^ r _ { f , \\tau } ) \\geq \\frac { \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ r } { d ^ { r - 1 } } , \\forall r \\in [ 1 , \\infty ) \\end{align*}"}
+{"id": "5743.png", "formula": "\\begin{align*} \\mathbf { x } ^ { A A ^ { \\prime } } \\rightarrow \\mathbf { x } _ { 1 } ^ { A A ^ { \\prime } } = - \\mu _ { 1 } ^ { A A ^ { \\prime } } \\mathbf { m } - \\mu _ { 1 } ^ { A A ^ { \\prime } } \\overline { \\mathbf { m } } + i \\nu _ { 1 } ^ { A A ^ { \\prime } } \\mathbf { m } \\overline { \\mathbf { m } } \\end{align*}"}
+{"id": "7506.png", "formula": "\\begin{align*} \\int _ { \\mathcal { O } _ K ^ { \\times } } { \\chi ( a c ( x ^ m ) ) | d x | } = { \\left \\{ \\begin{array} { r l } 1 - q ^ { - 1 } , \\ \\ \\ & { \\rm i f } \\ \\chi ^ m = \\chi _ { { \\rm t r i v } } , \\\\ 0 , \\ \\ \\ & { \\rm i f } \\ \\chi ^ m \\neq \\chi _ { { \\rm t r i v } } . \\end{array} \\right . } \\end{align*}"}
+{"id": "3179.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\bar { A } : D ^ 2 u & = 0 & & \\Omega , \\\\ u & = g & & \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3183.png", "formula": "\\begin{align*} c _ j ^ { k k } ( A ) = \\int _ { [ 0 , 1 ] ^ 3 } r a _ j \\partial _ j v ^ { k k } = \\int _ { [ 0 , 1 ] ^ 2 } r _ B \\ , b _ j \\ , \\partial _ j v ^ { k k } _ B = c _ j ^ { k k } ( B ) \\qquad \\forall j , k \\in \\{ 1 , 2 \\} , \\end{align*}"}
+{"id": "4548.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { s _ i } = 1 - \\frac { 1 } { \\prod _ { i = 1 } ^ n s _ i } \\end{align*}"}
+{"id": "3949.png", "formula": "\\begin{align*} R ^ * : = \\{ x \\in \\mathbf { R } ^ d ; - b _ i ^ { - 1 } \\leq x _ i \\leq a _ i ^ { - 1 } \\} . \\end{align*}"}
+{"id": "5830.png", "formula": "\\begin{align*} \\langle L u , \\phi \\rangle = \\int _ { \\Omega } \\Delta u \\Delta \\phi d x \\quad \\hbox { a n d } \\langle g v , \\phi \\rangle = \\int _ { \\Omega } a v \\phi d x , \\end{align*}"}
+{"id": "1427.png", "formula": "\\begin{align*} \\mathbb { P } ( \\tau _ h = N _ 2 ) \\leq \\frac { 3 ( \\sigma ^ 2 - 1 ) ^ { - 1 } } { N _ 2 } ( 2 h ^ 2 - z ^ 2 ) + \\mathbb { P } ( \\tau _ 1 \\leq N _ 2 ) . \\end{align*}"}
+{"id": "7325.png", "formula": "\\begin{align*} M ( \\gamma ) : = \\int _ { 1 } ^ { \\gamma } H ^ N ( \\infty ; x ) d m ( x ) . \\end{align*}"}
+{"id": "727.png", "formula": "\\begin{align*} d s = \\begin{cases} | d z | , z \\in \\Omega , \\\\ | d \\tilde { z } | , \\tilde { z } \\in \\tilde { \\Omega } . \\end{cases} \\end{align*}"}
+{"id": "58.png", "formula": "\\begin{align*} \\omega ( f ) = \\sum _ { i \\in V } { f ( i ) } \\end{align*}"}
+{"id": "4560.png", "formula": "\\begin{align*} \\frac { a '' } { b '' } = \\frac { a ' } { b ' } - \\frac { 1 } { n ' } \\end{align*}"}
+{"id": "5556.png", "formula": "\\begin{align*} S _ 3 ( x ^ 2 ) : = \\sum _ { n = \\ell } ^ { \\infty } M ( l ; n ) \\left ( e ^ { - \\frac { x ^ 2 } { n ^ 2 } } - e ^ { - \\frac { x ^ 2 } { ( n + 1 ) ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "2802.png", "formula": "\\begin{align*} x _ { \\sigma ( j ) } ^ { \\rho + \\Delta } x _ { \\sigma ( k ) } ^ { \\rho - \\Delta } + x _ { \\sigma ( j ) } ^ { \\rho - \\Delta } x _ { \\sigma ( k ) } ^ { \\rho + \\Delta } - x _ { \\sigma ( j ) } ^ { \\rho + \\delta } x _ { \\sigma ( k ) } ^ { \\rho - \\delta } - x _ { \\sigma ( j ) } ^ { \\rho - \\delta } x _ { \\sigma ( k ) } ^ { \\rho + \\delta } = 0 . \\end{align*}"}
+{"id": "1760.png", "formula": "\\begin{align*} - \\Delta u + u + \\lambda ( I _ { 2 } * | u | ^ 2 ) u = | u | ^ { p - 2 } u \\quad , \\end{align*}"}
+{"id": "4911.png", "formula": "\\begin{align*} \\mathcal { H } = \\int H ( z , z _ x ) \\mathrm { d } x , \\end{align*}"}
+{"id": "1029.png", "formula": "\\begin{align*} G _ { s } ( t ) \\lesssim \\left \\{ \\begin{aligned} & 1 + t ^ { s } , & \\ t \\rightarrow 0 ; \\\\ & e ^ { - c t } , & \\ t \\rightarrow \\infty . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5083.png", "formula": "\\begin{align*} \\det ( A _ n ) w _ { 1 } = \\big ( \\operatorname { a d j } ( A _ n ) A _ n \\vec w ) _ { 1 } = \\big ( \\operatorname { a d j } ( A _ n ) ( T _ n , \\ldots , T _ { n + k + p - 1 } ) ^ T \\big ) _ { 1 } = \\sum _ { j = 0 } ^ { k + p - 1 } \\gamma _ j T _ { n + j } \\end{align*}"}
+{"id": "2272.png", "formula": "\\begin{align*} \\int ^ { T } _ 0 \\int _ { \\Omega } ( \\rho \\partial _ t \\psi + \\rho u \\cdot \\nabla _ x \\psi ) \\ , d x d t + \\int _ { \\Omega } \\rho _ 0 \\psi ( 0 , x ) \\ , d x = \\int _ 0 ^ T \\int _ { \\Gamma _ { \\rm { i n } } } \\rho _ B u _ B \\cdot \\nu ( x ) \\varphi \\ , d \\sigma ( x ) d t ; \\end{align*}"}
+{"id": "5675.png", "formula": "\\begin{align*} f _ { Q _ j } ( y ; \\rho _ 1 , \\rho _ 2 ) = \\frac { \\sigma _ j ^ 2 } { \\rho _ 1 ^ 2 - \\rho _ 2 ^ 2 } \\left ( F _ { \\chi ^ 2 _ { n } ( \\frac { \\rho _ 2 ^ 2 } { \\sigma _ j ^ 2 } ) } ( y ) - F _ { \\chi ^ 2 _ { n } ( \\frac { \\rho _ 1 ^ 2 } { \\sigma _ j ^ 2 } ) } ( y ) \\right ) , \\end{align*}"}
+{"id": "1242.png", "formula": "\\begin{align*} \\bar \\nu _ k ( y ) : = \\frac { e ^ { - v _ k ( y ) } } { \\int _ { \\underline y } ^ { \\overline y } e ^ { - v _ k ( s ) } d s } . \\end{align*}"}
+{"id": "5477.png", "formula": "\\begin{align*} \\| \\mu - \\nu \\| _ { { \\rm V a r } , \\tilde V } \\leq \\sup _ { \\phi , \\psi \\in C _ b , \\atop \\phi - \\psi \\leq \\tilde V } ( \\nu ( \\phi ) - \\mu ( \\psi ) ) = W _ { \\tilde V } ( \\mu , \\nu ) \\end{align*}"}
+{"id": "7863.png", "formula": "\\begin{align*} f '' ( z ) - ( 2 e ^ { 2 z } + 1 ) f ' ( z ) + 2 n e ^ { 2 z } f ( z ) = 0 \\end{align*}"}
+{"id": "1451.png", "formula": "\\begin{align*} \\lambda = \\frac { k ( k - 1 ) ( k - 2 ) ( m - 1 ) ! ( n - 1 ) ! } { ( m n - 1 ) ( m n - 2 ) | K _ \\Delta | } . \\end{align*}"}
+{"id": "1066.png", "formula": "\\begin{align*} \\mathcal { R } _ { n , \\alpha } ( \\theta ( \\mathcal { P } _ \\varepsilon ) , \\Phi \\circ \\rho ) \\geq \\frac { \\Phi ( \\omega ( \\varepsilon ) / 2 ) } { 2 } \\left ( 1 - \\sqrt { 4 n \\alpha ^ 2 } \\mathrm { T V } ( \\widetilde { R } _ 0 , \\widetilde { R } _ 1 ) \\right ) = \\frac { \\Phi ( \\omega ( \\varepsilon ) / 2 ) } { 2 } , \\end{align*}"}
+{"id": "6737.png", "formula": "\\begin{align*} \\phi ( s ) = \\int _ { 0 } ^ { \\infty } y ^ { \\frac { s } { 2 } - 1 } \\Psi ( y ) d y \\end{align*}"}
+{"id": "8501.png", "formula": "\\begin{align*} \\widehat { C } ( p , t ) = l ( t ) A ( t ) C ( p ) + H ( t ) , \\end{align*}"}
+{"id": "8227.png", "formula": "\\begin{align*} Z = \\left [ \\begin{array} { c c c c c c c c } - 7 + 4 i & 9 - 3 i & - 6 + 2 i & 3 + 4 i & 7 + 6 i & 4 - 4 i & i & 5 - 8 i \\\\ 4 - 5 i & 1 + 4 i & - 8 - 2 i & - 7 + 4 i & 1 - 4 i & 1 - 8 i & 8 - 6 i & 1 - 3 i \\\\ \\end{array} \\right ] . \\end{align*}"}
+{"id": "8889.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { C } } \\mathcal { A } _ { m } ^ { \\nu } : = \\int _ { \\mathbb { C } ^ n } K _ { \\nu , m } ( z , z ) d \\mu _ { n } ( z ) = \\frac { ( 2 m + n + 2 \\nu ) \\Gamma ( m + n ) \\Gamma ( m + n + 2 \\nu ) } { n \\left ( \\Gamma ( n ) \\right ) ^ { 2 } \\Gamma ( m + 1 ) \\Gamma ( m + 2 \\nu + 1 ) } . \\end{align*}"}
+{"id": "2922.png", "formula": "\\begin{align*} \\mathcal { A } _ t ( \\varphi ) = \\lim _ { \\varepsilon \\to 0 } \\mathcal { A } ^ \\varepsilon _ { 0 , t } ( \\varphi ) \\end{align*}"}
+{"id": "1384.png", "formula": "\\begin{align*} \\psi = \\bigoplus _ \\rho \\left ( \\bigoplus _ { i \\in I _ \\rho } \\rho \\boxtimes S _ { a _ i } \\boxtimes S _ { b _ i } \\right ) , \\end{align*}"}
+{"id": "208.png", "formula": "\\begin{align*} \\frac { p _ { n } } { n } = \\frac { p _ { n } k _ { n } } { n k _ { n } } \\leq \\frac { 2 ( h _ { m + 1 } + c _ { m + 1 } ) } { n ( h _ { m } + c _ { m } ) } \\leq \\frac { 4 } { n } \\frac { ( r _ { m } + 1 ) ( h _ { m } + c _ { m } + r _ { m } ) } { ( h _ { m } + c _ { m } ) } = \\frac { 4 r _ { m } } { n } \\Big { ( } 1 + \\frac { r _ { m } } { h _ { m } + c _ { m } } \\Big { ) } \\to 0 \\end{align*}"}
+{"id": "3530.png", "formula": "\\begin{align*} \\mathbf { p } : = \\prod _ { 1 \\leq i < j \\leq n } \\mathbf { p } _ { i j } , \\end{align*}"}
+{"id": "8686.png", "formula": "\\begin{align*} \\tilde F _ \\lambda ( x _ 1 , \\dots , x _ n ; t ) = \\frac { ( 1 - t ) ^ n } { \\prod _ { i = 1 } ^ { n - l } ( 1 - t ^ i ) } \\sum _ { \\sigma \\in S _ n } \\sigma \\left ( f _ { \\lambda _ 1 } ( x _ { 1 } ) \\dots f _ { \\lambda _ n } ( x _ { n } ) \\prod _ { i = 1 } ^ { n } \\prod _ { i < j } \\frac { x _ { i } - t x _ { j } } { x _ { i } - x _ { j } } \\right ) , \\end{align*}"}
+{"id": "9037.png", "formula": "\\begin{align*} \\mathcal { W } _ p ^ p ( L _ n [ \\textbf { x } ] , L _ n [ \\textbf { y } ] ) \\le \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } | x _ j - y _ j | ^ p , \\end{align*}"}
+{"id": "5102.png", "formula": "\\begin{align*} \\sum _ { i \\in I _ \\tau } c _ { i , 0 } \\vec c _ i ^ { \\vec n } - \\sum _ { j \\in J _ \\tau } g _ j ( \\vec n ) = 0 \\end{align*}"}
+{"id": "4968.png", "formula": "\\begin{align*} q : = r ^ p - ( f ^ l g h ) ^ { p - 1 } r = ( f ^ { l p } x _ 2 ^ p + x _ 1 ^ { m p } ) - ( f ^ l g h ) ^ { p - 1 } ( f ^ l x _ 2 + x _ 1 ^ m ) \\end{align*}"}
+{"id": "5912.png", "formula": "\\begin{align*} \\widetilde C _ { p - 1 } = \\begin{pmatrix} C _ p \\\\ e _ { \\bar r } ^ T \\end{pmatrix} . \\end{align*}"}
+{"id": "850.png", "formula": "\\begin{align*} \\dot R = R \\widehat { \\Omega } \\end{align*}"}
+{"id": "6688.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\| g _ i ^ { k + 1 } - ( 1 - \\alpha ^ k ) g _ i ^ k \\right \\| \\\\ & \\leq \\left \\| g _ i ^ { k + 1 } - g _ i ^ k \\right \\| + \\| \\alpha ^ k g _ i ^ k \\| \\leq L \\left \\| x _ i ^ { k + 1 } - x _ i ^ k \\right \\| + \\alpha ^ k C , \\end{aligned} \\end{align*}"}
+{"id": "8234.png", "formula": "\\begin{align*} - \\Delta w + V \\left ( \\left | x \\right | \\right ) w - w \\left ( \\Delta w ^ 2 \\right ) = K ( | x | ) g ( w ) \\mathbb { R } ^ { N } \\end{align*}"}
+{"id": "2811.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ j \\lambda v _ i \\leq \\sum _ { i = 1 } ^ j \\lambda u _ i \\qquad \\sum _ { i = 1 } ^ j v _ i \\leq \\sum _ { i = 1 } ^ j u _ i . \\end{align*}"}
+{"id": "7706.png", "formula": "\\begin{align*} \\lambda ^ { - \\frac { 1 } { 2 } } = \\frac { 2 \\sqrt { \\tau } } { \\pi } \\left ( \\int _ 0 ^ 1 \\frac { 1 } { \\tau + \\lambda y ^ 2 } d y + \\int _ 0 ^ 1 \\frac { 1 } { \\tau y ^ 2 + \\lambda } d y \\right ) . \\end{align*}"}
+{"id": "3187.png", "formula": "\\begin{align*} \\bar { A } : = \\int _ { [ 0 , 1 ] ^ 3 } r A = \\mathrm { d i a g } ( 1 , 1 , 8 ) \\end{align*}"}
+{"id": "369.png", "formula": "\\begin{align*} T r _ i \\sum \\limits _ { k = 1 } ^ { n _ i } u _ k \\leq t < T r _ i \\sum \\limits _ { k = 1 } ^ { n _ i + 1 } u _ k \\textrm { f o r e a c h } i \\in \\{ 1 , 2 \\} \\ , . \\end{align*}"}
+{"id": "4837.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } r = d _ 0 F ^ { - 1 } \\big ( \\xi - g ( t ) - R _ 1 ( r , t ) \\big ) = h _ 1 ( r , t ) \\\\ t = f ^ { - 1 } \\big ( \\tau - R _ 2 ( r , t ) \\big ) = h _ 2 ( r , t ) . \\end{array} \\right . \\end{align*}"}
+{"id": "4791.png", "formula": "\\begin{align*} \\mathsf { H } ( Z ) & = \\mathsf { H } ( Z _ 1 ) + \\sum _ { i = 2 } ^ N \\mathsf { H } ( Z _ i | Z _ 1 , . . . , Z _ { i - 1 } ) \\\\ & \\leq \\sum _ { i = 1 } ^ { r } \\mathsf { H } ( Z _ i ) \\\\ & = r \\cdot h _ q ( \\alpha ) \\log ( q ) \\end{align*}"}
+{"id": "3602.png", "formula": "\\begin{align*} d _ { j } ^ + ( \\lambda ) = \\frac { ( - 1 ) ^ { \\sum _ { \\alpha = j } ^ n ( \\alpha + 1 ) ( \\lambda _ \\alpha - \\lambda _ { \\alpha + 1 } ) } } { \\lambda _ j + 1 } d _ { j } ^ + ( \\mu ) . \\end{align*}"}
+{"id": "1291.png", "formula": "\\begin{align*} \\big ( a , \\textstyle { \\sum _ { k = 1 } ^ n } b _ k \\beta _ k + \\Sigma ^ { - 1 } ( \\mu \\circ \\Phi ^ { - 1 } ) \\big ) \\end{align*}"}
+{"id": "1479.png", "formula": "\\begin{align*} \\lambda & = \\frac { 2 k ( k - 1 ) ( m - 1 ) ! ( m - 2 ) ! } { ( m + 1 ) | G _ \\Delta | } \\\\ & = \\frac { 2 k ( k - 1 ) ( k - 3 ) ! ( k - 4 ) ! } { ( k - 1 ) | G _ \\Delta | } \\\\ & = \\frac { 2 k ( k - 3 ) ! ( k - 4 ) ! } { | G _ \\Delta | } \\end{align*}"}
+{"id": "6674.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 2 \\lambda ^ k } { m } \\sum _ { i = 1 } ^ m \\left \\langle \\nabla f _ i ( \\bar { x } ^ k ) , \\bar x ^ k - \\theta ^ * \\right \\rangle & = 2 \\lambda ^ k \\left \\langle \\nabla F ( \\bar { x } ^ k ) , \\bar x ^ k - \\theta ^ * \\right \\rangle \\\\ & \\geq 2 \\lambda ^ k ( F ( \\bar { x } ^ k ) - F ( \\theta ^ * ) ) \\end{aligned} \\end{align*}"}
+{"id": "6328.png", "formula": "\\begin{align*} & \\nu W _ { l _ i } W _ { l _ j } \\sum _ { e \\in E ^ + } \\Big ( | A _ i ( e ) | | A _ j ( e ) | + | B _ i ( e ) | | B _ j ( e ) | - | A _ i ( e ) | | B _ j ( e ) | - | B _ i ( e ) | | A _ j ( e ) | \\Big ) \\\\ & = \\nu W _ { l _ i } W _ { l _ j } \\sum _ { e \\in E ^ + } \\Big ( ( | A _ i ( e ) | - | B _ i ( e ) | ) ( | A _ j ( e ) | - | B _ j ( e ) | ) \\Big ) = \\nu W _ { l _ i } W _ { l _ j } \\sum _ { e \\in E ^ + } t _ i ( e ) t _ j ( e ) , \\end{align*}"}
+{"id": "5953.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { s = 1 } ^ p \\beta _ { r s } ( E _ s , \\widehat { U } ) + ( C _ p ^ T Q _ r , \\widehat { U } ) = ( Q _ r , C _ p \\widehat { U } ) = 0 , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "3409.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } \\zeta _ { r } + v \\cdot \\nabla \\zeta _ { r } - \\mu ( \\Delta - \\frac { 1 } { r ^ { 2 } } ) \\zeta _ { r } = \\zeta _ { r } \\partial _ { r } v _ { r } + \\zeta _ { z } \\partial _ { z } v _ { r } , \\\\ { \\zeta _ { r } } _ { \\vert t = 0 } = \\zeta _ { r } ^ 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "7290.png", "formula": "\\begin{align*} \\liminf _ { x \\to 0 } \\frac { ( - 1 ) ^ k G ^ { k } _ m ( x ) } { x } \\geq \\int _ { 0 } ^ { 1 } ( - 1 ) ^ { k - 1 } G ^ { k - 1 } _ m ( z ) d m ( z ) = \\infty . \\end{align*}"}
+{"id": "4294.png", "formula": "\\begin{align*} ( \\mathbf { H } ^ ) _ { \\{ k , m \\} } = \\left \\{ \\begin{array} { l r } ( \\mathbf { H } ^ ) _ { \\{ k , m \\} } , & k \\in \\Omega _ m \\\\ 0 , & . \\end{array} \\right . \\end{align*}"}
+{"id": "8027.png", "formula": "\\begin{align*} A _ { n + 1 } = C ( A _ n - \\alpha _ 1 B _ n ) ^ { a _ 1 } \\cdots ( A _ n - \\alpha _ s B _ n ) ^ { a _ s } = p ( A _ n / B _ n ) \\cdot B _ n ^ { { \\rm d e g } ( p ( x ) ) } \\end{align*}"}
+{"id": "8656.png", "formula": "\\begin{align*} s _ \\lambda ( x _ 1 , x _ 2 , \\dots ) = \\sum _ { T } { \\bf x } ^ { T } , \\end{align*}"}
+{"id": "4313.png", "formula": "\\begin{align*} a _ \\alpha = - a _ \\gamma \\frac { M ^ \\alpha \\left ( \\lambda _ 0 , \\ldots , \\lambda _ { s - 1 } \\right ) } { M \\left ( \\lambda _ 0 , \\ldots , \\lambda _ { s - 1 } \\right ) } , \\ \\ \\ \\ \\alpha \\neq a _ \\gamma , \\end{align*}"}
+{"id": "6637.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u = \\nabla \\cdot A ( \\nabla u ) + \\xi \\\\ u | _ { t \\leq 0 } = 0 \\end{cases} \\end{align*}"}
+{"id": "3978.png", "formula": "\\begin{align*} g ( x , y _ 1 , v _ \\rho ( y _ 1 ) ) - g ( x , y _ \\lambda , v _ \\rho ( y _ \\lambda ) ) & = g _ y ( x , y _ \\tau , v _ \\tau ) \\cdot ( y _ 1 - y _ \\lambda ) \\\\ & \\quad \\quad + g _ z ( x , y _ \\tau , v _ \\tau ) ( v _ \\rho ( y _ 1 ) - v _ \\rho ( y _ \\lambda ) ) , \\end{align*}"}
+{"id": "4236.png", "formula": "\\begin{align*} L _ \\Omega ( f ) : = \\Omega \\int _ { \\R ^ N } \\overline { f } ( x ) L _ z f ( x ) d x \\end{align*}"}
+{"id": "575.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } p _ t = \\frac { \\sigma _ t } { \\gamma } \\ , ( A ^ { \\rm T } A ) : C \\left ( \\sigma _ t - 2 p _ t \\right ) = \\sigma _ t \\ , ( A ^ { \\rm T } A ) : ( A + A ^ { \\rm T } ) ^ { - 1 } \\left ( \\sigma _ t - 2 p _ t \\right ) \\end{align*}"}
+{"id": "6511.png", "formula": "\\begin{align*} \\Big [ \\frac { 1 } { \\sqrt { k } l } \\underset { i = 1 } { \\overset { k } { \\sum } } \\Big ( \\underset { j = 1 } { \\overset { l } { \\sum } } B ^ j _ i - l p _ i \\Big ) ^ 2 \\Big ] & = \\frac { 1 } { k l ^ 2 } \\sum _ { i = 1 } ^ k \\Big [ \\Big ( \\sum _ { j = 1 } ^ l B _ i ^ j - l p _ i \\Big ) ^ 2 \\Big ] \\leq 1 / 8 . \\end{align*}"}
+{"id": "767.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { | g | _ S = n } \\frac { 1 } { \\# S _ n } \\varphi ( g ) = \\Lambda . \\end{align*}"}
+{"id": "499.png", "formula": "\\begin{align*} { \\textstyle \\sum _ { \\overline { K } _ 1 \\ni j = k } ^ { \\nu } } \\ , \\big \\| x ^ { j + 1 } \\ ! - \\ ! x ^ j \\big \\| \\le ( 1 / 2 ) { \\textstyle \\sum _ { j = k } ^ { \\nu } } \\ , \\Xi _ j + b a ^ { - 1 } \\varphi ( \\Phi ( x ^ { \\ell ( k ) } ) \\ ! - \\ ! \\Phi ^ * ) . \\end{align*}"}
+{"id": "3668.png", "formula": "\\begin{align*} A u - f + \\Pi _ { \\rho } u & = 0 . \\end{align*}"}
+{"id": "6881.png", "formula": "\\begin{align*} \\rho _ { p _ i } \\left ( u \\right ) & = \\iint _ { \\R ^ { 2 N } \\setminus ( \\mathcal { C } \\Omega ) ^ 2 } \\frac { | u ( x ) - u ( y ) | ^ { p _ i ( x , y ) } } { | x - y | ^ { N + s _ i ( x , y ) p _ i ( x , y ) } } \\ , d x \\ , d y + \\int _ { \\Omega } { | u | ^ { \\overline { p } _ i ( x ) } } \\ , d x \\\\ & + \\int _ { \\mathcal { C } \\Omega } { \\beta ( x ) } | u | ^ { \\overline { p } _ i ( x ) } \\ , d x . \\end{align*}"}
+{"id": "6513.png", "formula": "\\begin{align*} \\eta _ { p , l , k } ' : = \\frac { l - 1 } { 2 k } \\left ( \\underset { i = 1 } { \\overset { k } { \\sum } } ( p _ i - \\frac { 1 } { 2 } ) \\right ) ^ 2 \\geq c _ { \\alpha , n } \\end{align*}"}
+{"id": "7256.png", "formula": "\\begin{align*} b _ \\pm & = \\int _ { 0 } ^ { \\infty } j _ \\pm ( d x ) \\int _ { 0 } ^ { x } m _ \\pm ( y , \\infty ) d y p = \\frac { b _ + } { b _ + + b _ - } . \\end{align*}"}
+{"id": "6483.png", "formula": "\\begin{align*} \\chi _ g ( h _ 1 B ) - \\chi _ g ( - h _ 1 B ) = \\chi _ g ( B ) - \\chi _ g ( - B ) , ~ ~ \\mbox { f o r a l l $ g \\in G $ } . \\end{align*}"}
+{"id": "1107.png", "formula": "\\begin{align*} \\theta _ { i + 1 } = [ \\theta _ i - \\eta _ i Z _ i ] _ r , \\end{align*}"}
+{"id": "3985.png", "formula": "\\begin{align*} u _ 1 ( x ) = \\begin{cases} \\overline { u } ( x ) & \\overline { \\Omega } \\\\ \\sup \\{ g ( x , y _ b , z _ b ) ; y _ b = T ( x _ b ) \\in \\partial \\Omega ^ * x _ b \\in \\partial \\Omega \\} & \\Omega _ \\delta \\setminus \\overline { \\Omega } . \\end{cases} \\end{align*}"}
+{"id": "5929.png", "formula": "\\begin{align*} t \\geqslant T : c _ { 3 } U = \\alpha _ 1 u _ 1 + \\alpha _ 2 u _ 2 = D _ 2 ^ T u \\equiv 0 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "7591.png", "formula": "\\begin{align*} N = \\bigg \\| \\int _ { 0 } ^ { s } \\big [ \\big ( G ( t - r , 0 , \\cdot ) - G ( s - r , 0 , \\cdot ) \\big ) * v ^ { 2 \\delta + 1 } ( r , \\cdot ) \\big ] \\d r \\bigg \\| _ { \\L ^ p } . \\end{align*}"}
+{"id": "5388.png", "formula": "\\begin{align*} & \\ll e ^ { O ( p ) } K \\prod _ { \\substack { i = 1 \\\\ i \\ne i _ 0 } } ^ { 2 p } \\left ( \\frac { K } { e ^ { h _ i } } \\right ) \\exp \\bigg ( 6 \\pi \\sum _ { \\substack { 1 \\le i < j \\le 2 p \\\\ h _ i , h _ j \\ne \\log K } } \\frac { 1 } { \\Vert t _ { \\sigma ( j ) } - t _ { \\sigma ( i ) } \\Vert e ^ { \\max ( h _ i , h _ j ) } } \\bigg ) . \\end{align*}"}
+{"id": "916.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } T _ n = \\lim \\limits _ { n \\to \\infty } T _ { n + 1 } \\end{align*}"}
+{"id": "3873.png", "formula": "\\begin{align*} \\psi ( \\cdot , u , D u ) = \\frac { f ( \\cdot ) } { f ^ * ( Y ( \\cdot , u , D u ) ) } \\end{align*}"}
+{"id": "3051.png", "formula": "\\begin{align*} \\bar { A } : = \\int _ Y r A ; \\end{align*}"}
+{"id": "6713.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\mu ( n + a ) e ^ { i 2 \\pi u n / q } = \\sum _ { n \\leq x } \\mu ( n + a ) e ^ { i 2 \\pi n / q } + O \\left ( \\frac { x } { ( \\log x ) ^ { D } } \\right ) , \\end{align*}"}
+{"id": "7765.png", "formula": "\\begin{align*} - \\frac { \\widetilde h _ { \\alpha , \\theta } ^ \\prime ( s | \\mu ) } { \\widetilde h _ { \\alpha , \\theta } ( s | \\mu ) } & = \\mu \\dfrac { ( 1 + s ) ^ { \\alpha - 1 } } { s ^ { \\alpha - \\theta } } \\equiv \\mu \\ , \\rho _ { \\alpha , \\theta } ( s ) \\\\ \\implies \\widetilde h _ { \\alpha , \\theta } ( s | \\mu ) & = \\exp \\left \\{ - \\mu \\int _ 0 ^ s \\rho _ { \\alpha , \\theta } ( t ) d t \\right \\} \\end{align*}"}
+{"id": "1643.png", "formula": "\\begin{align*} \\pi ^ * \\left ( i ^ { m n ^ 2 } \\sigma \\wedge \\Bar { \\sigma } \\right ) ^ { 1 / m } = \\prod _ i | f _ i | ^ { 2 a _ i } d V \\end{align*}"}
+{"id": "7276.png", "formula": "\\begin{align*} U \\bullet f ( x ) = \\int _ { 0 } ^ { x } f ( y ) d U ( y ) , \\end{align*}"}
+{"id": "8502.png", "formula": "\\begin{align*} \\partial _ t C = C _ { \\xi ^ 2 } = a ( C - P ) + f ( \\xi ) C _ { \\xi } \\end{align*}"}
+{"id": "8240.png", "formula": "\\begin{align*} - \\Delta u + V ( | x | ) f ( u ) f ' ( u ) = K ( | x | ) g ( f ( u ) ) f ' ( u ) \\quad \\mathbb { R } ^ N \\backslash \\{ 0 \\} . \\end{align*}"}
+{"id": "7304.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } c ^ d _ { m _ n } ( \\lambda ) = 0 . \\end{align*}"}
+{"id": "4303.png", "formula": "\\begin{align*} \\eta ^ \\prime = \\eta - \\left ( \\frac { K } { q - 1 } + 2 M \\right ) \\nu ^ { - 1 } _ 0 > 0 . \\end{align*}"}
+{"id": "1368.png", "formula": "\\begin{align*} 0 < k < k _ { 0 } = \\frac { 2 \\pi l e ^ { 3 } } { c _ { p } } , \\end{align*}"}
+{"id": "5933.png", "formula": "\\begin{align*} u = ( \\widetilde E , \\widetilde C _ 1 U ) , { \\mathcal L } \\theta = - ( \\widetilde E , \\widetilde C _ 1 A U ) , { \\mathcal R } \\theta = - ( \\widetilde E , \\widetilde C _ 1 B U ) . \\end{align*}"}
+{"id": "437.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\right \\| = b \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "3384.png", "formula": "\\begin{align*} \\gamma ( [ 0 , u _ n - 1 ] ) & = 2 ^ { n - 1 } - 1 \\enspace , \\\\ \\gamma ( [ 0 , 2 u _ n - 1 ] ) & = 2 ^ n - 2 \\enspace , \\\\ \\gamma ( [ 0 , 2 u _ n ] ) & = \\gamma ( [ 0 , + \\infty [ ) = 2 ^ n - 1 \\enspace . \\end{align*}"}
+{"id": "7211.png", "formula": "\\begin{align*} \\omega w + w \\omega = - 2 \\frac { | x | ' } { | x ' | } . \\end{align*}"}
+{"id": "294.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\chi _ { \\alpha } } { \\chi ' _ { \\alpha } } ( \\iota _ { F _ { \\alpha } } \\alpha ( \\mathrm { N m } _ { S ( E ) / S ( F ) } t ) ) & = \\frac { \\chi _ { \\alpha } \\circ \\mathrm { N m } _ { E _ { \\alpha } / F _ { \\alpha } } } { \\chi ' _ { \\alpha } \\circ \\mathrm { N m } _ { E _ { \\alpha } / F _ { \\alpha } } } ( \\iota _ { E _ { \\alpha } } \\alpha ( t ) ) \\\\ & = \\frac { \\chi _ { \\mathrm { B C } , \\alpha } } { \\chi ' _ { \\mathrm { B C } , \\alpha } } ( \\iota _ { E _ { \\alpha } } \\alpha ( t ) ) , \\end{aligned} \\end{align*}"}
+{"id": "4345.png", "formula": "\\begin{align*} \\sigma ( x , y ) = ( x \\wedge y = 0 ) \\wedge ( x \\vee y = 1 ) \\end{align*}"}
+{"id": "8902.png", "formula": "\\begin{align*} \\vartheta _ { 2 } ( t ) = \\sum \\limits _ { j = 0 } ^ { + \\infty } \\left ( 2 j + 1 \\right ) e ^ { - \\left ( j + \\frac { 1 } { 2 } \\right ) ^ { 2 } t } \\end{align*}"}
+{"id": "6327.png", "formula": "\\begin{align*} \\sum _ { x = 1 } ^ { | l _ { i } | } \\sum _ { y = 1 } ^ { | l _ { j } | } \\Big ( \\ 1 _ { e _ { y } = e _ { x } \\in E ^ + } J _ s ^ { ( 1 ) } + \\ 1 _ { e _ { y } ^ { - 1 } = e _ { x } ^ { - 1 } \\in E ^ + } J _ s ^ { ( 2 ) } + \\ 1 _ { e _ { y } ^ { - 1 } = e _ { x } \\in E ^ + } J _ s ^ { ( 3 ) } + \\ 1 _ { e _ { y } = e _ { x } ^ { - 1 } \\in E ^ + } J _ s ^ { ( 4 ) } \\Big ) \\end{align*}"}
+{"id": "3468.png", "formula": "\\begin{align*} \\widetilde { f } _ n ( \\cdot , w , z , x ; t ) = \\frac { 1 } { n ( n - 1 ) } \\sum _ { i , j = 1 , i \\not = j } ^ { n } h _ { i j } ^ { ( n ) } ( \\cdot , w , z , x ; t ) , \\end{align*}"}
+{"id": "4413.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 3 0 } = \\frac { 8 } { 1 5 } < \\frac { 3 1 } { 5 8 } = \\frac { 1 } { 2 } + \\frac { 1 } { 2 9 } \\end{align*}"}
+{"id": "7868.png", "formula": "\\begin{align*} \\lambda ( f ) = \\rho ( f ) = 1 + \\frac { 1 } { 2 } \\deg ( a _ 0 ) , \\end{align*}"}
+{"id": "8516.png", "formula": "\\begin{align*} \\omega = - Q ( z , w ) d z + P ( z , w ) d w . \\end{align*}"}
+{"id": "4468.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n u ' _ i & = \\left ( \\sum _ { \\substack { i = 1 \\\\ i \\neq \\ell - 1 , \\ell } } ^ n u ' _ i \\right ) + u ' _ { \\ell - 1 } + u ' _ { \\ell } \\\\ & < \\left ( \\sum _ { \\substack { i = 1 \\\\ i \\neq \\ell - 1 , \\ell } } ^ n u ' _ i \\right ) + u _ { \\ell - 1 } + u _ { \\ell } = \\sum _ { i = 1 } ^ n u _ i . \\end{align*}"}
+{"id": "1827.png", "formula": "\\begin{align*} \\lambda : = \\sup \\{ t \\ > 0 : D + t ( K _ X + \\Delta ) \\} . \\end{align*}"}
+{"id": "9211.png", "formula": "\\begin{align*} B = \\Gamma . \\end{align*}"}
+{"id": "6480.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma ( \\lambda _ g ) & = \\sigma ( \\chi _ g ( A ) ) = \\sigma \\left ( \\sum _ { a \\in A } \\prod _ { i = 1 } ^ r \\zeta _ { n _ i } ^ { g _ i a _ i } \\right ) = \\sum _ { a \\in A } \\prod _ { i = 1 } ^ r \\sigma ( \\zeta _ { n _ i } ^ { g _ i a _ i } ) \\\\ & = \\sum _ { a \\in A } \\prod _ { i = 1 } ^ r \\zeta _ { n _ i } ^ { \\eta ( \\sigma ) g _ i a _ i } = \\chi _ g ( \\eta ( \\sigma ) A ) \\end{aligned} \\end{align*}"}
+{"id": "233.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\tau ^ 2 - \\beta \\tau \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "5777.png", "formula": "\\begin{align*} r a n k ( \\widetilde C _ q D ) = r a n k ( D \\widetilde C _ q ^ T ) < N - q = r a n k ( \\widetilde C _ q ^ T ) . \\end{align*}"}
+{"id": "4673.png", "formula": "\\begin{align*} \\sum _ { m = n } ^ { \\infty } \\sum _ { k = n } ^ { \\min \\{ \\ell , m \\} } A _ { m , k } = \\sum _ { k = n } ^ { \\ell } \\sum _ { m = k } ^ { \\ell } A _ { m , k } + \\sum _ { k = n } ^ { \\ell } \\sum _ { m = \\ell + 1 } ^ { \\infty } A _ { m , k } = \\sum _ { k = n } ^ { \\ell } \\sum _ { m = k } ^ { \\infty } A _ { m , k } , \\end{align*}"}
+{"id": "1556.png", "formula": "\\begin{align*} & t _ { 1 } ^ { p ^ { + } - r ^ { - } } \\int _ { M } | D \\mathrm { u } ( z ) | ^ { p ( z ) } \\ , \\ , d v _ { g } ( z ) + t _ { 1 } ^ { q ^ { + } - r ^ { - } } \\int _ { M } \\mu ( z ) \\ , | D \\mathrm { u } ( z ) | ^ { q ( z ) } \\ , \\ , d v _ { g } ( z ) \\\\ & - t _ { 1 } ^ { - r ^ { - } - \\gamma ^ { - } + 1 } \\int _ { M } g ( z ) | \\mathrm { u } ( z ) | ^ { 1 - \\gamma ( z ) } \\ , \\ , d v _ { g } ( z ) = \\lambda \\int _ { M } | \\mathrm { u } ( z ) | ^ { r ( z ) } \\ , \\ , d v _ { g } ( z ) , \\end{align*}"}
+{"id": "2040.png", "formula": "\\begin{align*} & C ( h ) : = \\displaystyle \\sum _ { n = 1 } ^ { + \\infty } \\mathbb P [ S _ n = h , \\min _ { 1 \\leq i \\leq n } S _ i > 0 ] , \\\\ & C ' ( - h ) : = \\displaystyle \\sum _ { n = 1 } ^ { + \\infty } \\mathbb P [ S _ n ' = - h , \\min _ { 1 \\leq j \\leq n } S _ j ' > 0 ] . \\end{align*}"}
+{"id": "729.png", "formula": "\\begin{align*} z = \\begin{cases} z & z \\in \\Omega \\cup \\partial \\Omega , \\\\ \\overline { S ( { \\tilde { z } } ) } \\quad & \\tilde { z } \\in \\tilde { \\Omega } , \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "583.png", "formula": "\\begin{align*} \\mathbb { X } _ { t _ n , t _ { n + 1 } } ^ \\dagger : = \\int _ { t _ n } ^ { t _ { n + 1 } } ( X ^ \\dagger _ t - X ^ \\dagger _ { t _ n } ) \\otimes { \\rm d } X _ t ^ \\dagger . \\end{align*}"}
+{"id": "6056.png", "formula": "\\begin{align*} \\frac { p ( x ) } { x - z _ l } - \\sum _ { i = 1 } ^ g l _ i ( x ) \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac { p ( y ) } { y - z _ l } \\frac { \\dd y } { w ( y ) } \\equiv 0 \\end{align*}"}
+{"id": "5746.png", "formula": "\\begin{align*} D \\mu ^ { A A ^ { \\prime } } & = 0 , \\\\ \\nu ^ { A A ^ { \\prime } } & = i ( \\omega _ { B } ^ { A } \\mu ^ { B A ^ { \\prime } } - \\omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\mu ^ { A B ^ { \\prime } } ) \\\\ \\Omega _ { B } ^ { A } \\mu ^ { B A ^ { \\prime } } - \\Omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\mu ^ { A B ^ { \\prime } } & = 0 , \\end{align*}"}
+{"id": "3861.png", "formula": "\\begin{align*} A _ n - F _ { n + 1 } ^ 2 = \\sum _ { k = 3 } ^ { n } \\sum _ { l = 3 } ^ k ( \\mu _ l + \\delta _ { l , 3 } - 2 ) A _ { k - l } F ^ 2 _ { n - k + 1 } . \\end{align*}"}
+{"id": "4437.png", "formula": "\\begin{align*} a _ 2 = \\left ( \\frac { 1 } { 3 } + \\frac { 1 } { 3 } - \\frac { 1 } { 2 } \\right ) ^ { - 1 } = 6 \\end{align*}"}
+{"id": "2401.png", "formula": "\\begin{align*} \\Phi ' = U \\left [ \\begin{array} { c } 0 \\\\ I _ { a } \\end{array} \\right ] \\end{align*}"}
+{"id": "5107.png", "formula": "\\begin{align*} \\left [ \\partial _ { t } \\gamma ( t , \\theta ) - \\mathbf { v } ( t , \\gamma ( t , \\theta ) ) \\right ] \\cdot \\mathbf { n } ( t , \\gamma ( t , \\theta ) ) = 0 , \\end{align*}"}
+{"id": "2529.png", "formula": "\\begin{align*} & \\tilde \\varrho _ N ^ \\varphi ( x ) = \\frac 1 N \\sum _ { j = 1 } ^ N \\varphi ( x - y _ j ) \\ , , \\tilde \\varrho _ N ^ \\varphi \\tilde u _ N ^ \\varphi ( x ) = \\frac 1 N \\sum _ { j = 1 } ^ N \\varphi ( x - y _ j ) w _ j \\ , , \\\\ & \\tilde \\varrho _ N ^ \\varphi \\tilde T _ N ^ \\varphi ( x ) = \\frac 1 { N d } \\sum _ { j = 1 } ^ N \\varphi ( x - y _ j ) | w _ j - \\tilde u _ N ^ \\varphi ( x ) | ^ 2 \\ , , \\end{align*}"}
+{"id": "3706.png", "formula": "\\begin{align*} \\sigma _ i ( h ) : = \\mathcal { F } _ 2 \\circ L ( h ) \\circ \\mathcal { F } _ 2 ^ { - 1 } . \\end{align*}"}
+{"id": "8541.png", "formula": "\\begin{align*} S ( n - i + 1 ) - S ( n ) & = \\frac { 1 } { 6 } ( ( n - i + 1 ) ^ 3 - ( n - i + 1 ) - n ^ 3 + n ) \\\\ & = \\frac { 1 } { 6 } ( ( n - i + 1 - n ) ( ( n - i + 1 ) ^ 2 + n ( n - i + 1 ) + n ^ 2 ) + i - 1 ) \\\\ & = \\frac { 1 } { 6 } ( - ( i - 1 ) ( n ^ 2 - 2 ( i - 1 ) n + ( i - 1 ) ^ 2 + n ^ 2 - ( i - 1 ) n + n ^ 2 ) + i - 1 ) \\\\ & = \\frac { 1 } { 6 } ( - 3 ( i - 1 ) n ^ 2 + 3 ( i - 1 ) ^ 2 n - ( i - 1 ) ^ 3 + i - 1 ) . \\end{align*}"}
+{"id": "438.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\right \\| \\leq b \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "3522.png", "formula": "\\begin{align*} \\mathfrak { t } = \\mathfrak { t } ^ + \\oplus \\mathfrak { h } \\oplus \\mathfrak { t } ^ - , \\end{align*}"}
+{"id": "3855.png", "formula": "\\begin{align*} T _ { 2 ( n + 1 ) + j } ^ 2 = 1 + \\sum _ { k = 1 } ^ n \\left \\{ T _ { 2 k + j + 1 } ^ 2 + 2 T _ { 2 k + j } ^ 2 + 9 T _ { 2 k + j - 1 } ^ 2 + 4 \\ ! \\ ! \\ ! \\sum _ { i = 0 } ^ { 2 k + j - 4 } ( T _ { 2 k + j - i } + T _ { 2 k + j - i - 1 } ) T _ { i + 2 } ^ 2 \\right \\} . \\end{align*}"}
+{"id": "5701.png", "formula": "\\begin{align*} \\mathbf { L } _ { b } ^ { a } = \\delta _ { b } ^ { a } - \\frac { 1 } { 2 } ( \\alpha _ { b } ^ { a } + i \\lambda _ { 2 b } ^ { a } ) \\mathbf { m - } \\frac { 1 } { 2 } ( \\alpha _ { b } ^ { a } - i \\lambda _ { 2 b } ^ { a } ) \\overline { \\mathbf { m } } + \\frac { 1 } { 2 } ( \\beta _ { b } ^ { a } - \\overline { \\beta } _ { b } ^ { a } - i d \\lambda _ { 2 b } ^ { a } - i \\lambda _ { 2 c } ^ { a } \\alpha _ { b } ^ { c } ) \\mathbf { m } \\overline { \\mathbf { m } } , \\end{align*}"}
+{"id": "8991.png", "formula": "\\begin{align*} \\| \\nabla & ( d i s t _ N ( u ) - d i s t _ N ( v ) ) \\| ^ 2 _ { L ^ 2 ( B ) } \\\\ & \\le C \\int _ B ( | w | ^ 2 | \\nabla u | ^ 2 + ( | \\nabla u | + | \\nabla v | ) | \\nabla w | | w | ) d z \\\\ & \\le \\varepsilon \\| \\nabla w \\| ^ 2 _ { L ^ 2 ( B ) } + C ( \\varepsilon ) \\| w \\| ^ 2 _ { L ^ 4 ( B ) } ( \\| \\nabla u \\| ^ 2 _ { L ^ 4 ( B ) } + \\| \\nabla v \\| ^ 2 _ { L ^ 4 ( B ) } ) . \\end{align*}"}
+{"id": "4625.png", "formula": "\\begin{align*} \\frac { 1 } { x _ 2 } < \\theta - \\frac { 1 } { k } < \\frac { 1 } { k - 1 } - \\frac { 1 } { k } = \\frac { 1 } { k ( k - 1 ) } \\end{align*}"}
+{"id": "8186.png", "formula": "\\begin{align*} u ( t ) & = u _ 0 + \\int _ 0 ^ t k ( t , s ) L ^ f u ( s ) d s , t > 0 \\\\ \\lim _ { t \\searrow 0 } u ( t ) & = u _ 0 . \\end{align*}"}
+{"id": "7200.png", "formula": "\\begin{align*} q = z _ 0 + \\overline \\zeta \\ , k \\qquad \\mbox { w h e r e } \\zeta = x _ 3 + x _ 2 i , \\end{align*}"}
+{"id": "4293.png", "formula": "\\begin{align*} \\langle \\mathrm { d i v } \\ , \\widetilde { \\textbf { \\textit { u } } } , \\phi \\rangle _ { \\mathcal { D ' } ( \\R ^ 3 ) \\times \\mathcal { D } ( \\R ^ 3 ) } = \\displaystyle \\int _ { \\Omega } \\chi \\ , \\phi d \\textbf { \\textit { x } } . \\end{align*}"}
+{"id": "6849.png", "formula": "\\begin{align*} \\mathcal P ( \\tilde Y _ i , \\tilde Y _ j ) = F _ { \\mathcal M } \\ ; w ^ T _ { i , j } \\ , \\tilde X _ i ^ { \\dag } \\ , \\tilde X _ j \\ ; F _ { \\mathcal M } , \\mathcal M = F _ { \\tilde X _ i } \\ ; w _ { i , j } , w _ { i , j } : = w ( Y _ i , Y _ j ) , \\end{align*}"}
+{"id": "1145.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\binom { n } { i } p ^ { i - 1 - \\nu _ p ( n ) } v _ 1 ^ { n - i } t _ 1 ^ { i } . \\end{align*}"}
+{"id": "4836.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } d _ 0 F ( r ) + g ( t ) + R _ 1 ( r , t ) = \\xi \\\\ f ( t ) + R _ 2 ( r , t ) = \\tau , \\end{array} \\right . \\end{align*}"}
+{"id": "7349.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\infty } \\int _ { 0 } ^ { 1 } G ^ { d } _ { m _ \\gamma } ( x ) d m _ \\gamma ( x ) = 0 \\end{align*}"}
+{"id": "4268.png", "formula": "\\begin{align*} A _ \\lambda ^ { - 2 } = \\frac { 1 } { c } \\int _ { \\R ^ N } \\varphi ^ 2 ( \\lambda ^ { - 1 } x ) Q _ 0 ( x ) d x . \\end{align*}"}
+{"id": "7954.png", "formula": "\\begin{align*} ( m + \\ell _ 2 ) + \\ell _ 3 = x + \\ell _ 3 + ( y + \\ell _ 1 ) = m ' + \\ell _ 4 + ( y + \\ell _ 1 ) = y + ( m ' + \\ell _ 1 + \\ell _ 4 ) . \\end{align*}"}
+{"id": "15.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { E } \\int _ 0 ^ \\infty e ^ { - \\beta t } \\Big ( | \\hat { m } | ^ 2 + | \\hat { \\tilde { m } } | ^ 2 + | | \\hat { n } | | ^ 2 + | \\hat { q } | ^ 2 \\Big ) d t \\leq \\frac { 1 } { \\delta } \\mathbb { E } \\int _ 0 ^ \\infty e ^ { - \\beta t } | \\kappa ^ i _ t - \\kappa ^ j _ t | ^ 2 d t \\xrightarrow { i , j \\rightarrow \\infty } 0 . \\end{aligned} \\end{align*}"}
+{"id": "4885.png", "formula": "\\begin{align*} C _ { \\tau s _ 3 } C _ { s _ 0 s _ 2 x _ l } = C _ { x _ { l + 1 } } + C _ { x _ { l - 1 } } + \\Box . \\end{align*}"}
+{"id": "4396.png", "formula": "\\begin{align*} u _ n ( \\theta ) = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : \\left ( x _ i \\right ) _ { i = 1 } ^ n \\in U _ n ( \\theta ) x _ i < x _ i ^ * i = 1 , \\ldots , k \\right \\} . \\end{align*}"}
+{"id": "2771.png", "formula": "\\begin{align*} a _ { ( 1 ) } \\cdot S ( a _ { ( 2 ) } ) = \\varepsilon ( a ) = S ( a _ { ( 1 ) } ) \\cdot a _ { ( 2 ) } . \\end{align*}"}
+{"id": "7957.png", "formula": "\\begin{align*} \\beta ( x ) = \\beta ( y ) . \\end{align*}"}
+{"id": "6649.png", "formula": "\\begin{align*} \\partial _ t \\delta _ y w - \\nabla \\cdot a ( t ' , x ' ) \\nabla \\delta _ y w = \\nabla \\cdot g \\end{align*}"}
+{"id": "1816.png", "formula": "\\begin{align*} \\ddot { u } = - \\alpha \\frac { u } { \\lvert u \\rvert ^ { ( 2 + \\alpha ) } } , \\end{align*}"}
+{"id": "6753.png", "formula": "\\begin{align*} z ( 1 - s ) = \\lim _ { N \\rightarrow \\infty } z _ N ( 1 - s ) = \\sqrt { \\frac { 1 6 } { \\pi } } ( 1 - s ) \\left [ \\int _ { 0 } ^ { 1 } y ^ { \\frac { 1 - s } { 2 } - 1 } \\Psi ( y ) d y + \\frac { 1 } { 1 - s } \\right ] . \\end{align*}"}
+{"id": "1441.png", "formula": "\\begin{align*} \\eta ^ * _ t = _ d ( U _ { t - 1 } - \\mathbb { 1 } _ { \\{ R _ { t - 1 } = 0 \\} } , 1 - e ^ { - J _ t / ( \\lambda n ) } ) , \\end{align*}"}
+{"id": "5765.png", "formula": "\\begin{align*} t _ 1 - x _ 1 = 0 , \\cdots , t _ { r _ 0 } - x _ { r _ 0 } = 0 , t _ { r _ 0 + 1 } - f _ { r _ 0 + 1 } = 0 , \\cdots , t _ r - f _ r = 0 , \\end{align*}"}
+{"id": "373.png", "formula": "\\begin{align*} \\mathcal { J } : = \\begin{cases} ( \\{ 1 , 2 , \\ldots , n \\} \\setminus \\mathcal { I } ) \\cup { \\{ n _ 2 \\} } & \\textrm { i f $ n _ 2 - n _ 1 $ i s a n e v e n n u m b e r } \\ , , \\\\ \\{ 1 , 2 , \\ldots , n \\} \\setminus \\mathcal { I } & \\textrm { i f $ n _ 2 - n _ 1 $ i s a n o d d n u m b e r } \\ , . \\end{cases} \\end{align*}"}
+{"id": "1105.png", "formula": "\\begin{align*} R ( \\cdot ) = \\mathbb { E } _ { X \\sim P } \\{ | X - \\cdot | \\} \\end{align*}"}
+{"id": "5451.png", "formula": "\\begin{align*} b ^ n ( t , x , \\mu , \\alpha ) & : = b ( t , \\phi _ n ( x ) , \\mu \\circ \\phi ^ { - 1 } _ n , \\alpha ) , \\\\ \\sigma ^ n ( t , x , \\mu , \\alpha ) & : = \\sigma ( t , \\phi _ n ( x ) , \\mu \\circ \\phi ^ { - 1 } _ n , \\alpha ) . \\end{align*}"}
+{"id": "4095.png", "formula": "\\begin{align*} k _ t = - B _ t - i _ { X _ t } H _ 0 + d ( i _ { X _ t } b ) . \\end{align*}"}
+{"id": "6640.png", "formula": "\\begin{align*} \\delta _ y f ( t , x ) : = f ( t , x + y ) - f ( t , x ) \\end{align*}"}
+{"id": "9195.png", "formula": "\\begin{align*} \\sum _ { x \\in B } \\nabla ^ { e _ 1 } \\varphi ( x ) \\nabla ^ { e _ 1 } \\xi _ o ( x ) = - \\sum _ { x \\in l _ 3 } \\xi _ o ( x ) \\nabla ^ { - e _ 1 } \\varphi ( x ) - \\sum _ { x \\in l _ 1 } \\xi _ o ( x + e _ 1 ) \\nabla ^ { e _ 1 } \\varphi ( x ) + \\sum _ { x \\in B } \\xi _ o ( x ) \\nabla ^ { e _ 1 } \\nabla ^ { - e _ 1 } \\varphi ( x ) . \\end{align*}"}
+{"id": "1803.png", "formula": "\\begin{align*} L _ { \\dot { u } } ( 0 , u _ 0 ( 0 ) , \\dot { u } _ 0 ( 0 ) ) = L _ { \\dot { u } } ( T , u _ 0 ( T ) , \\dot { u } _ 0 ( T ) ) , \\end{align*}"}
+{"id": "955.png", "formula": "\\begin{align*} \\textrm { G } _ C \\textrm { - d i m } _ R ( \\lambda M \\otimes _ R C ) \\ , = \\ , \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { T r } _ C ( M ) ) - 1 . \\end{align*}"}
+{"id": "4788.png", "formula": "\\begin{align*} \\Pr _ { c \\sim \\mathcal { D } ( C ) } \\Big [ | c | = \\alpha N \\Big ] & = q ^ { - \\eta N } \\cdot \\binom { \\eta N } { ( 1 - \\alpha ) \\eta N } ( q - 1 ) ^ { \\alpha \\eta N } \\\\ & \\geq q ^ { - \\eta N } \\cdot \\sqrt { \\frac { 8 \\pi } { e ^ 4 \\eta N } } \\cdot 2 ^ { h ( \\alpha ) \\eta N } \\cdot q ^ { \\alpha \\eta N \\log _ q ( q - 1 ) } \\\\ & = \\sqrt { \\frac { 8 \\pi } { e ^ 4 \\eta N } } \\cdot q ^ { - ( 1 - h _ q ( \\alpha ) ) \\eta N } \\end{align*}"}
+{"id": "8252.png", "formula": "\\begin{align*} P ( G ) = p ^ M ( 1 - p ) ^ { \\binom { n } { 2 } - M } . \\end{align*}"}
+{"id": "1647.png", "formula": "\\begin{align*} \\chi \\mid _ { T ( F _ \\infty ) } ( t ) = \\chi _ 0 ( t ) \\prod _ { \\tilde \\sigma : F \\hookrightarrow \\C } t _ { \\tilde \\sigma } ^ { m _ { \\tilde \\sigma } } , \\underline { m } = ( m _ { \\tilde \\sigma } ) \\in \\Z ^ { [ F : \\Q ] } , \\end{align*}"}
+{"id": "1834.png", "formula": "\\begin{align*} D '^ 2 \\cdot \\omega _ 0 & = ( D + \\epsilon _ 0 \\Delta ) ^ 2 \\cdot \\omega _ 0 \\\\ & = D \\cdot ( D + \\epsilon _ 0 \\Delta ) \\cdot \\omega _ 0 + \\epsilon _ 0 ( D + \\epsilon _ 0 \\Delta ) \\cdot \\Delta \\cdot \\omega _ 0 \\\\ & = \\underbrace { D ^ 2 \\cdot \\omega _ 0 } _ { > 0 } + \\epsilon _ 0 \\underbrace { ( D \\cdot \\omega _ 0 \\cdot \\Delta ) } _ { \\ > 0 } + \\epsilon _ 0 \\underbrace { ( D ' \\cdot \\omega _ 0 \\cdot \\Delta ) } _ { \\ > 0 } \\\\ & > 0 . \\end{align*}"}
+{"id": "7332.png", "formula": "\\begin{align*} N ( \\gamma ) \\sim \\left \\{ \\begin{aligned} & \\frac { \\alpha } { ( \\alpha - 1 ) ( \\alpha - 2 ) } \\int _ { 1 } ^ { \\gamma } \\frac { K ( x ) ^ 2 L ( x ) } { x } d x & ( \\alpha > 2 ) , \\\\ & \\int _ { 1 } ^ { \\gamma } \\frac { K ( x ) ^ 2 } { x } d x \\int _ { 1 } ^ { x } \\frac { L ( y ) } { y } d y & ( \\alpha = 2 ) , \\end{aligned} \\right . ( \\gamma \\to \\infty ) . \\end{align*}"}
+{"id": "2914.png", "formula": "\\begin{align*} \\| \\mathcal { B } f \\| _ 0 = \\sup _ { k \\in \\mathbb { N } } \\sup _ { 0 < r < 1 } \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | ( \\mathcal { B } _ k f ) _ { [ r ] } | ) d m _ \\infty . \\end{align*}"}
+{"id": "5057.png", "formula": "\\begin{align*} F = \\sum _ { I \\subseteq [ 1 , l ] } \\frac { P _ I } { Q _ I } , \\end{align*}"}
+{"id": "268.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = & \\alpha ( w / u ) + u ( 1 + \\beta v ) , \\psi _ 2 ( u , v , w ) = \\alpha ( w / u ) + v ( 1 + \\beta u ) , \\\\ \\psi _ 3 ( u , v , w ) = & \\alpha ( w / v ) + v ( 1 + \\beta u ) \\psi _ 4 ( v , w ) = \\alpha ( w / v ) + u ( 1 + \\beta v ) . \\end{align*}"}
+{"id": "1332.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j , k \\leq n } f _ j ( \\tau _ k ) ^ m f _ k ( \\tau _ j ) ^ m & = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n f _ j ( \\tau _ k ) ^ m f _ k ( \\tau _ j ) ^ m \\geq \\frac { 1 } { { d + m - 1 \\choose m } } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) ^ m \\right ) ^ 2 \\\\ & = \\frac { 1 } { ( ^ m ( \\mathcal { X } ) ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) ^ m \\right ) ^ 2 . \\end{align*}"}
+{"id": "2092.png", "formula": "\\begin{align*} \\dim _ H ( \\mathcal { S } _ K ) \\leq \\frac { 2 } { 3 } \\dim ( K ) = \\frac { 4 \\log 2 } { 3 \\log 3 } . \\end{align*}"}
+{"id": "7871.png", "formula": "\\begin{align*} f ( z ) = S ( z ) + Q ( z ) e ^ { P ( z ) } \\quad a _ \\mu ( z ) = R ( z ) e ^ { - P ( z ) } , \\end{align*}"}
+{"id": "538.png", "formula": "\\begin{align*} \\Z _ 3 , \\ ; \\Z _ 9 , \\ ; \\Z _ 3 \\times \\Z _ 3 , \\mbox { o r } \\Z _ 3 \\times \\Z _ 9 & \\mbox { i f $ p = 3 $ } \\\\ \\Z _ 7 , \\mbox { o r } \\Z _ 7 \\times \\Z _ 7 & \\mbox { i f $ p = 7 $ } \\end{align*}"}
+{"id": "5622.png", "formula": "\\begin{align*} _ { t } ^ { C F } I _ { t _ { 0 } } ^ { \\alpha } f ( t ) = \\dfrac { 2 ( 1 - \\alpha ) } { B ( \\alpha ) ( 2 - \\alpha ) } f ( t ) + \\dfrac { 2 \\alpha } { B ( \\alpha ) ( 2 - \\alpha ) } \\ , _ { t } ^ { R L } I _ { t _ { 0 } } ^ { 1 } f ( t ) . \\end{align*}"}
+{"id": "7220.png", "formula": "\\begin{align*} \\frac { 1 } { R ^ 2 } \\int _ 0 ^ { \\pi / 2 } \\big \\langle p j , \\ , p ' \\big \\rangle d \\vartheta \\ , = \\ , \\frac { \\pi } { 2 } . \\end{align*}"}
+{"id": "2501.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty A _ k ^ 2 \\int _ { \\tau _ k } | \\varphi _ 1 | ^ 2 { \\mathrm d } m < \\infty \\end{align*}"}
+{"id": "5019.png", "formula": "\\begin{align*} g = \\mu ^ 2 e ^ 1 \\otimes e ^ 1 + a ^ 2 \\left ( e ^ 2 \\otimes e ^ 2 + e ^ 3 \\otimes e ^ 3 \\right ) + b ^ 2 \\left ( e ^ 4 \\otimes e ^ 4 + e ^ 5 \\otimes e ^ 5 \\right ) , \\end{align*}"}
+{"id": "4623.png", "formula": "\\begin{align*} ( 0 , 1 ] = \\bigcup _ { k = 1 } ^ { \\infty } \\left ( \\frac { 1 } { k + 1 } , \\frac { 1 } { k } \\right ] \\end{align*}"}
+{"id": "8892.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { C } } \\mathcal { A } _ { m } ^ { \\nu } = \\frac { 2 m + n + 2 \\nu } { n ! ( n - 1 ) ! } \\prod \\limits _ { j = 1 } ^ { n - 1 } ( m + j ) ( m + 2 \\nu + j ) . \\end{align*}"}
+{"id": "7401.png", "formula": "\\begin{align*} X _ { \\alpha } ( \\chi ) : = \\chi ( h _ { \\alpha } ^ { \\vee } ) \\end{align*}"}
+{"id": "8596.png", "formula": "\\begin{align*} f ( x ) : = \\frac { F ( x ) } { d ( x ) } \\mbox { , w h e r e } d ( x ) : = \\gcd ( c _ { \\mathcal { H } } ( x ) , F ( x ) ) , \\end{align*}"}
+{"id": "2554.png", "formula": "\\begin{align*} v _ { \\mathbf { y } } : = \\left \\{ \\prod _ { i = 1 } ^ { p - 1 } z _ { c _ { i - 1 } } ( k _ i + \\mathbf { y } _ { k _ i - c _ { i } } ^ { ( i ) } ) \\right \\} z _ { c _ { p - 1 } } ( k _ p + \\mathbf { y } _ { k _ p - 2 } ^ { ( p ) } ) , \\end{align*}"}
+{"id": "3032.png", "formula": "\\begin{align*} \\Omega _ { 1 / { 1 2 } } \\subset \\bigcup \\limits _ { \\widetilde Q _ l \\subset \\Omega _ { 1 / 4 } } Q _ l = \\bigcup _ { j = 0 } ^ { s _ 0 } \\bigcup _ { s = s _ 0 - j - 6 } ^ { { \\infty } } \\mathcal { F } ^ { s , j } _ { Q _ k } . \\end{align*}"}
+{"id": "5626.png", "formula": "\\begin{align*} H ( t ) & = \\left [ E _ \\alpha \\left [ - \\dfrac { \\alpha } { 1 - \\alpha } ( t - x ) ^ { \\alpha } \\right ] w ( x , t ) \\right ] ^ { x = t } _ { x = t _ 0 } \\\\ & - \\int ^ { t } _ { t _ { 0 } } \\dfrac { \\alpha ^ 2 ( t - x ) ^ { \\alpha - 1 } } { 1 - \\alpha } E _ { \\alpha , \\alpha + 1 } ^ { 2 } \\left [ - \\dfrac { \\alpha } { 1 - \\alpha } ( t - x ) ^ { \\alpha } \\right ] w ( x , t ) d x . \\end{align*}"}
+{"id": "4282.png", "formula": "\\begin{align*} j = \\begin{cases} [ m - ( 3 / p + k ) ] \\quad \\quad \\mathrm { i f } 3 / p + k \\notin \\Z ^ { - } , \\\\ m - 3 / p - k - 1 \\qquad \\mathrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "986.png", "formula": "\\begin{align*} h ( \\delta _ x ) & = \\underset { n } { \\lim } \\ , h _ n ( \\delta _ x ) \\\\ & = \\underset { n } { \\lim } f _ n ( x ) \\\\ & = f ( x ) = \\overline { f } ( \\delta _ x ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; ( x \\in X ) , \\end{align*}"}
+{"id": "6318.png", "formula": "\\begin{align*} \\inf \\{ \\| 1 / x - p ( x ) \\| _ { [ a , b ] } : p \\in \\Pi _ n \\} = \\frac { ( 1 + \\sqrt { r } ) ^ 2 } { b } \\rho ^ { n + 1 } . \\end{align*}"}
+{"id": "3201.png", "formula": "\\begin{align*} L ( E _ { r + 2 } ( \\psi , \\tau ) , g _ { \\ell } , h _ m , 2 + \\frac { r + \\ell + m } { 2 } ) = L ( g _ { \\ell } , h _ m \\times \\psi , 2 + \\frac { r + \\ell + m } { 2 } ) ^ 2 , \\end{align*}"}
+{"id": "3328.png", "formula": "\\begin{align*} \\dd s ^ 2 _ { M } ( z , \\bar { z } ) = \\sum _ { \\alpha , \\beta = 1 } ^ n h _ { \\alpha \\bar { \\beta } } \\dd z _ { \\alpha } \\otimes \\dd \\bar { z } _ { \\beta } . \\end{align*}"}
+{"id": "7020.png", "formula": "\\begin{align*} \\underline { M } ( C _ 2 / e ) = E _ m = [ C _ 2 / e _ + \\wedge S ^ { m } , E _ { C _ 2 } ] ^ { C _ 2 } , \\end{align*}"}
+{"id": "451.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { m } a _ k ^ 2 \\leq \\sum _ { k = 1 } ^ { m } \\lambda _ k \\sum _ { j = 1 } ^ { n } a _ j ^ 2 = \\sum _ { k = 1 } ^ { d } \\lambda _ k . \\end{align*}"}
+{"id": "4899.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\partial _ s u ( s , t ) + J _ t ( u ( s , t ) ) \\Big ( \\partial _ t u ( s , t ) - \\tau ( s ) X _ H ( u ( s , t ) ) \\Big ) = 0 \\\\ \\partial _ s \\tau ( s ) + \\int _ 0 ^ 1 H ( u ( s , t ) d t = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "2202.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\emph { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m - 1 } n + 5 ^ { 2 m - 1 } \\right ) } q ^ n & = \\gamma \\sum _ { i = 1 } ^ \\infty x _ { 2 m - 1 , i } \\xi ^ { i - 1 } , \\\\ \\sum _ { n = 0 } ^ \\infty \\overline { \\emph { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m } n + 5 ^ { 2 m } \\right ) } q ^ n & = \\delta \\sum _ { i = 1 } ^ \\infty x _ { 2 m , i } \\xi ^ { i - 1 } , \\end{align*}"}
+{"id": "4959.png", "formula": "\\begin{align*} \\sum _ { \\substack { n _ 0 + \\cdots + n _ r = n - r \\\\ m _ 0 + \\cdots + m _ r = m - 1 } } D _ { n _ 0 , m _ 0 } \\cdots D _ { n _ r , m _ r } . \\end{align*}"}
+{"id": "4436.png", "formula": "\\begin{align*} 3 = x _ 1 \\leq x _ 2 < 6 \\end{align*}"}
+{"id": "5038.png", "formula": "\\begin{align*} \\frac { d } { d t } ( M _ t , A _ t , A _ t ) = f ( M _ t , A _ t , B _ t ) , \\end{align*}"}
+{"id": "9260.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots v _ { i } , \\ldots , v _ { h ( i ) } , N v _ i , \\ldots \\widehat { v _ { r } } , \\ldots v _ n ) } \\end{aligned} \\end{align*}"}
+{"id": "3380.png", "formula": "\\begin{align*} ( t ^ { - u _ { i + 1 } + 1 } - t ^ { - u _ { i + 1 } } ) x _ i \\leq \\sum _ { j = i + 2 } ^ { n - 1 } t ^ { - u _ j } x _ j + x _ n + t ^ { - u _ n } \\ ; . \\end{align*}"}
+{"id": "4212.png", "formula": "\\begin{align*} S _ { t _ 0 } : = \\{ ( t , x ) : \\ ; t = t _ 0 \\} . \\end{align*}"}
+{"id": "820.png", "formula": "\\begin{align*} \\sigma _ \\ast ^ k \\widehat { \\nu } | _ { [ v ] } = \\alpha _ v ^ k \\mu | _ { [ v ] } . \\end{align*}"}
+{"id": "2971.png", "formula": "\\begin{align*} W _ j = \\eta _ j + \\frac { g ^ { \\prime \\prime } ( 0 ) } { 2 g ^ \\prime ( 0 ) } \\frac { \\eta _ j ^ 2 } { \\sqrt { n } } + \\frac { g ^ { ( 3 ) } ( 0 ) } { 6 g ^ \\prime ( 0 ) } \\frac { \\eta _ j ^ 3 } { n } + O ( n ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "2805.png", "formula": "\\begin{align*} c _ i = a _ i - b _ i . \\end{align*}"}
+{"id": "7824.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { k d _ 1 } \\prod _ { i = 1 } ^ { d _ 1 - 2 } \\prod _ { t = 1 } ^ { i } \\prod _ { s = 1 } ^ { t } \\begin{bmatrix} 3 \\\\ 1 \\end{bmatrix} _ { 1 , \\frac { - 2 k d _ 1 + 3 k + 1 - 2 j + 2 k i + 2 k t + 2 k s } { k d _ 1 } } \\\\ = \\prod _ { i = 1 } ^ { d _ 1 - 2 } \\prod _ { t = 1 } ^ { i } \\prod _ { s = 1 } ^ { t } \\begin{bmatrix} 3 \\\\ 1 \\end{bmatrix} _ { \\frac { 1 } { k d _ 1 } , \\frac { - 3 d _ 1 + 3 + 2 i + 2 t + 2 s } { d _ 1 } } . \\end{align*}"}
+{"id": "7720.png", "formula": "\\begin{align*} \\sqrt { 1 + x } = 1 + \\frac { x } { 2 } + \\mathcal { O } \\left ( x ^ 2 \\right ) , { \\rm f o r } x \\rightarrow 0 , \\end{align*}"}
+{"id": "6446.png", "formula": "\\begin{align*} w _ j & : \\bar U _ X \\times \\R ^ m \\times [ 0 , T ) \\times \\bar U _ X \\times \\R ^ m \\times [ 0 , T ] \\to \\R , \\\\ w _ j ( X , Y , t , \\tilde X , \\tilde Y , \\tilde t ) & : = \\tilde u ( X , Y , t ) - v ( \\tilde X , \\tilde Y , \\tilde t ) - \\bigl ( \\frac { j ^ 4 } { 4 } | X - \\tilde X | ^ 4 + \\frac { j ^ 4 } { 4 } | Y - \\tilde Y | ^ 4 + \\frac j 2 | t - \\tilde t | ^ 2 \\bigr ) . \\end{align*}"}
+{"id": "2279.png", "formula": "\\begin{align*} & \\int _ 0 ^ T \\int _ { \\Omega } \\nabla \\Phi _ { N } \\cdot \\nabla \\Psi \\ , d x d t + \\varepsilon \\int _ 0 ^ T \\int _ \\Omega \\nabla ^ { 2 m + 1 } \\Phi _ { N } \\cdot \\nabla ^ { 2 m + 1 } \\Psi \\ , d x d t \\\\ & = \\int _ 0 ^ T \\int _ \\Omega \\big ( n _ { N } - c ( x ) \\big ) \\Psi \\ , d x d t , \\end{align*}"}
+{"id": "3919.png", "formula": "\\begin{align*} \\int _ { X v ( E ^ * ) } f = \\int _ { Y u ^ { - 1 } ( E ^ * ) } f . \\end{align*}"}
+{"id": "1295.png", "formula": "\\begin{align*} p ( x _ 1 , \\ldots , x _ r ) = x _ r ^ 2 q _ 2 ( \\vec { x ' } ) + x _ r q _ 1 ( \\vec { x ' } ) + q _ 0 ( \\vec { x ' } ) . \\end{align*}"}
+{"id": "6876.png", "formula": "\\begin{align*} \\mathcal { N } ^ { s _ i ( \\cdot ) } _ { p _ i ( \\cdot ) } u ( x ) = \\int _ { \\Omega } \\frac { | u ( x ) - u ( y ) | ^ { p _ i ( x , y ) - 2 } ( u ( x ) - u ( y ) ) } { | x - y | ^ { N + s _ i ( x , y ) p _ i ( x , y ) } } \\ , d y x \\in \\R ^ N \\setminus \\overline { \\Omega } . \\end{align*}"}
+{"id": "2231.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m } n + 5 ^ { 2 m } \\right ) } q ^ n \\equiv x _ { 2 m , 1 } E ( q ^ 2 ) E ( q ^ 5 ) ^ 2 \\pmod { 5 ^ { 2 m } } , \\end{align*}"}
+{"id": "6920.png", "formula": "\\begin{align*} \\Gamma = \\{ u ( \\cdot ) \\in L ^ \\infty ( [ 0 , t _ f ] , \\mathbb { R } ) \\ : \\ 0 \\leq u ( t ) \\leq 1 , \\ t \\in [ 0 , t _ f ] \\} , \\end{align*}"}
+{"id": "3339.png", "formula": "\\begin{align*} ( g _ { x \\kappa } , g _ { y \\kappa } , g _ { q \\kappa } , g _ { p \\kappa } , g _ { \\kappa \\kappa } ) = ( \\frac { k \\sqrt { \\delta } } { y } , 0 , - \\nu \\sqrt { \\delta } p , \\nu \\sqrt { \\delta } q , \\delta ) . \\end{align*}"}
+{"id": "4902.png", "formula": "\\begin{align*} \\partial _ s \\rho ( s ) = \\int _ { s _ 0 } ^ s \\partial ^ 2 _ s \\rho ( \\sigma ) d \\sigma \\geq - \\int _ { s _ 0 } ^ s b ( \\sigma ) d \\sigma \\geq - \\int _ { - \\infty } ^ \\infty b ( \\sigma ) d \\sigma = - | | b | | _ { L ^ 1 } . \\end{align*}"}
+{"id": "2016.png", "formula": "\\begin{align*} A _ { k , \\psi ( P ) } : = \\frac { \\sum _ { v \\in V } \\psi ( P ( v ) ) } { q ^ { \\dim V } } . \\end{align*}"}
+{"id": "3368.png", "formula": "\\begin{align*} g = \\eta \\otimes \\eta - 4 \\sum _ { j , \\bar { k } = 1 } ^ n K _ { j \\bar { k } } \\dd z ^ j \\dd \\bar { z } ^ k . \\end{align*}"}
+{"id": "2348.png", "formula": "\\begin{align*} \\forall 1 \\le j \\le n , \\phi _ { n + j } = C \\phi _ j . \\end{align*}"}
+{"id": "7159.png", "formula": "\\begin{align*} \\displaystyle { [ D _ i ^ { [ 0 ] } , D _ j ^ { [ 1 ] } ] + [ D _ i ^ { [ 1 ] } , D _ j ^ { [ 0 ] } ] = 0 \\ , , i , j = 1 , . . . , N } \\end{align*}"}
+{"id": "8316.png", "formula": "\\begin{align*} \\mathbf { P } _ 1 = \\frac { l \\otimes l } { | l | ^ 2 } , \\mathbf { P } _ 2 = \\mathbf { I } - \\mathbf { P } _ 1 . \\end{align*}"}
+{"id": "753.png", "formula": "\\begin{align*} \\gamma \\cdot P ( X , Y ) : = P \\left ( ( X , Y ) \\gamma \\right ) , \\end{align*}"}
+{"id": "9058.png", "formula": "\\begin{align*} w _ 2 ( x ) : = \\sum _ { n = 1 } ^ { n _ { \\max } } c _ n G ( x , x _ n ) \\ , , \\end{align*}"}
+{"id": "576.png", "formula": "\\begin{align*} \\sigma _ t = \\frac { \\sigma _ 0 } { 1 + ( A ^ { \\rm T } A ) : ( A ^ { \\rm T } + A ) ^ { - 1 } \\ , \\sigma _ 0 t } \\end{align*}"}
+{"id": "527.png", "formula": "\\begin{align*} \\liminf _ { k \\to \\infty } \\Upsilon _ { \\delta } ( z ^ { \\ell ( k ) - j } ) \\ge \\lim _ { k \\to \\infty } \\Upsilon _ { \\delta } ( z ^ { \\ell ( k ) } ) \\ \\ { \\rm a n d } \\ \\ \\lim _ { k \\rightarrow \\infty } \\| z ^ { \\ell ( k ) - j } - z ^ { \\ell ( k ) - j - 1 } \\| = 0 . \\end{align*}"}
+{"id": "242.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - \\beta \\tau \\eta \\leqslant - \\frac { \\alpha } { 2 } ( 1 + 3 \\rho ^ 2 ) \\leqslant - \\frac { \\alpha } { 2 } , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - \\beta \\rho \\eta \\leqslant - \\frac { \\alpha ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\alpha } { 2 } , \\end{align*}"}
+{"id": "5939.png", "formula": "\\begin{align*} \\phi _ r = ( E _ r , U ) . \\end{align*}"}
+{"id": "1978.png", "formula": "\\begin{align*} - A _ { m } ( z ) = \\sum _ { j = 0 , j \\neq m } ^ { k } \\frac { A _ { j } ( z ) f ( z + c _ { j } ) } { f ( z + c _ { m } ) } - \\frac { F ( z ) } { f ( z + c _ { m } ) } . \\end{align*}"}
+{"id": "2935.png", "formula": "\\begin{align*} \\mathcal { M } ^ n _ t ( \\varphi ) = \\mathcal { X } ^ n _ t ( \\varphi ) - \\mathcal { X } ^ n _ 0 ( \\varphi ) - \\int _ 0 ^ t ( \\partial _ s + L _ n ) \\mathcal { X } ^ n _ s ( \\varphi ) d s \\end{align*}"}
+{"id": "3300.png", "formula": "\\begin{align*} A ^ { 1 / 2 } S ( \\tau ) [ A ^ { - 1 / 2 } \\mathbb P { \\rm d i v } F ] = S ( \\tau ) { \\mathbb P } { \\rm d i v } F . \\end{align*}"}
+{"id": "3631.png", "formula": "\\begin{align*} \\mbox { d } ( x , 0 ) = \\ln \\left ( \\frac { 1 + | x | } { 1 - | x | } \\right ) . \\end{align*}"}
+{"id": "2592.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ r L _ i = \\bigcup _ { i \\in I \\subset [ r ] , \\mid I \\mid = r - 1 } L _ i \\mbox { a n d } L _ i \\cap L _ j \\neq \\emptyset \\end{align*}"}
+{"id": "4060.png", "formula": "\\begin{align*} D v ( y ) = g ^ * _ y \\big ( Y u ^ { - 1 } ( y ) , y , u ( Y u ^ { - 1 } ( y ) ) \\big ) , \\end{align*}"}
+{"id": "2008.png", "formula": "\\begin{align*} \\sum _ { \\phi \\neq 1 } N _ { i j } ( \\phi ) = - 1 , \\end{align*}"}
+{"id": "4337.png", "formula": "\\begin{align*} \\omega ( j , k , i ) c ( i , j + k ) \\omega ( i , j , k ) = c ( i , k ) \\omega ( j , i , k ) c ( i , j ) \\end{align*}"}
+{"id": "4157.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda ( h , 0 ) = \\int _ M \\langle \\frac { 1 } { 2 } \\triangle h , h \\rangle - | h | ^ 2 d V _ g = 0 \\Longleftrightarrow h = 0 . \\end{align*}"}
+{"id": "1462.png", "formula": "\\begin{align*} \\lambda = \\frac { 2 k ( k - 1 ) ( m - 1 ) ! ( m - 2 ) ! } { ( m + 1 ) | G _ \\Delta | } . \\end{align*}"}
+{"id": "2675.png", "formula": "\\begin{align*} a ( z ) & = \\sum _ { k = 1 } ^ { \\infty } \\sum \\limits _ { n = k } ^ \\infty ( - 1 ) ^ { k - 1 } { n - 1 \\choose k - 1 } { \\Gamma ( k / q + 1 ) \\over k ! \\ , \\Gamma ( k / q - k + 2 ) } \\ , q ^ k z ^ n \\\\ & = \\sum \\limits _ { n = 1 } ^ \\infty z ^ n \\sum _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } { n - 1 \\choose k - 1 } { \\Gamma ( k / q + 1 ) \\over k ! \\ , \\Gamma ( k / q - k + 2 ) } \\ , q ^ k . \\end{align*}"}
+{"id": "1911.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ J \\left ( \\sum _ { p _ 1 < \\cdots < p _ h \\in P _ j } \\frac { 1 } { p _ 1 \\cdots p _ h } \\right ) \\cdot \\left ( \\sum _ { p _ 1 < \\cdots < p _ { k - J h } \\in P _ { J + 1 } } \\frac { 1 } { p _ 1 \\cdots p _ { k - J h } } \\right ) . \\end{align*}"}
+{"id": "5798.png", "formula": "\\begin{align*} W '' + \\mathcal L W + \\overline A _ p W + \\overline D _ { p } G W ' = 0 . \\end{align*}"}
+{"id": "8381.png", "formula": "\\begin{align*} [ B , \\epsilon ] ^ { s w } = \\{ ( f , g ) : \\exists ~ \\delta > 0 \\ \\forall x \\in B ^ \\delta , | f ( x ) - g ( x ) | < \\epsilon ( x ) \\} \\end{align*}"}
+{"id": "1582.png", "formula": "\\begin{align*} B _ s ( \\mu , \\nu ) : = \\Big \\{ \\xi \\in \\mathcal { W } _ p ( X ) \\ , \\Big | \\ , d _ { p } ( \\mu , \\xi ) \\le \\sqrt [ p ] { s } d _ p ( \\mu , \\nu ) , \\ d _ { p } ( \\xi , \\nu ) \\le \\sqrt [ p ] { ( 1 - s ) } d _ p ( \\mu , \\nu ) \\Big \\} . \\end{align*}"}
+{"id": "6837.png", "formula": "\\begin{align*} \\begin{aligned} \\nu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) & = \\sum \\limits _ { j = 2 } ^ { m - 1 } \\nu _ c ^ { \\pm } ( Y _ 1 , Y _ j , Y _ { j + 1 } ) = \\sum \\limits _ { j = 1 } ^ { m - 2 } \\nu _ c ^ { \\pm } ( Y _ j , Y _ { j + 1 } , Y _ m ) \\end{aligned} \\end{align*}"}
+{"id": "4252.png", "formula": "\\begin{align*} \\gamma ^ 2 - \\frac { \\Omega ^ 2 } { 2 \\delta } = \\frac { \\gamma ^ 2 - \\Omega ^ 2 } { 2 } \\delta = \\frac { 1 } { 2 } - \\frac { \\gamma ^ 2 - \\Omega ^ 2 } { 2 ( \\gamma ^ 2 + \\Omega ^ 2 ) } > 0 . \\end{align*}"}
+{"id": "1081.png", "formula": "\\begin{align*} \\omega ^ 2 ( \\varepsilon / 2 ) = \\omega ' \\left ( \\frac { \\varepsilon } { 2 - \\varepsilon } \\right ) \\gtrsim \\varepsilon ^ { 2 - 2 / k } , \\end{align*}"}
+{"id": "1262.png", "formula": "\\begin{align*} c _ f ( r ) & = \\begin{dcases} \\sqrt { s ( \\alpha , a ) } , & a _ f ( r ) > 1 - \\frac { 4 \\alpha } { ( 1 - \\alpha ) ( 1 + 6 \\alpha + \\alpha ^ 2 ) } , \\\\ a _ f ( r ) - 1 + \\dfrac { 1 } { 1 - \\alpha } , & 1 - \\frac { 4 \\alpha } { ( 1 - \\alpha ) ( 1 + 6 \\alpha + \\alpha ^ 2 ) } \\geq a _ f ( r ) , \\end{dcases} \\end{align*}"}
+{"id": "8616.png", "formula": "\\begin{align*} \\tilde { T } x = \\begin{cases} 0 , & x = 0 ; \\\\ T x , & . \\end{cases} \\end{align*}"}
+{"id": "4854.png", "formula": "\\begin{align*} D _ 0 ^ n G ( w _ { t _ 0 , s } ) = c _ n s ^ n \\int _ { \\Sigma _ n } [ g _ { v ( t _ n ) } ^ { t _ 0 } , \\ . , g _ { v ( t _ 1 ) } ^ { t _ 0 } ] d \\L ^ n + O ( s ^ { n + 1 } ) . \\end{align*}"}
+{"id": "7427.png", "formula": "\\begin{align*} K ( Z , W ) ( P ) = H ( Z ) \\left ( ( { \\rm i d } _ { { \\mathbb C } ^ { n \\times m } } \\otimes \\pi ) ( P ) \\right ) H ( W ) ^ * \\end{align*}"}
+{"id": "2941.png", "formula": "\\begin{align*} E ^ n _ t ( \\varphi ) = \\frac { 1 } { \\sqrt { n } } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\frac { f _ n } { n } ( \\eta ^ n _ j ( s ) - \\rho ) ( \\partial _ x \\varphi ^ n _ j ( s ) - \\nabla ^ n \\varphi ^ n _ j ( s ) ) d s . \\end{align*}"}
+{"id": "4494.png", "formula": "\\begin{align*} \\frac { 1 } { d n } = \\frac { 1 } { d n _ 1 } + \\cdots + \\frac { 1 } { d n _ k } \\end{align*}"}
+{"id": "4781.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ^ t } \\sum _ { x \\in \\mathbb { F } _ 2 ^ t } f ( x ) ^ 2 = \\sum _ { y \\in \\mathbb { F } _ 2 ^ t } \\hat { f } ( y ) ^ 2 . \\end{align*}"}
+{"id": "4681.png", "formula": "\\begin{align*} \\mathfrak { R } ^ { G _ { I } } = \\mathbb { C } [ \\varphi _ 2 , \\varphi _ 8 ] \\end{align*}"}
+{"id": "8728.png", "formula": "\\begin{align*} { G } _ \\lambda ( x _ 1 , \\dots , x _ n ; \\beta ) = \\frac { \\det [ x _ i ^ { \\lambda _ j + n - j } ( 1 + \\beta x _ i ) ^ { j - 1 } ] } { \\prod _ { i < j } ( x _ i - x _ j ) } . \\end{align*}"}
+{"id": "4411.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } \\frac { 1 } { a _ i } & \\leq \\sum _ { i = 1 } ^ { m } \\frac { 1 } { x _ i } - \\left ( \\sum _ { i = m + 1 } ^ { n } \\frac { 1 } { a _ i } - \\sum _ { i = m + 1 } ^ { n } \\frac { 1 } { x _ i } \\right ) \\\\ & < \\sum _ { i = 1 } ^ { m } \\frac { 1 } { x _ i } \\leq \\sum _ { i = 1 } ^ { n } \\frac { 1 } { x _ i } < \\frac { p } { q } . \\end{align*}"}
+{"id": "7804.png", "formula": "\\begin{align*} \\tilde N = N \\oplus M ^ { \\circ } , \\tilde M ^ { \\circ } = M ^ { \\circ } \\oplus N , \\end{align*}"}
+{"id": "716.png", "formula": "\\begin{align*} \\frac { \\partial R _ { \\rm r o b i n } ( w _ k ) } { \\partial w _ k } = \\frac { 1 } { 4 \\pi } \\big ( r _ { \\rm m e t r i c } ( w _ k ) - r _ { \\rm r o b i n } ( w _ k ) \\big ) = \\end{align*}"}
+{"id": "5685.png", "formula": "\\begin{align*} d \\mathbf { m } \\mathbf { = d } \\overline { \\mathbf { m } } = 1 , d ^ { 2 } = 0 . \\end{align*}"}
+{"id": "6492.png", "formula": "\\begin{align*} \\rho ^ 2 \\asymp \\begin{cases} n ^ { - \\frac { 2 s } { 2 s + 1 / 2 } } & b \\geq n ^ { \\frac { 1 } { 2 s + 1 / 2 } } , \\\\ \\left ( { b n } \\right ) ^ { - \\frac { 2 s } { 2 s + 3 / 2 } } & n ^ { \\frac { 1 } { 2 s + 1 / 2 } } / m ^ { \\frac { s + 3 / 4 } { 2 s + 1 / 2 } } \\leq b < n ^ { \\frac { 1 } { 2 s + 1 / 2 } } , \\\\ ( n / \\sqrt { m } ) ^ { - \\frac { 2 s } { 2 s + 1 / 2 } } & b < n ^ { \\frac { 1 } { 2 s + 1 / 2 } } / m ^ { \\frac { s + 3 / 4 } { 2 s + 1 / 2 } } , \\end{cases} \\end{align*}"}
+{"id": "1268.png", "formula": "\\begin{align*} R _ { \\mathcal { B S } ( \\alpha ) } ( \\mathcal { S } ^ * [ \\beta ] ) & = \\min \\left \\{ 1 ; \\dfrac { 1 } { ( 1 + \\alpha ) ( 1 - \\beta ) } \\right \\} . \\end{align*}"}
+{"id": "1046.png", "formula": "\\begin{align*} Z _ i ^ { ( \\ell ) } = [ R _ i ^ { ( \\ell ) } ] _ 0 ^ M + \\frac { M } { \\alpha } W _ i ^ { ( \\ell ) } , \\end{align*}"}
+{"id": "7885.png", "formula": "\\begin{align*} h '' ( z ) - 2 \\gamma e ^ { z / 2 } h ' ( z ) - \\left ( \\frac { \\gamma } { 2 } e ^ { z / 2 } - \\frac { 1 } { 4 } \\right ) h ( z ) = 0 \\end{align*}"}
+{"id": "1893.png", "formula": "\\begin{align*} u ( t ) = u _ 0 ( t ) + k ( 1 , 1 ) y ( 1 , t ) + \\int _ 0 ^ 1 k _ x ( 1 , \\zeta ) y ( \\zeta , t ) d \\zeta , \\end{align*}"}
+{"id": "8618.png", "formula": "\\begin{align*} x _ { n + 1 } = \\alpha _ n u + ( 1 - \\alpha _ n ) [ \\beta _ n x _ n + ( 1 - \\beta _ n ) T x _ n ] \\end{align*}"}
+{"id": "5659.png", "formula": "\\begin{align*} \\lambda _ 0 ^ { ( 1 ) } = \\frac { 4 \\pi D } { | \\Omega | } \\sum _ { j = 1 } ^ N \\ell _ j . \\end{align*}"}
+{"id": "5647.png", "formula": "\\begin{align*} M T ( s _ 1 , \\cdots , s _ r | s ) = \\sum _ { i = 1 } ^ r M T ( s _ 1 , \\cdots , s _ i - 1 , \\cdots , s _ r | s + 1 ) \\end{align*}"}
+{"id": "287.png", "formula": "\\begin{align*} \\begin{aligned} H ^ 1 ( W _ F , \\widehat { S } ) \\times \\{ \\prescript { L } { } { j } : \\prescript { L } { } { S } \\rightarrow \\prescript { L } { } { G } \\} _ { \\phi _ 0 } / \\widehat { S } _ 0 & \\rightarrow H ^ 1 ( W _ F , \\widehat { G } ) \\\\ ( \\theta , \\theta ' ) & \\mapsto \\mathrm { I n d } _ { W _ L } ^ { W _ F } ( \\theta \\cdot \\theta ' ) . \\end{aligned} \\end{align*}"}
+{"id": "3132.png", "formula": "\\begin{align*} - \\tilde { A } : D ^ 2 \\phi = - \\frac { A : D ^ 2 \\phi } { 1 + A : D ^ 2 \\phi } = \\gamma - 1 = \\gamma - \\bar { \\gamma } . \\end{align*}"}
+{"id": "1943.png", "formula": "\\begin{gather*} \\bar { \\mathbf { f } } ^ { ( Q ^ 1 . . Q ^ { k + 1 } ) } ( u , \\eta ) = ( \\bar { f } ^ { Q ^ 1 } _ 1 ( u , \\eta _ 1 ) , \\bar { f } ^ { Q ^ 2 } _ 2 ( u , \\eta _ 2 ) , \\cdots , \\bar { f } _ k ^ { Q ^ k } ( u , \\eta _ k ) , \\bar { f } _ { k + 1 } ^ { Q ^ { k + 1 } } ( u ) ) ^ { \\top } \\\\ \\mathbf { f } ( u , \\eta , x ) = ( f _ 1 ( u , \\eta _ 1 , x ) , f _ 2 ( u , \\eta _ 2 , x ) , \\cdots , f _ k ( u , \\eta _ k , x ) , f _ { k + 1 } ( u , x ) ) ^ { \\top } \\ \\end{gather*}"}
+{"id": "3585.png", "formula": "\\begin{align*} v _ A : = y _ { 2 1 } ^ { \\gamma _ { 2 1 } } y _ { 3 1 } ^ { \\gamma _ { 3 1 } } y _ { 3 2 } ^ { \\gamma _ { 3 2 } } ( z _ 3 ^ + ) ^ { \\lambda _ 3 } ( z _ 2 ^ + ) ^ { \\lambda _ 2 } ( z _ 1 ^ + ) ^ { \\lambda _ 1 } v _ 0 , \\end{align*}"}
+{"id": "6576.png", "formula": "\\begin{align*} E ( t ) : = \\sup _ { y \\in B _ { \\frac 1 4 } , r \\leq \\frac 1 4 } \\frac { \\mu _ t ( B _ r ( y ) ) } { r ^ { n - 1 } } . \\end{align*}"}
+{"id": "9241.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ^ C \\partial ^ { \\alpha } _ t X ( t ) + A X ( t ) = F ( X ( t ) ) + I _ t ^ { 1 - \\alpha } \\left [ G ( X ) \\frac { d W ( t ) } { d t } + \\Phi \\frac { d B ^ H ( t ) } { d t } \\right ] , \\\\ X ( 0 ) = X _ 0 , \\ , \\ , \\ , \\ , t \\in [ 0 , T ] . \\end{array} \\right . \\end{align*}"}
+{"id": "5831.png", "formula": "\\begin{align*} U '' + \\mathcal L U + A U + D \\mathcal G U ' = 0 . \\end{align*}"}
+{"id": "2141.png", "formula": "\\begin{align*} e ^ { - t } \\sum _ { i = 1 } ^ { M ( t ) } \\frac { t ^ i } { i ! } \\leq e ^ { - t } \\sum _ { i = 1 } ^ { M ( t ) } \\exp \\left ( i + i \\ln t - i \\ln i \\right ) \\leq e ^ { - t } M ( t ) \\left ( \\frac { e t } { M ( t ) } \\right ) ^ { M ( t ) } . \\end{align*}"}
+{"id": "5625.png", "formula": "\\begin{align*} \\Psi ( u ) & = \\int ^ { u } _ { u ^ { * } } \\dfrac { g ( s ) - g ( u ^ { * } ) } { g ( s ) } d s \\\\ & = u - u ^ { * } - \\int ^ { u } _ { u ^ { * } } \\dfrac { g ( u ^ { * } ) } { g ( s ) } d s , \\end{align*}"}
+{"id": "66.png", "formula": "\\begin{align*} x ' = x '' - y '' \\textnormal { a n d } y ' = y '' . \\end{align*}"}
+{"id": "8390.png", "formula": "\\begin{align*} w ( C ( X ) , \\tau ^ { w } ) & = i b ( C ( X ) , \\tau ^ { w } ) = c ( C ( X ) , \\tau ^ { w } ) = L ( C ( X ) , \\tau ^ { w } ) \\\\ & = d ( C ( X ) , \\tau ^ { w } ) = n w ( C ( X ) , \\tau ^ { w } ) . \\end{align*}"}
+{"id": "7861.png", "formula": "\\begin{align*} H _ n '' ( x ) - 2 x H _ n ' ( x ) + 2 n H _ n ( x ) = 0 \\end{align*}"}
+{"id": "282.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { \\alpha \\in H ^ 1 ( \\mathrm { G a l } ( E / F ) , G ( E ) ) } \\mathrm { d i m } \\mathrm { H o m } _ { G _ { \\alpha } ( F ) } ( \\pi , \\omega _ { G _ { \\alpha } ( F ) , E } ) = \\sum \\limits _ { \\tilde { \\phi } } m ( \\lambda _ { \\pi } , \\tilde { \\phi } ) , \\end{aligned} \\end{align*}"}
+{"id": "1819.png", "formula": "\\begin{align*} u \\bigg ( t + \\frac { T } { M } \\bigg ) = R u ( t ) , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "4887.png", "formula": "\\begin{align*} t _ { s _ 1 s _ 2 u _ 0 } t _ { u _ l } = t _ { s _ 1 s _ 2 u _ l } + t _ { s _ 1 s _ 2 u _ { l - 1 } } , { } \\ l > 0 . \\end{align*}"}
+{"id": "4225.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { I } ( t ) = \\mathcal { J } _ 1 + \\mathcal { J } _ 2 , \\end{align*}"}
+{"id": "8431.png", "formula": "\\begin{align*} & G ( \\mu ) : = 2 \\mu ( X ) - \\kappa ( \\mu , \\mu ) \\in ( - \\infty , \\infty ] , \\\\ & \\widehat { \\mathcal E } ^ + _ A : = \\bigl \\{ \\nu \\in \\mathcal E ^ + _ A : \\ \\kappa \\nu \\leqslant 1 \\bigr \\} . \\end{align*}"}
+{"id": "2620.png", "formula": "\\begin{align*} ( a _ { 1 , 2 } , a _ { 1 , 3 } , a _ { 1 , 4 } ) = ( 2 , 2 , 1 ) . \\end{align*}"}
+{"id": "2057.png", "formula": "\\begin{align*} \\alpha = \\frac { \\rho _ 2 - \\rho _ 1 } { \\rho _ 2 + \\rho _ 1 } \\end{align*}"}
+{"id": "7652.png", "formula": "\\begin{align*} \\nabla _ { \\omega } ( Q _ { I } \\eta _ { I } ) = d ( Q _ { I } \\eta _ { I } ) + \\omega _ { \\lambda } \\wedge Q _ { I } \\eta _ { I } = d ( Q _ { I } ) \\wedge \\eta _ { I } + Q _ { I } \\wedge d ( \\eta _ { I } ) + Q _ { I } \\omega _ { \\lambda } \\wedge \\eta _ { I } . \\end{align*}"}
+{"id": "9127.png", "formula": "\\begin{align*} G ( f ) \\ ; = \\ ; h ^ { | \\eta | + | \\gamma | } \\prod _ { ( x , y ) \\in E ( f ) } \\nu ( x - y ) , \\end{align*}"}
+{"id": "1985.png", "formula": "\\begin{align*} { \\mathcal { I } } _ { L } = I \\left ( { \\textbf { Y } } _ { } ; { \\textbf { G } } | { \\textbf { S } } \\right ) = N \\log _ 2 \\det \\left ( { \\textbf { I } } + { \\textbf { S } } ^ { \\mathsf { H } } { \\textbf { R } } _ { } { \\textbf { S } } \\right ) , \\end{align*}"}
+{"id": "5935.png", "formula": "\\begin{align*} \\hbox { K e r } ( C _ p ) \\cap \\hbox { I m } ( D ) = \\hbox { K e r } ( C _ p ) \\cap V ^ \\perp = \\{ 0 \\} , \\end{align*}"}
+{"id": "3473.png", "formula": "\\begin{align*} & g _ { n , R } ( x _ 1 , \\ldots , x _ n ; t , s ) = \\int _ { B _ R } \\big ( f _ n ( x _ 1 , \\ldots , x _ n , x ; t ) - f _ n ( x _ 1 , \\ldots , x _ n , x ; s ) \\big ) d x \\\\ & = \\int _ { B _ R } \\int _ { T _ n ( t ) } \\prod _ { j = 1 } ^ { n - 1 } G _ { t _ { j + 1 } - t _ j } ( x _ { j + 1 } - x _ j ) \\big ( G _ { t - t _ n } ( x - x _ n ) - G _ { s - t _ n } ( x - x _ n ) \\big ) d \\pmb { t } d x \\end{align*}"}
+{"id": "936.png", "formula": "\\begin{align*} x _ { + } ( h ) = x \\theta _ { C } + \\psi ( x ) \\theta _ { C } ^ { \\perp } x _ { + } ( \\tilde { h } ) = \\tilde { x } \\theta _ { C } + \\psi ( x ) \\theta _ { C } ^ { \\perp } , \\end{align*}"}
+{"id": "874.png", "formula": "\\begin{align*} \\mathcal { X } ^ { t + 1 } = \\mathcal { X } ^ { t } - ( \\mathcal { A } _ { ( i , : , : ) } ) ^ { T } * \\left ( \\mathcal { A } _ { ( i , : , : ) } * ( \\mathcal { A } _ { ( i , : , : ) } ) ^ { T } \\right ) ^ { \\dag } * \\left ( \\mathcal { A } _ { ( i , : , : ) } * \\mathcal { X } ^ { t } - \\mathcal { B } _ { ( i , : , : ) } \\right ) . \\end{align*}"}
+{"id": "7368.png", "formula": "\\begin{align*} [ f ] _ { L ^ { p , r } ( \\nu ) } = \\| f \\| _ { L ^ p ( \\nu ) } . \\end{align*}"}
+{"id": "5024.png", "formula": "\\begin{align*} * _ g H & = \\frac { h _ 1 \\left ( a ^ 4 + c ^ 2 \\right ) - 2 a ^ 2 c h _ 3 } { \\mu \\left ( a ^ 2 b ^ 2 - c ^ 2 \\right ) } \\ , e ^ { 2 3 } + \\frac { h _ 1 \\left ( b ^ 4 + c ^ 2 \\right ) - 2 b ^ 2 c h _ 3 } { \\mu \\left ( a ^ 2 b ^ 2 - c ^ 2 \\right ) } \\ , e ^ { 4 5 } \\\\ & - \\frac { h _ 4 } { \\mu } \\ , \\left ( e ^ { 2 4 } + e ^ { 3 5 } \\right ) - \\frac { h _ 3 \\left ( a ^ 2 b ^ 2 + c ^ 2 \\right ) - c h _ 1 \\left ( a ^ 2 + b ^ 2 \\right ) } { \\mu \\left ( a ^ 2 b ^ 2 - c ^ 2 \\right ) } \\left ( e ^ { 2 5 } - e ^ { 3 4 } \\right ) , \\end{align*}"}
+{"id": "4859.png", "formula": "\\begin{align*} \\frac { b _ 1 ^ n + \\dots + b _ { k - 1 } ^ n } { | b | ^ { n } } - | b | ^ { n } \\leq \\frac { 1 + ( k - 2 ) \\delta ^ n } { ( 1 + ( k - 2 ) \\delta ^ 2 ) ^ { n / 2 } } - ( ( k - 1 ) \\delta ^ 2 ) ^ { n / 2 } = \\psi ( \\delta ) . \\end{align*}"}
+{"id": "1982.png", "formula": "\\begin{align*} y _ { k , i } \\ ! = \\ ! { \\textbf { w } } _ k ^ { \\mathsf { H } } \\left ( \\sqrt { \\alpha _ k } { \\textbf { h } } _ k x _ { k , i } \\ ! + \\ ! \\sqrt { \\alpha _ { k ' } } { \\textbf { h } } _ { k ' } x _ { { k ' } , i } \\right ) \\ ! + \\ ! r _ { k , i } \\ ! + \\ ! n _ { k , i } . \\end{align*}"}
+{"id": "1456.png", "formula": "\\begin{align*} \\lambda = \\frac { k ( k - 1 ) ( m - 1 ) ! ( m - 2 ) ! } { ( m + 1 ) | K _ \\Delta | } . \\end{align*}"}
+{"id": "2785.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } f _ t ( \\beta ) = f ( \\beta ) \\coloneqq \\left \\{ \\begin{array} { l l } 1 + ( \\beta / \\beta _ c ) ^ 2 , & \\beta \\leq \\beta _ c , \\\\ 2 \\beta / \\beta _ c , & \\beta > \\beta _ c . \\end{array} \\right . \\end{align*}"}
+{"id": "6621.png", "formula": "\\begin{align*} \\norm { ( f _ \\epsilon ( \\abs { x } ) ) ( t ) \\psi } \\leq \\norm { x ^ m \\psi } + \\R { D } m \\sum _ { q = 0 } ^ { m - 1 } \\binom { m - 1 } { q } 2 ^ { m - 1 - q } \\sum _ { r = 0 } ^ q \\tilde { p } _ { q - r + 1 } ( t ) \\norm { x ^ r \\psi } \\ , . \\end{align*}"}
+{"id": "1757.png", "formula": "\\begin{align*} \\Upsilon = \\iota ^ { - 1 } \\left | \\begin{array} { c c } \\underline x _ 1 & \\underline y _ 1 \\\\ \\underline x _ 2 & \\underline y _ 2 \\end{array} \\right | ^ { \\underline k - 2 } \\in V ( \\underline { k } - 2 ) \\otimes V ( \\underline { k } - 2 ) , \\end{align*}"}
+{"id": "1294.png", "formula": "\\begin{align*} I ( p ) : = \\int _ { \\R ^ { r } } \\frac { p ( x _ 1 , \\ldots , x _ { r } ) d x _ 1 \\ldots d x _ { r } } { \\big ( 1 + x _ 1 ^ 2 \\big ) \\ldots \\big ( 1 + x _ { r } ^ 2 \\big ) \\big ( 1 + ( \\sum _ { j = 1 } ^ { r } b _ j x _ j ) ^ 2 \\big ) } \\ ; < \\infty . \\end{align*}"}
+{"id": "615.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } m _ t = \\frac { \\sigma _ t } { \\gamma } ( A ^ { \\rm T } A ) : \\tilde C \\ , ( 1 - m _ t ) . \\end{align*}"}
+{"id": "2869.png", "formula": "\\begin{align*} \\theta _ 2 : = \\frac { 7 } { 6 4 } , \\theta _ 3 : = \\frac { 5 } { 1 4 } , \\theta _ 4 : = \\frac { 9 } { 2 2 } , \\theta _ n : = \\frac { 1 } { 2 } - \\frac { 1 } { n ^ 2 + 1 } ( n \\geq 5 ) . \\\\ \\end{align*}"}
+{"id": "6778.png", "formula": "\\begin{align*} \\chi ( - n ) = - \\chi ( n ) \\end{align*}"}
+{"id": "7331.png", "formula": "\\begin{align*} \\nu ( d s ) : = \\left ( \\int _ { s } ^ { \\infty } n _ { m , j } [ T _ 0 > u ] d u \\right ) d s \\hat { \\nu } ( \\lambda ) : = \\int _ { 0 } ^ { \\infty } \\mathrm { e } ^ { - \\lambda s } \\nu ( d s ) = - \\frac { \\tilde { \\chi } ( \\lambda ) } { \\lambda ^ 2 } , \\end{align*}"}
+{"id": "3756.png", "formula": "\\begin{align*} \\kappa _ 0 : = \\mathrm { R e s } _ { s = 1 } \\zeta ( s ) , \\kappa _ 1 : = \\lim _ { s \\to 0 } \\frac { d } { d s } s \\zeta ( s ) . \\end{align*}"}
+{"id": "4368.png", "formula": "\\begin{align*} s _ 1 & = 2 \\\\ s _ 2 & = 3 \\\\ s _ 3 & = 7 \\\\ s _ 4 & = 4 3 \\\\ s _ 5 & = 1 8 0 7 \\\\ s _ 6 & = 3 2 6 3 4 4 3 \\\\ s _ 7 & = 1 0 6 5 0 0 5 6 9 5 0 8 0 7 \\\\ s _ 8 & = 1 1 3 4 2 3 7 1 3 0 5 5 4 2 1 8 4 4 3 6 1 0 0 0 4 4 3 \\\\ s _ 9 & = 1 2 8 6 4 9 3 8 6 8 3 2 7 8 6 7 1 7 4 0 5 3 7 1 4 5 9 9 8 3 6 0 9 6 1 5 4 6 6 5 3 2 5 9 4 8 5 1 9 5 8 0 7 . \\end{align*}"}
+{"id": "6111.png", "formula": "\\begin{align*} \\lim _ { n _ i \\to \\infty } \\frac { 1 } { 2 n _ i } m ( j _ i ) _ { n _ i } = \\lambda ( p ) . \\end{align*}"}
+{"id": "7463.png", "formula": "\\begin{align*} [ W ] ( e _ { i _ 1 } , \\dots , e _ { i _ { d k } } ) = { \\rm s i g n } ( { \\bf w } ) a ( W , { \\bf w } ) \\end{align*}"}
+{"id": "7130.png", "formula": "\\begin{align*} d = \\sum _ { i = 1 } ^ n \\left \\langle d , x ^ { \\rho _ 1 + \\cdots + \\rho _ { i - 1 } } \\right \\rangle \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) \\end{align*}"}
+{"id": "1879.png", "formula": "\\begin{align*} \\alpha _ { z _ i } ( \\sigma ) = \\frac { \\pi } { 2 } . \\end{align*}"}
+{"id": "1620.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\lim _ { n \\to \\infty } ( \\hat S ^ { ( t ) } ) \\leq \\frac { 1 } { C } \\sum _ { c = 1 } ^ C M \\psi ( \\tau ^ { ( \\infty ) } _ c , \\theta _ c , M ) \\end{align*}"}
+{"id": "6322.png", "formula": "\\begin{align*} \\mathcal { A } { : = } \\mathcal { K } \\ , \\ell ^ 2 _ { \\rm S G } , \\mathcal { S } { : = } \\mathcal { K } \\ , \\ell ^ { 1 / p ' } _ { \\rm S G } \\ , \\ell ^ { 1 / q ' } _ { \\rm c h i r a l } \\frac { 1 } { p ' } + \\frac { 1 } { q ' } = 1 , \\ , \\ , p ' , q ' \\in [ 1 , \\infty ) . \\end{align*}"}
+{"id": "2503.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - { \\mathbf u } ( z ) } = \\int _ { \\mathbb T } \\frac { 1 } { 1 - z \\overline \\zeta } \\sigma ( \\zeta ) , \\ \\ z \\in \\mathbb D , \\end{align*}"}
+{"id": "220.png", "formula": "\\begin{align*} x _ { X } : = ( X , \\lambda _ { X } , \\omega _ { X } , \\alpha _ { X } ) \\in M _ { i _ { X } , D _ { i _ { X } } ( m ) } \\end{align*}"}
+{"id": "7437.png", "formula": "\\begin{align*} K ( Z , W ) ( P ) = H ( Z ) \\left ( ( { \\rm i d } _ { n \\times m } \\otimes \\pi ) ( P ) \\cdot ( { \\rm i d } _ { n \\times m } \\otimes J ) \\right ) H ( W ) ^ * \\end{align*}"}
+{"id": "5443.png", "formula": "\\begin{align*} M _ 1 ( t ) & = \\int _ 0 ^ t \\langle \\nabla V ( X _ s , \\alpha _ s ) , \\sigma ( s , X _ s , \\mathcal { L } _ { X _ s } , \\alpha _ s ) d W _ s \\rangle , \\\\ M _ 2 ( t ) & = \\int _ 0 ^ t \\int _ { \\mathbb R } V ( X _ s , \\alpha _ 0 + h ( X _ s , \\alpha _ { s - } , z ) ) - V ( X _ s , \\alpha _ s ) \\mu ( d s , d z ) , \\end{align*}"}
+{"id": "6826.png", "formula": "\\begin{align*} \\begin{aligned} \\nu _ c ^ - ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) : = \\mu ( Y _ 1 , Y _ 2 ) + \\mu ( Y _ 2 , Y _ 3 ) + \\dots + \\mu ( Y _ { m - 1 } , Y _ m ) - \\mu ( Y _ 1 , Y _ m ) , \\\\ \\nu _ c ^ + ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) : = \\mu ^ * ( Y _ 1 , Y _ 2 ) + \\mu ^ * ( Y _ 2 , Y _ 3 ) + \\dots + \\mu ^ * ( Y _ { m - 1 } , Y _ m ) - \\mu ^ * ( Y _ 1 , Y _ m ) . \\end{aligned} \\end{align*}"}
+{"id": "2798.png", "formula": "\\begin{align*} x _ { \\sigma ( i ) } ^ { a _ i } = x _ { \\sigma ( i ) } ^ { a _ { \\sigma ^ { - 1 } ( \\sigma ( i ) ) } } = x _ j ^ { a _ { \\sigma ^ { - 1 } ( j ) } } \\end{align*}"}
+{"id": "8871.png", "formula": "\\begin{align*} \\varphi ( t , z ) = \\int _ { \\mathbb { C } ^ n } H _ { \\nu } ( t , z , w ) g ( w ) d \\mu _ n ( w ) \\end{align*}"}
+{"id": "5972.png", "formula": "\\begin{align*} \\langle \\phi _ 1 , \\dots , \\phi _ r \\rangle _ { T , ( d _ 1 , \\dots , d _ m ) } ^ X : = \\chi _ T \\left ( e v _ 1 ^ * E _ 1 \\otimes \\dots \\otimes e v _ r ^ * E _ r \\otimes \\O _ { \\overline { \\mathcal { M } _ { 0 , r } } ( X , \\d ) } \\right ) , \\end{align*}"}
+{"id": "4280.png", "formula": "\\begin{align*} - \\nu \\Delta \\textit { \\textbf { u } } + \\nabla \\pi & = \\textit { \\textbf { f } } \\quad \\quad \\Omega , \\\\ \\mathrm { d i v } \\ , \\textit { \\textbf { u } } & = 0 \\quad \\quad \\Omega , \\end{align*}"}
+{"id": "4703.png", "formula": "\\begin{align*} \\mathbb { R } ^ { G _ { I } } = \\mathbb { C } [ \\varphi _ 2 , \\varphi _ 8 ] \\end{align*}"}
+{"id": "3262.png", "formula": "\\begin{align*} \\frac { \\mathbf { d } ( b ) } { P _ { \\mathbf { d } , \\mathcal { A } } ( a b ) } = \\sum \\limits _ { c \\in \\mathcal { A } ^ * ( b ) } \\frac { P _ { \\mathbf { d } , \\mathcal { A } } ( b c ) } { P _ { \\mathbf { d } , \\mathcal { A } } ( a b ) } . \\end{align*}"}
+{"id": "5114.png", "formula": "\\begin{align*} \\mathbf { v } ( t , z ) = \\frac { 1 } { 2 \\pi } \\int _ { \\partial D _ { t } } K _ { 0 } ( \\lambda | z - \\xi | ) d \\xi , \\end{align*}"}
+{"id": "227.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = & v ( \\alpha u + ( 1 - \\alpha ) u ^ 2 + \\beta w ) , \\\\ \\psi _ 2 ( u , v , w ) = & u ( \\alpha v + ( 1 - \\alpha ) v ^ 2 + \\beta w ) , \\\\ \\psi _ 3 ( u , v , w ) = & u ( \\alpha v + ( 1 - \\alpha ) u ^ 2 + \\beta w ) \\\\ \\shortintertext { a n d } \\psi _ 4 ( v , w ) = & v ( \\alpha u + ( 1 - \\alpha ) v ^ 2 + \\beta w ) . \\end{align*}"}
+{"id": "6057.png", "formula": "\\begin{align*} \\frac { p ( x ) } { x - z } - \\sum _ { i = 1 } ^ g l _ i ( x ) \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac { p ( y ) } { y - z } \\frac { \\dd y } { w ( y ) } = \\mathcal O \\left ( z ^ { d - g - 1 } \\right ) , z \\to \\infty . \\end{align*}"}
+{"id": "8651.png", "formula": "\\begin{align*} 2 \\mu & \\sum _ { k = 1 } ^ N \\sum _ { j > i \\geq k } \\int _ { X \\times X } \\abs { \\epsilon _ { i k } \\epsilon _ { j k } } f ( t _ i ) f ( t _ j ) f ' ( t _ i ) f ' ( t _ j ) \\d x _ i \\d x _ j \\\\ & \\leq 4 \\mu \\sum _ { k = 1 } ^ N \\sum _ { j > i > k } \\int _ { X \\times X } f ( d ( x _ i , x _ k ) ) f ( d ( x _ j , x _ k ) ) f ' ( d ( x _ i , x _ k ) ) f ' ( d ( x _ j , x _ k ) ) \\d x _ i \\d x _ j \\\\ & = \\frac { 2 } { 3 } \\mu N ( N - 1 ) ( N - 2 ) K ( f ) ^ 2 . \\end{align*}"}
+{"id": "6015.png", "formula": "\\begin{align*} e v _ 3 ( X _ { 1 , w _ 0 } ) = h \\cdot \\{ ( [ x _ 1 \\dots x _ { i _ 1 + i _ 2 - n } : 0 : \\dots : 0 ] , [ y _ 1 \\dots y _ n ] ) \\in X \\} , \\end{align*}"}
+{"id": "4773.png", "formula": "\\begin{align*} \\langle \\chi _ y , \\chi _ { y ' } \\rangle = \\begin{cases} 1 & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "3824.png", "formula": "\\begin{align*} a _ 2 ( f ) & = 2 1 6 \\\\ a _ { 1 , 1 } ( f ) & = 1 9 1 4 , \\end{align*}"}
+{"id": "537.png", "formula": "\\begin{align*} \\Phi _ u ( X ) = & ( X - e ^ { 2 i \\pi / u } ) ( X - e ^ { 2 i \\pi ( u - 1 ) / u } ) ( X - e ^ { 2 i \\pi a / u } ) ( X - e ^ { 2 i \\pi ( u - a ) / u } ) \\\\ & ( X - e ^ { 2 i \\pi b / u } ) ( X - e ^ { 2 i \\pi ( u - b ) / u } ) ( X - e ^ { 2 i \\pi c / u } ) ( X - e ^ { 2 i \\pi ( u - c ) / u } ) \\end{align*}"}
+{"id": "7643.png", "formula": "\\begin{align*} , \\ \\ \\ \\ < \\Delta _ { V , h } \\Psi _ { j } , \\Psi _ { k } \\ > = \\delta _ { j , k } \\kappa _ { j , h } + O \\big ( h ^ \\infty \\sqrt { \\kappa _ j ( h ) \\kappa _ k ( h ) } \\big ) \\end{align*}"}
+{"id": "3875.png", "formula": "\\begin{align*} \\int _ \\Omega f = \\int _ { \\Omega ^ * } f ^ * . \\end{align*}"}
+{"id": "3499.png", "formula": "\\begin{align*} [ \\{ B _ i ^ \\xi , B _ j ^ \\eta \\} , B _ l ^ \\epsilon ] = ( \\epsilon - \\eta ) \\delta _ { j l } B _ i ^ \\xi + ( \\epsilon - \\xi ) \\delta _ { i l } B _ j ^ \\eta , \\end{align*}"}
+{"id": "6773.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\frac { 1 } { P } } y ^ { \\frac { s } { 2 } - 1 } \\Psi _ { \\chi } ( y ) d y = P ^ { - \\frac { s } { 2 } } \\int _ { 0 } ^ { 1 } y ^ { \\frac { s } { 2 } - 1 } \\Psi _ { \\chi } \\left ( \\frac { y } { P } \\right ) d y . \\end{align*}"}
+{"id": "3253.png", "formula": "\\begin{align*} ( \\mathfrak { d } ( Z ) , \\overline { \\mathfrak { d } ( Z ) } ) = \\lbrace ( W ^ { \\prime } , W ^ { \\prime \\prime } ) : W \\in Z \\rbrace \\end{align*}"}
+{"id": "7760.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n \\ , x \\ , \\{ g _ { \\mu / n , \\lambda _ 1 } \\star g _ { \\mu / n , \\lambda _ 2 } \\} ( x ) & = \\mu \\ , e ^ { - ( \\lambda _ 1 + \\lambda _ 2 ) x / 2 } \\ , \\sqrt { \\pi ( \\lambda _ 2 - \\lambda _ 1 ) x } \\ ; I _ { - \\tfrac { 1 } { 2 } } \\left ( \\frac { \\lambda _ 2 - \\lambda _ 1 } { 2 } \\ , x \\right ) \\\\ & = \\mu \\ , e ^ { - \\lambda _ 1 x } + \\mu \\ , e ^ { - \\lambda _ 2 x } \\end{align*}"}
+{"id": "340.png", "formula": "\\begin{align*} a _ 1 G _ { \\phi } ( l ) + a _ 2 \\bar { G } _ { \\phi } ( l ) = b _ 1 G _ { \\phi ' } ( l ) + b _ 2 \\bar { G } _ { \\phi ' } ( l ) \\end{align*}"}
+{"id": "5116.png", "formula": "\\begin{align*} \\Omega \\mbox { R e } \\left \\lbrace \\gamma ( 0 , s ) \\overline { \\partial _ { s } \\gamma ( 0 , s ) } \\right \\rbrace = \\mbox { I m } \\left \\lbrace \\mathbf { v } ( 0 , \\gamma ( 0 , s ) ) \\overline { \\partial _ { s } \\gamma ( 0 , s ) } \\right \\rbrace . \\end{align*}"}
+{"id": "37.png", "formula": "\\begin{align*} \\prod _ { n \\in \\Omega _ o } \\left ( 3 + \\frac { 1 } { n } \\right ) & = \\left ( \\sqrt [ K ] { \\prod _ { n \\in \\Omega _ o } \\left ( 3 + \\frac { 1 } { n } \\right ) } \\right ) ^ K \\leq \\left ( \\frac { 1 } { K } \\cdot \\sum _ { n \\in \\Omega _ o } \\left ( 3 + \\frac { 1 } { n } \\right ) \\right ) ^ K \\\\ & = ( 3 + \\mu ) ^ K , \\end{align*}"}
+{"id": "4354.png", "formula": "\\begin{align*} ( a , s ) ( b , t ) = ( a \\wedge b , t \\rceil ^ b _ { a \\wedge b } ) . \\end{align*}"}
+{"id": "7004.png", "formula": "\\begin{align*} R _ c ( R _ p ) = \\frac { 1 } { 2 } \\log ^ { + } { \\frac { N ^ 2 ( X , Y ) } { \\Delta ^ 2 e ^ { 2 R _ p } } } , \\end{align*}"}
+{"id": "3920.png", "formula": "\\begin{align*} | \\{ y \\in \\Omega ^ * ; f ^ * ( y ) > 0 \\ \\ X v ( y ) \\setminus \\overline { \\Omega } \\neq \\emptyset \\} | = 0 . \\end{align*}"}
+{"id": "1536.png", "formula": "\\begin{align*} j ^ { b j } _ A = p ^ { b j } _ A \\end{align*}"}
+{"id": "1148.png", "formula": "\\begin{align*} T _ { \\ell } ( f ) + p ^ { j - 1 - \\nu _ p ( n ) } \\left ( T _ { \\ell } ( g E _ { p - 1 } ^ n ) - \\lambda _ { \\ell } ( g ) E _ { p - 1 } ^ n \\right ) & = \\lambda ( \\ell ) f , \\mbox { \\ \\ \\ i . e . , } \\\\ T _ { \\ell } ( f + p ^ { j - 1 - \\nu _ p ( n ) } g E _ { p - 1 } ^ n ) & = \\lambda ( \\ell ) ( f + p ^ { j - 1 - \\nu _ p ( n ) } g E _ { p - 1 } ^ n ) , \\end{align*}"}
+{"id": "6562.png", "formula": "\\begin{align*} \\Psi ( Q ) = \\sum _ { q = 1 } ^ Q \\frac { 2 \\varphi ( q ) \\psi ( q ) } { q } . \\end{align*}"}
+{"id": "4484.png", "formula": "\\begin{align*} 2 N = \\sum _ { \\substack { d | n \\\\ d \\geq 1 } } d . \\end{align*}"}
+{"id": "3245.png", "formula": "\\begin{align*} I _ D ^ { + , i _ 1 } ( z ) : = & \\frac { 1 } { i _ 1 ! j _ 1 ! k _ 1 ! } \\int _ D \\prod _ { l = 1 } ^ { i _ 1 } \\frac { d u _ l } { 1 - u _ l } \\prod _ { m = 1 } ^ { j _ 1 } \\frac { - d v _ m } { 1 + v _ m } \\prod _ { n = 1 } ^ { k _ 1 } \\frac { d w _ n } { w _ n } \\frac { d t } { 1 - t } , \\\\ I _ D ^ { - , i _ 1 } ( z ) = & \\frac { 1 } { i _ 1 ! j _ 1 ! k _ 1 ! } \\int _ D \\prod _ { l = 1 } ^ { i _ 1 } \\frac { d u _ l } { 1 - u _ l } \\prod _ { m = 1 } ^ { j _ 1 } \\frac { - d v _ m } { 1 + v _ m } \\prod _ { n = 1 } ^ { k _ 1 } \\frac { d w _ n } { w _ n } \\frac { - d t } { 1 + t } \\end{align*}"}
+{"id": "8703.png", "formula": "\\begin{align*} \\Omega \\ , ( \\tau \\otimes \\tau ) = 0 \\end{align*}"}
+{"id": "8841.png", "formula": "\\begin{align*} \\cos ^ { 2 } d _ { F S } ( z , w ) = \\frac { | 1 + \\langle z , w \\rangle | ^ { 2 } } { ( 1 + \\langle z , z \\rangle ) ( 1 + \\langle w , w \\rangle ) } . \\end{align*}"}
+{"id": "3343.png", "formula": "\\begin{align*} \\zeta ( \\Phi _ { x p } \\Phi _ { y q } - \\Phi _ { x q } \\Phi _ { y p } ) + 1 = - \\Phi ^ 2 _ { x x } - \\Phi _ { x y } \\Phi _ { y x } = - \\Phi ^ 2 _ { q q } - \\Phi _ { q p } \\Phi _ { p q } . \\end{align*}"}
+{"id": "654.png", "formula": "\\begin{align*} A _ { c _ { 1 } \\cdots c _ { n } } : = A _ { c _ { 1 } } \\cap \\ldots \\cap A _ { c _ { n } } B _ { c _ { 1 } \\cdots c _ { n } } : = B _ { c _ { 1 } } \\cap \\ldots \\cap B _ { c _ { n } } , 1 \\leq n \\leq N . \\end{align*}"}
+{"id": "8606.png", "formula": "\\begin{align*} e _ 1 \\cdot ( x _ 1 , x _ 2 ) & = ( x _ 1 + a _ 1 , x _ 2 + b _ 1 ) , \\\\ e _ 2 \\cdot ( x _ 1 , x _ 2 ) & = ( x _ 1 + a _ 2 , - x _ 2 ) , \\\\ e _ 3 \\cdot ( x _ 1 , x _ 2 ) & = ( - x _ 1 , x _ 2 + b _ 3 ) . \\end{align*}"}
+{"id": "1210.png", "formula": "\\begin{align*} \\frac { \\nabla V } { d t } & = \\frac { \\nabla } { \\partial t } \\biggl ( \\frac { \\nabla } { \\partial s _ 1 } [ ( \\nabla ^ { k - 1 } \\Phi ^ t _ * ) ( \\gimel _ { s _ 2 } , \\ldots , \\gimel _ { s _ { k + 1 } } ) ] \\biggr ) - \\frac { \\nabla } { d t } \\biggl ( \\sum _ { \\ell = 2 } ^ { k + 1 } ( \\nabla ^ { k - 1 } \\Phi ^ t _ * ) ( v _ 2 , \\ldots , \\frac { \\nabla \\gimel _ { s _ \\ell } } { \\partial s _ 1 } , \\ldots , v _ { k + 1 } ) \\biggr ) . \\end{align*}"}
+{"id": "120.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ n } \\int _ { \\mathbb R ^ n } g ( x , y ) d \\mathcal H ^ n ( x ) d \\mathcal H ^ n ( y ) = \\int _ { ( n , 1 ) } \\int _ L \\int _ L g ( x , y ) | x - y | ^ { n - 1 } d \\mathcal H ^ 1 ( x ) d \\mathcal H ^ 1 ( y ) d L , \\end{align*}"}
+{"id": "1600.png", "formula": "\\begin{align*} 0 < \\mu ( x _ { n + 1 } ) = \\mu ( x _ { n + 1 } ) + \\mu ( u ) = \\mu ^ * ( u ) = \\mu ^ * ( v ) = \\mu ( v ) \\end{align*}"}
+{"id": "3188.png", "formula": "\\begin{align*} \\sum _ { j , k , l = 1 } ^ 3 c _ j ^ { k l } ( A ) \\ , \\partial _ { j k l } ^ 3 u = - \\frac { 1 } { 1 2 8 \\pi } \\partial _ { 1 1 1 } ^ 3 u - \\frac { 1 } { 1 2 8 \\pi } \\partial _ { 1 2 2 } ^ 3 u + \\frac { 1 } { 6 4 \\pi } \\partial _ { 1 3 3 } ^ 3 u \\equiv - \\frac { 1 5 } { 3 2 \\pi } . \\end{align*}"}
+{"id": "6709.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\leq x \\\\ n \\equiv a \\bmod q } } \\mu ( n ) = O \\left ( \\frac { x } { \\log ^ { C } x } \\right ) , \\end{align*}"}
+{"id": "8106.png", "formula": "\\begin{align*} \\abs { \\phi ' ( \\xi ) } \\geq \\frac { T } { M } - \\frac { x } { 2 M } \\sinh \\frac { \\xi \\pi } { M } \\gg \\frac { T } { M } - \\frac { x } { 2 M } \\frac { \\xi \\pi } { M } = \\frac { T } { M } - \\frac { x \\xi \\pi } { 2 M ^ 2 } \\end{align*}"}
+{"id": "5007.png", "formula": "\\begin{align*} \\phi ( s _ 1 , s _ 2 ) = \\begin{cases} s _ 1 , & s _ 1 , s _ 2 \\le M , \\\\ M + 1 , & s _ 1 , s _ 2 \\ge M + 1 \\end{cases} . \\end{align*}"}
+{"id": "3404.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } f + v \\cdot \\nabla f - \\mu \\Delta f = g , \\\\ g _ { | t = 0 } = g ^ 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "413.png", "formula": "\\begin{align*} V U x = V \\left ( \\sum _ { n = 1 } ^ { \\infty } g _ n ( x ) \\tau _ n \\right ) = \\sum _ { n = 1 } ^ { \\infty } g _ n ( x ) V \\tau _ n = \\sum _ { n = 1 } ^ { \\infty } g _ n ( x ) \\omega _ n = x , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "7270.png", "formula": "\\begin{align*} \\lim _ { x \\to + 0 } \\left ( u ^ + ( x ) - \\left ( \\lambda m ( x ) + \\sum _ { k = 1 } ^ { d - 1 } \\lambda ^ k \\int _ { x } ^ { 1 } G ^ k _ m ( y ) d m ( y ) \\right ) \\right ) = 0 . \\end{align*}"}
+{"id": "8046.png", "formula": "\\begin{align*} \\eta ( n , s ) = \\sum _ { a d = \\abs { n } } \\Bigl ( \\frac { a } { d } \\Bigr ) ^ { s - \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "2720.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { \\partial h } { \\partial t } & = \\frac { \\partial U } { \\partial t } + \\sum _ { j = 1 } ^ J \\lambda _ j ' ( t ) \\left ( \\Lambda W \\right ) _ { [ \\lambda _ j ( t ) ] } \\\\ \\frac { \\partial } { \\partial t } \\left ( \\frac { \\partial U } { \\partial t } \\right ) & = \\Delta h + 2 M h + 2 M v _ L + ( h + v _ L ) ^ 2 + M ^ 2 + \\Delta M . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7571.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 7 ) = & \\sum _ { a = 0 } ^ { \\omega - 1 } { Z _ f ( s , \\chi , A _ 7 ^ a ) } \\\\ = & \\sum _ { a = 0 } ^ { \\omega - 1 } q ^ { - a } { Z _ { f _ { 7 , a } } ( s , \\chi , B _ 7 ^ a ) } , \\end{align*}"}
+{"id": "196.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n - 1 } \\log ( r _ { j } ) & = \\sum _ { j = 1 } ^ { n - 1 } ( \\log ( j + 1 ) + \\log ( \\log ( j + 1 ) ) + \\log ( z _ { j } ) ) \\leq \\sum _ { j = 1 } ^ { n - 1 } 3 \\log ( j + 1 ) \\leq 3 n \\log ( n ) . \\end{align*}"}
+{"id": "3826.png", "formula": "\\begin{align*} \\begin{aligned} a _ 2 ( h ) = & ~ 4 h _ * ( \\alpha _ 1 ) \\beta _ 1 ^ 2 + 2 h _ * ( \\alpha _ 1 ^ 2 + \\alpha _ 2 ) \\beta _ 1 - 2 h _ * ( \\alpha _ 1 ) \\beta _ 2 + 2 h _ * ( \\alpha _ 1 \\alpha _ 2 ) \\\\ 2 ( a _ 2 ( h ) + a _ { 1 , 1 } ( h ) ) = & ~ \\delta ^ 2 - \\beta _ 1 \\delta + ( - 2 h _ * ( \\alpha _ 1 \\alpha _ 2 ) - h _ * ( \\alpha _ 3 ) - 2 h _ * ( \\alpha _ 1 ^ 2 + \\alpha _ 2 ) \\beta _ 1 \\\\ & + h _ * ( \\alpha _ 1 ) ( - 4 \\beta _ 1 ^ 2 + 3 \\beta _ 2 ) ) \\end{aligned} \\end{align*}"}
+{"id": "8018.png", "formula": "\\begin{align*} \\begin{aligned} \\dot x _ 1 & = x _ 2 + x _ 3 , \\\\ \\dot x _ 2 & = x _ 3 + \\Delta _ { - \\tau } x _ 2 , \\\\ \\dot x _ 3 & = x _ 2 + \\Delta _ { - \\tau } x _ 3 , \\\\ 0 & = x _ 1 + x _ 2 + x _ 3 + \\Delta _ { - \\tau } x _ 4 . \\end{aligned} \\end{align*}"}
+{"id": "2763.png", "formula": "\\begin{align*} \\hat { \\partial } _ { 0 } \\ , \\bar { \\triangleright } \\ , f ( \\mathbf { x } , t ) = \\frac { \\partial f ( \\mathbf { x } , t ) } { \\partial t } . \\end{align*}"}
+{"id": "343.png", "formula": "\\begin{align*} C ( p ^ { h } , 1 ) = B ( p ^ { h } , 1 ) - A ( 1 , p ) B ( p ^ { h - 1 } , 1 ) + A ( p , 1 ) B ( p ^ { h - 2 } , 1 ) - B ( p ^ { h - 3 } , 1 ) , \\end{align*}"}
+{"id": "1068.png", "formula": "\\begin{align*} r _ 1 = \\frac { \\mathrm { d } R _ 1 } { \\mathrm { d } ( R _ 1 + R _ 2 ) } r _ 2 = \\frac { \\mathrm { d } R _ 2 } { \\mathrm { d } ( R _ 1 + R _ 2 ) } . \\end{align*}"}
+{"id": "8784.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| f ^ m _ i - f ^ { m , n } _ i \\| _ 1 = 0 . \\end{align*}"}
+{"id": "5859.png", "formula": "\\begin{align*} \\psi | _ { \\Sigma } \\in H ^ { 2 \\alpha - 1 } ( \\Sigma ) = H ^ { 2 \\alpha - 1 } ( 0 , T ; L ^ 2 ( \\Gamma ) ) \\cap L ^ 2 ( 0 , T ; H ^ { 2 \\alpha - 1 } ( \\Gamma ) ) , \\end{align*}"}
+{"id": "902.png", "formula": "\\begin{align*} \\pi = \\sqrt [ t ] { ( a / b ) ^ s } . \\end{align*}"}
+{"id": "642.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } \\cdot \\delta ^ { \\epsilon } | B | \\leq | B _ { c } \\cap B _ { 0 } | = \\sum _ { j = 1 } ^ { N } | B _ { c } \\cap B _ { j } | \\leq \\sum _ { j \\notin \\mathcal { J } } \\delta ^ { \\bar { \\epsilon } } | B _ { j } | + \\sum _ { j \\in \\mathcal { J } } | B _ { j } | \\leq \\tfrac { 1 } { 4 } \\cdot \\delta ^ { \\epsilon } | B | + \\sum _ { j \\in \\mathcal { J } } | B _ { j } | . \\end{align*}"}
+{"id": "7530.png", "formula": "\\begin{align*} { \\rm o r d } ( \\beta _ j \\pi ^ { a j } ) = & { \\rm o r d } ( \\beta _ j ) + { \\rm o r d } ( \\pi ^ { a j } ) > { \\rm o r d } ( \\beta _ { j _ 0 } ) + { \\rm o r d } ( \\pi ^ { a j _ 0 } ) = { \\rm o r d } ( \\beta _ { j _ 0 } \\pi ^ { a j _ 0 } ) , \\end{align*}"}
+{"id": "1963.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } i ^ 2 n _ i ( v ) + \\sum _ { i = k + 1 } ^ { 2 k - 1 } ( 2 k - i ) ^ 2 n _ i ( v ) , \\end{align*}"}
+{"id": "8290.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { E } _ i = \\{ x , y , z \\ | \\ H _ i > 0 \\} \\end{array} \\right . \\end{align*}"}
+{"id": "8064.png", "formula": "\\begin{align*} V _ \\mp ( y , t ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 0 0 0 ) } y ^ { - u } F ( u ) \\frac { \\gamma _ \\mp \\Bigl ( \\dfrac { 1 } { 2 } + u , t \\Bigr ) } { \\gamma _ - \\Bigl ( \\dfrac { 1 } { 2 } , t \\Bigr ) } \\frac { d u } { u } \\end{align*}"}
+{"id": "5952.png", "formula": "\\begin{align*} \\displaystyle B ^ T E _ r = \\sum _ { s = 1 } ^ p \\beta _ { r s } E _ s + C _ p ^ T Q _ r , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "6356.png", "formula": "\\begin{align*} ( - m _ 0 '' / m _ 0 ) ' ( x ) = { 4 x ( x ^ 2 - 7 ) } / { ( 1 + x ^ 2 ) ^ 3 } . \\end{align*}"}
+{"id": "3367.png", "formula": "\\begin{align*} N ^ 1 : = [ \\Phi , \\Phi ] + \\dd \\eta \\otimes \\xi , \\end{align*}"}
+{"id": "7361.png", "formula": "\\begin{align*} ( 3 ^ d - 1 ) \\cdot ( d _ i + 1 ) ^ d \\cdot \\sum _ { j = i } ^ \\infty \\| g _ { j + 1 } - g _ { j } \\| _ \\infty < 1 / 2 . \\end{align*}"}
+{"id": "4090.png", "formula": "\\begin{align*} \\ker \\textbf { A } _ f ^ * = \\ker ( \\textbf { A } ' _ f ) ^ * \\oplus \\widetilde { \\mathcal { H } } . \\end{align*}"}
+{"id": "2224.png", "formula": "\\begin{align*} \\nu ( x _ { 2 m + 1 , i } ) & = \\nu { \\left ( \\sum _ { k = 1 } ^ \\infty x _ { 2 m , k } \\beta _ { k , i } \\right ) } \\\\ & \\geq \\min _ { k \\geq 2 } \\{ \\nu ( x _ { 2 m , k } ) + \\nu ( \\beta _ { k , i } ) \\} \\\\ & = \\min _ { k \\geq 4 } { \\left \\{ \\big ( \\nu ( x _ { 2 m , 2 } ) + \\nu ( \\beta _ { 2 , i } ) \\big ) , \\big ( \\nu ( x _ { 2 m , 3 } ) + \\nu ( \\beta _ { 3 , i } ) \\big ) , \\big ( \\nu ( x _ { 2 m , k } ) + \\nu ( \\beta _ { k , i } ) \\big ) \\right \\} } . \\end{align*}"}
+{"id": "6566.png", "formula": "\\begin{align*} 2 \\varphi ( m ) \\frac { \\varphi ( l ) ^ 2 } { l } \\prod _ { \\substack { p \\mid l } } \\left ( 1 - \\frac { 1 } { ( p - 1 ) ^ 2 } \\right ) \\sum _ { \\substack { 1 \\leq c \\leq l n , \\\\ \\gcd ( c , n ^ * ) = 1 } } w ^ * \\left ( \\frac { c } { l n } \\right ) \\prod _ { \\substack { p \\mid \\gcd ( l , c ) } } \\left ( 1 + \\frac { 1 } { p - 2 } \\right ) . \\end{align*}"}
+{"id": "9188.png", "formula": "\\begin{align*} ( E _ { j + 1 } , s _ { j + 1 } , z _ { j + 1 } , K _ { j + 1 } ) = \\Phi ^ { \\Lambda _ N } _ { j + 1 } ( E _ j , s _ j , z _ j , K _ j ) \\end{align*}"}
+{"id": "6757.png", "formula": "\\begin{align*} \\alpha _ n \\left ( \\frac { 1 } { 2 } + j \\left ( \\omega + j \\sigma \\right ) , \\rho \\right ) = n \\sqrt { 2 \\pi } \\exp \\left ( \\frac { 1 } { 2 } \\left [ 1 + \\frac { 1 } { \\left ( \\frac { 1 } { 2 } - \\sigma \\right ) + j \\omega } \\right ] \\log \\left ( 1 - \\rho \\right ) \\right ) \\end{align*}"}
+{"id": "5330.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { c u r l } \\operatorname { c u r l } E = \\lambda \\ , \\varepsilon E , & \\mbox { i n } \\Omega , \\\\ \\mathrm { d i v } \\ , \\varepsilon E = 0 , & \\mbox { i n } \\Omega , \\\\ \\nu \\times E = 0 , & \\mbox { o n } \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "8454.png", "formula": "\\begin{align*} \\kappa \\gamma _ A = \\sup _ { K \\in \\mathfrak C _ A } \\ , \\kappa \\gamma _ K \\end{align*}"}
+{"id": "6861.png", "formula": "\\begin{gather*} ( Q u - Q v , Q u - Q v ) = 0 \\Leftrightarrow Q ( u - v ) = 0 \\Leftrightarrow u = v , \\end{gather*}"}
+{"id": "6729.png", "formula": "\\begin{align*} T _ { 0 } ( x ) = \\sum _ { t \\leq x _ 1 } \\mu ( t ) d ( t ) \\sum _ { d ^ 2 \\leq x _ 0 } \\mu ( d ) \\sum _ { \\substack { k \\leq x / t \\\\ d ^ 2 \\mid k } } \\mu ( k t + a ) = O \\left ( \\frac { x } { ( \\log x ) ^ { ( C - 3 ) / 2 } } \\right ) , \\end{align*}"}
+{"id": "2414.png", "formula": "\\begin{align*} U ^ H \\Phi = \\left [ \\begin{array} { c } I _ { p + q } \\\\ 0 \\end{array} \\right ] . \\end{align*}"}
+{"id": "8818.png", "formula": "\\begin{align*} B _ { 1 1 } P _ 1 & = P _ 1 B _ { 1 1 } & B _ { 1 2 } P _ 2 = P _ 1 B _ { 1 2 } \\ , , \\\\ B _ { 2 1 } P _ 1 & = P _ 2 B _ { 2 1 } & B _ { 2 2 } P _ 2 = P _ 2 B _ { 2 2 } \\ , . \\end{align*}"}
+{"id": "7529.png", "formula": "\\begin{align*} Z _ g \\big ( s , \\chi , S ( \\Delta _ { \\gamma _ 1 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) ) \\big ) = \\sum _ { a = 1 } ^ { \\infty } q ^ { - a } Z _ { g _ { 1 , a } } ( s , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) , \\end{align*}"}
+{"id": "7375.png", "formula": "\\begin{align*} f ( k g k ' ) = \\tilde \\rho ( k ) f ( g ) \\tilde \\rho ( k ' ) , \\end{align*}"}
+{"id": "498.png", "formula": "\\begin{align*} { \\textstyle \\sum _ { k = \\widehat { k } } ^ { \\nu } } \\ , \\| x ^ { k + 1 } \\ ! - \\ ! x ^ k \\| < ( 1 / 4 ) { \\textstyle \\sum _ { k = \\widehat { k } } ^ \\nu } \\ , \\Xi _ k + b \\gamma ^ { - 1 } \\varphi ( \\Phi ( x ^ { \\ell ( \\widehat { k } ) } ) \\ ! - \\ ! \\Phi ^ * ) . \\end{align*}"}
+{"id": "8311.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\frac { 1 } { \\Lambda } \\big ( [ x _ 1 ( y _ 2 \\ ! - \\ ! y _ 3 ) + x _ 2 ( y _ 3 \\ ! - \\ ! y _ 1 ) + x _ 3 ( y _ 1 \\ ! - \\ ! y _ 2 ) ] R _ E ^ 2 \\\\ ~ ~ - [ x _ 2 ( y _ 3 \\ ! - \\ ! y _ E ) + x _ 3 ( y _ E \\ ! - \\ ! y _ 2 ) + x _ E ( y _ 2 \\ ! - \\ ! y _ 3 ) ] R _ 1 ^ 2 \\\\ ~ ~ - [ x _ 3 ( y _ 1 \\ ! - \\ ! y _ E ) + x _ 1 ( y _ E \\ ! - \\ ! y _ 3 ) + x _ E ( y _ 3 \\ ! - \\ ! y _ 1 ) ] R _ 2 ^ 2 \\\\ ~ ~ - [ x _ 1 ( y _ 2 \\ ! - \\ ! y _ E ) + x _ 2 ( y _ E \\ ! - \\ ! y _ 1 ) + x _ E ( y _ 1 \\ ! - \\ ! y _ 2 ) ] R _ 3 ^ 2 \\big ) = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1403.png", "formula": "\\begin{align*} \\pi _ 1 ^ { ( - , + , - ) } = L ( \\Delta [ 0 , - 2 ] , \\Delta [ 1 , - 3 ] ; \\pi ( 0 ^ - , 1 ^ + , 2 ^ - ) ) . \\end{align*}"}
+{"id": "3022.png", "formula": "\\begin{align*} \\hbox { d } ^ \\mathbf { A } p _ I + \\frac { 1 } { 2 } \\hat { e } ^ { ( N - 2 ) } _ { I J K } \\wedge \\Omega ^ { J K } - \\mathbf { F } { ^ J } _ { I b } \\ p { _ J } ^ { j b } \\ \\hat { e } ^ { ( N ) } _ j = \\Lambda _ 0 \\hat { e } ^ { ( N ) } _ I \\end{align*}"}
+{"id": "7670.png", "formula": "\\begin{align*} \\sum _ { m \\in M } ( - 1 ) ^ { \\sigma _ { M \\setminus \\{ m \\} } ( m ) } \\eta _ { M \\setminus \\{ m \\} } & = \\sum _ { m \\in M } ( - 1 ) ^ { \\sigma _ { M \\setminus \\{ m \\} } ( m ) } \\sum _ { p } \\eta _ { M \\setminus \\{ m \\} , p } \\\\ & = \\sum _ { p } \\sum _ { m \\in M } ( - 1 ) ^ { \\sigma _ { M \\setminus \\{ m \\} } ( m ) } \\eta _ { M \\setminus \\{ m \\} , p } \\\\ & = 0 . \\end{align*}"}
+{"id": "8103.png", "formula": "\\begin{align*} H _ { m , n } ^ { + , 1 } ( x ) = \\frac { 4 M T } { \\pi } \\int _ { - \\infty } ^ \\infty \\widehat { k ^ * } \\Bigl ( - \\frac { M \\zeta } { \\pi } \\Bigr ) \\cos ( x \\cosh \\zeta ) e \\Bigl ( \\frac { T \\zeta } { \\pi } \\Bigr ) \\ , d \\zeta , \\end{align*}"}
+{"id": "1565.png", "formula": "\\begin{align*} g ( X ) : = X ^ { Q + R } \\frac { A ^ { ( q ) } ( 1 / X ) } { A ( X ) } = \\frac { X ^ { Q + R } + X ^ Q + 1 } { X ^ { Q + R } + X ^ R + 1 } . \\end{align*}"}
+{"id": "6898.png", "formula": "\\begin{align*} 1 - \\chi ( m \\underline { d } , m \\underline { d } ) = 1 - m ^ 2 \\chi ( \\underline { d } , \\underline { d } ) = m ^ 2 ( l ^ 2 + 2 l - 4 ) + 1 . \\end{align*}"}
+{"id": "8519.png", "formula": "\\begin{align*} \\begin{cases} P ( z , w ) = w + z ( z ^ 2 + w ^ 2 + 1 ) \\\\ Q ( z , w ) = - z + w ( z ^ 2 + w ^ 2 + 1 ) \\end{cases} \\end{align*}"}
+{"id": "870.png", "formula": "\\begin{align*} \\mathcal { X } & = ^ { - 1 } ( ( \\mathcal { X } ) ) = ^ { - 1 } ( ( \\mathcal { X } ) ^ { \\frac { 1 } { 2 } } ( \\mathcal { X } ) ^ { \\frac { 1 } { 2 } } ) \\\\ & = ^ { - 1 } ( ( \\mathcal { X } ) ^ { \\frac { 1 } { 2 } } ) * ^ { - 1 } ( ( \\mathcal { X } ) ^ { \\frac { 1 } { 2 } } ) = \\mathcal { X } ^ { \\frac { 1 } { 2 } } * \\mathcal { X } ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "1328.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | \\geq \\frac { \\frac { n ^ 2 } { d } - n } { n ^ 2 - n } = \\frac { n - d } { d ( n - 1 ) } . \\end{align*}"}
+{"id": "1133.png", "formula": "\\begin{align*} ( r S ) ' & = r ( S ' ) \\mbox { \\ \\ \\ f o r \\ a l l \\ } r \\in R , \\\\ ( S T ) ' & = S ' T + S T ' , \\mbox { \\ \\ \\ a n d } \\\\ ( S + T ) ' & = S ' + T ' . \\end{align*}"}
+{"id": "5218.png", "formula": "\\begin{align*} c _ 2 & = ( c _ { 1 2 } - c _ { 5 6 } ) \\cos ( d ) \\tanh ( 2 s ) \\\\ c _ 1 & = ( c _ { 5 6 } - c _ { 1 2 } ) \\sin ( d ) \\tanh ( 2 s ) \\ , . \\end{align*}"}
+{"id": "2580.png", "formula": "\\begin{align*} \\frac { 1 } { - \\frac { \\pi } { 2 } + 2 m \\pi } e _ i = ( - 1 ) \\times \\frac { 1 } { - \\frac { 3 } { 2 } \\pi + 2 ( - m + 1 ) \\pi } e _ i , \\end{align*}"}
+{"id": "7140.png", "formula": "\\begin{align*} \\delta _ i ( x _ i ) = 0 \\ \\mbox { a n d } \\ ( - 1 ) ^ { i - 1 } ( x _ { i - 1 } + \\alpha _ i ( x _ i ) ) + \\delta _ i ( y _ i ) = 0 . \\end{align*}"}
+{"id": "2165.png", "formula": "\\begin{align*} R _ { \\mathcal { S } ^ { * } _ { c } } ( F ) & = \\begin{dcases} \\sigma _ { 0 } & \\ 2 \\alpha + \\beta - 2 \\geqslant 0 \\\\ \\sigma _ { 0 } & \\ 2 \\alpha + \\beta - 2 < 0 \\ \\ X ( \\alpha , \\beta ) \\leqslant 0 \\\\ \\tilde { \\sigma _ { 0 } } & \\ 2 \\alpha + \\beta - 2 < 0 \\ X ( \\alpha , \\beta ) > 0 , \\end{dcases} \\end{align*}"}
+{"id": "4738.png", "formula": "\\begin{align*} c ^ m , c ^ { m + 1 } \\in ( \\mathfrak { q } + y A ) B = \\mathfrak { q } B + y A B = \\mathfrak { q } + y B \\subseteq A + y B , \\end{align*}"}
+{"id": "5562.png", "formula": "\\begin{align*} | S _ 1 ( x ^ 2 ) | & = \\left | \\sum _ { n = 1 } ^ { \\ell - 1 } \\frac { \\mu ( n ) } { n ^ k } \\exp \\left ( { - \\frac { x ^ 2 } { n ^ 2 } } \\right ) \\right | \\leq \\sum _ { n = 1 } ^ { \\ell - 1 } \\exp \\left ( { - \\frac { x ^ 2 } { l ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "8210.png", "formula": "\\begin{align*} V ( t ) : = \\frac { a _ 1 + a _ 2 } { 2 } + \\Bigl ( V _ 0 - \\frac { a _ 1 + a _ 2 } { 2 } \\Bigr ) ( - 1 ) ^ { N ( t ) } , \\end{align*}"}
+{"id": "4626.png", "formula": "\\begin{align*} \\frac { 1 } { k } < \\frac { 1 } { y _ 1 } + \\frac { 1 } { y _ 2 } \\leq \\frac { 2 } { y _ 1 } \\leq \\frac { 2 } { 2 k } = \\frac { 1 } { k } \\end{align*}"}
+{"id": "2129.png", "formula": "\\begin{align*} f ( u ) = 0 , \\ ; \\forall u \\in \\{ 0 , \\theta , 1 \\} , \\end{align*}"}
+{"id": "2502.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty A _ k ^ 2 \\int _ { \\tau _ k } | \\varphi _ 1 | ^ 2 | g | ^ 2 { \\mathrm d } m \\leq C \\| g \\| ^ 2 \\ g \\in \\mathcal K _ { \\mathbf v } . \\end{align*}"}
+{"id": "1017.png", "formula": "\\begin{align*} W = \\sum _ { n \\in \\Z } A _ n \\delta ( \\cdot - Y _ n ) \\end{align*}"}
+{"id": "6154.png", "formula": "\\begin{align*} C _ 1 A = \\overline A _ 1 C _ 1 . \\end{align*}"}
+{"id": "7676.png", "formula": "\\begin{align*} \\phi ( \\bigoplus _ { I } \\sum _ { p } ( p + \\iota _ { E } ( \\omega ) ) \\eta _ { I , p } - \\nabla _ { \\omega } ( \\iota _ { E } ( \\bigoplus _ { I } \\eta _ { I } ) ) ) = & \\bigoplus _ { I } \\sum _ { p \\neq - \\iota _ { E } ( \\omega ) } \\eta _ { I , p } \\\\ & - \\nabla _ { \\omega } ( \\phi ( ( \\iota _ { E } ( \\bigoplus _ { I } \\eta _ { I } ) ) ) \\end{align*}"}
+{"id": "6380.png", "formula": "\\begin{align*} & \\gamma _ I ^ 2 = 1 \\\\ & \\gamma _ I \\gamma _ J = \\gamma _ J \\gamma _ { w _ J ( I ) } & \\mathrm { i f } \\ , \\ , I \\subset J \\ , \\ , \\mathrm { o r } \\ , \\ , W _ { I \\cup J } = W _ I \\times W _ J . \\end{align*}"}
+{"id": "4909.png", "formula": "\\begin{align*} \\chi = 0 . \\end{align*}"}
+{"id": "3378.png", "formula": "\\begin{align*} t x _ { n - 1 } \\leq t ^ { u _ n } x _ n + 1 = x _ { n - 1 } + 1 \\ , . \\end{align*}"}
+{"id": "8973.png", "formula": "\\begin{align*} \\rho \\int _ { C _ { \\rho } } & | \\nabla u | ^ 2 d s \\le 2 \\int _ { R ^ 2 } ^ R \\int _ { C _ { \\rho } } | \\nabla u | ^ 2 d s \\ , d \\rho \\Big / \\int _ { R ^ 2 } ^ R \\frac { d \\rho } { \\rho } \\\\ & \\le 2 \\int _ B | \\nabla u | ^ 2 d z \\Big / | \\log ( R ) | = 4 E ( u ) / | \\log ( R ) | . \\end{align*}"}
+{"id": "8320.png", "formula": "\\begin{align*} t = \\frac { s } { | y | ^ 2 - s ^ 2 } , x = \\frac { y } { | y | ^ 2 - s ^ 2 } , \\end{align*}"}
+{"id": "3411.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } \\zeta _ { \\theta } + v \\cdot \\nabla \\zeta _ { \\theta } - \\mu ( \\Delta - \\frac { 1 } { r ^ 2 } ) \\zeta _ { \\theta } = \\frac { v _ { r } } { r } \\zeta _ { \\theta } - \\partial _ z \\frac { ( B _ { \\theta } ) ^ 2 } { r } , \\\\ { \\zeta _ { \\theta } } _ { \\vert t = 0 } = \\zeta _ { \\theta } ^ { 0 } , \\end{array} \\right . \\end{align*}"}
+{"id": "931.png", "formula": "\\begin{align*} \\tau _ { E , \\alpha } ^ { \\prime } ( h ) = \\sum _ { x : f ( x ) - m _ { \\alpha } ( x ) = c _ { h } } \\frac { \\sqrt { 1 + | m _ { \\alpha } | ^ { 2 } } } { ( \\theta _ { L } ^ { \\perp } \\cdot e _ { 2 } ) M ( f ^ { \\prime } ( x ) , \\alpha ) } \\end{align*}"}
+{"id": "7926.png", "formula": "\\begin{align*} \\Delta v = - e ^ { - 2 K \\lambda } \\frac { K } { t } L _ { t , \\lambda } ^ 2 + \\theta _ 0 \\ , , \\quad \\ , , \\end{align*}"}
+{"id": "781.png", "formula": "\\begin{align*} \\P _ x \\left \\{ ( x _ i ) : \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\phi _ k ( ( x _ i ) ) \\geq \\epsilon \\right \\} \\leq C _ 1 e ^ { - \\alpha _ 1 n } . \\end{align*}"}
+{"id": "3290.png", "formula": "\\begin{align*} \\Theta ( t , x ) = \\frac { 1 } { 2 \\pi } \\frac { x ^ \\perp } { \\abs { x } ^ 2 } \\Bigl ( 1 - e ^ { - \\frac { \\abs { x } ^ 2 } { 4 ( 1 + t ) } } \\Bigr ) , x \\in \\mathbb { R } ^ 2 \\setminus \\{ ( 0 , 0 ) \\} , t \\geq 0 , \\end{align*}"}
+{"id": "9061.png", "formula": "\\begin{align*} v ' : = \\sum _ { l = 0 } ^ { l _ 0 } \\sum _ { m = 1 } ^ { d _ l } w _ { l m } ( \\rho ) Y _ { l m } ( \\theta ) \\end{align*}"}
+{"id": "7315.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { 0 } ^ { 1 } ( 1 - g _ { m _ n } + \\lambda G _ { m _ n } ) d j _ n = - \\kappa \\lambda ^ 2 . \\end{align*}"}
+{"id": "4904.png", "formula": "\\begin{align*} \\partial ^ 2 _ s ( \\rho + e ^ { - \\rho } ) = ( 1 - e ^ { - \\rho } ) \\partial ^ 2 _ s \\rho + e ^ { - \\rho } ( \\partial _ s \\rho ) ^ 2 = ( 1 - e ^ { - \\rho } ) ^ 2 + e ^ { - \\rho } \\big ( ( \\partial _ s \\rho ) ^ 2 + b \\big ) - b . \\end{align*}"}
+{"id": "7826.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { p } \\sum _ { s = 1 } ^ { j - 1 } x ^ { - 2 j - 2 s } & = \\frac { x ^ { - 4 p + 2 } ( 1 - x ^ { 2 p - 2 } ) ( 1 - x ^ { 2 p } ) } { ( 1 - x ^ 2 ) ( 1 - x ^ 4 ) } \\\\ & = \\begin{cases} \\displaystyle \\sum _ { j = 1 } ^ { p / 2 } \\sum _ { s = 1 } ^ { p - 1 } x ^ { - 2 p - 4 j + 2 s } & ( p : ) , \\\\ \\displaystyle \\sum _ { j = 1 } ^ { p } \\sum _ { s = 1 } ^ { ( p - 1 ) / 2 } x ^ { - 2 p - 2 - 2 j + 4 s } & ( p : ) . \\end{cases} \\end{align*}"}
+{"id": "3772.png", "formula": "\\begin{align*} ( \\varepsilon ' _ \\# \\alpha _ \\# ) ( v ) & = \\begin{cases} \\alpha ( v ) & \\alpha ( v ) \\in V _ 1 ( \\varepsilon ' ) ; \\\\ x _ { e ' } ^ { - 1 } \\alpha ( v ) x _ { e ' } & \\alpha ( v ) \\not \\in V _ 1 ( \\varepsilon ' ) . \\end{cases} \\\\ ( \\alpha _ \\# \\varepsilon _ \\# ) ( v ) & = \\begin{cases} \\alpha ( v ) & v \\in V _ 1 ( \\varepsilon ) ; \\\\ \\alpha _ \\# ( x _ { e } ^ { - 1 } v x _ { e } ) & v \\not \\in V _ 1 ( \\varepsilon ) . \\end{cases} \\end{align*}"}
+{"id": "8217.png", "formula": "\\begin{align*} P \\{ \\mathcal { T } ( t ) = x + v _ 0 ( t - s ) \\ | \\ \\mathcal { T } ( s ) = x , N ( s ) = 2 k + 1 \\} = P \\{ \\mathcal { T } ( t - s ) = v _ 0 ( t - s ) \\} . \\end{align*}"}
+{"id": "3773.png", "formula": "\\begin{align*} \\alpha _ \\# ( x _ { e } ^ { - 1 } v x _ { e } ) & = \\alpha _ \\# ( x _ { e } ) ^ { - 1 } \\alpha ( v ) \\alpha _ \\# ( x _ e ) = x _ { e ' } ^ { - 1 } \\alpha ( v ) x _ { e ' } \\end{align*}"}
+{"id": "7387.png", "formula": "\\begin{align*} R ^ + = \\left \\{ \\alpha , \\beta , \\alpha + \\beta , 2 \\alpha + \\beta , 3 \\alpha + \\beta , 3 \\alpha + 2 \\beta \\right \\} \\end{align*}"}
+{"id": "2367.png", "formula": "\\begin{align*} E \\dot x & = ( J - R ) x + G u , \\\\ y & = G ^ T x , \\end{align*}"}
+{"id": "5729.png", "formula": "\\begin{align*} \\mathbf { A } _ { B } ^ { A } = \\omega _ { B } ^ { A } - \\Omega _ { B } ^ { A } \\mathbf { m } \\mathbf { A } _ { B ^ { \\prime } } ^ { A ^ { \\prime } } = \\omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } - \\Omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\overline { \\mathbf { m } } \\end{align*}"}
+{"id": "3637.png", "formula": "\\begin{align*} \\Delta _ { M } = \\frac { 1 } { \\sqrt { a } } \\frac { \\partial } { \\partial r } \\left ( \\sqrt { a } \\frac { \\partial } { \\partial r } \\right ) + \\Delta _ { \\partial B ( o , r ) } = \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + m ( r , \\theta ) \\frac { \\partial } { \\partial r } + \\Delta _ { \\partial B ( o , r ) } , \\end{align*}"}
+{"id": "309.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta _ { ( { \\chi _ { \\mathrm { B C } } } _ { \\mu , \\alpha } , \\chi _ { \\mu \\circ \\mathrm { N m } _ { T ( E ) / S ^ { \\mathrm { o p } } ( F ) } , \\alpha } ) } | _ { F _ { \\alpha } ^ { \\times } } ( \\iota _ { F _ \\alpha } \\alpha ( t ) ) = 1 . \\end{aligned} \\end{align*}"}
+{"id": "7314.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( \\int _ { 0 } ^ { \\infty } ( 1 - g _ { m _ n } ( \\lambda ; x ) ) j _ n ( d x ) + \\lambda \\int _ { 0 } ^ { 1 } G _ { m _ n } ( x ) j _ n ( d x ) \\right ) = - \\kappa \\lambda ^ 2 . \\end{align*}"}
+{"id": "6051.png", "formula": "\\begin{align*} \\int _ { b _ k } ^ { a _ { k + 1 } } \\frac { l _ i ( x ) \\dd x } { w ( x ) } = \\delta _ { k i } , i , k \\in \\{ 1 , \\ldots , g \\} , \\end{align*}"}
+{"id": "2740.png", "formula": "\\begin{align*} [ C , D ] _ t = C D - t D C , \\end{align*}"}
+{"id": "7606.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\langle \\partial _ t v ^ n ( t ) , \\varphi \\rangle & = - \\langle \\mathfrak { A } _ n ( v ^ n ( t ) ) , \\varphi \\rangle , \\\\ ( v ^ n ( 0 ) , \\varphi ) & = ( v _ 0 ^ n , \\varphi ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7564.png", "formula": "\\begin{align*} Z _ { f _ { 5 , a } } ( s , \\chi , B _ 5 ^ a ) = { \\left \\{ \\begin{array} { r l } ( 1 - q ^ { - 1 } ) Z _ { \\tilde { f } _ { 5 , a } } ( s , \\chi , C _ 5 ^ a ) , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } = \\chi _ { \\rm t r i v } , \\\\ 0 , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } \\neq \\chi _ { \\rm t r i v } , \\end{array} \\right . } \\end{align*}"}
+{"id": "6105.png", "formula": "\\begin{align*} \\lambda ( p ) = \\frac { 1 } { 2 } \\left [ p \\ln ( r ) - p \\ln ( p ) - 2 ( 1 - p ) \\ln ( 1 - p ) - p \\right ] + \\sum _ { k = 2 } ^ \\infty d _ k p ^ k \\end{align*}"}
+{"id": "2490.png", "formula": "\\begin{align*} \\mathcal P _ n = \\sum _ { k = 0 } ^ n \\mathcal Q _ k . \\end{align*}"}
+{"id": "965.png", "formula": "\\begin{align*} D ( \\varphi ) : = \\Bigl \\{ x \\in X : \\mathrm { r k } \\bigl \\{ \\varphi ( x ) : \\mathcal A ( x ) \\longrightarrow \\mathcal B ( x ) \\bigr \\} \\le r - 1 \\Bigr \\} . \\end{align*}"}
+{"id": "1838.png", "formula": "\\begin{align*} H ^ * \\begin{bmatrix} A & 0 \\\\ 0 & B \\end{bmatrix} H = \\begin{bmatrix} X ^ * A X + Y ^ * B Y & \\star \\\\ \\star & Y ^ * A Y + X ^ * B X \\end{bmatrix} \\end{align*}"}
+{"id": "1634.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ^ { 2 } _ { \\mathbb { R } ^ d } ) u = \\mu u | u | ^ { d } , u ( 0 , x ) = u _ 0 ( x ) \\in H ^ 2 ( \\mathbb { R } ^ d ) , \\end{align*}"}
+{"id": "3986.png", "formula": "\\begin{align*} u _ 0 ( x ) : = \\begin{cases} u _ 1 ( x ) x \\in \\Omega , \\\\ u _ 1 ( x ) + t d ( x ) ^ 2 x \\in \\Omega _ \\delta \\setminus \\Omega . \\end{cases} \\end{align*}"}
+{"id": "9064.png", "formula": "\\begin{align*} E _ { X } = \\bigoplus _ { i = 1 } ^ { n } p _ { i } \\big ( E \\cap \\bigoplus _ { j = 1 } ^ { i } V _ { j } \\big ) . \\end{align*}"}
+{"id": "353.png", "formula": "\\begin{align*} D = \\sum _ s f _ s V ^ s + D ' + D '' \\end{align*}"}
+{"id": "6164.png", "formula": "\\begin{align*} D ^ T E _ 1 = 0 , A ^ T E _ 1 = \\alpha E _ 1 , B ^ T E _ 1 = \\beta E _ 1 . \\end{align*}"}
+{"id": "6794.png", "formula": "\\begin{align*} d ( \\mathcal { U } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) , \\mathcal { U } ( ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) \\leq ~ \\beta ( \\mathcal { M } ^ { * } _ { \\eta } ( \\alpha , ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } , ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) ) \\mathcal { M } ^ { * } _ { \\eta } ( \\alpha , ( \\overline { w } _ { \\i } ) _ { i = 1 } ^ { \\eta } , ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) \\end{align*}"}
+{"id": "425.png", "formula": "\\begin{align*} V \\omega _ n = \\theta _ \\tau \\theta _ g \\omega _ n = \\theta _ \\tau \\left ( \\sum _ { m = 1 } ^ \\infty g _ m ( \\omega _ n ) e _ m \\right ) = \\theta _ \\tau e _ n = \\tau _ n , \\forall n \\in \\mathbb { N } . \\end{align*}"}
+{"id": "2180.png", "formula": "\\begin{align*} \\sigma _ { 0 } & = \\dfrac { 4 } { 3 ( 2 - 2 \\alpha + \\beta ) + \\sqrt { 9 ( - 2 + 2 \\alpha - \\beta ) ^ 2 - 8 ( 4 - 6 \\alpha - 3 \\beta ) } } , \\intertext { a n d } \\tilde { \\sigma _ { 0 } } & = \\dfrac { 4 } { 3 ( 2 - 2 \\alpha + \\beta ) + \\sqrt { 9 ( - 2 + 2 \\alpha - \\beta ) ^ 2 - 8 ( 6 \\alpha - 8 + 3 \\beta ) } } . \\end{align*}"}
+{"id": "210.png", "formula": "\\begin{align*} \\sum _ { b = 0 } ^ { h _ { n } - 1 } | \\sum _ { i = r _ { n } - k _ { n } + 2 } ^ { r _ { n } } \\mu ( T ^ { k _ n ( h _ n + c _ n ) } ( I _ { n , b } ^ { [ i ] } ) \\cap B ) - \\mu ( I _ { n , b } ^ { [ i ] } ) \\mu ( B ) | \\leq \\sum _ { y = 0 } ^ { h _ { n + 1 } - 1 } | \\mu ( T ^ { h _ { n + 1 } } ( I _ { n + 1 , y } ) \\cap B ) - \\mu ( I _ { n + 1 , y } ) \\mu ( B ) | \\end{align*}"}
+{"id": "2067.png", "formula": "\\begin{align*} q _ H ( z ) = \\lim _ { \\varepsilon \\to 0 } q _ { H _ \\varepsilon } ( z ) = i \\frac { 2 \\kappa } { \\kappa _ 2 } \\cdot \\frac { | \\Gamma ( 1 ) | ^ 2 } { ( \\Gamma ( 1 ) ) ^ 2 } \\cdot \\frac { 1 } { 2 } \\Bigl ( 1 - i \\frac { \\kappa _ 3 } { \\kappa } \\Bigr ) \\Bigl ( \\frac { z } { i } \\Bigr ) ^ 0 = i \\frac { \\kappa - i \\kappa _ 3 } { \\kappa _ 2 } \\ , . \\end{align*}"}
+{"id": "1026.png", "formula": "\\begin{align*} \\widehat { p } _ t ^ s ( \\xi ) = C _ { n , s } G _ s ( t | \\xi | ) , \\end{align*}"}
+{"id": "1243.png", "formula": "\\begin{align*} \\nu _ k ( \\underline y , y ) : = \\mu ( X _ \\geq ( y , \\tilde k ( y ) ) ) . \\end{align*}"}
+{"id": "5231.png", "formula": "\\begin{align*} g ' ( F _ \\ast X , F _ \\ast Y ) = \\lambda ^ 2 g ( X , Y ) ~ ~ X , Y \\in \\Gamma ( k e r F _ \\ast ) ^ \\bot . \\end{align*}"}
+{"id": "2027.png", "formula": "\\begin{align*} C ( x , y ) = \\left \\{ \\begin{array} { l l l } \\displaystyle \\sum _ { t = 0 } ^ { - x - 1 } \\mu ( y - x - t ) \\ , \\mathcal { U } ( t ) & x < 0 y \\geq 0 , \\\\ \\displaystyle \\sum _ { t = - x } ^ { 0 } \\mu ' ( y - x - t ) \\ , \\mathcal { U ' } ( t ) & x \\geq 0 y < 0 , \\\\ 0 & { \\rm o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
+{"id": "4389.png", "formula": "\\begin{align*} u _ n ( \\theta ) = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ( \\theta ) \\right \\} . \\end{align*}"}
+{"id": "491.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb { X } } F ( x ) : = f ( x ) + g ( x ) , \\end{align*}"}
+{"id": "3159.png", "formula": "\\begin{align*} \\tilde B = ( b _ 1 + b _ 2 ) A _ m = \\frac { 1 } { C : A _ m } A _ m , \\quad C : = \\begin{pmatrix} 0 & \\frac { 1 } { 2 \\delta } \\\\ \\frac { 1 } { 2 \\delta } & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "8566.png", "formula": "\\begin{align*} i ! ( t - 1 - i ) ! = d _ { J , w } + d _ { I , w } . \\end{align*}"}
+{"id": "1030.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n + 1 } _ + } \\left | \\frac { \\partial u ( x , t ) } { \\partial t } \\right | ^ 2 t ^ { 1 - \\beta } d x d t = a ( n , s , \\beta ) \\int _ { \\mathbb { R } ^ n } | \\xi | ^ { \\beta } ( \\widehat { f } ( \\xi ) ) ^ 2 d \\xi . \\end{align*}"}
+{"id": "2036.png", "formula": "\\begin{align*} \\rho ( n ) : = \\left \\{ \\begin{array} { l l l } \\displaystyle \\sum _ { k = 0 } ^ { + \\infty } \\mathcal { A } ( k ) \\ , \\nu ( n - k ) & { \\rm i f } & n \\leq - 1 , \\\\ \\displaystyle \\sum _ { k = 0 } ^ { + \\infty } \\mathcal { A } ' ( - k ) \\ , \\nu ( n + k ) & { \\rm i f } & n \\geq 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1854.png", "formula": "\\begin{align*} H = \\biggl ( - \\frac { 2 \\tilde { \\kappa } ( \\gamma - 1 ) \\delta ^ { i j } u _ i u _ j } { \\sqrt { 6 } u + \\beta + 2 \\tau ^ { \\frac { 2 } { 3 } } } , 0 , 0 \\biggr ) ^ T . \\end{align*}"}
+{"id": "2327.png", "formula": "\\begin{align*} \\delta _ 1 ^ { - \\frac { N - p } { p - 1 } } - ( 2 - \\delta _ 1 ) ^ { - \\frac { N - p } { p - 1 } } = C ( N , p ) ^ { - 1 } M , \\delta _ 2 = \\ ( \\frac { M } { C ( N , p ) } \\ ) ^ { \\frac { p - 1 } { N - p } } . \\end{align*}"}
+{"id": "8862.png", "formula": "\\begin{align*} \\mathfrak { S } _ { n ; p , q } ^ { z , w } = ( 2 \\pi ^ { n } ) ^ { - 1 } \\Gamma ( n ) d ( n , p , q ) ( | z | | w | ) ^ { p + q } R _ { p , q } ^ { n - 2 } \\left ( \\left \\langle \\frac { z } { | z | } , \\frac { w } { | w | } \\right \\rangle \\right ) . \\end{align*}"}
+{"id": "8613.png", "formula": "\\begin{align*} x _ { n + 1 } = \\alpha _ n u + ( 1 - \\alpha _ n ) [ \\beta _ n x _ n + ( 1 - \\beta _ n ) T x _ n ] \\end{align*}"}
+{"id": "6420.png", "formula": "\\begin{align*} & \\phi ( \\tilde X _ 1 , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) + \\phi ( X _ 1 , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) - 2 \\phi ( X , Y , t ) \\\\ & = \\langle \\nabla _ X ^ 2 \\phi ( X , Y , t ) ( X _ 1 - X ) , ( X _ 1 - X ) \\rangle - 2 ( X \\cdot \\nabla _ Y - \\partial _ t ) \\phi ( X , Y , t ) ( \\tilde t - t ) + { o } ( \\epsilon ^ 2 ) , \\mbox { a s } \\epsilon \\to 0 . \\end{align*}"}
+{"id": "3034.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } \\Delta u = f \\ \\ \\ \\mathrm { i n } \\ \\ \\ \\ \\Omega _ 1 , \\\\ \\ \\ u = 0 \\ \\ \\ \\mathrm { o n } \\ \\ \\ ( \\partial \\Omega ) _ 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "4627.png", "formula": "\\begin{align*} A ^ { G } : = \\left \\{ \\omega : G \\to A \\right \\} \\end{align*}"}
+{"id": "3151.png", "formula": "\\begin{align*} r ( y _ 1 , y _ 2 ) & : = 1 + \\frac { 1 } { 4 } ( \\cos ( 2 \\pi y _ 1 ) - 2 \\sin ( 2 \\pi y _ 1 ) ) \\sin ( 2 \\pi y _ 2 ) = r _ 1 ( y _ 1 + y _ 2 ) + r _ 2 ( y _ 1 - y _ 2 ) , \\\\ a ( y _ 1 , y _ 2 ) & : = 1 - \\frac { 1 } { 2 } \\sin ( 2 \\pi y _ 1 ) \\sin ( 2 \\pi y _ 2 ) \\end{align*}"}
+{"id": "5132.png", "formula": "\\begin{align*} \\varphi _ { n } ( \\lambda ) - \\varphi _ { n } ( \\lambda b ) & \\underset { n \\rightarrow \\infty } { = } \\lambda ^ { 2 } ( b ^ { 2 } - 1 ) \\frac { ( n + 1 ) ^ { 3 } - n ^ { 3 } } { 4 n ^ { 3 } ( n + 1 ) ^ { 3 } } + o _ { \\lambda , b } \\left ( \\frac { 1 } { n ^ { 4 } } \\right ) \\\\ & \\underset { n \\rightarrow \\infty } { = } \\frac { 3 \\lambda ^ { 2 } ( b ^ { 2 } - 1 ) } { 4 n ^ { 4 } } + o _ { \\lambda , b } \\left ( \\frac { 1 } { n ^ { 4 } } \\right ) . \\end{align*}"}
+{"id": "4447.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 1 5 } = \\frac { 1 } { 5 } + \\frac { 1 } { 5 } = \\frac { 2 } { 5 } \\end{align*}"}
+{"id": "8807.png", "formula": "\\begin{align*} \\int _ { w \\wedge v \\in ( - b , - a ) } \\| R _ w \\| _ 2 \\| R _ v \\| _ 2 \\mathbf { P } _ { a , b } ( w , v ) d w d v = 2 \\int _ { - b } ^ { - a } \\int _ 0 ^ \\infty \\| R _ { x + v } \\| _ 2 \\| R _ v \\| _ 2 \\mathbf { P } _ { a , b } ( x + v , v ) d x d v . \\end{align*}"}
+{"id": "1440.png", "formula": "\\begin{align*} \\mathbb { S } ^ v _ { \\beta } \\setminus \\mathcal { P } _ v = \\bigcup _ { j = 1 } ^ { k _ v } I ^ v _ j , \\end{align*}"}
+{"id": "1691.png", "formula": "\\begin{align*} C ( n ) = \\sum _ { j } ( - 1 ) ^ { n + j + m } \\binom { \\frac { k - 2 } { 2 } - m } { j } \\binom { \\frac { k - 2 } { 2 } + m } { m - n + j } . \\end{align*}"}
+{"id": "5750.png", "formula": "\\begin{align*} \\theta ^ { A A ^ { \\prime } } = ( \\mathbf { L } ^ { - 1 } \\mathbf { ) } _ { B ^ { \\prime } } ^ { A ^ { \\prime } } ( \\mathbf { L } ^ { - 1 } \\mathbf { ) } _ { B } ^ { A } \\mathbf { d x } ^ { B B ^ { \\prime } } = ( \\delta _ { B ^ { \\prime } } ^ { A ^ { \\prime } } + \\omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\overline { \\mathbf { m } } \\mathbf { ) } ( \\delta _ { B } ^ { A } + \\omega _ { B } ^ { A } \\mathbf { m ) d x } ^ { B B ^ { \\prime } } \\end{align*}"}
+{"id": "7569.png", "formula": "\\begin{align*} Z _ { \\tilde { f } _ { 6 , a } } ( s , \\chi , C _ 6 ^ a ) = { \\left \\{ \\begin{array} { r l } F _ { 6 , a } ( q ^ { - s } ) , \\ \\ & 0 \\le a < \\dfrac { r } { p } , \\\\ \\dfrac { \\tilde F _ { 6 , a } ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\ \\ & { \\dfrac { r } { p } \\le a < \\omega } , \\end{array} \\right . } \\end{align*}"}
+{"id": "7970.png", "formula": "\\begin{align*} w + q ' = \\sum a _ i , q ' = \\sum b _ i . \\end{align*}"}
+{"id": "7409.png", "formula": "\\begin{align*} \\mu ( s \\widetilde { \\alpha } , \\sigma ) = \\gamma ( G / P ) ^ 2 q _ F ^ { n ( \\sigma \\times \\Pi ( \\sigma ) ) ) - n ( \\sigma ) } \\end{align*}"}
+{"id": "886.png", "formula": "\\begin{align*} \\mathbf { E } [ \\| \\mathcal { X } ^ { t + 1 } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } | \\mathcal { X } ^ { t } ] & = \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } - \\mathbf { E } _ { i ^ { t } \\sim \\mathbf { p } } [ f _ { i ^ { t } } ( \\mathcal { X } ^ { t } ) ] \\leq ( 1 - \\delta _ { \\mathbf { p } } ^ { 2 } ( \\mathcal { Q } , \\boldsymbol { \\mathcal { S } } ) ) \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } . \\end{align*}"}
+{"id": "1047.png", "formula": "\\begin{align*} \\delta = T ^ { - 2 } ( n \\alpha ^ 2 ) ^ { - 1 } \\mbox { a n d } \\tau = \\epsilon + ( 1 - \\epsilon ) ( 6 / M ) ^ k + 4 \\sqrt { 2 \\log ( 1 2 T / ( M \\delta ) ) / ( n \\alpha ^ 2 ) } , \\end{align*}"}
+{"id": "6458.png", "formula": "\\begin{align*} \\partial _ x ( M _ 2 \\partial _ x M _ 3 - M _ 3 \\partial _ x M _ 2 ) & = M _ 2 \\partial _ { x x } M _ 3 - M _ 3 \\partial _ { x x } M _ 2 \\\\ & \\stackrel { \\eqref { E L _ i n _ M } } { = } - M _ 2 ( \\Lambda ( x ) M _ 3 ) + M _ 3 ( \\Lambda ( x ) M _ 2 ) = 0 , \\end{align*}"}
+{"id": "165.png", "formula": "\\begin{align*} \\nu _ { j } ( \\theta , \\varkappa ) = \\theta \\varkappa , \\ \\mbox { f o r } \\ \\theta \\in ( \\theta _ j , \\lambda _ j ] , \\ j = 1 , \\dots , n . \\end{align*}"}
+{"id": "9198.png", "formula": "\\begin{align*} Z _ j ( \\varphi , \\Psi _ j ; ( \\Psi _ k ) _ { k < j } ) = \\sum _ { X \\in \\mathcal { P } _ { j } } e ^ { U _ j ( \\Lambda \\backslash X ) } ( K _ j ( X ; ( \\Psi _ k ) _ { k < j } ) + \\Psi _ j ( X ) ) = \\sum _ { X \\in \\mathcal { P } _ { j + 1 } } e ^ { U _ j ( \\Lambda \\backslash X ) } \\bar { K } ^ { \\Psi } _ j ( X ) \\end{align*}"}
+{"id": "5311.png", "formula": "\\begin{align*} \\mathrm { R i c } ^ { * } ( X , Y ) = - \\mathrm { t r a c e } ( A _ { \\xi } \\circ A _ { \\eta } ) \\ , g ( X , P Y ) - ( n - 2 ) \\lambda \\ , g ( X , A _ { \\eta } Y ) . \\end{align*}"}
+{"id": "738.png", "formula": "\\begin{align*} G ( z , w ) = \\frac { 1 } { 2 \\pi } ( - \\log | z - w | + H ( z , w ) ) \\end{align*}"}
+{"id": "6355.png", "formula": "\\begin{align*} ( - m _ 0 '' / m _ 0 ) ( x ) = 2 ( 3 - x ^ 2 ) / ( 1 + x ^ 2 ) ^ 2 . \\end{align*}"}
+{"id": "7096.png", "formula": "\\begin{align*} \\nu = \\bigoplus _ { i = 1 } ^ k \\rho _ i ^ { - 1 } \\gamma ( V ) . \\end{align*}"}
+{"id": "4680.png", "formula": "\\begin{align*} \\sum _ { m = \\ell _ 2 } ^ { \\infty } \\sum _ { k = \\max \\{ 0 , \\ell _ 1 + \\ell _ 2 - m \\} } ^ { \\ell _ 1 } A _ { m , k } = \\sum _ { k = 0 } ^ { \\ell _ 1 } \\sum _ { m = \\ell _ 1 + \\ell _ 2 - k } ^ { \\ell _ 1 + \\ell _ 2 } A _ { m , k } + \\sum _ { k = 0 } ^ { \\ell _ 1 } \\sum _ { m = \\ell _ 1 + \\ell _ 2 + 1 } ^ { \\infty } A _ { m , k } , \\end{align*}"}
+{"id": "4342.png", "formula": "\\begin{align*} p _ { i } ^ { \\mathbf { A } } ( \\vec { 0 } ^ { \\mathbf { A } } , \\vec { 1 } ^ { \\mathbf { A } } ) = q _ { i } ^ { \\mathbf { A } } ( \\vec { 0 } ^ { \\mathbf { A } } , \\vec { 1 } ^ { \\mathbf { A } } ) . \\end{align*}"}
+{"id": "2403.png", "formula": "\\begin{align*} E _ { 1 1 } ^ T = - E _ { 1 1 } , E _ { 2 2 } ^ T = - E _ { 2 2 } , 0 = \\dot E _ { 1 1 } , 0 = A _ { 1 2 } + \\dot E _ { 1 2 } , A _ { 2 2 } ^ T = A _ { 2 2 } + \\dot E _ { 2 2 } . \\end{align*}"}
+{"id": "3604.png", "formula": "\\begin{align*} d _ { j } ^ - ( \\mu + \\epsilon _ j ) = ( - 1 ) ^ { j + 1 } j ( p - j + 1 ) . \\end{align*}"}
+{"id": "7686.png", "formula": "\\begin{align*} \\omega ^ { \\prime } = \\sum _ { \\{ 1 \\leq k \\leq k \\mid W \\subseteq V ( f _ { k } ) \\} } \\lambda _ { k } \\frac { d f _ { k } } { f _ { k } } , \\end{align*}"}
+{"id": "5044.png", "formula": "\\begin{align*} g _ 1 ( \\omega ) + \\cdots + g _ m ( \\omega ) = 0 \\qquad . \\end{align*}"}
+{"id": "8222.png", "formula": "\\begin{align*} ( a ) \\ \\ \\tau \\left ( \\frac { 1 } { \\bar { z } } \\right ) = \\frac { 1 } { \\overline { \\tau ( z ) } } ( b ) \\ \\ \\tau ^ { - 1 } \\left ( \\frac { 1 } { \\bar { z } } \\right ) = \\frac { 1 } { \\overline { \\tau ^ { - 1 } ( z ) } } . \\end{align*}"}
+{"id": "1115.png", "formula": "\\begin{align*} R _ 0 = ( 1 - \\varepsilon ) R _ 1 + \\varepsilon P _ { r } , \\end{align*}"}
+{"id": "3412.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } \\omega + v \\cdot \\nabla \\omega - \\mu \\Delta \\omega = \\dfrac { v _ { r } } { r } \\omega - \\partial _ z \\big ( \\Sigma B \\big ) , \\\\ { \\omega } _ { \\vert t = 0 } = \\omega ^ { 0 } . \\end{array} \\right . \\end{align*}"}
+{"id": "2976.png", "formula": "\\begin{align*} \\frac { W _ { j - 1 } W _ j } { \\sqrt { n } } = \\frac { W _ { j - 1 } \\eta _ j } { \\sqrt { n } } + \\frac { g ^ { \\prime \\prime } ( 0 ) } { 2 g ^ \\prime ( 0 ) } \\frac { W _ { j - 1 } \\eta _ j ^ 2 } { n } + O ( n ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "1481.png", "formula": "\\begin{align*} \\lambda & = \\frac { k ( k - 1 ) ( k - 2 ) ( m - 1 ) ! ( m - 2 ) ! } { ( m + 1 ) ( m ^ 2 - 2 ) | K _ \\Delta | } \\\\ & = \\frac { 1 0 5 \\cdot 1 0 4 \\cdot 1 0 3 \\cdot 3 7 ! \\cdot 3 6 ! } { 3 9 \\cdot ( 3 8 ^ 2 - 2 ) \\cdot 2 ^ 9 \\cdot 3 ! \\cdot 4 ! \\cdot 5 ! ^ 2 } \\\\ & \\approx 9 . 6 \\cdot 1 0 ^ { 7 6 } . \\end{align*}"}
+{"id": "4655.png", "formula": "\\begin{align*} \\mathcal { T } _ { i j } = \\sum _ { m = 0 } ^ { \\infty } \\frac { C _ { i j } ^ m } { m ! } \\partial _ i ^ m \\partial _ j ^ m , \\ \\ i , j = 1 , \\ldots , N , \\ j \\neq i . \\end{align*}"}
+{"id": "9079.png", "formula": "\\begin{align*} \\int _ { Z } D ' D ^ { * } \\phi ( z ) \\overline { \\psi ( z ) } \\ , d z & = \\int _ { Z } \\phi ( z ) \\overline { D D '^ { * } \\psi ( z ) } \\ , d z = \\int _ { Z } \\phi ( z ) \\overline { D '^ { * } D \\psi ( z ) } \\ , d z \\\\ & = \\int _ { Z } D ^ { * } D ' \\phi ( z ) \\overline { \\psi ( z ) } \\ , d z , \\end{align*}"}
+{"id": "587.png", "formula": "\\begin{align*} \\mathbb { E } ^ \\dagger [ J ^ \\dagger _ { t _ n , t _ { n + 1 } } ] = ( A ^ { \\rm T } A ) : C \\Delta t + \\mathcal { O } ( \\Delta t ^ 2 ) . \\end{align*}"}
+{"id": "5405.png", "formula": "\\begin{align*} v _ 1 = \\tfrac { x _ 1 + y _ 1 } { 2 } \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} + \\tfrac { x _ 1 - y _ 1 } { 2 } \\begin{pmatrix} 1 \\\\ - 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "7355.png", "formula": "\\begin{align*} \\varepsilon : = ( d \\epsilon - k ) / k . \\end{align*}"}
+{"id": "7016.png", "formula": "\\begin{align*} K _ { ( N _ X , N _ Y ) } = \\begin{pmatrix} 1 - \\sigma ^ 2 _ { \\theta } - \\alpha & \\rho - \\sigma ^ 2 _ { \\theta } - \\alpha \\\\ \\rho - \\sigma ^ 2 _ { \\theta } - \\alpha & 1 - \\sigma ^ 2 _ { \\theta } - \\alpha \\end{pmatrix} . \\end{align*}"}
+{"id": "8797.png", "formula": "\\begin{align*} s _ 6 = \\frac { 1 } { \\eta _ 3 - \\eta _ 2 } \\left ( s _ 4 \\overline { \\left ( \\frac { { b _ 3 } } { { b _ 1 } } \\right ) } ( \\eta _ 1 - \\eta _ 2 ) + s _ 5 \\overline { \\left ( \\frac { { b _ 2 } } { { b _ 1 } } \\right ) } ( \\eta _ 3 - \\eta _ 1 ) \\right ) . \\end{align*}"}
+{"id": "7807.png", "formula": "\\begin{align*} \\begin{matrix} p _ 1 ^ * : & N & \\rightarrow & \\tilde M ^ { \\circ } \\\\ & n & \\mapsto & \\{ \\cdot , n \\} _ 1 \\end{matrix} , \\{ ( n ' , m ' ) , n \\} _ 1 = \\{ n ' , n \\} + \\langle n , m ' \\rangle . \\end{align*}"}
+{"id": "5587.png", "formula": "\\begin{align*} 0 = \\nabla _ 0 ( \\nabla S ) _ { 0 i j } = - \\kappa ^ m \\nabla _ m S _ { i j } + \\kappa _ i \\nabla _ 0 S _ { 1 j } + \\kappa _ j \\nabla _ 0 S _ { i 1 } = \\kappa _ i \\kappa _ j S _ { 1 1 } , \\end{align*}"}
+{"id": "5219.png", "formula": "\\begin{align*} & \\ell _ 1 = k _ 1 + k _ 0 \\cos ( \\theta ) , \\ell _ 2 = k _ 2 + k _ 0 \\sin ( \\theta ) \\\\ & \\theta ' b _ { 1 1 } b _ { 2 2 } = \\gamma \\ . \\end{align*}"}
+{"id": "1712.png", "formula": "\\begin{align*} F ( \\infty ) - F ( 0 ^ + ) = \\frac { ( - 1 ) ^ { m } ( k - 1 ) } { \\binom { k - 2 } { \\frac { k - 2 } { 2 } - m } } \\sum _ { s \\in \\Z } \\binom { k } { \\frac { k - 2 } { 2 } - m + 2 s } = \\frac { ( - 1 ) ^ { m } \\frac { ( k - 1 ) } { 2 } } { \\binom { k - 2 } { \\frac { k - 2 } { 2 } - m } } \\left ( ( 1 + 1 ) ^ k + \\frac { ( 1 - 1 ) ^ k } { ( - 1 ) ^ { \\frac { k - 2 } { 2 } + m } } \\right ) = \\frac { 2 ^ { k - 1 } ( - 1 ) ^ { m } ( k - 1 ) } { \\binom { k - 2 } { \\frac { k - 2 } { 2 } - m } } , \\end{align*}"}
+{"id": "1364.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { d } \\sum _ { k = 1 } ^ d \\lambda _ k \\right ) ^ r \\leq \\frac { 1 } { d } \\sum _ { k = 1 } ^ d \\lambda _ k ^ r . \\end{align*}"}
+{"id": "6393.png", "formula": "\\begin{align*} ( \\tilde X , \\tilde Y , \\tilde t ) \\circ ( X , Y , t ) = ( \\tilde X + X , \\tilde Y + Y - t \\tilde X , \\tilde t + t ) , \\end{align*}"}
+{"id": "7655.png", "formula": "\\begin{align*} L _ { \\omega } ( \\eta ) = \\nabla _ { \\omega } ( \\iota _ { E } ( \\eta ) ) + \\iota _ { E } ( ( \\nabla _ { \\omega } ) ( \\eta ) ) \\end{align*}"}
+{"id": "1851.png", "formula": "\\begin{align*} B ^ 0 \\partial _ { \\tau } \\mathbf { U } + \\tau ^ { \\gamma - \\frac { 7 } { 3 } } B ^ i \\partial _ { i } \\mathbf { U } = \\frac { 1 } { \\tau } \\mathcal { B } \\mathbb { P } \\mathbf { U } , \\end{align*}"}
+{"id": "3166.png", "formula": "\\begin{align*} A ( y ) = \\mathrm { d i a g } ( a _ 1 ( y ) , \\dots , a _ n ( y ) ) \\quad y \\in \\R ^ n \\end{align*}"}
+{"id": "3586.png", "formula": "\\begin{align*} d ( \\lambda ) : = \\prod _ { j = 1 } ^ 3 \\frac { ( - 1 ) ^ { \\frac { \\lambda _ j ( \\lambda _ j + 1 ) } { 2 } } } { \\lambda _ j ! j ^ { \\lambda _ j } } \\prod _ { k = 0 } ^ { \\lambda _ j - 1 } \\prod _ { \\ell = 1 } ^ { j - 1 } ( k - \\lambda _ \\ell - j + \\ell + 1 - [ k - \\lambda _ \\ell ] _ 2 ) . \\end{align*}"}
+{"id": "1796.png", "formula": "\\begin{align*} Z : = \\{ \\textstyle { \\frac { 1 } { n } } \\ | \\ n \\in \\mathbb { N } \\} \\cup \\{ 0 \\} \\subset [ 0 , 1 ] \\end{align*}"}
+{"id": "7499.png", "formula": "\\begin{align*} b _ 0 = 1 - \\Big ( \\dfrac { j _ 0 m } { i _ 0 } - \\Big \\lfloor \\dfrac { j _ 0 m } { i _ 0 } \\Big \\rfloor \\Big ) : = \\dfrac { n _ m ^ { ' } } { i _ 0 } , \\end{align*}"}
+{"id": "1541.png", "formula": "\\begin{align*} \\epsilon ^ A = \\epsilon ^ A _ B \\lambda ^ B \\end{align*}"}
+{"id": "2356.png", "formula": "\\begin{align*} A _ s ( 0 ) : = V _ s ( 0 ) ^ T J _ n V _ s ( 0 ) . \\end{align*}"}
+{"id": "7952.png", "formula": "\\begin{align*} h = f g : L \\to N = L / \\pi ^ { - 1 } ( H ) \\end{align*}"}
+{"id": "4719.png", "formula": "\\begin{align*} u _ { i 1 } ' : = ( v _ i ' , w _ i ' ) u ' _ { i 2 } = ( x ' _ i , y ' _ i ) i = 1 , 2 , \\ldots , e , \\ z ' _ j = ( v ' _ { e + j } , w ' _ { e + j } ) j = 1 , 2 \\ldots , d . \\end{align*}"}
+{"id": "3992.png", "formula": "\\begin{align*} D _ i \\phi = 2 K d d _ i - d _ i \\\\ D _ { i j } \\phi = 2 K d d _ { i j } + 2 K d _ i d _ j - d _ { i j } \\end{align*}"}
+{"id": "3186.png", "formula": "\\begin{align*} u ( x ) : = g ( x ) = 8 x _ 1 ^ 3 - 3 x _ 1 x _ 3 ^ 2 \\end{align*}"}
+{"id": "2682.png", "formula": "\\begin{align*} z + \\lim \\limits _ { t \\to \\infty } { Q \\big ( z + ( 1 - z ) ( 1 - p _ t ) \\big ) - ( 1 - p _ t ) - z p _ t \\over p _ t \\big ( 1 - Q ' ( 1 - p _ t ) \\big ) } & = z + \\lim \\limits _ { x \\rightarrow 1 - } { Q \\big ( z + ( 1 - z ) x \\big ) - \\big ( z + ( 1 - z ) x \\big ) \\over ( 1 - x ) ( 1 - Q ' ( x ) ) } \\\\ & = z + ( 1 - z ) ~ = 1 \\end{align*}"}
+{"id": "4400.png", "formula": "\\begin{align*} u _ n ( \\theta ) & = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ( \\theta ) \\right \\} \\\\ & = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ( \\theta ) x _ i < x _ i ^ * i = 1 , \\ldots , n \\right \\} \\\\ & = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ^ { ( n ) } ( \\theta ) \\right \\} \\end{align*}"}
+{"id": "7250.png", "formula": "\\begin{align*} u ( \\gamma ) = S ( \\gamma ) ^ { p / 2 } K ( \\gamma ) , v ( \\gamma ) = S ( \\gamma ) ^ { 1 - p } L ( \\gamma ) \\end{align*}"}
+{"id": "6622.png", "formula": "\\begin{align*} \\norm { D ^ m \\phi } ^ 2 \\leq C _ { m , V } \\sum _ { k = 0 } ^ m \\norm { H ^ k \\phi } . \\end{align*}"}
+{"id": "1316.png", "formula": "\\begin{align*} \\operatorname { T r a } ( S _ { f , \\tau } ) = \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) . \\end{align*}"}
+{"id": "3741.png", "formula": "\\begin{align*} ( \\tilde { d } _ 1 ' , \\tilde { d } _ 1 ) ( I ( f ) ) & : = \\frac { \\zeta ( 2 ) q ^ n } { \\zeta ( - 1 ) } \\big ( I ( f ) ( \\varpi ^ n ( 1 , 0 ) ) - I ( f ) ( 0 ) , I ( f ) ( \\varpi ^ n ( 0 , 1 ) ) - I ( f ) ( 0 ) \\big ) \\end{align*}"}
+{"id": "2329.png", "formula": "\\begin{align*} \\phi _ R ( | x | ) = \\begin{cases} 1 \\quad & \\ , \\ , | x | \\le 1 , \\\\ \\frac { \\log \\frac { R } { | x | } } { \\log R } & \\ , \\ , 1 < | x | < R , \\\\ 0 & \\ , \\ , | x | \\ge R . \\end{cases} \\end{align*}"}
+{"id": "6462.png", "formula": "\\begin{align*} \\frac { d } { d t } E ( m ) = - \\alpha \\int ( | \\delta E ( m ) | ^ 2 - | m \\cdot \\delta E ( m ) | ^ 2 ) \\ , d x + \\alpha h ( t ) \\int ( m \\wedge e _ 1 ) \\cdot ( m \\wedge \\delta E ( m ) ) \\ , d x . \\end{align*}"}
+{"id": "1154.png", "formula": "\\begin{align*} & \\Omega ( x , . . . , x , \\Omega ( x , . . . , x , y ) ) = \\Omega ( x , . . . , x , y ) , \\\\ & \\underbrace { t x + . . . + t x } _ { m - 1 } + t \\Omega ( x , . . . , x , y ) = \\underbrace { t x + . . . + t x } _ { m - 1 } + t y , \\\\ & t ( \\underbrace { t x + . . . + t x } _ { m - 1 } + t y ) = t y , \\\\ & \\underbrace { t x + . . . + t x } _ { m - 1 } + t y + t x = y + t x \\\\ & x + t y = y + t x \\implies t = 1 . \\end{align*}"}
+{"id": "4718.png", "formula": "\\begin{gather*} v _ i ' = ( v _ i , - b _ i ) , \\ w _ i ' = ( w _ i , c _ i ) , \\ x _ i ' = ( x _ i , - r _ i ) , \\ y _ i ' = ( y _ i , s _ i ) \\ i = 1 , 2 , \\ldots , e , \\\\ v _ i ' = ( v _ i , 0 ) , w _ i ' = ( w _ i , 0 ) i = e + 1 , \\ldots , e + d , \\end{gather*}"}
+{"id": "190.png", "formula": "\\begin{align*} \\frac { r _ { n } ^ { 2 } } { h _ { n } } & \\leq \\frac { ( f ( c _ { n } ) ) ^ { 2 } } { h _ { n } t _ { n - 1 } ^ { 2 } ( c _ { n } - c _ { n - 1 } ) ^ { 2 } } \\leq \\frac { ( c _ { n } ^ { 3 / 2 } ) ^ { 2 } } { h _ { n } c _ { n } ^ { 2 } } \\Big { ( } \\frac { c _ { n } } { c _ { n } - c _ { n - 1 } } \\Big { ) } ^ { 2 } \\frac { 1 } { t _ { n - 1 } ^ { 2 } } = \\frac { c _ { n } } { h _ { n } } \\Big { ( } \\frac { 1 } { 1 - \\frac { c _ { n - 1 } } { c _ { n } } } \\Big { ) } ^ { 2 } \\frac { 1 } { t _ { n - 1 } ^ { 2 } } \\to 0 . \\end{align*}"}
+{"id": "2807.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ n } \\prod _ { i = 1 } ^ n x _ { \\sigma ( i ) } ^ { a _ i } & = \\sum _ { \\sigma \\in S _ n } \\prod _ { i = 1 } ^ n x _ { \\sigma ( i ) } ^ { b _ i + c _ i } = \\sum _ { \\sigma \\in S _ n } \\prod _ { i = 1 } ^ n x _ { \\sigma ( i ) } ^ { b _ i } \\prod _ { i = 1 } ^ n x _ { \\sigma ( i ) } ^ { c _ i } . \\end{align*}"}
+{"id": "3442.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\to \\infty } \\mu _ { t } ( [ a ] ) = 0 , \\qquad \\forall a \\geq 2 . \\end{align*}"}
+{"id": "6533.png", "formula": "\\begin{align*} & \\int \\phi _ m ( x _ 1 ) \\phi _ m ( x _ 2 ) q \\left ( \\frac { x _ 1 - x _ 2 } { \\sqrt { 2 } } \\right ) q \\left ( \\frac { x _ 1 + x _ 2 } { \\sqrt { 2 } } \\right ) p _ 0 ^ m ( x _ 1 ) p _ 0 ^ m ( x _ 2 ) d ( x _ 1 , x _ 2 ) = \\\\ & \\int \\phi _ m \\Big ( \\frac { w _ 1 + w _ 2 } { \\sqrt { 2 } } \\Big ) \\phi _ m \\Big ( \\frac { w _ 1 - w _ 2 } { \\sqrt { 2 } } \\Big ) q ( w _ 1 ) q ( w _ 2 ) p _ 0 ^ m \\left ( \\frac { w _ 1 - w _ 2 } { \\sqrt { 2 } } \\right ) p _ 0 ^ m \\left ( \\frac { w _ 1 + w _ 2 } { \\sqrt { 2 } } \\right ) d ( w _ 1 , w _ 2 ) . \\end{align*}"}
+{"id": "297.png", "formula": "\\begin{align*} \\begin{aligned} \\widehat { \\iota } : \\mathrm { I n d } _ { W _ L } ^ { W _ F } \\widehat { S } & \\rightarrow \\widehat { S } , \\\\ f & \\mapsto \\sum _ { g _ i \\in W _ L \\backslash W _ F } g _ { i } ^ { - 1 } f ( g _ { i } ) . \\end{aligned} \\end{align*}"}
+{"id": "3828.png", "formula": "\\begin{align*} \\begin{aligned} \\chi _ { 1 0 } ( \\tau ) & = \\tilde { q } p q \\prod \\limits _ { ( r , s , t ) > 0 } ( 1 - \\tilde { q } ^ r p ^ s q ^ t ) ^ { c ( 4 r t - s ^ 2 ) } \\\\ & = \\tilde { q } p q - 2 \\tilde { q } q - 1 6 \\tilde { q } p q ^ 2 + \\cdots \\end{aligned} \\end{align*}"}
+{"id": "1184.png", "formula": "\\begin{align*} \\{ ( p , s ) \\in C ( \\overline { V } ) \\times \\C ^ * : w = s \\} \\end{align*}"}
+{"id": "6219.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p \\beta _ r u _ r = ( x , C _ p U _ n ) \\rightarrow 0 \\hbox { i n } C ^ 0 _ { l o c } ( [ T , + \\infty ) ; \\mathcal H _ { 0 } ) \\cap C ^ 1 _ { l o c } ( [ T , + \\infty ) ; \\mathcal H _ { - 1 } ) \\end{align*}"}
+{"id": "7050.png", "formula": "\\begin{align*} ( E _ { C _ 2 } \\wedge \\widetilde { E C } _ 2 ) ^ { C _ 2 } ( V ) = _ { W \\supset V } \\Omega ^ { ( W - V ) ^ { C _ 2 } } E _ { C _ 2 } ( W ) ^ { C _ 2 } . \\end{align*}"}
+{"id": "758.png", "formula": "\\begin{align*} \\mathsf { H } _ { u n i v } : = & \\tilde { \\Lambda } [ T _ q , U _ l , \\langle d \\rangle , q \\nmid N p ^ n , l \\mid N p \\ell ^ { 2 r } , d \\in \\Delta _ { N p ^ n \\ell ^ { 2 r } } ] , \\\\ \\mathsf { H } _ { u n i v } ^ B : = & \\tilde { \\Lambda } [ \\tilde { U } _ \\ell , T _ q , U _ l , \\langle d \\rangle , q \\nmid N p ^ n , l \\mid N p , d \\in \\Delta _ { N p ^ n \\ell ^ { 2 r } } ] \\end{align*}"}
+{"id": "5017.png", "formula": "\\begin{align*} e _ 1 = ( q H , - p H ) , e _ 2 = ( E , 0 ) , e _ 3 = ( V , 0 ) , e _ 4 = ( 0 , E ) , e _ 5 = ( 0 , V ) , e _ 6 = ( p H , q H ) , \\end{align*}"}
+{"id": "7503.png", "formula": "\\begin{align*} \\frac { \\partial g } { \\partial u } ( P ) = & \\sum \\limits _ { i = i _ 0 } ^ { \\infty } i \\alpha _ i u _ 0 ^ { i - 1 } = \\pi ^ { { \\rm o r d } ( \\alpha _ { i _ 0 } ) } \\sum \\limits _ { i = i _ 0 } ^ { \\infty } i \\pi ^ { - { \\rm o r d } ( \\alpha _ { i _ 0 } ) } \\alpha _ i u _ 0 ^ { i - 1 } \\\\ = & \\pi ^ { { \\rm o r d } ( \\alpha _ { i _ 0 } ) } ( \\pi ^ { - { \\rm o r d } ( \\alpha _ { i _ 0 } ) } \\alpha _ { i _ 0 } + \\sum \\limits _ { i = i _ 0 + 1 } ^ { \\infty } i \\pi ^ { - { \\rm o r d } ( \\alpha _ { i _ 0 } ) } \\alpha _ i u _ 0 ^ { i - 1 } ) . \\end{align*}"}
+{"id": "6479.png", "formula": "\\begin{align*} \\sigma ( \\mathbf { i } ) = \\sigma ( \\zeta _ 4 ) = \\sigma ( \\zeta _ { 4 n } ^ n ) = \\zeta _ { 4 n } ^ { \\eta ( \\sigma ) \\cdot n } = \\zeta _ { 4 } ^ { \\eta ( \\sigma ) } = \\mathbf { i } ^ { \\eta ( \\sigma ) } . \\end{align*}"}
+{"id": "1346.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j , k \\leq n } f _ j ( \\tau _ k ) ^ m f _ k ( \\tau _ j ) ^ m = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n f _ j ( \\tau _ k ) ^ m f _ k ( \\tau _ j ) ^ m \\geq \\frac { 1 } { ( G _ { f , \\tau } ^ { \\circ ^ m } ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) ^ m \\right ) ^ 2 . \\end{align*}"}
+{"id": "8679.png", "formula": "\\begin{align*} i _ { u _ i , t u _ j } \\left ( \\frac { u _ i - u _ j } { u _ i - t u _ j } \\right ) = 1 + \\sum _ { s \\ge 1 } ( t ^ s - t ^ { s - 1 } ) \\left ( \\frac { u _ j } { u _ i } \\right ) ^ s . \\end{align*}"}
+{"id": "2319.png", "formula": "\\begin{align*} | { \\bf B } ( x , y ) | ^ 2 = \\frac { | x | ^ 2 + ( y - 1 ) ^ 2 } { | x | ^ 2 + ( y + 1 ) ^ 2 } , \\end{align*}"}
+{"id": "6646.png", "formula": "\\begin{align*} a _ { i j } ( l ) = a _ { i j } ( 2 ^ { - m } l ) \\end{align*}"}
+{"id": "8152.png", "formula": "\\begin{align*} u = \\frac { \\dfrac { 4 \\pi \\sqrt { y p } } { c } - 2 T } { 2 M } \\end{align*}"}
+{"id": "8593.png", "formula": "\\begin{align*} g ( x ) & = x ^ { 1 0 } + x ^ 9 + x ^ 7 + x ^ 5 + x ^ 4 + 1 , \\\\ x \\ , g ( x ) & = x ^ { 1 1 } + x ^ { 1 0 } + x ^ 8 + x ^ 6 + x ^ 5 + x , \\\\ x ^ 2 \\ , g ( x ) & = x ^ { 1 2 } + x ^ { 1 1 } + x ^ 9 + x ^ 7 + x ^ 6 + x ^ 2 , \\\\ x ^ 3 \\ , g ( x ) & = x ^ { 1 3 } + x ^ { 1 2 } + x ^ { 1 0 } + x ^ 8 + x ^ 7 + x ^ 3 , \\\\ x ^ 4 \\ , g ( x ) & = x ^ { 1 2 } + x ^ { 1 0 } + x ^ 7 + x ^ 6 + x ^ 5 + x ^ 3 + x ^ 2 + x + 1 . \\end{align*}"}
+{"id": "4082.png", "formula": "\\begin{align*} \\mathfrak { g d i f f } _ H = \\{ ( X , \\omega ) \\in \\Gamma ( T M ) \\times \\Omega ^ 2 : d ( i _ X H + \\omega ) = 0 \\} . \\end{align*}"}
+{"id": "63.png", "formula": "\\begin{align*} f ( i ) = \\begin{cases} - 1 , & x _ i = y _ i = 0 \\\\ 1 , & x _ i = 1 , y _ i = 0 \\\\ 2 , & x _ i = 1 , y _ i = 1 . \\end{cases} \\end{align*}"}
+{"id": "4498.png", "formula": "\\begin{align*} \\frac { 2 } { 2 1 } = \\frac { 1 } { 1 1 } + \\frac { 1 } { 2 3 1 } = \\frac { 1 } { 1 2 } + \\frac { 1 } { 8 4 } = \\frac { 1 } { 1 4 } + \\frac { 1 } { 4 2 } = \\frac { 1 } { 1 5 } + \\frac { 1 } { 3 5 } \\end{align*}"}
+{"id": "8407.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { \\infty } | \\partial ^ 2 _ x [ ( h _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) ( \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) G _ { \\alpha , n } ] | \\leq C \\sum _ { n = 1 } ^ { \\infty } ( n + 1 ) n ^ { - \\left ( 1 + \\frac { 1 } { \\alpha } \\right ) } < \\infty , \\end{align*}"}
+{"id": "4037.png", "formula": "\\begin{align*} \\int _ { Y v ( \\Omega ' ) } f ^ * & = \\int _ { Y u _ 0 ( Y u _ 0 ^ { - 1 } ( Y v ( \\Omega ' ) ) ) } f ^ * \\\\ & = \\int _ { Y u _ 0 ^ { - 1 } ( Y v ( \\Omega ' ) ) } f ^ * ( Y ( \\cdot , u _ 0 , D u _ 0 ) ) \\det D Y u _ 0 \\\\ & \\leq \\int _ { \\Omega ' } f ^ * ( Y ( \\cdot , u _ 0 , D u _ 0 ) ) \\det D Y u _ 0 = \\int _ { Y u _ 0 ( \\Omega ' ) } f ^ * . \\end{align*}"}
+{"id": "9213.png", "formula": "\\begin{align*} \\begin{gathered} \\forall b \\in \\Gamma , \\ \\forall 0 < r < \\widetilde { r } , X _ { \\sigma } \\cap B ( b , r ) \\\\ A _ { \\pm } ( b , r ) = \\varphi ^ { - 1 } \\big ( \\{ - y _ { - } ^ { 2 } + y _ { + } ^ { 2 } < 0 \\} \\cap \\{ \\pm y _ { - } > 0 \\} \\big ) \\cap B ( b , r ) . \\end{gathered} \\end{align*}"}
+{"id": "6121.png", "formula": "\\begin{align*} \\mathcal R _ { ( p , q , \\cdots , r , s ) } = A ^ p B ^ q \\cdots A ^ r B ^ s D . \\end{align*}"}
+{"id": "7420.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ N \\langle K ( z _ i , z _ j ) ( a _ i ^ * a _ j ) y _ j , y _ i \\rangle \\ge 0 \\end{align*}"}
+{"id": "1824.png", "formula": "\\begin{align*} T _ 1 ( \\eta ) = T _ 2 ( \\eta ) \\end{align*}"}
+{"id": "7232.png", "formula": "\\begin{align*} W _ 1 W _ 1 ^ \\top + W _ 2 W _ 2 ^ \\top & = W _ 1 ^ \\top W _ 1 + W _ 2 ^ \\top W _ 2 = \\frac { 1 } { 2 } ( 2 k I _ v + \\lambda A + \\mu ( J _ v - I _ v - A ) ) , \\\\ W _ 1 W _ 2 ^ \\top + W _ 2 W _ 1 ^ \\top & = W _ 1 ^ \\top W _ 2 + W _ 2 ^ \\top W _ 1 = \\frac { 1 } { 2 } ( \\lambda A + \\mu ( J _ v - I _ v - A ) ) . \\end{align*}"}
+{"id": "7949.png", "formula": "\\begin{align*} \\pi ( m _ 1 ) + k _ 1 = \\pi ( m _ 2 ) + k _ 2 . \\end{align*}"}
+{"id": "3692.png", "formula": "\\begin{align*} \\widehat { L } \\coloneqq \\{ ( x , t ; \\xi , \\tau ) \\mid \\tau > 0 , ( x ; \\xi / \\tau ) \\in L , t = - f ( x ; \\xi / \\tau ) \\} . \\end{align*}"}
+{"id": "8194.png", "formula": "\\begin{align*} k ( t , s ) : = \\frac { \\alpha } { \\beta \\Gamma ( \\beta ) } s ^ { \\frac { \\alpha } { \\beta } - 1 } \\left ( t ^ { \\frac { \\alpha } { \\beta } } - s ^ { \\frac { \\alpha } { \\beta } } \\right ) ^ { \\beta - 1 } , \\qquad \\beta \\in ( 0 , 1 ] , \\quad \\alpha \\in ( 0 , 2 ) . \\end{align*}"}
+{"id": "6850.png", "formula": "\\begin{align*} \\begin{aligned} l ( \\mathcal Y ^ { [ 0 ] } , 0 , N + 1 ) & \\le l ( \\mathcal Y , 0 , N + 1 ) \\le l ( \\mathcal Y ^ { [ N + 1 ] } , 0 , N + 1 ) , \\\\ l ^ * ( \\mathcal Y ^ { [ N + 1 ] } , 0 , N + 1 ) & \\le l ^ * ( \\mathcal Y , 0 , N + 1 ) \\le l ^ * ( \\mathcal Y ^ { [ 0 ] } , 0 , N + 1 ) , \\end{aligned} \\end{align*}"}
+{"id": "1884.png", "formula": "\\begin{align*} a _ { i } ( ( z _ { 1 } , \\dots , z _ { n } ) ) = ( z _ { 1 } ( A ^ { i } _ { 1 } ) , \\dots , z _ { i - 1 } ( A ^ { i } _ { i - 1 } ) , - z _ { i } , z _ { i + 1 } ( A ^ { i } _ { i + 1 } ) , \\dots , z _ { n } ( A ^ { i } _ { n } ) ) . \\end{align*}"}
+{"id": "8486.png", "formula": "\\begin{align*} \\frac { \\partial C } { \\partial t } = W C _ { \\xi } + U C _ { \\xi ^ 2 } , \\end{align*}"}
+{"id": "8470.png", "formula": "\\begin{align*} c _ * ( A ) = \\| \\gamma _ A \\| ^ 2 = \\max _ { \\nu \\in \\mathcal E _ A ' } \\ , \\Bigl ( 2 \\int \\kappa \\gamma _ A \\ , d \\nu - \\| \\nu \\| ^ 2 \\Bigr ) , \\end{align*}"}
+{"id": "2031.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\bf C } ^ - ( r ) = \\mathcal { I } ^ - ( r ) \\cap \\bigcup _ { \\ell \\geq 1 } B _ { \\ell } B _ { \\ell } : = \\{ - b _ { \\ell } + r , \\ldots , - b _ { \\ell } + D + r - \\delta \\} . \\end{align*}"}
+{"id": "8836.png", "formula": "\\begin{align*} H _ 0 ( x , y , t ) = \\frac { e ^ { n ^ { 2 } t } } { 2 ^ { n - 2 } \\pi ^ { n + 1 } } \\int _ { \\rho } ^ { \\frac { \\pi } { 2 } } \\frac { - d ( \\cos u ) } { \\sqrt { \\cos ^ { 2 } \\rho - \\cos ^ { 2 } u } } \\left ( - \\frac { 1 } { \\sin u } \\frac { d } { d u } \\right ) ^ { n } \\left [ \\Theta _ { n + 1 } ( t , u ) \\right ] \\end{align*}"}
+{"id": "4662.png", "formula": "\\begin{align*} \\binom { m } { \\ell _ i } \\binom { m } { \\ell _ j } = \\sum _ { k = 0 } ^ { \\ell _ i } \\binom { m } { \\ell _ j } \\binom { \\ell _ j } { k } \\binom { m - \\ell _ j } { \\ell _ i - k } = \\sum _ { k = 0 } ^ { \\ell _ i } \\binom { \\ell _ i + \\ell _ j - k } { k , \\ell _ i - k , \\ell _ j - k } \\binom { m } { \\ell _ i + \\ell _ j - k } . \\end{align*}"}
+{"id": "3512.png", "formula": "\\begin{align*} \\left [ E _ { i j } , [ B _ { A ( 1 , l ) } ^ + , \\dots , B _ { A ( \\lambda _ l ' , l ) } ^ + ] \\right ] = \\sum _ { k = 1 } ^ { \\lambda _ l ' } \\delta _ { j , A ( k , l ) } [ B _ { A ( 1 , l ) } ^ + , \\dots , B _ { A ( k - 1 , l ) } ^ + , B _ i ^ + , B _ { A ( k + 1 , l ) } ^ + , \\dots , B _ { A ( \\lambda _ l ' , l ) } ^ + ] , \\end{align*}"}
+{"id": "7023.png", "formula": "\\begin{align*} \\dfrac { 1 } { F _ { M U } ( u , x ) } = d _ 0 + d _ 1 x + d _ 2 x ^ 2 + \\cdots \\in M U _ * ( ( u ) ) [ [ x ] ] \\end{align*}"}
+{"id": "520.png", "formula": "\\begin{align*} w ^ k : = \\ ! \\left ( \\begin{matrix} \\nabla \\ ! f ( x ^ { k } ) \\ ! - \\ ! \\nabla \\ ! f ( y ^ { k - 1 } ) - \\frac { 1 } { \\tau _ { k - 1 } } ( x ^ { k } \\ ! - \\ ! y ^ { k - 1 } ) + \\delta ( x ^ { k } \\ ! - \\ ! x ^ { k - 1 } ) \\\\ \\delta ( x ^ { k - 1 } - x ^ { k } ) \\end{matrix} \\right ) \\in \\partial H _ { \\delta } ( z ^ { k } ) . \\end{align*}"}
+{"id": "5117.png", "formula": "\\begin{align*} G _ { j } ( \\lambda , b , \\Omega , f _ { 1 } , f _ { 2 } ) ( w ) : = \\mbox { I m } \\left \\lbrace \\Big ( \\Omega \\Phi _ { j } ( w ) + S ( \\lambda , \\Phi _ { 2 } , \\Phi _ { j } ) ( w ) - S ( \\lambda , \\Phi _ { 1 } , \\Phi _ { j } ) ( w ) \\Big ) \\overline { w } \\overline { \\Phi _ { j } ' ( w ) } \\right \\rbrace , \\end{align*}"}
+{"id": "5826.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( \\ ! ( U '' , \\Phi ) \\ ! ) d x + \\int _ { \\Omega } ( \\ ! ( \\nabla U , \\nabla \\Phi ) \\ ! ) d x + \\int _ { \\Omega } ( \\ ! ( A U , \\Phi ) \\ ! ) d x + \\int _ { \\Omega } a ( \\ ! ( D U ' , \\Phi ) \\ ! ) d \\Gamma = 0 . \\end{align*}"}
+{"id": "5128.png", "formula": "\\begin{align*} B _ { n } ( \\lambda , b ) & : = ( 1 - b ^ { 2 } ) \\Lambda _ { 1 } ( \\lambda , b ) + b \\big [ \\Omega _ { n } ( \\lambda ) - \\Omega _ { n } ( \\lambda b ) \\big ] , \\\\ C _ { n } ( \\lambda , b ) & : = b \\left [ \\left ( \\Lambda _ { 1 } ( \\lambda , b ) - \\frac { 1 } { b } \\Omega _ { n } ( \\lambda ) \\right ) \\Big ( b \\Omega _ { n } ( \\lambda b ) - \\Lambda _ { 1 } ( \\lambda , b ) \\Big ) + \\Lambda _ { n } ^ { 2 } ( \\lambda , b ) \\right ] . \\end{align*}"}
+{"id": "3605.png", "formula": "\\begin{align*} R _ { \\mu } & = \\mu ^ { \\frac 1 3 } , & & L _ \\mu = \\mu ^ { \\frac 2 3 } & & \\\\ R _ { \\mu } & = \\mu ^ { \\frac 2 7 } ( \\ln \\mu ) ^ { - \\frac { 1 } { 7 } } , & & L _ \\mu = \\mu ^ { \\frac 3 7 } ( \\ln \\mu ) ^ { \\frac { 2 } { 7 } } , & & \\\\ R _ { \\mu } & = \\mu ^ { \\frac { 2 } { 2 n + 1 } } , & & L _ \\mu = \\mu ^ { \\frac { 3 } { 2 n + 1 } } , & & \\end{align*}"}
+{"id": "5386.png", "formula": "\\begin{align*} 2 \\sum _ { \\substack { k \\le e ^ { W + C } \\\\ N _ \\sigma < k \\le N } } \\frac { 1 } { k e ^ { 2 k \\sigma } } = 2 \\sum _ { k \\le e ^ { W + C } } \\frac { 1 } { k } \\left ( 1 + O \\left ( \\frac { k } { N } \\right ) \\right ) + O ( 1 ) = 2 W + O ( 1 ) . \\end{align*}"}
+{"id": "1798.png", "formula": "\\begin{align*} [ H \\backslash i ] _ { \\ast } R c _ { [ H \\backslash Y ] , \\ast } ( [ H \\backslash i ] ^ { \\diamondsuit , \\ast } A ) = 0 , \\end{align*}"}
+{"id": "7574.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 7 ) = \\dfrac { F _ 7 ( q ^ { - s } ) } { ( 1 - q ^ { - 1 - s } ) ( 1 - q ^ { - p - ( k + 1 ) - p ( k + 1 ) s } ) } , \\end{align*}"}
+{"id": "4106.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } g _ t = h , g _ 0 = g , \\\\ & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } H _ t = d K , H _ 0 = H , \\\\ & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } f _ t = \\phi , f _ 0 = f \\end{align*}"}
+{"id": "3341.png", "formula": "\\begin{align*} \\Phi _ { y x } = - \\frac { 1 } { \\Phi _ { x q } \\Phi _ { x p } } [ \\Phi _ { y q } ( \\Phi _ { x y } \\Phi _ { y p } + \\Phi _ { x q } \\Phi _ { q p } ) + \\Phi _ { y p } \\Phi _ { p q } \\Phi _ { x q } ] . \\end{align*}"}
+{"id": "3669.png", "formula": "\\begin{align*} [ \\phi ] _ \\rho & : = \\phi + \\rho \\log ( 1 + \\exp ( - \\phi / \\rho ) ) , & [ \\phi ] _ \\rho ' & = ( 1 + \\exp ( - \\phi / \\rho ) ) ^ { - 1 } . \\end{align*}"}
+{"id": "5917.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } U '' - \\Delta U = 0 & \\hbox { i n } ( T , + \\infty ) \\times \\Omega , \\\\ \\partial _ \\nu U + B U = 0 & \\hbox { o n } ( T , + \\infty ) \\times \\Gamma . \\end{array} \\right . \\end{align*}"}
+{"id": "5606.png", "formula": "\\begin{align*} \\nabla _ { 0 } S _ { 1 3 } = S _ { 1 1 } \\nabla _ { 0 } k _ { 3 } , \\nabla _ { 2 } S _ { 1 3 } = S _ { 1 1 } \\nabla _ { 2 } k _ { 3 } , \\nabla _ { 3 } S _ { 1 3 } = S _ { 1 1 } \\nabla _ { 3 } k _ { 3 } . \\end{align*}"}
+{"id": "6448.png", "formula": "\\begin{align*} w _ j ( X , Y , t , \\tilde X , \\tilde Y , \\tilde t ) = & v ( X , Y , t ) - v ( \\tilde X , \\tilde Y , \\tilde t ) - w ( Y , t ) \\\\ & - \\bigl ( \\frac { j ^ 4 } { 4 } | X - \\tilde X | ^ 4 + \\frac { j ^ 4 } { 4 } | Y - \\tilde Y | ^ 4 + \\frac j 2 | t - \\tilde t | ^ 2 \\bigr ) . \\end{align*}"}
+{"id": "5392.png", "formula": "\\begin{align*} A _ 0 ( \\C ^ n ) : = \\{ f \\in U _ 0 ( \\C ^ n ) : f \\hbox { \\ i s s u r j e c t i v e } \\} . \\end{align*}"}
+{"id": "8276.png", "formula": "\\begin{align*} \\begin{cases} \\dot x = y \\\\ \\dot y = z \\\\ \\dot z = - y - \\beta z + \\mu x ( 1 - x ) , \\end{cases} \\end{align*}"}
+{"id": "8136.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 ^ - = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { c \\leq \\frac { C } { m } } \\frac { S ( - n , p ; c ) } { c } H _ { m , n } ^ - \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) . \\end{align*}"}
+{"id": "8334.png", "formula": "\\begin{align*} K ( Q ) = \\sup _ { x \\in \\mathbb { R } ^ d } \\left \\{ | x | ^ { d - 2 } | Q ( x ) | + | x | ^ { d - 1 } | \\nabla Q ( x ) | \\right \\} < + \\infty . \\end{align*}"}
+{"id": "489.png", "formula": "\\begin{align*} [ v ] _ h ( \\cdot , t ) = e ^ { - \\frac { t } { h } } v _ 0 + \\frac { 1 } { h } \\int _ 0 ^ t e ^ { \\frac { s - t } { h } } v ( \\cdot , s ) \\ , d s , \\end{align*}"}
+{"id": "7617.png", "formula": "\\begin{align*} \\eta _ n ( x ) = \\begin{cases} 1 , & \\ x \\leq n , \\\\ n + 1 - x , & \\ n < x \\leq n + 1 \\\\ 0 , & \\ x > n + 1 . \\end{cases} , \\end{align*}"}
+{"id": "8070.png", "formula": "\\begin{align*} \\mathcal { O ^ + } = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } \\sum _ { c > 0 } \\frac { S ( n , p ; c ) } { c } H _ { m , n } ^ + \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) , \\end{align*}"}
+{"id": "4299.png", "formula": "\\begin{align*} \\| ( f , g ) \\| _ { \\gamma ^ s _ { \\eta , L ^ 2 } \\times \\gamma ^ s _ { \\eta , L ^ 2 } } = \\sqrt { \\| f \\| ^ 2 _ { \\gamma ^ s _ { \\eta , L ^ 2 } } + \\| g \\| ^ 2 _ { \\gamma ^ s _ { \\eta , L ^ 2 } } } . \\end{align*}"}
+{"id": "7946.png", "formula": "\\begin{align*} M = \\{ n \\in N \\mid g ( n ) = 0 \\} \\end{align*}"}
+{"id": "2119.png", "formula": "\\begin{align*} & \\textstyle C _ 1 = C _ 2 = F H e _ j , C _ { 1 2 } = F H e ' ~ ~ \\mbox { w h e r e } ~ e ' = \\sum _ { \\epsilon \\in E - \\{ e _ j , \\overline e _ j \\} } \\epsilon ; \\\\ & \\widehat C _ { 1 2 } = \\{ ( c , c ) \\ , | \\ , c \\in C _ { 1 2 } \\} ( \\mbox { i . e . , ~ t a k e $ g = e ' $ i n T h e o r e m \\ref { G o u r s a t C } } ) . \\end{align*}"}
+{"id": "7441.png", "formula": "\\begin{align*} \\phi ^ { ( n ) } \\big ( A ^ { - 1 } { \\mathbb P } B ^ { * - 1 } \\big ) = A ^ { - 1 } \\cdot \\phi ^ { ( n ) } ( { \\mathbb P } ) \\cdot B ^ { * - 1 } . \\end{align*}"}
+{"id": "6478.png", "formula": "\\begin{align*} \\sigma ( \\zeta _ { n _ i } ^ { g _ i } ) = \\sigma ( \\zeta _ { 4 n } ^ { 4 n g _ i / n _ i } ) = \\zeta _ { 4 n } ^ { \\eta ( \\sigma ) \\cdot 4 n g _ i / n _ i } = \\zeta _ { n _ i } ^ { \\eta ( \\sigma ) g _ i } . \\end{align*}"}
+{"id": "7234.png", "formula": "\\begin{align*} ( W _ 1 - W _ 2 ) ( W _ 1 ^ \\top - W _ 2 ^ \\top ) & = k I . \\end{align*}"}
+{"id": "35.png", "formula": "\\begin{align*} T ( n _ i ) & < \\frac { 1 } { n _ i } \\cdot \\left ( 1 + \\frac { 2 } { 3 } + \\frac { 4 } { 9 } \\right ) = \\frac { 1 } { n _ i } \\cdot \\frac { 1 9 } { 9 } \\\\ & < \\frac { 1 } { \\tfrac { 3 2 } { 2 7 } \\cdot X _ 0 } \\cdot \\frac { 1 9 } { 9 } = \\frac { 5 7 } { 3 2 } \\cdot \\frac { 1 } { X _ 0 } < \\frac { 9 7 } { 5 4 } \\cdot \\frac { 1 } { X _ 0 } . \\end{align*}"}
+{"id": "6218.png", "formula": "\\begin{align*} t = 0 : u _ r = ( E _ r , \\widehat U _ 0 ) , \\ u _ r ' = ( E _ r , \\widehat U _ 1 ) . \\end{align*}"}
+{"id": "5977.png", "formula": "\\begin{align*} \\mathrm { d i m } h ( Z _ 0 ) & = \\mathrm { d i m } Z _ 0 - \\mathrm { d i m } M + \\mathrm { d i m } ( h \\times f ) ( M ) \\\\ & = \\mathrm { d i m } f ^ { - 1 } ( z ) - \\mathrm { d i m } M + \\mathrm { d i m } ( h \\times f ) ( M ) , \\end{align*}"}
+{"id": "2893.png", "formula": "\\begin{align*} V _ { r , M } = \\left \\{ \\zeta \\in \\ell ^ 1 : \\| \\zeta \\| _ 1 < M \\ ; \\ ; n \\in \\mathbb { N } , \\ ; | \\zeta _ n | < r \\right \\} , \\end{align*}"}
+{"id": "2882.png", "formula": "\\begin{align*} | L _ U | = q ^ { k ' + 1 } + q ^ { k ' } - q ^ { j + 1 } + 1 . \\end{align*}"}
+{"id": "6284.png", "formula": "\\begin{align*} \\Phi ^ X ( t _ { x , j } , t _ 0 , x , p ) = \\Phi ^ X _ { t _ { x , j _ m } , t _ { x , j _ { m - 1 } } } \\circ \\Phi ^ X _ { t _ { x , j _ 1 } , t _ 0 } \\end{align*}"}
+{"id": "6357.png", "formula": "\\begin{align*} m _ { \\alpha } ( x ) : = m _ 0 ( x ) + \\alpha \\int _ 0 ^ x \\phi _ 1 ( x ) d x , \\end{align*}"}
+{"id": "6260.png", "formula": "\\begin{align*} f _ 0 | _ { \\mathcal { U } } = h _ 0 ' . \\end{align*}"}
+{"id": "2716.png", "formula": "\\begin{align*} J E ( W , 0 ) = \\sum _ { k = 1 } ^ K E \\left ( \\vec { V } ^ k ( 0 ) \\right ) \\end{align*}"}
+{"id": "8337.png", "formula": "\\begin{align*} \\Box _ { t , x } w = 3 Q _ l ^ 2 w + 3 Q _ l w ^ 2 + w ^ 3 . \\end{align*}"}
+{"id": "7963.png", "formula": "\\begin{align*} \\gamma ( \\vee y _ i ) = \\gamma ( \\vee ( x _ i \\vee 1 _ L ) ) = \\beta ( \\vee ( x _ i \\vee 1 _ L ) ) = \\vee ( \\beta ( x _ i \\vee 1 _ L ) ) = \\vee \\beta ( y _ i ) = \\vee \\gamma ( y _ i ) , \\end{align*}"}
+{"id": "2606.png", "formula": "\\begin{align*} \\sum _ { j \\in [ r ] \\setminus \\{ 1 \\} } a _ { 1 , j } = \\sum _ { j \\in [ r ] \\setminus \\{ 1 \\} } \\left ( k + 1 - \\sum _ { l \\in [ r ] \\setminus \\{ 1 , j \\} } a _ { l , j } \\right ) = \\sum _ { j \\in [ r ] \\setminus \\{ 1 \\} } ( k + 1 ) - \\sum _ { j \\in [ r ] \\setminus \\{ 1 \\} } \\sum _ { l \\in [ r ] \\setminus \\{ 1 , j \\} } a _ { l , j } . \\end{align*}"}
+{"id": "3529.png", "formula": "\\begin{align*} \\mathbf { p } _ { i j } : = \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { k ! } E _ { j i } ^ k E _ { i j } ^ k \\frac { 1 } { ( h _ i - h _ j + 1 ) _ k } , \\end{align*}"}
+{"id": "6847.png", "formula": "\\begin{align*} \\tau ( L _ 1 , L _ 2 , L _ 3 ) = \\mu _ c ^ { + } ( Y _ 1 , Y _ 2 , Y _ 3 ) - \\mu _ c ^ { - } ( Y _ 1 , Y _ 2 , Y _ 3 ) = \\nu _ c ^ { + } ( Y _ 1 , Y _ 2 , Y _ 3 ) - \\nu _ c ^ { - } ( Y _ 1 , Y _ 2 , Y _ 3 ) . \\end{align*}"}
+{"id": "6893.png", "formula": "\\begin{align*} y _ { i + 1 } - y ^ 2 _ i = \\frac { i + 1 } { 2 ^ { 2 ^ n } } - \\left ( \\frac { i } { 2 ^ { 2 ^ n } } \\right ) ^ 2 = \\frac { 2 ^ { 2 ^ n } ( i + 1 ) - i ^ 2 } { \\left ( 2 ^ { 2 ^ n } \\right ) ^ 2 } = \\frac { 2 ^ { 2 ^ n } + 2 ^ { 2 ^ n } i \\left ( 1 - \\frac { i } { 2 ^ { 2 ^ n } } \\right ) } { \\left ( 2 ^ { 2 ^ n } \\right ) ^ 2 } > 0 . \\end{align*}"}
+{"id": "5444.png", "formula": "\\begin{align*} \\mu ( d s , d z ) = p ( d s , d z ) - d s \\times \\tilde m ( d z ) . \\end{align*}"}
+{"id": "3895.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } = g _ z E ^ { m , i } q _ m D _ { x _ i } . \\end{align*}"}
+{"id": "5226.png", "formula": "\\begin{align*} \\ell _ 1 = k _ 2 \\ell _ 2 + k _ 1 + k _ 0 / \\ell _ 2 \\ . \\end{align*}"}
+{"id": "4469.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k \\frac { 1 } { u _ i } \\leq \\prod _ { i = 1 } ^ k \\frac { 1 } { v _ i } \\end{align*}"}
+{"id": "9060.png", "formula": "\\begin{align*} w ' _ 2 = \\sum _ { l = 0 } ^ \\infty \\sum _ { m = 1 } ^ { d _ l } w _ { l m } ( \\rho ) Y _ { l m } ( \\omega ) \\end{align*}"}
+{"id": "7930.png", "formula": "\\begin{align*} { } \\mathcal { L } _ { \\mathcal { P } , t _ 0 } : = \\Big \\langle \\big \\{ M ( f _ c ) ( t _ 0 ) : M \\in ( \\mathcal { P } ) \\big \\} \\Big \\rangle . \\end{align*}"}
+{"id": "7934.png", "formula": "\\begin{align*} \\textbf { a } ^ C _ { A , B } = \\# \\{ D \\subseteq C \\mid D \\simeq B \\textrm { a n d } C / D \\simeq A \\} . \\end{align*}"}
+{"id": "6131.png", "formula": "\\begin{align*} t = 0 : U = \\widehat U _ 0 , \\ U ' = \\widehat U _ 1 \\hbox { i n } \\Omega , \\end{align*}"}
+{"id": "6888.png", "formula": "\\begin{align*} c _ { \\mu } = \\inf \\limits _ { \\nu \\in \\Gamma } \\max \\limits _ { t \\in [ 0 , 1 ] } J _ { \\mu } ( \\nu ( t ) ) , \\end{align*}"}
+{"id": "5664.png", "formula": "\\begin{align*} U = U _ 0 + \\epsilon U _ 1 + \\ldots \\end{align*}"}
+{"id": "6336.png", "formula": "\\begin{align*} h ' ( t ) = - { f _ \\nu ( t , \\nu _ 0 ) } / { f _ \\nu ( h ( t ) , \\nu _ 0 ) } . \\end{align*}"}
+{"id": "7236.png", "formula": "\\begin{align*} ( A _ 0 + A _ 2 ) A _ 4 = A _ 4 ( A _ 0 + A _ 2 ) = \\frac { k } { m } ( A _ 4 + A _ 5 + A _ 6 ) \\end{align*}"}
+{"id": "1660.png", "formula": "\\begin{align*} \\frac { \\ell ( f _ 1 , \\chi ) \\cdot \\ell ( f _ 2 , \\chi ^ { - 1 } ) } { \\langle f _ 1 , f _ 2 \\rangle } = \\frac { \\Lambda _ F ( 2 ) \\cdot \\Lambda ( 1 / 2 , \\Pi , \\chi ) } { 2 \\Lambda ( 1 , \\Pi , { \\rm a d } ) } \\cdot \\prod _ v \\beta _ v ( f _ { 1 , v } , f _ { 2 , v } ) , \\end{align*}"}
+{"id": "3425.png", "formula": "\\begin{align*} L _ \\phi ( \\psi ) ( x ) = L _ \\phi ( \\psi ) ( x _ 0 ) = \\sum _ { \\substack { a \\in S \\\\ A ( a , x _ 0 ) = 1 } } \\exp ( { \\phi ( a x _ 0 ) } ) \\psi ( a ) . \\end{align*}"}
+{"id": "5476.png", "formula": "\\begin{align*} W _ { \\tilde V } \\left ( P _ t ^ * \\mu _ { \\mathcal I } , \\mu _ { \\mathcal I } \\right ) \\leq \\varliminf _ { s \\to \\infty } W _ { \\tilde V } \\left ( P _ t ^ * P _ s ^ * \\nu _ 0 , \\mu _ { \\mathcal I } \\right ) = 0 . \\end{align*}"}
+{"id": "314.png", "formula": "\\begin{align*} \\begin{aligned} ( \\chi _ { \\alpha } \\circ \\mathrm { N m } _ { K / E _ 2 } ) | _ { E _ { 1 } ^ { \\times } } & = \\chi _ { \\alpha } | _ { F ^ { \\times } } \\circ \\mathrm { N m } _ { E _ 1 / F } \\\\ & = \\omega _ { E _ 2 / F } ( \\mathrm { N m } _ { E _ 1 / F } ) = \\omega _ { K / E _ 1 } , \\end{aligned} \\end{align*}"}
+{"id": "8544.png", "formula": "\\begin{align*} n ^ { 3 n } 7 ^ n 2 ^ { - n ^ 2 + 2 n + 1 } = 2 ^ { 3 n \\log _ 2 ( n ) + \\log _ 2 ( 7 ) n - n ^ 2 + 2 n - 1 } , \\end{align*}"}
+{"id": "8345.png", "formula": "\\begin{align*} Q ^ { ( M _ 0 ) } ( x _ 1 , x _ 2 , x _ 3 ) = M _ 0 ^ { \\frac { d } { 2 } - 1 } Q \\left ( M _ 0 \\left ( x _ 1 - \\frac { l _ 1 } { \\sqrt { 1 - l _ 1 ^ 2 } } \\right ) , M _ 0 x _ 2 , M _ 0 x _ 3 \\right ) . \\end{align*}"}
+{"id": "4682.png", "formula": "\\begin{align*} \\mathfrak { R } ^ { G _ { I I } } = \\mathbb { C } [ \\varphi _ 8 , \\varphi _ { 2 4 } ] \\end{align*}"}
+{"id": "2435.png", "formula": "\\begin{align*} K _ { h , \\tau } q ( x ) : = P _ { [ 0 , M _ 1 ] } \\Big ( \\frac { f ( x ) - \\bar \\partial _ \\tau ^ \\alpha u _ h ^ N ( x ; q ) + \\psi _ h ( x ) } { g _ \\delta ( x ) } \\Big ) , \\end{align*}"}
+{"id": "3934.png", "formula": "\\begin{align*} & \\overline { g } ( q , p , \\overline { z } ) = q \\cdot p - z + a _ { i j , k l } ( q , p ) q _ i q _ j p _ k p _ l \\\\ & + z [ b _ { i j } ( q , p ) q _ i q _ j + c _ { i j } ( q , p ) q _ i p _ j + d _ { i j } ( q , p ) p _ i p _ j ] + f ( x , y , z ) z ^ 2 . \\end{align*}"}
+{"id": "606.png", "formula": "\\begin{align*} \\mathbb { X } ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 2 } } = \\mathbb { X } ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } + \\mathbb { X } ^ { ( \\epsilon ) } _ { t _ { n + 1 } , t _ { n + 2 } } + X ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } \\otimes X ^ { ( \\epsilon ) } _ { t _ { n + 1 } , t _ { n + 2 } } . \\end{align*}"}
+{"id": "5277.png", "formula": "\\begin{align*} \\tilde { s e c } ( U , V ) = \\hat { \\tilde { s e c } } ( U , V ) - \\lambda ^ 2 \\mid \\nabla ( f + \\log \\lambda ) \\mid ^ 2 , \\end{align*}"}
+{"id": "7717.png", "formula": "\\begin{align*} \\Phi ^ { ( 1 ) } \\left ( \\tau , 1 + \\rho ^ { \\star } e ^ { i \\beta \\pi } \\right ) = \\Phi ^ { ( 2 ) } ( \\tau , 1 + \\bar { \\rho } e ^ { - i \\beta \\pi } ) , \\end{align*}"}
+{"id": "7242.png", "formula": "\\begin{align*} \\R ^ n = \\bigcup _ { [ X _ { j _ 1 } , \\ldots , X _ { j _ n } ] \\in \\mathcal { F } _ { n - 1 } ( Q _ { n , N } ^ f ) } { \\rm c o n e } ( X _ { j _ 1 } , \\ldots , X _ { j _ n } ) , \\end{align*}"}
+{"id": "3844.png", "formula": "\\begin{align*} C _ { p } \\left ( x , y \\right ) : = \\left \\vert x \\right \\vert ^ { p } - \\left \\vert x - y \\right \\vert ^ { p } - p \\left \\vert x - y \\right \\vert ^ { p - 2 } \\operatorname { R e } \\left ( x - y \\right ) \\cdot \\overline { y } . \\end{align*}"}
+{"id": "933.png", "formula": "\\begin{align*} e _ { 2 } \\cdot \\left [ f _ { \\delta } ( x _ { 0 } + t \\theta _ { x _ { 0 } } ) - f ( x _ { 0 } + t \\theta _ { x _ { 0 } } ) \\right ] = e _ { 2 } \\cdot \\left [ \\frac { \\delta } { ( e _ { 2 } \\cdot \\theta _ { x _ { 0 } } ^ { \\perp } ) } ( \\theta _ { C } ^ { \\perp } ) \\right ] = \\delta . \\end{align*}"}
+{"id": "3027.png", "formula": "\\begin{align*} \\begin{array} { c c l c c l } \\omega { ^ a } _ b & = & \\gamma { ^ a } _ b - \\frac { 1 } { 2 } \\textsf { k } _ { i j } \\eta ^ { a c } \\mathbf { F } { ^ j } _ { c b } e ^ i & \\omega { ^ a } _ i & = & \\frac { 1 } { 2 } \\textsf { k } _ { i j } \\eta ^ { a c } \\mathbf { F } { ^ j } _ { b c } e ^ b \\\\ \\omega { ^ i } _ a & = & \\frac { 1 } { 2 } \\mathbf { F } { ^ i } _ { a b } e ^ b & \\omega { ^ i } _ j & = & \\frac { 1 } { 2 } c ^ i _ { j k } ( e ^ k - 2 \\mathbf { A } ^ k ) \\end{array} \\end{align*}"}
+{"id": "922.png", "formula": "\\begin{align*} F ( u ( z ) , d u ( z ) ) = 0 \\end{align*}"}
+{"id": "5099.png", "formula": "\\begin{align*} \\frac { a c ^ { n } } { b g _ b ( n ) } = \\beta , \\end{align*}"}
+{"id": "6587.png", "formula": "\\begin{align*} \\psi _ j ( g ) ( i ) = \\left \\{ \\begin{array} { l l } g ( i ) & i < j g ( i ) < g ( j ) \\\\ g ( i ) - 1 & i < j g ( i ) > g ( j ) \\\\ g ( i + 1 ) & i \\ge j g ( i + 1 ) < g ( j ) \\\\ g ( i + 1 ) - 1 & i \\ge j g ( i + 1 ) \\ge g ( j ) \\end{array} \\right . \\end{align*}"}
+{"id": "9047.png", "formula": "\\begin{align*} \\Theta : = ( Y , Z , U , W ) , \\Theta ^ { i } : = ( Y ^ i , Z ^ i , U ^ i , W ^ i ) , \\Theta ^ { i , n } : = ( Y ^ { i , n } , Z ^ { i , n } , U ^ { i , n } , W ^ { i , n } ) . \\end{align*}"}
+{"id": "2940.png", "formula": "\\begin{align*} g _ n ( k ) = n ^ { 1 / 2 } g ( n ^ { - 1 / 2 } k ) = g ^ \\prime ( 0 ) k + \\frac { 1 } { 2 } g ^ { \\prime \\prime } ( 0 ) k ^ 2 n ^ { - 1 / 2 } + O ( n ^ { - 1 } ) \\end{align*}"}
+{"id": "1583.png", "formula": "\\begin{align*} d _ p ^ p ( \\mu , \\xi _ s ^ { \\mu , \\nu } ) & = \\inf _ { \\pi \\in \\Pi ( \\mu , \\xi _ s ) } \\sum _ { ( x , y ) \\in X \\times X } \\varrho ^ p ( x , y ) \\cdot \\pi ( x , y ) \\leq \\sum _ { ( x , y ) \\in X \\times X } \\varrho ^ p ( x , y ) \\cdot \\pi _ s ( x , y ) \\\\ & = ( 1 - s ) \\sum _ { ( x , y ) \\in X \\times X } \\varrho ^ p ( x , y ) \\cdot \\pi _ { \\mu } ( x , y ) + s \\sum _ { ( x , y ) \\in X \\times X } \\varrho ^ p ( x , y ) \\cdot \\widetilde { \\pi } ( x , y ) \\\\ & = s d _ p ^ p ( \\mu , \\nu ) , \\end{align*}"}
+{"id": "1596.png", "formula": "\\begin{align*} d _ p ( \\Phi ( \\xi _ s ^ { \\delta _ { x _ 1 } , \\delta _ { x _ 2 } } ) , \\delta _ { x _ 2 } ) = d _ p ( \\Phi ( \\xi _ s ^ { \\delta _ { x _ 1 } , \\delta _ { x _ 2 } } ) , \\Phi ( \\delta _ { x _ 2 } ) ) = d _ p ( \\xi _ s ^ { \\delta _ { x _ 1 } , \\delta _ { x _ 2 } } , \\delta _ { x _ 2 } ) \\leq \\sqrt [ p ] { 1 - s } d _ p ( \\delta _ { x _ 1 } , \\delta _ { x _ 2 } ) . \\end{align*}"}
+{"id": "1021.png", "formula": "\\begin{align*} P ^ { p , q } _ \\beta ( E ) = \\left ( \\int _ E \\left ( \\int _ { E ^ c } \\frac { d x } { | x - y | ^ { \\frac { ( n + p \\beta ) p } { q } } } \\right ) ^ { q / p } d y \\right ) ^ { 1 / q } . \\end{align*}"}
+{"id": "7083.png", "formula": "\\begin{align*} \\left ( E ^ { C _ 2 } _ { * + | n \\sigma | - n \\sigma } \\overset { u } { \\longrightarrow } E ^ { C _ 2 } _ { * + | n \\sigma | - ( n + 1 ) \\sigma } \\right ) & = \\left ( E ^ { C _ 2 } _ * \\{ 1 , \\dots , u ^ n \\} \\overset { u } { \\longrightarrow } E ^ { C _ 2 } _ { * + | n \\sigma | - ( n + 1 ) \\sigma } \\right ) \\\\ & = \\left ( E ^ { C _ 2 } _ { * } \\{ u , \\dots , u ^ { n + 1 } \\} \\to E ^ { C _ 2 } _ { * + | n \\sigma | - ( n + 1 ) \\sigma } \\right ) , \\end{align*}"}
+{"id": "5208.png", "formula": "\\begin{align*} & \\ell _ 1 ' = - \\frac { c _ { 1 6 } } { b _ { 1 1 } ^ 2 } \\quad , \\ell _ 2 ' = - \\frac { c _ { 2 4 } } { b _ { 2 2 } ^ 2 } \\\\ & c _ { 3 5 } \\ell _ { 1 } ^ { 2 } \\ell _ { 2 } ^ { 2 } b _ { 1 1 } ^ { 4 } b _ { 2 2 } ^ { 4 } + c _ { 1 6 } b _ { 2 2 } ^ { 2 } \\ell _ { 2 } + c _ { 2 4 } b _ { 1 1 } ^ { 2 } \\ell _ { 1 } = 0 \\\\ & b _ { 1 2 } = b _ { 1 3 } = b _ { 2 3 } = 0 \\ . \\end{align*}"}
+{"id": "411.png", "formula": "\\begin{align*} U : \\mathcal { X } \\ni \\sum _ { n = 1 } ^ { \\infty } g _ n ( x ) \\omega _ n \\mapsto \\sum _ { n = 1 } ^ { \\infty } g _ n ( x ) \\tau _ n \\in \\mathcal { X } . \\end{align*}"}
+{"id": "3119.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( A ) = \\bar { \\gamma } \\left ( c _ 1 ^ { 1 1 } ( A ^ 1 ) + \\bar { a } _ { 1 1 } ^ 1 \\int _ Y r ^ 1 A ^ 1 e _ 1 \\cdot \\nabla w \\right ) = s \\ , \\bar { a } _ { 1 1 } ^ 1 \\int _ Y r ^ 1 A ^ 1 e _ 1 \\cdot \\nabla q \\neq 0 , \\end{align*}"}
+{"id": "3566.png", "formula": "\\begin{align*} \\mathbf { p } E _ { j m } = E _ { j m } + \\sum _ { i = 1 } ^ { j - 1 } \\sum _ { s = 2 } ^ { j - i + 1 } \\sum _ { I \\in \\mathcal { I } _ { i j } ( s ) } E _ { i _ s i _ { s - 1 } } \\cdots E _ { i _ 2 i _ 1 } E _ { i _ 1 m } \\frac { 1 } { \\prod _ { \\ell \\in I , \\ell \\neq j } ( h _ j - h _ \\ell ) } \\end{align*}"}
+{"id": "145.png", "formula": "\\begin{align*} \\mathbb { W } : = \\{ \\varkappa \\in \\mathcal { Z } : \\| \\cdot \\| _ { P C } \\leq \\alpha _ { 1 } \\} , \\end{align*}"}
+{"id": "8677.png", "formula": "\\begin{align*} \\Gamma ^ + ( u ) & = E ( - t u ) H ( u ) E ^ { \\perp } ( - u ) , \\\\ \\Gamma ^ - ( u ) & = H ( t u ) E ( - u ) H ^ { \\perp } ( u ) . \\end{align*}"}
+{"id": "6076.png", "formula": "\\begin{align*} \\oint _ { \\partial U } g ( s ) ( \\partial _ { \\boldsymbol n } h ) ( s ) | \\dd s | = \\oint _ { \\partial U } h ( s ) ( \\partial _ { \\boldsymbol n } g ) ( s ) | \\dd s | , \\end{align*}"}
+{"id": "4838.png", "formula": "\\begin{align*} | h _ 1 ( r , t ) | & \\leq \\| d _ 0 F ^ { - 1 } \\| ( | \\xi | + | g ( t ) ) | + | R _ 1 ( r , t ) | ) \\\\ & \\leq C _ 2 ( | \\xi | + | t | + \\sigma ( | r | + | t | ) ) \\\\ & \\leq C _ 2 ( \\delta + 2 \\lambda _ 0 \\varepsilon + \\sigma \\varepsilon ) . \\end{align*}"}
+{"id": "1025.png", "formula": "\\begin{align*} c ^ \\beta _ { p , q } ( \\mu ; \\lambda ) : = \\inf \\Big \\{ C ^ { p , q } _ \\beta ( O ) : \\ \\hbox { o p e n } \\ O \\subseteq \\mathbb R ^ { n } , \\mu ( T ( O ) ) > \\lambda \\Big \\} \\end{align*}"}
+{"id": "7217.png", "formula": "\\begin{align*} q ( t ) = R \\Big ( e ^ { i \\phi } + e ^ { i \\psi } j \\Big ) \\ , + \\ , R \\Big [ \\cos \\vartheta e ^ { i \\phi } + \\sin \\vartheta e ^ { i \\psi } j \\Big ] , \\vartheta \\in \\left [ 0 , \\ , 2 n \\pi \\right ] \\end{align*}"}
+{"id": "5362.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\xi \\ , E ^ { ( i ) } _ h E ^ { ( j ) } _ h \\ , d x = 0 \\forall i , j \\in \\{ 1 , \\ldots , m \\} , i \\neq j , \\forall h = 1 , 2 , 3 , \\end{align*}"}
+{"id": "6907.png", "formula": "\\begin{align*} \\limsup _ { t \\rightarrow + \\infty } \\dfrac { M _ t } { t } = 0 . \\end{align*}"}
+{"id": "5947.png", "formula": "\\begin{align*} \\displaystyle A ^ T E _ r = \\sum _ { s = 1 } ^ p \\alpha _ { r s } E _ s + C _ p ^ T P _ r , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "2860.png", "formula": "\\begin{align*} P = P _ 1 \\bigoplus I _ { n - 2 } \\end{align*}"}
+{"id": "2765.png", "formula": "\\begin{align*} \\varepsilon ( X ^ { i } ) = \\underline { \\bar { \\varepsilon } } ( X ^ { i } ) = 0 . \\end{align*}"}
+{"id": "4896.png", "formula": "\\begin{align*} C _ { s _ 1 } C _ { s _ 2 u _ l s _ 2 s _ 1 } = C _ { s _ 1 s _ 2 u _ l s _ 2 s _ 1 } + C _ { u _ l s _ 2 s _ 1 } + C _ { s _ 1 s _ 2 u _ { l - 1 } s _ 2 s _ 1 } + \\Box . \\end{align*}"}
+{"id": "830.png", "formula": "\\begin{align*} p _ i = \\frac { e _ i ^ T A ^ { n p } 1 } { \\lambda ^ { n p } } + O \\left ( \\lambda ^ { - \\delta n } \\right ) \\ \\ u _ i = \\frac { e _ \\ast ^ T A ^ { n p } e _ i } { \\lambda ^ { n p } } + O \\left ( \\lambda ^ { - \\delta n } \\right ) \\end{align*}"}
+{"id": "6077.png", "formula": "\\begin{align*} I = \\frac { - 1 } { 4 \\pi \\log \\rho } \\int _ \\Delta \\log \\left ( \\frac { \\lambda ( x ) } { G _ \\lambda } \\right ) \\big ( \\partial _ { \\boldsymbol n + } g + \\partial _ { \\boldsymbol n - } g ) ( x ) \\dd x . \\end{align*}"}
+{"id": "7757.png", "formula": "\\begin{align*} \\widetilde f _ \\alpha ( s | \\mu ) & = \\frac { 1 } { \\pi } \\ , { \\rm I m } \\int _ 0 ^ \\infty \\frac { \\widetilde f _ \\alpha ( e ^ { - i \\pi } t | \\mu ) } { s + t } \\ ; d t \\end{align*}"}
+{"id": "8010.png", "formula": "\\begin{align*} V _ H & = \\{ F _ 1 , F _ 2 , F _ 3 \\} , \\\\ E _ H & = \\left \\{ ( F _ 2 , F _ 1 ) , ( F _ 3 , F _ 1 ) , ( F _ 3 , F _ 2 ) \\right \\} , \\end{align*}"}
+{"id": "6242.png", "formula": "\\begin{align*} \\delta _ \\sigma = \\begin{cases} \\delta , & \\sigma _ 1 \\sigma _ 2 \\sigma _ 3 \\neq 0 \\\\ \\delta ^ * , & ( \\sigma _ 1 , \\sigma _ 2 , \\sigma _ 3 ) = ( \\pm , \\mp , 0 ) , ( \\pm , 0 , \\mp ) \\\\ 2 \\max \\{ \\eta , 1 \\} , & \\sigma _ 1 = 0 , \\sigma _ 2 \\sigma _ 3 \\neq 0 \\\\ \\min \\{ \\eta , 1 \\} - \\epsilon , & \\vec \\sigma \\neq \\vec 0 \\end{cases} \\end{align*}"}
+{"id": "5260.png", "formula": "\\begin{align*} - g ( T _ U U , X ) = g ( U , U ) \\frac { d f } { d t } - g ( A _ X X , U ) . \\end{align*}"}
+{"id": "7497.png", "formula": "\\begin{align*} \\mathbb { N } ^ n = \\{ 0 \\} \\bigcup \\bigcup _ { \\gamma } \\big ( \\Delta _ { \\gamma } \\bigcap ( \\mathbb { N } ^ n \\setminus \\{ 0 \\} ) \\big ) . \\end{align*}"}
+{"id": "5542.png", "formula": "\\begin{align*} \\frac { 1 } { \\zeta ( k - 2 s ) } = \\frac { \\pi ^ { \\frac { 1 } { 2 } - k + 2 s } \\Gamma \\left ( \\frac { k - 2 s } { 2 } \\right ) } { \\Gamma \\left ( \\frac { 1 - k + 2 s } { 2 } \\right ) \\zeta ( 1 - k + 2 s ) } . \\end{align*}"}
+{"id": "492.png", "formula": "\\begin{align*} \\widehat { \\partial } h ( x ) : = \\bigg \\{ v \\in \\mathbb { X } \\ \\big | \\ \\liminf _ { x \\ne x ' \\to x } \\frac { h ( x ' ) - h ( x ) - \\langle v , x ' - x \\rangle } { \\| x ' - x \\| } \\ge 0 \\bigg \\} ; \\end{align*}"}
+{"id": "6874.png", "formula": "\\begin{gather*} J ( u ; F ) = \\norm { Q u - F } _ Y ^ 2 = \\int _ Y \\norm { Q u - F } _ y \\ ; d \\vec { x } = \\\\ \\int _ \\Omega \\norm { \\mathcal { L } u - f } _ a \\ ; d \\vec { x } + \\int _ { \\partial \\Omega } \\norm { \\mathcal { B } u - g } _ b \\ ; d \\vec { x } \\end{gather*}"}
+{"id": "3629.png", "formula": "\\begin{align*} f ( s ) : = \\left \\{ \\begin{array} { l l } \\dfrac { 1 } { | s | ^ 3 } \\ , e ^ { s ^ 2 } , & \\mbox { f o r } \\ | s | > R \\\\ \\kappa s ^ 2 , & \\mbox { i f } \\ | s | \\leq R , \\end{array} \\right . \\end{align*}"}
+{"id": "5910.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } \\displaystyle \\sum _ { r = 1 } ^ p ( u '' _ r e _ r - \\Delta u _ r e _ r + u _ r A e _ r ) = 0 & \\hbox { i n } ( T , + \\infty ) \\times \\Omega , \\\\ \\displaystyle \\sum _ { r = 1 } ^ p ( \\partial _ \\nu u _ r e _ r + u _ r B e _ r ) = 0 & \\hbox { o n } ( T , + \\infty ) \\times \\Gamma . \\end{array} \\right . \\end{align*}"}
+{"id": "712.png", "formula": "\\begin{align*} \\dot { a } _ j = - \\frac { \\Gamma } { V } \\frac { \\partial \\mathcal { H } } { \\partial { b } _ j } , \\dot { b } _ j = + \\frac { \\Gamma } { V } \\frac { \\partial \\mathcal { H } } { \\partial { a } _ j } . \\end{align*}"}
+{"id": "5744.png", "formula": "\\begin{align*} \\mu _ { 1 } ^ { A A ^ { \\prime } } & = ( L ^ { - 1 } ) _ { B ^ { \\prime } } ^ { A ^ { \\prime } } ( L ^ { - 1 } ) _ { B } ^ { A } \\mu ^ { B B ^ { \\prime } } \\\\ \\nu _ { 1 } ^ { A A ^ { \\prime } } & = ( L ^ { - 1 } ) _ { B ^ { \\prime } } ^ { A ^ { \\prime } } ( L ^ { - 1 } ) _ { B } ^ { A } \\nu ^ { B B ^ { \\prime } } + i ( L ^ { - 1 } ) _ { B ^ { \\prime } } ^ { A ^ { \\prime } } d [ ( L ^ { - 1 } ) _ { B } ^ { A } ] \\mu ^ { B B ^ { \\prime } } \\\\ & - i [ d ( L ^ { - 1 } ) _ { B ^ { \\prime } } ^ { A ^ { \\prime } } ] ( L ^ { - 1 } ) _ { B } ^ { A } \\mu ^ { B B ^ { \\prime } } . \\end{align*}"}
+{"id": "2323.png", "formula": "\\begin{align*} \\int _ { h ^ { - 1 } ( \\{ t \\} ) \\cap \\Omega } | \\nabla G ( y ) | ^ { p - 1 } \\ , d \\mathcal { H } ^ { N - 1 } ( y ) = 1 ( { } ^ { \\forall } t \\in ( 0 , 1 ) ) , \\end{align*}"}
+{"id": "1990.png", "formula": "\\begin{align*} \\left [ \\begin{matrix} | A | A ^ * & \\\\ & I _ k \\end{matrix} \\right ] \\left [ \\begin{matrix} A & C \\\\ & B \\end{matrix} \\right ] \\left [ \\begin{matrix} I _ m & - f ( | A | ) A ^ * C \\\\ & I _ l \\end{matrix} \\right ] = \\left [ \\begin{matrix} | A | ^ 3 & ( | A | - | A | ^ 3 f ( | A | ) ) A ^ * C \\\\ & B \\end{matrix} \\right ] . \\end{align*}"}
+{"id": "8958.png", "formula": "\\begin{align*} { } _ { 2 } F { } _ { 1 } \\left ( \\begin{array} { c } - s , - t \\\\ \\gamma + 1 \\end{array} \\big | y \\right ) = ( 1 - y ) ^ { \\frac { s + t } { 2 } } R _ { s , t } ^ { \\gamma } \\left ( ( 1 - y ) ^ { - \\frac { 1 } { 2 } } \\right ) . \\end{align*}"}
+{"id": "5373.png", "formula": "\\begin{align*} f _ 1 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) & = c _ 1 , \\\\ f _ 2 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) & = c _ 2 , \\\\ f _ 3 ( a x _ 1 + b x _ 2 ) & = c _ 3 , \\\\ f _ 4 ( a x _ 1 + b x _ 2 ) & = c _ 4 . \\end{align*}"}
+{"id": "4128.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\lambda ( u g , K ) = \\int _ M \\frac { n - 2 } { 2 } u \\triangle u - \\frac { u ^ 2 } { 2 } | H | ^ 2 + u \\langle d K , H \\rangle - \\frac { 1 } { 6 } | d K | ^ 2 - \\frac { 1 } { 2 } | \\nabla v | ^ 2 d V _ g \\end{align*}"}
+{"id": "1410.png", "formula": "\\begin{align*} \\bigsqcup _ { j = 1 } ^ t \\{ \\underbrace { [ - x + j - 1 , x ] _ \\rho , \\dots , [ - x + j - 1 , x ] _ \\rho } _ { k _ { j - 1 } - k _ j } \\} \\end{align*}"}
+{"id": "2289.png", "formula": "\\begin{align*} \\frac { d u _ { i } ( t ) } { d t } = \\frac { g _ { i + 1 / 2 } - g _ { i - 1 / 2 } } { \\Delta x ^ 2 } . \\end{align*}"}
+{"id": "1546.png", "formula": "\\begin{align*} \\gamma _ 5 : = \\gamma _ 1 \\gamma _ 2 \\gamma _ 3 \\gamma _ 4 \\end{align*}"}
+{"id": "702.png", "formula": "\\begin{align*} \\frac { d b _ k } { d t } = \\frac { \\Gamma } { V } \\oint _ { \\beta _ k } * \\nu = \\frac { \\Gamma } { V } \\oint _ { \\beta _ k } * \\eta . \\end{align*}"}
+{"id": "1374.png", "formula": "\\begin{align*} E ( t ) \\geq Q ( \\tilde { \\rho } ) = \\frac { 1 } { 2 } Q _ { 0 } \\tilde { \\rho } ^ { 2 } - \\frac { k } { 4 } \\tilde { \\rho } ^ { 2 } \\ln \\tilde { \\rho } ^ { 2 } \\end{align*}"}
+{"id": "8247.png", "formula": "\\begin{align*} y _ 1 + \\dots + \\hat { y _ i } + \\dots + y _ k + z _ 1 + \\dots + z _ m = \\sum _ { j = 1 } ^ s b _ j w _ j \\end{align*}"}
+{"id": "6398.png", "formula": "\\begin{align*} \\sup _ { \\tilde X \\in { { B _ \\epsilon ( X ) } } } u ( \\tilde X , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) & = \\max _ { \\tilde X \\in \\overline { B _ \\epsilon ( X ) } } u ( \\tilde X , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) , \\\\ \\inf _ { \\tilde X \\in { { B _ \\epsilon ( X ) } } } u ( \\tilde X , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) & = \\min _ { \\tilde X \\in \\overline { B _ \\epsilon ( X ) } } u ( \\tilde X , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) , \\end{align*}"}
+{"id": "2003.png", "formula": "\\begin{align*} m _ { \\chi ( X ) , \\chi ( Y ) } = \\prod _ { r = 1 } ^ n a _ { ( \\chi ( X ) ) _ r , ( \\chi ( Y ) ) _ r } = ( - 1 ) ^ { | \\{ r \\in \\{ 1 , \\dots , n \\} | a _ { ( \\chi ( X ) ) _ r , ( \\chi ( Y ) ) _ r } = - 1 \\} | } = ( - 1 ) ^ { | X \\cap Y | } . \\end{align*}"}
+{"id": "5742.png", "formula": "\\begin{align*} \\mathbf { x } ^ { A A ^ { \\prime } } = - \\mu ^ { A A ^ { \\prime } } \\mathbf { m } - \\mu ^ { A A ^ { \\prime } } \\overline { \\mathbf { m } } + i \\nu ^ { A A ^ { \\prime } } \\mathbf { m } \\overline { \\mathbf { m } } , \\end{align*}"}
+{"id": "8180.png", "formula": "\\begin{align*} k ( t , t s ) = t ^ { \\theta - 1 } k ( 1 , s ) , \\qquad \\qquad \\forall \\ , \\ , t \\in ( 0 , \\infty ) , s \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "3551.png", "formula": "\\begin{align*} L _ \\mu ^ + : = \\big \\{ v \\in L ^ + : E _ { i i } v = \\mu _ i v , \\ i \\in \\{ 1 , \\dots , n \\} \\big \\} . \\end{align*}"}
+{"id": "5434.png", "formula": "\\begin{align*} - \\left ( \\left ( 1 + 2 t f ' ( \\bar { \\alpha } \\alpha ) \\bar { \\alpha } \\alpha \\right ) G ^ { - 1 } \\right ) ^ { a b } _ { , a b } = A _ 0 . \\end{align*}"}
+{"id": "8974.png", "formula": "\\begin{align*} \\Delta u = \\frac { 1 } { r } ( r u _ r ) _ r + \\frac { 1 } { r ^ 2 } u _ { \\phi \\phi } \\end{align*}"}
+{"id": "125.png", "formula": "\\begin{align*} \\mathcal L ^ { 2 n } ( E ( F ) ) \\leq & C \\frac { 5 ^ n } { n } \\int _ { ( n , 1 ) } \\int _ { \\mathbb R } \\left | F _ L ( \\hat { x } + s \\omega ) \\right | d \\mathcal H ^ 1 ( s ) d L \\\\ = & C \\frac { 5 ^ n } { 2 n } \\int _ { S ^ { n - 1 } } \\int _ { \\omega ^ { \\bot } } \\int _ { \\mathbb R } \\left | F _ { \\hat { x } + L _ { \\omega } } ( \\hat { x } + s \\omega ) \\right | d \\mathcal H ^ 1 ( s ) d \\mathcal H ^ { n - 1 } ( \\hat { x } ) d \\mathcal H ^ { n - 1 } ( \\omega ) , \\end{align*}"}
+{"id": "5944.png", "formula": "\\begin{align*} \\begin{cases} \\widehat { U } '' - { \\Delta } \\widehat { U } + A \\widehat { U } = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\partial _ \\nu \\widehat { U } + B \\widehat { U } = D \\widehat H & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma \\\\ t = 0 : \\widehat { U } = \\widehat { U } ' = 0 & \\hbox { i n } \\Omega . \\end{cases} \\end{align*}"}
+{"id": "3369.png", "formula": "\\begin{align*} \\det ( x ^ 1 , \\dots , x ^ { n + 1 } ) = \\sum _ { \\sigma \\in S _ { n + 1 } } \\textnormal { s i g n } ( \\sigma ) \\prod _ { j = 1 } ^ { n + 1 } x ^ j _ { \\sigma ( j ) } \\ , . \\end{align*}"}
+{"id": "4891.png", "formula": "\\begin{align*} C _ { s _ 1 } C _ { s _ 2 u _ l } = C _ { s _ 1 s _ 2 u _ l } + C _ { u _ l } + C _ { s _ 1 s _ 2 u _ { l - 1 } } + \\Box . \\end{align*}"}
+{"id": "5874.png", "formula": "\\begin{align*} 0 = n _ 0 < n _ 1 < n _ 2 < \\cdots < n _ p = N \\end{align*}"}
+{"id": "3342.png", "formula": "\\begin{align*} 2 \\Phi _ { x x } & = - \\frac { 1 } { \\Phi _ { x q } } ( \\Phi _ { x y } \\Phi _ { y q } + \\Phi _ { p q } \\Phi _ { x p } ) - \\frac { 1 } { \\Phi _ { x p } } ( \\Phi _ { x y } \\Phi _ { y p } + \\Phi _ { x q } \\Phi _ { q p } ) , \\\\ 2 \\Phi _ { q q } & = - \\frac { 1 } { \\Phi _ { x q } } ( \\Phi _ { x y } \\Phi _ { y q } + \\Phi _ { p q } \\Phi _ { x p } ) + \\frac { 1 } { \\Phi _ { x p } } ( \\Phi _ { x y } \\Phi _ { y p } + \\Phi _ { x q } \\Phi _ { q p } ) . \\end{align*}"}
+{"id": "7195.png", "formula": "\\begin{align*} \\lambda = \\frac { D ( { \\Sigma _ 0 } ) - B _ { \\alpha } ( { \\Sigma _ 0 } ) y ^ 2 _ { \\alpha } } { B _ { \\alpha } ( { \\Sigma _ 0 } ) \\left [ y ^ 1 _ { \\alpha } - y ^ 2 _ { \\alpha } \\right ] } , \\end{align*}"}
+{"id": "6041.png", "formula": "\\begin{align*} \\dd \\mu ( x ) = \\frac { \\dot { \\mu } ( x ) \\dd x } { \\pi \\sqrt { ( x - a ) ( b - x ) } } + \\dd \\mu _ s ( x ) , \\end{align*}"}
+{"id": "7263.png", "formula": "\\begin{align*} \\frac { d } { d m } \\frac { d ^ + } { d x } u = \\lambda u , u ( 0 ) = 1 , u ^ + ( 0 ) = 0 , \\end{align*}"}
+{"id": "4783.png", "formula": "\\begin{align*} \\Pr _ { \\rho \\sim P _ \\epsilon } [ \\rho \\notin D _ k ( H \\rho ^ \\intercal ) ] & \\leq 2 e ^ { - \\frac { \\sqrt { N } } { 3 \\epsilon } } + \\frac { N } { k ^ * } \\max _ { j \\in B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } [ | v H | = j ] \\cdot \\frac { 2 ^ N } { \\binom { N } { j } } - 1 \\right \\} \\\\ & + \\frac { 2 ^ { h ( \\epsilon ) N + N ^ { \\frac { 4 } { 5 } } } } { k ^ * } \\max _ { j \\notin B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } \\big [ | v H | = j \\big ] \\cdot 2 ^ { 2 \\epsilon N \\log | 1 - \\frac { 2 j } { N } | } \\right \\} . \\end{align*}"}
+{"id": "9105.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\frac { 1 } { n _ 1 } \\tilde { i } ( x ^ { n _ 1 } ; Y _ 1 ^ { n _ 1 } , Y _ 2 ^ { n _ 2 } ) \\right ] = \\mathsf { C } _ { \\rho , 1 } + \\frac { \\mathsf { C } _ { \\rho , 2 } } { n _ 1 } \\cdot \\sum _ { i = n _ 2 + 1 } ^ { n _ 1 } x _ i ^ 2 , \\end{align*}"}
+{"id": "7507.png", "formula": "\\begin{align*} S _ { k , r } ( j ) : = { \\left \\{ \\begin{array} { r l } \\sum \\limits _ { i = k - j } ^ k ( - 1 ) ^ { k - i } \\binom { k + r } { i + r } \\binom { i + r } { i + j - k } , \\ \\ & 1 \\le j \\le k , \\\\ \\sum \\limits _ { i = 0 } ^ k ( - 1 ) ^ { k - i } \\binom { k + r } { i + r } \\binom { i + r } { i + j - k } , \\ \\ & k + 1 \\le j \\le k + r . \\end{array} \\right . } \\end{align*}"}
+{"id": "8158.png", "formula": "\\begin{align*} H = \\frac { 1 } { \\pi } \\int _ { - \\infty } ^ \\infty h ( t ) \\tanh ( \\pi t ) t \\ , d t , \\end{align*}"}
+{"id": "8664.png", "formula": "\\begin{align*} q _ \\lambda ( x _ 1 , x _ 2 , \\dots ) = \\ , M _ \\lambda . \\end{align*}"}
+{"id": "4068.png", "formula": "\\begin{align*} \\lambda ( g , H ) \\coloneqq \\inf \\Big \\{ \\mathcal { F } ( g , H , f ) \\big | f \\in C ^ \\infty ( M ) , \\ , \\int _ M e ^ { - f } d V _ g = 1 \\Big \\} . \\end{align*}"}
+{"id": "7371.png", "formula": "\\begin{align*} L _ 1 = [ - ( \\tfrac { 1 } { 2 } - \\varepsilon ) , \\tfrac { 1 } { 2 } ] , L _ 2 = [ \\tfrac { 1 } { 2 } , \\tfrac { 3 } { 2 } - \\varepsilon ] , \\end{align*}"}
+{"id": "4977.png", "formula": "\\begin{align*} \\widetilde { q } _ 1 : = b ^ { 1 - p } ( q _ 1 ^ p - \\rho ^ p ( q ) ) = b ^ { 1 - p } \\xi ^ p ( q ) \\in b w ^ p + \\rho ' ( y ) ^ p { \\cdot } ( d F ) ^ { p - 1 } y + b ^ { p - 1 } R [ b x , y ] , \\end{align*}"}
+{"id": "1215.png", "formula": "\\begin{align*} \\tilde { E } _ { h _ 0 } : \\tilde { U } \\times S ^ 1 & \\to \\C ^ * , \\\\ ( \\tilde { u } , h ) & \\mapsto e ^ { \\tilde { f } ( ( \\tilde { u } , h _ 0 ) , h _ 0 ^ { - 1 } h ) - C _ { h _ 0 } ( \\tilde { u } ) } , \\end{align*}"}
+{"id": "4928.png", "formula": "\\begin{align*} \\alpha u ( t + \\Delta t ) + \\beta u ( t ) = \\tfrac { \\Delta t } 2 ( u ' ( t + \\Delta t ) + u ' ( t ) ) , \\end{align*}"}
+{"id": "4635.png", "formula": "\\begin{align*} \\mathrm { K } ( E _ 0 ) : = \\left ( \\mathrm { d e t } \\ , E ^ { k , 0 } _ 0 \\right ) ^ { \\otimes k } \\otimes \\left ( \\mathrm { d e t } \\ , E ^ { k - 1 , 1 } _ 0 \\right ) ^ { \\otimes ( k - 1 ) } \\otimes \\dots \\otimes \\left ( \\mathrm { d e t } \\ , E ^ { 1 , k - 1 } _ 0 \\right ) . \\end{align*}"}
+{"id": "3329.png", "formula": "\\begin{align*} \\theta & = \\sqrt { \\delta } ( \\dd \\kappa - p \\dd q + q \\dd p ) , \\delta > 0 , \\\\ \\omega & = \\frac { k } { y ^ 2 } \\dd x \\wedge \\dd y + 2 \\nu \\dd q \\wedge \\dd p , y > 0 . \\end{align*}"}
+{"id": "3663.png", "formula": "\\begin{align*} M ( \\phi , \\varphi ) & = 0 & & \\Leftrightarrow & \\phi & \\ge 0 , \\ \\varphi \\le 0 , \\ \\langle \\phi , \\varphi \\rangle = 0 . \\end{align*}"}
+{"id": "7737.png", "formula": "\\begin{align*} \\mathcal { L } u = - u { '' } + c u { ' } , c \\geq 0 , \\end{align*}"}
+{"id": "3781.png", "formula": "\\begin{align*} \\tilde \\alpha g \\tilde \\alpha ^ { - 1 } & = ( \\mathcal { E } _ \\alpha \\alpha ) g ( \\alpha ^ { - 1 } \\mathcal { E } _ \\alpha ^ { - 1 } ) = \\mathcal { E } _ \\alpha \\alpha _ * ( g ) \\mathcal { E } _ \\alpha ^ { - 1 } = \\alpha _ * ( g ) . \\end{align*}"}
+{"id": "6522.png", "formula": "\\begin{align*} \\Phi ( x ) - \\frac { 1 } { 2 } = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ 0 ^ x e ^ { - \\frac { 1 } { 2 } z ^ 2 } d z = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ 0 ^ x \\underset { i = 0 } { \\overset { \\infty } { \\sum } } \\frac { ( - 1 ) ^ i z ^ { 2 i } } { 2 ^ i i ! } d z = \\frac { x } { \\sqrt { 2 \\pi } } \\Big ( \\underset { i = 0 } { \\overset { \\infty } { \\sum } } \\frac { ( - 1 ) ^ i ( x / \\sqrt { 2 } ) ^ { 2 i } } { ( 2 i + 1 ) i ! } \\Big ) , \\end{align*}"}
+{"id": "4802.png", "formula": "\\begin{align*} \\max _ { j \\in B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } \\big [ | v H | = j \\big ] \\cdot \\frac { 2 ^ N } { \\binom { N } { j } } \\right \\} & \\leq \\sqrt { \\frac { e ^ 4 N } { 8 \\pi } } \\cdot 2 ^ { ( 1 - h ( \\beta ) ) \\eta N } \\\\ & \\leq \\sqrt { \\frac { e ^ 4 N } { 8 \\pi } } \\cdot 2 ^ { 4 \\Tilde { \\epsilon } ( 1 - \\Tilde { \\epsilon } ) \\eta N } . \\end{align*}"}
+{"id": "4821.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial \\epsilon ^ 2 } f ( \\epsilon , \\frac { 2 } { \\ln 2 } ) = - \\frac { 1 } { ( 1 - \\epsilon ) ^ 2 \\ln 2 } < 0 . \\end{align*}"}
+{"id": "345.png", "formula": "\\begin{align*} e ( G ) \\geq f ( m , n ; p , \\left \\lfloor k _ \\ell / 2 \\right \\rfloor - 1 ) = p \\left ( n - \\left \\lfloor k _ \\ell / 2 \\right \\rfloor + 1 \\right ) + ( m - p ) ( \\left \\lfloor k _ \\ell / 2 \\right \\rfloor - 1 ) . \\end{align*}"}
+{"id": "6992.png", "formula": "\\begin{align*} \\lambda ^ \\prime ( y ) = \\lambda ( y ) \\cdot \\exp ( - J ( y , \\pi ) - J ( \\pi , y ) ) = \\left ( \\lambda + \\tau ( \\pi ) \\right ) ( y ) . \\end{align*}"}
+{"id": "1252.png", "formula": "\\begin{align*} \\beta < \\min \\left \\{ \\beta _ 0 , \\tilde { \\beta } _ 0 \\right \\} = \\beta _ 0 , \\beta & > \\max \\left \\{ \\beta _ 0 , \\tilde { \\beta } _ 0 \\right \\} = \\tilde { \\beta } _ 0 . \\end{align*}"}
+{"id": "4895.png", "formula": "\\begin{align*} C _ { s _ 2 } C _ { u _ l s _ 2 s _ 1 } = C _ { s _ 2 u _ l s _ 2 s _ 1 } + \\Box . \\end{align*}"}
+{"id": "1217.png", "formula": "\\begin{align*} \\tilde { f } ( ( \\gamma \\tilde { u } , h _ 0 ) , h _ 0 ^ { - 1 } \\iota ( \\gamma ) h ) - C _ { h _ 0 } ( \\gamma \\tilde { u } ) & = \\tilde { f } ( ( \\tilde { u } , h _ 0 ) , h _ 0 ^ { - 1 } h ) - C _ { h _ 0 } ( \\tilde { u } ) . \\end{align*}"}
+{"id": "2334.png", "formula": "\\begin{align*} { \\bf B } ( z ) = R \\circ J \\circ T _ { e _ N } \\circ S _ 2 \\circ J \\circ T _ { - e _ N } ( z ) , \\ , \\ , R = \\begin{pmatrix} 1 & & & \\\\ & \\ddots & & \\\\ & & 1 & \\\\ & & & - 1 \\end{pmatrix} , \\ , e _ N = \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ 1 \\end{pmatrix} \\end{align*}"}
+{"id": "1108.png", "formula": "\\begin{align*} Z _ i = \\frac { e ^ { \\alpha } + 1 } { e ^ { \\alpha } - 1 } W _ i s ( \\theta _ i - Y _ i ) \\mbox { a n d } s ( \\theta _ i - Y _ i ) \\begin{cases} = \\mathrm { s i g n } ( \\theta _ i - Y _ i ) , & \\theta _ i \\neq Y _ i , \\\\ \\sim \\mathrm { U n i f } \\{ - 1 , 1 \\} , & \\theta _ i = Y _ i , \\end{cases} \\end{align*}"}
+{"id": "5510.png", "formula": "\\begin{align*} \\zeta ( \\sigma ) = { \\sigma \\over \\sigma - 1 } - \\sigma \\int _ 1 ^ \\infty { \\{ x \\} \\over x ^ { \\sigma + 1 } } \\mathrm d x \\le { \\sigma \\over \\sigma - 1 } \\end{align*}"}
+{"id": "1593.png", "formula": "\\begin{align*} \\xi : = \\xi _ { \\frac { 1 } { 2 } } ^ { \\mu , \\nu } + \\widetilde { c } \\delta _ { x _ 1 } + \\widetilde { c } \\delta _ { y _ 1 } - \\widetilde { c } \\delta _ { x _ 2 } - \\widetilde { c } \\delta _ { y _ 2 } . \\end{align*}"}
+{"id": "863.png", "formula": "\\begin{align*} t _ 0 = \\max \\left \\{ \\frac 1 \\epsilon , \\ ; \\exp \\left ( \\frac { 3 ( 1 + \\eta ) } { \\delta ^ { 1 6 } } \\right ) \\right \\} . \\end{align*}"}
+{"id": "4553.png", "formula": "\\begin{align*} \\frac { p } { q } = \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 } + \\cdots + \\frac { 1 } { a _ { n - 1 } } + \\frac { 1 } { a _ n } \\end{align*}"}
+{"id": "8996.png", "formula": "\\begin{align*} \\int _ { t _ 0 } ^ 0 \\int _ { - R } ^ R & | \\partial _ t v _ k | ^ 2 d x \\ ; d t \\\\ & \\le C \\int _ { t _ 0 } ^ 0 \\int _ { - R } ^ R | d \\pi _ N ( v _ k ) \\partial _ y v _ k | ^ 2 d x \\ ; d t \\to 0 \\ \\hbox { a s } k \\to \\infty , \\end{align*}"}
+{"id": "5176.png", "formula": "\\begin{align*} \\begin{aligned} p _ { ( v _ i , k ) } & = \\sum _ { i _ 1 = i } ^ n s _ { ( e _ { i i _ 1 } , m _ i + k ) } s _ { ( e _ { i i _ 1 } , m _ i + k ) } ^ * \\\\ & = \\sum _ { i _ 1 = i } ^ n s _ { ( e _ { i i _ 1 } , m _ i + k ) } p _ { r ( ( e _ { i i _ 1 } m _ i + k ) ) } s _ { ( e _ { i i _ 1 } , m _ i + k ) } ^ * \\\\ & = \\sum _ { i _ 1 = i } ^ n s _ { ( e _ { i i _ 1 } , m _ i + k ) } p _ { ( v _ { i _ 1 } , m _ i + k ) } s _ { ( e _ { i i _ 1 } , m _ i + k ) } ^ * . \\end{aligned} \\end{align*}"}
+{"id": "7479.png", "formula": "\\begin{align*} S _ A = \\min _ { ( u , v ) \\in E _ A } S _ { u v } . \\end{align*}"}
+{"id": "5717.png", "formula": "\\begin{align*} \\theta _ { 1 } ^ { a } = ( \\mathbf { L } _ { 1 } ^ { - 1 } ) _ { b } ^ { a } d \\mathbf { x } _ { 1 } ^ { b } . \\end{align*}"}
+{"id": "2523.png", "formula": "\\begin{align*} M _ f ( x , v , t ) = M _ { u _ f ( x , t ) , T _ f ( x , t ) } ( v ) \\ , , \\end{align*}"}
+{"id": "6259.png", "formula": "\\begin{align*} H \\big | _ { \\mathcal { U } _ { 0 } \\times ( - \\delta ' , 1 + \\delta ' ) } = h \\big | _ { \\mathcal { U } _ { 0 } \\times ( - \\delta ' , 1 + \\delta ' ) } . \\end{align*}"}
+{"id": "9163.png", "formula": "\\begin{align*} \\langle F ( \\sigma ) \\rangle _ { \\beta } = \\lim _ { m ^ 2 \\downarrow 0 } \\langle F ( \\sigma ) \\rangle _ { \\beta , m ^ 2 } , \\end{align*}"}
+{"id": "4771.png", "formula": "\\begin{align*} \\| f \\| _ 2 ^ 2 = \\sum _ { j = 0 } ^ N \\Pr [ | c ^ \\perp | = j ] \\cdot K _ { \\epsilon N } ( j ) ^ 2 , \\end{align*}"}
+{"id": "618.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } Z _ t ^ { ( \\epsilon ) } = \\frac { 1 } { \\delta } ( X _ t ^ { ( \\epsilon ) } - Z _ t ^ { ( \\epsilon ) } ) . \\end{align*}"}
+{"id": "7029.png", "formula": "\\begin{align*} d _ { i , j + 1 } ( d _ { k , \\ell } - c _ { k , \\ell } ) & = ( d _ { i , j } - c _ { i , j } ) d _ { k , \\ell + 1 } \\\\ d _ { i , j + 1 } ( q _ \\ell - p _ \\ell ) & = ( d _ { i , j } - c _ { i , j } ) q _ { \\ell + 1 } \\\\ q _ { j + 1 } ( q _ \\ell - p _ \\ell ) & = ( q _ j - p _ j ) q _ { \\ell + 1 } \\\\ q _ 0 & = 0 , \\end{align*}"}
+{"id": "4881.png", "formula": "\\begin{align*} a _ { 1 1 } = a _ { 3 3 } , \\ a _ { 1 3 } = a _ { 3 1 } , \\ a _ { 2 2 } = a _ { 1 1 } + a _ { 1 3 } , \\ a _ { 1 2 } = a _ { 2 1 } = a _ { 2 3 } = a _ { 3 2 } . \\end{align*}"}
+{"id": "2535.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { j = 1 } ^ N | p _ { 1 , j } - q _ { 1 , j } | | w _ j | \\bigg ) ^ 2 \\le \\bigg ( \\sum _ { j = 1 } ^ N ( p _ { 1 , j } + q _ { 1 , j } ) | w _ j | \\bigg ) ^ 2 \\le 2 \\sum _ { j = 1 } ^ N ( p _ { 1 , j } + q _ { 1 , j } ) | w _ j | ^ 2 \\ , , \\end{align*}"}
+{"id": "2866.png", "formula": "\\begin{align*} \\prod _ { \\sigma \\in S _ n } x _ { \\sigma ( 1 ) } ^ { c _ 1 } x _ { \\sigma ( 2 ) } ^ { c _ 2 } \\cdots x _ { \\sigma ( n ) } ^ { c _ n } = \\prod _ { j = 1 } ^ n x _ j ^ { ( n - 1 ) ! \\sum _ { i = 1 } ^ n c _ i } = 1 . \\end{align*}"}
+{"id": "3420.png", "formula": "\\begin{align*} P _ \\mu = h ( \\mu ) + \\mu ( \\phi ) = P _ { t o p } ( \\phi ) . \\end{align*}"}
+{"id": "588.png", "formula": "\\begin{align*} \\mu _ { n + 1 } = \\mu _ n + \\frac { \\sigma _ n } { \\gamma } \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ \\dagger ) ^ { \\rm T } { \\rm d } X _ t ^ { \\dagger } - K _ n A X _ { t _ n } ^ \\dagger \\mu _ n \\Delta t . \\end{align*}"}
+{"id": "8698.png", "formula": "\\begin{align*} \\mathcal M _ { i , j } = \\begin{cases} \\mathcal Q ( u _ i , u _ j ) , & i < j , \\\\ 0 , & i = j , \\\\ - \\mathcal Q ( u _ j , u _ i ) , & i > j . \\end{cases} \\end{align*}"}
+{"id": "6455.png", "formula": "\\begin{align*} H & = H ( m ) = - \\delta E ( m ) + H _ a \\\\ \\textrm { w i t h } \\delta E ( m ) & = - \\partial _ { x x } m - 2 \\gamma e _ 1 \\wedge \\partial _ x m + m _ 2 e _ 2 + m _ 3 e _ 3 . \\end{align*}"}
+{"id": "3017.png", "formula": "\\begin{align*} \\begin{array} { c c l c } \\Theta ^ a & : = & \\hbox { d } \\theta ^ a & \\hbox { f o r } 0 \\leq a \\leq 3 \\\\ \\Theta ^ i & : = & \\hbox { d } \\theta ^ i + \\frac { 1 } { 2 } c ^ i _ { j k } \\theta ^ j \\wedge \\theta ^ k & \\hbox { f o r } 3 < i \\leq N \\end{array} \\end{align*}"}
+{"id": "5009.png", "formula": "\\begin{align*} g ( \\nabla _ X Y , Z ) = g ( \\nabla ^ g _ X Y , Z ) + \\frac 1 2 H ( X , Y , Z ) , \\end{align*}"}
+{"id": "2424.png", "formula": "\\begin{align*} \\dot { \\tilde v } _ 2 = \\tilde J _ { 2 2 } ( t ) \\tilde v _ 2 + \\tilde f _ 2 ( t ) , \\end{align*}"}
+{"id": "4897.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\partial _ s x ( s ) + \\nabla f ( x ( s ) ) + \\tau ( s ) \\nabla h ( x ( s ) ) = 0 \\\\ \\partial _ s \\tau ( s ) + h ( x ( s ) ) = 0 \\end{array} \\right . \\end{align*}"}
+{"id": "1344.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | \\geq \\frac { \\frac { 1 } { ( G _ { f , \\tau } ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ 2 } { n ^ 2 - n } , \\end{align*}"}
+{"id": "7451.png", "formula": "\\begin{align*} \\widehat M _ \\chi = ( \\widehat M _ { \\chi _ 1 } , \\dots , \\widehat M _ { \\chi _ d } ) = \\widehat Z ^ { ( 0 ) } . \\end{align*}"}
+{"id": "1777.png", "formula": "\\begin{align*} \\beta _ { n + 1 } = m \\alpha ^ { k + n } ( q ( \\alpha ) - 1 ) + ( m \\alpha ^ { k + n } + \\beta ' _ { n + 1 } ) = m \\alpha ^ { k + n } ( q ( \\alpha ) - 1 ) + \\beta _ n , \\end{align*}"}
+{"id": "2749.png", "formula": "\\begin{align*} \\overline { \\varphi _ { R } } = \\varphi _ { L } . \\end{align*}"}
+{"id": "8855.png", "formula": "\\begin{align*} d \\mu _ { n } ( z ) : = ( 1 + \\langle z , z \\rangle ) ^ { - ( n + 1 ) } d \\mu \\left ( z \\right ) , \\end{align*}"}
+{"id": "1580.png", "formula": "\\begin{align*} \\big ( \\psi _ \\# ( \\mu ) \\big ) ( x ) : = \\mu \\big ( \\psi ^ { - 1 } ( x ) \\big ) \\qquad ( x \\in X ) \\end{align*}"}
+{"id": "1589.png", "formula": "\\begin{align*} ( 1 - s ) \\alpha \\geq d _ p ^ p ( \\nu , \\xi ) \\geq | \\xi ( v ) - \\nu ( v ) | \\geq \\big | \\eta ( v ) + s \\alpha - \\big ( \\eta ( v ) + \\alpha \\delta _ v ( v ) \\big ) \\big | = ( 1 - s ) \\alpha . \\end{align*}"}
+{"id": "3374.png", "formula": "\\begin{align*} \\allowdisplaybreaks z ' _ { n - 1 } & = ( u _ n - 1 ) - \\lambda ) \\vee ( - 1 ) \\\\ & = u _ n + ( ( - \\lambda - 1 ) \\vee ( - u _ n - 1 ) ) \\\\ & = u _ n + ( ( - \\lambda - 1 ) \\vee ( - \\lambda ) ) & & \\\\ & \\leq u _ n - \\lambda = u _ n + z ' _ n \\ , . \\end{align*}"}
+{"id": "1989.png", "formula": "\\begin{align*} \\left \\{ \\left ( { \\mathcal { R } } ^ { } , { \\mathcal { R } } ^ { } \\right ) | 0 \\leq { \\mathcal { R } } ^ { } \\leq { L } ^ { - 1 } \\sum \\nolimits _ { i = 1 } ^ { L } \\bar { \\mathcal { R } } _ i , 0 \\leq { \\mathcal { R } } ^ { } \\leq { \\mathcal { R } } _ { } \\right \\} , \\end{align*}"}
+{"id": "3050.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\bar { A } : D ^ 2 u & = f & & \\Omega , \\\\ u & = g & & \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "2909.png", "formula": "\\begin{align*} \\sup _ { 0 < r < 1 } \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | ( \\mathcal { B } _ k f ) _ { [ r ] } | ) d m _ \\infty = \\sup _ { \\sigma > 0 } \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | \\mathcal { B } _ k f _ \\sigma | ) d m _ \\infty \\leq \\| f \\| _ 0 < \\infty , \\end{align*}"}
+{"id": "7175.png", "formula": "\\begin{align*} \\mathcal { S } = \\mathbb { R } \\times \\Sigma . \\end{align*}"}
+{"id": "3007.png", "formula": "\\begin{align*} p _ i \\ ! = \\ ! \\frac { 1 } { 2 } p { _ i } ^ { a b } e ^ { ( 2 ) } _ { a b } \\wedge \\bar { e } ^ { ( r ) } - p { _ i } ^ { a k } e ^ { ( 3 ) } _ { a } \\wedge \\bar { e } ^ { ( r - 1 ) } _ k + \\frac { 1 } { 2 } p { _ i } ^ { j k } e ^ { ( 4 ) } \\wedge \\bar { e } ^ { ( r - 2 ) } _ { j k } \\end{align*}"}
+{"id": "8108.png", "formula": "\\begin{align*} \\mathcal { R } ^ + _ 3 = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { c \\leq \\frac { C _ 2 } { m } } \\frac { S ( n , p ; c ) } { c } H _ { m , n } ^ + \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) . \\end{align*}"}
+{"id": "2753.png", "formula": "\\begin{align*} \\varphi _ { R } ^ { \\ast } & = ( \\varphi _ { R } ^ { \\ast } ) ^ { [ + ] } + ( \\varphi _ { R } ^ { \\ast } ) ^ { [ - ] } , \\\\ \\varphi _ { L } ^ { \\ast } & = ( \\varphi _ { L } ^ { \\ast } ) ^ { [ + ] } + ( \\varphi _ { L } ^ { \\ast } ) ^ { [ - ] } . \\end{align*}"}
+{"id": "6466.png", "formula": "\\begin{align*} d w _ * & = n _ * d \\theta _ * + \\sin \\theta _ * p _ * d \\varphi _ * , \\\\ d n _ * & = - w _ * d \\theta _ * + \\cos \\theta _ * p _ * d \\varphi _ * , \\\\ d p _ * & = - ( \\sin \\theta _ * w _ * + \\cos \\theta _ * n _ * ) d \\varphi _ * . \\end{align*}"}
+{"id": "3549.png", "formula": "\\begin{align*} M ( \\mathfrak { g } , \\mathfrak { t } ) w _ 0 = U ( \\mathfrak { g } ) w _ 0 = L . \\end{align*}"}
+{"id": "9026.png", "formula": "\\begin{align*} _ k ( H ^ 2 ( \\Q _ \\Sigma / \\Q _ { n } , \\Phi ) ) - _ k ( N / N _ { } ) = ( h _ 1 - d ^ - ( \\Phi ) ) p ^ n . \\end{align*}"}
+{"id": "8495.png", "formula": "\\begin{align*} l '' ( t _ 0 ) & = - \\frac { 1 } { 2 } \\int ^ 1 _ 0 \\bigg ( \\frac { 9 } { 2 } U _ { \\xi ^ 4 } ^ 2 + 3 ( \\varphi ^ 2 - \\epsilon ) U _ { \\xi ^ 3 } ^ 2 + \\frac { ( \\varphi ^ 2 - \\epsilon ) ^ 2 } { 2 } U _ { \\xi ^ 2 } ^ 2 \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\frac { ( \\varphi ^ 2 - 4 \\epsilon ) ( 2 \\varphi ^ 2 + \\epsilon ) ^ 2 } { 1 6 2 } U _ { \\xi } ^ 2 \\bigg ) g d p . \\end{align*}"}
+{"id": "330.png", "formula": "\\begin{align*} P ( l ) = \\frac { 2 } { 3 } \\frac { 1 } { \\zeta ( 2 ) \\sqrt { l _ 1 } } \\prod _ { p \\mid l } \\frac { p } { p + 1 } I ( l ) + O ( l _ 1 ^ { \\theta _ 3 - \\frac { 1 } { 2 } + \\varepsilon } Y ^ { - 1 } ) , \\end{align*}"}
+{"id": "6570.png", "formula": "\\begin{align*} S \\geq \\alpha _ k ( 1 - \\beta _ k ) + \\beta _ { l } ( 1 - \\alpha _ { l } ) \\geq \\alpha _ k \\beta _ { l } + \\beta _ { l } \\alpha _ k = 2 \\alpha _ k \\beta _ { l } . \\end{align*}"}
+{"id": "8659.png", "formula": "\\begin{align*} f ^ \\perp = F ( \\partial / \\partial p _ 1 , 2 \\partial / \\partial p _ 2 , 3 \\partial / \\partial p _ 3 , \\dots ) . \\end{align*}"}
+{"id": "2274.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla _ x f + { \\rm { d i v } } _ v ( ( \\nabla \\Phi ^ * + \\tilde u \\chi _ { \\{ | \\tilde u | \\leq N \\} } - v ) f ) - \\Delta _ v f = 0 , \\end{align*}"}
+{"id": "8927.png", "formula": "\\begin{align*} \\sum \\limits _ { p = 0 } ^ { n - 1 } \\tau _ { p } ^ { ( \\nu , n ) } \\mu ^ { 2 p } = 0 , \\end{align*}"}
+{"id": "5646.png", "formula": "\\begin{align*} s + \\sum _ { i = 1 } ^ k s _ i > k \\end{align*}"}
+{"id": "982.png", "formula": "\\begin{align*} \\Vert e _ { \\alpha } a - a \\Vert _ { \\mathcal { A } } & = \\Vert f _ { \\alpha } ( x _ 0 ) g ( x _ 0 ) a - g ( x _ 0 ) a \\Vert _ { \\mathcal { A } } \\\\ & \\leq \\Vert f _ { \\alpha } \\cdot f _ { a } - f _ { a } \\Vert _ { \\infty , \\mathcal { A } } < \\epsilon . \\end{align*}"}
+{"id": "8550.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } 2 ^ { m \\log _ 2 ( n ) + n \\log _ 2 ( 1 - 2 ^ { - ( m ^ 2 + 3 m + 2 ) / 2 } ) } = 0 . \\end{align*}"}
+{"id": "2569.png", "formula": "\\begin{align*} p ^ { G \\times G } \\circ \\Psi ^ { G \\times G } \\circ \\Omega ( g ) & = p ^ { G \\times G } \\circ \\Psi ^ { G \\times G } ( \\tilde { g } ) = p ^ { G \\times G } ( \\tilde { g } ( 0 ) , \\tilde { g } ( 1 ) ) \\\\ & = \\tilde { g } ( 0 ) = ( \\tilde { g } _ 1 ( 0 ) , \\tilde { g } _ 2 ( 0 ) ) = ( g ( 0 ) , g ( 1 ) ) = \\Psi ^ G ( g ) . \\end{align*}"}
+{"id": "5438.png", "formula": "\\begin{align*} d X _ t = b ( t , X _ t , \\mathcal { L } _ { X _ t } , \\alpha _ t ) d t + \\sigma ( t , X _ t , \\mathcal { L } _ { X _ t } , \\alpha _ t ) d W _ t \\end{align*}"}
+{"id": "1570.png", "formula": "\\begin{align*} R i c + \\nabla ^ 2 f = \\mu g , \\end{align*}"}
+{"id": "5504.png", "formula": "\\begin{align*} f ( s ) = \\zeta ( s ) L ( s , \\chi _ 1 ) L ( s , \\chi _ 2 ) L ( s , \\chi _ 1 \\chi _ 2 ) \\end{align*}"}
+{"id": "6264.png", "formula": "\\begin{align*} \\mathcal { S } _ { 0 } = \\{ ( x , y , z ) \\in \\mathcal { M } _ { 0 } : x = 0 , z = 1 \\} . \\end{align*}"}
+{"id": "6158.png", "formula": "\\begin{align*} \\hbox { d i m I m } ( C _ 1 ^ T ) + \\hbox { d i m K e r } ( \\mathcal R ^ T ) = N - 1 + d > N , \\end{align*}"}
+{"id": "595.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\rm s t a t } \\left [ P _ t ^ { ( \\epsilon ) } \\otimes P _ t ^ { ( \\epsilon ) } \\right ] = \\epsilon \\ , ( M + M ^ { \\rm T } ) ^ { - 1 } = \\frac { \\epsilon } { 2 } I . \\end{align*}"}
+{"id": "4410.png", "formula": "\\begin{align*} \\prod _ { i = m + k + 1 } ^ n a _ i \\leq \\frac { \\prod _ { i = m + 1 } ^ n x _ i } { \\prod _ { i = m + 1 } ^ { m + k } a _ i } = \\left ( \\frac { \\prod _ { i = m + 1 } ^ { m + k } x _ i } { \\prod _ { i = m + 1 } ^ { m + k } a _ i } \\right ) \\prod _ { i = m + k + 1 } ^ n x _ i < \\prod _ { i = m + k + 1 } ^ n x _ i \\end{align*}"}
+{"id": "6411.png", "formula": "\\begin{align*} ( \\nabla \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) , \\nabla ^ 2 \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) ) = ( q _ k , - H _ k ) . \\end{align*}"}
+{"id": "958.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( \\Omega ^ { n - 1 } M , C ) \\ , = \\ , \\textrm { r . g r a d e } _ R ( \\textrm { T r } _ C ( M ' ) , C ) . \\end{align*}"}
+{"id": "1642.png", "formula": "\\begin{align*} \\min \\{ x \\in \\bar { D } : v ( x ) - u ( x ) \\} = \\min \\{ x \\in \\partial D : v ( x ) - u ( x ) \\} . \\end{align*}"}
+{"id": "8285.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { B } : = \\big \\{ \\ \\textbf { x } \\ | \\ B ( \\textbf { x } ) = 0 \\big \\} \\end{array} \\right . \\end{align*}"}
+{"id": "7116.png", "formula": "\\begin{align*} A ^ { G ^ \\vee } = \\prod _ { G ^ \\vee } A \\end{align*}"}
+{"id": "1365.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { d } \\int _ \\Omega f _ \\alpha ( \\tau _ \\alpha ) \\ , d \\mu ( \\alpha ) \\right ) ^ r = \\left ( \\frac { 1 } { d } \\operatorname { T r a } ( S _ { f , \\tau } ) \\right ) ^ r \\leq \\frac { 1 } { d } \\operatorname { T r a } ( S _ { f , \\tau } ) ^ r . \\end{align*}"}
+{"id": "5937.png", "formula": "\\begin{align*} V = \\{ E _ 1 , \\cdots , E _ p \\} ~ ( e _ r , E _ s ) = \\delta _ { r s } ( r , s = 1 , \\cdots , p ) . \\end{align*}"}
+{"id": "1964.png", "formula": "\\begin{align*} n _ i ( v ) = | \\{ j \\in \\{ 1 , 2 , \\ldots , n \\} \\mid \\rho ( v _ j ) \\equiv i \\pmod { 2 k } \\} | \\ ( i = 0 , 1 , \\ldots , 2 k - 1 ) . \\end{align*}"}
+{"id": "2821.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 ) = x _ 1 \\end{align*}"}
+{"id": "8093.png", "formula": "\\begin{align*} \\mathcal { R } ^ + _ 2 = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { \\frac { C _ 2 } { m } \\leq c \\leq \\frac { C _ 1 } { m } } \\frac { S ( n , p ; c ) } { c } H _ { m , n } ^ + \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) \\end{align*}"}
+{"id": "961.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\mathfrak { K o s z } } = ( n - 2 ) \\cdot \\mathcal { D } _ { \\mathfrak { R e s } } \\end{align*}"}
+{"id": "5896.png", "formula": "\\begin{align*} t \\geq T : \\sum _ { r = 1 } ^ p C _ p A e _ r u _ r = 0 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "6720.png", "formula": "\\begin{align*} \\mu ( q + a ) = \\mu ( 2 q + a ) = \\mu ( 3 q + a ) = \\mu ( 4 q + a ) = \\cdots = \\mu ( [ x / q ] q + a ) = 1 \\end{align*}"}
+{"id": "3266.png", "formula": "\\begin{align*} \\tilde { P } _ { \\mathbf { d } , \\mathcal { A } } ( a b ) = \\frac { ( n - 1 ) d _ a d _ b } { d n _ a n _ b } \\left ( 1 + \\mathcal { O } \\left ( \\frac { t ^ 2 } { n d } \\right ) \\right ) , \\end{align*}"}
+{"id": "1405.png", "formula": "\\begin{align*} \\psi = S _ 1 + S _ 2 \\boxtimes S _ 2 + S _ 4 \\boxtimes S _ 4 . \\end{align*}"}
+{"id": "1248.png", "formula": "\\begin{align*} \\sqrt { H ( x _ 2 ) } & = \\sqrt { \\dfrac { \\sqrt { \\alpha [ \\alpha - ( 1 - a ) ^ 2 ( 1 - \\alpha ^ 2 ) ^ 2 ] } + \\alpha ( 1 + 2 ( 1 + \\alpha ) ^ 2 ( 1 - a ) ^ 2 ) } { 2 \\alpha ( 1 + \\alpha ) ^ 2 } } . \\end{align*}"}
+{"id": "8369.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\int _ { | x | > - R + t } \\left ( | \\nabla u | ^ 2 + | \\partial _ t u | ^ 2 \\right ) = 0 . \\end{align*}"}
+{"id": "7400.png", "formula": "\\begin{align*} q _ { \\alpha ^ * } = 1 , q _ { \\alpha } = q _ L = q _ F ^ { f ( L / F ) } \\end{align*}"}
+{"id": "3803.png", "formula": "\\begin{align*} & \\Delta ( ( f + 1 0 0 ) ^ { - 8 } ) + \\langle X , \\nabla ( ( f + 1 0 0 ) ^ { - 8 } ) \\rangle \\\\ & = - 8 \\ , ( f + 1 0 0 ) ^ { - 9 } \\ , ( \\Delta f + \\langle X , \\nabla f \\rangle ) + 7 2 \\ , ( f + 1 0 0 ) ^ { - 1 0 } \\ , | \\nabla f | ^ 2 \\\\ & \\leq - 8 \\ , ( f + 1 0 0 ) ^ { - 9 } + 7 2 \\ , ( f + 1 0 0 ) ^ { - 1 0 } \\\\ & \\leq - ( f + 1 0 0 ) ^ { - 9 } . \\end{align*}"}
+{"id": "6373.png", "formula": "\\begin{align*} m ' _ \\lambda ( r _ 2 ) + m ' _ \\lambda ( r _ 1 ) = \\alpha \\cos u _ 2 \\left ( \\frac { 1 } { \\sqrt { x ' _ \\lambda ( u _ 2 ) ^ 2 + y ' ( u _ 2 ) ^ 2 } } - \\frac { 1 } { \\sqrt { x ' _ \\lambda ( u _ 1 ) ^ 2 + y ' ( u _ 2 ) ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "979.png", "formula": "\\begin{align*} N ( n , m ) = 3 n - 2 + m ( m - 1 ) . \\end{align*}"}
+{"id": "8829.png", "formula": "\\begin{align*} \\mathcal F _ { \\mathbb N } = \\cup _ { n \\in \\mathbb N \\cup \\{ 0 \\} } \\ , \\mathcal F _ n \\ , . \\end{align*}"}
+{"id": "1436.png", "formula": "\\begin{align*} \\mathbb { E } [ \\eta _ { t } | R _ { t - 1 } ] \\geq \\mathbb { E } [ \\eta ' _ { t } | R _ { t - 1 } ] = ( d - 1 ) p - \\mathbb { E } [ X ^ { ( 1 ) } _ t | R _ { t - 1 } ] - \\mathbb { E } [ X ^ { ( 2 ) } _ t | R _ { t - 1 } ] . \\end{align*}"}
+{"id": "5990.png", "formula": "\\begin{align*} \\chi _ H & \\left ( e v _ 1 ^ * \\alpha _ 1 \\otimes \\dots \\otimes e v _ r ^ * \\alpha _ r \\otimes [ \\O _ { \\overline { { \\mathcal { M } } _ { 0 , r } } ( X , \\d ) } ] \\right ) & = \\sum _ { e _ 1 , \\dots , e _ r \\in I } \\chi _ H \\left ( a _ { 1 , e _ 1 } ( e v _ 1 ^ * \\phi _ { e _ 1 } ) \\otimes \\dots \\otimes a _ { r , e _ r } ( e v _ r ^ * \\phi _ { e _ r } ) \\right ) \\end{align*}"}
+{"id": "8077.png", "formula": "\\begin{align*} H _ { m , p } = \\frac { 2 } { \\pi } \\int _ 0 ^ \\infty k ( t ) V ( m ^ 2 p , t ) \\tanh ( \\pi t ) t \\ , d t \\end{align*}"}
+{"id": "8267.png", "formula": "\\begin{align*} S : = S ( \\rho _ 0 ) : = U ( \\rho _ 0 ) + U ( \\overline { \\rho _ 0 } ) , \\end{align*}"}
+{"id": "5053.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ d ( 1 - \\lambda _ i x _ i ) ^ { - k _ i } = \\sum _ { n _ 1 , \\ldots , n _ d = 0 } ^ \\infty { \\binom { k _ 1 + n _ 1 - 1 } { n _ 1 } } \\cdots { \\binom { k _ d + n _ d - 1 } { n _ d } } \\lambda _ 1 ^ { n _ 1 } \\cdots \\lambda _ d ^ { n _ d } x _ 1 ^ { n _ 1 } \\cdots x _ d ^ { n _ d } , \\end{align*}"}
+{"id": "8490.png", "formula": "\\begin{align*} C _ t = W C _ \\xi + U C _ { \\xi ^ 2 } , \\end{align*}"}
+{"id": "7227.png", "formula": "\\begin{align*} A _ j = \\sum _ { i = 0 } ^ d P _ { i j } E _ i , E _ j = \\frac { 1 } { | X | } \\sum _ { i = 0 } ^ { d } Q _ { i j } A _ i . \\end{align*}"}
+{"id": "7997.png", "formula": "\\begin{align*} \\overline { \\lim _ { N \\rightarrow \\infty } } D _ { N } \\left ( \\varepsilon ^ 0 \\right ) = + \\infty , \\end{align*}"}
+{"id": "2225.png", "formula": "\\begin{align*} \\nu ( x _ { 2 m , 2 } ) & = \\nu { \\left ( \\sum _ { k = 1 } ^ \\infty x _ { 2 m - 1 , k } \\alpha _ { k , 2 } \\right ) } \\\\ & = \\min _ { k \\geq 4 } { \\left \\{ \\big ( \\nu ( x _ { 2 m - 1 , 2 } ) + \\nu ( \\alpha _ { 2 , 2 } ) \\big ) , \\big ( \\nu ( x _ { 2 m - 1 , 3 } ) + \\nu ( \\alpha _ { 3 , 2 } ) \\big ) \\right . } , \\\\ & \\qquad \\qquad \\left . \\big ( \\nu ( x _ { 2 m - 1 , k } ) + \\nu ( \\alpha _ { k , 2 } ) \\big ) \\right \\} \\\\ & \\geq 2 m . \\end{align*}"}
+{"id": "845.png", "formula": "\\begin{align*} J = \\frac { 1 } { 2 } \\int _ \\textit { B } \\rho ( X ) \\widehat { X } \\widehat { X } ^ T d ^ 3 X \\end{align*}"}
+{"id": "2950.png", "formula": "\\begin{align*} \\nu ^ n _ \\rho ( \\eta ^ { j , j + 1 } ) = \\frac { g _ n ( \\eta _ j ) } { g _ n ( \\eta _ { j + 1 } + 1 ) } \\nu ^ n _ \\rho ( \\eta ) \\end{align*}"}
+{"id": "3008.png", "formula": "\\begin{align*} p { _ i } ^ { a b } + \\mathbf { F } { _ i } ^ { a b } = 0 \\end{align*}"}
+{"id": "3313.png", "formula": "\\begin{align*} 2 \\ , x = l \\end{align*}"}
+{"id": "1876.png", "formula": "\\begin{align*} \\beta = \\sin ^ { - 1 } \\left ( \\frac { - ( p _ { z _ i } \\sigma + q _ { z _ i } ) } { \\sqrt { \\dot { z } _ i ^ 2 + ( p _ { z _ i } \\sigma + q _ { z _ i } ) ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "6133.png", "formula": "\\begin{align*} t = 0 : \\Phi = \\Phi _ 0 , \\ \\Phi ' = \\Phi _ 1 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "766.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { \\# S _ n } \\# \\left \\{ g \\in S _ n : \\left | \\frac { \\varphi ( g ) } { n } - \\Lambda \\right | > \\epsilon \\right \\} = 0 . \\end{align*}"}
+{"id": "8763.png", "formula": "\\begin{align*} W _ { u , v , w , k } = \\begin{bmatrix} & & 0 & w _ { u , v , w , 2 T - 1 } ^ * & w _ { u , v , w , 2 T - 2 } ^ * & \\cdots & w _ { u , v , w , 1 } ^ * & & \\end{bmatrix} , \\end{align*}"}
+{"id": "4428.png", "formula": "\\begin{align*} a _ 2 = 2 4 > \\left ( \\frac { 1 } { 7 } + \\frac { 1 } { 1 0 } - \\frac { 1 } { 5 } \\right ) ^ { - 1 } = \\frac { 7 0 } { 3 } > 2 3 \\end{align*}"}
+{"id": "1064.png", "formula": "\\begin{align*} ( 1 - \\varepsilon ) R _ 0 + \\varepsilon G _ 0 = ( 1 - \\varepsilon ) R _ 1 + \\varepsilon G _ 1 . \\end{align*}"}
+{"id": "940.png", "formula": "\\begin{align*} \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { T r } _ C ( M ) ) \\ , = \\ , \\sup \\{ j \\geq 0 \\mid \\textrm { E x t } _ { R } ^ j ( \\textrm { T r } _ C ( M ) , \\ , C ) \\neq 0 \\} . \\end{align*}"}
+{"id": "4737.png", "formula": "\\begin{align*} ( A _ { \\mathfrak { p } } : _ { A _ { \\mathfrak { p } } } r x ) = ( A : _ A r x ) A _ { \\mathfrak { p } } = \\mathfrak { q } A _ { \\mathfrak { p } } \\subsetneq A _ { \\mathfrak { p } } . \\end{align*}"}
+{"id": "2864.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n c _ i = 0 \\end{align*}"}
+{"id": "473.png", "formula": "\\begin{align*} \\phi ( u ) : = \\int _ 0 ^ u \\ , b ( s ) \\ , d s , \\mbox { f o r a l l } u \\geq 0 . \\end{align*}"}
+{"id": "1870.png", "formula": "\\begin{align*} \\alpha _ { s _ i } ( \\sigma ) = \\left \\{ \\begin{array} { l l } \\frac { \\pi } { 2 } & \\mbox { i f $ s _ i - \\phi _ k + p _ { s _ i } \\sigma + q _ { s _ i } \\ge 0 $ } \\\\ \\pi - \\sin ^ { - 1 } \\left ( \\frac { - ( s _ i - \\phi _ k + p _ { s _ i } \\sigma + q _ { s _ i } ) } { \\sqrt { \\dot { s } _ i ^ 2 + ( p _ { s _ i } \\sigma + q _ { s _ i } ) ^ 2 } } \\right ) - \\beta & \\mbox { i f $ s _ i - \\phi _ k + p _ { s _ i } \\sigma + q _ { s _ i } \\le 0 $ } \\end{array} \\right . \\end{align*}"}
+{"id": "1190.png", "formula": "\\begin{align*} d \\Psi _ i | _ p ( v ) + T ^ { 1 , 0 } D _ i = v . \\end{align*}"}
+{"id": "7899.png", "formula": "\\begin{align*} f ( z ) ^ 2 + a _ { 1 } f ( z ) + q ( z ) e ^ { Q ( z ) } f ^ { ( k ) } ( z + c ) = P ( z ) , \\end{align*}"}
+{"id": "8905.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { m } ( k + a ) ^ q = \\frac { 1 } { q + 1 } \\left [ B _ { q + 1 } ( m + 1 + a ) - B _ { q + 1 } ( a ) \\right ] , q = 1 , 2 , \\cdots , \\end{align*}"}
+{"id": "337.png", "formula": "\\begin{align*} \\Re ( A ( p , 1 ) ) = \\Re ( A ' ( p , 1 ) ) . \\end{align*}"}
+{"id": "1944.png", "formula": "\\begin{align*} h ^ { ( Q _ N ^ 1 . . Q _ N ^ { k + 1 } ) } ( u , \\eta ) = \\big ( \\bar { f } _ 1 ^ { Q _ N ^ 1 } ( u , \\eta _ 1 ) , \\bar { f } ^ { Q _ N ^ 2 } _ 2 ( u , \\eta _ 2 ) , \\dots , \\bar { f } ^ { Q _ N ^ k } _ k ( u , \\eta _ k ) , \\bar { f } ^ { Q _ N ^ { k + 1 } } _ { k + 1 } ( u ) \\big ) ^ { \\top } \\end{align*}"}
+{"id": "8900.png", "formula": "\\begin{align*} \\mathcal { S } = 2 \\sum \\limits _ { p = 0 } ^ { n - 1 } \\gamma _ { p } ^ { ( \\nu , n ) } \\mathcal { R } _ { p } ^ { \\left ( \\nu \\right ) } \\left ( t \\right ) \\end{align*}"}
+{"id": "1084.png", "formula": "\\begin{align*} \\mathbb { E } \\{ ( \\hat { \\mu } - \\mu ) ^ 2 \\} = ( \\mathbb { E } \\hat { \\mu } - \\mu ) ^ 2 + \\mathrm { V a r } \\bigl ( \\mathbb { E } \\bigl \\{ \\hat { \\mu } | J \\bigr \\} \\bigr ) + \\mathbb { E } \\bigl \\{ \\mathrm { V a r } ( \\hat { \\mu } | J ) \\bigr \\} . \\end{align*}"}
+{"id": "4000.png", "formula": "\\begin{align*} w ^ { i i } u _ { k i } u _ { k i } = w ^ { i i } ( w _ { k i } + A _ { k i } ) ( w _ { k i } + A _ { k i } ) \\geq w _ { i i } - C ( 1 + w ^ { i i } ) , \\end{align*}"}
+{"id": "2172.png", "formula": "\\begin{align*} r _ a & = \\begin{dcases} a - 2 ( \\sqrt { 2 } - 1 ) & \\ 2 ( \\sqrt { 2 } - 1 ) < a \\leq \\sqrt { 2 } \\\\ 2 - a & \\ \\sqrt { 2 } \\leq a < 2 . \\end{dcases} \\end{align*}"}
+{"id": "1623.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\lim _ { n \\to \\infty } ( \\hat S ^ { ( t ) } ) \\leq \\frac { 1 } { C } \\sum _ { c = 1 } ^ C M \\tilde \\epsilon ^ * ( \\tau ^ { ( \\infty ) } _ c , M ) \\end{align*}"}
+{"id": "2140.png", "formula": "\\begin{align*} \\tilde { g } ( w ) : = r ^ - ( w - \\theta ) . \\end{align*}"}
+{"id": "3128.png", "formula": "\\begin{align*} - \\tilde { A } : D ^ 2 w _ { \\gamma } = \\gamma - \\bar { \\gamma } \\quad Y , w _ { \\gamma } Y , \\int _ Y w _ { \\gamma } = 0 , \\end{align*}"}
+{"id": "1465.png", "formula": "\\begin{align*} \\mathcal { T } : = \\{ ( P , E ) \\mid P \\mbox { i s a $ 2 $ - p a t h o f $ \\Delta $ , a n d $ E $ i s a n e d g e o f $ \\Delta $ n o t i n $ P $ } \\} . \\end{align*}"}
+{"id": "7489.png", "formula": "\\begin{align*} f ( x , y , z ) = & x ^ p + y ^ r z ^ l \\sum _ { i = 0 } ^ { k } { \\dbinom { k + r } { i + r } y ^ i ( t z - y ) ^ { k - i } } \\\\ : = & x ^ p + y ^ r z ^ l \\mathbb { H } _ r ^ k ( y , t z - y ) \\end{align*}"}
+{"id": "7131.png", "formula": "\\begin{align*} d ' \\otimes d '' = \\sum _ { i , j \\geq 1 } \\left \\langle d ' \\otimes d '' , x ^ { \\rho _ 1 + \\cdots + \\rho _ { i - 1 } } \\otimes x ^ { \\rho _ i + \\cdots + \\rho _ { i + j - 1 } } ) \\right \\rangle \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) \\otimes \\beta ( \\rho _ i , \\dots , \\rho _ j ) . \\end{align*}"}
+{"id": "2115.png", "formula": "\\begin{align*} n = \\dim _ F C & = \\dim _ F C _ 1 + \\dim _ F C _ 2 + \\dim _ F C _ { 1 2 } \\\\ & \\le \\dim _ F C _ 1 + \\dim _ F \\overline C _ 1 + \\dim _ F F H e ' = n . \\end{align*}"}
+{"id": "3002.png", "formula": "\\begin{align*} \\pi _ i = \\frac { 1 } { 2 } \\pi { _ i } ^ { a b } e _ { a b } ^ { ( 2 ) } \\wedge \\bar { \\theta } ^ { ( r ) } - \\pi { _ i } ^ { a k } e _ a ^ { ( 3 ) } \\wedge \\bar { \\theta } ^ { ( r - 1 ) } _ k + \\frac { 1 } { 2 } \\pi { _ i } ^ { j k } e ^ { ( 4 ) } \\wedge \\bar { \\theta } ^ { ( r - 2 ) } _ { j k } \\end{align*}"}
+{"id": "6178.png", "formula": "\\begin{align*} \\hbox { r a n k } ( C _ 1 D ) = \\hbox { r a n k } ( D ) = N - 1 . \\end{align*}"}
+{"id": "1353.png", "formula": "\\begin{align*} P F P ( \\{ f _ j \\} _ { j = 1 } ^ n , \\{ \\tau _ j \\} _ { j = 1 } ^ n ) \\coloneqq \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) . \\end{align*}"}
+{"id": "6369.png", "formula": "\\begin{align*} x _ \\lambda ' ( u ) = - \\sin u \\cdot ( 1 + F _ \\lambda ' ( \\cos u ) ) , \\end{align*}"}
+{"id": "2857.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ j b _ i \\leq \\sum _ { i = 1 } ^ j a _ i = \\sum _ { i = 1 } ^ j c _ i . \\end{align*}"}
+{"id": "5342.png", "formula": "\\begin{align*} H ^ 1 _ 0 ( \\Omega , \\mathrm { d i v } ( \\varepsilon \\nabla \\cdot ) ) : = \\{ u \\in H ^ 1 _ 0 ( \\Omega ) : \\mathrm { d i v } ( \\varepsilon \\nabla u ) \\in L ^ 2 ( \\Omega ) \\} . \\end{align*}"}
+{"id": "6657.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty ( \\gamma ^ k ) ^ 2 \\ , \\max _ { i \\in [ m ] } ( \\sigma _ { i } ^ k ) ^ 2 < \\infty \\end{align*}"}
+{"id": "6573.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( \\tilde { M } _ t \\backslash M _ t ) = 0 \\end{align*}"}
+{"id": "7481.png", "formula": "\\begin{align*} S _ { u , A } : = \\min _ { v : \\ ; ( u , v ) \\in E _ A } S _ { u v } \\quad u \\in V ^ { ' \\star } _ A \\ , . \\end{align*}"}
+{"id": "7609.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\int _ 0 ^ T \\langle \\mathfrak { A } _ n ( v ^ n ( t ) ) , v ^ n ( t ) \\rangle \\d t & = - \\frac { 1 } { 2 } \\limsup _ { n \\to \\infty } \\| v ^ n ( T ) \\| _ { \\L ^ 2 } ^ 2 + \\frac { 1 } { 2 } \\liminf _ { n \\to \\infty } \\| v _ 0 ^ n \\| _ { \\L ^ 2 } ^ 2 \\\\ & \\leq - \\frac { 1 } { 2 } \\| v ( T ) \\| _ { \\L ^ 2 } ^ 2 + \\frac { 1 } { 2 } \\| v _ 0 \\| _ { \\L ^ 2 } ^ 2 = \\int _ 0 ^ T \\langle \\mathfrak { A } _ 0 ( t ) , v ( t ) \\rangle \\d t . \\end{align*}"}
+{"id": "3185.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( A ) = c _ 1 ^ { 2 2 } ( A ) = - \\frac { 1 } { 1 2 8 \\pi } , c _ 1 ^ { 3 3 } ( A ) = \\frac { 1 } { 6 4 \\pi } , c _ j ^ { k l } ( A ) = 0 \\ ; \\ ; j \\neq 1 . \\end{align*}"}
+{"id": "7448.png", "formula": "\\begin{align*} \\langle f ^ * , g ^ * \\rangle = \\operatorname { t r } ( f ( Z ^ { ( 0 ) } ) ^ * g ( Z ^ { ( 0 ) } ) ) . \\end{align*}"}
+{"id": "8930.png", "formula": "\\begin{align*} \\varrho _ { p } ^ { \\left ( \\nu \\right ) } \\left ( \\ell \\right ) = \\frac { ( - 1 ) \\ell } { \\ell ! } \\sum \\limits _ { \\mu = 1 } ^ { \\nu - 1 } \\mu ^ { 2 ( p + \\ell ) + 1 } . \\end{align*}"}
+{"id": "5638.png", "formula": "\\begin{align*} \\emptyset \\sqcup w = w \\sqcup \\emptyset & = w , \\\\ ( \\omega _ 1 \\cdots \\omega _ k ) \\sqcup ( \\omega _ 1 ' \\cdots \\omega _ n ' ) & = ( \\omega _ 1 \\cdots \\omega _ k \\omega _ 1 ' \\cdots \\omega _ n ' ) \\end{align*}"}
+{"id": "1842.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { \\rho _ 0 } + 3 H \\rho _ { 0 } = 0 , \\\\ & \\dot { H } + H ^ { 2 } = - \\frac { 4 \\pi G } { 3 } \\rho _ 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7691.png", "formula": "\\begin{align*} f _ { E , k } = \\prod _ { j \\in S _ { k } \\cap J ( E ) } \\ell _ { j } . \\end{align*}"}
+{"id": "814.png", "formula": "\\begin{align*} S _ n \\xi ( t _ 1 , \\ldots , t _ d ) : = \\left ( S _ n \\xi ( t _ 1 ) , S _ n \\xi ( t _ 2 ) - S _ n \\xi ( t _ 1 ) , \\ldots , S _ n \\xi ( t _ d ) - S _ n \\xi ( t _ { d } ) \\right ) . \\end{align*}"}
+{"id": "9033.png", "formula": "\\begin{align*} \\overline { V } ^ { ( 1 ) } = \\overline { \\Psi } ( \\overline { V } ^ { ( 0 ) } ) = \\underline { \\Psi } ( \\overline { V } ^ { ( 0 ) } ) = \\underline { \\Psi } ( \\underline { V } ^ { ( 0 ) } ) = \\underline { V } ^ { ( 1 ) } . \\end{align*}"}
+{"id": "3900.png", "formula": "\\begin{align*} h '' ( \\theta ) & = \\big [ D _ { i j } u ( x _ { \\theta } ) - g _ { i j } ( x _ { \\theta } , y _ 0 , z _ 0 ) \\\\ & \\quad - D _ { p _ k } g _ { i j } ( x _ { \\theta } , y _ 0 , z _ 0 ) ( D _ k u ( x _ \\theta ) - D _ k g ( x _ { \\theta } , y _ 0 , z _ 0 ) ) \\big ] ( \\dot { x _ \\theta } ) _ i ( \\dot { x _ \\theta } ) _ j \\\\ & \\quad \\quad + g _ { j , z } ( E ^ { m , j } q _ m h ' + E ^ { n , j } q _ n h ' ) . \\end{align*}"}
+{"id": "3976.png", "formula": "\\begin{align*} \\overline { u } ( x ) = \\sup _ { y \\in B _ \\rho ( y _ 0 ) } g ( x , y , v _ \\rho ( y ) ) x \\in \\overline { \\Omega } . \\end{align*}"}
+{"id": "4986.png", "formula": "\\begin{align*} \\iint _ { \\R ^ d \\times \\R ^ d } \\left | ( G _ { p , q } , f ) \\right | ^ 2 \\ , \\frac { d p \\ , d q } { ( 2 \\pi ) ^ d } = \\| f \\| ^ 2 \\ , . \\end{align*}"}
+{"id": "686.png", "formula": "\\begin{align*} \\oint _ { \\alpha _ k } d U _ { \\alpha _ j } = 0 , \\oint _ { \\beta _ k } d U _ { \\alpha _ j } = \\delta _ { k j } . \\end{align*}"}
+{"id": "6061.png", "formula": "\\begin{align*} f ( z ) - \\frac { p _ n ( z ) } { q _ n ( z ) } = ( 2 + o ( 1 ) ) ( T _ n \\psi _ n ) ( z ) \\frac { \\big ( m S _ { \\dot \\mu } ^ 2 \\big ) ( z ) } { w ( z ) } , \\end{align*}"}
+{"id": "2744.png", "formula": "\\begin{align*} D _ { \\circ } ( a b ) = \\lim _ { \\alpha } D _ { \\alpha } ( a b ) = & \\lim _ { \\alpha } ( \\varphi \\circ \\psi ( a ) D _ { \\alpha } ( b ) + \\varphi \\circ \\psi ( b ) D _ { \\alpha } ( a ) ) \\\\ = & \\lim _ { \\alpha } \\varphi \\circ \\psi ( a ) D _ { \\alpha } ( b ) + \\lim _ { \\alpha } \\varphi \\circ \\psi ( b ) D _ { \\alpha } ( a ) \\\\ = & \\varphi \\circ \\psi ( a ) D _ { \\circ } ( b ) + \\varphi \\circ \\psi ( b ) D _ { \\circ } ( a ) . \\end{align*}"}
+{"id": "2729.png", "formula": "\\begin{align*} A ' ( t ) \\geq \\max \\left ( ( \\kappa _ 2 + \\frac 1 C ) \\sum _ { j = 1 } ^ { J _ 0 } \\theta _ j \\beta _ j ^ 2 ( t ) \\ , \\ ( \\frac { 1 } { \\kappa _ 2 } + \\frac 1 C ) \\sum _ { j = 1 } ^ { J _ 0 } \\theta _ j ( \\lambda _ j ' ( t ) ) ^ 2 \\right ) . \\end{align*}"}
+{"id": "4028.png", "formula": "\\begin{align*} \\overline { L } v & = w ^ { i j } F _ { p _ l p _ m } D _ { l i } u D _ { m j } u + F _ { p _ l } w ^ { i j } ( D _ { l i j } u - D _ { p _ k } A _ { i j } D _ { l k } u ) \\\\ & + w ^ { i j } ( F _ { u p _ l } u _ j D _ { l i } u + F _ { u p _ l } u _ i D _ { l j } u + F _ z u _ { i j } + F _ { j p _ l } D _ { l i } u + F _ { i p _ l } D _ { l j } u ) \\\\ & + w ^ { i j } \\big ( F _ { u u } u _ { i } u _ j + F _ { i u } u _ j + F _ { j u } u _ i + F _ { i j } - D _ { p _ k } A _ { i j } ( F _ k + F _ z u _ k ) \\big ) . \\end{align*}"}
+{"id": "3501.png", "formula": "\\begin{align*} \\mathbb { Y } ( \\lambda ) : = \\{ A \\in \\mathbb { Y } : \\lambda _ A = \\lambda \\} . \\end{align*}"}
+{"id": "3430.png", "formula": "\\begin{align*} \\Sigma ( a ) \\cap [ b ] = \\bigcup _ { i = 1 } ^ { \\infty } \\bigcup _ { k = 1 } ^ { l ( \\underline { \\gamma } _ i ) } \\sigma ^ k [ \\underline { \\gamma } _ i ] \\cap [ b ] . \\end{align*}"}
+{"id": "377.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { T _ * } { n } \\right ) ^ 2 \\frac { n } { T _ * } = \\left ( \\frac { n } { T _ * } - 1 \\right ) + \\left ( \\frac { T _ * } { n } - 1 \\right ) \\leq \\frac { 1 } { T _ * } \\left | n - T _ * \\right | + \\frac { 1 } { n } \\left | T _ * - n \\right | \\ , . \\end{align*}"}
+{"id": "8207.png", "formula": "\\begin{align*} ( \\Phi ( t , L ) - I d ) u _ 0 & = \\Psi ( t , L ) L u _ 0 = \\int _ 0 ^ t k ( t , s ) \\Phi ( s , L ) L u _ 0 d s = \\int _ 0 ^ t k ( t , s ) L \\Phi ( s , L ) u _ 0 d s \\\\ \\Leftrightarrow \\ \\Phi ( t , L ) u _ 0 & = u _ 0 + \\int _ 0 ^ t k ( t , s ) L \\Phi ( s , L ) u _ 0 d s . \\end{align*}"}
+{"id": "6631.png", "formula": "\\begin{align*} \\partial _ { x _ j } ^ n \\frac { 1 } { ( 1 + \\epsilon x ^ { 2 m } ) ^ 2 } = \\partial _ { x _ j } ^ n f _ \\epsilon ( g ( x ) ) = \\sum _ { \\substack { ( m _ 1 , \\dots , m _ n ) \\\\ \\sum _ { i = 1 } ^ n i m _ i = n } } c _ { n , m _ i } f _ \\epsilon ^ { ( m _ 1 + \\dots + m _ n ) } ( g ( x ) ) \\prod _ { i = 1 } ^ n ( \\partial _ { x _ j } ^ i g ( x ) ) ^ { m _ i } \\ , . \\end{align*}"}
+{"id": "3728.png", "formula": "\\begin{align*} \\mathcal { S } ( X _ { P _ 1 } ( F ) ) : = I ( \\mathcal { S } ( X _ 1 ( F ) ) ) . \\end{align*}"}
+{"id": "5030.png", "formula": "\\begin{align*} \\mathcal P ' \\coloneqq \\left \\{ ( g , H ) \\in \\mathcal P | \\ g ( e _ 2 , e _ 5 ) = 0 \\right \\} . \\end{align*}"}
+{"id": "888.png", "formula": "\\begin{align*} \\mathbf { V a r } _ { i \\sim \\mathbf { u } } \\left [ f _ { i } ( \\mathcal { X } ^ { t } ) \\right ] & = \\mathbf { E } _ { i \\sim \\mathbf { u } } \\left [ f _ { i } ( \\mathcal { X } ^ { t } ) ^ { 2 } \\right ] - \\mathbf { E } _ { i \\sim \\mathbf { u } } \\left [ f _ { i } ( \\mathcal { X } ^ { t } ) \\right ] ^ { 2 } = \\frac { 1 } { q } \\sum _ { i = 1 } ^ { q } \\left ( f _ { i } ( \\mathcal { X } ^ { t } ) \\right ) ^ { 2 } - \\frac { 1 } { q ^ { 2 } } \\left ( \\sum _ { i = 1 } ^ { q } f _ { i } ( \\mathcal { X } ^ { t } ) \\right ) ^ { 2 } . \\end{align*}"}
+{"id": "3842.png", "formula": "\\begin{align*} \\nabla _ A u : = \\nabla u + i A ( x ) u ; \\textrm { d i v } _ { A } F : = \\textrm { d i v } F + i A \\cdot F , \\end{align*}"}
+{"id": "3332.png", "formula": "\\begin{align*} q ^ 1 = k x , p _ 1 = - \\frac { 1 } { y } , q ^ 2 = 2 \\nu q , p _ 2 = p , n = 2 . \\end{align*}"}
+{"id": "822.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\nu ( \\mathcal { A } _ n ( x ) ) = \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma } \\int _ { - \\infty } ^ x e ^ { - t ^ 2 / 2 \\sigma ^ 2 } \\ d t \\ \\end{align*}"}
+{"id": "3946.png", "formula": "\\begin{align*} \\vee ( x ) & = \\sup \\{ \\phi _ y ( x ) : = g ( x , y , g ^ * ( x _ 0 , y , u _ 0 ) ) ; \\\\ & \\quad \\quad \\phi _ y ( x ) \\leq g ( x , y _ 0 , z _ h ) \\partial D \\} . \\end{align*}"}
+{"id": "4281.png", "formula": "\\begin{align*} { W _ { k } ^ { 1 , p } ( \\Omega ) } : = { \\{ } & u \\in \\mathcal { D } ' ( \\Omega ) ; \\ , \\rho ^ { k - 1 } u \\in L ^ { p } ( \\Omega ) , \\ , \\rho ^ { k } \\ , \\nabla \\ , u \\in L ^ { p } ( \\Omega ) \\} \\end{align*}"}
+{"id": "1521.png", "formula": "\\begin{align*} v ^ A _ { I , J } ( \\varpi ) = \\dd _ J \\lambda ^ A _ I ( \\varpi ) \\end{align*}"}
+{"id": "2581.png", "formula": "\\begin{align*} \\frac { 1 } { - \\frac { \\pi } { 4 } + 2 m \\pi } e _ i & = ( - 1 ) \\times \\frac { 1 } { - \\frac { 7 } { 4 } \\pi + 2 ( - m + 1 ) \\pi } e _ i , \\\\ \\frac { 1 } { - \\frac { 5 } { 4 } \\pi + 2 m \\pi } e _ i & = ( - 1 ) \\times \\frac { 1 } { - \\frac { 3 } { 4 } \\pi + 2 ( - m + 1 ) \\pi } e _ i , \\end{align*}"}
+{"id": "1289.png", "formula": "\\begin{align*} \\sigma = \\Sigma ( \\nu ) \\quad \\nu = \\Sigma ^ { - 1 } ( \\sigma ) \\end{align*}"}
+{"id": "465.png", "formula": "\\begin{align*} g _ { n } \\left ( k \\right ) : = \\left \\{ \\begin{array} { l l } 2 ^ { n } + k & 0 \\leq k \\leq 2 ^ { n } \\ , , \\\\ k - 2 ^ { n } & 2 ^ { n } < k \\leq 2 ^ { n + 1 } \\ , , \\\\ k & 2 ^ { n + 1 } < k \\ , , \\end{array} \\right . \\end{align*}"}
+{"id": "5028.png", "formula": "\\begin{align*} \\mu = 2 \\sqrt { 2 } \\ , t , a = b = t , h _ 1 = \\pm 2 t ^ 2 , \\end{align*}"}
+{"id": "50.png", "formula": "\\begin{align*} T ( n _ i ) + T ( n _ { i + 1 } ) + T ( n _ { i + 2 } ) & < \\left ( \\frac { 8 5 } { 2 7 } + \\frac { 6 4 } { 2 7 } \\right ) \\cdot \\frac { 1 } { X _ 0 } = \\frac { 1 4 9 } { 2 7 } \\cdot \\frac { 1 } { X _ 0 } \\\\ & < 8 \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } = ( k _ i + k _ { i + 1 } + k _ { i + 2 } ) \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } , \\end{align*}"}
+{"id": "500.png", "formula": "\\begin{align*} { \\textstyle \\sum _ { K _ 1 \\ni j = k } ^ { \\nu } } \\| x ^ { j + 1 } \\ ! - \\ ! x ^ { j } \\| \\le \\sqrt { 2 a ^ { - 1 } } { \\textstyle \\sum _ { K _ 1 \\ni j = k } ^ { \\nu } } \\sqrt { \\Phi ( x ^ { \\ell ( j + 1 ) } ) - \\Phi ( x ^ { j + 1 } ) } , \\end{align*}"}
+{"id": "4265.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } E _ \\gamma ( g _ n - \\phi ) & = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { \\lambda _ n ^ 2 } E _ \\gamma ( \\lambda _ n ( g _ n - \\phi ) ) + \\frac { 2 ( \\lambda _ n ^ { p - 1 } - 1 ) } { p + 1 } \\| g _ n - \\phi \\| ^ { p + 1 } _ { L ^ { p + 1 } } \\\\ & \\geq \\frac { c - \\| \\phi \\| ^ 2 _ { L ^ 2 } } { c } I _ \\gamma ( c ) . \\end{align*}"}
+{"id": "956.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( \\lambda M \\otimes _ R C , C ) \\ , = \\ , \\textrm { r . g r a d e } _ R ( \\textrm { T r } _ C ( M ) , C ) - 1 . \\end{align*}"}
+{"id": "1578.png", "formula": "\\begin{align*} \\sum \\limits _ { x ' \\in X } \\pi ( x , x ' ) = \\mu ( x ) \\qquad \\mbox { a n d } \\qquad \\sum \\limits _ { x \\in X } \\pi ( x , x ' ) = \\nu ( x ' ) . \\end{align*}"}
+{"id": "4644.png", "formula": "\\begin{align*} \\mathsf { E } \\left [ g ( X ) X \\right ] = \\mu \\mathsf { E } [ g ( X ) ] + \\sigma ^ 2 \\mathsf { E } \\left [ g ' ( X ) \\right ] , \\end{align*}"}
+{"id": "2520.png", "formula": "\\begin{align*} & \\varrho ( x ) = \\frac 1 N \\sum _ { j = 1 } ^ N \\varphi ( x - x _ j ) \\ , , \\varrho u ( x ) = \\frac 1 N \\sum _ { j = 1 } ^ N \\varphi ( x - x _ j ) v _ j \\ , , \\\\ & \\varrho ( | u | ^ 2 + 3 T ) ( x ) = \\frac 1 N \\sum _ { j = 1 } ^ N \\varphi ( x - x _ j ) v _ j ^ 2 \\ , , \\end{align*}"}
+{"id": "8874.png", "formula": "\\begin{align*} \\sum \\limits _ { m = 0 } ^ { + \\infty } \\sup \\limits _ { ( z , w ) \\in \\mathbb { C } ^ n \\times \\mathbb { C } ^ n } \\left | \\partial _ t ^ k D _ { z , w } ^ { \\alpha , \\beta } \\left [ e ^ { \\beta _ m t } K _ { \\nu , m } ( z , w ) \\right ] \\right | \\leq C ( t ) < + \\infty \\end{align*}"}
+{"id": "6072.png", "formula": "\\begin{align*} K ( z ; x ) : = \\frac 1 { x - z } + \\frac x { 1 - x z } = \\frac { 1 - 2 x z + x ^ 2 } { ( x - z ) ( 1 - x z ) } \\end{align*}"}
+{"id": "7831.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { j = 1 } ^ n P _ j ( z ) e ^ { w _ j z } , \\end{align*}"}
+{"id": "3589.png", "formula": "\\begin{align*} v _ { A ( 3 ) } & = - 2 \\Omega _ { A ( 3 ) } \\\\ & = - 1 6 [ B _ 1 ^ + , B _ 3 ^ + ] ^ 2 ( B _ 2 ^ + ) ^ 2 v _ 0 - 1 6 [ B _ 2 ^ + , B _ 3 ^ + ] ^ 2 ( B _ 1 ^ + ) ^ 2 v _ 0 \\\\ & \\qquad - 1 6 [ B _ 1 ^ + , B _ 3 ^ + ] [ B _ 2 ^ + , B _ 3 ^ + ] B _ 1 ^ + B _ 2 ^ + v _ 0 - 1 6 [ B _ 1 ^ + , B _ 3 ^ + ] [ B _ 2 ^ + , B _ 3 ^ + ] B _ 2 ^ + B _ 1 ^ + v _ 0 \\\\ & \\qquad - 1 6 [ B _ 2 ^ + , B _ 3 ^ + ] [ B _ 1 ^ + , B _ 3 ^ + ] B _ 1 ^ + B _ 2 ^ + v _ 0 - 1 6 [ B _ 2 ^ + , B _ 3 ^ + ] [ B _ 1 ^ + , B _ 3 ^ + ] B _ 2 ^ + B _ 1 ^ + v _ 0 . \\end{align*}"}
+{"id": "4381.png", "formula": "\\begin{align*} \\frac { 1 } { a _ n } < \\theta - \\sum _ { i = 1 } ^ { n - 1 } \\frac { 1 } { a _ i } \\leq \\frac { 1 } { a _ n - 1 } \\end{align*}"}
+{"id": "7903.png", "formula": "\\begin{align*} a _ j \\cos \\theta _ { j + 1 } ^ \\bot + b _ j \\sin \\theta _ { j + 1 } ^ \\bot & = a _ { j + 1 } \\cos \\theta _ { j + 1 } ^ \\bot + b _ { j + 1 } \\sin \\theta _ { j + 1 } ^ \\bot \\\\ & > \\max _ { k \\neq j , j + 1 } \\{ a _ k \\cos \\theta _ { j + 1 } ^ \\bot + b _ k \\sin \\theta _ { j + 1 } ^ \\bot \\} , \\end{align*}"}
+{"id": "2789.png", "formula": "\\begin{align*} 2 x _ 1 ^ { a + b } = x _ 1 ^ a x _ 2 ^ b + x _ 1 ^ b x _ 2 ^ a = x _ 1 ^ { a + b } + x _ 2 ^ { a + b } . \\end{align*}"}
+{"id": "8008.png", "formula": "\\begin{align*} E _ H : = \\big \\{ & \\left . ( F _ i , F _ \\ell ) \\in V _ H \\times V _ H \\ ; \\right | \\ ; ( F _ i , v _ k , F _ \\ell ) \\textnormal { i s a n a l t e r n a t i n g } \\\\ & \\{ F _ i , v _ k \\} \\in E \\setminus \\mathcal { M } , \\{ v _ k , F _ \\ell \\} \\in \\mathcal { M } \\big \\} . \\end{align*}"}
+{"id": "1321.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | & \\geq \\sqrt { \\frac { \\frac { 1 } { d } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ 2 } { n ^ 2 - n } } \\\\ & = \\sqrt { \\frac { \\frac { 1 } { ( \\mathcal { X } ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ 2 } { n ^ 2 - n } } . \\end{align*}"}
+{"id": "6680.png", "formula": "\\begin{align*} \\begin{aligned} & x ^ { k + 1 } - { \\bf 1 } \\otimes \\bar { x } ^ { k + 1 } = \\hat { W } _ k x ^ k + \\gamma ^ k \\Xi \\zeta _ w ^ k - \\lambda ^ k \\Xi g ^ k \\end{aligned} \\end{align*}"}
+{"id": "25.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\mathbf { D } _ { 2 5 } = \\mathbb { E } \\int _ 0 ^ \\infty e ^ { - \\beta t } \\bar { \\mathcal { Z } } \\bar { f } _ { \\gamma } \\int _ { \\mathcal { E } } \\gamma _ 1 \\nu ( d e ) d t . \\end{align*}"}
+{"id": "3021.png", "formula": "\\begin{align*} \\begin{array} { l l l l } e ^ I & = & S ^ I _ J \\theta ^ J , & \\hbox { i . e . } e ^ a : = \\theta ^ a \\hbox { a n d } e ^ i = S ^ i _ j \\theta ^ j \\\\ p _ I & = & ( S ^ { - 1 } ) ^ J _ I \\pi _ J & \\hbox { i . e . } p _ a : = \\pi _ a \\hbox { a n d } p _ i = ( S ^ { - 1 } ) ^ j _ i \\pi _ j \\\\ \\omega ^ { I J } & = & S ^ I _ K S ^ J _ L \\varphi ^ { K L } - \\hbox { d } S ^ I _ K S ^ J _ L \\textsf { h } ^ { K L } \\\\ \\Omega ^ { I J } & = & S ^ I _ K S ^ J _ L \\Phi ^ { K L } \\end{array} \\end{align*}"}
+{"id": "2354.png", "formula": "\\begin{align*} J _ n Q ^ T ( - k ) J _ n = - Q ( k ) . \\end{align*}"}
+{"id": "8863.png", "formula": "\\begin{align*} { } _ { 2 } F { } _ { 1 } \\left ( \\begin{array} { c } p - m , q - m - 2 \\nu \\\\ n + p + q \\end{array} \\big | - | z | ^ { 2 } \\right ) = \\left ( 1 + | z | ^ { 2 } \\right ) ^ { m + \\nu - \\frac { p + q } { 2 } } R _ { m - p , m - q + 2 \\nu } ^ { n + p + q - 1 } \\left ( \\left ( 1 + | z | ^ { 2 } \\right ) ^ { - \\frac { 1 } { 2 } } \\right ) . \\end{align*}"}
+{"id": "2275.png", "formula": "\\begin{align*} \\big ( f ( 0 , x , v ) , \\rho ( 0 , x ) , u ( 0 , x ) \\big ) = v ( f _ { 0 \\varepsilon } ( x , v ) , \\rho _ 0 ( x ) , u _ 0 ( x ) \\big ) , \\end{align*}"}
+{"id": "5991.png", "formula": "\\begin{align*} X ( i , j ) : & = X ( w _ { i , j } ) = \\overline { B w _ { i , j } P / P } \\\\ & = \\left \\{ ( L , H ) \\in X \\bigr \\rvert L \\subset < e _ 1 , \\cdots , e _ i > , \\ : < e _ 1 , \\cdots , e _ { j - 1 } > \\subset H \\right \\} . \\end{align*}"}
+{"id": "2793.png", "formula": "\\begin{align*} [ f ] _ G = \\frac { 1 } { | G | } \\sum _ { \\sigma \\in G } \\sigma f . \\end{align*}"}
+{"id": "8374.png", "formula": "\\begin{align*} w = \\partial _ t w = 0 , \\mathrm { a t } \\ t = R . \\end{align*}"}
+{"id": "1896.png", "formula": "\\begin{align*} d \\eta ( t ) = ( A + L C ) \\eta ( t ) d t + \\sigma L Z ( t ) d B ( t ) . \\end{align*}"}
+{"id": "5971.png", "formula": "\\begin{align*} t \\geqslant T : \\psi _ r = ( E _ r , U ) = \\sum _ { s = 1 } ^ p ( E _ r , e _ s ) u _ s = u _ r , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "7309.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } g _ { m _ n } ( \\lambda ; x ) = 1 . \\end{align*}"}
+{"id": "605.png", "formula": "\\begin{align*} \\frac { \\Delta t \\ , \\gamma } { 2 } M _ { \\rm e s t } = \\mathbb { E } ^ \\dagger [ \\mathbb { X } ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } ] , \\end{align*}"}
+{"id": "348.png", "formula": "\\begin{align*} | N _ { G } ( v ) \\cap ( A \\cup B ) | = | N _ { G } ( v ) \\cap B | + | N _ { G } ( v ) \\cap A | \\leq 2 + p = p + 2 . \\end{align*}"}
+{"id": "6985.png", "formula": "\\begin{gather*} H _ { 2 , n } ^ { ( \\{ 1 , 1 \\} ) } ( x ) = x \\ , \\frac { 2 ^ { n + 3 } n ( n - 1 ) ( n - 2 ) } { \\prod _ { 1 \\leq i < j \\leq n - 1 } ( U _ { j , n } - U _ { i , n } ) } \\det \\big ( M _ n ^ { \\{ 1 , 1 \\} } \\big ) + 1 6 ( n - 1 ) ( n - 2 ) H _ { n - 2 } ( 0 ) . \\end{gather*}"}
+{"id": "5340.png", "formula": "\\begin{align*} \\begin{cases} - \\operatorname { d i v } ( \\varepsilon \\nabla f ) = \\rho f & \\Omega , \\\\ f = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "4613.png", "formula": "\\begin{align*} \\frac { r } { \\prod _ { i = 1 } ^ n b _ i } \\leq \\frac { 1 } { q } \\left ( 1 - \\frac { 1 } { \\prod _ { i = 1 } ^ n b _ i } \\right ) \\end{align*}"}
+{"id": "4686.png", "formula": "\\begin{align*} G _ { I I } = \\left \\langle \\frac { 1 } { \\sqrt { 2 } } \\begin{pmatrix} 1 & 1 \\\\ 1 & - 1 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ 0 & i \\end{pmatrix} \\right \\rangle , \\end{align*}"}
+{"id": "7162.png", "formula": "\\begin{align*} \\displaystyle { [ H _ i , H _ j ] = 0 \\ , , i , j = 1 , . . . , N - 1 } \\end{align*}"}
+{"id": "2191.png", "formula": "\\begin{align*} \\tilde { \\sigma _ { 0 } } & = 2 ( \\mathit { e } - 1 ) \\big ( ( 2 - 2 \\alpha + \\beta ) ( \\mathit { e } + 1 ) \\\\ & + \\sqrt { ( ( \\mathit { e } + 1 ) ( 2 - 2 \\alpha + \\beta ) ) ^ 2 + 4 ( \\mathit { e } - 1 ) ( 1 + 3 \\mathit { e } - 2 \\alpha - 2 \\alpha \\mathit { e } - \\beta - \\beta \\mathit { e } ) } \\big ) ^ { - 1 } \\end{align*}"}
+{"id": "881.png", "formula": "\\begin{align*} \\mathbf { E } [ \\| \\mathcal { X } ^ { t + 1 } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } | \\mathcal { X } ^ { t } ] = \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } - \\mathbf { E } _ { i \\sim \\mathbf { p } ^ { t } } [ f _ { i } ( \\mathcal { X } ^ { t } ) ] , \\end{align*}"}
+{"id": "7854.png", "formula": "\\begin{align*} \\sum _ { | z _ k | \\leq r } r _ k = o ( r ) \\end{align*}"}
+{"id": "5452.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\d X ^ n _ t & = b ^ n ( t , X ^ n _ t , \\mathcal { L } _ { X ^ n _ t } , \\alpha ^ { n } _ t ) d t + \\sigma ^ n ( t , X ^ n _ t , \\mathcal { L } _ { X ^ n _ t } , \\alpha ^ { n } _ t ) d W _ t \\\\ X ^ n _ 0 & = X _ 0 , \\alpha ^ { n } _ 0 = \\alpha _ 0 \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "2739.png", "formula": "\\begin{align*} T ' _ { i , e } ( u v ) = T ' _ { i , e } ( u ) T ' _ { i , e } ( v ) , T '' _ { i , e } ( u v ) = T '' _ { i , e } ( u ) T '' _ { i , e } ( v ) . \\end{align*}"}
+{"id": "2044.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { n = 0 } ^ { + \\infty } H ( n ) \\ , H ' ( - n ) < + \\infty , \\end{align*}"}
+{"id": "3072.png", "formula": "\\begin{align*} v ^ { k l } = \\eta ^ { k l } = m _ { k l } w \\forall \\ , 1 \\leq k , l \\leq n . \\end{align*}"}
+{"id": "2431.png", "formula": "\\begin{align*} \\begin{aligned} F _ \\tau ( n ; q ) & = \\frac { 1 } { 2 \\pi \\mathrm { i } } \\int _ { \\Gamma _ { \\theta , \\sigma } ^ \\tau } e ^ { z t _ n } { e ^ { - z \\tau } } \\delta _ \\tau ( e ^ { - z \\tau } ) ^ { \\alpha - 1 } ( { \\delta _ \\tau ( e ^ { - z \\tau } ) ^ \\alpha } + A ( q ) ) ^ { - 1 } \\ , \\d z , \\\\ E _ \\tau ( n ; q ) & = \\frac { 1 } { 2 \\pi \\mathrm { i } } \\int _ { \\Gamma _ { \\theta , \\sigma } ^ \\tau } e ^ { z t _ n } e ^ { - z \\tau } ( { \\delta _ \\tau ( e ^ { - z \\tau } ) ^ \\alpha } + A ( q ) ) ^ { - 1 } \\ , \\d z , \\end{aligned} \\end{align*}"}
+{"id": "3250.png", "formula": "\\begin{align*} u ( x , t ) = o \\left ( | x | + | t | ^ { \\frac { 1 } { ^ { \\sigma } } } \\right ) ^ { \\sigma } , \\mbox { a s } | x | + | t | \\to \\infty , \\end{align*}"}
+{"id": "6487.png", "formula": "\\begin{align*} X ^ { j } = f + \\sqrt { \\frac { m } { n } } Z ^ { j } , \\end{align*}"}
+{"id": "5406.png", "formula": "\\begin{align*} & \\Big ( \\prod _ { k \\in \\N _ { \\ge 1 } } \\big ( 1 + \\tfrac { 1 } { ( \\alpha \\beta ) ^ { 2 ^ { k - 1 } } } \\big ) \\Big ) \\Big ( \\prod _ { k \\in \\N _ { \\ge 1 } } \\big ( 1 - \\tfrac { 1 } { ( \\alpha \\beta ) ^ { 2 ^ { k - 1 } } } \\big ) \\Big ) \\\\ = & \\prod _ { k \\in \\N _ { \\ge 1 } } ^ \\Big ( \\big ( 1 + \\tfrac { 1 } { ( \\alpha \\beta ) ^ { 2 ^ { k - 1 } } } \\big ) \\big ( 1 - \\tfrac { 1 } { ( \\alpha \\beta ) ^ { 2 ^ { k - 1 } } } \\big ) \\Big ) \\\\ = & \\prod _ { k = 2 } ^ \\infty \\big ( 1 - \\tfrac { 1 } { ( \\alpha \\beta ) ^ { 2 ^ { k - 1 } } } \\big ) . \\end{align*}"}
+{"id": "770.png", "formula": "\\begin{align*} \\nu = \\lim _ { n \\to \\infty } \\frac { \\sum _ { | g | _ S \\leq n } \\lambda ^ { - | g | _ S } \\delta _ g } { \\sum _ { | g | _ S \\leq n } \\lambda ^ { - | g | _ S } } . \\end{align*}"}
+{"id": "4267.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| u _ { 0 , n } - \\phi _ 0 \\| _ { \\Sigma _ \\gamma } = 0 \\end{align*}"}
+{"id": "4403.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { s _ i } \\leq \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < 1 \\end{align*}"}
+{"id": "205.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu ( T ^ { n } A \\cap B ) = \\mu ( A ) \\mu ( B ) . \\end{align*}"}
+{"id": "2369.png", "formula": "\\begin{align*} \\partial _ t { v } & = \\nu \\Delta v - ( v \\cdot \\nabla ) v - \\nabla p + f \\Omega \\times \\mathbb T , \\\\ 0 & = \\nabla ^ T v \\Omega \\times \\mathbb T , \\end{align*}"}
+{"id": "495.png", "formula": "\\begin{align*} \\Phi ( x ^ { \\ell ( k + 1 ) } ) \\le \\ ! \\max \\{ \\Phi ( x ^ { \\ell ( k ) } ) , \\Phi ( x ^ { k + 1 } ) \\} \\le \\ ! \\max \\{ \\Phi ( x ^ { \\ell ( k ) } ) , \\Phi ( x ^ { \\ell ( k ) } ) \\ ! - \\ ! a \\| x ^ { k + 1 } \\ ! - \\ ! x ^ k \\| ^ 2 \\} = \\ ! \\Phi ( x ^ { \\ell ( k ) } ) . \\end{align*}"}
+{"id": "4189.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ( \\alpha \\partial _ t \\Lambda ) - \\partial _ x ( \\alpha \\partial _ x \\Lambda ) = 2 \\alpha \\sinh { 2 \\Lambda } ( ( \\partial _ t \\phi ) ^ 2 - ( \\partial _ x \\phi ) ^ 2 ) \\\\ \\partial _ t ( \\alpha \\sinh ^ 2 \\Lambda \\partial _ t \\phi ) - \\partial _ x ( \\alpha \\sinh ^ 2 \\Lambda \\partial _ x \\phi ) = 0 , \\end{cases} \\end{align*}"}
+{"id": "5127.png", "formula": "\\begin{align*} \\det \\big ( M _ { n } ( \\lambda , b , \\Omega ) \\big ) & = \\Big ( \\Omega _ { n } ( \\lambda ) - \\Omega - b \\Lambda _ { 1 } ( \\lambda , b ) \\Big ) \\Big ( \\Lambda _ { 1 } ( \\lambda , b ) - b \\big [ \\Omega _ { n } ( \\lambda b ) + \\Omega \\big ] \\Big ) + b \\Lambda _ { n } ^ { 2 } ( \\lambda , b ) \\\\ & = b \\Omega ^ { 2 } - B _ { n } ( \\lambda , b ) \\Omega + C _ { n } ( \\lambda , b ) , \\end{align*}"}
+{"id": "6340.png", "formula": "\\begin{align*} \\varphi ' ( \\nu _ 0 ) + \\int _ { 1 } ^ { t } f _ { \\nu } ( x , \\nu _ 0 ) d x - \\int _ { h ( t ) } ^ { 1 } f _ \\nu ( x , \\nu _ 0 ) d x = 0 , \\end{align*}"}
+{"id": "664.png", "formula": "\\begin{align*} i ( { \\bf v } ) { \\rm v o l } = * \\nu , \\end{align*}"}
+{"id": "6582.png", "formula": "\\begin{align*} \\eta ( x ) \\in C ^ \\infty _ c ( B _ { d } ) \\ \\mathrm { w i t h } \\ \\eta = 1 \\ \\mathrm { o n } \\ B _ { d / 2 } , \\ 0 \\leq \\eta \\leq 1 , \\end{align*}"}
+{"id": "3441.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ { t } ( [ a ] ) & \\leq \\exp \\left ( - { ( a + 2 ) ( a - 1 ) } t \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "4915.png", "formula": "\\begin{align*} \\widetilde H ( \\mathbf { Z } ) _ m = \\sum _ \\ell P _ \\ell ( Z _ m ) Q ^ + _ \\ell ( Z _ m ) Q ^ - _ \\ell ( Z _ m ) \\approx H ( z , z _ x ) \\vert _ { x = x _ m } , \\end{align*}"}
+{"id": "5563.png", "formula": "\\begin{align*} S _ 1 ( x ^ 2 ) = O \\left ( x ^ { 1 - \\epsilon } \\exp ( - x ^ { 2 \\epsilon } ) \\right ) , \\end{align*}"}
+{"id": "3495.png", "formula": "\\begin{align*} { { \\bf { \\bar f } } ^ { { \\rm { M M S E } } } } = \\sqrt P { { { \\bf { C } } ^ { - 1 } } { \\bf { \\bar h } } } / { \\left \\| { { { \\bf { C } } ^ { - 1 } } { \\bf { \\bar h } } } \\right \\| } , \\end{align*}"}
+{"id": "8768.png", "formula": "\\begin{align*} G _ l & = [ C _ 0 A _ 0 ^ { 2 T - 1 + l } , C _ 0 A _ 0 ^ { 2 T + l } , \\dots , C _ 0 , 0 , \\dots , 0 ] , \\end{align*}"}
+{"id": "7504.png", "formula": "\\begin{align*} g ( u , v ) = & \\tilde { g } ( u , v ) + \\sum \\limits _ { i = i _ 0 + 1 } ^ { \\infty } \\alpha _ i u ^ i + \\sum \\limits _ { j = j _ 0 + 1 } ^ { \\infty } \\beta _ j v ^ j \\\\ = & \\tilde { g } ( u , v ) + \\pi ^ { e _ 0 + 1 } \\Big ( \\sum \\limits _ { i = i _ 0 + 1 } ^ { \\infty } \\pi ^ { - ( e _ 0 + 1 ) } \\alpha _ i u ^ i + \\sum \\limits _ { j = j _ 0 + 1 } ^ { \\infty } \\pi ^ { - ( e _ 0 + 1 ) } \\beta _ j v ^ j \\Big ) , \\end{align*}"}
+{"id": "3837.png", "formula": "\\begin{align*} I _ 1 ( v ) \\sim & ( 1 - \\theta ^ 2 ) w _ 2 ^ 2 \\int _ 1 ^ \\infty \\frac { x ^ 3 v ^ 2 } { \\pi \\bigl ( w _ 2 ^ 2 x ^ 4 + ( 1 - \\theta ^ 2 ) v ^ 2 x ^ 2 \\bigr ) ^ { 3 / 2 } } \\ , d x \\\\ = & w _ 2 \\int _ { w _ 2 / ( v ( 1 - \\theta ^ 2 ) ^ { 1 / 2 } ) } ^ \\infty \\frac { 1 } { \\pi ( x ^ 2 + 1 ) ^ { 3 / 2 } } \\ , d x \\to w _ 2 \\int _ 0 ^ \\infty \\frac { 1 } { \\pi ( x ^ 2 + 1 ) ^ { 3 / 2 } } \\ , d x = w _ 2 / \\pi . \\end{align*}"}
+{"id": "7084.png", "formula": "\\begin{align*} E ^ { C _ 2 } _ { * + | n \\sigma | - n \\sigma } ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) \\to \\widehat { E } ^ { C _ 2 } _ * ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) \\cong \\bigoplus _ { i = 1 } ^ \\infty \\widehat { E } ^ { C _ 2 } _ * \\{ \\Pi _ { \\rho _ 1 + \\cdots + \\rho _ i } \\} \\end{align*}"}
+{"id": "2312.png", "formula": "\\begin{align*} \\nu ( u ) = \\begin{cases} 0 , & | u | \\leqslant 0 . 2 5 , \\\\ 1 , & | u | > 0 . 2 5 . \\end{cases} \\end{align*}"}
+{"id": "3905.png", "formula": "\\begin{align*} \\int _ { Y u ( E ) } f ^ * = \\int _ E f . \\end{align*}"}
+{"id": "8079.png", "formula": "\\begin{align*} & \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } H _ { m , p } \\\\ & \\quad = \\frac { 2 } { \\pi } \\int _ 0 ^ \\infty k ( t ) \\tanh ( \\pi t ) t \\biggl ( \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 0 0 0 ) } p ^ { - u } \\Bigl ( \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m ^ { 1 + 2 u } } \\Bigr ) F ( u ) \\frac { \\gamma \\Bigl ( \\dfrac { 1 } { 2 } + u , t \\Bigr ) } { \\gamma \\Bigl ( \\dfrac { 1 } { 2 } , t \\Bigr ) } \\frac { d u } { u } \\biggr ) \\ , d t . \\end{align*}"}
+{"id": "6094.png", "formula": "\\begin{align*} W _ k ( x ) = \\Re \\big ( \\Gamma ( 1 - k , - 2 x ) \\big ) = \\Gamma ( 1 - k , - 2 x ) + \\begin{cases} \\frac { ( - 1 ) ^ { 1 - k } \\pi } { ( k - 1 ) ! } & x > 0 , \\\\ 0 & x < 0 , \\end{cases} \\end{align*}"}
+{"id": "1437.png", "formula": "\\begin{align*} \\mathbb { P } ( | \\mathcal { C } _ { \\max } ( \\mathbb { G } ( n , m , p ) ) | < n ^ { 2 / 3 } / A ) \\leq \\mathbb { P } ( | \\mathcal { C } _ { \\max } ( \\mathbb { G } ( n , m , p ) ) | < T ) = \\mathbb { P } ( t _ i - t _ { i - 1 } < T \\forall i ) . \\end{align*}"}
+{"id": "3077.png", "formula": "\\begin{align*} A ( y ) = C + a ( y ) M \\end{align*}"}
+{"id": "115.png", "formula": "\\begin{align*} [ g ] ^ p _ { M ^ p ( \\mathbb R ^ n \\times \\mathbb R ^ n , \\mathcal L ^ { 2 n } ) } : = \\sup _ { \\lambda > 0 } \\lambda ^ p \\mathcal L ^ { 2 n } \\left ( \\{ x \\in \\mathbb R ^ n \\times \\mathbb R ^ n : \\ | g ( x ) | \\ge \\lambda \\} \\right ) , \\end{align*}"}
+{"id": "7951.png", "formula": "\\begin{align*} \\overline { m _ 1 } = \\overline { m _ 1 + d _ 1 } = \\overline { m _ 2 + d _ 2 } = \\overline { m _ 2 } , \\end{align*}"}
+{"id": "5553.png", "formula": "\\begin{align*} P _ { k } ( x ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n ^ k } \\exp \\left ( { - \\frac { x } { n ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "1055.png", "formula": "\\begin{align*} \\hat { f } = \\sum _ { j = 1 } ^ k \\overline { Z } _ j \\varphi _ j = \\sum _ { j = 1 } ^ k \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Z _ { i , j } \\right ) \\varphi _ j . \\end{align*}"}
+{"id": "5288.png", "formula": "\\begin{align*} \\tau ^ { k e r F _ \\ast } ( F ) = - ( m - n ) F _ \\ast ( H ) . \\end{align*}"}
+{"id": "8072.png", "formula": "\\begin{align*} \\mathcal { D } & = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } \\delta ( n , p ) H _ { m , n } \\\\ & = \\frac { L ( 1 , \\tilde f ) \\bigl ( A ( p , 1 ) p - 1 \\bigr ) } { p ^ { \\frac { 3 } { 2 } } \\pi } \\int _ 0 ^ \\infty k ( t ) \\tanh ( \\pi t ) t \\ , d t + O ( M T ^ { \\frac { 1 } { 7 } + \\varepsilon } p ^ { \\varepsilon } ) . \\end{align*}"}
+{"id": "6317.png", "formula": "\\begin{align*} I _ 2 & = \\sum _ { j = - y } ^ { - 1 } \\zeta ( j ) ^ p \\zeta ( j + y ) ^ p = 2 \\zeta ( y - 1 ) ^ p + \\sum _ { j = - y + 2 } ^ { - y / 2 } \\zeta ( j ) ^ p \\zeta ( j + y ) ^ p + \\sum _ { j = - y / 2 + 1 } ^ { - 2 } \\zeta ( j ) ^ p \\zeta ( j + y ) ^ p \\\\ & \\leq 2 \\zeta ( y - 1 ) ^ p + 1 2 \\zeta ( y ) \\\\ & \\leq 1 4 \\zeta ( y - 1 ) ^ p . \\end{align*}"}
+{"id": "4291.png", "formula": "\\begin{align*} 2 \\int _ { \\Gamma } \\boldsymbol { \\lambda } \\cdot \\mathrm { \\textbf { D } } ( { \\textbf { \\textit { u ' } } } ) \\textbf { \\textit { n } } d \\boldsymbol { \\sigma } - 2 \\int _ { \\Gamma } \\textbf { \\textit { u ' } } \\cdot \\mathrm { \\textbf { D } } ( \\boldsymbol { \\lambda } ) \\textbf { \\textit { n } } d \\boldsymbol { \\sigma } - \\int _ { \\Gamma } ( \\boldsymbol { \\lambda } \\cdot \\textbf { \\textit { n } } ) \\pi ' d \\boldsymbol { \\sigma } = 0 . \\end{align*}"}
+{"id": "4748.png", "formula": "\\begin{align*} t - \\sum _ { i \\le h } \\rho _ i t _ i = c + y b \\end{align*}"}
+{"id": "4966.png", "formula": "\\begin{align*} \\widetilde { q } _ 1 : = a ^ { 1 - p } \\xi ^ p ( q ) & = a ^ { 1 - p } \\Bigl ( \\xi ^ p ( c ^ p ) - ( \\xi ^ p ) _ 1 ( c ^ p ) { \\cdot } ( a b ) ^ { p - 1 } c + \\sum _ { i \\ge 2 } ( \\xi ^ p ) _ i ( c ^ p ) { \\cdot } ( - ( a b ) ^ { p - 1 } c ) ^ i \\Bigr ) \\\\ & \\in a w ^ p - \\xi ' ( c ) ^ p { \\cdot } b ^ { p - 1 } c + a ^ { p - 1 } b ^ { 2 ( p - 1 ) } c ^ 2 S [ a b , c ] \\subset B . \\end{align*}"}
+{"id": "349.png", "formula": "\\begin{align*} T M = H _ { - r } \\supset \\cdots \\supset H _ { - 1 } \\supset 0 , \\end{align*}"}
+{"id": "5096.png", "formula": "\\begin{align*} [ v u _ 1 ^ n ] H = a c ^ n \\in B G _ 0 \\end{align*}"}
+{"id": "4489.png", "formula": "\\begin{align*} \\frac { 1 } { d ' _ 1 } + \\frac { 1 } { d ' _ 2 } + \\frac { 1 } { d ' _ 3 } + \\frac { 1 } { d ' _ 4 } + \\frac { 1 } { d ' _ 5 } = 1 \\end{align*}"}
+{"id": "5752.png", "formula": "\\begin{align*} d \\log m _ { \\mathbf { w } } = w _ 1 \\frac { d x _ 1 } { x _ 1 } + \\cdots + w _ n \\frac { d x _ n } { x _ n } \\end{align*}"}
+{"id": "355.png", "formula": "\\begin{align*} [ X ^ { ( - i ) } , Y ^ { ( - j ) } ] = [ X , Y ] ^ { ( - i - j ) } , \\ \\ ( f X ) ^ { ( - i ) } = \\sum _ { j = 0 } ^ { r - i } f ^ { ( j ) } X ^ { ( - i - j ) } . \\end{align*}"}
+{"id": "6734.png", "formula": "\\begin{align*} S _ 1 ( x ) = \\sum _ { t \\leq x } \\mu ( t ) d ( t ) \\sum _ { x _ 0 < d ^ 2 \\leq x } \\mu ( d ) \\sum _ { \\substack { n \\leq x \\\\ t \\mid n \\\\ d ^ 2 \\mid n } } \\mu ( n + a ) \\ll \\frac { x } { ( \\log x ) ^ { C } , } \\end{align*}"}
+{"id": "8298.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } y ^ * ( \\textbf { x } ) \\ ! \\ ! \\ ! \\ ! & = \\frac { \\eta _ E R _ E ^ 2 - \\eta _ 1 R _ 1 ^ 2 - \\eta _ 2 R _ 2 ^ 2 - \\eta _ 3 R _ 3 ^ 2 } { 2 \\Lambda } \\end{array} \\right . \\end{align*}"}
+{"id": "6901.png", "formula": "\\begin{align*} - ( a + 1 ) & \\leq m i n \\{ b _ 1 , b _ 2 , b _ 1 + b _ 2 \\} \\\\ b _ 1 , b _ 2 & \\leq 1 \\end{align*}"}
+{"id": "8176.png", "formula": "\\begin{align*} u ( t , x ) = u _ 0 ( x ) - \\int _ 0 ^ t \\frac { ( t - s ) ^ { \\beta - 1 } } { \\Gamma ( \\beta ) } \\left ( - \\frac 1 2 \\Delta \\right ) ^ \\gamma u ( s , x ) d s , \\beta \\in ( 0 , 1 ] , \\quad \\gamma \\in ( 0 , 1 ] . \\end{align*}"}
+{"id": "3876.png", "formula": "\\begin{align*} g ( x , Y ( x , u , D u ) , Z ( x , u , D u ) ) & = u , \\\\ g _ { i } ( x , Y ( x , u , D u ) , Z ( x , u , D u ) ) & = D _ i u . \\end{align*}"}
+{"id": "9066.png", "formula": "\\begin{align*} L ^ { \\vee } ( m a n ) \\mu = a ^ { \\lambda - \\rho _ { S } } \\xi ^ { \\vee } ( m ^ { - 1 } ) \\mu \\qquad ( m \\in M _ { S } , a \\in A _ { S } , n \\in N _ { S } ) . \\end{align*}"}
+{"id": "3335.png", "formula": "\\begin{align*} \\Phi = ( \\Phi ) _ { i j } , i , j = x , y , q , p , \\kappa . \\end{align*}"}
+{"id": "3710.png", "formula": "\\begin{align*} Z _ { r _ i } ( f , s ) & = Z ( \\Psi _ { f } , \\chi | \\cdot | ^ { s } ) : = \\int _ { F ^ \\times } \\Psi _ f ( a ) \\chi ( a ) | a | ^ s d ^ \\times a . \\end{align*}"}
+{"id": "9214.png", "formula": "\\begin{align*} X _ { \\sigma } \\cap ( \\Gamma + B ( 0 , r ) ) = A _ { + } ( r ) \\cup A _ { - } ( r ) . \\end{align*}"}
+{"id": "4378.png", "formula": "\\begin{align*} \\frac { p } { q } - \\sum _ { i = 1 } ^ k \\frac { 1 } { a _ i } = \\frac { 1 } { q \\prod _ { i = 1 } ^ k a _ i } \\end{align*}"}
+{"id": "5122.png", "formula": "\\begin{align*} D _ { f _ { 1 } } \\mathcal { I } _ { 1 } ( \\lambda , b , 0 , 0 ) h _ { 1 } ( w ) = - \\sum _ { n = 0 } ^ { \\infty } b ( n + 1 ) I _ { 1 } ( \\lambda b ) K _ { 1 } ( \\lambda ) a _ { n } e _ { n + 1 } ( w ) . \\end{align*}"}
+{"id": "8209.png", "formula": "\\begin{align*} g _ \\sigma ( B _ t + w t , x ) = x + \\int _ 0 ^ t \\sigma \\big ( g _ \\sigma ( B _ s + w s , x ) \\big ) \\circ d B _ s + \\int _ 0 ^ t w \\sigma \\big ( g _ \\sigma ( B _ s + w s , x ) \\big ) d s . \\end{align*}"}
+{"id": "9055.png", "formula": "\\begin{align*} C _ p \\mathbb { E } \\left [ \\left | \\int _ 0 ^ T Y _ s \\sum _ { j = 1 } ^ n Z _ s ^ { j } d B ^ j _ s \\right | ^ { \\frac { p } { 2 } } \\right ] & \\leq d _ p \\mathbb { E } \\left [ \\left ( \\int _ 0 ^ T Y _ s ^ 2 | Z _ s | ^ 2 d s \\right ) ^ { \\frac { p } { 4 } } \\right ] \\\\ & \\leq \\frac { d ^ 2 _ p } { 2 } \\| Y _ s \\| ^ p _ { \\mathcal { S } ^ p } + \\frac { 1 } { 2 } \\mathbb { E } \\left [ \\left ( \\int _ 0 ^ T | Z _ s | ^ 2 d s \\right ) ^ { \\frac { p } { 2 } } \\right ] \\end{align*}"}
+{"id": "1713.png", "formula": "\\begin{align*} I ( \\chi , m ) : = \\int _ { T ( \\R ) } \\chi ( t ) \\langle \\imath ( t ) \\delta s ( \\mu _ m ) , \\delta s ( \\mu _ { - m } ) \\rangle t ^ m d ^ \\times t , \\end{align*}"}
+{"id": "2601.png", "formula": "\\begin{align*} \\dim \\sum _ { \\{ a , b \\} \\in I } V _ { a , b } ^ \\perp = | I | , \\ \\ v _ { i , k } \\notin \\left < v _ { i , j } \\right > \\end{align*}"}
+{"id": "9222.png", "formula": "\\begin{align*} 0 & = \\vert \\nabla f + \\ell _ { 0 } \\nabla \\ell _ { 0 } \\vert ^ { 2 } - \\vert \\nabla f \\vert ^ { 2 } \\\\ & = \\vert \\nabla f \\vert ^ { 2 } + 2 \\ell _ { 0 } \\nabla f \\cdot \\nabla \\ell _ { 0 } + \\ell _ { 0 } ^ { 2 } \\vert \\nabla \\ell _ { 0 } \\vert ^ { 2 } - \\vert \\nabla f \\vert ^ { 2 } \\\\ & = \\ell _ { 0 } \\big ( 2 \\nabla f \\cdot \\nabla \\ell _ { 0 } + \\vert \\nabla \\ell _ { 0 } \\vert ^ { 2 } \\ell _ { 0 } \\big ) . \\end{align*}"}
+{"id": "6816.png", "formula": "\\begin{align*} B U = \\left [ \\begin{array} { c c } 1 & 0 \\\\ 0 & - \\mu \\end{array} \\right ] . \\end{align*}"}
+{"id": "6610.png", "formula": "\\begin{align*} ( U \\psi _ N ) _ { \\theta } - ( U \\psi ) _ { \\theta } = \\sum _ { n = 1 } ^ N P _ n ( \\theta ) ( U \\psi ) _ { \\theta } - ( U \\psi ) _ { \\theta } = - \\sum _ { n = N + 1 } ^ \\infty P _ n ( \\theta ) ( U \\psi ) _ { \\theta } . \\end{align*}"}
+{"id": "8989.png", "formula": "\\begin{align*} & \\partial _ t w + \\partial _ r w = \\nu ( u ) \\partial _ r ( d i s t _ N ( u ) ) - \\nu ( v ) \\partial _ r ( d i s t _ N ( v ) ) \\\\ & \\quad = ( \\nu ( u ) - \\nu ( v ) ) \\partial _ r ( d i s t _ N ( u ) ) + \\nu ( v ) \\partial _ r ( d i s t _ N ( u ) - d i s t _ N ( v ) ) \\end{align*}"}
+{"id": "6872.png", "formula": "\\begin{gather*} \\norm { \\cdot } _ { \\mathcal { K } ( \\mathbb { D } ) } = \\inf \\Big \\{ c > 0 \\ ; : \\ ; c f \\in B _ 1 ( \\mathbb { D } ) \\Big \\} , \\\\ \\mathcal { K } ( \\mathbb { D } ) = \\Big \\{ f \\in X \\ ; : \\ ; \\norm { f } _ { \\mathcal { K } ( \\mathbb { D } ) } < \\infty \\Big \\} . \\end{gather*}"}
+{"id": "7208.png", "formula": "\\begin{align*} I = [ a , \\ , b ] , \\quad \\mbox { a n d t h e i r v a r i a b l e } t \\in I . \\end{align*}"}
+{"id": "1395.png", "formula": "\\begin{align*} \\psi = ( S _ 2 \\boxtimes S _ 1 ) ^ { \\oplus 2 } + S _ 4 \\boxtimes S _ 3 . \\end{align*}"}
+{"id": "5537.png", "formula": "\\begin{align*} R _ { 0 } = \\lim _ { s \\rightarrow 0 } \\frac { s \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } = \\frac { 1 } { \\zeta ( k ) } . \\end{align*}"}
+{"id": "2974.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { L } _ n ( \\eta _ j ^ 2 - \\eta _ j ) = 2 ( W _ { j - 1 } - W _ j ) \\eta _ j + 2 W _ j , \\\\ & \\tilde { L } _ n \\big ( \\eta _ j ^ 3 + \\frac { 3 } { 2 } \\eta _ j ^ 2 + \\frac { 5 } { 2 } \\eta _ j \\big ) = 3 ( W _ { j - 1 } - W _ j ) \\eta _ j ^ 2 + 6 W _ { j - 1 } \\eta _ j + 3 W _ j \\end{aligned} \\end{align*}"}
+{"id": "8735.png", "formula": "\\begin{align*} \\mathcal J ( u _ 1 , \\dots , u _ l ) = \\prod _ { i = 1 } ^ { l } \\left ( 1 + { \\beta } { u _ i } \\right ) ^ { 1 - i } \\mathcal S ( u _ 1 , \\dots , u _ l ) , \\end{align*}"}
+{"id": "6014.png", "formula": "\\begin{align*} & & \\mathbb { A } ^ 1 \\times L _ { g _ 1 } \\setminus \\{ z _ 2 \\} & \\rightarrow \\P ^ { n - 1 } \\times \\P ^ { n - 1 } \\\\ f : & & ( t , x = [ x ^ 1 : \\dots : x ^ n ] ) & \\rightarrow ( x , [ x ^ 1 + t z ^ 1 : \\dots : x ^ n + t z ^ n ] ) . \\end{align*}"}
+{"id": "3255.png", "formula": "\\begin{align*} F ^ { \\ast } = \\sum \\limits _ { e \\in ( S , \\overline { S } ) } k ( e ) . \\end{align*}"}
+{"id": "7526.png", "formula": "\\begin{align*} Z _ f ( s , \\chi ) = Z _ f \\big ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ n \\big ) + \\sum _ { \\gamma } Z _ f \\big ( s , \\chi , S ( \\Delta _ { \\gamma } \\bigcap ( \\mathbb { N } ^ n \\setminus \\{ 0 \\} ) ) \\big ) , \\end{align*}"}
+{"id": "7251.png", "formula": "\\begin{align*} f ( \\gamma ) = \\frac { 1 } { u ( \\gamma ^ { \\alpha / 2 } w ^ { \\sharp } ( \\gamma ) ^ { \\alpha / 2 } ) } \\sqrt { \\frac { \\gamma } { w ^ { \\sharp } ( \\gamma ) } } , \\end{align*}"}
+{"id": "2418.png", "formula": "\\begin{align*} E _ { 2 3 } V = [ \\ > I _ r \\ > \\ > 0 \\ > ] \\end{align*}"}
+{"id": "4051.png", "formula": "\\begin{align*} D _ { x _ r , y _ l } Q _ k & = - \\frac { 1 } { g _ z } \\left ( D _ { y _ l } E _ { r k } - E _ { r k } \\frac { g _ { , l , z } } { g _ z } \\right ) , \\\\ D _ { x _ r , z } Q _ k & = - \\frac { 1 } { g _ z } \\left ( D _ { z } E _ { r k } - E _ { r k } \\frac { g _ { , z z } } { g _ z } \\right ) . \\end{align*}"}
+{"id": "2195.png", "formula": "\\begin{align*} \\dfrac { ( \\phi \\ast ( H _ { k } F _ n ) ) ( z ) } { ( \\phi \\ast F _ n ) ( z ) } = \\dfrac { ( \\phi \\ast z f ' _ { k } ) ( z ) } { ( \\phi \\ast F _ { n } ) ( z ) } = \\dfrac { z ( \\phi \\ast f _ { k } ) ' ( z ) } { ( \\phi \\ast F ) _ { n } ( z ) } \\end{align*}"}
+{"id": "1347.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | ^ m \\geq \\frac { \\frac { 1 } { ( G _ { f , \\tau } ^ { \\circ ^ m } ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) ^ m \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ { 2 m } } { n ^ 2 - n } , \\end{align*}"}
+{"id": "1811.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 ^ + } w \\cdot W _ \\varepsilon ( T ) w & = w \\cdot W _ 0 ( T ) w \\in \\R , \\\\ \\lim _ { \\varepsilon \\to 0 ^ + } w \\cdot W _ \\varepsilon ( 0 ) w & = \\lim _ { \\varepsilon \\to 0 ^ + } \\frac { 1 } { \\varepsilon } = - \\infty . \\end{align*}"}
+{"id": "4579.png", "formula": "\\begin{align*} 1 = \\frac { 1 } { 3 } + \\frac { 1 } { 4 } + \\frac { 1 } { 4 } + \\frac { 1 } { 8 } + \\frac { 1 } { 2 4 } \\end{align*}"}
+{"id": "7498.png", "formula": "\\begin{align*} \\gamma _ 1 & = \\{ ( u , 0 ) | u \\ge i _ 0 \\} , \\\\ \\gamma _ 2 & = ( i _ 0 , 0 ) , \\\\ \\gamma _ 3 & = \\{ ( u , v ) | j _ 0 u + i _ 0 v = i _ 0 j _ 0 , 0 \\le u \\le i _ 0 , 0 \\le v \\le j _ 0 \\} , \\\\ \\gamma _ 4 & = ( 0 , j _ 0 ) , \\\\ \\gamma _ 5 & = \\{ ( 0 , v ) | v \\ge j _ 0 \\} \\end{align*}"}
+{"id": "8685.png", "formula": "\\begin{align*} \\tilde F _ { \\lambda } = \\tilde \\Gamma ^ + _ { - \\lambda _ 1 } \\dots \\tilde \\Gamma ^ + _ { - \\lambda _ l } ( 1 ) . \\end{align*}"}
+{"id": "3854.png", "formula": "\\begin{align*} A _ { n + 2 } - 1 = \\sum _ { k = 0 } ^ n \\sum _ { i = 0 } ^ k ( \\mu _ { k + 2 - i } + \\delta _ { i , k } + \\delta _ { i , k - 1 } ) A _ i . \\end{align*}"}
+{"id": "3016.png", "formula": "\\begin{align*} \\mathcal { A } _ 0 [ \\theta , \\varphi , \\pi ] = \\int _ { \\mathcal { Y } ^ { N + 1 } } \\frac { 1 } { 2 } \\hat { \\theta } ^ { ( N - 1 ) } _ { I J } \\wedge \\Phi ^ { I J } + \\pi _ I \\wedge \\Theta ^ I \\end{align*}"}
+{"id": "8975.png", "formula": "\\begin{align*} \\int _ { \\partial B } | \\nabla ^ k u | ^ 2 d \\phi = 2 \\int _ { \\partial B } | \\nabla ^ { k - 1 } u _ { \\phi } | ^ 2 d \\phi = \\dots = 2 ^ k \\int _ { \\partial B } | \\partial _ { \\phi } ^ k u | ^ 2 d \\phi \\end{align*}"}
+{"id": "6896.png", "formula": "\\begin{align*} \\langle \\alpha , \\beta \\rangle = \\sum _ { i \\in Q _ 0 } \\alpha _ i \\beta _ i - \\sum _ { a \\in Q _ 1 } \\alpha _ { s ( a ) } \\beta _ { t ( a ) } . \\end{align*}"}
+{"id": "7672.png", "formula": "\\begin{align*} [ \\bigoplus _ { I } \\sum _ { p \\in \\mathbb { Z } } ( p + \\iota _ { E } ( \\omega ) ) \\eta _ { I , p } ] = L _ { \\omega } ( [ \\bigoplus _ { I } \\eta _ { I } ] ) = \\nabla _ { \\omega } ( \\iota _ { E } ( [ \\bigoplus _ { I } \\eta _ { I } ] ) ) = \\nabla _ { \\omega } ( [ \\bigoplus _ { I } \\iota _ { E } ( \\eta _ { I } ) ] ) . \\end{align*}"}
+{"id": "1103.png", "formula": "\\begin{align*} Z \\sim \\begin{cases} \\mathrm { U n i f o r m } \\Big ( z \\in \\{ - B , B \\} ^ k \\Big | \\langle z , \\tilde { V } \\rangle \\geq 0 \\Big ) , & T = 1 , \\\\ \\mathrm { U n i f o r m } \\Big ( z \\in \\{ - B , B \\} ^ k \\Big | \\langle z , \\tilde { V } \\rangle \\leq 0 \\Big ) , & T = 0 , \\end{cases} \\end{align*}"}
+{"id": "5068.png", "formula": "\\begin{align*} 0 = \\sum _ { j = 1 } ^ n \\sum _ { s = 0 } ^ m q _ { j , s } { y } ^ { s } f _ j ( \\omega ) = \\sum _ { j = 1 } ^ n Q _ j ( y + b ) f _ j ( \\omega ) \\end{align*}"}
+{"id": "8295.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\textbf { u } ^ * _ E = \\frac { 1 } { d _ E } [ x ^ * ( \\textbf { x } ) - x _ E , \\ y ^ * ( \\textbf { x } ) - y _ E , \\ z ^ * ( \\textbf { x } ) - z _ E ] \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "6269.png", "formula": "\\begin{align*} X _ { 0 } \\big | _ { ( 0 , y , - 1 ) } = Y _ { 0 } \\big | _ { ( 0 , y , - 1 ) } = e ^ { - \\frac { 1 } { y } } \\frac { \\partial } { \\partial y } \\Big | _ { ( 0 , y , - 1 ) } , \\end{align*}"}
+{"id": "2519.png", "formula": "\\begin{align*} \\frac 1 \\tau ( P - 1 ) f = \\varrho M _ f - f \\ , , \\end{align*}"}
+{"id": "3428.png", "formula": "\\begin{align*} \\mu _ t ( \\underline { \\gamma } ) = \\mu _ t ( x _ 0 ) \\exp \\left ( t \\phi ( \\underline { \\gamma } ) - n P _ G ( t \\phi ) \\right ) , \\end{align*}"}
+{"id": "7692.png", "formula": "\\begin{align*} ( \\tau ^ { \\sharp } ) ^ { - j } ( \\textbf { a } ) = ( a _ { 1 } - j , \\dots , a _ { r } - j ) . \\end{align*}"}
+{"id": "5784.png", "formula": "\\begin{align*} ( \\mathcal L + A ) \\Phi = \\beta ^ 2 \\Phi \\hbox { a n d } \\mathcal G \\Phi = 0 . \\end{align*}"}
+{"id": "8626.png", "formula": "\\begin{align*} M _ k ( S ) ( x , \\omega ) = \\sum _ { j \\in \\N \\cap b ^ k S } I _ j ( x , \\omega ) , \\end{align*}"}
+{"id": "9258.png", "formula": "\\begin{align*} \\begin{aligned} ( s _ { j k } w ) ^ { - 1 } ( j ) & = ( w ^ { - 1 } s _ { j k } ) ( j ) = w ^ { - 1 } ( k ) \\leq h ( ( s _ { j k } w ) ^ { - 1 } ( j + 1 ) ) = h ( w ^ { - 1 } ( j + 1 ) ) \\\\ \\end{aligned} \\end{align*}"}
+{"id": "458.png", "formula": "\\begin{align*} | \\langle x , \\tau _ n \\rangle | = | \\langle y , \\tau _ n \\rangle | , \\forall n \\in \\mathbb { N } , \\end{align*}"}
+{"id": "1040.png", "formula": "\\begin{align*} n \\gtrsim \\frac { k _ 0 } { \\delta ^ 2 \\alpha ^ 2 } \\asymp \\frac { D / \\delta } { \\delta ^ 2 \\alpha ^ 2 } = \\frac { D } { \\delta ^ 3 \\alpha ^ 2 } , \\mbox { i . e . } \\delta \\gtrsim \\left ( \\frac { D } { n \\alpha ^ 2 } \\right ) ^ { 1 / 3 } . \\end{align*}"}
+{"id": "9021.png", "formula": "\\begin{align*} \\mu \\big ( H ^ 1 ( \\Q _ \\Sigma / \\Q _ { \\infty } , A ) \\big ) - \\mu \\big ( H ^ 1 ( \\Q _ \\Sigma / \\Q _ { \\infty } , A _ 1 ) \\big ) = \\underset { \\ell \\in \\Sigma } { \\sum } \\Big ( \\ , \\mu \\big ( \\mathcal { H } ^ 1 _ { \\ell } ( \\Q _ { \\infty } , A ) \\big ) - \\ , \\mu \\big ( \\mathcal { H } ^ 1 _ { \\ell } ( \\Q _ { \\infty } , A _ 1 ) \\big ) \\Big ) . \\end{align*}"}
+{"id": "2596.png", "formula": "\\begin{align*} V _ { [ 4 ] \\setminus \\{ l \\} } = \\left < v _ { a , b } ^ t \\ \\middle | \\ a , b \\in [ 4 ] \\setminus \\{ l \\} , t = 1 , \\dots , d \\right > \\end{align*}"}
+{"id": "7956.png", "formula": "\\begin{align*} 0 _ K = \\alpha \\pi ( \\ell ) = \\beta i ( \\ell ) = \\beta ( \\ell ) . \\end{align*}"}
+{"id": "4808.png", "formula": "\\begin{align*} \\Pr _ { y \\sim \\mathcal { D } ( C ^ \\perp ) } \\big [ | y | = j \\big ] \\leq \\big ( 1 + o ( N ^ { - 1 } ) \\big ) \\frac { \\binom { N } { j } } { 2 ^ N } , \\end{align*}"}
+{"id": "3970.png", "formula": "\\begin{align*} u ( x _ 0 ) & = v ( x _ 0 ) = g ( x _ 0 , Y v ( x _ 0 ) , Z v ( x _ 0 ) ) , \\\\ \\quad u ( x ) & \\geq g ( x , Y v ( x _ 0 ) , Z v ( x _ 0 ) ) x _ 0 . \\end{align*}"}
+{"id": "3152.png", "formula": "\\begin{align*} r _ 1 ( t ) : = 1 - r _ 2 ( t ) : = \\frac { 1 } { 8 } \\left ( \\sin ( 2 \\pi t ) + 2 \\cos ( 2 \\pi t ) \\right ) \\quad t \\in \\R . \\end{align*}"}
+{"id": "2496.png", "formula": "\\begin{align*} Z Y = \\psi ( S ) , \\end{align*}"}
+{"id": "6577.png", "formula": "\\begin{align*} K ( \\mu ) : = \\sup _ { B _ r ( x ) \\subset \\R ^ n } \\frac { 1 } { r ^ { n - 1 } } \\mu ( B _ r ( x ) ) < \\infty . \\end{align*}"}
+{"id": "1604.png", "formula": "\\begin{align*} 0 \\leq \\Phi ( \\mu ^ * ) ( v ) = \\mu ^ * ( v ) - c \\delta _ v ( v ) + c \\delta _ w ( v ) = \\mu ^ * ( v ) - c . \\end{align*}"}
+{"id": "7796.png", "formula": "\\begin{align*} \\Psi _ { a _ 2 , b _ 2 } [ n _ 2 ] \\Psi _ { a _ 1 , b _ 1 } [ n _ 1 ] = \\Psi _ { a _ 1 , b _ 1 } [ n _ 1 ] \\Psi _ { a _ 2 , b _ 2 } [ n _ 2 ] . \\end{align*}"}
+{"id": "7695.png", "formula": "\\begin{align*} \\beta : \\mathbb { C } [ s _ { 1 } , \\dots , s _ { d } ] \\to \\frac { \\mathbb { C } [ s _ { 1 } , \\dots , s _ { d } ] } { ( \\{ s _ { t } - s _ { j } \\mid t , j \\in S _ { k } , 1 \\leq k \\leq r \\} ) } = \\mathbb { C } [ s _ { 1 } , \\dots , s _ { r } ] . \\end{align*}"}
+{"id": "7144.png", "formula": "\\begin{align*} \\Phi ^ { s , m } _ t = \\begin{cases} 0 , & \\mbox { i f $ s < t $ } ; \\\\ \\sum _ { i = 0 } ^ { r } a ^ { m - s } _ { r , i } \\Phi _ { t } ^ { s , s - i } , & \\mbox { i f $ 0 \\leq s - t = r $ , $ 0 \\leq s \\leq m $ } ; \\\\ ( - 1 ) ^ { t } \\gamma _ { t - 1 } ^ { s , m } + ( - 1 ) ^ s \\gamma _ { t - 1 } ^ { s , m - 1 } , & \\mbox { i f $ 0 \\leq s - t $ , $ m \\leq s \\leq m + n $ } . \\end{cases} \\end{align*}"}
+{"id": "6744.png", "formula": "\\begin{align*} \\left | x \\right | = \\left \\{ \\begin{array} { l c c } x & ; & x \\ge 0 \\\\ - x & ; & x < 0 \\end{array} \\right . \\end{align*}"}
+{"id": "4432.png", "formula": "\\begin{align*} \\frac { 1 } { 5 } + \\frac { 1 } { 4 9 5 } = \\frac { 1 } { 9 } + \\frac { 1 } { 1 1 } = \\frac { 2 0 } { 9 9 } < \\theta \\leq \\frac { 4 9 9 } { 2 4 7 0 } = \\frac { 1 } { 5 } + \\frac { 1 } { 4 9 4 } . \\end{align*}"}
+{"id": "4161.png", "formula": "\\begin{align*} \\| g - g _ c \\| _ { C ^ { 2 , \\alpha } } < C , \\| b \\| _ { C ^ { 2 , \\alpha } } < C ( \\| H - H _ c \\| _ { C ^ { 1 , \\alpha } } = \\| d b \\| _ { C ^ { 1 , \\alpha } } < C ) . \\end{align*}"}
+{"id": "352.png", "formula": "\\begin{align*} ( V ^ s f ) | _ N = 0 \\end{align*}"}
+{"id": "4433.png", "formula": "\\begin{align*} a _ 2 = 1 3 = \\left \\lceil \\frac { 9 0 } { 7 } \\right \\rceil = \\left \\lceil \\left ( \\frac { 1 } { 6 } + \\frac { 1 } { 9 } - \\frac { 1 } { 5 } \\right ) ^ { - 1 } \\right \\rceil > \\frac { 9 0 } { 7 } . \\end{align*}"}
+{"id": "4949.png", "formula": "\\begin{align*} \\widetilde { G } _ 1 = \\alpha ( u _ { m , n } ^ 2 + v _ { m , n } ^ 2 ) , \\end{align*}"}
+{"id": "3334.png", "formula": "\\begin{align*} \\xi ^ t = ( 0 , 0 , 0 , 0 , \\xi _ { \\kappa } ) , \\end{align*}"}
+{"id": "203.png", "formula": "\\begin{align*} | \\mu ( T ^ { k ( h _ n + c _ n ) } ( I _ { n , a } ) \\cap B ) - \\mu ( I _ { n , a } ) \\mu ( B ) | \\leq \\int \\limits _ { I _ { n , a } } \\Big { | } \\frac { 1 } { r _ n + 1 } \\sum \\limits _ { i = 0 } ^ { r _ n } \\chi _ { B } \\circ T ^ { - k i - \\frac { 1 } { 2 } k ( k - 1 ) } - \\mu ( B ) \\Big { | } d \\mu + \\frac { 2 k + 2 } { r _ n + 1 } \\mu ( I _ { n } ) . \\end{align*}"}
+{"id": "3355.png", "formula": "\\begin{align*} \\eta = \\dd \\kappa - p _ i \\dd q ^ i , \\dd \\eta = \\dd q ^ i \\wedge \\dd p _ i . \\end{align*}"}
+{"id": "7762.png", "formula": "\\begin{align*} \\left ( \\sqrt { 1 + s } + \\sqrt { s } \\right ) ^ { - 2 \\mu } = \\left ( \\sqrt { 1 + s } - \\sqrt { s } \\right ) ^ { 2 \\mu } & = \\left ( 1 + 2 s + 2 \\sqrt { s ( 1 + s ) } \\right ) ^ { - \\mu } = \\left ( 1 + 2 s - 2 \\sqrt { s ( 1 + s ) } \\right ) ^ { \\mu } \\end{align*}"}
+{"id": "2618.png", "formula": "\\begin{align*} L _ i = \\bigcup _ { j \\in [ r ] \\setminus \\{ i \\} } A _ { i , j } , \\end{align*}"}
+{"id": "6598.png", "formula": "\\begin{align*} \\int | V _ \\phi f ( x , \\xi ) | \\ , d \\xi & \\geq ( 2 \\pi ) ^ { - n / 2 } \\left | \\int e ^ { i \\langle x , \\eta \\rangle } \\hat { f } ( \\eta ) \\int \\overline { \\hat { \\phi } ( \\xi - \\eta ) } \\ , d \\xi \\ , d \\eta \\right | \\\\ & = C _ \\phi | f ( x ) | , \\end{align*}"}
+{"id": "8192.png", "formula": "\\begin{align*} \\Phi ( t , \\lambda ) = \\Gamma ( q _ 2 ) E _ { q _ 1 , q _ 2 } ^ { q _ 3 } ( \\lambda t ^ b ) , \\end{align*}"}
+{"id": "7897.png", "formula": "\\begin{align*} f ( z ) ^ { n } + q ( z ) e ^ { Q ( z ) } f ( z + c ) = P ( z ) , \\end{align*}"}
+{"id": "1498.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ { \\frac \\pi 2 } \\theta \\log \\left ( \\cos \\frac \\theta 2 \\right ) d \\theta & = \\frac { \\pi ^ 2 } 8 \\log \\frac 1 { \\sqrt 2 } - 2 \\pi ^ 2 \\log \\C _ 3 \\left ( \\frac 1 4 \\right ) \\\\ & = \\frac { \\pi ^ 2 } 8 \\log \\frac 1 { \\sqrt 2 } - \\log \\left ( \\prod _ { n = 1 , n } ^ \\infty \\left ( 1 - \\frac { 1 } { 4 n ^ 2 } \\right ) ^ { \\frac { n ^ 2 } 4 } e ^ { \\frac 1 { 1 6 } } \\right ) ^ { 2 \\pi ^ 2 } , \\end{aligned} \\end{align*}"}
+{"id": "8393.png", "formula": "\\begin{align*} \\sigma ^ 2 ( \\alpha ) = \\frac { \\tilde { \\sigma } ^ 2 ( \\alpha ) } { \\int \\tau _ { \\alpha } d \\mu _ { \\alpha } } , \\end{align*}"}
+{"id": "2717.png", "formula": "\\begin{align*} u ( s _ n + \\tau ) = v _ L ( s _ n + \\tau ) + \\sum _ { k = 1 } ^ K V _ n ^ k ( \\tau ) + r _ n ( \\tau ) , \\end{align*}"}
+{"id": "7254.png", "formula": "\\begin{align*} m ( \\gamma , \\infty ) : = w _ + ^ { - 1 } m _ + ( \\gamma , \\infty ) j ( \\gamma , \\infty ) : = r _ + ^ { - 2 } j _ + ( \\gamma , \\infty ) . \\end{align*}"}
+{"id": "8259.png", "formula": "\\begin{align*} \\psi ( x ) = x + O ( x \\exp ( - c \\sqrt { \\log x } ) \\end{align*}"}
+{"id": "7190.png", "formula": "\\begin{align*} A _ { \\beta \\alpha } ( { \\Sigma _ 0 } ) y ^ 1 _ { \\alpha } = C _ \\beta ( { \\Sigma _ 0 } ) , \\end{align*}"}
+{"id": "8128.png", "formula": "\\begin{align*} q ( y ) = y ^ { - \\frac { 4 } { 3 } } g \\Bigl ( \\frac { y ^ 2 n _ 1 ^ 4 } { N p } \\Bigr ) \\widehat { k ^ * } \\Bigl ( \\frac { M T c m p } { 2 \\pi ^ 2 y n _ 1 ^ 2 } \\Bigr ) . \\end{align*}"}
+{"id": "5423.png", "formula": "\\begin{align*} { r ^ m _ i } ( x y ) = \\sum _ { t = 0 } ^ { m } v ^ { ( t i , \\nu - ( m - t ) i ) + t ( m - t ) } \\frac { [ m ] _ { v } ! } { [ t ] _ { v } ! [ m - t ] _ { v } ! } { r _ i ^ t } ( x ) { r _ i ^ { m - t } } ( y ) \\end{align*}"}
+{"id": "6059.png", "formula": "\\begin{align*} \\theta \\left ( \\vec u \\right ) : = \\sum _ { \\vec n \\in \\Z ^ g } \\exp \\bigg \\{ \\pi \\mathrm { i } \\vec n ^ \\mathsf { T } \\mathsf B \\vec n + 2 \\pi \\mathrm { i } \\vec n ^ \\mathsf { T } \\vec u \\bigg \\} , \\vec u \\in \\C ^ g . \\end{align*}"}
+{"id": "5754.png", "formula": "\\begin{align*} \\alpha = \\sum _ { j = 0 } ^ r \\xi ^ j \\cap { \\bar \\pi } ^ * \\alpha _ j , \\end{align*}"}
+{"id": "6875.png", "formula": "\\begin{gather*} ( \\cdot , \\cdot ) _ h : X _ h \\times X _ h \\to \\mathbb { R } \\\\ ( u _ h , v _ h ) _ h \\mapsto \\frac { 1 } { N _ { \\Omega } ^ { \\gamma _ 1 } } \\sum _ { i = 1 } ^ { N _ \\Omega } \\Big ( u _ h ( { \\vec { \\omega } _ i } ) , v _ h ( { \\vec { \\omega } _ i } ) \\Big ) _ a + \\\\ \\frac { 1 } { N _ { \\partial \\Omega } ^ { \\gamma _ 2 } } \\sum _ { i = 1 } ^ { N _ { \\partial \\Omega } } \\Big ( u _ h ( { \\vec { \\beta } _ i } ) , v _ h ( { \\vec { \\beta } _ i } ) \\Big ) _ b . \\end{gather*}"}
+{"id": "5547.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { \\lambda _ 1 - i \\infty } ^ { \\lambda _ 1 - i \\infty } \\frac { \\Gamma ( s ) \\Gamma \\left ( \\frac { k } { 2 } - s \\right ) } { \\Gamma \\left ( \\frac { 1 - k } { 2 } + s \\right ) } X _ n ^ { - s } { \\rm d } s = G _ { 1 , 2 } ^ { 1 , 1 } \\left ( \\begin{matrix} 1 \\\\ \\frac { k } { 2 } , \\frac { k + 1 } { 2 } \\end{matrix} \\Big | \\frac { 1 } { X _ n } \\right ) . \\end{align*}"}
+{"id": "836.png", "formula": "\\begin{align*} \\pi _ { p n - 2 p \\lfloor c \\log n \\rfloor } ( \\widehat { E } ( t \\pm C n ^ { - 1 / 2 } \\log n ) ) & = N ( t \\pm C n ^ { - 1 / 2 } \\log n , \\sigma ) + O \\left ( \\frac { \\log n } { \\sqrt { n } } \\right ) \\\\ & = N ( t , \\sigma ) + O \\left ( \\frac { \\log n } { \\sqrt { n } } \\right ) \\end{align*}"}
+{"id": "2615.png", "formula": "\\begin{align*} 6 & = a _ { 2 , 3 } + a _ { 2 , 4 } + a _ { 2 , 5 } + a _ { 3 , 4 } + a _ { 3 , 5 } + a _ { 4 , 5 } = 1 + 1 + 1 + 1 + 1 + 1 \\\\ & \\leftrightarrow ( a _ { 1 , 2 } , a _ { 1 , 3 } , a _ { 1 , 4 } , a _ { 1 , 5 } , a _ { 2 , 3 } , a _ { 2 , 4 } , a _ { 2 , 5 } , a _ { 3 , 4 } , a _ { 3 , 5 } , a _ { 4 , 5 } ) = ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) . \\end{align*}"}
+{"id": "4602.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { a _ i } < \\frac { 1 } { q } \\end{align*}"}
+{"id": "6055.png", "formula": "\\begin{align*} x ^ j - \\sum _ { i = 1 } ^ g l _ i ( x ) \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac { y ^ j \\dd y } { w ( y ) } \\equiv 0 \\end{align*}"}
+{"id": "433.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\right \\| = b \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "5366.png", "formula": "\\begin{align*} E ^ { ( i ) } \\cdot E ^ { ( j ) } = 0 \\forall i , j \\in \\{ 1 , \\ldots , m \\} , i \\neq j . \\end{align*}"}
+{"id": "8261.png", "formula": "\\begin{align*} d ( t ) : = \\frac { ( b - \\theta ) ^ 2 } { 4 n \\left ( 1 2 \\log ( | t | + B + 5 ) + 5 1 \\log ( A + \\kappa ) + 3 1 \\log \\frac { 1 } { b - \\theta } + 1 1 3 \\right ) } . \\end{align*}"}
+{"id": "7583.png", "formula": "\\begin{align*} u ( 0 , x ) = u _ 0 ( x ) . \\end{align*}"}
+{"id": "2904.png", "formula": "\\begin{align*} \\hat { K } ( - \\alpha ) = \\int _ { \\mathbb { T } ^ \\infty } K e _ \\alpha d m _ \\infty = \\int _ { \\mathbb { T } ^ \\infty } 1 \\cdot e _ \\alpha K d m _ \\infty = 0 . \\end{align*}"}
+{"id": "1238.png", "formula": "\\begin{align*} \\mu ( X ^ c _ \\geq ( y , k ) ) = \\nu ( ( \\underline y , y ) ) . \\end{align*}"}
+{"id": "6415.png", "formula": "\\begin{align*} ( p - 2 ) & \\Delta ^ N _ { \\infty , X } \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) + \\Delta _ X \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) \\\\ & \\geq ( p - 2 ) \\Lambda ( \\nabla _ X ^ 2 \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) ) + \\Delta _ X \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) \\\\ & = ( p - 2 ) \\Lambda ( - H ^ X _ k ) + \\textrm { t r } ( - H ^ X _ k ) \\\\ & = ( p - 1 ) \\Lambda ( - H ^ X _ k ) + \\sum ^ { m - 1 } _ { i = 1 } \\lambda _ i ( - H ^ X _ k ) . \\end{align*}"}
+{"id": "7600.png", "formula": "\\begin{align*} \\| \\pi _ n u - \\pi _ n v \\| _ { \\L ^ p } & = \\bigg \\| u - \\frac { n } { \\| v \\| _ { \\L ^ p } } v \\bigg \\| _ { \\L ^ p } = \\frac { 1 } { \\| v \\| _ { \\L ^ p } } \\| ( u - v ) \\| v \\| _ { \\L ^ p } + v ( \\| v \\| _ { \\L ^ p } - n ) \\| _ { \\L ^ p } \\\\ & \\leq \\| u - v \\| _ { \\L ^ p } + \\| v \\| _ { \\L ^ p } - n \\leq \\| u - v \\| _ { \\L ^ p } + ( \\| v \\| _ { \\L ^ p } - \\| u \\| _ { \\L ^ p } ) \\\\ & \\leq 2 \\| u - v \\| _ { \\L ^ p } , \\end{align*}"}
+{"id": "4997.png", "formula": "\\begin{align*} \\alpha \\otimes e \\cdot \\beta \\otimes f = \\sum _ { g \\in G } \\sum _ { p \\in \\alpha \\cap \\beta g } \\epsilon _ p ( \\alpha , \\beta g ) \\ , \\ , p \\otimes e \\otimes g ^ { - 1 } f . \\end{align*}"}
+{"id": "9069.png", "formula": "\\begin{align*} \\mu _ { z , X } ( \\phi ) = \\sum _ { \\substack { \\beta \\\\ \\kappa _ { \\beta } = \\kappa } } \\int _ { M } \\int _ { A } \\int _ { N _ { P } } \\int _ { w \\overline { N } _ { P } w ^ { - 1 } \\cap \\overline { N } _ { P } } a ^ { - \\lambda + \\rho _ { P } } \\Big ( \\sigma ^ { \\vee } ( m ) c _ { \\beta } , \\partial ^ { \\beta } \\phi ( m a n \\overline { n } ) \\Big ) \\ , d \\overline { n } \\ , d n \\ , d a \\ , d m . \\end{align*}"}
+{"id": "8535.png", "formula": "\\begin{align*} \\binom { n - 1 } { 2 } ^ p \\le \\bigg ( \\frac { n ^ 2 } { 2 } \\bigg ) ^ p . \\end{align*}"}
+{"id": "8441.png", "formula": "\\begin{align*} \\gamma _ K : = \\lambda _ K / w ( K ) = c ( K ) \\lambda _ K \\in \\mathcal E ^ + _ K \\end{align*}"}
+{"id": "376.png", "formula": "\\begin{align*} \\hat { s } _ \\ell : = s _ { 2 \\ell } + s _ { 2 \\ell - 1 } \\textrm { a n d } \\omega _ \\ell : = \\frac { s _ { 2 \\ell } - s _ { 2 \\ell - 1 } } { \\hat { s } _ \\ell } \\ , \\end{align*}"}
+{"id": "4261.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\| ( \\nabla - i A ) \\phi \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } V _ \\gamma ( x ) | \\phi ( x ) | ^ 2 d x & \\leq \\liminf _ { n \\rightarrow \\infty } \\frac { 1 } { 2 } \\| ( \\nabla - i A ) g _ n \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } V _ \\gamma ( x ) | g _ n ( x ) | ^ 2 d x \\\\ & = \\liminf _ { n \\rightarrow \\infty } \\frac { 1 } { 2 } \\| ( \\nabla - i A ) f _ n \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } V _ \\gamma ( x ) | f _ n ( x ) | ^ 2 d x . \\end{align*}"}
+{"id": "6918.png", "formula": "\\begin{align*} \\min _ { u } J ( u ) = \\int \\limits _ { 0 } ^ { t _ f } \\left ( w _ 1 I ^ 2 ( t ) + w _ 2 u ^ 2 ( t ) \\right ) d t , \\end{align*}"}
+{"id": "4845.png", "formula": "\\begin{align*} \\int _ 0 ^ t v ( s ) d s = - \\int _ t ^ 1 v ( s ) d s , t \\in I , \\end{align*}"}
+{"id": "6916.png", "formula": "\\begin{align*} \\kappa ( t ) = \\frac { \\kappa _ 1 } { \\exp ( \\kappa _ 2 ( t - \\kappa _ 3 ) ) + \\exp ( - \\kappa _ 2 ( t - \\kappa _ 3 ) ) } , \\end{align*}"}
+{"id": "8744.png", "formula": "\\begin{align*} \\check { d } _ { \\xi } = d _ 0 . \\end{align*}"}
+{"id": "5523.png", "formula": "\\begin{align*} | \\Gamma ( \\sigma + i T ) | = \\sqrt { 2 \\pi } | T | ^ { \\sigma - 1 / 2 } e ^ { - \\frac { 1 } { 2 } \\pi | T | } \\left ( 1 + O \\left ( \\frac { 1 } { | T | } \\right ) \\right ) { \\rm a s } | T | \\rightarrow \\infty . \\end{align*}"}
+{"id": "4336.png", "formula": "\\begin{align*} \\delta ( f ) ( g _ 1 , g _ 2 , \\ldots , g _ { n + 1 } ) : = & g _ 1 f ( g _ 2 , \\ldots , g _ { n + 1 } ) + \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ { i } f ( g _ 1 , \\ldots , g _ i g _ { i + 1 } , \\ldots , g _ { n + 1 } ) \\\\ & + ( - 1 ) ^ { n + 1 } f ( g _ 1 , \\ldots , g _ n ) . \\end{align*}"}
+{"id": "875.png", "formula": "\\begin{align*} \\mathcal { X } ^ { t + 1 } - \\mathcal { X } ^ { \\star } = ( \\mathcal { I } - \\mathcal { Q } ^ { - 1 } * \\mathcal { W } ) * ( \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } ) . \\end{align*}"}
+{"id": "3511.png", "formula": "\\begin{align*} \\omega _ A = [ B _ { A ( 1 , 1 ) } ^ + , \\dots , B _ { A ( \\lambda _ 1 ' , 1 ) } ^ + ] \\cdots [ B _ { A ( 1 , \\ell ( \\lambda ' ) ) } ^ + , \\dots , B _ { A ( \\lambda _ { \\ell ( \\lambda ' ) } ' , \\ell ( \\lambda ' ) ) } ^ + ] v _ 0 , \\end{align*}"}
+{"id": "8310.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } B ( \\textbf { x } ) \\ ! \\ ! \\ ! & = ( x _ E - a ( \\textbf { x } ) ) ^ 2 + ( y _ E - b ( \\textbf { x } ) ) ^ 2 + z _ E ^ 2 \\\\ & ~ ~ - ( x _ 1 - a ( \\textbf { x } ) ) ^ 2 - ( y _ 1 - b ( \\textbf { x } ) ) ^ 2 - z _ 1 ^ 2 . \\end{array} \\right . \\end{align*}"}
+{"id": "885.png", "formula": "\\begin{align*} 0 < \\lambda _ { \\min } ( \\mathbf { E } _ { i \\sim \\mathbf { p } } [ b c i r c ( \\mathcal { Z } _ { i } ) ] ) = \\delta _ { \\mathbf { p } } ^ { 2 } ( \\mathcal { Q } , \\boldsymbol { \\mathcal { S } } ) \\leq \\delta _ { \\infty } ^ { 2 } ( \\mathcal { Q } , \\boldsymbol { \\mathcal { S } } ) \\leq 1 . \\end{align*}"}
+{"id": "8056.png", "formula": "\\begin{align*} S ( a , b ; c ) = \\sum _ { d \\bar d \\equiv 1 \\pmod * { c } } e \\Bigl ( \\frac { d a + \\bar d b } { c } \\Bigr ) \\end{align*}"}
+{"id": "5890.png", "formula": "\\begin{align*} \\hbox { r a n k } ( C _ p D ) = \\hbox { r a n k } ( D ^ T C _ p ^ T ) = \\hbox { r a n k } ( C _ p ^ T ) = N - p . \\end{align*}"}
+{"id": "2858.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ j b _ i \\leq \\sum _ { i = 1 } ^ j a _ i = a _ k + a _ { \\ell } + \\sum _ { \\substack { i = 1 \\\\ i \\neq k , \\ell } } ^ j a _ i = c _ k + c _ { \\ell } + \\sum _ { \\substack { i = 1 \\\\ i \\neq k , \\ell } } ^ j c _ i = \\sum _ { i = 1 } ^ j c _ i . \\end{align*}"}
+{"id": "5497.png", "formula": "\\begin{align*} g ( x , \\lambda ) : = \\pi ( \\chi ' ( \\lambda ) \\| x \\| _ 2 ^ 2 - 1 ) . \\end{align*}"}
+{"id": "6261.png", "formula": "\\begin{align*} F \\big | _ { \\mathcal { U } _ { 0 } \\times ( - \\delta ' , 1 + \\delta ' ) } = h ' \\big | _ { \\mathcal { U } _ { 0 } \\times ( - \\delta ' , 1 + \\delta ' ) } . \\end{align*}"}
+{"id": "2732.png", "formula": "\\begin{align*} W ^ \\theta = \\{ w \\in W \\mid w \\theta = \\theta w \\} . \\end{align*}"}
+{"id": "6939.png", "formula": "\\begin{align*} H _ n ( x ) = 2 x H _ { n - 1 } ( x ) - H _ { n - 1 } ' ( x ) . \\end{align*}"}
+{"id": "3791.png", "formula": "\\begin{align*} R _ 0 & = \\{ ( s , t ; m ( e ) ) = ( t , s ; m ( e ) ) \\mid e = \\{ s , t \\} \\in E \\} \\\\ & \\mathrel { \\hphantom { = } } \\cup \\{ \\tilde \\alpha _ i v = ( \\tilde \\alpha _ i ) _ \\# ( v ) \\tilde \\alpha _ i \\mid 0 \\le i \\le 4 , v \\in V \\} \\\\ & \\mathrel { \\hphantom { = } } \\cup \\{ \\varepsilon _ i ^ 2 v = ( \\varepsilon _ i ) _ \\# ^ 2 ( v ) \\varepsilon _ i ^ 2 \\mid 1 \\le i \\le 4 , v \\in V \\} \\\\ & \\mathrel { \\hphantom { = } } \\cup \\{ \\iota v = v ^ { - 1 } \\iota \\mid v \\in V \\} . \\end{align*}"}
+{"id": "8256.png", "formula": "\\begin{align*} \\mathcal { I } _ n = \\frac { 1 } { V _ n ( \\hat { \\theta } ) } \\left ( \\binom { n } { 2 } ^ { - 1 } \\sum _ { t = 1 } ^ { \\binom { n } { 2 } } \\left ( U _ t ^ 2 - 2 U _ t \\frac { e ^ { \\hat { \\theta } } } { 1 + e ^ { \\hat { \\theta } } } + e ^ { \\hat { \\theta } } \\left ( \\frac { e ^ { \\hat { \\theta } } - 1 } { ( 1 + e ^ { \\hat { \\theta } } ) ^ 2 } \\right ) \\right ) \\right ) ^ 2 \\end{align*}"}
+{"id": "1566.png", "formula": "\\begin{align*} 0 = \\alpha ^ { Q + R } A ( \\alpha ) ^ q = \\alpha ^ { Q + R } A ( \\alpha ^ q ) = \\alpha ^ { Q + R } A \\bigl ( \\frac 1 { \\alpha } \\bigr ) = \\alpha ^ Q + \\alpha ^ { Q + R } + 1 . \\end{align*}"}
+{"id": "2842.png", "formula": "\\begin{align*} ( 1 - \\lambda ) a _ k + \\lambda a _ { \\ell } & = \\frac { 1 } { 2 \\tau } \\left ( ( \\tau - \\sigma ) ( \\rho + \\tau ) + ( \\tau + \\sigma ) ( \\rho - \\tau ) \\right ) \\\\ & = \\rho - \\sigma = b _ { \\ell } \\end{align*}"}
+{"id": "2132.png", "formula": "\\begin{align*} \\psi ( t , \\cdot ) : = e ^ { - t } \\sum _ { i = 1 } ^ { \\infty } \\frac { t ^ i } { i ! } J ^ { \\ast ( i ) } , \\end{align*}"}
+{"id": "6967.png", "formula": "\\begin{align*} L _ { m , n } ^ { I I I , ( \\alpha ) } ( x ) = { } & \\frac { x ( - 1 ) ^ { n - m - 1 } n \\det \\big ( M _ n ^ { I I I } \\big ) } { m ! ( n - m - 1 ) ! \\prod _ { 1 \\leq i < j \\leq n - 1 } \\big ( Z ^ { ( \\alpha ) } _ { j , m , n } - Z ^ { ( \\alpha ) } _ { i , m , n } \\big ) } \\\\ & + ( m + 1 ) \\binom { n - m + \\alpha } { n - m - 1 } \\binom { m - \\alpha - 1 } { m + 1 } . \\end{align*}"}
+{"id": "7748.png", "formula": "\\begin{align*} \\widetilde f ( s | \\mu ) & = \\exp \\left ( - \\mu \\ , \\psi ( s ) \\right ) \\end{align*}"}
+{"id": "1950.png", "formula": "\\begin{align*} f _ { j - 1 } \\left ( u , \\sum _ { i = 1 } ^ N \\frac { 1 } { N } \\int f _ j ( u , \\eta _ j , x ) K ( \\frac { x - X _ i } { h _ N } ) \\frac { 1 } { h _ N } d x , X \\right ) \\geq f _ { j - 1 } \\left ( u , \\frac { 1 } { N } \\sum _ { i = 1 } ^ N f _ j ( u , \\eta _ j , X _ i ) , X \\right ) . \\end{align*}"}
+{"id": "652.png", "formula": "\\begin{align*} \\nu ^ { N - n } ( \\Omega _ { c _ { 1 } \\cdots c _ { n } } ) = \\int \\nu ^ { N - n - 1 } ( \\Omega _ { c _ { 1 } \\cdots c _ { n } c } ) \\ , d \\nu ( c ) \\end{align*}"}
+{"id": "5593.png", "formula": "\\begin{align*} \\begin{aligned} & \\Psi _ 0 = W _ { a b c d } k ^ a m ^ b k ^ c m ^ d , \\Psi _ 1 = W _ { a b c d } k ^ a l ^ b k ^ c m ^ d , \\Psi _ 2 = W _ { a b c d } k ^ a m ^ b l ^ c \\overline { m } ^ d , \\\\ & \\Psi _ 3 = W _ { a b c d } l ^ a k ^ b \\overline { m } ^ c l ^ d , \\Psi _ 4 = W _ { a b c d } l ^ a \\overline { m } ^ b l ^ c \\overline { m } ^ d . \\end{aligned} \\end{align*}"}
+{"id": "1892.png", "formula": "\\begin{align*} y ( x , t ) = z ( x , t ) + \\int _ 0 ^ x l ( x , \\zeta ) z ( \\zeta , t ) d \\zeta , \\end{align*}"}
+{"id": "7291.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\frac { G ^ { k + 1 } _ m ( x ) } { G ^ { k } _ m ( x ) } = 0 . \\end{align*}"}
+{"id": "5290.png", "formula": "\\begin{align*} \\tau ^ { ( k e r F _ \\ast ) ^ \\bot } ( F ) = ( n - 2 ) \\frac { \\lambda ^ 2 } { 2 } F _ \\ast \\left ( \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\right ) . \\end{align*}"}
+{"id": "3294.png", "formula": "\\begin{align*} W ( t ) = S ( t - t _ 0 ) W _ 0 + \\int _ { t _ 0 } ^ { t } S ( t - s ) F _ { \\alpha } ( s ) { \\rm d } s \\end{align*}"}
+{"id": "1673.png", "formula": "\\begin{align*} \\varphi \\left ( \\chi \\cup { \\rm E S } _ \\lambda ( \\Phi ) \\right ) \\cap \\eta = h \\cdot \\ell ( \\Phi ( \\underline { \\delta s } ( \\mu _ { \\underline { m } } ) ) , \\chi ) \\in \\C , \\end{align*}"}
+{"id": "5635.png", "formula": "\\begin{align*} { \\psi ( x ) = \\lim \\psi _ n ( x ) } \\end{align*}"}
+{"id": "2951.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } _ n \\bigg [ \\sup _ { 0 \\le t \\le T } \\bigg | \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\overline { W } _ { j - 1 } ( s ) ( \\overline { W } _ j ( s ) - \\overrightarrow { W } ^ \\ell _ j ( s ) ) \\varphi _ j ( s ) d s \\bigg | ^ 2 \\bigg ] \\le C \\frac { \\ell } { n ^ 2 } \\int _ 0 ^ T \\sum _ { j \\in \\mathbb { Z } } ( \\varphi _ j ( t ) ) ^ 2 d t . \\end{aligned} \\end{align*}"}
+{"id": "7749.png", "formula": "\\begin{align*} \\widetilde f \\ , ^ \\prime ( s | \\mu ) & = - \\mu \\widetilde \\rho ( s ) \\widetilde f ( s | \\mu ) \\\\ \\implies \\mu \\ , \\widetilde \\rho ( s ) & = - \\frac { \\widetilde f \\ , ^ \\prime ( s | \\mu ) } { \\widetilde f ( s | \\mu ) } \\end{align*}"}
+{"id": "7039.png", "formula": "\\begin{align*} \\Omega ^ { C _ 2 } _ * \\to \\Phi M U ^ { C _ 2 } _ * \\to \\widetilde { H } _ * ( M U \\wedge B U _ + ) = \\mathbb { Z } [ b _ i , b _ i ' : i \\geq 1 ] [ u ^ { \\pm 1 } ] \\end{align*}"}
+{"id": "1689.png", "formula": "\\begin{align*} \\frac { x ^ { \\frac { k - 2 } { 2 } - m } y ^ { \\frac { k - 2 } { 2 } + m } } { 2 ^ { 2 - k } } = \\sum _ { \\frac { 2 - k } { 2 } \\leq n \\leq \\frac { k - 2 } { 2 } } \\binom { k - 2 } { n + \\frac { k - 2 } { 2 } } \\mu _ m \\left ( P _ { n + \\frac { k - 2 } { 2 } } \\right ) z ^ { n + \\frac { k - 2 } { 2 } } \\bar z ^ { \\frac { k - 2 } { 2 } - n } . \\end{align*}"}
+{"id": "3004.png", "formula": "\\begin{align*} \\begin{array} { c c c c } \\pi { ^ i } _ { a b } + \\Theta { ^ i } _ { a b } & = & 0 & \\hbox { ( a ) } \\\\ \\Theta { ^ i } _ { a k } & = & 0 & \\hbox { ( b ) } \\\\ \\Theta { ^ i } _ { j k } & = & 0 & \\hbox { ( c ) } \\\\ \\hbox { d } ^ \\theta \\pi _ i & = & \\frac { 1 } { 2 } \\| \\pi \\| ^ 2 e ^ { ( 4 ) } \\wedge \\bar { \\theta } ^ { ( r - 1 ) } _ i & \\hbox { ( d ) } \\end{array} \\end{align*}"}
+{"id": "3860.png", "formula": "\\begin{align*} T _ { n + 2 } ^ 2 = F _ { n + 1 } ^ 2 + \\sum _ { k = 3 } ^ n \\sum _ { l = 3 } ^ k \\{ 4 ( T _ l + T _ { l - 1 } ) + \\delta _ { l , 3 } - 2 \\} T _ { k - l + 2 } ^ 2 F _ { n - k + 1 } ^ 2 . \\end{align*}"}
+{"id": "6384.png", "formula": "\\begin{align*} s \\le \\lfloor \\log _ X { u } \\rfloor \\quad \\sum _ { i = 1 } ^ s \\frac 1 { p _ i } \\le \\frac { \\lfloor \\log _ X { u } \\rfloor } { X } . \\end{align*}"}
+{"id": "7914.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } \\frac { e ^ { - \\sqrt { 2 } x } } { ( - x ) } t ^ { 3 / 2 } e ^ { x ^ 2 / 2 t } u _ { f } ( t , \\sqrt { 2 } t - x ) = \\gamma ( f ) , \\end{align*}"}
+{"id": "3213.png", "formula": "\\begin{align*} t \\hat \\mu ( K _ { s + t } ) \\le C \\hat \\mu ( K _ s ) ^ { ( 1 - \\frac { 1 } { p } ) \\frac { k + 1 } { k } } = C \\hat \\mu ( K _ s ) ^ { 1 + \\delta } . \\end{align*}"}
+{"id": "8145.png", "formula": "\\begin{align*} \\phi ' ( \\xi ) = - \\frac { T } { M } \\pm \\frac { x } { 2 M } \\cosh \\frac { \\xi \\pi } { M } \\end{align*}"}
+{"id": "7178.png", "formula": "\\begin{align*} \\hat E _ t = \\left \\{ z _ { \\alpha , j k } ( x _ { j } , \\ , t ) \\right \\} , \\end{align*}"}
+{"id": "2212.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m } n + 5 ^ { 2 m } \\right ) } q ^ n & = \\delta \\sum _ { i = 1 } ^ \\infty x _ { 2 m , i } \\xi ^ { i - 1 } . \\end{align*}"}
+{"id": "5486.png", "formula": "\\begin{align*} L V ( x , 1 ) & = { \\rm s i g n } x \\cdot ( - x ^ 3 - x ) + ( \\frac { 1 } { 3 } + \\frac { 1 } { 4 } \\cos x ) ( 2 - 1 ) | x | \\\\ & \\leq - | x | + \\frac { 7 } { 1 2 } | x | = - \\frac { 5 } { 1 2 } | x | = - \\frac { 5 } { 1 2 } V ( x , 1 ) , \\\\ L V ( x , 2 ) & = 2 { \\rm s i g n } x \\times ( - \\frac { 1 } { 2 } x ) + ( \\frac { 7 } { 3 } + \\frac { 1 } { 2 } \\sin x ) ( 1 - 2 ) | x | \\\\ & \\leq - | x | - \\frac { 1 1 } { 6 } | x | = - \\frac { 1 7 } { 1 2 } V ( x , 2 ) . \\end{align*}"}
+{"id": "2743.png", "formula": "\\begin{align*} \\Big [ B _ i ^ \\sigma , \\big [ B _ i ^ \\sigma , [ B _ i ^ \\sigma , F _ j ] _ { q ^ 3 } \\big ] _ q \\Big ] _ { q ^ { - 1 } } = \\Big [ B _ i ^ \\sigma , \\big [ F _ i , [ F _ i , F _ j ] _ { q ^ 3 } \\big ] _ q \\Big ] _ { q ^ { - 1 } } + \\Big [ B _ i ^ \\sigma , \\big [ K _ i E _ i , [ F _ i , F _ j ] _ { q ^ 3 } \\big ] _ q \\Big ] _ { q ^ { - 1 } } . \\end{align*}"}
+{"id": "1940.png", "formula": "\\begin{align*} \\vartheta = \\min _ { u \\in U } \\ ; \\varrho [ u , X ] , \\end{align*}"}
+{"id": "1806.png", "formula": "\\begin{align*} \\Q ( v ) = \\int _ { 0 } ^ { T } \\big ( v ( t ) \\cdot P ( t ) v ( t ) + 2 \\dot { v } ( t ) \\cdot Q ( t ) v ( t ) + \\dot { v } ( t ) \\cdot R ( t ) \\dot { v } ( t ) \\big ) d t , \\end{align*}"}
+{"id": "536.png", "formula": "\\begin{align*} C _ F ( H ) & : = \\{ f \\in F \\mid \\forall \\ , h \\in H , \\ , f h = h f \\} \\\\ N _ F ( H ) & : = \\{ f \\in F \\mid f H = H f \\} \\end{align*}"}
+{"id": "3550.png", "formula": "\\begin{align*} L ^ + : = \\{ v \\in L : \\mathfrak { t } ^ + v = 0 \\} = S ( \\mathfrak { g } , \\mathfrak { t } ) w _ 0 \\end{align*}"}
+{"id": "1318.png", "formula": "\\begin{align*} \\operatorname { T r a } ( S ^ 2 _ { f , \\tau } ) & = \\sum _ { j = 1 } ^ n f _ j ( S _ { f , \\tau } \\tau _ j ) = \\sum _ { j = 1 } ^ n f _ j \\left ( \\sum _ { k = 1 } ^ n f _ k ( \\tau _ j ) \\tau _ k \\right ) \\\\ & = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) . \\end{align*}"}
+{"id": "8293.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\nu _ E = x _ 1 ( y _ 2 \\ ! - \\ ! y _ 3 ) + x _ 2 ( y _ 3 \\ ! - \\ ! y _ 1 ) + x _ 3 ( y _ 1 \\ ! - \\ ! y _ 2 ) \\\\ \\nu _ 1 = x _ 2 ( y _ 3 \\ ! - \\ ! y _ E ) + x _ 3 ( y _ E \\ ! - \\ ! y _ 2 ) + x _ E ( y _ 2 \\ ! - \\ ! y _ 3 ) \\\\ \\nu _ 2 = x _ 3 ( y _ 1 \\ ! - \\ ! y _ E ) + x _ 1 ( y _ E \\ ! - \\ ! y _ 3 ) + x _ E ( y _ 3 \\ ! - \\ ! y _ 1 ) \\\\ \\nu _ 3 = x _ 1 ( y _ 2 \\ ! - \\ ! y _ E ) + x _ 2 ( y _ E \\ ! - \\ ! y _ 1 ) + x _ E ( y _ 1 \\ ! - \\ ! y _ 2 ) \\end{array} \\right . \\end{align*}"}
+{"id": "3744.png", "formula": "\\begin{align*} \\tilde { a } _ 2 ( f ) : = a _ 0 ( f ) ( ( 0 : 0 : 0 : 1 ) ) \\end{align*}"}
+{"id": "4616.png", "formula": "\\begin{align*} \\prod _ { i = m } ^ { m + k - 1 } \\frac { a _ i } { b _ i } \\leq \\prod _ { i = m + k } ^ { n } \\frac { b _ i } { a _ i } < 1 \\end{align*}"}
+{"id": "7689.png", "formula": "\\begin{align*} \\delta ( g f ^ { - \\textbf { p } } ) = \\delta ( g ) f ^ { - \\textbf { p } } + \\sum _ { 1 \\leq k \\leq d } g ( - p _ { k } + \\lambda _ { k } ) \\delta ( f _ { k } ) f _ { k } ^ { - 1 } f ^ { - \\textbf { p } } . \\end{align*}"}
+{"id": "3325.png", "formula": "\\begin{align*} \\frac { \\dd q ^ i } { \\dd t } - A _ i = 0 , \\frac { \\dd p _ i } { \\dd t } - B _ i = 0 , \\frac { \\dd \\kappa } { \\dd t } - C = 0 , i = 1 , \\dots , n , \\end{align*}"}
+{"id": "1592.png", "formula": "\\begin{align*} \\xi : = \\xi _ { \\frac { 1 } { 2 } } ^ { \\mu , \\nu } - c \\delta _ { x _ 0 } + c \\delta _ { x _ 1 } + c \\delta _ { x _ { k - 1 } } - c \\delta _ { x _ k } . \\end{align*}"}
+{"id": "5543.png", "formula": "\\begin{align*} V ( x , k ) : = \\frac { \\pi ^ { \\frac { 1 } { 2 } - k } } { 2 \\pi i } \\int _ { \\lambda - i \\infty } ^ { \\lambda + i \\infty } \\frac { \\Gamma ( s ) \\Gamma \\left ( \\frac { k } { 2 } - s \\right ) } { \\Gamma \\left ( \\frac { 1 - k } { 2 } + s \\right ) \\zeta ( 1 - k + 2 s ) } \\left ( \\frac { x } { \\pi ^ 2 } \\right ) ^ { - s } { \\rm d } s . \\end{align*}"}
+{"id": "6663.png", "formula": "\\begin{align*} \\| { \\bf x } ^ k \\| _ R = \\left \\| \\left [ \\| { \\bf x } ^ k _ { ( 1 ) } \\| _ R , \\ , \\| { \\bf x } ^ k _ { ( 2 ) } \\| _ R , \\cdots , \\| { \\bf x } ^ k _ { ( d ) } \\| _ R \\right ] \\right \\| _ 2 \\end{align*}"}
+{"id": "8197.png", "formula": "\\begin{align*} u ( t , x ) = u _ 0 ( x ) - \\int _ 0 ^ t k ( t , s ) \\left ( - L _ { ( \\sigma , w ) } - c \\right ) ^ \\gamma u ( s , x ) d s . \\end{align*}"}
+{"id": "4419.png", "formula": "\\begin{align*} J ( a _ 1 , a _ 2 ) = \\left ( \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 } , \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 - 1 } \\right ] . \\end{align*}"}
+{"id": "1895.png", "formula": "\\begin{align*} d Z ( t ) = - \\big [ ( c + \\pi ^ 2 ) Z ( t ) + u _ 0 ( t ) + w ( t ) \\big ] d t + \\sigma Z ( t ) d B ( t ) . \\end{align*}"}
+{"id": "7185.png", "formula": "\\begin{align*} A _ { \\beta \\alpha } \\equiv \\left [ \\frac { \\partial U _ { \\beta } } { \\partial z _ \\alpha } , \\ , \\ , \\ , \\ , \\frac { \\partial U _ { \\beta } } { \\partial z _ { \\alpha , j } } , \\ , \\ , \\ , \\ , \\left ( \\frac { \\partial U _ { \\beta } } { \\partial z _ { \\alpha , k } } v _ j + \\frac { \\partial \\Phi ^ \\beta _ j } { \\partial z _ { \\alpha , k } } \\right ) \\right ] \\ , \\ , \\ , \\ , \\ , ( j , \\ , k = 1 , 2 , 3 ) , \\end{align*}"}
+{"id": "6973.png", "formula": "\\begin{gather*} R _ n ( x ; \\beta ) = \\sum _ { j = 0 } ^ n \\frac { \\beta ^ { ( j ) } ( - x ) ^ { j } } { j ! } . \\end{gather*}"}
+{"id": "5294.png", "formula": "\\begin{align*} g ( \\mathcal { H } \\nabla _ U V , Y ) = - \\frac { Y } { 2 } f _ 1 ^ 2 g _ 2 ( \\sigma _ \\ast U , \\sigma _ \\ast V ) . \\end{align*}"}
+{"id": "6181.png", "formula": "\\begin{align*} C _ p = \\begin{pmatrix} S _ 1 \\\\ & S _ 2 \\\\ & & \\ddots \\\\ & & & S _ p \\end{pmatrix} . \\end{align*}"}
+{"id": "3611.png", "formula": "\\begin{align*} t + 1 & = \\frac { R _ * ^ { \\frac 3 2 } } { L _ * \\beta _ 0 ^ { \\frac 3 2 } } \\Big [ \\frac 7 { 2 } \\Big ( \\arctan \\Big ( \\frac { 7 } { 2 \\sqrt { \\alpha _ 0 } } \\Big ) - \\arctan \\Big ( \\frac { 7 - 2 \\beta _ 0 R } { 2 \\sqrt { g ( R ) } } \\Big ) \\Big ) + \\left ( \\sqrt { \\alpha _ 0 } - \\sqrt { g ( R ) } \\right ) \\Big ] . \\end{align*}"}
+{"id": "8690.png", "formula": "\\begin{align*} f _ 2 ( x ) = x ^ 2 + a , f _ k ( x ) = x ^ k k \\ne 2 . \\end{align*}"}
+{"id": "2290.png", "formula": "\\begin{align*} p ( x ) = c _ 0 + c _ 1 x + c _ 2 x ^ 2 + c _ 3 x ^ 3 + c _ 4 x ^ 4 , \\end{align*}"}
+{"id": "9092.png", "formula": "\\begin{align*} \\mathfrak { p } _ s & = [ \\mathfrak { p } _ { s - 1 } , \\mathfrak { n } ] + [ J \\mathfrak { p } _ { s - 1 } , \\mathfrak { n } ] \\\\ & \\subseteq [ \\mathfrak { d } ^ { j _ 0 - s } , \\mathfrak { n } ] + [ J \\mathfrak { d } ^ { j _ 0 - s } , \\mathfrak { n } ] \\\\ & \\subseteq \\mathfrak { d } ^ { j _ 0 - s - 1 } . \\end{align*}"}
+{"id": "5045.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d a _ i \\epsilon _ i \\le \\sum _ { i = 1 } ^ d b _ i \\epsilon _ i . \\end{align*}"}
+{"id": "4864.png", "formula": "\\begin{align*} f ( s _ j ) & = u ^ j ( s _ j ) \\int _ { B _ { n - j } ( s _ j ) } u ^ { j + 1 } ( s _ { j + 1 } ) \\ . u ^ { n } ( s _ { n } ) d \\L ^ { n - j } , \\\\ g ( s _ j ) & = \\int _ { A _ { j - 1 } ( s _ j ) } u ^ { j - 1 } ( s _ { j - 1 } ) \\dots u ^ 1 ( s _ 1 ) d \\L ^ { j - 1 } , \\end{align*}"}
+{"id": "107.png", "formula": "\\begin{align*} ( x y ) \\circ z - x ( y \\circ z ) = ( x z ) \\circ y - x ( z \\circ y ) . \\end{align*}"}
+{"id": "332.png", "formula": "\\begin{align*} R ( l ) = \\frac { 1 } { 2 l } \\sum _ { \\substack { \\alpha \\leq Y \\\\ ( \\alpha , 2 l ) = 1 } } \\frac { \\mu ( \\alpha ) } { \\alpha ^ 2 } \\sum _ { \\substack { k = - \\infty \\\\ k \\neq 0 } } ^ { \\infty } \\frac { ( - 1 ) ^ k } { 2 \\pi i } \\int _ { ( c ) } \\sum _ { \\substack { n = 1 \\\\ ( n , 2 \\alpha ) = 1 } } ^ \\infty \\frac { A ( n , 1 ) } { n ^ { \\frac { 3 } { 2 } + w } } G _ { 4 k } ( l n ) \\phi ( \\frac { k X } { 2 \\alpha ^ 2 l } , w ) \\dd w \\end{align*}"}
+{"id": "1187.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\tilde { K } } r ^ k | \\nabla _ { g _ 0 } ^ k ( [ H _ { 1 , i } ^ { - 1 } \\circ \\Phi _ { \\overline { V } _ i } \\circ H _ { 0 , i } ] ^ * g _ i - g _ 0 ) | _ { g _ 0 } = O _ { \\tilde { K } , \\tilde { \\epsilon } , i } ( r ^ { \\max \\{ \\lambda , - 2 \\} + 2 \\tilde { \\epsilon } } ) \\ ; \\ , \\ ; \\ , r \\to \\infty , \\end{align*}"}
+{"id": "9230.png", "formula": "\\begin{align*} v _ { \\Gamma } ( x , h ) = C _ { \\Gamma } ^ { - 1 } \\int _ { 0 } ^ { \\ell _ { \\Gamma } ( x , h ) } \\zeta ( s / \\tau ) e ^ { - s ^ { 2 } / 2 h } d s , \\end{align*}"}
+{"id": "8668.png", "formula": "\\begin{align*} \\psi ^ \\pm ( u ) \\psi ^ \\pm ( v ) + \\psi ^ \\pm ( v ) \\psi ^ \\pm ( u ) & = 0 , \\\\ \\psi ^ + ( u ) \\psi ^ - ( v ) + \\psi ^ - ( v ) \\psi ^ + ( u ) & = \\delta ( u , v ) , \\end{align*}"}
+{"id": "74.png", "formula": "\\begin{align*} P _ n = [ n ] ( Q ) + P _ 0 ; \\end{align*}"}
+{"id": "6708.png", "formula": "\\begin{align*} \\sum _ { x \\leq n \\leq x + H } \\mu ( n ) = O \\left ( \\frac { H } { \\log ^ { C } x } \\right ) . \\end{align*}"}
+{"id": "4448.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 2 4 } = \\frac { 1 } { 4 } + \\frac { 1 } { 8 } = \\frac { 3 } { 8 } \\end{align*}"}
+{"id": "1732.png", "formula": "\\begin{align*} \\delta : P : = \\left \\{ \\left ( \\begin{array} { c c } r & x \\\\ & r ^ { - 1 } \\end{array} \\right ) , r \\in \\R _ { > 0 } , \\ ; x \\in \\C \\right \\} \\longrightarrow \\R , \\delta \\left ( \\begin{array} { c c } r & x \\\\ & r ^ { - 1 } \\end{array} \\right ) = r ^ 4 . \\end{align*}"}
+{"id": "947.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( M , C ) \\ , = \\ , \\textrm { g r a d e } _ R ( \\textrm { E x t } _ { R } ^ n ( M ' , \\ , C ) ) . \\end{align*}"}
+{"id": "5585.png", "formula": "\\begin{align*} 0 = \\nabla _ 0 S _ { i j } = \\kappa _ i v _ j + \\kappa _ j v _ i , 0 = \\nabla _ 0 S _ { 0 1 } = - \\kappa ^ j v _ j . \\end{align*}"}
+{"id": "8984.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\big ( \\| \\nabla & u ^ { ( m ) } \\| ^ 2 _ { L ^ 2 ( B ) } \\big ) = \\int _ B \\nabla u ^ { ( m ) } \\nabla u ^ { ( m ) } _ t \\ ; d z = ( u ^ { ( m ) } _ r , u ^ { ( m ) } _ t ) _ { L ^ 2 ( S ^ 1 ) } \\\\ & = - ( u ^ { ( m ) } _ r , ( \\varepsilon + d \\pi _ N ( v ) ) u ^ { ( m ) } _ r ) _ { L ^ 2 ( S ^ 1 ) } \\\\ & = - \\varepsilon \\| u ^ { ( m ) } _ r \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } - \\| d \\pi _ N ( v ) u ^ { ( m ) } _ r \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } \\\\ & \\le - \\frac 1 2 \\| u ^ { ( m ) } _ t \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } \\le 0 , \\end{align*}"}
+{"id": "4341.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\mathbf { C g } ^ { \\mathbf { A } } ( \\vec { 0 } , \\varphi _ { \\mathbf { A } } ( \\mu _ { \\mathbf { A } } ( \\vec { e } ) ) ) = \\mathbf { C g } ^ { \\mathbf { A } } ( \\vec { 0 } , \\vec { e } ) \\\\ \\mathbf { C g } ^ { \\mathbf { A } } ( \\vec { 1 } , \\varphi _ { \\mathbf { A } } ( \\mu _ { \\mathbf { A } } ( \\vec { e } ) ) ) = \\mathbf { C g } ^ { \\mathbf { A } } ( \\vec { 1 } , \\vec { e } ) , \\end{array} \\end{align*}"}
+{"id": "1534.png", "formula": "\\begin{align*} \\zeta ^ \\alpha _ I \\circ \\psi = \\dd _ I \\psi ^ \\alpha \\end{align*}"}
+{"id": "7585.png", "formula": "\\begin{align*} b ( u , u , v ) = - \\frac { 1 } { \\delta + 1 } b ( u , v , u ) , \\end{align*}"}
+{"id": "927.png", "formula": "\\begin{align*} \\tau _ { E , \\alpha } ( h ) : = \\int _ { - 1 } ^ { 0 } \\chi _ E ( Y _ \\alpha ^ x ( t ) ) \\mathrm { d } t . \\end{align*}"}
+{"id": "1514.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d x } \\left ( \\frac 1 { \\pi x ^ { r - 1 } } \\frac { \\C _ r ' ( x ) } { \\C _ r ( x ) } \\right ) & = - \\pi \\sec ^ 2 ( \\pi x ) \\\\ & = - \\pi ( \\tan ^ 2 ( \\pi x ) + 1 ) \\\\ & = - \\pi \\left ( \\left ( - \\frac 1 { \\pi x ^ { r - 1 } } \\frac { \\C _ r ' ( x ) } { \\C _ r ( x ) } \\right ) ^ 2 + 1 \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "2929.png", "formula": "\\begin{align*} \\partial _ t u = \\frac { 1 } { 2 } g ^ \\prime ( 0 ) \\partial _ x ^ 2 u - \\frac { 1 } { 2 } g ^ { \\prime \\prime } ( 0 ) \\partial _ x u ^ 2 + \\sqrt { g ^ \\prime ( 0 ) \\rho } \\partial _ x \\dot { W } ( t , x ) . \\end{align*}"}
+{"id": "8852.png", "formula": "\\begin{align*} d ( n , p , 0 ) = \\delta ( n , p , 0 ) \\mathrm { a n d } d ( n , 0 , q ) = \\delta ( n , 0 , q ) . \\end{align*}"}
+{"id": "8572.png", "formula": "\\begin{align*} \\Omega : = \\{ 0 \\neq h ( x ) : h ( x ) \\in \\mathbb { F } _ 2 [ x ] \\mbox { w i t h } \\deg ( h ( x ) ) < n \\} . \\end{align*}"}
+{"id": "804.png", "formula": "\\begin{align*} U _ n ( \\epsilon ) = \\left \\{ \\xi \\in \\partial \\Gamma : \\left | \\frac { \\log \\| \\rho ( \\xi _ n ) \\| } { n } - \\Lambda \\right | > \\epsilon \\right \\} \\end{align*}"}
+{"id": "5782.png", "formula": "\\begin{align*} u '' + L u + g u ' = 0 \\end{align*}"}
+{"id": "4601.png", "formula": "\\begin{align*} \\frac { 1 } { q } - \\sum _ { k = 1 } ^ n \\frac { 1 } { a _ k } + \\frac { 1 } { q \\prod _ { i = 1 } ^ n a _ k } . \\end{align*}"}
+{"id": "2815.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ { n + 1 } } \\prod _ { i = 1 } ^ { n + 1 } x _ { \\sigma ( i ) } ^ { \\log v _ i } = \\sum _ { \\sigma \\in S _ { n + 1 } } \\prod _ { i = 1 } ^ { n + 1 } x _ { \\sigma ( i ) } ^ { b _ i } < \\sum _ { \\sigma \\in S _ { n + 1 } } \\prod _ { i = 1 } ^ { n + 1 } x _ { \\sigma ( i ) } ^ { a _ i } = \\sum _ { \\sigma \\in S _ { n + 1 } } \\prod _ { i = 1 } ^ { n + 1 } x _ { \\sigma ( i ) } ^ { \\log u _ i } . \\end{align*}"}
+{"id": "7239.png", "formula": "\\begin{align*} \\int _ { \\partial B _ n } f ( x ) ^ { - \\frac { 2 } { n - 1 } } \\ , d \\mu _ { \\partial B _ n } ( x ) & \\geq \\mu _ { \\partial B _ n } ( \\partial B _ n ) ^ { \\frac { n + 1 } { n - 1 } } = \\int _ { \\partial B _ n } f _ { \\rm u n i f } ( x ) ^ { - \\frac { 2 } { n - 1 } } \\ , d \\mu _ { \\partial B _ n } ( x ) . \\end{align*}"}
+{"id": "5375.png", "formula": "\\begin{align*} | | s _ 1 - u _ 1 | | ^ 2 = c _ 1 , \\\\ | | s _ 2 - u _ 2 | | ^ 2 = c _ 2 , \\\\ | | u _ 2 - u _ 1 | | ^ 2 = c _ 3 . \\end{align*}"}
+{"id": "7560.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 3 ) = \\dfrac { F _ 3 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "5788.png", "formula": "\\begin{align*} C _ p = \\begin{pmatrix} S _ 1 \\cr & S _ 2 \\cr & & \\ddots \\cr & & & S _ p \\end{pmatrix} . \\end{align*}"}
+{"id": "3542.png", "formula": "\\begin{align*} - s _ 1 \\alpha _ 1 - \\cdots - s _ m \\alpha _ m + \\epsilon _ i = \\epsilon _ j \\end{align*}"}
+{"id": "6948.png", "formula": "\\begin{gather*} L _ { m , n } ^ { I , ( \\alpha ) } ( x ) = \\frac { \\alpha + n } { x ^ { \\alpha } } \\int _ 0 ^ x t ^ { \\alpha - 1 } L _ m ^ { ( \\alpha - 1 ) } ( - t ) L _ { n - m } ^ { ( \\alpha - 1 ) } ( t ) \\ , { \\rm d } t . \\end{gather*}"}
+{"id": "4348.png", "formula": "\\begin{align*} x = \\sum _ { i _ { 1 } , i _ { 2 } \\ldots , i _ { j - 1 } = 1 } ^ { n } \\prod _ { k = 1 } ^ { j } a _ { i _ { k - 1 } , i _ { k } } , \\end{align*}"}
+{"id": "6196.png", "formula": "\\begin{align*} \\overline A _ p \\overline x _ l ^ { ( k ) } = C _ p ( \\lambda _ k x _ l ^ { ( k ) } + x _ { l + 1 } ^ { ( k ) } ) = \\lambda _ k \\overline x _ l ^ { ( k ) } + \\overline x _ { l + 1 } ^ { ( k ) } . \\end{align*}"}
+{"id": "7698.png", "formula": "\\begin{align*} \\beta : \\mathbb { C } [ s _ { 1 } , \\dots , s _ { d } ] \\to \\frac { \\mathbb { C } [ s _ { 1 } , \\dots , s _ { d } ] } { ( \\{ s _ { t } - s _ { j } \\mid t , j \\in S _ { k } , 1 \\leq k \\leq r \\} ) } = \\mathbb { C } [ s _ { 1 } , \\dots , s _ { r } ] . \\end{align*}"}
+{"id": "344.png", "formula": "\\begin{align*} A ( p ^ { 2 h } , 1 ) = C ( p ^ { h } , 1 ) + A ( 1 , p ) C ( p ^ { h - 1 } , 1 ) . \\end{align*}"}
+{"id": "8387.png", "formula": "\\begin{align*} n w ( X , \\mathcal { B } ) = \\aleph _ 0 + \\min \\{ | \\varLambda | : \\varLambda \\mathcal { B } X \\} . \\end{align*}"}
+{"id": "6103.png", "formula": "\\begin{align*} \\lambda ( p ) = \\frac { 1 } { 2 } \\left [ p \\ln ( r ) - p \\ln ( p ) - 2 ( 1 - p ) \\ln ( 1 - p ) - p \\right ] + \\sum _ { k = 2 } ^ \\infty d _ k p ^ k \\end{align*}"}
+{"id": "991.png", "formula": "\\begin{align*} M _ b ( G , w ) & = \\mathcal M ( M _ b ( G , w ) ) \\\\ & = C _ { B S E } \\big ( \\Delta ( M _ b ( G , w ) ) \\big ) \\\\ & = C _ b ( \\Delta ( M _ b ( G , w ) ) ) . \\end{align*}"}
+{"id": "1141.png", "formula": "\\begin{align*} T _ p \\ , f _ { \\ell } - \\lambda ( T _ p ) \\ , f _ { \\ell } = T _ { \\ell } \\ , f _ p - \\lambda ( T _ { \\ell } ) \\ , f _ p \\end{align*}"}
+{"id": "1457.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\binom { x _ i } { 3 } = \\sum _ { j = 1 } ^ m \\binom { y _ j } { 3 } = \\frac { k ( k - 1 ) ( k - 2 ) ( m - 2 ) } { 6 ( m + 1 ) ( m ^ 2 - 2 ) } ; \\end{align*}"}
+{"id": "6636.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v _ { a ( t ' , x ' ) } = \\nabla \\cdot a ( t ' , x ' ) \\nabla v _ { a ( t ' , x ' ) } + \\xi \\\\ v _ { a ( t ' , x ' ) } | _ { t \\leq 0 } = 0 \\end{cases} \\end{align*}"}
+{"id": "7777.png", "formula": "\\begin{align*} m _ { \\alpha , \\lambda } ( x | \\mu ) & = \\frac { \\lambda ^ \\mu } { \\Gamma ( \\mu ) } \\int _ 0 ^ \\infty f _ \\alpha ( x | y ) \\ , y ^ { \\mu - 1 } \\ , e ^ { - \\lambda y } \\ , d y \\\\ r _ { \\alpha , \\lambda } ( x ) & = x \\int _ 0 ^ \\infty f _ \\alpha ( x | y ) \\ , y ^ { - 1 } \\ , e ^ { - \\lambda y } \\ , d y \\end{align*}"}
+{"id": "6176.png", "formula": "\\begin{align*} \\begin{cases} \\Psi '' - \\Delta \\Psi + \\overline A _ 1 ^ T \\Psi = 0 & \\hbox { i n } ( 0 , T ) \\times \\Omega , \\\\ \\Psi = 0 & \\hbox { o n } ( 0 , T ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu \\Psi + \\overline B _ 1 ^ T \\Psi = 0 & \\hbox { o n } ( 0 , T ) \\times \\Gamma _ 1 \\end{cases} \\end{align*}"}
+{"id": "2794.png", "formula": "\\begin{align*} [ f ] _ { G } ( x _ 1 , x _ 2 , \\ldots , x _ n ) & = \\frac { 1 } { | G | } \\sum _ { \\sigma \\in G } f \\left ( x _ { \\sigma ( 1 ) } , x _ { \\sigma ( 2 ) , } \\ldots , x _ { \\sigma ( n ) } \\right ) . \\end{align*}"}
+{"id": "4813.png", "formula": "\\begin{align*} \\frac { 1 } { \\binom { N } { S } } \\sum _ { j \\notin A _ \\epsilon } \\Pr _ { c \\sim C ^ \\perp } \\big [ | c | = j \\big ] K _ S ( j ) ^ 2 & \\leq \\binom { N } { S } \\Pr \\big [ | y | \\notin A _ \\epsilon \\big ] . \\end{align*}"}
+{"id": "4477.png", "formula": "\\begin{align*} u _ { k + 1 } = u _ k ( u _ k + 1 ) . \\end{align*}"}
+{"id": "1408.png", "formula": "\\begin{align*} \\psi = S _ 1 + S _ 3 \\boxtimes S _ 3 + S _ 3 \\boxtimes S _ 3 . \\end{align*}"}
+{"id": "6207.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ d \\beta _ r u _ r ( T ) = \\sum _ { r = 1 } ^ d \\beta _ r u _ r ' ( T ) = 0 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "2571.png", "formula": "\\begin{align*} \\mathfrak { g } _ 0 & = \\{ x \\in \\mathfrak { g } \\mid \\operatorname { a d } ( \\eta ) x = 0 \\ \\ \\ \\eta \\in \\mathfrak { t } \\} , \\\\ \\mathfrak { g } _ \\alpha & = \\{ x \\in \\mathfrak { g } \\mid \\operatorname { a d } ( \\eta ) ^ 2 x = - \\langle \\alpha , \\eta \\rangle ^ 2 x \\ \\ \\ \\eta \\in \\mathfrak { t } \\} . \\end{align*}"}
+{"id": "2461.png", "formula": "\\begin{align*} \\eta ^ { k + 1 } | G _ k ( s ) | = \\eta \\Big | \\sum _ { n = 1 } ^ { \\infty } \\frac { \\Lambda _ { \\pi } ( n ) + \\Lambda _ { \\pi \\otimes \\chi } ( n ) n ^ { \\beta _ { \\chi } - 1 } } { n ^ { 1 + \\eta + i \\tau } } \\frac { ( \\eta \\log n ) ^ k } { k ! } \\Big | . \\end{align*}"}
+{"id": "4451.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 6 0 } = \\frac { 1 } { 4 } + \\frac { 1 } { 1 0 } = \\frac { 7 } { 2 0 } \\end{align*}"}
+{"id": "3815.png", "formula": "\\begin{align*} \\sigma ( H ) = \\sigma ( H _ 1 ) + \\sigma ( H _ 2 ) + \\sigma ( H _ 3 ) , \\end{align*}"}
+{"id": "77.png", "formula": "\\begin{align*} \\langle \\eta \\rangle _ j = T r _ { F / K } ( \\eta \\cdot \\lambda _ j ^ { \\prime } ) . \\end{align*}"}
+{"id": "1671.png", "formula": "\\begin{align*} \\delta _ \\tau \\bar \\varphi _ \\Sigma ( ( f \\otimes \\mu ) \\otimes \\Phi ) ( z , t ) = f ( z , t ) \\cdot \\left ( \\frac { \\partial F } { \\partial x } ( 0 , 0 ) + \\frac { \\partial F } { \\partial y } ( 0 , 0 ) \\right ) = f ( z , t ) \\cdot \\left ( \\Phi ( \\underline { \\delta s } ( z ^ { - 1 } \\mu ) ) ( z , t ) \\right ) = \\varphi _ \\Sigma ( ( f \\otimes \\mu ) \\otimes \\Phi ) ( z , t ) . \\end{align*}"}
+{"id": "7393.png", "formula": "\\begin{align*} \\zeta _ \\gamma \\left ( \\left ( \\begin{matrix} t & 0 \\cr 0 & 1 \\end{matrix} \\right ) \\right ) : = \\zeta _ { \\gamma ' } \\left ( \\left ( \\begin{matrix} t & 0 \\cr 0 & t ^ { - 1 } \\end{matrix} \\right ) \\right ) , \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
+{"id": "5720.png", "formula": "\\begin{align*} d s ^ { 2 } = \\eta _ { a b } \\theta ^ { a } \\otimes \\theta ^ { b } . \\end{align*}"}
+{"id": "4347.png", "formula": "\\begin{align*} s _ { k } ( \\Lambda ) ^ { m } \\leq n ^ { m - 1 } s _ { k m } ( \\Lambda ) , ~ ~ k , m , = 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "1849.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\varrho ( t , \\boldsymbol { q } ) = & t ^ { \\frac { 2 } { 3 } } f ( t ) g ( \\boldsymbol { q } ) , \\\\ \\varrho _ { 0 } ( t , \\boldsymbol { q } ) = & f _ { 0 } ( t ) g ( \\boldsymbol { q } ) t ^ { \\frac { 2 } { 3 } } + \\frac { 2 } { 3 } t ^ { - \\frac { 1 } { 3 } } f ( t ) g ( \\boldsymbol { q } ) , \\\\ \\varrho _ i ( t , \\boldsymbol { q } ) = & t ^ { \\frac { 2 } { 3 } } f ( t ) g _ i ( \\boldsymbol { q } ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8914.png", "formula": "\\begin{align*} \\sum \\limits _ { d = 0 } ^ { n - 1 } \\frac { \\gamma _ { n - 1 - d } ^ { ( \\nu , n ) } } { ( n - 1 ) ! } \\left ( n - 1 - d \\right ) ! t ^ { d } \\end{align*}"}
+{"id": "6590.png", "formula": "\\begin{align*} [ M _ - , \\sigma ( g _ 1 , \\dots , g _ n ) , M _ + ] [ M _ + , \\tau ( h _ 1 , \\dots , h _ n ) , N _ + ] = [ M _ - , ( \\sigma \\circ \\tau ) ( g _ 1 h _ 1 , \\dots , g _ n h _ n ) , N _ + ] \\end{align*}"}
+{"id": "523.png", "formula": "\\begin{align*} \\beta _ { \\rm m a x } = 1 , \\ , \\alpha = 1 0 ^ { - 5 } , \\ , \\eta _ 2 = 0 . 1 , \\ , \\tau _ { \\rm m i n } = { 1 0 ^ { - 3 } } / { [ 2 ( \\alpha \\ ! + \\ ! \\delta ) \\ ! + \\ ! L _ { \\ ! f } ] } , \\ , \\tau _ { \\rm m a x } = 1 0 ^ { 6 } , \\ \\tau _ { 0 , 0 } = { 1 0 } / { \\| \\widetilde { A } \\| } . \\end{align*}"}
+{"id": "1735.png", "formula": "\\begin{align*} \\rho ^ \\vee ( \\mu ) \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) = \\mu \\left ( \\left | \\begin{array} { c c } X & Y \\\\ c & d \\end{array} \\right | ^ { \\underline { k } - 2 } \\right ) \\cdot ( a d - b c ) ^ { \\frac { 2 - \\underline { k } } { 2 } } . \\end{align*}"}
+{"id": "1012.png", "formula": "\\begin{align*} \\lVert \\varphi \\rVert _ { \\infty , \\alpha , \\beta } = \\sup _ { x \\in \\R } | x | ^ \\alpha | \\mathrm { D } ^ \\beta \\varphi ( x ) | \\end{align*}"}
+{"id": "1466.png", "formula": "\\begin{align*} N = 3 \\times \\frac { k ( k - 1 ) ( k - 2 ) ( m - 2 ) } { 3 ( m + 1 ) ( m ^ 2 - 2 ) } + 2 \\times \\frac { k ( k - 1 ) ( k - 2 ) ( m - 1 ) } { ( m + 1 ) ( m ^ 2 - 2 ) } . \\end{align*}"}
+{"id": "2053.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial } { \\partial t } W ( t , z ) J = z W ( t , z ) H ( t ) , t \\in [ 0 , L ) , \\\\ [ 1 e x ] W ( 0 , z ) = I . \\end{cases} \\end{align*}"}
+{"id": "5725.png", "formula": "\\begin{align*} \\Omega _ { B } ^ { A } \\theta ^ { B A ^ { \\prime } } + \\Omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\theta ^ { A B ^ { \\prime } } & = 0 , \\\\ D \\Omega _ { B } ^ { A } & = D \\Omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } = 0 , \\end{align*}"}
+{"id": "3922.png", "formula": "\\begin{align*} v _ { \\mathbf { u } } & : \\overline { \\Omega ^ * } \\rightarrow \\mathbf { R } \\\\ v _ \\mathbf { u } ( y ) & : = \\sup \\{ g ^ * ( x _ i , y , u _ i ) ; i = 0 , \\dots , N \\} . \\end{align*}"}
+{"id": "5884.png", "formula": "\\begin{align*} ( D ^ T ) = \\hbox { S p a n } \\{ e _ 1 , \\cdots , e _ p \\} = ( C _ p ) . \\end{align*}"}
+{"id": "1848.png", "formula": "\\begin{align*} \\varrho ( t , \\boldsymbol { q } ) = & \\Bigl ( \\frac { 1 } { 2 } - \\frac { 5 } { 2 \\sqrt { 3 6 \\lambda \\tilde { \\kappa } + 2 5 } } \\Bigr ) t ^ { \\frac { 2 } { 3 } - \\frac { 5 + \\sqrt { 2 5 + 3 6 \\lambda \\tilde { \\kappa } } } { 6 } } g ( \\boldsymbol { q } ) + \\Bigl ( \\frac { 5 } { 2 \\sqrt { 3 6 \\lambda \\tilde { \\kappa } + 2 5 } } + \\frac { 1 } { 2 } \\Bigr ) t ^ { \\frac { 2 } { 3 } - \\frac { 5 - \\sqrt { 2 5 + 3 6 \\lambda \\tilde { \\kappa } } } { 6 } } g ( \\boldsymbol { q } ) , \\end{align*}"}
+{"id": "8952.png", "formula": "\\begin{align*} P _ { k } ^ { ( a , b ) } ( x ) = 2 ^ { - k } \\sum _ { j = 0 } ^ { k } \\binom { k + a } { j } \\binom { k + b } { k - j } ( x + 1 ) ^ { j } ( x - 1 ) ^ { k - j } , a , b > - 1 . \\end{align*}"}
+{"id": "3291.png", "formula": "\\begin{align*} v _ 0 = \\alpha \\Theta ( t _ 0 , \\cdot ) + w _ 0 \\ell _ 0 = \\ell ^ 0 _ { w } \\omega _ 0 = \\dfrac { \\alpha } { 2 \\pi } \\left ( 1 - \\exp ( - 1 / 4 ( 1 + t _ 0 ) ) \\right ) + \\omega ^ { 0 } _ { w } \\end{align*}"}
+{"id": "2777.png", "formula": "\\begin{align*} x _ i = \\ ! \\left \\{ \\ ! \\begin{aligned} & x ' _ i , ~ ~ ~ ~ ~ i \\in [ 1 , d _ 1 - 1 ] \\backslash \\{ \\lambda _ 1 \\} , \\\\ & x ' _ { i - 1 } , ~ ~ i \\in [ d _ 1 + 1 , d _ 2 ] \\backslash \\{ \\lambda _ 2 \\} , \\\\ & x ' _ i , ~ ~ ~ ~ ~ i \\in [ d _ 2 + 1 , n ] . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6949.png", "formula": "\\begin{align*} L _ { m , n } ^ { I I , ( \\alpha ) } ( x ) = - ( \\alpha + n + 1 - 2 m ) { \\rm e } ^ x \\int _ { x } ^ { \\infty } { \\rm e } ^ { - t } L _ m ^ { ( - \\alpha - 1 ) } ( t ) L _ { n - m } ^ { ( \\alpha + 1 ) } ( t ) \\ , { \\rm d } t . \\end{align*}"}
+{"id": "3708.png", "formula": "\\begin{align*} \\sigma _ i ( u ( x ) ) f ( \\xi , \\xi ' _ 1 , \\xi ' _ 2 ) & = \\int _ F \\left ( \\int _ { F } f ( \\xi - J x \\xi _ 1 ' , \\xi _ 1 ' , v ) \\bar { \\psi } ( ( x ^ t \\xi - Q _ i ( x ) \\xi ' _ 1 + u ) v ) d v \\right ) \\psi ( \\xi _ 2 ' u ) d u \\\\ & = \\int _ F \\left ( \\int _ { F } f ( \\xi - J x \\xi _ 1 ' , \\xi _ 1 ' , v ) \\bar { \\psi } ( u v ) d v \\right ) \\psi ( \\xi _ 2 ' ( - x ^ t \\xi + Q _ i ( x ) \\xi ' _ 1 + u ) ) d u \\\\ & = f ( \\xi - J x \\xi _ 1 ' , \\xi _ 1 ' , \\xi ' _ 2 ) \\psi ( \\xi _ 2 ' ( - x ^ t \\xi + Q _ i ( x ) \\xi ' _ 1 ) ) . \\end{align*}"}
+{"id": "9248.png", "formula": "\\begin{align*} \\tilde { d } = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots v _ { w ^ { - 1 } ( k ) } , v _ { w ^ { - 1 } ( k ) + 1 } , X v _ { w ^ { - 1 } ( k ) } , \\ldots \\widehat { v _ { r } } , \\ldots v _ n ) } = 0 . \\end{align*}"}
+{"id": "1591.png", "formula": "\\begin{align*} \\xi = \\eta + ( 1 - s ) \\alpha \\delta _ u + s \\alpha \\delta _ v = \\xi _ { s } ^ { \\delta _ u , \\delta _ v } . \\end{align*}"}
+{"id": "7247.png", "formula": "\\begin{align*} u ( \\gamma ) ^ 2 v ( \\gamma ) \\sim N ( \\gamma ) ( \\gamma \\to \\infty ) \\lim _ { \\gamma \\to \\infty } \\frac { K ( \\gamma ) } { u ( \\gamma ) } = \\lim _ { \\gamma \\to \\infty } \\frac { L ( \\gamma ) } { v ( \\gamma ) } = 0 . \\end{align*}"}
+{"id": "3888.png", "formula": "\\begin{align*} v ( y ) = \\sup _ { x \\in \\Omega } g ^ * ( x , y , u ( x ) ) . \\end{align*}"}
+{"id": "867.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { r ' } { \\left | { { I _ { { R _ i } } } } \\right | } = \\sum \\limits _ { i = 1 } ^ { r } { \\left | { { I _ { { R _ i } } } } \\right | } \\le \\sum \\limits _ { j = 1 } ^ { b ' } { \\left | { { S _ { { B _ j } } } } \\right | } , \\ ; \\ ; \\ ; \\ ; \\sum \\limits _ { j = 1 } ^ { b ' } { \\left | { { I _ { { B _ j } } } } \\right | } \\le \\sum \\limits _ { i = 1 } ^ { r ' } { \\left | { { S _ { { R _ i } } } } \\right | } , \\end{align*}"}
+{"id": "6840.png", "formula": "\\begin{align*} \\begin{aligned} \\nu ^ { \\pm } _ c ( Y _ 1 , Y _ 2 , Y _ 4 ) - \\nu ^ { \\pm } _ c ( Y _ 1 , Y _ 2 , Y _ 3 ) = \\nu ^ { \\pm } _ c ( Y _ 1 , Y _ 3 , Y _ 4 ) - \\nu ^ { \\pm } _ c ( Y _ 2 , Y _ 3 , Y _ 4 ) . \\end{aligned} \\end{align*}"}
+{"id": "8573.png", "formula": "\\begin{align*} F ( x ) : = \\sum _ { i = 0 } ^ { 2 ^ n - 2 } x ^ i = 1 + x + \\ldots + x ^ { 2 ^ n - 3 } + x ^ { 2 ^ n - 2 } \\end{align*}"}
+{"id": "2766.png", "formula": "\\begin{align*} f ( ( \\ominus \\ , \\mathbf { x } ) \\oplus \\mathbf { x } ) & = f ( \\mathbf { x } \\oplus ( \\ominus \\ , \\mathbf { x } ) ) = f ( 0 ) , \\\\ f ( ( \\bar { \\ominus } \\ , \\mathbf { x } ) \\ , \\bar { \\oplus } \\ , \\mathbf { x } ) & = f ( \\mathbf { x } \\ , \\bar { \\oplus } \\ , ( \\bar { \\ominus } \\ , \\mathbf { x } ) ) = f ( 0 ) , \\end{align*}"}
+{"id": "5821.png", "formula": "\\begin{align*} ( \\widetilde U _ 0 , \\widetilde U _ 1 ) = ( U _ 0 , U _ 1 ) + C _ p ^ T ( W _ 0 , W _ 1 ) , \\forall ( W _ 0 , W _ 1 ) \\in ( V \\times H ) ^ { N - p } . \\end{align*}"}
+{"id": "7539.png", "formula": "\\begin{align*} Z _ { g _ 3 } ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) = \\dfrac { \\tilde { G } _ 3 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "1251.png", "formula": "\\begin{align*} a _ f ( r ) : = \\dfrac { 1 + ( 1 - 2 \\beta ) r ^ 2 } { 1 - r ^ 2 } c _ f ( r ) : = \\dfrac { 2 ( 1 - \\beta ) r } { 1 - r ^ 2 } . \\end{align*}"}
+{"id": "326.png", "formula": "\\begin{align*} P ( l ) = \\frac { 1 } { 2 l } \\sum _ { n = 1 } ^ \\infty \\frac { A ( n , 1 ) } { n ^ \\frac { 3 } { 2 } } ( \\frac { 1 6 } { l n } ) \\sum _ { \\substack { \\alpha \\leq Y \\\\ ( \\alpha , 2 l n ) = 1 } } \\frac { \\mu ( \\alpha ) } { \\alpha ^ 2 } G _ 0 ( l n ) \\widetilde \\Phi _ { n } ( 0 ) , \\end{align*}"}
+{"id": "4256.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\| ( \\nabla - i A ) \\Phi \\| ^ 2 _ { L ^ 2 ( \\R ^ N ) } + \\int _ { \\R ^ N } V _ \\gamma ( x ) | \\Phi ( x ) | ^ 2 d x = \\frac { 1 } { 2 } \\| \\nabla \\Phi \\| ^ 2 _ { L ^ 2 ( \\R ^ N ) } + \\int _ { \\R ^ N } V ( x ) | \\Phi ( x ) | ^ 2 d x = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ N \\gamma _ j \\end{align*}"}
+{"id": "2187.png", "formula": "\\begin{align*} \\phi ( r ) : = ( ( 2 - 2 \\alpha - \\beta ) ( \\mathit { e } + 1 ) - ( \\mathit { e } - 1 ) ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) ( \\mathit { e } + 1 ) r + ( \\mathit { e } - 1 ) . \\end{align*}"}
+{"id": "3785.png", "formula": "\\begin{align*} R _ 0 & = \\left \\{ ( s , t ; m ( e ) ) = ( t , s ; m ( e ) ) \\mid e = \\{ s , t \\} \\in E \\right \\} \\\\ & \\mathrel { \\hphantom { = } } \\cup \\{ \\alpha v = ( \\alpha ( v ) ) \\alpha \\mid v \\in V \\} \\cup \\{ \\varepsilon _ i v = ( \\varepsilon _ i ( v ) ) \\varepsilon _ i \\mid v \\in V , i = 1 , 2 \\} \\cup \\{ \\iota v = v ^ { - 1 } \\iota \\mid v \\in V \\} \\end{align*}"}
+{"id": "2728.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 / L } A ' ( t ) \\lambda _ 1 ^ 4 ( t ) = A \\left ( \\frac { 1 } { L } \\right ) \\lambda _ 1 ^ 4 \\left ( \\frac { 1 } { L } \\right ) - A ( 0 ) - 4 \\int _ 0 ^ { 1 / L } A ( t ) \\lambda _ 1 ^ 3 ( t ) \\lambda _ 1 ' ( t ) d t . \\end{align*}"}
+{"id": "7613.png", "formula": "\\begin{align*} \\tau ^ n : = \\inf _ { t \\geq 0 } \\{ t : \\| u ^ n ( t ) \\| _ { \\L ^ p } \\geq n \\} \\wedge T , \\end{align*}"}
+{"id": "4775.png", "formula": "\\begin{align*} D _ k ( x ) & = \\mathop { \\textnormal { a r g m a x } } _ { \\substack { \\{ z _ 1 , . . . , z _ k \\} \\subseteq \\mathbb { F } _ 2 ^ N \\\\ { H z _ i } ^ \\intercal = x \\textnormal { f o r a l l } i } } \\{ P _ \\epsilon ( z _ 1 ) + . . . + P _ \\epsilon ( z _ k ) \\} \\\\ & = \\mathop { \\textnormal { a r g m i n } } _ { \\substack { \\{ z _ 1 , . . . , z _ k \\} \\subseteq \\mathbb { F } _ 2 ^ N \\\\ H z _ i ^ \\intercal = x \\textnormal { f o r a l l } i } } \\{ | z _ 1 | + . . . + | z _ k | \\} . \\end{align*}"}
+{"id": "2151.png", "formula": "\\begin{align*} F ( z ) : = f ( z ) ( Q ( z ) ) ^ { \\frac { \\beta } { n } } , \\beta > 0 \\end{align*}"}
+{"id": "5988.png", "formula": "\\begin{align*} i _ * ( \\Pi \\times e v ) _ * \\left [ \\O _ { \\overline { { \\mathcal { M } } _ { 0 , r } } ( X , \\d ) } \\right ] = \\sum _ { e _ 1 , \\dots , e _ r \\in I } \\left ( ( e v _ 1 ^ * \\phi _ { e _ 1 } ) \\otimes \\dots \\otimes ( e v _ r ^ * \\phi _ { e _ r } ) \\right ) \\boxtimes \\left ( \\pi ^ * ( \\phi _ { e _ 1 } ^ { \\vee } ) \\boxtimes \\dots \\boxtimes \\pi ^ * ( \\phi _ { e _ r } ^ { \\vee } ) \\right ) \\end{align*}"}
+{"id": "7359.png", "formula": "\\begin{align*} \\Tilde { K } _ i = K _ i \\setminus \\left ( \\bigcup _ { j = i + 1 } ^ \\infty J _ j \\right ) , \\end{align*}"}
+{"id": "1213.png", "formula": "\\begin{align*} F ( \\ell , g h ) = F ( \\ell g , h ) F ( \\ell , g ) . \\end{align*}"}
+{"id": "2605.png", "formula": "\\begin{align*} a _ { 1 , j } = k + 1 - \\sum _ { l \\in [ r ] \\setminus \\{ 1 , j \\} } a _ { l , j } . \\end{align*}"}
+{"id": "4849.png", "formula": "\\begin{align*} I _ i ( t _ 0 , s ; v _ { t _ 0 , s } ) = s ^ i I _ i ( 0 , 1 ; v ) = 0 . \\end{align*}"}
+{"id": "2724.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\gamma ( \\tau _ n ) = 0 . \\end{align*}"}
+{"id": "8292.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } V ( \\textbf { x } ) = \\frac { \\nu _ E R _ E ^ 2 - \\nu _ 1 R _ 1 ^ 2 - \\nu _ 2 R _ 2 ^ 2 - \\nu _ 3 R _ 3 ^ 2 } { 2 \\Lambda } \\end{array} \\right . \\end{align*}"}
+{"id": "5261.png", "formula": "\\begin{align*} g ( T _ U U , X ) = - g ( U , U ) g ( X , \\nabla f ) - \\frac { \\lambda ^ 2 } { 2 } g ( X , X ) g \\left ( U , \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } \\right ) . \\end{align*}"}
+{"id": "8622.png", "formula": "\\begin{align*} \\binom { N } { i } p ^ i ( 1 - p ) ^ { N - i } . \\end{align*}"}
+{"id": "2916.png", "formula": "\\begin{align*} \\partial _ t h = \\nu \\partial _ x ^ 2 h + \\lambda ( \\partial _ x h ) ^ 2 + \\sqrt { D } \\dot { W } ( t , x ) . \\end{align*}"}
+{"id": "1491.png", "formula": "\\begin{align*} \\C _ 4 ( x ) = \\prod _ { n = 1 , n } ^ \\infty \\left \\{ \\left ( \\frac { 1 - \\frac x { ( \\frac n 2 ) } } { 1 + \\frac x { ( \\frac n 2 ) } } \\right ) ^ { ( \\frac n 2 ) ^ 3 } e ^ { \\frac { n ^ 2 } 2 x + \\frac 2 3 x ^ 3 } \\right \\} \\end{align*}"}
+{"id": "38.png", "formula": "\\begin{align*} 3 ^ K & < 2 ^ { K + L } \\leq ( 3 + \\mu ) ^ K , \\\\ K \\cdot \\log ( 3 ) & < ( K + L ) \\cdot \\log ( 2 ) \\leq K \\cdot \\log ( 3 + \\mu ) \\\\ & = K \\cdot \\left ( \\log ( 3 ) + \\log \\left ( 1 + \\frac { \\mu } { 3 } \\right ) \\right ) \\\\ & < K \\cdot \\left ( \\log ( 3 ) + \\frac { \\mu } { 3 } \\right ) , \\\\ \\delta = \\frac { \\log ( 3 ) } { \\log ( 2 ) } & < \\frac { K + L } { K } < \\frac { \\log ( 3 ) } { \\log ( 2 ) } + \\frac { \\mu } { 3 \\cdot \\log ( 2 ) } . \\end{align*}"}
+{"id": "3479.png", "formula": "\\begin{align*} \\begin{aligned} y \\left [ n \\right ] = \\underbrace { { \\bf { h } } _ 1 ^ H { \\bf { f } } s \\left [ { n - { n _ 1 } } \\right ] } _ { { \\rm { d e s i r e d } } \\ ; { \\rm { s i g n a l } } } + \\underbrace { \\sum \\nolimits _ { l \\ne 1 } ^ L { { \\bf { h } } _ l ^ H { \\bf { f } } s \\left [ { n - { n _ l } } \\right ] } } _ { { \\rm { I S I } } } + z \\left [ n \\right ] , \\end{aligned} \\end{align*}"}
+{"id": "5774.png", "formula": "\\begin{align*} W '' + \\mathcal L W + \\overline A _ p W + \\overline D _ p \\mathcal G W ' = 0 . \\end{align*}"}
+{"id": "4603.png", "formula": "\\begin{align*} a _ 1 = q + 1 \\end{align*}"}
+{"id": "8324.png", "formula": "\\begin{align*} \\square _ { s , y } ( \\bar \\rho \\bar \\sigma ) ^ { \\frac { d - 1 } { 2 } } = - 2 ( d - 1 ) ^ 2 ( \\bar \\rho \\bar \\sigma ) ^ { \\frac { d - 3 } { 2 } } \\end{align*}"}
+{"id": "1116.png", "formula": "\\begin{align*} A _ j = 2 ^ { - 1 } ( \\theta _ j - \\theta ) ^ 2 a _ j = \\mathbb { E } _ Y \\{ A _ j \\} = 2 ^ { - 1 } \\mathbb { E } \\{ ( \\theta _ j - \\theta ) ^ 2 \\} . \\end{align*}"}
+{"id": "6937.png", "formula": "\\begin{align*} H _ { 2 n } ( x ) = ( - 4 ) ^ n n ! L _ n ^ { - 1 / 2 } \\big ( x ^ 2 \\big ) , H _ { 2 n + 1 } ( x ) = 2 ( - 4 ) ^ n n ! x L _ n ^ { 1 / 2 } \\big ( x ^ 2 \\big ) , \\end{align*}"}
+{"id": "5699.png", "formula": "\\begin{align*} \\mathbf { A } _ { b } ^ { a } = \\alpha _ { b } ^ { a } - \\frac { 1 } { 2 } ( \\digamma _ { b } ^ { a } + \\beta _ { b } ^ { a } - \\overline { \\beta } _ { b } ^ { a } ) \\mathbf { m } - \\frac { 1 } { 2 } ( \\digamma _ { b } ^ { a } - \\beta _ { b } ^ { a } + \\overline { \\beta } _ { b } ^ { a } ) \\overline { \\mathbf { m } } + \\frac { 1 } { 2 } D ( \\beta _ { b } ^ { a } - \\overline { \\beta } _ { b } ^ { a } ) \\mathbf { m } \\overline { \\mathbf { m } } . \\end{align*}"}
+{"id": "5775.png", "formula": "\\begin{align*} u '' + L u + g u ' = 0 \\end{align*}"}
+{"id": "7849.png", "formula": "\\begin{align*} D _ n = \\{ z : | z - z _ n | \\leq 1 / \\kappa ( | z _ n | ) \\} , \\end{align*}"}
+{"id": "9049.png", "formula": "\\begin{align*} \\underset { n \\to \\infty } \\lim \\ , \\underset { 0 \\le t \\le T } { \\sup } \\ , E [ | Y ^ { i , n } _ t - Y ^ i _ t | ^ p ] = 0 . \\end{align*}"}
+{"id": "7682.png", "formula": "\\begin{align*} \\omega = \\sum _ { k } \\lambda _ { k } \\frac { d f _ { k } } { f _ { k } } . \\end{align*}"}
+{"id": "5921.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } w '' - \\Delta w = 0 & \\hbox { i n } ( T , + \\infty ) \\times \\Omega , \\\\ \\partial _ \\nu w + \\widehat { \\Lambda } w = 0 & \\hbox { o n } ( T , + \\infty ) \\times \\Gamma , \\end{array} \\right . \\end{align*}"}
+{"id": "5825.png", "formula": "\\begin{align*} U '' + \\mathcal L U + A U + D \\mathcal G U ' = 0 . \\end{align*}"}
+{"id": "81.png", "formula": "\\begin{align*} B ( w , \\tau ) = ( \\Sigma _ { d _ { n s } } ( w ) ) ( \\tau ) . \\end{align*}"}
+{"id": "1871.png", "formula": "\\begin{align*} \\alpha _ { s _ i } ( \\sigma ) = \\frac { \\pi } { 2 } . \\end{align*}"}
+{"id": "7792.png", "formula": "\\begin{align*} \\frac { q ^ { 2 j \\{ n , n ' \\} } - 1 } { q ^ { 2 j a } - 1 } & = \\frac { q ^ { 2 \\alpha j / s d ( n ) } - 1 } { q ^ { 2 j / s d ( n ) } - 1 } = \\begin{cases} \\displaystyle \\sum _ { p = 0 } ^ { \\alpha - 1 } q ^ { 2 j p / s d ( n ) } & \\alpha > 0 , \\\\ 0 & \\alpha = 0 , \\\\ \\displaystyle - \\sum _ { p = 1 } ^ { \\alpha } q ^ { - 2 j p / s d ( n ) } & \\alpha < 0 . \\end{cases} \\end{align*}"}
+{"id": "9024.png", "formula": "\\begin{align*} _ k ( H ^ 1 ( \\Q _ \\Sigma / \\Q _ { n } , \\Phi ) ) = h _ 1 p ^ n + _ k ( N / N _ { } ) . \\end{align*}"}
+{"id": "6885.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow \\infty } \\langle \\rho ' ( u _ n ) - \\rho ' ( u ) , u _ n - u \\rangle = 0 . \\end{align*}"}
+{"id": "1533.png", "formula": "\\begin{align*} i ^ * \\omega _ R = - \\omega _ L \\end{align*}"}
+{"id": "4700.png", "formula": "\\begin{align*} \\mathfrak { E } ^ { G _ { I I } } = \\mathbb { C } [ \\varphi _ 8 , \\varphi _ { 2 4 } ] \\end{align*}"}
+{"id": "8753.png", "formula": "\\begin{align*} \\gamma _ \\alpha ( \\mathcal { A } , d ) : = \\inf \\sup \\limits _ { t \\in \\mathcal { A } } \\sum \\limits _ { r = 0 } ^ { \\infty } 2 ^ { r / \\alpha } d ( t , \\mathbb { A } _ r ) \\lesssim \\int _ { 0 } ^ { \\Delta ( \\mathcal { A } ) } ( \\log N ( \\mathcal { A } , d , u ) ) ^ { 1 / \\alpha } d u , \\end{align*}"}
+{"id": "1823.png", "formula": "\\begin{align*} R = \\begin{pmatrix} O _ { 2 } & G & O _ { 2 } \\\\ O _ { 2 } & O _ { 2 } & G \\\\ G & O _ { 2 } & O _ { 2 } \\end{pmatrix} \\in O ( 6 ) , \\end{align*}"}
+{"id": "2035.png", "formula": "\\begin{align*} \\mathcal { A } * \\mathcal { U } = \\mathcal { U } * \\mathcal { A } = \\delta _ { 0 } \\mathcal { A } ' * \\mathcal { U } ' = \\mathcal { U } ' * \\mathcal { A } ' = \\delta _ { 0 } . \\end{align*}"}
+{"id": "7275.png", "formula": "\\begin{align*} G _ m ( x ) & = \\int _ { 0 } ^ { x } m ( y ) d y ( x \\geq 0 ) , \\\\ \\tilde { m } ( x ) & = m ( x ) - m ( 1 ) ( x > 0 ) , \\\\ G ^ 1 _ m ( x ) & = \\int _ { 0 } ^ { x } \\tilde { m } ( y ) d y ( x \\geq 0 ) . \\end{align*}"}
+{"id": "6960.png", "formula": "\\begin{gather*} \\vec { a } = \\begin{pmatrix} a _ { 0 , n } \\\\ a _ { 1 , n } \\\\ \\vdots \\\\ a _ { n , n } \\end{pmatrix} \\ ! , \\vec { b } _ I = \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ L _ { m , n } ^ { I , ( \\alpha ) } ( x ) \\end{pmatrix} \\ ! , \\end{gather*}"}
+{"id": "4711.png", "formula": "\\begin{align*} \\dim T = \\sum _ { i = 0 } ^ d \\frac { | G _ { I V } | } { | C _ i | } \\end{align*}"}
+{"id": "4272.png", "formula": "\\begin{align*} H _ \\gamma ( f ) : = \\| ( \\nabla - i A ) f \\| ^ 2 _ { L ^ 2 } + 2 \\int _ { \\R ^ N } V _ \\gamma ( x ) | f ( x ) | ^ 2 d x . \\end{align*}"}
+{"id": "572.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } m _ t = \\frac { \\sigma _ t } { \\gamma } \\ , ( A ^ { \\rm T } A ) : \\mathbb { E } ^ \\dagger \\left [ X _ t ^ \\dagger \\otimes X _ t ^ \\dagger \\right ] - \\frac { \\sigma _ t } { \\gamma } \\ , ( A ^ { \\rm T } A ) : C \\ , m _ t , \\end{align*}"}
+{"id": "688.png", "formula": "\\begin{align*} \\int _ b ^ a d U _ { \\beta _ j } = \\oint _ { \\beta _ j } * d G ^ { \\delta _ a - \\delta _ b } . \\end{align*}"}
+{"id": "1305.png", "formula": "\\begin{align*} \\int _ { \\mathbb { C } \\mathbb { P } ^ { n - 1 } \\times \\mathbb { C } \\mathbb { P } ^ { n - 1 } } | \\langle \\tau _ \\alpha , \\tau _ \\beta \\rangle | ^ { 2 m } \\ , d ( \\mu \\times \\mu ) ( \\alpha , \\beta ) = \\int _ { \\mathbb { C } \\mathbb { P } ^ { n - 1 } } \\int _ { \\mathbb { C } \\mathbb { P } ^ { n - 1 } } | \\langle \\tau _ \\alpha , \\tau _ \\beta \\rangle | ^ { 2 m } \\ , d \\mu ( \\alpha ) \\ , d \\mu ( \\beta ) \\geq \\frac { 1 } { { d + m - 1 \\choose m } } , \\forall m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "4467.png", "formula": "\\begin{align*} u ' _ { \\ell - 1 } + u ' _ { \\ell } = \\frac { u _ { \\ell - 1 } } { t } + t u _ { \\ell } < u _ { \\ell - 1 } + u _ { \\ell } \\end{align*}"}
+{"id": "446.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n \\in \\mathbb { M } } f _ n ( x ) \\tau _ n - \\sum _ { n = 1 } ^ { \\infty } r _ n f _ n ( x ) \\tau _ n \\right \\| \\leq c \\varepsilon ^ \\frac { 1 } { d } \\| x \\| , \\forall x \\in \\mathcal { X } , \\end{align*}"}
+{"id": "1781.png", "formula": "\\begin{align*} \\frac { s } { t } = \\frac { L + c } { \\ell + c } \\in S . \\end{align*}"}
+{"id": "746.png", "formula": "\\begin{align*} \\big ( \\frac { \\partial \\tilde { h } _ 1 ( \\tilde { w } ) } { \\partial \\tilde { w } } - 2 \\tilde { h } _ 2 ( \\tilde { w } ) \\big ) d \\tilde { w } ^ 2 = \\big ( \\frac { \\partial h _ 1 ( w ) } { \\partial w } - 2 { h } _ 2 ( w ) \\big ) d { w } ^ 2 + \\frac { 1 } { 6 } \\{ \\tilde { w } , w \\} _ 2 d w ^ 2 , \\end{align*}"}
+{"id": "6980.png", "formula": "\\begin{align*} H _ { 2 , n } ^ { ( \\{ 1 , 1 \\} ) } ( x ) & = 8 n ( n - 1 ) ( n - 2 ) \\int _ 0 ^ x H _ { \\{ 1 , 1 \\} } ( t ) H _ { n - 3 } ( t ) \\ , { \\rm d } t + 1 6 ( n - 1 ) ( n - 2 ) H _ { n - 2 } ( 0 ) \\\\ & = \\int _ 0 ^ x H _ { \\{ 1 , 1 \\} } ( t ) H ''' _ { n } ( t ) \\ , { \\rm d } t + 8 ( n - 2 ) H _ { n - 1 } ' ( 0 ) . \\end{align*}"}
+{"id": "4507.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n e ^ { b _ i } < \\sum _ { i = 1 } ^ n e ^ { a _ i } . \\end{align*}"}
+{"id": "3223.png", "formula": "\\begin{align*} f ' ( x ) + f ' ( y ) \\geq f ( x ) + f ( y ) - 2 > 2 k _ 3 + 2 k _ 2 + k _ 1 - 2 = 2 k _ 3 ' + 2 k ' _ 2 + k _ 1 ' - 1 , \\end{align*}"}
+{"id": "5654.png", "formula": "\\begin{align*} { \\mathcal I } _ 1 ( \\Lambda ) \\equiv \\left ( \\frac { \\Lambda } { \\bar { \\ell } } \\sum _ { n \\geq 0 } ( - 1 ) ^ n \\Lambda ^ n \\right ) \\sum _ { j = 1 } ^ N \\ell _ j \\widehat { \\Phi } _ j = \\frac { 1 } { \\bar { \\ell } } \\frac { \\Lambda } { 1 + \\Lambda } \\sum _ { j = 1 } ^ N \\ell _ j \\widehat { \\Phi } _ j = \\frac { \\Lambda } { 1 + \\Lambda } [ \\overline { \\Phi } - \\Gamma / | \\Omega | ] . \\end{align*}"}
+{"id": "1852.png", "formula": "\\begin{align*} \\ddot { \\varrho } + \\frac { 4 } { 3 t } \\dot { \\varrho } - \\tilde { \\kappa } t ^ { - 2 \\gamma + \\frac { 2 } { 3 } } \\Delta \\varrho - \\frac { 2 } { 3 t ^ 2 } \\varrho = ( \\gamma - 1 ) \\tilde { \\kappa } t ^ { - 2 \\gamma + \\frac { 2 } { 3 } } \\frac { D ^ i \\varrho D _ i \\varrho } { 1 + \\varrho } . \\end{align*}"}
+{"id": "6221.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p \\beta _ r ( E _ r , \\widehat U _ 0 ) = \\sum _ { r = 1 } ^ p \\beta _ r ( E _ r , \\widehat U _ 1 ) = 0 \\hbox { i n } \\Omega \\end{align*}"}
+{"id": "3636.png", "formula": "\\begin{align*} { \\rm d } s ^ 2 = { \\rm d } r ^ 2 + a _ { i , j } ( r , \\theta ) { \\rm d } \\theta _ { i } { \\rm d } \\theta _ { j } , \\end{align*}"}
+{"id": "6453.png", "formula": "\\begin{align*} m = \\begin{pmatrix} \\cos \\theta \\\\ \\sin \\theta \\cos \\varphi \\\\ \\sin \\theta \\sin \\varphi \\end{pmatrix} . \\end{align*}"}
+{"id": "2163.png", "formula": "\\begin{align*} \\zeta ( \\tilde { \\sigma _ { 0 } } ) = & \\dfrac { 1 } { 4 ( - 1 - \\mathit { e } + 2 \\alpha + \\beta ) ^ 2 } \\big ( 2 ( 2 + \\beta - 2 \\alpha ) ( 1 + \\mathit { e } ^ 2 - 2 \\mathit { e } ( 2 \\alpha + \\beta - 1 ) ) \\\\ & \\quad \\times ( ( - 2 \\alpha + 2 + \\beta ) - \\sqrt { ( - 2 \\alpha + 2 + \\beta ) ^ 2 - 4 ( \\mathit { e } - 1 ) ( 2 \\alpha - 1 + \\beta - \\mathit { e } ) } ) \\\\ & \\quad { } + 4 ( 2 \\alpha + \\beta - 1 - \\mathit { e } ) ( \\mathit { e } ^ 2 - 1 ) ( 2 \\alpha + \\beta - 2 ) \\big ) , \\end{align*}"}
+{"id": "6792.png", "formula": "\\begin{align*} d ( \\mathcal { U } ( ( \\overline { w } _ { \\i , \\widehat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } ) , \\mathcal { U } ( ( \\overline { v } _ { \\i , \\widehat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } ) ) \\leq ~ c F _ { \\eta } ( ( \\overline { w } _ { \\i , \\widehat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } , ( \\overline { v } _ { \\i , \\widehat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } ) ) , \\end{align*}"}
+{"id": "5414.png", "formula": "\\begin{align*} \\delta ' ( q _ { i , ( i + m _ i + n _ i - 1 ) \\bmod p , m _ i + n _ i - 1 } , \\ell _ { m _ i + n _ i - 1 } ^ { ( i ) } ) & = q _ { i , ( i + m _ i + n _ i ) \\bmod p , m _ i } \\end{align*}"}
+{"id": "777.png", "formula": "\\begin{align*} \\ell _ \\varphi ^ + = \\sup _ { \\eta } \\int \\varphi \\ d \\eta \\ell _ \\varphi ^ - = \\inf _ { \\eta } \\int \\varphi \\ d \\eta \\end{align*}"}
+{"id": "2828.png", "formula": "\\begin{align*} x ^ 2 y ^ 3 = \\sqrt [ 5 ] { x ^ { 1 0 } y ^ { 1 5 } } \\leq \\frac { 2 x ^ 5 + 3 y ^ 5 } { 5 } . \\end{align*}"}
+{"id": "6209.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ d \\beta _ r E _ r = 0 , \\end{align*}"}
+{"id": "2915.png", "formula": "\\begin{align*} \\| \\mathcal { B } f \\| _ 0 = \\sup _ { k \\in \\mathbb { N } } \\sup _ { \\sigma > 0 } \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | \\mathcal { B } _ k f _ \\sigma | ) d m _ \\infty = \\sup _ { \\sigma > 0 } \\sup _ { k \\in \\mathbb { N } } \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | \\mathcal { B } _ k f _ \\sigma | ) d m _ \\infty . \\end{align*}"}
+{"id": "2489.png", "formula": "\\begin{align*} \\mathcal H = \\vee _ { n \\geq 0 } x ' _ n . \\end{align*}"}
+{"id": "5278.png", "formula": "\\begin{align*} \\hat { \\tilde { s e c } } ( U , V ) = \\frac { \\tilde { g } ( \\hat { \\tilde { R } } ( U , V ) V , U ) } { \\tilde { g } ( U , U ) \\tilde { g } ( V , V ) - \\tilde { g } ( U , V ) ^ 2 } = \\frac { 1 } { \\lambda ^ 2 } g ( \\hat { \\tilde { R } } ( U , V ) V , U ) . \\end{align*}"}
+{"id": "9234.png", "formula": "\\begin{align*} S _ { j } : = S ( m _ { j } ) , \\varphi _ { j } : = \\frac { \\psi _ { m _ { j } } } { \\Vert \\psi _ { m _ { j } } \\Vert _ { L ^ { 2 } } } \\mu _ { j } : = \\ < \\Delta _ { f } \\varphi _ { j } , \\varphi _ { j } \\ > . \\end{align*}"}
+{"id": "1501.png", "formula": "\\begin{align*} ~ \\zeta ( s , a ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { ( n + a ) ^ { s } } \\end{align*}"}
+{"id": "1653.png", "formula": "\\begin{align*} \\delta _ \\sigma f ( r e ^ { i \\theta } ) = \\left . \\frac { d } { d t } f ( r e ^ { i \\theta } e ^ { t } ) \\right | _ { t = 0 } = r \\frac { \\partial } { \\partial r } f ( r e ^ { i \\theta } ) , f \\in C ^ \\infty ( T ( F _ \\sigma ) , \\C ) . \\end{align*}"}
+{"id": "3376.png", "formula": "\\begin{align*} \\bigvee _ { j = 1 } ^ { i - 1 } ( - u _ i + z ' _ j ) \\vee ( - u _ { i + 1 } + 1 + z ' _ i ) & \\leq \\bigvee _ { j = i + 1 } ^ { n - 1 } ( - u _ j + z ' _ j ) \\vee ( - u _ n ) & & \\enspace i = 1 , \\dots , n - 2 \\ , . \\end{align*}"}
+{"id": "1162.png", "formula": "\\begin{align*} & I _ { n , i , q } ( i , a , b ) \\wedge L i n k e d ( a , a ' , i , j , \\Theta ) \\wedge L i n k e d ( b , b ' , i , j , \\Theta ) \\rightarrow I _ { n , i , q } ( j , a ' , b ' ) . \\end{align*}"}
+{"id": "8422.png", "formula": "\\begin{align*} c _ * ( A ) = \\lim _ { K \\uparrow A } \\ , c ( K ) . \\end{align*}"}
+{"id": "8045.png", "formula": "\\begin{align*} \\phi ( n , s ) = \\pi ^ s \\Gamma ( s ) ^ { - 1 } \\zeta ( 2 s ) ^ { - 1 } \\abs { n } ^ { - \\frac { 1 } { 2 } } \\eta ( n , s ) , \\end{align*}"}
+{"id": "7800.png", "formula": "\\begin{align*} & \\Psi _ { a , b } [ n ] = \\prod _ { t = 1 } ^ p \\Psi _ { p a , b \\pm ( 2 t - p - 1 ) a } [ n ] , \\\\ & \\prod _ { t = 1 } ^ p \\Psi _ { a , b \\pm ( 2 t - p - 1 ) a / p } [ n ] = \\Psi _ { a / p , b } [ n ] , \\end{align*}"}
+{"id": "8118.png", "formula": "\\begin{align*} a ( y ) = \\psi ( y ) y ^ { - \\frac { 1 } { 3 } } = g \\Bigl ( \\frac { m ^ 2 y } { N } \\Bigr ) \\widehat { k ^ * } \\Bigl ( \\frac { M T c } { 2 \\pi ^ 2 \\sqrt { y p } } \\Bigr ) e \\Bigl ( - \\frac { T ^ 2 c } { 4 \\pi ^ 2 \\sqrt { y p } } \\Bigr ) y ^ { - \\frac { 1 3 } { 1 2 } } . \\end{align*}"}
+{"id": "5357.png", "formula": "\\begin{align*} \\{ \\lambda _ n [ \\varepsilon _ t ] , \\ldots , \\lambda _ { n + m - 1 } [ \\varepsilon _ t ] \\} = \\{ g _ 1 ( t ) , \\ldots , g _ m ( t ) \\} \\forall t \\in I . \\end{align*}"}
+{"id": "5313.png", "formula": "\\begin{align*} \\mathrm { R i c } ( X , Y ) = n g ( A _ { \\mathbf { H } } X , Y ) + 2 g ( X , A _ { \\eta } Y ) . \\end{align*}"}
+{"id": "7216.png", "formula": "\\begin{align*} \\big \\langle p j , \\ , p ' \\big \\rangle \\ , = \\ , \\rho ^ 2 \\left [ \\ , \\vartheta ' \\cos \\big ( \\phi - \\psi \\big ) \\ , + \\ , \\sin \\vartheta \\cos \\vartheta \\sin \\big ( \\phi - \\psi \\big ) \\big ( \\phi + \\psi \\big ) ' \\ , \\right ] , \\end{align*}"}
+{"id": "8472.png", "formula": "\\begin{align*} c ^ * ( A ) = \\sup _ { \\nu \\in \\mathcal E ^ + _ A } \\ , G ( \\nu ) = \\sup _ { \\nu \\in \\widehat { \\mathcal E } ^ + _ A } \\ , \\nu ( X ) . \\end{align*}"}
+{"id": "4478.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i + \\varepsilon _ i } : \\varepsilon _ i = 0 , \\pm 1 \\right \\} . \\end{align*}"}
+{"id": "7257.png", "formula": "\\begin{align*} b _ \\pm & = \\int _ { 0 } ^ { \\infty } j _ \\pm ( d x ) \\int _ { 0 } ^ { x } m _ \\pm ( y , \\infty ) d y p = \\frac { b _ + } { b _ + + b _ - } . \\end{align*}"}
+{"id": "1743.png", "formula": "\\begin{align*} \\frac { \\partial P _ n } { \\partial y } ( - \\bar \\beta , \\bar \\alpha ) = \\left ( ( \\lambda - n ) ( \\alpha y - \\beta x ) \\bar \\beta - ( n + \\lambda ) ( \\bar \\beta y + \\bar \\alpha x ) \\alpha \\right ) P _ { n - 1 } ( - \\bar \\beta , \\bar \\alpha ) = \\left ( 2 n ( \\beta x - \\alpha y ) \\bar \\beta - ( n + \\lambda ) x \\right ) P _ { n - 1 } ( - \\bar \\beta , \\bar \\alpha ) . \\end{align*}"}
+{"id": "22.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\mathbf { D } _ 1 = \\mathbb { E } \\big [ \\bar { \\phi } _ y y _ { ( 1 , 0 ) } \\big ] . \\end{align*}"}
+{"id": "8162.png", "formula": "\\begin{align*} H _ { m , n } ^ + ( x ) = 2 i \\int _ { - \\infty } ^ \\infty J _ { 2 i t } ( x ) \\frac { k ( t ) U ( m ^ 2 n , t ) t } { \\cosh ( \\pi t ) } \\ , d t , \\end{align*}"}
+{"id": "4089.png", "formula": "\\begin{align*} i _ X H + \\frac { d } { d t } \\Big | _ { t = 0 } B _ t = d \\alpha . \\end{align*}"}
+{"id": "5521.png", "formula": "\\begin{align*} P _ { k } ( x ) = O _ { \\epsilon } \\bigg ( x ^ { - \\frac { k } { 2 } + \\frac { 1 } { 4 } + \\epsilon } \\bigg ) , \\mathrm { a s } \\ , \\ , x \\rightarrow \\infty , \\end{align*}"}
+{"id": "5258.png", "formula": "\\begin{align*} - a \\cos \\omega \\sin \\omega \\frac { d \\omega } { d t } = a \\sin ^ 2 \\omega \\frac { d f } { d t } . \\end{align*}"}
+{"id": "9124.png", "formula": "\\begin{align*} Q _ { i } : = \\{ q ~ : q \\not \\equiv 1 \\pmod { p _ { i } } , ~ ~ g ( x ) ~ ~ q \\} \\end{align*}"}
+{"id": "977.png", "formula": "\\begin{align*} \\sigma _ { m + 1 } ( s ) = & \\bigg ( \\sum _ { j = 1 } ^ { m } R _ j + ( 1 - e ^ { - i s } ) R _ { m + 1 } , 2 \\big ( \\frac { \\pi } { 2 } - 1 \\big ) \\big ( \\sum _ { j = 1 } ^ { m } \\| R _ j \\| ^ 2 \\big ) + 2 ( s - \\sin s ) \\| R _ { m + 1 } \\| ^ 2 \\\\ & - 2 \\langle R _ 1 , R _ 2 \\rangle - 2 \\langle R _ 1 + R _ 2 , R _ 3 \\rangle - . . . - 2 \\langle R _ 1 + . . . + R _ { m - 1 } , R _ { m } \\rangle \\\\ & - 2 \\sin s \\langle R _ 1 + . . . + R _ { m } , R _ { m + 1 } \\rangle \\bigg ) \\end{align*}"}
+{"id": "2436.png", "formula": "\\begin{align*} P _ { [ 0 , M _ 1 ] } ( a ) : = \\max ( \\min ( M _ 1 , a ) , 0 ) . \\end{align*}"}
+{"id": "2463.png", "formula": "\\begin{align*} \\# \\{ \\rho = \\beta + i \\gamma \\colon L ( \\rho , \\pi ) = 0 , ~ \\beta \\geq 1 - \\alpha , ~ | \\gamma - t | \\leq 6 \\} \\ll N _ { \\pi } ( 1 - \\alpha , | t | + 6 ) . \\end{align*}"}
+{"id": "9111.png", "formula": "\\begin{align*} \\frac { d } { d \\rho } h _ { \\rho } = \\frac { - 2 \\sqrt { h _ 1 h _ 2 } \\rho ^ 2 + 2 ( h _ 1 + h _ 2 ) \\rho - 2 \\sqrt { h _ 1 h _ 2 } } { ( 1 - \\rho ^ 2 ) ^ 2 } \\end{align*}"}
+{"id": "5164.png", "formula": "\\begin{align*} \\mathcal { P } ( D L G ( 1 , 2 ) , n ) & = \\binom { n } { 2 } \\cdot \\binom { n } { 2 } \\cdot \\binom { n } { 2 } = \\dfrac { n ^ { 3 } ( n - 1 ) ^ { 3 } } { 8 } . \\end{align*}"}
+{"id": "68.png", "formula": "\\begin{align*} ( z _ i \\neq - 1 ) \\vee \\bigvee \\limits _ { j \\in N ( i ) } ( z _ j = 2 ) , i \\in \\{ 1 , . . . , n \\} \\end{align*}"}
+{"id": "5674.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\left ( F _ { \\chi ^ 2 _ { n } ( \\frac { \\rho _ 2 ^ 2 } { \\sigma _ j ^ 2 } ) } ( y ) - F _ { \\chi ^ 2 _ { n } ( \\frac { \\rho _ 1 ^ 2 } { \\sigma _ j ^ 2 } ) } ( y ) \\right ) d y = \\frac { \\rho _ 1 ^ 2 - \\rho _ 2 ^ 2 } { \\sigma _ j ^ 2 } . \\end{align*}"}
+{"id": "8665.png", "formula": "\\begin{align*} R ( u ) ( z ^ m f ) = { z ^ { m + 1 } } { u } ^ { m + 1 } f , R ^ { - 1 } ( u ) ( z ^ m f ) = { z } ^ { m - 1 } u ^ { - m } f . \\end{align*}"}
+{"id": "6470.png", "formula": "\\begin{align*} \\delta E ( \\eta ) & = O _ 2 ^ 2 ( \\eta ) + 2 ( \\partial _ { x x } \\theta _ * \\nu + \\partial _ x \\theta _ * \\partial _ x \\nu ) w _ * + \\left ( - \\partial _ { x x } \\nu + ( 1 - \\gamma ^ 2 ) \\nu \\right ) n _ * + ( - \\partial _ { x x } \\rho + ( 1 - \\gamma ^ 2 ) \\rho ) p _ * . \\end{align*}"}
+{"id": "8861.png", "formula": "\\begin{align*} \\mathfrak { S } _ { n ; p , q } ^ { z , w } & : = \\sum _ { \\substack { 1 \\leq j \\leq d ( n , p , q ) } } h _ { p , q } ^ { j } ( z , \\bar { z } ) \\overline { h _ { p , q } ^ { j } ( w , \\bar { w } ) } \\\\ & = \\left ( | z | | w | \\right ) ^ { p + q } \\sum _ { \\substack { 1 \\leq j \\leq d ( n , p , q ) } } h _ { p , q } ^ { j } \\left ( \\frac { z } { | z | } , \\frac { \\bar { z } } { | z | } \\right ) \\overline { h _ { p , q } ^ { j } \\left ( \\frac { w } { | w | } , \\frac { \\bar { w } } { | w | } \\right ) } . \\end{align*}"}
+{"id": "5579.png", "formula": "\\begin{align*} u _ { ( x , 0 ) , r } ( v , w ) = \\max _ { \\partial B _ { r _ 1 } ^ n ( 0 , 0 ) } u _ { ( x , 0 ) , r } \\ge \\frac { C _ 1 ( n ) } { 4 ^ { 1 + \\gamma } } . \\end{align*}"}
+{"id": "4543.png", "formula": "\\begin{align*} s _ { i + 1 } = s _ i ( s _ i - 1 ) + 1 \\end{align*}"}
+{"id": "3808.png", "formula": "\\begin{align*} C _ 1 ( m , \\lambda ) = \\prod _ { k = 0 } ^ { m - 1 } ( \\lambda + 2 k ) . \\end{align*}"}
+{"id": "4809.png", "formula": "\\begin{align*} \\Pr _ { y \\sim \\mathcal { D } ( C ^ \\perp ) } \\big [ | y | \\notin A _ \\epsilon \\big ] \\leq 2 ^ { N ^ { \\frac { 3 } { 4 } } } \\cdot \\frac { \\sum _ { i \\notin A _ \\epsilon } \\binom { N } { i } } { 2 ^ N } . \\end{align*}"}
+{"id": "1041.png", "formula": "\\begin{align*} \\mathcal { P } = \\mathcal { P } _ k = \\left \\{ P : \\ , \\mu = \\mathbb { E } _ { X \\sim P } ( X ) \\in [ - D , D ] , \\ , \\sigma ^ k = \\mathbb { E } _ { X \\sim P } [ | X - \\mu | ^ k ] \\leq 1 \\right \\} , \\end{align*}"}
+{"id": "1977.png", "formula": "\\begin{align*} - A _ { m } ( z ) = \\sum _ { j = 0 , j \\neq m } ^ { k } \\frac { A _ { j } ( z ) f ( z + c _ { j } ) } { f ( z + c _ { m } ) } . \\end{align*}"}
+{"id": "5969.png", "formula": "\\begin{align*} A ^ T E _ r - \\sum _ { s = 1 } ^ p \\alpha _ { r s } E _ s \\in \\{ ( C _ p ) \\} ^ \\bot = ( C _ p ^ T ) , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "6944.png", "formula": "\\begin{align*} L _ { m , n } ^ { I I I , ( \\alpha ) } ( x ) = n \\int _ 0 ^ x L _ { n - m - 1 } ^ { ( \\alpha + 1 ) } ( t ) L _ m ^ { ( - \\alpha - 1 ) } ( - t ) { \\rm d } t + ( m + 1 ) \\binom { n - m + \\alpha } { n - m - 1 } \\binom { m - \\alpha - 1 } { m + 1 } , \\end{align*}"}
+{"id": "1104.png", "formula": "\\begin{align*} \\mathcal { P } _ r = \\{ \\mbox { d i s t r i b u t i o n } P \\mbox { s u c h t h a t } | \\mathrm { m e d } ( P ) | \\leq r , \\ , \\mathbb { E } _ P \\{ | X | \\} < \\infty \\} , \\end{align*}"}
+{"id": "3628.png", "formula": "\\begin{align*} \\sup _ { u \\in C _ { 0 } ^ { \\infty } ( \\mathbb { B } ^ n ) , \\int _ { \\mathbb { B } ^ n } | \\nabla _ { h } u | ^ n \\ , { \\rm d } v _ h \\leq 1 } \\int _ { \\mathbb { B } ^ n } \\left ( e ^ { \\alpha _ n u ^ { \\frac { n } { n - 1 } } } - \\sum _ { j = 0 } ^ { n - 2 } \\frac { 1 } { j ! } \\alpha _ n ^ j | u | ^ { \\frac { j n } { n - 1 } } \\right ) \\ , { \\rm d } v _ { h } < \\infty , \\end{align*}"}
+{"id": "1499.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ { \\frac \\pi 2 } \\theta ^ 2 \\log \\left ( \\cos \\frac \\theta 2 \\right ) d \\theta & = \\frac { \\pi ^ 3 } { 2 4 } \\log \\frac 1 { \\sqrt 2 } - \\frac { 8 \\pi ^ 3 } 3 \\log \\C _ 4 \\left ( \\frac 1 4 \\right ) \\\\ & = \\frac { \\pi ^ 3 } { 2 4 } \\log \\frac 1 { \\sqrt 2 } - \\log \\left ( \\prod _ { n = 1 , n } ^ \\infty \\left ( \\frac { 2 n - 1 } { 2 n + 1 } \\right ) ^ { \\frac { n ^ 3 } 8 } e ^ { \\frac { n ^ 2 } { 8 } + \\frac 1 { 9 6 } } \\right ) ^ { \\frac { 8 \\pi ^ 3 } 3 } . \\end{aligned} \\end{align*}"}
+{"id": "5483.png", "formula": "\\begin{align*} b ( x , \\mu , 1 ) & = - x ^ 3 - 2 \\int _ { \\mathbb R } ( x + \\beta y ) \\mu ( d y ) , ~ b ( x , \\mu , 2 ) = - 2 x , \\\\ \\sigma ( x , \\mu , 1 ) & = \\int _ { \\mathbb R } ( x + \\beta y ) \\mu ( d y ) , ~ \\sigma ( x , \\mu , 2 ) = x , \\end{align*}"}
+{"id": "9149.png", "formula": "\\begin{align*} E ^ \\rho ( i ) = \\bigoplus _ { \\langle m , u _ \\rho \\rangle \\geq i } E ^ \\sigma _ { m } . \\end{align*}"}
+{"id": "4149.png", "formula": "\\begin{align*} d K = 3 \\Lambda \\end{align*}"}
+{"id": "3575.png", "formula": "\\begin{align*} v _ A = \\sum _ { B \\geq A } T _ { A B } E ^ { \\gamma _ B } \\Omega _ { \\lambda _ B } , \\end{align*}"}
+{"id": "2014.png", "formula": "\\begin{align*} A ( s ) = J - s P , \\end{align*}"}
+{"id": "4979.png", "formula": "\\begin{align*} \\Lambda : = b ^ p y _ 2 + \\rho ^ p ( y _ 0 ) - y _ 1 ^ p + b ^ { p - 1 } \\lambda ( y _ 0 , y _ 1 ) \\nu ( y _ 0 , y _ 1 ) . \\end{align*}"}
+{"id": "8982.png", "formula": "\\begin{align*} \\varphi _ { 2 l - 1 } ( r e ^ { i \\theta } ) = \\frac { 1 } { \\sqrt { \\pi } } r ^ l s i n ( l \\theta ) , \\ \\varphi _ { 2 l } ( r e ^ { i \\theta } ) = \\frac { 1 } { \\sqrt { \\pi } } r ^ l c o s ( l \\theta ) , \\ l \\in \\N . \\end{align*}"}
+{"id": "4017.png", "formula": "\\begin{align*} G _ p \\cdot \\gamma = \\phi ^ * _ k Y ^ k _ { p _ i } \\gamma _ i . \\end{align*}"}
+{"id": "1814.png", "formula": "\\begin{gather*} p ( t ) = L _ { \\dot { u } } ( t , u _ 0 ( t ) , \\dot { u } _ 0 ( t ) ) , \\\\ V ( t , y ) = p ( t ) \\cdot y + \\frac { 1 } { 2 } ( y - u _ 0 ( t ) ) \\cdot \\widetilde { W } ( t ) ( y - u _ 0 ( t ) ) . \\end{gather*}"}
+{"id": "6842.png", "formula": "\\begin{align*} \\det \\tilde X _ k \\ne 0 , k = 1 , 2 , \\dots , m , \\ ; \\tilde X _ k = \\cos ( \\alpha ) X _ k - \\sin ( \\alpha ) U _ k \\end{align*}"}
+{"id": "8879.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\to 0 } \\Vert \\varphi ( t , \\cdot ) - g ( \\cdot ) \\Vert _ { L ^ 2 ( \\mathbb { C } ^ n , d \\mu _ n ( z ) ) } = 0 . \\end{align*}"}
+{"id": "1948.png", "formula": "\\begin{align*} \\phi ( x ) = \\begin{cases} 0 & | x | \\ge 3 / 2 , \\\\ 0 . 5 ( 1 . 5 + x ) ^ 2 , & - 3 / 2 < x \\le - 1 / 2 , \\\\ 1 + x - ( x + 0 . 5 ) ^ 2 , & - 0 . 5 < x < 0 . 5 , \\\\ 0 . 5 ( 1 . 5 - x ) ^ 2 , & 1 / 2 \\le x \\le 3 / 2 . \\end{cases} \\end{align*}"}
+{"id": "2783.png", "formula": "\\begin{align*} x _ i = \\ ! \\left \\{ \\ ! \\begin{aligned} & x ' _ i , ~ ~ ~ ~ ~ i \\in [ 1 , d _ 1 - 1 ] , \\\\ & x ' _ { i - 1 } , ~ ~ i \\in [ d _ 1 + 1 , d _ 2 ] , \\\\ & x ' _ i , ~ ~ ~ ~ ~ i \\in [ d _ 2 + 1 , n ] \\backslash \\{ \\lambda _ 1 , \\lambda _ 2 \\} . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5820.png", "formula": "\\begin{align*} E ( t ) = E ( 0 ) , t \\geqslant 0 . \\end{align*}"}
+{"id": "5766.png", "formula": "\\begin{align*} c _ * ^ { T ^ * X } \\left ( \\phi ^ * \\left ( \\overline { \\Lambda _ { k } } ^ { \\ , \\log } \\right ) \\right ) = c _ * ^ { T ^ * ( X , D ) } \\left ( \\overline { \\Lambda _ { k } } ^ { \\ , \\log } \\right ) . \\end{align*}"}
+{"id": "4668.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { \\min \\{ 2 p + 1 , 2 m \\} } A _ { m , \\ell } & = \\sum _ { m = 0 } ^ { p } \\sum _ { \\ell = 0 } ^ { 2 m } A _ { m , \\ell } + \\sum _ { m = p + 1 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { 2 p + 1 } A _ { m , \\ell } = \\sum _ { m = 0 } ^ { p } \\sum _ { \\ell = 0 } ^ { 2 m } A _ { m , \\ell } + \\sum _ { m = p + 1 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { 2 p } A _ { m , \\ell } + \\sum _ { m = p + 1 } ^ { \\infty } A _ { m , 2 p + 1 } . \\end{align*}"}
+{"id": "5502.png", "formula": "\\begin{align*} ( s \\cdot \\jmath ^ * \\circ ^ { - 1 } + ( 1 - s ) \\cdot ^ { - 1 } \\circ \\jmath ^ * ) e _ k = e _ { k - 1 } . \\end{align*}"}
+{"id": "44.png", "formula": "\\begin{align*} & \\phantom { = } \\frac { \\delta ^ { m _ 2 } - 1 } { \\delta - 1 } \\cdot \\frac { \\log \\left ( \\frac { 1 6 2 } { 9 7 } \\cdot X _ 0 \\right ) } { \\log ( 2 ) } . \\end{align*}"}
+{"id": "331.png", "formula": "\\begin{align*} C _ \\phi ( l ) = G _ \\phi ( l ) ( \\frac { 2 } { 3 } a - \\log \\pi + \\frac { \\check { \\Phi } ^ { ' } ( 0 ) } { \\check { \\Phi } ( 0 ) } ) + \\frac { 4 } { 3 } G ^ { ' } _ \\phi ( 1 ; l ) . \\end{align*}"}
+{"id": "4508.png", "formula": "\\begin{align*} \\log \\prod _ { i = 1 } ^ k v _ i = \\sum _ { i = 1 } ^ k \\log v _ i = \\sum _ { i = 1 } ^ k b _ i \\leq \\sum _ { i = 1 } ^ k a _ i = \\sum _ { i = 1 } ^ k \\log u _ i = \\log \\prod _ { i = 1 } ^ k u _ i \\end{align*}"}
+{"id": "8113.png", "formula": "\\begin{align*} k _ 0 ^ * ( \\xi ) = \\widehat { k ^ * } ( \\xi ) e \\Bigl ( - \\frac { \\pi ^ 3 x \\xi ^ 4 } { 4 8 M ^ 4 } \\Bigr ) \\end{align*}"}
+{"id": "7063.png", "formula": "\\begin{align*} X ^ \\xi \\to ( \\xi ^ { \\mathbf { C } ^ \\infty } ( \\mathbf { C } ^ n ) \\to ^ { \\mathbf { C } ^ \\infty } ( \\mathbf { C } ^ n ) ) = \\Omega _ { C _ 2 } ( \\mathbf { C } ^ n ) \\end{align*}"}
+{"id": "213.png", "formula": "\\begin{align*} L _ { N } : = - ( \\mathbb { E } _ { 8 } ^ { \\oplus ( \\lceil \\frac { b _ { 2 } - 3 } { 8 } \\rceil + \\lceil \\frac { b _ { 2 } - 1 } { 8 } \\rceil ) } \\oplus \\mathbb { H } ^ { \\oplus 2 } ) , \\end{align*}"}
+{"id": "2233.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m } n + 5 ^ { 2 m } \\right ) } q ^ n & \\equiv x _ { 2 m , 1 } E ( q ^ 5 ) ^ 2 E ( q ^ { 5 0 } ) \\\\ & \\quad \\times \\left ( \\dfrac { 1 } { R ( q ^ { 1 0 } ) } - q ^ 2 - q ^ 4 R ( q ^ { 1 0 } ) \\right ) \\pmod { 5 ^ { 2 m } } , \\end{align*}"}
+{"id": "2884.png", "formula": "\\begin{align*} \\mathrm { I m } ( \\Psi _ { s _ 1 } ) \\cap \\mathrm { I m } ( \\Psi _ { s _ 2 } ) = \\emptyset , \\end{align*}"}
+{"id": "5787.png", "formula": "\\begin{align*} S _ r = \\begin{pmatrix} 1 & - 1 \\\\ & 1 & - 1 & \\cr & & \\ddots & \\ddots \\\\ & & & 1 & - 1 \\end{pmatrix} , 1 \\leqslant r \\leqslant p . \\end{align*}"}
+{"id": "2083.png", "formula": "\\begin{align*} \\dim _ H ( \\mathcal { W } _ m ^ { ( b ) } ( w ) ) = \\frac { m } { w + 1 } , \\ ; w \\in [ 0 , \\infty ] , \\dim _ H ( \\mathcal { W } _ m ( w ) ) = \\frac { m + 1 } { w + 1 } , \\ ; w \\in [ 1 / m , \\infty ] , \\end{align*}"}
+{"id": "4136.png", "formula": "\\begin{align*} R m ( X , Y , Z , W ) = \\frac { 1 } { 4 } \\langle H ( X , W ) , H ( Y , Z ) \\rangle - \\frac { 1 } { 4 } \\langle H ( Y , W ) , H ( X , Z ) \\rangle \\nabla H = 0 \\end{align*}"}
+{"id": "5641.png", "formula": "\\begin{align*} \\zeta ( F ) = \\sum _ { \\substack { n _ v = 1 , \\\\ v \\in V _ y ( F ) | s _ v \\in S _ 0 ( F ) } } ^ { + \\infty } \\prod _ { v \\in V _ y ( F ) | s _ v \\in S _ 0 ( F ) } ( n _ v ) ^ { - | s _ v | } \\int _ { \\Delta _ F \\setminus S _ 0 ( F ) } \\prod _ { n = 1 } ^ { N _ F } \\prod _ { s _ v \\in S _ n ( F ) } d \\omega _ { s _ v } \\prod _ { v \\in V _ y ( F ) | s _ v \\in S _ 1 ( F ) } ( z _ v ) ^ { \\sum _ { \\substack { v ' \\in V _ y ( F ) \\\\ v ' \\succ v } } n _ { v ' } } \\end{align*}"}
+{"id": "7974.png", "formula": "\\begin{align*} S ( M / L ) = S ( M ) / S ( L ) . \\end{align*}"}
+{"id": "5806.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ p g u _ s ' C _ p D e _ s + \\sum _ { s = 1 } ^ p u _ s C _ p A e _ s = 0 . \\end{align*}"}
+{"id": "7980.png", "formula": "\\begin{align*} [ m ] + [ n ] = [ m + n ] . \\end{align*}"}
+{"id": "1309.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\| \\langle \\tau _ j , \\tau _ k \\rangle \\| ^ { 2 m } \\geq \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , \\tau _ k \\rangle ^ { m } \\langle \\tau _ k , \\tau _ j \\rangle ^ { m } \\geq \\frac { n ^ 2 } { { d + m - 1 \\choose m } } , \\forall m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "3576.png", "formula": "\\begin{align*} ( \\gamma _ A ) _ { i j } = \\# \\{ \\} . \\end{align*}"}
+{"id": "3131.png", "formula": "\\begin{align*} w _ { \\gamma } = \\phi = s \\ , \\partial _ 1 w . \\end{align*}"}
+{"id": "690.png", "formula": "\\begin{align*} \\psi = G ^ \\omega + \\psi _ 0 \\end{align*}"}
+{"id": "6449.png", "formula": "\\begin{align*} \\tilde \\phi _ 2 ( X , Y , t ) & : = u ( X _ { j _ l } , Y _ { j _ l } , t _ { j _ l } ) - w ( Y _ { j _ l } , t _ { j _ l } ) - \\varphi _ { j _ l } ( X _ { j _ l } , Y _ { j _ l } , t _ { j _ l } , \\tilde X _ { j _ l } , \\tilde Y _ { j _ l } , \\tilde t _ { j _ l } ) \\\\ & + \\varphi _ { j _ l } ( X , Y , t , \\tilde X _ { j _ l } , \\tilde Y _ { j _ l } , \\tilde t _ { j _ l } ) + w ( Y , t ) , \\end{align*}"}
+{"id": "525.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb { X } , y \\in \\mathbb { Y } } \\Psi ( x , y ) : = f ( x ) + g ( y ) + H ( x , y ) , \\end{align*}"}
+{"id": "9003.png", "formula": "\\begin{align*} \\forall y \\in G \\colon \\vert \\{ x \\in G \\mid \\exists g \\in K \\colon \\ , \\gamma ( g , x ) = y \\} \\vert \\ , = \\ , \\vert K \\vert . \\end{align*}"}
+{"id": "5094.png", "formula": "\\begin{align*} ( 1 - \\alpha _ 1 ^ n u _ 1 ^ n ) = ( 1 - \\alpha _ 1 u _ 1 ) \\bigg ( \\sum _ { j = 0 } ^ { n - 1 } \\alpha _ 1 ^ j u _ 1 ^ j \\bigg ) . \\end{align*}"}
+{"id": "4139.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\lambda ( h , 0 ) & \\leq \\int _ M \\langle \\frac { 1 } { 2 } \\triangle _ L h + ^ * h , h \\rangle - \\frac { 1 } { 2 } h _ { i j } h _ { a c } H _ { i a b } H _ { j c b } d V _ g \\\\ & = \\int _ M \\langle \\frac { 1 } { 2 } \\triangle h + ^ * h , h \\rangle - \\langle \\mathring { R } h , h \\rangle - R _ { i j } h _ { i k } h _ { j k } d V _ g . \\end{align*}"}
+{"id": "6311.png", "formula": "\\begin{align*} X ^ i _ { ( k + 1 ) c } = \\begin{cases} X ^ i _ { k c } + N ^ s _ { ( k + 1 ) c } - N ^ s _ { k c } , & \\\\ N ^ s _ { ( k + 1 ) c } - N ^ s _ { k c } , & \\\\ \\end{cases} \\end{align*}"}
+{"id": "6917.png", "formula": "\\begin{align*} \\frac { d W } { d t } ( t ) = u ( t ) S ( t ) , \\ \\ \\ W ( 0 ) = 0 , \\\\ \\end{align*}"}
+{"id": "6028.png", "formula": "\\begin{align*} P _ { l _ 1 + l _ 2 } ( h _ 1 , w _ { k , p } ) = \\left \\{ \\begin{array} { l r } 0 & \\mathrm { i f } \\ : ( k , p ) \\neq ( 1 , n ) \\\\ Q _ 1 Q _ 2 \\left ( [ \\O _ X ] - \\O _ { h _ 1 } \\right ) & \\mathrm { i f } \\ : ( k , p ) = ( 1 , n ) \\end{array} \\right . \\end{align*}"}
+{"id": "5484.png", "formula": "\\begin{align*} L V ( x , 1 ) & = \\sigma ^ 2 ( x , \\mu , 1 ) + b ( x , \\mu , 1 ) \\cdot 2 x \\\\ & \\leq ( \\int _ { \\mathbb R } ( x + \\beta y ) \\mu ( d y ) ) ^ 2 - 4 x \\int _ { \\mathbb R } ( x + \\beta y ) \\mu ( d y ) \\\\ & = - 3 x ^ 2 - 2 \\beta x \\int _ { \\mathbb R } y \\mu ( d y ) + \\beta ^ 2 ( \\int _ { \\mathbb R } y \\mu ( d y ) ) ^ 2 \\\\ & \\leq - 2 x ^ 2 + 2 \\beta ^ 2 \\int _ { \\mathbb R } x ^ 2 \\mu ( d x ) , \\end{align*}"}
+{"id": "4918.png", "formula": "\\begin{align*} \\mathcal { F } ( \\mathbf { z } _ n , \\mathbf { z } _ { n + 1 } ) _ m = \\left ( \\frac { \\delta \\widetilde H } { \\delta ( \\mathbf { z _ { n + 1 } ^ * } , \\mathbf { z _ { n } ^ * } ) } \\right ) _ m \\end{align*}"}
+{"id": "5051.png", "formula": "\\begin{align*} A _ \\nu ( x _ 1 , \\ldots , x _ d ) \\coloneqq B _ \\nu \\Big ( p _ 1 / N + \\sum _ { j = 1 } ^ d x _ j q _ { 1 , j } / N , \\ , \\ldots , \\ , p _ s / N + \\sum _ { j = 1 } ^ d x _ j q _ { s , j } / N \\Big ) \\prod _ { i = 1 } ^ s \\gamma _ { \\nu , i } ^ { p _ i } . \\end{align*}"}
+{"id": "202.png", "formula": "\\begin{align*} \\mu ( I _ { n , a } ^ { [ i ] } \\cap B ) - \\mu ( I _ { n , a } ^ { [ i ] } ) \\mu ( B ) = \\frac { 1 } { r _ { n } + 1 } \\big { ( } \\mu ( I _ { n , a } \\cap B ) - \\mu ( I _ { n , a } ) \\mu ( B ) \\big { ) } . \\end{align*}"}
+{"id": "5163.png", "formula": "\\begin{align*} \\left ( \\prod ^ { k - 1 } _ { j = 0 } \\dfrac { 1 } { P ( D _ { j } ) } \\right ) ^ { \\frac { 1 } { k ( k - 1 ) } } & = \\prod ^ { k - 1 } _ { j = 0 } \\left ( n _ { 0 , j } \\frac { n _ { 1 , j } } { 3 } \\frac { n _ { 2 , j } } { 3 } { n _ { 3 , j } } \\ldots n _ { k - 2 , j } \\frac { n _ { k - 1 , j } } { 3 } \\right ) ^ { \\frac { 1 } { k ( k - 1 ) } } \\end{align*}"}
+{"id": "14.png", "formula": "\\begin{align*} \\begin{aligned} - d q ^ i _ t & = \\bigg ( \\bar { \\tilde { b } } _ x q ^ i _ t + \\bar { \\sigma } _ x m ^ i _ t + \\bar { \\tilde { \\sigma } } _ x \\tilde { m } ^ i _ t + \\int _ { \\mathcal { E } } \\bar { l } _ x n ^ i _ { ( t , e ) } \\nu ( d e ) - \\beta q ^ i _ t + \\kappa ^ i _ t \\bigg ) d t \\\\ & \\quad - m ^ i _ t d W _ t - \\tilde { m } ^ i _ t d \\xi _ t - \\int _ { \\mathcal { E } } n ^ i _ { ( t , e ) } \\tilde { N } ( d e , d t ) , \\\\ \\end{aligned} \\end{align*}"}
+{"id": "5838.png", "formula": "\\begin{align*} \\begin{cases} U '' - { \\Delta } U + A U = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ U = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\cr \\partial _ \\nu U = D H & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{cases} \\end{align*}"}
+{"id": "2967.png", "formula": "\\begin{align*} \\tilde { \\mathcal { X } } ^ n _ t ( \\varphi ) = \\frac { 1 } { \\sqrt { n } } \\sum _ { j \\in \\mathbb { Z } } \\overline { W } _ j ( t ) \\varphi ^ n _ j ( t ) \\end{align*}"}
+{"id": "8289.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } H _ i \\ ! = \\ ! ( x _ E \\ ! - \\ ! x _ i ) x \\ ! + \\ ! ( y _ E \\ ! - \\ ! y _ i ) y \\ ! + \\ ! ( z _ E \\ ! - \\ ! z _ i ) z - \\frac { R _ E ^ 2 - R _ i ^ 2 } { 2 } \\end{array} \\right . \\end{align*}"}
+{"id": "5881.png", "formula": "\\begin{align*} C _ p A = \\overline A _ p C _ p , C _ p B = \\overline B _ p C _ p . \\end{align*}"}
+{"id": "1062.png", "formula": "\\begin{align*} \\hat { \\beta } _ { j k } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Z _ { i j k } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\left \\{ \\psi _ { j k } ( X _ i ) + \\sigma _ J W _ { i j k } \\right \\} . \\end{align*}"}
+{"id": "8542.png", "formula": "\\begin{align*} i = 2 & \\implies S ( n - i + 1 ) - S ( n ) = - \\frac { 1 } { 2 } n ^ 2 + \\frac { 1 } { 2 } n , \\\\ i = 3 & \\implies S ( n - i + 1 ) - S ( n ) = - n ^ 2 + 2 n - 1 , \\\\ i = 4 & \\implies S ( n - i + 1 ) - S ( n ) = - \\frac { 3 } { 2 } n ^ 2 + \\frac { 9 } { 2 } n - 4 . \\end{align*}"}
+{"id": "3610.png", "formula": "\\begin{align*} A ( t ) \\ = \\ \\frac { 2 } { \\pi } \\sqrt { 1 - t ^ 2 } \\ \\chi _ { I } ( t ) \\ , . \\end{align*}"}
+{"id": "2621.png", "formula": "\\begin{align*} \\frac { ( 3 - 2 ) ( k + 1 ) } { 2 } = s = a _ { 2 , 3 } \\leftrightarrow ( a _ { 1 , 2 } , a _ { 1 , 3 } , a _ { 2 , 3 } ) = ( s , s , s ) . \\end{align*}"}
+{"id": "3069.png", "formula": "\\begin{align*} r = \\xi = 1 + M : D ^ 2 w . \\end{align*}"}
+{"id": "7055.png", "formula": "\\begin{align*} d _ { i , j } - c _ { i , j } & = u d _ { i , j + 1 } \\\\ q _ j - p _ j & = u q _ { j + 1 } \\\\ q _ 0 & = 0 \\end{align*}"}
+{"id": "5892.png", "formula": "\\begin{align*} D ^ T C _ p ^ T E = 0 . \\end{align*}"}
+{"id": "7648.png", "formula": "\\begin{align*} \\nabla _ { \\omega } ( \\eta ) = d ( \\eta ) + \\omega _ { \\lambda } \\wedge \\eta . \\end{align*}"}
+{"id": "2368.png", "formula": "\\begin{align*} \\mathcal H ( I , V _ 1 , V _ 2 ) = \\frac 1 2 L I ^ 2 + \\frac 1 2 C _ 1 V _ 1 ^ 2 + \\frac 1 2 C _ 2 V _ 2 ^ 2 . \\end{align*}"}
+{"id": "7406.png", "formula": "\\begin{align*} s _ { 2 \\alpha + \\beta } ( \\chi ^ 2 \\chi '^ { - 1 } ( \\varpi _ L ) q _ L ^ { - s } ) = \\chi ^ 2 \\chi '^ { - 1 } ( \\varpi _ L ) q _ L ^ { - s } \\end{align*}"}
+{"id": "1695.png", "formula": "\\begin{align*} \\delta f _ n = \\frac { 1 } { 2 } \\left ( \\frac { k } { 2 } + n \\right ) f _ { n + 1 } + \\frac { 1 } { 2 } \\left ( \\frac { k } { 2 } - n \\right ) f _ { n - 1 } . \\end{align*}"}
+{"id": "17.png", "formula": "\\begin{align*} \\begin{aligned} & H ( t , x , y , z , \\tilde { z } , \\gamma , \\mathcal { Z } , u ; p , q , m , \\tilde { m } , n , \\tilde { M } ) : = g ( x , y , z , \\tilde { z } , \\gamma , u ) p + \\tilde { b } ( x , u ) q + \\sigma ( x , u ) m \\\\ & + \\tilde { \\sigma } ( x , u ) \\tilde { m } + \\int _ { \\mathcal { E } } l ( x , u , e ) n _ { e } \\nu ( d e ) + e ^ { - \\frac { \\beta } { 2 } t } h ( x , u ) \\tilde { M } \\mathcal { Z } + f ( x , y , z , \\tilde { z } , \\gamma , u ) \\mathcal { Z } . \\end{aligned} \\end{align*}"}
+{"id": "2979.png", "formula": "\\begin{align*} ( \\overrightarrow { W } _ { j + 1 } ^ \\ell ) ^ 2 ( \\overline { W } _ { j - 1 } - \\overrightarrow { W } ^ \\ell _ j ) = ( \\overrightarrow { W } _ { j + 1 } ^ \\ell ) ^ 2 \\sum _ { i = 0 } ^ { \\ell - 1 } \\psi _ i ( W _ { j + i - 1 } - W _ { j + i } ) \\end{align*}"}
+{"id": "2038.png", "formula": "\\begin{align*} \\nu _ { x _ 0 } ( n ) : = \\mathbb { E } _ { \\rho _ { x _ { 0 } } } \\left ( \\displaystyle \\sum _ { j = 0 } ^ { C _ 1 - 1 } \\mathbb { 1 } _ { n } ( X _ j ^ { ( 0 ) } ) \\right ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , n \\in \\mathcal { I } ( r _ { x _ 0 } ) \\end{align*}"}
+{"id": "4235.png", "formula": "\\begin{align*} V ( x ) = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { N } \\gamma _ j ^ 2 x _ j ^ 2 , \\gamma _ j > 0 , j = 1 , \\cdots , N \\end{align*}"}
+{"id": "7022.png", "formula": "\\begin{align*} F _ { M U } ( u , x ) & = u + ( \\sum a _ { 1 , j } u ^ j ) x + ( \\sum a _ { 2 , j } u ^ j ) x ^ 2 + ( \\sum a _ { 3 , j } u ^ j ) x ^ 3 + \\cdots \\in M U _ * [ [ u ] ] [ [ x ] ] \\end{align*}"}
+{"id": "1163.png", "formula": "\\begin{align*} \\forall i \\in \\{ i _ 0 , i _ 1 , . . . , i _ g \\} , \\ , H \\restriction _ { X ' } ( i ) = \\langle i , a \\rangle \\iff H ( i ) = \\langle i , a \\rangle . \\end{align*}"}
+{"id": "1253.png", "formula": "\\begin{align*} ( ( u - 1 ) ^ 2 + v ^ 2 ) ^ 2 = \\left ( \\dfrac { u - 1 } { 1 - \\alpha } \\right ) ^ 2 + \\left ( \\dfrac { v } { 1 + \\alpha } \\right ) ^ 2 . \\end{align*}"}
+{"id": "7112.png", "formula": "\\begin{align*} f = \\sum _ { t = 1 } ^ m q _ t ( u x _ { i _ t } + p _ { i _ t } ) \\in R [ x _ i ] . \\end{align*}"}
+{"id": "5203.png", "formula": "\\begin{align*} \\begin{cases} g _ { 1 3 4 } + g _ { 2 3 5 } = 0 \\\\ g _ { 1 3 5 } - g _ { 2 3 4 } = 0 \\end{cases} \\ , \\ \\begin{cases} g _ { 1 3 4 } = 0 \\\\ g _ { 2 3 5 } = 0 \\end{cases} \\ \\mathrm { a n d } \\ \\begin{cases} g _ { 1 3 4 } + g _ { 2 3 5 } = 0 \\\\ g _ { 1 3 5 } = 0 \\end{cases} \\ . \\end{align*}"}
+{"id": "8915.png", "formula": "\\begin{align*} \\sum \\limits _ { d = n } ^ { + \\infty } \\left ( \\sum \\limits _ { p = 0 } ^ { n - 1 } \\frac { ( - 1 ) ^ { d - n + 1 } \\gamma _ { p } ^ { ( \\nu , n ) } B _ { 2 ( d - n + p + 1 ) } \\left ( \\nu + \\frac { 1 } { 2 } \\right ) } { ( n - 1 ) ! ( d - n + p + 1 ) ( d - n ) ! } \\right ) t ^ { d } . \\end{align*}"}
+{"id": "7385.png", "formula": "\\begin{align*} \\mu ^ { M _ \\alpha } ( \\sigma \\otimes \\cdot ) = c _ { s _ { \\alpha } } ' \\ , \\frac { ( 1 - X _ { \\alpha } ) ( 1 - X _ { \\alpha } ^ { - 1 } ) } { ( 1 - q _ { \\alpha } ^ { - 1 } X _ { \\alpha } ) ( 1 - q _ { \\alpha } ^ { - 1 } X _ { \\alpha } ^ { - 1 } ) } \\cdot \\frac { ( 1 + X _ { \\alpha } ) ( 1 + X _ { \\alpha } ^ { - 1 } ) } { ( 1 + q _ { \\alpha ^ * } ^ { - 1 } X _ { \\alpha } ) ( 1 + q _ { \\alpha ^ * } ^ { - 1 } X _ { \\alpha } ^ { - 1 } ) } . \\end{align*}"}
+{"id": "5027.png", "formula": "\\begin{align*} E _ 1 & = \\frac { 1 } { \\mu } \\ , e _ 1 , E _ i = \\frac { 1 } { a } \\ , e _ i , ~ i = 2 , 3 , \\\\ E _ j & = - \\frac { c } { a \\sqrt { a ^ 2 b ^ 2 - c ^ 2 } } \\ , e _ { j - 2 } + \\frac { a } { \\sqrt { a ^ 2 b ^ 2 - c ^ 2 } } \\ , e _ j , ~ j = 4 , 5 . \\end{align*}"}
+{"id": "5864.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } w '' - \\Delta w = { \\cal L } \\theta & \\hbox { i n } ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu w = { \\cal R } \\theta & \\hbox { o n } ( 0 , T ) \\times \\Gamma , \\\\ t = 0 : w = 0 , w ' = \\theta & \\hbox { i n } ~ \\Omega \\end{array} \\right . \\end{align*}"}
+{"id": "3880.png", "formula": "\\begin{align*} \\det [ D ^ 2 u - A ( \\cdot , u , D u ) ] = B ( \\cdot , u , D u ) , \\end{align*}"}
+{"id": "7324.png", "formula": "\\begin{align*} H ^ k ( \\gamma ; x ) = \\left \\{ \\begin{aligned} & - G _ m ( x ) & ( k = 1 ) , \\\\ & \\int _ { 0 } ^ { x } d y \\int _ { y } ^ { \\infty } H ^ { k - 1 } ( \\infty ; z ) d m ( z ) & ( 2 \\leq k \\leq N ) , \\\\ & \\int _ { 0 } ^ { x } d y \\int _ { y } ^ { \\gamma } H ^ { k - 1 } ( \\gamma ; z ) d m ( z ) & ( k > N ) \\end{aligned} \\right . \\end{align*}"}
+{"id": "6869.png", "formula": "\\begin{gather*} ( u _ h ^ { \\bot } - u _ h , v ) _ h = e ( u , v ) v \\in X _ h \\\\ \\norm { u _ h ^ { \\bot } - u _ h } _ h = \\underset { v \\in X _ h } { \\sup } \\frac { ( u _ h ^ { \\bot } - u _ h , v ) _ h } { \\norm { v } _ h } = \\underset { v \\in X _ h } { \\sup } \\frac { e ( u , v ) _ h } { \\norm { v } _ h } . \\end{gather*}"}
+{"id": "1198.png", "formula": "\\begin{align*} \\Psi \\circ S _ \\lambda = \\Psi ^ { \\log \\lambda } \\circ \\Psi . \\end{align*}"}
+{"id": "5138.png", "formula": "\\begin{align*} C _ { 1 } ( \\lambda , b , \\mathbf { m } ) & : = \\displaystyle \\frac { \\Lambda _ { 1 } ( \\lambda , b ) - b \\Omega _ { \\mathbf { m } } ^ { \\pm } ( \\lambda , b ) - b I _ { 1 } ( \\lambda b ) K _ { 1 } ( \\lambda b ) } { \\mathbf { m } } , \\\\ C _ { 2 } ( b , \\mathbf { m } ) & : = \\displaystyle - \\frac { b } { \\mathbf { m } } . \\end{align*}"}
+{"id": "8945.png", "formula": "\\begin{align*} ( - n ) _ { k } = \\left \\{ \\begin{array} { c l } \\frac { ( - 1 ) ^ { k } n ! } { ( n - k ) ! } , & 0 \\leq k \\leq n , \\\\ 0 , & k > n . \\end{array} \\right . \\end{align*}"}
+{"id": "7149.png", "formula": "\\begin{align*} \\{ v _ i , z _ j \\} = \\delta _ { i j } \\ , , \\{ v _ i , v _ j \\} = \\{ z _ i , z _ j \\} = 0 \\ , . \\end{align*}"}
+{"id": "301.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { \\alpha \\in H ^ 1 ( \\mathrm { G a l } ( E / F ) , G ( E ) ) } \\mathrm { d i m } \\mathrm { H o m } _ { G _ { \\alpha } ( F ) } ( \\pi , \\omega _ { G _ { \\alpha } ( F ) , E } ) = \\sum \\limits _ { \\tilde { \\phi } } m ( \\lambda _ { \\pi } , \\tilde { \\phi } ) , \\end{aligned} \\end{align*}"}
+{"id": "6125.png", "formula": "\\begin{align*} D ^ T ( B ^ T ) ^ s ( A ^ T ) ^ r \\cdots ( B ^ T ) ^ q ( A ^ T ) ^ p y = 0 \\end{align*}"}
+{"id": "3453.png", "formula": "\\begin{align*} 4 p = ( 2 \\lambda - \\delta ) ^ 2 + ( 2 \\mu - \\delta ) ^ 2 + 1 0 \\delta ^ 2 . \\end{align*}"}
+{"id": "6303.png", "formula": "\\begin{align*} s _ i s _ { i + 1 } s _ i ( T ) = s _ { i + 1 } s _ i s _ { i + 1 } ( T ) , \\end{align*}"}
+{"id": "1584.png", "formula": "\\begin{align*} \\mu = \\eta + \\alpha \\delta _ u \\quad \\mbox { a n d } \\nu = \\eta + \\alpha \\delta _ v . \\end{align*}"}
+{"id": "2051.png", "formula": "\\begin{align*} \\alpha = \\frac { \\rho _ 2 - \\rho _ 1 } { \\rho _ 2 + \\rho _ 1 } , \\omega ' = c _ 1 ^ { \\frac { \\alpha + 1 } { 2 } } c _ 2 ^ { \\frac { \\alpha - 1 } { 2 } } \\omega , \\end{align*}"}
+{"id": "7513.png", "formula": "\\begin{align*} | g ( u , v ) | = & | \\alpha _ { i _ 0 } u ^ { i _ 0 } + \\sum _ { i = i _ 0 + 1 } ^ { \\infty } \\alpha _ i u ^ i + \\sum \\limits _ { j = j _ 0 } ^ { \\infty } \\beta _ j v ^ j | \\\\ = & q ^ { - e _ 1 } | \\pi ^ { - e _ 1 } \\alpha _ { i _ 0 } u ^ { i _ 0 } + \\pi ^ { - e _ 1 } \\big ( \\sum _ { i = i _ 0 + 1 } ^ { \\infty } \\alpha _ i u ^ i + \\sum \\limits _ { j = j _ 0 } ^ { \\infty } \\beta _ j v ^ j \\big ) | \\\\ = & q ^ { - e _ 1 } . \\end{align*}"}
+{"id": "683.png", "formula": "\\begin{align*} { \\bf V } = \\dot { x } \\frac { \\partial } { \\partial x } + \\dot { y } \\frac { \\partial } { \\partial y } = \\dot { z } \\frac { \\partial } { \\partial z } + \\dot { \\bar { z } } \\frac { \\partial } { \\partial \\bar { z } } , \\end{align*}"}
+{"id": "8530.png", "formula": "\\begin{align*} E ( u _ 0 , d _ 0 ) : = \\frac 1 2 \\int _ { B _ 1 ^ 3 } ( | u _ 0 | ^ 2 + | \\nabla d _ 0 | ^ 2 ) \\le \\epsilon _ 0 ^ 2 . \\end{align*}"}
+{"id": "2699.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u - \\Delta u = u ^ 2 , \\end{align*}"}
+{"id": "8937.png", "formula": "\\begin{align*} T r \\left ( e ^ { \\frac { 1 } { 4 } t \\Delta _ { \\nu } } \\right ) \\simeq \\frac { \\omega _ { n } } { ( 4 \\pi t ) ^ { n } } \\left [ \\sum \\limits _ { k = 0 } ^ { + \\infty } c _ { k } ^ { \\left ( \\nu , n \\right ) } \\ t ^ { k } \\right ] \\left [ \\sum \\limits _ { j = 0 } ^ { + \\infty } \\frac { 1 } { j ! } \\left ( \\frac { n ^ { 2 } } { 4 } + \\nu ^ { 2 } \\right ) ^ { j } \\right ] . \\end{align*}"}
+{"id": "6670.png", "formula": "\\begin{align*} x ^ { k + 1 } = ( I + \\gamma ^ k W \\otimes I _ d ) x ^ k + \\gamma ^ k \\zeta _ w ^ k - \\lambda ^ k g ^ k \\end{align*}"}
+{"id": "5257.png", "formula": "\\begin{align*} g ( A _ X X + T _ U X , U ) = - a \\sin \\omega ( t ) \\cos \\omega ( t ) \\frac { d \\omega } { d t } . \\end{align*}"}
+{"id": "2088.png", "formula": "\\begin{align*} \\dim ( K ) = \\sum _ { i = 1 } ^ { m } \\dim ( C _ { b , W } ) = \\frac { \\sum _ { i = 1 } ^ { m } \\log | W _ i | } { \\log b } . \\end{align*}"}
+{"id": "3568.png", "formula": "\\begin{align*} y _ { j m } & = \\mathbf { p } E _ { j m } ( h _ j - h _ 1 ) \\cdots ( h _ j - h _ { j - 1 } ) \\\\ y _ { m j } & = \\mathbf { p } E _ { m j } ( h _ j - h _ { j + 1 } ) \\cdots ( h _ j - h _ { m - 1 } ) , \\end{align*}"}
+{"id": "398.png", "formula": "\\begin{align*} a \\| h \\| ^ 2 \\leq \\sum _ { n = 1 } ^ \\infty | \\langle h , \\tau _ n \\rangle | ^ 2 \\leq b \\| h \\| ^ 2 , \\forall h \\in \\mathcal { H } . \\end{align*}"}
+{"id": "3716.png", "formula": "\\begin{align*} \\Theta _ f ( g ) = \\sum _ { \\xi \\in V _ { i + 1 } ( F ) } r _ i ( g ) \\mathcal { F } _ 2 ( f ) ( \\xi ) . \\end{align*}"}
+{"id": "6593.png", "formula": "\\begin{align*} s _ 1 ( k , i ) = \\left \\{ \\begin{array} { l l } ( k + 1 , i ) & k < n \\\\ ( 1 , i + 1 ) & k = n \\end{array} \\right . \\end{align*}"}
+{"id": "412.png", "formula": "\\begin{align*} U V x = U \\left ( \\sum _ { n = 1 } ^ { \\infty } f _ n ( x ) \\omega _ n \\right ) = \\sum _ { n = 1 } ^ { \\infty } f _ n ( x ) U \\omega _ n = \\sum _ { n = 1 } ^ { \\infty } f _ n ( x ) \\tau _ n = x , \\forall x \\in \\mathcal { X } , \\end{align*}"}
+{"id": "4898.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\partial _ s x ( s ) + \\nabla f ( x ( s ) ) + \\tau ( s ) \\nabla h ( x ( s ) ) = 0 \\\\ h ( x ( s ) ) = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "2259.png", "formula": "\\begin{align*} x = ( k _ 1 - 1 ) y _ 1 + \\cdots + ( k _ r - 1 ) y _ r . \\end{align*}"}
+{"id": "1412.png", "formula": "\\begin{align*} M _ { \\rho , x } = \\sum _ { \\substack { z \\in ( 1 / 2 ) \\Z \\\\ z \\geq x } } k _ { \\rho , z } . \\end{align*}"}
+{"id": "8460.png", "formula": "\\begin{align*} \\| \\gamma \\| ^ 2 = \\lim _ { t \\in T } \\ , \\| \\gamma _ { A _ t } \\| ^ 2 = \\lim _ { t \\in T } \\ , c _ * ( A _ t ) \\geqslant c _ * ( A ) . \\end{align*}"}
+{"id": "4013.png", "formula": "\\begin{align*} W [ u ] & : = [ D _ { \\alpha \\beta } u - A _ { \\alpha \\beta } ( x , u , D u ) ] \\xi _ \\alpha \\xi _ \\beta , \\\\ & = [ D _ { \\alpha \\beta } \\phi - A _ { \\alpha \\beta } ( x , \\phi , D ' \\phi , D _ n u ) ] \\xi _ \\alpha \\xi _ \\beta . \\end{align*}"}
+{"id": "5281.png", "formula": "\\begin{align*} s e c ( U , V ) & = \\hat { s } e c ( U , V ) - \\lambda ^ 2 \\mid \\nabla ( f + \\log \\lambda ) \\mid ^ 2 + \\frac { \\lambda ^ 4 } { 4 } \\mid g r a d _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\mid ^ 2 - \\lambda ^ 2 g \\left ( \\nabla f , \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\right ) . \\end{align*}"}
+{"id": "8518.png", "formula": "\\begin{align*} \\alpha = ( \\frac { P _ z + Q _ w } { P ^ 2 + Q ^ 2 } ) ( P d z + Q d w ) . \\end{align*}"}
+{"id": "7056.png", "formula": "\\begin{align*} ( F / u ) G = F ( G / u ) \\end{align*}"}
+{"id": "2590.png", "formula": "\\begin{align*} \\hat { \\mathfrak { k } } & = \\{ ( u ( t ) , \\sigma ( u ( - t ) ) ) \\mid u \\in L ( \\mathfrak { g } , \\sigma ) \\} , \\\\ \\hat { \\mathfrak { m } } & = \\{ ( u ( t ) , - \\sigma u ( - t ) ) + \\alpha c + \\beta d \\mid u \\in L ( \\mathfrak { g } , \\sigma ) , \\ \\alpha , \\beta \\in \\mathbb { R } \\} . \\end{align*}"}
+{"id": "1109.png", "formula": "\\begin{align*} \\widehat { \\theta } = \\frac { 1 } { \\sum _ { j = 1 } ^ n \\eta _ j } \\sum _ { i = 1 } ^ n \\eta _ i \\theta _ i . \\end{align*}"}
+{"id": "8164.png", "formula": "\\begin{align*} \\mathcal { D } = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } \\delta ( n , p ) H _ { m , n } , \\end{align*}"}
+{"id": "4917.png", "formula": "\\begin{align*} \\delta _ n ^ + z _ { m , n } = - \\mathrm { i } \\mathcal { F } ( \\mathbf { z } _ n , \\mathbf { z } _ { n + 1 } ) _ m , \\end{align*}"}
+{"id": "8047.png", "formula": "\\begin{align*} L ( s , f \\times E ) = \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { \\bar \\eta \\Bigl ( n , \\dfrac { 1 } { 2 } + i t \\Bigr ) A ( n , m ) } { ( m ^ 2 n ) ^ s } \\end{align*}"}
+{"id": "6163.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ d \\alpha _ r E _ r = 0 . \\end{align*}"}
+{"id": "8260.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } e ^ { a s ^ 2 + b s } d s = \\frac { 1 } { 2 \\sqrt { \\pi a } } \\exp \\left ( - \\frac { b ^ 2 } { 4 a } \\right ) . \\end{align*}"}
+{"id": "2303.png", "formula": "\\begin{align*} u ^ { ( 1 ) } & = u ^ n + \\Delta t L ( u ^ n ) , \\\\ u ^ { ( 2 ) } & = \\frac { 3 } { 4 } u ^ n + \\frac { 1 } { 4 } u ^ { ( 1 ) } + \\frac { 1 } { 4 } \\Delta t L \\left ( u ^ { ( 1 ) } \\right ) , \\\\ u ^ { n + 1 } & = \\frac { 1 } { 3 } u ^ n + \\frac { 2 } { 3 } u ^ { ( 2 ) } + \\frac { 2 } { 3 } \\Delta t L \\left ( u ^ { ( 2 ) } \\right ) , \\end{align*}"}
+{"id": "4006.png", "formula": "\\begin{align*} L ( \\log \\eta ) = \\frac { L \\eta } { \\eta } - \\sum _ { i = 1 } ^ n w ^ { i i } \\left ( \\frac { D _ i \\eta } { \\eta } \\right ) ^ 2 . \\end{align*}"}
+{"id": "8525.png", "formula": "\\begin{align*} X \\sigma = \\{ 1 3 2 5 4 , 5 2 3 1 4 , 2 4 3 1 5 , 1 5 4 3 2 , 3 4 5 1 2 , 4 2 5 1 3 \\} \\end{align*}"}
+{"id": "1924.png", "formula": "\\begin{align*} - a \\Delta _ \\lambda u _ { \\lambda , \\sigma } + ( R _ \\lambda - \\chi _ \\lambda R _ g ) u _ { \\lambda , \\sigma } & = 0 \\quad M _ \\sigma , \\\\ \\nu _ g ( u _ { \\lambda , \\sigma } ) & = 0 \\quad \\partial M _ \\sigma , \\\\ u _ { \\lambda , \\sigma } & \\to 1 \\quad , \\end{align*}"}
+{"id": "7177.png", "formula": "\\begin{align*} \\mathcal { H } = \\bigcup _ { t \\in [ 0 , \\ , \\infty ] } \\{ t \\} \\times H _ t . \\end{align*}"}
+{"id": "8307.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } a ( \\textbf { x } ) = \\frac { 1 } { 2 } \\frac { ( y _ 2 - y _ 3 ) R _ 1 ^ 2 + ( y _ 3 - y _ 1 ) R _ 2 ^ 2 + ( y _ 1 - y _ 2 ) R _ 3 ^ 2 } { x _ 1 ( y _ 2 - y _ 3 ) + x _ 2 ( y _ 3 - y _ 1 ) + x _ 3 ( y _ 1 - y _ 2 ) } \\\\ b ( \\textbf { x } ) = \\frac { 1 } { 2 } \\frac { ( x _ 3 - x _ 2 ) R _ 1 ^ 2 + ( x _ 1 - x _ 3 ) R _ 2 ^ 2 + ( x _ 2 - x _ 1 ) R _ 3 ^ 2 } { x _ 1 ( y _ 2 - y _ 3 ) + x _ 2 ( y _ 3 - y _ 1 ) + x _ 3 ( y _ 1 - y _ 2 ) } . \\end{array} \\right . \\end{align*}"}
+{"id": "2099.png", "formula": "\\begin{align*} \\dim _ H ( \\mathcal { W } _ m ( \\infty ) + A ) \\leq \\dim _ H ( \\mathcal { W } _ m ( \\infty ) ) + \\dim _ P ( A ) = \\dim _ P ( A ) , \\end{align*}"}
+{"id": "2126.png", "formula": "\\begin{align*} \\gamma _ k \\tau _ p ^ { 2 \\mu _ k } - 2 \\mu _ k \\zeta _ k \\tau _ p = - ( 2 \\mu _ k - 1 ) \\gamma _ k \\left ( \\frac { | \\zeta _ k | } { \\gamma _ k } \\right ) ^ \\frac { 2 \\mu _ k } { 2 \\mu _ k - 1 } \\leq 0 . \\end{align*}"}
+{"id": "1191.png", "formula": "\\begin{align*} \\Upsilon ^ { * } w _ { n } = w _ { n } + A _ { n , 1 } ^ { [ 4 ] } w _ { n } ^ { 2 } + A _ { n , 2 } ^ { [ 4 ] } w _ { n } \\overline { w } _ { n } + A _ { n , 3 } ^ { [ 4 ] } \\overline { w } _ { n } ^ { 2 } , \\end{align*}"}
+{"id": "7523.png", "formula": "\\begin{align*} Z _ h ( s , \\chi , D _ { S ( h , D ) } ) = q ^ { - 2 } Z _ { h _ 1 } ( s , \\chi ) = \\dfrac { H _ 2 ( q ^ { - s } ) } { ( 1 - q ^ { - 1 - s } ) ( 1 - q ^ { - p - ( k + 1 ) - p ( k + 1 ) s } ) } , \\end{align*}"}
+{"id": "3830.png", "formula": "\\begin{align*} \\begin{aligned} E _ 4 ^ { ( 2 ) } E _ 6 ^ { ( 2 ) } = ~ & ( 1 + 2 4 0 \\tilde { q } + 2 4 0 q + \\cdots + 1 3 4 4 0 \\tilde { q } p q + 3 0 2 4 0 \\tilde { q } q + \\cdots ) \\times \\\\ & ( 1 - 5 0 4 \\tilde { q } - 5 0 4 q + \\cdots + 4 4 3 5 2 \\tilde { q } p q + 1 6 6 3 2 0 \\tilde { q } q + \\cdots ) \\\\ = ~ & 1 - 2 6 4 \\tilde { q } - 2 6 4 q + \\cdots + 5 7 7 9 2 \\tilde { q } p q - 4 5 3 6 0 \\tilde { q } q + \\cdots . \\end{aligned} \\end{align*}"}
+{"id": "4241.png", "formula": "\\begin{align*} L _ \\Omega ( u ( t ) ) + \\Omega \\int _ 0 ^ t \\int _ { \\R ^ N } i | u ( s , x ) | ^ 2 L _ z V ( x ) d x d s = L _ \\Omega ( u _ 0 ) \\end{align*}"}
+{"id": "4223.png", "formula": "\\begin{align*} \\lambda ( t ) : = \\dfrac { t } { \\log ^ 2 t } , \\frac { \\lambda ' ( t ) } { \\lambda ( t ) } = \\frac { 1 } { t } \\left ( 1 - \\dfrac { 2 } { \\log t } \\right ) . \\end{align*}"}
+{"id": "3044.png", "formula": "\\begin{align*} E _ 1 = o ( n ^ 3 ) E _ 2 = o ( n ^ 3 ) . \\end{align*}"}
+{"id": "3192.png", "formula": "\\begin{align*} r _ 1 ( t ) : = 1 + \\frac { 1 } { 3 } \\sin ( 2 \\pi t ) , r _ 2 ( t ) : = \\frac { 1 } { 3 } \\cos ( 2 \\pi t ) \\quad t \\in \\R . \\end{align*}"}
+{"id": "5095.png", "formula": "\\begin{align*} P = Q ( 1 - c u _ 1 ) + R \\end{align*}"}
+{"id": "7391.png", "formula": "\\begin{align*} x _ \\gamma ( u ) = \\zeta _ \\gamma \\left ( \\begin{matrix} 1 & u \\cr 0 & 1 \\end{matrix} \\right ) , \\ ; \\ ; \\ ; \\ ; x _ { - \\gamma } ( u ) = \\zeta _ { - \\gamma } \\left ( \\begin{matrix} 1 & 0 \\cr u & 1 \\end{matrix} \\right ) \\ ; \\ ; \\ ; \\ ; \\gamma ^ \\vee ( t ) = \\zeta _ \\gamma \\left ( \\begin{matrix} t & 0 \\cr 0 & t ^ { - 1 } \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "6364.png", "formula": "\\begin{align*} g = d r ^ 2 + m ( r ) ^ 2 d \\theta ^ 2 \\end{align*}"}
+{"id": "4030.png", "formula": "\\begin{align*} \\phi ^ * _ k D _ i Y ^ k \\tau _ i = 0 . \\end{align*}"}
+{"id": "6572.png", "formula": "\\begin{align*} 2 < q < \\infty , \\ \\frac { n q } { 2 ( q - 1 ) } < p < \\infty \\ \\left ( \\frac { 4 } { 3 } \\leq p < \\infty \\ \\mathrm { i n \\ a d d i t i o n \\ i f } \\ n = 2 \\right ) . \\end{align*}"}
+{"id": "8932.png", "formula": "\\begin{align*} \\varrho _ { p } ^ { \\left ( \\nu \\right ) } \\left ( \\ell \\right ) = \\frac { ( - 1 ) ^ { \\ell } } { 2 ( p + \\ell + 1 ) \\ell ! } \\left [ B _ { 2 ( p + \\ell + 1 ) } \\left ( \\nu \\right ) - B _ { 2 ( p + \\ell + 1 ) } \\right ] . \\end{align*}"}
+{"id": "8161.png", "formula": "\\begin{align*} H _ { m , n } = \\frac { 1 } { \\pi } \\int _ { - \\infty } ^ \\infty k ( t ) U ( m ^ 2 n , t ) \\tanh ( \\pi t ) t \\ , d t , \\end{align*}"}
+{"id": "1156.png", "formula": "\\begin{align*} & \\Omega ( a , a , \\Omega ( a , a , b ) ) = \\Omega ( a , a , b ) = b , \\\\ & \\Omega ( b , b , \\Omega ( b , b , a ) ) = \\Omega ( b , b , a ) = a . \\\\ \\end{align*}"}
+{"id": "841.png", "formula": "\\begin{align*} D _ 2 L _ d ( q _ { k - 1 } , q _ k ) + D _ 1 L _ d ( q _ k , q _ { k + 1 } ) = 0 \\ ; \\forall \\ ; k = 0 , 1 , . . . , N - 1 \\end{align*}"}
+{"id": "7738.png", "formula": "\\begin{align*} \\delta = \\Im \\left \\lbrace \\frac { 1 } { 2 } \\left ( s _ 0 e ^ { i \\frac { \\pi } { 2 } } + \\frac { 1 } { s _ 0 } e ^ { - i \\frac { \\pi } { 2 } } \\right ) \\right \\rbrace = \\frac { 1 } { 2 } \\left ( s _ 0 - \\frac { 1 } { s _ 0 } \\right ) , \\end{align*}"}
+{"id": "8147.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } A ( n , m ) S ( - n , p ; c ) \\psi ( n ) = \\sum _ { d \\bar d \\equiv 1 \\pmod * { c } } e \\Bigl ( \\frac { - p \\bar d } { c } \\Bigr ) \\sum _ { n \\geq 1 } A ( n , m ) e \\Bigl ( \\frac { n d } { c } \\Bigr ) \\psi ( n ) \\end{align*}"}
+{"id": "6531.png", "formula": "\\begin{align*} \\bar { G } = \\underset { t \\to \\infty } { \\lim } \\frac { \\int \\phi _ m \\left ( x \\right ) \\ , q _ t ( x ) d P _ 0 ^ m ( x ) } { \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( x ^ j \\right ) \\ , q _ t ( x ) d P _ 0 ^ m ( x ) } = \\frac { \\int \\phi _ m \\left ( x \\right ) \\ , q ( x ) d P _ 0 ^ m ( x ) } { \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( x ^ j \\right ) \\ , q ( x ) d P _ 0 ^ m ( x ) } . \\end{align*}"}
+{"id": "7064.png", "formula": "\\begin{align*} \\widehat { E } _ { C _ 2 } & = E _ { C _ 2 } [ 1 / \\tau ] = \\left ( E _ { C _ 2 } \\overset { \\tau } { \\longrightarrow } \\Sigma ^ { \\sigma - | \\sigma | } E _ { C _ 2 } \\overset { \\tau } { \\longrightarrow } \\Sigma ^ { 2 \\sigma - | 2 \\sigma | } E _ { C _ 2 } \\overset { \\tau } { \\longrightarrow } \\cdots \\right ) \\end{align*}"}
+{"id": "6664.png", "formula": "\\begin{align*} \\| { \\bf y } ^ k \\| _ C = \\left \\| \\left [ \\| { \\bf y } ^ k _ { ( 1 ) } \\| _ C , \\ , \\| { \\bf y } ^ k _ { ( 2 ) } \\| _ C , \\cdots , \\| { \\bf y } ^ k _ { ( d ) } \\| _ C \\right ] \\right \\| _ 2 \\end{align*}"}
+{"id": "6805.png", "formula": "\\begin{align*} \\delta A _ d [ q _ d ] & = \\delta \\sum _ { m = 0 } ^ { N - 1 } L _ d ( q _ m , q _ { m + 1 } ) , \\\\ & = \\sum _ { m = 0 } ^ { N - 1 } [ D _ 1 L _ d ( q _ m , q _ { m + 1 } ) \\cdot \\delta { q _ m } + D _ 2 L _ d ( q _ m , q _ { m + 1 } ) \\cdot \\delta { q _ { m + 1 } } ] . \\end{align*}"}
+{"id": "4558.png", "formula": "\\begin{align*} \\frac { a ' } { b ' } = \\frac { a n - b } { b n } \\qquad a ' b n = b ' ( a n - b ) \\end{align*}"}
+{"id": "4844.png", "formula": "\\begin{align*} I _ n ^ \\flat ( 0 , t ; v ) = ( - 1 ) ^ n I _ n ( t , 1 - t ; v ) . \\end{align*}"}
+{"id": "8962.png", "formula": "\\begin{align*} u | _ { t = 0 } = u _ 0 \\in H ^ { 1 / 2 } ( S ^ 1 ; N ) . \\end{align*}"}
+{"id": "8296.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\textbf { u } ^ * _ i = \\frac { 1 } { d _ i } [ x ^ * ( \\textbf { x } ) - x _ i , \\ y ^ * ( \\textbf { x } ) - y _ i , \\ z ^ * ( \\textbf { x } ) - z _ i ] \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "8585.png", "formula": "\\begin{align*} \\mathbf { a } : = a _ { n - 1 } , a _ { n - 2 } , \\ldots , a _ { 1 } , a _ 0 \\end{align*}"}
+{"id": "8980.png", "formula": "\\begin{align*} u _ t = - ( \\varepsilon + d \\pi _ N ( u ) ) u _ r \\hbox { o n } \\partial B . \\end{align*}"}
+{"id": "5773.png", "formula": "\\begin{align*} ( C _ p C _ p ^ T ) ^ { - 1 / 2 } C _ p A = \\overline A _ p ( C _ p C _ p ^ T ) ^ { - 1 / 2 } C _ p . \\end{align*}"}
+{"id": "8363.png", "formula": "\\begin{align*} T _ \\tau w = 0 , \\mathrm { f o r } \\ | x | > k | t | + R + k | \\tau | , \\ | t | < \\nu ( k ) . \\end{align*}"}
+{"id": "3241.png", "formula": "\\begin{align*} L _ 1 ( t _ 1 , t _ 2 ) : = & \\int _ { t _ 1 } ^ { t _ 2 } \\frac { d t } { 1 - t } = \\log \\frac { 1 - t _ 1 } { 1 - t _ 2 } \\\\ L _ { - 1 } ( t _ 1 , t _ 2 ) : = & \\int _ { t _ 1 } ^ { t _ 2 } \\frac { - d t } { 1 + t } = \\log \\frac { 1 + t _ 1 } { 1 + t _ 2 } \\\\ L _ 0 ( t _ 1 , t _ 2 ) : = & \\int _ { t _ 1 } ^ { t _ 2 } \\frac { d t } { t } = \\log \\frac { t _ 2 } { t _ 1 } . \\end{align*}"}
+{"id": "663.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\bf { v } } = d \\circ i ( { \\bf { v } } ) + i ( { \\bf { v } } ) \\circ d . \\end{align*}"}
+{"id": "2703.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u - \\partial _ r ^ 2 u - \\frac { 1 } { r } \\partial _ r u + k ^ 2 \\frac { \\sin ( 2 u ) } { 2 r ^ 2 } = 0 . \\end{align*}"}
+{"id": "3525.png", "formula": "\\begin{align*} \\mu = ( t _ 1 - s _ 1 ) \\alpha _ 1 + \\cdots + ( t _ m - s _ m ) \\alpha _ m . \\end{align*}"}
+{"id": "4547.png", "formula": "\\begin{align*} \\frac { 1 } { s _ 1 } + \\frac { 1 } { s _ 2 } = \\frac { 1 } { 2 } + \\frac { 1 } { 3 } = 1 - \\frac { 1 } { 6 } = 1 - \\frac { 1 } { s _ 1 s _ 2 } . \\end{align*}"}
+{"id": "4705.png", "formula": "\\begin{align*} \\mathbb { R } ^ { G _ { I I I } } = \\mathbb { C } [ \\varphi _ 4 , \\varphi _ { 1 2 } ] \\end{align*}"}
+{"id": "1448.png", "formula": "\\begin{align*} \\lambda = \\frac { k ( k - 1 ) ( m - 1 ) ! ( n - 1 ) ! } { ( m n - 1 ) | K _ \\Delta | } . \\end{align*}"}
+{"id": "8533.png", "formula": "\\begin{align*} \\begin{cases} v _ t = 2 v _ { x x } + \\varphi _ { t } , \\\\ 2 \\varphi _ t = \\varphi _ { x x } - v _ { x x } . \\end{cases} \\end{align*}"}
+{"id": "8671.png", "formula": "\\begin{align*} & \\varphi ( u ) = E ( u ) H ( u ) E ^ { \\perp } ( - u ) = Q ( u ) S _ { o d d } ^ { \\perp } ( - u ) . \\end{align*}"}
+{"id": "5974.png", "formula": "\\begin{align*} \\psi _ * \\O _ { \\widetilde { X } } = \\O _ X \\ : \\mathrm { a n d } \\ : \\ : \\forall i > 0 , \\ : R ^ i \\pi _ * \\O _ { \\widetilde { X } } = 0 . \\end{align*}"}
+{"id": "5645.png", "formula": "\\begin{align*} M T ( s _ 1 , \\cdots , s _ r | s ) : = \\sum _ { n _ 1 , \\cdots , n _ r \\geq 1 } \\frac { 1 } { n _ 1 ^ { s _ 1 } \\cdots n _ r ^ { s _ r } ( n _ 1 + \\cdots + n _ r ) ^ s } \\end{align*}"}
+{"id": "4950.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } _ n = \\ , \\Delta x \\sum _ m ( u _ { m , n } ^ 2 + v _ { m , n } ^ 2 ) , \\\\ \\end{align*}"}
+{"id": "670.png", "formula": "\\begin{align*} \\phi = \\frac { 1 } { 2 } | { \\bf v } | ^ 2 - { p } + { \\rm c o n s t a n t } . \\end{align*}"}
+{"id": "8474.png", "formula": "\\begin{align*} \\Gamma _ { A , \\mu } ^ * : = \\bigl \\{ \\nu \\in \\mathcal E ^ + : \\ \\kappa \\nu \\geqslant \\kappa \\mu \\bigr \\} . \\end{align*}"}
+{"id": "2426.png", "formula": "\\begin{align*} \\begin{aligned} u ( t ) & = F ( t ; q ) v + A ( q ) \\int _ 0 ^ t E ( s ; q ) D ( q ) b \\d s + \\int _ 0 ^ t E ( s ; q ) f \\d s \\\\ & = F ( t ; q ) v + ( I - F ( t ; q ) ) D ( q ) b + ( I - F ( t ; q ) ) A ( q ) ^ { - 1 } f , \\end{aligned} \\end{align*}"}
+{"id": "7984.png", "formula": "\\begin{align*} \\beta ( x ) \\in \\beta ( y + n ) \\subseteq \\beta ( y ) + \\beta ( n ) = \\beta ( y ) , \\end{align*}"}
+{"id": "6341.png", "formula": "\\begin{align*} h ' ( t ) = - { f _ \\nu ( t , \\nu _ 0 ) } / { f _ \\nu ( h ( t ) , \\nu _ 0 ) } . \\end{align*}"}
+{"id": "5190.png", "formula": "\\begin{align*} \\left ( n _ 2 ^ { - 1 } n _ 1 - m _ 2 ^ { - 1 } m _ 1 \\right ) \\frac { r ( 2 r - K ) ( r - K ) } { 3 K } = t \\cdot r , \\end{align*}"}
+{"id": "5180.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\frac { r - K } { K } } ( r - j K ) = \\frac { r ( r - K ) } { 2 K } . \\end{align*}"}
+{"id": "8012.png", "formula": "\\begin{align*} \\begin{aligned} e & = ( F _ 3 , F _ 1 ) , \\\\ T & = \\left ( \\{ F _ 3 , F _ 1 \\} , \\{ ( F _ 3 , F _ 1 ) \\} \\right ) , \\\\ F & = \\left [ ( F _ 3 , F _ 2 ) \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "7658.png", "formula": "\\begin{align*} \\omega \\wedge \\iota _ { E } ( \\eta ) + \\iota _ { E } ( \\omega \\wedge \\eta ) & = \\omega \\wedge \\iota _ { E } ( \\eta ) + \\iota _ { E } ( \\omega ) \\wedge \\eta - \\omega \\wedge \\iota _ { E } ( \\eta ) \\\\ & = \\iota _ { E } ( \\omega ) \\eta = ( \\sum \\lambda _ { k } \\iota _ { E } ( d \\log f _ { k } ) ) \\eta = ( \\sum \\lambda _ { k } ) \\eta . \\end{align*}"}
+{"id": "4545.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { s _ i } = 1 - \\frac { 1 } { \\prod _ { i = 1 } ^ n s _ i } . \\end{align*}"}
+{"id": "3163.png", "formula": "\\begin{align*} c _ j ^ { k l } ( \\lambda A ) = \\lambda \\ , c _ j ^ { k l } ( A ) + \\lambda \\ , \\bar { a } _ { k l } \\int _ Y r A e _ j \\cdot \\nabla w \\forall j , k , l \\in \\{ 1 , \\dots , n \\} , \\end{align*}"}
+{"id": "4636.png", "formula": "\\begin{align*} \\Theta ( \\mathrm { K } ( E _ 0 ) ) = k \\ , \\mathrm { T r } \\ , \\Theta ( E ^ { k , 0 } _ 0 ) + ( k - 1 ) \\ , \\mathrm { T r } \\ , \\Theta ( E ^ { k - 1 , 1 } _ 0 ) + \\dots + \\mathrm { T r } \\ , \\Theta ( E ^ { 1 , k - 1 } _ 0 ) . \\end{align*}"}
+{"id": "3297.png", "formula": "\\begin{align*} A _ { q , 2 } = \\biggl ( \\frac { ( p + 2 ) ( p - 2 ) } { 1 6 \\pi } \\biggr ) ^ { \\frac { 1 } { 4 } - \\frac { 1 } { 2 p } } \\biggl ( \\frac { 4 } { p + 2 } \\biggr ) ^ \\frac { 1 } { p } \\biggl ( \\frac { 4 } { p - 2 } \\biggr ) ^ { \\frac { 1 } { 4 } - \\frac { 1 } { 2 p } } \\leq C \\ , p ^ \\frac { 1 } { 4 } . \\end{align*}"}
+{"id": "8806.png", "formula": "\\begin{align*} \\| \\mathbf { O } _ { a , b } \\| _ 1 \\le \\int \\| \\mathbf { R } _ { w , v } \\| _ 1 \\mathbf { P } _ { a , b } ( w , v ) d w d v = \\int \\| R _ w \\| _ 2 \\| R _ v \\| _ 2 \\mathbf { P } _ { a , b } ( w , v ) d w d v . \\end{align*}"}
+{"id": "891.png", "formula": "\\begin{align*} \\mathbf { E } \\left [ \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } | \\mathcal { X } ^ { 0 } \\right ] \\leq \\left ( 1 - \\mathop { \\min } _ { k = 1 , 2 , \\cdots , l } \\lambda _ { \\min } ( \\mathbf { E } [ \\widehat { \\mathcal { Z } } _ { ( k ) } ] ) \\right ) ^ { t } \\| \\mathcal { X } ^ { 0 } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } . \\end{align*}"}
+{"id": "2841.png", "formula": "\\begin{align*} \\lambda a _ k + ( 1 - \\lambda ) a _ { \\ell } & = \\frac { 1 } { 2 \\tau } \\left ( ( \\tau + \\sigma ) ( \\rho + \\tau ) + ( \\tau - \\sigma ) ( \\rho - \\tau ) \\right ) \\\\ & = \\rho + \\sigma = b _ k \\end{align*}"}
+{"id": "1266.png", "formula": "\\begin{align*} R _ { \\mathcal { B S } ( \\alpha ) } ( \\mathcal { S ^ { * } } [ A , B ] ) & = \\begin{dcases} \\min \\left \\{ 1 , \\dfrac { 2 \\sqrt { \\alpha } } { ( 1 + \\alpha ) \\sqrt { 4 \\alpha ( A - B ) ^ 2 + B ^ 2 } } \\right \\} , & 4 A \\alpha \\geq ( 3 \\alpha + 1 ) B , \\\\ \\min \\left \\{ 1 , \\dfrac { 1 } { ( 1 - \\alpha ) ( A - B ) + B } \\right \\} , & 4 A \\alpha \\leq ( 3 \\alpha + 1 ) B . \\end{dcases} \\end{align*}"}
+{"id": "6908.png", "formula": "\\begin{align*} \\dfrac { S ( t ) - S ( 0 ) } { t } & = \\frac { 1 } { t } \\int _ 0 ^ t ( \\Lambda - \\beta I S - \\mu S ) d s - \\frac { 1 } { t } \\int _ 0 ^ t \\sigma I S d W _ s - \\frac { 1 } { t } \\int _ 0 ^ t \\int _ U J ( u ) I S \\check { N } ( d s , d u ) \\\\ & \\geq \\frac { 1 } { t } \\int _ 0 ^ t ( \\Lambda - \\beta \\frac { \\Lambda } { \\mu } S - \\mu S ) d s - \\frac { 1 } { t } \\int _ 0 ^ t \\sigma I S d W _ s - \\frac { 1 } { t } \\int _ 0 ^ t \\int _ U J ( u ) I S \\check { N } ( d s , d u ) \\end{align*}"}
+{"id": "9208.png", "formula": "\\begin{align*} \\Lambda ( m , h ) \\sim \\lambda ( m , h ) \\times \\left \\{ \\begin{aligned} & \\frac { r _ { m } ^ { d - 1 } } { s _ { m } ^ { d - 1 } } & & m \\neq \\{ 0 \\} , \\\\ & \\pi ^ { \\frac { d - 1 } { 2 } } F ^ { \\prime \\prime } ( r _ { m } ) ^ { \\frac { 1 - d } { 2 } } s _ { m } ^ { 1 - d } \\vert \\S ^ { d - 1 } \\vert ^ { - 1 } h ^ { \\frac { d - 1 } { 2 } } & & m = \\{ 0 \\} . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6535.png", "formula": "\\begin{align*} & \\bar { G } ^ 2 \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( w ^ j \\right ) \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( x ^ j \\right ) \\ , Q ( x , w ) ( d P _ 0 ^ m \\times P _ 0 ^ m ) ( x , w ) \\\\ & \\geq \\bar { G } \\int \\Pi _ { j = 1 } ^ { m } \\phi _ 1 \\left ( x ^ j \\right ) \\int \\phi _ m ( w ) \\ , Q ( x , w ) d P _ 0 ^ m ( w ) d P _ 0 ^ m ( x ) . \\end{align*}"}
+{"id": "2594.png", "formula": "\\begin{align*} v _ { i , j } \\coloneqq \\sum _ { p = i } ^ { j - 1 } v _ { p , p + 1 } = - \\sum _ { p = j } ^ { i - 1 } v _ { p , p + 1 } \\in H _ { i , j } ^ 0 , v _ { p , p + 1 } \\in H _ { p , p + 1 } ^ 0 , \\end{align*}"}
+{"id": "5077.png", "formula": "\\begin{align*} \\alpha _ i = c _ i x ^ { h _ i } \\qquad c _ i \\in K ^ * h _ i \\in H . \\end{align*}"}
+{"id": "6054.png", "formula": "\\begin{align*} \\psi _ n ( z ) : = \\exp \\left \\{ \\int _ { b _ { g + 1 } } ^ z \\left ( \\sum _ { e \\in E _ n , | e | < \\infty } \\frac { m _ e ( s ) } { s - e } + \\sum _ { e \\in E _ n , | e | = \\infty } m _ \\infty ( s ) \\right ) \\frac { \\dd s } { w ( s ) } \\right \\} , \\end{align*}"}
+{"id": "1992.png", "formula": "\\begin{align*} \\zeta ( k _ 1 , \\ldots , k _ r ) = \\underset { 1 > x _ 1 > \\cdots > x _ k > 0 } { \\int } \\prod _ { i = 1 } ^ k \\omega _ i ( x _ i ) , \\end{align*}"}
+{"id": "8653.png", "formula": "\\begin{align*} \\int _ a ^ R f _ \\infty ( r ) ^ 2 \\sinh ^ { d - 1 } r \\d r & = [ f _ \\infty ( r ) ^ 2 u ( r ) ] _ a ^ R - \\int _ a ^ R 2 f _ \\infty ( r ) f _ \\infty ' ( r ) u ( r ) d r \\\\ & = f _ \\infty ( R ) ^ 2 \\int _ 0 ^ R \\sinh ^ { d - 1 } r \\d r - \\int _ a ^ R 2 f _ \\infty ( r ) f _ \\infty ' ( r ) u ( r ) d r . \\end{align*}"}
+{"id": "1694.png", "formula": "\\begin{align*} f _ n \\left ( \\begin{array} { c c } x & y \\\\ m & n \\end{array} \\right ) = \\frac { \\Delta ^ { \\frac { k } { 2 } } ( m ^ 2 + n ^ 2 ) ^ { n - \\frac { k } { 2 } } } { ( m i + n ) ^ { 2 n } } . \\end{align*}"}
+{"id": "6575.png", "formula": "\\begin{align*} & \\eta ( x ) \\in C ^ \\infty _ c ( B _ { \\frac 1 4 } ) \\ \\mathrm { w i t h } \\ \\eta = 1 \\ \\mathrm { o n } \\ B _ \\frac 1 8 , \\ 0 \\leq \\eta \\leq 1 , \\\\ & \\tilde { \\rho } _ { ( y , s ) } ( x , t ) : = \\rho _ { ( y , s ) } ( x , t ) \\eta ( x - y ) = \\frac { 1 } { ( 4 \\pi ( s - t ) ) ^ { \\frac { n - 1 } { 2 } } } e ^ { - \\frac { | x - y | ^ 2 } { 4 ( s - t ) } } \\eta ( x - y ) , \\end{align*}"}
+{"id": "7439.png", "formula": "\\begin{align*} & { \\mathbb K } ( Z , W ) ( P ) = \\\\ & { \\mathbb H } _ 0 ( Z ) \\begin{bmatrix} ( { \\rm i d } _ { n \\times m } \\otimes \\pi ) ( P ) & ( { \\rm i d } _ { n \\times m } \\otimes \\pi ) ( P ) \\cdot { \\rm i d } _ { n \\times m } \\otimes J \\\\ ( { \\rm i d } _ { n \\times m } \\otimes \\pi ) ( P ) \\cdot { \\rm i d } _ { n \\times m } \\otimes J & ( { \\rm i d } _ { n \\times m } \\otimes \\pi ) ( P ) \\end{bmatrix} { \\mathbb H } _ 0 ( W ) ^ * \\end{align*}"}
+{"id": "5285.png", "formula": "\\begin{align*} \\small \\begin{array} { l l } \\frac { 1 } { ( m - n - 1 ) } R i c ( U _ i , U _ i ) & = \\frac { 1 } { ( m - n - 1 ) } \\hat { R } i c ( U _ i , U _ i ) - \\lambda ^ 2 ( m - n - 1 ) \\| \\nabla ( f + \\log \\lambda ) \\| ^ 2 \\\\ & + \\frac { \\lambda ^ 4 } { 4 } ( m - n - 1 ) \\| g r a d _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\| ^ 2 - \\lambda ^ 2 ( m - n - 1 ) g \\left ( \\nabla f , \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\right ) . \\end{array} \\end{align*}"}
+{"id": "5501.png", "formula": "\\begin{align*} ( s \\cdot \\jmath ^ * \\circ ^ { - 1 } + ( 1 - s ) \\cdot ^ { - 1 } \\circ \\jmath ^ * ) e _ k = \\Big ( \\frac { s } { | k - 1 | } + \\frac { 1 - s } { | k | } \\Big ) e _ { k - 1 } \\neq 0 , \\forall s \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "7001.png", "formula": "\\begin{align*} X & = \\theta + Z _ x , \\\\ Y & = \\theta + Z _ y , \\end{align*}"}
+{"id": "8840.png", "formula": "\\begin{align*} \\Delta _ { 0 } = 4 ( 1 + \\langle z , z \\rangle ) \\sum _ { i , j = 1 } ^ { n } ( \\delta _ { i j } + z _ { i } \\bar { z } _ { j } ) \\frac { \\partial ^ { 2 } } { \\partial z _ { i } \\partial \\bar { z } _ { j } } . \\end{align*}"}
+{"id": "108.png", "formula": "\\begin{align*} | | f | | ^ p _ { W ^ { s , p } ( \\mathbb R ^ n ) } : = \\int _ { \\mathbb R ^ n } \\int _ { \\mathbb R ^ n } \\frac { | f ( x ) - f ( y ) | ^ p } { | x - y | ^ { n + s p } } d x d y , \\end{align*}"}
+{"id": "540.png", "formula": "\\begin{align*} S _ 2 & \\leq \\zeta ( 2 - 2 \\theta _ 3 - 0 . 0 1 ) ^ { 3 5 } B X \\prod _ { i = 1 } ^ { R } \\exp \\Big ( \\frac { k ^ 2 - 1 } { 2 } \\sum _ { q \\in P _ i } \\frac { A ( q , 1 ) ^ 2 } { q } \\Big ) \\sum _ { p \\in \\cup P _ j } \\frac { A ( p , 1 ) ^ 2 \\log p } { p } \\\\ & \\leq \\zeta ( 2 - 2 \\theta _ 3 - 0 . 0 1 ) ^ { 3 5 } B X ( \\frac { \\log X } { 1 0 ^ { 2 M } } + O ( 1 ) ) \\exp \\Big ( \\frac { k ^ 2 - 1 } { 2 } \\sum _ { q \\in P _ i } \\frac { A ( q , 1 ) ^ 2 } { q } \\Big ) . \\end{align*}"}
+{"id": "6053.png", "formula": "\\begin{align*} m _ e ( x ) = c _ e \\left ( 1 - \\sum _ { i = 1 } ^ g \\left ( \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac 1 { y - e } \\frac { \\dd y } { w ( y ) } \\right ) ( x - e ) l _ i ( x ) \\right ) , \\end{align*}"}
+{"id": "8988.png", "formula": "\\begin{align*} \\int _ 0 ^ { T _ 0 } \\int _ { \\partial B } & ( u _ t + d \\pi _ N ( u ) u _ r ) \\cdot \\varphi d \\phi \\ , d t \\\\ & = \\int _ 0 ^ { T _ 0 } \\int _ { \\partial B } u _ t \\cdot \\varphi d \\phi \\ , d t + \\int _ 0 ^ { T _ 0 } \\int _ B \\nabla u \\cdot \\nabla \\big ( d \\pi _ N ( u ) \\varphi \\big ) d z \\ , d t = 0 \\end{align*}"}
+{"id": "271.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\tau } - \\beta \\rho \\tau \\leqslant - \\frac { \\beta ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\beta } { 2 } , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\tau } - \\beta \\rho \\tau \\leqslant - \\frac { \\beta ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\beta } { 2 } , \\end{align*}"}
+{"id": "6422.png", "formula": "\\begin{align*} \\phi ( X _ 1 , \\tilde Y - ( \\tilde t - t ) X _ 1 , \\tilde t ) & = \\phi ( X , Y , t ) + \\nabla _ X \\phi ( X , Y , t ) \\cdot ( X _ 1 - X ) \\\\ & + \\frac 1 2 \\langle \\nabla _ X ^ 2 \\phi ( X , Y , t ) ( X _ 1 - X ) , ( X _ 1 - X ) \\rangle \\\\ & - ( \\tilde t - t ) \\bigl ( X \\cdot \\nabla _ Y - \\partial _ t \\bigr ) \\phi ( X , Y , t ) + { o } ( \\epsilon ^ 2 ) , \\end{align*}"}
+{"id": "291.png", "formula": "\\begin{align*} \\begin{aligned} K _ { \\chi } : = \\prescript { L } { } { j } _ { \\chi } : H ^ 1 ( W _ F , \\widehat { S } ) ^ { \\mathrm { r e g } } & \\rightarrow H ^ 1 ( W _ F , \\widehat { G } ) , \\\\ \\phi _ { \\mu } & \\mapsto \\phi _ { \\pi _ { ( S , \\mu ) } } . \\end{aligned} \\end{align*}"}
+{"id": "6185.png", "formula": "\\begin{align*} \\overline A _ p = C _ p A C _ p ^ T ( C _ p C _ p ^ T ) ^ { - 1 } \\end{align*}"}
+{"id": "7449.png", "formula": "\\begin{align*} \\widehat M ^ * _ Q = Q ( \\widehat Z ^ { ( 0 ) } ) ^ * . \\end{align*}"}
+{"id": "4380.png", "formula": "\\begin{align*} \\frac { p } { q } - \\sum _ { i = 1 } ^ { k + 1 } \\frac { 1 } { a _ i } & = \\frac { p } { q } - \\sum _ { i = 1 } ^ { k } \\frac { 1 } { a _ i } - \\frac { 1 } { a _ { k + 1 } } = \\frac { 1 } { q \\prod _ { i = 1 } ^ k a _ i } - \\frac { 1 } { q \\prod _ { i = 1 } ^ k a _ i + 1 } \\\\ & = \\frac { 1 } { q \\prod _ { i = 1 } ^ k a _ i \\left ( q \\prod _ { i = 1 } ^ k a _ i + 1 \\right ) } \\\\ & = \\frac { 1 } { q \\prod _ { i = 1 } ^ { k + 1 } a _ i } . \\end{align*}"}
+{"id": "7429.png", "formula": "\\begin{align*} \\phi ( \\alpha \\cdot P \\cdot \\beta ^ * ) = \\alpha \\cdot \\phi ( P ) \\cdot \\beta ^ * \\end{align*}"}
+{"id": "5503.png", "formula": "\\begin{align*} \\Omega & : = A ^ { - N - 1 , N } ( r _ 0 , R _ 0 ) \\times [ \\lambda _ 0 , \\lambda _ 0 + 1 ] \\\\ & : = \\big \\{ ( y , z ) \\in E ^ { - N , N } \\times E _ { - N - 1 } \\ \\big | \\ r _ 0 \\leq \\| y \\| ^ 2 + \\| z \\| ^ 2 \\leq R _ 0 \\big \\} \\times [ \\lambda _ 0 , \\lambda _ 0 + 1 ] \\end{align*}"}
+{"id": "1577.png", "formula": "\\begin{align*} \\mu ( A ) = \\sum _ { x \\in A } \\mu ( \\{ x \\} ) \\qquad \\mbox { f o r a l l } A \\subseteq X , \\end{align*}"}
+{"id": "2851.png", "formula": "\\begin{align*} c _ { k - 1 } & = a _ { k - 1 } \\geq a _ k = \\rho + \\tau > \\rho + \\sigma = c _ k \\\\ & > c _ { k + 1 } = \\rho - \\sigma > \\rho - \\tau = a _ { k + 1 } \\geq a _ { k + 2 } = c _ { k + 2 } = c _ { \\ell + 1 } . \\end{align*}"}
+{"id": "1791.png", "formula": "\\begin{align*} \\deg _ P ( \\lambda ) = \\langle 2 \\rho _ G - 2 \\rho _ M , \\lambda \\rangle , \\end{align*}"}
+{"id": "6882.png", "formula": "\\begin{align*} \\rho ( u ) = \\rho _ { p _ 1 } ( u ) + \\rho _ { p _ 2 } ( u ) \\end{align*}"}
+{"id": "531.png", "formula": "\\begin{align*} \\underline { \\tau } = \\ ! 1 0 ^ { - 8 } , \\overline { \\tau } \\ ! = 1 0 ^ { 8 } , \\beta _ { \\rm m a x } \\ ! = 1 , \\eta = 0 . 0 1 , \\eta _ 1 = \\eta _ 2 = 0 . 5 , \\delta = 0 . 0 1 , \\alpha = 1 0 ^ { - 5 } , \\tau _ { 1 , 0 } ^ 0 = \\ ! \\frac { 1 0 0 } { \\| V ^ 0 \\| ^ 2 } , \\tau _ { 2 , 0 } ^ 0 = \\ ! \\frac { 1 0 0 } { \\| U ^ 0 \\| ^ 2 } . \\end{align*}"}
+{"id": "1661.png", "formula": "\\begin{align*} \\beta _ v ( f _ { 1 , v } , f _ { 2 , v } ) : = \\frac { L ( 1 , \\eta _ { T , v } ) \\cdot L ( 1 , \\Pi _ v , { \\rm a d } ) } { \\zeta _ { v } ( 2 ) \\cdot L ( 1 / 2 , \\Pi _ v , \\chi _ v ) } \\int _ { T ( F _ v ) } \\chi _ v ( t _ v ) \\frac { \\langle \\pi _ v ( t _ v ) f _ { 1 , v } , f _ { 2 , v } \\rangle _ v } { \\langle f _ { 1 , v } , f _ { 2 , v } \\rangle _ v } d ^ \\times t _ v \\end{align*}"}
+{"id": "2925.png", "formula": "\\begin{align*} \\overline { \\nu } _ \\alpha ( \\eta _ j = k ) = \\frac { 1 } { Z _ n ( \\alpha ) } \\frac { \\alpha ^ k } { g _ n ! ( k ) } , Z _ n ( \\alpha ) = \\sum _ { k \\ge 0 } \\frac { \\alpha ^ k } { g _ n ! ( k ) } \\end{align*}"}
+{"id": "5860.png", "formula": "\\begin{align*} t = T : U = U ' = 0 . \\end{align*}"}
+{"id": "6153.png", "formula": "\\begin{align*} ( C _ 1 ) = \\hbox { S p a n } \\{ e _ 1 \\} \\hbox { w i t h } e _ 1 = ( 1 , \\cdots , 1 ) ^ T . \\end{align*}"}
+{"id": "2902.png", "formula": "\\begin{align*} \\hat { K } ( \\alpha ) = \\hat { K } ( - \\alpha ) = 0 , 0 \\neq \\alpha \\in \\mathbb { N } _ { 0 } ^ { \\infty } , \\end{align*}"}
+{"id": "7388.png", "formula": "\\begin{align*} ( \\alpha | \\alpha ) = 2 , \\ ; \\ ; \\ ; ( \\beta | \\beta ) = 6 \\ ; \\ ; \\ ; \\ ; ( \\alpha | \\beta ) = - 3 . \\end{align*}"}
+{"id": "9240.png", "formula": "\\begin{align*} \\lambda _ { j } ( h ) = \\mu _ { n _ { 0 } - j + 1 } ( 1 + \\mathcal { O } ( h ^ { \\infty } ) ) = \\ < \\Delta _ { f } \\varphi _ { n _ { 0 } - j + 1 } , \\varphi _ { n _ { 0 } - j + 1 } \\ > ( 1 + \\mathcal { O } ( h ^ { \\infty } ) ) . \\end{align*}"}
+{"id": "4160.png", "formula": "\\begin{align*} \\lambda ( g , b ) = \\inf \\Big \\{ \\mathcal { F } ( g , b , f ) \\big | \\ , f \\in C ^ \\infty ( M ) , \\ , \\int _ M e ^ { - f } d V _ g = 1 \\Big \\} . \\end{align*}"}
+{"id": "3510.png", "formula": "\\begin{align*} E _ { i j } \\Omega _ { D ( \\gamma ) } = \\delta _ { i j } \\frac { p } { 2 } \\Omega _ { D ( \\gamma ) } + \\sum _ { k = 1 } ^ n \\gamma _ { j k } \\Omega _ { D ( \\gamma + e _ { i k } - e _ { j k } ) } , \\end{align*}"}
+{"id": "619.png", "formula": "\\begin{align*} \\tilde J _ { t _ n , t _ { n + 1 } } ^ { ( \\epsilon ) } : = \\int _ { t _ n } ^ { t _ { n + 1 } } ( A Z _ t ^ { ( \\epsilon ) } ) ^ { \\rm T } { \\rm d } X _ t ^ { ( \\epsilon ) } \\end{align*}"}
+{"id": "1979.png", "formula": "\\begin{align*} T ( r , F ) \\leq \\sum _ { j = 0 , j \\neq m } ^ { k } T ( r , A _ { j } ) + T ( r , A _ { m } ) + ( k + 2 ) T ( 2 r , f ) \\end{align*}"}
+{"id": "8929.png", "formula": "\\begin{align*} S _ 2 & = 2 \\sum \\limits _ { p = 0 } ^ { n - 1 } \\tau _ { p } ^ { ( \\nu , n ) } \\sum \\limits _ { \\mu = 1 } ^ { \\nu - 1 } e ^ { - \\mu ^ { 2 } t } \\mu ^ { 2 p + 1 } \\\\ & = 2 \\sum \\limits _ { p = 0 } ^ { n - 1 } \\tau _ { p } ^ { ( \\nu , n ) } \\sum \\limits _ { \\ell = 0 } ^ { + \\infty } \\varrho _ { p } ^ { ( \\nu ) } \\left ( \\ell \\right ) \\ t ^ { \\ell } \\end{align*}"}
+{"id": "7053.png", "formula": "\\begin{align*} \\Omega ^ { C _ 2 } _ * \\to M U ^ { C _ 2 } _ * = M U _ * [ u , d _ { i , j } , q _ j ] / J \\end{align*}"}
+{"id": "627.png", "formula": "\\begin{align*} J _ h = ( m \\in G ( I ^ h ) : m m _ h ) , \\end{align*}"}
+{"id": "4735.png", "formula": "\\begin{align*} ( A : _ A x ) & = \\mathfrak { q } _ 1 \\cap \\mathfrak { q } _ 2 \\cap \\cdots \\cap \\mathfrak { q } _ n . \\end{align*}"}
+{"id": "8124.png", "formula": "\\begin{align*} b ( y ) = y ^ { - 1 } g \\Bigl ( \\frac { y ^ 2 n _ 1 ^ 4 } { N p } \\Bigr ) \\widehat { k ^ * } \\Bigl ( \\frac { M T c m p } { 2 \\pi ^ 2 y n _ 1 ^ 2 } \\Bigr ) e \\Bigl ( \\frac { - T ^ 2 c m p } { 4 \\pi ^ 2 y n _ 1 ^ 2 } \\Bigr ) . \\end{align*}"}
+{"id": "2379.png", "formula": "\\begin{align*} U ^ H \\Phi = \\left [ \\begin{array} { c } I _ { d } \\\\ 0 \\end{array} \\right ] . \\end{align*}"}
+{"id": "5792.png", "formula": "\\begin{align*} \\overline D _ p = ( C _ p C _ p ^ T ) ^ { 1 / 2 } R ( C _ p C _ p ^ T ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "7603.png", "formula": "\\begin{align*} \\langle \\mathfrak { A } ( \\psi ) , \\zeta \\rangle & = \\nu \\int _ 0 ^ 1 \\frac { \\partial \\psi ( x ) } { \\partial x } \\frac { \\partial \\zeta ( x ) } { \\partial x } \\d x - \\frac { \\alpha } { \\delta + 1 } \\int _ 0 ^ 1 \\mathfrak { p } ( \\psi ( x ) + \\varphi ( x ) ) \\frac { \\partial \\zeta ( x ) } { \\partial x } \\d x \\\\ & \\quad - \\beta \\int _ 0 ^ 1 \\mathfrak { c } ( \\psi ( x ) + \\varphi ( x ) ) \\zeta ( x ) \\d x , \\end{align*}"}
+{"id": "6325.png", "formula": "\\begin{align*} - \\partial _ { x _ q } ( \\sigma ^ { \\rm e f f } _ { p q } - \\partial _ { x _ r } \\mu ^ { \\rm e f f } _ { p q r } ) = g _ p \\mathcal { D } ( \\Omega ) , \\end{align*}"}
+{"id": "2471.png", "formula": "\\begin{align*} D ^ n f ( x ) \\left [ ( y - x ) ^ n \\right ] = D ^ n f ( x ) [ y - x , \\ldots , y - x ] . \\end{align*}"}
+{"id": "7793.png", "formula": "\\begin{align*} & \\ \\Psi _ { 1 / s d ( n ) , b } [ n ] ( y ^ { n ' } ) = \\begin{cases} \\displaystyle y ^ { n ' } \\prod _ { p = 1 } ^ { \\alpha } ( 1 + q ^ { ( 2 p - 1 ) / s d ( n ) } q ^ b y ^ n ) & \\alpha > 0 , \\\\ y ^ { n ' } & \\alpha = 0 , \\\\ \\displaystyle y ^ { n ' } \\prod _ { p = 1 } ^ { - \\alpha } ( 1 + q ^ { - ( 2 p - 1 ) / s d ( n ) } q ^ b y ^ n ) ^ { - 1 } & \\alpha < 0 . \\end{cases} \\end{align*}"}
+{"id": "6227.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p C _ p B e _ r u _ r ( T ) \\equiv 0 \\hbox { o n } \\Gamma _ 1 . \\end{align*}"}
+{"id": "84.png", "formula": "\\begin{align*} k = \\sum _ { i = 1 } ^ { \\infty } k ( i ) p ^ { i - 1 } , \\ , \\ k ( i ) \\in \\{ 0 , 1 , \\cdots , p - 1 \\} \\end{align*}"}
+{"id": "515.png", "formula": "\\begin{align*} \\frac { 2 m \\ ! + \\ ! 1 } { 4 ( m \\ ! + \\ ! 1 ) } \\sqrt { \\frac { a } { 2 } } \\ ! \\sum _ { K _ 1 \\ni j = k } ^ { \\infty } \\| x ^ { j + 1 } \\ ! - \\ ! x ^ j \\| \\le c _ 1 \\sum _ { j = k } ^ { \\infty } \\ ! \\Xi _ { j } + c _ 2 \\sum _ { j = k } ^ { \\infty } \\ ! \\Xi _ { j } ^ { \\frac { 1 - \\theta } { \\theta } } + a _ 1 ( m \\ ! + \\ ! 1 ) ( \\Delta _ { k - m } - \\Delta _ { k } ) . \\end{align*}"}
+{"id": "8335.png", "formula": "\\begin{align*} Q _ l ( t , x ) = Q \\left ( \\frac { x _ 1 - l _ 1 t } { \\sqrt { 1 - l _ 1 ^ 2 } } , x _ 2 , x _ 3 \\right ) . \\end{align*}"}
+{"id": "3316.png", "formula": "\\begin{align*} \\implies \\left \\{ P _ { f } \\left ( \\mathbf { x } \\right ) \\right \\} = \\left ( G _ { f } \\right ) . \\end{align*}"}
+{"id": "2473.png", "formula": "\\begin{align*} y ^ 2 + x y = x ^ 3 - x ^ 2 - 9 2 7 x + 1 1 0 9 7 \\end{align*}"}
+{"id": "5823.png", "formula": "\\begin{align*} \\int _ \\Omega ( \\ ! ( U '' , \\Phi ) \\ ! ) d x + \\int _ \\Omega ( \\ ! ( \\nabla U , \\nabla \\Phi ) \\ ! ) d x + \\int _ \\Omega ( \\ ! ( A U , \\Phi ) \\ ! ) d x + \\int _ { \\Gamma _ 1 } ( \\ ! ( D U ' , \\Phi ) \\ ! ) d \\Gamma = 0 . \\end{align*}"}
+{"id": "1631.png", "formula": "\\begin{align*} \\begin{aligned} \\| F _ { 1 } ( t ) \\| _ { L _ { x } ^ { \\infty } ( \\R ^ d ) } & \\leq \\int _ { 0 } ^ { M } \\| e ^ { i ( t - s ) \\Delta ^ 2 } | u | ^ { 2 } u ( s ) \\| _ { L _ { x } ^ { \\infty } ( \\R ^ d ) } \\ , d s \\\\ & \\lesssim M ( t - M ) ^ { - \\frac { d } { 4 } } \\sup _ { s } \\| u ^ { 3 } ( s ) \\| _ { L _ { x } ^ { 1 } ( \\R ^ d ) } \\\\ & \\lesssim M t ^ { - \\frac { d } { 4 } } \\sup _ { s } \\| u ( s ) \\| _ { H _ { x } ^ { 3 } ( \\R ^ d ) } ^ { 3 } \\\\ & \\lesssim M M _ { 1 } ^ { 3 } t ^ { - \\frac { d } { 4 } } . \\end{aligned} \\end{align*}"}
+{"id": "8623.png", "formula": "\\begin{align*} \\lim _ { \\substack { N \\to \\infty \\\\ \\lambda = N p } } \\binom { N } { i } p ^ i ( 1 - p ) ^ { N - i } & = \\lim _ { N \\to \\infty } \\frac { N ( N - 1 ) \\cdots ( N - i + 1 ) } { N ^ i } \\left ( 1 - p \\right ) ^ { N } \\frac { \\lambda ^ i } { i ! } = e ^ { - \\lambda } \\frac { \\lambda ^ i } { i ! } \\end{align*}"}
+{"id": "5893.png", "formula": "\\begin{align*} w = ( E , C _ p U ) , { \\cal L } \\theta = - ( E , C _ p A U ) , { \\cal R } \\theta = - ( E , C _ p B U ) . \\end{align*}"}
+{"id": "1614.png", "formula": "\\begin{align*} B _ { r , c } = \\begin{cases} \\frac { 1 - \\rho } { \\omega } , & c \\leq r \\leq c + \\omega - 1 \\\\ \\frac { \\rho } { \\Lambda - 1 } , & \\mathrm { o / w } \\end{cases} \\end{align*}"}
+{"id": "1464.png", "formula": "\\begin{align*} \\lambda = \\frac { 2 k ( k - 1 ) ( k - 2 ) ( m - 1 ) ! ( m - 2 ) ! } { ( m + 1 ) ( m ^ 2 - 2 ) | G _ \\Delta | } . \\end{align*}"}
+{"id": "705.png", "formula": "\\begin{align*} A = \\oint _ \\alpha \\eta = \\oint _ \\alpha \\nu + \\oint _ \\alpha * d G ^ \\omega = a + \\oint _ \\alpha * d G ^ \\omega , \\end{align*}"}
+{"id": "7545.png", "formula": "\\begin{align*} f ( x , y , z ) = & x ^ p + \\sum _ { i = 0 } ^ k ( - 1 ) ^ i a _ { k , r } ( i ) t ^ { k - i } y ^ { r + i } z ^ { k + l - i } \\\\ = & x ^ p + \\sum _ { i = 0 \\atop p \\nmid a _ { k , r } ( i ) } ^ k \\Big ( ( - 1 ) ^ i a _ { k , r } ( i ) t ^ { ( k - i ) / p } y ^ { ( r + i ) / p } z ^ { ( k + l - i ) / p } \\Big ) ^ p \\\\ = & x ^ p + \\Big ( \\sum _ { i = 0 \\atop p \\nmid a _ { k , r } ( i ) } ^ k ( - 1 ) ^ i a _ { k , r } ( i ) t ^ { ( k - i ) / p } y ^ { ( r + i ) / p } z ^ { ( k + l - i ) / p } \\Big ) ^ p \\\\ : = & x ^ p + h ( y , z ) ^ p \\end{align*}"}
+{"id": "9223.png", "formula": "\\begin{align*} \\lll u = r , \\end{align*}"}
+{"id": "8491.png", "formula": "\\begin{align*} \\varphi _ { \\xi ^ 3 } + \\varphi \\varphi _ { \\xi ^ 2 } + \\frac { 2 \\varphi ^ 2 + 3 \\varphi _ { \\xi } + \\epsilon } { 9 } \\varphi _ { \\xi } = 0 \\end{align*}"}
+{"id": "6594.png", "formula": "\\begin{align*} V _ \\phi [ D ^ \\beta f ] ( x , \\xi ) = \\sum _ { \\alpha \\leq \\beta } C ' _ { \\alpha , \\beta } \\xi ^ { \\beta - \\alpha } \\iint V _ \\phi f ( x - y , \\xi - \\eta ) V _ { D ^ \\beta \\phi } \\phi ( y , \\eta ) e ^ { - i \\langle x - y , \\eta \\rangle } d y \\ , d \\eta , \\end{align*}"}
+{"id": "9172.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\lim _ { N \\to \\infty } \\lim _ { m ^ 2 \\downarrow 0 } ( f _ \\epsilon , \\tilde { C } ^ { \\Lambda _ N } ( s , m ^ 2 ) f _ \\epsilon ) = \\frac { 1 } { v _ J ^ 2 + s } ( f , ( - \\Delta _ { \\R ^ 2 } ) ^ { - 1 } f ) _ { \\R ^ 2 } , \\end{align*}"}
+{"id": "6278.png", "formula": "\\begin{align*} \\mathcal { D } = \\tilde { \\mathcal { D } } _ { 0 } \\cup \\{ \\big ( 0 , y , - \\sqrt { r ( y ) } \\big ) : y > 0 \\} . \\end{align*}"}
+{"id": "1315.png", "formula": "\\begin{align*} & \\operatorname { T r a } ( S _ { f , \\tau } ) = \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) , \\\\ & \\operatorname { T r a } ( S ^ 2 _ { f , \\tau } ) = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) . \\end{align*}"}
+{"id": "6801.png", "formula": "\\begin{align*} ( \\overline { w } _ { i , \\hat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } ( t ) = \\lambda \\int _ { a } ^ { t } \\mathbb { K } ( t , ( \\overline { w } _ { \\i , \\hat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } ( s ) ) d s ; \\end{align*}"}
+{"id": "4097.png", "formula": "\\begin{align*} 0 = ( d H ) _ { j i k l } = \\nabla _ j H _ { i k l } - \\nabla _ i H _ { j k l } + \\nabla _ k H _ { j i l } - \\nabla _ l H _ { j i k } . \\end{align*}"}
+{"id": "8592.png", "formula": "\\begin{align*} M : = \\begin{bmatrix*} [ r ] 3 & - 2 & - 1 \\\\ - 2 & 4 & - 2 \\\\ - 1 & - 2 & 3 \\end{bmatrix*} . \\end{align*}"}
+{"id": "8476.png", "formula": "\\begin{align*} \\mu ^ { * A } = \\mu ^ A , \\end{align*}"}
+{"id": "6607.png", "formula": "\\begin{align*} ( U D \\phi ) _ { \\theta } = ( D + \\theta ) ( U \\phi ) _ { \\theta } , \\end{align*}"}
+{"id": "3776.png", "formula": "\\begin{align*} \\alpha _ \\# \\iota & = \\iota \\alpha _ \\# , & \\varepsilon _ \\# ^ { - 1 } \\iota & = \\iota \\varepsilon _ \\# , & \\bar { g } _ \\# ^ { - 1 } \\iota & = \\iota g _ \\# , \\end{align*}"}
+{"id": "2680.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\rightarrow 1 - } & { Q \\big ( z + ( 1 - z ) x \\big ) - \\big ( z + ( 1 - z ) x \\big ) \\over ( 1 - x ) ( 1 - Q ' ( x ) ) } = { 1 \\over 1 - Q ' ( 1 ) } \\lim \\limits _ { x \\rightarrow 1 - } { Q \\big ( z + ( 1 - z ) x \\big ) - \\big ( z + ( 1 - z ) x \\big ) \\over 1 - x } \\\\ & = { 1 - z \\over 1 - Q ' ( 1 ) } \\lim \\limits _ { x \\rightarrow 1 - } { Q \\big ( 1 - ( 1 - z ) ( 1 - x ) \\big ) - Q ( 1 ) + ( 1 - z ) ( 1 - x ) \\over ( 1 - z ) ( 1 - x ) } ~ = 1 - z . \\end{align*}"}
+{"id": "5209.png", "formula": "\\begin{align*} ( \\ell _ 1 ' ) ^ 3 = & k _ 0 \\ , \\ell _ 1 ^ 2 ( m _ 1 \\ell _ 1 + m _ 0 ) ^ 2 \\quad , \\ell _ 2 = \\frac { 1 } { m _ 1 \\ell _ 1 + m _ 0 } \\\\ b _ { 1 1 } ^ 3 = & \\frac { k _ 1 } { \\ell _ 1 ( m _ 1 \\ell _ 1 + m _ 0 ) } \\quad , b _ { 2 2 } = k _ 2 b _ { 1 1 } ( m _ 1 \\ell _ 1 + m _ 0 ) \\\\ b _ { 1 2 } = & k _ 3 b _ { 1 1 } ( m _ 1 \\ell _ 1 + m _ 0 ) \\quad , b _ { 1 3 } = k _ 4 b _ { 1 1 } \\ell _ 1 \\quad , b _ { 2 3 } = k _ 5 b _ { 1 1 } \\ell _ 1 \\ . \\end{align*}"}
+{"id": "3730.png", "formula": "\\begin{align*} I ( f ) ( \\varpi ^ n \\xi ) & = \\sum _ { j = - B } ^ { \\infty } q ^ { j ( \\ell - 2 ) } f ( \\varpi ^ { n - j } \\xi , 0 , \\varpi ^ j ) \\\\ & = \\sum _ { j = - B } ^ { A - 1 } q ^ { j ( \\ell - 2 ) } f ( 0 , 0 , \\varpi ^ j ) + \\sum _ { j = A } ^ { n - A } q ^ { j ( \\ell - 2 ) } f ( 0 ) + \\sum _ { j = n - A + 1 } ^ { n + B } q ^ { j ( \\ell - 2 ) } f ( \\varpi ^ { n - j } \\xi , 0 , 0 ) . \\end{align*}"}
+{"id": "8718.png", "formula": "\\begin{align*} \\prod _ { i } \\frac { u - { x _ i t } } { u - { x _ i } } = 1 + ( 1 - t ) \\sum _ { r = 1 } ^ \\infty \\sum _ { i } \\frac { x _ i ( x _ i | a ) _ { r - 1 } } { ( u | a ) _ { r } } \\prod _ { i \\ne j } \\frac { x - t x _ j } { x _ i - x _ j } . \\end{align*}"}
+{"id": "1337.png", "formula": "\\begin{align*} S _ \\tau : ^ m ( \\mathcal { H } ) \\ni h \\mapsto \\sum _ { j = 1 } ^ { n } f _ j ^ { \\otimes m } ( h ) \\tau _ j ^ { \\otimes m } = \\sum _ { j = 1 } ^ { n } \\langle h , \\tau _ j ^ { \\otimes m } \\rangle \\tau _ j ^ { \\otimes m } \\in ^ m ( \\mathcal { H } ) \\end{align*}"}
+{"id": "8423.png", "formula": "\\begin{align*} & \\gamma _ K ( X ) = \\kappa ( \\gamma _ K , \\gamma _ K ) = c ( K ) , \\\\ & \\kappa \\gamma _ K \\geqslant 1 K , \\\\ & \\kappa \\gamma _ K \\leqslant 1 S ( \\gamma _ K ) , \\\\ & \\kappa \\gamma _ K = 1 \\end{align*}"}
+{"id": "9006.png", "formula": "\\begin{align*} \\alpha \\left ( A _ { i _ n } , K \\cup \\left ( \\bigcup \\nolimits _ { m = n + 1 } ^ { { { N } } } A _ { i _ m } ^ { - 1 } \\right ) \\right ) \\leq \\epsilon ^ { 2 ( N - n ) + 4 } . \\end{align*}"}
+{"id": "7626.png", "formula": "\\begin{align*} \\mathcal { A } ^ { ( 3 ) } & = - \\bigg ( v _ n ^ { \\delta + 1 } , \\psi _ k '' ( w _ n ) \\frac { \\partial } { \\partial x } w _ n \\bigg ) \\leq C ( \\delta , n ) \\int _ { 0 } ^ { 1 } 1 \\cdot \\psi _ k '' ( w _ n ) \\frac { \\partial } { \\partial x } w _ n \\d x \\\\ & \\leq C ( \\nu , \\delta , n ) \\int _ { 0 } ^ { 1 } | w _ n ( x ) | _ + ^ 2 \\d x + \\frac { \\nu } { 4 } \\int _ { 0 } ^ { 1 } \\psi _ k '' ( w _ n ( x ) ) \\bigg ( \\frac { \\partial } { \\partial x } w _ n ( x ) \\bigg ) ^ 2 \\d x . \\end{align*}"}
+{"id": "2891.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ \\infty } f \\circ \\Phi _ { \\zeta } d m _ \\infty = \\int _ { \\mathbb { T } ^ \\infty } f \\mathbf { P } _ \\zeta d m _ \\infty \\leq \\| \\mathbf { P } _ \\zeta \\| _ \\infty \\int _ { \\mathbb { T } ^ \\infty } f d m _ \\infty . \\end{align*}"}
+{"id": "8574.png", "formula": "\\begin{align*} \\mathcal { A } : = \\bigcup _ { \\substack { \\delta \\mid n \\\\ \\delta \\neq 1 } } \\{ \\mbox { i r r e d u c i b l e } r ( x ) \\in \\mathbb { F } _ 2 [ x ] : \\deg ( r ( x ) ) = \\delta \\} . \\end{align*}"}
+{"id": "2704.png", "formula": "\\begin{gather*} \\| u \\| ^ p _ { L ^ p _ R } = \\int _ { R } ^ { \\infty } ( u ( r ) ) ^ p r ^ 5 d r , \\| u \\| ^ 2 _ { \\dot { H } ^ 1 _ R } = \\int _ { R } ^ { \\infty } ( \\partial _ r u ( r ) ) ^ 2 r ^ 5 d r , \\\\ \\| \\psi \\| _ { L ^ 2 ( R , \\infty ) } = \\int _ { R } ^ { \\infty } ( \\psi ( r ) ) ^ 2 d r . \\end{gather*}"}
+{"id": "9180.png", "formula": "\\begin{align*} \\exp \\Big ( c _ w \\kappa _ L w _ j ( X , \\varphi ) ^ 2 \\Big ) , g _ { j } ( X , \\varphi ) = \\exp \\Big ( c _ 4 \\kappa _ L \\sum _ { a = 0 , 1 , 2 } W _ { j } ( X , \\nabla ^ a _ j \\varphi ) ^ 2 \\Big ) , \\end{align*}"}
+{"id": "5723.png", "formula": "\\begin{align*} D \\theta ^ { A A ^ { \\prime } } \\equiv d \\theta ^ { A A ^ { \\prime } } + \\omega _ { B } ^ { A } \\theta ^ { B A ^ { \\prime } } + \\omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\theta ^ { A B ^ { \\prime } } = 0 . \\end{align*}"}
+{"id": "6232.png", "formula": "\\begin{align*} \\alpha = \\begin{cases} 3 / 5 - \\epsilon , & \\Omega \\hbox { i s a b o u n d e d s m o o t h d o m a i n } , \\\\ 3 / 4 - \\epsilon , & \\Omega \\hbox { i s a p a r a l l e l e p i p e d } , \\end{cases} \\end{align*}"}
+{"id": "7277.png", "formula": "\\begin{align*} g _ m \\psi ^ + _ { m } - g ^ + _ m \\psi _ m & = g _ m \\psi ^ + _ { m } - \\psi ^ + _ { m } \\left ( \\int _ { x } ^ { \\infty } \\frac { d y } { \\psi _ { m } ( y ) ^ 2 } \\right ) \\psi _ { m } + 1 = 1 , \\end{align*}"}
+{"id": "5293.png", "formula": "\\begin{align*} g ( X , X ) = \\lambda ^ 2 ( q ) g _ 1 ( \\pi _ \\ast X , \\pi _ \\ast X ) + f _ 1 ^ 2 ( p ) g _ 2 ( \\sigma _ \\ast X , \\sigma _ \\ast X ) . \\end{align*}"}
+{"id": "5390.png", "formula": "\\begin{align*} F ^ * ( x ) = \\{ y \\in F ( x ) \\mid & \\mbox { t h e r e e x i s t s a n a l l d i f f e r e n t s e l e c t i o n } s \\mbox { o f } F \\\\ & \\mbox { s u c h t h a t } y = s ( x ) \\} \\end{align*}"}
+{"id": "7837.png", "formula": "\\begin{align*} | H _ k | e ^ { w _ k x } > \\sum _ { j \\neq k } ^ n | H _ j | e ^ { w _ j x } , k = 0 , 1 , \\ldots , n , \\end{align*}"}
+{"id": "5398.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\sum _ { x = 0 } ^ { x = w } A _ x \\\\ = & \\frac { 1 } { 2 } \\left ( \\frac { A _ 0 + A _ w } { 2 } + \\frac { A _ 1 + A _ { w - 1 } } { 2 } + \\dots + \\frac { A _ w + A _ 0 } { 2 } \\right ) \\\\ = & \\frac { 1 } { 2 } \\sum _ { x = 0 } ^ { x = w } \\frac { C _ x C _ { w - x } } { 2 ^ { 2 w + 2 } } . \\\\ \\end{align*}"}
+{"id": "338.png", "formula": "\\begin{align*} ( p ^ 2 + 2 p \\Re ( A ( p , 1 ) ) + 1 ) ( \\Im ( A ( p , 1 ) ) - \\Im ( A ' ( p , 1 ) ) ) = 0 . \\end{align*}"}
+{"id": "2428.png", "formula": "\\begin{align*} \\bar \\partial _ \\tau ^ \\alpha u ^ n ( q ) - \\Delta u ^ n ( q ) + q u ^ n ( q ) = f ~ ~ n = 1 , 2 , \\ldots , N , \\end{align*}"}
+{"id": "2685.png", "formula": "\\begin{align*} \\mathcal { H } ^ s | _ K = \\mathcal { H } ^ s ( K ) \\mu , \\end{align*}"}
+{"id": "5010.png", "formula": "\\begin{align*} \\xi ( [ Z , X ] , Y ) + \\xi ( X , [ Z , Y ] ) = 0 , \\end{align*}"}
+{"id": "1135.png", "formula": "\\begin{align*} ( \\mu \\delta ) ^ n T & = \\sum _ { i = 0 } ^ n ( - 1 ) ^ i \\binom { n } { i } E ^ i T E ^ { n - i } \\end{align*}"}
+{"id": "7883.png", "formula": "\\begin{align*} f '' - \\left ( \\gamma ^ 2 e ^ z - \\frac { \\gamma } { 2 } e ^ { z / 2 } + \\frac { 1 } { 4 } \\right ) f = 0 \\end{align*}"}
+{"id": "1767.png", "formula": "\\begin{align*} \\alpha _ { 1 , 1 } X _ n ^ { ( 1 , 1 ) } + \\cdots + \\alpha _ { n - 3 , n - 3 } X _ n ^ { ( n - 3 , n - 3 ) } = \\mathbf { 0 } . \\end{align*}"}
+{"id": "8156.png", "formula": "\\begin{align*} \\omega _ j = \\frac { 4 \\pi \\abs { \\rho _ j ( 1 ) } ^ 2 } { \\cosh ( \\pi t _ j ) } , \\end{align*}"}
+{"id": "8903.png", "formula": "\\begin{align*} \\sum \\limits _ { \\ell = 0 } ^ { + \\infty } \\sigma _ { p } ^ { ( \\nu ) } \\left ( \\ell \\right ) \\ t ^ { \\ell } \\end{align*}"}
+{"id": "4567.png", "formula": "\\begin{align*} \\frac { 1 } { a _ { n + 1 } } < 1 - \\sum _ { i = 1 } ^ { n } \\frac { 1 } { a _ i } . \\end{align*}"}
+{"id": "4828.png", "formula": "\\begin{align*} \\phi ( v _ 1 ) = - ( d _ 0 F ) ^ { - 1 } ( d ^ 2 _ 0 F ( v _ 1 , v _ 1 ) ) . \\end{align*}"}
+{"id": "2610.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i < j \\leq r } a _ { i , j } = \\sum _ { j \\in [ r ] \\setminus \\{ 1 \\} } a _ { 1 , j } + \\sum _ { l , t \\in [ r ] \\setminus \\{ 1 \\} , l < t } a _ { l , t } \\geq r - 1 + \\frac { ( r - 1 ) ( r - 2 ) } { 2 } = \\binom { r } { 2 } . \\end{align*}"}
+{"id": "1866.png", "formula": "\\begin{align*} \\alpha _ { s _ i } ( \\sigma ) = \\left \\{ \\begin{array} { l l } \\frac { \\pi } { 2 } & \\mbox { i f $ \\dot { s } _ i \\le s _ i - \\phi _ k $ } \\\\ \\sin ^ { - 1 } \\left ( \\frac { s _ i - \\phi _ k } { \\dot { s } _ i } \\right ) & \\mbox { i f $ \\dot { s } _ i \\ge s _ i - \\phi _ k $ } \\end{array} \\right . \\end{align*}"}
+{"id": "6754.png", "formula": "\\begin{align*} \\sqrt { \\frac { \\pi } { 1 6 } } \\left [ \\frac { z ( s ) } { s } + \\frac { z ( 1 - s ) } { 1 - s } \\right ] = \\frac { 1 } { s } + \\frac { 1 } { 1 - s } + \\int _ { 0 } ^ { 1 } y ^ { \\frac { s } { 2 } - 1 } \\Psi ( y ) d y + \\int _ { 0 } ^ { 1 } y ^ { \\frac { 1 - s } { 2 } - 1 } \\Psi ( y ) d y \\end{align*}"}
+{"id": "1748.png", "formula": "\\begin{align*} \\langle \\mu _ { \\underline { m } } , \\mu _ { \\underline { m } ' } \\rangle ' = \\left \\{ \\begin{array} { l l } ( - 1 ) ^ { M + m } \\binom { k _ { \\rm i d } - 2 } { \\frac { k _ { \\rm i d } - 2 } { 2 } - m _ { \\rm i d } } ^ { - 1 } \\binom { k _ { c } - 2 } { \\frac { k _ { c } - 2 } { 2 } - m _ { c } } ^ { - 1 } , & \\underline { m } = - \\underline { m } ' \\\\ 0 , & \\underline { m } \\neq - \\underline { m } ' \\end{array} \\right . \\end{align*}"}
+{"id": "8304.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\frac { \\partial V ( \\textbf { x } ) } { \\partial \\textbf { x } _ E } \\ ! \\ ! \\ ! \\ ! & = \\frac { \\nu _ E } { \\Lambda } [ x _ E - x ^ * ( \\textbf { x } ) , \\ y _ E - y ^ * ( \\textbf { x } ) , \\ z _ E - z ^ * ( \\textbf { x } ) ] . \\end{array} \\right . \\end{align*}"}
+{"id": "8504.png", "formula": "\\begin{align*} \\varphi _ { \\xi ^ 3 } & = \\ ; \\frac { 1 } { 9 } ( 3 \\varphi \\varphi _ { \\xi ^ 2 } + 6 \\varphi _ { \\xi } ^ 2 + 2 \\varphi ^ 2 \\varphi _ { \\xi } ) , \\\\ \\varphi _ { \\xi ^ 4 } & = \\ ; \\frac { \\varphi ^ 2 + 5 \\varphi _ { \\xi } } { 3 } \\varphi _ { \\xi ^ 2 } + \\frac { 2 \\varphi ( \\varphi ^ 2 + 9 \\varphi _ { \\xi } ) } { 2 7 } \\varphi _ { \\xi } . \\end{align*}"}
+{"id": "7859.png", "formula": "\\begin{align*} f ( z ) = 1 + C _ 1 e ^ z + \\cdots + C _ m e ^ { m z } , C _ j \\in \\mathbb { C } . \\end{align*}"}
+{"id": "304.png", "formula": "\\begin{align*} \\begin{aligned} \\theta = \\mu \\circ \\mathrm { N m } _ { T ( E ) / S ^ { \\mathrm { o p } } ( F ) } \\cdot \\zeta _ { ( { \\chi _ { \\mu } } _ { \\mathrm { B C } } , \\chi _ { \\mu \\circ \\mathrm { N m } _ { T ( E ) / S ^ { \\mathrm { o p } } ( F ) } } ) } . \\end{aligned} \\end{align*}"}
+{"id": "6702.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\omega ( n ) } = \\sum _ { t \\mid n } \\mu ( t ) d ( t ) , \\end{align*}"}
+{"id": "225.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - \\beta \\tau \\eta \\leqslant - \\alpha \\frac { ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\alpha } { 2 } , \\end{align*}"}
+{"id": "1304.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j , k \\leq n } | \\langle \\tau _ j , \\tau _ k \\rangle | ^ { 2 } = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n | \\langle \\tau _ j , \\tau _ k \\rangle | ^ { 2 } \\geq \\frac { n ^ 2 } { { d } } . \\end{align*}"}
+{"id": "2855.png", "formula": "\\begin{align*} a _ k + a _ { \\ell } = ( \\rho + \\tau ) + ( \\rho - \\tau ) = ( \\rho + \\sigma ) + ( \\rho - \\sigma ) = c _ k + c _ { \\ell } \\end{align*}"}
+{"id": "6968.png", "formula": "\\begin{gather*} L _ { m , n } ^ { I I I , ( \\alpha ) } ( x ) = \\sum _ { j = 0 } ^ n h _ { j , n } x ^ j . \\end{gather*}"}
+{"id": "1218.png", "formula": "\\begin{align*} A = \\begin{pmatrix} \\Lambda & { \\bf c } \\\\ { \\bf 0 } ^ T & d \\end{pmatrix} \\end{align*}"}
+{"id": "4637.png", "formula": "\\begin{align*} N _ { \\mathrm { r a m } \\ , p _ Y } ( r ) : = \\frac { 1 } { \\mathrm { d e g } \\ , p _ Y } \\int ^ r _ 0 \\# \\left ( \\mathrm { R a m } ( p _ Y ) \\cap Y ( t ) \\right ) \\frac { d t } { t } . \\end{align*}"}
+{"id": "809.png", "formula": "\\begin{align*} \\alpha _ v ^ k = \\frac { \\widehat { \\nu } ( \\sigma ^ { - k } [ v ] ) } { \\mu ( [ v ] ) } \\le \\max _ { v \\in V _ { \\max } } \\frac { 1 } { \\mu ( [ v ] ) } < \\infty \\end{align*}"}
+{"id": "8968.png", "formula": "\\begin{align*} \\partial _ r u = d \\pi _ N ( u ) \\partial _ r u + d \\pi _ N ^ { \\perp } ( u ) \\partial _ r u = d \\pi _ N ( u ) \\partial _ r u + \\nu ( u ) \\partial _ r ( d i s t _ N ( u ) ) \\end{align*}"}
+{"id": "4841.png", "formula": "\\begin{align*} ( v \\otimes w ) ^ k = v ^ i w ^ j , k = m ( i - 1 ) + j , \\end{align*}"}
+{"id": "4598.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n v _ i \\leq \\sum _ { i = 1 } ^ n u _ i . \\end{align*}"}
+{"id": "9108.png", "formula": "\\begin{align*} \\mathrm { V a r } \\left [ \\frac { 1 } { n _ 1 } \\tilde { i } ( x ^ { n _ 1 } ; Y _ 1 ^ { n _ 1 } , Y _ 2 ^ { n _ 2 } ) \\right ] & = \\frac { \\mathsf { V } _ { \\rho , 1 } } { n _ 1 } + \\frac { \\mathsf { V } _ { \\rho , 2 } } { n _ 1 ^ 2 } \\cdot \\sum _ { i = n _ 2 + 1 } ^ { n _ 1 } x _ i ^ 2 , \\end{align*}"}
+{"id": "3220.png", "formula": "\\begin{align*} f ' ( w ) = f ( w ) \\geq ( 4 k _ 3 + 3 k _ 2 + 2 k _ 1 + d - 1 ) - ( 2 k _ 3 + 2 k _ 2 + k _ 1 ) = 2 k _ 3 + k _ 2 + k _ 1 + d - 1 \\ge 2 k _ 3 ' + k _ 1 ' + k ' _ 2 + 1 , \\end{align*}"}
+{"id": "2333.png", "formula": "\\begin{align*} \\ ( ( v \\otimes v ) v \\ ) _ i & = \\sum _ { j = 1 } ^ N v _ i v _ j v _ j = \\ ( \\sum _ { j = 1 } ^ N v _ j v _ j \\ ) v _ i = v _ i \\end{align*}"}
+{"id": "8793.png", "formula": "\\begin{align*} 0 = \\left ( \\sum _ { i = 1 } ^ 3 \\alpha _ i ( k _ { t _ i } \\otimes k _ { t _ i } ) + \\sum _ { i = 4 } ^ 5 ( \\alpha _ i k _ { \\lambda _ i } \\otimes C _ { \\theta } k _ { \\lambda _ i } ) \\right ) ( f ) = { \\alpha _ 1 } f ( t _ 1 ) k _ { t _ 1 } + \\sum _ { i = 4 } ^ 5 { \\alpha _ i } k _ { \\lambda _ i } \\langle f , C _ { \\theta } k _ { \\lambda _ i } \\rangle . \\end{align*}"}
+{"id": "3614.png", "formula": "\\begin{align*} 0 = \\pi R _ * L _ * \\int R - \\frac { 6 \\pi } { 7 } \\frac { R _ * ^ 4 } { L _ * } \\int R ^ 2 ( R ' ) ^ 2 \\end{align*}"}
+{"id": "574.png", "formula": "\\begin{align*} p _ t : = \\mathbb { E } ^ \\dagger [ ( \\mu _ t - m _ t ) ^ 2 ] . \\end{align*}"}
+{"id": "6516.png", "formula": "\\begin{align*} \\Big [ \\frac { 1 } { l k } \\Big ( \\underset { i = 1 } { \\overset { k } { \\sum } } \\underset { j = 1 } { \\overset { l } { \\sum } } ( B ^ j _ i - p _ i ) \\Big ) ^ 2 \\Big ] & = \\frac { 1 } { l ^ 2 k ^ 2 } \\Big [ \\Big ( \\sum _ { j = 1 } ^ l \\sum _ { i = 1 } ^ k ( B _ i ^ j - p _ i ) \\Big ) ^ 2 \\Big ] \\leq 1 / 8 . \\end{align*}"}
+{"id": "8821.png", "formula": "\\begin{align*} P _ 2 ^ * P _ 2 A _ { 2 1 } = P _ 2 ^ * A _ { 2 1 } P _ 1 = B _ { 1 2 } ^ * P _ 1 = A _ { 2 1 } . \\end{align*}"}
+{"id": "1019.png", "formula": "\\begin{align*} W \\in \\S ' ( \\R ) \\Longleftrightarrow \\ \\beta _ 0 > 0 W \\notin \\S ' ( \\R ) \\Longleftrightarrow \\ \\beta _ 0 = 0 . \\end{align*}"}
+{"id": "2538.png", "formula": "\\begin{align*} p _ { 1 , j } - q _ { 1 , j } & = \\frac { \\varphi ( x _ 1 + \\xi - x _ j ) - \\varphi ( y _ 1 + \\xi - y _ j ) } { N \\varrho _ N ^ \\varphi ( x _ 1 + \\xi ) } \\\\ & + \\varphi ( y _ 1 + \\xi - y _ j ) \\frac { \\sum _ k [ \\varphi ( y _ 1 + \\xi - y _ k ) - \\varphi ( x _ 1 + \\xi - x _ k ) ] } { N ^ 2 \\varrho _ N ^ \\varphi ( x _ 1 + \\xi ) \\tilde \\varrho _ N ^ \\varphi ( y _ 1 + \\xi ) } \\ , , \\end{align*}"}
+{"id": "6777.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\left ( \\frac { \\left | \\sum _ { n = 1 } ^ N n \\chi ( n ) \\right | } { \\left | \\sum _ { n = 1 } ^ N n \\chi ^ { - 1 } ( n ) \\right | } \\right ) = A \\end{align*}"}
+{"id": "592.png", "formula": "\\begin{align*} \\Theta _ { n + 1 } ^ { ( \\epsilon ) } = \\Theta _ n ^ { ( \\epsilon ) } + K _ n ^ { ( \\epsilon ) } \\left \\{ \\left ( X _ { t _ { n + 1 } } ^ { ( \\epsilon ) } - X _ { t _ n } ^ { ( \\epsilon ) } \\right ) - \\frac { 1 } { 2 } \\left ( \\Theta _ n ^ { ( \\epsilon ) } + \\pi _ n ^ { ( \\epsilon ) } [ \\theta ] \\right ) A X _ { t _ n } ^ { ( \\epsilon ) } \\Delta t \\right \\} \\end{align*}"}
+{"id": "8788.png", "formula": "\\begin{align*} \\langle C f , C g \\rangle = \\langle g , f \\rangle \\end{align*}"}
+{"id": "549.png", "formula": "\\begin{align*} { \\rm d } I _ t ^ { ( i ) } = { \\rm d } X _ t ^ \\dagger - f ( X _ t ^ \\dagger , \\Theta _ t ^ { ( i ) } ) { \\rm d } t - \\gamma ^ { 1 / 2 } { \\rm d } W _ t ^ { ( i ) } \\end{align*}"}
+{"id": "8139.png", "formula": "\\begin{align*} I _ \\nu ( z ) = \\frac { \\Bigl ( \\dfrac { z } { 2 } \\Bigr ) ^ \\nu } { \\Gamma \\Bigl ( \\nu + \\dfrac { 1 } { 2 } \\Bigr ) \\Gamma \\Bigl ( \\dfrac { 1 } { 2 } \\Bigr ) } \\int _ 0 ^ \\pi e ^ { \\pm z \\cos ( \\theta ) } \\sin ( \\theta ) ^ { 2 \\nu } \\ , d \\theta \\end{align*}"}
+{"id": "3321.png", "formula": "\\begin{align*} { \\rm d } s ^ 2 _ { G ^ J _ 1 ( \\R ) } & = \\sum _ { i = 1 } ^ 6 \\lambda _ i ^ 2 = \\alpha \\frac { { \\rm d } x ^ 2 + { \\rm d } y ^ 2 } { y ^ 2 } + \\beta \\left ( \\frac { { \\rm d } x } { y } + 2 { \\rm d } \\theta \\right ) ^ 2 , \\\\ & + \\frac { \\gamma } { y } \\big ( { \\rm d } q ^ 2 + S { \\rm d } p ^ 2 + 2 x { \\rm d } p { \\rm d } q \\big ) + \\delta ( { \\rm d } \\kappa - p { \\rm d } q + q { \\rm d } p ) ^ 2 , ~ ~ S : = x ^ 2 + y ^ 2 . \\end{align*}"}
+{"id": "4086.png", "formula": "\\begin{align*} \\Big ( ( h _ 1 , k _ 1 ) , ( h _ 2 , k _ 2 ) \\Big ) = \\int _ M \\langle h _ 1 , h _ 2 \\rangle _ g + \\langle k _ 1 , k _ 2 \\rangle _ g d V _ g ( h _ 1 , k _ 1 ) , ( h _ 2 , k _ 2 ) \\in \\Gamma ( S ^ 2 M ) \\times \\Omega ^ 2 . \\end{align*}"}
+{"id": "7202.png", "formula": "\\begin{align*} \\big \\langle p , \\ , q \\big \\rangle \\ , = \\ , \\mathfrak { R e } \\big [ \\ , p \\overline q \\ , \\big ] , \\end{align*}"}
+{"id": "803.png", "formula": "\\begin{align*} \\frac { \\# ( S _ n \\cap A _ \\epsilon ) } { \\# S _ n } & \\le C _ 2 \\ \\nu \\left ( \\bigcup _ { x \\in S _ n \\cap A _ \\epsilon } O ( x , R ) \\right ) \\\\ & \\le C _ 2 \\ \\nu \\left ( \\xi \\in \\partial \\Gamma : \\left | \\frac { \\log \\| \\rho ( \\xi _ n ) \\| } { n } - \\Lambda \\right | > \\frac { \\epsilon } { 2 } \\right ) . \\end{align*}"}
+{"id": "2801.png", "formula": "\\begin{align*} 0 \\leq a _ k = \\rho - \\Delta < \\rho - \\delta = b _ k \\leq b _ j = \\rho + \\delta < \\rho + \\Delta = a _ j \\end{align*}"}
+{"id": "7898.png", "formula": "\\begin{align*} f ( z ) ^ 2 - 2 e ^ z f ( z - \\log 2 ) = 1 . \\end{align*}"}
+{"id": "8078.png", "formula": "\\begin{align*} V ( m ^ 2 p , t ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 0 0 0 ) } ( m ^ 2 p ) ^ { - u } F ( u ) \\frac { \\gamma \\Bigl ( \\dfrac { 1 } { 2 } + u , t \\Bigr ) } { \\gamma \\Bigl ( \\dfrac { 1 } { 2 } , t \\Bigr ) } \\frac { d u } { u } , \\end{align*}"}
+{"id": "6749.png", "formula": "\\begin{align*} y = \\exp \\left ( \\frac { 1 } { s } \\log \\left ( 1 - \\rho \\right ) \\right ) \\end{align*}"}
+{"id": "1558.png", "formula": "\\begin{align*} \\psi '' _ { \\mathrm { u } } ( t _ { 1 } ) = & ( p ^ { + } + \\gamma ^ { - } - 1 ) t _ { 1 } ^ { p ^ { + } } \\int _ { M } | D \\mathrm { u } ( z ) | ^ { p ( z ) } \\ , \\ , d v _ { g } ( z ) \\\\ & + ( q ^ { + } + \\gamma ^ { - } - 1 ) t _ { 1 } ^ { q ^ { + } } \\int _ { M } \\mu ( z ) \\ , | D \\mathrm { u } ( z ) | ^ { q ( z ) } \\ , \\ , d v _ { g } ( z ) \\\\ & - \\lambda ( r ^ { - } + \\gamma ^ { - } - 1 ) t _ { 1 } ^ { r ^ { - } } \\int _ { M } | \\mathrm { u } ( z ) | ^ { r ( z ) } \\ , \\ , d v _ { g } ( z ) . \\end{align*}"}
+{"id": "5880.png", "formula": "\\begin{align*} M = \\hbox { r a n k } ( D ) = \\hbox { r a n k } ( C _ p D ) = N - p , \\end{align*}"}
+{"id": "4116.png", "formula": "\\begin{align*} \\mathcal { G } _ t = \\mathcal { G } _ t ( g _ t , b _ t ) \\end{align*}"}
+{"id": "3018.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\Lambda _ 0 } [ \\theta , \\varphi , \\pi ] = \\mathcal { A } _ 0 [ \\theta , \\varphi , \\pi ] - \\int _ { \\mathcal { Y } ^ { N + 1 } } \\Lambda _ 0 \\hat { \\theta } ^ { ( N + 1 ) } \\end{align*}"}
+{"id": "2643.png", "formula": "\\begin{align*} \\lim _ { X \\rightarrow \\infty } \\frac { S ( X , Y ; \\hat { \\phi } , \\Phi ) } { X \\log X } = 0 . \\end{align*}"}
+{"id": "950.png", "formula": "\\begin{align*} \\textrm { G } _ C \\textrm { - d i m } _ R ( N ) + \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { T r } _ C ( N ) ) \\ , = \\ , \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { E x t } _ { R } ^ t ( N , \\ , C ) ) + 1 \\end{align*}"}
+{"id": "59.png", "formula": "\\begin{align*} x _ i = \\begin{cases} 1 , & f ( i ) \\geq 1 \\\\ 0 , & o t h e r w i s e . \\end{cases} \\end{align*}"}
+{"id": "4941.png", "formula": "\\begin{align*} \\widetilde { H } ( \\mathbf Z ) _ m = \\frac { | \\delta _ m ^ + Z _ m | ^ 2 + | \\delta _ m ^ - Z _ m | ^ 2 } 2 - \\frac { 1 } 2 | Z _ m | ^ 4 . \\end{align*}"}
+{"id": "4131.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda = \\int _ M \\frac { u \\triangle u } { 2 } - 1 2 u ^ 2 + 1 2 u \\triangle \\omega - ( \\triangle \\omega ) ^ 2 - \\frac { 1 } { 2 } | \\nabla v | ^ 2 d V _ g \\end{align*}"}
+{"id": "7095.png", "formula": "\\begin{align*} \\pi _ { 2 + 2 \\sigma } = 4 d _ { 1 , 1 } + 2 q _ 4 - 2 q _ 2 q _ 3 - q _ 2 ^ 3 + ( 6 b _ 1 ^ 3 - 1 8 b _ 1 b _ 2 + 6 b _ 3 ) q _ 1 \\end{align*}"}
+{"id": "5069.png", "formula": "\\begin{align*} P ( n ) - g _ 1 ( n ) - \\cdots - g _ r ( n ) = 0 \\qquad \\end{align*}"}
+{"id": "3155.png", "formula": "\\begin{align*} - A : D ^ 2 ( v ^ { 1 1 } + v ^ { 2 2 } ) = \\left ( a _ { 1 1 } - \\bar { a } _ { 1 1 } \\right ) + \\left ( a _ { 2 2 } - \\bar { a } _ { 2 2 } \\right ) = \\mathrm { t r } ( A ) - \\mathrm { t r } ( \\bar { A } ) \\equiv 0 , \\end{align*}"}
+{"id": "7019.png", "formula": "\\begin{align*} \\underline { M } ( C _ 2 / C _ 2 ) = E ^ { C _ 2 } _ m = [ S ^ { m } , E _ { C _ 2 } ] ^ { C _ 2 } \\end{align*}"}
+{"id": "4535.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 1 2 } = \\frac { 1 } { 3 } + \\frac { 1 } { 4 } = \\frac { 7 } { 1 2 } < \\theta \\leq \\frac { 1 } { 2 } + \\frac { 1 } { 1 1 } = \\frac { 1 3 } { 2 2 } . \\end{align*}"}
+{"id": "6488.png", "formula": "\\begin{align*} \\eta _ { p , l , k } : = \\frac { l - 1 } { 2 \\sqrt { k } } \\underset { i = 1 } { \\overset { k } { \\sum } } \\left ( p _ i - \\frac { 1 } { 2 } \\right ) ^ 2 \\geq \\kappa _ \\alpha , \\end{align*}"}
+{"id": "2117.png", "formula": "\\begin{align*} ( e _ { C _ { 1 2 } } \\overline e _ { C _ { 1 2 } } + g \\overline g ) ( P ) = 0 , \\end{align*}"}
+{"id": "4392.png", "formula": "\\begin{align*} x _ 1 \\geq n c ^ { ( 1 ) } _ 1 = x _ 1 ^ * \\end{align*}"}
+{"id": "8375.png", "formula": "\\begin{align*} \\alpha _ r n + a _ r \\neq 0 , - 1 , - 2 , \\ldots ( n = 0 , 1 , 2 , \\ldots \\ ; \\ , 1 \\leq r \\leq p ) \\end{align*}"}
+{"id": "429.png", "formula": "\\begin{align*} \\| \\tau _ j \\| = 1 \\| f _ j \\| = 1 , \\forall 1 \\leq j \\leq n \\end{align*}"}
+{"id": "1752.png", "formula": "\\begin{align*} \\frac { 2 - \\underline k } { 2 } \\leq \\underline { m } _ N : = \\left ( N - \\frac { k _ c - 2 } { 2 } - \\bar m , N - \\frac { k _ c - 2 } { 2 } \\right ) \\leq \\frac { \\underline k - 2 } { 2 } . \\end{align*}"}
+{"id": "9183.png", "formula": "\\begin{align*} G _ j ^ { \\Psi } ( X , \\varphi ' + \\zeta ; \\ , u _ j ) \\leq \\prod _ { k = 1 } ^ M \\sup _ { t \\in [ 0 , 1 ] } g _ { j + \\frac { k - 1 } { M } } ( X _ { k / M } , \\xi _ k + 1 _ { k = 1 } t u _ j ) G _ { j + 1 } ( \\bar { X } , \\varphi ' ) \\end{align*}"}
+{"id": "6409.png", "formula": "\\begin{align*} ( \\nabla ^ 2 \\Psi ( P _ j ) ) ^ 2 = 2 \\begin{pmatrix} M _ 1 ^ 2 & 0 & 0 & - M _ 1 ^ 2 & 0 & 0 \\\\ 0 & M _ 2 ^ 2 & 0 & 0 & - M _ 2 ^ 2 & 0 \\\\ 0 & 0 & j ^ 2 & 0 & 0 & - j ^ 2 \\\\ - M _ 1 ^ 2 & 0 & 0 & M _ 1 ^ 2 & 0 & 0 \\\\ 0 & - M _ 2 ^ 2 & 0 & 0 & M _ 2 ^ 2 & 0 \\\\ 0 & 0 & - j ^ 2 & 0 & 0 & j ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "1812.png", "formula": "\\begin{align*} E ( t , u , v , w ) : = L ( t , u , w ) - L ( t , u , v ) - ( w - v ) \\cdot L _ { \\dot { u } } ( t , u , v ) . \\end{align*}"}
+{"id": "4570.png", "formula": "\\begin{align*} a _ { n + 1 , r } = a _ { n , r } ^ 2 - r a _ { n , r } + r \\qquad \\end{align*}"}
+{"id": "7407.png", "formula": "\\begin{align*} \\mu ( s \\widetilde { \\alpha } , \\sigma ) = \\gamma ( G / P ) ^ 2 q _ F ^ { n ( \\omega ) + n ( \\sigma \\times \\Pi ( \\sigma ) ) - n ( \\sigma ) } \\dfrac { ( 1 - \\omega ( \\varpi ) q _ F ^ { - 2 s } ) ( 1 - \\omega ^ { - 1 } ( \\varpi ) q _ F ^ { 2 s } ) } { ( 1 - \\omega ^ { - 1 } ( \\varpi ) q _ F ^ { - 1 + 2 s } ) ( 1 - \\omega ( \\varpi ) q _ F ^ { - 1 - 2 s } ) } \\end{align*}"}
+{"id": "558.png", "formula": "\\begin{align*} K _ n = \\sigma _ n ( A X _ { t _ n } ^ \\dagger ) ^ { \\rm T } \\left ( \\gamma + \\Delta t \\sigma _ n ( A X _ { t _ n } ^ \\dagger ) ^ { \\rm T } A X _ { t _ n } ^ \\dagger \\right ) ^ { - 1 } \\ , . \\end{align*}"}
+{"id": "7241.png", "formula": "\\begin{align*} \\Phi ^ { a , b } _ { j _ 1 , \\ldots , j _ n } ( X _ 1 , \\ldots , X _ N ) : = 0 \\end{align*}"}
+{"id": "5894.png", "formula": "\\begin{align*} w ( T ) = w ' ( T ) = 0 . \\end{align*}"}
+{"id": "5291.png", "formula": "\\begin{align*} \\tau ( F ) = ( n - 2 ) \\frac { \\lambda ^ 2 } { 2 } F _ \\ast \\left ( \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\right ) - ( m - n ) F _ \\ast ( H ) . \\end{align*}"}
+{"id": "2310.png", "formula": "\\begin{align*} f ( u ) = \\frac { u ^ 2 } { u ^ 2 + ( 1 - u ) ^ 2 } ( 1 - 5 ( 1 - u ) ^ 2 ) , \\end{align*}"}
+{"id": "3839.png", "formula": "\\begin{align*} g _ V ( v ) = \\int _ { - \\infty } ^ \\infty \\int _ { - \\infty } ^ \\infty h _ { y _ 1 , y _ 2 } ( v ) f _ { Y _ 1 , Y _ 2 } ( y _ 1 , y _ 2 ) \\ , d y _ 1 d y _ 2 , \\end{align*}"}
+{"id": "3720.png", "formula": "\\begin{align*} E ( g , \\Phi _ s ) : = \\sum _ { \\xi ' \\in F ^ 2 - \\{ 0 \\} } \\Phi _ s ( \\xi ' g ) . \\end{align*}"}
+{"id": "5640.png", "formula": "\\begin{align*} \\int _ { 0 \\leq z _ p \\leq \\cdots \\leq z _ 1 \\leq Z } \\frac { d z _ 1 } { z _ 1 } \\cdots \\frac { d z _ { p - 1 } } { z _ { p - 1 } } \\frac { d z _ p } { 1 - z _ p } ( z _ p ) ^ A = \\sum _ { n = 1 } ^ { + \\infty } \\frac { Z ^ { n + A } } { ( n + A ) ^ p } \\end{align*}"}
+{"id": "7911.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\mathcal { \\hat { E } } ^ { ( k ) } _ t = \\mathcal { \\hat { E } } ^ { ( k ) } _ \\infty , \\end{align*}"}
+{"id": "2343.png", "formula": "\\begin{align*} \\langle \\psi _ 2 - T \\psi _ 2 , \\phi _ 1 \\rangle = - \\langle T \\psi _ 2 , \\phi _ 1 \\rangle = - \\langle T \\phi _ 1 , T ^ 2 \\psi _ 2 \\rangle = - \\langle \\phi _ 1 , \\psi _ 2 \\rangle = 0 , \\end{align*}"}
+{"id": "2930.png", "formula": "\\begin{align*} \\mathcal { L } _ q f ( \\xi ) = \\sum _ { j \\in \\mathbb { Z } } ( 1 - q ^ { \\xi _ j } ) \\nabla _ { j , j + 1 } f ( \\xi ) \\end{align*}"}
+{"id": "4066.png", "formula": "\\begin{align*} 1 & < \\exp \\Big ( \\frac { u _ 0 ( x ) - u ( x , t ) } { t } \\Big ) \\\\ & = \\exp \\Big ( \\frac { 1 } { t } \\int _ { 0 } ^ t - \\frac { \\partial u } { \\partial t } ( x , \\tau ) \\ d \\tau \\Big ) . \\end{align*}"}
+{"id": "6280.png", "formula": "\\begin{align*} \\mathcal { W } ^ { c } = \\{ ( x , y , z ) \\in \\mathcal { M } : z \\leq 0 , y \\leq 1 \\} = \\mathcal { C } \\cup \\mathcal { K } \\end{align*}"}
+{"id": "169.png", "formula": "\\begin{align*} F ( \\theta , z ( \\theta - \\beta , \\hslash ) ) = \\int _ { 0 } ^ { \\theta } ( \\theta - s ) \\rho ( s , z ( s - \\beta , \\hslash ) ) d s , \\end{align*}"}
+{"id": "3770.png", "formula": "\\begin{align*} ( s , t ; 1 ) & = s , & ( s , t ; 2 ) & = s t , & ( s , t ; 3 ) & = s t s , & & \\dots \\\\ ( t , s ; 1 ) & = t , & ( t , s ; 2 ) & = t s , & ( t , s ; 3 ) & = t s t , & & \\dots \\end{align*}"}
+{"id": "5071.png", "formula": "\\begin{align*} \\frac { P } { Q } = B _ 2 + \\frac { B _ 1 } { ( 1 - \\alpha x _ 1 ) } + \\frac { B _ 0 } { ( 1 - \\alpha x _ 1 ) ^ 2 } = B _ 2 + \\sum _ { n = 0 } ^ \\infty B _ 1 \\alpha ^ n x _ 1 ^ n + ( n + 1 ) B _ 0 \\alpha ^ n x _ 1 ^ n . \\end{align*}"}
+{"id": "8039.png", "formula": "\\begin{align*} \\frac { 3 L ( 1 , f ) } { \\pi } \\Bigl ( \\frac { A _ f ( p , 1 ) } { p ^ { \\frac { 1 } { 2 } } } - \\frac { 1 } { p ^ { \\frac { 3 } { 2 } } } \\Bigr ) T M \\log T = \\frac { 3 c L ( 1 , g ) } { \\pi } \\Bigl ( \\frac { A _ g ( p , 1 ) } { p ^ { \\frac { 1 } { 2 } } } - \\frac { 1 } { p ^ { \\frac { 3 } { 2 } } } \\Bigr ) T M \\log T + O _ p ( T M ) . \\end{align*}"}
+{"id": "1648.png", "formula": "\\begin{align*} C ( \\underline k , \\underline m ) = ( - 1 ) ^ { \\left ( \\sum _ { \\sigma \\not \\in \\Sigma _ B } \\frac { k _ { \\sigma } - 2 } { 2 } \\right ) } 4 ^ { r _ \\R } \\cdot ( 3 2 \\pi ) ^ { r _ \\C } \\left ( \\frac { 1 } { \\pi } \\right ) ^ { d - r } \\prod _ { \\tilde \\sigma : F \\hookrightarrow \\C } \\frac { \\Gamma ( \\frac { k _ { \\tilde \\sigma } } { 2 } - m _ { \\tilde \\sigma } ) \\Gamma ( \\frac { k _ { \\tilde \\sigma } } { 2 } + m _ { \\tilde \\sigma } ) } { ( - 1 ) ^ { m _ { \\tilde \\sigma } } . ( 2 \\pi ) ^ { k _ { \\tilde \\sigma } } } . \\end{align*}"}
+{"id": "3630.png", "formula": "\\begin{align*} { \\rm d } s ^ 2 = \\left ( \\frac { 2 } { 1 - | x | ^ 2 } \\right ) ^ 2 \\ , { \\rm d } x ^ 2 \\end{align*}"}
+{"id": "2859.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ j b _ i = b _ k + \\sum _ { i = 1 } ^ { k - 1 } b _ i + \\sum _ { i = k + 1 } ^ j b _ i \\leq c _ k + \\sum _ { i = 1 } ^ { k - 1 } a _ i + \\sum _ { i = k + 1 } ^ j c _ i = \\sum _ { i = 1 } ^ j c _ i . \\end{align*}"}
+{"id": "4117.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } \\mathcal { G } _ t = ( h , K ) \\in \\mathcal { V } ( g _ t , b _ t ) = ( f _ t ^ * g , - B _ t ) . \\end{align*}"}
+{"id": "3413.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } \\widetilde \\omega _ { q } - \\mu \\Delta \\widetilde \\omega _ { q } + ( v \\cdot \\nabla ) \\widetilde \\omega _ { q } = \\widetilde \\omega _ { q } \\cdot \\nabla v - \\partial _ { z } \\Big ( \\Delta _ { q } \\Big ( \\frac { B _ { \\theta } } { r } B \\Big ) \\Big ) . \\\\ \\widetilde \\omega _ { q } \\vert _ { t = 0 } = \\Delta _ { q } \\omega ^ { 0 } . \\end{array} \\right . \\end{align*}"}
+{"id": "6321.png", "formula": "\\begin{align*} \\mathcal { K } \\mathsf { K } ^ * = \\mathsf { K } , \\mathcal { S } \\mathsf { S } ^ * = \\mathsf { S } , \\mathcal { A } \\mathsf { A } ^ * = \\mathsf { A } . \\end{align*}"}
+{"id": "8299.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mu _ E = y _ 1 ( z _ 2 \\ ! - \\ ! z _ 3 ) + y _ 2 ( z _ 3 \\ ! - \\ ! z _ 1 ) + y _ 3 ( z _ 1 \\ ! - \\ ! z _ 2 ) \\\\ \\mu _ 1 = y _ 2 ( z _ 3 \\ ! - \\ ! z _ E ) + y _ 3 ( z _ E \\ ! - \\ ! z _ 2 ) + y _ E ( z _ 2 \\ ! - \\ ! z _ 3 ) \\\\ \\mu _ 2 = y _ 3 ( z _ 1 \\ ! - \\ ! z _ E ) + y _ 1 ( z _ E \\ ! - \\ ! z _ 3 ) + y _ E ( z _ 3 \\ ! - \\ ! z _ 1 ) \\\\ \\mu _ 3 = y _ 1 ( z _ 2 \\ ! - \\ ! z _ E ) + y _ 2 ( z _ E \\ ! - \\ ! z _ 1 ) + y _ E ( z _ 1 \\ ! - \\ ! z _ 2 ) \\end{array} \\right . \\end{align*}"}
+{"id": "5926.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } w '' - \\Delta w = 0 & \\hbox { i n } ( T , + \\infty ) \\times \\Omega , \\\\ \\partial _ \\nu w + \\mu w = 0 & \\hbox { o n } ( T , + \\infty ) \\times \\Gamma . \\end{array} \\right . \\end{align*}"}
+{"id": "6772.png", "formula": "\\begin{align*} \\phi _ { \\chi } ^ e ( s ) = \\int _ { \\frac { 1 } { P } } ^ { \\infty } y ^ { \\frac { s } { 2 } - 1 } \\Psi _ { \\chi } ( y ) d y + \\frac { P ^ { 1 - s } } { G \\left ( \\overline { \\chi } , P \\right ) } \\int _ { \\frac { 1 } { P } } ^ { \\infty } y ^ { \\frac { 1 - s } { 2 } - 1 } \\Psi _ { \\overline { \\chi } } ( y ) d y \\end{align*}"}
+{"id": "4555.png", "formula": "\\begin{align*} \\frac { 1 } { 2 m - 1 } = \\frac { 1 } { 2 m } + \\frac { 1 } { 2 m ( 2 m - 1 ) ) } \\end{align*}"}
+{"id": "6991.png", "formula": "\\begin{align*} \\gamma ( y ) = \\lambda ( y ) \\cdot \\exp ( J ( y , x ) ) , \\end{align*}"}
+{"id": "3736.png", "formula": "\\begin{align*} \\tilde { c } _ \\ell ( I ( f ) ) = Z _ { r _ \\ell } ( f , 2 - \\ell ) = : c _ \\ell ( f ) . \\end{align*}"}
+{"id": "4731.png", "formula": "\\begin{align*} \\Tilde { f } = a _ 0 + a _ 1 X + \\cdots + a _ m X ^ m . \\end{align*}"}
+{"id": "8274.png", "formula": "\\begin{align*} \\bigg | \\sin ( ( 2 n _ k + 1 ) v y ) & \\left ( \\frac { | \\rho _ 0 | } { | \\rho _ k | } - \\frac { 1 } { 2 n _ k + 1 } \\right ) \\bigg | \\le \\frac { ( 2 n _ k + 1 ) | \\rho _ 0 | - | \\rho | _ k } { | \\rho _ k | ( 2 n _ k + 1 ) } \\\\ & = \\frac { ( 2 n _ k + 1 ) ^ 2 | \\rho _ 0 | ^ 2 - | \\rho | _ k ^ 2 } { | \\rho _ k | ( 2 n _ k + 1 ) ( ( 2 n _ k + 1 ) | \\rho _ 0 | + | \\rho _ k | ) } < \\frac { ( 2 n _ k + 1 ) ^ 2 \\beta _ 0 ^ 2 } { ( 2 n _ k + 1 ) ^ 2 | \\rho _ 0 | | \\rho _ k | } < \\frac { 1 } { v } . \\end{align*}"}
+{"id": "837.png", "formula": "\\begin{align*} \\tau _ { n p } ^ { v } \\left ( g \\in \\Omega : \\frac { \\log \\| \\rho ( \\overline { g } ) \\| - \\Lambda | \\overline { g } | } { \\sqrt { | \\overline { g } | } } \\le x \\right ) = N ( x , \\sigma ) + O \\left ( \\frac { \\log n } { \\sqrt { n } } \\right ) \\end{align*}"}
+{"id": "453.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { m } b _ k ^ q \\leq \\sum _ { k = 1 } ^ { m } \\lambda _ k \\sum _ { j = 1 } ^ { n } b _ j ^ q = \\sum _ { k = 1 } ^ { d } \\lambda _ k , \\end{align*}"}
+{"id": "2509.png", "formula": "\\begin{align*} \\ddot y ( c ) \\ , \\mathrm { \\ e q u a l s \\ t h e \\ o r d i n a r y \\ s e c o n d \\ d e r i v a t i v e \\ o f \\ } y ( t ) \\ , \\mathrm { \\ a t \\ } \\ , t = c \\end{align*}"}
+{"id": "526.png", "formula": "\\begin{align*} \\Upsilon _ { \\ ! \\delta } ( z ) : = f ( x ) + g ( y ) + H ( x , y ) + ( { \\delta } / { 2 } ) \\| x - u \\| ^ 2 + ( { \\delta } / { 2 } ) \\| y - v \\| ^ 2 , \\end{align*}"}
+{"id": "5688.png", "formula": "\\begin{align*} \\mathbf { D \\mathbf { e } } ^ { a } & = d \\mathbf { \\mathbf { e } } ^ { a } + \\mathbf { A } _ { b } ^ { a } \\mathbf { \\mathbf { e } } ^ { b } = 0 , \\\\ \\mathbf { F } _ { b } ^ { a } & = d \\mathbf { A } _ { b } ^ { a } + \\mathbf { A } _ { c } ^ { a } \\mathbf { A } _ { b } ^ { c } , \\end{align*}"}
+{"id": "2098.png", "formula": "\\begin{align*} \\dim _ P ( \\mathcal { S } _ m ^ { ( b ) \\ast } ( w ) ) & \\geq \\dim _ H ( A ^ { \\ast } ( w ) + B ) - \\dim _ H ( B ) \\\\ & = \\dim ( K ) - \\dim _ H ( B ) = \\frac { \\sum _ { h = 1 } ^ { m } \\log | W _ h | } { \\log b } - \\dim _ H ( B ) . \\end{align*}"}
+{"id": "7071.png", "formula": "\\begin{align*} C _ 2 ^ \\vee = \\mathbf { C P } ( 1 ) \\amalg \\mathbf { C P } ( \\sigma ) \\to \\mathbf { C P } ^ \\infty _ { C _ 2 } \\end{align*}"}
+{"id": "6093.png", "formula": "\\begin{gather*} f ( \\tau ) = f ^ + ( \\tau ) + f ^ - ( \\tau ) = \\sum _ { n \\gg - \\infty } c _ f ^ + ( n ) q ^ n + c _ f ^ - ( 0 ) v ^ { 1 - k } + \\sum _ { n \\ll \\infty } c _ f ^ - ( n ) W _ k ( 4 \\pi n v ) q ^ n \\end{gather*}"}
+{"id": "5745.png", "formula": "\\begin{align*} \\mathbf { e } ^ { A A ^ { \\prime } } = 2 \\mu ^ { A A ^ { \\prime } } \\end{align*}"}
+{"id": "1363.png", "formula": "\\begin{align*} \\left ( \\int _ \\Omega f _ \\alpha ( \\tau _ \\alpha ) \\ , d \\mu ( \\alpha ) \\right ) ^ 2 & = ( \\operatorname { T r a } ( S _ { f , \\tau } ) ) ^ 2 = \\left ( \\sum _ { k = 1 } ^ d \\lambda _ k \\right ) ^ 2 \\leq d \\sum _ { k = 1 } ^ d \\lambda _ k ^ 2 \\\\ & = d \\operatorname { T r a } ( S ^ 2 _ { f , \\tau } ) = d \\int _ \\Omega \\int _ \\Omega f _ \\beta ( \\tau _ \\alpha ) f _ \\alpha ( \\tau _ \\beta ) \\ , d \\mu ( \\alpha ) \\ , d \\mu ( \\beta ) . \\end{align*}"}
+{"id": "8249.png", "formula": "\\begin{align*} x = _ T y \\enspace \\to \\enspace \\forall P , \\left ( P ( x ) \\leftrightarrow P ( y ) \\right ) \\enspace . \\end{align*}"}
+{"id": "5121.png", "formula": "\\begin{align*} D _ { f _ { 2 } } \\mathcal { S } _ { 2 } ( \\lambda , b , \\Omega , 0 ) h _ { 2 } ( w ) = - \\sum _ { n = 0 } ^ { \\infty } ( n + 1 ) b \\left ( \\Omega _ { n + 1 } ( \\lambda b ) + \\Omega \\right ) b _ { n } e _ { n + 1 } ( w ) . \\end{align*}"}
+{"id": "6484.png", "formula": "\\begin{align*} \\mathcal { H } = \\{ h \\in \\mathbb { Z } _ { 4 n } ^ * \\mid h A = A ; ~ h B = B ~ \\mbox { i f } ~ h \\in F ~ \\mbox { a n d } ~ h B = - B ~ \\mbox { i f } ~ h \\in - F \\} . \\end{align*}"}
+{"id": "8761.png", "formula": "\\begin{align*} \\mathcal { H } _ { i , j } = \\begin{cases} 1 / | \\bar { l } | ( U _ l \\bar { X } \\bar { g } ) ^ * ( i , j ) \\in \\mathbb { N } ^ 2 , \\ \\ 1 \\leq | i - j | = l \\leq 2 T - 1 , \\\\ 0 . \\end{cases} \\end{align*}"}
+{"id": "764.png", "formula": "\\begin{align*} \\mathbb { V } : = \\varinjlim \\left ( S _ 2 ^ { n e w } ( U _ n , \\widetilde { \\psi } , F / \\mathcal { O } ) ^ { o r d } \\oplus S _ k ^ { n e w } ( \\Gamma _ 1 ( N p ^ n \\ell ) , \\psi , F / \\mathcal { O } ) ^ { \\left ( \\frac { - } { \\ell } \\right ) , o r d } \\right ) , \\end{align*}"}
+{"id": "3114.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( A ) = \\bar { \\gamma } \\left ( c _ 1 ^ { 1 1 } ( A ^ 1 ) + \\bar { a } _ { 1 1 } ^ 1 \\int _ Y r ^ 1 A ^ 1 e _ 1 \\cdot \\nabla w \\right ) , \\end{align*}"}
+{"id": "7588.png", "formula": "\\begin{align*} G ( t , x , y ) = \\frac { 1 } { \\sqrt { 4 \\pi t } } \\sum _ { n = - \\infty } ^ { \\infty } \\bigg [ e ^ { - \\frac { ( y - x - 2 n ) ^ 2 } { 4 t } } - e ^ { - \\frac { ( y + x - 2 n ) ^ 2 } { 4 t } } \\bigg ] , \\end{align*}"}
+{"id": "5185.png", "formula": "\\begin{align*} ( a _ 2 ' - a _ 2 ) \\frac { r } { K } = u _ { 2 3 } \\frac { r } { K } + k r . \\end{align*}"}
+{"id": "8909.png", "formula": "\\begin{align*} B _ { d } = \\frac { \\left ( - 1 \\right ) ^ { d } } { d + 1 } \\left ( 1 - 2 ^ { - 2 d - 1 } \\right ) B _ { 2 d + 2 } , d = 0 , 1 , 2 , . . . \\end{align*}"}
+{"id": "5325.png", "formula": "\\begin{align*} \\mathrm { A d } _ U \\circ \\Phi = \\mathrm { A d } _ { T ^ { - 1 } } \\circ \\mathrm { A d } _ Z \\circ \\Phi = \\mathrm { A d } _ { T ^ { - 1 } } \\circ \\Psi = \\Phi _ i , \\end{align*}"}
+{"id": "164.png", "formula": "\\begin{align*} { \\varkappa } ( \\theta ) ( \\hslash ) & = z ( \\theta , \\hslash ) , \\ \\hslash \\in [ 0 , \\pi ] , \\\\ { \\phi } ( \\theta ) ( \\hslash ) & = \\phi ( \\theta , \\hslash ) , \\ \\hslash \\in [ 0 , \\pi ] . \\end{align*}"}
+{"id": "4404.png", "formula": "\\begin{align*} \\prod _ { i = m + 1 } ^ { m + k } a _ i \\leq \\prod _ { i = m + 1 } ^ { m + k } x _ i \\end{align*}"}
+{"id": "1721.png", "formula": "\\begin{align*} \\langle \\delta s ( \\mu _ { 0 } ) , \\delta s ( \\mu _ { 0 } ) \\rangle = \\frac { ( k - 1 ) ^ 2 } { 2 } . \\end{align*}"}
+{"id": "5945.png", "formula": "\\begin{align*} ( E _ r , \\widehat { U } ) \\equiv 0 , \\forall ~ \\widehat H \\in L ^ 2 ( 0 , T ; ( L ^ 2 ( \\Gamma ) ) ^ M ) , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "8325.png", "formula": "\\begin{align*} \\square _ { s , y } v = ( | y | ^ 2 - s ^ 2 ) ^ \\frac { 2 } { d - 2 } | v | ^ { \\frac { 4 } { d - 2 } } v . \\end{align*}"}
+{"id": "4782.png", "formula": "\\begin{align*} f ( x ) = \\Pr _ { z \\sim \\lambda _ S } [ H z ^ \\intercal = x ] = \\frac { 1 } { \\binom { N } { S } } \\sum _ { \\substack { z \\in \\mathbb { F } _ 2 ^ N \\\\ H z ^ \\intercal = x } } L _ S ( z ) . \\end{align*}"}
+{"id": "8076.png", "formula": "\\begin{align*} \\mathcal { D } = \\frac { A ( p , 1 ) } { 2 p ^ { \\frac { 1 } { 2 } } } \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } H _ { m , p } - \\frac { 1 } { 2 p ^ { \\frac { 3 } { 2 } } } \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } H _ { m p , p } . \\end{align*}"}
+{"id": "6568.png", "formula": "\\begin{align*} F ( x ) = F _ C ( x ) : = \\exp \\left ( \\frac { \\log ( x + 1 0 0 ) \\log \\log \\log ( x + 1 0 0 ) } { 8 C \\log \\log ( x + 1 0 0 ) } + 1 \\right ) . \\end{align*}"}
+{"id": "161.png", "formula": "\\begin{align*} R e ( < A ^ { * } v , v > ) = - \\frac { 1 } { 2 } | z ( \\pi ) | ^ { 2 } \\le 0 . \\end{align*}"}
+{"id": "1267.png", "formula": "\\begin{align*} R _ { \\mathcal { B S } ( \\alpha ) } ( \\mathcal { S } ^ * _ \\beta ) & = \\begin{dcases} \\min \\left \\{ 1 , \\dfrac { 1 } { \\beta ( 3 - 2 \\alpha ) } \\right \\} , & \\ 0 \\leq \\alpha \\leq \\frac { 1 } { 9 } , \\\\ \\min \\left \\{ 1 , \\dfrac { 2 \\sqrt { \\alpha } } { \\beta ( 1 + \\alpha ) \\sqrt { 1 + 1 6 \\alpha } } \\right \\} , & \\frac { 1 } { 9 } \\leq \\alpha \\leq 1 . \\end{dcases} \\end{align*}"}
+{"id": "2287.png", "formula": "\\begin{align*} \\frac { d u _ { i } ( t ) } { d t } = \\left . \\frac { \\partial ^ 2 b } { \\partial x ^ 2 } \\right | _ { x = x _ i } , \\end{align*}"}
+{"id": "4572.png", "formula": "\\begin{align*} a _ { n + 1 , r } - \\rho = \\left ( a _ { n , r } - \\rho \\right ) ^ 2 + \\rho ( 1 - \\rho ) \\end{align*}"}
+{"id": "999.png", "formula": "\\begin{align*} g ' \\big ( \\tfrac 1 3 + ) - g ' \\big ( \\tfrac 1 3 - ) = \\sigma g \\big ( \\tfrac 1 3 ) , g ' \\big ( \\tfrac 2 3 + ) - g ' \\big ( \\tfrac 2 3 - ) = \\sigma g \\big ( \\tfrac 2 3 ) . \\end{align*}"}
+{"id": "4145.png", "formula": "\\begin{align*} \\Lambda _ { a i b } = \\frac { 1 } { 3 } ( h _ { a j } H _ { j i b } + h _ { b j } H _ { j a i } + h _ { i j } H _ { j b a } ) \\end{align*}"}
+{"id": "6342.png", "formula": "\\begin{align*} \\Phi _ { \\nu _ 0 } ( u , v ) : = \\varphi ( \\nu _ 0 ) + \\int _ 1 ^ { u } f ( x , \\nu _ 0 ) d x - \\int ^ 1 _ { v } f ( x , \\nu _ 0 ) d x \\end{align*}"}
+{"id": "6042.png", "formula": "\\begin{align*} \\psi ( z ) = \\frac 2 { b - a } \\left ( z - \\frac { b + a } 2 - w ( z ) \\right ) . \\end{align*}"}
+{"id": "1463.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\binom { x _ i } { 3 } + \\sum _ { j = 1 } ^ m \\binom { y _ j } { 3 } = \\frac { k ( k - 1 ) ( k - 2 ) ( m - 2 ) } { 3 ( m + 1 ) ( m ^ 2 - 2 ) } ; \\end{align*}"}
+{"id": "9004.png", "formula": "\\begin{align*} \\vert H \\vert \\ , \\leq \\ , \\sum \\nolimits _ { g \\in K } \\left \\lvert g ^ { - 1 } \\phi _ { g } ^ { - 1 } ( F ) \\right \\rvert \\ , = \\ , \\vert K \\vert \\vert F \\vert . \\end{align*}"}
+{"id": "2256.png", "formula": "\\begin{align*} h _ j ' = [ y _ j , z _ j ] , \\ \\ [ y _ i , z _ j ] = 0 \\ \\mathrm { i f } \\ i \\neq j , \\ \\ [ y _ i , y _ j ] = [ z _ i , z _ j ] = [ y _ i , h _ j ' ] = [ z _ i , h _ j ' ] = 0 . \\end{align*}"}
+{"id": "5564.png", "formula": "\\begin{align*} P _ k ( x ^ 2 ) = O _ { \\epsilon } \\left ( x ^ { \\frac { 1 } { 2 } - k + \\epsilon } \\right ) . \\end{align*}"}
+{"id": "7906.png", "formula": "\\begin{align*} & \\partial _ t { u } = \\frac { 1 } { 2 } \\Delta u - u ( 1 - u ) . \\end{align*}"}
+{"id": "7399.png", "formula": "\\begin{align*} \\mu ( s \\widetilde { \\alpha } , \\sigma ) = \\gamma ( G / P ) ^ 2 q _ F ^ { n ( \\sigma \\times \\Pi ( \\sigma ) ) - n ( \\sigma ) } \\dfrac { ( 1 - \\chi ^ 2 \\chi '^ { - 1 } ( \\varpi _ L ) q _ L ^ { - s } ) ( 1 - \\chi ^ { - 2 } \\chi ' ( \\varpi _ L ) q _ L ^ s ) } { ( 1 - \\chi ^ 2 \\chi '^ { - 1 } ( \\varpi _ L ) q _ L ^ { - 1 - s } ) ( 1 - \\chi ^ { - 2 } \\chi ' ( \\varpi _ L ) q _ L ^ { - 1 + s } ) } \\end{align*}"}
+{"id": "3117.png", "formula": "\\begin{align*} w = \\phi = s q . \\end{align*}"}
+{"id": "86.png", "formula": "\\begin{align*} w _ { \\mathbf { k } } ( \\stackrel { \\rightarrow } { \\xi } ) = \\prod _ { \\iota = 1 } ^ h w _ { k ^ { ( \\iota ) } } ( \\xi ^ { ( \\iota ) } ) . \\end{align*}"}
+{"id": "3086.png", "formula": "\\begin{align*} \\frac { c _ j ^ { k l } ( A ) } { \\bar { a } } = \\int _ Y \\frac { r } { \\bar { a } } A e _ j \\cdot \\nabla v ^ { k l } = \\int _ Y r _ B B e _ j \\cdot \\nabla \\left ( v ^ { k l } _ B + \\bar { b } _ { k l } w \\right ) = c _ j ^ { k l } ( B ) + \\bar { b } _ { k l } \\int _ Y r _ B B e _ j \\cdot \\nabla w \\end{align*}"}
+{"id": "8538.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { p = 0 } ^ { \\lfloor ( n - 1 ) / 2 \\rfloor } n \\binom { n - 1 } { 2 } ^ p 2 ^ { Q ( n , n - 2 p ) - Q ( n , n ) } & \\le \\sum _ { p = 0 } ^ { \\lfloor ( n - 1 ) / 2 \\rfloor } \\bigg ( \\frac { n ^ 2 } { 2 ^ n } \\bigg ) ^ p \\\\ & = \\frac { 1 - ( n ^ 2 / 2 ^ n ) ^ { \\lfloor ( n - 1 ) / 2 \\rfloor + 1 } } { 1 - n ^ 2 / 2 ^ n } . \\end{align*}"}
+{"id": "8617.png", "formula": "\\begin{align*} \\abs { T ( 1 / 2 ) - 1 } = \\abs { - 1 / 2 - 1 } = 3 / 2 > 1 / 2 = \\abs { 1 / 2 - 1 } , \\end{align*}"}
+{"id": "1011.png", "formula": "\\begin{align*} \\S ' ( \\Z ) = \\bigcup _ { \\alpha \\geq 0 } \\ell _ { \\infty , - \\alpha } ( \\Z ) = \\lim _ { \\alpha \\rightarrow \\infty } \\ell _ { \\infty , - \\alpha } ( \\Z ) . \\end{align*}"}
+{"id": "6081.png", "formula": "\\begin{align*} f ( z ) - r _ n ( z ) = \\big ( 2 G _ { \\lambda _ n } + o ( 1 ) \\big ) \\frac { m _ n ( z ) } { B _ n ^ 2 ( z ) } \\frac { D _ { \\lambda _ n } ^ 2 ( z ) } { w ( z ) } \\left ( \\frac \\rho { \\varphi ( z ) } \\right ) ^ { 2 ( n - d _ n ) } \\end{align*}"}
+{"id": "7148.png", "formula": "\\begin{align*} h _ k = \\sum \\limits _ { \\substack { | I | = k } } \\prod \\limits _ { \\substack { i \\in I \\\\ j \\notin I } } \\phi ( z _ j - z _ i ) \\prod _ { i \\in I } e ^ { - v _ i / c } k = 1 , \\dots , N \\ , , \\end{align*}"}
+{"id": "2973.png", "formula": "\\begin{align*} W _ j - \\eta _ j = \\frac { g ^ { \\prime \\prime } ( 0 ) } { 2 g ^ \\prime ( 0 ) } \\frac { W _ j \\eta _ j } { \\sqrt { n } } + \\bigg ( \\frac { g ^ { ( 3 ) } ( 0 ) } { 6 g ^ \\prime ( 0 ) } - \\frac { g ^ { \\prime \\prime } ( 0 ) ^ 2 } { 4 g ^ \\prime ( 0 ) ^ 2 } \\bigg ) \\frac { W _ j \\eta _ j ^ 2 } { n } + O ( n ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "9022.png", "formula": "\\begin{align*} \\mu \\big ( H ^ 1 ( \\Q _ \\Sigma / \\Q _ { \\infty } , A ) \\big ) - \\mu \\big ( H ^ 1 ( \\Q _ \\Sigma / \\Q _ { \\infty } , A _ 1 ) \\big ) = \\mu \\big ( \\mathcal { H } ^ 1 _ { p } ( \\Q _ { \\infty } , A ) \\big ) - \\mu \\big ( \\mathcal { H } ^ 1 _ { p } ( \\Q _ { \\infty } , A _ 1 ) \\big ) . \\end{align*}"}
+{"id": "6740.png", "formula": "\\begin{align*} \\xi ( s ) = s ( s - 1 ) \\phi ( s ) . \\end{align*}"}
+{"id": "2118.png", "formula": "\\begin{align*} & \\big \\langle C _ 1 \\times 0 , \\ , C _ 1 \\times 0 \\big \\rangle _ { ( F \\ ! H ) ^ 2 } = 0 , \\big \\langle C _ 1 \\times 0 , \\ , \\widehat C _ { 1 2 } \\big \\rangle _ { ( F \\ ! H ) ^ 2 } = 0 ; \\\\ & \\big \\langle 0 \\times C _ 2 , \\ , 0 \\times C _ 2 \\big \\rangle _ { ( F \\ ! H ) ^ 2 } = 0 , \\big \\langle 0 \\times C _ 2 , \\ , \\widehat C _ { 1 2 } \\big \\rangle _ { ( F \\ ! H ) ^ 2 } = 0 . \\end{align*}"}
+{"id": "8990.png", "formula": "\\begin{align*} | \\Delta ( d i s t _ N ( u ) & - d i s t _ N ( v ) ) | = | \\nabla u \\cdot d \\nu ( u ) \\nabla u - \\nabla v \\cdot d \\nu ( v ) \\nabla v | \\\\ & \\le C ( | w | | \\nabla u | ^ 2 + ( | \\nabla u | + | \\nabla v | ) | \\nabla w | ) \\hbox { i n } B . \\end{align*}"}
+{"id": "8001.png", "formula": "\\begin{align*} \\sum _ { m + n = k } ^ { \\infty } \\dfrac { ( - 1 ) ^ { m } } { m ! n ! } \\phi ( z a ^ n , a ^ m ) = 0 \\end{align*}"}
+{"id": "5197.png", "formula": "\\begin{align*} v ( z ) = \\big ( z _ 1 , z _ 2 , z _ 3 , f ^ 1 ( z ) , \\dots , f ^ { m - 3 } ( z ) \\big ) \\ , . \\end{align*}"}
+{"id": "5080.png", "formula": "\\begin{align*} A _ n = \\begin{pmatrix} \\alpha _ 1 ^ n & \\cdots & \\alpha _ k ^ n & c _ { k _ { 1 } } ^ n & \\cdots & c _ { k _ p } ^ { n } \\\\ \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha _ 1 ^ { n + k - 1 + p } & \\cdots & \\alpha _ k ^ { n + k - 1 + p } & c _ { k _ { 1 } } ^ { n + k - 1 + p } & \\cdots & c _ { k _ p } ^ { n + k - 1 + p } \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "8510.png", "formula": "\\begin{align*} \\frac { \\partial C } { \\partial t } = \\alpha C _ { \\sigma } + \\beta C _ { \\sigma ^ 2 } , \\end{align*}"}
+{"id": "3987.png", "formula": "\\begin{align*} D _ i u _ \\epsilon ( x ) & = \\int _ { B _ \\epsilon ( x ) } D _ i u _ 0 ( x - y ) \\rho _ \\epsilon ( y ) \\ d y . \\end{align*}"}
+{"id": "3298.png", "formula": "\\begin{align*} \\| A ^ { \\frac 1 2 } V \\| _ { \\mathcal L ^ 2 } = \\sqrt { 2 } \\| D ( v ) \\| _ { L ^ 2 ( \\mathcal F _ 0 ) } . \\end{align*}"}
+{"id": "7578.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l r } v _ t - \\nabla \\cdot ( A _ 0 ( x ) \\nabla v ) = f ( x , t ) , & ( x , t ) \\in Q _ T , & \\\\ v = 0 , & ( x , t ) \\in S _ T , & \\\\ v = u _ 0 , & x \\in \\Omega , t = 0 , & \\end{array} \\right . \\end{align*}"}
+{"id": "2270.png", "formula": "\\begin{align*} & \\int ^ { T } _ 0 \\int _ { \\Omega \\times \\mathbb { R } ^ 3 } f \\big ( \\partial _ t \\varphi + v \\cdot \\nabla _ x \\varphi + ( u - v ) \\cdot \\nabla _ v \\varphi - \\nabla _ x \\Phi \\cdot \\nabla _ v \\varphi + \\Delta _ v \\varphi \\big ) \\ , d x d v d t \\\\ & \\qquad + \\int _ { \\Omega \\times \\mathbb { R } ^ 3 } f _ 0 \\varphi ( 0 , x , v ) \\ , d x d v = \\int _ 0 ^ T \\int _ { \\Sigma ^ - } ( v \\cdot \\nu ( x ) ) g \\varphi \\ , d \\sigma ( x ) d v d t . \\end{align*}"}
+{"id": "2879.png", "formula": "\\begin{align*} N _ 1 + N _ 2 ( q + 1 ) + \\ldots + N _ k ( q ^ { k - 1 } + \\ldots + q + 1 ) = q ^ { k - 1 } + \\ldots + q + 1 . \\end{align*}"}
+{"id": "911.png", "formula": "\\begin{align*} a _ n = ( 1 - \\frac 1 2 ) ^ { ( \\frac 1 2 - \\frac 1 3 ) ^ { . . . ^ { ( \\frac { 1 } { n } - \\frac { 1 } { n + 1 } ) } } } \\end{align*}"}
+{"id": "3292.png", "formula": "\\begin{align*} w _ 0 = \\ell ^ 0 _ w + \\omega _ w ^ 0 x ^ { \\bot } . \\end{align*}"}
+{"id": "3947.png", "formula": "\\begin{align*} \\overline { \\vee } ( q ) = \\sup \\{ \\phi _ p ( q ) : = \\overline { g } ( q , p , h ) ; \\phi _ p \\leq 0 D _ q \\} , \\end{align*}"}
+{"id": "5491.png", "formula": "\\begin{align*} \\dot x ( t ) = J \\nabla H ( t , x ( t ) ) \\end{align*}"}
+{"id": "945.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( \\Omega ^ { n - 1 } M , C ) \\ , = \\ , \\textrm { r . g r a d e } _ R ( \\textrm { T r } _ C ( M ' ) , C ) . \\end{align*}"}
+{"id": "8589.png", "formula": "\\begin{align*} \\mathcal { C } = ( 1 , \\ldots , 2 ^ n - 1 , 2 ^ n - 2 , \\ldots , 2 ^ { n - 1 } - 1 ) . \\end{align*}"}
+{"id": "4660.png", "formula": "\\begin{align*} m ^ { \\underline { \\ell _ i } } m ^ { \\underline { \\ell _ j } } = \\sum _ { k = \\max \\{ 0 , \\ell _ i + \\ell _ j - m \\} } ^ { \\min \\{ \\ell _ i , \\ell _ j \\} } G _ { \\ell _ i , \\ell _ j , k } m ^ { \\underline { \\ell _ i + \\ell _ j - k } } . \\end{align*}"}
+{"id": "3810.png", "formula": "\\begin{align*} \\norm { f } _ { L ^ 2 ( Q _ k \\cap \\omega ) } ^ 2 \\geq a _ k \\norm { f } _ { L ^ 2 ( Q _ k ) } ^ 2 \\quad a _ k = 1 2 \\Bigl ( \\frac { \\abs { Q _ k \\cap \\omega } } { 2 4 \\cdot 2 ^ d d ^ { 1 + d } \\abs { Q _ k } } \\Bigr ) ^ { 4 \\frac { \\log M _ k } { \\log 2 } + 1 } \\end{align*}"}
+{"id": "2583.png", "formula": "\\begin{align*} \\hat { \\rho } ( c ) = - c , \\hat { \\rho } ( d ) = - d , \\hat { \\rho } ( u , v ) ( t ) = ( v ( - t ) , u ( - t ) ) . \\end{align*}"}
+{"id": "8785.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } F _ j '' ( t ) \\varepsilon ^ j = \\sum _ { j = 0 } ^ { \\infty } \\left ( F _ j ( t ) \\varepsilon ^ { j + 1 } + F ' _ j ( t ) \\varepsilon ^ j ( \\varepsilon + 1 - e ^ { - \\varepsilon t } \\sum _ { k = 0 } ^ { \\infty } \\varepsilon ^ { k + 1 } F _ k ) \\right ) , \\end{align*}"}
+{"id": "5928.png", "formula": "\\begin{align*} t \\geqslant T : z = \\widehat D ^ T w = D _ 2 ^ T u \\equiv 0 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "5369.png", "formula": "\\begin{align*} B _ n : = \\left \\{ \\eta \\in B _ 0 : \\mbox { t h e f i r s t $ n $ p o s i t i v e M a x w e l l e i g e n v a l u e s w i t h $ \\varepsilon = \\tilde \\varepsilon + \\eta $ a r e s i m p l e } \\right \\} \\end{align*}"}
+{"id": "8125.png", "formula": "\\begin{align*} e \\Bigl ( \\frac { \\pm n _ 2 \\bar u } { m c n _ 1 ^ { - 1 } } \\Bigr ) e \\Bigl ( \\frac { - n _ 2 n _ 1 ^ 2 } { c m p } \\Bigr ) & = e \\Bigl ( \\frac { \\pm n _ 2 \\bar u } { m c n _ 1 ^ { - 1 } } - \\frac { n _ 2 n _ 1 ^ 2 } { c m p } \\Bigr ) = e \\Bigl ( \\frac { \\pm n _ 2 \\bar u p - n _ 2 n _ 1 } { m c n _ 1 ^ { - 1 } p } \\Bigr ) \\\\ & = e \\Bigl ( \\frac { n _ 2 ( \\pm \\bar u p - n _ 1 ) } { m c n _ 1 ^ { - 1 } p } \\Bigr ) . \\end{align*}"}
+{"id": "3885.png", "formula": "\\begin{align*} g ( x , y , g ^ * ( x , y , u ) ) = u , \\end{align*}"}
+{"id": "4998.png", "formula": "\\begin{align*} \\alpha \\otimes e \\cdot \\beta \\otimes f = \\sum _ { g \\in G } \\sum _ { p \\in \\alpha \\cap \\beta g } \\epsilon _ p ( \\alpha , \\beta g ) p \\otimes e \\otimes g ^ { - 1 } f \\end{align*}"}
+{"id": "2175.png", "formula": "\\begin{align*} \\sigma _ { 0 } : = & \\dfrac { 2 ( 3 - 2 \\sqrt { 2 } ) } { ( 2 - 2 \\alpha + \\beta ) + \\sqrt { ( - 2 + 2 \\alpha - \\beta ) ^ 2 - 4 ( 3 - 2 \\sqrt { 2 } ) ( 2 \\sqrt { 2 } - 1 - 2 \\alpha - \\beta ) } } , \\intertext { a n d } \\tilde { \\sigma _ { 0 } } : = & \\dfrac { 2 } { ( 2 - 2 \\alpha + \\beta ) + \\sqrt { ( - 2 + 2 \\alpha - \\beta ) ^ 2 - 4 ( 2 \\alpha - 3 + \\beta ) } } . \\end{align*}"}
+{"id": "2416.png", "formula": "\\begin{align*} ( \\widetilde E , \\widetilde A ) = ( Q ^ T E Q , Q ^ T A Q - Q ^ T E \\dot Q ) \\end{align*}"}
+{"id": "6797.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } d \\left ( \\overline { x } _ { n + \\eta } , \\overline { x } _ { n + \\eta + 1 } \\right ) = 0 . \\end{align*}"}
+{"id": "8810.png", "formula": "\\begin{align*} & V _ 1 ( h _ 0 , h _ 1 , h _ 2 , \\dots ) = ( A h _ 0 , F _ 2 ^ * D _ P h _ 0 + F _ 1 h _ 1 , F _ 2 ^ * h _ 1 + F _ 1 h _ 2 , F _ 2 ^ * h _ 2 + F _ 1 h _ 3 , \\dots ) \\\\ & V _ 2 ( h _ 0 , h _ 1 , h _ 2 , \\dots ) = ( B h _ 0 , F _ 1 ^ * D _ P h _ 0 + F _ 2 h _ 1 , F _ 1 ^ * h _ 1 + F _ 2 h _ 2 , F _ 1 ^ * h _ 2 + F _ 2 h _ 3 , \\dots ) \\\\ & V _ 3 ( h _ 0 , h _ 1 , h _ 2 , \\dots ) = ( P h _ 0 , D _ P h _ 0 , h _ 1 , h _ 2 , \\dots ) . \\end{align*}"}
+{"id": "285.png", "formula": "\\begin{align*} \\begin{aligned} \\prescript { L } { } { j } ^ { \\theta } : \\prescript { L } { } { S } & \\rightarrow \\prescript { L } { } { G } \\\\ t \\rtimes w & \\mapsto \\widehat { j } ( t \\theta ( w ) ) \\varphi _ { \\prescript { L } { } { j } } ( w ) \\rtimes w . \\end{aligned} \\end{align*}"}
+{"id": "860.png", "formula": "\\begin{align*} D f ( w ) = [ \\frac { \\partial f ( w ) } { \\partial w _ 1 } \\ ; \\frac { \\partial f ( w ) } { \\partial w _ 2 } \\ ; \\frac { \\partial f ( w ) } { \\partial w _ 3 } ] ^ T \\end{align*}"}
+{"id": "7881.png", "formula": "\\begin{align*} f '' + A ( z ) f = 0 \\end{align*}"}
+{"id": "7680.png", "formula": "\\begin{align*} H ^ { 1 } _ { \\mathfrak { m } } ( \\Omega _ { R } ^ { j } ) _ { \\ell } = 0 \\ell \\notin \\{ - 1 , - 2 , - 3 , \\dots \\} . \\end{align*}"}
+{"id": "2649.png", "formula": "\\begin{align*} a _ \\mu ( f ) = \\sum _ { j \\in \\N } a _ { \\mu _ j } ( f _ j ) , \\forall f = ( f _ j ) _ { j \\in \\N } \\in X ^ * . \\end{align*}"}
+{"id": "3849.png", "formula": "\\begin{align*} \\mu _ l = \\mu _ { l - 1 } + \\mu _ { l - 2 } + \\mu _ { l - 3 } + 6 \\delta _ { l , 3 } + 2 ( \\delta _ { l , 2 } + \\delta _ { l , 4 } + \\delta _ { l , 5 } ) , \\mu _ { l < 2 } = 0 . \\end{align*}"}
+{"id": "8782.png", "formula": "\\begin{align*} \\dot { \\phi } _ i = & \\omega _ i + \\frac { 1 } { N } \\sum _ { j = 1 } ^ N W _ { i j } ( t ) g ( \\phi _ j - \\phi _ i ) , \\\\ \\dot { W } _ { i j } = & - \\varepsilon ( W _ { i j } + \\sum _ { k = 1 } ^ N a _ { j k } h _ 1 ( \\phi _ j - \\phi _ k ) + \\sum _ { k = 1 } ^ N b _ { i k } h _ 2 ( \\phi _ j - \\phi _ k ) ) , \\end{align*}"}
+{"id": "6414.png", "formula": "\\begin{align*} \\lambda ( ( p - 2 ) \\nabla _ X ^ 2 \\phi ( \\tilde X _ j ^ k , \\tilde Y _ j ^ k , \\tilde t _ j ^ k ) ) + \\Delta _ X \\phi ( \\tilde X _ j ^ k , \\tilde Y _ j ^ k , \\tilde t _ j ^ k ) \\leq ( p - 1 ) \\Lambda ( E ^ X _ k ) + \\sum ^ { m - 1 } _ { i = 1 } \\lambda _ i ( E ^ X _ k ) . \\end{align*}"}
+{"id": "3872.png", "formula": "\\begin{align*} Y ( \\cdot , u , D u ) ( \\Omega ) = \\Omega ^ * . \\end{align*}"}
+{"id": "5070.png", "formula": "\\begin{align*} P ( 0 ) = g _ 1 ( l ^ e ) + \\cdots + g _ r ( l ^ e ) . \\end{align*}"}
+{"id": "812.png", "formula": "\\begin{align*} \\widehat { \\nu } ( E _ n ( \\epsilon / 2 , \\delta ) ) \\leq \\widetilde { \\nu } _ { m } ( E _ n ( \\epsilon / 2 , \\delta ) ) + C _ 0 \\theta ^ m = \\sum _ { k = 1 } ^ { m } \\widehat { \\nu } ( E _ n ( \\epsilon / 2 , \\delta ) \\cap A _ k ) + C _ 0 \\theta ^ m , \\end{align*}"}
+{"id": "3956.png", "formula": "\\begin{align*} \\lambda \\chi _ { \\Omega } \\leq \\det D Y \\tilde { u } & \\leq \\Lambda \\chi _ { \\Omega } , \\\\ Y \\tilde { u } ( U ) & = \\overline { \\Omega ^ * } . \\end{align*}"}
+{"id": "1605.png", "formula": "\\begin{align*} K : = \\max \\{ \\varrho ( u _ i , u _ j ) \\ , | \\ , 1 \\leq i , j \\leq L \\} , \\end{align*}"}
+{"id": "7114.png", "formula": "\\begin{align*} \\dfrac { M ( q _ t ) } { x _ { i _ s } } = \\dfrac { M ( q _ s ) } { x _ { i _ t } } \\end{align*}"}
+{"id": "6437.png", "formula": "\\begin{align*} \\tilde u _ \\epsilon ( Z , W , \\tau ) - u _ \\epsilon ( Z , W , \\tau ) = u _ \\epsilon ( ( \\hat Z , \\hat W , \\hat \\tau ) \\circ ( Z , W , \\tau ) ) - u _ \\epsilon ( Z , W , \\tau ) + \\eta > 0 , \\end{align*}"}
+{"id": "2997.png", "formula": "\\begin{align*} \\begin{array} { c c c c } \\partial _ b ^ \\gamma \\textbf { F } ^ { a b } ( x ) & = & \\partial _ s p ^ a ( x , s ) & \\hbox { ( a ) } \\\\ \\partial _ a ^ \\gamma p ^ a ( x , s ) & = & - \\frac { 1 } { 2 } \\| \\textbf { F } \\| ^ 2 ( x ) & \\hbox { ( b ) } \\end{array} \\end{align*}"}
+{"id": "2395.png", "formula": "\\begin{align*} \\dot x _ 2 = J ^ { - 1 } C ( t ) x _ 2 + J ^ { - 1 } \\widetilde f _ 2 ( t ) . \\end{align*}"}
+{"id": "8513.png", "formula": "\\begin{align*} C _ { \\sigma ^ 3 t } & = \\ ; C _ { \\sigma ^ 2 t \\sigma } - \\frac { \\bar { g } _ t } { \\bar { g } } C _ { \\sigma ^ 3 } \\\\ & = \\ ; - \\frac { 1 } { 3 } \\Big ( \\beta \\mu _ { \\sigma ^ 2 } + 4 \\mu \\beta _ { \\sigma ^ 2 } + 3 \\beta \\mu ^ 2 + 5 \\beta _ { \\sigma } \\mu _ { \\sigma } + 3 \\alpha \\mu _ { \\sigma } + \\beta _ { \\sigma ^ 4 } \\Big ) C _ { \\sigma } \\\\ & \\qquad \\qquad - \\mu \\Big ( \\alpha + \\beta _ { \\sigma } \\Big ) C _ { \\sigma ^ 2 } . \\end{align*}"}
+{"id": "5923.png", "formula": "\\begin{align*} \\hbox { r a n k } ( \\widehat D , \\widehat { \\Lambda } \\widehat D ) = 2 \\end{align*}"}
+{"id": "1886.png", "formula": "\\begin{align*} | \\mathrm { s u p p } ( \\chi _ { T } ) | = \\begin{cases} 2 | T | & \\\\ [ 1 0 p t ] 2 | T | - 1 & \\\\ [ 1 0 p t ] 2 | T | & \\\\ [ 1 0 p t ] 2 | T | + 1 & \\\\ \\end{cases} \\end{align*}"}
+{"id": "703.png", "formula": "\\begin{align*} = \\left ( \\begin{array} { c c } ( \\int _ M d U _ { \\beta _ k } \\wedge * d U _ { \\beta _ j } ) & ( - \\int _ M d U _ { \\beta _ k } \\wedge * d U _ { \\alpha _ j } ) \\\\ ( - \\int _ M d U _ { \\alpha _ k } \\wedge * d U _ { \\beta _ j } ) & ( \\int _ M d U _ { \\alpha _ k } \\wedge * d U _ { \\alpha _ j } ) \\\\ \\end{array} \\right ) \\end{align*}"}
+{"id": "7258.png", "formula": "\\begin{align*} \\chi _ { m , j } ( \\lambda ) = \\int _ { 0 } ^ { \\infty } ( 1 - g _ m ( \\lambda ; x ) ) j ( d x ) , \\end{align*}"}
+{"id": "6239.png", "formula": "\\begin{align*} \\begin{cases} C _ p ^ T \\Psi '' - C _ p ^ T \\Delta \\Psi + A ^ T C _ p ^ T \\Psi = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ C _ p ^ T \\Psi = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ C _ p ^ T \\partial _ \\nu \\Psi + B ^ T C _ p ^ T \\Psi = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 . \\end{cases} \\end{align*}"}
+{"id": "4789.png", "formula": "\\begin{align*} \\Pr _ { Z \\sim \\mathcal { D } ( C ^ { ( \\alpha ) } ) } [ Z _ j = 0 ] = 1 - \\alpha . \\end{align*}"}
+{"id": "4932.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { H } } = \\Delta x \\sum _ m \\widetilde { H } ( \\mathbf { Z } ) _ m , \\end{align*}"}
+{"id": "2349.png", "formula": "\\begin{align*} T \\phi _ { n + j } = T C \\phi _ j = C T \\phi _ j = C \\phi _ j = \\phi _ { n + j } , C \\phi _ { n + j } = C ^ 2 \\phi _ j = \\phi _ j . \\end{align*}"}
+{"id": "8394.png", "formula": "\\begin{align*} \\tilde { \\sigma } ^ 2 ( \\alpha ) = \\int \\hat \\Psi _ { \\alpha } ^ 2 \\ : d \\mu _ { \\alpha } + 2 \\sum _ { k \\geq 1 } \\int \\hat \\Psi _ { \\alpha } \\cdot \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha } ^ k \\ : d \\mu _ { \\alpha } . \\end{align*}"}
+{"id": "2485.png", "formula": "\\begin{align*} \\begin{aligned} & c > 0 c ^ 2 \\sum _ n \\| x _ n \\| ^ 2 \\leq \\Bigl \\| \\sum _ n x _ n \\Bigr \\| ^ 2 \\\\ & \\{ x _ n \\} _ n x _ n \\in \\mathcal H _ n n . \\end{aligned} \\end{align*}"}
+{"id": "8367.png", "formula": "\\begin{align*} \\lim _ { t \\to - \\infty } \\int _ { | x | > - \\varepsilon - t } \\left ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 \\right ) \\mathrm { d } x = 0 . \\end{align*}"}
+{"id": "3164.png", "formula": "\\begin{align*} A ( y ) = \\mathrm { d i a g } ( a _ 1 ( y _ 1 ) , a _ 2 ( y _ 2 ) , \\dots , a _ n ( y _ n ) ) \\quad y = ( y _ 1 , \\dots , y _ n ) \\in \\R ^ n \\end{align*}"}
+{"id": "625.png", "formula": "\\begin{align*} I _ \\Delta \\ ! = \\ ! ( x _ 1 x _ 2 x _ 3 , \\ ! x _ 1 x _ 2 x _ 4 , x _ 1 x _ 3 x _ 5 , x _ 1 x _ 4 x _ 6 , x _ 1 x _ 5 x _ 6 , x _ 2 x _ 3 x _ 6 , x _ 2 x _ 4 x _ 5 , x _ 2 x _ 5 x _ 6 , x _ 3 x _ 4 x _ 5 , x _ 3 x _ 4 x _ 6 ) \\end{align*}"}
+{"id": "5141.png", "formula": "\\begin{align*} r _ { n } ( \\lambda , b , \\mathbf { m } ) : = \\det \\left ( M _ { n \\mathbf { m } } ( \\lambda , b , \\Omega _ { \\mathbf { m } } ^ { \\pm } ( \\lambda , b ) ) \\right ) - d _ { \\infty } ( \\lambda , b , \\mathbf { m } ) \\underset { n \\rightarrow \\infty } { = } \\frac { \\widetilde { d } _ { \\infty } ( \\lambda , b , \\mathbf { m } ) } { n } + O _ { \\lambda , b , \\mathbf { m } } \\left ( \\frac { 1 } { n ^ { 3 } } \\right ) . \\end{align*}"}
+{"id": "4055.png", "formula": "\\begin{align*} & y _ 0 = Y ( x _ 0 , u _ 0 , p _ 0 ) & & y _ 1 = Y ( x _ 0 , u _ 0 , p _ 1 ) , \\\\ & z _ 0 = Z ( x _ 0 , u _ 0 , p _ 0 ) & & z _ 1 = Z ( x _ 0 , u _ 0 , p _ 1 ) . \\end{align*}"}
+{"id": "8365.png", "formula": "\\begin{align*} ( \\tilde { u } _ 0 ( x ) , \\tilde { u } _ 1 ( x ) ) = ( u ( M _ 0 , x ) , u _ 1 ( M _ 0 , x ) ) , \\mathrm { f o r } \\ | x | > M _ 0 , \\end{align*}"}
+{"id": "6953.png", "formula": "\\begin{gather*} \\alpha L _ m ^ { ( \\alpha ) } ( t ) - t L _ { m - 1 } ^ { ( \\alpha + 1 ) } ( t ) = ( m + \\alpha ) L _ { m } ^ { ( \\alpha - 1 ) } ( t ) , \\end{gather*}"}
+{"id": "790.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { f _ n ( x ) } { n } = \\Lambda \\ \\end{align*}"}
+{"id": "1036.png", "formula": "\\begin{align*} & \\mathrm { H } _ 0 : P \\in \\mathcal { P } _ \\varepsilon ( P _ 0 ) = \\{ P _ \\varepsilon : \\ , ( 1 - \\varepsilon ) P _ 0 + \\varepsilon G , \\ , G \\in \\mathcal { G } \\} \\\\ \\mathrm { v s . } & \\mathrm { H } _ 1 : P \\in \\mathcal { P } _ \\varepsilon ( P _ 1 ) = \\{ P _ \\varepsilon : \\ , ( 1 - \\varepsilon ) P _ 1 + \\varepsilon G , \\ , G \\in \\mathcal { G } \\} , \\end{align*}"}
+{"id": "7476.png", "formula": "\\begin{align*} E _ A : = \\{ ( u , v ) \\in E ' : u \\notin A , ~ v \\in A \\} \\ , , \\end{align*}"}
+{"id": "19.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\mathbb { E } \\big [ e ^ { - \\beta t } p _ t y _ { ( 1 , t ) } \\big ] = 0 , \\lim _ { t \\rightarrow \\infty } \\mathbb { E } \\big [ e ^ { - \\beta t } Q _ t \\mathcal { Z } _ { ( 1 , t ) } \\big ] = 0 , \\lim _ { t \\rightarrow \\infty } \\mathbb { E } \\big [ e ^ { - \\beta t } q _ t x _ { ( 1 , t ) } \\big ] = 0 . \\end{align*}"}
+{"id": "8547.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Bigg ( \\frac { 1 } { ( \\sum _ { i = 1 } ^ { n } \\sum _ { p = 1 } ^ { \\lfloor ( n - i ) / 2 \\rfloor } \\binom { n } { i } \\binom { n - i } { 2 } ^ p ( 2 ^ i - 1 ) ^ n 2 ^ { Q ( n - i + 1 , n - i + 1 - 2 p ) - Q ( n , n ) } ) / n } \\Bigg ) \\ge 1 . \\end{align*}"}
+{"id": "8120.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ( u _ 2 ( y ) ) a ( y ) \\ , d y = \\frac { a ( y _ 0 ) e \\Bigl ( - x c ^ 2 p ^ { - 1 } + \\dfrac { 1 } { 8 } \\Bigr ) } { \\sqrt { u _ 2 '' ( y _ 0 ) } } + O ( c ^ { \\frac { 7 } { 2 } } T ^ 4 N ^ { - \\frac { 1 1 } { 6 } } m ^ { \\frac { 1 1 } { 3 } } p ^ { - \\frac { 7 } { 4 } } ) . \\end{align*}"}
+{"id": "1155.png", "formula": "\\begin{align*} & \\Omega ( a , a , a ) = a , \\ , \\ , \\ , \\Omega ( b , b , b ) = b , \\\\ & \\Omega ( b , a , a ) = \\Omega ( a , b , a ) = \\Omega ( a , a , b ) = b , \\\\ & \\Omega ( a , b , b ) = \\Omega ( b , a , b ) = \\Omega ( b , b , a ) = a . \\\\ \\end{align*}"}
+{"id": "4829.png", "formula": "\\begin{align*} \\Phi ( v _ 1 , v _ 2 ) = ( v _ 1 , v _ 2 + \\phi ( v _ 1 ) ) , \\end{align*}"}
+{"id": "4011.png", "formula": "\\begin{align*} v & = 0 \\partial \\Omega \\\\ L v & \\leq - \\epsilon w ^ { i i } + C \\Omega \\end{align*}"}
+{"id": "5449.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { t \\in [ 0 , T ] } { | b ( t , x _ n , \\mu _ n , \\alpha ) - b ( t , x , \\mu , \\alpha ) | + | ( \\sigma ( t , x _ n , \\mu _ n , \\alpha ) - \\sigma ( t , x , \\mu , \\alpha ) | } = 0 . \\end{align*}"}
+{"id": "2260.png", "formula": "\\begin{align*} ( 0 , x ) = \\big ( \\sum _ { i = 1 } ^ r ( k _ i - 1 ) z _ i , \\sum _ { i = 1 } ^ r ( k _ i - 1 ) y _ i \\big ) = \\sum _ { i = 1 } ^ r ( k _ i - 1 ) v _ i , \\end{align*}"}
+{"id": "3372.png", "formula": "\\begin{align*} \\bigvee _ { j = 1 } ^ { i - 1 } ( - u _ i + z _ j ) \\vee ( - u _ { i + 1 } + 1 + z _ i ) \\leq \\bigvee _ { j = i + 1 } ^ { n - 2 } ( - u _ j + z _ j ) \\vee x _ n \\vee ( - u _ { n - 1 } ) \\ ; i = 1 , \\dots , n - 2 \\ , . \\end{align*}"}
+{"id": "5812.png", "formula": "\\begin{align*} U = \\sum _ { r = 1 } ^ p u _ r e _ r / \\| e _ r \\| + C _ p ^ T ( C _ p C _ p ^ T ) ^ { - 1 } C _ p U = u + C _ p ^ T ( C _ p C _ p ^ T ) ^ { - 1 } C _ p U . \\end{align*}"}
+{"id": "6445.png", "formula": "\\begin{align*} ( \\partial _ t - X \\cdot \\nabla _ Y ) w ( X , Y , t ) & = \\frac \\theta 2 e ^ { - \\lambda ( T - t ) } \\bigl ( - 2 X \\cdot Y + \\lambda ( | Y | ^ 2 + A ) \\bigr ) + \\eta / ( T - t ) ^ 2 \\\\ & \\geq \\frac \\theta 2 e ^ { - \\lambda T } ( \\lambda ( | Y | ^ 2 + A ) - 2 R | Y | ) , \\end{align*}"}
+{"id": "253.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = - ( 1 - \\alpha ) \\rho ^ 2 - \\alpha \\rho \\tau - \\beta \\tau \\eta \\leqslant - \\frac { \\alpha } { 2 } ( 1 + 3 \\rho ^ 2 ) \\leqslant - \\frac { \\alpha } { 2 } , \\end{align*}"}
+{"id": "4514.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n \\lambda v _ i = \\prod _ { i = 1 } ^ n \\lambda u _ i . \\end{align*}"}
+{"id": "7088.png", "formula": "\\begin{align*} M U _ * \\to H _ * M U = \\mathbb { Z } [ b _ 1 , b _ 2 , \\dots ] , \\end{align*}"}
+{"id": "2383.png", "formula": "\\begin{align*} x ( t ) = \\left [ \\begin{array} { c } \\log ( \\vert t - \\hat t \\vert ) w \\\\ 0 \\end{array} \\right ] \\end{align*}"}
+{"id": "3935.png", "formula": "\\begin{align*} \\tilde { g } _ x ( x , y , z ) & = \\frac { - g _ { x z } ( x , 0 , 0 ) } { g _ z ( x , 0 , 0 ) } \\tilde { g } ( x , y , z ) \\\\ & \\quad \\quad + \\frac { g _ z ( 0 , 0 , 0 ) } { g _ z ( x , 0 , 0 ) } [ g _ x ( x , y , g ^ * ( 0 , y , h - z ) ) - g _ x ( x , 0 , 0 ) ] , \\end{align*}"}
+{"id": "5142.png", "formula": "\\begin{align*} \\Big ( \\Lambda _ { 1 } ( \\lambda , b ) - b \\big [ \\Omega _ { \\mathbf { m } } ( \\lambda b ) + \\Omega _ { \\mathbf { m } } ^ { \\pm } ( \\lambda , b ) \\big ] \\Big ) ^ { 2 } - b ^ { 2 } \\Lambda _ { \\mathbf { m } } ^ { 2 } ( \\lambda , b ) = 0 . \\end{align*}"}
+{"id": "3887.png", "formula": "\\begin{align*} v ( y _ 0 ) & = g ^ * ( x _ 0 , y _ 0 , u _ 0 ) , \\\\ v ( y ) & \\geq g ^ * ( x _ 0 , y , u _ 0 ) y \\in \\Omega ^ * . \\end{align*}"}
+{"id": "6697.png", "formula": "\\begin{align*} \\sum _ { p \\leq x } \\lambda ( p ) \\lambda ( p + a ) = \\sum _ { p \\leq x } \\lambda ( p + a ) . \\end{align*}"}
+{"id": "3723.png", "formula": "\\begin{align*} & c _ i ( f ) \\Phi ( 0 , 0 ) + c _ { i - 1 } ( d _ { i , i - 1 } ( f ) ) \\Phi ( 0 , 0 ) \\\\ & - c _ i ( f ) \\mathcal { F } _ \\wedge ( \\Phi ) ( 0 , 0 ) - c _ { i - 1 } ( d _ { i , i - 1 } ( f ) ) \\mathcal { F } _ \\wedge ( \\Phi ) ( 0 , 0 ) \\\\ & = c _ i ( f ) + c _ { i - 1 } ( d _ { i , i - 1 } ( f ) ) \\end{align*}"}
+{"id": "4441.png", "formula": "\\begin{align*} a _ 2 = \\left ( \\frac { 1 } { 3 } + \\frac { 1 } { 5 } - \\frac { 1 } { 2 } \\right ) ^ { - 1 } = 3 0 \\end{align*}"}
+{"id": "1700.png", "formula": "\\begin{align*} \\left \\langle \\mu _ { \\frac { k - 2 } { 2 } } , ( y ^ { k - 2 } ) ^ \\vee \\right \\rangle ' = ( y ^ { k - 2 } ) ^ \\vee \\mu _ { \\frac { k - 2 } { 2 } } \\left ( \\left | \\begin{array} { c c } X & Y \\\\ x & y \\end{array} \\right | ^ { k - 2 } \\right ) = ( y ^ { k - 2 } ) ^ \\vee \\left ( y ^ { k - 2 } \\right ) = 1 . \\end{align*}"}
+{"id": "7921.png", "formula": "\\begin{align*} \\Pi \\left ( \\pi _ 1 ^ { - 1 } ( A ) \\right ) = \\mu ( A ) \\ , , \\Pi \\left ( \\pi _ 2 ^ { - 1 } ( B ) \\right ) = \\nu ( B ) \\ , , \\end{align*}"}
+{"id": "2302.png", "formula": "\\begin{align*} & L ^ 1 = \\frac { 1 } { N + 1 } \\sum _ { i = 0 } ^ N \\left | u _ i ( T ) - u ( x _ i , T ) \\right | , \\\\ & L ^ 2 = \\sqrt { \\frac { 1 } { N + 1 } \\sum _ { i = 0 } ^ N \\left ( u _ i ( T ) - u ( x _ i , T ) \\right ) ^ 2 } , \\\\ & L ^ { \\infty } = \\max _ { 0 \\leqslant i \\leqslant N } \\left | u _ i ( T ) - u ( x _ i , T ) \\right | , \\end{align*}"}
+{"id": "3711.png", "formula": "\\begin{align*} Z _ { r _ i } ( f , s ) & = \\int _ { F ^ \\times \\times K } \\chi ( a ) | a | ^ { - s } ( r _ i ( k ) f ) \\left ( 0 _ { V _ i } , 0 , a ^ { - 1 } \\right ) d k d ^ \\times a \\\\ & = \\int _ { F ^ \\times \\times K } \\chi ( a ) | a | ^ { s } ( r _ i ( k ) f ) \\left ( 0 _ { V _ i } , 0 , a \\right ) d k d ^ \\times a . \\end{align*}"}
+{"id": "4418.png", "formula": "\\begin{align*} ( 0 , 1 ] = \\bigcup _ { a _ 1 = 2 } ^ { \\infty } \\left ( \\frac { 1 } { a _ 1 } , \\frac { 1 } { a _ 1 - 1 } \\right ] . \\end{align*}"}
+{"id": "553.png", "formula": "\\begin{align*} { \\rm d } { \\bf u } _ t = - ( U D + \\rho D D ^ { \\rm T } ) { \\bf u } _ t { \\rm d } t + \\Delta y ^ { - 1 / 2 } { \\rm d } W _ t , \\end{align*}"}
+{"id": "9131.png", "formula": "\\begin{align*} Q _ { h , \\nu } ( \\eta | \\gamma ) \\ ; = \\ ; 0 , \\ ; \\ ; \\ ; \\eta \\cap \\gamma \\neq \\emptyset . \\end{align*}"}
+{"id": "7636.png", "formula": "\\begin{align*} M _ \\psi = \\begin{pmatrix} 2 & 2 & 2 \\\\ 3 & 1 & 0 \\\\ 4 & 0 & 0 \\\\ \\end{pmatrix} \\lambda _ 1 \\approx 5 . 0 5 9 3 , \\ \\lambda _ 2 \\approx - 2 . 6 5 4 9 , \\ \\lambda _ 3 \\approx 0 . 5 9 5 6 . \\end{align*}"}
+{"id": "8848.png", "formula": "\\begin{align*} u ( z , \\bar { z } ) = \\sum _ { | \\alpha | \\leq k } \\sum _ { | \\beta | \\leq l } c _ { \\alpha , \\beta } z ^ { \\alpha } \\bar { z } ^ { \\beta } , c _ { \\alpha , \\beta } \\in \\mathbb { C } , \\alpha , \\beta \\in \\mathbb { Z } _ { + } ^ { n } , \\end{align*}"}
+{"id": "8849.png", "formula": "\\begin{align*} \\delta ( n , p , q ) = \\begin{pmatrix} p + n - 1 \\\\ p - 1 \\end{pmatrix} \\begin{pmatrix} q + n - 1 \\\\ q - 1 \\end{pmatrix} \\end{align*}"}
+{"id": "7412.png", "formula": "\\begin{align*} \\mu ( s \\widetilde { \\alpha } , \\sigma ) = \\gamma ( G / P ) ^ 2 q _ F ^ { n ( \\sigma ) + n ( \\sigma \\otimes \\omega ) } \\frac { ( 1 - \\omega ( \\varpi ) q _ F ^ { - 2 s } ) ( 1 - \\omega ( \\varpi ) ^ { - 1 } q _ F ^ { 2 s } ) } { ( 1 - \\omega ( \\varpi ) q _ F ^ { - 1 - 2 s } ) ( 1 - \\omega ( \\varpi ) ^ { - 1 } q _ F ^ { - 1 + 2 s } ) } \\end{align*}"}
+{"id": "1610.png", "formula": "\\begin{align*} Y ^ n = \\sum _ { u = 1 } ^ K H _ u X _ u ^ n + Z ^ n \\end{align*}"}
+{"id": "8523.png", "formula": "\\begin{align*} \\displaystyle { g ( n , k ) \\ge \\frac { \\binom { n } { k / 2 } - \\binom { n } { k / 2 - 1 } } { k ! } . } \\end{align*}"}
+{"id": "6344.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Phi _ { \\nu _ 0 } ( t , \\xi \\circ m ( t ) ) = f ( t , \\nu _ 0 ) ( 1 + ( \\xi \\circ m ) ' ( t ) ) . \\end{align*}"}
+{"id": "4324.png", "formula": "\\begin{align*} \\psi ( \\partial S \\times \\{ 0 \\} ) = \\phi ( \\partial S \\times \\{ 0 \\} ) . \\end{align*}"}
+{"id": "6319.png", "formula": "\\begin{align*} \\sigma _ { i j } { = } \\mathsf { K } _ { i j k \\ell } \\frac { \\partial u _ k } { \\partial x _ \\ell } + \\mathsf { S } _ { i j } ^ { k \\ell m } \\frac { \\partial ^ 2 u _ k } { \\partial x _ m \\partial x _ \\ell } , \\end{align*}"}
+{"id": "6950.png", "formula": "\\begin{align*} x y _ n ' ( x ) + \\alpha y _ n ( x ) = Q ^ I _ { n - m } ( x ) L _ { m } ^ { ( \\alpha - 1 ) } ( - x ) . \\end{align*}"}
+{"id": "2808.png", "formula": "\\begin{align*} 0 < u _ { n + 1 } = \\min ( v _ n , u _ n ) \\left ( \\frac { \\prod _ { i = 1 } ^ n v _ i } { \\prod _ { i = 1 } ^ n u _ i } \\right ) . \\leq \\min ( v _ n , u _ n ) \\leq u _ n \\end{align*}"}
+{"id": "1367.png", "formula": "\\begin{align*} b ( 0 ) > 0 , \\ , 1 - \\int _ { 0 } ^ { t } b ( \\tau ) d \\tau = l > 0 . \\end{align*}"}
+{"id": "1362.png", "formula": "\\begin{align*} & ( S _ { f , \\tau } ) = \\int _ \\Omega f _ \\alpha ( \\tau _ \\alpha ) \\ , d \\mu ( \\alpha ) , \\\\ & ( S _ { f , \\tau } ^ 2 ) = \\int _ \\Omega \\int _ \\Omega f _ \\beta ( \\tau _ \\alpha ) f _ \\alpha ( \\tau _ \\beta ) \\ , d \\mu ( \\alpha ) \\ , d \\mu ( \\beta ) . \\end{align*}"}
+{"id": "3444.png", "formula": "\\begin{align*} \\mu _ { \\infty } ( [ 1 \\ldots 1 ] ) = 1 , \\end{align*}"}
+{"id": "742.png", "formula": "\\begin{align*} \\{ \\frac { \\partial ^ 2 H ( z , w ) } { \\partial z \\partial \\bar { z } } \\} _ { z = w } = h _ { 1 1 } ( w ) = \\frac { \\pi } { 2 V } \\lambda ( w ) ^ 2 , \\end{align*}"}
+{"id": "1526.png", "formula": "\\begin{align*} \\nabla = \\nabla ^ { L C } + K \\end{align*}"}
+{"id": "387.png", "formula": "\\begin{align*} J ^ { ( 1 , 1 ) } _ 2 & \\leq C ^ { ( 1 ) } _ 2 T _ * L ^ { - 2 } _ * T ^ { - 1 / 2 } _ * ( 1 + T ^ { - 3 / 2 } _ * ) \\ , . \\end{align*}"}
+{"id": "4093.png", "formula": "\\begin{align*} R ^ + = R - \\frac { 1 } { 4 } | H | ^ 2 , R ^ - = R - \\frac { 1 } { 4 } | H | ^ 2 \\end{align*}"}
+{"id": "7580.png", "formula": "\\begin{align*} \\eta = \\eta ( r ) = \\begin{cases} 1 , & | r | \\leq \\frac { d } { 2 } , \\\\ 0 , & | r | \\geq \\frac { 3 d } { 4 } , \\end{cases} \\end{align*}"}
+{"id": "7978.png", "formula": "\\begin{align*} [ x ] + [ y ] : = \\{ [ z ] \\mid z \\in x ' + y ' \\emph { s u c h t h a t } [ x ' ] = [ x ] , [ y ' ] = [ y ] \\} , \\end{align*}"}
+{"id": "4125.png", "formula": "\\begin{align*} d K = \\psi d V _ g \\psi \\in C ^ \\infty ( M ) . \\end{align*}"}
+{"id": "3523.png", "formula": "\\begin{align*} M ( \\mathfrak { g } , \\mathfrak { t } ) : = U ( \\mathfrak { g } ) / \\mathfrak { J } , \\end{align*}"}
+{"id": "6075.png", "formula": "\\begin{align*} p ( x ) - \\sum _ { i = 1 } ^ g \\ell _ i ( x ) \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac { p ( y ) \\dd y } { ( w \\tilde w ) ( y ) } = p _ 0 \\left ( x ^ g - \\sum _ { i = 1 } ^ g \\ell _ i ( x ) \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac { y ^ g \\dd y } { ( w \\tilde w ) ( y ) } \\right ) = \\frac { p _ 0 } { u _ 0 } u ( x ) , \\end{align*}"}
+{"id": "573.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } m _ t = \\frac { \\sigma _ t } { \\gamma } \\ , ( A ^ { \\rm T } A ) : C \\ , ( 1 - m _ t ) = \\sigma _ t \\ , ( A ^ { \\rm T } A ) : ( A + A ^ { \\rm T } ) ^ { - 1 } \\ , ( 1 - m _ t ) . \\end{align*}"}
+{"id": "924.png", "formula": "\\begin{align*} \\int _ { T M } D \\varphi ( \\pi ( v ) ) v \\ d \\mu ( v ) = 0 \\end{align*}"}
+{"id": "4795.png", "formula": "\\begin{align*} \\Pr _ { v \\sim \\mu _ t } \\big [ | v H | = j \\big ] \\cdot \\frac { 2 ^ N } { \\binom { N } { j } } & \\leq 2 ^ { - ( 1 - h ( j / N ) ) ( 1 - \\eta ) N } \\cdot \\frac { 2 ^ N } { \\sqrt { \\frac { 8 \\pi } { e ^ 4 N } } \\cdot 2 ^ { h ( j / N ) N } } \\\\ & = \\sqrt { \\frac { e ^ 4 N } { 8 \\pi } } \\cdot 2 ^ { ( 1 - h ( j / N ) ) \\eta N } . \\end{align*}"}
+{"id": "2338.png", "formula": "\\begin{align*} & \\frac { \\Gamma \\ ( \\frac { q } { q - N } \\ ) \\Gamma \\ ( \\frac { N ( q - 1 ) } { q - N } \\ ) } { \\Gamma \\ ( \\frac { N q } { q - N } \\ ) } = \\frac { \\Gamma ( s ) \\Gamma ( \\frac { ( q - 1 ) N } { q } s ) } { \\Gamma ( N s ) } \\sim \\frac { \\Gamma ( s ) \\Gamma ( ( N - 1 ) s ) } { \\Gamma ( N s ) } \\\\ & \\sim \\sqrt { 2 \\pi } \\frac { s ^ { s - 1 / 2 } e ^ { - s } ( ( N - 1 ) s ) ) ^ { ( N - 1 ) s - 1 / 2 } e ^ { - ( N - 1 ) s } } { ( N s ) ^ { N s - 1 / 2 } e ^ { - N s } } = \\sqrt { 2 \\pi } \\frac { ( N - 1 ) ^ { ( N - 1 ) s - 1 / 2 } } { N ^ { N s - 1 / 2 } } s ^ { - 1 / 2 } \\end{align*}"}
+{"id": "8529.png", "formula": "\\begin{align*} \\begin{cases} \\ u ( { \\bf x } , t ) = u _ 0 ( { \\bf x } ) & \\ { \\bf x } \\in \\partial \\Omega , \\ t > 0 , \\\\ \\ d ( { \\bf x } , t ) = d _ 0 ( { \\bf x } ) & \\ { \\bf x } \\in \\partial B _ 1 ^ 2 \\times [ 0 , 1 ] , \\ t > 0 , \\\\ \\frac { \\partial d } { \\partial z } ( { \\bf x } , t ) = 0 & \\ { \\bf x } \\in B _ 1 ^ 2 \\times \\{ 0 , 1 \\} , \\ t > 0 . \\end{cases} \\end{align*}"}
+{"id": "3757.png", "formula": "\\begin{align*} \\tilde { c } _ { 2 1 } ( f ) : & = \\bigg ( 2 | D | ^ { - 1 / 2 } \\kappa _ 1 - ( \\log | D | ) \\kappa _ 0 \\bigg ) \\tilde { c } _ 1 ( \\tilde { d } _ 2 ( f ) ) \\\\ & + | D | ^ { 1 / 2 } \\kappa _ 0 \\sum _ { v } ( \\log q _ v ) \\bigg ( ( e ( \\psi _ v ) + 1 ) \\tilde { c } _ { 2 , v } ( I ( f _ v ) ) - \\tilde { a } _ { 2 , v } ( f _ v ) \\bigg ) \\prod _ { v ' \\neq v } \\tilde { c } _ { 2 , v ' } ( f _ { v ' } ) , \\end{align*}"}
+{"id": "6170.png", "formula": "\\begin{align*} \\begin{pmatrix} C _ 1 \\\\ E _ 1 ^ T \\end{pmatrix} U _ n = \\begin{pmatrix} C _ 1 U _ n \\\\ ( E _ 1 , U _ n ) \\end{pmatrix} \\rightarrow \\begin{pmatrix} 0 \\\\ u \\end{pmatrix} = u \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\end{align*}"}
+{"id": "6173.png", "formula": "\\begin{align*} C _ 1 B e _ 1 u = 0 \\hbox { o n } [ T , + \\infty ) \\times \\Gamma _ 1 . \\end{align*}"}
+{"id": "2125.png", "formula": "\\begin{align*} \\forall \\tau _ p \\in \\left [ - \\eta , \\varepsilon \\right ] , \\forall t \\in [ - \\ell _ { k , p } , \\ell _ { k , p } ] , \\Im ( \\gamma _ { k , p } ( t ) ) = t + \\theta _ k , \\Re ( \\gamma _ { k , p } ( t ) ) = h _ { k , p } ( t ) : = \\Psi _ k ^ { - 1 } \\left ( \\Psi _ k ( \\tau _ p ) - A _ I t ^ { 2 \\mu _ k } \\right ) . \\end{align*}"}
+{"id": "1786.png", "formula": "\\begin{align*} t L _ x - s \\ell _ x & = t \\sum _ { j = 1 } ^ { s - t } L _ { k _ j } - s \\sum _ { j = 1 } ^ { s - t } \\ell _ { k _ j } = \\sum _ { j = 1 } ^ { s - t } ( t L _ { k _ j } - s \\ell _ { k _ j } ) \\\\ & \\equiv \\sum _ { j = 1 } ^ { s - t } r _ j \\equiv ( s - t ) r \\equiv 0 \\ ! \\ ! \\ ! \\pmod { s - t } . \\end{align*}"}
+{"id": "1297.png", "formula": "\\begin{align*} h _ { \\vec { \\zeta } } ( \\vec { w } ) : = \\sum _ { \\vec { k } \\in \\Z ^ m } \\vec { \\zeta } ^ { \\vec { k } } H _ { \\vec { k } } ( \\vec { w } ) \\qquad \\textrm { a n d } g _ { \\vec { \\zeta } } ( \\vec { w } ) : = \\sum _ { \\vec { k } \\in \\Z ^ m } \\vec { \\zeta } ^ { \\vec { k } } G _ { \\vec { k } } ( \\vec { w } ) \\end{align*}"}
+{"id": "219.png", "formula": "\\begin{align*} \\omega _ { \\mathcal { X } / R } : = \\bigwedge \\Omega _ { \\mathcal { X } / R } ^ { 1 } \\end{align*}"}
+{"id": "1805.png", "formula": "\\begin{gather*} \\hat { L } _ { u u } ( t ) = L _ { u u } ( t , u _ 0 ( t ) , \\dot { u } _ 0 ( t ) ) , \\\\ \\hat { L } _ { u \\dot { u } } ( t ) = L _ { u \\dot { u } } ( t , u _ 0 ( t ) , \\dot { u } _ 0 ( t ) ) , \\\\ \\hat { L } _ { \\dot { u } \\dot { u } } ( t ) = L _ { \\dot { u } \\dot { u } } ( t , u _ 0 ( t ) , \\dot { u } _ 0 ( t ) ) , \\end{gather*}"}
+{"id": "4819.png", "formula": "\\begin{align*} f ( \\epsilon , \\frac { 2 } { \\ln 2 } ) = \\log ( 1 - \\epsilon ) + 2 \\epsilon \\log ( \\ln 2 ) + \\frac { 2 \\epsilon } { \\ln 2 } \\geq 0 . \\end{align*}"}
+{"id": "5160.png", "formula": "\\begin{align*} P ( D ) = \\prod ^ { k - 1 } _ { i = 0 } P _ { i } ( v _ { i } | v _ { i - 1 } ) . \\end{align*}"}
+{"id": "1704.png", "formula": "\\begin{align*} f _ n \\left ( \\begin{array} { c c } x & y \\\\ r & s \\end{array} \\right ) = \\Delta ^ { \\frac { 2 - k } { 2 } } \\frac { ( s - r i ) ^ { n - \\frac { 2 - k } { 2 } } } { ( s + r i ) ^ { n + \\frac { 2 - k } { 2 } } } , \\Delta = ( x s - r y ) > 0 , \\end{align*}"}
+{"id": "5237.png", "formula": "\\begin{align*} T _ U V = g ( U , V ) H ~ ~ T _ U X = - g ( H , X ) U , \\end{align*}"}
+{"id": "6062.png", "formula": "\\begin{align*} \\tilde w ( z ) : = z ^ { g + 1 } w ( 1 / z ) , \\end{align*}"}
+{"id": "3299.png", "formula": "\\begin{align*} \\langle \\mathbb P { \\rm d i v } F , A ^ { - \\frac 1 2 } w \\rangle & = \\dfrac { m } { \\pi } \\int _ { B _ 0 } { \\rm d i v } F \\cdot v + \\int _ { \\mathcal F _ 0 } { \\rm d i v } F \\cdot v \\\\ & = - \\int _ { \\mathcal F _ 0 } F : \\nabla v . \\end{align*}"}
+{"id": "4900.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\partial _ s v ( s , t ) + J _ t ( u ( s , t ) ) \\Big ( \\partial _ t v ( s , t ) - \\tau ( s ) X _ H ( v ( s , t ) ) \\Big ) = 0 \\\\ \\int _ 0 ^ 1 H ( v ( s , t ) d t = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "9221.png", "formula": "\\begin{align*} 2 \\nabla f \\cdot \\nabla \\ell _ { 0 } + \\vert \\nabla \\ell _ { 0 } \\vert ^ { 2 } \\ell _ { 0 } = 0 , \\end{align*}"}
+{"id": "1804.png", "formula": "\\begin{align*} \\dot { u } _ 0 ( 0 ) = \\dot { u } _ 0 ( T ) , \\end{align*}"}
+{"id": "4014.png", "formula": "\\begin{align*} W [ \\underline { u } ] : = [ D _ { \\alpha \\beta } \\phi - A _ { \\alpha \\beta } ( x , \\phi , D ' \\phi , D _ n \\underline { u } ) ] \\xi _ \\alpha \\xi _ \\beta \\geq \\delta . \\end{align*}"}
+{"id": "8720.png", "formula": "\\begin{align*} \\tilde s _ \\lambda ( x _ 1 , \\dots , x _ n ) = \\frac { \\det [ x _ i ^ { n - j + 1 } ( x _ i | a ) _ { \\lambda _ j - 1 } ] } { \\det [ x ^ { n - j } _ i ] } . \\end{align*}"}
+{"id": "8380.png", "formula": "\\begin{align*} [ B , \\epsilon ] ^ s = \\{ ( f , g ) : \\ \\exists ~ \\delta > 0 \\ \\forall x \\in B ^ \\delta , | f ( x ) - g ( x ) | < \\epsilon \\} \\ \\ ( B \\in \\mathcal { B } , \\epsilon > 0 ) . \\end{align*}"}
+{"id": "3948.png", "formula": "\\begin{align*} R = \\{ x \\in \\mathbf { R } ^ d ; - b _ i \\leq x _ i \\leq a _ i \\} , \\end{align*}"}
+{"id": "8764.png", "formula": "\\begin{align*} q _ { u , v , w } ( z ) & = \\sum \\limits _ { l = 1 } ^ { 2 T - 1 } \\frac { \\exp { ( i 2 \\pi \\bar { l } ( u + z ) ) } } { | \\bar { l } | } v w ^ * . \\end{align*}"}
+{"id": "1833.png", "formula": "\\begin{align*} K _ X \\cdot C = K _ X | _ { S _ j } \\cdot C > ( K _ X + S _ j ) | _ { S _ j } \\cdot C \\ > K _ { S _ j } \\cdot C . \\end{align*}"}
+{"id": "2153.png", "formula": "\\begin{align*} a _ f ( r ) = \\dfrac { 1 + ( 1 - 2 \\alpha ) r ^ 2 } { 1 - r ^ 2 } , c _ f ( r ) = \\dfrac { 2 ( 1 - \\alpha ) r } { 1 - r ^ 2 } . \\end{align*}"}
+{"id": "443.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\right \\| = b \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "3158.png", "formula": "\\begin{align*} \\tilde { B } ( y ) : = \\begin{pmatrix} b _ { 1 } ( y ) & \\delta \\\\ \\delta & b _ { 2 } ( y ) \\end{pmatrix} \\quad y \\in \\R ^ 2 , \\end{align*}"}
+{"id": "468.png", "formula": "\\begin{align*} L ' _ { g , d , k } = | L ' _ { + } \\setminus \\psi ( L ' _ { - } ) | . \\end{align*}"}
+{"id": "2179.png", "formula": "\\begin{align*} R _ { \\mathcal { S } ^ { * } _ { N e } } ( F ) & = \\begin{dcases} \\sigma _ { 0 } & \\ 2 \\alpha + \\beta - 2 \\geqslant 0 \\\\ \\tilde { \\sigma _ { 0 } } & \\ 2 \\alpha + \\beta - 2 < 0 , \\end{dcases} \\end{align*}"}
+{"id": "7701.png", "formula": "\\begin{align*} Q ( \\textbf { s } - \\textbf { z } ) f ^ { \\textbf { s } + \\textbf { 1 } - \\textbf { z } } = b ( \\textbf { s } - \\textbf { s } ) f ^ { \\textbf { s } - \\textbf { z } } \\end{align*}"}
+{"id": "394.png", "formula": "\\begin{align*} \\pi ^ * \\left ( \\max \\limits _ { 1 \\leq m \\leq n } \\left | \\sum _ { j = 1 } ^ m X _ j - \\sum _ { j = 1 } ^ m Y _ j \\right | \\geq C \\ln ( n ) + x \\right ) \\leq K e ^ { - \\vartheta x } \\ , . \\end{align*}"}
+{"id": "4481.png", "formula": "\\begin{align*} \\frac { 2 } { 1 1 } = \\frac { 1 } { 1 1 } + \\frac { 1 } { 1 2 } + \\frac { 1 } { 1 3 2 } . \\end{align*}"}
+{"id": "7864.png", "formula": "\\begin{align*} f '' ( z ) - ( e ^ { z } - \\alpha ) f ' ( z ) + n e ^ { z } f ( z ) = 0 \\end{align*}"}
+{"id": "4298.png", "formula": "\\begin{gather*} \\Lambda : = \\nu _ { 0 } ^ { - 1 } \\left ( \\int _ { 0 } ^ { \\| \\nabla u _ { 0 } \\| _ { L ^ { 2 } } ^ { 2 } } \\varphi ( \\rho ) d \\rho + \\| \\partial _ { t } u _ { 1 } \\| _ { L ^ { 2 } } ^ { 2 } \\right ) , \\\\ M : = \\sup _ { \\rho \\in [ 0 , \\Lambda ] } \\varphi ( \\rho ) , L : = \\sup _ { \\rho _ { 1 } , \\rho _ { 2 } \\in [ 0 , \\Lambda ] } \\frac { | \\varphi ( \\rho _ { 2 } ) - \\varphi ( \\rho _ { 1 } ) | } { | \\rho _ { 2 } - \\rho _ { 1 } | } , \\end{gather*}"}
+{"id": "8172.png", "formula": "\\begin{align*} \\mathcal { D } & = \\frac { 3 L ( 1 , \\tilde f ) \\bigl ( A ( p , 1 ) p - 1 \\bigr ) } { 2 p ^ { \\frac { 3 } { 2 } } \\pi } \\int _ { - \\infty } ^ { \\infty } k ( t ) \\tanh ( \\pi t ) t \\log \\abs { t } \\ , d t + { } \\\\ & { } + \\frac { K \\bigl ( A ( p , 1 ) p - 1 \\bigr ) } { 2 p ^ { \\frac { 3 } { 2 } } \\pi } \\int _ { - \\infty } ^ { \\infty } k ( t ) \\tanh ( \\pi t ) t \\ , d t + O ( T ^ { \\frac { 1 } { 7 } + \\varepsilon } M p ^ { \\varepsilon } ) \\end{align*}"}
+{"id": "7776.png", "formula": "\\begin{align*} \\widetilde m _ { \\alpha , \\lambda } ( s ) & = \\frac { \\lambda } { \\lambda + s ^ \\alpha } = \\frac { \\lambda / s ^ \\alpha } { 1 + \\lambda / s ^ \\alpha } = - \\sum _ { k = 1 } ^ \\infty \\left ( - \\frac { \\lambda } { s ^ \\alpha } \\right ) ^ { k } \\\\ \\implies m _ { \\alpha , \\lambda } ( x ) & = - \\sum _ { k = 1 } ^ \\infty ( - \\lambda ) ^ k \\frac { x ^ { \\alpha k - 1 } } { \\Gamma ( \\alpha k ) } \\end{align*}"}
+{"id": "785.png", "formula": "\\begin{align*} S _ { n } ( t ) = \\frac { 1 } { ( n \\sigma ^ 2 ) ^ { 1 / 2 } } \\left ( \\log \\| M _ { \\lfloor t n \\rfloor } \\| - n t \\Lambda + ( n t - \\lfloor n t \\rfloor ) ( \\log \\| M _ { \\lfloor t n \\rfloor + 1 } \\| - \\log \\| M _ { \\lfloor t n \\rfloor } \\| ) \\right ) \\end{align*}"}
+{"id": "4770.png", "formula": "\\begin{align*} \\Pr _ { y \\sim \\mathcal { D } ( C ^ \\perp ) } \\big [ | y | \\notin A _ \\epsilon \\big ] \\leq 2 ^ { N ^ { \\frac { 3 } { 4 } } } \\cdot \\frac { \\sum _ { i \\notin A _ \\epsilon } \\binom { N } { i } } { 2 ^ N } . \\end{align*}"}
+{"id": "501.png", "formula": "\\begin{align*} \\sum _ { j = k } ^ { \\nu } \\| x ^ { j + 1 } \\ ! - \\ ! x ^ { j } \\| \\le \\frac { 1 } { 2 } \\sum _ { j = k } ^ { \\nu } \\Xi _ j + \\frac { b \\varphi ( \\Phi ( x ^ { \\ell ( k ) } ) \\ ! - \\ ! \\Phi ^ * ) } { a } + \\ ! \\sqrt { \\frac { 2 } { a } } \\ ! \\sum _ { K _ 1 \\ni j = k } ^ { \\nu } \\ ! \\sqrt { \\Phi ( x ^ { \\ell ( j + 1 ) } ) - \\Phi ( x ^ { j + 1 } ) } . \\end{align*}"}
+{"id": "7693.png", "formula": "\\begin{align*} ( \\tau ^ { \\sharp } ) ^ { - j } ( \\textbf { a } ) = ( a _ { 1 } - j , \\dots , a _ { r } - j ) . \\end{align*}"}
+{"id": "1899.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } d z ( x , t ) = z _ { x x } ( x , t ) d t - c z ( x , t ) d t + { \\sigma } z ( x , t ) d B ( t ) , \\cr z _ x ( 0 , t ) = 0 , \\ ; t \\geq 0 , \\cr z _ x ( 1 , t ) = - \\widehat { w } ( t ) + w ( t ) \\triangleq \\tilde { w } ( t ) , \\ ; t \\geq 0 , \\cr z ( x , 0 ) = z _ 0 ( x ) , \\ ; 0 \\leq x \\leq 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "4479.png", "formula": "\\begin{align*} \\frac { 3 } { 8 5 } = \\frac { 1 } { 2 9 } + \\frac { 1 } { 1 2 3 3 } + \\frac { 1 } { 3 0 3 9 3 4 5 } . \\end{align*}"}
+{"id": "3896.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d \\theta ^ 2 } & = g _ z E ^ { n , j } D _ { x _ j } ( g _ z E ^ { m , i } D _ { x _ i } ) q _ m q _ n \\\\ & = g _ { z } ^ 2 E ^ { n , j } E ^ { m , i } q _ m q _ n D _ { x _ i x _ j } + g _ z ^ 2 q _ m q _ n E ^ { n , j } D _ { x _ j } ( E ^ { m , i } ) D _ { x _ i } \\\\ & \\quad \\quad + g _ z g _ { j , z } E ^ { n , j } E ^ { m , i } q _ m q _ n D _ { x _ i } . \\end{align*}"}
+{"id": "7993.png", "formula": "\\begin{align*} a _ i + a _ j = \\begin{cases} \\{ 0 , a _ i \\} \\textrm { i f } i = j , \\\\ \\{ a _ k \\mid k \\neq j \\textrm { a n d } k \\neq i \\} \\textrm { i f } i \\neq j . \\end{cases} \\end{align*}"}
+{"id": "4285.png", "formula": "\\begin{align*} - \\Delta \\ , \\textbf { \\textit { u } } + \\nabla \\ , { \\pi } = \\textbf { \\textit { f } } \\quad \\mathrm { d i v } \\ , \\textbf { \\textit { u } } = { \\chi } \\quad \\quad \\R ^ 3 \\ , . \\end{align*}"}
+{"id": "5592.png", "formula": "\\begin{align*} \\rho ^ q \\rho _ q ( \\rho _ i \\Psi _ { j k } - \\rho _ j \\Psi _ { i k } ) & = 2 \\rho _ k \\rho ^ q ( \\rho _ i \\Psi _ { j q } - \\rho _ j \\Psi _ { i q } ) , \\\\ \\rho ^ q \\rho _ q \\rho ^ k \\Psi _ { i k } & = 2 \\rho _ i ( \\rho ^ j \\rho ^ k \\Psi _ { j k } ) \\end{align*}"}
+{"id": "8155.png", "formula": "\\begin{align*} & \\sideset { } { ^ * } \\sum _ j h ( t _ j ) \\omega _ j \\lambda _ j ( m ) \\lambda _ j ( n ) \\\\ & = \\frac { 1 } { 2 } \\delta ( m , n ) H + \\sum _ { c > 0 } \\frac { 1 } { 2 c } \\left ( S ( m , n ; c ) H ^ + \\Bigl ( \\frac { 4 \\pi \\sqrt { m n } } { c } \\Bigr ) - S ( - m , n ; c ) H ^ - \\Bigl ( \\frac { 4 \\pi \\sqrt { m n } } { c } \\Bigr ) \\right ) , \\end{align*}"}
+{"id": "4114.png", "formula": "\\begin{align*} \\int _ M \\Big ( \\langle C ( K _ 1 ) , h _ 2 \\rangle + \\langle K _ 2 , D ( h _ 1 ) \\rangle \\Big ) e ^ { - f } d V _ g = \\int _ M \\Big ( \\langle C ( K _ 2 ) , h _ 1 \\rangle + \\langle K _ 1 , D ( h _ 2 ) \\rangle \\Big ) e ^ { - f } d V _ g . \\end{align*}"}
+{"id": "3327.png", "formula": "\\begin{align*} X _ H \\lrcorner \\theta = - H , X _ H \\lrcorner \\Omega = \\dd H - R ( H ) \\theta . \\end{align*}"}
+{"id": "4197.png", "formula": "\\begin{align*} Q _ 0 ( \\phi , \\tilde { \\Lambda } ) = m ^ { \\alpha \\beta } \\partial _ { \\alpha } \\phi \\partial _ { \\beta } \\tilde { \\Lambda } , \\end{align*}"}
+{"id": "8966.png", "formula": "\\begin{align*} \\int _ { B _ R ( z _ 0 ) \\cap B } & | u _ t | ^ 2 d z \\le 2 R \\int _ { \\partial B } | u _ t | ^ 2 d \\phi , \\end{align*}"}
+{"id": "3718.png", "formula": "\\begin{align*} \\sum _ { s _ i \\in \\left \\{ \\frac { \\dim V _ i } { 2 } - 1 , \\ , \\frac { \\dim V _ i } { 2 } - 2 , \\ , 0 \\right \\} } \\mathrm { R e s } _ { s = s _ i } \\frac { e ^ { T s } Z _ { r _ i } ( \\mathcal { F } _ 2 ( f ) , s + 2 - \\frac { \\dim V _ i } { 2 } ) } { s } + o _ f ( 1 ) \\end{align*}"}
+{"id": "7687.png", "formula": "\\begin{align*} ( q + \\iota _ { E } ( \\omega ) ) \\eta _ { q } = L _ { \\omega } ( \\eta ) = \\nabla _ { \\omega } ( \\iota _ { E } ( \\eta _ { q } ) ) + \\iota _ { E } ( \\nabla _ { \\omega } ) ( \\eta _ { q } ) = \\nabla _ { \\omega } ( \\iota _ { E } ( \\eta _ { q } ) ) . \\end{align*}"}
+{"id": "6829.png", "formula": "\\begin{align*} \\tau ( L _ 1 , L _ 2 , \\dots , L _ m ) = \\sum \\limits _ { j = 2 } ^ { m - 1 } \\tau ( L _ 1 , L _ j , L _ { j + 1 } ) . \\end{align*}"}
+{"id": "7587.png", "formula": "\\begin{align*} ( \\c ( u ) , u ) & = ( u ( 1 - u ^ { \\delta } ) ( u ^ { \\delta } - \\gamma ) , u ) = ( ( 1 + \\gamma ) u ^ { \\delta + 1 } - \\gamma u - u ^ { 2 \\delta + 1 } , u ) \\\\ & = ( 1 + \\gamma ) ( u ^ { \\delta + 1 } , u ) - \\gamma \\| u \\| _ { \\L ^ 2 } ^ 2 - \\| u \\| _ { \\L ^ { 2 ( \\delta + 1 ) } } ^ { 2 ( \\delta + 1 ) } , \\end{align*}"}
+{"id": "1747.png", "formula": "\\begin{align*} s ' \\delta s ( \\mu _ { - \\underline { m } } ) \\left ( \\begin{array} { c c } \\ast & \\ast \\\\ R & S \\end{array} \\right ) = - 2 \\binom { 2 M } { k _ { \\rm i d } - 2 } \\frac { t _ M ( \\mu _ { - \\underline { m } } ) ^ \\ast \\left ( \\left | \\begin{array} { c c } \\bar S & - \\bar R \\\\ x & y \\end{array} \\right | ^ { k _ c - 1 } \\left | \\begin{array} { c c } R & S \\\\ x & y \\end{array} \\right | ^ { k _ { \\rm i d } - 1 } \\right ) } { \\Delta ^ { \\frac { \\underline { k } } { 2 } - 1 } \\cdot \\left ( \\left | R \\right | ^ 2 + \\left | S \\right | ^ 2 \\right ) } , \\end{align*}"}
+{"id": "807.png", "formula": "\\begin{align*} \\widetilde { \\nu } _ n ( R ) = \\widehat { \\nu } \\left ( R \\cap \\bigcup _ { i = 1 } ^ n A _ i \\right ) . \\end{align*}"}
+{"id": "6250.png", "formula": "\\begin{align*} A ^ F : = \\{ ( c _ k , c _ m ) \\in [ - 1 , 1 ] ^ 2 : \\min _ { ( \\sigma _ 1 , \\sigma _ 2 ) \\in \\{ \\pm \\} ^ 2 } | \\sigma _ 1 { \\omega _ { n } } + \\sigma _ 2 { \\omega _ k } + \\omega _ m | \\leq \\delta \\} . \\end{align*}"}
+{"id": "565.png", "formula": "\\begin{align*} \\Theta _ { n + 1 } = \\Theta _ n + \\frac { \\sigma _ n } { \\gamma } \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ \\dagger ) ^ { \\rm T } { \\rm d } X _ t ^ { \\dagger } - \\frac { 1 } { 2 } K _ n A X _ { t _ n } ^ \\dagger \\left ( \\Theta _ n + \\pi _ n [ \\theta ] \\right ) \\Delta t , \\end{align*}"}
+{"id": "8682.png", "formula": "\\begin{align*} F _ \\lambda = \\Gamma _ { - \\lambda _ 1 } ^ { + } \\Gamma _ { - \\lambda _ 2 } ^ { + } \\dots \\Gamma _ { - \\lambda _ l } ^ { + } ( 1 ) \\ , \\in \\Lambda [ t ] . \\end{align*}"}
+{"id": "6867.png", "formula": "\\begin{align*} ( u _ h - u , v ) _ h = e ( u , v ) \\forall v \\in X _ h \\end{align*}"}
+{"id": "5845.png", "formula": "\\begin{align*} t = 0 : \\Phi = \\widehat { \\Phi } _ 0 , \\Phi ' = \\widehat { \\Phi } _ 1 \\hbox { i n } ~ ~ \\Omega , \\end{align*}"}
+{"id": "1414.png", "formula": "\\begin{align*} M _ { \\rho , x } = \\sum _ { \\substack { z \\in ( 1 / 2 ) \\Z \\\\ z \\geq x } } ( z - x + 1 ) m _ { \\rho , z } . \\end{align*}"}
+{"id": "507.png", "formula": "\\begin{align*} \\Lambda _ { k + m + 1 } \\le M { \\textstyle \\sum _ { j = k } ^ { k + m } } \\ , \\Xi _ j ^ { \\frac { 1 - \\theta } { \\theta } } \\le M ( m \\ ! + \\ ! 1 ) ^ { \\frac { 2 \\theta - 1 } { \\theta } } \\big [ { \\textstyle \\sum _ { j = k } ^ { k + m } } \\ , \\Xi _ j \\big ] ^ { \\frac { 1 - \\theta } { \\theta } } \\le M ( m \\ ! + \\ ! 1 ) ^ { \\frac { 2 \\theta - 1 } { \\theta } } \\big ( \\Lambda _ { k } \\ ! - \\ ! \\Lambda _ { k + m + 1 } \\big ) ^ { \\frac { 1 - \\theta } { \\theta } } , \\end{align*}"}
+{"id": "7421.png", "formula": "\\begin{align*} K ( z , w ) ( a ) = H ( z ) \\pi ( a ) H ( w ) ^ * . \\end{align*}"}
+{"id": "445.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n \\in \\mathbb { M } } f _ n ( \\cdot ) \\tau _ n - \\sum _ { n = 1 } ^ { \\infty } r _ n f _ n ( \\cdot ) \\tau _ n \\right \\| \\leq c \\varepsilon ^ \\frac { 1 } { d } . \\end{align*}"}
+{"id": "1052.png", "formula": "\\begin{align*} \\| g _ 1 - g _ 2 \\| _ 2 ^ 2 = \\int _ { [ 0 , 1 ] } \\{ g _ 1 ( x ) - g _ 2 ( x ) \\} ^ 2 \\ , \\mathrm { d } x \\| g _ 1 - g _ 2 \\| _ \\infty = \\sup _ { x \\in [ 0 , 1 ] } | g _ 1 ( x ) - g _ 2 ( x ) | , \\end{align*}"}
+{"id": "2510.png", "formula": "\\begin{align*} - \\tfrac 1 a \\Delta _ g f + f = f ^ { q - 1 } . \\end{align*}"}
+{"id": "2995.png", "formula": "\\begin{align*} \\begin{array} { c c l } p ^ a ( x , f ( x , y ) ) & = & \\pi ^ a ( x , y ) - \\pi ^ { a b } ( x , y ) \\textbf { A } _ b ( x , f ( x , y ) ) \\\\ p ^ { a b } ( x , f ( x , y ) ) & = & \\pi ^ { a b } ( x , y ) \\end{array} \\end{align*}"}
+{"id": "3548.png", "formula": "\\begin{align*} z _ j ^ + & : = \\mathbf { p } B _ j ^ + ( h _ 1 - h _ j ) \\cdots ( h _ { j - 1 } - h _ j ) , ( 1 \\leq j \\leq n ) , \\\\ z _ j ^ - & : = \\mathbf { p } B _ j ^ - ( h _ j - h _ { j + 1 } ) \\cdots ( h _ j - h _ n ) , ( 1 \\leq j \\leq n ) , \\\\ z _ { i j } ^ + & : = \\mathbf { p } \\{ B _ i ^ + , B _ j ^ + \\} \\pi _ { i j } ^ + \\\\ z _ { i j } ^ - & : = \\mathbf { p } \\{ B _ i ^ - , B _ j ^ - \\} \\pi _ { i j } ^ - , \\end{align*}"}
+{"id": "1383.png", "formula": "\\begin{align*} K _ k ^ { + } ( u ) = M _ k K _ k ^ { - } ( - u + 4 \\eta + i \\pi ) | _ { \\epsilon \\rightarrow \\ , \\epsilon ' } , \\end{align*}"}
+{"id": "6788.png", "formula": "\\begin{align*} K _ { \\mathcal { U } _ \\eta } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } , ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) = ~ d ( \\mathcal { U } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) , \\overline { w } _ \\eta ) + d ( \\mathcal { U } ( ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) , \\overline { v } _ \\eta ) \\end{align*}"}
+{"id": "5326.png", "formula": "\\begin{align*} \\Phi ( X ) = \\mathrm { A d } _ { Z ^ { - 1 } } \\circ \\Psi ( X ) = \\sum _ { j = 1 } ^ k ( V _ { i _ j } Z ^ { - 1 } ) ^ * X ( V _ { i _ j } Z ^ { - 1 } ) . \\end{align*}"}
+{"id": "5771.png", "formula": "\\begin{align*} U '' + \\mathcal L U + A U + D \\mathcal G U ' = 0 , \\end{align*}"}
+{"id": "3552.png", "formula": "\\begin{align*} B _ j ^ + \\Omega _ \\lambda = \\sum _ { i = 1 } ^ { j } \\sum _ { s = 1 } ^ { j - i + 1 } \\sum _ { I \\in \\mathcal { I } _ { i j } ( s ) } d _ { i } ^ + ( \\lambda ) \\frac { \\prod _ { \\ell \\in I ^ \\complement } ( \\lambda _ i - \\lambda _ \\ell - i + \\ell + 1 ) } { \\prod _ { \\ell = i + 1 } ^ { j } ( \\lambda _ i - \\lambda _ \\ell - i + \\ell ) } E ^ { e _ I } \\Omega _ { \\lambda + \\epsilon _ i } \\end{align*}"}
+{"id": "7036.png", "formula": "\\begin{align*} F _ n \\widehat { E } ^ { C _ 2 } _ * ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) = E ^ { C _ 2 } _ { * + | n \\sigma | - n \\sigma } ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) \\end{align*}"}
+{"id": "3056.png", "formula": "\\begin{align*} A ( y ) & : = \\frac { 1 } { r ( y ) } \\ , \\mathrm { d i a g } \\left ( 1 - \\frac { 1 } { 2 } \\sin ( 2 \\pi y _ 1 ) \\sin ( 2 \\pi y _ 2 ) , 1 + \\frac { 1 } { 2 } \\sin ( 2 \\pi y _ 1 ) \\sin ( 2 \\pi y _ 2 ) \\right ) , \\\\ r ( y ) & : = 1 + \\frac { 1 } { 4 } ( \\cos ( 2 \\pi y _ 1 ) - 2 \\sin ( 2 \\pi y _ 1 ) ) \\sin ( 2 \\pi y _ 2 ) \\end{align*}"}
+{"id": "840.png", "formula": "\\begin{align*} A _ d : = \\sum _ { k = 0 } ^ { N - 1 } L _ d ( q _ k , q _ { k + 1 } ) \\end{align*}"}
+{"id": "2286.png", "formula": "\\begin{align*} u _ t = ( u ^ m ) _ { x x } , ~ ~ m > 1 , \\end{align*}"}
+{"id": "2577.png", "formula": "\\begin{align*} U ( 1 ) _ \\alpha ^ \\top & = \\{ \\epsilon \\in U ( 1 ) \\mid \\mathfrak { g } _ { \\alpha , \\epsilon } \\neq \\{ 0 \\} , \\ \\langle \\alpha , w \\rangle + \\arg \\epsilon \\notin 2 \\pi \\mathbb { Z } \\} , \\\\ U ( 1 ) _ \\alpha ^ \\perp & = \\{ \\epsilon \\in U ( 1 ) \\mid \\mathfrak { g } _ { \\alpha , \\epsilon } \\neq \\{ 0 \\} , \\ \\langle \\alpha , w \\rangle + \\arg \\epsilon \\in 2 \\pi \\mathbb { Z } \\} . \\end{align*}"}
+{"id": "773.png", "formula": "\\begin{align*} d _ S ( g , h ) = | g ^ { - 1 } h | _ S \\ \\ \\ \\ d _ R ( g , h ) : = | g h ^ { - 1 } | _ S . \\end{align*}"}
+{"id": "415.png", "formula": "\\begin{align*} f _ n ( V ^ { - 1 } x ) = f _ n ( U x ) = f _ n \\left ( \\sum _ { m = 1 } ^ { \\infty } g _ m ( x ) \\tau _ m \\right ) = \\sum _ { m = 1 } ^ { \\infty } g _ m ( x ) f _ n ( \\tau _ m ) = g _ n ( x ) , \\forall n \\in \\mathbb { N } , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "9196.png", "formula": "\\begin{align*} \\norm { \\nabla _ { j + s } ( \\varphi + \\xi _ o ) } ^ 2 _ { L ^ 2 _ { j + s } ( X ) } & \\leq \\norm { \\nabla _ { j + s ' } \\varphi } ^ 2 _ { L ^ 2 _ { j + s ' } ( X ) } + \\norm { \\nabla _ { j + s } \\xi _ o } ^ 2 _ { L ^ 2 _ { j + s } ( X ) } \\\\ & + 2 \\tau W _ { j + s } ( X , \\xi _ o ) ^ 2 + \\tau ^ { - 1 } \\big ( \\norm { \\nabla _ { j + s } \\varphi } ^ 2 _ { L ^ 2 _ { j + s } ( \\partial X ) } + W _ { j + s } ( X , \\nabla ^ 2 _ { j + s } \\varphi ) ^ 2 \\big ) . \\end{align*}"}
+{"id": "2143.png", "formula": "\\begin{align*} e ^ { - t } \\sum _ { i = 1 } ^ { + \\infty } \\frac { t ^ i } { i ! } R _ i \\left ( ( 1 - m ) L \\right ) = e ^ { - t } \\sum _ { i = M ( t ) } ^ { N ( t ) } \\frac { t ^ i } { i ! } R _ i \\left ( ( 1 - m ) L \\right ) + o \\left ( e ^ { - r ^ - t } \\right ) , t \\to + \\infty , \\end{align*}"}
+{"id": "8411.png", "formula": "\\begin{align*} ( I I ) & = \\int P _ { \\alpha } ^ k ( \\partial _ { \\alpha } [ \\hat \\Psi _ { \\alpha } h _ { \\alpha } ] ) \\hat \\Psi _ { \\alpha } \\ : d m + \\int \\partial _ { \\alpha } [ P _ { \\alpha } ^ k ] ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\hat \\Psi _ { \\alpha } \\ : d m \\\\ & = ( I I a ) + ( I I b ) \\end{align*}"}
+{"id": "8826.png", "formula": "\\begin{align*} P ( D _ Q ^ 2 ) ^ n = ( D _ Q ^ 2 ) ^ n P n = 0 , 1 , 2 , \\dots \\ ; . \\end{align*}"}
+{"id": "1088.png", "formula": "\\begin{align*} \\mathbb { E } _ { P _ k } [ | X | ^ k ] ^ { 1 / k } = \\mathbb { E } _ { P _ k } [ | X - \\mu + \\mu | ^ k ] ^ { 1 / k } \\leq 2 ( \\mathbb { E } _ { P _ k } | X - \\mu | ^ k + | \\mu | ^ k ) ^ { 1 / k } \\leq 2 ( 1 + D ) \\end{align*}"}
+{"id": "1669.png", "formula": "\\begin{align*} \\prod _ { v \\nmid \\infty } \\left ( \\beta _ v ( \\Phi _ { 1 , v } , \\Phi _ { 2 , v } ) \\cdot \\alpha _ { v } ( W _ { \\mathfrak { f } , v } , W ^ - _ { \\mathfrak { f } , v } ) \\right ) = \\frac { { \\rm v o l } ( U _ 0 ( N ) ) } { { \\rm v o l } ( U ) } | c _ 1 ^ 2 D | ^ { - \\frac { 1 } { 2 } } 2 ^ { \\# S _ D } L _ { c _ 1 } ( 1 , \\eta _ { T } ) ^ 2 \\prod _ { v \\in S } L ( 1 / 2 , \\Pi _ v , \\chi _ v ) ^ { - 1 } , \\end{align*}"}
+{"id": "3665.png", "formula": "\\begin{align*} A u + \\lambda & = f , \\\\ M ( \\lambda , u - g ) & = 0 . \\end{align*}"}
+{"id": "8639.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d r } \\sqrt { f ^ 2 + g ^ 2 } \\right ) ^ 2 + \\frac { ( f g ' - g f ' ) ^ 2 } { f ^ 2 + g ^ 2 } = ( f ' ) ^ 2 + ( g ' ) ^ 2 \\end{align*}"}
+{"id": "2910.png", "formula": "\\begin{align*} \\sum _ { n = k _ { s _ j } + 1 } ^ { k _ { s _ { j + 1 } } } | a _ n | \\leq \\frac { 6 A } { \\pi ^ 2 } \\sum _ { n = s _ j + 1 } ^ { s _ { j + 2 } } \\frac { 1 } { n ^ 2 } . \\end{align*}"}
+{"id": "8695.png", "formula": "\\begin{align*} \\tilde s _ \\lambda ( x _ 1 , \\dots , x _ n ) = \\frac { \\det [ f _ { \\lambda _ j } ( x _ i ) x _ i ^ { n - j } ] } { \\det [ x _ i ^ { n - j } ] } , \\end{align*}"}
+{"id": "6197.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } W '' - \\Delta W + \\overline A _ p W = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ W = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu W + \\overline B _ p W = C _ p D H & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "1034.png", "formula": "\\begin{align*} \\omega ( \\varepsilon ) = \\sup \\{ \\rho ( \\theta ( R _ 0 ) , \\theta ( R _ 1 ) ) : \\ , \\mathrm { T V } ( R _ 0 , R _ 1 ) \\leq \\varepsilon / ( 1 - \\varepsilon ) , R _ 0 , R _ 1 \\in \\mathcal { P } \\} . \\end{align*}"}
+{"id": "1500.png", "formula": "\\begin{align*} ~ \\zeta ( s ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { s } } \\end{align*}"}
+{"id": "9031.png", "formula": "\\begin{align*} \\overline { V } _ t = \\overline { \\Psi } ( \\overline { V } ) _ t . \\end{align*}"}
+{"id": "9043.png", "formula": "\\begin{align*} \\overline { V } _ \\cdot ^ { ( m ) , i , n } = \\overline { \\Psi } _ \\cdot ^ { i } ( \\overline { \\textbf { V } } ^ { ( m - 1 ) , n } ) = \\underline { \\Psi } _ \\cdot ^ { i } ( \\overline { \\textbf { V } } ^ { ( m - 1 ) , n } ) = \\underline { \\Psi } _ \\cdot ^ { i } ( \\underline { \\textbf { V } } ^ { ( m - 1 ) , n } ) = \\underline { V } _ \\cdot ^ { ( m ) , i , n } . \\end{align*}"}
+{"id": "8462.png", "formula": "\\begin{align*} c _ * ( A ) & = \\sup _ { \\nu \\in \\mathcal E ' _ A } \\ , G ( \\nu ) , \\\\ c _ * ( A ) & = \\sup _ { \\nu \\in \\widehat { \\mathcal E } ' _ A } \\ , \\nu ( X ) . \\end{align*}"}
+{"id": "3615.png", "formula": "\\begin{align*} \\int R ^ 2 ( R ' ) ^ 2 = \\frac { L ^ 2 } { R _ * ^ 3 } \\left ( 7 \\int R - \\beta \\int R ^ 2 + \\int \\frac { \\alpha } { \\beta } \\right ) \\end{align*}"}
+{"id": "6736.png", "formula": "\\begin{align*} \\Gamma ( s ) = \\int _ { 0 } ^ { \\infty } y ^ { s - 1 } \\exp \\left ( - y \\right ) d y \\end{align*}"}
+{"id": "1951.png", "formula": "\\begin{align*} \\begin{gathered} \\varrho _ { E } ^ { ( N ) } [ u , X ] \\leq { \\varrho } _ { w } ^ { ( N , J ) } [ u , X ] \\ ; \\ ; u \\in U , \\\\ \\vartheta _ { E } ^ { ( N ) } \\leq \\vartheta _ { w } ^ { ( N , J ) } \\end{gathered} \\end{align*}"}
+{"id": "8119.png", "formula": "\\begin{align*} u _ 2 ' ( y ) = p ^ { \\frac { 1 } { 2 } } c ^ { - 1 } y ^ { - \\frac { 1 } { 2 } } - x ^ { \\frac { 1 } { 3 } } y ^ { - \\frac { 2 } { 3 } } , \\end{align*}"}
+{"id": "2536.png", "formula": "\\begin{align*} \\big ( \\bar T _ N ^ \\varphi - \\tilde T _ N ^ \\varphi \\big ) ^ 2 \\le \\bigg ( \\sum _ { j = 1 } ^ N ( p _ { 1 , j } + q _ { 1 , j } ) | w _ j - \\tilde u _ N ^ \\varphi | ^ 2 \\bigg ) ^ 2 \\le C \\sum _ { j = 1 } ^ N ( p _ { 1 , j } + q _ { 1 , j } ) | w _ j | ^ 4 + C | \\tilde u _ N ^ \\varphi \\big | ^ 4 \\ , , \\end{align*}"}
+{"id": "9182.png", "formula": "\\begin{align*} G _ j ^ { \\Psi } ( X , \\varphi ; \\ , u _ j ) = \\sup _ { t \\in [ 0 , 1 ] } \\exp \\Big ( \\kappa _ L \\norm { \\nabla _ j ( \\varphi + t u _ j ) } ^ 2 _ { L ^ 2 _ j ( X ) } + c _ 2 \\kappa _ L \\norm { \\nabla _ j ( \\varphi + t u _ j ) } ^ 2 _ { L ^ 2 _ j ( \\partial X ) } + \\kappa _ L W _ j ^ { \\Psi } ( X , \\varphi ; \\ , u _ j ) ^ 2 \\Big ) \\end{align*}"}
+{"id": "8095.png", "formula": "\\begin{align*} H _ { m , n } ^ + ( x ) = 2 i \\int _ 0 ^ \\infty \\frac { J _ { 2 i t } ( x ) - J _ { - 2 i t } ( x ) } { \\cosh ( \\pi t ) } k ( t ) V ( m ^ 2 n , t ) t \\ , d t , \\end{align*}"}
+{"id": "4177.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde { g } ^ s _ i ( t ) = \\widetilde { g } _ i ( t + t _ i ) \\\\ \\widetilde { b } ^ s _ i ( t ) = \\widetilde { b } _ i ( t + t _ i ) \\end{cases} t \\in [ T _ i , 0 ] . \\end{align*}"}
+{"id": "1455.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\binom { x _ i } { 2 } = \\sum _ { j = 1 } ^ m \\binom { y _ j } { 2 } = \\frac { k ( k - 1 ) } { 2 ( m + 1 ) } . \\end{align*}"}
+{"id": "2963.png", "formula": "\\begin{align*} \\mathcal { S } ^ n _ t ( \\varphi ) = \\frac { 1 } { 2 \\sqrt { n } } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } ( g _ n ( \\eta ^ n _ j ( s ) ) - \\Phi _ n ( \\rho ) ) \\Delta ^ n \\varphi ^ n _ j ( s ) d s \\end{align*}"}
+{"id": "633.png", "formula": "\\begin{align*} \\beta _ { i } ( ( z I + ( w ) ) ^ h ) = \\beta _ { i } ( I ^ h ) + \\beta _ { i - 1 } ( I ^ h ) + \\beta _ { i } ( w ( z I + ( w ) ) ^ { h - 1 } ) . \\end{align*}"}
+{"id": "2678.png", "formula": "\\begin{align*} r ( y + \\varepsilon ) = r ( y ) + r ( \\varepsilon ) \\forall y , \\varepsilon \\in ( - \\infty , 0 ] . \\end{align*}"}
+{"id": "486.png", "formula": "\\begin{align*} H \\ge 0 \\mbox { a n d } H ( x , y , 0 ) = 0 \\mbox { a . e . } x , y \\in \\mathbb { R } ^ N \\times \\mathbb { R } ^ N \\end{align*}"}
+{"id": "3454.png", "formula": "\\begin{align*} G _ t ( x ) : = \\begin{cases} \\dfrac { 1 } { 2 } \\mathbf { 1 } _ { \\{ | x | < t \\} } & ; \\\\ \\dfrac { 1 } { 2 \\pi \\sqrt { t ^ 2 - | x | ^ 2 } } \\mathbf { 1 } _ { \\{ | x | < t \\} } & , \\end{cases} \\end{align*}"}
+{"id": "1020.png", "formula": "\\begin{align*} \\left ( \\int _ { \\mathbb { R } ^ n } \\frac { | f ^ { \\# } ( x ) | ^ p } { | x | ^ { p \\beta } } d x \\right ) ^ { 1 / p } = C \\| f \\| _ { L ^ { \\frac { n p } { n - p \\beta } , p } } , \\end{align*}"}
+{"id": "3193.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\int _ 0 ^ 1 a ( y _ 1 , y _ 2 ) R _ 1 ' ( y _ 1 + y _ 2 ) \\ , \\mathrm { d } y _ 1 \\ , \\mathrm { d } y _ 2 = \\int _ 0 ^ 1 \\int _ 0 ^ 1 a ( y _ 1 , y _ 2 ) R _ 2 ' ( y _ 1 - y _ 2 ) \\ , \\mathrm { d } y _ 1 \\ , \\mathrm { d } y _ 2 = 0 , \\end{align*}"}
+{"id": "513.png", "formula": "\\begin{align*} { \\textstyle \\sum _ { K _ { 3 2 } \\ni j = k } ^ { \\nu } } \\ , \\| x ^ { j + 1 } \\ ! - \\ ! x ^ j \\| \\le \\widehat { c } _ 2 ( \\theta ) { \\textstyle \\sum _ { K _ { 3 2 } \\ni j = k } ^ { \\nu } } \\ , \\Xi _ { j + 1 } \\ \\ { \\rm w i t h } \\ \\ \\widehat { c } _ 2 ( \\theta ) : = b c ( 1 \\ ! - \\ ! \\theta ) ( 4 / a ) ^ { \\theta } . \\end{align*}"}
+{"id": "6495.png", "formula": "\\begin{align*} \\tilde { X } ^ { j } _ { 0 : L } = \\tilde { f } ^ L + \\sqrt { \\frac { m } { n } } Z ^ { j } , \\end{align*}"}
+{"id": "6274.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ ( x , y , z ) \\in \\mathcal { M } : x = 0 , z = \\sqrt { r ( y ) } \\} . \\end{align*}"}
+{"id": "1883.png", "formula": "\\begin{align*} \\chi = \\begin{bmatrix} 1 & 0 & \\cdots & 0 & 1 & 0 & \\cdots & 0 \\\\ 0 & 1 & \\cdots & 0 & 1 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & 0 \\\\ 0 & 0 & \\cdots & 1 & 1 & 0 & \\cdots & 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "9099.png", "formula": "\\begin{align*} Y _ { k , i } = \\sqrt { h _ k } X _ { i } + Z _ { k , i } , \\end{align*}"}
+{"id": "3758.png", "formula": "\\begin{align*} & c _ 2 ( f ) + c _ 1 ( d _ { 2 } ( f ) ) = \\tilde { c } _ { 2 1 } ( I ( f ) ) . \\end{align*}"}
+{"id": "709.png", "formula": "\\begin{align*} \\eta = - A ^ T d U _ \\beta + B ^ T d U _ \\alpha = \\left ( \\begin{array} { c c } B ^ T , & - A ^ T \\end{array} \\right ) \\left ( \\begin{array} { c } d U _ \\alpha \\\\ d U _ \\beta \\end{array} \\right ) , \\end{align*}"}
+{"id": "3643.png", "formula": "\\begin{align*} a ^ { \\prime } ( t ) & = h ( t ) \\ , a ( t ) \\ , ( e ^ { \\beta ( C a ( t ) ) ^ p } - 1 ) . \\end{align*}"}
+{"id": "5385.png", "formula": "\\begin{align*} & \\mathcal { R } ' : = \\bigg \\{ ( \\mathbf { x } ' , \\mathbf { y } ' ) \\in \\mathbb { R } ^ { M - M _ 2 } \\times \\mathbb { R } ^ { M - M _ 2 } : \\\\ & - C ( j + M _ 2 + 1 ) + 3 \\log M _ 2 \\le \\sum _ { m ' = 1 } ^ j x _ { m ' } ' , \\sum _ { m ' = 1 } ^ j y _ { m ' } ' \\le C ( M _ 2 + 1 ) - 3 \\log ( j + M _ 2 ) \\ ; \\forall \\ ; 1 \\le j \\le M - M _ 2 \\bigg \\} . \\end{align*}"}
+{"id": "7913.png", "formula": "\\begin{align*} & \\P \\left ( \\exists s \\geq 0 , u \\in \\mathcal { N } _ s : X _ u ( s ) \\geq \\sqrt { 2 } s + K \\right ) \\\\ & \\leq C ( K + 1 ) e ^ { - \\sqrt { 2 } K } \\sum _ { l = 1 } ^ { \\infty } \\frac { 1 } { ( l + 1 ) ^ { 3 / 2 } } \\leq C ( K + 1 ) e ^ { - \\sqrt { 2 } K } . \\end{align*}"}
+{"id": "7892.png", "formula": "\\begin{align*} \\deg ( Q ) + 2 < 2 \\deg ( P ) , & \\ \\deg ( P ) \\geq 2 , \\\\ Q ( z ) \\equiv 0 , & \\ \\deg ( P ) = 1 , \\end{align*}"}
+{"id": "4416.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 1 7 } < \\frac { 1 } { 4 } + \\frac { 1 } { 7 } = \\frac { 1 1 } { 2 8 } < \\theta \\leq \\frac { 1 9 } { 4 8 } \\end{align*}"}
+{"id": "8445.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\ , c ( K _ j ) = c _ * ( A ) , \\end{align*}"}
+{"id": "4366.png", "formula": "\\begin{align*} \\frac { 1 } { a _ { i + 1 } } & < \\theta - \\sum _ { j = 1 } ^ i \\frac { 1 } { a _ j } = \\left ( \\theta - \\sum _ { j = 1 } ^ { i - 1 } \\frac { 1 } { a _ j } \\right ) - \\frac { 1 } { a _ { i } } \\\\ & \\leq \\frac { 1 } { a _ { i } - 1 } - \\frac { 1 } { a _ { i } } = \\frac { 1 } { a _ { i } ( a _ { i } - 1 ) } \\end{align*}"}
+{"id": "2131.png", "formula": "\\begin{align*} \\partial _ t u = J * u - u \\end{align*}"}
+{"id": "8170.png", "formula": "\\begin{align*} \\sum _ { m \\geq 1 } & \\frac { A ( 1 , m ) } { m } U ( m ^ 2 p , t ) \\\\ & = 3 L ( 1 , \\tilde f ) \\log \\abs { t } + 2 L ' ( 1 , \\tilde f ) - 3 L ( 1 , \\tilde f ) \\log ( 2 \\pi ) - L ( 1 , \\tilde f ) \\log p + { } \\\\ & \\qquad { } + O ( \\abs { t } ^ { - 1 } p ^ { \\varepsilon } ) + O ( p ^ { \\frac { 1 } { 7 } - \\varepsilon } \\abs { t } ^ { - \\frac { 6 } { 7 } + \\varepsilon } ) \\end{align*}"}
+{"id": "8277.png", "formula": "\\begin{align*} \\begin{cases} \\dot x = x ( \\beta x - f y - z + g ) , \\\\ \\dot y = y ( x + s z - \\alpha ) , \\\\ \\dot z = ( x - \\alpha z ^ 3 + b z ^ 2 - c z ) / \\varepsilon , \\end{cases} \\end{align*}"}
+{"id": "2301.png", "formula": "\\begin{align*} \\hat { g } ^ \\C _ { i - 1 / 2 } = g _ { i - 1 / 2 } + \\frac { 2 3 } { 3 6 0 } h _ i ^ { ( 5 ) } \\Delta x ^ 5 + O ( \\Delta x ^ 6 ) . \\end{align*}"}
+{"id": "6389.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { a - 1 } \\bigl ( A _ k f _ 1 ^ { ( 2 a - 1 - k ) } f _ { 1 2 } ^ { ( k + 1 ) } f _ 2 ^ { ( 2 a - 1 - k ) } + B _ k f _ 1 ^ { ( 2 a - 2 - k ) } f _ { 1 2 } ^ { ( k + 2 ) } f _ 2 ^ { ( 2 a - 2 - k ) } + C _ k f _ 1 ^ { ( 2 a - k ) } f _ { 1 2 } ^ { ( k ) } f _ 2 ^ { ( 2 a - k ) } \\bigr ) , \\end{align*}"}
+{"id": "1325.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | \\geq \\frac { \\frac { 1 } { d } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ 2 } { n ^ 2 - n } . \\end{align*}"}
+{"id": "6240.png", "formula": "\\begin{align*} \\begin{cases} C _ p ^ T ( \\Psi '' - \\Delta \\Psi + \\overline A _ p ^ T \\Psi ) = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ C _ p ^ T \\Psi = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 \\\\ C _ p ^ T ( \\partial _ \\nu \\Psi + \\overline B _ p ^ T \\Psi ) = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 . \\end{cases} \\end{align*}"}
+{"id": "5259.png", "formula": "\\begin{align*} g ( U , U ) \\frac { d f } { d t } = - a \\cos \\omega \\sin \\omega \\frac { d \\omega } { d t } . \\end{align*}"}
+{"id": "8400.png", "formula": "\\begin{align*} \\partial _ { \\alpha } ( h _ { \\alpha } \\circ F _ { \\alpha , ( \\underline { n } , n _ 0 ) } ^ { - 1 } ) = ( \\partial _ { \\alpha } h _ { \\alpha } ) \\circ F _ { \\alpha , ( \\underline { n } , n _ 0 ) } ^ { - 1 } + h ' _ { \\alpha } \\circ F _ { \\alpha , ( \\underline { n } , n _ 0 ) } ^ { - 1 } \\partial _ { \\alpha } F _ { \\alpha , ( \\underline { n } , n _ 0 ) } ^ { - 1 } \\end{align*}"}
+{"id": "715.png", "formula": "\\begin{align*} r _ { \\rm r o b i n } ( w ) \\ = - 2 h _ 1 ( w ) = - 2 \\frac { \\partial } { \\partial w } h _ 0 ( w ) , \\end{align*}"}
+{"id": "6695.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\lambda ( n ) \\lambda ( n + a ) = O \\left ( \\frac { x } { ( \\log x ) ^ { c } } \\right ) . \\end{align*}"}
+{"id": "8979.png", "formula": "\\begin{align*} \\frac { d } { d t } \\big ( \\| \\partial ^ 2 _ { \\phi } & u \\varphi _ 2 \\| ^ 2 _ { L ^ 2 ( \\partial B ) } \\big ) + \\frac 1 2 \\| \\nabla \\partial ^ 2 _ { \\phi } u \\varphi _ 2 \\| ^ 2 _ { L ^ 2 ( B ) } \\\\ & \\le C ( 1 + \\| \\nabla \\partial _ { \\phi } u \\varphi _ 2 \\| ^ 2 _ { L ^ 2 ( B ) } ) \\| \\nabla \\partial _ { \\phi } u \\varphi _ 1 \\| ^ 2 _ { L ^ 2 ( B ) } + C . \\end{align*}"}
+{"id": "8687.png", "formula": "\\begin{align*} \\tilde { \\tilde { F } } _ \\lambda ( x _ 1 , \\dots , x _ n ; t ) = \\frac { ( 1 - t ) ^ n } { \\prod _ { i = 1 } ^ { n - l } ( 1 - t ^ i ) } \\sum _ { \\sigma \\in S _ n } \\sigma \\left ( f _ { \\lambda _ 1 } ( x _ { 1 } ) \\dots f _ { \\lambda _ n } ( x _ { n } ) \\prod _ { i = 1 } ^ { n } \\prod _ { i < j } \\frac { x _ { i } - t x _ { j } } { x _ { i } - x _ { j } } \\right ) , \\end{align*}"}
+{"id": "5078.png", "formula": "\\begin{align*} [ 1 , k ] = [ k _ 1 , k _ 2 - 1 ] \\uplus [ k _ 2 , k _ 3 - 1 ] \\uplus \\cdots \\uplus [ k _ { p } , k _ { p + 1 } - 1 ] , \\end{align*}"}
+{"id": "3908.png", "formula": "\\begin{align*} \\mu ( A ) + \\mu ( B ) & \\leq \\int _ { Y u ( A _ \\delta ) \\setminus \\mathcal { Z } } f ^ * + \\int _ { Y u ( B _ \\delta ) \\setminus \\mathcal { Z } } f ^ * \\\\ & = \\int _ { Y u ( A _ \\delta ) \\cup Y u ( B _ \\delta ) } f ^ * \\leq \\int _ { Y u ( C ) } f ^ * \\leq \\mu ( A \\cup B ) + \\epsilon . \\end{align*}"}
+{"id": "5318.png", "formula": "\\begin{align*} \\Phi = \\mathrm { A d } _ S \\circ \\Psi + \\mathrm { A d } _ { T _ { m + 1 } } \\circ \\Phi _ { m + 1 } . \\end{align*}"}
+{"id": "3583.png", "formula": "\\begin{align*} E _ { 3 1 } \\Omega _ { \\lambda - \\epsilon _ 3 } = E ^ { ( \\lambda _ 1 - 1 ) e _ { 1 1 } + \\lambda _ 2 e _ { 2 2 } + ( \\lambda _ 3 - 1 ) e _ { 3 3 } + e _ { 3 1 } } \\Omega _ { \\lambda - \\epsilon _ 3 } \\end{align*}"}
+{"id": "626.png", "formula": "\\begin{align*} I ( G ) = & \\ ( x _ 1 x _ 2 , x _ 2 x _ 3 , x _ 2 x _ 4 , x _ 2 x _ 5 , x _ 2 x _ 6 , x _ 2 x _ { 1 2 } , x _ 1 x _ 4 , x _ 1 x _ 6 , x _ 1 x _ 7 , x _ 1 x _ 8 , x _ 2 x _ { 1 2 } , x _ 3 x _ 5 , \\\\ & \\ \\ x _ 3 x _ 8 , x _ 3 x _ { 1 1 } , x _ 3 x _ { 1 2 } , x _ 4 x _ 5 , x _ 4 x _ 9 , x _ 4 x _ { 1 0 } , x _ 5 x _ 7 , x _ 5 x _ 9 , x _ 6 x _ 7 , x _ 6 x _ { 1 0 } , x _ 6 x _ { 1 1 } , x _ 7 x _ 8 , \\\\ & \\ \\ x _ 7 x _ 9 , x _ 7 x _ { 1 2 } , x _ 8 x _ { 1 1 } , x _ 9 x _ { 1 0 } , x _ 9 x _ { 1 2 } , x _ { 1 0 } x _ { 1 1 } , x _ { 1 0 } x _ { 1 2 } , x _ { 1 1 } x _ { 1 2 } ) . \\end{align*}"}
+{"id": "483.png", "formula": "\\begin{align*} T _ N ( s ) : = \\begin{cases} - N & \\mbox { i f } s < - N , \\\\ s & \\mbox { i f } - N \\leq s \\leq N , \\\\ N & \\mbox { i f } s > N . \\end{cases} \\end{align*}"}
+{"id": "8762.png", "formula": "\\begin{align*} G _ l & = [ C _ 0 A _ 0 ^ { 2 T - 1 + l } B _ 0 , C _ 0 A _ 0 ^ { 2 T + l } B _ 0 , \\dots , C _ 0 A _ 0 ^ { 2 T - 1 } B _ 0 , 0 , \\dots , 0 ] , \\end{align*}"}
+{"id": "6388.png", "formula": "\\begin{align*} K = \\begin{cases} \\lfloor { q / 4 } \\rfloor & q \\equiv 1 \\pmod 4 , \\\\ \\lfloor { q / 2 } \\rfloor & . \\end{cases} \\end{align*}"}
+{"id": "2174.png", "formula": "\\begin{align*} \\phi ( r ) : & = ( 2 ( \\sqrt { 2 } - 1 ) - ( 2 \\alpha - 1 + \\beta ) ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + 3 - 2 \\sqrt { 2 } , \\intertext { a n d } \\psi ( r ) : & = ( 2 \\alpha - 3 + \\beta ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + 1 \\end{align*}"}
+{"id": "5978.png", "formula": "\\begin{align*} \\dim \\mathcal { W } _ 0 - \\dim ( h \\times f ) ( \\mathcal { W } _ 0 ) & = \\dim { ( h \\times f ) } ^ { - 1 } ( \\nu ) \\cap \\mathcal { W } _ 0 ^ \\circ \\\\ & = \\mathrm { d i m } M - \\mathrm { d i m } ( h \\times f ) ( M ) \\end{align*}"}
+{"id": "7230.png", "formula": "\\begin{align*} W _ { i j } & = \\begin{cases} k I _ q & i = j \\\\ C H _ { i j } R & i \\ne j . \\end{cases} \\end{align*}"}
+{"id": "5305.png", "formula": "\\begin{align*} \\mathrm { I I } ( X , Y ) = - g ( A _ { \\eta } X , Y ) \\xi - g ( A _ { \\xi } X , Y ) \\eta , \\end{align*}"}
+{"id": "6781.png", "formula": "\\begin{align*} \\phi _ { \\chi } ^ o ( s ) = \\int _ { \\frac { 1 } { P } } ^ { \\infty } y ^ { \\frac { s } { 2 } - 1 } \\tilde { \\Psi } _ { \\chi } ( y ) d y + \\frac { - j P ^ { 1 - s } } { G \\left ( \\overline { \\chi } , P \\right ) } \\int _ { \\frac { 1 } { P } } ^ { \\infty } y ^ { \\frac { 1 - s } { 2 } - 1 } \\tilde { \\Psi } _ { \\overline { \\chi } } ( y ) d y \\end{align*}"}
+{"id": "2546.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { \\begin{subarray} { c } 0 { \\le } m _ 1 < _ { c _ 1 } \\cdots < _ { c _ { p - 1 } } m _ p < \\infty \\\\ m _ i \\in \\mathbb { Z } \\end{subarray} } \\frac { ( \\alpha ) _ { m _ 1 } } { { m _ 1 } ! } \\frac { { m _ p } ! } { ( \\alpha ) _ { m _ p } } \\left \\{ \\prod _ { i = 1 } ^ { p } \\frac { 1 } { ( m _ i + \\alpha ) ^ { a _ i } ( m _ i + \\beta ) ^ { b _ i } } \\right \\} , \\end{aligned} \\end{align*}"}
+{"id": "5415.png", "formula": "\\begin{align*} \\delta ( q _ { i _ 1 , ( i _ 1 + k _ 1 ) \\bmod p , \\ , k _ 1 } , a ) = q _ { i _ 2 , ( i _ 2 + k _ 2 ) \\bmod p , \\ , k _ 2 } \\end{align*}"}
+{"id": "8614.png", "formula": "\\begin{align*} x _ { n + 1 } = \\alpha _ n x _ n + ( 1 - \\alpha _ n ) T x _ n \\end{align*}"}
+{"id": "3505.png", "formula": "\\begin{align*} E ^ { \\gamma _ A } \\Omega _ { \\lambda _ A } = \\frac { ( \\lambda _ A ) _ 1 ! \\dots ( \\lambda _ A ) _ n ! } { ( \\gamma _ A ) _ { 1 1 } ! \\cdots ( \\gamma _ A ) _ { n n } ! } \\Omega _ A , \\end{align*}"}
+{"id": "2025.png", "formula": "\\begin{align*} \\nu ^ * ( m ) : = \\left \\{ \\begin{array} { l l l } \\mu ' ] - \\infty , m - 1 ] & { \\rm i f } & m \\leq 0 \\\\ \\mu [ m , + \\infty [ & { \\rm i f } & m \\geq 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "8684.png", "formula": "\\begin{align*} F _ \\lambda = F _ \\lambda ( x _ 1 , \\dots , x _ n ; t ) = \\frac { ( 1 - t ) ^ n } { \\prod _ { i = 1 } ^ { n - l } ( 1 - t ^ i ) } \\sum _ { \\sigma \\in S _ n } \\sigma \\left ( x _ { 1 } ^ { \\lambda _ 1 } \\dots x _ { n } ^ { \\lambda _ n } \\prod _ { i = 1 } ^ { n } \\prod _ { i < j } \\frac { x _ { i } - t x _ { j } } { x _ { i } - x _ { j } } \\right ) . \\end{align*}"}
+{"id": "5900.png", "formula": "\\begin{align*} t \\geq T : \\widetilde C _ { p - 1 } U = 0 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "6547.png", "formula": "\\begin{align*} \\varepsilon \\left ( { { M } } \\right ) \\buildrel \\Delta \\over = { F _ Y } ( \\infty ) - { { \\hat F } _ Y } ( \\infty ) . \\end{align*}"}
+{"id": "6282.png", "formula": "\\begin{align*} p ^ \\nu _ K = \\left \\{ \\begin{array} { l c l } p ^ m _ K , \\ \\ \\ \\ \\ \\ \\ \\nu = m \\\\ p ^ { m + } _ K , \\ \\ \\ \\nu = m + , \\\\ p ^ \\infty _ { K , m } , \\ \\ \\ \\ \\nu = \\infty , \\\\ p ^ \\omega _ { K , \\boldsymbol { a } } , \\ \\ \\ \\ \\ \\nu = \\omega , \\\\ p ^ { } _ K , \\ \\ \\ \\ \\ \\ \\nu = \\end{array} \\right . \\end{align*}"}
+{"id": "7996.png", "formula": "\\begin{align*} D _ N \\left ( \\varepsilon \\right ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N - 1 } \\left | \\int _ { 0 } ^ { \\frac { i } { N } } Q _ N \\left ( x , \\varepsilon \\right ) d x \\right | . \\end{align*}"}
+{"id": "7675.png", "formula": "\\begin{align*} [ \\bigoplus _ { I } \\sum _ { p } ( p + \\iota _ { E } ( \\omega ) ) \\eta _ { I , p } ] = L _ { \\omega } [ \\bigoplus _ { I } \\eta _ { I } ] = \\nabla _ { \\omega } ( \\iota _ { E } ( [ \\bigoplus _ { I } \\eta _ { I } ] ) ) = \\nabla _ { \\omega } [ \\iota _ { E } ( \\bigoplus _ { I } \\eta _ { I } ) ] . \\end{align*}"}
+{"id": "187.png", "formula": "\\begin{align*} \\mu ( [ w ] ) : = \\lim _ { n \\rightarrow \\infty } \\frac { | \\{ i \\ : \\ B _ n ( i ) \\ldots B _ n ( i + \\ell - 1 ) = w \\} | } { | B _ n | } \\end{align*}"}
+{"id": "1303.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j , k \\leq n } | \\langle \\tau _ j , \\tau _ k \\rangle | ^ { 2 m } = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n | \\langle \\tau _ j , \\tau _ k \\rangle | ^ { 2 m } \\geq \\frac { n ^ 2 } { { d + m - 1 \\choose m } } , \\forall m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "5672.png", "formula": "\\begin{align*} \\mathsf { h } _ { \\nu } ( x ) = \\frac { x } { \\frac { 2 \\nu - 1 } { 2 } + \\sqrt { \\frac { ( 2 \\nu - 1 ) ^ 2 } { 4 } + x ^ 2 } } \\cdot g _ \\nu ( x ) , \\end{align*}"}
+{"id": "2042.png", "formula": "\\begin{align*} \\displaystyle \\int _ { - \\epsilon } ^ { \\epsilon } \\left ( \\dfrac { 1 } { ( 1 - \\mathbb E [ e ^ { i t S _ { \\ell _ + } } ] ) \\ , ( 1 - \\mathbb E [ e ^ { i t S _ { \\ell _ - ' } ' } ] ) } \\right ) \\ , d t = + \\infty , \\end{align*}"}
+{"id": "4752.png", "formula": "\\begin{align*} x u = \\sum _ { i \\leq k } y ^ { f _ i } \\gamma _ i ( \\alpha _ i + y ^ { e _ i } s _ i ) = \\sum _ { i \\leq k } y ^ { f _ i } \\gamma _ i \\alpha _ i + \\sum _ { i \\leq k } y ^ { e _ k - e _ i } y ^ { e _ i } \\gamma _ i s _ i = \\sum _ { i \\leq k } y ^ { f _ i } \\gamma _ i \\alpha _ i + y ^ { e _ k } s = \\delta + y ^ { e _ k } s , \\end{align*}"}
+{"id": "3162.png", "formula": "\\begin{align*} c _ j ^ { k l } ( \\lambda A ) = \\lambda \\ , c _ j ^ { k l } ( A ) \\forall j , k , l \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "9206.png", "formula": "\\begin{align*} \\lambda ( m , h ) = \\mathcal { D } ( m ) h ^ { \\frac { d _ { m } - d _ { m } ^ { \\rm m a x } } { 2 } + 1 } e ^ { - \\frac { 2 S ( m ) } { h } } \\alpha ( h ) , \\end{align*}"}
+{"id": "2818.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } v _ i = \\sum _ { i = 1 } ^ { n + 1 } v _ i - v _ { n + 1 } < \\sum _ { i = 1 } ^ { n + 1 } u _ i - u _ { n + 1 } = \\sum _ { i = 1 } ^ n u _ i . \\end{align*}"}
+{"id": "2064.png", "formula": "\\begin{align*} \\begin{aligned} w _ { 1 1 } ( x , z ) & = \\exp \\Bigl ( \\frac { \\kappa i z } { \\rho _ 3 } x ^ { \\rho _ 3 } \\Bigr ) M \\Bigl ( a _ - , \\ , b _ - , \\ , - \\frac { 2 \\kappa i z } { \\rho _ 3 } x ^ { \\rho _ 3 } \\Bigr ) , \\\\ [ 1 e x ] w _ { 2 1 } ( x , z ) & = - \\frac { \\kappa _ 2 } { \\rho _ 2 } z x ^ { \\rho _ 2 } \\exp \\Bigl ( \\frac { \\kappa i z } { \\rho _ 3 } x ^ { \\rho _ 3 } \\Bigr ) M \\Bigl ( a _ + , \\ , b _ + , \\ , - \\frac { 2 \\kappa i z } { \\rho _ 3 } x ^ { \\rho _ 3 } \\Bigr ) , \\end{aligned} \\end{align*}"}
+{"id": "6368.png", "formula": "\\begin{align*} \\frac { - m _ \\lambda '' ( r ) } { m _ \\lambda ( r ) } = \\frac { Q _ \\lambda ( u ( r ) ) } { l _ \\lambda ^ 2 ( x _ \\lambda ' ( u ( r ) ) ^ 2 + y ' ( u ( r ) ) ^ 2 ) ^ 2 } , \\end{align*}"}
+{"id": "5370.png", "formula": "\\begin{align*} B : = \\bigcap _ { n \\in \\mathbb { N } } B _ n = \\left \\{ \\eta \\in B _ 0 : \\mbox { a l l t h e p o s i t i v e M a x w e l l e i g e n v a l u e s w i t h $ \\varepsilon = \\tilde \\varepsilon + \\eta $ a r e s i m p l e } \\right \\} \\end{align*}"}
+{"id": "3680.png", "formula": "\\begin{align*} \\vert U _ 1 ( E , F _ Y ) \\vert \\leq \\frac { 2 q ^ { Y / 2 } } { \\sqrt { q } Y ( 1 - q ^ { - 1 / 2 } ) ^ 2 } + \\frac { ( 2 g + 1 ) } { ( 1 - q ^ { - 1 } ) } Y . \\end{align*}"}
+{"id": "6690.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\bar { y } ^ { k + 1 } - \\bar { g } ^ { k + 1 } \\| ^ 2 = & ( 1 - \\alpha ^ k ) ^ 2 \\| \\bar { y } ^ k - \\bar { g } ^ k \\| ^ 2 + ( \\gamma _ 2 ^ k ) ^ 2 \\| \\bar { \\xi } _ w ^ k \\| ^ 2 \\\\ & + 2 \\left \\langle ( 1 - \\alpha ^ k ) ( \\bar { y } ^ k - \\bar { g } ^ k ) , \\gamma _ 2 ^ k \\bar { \\xi } _ w ^ k \\right \\rangle \\end{aligned} \\end{align*}"}
+{"id": "3701.png", "formula": "\\begin{align*} N _ { i + 1 } ( R ) : = \\{ u ( x ) : x \\in R ^ { \\dim V _ i } \\} . \\end{align*}"}
+{"id": "3531.png", "formula": "\\begin{align*} \\mathbf { p } = \\mathbf { p } _ { 1 2 } \\mathbf { p } _ { 1 3 } \\mathbf { p } _ { 2 3 } \\cdots \\mathbf { p } _ { 1 n } \\cdots \\mathbf { p } _ { n - 1 , n } . \\end{align*}"}
+{"id": "7469.png", "formula": "\\begin{align*} \\mu = \\sum _ { e \\in E ' } \\mu _ e \\ , . \\end{align*}"}
+{"id": "2629.png", "formula": "\\begin{align*} \\dim \\sum _ { l = 2 } ^ r V _ { [ r ] \\setminus \\{ l \\} } ^ \\perp = ( r - 1 ) s - 1 \\iff \\dim \\bigcap _ { l = 2 } ^ r V _ { [ r ] \\setminus \\{ l \\} } = 0 . \\end{align*}"}
+{"id": "3827.png", "formula": "\\begin{align*} \\begin{aligned} \\Theta _ \\psi ( \\tau ) : & = \\sum \\limits _ { \\beta \\geq 0 } ( \\deg ( \\psi ^ \\ast Z ( \\beta ) ) ) e ^ { 2 \\pi i \\mathrm { t r } ( \\beta \\tau ) } \\\\ & = \\sum \\limits _ { k , l , m \\geq 0 } N _ { k , l , m } \\tilde { q } ^ k p ^ l q ^ m \\end{aligned} \\end{align*}"}
+{"id": "1613.png", "formula": "\\begin{align*} \\mathcal { E } ^ * = \\limsup _ { n \\to \\infty } \\inf \\{ \\mathcal { E } : \\exists ( n , M , \\epsilon , \\mathcal { E } , K = \\mu n ) - \\mathrm { c o d e } \\} \\end{align*}"}
+{"id": "2184.png", "formula": "\\begin{align*} 3 ( ( 1 - 2 \\alpha - \\beta ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + 1 ) = ( 1 - r ^ 2 ) . \\end{align*}"}
+{"id": "4019.png", "formula": "\\begin{align*} D _ j Y ^ i = D _ { p _ k } Y ^ i w _ { k j } . \\end{align*}"}
+{"id": "848.png", "formula": "\\begin{align*} J _ d = \\frac { 1 } { 2 } t r ( J ) I _ { 3 \\times 3 } - J \\end{align*}"}
+{"id": "6310.png", "formula": "\\begin{align*} S = S ^ + \\cup S ^ \\circ \\cup S ^ - \\end{align*}"}
+{"id": "4122.png", "formula": "\\begin{align*} K _ { i j } = \\frac { 1 } { 4 } \\sum _ { m = 1 } ^ n \\alpha ^ 2 _ { i j m } \\geq 0 \\end{align*}"}
+{"id": "6648.png", "formula": "\\begin{align*} \\partial _ t \\delta _ y w - \\nabla \\cdot a _ y \\nabla \\delta _ y w = \\nabla \\cdot ( a _ y - a ( t ' , x ' ) ) \\nabla \\delta _ y v . \\end{align*}"}
+{"id": "643.png", "formula": "\\begin{align*} \\epsilon _ { 0 } = \\bar { \\epsilon } _ { 0 } . \\end{align*}"}
+{"id": "5023.png", "formula": "\\begin{align*} \\mu = \\sqrt { 2 \\left ( p ^ 2 + q ^ 2 \\right ) } , a = \\sqrt { \\frac { q ^ 2 } { p ^ 2 + q ^ 2 } } , b = \\sqrt { \\frac { p ^ 2 } { p ^ 2 + q ^ 2 } } . \\end{align*}"}
+{"id": "5364.png", "formula": "\\begin{align*} E ^ { ( i ) } _ h E ^ { ( j ) } _ h = 0 \\forall i , j \\in \\{ 1 , \\ldots , m \\} , \\ , i \\neq j , \\forall h = 1 , 2 , 3 , \\end{align*}"}
+{"id": "601.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } K _ n ^ { ( \\epsilon ) } = K _ n . \\end{align*}"}
+{"id": "4350.png", "formula": "\\begin{align*} x y t = ( s y t ) y t = s ( y t ) ( y t ) = s y t = x \\end{align*}"}
+{"id": "5619.png", "formula": "\\begin{align*} ^ { R L } I _ { t _ { 0 } } ^ { \\alpha } f ( t ) = \\dfrac { 1 } { \\Gamma ( \\alpha ) } \\int ^ { t } _ { t _ { 0 } } ( t - x ) ^ { \\alpha - 1 } f ( x ) d x , \\end{align*}"}
+{"id": "8877.png", "formula": "\\begin{align*} \\int _ { \\mathbb { C } ^ n } K _ { \\nu , m } ( z , w ) g ( w ) d \\mu _ n ( w ) = g _ m ( z ) . \\end{align*}"}
+{"id": "8791.png", "formula": "\\begin{align*} \\textbf { c b } _ i ( z ) = \\left ( \\overline { \\eta _ { i } } \\frac { \\alpha + \\theta ( t ) } { 1 + \\overline { \\theta ( t ) } \\alpha } \\right ) ^ { \\frac { 1 } { 2 } } \\frac { k _ { \\eta _ { i } } } { \\| k _ { \\eta _ { i } } \\| } , \\end{align*}"}
+{"id": "5911.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } u '' _ r e _ r - \\Delta u _ r e _ r + u _ r A e _ r = 0 & \\hbox { i n } ( T , + \\infty ) \\times \\Omega , \\\\ \\noalign { \\medskip } \\partial _ \\nu u _ r e _ r + u _ r B e _ r = 0 & \\hbox { o n } ( T , + \\infty ) \\times \\Gamma . \\end{array} \\right . \\end{align*}"}
+{"id": "6954.png", "formula": "\\begin{gather*} x ^ { \\alpha } y _ n ( x ) = ( \\alpha + n ) \\int _ c ^ x t ^ { \\alpha - 1 } L _ m ^ { ( \\alpha - 1 ) } ( - t ) L _ { n - m } ^ { ( \\alpha - 1 ) } ( t ) \\ , { \\rm d } t \\end{gather*}"}
+{"id": "6841.png", "formula": "\\begin{align*} R _ { \\alpha } = \\begin{pmatrix} \\cos ( \\alpha ) \\ , I & \\sin ( \\alpha ) \\ , I \\\\ - \\sin ( \\alpha ) \\ , I & \\cos ( \\alpha ) \\ , I \\\\ \\end{pmatrix} \\end{align*}"}
+{"id": "1424.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( | \\mathcal C _ { \\max } | < n ^ { 2 / 3 } / A \\right ) = O ( A ^ { - 1 / 2 } ) . \\end{align*}"}
+{"id": "6016.png", "formula": "\\begin{align*} [ C ' ] & = [ X ] \\cup [ L _ 1 \\times L _ 2 ] \\\\ & = ( [ h _ 1 ] + [ h _ 2 ] ) \\cup [ L _ 1 ] \\boxtimes [ L _ 2 ] \\\\ & = ( [ h _ 1 ] + [ h _ 2 ] ) \\cup ( l _ 1 + l _ 2 ) \\\\ & = l _ 1 + l _ 2 \\end{align*}"}
+{"id": "1987.png", "formula": "\\begin{align*} { \\mathcal { R } } _ { } = \\frac { N } { L } \\sum \\nolimits _ { m = 1 } ^ { M } \\log _ 2 \\left ( 1 + \\lambda _ m s _ m ^ { \\star } \\right ) , \\end{align*}"}
+{"id": "2159.png", "formula": "\\begin{align*} R _ { \\mathcal { S } ^ { * } _ { \\mathit { e } } } ( F ) & = \\begin{dcases} \\sigma _ { 0 } & \\ 2 \\alpha + \\beta - 2 \\geqslant 0 \\\\ \\sigma _ { 0 } & \\ 2 \\alpha + \\beta - 2 < 0 \\ \\ X ( \\alpha , \\beta ) \\leqslant 0 \\\\ \\tilde { \\sigma _ { 0 } } & \\ 2 \\alpha + \\beta - 2 < 0 \\ X ( \\alpha , \\beta ) > 0 , \\end{dcases} \\end{align*}"}
+{"id": "6116.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } U '' - \\Delta U + A U = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ U = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu U + B U = D H & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "1393.png", "formula": "\\begin{align*} \\psi = S _ 2 \\boxtimes S _ 1 + S _ 3 \\boxtimes S _ 2 + S _ 3 \\boxtimes S _ 2 . \\end{align*}"}
+{"id": "2952.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j \\in \\mathbb { Z } } \\overline { W } _ { j - 1 } ( \\overline { W } _ j - \\overrightarrow { W } ^ \\ell _ j ) \\varphi _ j & = \\sum _ { j \\in \\mathbb { Z } } \\overline { W } _ { j - 1 } \\sum _ { i = 0 } ^ { \\ell - 2 } ( W _ { j + i } - W _ { j + i + 1 } ) \\psi _ { i + 1 } \\varphi _ j \\\\ & = \\sum _ { k \\in \\mathbb { Z } } F _ k ( W _ k - W _ { k + 1 } ) \\end{aligned} \\end{align*}"}
+{"id": "4401.png", "formula": "\\begin{align*} U _ n ^ { ( n ) } ( \\theta ) = \\left \\{ ( x _ i ) _ { i = 1 } ^ n \\in U _ n ( \\theta ) : x _ i < x _ i ^ * i = 1 , \\ldots , n \\right \\} . \\end{align*}"}
+{"id": "3042.png", "formula": "\\begin{align*} \\mu \\left ( \\left ( A ^ { [ k _ n ] } \\cap T ^ { - n } ( ( V _ k ^ { ( A ) } ) ^ { [ k _ n ] } ) \\right ) \\Delta ( A \\cap E ^ { k } _ { 0 , n } ) \\right ) = O ( n ^ { - 1 3 / 2 0 } ) \\end{align*}"}
+{"id": "2266.png", "formula": "\\begin{align*} \\big ( f ( 0 , x , v ) , \\rho ( 0 , x ) , u ( 0 , x ) \\big ) = \\big ( f _ 0 ( x , v ) , \\rho _ 0 ( x ) , u _ 0 ( x ) \\big ) . \\end{align*}"}
+{"id": "4046.png", "formula": "\\begin{align*} D _ { x _ j } E _ { i r } - D _ { x _ j } ( Q _ r ) g _ { i , z } & = D _ { x _ j } \\left ( g _ { i , r } - \\frac { g _ { i , z } g _ { , r } } { g _ z } \\right ) + D _ { x _ j } \\left ( \\frac { g _ { , r } } { g _ z } \\right ) g _ { i , z } \\\\ & = g _ { i j , r } - g _ { i j , z } \\frac { g _ { , r } } { g _ z } . \\end{align*}"}
+{"id": "3308.png", "formula": "\\begin{align*} F _ { \\lambda } ^ { ( 0 ) } [ W ] : = \\dfrac { 1 } { m } \\left ( \\int _ { \\partial B _ 0 } \\partial _ n v ^ { ( 0 ) } _ { \\lambda } [ W ] { \\rm d } \\sigma \\right ) , \\tau _ { \\lambda } ^ { ( 0 ) } [ W ] : = \\mathcal { J } ^ { - 1 } \\left ( \\int _ { \\partial B _ 0 } z ^ { \\bot } \\cdot \\partial _ n v ^ { ( 0 ) } _ { \\lambda } [ W ] { \\rm d } \\sigma \\right ) , \\end{align*}"}
+{"id": "3961.png", "formula": "\\begin{align*} \\mathcal { C } : = \\{ t \\xi ; 0 < t \\leq t _ 0 , \\xi \\in B _ r ' \\} , \\end{align*}"}
+{"id": "1485.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\frac \\pi 2 } \\log ( \\sin \\theta ) d \\theta = - \\frac \\pi 2 \\log 2 . \\end{align*}"}
+{"id": "2066.png", "formula": "\\begin{align*} \\bigl ( x ^ { - \\rho _ 2 + 1 } y _ 1 ' ( x ) \\bigr ) ' + \\Bigl [ z \\kappa _ 3 \\Bigl ( x ^ { \\frac { \\rho _ 1 - \\rho _ 2 } { 2 } } \\Bigr ) ' + z ^ 2 \\bigl ( \\kappa _ 1 \\kappa _ 2 - \\kappa _ 3 ^ 2 \\bigr ) x ^ { \\rho _ 1 - 1 } \\Bigr ] y _ 1 ( x ) = 0 . \\end{align*}"}
+{"id": "9128.png", "formula": "\\begin{align*} \\sum _ { f \\in \\mathfrak { F } _ { \\eta ; \\gamma } } \\prod _ { ( x ' , y ' ) \\in E ( f ) } \\nu ( x ' - y ' ) = \\sum _ { \\xi \\subseteq \\gamma } \\prod _ { ( y \\in \\xi ) } \\nu ( x - y ) \\sum _ { f \\in \\mathfrak { F } _ { \\eta ^ { \\hat { x } } \\cup \\xi ; \\gamma \\setminus \\xi } } \\prod _ { ( x ' , y ' ) \\in E ( f ) } \\nu ( x ' - y ' ) . \\end{align*}"}
+{"id": "5067.png", "formula": "\\begin{align*} l ^ { \\varepsilon ( \\omega ) s } \\sum _ { j = 1 } ^ n q _ { j , s } f _ j ( \\omega ) = 0 , \\end{align*}"}
+{"id": "5671.png", "formula": "\\begin{align*} M ( x ) = \\int \\eta ( \\omega ) f ( x , \\omega ) { \\rm d } \\omega , \\end{align*}"}
+{"id": "4180.png", "formula": "\\begin{align*} \\partial _ t \\left ( \\partial _ t g g ^ { - 1 } \\right ) - \\partial _ x \\left ( \\partial _ x g g ^ { - 1 } \\right ) = 0 , ( t , x ) \\in \\mathbb R \\times \\mathbb R , \\end{align*}"}
+{"id": "9008.png", "formula": "\\begin{align*} \\alpha ( R _ { n + 1 } , K _ { n + 1 } ) \\ , & = \\ , \\alpha \\left ( R _ { n } \\setminus A _ { i _ { n + 1 } } C _ { n + 1 } , K _ { n + 1 } \\right ) \\\\ & \\leq \\ , \\frac { \\alpha ( R _ { n } , K _ { n + 1 } ) + \\alpha ( A _ { i _ { n + 1 } } C _ { n + 1 } , K _ { n + 1 } ) } { \\epsilon } \\\\ & \\leq \\ , 2 \\epsilon ^ { 2 ( N - n ) } \\ , \\leq \\ , \\epsilon ^ { 2 ( N - n ) - 1 } \\ , = \\ , \\epsilon ^ { 2 ( N - ( n + 1 ) ) + 1 } , \\end{align*}"}
+{"id": "622.png", "formula": "\\begin{align*} A = \\frac { - 1 } { 2 } \\left ( \\begin{array} { c c } 1 & - 1 \\\\ 1 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "4671.png", "formula": "\\begin{align*} \\sum _ { m = n } ^ { \\infty } \\sum _ { k = n } ^ { \\min \\{ \\ell , m \\} } A _ { m , k } = \\sum _ { k = n } ^ { \\ell } \\sum _ { m = k } ^ { \\infty } A _ { m , k } . \\end{align*}"}
+{"id": "2853.png", "formula": "\\begin{align*} c _ { \\ell } = \\rho - \\sigma \\leq \\rho - | \\rho - b _ { \\ell } | \\leq b _ { \\ell } \\end{align*}"}
+{"id": "8471.png", "formula": "\\begin{align*} c _ * ( A ) = \\gamma _ A ( X ) \\leqslant \\sup _ { \\nu \\in \\widehat { \\mathcal E } ' _ A } \\ , \\nu ( X ) \\leqslant \\sup _ { \\nu \\in \\widehat { \\mathcal E } ' _ A } \\ , G ( \\nu ) \\leqslant \\max _ { \\nu \\in \\mathcal E ' _ A } \\ , G ( \\nu ) = G ( \\gamma _ A ) = c _ * ( A ) , \\end{align*}"}
+{"id": "6213.png", "formula": "\\begin{align*} \\hbox { K e r } ( \\mathcal R ^ T ) \\cap \\{ \\hbox { K e r } ( C _ p ) \\} ^ \\perp = \\hbox { K e r } ( \\mathcal R ^ T ) \\cap \\hbox { I m } ( C _ p ^ T ) = \\{ 0 \\} , \\end{align*}"}
+{"id": "1138.png", "formula": "\\begin{align*} a \\chi ( f \\otimes m ) & = a f ( m ) = \\chi ( a f \\otimes m ) , \\mbox { \\ \\ \\ a n d } \\\\ ( \\chi ( f \\otimes m ) ) a & = \\lambda ( a ) \\chi ( f \\otimes m ) = \\chi ( f \\otimes \\lambda ( a ) m ) = \\chi ( ( f \\otimes m ) a ) . \\end{align*}"}
+{"id": "2440.png", "formula": "\\begin{align*} f _ 1 ( x ) & = 1 + x ^ e , & f _ 2 ( x ) & = 1 - x ^ e , & f _ 3 ( x ) & = 1 + x ^ { 2 e } , & f _ 4 ( x ) & = 1 - x ^ { 2 e } , \\\\ f _ 5 ( x ) & = 1 + x ^ e + x ^ { 2 e } , & f _ 6 ( x ) & = 1 - x ^ e + x ^ { 2 e } , & f _ 7 ( x ) & = 1 + x ^ e - x ^ { 2 e } , & f _ 8 ( x ) & = 1 - x ^ e - x ^ { 2 e } . \\end{align*}"}
+{"id": "1504.png", "formula": "\\begin{align*} \\log \\C _ 3 \\left ( \\frac 1 4 \\right ) = \\frac 1 { 2 ^ 5 } \\log 2 - \\frac { G } { 4 \\pi } + \\frac { 2 1 \\zeta ( 3 ) } { 2 ^ 6 \\pi ^ 2 } , \\end{align*}"}
+{"id": "6129.png", "formula": "\\begin{align*} \\mathcal R = ( D , A D , \\cdots , A ^ { N - 1 } D ) . \\end{align*}"}
+{"id": "4699.png", "formula": "\\begin{align*} \\mathfrak { E } ^ { G _ I } = \\mathbb { C } [ \\varphi _ 2 , \\varphi _ 8 ] \\end{align*}"}
+{"id": "5668.png", "formula": "\\begin{align*} { \\mathcal I } _ 1 ( \\Lambda ) \\equiv \\left ( \\frac { \\Lambda } { \\bar { \\ell } } \\sum _ { n \\geq 0 } ( - 1 ) ^ n \\Lambda ^ n \\right ) \\sum _ { j = 1 } ^ N \\ell _ j \\widehat { \\Phi } _ j = \\frac { 1 } { \\bar { \\ell } } \\frac { \\Lambda } { 1 + \\Lambda } \\sum _ { j = 1 } ^ N \\ell _ j \\widehat { \\Phi } _ j = \\frac { \\Lambda } { 1 + \\Lambda } [ \\overline { \\Phi } - \\Gamma / | \\Omega | ] . \\end{align*}"}
+{"id": "978.png", "formula": "\\begin{align*} \\| R _ m \\| ^ 2 & = \\frac { 1 } { 4 \\pi } \\big [ 2 \\| R _ { m - 1 } \\| ^ 2 ( \\pi + 2 ) + 4 \\langle R _ 1 + . . . + R _ { m - 2 } , R _ { m - 1 } \\rangle \\big ] \\\\ & = \\| R _ { m - 1 } \\| ^ 2 \\big ( \\frac { \\pi + 2 } { 2 \\pi } \\big ) + \\frac { 1 } { \\pi } \\langle R _ 1 + . . . + R _ { m - 2 } , R _ { m - 1 } \\rangle \\end{align*}"}
+{"id": "3974.png", "formula": "\\begin{align*} 0 = \\log \\det [ D ^ 2 v & - A ( \\cdot , v , D v ) ] - \\log \\det [ D ^ 2 u - A ( \\cdot , u , D u ) ] \\\\ & + \\log B ( \\cdot , u , D u ) - \\log B ( \\cdot , v , D v ) . \\end{align*}"}
+{"id": "1290.png", "formula": "\\begin{align*} \\nu : = \\Sigma ^ { - 1 } ( \\mu \\circ \\Phi ^ { - 1 } ) , \\end{align*}"}
+{"id": "6044.png", "formula": "\\begin{align*} f ( z ) - \\frac { p _ n ( z ) } { q _ n ( z ) } = ( 2 + o ( 1 ) ) \\frac { S _ { \\dot \\mu } ^ 2 ( z ) } { w ( z ) } \\prod _ { e \\in E _ n } \\frac { \\psi ( z ) - \\psi ( e ) } { 1 - \\psi ( z ) \\overline { \\psi ( e ) } } \\end{align*}"}
+{"id": "7248.png", "formula": "\\begin{align*} n _ { m , j } [ T _ 0 > s ] = o ( s ^ { - 2 } U ^ { \\sharp } ( s ^ 2 ) ^ { - 1 } v ( s ^ { \\alpha } U ^ { \\sharp } ( s ^ 2 ) ^ { \\alpha / 2 } ) ) ( s \\to \\infty ) , \\end{align*}"}
+{"id": "9253.png", "formula": "\\begin{align*} N v _ { w ^ { - 1 } ( k ) } = \\begin{cases} e _ { k - 1 } + c e _ { j - 1 } & j , k \\neq 1 , \\\\ e _ { k - 1 } & j = 1 , \\ ; \\end{cases} \\end{align*}"}
+{"id": "6810.png", "formula": "\\begin{align*} D _ 1 L _ d ( q _ { m } , q _ { m + 1 } ) & = \\frac { h } { 2 } [ \\nabla \\alpha ( q _ { m } ) \\cdot \\frac { q _ { m + 1 } - q _ { m } } { h } - \\frac { \\alpha ( q _ { m } ) } { h } - \\frac { \\alpha ( q _ { m + 1 } ) } { h } - \\nabla H ( q _ { m } ) ] , \\\\ D _ 2 L _ d ( q _ { m - 1 } , q _ { m } ) & = \\frac { h } { 2 } [ \\nabla \\alpha ( q _ { m } ) . \\frac { q _ { m } - q _ { m - 1 } } { h } + \\frac { \\alpha ( q _ { m } ) } { h } + \\frac { \\alpha ( q _ { m - 1 } ) } { h } - \\nabla H ( q _ { m } ) ] . \\end{align*}"}
+{"id": "6058.png", "formula": "\\begin{align*} \\left [ \\vec V _ n \\right ] = \\left [ \\sum _ { i \\in I _ 1 \\cup I _ 2 \\cup I _ 3 \\cup I _ 5 } \\vec V _ { n , i } + \\vec U \\right ] , \\vec U : = \\frac 1 2 \\left ( \\sum _ { i \\in I _ 1 } d _ i \\big [ \\mathsf B \\big ] _ i + \\sum _ { i \\in I _ 2 \\cup I _ 3 \\cup I _ 4 } \\big [ \\mathsf B \\big ] _ i \\right ) . \\end{align*}"}
+{"id": "2994.png", "formula": "\\begin{align*} \\theta = \\mathbf { A } _ a ( x ) e ^ a + \\hbox { d } s \\end{align*}"}
+{"id": "2933.png", "formula": "\\begin{align*} \\nabla ^ n \\varphi ^ n _ j = \\frac { n } { 2 } ( \\varphi ^ n _ { j + 1 } - \\varphi ^ n _ { j - 1 } ) , \\Delta ^ n \\varphi ^ n _ j = n ^ 2 ( \\varphi ^ n _ { j + 1 } + \\varphi ^ n _ { j - 1 } - 2 \\varphi ^ n _ j ) . \\end{align*}"}
+{"id": "5905.png", "formula": "\\begin{align*} \\sum _ { j = n _ { r - 1 } + 1 } ^ { n _ { r } } a _ { i j } = \\alpha _ { s r } , n _ { s - 1 } + 1 \\leq i \\leq n _ { s } , 1 \\leq r , s \\leq p . \\end{align*}"}
+{"id": "7775.png", "formula": "\\begin{align*} \\frac { \\mu \\alpha } { \\pi } { \\rm I m } \\left \\{ \\frac { ( e ^ { - i \\pi } t ) ^ { \\alpha - 1 } } { \\lambda + e ^ { - i \\pi \\alpha } \\ , t ^ \\alpha } \\right \\} & = \\frac { \\mu \\alpha } { \\pi } \\frac { \\lambda \\ , t ^ { \\alpha - 1 } \\sin \\pi \\alpha } { \\lambda ^ 2 + 2 \\lambda \\ , t ^ \\alpha \\cos \\pi \\alpha + t ^ { 2 \\alpha } } \\end{align*}"}
+{"id": "8283.png", "formula": "\\begin{align*} \\mathcal { T } _ e : = \\big \\{ \\textbf { x } \\ | \\ z _ E = 0 \\big \\} \\end{align*}"}
+{"id": "6289.png", "formula": "\\begin{align*} \\int _ { \\mathbb { I ' } } p ^ { \\nu } _ { K ' } \\left ( \\frac { d } { d \\tau } \\big | _ { \\tau = s } f _ j \\circ \\Psi _ { \\tau , t _ 0 } \\circ \\Psi ^ { - 1 } _ { s , t _ 0 } ( x ) - \\frac { d } { d \\tau } \\big | _ { \\tau = s } f _ j \\circ \\Psi _ { \\tau , t _ 0 } \\circ \\Phi ^ { - 1 } _ { s , t _ 0 } ( x ) \\right ) \\d s < \\frac { r } { 2 } . \\end{align*}"}
+{"id": "6892.png", "formula": "\\begin{align*} \\begin{bmatrix} A & 0 \\\\ C & D \\end{bmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\geq \\begin{pmatrix} a \\\\ b \\end{pmatrix} \\end{align*}"}
+{"id": "2612.png", "formula": "\\begin{align*} \\frac { ( r - 2 ) ( k + 1 ) } { 2 } = \\sum _ { l , t \\in [ r ] \\setminus \\{ 1 \\} , l < t } a _ { l , t } . \\end{align*}"}
+{"id": "2222.png", "formula": "\\begin{align*} h ( i , k ) = \\left \\lfloor \\dfrac { 5 k - 1 0 } { 6 } \\right \\rfloor + \\left \\lfloor \\dfrac { 5 i - k - 1 } { 6 } \\right \\rfloor . \\end{align*}"}
+{"id": "8004.png", "formula": "\\begin{align*} x ( t ) = \\phi ( t ) t \\in [ - \\tau , 0 ] . \\end{align*}"}
+{"id": "8086.png", "formula": "\\begin{align*} \\mathcal { R } ^ + _ 1 = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { c \\geq \\frac { C _ 1 } { m } } \\frac { S ( n , p ; c ) } { c } H _ { m , n } ^ + \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) , \\end{align*}"}
+{"id": "4549.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n + 1 } \\frac { 1 } { s _ i } & = \\sum _ { i = 1 } ^ n \\frac { 1 } { s _ i } + \\frac { 1 } { s _ { n + 1 } } = 1 - \\frac { 1 } { \\prod _ { i = 1 } ^ n s _ i } + \\frac { 1 } { s _ { n + 1 } } \\\\ & = 1 - \\frac { s _ { n + 1 } - \\prod _ { i = 1 } ^ n s _ i } { \\prod _ { i = 1 } ^ { n + 1 } s _ i } = 1 - \\frac { 1 } { \\prod _ { i = 1 } ^ { n + 1 } s _ i } . \\end{align*}"}
+{"id": "5571.png", "formula": "\\begin{align*} \\epsilon ' = \\epsilon _ 0 \\max \\{ r ^ { - n } , r ^ { - ( n - 2 \\gamma ) } \\} . \\end{align*}"}
+{"id": "3040.png", "formula": "\\begin{align*} \\begin{aligned} | | D ^ 2 w | | _ { L ^ p ( \\Omega _ { 1 / { 1 2 } } ) } ^ { p } \\leq \\sum \\limits _ { s _ 0 = 2 } ^ { \\infty } \\sum \\limits _ { Q _ k \\in \\mathcal { F } ^ { s _ 0 } } \\left ( \\sum \\limits _ { j = 0 } ^ { s _ 0 } \\sum \\limits _ { s = s _ 0 - j - 6 } ^ { { \\infty } } | | D ^ 2 w ^ { s , j } _ { k } | | _ { L ^ p ( Q _ k ) } \\right ) ^ p . \\end{aligned} \\end{align*}"}
+{"id": "5932.png", "formula": "\\begin{align*} D ^ T \\widetilde C _ 1 ^ T \\widetilde E = 0 . \\end{align*}"}
+{"id": "5487.png", "formula": "\\begin{align*} \\tilde L ^ { ( 1 ) } \\tilde V ( x - y ) & = { \\rm s i g n } ( x - y ) \\cdot ( - x ^ 3 - x + y ^ 3 + y ) \\\\ & \\leq - | x - y | = - \\tilde V ( x - y ) , \\\\ \\tilde L ^ { ( 2 ) } \\tilde V ( x - y ) & = { \\rm s i g n } \\left ( x - y \\right ) ( - \\frac { 1 } { 2 } x + \\frac { 1 } { 2 } y ) \\\\ & = - \\frac { 1 } { 2 } | x - y | = - \\frac { 1 } { 2 } \\tilde V ( x - y ) , \\end{align*}"}
+{"id": "4657.png", "formula": "\\begin{align*} \\mathsf { E } \\left [ g [ X ] X ( t ) \\right ] = \\mu ( t ) \\mathsf { E } \\left [ g ( X ) \\right ] + \\int _ { t _ 0 } ^ t C ( t , s ) \\mathsf { E } \\left [ \\frac { \\delta g [ X ] } { \\delta X ( s ) } \\right ] \\mathrm { d } s , \\end{align*}"}
+{"id": "1719.png", "formula": "\\begin{align*} \\frac { \\langle \\delta s ( \\Upsilon ) \\rangle } { \\langle \\Upsilon \\rangle ' } = ( k - 1 ) \\cdot 2 ^ { k - 3 } . \\end{align*}"}
+{"id": "1031.png", "formula": "\\begin{align*} \\mathcal { P } _ \\varepsilon ( \\mathcal { P } ) = \\{ P _ \\varepsilon = ( 1 - \\varepsilon ) P + \\varepsilon G : \\ , \\varepsilon \\in [ 0 , 1 ] , \\ , P \\in \\mathcal { P } , \\ , G \\in \\mathcal { G } \\} . \\end{align*}"}
+{"id": "5192.png", "formula": "\\begin{align*} \\begin{aligned} u _ t + u \\nabla u + \\nabla p = 0 \\\\ \\nabla \\cdot u = 0 \\end{aligned} \\end{align*}"}
+{"id": "1550.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\lambda } = \\{ \\mathrm { u } \\in W _ { 0 } ^ { 1 , q ( z ) } ( M ) \\backslash \\{ 0 \\} ; \\ , & | | D \\mathrm { u } | | _ { p ( z ) } ^ { p ( z ) } + \\int _ { M } \\mu ( z ) \\ , | D \\mathrm { u } ( z ) | ^ { q ( z ) } \\ , \\ , d v _ { g } ( x ) \\\\ & = \\int _ { M } g ( z ) | \\mathrm { u } ( z ) | ^ { 1 - \\gamma ( z ) } \\ , \\ , d v _ { g } ( z ) + \\lambda | | \\mathrm { u } | | _ { r ( z ) } ^ { r ( z ) } \\ , \\} . \\end{align*}"}
+{"id": "5048.png", "formula": "\\begin{align*} f ( m , n ) = \\begin{cases} 2 ^ m + 3 ^ m & , \\\\ \\frac { 1 } { 5 } ( m + 1 ) ( \\frac { 3 } { 5 } ) ^ m 5 ^ n & \\\\ 1 & . \\end{cases} \\end{align*}"}
+{"id": "7198.png", "formula": "\\begin{align*} q \\ , = \\ , | q | \\big ( c o s \\theta + \\sin \\theta \\ , \\omega \\big ) , \\qquad \\mbox { w h e r e } \\qquad \\theta \\in [ 0 , \\ , \\pi ] , \\end{align*}"}
+{"id": "5323.png", "formula": "\\begin{align*} \\mathrm { A d } _ Z \\circ \\Phi ( X ) & = Z ^ * V ^ * \\pi ( X ) ( U U ^ * ) V Z \\\\ & = ( U ^ * V Z ) ^ * \\pi ( X ) ( U ^ * V Z ) \\\\ & = ( T ^ { \\frac { 1 } { 2 } } V ) ^ * \\pi ( X ) ( T ^ { \\frac { 1 } { 2 } } V ) \\\\ & = V ^ * T \\pi ( X ) V \\\\ & = \\Psi ( X ) \\end{align*}"}
+{"id": "247.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = - ( 1 - \\alpha ) \\rho ^ 2 - \\beta \\rho \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "5834.png", "formula": "\\begin{align*} \\mathcal H _ 0 = L ^ 2 ( \\Omega ) , \\mathcal H _ 1 = H ^ 1 _ { \\Gamma _ 0 } ( \\Omega ) , \\end{align*}"}
+{"id": "3944.png", "formula": "\\begin{align*} \\overline { g } ( q , p , 0 ) = c ( q , p ) = q \\cdot p + a _ { i j , k l } q _ i q _ j p _ k p _ l . \\end{align*}"}
+{"id": "2787.png", "formula": "\\begin{align*} x _ 1 ^ a x _ 2 ^ b + x _ 1 ^ b x _ 2 ^ a = x _ 1 ^ { a + b } + x _ 2 ^ { a + b } . \\end{align*}"}
+{"id": "6698.png", "formula": "\\begin{align*} \\omega ( n ) = \\sum _ { p \\mid n } 1 \\Omega ( n ) = \\sum _ { p ^ v \\mid n } 1 \\end{align*}"}
+{"id": "6457.png", "formula": "\\begin{align*} m \\wedge \\delta E ( m ) = 0 \\textrm { w i t h } E ( m ) < \\infty \\textrm { a n d } m ( \\pm \\infty ) = ( \\pm 1 , 0 , 0 ) , \\end{align*}"}
+{"id": "5633.png", "formula": "\\begin{align*} 1 = \\| T ^ { - 1 } ( T ^ { - 1 } ) ^ * T ^ * T \\xi \\| \\leq ( 1 + \\delta ) \\| T ^ * T \\xi \\| \\leq ( 1 + \\delta ) ^ 2 \\end{align*}"}
+{"id": "7132.png", "formula": "\\begin{align*} \\Delta \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) = \\sum _ { i = 1 } ^ n \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) \\otimes \\beta ( \\rho _ i , \\dots , \\rho _ n ) . \\end{align*}"}
+{"id": "6472.png", "formula": "\\begin{align*} 1 - ( \\gamma ^ 2 + \\beta _ * ) = 1 - \\gamma ^ 2 - 2 ( 1 - \\gamma ^ 2 ) \\sin ^ 2 \\theta _ * = ( 1 - \\gamma ^ 2 ) ( \\cos ^ 2 \\theta _ * - \\sin ^ 2 \\theta _ * ) , \\end{align*}"}
+{"id": "94.png", "formula": "\\begin{align*} ( \\beta ^ { \\delta \\tau _ { i , 0 } } , \\beta ^ { \\delta \\tau _ { i , 1 } } , \\cdots , \\beta ^ { \\delta \\tau _ { i , a - 1 } } ) = \\sum _ { s = 0 } ^ { \\delta } \\eta _ { i , s } ( 1 , \\gamma ^ { s } , \\cdots , \\gamma ^ { s ( a - 1 ) } ) \\end{align*}"}
+{"id": "5461.png", "formula": "\\begin{align*} & E ( \\sup _ { s , t \\in [ 0 , T ] , | t - s | \\leq \\epsilon } | X ^ n _ { t \\wedge \\tau _ N ^ n } - X ^ n _ { s \\wedge \\tau _ N ^ n } | ^ { 2 l } ) \\\\ \\leq & C ( l ) \\sum _ { j = 1 } ^ { k } E ( \\sup _ { t \\in [ ( j - 1 ) \\epsilon , j \\epsilon \\wedge T ] } | X ^ n _ { t \\wedge \\tau _ N ^ n } - X ^ n _ { ( j - 1 ) \\epsilon \\wedge \\tau _ N ^ n } | ^ { 2 l } ) \\\\ \\leq & C ( N , l ) ( T + \\epsilon ) \\epsilon ^ { l - 1 } . \\end{align*}"}
+{"id": "4687.png", "formula": "\\begin{align*} G _ { I I I } = \\left \\langle \\frac { 1 } { \\sqrt { 3 } } \\begin{pmatrix} 1 & 2 \\\\ 1 & - 1 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ 0 & \\exp ( 2 \\pi i / 3 ) \\end{pmatrix} \\right \\rangle , \\end{align*}"}
+{"id": "497.png", "formula": "\\begin{align*} \\sum _ { K _ 1 \\ni k = 0 } ^ { \\infty } \\ ! \\sqrt { \\Phi ( x ^ { \\ell ( k + 1 ) } ) \\ ! - \\ ! \\Phi ( x ^ { k + 1 } ) } < \\infty \\ \\ { \\rm w h e n } \\ \\liminf _ { K _ 1 \\ni k \\to \\infty } \\frac { \\Phi ( x ^ { \\ell ( k ) } ) - \\Phi ( x ^ { \\ell ( k + 1 ) } ) } { \\| x ^ { k + 1 } - x ^ k \\| ^ 2 } = 0 , \\end{align*}"}
+{"id": "749.png", "formula": "\\begin{align*} \\widetilde { \\chi } ( b ) : = \\prod _ { q \\mid N } \\widetilde { \\chi _ q } ( b _ { q } ) , \\end{align*}"}
+{"id": "3991.png", "formula": "\\begin{align*} \\frac { - g _ { y _ a } } { g _ z } ( X ( y , z , q ) , y , z ) = - q _ a , \\end{align*}"}
+{"id": "336.png", "formula": "\\begin{align*} p ^ 2 A ( p , 1 ) + p A ( p , 1 ) \\overline { A ' ( p , 1 ) } + \\overline { A ' ( p , 1 ) } = p ^ 2 A ' ( p , 1 ) + p A ' ( p , 1 ) \\overline { A ( p , 1 ) } + \\overline { A ( p , 1 ) } . \\end{align*}"}
+{"id": "4213.png", "formula": "\\begin{align*} D _ { t _ o } : = \\{ ( t , x ) : \\ ; 0 \\leq t \\leq t _ 0 \\} , D _ { t _ 0 } = \\bigcup _ { 0 \\leq t \\leq t _ 0 } S _ { t _ 0 } . \\end{align*}"}
+{"id": "1274.png", "formula": "\\begin{align*} \\mathcal { C } _ 2 = \\left \\{ \\big ( c ^ { ( y ^ 0 ) } | c ^ { ( y ^ 1 ) } | \\cdots | c ^ { ( y ^ { \\ell - 1 } ) } \\big ) : c ( x , y , z ) \\in \\mathcal { C } \\right \\} , \\end{align*}"}
+{"id": "8305.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\frac { \\partial V ( \\textbf { x } ) } { \\partial \\textbf { x } _ i } \\ ! \\ ! \\ ! \\ ! & = - \\frac { \\nu _ i } { \\Lambda } [ x _ i - x ^ * ( \\textbf { x } ) , \\ y _ i - y ^ * ( \\textbf { x } ) , \\ z _ i - z ^ * ( \\textbf { x } ) ] \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "8081.png", "formula": "\\begin{align*} \\mathcal { D } & = \\frac { L ( 1 , \\tilde f ) \\bigl ( A ( p , 1 ) p - 1 \\bigr ) } { p ^ { \\frac { 3 } { 2 } } \\pi } \\int _ 0 ^ { \\infty } k ( t ) \\tanh ( \\pi t ) t \\ , d t + O ( M T ^ { \\frac { 1 } { 7 } + \\varepsilon } p ^ { \\varepsilon } ) \\end{align*}"}
+{"id": "7482.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda } + \\sum _ { j = 0 } ^ { n } \\frac { 1 } { \\mu _ { v _ { j } v _ { j + 1 } } } \\ , . \\end{align*}"}
+{"id": "7080.png", "formula": "\\begin{align*} u ^ k = \\left [ S ^ 0 \\subset S ^ { k \\sigma } \\to \\Sigma ^ { k \\sigma } E _ { C _ 2 } \\simeq \\Sigma ^ { | k \\sigma | } \\Sigma ^ { k \\sigma - | k \\sigma | } E _ { C _ 2 } \\to \\Sigma ^ { | k \\sigma | } \\widehat { E } _ { C _ 2 } \\right ] \\in \\widehat { E } ^ { C _ 2 } _ { - | k \\sigma | } \\end{align*}"}
+{"id": "4801.png", "formula": "\\begin{align*} \\max _ { j \\in B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } \\big [ | v H | = j \\big ] \\cdot \\frac { 2 ^ N } { \\binom { N } { j } } \\right \\} & \\leq \\max _ { j \\in B } \\left \\{ 2 ^ { - ( 1 - h ( j / N ) ) ( 1 - \\eta ) N } \\cdot \\frac { 2 ^ N } { \\sqrt { \\frac { 8 \\pi } { e ^ 4 N } } \\cdot 2 ^ { h ( j / N ) N } } \\right \\} \\\\ & = \\max _ { j \\in B } \\left \\{ \\sqrt { \\frac { e ^ 4 N } { 8 \\pi } } \\cdot 2 ^ { ( 1 - h ( j / N ) ) \\eta N } \\right \\} . \\end{align*}"}
+{"id": "1122.png", "formula": "\\begin{align*} \\Theta = \\left [ \\begin{smallmatrix} \\alpha _ 1 & \\alpha _ 2 & \\dots & \\alpha _ l \\\\ \\beta _ 1 & \\beta _ 2 & \\dots & \\beta _ l \\\\ \\gamma _ 1 & \\gamma _ 2 & \\dots & \\gamma _ l \\end{smallmatrix} \\right ] \\end{align*}"}
+{"id": "6217.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } u _ r '' - \\Delta u _ r + \\sum _ { s = 1 } ^ p \\alpha _ { r s } u _ s = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\cr u _ r = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\cr \\partial _ \\nu u _ r + \\sum _ { s = 1 } ^ p \\beta _ { r s } u _ s = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "6710.png", "formula": "\\begin{align*} \\sup _ { \\alpha \\in \\R } \\sum _ { n \\leq x } \\mu ( n ) e ^ { i 2 \\pi \\alpha n } = O \\left ( \\frac { x } { ( \\log x ) ^ { D } } \\right ) , \\end{align*}"}
+{"id": "8465.png", "formula": "\\begin{align*} \\gamma _ A ( X ) = \\min _ { \\nu \\in \\Gamma _ A ^ + } \\ , \\nu ( X ) . \\end{align*}"}
+{"id": "4519.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n v _ i \\leq \\sum _ { i = 1 } ^ n u _ i . \\end{align*}"}
+{"id": "3102.png", "formula": "\\begin{align*} \\gamma ( y ) : = \\frac { 1 } { C : A ( y ) } \\quad y \\in \\R ^ n \\end{align*}"}
+{"id": "2450.png", "formula": "\\begin{align*} L ( s , \\pi _ { \\infty } ) = \\pi ^ { - \\frac { m s } { 2 } } \\prod _ { j = 1 } ^ { m } \\Gamma \\Big ( \\frac { s + \\mu _ { \\pi } ( j ) } { 2 } \\Big ) . \\end{align*}"}
+{"id": "5003.png", "formula": "\\begin{align*} \\begin{cases} z _ t = \\Delta z + f ( x , t ) & , \\\\ \\partial _ \\nu z = 0 & , \\\\ z ( \\cdot , 0 ) = z _ 0 & , \\end{cases} \\end{align*}"}
+{"id": "9154.png", "formula": "\\begin{align*} f ( \\lambda _ * ) = \\left ( 1 - \\frac { \\tau } { m + \\tau } \\right ) ^ { \\frac { m + \\tau } { \\tau } \\cdot \\tau } \\cdot \\left ( 1 - \\frac { \\tau } { m + \\tau } \\right ) ^ { - \\tau } \\cdot \\left ( \\frac { \\tau } { m + \\tau } \\right ) ^ \\tau . \\\\ \\end{align*}"}
+{"id": "6571.png", "formula": "\\begin{align*} 2 < q < \\infty , \\ \\frac { n q } { 2 ( q - 1 ) } < p < \\infty \\ \\left ( \\frac { 4 } { 3 } \\leq p < \\infty \\ \\mathrm { i n \\ a d d i t i o n \\ i f } \\ n = 2 \\right ) , \\end{align*}"}
+{"id": "8767.png", "formula": "\\begin{align*} \\mathcal { H } _ { i , j } = \\begin{cases} 1 / | \\bar { l } | ( U _ l W h ^ * ) ^ * ( i , j ) \\in \\mathbb { N } ^ 2 \\ \\ 1 \\leq | i - j | = l \\leq 2 T - 1 , \\\\ 0 . \\end{cases} \\end{align*}"}
+{"id": "6955.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } x } L _ { m , n } ^ { I I , ( \\alpha ) } ( x ) - L _ { m , n } ^ { I I , ( \\alpha ) } ( x ) = ( \\alpha + n + 1 - 2 m ) L _ { m } ^ { ( - \\alpha - 1 ) } ( x ) L _ { n - m } ^ { ( \\alpha + 1 ) } ( x ) \\end{align*}"}
+{"id": "3396.png", "formula": "\\begin{align*} \\partial _ { t } \\omega + v \\cdot \\nabla \\omega = 0 , \\end{align*}"}
+{"id": "80.png", "formula": "\\begin{align*} \\mathcal { X } ^ T ( \\omega , t ) = T ^ { 1 / 2 } \\cdot \\mathcal { X } ( \\omega , t / T ) , \\ , \\ \\omega \\in \\Omega , \\ , \\ t \\in [ 0 , T ] \\end{align*}"}
+{"id": "3509.png", "formula": "\\begin{align*} B _ A ^ + : = \\prod _ { l = 1 , \\dots , \\ell ( \\lambda ' ) } ^ \\rightarrow B _ { A ( 1 , l ) } ^ + \\cdots B _ { A ( \\lambda _ l ' , l ) } ^ + \\end{align*}"}
+{"id": "5959.png", "formula": "\\begin{align*} ( \\sum _ { r = 1 } ^ p a _ r e _ r , E _ s ) = ( \\sum _ { r = 1 } ^ p a _ r P e _ r , P e _ s ) = 0 , s = 1 , \\cdots , p . \\end{align*}"}
+{"id": "7454.png", "formula": "\\begin{align*} & \\alpha \\preceq \\beta \\beta \\preceq \\alpha \\Rightarrow \\alpha = \\alpha , \\\\ & \\alpha \\preceq \\beta \\beta \\preceq \\gamma \\Rightarrow \\alpha \\preceq \\gamma . \\end{align*}"}
+{"id": "2376.png", "formula": "\\begin{align*} E _ { 2 2 } ( t ) \\dot x _ 2 = A _ { 2 2 } ( t ) x _ 2 + f _ 2 ( t ) \\end{align*}"}
+{"id": "2362.png", "formula": "\\begin{align*} V = { \\rm d i a g } ( 1 , 1 , \\cdots , 1 , - 1 ) U . \\end{align*}"}
+{"id": "1056.png", "formula": "\\begin{align*} \\tilde { f } = \\sum _ { j = 1 } ^ k \\left \\{ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\varphi _ j ( X _ i ) \\right \\} \\varphi _ j , \\end{align*}"}
+{"id": "5460.png", "formula": "\\begin{align*} & E V ( X ^ n _ { T \\wedge \\tau _ N ^ n } , \\alpha ^ n _ { T \\wedge \\tau _ N ^ n } ) \\leq e ^ { ( \\lambda _ 1 + \\lambda _ 2 ) T } E V ( X ^ n _ 0 , \\alpha ^ n _ 0 ) = : \\delta . \\end{align*}"}
+{"id": "3625.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } u = \\Delta _ { \\mathbb { R } ^ { n } } u \\ , + \\ , t ^ q \\ , | u | ^ { p - 1 } u & \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ u = 0 & \\hbox { o n } ~ \\partial \\Omega \\times ( 0 , T ) , \\\\ u = u _ 0 \\geq 0 & \\hbox { i n } ~ \\Omega \\times \\{ 0 \\} , \\end{array} \\right . \\end{align*}"}
+{"id": "946.png", "formula": "\\begin{align*} \\textrm { r . g r a d e } _ R ( \\Omega ^ { n - 1 } M , C ) \\ , = \\ , \\textrm { r . g r a d e } _ R ( M , C ) - n + 1 . \\end{align*}"}
+{"id": "3258.png", "formula": "\\begin{align*} \\frac { 1 } { f } = \\frac { 1 } { 1 - ( 1 - f ) } = \\sum \\limits _ { i = 0 } ^ { \\infty } ( 1 - f ) ^ i . \\end{align*}"}
+{"id": "5876.png", "formula": "\\begin{align*} S _ { r } = \\left ( \\begin{array} { c c c c c } 1 & - 1 & & & \\\\ & 1 & - 1 & & \\\\ & & \\ddots & \\ddots & \\\\ & & & 1 & - 1 \\end{array} \\right ) , 1 \\leq r \\leq p , \\end{align*}"}
+{"id": "3277.png", "formula": "\\begin{align*} \\lambda _ { 1 } = \\inf \\left \\{ \\frac { \\left \\Vert \\nabla u \\right \\Vert _ { p } ^ { p } } { \\underset { \\Omega } { \\int } \\left \\vert u \\right \\vert ^ { p _ { 0 } } \\left \\vert \\nabla u \\right \\vert ^ { p _ { 1 } } d x } \\left \\vert \\ u \\in B _ { 1 } ^ { \\overset { 0 } { W } \\ ^ { 1 . p } } \\left ( 0 \\right ) \\right . \\right \\} . \\end{align*}"}
+{"id": "23.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\mathbf { D } _ { 2 1 } = \\mathbb { E } \\int _ 0 ^ \\infty e ^ { - \\beta t } \\bar { \\mathcal { Z } } \\bar { f } _ x x _ 1 d t . \\end{align*}"}
+{"id": "7330.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\infty } \\frac { m ( \\gamma ' ) } { u ( \\gamma ' ) ^ 2 v ( \\gamma ' ) } \\int _ { 0 } ^ { \\gamma ' } f ( \\gamma '^ { - 1 } x ) j ( d x ) \\int _ { 0 } ^ { x } d y \\int _ { y } ^ { \\gamma ' } z d m ( z ) = 0 . \\end{align*}"}
+{"id": "9246.png", "formula": "\\begin{align*} J _ { X , j , h ( j ) } & = \\left \\langle d _ { R , C } : R \\subset \\left \\lbrace 1 , 2 , \\ldots n \\right \\rbrace C \\subset \\left \\lbrace u _ 1 , \\ldots u _ { h ( j ) } , X u _ 1 , \\ldots , X u _ j \\right \\rbrace \\right . \\\\ & \\qquad \\qquad \\left . \\ ; | R | = | C | = h ( j ) + 1 \\right \\rangle , \\\\ J ' _ { X , j , h ( j ) + 1 } & = \\left \\langle d _ { R , C } : R = \\{ 1 , \\ldots , h ( j ) + 1 \\} C = \\{ u _ 1 , \\ldots u _ { h ( j ) } , X u _ j \\} \\right \\rangle , \\end{align*}"}
+{"id": "8560.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { v - 1 } j ^ { i } d _ { w } ( j ) = \\frac { p ! \\lambda } { v } \\sum _ { j = 0 } ^ { v - 1 } j ^ { i } . \\end{align*}"}
+{"id": "6676.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { m ^ 2 } \\left \\| \\sum _ { i = 1 } ^ m g _ i ^ k \\right \\| ^ 2 & = \\frac { 1 } { m ^ 2 } \\left \\| \\sum _ { i = 1 } ^ m \\left ( g _ i ^ k - \\nabla f _ i ( \\theta ^ { \\ast } ) \\right ) \\right \\| ^ 2 \\\\ & \\leq \\frac { L ^ 2 } { m } \\sum _ { i = 1 } ^ m \\left \\| x _ i ^ k - \\theta ^ { \\ast } \\right \\| ^ 2 = \\frac { L ^ 2 } { m } \\| x ^ k - x ^ { \\ast } \\| ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "828.png", "formula": "\\begin{align*} \\widetilde { \\tau } _ { n p } ^ c ( \\gamma ) = \\frac { e _ \\ast ^ T A ^ { p \\lfloor c \\log n \\rfloor } e _ i \\ e _ j ^ T A ^ { p \\lfloor c \\log n \\rfloor } 1 } { e _ \\ast ^ T A ^ { n p } 1 } . \\end{align*}"}
+{"id": "6309.png", "formula": "\\begin{align*} \\Delta F _ \\alpha = \\sum _ { \\substack { ( \\beta , \\gamma ) \\\\ \\beta \\cdot \\gamma = \\alpha \\\\ \\beta \\odot \\gamma = \\alpha } } F _ \\beta \\otimes F _ \\gamma \\end{align*}"}
+{"id": "262.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\tau } \\leqslant 0 , \\end{align*}"}
+{"id": "9075.png", "formula": "\\begin{align*} L ^ { \\vee } ( n ) \\chi _ { z } ( P : \\sigma : \\lambda : \\eta ) = \\chi _ { z } ( P : \\sigma : \\lambda : \\eta ) \\qquad ( n \\in N _ { P } ) . \\end{align*}"}
+{"id": "3622.png", "formula": "\\begin{align*} g ( s ) : = \\left \\{ \\begin{array} { l l } s | \\ln s | ^ { - \\alpha } , & \\mbox { f o r } \\ s \\sim 0 \\\\ h ( s ) , & \\mbox { f o r } ~ s > \\frac { 1 } { 2 } , \\end{array} \\right . \\end{align*}"}
+{"id": "5295.png", "formula": "\\begin{align*} T _ U V = - g ( U , V ) \\nabla _ { \\mathcal { H } } \\log ( f _ 1 ) , \\end{align*}"}
+{"id": "3288.png", "formula": "\\begin{align*} { \\rm d i v } v _ 0 = 0 v _ 0 \\cdot n = ( \\ell _ 0 + \\omega _ 0 x ^ { \\bot } ) \\cdot n \\end{align*}"}
+{"id": "8792.png", "formula": "\\begin{align*} \\theta \\left ( \\eta _ { i } \\right ) = \\frac { \\alpha + \\theta ( t ) } { 1 + \\overline { \\theta ( t ) } \\alpha } , \\end{align*}"}
+{"id": "1914.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k e ^ { t _ j } \\leq e ^ { M } . \\end{align*}"}
+{"id": "1772.png", "formula": "\\begin{align*} \\Delta ^ { \\leq n } ( \\Sigma ) \\ : = \\ \\bigcup _ { k = 0 } ^ n \\Delta ^ k ( \\Sigma ) \\ = \\ \\big \\{ \\Delta ^ i u \\ , | \\ , u \\in \\Sigma , \\ , 0 \\leq k \\leq n \\big \\} \\ , \\end{align*}"}
+{"id": "6107.png", "formula": "\\begin{align*} P ( p ) = p / 2 + \\sum _ { k = 2 } ^ \\infty m _ k p ^ k \\end{align*}"}
+{"id": "374.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 \\leq \\frac { \\sqrt { 2 } C v ^ 2 _ 0 L ^ { - 2 } T ^ 2 } { 3 } \\sum _ { n = 1 } ^ \\infty p _ n ( T _ * ) \\ , \\frac { \\ln ( n + 1 ) } { n ^ { 3 / 2 } } \\ , . \\end{align*}"}
+{"id": "4492.png", "formula": "\\begin{align*} \\frac { 2 } { 1 3 } = \\frac { 1 } { 7 } + \\frac { 1 } { 9 1 } = \\frac { 1 } { 1 3 } + \\frac { 1 } { 2 6 } + \\frac { 1 } { 3 9 } + \\frac { 1 } { 7 8 } . \\end{align*}"}
+{"id": "6760.png", "formula": "\\begin{align*} \\beta _ n ( t ) = \\beta _ n \\left ( \\frac { 1 } { 2 } - j \\left ( \\omega + j \\sigma \\right ) , \\rho ( t ) \\right ) = n \\sqrt { 2 \\pi } \\exp \\left ( - \\frac { 1 } { 2 } \\left [ 1 + \\gamma _ { \\beta } \\right ] t \\right ) \\exp \\left ( \\frac { j } { 2 } \\omega _ { \\beta } t \\right ) \\end{align*}"}
+{"id": "4973.png", "formula": "\\begin{align*} R [ x , y ] ^ { \\phi } = R _ a [ f ] [ y ] ^ { \\widetilde { \\phi } } \\cap R [ x , y ] = R _ a [ f , q ] \\cap R [ x , y ] . \\end{align*}"}
+{"id": "3983.png", "formula": "\\begin{align*} g _ x ( x _ b , y _ b , z _ b ) = D \\overline { u } ( x _ b ) + s \\gamma , \\end{align*}"}
+{"id": "6706.png", "formula": "\\begin{align*} \\lambda ( n ) = \\sum _ { d ^ 2 \\mid n } \\mu ( n / d ^ 2 ) . \\end{align*}"}
+{"id": "5680.png", "formula": "\\begin{align*} \\Omega _ { b } ^ { a } = d \\omega _ { b } ^ { a } + \\omega _ { c } ^ { a } \\omega _ { b } ^ { c } . \\end{align*}"}
+{"id": "8140.png", "formula": "\\begin{align*} C = \\sqrt { N } + T , \\end{align*}"}
+{"id": "3864.png", "formula": "\\begin{align*} T _ { n + 1 } T _ n = \\sum _ { l = 2 } ^ n ( T _ l + T _ { l - 2 } ) T _ { n - l + 2 } ^ 2 . \\end{align*}"}
+{"id": "4772.png", "formula": "\\begin{align*} | T | = 2 ^ { \\epsilon N \\log \\frac { 4 \\epsilon ( 1 - \\epsilon ) } { ( 1 - \\eta ) ^ { 4 \\ln 2 } } + o ( N ) } \\end{align*}"}
+{"id": "5399.png", "formula": "\\begin{align*} x = \\sum _ { n \\in \\N } \\frac { a _ n } { \\prod _ { i = 0 } ^ n b _ i } . \\end{align*}"}
+{"id": "384.png", "formula": "\\begin{align*} b _ j : = \\left ( \\mathbb { E } _ { \\pi ^ * } \\left [ \\sum \\limits _ { \\ell = 1 } ^ { j - 1 } \\textsf { N } _ \\ell ( 0 , 2 L ^ { - 2 } _ * ) | ^ { 4 } \\right ] \\right ) ^ { 1 / 2 } = \\left ( \\mathbb { E } _ { \\pi ^ * } \\left [ \\textsf { N } _ \\ell ( 0 , 2 ( j - 1 ) L ^ { - 2 } _ * ) | ^ { 4 } \\right ] \\right ) ^ { 1 / 2 } = 2 ( j - 1 ) L ^ { - 2 } _ * \\end{align*}"}
+{"id": "3301.png", "formula": "\\begin{align*} \\| \\tilde { \\mathcal A } ^ { \\frac 1 2 } W \\| ^ 2 _ { L ^ 2 [ B _ 0 ] } = \\int _ { \\mathbb R ^ 2 } | \\nabla W | ^ 2 \\forall \\ , W \\in \\mathcal { D } ( \\tilde { \\mathcal A } ^ { \\frac 1 2 } ) . \\end{align*}"}
+{"id": "1836.png", "formula": "\\begin{align*} R \\begin{bmatrix} A & 0 \\\\ 0 & B \\end{bmatrix} R = \\begin{bmatrix} ( 1 - x ) A + x B & \\star \\\\ \\star & x A + ( 1 - x ) B \\end{bmatrix} \\end{align*}"}
+{"id": "4463.png", "formula": "\\begin{align*} u ' _ { \\ell - 1 } u ' _ { \\ell } = \\left ( \\frac { u _ { \\ell - 1 } } { t } \\right ) \\left ( t u _ { \\ell } \\right ) = u _ { \\ell - 1 } u _ { \\ell } \\end{align*}"}
+{"id": "7194.png", "formula": "\\begin{align*} = B _ { \\alpha } ( { \\Sigma _ 0 } ) \\left [ \\lambda y ^ 1 _ { \\alpha } + ( 1 - \\lambda ) y ^ 2 _ { \\alpha } \\right ] - D ( { \\Sigma _ 0 } ) . \\end{align*}"}
+{"id": "4.png", "formula": "\\begin{align*} \\begin{aligned} & \\leq \\mathbb { E } \\int _ t ^ \\infty e ^ { - \\beta s } \\big ( e ^ { - K _ 0 s } | y _ n - y | ^ 2 + e ^ { - K _ 1 s } | z _ n - z | ^ 2 + e ^ { - K _ 2 s } | \\tilde { z } _ n - \\tilde { z } | ^ 2 \\\\ & + e ^ { - K _ 3 s } | | \\gamma _ n - \\gamma | | ^ 2 \\big ) d s \\rightarrow 0 , \\ \\ n \\rightarrow \\infty , \\end{aligned} \\end{align*}"}
+{"id": "7160.png", "formula": "\\begin{align*} \\displaystyle { [ D _ i ^ { [ 0 ] } , D _ j ^ { [ 2 ] } ] + [ D _ i ^ { [ 1 ] } , D _ j ^ { [ 1 ] } ] + [ D _ i ^ { [ 2 ] } , D _ j ^ { [ 0 ] } ] = 0 \\ , , i , j = 1 , . . . , N \\ , . } \\end{align*}"}
+{"id": "5400.png", "formula": "\\begin{align*} x = \\sum _ { n \\in \\N } \\frac { a _ n } { \\beta ^ { i + 1 } } . \\end{align*}"}
+{"id": "8035.png", "formula": "\\begin{align*} ( p _ 1 , \\ldots , p _ 5 ) = \\| p _ 2 - p _ 1 \\| + \\| p _ 3 - p _ 2 \\| + \\| p _ 4 - p _ 3 \\| + \\| p _ 5 - p _ 4 \\| + \\| p _ 1 - p _ 5 \\| , \\end{align*}"}
+{"id": "6157.png", "formula": "\\begin{align*} b = \\sum _ { j = 1 } ^ N b _ { i j } , i = 1 , \\cdots , N , \\end{align*}"}
+{"id": "4248.png", "formula": "\\begin{align*} \\lim _ { t \\nearrow T ^ * } \\| u ( t ) \\| _ { \\Sigma _ \\gamma } = \\infty \\left ( \\lim _ { t \\searrow - T _ * } \\| u ( t ) \\| _ { \\Sigma _ \\gamma } = \\infty \\right ) . \\end{align*}"}
+{"id": "666.png", "formula": "\\begin{align*} * \\nu = d \\psi \\end{align*}"}
+{"id": "9197.png", "formula": "\\begin{align*} \\varphi = \\varphi ' + \\zeta \\end{align*}"}
+{"id": "421.png", "formula": "\\begin{align*} f _ n = g _ n U , ~ \\tau _ n = V \\omega _ n , \\forall n \\in \\mathbb { N } , \\end{align*}"}
+{"id": "215.png", "formula": "\\begin{align*} \\mathbb { K } _ { i , m } ^ { \\mathrm { s p } } : = \\{ g \\in \\mathrm { G S p i n } _ { L _ { i } } ( \\widehat { \\Z } ) \\mid g = 1 \\textup { i n } C ^ { + } ( L _ { i , \\widehat { \\Z } / m \\widehat { \\Z } } ) \\} , \\end{align*}"}
+{"id": "1632.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ^ { 2 } _ { \\mathbb { R } ^ d } ) u = - u | u | ^ { \\frac { 8 } { d } } , u ( 0 , x ) = u _ 0 ( x ) \\in L ^ 2 ( \\mathbb { R } ^ d ) , \\end{align*}"}
+{"id": "5897.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p \\alpha _ r u _ r = 0 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "8894.png", "formula": "\\begin{align*} \\mathcal { \\wp } = \\prod \\limits _ { l = \\frac { 1 } { 2 } + \\nu } ^ { \\frac { n } { 2 } + \\nu - 1 } \\left ( r ^ { 2 } - l ^ { 2 } \\right ) \\prod \\limits _ { s = \\frac { 1 } { 2 } - \\nu } ^ { \\frac { n } { 2 } - \\nu - 1 } \\left ( r ^ { 2 } - s ^ { 2 } \\right ) \\end{align*}"}
+{"id": "2148.png", "formula": "\\begin{align*} \\eta ( 0 ) = 0 , \\ ; \\eta ' ( t ) > 0 , \\ ; \\eta ( t ) \\leq \\eta _ 0 \\leq - \\xi _ 0 . \\end{align*}"}
+{"id": "6662.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { x } ^ { k + 1 } & = \\bar { x } ^ k + \\gamma _ 1 ^ k \\bar { \\zeta } _ w ^ k - \\lambda ^ k \\frac { ( u \\otimes { I _ d } ) ^ T } { m } \\left ( y ^ k - ( v \\otimes { I _ d } ) \\bar { y } ^ k \\right ) \\\\ & \\qquad - \\lambda ^ k \\frac { ( u \\otimes { I _ d } ) ^ T } { m } ( v \\otimes { I _ d } ) \\bar { y } ^ k \\end{aligned} \\end{align*}"}
+{"id": "3802.png", "formula": "\\begin{align*} \\Delta U ^ { ( a ) } + D _ X U ^ { ( a ) } & = \\Delta U ^ { ( a ) } + D _ { U ^ { ( a ) } } X - [ U ^ { ( a ) } , X ] \\\\ & = \\Delta U ^ { ( a ) } + ( U ^ { ( a ) } ) - [ U ^ { ( a ) } , X ] , \\end{align*}"}
+{"id": "3408.png", "formula": "\\begin{align*} ( \\mathcal { R } _ { \\alpha } ^ { - 1 } ( v \\cdot \\nabla \\zeta ) ) ( t , \\mathcal { R } _ { \\alpha } x ) = v ( t , x ) \\cdot \\nabla \\zeta _ { \\alpha } ( t , x ) . \\end{align*}"}
+{"id": "119.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow 0 ^ + } \\lambda ^ p \\mathcal L ^ { 2 n } ( \\widetilde E _ { \\lambda , K } ) = 2 | K | | | f | | ^ p _ { L ^ p ( \\mathbb R ^ n ) } . \\end{align*}"}
+{"id": "5307.png", "formula": "\\begin{align*} g ( R ( X , Y ) Z , V ) = \\widetilde { g } ( \\widetilde { R } ( X , Y ) Z , V ) - \\widetilde { g } ( \\mathrm { I I } ( X , Z ) , \\mathrm { I I } ( Y , V ) ) + \\widetilde { g } ( \\mathrm { I I } ( X , V ) , \\mathrm { I I } ( Y , Z ) ) , \\end{align*}"}
+{"id": "3243.png", "formula": "\\begin{align*} T _ 1 & : = \\{ ( s , t ) \\in \\mathbb { R } ^ 2 \\mid 0 < s < t < z \\} \\\\ T _ 2 & : = \\{ ( s , t ) \\in \\mathbb { R } ^ 2 \\mid 0 < t < s < z \\} \\\\ T _ 3 & : = \\{ ( s , t ) \\in \\mathbb { R } ^ 2 \\mid 0 < s = t < z \\} . \\end{align*}"}
+{"id": "8938.png", "formula": "\\begin{align*} b _ { j } ^ { \\left ( \\nu , 1 \\right ) } = 4 \\pi \\left [ \\frac { \\left ( \\frac { 1 } { 4 } + \\nu ^ { 2 } \\right ) ^ { j } } { j ! } + \\sum \\limits _ { i = 1 } ^ { j } \\frac { ( - 1 ) ^ i \\left ( \\frac { 1 } { 4 } + \\nu ^ { 2 } \\right ) ^ { j - i } } { \\left ( j - i \\right ) ! i ! } B _ { 2 i } ( \\nu + 1 / 2 ) \\right ] \\end{align*}"}
+{"id": "3071.png", "formula": "\\begin{align*} - A : D ^ 2 \\eta ^ { k l } = a _ { k l } - \\int _ Y r a _ { k l } \\quad Y , \\eta ^ { k l } Y , \\int _ Y \\eta ^ { k l } = 0 , \\end{align*}"}
+{"id": "7054.png", "formula": "\\begin{align*} \\Omega ^ { C _ 2 } _ * = \\{ f \\in M U ^ { C _ 2 } _ * : _ u \\phi ( f ) \\leq 0 \\} . \\end{align*}"}
+{"id": "6738.png", "formula": "\\begin{align*} \\Psi ( y ) = \\sum _ { n = 1 } ^ { \\infty } \\exp \\left ( - n ^ 2 \\pi y \\right ) . \\end{align*}"}
+{"id": "6803.png", "formula": "\\begin{align*} A [ q ( t ) ] = \\int { L ( q ( t ) , \\dot { q } ( t ) } ) d t . \\end{align*}"}
+{"id": "116.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\lambda ^ p \\mathcal L ^ { 2 n } ( E _ { \\lambda } ) = \\frac { 1 } { n } k ( p , n ) | | \\nabla f | | ^ p _ { L ^ p ( \\mathbb R ^ n ) } . \\end{align*}"}
+{"id": "7753.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n \\ , x \\ , f _ \\lambda ( x | \\tfrac { \\mu } { n } ) & = \\mu \\rho _ \\lambda ( x ) = \\mu \\ , e ^ { - \\lambda x } \\end{align*}"}
+{"id": "8765.png", "formula": "\\begin{align*} d _ \\infty ( ( u _ { 1 } , v _ 1 , w _ 2 ) , ( u _ { 1 } , v _ 1 , w _ 2 ) ) & \\leq \\sum \\limits _ { l = 1 } ^ { T - 1 } \\frac { 1 } { | \\bar { l } | } | v _ 1 w _ 1 ^ * - v _ 2 w _ 2 ^ * | _ { S _ 2 } \\\\ & + \\left | \\sum \\limits _ { l = 1 } ^ { T - 1 } \\frac { 1 } { | \\bar { l } | } ( e ^ { i 2 \\pi l ( u _ 1 - z ) } - e ^ { i 2 \\pi l ( u _ 2 - z ) } ) \\right | \\\\ & \\lesssim \\log ( T ) | v _ 1 - v _ 2 | _ 2 + \\log ( T ) | w _ 1 - w _ 2 | _ 2 + T | u _ 1 - u _ 2 | . \\end{align*}"}
+{"id": "9120.png", "formula": "\\begin{align*} f ( x ) = x ^ { e _ { 0 } } \\Phi _ { d _ { 1 } } ( x ) ^ { e _ { 1 } } \\Phi _ { d _ { 2 } } ( x ) ^ { e _ { 2 } } \\cdots \\Phi _ { d _ { m } } ( x ) ^ { e _ { m } } , \\end{align*}"}
+{"id": "8.png", "formula": "\\begin{align*} \\tilde { \\Phi } ^ \\epsilon : = \\frac { \\Phi ^ \\epsilon - \\bar { \\Phi } } { \\epsilon } - \\Phi _ 1 , \\ \\Phi = x , y , z , \\tilde { z } , \\gamma , \\mathcal { Z } , \\end{align*}"}
+{"id": "2850.png", "formula": "\\begin{align*} \\sigma = \\max ( | b _ k - \\rho | , | b _ { \\ell } - \\rho | \\} . \\end{align*}"}
+{"id": "5567.png", "formula": "\\begin{align*} J _ { Q } ( v , \\Omega ) : = \\int _ { \\Omega } | \\nabla v | ^ 2 + Q ^ 2 ( x ) \\chi _ { \\{ v > 0 \\} } d x \\end{align*}"}
+{"id": "7653.png", "formula": "\\begin{align*} \\iota _ { E } ( Q _ { I } \\eta _ { I } ) = Q _ { I } \\iota _ { E } ( \\eta _ { I } ) . \\end{align*}"}
+{"id": "6023.png", "formula": "\\begin{align*} P _ { l _ 2 } ( h _ 1 , v ) = 0 . \\end{align*}"}
+{"id": "548.png", "formula": "\\begin{align*} { \\rm d } I _ t = { \\rm d } X _ t ^ \\dagger - f ( X _ t ^ \\dagger , \\Theta _ t ) { \\rm d } t - \\gamma ^ { 1 / 2 } { \\rm d } W _ t , \\end{align*}"}
+{"id": "6177.png", "formula": "\\begin{align*} \\hbox { I m } ( D ) = \\hbox { S p a n } \\{ E _ 1 \\} ^ \\perp . \\end{align*}"}
+{"id": "7955.png", "formula": "\\begin{align*} \\pi ' i ( \\ell ) = i ' \\pi ( \\ell ) = 0 _ { N / L } . \\end{align*}"}
+{"id": "207.png", "formula": "\\begin{align*} \\int \\Big { | } \\frac { 1 } { n } \\sum \\limits _ { j = 0 } ^ { n - 1 } \\chi \\circ T ^ { - j k _ n } \\Big { | } d \\mu \\rightarrow 0 . \\end{align*}"}
+{"id": "4283.png", "formula": "\\begin{align*} \\Delta \\ , \\textbf { \\textit { v } } = 2 \\mathrm { d i v } \\mathrm { \\textbf { D } } ( \\textbf { \\textit { v } } ) - \\mathbf { g r a d } ( \\textbf { \\textit { v } } ) \\end{align*}"}
+{"id": "4328.png", "formula": "\\begin{align*} & 2 ^ { n _ 1 } ( 1 + o ( 1 ) ) \\left \\lfloor \\frac { C _ { t - k } } { \\binom { n _ 1 - w + k } { k } } \\right \\rfloor \\\\ & \\ge \\frac { 2 ^ n } { n _ 2 ^ { 2 ( t - k ) } \\binom { n _ 1 / 2 + \\sqrt { n _ 1 \\ln n _ 1 } + k } { k } } ( 1 + o ( 1 ) ) \\\\ & = \\frac { 2 ^ { n + k } k ! } { n _ 1 ^ k n _ 2 ^ { 2 ( t - k ) } } ( 1 + o ( 1 ) ) \\end{align*}"}
+{"id": "2391.png", "formula": "\\begin{align*} Q ^ T E Q = \\begin{bmatrix} \\Sigma & 0 \\\\ 0 & 0 \\end{bmatrix} \\end{align*}"}
+{"id": "7164.png", "formula": "\\begin{align*} \\displaystyle { u '^ { \\{ k \\} } _ m = c \\frac { d z _ m } { d t _ k ' } \\Big | _ { \\rm e q } } \\end{align*}"}
+{"id": "8098.png", "formula": "\\begin{align*} H _ { m , n } ^ + ( x ) & = \\frac { 4 M } { \\pi } \\int _ { u = - \\frac { T } { M } } ^ \\infty \\int _ { \\zeta = - \\infty } ^ \\infty \\cos ( x \\cosh \\zeta ) e \\Bigl ( \\frac { ( M u + T ) \\zeta } { \\pi } \\Bigr ) ( M u + T ) e ^ { - u ^ 2 } \\times { } \\\\ & \\qquad \\qquad { } \\times V ( m ^ 2 n , M u + T ) \\ , d \\zeta \\ , d u + O ( T ^ { - A } ) . \\end{align*}"}
+{"id": "3211.png", "formula": "\\begin{align*} z _ n : \\Omega \\to X , \\mathrm { v } _ n : \\Omega \\to Y ; \\\\ \\tilde { z } _ n : \\Omega \\to X , \\tilde { \\mathrm { v } } _ n : \\Omega \\to Y \\end{align*}"}
+{"id": "4417.png", "formula": "\\begin{align*} I ( a _ 1 ) = \\left ( \\frac { 1 } { a _ 1 } , \\frac { 1 } { a _ 1 - 1 } \\right ] = \\left ( \\frac { 1 } { a _ 1 } , \\frac { 1 } { a _ 1 } + \\frac { 1 } { a ^ 2 _ 1 - a _ 1 } \\right ] . \\end{align*}"}
+{"id": "2194.png", "formula": "\\begin{align*} F _ { n } ( z ) : = \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } \\varepsilon ^ { - j } F ( \\varepsilon ^ { j } z ) \\end{align*}"}
+{"id": "8088.png", "formula": "\\begin{align*} \\mathcal { R } ^ + _ 3 = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { c \\leq \\frac { C _ 2 } { m } } \\frac { S ( n , p ; c ) } { c } H _ { m , n } ^ + \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) . \\end{align*}"}
+{"id": "5670.png", "formula": "\\begin{align*} \\frac { d } { d s } \\frac { \\Lambda ^ n } { ( 1 + \\Lambda ) ^ n } = - \\frac { 1 } { s } \\frac { n { \\Lambda } ^ n } { ( 1 + { \\Lambda } ) ^ { n + 1 } } \\rightarrow - \\frac { n | \\Omega | } { 4 \\pi \\epsilon D \\bar { \\ell } } \\mbox { a s } \\ { s \\rightarrow 0 } . \\end{align*}"}
+{"id": "6019.png", "formula": "\\begin{align*} \\O _ u \\star \\O _ v = \\sum _ { \\gamma , w \\in W ^ P } \\delta ^ u \\delta ^ v \\delta ^ \\gamma \\mathcal { G } ( 0 ) \\ : \\mathcal { G } ^ { \\gamma , w } \\O _ w . \\end{align*}"}
+{"id": "1714.png", "formula": "\\begin{align*} I ( \\chi , m ) = \\left \\{ \\begin{array} { l l } ( - 1 ) ^ { m } 2 ^ { k } ( k - 1 ) \\binom { k - 2 } { \\frac { k - 2 } { 2 } - m } ^ { - 1 } , & \\chi ( - 1 ) = \\epsilon , \\\\ 0 , & \\chi ( - 1 ) = \\epsilon . \\end{array} \\right . \\end{align*}"}
+{"id": "8413.png", "formula": "\\begin{align*} ( \\partial _ { \\alpha } [ \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ] ) ' & = \\psi ' \\left ( \\frac { z _ { n } + 1 } { 2 } \\right ) z _ { n } ' \\partial _ { \\alpha } z _ { n } + \\psi \\left ( \\frac { z _ { n } + 1 } { 2 } \\right ) \\partial _ { \\alpha } z _ { n } ' \\\\ & + \\sum _ { i = 1 } ^ { n - 1 } \\psi ' \\left ( z _ { i } \\right ) z _ { i } ' \\partial _ { \\alpha } z _ i + \\psi \\left ( z _ i \\right ) \\partial _ { \\alpha } z _ i ' , \\end{align*}"}
+{"id": "8696.png", "formula": "\\begin{align*} \\mathcal { Q } ( u _ 1 , \\dots , u _ l ) & = \\mathcal F ( u _ 1 , . . , u _ l ; t ) | _ { t = - 1 } = \\Gamma ^ + ( u _ 1 ) | _ { t = - 1 } \\dots \\Gamma ^ + ( u _ l ) | _ { t = - 1 } ( 1 ) \\\\ & = \\prod _ { 1 \\le i < j \\le l } i _ { u _ i , u _ j } \\left ( \\frac { u _ i - u _ j } { u _ i + u _ j } \\right ) \\prod _ { i = 1 } ^ { l } Q ( u ) . \\end{align*}"}
+{"id": "2128.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) = \\partial _ { x x } u ( t , x ) + f \\left ( u ( t , x ) \\right ) , t > 0 , \\ , x \\in \\R . \\end{align*}"}
+{"id": "3470.png", "formula": "\\begin{align*} & f _ { i j } ^ { ( n ) } ( x _ 1 , \\ldots , x _ { n - 2 } , w , z , x ; t ) \\\\ = & \\int _ { \\{ 0 < t _ 1 < \\ldots < t _ { i - 1 } < \\theta < t _ i < \\ldots < t _ { j - 2 } < r < t _ { j - 1 } < \\ldots < t _ { n - 2 } < t \\} } g _ { n - j } ( t _ { j - 1 } , x _ { j - 1 } , \\ldots , t _ { n - 2 } , x _ { n - 2 } , r , z , t , x ) \\\\ & g _ { j - i - 1 } ( t _ { i } , x _ { i } , \\ldots , t _ { j - 2 } , x _ { j - 2 } , \\theta , w , r , z ) f _ { i - 1 } ( t _ 1 , x _ 1 , \\ldots , t _ { i - 1 } , x _ { i - 1 } , \\theta , w ) d t _ 1 \\ldots d t _ { n - 2 } d r d \\theta , \\end{align*}"}
+{"id": "7828.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ { p - 2 } \\sum _ { t = 1 } ^ { p - j - 1 } \\sum _ { s = 1 } ^ { t } x ^ { - 2 j + 2 t + 2 s } = \\frac { x ^ { - 2 p + 8 } ( 1 - x ^ { 2 p - 4 } ) ( 1 - x ^ { 2 p - 2 } ) ( 1 - x ^ { 2 p } ) } { ( 1 - x ^ 2 ) ( 1 - x ^ 4 ) ( 1 - x ^ 6 ) } \\end{align*}"}
+{"id": "3962.png", "formula": "\\begin{align*} \\tilde { g } ( x , y , z ) & = \\frac { g _ z ( x _ 0 , 0 , 0 ) } { g _ z ( x , 0 , 0 ) } [ g ( x , y , g ^ * ( x _ 0 , y , u ( x _ 0 ) - z ) ) - g ( x , 0 , 0 ) ] , \\end{align*}"}
+{"id": "8540.png", "formula": "\\begin{align*} S ( m ) & = \\frac { 1 } { 6 } ( m - 1 ) ( ( m - 1 ) ^ 2 + 3 m - 1 ) \\\\ & = \\frac { 1 } { 6 } ( m - 1 ) ( m ^ 2 - 2 m + 1 + 3 m - 1 ) \\\\ & = \\frac { 1 } { 6 } ( m - 1 ) ( m ^ 2 + m ) \\\\ & = \\frac { 1 } { 6 } ( m ^ 3 - m ) , \\end{align*}"}
+{"id": "7651.png", "formula": "\\begin{align*} H _ { Q } ^ { t } ( M ) = H ^ { t } ( C ( q _ { 1 } , \\dots , q _ { p } ; M ) ) . \\end{align*}"}
+{"id": "6921.png", "formula": "\\begin{align*} \\begin{gathered} \\min \\limits _ { u \\in \\Gamma } J ( u ) = \\int \\limits _ { 0 } ^ { t _ f } \\left ( w _ 1 x _ 3 ^ 2 ( t ) + w _ 2 u ^ 2 ( t ) \\right ) d t , \\\\ \\dot { x } ( t ) = A ( t ) x ( t ) + B ( x ( t ) ) u ( t ) + f ( x ( t ) ) , x ( 0 ) = ( S _ 0 , E _ 0 , I _ 0 , Q _ 0 , R _ 0 , D _ 0 , P _ 0 , W _ 0 ) , \\end{gathered} \\end{align*}"}
+{"id": "7928.png", "formula": "\\begin{align*} \\Delta g = \\frac { \\partial g } { \\partial \\lambda } \\ , , \\quad \\ , . \\end{align*}"}
+{"id": "7343.png", "formula": "\\begin{align*} \\ell ( x ) = \\left ( \\frac { \\log x } { \\log x _ 0 } \\right ) ^ { s - 1 } ( x > 0 ) . \\end{align*}"}
+{"id": "2892.png", "formula": "\\begin{align*} \\log ( 1 + | F _ { [ r ] } ( \\zeta ) | ) = \\log ( 1 + | ( F _ { [ r ] } \\circ \\Phi _ { \\zeta } ) ( 0 ) | ) \\leq \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | F _ { [ r ] } \\circ \\Phi _ { \\zeta } | ) d m _ \\infty . \\end{align*}"}
+{"id": "7795.png", "formula": "\\begin{align*} & \\ \\Psi _ { a , b } [ n ] ( y ^ { n ' } ) = \\begin{cases} \\displaystyle y ^ { n ' } ( 1 + q ^ { a } q ^ b y ^ n ) & \\{ n , n ' \\} > 0 , \\\\ \\displaystyle y ^ { n ' } ( 1 + q ^ { - a } q ^ b y ^ n ) ^ { - 1 } & \\{ n , n ' \\} < 0 . \\end{cases} \\end{align*}"}
+{"id": "2917.png", "formula": "\\begin{align*} \\partial _ t u = \\nu \\partial _ x ^ 2 u + \\lambda \\partial _ x u ^ 2 + \\sqrt { D } \\partial _ x \\dot { W } ( t , x ) . \\end{align*}"}
+{"id": "6501.png", "formula": "\\begin{align*} \\int \\left ( \\begin{matrix} x _ 1 \\\\ x _ 2 \\end{matrix} \\right ) \\left ( \\begin{matrix} x _ 1 ^ \\top & x _ 2 ^ \\top \\end{matrix} \\right ) q _ u ( x _ 1 , x _ 2 ) d ( P _ 0 \\times P _ 0 ) ( x _ 1 , x _ 2 ) = : \\Sigma \\in \\R ^ { 2 m d \\times 2 m d } , \\end{align*}"}
+{"id": "3863.png", "formula": "\\begin{align*} T _ { n + 2 } ^ 2 = p _ n ^ 2 + \\sum _ { k = 1 } ^ n \\sum _ { l = 1 } ^ k \\{ 4 ( T _ l + T _ { l - 1 } ) + \\delta _ { l , 1 } - 2 \\delta _ { l , 2 } - 2 c _ { l - 5 } \\} T _ { k - l + 2 } ^ 2 p _ { n - k } ^ 2 . \\end{align*}"}
+{"id": "4691.png", "formula": "\\begin{align*} C ^ { \\perp } : = \\lbrace \\mathbf { y } \\in \\mathbb { F } _ q ^ n \\ \\vert \\ ( \\mathbf { x } , \\mathbf { y } ) = 0 , \\forall \\mathbf { x } \\in C \\rbrace . \\end{align*}"}
+{"id": "6739.png", "formula": "\\begin{align*} \\phi ( s ) = - \\frac { 1 } { s } - \\frac { 1 } { 1 - s } + \\int _ { 1 } ^ { \\infty } \\left ( y ^ { \\frac { s } { 2 } - 1 } + y ^ { \\frac { 1 - s } { 2 } - 1 } \\right ) \\Psi ( y ) d y \\end{align*}"}
+{"id": "8383.png", "formula": "\\begin{align*} d n _ { \\mathcal { B } } ^ s ( X ) = \\aleph _ 0 + \\min \\{ & | F _ { \\mathcal { B } } ^ s | : F _ { \\mathcal { B } } ^ s C ( X ) \\\\ & \\mathcal { B } \\} . \\end{align*}"}
+{"id": "4870.png", "formula": "\\begin{align*} w _ { z ; k } w _ { \\zeta ; h } = \\frac 1 2 \\big ( w _ { z \\zeta ; k + h } + w _ { \\bar z \\zeta ; h - k } \\big ) . \\end{align*}"}
+{"id": "6922.png", "formula": "\\begin{align*} H ( t , x , \\psi , u ) = w _ 1 x _ 3 ^ 2 + w _ 2 u ^ 2 + \\psi ^ T \\left ( A ( t ) x + B ( x ) u + f ( x ) \\right ) . \\end{align*}"}
+{"id": "6743.png", "formula": "\\begin{align*} \\mathcal { E } \\left [ f ; ( m _ 1 , m _ 2 , \\rho ) \\right ] = \\int _ { - \\infty } ^ { \\infty } \\int _ { - \\infty } ^ { \\infty } f ( x _ 1 , x _ 2 ) p \\left [ x _ 1 , x _ 2 ; ( m _ 1 , m _ 2 , \\rho ) \\right ] d x _ 1 d x _ 2 . \\end{align*}"}
+{"id": "3645.png", "formula": "\\begin{align*} \\partial _ { t } u - \\Delta _ { \\mathbb { H } ^ { n } } u = \\left ( \\frac { \\beta _ { 0 } } { \\beta } \\right ) ^ { \\frac { 1 } { p } } \\big ( \\partial _ { t } v - \\Delta _ { \\mathbb { H } ^ { n } } v \\big ) = e ^ { \\mu t } u \\big ( e ^ { \\beta u ^ { p } } - 1 \\big ) ~ \\hbox { i n } \\mathbb { H } ^ { n } \\times ( 0 , + \\infty ) . \\\\ \\end{align*}"}
+{"id": "7714.png", "formula": "\\begin{align*} S ^ { ( i ) } = \\Bigg | t _ 0 ^ { ( i ) } + \\sqrt { \\left ( t _ 0 ^ { ( i ) } \\right ) ^ 2 - 1 } \\Bigg | , i = 1 , 2 . \\end{align*}"}
+{"id": "9117.png", "formula": "\\begin{align*} \\mathsf { K } _ n : = \\sup _ { y ^ n \\in \\mathcal { Y } ^ n } \\frac { d P _ { Y ^ n } ( y ^ n ) } { d Q _ { Y ^ n } ( y ^ n ) } . \\end{align*}"}
+{"id": "8569.png", "formula": "\\begin{align*} X = \\bigcup _ { i = 0 } ^ { 5 } G f ^ { i } \\end{align*}"}
+{"id": "4983.png", "formula": "\\begin{align*} \\| G _ { p , q } \\| = 1 \\ , . \\end{align*}"}
+{"id": "8746.png", "formula": "\\begin{align*} | \\hat { \\mathcal { H } } _ d - H g _ 0 ^ * | _ { S _ 2 } & = | H \\hat { g } - H g _ 0 ^ * - \\hat { \\mathcal { H } } _ { \\bar { d } } | _ { S _ 2 } \\leq | H \\Delta g | _ { S _ 2 } + | \\hat { \\mathcal { H } } _ { \\bar { d } } | _ { S _ 2 } \\\\ & \\leq 2 | H \\Delta g ^ * | _ { S _ 2 } \\leq \\frac { 1 0 \\sqrt { 2 } } { 3 \\sigma _ u ^ 2 } d _ 0 ^ { 1 / 2 } \\lambda T , \\end{align*}"}
+{"id": "648.png", "formula": "\\begin{align*} | G | \\geq \\delta ^ { \\xi } | A | | B | | \\{ x + c y : ( x , y ) \\in G \\} | _ { \\delta } = | \\pi _ { c } ( G ) | _ { \\delta } \\leq \\delta ^ { - \\xi } | A | . \\end{align*}"}
+{"id": "1003.png", "formula": "\\begin{align*} \\phi _ 1 \\big ( \\tfrac 1 3 , y \\big ) = \\ & \\phi _ 1 \\big ( \\tfrac 2 3 , y \\big ) = \\frac { \\sin ( 2 \\pi y ) } { \\sin ( \\pi y ) } , \\\\ \\phi _ 2 \\big ( \\tfrac 1 3 , y \\big ) = - & \\phi _ 2 \\big ( \\tfrac 2 3 , y \\big ) = \\frac { \\sin ( 2 \\pi y ) } { \\sin ( \\pi y ) } , \\\\ \\phi _ 3 \\big ( \\tfrac 1 3 , y \\big ) = \\ & \\phi _ 3 \\big ( \\tfrac 2 3 , y \\big ) = \\frac { \\sin ( 3 \\pi y ) } { \\sin ( \\pi y ) } . \\end{align*}"}
+{"id": "6363.png", "formula": "\\begin{align*} c = - \\int ^ \\infty _ 0 \\tilde m '' ( r ) d r \\int ^ { 2 \\pi } _ 0 d \\theta = 2 \\pi ( 1 - \\lim _ { r \\to \\infty } \\tilde m ' ( r ) ) . \\end{align*}"}
+{"id": "6003.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( [ 1 : 0 ] ) & = \\{ [ 1 : 0 ] \\} \\times g \\cdot X _ { i , j } ^ { k , l } \\\\ \\pi ^ { - 1 } ( [ 0 : 1 ] ) & = \\{ [ 0 : 1 ] \\} \\times Y _ { i + k - n - 1 , j + l - 1 - i - k + n } ^ { i + k - n - 1 } \\cup \\{ [ 0 : 1 ] \\} \\times Y _ { i + k - n , j + l - 1 - i - k + n } ^ { i + k - n } \\end{align*}"}
+{"id": "4817.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial c } f ( \\epsilon , c ) & = - \\frac { 2 \\epsilon } { c \\ln 2 } + \\epsilon \\end{align*}"}
+{"id": "3563.png", "formula": "\\begin{align*} \\gamma ^ + : = \\gamma - e _ J + \\sum _ { u = 2 } ^ s e _ { v _ u i _ { u - 1 } } - e _ { v _ u i _ u } \\end{align*}"}
+{"id": "6299.png", "formula": "\\begin{align*} \\pi _ i ( T ) = \\begin{cases} T & , \\\\ s _ i ( T ) & s _ i ( T ) \\in S T ( \\alpha ) , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "4146.png", "formula": "\\begin{align*} d K = 3 \\Lambda + N \\end{align*}"}
+{"id": "1452.png", "formula": "\\begin{align*} b = \\frac { | K | } { | K _ \\Delta | } = \\frac { m ! n ! } { | K _ \\Delta | } . \\end{align*}"}
+{"id": "9130.png", "formula": "\\begin{align*} Q _ { h , \\nu } ( \\emptyset | \\emptyset ) \\ ; = \\ ; 1 , \\ ; \\ ; Q _ { h , \\nu } ( \\emptyset | \\gamma ) \\ ; = \\ ; 0 , \\ ; \\ ; \\ ; \\ ; \\gamma \\neq \\emptyset , \\end{align*}"}
+{"id": "1916.png", "formula": "\\begin{align*} m _ \\mathrm { A D M } ( \\mathcal E , g ) = \\lim _ { r \\to \\infty } \\frac { 1 } { 2 ( n - 1 ) \\omega _ { n - 1 } } \\int _ { | x | = r } ( g _ { i j , i } - g _ { i i , j } ) \\frac { x ^ j } { | x | } \\ , d \\mu _ { S _ r , \\overline g } . \\end{align*}"}
+{"id": "293.png", "formula": "\\begin{align*} \\begin{aligned} { \\zeta _ { S } } _ { \\chi _ { \\mathrm { B C } } , { \\chi ' } _ { \\mathrm { B C } } } = { \\zeta _ { S } } _ { \\chi , \\chi ' } \\circ \\mathrm { N m } _ { S ( E ) / S ( F ) } . \\end{aligned} \\end{align*}"}
+{"id": "152.png", "formula": "\\begin{align*} \\mathbb { Y } : = \\{ \\varkappa \\in \\mathcal { A } : \\| \\cdot \\| _ { P C } \\leq \\alpha _ { 2 } \\} , \\end{align*}"}
+{"id": "849.png", "formula": "\\begin{align*} L ( R , \\Omega ) = K = \\frac { 1 } { 2 } t r ( \\widehat { \\Omega } J _ d \\widehat { \\Omega } ^ T ) \\end{align*}"}
+{"id": "5976.png", "formula": "\\begin{align*} \\dim Z _ 0 - \\dim h ( Z _ 0 ) & = \\dim { h } ^ { - 1 } ( \\nu ) \\cap Z _ 0 ^ \\circ \\\\ & = \\mathrm { d i m } M - \\mathrm { d i m } ( h \\times f ) ( M ) \\end{align*}"}
+{"id": "609.png", "formula": "\\begin{align*} h ( \\Delta t ) : = \\Delta t ^ { - 2 } \\left \\| \\mathbb { E } ^ \\dagger \\left [ X ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } \\otimes X ^ { ( \\epsilon ) } _ { t _ { n + 1 } , t _ { n + 2 } } \\right ] \\right \\| \\end{align*}"}
+{"id": "2134.png", "formula": "\\begin{align*} \\int _ { \\R } \\psi ( t , x ) d x = 1 - e ^ { - t } , \\forall t \\ge 0 . \\end{align*}"}
+{"id": "8554.png", "formula": "\\begin{align*} x \\left ( n ; \\beta \\right ) : = \\left \\{ \\begin{array} { l l } \\pi _ { \\left ( n ; \\beta \\right ) } ^ { \\left ( k , m ; \\gamma _ { k } , \\tau \\right ) } t _ { k } & \\left ( n ; \\beta \\right ) \\leq \\left ( k , m ; \\gamma _ { k } , \\tau \\right ) k \\in \\mathbb { N } \\\\ 0 & \\end{array} \\right . \\end{align*}"}
+{"id": "2005.png", "formula": "\\begin{align*} a _ { j l } = \\phi ^ { j l } , \\end{align*}"}
+{"id": "3362.png", "formula": "\\begin{align*} \\eta \\lrcorner \\xi = 1 , \\Phi ^ 2 X = - X + \\eta ( X ) \\xi . \\end{align*}"}
+{"id": "2932.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty x ^ k \\frac { ( a ; q ) _ k } { ( q ; q ) _ k } = \\frac { ( a x ; q ) _ \\infty } { ( x ; q ) _ \\infty } \\end{align*}"}
+{"id": "5584.png", "formula": "\\begin{align*} 0 = \\nabla _ { j } \\nabla _ { i } k _ 0 = \\nabla _ j ( \\nabla k ) _ { i 0 } \\stackrel { ( i ) } { = } \\nabla _ j k _ i \\nabla _ 1 k _ 0 - \\nabla ^ l k _ j { } \\nabla _ i k _ l \\stackrel { ( i i ) } { = } - \\rho ^ l { } _ j \\rho _ { l i } \\end{align*}"}
+{"id": "3599.png", "formula": "\\begin{align*} \\Omega _ \\lambda = \\lambda ! \\omega _ \\lambda & = \\lambda ! [ B _ 1 ^ + , \\dots , B _ n ^ + ] ^ { \\lambda _ n - \\lambda _ { n + 1 } } \\cdots [ B _ 1 ^ + , B _ 2 ^ + ] ^ { \\lambda _ 2 - \\lambda _ 3 } ( B _ 1 ^ + ) ^ { \\lambda _ 1 - \\lambda _ 2 } v _ 0 \\\\ & = \\frac { \\lambda ! } { \\mu ! } [ B _ 1 ^ + , \\dots , B _ n ^ + ] ^ { \\lambda _ n - \\lambda _ { n + 1 } } \\cdots [ B _ 1 ^ + , \\dots , B _ j ^ + ] ^ { \\lambda _ j - \\lambda _ { j + 1 } } \\Omega _ \\mu . \\end{align*}"}
+{"id": "4824.png", "formula": "\\begin{align*} d _ 0 ^ n F ( v ) = \\frac { d ^ n } { d t ^ n } F ( t v ) \\big | _ { t = 0 } , v \\in X . \\end{align*}"}
+{"id": "4176.png", "formula": "\\begin{align*} \\| ( \\widetilde { g } _ i ( t _ i ) , \\widetilde { b } _ i ( t _ i ) ) - ( g _ c , 0 ) \\| _ { C ^ { k - 2 } } = \\epsilon \\end{align*}"}
+{"id": "8992.png", "formula": "\\begin{align*} \\int _ { B _ 1 ( 0 ) \\cap \\Omega _ k } & | \\nabla u _ k ( 0 ) | ^ 2 d z \\\\ & = \\sup _ { z _ k + r _ k z ' \\in B _ { r _ 0 } ( z _ 0 ) , - t _ k / r _ k \\le t < 0 } \\int _ { B _ 1 ( z ' ) \\cap \\Omega _ k } | \\nabla u _ k ( t ) | ^ 2 d z = \\delta . \\end{align*}"}
+{"id": "7041.png", "formula": "\\begin{align*} \\tau = [ S ^ 2 \\to \\Omega _ { C _ 2 } ( \\mathbf { C } ^ \\sigma ) ] \\in \\Omega ^ { C _ 2 } _ { 2 - \\sigma } . \\end{align*}"}
+{"id": "5996.png", "formula": "\\begin{align*} Y _ { r , p } ^ h : = \\{ ( L , H ) \\in X | L \\subset < e _ 1 , \\dots e _ h > , \\ : < e _ 1 , \\dots e _ r , e _ { n - p + 1 } , \\dots , e _ n > \\subset H \\} . \\end{align*}"}
+{"id": "8719.png", "formula": "\\begin{align*} \\tilde F _ \\lambda ( x _ 1 , \\dots , x _ n ; t ) = \\frac { ( 1 - t ) ^ n } { \\prod _ { i = 1 } ^ { n - l } ( 1 - t ^ i ) } \\sum _ { \\sigma \\in S _ n } \\sigma \\left ( ( x _ { 1 } | a ) _ { \\lambda _ 1 - 1 } \\dots ( x _ { l } | a ) _ { \\lambda _ n - 1 } \\prod _ { i = 1 } ^ { n } { x _ i } \\prod _ { i < j } \\frac { x _ { i } - t x _ { j } } { x _ { i } - x _ { j } } \\right ) , \\end{align*}"}
+{"id": "1547.png", "formula": "\\begin{align*} \\{ [ \\gamma _ \\mu , \\gamma _ \\nu ] , \\gamma _ \\tau \\} = [ \\gamma _ \\mu , \\{ \\gamma _ \\nu , \\gamma _ \\tau \\} ] - \\{ \\gamma _ \\nu , [ \\gamma _ \\mu , \\gamma _ \\tau ] \\} = \\{ \\gamma _ \\nu , [ \\gamma _ \\tau , \\gamma _ \\mu ] \\} \\end{align*}"}
+{"id": "5498.png", "formula": "\\begin{align*} F _ { \\lambda _ 0 } ( ( x ) ) & = - J \\dot { ( ( x ) ) } - \\nabla H ( ( x ) ) + 2 \\pi \\lambda _ 0 ( x ) \\\\ & = ( - J \\dot x ) + 2 \\pi - ( \\nabla H ( x ) ) + 2 \\pi \\lambda _ 0 ( x ) \\\\ & = ( - J \\dot x - \\nabla H ( x ) + 2 \\pi ( \\lambda _ 0 + 1 ) x ) \\\\ & = ( F _ { \\lambda _ 0 + 1 } ( x ) ) , \\end{align*}"}
+{"id": "3234.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { c \\in F } m _ 2 ( c ) & = \\sum _ { c \\in C } m _ 2 ( c ) - \\sum _ { c \\in C - F } m _ 2 ( c ) = 2 k - m _ 2 \\ge k + | S _ { f _ 1 , 2 } | . \\end{aligned} \\end{align*}"}
+{"id": "4330.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ s \\binom { n - w + i } { i } z _ { w - i } + \\sum \\limits _ { j = 0 } ^ { t - s } \\binom { w + j } { j } z _ { w + j } \\le \\binom { n } { w } \\end{align*}"}
+{"id": "8223.png", "formula": "\\begin{align*} h ( A ) = \\prod _ { i = 1 } ^ n a _ { i , i } . \\end{align*}"}
+{"id": "1350.png", "formula": "\\begin{align*} \\mathcal { Z } ( d , \\mathbb { K } ) \\coloneqq \\left \\{ \\begin{array} { c c } d ^ 2 & \\mathbb { K } = \\mathbb { C } \\\\ \\frac { d ( d + 1 ) } { 2 } & \\mathbb { K } = \\mathbb { R } . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "5608.png", "formula": "\\begin{align*} \\nabla _ { X } S _ { 1 1 } = - 2 S _ { 0 1 } \\nabla _ { X } l _ { 1 } = 0 , \\end{align*}"}
+{"id": "706.png", "formula": "\\begin{align*} B = \\oint _ \\beta \\eta = \\oint _ \\beta \\nu + \\oint _ \\beta * d G ^ \\omega = b + \\oint _ \\beta * d G ^ \\omega . \\end{align*}"}
+{"id": "8216.png", "formula": "\\begin{align*} P \\{ V ( 0 ) = v _ 0 \\ | \\ \\mathcal { T } ( t ) = x , N ( t ) = n \\} = \\begin{cases} \\begin{array} { l l } \\frac { | v _ 1 t - x | } { ( a _ 1 - a _ 2 ) t } & \\ n \\ \\\\ \\frac { 1 } { 2 } & \\ n \\ \\end{array} \\end{cases} \\end{align*}"}
+{"id": "5412.png", "formula": "\\begin{align*} \\delta ' ( q _ { i , j , k } , s ) = q _ { ( j + 1 ) \\bmod p , ( j + 1 ) \\bmod p , 0 } . \\end{align*}"}
+{"id": "1502.png", "formula": "\\begin{align*} G = \\frac 1 { 4 ^ 2 } \\left ( \\zeta \\left ( 2 , \\frac 1 4 \\right ) - \\zeta \\left ( 2 , \\frac 3 4 \\right ) \\right ) \\end{align*}"}
+{"id": "3381.png", "formula": "\\begin{align*} ( i t ^ { - u _ { i + 1 } } + t ^ { - u _ { i + 2 } + 1 } ) x _ i \\geq \\sum _ { j = i + 2 } ^ { n - 1 } t ^ { - u _ j } x _ j + x _ n + t ^ { - u _ n } \\ , . \\end{align*}"}
+{"id": "7866.png", "formula": "\\begin{align*} f ^ { ( n ) } + P _ { n - 1 } ( z ) f ^ { ( n - 1 ) } + \\cdots + P _ 0 ( z ) f = h ( z ) e ^ { Q ( z ) } , \\end{align*}"}
+{"id": "6987.png", "formula": "\\begin{gather*} \\vec { t } = \\begin{pmatrix} t _ { 1 , n } \\\\ t _ { 2 , n } \\\\ \\vdots \\\\ t _ { n , n } \\end{pmatrix} \\ ! , \\vec { b } _ { \\{ 1 , 1 \\} } = \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ Q _ n ( x ) \\end{pmatrix} \\ ! , \\end{gather*}"}
+{"id": "754.png", "formula": "\\begin{align*} \\mu | _ { \\gamma } ( P ( X , Y ) ) : = \\mu ( \\gamma \\cdot P ( ( X , Y ) ) ) , \\end{align*}"}
+{"id": "2827.png", "formula": "\\begin{align*} x ^ 3 y ^ 2 = \\sqrt [ 5 ] { x ^ { 1 5 } y ^ { 1 0 } } \\leq \\frac { 3 x ^ 5 + 2 y ^ 5 } { 5 } \\end{align*}"}
+{"id": "1865.png", "formula": "\\begin{align*} \\alpha _ { s _ i } ( \\sigma ) = \\left \\{ \\begin{array} { l l } \\frac { \\pi } { 2 } & \\mbox { i f $ s _ i + ( p _ { s _ i } \\sigma + q _ { s _ i } ) - \\phi _ k \\ge 0 $ } \\\\ \\cos ^ { - 1 } \\left ( \\frac { s _ i - \\phi _ k + p _ { s _ i } \\sigma + q _ { s _ i } } { p _ { s _ i } \\sigma + q _ { s _ i } } \\right ) & \\mbox { i f $ s _ i + ( p _ { s _ i } \\sigma + q _ { s _ i } ) - \\phi _ k \\le 0 $ } . \\end{array} \\right . \\end{align*}"}
+{"id": "4263.png", "formula": "\\begin{align*} \\| ( \\nabla - i A ) g _ n \\| ^ 2 _ { L ^ 2 } & = \\| ( \\nabla - i A ) \\phi \\| ^ 2 _ { L ^ 2 } + \\| ( \\nabla - i A ) ( g _ n - \\phi ) \\| ^ 2 _ { L ^ 2 } + o _ n ( 1 ) , \\\\ \\int _ { \\R ^ N } V _ \\gamma ( x ) | g _ n ( x ) | ^ 2 d x & = \\int _ { \\R ^ N } V _ \\gamma ( x ) | \\phi ( x ) | ^ 2 d x + \\int _ { \\R ^ N } V _ \\gamma ( x ) | g _ n ( x ) - \\phi ( x ) | ^ 2 d x + o _ n ( 1 ) . \\end{align*}"}
+{"id": "7379.png", "formula": "\\begin{align*} ( \\varphi _ { \\chi \\otimes \\sigma } , \\varrho _ { \\chi \\otimes \\sigma } ) = \\chi ^ \\vee \\cdot ( \\varphi _ \\sigma , \\varrho _ \\sigma ) , \\end{align*}"}
+{"id": "1696.png", "formula": "\\begin{align*} \\delta s ( \\mu _ m ) = \\sum _ { \\frac { 2 - k } { 2 } \\leq n \\leq \\frac { k - 2 } { 2 } } \\mu _ m ( P _ { n + \\frac { k - 2 } { 2 } } ) \\left ( \\frac { 1 } { 2 } \\left ( \\frac { k } { 2 } + n \\right ) f _ { n + 1 } + \\frac { 1 } { 2 } \\left ( \\frac { k } { 2 } - n \\right ) f _ { n - 1 } + m f _ n \\right ) . \\end{align*}"}
+{"id": "8638.png", "formula": "\\begin{align*} f _ { R } ( r ) : = \\frac { f _ \\infty ( r ) } { f _ \\infty ( R ) } , f _ \\infty ( r ) : = \\begin{cases} \\ln \\left ( \\frac { \\tanh ( r / 2 ) } { \\tanh ( a / 2 ) } \\right ) & : d = 2 , \\\\ 1 - \\frac { \\tanh a } { \\tanh r } & : d = 3 , \\end{cases} \\end{align*}"}
+{"id": "2304.png", "formula": "\\begin{align*} B _ m ( x , t ) = t ^ { - q } \\left [ \\left ( 1 - \\frac { q ( m - 1 ) } { 2 m } \\frac { x ^ 2 } { t ^ { 2 q } } \\right ) _ + \\right ] ^ { 1 / ( m - 1 ) } , ~ ~ m > 1 \\end{align*}"}
+{"id": "6402.png", "formula": "\\begin{align*} \\eta & < - ( m + p ) ( \\partial _ t - \\tilde X _ { j _ l } \\cdot \\nabla _ Y ) \\varphi _ { j _ l } ( \\tilde X _ { j _ l } , \\tilde Y _ { j _ l } , \\tilde t _ { j _ l } ) + ( m + p ) ( \\partial _ t - X _ { j _ l } \\cdot \\nabla _ Y ) \\psi _ { j _ l } ( X _ { j _ l } , Y _ { j _ l } , t _ { j _ l } ) \\\\ & = { { j _ l ^ 4 } } ( \\tilde X _ { j _ l } - X _ { j _ l } ) \\cdot ( \\tilde Y _ { j _ l } - Y _ { j _ l } ) | \\tilde Y _ { j _ l } - Y _ { j _ l } | ^ 2 \\\\ & = 0 , \\end{align*}"}
+{"id": "4088.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = 0 } \\mathcal { G } _ t = ( L _ X g , L _ X b - \\frac { d } { d t } \\Big | _ { t = 0 } B _ t ) . \\end{align*}"}
+{"id": "5383.png", "formula": "\\begin{align*} \\exp \\bigg ( 2 \\sum _ { m = 1 } ^ M \\sigma _ { m , W } ^ 2 \\bigg ) \\prod _ { m = 1 } ^ M \\frac { 1 } { \\sqrt { 2 \\pi \\sigma _ { m , W } ^ 2 } } \\idotsint \\limits _ { \\mathcal { R } ' } \\exp \\bigg ( - \\sum _ { m = 1 } ^ M \\frac { z _ m '^ 2 } { 2 \\sigma _ { m , W } ^ 2 } \\bigg ) \\ , d \\mathbf { z } ' , \\end{align*}"}
+{"id": "2152.png", "formula": "\\begin{align*} \\dfrac { z F ' ( z ) } { F ( z ) } = \\dfrac { z f ' ( z ) } { f ( z ) } + \\dfrac { \\beta } { n } \\dfrac { z Q ' ( z ) } { Q ( z ) } . \\end{align*}"}
+{"id": "214.png", "formula": "\\begin{align*} D _ { i } ( m ) : = f _ { i } ( \\mathbb { K } _ { i , m } ^ { \\mathrm { s p } } ) , \\end{align*}"}
+{"id": "5489.png", "formula": "\\begin{align*} F = L + K \\end{align*}"}
+{"id": "7865.png", "formula": "\\begin{align*} t ^ 2 v '' + t \\bigl ( 1 + p ( t ) \\bigr ) v ' + q ( t ) v = 0 . \\end{align*}"}
+{"id": "4148.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\lambda ( u g + h , d K ) & \\leq \\int _ M \\langle \\frac { 1 } { 2 } \\triangle h , h \\rangle - \\langle \\mathring { R } h , h \\rangle - \\mu | h | ^ 2 + \\frac { 3 } { 2 } | \\Lambda | ^ 2 d V _ g \\\\ & = \\int _ M \\langle \\frac { 1 } { 2 } \\triangle h , h \\rangle + 3 \\langle \\mathring { R } h , h \\rangle + \\mu | h | ^ 2 d V _ g = \\int _ M \\langle \\triangle _ G h , h \\rangle d V _ g \\leq 0 \\end{align*}"}
+{"id": "2398.png", "formula": "\\begin{align*} E _ { 3 3 } ^ T = - E _ { 3 3 } \\ . , A _ { 2 2 } ^ T = A _ { 2 2 } \\ . , A _ { 3 2 } ^ T = A _ { 2 3 } \\ . , A _ { 3 3 } ^ T = A _ { 3 3 } + \\dot E _ { 3 3 } . \\end{align*}"}
+{"id": "7142.png", "formula": "\\begin{align*} E _ { i + 1 } \\tilde { f } ( B _ { i + 1 } \\otimes B _ i ) B _ { i - 1 } & = B _ { i + 1 } \\tilde { f } ( B _ { i } \\otimes B _ { i - 1 } ) E _ { i - 2 } = B _ { i + 1 } C _ i B _ { i - 1 } . \\end{align*}"}
+{"id": "284.png", "formula": "\\begin{align*} \\begin{aligned} \\{ \\prescript { L } { } { j } : \\prescript { L } { } { S } \\rightarrow \\prescript { L } { } { G } \\} / \\widehat { j } ( \\widehat { S } ) & \\leftrightarrow H ^ 1 ( W _ F , \\widehat { S } ) \\\\ \\prescript { L } { } { j } & \\mapsto w \\mapsto \\varphi _ { \\prescript { L } { } { j } } ( w ) \\cdot \\varphi _ { \\prescript { L } { } { j _ 0 } } ( w ) ^ { - 1 } . \\end{aligned} \\end{align*}"}
+{"id": "241.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - \\beta \\tau \\eta \\leqslant - \\frac { \\alpha } { 2 } ( 1 + 3 \\rho ^ 2 ) \\leqslant - \\frac { \\alpha } { 2 } , \\end{align*}"}
+{"id": "4187.png", "formula": "\\begin{align*} \\partial _ { t } ^ 2 ( \\ln f ) - \\partial _ { x } ^ 2 ( \\ln f ) = G , \\end{align*}"}
+{"id": "2413.png", "formula": "\\begin{align*} E ( t ) \\dot x = A ( t ) x \\end{align*}"}
+{"id": "6088.png", "formula": "\\begin{align*} h _ n ( \\tau ) = \\log \\left ( \\frac { \\varkappa _ n \\gamma _ n \\gamma _ n ^ * } { G _ { \\lambda _ n } \\rho ^ { 2 ( n - d _ n ) } } \\right ) + o ( 1 ) \\end{align*}"}
+{"id": "2091.png", "formula": "\\begin{align*} \\dim _ P ( \\mathcal { S } _ K ^ { \\ast } ( w ) ) \\geq ( 1 - w ) \\dim ( K ) = ( 1 - w ) \\frac { \\sum _ { i = 1 } ^ { m } \\log | W _ i | } { \\log b } > 0 . \\end{align*}"}
+{"id": "365.png", "formula": "\\begin{align*} C ^ 2 _ 1 \\leq \\frac { \\widetilde { C } T ^ { p - 1 } _ * } { p L ^ { 2 ( p - 1 ) } _ * } + \\frac { { C ^ \\prime } T ^ { p - 2 } _ * } { ( p - 1 ) L ^ { 2 ( p - 1 ) } _ * } , C ^ 2 _ 2 = \\frac { 2 ^ { p - 1 } \\Gamma ( \\frac { 2 p - 1 } { 2 } ) } { p \\sqrt { \\pi } } \\frac { T ^ { p - 1 } _ * } { L ^ { 2 ( p - 1 ) } _ * } \\ , , \\end{align*}"}
+{"id": "4473.png", "formula": "\\begin{align*} v = w = \\frac { n ( n + 1 ) } { 2 } . \\end{align*}"}
+{"id": "8939.png", "formula": "\\begin{align*} u _ 0 ^ 1 = 1 , u _ i ^ 1 = \\frac { B _ { i - 1 } } { ( i - 1 ) ! } , \\ i \\geq 1 \\end{align*}"}
+{"id": "1234.png", "formula": "\\begin{align*} \\bar \\mu ^ y ( x ) = \\bar \\mu ( x ) / [ \\bar \\nu ( y ) J T ^ y ( x ) ] \\end{align*}"}
+{"id": "1361.png", "formula": "\\begin{align*} \\int _ { \\Omega } f _ \\alpha ( x ) \\tau _ \\alpha \\ , d \\mu ( \\alpha ) & = \\sum _ { j = 1 } ^ n \\int _ { \\Omega _ j } f _ \\alpha ( x ) \\tau _ \\alpha \\ , d \\mu ( \\alpha ) = \\sum _ { j = 1 } ^ n \\int _ { \\Omega _ j } \\frac { g _ j ( x ) } { \\sqrt { \\mu ( \\Omega _ j ) } } \\frac { \\omega _ j } { \\sqrt { \\mu ( \\Omega _ j ) } } \\ , d \\mu ( \\alpha ) \\\\ & = \\sum _ { j = 1 } ^ n g _ j ( x ) \\omega _ j , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "8208.png", "formula": "\\begin{align*} d \\xi _ t = \\sigma ( \\xi _ s ) \\circ d B _ t + w \\sigma ( \\xi _ t ) d t \\end{align*}"}
+{"id": "7010.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\ell } { \\partial \\nu ^ 2 } = - \\frac { 1 } { \\nu ( 2 \\nu - 1 ) } < 0 , \\end{align*}"}
+{"id": "1118.png", "formula": "\\begin{align*} \\mathbb { E } _ Y \\{ ( \\theta _ j - \\theta ) Z _ j \\} & = \\mathbb { E } _ { Y } \\{ ( \\theta _ j - \\theta ) s ( \\theta _ j - Y _ j ) \\} = \\mathbb { E } _ Y \\{ ( \\theta _ j - \\theta ) \\mathbb { E } [ s ( \\theta _ j - Y _ j ) | \\theta _ j ] \\} \\\\ & = \\mathbb { E } _ Y [ ( \\theta _ j - \\theta ) \\{ \\mathbb { P } _ { Y _ j | \\theta _ j } ( Y _ j < \\theta _ j ) - \\mathbb { P } _ { Y _ j | \\theta _ j } ( Y _ j > \\theta _ j ) \\} ] . \\end{align*}"}
+{"id": "8153.png", "formula": "\\begin{align*} \\abs [ \\Bigg ] { \\frac { \\dfrac { 4 \\pi \\sqrt { y p } } { c } - 2 T } { 2 M } } = \\abs [ \\Bigg ] { \\frac { \\dfrac { 2 \\pi \\sqrt { y p } } { c } - T } { M } } \\leq T ^ \\varepsilon . \\end{align*}"}
+{"id": "4064.png", "formula": "\\begin{align*} \\int _ { \\Omega } f \\ d x & < \\int _ { \\Omega } f \\exp \\Big ( \\frac { u ( x , t ) - u _ 0 ( x ) } { t } \\Big ) \\ d x \\\\ & = \\int _ { \\Omega } f \\exp \\Big ( \\frac { 1 } { t } \\int _ { 0 } ^ t \\frac { \\partial u } { \\partial t } d \\tau \\Big ) \\ d x . \\end{align*}"}
+{"id": "547.png", "formula": "\\begin{align*} \\pi _ t ^ M [ g ] = \\frac { 1 } { M } \\sum _ { i = 1 } ^ M g ( \\Theta _ t ^ { ( i ) } ) \\end{align*}"}
+{"id": "7763.png", "formula": "\\begin{align*} \\{ g _ { 1 - \\alpha , 0 } \\star g _ { \\alpha - \\theta , 1 } \\} ( x ) & = g _ { 1 - \\theta , 0 } ( x ) \\ , \\widetilde \\beta _ { \\alpha - \\theta , 1 - \\alpha } ( x ) \\end{align*}"}
+{"id": "5155.png", "formula": "\\begin{align*} \\mathcal { I } ( H , G ) = \\dfrac { \\mathcal { P } ( H , G ) } { \\binom { n } { k } } . \\end{align*}"}
+{"id": "8318.png", "formula": "\\begin{align*} \\lim _ { t \\to \\pm \\infty } \\int _ { | x | > A + | t | } \\left ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 \\right ) \\ , \\mathrm { d } x = 0 . \\end{align*}"}
+{"id": "7346.png", "formula": "\\begin{align*} L ( x ) = ( \\log x ) ^ { t - 1 } . \\end{align*}"}
+{"id": "5790.png", "formula": "\\begin{align*} K e r ( C _ p ) = S p a n \\{ e _ 1 , \\cdots , e _ p \\} . \\end{align*}"}
+{"id": "418.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n = 1 } ^ \\infty a _ n \\omega _ n \\right \\| ^ p & = \\left \\| \\sum _ { n = 1 } ^ \\infty a _ n V \\tau _ n \\right \\| ^ p = \\left \\| V \\left ( \\sum _ { n = 1 } ^ \\infty a _ n \\tau _ n \\right ) \\right \\| ^ p \\\\ & = \\left \\| \\sum _ { n = 1 } ^ \\infty a _ n \\tau _ n \\right \\| ^ p = \\sum _ { n = 1 } ^ \\infty | a _ n | ^ p , \\{ a _ n \\} _ n \\in \\ell ^ p ( \\mathbb { N } ) . \\end{align*}"}
+{"id": "1101.png", "formula": "\\begin{align*} B ^ { \\beta - 1 / 2 , \\infty } _ \\infty = \\left \\{ f = \\sum _ { j \\geq - 1 } \\sum _ { k \\in \\mathcal { N } _ j } \\beta _ { j k } \\psi _ { j k } , \\ , \\sup _ { j , k } 2 ^ { j \\beta } | \\beta _ { j k } | < \\infty \\right \\} . \\end{align*}"}
+{"id": "2339.png", "formula": "\\begin{align*} \\boxed { T ^ { - 1 } P ( k ) T = P ( - k ) , . } \\end{align*}"}
+{"id": "559.png", "formula": "\\begin{align*} ( A X _ t ^ \\dagger ) ^ { \\rm T } A X _ t ^ \\dagger = ( A ^ { \\rm T } A ) : ( X _ t ^ \\dagger \\otimes X _ t ^ \\dagger ) \\approx ( A ^ { \\rm T } A ) : C \\end{align*}"}
+{"id": "1087.png", "formula": "\\begin{align*} Z _ i = [ X _ i ] _ M + \\frac { 2 M } { \\alpha } W _ i , \\end{align*}"}
+{"id": "3246.png", "formula": "\\begin{align*} & \\int _ { t _ 1 } ^ z \\frac { d t } { 1 - t } = \\log \\frac { 1 - t _ 1 } { 1 - z } = L _ 1 ( t _ 1 , z ) , \\\\ & \\int _ { t _ 1 } ^ z \\frac { - d t } { 1 + t } = \\log \\frac { 1 + t _ 1 } { 1 + z } = L _ { - 1 } ( t _ 1 , z ) , \\\\ & \\int _ { t _ 1 } ^ z \\frac { d t } { t } = \\log \\frac { z } { t _ 1 } = L _ 0 ( t _ 1 , z ) . \\end{align*}"}
+{"id": "7501.png", "formula": "\\begin{align*} \\Delta _ { \\gamma _ 2 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) = \\bigcup \\limits _ { m = 0 } ^ { j _ 0 - 1 } \\Big \\{ ( m + b j _ 0 , \\frac { m i _ 0 + n _ m } { j _ 0 } + b i _ 0 + a ) | a , b \\in \\mathbb { Z } ^ { + } \\Big \\} \\end{align*}"}
+{"id": "4483.png", "formula": "\\begin{align*} \\frac { 2 } { 1 1 } = \\frac { 1 } { 1 1 } + \\frac { 1 } { 2 2 } + \\frac { 1 } { 3 3 } + \\frac { 1 } { 6 6 } . \\end{align*}"}
+{"id": "8048.png", "formula": "\\begin{align*} L \\Bigl ( \\frac { 1 } { 2 } , f \\times E \\Bigr ) = \\abs [ \\Big ] { L \\Bigl ( \\frac { 1 } { 2 } - i t , f \\Bigr ) } ^ 2 , \\end{align*}"}
+{"id": "5676.png", "formula": "\\begin{align*} \\frac { d } { d y } f _ { \\chi ^ 2 _ { n } ( \\rho ^ 2 ) } ( y ) = \\frac { 1 } { 2 } f _ { \\chi ^ 2 _ { n - 2 } ( \\rho ^ 2 ) } ( y ) - \\frac { 1 } { 2 } f _ { \\chi ^ 2 _ { n } ( \\rho ^ 2 ) } ( y ) ; \\end{align*}"}
+{"id": "1918.png", "formula": "\\begin{align*} - \\Delta _ g w + V w & = 0 \\quad M _ \\sigma \\\\ \\nu _ g ( w ) & = 0 \\quad \\partial M _ \\sigma . \\end{align*}"}
+{"id": "1458.png", "formula": "\\begin{align*} \\lambda = \\frac { k ( k - 1 ) ( k - 2 ) ( m - 1 ) ! ( m - 2 ) ! } { ( m + 1 ) ( m ^ 2 - 2 ) | K _ \\Delta | } . \\end{align*}"}
+{"id": "7890.png", "formula": "\\begin{align*} f '' + \\left ( e ^ { P _ 1 ( z ) } + e ^ { P _ 2 ( z ) } + Q ( z ) \\right ) f = 0 , \\end{align*}"}
+{"id": "7604.png", "formula": "\\begin{align*} v ( t ) = u _ 0 - \\int _ 0 ^ t \\mathfrak { A } ( v ( s ) ) \\d s , \\end{align*}"}
+{"id": "3650.png", "formula": "\\begin{align*} \\Phi ( x , 0 ) = \\int \\limits _ { \\mathbb { H } ^ { n } } \\mathcal { K } ( x , y , T ) u _ { 0 } ( y ) ~ { \\rm d } v _ { \\mathbb { H } ^ { n } } ( y ) = \\left ( e ^ { T \\Delta _ { \\mathbb { H } ^ { n } } } u _ { 0 } \\right ) ( x ) . \\end{align*}"}
+{"id": "3514.png", "formula": "\\begin{align*} A _ { ( k , l ) \\to i } ( s , t ) = \\begin{cases} A ( s , t ) , & ( s , t ) \\neq ( k , l ) , \\\\ i , & ( s , t ) = ( k , l ) . \\end{cases} \\end{align*}"}
+{"id": "714.png", "formula": "\\begin{align*} r _ { \\rm m e t r i c } ( w ) = 2 \\frac { \\partial } { \\partial w } \\log \\lambda ( w ) , \\qquad \\end{align*}"}
+{"id": "2819.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 ) = x _ 1 \\end{align*}"}
+{"id": "7745.png", "formula": "\\begin{align*} \\Pr ( x , n | \\mu ) = \\Pr ( x | n ) \\Pr ( n | \\mu ) & = \\ell ^ { n \\star } ( x ) \\ , e ^ { - \\mu } \\ , \\frac { \\mu ^ n } { n ! } \\intertext { H e n c e t h e u n c o n d i t i o n a l d i s t r i b u t i o n o f $ X $ i s } \\Pr ( x | \\mu ) = \\sum _ { n = 0 } ^ \\infty \\Pr ( x | n ) \\Pr ( n | \\mu ) & = e ^ { - \\mu } \\sum _ { n = 0 } ^ \\infty \\frac { \\mu ^ n } { n ! } \\ , \\ell ^ { n \\star } ( x ) \\end{align*}"}
+{"id": "2411.png", "formula": "\\begin{align*} E _ { 3 3 } ( t ) \\dot x _ 3 = A _ { 3 3 } ( t ) x _ 3 + f _ 3 ( t ) \\end{align*}"}
+{"id": "2160.png", "formula": "\\begin{align*} \\sigma _ { 0 } : = & \\dfrac { 2 ( \\mathit { e } - 1 ) } { \\mathit { e } ( 2 - 2 \\alpha + \\beta ) + \\sqrt { ( \\mathit { e } ( 2 - 2 \\alpha + \\beta ) ) ^ 2 - 4 ( \\mathit { e } - 1 ) ( 1 - \\mathit { e } ( 2 \\alpha - 1 + \\beta ) ) } } , \\intertext { a n d } \\tilde { \\sigma _ { 0 } } : = & \\dfrac { 2 ( \\mathit { e } - 1 ) } { ( 2 - 2 \\alpha + \\beta ) + \\sqrt { ( 2 - 2 \\alpha + \\beta ) ^ 2 - 4 ( \\mathit { e } - 1 ) ( 2 \\alpha - 1 + \\beta - \\mathit { e } ) } } . \\end{align*}"}
+{"id": "2734.png", "formula": "\\begin{align*} B _ { i } \\Omega _ i = \\Omega _ i B _ { i } . \\end{align*}"}
+{"id": "251.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 7 ( i \\rho , i \\tau , \\xi + i \\eta ) = - ( 1 - \\alpha ) \\tau ^ 2 - \\beta \\tau \\eta \\leqslant 0 , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 8 ( i \\rho , i \\tau , \\xi + i \\eta ) = - ( 1 - \\alpha ) \\tau ^ 2 - \\beta \\rho \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "4327.png", "formula": "\\begin{align*} & 2 ^ { n _ 1 } \\cdot \\textbf { P r } ( \\xi _ n \\ge w ) = 2 ^ { n _ 1 } \\cdot \\textbf { P r } ( \\xi _ n - \\textbf { E } \\xi _ n \\ge - \\sqrt { n _ 1 } \\ln n _ 1 ) \\\\ & \\ge 2 ^ { n _ 1 } ( 1 - \\exp ( - 2 ( \\sqrt { n _ 1 \\ln n _ 1 } ) ^ 2 / n _ 1 ) = 2 ^ { n _ 1 } ( 1 - n _ 1 ^ { - 2 } ) \\\\ & = 2 ^ { n _ 1 } ( 1 + o ( 1 ) ) \\end{align*}"}
+{"id": "3415.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } \\widetilde \\omega _ { q } ^ 1 - \\mu \\Delta \\widetilde \\omega _ { q } ^ 1 + ( v \\cdot \\nabla ) \\widetilde \\omega _ { q } ^ 1 = \\dfrac { v ^ 2 } { x _ 2 } \\widetilde \\omega _ { q } ^ 1 - \\partial _ z \\Big ( \\Delta _ { q } \\Big ( \\frac { B _ { \\theta } } { r } B ^ 1 \\Big ) \\Big ) , \\\\ \\widetilde \\omega _ { q } ^ 1 \\vert _ { \\vert t = 0 } = \\Delta _ { q } \\omega _ { 0 } ^ 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "2136.png", "formula": "\\begin{align*} \\lim _ { i \\to + \\infty } \\sup _ { L \\geq \\gamma _ i } \\left \\vert \\frac { R _ i ( L ) } { i R _ 1 ( L ) } - 1 \\right \\vert = 0 . \\end{align*}"}
+{"id": "2980.png", "formula": "\\begin{align*} \\begin{aligned} & E _ { \\nu ^ n _ \\rho } [ f ( \\overrightarrow { W } _ { j + 1 } ^ \\ell ) ^ 2 ( W _ { j + i - 1 } - W _ { j + i } ) ] \\\\ & = - E _ { \\nu ^ n _ \\rho } [ \\sigma _ { j + i - 1 , j + i } ( \\overrightarrow { W } _ { j + 1 } ^ \\ell ) ^ 2 ( \\nabla _ { j + i - 1 , j + i } f ) W _ { j + i - 1 } ] - E _ { \\nu ^ n _ \\rho } [ f ( \\nabla _ { j + i - 1 , j + i } ( \\overrightarrow { W } _ { j + 1 } ^ \\ell ) ^ 2 ) W _ { j + i - 1 } ] \\end{aligned} \\end{align*}"}
+{"id": "1449.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\binom { x _ i } { 3 } = \\frac { k ( k - 1 ) ( k - 2 ) ( n - 1 ) ( n - 2 ) } { 6 ( m n - 1 ) ( m n - 2 ) } ; \\end{align*}"}
+{"id": "1898.png", "formula": "\\begin{align*} u ( t ) = k ( 1 , 1 ) y ( 1 , t ) + \\int _ 0 ^ 1 k _ x ( 1 , \\zeta ) y ( \\zeta , t ) d \\zeta - \\widehat { w } ( t ) , \\end{align*}"}
+{"id": "554.png", "formula": "\\begin{align*} f ( x , \\theta ) = \\theta A x \\ , , \\end{align*}"}
+{"id": "5196.png", "formula": "\\begin{align*} \\sum \\gamma _ { i j k } g _ { i j k } = 0 \\textrm { a n d } \\sum \\gamma _ { i j } G _ { i j } = 0 \\ , , \\end{align*}"}
+{"id": "12.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d Q ^ k _ t & = \\Big ( \\varphi _ t ^ k + e ^ { - \\frac { \\beta } { 2 } t } \\bar { h } \\tilde { M } ^ k _ t - \\beta Q ^ k _ t \\Big ) d t - \\tilde { M } ^ k _ t d \\xi _ t , \\\\ Q _ k ^ k & = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2476.png", "formula": "\\begin{align*} | r _ p ( R _ S ( E [ p ] / K _ n ) ) - r _ p ( R ( E [ p ] / K _ n ) ) | = O ( 1 ) \\end{align*}"}
+{"id": "8218.png", "formula": "\\begin{align*} P \\{ \\mathcal { T } ( t ) = x + v _ 0 ( t - s ) \\ | \\ \\mathcal { T } ( s ) = x , N ( s ) = 2 k \\} = \\frac { | v _ 1 s - x | } { ( a _ 1 - a _ 2 ) s } P \\{ \\mathcal { T } ( t - s ) = v _ 0 ( t - s ) \\} . \\end{align*}"}
+{"id": "5697.png", "formula": "\\begin{align*} 2 \\gamma _ { b } ^ { a } \\theta ^ { b } = ( \\overline { \\tau } ^ { b } - \\tau ^ { b } ) \\mathfrak { f } _ { b } ^ { a } + D ( \\beta _ { b } ^ { a } - \\overline { \\beta } _ { b } ^ { a } ) \\theta ^ { b } . \\end{align*}"}
+{"id": "8450.png", "formula": "\\begin{align*} \\lim _ { ( s , t ) \\in S \\times T } \\ , \\| \\mu _ s - \\nu _ t \\| = 0 , \\end{align*}"}
+{"id": "1254.png", "formula": "\\begin{align*} \\dfrac { z k _ \\beta ' ( z ) } { k _ \\beta ( z ) } & = \\dfrac { 1 + ( 1 - 2 \\beta ) z } { 1 - z } = \\frac { 1 - \\overline { z } + ( 1 - 2 \\beta ) z - ( 1 - 2 \\beta ) | z | ^ 2 } { | 1 - z | ^ 2 } \\\\ & = \\dfrac { 1 - 2 \\beta { \\rho } _ 0 \\cos t - ( 1 - 2 \\beta ) ( { \\rho } _ 0 ) ^ 2 } { 1 + ( { \\rho } _ 0 ) ^ 2 - 2 { \\rho } _ 0 \\cos t } + i \\dfrac { 2 ( 1 - \\beta ) { \\rho } _ 0 \\sin t } { 1 + ( { \\rho } _ 0 ) ^ 2 - 2 { \\rho } _ 0 \\cos t } . \\end{align*}"}
+{"id": "2252.png", "formula": "\\begin{align*} \\mu \\cdot Y _ j : = [ \\mu ] Y _ j [ \\mu ] ^ { - 1 } = \\alpha _ j ( \\mu ) Y _ j \\end{align*}"}
+{"id": "7176.png", "formula": "\\begin{align*} H _ t = \\left \\{ z _ { \\alpha , t } ( x _ { j } , \\ , t ) , \\ , z _ { \\alpha , j t } ( x _ { j } , \\ , t ) , \\ , z _ { \\alpha , j k } ( x _ { j } , \\ , t ) \\right \\} , \\end{align*}"}
+{"id": "3392.png", "formula": "\\begin{align*} f _ n \\circ p \\circ f _ n = f _ n \\circ ( \\mbox { i d } - \\alpha ) = f _ n - f _ n \\circ \\alpha = f _ n = \\mbox { i d } \\circ f _ n . \\end{align*}"}
+{"id": "1270.png", "formula": "\\begin{align*} & 2 | L ( T ) | \\geq | L ( T ) | + 2 + | B ( T ) | \\geq k + m + 5 \\geq m - 1 + m + 5 = 2 m + 4 \\\\ & \\Rightarrow | L ( T ) | \\geq m + 2 . \\end{align*}"}
+{"id": "4672.png", "formula": "\\begin{align*} \\sum _ { m = n } ^ { \\infty } \\sum _ { k = n } ^ { \\min \\{ \\ell , m \\} } A _ { m , k } = \\sum _ { m = n } ^ { \\ell } \\sum _ { k = n } ^ m A _ { m , k } + \\sum _ { m = \\ell + 1 } ^ { \\infty } \\sum _ { k = n } ^ { \\ell } A _ { m , k } . \\end{align*}"}
+{"id": "3254.png", "formula": "\\begin{align*} F ^ { \\ast } = \\sum \\limits _ { e \\in ( S , \\overline { S } ) } f ^ { \\ast } ( e ) - \\sum \\limits _ { e \\in ( \\overline { S } , S ) } f ^ { \\ast } ( e ) . \\end{align*}"}
+{"id": "1166.png", "formula": "\\begin{align*} c = a ^ { - 1 } \\longleftrightarrow c < p \\wedge c \\neq 0 \\wedge c \\cdot _ { ( m o d \\ , p ) } a = a \\cdot _ { ( m o d \\ , p ) } c = 1 . \\end{align*}"}
+{"id": "7640.png", "formula": "\\begin{align*} \\sigma ( \\Delta _ { V , h } ) \\cap [ 0 , \\varepsilon _ { * } h ] = \\{ 0 , \\mu _ { 2 , h } ^ \\Delta , \\ldots , \\mu _ { n _ 0 , h } ^ \\Delta \\} , \\end{align*}"}
+{"id": "3780.png", "formula": "\\begin{align*} R _ C & = \\{ s ( t ) g s ( t ) ^ { - 1 } w ^ { - 1 } \\mid g \\in S _ N , t \\in S _ Q , w \\in N , [ s ( t ) g s ( t ) ^ { - 1 } ] = i ( w ) \\in G \\} \\\\ \\tilde { R } _ Q & = \\{ s ( r ) h ^ { - 1 } \\mid r \\in R _ Q , h \\in N , [ s ( r ) ] = i ( h ) \\in G \\} . \\end{align*}"}
+{"id": "5100.png", "formula": "\\begin{align*} \\frac { a c ^ { n _ 1 } } { b g _ b ( n _ 1 ) } = \\frac { a c ^ { n _ 2 } } { b g _ b ( n _ 2 ) } \\end{align*}"}
+{"id": "9204.png", "formula": "\\begin{align*} f ( y ) = - \\vert y _ { - } \\vert ^ { 2 } + \\vert y _ { + } \\vert ^ { 2 } . \\end{align*}"}
+{"id": "7729.png", "formula": "\\begin{align*} t _ 0 ^ { ( 2 ) } & = \\frac { 2 } { \\sqrt { \\tau } } \\left ( 1 + \\rho _ 0 \\cos ( \\beta \\pi ) - i \\rho _ 0 \\sin ( \\beta \\pi ) \\right ) ^ { \\frac { 1 } { 2 } } - 1 \\\\ & = \\frac { 2 } { \\sqrt { \\tau } } \\left ( i \\sqrt { A ^ + _ { \\beta } } + \\sqrt { A ^ - _ { \\beta } } \\right ) - 1 , \\end{align*}"}
+{"id": "5416.png", "formula": "\\begin{align*} \\delta ( q _ { i , ( i + k ) \\bmod p , \\ , k } , a ) = \\begin{cases} q _ { i , ( i + k + 1 ) \\bmod p , \\ , k + 1 } & a = t _ { k } ^ { ( i ) } k \\ne m _ i + n _ i - 1 \\\\ q _ { i , ( i + m _ i ) \\bmod p , \\ , m _ i } & a = t _ { k } ^ { ( i ) } k = m _ i + n _ i - 1 \\\\ q _ { ( i + k + 1 ) \\bmod p , ( i + k + 1 ) \\bmod p , \\ , 0 } & a \\in [ \\ ! [ 0 , t _ k ^ { ( i ) } - 1 ] \\ ! ] . \\end{cases} \\end{align*}"}
+{"id": "3176.png", "formula": "\\begin{align*} a _ 1 ( y ) : = b _ 1 ( y _ 1 , y _ 2 ) , \\ ; \\ ; a _ 2 ( y ) : = b _ 2 ( y _ 1 , y _ 2 ) , \\ ; \\ ; a _ 3 ( y ) : = 1 0 - b _ 1 ( y _ 1 , y _ 2 ) - b _ 2 ( y _ 1 , y _ 2 ) \\end{align*}"}
+{"id": "7822.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { p - 1 } \\sum _ { t = 1 } ^ { p - j } x ^ { - 2 j + 2 t } & = \\frac { x ^ { - 2 p + 4 } ( 1 - x ^ { 2 p - 2 } ) ( 1 - x ^ { 2 p } ) } { ( 1 - x ^ 2 ) ( 1 - x ^ 4 ) } \\\\ & = \\begin{cases} \\displaystyle \\sum _ { j = 1 } ^ { p / 2 } \\sum _ { t = 1 } ^ { p - 1 } x ^ { 2 - 4 j + 2 t } & ( p : ) , \\\\ \\displaystyle \\sum _ { j = 1 } ^ { p } \\sum _ { t = 1 } ^ { ( p - 1 ) / 2 } x ^ { - 2 j + 4 t } & ( p : ) . \\end{cases} \\end{align*}"}
+{"id": "4137.png", "formula": "\\begin{align*} R _ { i j k l } = \\frac { 1 } { 4 } H _ { i l b } H _ { j k b } - \\frac { 1 } { 4 } H _ { j l b } H _ { i k b } \\end{align*}"}
+{"id": "7877.png", "formula": "\\begin{align*} f ^ n + P ( z , f ) = h ( z ) , \\end{align*}"}
+{"id": "8933.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d t } \\right ) ^ { \\ell } \\vartheta _ { 3 } ( t ) \\simeq \\frac { ( - 1 ) ^ { \\ell } \\ell ! } { t ^ { 1 + \\ell } } + \\sum \\limits _ { j = \\ell } ^ { + \\infty } \\frac { \\left ( - 1 \\right ) ^ { j } B _ { 2 j + 2 } } { ( j + 1 ) ( j - \\ell ) ! } t ^ { j - \\ell } \\end{align*}"}
+{"id": "2020.png", "formula": "\\begin{align*} X _ { n + 1 } ^ { ( \\alpha ) } : = X _ n ^ { ( \\alpha ) } + \\left ( \\xi _ { n + 1 } \\mathbb 1 _ { \\{ X _ n ^ { ( \\alpha ) } \\leq - 1 \\} } + \\eta _ { n + 1 } \\mathbb 1 _ { \\{ X _ n ^ { ( \\alpha ) } = 0 \\} } + \\xi _ { n + 1 } ' \\mathbb 1 _ { \\{ X _ n ^ { ( \\alpha ) } \\geq 1 \\} } \\right ) , \\end{align*}"}
+{"id": "1807.png", "formula": "\\begin{align*} \\frac { d } { d t } \\big ( R \\dot { y } + Q y \\big ) = Q ^ T \\dot { y } + P y , \\end{align*}"}
+{"id": "5756.png", "formula": "\\begin{align*} \\check { c } _ { * } ( \\varphi ) = \\sum _ { j \\geq 0 } \\check { c } _ j ( \\varphi ) : = \\sum _ { j \\geq 0 } ( - 1 ) ^ j { c } _ j ( \\varphi ) . \\end{align*}"}
+{"id": "7968.png", "formula": "\\begin{align*} z + q = \\sum x _ i , q = \\sum y _ i . \\end{align*}"}
+{"id": "9235.png", "formula": "\\begin{align*} \\Vert \\Delta _ { f } \\varphi _ { j } \\Vert = \\mathcal { O } \\Big ( h ^ { \\infty } \\sqrt { \\ < \\Delta _ { f } \\varphi _ { j } , \\varphi _ { j } \\ > } \\Big ) = \\mathcal { O } ( h ^ { \\infty } \\sqrt { \\mu _ { j } } ) . \\end{align*}"}
+{"id": "8579.png", "formula": "\\begin{align*} s ( x ) = \\sum _ { i = 0 } ^ { \\infty } s _ i \\ , x ^ i = \\frac { g ( x ) } { f ( x ) } , \\end{align*}"}
+{"id": "794.png", "formula": "\\begin{align*} S _ { n } ^ j ( t ) = \\frac { 1 } { ( n \\sigma ^ 2 ) ^ { 1 / 2 } } \\left ( \\log \\| M ^ j _ { \\lfloor t n \\rfloor } \\| - n t \\Lambda + ( n t - \\lfloor n t \\rfloor ) ( \\log \\| M ^ j _ { \\lfloor t n \\rfloor + 1 } \\| - \\log \\| M ^ j _ { \\lfloor t n \\rfloor } \\| ) \\right ) . \\end{align*}"}
+{"id": "8386.png", "formula": "\\begin{align*} L ^ s ( X , \\mathcal { B } ) = & \\aleph _ 0 + \\min \\{ \\mathfrak { m } : \\mathcal { B } X \\\\ & \\mathcal { B } X \\leq \\mathfrak { m } \\} . \\end{align*}"}
+{"id": "3609.png", "formula": "\\begin{align*} | \\hat u | ^ 2 \\ & = \\ | \\widehat u ( \\xi _ 1 , 0 ) | ^ 2 \\chi _ { D ' _ { \\eta } ( \\xi _ 1 ) } ( \\xi ' ) + | \\hat u - \\widehat u ( \\xi _ 1 , 0 ) \\chi _ { D ' _ { \\eta } ( \\xi _ 1 ) } ( \\xi ' ) | ^ 2 \\\\ & + 2 \\Re \\Big [ ( \\hat u - \\widehat u ( \\xi _ 1 , 0 ) ) \\widehat u ( \\xi _ 1 , 0 ) \\chi _ { D ' _ { \\eta } ( \\xi _ 1 ) } ( \\xi ' ) \\Big ] , \\end{align*}"}
+{"id": "4202.png", "formula": "\\begin{align*} \\Lambda ^ { ( 0 ) } _ { \\varepsilon } : = \\lambda + \\varepsilon \\theta ( t + x ) , \\phi ^ { ( 0 ) } : = 0 . \\end{align*}"}
+{"id": "2457.png", "formula": "\\begin{align*} \\sum _ { \\rho } \\frac { 1 + \\eta - \\beta } { | 1 + \\eta + i t - \\rho | ^ 2 } & \\leq \\frac { 1 } { 5 \\eta } \\# \\{ \\rho \\colon | \\rho - ( 1 + i t ) | \\leq \\eta , ~ L ( \\rho , \\pi \\otimes \\chi ) = 0 \\} \\\\ & \\leq \\frac { 1 } { 2 } \\log q _ { \\pi \\otimes \\chi } + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { m } \\mathrm { R e } \\frac { \\Gamma ' } { \\Gamma } \\Big ( \\frac { 1 + \\eta + i t + \\mu _ { \\pi \\otimes \\chi } ( j ) } { 2 } \\Big ) + \\sum _ { n = 1 } ^ { \\infty } \\frac { | \\Lambda _ { \\pi \\otimes \\chi } ( n ) | } { n ^ { 1 + \\eta } } \\end{align*}"}
+{"id": "2871.png", "formula": "\\begin{align*} \\abs { L _ f ( 2 + i t ) } & = \\prod _ p \\prod _ { i = 1 } ^ n \\abs { \\left ( 1 - \\frac { \\alpha _ { p , i } } { p ^ { 2 + i t } } \\right ) } ^ { - 1 } \\\\ & \\leq \\prod _ p \\prod _ { i = 1 } ^ n \\left ( 1 - \\frac { 1 } { p ^ { \\frac { 3 } { 2 } } } \\right ) ^ { - 1 } \\\\ & \\leq \\left ( \\zeta \\left ( \\frac { 3 } { 2 } \\right ) \\right ) ^ n \\\\ & \\ll _ n 1 . \\\\ \\end{align*}"}
+{"id": "6581.png", "formula": "\\begin{align*} \\frac { 3 l ^ { 1 - n } } { 4 } E _ 1 \\leq \\frac { \\mu _ { t _ 0 } ( B _ { r _ 0 } ( x _ 0 ) ) } { r _ 0 ^ { n - 1 } } = \\frac { \\tilde { \\mu } _ 1 ( B _ { \\tilde { r } } ) } { \\tilde { r } ^ { n - 1 } _ 0 } \\leq e ^ { \\frac { 1 } { 1 6 } } ( \\sqrt { 4 \\pi } ) ^ { n - 1 } \\int _ { B _ { \\frac 1 4 } } \\tilde { \\rho } \\ d \\tilde { \\mu } _ 1 . \\end{align*}"}
+{"id": "5862.png", "formula": "\\begin{align*} \\partial _ \\nu U + B U = D ( \\widehat { H } + D ^ { - 1 } B U ) \\overset { } { = } D H ~ ( 0 , T ) \\times \\Gamma . \\end{align*}"}
+{"id": "5478.png", "formula": "\\begin{align*} \\tilde L ^ { ( i ) } f ( x - y ) : = & ( b ( t , x , \\mu , i ) - b ( t , y , \\nu , i ) ) \\nabla f ( x - y ) + \\frac { 1 } { 2 } t r ( \\nabla ^ 2 f ( x - y ) A ( t , x , y , \\mu , \\nu , i ) ) \\end{align*}"}
+{"id": "8564.png", "formula": "\\begin{align*} d _ { w } \\left ( \\frac { t } { 2 } - i \\right ) & = \\frac { \\lambda t ! - \\left ( \\frac { t } { 2 } - i + 1 \\right ) d _ { w } \\left ( \\frac { t } { 2 } - i + 1 \\right ) } { \\frac { t } { 2 } + i } \\\\ d _ { w } \\left ( \\frac { t } { 2 } + i \\right ) & = \\frac { \\lambda t ! - \\left ( \\frac { t } { 2 } - i + 1 \\right ) d _ { w } \\left ( \\frac { t } { 2 } + i - 1 \\right ) } { \\frac { t } { 2 } + i } \\end{align*}"}
+{"id": "3448.png", "formula": "\\begin{align*} L ^ 2 = ( 2 \\lambda - \\delta ) ^ 2 + ( 2 \\mu - \\delta ) ^ 2 + 1 0 \\delta ^ 2 . \\end{align*}"}
+{"id": "1995.png", "formula": "\\begin{align*} \\zeta _ { r , d } ^ \\bullet ( \\underbar { { \\bf s } } , A _ r ) = \\left ( \\sum _ { m _ 1 = 0 } ^ \\infty \\cdots \\sum _ { m _ d = 0 } ^ \\infty \\right ) ' \\left ( \\sum _ { m _ { d + 1 } = 1 } ^ \\infty \\cdots \\sum _ { m _ r = 1 } ^ \\infty \\right ) \\prod _ { 1 \\le i < j \\le r + 1 } ( m _ i + \\cdots + m _ { j - 1 } ) ^ { - s ( i , j ) } . \\end{align*}"}
+{"id": "878.png", "formula": "\\begin{align*} \\mathbf { E } [ \\| \\Gamma ^ { t + 1 } \\| _ { F } ^ { 2 } | \\mathcal { X } ^ { t } ] & = \\| \\Gamma ^ { t } \\| _ { F } ^ { 2 } - \\mathbf { E } [ \\| \\mathcal { Z } * \\Gamma ^ { t } \\| ^ { 2 } _ { F } ] . \\end{align*}"}
+{"id": "7947.png", "formula": "\\begin{align*} m _ 1 + c _ 1 = m _ 2 + c _ 2 \\textrm { f o r s o m e } c _ 1 , c _ 2 \\in \\pi ^ { - 1 } ( K ) . \\end{align*}"}
+{"id": "7444.png", "formula": "\\begin{align*} Z = ( Z _ 1 , \\dots , Z _ d ) Z _ i \\in { \\mathbb C } ^ { n \\times n } Z \\in \\Omega _ { n } . \\end{align*}"}
+{"id": "5753.png", "formula": "\\begin{align*} \\mathfrak { X } ^ \\circ ( Z ) = \\{ ( z , \\mathbf { w } ) \\in Z _ { \\textrm { r e g } } \\times \\C ^ n \\mid z m _ { \\mathbf { w } } | _ { Z _ { \\textrm { r e g } } } \\} . \\end{align*}"}
+{"id": "7979.png", "formula": "\\begin{align*} x ' + A = x + A , \\end{align*}"}
+{"id": "3642.png", "formula": "\\begin{align*} \\lim _ { t \\uparrow T _ { { \\tiny \\mbox { m a x } } } } | | u ( \\cdot , t ) | | _ { L ^ { \\infty } ( \\mathbb { H } ^ n ) } = \\infty . \\end{align*}"}
+{"id": "5441.png", "formula": "\\begin{align*} P ( \\alpha _ { t + \\Delta t } = j | \\alpha _ { t } = i , ( X _ s , \\alpha _ s ) , s \\leq t ) = q _ { i j } ( X _ t ) \\Delta t + o ( \\Delta t ) , \\end{align*}"}
+{"id": "4811.png", "formula": "\\begin{align*} \\frac { 1 } { \\binom { N } { S } } \\sum _ { j \\in A _ \\epsilon } \\Pr _ { c \\sim C ^ \\perp } \\big [ | c | = j \\big ] K _ S ( j ) ^ 2 & \\leq 1 + o \\big ( \\frac { 1 } { N } \\big ) . \\end{align*}"}
+{"id": "8778.png", "formula": "\\begin{align*} \\phi ^ { N } ( t , x ) = \\phi ^ N _ i ( t ) , \\varphi ^ N ( x ) = \\varphi ^ N _ i , \\omega ^ N ( t , x ) = \\omega _ i ^ N ( t ) , \\end{align*}"}
+{"id": "4312.png", "formula": "\\begin{align*} \\sum _ \\alpha \\ B _ { \\alpha , n } \\left ( \\lambda _ 0 , \\ldots , \\lambda _ { s - 1 } \\right ) a _ \\alpha = 0 , \\ \\ n \\ge 1 , \\end{align*}"}
+{"id": "188.png", "formula": "\\begin{align*} p ( q ) = p ( c _ { n } ) + \\sum _ { \\ell = c _ { n } } ^ { q - 1 } ( p ( \\ell + 1 ) - p ( \\ell ) ) \\leq p ( c _ { n } ) + ( q - c _ { n } ) ( r _ { n } + 1 ) \\end{align*}"}
+{"id": "2441.png", "formula": "\\begin{align*} f _ 3 ( x ) & = 1 + x ^ { 2 e } = ( x ^ e + 2 ) ( x ^ e - 2 ) , \\\\ f _ 7 ( x ) & = 1 + x ^ e - x ^ { 2 e } = - ( x ^ e + 2 ) ^ 2 , \\\\ f _ 8 ( x ) & = 1 - x ^ e + x ^ 2 e = - ( x ^ e - 2 ) ^ 2 . \\end{align*}"}
+{"id": "4390.png", "formula": "\\begin{align*} u _ n ( \\theta ) = \\sum _ { i = 1 } ^ n \\frac { 1 } { b _ i } < \\theta . \\end{align*}"}
+{"id": "3148.png", "formula": "\\begin{align*} \\int _ Y ( \\partial _ 1 w _ A ) ( \\partial _ { 2 2 } ^ 2 w _ B ) = - \\frac { 1 } { 2 c } ( Q _ 1 + Q _ 2 ) = 0 , \\int _ Y ( \\partial _ 2 w _ A ) ( \\partial _ { 1 1 } ^ 2 w _ B ) & = \\frac { 1 } { 2 c } ( Q _ 2 - Q _ 1 ) = 0 . \\end{align*}"}
+{"id": "5895.png", "formula": "\\begin{align*} C _ p A e _ r = 0 , 1 \\leq r \\leq p . \\end{align*}"}
+{"id": "9114.png", "formula": "\\begin{align*} \\log e \\frac { 1 - p } { 2 } \\left ( \\frac { h _ 1 + h _ 2 - 2 \\rho \\sqrt { h _ 1 h _ 2 } } { 1 - \\rho ^ 2 + \\mathsf { P } ( h _ 1 + h _ 2 - 2 \\rho \\sqrt { h _ 1 h _ 2 } ) } - \\frac { h _ 1 } { h _ 1 \\mathsf { P } + 1 } \\right ) \\mathsf { P } = 0 , \\end{align*}"}
+{"id": "3043.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ D k \\mu ( A ^ { [ k _ n ] } ) \\mu ( V _ k ^ { ( x _ A ) } ) - \\int _ A \\sum _ { k = 1 } ^ D k \\mu ( V _ k ^ { ( x ) } ) d \\mu ( x ) \\right | & \\leq \\sum _ { k = 1 } ^ D k \\left ( \\mu ( ( \\partial A ) ^ { [ k _ n ] } ) + \\mu ( A ^ { [ k _ n ] } ) \\mu ( B _ A ^ { [ k _ n ] } ) \\right ) \\\\ & \\leq C \\mu ( A ) n ^ { - 1 / 2 0 } . \\end{align*}"}
+{"id": "6712.png", "formula": "\\begin{align*} \\mathcal { K } ( f ( n ) ) = \\frac { 1 } { p } \\sum _ { 0 \\leq t \\leq p - 1 , } \\sum _ { 0 \\leq s \\leq p - 1 } \\omega ^ { t ( n - s ) } f ( s ) = f ( n ) . \\end{align*}"}
+{"id": "2176.png", "formula": "\\begin{align*} \\dfrac { z F ' ( z ) } { F ( z ) } = & \\dfrac { ( 1 - 2 \\alpha - \\beta ) z ^ 2 + ( 2 - 2 \\alpha + \\beta ) z + 1 } { 1 - z ^ 2 } . \\end{align*}"}
+{"id": "5020.png", "formula": "\\begin{align*} * _ g H = \\frac { a ^ 2 } { \\mu \\ , b ^ 2 } \\ , h _ 2 \\ , e ^ { 2 3 } + \\frac { b ^ 2 } { \\mu \\ , a ^ 2 } \\ , h _ 1 \\ , e ^ { 4 5 } , \\end{align*}"}
+{"id": "4127.png", "formula": "\\begin{align*} \\Gamma ( S ^ 2 M ) \\times \\Omega ^ 2 = ( \\mathcal { V } \\oplus \\mathcal { V } _ 1 ) \\oplus ( \\mathcal { V } ^ \\perp \\cap \\mathcal { V } _ 1 ^ \\perp ) \\end{align*}"}
+{"id": "5088.png", "formula": "\\begin{align*} Q = \\mu \\prod _ { i = 1 } ^ s ( 1 - \\lambda _ i x _ d ) \\end{align*}"}
+{"id": "8007.png", "formula": "\\begin{align*} C _ { F _ j } : = \\big \\{ \\left . F _ k \\in V _ E \\ ; \\right | \\ ; \\exists F _ j F _ k G \\big \\} \\end{align*}"}
+{"id": "7044.png", "formula": "\\begin{align*} [ 2 ] u = \\sum _ { i , j \\geq 0 } a _ { i , j } u ^ { i + j } = p _ 0 + p _ 1 u + p _ 2 u ^ 2 + \\dots \\end{align*}"}
+{"id": "7485.png", "formula": "\\begin{align*} \\dim Y ' = \\dim Y + \\kappa ( F ) \\geq \\dim Y , \\end{align*}"}
+{"id": "4956.png", "formula": "\\begin{align*} D _ { n , m } : = \\frac { 1 } { n + 1 } \\binom { n - 1 } { m - 1 } \\binom { n + m } { m } , \\end{align*}"}
+{"id": "8680.png", "formula": "\\begin{align*} \\left ( u - { v t } \\right ) \\Gamma ^ \\pm ( u ) \\Gamma ^ \\pm ( v ) + \\left ( v - u t \\right ) \\Gamma ^ \\pm ( v ) \\Gamma ^ \\pm ( u ) & = 0 , \\\\ \\left ( v - u t \\right ) \\Gamma ^ + ( u ) \\Gamma ^ - ( v ) + \\left ( u - v t \\right ) \\Gamma ^ - ( v ) \\Gamma ^ + ( u ) & = \\delta ( u , v ) ( 1 - t ) ^ 2 . \\end{align*}"}
+{"id": "9238.png", "formula": "\\begin{align*} \\ < v _ { j } , v _ { k } \\ > = \\delta _ { j , k } + \\mathcal { O } ( e ^ { - c / h } ) , \\end{align*}"}
+{"id": "409.png", "formula": "\\begin{align*} \\operatorname { s p a n } \\{ f _ 1 , \\dots , f _ j \\} = \\operatorname { s p a n } \\{ g _ 1 , \\dots , g _ j \\} ~ \\operatorname { s p a n } \\{ \\tau _ 1 , \\dots , \\tau _ j \\} = \\operatorname { s p a n } \\{ \\omega _ 1 , \\dots , \\omega _ j \\} , \\forall 1 \\leq j \\leq n . \\end{align*}"}
+{"id": "7240.png", "formula": "\\begin{align*} s & = \\int _ { \\{ g ( x ) \\geq h \\} } f ( x ) \\sqrt { 1 + ( \\nabla g ( x ) ) ^ 2 } \\ , d x \\\\ & = \\int _ { \\partial B _ { n - 1 } } \\int _ 0 ^ { \\sqrt { 1 - h ^ 2 } } f ( r u ) \\sqrt { 1 + ( \\nabla g ( r u ) ) ^ 2 } \\ , r ^ { n - 2 } \\ , d r \\ , \\mu _ { \\partial B _ n } ( d u ) . \\end{align*}"}
+{"id": "6467.png", "formula": "\\begin{align*} L _ \\gamma = - \\partial _ { x x } + ( 1 - \\gamma ^ 2 ) ( \\cos ^ 2 \\theta _ * - \\sin ^ 2 \\theta _ * ) . \\end{align*}"}
+{"id": "5356.png", "formula": "\\begin{align*} \\Lambda _ { F , s } [ \\varepsilon ] = \\sum _ { \\substack { 0 \\leq s _ 1 \\leq | F _ 1 | \\\\ \\ldots \\\\ 0 \\leq s _ n \\leq | F _ n | \\\\ s _ 1 + \\ldots + s _ n = s } } \\prod _ { k = 1 } ^ n \\Lambda _ { F _ k , s _ k } [ \\varepsilon ] \\forall \\varepsilon \\in \\mathcal { W } . \\end{align*}"}
+{"id": "4551.png", "formula": "\\begin{align*} \\frac { 1 } { 5 } = \\frac { 1 } { 6 } + \\frac { 1 } { 3 0 } \\end{align*}"}
+{"id": "2736.png", "formula": "\\begin{align*} w '' ( \\alpha _ i ) = r \\alpha _ i + s \\alpha _ j + \\alpha _ \\bullet \\end{align*}"}
+{"id": "4924.png", "formula": "\\begin{align*} u ( t + \\Delta t ) = & \\ , u ( t + \\tfrac { \\Delta t } 2 ) + \\tfrac { \\Delta t } 2 u ' ( t + \\tfrac { \\Delta t } 2 ) + \\mathcal { O } ( \\Delta t ^ 2 ) , \\\\ u ( t ) = & \\ , u ( t + \\tfrac { \\Delta t } 2 ) - \\tfrac { \\Delta t } 2 u ' ( t + \\tfrac { \\Delta t } 2 ) + \\mathcal { O } ( \\Delta t ^ 2 ) , \\end{align*}"}
+{"id": "8238.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { N } } \\nabla u \\nabla h \\ , d x + \\int _ { \\mathbb { R } ^ { N } } V ( | x | ) f ( u ) f ' ( u ) h \\ , d x - \\int _ { \\mathbb { R } ^ { N } } K ( | x | ) g ( f ( u ) ) f ' ( u ) h \\ , d x = 0 \\end{align*}"}
+{"id": "8404.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { \\infty } | ( h _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) ( \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha , n } ^ { - 1 } ) G _ { \\alpha , n } | \\leq C \\sum _ { n = 1 } ^ { \\infty } ( n + 1 ) n ^ { - \\left ( 1 + \\frac { 1 } { \\alpha } \\right ) } < \\infty \\end{align*}"}
+{"id": "4860.png", "formula": "\\begin{align*} v _ i = ( 0 , \\ . , 0 , u ^ i , 0 , \\ . , 0 ) , \\textrm { w i t h } u ^ i = w _ i ^ { \\j _ i } , \\end{align*}"}
+{"id": "1765.png", "formula": "\\begin{align*} \\mathcal { G } ( H ) = \\{ G _ k \\mid k \\ge m _ { H } ( P ) \\} \\end{align*}"}
+{"id": "5995.png", "formula": "\\begin{align*} X _ { i , j } ^ { k , l } & : = X _ { w _ { i , j } } ^ { w _ 0 w _ { k , l } } = X ( w _ { i , j } ) \\cap w _ 0 X ( w _ { k , l } ) \\\\ & = \\{ ( L , H ) \\in X | L \\subset < e _ { n - k + 1 } , \\dots , e _ i > ; \\ : < e _ 1 , \\dots e _ { j - 1 } , e _ { n - l + 2 } , \\dots , e _ n > \\subset H \\} , \\end{align*}"}
+{"id": "3377.png", "formula": "\\begin{align*} t ^ { - u _ { n - 1 } } \\Big ( \\sum _ { j = 1 } ^ { n - 2 } x _ j \\Big ) + t ^ { - u _ n + 1 } x _ { n - 1 } \\leq x _ n + t ^ { - u _ n } \\ , . \\end{align*}"}
+{"id": "3767.png", "formula": "\\begin{align*} \\mathcal { E } _ { * _ \\Gamma } = \\varepsilon _ 1 \\varepsilon _ 2 \\cdots \\varepsilon _ N . \\end{align*}"}
+{"id": "8507.png", "formula": "\\begin{align*} \\mu _ t \\ ; = & - \\frac { 1 } { 2 7 \\mu ^ 2 } \\mu _ { \\sigma ^ 4 } + \\frac { 8 } { 2 7 \\mu ^ 3 } \\mu _ { \\sigma } \\mu _ { \\sigma ^ 3 } + \\frac { 2 } { 9 \\mu ^ 3 } \\mu _ { \\sigma ^ 2 } ^ 2 - \\frac { 4 } { 2 7 \\mu ^ 4 } \\left ( 9 \\mu _ { \\sigma } ^ 2 + \\mu ^ 3 \\right ) \\mu _ { \\sigma ^ 2 } \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad + \\frac { 8 \\mu _ { \\sigma } ^ 4 } { 9 \\mu ^ 5 } + \\frac { 2 \\mu _ { \\sigma } ^ 2 } { 9 \\mu ^ 2 } + \\frac { 2 } { 2 7 } ( \\sigma \\mu _ { \\sigma } + 2 \\mu ) . \\end{align*}"}
+{"id": "1983.png", "formula": "\\begin{align*} & P _ { k , i } = \\Pr \\left ( \\gamma _ { k , i } < \\bar \\gamma _ { k , i } \\right ) , \\\\ & P _ { k ' , i } = 1 - \\Pr \\left ( \\gamma _ { k , i } > \\bar \\gamma _ { k , i } , \\gamma _ { k ' , i } > \\bar \\gamma _ { k ' , i } \\right ) , \\\\ & P _ { { } , { k } , i } = \\Pr \\left ( \\gamma _ { { } , { k } , i } < \\bar \\gamma _ { { } , { k } , i } \\right ) , \\end{align*}"}
+{"id": "974.png", "formula": "\\begin{align*} \\sigma _ m ( s ) = & \\bigg ( \\sum _ { j = 1 } ^ { m - 1 } R _ j + ( 1 - e ^ { - i s } ) R _ m , t _ m ( s ) \\bigg ) \\mbox { w h e r e } \\\\ t _ m ( s ) = & 2 \\big ( \\frac { \\pi } { 2 } - 1 \\big ) \\big ( \\sum _ { j = 1 } ^ { m - 1 } \\| R _ j \\| ^ 2 \\big ) + 2 ( s - \\sin s ) \\| R _ m \\| ^ 2 - 2 \\sum \\limits _ { j = 2 } ^ { m - 1 } ( \\langle \\sum \\limits _ { k = 1 } ^ { j - 1 } R _ k , R _ j \\rangle ) - 2 \\sin s \\langle \\sum _ { j = 1 } ^ { m - 1 } R _ j , R _ m \\rangle . \\end{align*}"}
+{"id": "551.png", "formula": "\\begin{align*} \\int _ 0 ^ T g ( X _ t ^ \\dagger , t ) \\ , { \\rm d } X _ t ^ \\dagger = \\lim _ { \\Delta t \\to 0 } \\sum _ { i = 1 } ^ L g ( X _ { t _ n } ^ \\dagger , t _ n ) ( X _ { t _ { n + 1 } } ^ \\dagger - X _ { t _ n } ^ \\dagger ) \\end{align*}"}
+{"id": "9023.png", "formula": "\\begin{align*} _ k ( X ^ { \\Gamma _ n } ) = h p ^ n + O ( 1 ) \\end{align*}"}
+{"id": "7756.png", "formula": "\\begin{align*} \\widetilde f ( s ) & = \\int _ 0 ^ \\infty d t \\ ; e ^ { - s x } \\int _ 0 ^ \\infty e ^ { - x t } u ( t ) d x = \\int _ 0 ^ \\infty u ( t ) \\int _ 0 ^ \\infty e ^ { - ( s + t ) x } d x \\ ; d t \\\\ & = \\int _ 0 ^ \\infty \\frac { u ( t ) } { s + t } \\ ; d t \\end{align*}"}
+{"id": "2487.png", "formula": "\\begin{align*} ( x ' _ n , x _ n ) = 1 \\ \\ ( x ' _ n , x _ k ) = 0 , \\ \\ k \\neq n , \\ n , k \\geq 0 , \\end{align*}"}
+{"id": "1432.png", "formula": "\\begin{align*} \\mathbb { P } ( | \\mathcal { C } _ { \\max } ( \\mathbb { G } ) | > k ) \\leq \\mathbb { P } ( Z _ { > k } \\geq k ) \\leq \\frac { \\mathbb { E } [ Z _ { > k } ] } { k } & = k ^ { - 1 } n \\mathbb { P } ( | \\mathcal { C } ( V _ n ) | > k ) \\\\ & \\leq k ^ { - 1 } n c _ 0 \\mathbb { P } ( r + S _ t > 0 \\forall t \\leq k ) . \\end{align*}"}
+{"id": "2085.png", "formula": "\\begin{align*} A \\circ B = \\{ ( a _ 1 \\circ b _ 1 , \\ldots , a _ m \\circ b _ m ) : \\ ; \\ ; ( a _ 1 , \\ldots , a _ m ) \\in A , \\ ; ( b _ 1 , \\ldots , b _ m ) \\in B \\} , \\end{align*}"}
+{"id": "4053.png", "formula": "\\begin{align*} h ( \\theta ) & = g ( x _ \\theta , y _ 1 , g ^ * ( x _ 0 , y _ 1 , u _ 0 ) ) - g ( x _ \\theta , y _ 0 , g ^ * ( x _ 0 , y _ 0 , u _ 0 ) ) \\\\ & = g ( x _ \\theta , y _ 1 , z _ 1 ) - g ( x _ \\theta , y _ 0 , z _ 0 ) . \\end{align*}"}
+{"id": "7340.png", "formula": "\\begin{align*} A = \\frac { d } { d \\tilde { m } } \\frac { d } { d \\tilde { s } } . \\end{align*}"}
+{"id": "2205.png", "formula": "\\begin{align*} \\emph { \\textbf { x } } _ { 2 m } & = \\emph { \\textbf { x } } _ { 2 m - 1 } A , \\\\ \\emph { \\textbf { x } } _ { 2 m + 1 } & = \\emph { \\textbf { x } } _ { 2 m } B , \\end{align*}"}
+{"id": "9010.png", "formula": "\\begin{align*} \\frac { f \\left ( \\bigcup _ { i \\in F } A _ { i } C _ i \\right ) } { \\theta ( A ) } & \\leq \\sum _ { i \\in F } \\sum _ { g \\in C _ i } \\frac { f ( A _ i g ) } { \\theta ( A ) } \\\\ & = \\sum _ { i \\in F } \\sum _ { g \\in C _ i } \\frac { f ( A _ { i } ) } { \\theta ( A _ { i } ) } \\frac { \\theta ( A _ { i } ) } { \\theta ( A ) } \\\\ & \\leq ( \\lambda + \\epsilon ) \\sum _ { i \\in F } \\sum _ { g \\in C _ i } \\frac { \\theta ( A _ { i } ) } { \\theta ( A ) } . \\end{align*}"}
+{"id": "5034.png", "formula": "\\begin{align*} \\begin{cases} 4 c ^ 4 = \\lambda h ^ 2 , \\\\ 4 c ^ 2 = \\frac { h ^ 2 } { \\mu } . \\end{cases} \\end{align*}"}
+{"id": "3704.png", "formula": "\\begin{align*} \\gamma : = \\gamma ( Q _ i ) \\end{align*}"}
+{"id": "1705.png", "formula": "\\begin{align*} F ( t ) = ( - 1 ) ^ { \\frac { k } { 2 } } ( i t ) ^ { \\frac { 2 - k } { 2 } + m } ( k - 1 ) \\sum _ { 0 \\leq j \\equiv \\frac { k - 2 } { 2 } - m \\ ; ( { \\rm m o d } \\ ; 2 ) } \\binom { k } { j } \\int _ { S ^ 1 } s ( \\mu _ m ) ( \\kappa ( \\theta ) ) \\frac { ( i t ) ^ j ( - \\sin \\theta ) ^ { k - j } \\cos \\theta ^ j } { ( \\sin ^ 2 \\theta + t ^ 2 \\cos ^ 2 \\theta ) } d \\theta . \\end{align*}"}
+{"id": "811.png", "formula": "\\begin{align*} E _ n ( \\epsilon , \\delta ) = \\left \\{ x \\in Y _ \\infty : \\sup _ { | t - s | < \\delta } | \\widehat { S } _ n x ( t ) - \\widehat { S } _ n x ( s ) | > \\epsilon \\right \\} . \\end{align*}"}
+{"id": "7853.png", "formula": "\\begin{align*} f ^ { ( n ) } + A _ { n - 1 } ( z ) f ^ { ( n - 1 ) } + \\cdots + A _ 1 ( z ) f ' + A _ 0 ( z ) f = 0 \\end{align*}"}
+{"id": "4205.png", "formula": "\\begin{align*} \\Psi ^ { ( 1 ) } = ( 0 , 0 ) , \\end{align*}"}
+{"id": "5811.png", "formula": "\\begin{align*} u = \\sum _ { r = 1 } ^ p u _ r e _ r / \\| e _ r \\| , \\end{align*}"}
+{"id": "3033.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { D } | x - y | ^ { 2 - n } d y \\leq \\int _ { B _ R ( x ) } | x - y | ^ { 2 - n } d y \\leq C _ n R ^ { 2 } \\leq C _ n | D | ^ { \\frac { 2 } { n } } , \\ \\ \\forall x \\in \\mathbb { R } ^ n . \\end{aligned} \\end{align*}"}
+{"id": "6123.png", "formula": "\\begin{align*} D ^ T ( B ^ T ) ^ s ( A ^ T ) ^ r \\cdots ( B ^ T ) ^ q ( A ^ T ) ^ p x = 0 \\end{align*}"}
+{"id": "106.png", "formula": "\\begin{align*} x \\circ ( y z ) = ( x \\circ y ) z \\end{align*}"}
+{"id": "5637.png", "formula": "\\begin{align*} \\omega _ v ( z _ v ) = \\begin{cases} & \\frac { d z _ v } { z _ v } \\quad d _ F ( v ) = x \\\\ & \\frac { d z _ v } { 1 - z _ v } \\quad d _ F ( v ) = y \\end{cases} \\end{align*}"}
+{"id": "4269.png", "formula": "\\begin{align*} E _ \\gamma ( f ) & \\geq \\frac { 1 } { 2 } \\| ( \\nabla - i A ) f \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } V _ \\gamma ( x ) | f ( x ) | ^ 2 d x - \\frac { 1 } { 2 } \\left ( \\frac { c } { M ( Q ) } \\right ) ^ { \\frac { 2 } { N } } \\| ( \\nabla - i A ) f \\| ^ 2 _ { L ^ 2 } \\\\ & = \\int _ { \\R ^ N } V _ \\gamma ( x ) | f ( x ) | ^ 2 d x \\geq 0 . \\end{align*}"}
+{"id": "695.png", "formula": "\\begin{align*} H ( z , w ) = h _ 0 ( w ) + \\frac { 1 } { 2 } \\left ( h _ 1 ( w ) ( z - w ) + \\overline { h _ 1 ( w ) } ( \\bar { z } - \\bar { w } ) \\right ) + \\end{align*}"}
+{"id": "7953.png", "formula": "\\begin{align*} x + y + \\ell _ 1 = m + \\ell _ 2 , x + \\ell _ 3 = m ' + \\ell _ 4 , \\ell _ i \\in L . \\end{align*}"}
+{"id": "4632.png", "formula": "\\begin{align*} v = P ^ { n } u \\end{align*}"}
+{"id": "5858.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } \\psi '' - \\Delta \\psi + b ( \\psi ) = 0 & \\hbox { i n } ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu \\psi = e ^ { \\lambda h } ( e , D H ) & \\hbox { o n } ( 0 , T ) \\times \\Gamma , \\\\ t = 0 : \\psi = 0 , \\psi ' = 0 & \\hbox { i n } ~ \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "7358.png", "formula": "\\begin{align*} \\| f _ \\infty - f _ m \\| _ \\infty \\le \\sum _ { i = m } ^ { \\infty } \\| f _ { i + 1 } - f _ i \\| _ \\infty \\le \\sum _ { i = m } ^ { \\infty } \\frac { 2 c } { 2 ^ { d + \\varepsilon i } } = \\frac { c \\ , 2 ^ { 1 + \\varepsilon } } { 2 ^ { d + \\varepsilon m } ( 2 ^ \\varepsilon - 1 ) } . \\end{align*}"}
+{"id": "6124.png", "formula": "\\begin{align*} D ^ T y = 0 , A ^ T y \\in V , B ^ T y \\in V . \\end{align*}"}
+{"id": "5191.png", "formula": "\\begin{align*} & p _ { 1 2 3 } p _ { 1 4 5 } - p _ { 1 2 4 } p _ { 1 3 5 } + p _ { 1 2 5 } p _ { 1 3 4 } = 0 \\ , , \\\\ & p _ { 1 2 3 } p _ { 4 5 6 } - p _ { 1 2 4 } p _ { 3 5 6 } + p _ { 1 2 5 } p _ { 3 4 6 } - p _ { 1 2 6 } p _ { 3 4 5 } = 0 \\\\ & p _ { 2 3 4 } A _ 1 - p _ { 1 3 4 } A _ 2 + p _ { 1 2 4 } A _ 3 - p _ { 1 2 3 } A _ 4 = 0 \\\\ & p _ { 1 2 3 } ^ 2 p _ { 4 5 6 } + \\det { \\begin{pmatrix} p _ { 2 3 4 } & p _ { 2 3 5 } & p _ { 2 3 6 } \\\\ p _ { 1 3 4 } & p _ { 1 3 5 } & p _ { 1 3 6 } \\\\ p _ { 1 2 4 } & p _ { 1 2 5 } & p _ { 1 2 6 } \\end{pmatrix} } = 0 \\ , . \\end{align*}"}
+{"id": "928.png", "formula": "\\begin{align*} \\lambda _ { 2 } ( E ) = \\int _ 0 ^ 1 \\tau _ { E , \\alpha } ( h ) \\mathrm { d } h . \\end{align*}"}
+{"id": "3729.png", "formula": "\\begin{align*} I ( f ) ( \\varpi ^ n \\xi ) = n \\Psi _ f ( 0 ) + a _ 0 ( f ) ( \\xi ) \\end{align*}"}
+{"id": "7663.png", "formula": "\\begin{align*} \\eta = \\bigoplus _ { \\substack { I \\subseteq [ n ] \\\\ | I | = t } } \\eta _ { I } = \\bigoplus _ { \\substack { I \\subseteq [ n ] \\\\ | I | = t } } \\sum _ { p \\in \\mathbb { Z } } \\eta _ { I , p } \\in \\bigoplus _ { \\substack { I \\subseteq [ n ] \\\\ | I | = t } } \\Gamma ( D ( x _ { I } ) , \\Omega _ { X } ^ { j } ( \\log f ) ) \\end{align*}"}
+{"id": "7718.png", "formula": "\\begin{align*} t _ 0 ^ { ( 1 ) } & = 2 \\left ( \\frac { \\tau } { 1 + \\rho e ^ { i \\beta \\pi } } \\right ) ^ { \\frac { 1 } { 2 } } i - 1 \\\\ & \\sim 2 \\left ( \\frac { \\tau } { \\rho } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { i \\frac { \\pi } { 2 } ( 1 - \\beta ) } - 1 , \\rho \\rightarrow \\infty . \\end{align*}"}
+{"id": "655.png", "formula": "\\begin{align*} \\bar { A } : = H _ { n } , \\bar { B } : = B _ { c _ { 1 } \\cdots c _ { n } } , \\bar { \\nu } : = \\nu ( C _ { n + 1 } ) ^ { - 1 } \\nu | _ { C _ { n + 1 } } . \\end{align*}"}
+{"id": "4588.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\frac { 1 } { b _ i } \\leq \\sum _ { i = 1 } ^ m \\frac { 1 } { a _ i } . \\end{align*}"}
+{"id": "198.png", "formula": "\\begin{align*} \\liminf \\frac { r _ { n } + 1 } { \\log ( h _ { n } ) } \\geq \\liminf \\frac { r _ { n } + 1 } { n \\log ( r _ { n } + 1 ) } \\geq \\liminf \\frac { r _ { n } } { n r _ { n } ^ { 1 / 3 } } = \\liminf \\frac { r _ { n } ^ { 2 / 3 } } { n } \\geq \\liminf \\frac { n ^ { 4 / 3 } } { n } = \\infty . \\end{align*}"}
+{"id": "9243.png", "formula": "\\begin{align*} I _ { X , H _ h } = \\left \\langle d _ { ( 1 , \\ldots , n ) ( u _ 1 , \\ldots , u _ { i - 1 } , X u _ j , u _ { i + 1 } , \\ldots , u _ n ) } : i > h ( j ) \\right \\rangle \\end{align*}"}
+{"id": "2249.png", "formula": "\\begin{align*} [ g U ^ { i + 1 } _ D , g ' U ^ { j + 1 } _ D ] : = g g ' g ^ { - 1 } g '^ { - 1 } U ^ { i + j + 1 } _ D , ~ ~ g \\in U ^ { i } _ D , ~ g ' \\in U ^ { j } _ D , \\end{align*}"}
+{"id": "5870.png", "formula": "\\begin{align*} w = ( e , U ) , { \\cal L } \\theta = - ( e , A U ) , { \\cal R } \\theta = - ( e , B U ) | _ { \\Sigma } . \\end{align*}"}
+{"id": "5337.png", "formula": "\\begin{align*} \\operatorname { d i v } ( \\varepsilon v ) = \\operatorname { t r } ( \\varepsilon D v ) + \\operatorname { d i v } \\varepsilon \\cdot v \\mbox { a . e . i n } \\Omega . \\end{align*}"}
+{"id": "1413.png", "formula": "\\begin{align*} k _ { \\rho , x } = M _ { \\rho , x } - M _ { \\rho , x + 1 } . \\end{align*}"}
+{"id": "6877.png", "formula": "\\begin{align*} \\Phi ^ { - } : = \\inf _ { \\mathcal { D } } \\Phi ( x ) \\quad \\Phi ^ { + } : = \\sup _ { \\mathcal { D } } \\Phi ( x ) . \\end{align*}"}
+{"id": "9265.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , v _ 2 , \\ldots , v _ { h ( i ) } , N v _ i , \\ldots v _ n ) } , \\end{aligned} \\end{align*}"}
+{"id": "3478.png", "formula": "\\begin{align*} { { \\bf { h } } ^ H } \\left [ n \\right ] = \\sum \\nolimits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H \\delta \\left [ { n - { n _ l } } \\right ] } , \\end{align*}"}
+{"id": "1968.png", "formula": "\\begin{align*} & x = ( 1 , 1 , 3 , 0 , 1 , 3 , 1 , 0 , 3 , 2 , 3 , 0 , 2 , 2 , 1 , 1 , 1 , 2 , 1 , 3 , 2 , 2 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 ) , \\\\ & y = ( 1 , 3 , 2 , 0 , 0 , 3 , 2 , 2 , 0 , 2 , 0 , 3 , 1 , 2 , 1 , 0 , 2 , 0 , 2 , 2 , 0 , 2 , 2 , 3 , 1 , 1 , 1 , 1 , 2 , 1 , 2 , 2 ) . \\end{align*}"}
+{"id": "7632.png", "formula": "\\begin{align*} \\varrho ( H , \\vec { y } ) = { | \\vec { a } ^ { \\ , \\mathrm { T } } \\vec { y } | } . \\end{align*}"}
+{"id": "2737.png", "formula": "\\begin{align*} M = \\bigoplus _ { n \\in \\Z } M _ n , M _ n = \\{ v \\in M | K _ i v = q _ i ^ n v \\} . \\end{align*}"}
+{"id": "7848.png", "formula": "\\begin{align*} f ( z ) - a = \\frac { g ( z ) - a h ( z ) } { h ( z ) } = \\frac { \\sum _ { j = 0 } ^ n ( G _ j ( z ) - a H _ j ( z ) ) e ^ { w _ j z ^ q } } { \\sum _ { j = 0 } ^ n H _ j ( z ) e ^ { w _ j z ^ q } } . \\end{align*}"}
+{"id": "661.png", "formula": "\\begin{align*} \\frac { \\partial { \\bf v } } { \\partial t } + ( { \\bf { v } } \\cdot \\nabla ) { \\bf v } = \\nabla ( { \\rm s o m e t h i n g } ) , \\end{align*}"}
+{"id": "7003.png", "formula": "\\begin{align*} R _ c ( R _ p ) = \\frac { 1 } { 2 } \\log ^ { + } { \\frac { N ^ 2 ( X , Y ) } { ( 1 - \\rho ) \\left ( 2 \\Delta e ^ { R _ p } + \\rho - 1 \\right ) } } , \\end{align*}"}
+{"id": "2229.png", "formula": "\\begin{align*} \\nu ( x _ { 2 m + 1 , 1 } ) = \\nu { \\left ( x _ { 2 m , 1 } \\beta _ { 1 , 1 } + \\sum _ { k = 2 } ^ \\infty x _ { 2 m , k } \\beta _ { k , 1 } \\right ) } . \\end{align*}"}
+{"id": "7360.png", "formula": "\\begin{align*} \\Tilde { L } _ i = L _ i \\setminus \\left ( \\bigcup _ { j = i + 1 } ^ \\infty K _ j \\right ) . \\end{align*}"}
+{"id": "8884.png", "formula": "\\begin{align*} \\left ( - \\frac { d } { \\sin u d u } \\right ) ^ { l } \\left ( \\frac { \\sin ( k + l + 1 ) u } { \\sin u } \\right ) = 2 ^ { l } l ! C _ { k } ^ { l + 1 } ( \\cos u ) , \\end{align*}"}
+{"id": "8620.png", "formula": "\\begin{align*} x _ { n + 1 } = \\alpha _ n x _ n + ( 1 - \\alpha _ n ) T x _ n \\end{align*}"}
+{"id": "9113.png", "formula": "\\begin{align*} \\left . \\frac { d ^ 2 } { d \\rho ^ 2 } h _ { \\rho } \\right | _ { \\rho = \\rho _ 1 } > 0 , \\end{align*}"}
+{"id": "1010.png", "formula": "\\begin{align*} - \\log ( 1 - F ( C | n | ^ \\alpha ) ) \\underset { C \\rightarrow \\infty } { \\longrightarrow } - \\log ( 1 - F ( \\infty ) ) = - \\log ( 1 - 0 ) = 0 . \\end{align*}"}
+{"id": "16.png", "formula": "\\begin{align*} \\mu _ 1 + \\mu _ 2 < \\frac { 1 } { 2 } \\lambda _ { m i n } , \\beta = \\frac { \\mu _ 1 - \\mu _ 2 } { 2 } + \\frac { B _ { \\bar { \\tilde { b } } _ x } - B _ { \\bar { g } _ y } } { 2 } , \\end{align*}"}
+{"id": "8288.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { H } \\ ! \\ ! \\ ! & = \\lambda _ { x _ E } u _ { x _ E } + \\lambda _ { y _ E } u _ { y _ E } + \\lambda _ { z _ E } u _ { z _ E } \\\\ & ~ ~ + \\sum _ { i = 1 } ^ 3 ( \\lambda _ { x _ i } u _ { x _ i } + \\lambda _ { y _ i } u _ { y _ i } + \\lambda _ { z _ i } u _ { z _ i } ) \\end{array} \\right . \\end{align*}"}
+{"id": "503.png", "formula": "\\begin{align*} \\sum _ { K _ 1 \\ni j = k } ^ { \\nu } \\ ! \\sqrt { \\Phi ( x ^ { \\ell ( j + 1 ) } ) - \\Phi ( x ^ { j + 1 } ) } \\le \\frac { \\sqrt { 2 } ( m \\ ! + \\ ! 1 ) ^ 2 b } { \\sqrt { a } } \\sum _ { j = k } ^ { \\nu } \\ ! \\Xi _ { j + 1 } + a _ 1 \\sum _ { K _ 1 \\ni j = k } ^ { \\nu } \\sum _ { l = j - m } ^ { j } \\| x ^ { l + 1 } \\ ! - \\ ! x ^ { l } \\| . \\end{align*}"}
+{"id": "3792.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } g ( t ) = - 2 \\ , _ { g ( t ) } . \\end{align*}"}
+{"id": "2955.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla _ { j + i - 1 , j + i } \\overrightarrow { W } ^ \\ell _ j = \\ell ^ { - 1 } ( W ^ - _ { j + i - 1 } - W _ { j + i - 1 } + W ^ + _ { j + i } - W _ { j + i } ) \\end{aligned} \\end{align*}"}
+{"id": "937.png", "formula": "\\begin{align*} \\alpha \\cdot ( x _ { + } ( h ) - x _ { + } ( \\tilde { h } ) ) = \\sqrt { 1 + | m _ { \\alpha } | ^ { 2 } } \\left [ g \\left ( \\frac { h - h _ { 0 } } { e _ { 2 } \\cdot \\theta _ { x _ { 0 } } ^ { \\perp } } \\right ) - g \\left ( \\frac { \\tilde { h } - h _ { 0 } } { e _ { 2 } \\cdot \\theta _ { x _ { 0 } } ^ { \\perp } } \\right ) \\right ] . \\end{align*}"}
+{"id": "1214.png", "formula": "\\begin{align*} f ( \\ell , g h ) = f ( \\ell g , h ) + f ( \\ell , g ) . \\end{align*}"}
+{"id": "4287.png", "formula": "\\begin{align*} \\Delta \\ , w ( p ) = 0 \\quad \\mathrm { i n } \\ , \\ , \\Omega \\quad \\mathrm { a n d } \\quad \\dfrac { \\partial w ( p ) } { \\partial \\textbf { \\textit { n } } } = \\dfrac { \\partial p } { \\partial \\textbf { \\textit { n } } } \\quad \\mathrm { o n } \\ , \\ , \\Gamma . \\end{align*}"}
+{"id": "8953.png", "formula": "\\begin{align*} P _ { k } ^ { ( a , b ) } ( 1 ) = \\frac { ( a + 1 ) _ k } { k ! } . \\end{align*}"}
+{"id": "1938.png", "formula": "\\begin{align*} \\tilde { m } _ { { x } } = \\lim _ { n \\to \\infty } \\left ( \\hat { g } ^ n _ { P ^ s ( { \\sigma } ^ { - n } ( { x } ) ) } \\right ) _ * \\hat m _ { P ^ s ( { \\sigma } ^ { - n } ( { x } ) ) } , \\end{align*}"}
+{"id": "2681.png", "formula": "\\begin{align*} z + \\lim \\limits _ { t \\to \\infty } { Q \\big ( z + ( 1 - z ) ( 1 - p _ t ) \\big ) - ( 1 - p _ t ) - z p _ t \\over p _ t \\big ( 1 - Q ' ( 1 - p _ t ) \\big ) } & = z + \\lim \\limits _ { x \\rightarrow 1 - } { Q \\big ( z + ( 1 - z ) x \\big ) - \\big ( z + ( 1 - z ) x \\big ) \\over ( 1 - x ) ( 1 - Q ' ( x ) ) } \\\\ & = z + { 1 \\over 2 - L } ( 1 - z ) ^ { 2 - L } , \\end{align*}"}
+{"id": "3681.png", "formula": "\\begin{align*} \\textrm { E x t } _ { R ' } ^ { i } ( M ' , N ' ) \\cong \\left \\{ \\begin{array} { c c c c c c c } \\textrm { H o m } _ { R } ( M , N ) / y _ 1 \\textrm { H o m } _ { R } ( M , N ) , \\textrm { i f } i = 0 , \\\\ 0 , \\textrm { i f } i = 1 , \\ldots , n - 2 . \\end{array} \\right . \\end{align*}"}
+{"id": "1797.png", "formula": "\\begin{align*} g _ n ( 1 / m ) = 1 / \\pi _ C ^ m , h _ n ( 1 / m ) = 1 \\end{align*}"}
+{"id": "3573.png", "formula": "\\begin{align*} | m _ A ) : = \\frac { 1 } { \\sqrt { \\langle v _ A , v _ A \\rangle } } v _ A , \\end{align*}"}
+{"id": "8522.png", "formula": "\\begin{align*} U ' = \\{ 3 1 2 4 , 1 3 2 4 , 1 2 4 3 , 2 1 3 4 , 1 3 4 2 , 2 3 1 4 , 2 3 4 1 \\} . \\end{align*}"}
+{"id": "1847.png", "formula": "\\begin{align*} \\ddot { \\varrho } + \\frac { 4 } { 3 t } \\dot { \\varrho } - \\tilde { \\kappa } t ^ { - 2 \\gamma + \\frac { 2 } { 3 } } \\Delta \\varrho - \\frac { 2 } { 3 t ^ 2 } \\varrho = ( \\gamma - 1 ) \\tilde { \\kappa } t ^ { - 2 \\gamma + \\frac { 2 } { 3 } } \\frac { D ^ i \\varrho D _ i \\varrho } { 1 + \\varrho } . \\end{align*}"}
+{"id": "6182.png", "formula": "\\begin{align*} ( e _ r ) _ j = \\left \\lbrace \\begin{array} { l } 1 , n _ { r - 1 } + 1 \\leqslant j \\leqslant n _ { r } , \\\\ 0 , \\hbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
+{"id": "8548.png", "formula": "\\begin{align*} \\nu ( x ) = \\begin{cases} \\mu ( x ) & \\rm { i f } x \\in U \\\\ y & \\rm { i f } x = v \\end{cases} \\end{align*}"}
+{"id": "1561.png", "formula": "\\begin{align*} \\tilde { \\zeta ' } _ { \\tilde { v } ^ { * } } ( t _ { 2 } ) < 0 \\ , \\ , \\ , \\mbox { a n d } \\ , \\ , \\ , \\tilde { \\zeta } _ { \\tilde { v } ^ { * } } ( t _ { 2 } ) = \\lambda | | \\tilde { v } ^ { * } | | _ { r ( z ) } ^ { r ^ { - } } . \\end{align*}"}
+{"id": "34.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m _ 1 } T ( n _ i ) & = T ( n _ { m _ 1 } ) + \\sum _ { i = 1 } ^ { m _ 1 - 1 } T ( n _ i ) < T ( n _ { m _ 1 } ) + ( m _ 1 - 1 ) \\cdot \\frac { 3 5 } { 1 8 } \\cdot \\frac { 1 } { X _ 0 } \\\\ & < \\frac { 3 } { X _ 0 } + ( m _ 1 - 1 ) \\cdot \\frac { 3 5 } { 1 8 } \\cdot \\frac { 1 } { X _ 0 } = \\frac { 3 5 m _ 1 + 1 9 } { 1 8 } \\cdot \\frac { 1 } { X _ 0 } , \\end{align*}"}
+{"id": "620.png", "formula": "\\begin{align*} \\tilde J ^ \\dagger _ { t _ n , t _ { n + 1 } } : = \\int _ { t _ n } ^ { t _ { n + 1 } } ( A Z _ t ^ \\dagger ) ^ { \\rm T } { \\rm d } X _ t ^ \\dagger \\end{align*}"}
+{"id": "3909.png", "formula": "\\begin{align*} Y u ( K ) \\supset \\bigcap _ { i = 1 } ^ \\infty \\bigcup _ { k = i } ^ \\infty Y u _ k ( K ) . \\end{align*}"}
+{"id": "6390.png", "formula": "\\begin{align*} f _ 2 f _ { 2 ' } ^ { ( 3 ) } f _ 2 + f _ { 2 ' } ^ { ( 3 ) } f _ 2 = f _ { 2 ' } ^ { ( 2 ) } f _ 2 ^ { ( 2 ) } f _ { 2 ' } , \\\\ f _ 2 f _ { 2 ' } ^ { ( 4 ) } f _ 2 ^ { ( 2 ) } + f _ 2 ^ { ( 2 ) } f _ { 2 ' } ^ { ( 4 ) } f _ 2 = f _ { 2 ' } ^ { ( 2 ) } f _ 2 ^ { ( 3 ) } f _ { 2 ' } ^ { ( 2 ) } . \\end{align*}"}
+{"id": "8947.png", "formula": "\\begin{align*} \\Gamma ( \\xi ) = \\int \\limits _ { 0 } ^ { \\infty } t ^ { \\xi - 1 } e ^ { - t } d t , \\mathrm { R e } \\ , \\xi > 0 . \\end{align*}"}
+{"id": "1954.png", "formula": "\\begin{align*} \\begin{gathered} \\varrho _ { K } ^ { ( N , J ) } [ u , X ] \\leq { \\varrho } _ E ^ { ( N ) } [ u , X ] + L h _ N ^ \\alpha \\ ; \\ ; u \\in U , \\\\ { \\vartheta } _ { K } ^ { ( N , J ) } \\leq { \\vartheta } _ E ^ { ( N ) } + L h _ N ^ \\alpha \\ ; \\ ; . \\end{gathered} \\end{align*}"}
+{"id": "7927.png", "formula": "\\begin{align*} \\Delta _ x g ( \\cdot , \\lambda ) = \\frac { \\partial g } { \\partial \\lambda } ( \\cdot , \\lambda ) \\ , , \\quad \\end{align*}"}
+{"id": "7298.png", "formula": "\\begin{align*} g _ m ( \\lambda ; x ) = P ^ m _ x [ \\mathrm { e } ^ { - \\lambda T _ 0 } ] . \\end{align*}"}
+{"id": "8669.png", "formula": "\\begin{align*} [ \\alpha ( z ) , \\alpha ( w ) ] = \\partial _ w \\delta ( z , w ) \\cdot 1 , \\end{align*}"}
+{"id": "9168.png", "formula": "\\begin{align*} \\tilde C ( s , m ^ 2 ) = \\gamma ( 1 + s \\gamma \\Delta ) + ( 1 + s \\gamma \\Delta ) C ( s , m ^ 2 ) ( 1 + s \\gamma \\Delta ) , \\end{align*}"}
+{"id": "7066.png", "formula": "\\begin{align*} H \\underline { R } [ 1 / \\tau ] ^ { C _ 2 } _ * = H \\underline { R } ^ { C _ 2 } _ \\diamond / ( \\tau - 1 ) & = R [ \\mu , \\tau ] / ( 2 \\mu , \\tau - 1 ) \\\\ & \\cong R [ u ] / ( 2 u ) \\\\ & = F ( E C _ { 2 + } , H R ) ^ { C _ 2 } _ * , \\end{align*}"}
+{"id": "546.png", "formula": "\\begin{align*} K _ t ( \\pi _ t ) = \\gamma ^ { - 1 } \\pi _ t \\left [ ( \\theta - \\pi _ t [ \\theta ] ) \\otimes f ( X _ t ^ \\dagger , \\theta ) \\right ] , \\end{align*}"}
+{"id": "2422.png", "formula": "\\begin{align*} p = B _ 1 ^ { - 1 } ( J _ { 1 2 } ( t ) v _ 2 - M _ { 1 2 } \\dot v _ 2 + f _ 1 ( t ) ) , \\end{align*}"}
+{"id": "4784.png", "formula": "\\begin{align*} \\Pr _ { \\rho \\sim P _ \\epsilon } [ \\rho \\notin D _ k ( H \\rho ^ \\intercal ) ] & \\leq 2 e ^ { - \\frac { \\sqrt { N } } { 3 \\epsilon } } + \\frac { N } { k ^ * } \\mathop { \\max } _ { \\substack { S \\subseteq \\{ \\epsilon N \\pm N ^ { 3 / 4 } \\} \\\\ 1 \\leq | S | \\leq 2 } } \\Big \\{ \\frac { 1 } { \\binom { N } { S } } \\sum _ { j = 0 } ^ N \\Pr _ { v \\sim \\mu _ { t } } \\big [ | v H | = j \\big ] K _ S ( j ) ^ 2 - 1 \\Big \\} . \\end{align*}"}
+{"id": "2705.png", "formula": "\\begin{gather*} \\partial _ t ^ 2 u _ L - \\Delta u _ L + V u _ L = 0 \\\\ \\vec { u } _ { L \\restriction t = 0 } = ( u _ 0 , u _ 1 ) , \\end{gather*}"}
+{"id": "7373.png", "formula": "\\begin{align*} { } ^ L \\natural _ { \\chi ^ \\vee } = \\natural _ { \\chi } \\quad . \\end{align*}"}
+{"id": "6634.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u = \\nabla \\cdot A ( \\nabla u ) + \\xi \\\\ u | _ { t \\leq 0 } = 0 \\end{cases} \\end{align*}"}
+{"id": "3242.png", "formula": "\\begin{align*} L _ j ( t _ 1 , t _ 2 ) = L _ j ( t _ 1 , u ) + L _ j ( u , t _ 2 ) \\ ; \\ ; ( j \\in \\{ \\pm 1 , 0 \\} ) . \\end{align*}"}
+{"id": "3766.png", "formula": "\\begin{align*} \\alpha _ i \\alpha _ j = \\begin{cases} \\operatorname { I d } & i = j ; \\\\ \\alpha _ 3 \\alpha _ 0 & i = 0 , j = 2 ; \\\\ \\alpha _ 2 \\alpha _ 0 & i = 0 , j = 3 ; \\\\ \\alpha _ j \\alpha _ i & \\end{cases} \\end{align*}"}
+{"id": "980.png", "formula": "\\begin{align*} \\big \\vert \\sum _ { j = 1 } ^ { n } c _ j \\sigma ( \\varphi _ j ) \\big \\vert \\leq \\beta \\Vert \\sum _ { j = 1 } ^ { n } c _ j \\varphi _ j \\Vert _ { \\mathcal { A ^ { \\ast } } } \\end{align*}"}
+{"id": "8790.png", "formula": "\\begin{align*} U _ { t , \\alpha } = S _ { t } + ( \\alpha + \\theta ( t ) ) \\left ( k _ { t } \\otimes C _ { \\theta } k _ { t } \\right ) , \\end{align*}"}
+{"id": "3295.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } t ^ { \\frac 1 2 - \\frac 1 p } \\| v ( t ) - \\alpha \\Theta ( t , \\cdot ) \\| _ { L ^ p ( \\mathcal F _ 0 ) } = 0 \\forall \\ , p \\in ( 2 , \\infty ) \\\\ & \\sup _ { t > t _ 0 } ( t - t _ 0 ) ^ { \\frac 1 q } | \\ell ( t ) | < \\infty \\end{align*}"}
+{"id": "7627.png", "formula": "\\begin{align*} \\mathcal { A } ^ { ( 2 ) } & = - \\big ( ( n + 1 - | v _ n | ) \\c ( v _ n ) , \\psi _ k ' ( w _ n ) \\big ) \\leq C ( \\gamma , \\delta , n ) \\int _ { 0 } ^ { 1 } 1 \\cdot \\psi _ k ' ( w _ n ) \\d x \\\\ & \\leq C ( \\gamma , \\delta , n ) \\int _ { 0 } ^ { 1 } | w _ n ( x ) | _ + ^ 2 \\d x . \\end{align*}"}
+{"id": "698.png", "formula": "\\begin{align*} R _ { \\rm r o b i n } ( w ) = \\frac { 1 } { 2 \\pi } ( h _ 0 ( w ) + \\log \\lambda ( w ) ) . \\end{align*}"}
+{"id": "7531.png", "formula": "\\begin{align*} Z _ { \\tilde { g } _ { 1 , a } } ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) = \\dfrac { G _ { 1 , a } ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "4969.png", "formula": "\\begin{align*} q _ 1 : = g ^ { - 1 } ( f ^ { l t + 1 } + q ^ { t / p } ) \\in x _ 1 + g ^ { p - 2 } ( f ^ l h ) ^ { p - 1 } r k [ f , f ^ l g h , r ] . \\end{align*}"}
+{"id": "1518.png", "formula": "\\begin{align*} \\int _ 0 ^ x \\theta ^ { r - 2 } \\log \\left ( \\cos \\frac \\theta 2 \\right ) d \\theta = - \\sum _ { m = 1 } ^ \\infty \\frac { ( 1 - 2 ^ { - 2 m } ) \\zeta ( 2 m ) } { \\pi ^ { 2 m } m } \\frac { x ^ { 2 m + r - 1 } } { 2 m + r - 1 } , \\end{align*}"}
+{"id": "6036.png", "formula": "\\begin{align*} \\mathcal R _ n : = \\left \\{ \\frac { p ( z ) } { q ( z ) } = \\frac { p _ { n - 1 } z ^ { n - 1 } + p _ { n - 2 } z ^ { n - 2 } + \\cdots + p _ 0 } { z ^ n + q _ { n - 1 } z ^ { n - 1 } + \\cdots + q _ 0 } : ~ p \\in \\mathcal P _ { n - 1 } , ~ q \\in \\mathcal Q _ n \\right \\} \\end{align*}"}
+{"id": "5601.png", "formula": "\\begin{align*} \\nabla _ 3 S _ { 3 3 } = \\nabla _ { 1 } S _ { 0 3 } = \\nabla _ { 0 } S _ { 1 3 } = \\nabla _ { 3 } S _ { 0 1 } = 0 . \\end{align*}"}
+{"id": "1737.png", "formula": "\\begin{align*} s = \\sum _ { | \\lambda | \\leq n \\leq M } C _ n \\cdot \\varphi _ n \\circ t _ n . \\end{align*}"}
+{"id": "3714.png", "formula": "\\begin{align*} \\Theta _ { \\mathcal { F } _ 2 ( f ) ( \\cdot , 0 , 0 ) } ( g ) : & = \\sum _ { \\xi \\in V _ i ( F ) } \\rho _ i ( g ) \\mathcal { F } _ 2 ( f ) ( \\xi , 0 , 0 ) \\end{align*}"}
+{"id": "2385.png", "formula": "\\begin{align*} P E _ 1 = \\left [ \\begin{array} { c } I _ { d } \\\\ 0 \\end{array} \\right ] , \\end{align*}"}
+{"id": "5467.png", "formula": "\\begin{align*} E V ( X _ t , \\alpha _ t ) - E V ( X _ 0 , \\alpha _ 0 ) = E \\int _ 0 ^ t L V ( X _ s , \\alpha _ s ) d s \\leq ( \\lambda _ 1 + \\lambda _ 2 ) E \\int _ 0 ^ t V ( X _ s , \\alpha _ s ) d s . \\end{align*}"}
+{"id": "535.png", "formula": "\\begin{align*} \\Z [ \\zeta _ { k } ] \\otimes \\C = \\bigoplus _ { \\xi \\in Z ( \\Phi _ k ) } E _ { \\zeta _ { k } \\otimes 1 } ( \\xi ) \\end{align*}"}
+{"id": "7342.png", "formula": "\\begin{align*} m ' ( x ) = \\tilde { m } ' ( \\tilde { s } ^ { - 1 } ( x ) ) ( \\tilde { s } ^ { - 1 } ( x ) ) ' = 2 W ( \\tilde { s } ^ { - 1 } ( x ) ) ^ 2 . \\end{align*}"}
+{"id": "8531.png", "formula": "\\begin{align*} \\displaystyle \\frac { 1 } { r } ( r v ) _ r + w _ z = 0 , \\ ( r , z ) \\in [ 0 , 1 ] \\times [ 0 , 1 ] , \\end{align*}"}
+{"id": "324.png", "formula": "\\begin{align*} \\frac { 1 } { X } \\sum _ { 2 \\nmid d } \\nolimits ^ \\flat L ( \\frac { 1 } { 2 } , \\phi \\otimes \\chi _ { 8 d } ) \\chi _ { 8 d } ( l ) \\Phi ( \\frac { d } { X } ) = S ( \\chi _ { 8 d } ( l ) B ( d ) ; \\Phi ) + \\prod _ { i = 1 } ^ 3 \\frac { \\Gamma ( \\frac { \\frac { 1 } { 2 } + \\gamma _ i } { 2 } ) } { \\Gamma ( \\frac { \\frac { 1 } { 2 } - \\gamma _ i } { 2 } ) } \\overline { S ( \\chi _ { 8 d } ( l ) B ( d ) ; \\Phi ) } , \\end{align*}"}
+{"id": "5166.png", "formula": "\\begin{align*} \\mathcal { I } ( D L G ( 1 , 2 ) ) & \\approx \\dfrac { 6 ! \\cdot n ^ { 6 } } { 3 ^ { 6 } n ^ { 6 } } = \\dfrac { 1 0 } { 8 1 } . \\end{align*}"}
+{"id": "7079.png", "formula": "\\begin{align*} F _ n E ^ { C _ 2 } _ * ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) = E ^ { C _ 2 } _ { * + | n \\sigma | - n \\sigma } ( \\mathbf { C P } ^ \\infty _ { C _ 2 } ) \\end{align*}"}
+{"id": "3423.png", "formula": "\\begin{align*} \\alpha ( \\phi ) : = \\sup \\left \\{ \\nu ( \\phi ) \\colon \\nu \\in \\mathcal { M } _ { \\sigma } ( \\Sigma ) \\right \\} . \\end{align*}"}
+{"id": "9186.png", "formula": "\\begin{align*} \\Psi _ j ( X , \\varphi ) = \\sum _ { q \\in \\Z } \\hat { \\Psi } _ { j , q } ( X , \\varphi ) \\end{align*}"}
+{"id": "8658.png", "formula": "\\begin{align*} f = F ( p _ 1 , p _ 2 , p _ 3 , \\dots ) , \\end{align*}"}
+{"id": "3481.png", "formula": "\\begin{align*} \\begin{aligned} y \\left [ n \\right ] & = \\sum \\nolimits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ l } s \\left [ { n - { \\kappa _ l } - { n _ l } } \\right ] } + \\\\ & \\sum \\nolimits _ { l = 1 } ^ L { \\sum \\nolimits _ { l ' \\ne l } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ { l ' } } s \\left [ { n - { \\kappa _ { l ' } } - { n _ l } } \\right ] } } + z \\left [ n \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "8073.png", "formula": "\\begin{align*} \\mathcal { D } = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\frac { A ( p , m ) } { m p ^ { \\frac { 1 } { 2 } } } H _ { m , p } . \\end{align*}"}
+{"id": "8446.png", "formula": "\\begin{align*} c _ * ( A ' ) = c _ * ( H ) = c _ * ( A ) . \\end{align*}"}
+{"id": "4876.png", "formula": "\\begin{align*} \\Delta _ n ( q ) = \\mathrm { s p a n } _ \\R \\big \\{ [ f _ { \\j _ n } , \\dots , f _ { \\j _ 1 } ] ( q ) \\mid \\j _ 1 , \\dots , \\j _ n \\in \\{ 1 , \\dots , d \\} \\big \\} \\subset T _ q M . \\end{align*}"}
+{"id": "8182.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - \\lambda a } \\mathcal { P } _ { A } ( d a ) = \\Phi ( 1 , - \\lambda ) \\forall \\lambda \\in \\mathbb { C } , \\quad \\Re \\lambda \\geq 0 . \\end{align*}"}
+{"id": "8514.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t + a ( x , u , u _ x ) u _ { x x x x } + b ( x , u , u _ x , u _ { x x } ) u _ { x x x } + c ( x , u , u _ x , u _ { x x } ) = 0 , \\\\ & u ( x , 0 ) = u _ 0 ( x ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "997.png", "formula": "\\begin{align*} \\lambda ( \\varphi _ t ( P ) ) = \\sum _ { j = 1 } ^ k a _ j ^ 2 \\lambda _ 1 ( \\varphi _ t ( \\Omega _ j ) ) , \\end{align*}"}
+{"id": "7876.png", "formula": "\\begin{align*} f ' = \\sum _ { k = 0 } ^ { n } \\exp ( - q _ k z ) P _ k ( e ^ z ) f ^ k , \\end{align*}"}
+{"id": "6515.png", "formula": "\\begin{align*} \\mu _ p ' \\leq 1 / 4 - \\frac { 1 } { k } \\underset { i = 1 } { \\overset { k } { \\sum } } ( p _ i - \\frac { 1 } { 2 } ) ^ 2 = \\frac { 1 } { k } \\underset { i = 1 } { \\overset { k } { \\sum } } p _ i ( 1 - p _ i ) = : \\mu _ p . \\end{align*}"}
+{"id": "4279.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| u _ { 0 , n } - \\phi _ 0 \\| _ { \\Sigma _ \\gamma } = 0 , \\end{align*}"}
+{"id": "5222.png", "formula": "\\begin{align*} & Q _ { 5 6 } = b _ { 3 5 } ^ { 2 } \\left ( k \\cos \\ ! \\left ( \\theta \\right ) + 1 \\right ) \\theta ' = c _ { 5 6 } \\\\ & b _ { 1 1 } b _ { 2 2 } b _ { 3 5 } \\big ( \\ell _ 1 ^ 2 + \\ell _ 2 ^ 2 \\big ) + 1 = \\frac { \\theta ' + \\gamma b _ { 3 5 } \\big ( k ^ { 2 } + 2 \\ , k \\ , \\cos \\ ! \\left ( \\theta \\right ) + 1 \\big ) } { \\theta ' } = 0 \\ . \\end{align*}"}
+{"id": "1096.png", "formula": "\\begin{align*} \\sum _ { j > k } \\theta _ j ^ 2 = \\sum _ { j > k } j ^ { 2 \\beta } \\frac { \\theta _ j ^ 2 } { j ^ { 2 \\beta } } \\leq \\frac { 1 } { k ^ { 2 \\beta } } \\sum _ { j > k } j ^ { 2 \\beta } \\theta _ j ^ 2 \\leq r ^ 2 k ^ { - 2 \\beta } . \\end{align*}"}
+{"id": "6297.png", "formula": "\\begin{align*} \\pi _ i ^ { r e v B S } ( T ) = \\begin{cases} T , & , \\\\ 0 , & , \\\\ s _ i ( T ) , & . \\end{cases} \\end{align*}"}
+{"id": "3835.png", "formula": "\\begin{align*} g _ V ( 0 ) = & \\int _ { 0 } ^ { \\infty } \\frac { ( 1 - \\theta ^ 2 ) ( w _ 2 ^ 2 x ^ 2 - 2 \\theta w _ 1 w _ 2 x + w _ 1 ^ 2 ) x } { 2 \\pi \\big ( w _ 2 ^ 2 x ^ 2 - 2 \\theta w _ 1 w _ 2 x + w _ 1 ^ 2 \\big ) ^ { 3 / 2 } \\bigl ( x ^ 2 + 2 \\theta x + 1 \\big ) ^ { 3 / 2 } } d x \\\\ & + \\int _ { 0 } ^ { \\infty } \\frac { ( 1 - \\theta ^ 2 ) ( w _ 2 ^ 2 x ^ 2 - 2 \\theta w _ 1 w _ 2 x + w _ 1 ^ 2 ) x } { 2 \\pi \\big ( w _ 2 ^ 2 x ^ 2 - 2 \\theta w _ 1 w _ 2 x + w _ 1 ^ 2 \\bigr ) ^ { 3 / 2 } \\bigl ( x ^ 2 - 2 \\theta x + 1 \\big ) ^ { 3 / 2 } } d x , \\end{align*}"}
+{"id": "8965.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\nu _ i ( u ) \\partial _ r d i s t ^ i _ N ( u ) = \\sum _ { i = 1 } ^ m \\nu _ i ( u ) \\nu _ i ( u ) \\cdot u _ r = d \\pi _ N ^ { \\perp } ( u ) u _ r \\ \\hbox { o n } \\partial B = S ^ 1 , \\end{align*}"}
+{"id": "5851.png", "formula": "\\begin{align*} \\| ( U ' ( t ) , - U ( t ) ) \\| _ { ( \\mathcal H _ { - 1 } ) ^ N \\times ( \\mathcal H _ 0 ) ^ N } = \\| L _ t \\circ S _ t ^ { - 1 } \\| \\qquad \\qquad \\qquad \\\\ \\leq c ( \\| ( \\widehat { U } _ 0 , \\widehat { U } _ 1 ) \\| _ { ( \\mathcal H _ 0 ) ^ N \\times ( \\mathcal H _ { - 1 } ) ^ N } + \\| H \\| _ { L ^ 2 ( 0 , T ; ( L ^ 2 ( \\Gamma ) ) ^ M ) } ) \\end{align*}"}
+{"id": "1444.png", "formula": "\\begin{align*} \\mathbb { P } ( | \\mathcal { C } ( V _ n ) | > k ) = \\mathbb { P } ( | \\mathcal { A } _ t | > 0 \\forall t \\leq k ) & = \\mathbb { P } ( 1 + \\sum _ { i = 1 } ^ { t } ( \\eta _ i - 1 ) > 0 \\forall t \\leq k ) \\\\ & \\leq \\mathbb { P } ( 1 + \\sum _ { i = 1 } ^ { t } ( X _ i - 1 ) > 0 \\forall t \\leq k ) . \\end{align*}"}
+{"id": "4364.png", "formula": "\\begin{align*} a _ { i + 1 } = G \\left ( \\theta - \\sum _ { j = 1 } ^ i \\frac { 1 } { a _ j } \\right ) . \\end{align*}"}
+{"id": "8133.png", "formula": "\\begin{align*} \\mathcal { R } ^ - = O ( M ^ { - 1 } T ^ { \\frac { 3 } { 2 } + \\varepsilon } p ^ { \\varepsilon } ) . \\end{align*}"}
+{"id": "6193.png", "formula": "\\begin{align*} \\overline x _ l ^ { ( k ) } = C _ p \\widehat x _ l ^ { ( k ) } , 1 \\leqslant k \\leqslant \\overline d , 1 \\leqslant l \\leqslant \\overline r _ k . \\end{align*}"}
+{"id": "4426.png", "formula": "\\begin{align*} a _ 2 = \\left ( \\frac { 1 } { x _ 1 } + \\frac { 1 } { x _ 2 } - \\frac { 1 } { a _ 1 } \\right ) ^ { - 1 } . \\end{align*}"}
+{"id": "4554.png", "formula": "\\begin{align*} \\frac { p } { q } = \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 } + \\cdots + \\frac { 1 } { a _ { n - 1 } } + \\frac { 1 } { a _ n + 1 } + \\frac { 1 } { a _ n ( a _ n + 1 ) } . \\end{align*}"}
+{"id": "6306.png", "formula": "\\begin{align*} K _ { \\alpha , \\alpha } = 1 K _ { \\alpha , \\beta } \\ne 0 \\Longrightarrow \\beta \\le _ \\ell \\ , \\alpha . \\end{align*}"}
+{"id": "4129.png", "formula": "\\begin{align*} - \\frac { 1 } { 6 } | d K - \\sqrt { 3 } u H | ^ 2 = - \\frac { 1 } { 6 } | d K | ^ 2 + \\frac { \\sqrt { 3 } u } { 3 } \\langle d K , H \\rangle - \\frac { u ^ 2 } { 2 } | H | ^ 2 , \\end{align*}"}
+{"id": "7111.png", "formula": "\\begin{align*} x _ i p _ j - x _ j p _ i = x _ i ( u x _ j + p _ j ) - x _ j ( u x _ i + p _ i ) \\in I . \\end{align*}"}
+{"id": "7302.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { x \\in [ 0 , 1 ] } | G ^ { k + 1 } _ { m _ n } ( x ) | = 0 . \\end{align*}"}
+{"id": "757.png", "formula": "\\begin{align*} & \\mathsf { H } ^ { B } _ \\infty ( A ) = \\varprojlim \\mathsf { H } ^ { B } _ n ( A ) & & & & \\mathsf { h } ^ { B } _ \\infty ( A ) = \\varprojlim \\mathsf { h } ^ { B } _ n ( A ) , \\end{align*}"}
+{"id": "5509.png", "formula": "\\begin{align*} \\Gamma ( s ) = \\int _ 0 ^ \\infty y ^ { s - 1 } e ^ { - y } \\mathrm d y \\end{align*}"}
+{"id": "5215.png", "formula": "\\begin{align*} A = \\begin{pmatrix} \\cos ( \\theta _ 0 t ) & - \\sin ( \\theta _ 0 t ) & \\cos ( \\theta _ 0 t ) & \\sin ( \\theta _ 0 t ) & 0 & 0 \\\\ \\sin ( \\theta _ 0 t ) & \\cos ( \\theta _ 0 t ) & - \\sin ( \\theta _ 0 t ) & \\cos ( \\theta _ 0 t ) & 0 & 0 \\\\ 0 & 0 & 0 & 0 & \\cos ( 2 \\theta _ 0 t ) & \\sin ( 2 \\theta _ 0 t ) \\end{pmatrix} \\ , \\end{align*}"}
+{"id": "4003.png", "formula": "\\begin{align*} D _ i \\log ( w _ { 1 1 } ) = \\frac { w _ { 1 1 , i } } { w _ { 1 1 } } \\\\ D _ { i i } \\log ( w _ { 1 1 } ) = \\frac { w _ { 1 1 , i i } } { w _ { 1 1 } } - \\frac { w _ { 1 1 , i } ^ 2 } { w _ { 1 1 } ^ 2 } , \\end{align*}"}
+{"id": "401.png", "formula": "\\begin{align*} a \\left ( \\sum _ { n = 1 } ^ { m } | c _ n | ^ 2 \\right ) ^ \\frac { 1 } { 2 } \\leq \\left \\| \\sum _ { n = 1 } ^ { m } c _ n \\tau _ n \\right \\| \\leq b \\left ( \\sum _ { n = 1 } ^ { m } | c _ n | ^ 2 \\right ) ^ \\frac { 1 } { 2 } , \\forall c _ 1 , \\dots , c _ m \\in \\mathbb { R } \\mathbb { C } \\end{align*}"}
+{"id": "167.png", "formula": "\\begin{align*} ( \\nu ( \\varkappa ) ) ( \\theta ) ( \\hslash ) : = \\nu ( z ( \\theta , \\hslash ) ) , \\ \\theta \\in [ 0 , \\tau ] . \\end{align*}"}
+{"id": "4087.png", "formula": "\\begin{align*} \\mathcal { G } _ t = \\rho _ { \\mathcal { G M } } ( ( f _ t , B _ t ) , ( g , b ) ) = ( f _ t ^ * g , f _ t ^ * b - B _ t ) \\end{align*}"}
+{"id": "8269.png", "formula": "\\begin{align*} U ( w ) & = \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma - u } D ( s + w ) e ^ { m s ^ 2 + M s } d s + \\sum ^ { \\star } _ { \\rho } \\exp \\left ( m ( \\rho - w ) ^ 2 + M ( \\rho - w ) \\right ) , \\end{align*}"}
+{"id": "5247.png", "formula": "\\begin{align*} \\mathcal { H } \\nabla _ X X + 2 A _ X U + T _ U U = 0 . \\end{align*}"}
+{"id": "5934.png", "formula": "\\begin{align*} \\hbox { K e r } ( D ^ T ) = V . \\end{align*}"}
+{"id": "6724.png", "formula": "\\begin{align*} \\frac { 1 } { q } \\sum _ { 0 \\leq u < q } e ^ { i 2 \\pi u ( n - a ) / q } = \\begin{cases} 1 & n = m q + a , \\\\ 0 & n \\ne m q + a , \\end{cases} \\end{align*}"}
+{"id": "8532.png", "formula": "\\begin{align*} \\begin{cases} w _ t = 2 w _ { x x } + \\varphi _ { t x } , \\\\ 2 \\varphi _ t = \\varphi _ { x x } - w _ x . \\end{cases} \\end{align*}"}
+{"id": "2077.png", "formula": "\\begin{align*} & \\frac { ( p + ( 1 - p ) \\xi ) ^ { 1 + \\nu } } { ( p + ( 1 - p ) \\xi ^ { 1 + \\nu } ) } = 1 - \\frac { \\Phi ( \\xi , p , \\nu ) } { ( p + ( 1 - p ) \\xi ^ { 1 + \\nu } ) } < 1 - c _ 1 \\nu . \\end{align*}"}
+{"id": "671.png", "formula": "\\begin{align*} ( \\frac { \\partial } { \\partial t } + \\mathcal { L } _ { \\bf { v } } ) ( \\omega ) = 0 . \\end{align*}"}
+{"id": "4361.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < \\theta . \\end{align*}"}
+{"id": "390.png", "formula": "\\begin{align*} \\frac { 1 } { j ^ p } \\leq \\prod _ { k = 1 } ^ { p } \\frac { 2 ^ k } { j + k } \\textrm { f o r a n y } j \\in \\mathbb { N } \\ , . \\end{align*}"}
+{"id": "7782.png", "formula": "\\begin{align*} b _ { i j } = \\{ d _ i e _ i , e _ j \\} . \\end{align*}"}
+{"id": "1639.png", "formula": "\\begin{align*} \\lambda _ s : = \\frac { | s - 1 | } { s } , \\quad \\alpha _ s = s ( 1 - \\lambda _ s ) ( 1 - A _ 1 | s - 1 | ) \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "2335.png", "formula": "\\begin{align*} R \\circ J \\circ T _ { e _ N } \\circ S _ 2 \\circ J \\circ T _ { - e _ N } ( z ) = \\frac { ( 2 x , 1 - | x | ^ 2 - y ^ 2 ) } { | x | ^ 2 + ( 1 + y ) ^ 2 } = { \\bf B } ( z ) . \\end{align*}"}
+{"id": "7490.png", "formula": "\\begin{align*} g ( u , v ) : = \\sum _ { i = i _ 0 } ^ { \\infty } \\alpha _ i u ^ i + \\sum _ { j = j _ 0 } ^ { \\infty } \\beta _ j v ^ j \\in \\mathcal { O } _ K [ u , v ] , \\end{align*}"}
+{"id": "4931.png", "formula": "\\begin{align*} - \\mathrm { i } \\frac { \\Delta t \\Delta x } { \\alpha } \\sum _ m \\left \\{ \\mathcal { F } ( \\mathbf { z } _ n ^ * , \\mathbf { z } _ { n + 1 } ^ * ) _ m \\mathcal { F } ( \\mathbf { z } _ n , \\mathbf { z } _ { n + 1 } ) _ m - \\mathcal { F } ( \\mathbf { z } _ n , \\mathbf { z } _ { n + 1 } ) _ m \\mathcal { F } ( \\mathbf { z } ^ * _ n , \\mathbf { z } ^ * _ { n + 1 } ) _ m \\right \\} = 0 \\end{align*}"}
+{"id": "3037.png", "formula": "\\begin{align*} m = s - s _ 0 \\ \\ \\mathrm { a n d } \\ \\ \\beta = \\alpha + \\frac { n } { p } + \\frac { 2 } { n p ' } - 1 . \\end{align*}"}
+{"id": "6296.png", "formula": "\\begin{align*} \\pi _ i ^ { T v W } ( T ) = \\begin{cases} T , & , \\\\ 0 , & , \\\\ s _ i ( T ) , & . \\end{cases} \\end{align*}"}
+{"id": "1936.png", "formula": "\\begin{align*} \\tilde { m } _ { { x } } = ( h ^ s _ { { x } , \\varphi ( { x } ) } ) _ * { m } _ { { x } } , \\end{align*}"}
+{"id": "7051.png", "formula": "\\begin{align*} \\dfrac { 1 } { u + u b _ 1 ' x + u b _ 2 ' x ^ 2 + \\cdots } = u ^ { - 1 } + d _ 1 x + d _ 2 x ^ 2 + \\cdots . \\end{align*}"}
+{"id": "6090.png", "formula": "\\begin{align*} h _ n ( \\tau ) = \\log \\left ( \\frac { \\varkappa _ n \\gamma _ n \\gamma _ n ^ * } { G _ { \\lambda _ n } \\rho ^ { 2 ( n - d _ n ) } } \\right ) + o ( 1 ) = o ( 1 ) . \\end{align*}"}
+{"id": "1910.png", "formula": "\\begin{align*} \\pi ( x ; q , a ) = \\frac { x } { \\phi ( q ) \\log x } + O \\left ( \\frac { x } { \\phi ( q ) ( \\L x ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "2920.png", "formula": "\\begin{align*} \\partial _ t u = \\nu \\partial _ x ^ 2 u + \\lambda \\partial _ x u ^ 2 + \\sqrt { D } \\partial _ x \\dot { W } ( t , x ) \\mathbb { R } _ + \\times \\mathbb { R } . \\end{align*}"}
+{"id": "8508.png", "formula": "\\begin{align*} \\frac { g _ t } { g } & = \\ ; P ^ 3 _ 1 ( \\varphi ) + P _ 2 ^ 2 ( \\varphi ) + P _ 3 ^ 1 ( \\varphi ) + P _ 1 ^ 1 ( \\varphi ) , \\\\ \\varphi _ t & = \\ ; - 3 \\varphi _ { \\xi ^ 5 } + P _ 2 ^ 3 ( \\varphi ) + P _ 3 ^ 2 ( \\varphi ) + P _ 1 ^ 2 ( \\varphi ) + P _ 4 ^ 1 ( \\varphi ) + P _ 2 ^ 1 ( \\varphi ) . \\end{align*}"}
+{"id": "9261.png", "formula": "\\begin{align*} N v _ 1 = v _ { k _ 1 } & w ( 1 ) - 1 = w ( k _ 1 ) 2 \\leq k _ 1 \\leq h ( 1 ) . \\end{align*}"}
+{"id": "8524.png", "formula": "\\begin{align*} X = \\{ 1 2 3 4 5 , 4 3 2 1 5 , 3 5 2 1 4 , 1 4 5 2 3 , 2 5 4 1 3 , 5 3 4 1 2 \\} . \\end{align*}"}
+{"id": "3100.png", "formula": "\\begin{align*} c _ 1 ^ { 2 2 } ( A ) & = \\bar { a } ( 1 - \\bar { b } ) \\int _ Y r _ B b _ { 1 1 } \\partial _ 1 w _ A = - 2 \\bar { a } ( 1 - \\bar { b } ) \\int _ Y ( \\partial _ 1 w _ A ) ( \\partial _ { 2 2 } ^ 2 w _ B ) , \\\\ c _ 2 ^ { 2 2 } ( A ) & = \\bar { a } ( 1 - \\bar { b } ) \\int _ Y r _ B b _ { 2 2 } \\partial _ 2 w _ A = 2 \\bar { a } ( 1 - \\bar { b } ) \\int _ Y ( \\partial _ 2 w _ A ) ( \\partial _ { 1 1 } ^ 2 w _ B ) . \\end{align*}"}
+{"id": "9229.png", "formula": "\\begin{align*} \\widehat { u } ( x ) = \\int _ { - \\infty } ^ { 0 } e ^ { - \\int _ { t } ^ { 0 } \\vert \\nabla \\ell _ { 0 } \\vert ^ { 2 } ( \\varphi _ { s } ( x ) ) \\ , d s } \\widehat { r } ( \\varphi _ { t } ( x ) ) \\ , d t . \\end{align*}"}
+{"id": "461.png", "formula": "\\begin{align*} \\int _ \\Omega f _ \\alpha ( x , y ) \\tau _ \\alpha \\ , d \\mu ( \\alpha ) & = \\int _ { 0 } ^ { 2 \\pi } ( x \\cos \\alpha + y \\sin \\alpha ) ( \\cos \\alpha , \\sin \\alpha ) \\ , d \\alpha \\\\ & = \\left ( \\int _ { 0 } ^ { 2 \\pi } ( x \\cos \\alpha + y \\sin \\alpha ) \\cos \\alpha \\ , d \\alpha , \\int _ { 0 } ^ { 2 \\pi } ( x \\cos \\alpha + y \\sin \\alpha ) \\sin \\alpha \\ , d \\alpha \\right ) \\\\ & = \\pi ( x , y ) , \\forall ( x , y ) \\in \\mathbb { R } ^ 2 , \\end{align*}"}
+{"id": "4960.png", "formula": "\\begin{align*} Q ( z , w ) = 1 + \\frac { w z P ^ 2 } { 1 - z ^ 3 P ^ 3 } = 1 + w z P ^ 2 + w z ^ 4 P ^ 5 + w z ^ 7 P ^ 8 + \\cdots . \\end{align*}"}
+{"id": "8396.png", "formula": "\\begin{align*} \\hat \\Psi _ { \\alpha } ( x ) & = \\hat \\Psi _ { \\alpha } ( x ) - \\tau _ { \\alpha } ( x ) \\hat \\psi _ { \\alpha } ( 0 ) + \\tau _ { \\alpha } ( x ) \\hat \\psi _ { \\alpha } ( 0 ) \\\\ & = \\tau _ { \\alpha } ( x ) \\hat \\psi _ { \\alpha } ( 0 ) + \\sum _ { k = 0 } ^ { \\tau _ { \\alpha } ( x ) - 1 } \\hat \\psi _ { \\alpha } ( f _ { \\alpha } ^ k ( x ) ) - \\hat \\psi _ { \\alpha } ( 0 ) , \\end{align*}"}
+{"id": "5213.png", "formula": "\\begin{align*} & g ^ 1 _ { 1 0 } g ^ 3 _ { 0 1 } - g ^ 1 _ { 0 1 } g ^ 3 _ { 1 0 } = 0 \\\\ & g ^ 1 _ { 0 1 } + g ^ 3 _ { 1 0 } = 0 \\\\ & g ^ 1 g ^ 2 _ { 0 1 } + g ^ 3 g ^ 2 _ { 1 0 } = 0 \\ . \\end{align*}"}
+{"id": "7103.png", "formula": "\\begin{align*} \\Omega ^ { C _ 2 } _ \\diamond = \\bigoplus _ { n \\geq 0 } \\Omega ^ { C _ 2 } _ { * - n \\sigma } \\end{align*}"}
+{"id": "5217.png", "formula": "\\begin{align*} A _ 4 & = \\ell _ 1 A _ 1 + \\ell _ 2 A _ 2 \\\\ A _ 5 & = \\cos { \\theta } A _ 1 - \\sin { \\theta } A _ 2 \\\\ A _ 6 & = \\sin { \\theta } A _ 1 + \\cos { \\theta } A _ 2 \\ , . \\end{align*}"}
+{"id": "8824.png", "formula": "\\begin{align*} \\| T ^ n h \\| = \\| h \\| = \\| T ^ { * n } h \\| \\ , , ( n = 1 , 2 , \\dots ) \\ , ; \\end{align*}"}
+{"id": "8888.png", "formula": "\\begin{align*} T r \\left ( \\exp \\left ( \\frac { 1 } { 4 } t \\Delta _ { \\nu } \\right ) \\right ) = e ^ { \\left ( \\frac { n ^ { 2 } } { 4 } + \\nu ^ { 2 } \\right ) t } \\sum \\limits _ { m = 0 } ^ { + \\infty } \\left ( \\dim _ { \\mathbb { C } } \\mathcal { A } _ { m } ^ { \\nu } \\right ) e ^ { - \\left ( m + \\frac { n } { 2 } + \\nu \\right ) ^ { 2 } t } , \\end{align*}"}
+{"id": "4697.png", "formula": "\\begin{align*} \\mathcal { I } ( G _ { I V } ) = \\frac { 1 } { ( 1 - t ^ 2 ) ( 1 - t ^ 6 ) } . \\end{align*}"}
+{"id": "3196.png", "formula": "\\begin{align*} a _ 1 ( y _ 1 , y _ 2 ) & : = 1 - \\frac { 1 } { 2 } \\sin ( 2 \\pi ( y _ 1 + y _ 2 ) ) + \\frac { 1 } { 4 } \\sin ( 4 \\pi y _ 2 ) , \\\\ a _ 2 ( y _ 1 , y _ 2 ) & : = 1 + \\frac { 1 } { 2 } \\sin ( 2 \\pi ( y _ 1 + y _ 2 ) ) + \\frac { 1 } { 4 } \\cos ( 2 \\pi y _ 1 ) \\end{align*}"}
+{"id": "2990.png", "formula": "\\begin{align*} \\mathcal { A } [ \\theta , \\pi ] = \\int _ { \\mathcal { Y } ^ 5 } \\frac { 1 } { 2 } \\| \\pi \\| ^ 2 e ^ { ( 4 ) } \\wedge \\theta + \\pi \\wedge \\hbox { d } \\theta \\end{align*}"}
+{"id": "3938.png", "formula": "\\begin{align*} \\overline { g } & ( q , p , z ) = \\overline { g } ( q , p , 0 ) + \\overline { g } _ z ( q , p , 0 ) z + \\frac { 1 } { 2 } \\overline { g } _ { z z } ( q , p , \\tau z ) z ^ 2 \\\\ & = \\overline { g } ( q , p , 0 ) + \\overline { g } _ z ( 0 , 0 , 0 ) z + \\overline { g } _ { q _ i , z } ( 0 , 0 , 0 ) q _ i z + \\overline { g } _ { p _ i , z } ( 0 , 0 , 0 ) p _ i z \\\\ & + z [ b _ { i j } ( q , p ) q _ i p _ j + c _ { i j } ( q , p ) q _ j q _ i + d _ { i j } ( q , p ) p _ j p _ i ] + f ( q , p , z ) z ^ 2 . \\end{align*}"}
+{"id": "8245.png", "formula": "\\begin{align*} a _ 1 x _ 1 + \\dots + a _ r x _ r - b _ 1 y _ 1 - \\dots - b _ s y _ s = 0 \\end{align*}"}
+{"id": "5993.png", "formula": "\\begin{align*} X ( i , j ) = \\left \\{ \\left ( [ x _ 1 : \\dots : x _ i : 0 : \\dots : 0 ] , [ 0 : \\dots 0 : y _ j : \\dots : y _ n ] \\right ) \\in \\mathbb { P } ^ { n - 1 } \\times \\mathbb { P } ^ { n - 1 } \\right \\} . \\end{align*}"}
+{"id": "2883.png", "formula": "\\begin{align*} \\lvert L _ U \\rvert - 1 = q ^ { \\ell t } ( \\lvert L _ { U ' } \\rvert - 1 ) , \\end{align*}"}
+{"id": "1476.png", "formula": "\\begin{align*} \\lambda = \\frac { ( k - 1 ) ! ( k - 3 ) ! } { ( ( k - 3 ) / 2 ) ! ^ 2 } ; \\end{align*}"}
+{"id": "4610.png", "formula": "\\begin{align*} 0 < \\sum _ { i = 1 } ^ n \\frac { 1 } { b _ i } = \\frac { r } { \\prod _ { i = 1 } ^ n b _ i } < \\frac { 1 } { q } \\end{align*}"}
+{"id": "4217.png", "formula": "\\begin{align*} \\square \\partial _ x \\tilde { \\Lambda } = - 2 \\Big [ \\sinh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) \\left ( Q _ 0 ( \\partial _ x \\phi , \\phi ) + Q _ 0 ( \\phi , \\partial _ x \\phi ) \\right ) + 2 \\partial _ x \\tilde { \\Lambda } \\cosh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) Q _ 0 ( \\phi , \\phi ) \\Big ] . \\end{align*}"}
+{"id": "9052.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( \\int _ 0 ^ T \\int _ { \\R ^ * } | \\delta U ^ { i , n } _ s ( e ) | ^ 2 \\nu ( d e ) d s \\right ) ^ { \\frac { p } { 2 } } \\right ] \\leq e _ p \\mathbb { E } \\left [ \\left ( \\int _ 0 ^ T \\int _ { \\R ^ * } \\sum _ { j = 1 } ^ n | \\delta U ^ { i , j , n } _ s ( e ) | ^ 2 N ^ j ( d e , d s ) \\right ) ^ { \\frac { p } { 2 } } \\right ] . \\end{align*}"}
+{"id": "2756.png", "formula": "\\begin{align*} \\partial _ { t } \\triangleright i ^ { A } + \\partial ^ { D } \\triangleright T _ { D } ^ { A } = 0 , \\end{align*}"}
+{"id": "6108.png", "formula": "\\begin{align*} m _ k = ( k - 1 ) \\left ( \\frac { 1 } { ( k ) ( k - 1 ) } - d _ k \\right ) . \\end{align*}"}
+{"id": "2934.png", "formula": "\\begin{align*} L ^ { * } _ n f ( \\eta ) = n ^ 2 \\sum _ { j \\in \\mathbb { Z } } g _ n ( \\eta _ j ) \\nabla _ { j , j - 1 } f ( \\eta ) . \\end{align*}"}
+{"id": "4186.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ( \\alpha \\partial _ t \\Lambda ) - \\partial _ x ( \\alpha \\partial _ x \\Lambda ) = 2 \\alpha \\sinh { 2 \\Lambda } ( ( \\partial _ t \\phi ) ^ 2 - ( \\partial _ x \\phi ) ^ 2 ) , \\\\ \\partial _ t ( \\alpha \\sinh ^ 2 \\Lambda \\partial _ t \\phi ) - \\partial _ x ( \\alpha \\sinh ^ 2 \\Lambda \\partial _ x \\phi ) = 0 , \\\\ \\partial _ { t } ^ 2 \\alpha - \\partial _ { x } ^ 2 \\alpha = 0 , \\end{cases} \\end{align*}"}
+{"id": "4452.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 1 0 5 } = \\frac { 1 } { 5 } + \\frac { 1 } { 7 } = \\frac { 1 2 } { 3 5 } . \\end{align*}"}
+{"id": "521.png", "formula": "\\begin{align*} F ( x ^ k ) - \\omega ^ * \\le \\left \\{ \\begin{array} { c l } \\gamma ' \\widehat { \\varrho } ^ { \\lceil \\frac { k - 1 } { m + 1 } \\rceil } & { \\rm i f } \\ \\theta = 1 / 2 , \\\\ \\gamma ' { k } ^ { \\frac { 1 - \\theta } { 1 - 2 \\theta } } & { \\rm i f } \\ \\theta \\in ( 1 / 2 , 1 ) . \\end{array} \\right . \\end{align*}"}
+{"id": "3639.png", "formula": "\\begin{align*} - \\Delta _ M \\phi = \\lambda _ 1 ( M ) \\phi \\ \\ i n \\ M . \\end{align*}"}
+{"id": "240.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\beta \\rho \\eta - \\alpha \\rho \\tau \\leqslant - \\frac { \\alpha } { 2 } ( 1 + 3 \\rho ^ 2 ) \\leqslant - \\frac { \\alpha } { 2 } , \\end{align*}"}
+{"id": "6545.png", "formula": "\\begin{align*} { y _ k } = \\sqrt { D _ { a k } ^ { - { \\alpha _ { a k } } } { P _ { a k } } } { h _ { a k } } { s _ k } + n + \\sqrt { D _ { j k } ^ { - { \\alpha _ { j k } } } { P _ { j k } } } { h _ { j k } } , \\end{align*}"}
+{"id": "9177.png", "formula": "\\begin{align*} \\lambda ( p ) | \\hat { f } _ { \\epsilon } ( p ) | = | \\widehat { \\Delta f } _ { \\epsilon } ( p ) | = | \\sum _ { x \\in \\Z ^ 2 } e ^ { - i p \\cdot x } \\Delta f _ { \\epsilon } ( x ) | \\leq \\norm { \\Delta f _ { \\epsilon } } _ { \\ell ^ 1 ( \\Z ^ 2 ) } . \\end{align*}"}
+{"id": "4123.png", "formula": "\\begin{align*} R _ { k l } = \\sum _ { i = 1 } ^ n \\langle R ( e _ i , e _ k ) e _ l , e _ i \\rangle = \\frac { 1 } { 4 } \\sum _ { i = 1 } ^ n \\langle [ e _ i , e _ k ] , [ e _ i , e _ l ] \\rangle = - \\frac { 1 } { 4 } \\mathcal { B } _ { k l } \\end{align*}"}
+{"id": "8928.png", "formula": "\\begin{align*} S _ 2 = 2 \\sum \\limits _ { \\mu = 1 } ^ { \\nu - 1 } e ^ { - \\mu ^ { 2 } t } \\sum \\limits _ { p = 0 } ^ { n - 1 } \\tau _ { p } ^ { ( \\nu , n ) } \\mu ^ { 2 p + 1 } . \\end{align*}"}
+{"id": "308.png", "formula": "\\begin{align*} \\zeta _ { ( { \\chi _ { \\mathrm { B C } } } _ { \\mu , \\alpha } , \\chi _ { \\mu \\circ \\mathrm { N m } _ { T ( E ) / S ^ { \\mathrm { o p } } ( F ) } , \\alpha } ) } | _ { F _ { \\alpha } ^ { \\times } } ( \\iota _ { F _ \\alpha } \\alpha ( t ) ) = 1 . \\end{align*}"}
+{"id": "543.png", "formula": "\\begin{align*} \\kappa ( j ) & : = \\mu ( j \\to j _ * ) = \\max \\{ x \\in L \\mid j \\wedge x = j _ * \\} , \\\\ \\kappa ^ d ( m ) & : = \\gamma ( m ^ * \\to m ) = \\min \\{ x \\in L \\mid m \\vee x = m ^ * \\} . \\end{align*}"}
+{"id": "5719.png", "formula": "\\begin{align*} ( _ { c } \\mathbf { L } ) _ { b } ^ { a } & = [ \\delta _ { c } ^ { a } + ( d L _ { d } ^ { a } ) ( L ^ { - 1 } ) _ { c } ^ { d } \\mathbf { m } ] ( L ) _ { b } ^ { c } , \\\\ ( _ { c } \\mathbf { L } ^ { - 1 } ) _ { b } ^ { a } & = [ \\delta _ { c } ^ { a } - ( L ^ { - 1 } ) _ { d } ^ { a } d L _ { c } ^ { d } \\mathbf { m } ] ( L ^ { - 1 } ) _ { b } ^ { c } . \\end{align*}"}
+{"id": "5189.png", "formula": "\\begin{align*} \\begin{aligned} x _ t ' - x _ t \\equiv \\left ( m _ 2 ^ { - 1 } m _ 1 - n _ 2 ^ { - 1 } n _ 1 \\right ) \\frac { r ( r - 1 ) ( r - 2 ) } { 3 K } + \\sum _ { \\ell = 1 } ^ { r - 2 } \\ell \\frac { r } { K } ( k _ \\ell ' - k _ \\ell ) \\pmod { r } \\end{aligned} \\end{align*}"}
+{"id": "3672.png", "formula": "\\begin{align*} A u + \\lambda & = f , \\\\ M _ \\rho ( \\lambda , u - g ) & = 0 . \\end{align*}"}
+{"id": "3572.png", "formula": "\\begin{align*} \\langle v _ 0 , v _ 0 \\rangle = 1 \\langle B _ j ^ \\pm v _ A , v _ B \\rangle = \\langle v _ A , B _ j ^ \\mp v _ B \\rangle , \\end{align*}"}
+{"id": "5627.png", "formula": "\\begin{align*} _ { t _ { 0 } } ^ { A B C } D _ { t } ^ { \\alpha } \\left [ u ( t ) - u ^ * - u ^ * \\ln \\dfrac { u ( t ) } { u ^ * } \\right ] & \\leq \\left ( 1 - \\dfrac { u ^ * } { u ( t ) } \\right ) \\ : _ { t _ { 0 } } ^ { A B C } D _ { t } ^ { \\alpha } u ( t ) , \\\\ _ { t _ { 0 } } ^ { C F } D _ { t } ^ { \\alpha } \\left [ u ( t ) - u ^ * - u ^ * \\ln \\dfrac { u ( t ) } { u ^ * } \\right ] & \\leq \\left ( 1 - \\dfrac { u ^ * } { u ( t ) } \\right ) \\ : _ { t _ { 0 } } ^ { C F } D _ { t } ^ { \\alpha } u ( t ) . \\end{align*}"}
+{"id": "5447.png", "formula": "\\begin{align*} P ( \\alpha ^ { n } _ { t + \\Delta t } = j | \\alpha ^ { n } _ t = i , ( X ^ n _ s , \\alpha ^ { n } _ s ) , s \\leq t ) = q _ { i j } ( X ^ { n } _ t ) \\Delta t + o ( \\Delta t ) . \\end{align*}"}
+{"id": "7030.png", "formula": "\\begin{align*} E _ { C _ 2 } ^ * ( X ) = \\widetilde { E } _ { C _ 2 } ^ { * + 2 \\dim \\xi } ( X ^ \\xi ) . \\end{align*}"}
+{"id": "476.png", "formula": "\\begin{align*} \\mathfrak { b } [ u , v ] & : = \\phi ( v ) - \\phi ( u ) - b ( u ) ( v - u ) \\\\ & = \\phi ( v ) + \\phi ^ * ( b ( u ) ) - b ( u ) v \\mbox { f o r a l l } u , v \\geq 0 \\end{align*}"}
+{"id": "2447.png", "formula": "\\begin{align*} \\mu _ { \\varphi } ( \\Psi ) = \\frac { 3 } { \\pi } \\mu ( \\Psi ) + O _ { \\Psi } \\Big ( \\frac { \\log \\lambda _ { \\varphi } } { g ( \\lambda _ { \\varphi } ) } \\Big ) . \\end{align*}"}
+{"id": "6705.png", "formula": "\\begin{align*} f ( n ) = \\sum _ { d \\mid n } g ( d ) g ( n ) = \\sum _ { d \\mid n } \\mu ( d ) f ( n / d ) , \\end{align*}"}
+{"id": "1563.png", "formula": "\\begin{align*} g ( X ) : = \\frac { X ^ { Q + R } + X ^ Q + X } { X ^ { Q + R } + X ^ R + X } \\end{align*}"}
+{"id": "21.png", "formula": "\\begin{align*} F ( x , y , z , \\tilde { z } , \\gamma , \\mathcal { Z } , u ) : = \\mathcal { Z } f ( x , y , z , \\tilde { z } , \\gamma , u ) . \\end{align*}"}
+{"id": "598.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\mathbb { X } ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } = \\mathbb { X } ^ \\dagger _ { t _ n , t _ { n + 1 } } + \\frac { \\Delta t \\ , \\gamma } { 2 } M \\end{align*}"}
+{"id": "8020.png", "formula": "\\begin{align*} T & = ( \\{ F _ 4 \\} , \\emptyset ) , \\\\ F & = \\left [ ( F _ 4 , F _ 3 ) , ( F _ 4 , F _ 2 ) , ( F _ 4 , F _ 1 ) \\right ] , \\end{align*}"}
+{"id": "6787.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\beta ( t _ n ) = 1 ~ i m p l i e s \\lim \\limits _ { n \\to \\infty } t _ n = 0 . \\end{align*}"}
+{"id": "621.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\lim _ { \\epsilon \\to 0 } \\Theta _ t ^ { ( \\epsilon ) } = \\theta ^ \\dagger . \\end{align*}"}
+{"id": "4679.png", "formula": "\\begin{align*} \\sum _ { m = \\ell _ 2 } ^ { \\infty } \\sum _ { k = \\max \\{ 0 , \\ell _ 1 + \\ell _ 2 - m \\} } ^ { \\ell _ 1 } A _ { m , k } = \\sum _ { m = \\ell _ 2 } ^ { \\ell _ 1 + \\ell _ 2 } \\sum _ { k = \\ell _ 1 + \\ell _ 2 - m } ^ { \\ell _ 1 } A _ { m , k } + \\sum _ { m = \\ell _ 1 + \\ell _ 2 + 1 } ^ { \\infty } \\sum _ { k = 0 } ^ { \\ell _ 1 } A _ { m , k } . \\end{align*}"}
+{"id": "2307.png", "formula": "\\begin{align*} u _ t + f ( u ) _ x = \\epsilon \\left ( \\nu ( u ) u _ x \\right ) _ x , ~ ~ \\epsilon \\nu ( u ) \\geqslant 0 , \\end{align*}"}
+{"id": "8567.png", "formula": "\\begin{align*} \\delta _ k = \\sum _ { i = 0 } ^ k \\big ( d ' _ w ( i ) - d _ w ( i ) \\big ) \\end{align*}"}
+{"id": "423.png", "formula": "\\begin{align*} \\theta _ f S _ { f , \\tau } ^ { - 1 } \\theta _ \\tau = I _ { \\ell ^ p ( \\mathbb { N } ) } . \\end{align*}"}
+{"id": "6481.png", "formula": "\\begin{align*} H = \\eta ( \\mathrm { G a l } ( \\mathbb { Q } ( \\zeta _ { 4 n } ) / \\mathbb { K } ) ) , ~ ~ H _ 1 = H \\cap F ~ ~ \\mbox { a n d } ~ ~ H _ 2 = H \\cap ( - F ) . \\end{align*}"}
+{"id": "1227.png", "formula": "\\begin{align*} \\tilde W _ \\nu ( \\pi _ 0 , \\pi _ 1 ) = \\sqrt { \\inf _ { \\gamma \\in \\mathcal P ( X \\times X \\times X ) | \\gamma _ { y x _ i } = \\pi _ i , i = 0 , 1 } \\int _ { X \\times X \\times X } | x _ 0 - x _ 1 | ^ 2 d \\gamma ( y , x _ 0 , x _ 1 ) } \\end{align*}"}
+{"id": "8354.png", "formula": "\\begin{align*} \\Sigma _ 1 & = \\{ s = T ' , | y | < T ' \\} , \\\\ \\Sigma _ 2 & = \\{ s = ( 2 R ) ^ { - 1 } , | y | < ( 2 R ) ^ { - 1 } \\} , \\\\ \\Sigma _ 3 & = \\{ ( 2 R ) ^ { - 1 } < s = | y | < T ' \\} , \\end{align*}"}
+{"id": "3461.png", "formula": "\\begin{align*} \\widetilde { f } _ n ( \\cdot , z , x ; t ) = \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } h _ j ^ { ( n ) } ( \\cdot , z , x ; t ) , \\end{align*}"}
+{"id": "7473.png", "formula": "\\begin{align*} F _ { A ' } \\left ( s \\right ) & = 1 \\ , . \\end{align*}"}
+{"id": "3862.png", "formula": "\\begin{align*} T _ { n + 2 } ^ 2 = c _ n ^ 2 + \\sum _ { k = 2 } ^ n \\sum _ { l = 2 } ^ k \\{ 4 ( T _ l + T _ { l - 1 } ) - \\delta _ { l , 2 } - 2 p _ { l - 1 } \\} T _ { k - l + 2 } ^ 2 c _ { n - k } ^ 2 . \\end{align*}"}
+{"id": "3936.png", "formula": "\\begin{align*} - \\frac { \\tilde { g } _ y } { \\tilde { g } _ z } ( x , y , z ) = \\frac { 1 } { g ^ * _ u ( 0 , y , h - z ) } \\frac { g _ y } { g _ z } ( x , y , g ^ * ( 0 , y , h - z ) ) + \\frac { g ^ * _ y } { g ^ * _ u } ( 0 , y , h - z ) , \\end{align*}"}
+{"id": "7597.png", "formula": "\\begin{align*} Q & = \\int _ { 0 } ^ { t } \\bigg \\{ \\int _ { 0 } ^ { 1 } \\chi _ { \\{ x + z \\in [ 0 , 1 ] \\} } \\bigg | \\int _ { 0 } ^ { 1 } \\chi _ { \\{ | x - y | > | z | \\} } \\bigg ( \\frac { \\partial G } { \\partial y } ( t - s , x + z , y ) - \\frac { \\partial G } { \\partial y } ( t - s , x , y ) \\bigg ) \\\\ & \\times v ^ { \\delta + 1 } ( s , y ) \\d y \\bigg | ^ p \\d x \\bigg \\} ^ { \\frac { 1 } p } \\d s . \\end{align*}"}
+{"id": "1246.png", "formula": "\\begin{align*} W ^ 2 _ { \\nu _ 1 , . . . \\nu _ k } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\gamma _ { y _ 1 . . . y _ k x _ i } \\in \\Pi ^ k _ { o p t } ( \\nu _ 1 , . . . , \\nu _ k , \\mu _ i ) , i = 0 , 1 } \\int _ { X ^ { k + 2 } } | x _ 0 - x _ 1 | ^ 2 d \\gamma ( y _ 1 , . . . . , y _ k , x _ 0 , x _ 1 ) . \\end{align*}"}
+{"id": "2735.png", "formula": "\\begin{align*} B _ { j } \\Omega _ j = \\Omega _ j B _ { j } , \\end{align*}"}
+{"id": "3621.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } u = \\Delta _ { \\mathbb { H } ^ { n } } u + f ( u , t ) & \\hbox { i n } ~ \\mathbb { H } ^ { n } \\times ( 0 , T ) , \\\\ \\\\ u = u _ { 0 } \\in C ( \\mathbb { H } ^ { n } ) \\cap L ^ { \\infty } ( \\mathbb { H } ^ { n } ) & \\hbox { i n } ~ \\mathbb { H } ^ { n } \\times \\{ 0 \\} , \\end{array} \\right . \\end{align*}"}
+{"id": "920.png", "formula": "\\begin{align*} b _ n = 2 ^ { - 6 ^ { \\dots { - ( ( n - 1 ) n ) ^ { - t _ n } } } } \\end{align*}"}
+{"id": "2846.png", "formula": "\\begin{align*} a _ k = \\rho + \\tau , a _ { \\ell } = \\rho - \\tau , \\end{align*}"}
+{"id": "490.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\int _ 0 ^ T \\iint \\limits _ { \\mathbb { R } ^ N \\times \\mathbb { R } ^ N } \\frac { H \\left ( x , y , [ v ] _ h ( x , t ) - [ v ] _ h ( y , t ) \\right ) } { | x - y | ^ N } \\ , d x \\ , d y \\ , d t = \\int _ 0 ^ T \\iint \\limits _ { \\mathbb { R } ^ N \\times \\mathbb { R } ^ N } \\frac { H \\left ( x , y , v ( x , t ) - v ( y , t ) \\right ) } { | x - y | ^ N } \\ , d x \\ , d y \\ , d t \\end{align*}"}
+{"id": "930.png", "formula": "\\begin{align*} \\tau _ { E , \\alpha } ^ { \\prime } ( h ) = \\frac { \\sqrt { 1 + | m _ { \\alpha } | ^ { 2 } } } { e _ { 2 } \\cdot \\theta _ { L } ^ { \\perp } } ( G ^ { - 1 } ) ^ { \\prime } ( h + y _ { 0 } ) . \\end{align*}"}
+{"id": "2889.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\mathbb { T } ^ \\infty } \\left ( \\sup \\limits _ { 0 < r < 1 } I _ { w , r } \\right ) d m _ \\infty ( w ) & = \\sup \\limits _ { 0 < r < 1 } \\int _ { \\mathbb { T } ^ \\infty } \\int _ { \\mathbb { T } } \\log ^ { + } | F _ { w } ( r \\lambda ) | d m _ { 1 } ( \\lambda ) d m _ \\infty ( w ) \\\\ & = \\sup \\limits _ { 0 < r < 1 } \\int _ { \\mathbb { T } ^ \\infty } \\int _ { \\mathbb { T } } \\log ^ { + } | F _ { [ r ] } ( \\lambda \\star w ) | d m _ { 1 } ( \\lambda ) d m _ \\infty ( w ) . \\end{aligned} \\end{align*}"}
+{"id": "7264.png", "formula": "\\begin{align*} u ( 0 ) = 1 , \\lim _ { x \\to + 0 } ( u ^ + ( x ) - \\lambda ( m ( x ) - m ( 1 ) ) ) = 0 . \\end{align*}"}
+{"id": "5580.png", "formula": "\\begin{align*} ( \\nabla _ { a } T _ { c _ { 1 } \\cdots c _ { r } } ) Q ^ { c _ { 1 } \\dots c _ { r } } = \\nabla _ { a } k ^ { b } \\ , \\langle T \\vert k \\vert Q \\rangle _ { b } . \\end{align*}"}
+{"id": "9084.png", "formula": "\\begin{align*} \\mathfrak { c } ^ { j } ( \\mathfrak { g } ) / \\mathfrak { c } ^ { j - 1 } ( \\mathfrak { g } ) = \\mathfrak { Z } \\left ( \\mathfrak { g } / \\mathfrak { c } ^ { j - 1 } ( \\mathfrak { g } ) \\right ) j \\geq 1 , \\end{align*}"}
+{"id": "5514.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n } \\exp \\left ( { - \\frac { x } { n ^ 2 } } \\right ) = \\sqrt { \\frac { \\pi } { x } } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n } \\exp \\left ( - \\frac { \\pi ^ 2 } { n ^ 2 x } \\right ) - \\frac { 1 } { 2 \\sqrt { \\pi } } \\sum _ { \\rho } \\left ( \\frac { \\pi } { \\sqrt { x } } \\right ) ^ { \\rho } \\frac { \\Gamma \\left ( \\frac { 1 - \\rho } { 2 } \\right ) } { \\zeta ' ( \\rho ) } , \\end{align*}"}
+{"id": "1421.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( | \\mathcal C _ { \\max } | < n ^ { 2 / 3 } / A \\right ) = O ( A ^ { - 1 / 2 } ) \\mathbb { P } \\left ( | \\mathcal C _ { \\max } | > A n ^ { 2 / 3 } \\right ) = O ( A ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "7759.png", "formula": "\\begin{align*} \\{ g _ { \\mu , \\lambda _ 1 } \\star g _ { \\mu , \\lambda _ 2 } \\} ( x ) & = \\frac { \\sqrt { \\pi } } { \\Gamma ( \\mu ) } \\ , e ^ { - ( \\lambda _ 1 + \\lambda _ 2 ) x / 2 } \\ , \\left ( \\frac { x } { \\lambda _ 2 - \\lambda _ 1 } \\right ) ^ { \\mu - \\tfrac { 1 } { 2 } } \\ , I _ { \\mu - \\tfrac { 1 } { 2 } } \\left ( \\frac { \\lambda _ 2 - \\lambda _ 1 } { 2 } \\ , x \\right ) \\end{align*}"}
+{"id": "5347.png", "formula": "\\begin{align*} \\mathcal { R } [ u ] : = \\int _ \\Omega | \\operatorname { c u r l } u | ^ 2 d x , \\mathcal { D } _ \\varepsilon [ u ] : = \\int _ \\Omega | \\operatorname { d i v } ( \\varepsilon u ) | ^ 2 d x . \\end{align*}"}
+{"id": "3216.png", "formula": "\\begin{align*} S = \\bigcup \\limits _ { j = 1 } ^ { k } \\eta _ { j } \\left ( A _ { j } \\right ) . \\end{align*}"}
+{"id": "1171.png", "formula": "\\begin{align*} z = z ( n , \\omega ) : = \\omega \\sqrt { n } . \\end{align*}"}
+{"id": "6873.png", "formula": "\\begin{align*} \\Sigma _ { N , M } ( \\mathbb { D } ) = \\Big \\{ \\sum _ { j = 1 } ^ { N } a _ j d _ j \\ ; : d _ j \\in \\mathbb { D } \\ ; a n d \\ ; \\norm { \\{ a _ j \\} _ { j = 1 } ^ N } _ { \\ell _ 1 } \\leq M \\Big \\} , \\end{align*}"}
+{"id": "6137.png", "formula": "\\begin{align*} A ^ T E _ r = \\sum _ { s = 1 } ^ d \\alpha _ { r s } E _ s , B ^ T E _ r = \\sum _ { s = 1 } ^ d \\beta _ { r s } E _ s , 1 \\leqslant r \\leqslant d . \\end{align*}"}
+{"id": "3309.png", "formula": "\\begin{align*} \\tilde { V } _ { \\lambda } [ W ] : = V ^ { ( 0 ) } _ { \\lambda } [ W ] + V _ { \\lambda } [ \\ell _ { W } - F _ { \\lambda } ^ { ( 0 ) } [ W ] , \\omega _ { W } - \\tau _ { \\lambda } ^ { ( 0 ) } [ W ] ] . \\end{align*}"}
+{"id": "8273.png", "formula": "\\begin{align*} Z ( s ) = L ( s ) + R ( s ) + \\frac 1 2 ( L ( 2 s ) + R ( 2 s ) ) + P ^ * ( s ) = L ( s ) + \\frac 1 2 L ( 2 s ) + R ^ * ( s ) , \\end{align*}"}
+{"id": "6134.png", "formula": "\\begin{align*} \\int _ \\Omega ( \\Phi '' , \\widehat \\Phi ) d x + \\int _ \\Omega \\langle \\nabla \\Phi , \\nabla \\widehat \\Phi \\rangle d x + \\int _ { \\Gamma _ 1 } ( B \\Phi , \\widehat \\Phi ) d \\Gamma + \\int _ \\Omega ( A \\Phi , \\widehat \\Phi ) d x = 0 \\end{align*}"}
+{"id": "2192.png", "formula": "\\begin{align*} ( 1 - 2 \\alpha - \\beta ) r ^ 2 + ( 2 - 2 \\alpha + \\beta ) r + 1 = ( ( - 3 + 2 \\alpha + \\beta ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + 1 ) \\mathit { e } . \\end{align*}"}
+{"id": "49.png", "formula": "\\begin{align*} T ( n _ i ) + T ( n _ { i + 1 } ) + T ( n _ { i + 2 } ) & < \\left ( \\frac { 8 5 } { 2 7 } + \\frac { 1 2 1 6 } { 7 2 9 } \\right ) \\cdot \\frac { 1 } { X _ 0 } = \\frac { 3 5 1 1 } { 7 2 9 } \\cdot \\frac { 1 } { X _ 0 } \\\\ & < 7 \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } = ( k _ i + k _ { i + 1 } + k _ { i + 2 } ) \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } . \\end{align*}"}
+{"id": "4165.png", "formula": "\\begin{align*} \\mathcal { L } ( g , b , f ) = ( 0 , 0 ) \\Longleftrightarrow S ( g , b , f ) = \\int _ M e ^ { - f } d V _ g = 1 . \\end{align*}"}
+{"id": "2137.png", "formula": "\\begin{align*} R _ i ( L ) \\leq C \\left [ \\exp \\left ( - \\frac { L ^ 2 } { 2 0 i } \\right ) + i R _ 1 \\left ( \\frac L 2 \\right ) \\right ] . \\end{align*}"}
+{"id": "3345.png", "formula": "\\begin{align*} \\Phi _ { x p } = \\Phi _ { x q } . \\end{align*}"}
+{"id": "2645.png", "formula": "\\begin{align*} g ( \\omega , x ) = \\sum _ { j = 1 } ^ J y _ j ( \\omega ) \\psi _ j ( x ) \\ , ; , \\end{align*}"}
+{"id": "8919.png", "formula": "\\begin{align*} \\gamma _ { p } ^ { ( \\nu , n ) } = \\frac { ( n - 1 ) ! } { p ! } c _ { n - p - 1 } ^ { \\left ( \\nu , n \\right ) } , 0 \\leq p \\leq n - 1 \\end{align*}"}
+{"id": "6521.png", "formula": "\\begin{align*} \\underset { j = 1 } { \\overset { m } { \\sum } } N ^ { j } \\sim \\left ( 1 / 2 , m d C _ { b , d } \\right ) . \\end{align*}"}
+{"id": "6965.png", "formula": "\\begin{gather*} \\vec { c } = \\begin{pmatrix} c _ { 0 , n } \\\\ c _ { 1 , n } \\\\ \\vdots \\\\ c _ { n , n } \\end{pmatrix} \\ ! , \\vec { b } _ { I I } = \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ L _ { m , n } ^ { I I , ( \\alpha ) } ( x ) \\end{pmatrix} \\ ! , \\end{gather*}"}
+{"id": "7488.png", "formula": "\\begin{align*} \\kappa ( X ) = \\kappa ( G ) + \\dim Z = \\kappa ( F ) + \\dim Y . \\end{align*}"}
+{"id": "9039.png", "formula": "\\begin{align*} \\mathcal { W } _ p ^ p ( L _ { n } [ \\textbf { Y } ^ n _ s ] , L _ { n } [ \\bar { \\textbf { Y } } ^ n _ s ] ) ) \\le \\frac { 1 } { n } \\sum _ { j = 1 } ^ n | Y _ s ^ { j , n } - \\bar { Y } _ s ^ { j , n } | ^ p . \\end{align*}"}
+{"id": "2814.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n + 1 } b _ i = \\sum _ { i = 1 } ^ { n + 1 } \\log v _ i = \\log \\prod _ { i = 1 } ^ { n + 1 } v _ i = \\log \\prod _ { i = 1 } ^ { n + 1 } u _ i = \\sum _ { i = 1 } ^ { n + 1 } \\log u _ i = \\sum _ { i = 1 } ^ { n + 1 } a _ i . \\end{align*}"}
+{"id": "307.png", "formula": "\\begin{align*} \\zeta _ { ( { \\chi _ { \\mathrm { B C } } } _ { \\mu , \\alpha } , \\chi _ { \\mu \\circ \\mathrm { N m } _ { T ( E ) / S ^ { \\mathrm { o p } } ( F ) } , \\alpha } ) } | _ { F _ { \\alpha } ^ { \\times } } ( \\iota _ { F _ \\alpha } \\alpha ( t ) ) = \\omega _ { E _ { \\alpha } / F _ { \\alpha } } ( \\iota _ { F _ \\alpha } \\alpha ( t ) ) . \\end{align*}"}
+{"id": "5056.png", "formula": "\\begin{align*} F = \\frac { P } { ( 1 - c _ 1 u _ 1 ) ^ { e _ 1 } \\cdots ( 1 - c _ l u _ l ) ^ { e _ l } } , \\end{align*}"}
+{"id": "5748.png", "formula": "\\begin{align*} d s ^ { 2 } = \\epsilon _ { A B } \\epsilon _ { A ^ { \\prime } B ^ { \\prime } } \\mathbf { e } ^ { A A ^ { \\prime } } \\otimes \\mathbf { e } ^ { B B ^ { \\prime } } = \\epsilon _ { A B } \\epsilon _ { A ^ { \\prime } B ^ { \\prime } } \\theta ^ { A A ^ { \\prime } } \\otimes \\theta ^ { B B ^ { \\prime } } = \\epsilon _ { A B } \\epsilon _ { A ^ { \\prime } B ^ { \\prime } } \\mathbf { d x } ^ { A A ^ { \\prime } } \\otimes \\mathbf { d x } ^ { B B ^ { \\prime } } . \\end{align*}"}
+{"id": "2901.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ \\infty } \\log | 1 - \\varphi | ^ p d m _ { \\infty } \\geq \\log | 1 - \\tilde { \\varphi } ( 0 ) | ^ p = 0 . \\end{align*}"}
+{"id": "7125.png", "formula": "\\begin{align*} d & = \\sum _ { i \\geq 1 } \\left \\langle d , x ^ { \\rho _ 1 + \\cdots + \\rho _ { i - 1 } } \\right \\rangle \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) . \\end{align*}"}
+{"id": "8214.png", "formula": "\\begin{align*} n = 2 k + 1 \\implies \\mathcal { T } ( t ) & = v _ 0 T _ 1 + v _ 1 \\bigl ( T _ 2 - T _ 1 \\bigr ) + \\dots + v _ 0 \\bigl ( T _ { 2 k + 1 } - T _ { 2 k } \\bigr ) + v _ 1 \\bigl ( t - T _ { 2 k + 1 } \\bigr ) \\\\ & = ( v _ 0 - v _ 1 ) \\bigl ( T _ 1 - T _ 2 + T _ 3 - \\dots - T _ { 2 k } + T _ { 2 k + 1 } \\bigr ) + v _ 1 t \\end{align*}"}
+{"id": "1961.png", "formula": "\\begin{align*} \\langle x , y \\rangle = x _ 1 y _ 1 + x _ 2 y _ 2 + \\cdots + x _ n y _ n . \\end{align*}"}
+{"id": "2537.png", "formula": "\\begin{align*} | \\tilde u _ N ^ \\varphi \\big | ^ 4 \\le \\bigg ( \\sum _ { j = 1 } ^ N q _ { 1 , j } | w _ j | \\bigg ) ^ 4 \\le \\sum _ { j = 1 } ^ N q _ { 1 , j } | w _ j | ^ 4 \\ , . \\end{align*}"}
+{"id": "6589.png", "formula": "\\begin{align*} \\theta ( \\ell , i ) = \\beta _ { \\ell , M _ - ' } \\circ g _ \\ell \\circ \\beta _ { \\ell , M _ + ' } ^ { - 1 } ( \\ell , i ) = \\beta _ { \\ell , M _ - } \\circ g _ \\ell \\circ \\beta _ { \\ell , M _ + } ^ { - 1 } ( \\ell , i ) = \\eta ( \\ell , i ) \\end{align*}"}
+{"id": "8536.png", "formula": "\\begin{align*} Q ( n , n - 2 p ) - Q ( n , n ) & = \\frac { 1 } { 6 } ( n - 1 ) ( ( n - 1 ) ^ 2 + 3 ( n - 2 p ) - 1 ) - \\frac { 1 } { 6 } ( n - 1 ) ( ( n - 1 ) ^ 2 + 3 n - 1 ) \\\\ & = \\frac { 1 } { 6 } ( n - 1 ) ( ( n - 1 ) ^ 2 + 3 n - 6 p - 1 - ( ( n - 1 ) ^ 2 + 3 n - 1 ) ) \\\\ & = \\frac { 1 } { 6 } ( n - 1 ) ( 3 n - 6 p - 3 n ) \\\\ & = ( 1 - n ) p , \\end{align*}"}
+{"id": "1957.png", "formula": "\\begin{align*} \\varrho [ X ] = \\min _ { u \\in \\mathbb { R } } \\Big \\{ u + \\frac { 1 } { \\alpha } [ \\mathbb { E } ( \\max ( 0 , X - u ) ^ q ) ] ^ { 1 / q } \\Big \\} , \\end{align*}"}
+{"id": "3257.png", "formula": "\\begin{align*} \\chi _ S ( f g ) = \\sum \\limits _ { S _ 1 \\cup S _ 2 = S } \\chi _ { S _ 1 } ( f ) \\chi _ { S _ 2 } ( g ) . \\end{align*}"}
+{"id": "3087.png", "formula": "\\begin{align*} B ( y ) : = \\mathrm { d i a g } ( 1 + b ( y ) , 1 - b ( y ) ) \\quad y \\in \\R ^ 2 , \\end{align*}"}
+{"id": "5420.png", "formula": "\\begin{align*} { _ { m i } \\mathcal { R } } ( A \\ast B ) \\simeq \\bigoplus _ { t = a } ^ { b } \\mathbf { 1 } _ { p t } ^ { f _ { m , t } ( v ) } \\boxtimes ( { _ { t i } \\mathcal { R } } ( A ) \\ast { _ { ( m - t ) i } \\mathcal { R } } ( B ) ) [ - P _ t ] ( - \\frac { P _ t } { 2 } ) , \\end{align*}"}
+{"id": "8528.png", "formula": "\\begin{align*} \\begin{cases} u _ t + u \\cdot \\nabla u + \\nabla P = - \\nabla \\cdot \\big ( \\nabla d \\odot \\nabla d \\big ) + \\nabla \\cdot ( \\sigma ^ L ( u , d ) ) , \\\\ \\nabla \\cdot u = 0 , \\\\ \\lambda _ 1 ( d _ t + u \\cdot \\nabla d - \\Lambda d ) + \\lambda _ 2 A d = \\Delta d + | \\nabla d | ^ 2 d + \\lambda _ 2 ( d ^ T A d ) d , \\end{cases} \\end{align*}"}
+{"id": "1067.png", "formula": "\\begin{align*} ( 1 - \\varepsilon ) R _ 1 + \\varepsilon G _ 1 = ( 1 - \\varepsilon ) R _ 2 + \\varepsilon G _ 2 . \\end{align*}"}
+{"id": "29.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial s } & = \\frac { 1 } { \\delta } \\frac { \\partial u } { \\partial z } , \\\\ & = c \\left [ 1 - \\left ( u ^ 2 / 2 i _ 0 \\right ) \\right ] ^ { - 6 / 2 5 } , \\\\ & = c \\left [ 1 + \\frac { 6 } { 2 5 } \\cdot \\frac { u ^ 2 } { 2 i _ 0 } + \\frac { 6 } { 2 5 } \\cdot \\frac { u ^ 4 } { 2 ( 2 i _ 0 ) ^ 2 } + \\frac { 1 } { 2 } \\cdot \\frac { 6 } { 2 5 } \\cdot \\frac { u ^ 4 } { ( 2 i _ 0 ) ^ 2 } + o ( u ^ 6 ) \\right ] . \\\\ \\end{align*}"}
+{"id": "1033.png", "formula": "\\begin{align*} \\mathcal { R } _ { n , \\alpha } ( \\theta ( \\mathcal { P } ) , \\Phi \\circ \\rho , \\varepsilon ) = \\inf _ { Q \\in \\mathcal { Q } _ \\alpha } \\inf _ { \\hat { \\theta } } \\sup _ { P _ \\varepsilon \\in \\mathcal { P } _ \\varepsilon ( \\mathcal { P } ) } \\mathbb { E } _ { P _ \\varepsilon , Q } \\left [ \\Phi \\circ \\rho \\left \\{ \\hat { \\theta } , \\ , \\theta ( P ) \\right \\} \\right ] , \\end{align*}"}
+{"id": "1370.png", "formula": "\\begin{align*} u _ { 0 } ^ { m } ( x ) = \\sum _ { j = 1 } ^ { m } g _ { j } ^ { m } ( t ) w _ { j } ( x ) , \\end{align*}"}
+{"id": "2449.png", "formula": "\\begin{align*} L ( s , \\pi ) = \\prod _ { p } L ( s , \\pi _ { p } ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda _ { \\pi } ( n ) } { n ^ s } , \\mathrm { R e } ( s ) > 1 . \\end{align*}"}
+{"id": "7553.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 1 ) = { \\left \\{ \\begin{array} { r l } ( 1 - q ^ { - 1 } ) q ^ { - ( \\omega + 1 ) - l s } Z _ { f _ 1 } ( s , \\chi ) , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } = \\chi _ { \\rm t r i v } , \\\\ 0 , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } \\neq \\chi _ { \\rm t r i v } . \\end{array} \\right . } \\end{align*}"}
+{"id": "3626.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } u = \\Delta _ { \\mathbb { H } ^ { n } } u + h ( t ) u ^ { p } & \\hbox { i n } ~ \\mathbb { H } ^ { n } \\times ( 0 , T ) , \\\\ \\\\ u = u _ { 0 } \\in C ( \\mathbb { H } ^ { n } ) \\cap L ^ { \\infty } _ + ( \\mathbb { H } ^ { n } ) & \\hbox { i n } ~ \\mathbb { H } ^ { n } \\times \\{ 0 \\} , \\end{array} \\right . \\end{align*}"}
+{"id": "2110.png", "formula": "\\begin{align*} y \\big ( a ( x ) + a ' ( x ) y \\big ) = y a ( x ) + y a ' ( x ) y = \\overline { a ' } ( x ) + \\overline a ( x ) y . \\end{align*}"}
+{"id": "123.png", "formula": "\\begin{align*} \\mathcal L ^ { 2 n } ( E ( F ) ) = \\int _ { \\mathbb R ^ n } \\int _ { \\mathbb R ^ n } \\textbf { 1 } _ { E ( F ) } d \\mathcal H ^ n ( x ) d \\mathcal H ^ n ( y ) = \\int _ { ( n , 1 ) } \\int _ L \\int _ L \\textbf { 1 } _ { E ( F ) } \\cdot | x - y | ^ { n - 1 } d \\mathcal H ^ 1 ( x ) d \\mathcal H ^ 1 ( y ) d L . \\end{align*}"}
+{"id": "2532.png", "formula": "\\begin{align*} p _ { i , j } & = p _ { i , j } ( \\xi ) : = \\frac { \\varphi ( x _ i + \\xi - x _ j ) } { \\sum _ k \\varphi ( x _ i + \\xi - x _ k ) } = \\frac { \\varphi ( x _ i + \\xi - x _ j ) } { N \\varrho _ N ^ \\varphi ( x _ i + \\xi ) } \\ , , \\\\ q _ { i , j } & = q _ { i , j } ( \\xi ) : = \\frac { \\varphi ( y _ i + \\xi - y _ j ) } { \\sum _ k \\varphi ( y _ i + \\xi - y _ k ) } = \\frac { \\tilde \\varphi ( y _ i + \\xi - y _ j ) } { N \\varrho _ N ^ \\varphi ( y _ i + \\xi ) } \\ , . \\end{align*}"}
+{"id": "2382.png", "formula": "\\begin{align*} E _ 1 ( \\hat t ) w = 0 . \\end{align*}"}
+{"id": "7158.png", "formula": "\\begin{align*} \\displaystyle { D _ k ^ { [ 0 ] } = \\sum _ { | I | = k } \\prod \\limits _ { \\substack { i \\in I \\\\ j \\notin I } } \\phi ( z _ j - z _ i ) \\ , , k = 1 , . . . , N } \\end{align*}"}
+{"id": "8755.png", "formula": "\\begin{align*} L ^ * L = \\begin{bmatrix} L _ 1 ^ * L _ 1 & L _ 1 ^ * L _ 2 & & L _ 1 ^ * L _ { 2 T - 1 } \\\\ L _ 2 ^ * L _ 1 & L _ 1 ^ * L _ 1 & & L _ 1 ^ * L _ { 2 T - 2 } \\\\ & & & \\\\ L _ { 2 T - 1 } ^ * L _ 1 & L _ { 2 T - 2 } ^ * L _ 1 & & L _ 1 ^ * L _ 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "783.png", "formula": "\\begin{align*} \\eta \\left \\{ ( x , c ) \\in Y : \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } Q ^ k \\sigma ( x , c ) \\underset { n \\to \\infty } { \\longrightarrow } \\int \\sigma \\ d \\eta \\right \\} = 1 . \\end{align*}"}
+{"id": "5872.png", "formula": "\\begin{align*} U ( T ) = U ' ( T ) = 0 . \\end{align*}"}
+{"id": "6067.png", "formula": "\\begin{align*} \\ell ( t ) = h \\prod _ { i = 1 } ^ g \\left ( \\frac { 1 - x _ 0 ^ 2 } { 1 - x _ i x _ 0 } \\right ) ^ 2 \\frac { \\prod _ { i = 1 } ^ g ( x - x _ i ) ( 1 - x x _ i ) } { ( 1 - x x _ 0 ) ^ { 2 g } } , \\end{align*}"}
+{"id": "7543.png", "formula": "\\begin{align*} f ( x , y , z ) = & x ^ p + \\sum _ { i = 0 } ^ k ( - 1 ) ^ i \\dbinom { k + r } { k - i } \\dbinom { i + r - 1 } { i } t ^ { k - i } y ^ { r + i } z ^ { k + l - i } \\\\ = & x ^ p + \\sum _ { i = 0 } ^ k ( - 1 ) ^ i a _ { k , r } ( i ) t ^ { k - i } y ^ { r + i } z ^ { k + l - i } , \\end{align*}"}
+{"id": "6741.png", "formula": "\\begin{align*} p \\left [ x _ 1 , x _ 2 ; ( m _ 1 , m _ 2 , \\rho ) \\right ] = \\frac { 1 } { 2 \\pi \\sqrt { 1 - \\rho ^ 2 } } \\exp \\left [ - \\frac { \\left ( x _ 1 - m _ 1 \\right ) ^ 2 + \\left ( x _ 2 - m _ 2 \\right ) ^ 2 - 2 \\rho \\left ( x _ 1 - m _ 1 \\right ) \\left ( x _ 2 - m _ 2 \\right ) } { 2 \\left ( 1 - \\rho ^ 2 \\right ) } \\right ] \\end{align*}"}
+{"id": "6475.png", "formula": "\\begin{align*} \\MoveEqLeft \\int | \\delta E ( m ) | ^ 2 d x - \\int | m \\cdot \\delta E ( m ) | ^ 2 d x = \\int | \\delta E ( \\eta ) | ^ 2 d x \\\\ & - 2 \\int \\beta _ * ^ 2 \\mu d x - 2 \\int \\beta _ * \\eta \\cdot \\delta E ( \\eta ) d x - \\int | w _ * \\cdot \\delta E ( \\eta ) | ^ 2 d x + O ( \\| \\eta \\| _ { H ^ 1 } \\| \\eta \\| _ { H ^ 2 } ^ 2 ) . \\end{align*}"}
+{"id": "3910.png", "formula": "\\begin{align*} \\int _ { Y u ( K ) } f ^ * & \\geq \\int _ { \\cap _ { i = 1 } ^ \\infty \\cup _ { k = i } ^ \\infty Y u _ k ( K ) } f ^ * \\\\ & = \\lim _ { i \\rightarrow \\infty } \\int _ { \\cup _ { k = i } ^ \\infty Y u _ k ( K ) } f ^ * \\\\ & \\geq \\lim _ { i \\rightarrow \\infty } \\limsup _ { k \\rightarrow \\infty } \\int _ { Y u _ k ( K ) } f ^ * = \\limsup _ { k \\rightarrow \\infty } \\int _ { Y u _ k ( K ) } f ^ * . \\end{align*}"}
+{"id": "1351.png", "formula": "\\begin{align*} I _ { } ( \\{ f _ j \\} _ { j = 1 } ^ n , \\{ \\tau _ j \\} _ { j = 1 } ^ n ) \\coloneqq \\left ( \\frac { 1 } { n ( n - 1 ) } \\sum _ { 1 \\leq j , k \\leq n , j \\neq k } f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "5039.png", "formula": "\\begin{align*} \\mathrm { J } ( f ) _ { x _ { o , \\lambda } } = \\begin{pmatrix} - \\frac { \\left ( p ^ 2 + q ^ 2 \\right ) ^ 2 } { \\lambda p ^ 2 q ^ 2 } & 0 & 0 \\\\ 0 & - { \\frac { p ^ 2 + q ^ 2 } { \\lambda q ^ 2 } } & 0 \\\\ 0 & 0 & - { \\frac { p ^ 2 + q ^ 2 } { \\lambda p ^ 2 } } \\end{pmatrix} . \\end{align*}"}
+{"id": "1813.png", "formula": "\\begin{align*} V _ t ( t , y ) + & V _ y ( t , y ) \\cdot v - L ( t , y , v ) \\leq \\\\ & V _ t ( t , u _ 0 ( t ) ) + V _ y ( t , u _ 0 ( t ) ) \\cdot \\dot { u } _ 0 ( t ) - L ( t , u _ 0 ( t ) , \\dot { u } _ 0 ( t ) ) . \\end{align*}"}
+{"id": "3804.png", "formula": "\\begin{align*} & \\lambda _ 1 ( D ( T ( a , b ) ) ) \\geq \\lambda _ 1 ( D ( T ( 1 , 1 ) ) = 8 . 2 8 8 2 , \\\\ & \\lambda _ 2 ( D ( T ( a , b ) ) ) \\geq \\lambda _ 2 ( D ( T ( 1 , 1 ) ) = - 0 . 5 5 7 8 , \\\\ & \\lambda _ 3 ( D ( T ( a , b ) ) ) \\geq \\lambda _ 3 ( D ( T ( 1 , 1 ) ) = - 0 . 7 6 3 9 , \\\\ & \\lambda _ 4 ( D ( T ( a , b ) ) ) \\geq \\lambda _ 4 ( D ( T ( 1 , 1 ) ) = - 1 . 7 3 0 4 , \\\\ & \\lambda _ n ( D ( T ( a , b ) ) ) \\leq \\lambda _ 5 ( D ( T ( 1 , 1 ) ) = - 5 . 2 3 6 1 . \\end{align*}"}
+{"id": "8417.png", "formula": "\\begin{align*} F _ { s c e } & : = F ^ 0 \\setminus \\overline { r ( E ^ 1 ) } , \\\\ F ^ 0 _ { f i n } & : = \\{ v \\in F ^ 0 : \\exists V ~ ~ v ~ ~ r ^ { - 1 } ( V ) ~ \\} , \\\\ F ^ 0 _ { r g } & : = F ^ 0 _ { f i n } \\setminus \\overline { F ^ 0 _ { s c e } } , \\\\ F ^ 0 _ { s g } & : = F ^ 0 \\setminus F ^ 0 _ { r g } . \\end{align*}"}
+{"id": "834.png", "formula": "\\begin{align*} \\tau _ { n p } \\left ( g \\in \\Gamma : \\frac { \\log \\| \\rho ( g ) \\| - \\Lambda | g | } { \\sqrt { | g | } } \\le t \\right ) = N ( t , \\sigma ) + O \\left ( \\frac { \\log n } { \\sqrt { n } } \\right ) \\end{align*}"}
+{"id": "7520.png", "formula": "\\begin{align*} h _ 1 ( u , v ) = & ( - t ^ { ( k + r ) / p } + \\pi u ) ^ p + \\sum \\limits _ { i = 0 } ^ k \\dbinom { k + r } { i + r } ( t + \\pi v ) ^ { r + i } ( - \\pi v ) ^ { k - i } \\\\ = & \\pi ^ p u ^ p + ( - t ^ { ( k + r ) / p } ) ^ p + t ^ { k + r } + \\sum _ { j = 1 } ^ { k + r } S _ { k , r } ( j ) t ^ { k + r - j } \\pi ^ j v ^ j \\\\ = & \\pi ^ p u ^ p + \\sum _ { j = 1 } ^ { k + r } S _ { k , r } ( j ) t ^ { k + r - j } \\pi ^ j v ^ j , \\end{align*}"}
+{"id": "8157.png", "formula": "\\begin{align*} \\omega ( t ) = \\frac { 4 \\pi \\abs [ \\Big ] { \\phi \\Bigl ( 1 , \\dfrac { 1 } { 2 } + i t \\Bigr ) } ^ 2 } { \\cosh ( \\pi t ) } . \\end{align*}"}
+{"id": "941.png", "formula": "\\begin{align*} \\textrm { G } _ C \\textrm { - d i m } _ R ( M ) + \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { T r } _ C ( M ) ) - 1 \\ , = \\ , \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { E x t } _ { R } ^ n ( M , \\ , C ) ) . \\end{align*}"}
+{"id": "6894.png", "formula": "\\begin{align*} y _ n - y ^ 2 _ { n - 1 } & = \\frac { 1 } { 2 } - \\frac { 1 } { 2 ^ { 2 ^ n } } - \\left ( \\frac { n - 1 } { 2 ^ { 2 ^ n } } \\right ) ^ 2 = \\frac { \\left ( 2 ^ { 2 ^ n } \\right ) ^ 2 - 2 \\cdot 2 ^ { 2 ^ n } - 2 ( n - 1 ) ^ 2 } { 2 \\cdot \\left ( 2 ^ { 2 ^ n } \\right ) ^ 2 } \\\\ & = \\frac { 2 ^ { 2 ^ n } \\left ( 2 ^ { 2 ^ n } - 2 - \\frac { 2 ( n - 1 ) ^ 2 } { 2 ^ { 2 ^ n } } \\right ) } { 2 \\cdot \\left ( 2 ^ { 2 ^ n } \\right ) ^ 2 } > 0 . \\end{align*}"}
+{"id": "5026.png", "formula": "\\begin{align*} h _ 3 = \\frac { c h _ 1 \\left ( a ^ 2 + b ^ 2 \\right ) } { a ^ 2 b ^ 2 + c ^ 2 } , h _ 4 = 0 . \\end{align*}"}
+{"id": "2868.png", "formula": "\\begin{align*} \\Lambda _ f ( s ) & : = \\prod _ { i = 1 } ^ n \\pi ^ { \\frac { - s + \\lambda _ i ( v _ f ) } { 2 } } \\Gamma \\left ( \\frac { s - \\lambda _ i ( v _ f ) } { 2 } \\right ) L _ f ( s ) \\\\ & = \\Lambda _ { \\tilde { f } } ( 1 - s ) , \\end{align*}"}
+{"id": "2661.png", "formula": "\\begin{align*} \\Phi ( u ) - \\Phi ( v ) = \\int _ { 0 } ^ 1 D \\Phi ( v + t w ) w \\dd t . \\end{align*}"}
+{"id": "3336.png", "formula": "\\begin{align*} \\xi ^ t = ( 0 , 0 , 0 , 0 , \\frac { 1 } { \\sqrt { \\delta } } ) . \\end{align*}"}
+{"id": "5445.png", "formula": "\\begin{align*} X _ { s , t } ^ { \\alpha } ( \\xi ) = X ^ { \\alpha _ r } _ { r , t } ( X _ { s , r } ^ { \\alpha } ( \\xi ) ) , ~ 0 \\leq s \\leq r \\leq t . \\end{align*}"}
+{"id": "4369.png", "formula": "\\begin{align*} a _ 1 = \\frac { q + 1 } { p } \\end{align*}"}
+{"id": "8364.png", "formula": "\\begin{align*} 0 & = \\liminf _ { t \\to + \\infty } \\int _ { | x | > t } \\left ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 \\right ) \\mathrm { d } x \\\\ & = \\liminf _ { t \\to - \\infty } \\int _ { | x | > - \\varepsilon - t } \\left ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 \\right ) \\mathrm { d } x . \\end{align*}"}
+{"id": "2062.png", "formula": "\\begin{align*} u _ + ( x ) & = x \\exp \\Bigl ( - \\frac { d } { \\gamma } x ^ \\gamma \\Bigr ) M \\biggl ( \\frac { \\gamma + 1 - \\frac { c } { d } } { 2 \\gamma } , \\frac { \\gamma + 1 } { \\gamma } , \\frac { 2 d } { \\gamma } x ^ \\gamma \\biggr ) , \\\\ [ 1 e x ] u _ - ( x ) & = \\exp \\Bigl ( - \\frac { d } { \\gamma } x ^ \\gamma \\Bigr ) M \\biggl ( \\frac { \\gamma - 1 - \\frac { c } { d } } { 2 \\gamma } , \\frac { \\gamma - 1 } { \\gamma } , \\frac { 2 d } { \\gamma } x ^ \\gamma \\biggr ) , \\end{align*}"}
+{"id": "1173.png", "formula": "\\begin{align*} \\Phi _ G ( m ) & = \\frac { \\Phi _ G ( m _ 0 + i ) } { \\Phi _ G ( m _ 0 ) } = \\frac { \\binom { y - 2 m _ 0 } { 2 } } { \\binom { y - 2 m _ 0 - 2 i } { 2 } } \\frac { m _ 0 + i + 1 } { m _ 0 + 1 } \\\\ & = \\left ( 1 + \\frac { 2 i } { y - 2 m _ 0 - 2 i + O ( 1 ) } \\right ) ^ 2 \\left ( 1 + \\frac { i } { m _ 0 + 1 } \\right ) = 1 + \\frac { \\Theta ( i ) } { \\lambda _ 1 } . \\end{align*}"}
+{"id": "310.png", "formula": "\\begin{align*} \\omega _ { G ( F ) , E } | _ { S ( F ) } ( t ) = \\prod \\limits _ { \\substack { \\alpha \\in \\Phi _ { \\mathrm { s y m , r a m } } / \\Gamma _ F \\\\ \\alpha \\in \\Phi _ { \\mathrm { s y m , r a m } } / \\Gamma _ E } } \\omega _ { E _ { \\alpha } / F _ \\alpha } ( \\iota _ { \\alpha } \\alpha ( t ) ) , \\end{align*}"}
+{"id": "8997.png", "formula": "\\begin{align*} L _ 1 \\int _ { B ^ + _ 4 ( 0 ) } & | \\nabla w _ k | ^ 2 d z + o ( 1 ) = L _ 1 \\int _ { B ^ + _ 4 ( 0 ) } | \\nabla v _ k ( s _ k ) | ^ 2 d z + o ( 1 ) \\\\ & \\ge L _ 1 \\int _ { B } | \\nabla u ( t _ k + r _ k s _ k ) | ^ 2 \\varphi _ k ^ 2 d z + o ( 1 ) \\ge L _ 1 \\int _ { B } | \\nabla u ( t _ k ) | ^ 2 \\varphi _ k ^ 2 d z \\\\ & \\ge L _ 1 \\int _ { B ^ + _ 2 ( 0 ) } | \\nabla v _ k ( 0 ) | ^ 2 d z \\ge L _ 1 \\delta . \\end{align*}"}
+{"id": "3351.png", "formula": "\\begin{align*} X \\rightarrow X ^ { \\flat } = X \\lrcorner \\Omega + ( X \\lrcorner \\theta ) \\theta \\end{align*}"}
+{"id": "4796.png", "formula": "\\begin{align*} \\max _ { j \\in B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } \\big [ | v H | = j \\big ] \\cdot \\frac { 2 ^ N } { \\binom { N } { j } } \\right \\} & \\leq \\sqrt { \\frac { e ^ 4 N } { 8 \\pi } } \\cdot 2 ^ { ( 1 - h ( \\beta ) ) \\eta N } \\\\ & \\leq \\sqrt { \\frac { e ^ 4 N } { 8 \\pi } } \\cdot 2 ^ { 4 \\tilde { \\epsilon } ( 1 - \\tilde { \\epsilon } ) \\eta N } . \\end{align*}"}
+{"id": "3913.png", "formula": "\\begin{align*} Y u ( \\Omega ) = \\Omega ^ * \\end{align*}"}
+{"id": "6126.png", "formula": "\\begin{align*} \\hbox { r a n k } ( \\mathcal R ) = N - d \\end{align*}"}
+{"id": "787.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\sigma ^ k _ \\ast \\widehat { \\nu } = \\mu , \\end{align*}"}
+{"id": "1636.png", "formula": "\\begin{align*} \\Omega : = X \\setminus \\{ \\rho = - \\infty \\} . \\end{align*}"}
+{"id": "6452.png", "formula": "\\begin{align*} \\hat d ( ( X , Y , t ) , ( \\tilde Y , \\tilde X , \\tilde t ) ) : = | X - \\tilde X | + | Y - \\tilde Y | ^ \\frac 1 2 + | t - \\tilde t | ^ \\frac 1 2 , \\end{align*}"}
+{"id": "7468.png", "formula": "\\begin{align*} \\Delta _ v = \\tilde { \\Delta } _ v \\end{align*}"}
+{"id": "2459.png", "formula": "\\begin{align*} G _ k ( s ) = \\sum _ { \\substack { L ( \\rho , \\pi ) = 0 \\\\ | s - \\rho | \\leq 2 0 0 \\eta } } \\frac { 1 } { ( s - \\rho ) ^ { k + 1 } } + \\sum _ { \\substack { L ( \\rho ' , \\pi \\otimes \\chi ) = 0 \\\\ | s - \\rho ' | \\leq 2 0 0 \\eta } } \\frac { 1 } { ( s + 1 - \\beta _ { \\chi } - \\rho ' ) ^ { k + 1 } } + O \\Big ( \\frac { m ^ 2 \\eta \\log ( q Q T ) } { ( 2 0 0 \\eta ) ^ k } \\Big ) . \\end{align*}"}
+{"id": "8505.png", "formula": "\\begin{align*} C _ { \\xi ^ 2 } = \\frac { 1 } { 9 \\mu } C _ { \\sigma ^ 2 } - \\frac { \\mu _ { \\sigma } } { 1 8 \\mu ^ 2 } C _ { \\sigma } , \\end{align*}"}
+{"id": "1022.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\hbox { d i v } ( t ^ { 1 - s } \\nabla u ) & = 0 , \\hbox { i n } \\mathbb { R } ^ { n + 1 } _ { + } ; \\\\ u & = f , \\hbox { o n } \\mathbb { R } ^ n \\times \\{ t = 0 \\} . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5409.png", "formula": "\\begin{align*} \\delta ( q _ { i , j , k } , t _ k ^ { ( i ) } ) = \\begin{cases} q _ { i , ( j + 1 ) \\bmod p , k + 1 } & k \\ne m _ i + n _ i - 1 \\\\ q _ { i , ( j + 1 ) \\bmod p , m _ i } & \\end{cases} \\end{align*}"}
+{"id": "4340.png", "formula": "\\begin{align*} c ( i + j , k ) \\omega ( i , j , k ) ^ { - 1 } = c ( i , k ) c ( j , k ) . \\end{align*}"}
+{"id": "3966.png", "formula": "\\begin{align*} 0 \\geq h _ \\sigma ( \\theta ) \\geq \\inf _ { \\theta \\in [ - 1 , 1 ] } h _ \\sigma ( \\theta ) = : - H . \\end{align*}"}
+{"id": "5835.png", "formula": "\\begin{align*} \\mathcal H _ 0 = L ^ 2 ( \\Omega ) , \\mathcal H _ 1 = H ^ 1 ( \\Omega ) , \\end{align*}"}
+{"id": "8539.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\bigg ( \\frac { 1 - ( n ^ 2 / 2 ^ n ) ^ { \\lfloor ( n - 1 ) / 2 \\rfloor + 1 } } { 1 - n ^ 2 / 2 ^ n } \\bigg ) = 1 . \\end{align*}"}
+{"id": "3219.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\sum _ { i \\in \\tilde { W } } r _ { i } ^ { s } \\right ) = 1 . \\end{align*}"}
+{"id": "5304.png", "formula": "\\begin{align*} h ^ { \\sigma } ( X , Y ) = h ( \\beta _ { \\sigma } ( X ) , \\beta _ { \\sigma } ( Y ) ) , \\end{align*}"}
+{"id": "3831.png", "formula": "\\begin{align*} \\Theta _ { \\psi _ 3 } = E _ 4 ^ { ( 2 ) } E _ 6 ^ { ( 2 ) } - 5 6 1 6 0 \\chi _ { 1 0 } . \\end{align*}"}
+{"id": "1840.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial \\rho } { \\partial t } + \\nabla ( \\rho \\boldsymbol v ) = 0 , \\\\ & \\frac { \\partial \\boldsymbol v } { \\partial t } + ( \\boldsymbol v \\cdot \\nabla ) \\boldsymbol v + \\frac { \\nabla p } { \\rho } + \\nabla \\phi = 0 , \\\\ & \\Delta \\phi = 4 \\pi G \\rho , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4434.png", "formula": "\\begin{align*} J ( 2 , a _ 2 ) = \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { a _ 2 } , \\frac { 1 } { 2 } + \\frac { 1 } { a _ 2 - 1 } \\right ] . \\end{align*}"}
+{"id": "7635.png", "formula": "\\begin{align*} M _ \\psi = \\begin{pmatrix} 0 & 1 & 1 \\\\ 3 & 2 & 3 \\\\ 3 & 2 & 1 \\\\ \\end{pmatrix} \\lambda _ 1 = 2 + \\sqrt { 1 0 } , \\ \\lambda _ 2 = 2 - \\sqrt { 1 0 } , \\ \\lambda _ 3 = - 1 . \\end{align*}"}
+{"id": "6262.png", "formula": "\\begin{align*} h & : \\mathcal { U } \\times ( - \\delta , 1 + \\delta ) \\rightarrow N _ { \\mathcal { S } } \\\\ h ( x , s ) & = f ^ { - 1 } \\circ \\varphi ^ { s \\cdot T } \\circ f ( x ) , \\end{align*}"}
+{"id": "1005.png", "formula": "\\begin{align*} \\langle u , v \\rangle = \\sum _ { n \\in \\Z } u [ n ] v [ n ] \\end{align*}"}
+{"id": "5035.png", "formula": "\\begin{align*} g _ t = \\mu _ t ^ 2 e ^ 1 \\otimes e ^ 1 + a _ t ^ 2 \\left ( e ^ 2 \\otimes e ^ 2 + e ^ 3 \\otimes e ^ 3 \\right ) + b _ t ^ 2 \\left ( e ^ 4 \\otimes e ^ 4 + e ^ 5 \\otimes e ^ 5 \\right ) , \\end{align*}"}
+{"id": "1520.png", "formula": "\\begin{align*} v ^ { ( k ) } _ { y _ l } = v ^ { ( l ) } _ { y _ k } . \\end{align*}"}
+{"id": "7269.png", "formula": "\\begin{align*} \\frac { d } { d m } \\frac { d ^ + } { d x } u = \\lambda u , u ( 0 ) = 1 \\end{align*}"}
+{"id": "8898.png", "formula": "\\begin{align*} \\mathcal { S } = 2 \\sum \\limits _ { \\mu = \\nu + \\frac { 1 } { 2 } } ^ { + \\infty } \\mu e ^ { - \\mu ^ { 2 } t } \\sum \\limits _ { p = 0 } ^ { n - 1 } \\gamma _ { p } ^ { ( \\nu , n ) } \\mu ^ { 2 p } - 2 \\sum \\limits _ { \\mu = \\nu + \\frac { 1 } { 2 } } ^ { \\nu + \\frac { n } { 2 } - 1 } \\mu e ^ { - \\mu ^ { 2 } t } \\sum \\limits _ { p = 0 } ^ { n - 1 } \\gamma _ { p } ^ { ( \\nu , n ) } \\mu ^ { 2 p } \\end{align*}"}
+{"id": "8352.png", "formula": "\\begin{align*} T _ { M , B } : = \\int _ { \\Sigma _ { 1 , M , B } } \\left ( \\frac 1 2 \\left | ( \\partial _ t - \\partial _ r ) w \\right | ^ 2 + \\frac 1 2 \\left | \\frac { \\partial _ \\omega w } { r } \\right | ^ 2 \\right ) \\mathrm { d } t \\mathrm { d } S \\end{align*}"}
+{"id": "7697.png", "formula": "\\begin{align*} ( \\tau ^ { \\sharp } ) ^ { - j } ( \\Gamma ) \\cap Z ( B _ { F , 0 } ) = \\emptyset j \\in \\{ 2 , 3 , \\dots , \\} . \\end{align*}"}
+{"id": "9045.png", "formula": "\\begin{align*} D _ G ( P , Q ) = \\inf \\left \\{ \\left ( \\int _ { G \\times G } \\delta ( \\theta , \\theta ^ { \\prime } ) ^ p R ( d \\theta , d \\theta ^ { \\prime } ) \\right ) ^ { 1 / p } \\right \\} , \\end{align*}"}
+{"id": "1446.png", "formula": "\\begin{align*} x _ i & = \\# \\{ j \\mid \\{ R _ i , C _ j \\} \\ \\mbox { i s a n e d g e o f $ \\Delta $ } \\} \\\\ & = \\# \\{ j \\mid ( R _ i , C _ j ) \\in B ( \\Delta ) \\} , \\\\ \\mbox { a n d } y _ j & = \\# \\{ i \\mid \\{ R _ i , C _ j \\} \\ \\mbox { i s a n e d g e o f $ \\Delta $ } \\} \\\\ & = \\# \\{ i \\mid ( R _ i , C _ j ) \\in B ( \\Delta ) \\} . \\end{align*}"}
+{"id": "7403.png", "formula": "\\begin{align*} q _ F ^ { \\lambda ( \\alpha ) } = q _ { \\alpha } q _ { \\alpha ^ * } \\in \\R _ { > 1 } . \\end{align*}"}
+{"id": "2508.png", "formula": "\\begin{align*} \\begin{pmatrix} I _ { H ^ 2 _ \\infty } & \\mathbb O \\\\ \\mathbb O & X _ 0 \\end{pmatrix} T = V \\begin{pmatrix} I _ { H ^ 2 _ \\infty } & \\mathbb O \\\\ \\mathbb O & X _ 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "5751.png", "formula": "\\begin{align*} \\mathbf { x } ^ { A A ^ { \\prime } } = - \\frac { 1 } { 2 } \\theta ^ { A A ^ { \\prime } } \\mathbf { m } - \\frac { 1 } { 2 } \\theta ^ { A A ^ { \\prime } } \\overline { \\mathbf { m } } + \\frac { 1 } { 2 } [ \\omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\theta ^ { A B ^ { \\prime } } - \\omega _ { B } ^ { A } \\theta ^ { B A ^ { \\prime } } ] \\mathbf { m } \\overline { \\mathbf { m } } . \\end{align*}"}
+{"id": "8549.png", "formula": "\\begin{align*} 1 + m + \\frac { m ^ 2 + m } { 2 } = \\frac { m ^ 2 + 3 m + 2 } { 2 } \\end{align*}"}
+{"id": "1953.png", "formula": "\\begin{align*} \\begin{gathered} \\varrho _ { K } ^ { ( N , J ) } [ u , X ] \\leq { \\varrho } _ E ^ { ( N ) } [ u , X ] + \\varepsilon \\ ; \\ ; u \\in U , \\\\ { \\vartheta } _ { K } ^ { ( N , J ) } \\leq { \\vartheta } _ E ^ { ( N ) } + \\varepsilon \\ ; \\ ; . \\end{gathered} \\end{align*}"}
+{"id": "892.png", "formula": "\\begin{align*} \\mathbf { E } \\left [ \\| \\mathcal { X } ^ { t + 1 } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } | \\mathcal { X } ^ { t } \\right ] \\leq \\left ( 1 - \\mathop { \\min } _ { k = 1 , 2 , \\cdots , l } \\lambda _ { \\min } ( \\mathbf { E } [ \\widehat { \\mathcal { Z } } _ { ( k ) } ] ) \\right ) \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } . \\end{align*}"}
+{"id": "1885.png", "formula": "\\begin{align*} B = \\begin{bmatrix} 0 & 1 & \\cdots & 1 \\\\ 0 & 0 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "4653.png", "formula": "\\begin{align*} \\mathcal { T } _ { i j } = \\exp \\left ( C _ { i j } \\partial _ i \\partial _ j \\right ) , \\ \\ i , j = 1 , \\ldots , N , \\ j \\neq i , \\end{align*}"}
+{"id": "2054.png", "formula": "\\begin{align*} H _ 1 ( t ) = H _ 2 ( \\varphi ( t ) ) \\varphi ' ( t ) , t \\in ( 0 , L _ 1 ) \\ a . e . \\end{align*}"}
+{"id": "234.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho ^ 2 - \\beta \\rho \\eta \\leqslant 0 , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\tau ^ 2 - \\beta \\tau \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "6830.png", "formula": "\\begin{align*} ( I \\ ; 0 ) Z _ i ^ { - 1 } Y _ j = - w _ { i , j } , \\ , w _ { i , j } : = w ( Y _ i , Y _ j ) , \\end{align*}"}
+{"id": "3834.png", "formula": "\\begin{align*} \\lim _ { v \\to \\infty } v ^ 2 g _ V ( v ) = 1 / \\pi , \\end{align*}"}
+{"id": "1400.png", "formula": "\\begin{align*} \\pi _ 1 ^ { ( - , - , + ) } = L ( \\Delta [ 0 , - 2 ] , \\Delta [ 1 , - 3 ] ; \\pi ( 0 ^ - , 1 ^ - , 2 ^ + ) ) . \\end{align*}"}
+{"id": "7942.png", "formula": "\\begin{align*} a \\odot b = a + b , a \\oplus b = \\begin{cases} \\max \\{ a , b \\} \\textrm { i f $ a \\neq b $ , } \\\\ [ - \\infty , a ] \\textrm { i f $ a = b $ , } \\end{cases} \\end{align*}"}
+{"id": "8143.png", "formula": "\\begin{align*} H _ { m , n } ^ { - , 1 } ( x ) = 4 T \\int _ { - \\infty } ^ { \\infty } \\widehat { k ^ * } ( \\xi ) \\cos \\Bigl ( x \\sinh \\frac { \\xi \\pi } { M } \\Bigr ) e \\Bigl ( - \\frac { T \\xi } { M } \\Bigr ) \\ , d \\xi . \\end{align*}"}
+{"id": "1685.png", "formula": "\\begin{align*} \\delta s ( \\mu _ m ) = \\frac { ( k - 1 ) } { 2 } \\left ( ( - i ) ^ { \\frac { k - 2 } { 2 } + m } f _ { \\frac { k } { 2 } } + i ^ { \\frac { k - 2 } { 2 } + m } f _ { - \\frac { k } { 2 } } \\right ) . \\end{align*}"}
+{"id": "481.png", "formula": "\\begin{align*} \\lim _ { h \\downarrow 0 } \\delta _ 1 ( h ) = 0 \\mbox { a n d } \\lim _ { h \\downarrow 0 } \\delta _ 2 ( h ) = 0 . \\end{align*}"}
+{"id": "1300.png", "formula": "\\begin{align*} H ^ i ( \\lambda ) = 0 \\lambda \\in X i > N . \\end{align*}"}
+{"id": "4351.png", "formula": "\\begin{align*} x y x = x y ( x y t ) = ( x y ) ( x y ) t = x y t = x . \\end{align*}"}
+{"id": "8415.png", "formula": "\\begin{align*} ( I I b ) \\leq \\begin{cases} C _ 3 & k = 1 \\\\ C _ 3 k ( 1 - \\theta ) ^ { k - 2 } & k \\geq 2 \\end{cases} \\end{align*}"}
+{"id": "2495.png", "formula": "\\begin{align*} \\operatorname { c l o s } \\varphi _ 0 \\mathcal K _ { \\mathbf u } = \\mathcal K _ { \\mathbf v } . \\end{align*}"}
+{"id": "7471.png", "formula": "\\begin{align*} \\Pr [ E _ n = e ] = \\frac { \\mu _ e } { \\mu } \\ , , \\end{align*}"}
+{"id": "4930.png", "formula": "\\begin{align*} \\alpha \\delta _ n ^ { + } z _ { m , n } = - \\mathrm { i } \\mathcal { F } ( \\mathbf { z } _ n , \\mathbf { z } _ { n + 1 } ) _ m \\end{align*}"}
+{"id": "2226.png", "formula": "\\begin{align*} \\nu ( x _ { 2 m + 1 , 2 } ) & = \\nu { \\left ( \\sum _ { k = 1 } ^ \\infty x _ { 2 m , k } \\beta _ { k , 2 } \\right ) } \\\\ & = \\min _ { k \\geq 4 } { \\left \\{ \\big ( \\nu ( x _ { 2 m , 2 } ) + \\nu ( \\beta _ { 2 , 2 } ) \\big ) , \\big ( \\nu ( x _ { 2 m , 3 } ) + \\nu ( \\beta _ { 3 , 2 } ) \\big ) \\right . } , \\\\ & \\qquad \\qquad \\left . \\big ( \\nu ( x _ { 2 m , k } ) + \\nu ( \\beta _ { k , 2 } ) \\big ) \\right \\} \\\\ & \\geq 2 m + 1 . \\end{align*}"}
+{"id": "8831.png", "formula": "\\begin{align*} \\left ( \\partial _ t + \\Delta \\right ) H = 0 \\lim \\limits _ { t \\to 0 } \\int _ { \\mathcal { M } } H ( x , y , t ) f ( y ) d y = f ( x ) \\end{align*}"}
+{"id": "5419.png", "formula": "\\begin{align*} { r ^ m _ i } ( x y ) = \\sum _ { t = 0 } ^ { m } v ^ { ( t i , \\nu ' - ( m - t ) i ) + t ( m - t ) } \\frac { [ m ] _ { v } ! } { [ t ] _ { v } ! [ m - t ] _ { v } ! } { r _ i ^ t } ( x ) { r _ i ^ { m - t } } ( y ) \\end{align*}"}
+{"id": "479.png", "formula": "\\begin{align*} [ v ] _ h ( \\cdot , t ) = e ^ { - \\frac { t } { h } } v _ 0 + \\frac { 1 } { h } \\int _ 0 ^ t e ^ { \\frac { s - t } { h } } v ( \\cdot , s ) \\ , d s , \\end{align*}"}
+{"id": "5227.png", "formula": "\\begin{align*} \\left ( \\ell _ { 2 } ' \\right ) ^ { 3 } = & \\frac { k _ 3 \\ell _ { 2 } ^ { 4 } } { k _ 0 - k _ 2 \\ell _ { 2 } ^ { 2 } } & l _ { 1 } = & k _ 2 \\ell _ 2 + k _ 1 + k _ 0 / \\ell _ 2 \\\\ b _ { 2 5 } ^ 6 = & \\frac { c _ { 4 5 } ^ 3 ( k _ 0 - k _ 2 \\ell _ { 2 } ^ { 2 } ) } { k _ 3 \\ell _ { 2 } ^ { 4 } } & b _ { 1 3 } = & \\ , b _ { 1 5 } = 0 \\\\ b _ { 2 3 } = & ( k _ 4 \\ell _ 2 - k _ 0 ) b _ { 2 5 } & b _ { 1 2 } ^ 2 = & \\frac { c _ { 1 2 } b _ { 2 5 } ^ 2 \\ell _ 2 ^ 2 } { c _ { 4 5 } ( k _ 0 - k _ 2 \\ell _ { 2 } ^ { 2 } ) } \\end{align*}"}
+{"id": "4708.png", "formula": "\\begin{align*} f = & 3 \\varphi _ 4 , \\\\ g = & \\frac { 1 } { 1 0 2 4 } \\left ( 1 6 4 7 \\varphi _ 4 ^ 3 - 2 4 3 \\varphi _ { 1 2 } \\right ) . \\end{align*}"}
+{"id": "1086.png", "formula": "\\begin{align*} \\mathrm { V a r } \\bigl ( \\mathbb { E } \\bigl \\{ \\hat { \\mu } | J \\bigr \\} \\bigr ) \\leq \\mathbb { E } \\Bigl [ \\Bigl \\{ \\mathbb { E } ( \\hat { \\mu } | J ) - \\mathbb { E } ( \\hat { \\mu } | J = j _ 0 - 1 ) \\Bigr \\} ^ 2 \\Bigr ] \\leq ( T + M ) ^ 2 \\delta + 8 ( T + M ) ^ 2 \\exp ( - n \\alpha ^ 2 / 5 1 2 ) . \\end{align*}"}
+{"id": "607.png", "formula": "\\begin{align*} \\mathbb { E } ^ \\dagger \\left [ X ^ { ( \\epsilon ) } _ { t _ n , t _ { n + 1 } } \\otimes X ^ { ( \\epsilon ) } _ { t _ { n + 1 } , t _ { n + 2 } } \\right ] = \\mathcal { O } ( \\Delta t ^ 2 ) \\ , , \\end{align*}"}
+{"id": "8584.png", "formula": "\\begin{align*} a ( x ) = a _ 0 + a _ 1 \\ , x + \\ldots + a _ { n - 2 } \\ , x ^ { n - 2 } + a _ { n - 1 } \\ , x ^ { n - 1 } \\in \\Omega \\end{align*}"}
+{"id": "8934.png", "formula": "\\begin{align*} T _ 1 = \\sum \\limits _ { p = 0 } ^ { n - 2 } \\frac { \\tau _ { p } ^ { ( \\nu , n ) } p ! } { ( n - 1 ) ! } t ^ { n - p - 1 } = \\sum \\limits _ { d = 1 } ^ { n - 1 } \\frac { \\tau _ { n - d - 1 } ^ { ( \\nu , n ) } \\left ( n - d - 1 \\right ) ! } { ( n - 1 ) ! } t ^ { d } . \\end{align*}"}
+{"id": "245.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\tau ^ 2 - \\beta \\rho \\eta \\leqslant 0 , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\tau ^ 2 - \\beta \\tau \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "6444.png", "formula": "\\begin{align*} u ( X , Y , t ) - w _ { \\lambda , A , \\theta , \\eta } ( Y , t ) = \\tilde u ( X , Y , t ) \\leq v ( X , Y , t ) , \\end{align*}"}
+{"id": "852.png", "formula": "\\begin{align*} L _ d ( R _ k , R _ { k + 1 } ) & \\approx h L ( R _ k , \\frac { R _ { k + 1 } - R _ k } { h } ) \\\\ & = \\frac { h } { 2 } t r ( \\frac { R _ k ^ T ( R _ { k + 1 } - R _ k ) } { h } J _ d \\frac { ( R _ { k + 1 } - R _ k ) ^ T R _ k } { h } ) \\\\ & = \\frac { 1 } { 2 h } t r ( ( I _ { 3 \\times 3 } - F _ k ) J _ d ) \\end{align*}"}
+{"id": "5936.png", "formula": "\\begin{align*} \\hbox { r a n k } ( C _ p D ) = \\hbox { r a n k } ( D ) = M = N - p . \\end{align*}"}
+{"id": "1219.png", "formula": "\\begin{align*} ( \\lambda _ i - d ) r _ { i } = c _ i \\quad ( 1 \\leq i \\leq n - 1 ) \\end{align*}"}
+{"id": "7568.png", "formula": "\\begin{align*} Z _ { f _ { 6 , a } } ( s , \\chi , B _ 6 ^ a ) = { \\left \\{ \\begin{array} { r l } ( 1 - q ^ { - 1 } ) Z _ { \\tilde { f } _ { 6 , a } } ( s , \\chi , C _ 6 ^ a ) , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } = \\chi _ { \\rm t r i v } , \\\\ 0 , \\ \\ \\ & { \\rm i f } \\ \\chi ^ { k + r + l } \\ne \\chi _ { \\rm t r i v } , \\end{array} \\right . } \\end{align*}"}
+{"id": "5536.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { c - i T } ^ { c + i T } \\frac { \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } { \\rm d } s = \\frac { 1 } { 2 \\pi i } \\int _ { \\lambda - i T } ^ { \\lambda + i T } \\frac { \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } { \\rm d } s - R _ 0 - \\mathcal { R } ( x ) , \\end{align*}"}
+{"id": "5997.png", "formula": "\\begin{align*} Y _ { r , p } ^ h : = \\{ [ x _ 1 , \\dots , x _ h , 0 , \\dots , 0 ] ; [ 0 : \\dots : 0 : y _ { r + 1 } : \\dots : y _ { n - p } : 0 : \\dots : 0 ] \\in \\mathbb { P } ^ { n - 1 } \\times \\mathbb { P } ^ { n - 1 } | \\sum _ a x _ a y _ a = 0 \\} . \\end{align*}"}
+{"id": "306.png", "formula": "\\begin{align*} \\zeta _ { ( { \\chi _ { \\mathrm { B C } } } _ { \\mu , \\alpha } , \\chi _ { \\mu \\circ \\mathrm { N m } _ { T ( E ) / S ^ { \\mathrm { o p } } ( F ) } , \\alpha } ) } | _ { F _ { \\alpha } ^ { \\times } } ( \\iota _ { F _ \\alpha } \\alpha ( t ) ) = \\omega _ { E _ { \\alpha } / F _ { \\alpha } } ( \\iota _ { F _ \\alpha } \\alpha ( t ) ) . \\end{align*}"}
+{"id": "2166.png", "formula": "\\begin{align*} \\phi ( r ) : & = ( 3 ( 2 \\alpha - 1 + \\beta ) - 1 ) r ^ 2 + 3 ( 2 - 2 \\alpha + \\beta ) r - 2 , \\intertext { a n d } \\psi ( r ) : & = ( 2 \\alpha - 1 + \\beta - 3 ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + 2 . \\end{align*}"}
+{"id": "8630.png", "formula": "\\begin{align*} \\mu ^ \\N \\Big ( x \\in \\Omega ^ N _ { } : x [ i , i + k ) = x [ j , j + k ) = \\omega \\Big ) = b ^ { - k - ( j - i ) } . \\end{align*}"}
+{"id": "844.png", "formula": "\\begin{align*} K = \\frac { 1 } { 2 } \\Omega ^ T J \\Omega = \\frac { 1 } { 2 } t r ( \\widehat { \\Omega } J _ d \\widehat { \\Omega } ^ T ) \\end{align*}"}
+{"id": "2655.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N b _ j \\varrho _ j \\leq \\xi \\ , . \\end{align*}"}
+{"id": "2830.png", "formula": "\\begin{align*} P _ { \\sigma } P _ { \\tau } = P _ { \\sigma \\tau } . \\end{align*}"}
+{"id": "2552.png", "formula": "\\begin{align*} Z ( v ; ( \\alpha , \\beta ) ) = Z ( \\tau ( v ) ; ( \\beta , \\alpha ) ) \\end{align*}"}
+{"id": "2299.png", "formula": "\\begin{align*} \\hat { g } ^ \\C _ { i + 1 / 2 } = p _ \\C ( x _ { i + 1 / 2 } ) = - \\frac { 3 } { 4 0 } b _ { i - 2 } + \\frac { 1 1 } { 2 4 } b _ { i - 1 } - 2 b _ i + 2 b _ { i + 1 } - \\frac { 1 1 } { 2 4 } b _ { i + 2 } + \\frac { 3 } { 4 0 } b _ { i + 3 } , \\end{align*}"}
+{"id": "5145.png", "formula": "\\begin{align*} \\forall n \\in \\mathbb { N } , \\quad \\forall \\lambda > 0 , I _ { - n } ( \\lambda ) = I _ { n } ( \\lambda ) > 0 \\quad \\mbox { a n d } K _ { - n } ( \\lambda ) = K _ { n } ( \\lambda ) > 0 . \\end{align*}"}
+{"id": "404.png", "formula": "\\begin{align*} \\left \\| \\sum _ { { n \\in \\mathbb { M } } } a _ n \\tau _ n \\right \\| ^ p = \\sum _ { { n \\in \\mathbb { M } } } | a _ n | ^ p . \\end{align*}"}
+{"id": "4822.png", "formula": "\\begin{align*} v \\in C ^ \\perp & \\implies \\sum _ k v _ k c _ { \\pi ( k ) } = 0 \\textnormal { f o r a l l } c \\in C \\\\ & \\implies \\sum _ k v _ { \\pi ^ { - 1 } ( k ) } c _ k = 0 \\textnormal { f o r a l l } c \\in C \\\\ & \\implies v _ { \\pi ^ { - 1 } } \\in C ^ \\perp . \\end{align*}"}
+{"id": "6735.png", "formula": "\\begin{align*} \\phi ( s ) = \\pi ^ { - \\frac { s } { 2 } } \\Gamma ( \\frac { s } { 2 } ) \\zeta ( s ) \\end{align*}"}
+{"id": "3477.png", "formula": "\\begin{align*} \\left ( \\frac { F _ R ( t _ 1 ) } { R ^ { d - \\beta / 2 } } , \\ldots , \\frac { F _ R ( t _ m ) } { R ^ { d - \\beta / 2 } } \\right ) = \\left ( \\frac { J _ { 1 , R } ( t _ 1 ) } { R ^ { d - \\beta / 2 } } , \\ldots , \\frac { J _ { 1 , R } ( t _ m ) } { R ^ { d - \\beta / 2 } } \\right ) + \\frac { 1 } { R ^ { d - \\beta / 2 } } \\sum _ { n \\geq 2 } ( J _ { n , R } ( t _ 1 ) , \\ldots , J _ { n , R } ( t _ m ) ) . \\end{align*}"}
+{"id": "8040.png", "formula": "\\begin{align*} L ( 1 , f ) T M \\log T = c L ( 1 , g ) T M \\log T + O _ p ( T M ) . \\end{align*}"}
+{"id": "1750.png", "formula": "\\begin{align*} p ( N ) : = \\int _ { S ^ 3 } s ( \\mu _ { \\underline { m } } ) ( \\kappa ( \\alpha , \\beta ) ) \\cdot \\alpha ^ { N } ( - \\beta ) ^ { k _ c - 2 - N } ( - \\bar \\alpha ) ^ { N + \\lambda - \\bar m } ( - \\bar \\beta ) ^ { M + \\bar m - N } d ( \\alpha , \\beta ) \\end{align*}"}
+{"id": "939.png", "formula": "\\begin{align*} \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { E x t } _ { R } ^ n ( M , \\ , C ) ) \\ , = \\ , \\sup \\{ j \\geq 0 \\mid \\textrm { E x t } _ { R } ^ j ( \\textrm { E x t } _ { R } ^ n ( M , \\ , C ) , \\ , C ) \\neq 0 \\} , \\end{align*}"}
+{"id": "890.png", "formula": "\\begin{align*} \\mathbf { E } [ \\| \\mathcal { X } ^ { t + 1 } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } | \\mathcal { X } ^ { t } ] & = \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } - \\mathbf { E } _ { i \\sim \\mathbf { p } ^ { t } } [ f _ { i } ( \\mathcal { X } ^ { t } ) ] = \\| \\mathcal { X } ^ { t } - \\mathcal { X } ^ { \\star } \\| _ { F ( \\mathcal { Q } ) } ^ { 2 } - \\sum _ { i \\in \\mathfrak { W } _ { t } } p _ { i } ^ { t } f _ { i } ( \\mathcal { X } ^ { t } ) . \\end{align*}"}
+{"id": "7090.png", "formula": "\\begin{align*} ( m + n ) m _ { m + n - 1 } = _ { x ^ { m + n - 1 } } \\dfrac { 1 } { ( 1 + b _ 1 x + b _ 2 x ^ 2 + \\cdots ) ^ { m + n } } , \\end{align*}"}
+{"id": "9087.png", "formula": "\\begin{align*} \\mathfrak { d } ^ j / \\mathfrak { d } ^ { j - 1 } = \\mathfrak { Z } ( \\mathfrak { n } / \\mathfrak { d } ^ { j - 1 } ) \\cap J \\mathfrak { Z } ( \\mathfrak { n } / \\mathfrak { d } ^ { j - 1 } ) j \\geq 1 . \\end{align*}"}
+{"id": "8904.png", "formula": "\\begin{align*} \\sigma _ { p } ^ { \\left ( \\nu \\right ) } \\left ( \\ell \\right ) = \\frac { ( - 1 ) \\ell } { \\ell ! } \\sum \\limits _ { j = 0 } ^ { \\nu - 1 } \\left ( j + \\frac { 1 } { 2 } \\right ) ^ { 2 ( p + \\ell ) + 1 } . \\end{align*}"}
+{"id": "1831.png", "formula": "\\begin{align*} K _ Y + \\Delta _ Y = \\mu ^ * ( K _ X + \\Delta ) + \\sum r _ i E _ i , r _ i > 0 i . \\end{align*}"}
+{"id": "5650.png", "formula": "\\begin{align*} & \\sum _ { n _ 1 , \\cdots , n _ 7 \\geq 1 } \\frac { 1 } { ( n _ 1 + \\cdots + n _ 7 ) ^ 5 ( n _ 2 + \\cdots + n _ 7 ) ^ 2 n _ 3 ( n _ 4 ) ^ 2 ( n _ 5 + n _ 6 + n _ 7 ) ^ 2 n _ 6 n _ 7 } \\\\ = ~ & 8 \\zeta ( 5 , 2 , 1 , 2 , 2 , 1 , 1 ) + 1 6 \\zeta ( 5 , 2 , 1 , 3 , 1 , 1 , 1 ) + 2 \\zeta ( 5 , 2 , 1 , 2 , 1 , 1 , 2 ) + 4 \\zeta ( 5 , 2 , 1 , 2 , 1 , 2 , 1 ) + 4 8 \\zeta ( 5 , 2 , 2 , 2 , 1 , 1 , 1 ) \\\\ + ~ & 2 8 \\zeta ( 5 , 2 , 2 , 1 , 2 , 1 , 1 ) + 8 \\zeta ( 5 , 2 , 2 , 1 , 1 , 1 , 2 ) + 1 6 \\zeta ( 5 , 2 , 2 , 1 , 1 , 2 , 1 ) + 4 0 \\zeta ( 5 , 2 , 3 , 1 , 1 , 1 , 1 ) . \\end{align*}"}
+{"id": "31.png", "formula": "\\begin{align*} T ( n _ i ) = \\sum _ { t = 0 } ^ { k _ i - 1 } \\frac { 1 } { C ^ t ( n _ i ) } < \\sum _ { t = 0 } ^ { \\infty } \\frac { 1 } { n _ i \\cdot \\left ( \\tfrac { 3 } { 2 } \\right ) ^ t } = \\frac { 1 } { n _ i } \\cdot \\sum _ { t = 0 } ^ { \\infty } \\left ( \\frac { 2 } { 3 } \\right ) ^ t = \\frac { 3 } { n _ i } < \\frac { 3 } { X _ 0 } . \\end{align*}"}
+{"id": "7812.png", "formula": "\\begin{align*} X _ n ( x ^ { \\tilde m } ) : = & \\ [ \\langle n , \\tilde m \\rangle _ 1 ] _ q x ^ { \\tilde m + p _ 1 ^ * ( n ) } \\\\ = & \\ \\frac { q ^ { 2 \\langle n , \\tilde m \\rangle _ 1 } - 1 } { q - q ^ { - 1 } } x ^ { \\tilde m } x ^ { p _ 1 ^ * ( n ) } , \\end{align*}"}
+{"id": "6304.png", "formula": "\\begin{align*} s _ i s _ { i + 1 } s _ i ( T ) = s _ { i + 1 } s _ i s _ { i + 1 } ( T ) , \\end{align*}"}
+{"id": "4431.png", "formula": "\\begin{align*} a _ 2 = 4 9 5 = \\left ( \\frac { 1 } { 9 } + \\frac { 1 } { 1 1 } - \\frac { 1 } { 5 } \\right ) ^ { - 1 } \\end{align*}"}
+{"id": "380.png", "formula": "\\begin{align*} J ^ { ( 1 , 2 ) } _ 2 \\leq 4 T _ * L ^ { - 2 } _ * \\left ( \\sum \\limits _ { n = 2 } ^ { \\infty } p _ n ( T _ * ) \\left | \\frac { 2 \\widetilde { n } } { T _ * } - 1 \\right | ^ 2 \\right ) ^ { 1 / 2 } \\leq 4 \\sqrt { 3 5 } T _ * L ^ { - 2 } _ * T ^ { - 1 / 2 } _ * . \\end{align*}"}
+{"id": "3809.png", "formula": "\\begin{align*} \\sum _ { \\abs { \\alpha } = m } \\frac { 1 } { \\alpha ! } \\abs { \\partial ^ \\alpha f ( x ) } ^ 2 > \\frac { 4 ^ { m + 1 } \\kappa C _ B ( m , \\lambda ) } { m ! \\abs { Q _ k } } \\norm { f } _ { L ^ 2 ( Q _ k ) } ^ 2 . \\end{align*}"}
+{"id": "2659.png", "formula": "\\begin{align*} R : = \\min \\left \\{ { \\alpha } , \\frac { { \\alpha } ( p _ 1 ^ { - 1 } - 3 / 2 ) } { { \\alpha } + p _ 1 ^ { - 1 } - p _ 2 ^ { - 1 } } \\right \\} . \\end{align*}"}
+{"id": "1549.png", "formula": "\\begin{align*} & \\int _ { M } \\bigg ( | D \\mathrm { u } ( z ) | ^ { p ( z ) - 2 } D \\mathrm { u } ( z ) + \\mu ( z ) \\ , | D \\mathrm { u } ( z ) | ^ { q ( z ) - 2 } D \\mathrm { u } ( z ) \\ , \\bigg ) \\ , D \\varphi ( z ) \\ , \\ , d v _ { g } ( z ) \\\\ & = \\int _ { M } g ( z ) \\ , | \\mathrm { u } ( z ) | ^ { - \\gamma ( z ) } \\ , \\varphi ( z ) \\ , \\ , d v _ { g } ( z ) + \\lambda \\int _ { M } | \\mathrm { u } ( z ) | ^ { r ( z ) - 1 } \\varphi ( z ) \\ , \\ , d v _ { g } ( z ) , \\end{align*}"}
+{"id": "3793.png", "formula": "\\begin{align*} \\mathcal { E } ( g ) = \\int _ { S ^ 2 } R \\ , \\log \\Big ( \\frac { A R } { 8 \\pi } \\Big ) \\ , d \\mu , \\end{align*}"}
+{"id": "4083.png", "formula": "\\begin{align*} \\mathfrak { g d i f f } _ H ^ e = \\{ ( X , \\omega ) \\in \\Gamma ( T M ) \\times \\Omega ^ 2 : i _ X H + \\omega = d \\alpha \\} \\end{align*}"}
+{"id": "8330.png", "formula": "\\begin{align*} \\Box _ { t , x } w = 5 Q _ l ^ 4 w + 1 0 Q _ l ^ 3 w ^ 2 + 1 0 Q _ l ^ 2 w ^ 3 + 5 Q _ l w ^ 4 + w ^ 5 . \\end{align*}"}
+{"id": "1956.png", "formula": "\\begin{align*} \\min _ { u \\in U } \\varrho ^ { ( N , J ) } _ K [ u , X ] & \\leq \\varrho ^ { ( N , J ) } _ K [ u _ { E } , X ] \\leq \\varrho ^ { ( N ) } _ E [ u _ { E } , X ] + L h _ N ^ \\alpha \\\\ & = \\min _ { u \\in U } \\varrho ^ { ( N ) } _ E [ u , X ] + L h _ N ^ \\alpha \\quad & \\end{align*}"}
+{"id": "7789.png", "formula": "\\begin{align*} \\Psi [ n ] = \\exp \\Biggl ( \\sum _ { j = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { j + 1 } } { j ^ 2 } X _ { j n } \\Biggr ) \\end{align*}"}
+{"id": "725.png", "formula": "\\begin{align*} \\arg u = \\arg \\frac { d w } { d t } \\pm \\frac { \\pi } { 2 } , \\end{align*}"}
+{"id": "8993.png", "formula": "\\begin{align*} \\int _ { \\Omega _ k } | \\nabla u _ k ( t ) | ^ 2 d z = \\int _ { B } | \\nabla u ( t _ k + r _ k t ) | ^ 2 d z \\le 2 E ( u _ 0 ) , \\end{align*}"}
+{"id": "159.png", "formula": "\\begin{align*} A ^ { * } v = - \\frac { \\partial } { \\partial \\hslash } v , \\ D ( A ^ { * } ) : = \\{ v \\in H ^ { 1 } ( ( 0 , \\pi ) ; \\mathbb { R } ) : v ( 0 ) = 0 \\} . \\end{align*}"}
+{"id": "6601.png", "formula": "\\begin{align*} | H ( x , \\xi ) | & = ( 2 \\pi ) ^ { - n } | V _ { \\phi _ 1 } f ( x , \\xi ) | \\cdot | V _ { \\phi _ 2 } g ( x , \\xi ) | \\\\ & \\leq C _ { N , r } ( 1 + | \\xi | ^ 2 ) ^ { - ( N - N _ 0 ) } e ^ { - ( r _ 0 - r ) | x | ^ { 1 / s } } . \\end{align*}"}
+{"id": "1971.png", "formula": "\\begin{align*} A _ { k } ( z ) f ( z + c _ { k } ) + A _ { k - 1 } ( z ) f ( z + c _ { k - 1 } ) + \\cdots + A _ { 1 } ( z ) f ( z + c _ { 1 } ) + A _ { 0 } ( z ) f ( z ) = 0 \\end{align*}"}
+{"id": "8237.png", "formula": "\\begin{align*} I ' ( u ) h = \\int _ { \\mathbb { R } ^ { N } } \\nabla u \\nabla h d x + \\int _ { \\mathbb { R } ^ { N } } V ( | x | ) f ( u ) f ' ( u ) h d x - \\int _ { \\mathbb { R } ^ { N } } K ( | x | ) g ( f ( u ) ) f ' ( u ) h \\ , d x \\end{align*}"}
+{"id": "4339.png", "formula": "\\begin{align*} \\omega ( j , k , i ) c ( i , j + k ) = c ( i , k ) c ( i , j ) \\end{align*}"}
+{"id": "2308.png", "formula": "\\begin{align*} f ( u ) = \\frac { u ^ 2 } { u ^ 2 + ( 1 - u ) ^ 2 } , \\end{align*}"}
+{"id": "7061.png", "formula": "\\begin{align*} ( S ^ { \\rho ^ { - 1 } } , * ) \\simeq ( \\mathbf { C P } ( 1 + \\rho ) , \\mathbf { C P } ( 1 ) ) \\subset ( \\mathbf { C P } ^ \\infty _ { C _ 2 } , \\mathbf { C P } ( 1 ) ) = ( \\mathbf { C P } ^ \\infty _ { C _ 2 } , * ) . \\end{align*}"}
+{"id": "7110.png", "formula": "\\begin{align*} J = ( x _ i p _ j - x _ j p _ i : i , j \\geq 1 ) . \\end{align*}"}
+{"id": "6035.png", "formula": "\\begin{align*} \\forall u , \\ : v , \\ : w \\ : \\in W ^ P , \\ : \\Delta ( u , v , w ) & = 1 \\ : & \\mathrm { i f } \\ : ( - 1 ) ^ { \\ell ( w _ 0 w ) - \\ell ( w _ 0 u ) - \\ell ( w _ 0 v ) + \\int _ { d _ 1 ( u , v , w ) l _ 1 + d _ 2 ( u , v , w ) l _ 2 } c _ 1 ( T _ X ) } \\geq 0 \\\\ & = 0 & \\mathrm { i f } \\ : ( - 1 ) ^ { \\ell ( w _ 0 w ) - \\ell ( w _ 0 u ) - \\ell ( w _ 0 v ) + \\int _ { d _ 1 ( u , v , w ) l _ 1 + d _ 2 ( u , v , w ) l _ 2 } c _ 1 ( T _ X ) } < 0 \\end{align*}"}
+{"id": "5374.png", "formula": "\\begin{align*} D F = \\left [ \\begin{array} { c c c c } f _ { 1 1 } & f _ { 1 2 } & f _ { 1 3 } & f _ { 1 4 } \\\\ f _ { 2 1 } & f _ { 2 2 } & f _ { 2 3 } & f _ { 2 4 } \\\\ a f ' _ { 3 } & b f ' _ { 3 } & 0 & 0 \\\\ a f ' _ { 4 } & b f ' _ { 4 } & 0 & 0 \\end{array} \\right ] \\end{align*}"}
+{"id": "7457.png", "formula": "\\begin{align*} \\| \\Gamma ( Z _ i , Z _ j ) \\| & \\le \\bigg \\| \\Gamma \\bigg ( \\begin{bmatrix} Z _ i & 0 \\\\ 0 & Z _ j \\end{bmatrix} , \\begin{bmatrix} Z _ i & 0 \\\\ 0 & Z _ j \\end{bmatrix} \\bigg ) \\bigg \\| \\le \\| \\Gamma ( Z ^ { ( 0 ) } , Z ^ { ( 0 ) } ) \\| \\\\ & = \\| \\Gamma ( Z ^ { ( 0 ) } , Z ^ { ( 0 ) } ) \\| = \\| \\Gamma ( Z ^ { ( 0 ) } , Z ^ { ( 0 ) } ) ( I _ { N _ 0 } ) \\| \\\\ & = \\| K ( Z ^ { ( 0 ) } , Z ^ { ( 0 ) } ) ( I _ { N _ 0 } ) \\| \\end{align*}"}
+{"id": "2211.png", "formula": "\\begin{align*} v _ i = U ( q ^ { - 2 } \\gamma \\xi ^ { i - 1 } ) = \\delta \\sum _ { j = 1 } ^ { 5 i } \\alpha _ { i , j } \\xi ^ { j - 1 } , \\end{align*}"}
+{"id": "5760.png", "formula": "\\begin{align*} q : T ^ * V | _ U \\rightarrow T ^ * U , \\xi + \\sum _ { i = 1 } ^ r \\lambda _ i d t _ i \\mapsto \\xi + \\sum _ { i = 1 } ^ r \\lambda _ i d f _ i , \\end{align*}"}
+{"id": "1324.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 & = ( \\operatorname { T r a } ( S _ { f , \\tau } ) ) ^ 2 = \\left ( \\sum _ { k = 1 } ^ d \\lambda _ k \\right ) ^ 2 \\leq d \\sum _ { k = 1 } ^ d \\lambda _ k ^ 2 \\\\ & = d \\operatorname { T r a } ( S ^ 2 _ { f , \\tau } ) = d \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) . \\end{align*}"}
+{"id": "226.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho ^ 2 - \\beta \\rho \\eta \\leqslant 0 , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\tau ^ 2 - \\beta \\tau \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "7169.png", "formula": "\\begin{align*} U _ { \\beta , t } + U _ { \\beta , j } v _ { j } + \\Phi ^ \\beta _ { k , k } = r _ \\beta , \\ , \\ , \\ , \\ , \\ , \\beta = 1 \\dots \\omega , \\end{align*}"}
+{"id": "7461.png", "formula": "\\begin{align*} { \\bf w } ( \\ell ) = 1 2 3 3 4 5 5 6 6 7 7 4 8 9 9 8 2 1 . \\end{align*}"}
+{"id": "1868.png", "formula": "\\begin{align*} \\beta = \\sin ^ { - 1 } \\left ( \\frac { - \\ddot { s } _ i } { \\sqrt { \\dot { s } _ i ^ 2 + \\ddot { s } _ i ^ 2 } } \\right ) = \\sin ^ { - 1 } \\left ( \\frac { - ( p _ { s _ i } \\sigma + q _ { s _ i } ) } { \\sqrt { \\dot { s } _ i ^ 2 + ( p _ { s _ i } \\sigma + q _ { s _ i } ) ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "4678.png", "formula": "\\begin{align*} \\sum _ { m = \\ell _ 2 } ^ { \\infty } \\sum _ { k = \\max \\{ 0 , \\ell _ 1 + \\ell _ 2 - m \\} } ^ { \\ell _ 1 } A _ { m , k } = \\sum _ { k = 0 } ^ { \\ell _ 1 } \\sum _ { m = \\ell _ 1 + \\ell _ 2 - k } ^ { \\infty } A _ { m , k } . \\end{align*}"}
+{"id": "204.png", "formula": "\\begin{align*} | \\mu ( T ^ { k ( h _ n + c _ n ) } ( I _ { n , a } ) \\cap B ) & - \\mu ( I _ { n , a } ) \\mu ( B ) | = | \\sum \\limits _ { i = 0 } ^ { r _ { n } } \\mu ( T ^ { k ( h _ n + c _ n ) } ( I ^ { [ i ] } _ { n , a } ) \\cap B ) - \\mu ( I ^ { [ i ] } _ { n , a } ) \\mu ( B ) | . \\end{align*}"}
+{"id": "6412.png", "formula": "\\begin{align*} ( m + p ) ( \\partial _ t - X ^ k _ j \\cdot \\nabla _ Y ) \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) & \\geq ( p - 2 ) \\Delta ^ N _ { \\infty , X } \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) + \\Delta _ X \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) \\\\ & \\geq ( p - 2 ) \\lambda ( \\nabla ^ 2 _ X \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) ) + \\Delta _ X \\zeta ^ k _ j ( X ^ k _ j , Y ^ k _ j , t ^ k _ j ) \\\\ & = ( p - 2 ) \\lambda ( - H _ k ^ X ) - \\textrm { t r } ( H _ k ^ X ) . \\end{align*}"}
+{"id": "3074.png", "formula": "\\begin{align*} c _ j ^ { k l } ( A ) & = \\int _ Y r A e _ j \\cdot \\nabla v ^ { k l } \\\\ & = \\int _ Y ( r - 1 ) ( C e _ j \\cdot \\nabla v ^ { k l } ) + \\int _ Y ( r a - \\bar { a } ) ( M e _ j \\cdot \\nabla v ^ { k l } ) \\\\ & = m _ { k l } \\int _ Y ( M : D ^ 2 w ) ( C e _ j \\cdot \\nabla w ) - m _ { k l } \\int _ Y ( C : D ^ 2 w ) ( M e _ j \\cdot \\nabla w ) \\\\ & = 0 , \\end{align*}"}
+{"id": "1974.png", "formula": "\\begin{align*} - A _ { l } ( z ) = A _ { k } ( z ) \\frac { f ( z + c _ { k } ) } { f ( z + c _ { l } ) } + . . . + A _ { l - 1 } ( z ) \\frac { f ( z + c _ { l - 1 } ) } { f ( z + c _ { l } ) } + . . . + A _ { 1 } ( z ) \\frac { f ( z + c _ { 1 } ) } { f ( z + c _ { l } ) } + A _ { 0 } ( z ) \\frac { f ( z ) } { f ( z + c _ { l } ) } . \\end{align*}"}
+{"id": "4524.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n + 1 } v _ i \\leq \\sum _ { i = 1 } ^ { n + 1 } u _ i \\end{align*}"}
+{"id": "4571.png", "formula": "\\begin{align*} a _ { n + 1 , r } - 1 = \\left ( a _ { n , r } - 1 \\right ) \\left ( a _ { n , r } - ( r - 1 ) \\right ) \\end{align*}"}
+{"id": "5925.png", "formula": "\\begin{align*} \\widehat { \\Lambda } w = \\widehat { \\Lambda } \\alpha E = \\mu \\alpha E = \\mu w \\Gamma . \\end{align*}"}
+{"id": "8342.png", "formula": "\\begin{align*} \\int _ { | x | > A + t , t = M _ 0 } \\left ( | \\partial _ t w | ^ 2 + | \\nabla w | ^ 2 \\right ) \\mathrm { d } x \\le 2 \\delta _ 0 . \\end{align*}"}
+{"id": "8089.png", "formula": "\\begin{align*} H _ { m , n } ^ + ( x ) = 2 i \\int _ { - \\infty } ^ \\infty J _ { 2 i t } ( x ) \\frac { k ( t ) V ( m ^ 2 n , t ) t } { \\cosh ( \\pi t ) } \\ , d t \\end{align*}"}
+{"id": "5863.png", "formula": "\\begin{align*} H = \\widehat { H } + D ^ { - 1 } B U ~ ( 0 , T ) \\times \\Gamma , \\end{align*}"}
+{"id": "3496.png", "formula": "\\begin{align*} { \\gamma _ { { \\rm { M M S E } } } } = { { { \\bf { \\bar h } } } ^ H } { { \\bf { C } } ^ { - 1 } } { \\bf { \\bar h } } . \\end{align*}"}
+{"id": "2246.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 2 k - 2 } | b _ j ^ { + , n } - b _ j ^ { - , n } | \\to 0 , \\end{align*}"}
+{"id": "6963.png", "formula": "\\begin{gather*} L _ { m , n } ^ { I I , ( \\alpha ) } ( x ) = \\frac { ( - 1 ) ^ { n + 1 } - ( m - \\alpha - 1 ) / ( n - m ) } { m ! ( n - m - 1 ) ! \\prod _ { 1 \\leq i < j \\leq n } \\big ( Y ^ { ( \\alpha ) } _ { j , m , n } - Y ^ { ( \\alpha ) } _ { i , m , n } \\big ) } \\det \\big ( M _ n ^ { I I } \\big ) . \\end{gather*}"}
+{"id": "6786.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\left ( \\frac { \\left | \\sum _ { n = 1 } ^ N n ^ 2 \\chi ( n ) \\right | } { \\left | \\sum _ { n = 1 } ^ N n ^ 2 \\overline { \\chi } ( n ) \\right | } \\right ) = A \\end{align*}"}
+{"id": "7893.png", "formula": "\\begin{align*} f '' + \\left ( H _ 1 e ^ { z } + H _ 2 e ^ { \\rho z } + q ( z ) \\right ) f = 0 , \\end{align*}"}
+{"id": "893.png", "formula": "\\begin{align*} 0 \\leq 1 - \\mathop { \\min } _ { k = 1 , 2 , \\cdots , l } \\frac { \\mathbf { E } [ d _ { k } ] } { n } \\leq 1 - \\mathop { \\min } _ { k = 1 , 2 , \\cdots , l } \\lambda _ { \\min } ( \\mathbf { E } [ \\widehat { \\mathcal { Z } } _ { ( k ) } ] ) < 1 , \\end{align*}"}
+{"id": "6267.png", "formula": "\\begin{align*} \\mathcal { D } _ { 0 } = \\{ ( x , y , z ) \\in \\mathcal { M } _ { 0 } : z > - 1 \\} . \\end{align*}"}
+{"id": "274.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\tau } - \\beta \\rho ^ 2 \\leqslant 0 , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\rho } - \\beta \\tau ^ 2 \\leqslant 0 , \\end{align*}"}
+{"id": "5954.png", "formula": "\\begin{align*} B ^ T E _ r = \\sum _ { s = 1 } ^ p \\beta _ { r s } E _ s , r = 1 , \\cdots , p , \\end{align*}"}
+{"id": "5182.png", "formula": "\\begin{align*} P _ \\ell = \\frac { r - ( m _ 2 ^ { - 1 } m _ 1 ( \\ell + 1 ) + r k _ \\ell - h ) } { K } \\end{align*}"}
+{"id": "78.png", "formula": "\\begin{align*} \\phi _ i ( \\zeta ) = T r _ { K / \\mathbf { F } _ p } ( \\zeta \\cdot \\kappa _ i ^ { \\prime } ) \\end{align*}"}
+{"id": "6755.png", "formula": "\\begin{align*} \\phi ( s ) = \\sqrt { \\frac { \\pi } { 1 6 } } \\left [ \\frac { z ( s ) } { s } + \\frac { z ( 1 - s ) } { 1 - s } \\right ] ; \\mathfrak { R e } ( s ) \\in ( 0 , 1 ) \\end{align*}"}
+{"id": "7268.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } ( - 1 ) ^ k G ^ k _ m ( x ) d m ( x ) \\begin{cases} = \\infty & k < d ( m ) , \\\\ < \\infty & k \\geq d ( m ) . \\end{cases} \\end{align*}"}
+{"id": "4715.png", "formula": "\\begin{align*} M _ { i j } : = v _ i ( j ) . \\end{align*}"}
+{"id": "2825.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 ) = x _ 1 ^ 5 x _ 2 ^ 7 . \\end{align*}"}
+{"id": "7940.png", "formula": "\\begin{align*} 1 \\cdot 1 = ( - 1 ) \\cdot ( - 1 ) = 1 , 1 \\cdot ( - 1 ) = - 1 , 0 \\cdot ( - 1 ) = 0 \\cdot 0 = 0 \\cdot 1 = 0 , \\end{align*}"}
+{"id": "1607.png", "formula": "\\begin{align*} \\big ( \\zeta \\times \\zeta ' \\big ) ( u _ i , u _ j ) = \\zeta ( u _ i ) \\zeta ' ( u _ j ) \\leq \\zeta ( u _ i ) = a _ i - m _ i \\leq | a _ i - c _ i | \\leq \\frac { \\widetilde \\varepsilon ^ p } { K ^ p L ^ 2 } . \\end{align*}"}
+{"id": "185.png", "formula": "\\begin{align*} p ( n ) = p ( m ) + \\sum _ { \\ell = m } ^ { n - 1 } | \\mathcal { L } ^ { R S } _ \\ell ( X ) | . \\end{align*}"}
+{"id": "2262.png", "formula": "\\begin{align*} ( a + \\varpi _ D b ) \\cdot X = a X + p \\sigma ( b ) Y , \\ \\ \\ ( a + \\varpi _ D b ) \\cdot Y = b X + \\sigma ( a ) Y . \\end{align*}"}
+{"id": "3445.png", "formula": "\\begin{align*} \\mu _ { \\infty } ( [ \\underline { \\omega } ] ) = \\delta _ { \\overline { 1 } } ( [ \\underline { \\omega } ] ) , \\mbox { f o r a l l } \\underline { \\omega } \\in \\mathcal { W } . \\end{align*}"}
+{"id": "4639.png", "formula": "\\begin{align*} N _ { \\mathrm { r a m } \\ , p _ Y } ( r ) = N ( r , R _ 1 ) + N ( r , R _ 2 ) \\end{align*}"}
+{"id": "9267.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { g } _ i & = ( x _ i - i ) \\prod _ { k = i + 1 } ^ { h ( i ) } ( x _ i - x _ k - 1 ) \\cdot \\prod _ { \\ell = i } ^ { h ( i - 1 ) } \\left ( \\prod _ { j = j _ { \\ell } } ^ { i - 1 } ( x _ j - x _ { \\ell } - 1 ) \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "5581.png", "formula": "\\begin{align*} X ^ a ( \\nabla _ { a } T _ { c _ { 1 } \\cdots c _ { r } } ) Q ^ { c _ { 1 } \\dots c _ { r } } = \\pi ( X ^ a \\nabla _ { a } k ^ { b } ) \\ , \\langle T \\vert k \\vert Q \\rangle _ { b } . \\end{align*}"}
+{"id": "7917.png", "formula": "\\begin{align*} & \\lim _ { \\epsilon \\to 0 } \\limsup _ { t \\to \\infty } \\mathbb { P } \\left ( \\exists u \\in \\mathcal { N } ^ 2 _ t : X _ u ( t ) \\geq m _ t - A , X _ { u } ( T ( u ) ) - \\sqrt { 2 } T ( u ) \\notin [ - \\frac { 1 } { \\epsilon } \\sqrt { t } , - \\epsilon \\sqrt { t } ] \\right ) = 0 \\end{align*}"}
+{"id": "266.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\tau } - \\beta \\rho ^ 2 \\leqslant 0 , \\end{align*}"}
+{"id": "8648.png", "formula": "\\begin{align*} \\nabla _ k \\Psi & = \\sum _ { i \\geq k } \\frac { \\Psi } { F _ i } f ' ( t _ i ) \\nabla _ k t _ i = \\sum _ { i \\geq k } \\frac { \\Psi } { F _ i } f ' ( t _ i ) \\nabla _ k d ( x _ i , x _ { i ^ * } ) , \\end{align*}"}
+{"id": "1722.png", "formula": "\\begin{align*} \\kappa ( a , b ) : = \\left ( \\begin{array} { c c } a & b \\\\ - \\bar b & \\bar a \\end{array} \\right ) . \\end{align*}"}
+{"id": "4023.png", "formula": "\\begin{align*} G _ p \\cdot \\gamma = \\chi ^ { - 1 } \\phi ^ * _ k \\phi ^ * _ l Y ^ l _ { p _ j } Y ^ k _ { p _ i } w _ { i j } . \\end{align*}"}
+{"id": "2623.png", "formula": "\\begin{align*} ( a _ { 2 , 3 } , a _ { 2 , 4 } , a _ { 3 , 4 } , a _ { 1 , 2 , 3 } , a _ { 1 , 2 , 4 } , a _ { 1 , 3 , 4 } , a _ { 2 , 3 , 4 } ) = ( 6 , 6 , 6 , 3 , 3 , 3 , 3 ) . \\end{align*}"}
+{"id": "2482.png", "formula": "\\begin{align*} \\delta _ n = \\inf _ { 0 \\neq x \\in \\mathcal H _ n } \\frac { \\| X _ T x \\| } { \\| x \\| } , \\ \\ \\ n \\geq 0 . \\end{align*}"}
+{"id": "8603.png", "formula": "\\begin{align*} D _ a = \\ell _ 1 + \\ell _ 1 ' , ~ D _ b = \\ell _ 2 + \\ell _ 2 ' , ~ D _ c = \\ell _ 3 + \\ell _ 3 ' . \\end{align*}"}
+{"id": "8633.png", "formula": "\\begin{align*} O _ k = \\bigcup _ { \\lambda \\in L _ k } \\bigcup _ { i \\in J _ k } B a d ( \\lambda , k , i ) \\end{align*}"}
+{"id": "1872.png", "formula": "\\begin{align*} \\alpha _ { s _ i } ( \\sigma ) = \\frac { \\pi } { 2 } . \\end{align*}"}
+{"id": "579.png", "formula": "\\begin{align*} \\mu _ { n + 1 } = \\mu _ n + K _ n \\left \\{ ( X _ { t _ { n + 1 } } ^ \\dagger - X _ { t _ n } ^ \\dagger ) - \\mu _ n A X _ { t _ n } ^ \\dagger \\Delta t \\right \\} \\end{align*}"}
+{"id": "7816.png", "formula": "\\begin{align*} \\frac { q ^ { 2 j \\langle n , \\tilde m \\rangle _ 1 } - 1 } { q ^ { 2 j a } - 1 } & = \\frac { q ^ { 2 \\alpha j / s d ( n ) } - 1 } { q ^ { 2 j / s d ( n ) } - 1 } = \\begin{cases} \\displaystyle \\sum _ { p = 0 } ^ { \\alpha - 1 } q ^ { 2 j p / s d ( n ) } & \\alpha > 0 , \\\\ 0 & \\alpha = 0 , \\\\ \\displaystyle - \\sum _ { p = 1 } ^ { \\alpha } q ^ { - 2 j p / s d ( n ) } & \\alpha < 0 . \\end{cases} \\end{align*}"}
+{"id": "9019.png", "formula": "\\begin{align*} C _ { \\Z _ p [ [ \\Gamma ] ] } ( X ( T _ f / \\Q _ \\infty ) ) = C _ { \\Z _ p [ [ \\Gamma ] ] } ( X ( T _ f ^ \\ast / \\Q _ \\infty ) ^ \\iota ) , \\end{align*}"}
+{"id": "30.png", "formula": "\\begin{align*} P ( \\mathbf N ) : = \\bigl \\{ \\mathbf n = ( n _ 1 , \\dots , n _ d ) : \\ , n _ j \\in \\N _ 0 , 0 \\le n _ j \\le 2 N _ j , \\ , j = 1 , \\dots , d \\bigr \\} , \\end{align*}"}
+{"id": "6419.png", "formula": "\\begin{align*} \\phi ( X _ 1 , \\tilde Y - ( \\tilde t - t ) X , \\tilde t ) & = \\phi ( X , Y , t ) + \\nabla _ X \\phi ( X , Y , t ) \\cdot ( X _ 1 - X ) \\\\ & + \\frac 1 2 \\langle \\nabla _ X ^ 2 \\phi ( X , Y , t ) ( X _ 1 - X ) , ( X _ 1 - X ) \\rangle \\\\ & - ( \\tilde t - t ) \\bigl ( X \\cdot \\nabla _ Y - \\partial _ t \\bigr ) \\phi ( X , Y , t ) + { o } ( \\epsilon ^ 2 ) , \\end{align*}"}
+{"id": "6156.png", "formula": "\\begin{align*} C _ 1 B = \\overline B _ 1 C _ 1 , \\end{align*}"}
+{"id": "4188.png", "formula": "\\begin{align*} \\begin{aligned} & ~ { } G : = - \\left ( \\partial _ { t } ^ 2 ( \\ln \\alpha ) - \\partial _ { x } ^ 2 ( \\ln \\alpha ) \\right ) - \\dfrac { 1 } { 2 \\alpha ^ 2 } ( ( \\partial _ t \\alpha ) ^ 2 - ( \\partial _ x \\alpha ) ^ 2 ) \\\\ & ~ { } - \\dfrac { 1 } { 2 } ( ( \\partial _ t \\Lambda ) ^ 2 - ( \\partial _ x \\Lambda ) ^ 2 ) - 2 \\sinh ^ 2 \\Lambda ( ( \\partial _ t \\phi ) ^ 2 - ( \\partial _ x \\phi ) ^ 2 ) . \\end{aligned} \\end{align*}"}
+{"id": "9096.png", "formula": "\\begin{align*} \\psi ( X , Y ) = \\phi ( X , Y ) + \\phi ( J X , J Y ) , X , Y \\in \\mathfrak { n } , \\end{align*}"}
+{"id": "1044.png", "formula": "\\begin{align*} A _ j = [ ( j - 1 ) M / 3 , j M / 3 ) , j \\in \\mathcal { J } = \\{ - 3 T / M , - 3 T / M + 1 , \\ldots , 3 T / M , 3 T / M + 1 \\} . \\end{align*}"}
+{"id": "7721.png", "formula": "\\begin{align*} \\Phi ^ { ( 1 ) } ( \\tau , 1 + \\rho e ^ { i \\beta \\pi } ) \\sim \\frac { \\pi } { \\sqrt { \\tau } } \\rho ^ { - \\frac { 1 } { 2 } } \\left [ 1 + \\sqrt { 2 } C _ { \\beta } \\left ( \\frac { \\tau } { \\rho } \\right ) ^ { \\frac { 1 } { 4 } } \\right ] ^ { - 2 n } = : g ^ { ( 1 ) } ( \\tau , \\rho ) , \\end{align*}"}
+{"id": "60.png", "formula": "\\begin{align*} y _ i = \\begin{cases} 1 , & f ( i ) = 2 \\\\ 0 , & o t h e r w i s e . \\end{cases} \\end{align*}"}
+{"id": "2451.png", "formula": "\\begin{align*} C ( \\pi , t ) = q _ { \\pi } \\prod _ { j = 1 } ^ { m } ( 3 + | i t + \\mu _ { \\pi } ( j ) | ) , C ( \\pi ) = C ( \\pi , 0 ) . \\end{align*}"}
+{"id": "7447.png", "formula": "\\begin{align*} \\{ \\| \\Gamma _ { 1 , k } ( Z , Z ) \\| , \\ , \\| \\Gamma _ { 2 , k } ( Z , Z ) \\| \\colon Z \\in \\Omega , \\ , k = 1 , 2 , \\dots \\} \\end{align*}"}
+{"id": "2464.png", "formula": "\\begin{align*} \\alpha = \\frac { \\log ( \\frac { e \\log q } { 1 0 0 \\lambda c _ m } ) } { 4 \\times 1 0 ^ 7 m ^ 4 \\log ( C ( \\pi ) ( | t | + 6 ) ^ m ) } . \\end{align*}"}
+{"id": "4509.png", "formula": "\\begin{align*} \\log \\prod _ { i = 1 } ^ n v _ i = \\sum _ { i = 1 } ^ n \\log v _ i = \\sum _ { i = 1 } ^ n b _ i = \\sum _ { i = 1 } ^ n a _ i = \\sum _ { i = 1 } ^ n \\log u _ i = \\log \\prod _ { i = 1 } ^ n u _ i . \\end{align*}"}
+{"id": "174.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq n \\\\ p \\notin \\mathcal { S } } } \\beta _ p \\log p \\leq \\log ( 2 n ^ 2 + l ) \\sum _ { \\substack { p \\leq n \\\\ p \\in \\mathcal { S } } } 1 + \\dfrac { 1 } { \\lambda } \\sum _ { i = 1 } ^ { r } e _ i \\log p _ i + n \\log 4 . \\end{align*}"}
+{"id": "1545.png", "formula": "\\begin{align*} \\frac 1 2 \\rho _ { i } ^ { a b } \\sigma _ { a b } = \\sigma _ i \\end{align*}"}
+{"id": "6555.png", "formula": "\\begin{align*} \\mathbb { P } _ { M D } \\ ! = \\ ! \\Pr \\left ( { { \\kappa ^ 2 } \\ ! + \\ ! { C _ { 1 w } } X _ w \\ ! + \\ ! { C _ { 2 w } } Z _ w \\ ! < \\ ! { \\varepsilon _ { k } } } \\right ) \\ ! = \\Pr \\left ( { T _ w \\ ! + \\ ! { C _ { 1 w } } X _ w \\ ! < \\ ! { \\varepsilon _ { k } } } \\right ) . \\end{align*}"}
+{"id": "2527.png", "formula": "\\begin{align*} M _ g ^ \\varphi ( x , v , t ) = M _ { u _ g ^ \\varphi ( x , t ) , T _ g ^ \\varphi ( x , t ) } ( v ) \\ , , \\end{align*}"}
+{"id": "8313.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { C } _ s = \\{ x , y , z \\ | & ( x - a ) ^ 2 + ( y - b ) ^ 2 + z ^ 2 \\\\ & = ( x _ 1 - a ) ^ 2 + ( y _ 1 - b ) ^ 2 + z _ 1 ^ 2 \\} , \\end{array} \\right . \\end{align*}"}
+{"id": "3414.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } \\tilde \\omega _ { q } - \\mu \\Delta \\widetilde \\omega _ { q } + ( v \\cdot \\nabla ) \\widetilde \\omega _ { q } = \\frac { v ^ r } { r } \\widetilde \\omega _ { q } - \\partial _ { z } \\Big ( \\Delta _ { q } \\Big ( \\frac { B _ { \\theta } } { r } B \\Big ) \\Big ) , \\\\ \\widetilde \\omega _ { q } \\vert _ { t = 0 } = \\Delta _ { q } \\omega _ { 0 } . \\end{array} \\right . \\end{align*}"}
+{"id": "7060.png", "formula": "\\begin{align*} G r _ n M U ^ { C _ 2 } _ * = \\Omega ^ { C _ 2 } _ { * - n \\sigma } / \\Omega ^ { C _ 2 } _ { * - ( n - 1 ) \\sigma } & \\cong \\dfrac { \\Omega ^ { C _ 2 } _ * \\{ u ^ n \\} } { \\begin{matrix} u ^ n d _ { i , j + 1 } = 0 \\\\ u ^ n q _ { j + 1 } = 0 \\end{matrix} } \\\\ & \\cong M U _ * [ d _ { i , 0 } : i \\geq 1 ] \\{ u ^ n \\} , \\end{align*}"}
+{"id": "5942.png", "formula": "\\begin{align*} \\phi _ r = ( E _ r , U ) , r = 1 , \\cdots , p \\end{align*}"}
+{"id": "957.png", "formula": "\\begin{align*} \\textrm { g r a d e } _ R ( \\textrm { E x t } _ { R } ^ n ( M ' , \\ , C ) ) \\ , = \\ , n + \\textrm { r . g r a d e } _ R ( \\textrm { T r } _ C ( M ' ) , C ) - 1 . \\end{align*}"}
+{"id": "8587.png", "formula": "\\begin{align*} \\Theta : = \\{ \\alpha _ 1 , \\ , \\ldots , \\ , \\alpha _ { 2 ^ n - 1 } \\ ; : \\ ; \\alpha _ i \\in V \\mbox { a n d } \\alpha _ i \\ne \\alpha _ j \\mbox { f o r } i \\ne j \\} , \\end{align*}"}
+{"id": "3591.png", "formula": "\\begin{align*} d _ { j } ^ - ( \\lambda ) = \\frac { ( \\lambda _ j + 1 ) j ( \\lambda _ j + n - j + [ \\lambda _ j ] _ 2 ( p - n ) ) } { ( - 1 ) ^ { \\sum _ { \\alpha = j } ^ n ( \\alpha + 1 ) ( \\lambda _ \\alpha - \\lambda _ { \\alpha + 1 } ) } } \\bigg ( \\prod _ { \\ell = j + 1 } ^ n \\frac { \\lambda _ j - \\lambda _ \\ell - j + \\ell - 1 } { \\lambda _ j - \\lambda _ \\ell - j + \\ell - [ \\lambda _ j - \\lambda _ \\ell ] _ 2 } \\bigg ) . \\end{align*}"}
+{"id": "7515.png", "formula": "\\begin{align*} Z _ { g _ 1 } ( s , \\chi , D _ { S ( g _ 1 , D ) } ) = q ^ { - 1 } Z _ { g _ 2 } ( s , \\chi , D ) , \\end{align*}"}
+{"id": "733.png", "formula": "\\begin{align*} * d U _ \\alpha = Q \\ , d U _ \\beta . \\end{align*}"}
+{"id": "819.png", "formula": "\\begin{align*} \\frac { 1 } { n ^ { 1 / 4 } } \\sum _ { k = 0 } ^ { n ^ { 1 / 4 } } \\widetilde { \\nu } _ k ( E _ { n , t _ 1 , \\ldots , t _ d } ( x ) ) = F _ { t _ 1 , \\ldots , t _ d } ( x ) . \\end{align*}"}
+{"id": "2465.png", "formula": "\\begin{align*} \\omega ( R ) = \\left ( \\ln \\ln \\frac { 1 0 0 } { R } \\right ) ^ { - 1 } . \\end{align*}"}
+{"id": "5899.png", "formula": "\\begin{align*} \\widetilde C _ { p - 1 } = \\begin{pmatrix} C _ p \\\\ c _ { p + 1 } \\end{pmatrix} . \\end{align*}"}
+{"id": "4521.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n v _ i \\leq \\sum _ { i = 1 } ^ n u _ i . \\end{align*}"}
+{"id": "6034.png", "formula": "\\begin{align*} \\forall 1 \\leq i , j , k \\leq n , \\ : i \\neq j , \\ : k \\neq p , \\ : \\ : \\ : \\ : & t _ 0 ( w _ { i , j } , w _ { k , p } ) = w _ { ( i + k - 1 ) m o d ( n ) + 1 , ( j + p - 2 ) m o d ( n ) + 1 } \\\\ & t _ 1 ( w _ { i , j } , w _ { k , p } ) = w _ { ( i + k - 2 ) m o d ( n ) + 1 , ( j + p - 2 ) m o d ( n ) + 1 } \\\\ & t _ 2 ( w _ { i , j } , w _ { k , p } ) = w _ { ( i + k - 1 ) m o d ( n ) + 1 , ( j + p - 1 ) m o d ( n ) + 1 } \\\\ & t _ 3 ( w _ { i , j } , w _ { k , p } ) = w _ { ( i + k - 2 ) m o d ( n ) + 1 , ( j + p - 1 ) m o d ( n ) + 1 } . \\end{align*}"}
+{"id": "9001.png", "formula": "\\begin{align*} \\| v \\| ^ 4 _ { L ^ 4 ( B ) } \\le C \\| v \\| ^ 2 _ { H ^ 1 ( B ) } \\| v \\| ^ 2 _ { L ^ 2 ( B ) } \\end{align*}"}
+{"id": "4971.png", "formula": "\\begin{align*} \\xi : = \\left \\{ \\begin{array} { l l } q _ 1 ^ { m p } & h ^ { p - 1 } \\in f ^ l k [ f , g ] \\\\ q _ 1 ^ { m p } - f ^ { l ( p - 1 ) } \\eta ( q _ 1 ^ { m t - 1 } ) q _ 1 ^ m & \\end{array} \\right . \\end{align*}"}
+{"id": "2093.png", "formula": "\\begin{align*} & \\chi _ i : \\mathcal { B } \\to W _ i , \\\\ & \\chi _ i ( n ) = c _ { i , n } \\end{align*}"}
+{"id": "5198.png", "formula": "\\begin{align*} Q _ { i j } = 4 \\langle H _ a a ' , B _ i \\times B _ j \\rangle + \\langle B _ i ' , B _ j \\rangle - \\langle B _ i , B _ j ' \\rangle \\ . \\end{align*}"}
+{"id": "8998.png", "formula": "\\begin{align*} d \\pi _ N ( w _ { \\infty } ) \\partial _ y w _ { \\infty } = 0 \\ \\hbox { o n } \\partial \\R ^ 2 _ + . \\end{align*}"}
+{"id": "2257.png", "formula": "\\begin{align*} x = ( k _ 1 - 1 ) y _ 1 + \\cdots + ( k _ r - 1 ) y _ r . \\end{align*}"}
+{"id": "7547.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A ) = q ^ { - i - j - k } \\int _ { \\tilde A } \\chi ( a c f ( \\pi ^ i x , \\pi ^ j y , \\pi ^ k z ) ) | f ( \\pi ^ i x , \\pi ^ j y , \\pi ^ k z ) | ^ s | d x d y d z | , \\end{align*}"}
+{"id": "9007.png", "formula": "\\begin{align*} \\alpha ( A _ { i _ { n + 1 } } C _ { n + 1 } , K _ { n + 1 } ) \\ , & \\leq \\ , \\frac { 1 } { 1 - \\epsilon } \\max _ { g \\in C _ { n + 1 } } \\alpha ( A _ { i _ { n + 1 } } g , K _ { n + 1 } ) \\\\ & = \\ , \\frac { \\alpha ( A _ { i _ { n + 1 } } , K _ { n + 1 } ) } { 1 - \\epsilon } \\\\ & \\leq \\ , \\frac { \\epsilon ^ { 2 ( N - n ) + 4 } } { 1 - \\epsilon } \\ , \\leq \\ , \\epsilon ^ { 2 ( N - n ) + 1 } . \\end{align*}"}
+{"id": "6567.png", "formula": "\\begin{align*} L _ s ( q , r ) : = \\sum _ { \\substack { p \\mid \\frac { q r } { \\gcd ( q , r ) ^ 2 } , \\\\ p \\geq s } } \\frac { 1 } { p } , \\end{align*}"}
+{"id": "5871.png", "formula": "\\begin{align*} t = 0 : U = 0 , U ' = e \\theta \\end{align*}"}
+{"id": "6428.png", "formula": "\\begin{align*} v _ { k + 2 } ( X , Y , t ) - v _ { k + 1 } ( X , Y , t ) = \\mathcal { T } ( \\mathcal { T } v _ { k } ) ( X , Y , t ) - ( \\mathcal { T } v _ { k } ) ( X , Y , t ) . \\end{align*}"}
+{"id": "1789.png", "formula": "\\begin{align*} x : = [ a ] + [ b ] \\xi \\end{align*}"}
+{"id": "512.png", "formula": "\\begin{align*} \\sum _ { K _ 2 \\cup K _ { 3 1 } \\ni j = k } ^ { \\nu } \\ ! \\big \\| x ^ { j + 1 } \\ ! - \\ ! x ^ { j } \\big \\| \\le \\left \\{ \\begin{array} { c l } \\sqrt { 2 { a } ^ { - 1 } } \\ , \\widetilde { \\gamma } \\widetilde { \\tau } ^ k & { \\rm i f } \\ \\theta \\in ( 0 , 1 / 2 ] , \\\\ \\ ! \\sqrt { { 2 } { a } ^ { - 1 } } \\ , \\widetilde { \\gamma } k ^ { \\frac { 1 - \\theta } { 1 - 2 \\theta } } & { \\rm i f } \\ \\theta \\in ( 1 / 2 , 1 ) . \\end{array} \\right . \\end{align*}"}
+{"id": "6423.png", "formula": "\\begin{align*} & \\phi ( \\tilde X _ 1 , \\tilde Y - ( \\tilde t - t ) X _ 1 , \\tilde t ) + \\phi ( X _ 1 , \\tilde Y - ( \\tilde t - t ) X _ 1 , \\tilde t ) - 2 \\phi ( X , Y , t ) \\\\ & = \\langle \\nabla _ X ^ 2 \\phi ( X , Y , t ) ( X _ 1 - X ) , ( X _ 1 - X ) \\rangle - 2 ( X \\cdot \\nabla _ Y - \\partial _ t ) \\phi ( X , Y , t ) ( \\tilde t - t ) + { o } ( \\epsilon ^ 2 ) , \\mbox { a s } \\epsilon \\to 0 . \\end{align*}"}
+{"id": "2817.png", "formula": "\\begin{align*} u _ { n + 1 } = \\min ( v _ n , u _ n ) \\frac { \\prod _ { i = 1 } ^ n v _ i } { \\prod _ { i = 1 } ^ n u _ i } \\leq \\min ( v _ n , u _ n ) = v _ { n + 1 } \\end{align*}"}
+{"id": "2340.png", "formula": "\\begin{align*} \\Omega : = [ - \\tfrac 1 2 , \\tfrac 1 2 ] \\times [ 0 , 1 ] \\ni ( k , s ) . \\end{align*}"}
+{"id": "6092.png", "formula": "\\begin{gather*} E _ { \\frac { 3 } { 2 } } ^ \\ast ( \\tau ) = \\sum _ { D \\geq 0 } H ( D ) e ^ { 2 \\pi i D \\tau } + \\frac { 1 } { 1 6 \\pi } \\sum _ { n \\in \\Z } v ^ { - \\frac { 1 } { 2 } } \\beta _ { \\frac { 3 } { 2 } } ( 4 \\pi n ^ { 2 } v ) e ^ { - 2 \\pi i n ^ { 2 } \\tau } , \\ , v = \\Im ( \\tau ) , \\end{gather*}"}
+{"id": "8641.png", "formula": "\\begin{align*} H _ N = \\sum _ { i = 1 } ^ N ( - \\Delta _ { x _ i } ) + \\sum _ { 1 \\leq i < j \\leq N } V ( x _ i - x _ j ) . \\end{align*}"}
+{"id": "7266.png", "formula": "\\begin{align*} G ^ k _ m ( x ) = - \\int _ { 0 } ^ { x } d y \\int _ { y } ^ { 1 } G ^ { k - 1 } _ m ( z ) d m ( z ) . \\end{align*}"}
+{"id": "8173.png", "formula": "\\begin{align*} \\tilde { \\mathcal { D } } & = \\frac { 3 L ( 1 , f ) \\bigl ( A ( 1 , p ) p - 1 \\bigr ) } { 2 p ^ { \\frac { 3 } { 2 } } \\pi } \\int _ { - \\infty } ^ { \\infty } k ( t ) \\tanh ( \\pi t ) t \\log \\abs { t } \\ , d t + { } \\\\ & { } + \\frac { \\tilde K \\bigl ( A ( p , 1 ) p - 1 \\bigr ) } { 2 p ^ { \\frac { 3 } { 2 } } \\pi } \\int _ { - \\infty } ^ { \\infty } k ( t ) \\tanh ( \\pi t ) t \\ , d t + O ( T ^ { \\frac { 1 } { 7 } + \\varepsilon } M p ^ { \\varepsilon } ) \\end{align*}"}
+{"id": "7628.png", "formula": "\\begin{align*} J _ 0 ( t ) = \\int _ { 0 } ^ { 1 } G ( t , x , y ) u _ 0 ( y ) \\d y , \\ ; t \\in [ 0 , T ] \\end{align*}"}
+{"id": "635.png", "formula": "\\begin{align*} B = \\{ 1 \\leq s \\leq h : \\beta _ i ^ { \\mathbb { Z } _ p } ( I ^ s ) \\neq \\beta _ i ^ \\mathbb { Q } ( I ^ s ) \\} . \\end{align*}"}
+{"id": "236.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\beta \\rho \\eta - \\alpha \\tau ^ 2 + \\alpha \\leqslant \\alpha , \\end{align*}"}
+{"id": "3065.png", "formula": "\\begin{align*} \\bar { a } : = \\int _ Y r a , \\end{align*}"}
+{"id": "7392.png", "formula": "\\begin{align*} \\gamma ' : = \\begin{cases} 3 \\alpha + \\beta & \\cr 3 \\alpha + 2 \\beta & \\end{cases} \\end{align*}"}
+{"id": "4599.png", "formula": "\\begin{align*} a _ 1 = q + 1 \\end{align*}"}
+{"id": "5109.png", "formula": "\\begin{align*} \\Omega _ { \\mathbf { m } } ^ { \\pm } ( b ) = \\frac { 1 - b ^ { 2 } } { 4 } \\pm \\frac { 1 } { 2 \\mathbf { m } } \\sqrt { \\left ( \\frac { \\mathbf { m } ( 1 - b ^ { 2 } ) } { 2 } - 1 \\right ) ^ { 2 } - b ^ { 2 \\mathbf { m } } } . \\end{align*}"}
+{"id": "722.png", "formula": "\\begin{align*} \\mathcal { H } ( w + u , w - u ) = \\frac { \\Gamma _ 1 ^ 2 } { 2 \\pi } \\big ( \\log \\lambda ( w ) + \\log | 2 u | \\big ) + \\mathcal { O } ( 1 ) ( u \\to 0 ) . \\end{align*}"}
+{"id": "7108.png", "formula": "\\begin{align*} f = \\alpha _ 1 + \\sum _ { | m | < n } \\alpha _ m m \\end{align*}"}
+{"id": "8956.png", "formula": "\\begin{align*} R _ { k } ^ { ( \\alpha , \\beta ) } ( u ) = \\left ( \\frac { 1 + u } { 2 } \\right ) ^ { k } { } _ { 2 } F { } _ { 1 } \\left ( \\begin{array} { c } - k , - k - \\beta \\\\ \\alpha + 1 \\end{array} \\big | \\frac { 1 - u } { 1 + u } \\right ) . \\end{align*}"}
+{"id": "1857.png", "formula": "\\begin{align*} \\| f _ 1 \\cdots f _ i \\| _ { H ^ s } \\leq C \\prod ^ l _ { i = 1 } \\| f _ i \\| _ { H ^ s } . \\end{align*}"}
+{"id": "9192.png", "formula": "\\begin{align*} \\frac { \\tilde { Z } _ N ( u , 0 ) } { \\tilde { Z } _ N ( 0 , 0 ) } = \\exp ( e _ N ) \\frac { 1 + C ( M ) O ( \\norm { W _ N } _ { \\Omega _ N ^ U } + \\norm { K _ N ( \\cdot ; ( \\Psi _ k ) _ { k < N } ) } _ { \\Omega _ N ^ K } ) } { 1 + C ( M ) O ( \\norm { W _ N } _ { \\Omega _ N ^ U } + \\norm { K _ N ( \\cdot ; 0 ) } _ { \\Omega _ N ^ K } ) } . \\end{align*}"}
+{"id": "8948.png", "formula": "\\begin{align*} ( a ) _ { k } = a ( a + 1 ) \\cdots ( a + k - 1 ) , \\end{align*}"}
+{"id": "6833.png", "formula": "\\begin{align*} \\mu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) = \\mu _ c ^ { \\mp } ( Y _ m , Y _ { m - 1 } , \\dots , Y _ 1 ) , \\nu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) = \\nu _ c ^ { \\mp } ( Y _ m , Y _ { m - 1 } , \\dots , Y _ 1 ) . \\end{align*}"}
+{"id": "9020.png", "formula": "\\begin{align*} \\dim H ^ 1 _ g ( G _ { v _ n } , V ( \\varphi ) ) = \\dim H ^ 1 ( G _ { v _ n } / I _ { v _ n } , V ( \\varphi ) ^ { I _ { v _ n } } ) = H ^ 0 ( G _ { v _ n } , V ( \\varphi ) ) . \\end{align*}"}
+{"id": "800.png", "formula": "\\begin{align*} \\mu _ j ( F ( t , n ) ) = \\P _ { \\mu _ { e , j } } \\left \\{ ( ( x _ 0 , x _ 1 ) , \\ldots ) \\in E _ j : \\frac { \\log \\| \\rho ( \\lambda ( x _ 0 , x _ 1 ) \\ldots \\lambda ( x _ { n - 1 } , x _ n ) ) \\| - \\Lambda n } { \\sqrt { n } } < t \\right \\} . \\end{align*}"}
+{"id": "3724.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\textrm { B K } } ( X _ P ( F ) ) : = \\mathcal { S } ( X _ P ^ { \\circ } ( F ) ) + \\mathcal { F } _ { P ^ { \\mathrm { o p } } | P } ( \\mathcal { S } ( X _ { P ^ { \\mathrm { o p } } } ^ { \\circ } ( F ) ) ) , \\end{align*}"}
+{"id": "8964.png", "formula": "\\begin{align*} d \\pi _ N ( \\bar { u } ^ { ( i ) } ) \\partial _ r \\bar { u } ^ { ( i ) } = 0 , \\ , 1 \\le i \\le i _ 0 . \\end{align*}"}
+{"id": "8154.png", "formula": "\\begin{align*} U _ { \\mp } ( y , t ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 0 0 0 ) } y ^ { - u } F ( u ) \\frac { \\gamma _ { \\mp } \\Bigl ( \\dfrac { 1 } { 2 } + u , t \\Bigr ) } { \\gamma _ - \\Bigl ( \\dfrac { 1 } { 2 } , t \\Bigr ) } \\frac { d u } { u ^ 2 } . \\end{align*}"}
+{"id": "4831.png", "formula": "\\begin{align*} \\Phi ( 0 , s t ) = \\sum _ { j = 1 } ^ { n - 1 } \\frac { s ^ j } { j ! } w _ j ( t ) , \\end{align*}"}
+{"id": "7172.png", "formula": "\\begin{align*} \\mathcal { C } = \\mathbb { R } \\times C . \\end{align*}"}
+{"id": "2239.png", "formula": "\\begin{align*} \\mathcal { S } : = \\{ \\theta = \\pi \\} = \\{ x _ 1 \\leq 0 , x _ { 2 } = 0 \\} . \\end{align*}"}
+{"id": "6793.png", "formula": "\\begin{align*} d ( \\mathcal { U } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) , \\mathcal { U } ( ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) \\leq ~ \\beta ( L ^ { * } _ { \\eta } ( \\alpha , ( w _ { \\i } ) _ { \\i = 1 } ^ { \\eta } , ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) ) L ^ { * } _ { \\eta } ( \\alpha , ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } , ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) \\end{align*}"}
+{"id": "1530.png", "formula": "\\begin{align*} p ^ i _ A = \\dd _ { v ^ A _ i } L \\end{align*}"}
+{"id": "6370.png", "formula": "\\begin{align*} Q _ \\lambda ( u ) = ( 1 + F _ \\lambda ' ( \\cos u ) ) ^ 2 + \\lambda \\cos u \\cdot \\sin ^ 2 2 u \\cdot ( 1 + F _ \\lambda ' ( \\cos u ) ) / 4 . \\end{align*}"}
+{"id": "5.png", "formula": "\\begin{align*} J ( u ) = \\mathbb { E } \\bigg [ \\int _ 0 ^ \\infty e ^ { - \\beta t } \\mathcal { Z } _ t f ( x _ t ^ u , y _ t ^ u , z _ t ^ u , \\tilde { z } _ t ^ u , \\int _ { \\mathcal { E } } \\gamma _ { ( t , e ) } ^ u \\nu ( d e ) , u _ t ) d t + \\phi ( y _ 0 ^ u ) \\bigg ] , \\end{align*}"}
+{"id": "611.png", "formula": "\\begin{align*} { \\rm d } Z _ t ^ \\dagger = \\frac { 1 } { \\delta } ( X _ t ^ \\dagger - Z _ t ^ \\dagger ) { \\rm d } t , \\end{align*}"}
+{"id": "9150.png", "formula": "\\begin{align*} \\varphi _ \\kappa \\ , \\le \\varphi _ \\beta ^ p \\ , \\varphi _ \\alpha ^ { 1 - p } = \\varphi _ \\beta ^ { \\frac { \\alpha - \\kappa } { \\alpha - \\beta } } \\ , \\varphi _ \\alpha ^ { \\frac { \\kappa - \\beta } { \\alpha - \\beta } } \\qquad \\varphi _ \\beta \\ge \\varphi _ \\kappa ^ { \\frac { 1 } { p } } \\varphi _ \\alpha ^ { 1 - \\frac { 1 } { p } } = \\varphi _ \\kappa ^ { 1 + \\frac { \\kappa - \\beta } { \\alpha - \\kappa } } \\varphi _ \\alpha ^ { - \\frac { \\kappa - \\beta } { \\alpha - \\kappa } } . \\end{align*}"}
+{"id": "4768.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} 4 & - 1 \\\\ - 1 & 2 \\end{matrix} \\right ) \\left ( \\begin{matrix} \\omega [ m _ 1 ] \\\\ \\omega [ m _ 2 ] \\end{matrix} \\right ) = \\left ( \\begin{matrix} 1 \\\\ 1 \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "6796.png", "formula": "\\begin{align*} & B _ \\eta \\left ( \\left ( \\overline { x } _ { n + \\i , \\widehat { n + \\eta } } \\right ) _ { \\i = 1 } ^ { \\infty } , \\left ( \\overline { x } _ { n + \\i + 1 , \\widehat { n + \\eta + 1 } } \\right ) _ { \\i = 1 } ^ { \\infty } \\right ) \\\\ & = d \\left ( \\overline { x } _ { n + \\eta } , \\overline { x } _ { n + \\eta + 1 } \\right ) \\end{align*}"}
+{"id": "6047.png", "formula": "\\begin{align*} w ( z ) : = \\sqrt { ( z - a _ 1 ) ( z - b _ 1 ) \\cdots ( z - a _ { g + 1 } ) ( z - b _ { g + 1 } ) } \\end{align*}"}
+{"id": "4306.png", "formula": "\\begin{align*} k ( t , \\xi ) = \\exp \\left ( - \\int ^ t _ 0 \\alpha ( \\tau , \\xi ) \\ , d \\tau + \\eta | \\xi | ^ { \\frac { 1 } { s } } \\right ) . \\end{align*}"}
+{"id": "3784.png", "formula": "\\begin{align*} ( \\iota \\tilde \\alpha ) ( v ) & = \\iota ( \\mathcal { E } _ \\alpha ) _ \\# \\alpha ( v ) = ( \\mathcal { E } _ \\alpha ) _ \\# ^ { - 1 } ( \\iota \\alpha ) ( v ) = ( \\mathcal { E } _ \\alpha ) _ \\# ^ { - 1 } ( \\alpha \\iota ) ( v ) \\end{align*}"}
+{"id": "5828.png", "formula": "\\begin{align*} U '' + \\mathcal L U + A U + D \\mathcal G U ' = 0 . \\end{align*}"}
+{"id": "3881.png", "formula": "\\begin{align*} A ( x , u , D u ) & = g _ { x x } ( x , Y ( x , u , D u ) , Z ( x , u , D u ) ) , \\\\ B ( x , u , D u ) & = \\det E \\ \\psi ( x , u , D u ) . \\end{align*}"}
+{"id": "2000.png", "formula": "\\begin{align*} ( A \\chi _ I , \\chi _ I ) = | I | ^ 2 . \\end{align*}"}
+{"id": "8080.png", "formula": "\\begin{align*} \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } H _ { m p , p } & = L ( 1 , \\tilde f ) \\frac { 2 } { \\pi } \\int _ 0 ^ \\infty k ( t ) \\tanh ( \\pi t ) t \\ , d t + { } \\\\ & \\qquad { } + O \\Bigl ( p ^ { \\frac { 3 } { 7 } - \\varepsilon } \\int _ 0 ^ \\infty k ( t ) \\tanh ( \\pi t ) t ^ { \\frac { 1 } { 7 } + \\varepsilon } \\ , d t \\Bigr ) . \\end{align*}"}
+{"id": "20.png", "formula": "\\begin{align*} - d Y _ t = g ( t , Y _ t , Z _ t ) d t - Z _ t d B _ t , \\ t \\in [ 0 , \\infty ) , \\end{align*}"}
+{"id": "3567.png", "formula": "\\begin{align*} \\mathbf { p } E _ { m j } = E _ { m j } + \\sum _ { i = j + 1 } ^ { m - 1 } \\sum _ { s = 2 } ^ { i - j + 1 } \\sum _ { I \\in \\mathcal { I } _ { j i } ( s ) } E _ { i _ 2 i _ 1 } \\cdots E _ { i _ s i _ { s - 1 } } E _ { m i _ s } \\frac { 1 } { \\prod _ { \\ell \\in I , \\ell \\neq j } ( h _ j - h _ \\ell ) } , \\end{align*}"}
+{"id": "6780.png", "formula": "\\begin{align*} \\tilde { \\Psi } _ { \\chi } ( y ) = \\sum _ { n = 1 } ^ { \\infty } \\chi ( n ) n \\sqrt { y } \\exp \\left [ - \\pi n ^ 2 y \\right ] . \\end{align*}"}
+{"id": "231.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = & u ( \\alpha u + ( 1 - \\alpha ) u ^ 2 + \\beta w ) , \\\\ \\psi _ 2 ( u , v , w ) = & v ( \\alpha v + ( 1 - \\alpha ) v ^ 2 + \\beta w ) , \\\\ \\psi _ 3 ( u , v , w ) = & u ( \\alpha u + ( 1 - \\alpha ) v ^ 2 + \\beta w ) \\shortintertext { a n d } \\psi _ 4 ( v , w ) = & v ( \\alpha v + ( 1 - \\alpha ) u ^ 2 + \\beta w ) . \\end{align*}"}
+{"id": "5292.png", "formula": "\\begin{align*} \\tau ( F ) = ( n - 2 ) \\frac { \\lambda ^ 2 } { 2 } F _ \\ast \\left ( \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\right ) + ( m - n ) F _ \\ast ( \\mathcal { H } \\nabla f ) . \\end{align*}"}
+{"id": "1688.png", "formula": "\\begin{align*} \\left | \\begin{array} { c c } X & Y \\\\ x & y \\end{array} \\right | ^ { k - 2 } = 2 ^ { 2 - k } \\left | \\begin{array} { c c } Y + i X & Y - i X \\\\ x - y i & - i y - x \\end{array} \\right | ^ { k - 2 } = 2 ^ { 2 - k } \\sum _ { j = 0 } ^ { k - 2 } \\binom { k - 2 } { j } P _ { j } ( X , Y ) z ^ j \\bar z ^ { k - 2 - j } , \\end{align*}"}
+{"id": "3811.png", "formula": "\\begin{align*} A = \\frac { 1 } { 2 } \\mathrm { T r } ( Q \\nabla _ x ^ 2 ) - \\frac { 1 } { 2 } R x \\cdot x + B x \\cdot \\nabla _ x \\end{align*}"}
+{"id": "7548.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A ) = Z _ f ( s , \\chi ) - Z _ f ( s , \\chi , A ^ c ) . \\end{align*}"}
+{"id": "8142.png", "formula": "\\begin{align*} K _ { 2 i t } ( x ) = \\frac { 1 } { 2 } \\cosh ( t \\pi ) ^ { - 1 } \\int _ { - \\infty } ^ \\infty \\cos ( x \\sinh \\zeta ) e \\Bigl ( - \\frac { t \\zeta } { \\pi } \\Bigr ) \\ , d \\zeta \\end{align*}"}
+{"id": "5387.png", "formula": "\\begin{align*} \\mathbb { E } [ Z _ t ( m ) ^ 2 ] = \\sigma _ m ^ 2 & : = \\sum _ { e ^ { W + C + m - 1 } < k \\le e ^ { W + C + m } } \\frac { 1 } { 2 k e ^ { 2 k \\sigma } } \\\\ \\mathbb { E } [ Z ( m ) Z _ t ( m ) ] = \\rho _ { m , t } \\sigma _ m ^ 2 & : = \\sum _ { e ^ { W + C + m - 1 } < k \\le e ^ { W + C + m } } \\frac { \\cos ( k t ) } { 2 k e ^ { 2 k \\sigma } } \\end{align*}"}
+{"id": "1092.png", "formula": "\\begin{align*} \\| f _ 1 - f _ 0 \\| _ t = \\gamma 2 ^ { j ( 1 / 2 - 1 / t ) } \\| \\psi \\| _ t \\asymp \\varepsilon ^ { \\beta / ( \\beta + 1 / 2 ) } \\varepsilon ^ { ( 1 / t - 1 / 2 ) / ( \\beta + 1 / 2 ) } = \\varepsilon ^ { ( \\beta + 1 / t - 1 / 2 ) / ( \\beta + 1 / 2 ) } , \\end{align*}"}
+{"id": "1891.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } k _ { x x } ( x , \\zeta ) - k _ { \\zeta \\zeta } ( x , \\zeta ) = ( c + a ( \\zeta ) ) k ( x , \\zeta ) , \\cr k _ \\zeta ( x , 0 ) = 0 , \\cr k ( x , x ) = - \\frac { 1 } { 2 } \\int _ 0 ^ x a ( \\zeta ) d \\zeta - \\frac { c } { 2 } x . \\end{array} \\right . \\end{align*}"}
+{"id": "7558.png", "formula": "\\begin{align*} f _ 2 ( u , v ) = & \\pi ^ { k + l } u ^ p + v ^ r \\sum _ { i = 0 } ^ k ( - 1 ) ^ i \\dbinom { k + r } { k - i } \\dbinom { i + r - 1 } { i } t ^ { k - i } \\pi ^ i v ^ i \\\\ = & \\pi ^ { k + l } u ^ p + t ^ k \\dbinom { k + r } { k } v ^ r + \\sum _ { i = 1 } ^ k ( - 1 ) ^ i \\dbinom { k + r } { k - i } \\dbinom { i + r - 1 } { i } t ^ { k - i } \\pi ^ i v ^ { r + i } . \\end{align*}"}
+{"id": "6667.png", "formula": "\\begin{align*} \\begin{aligned} & \\| y _ i ^ { k + 1 } - { y ' _ i } ^ { k + 1 } \\| _ 1 \\\\ & \\leq ( 1 - \\alpha ^ k - \\gamma ^ k _ 2 | C _ { i i } | ) \\| y _ i ^ k - { y ' _ i } ^ k \\| _ 1 \\\\ & + \\| g _ i ^ { k + 1 } - { g ' _ i } ^ { k + 1 } \\| _ 1 + ( 1 - \\alpha ^ k ) \\| g _ i ^ k - { g ' _ i } ^ k \\| _ 1 \\\\ & \\leq ( 1 - \\alpha ^ k - \\gamma ^ k _ 2 | C _ { i i } | ) \\| y _ i ^ k - { y ' _ i } ^ k \\| _ 1 + ( 2 - \\alpha ^ k ) 2 C , \\end{aligned} \\end{align*}"}
+{"id": "560.png", "formula": "\\begin{align*} K _ n = \\sigma _ n \\ , ( A X _ { t _ n } ^ \\dagger ) ^ { \\rm T } \\left ( \\gamma + \\Delta t \\sigma _ n \\ , ( A ^ { \\rm T } A ) : C \\right ) ^ { - 1 } . \\end{align*}"}
+{"id": "6996.png", "formula": "\\begin{align*} \\mathcal { E } _ r = \\left \\{ \\Delta _ x < P _ { e _ x } \\right \\} \\cup \\left \\{ \\Delta _ y < P _ { e _ y } \\right \\} . \\end{align*}"}
+{"id": "5541.png", "formula": "\\begin{align*} V ( x , k ) : = \\frac { 1 } { 2 \\pi i } \\int _ { \\lambda - i \\infty } ^ { \\lambda + i \\infty } \\frac { \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } { \\rm d } s , \\end{align*}"}
+{"id": "1211.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\sum _ { \\alpha \\in I _ { i , k - 1 } } \\biggl ( \\biggl [ \\nabla _ { v _ 1 } \\biggl \\{ \\eqref { a n e u r y s m } \\biggr \\} \\biggr ] ( v _ 2 \\otimes \\cdots \\otimes v _ { k + 1 } ) + \\sum _ { \\ell = 2 } ^ { k + 1 } \\biggl \\{ \\eqref { a n e u r y s m } \\biggr \\} \\biggl [ v _ 2 \\otimes \\cdots \\otimes \\frac { \\nabla \\gimel _ { s _ \\ell } } { \\partial s _ 1 } \\otimes \\cdots \\otimes v _ { k + 1 } \\biggr ] \\biggr ) . \\end{align*}"}
+{"id": "3580.png", "formula": "\\begin{align*} E _ { 3 1 } \\Omega _ { \\lambda - \\epsilon _ 3 } = E _ { 3 2 } E _ { 2 1 } \\Omega _ { \\lambda - \\epsilon _ 3 } - E _ { 2 1 } E _ { 3 2 } \\Omega _ { \\lambda - \\epsilon _ 3 } = - E _ { 2 1 } E _ { 3 2 } \\Omega _ { \\lambda - \\epsilon _ 3 } . \\end{align*}"}
+{"id": "2209.png", "formula": "\\begin{align*} \\xi ^ 5 & - \\big ( 2 0 5 X + 4 3 0 0 X ^ 2 + 3 4 0 0 0 X ^ 3 + 1 2 0 0 0 0 X ^ 4 + 1 6 0 0 0 0 X ^ 6 \\big ) \\xi ^ 4 \\\\ & - \\big ( 2 1 5 X + 4 4 7 5 X + 3 5 0 0 0 X ^ 3 + 1 2 2 0 0 0 X ^ 4 + 1 6 0 0 0 0 X ^ 5 \\big ) \\xi ^ 3 \\\\ & - \\big ( 8 5 X + 1 7 5 0 X ^ 2 + 1 3 5 2 5 X ^ 3 + 4 6 5 0 0 X ^ 4 + 6 0 0 0 0 X ^ 5 \\big ) \\xi ^ 2 \\\\ & - \\big ( 1 5 X + 3 0 5 X ^ 2 + 2 3 2 5 X ^ 3 + 7 8 7 5 X ^ 4 + 1 0 0 0 0 X ^ 5 \\big ) \\xi \\\\ & - \\big ( X + 2 0 X ^ 2 + 1 5 0 X ^ 3 + 5 0 0 X ^ 4 + 6 2 5 X ^ 5 \\big ) = 0 , \\end{align*}"}
+{"id": "3673.png", "formula": "\\begin{align*} ( ( I - \\alpha _ \\rho ^ k ) A + \\alpha _ \\rho ^ k c ) \\delta u & = M _ { \\rho } ( f - A u ^ k , u ^ k - g ) . \\end{align*}"}
+{"id": "7647.png", "formula": "\\begin{align*} \\omega = \\sum _ { k } \\lambda _ { k } \\frac { d f _ { k } } { f _ { k } } = \\sum _ { k } \\lambda _ { k } d \\log f _ { k } . \\end{align*}"}
+{"id": "6271.png", "formula": "\\begin{align*} \\dot { \\gamma } ( t ) = e ^ { - \\frac { 1 } { \\gamma ( t ) } } > 0 . \\end{align*}"}
+{"id": "2317.png", "formula": "\\begin{align*} ( A \\circ B ) ^ { \\# } ( z ) & = | \\det ( A \\circ B ) ' ( z ) | ^ { \\frac { N - p } { N p } } f ( ( A \\circ B ) ( z ) ) \\\\ & = | \\det A ' ( B ( z ) ) \\cdot \\det B ' ( z ) | ^ { \\frac { N - p } { N p } } f ( A ( B ( z ) ) ) \\\\ & = | \\det B ' ( z ) | ^ { \\frac { N - p } { N p } } ( A ^ { \\# } f ) ( B ( z ) ) \\\\ & = [ ( B ^ { \\# } \\circ A ^ { \\# } ) f ] ( z ) , \\end{align*}"}
+{"id": "1887.png", "formula": "\\begin{align*} \\beta _ { i } = \\begin{cases} 2 \\binom { n - 2 } { i - 1 } + \\binom { n - 2 } { i } & \\\\ [ 1 0 p t ] 2 \\binom { n - 2 } { i - 1 } + \\binom { n - 2 } { i - 2 } & . \\\\ \\end{cases} \\end{align*}"}
+{"id": "7713.png", "formula": "\\begin{align*} | e _ n ^ { ( i ) } ( \\lambda ) | & \\cong 4 \\pi | r ^ { ( i ) } | { S ^ { ( i ) } } ^ { - 2 n } \\\\ & = \\frac { \\pi } { \\sqrt { \\tau } } | \\lambda | ^ { - \\frac { 1 } { 2 } } { S ^ { ( i ) } } ^ { - 2 n } = : \\Phi ^ { ( i ) } ( \\tau , \\lambda ) , \\end{align*}"}
+{"id": "5805.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ p u _ s '' e _ s + \\sum _ { s = 1 } ^ p L u _ s e _ s + \\sum _ { s = 1 } ^ p g u _ s ' D e _ s + \\sum _ { s = 1 } ^ p u _ s A e _ s = 0 . \\end{align*}"}
+{"id": "7852.png", "formula": "\\begin{align*} g ^ d + \\phi _ 1 g ^ { d - 1 } + \\cdots + \\phi _ { d - 1 } g + \\phi _ d = 0 . \\end{align*}"}
+{"id": "9144.png", "formula": "\\begin{align*} M _ 1 \\ ; : = & \\ ; m \\sum _ { k = 0 } ^ n { n \\choose k } ( m + n - 1 ) ^ { n - k - 1 } = \\\\ = & m ( m + n - 1 ) ^ { - 1 } \\sum _ { k = 0 } ^ n { n \\choose k } ( m + n - 1 ) ^ { n - k } = m ( m + n - 1 ) ^ { - 1 } ( m + n ) ^ { n } , \\\\ \\end{align*}"}
+{"id": "4994.png", "formula": "\\begin{align*} c + \\nabla u ( q _ 0 ) = - \\partial _ q S ( q _ 0 , q _ 1 ) , c + \\nabla u ( q _ n ) = \\partial _ Q S ( q _ { n - 1 } , q _ n ) \\end{align*}"}
+{"id": "3386.png", "formula": "\\begin{align*} { \\rm p d } ( \\mathcal { C } ) : = { \\rm s u p } \\{ { \\rm p d } ( C ) \\mid \\ C \\in \\mathcal { C } \\} . \\end{align*}"}
+{"id": "1698.png", "formula": "\\begin{align*} \\langle \\ ; , \\ ; \\rangle ' : V ( k - 2 ) ( \\pm ) \\times V ( k - 2 ) ( \\pm ) \\longrightarrow \\C , \\langle \\mu _ 1 , \\mu _ 2 \\rangle ' = \\mu _ 2 \\mu _ 1 \\left ( \\left | \\begin{array} { c c } X _ 1 & Y _ 1 \\\\ X _ 2 & Y _ 2 \\end{array} \\right | ^ { k - 2 } \\right ) . \\end{align*}"}
+{"id": "7878.png", "formula": "\\begin{align*} f f '' - a ( z ) ( f ' ) ^ 2 = b ( z ) e ^ { 2 c ( z ) } , \\end{align*}"}
+{"id": "685.png", "formula": "\\begin{align*} \\oint _ { \\alpha _ k } ( - d U _ { \\beta _ j } ) = \\delta _ { k j } , \\oint _ { \\beta _ k } ( - d U _ { \\beta _ j } ) = 0 , \\end{align*}"}
+{"id": "1167.png", "formula": "\\begin{align*} & C = A + B \\longleftrightarrow M X _ { m \\times n , p } ( C ) \\wedge \\forall i < m , j < n \\ , \\ , C _ { i j } = A _ { i j } + _ { ( m o d \\ , p ) } B _ { i j } , \\end{align*}"}
+{"id": "1739.png", "formula": "\\begin{align*} \\mu _ { \\underline { m } } \\left ( \\left | \\begin{array} { c c } X & Y \\\\ x & y \\end{array} \\right | ^ { \\underline { k } - 2 } \\right ) = x ^ { \\frac { \\underline k - 2 } { 2 } - \\underline m } y ^ { \\frac { \\underline k - 2 } { 2 } + \\underline m } , \\left | \\begin{array} { c c } X & Y \\\\ x & y \\end{array} \\right | ^ { \\underline { k } - 2 } : = \\prod _ \\sigma \\left | \\begin{array} { c c } X _ \\sigma & Y _ \\sigma \\\\ x _ \\sigma & y _ \\sigma \\end{array} \\right | ^ { k _ \\sigma - 2 } . \\end{align*}"}
+{"id": "2943.png", "formula": "\\begin{align*} \\tilde { \\mathcal { B } } ^ n _ t ( \\varphi ) = \\frac { g ^ { \\prime \\prime } ( 0 ) } { 2 } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\overline { W } _ { j - 1 } ( s ) \\overline { W } _ j ( s ) \\nabla ^ n \\varphi ^ n _ j ( s ) d s \\end{align*}"}
+{"id": "8967.png", "formula": "\\begin{align*} d \\pi _ N ( u ) \\partial _ r u + f = 0 \\ \\hbox { o n } \\partial B = S ^ 1 , \\end{align*}"}
+{"id": "6000.png", "formula": "\\begin{align*} g \\cdot Y _ { r , p } ^ r & = \\{ ( L , H ) \\in X | L \\subset < e _ { 1 } , \\dots , e _ r > , \\ : < e _ { 1 } , \\dots e _ { r } , e _ { r + 1 } , \\dots , e _ { r + p } > \\subset H \\} \\\\ & = X ( r , r + p + 1 ) \\end{align*}"}
+{"id": "9156.png", "formula": "\\begin{align*} ( \\Delta _ J f ) ( x ) = \\frac { 1 } { | J | } \\sum _ { y \\in J } ( f ( x + y ) - f ( x ) ) , \\end{align*}"}
+{"id": "1165.png", "formula": "\\begin{align*} c = m \\cdot _ { ( m o d \\ , p ) } a \\longleftrightarrow c < p \\wedge c \\equiv ( m a ) \\ , ( m o d \\ , p ) . \\end{align*}"}
+{"id": "8469.png", "formula": "\\begin{align*} \\gamma _ A = ( \\gamma _ A ) ^ A , \\end{align*}"}
+{"id": "1402.png", "formula": "\\begin{align*} \\psi = S _ 1 + S _ 2 \\boxtimes S _ 2 + S _ 4 \\boxtimes S _ 4 . \\end{align*}"}
+{"id": "4985.png", "formula": "\\begin{align*} \\| \\nabla G _ { p , q } \\| ^ 2 = p ^ 2 + \\| \\nabla g \\| ^ 2 \\ , . \\end{align*}"}
+{"id": "6117.png", "formula": "\\begin{align*} t = 0 : U = \\widehat U _ 0 , U ' = \\widehat U _ 1 \\hbox { i n } \\Omega , \\end{align*}"}
+{"id": "2954.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j \\in \\mathbb { Z } } \\overrightarrow { W } ^ \\ell _ j ( \\overline { W } _ { j - 1 } - \\overrightarrow { W } ^ \\ell _ j ) \\varphi _ j = \\sum _ { j \\in \\mathbb { Z } } \\overrightarrow { W } ^ \\ell _ j \\sum _ { i = 0 } ^ { \\ell - 1 } ( \\overline { W } _ { j + i - 1 } - \\overline { W } _ { j + i } ) \\psi _ i \\varphi _ j \\end{aligned} \\end{align*}"}
+{"id": "3029.png", "formula": "\\begin{align*} \\mathcal { A } _ 0 [ \\theta , \\varphi , \\pi ] = \\int _ { \\mathcal { Y } ^ 5 } \\frac { 1 } { 2 } \\hat { \\theta } ^ { ( 3 ) } _ { I J } \\wedge \\Phi ^ { I J } + \\pi _ I \\wedge \\hbox { d } \\theta ^ I \\end{align*}"}
+{"id": "193.png", "formula": "\\begin{align*} \\frac { r _ { n } ^ { 3 } } { h _ { n } } \\leq \\frac { 2 ^ { 3 n ^ { \\alpha } } } { 2 ^ { d ( n - 1 ) ^ { 1 + \\alpha } } } \\to 0 \\quad \\quad \\quad \\frac { d ( n - 1 ) ^ { 1 + \\alpha } - 3 n ^ { \\alpha } } { n ^ { \\alpha } } = d \\Big { ( } 1 - \\frac { 1 } { n } \\Big { ) } ^ { \\alpha } ( n - 1 ) - 3 \\to \\infty . \\end{align*}"}
+{"id": "1373.png", "formula": "\\begin{align*} E ( t ) = \\frac { 1 } { \\rho + 2 } \\left \\| u _ { t } \\right \\| _ { \\rho + 2 } ^ { \\rho + 2 } + J ( t ) , \\end{align*}"}
+{"id": "542.png", "formula": "\\begin{align*} a _ * & : = \\bigvee \\{ x \\in L \\mid x < a \\} \\\\ a ^ * & : = \\bigwedge \\{ x \\in L \\mid x > a \\} \\end{align*}"}
+{"id": "5594.png", "formula": "\\begin{align*} Q _ { - 2 } = l \\overline { m } m \\overline { m } , Q _ { - 1 } = k m l m , Q _ { 0 } = k m m \\overline { m } , Q _ 1 = k m k m . \\end{align*}"}
+{"id": "6559.png", "formula": "\\begin{align*} \\frac { { { m _ w } ^ { { m _ w } } } } { { \\Gamma \\left ( { { m _ w } } \\right ) } } \\sum \\limits _ { j = 0 } ^ \\infty { \\frac { \\Gamma ^ { - 1 } \\ ! \\left ( { j \\ ! + \\ ! 1 } \\right ) { { K _ w } ^ j { \\alpha _ { w j } } { I _ E } } } { { j ! { { \\left ( { 2 { C _ { 1 w } } \\sigma _ w ^ 2 } \\right ) } ^ { j + 1 } } } } } > { \\left ( { { \\varepsilon _ { t h } } \\ ! - \\ ! { \\kappa ^ 2 } } \\right ) ^ { q + \\ ! { m _ { f w } } - 2 } } , \\end{align*}"}
+{"id": "8890.png", "formula": "\\begin{align*} T r \\left ( \\exp \\left ( \\frac { 1 } { 4 } t \\Delta _ { \\nu } \\right ) \\right ) \\simeq \\frac { 1 } { ( 4 \\pi t ) ^ { n } } \\sum \\limits _ { j = 0 } ^ { + \\infty } b _ { j } ^ { \\left ( \\nu , n \\right ) } \\ t ^ { j } , t \\searrow 0 ^ { + } \\end{align*}"}
+{"id": "5847.png", "formula": "\\begin{align*} \\langle \\ ! \\langle & ( U ' ( t ) , - U ( t ) ) , ( \\Phi ( t ) , \\Phi ' ( t ) ) \\rangle \\ ! \\rangle \\\\ & = \\langle \\ ! \\langle ( \\widehat { U } _ 1 , - \\widehat { U } _ 0 ) , ( \\widehat { \\Phi } _ 0 , \\widehat { \\Phi } _ 1 ) \\rangle \\ ! \\rangle + \\int _ 0 ^ t \\int _ { \\Gamma } ( D H ( \\tau ) , \\Phi ( \\tau ) ) d x d t , \\end{align*}"}
+{"id": "7995.png", "formula": "\\begin{align*} Q _ N \\left ( x , \\varepsilon \\right ) : = \\sum _ { k = 1 } ^ { N } d _ k k ^ { \\frac { 1 } { 2 } } \\varphi _ k \\left ( x \\right ) \\varepsilon _ k , \\end{align*}"}
+{"id": "6372.png", "formula": "\\begin{align*} 0 < u _ 2 : = u ( r _ 2 ) < \\pi / 2 = u ( r _ e ) < u _ 1 : = u ( r _ 1 ) < \\pi , { \\ : \\rm a n d \\ : } u _ 1 + u _ 2 = \\pi . \\end{align*}"}
+{"id": "4656.png", "formula": "\\begin{align*} G _ { \\ell _ 1 , \\ell _ 2 , k } = \\binom { \\ell _ 1 } { k } \\binom { \\ell _ 2 } { k } k ! , \\ \\ k = 0 , \\ldots , \\min \\{ \\ell _ 1 , \\ell _ 2 \\} . \\end{align*}"}
+{"id": "2155.png", "formula": "\\begin{align*} ( 1 - 2 \\alpha - \\beta + \\lambda ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + 1 - \\lambda = 0 \\end{align*}"}
+{"id": "8346.png", "formula": "\\begin{align*} \\tilde { K } ( Q ) = \\sup _ { x \\in \\mathbb { R } ^ d , | x - \\tilde { l } | \\ge c _ 0 } \\left \\{ | x | ^ { d - 2 } | Q ( x ) | + | x | ^ { d - 1 } | \\nabla Q ( x ) | \\right \\} < + \\infty . \\end{align*}"}
+{"id": "1112.png", "formula": "\\begin{align*} R _ 1 : \\ , \\mathbb { P } ( X = r ) = \\frac { 1 - \\delta } { 2 } = 1 - \\mathbb { P } ( X = - r ) . \\end{align*}"}
+{"id": "7840.png", "formula": "\\begin{align*} f ( z ) = F _ 0 ( z ) + F _ 1 ( z ) e ^ { w _ 1 z ^ q } + \\cdots + F _ n ( z ) e ^ { w _ n z ^ q } , \\end{align*}"}
+{"id": "5922.png", "formula": "\\begin{align*} t \\geq T : \\widehat { D } ^ T w = 0 \\qquad \\hbox { o n } \\Gamma , \\end{align*}"}
+{"id": "1341.png", "formula": "\\begin{align*} n ( n - 1 ) \\left ( \\frac { n - d } { d ( n - 1 ) } \\right ) ^ \\frac { p } { 2 } + n & = \\frac { 1 } { ( n ^ 2 - n ) ^ { \\frac { p } { 2 } - 1 } } \\left ( \\frac { n ^ 2 } { d } - n \\right ) ^ \\frac { p } { 2 } + n \\\\ & \\leq \\sum _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | ^ \\frac { p } { 2 } + \\sum _ { j = 1 } ^ { n } | f _ j ( \\tau _ j ) f _ j ( \\tau _ j ) | ^ \\frac { p } { 2 } \\\\ & = \\sum _ { 1 \\leq j , k \\leq n } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | ^ \\frac { p } { 2 } . \\end{align*}"}
+{"id": "1415.png", "formula": "\\begin{align*} k _ { \\rho , x } = M _ { \\rho , x } - M _ { \\rho , x + 1 } = \\sum _ { \\substack { z \\in ( 1 / 2 ) \\Z \\\\ z \\geq x } } m _ { \\rho , z } , \\end{align*}"}
+{"id": "4814.png", "formula": "\\begin{align*} \\Pr _ { \\rho \\sim P _ \\epsilon } \\big [ \\rho = c + x \\big ] & \\leq \\epsilon ^ { \\epsilon N - \\epsilon N ^ { 3 / 4 } } ( 1 - \\epsilon ) ^ { ( 1 - \\epsilon ) N + \\epsilon N ^ { 3 / 4 } } \\\\ & \\leq 2 ^ { - ( 1 - N ^ { - 1 / 4 } ) h ( \\epsilon ) N } \\end{align*}"}
+{"id": "162.png", "formula": "\\begin{align*} ( \\mathrm { T } ( \\theta ) v ) ( \\hslash ) = \\left \\{ \\begin{aligned} \\begin{array} { l l } v ( \\hslash + \\theta ) , & \\mbox { i f } \\ ; ( \\hslash + \\theta ) \\leq \\pi , \\\\ 0 , & \\mbox { i f } \\ ; ( \\hslash + \\theta ) \\textgreater \\pi , \\end{array} \\end{aligned} \\right . \\end{align*}"}
+{"id": "5149.png", "formula": "\\begin{align*} \\forall \\theta \\in \\mathbb { R } , K _ { 0 } \\left ( \\sqrt { a ^ { 2 } + b ^ { 2 } - 2 a b \\cos ( \\theta ) } \\right ) = \\sum _ { m = - \\infty } ^ { \\infty } I _ { m } ( b ) K _ { m } ( a ) \\cos ( m \\theta ) . \\end{align*}"}
+{"id": "4747.png", "formula": "\\begin{align*} x u \\in A + y ^ { e _ j } ( A + \\mathfrak { m } B ) = A + y ^ { e _ j } ( A + y B ) = A + y ^ { e _ j + 1 } B \\end{align*}"}
+{"id": "3153.png", "formula": "\\begin{align*} R _ 1 ( t ) : = - R _ 2 ( t ) : = - \\frac { 1 } { 3 2 \\pi ^ 2 } \\left ( \\sin ( 2 \\pi t ) + 2 \\cos ( 2 \\pi t ) \\right ) \\quad t \\in \\R . \\end{align*}"}
+{"id": "6214.png", "formula": "\\begin{align*} ( E _ r , e _ s ) = \\delta _ { r s } , r , s = 1 , \\cdots , p . \\end{align*}"}
+{"id": "3439.png", "formula": "\\begin{align*} \\nu _ { t } \\left ( L _ { \\phi } f \\right ) = \\lambda \\nu _ { t } \\left ( f \\right ) , \\qquad \\mbox { p a r a } f \\in L ^ 1 ( \\nu _ { t } ) . \\end{align*}"}
+{"id": "6685.png", "formula": "\\begin{align*} \\begin{aligned} x ^ { k + 1 } ( \\ell ) & = ( I + \\gamma _ 1 ^ k R ) x ^ k ( \\ell ) + \\gamma _ 1 ^ k \\zeta _ w ^ k ( \\ell ) - \\lambda ^ k y ^ k ( \\ell ) \\cr y ^ { k + 1 } ( \\ell ) & = ( I - \\alpha ^ k + \\gamma _ 2 ^ k C ) y ^ k ( \\ell ) + \\gamma _ 2 ^ k \\xi _ w ^ k ( \\ell ) \\\\ & \\quad + g ^ { k + 1 } ( \\ell ) - ( 1 - \\alpha ^ k ) g ^ k ( \\ell ) \\end{aligned} \\end{align*}"}
+{"id": "3677.png", "formula": "\\begin{align*} r _ { \\mathrm { a n } } f ( 0 ) + \\sum _ { \\theta \\in \\mathcal { Z } } f ( \\theta ) = c ( 0 ) ( n _ E + 4 g - 4 ) + 2 \\sum _ { m = 1 } ^ { Y } U _ m ( E , f ) , \\end{align*}"}
+{"id": "1755.png", "formula": "\\begin{align*} I ( \\chi , { \\underline m } ) : = \\int _ { T ( \\C ) } \\chi ( t ) \\langle \\imath ( t ) \\delta s ( \\mu _ { \\underline m } ) , \\delta s ( \\mu _ { - \\underline m } ) \\rangle t ^ { \\underline m } d ^ \\times t , \\end{align*}"}
+{"id": "6095.png", "formula": "\\begin{align*} D : = \\frac { 1 } { 2 \\pi i } \\partial _ \\tau . \\end{align*}"}
+{"id": "1050.png", "formula": "\\begin{align*} \\hat { \\mu } = \\frac { 1 } { n } \\sum _ { i = ( L + 1 ) n + 1 } ^ { ( L + 2 ) n } Z _ i ^ { ( L ) } + ( J - 1 ) M / 3 . \\end{align*}"}
+{"id": "8244.png", "formula": "\\begin{align*} 0 = I ^ { \\prime } \\left ( u \\right ) u ^ { - } = - \\int _ { \\mathbb { R } ^ { N } } | \\nabla u ^ { - } | ^ 2 d x + \\int _ { \\mathbb { R } ^ { N } } V ( | x | ) f ( u ) f ' ( u ) u ^ { - } d x , \\end{align*}"}
+{"id": "2966.png", "formula": "\\begin{align*} \\mathcal { S } _ t = \\frac { g ^ \\prime ( 0 ) } { 2 } \\int _ 0 ^ t \\mathcal { X } _ s ( \\partial _ x ^ 2 \\varphi ) d s . \\end{align*}"}
+{"id": "2553.png", "formula": "\\begin{align*} \\frac { ( \\beta ) _ { m _ 1 } } { ( \\beta ) _ { m _ q + 1 } } = \\frac { 1 } { ( m _ 1 + \\beta ) ^ { c ^ { ' } _ 1 } } \\left ( \\prod _ { i = 2 } ^ { q } \\frac { ( \\beta ) _ { m _ { i - 1 } + c ^ { ' } _ { i - 1 } } } { ( \\beta ) _ { m _ { i } + c ^ { ' } _ i } } \\right ) , \\end{align*}"}
+{"id": "5015.png", "formula": "\\begin{align*} 0 = \\xi ( Z , Z ) = c _ 2 B _ 2 ( Z _ 2 , Z _ 2 ) , \\end{align*}"}
+{"id": "5301.png", "formula": "\\begin{align*} g \\left ( U , \\nabla \\lambda \\right ) = 0 . \\end{align*}"}
+{"id": "7133.png", "formula": "\\begin{align*} \\left \\langle \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) , x ^ { \\rho _ 1 + \\cdots + \\rho _ { j - 1 } } \\right \\rangle = \\begin{cases} 1 & i - j \\\\ 0 & i \\neq j . \\end{cases} \\end{align*}"}
+{"id": "7909.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\mathcal { \\hat { E } } _ t = \\mathcal { \\hat { E } } _ \\infty , \\end{align*}"}
+{"id": "5707.png", "formula": "\\begin{align*} \\mathbf { A } _ { b } ^ { a } = \\alpha _ { b } ^ { a } - \\digamma _ { b } ^ { a } \\mathbf { m . } \\end{align*}"}
+{"id": "1707.png", "formula": "\\begin{align*} C ( n ) : = \\int _ { S ^ 1 } s ( \\mu _ m ) ( \\kappa ( \\theta ) ) ( - \\sin \\theta ) ^ { k - 2 - n } \\cos \\theta ^ { n } d \\theta . \\end{align*}"}
+{"id": "6206.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ d \\beta _ r u _ r = ( x , C _ p U _ n ) \\rightarrow 0 \\hbox { i n } C ^ 0 _ { l o c } ( [ T , + \\infty ) ; \\mathcal H _ { 0 } ) \\cap C ^ 1 _ { l o c } ( [ T , + \\infty ) ; \\mathcal H _ { - 1 } ) \\end{align*}"}
+{"id": "8911.png", "formula": "\\begin{align*} B _ { 2 ( d + 1 ) } \\left ( \\frac { 1 } { 2 } \\right ) = ( - 1 ) ^ { d + 1 } ( d + 1 ) B _ d . \\end{align*}"}
+{"id": "4022.png", "formula": "\\begin{align*} G _ p \\cdot \\gamma = \\chi w ^ { i j } \\gamma _ i \\gamma _ j . \\end{align*}"}
+{"id": "3340.png", "formula": "\\begin{align*} \\Phi _ { x x } + \\Phi _ { q q } & = - \\frac { 1 } { \\Phi _ { x q } } ( \\Phi _ { x y } \\Phi _ { y q } + \\Phi _ { x p } \\Phi _ { p q } ) , \\\\ \\Phi _ { x x } - \\Phi _ { q q } & = - \\frac { 1 } { \\Phi _ { x p } } ( \\Phi _ { x y } \\Phi _ { y p } + \\Phi _ { x q } \\Phi _ { q p } ) . \\end{align*}"}
+{"id": "3538.png", "formula": "\\begin{align*} \\mathbf { p } \\{ B _ i ^ - , B _ j ^ - \\} & = \\sum _ { k = i } ^ { n } \\sum _ { s = 1 } ^ { k - i + 1 } \\sum _ { I \\in \\mathcal { I } _ { i k } ( s ) } \\sum _ { l = j } ^ { n } \\sum _ { t = 1 } ^ { l - j + 1 } \\sum _ { J \\in \\mathcal { I } _ { j l } ( t ) } E ^ { e _ I + e _ J } \\{ B _ k ^ - , B _ l ^ - \\} H _ { I J } ^ - . \\end{align*}"}
+{"id": "7535.png", "formula": "\\begin{align*} Z _ g \\big ( s , \\chi , S ( \\Delta _ { \\gamma _ 2 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) ) \\big ) = \\sum _ { m = 0 } ^ { j _ 0 - 1 } \\sum _ { a = 1 } ^ { \\infty } \\sum _ { b = 1 } ^ { \\infty } Z _ { g } ( s , \\chi , D _ 2 ( a , b ) ) , \\end{align*}"}
+{"id": "9082.png", "formula": "\\begin{align*} \\big ( \\omega ^ { S } _ { \\xi , \\lambda } \\phi \\big ) ( f ) & = \\int _ { G } \\Big ( \\phi ( x ) , \\psi _ { 0 } ( x ) f ( x ) \\Big ) \\ , d x \\\\ & = \\int _ { M _ { S } } \\int _ { A _ { S } } \\int _ { N _ { S } } a ^ { 2 \\rho _ { S } } \\psi _ { 0 } ( m a n ) \\ , d n \\ , d a \\ , d m \\int _ { K } \\Big ( \\phi ( k ) , f ( k ) \\Big ) \\ , d k . \\end{align*}"}
+{"id": "156.png", "formula": "\\begin{align*} \\| h _ { 0 } ( \\varkappa ) - h _ { 0 } ( y ) \\| _ { \\mathbb { X } } & = K L _ { q } b \\kappa _ { b } \\| \\tilde { \\varkappa } _ { \\eta } - \\tilde { y } _ { \\eta } \\| _ { D } . \\end{align*}"}
+{"id": "103.png", "formula": "\\begin{align*} x \\ast y + y \\ast x = - x u y - y u x = - 2 x y u . \\end{align*}"}
+{"id": "741.png", "formula": "\\begin{align*} \\{ \\frac { \\partial ^ 2 H ( z , w ) } { \\partial z ^ 2 } \\} _ { z = w } = h _ { 2 } ( w ) , \\qquad \\qquad \\end{align*}"}
+{"id": "2570.png", "formula": "\\begin{align*} \\Phi _ { \\Delta G } ( \\Upsilon ( u ) ) & = \\Phi _ { \\Delta G } ( \\Omega ( h ) * ( \\hat { 0 } , \\hat { 0 } ) ) = ( p ^ { G \\times G } \\circ \\Psi ^ { G \\times G } ) ( h ) \\cdot \\Phi _ { \\Delta G } ( \\hat { 0 } , \\hat { 0 } ) \\\\ & = ( h ( 0 ) , h ( 1 ) ) \\Delta G = \\phi ^ { - 1 } ( h ( 0 ) h ( 1 ) ^ { - 1 } ) = \\phi ^ { - 1 } ( \\Phi ( u ) ) . \\end{align*}"}
+{"id": "488.png", "formula": "\\begin{align*} \\lim _ { s \\downarrow 0 } \\frac { 1 } { s } & \\int _ 0 ^ T \\iint \\limits _ { \\mathbb { R } ^ N \\times \\mathbb { R } ^ N } \\frac { H ( x , y , ( u + s \\psi ) ( x , t ) - ( u + s \\psi ) ( y , t ) ) - H ( x , y , u ( x , t ) - u ( y , t ) ) } { | x - y | ^ N } \\ , d x \\ , d y \\ , d t \\\\ & = \\int _ 0 ^ T \\iint \\limits _ { \\mathbb { R } ^ N \\times \\mathbb { R } ^ N } \\int _ 0 ^ 1 \\frac { D _ \\xi H ( x , y , u ( x , t ) - u ( y , t ) ) \\cdot ( \\psi ( x , t ) - \\psi ( y , t ) ) } { | x - y | ^ N } \\ , d \\sigma \\ , d x \\ , d y \\ , d t \\end{align*}"}
+{"id": "3805.png", "formula": "\\begin{align*} & ( - \\lambda - 2 ) ^ { 2 c - 2 } ( \\lambda ^ 5 + ( 4 c - 4 ) \\lambda ^ 4 + ( 1 2 c ^ 2 + 3 6 c + 2 ) \\lambda ^ 3 + ( 2 4 c ^ 2 + 8 8 c + 2 8 ) \\lambda ^ 2 + ( 8 c ^ 2 + 7 2 c + 4 0 ) \\lambda + 1 6 c + 1 6 ) \\\\ & = ( - \\lambda - 2 ) ^ { 2 c - 2 } ( \\lambda ^ 2 + ( 2 c + 4 ) \\lambda + 4 ) ( - \\lambda ^ 3 + 6 c \\lambda ^ 2 + ( 1 2 c + 6 ) \\lambda + 4 c + 4 ) . \\end{align*}"}
+{"id": "3560.png", "formula": "\\begin{align*} B _ \\ell ^ - E ^ \\gamma \\Omega _ \\lambda = \\sum _ { j = 1 } ^ \\ell \\gamma _ { \\ell j } ^ { 1 - \\delta _ { \\ell j } } E ^ { \\gamma - e _ { \\ell j } } B _ j ^ - \\Omega _ \\lambda \\end{align*}"}
+{"id": "7308.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\psi _ { m _ n } ( \\lambda ; x ) = x ( x \\geq 0 , \\lambda > 0 ) . \\end{align*}"}
+{"id": "567.png", "formula": "\\begin{align*} { \\rm d } \\mu _ t = \\frac { \\sigma _ t } { \\gamma } \\left ( ( A X _ t ^ \\dagger ) ^ { \\rm T } { \\rm d } X ^ \\dagger _ t - \\mu _ t \\ , ( A ^ { \\rm T } A ) : ( X _ t ^ \\dagger \\otimes X _ t ^ \\dagger ) \\ , { \\rm d } t \\right ) , \\end{align*}"}
+{"id": "4757.png", "formula": "\\begin{align*} \\sum _ { x : f ( x ) = y _ 1 } P ( x _ 0 , x ) = Q ( y _ 0 , y _ 1 ) . \\end{align*}"}
+{"id": "594.png", "formula": "\\begin{align*} \\Theta _ { n + 1 } ^ { ( \\epsilon ) } = \\Theta _ n ^ { ( \\epsilon ) } + \\frac { \\sigma _ n ^ { ( \\epsilon ) } } { \\gamma } \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ { ( \\epsilon ) } ) ^ { \\rm T } { \\rm d } X _ t ^ { ( \\epsilon ) } - \\frac { 1 } { 2 } K _ n ^ { ( \\epsilon ) } A X _ { t _ n } ^ { ( \\epsilon ) } \\left ( \\Theta _ n ^ { ( \\epsilon ) } + \\pi _ n ^ { ( \\epsilon ) } [ \\theta ] \\right ) \\Delta t . \\end{align*}"}
+{"id": "7452.png", "formula": "\\begin{align*} \\widehat M _ \\chi = ( \\widehat M _ { \\chi _ 1 } , \\dots , \\widehat M _ { \\chi _ d } ) \\in [ \\Omega ] _ { \\rm n c } . \\end{align*}"}
+{"id": "8266.png", "formula": "\\begin{align*} | U _ 3 | & < \\frac { 3 A ' } { \\sqrt { \\pi m } } e ^ { 5 m } \\int _ { Q } ^ { \\infty } \\exp \\left ( - \\left ( \\frac { ( \\log x - M ) } { 2 \\sqrt { m } } \\right ) ^ 2 \\right ) d x \\\\ & = \\frac { 6 A ' } { \\sqrt { \\pi } } e ^ { 5 m } \\int _ { 6 \\sqrt { m } } ^ { \\infty } \\exp ( - y ^ 2 ) ~ e ^ { 2 \\sqrt { m } y + M } ~ d y \\\\ & = \\frac { 6 A ' } { \\sqrt { \\pi } } e ^ { 5 m + M + m } \\int _ { 6 \\sqrt { m } } ^ \\infty e ^ { - ( y - \\sqrt { m } ) ^ 2 } d y < 4 m e ^ { 2 2 m } e ^ { - 2 5 m } < e ^ { - 2 m } . \\end{align*}"}
+{"id": "7992.png", "formula": "\\begin{align*} \\ell ( x , y ) = ( x + y ) \\cup \\{ x , y \\} . \\end{align*}"}
+{"id": "292.png", "formula": "\\begin{align*} \\begin{aligned} i ' : \\widehat { G } & \\rightarrow \\mathrm { I n d } _ { W _ E } ^ { W _ F } \\widehat { G } , \\\\ g & \\mapsto f _ g : u \\mapsto u \\cdot g , ~ \\forall u \\in W _ F . \\end{aligned} \\end{align*}"}
+{"id": "3356.png", "formula": "\\begin{align*} R \\lrcorner \\dd \\eta = 0 , R \\lrcorner \\eta = 1 , \\end{align*}"}
+{"id": "9054.png", "formula": "\\begin{align*} W _ T ^ { i } = Y _ 0 ^ { i } - \\xi ^ { i } - \\int _ 0 ^ T f ( s , Y _ s ^ { i } , Z _ s ^ { i } , U _ s ^ { i } , \\mathbb { P } _ { Y _ s ^ { i } } ) d s + \\int _ 0 ^ T Z _ s ^ { i } d B ^ i _ s + \\int _ 0 ^ T \\int _ { \\R ^ * } U _ s ^ { i } ( e ) \\Tilde { N } ^ i ( d s , d e ) , \\end{align*}"}
+{"id": "6226.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p C _ p A e _ r u _ r ( T ) \\equiv 0 \\hbox { i n } \\Omega \\end{align*}"}
+{"id": "846.png", "formula": "\\begin{align*} J _ d = \\frac { 1 } { 2 } \\int _ \\textit { B } \\rho ( X ) X X ^ T d ^ 3 X \\end{align*}"}
+{"id": "3127.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( \\tilde { A } ) = \\bar { \\gamma } \\left ( c _ 1 ^ { 1 1 } ( A ) + \\bar { a } _ { 1 1 } \\int _ Y r A e _ 1 \\cdot \\nabla w _ { \\gamma } \\right ) , \\end{align*}"}
+{"id": "5559.png", "formula": "\\begin{align*} \\sum _ { n = \\ell } ^ { [ x ] } \\frac { x ^ 2 } { y _ n ^ 3 } e ^ { - \\frac { x ^ 2 } { y _ n ^ 2 } } \\ll \\sum _ { n = \\ell } ^ \\infty \\frac { x ^ 2 } { n ^ 3 } \\ll \\frac { x ^ 2 } { l ^ 2 } \\ll x ^ { 2 \\epsilon } , \\end{align*}"}
+{"id": "8138.png", "formula": "\\begin{align*} H _ { m , n } ^ - ( x ) & = 2 \\int _ { - \\infty } ^ \\infty \\frac { I _ { - 2 i t } ( x ) - I _ { 2 i t } ( x ) } { \\sin ( 2 i t \\pi ) } \\sinh ( \\pi t ) k ( t ) V ( m ^ 2 n , t ) t \\ , d t \\\\ & = - 4 \\int _ { - \\infty } ^ \\infty \\frac { I _ { 2 i t } ( x ) } { \\sin ( 2 i t \\pi ) } \\sinh ( \\pi t ) k ( t ) V ( m ^ 2 n , t ) t \\ , d t \\end{align*}"}
+{"id": "6281.png", "formula": "\\begin{align*} \\mathbb { G } _ { M , \\pi , m } ( j _ m \\xi ( x ) , j _ m \\eta ( x ) ) = \\sum _ { j = 0 } ^ { m } \\mathbb { G } _ { M , \\pi } \\left ( \\frac { 1 } { j ! } D ^ j _ { \\nabla ^ M , \\nabla ^ \\pi } ( \\xi ) ( x ) , \\frac { 1 } { j ! } D ^ j _ { \\nabla ^ M , \\nabla ^ \\pi } ( \\eta ) ( x ) \\right ) . \\end{align*}"}
+{"id": "7699.png", "formula": "\\begin{align*} \\gamma = \\mathbb { C } [ s _ { 1 } , \\dots , s _ { d } ] \\xrightarrow [ ] { \\beta } \\mathbb { C } [ S ] \\xrightarrow [ ] { } \\iffalse \\mathbb { C } [ S ] \\otimes _ { \\mathbb { C } [ S ] } \\fi T _ { j } ^ { - 1 } \\mathbb { C } [ S ] . \\end{align*}"}
+{"id": "6564.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } f ( n ) & = \\sum _ { n \\leq x } \\sum _ { d \\mid n } g ( d ) \\\\ & = \\sum _ { d \\leq x } g ( d ) \\left \\lfloor \\frac { x } { d } \\right \\rfloor \\\\ & = x \\sum _ { d \\leq x } \\frac { g ( d ) } { d } + O \\left ( \\sum _ { d \\leq x } g ( d ) \\right ) \\\\ & = x \\sum _ { d = 1 } ^ \\infty \\frac { g ( d ) } { d } + O \\left ( x \\sum _ { d > x } \\frac { g ( d ) } { d } + \\sum _ { d \\leq x } g ( d ) \\right ) . \\end{align*}"}
+{"id": "4219.png", "formula": "\\begin{align*} \\begin{aligned} I _ { 3 5 } \\lesssim & ~ { } \\iint _ { D _ t } \\left ( ( 1 + \\left | \\underline { u } \\right | ^ 2 ) ^ { 1 + \\delta } | \\underline { L } \\partial _ x \\tilde { \\Lambda } | \\right ) \\left ( | \\partial _ x \\tilde { \\Lambda } | | L \\phi | | \\underline { L } \\phi | + | L \\partial _ x \\phi | | \\underline { L } \\phi | + | L \\phi | | \\underline { L } \\partial _ x \\phi | \\right ) \\\\ = : & ~ { } I _ { 3 5 , 1 } + I _ { 3 5 , 2 } + I _ { 3 5 , 3 } . \\end{aligned} \\end{align*}"}
+{"id": "514.png", "formula": "\\begin{align*} { \\textstyle \\sum _ { j = k } ^ { \\nu } } \\ , \\sqrt { \\Phi ( x ^ { \\ell ( j ) } ) \\ ! - \\ ! \\Phi ( x ^ j ) } \\le \\sqrt { b ^ 2 / 2 } \\ , \\Xi _ k + m \\sqrt { ( \\rho \\ ! + \\ ! 1 ) / 2 } \\ , { \\textstyle \\sum _ { j = k - \\widehat { k } _ 1 } ^ { \\nu } } \\ , \\| x ^ { j + 1 } \\ ! - \\ ! x ^ { j } \\| . \\end{align*}"}
+{"id": "2048.png", "formula": "\\begin{align*} \\mathcal { L } ( t _ { k + 1 } - t _ k \\mid X _ { t _ k } ^ { ( 0 ) } < 0 ) = \\mathcal { L } ( t _ 1 \\mid X _ 0 ^ { ( 0 ) } = x ) \\mathcal { L } ( t _ { k + 1 } - t _ k \\mid X _ { t _ k } ^ { ( 0 ) } \\geq 0 ) = \\mathcal { L } ( t _ 1 \\mid X _ 0 ^ { ( 0 ) } = y ) \\end{align*}"}
+{"id": "4130.png", "formula": "\\begin{align*} d K = \\triangle \\omega d V _ g \\end{align*}"}
+{"id": "6272.png", "formula": "\\begin{align*} r ( y ) = \\begin{cases} 1 - e ^ { - \\frac { 1 } { y } } & y > 0 \\\\ 1 & y \\leq 0 \\end{cases} . \\end{align*}"}
+{"id": "2295.png", "formula": "\\begin{align*} \\omega ^ { \\pm } _ k = \\frac { \\alpha ^ { \\pm } _ k } { \\sum ^ 2 _ { l = 0 } \\alpha ^ { \\pm } _ l } , ~ ~ \\alpha ^ { \\pm } _ k = \\frac { \\gamma ^ { \\pm } _ k } { ( \\beta _ k + \\epsilon ) ^ 2 } , ~ ~ k = 0 , 1 , 2 , \\end{align*}"}
+{"id": "3787.png", "formula": "\\begin{align*} R _ 4 = \\{ \\iota ^ 2 , \\iota \\alpha = \\alpha \\iota \\} \\cup \\{ \\iota \\varepsilon _ i = \\varepsilon _ i ^ { - 1 } \\iota \\mid i = 1 , 2 \\} \\end{align*}"}
+{"id": "3092.png", "formula": "\\begin{align*} B = C + b M \\qquad C : = I _ 2 , M : = \\mathrm { d i a g } ( 1 , - 1 ) , \\end{align*}"}
+{"id": "1503.png", "formula": "\\begin{align*} \\log \\C _ 2 \\left ( \\frac 1 4 \\right ) = \\frac 1 { 2 ^ 3 } \\log 2 - \\frac { G } { 2 \\pi } , \\end{align*}"}
+{"id": "7002.png", "formula": "\\begin{align*} K _ { ( Z _ x , Z _ y ) } = \\begin{pmatrix} 1 - \\sigma ^ 2 _ { \\theta } & \\rho - \\sigma ^ 2 _ { \\theta } \\\\ \\rho - \\sigma ^ 2 _ { \\theta } & 1 - \\sigma ^ 2 _ { \\theta } \\end{pmatrix} . \\end{align*}"}
+{"id": "9015.png", "formula": "\\begin{align*} \\operatorname { p } _ f ^ { ( O W ) } [ \\pi , \\eta ] = \\limsup _ { i \\in I } \\frac { P _ { f _ { A _ i } } [ \\eta _ { F _ i } ] } { \\theta ( A _ i ) } \\leq \\limsup _ { i \\in I } \\frac { P _ { f _ { A _ i } } ( \\mathcal { U } _ { F _ i } ) } { \\theta ( A _ i ) } \\leq \\operatorname { p } _ f ^ { ( O W ) } ( \\pi ) . \\end{align*}"}
+{"id": "2945.png", "formula": "\\begin{align*} \\mathcal { Q } ^ n _ \\rho ( \\ell ; t ) = \\frac { g ^ { \\prime \\prime } ( 0 ) } { 2 } \\bigg ( \\big ( \\overrightarrow { W } ^ \\ell _ 0 ( t ) \\big ) ^ 2 - \\frac { \\sigma ^ 2 _ n ( \\rho ) } { \\ell } \\bigg ) \\end{align*}"}
+{"id": "6659.png", "formula": "\\begin{align*} R _ { i i } = - \\sum _ { j \\in \\mathbb { N } _ { R , i } ^ { \\rm i n } } R _ { i j } , C _ { i i } = - \\sum _ { j \\in \\mathbb { N } _ { C , i } ^ { \\rm o u t } } C _ { j i } \\end{align*}"}
+{"id": "3091.png", "formula": "\\begin{align*} - B : D ^ 2 w _ B = b - \\bar { b } \\quad Y , w _ B Y , \\int _ Y w _ B = 0 . \\end{align*}"}
+{"id": "1638.png", "formula": "\\begin{align*} C = \\sup _ { X _ T } | F ( t , x , 0 ) | + ( \\lambda _ F + 1 ) \\sup _ X ( | \\phi | + | \\Phi | ) + \\max ( - c _ - , c _ + ) . \\end{align*}"}
+{"id": "2797.png", "formula": "\\begin{align*} [ \\tau f ] _ { G } & = \\frac { 1 } { | G | } \\sum _ { \\sigma \\in G } \\sigma ( \\tau f ) = \\frac { 1 } { | G | } \\sum _ { \\sigma \\in G } ( \\sigma \\tau ) f = \\frac { 1 } { | G | } \\sum _ { \\sigma \\in G } \\sigma f = [ f ] _ { G } . \\end{align*}"}
+{"id": "5566.png", "formula": "\\begin{align*} \\zeta ( 2 s + k ) \\left ( \\int _ { 0 } ^ { N } + \\int _ { N } ^ { \\infty } \\right ) x ^ { - s - 1 } P _ k ( x ) { \\rm d } x = \\Gamma ( - s ) \\end{align*}"}
+{"id": "7870.png", "formula": "\\begin{align*} f ( z ) = S ( z ) + F _ 1 ( z ) e ^ { w _ 1 z ^ q } + \\cdots + F _ m ( z ) e ^ { w _ m z ^ q } , \\end{align*}"}
+{"id": "4716.png", "formula": "\\begin{align*} ( \\varepsilon _ 1 , \\ldots , \\varepsilon _ s ) = ( e _ 1 , \\ldots , e _ s ) M ^ { - 1 } . \\end{align*}"}
+{"id": "6078.png", "formula": "\\begin{align*} v _ n ( z ) = \\varkappa _ n \\tilde q _ n ^ 2 ( z ) , \\end{align*}"}
+{"id": "4174.png", "formula": "\\begin{align*} \\| ( \\widetilde { g } ( t _ 1 ) , \\widetilde { H } ( t _ 1 ) ) - ( \\widetilde { g } ( t _ 2 ) , \\widetilde { H } ( t _ 2 ) ) \\| _ { C ^ { k - 1 } _ { g _ c } } & = \\int _ { t _ 1 } ^ { t _ 2 } \\| ( \\frac { \\partial \\widetilde { g } } { \\partial t } , \\frac { \\partial \\widetilde { H } } { \\partial t } ) \\| _ { C ^ { k - 1 } _ { g ( t ) } } d t \\\\ & \\leq C | \\lambda ( \\widetilde { g } ( t _ 1 ) , \\widetilde { H } ( t _ 1 ) ) - \\lambda ( g _ c , H _ c ) | ^ \\theta \\leq C ( t _ 1 + 1 ) ^ { \\frac { \\theta } { 1 - 2 \\alpha } } . \\end{align*}"}
+{"id": "5716.png", "formula": "\\begin{align*} \\mathbf { x } ^ { b } = \\xi ^ { b } - ( \\theta ^ { b } - d \\xi ^ { b } ) \\mathbf { m } , \\end{align*}"}
+{"id": "8104.png", "formula": "\\begin{align*} H _ { m , n } ^ { + , 1 } ( x ) = 4 T \\int _ { - \\infty } ^ \\infty \\widehat { k ^ * } ( \\xi ) \\cos \\Bigl ( x \\cosh \\frac { \\xi \\pi } { M } \\Bigr ) e \\Bigl ( - \\frac { T \\xi } { M } \\Bigr ) \\ , d \\xi . \\end{align*}"}
+{"id": "4774.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\Pr _ { \\rho _ N \\sim P _ \\epsilon } \\big [ c _ N \\notin d _ N ( c _ N + \\rho _ N ) \\big ] = 0 . \\end{align*}"}
+{"id": "482.png", "formula": "\\begin{align*} u ^ { ( k ) } ( \\cdot , t ) : = u ^ { ( k ) } _ i \\mbox { f o r } t \\in ( ( i - 1 ) h _ k , \\ , i h _ k ) \\mbox { w i t h } i = 0 , 1 , \\ldots , k , \\end{align*}"}
+{"id": "1192.png", "formula": "\\begin{align*} \\Upsilon ^ { * } ( w _ { j } - A _ j ^ { [ 5 ] } ( w _ 1 , \\ldots , w _ { n - 1 } ) w _ n ) = w _ { j } + A _ { j , 1 } ^ { [ 3 ] } w _ { n } ^ 2 + A _ { j , 2 } ^ { [ 3 ] } w _ n \\overline { w } _ { n } + A _ { j , 3 } ^ { [ 3 ] } \\overline { w } _ n ^ 2 , \\end{align*}"}
+{"id": "6270.png", "formula": "\\begin{align*} \\varphi _ { 0 } ^ { t } ( 0 , y , - 1 ) = ( 0 , \\gamma ( t ) , - 1 ) \\end{align*}"}
+{"id": "8357.png", "formula": "\\begin{align*} ( \\Box + V _ 2 ) v = 0 , V _ 2 = \\sum _ { j = 1 } ^ 5 G _ j ^ { ( 3 ) } v ^ { j - 1 } . \\end{align*}"}
+{"id": "2318.png", "formula": "\\begin{align*} ( A \\circ B ) ^ { \\# } = B ^ { \\# } \\circ A ^ { \\# } . \\end{align*}"}
+{"id": "3491.png", "formula": "\\begin{align*} { \\gamma _ { { \\rm { Z F } } } } \\to \\bar P \\sum \\nolimits _ { l = 1 } ^ L { { { \\left \\| { { { \\bf { h } } _ l } } \\right \\| } ^ 2 } } . \\end{align*}"}
+{"id": "2548.png", "formula": "\\begin{align*} Z ( v ; ( \\alpha , \\beta ) ) = Z ( \\tau ( v ) ; ( \\beta , \\alpha ) ) , \\mathrm { R e } ( \\alpha ) , \\mathrm { R e } ( \\beta ) > 0 \\end{align*}"}
+{"id": "3488.png", "formula": "\\begin{align*} { \\bf { f } } _ l ^ { \\rm { Z F } } = \\frac { { \\sqrt P { { \\bf { Q } } _ l } { { \\bf { h } } _ l } } } { { \\sqrt { \\sum \\nolimits _ { l = 1 } ^ L { { { \\left \\| { { { \\bf { Q } } _ l } { { \\bf { h } } _ l } } \\right \\| } ^ 2 } } } } } , \\ \\forall l , \\end{align*}"}
+{"id": "2281.png", "formula": "\\begin{align*} - \\Delta \\partial _ t \\Phi _ { N } - \\varepsilon \\Delta ^ { 2 m + 1 } \\partial _ t \\Phi _ { N } = - { \\rm { d i v } } _ x \\int _ { \\mathbb { R } ^ 3 } v f _ { N } \\ , d v . \\end{align*}"}
+{"id": "2604.png", "formula": "\\begin{align*} L _ i = \\bigcup _ { j \\in [ r ] \\setminus \\{ i \\} } A _ { i , j } , \\end{align*}"}
+{"id": "7678.png", "formula": "\\begin{align*} [ [ \\bigoplus _ { I } \\eta _ { I } ] ] = [ [ \\bigoplus _ { I } \\eta _ { I , - \\iota _ { E } ( \\omega ) } ] ] . \\end{align*}"}
+{"id": "2795.png", "formula": "\\begin{align*} [ \\tau f ] _ { G } = [ f ] _ { G } \\end{align*}"}
+{"id": "9122.png", "formula": "\\begin{align*} g ( x ) = q ( x ) h ( x ) + r ( x ) \\end{align*}"}
+{"id": "8759.png", "formula": "\\begin{align*} E & = \\gamma _ 2 ( [ 0 , 1 ] \\times \\mathbb { S } _ 2 ^ { r - 1 } \\times \\mathbb { S } _ 2 ^ { r - 1 } , d _ 2 ) + \\gamma _ 1 ( [ 0 , 1 ] \\times \\mathbb { S } _ 2 ^ { r - 1 } \\times \\mathbb { S } _ 2 ^ { r - 1 } , d _ \\infty ) , \\\\ V & = \\Delta _ 2 ( [ 0 , 1 ] \\times \\mathbb { S } _ 2 ^ { r - 1 } \\times \\mathbb { S } _ 2 ^ { r - 1 } , d _ 2 ) , U = \\Delta _ \\infty ( [ 0 , 1 ] \\times \\mathbb { S } _ 2 ^ { r - 1 } \\times \\mathbb { S } _ 2 ^ { r - 1 } , d _ \\infty ) . \\end{align*}"}
+{"id": "5187.png", "formula": "\\begin{align*} \\frac { r ( r - K ) } { 2 K } + a _ 2 ' t \\frac { r } { K } = y _ t ' = r v _ { 3 , t + 4 } + y _ t + u _ { 2 3 } \\frac { r } { K } = r v _ { 3 , t + 4 } + \\frac { r ( r - K ) } { 2 K } + a _ 2 t \\frac { r } { K } + u _ { 2 3 } \\frac { r } { K } \\end{align*}"}
+{"id": "2516.png", "formula": "\\begin{align*} ( \\partial _ t f + v \\cdot \\nabla _ x f ) ( x , v , t ) = \\lambda ( \\varrho ) \\big ( \\varrho ( x , t ) M _ f ( x , v , t ) - f ( x , v , t ) \\big ) \\ , , \\end{align*}"}
+{"id": "2774.png", "formula": "\\begin{align*} u _ i & = \\sum _ { \\ell = i } ^ n x _ \\ell - \\sum _ { \\ell = i } ^ n x ' _ \\ell \\\\ & = x _ { \\lambda _ 1 } + x _ { \\lambda _ 2 } + x _ { d _ 1 } - x ' _ { \\lambda _ 1 } - x ' _ { \\lambda _ 2 } - x ' _ { d _ 2 } \\end{align*}"}
+{"id": "3833.png", "formula": "\\begin{align*} \\Sigma ^ { - 1 } = \\left [ \\begin{matrix} 1 & \\theta \\\\ \\theta & 1 \\end{matrix} \\right ] \\ \\ \\end{align*}"}
+{"id": "7245.png", "formula": "\\begin{align*} 1 - \\phi _ f ( u , 1 ) = \\int _ { \\partial B _ n \\cap H ( u , 1 ) ^ + } f ( x ) \\ , d \\mu _ { \\partial B _ n } ( x ) \\geq \\frac { 1 } { 2 } c _ { \\min } ( f ) \\mu _ { \\partial B _ n } ( \\partial B _ n ) . \\end{align*}"}
+{"id": "8087.png", "formula": "\\begin{align*} \\mathcal { R } ^ + _ 2 = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { \\frac { C _ 2 } { m } \\leq c \\leq \\frac { C _ 1 } { m } } \\frac { S ( n , p ; c ) } { c } H _ { m , n } ^ + \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) , \\end{align*}"}
+{"id": "6552.png", "formula": "\\begin{align*} \\mathop { \\max } \\limits _ { \\{ { P _ { a k } } \\} } & \\sum \\limits _ { k = 1 } ^ K { \\ln \\left ( { { R _ k } - R _ k ^ { { \\rm { t h } } } } \\right ) } \\\\ { \\rm s . t . } & \\eqref { s t 1 } , \\eqref { s t 2 } , \\eqref { s t 3 } . \\end{align*}"}
+{"id": "2937.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { S } ^ n _ t ( \\varphi ) = \\int _ 0 ^ t S _ n \\mathcal { X } ^ n _ s ( \\varphi ) d s = \\frac { 1 } { 2 \\sqrt { n } } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } g _ n ( \\eta ^ n _ j ( s ) ) \\Delta ^ n \\varphi ^ n _ j ( s ) d s \\end{aligned} \\end{align*}"}
+{"id": "3695.png", "formula": "\\begin{align*} Q _ i ( x ) : = \\frac { x ^ t J _ i x } { 2 } . \\end{align*}"}
+{"id": "8783.png", "formula": "\\begin{align*} \\dot { \\phi } _ i ( t ) = & \\omega _ i + \\frac { 1 } { N } \\sum _ { j = 1 } ^ N W _ { i j } ( t ) g ( \\phi _ j ( t ) - \\phi _ i ( t ) ) , \\\\ \\dot { W } _ { i j } ( t ) = & - \\varepsilon ( W _ { i j } + h ( \\phi _ j - \\phi _ i ) ) , \\end{align*}"}
+{"id": "9016.png", "formula": "\\begin{align*} \\operatorname { p } ^ { ( O W ) } _ f [ \\pi , \\eta ] = \\lim _ { i \\in I } \\frac { P _ { f _ { K F _ i } } ( \\eta _ { K F _ i } ) } { \\theta ( K F _ i ) } = \\lim _ { i \\in I } \\frac { P _ { \\sum _ { F _ i } f _ { K } } ( ( \\eta _ K ) _ { F _ i } ) } { \\theta ( K ) | F _ i | } = \\operatorname { d e n s } ( \\omega ) \\operatorname { p } ^ { ( O W ) } _ { f _ K } [ \\phi , \\eta _ K ] . \\end{align*}"}
+{"id": "4394.png", "formula": "\\begin{align*} U _ n ^ { ( 1 ) } ( \\theta ) = \\left \\{ \\left ( x _ i \\right ) _ { i = 1 } ^ n \\in U _ n ( \\theta ) x _ 1 < x _ 1 ^ * \\right \\} . \\end{align*}"}
+{"id": "1906.png", "formula": "\\begin{align*} \\Big | [ N ] \\cdot [ N ] \\Big | = \\frac { N ^ { 2 } } { ( \\log N ) ^ { 2 \\theta + o ( 1 ) } } , \\end{align*}"}
+{"id": "3344.png", "formula": "\\begin{align*} \\Phi _ { y x } = - \\frac { 1 } { \\Phi _ { x q } \\Phi _ { x p } } [ \\Phi _ { y q } ( \\Phi _ { x y } \\Phi _ { y p } + \\Phi _ { x q } \\Phi _ { q p } ) + \\Phi _ { y p } \\Phi _ { p q } \\Phi _ { x p } ] . \\end{align*}"}
+{"id": "9239.png", "formula": "\\begin{align*} \\ < \\Delta _ { f } v _ { j } , v _ { k } \\ > = \\delta _ { j , k } \\mu _ { j } + \\mathcal { O } ( h ^ { \\infty } \\sqrt { \\mu _ { j } \\mu _ { k } } ) . \\end{align*}"}
+{"id": "2540.png", "formula": "\\begin{align*} d ( \\tilde T _ N ^ \\varphi - T _ g ^ \\varphi ) = \\sum _ { j = 1 } ^ N q _ { 1 , j } | w _ j | ^ 2 - ( d T _ g ^ \\varphi + | u _ g ^ \\varphi | ^ 2 ) + | u _ g ^ \\varphi | ^ 2 - | \\tilde u _ N ^ \\varphi | ^ 2 \\ , , \\end{align*}"}
+{"id": "4665.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { \\min \\{ n , 2 m \\} } A _ { m , \\ell } = \\sum _ { \\ell = 0 } ^ n \\sum _ { m = \\lceil \\ell / 2 \\rceil } ^ { \\infty } A _ { m , \\ell } . \\end{align*}"}
+{"id": "5666.png", "formula": "\\begin{align*} \\lambda _ 2 = - \\frac { 4 \\pi D } { | \\Omega | } \\sum _ { j = 1 } ^ N \\ell _ j \\chi _ j . \\end{align*}"}
+{"id": "2831.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ n } t _ { \\sigma } = 1 \\end{align*}"}
+{"id": "6215.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p \\beta _ r E _ r = C _ p ^ T x . \\end{align*}"}
+{"id": "3391.png", "formula": "\\begin{align*} \\iota \\circ \\sigma \\circ t \\circ \\sigma = f ^ * \\circ t \\circ \\sigma = f ^ * \\circ \\alpha ^ * = f ^ * = \\iota \\circ \\sigma = \\iota \\circ \\mbox { i d } \\circ \\sigma . \\end{align*}"}
+{"id": "6748.png", "formula": "\\begin{align*} z _ N ( s ) = \\frac { 4 } { \\sqrt { \\pi } } \\int _ { 0 } ^ { 1 } \\frac { 1 } { \\sqrt { 1 - \\rho } } \\left [ \\frac { 1 } { 2 } + \\sum _ { n = 1 } ^ { N } \\exp \\left [ - n ^ 2 \\pi \\left ( 1 - \\rho \\right ) ^ { \\frac { 1 } { s } } \\right ] \\right ] d \\rho . \\end{align*}"}
+{"id": "7783.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow 1 } [ \\alpha ] _ q = \\alpha . \\end{align*}"}
+{"id": "2466.png", "formula": "\\begin{align*} m _ { R } = \\inf _ { Q ( R ) \\backslash S } \\Gamma , M _ { R } = \\sup _ { Q ( R ) \\backslash S } \\Gamma , J _ { R } = M _ { R } - m _ { R } , \\end{align*}"}
+{"id": "8190.png", "formula": "\\begin{align*} k ( t , s ) : = \\frac { a } { \\Gamma ( b / a ) } ( t ^ a - s ^ a ) ^ { \\frac { b } { a } - 1 } t ^ { a - \\nu } s ^ { \\nu - 1 } F _ 3 \\left ( \\frac { \\nu } { a } - 1 , \\frac { b } { a } , 1 , \\mu , \\frac { b } { a } , 1 - \\left ( \\frac { s } { t } \\right ) ^ a , 1 - \\left ( \\frac { t } { s } \\right ) ^ a \\right ) , \\end{align*}"}
+{"id": "1794.png", "formula": "\\begin{align*} P ( [ \\pi _ E ^ { \\flat } ] ) = [ \\pi _ E ^ \\flat ] ^ d + a _ 1 [ \\pi _ E ^ \\flat ] ^ { d - 1 } + \\dots + a _ d . \\end{align*}"}
+{"id": "6605.png", "formula": "\\begin{align*} A = \\int _ M ^ \\oplus A ( m ) \\dd m , ( A f ) _ m = A ( m ) f _ m . \\end{align*}"}
+{"id": "2931.png", "formula": "\\begin{align*} \\overline { \\nu } ^ n _ \\alpha ( \\eta _ j = k ) = ( \\alpha / \\sqrt { n } ; q _ n ) _ \\infty \\frac { ( \\alpha / \\sqrt { n } ) ^ k } { ( q _ n ; q _ n ) _ k } , k \\in \\mathbb { Z } _ + \\end{align*}"}
+{"id": "4848.png", "formula": "\\begin{align*} v _ { t _ 0 , s } ( t ) = v \\left ( \\frac { t - t _ 0 } { s } \\right ) \\chi _ { [ t _ 0 , t _ 0 + s ] } ( t ) , t \\in I . \\end{align*}"}
+{"id": "7820.png", "formula": "\\begin{gather*} \\prod _ { j = 1 } ^ { k d _ 1 } \\prod _ { i = 1 } ^ { d _ 1 } \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} _ { 1 , \\frac { 2 k d _ 1 + 1 + k - 2 j - 2 k i } { k d _ 1 } } = \\prod _ { i = 1 } ^ { d _ 1 } \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} _ { \\frac { 1 } { k d _ 1 } , \\frac { d _ 1 + 1 - 2 i } { d _ 1 } } . \\end{gather*}"}
+{"id": "4278.png", "formula": "\\begin{align*} H _ \\gamma ( \\phi ) & = 2 E _ \\gamma ( \\phi ) + \\frac { 4 } { p + 1 } \\| \\phi \\| ^ { p + 1 } _ { L ^ { p + 1 } } \\\\ & \\leq 2 I ^ m _ \\gamma ( c ) + B m ^ { \\frac { N ( p - 1 ) } { 4 } } c ^ { \\frac { 4 - ( N - 2 ) ( p - 1 ) } { 4 } } \\\\ & \\leq 2 \\omega ^ 0 _ \\gamma c + B m ^ { \\frac { N ( p - 1 ) } { 4 } } c ^ { \\frac { 4 - ( N - 2 ) ( p - 1 ) } { 4 } } \\end{align*}"}
+{"id": "2975.png", "formula": "\\begin{align*} \\begin{aligned} & 2 n ^ { - 2 } ( | \\mathrm { s u p p } G _ j | + 1 ) \\sum _ { \\substack { j , k \\in \\mathbb { Z } , \\\\ k \\in \\mathrm { s u p p } G _ j \\cup ( \\mathrm { s u p p } G _ j + 1 ) } } E _ { \\nu ^ n _ \\rho } [ g ( \\eta _ k ) G _ j ( \\eta ) ^ 2 ] ( \\nabla ^ n \\varphi ^ n _ j ) ^ 2 \\\\ & + \\frac { n ^ 2 } { 2 } \\sum _ { k \\in \\mathbb { Z } } E _ { \\nu ^ n _ \\rho } [ g ( \\eta _ k ) ( \\nabla _ { k , k - 1 } f ( \\eta ) ) ^ 2 ] . \\end{aligned} \\end{align*}"}
+{"id": "6098.png", "formula": "\\begin{align*} \\int _ \\mathcal { F } f ( z ) \\overline { \\Theta ( \\varphi , \\tau , z ) } \\Im ( z ) ^ { - k } \\frac { d x d y } { y ^ 2 } , \\ , z = x + i y . \\end{align*}"}
+{"id": "2120.png", "formula": "\\begin{align*} x ^ n = 1 , \\tilde y ^ 2 = - 1 , \\tilde y x = x ^ { - 1 } \\tilde y . \\end{align*}"}
+{"id": "1800.png", "formula": "\\begin{align*} \\mathcal { A } ( u ) = \\int _ { 0 } ^ { T } L ( t , u , \\dot { u } ) d t , \\end{align*}"}
+{"id": "6358.png", "formula": "\\begin{gather*} | f | _ { 1 , n } : = \\sup \\{ | f ( x ) | + | f ' ( x ) | ; x \\in I _ n \\} \\mbox { a n d } \\\\ | f | _ { 2 , n } : = \\sup \\{ | f ( x ) | + | f ' ( x ) | + | f '' ( x ) | ; x \\in I _ n \\} , \\end{gather*}"}
+{"id": "8732.png", "formula": "\\begin{align*} \\mathcal G ( u _ 1 , \\dots , u _ l ) = B ^ + ( u _ 1 ) \\dots B ^ + ( u _ l ) \\ , ( 1 ) , \\end{align*}"}
+{"id": "8169.png", "formula": "\\begin{align*} \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } U ( m ^ 2 p , t ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 0 0 0 ) } p ^ { - u } L ( 1 + 2 u , \\tilde f ) F ( u ) \\frac { \\gamma \\Bigl ( \\dfrac { 1 } { 2 } + u , t \\Bigr ) } { \\gamma \\Bigl ( \\dfrac { 1 } { 2 } , t \\Bigr ) } \\frac { d u } { u ^ 2 } . \\end{align*}"}
+{"id": "403.png", "formula": "\\begin{align*} \\theta _ \\tau : \\ell ^ p ( \\mathbb { N } ) \\ni \\{ a _ n \\} _ n \\mapsto \\theta _ \\tau \\{ a _ n \\} _ n \\coloneqq \\sum _ { n = 1 } ^ \\infty a _ n \\tau _ n \\in \\mathcal { X } \\end{align*}"}
+{"id": "255.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\tau } { \\rho } + \\frac { \\beta \\eta } { \\rho } \\geqslant \\frac { \\alpha } { 2 } \\frac { ( 1 + 3 \\rho ^ 2 ) } { \\rho ^ 2 } \\geqslant \\frac { 3 \\alpha } { 2 } , \\end{align*}"}
+{"id": "9187.png", "formula": "\\begin{align*} Z _ j ( \\varphi , \\Psi _ j ; ( \\Psi _ k ) _ { k < j } | \\Lambda ) = e ^ { - E _ { j } | \\Lambda | + e _ j } \\sum _ { X \\in \\mathcal { P } _ j ( \\Lambda ) } e ^ { U _ j ( \\Lambda \\backslash X , \\varphi ) } \\prod _ { Y \\in \\textnormal { C o m p } _ j ( X ) } \\big ( K _ j ( Y , \\varphi ; ( \\Psi _ k ) _ { k < j } ) + \\Psi _ j ( Y , \\varphi ) \\big ) , \\end{align*}"}
+{"id": "6138.png", "formula": "\\begin{align*} \\Phi = \\sum _ { r = 1 } ^ d \\phi _ r E _ r , \\end{align*}"}
+{"id": "45.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m T ( n _ i ) & = \\sum _ { i = 1 } ^ { m - m _ 2 } T ( n _ i ) + T ( n _ { m - m _ 2 + 1 } ) + \\sum _ { i = m - m _ 2 + 1 + 1 } ^ m T ( n _ i ) \\\\ & < \\frac { 9 7 ( m - m _ 2 ) + 1 } { 5 4 \\cdot X _ 0 } + \\frac { 3 } { 2 ^ v - 1 } + \\frac { 3 \\cdot ( m _ 2 - 1 ) } { ( 2 ^ v - 1 ) ^ { \\delta } } , \\end{align*}"}
+{"id": "2708.png", "formula": "\\begin{align*} W ^ - ( r ) = - \\frac { c _ W } { r ^ 4 } - \\int _ { r } ^ { \\infty } \\frac { 1 } { \\rho ^ 5 } \\int _ { \\rho } ^ { \\infty } \\left ( W ^ - ( s ) \\right ) ^ 2 s ^ 5 d s \\end{align*}"}
+{"id": "5527.png", "formula": "\\begin{align*} \\zeta ( 2 s + k ) \\int _ { 0 } ^ { \\infty } x ^ { - s - 1 } P _ k ( x ) { \\rm d } x = \\int _ { 0 } ^ { \\infty } x ^ { - s - 1 } e ^ { - x } { \\rm d } x = \\Gamma ( - s ) , \\end{align*}"}
+{"id": "1358.png", "formula": "\\begin{align*} | f _ j ( \\tau _ k ) | ^ 2 = \\frac { 1 } { d + 1 } , \\forall 1 \\leq j , k \\leq d ^ 2 , j \\neq k . \\end{align*}"}
+{"id": "1904.png", "formula": "\\begin{align*} \\tau _ { n } = T \\wedge \\inf \\{ t \\geq 0 : | z ( \\cdot , t ) | _ { L ^ 2 ( 0 , 1 ) } ^ 2 \\geq n \\} , \\end{align*}"}
+{"id": "680.png", "formula": "\\begin{align*} G ( a , b ) = G ^ { \\delta _ a } ( b ) = \\mathcal { E } ( \\delta _ a , \\delta _ b ) . \\end{align*}"}
+{"id": "4764.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\int _ { \\Omega } \\ \\mathcal { H } ( x , | u _ { n } | \\ | \\nabla v | ) = 0 . \\end{align*}"}
+{"id": "8960.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { 1 / 2 } u = \\partial _ r U \\end{align*}"}
+{"id": "6048.png", "formula": "\\begin{align*} \\rho ( x ) = : \\dot \\mu ( x ) m ( x ) , x \\in \\Delta , \\end{align*}"}
+{"id": "1652.png", "formula": "\\begin{align*} ( g \\mu ) ( P ) = \\mu ( g ^ { - 1 } P ) , \\mu \\in V ( k ) . \\end{align*}"}
+{"id": "7827.png", "formula": "\\begin{gather*} \\prod _ { j = 1 } ^ { k d _ 1 - 2 } \\prod _ { i = 1 } ^ { d _ 1 } \\prod _ { t = 1 } ^ { k d _ 1 - j - 1 } \\prod _ { s = 1 } ^ { t } \\begin{bmatrix} 1 \\\\ 3 \\end{bmatrix} _ { 1 , \\frac { k + 1 - 2 j - 2 k i + 2 t + 2 s } { k d _ 1 } } , \\end{gather*}"}
+{"id": "2344.png", "formula": "\\begin{align*} T = \\begin{pmatrix} 0 & K _ M \\\\ - K _ M & 0 \\end{pmatrix} = J _ { 2 M } \\ , K _ { 2 M } . \\end{align*}"}
+{"id": "7459.png", "formula": "\\begin{align*} \\sum _ { i \\in [ s ] } d _ i = 2 { \\rm w t } ( \\mathfrak { d } ) \\end{align*}"}
+{"id": "7229.png", "formula": "\\begin{align*} \\abs { r _ i } \\abs { r _ j } ^ T & = ( J - I ) ( \\abs { H _ { j i } } R ) ^ T + ( \\abs { H _ { i j } } R ) ( J - I ) ^ T + ( q - 1 ) \\sum _ { \\ell = 0 } ^ q \\abs { N } ^ \\ell \\\\ & = ( q + 1 ) J - 2 \\abs { H _ { i j } } R , \\end{align*}"}
+{"id": "5150.png", "formula": "\\begin{align*} \\forall n \\in \\mathbb { N } ^ * , \\quad \\forall \\lambda > 0 , ( I _ { n } K _ { n } ) ( \\lambda ) = \\frac 1 2 \\int _ { 0 } ^ \\infty J _ 0 \\big ( 2 \\lambda \\sinh ( \\tfrac { t } { 2 } ) \\big ) e ^ { - n t } d t . \\end{align*}"}
+{"id": "5801.png", "formula": "\\begin{align*} K e r ( D ) \\cap \\{ K e r ( C _ p ) \\} ^ \\perp = K e r ( C _ p ) \\cap \\{ K e r ( D ^ T ) \\} ^ \\perp = \\{ 0 \\} . \\end{align*}"}
+{"id": "4476.png", "formula": "\\begin{align*} \\frac { 1 } { x _ 1 } + \\frac { 1 } { x _ 2 } + \\cdots + \\frac { 1 } { x _ n } = 1 \\end{align*}"}
+{"id": "184.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int \\big { | } \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } f \\circ T ^ { - i } \\big { | } ^ { 2 } \\ d \\mu = 0 . \\end{align*}"}
+{"id": "2617.png", "formula": "\\begin{align*} V _ { a , b } = \\left < v _ { a , b } ^ t \\ \\middle | \\ t = 1 , \\dots , d \\right > \\end{align*}"}
+{"id": "42.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m T ( n _ i ) & < \\frac { 9 7 } { 5 4 } \\cdot m \\cdot \\frac { 1 } { X _ 0 } + \\varepsilon \\\\ \\sum _ { i = 1 } ^ m T ( n _ i ) & \\leq \\frac { 9 7 } { 5 4 } \\cdot m \\cdot \\frac { 1 } { X _ 0 } , \\end{align*}"}
+{"id": "8912.png", "formula": "\\begin{align*} B _ { 2 ( p + s + 1 ) } \\left ( \\frac { 1 } { 2 } \\right ) = ( - 1 ) ^ { p + s + 1 } ( p + s + 1 ) B _ { p + s } . \\end{align*}"}
+{"id": "3109.png", "formula": "\\begin{align*} c _ j ^ { k l } ( A _ m ) = \\int _ Y r _ m A _ m e _ j \\cdot \\nabla v ^ { k l } _ m \\longrightarrow \\int _ Y r A e _ j \\cdot \\nabla v ^ { k l } = c _ j ^ { k l } ( A ) \\quad m \\rightarrow \\infty , \\end{align*}"}
+{"id": "2646.png", "formula": "\\begin{align*} \\tilde { H } _ k ( x / \\sqrt { 2 } ) : = 2 ^ { k / 2 } \\sqrt { k ! } H _ k ( x ) . \\end{align*}"}
+{"id": "8749.png", "formula": "\\begin{align*} \\hat { \\mathcal { O } } _ { r + 1 : r T , 1 : d _ 0 } : = \\hat { \\mathcal { O } } ^ - , \\ \\bar { \\mathcal { O } } _ { 1 : r ( T - 1 ) , 1 : d _ 0 } : = \\bar { \\mathcal { O } } ^ + , \\ \\bar { \\mathcal { O } } _ { r + 1 : r T , 1 : d _ 0 } : = \\bar { \\mathcal { O } } ^ - . \\end{align*}"}
+{"id": "7486.png", "formula": "\\begin{align*} \\kappa ( X ) = \\kappa ( F ) + \\dim Y . \\end{align*}"}
+{"id": "158.png", "formula": "\\begin{align*} A v = \\frac { \\partial } { \\partial \\hslash } v , \\ D ( A ) : = \\{ v \\in H ^ { 1 } ( ( 0 , \\pi ) ; \\mathbb { R } ) : v ( \\pi ) = 0 \\} . \\end{align*}"}
+{"id": "7464.png", "formula": "\\begin{align*} \\Delta _ I ( N ) : = \\sum _ { \\pi \\colon \\ , \\partial ( \\pi ) = I } { \\rm w t } _ N ( \\pi ) , \\end{align*}"}
+{"id": "8769.png", "formula": "\\begin{align*} W _ { u , v , w , k } = \\begin{bmatrix} & & 0 & w _ { u , v , w , 2 T - 1 } ^ * & w _ { u , v , w , 2 T - 2 } ^ * & \\cdots & w _ { u , v , w , 1 } ^ * & & \\end{bmatrix} , \\end{align*}"}
+{"id": "6912.png", "formula": "\\begin{align*} Z _ { \\sigma } ^ q & = ( u _ 0 x _ 1 ) ( u _ 1 x _ 2 ) \\dots ( u _ d x _ 0 ) Z _ { \\sigma } \\\\ & = \\varpi u _ 0 \\dots u _ d Z _ { \\sigma } = \\varpi Z _ { \\sigma } . \\end{align*}"}
+{"id": "8756.png", "formula": "\\begin{align*} | L ^ * L - \\sigma _ u ^ 2 ( N - 4 T + 3 ) I _ { ( 2 T - 1 ) r } | _ { S _ \\infty } \\leq | \\mathcal { T } | _ { 2 \\to 2 } = \\sup \\limits _ { x \\in [ 0 , 1 ] } | p ( x ) | _ { S _ \\infty } , \\end{align*}"}
+{"id": "8397.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\tau _ { \\alpha } ( x ) - 1 } | \\hat \\psi _ { \\alpha } ( f _ { \\alpha } ^ k ( x ) ) - \\hat \\psi _ { \\alpha } ( 0 ) | \\leq C \\sum _ { k = 0 } ^ { \\tau _ { \\alpha } ( x ) - 1 } | f _ { \\alpha } ^ k ( x ) | . \\end{align*}"}
+{"id": "6574.png", "formula": "\\begin{align*} \\theta ( x , t ) = \\begin{cases} \\mathrm { e v e n \\ i n t e g e r } \\geq 2 \\mathrm { i f } \\ x \\in M _ t \\backslash \\tilde { M } _ t , \\\\ \\mathrm { o d d \\ i n t e g e r } \\geq 1 \\mathrm { i f } \\ x \\in \\tilde { M } _ t \\end{cases} \\end{align*}"}
+{"id": "8418.png", "formula": "\\begin{align*} \\Gamma & : = \\Gamma ( \\partial E , \\sigma _ E ) \\\\ & = \\{ ( \\mu , k - l , \\nu ) \\in \\partial E \\times \\Z \\times \\partial E : \\mu \\in ( \\sigma ^ k ) , \\nu \\in ( \\sigma ^ l ) , \\sigma ^ k ( \\mu ) = \\sigma ^ l ( \\nu ) \\} . \\end{align*}"}
+{"id": "506.png", "formula": "\\begin{align*} \\frac { 3 } { 4 } \\Lambda _ { k + m + 1 } \\le \\frac { 1 } { 4 } { \\textstyle \\sum _ { j = k } ^ { k + m } } \\ , \\Xi _ j + { b c } a ^ { - 1 } \\big [ b c ( 1 - \\theta ) \\big ] ^ { \\frac { 1 - \\theta } { \\theta } } { \\textstyle \\sum _ { j = k } ^ { k + m } } \\ , \\Xi _ j ^ { \\frac { 1 - \\theta } { \\theta } } . \\end{align*}"}
+{"id": "8922.png", "formula": "\\begin{align*} \\mathcal { \\wp } ' = \\sum \\limits _ { p = 0 } ^ { n - 1 } \\tau _ { p } ^ { ( \\nu , n ) } r ^ { 2 p } . \\end{align*}"}
+{"id": "5949.png", "formula": "\\begin{align*} ( E _ r , B \\widehat { U } ) = ( E _ r , D \\widehat H ) \\hbox { o n } \\Gamma , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "8191.png", "formula": "\\begin{align*} F _ 3 \\left ( \\alpha , \\alpha ' , \\beta , \\beta ' , \\gamma , x , y \\right ) : = \\sum _ { m , n \\geq 0 } \\frac { ( \\alpha ) _ m ( \\beta ) _ m ( \\alpha ' ) _ n ( \\beta ' ) _ n } { ( \\gamma ) _ { m + n } n ! m ! } x ^ m y ^ n , \\end{align*}"}
+{"id": "2217.png", "formula": "\\begin{align*} A & = 2 0 5 \\xi + 4 3 0 0 \\xi ^ 2 + 3 4 0 0 0 \\xi ^ 3 + 1 2 0 0 0 0 \\xi ^ 4 + 1 6 0 0 0 0 \\xi ^ 5 , \\\\ B & = 2 1 5 \\xi + 4 4 7 5 \\xi ^ 2 + 3 5 0 0 0 \\xi ^ 3 + 1 2 2 0 0 0 \\xi ^ 4 + 1 6 0 0 0 0 \\xi ^ 5 , \\\\ C & = 8 5 \\xi + 1 7 5 0 \\xi ^ 2 + 1 3 5 2 5 \\xi ^ 3 + 4 6 5 0 0 \\xi ^ 4 + 6 0 0 0 0 \\xi ^ 5 , \\\\ D & = 1 5 \\xi + 3 0 5 \\xi ^ 2 + 2 3 2 5 \\xi ^ 3 + 7 8 7 5 \\xi ^ 4 + 1 0 0 0 0 \\xi ^ 5 , \\\\ E & = \\xi + 2 0 \\xi ^ 2 + 1 5 0 \\xi ^ 3 + 5 0 0 \\xi ^ 4 + 6 2 5 \\xi ^ 5 . \\end{align*}"}
+{"id": "6425.png", "formula": "\\begin{align*} \\Gamma _ \\epsilon ^ 1 & : = ( \\Gamma _ X ^ \\epsilon \\times \\mathbb R ^ m \\times ( - \\epsilon ^ 2 / 2 , T ] ) , \\ \\Gamma _ \\epsilon ^ 2 : = ( U _ X \\times \\mathbb R ^ m \\times ( - \\epsilon ^ 2 / 2 , 0 ] ) . \\end{align*}"}
+{"id": "4722.png", "formula": "\\begin{align*} \\mathcal { G } = \\{ ( v _ 1 , w _ 1 ) , ( v _ 2 , w _ 2 ) , \\ldots , ( v _ { c + d } , w _ { c + d } ) , ( x _ 1 , y _ 1 ) , ( x _ 2 , y _ 2 ) , \\ldots , ( x _ { c } , y _ { c } ) \\} \\end{align*}"}
+{"id": "4405.png", "formula": "\\begin{align*} \\sum _ { i = m + 1 } ^ n \\frac { 1 } { x _ i } < \\sum _ { i = m + 1 } ^ n \\frac { 1 } { a _ i } . \\end{align*}"}
+{"id": "8361.png", "formula": "\\begin{align*} w = 0 , \\mathrm { f o r } \\ | x | ^ 2 - t ^ 2 > R ^ 2 , t \\ge 0 . \\end{align*}"}
+{"id": "2372.png", "formula": "\\begin{align*} E \\dot x = A x + B u + f , x ( t _ 0 ) = { x _ 0 } \\in { \\mathbb R } ^ { n } , \\end{align*}"}
+{"id": "4334.png", "formula": "\\begin{align*} \\C = \\mathcal { M } \\C \\boxtimes \\C ( \\Z _ m , P ) . \\end{align*}"}
+{"id": "3707.png", "formula": "\\begin{align*} L ( u ( x ) ) \\mathcal { F } _ 2 ^ { - 1 } ( f ) ( \\xi , \\xi _ 1 ' , \\xi ' _ 2 ) & = \\int _ { F } f ( \\xi - J _ i x \\xi _ 1 ' , \\xi _ 1 ' , v ) \\bar { \\psi } ( ( x ^ t \\xi - Q _ i ( x ) \\xi ' _ 1 + \\xi _ 2 ' ) v ) d v . \\end{align*}"}
+{"id": "192.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { n - 1 } ( r _ { j } + 1 ) = \\prod _ { j = 1 } ^ { n - 1 } 2 ^ { j ^ { \\alpha } } = 2 ^ { \\sum _ { j = 1 } ^ { n - 1 } j ^ { \\alpha } } 2 ^ { d ( n - 1 ) ^ { 1 + \\alpha } } \\leq \\prod _ { j = 1 } ^ { n - 1 } ( r _ { j } + 1 ) \\leq 2 ^ { n ^ { 1 + \\alpha } } . \\end{align*}"}
+{"id": "4641.png", "formula": "\\begin{align*} T ( r , \\tilde { \\gamma } , L ) = O ( \\log r ) . \\end{align*}"}
+{"id": "3782.png", "formula": "\\begin{align*} \\iota \\varepsilon _ \\# ( v ) & = \\begin{cases} \\iota ( v ) & v \\in V _ 1 ( \\varepsilon ) ; \\\\ \\iota ( x _ e ^ { - 1 } v x _ e ) & v \\not \\in V _ 1 ( \\varepsilon ) . \\end{cases} \\\\ & = \\begin{cases} v & v \\in V _ 1 ( \\varepsilon ) ; \\\\ \\overline { x _ e } ^ { - 1 } v ^ { - 1 } \\overline { x _ e } & v \\not \\in V _ 1 ( \\varepsilon ) , \\end{cases} \\end{align*}"}
+{"id": "5488.png", "formula": "\\begin{align*} L : E \\to E , L x = L ( x ^ + + x ^ - + x ^ 0 ) : = x ^ + - x ^ - , \\end{align*}"}
+{"id": "6120.png", "formula": "\\begin{align*} \\hbox { r a n k } ( D , A D , \\cdots , A ^ { N - 1 } D ) = N \\end{align*}"}
+{"id": "244.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho ^ 2 - \\beta \\tau \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "8137.png", "formula": "\\begin{align*} K _ \\nu ( z ) = \\frac { \\pi } { 2 } \\frac { I _ { - \\nu } ( z ) - I _ \\nu ( z ) } { \\sin ( \\pi \\nu ) } , \\end{align*}"}
+{"id": "8492.png", "formula": "\\begin{align*} \\varphi _ { \\xi ^ 3 } ( \\xi , t _ 0 ) + \\varphi ( \\xi , t _ 0 ) \\varphi _ { \\xi ^ 2 } ( \\xi , t _ 0 ) + \\frac { 2 \\varphi ( \\xi , t _ 0 ) ^ 2 + 3 \\varphi _ { \\xi } ( \\xi , t _ 0 ) + \\epsilon } { 9 } \\varphi _ { \\xi } ( \\xi , t _ 0 ) = 0 . \\end{align*}"}
+{"id": "8440.png", "formula": "\\begin{align*} \\sup _ { \\nu \\in \\mathcal E ^ + _ A } \\ , G ( \\nu ) = \\Bigl [ \\inf _ { \\mu \\in \\breve { \\mathcal E } ^ + _ A } \\ , \\| \\mu \\| ^ 2 \\Bigr ] ^ { - 1 } = c _ * ( A ) , \\end{align*}"}
+{"id": "4422.png", "formula": "\\begin{align*} I ( a _ 1 ) & = \\bigcup _ { a _ 2 = a _ 1 ^ 2 - a _ 1 + 1 } ^ { \\infty } J ( a _ 1 , a _ 2 ) = \\bigcup _ { a _ 2 = a _ 1 ^ 2 - a _ 1 + 1 } ^ { \\infty } \\left ( \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 } , \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 - 1 } \\right ] . \\end{align*}"}
+{"id": "4370.png", "formula": "\\begin{align*} a _ { k + 1 } = q \\prod _ { i = 1 } ^ k a _ i + 1 \\end{align*}"}
+{"id": "5432.png", "formula": "\\begin{align*} \\alpha _ { t s } ( \\omega ' ) = \\gamma _ t ^ { - 1 } { } ^ * \\left ( \\alpha _ s ( \\gamma _ t ^ * \\omega ' ) \\right ) . \\end{align*}"}
+{"id": "3319.png", "formula": "\\begin{align*} e _ { k } ( x ) & = \\frac { 1 } { k } \\sum _ { 0 \\le \\ell \\le k } ( - 1 ) ^ { \\ell - 1 } e _ { k - \\ell } ( x ) p _ { i } ( x ) \\end{align*}"}
+{"id": "7752.png", "formula": "\\begin{align*} g _ { \\mu , \\lambda } ( x ) & = \\frac { \\lambda ^ \\mu } { \\Gamma ( \\mu ) } \\ , x ^ { \\mu - 1 } e ^ { - \\lambda x } = \\frac { \\mu \\ , \\lambda ^ \\mu } { \\Gamma ( 1 + \\mu ) } \\ , x ^ { \\mu - 1 } e ^ { - \\lambda x } \\mu , \\ , \\lambda > 0 \\end{align*}"}
+{"id": "3901.png", "formula": "\\begin{align*} Y u ( x _ 0 ) : = \\{ y ; v ( y ) = g ^ * ( x _ 0 , y , u ( x _ 0 ) ) \\} , \\end{align*}"}
+{"id": "7730.png", "formula": "\\begin{align*} \\Phi ^ { ( 2 ) } ( \\tau , 1 + \\rho _ 0 e ^ { - i \\beta \\pi } ) \\sim \\frac { \\pi } { \\sqrt { \\tau } D _ { \\beta } } \\left ( 1 + \\sqrt { 2 } G _ { \\beta } \\tau ^ { - \\frac { 1 } { 4 } } \\right ) ^ { - 2 n } = : g ^ { ( 2 ) } ( \\tau , \\rho _ 0 ) . \\end{align*}"}
+{"id": "8308.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { B } _ { x y } = \\big \\{ \\ \\textbf { x } \\ | \\ z _ E > 0 , \\ ( x _ E , y _ E ) \\in \\mathcal { H } _ { x y } \\big \\} . \\end{array} \\right . \\end{align*}"}
+{"id": "628.png", "formula": "\\begin{align*} J _ h = x _ 4 x _ 5 J _ { h - 1 } = ( x _ 4 x _ 5 ) ^ { h - 3 } J _ 3 . \\end{align*}"}
+{"id": "7519.png", "formula": "\\begin{align*} Z _ h ( s , \\chi , D ) = & v ( \\bar { h } , D , \\chi ) + \\sigma ( \\bar { h } , D , \\chi ) \\dfrac { ( 1 - q ^ { - 1 } ) q ^ { - s } } { 1 - q ^ { - 1 - s } } + Z _ h ( s , \\chi , D _ { S ( h , D ) } ) \\\\ : = & \\dfrac { H _ 1 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } + Z _ h ( s , \\chi , D _ { S ( h , D ) } ) \\end{align*}"}
+{"id": "2701.png", "formula": "\\begin{align*} \\sum _ { \\pm } \\lim _ { t \\to \\pm \\infty } \\int _ { | x | > R + | t | } \\left ( | \\nabla u ( t , x ) | ^ 2 + ( \\partial _ t u ( t , x ) ) ^ 2 \\right ) d x = 0 . \\end{align*}"}
+{"id": "8023.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 1 & \\cdots & 1 \\\\ \\vdots & & \\vdots \\\\ 1 & \\cdots & 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "1492.png", "formula": "\\begin{align*} \\zeta ( 3 ) = \\frac { 4 \\pi ^ 2 } { 2 1 } \\log \\left ( \\frac { e ^ { \\frac { 4 G } { \\pi } } \\C _ 3 \\left ( \\frac 1 4 \\right ) ^ { 1 6 } } { \\sqrt 2 } \\right ) , \\end{align*}"}
+{"id": "6889.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( v _ n ) - \\rho ( v _ n - v _ \\mu ) & = \\rho ( v _ \\mu ) + o _ n ( 1 ) \\\\ \\int _ { \\Omega } b ( x ) \\left ( | v _ { n } | ^ { \\alpha ( x ) } - | v _ { n } - v _ \\mu | ^ { \\alpha ( x ) } \\right ) d x & = \\int _ { \\Omega } b ( x ) | v _ \\mu | ^ { \\alpha ( x ) } d x + o _ n ( 1 ) , \\end{aligned} \\end{align*}"}
+{"id": "7735.png", "formula": "\\begin{align*} \\hat { \\tau } = \\left ( - \\frac { \\rho _ N ^ { \\frac { 1 } { 4 } } } { 8 \\sqrt { 2 } C _ { \\beta } n } \\ln \\left ( \\frac { \\sqrt { \\rho _ N } } { D _ { \\beta } } \\right ) + \\sqrt { \\left ( \\frac { \\rho _ N ^ { \\frac { 1 } { 4 } } } { 8 \\sqrt { 2 } C _ { \\beta } n } \\ln \\left ( \\frac { \\sqrt { \\rho _ N } } { D _ { \\beta } } \\right ) \\right ) ^ 2 + \\frac { G _ { \\beta } } { C _ { \\beta } } \\rho _ N ^ { \\frac { 1 } { 4 } } } \\right ) ^ 4 . \\end{align*}"}
+{"id": "7994.png", "formula": "\\begin{align*} \\textbf { a } _ { E _ 1 , E _ 2 } ^ E = \\# \\{ L \\subseteq E \\mid L \\simeq E _ 2 \\textrm { a n d } E / L \\simeq E _ 1 \\} . \\end{align*}"}
+{"id": "7404.png", "formula": "\\begin{align*} \\lambda ( \\alpha ) = f ( L / F ) , \\quad \\lambda ^ * ( \\alpha ) = f ( L / F ) . \\end{align*}"}
+{"id": "2244.png", "formula": "\\begin{align*} & \\sum _ { l = 1 } ^ { 2 k - 2 } \\biggl | \\tilde { a } _ l - \\sum _ { j = 0 } ^ { l } \\frac { \\alpha _ { l - j } } { j ! } \\tau ^ j \\biggr | + \\biggl | \\tilde { a } _ { 2 k - 1 } - \\sum _ { j = 1 } ^ { 2 k - 1 } \\frac { \\alpha _ { 2 k - 1 - j } } { j ! } \\tau ^ j \\biggr | \\\\ & + \\sum _ { j = 1 } ^ { 2 k - 2 } \\biggl | \\tilde { b } _ { j } ^ { + } - \\sum _ { l = 0 } ^ { 2 k - 2 } \\frac { \\alpha _ { 2 k - 2 - l } } { ( l + j + 1 ) ! } \\tau ^ { l + j + 1 } \\biggr | \\leq \\omega ( \\sum _ { j = 1 } ^ { 2 k - 2 } | b _ j ^ + - b _ j ^ - | ) \\end{align*}"}
+{"id": "8458.png", "formula": "\\begin{align*} & \\gamma _ { A _ j } \\to \\gamma _ A , \\\\ & \\kappa \\gamma _ A = \\liminf _ { j \\to \\infty } \\ , \\kappa \\gamma _ { A _ j } \\end{align*}"}
+{"id": "6183.png", "formula": "\\begin{align*} \\hbox { K e r } ( C _ p ) = \\hbox { S p a n } \\{ e _ 1 , e _ 2 , \\cdots , e _ p \\} . \\end{align*}"}
+{"id": "8562.png", "formula": "\\begin{align*} A \\begin{pmatrix} y _ { w } ( 1 ) \\\\ y _ { w } ( 2 ) \\\\ \\vdots \\\\ y _ { w } ( p - 1 ) \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} \\end{align*}"}
+{"id": "8578.png", "formula": "\\begin{align*} s ( x ) : = s _ 0 + s _ 1 \\ , x + \\ldots = \\sum _ { i = 0 } ^ { \\infty } s _ i \\ , x ^ i \\end{align*}"}
+{"id": "212.png", "formula": "\\begin{align*} D _ { i } : = \\{ g \\in \\mathrm { S O } _ { \\Lambda _ { i } } ( \\widehat { \\Z } ) \\mid g ( v _ { i } ) = v _ { i } \\} , \\end{align*}"}
+{"id": "989.png", "formula": "\\begin{align*} \\hat { f } ( \\chi _ i \\otimes \\varphi _ i ) & = \\chi _ i ( \\varphi _ i \\circ f ) \\\\ & = \\int _ G \\overline { \\chi _ i ( x ) } \\varphi _ i ( 1 _ G ( x ) a ) d x \\\\ & = \\varphi _ i ( a ) \\int _ G \\overline { \\chi _ i ( x ) } d x \\\\ & = \\varphi _ i ( a ) \\chi _ i ( 1 _ G ) \\\\ & = \\varphi _ i ( a ) \\\\ & = \\hat { a } ( \\varphi _ i ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; ( 1 \\leq i \\leq n ) . \\end{align*}"}
+{"id": "8477.png", "formula": "\\begin{align*} d s : = f ( p ) d p , \\end{align*}"}
+{"id": "463.png", "formula": "\\begin{align*} | \\phi ( \\tau _ \\alpha ) | = | \\psi ( \\tau _ \\alpha ) | | f _ \\alpha ( x ) | = | f _ \\alpha ( y ) | , \\forall \\alpha \\in \\Omega , \\end{align*}"}
+{"id": "2849.png", "formula": "\\begin{align*} t _ { k , \\ell } = t _ { \\ell , k } = \\frac { \\tau - \\sigma } { 2 \\tau } \\end{align*}"}
+{"id": "3633.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } u = \\Delta _ { \\mathbb { H } ^ { n } } u + f ( u , t ) & \\hbox { i n } ~ \\mathbb { H } ^ { n } \\times ( 0 , T ] , \\\\ \\\\ u ( x , 0 ) = u _ { 0 } ( x ) & \\hbox { i n } ~ \\mathbb { H } ^ { n } . \\end{array} \\right . \\end{align*}"}
+{"id": "568.png", "formula": "\\begin{align*} { \\rm d } \\mu _ t = \\frac { \\sigma _ t } { \\gamma } \\left ( ( A X _ t ^ \\dagger ) ^ { \\rm T } { \\rm d } X ^ \\dagger _ t - \\mu _ t \\ , ( A ^ { \\rm T } A ) : C \\ , { \\rm d } t \\right ) , \\end{align*}"}
+{"id": "1057.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\psi ( x ) \\ , \\mathrm { d } x = 0 , \\| \\psi \\| _ { \\infty } < \\infty \\| \\phi \\| _ { \\infty } < \\infty , \\end{align*}"}
+{"id": "6482.png", "formula": "\\begin{align*} \\chi _ g ( h A ) = \\chi _ g ( \\eta ( \\sigma ) A ) = \\sigma ( \\lambda _ g ) = \\lambda _ g = \\chi _ g ( A ) , ~ ~ \\mbox { f o r a l l $ g \\in G $ } . \\end{align*}"}
+{"id": "1692.png", "formula": "\\begin{align*} \\mu _ m ( P _ { n + \\frac { k - 2 } { 2 } } ) = i ^ { \\frac { 2 - k } { 2 } - m } C ( n ) \\binom { k - 2 } { n + \\frac { k - 2 } { 2 } } ^ { - 1 } = i ^ { \\frac { 2 - k } { 2 } - m } \\sum _ { j } \\frac { ( - 1 ) ^ { n + j + m } \\binom { \\frac { k - 2 } { 2 } - m } { j } \\binom { \\frac { k - 2 } { 2 } + m } { m - n + j } } { \\binom { k - 2 } { n + \\frac { k - 2 } { 2 } } } . \\end{align*}"}
+{"id": "2689.png", "formula": "\\begin{align*} \\mu _ t = \\sum _ { i = 1 } ^ { \\ell } r ^ s \\mu _ { t } \\circ ( \\phi _ { t , i } ) ^ { - 1 } . \\end{align*}"}
+{"id": "1090.png", "formula": "\\begin{align*} f _ 1 \\geq c _ 0 - \\gamma \\| \\psi _ { j k } \\| _ \\infty = c _ 0 - c 2 ^ { - j ( \\beta - 1 / 2 ) } \\| \\psi \\| _ { \\infty } \\geq c _ 0 - c \\| \\psi \\| _ { \\infty } \\geq 0 . \\end{align*}"}
+{"id": "6836.png", "formula": "\\begin{align*} \\begin{aligned} \\nu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) & = \\nu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ { l } ) + \\nu _ c ^ { \\pm } ( Y _ 1 , Y _ l , \\dots , Y _ { m } ) = \\nu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ { l } , Y _ m ) + \\nu _ c ^ { \\pm } ( Y _ l , \\dots , Y _ { m } ) , \\end{aligned} \\end{align*}"}
+{"id": "5333.png", "formula": "\\begin{align*} c _ \\varepsilon : = \\max \\Big \\{ c > 0 : \\varepsilon ( x ) \\ , \\xi \\cdot \\xi \\geq c \\ , \\abs { \\xi } ^ 2 x \\in \\Omega , \\xi \\in \\mathbb { R } ^ 3 \\Big \\} . \\end{align*}"}
+{"id": "7990.png", "formula": "\\begin{align*} i ' ( d ) \\in j ' i ( a ) + j ' i ( b ) = j ' i ( a + b ) . \\end{align*}"}
+{"id": "612.png", "formula": "\\begin{align*} ( A ^ { \\rm T } A ) : Z _ t ^ \\dagger \\otimes X _ t ^ \\dagger \\approx ( A ^ { \\rm T } A ) : \\Sigma _ \\infty ^ { z x } = ( A ^ { \\rm T } A ) : \\Sigma _ \\infty ^ { z z } \\end{align*}"}
+{"id": "5407.png", "formula": "\\begin{align*} \\prod _ { k \\in \\N _ { \\ge 1 } } \\big ( 1 + \\tfrac { 1 } { ( \\alpha \\beta ) ^ { 2 ^ { k - 1 } } } \\big ) = \\frac { 1 } { 1 - \\frac { 1 } { \\alpha \\beta } } . \\end{align*}"}
+{"id": "9174.png", "formula": "\\begin{align*} \\frac { \\tilde { Z } _ N ( u [ \\epsilon ] , 0 ) } { \\tilde { Z } _ N ( 0 , 0 ) } = 1 + O _ f ( e ^ { - \\alpha \\log ( 8 C _ f \\epsilon ^ { - 1 } ) } ) = 1 + O _ f ( \\epsilon ^ { \\alpha } ) \\end{align*}"}
+{"id": "1193.png", "formula": "\\begin{align*} \\Upsilon ^ { * } h = h + A _ { h , 1 } ^ { [ 5 ] } w _ { n } + A _ { h , 2 } ^ { [ 5 ] } \\overline { w } _ { n } . \\end{align*}"}
+{"id": "5946.png", "formula": "\\begin{align*} 0 = ( E _ { \\bar r } , \\widehat { U } ) = ( R _ { \\bar r } , C _ p \\widehat { U } ) , \\forall \\widehat H \\in L ^ 2 ( 0 , T ; ( L ^ 2 ( \\Gamma ) ) ^ M ) . \\end{align*}"}
+{"id": "5657.png", "formula": "\\begin{align*} \\lambda _ 0 = \\epsilon \\lambda _ 0 ^ { ( 1 ) } + \\epsilon ^ 2 \\lambda _ 0 ^ { ( 2 ) } + \\ldots \\end{align*}"}
+{"id": "5031.png", "formula": "\\begin{align*} F : \\mathcal { A } \\rightarrow \\R ^ 4 , F ( \\mu , a , b , c , h _ 1 ) = ( p _ 1 , p _ 2 , p _ 3 , p _ 4 ) . \\end{align*}"}
+{"id": "6234.png", "formula": "\\begin{align*} D ^ T E _ i = 0 , 1 \\leqslant i \\leqslant p . \\end{align*}"}
+{"id": "3275.png", "formula": "\\begin{align*} - \\nabla \\circ \\left ( \\left \\vert \\nabla u \\right \\vert ^ { p - 2 } \\nabla u \\right ) - \\lambda \\left \\vert u \\right \\vert ^ { p _ { 0 } - 2 } u \\left \\vert \\nabla u \\right \\vert ^ { p _ { 1 } } = 0 , u \\left \\vert _ { \\ \\partial \\Omega } \\right . = 0 , \\ \\lambda \\in \\mathbb { C } , \\end{align*}"}
+{"id": "3727.png", "formula": "\\begin{align*} \\mathcal { S } ( X _ { P _ 2 } ( F ) ) = \\mathcal { S } ( X _ { P _ 2 } ^ \\circ ( F ) ) + \\mathcal { F } _ { X _ { P _ 2 } } \\left ( \\mathcal { S } ( X _ { P _ 2 } ^ \\circ ( F ) ) \\right ) \\end{align*}"}
+{"id": "1471.png", "formula": "\\begin{align*} \\lambda = \\frac { k ( k - 1 ) ( k - 2 ) } { m ^ 2 ( m ^ 2 - 1 ) ( m ^ 2 - 2 ) } \\cdot b . \\end{align*}"}
+{"id": "6603.png", "formula": "\\begin{align*} { \\mathbf { T } } _ { * } ( x , y ) = \\pm \\frac { 1 } { \\ ; \\Vert \\ ; \\nabla f ( x , y ) \\ ; \\Vert \\ ; } \\nabla f ( x , y ) = \\begin{bmatrix} \\ ; \\cos { \\Theta } _ { * } ( x , y ) \\ ; \\\\ \\ ; \\sin { \\Theta } _ { * } ( x , y ) \\ ; \\end{bmatrix} , \\end{align*}"}
+{"id": "1880.png", "formula": "\\begin{align*} \\alpha _ { z _ i } ( \\sigma ) = \\frac { \\pi } { 2 } . \\end{align*}"}
+{"id": "9216.png", "formula": "\\begin{align*} q ( x , \\xi ) : = - p ( x , i \\xi ) = \\xi ^ { 2 } - \\vert \\nabla f ( x ) \\vert ^ { 2 } , \\end{align*}"}
+{"id": "4326.png", "formula": "\\begin{align*} x ^ { n _ 1 } = c _ 1 ( m ) ; \\ ; \\ ; ( x _ { n _ 1 + 1 } , \\ldots x _ { n } ) = c _ 2 ( m , y ^ { n _ 1 } ) . \\end{align*}"}
+{"id": "4175.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( \\lambda ( \\widetilde { g } _ i , \\widetilde { H } _ i ) - \\lambda ( g _ c , H _ c ) \\right ) ^ { 1 - 2 \\alpha } & = ( 1 - 2 \\alpha ) \\left ( \\lambda ( \\widetilde { g } _ i , \\widetilde { H } _ i ) - \\lambda ( g _ c , H _ c ) \\right ) ^ { - 2 \\alpha } \\frac { d } { d t } \\lambda ( \\widetilde { g } _ i , \\widetilde { H } _ i ) \\\\ & \\geq - C . \\end{align*}"}
+{"id": "1142.png", "formula": "\\begin{align*} \\left | g _ 2 \\right | & = 8 , \\mbox { \\ \\ a n d } \\\\ \\left | g _ 3 \\right | & = 1 2 , \\end{align*}"}
+{"id": "2775.png", "formula": "\\begin{align*} u _ i \\ ! = \\ ! \\left \\{ \\ ! \\begin{aligned} & 0 , ~ ~ ~ ~ ~ ~ ~ ~ \\ ! ~ ~ ~ ~ ~ i \\in [ 1 , \\lambda _ 1 ] ; \\\\ & x _ { \\lambda _ 2 } - x ' _ { \\lambda _ 2 } , ~ ~ i \\in [ \\lambda _ 1 + 1 , \\lambda _ 2 ] ; \\\\ & 0 , ~ ~ ~ ~ ~ ~ ~ ~ \\ ! ~ ~ ~ ~ ~ i \\in [ \\lambda _ 2 + 1 , d _ 1 ] ; \\\\ & x _ { i } - x ' _ { d _ 2 } , ~ ~ ~ ~ i \\in [ d _ 1 + 1 , d _ 2 ] ; \\\\ & 0 , ~ ~ ~ ~ ~ ~ ~ ~ \\ ! ~ ~ ~ ~ ~ i \\in [ d _ 2 + 1 , n ] . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3752.png", "formula": "\\begin{align*} \\epsilon _ v ( s , \\psi _ v ) = q ^ { ( \\frac { 1 } { 2 } - s ) e ( \\psi _ v ) } , \\end{align*}"}
+{"id": "5703.png", "formula": "\\begin{align*} D \\theta ^ { a } & = \\tau ^ { a } , \\\\ D \\tau ^ { a } & = \\beta _ { b } ^ { a } \\theta ^ { b } , \\\\ \\mathbf { F } _ { b } ^ { a } & = \\digamma _ { b } ^ { a } - \\beta _ { b } ^ { a } - D \\beta _ { b } ^ { a } \\mathbf { m , } \\\\ \\digamma _ { b } ^ { a } & = d \\alpha _ { b } ^ { a } + \\alpha _ { c } ^ { a } \\alpha _ { b } ^ { c } . \\end{align*}"}
+{"id": "6025.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l r } & P _ { l _ 1 } ( h _ 1 , w _ { k , n } ) = \\delta _ { k , 1 } Q _ 1 \\O _ { n - 1 , n } & \\\\ & P _ { l _ 1 } ( h _ 1 , w _ { 1 , p } ) = Q _ 1 \\O _ { n , p } & \\mathrm { i f } \\ : p < n \\\\ & P _ { l _ 1 } ( h _ 1 , w _ { 2 , 1 } ) = Q _ 1 ( \\O _ { n , 1 } - \\O _ { n , 2 } ) = Q _ 1 ( [ O _ X ] - \\O _ { h _ 2 } ) & \\\\ & P _ { l _ 1 } ( h _ 1 , w _ { k , p } ) = 0 & \\mathrm { i f } \\ : ( 2 < k \\ : \\mathrm { a n d } \\ : p < n ) \\ : \\mathrm { o r } \\ : ( k = 2 \\ : \\mathrm { a n d } \\ : p > 1 ) \\end{array} \\right . \\end{align*}"}
+{"id": "3500.png", "formula": "\\begin{align*} [ E _ { i j } , E _ { k l } ] = \\delta _ { j k } E _ { i l } - \\delta _ { i l } E _ { k j } , ( i , j , k , l \\in \\{ 1 , \\dots , n \\} ) , \\end{align*}"}
+{"id": "6602.png", "formula": "\\begin{align*} | \\hat { H } ( \\eta , y ) | & = ( 2 \\pi ) ^ { - n } | V _ { \\phi _ 2 } \\phi _ 1 ( y , - \\eta ) \\cdot V _ f g ( - y , \\eta ) | . \\end{align*}"}
+{"id": "4385.png", "formula": "\\begin{align*} \\frac { 1 } { a _ k } < \\theta - \\sum _ { i = 1 } ^ { k - 1 } \\frac { 1 } { a _ i } \\leq \\frac { 1 } { a _ k - 1 } \\end{align*}"}
+{"id": "8485.png", "formula": "\\begin{align*} C _ { \\xi ^ 3 } & = - \\lambda C _ \\xi - \\varphi C _ { \\xi ^ 2 } , \\\\ C _ { \\xi ^ 4 } & = ( \\varphi \\lambda - \\lambda _ \\xi ) C _ \\xi + ( \\varphi ^ 2 - \\lambda - \\varphi _ \\xi ) C _ { \\xi ^ 2 } . \\end{align*}"}
+{"id": "7541.png", "formula": "\\begin{align*} Z _ g \\big ( s , \\chi , S ( \\Delta _ { \\gamma _ 4 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) ) \\big ) = \\dfrac { G _ 4 ( q ^ { - s } ) } { ( 1 - q ^ { - 1 - s } ) ( 1 - q ^ { - i _ 0 - j _ 0 - i _ 0 j _ 0 s } ) } , \\end{align*}"}
+{"id": "2942.png", "formula": "\\begin{align*} W _ j ( t ) = g _ n ( \\eta ^ n _ j ( t ) ) / g ^ \\prime ( 0 ) , \\overline { W } _ j ( t ) = W _ j ( t ) - \\Phi _ n ( \\rho ) / g ^ \\prime ( 0 ) . \\end{align*}"}
+{"id": "7322.png", "formula": "\\begin{align*} & \\lim _ { \\gamma \\to \\infty } \\int _ { 1 } ^ { \\infty } G _ { m _ \\gamma } ( x ) j _ \\gamma ( d x ) = 0 . \\end{align*}"}
+{"id": "1928.png", "formula": "\\begin{align*} \\partial ^ 2 g _ \\lambda & = \\partial ^ 2 \\left ( \\chi _ \\lambda ( g - \\delta ) \\right ) = \\partial ^ 2 \\chi _ \\lambda ( g - \\delta ) + \\partial \\chi _ \\lambda \\partial ( g - \\delta ) + \\chi _ \\lambda \\partial ^ 2 g \\\\ & = \\chi _ \\lambda \\partial ^ 2 g + O ( | x | ^ { - q - 2 } ) . \\end{align*}"}
+{"id": "8053.png", "formula": "\\begin{align*} H = \\frac { 2 } { \\pi } \\int _ 0 ^ \\infty h ( t ) \\tanh ( \\pi t ) t \\ , d t , \\end{align*}"}
+{"id": "6546.png", "formula": "\\begin{align*} { \\gamma _ k } = \\frac { { D _ { a k } ^ { - { \\alpha _ { a k } } } { P _ { a k } } h _ { a k } ^ 2 } } { { { \\kappa ^ 2 } + D _ { j k } ^ { - { \\alpha _ { j k } } } { P _ { j k } } h _ { j k } ^ 2 } } \\triangleq \\frac { { { C _ { 1 k } } { X _ k } } } { { { \\kappa ^ 2 } + { C _ { 2 k } } { Z _ k } } } , \\end{align*}"}
+{"id": "1505.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\C _ 4 \\left ( \\frac 1 4 \\right ) = \\frac 1 { 2 ^ 7 } \\log 2 - \\frac { 3 G } { 2 ^ 5 \\pi } - \\frac { 3 \\zeta ( 3 ) } { 2 ^ 7 \\pi ^ 3 } + \\frac 3 { 2 ^ { 1 0 } \\pi ^ 3 } \\left ( \\zeta \\left ( 4 , \\frac 1 4 \\right ) - \\zeta \\left ( 4 , \\frac 3 4 \\right ) \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "1926.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\| R _ \\lambda - \\chi _ \\lambda R _ g \\| _ { L ^ p _ { - q ' - 2 } } = 0 . \\end{align*}"}
+{"id": "631.png", "formula": "\\begin{align*} \\beta _ { i , m } ( w ( z I ) ^ h ) \\neq 0 , \\beta _ { i , m } ( ( z I ) ^ h ) = \\beta _ { i , m } ( w ( z I + ( w ) ) ^ { h - 1 } ) = 0 . \\end{align*}"}
+{"id": "33.png", "formula": "\\begin{align*} T ( n _ i ) & = \\sum _ { t = 0 } ^ { k _ i - 1 } \\frac { 1 } { C ^ t ( n _ i ) } < \\sum _ { t = 0 } ^ { k _ i - 1 } \\frac { 1 } { n _ i \\cdot \\left ( \\tfrac { 3 } { 2 } \\right ) ^ t } \\\\ & = \\frac { 1 } { n _ i } \\cdot \\frac { 1 - \\left ( \\frac { 2 } { 3 } \\right ) ^ { k _ i } } { 1 - \\frac { 2 } { 3 } } \\\\ & = \\frac { 1 } { n _ i } \\cdot \\left ( 3 - 3 \\cdot \\left ( \\frac { 2 } { 3 } \\right ) ^ { k _ i } \\right ) \\end{align*}"}
+{"id": "8751.png", "formula": "\\begin{align*} \\mathbb { P } ( \\{ T _ 1 = d _ 0 + 1 \\} \\cap \\mathcal { A } ) & \\geq \\mathbb { P } ( \\{ T _ 1 = \\check { d } _ { \\xi } + 1 \\} \\cap \\{ \\check { d } _ { \\xi } = d _ 0 \\} \\cap \\mathcal { A } ) \\\\ & = \\mathbb { P } ( \\{ \\check { d } _ { \\xi } = d _ 0 \\} \\cap \\mathcal { A } ) \\geq \\mathbb { P } ( \\mathcal { A } \\cap \\mathcal { B } ) \\\\ & \\geq 1 - 4 \\delta , \\end{align*}"}
+{"id": "8789.png", "formula": "\\begin{align*} C ^ { 2 } = I . \\end{align*}"}
+{"id": "5737.png", "formula": "\\begin{align*} \\theta _ { 1 } ^ { A A ^ { \\prime } } = ( \\mathbf { L } _ { 1 } ^ { - 1 } ) _ { B ^ { \\prime } } ^ { A ^ { \\prime } } ( \\mathbf { L } _ { 1 } ^ { - 1 } ) _ { B } ^ { A } d \\mathbf { x } _ { 1 } ^ { B B ^ { \\prime } } = ( \\mathbf { L } _ { 1 } ^ { - 1 } ) _ { b } ^ { a } d \\mathbf { x } _ { 1 } ^ { b } . \\end{align*}"}
+{"id": "3285.png", "formula": "\\begin{align*} = \\frac { p } { 2 \\left ( p - 1 \\right ) } \\inf \\left \\{ \\ \\frac { \\left \\Vert \\left \\vert \\Delta u \\right \\vert ^ { \\frac { p } { 2 } } \\right \\Vert _ { 2 } } { \\left \\Vert \\nabla \\left ( \\left \\vert u \\right \\vert ^ { \\frac { p - 2 } { 2 } } \\left \\vert u \\right \\vert \\right ) \\right \\Vert _ { 2 } } \\left \\vert \\ u \\in W ^ { 2 , p } \\cap W _ { 0 } ^ { 1 , p } \\right . \\right \\} \\end{align*}"}
+{"id": "1189.png", "formula": "\\begin{align*} r ^ { k } | \\nabla _ { g _ { 0 } } ^ { k } ( \\Psi _ i ^ * g _ 0 - g _ 0 ) | _ { g _ 0 } = O _ { i } ( r ^ { \\nu _ i } ) \\ ; \\ , { \\rm a s } \\ ; \\ , r \\to \\infty \\end{align*}"}
+{"id": "9035.png", "formula": "\\begin{align*} \\tau _ { S } ^ \\star : = \\inf \\{ t \\geq S : \\ , \\ , V _ t = h _ 1 ( t , V _ t , \\mathbb { P } _ { V _ t } ) \\} \\ , \\ , \\ , \\ , \\ , \\ , \\sigma _ { S } ^ \\star : = \\inf \\{ t \\geq S : \\ , \\ , V _ t = h _ 2 ( t , V _ t , \\mathbb { P } _ { V _ t } ) \\} . \\end{align*}"}
+{"id": "3618.png", "formula": "\\begin{align*} a l p h a & = 1 8 . 8 7 9 9 9 4 6 7 \\\\ b e t a & = 9 . 0 7 9 4 2 6 0 2 \\\\ g a m m a & = 1 . 2 2 0 6 3 0 2 5 6 0 9 4 0 0 4 \\\\ d e l t a & = 2 2 . 5 3 7 6 0 3 9 7 3 7 9 3 4 3 4 \\end{align*}"}
+{"id": "4488.png", "formula": "\\begin{align*} \\frac { 1 } { d ' _ 1 } + \\frac { 1 } { d ' _ 2 } + \\frac { 1 } { d ' _ 3 } + \\frac { 1 } { d ' _ 4 } = 1 \\end{align*}"}
+{"id": "3846.png", "formula": "\\begin{align*} \\mu _ l ^ { [ m _ 1 m _ 2 ] } = \\mu _ l ^ { [ m _ 2 m _ 1 ] } , \\end{align*}"}
+{"id": "6584.png", "formula": "\\begin{align*} \\theta ( k , i ) = \\beta _ { k , M _ - } ( 1 ) = \\beta _ { k , M _ - } \\circ \\beta _ { k , M _ + } ^ { - 1 } ( k , i ) = \\eta ( k , i ) \\end{align*}"}
+{"id": "4179.png", "formula": "\\begin{align*} _ { \\mu \\nu } ( \\widetilde g ) = 0 . \\end{align*}"}
+{"id": "2727.png", "formula": "\\begin{align*} A ' ( t ) & \\geq \\kappa _ 2 \\sum _ { j = 1 } ^ { J _ 0 } \\theta _ j \\beta _ j ^ 2 ( t ) + \\frac { 1 } { C } \\gamma ^ 2 ( t ) , \\\\ A ' ( t ) & \\geq \\frac { 1 } { \\kappa _ 2 } \\sum _ { j = 1 } ^ { J _ 0 } \\theta _ j \\left ( \\lambda _ j ' ( t ) \\right ) ^ 2 + \\frac { 1 } { C } \\gamma ^ 2 ( t ) . \\end{align*}"}
+{"id": "6683.png", "formula": "\\begin{align*} \\begin{aligned} & \\kappa _ 1 \\lambda ^ k \\pi _ 1 + ( 1 - \\kappa _ 3 \\gamma _ 1 ^ k ) \\pi _ 2 \\leq \\pi _ 2 , \\\\ & \\kappa _ 2 \\lambda ^ k \\pi _ 1 + ( 1 - \\kappa _ 4 \\gamma _ 2 ^ k ) \\pi _ 3 \\leq \\pi _ 3 , \\\\ & \\left ( \\kappa _ 6 \\lambda ^ k - \\kappa _ 7 ( \\lambda ^ k ) ^ 2 \\right ) \\pi _ 1 - \\kappa _ 8 \\frac { ( \\lambda ^ k ) ^ 2 } { \\gamma _ 1 ^ k } \\pi _ 2 - \\kappa _ 9 \\frac { ( \\lambda ^ k ) ^ 2 } { \\gamma _ 2 ^ k } \\pi _ 3 \\geq 0 \\end{aligned} \\end{align*}"}
+{"id": "7141.png", "formula": "\\begin{align*} [ \\delta ( x ) ] & = [ \\delta ' ( x ) ] = [ x ] B , \\\\ [ \\alpha ( x ) ] & = \\tilde f ( [ x ] \\otimes B ) E ' + [ x ] C \\end{align*}"}
+{"id": "1958.png", "formula": "\\begin{gather*} \\vartheta _ K ^ { ( N ) } = \\min _ { u \\in \\mathbb { R } } \\left \\{ u + \\frac { 1 } { \\alpha } \\left [ \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int ( \\max ( 0 , x - u ) ) ^ q \\frac { 1 } { h } K ( \\frac { x - X _ i } { h } ) d x \\right ] ^ { 1 / q } \\right \\} \\\\ \\vartheta _ w ^ { ( N ) } = \\min _ { u \\in \\mathbb { R } } \\left \\{ u + \\frac { 1 } { \\alpha } \\left [ \\int ( \\max ( 0 , x - u ) ) ^ q \\tilde { d } _ { N , j } ( x ) d x \\right ] ^ { 1 / q } \\right \\} . \\end{gather*}"}
+{"id": "9028.png", "formula": "\\begin{align*} \\mathcal { W } _ p ( \\mu , \\nu ) : = \\inf \\left \\{ \\int _ { \\R \\times \\R } | x - y | ^ p \\pi ( d x , d y ) \\right \\} ^ { 1 / p } , \\end{align*}"}
+{"id": "992.png", "formula": "\\begin{align*} \\delta ( \\psi _ * ) : = \\ell ( \\psi _ * ) - \\nu ( \\psi _ * ) \\end{align*}"}
+{"id": "2697.png", "formula": "\\begin{align*} \\frac { d P } { d t } & = \\sum f ' ( x ( t ) ) / f ( x ( t ) ) \\\\ \\frac { d ^ 2 P } { d t ^ 2 } & = \\sum \\frac { f ( x ( t ) ) f '' ( x ( t ) ) - \\left [ f ' ( x ( t ) ) \\right ] ^ 2 } { \\left [ f ( x ( t ) ) \\right ] ^ 3 } \\end{align*}"}
+{"id": "8674.png", "formula": "\\begin{align*} \\tilde F _ { \\lambda } = \\tilde \\Gamma _ { - \\lambda _ 1 } \\dots \\tilde \\Gamma _ { - \\lambda _ l } ( f ) . \\end{align*}"}
+{"id": "6541.png", "formula": "\\begin{align*} { f _ X } \\ ! \\left ( x \\right ) = \\frac { { { m ^ m } } } { { \\Gamma \\left ( m \\right ) } } \\sum \\limits _ { j = 0 } ^ M { \\frac { { { K ^ j } { \\alpha _ j } } } { { j ! } } } \\frac { { { x ^ j } } } { { \\Gamma \\left ( { j \\ ! + \\ ! 1 } \\right ) { { \\left ( { 2 { \\sigma ^ 2 } } \\right ) } ^ { j + 1 } } } } \\exp \\ ! \\left ( \\ ! { - \\frac { x } { { 2 { \\sigma ^ 2 } } } } \\ ! \\right ) , \\end{align*}"}
+{"id": "5813.png", "formula": "\\begin{align*} A e _ r = \\sum _ { s = 1 } ^ p \\beta _ { r s } { \\| e _ r \\| \\over \\| e _ s \\| } e _ s , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "8605.png", "formula": "\\begin{gather*} R _ 1 , ~ R _ 2 , ~ R _ 3 , ~ R _ 4 , ~ R _ 5 , ~ R _ 7 , ~ R _ 9 , \\\\ A = \\frac { 1 } { 2 } ( R _ 2 + R _ 3 + R _ 5 + R _ 6 ) , ~ B = \\frac { 1 } { 2 } ( R _ 2 + R _ 3 + R _ { 1 1 } + R _ { 1 2 } ) , \\\\ C = \\frac { 1 } { 2 } ( R _ 1 + R _ 2 + R _ 4 + R _ 5 + R _ 7 + R _ 8 + R _ { 1 0 } + R _ { 1 1 } ) . \\end{gather*}"}
+{"id": "638.png", "formula": "\\begin{align*} \\beta ^ { \\mathbb { Z } _ p } _ { i + a } ( J ^ { h } ) - \\beta ^ { \\mathbb Q } _ { i + a } ( J ^ { h } ) \\geq \\sum _ { \\ell = 1 } ^ h \\left [ \\beta ^ { \\mathbb { Z } _ p } _ { i + a } ( T ^ { \\ell } ) - \\beta ^ { \\mathbb Q } _ { i + a } ( T ^ { \\ell } ) \\right ] \\geq \\sum _ { \\ell = 1 } ^ h \\binom { \\ell + r - 1 } { r } = \\binom { h + r } { r + 1 } . \\end{align*}"}
+{"id": "5698.png", "formula": "\\begin{align*} \\digamma _ { b } ^ { a } - ( \\beta _ { b } ^ { a } + \\overline { \\beta } _ { b } ^ { a } ) = 0 ; \\frac { 1 } { 2 } D ( \\beta _ { b } ^ { a } - \\overline { \\beta } _ { b } ^ { a } ) = \\gamma _ { b } ^ { a } . \\end{align*}"}
+{"id": "1206.png", "formula": "\\begin{align*} A \\sum _ { \\substack { d , m _ 1 , \\ldots , m _ d \\in \\N _ 0 \\\\ m _ 1 + 2 m _ 2 + \\cdots + d m _ d = k } } \\left ( \\sum _ { \\substack { \\ell _ 1 , \\ell _ 2 \\in \\N _ 0 \\\\ \\ell _ 1 + \\ell _ 2 = m _ 1 + \\cdots + m _ d } } r ^ \\epsilon \\right ) \\prod _ { j = 1 } ^ d r ^ { - j m _ j } \\leq A r ^ { - k + \\epsilon } . \\end{align*}"}
+{"id": "4424.png", "formula": "\\begin{align*} \\frac { x _ 1 - a _ 1 } { a _ 1 x _ 1 } = \\frac { 1 } { a _ 1 } - \\frac { 1 } { x _ 1 } < \\frac { 1 } { x _ 2 } \\end{align*}"}
+{"id": "5426.png", "formula": "\\begin{align*} f ( \\bar { \\alpha } \\alpha ) = \\sum _ a k ( \\lambda _ a ) \\end{align*}"}
+{"id": "1597.png", "formula": "\\begin{align*} \\mu _ * : = \\mu - \\mu ( u ) \\delta _ { u } + \\mu ( u ) \\delta _ { x _ { n + 1 } } \\quad \\mbox { a n d } \\mu ^ * : = \\mu - \\mu ( x _ { n + 1 } ) \\delta _ { x _ { n + 1 } } + \\mu ( x _ { n + 1 } ) \\delta _ { u } . \\end{align*}"}
+{"id": "861.png", "formula": "\\begin{align*} w ^ { n + 1 } = w ^ { n } - \\alpha D f ( w ^ n ) ^ { - 1 } f ( w ^ n ) \\end{align*}"}
+{"id": "7327.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } G ^ d _ { m _ \\gamma } ( y ) d m _ { \\gamma } ( y ) = O \\left ( \\frac { K ( \\gamma ^ { \\alpha / 2 } ) ^ { d + 1 } } { u ( \\gamma ^ { \\alpha / 2 } ) ^ { d + 1 } } \\right ) \\xrightarrow { \\gamma \\to \\infty } 0 . \\end{align*}"}
+{"id": "8581.png", "formula": "\\begin{align*} \\Delta ( \\mathbf { s } ) = \\sum _ { d \\mid n } a _ d \\cdot d \\mbox { , w i t h } 0 \\leq a _ d \\leq \\mathcal { I } ( d ) . \\end{align*}"}
+{"id": "5282.png", "formula": "\\begin{align*} R i c ( U _ i , U _ i ) & = \\lambda _ f - ( m - n - 1 ) ^ 2 \\left \\{ \\lambda ^ 2 \\| \\nabla ( f + \\log \\lambda ) \\| ^ 2 - \\frac { \\lambda ^ 4 } { 4 } \\| g r a d _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\| ^ 2 + \\lambda ^ 2 g ( \\nabla f , \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } ) \\right \\} , \\end{align*}"}
+{"id": "5480.png", "formula": "\\begin{align*} \\mathcal P _ { \\hat V } : = \\left \\{ \\mu \\in \\mathcal P ( \\mathbb R ^ d \\times \\mathcal M ) : \\int _ { \\mathbb R ^ d \\times \\mathcal M } \\hat V ( x ) \\mu ( d x \\times \\{ i \\} ) < \\infty \\right \\} . \\end{align*}"}
+{"id": "4384.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } \\frac { 1 } { a _ i } < \\theta \\leq \\sum _ { i = 1 } ^ { k - 1 } \\frac { 1 } { a _ i } + \\frac { 1 } { a _ k - 1 } . \\end{align*}"}
+{"id": "1343.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j , k \\leq n } f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) \\geq \\frac { 1 } { ( G _ { f , \\tau } ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 . \\end{align*}"}
+{"id": "600.png", "formula": "\\begin{align*} \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ { ( \\epsilon ) } ) ^ { \\rm T } { \\rm d } X _ t ^ { ( \\epsilon ) } = A ^ { \\rm T } : \\left ( X _ { t _ { n + 1 / 2 } } ^ { ( \\epsilon ) } \\otimes X _ { t _ n , t _ { n + 1 } } ^ { ( \\epsilon ) } - \\frac { 1 } { 2 } \\int _ { t _ n } ^ { t _ { n + 1 } } [ X _ { t _ n , t } ^ { ( \\epsilon ) } , { \\rm d } X _ { t } ^ { ( \\epsilon ) } ] \\right ) . \\end{align*}"}
+{"id": "3045.png", "formula": "\\begin{align*} & \\limsup _ { b \\rightarrow 0 } \\lim _ { n \\rightarrow \\infty } \\| n ^ { - 1 / 2 } N _ n ( \\lfloor n ^ { 1 / 2 } a \\rfloor ) - n ^ { - 1 / 2 } N _ n ( \\lfloor n ^ { 1 / 2 } ( a + b ) \\rfloor ) \\| _ { L ^ 2 } ^ 2 = \\\\ & \\limsup _ { b \\rightarrow 0 } \\lim _ { n \\rightarrow \\infty } O ( | n ^ { - 1 / 2 } \\lfloor n ^ { 1 / 2 } a \\rfloor - n ^ { - 1 / 2 } \\lfloor n ^ { 1 / 2 } ( a + b ) \\rfloor | ) = o ( 1 ) . \\end{align*}"}
+{"id": "3684.png", "formula": "\\begin{align*} a _ { t - 1 } m ' _ { t - 1 } + z _ t w _ { t , t - 1 } = z _ 1 w _ { t - 1 , 1 } + \\ldots + z _ { t - 2 } w _ { t - 1 , t - 2 } \\end{align*}"}
+{"id": "1181.png", "formula": "\\begin{align*} \\tau = t w ^ \\mu , \\ ; \\ , \\zeta _ n = z _ n w ^ { \\mu _ n } \\ ; \\ , ( n = 1 , \\ldots , N ) \\end{align*}"}
+{"id": "6851.png", "formula": "\\begin{align*} l ( \\mathcal Y ^ { [ 0 ] } , 0 , N + 1 ) = l ^ * ( \\mathcal Y ^ { [ N + 1 ] } , 0 , N + 1 ) , l ^ * ( \\mathcal Y ^ { [ 0 ] } , 0 , N + 1 ) = l ( \\mathcal Y ^ { [ N + 1 ] } , 0 , N + 1 ) . \\end{align*}"}
+{"id": "9074.png", "formula": "\\begin{align*} D _ { j } \\varphi ^ { \\lambda } = b _ { j } ( \\lambda ) \\varphi ^ { \\lambda - \\nu _ { j } } . \\end{align*}"}
+{"id": "4054.png", "formula": "\\begin{align*} & p _ 0 : = g _ x ( x _ 0 , y _ 0 , z _ 0 ) & & p _ 1 : = g _ x ( x _ 0 , y _ 1 , z _ 1 ) , \\end{align*}"}
+{"id": "3874.png", "formula": "\\begin{align*} \\det D Y ( \\cdot , u , D u ) = \\frac { f ( \\cdot ) } { f ^ * ( Y ( \\cdot , u , D u ) ) } . \\end{align*}"}
+{"id": "2388.png", "formula": "\\begin{align*} [ \\ > E _ 1 ( t ) \\ > \\ > E _ 2 ( t ) \\ > ] \\left [ \\begin{array} { c } \\dot x _ 1 ( t ) \\\\ \\dot x _ 2 ( t ) \\end{array} \\right ] = [ \\ > 0 \\ > \\ > A _ 2 ( t ) \\ > ] \\left [ \\begin{array} { c } x _ 1 ( t ) \\\\ x _ 2 ( t ) \\end{array} \\right ] . \\end{align*}"}
+{"id": "8111.png", "formula": "\\begin{align*} \\tilde H _ { m , n } ^ { + , 1 } ( x ) = W _ { m , n } ^ + ( x ) - \\frac { 2 \\pi ^ 6 x } { 1 4 4 0 M ^ 6 } W _ { m , n } ^ - ( x ) + O \\Bigl ( \\frac { T \\abs { x } } { M ^ 8 } \\Bigr ) \\end{align*}"}
+{"id": "8141.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 ^ - = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { c \\leq \\frac { C } { m } } \\frac { S ( - n , p ; c ) } { c } H _ { m , n } ^ - \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) \\end{align*}"}
+{"id": "4816.png", "formula": "\\begin{align*} f ( \\epsilon , c ) : = \\log ( 1 - \\epsilon ) + 2 \\epsilon - 2 \\epsilon \\log c + c \\epsilon > 0 . \\end{align*}"}
+{"id": "4740.png", "formula": "\\begin{align*} x ^ 2 u = \\delta x + x y ^ e s . \\end{align*}"}
+{"id": "6899.png", "formula": "\\begin{align*} 1 - \\chi ( m \\underline { d } , m \\underline { d } ) = 1 - m ^ 2 \\chi ( \\underline { d } , \\underline { d } ) = m ^ 2 ( 2 l ^ 2 + 4 l - 2 ) + 1 . \\end{align*}"}
+{"id": "8868.png", "formula": "\\begin{align*} K _ { \\nu , m } ( z , w ) = \\frac { ( 2 m + 2 \\nu + n ) \\Gamma ( m + 2 \\nu + n ) } { \\pi ^ { n } \\Gamma ( m + 2 \\nu + 1 ) } \\left ( \\frac { \\left ( 1 + \\langle w , z \\rangle \\right ) ^ { 2 } } { ( 1 + | z | ^ { 2 } ) ( 1 + | w | ^ { 2 } ) } \\right ) ^ { \\nu } P _ { m } ^ { ( n - 1 , 2 \\nu ) } \\left ( \\frac { 2 | 1 + \\langle z , w \\rangle | ^ { 2 } } { ( 1 + | z | ^ { 2 } ) ( 1 + | w | ^ { 2 } ) } - 1 \\right ) . \\end{align*}"}
+{"id": "6825.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ c ^ - ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) : = \\mu ( Y _ 1 , Y _ 2 ) + \\mu ( Y _ 2 , Y _ 3 ) + \\dots + \\mu ( Y _ { m - 1 } , Y _ m ) + \\mu ( Y _ m , Y _ 1 ) , \\\\ \\mu _ c ^ + ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) : = \\mu ^ * ( Y _ 1 , Y _ 2 ) + \\mu ^ * ( Y _ 2 , Y _ 3 ) + \\dots + \\mu ^ * ( Y _ { m - 1 } , Y _ m ) + \\mu ^ * ( Y _ m , Y _ 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "4778.png", "formula": "\\begin{align*} \\Pr _ { \\rho \\sim P _ \\epsilon } [ \\rho \\neq \\tilde { D } _ { 1 } ( H \\rho ^ \\intercal ) ] & \\leq 2 e ^ { - \\frac { l ^ 2 } { 3 \\epsilon N } } + ( 2 l + 1 ) \\mathop { \\max } _ { \\substack { w , w ' \\in \\{ \\epsilon N \\pm l \\} \\\\ w ' \\leq w } } \\left \\{ \\Pr _ { \\rho \\sim \\lambda _ { w , w ' } } \\big [ | \\Omega _ \\rho ^ { \\{ w , w ' \\} } | > 1 \\big | | \\rho | = w \\big ] \\right \\} . \\end{align*}"}
+{"id": "1970.png", "formula": "\\begin{align*} W _ { 5 6 } ( a ) = 1 + a y ^ 8 + ( 8 1 9 0 + 6 a ) y ^ { 1 2 } + ( 6 2 2 3 1 4 - 8 3 a ) y ^ { 1 6 } + \\cdots , \\end{align*}"}
+{"id": "2959.png", "formula": "\\begin{align*} \\begin{aligned} E _ { \\nu ^ n _ \\rho } [ f ( \\nabla _ { j - 1 , j } \\overrightarrow { W } ^ \\ell _ j ) W _ { j - 1 } ] = \\ell ^ { - 1 } E _ { \\nu ^ n _ \\rho } [ W _ j ^ 2 ( \\nabla _ { j , j - 1 } f ) ] + \\ell ^ { - 1 } E _ { \\nu ^ n _ \\rho } [ W _ j ( W _ j - W _ { j - 1 } ) f ] . \\end{aligned} \\end{align*}"}
+{"id": "5687.png", "formula": "\\begin{align*} \\mathbf { m } ^ { 1 } & = \\overline { \\mathbf { m } } ^ { 1 } , \\mathbf { m } ^ { 2 } = \\overline { \\mathbf { m } } ^ { 2 } \\mathbf { m = \\mathbf { m } ^ { 1 } + i \\mathbf { m } ^ { 2 } , } \\\\ d \\mathbf { m } ^ { 1 } & = 1 , d \\mathbf { m } ^ { 2 } = 0 . \\end{align*}"}
+{"id": "1185.png", "formula": "\\begin{align*} | \\xi _ i - \\xi | = O ( c _ i ^ { - 1 - \\frac { 1 } { d } } ) \\ ; \\ , \\ ; \\ , i \\to \\infty , \\end{align*}"}
+{"id": "3146.png", "formula": "\\begin{align*} - A : D ^ 2 w _ A = - \\frac { c } { 2 r } B : D ^ 2 w _ A = - \\frac { c } { r } \\partial _ { 1 1 } ^ 2 w _ A = - \\frac { c } { 2 r } ( r - 1 ) = \\frac { c } { 2 } \\left ( \\frac { 1 } { r } - 1 \\right ) . \\end{align*}"}
+{"id": "8432.png", "formula": "\\begin{align*} c _ * ( A ) = \\sup _ { \\nu \\in \\mathcal E ^ + _ A } \\ , G ( \\nu ) . \\end{align*}"}
+{"id": "8271.png", "formula": "\\begin{align*} Z ( s ) = P ( s ) + \\frac 1 2 P ( 2 s ) + P ^ * ( s ) ( \\Re s > 1 ) \\textrm { w i t h } | P ^ * ( s ) | = O ( 1 ) ~ ( \\Re s \\ge 1 / 2 ) . \\end{align*}"}
+{"id": "4118.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\lambda ( h , K ) & = \\int _ M \\Big \\langle \\frac { 1 } { 2 } \\triangle h + \\mathring { R } ( h ) , h \\Big \\rangle - \\frac { 1 } { 6 } | d K | ^ 2 d V _ g = \\int _ M \\Big \\langle \\frac { 1 } { 2 } \\triangle _ L h , h \\Big \\rangle - \\frac { 1 } { 6 } | d K | ^ 2 d V _ g . \\end{align*}"}
+{"id": "2539.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { j = 1 } ^ N | p _ { 1 , j } - q _ { 1 , j } | | w _ j | \\bigg ) ^ 2 \\le \\frac { C \\| \\nabla \\varphi \\| _ \\infty ^ 2 } { r ^ { 4 d } \\varphi _ 0 ^ 2 } \\bigg ( | x _ 1 - y _ 1 | ^ 2 + \\frac 1 N \\sum _ { k = 1 } ^ N | x _ k - y _ k | ^ 2 \\bigg ) \\frac 1 N \\sum _ { j = 1 } ^ N | w _ j | ^ 2 \\ , . \\end{align*}"}
+{"id": "6440.png", "formula": "\\begin{align*} f ( \\hat X , \\hat Y , \\hat t ) & = 0 , \\\\ f ( X , Y , t ) & > 0 , \\textrm { i f } ( X , Y , t ) \\neq ( \\hat X , \\hat Y , \\hat t ) . \\end{align*}"}
+{"id": "5267.png", "formula": "\\begin{align*} g ( T _ U V , X ) = - g ( U , V ) g ( X , \\nabla f ) , \\end{align*}"}
+{"id": "6613.png", "formula": "\\begin{align*} \\nabla _ { \\theta } ( P _ n H ( \\theta ) P _ n ) = \\nabla _ { \\theta } ( E _ n P _ n ) = ( \\nabla _ { \\theta } E _ n ) P _ n + E _ n \\nabla _ { \\theta } P _ n \\ , . \\end{align*}"}
+{"id": "7127.png", "formula": "\\begin{align*} \\left \\langle d , x ^ { \\rho _ 1 + \\cdots + \\rho _ i } \\right \\rangle = \\left \\langle d \\cap x ^ { \\rho _ 1 } , x ^ { \\rho _ 2 + \\cdots + \\rho _ i } \\right \\rangle = \\left \\langle 0 , x ^ { \\rho _ 2 + \\cdots + \\rho _ i } \\right \\rangle = 0 , \\end{align*}"}
+{"id": "5583.png", "formula": "\\begin{align*} \\langle S | k | k m _ i \\rangle _ j = \\lambda \\delta _ { i j } - S _ { i j } \\ , . \\end{align*}"}
+{"id": "6528.png", "formula": "\\begin{align*} \\left ( S _ n ( L ) \\geq c \\sqrt { \\log \\log ( n ) } \\right ) \\leq \\underset { t > 0 } { \\inf } e ^ { \\frac { t ^ 2 } { 2 } - c t \\sqrt { \\log \\log ( n ) } } = e ^ { - \\frac { c ^ 2 } { 2 } \\log \\log ( n ) } . \\end{align*}"}
+{"id": "856.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & - w _ 3 & w _ 2 \\\\ w _ 3 & 0 & - w _ 1 \\\\ - w _ 2 & w _ 1 & 0 \\end{bmatrix} ^ \\vee = \\begin{bmatrix} w _ 1 \\\\ w _ 2 \\\\ w _ 3 \\end{bmatrix} \\end{align*}"}
+{"id": "1727.png", "formula": "\\begin{align*} \\int _ { S ^ 3 } f ( a , b ) d ( a , b ) : = \\int _ { S ^ 1 } \\int _ { S ^ 1 } \\int _ 0 ^ { \\pi / 2 } \\sin 2 \\theta \\cdot f ( \\cos \\theta e ^ { i \\alpha } , \\sin \\theta e ^ { i \\beta } ) d \\theta d \\alpha d \\beta , \\end{align*}"}
+{"id": "5430.png", "formula": "\\begin{align*} \\left ( k _ \\psi ^ * \\sigma _ \\alpha \\right ) _ \\omega ( \\varphi ) = \\left ( \\sigma _ \\alpha \\right ) _ \\omega ( \\varphi ) . \\end{align*}"}
+{"id": "3460.png", "formula": "\\begin{align*} D _ { z } u ( t , x ) = \\sum _ { n \\geq 1 } n I _ { n - 1 } ( \\widetilde { f } _ n ( \\cdot , z , x ; t ) ) : = \\sum _ { n \\geq 1 } A _ n ( z , x ; t ) \\mbox { i n $ L ^ 2 ( \\Omega ) $ } . \\end{align*}"}
+{"id": "399.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty | \\langle h , \\tau _ n \\rangle | ^ 2 = \\| h \\| ^ 2 , \\forall h \\in \\mathcal { H } , \\end{align*}"}
+{"id": "5270.png", "formula": "\\begin{align*} K _ \\nu & = \\hat { K } - ( m - n ) ( m - n - 1 ) \\left \\{ \\lambda ^ 2 \\| \\nabla ( f + \\log \\lambda ) \\| ^ 2 - \\frac { \\lambda ^ 4 } { 4 } \\| \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\| ^ 2 + \\lambda ^ 2 g \\left ( \\nabla f , \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\right ) \\right \\} , \\end{align*}"}
+{"id": "6498.png", "formula": "\\begin{align*} T ^ { p r i v , a d a p t } = T _ { } ^ { p r i v , a d a p t } \\vee T _ { } ^ { p r i v , a d a p t } T ^ { p u b , a d a p t } = T _ { } ^ { p u b , a d a p t } \\vee T _ { } ^ { p u b , a d a p t } . \\end{align*}"}
+{"id": "8071.png", "formula": "\\begin{align*} \\mathcal { O } ^ - = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } \\sum _ { c > 0 } \\frac { S ( - n , p ; c ) } { c } H _ { m , n } ^ - \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) . \\end{align*}"}
+{"id": "1562.png", "formula": "\\begin{align*} \\rho ( X ) : = \\frac { X + \\omega } { \\omega X + 1 } \\end{align*}"}
+{"id": "1199.png", "formula": "\\begin{align*} J \\Psi _ * \\tilde \\xi = J \\tilde \\xi = - \\Psi _ * \\partial _ r = \\Psi _ * \\tilde { J } \\tilde \\xi \\ ; \\ , \\ ; \\ , J \\Psi _ * \\partial _ r = \\tilde \\xi = \\Psi _ * \\tilde \\xi = \\Psi _ * \\tilde { J } \\partial _ r \\ ; \\ , \\ ; \\ , r = 1 . \\end{align*}"}
+{"id": "6639.png", "formula": "\\begin{align*} P _ r ( z ) : = ( t ' - r ^ 2 , t ' ) \\times B _ r ( x ' ) \\end{align*}"}
+{"id": "5827.png", "formula": "\\begin{align*} \\langle L u , \\phi \\rangle = \\int _ \\Omega \\nabla u \\cdot \\nabla \\phi d x \\quad \\hbox { a n d } \\langle g v , \\phi \\rangle = \\int _ \\Omega a v \\phi d x , \\end{align*}"}
+{"id": "4863.png", "formula": "\\begin{align*} I _ k ^ \\flat ( u ) & = \\int _ { \\Sigma _ { k } ^ \\flat } u ^ { 1 } ( s _ 1 ) \\dots u ^ k ( s _ k ) d \\L ^ k , \\\\ I _ { n - k } ( u ) & = \\int _ { \\Sigma _ { n - k } } u ^ { k + 1 } ( s _ { k + 1 } ) \\dots u ^ { n } ( s _ n ) d \\L ^ { n - k } . \\end{align*}"}
+{"id": "5850.png", "formula": "\\begin{align*} L _ t \\circ S _ t ^ { - 1 } ( \\Phi ( t ) , \\Phi ' ( t ) ) = L _ t ( \\widehat { \\Phi } _ 0 , \\widehat { \\Phi } _ 1 ) \\end{align*}"}
+{"id": "3968.png", "formula": "\\begin{align*} \\{ u < h / 2 \\} \\subset G ^ h = \\{ u < h - p \\cdot x \\} \\subset \\{ u < 3 h / 2 \\} . \\end{align*}"}
+{"id": "6609.png", "formula": "\\begin{align*} \\frac { \\norm { x ^ m ( t ) ( \\psi _ N - \\psi ) } } { t ^ m } \\leq c _ m \\frac { \\norm { x ^ m ( \\psi _ N - \\psi ) } } { t ^ m } + C _ m \\sum _ { k = 0 } ^ m \\norm { H ^ k ( \\psi _ N - \\psi ) } . \\end{align*}"}
+{"id": "7455.png", "formula": "\\begin{align*} \\| \\Gamma \\| = \\underset { Z , W \\in \\Omega _ \\mu } \\sup \\| \\Gamma ( Z , W ) \\| , \\end{align*}"}
+{"id": "1281.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { k - 1 } \\sum _ { j = 0 } ^ { \\ell - 1 } m _ { t , j } ( x ) \\zeta _ { t } ( z ) \\eta _ j ( y ) p _ j ^ { ( t ) } ( x ) = a ( x , y , z ) ( x ^ s - \\alpha ) + b ( x , y , z ) ( y ^ \\ell - \\beta ) + c ( x , y , z ) ( z ^ k - \\gamma ) . \\end{align*}"}
+{"id": "5086.png", "formula": "\\begin{align*} [ x ^ h ] P _ n = \\sum _ { i \\in I } p _ { h , i } ( n ) c _ i ^ n , \\end{align*}"}
+{"id": "2314.png", "formula": "\\begin{align*} | \\Delta _ p U _ p | & = O \\ ( | \\nabla U _ p | ^ { p - 4 } U _ p ( | x | ^ 2 + ( y - 1 ) ^ 2 ) ^ { - \\frac { N - p } { 2 ( p - 1 ) } - 1 } \\ ) , \\\\ | \\nabla U _ p | ^ { p - 4 } & = O \\ ( ( | x | ^ 2 + ( y - 1 ) ^ 2 ) ^ { ( - \\frac { N - p } { p - 1 } - 1 ) ( \\frac { p - 4 } { 2 } ) } \\ ) , \\\\ | U _ p | & = O \\ ( ( | x | ^ 2 + ( y - 1 ) ^ 2 ) ^ { - \\frac { N - p } { 2 ( p - 1 ) } } \\ ) \\end{align*}"}
+{"id": "8085.png", "formula": "\\begin{align*} \\mathcal { R } ^ + = \\mathcal { R } ^ + _ 1 + \\mathcal { R } ^ + _ 2 + \\mathcal { R } ^ + _ 3 , \\end{align*}"}
+{"id": "5621.png", "formula": "\\begin{align*} ^ { C F } D _ { t } ^ { \\alpha } f ( t ) = \\dfrac { 1 } { 2 } \\dfrac { B ( \\alpha ) ( 2 - \\alpha ) } { 1 - \\alpha } \\int ^ { t } _ { t _ { 0 } } f ^ { ' } ( x ) \\exp \\left [ - \\dfrac { \\alpha } { 1 - \\alpha } ( t - x ) \\right ] d x , \\end{align*}"}
+{"id": "7221.png", "formula": "\\begin{align*} p ( t , \\ , 0 ) \\ , = \\ , q ( t ) . \\end{align*}"}
+{"id": "7576.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ 7 ) = & \\dfrac { ( 1 - 5 ^ { - 1 } ) ^ 2 } { 1 - 5 ^ { - 1 } t } \\Big ( - ( 5 ^ { - 1 } + 2 \\times 5 ^ { - 2 } - 5 ^ { - 3 } + 5 ^ { - 4 } ) t + 1 - 5 ^ { - 1 } - 2 \\times 5 ^ { - 2 } \\Big ) \\\\ + & ( 5 ^ { - 2 } - 5 ^ { - 3 } ) t ^ 5 Z _ { g _ 7 } ( s , \\chi ) , \\end{align*}"}
+{"id": "4701.png", "formula": "\\begin{align*} \\mathfrak { E } ^ { G _ { I I } } = \\mathbb { C } [ \\varphi _ 4 , \\varphi _ { 1 2 } ] \\end{align*}"}
+{"id": "4143.png", "formula": "\\begin{align*} \\| D h \\| _ { L ^ 2 } ^ 2 = \\| \\nabla h \\| _ { L ^ 2 } ^ 2 + 2 \\| h \\| _ { L ^ 2 } ^ 2 + 2 ( \\mathring { R } h , h ) _ { L ^ 2 } - 2 \\int _ M R _ { i j } h _ { i k } h _ { j k } d V _ g . \\end{align*}"}
+{"id": "2392.png", "formula": "\\begin{align*} \\widetilde E = Q ^ T E Q , \\quad \\widetilde A = Q ^ T A Q - Q ^ T E \\dot Q \\end{align*}"}
+{"id": "4376.png", "formula": "\\begin{align*} \\frac { p } { q } - \\frac { 1 } { a _ 1 } - \\frac { 1 } { a _ 2 } = \\frac { 1 } { q a _ 1 } - \\frac { 1 } { q a _ 1 + 1 } = \\frac { 1 } { q a _ 1 ( q a _ 1 + 1 ) } = \\frac { 1 } { q a _ 1 a _ 2 } \\end{align*}"}
+{"id": "5861.png", "formula": "\\begin{align*} \\partial _ \\nu U = D \\widehat { H } ~ ( 0 , T ) \\times \\Gamma \\end{align*}"}
+{"id": "5308.png", "formula": "\\begin{align*} \\beta ( X ) ( \\xi ) = \\widetilde { \\nabla } _ { X } \\xi + \\ , \\mathcal { T } ^ { 1 } = - A _ { \\xi } ( X ) + \\omega ( X ) \\xi + \\ , \\mathcal { T } ^ { 1 } = - A _ { \\xi } ( X ) + \\ , \\mathcal { T } ^ { 1 } , \\end{align*}"}
+{"id": "3952.png", "formula": "\\begin{align*} h & = g ( d ^ * e _ n , \\hat { y } , h ) - g ( 0 , \\hat { y } , h ) \\\\ & = d ^ * g _ { x _ n } ( \\tau d ^ * e _ n , \\hat { y } , h ) \\\\ & \\leq d ^ * | g _ x ( 0 , \\hat { y } , h ) | + C d ^ * ( d ^ * + h ) | \\hat { y } | + K h ( h + d ^ * ) . \\end{align*}"}
+{"id": "6798.png", "formula": "\\begin{align*} & B _ \\eta \\left ( \\left ( \\overline { x } _ { n ( p ) + \\i - 1 , \\widehat { n ( p ) + \\eta - 1 } } \\right ) _ { \\i = 1 } ^ { \\infty } , \\left ( \\overline { x } _ { m ( p ) + \\i - 1 , \\widehat { m ( p ) + \\eta - 1 } } \\right ) _ { \\i = 1 } ^ { \\infty } \\right ) \\\\ & = d \\left ( \\overline { x } _ { n ( p ) + \\eta - 1 } , \\overline { x } _ { m ( p ) + \\eta - 1 } \\right ) . \\\\ \\end{align*}"}
+{"id": "5058.png", "formula": "\\begin{align*} F = \\frac { 1 } { ( 1 - x ) ( 1 - y ) ( 1 - x y ) } = \\frac { x } { ( 1 - x ) ( 1 - x y ) ^ 2 } + \\frac { 1 } { ( 1 - y ) ( 1 - x y ) ^ 2 } . \\end{align*}"}
+{"id": "339.png", "formula": "\\begin{align*} \\Im ( A ( p , 1 ) ) = \\Im ( A ' ( p , 1 ) ) \\end{align*}"}
+{"id": "2792.png", "formula": "\\begin{align*} \\left ( \\tau \\sigma \\right ) f = \\tau ( \\sigma f ) \\end{align*}"}
+{"id": "5242.png", "formula": "\\begin{align*} \\begin{array} { l l } ( \\nabla F _ \\ast ) ( X , Y ) = - \\frac { \\lambda ^ 2 } { 2 } \\left \\{ X ( \\frac { 1 } { \\lambda ^ 2 } ) \\tilde { Y } + Y ( \\frac { 1 } { \\lambda ^ 2 } ) \\tilde { X } - g ( X , Y ) F _ \\ast ( g r a d _ \\mathcal { H } \\frac { 1 } { \\lambda ^ 2 } ) ) \\right \\} , \\end{array} \\end{align*}"}
+{"id": "7511.png", "formula": "\\begin{align*} S _ { k , r } ( j ) = & \\sum \\limits _ { i = 0 } ^ k ( - 1 ) ^ { k - i } \\dbinom { k + r } { k - i } \\dbinom { i + r } { j - ( k - i ) } \\\\ = & \\sum \\limits _ { i = 0 } ^ k ( - 1 ) ^ i \\dbinom { k + r } { i } \\dbinom { k - i + r } { j - i } . \\end{align*}"}
+{"id": "6052.png", "formula": "\\begin{align*} m _ \\infty ( x ) = - x ^ g + \\sum _ { i = 1 } ^ g \\left ( \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac { y ^ g \\dd y } { w ( y ) } \\right ) l _ i ( x ) . \\end{align*}"}
+{"id": "6660.png", "formula": "\\begin{align*} \\begin{aligned} x ^ { k + 1 } & = \\left ( ( I + \\gamma _ 1 ^ k R ) \\otimes I _ d \\right ) x ^ k + \\gamma _ 1 ^ k \\zeta _ w ^ k - \\lambda ^ k y ^ k \\\\ y ^ { k + 1 } & = \\left ( ( ( 1 - \\alpha ^ k ) I + \\gamma _ 2 ^ k C ) \\otimes I _ d \\right ) y ^ k + \\gamma _ 2 ^ k \\xi _ w ^ k + g ^ { k + 1 } \\\\ & \\qquad - ( 1 - \\alpha ^ k ) g ^ { k } \\end{aligned} \\end{align*}"}
+{"id": "4185.png", "formula": "\\begin{align*} g = \\alpha \\left ( \\begin{array} { c c } \\cosh { \\Lambda } + \\cos 2 \\phi \\sinh { \\Lambda } & \\sin 2 \\phi \\sinh { \\Lambda } \\\\ \\sin 2 \\phi \\sinh { \\Lambda } & \\cosh { \\Lambda } - \\cos 2 \\phi \\sinh { \\Lambda } \\end{array} \\right ) . \\end{align*}"}
+{"id": "5341.png", "formula": "\\begin{align*} p : = \\mathrm { d i v } ( \\varepsilon u ) = 0 , \\end{align*}"}
+{"id": "8689.png", "formula": "\\begin{align*} A = I d + a E _ { - 2 , 0 } = \\small { \\begin{array} { r r r r | c | r r r c } \\dots & 1 & 0 & 0 & 0 & 0 & 0 & 0 & \\dots \\\\ \\dots & 0 & 1 & 0 & a & 0 & 0 & 0 & \\dots \\\\ \\dots & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \\dots \\\\ \\hline \\dots & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \\dots \\\\ \\hline \\dots & 0 & 0 & 0 & 0 & 1 & 0 & 0 & \\dots \\\\ \\dots & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \\dots \\\\ \\dots & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \\dots \\\\ \\end{array} } \\end{align*}"}
+{"id": "289.png", "formula": "\\begin{align*} r _ { \\chi } ( w ) : = \\prod _ { \\alpha \\in \\Phi / \\Gamma \\times \\{ \\pm 1 \\} \\atop g _ { i } \\in \\Gamma _ { F _ { \\pm \\alpha } } \\backslash \\Gamma _ F } \\chi _ { \\alpha } ( v _ 0 ( u _ { g _ { i } } ( w ) ) ) ^ { \\widehat { g _ { i } ^ { - 1 } \\alpha } } . \\end{align*}"}
+{"id": "1329.png", "formula": "\\begin{align*} \\left ( \\sum _ { k = 1 } ^ d \\lambda _ k \\right ) ^ 2 = \\left ( \\sum _ { k = 1 } ^ d 1 \\right ) \\left ( \\sum _ { k = 1 } ^ d \\lambda _ k ^ 2 \\right ) \\end{align*}"}
+{"id": "7209.png", "formula": "\\begin{align*} \\begin{aligned} i . & \\qquad \\overline { \\omega } ' = - \\ , \\omega ' \\\\ i i . & \\qquad \\omega \\omega ' = - \\ , \\omega ' \\omega \\mbox { w h e r e } \\omega = \\frac { x } { | x | } \\\\ i i i . & \\qquad \\big ( \\omega ' \\big ) ^ 2 = - \\frac { 1 } { | x | ^ 2 } \\Big ( | x | ' - | x ' | \\Big ) ^ 2 \\\\ i v . & \\qquad \\omega \\omega '' + \\omega '' \\omega = 2 | \\omega ' | ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "1150.png", "formula": "\\begin{align*} \\langle x , y \\rangle = \\frac { ( x + y ) ( x + y + 1 ) } { 2 } + y . \\end{align*}"}
+{"id": "7152.png", "formula": "\\begin{align*} \\displaystyle { u ^ { \\{ k \\} } _ m = c \\frac { d z _ m } { d t _ k } \\Bigg | _ { \\rm e q } = - \\sum \\limits _ { \\substack { | I | = k \\\\ m \\in I } } \\prod \\limits _ { \\substack { i \\in I \\\\ j \\notin I } } \\phi ( x _ j - x _ i ) \\ , . } \\end{align*}"}
+{"id": "4937.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } u _ t \\\\ v _ t \\end{array} \\right ) = \\mathcal { J } \\left ( \\begin{array} { c } \\frac { \\delta \\mathcal H } { \\delta u } \\\\ \\frac { \\delta \\mathcal H } { \\delta v } \\end{array} \\right ) , \\mathcal { J } = \\left ( \\begin{array} { c c } 0 & \\tfrac { 1 } 2 \\\\ - \\tfrac { 1 } 2 & 0 \\end{array} \\right ) , \\mathcal { H } = \\int ( u _ x ^ 2 + v _ x ^ 2 - \\tfrac { 1 } 2 ( u ^ 2 + v ^ 2 ) ^ 2 ) \\ , \\mathrm { d } x . \\end{align*}"}
+{"id": "2474.png", "formula": "\\begin{align*} y ^ 2 + y = x ^ 3 - x ^ 2 \\end{align*}"}
+{"id": "2050.png", "formula": "\\begin{align*} y ' ( t ) = z J H ( t ) y ( t ) , t \\in ( 0 , L ) , \\end{align*}"}
+{"id": "3612.png", "formula": "\\begin{align*} \\arctan \\left ( \\frac { 7 - 2 \\beta _ 0 R } { 2 \\sqrt { \\alpha _ 0 + 7 \\beta _ 0 R - \\beta _ 0 ^ 2 R ^ 2 } } \\right ) \\ & = \\ \\arccos \\left ( \\frac { 2 \\sqrt { 7 \\beta ( r - 1 ) - \\beta ^ 2 ( r ^ 2 - 1 ) } } { 4 \\beta ^ 2 - 2 8 \\beta + 4 9 } \\right ) \\\\ \\ & = \\ \\arccos \\left ( \\frac { 2 \\sqrt { \\beta ( r - 1 ) ( 7 - \\beta ( r + 1 ) ) } } { 4 \\beta ^ 2 - 2 8 \\beta + 4 9 } \\right ) \\end{align*}"}
+{"id": "564.png", "formula": "\\begin{align*} X _ { t _ { n + 1 / 2 } } ^ \\dagger = \\frac { 1 } { 2 } ( X _ { t _ n } ^ \\dagger + X _ { t _ { n + 1 } } ^ \\dagger ) \\ , . \\end{align*}"}
+{"id": "2013.png", "formula": "\\begin{align*} A ^ * A = k E , \\end{align*}"}
+{"id": "2220.png", "formula": "\\begin{align*} \\nu ( x _ { 2 m , i } ) & = \\nu { \\left ( \\sum _ { k = 1 } ^ \\infty x _ { 2 m - 1 , k } \\alpha _ { k , i } \\right ) } \\\\ & = \\nu { \\left ( x _ { 2 m - 1 , 1 } \\alpha _ { 1 , i } + \\sum _ { k = 2 } ^ \\infty x _ { 2 m - 1 , k } \\alpha _ { k , i } \\right ) } \\\\ & \\geq \\min _ { k \\geq 2 } \\{ \\nu ( x _ { 2 m - 1 , k } ) + \\nu ( \\alpha _ { k , i } ) \\} . \\end{align*}"}
+{"id": "720.png", "formula": "\\begin{align*} \\lambda ( w \\pm u ) ^ 2 = \\lambda ( w ) ^ 2 \\Big ( 1 \\pm \\big ( r ( w ) u + \\overline { r ( w ) } \\bar { u } \\big ) + \\mathcal { O } ( | u | ^ 2 ) \\Big ) . \\end{align*}"}
+{"id": "779.png", "formula": "\\begin{align*} \\begin{aligned} E ^ \\N & \\to Y ^ \\N \\\\ ( x _ i ) & \\mapsto ( ( x _ 0 , c ) , ( x _ 1 , X ( x _ 0 ) c ) , ( x _ 2 , X ( x _ 1 ) X ( x _ 0 ) c ) , \\ldots , y _ n , \\ldots ) \\end{aligned} \\end{align*}"}
+{"id": "1333.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | ^ m & \\geq \\frac { \\frac { 1 } { { d + m - 1 \\choose m } } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) ^ m \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ { 2 m } } { n ^ 2 - n } \\\\ & = \\frac { \\frac { 1 } { ( ^ m ( \\mathcal { X } ) ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) ^ m \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ { 2 m } } { n ^ 2 - n } , \\end{align*}"}
+{"id": "2799.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n b _ i = \\sum _ { i = 1 } ^ n a _ i . \\end{align*}"}
+{"id": "7295.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & ( i ) \\ j ( 1 , \\infty ) + \\int _ { 0 } ^ { 1 } x j ( d x ) + \\int _ { 0 } ^ { 1 } | G _ m ( x ) | j ( d x ) < \\infty , \\\\ & ( i i ) \\ j ( 0 , 1 ) = \\infty \\end{aligned} \\right . \\end{align*}"}
+{"id": "4584.png", "formula": "\\begin{align*} 1 - \\sum _ { i = 1 } ^ n \\frac { 1 } { b _ i } = \\frac { r } { \\prod _ { i = 1 } ^ n b _ i } \\geq \\frac { 1 } { \\prod _ { i = 1 } ^ n b _ i } > 0 \\end{align*}"}
+{"id": "3432.png", "formula": "\\begin{align*} \\frac { \\mu _ { t } ( [ b ] ) } { \\mu _ { t } ( [ a ] ) } = \\sum _ { i = 1 } ^ { \\infty } \\exp \\left ( ( t \\phi - P _ { G } ( t \\phi ) ) ( \\underline { \\gamma } _ i ) \\right ) N ( b , \\underline { \\gamma } _ i ) . \\end{align*}"}
+{"id": "6345.png", "formula": "\\begin{align*} \\psi ' ( \\nu _ 0 ) - \\int ^ { r ( q ) } _ 1 f _ \\nu ( x , \\nu _ 0 ) d x + \\int _ { r ( q _ c ) } ^ 1 f _ \\nu ( x , \\nu _ 0 ) d x = 0 , \\end{align*}"}
+{"id": "8322.png", "formula": "\\begin{align*} \\begin{cases} \\rho = | x | + t , \\sigma = | x | - t , \\\\ \\bar { \\rho } = | y | + s , \\bar { \\sigma } = | y | - s . \\end{cases} \\end{align*}"}
+{"id": "1472.png", "formula": "\\begin{align*} x _ i = \\begin{cases} 1 & i = 1 \\\\ 2 & 2 \\leq i \\leq a + 1 \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "3890.png", "formula": "\\begin{align*} \\det D X v = \\frac { f ^ * ( \\cdot ) } { f ( X v ( \\cdot ) ) } , \\\\ X v ( \\Omega ^ * ) = \\Omega . \\end{align*}"}
+{"id": "2877.png", "formula": "\\begin{align*} \\int _ T ^ { 2 T } \\abs { \\frac { L _ f ' } { L _ f } ( \\sigma _ 0 + i t ) } ^ 2 d t & \\ll _ f \\int _ T ^ { 2 T } \\abs { \\sum _ { m = 1 } ^ { \\infty } \\frac { \\Lambda _ f ( m ) e ^ { - \\frac { m } { Y } } } { m ^ { \\sigma _ 0 + i t } } } ^ 2 d t + Y ^ { 1 - 2 \\sigma _ 0 } T ( \\log T ) ^ 2 . \\\\ \\end{align*}"}
+{"id": "4230.png", "formula": "\\begin{align*} \\tilde { \\Lambda } _ 0 ( x ) & : = \\frac { 1 } { \\varepsilon } \\left ( \\cosh ^ { - 1 } \\left ( | v | \\cosh ( \\lambda + \\varepsilon \\theta ( x ) ) - \\frac { 1 } { \\sqrt { c } } \\tanh ( \\beta ( \\lambda + \\varepsilon \\theta ( x ) ) ) \\sinh ( \\lambda + \\varepsilon \\theta ( x ) ) \\right ) \\right ) \\\\ & - \\frac { 1 } { \\varepsilon } \\left ( \\cosh ^ { - 1 } \\left ( | v | \\cosh ( \\lambda ) - \\frac { 1 } { \\sqrt { c } } \\tanh ( \\beta \\lambda ) \\sinh ( \\lambda ) \\right ) \\right ) . \\end{align*}"}
+{"id": "2863.png", "formula": "\\begin{align*} [ f _ i ] _ { S _ n } = \\frac { x _ 1 + \\cdots + x _ n } { n } . \\end{align*}"}
+{"id": "6543.png", "formula": "\\begin{align*} { f _ Z } \\left ( z \\right ) = \\frac { { { m _ f } ^ { { m _ f } } { { \\left ( { { m _ s } - 1 } \\right ) } ^ { { m _ s } } } { { \\bar z } ^ { { m _ s } } } { z ^ { { m _ f } - 1 } } } } { { B \\left ( { { m _ f } , { m _ s } } \\right ) { { \\left ( { { m _ f } z + \\left ( { { m _ s } - 1 } \\right ) \\bar z } \\right ) } ^ { { m _ f } + { m _ s } } } } } , \\end{align*}"}
+{"id": "1845.png", "formula": "\\begin{align*} p ( \\rho ) = \\kappa \\rho ^ \\gamma ( \\gamma > 1 ) \\end{align*}"}
+{"id": "3095.png", "formula": "\\begin{align*} - \\Delta w _ B = r _ B b - \\bar { b } \\quad Y , w _ B Y , \\int _ Y w _ B = 0 ; \\end{align*}"}
+{"id": "5565.png", "formula": "\\begin{align*} \\zeta ( 2 s + k ) \\int _ { 0 } ^ { \\infty } x ^ { - s - 1 } P _ k ( x ) { \\rm d } x = \\Gamma ( - s ) \\end{align*}"}
+{"id": "1993.png", "formula": "\\begin{align*} I _ 2 & = \\sum _ { \\substack { ( m _ { i j } ) \\in S S Y T ( \\lambda ) \\\\ ( m ) \\in S S Y T ( e ^ 1 ) } } \\frac { 1 } { m ( m + m _ { 1 1 } ) ^ 2 ( m + m _ { 2 1 } ) ^ 2 m _ { 1 2 } ^ { 3 } } , \\\\ I _ 3 = I _ 4 & = \\sum _ { \\substack { ( m _ { i j } ) \\in S S Y T ( \\lambda ) \\\\ ( m ) \\in S S Y T ( e ^ 1 ) } } \\frac { 1 } { m ( m + m _ { 1 1 } ) ( m + m _ { 2 1 } ) ^ 3 m _ { 1 2 } ^ { 3 } } . \\end{align*}"}
+{"id": "2063.png", "formula": "\\begin{align*} y '' ( x ) - 2 d x ^ { \\gamma - 1 } y ' ( x ) + \\bigl ( c - d ( \\gamma - 1 ) \\bigr ) x ^ { \\gamma - 2 } y ( x ) = 0 . \\end{align*}"}
+{"id": "7204.png", "formula": "\\begin{align*} \\big \\langle p , \\ , q \\big \\rangle \\ , = \\ , \\ , p \\overline q . \\end{align*}"}
+{"id": "9040.png", "formula": "\\begin{align*} \\alpha : & = \\max ( \\delta ^ { 1 + \\frac { p - 2 } { 2 } } 2 ^ { \\frac { p } { 2 } - 1 } \\eta ^ { p } C ^ { p } _ f + 2 ^ { \\frac { p } { 2 } - 1 } 4 ^ { p - 1 } \\gamma ^ p _ 1 , \\delta ^ { 1 + \\frac { p - 2 } { 2 } } \\eta ^ { p } C ^ { p } _ f + 2 ^ { \\frac { p } { 2 } - 1 } 4 ^ { p - 1 } \\gamma ^ p _ 2 , \\\\ & \\qquad \\delta ^ { 1 + \\frac { p - 2 } { 2 } } \\eta ^ { p } C ^ { p } _ f + 2 ^ { \\frac { p } { 2 } - 1 } 4 ^ { p - 1 } \\kappa ^ p _ 1 , \\delta ^ { 1 + \\frac { p - 2 } { 2 } } \\eta ^ { p } C ^ { p } _ f + 2 ^ { \\frac { p } { 2 } - 1 } 4 ^ { p - 1 } \\kappa ^ p _ 2 ) . \\end{align*}"}
+{"id": "1443.png", "formula": "\\begin{align*} \\mathbb { P } ( S _ t > 0 \\forall t \\in [ k ] ) & = \\mathbb { P } ( I _ d + \\sum _ { j = 1 } ^ { d - 1 } I _ j + \\sum _ { i = 2 } ^ { t } ( X _ i - 1 ) > 0 \\forall t \\in [ k ] ) \\\\ & = \\mathbb { P } ( 1 + I _ d + \\left ( \\sum _ { j = 1 } ^ { d - 1 } I _ j - 1 \\right ) + \\sum _ { i = 2 } ^ { t } ( X _ i - 1 ) > 0 \\forall t \\in [ k ] ) \\\\ & = \\mathbb { P } ( 1 + I _ d + \\sum _ { i = 1 } ^ { t } ( X ' _ i - 1 ) > 0 \\forall t \\in [ k ] ) , \\end{align*}"}
+{"id": "7502.png", "formula": "\\begin{align*} Z _ g ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) = Z _ { \\tilde { g } } ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) = { \\left \\{ \\begin{array} { r l } \\dfrac { G _ 0 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\ \\ & { \\rm o r d } ( \\alpha _ { i _ 0 } ) = { \\rm o r d } ( \\beta _ { j _ 0 } ) , \\\\ c ( \\chi ) q ^ { - e _ 0 s } , \\ \\ & { \\rm o t h e r w i s e } , \\end{array} \\right . } \\end{align*}"}
+{"id": "4450.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 3 6 } = \\frac { 1 } { 4 } + \\frac { 1 } { 9 } = \\frac { 1 3 } { 3 6 } \\end{align*}"}
+{"id": "439.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\right \\| = b \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "8723.png", "formula": "\\begin{align*} 1 + \\sum _ { k > 0 } \\frac { h ^ { [ O R V ] } _ { k ; \\tau ^ { - r } a } } { ( u - a _ { 1 - r } ) \\dots ( u - a _ { k - r } ) } = H ( 1 / u ) . \\end{align*}"}
+{"id": "9218.png", "formula": "\\begin{align*} K _ { \\pm } = \\Lambda _ { \\pm } \\cap \\Lambda _ { f } T _ { ( x , 0 ) } K _ { \\pm } = T _ { ( x , 0 ) } \\Lambda _ { \\pm } \\cap T _ { ( x , 0 ) } \\Lambda _ { f } , \\end{align*}"}
+{"id": "7226.png", "formula": "\\begin{align*} A _ { B , x } = \\begin{cases} 1 & x \\in B , \\\\ 0 & x \\not \\in B . \\end{cases} \\end{align*}"}
+{"id": "3371.png", "formula": "\\begin{align*} \\allowdisplaybreaks \\bigvee _ { j = 1 } ^ { i - 1 } ( - u _ i + x _ j ) \\vee ( - u _ { i + 1 } + 1 + x _ i ) & \\leq \\bigvee _ { j = i + 1 } ^ { n - 2 } ( - u _ j + x _ j ) \\vee x _ { n - 1 } \\vee ( - u _ { n - 1 } ) & & \\enspace i = 1 , \\dots , n - 2 \\\\ \\bigvee _ { j = 1 } ^ { n - 1 } x _ j & \\leq 0 \\\\ - \\infty \\leq x _ 1 \\leq x _ 2 & \\leq \\dots \\leq x _ { n - 2 } \\leq u _ { n - 1 } + x _ { n - 1 } \\ , . \\end{align*}"}
+{"id": "8839.png", "formula": "\\begin{align*} \\sum \\limits _ { i , j = 1 } ^ { n } g ^ { i j } \\left ( z \\right ) \\frac { \\partial ^ { 2 } } { \\partial z _ { i } \\partial \\overline { z } _ { j } } \\end{align*}"}
+{"id": "4991.png", "formula": "\\begin{align*} \\| ( - i \\nabla + A ) G _ { p , q } \\| ^ 2 = p ^ 2 + 2 p \\cdot \\langle A \\rangle ( q ) + \\langle A ^ 2 \\rangle ( q ) + \\| \\nabla g \\| ^ 2 \\ , , \\end{align*}"}
+{"id": "4168.png", "formula": "\\begin{align*} \\lambda ( g _ c , 0 ) & = \\int _ M ( R _ c - \\frac { 1 } { 1 2 } | H _ c | ^ 2 ) \\omega _ c ^ 2 + 4 | \\nabla \\omega _ c | ^ 2 d V _ { g _ c } \\\\ & \\leq \\int _ M ( R _ c - \\frac { 1 } { 1 2 } | H _ c | ^ 2 ) \\omega ^ 2 _ { \\infty } + 4 | \\nabla \\omega _ { \\infty } | ^ 2 d V _ { g _ c } = \\lim _ { i \\to \\infty } \\lambda ( g _ i , b _ i ) \\\\ & \\leq \\lim _ { i \\to \\infty } \\mathcal { F } ( g _ i , b _ i , f _ c ) = \\lambda ( g _ c , 0 ) . \\end{align*}"}
+{"id": "2216.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m + 1 } n + 5 ^ { 2 m + 1 } \\right ) } q ^ n & = \\gamma \\sum _ { i = 1 } ^ \\infty x _ { 2 m + 1 , i } \\xi ^ { i - 1 } . \\end{align*}"}
+{"id": "2924.png", "formula": "\\begin{align*} L _ n f ( \\eta ) = n ^ { 2 } \\sum _ { j \\in \\mathbb { Z } } g _ n ( \\eta _ j ) \\nabla _ { j , j + 1 } f ( \\eta ) , \\end{align*}"}
+{"id": "2494.png", "formula": "\\begin{align*} P _ { \\mathcal K _ { \\mathbf u } } S f = \\chi f - ( f , P _ { H ^ 2 } ( { \\mathbf u } \\overline \\chi ) ) { \\mathbf u } , \\ \\ \\ f \\in \\mathcal K _ { \\mathbf u } . \\end{align*}"}
+{"id": "5849.png", "formula": "\\begin{align*} L _ t \\circ S _ t ^ { - 1 } ( \\Phi ( t ) , \\Phi ' ( t ) ) = \\langle \\ ! \\langle ( U ' ( t ) , - U ( t ) ) , ( \\Phi ( t ) , \\Phi ' ( t ) ) \\rangle \\ ! \\rangle . \\end{align*}"}
+{"id": "3417.png", "formula": "\\begin{align*} P _ { t o p } ( \\phi ) : = \\sup \\{ h ( \\nu ) + \\nu ( \\phi ) \\colon \\nu \\in \\mathcal { M } _ { \\sigma } \\big ( \\Sigma \\big ) \\mbox { s . t . } - \\nu ( \\phi ) < \\infty \\} . \\end{align*}"}
+{"id": "430.png", "formula": "\\begin{align*} ( 1 - \\varepsilon ) \\left ( \\sum _ { n = 1 } ^ { m } | c _ n | ^ p \\right ) ^ \\frac { 1 } { p } \\leq \\left \\| \\sum _ { n = 1 } ^ { m } c _ n \\tau _ n \\right \\| \\leq ( 1 + \\varepsilon ) \\left ( \\sum _ { n = 1 } ^ { m } | c _ n | ^ p \\right ) ^ \\frac { 1 } { p } , \\forall c _ 1 , \\dots , c _ m \\in \\mathbb { K } . \\end{align*}"}
+{"id": "2121.png", "formula": "\\begin{align*} \\tilde y \\big ( a ( x ) + a ' ( x ) \\tilde y \\big ) = \\tilde y a ( x ) + \\tilde y a ' ( x ) \\tilde y = - \\overline { a ' } ( x ) + \\overline a ( x ) \\tilde y , \\end{align*}"}
+{"id": "8838.png", "formula": "\\begin{align*} d s _ { F S } ^ { 2 } : = \\sum \\limits _ { i , j = 1 } ^ { n } \\left ( \\left ( 1 + \\langle z , z \\rangle \\right ) \\delta _ { i j } - z _ { i } \\overline { z } _ { j } \\right ) d z _ { i } \\otimes d \\overline { z } _ { j } , \\langle z , z \\rangle = | z | ^ 2 , \\end{align*}"}
+{"id": "3068.png", "formula": "\\begin{align*} - D ^ 2 : ( \\xi A ) = 0 \\quad Y , \\xi Y , \\int _ Y \\xi = 1 , \\end{align*}"}
+{"id": "6774.png", "formula": "\\begin{align*} \\phi _ { \\chi } ^ e ( s ) = \\frac { \\sqrt { \\pi } } { 4 } \\left [ P ^ { - \\frac { s } { 2 } } \\frac { z ( s ) } { s } + \\frac { P ^ { - \\frac { 1 - s } { 2 } } } { G \\left ( \\overline { \\chi } , P \\right ) } \\frac { z ( 1 - s ) } { 1 - s } \\right ] ; \\mathfrak { R e } ( s ) \\in ( 0 , 1 ) \\end{align*}"}
+{"id": "2499.png", "formula": "\\begin{align*} E _ { \\mathbf v } = \\{ \\zeta \\in \\mathbb T \\ : \\ k _ { \\mathbf v , \\zeta } \\in H ^ 2 \\} , \\end{align*}"}
+{"id": "8033.png", "formula": "\\begin{align*} f _ 2 ( v ) = ( a _ 2 - x _ 3 ) ( y _ 4 - y _ 3 ) - ( x _ 4 - x _ 3 ) ( b _ 2 - y _ 3 ) = 0 , \\\\ f _ 3 ( v ) = ( a _ 3 - x _ 5 ) ( y _ 6 - y _ 5 ) - ( x _ 6 - x _ 5 ) ( b _ 3 - y _ 5 ) = 0 . \\end{align*}"}
+{"id": "6246.png", "formula": "\\begin{align*} k = M \\lambda _ k ' ( \\sin \\theta _ k \\cos \\phi _ k , \\sin \\theta _ k \\sin \\phi _ k , \\cos \\theta _ k ) , \\quad k \\in \\Z ^ 3 . \\end{align*}"}
+{"id": "2630.png", "formula": "\\begin{align*} \\left < v _ { 1 , i } ^ t \\ \\middle | \\ i = 2 , \\dots , m , \\ t = 1 , \\dots , d \\right > \\cap \\left < v _ { 1 , m + 1 } ^ t \\ \\middle | \\ t = 1 , \\dots , d \\right > = \\{ 0 \\} \\end{align*}"}
+{"id": "4295.png", "formula": "\\begin{align*} \\widehat { \\mathbf { H } } _ n = f _ ( \\widehat { \\mathbf { H } } _ { n , } ) , \\end{align*}"}
+{"id": "2322.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u | ^ p & = \\int _ { \\Omega } | v ' ( h ( y ) ) | ^ p | \\nabla h ( y ) | ^ { p - 1 } | \\nabla h ( y ) | \\ , d y \\\\ & = \\int _ 0 ^ 1 \\left [ \\ , \\int _ { h ^ { - 1 } ( \\{ t \\} ) \\cap \\Omega } | v ' ( h ( y ) ) | ^ p | \\nabla h ( y ) | ^ { p - 1 } \\ , d \\mathcal { H } ^ { N - 1 } ( y ) \\ , \\right ] \\ , d t . \\end{align*}"}
+{"id": "6400.png", "formula": "\\begin{align*} w _ j ( X _ j , Y _ j , t _ j , \\tilde X _ j , \\tilde Y _ j , \\tilde t _ j ) & \\geq u ( X _ j , Y _ j , t _ j ) - \\phi ( \\tilde X _ j , \\tilde Y _ j , \\tilde t _ j ) \\\\ & = u ( X _ j , Y _ j , t _ j ) - \\phi ( X _ j , Y _ j , t _ j ) + \\phi ( X _ j , Y _ j , t _ j ) - \\phi ( \\tilde X _ j , \\tilde Y _ j , \\tilde t _ j ) \\\\ & > 2 \\varepsilon - \\varepsilon \\\\ & = \\varepsilon . \\end{align*}"}
+{"id": "1473.png", "formula": "\\begin{align*} y _ j = \\begin{cases} 1 & j = a + 1 \\\\ 2 & 1 \\leq j \\leq a \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "6596.png", "formula": "\\begin{align*} I _ 1 & \\leq \\int e ^ { - r _ 2 ( | y - x | ^ { 1 / s } + | y | ^ { 1 / s } ) } d y \\\\ & \\leq e ^ { - r _ 2 c | x | ^ { 1 / s } } \\int e ^ { - r _ 2 c | y | ^ { 1 / s } } d y \\\\ & = B e ^ { - r _ 2 c | x | ^ { 1 / s } } , \\end{align*}"}
+{"id": "7414.png", "formula": "\\begin{align*} X _ { \\alpha } ( \\chi _ s ) = \\omega ( \\varpi ) q _ F ^ { - 2 s } . \\end{align*}"}
+{"id": "778.png", "formula": "\\begin{align*} Q ( y , A \\times B ) : = P ( x , A ) \\ 1 _ { B } ( X ( x ) \\cdot c ) . \\end{align*}"}
+{"id": "496.png", "formula": "\\begin{align*} ( { 3 } / { 4 } ) { \\textstyle \\sum _ { j = k + m + 1 } ^ { \\nu + m + 1 } } \\ , \\Xi _ j \\le ( { 1 } / { 4 } ) { \\textstyle \\sum _ { j = k } ^ { k + m } } \\ , \\Xi _ j + b a ^ { - 1 } { \\textstyle \\sum _ { j = k } ^ { k + m } } \\varphi \\big ( \\Phi ( x ^ { \\ell ( j ) } ) \\ ! - \\ ! \\Phi ^ * \\big ) . \\end{align*}"}
+{"id": "3218.png", "formula": "\\begin{align*} \\sum \\limits _ { i \\in T \\cap \\Pi } r _ { i } ^ { \\alpha } = \\sum \\limits _ { i \\in T \\cap \\Pi ^ { - } \\cap \\Pi } r _ { i } ^ { \\alpha } + \\sum _ { w \\in T \\cap \\Pi ^ { - } \\setminus \\Pi } r _ { w } ^ { \\alpha } \\cdot \\sum _ { v \\in W _ { T } \\left ( w \\right ) } r _ { v } ^ { \\alpha } . \\end{align*}"}
+{"id": "2364.png", "formula": "\\begin{align*} E ^ T = - E , A ^ T = A + \\dot E \\end{align*}"}
+{"id": "8341.png", "formula": "\\begin{align*} u ^ { ( M _ 0 ) } ( t , x ) & = M _ 0 ^ { \\frac { d } { 2 } - 1 } u ( M _ 0 t + M _ 0 , M _ 0 x ) , \\\\ Q _ l ^ { ( M _ 0 ) } ( t , x ) & = M _ 0 ^ { \\frac { d } { 2 } - 1 } Q _ l ( M _ 0 t + M _ 0 , M _ 0 x ) . \\end{align*}"}
+{"id": "7494.png", "formula": "\\begin{align*} \\dbinom { k + r } { i + 1 } ( i + 1 ) = & \\dfrac { ( k + r ) \\cdots ( k + r - i ) } { ( i + 1 ) ! } ( i + 1 ) \\\\ = & \\dfrac { ( k + r ) \\cdots ( k + r - i + 1 ) } { i ! } ( k + r - i ) \\\\ = & \\dbinom { k + r } { i } ( k + r - i ) \\end{align*}"}
+{"id": "3178.png", "formula": "\\begin{align*} g : \\bar { \\Omega } \\rightarrow \\R , g ( x _ 1 , x _ 2 , x _ 3 ) : = 8 x _ 1 ^ 3 - 3 x _ 1 x _ 3 ^ 2 . \\end{align*}"}
+{"id": "7612.png", "formula": "\\begin{align*} \\tau ^ n : = \\inf _ { t \\geq 0 } \\left \\{ t : \\| u _ 1 ( t ) \\| _ { \\L ^ p } \\geq n \\right \\} \\wedge T \\ \\mbox { a n d } \\ \\sigma ^ n : = \\inf _ { t \\geq 0 } \\left \\{ t : \\| u _ 2 ( t ) \\| _ { \\L ^ p } \\geq n \\right \\} \\wedge T . \\end{align*}"}
+{"id": "7516.png", "formula": "\\begin{align*} Z _ g ( s , \\chi , D ) = \\dfrac { M _ { 2 , 4 } ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } + q ^ { - d - 1 - d j _ 0 s - e _ 2 s } Z _ { g _ 3 } ( s , \\chi , D ) , \\end{align*}"}
+{"id": "1008.png", "formula": "\\begin{align*} \\S ' ( \\Z ) = \\bigcup _ { \\alpha \\geq 0 } \\ell _ { \\infty , - \\alpha } ( \\Z ) . \\end{align*}"}
+{"id": "1043.png", "formula": "\\begin{align*} \\omega ' ( \\eta ) = \\omega ( \\eta / ( 1 + \\eta ) ) = \\sup \\{ | \\mu _ 0 - \\mu _ 1 | : \\mathrm { T V } ( R _ 0 , R _ 1 ) \\leq \\eta , \\ , R _ 0 , R _ 1 \\in \\mathcal { P } _ k \\} , \\end{align*}"}
+{"id": "5984.png", "formula": "\\begin{align*} 0 \\rightarrow & E = \\mathcal { O } _ { \\mathbb { P } ^ 1 } ( - d _ 1 ) \\oplus \\dots \\oplus \\mathcal { O } _ { \\mathbb { P } ^ 1 } ( - d _ r ) \\rightarrow \\\\ & \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : F = \\mathcal { O } _ { \\mathbb { P } ^ 1 } ( - d _ 1 ) \\oplus \\dots \\oplus \\mathcal { O } _ { \\mathbb { P } ^ 1 } ( - d _ r ) \\oplus \\mathcal { O } _ { \\mathbb { P } ^ 1 } ( - d _ { r + 1 } ' ) \\oplus \\dots \\oplus \\mathcal { O } _ { \\mathbb { P } ^ 1 } ( - d _ { r + p } ' ) \\end{align*}"}
+{"id": "3465.png", "formula": "\\begin{align*} I _ { z , t , x } ^ { ( j , n ) } ( r ) \\leq \\left \\{ \\begin{array} { l l } C G _ { t - r } ^ { 1 - 1 / ( 2 q ) } ( x - r ) & \\mbox { i f $ n - j = 1 , 2 , 3 $ } \\\\ C ^ { n - j + 1 } \\frac { 1 } { [ ( n - j ) ! ] ^ { 1 / 2 } } \\frac { 1 } { [ ( n - j + 2 ) ! ] ^ { 1 / 2 } } 1 _ { \\{ | x - z | < t - r \\} } & \\mbox { i f $ n - j \\geq 4 $ } , \\end{array} \\right . \\end{align*}"}
+{"id": "1439.png", "formula": "\\begin{align*} \\mathbb { E } [ \\eta ^ 2 _ t ] = \\mathbb { E } [ \\mathbb { E } ( \\eta ^ 2 _ t | R _ { t - 1 } , N _ t ) ] & \\leq \\mathbb { E } [ \\mathbb { E } ( ^ 2 ( n , p N _ t ) | N _ t ) ] \\\\ & \\leq n p \\mathbb { E } [ N _ t ] + ( n p ) ^ 2 \\mathbb { E } [ N ^ 2 _ t ] . \\end{align*}"}
+{"id": "1612.png", "formula": "\\begin{align*} S E R = \\frac { 1 } { K } \\sum _ { u = 1 } ^ K 1 [ S _ { ( u - 1 ) M + 1 : u M } \\neq \\hat S _ { ( u - 1 ) M + 1 : u M } ] \\end{align*}"}
+{"id": "6435.png", "formula": "\\begin{align*} \\tilde u _ \\epsilon ( Z , W , \\tau ) & : = u _ \\epsilon ( ( \\hat Z , \\hat W , \\hat \\tau ) \\circ ( Z , W , \\tau ) ) + \\eta \\\\ & = u _ \\epsilon ( Z + \\hat Z , W + \\hat W - \\tau \\hat Z , \\hat \\tau + \\tau ) + \\eta , \\end{align*}"}
+{"id": "5588.png", "formula": "\\begin{align*} 0 = - ( \\rho _ { i l } \\rho ^ m { } _ k \\rho _ { m j } + \\rho ^ m { } _ l \\rho _ { i k } \\rho _ { m j } + \\rho ^ m { } _ l \\rho _ { m k } \\rho _ { i j } ) S _ { 1 1 } . \\end{align*}"}
+{"id": "8913.png", "formula": "\\begin{align*} \\mathcal { S } \\simeq \\sum \\limits _ { p = 0 } ^ { n - 1 } \\gamma _ { p } ^ { ( \\nu , n ) } \\left [ p ! t ^ { - 1 - p } + \\sum \\limits _ { s = 0 } ^ { + \\infty } \\frac { ( - 1 ) ^ { s + 1 } B _ { 2 ( p + s + 1 ) } \\left ( \\nu + \\frac { 1 } { 2 } \\right ) } { ( p + s + 1 ) s ! } t ^ { s } \\right ] . \\end{align*}"}
+{"id": "124.png", "formula": "\\begin{align*} & \\int _ L \\int _ L \\textbf { 1 } _ { E ( F ) } \\cdot | x - y | ^ { n - 1 } d \\mathcal H ^ 1 ( x ) d \\mathcal H ^ 1 ( y ) \\\\ = & \\iint _ { E ( \\widetilde { F } _ L , n ) } | s _ x - s _ y | ^ { n - 1 } d \\mathcal H ^ 1 ( s _ x ) d \\mathcal H ^ 1 ( s _ y ) \\\\ \\leq & C \\frac { 5 ^ n } { n } \\int _ { \\mathbb R } \\left | \\widetilde { F } _ L ( s ) \\right | d \\mathcal H ^ 1 ( s ) = C \\frac { 5 ^ n } { n } \\int _ { \\mathbb R } \\left | F _ L ( \\hat { x } + s \\omega ) \\right | d \\mathcal H ^ 1 ( s ) . \\end{align*}"}
+{"id": "3930.png", "formula": "\\begin{align*} a ^ { i j } & = [ D ^ 2 w - A ( \\cdot , w , D w ) ] ^ { i j } , \\\\ b ^ i & = - a ^ { i j } A _ { i j , p _ k } ( \\cdot , w _ \\tau , D w _ \\tau ) + \\tilde { B } _ { p _ k } ( \\cdot , w _ \\tau , D w _ \\tau ) , \\\\ c & = - a ^ { i j } A _ { i j , u } ( \\cdot , w _ \\tau , D w _ \\tau ) + \\tilde { B } _ { u } ( \\cdot , w _ \\tau , D w _ \\tau ) . \\end{align*}"}
+{"id": "3463.png", "formula": "\\begin{align*} f _ j ^ { ( n ) } ( x _ 1 , \\ldots , x _ { n - 1 } , z , x ; t ) & = \\int _ { \\{ 0 < t _ 1 < \\ldots < t _ { j - 1 } < r < t _ j < \\ldots < t _ { n - 1 } < t \\} } g _ { n - j } ( t _ j , x _ j , \\ldots , t _ { n - 1 } , x _ { n - 1 } , r , z , t , x ) \\\\ & f _ { j - 1 } ( t _ 1 , x _ 1 , \\ldots , t _ { j - 1 } , x _ { j - 1 } , r , z ) d t _ 1 \\ldots d t _ { n - 1 } d r \\end{align*}"}
+{"id": "5653.png", "formula": "\\begin{align*} \\chi _ j ^ { ( 1 ) } ( s ) = 4 \\pi D \\sum _ { i = 1 } ^ N \\ell _ i V _ i ( s ) { \\mathcal G } _ { i j } ( s ) . \\end{align*}"}
+{"id": "6456.png", "formula": "\\begin{align*} \\partial _ t m = m \\wedge H ( m ) - \\alpha m \\wedge ( m \\wedge H ( m ) ) , \\end{align*}"}
+{"id": "2585.png", "formula": "\\begin{align*} & \\hat { \\mathfrak { k } } = \\{ u \\in L ( \\mathfrak { g } , \\sigma ) \\mid \\rho ( u ( - t ) ) = u ( t ) \\} , \\\\ & \\hat { \\mathfrak { m } } = \\{ u + \\alpha c + \\beta d \\mid u \\in L ( \\mathfrak { g } , \\sigma ) , \\ \\rho ( u ( - t ) ) = - u ( t ) , \\ \\alpha , \\beta \\in \\mathbb { R } \\} . \\end{align*}"}
+{"id": "6128.png", "formula": "\\begin{align*} \\hbox { r a n k } ( \\mathcal R ) = N - \\hbox { d i m K e r } ( \\mathcal R ^ T ) \\geqslant N - d , \\end{align*}"}
+{"id": "2524.png", "formula": "\\begin{align*} & \\varrho _ N ^ \\varphi ( x ) = \\frac 1 N \\sum _ { j = 1 } ^ N \\varphi ( x - x _ j ) \\ , , \\varrho _ N ^ \\varphi u _ N ^ \\varphi ( x ) = \\frac 1 N \\sum _ { j = 1 } ^ N \\varphi ( x - x _ j ) v _ j \\ , , \\\\ & \\varrho _ N ^ \\varphi T _ N ^ \\varphi ( x ) = \\frac 1 { N d } \\sum _ { j = 1 } ^ N \\varphi ( x - x _ j ) | v _ j - u _ N ^ \\varphi ( x ) | ^ 2 \\ , , \\end{align*}"}
+{"id": "4036.png", "formula": "\\begin{align*} f ^ * ( Y ( \\cdot , v , D v ) ) \\det D Y ( \\cdot , v , D v ) & > f B _ a ( x _ 0 ) \\\\ f ^ * ( Y ( \\cdot , v , D v ) ) \\det D Y ( \\cdot , v , D v ) & = f \\Omega \\setminus \\overline { B _ a ( x _ 0 ) } . \\end{align*}"}
+{"id": "5199.png", "formula": "\\begin{align*} Q _ { i j } = \\langle w , B _ i \\times B _ j \\rangle + \\langle B _ i ' , B _ j \\rangle - \\langle B _ i , B _ j ' \\rangle \\ . \\end{align*}"}
+{"id": "8109.png", "formula": "\\begin{align*} \\tilde H _ { m , n } ^ { + , 1 } ( x ) = 4 T \\int _ { - \\infty } ^ \\infty \\widehat { k ^ * } ( \\xi ) e \\Bigl ( - \\frac { T \\xi } { M } - \\frac { x } { 2 \\pi } \\cosh \\frac { \\xi \\pi } { M } \\Bigr ) \\ , d \\xi . \\end{align*}"}
+{"id": "4690.png", "formula": "\\begin{align*} ( \\mathbf { x } , \\mathbf { y } ) : = x _ 1 y _ 1 + \\cdots + x _ n y _ n n . \\end{align*}"}
+{"id": "216.png", "formula": "\\begin{align*} D _ { N } ( m ) : = f _ { N } ( \\mathbb { K } _ { N , m } ^ { \\mathrm { s p } } ) , \\end{align*}"}
+{"id": "7536.png", "formula": "\\begin{align*} Z _ { \\tilde { g } _ { 2 , a } } ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) = \\dfrac { G _ { 2 , a , m } ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } , \\end{align*}"}
+{"id": "9132.png", "formula": "\\begin{align*} Q _ { h , \\nu } ( \\eta | \\gamma ) \\ ; = \\ ; \\sum _ { f \\in \\mathfrak { F } _ { \\eta , \\gamma } } G ( f ) , \\end{align*}"}
+{"id": "1576.png", "formula": "\\begin{align*} V ^ { k + 1 } : = \\{ v \\in V ^ k \\mid \\alpha ^ k _ v > 0 \\} \\cup \\{ \\hat x ^ k \\} . \\end{align*}"}
+{"id": "13.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d q _ t & = \\tilde { G } \\big ( q _ t , m _ t , \\tilde { m } _ t , n _ { ( t , e ) } \\big ) d t - m _ t d W _ t - \\tilde { m } _ t d \\xi _ t - \\int _ { \\mathcal { E } } n _ { ( t , e ) } \\tilde { N } ( d e , d t ) , \\\\ q _ T & = 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4696.png", "formula": "\\begin{align*} \\mathcal { I } ( G _ { I I I } ) = \\frac { 1 } { ( 1 - t ^ 4 ) ( 1 - t ^ { 1 2 } ) } , \\end{align*}"}
+{"id": "4974.png", "formula": "\\begin{align*} I : = a R [ x , y ] + \\theta ' ( y ) R [ x , y ] , \\ d : = \\gcd ( a , \\theta ' ( y ) ) , \\ b : = a d ^ { - 1 } , \\ \\rho ( y ) : = \\sum _ { p \\nmid i } t _ i y ^ i , \\end{align*}"}
+{"id": "342.png", "formula": "\\begin{align*} B ( p ^ { h } , 1 ) = \\lambda _ { \\phi \\otimes \\phi } ( p ^ { h } ) - A ( 1 , p ) \\lambda _ { \\phi \\otimes \\phi } ( p ^ { h - 1 } ) + A ( p , 1 ) \\lambda _ { \\phi \\otimes \\phi } ( p ^ { h - 2 } ) - \\lambda _ { \\phi \\otimes \\phi } ( p ^ { h - 3 } ) . \\end{align*}"}
+{"id": "7372.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ { p ^ * , p } } \\lesssim \\| \\nabla u \\| _ { L ^ p } , 1 \\leq p < n , \\frac { 1 } { p ^ * } = \\frac { 1 } { p } - \\frac { 1 } { n } \\end{align*}"}
+{"id": "4208.png", "formula": "\\begin{align*} F _ 1 ( x , p ) = 2 \\sinh ( 2 \\lambda + 2 x _ 1 ) \\left ( p _ 4 ^ 2 - p _ 3 ^ 2 \\right ) , F _ 2 ( x , p ) = \\dfrac { \\sinh ( 2 ( \\lambda + x _ 1 ) ) } { \\sinh ^ 2 ( \\lambda + x _ 1 ) } \\left ( p _ 3 p _ 1 - p _ 2 p _ 4 \\right ) . \\end{align*}"}
+{"id": "7492.png", "formula": "\\begin{align*} \\mathbb { H } _ r ^ k ( y , t z - y ) = \\sum _ { i = 0 } ^ k ( - 1 ) ^ i \\dbinom { k + r } { k - i } \\dbinom { i + r - 1 } { i } t ^ { k - i } y ^ i z ^ { k - i } . \\end{align*}"}
+{"id": "6324.png", "formula": "\\begin{align*} \\int _ { \\Sigma } \\mu ^ { \\rm e f f } _ { p q r } n _ r \\partial _ { x _ q } V _ p \\ , d s = \\int _ { \\Sigma } \\mu ^ { \\rm e f f } _ { p q r } n _ q n _ r n _ l \\partial _ { x _ l } V _ p \\ , d s + \\int _ { \\Sigma } \\mu ^ { \\rm e f f } _ { p q r } n _ r \\Pi _ { l q } \\partial _ { x _ l } V _ p \\ , d s . \\end{align*}"}
+{"id": "6331.png", "formula": "\\begin{align*} m ( 0 ) = 0 \\mbox { a n d } m ' ( 0 ) = 1 , \\end{align*}"}
+{"id": "8116.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } A ( n , m ) e \\Bigl ( \\frac { n d } { c } \\Bigr ) \\psi ( n ) = c \\sum _ { \\pm } \\sum _ { n _ 1 \\mid c m } \\sum _ { n _ 2 \\geq 1 } \\frac { A ( n _ 1 , n _ 2 ) } { n _ 1 n _ 2 } S ( m \\bar d , \\pm n _ 2 ; m c n _ 1 ^ { - 1 } ) \\Psi ^ { \\pm } \\Bigl ( \\frac { n _ 2 n _ 1 ^ 2 } { c ^ 3 m } \\Bigr ) . \\end{align*}"}
+{"id": "5040.png", "formula": "\\begin{align*} P _ { i , n } \\cdot ( \\partial / \\partial x _ i ) ^ n F + P _ { i , n - 1 } \\cdot ( \\partial / \\partial x _ i ) ^ { n - 1 } F + \\cdots + P _ { i , 1 } \\cdot ( \\partial / \\partial x _ i ) F + P _ { i , 0 } \\cdot F = 0 . \\end{align*}"}
+{"id": "4196.png", "formula": "\\begin{align*} \\begin{cases*} \\square \\tilde { \\Lambda } = - 2 \\sinh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) Q _ 0 ( \\phi , \\phi ) , \\\\ \\square \\phi = \\dfrac { \\sinh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) } { \\sinh ^ 2 ( \\lambda + \\tilde { \\Lambda } ) } Q _ 0 ( \\phi , \\tilde { \\Lambda } ) , \\end{cases*} \\end{align*}"}
+{"id": "283.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi _ { \\prescript { L } { } { j } } ( v ) v _ { \\widehat { G } } ( \\widehat { j } ( t ) ) \\varphi _ { \\prescript { L } { } { j } } ( v ) ^ { - 1 } & = \\widehat { j } ( v _ { \\widehat { S } } ( t ) ) \\\\ \\varphi _ { \\prescript { L } { } { j } } ( v w ) & = \\varphi _ { \\prescript { L } { } { j } } ( v ) v _ { \\widehat { G } } ( \\varphi _ { \\prescript { L } { } { j } } ( w ) ) , \\end{aligned} \\end{align*}"}
+{"id": "616.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } p _ t = \\frac { \\sigma _ t } { \\gamma } ( A ^ { \\rm T } A ) : \\tilde C \\ , ( \\sigma _ t - 2 p _ t ) . \\end{align*}"}
+{"id": "6763.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\exp \\left [ - \\frac { 1 } { 4 } e ^ { t } \\left ( \\alpha _ n ^ 2 ( t ) - \\mathfrak { R e } ( \\alpha _ n ( t ) ) ^ 2 \\right ) \\right ] = 1 \\end{align*}"}
+{"id": "4933.png", "formula": "\\begin{align*} \\delta _ n ^ + \\mathbf { z } _ n = \\widetilde { \\mathcal { J } } \\int _ 0 ^ 1 \\nabla \\widetilde { \\mathcal { H } } ( \\xi \\mathbf { z } _ { n + 1 } + ( 1 - \\xi ) \\mathbf { z } _ n ) \\ , \\mathrm { d } \\xi , \\end{align*}"}
+{"id": "5228.png", "formula": "\\begin{align*} l _ { 1 } = & k _ 2 k _ 4 t ^ 3 + k _ 1 & l _ { 2 } = & k _ 4 t ^ 3 & b _ { 1 2 } = & k _ 5 / t & b _ { 1 3 } = & k _ 6 t ^ 2 \\\\ b _ { 1 5 } = & 0 & b _ { 2 3 } = & k _ 7 t ^ 2 & b _ { 2 5 } = & k _ 3 / t \\end{align*}"}
+{"id": "3820.png", "formula": "\\begin{align*} \\mathcal { H P } _ { \\mathcal { P } _ i } ( q ) = \\frac { \\mathcal { H P } _ { \\textbf { K } [ x _ j , j \\geq 3 - i ] / < x _ j ^ 2 , x _ j x _ { j + 1 } , j \\geq 3 - i > } ( q ) } { \\prod _ { j \\geq 3 - i } ( 1 - q ^ j ) } . \\end{align*}"}
+{"id": "5410.png", "formula": "\\begin{align*} \\delta ( q _ { i , j , k } , s ) = q _ { ( j + 1 ) \\bmod p , ( j + 1 ) \\bmod p , 0 } . \\end{align*}"}
+{"id": "1997.png", "formula": "\\begin{align*} I ( ( X , Y ) , \\pmb k ) : = \\int _ { \\Delta ( X , Y ) } \\prod _ { \\alpha \\in X \\cup Y } \\omega _ { \\delta ( \\alpha ) } ( t _ \\alpha ) \\sum _ { m _ { 1 j } \\in S S Y T ( h _ { r - 1 } ) } \\frac { 1 - t _ { | X | } ^ { m _ { 1 2 } } } { m _ { 1 2 } ^ { k _ { 2 } } \\cdots m _ { 1 r } ^ { k _ r } } , \\end{align*}"}
+{"id": "5496.png", "formula": "\\begin{align*} \\inf _ { \\| x \\| = 1 } \\| \\mathcal T ( s , x , \\lambda ) \\| > \\epsilon , \\forall s \\in [ 0 , 1 ] , \\ \\forall \\lambda \\in [ \\lambda _ 0 , \\lambda _ 0 + \\epsilon ) \\cup ( \\lambda _ 0 + 1 - \\epsilon , \\lambda _ 0 + 1 ] . \\end{align*}"}
+{"id": "505.png", "formula": "\\begin{align*} \\varphi ( \\Phi ( x ^ { \\ell ( k ) } ) \\ ! - \\ ! \\Phi ^ * ) = c ( \\Phi ( x ^ { \\ell ( k ) } ) \\ ! - \\ ! \\Phi ^ * ) ^ { 1 - \\theta } \\le c \\big [ b c ( 1 \\ ! - \\ ! \\theta ) \\| x ^ { \\ell ( k ) } - x ^ { \\ell ( k ) - 1 } \\| \\big ] ^ { \\frac { 1 - \\theta } { \\theta } } . \\end{align*}"}
+{"id": "6313.png", "formula": "\\begin{align*} \\mathbb { E } [ Y ^ i _ t ] & = \\sum _ { m = 2 } ^ { k + 1 } m c ( 1 - p _ n ) p _ n ^ { m - 2 } + ( k + 2 ) c p _ n ^ { k } \\\\ & < \\frac { c } { p _ n } \\sum _ { m = 2 } ^ { k + 2 } m p _ n ^ { m - 1 } \\\\ & < \\frac { c ( 2 - p _ n ) } { ( 1 - p _ n ) ^ 2 } \\\\ & = \\frac { c } { ( 1 - p _ n ) ^ 2 } + \\frac { c } { ( 1 - p _ n ) } \\end{align*}"}
+{"id": "7755.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - x t } \\ , \\delta ( t - a ) d t & = e ^ { - x a } \\end{align*}"}
+{"id": "8459.png", "formula": "\\begin{align*} \\lim _ { t \\in T } \\ , c _ * ( A _ t ) = c _ * ( A ) . \\end{align*}"}
+{"id": "1480.png", "formula": "\\begin{align*} \\lambda & = \\frac { 2 k ( k - 1 ) ( k - 2 ) ( m - 1 ) ! ( m - 2 ) ! } { ( m + 1 ) ( m ^ 2 - 2 ) | G _ \\Delta | } \\\\ & = \\frac { 7 2 \\cdot 3 5 \\cdot 3 4 \\cdot 1 0 ! \\cdot 9 ! } { 1 2 \\cdot 1 1 9 \\cdot ( 2 4 ) ^ 2 } \\\\ & = \\frac { ( 1 0 ! ) ^ 2 } { 9 6 } = 1 3 7 , 1 6 8 , 6 4 0 , 0 0 0 . \\end{align*}"}
+{"id": "2445.png", "formula": "\\begin{align*} \\mu _ { \\varphi } ( \\Psi ) - \\frac { 3 } { \\pi } \\mu ( \\Psi ) = \\sum _ { k = 1 } ^ { \\infty } \\langle \\Psi , u _ { k } \\rangle \\langle u _ k , | \\varphi | ^ 2 \\rangle + \\frac { 1 } { 4 \\pi } \\int _ { \\mathbb { R } } \\langle \\Psi , E ( \\cdot , \\tfrac { 1 } { 2 } + i t ) \\rangle \\langle E ( \\cdot , \\tfrac { 1 } { 2 } + i t ) , | \\varphi | ^ 2 \\rangle d t . \\end{align*}"}
+{"id": "8041.png", "formula": "\\begin{align*} E ( z , s ) = \\frac { 1 } { 2 } \\sum _ { \\substack { c , d \\in \\Z \\\\ ( c , d ) = 1 } } \\frac { y ^ s } { \\abs { c z + d } ^ { 2 s } } \\end{align*}"}
+{"id": "8867.png", "formula": "\\begin{align*} K _ { \\nu , m } ( z , w ) = \\beta _ { \\nu , n , m } R _ { m , m + 2 \\nu } ^ { n - 1 } \\left ( \\frac { 1 + \\langle z , w \\rangle } { ( 1 + | z | ^ { 2 } ) ^ { \\frac { 1 } { 2 } } ( 1 + | w | ^ { 2 } ) ^ { \\frac { 1 } { 2 } } } \\right ) . \\end{align*}"}
+{"id": "1386.png", "formula": "\\begin{align*} \\Pi _ \\psi = \\left \\{ \\tau _ { \\psi _ 1 } \\rtimes \\pi _ 0 \\ ; \\middle | \\ ; \\pi _ 0 \\in \\Pi _ { \\psi _ 0 } \\right \\} . \\end{align*}"}
+{"id": "3541.png", "formula": "\\begin{align*} \\mathbf { p } B _ j ^ + = \\sum _ { i = 1 } ^ j \\sum _ { \\substack { s _ 1 , \\dots , s _ m \\in \\N _ 0 , \\\\ - s _ 1 \\alpha _ 1 - \\cdots - s _ m \\alpha _ m + \\epsilon _ i = \\epsilon _ j } } E _ { - \\alpha _ 1 } ^ { s _ 1 } \\cdots E _ { - \\alpha _ m } ^ { s _ m } B _ i ^ + \\hat { H } _ { s _ 1 , \\dots , s _ m } ( i ) , \\end{align*}"}
+{"id": "6671.png", "formula": "\\begin{align*} \\bar x ^ { k + 1 } = \\bar x ^ k + \\gamma ^ k \\bar \\zeta _ w ^ k - \\frac { \\lambda ^ k } { m } \\sum _ { i = 1 } ^ m g _ i ^ k \\end{align*}"}
+{"id": "5689.png", "formula": "\\begin{align*} \\mathbf { F } _ { b } ^ { a } \\mathbf { \\mathbf { e } } ^ { b } = 0 ; \\mathbf { D F } _ { b } ^ { a } = 0 . \\end{align*}"}
+{"id": "1317.png", "formula": "\\begin{align*} S _ { f , \\tau } ^ 2 x = \\sum _ { j = 1 } ^ n f _ j ( x ) ( S _ { f , \\tau } \\tau _ j ) , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "6199.png", "formula": "\\begin{align*} \\begin{cases} \\Psi '' - \\Delta \\Psi + \\overline A _ p ^ T \\Psi = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\Psi = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu \\Psi + \\overline B _ p ^ T \\Psi = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{cases} \\end{align*}"}
+{"id": "5511.png", "formula": "\\begin{align*} A ( x , \\beta ) = \\sum _ { n \\le x } { a _ n \\over n ^ \\beta } \\end{align*}"}
+{"id": "6711.png", "formula": "\\begin{align*} \\hat { f } ( s ) = \\frac { 1 } { p } \\sum _ { 0 \\leq t \\leq p - 1 } f ( t ) e ^ { i \\pi s t / p } \\end{align*}"}
+{"id": "6831.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) = & \\mu _ c ^ { \\pm } ( Y _ 1 C _ 1 , Y _ 2 C _ 2 , \\dots , Y _ m C _ m ) , \\nu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) = \\nu _ c ^ { \\pm } ( Y _ 1 C _ 1 , Y _ 2 C _ 2 , \\dots , Y _ m C _ m ) . \\end{aligned} \\end{align*}"}
+{"id": "8339.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\int _ { | x | > A + t } \\left ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 \\right ) \\mathrm { d } x = 0 , \\end{align*}"}
+{"id": "1682.png", "formula": "\\begin{align*} s ( \\mu ) = \\sum _ { \\frac { 2 - k } { 2 } \\leq n \\leq \\frac { k - 2 } { 2 } } \\mu \\left ( P _ { n + \\frac { k - 2 } { 2 } } \\right ) f _ n . \\end{align*}"}
+{"id": "4250.png", "formula": "\\begin{align*} \\omega ^ 0 _ \\gamma : = \\inf \\left \\{ \\frac { 1 } { 2 } \\| ( \\nabla - i A ) f \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } V _ \\gamma ( x ) | f ( x ) | ^ 2 d x \\ : \\ f \\in \\Sigma _ \\gamma , M ( f ) = 1 \\right \\} . \\end{align*}"}
+{"id": "4622.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { k + 1 } , \\frac { 1 } { k } \\right ] = \\left ( \\frac { 1 } { k + 1 } , \\frac { 1 } { k + 1 } + \\frac { 1 } { k ( k + 1 ) } \\right ] \\end{align*}"}
+{"id": "5490.png", "formula": "\\begin{align*} \\dot x ( t ) = J \\nabla H _ 0 ( t , x ( t ) ) \\end{align*}"}
+{"id": "4004.png", "formula": "\\begin{align*} \\det [ D ^ 2 u - A ( \\cdot , u , D u ) ] = B ( \\cdot , u , D u ) \\Omega , \\\\ u = g ( \\cdot , y , z ) \\partial \\Omega , \\end{align*}"}
+{"id": "3801.png", "formula": "\\begin{align*} \\langle X , \\nabla ( R ^ { - 1 } - f ) \\rangle & = - R ^ { - 2 } \\ , \\langle X , \\nabla R \\rangle - | \\nabla f | ^ 2 \\\\ & = R ^ { - 2 } \\ , \\Delta R + 2 R ^ { - 2 } \\ , | | ^ 2 - | \\nabla f | ^ 2 \\end{align*}"}
+{"id": "199.png", "formula": "\\begin{align*} \\frac { p ( q ) } { q } \\geq \\frac { ( h _ { n } + c _ { n } - r _ { n } ) r _ { n } } { m _ { n } } = \\frac { 1 + \\frac { c _ { n } - r _ { n } } { h _ { n } } } { 1 + 2 \\frac { c _ { n } + r _ { n } - 1 } { h _ { n } } } \\cdot r _ { n } > \\frac { 1 - \\epsilon } { 1 + 2 \\epsilon } \\cdot \\frac { 1 } { \\epsilon } \\geq \\frac { 1 } { 2 \\epsilon } . \\end{align*}"}
+{"id": "2328.png", "formula": "\\begin{align*} \\ ( \\frac { p - 1 } { p } \\ ) ^ p \\frac { | \\nabla \\rho | ^ p } { \\rho ^ p } & = \\ ( \\frac { p - 1 } { p } \\ ) ^ p \\frac { | \\nabla U _ p | ^ p } { U _ p ^ p } \\\\ & = \\ ( \\frac { p - 1 } { p } \\ ) ^ p \\ ( \\frac { N - p } { p - 1 } \\ ) ^ p \\frac { | \\nabla h ( x , y ) | ^ p } { h ( x , y ) ^ p } \\\\ & = \\ ( \\frac { N - p } { p } \\ ) ^ p \\frac { V _ p ( x , y ) ^ { \\frac { p } { 2 } } } { \\left [ | x | ^ 2 + ( 1 - y ) ^ 2 \\right ] ^ { \\frac { p } { 2 } } } , \\end{align*}"}
+{"id": "4408.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n a _ i \\leq \\prod _ { i = 1 } ^ n x _ i . \\end{align*}"}
+{"id": "6719.png", "formula": "\\begin{align*} \\mu ( 1 + a ) = \\mu ( 2 + a ) = \\mu ( 3 + a ) = \\mu ( 4 + a ) = \\cdots = \\mu ( q + a ) = 1 , \\end{align*}"}
+{"id": "1117.png", "formula": "\\begin{align*} A _ { j + 1 } & = 2 ^ { - 1 } \\{ \\Pi _ r ( \\theta _ j - \\eta _ j Z _ j ) - \\theta \\} ^ 2 = 2 ^ { - 1 } \\{ \\Pi _ r ( \\theta _ j - \\eta _ j Z _ j ) - \\Pi _ r ( \\theta ) \\} ^ 2 \\\\ & \\leq 2 ^ { - 1 } ( \\theta _ j - \\eta _ j Z _ j - \\theta ) ^ 2 = A _ j + 2 ^ { - 1 } \\eta _ j ^ 2 Z _ j ^ 2 - \\eta _ j ( \\theta _ j - \\theta ) Z _ j . \\end{align*}"}
+{"id": "1912.png", "formula": "\\begin{align*} \\Pr \\left ( \\sum _ { i = 1 } ^ n \\mathbf { 1 } _ { U _ i \\leq t } \\leq ( n + w - u ) t + u \\ ; \\forall \\ ; t \\in [ 0 , 1 ] \\right ) = 1 - e ^ { - \\frac { 2 u w } { n } } + O \\left ( \\frac { u + w } { n } \\right ) . \\end{align*}"}
+{"id": "1994.png", "formula": "\\begin{align*} I ( X , \\pmb k ) : = \\int _ { \\Delta ( X ) } \\prod _ { x \\in X } \\omega _ { \\delta _ X ( x ) } ( t _ x ) \\sum _ { m _ { 1 j } \\in S S Y T ( h _ { r - 1 } ) } \\frac { 1 - t _ { x } ^ { m _ { 1 2 } } } { m _ { 1 2 } ^ { k _ { 2 } } \\cdots m _ { 1 r } ^ { k _ r } } , \\end{align*}"}
+{"id": "372.png", "formula": "\\begin{align*} \\hat u _ k : = u _ { 2 k + n _ 1 } + u _ { 2 k + n _ 1 + 1 } \\textrm { a n d } \\omega _ k : = \\frac { u _ { 2 k + n _ 1 } - u _ { 2 k + n _ 1 + 1 } } { \\hat u _ k } \\ , \\end{align*}"}
+{"id": "5284.png", "formula": "\\begin{align*} \\small \\sum \\limits _ { j = 1 , j \\neq i } ^ { m - n } s e c ( U _ i , U _ j ) & = \\sum \\limits _ { j = 1 , j \\neq i } ^ { m - n } \\frac { g ( R ( U _ i , U _ j ) U _ j , U _ i ) } { g ( U _ i , U _ i ) g ( U _ j , U _ j ) - g ( U _ i , U _ j ) ^ 2 } \\\\ & = \\frac { 1 } { ( m - n - 1 ) } R i c ( U _ i , U _ i ) . \\end{align*}"}
+{"id": "5271.png", "formula": "\\begin{align*} T _ U V = - g ( U , V ) \\nabla f ; ~ \\nu ( \\nabla \\lambda ) = 0 , \\end{align*}"}
+{"id": "1716.png", "formula": "\\begin{align*} I ( \\chi , m ) = \\left ( 1 + \\chi ( - 1 ) \\epsilon \\right ) \\int _ 0 ^ \\infty t ^ m \\langle \\imath ( t ) \\delta s ( \\mu _ m ) , J _ T \\delta s ( \\mu _ m ) \\rangle d ^ \\times t . \\end{align*}"}
+{"id": "6312.png", "formula": "\\begin{align*} Y ^ i _ t = 2 c Z ^ i _ { k - 1 } + ( 1 - Z ^ i _ { k - 1 } ) ( Y ^ i _ { k c - 1 } + c ) \\end{align*}"}
+{"id": "2968.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { Z } } \\tau _ j \\mathcal { Q } ^ n _ \\rho ( \\varepsilon n ; t ) \\nabla ^ n \\varphi ^ n _ j ( t ) = \\frac { g ^ { \\prime \\prime } ( 0 ) } { 2 n } \\sum _ { j \\in \\mathbb { Z } } \\big ( \\tilde { \\mathcal { X } } ^ n _ t ( { \\textstyle \\iota _ { \\varepsilon } ( \\frac { j - f _ n t } { n } ; \\cdot ) } ) \\big ) ^ 2 \\nabla ^ n \\varphi ^ n _ j ( t ) . \\end{align*}"}
+{"id": "362.png", "formula": "\\begin{align*} T _ * : = \\lambda T , { L } _ * = | v _ 0 | ^ { - 1 } \\lambda L \\ , , \\end{align*}"}
+{"id": "752.png", "formula": "\\begin{align*} \\# R ^ \\times = \\begin{cases} 2 & \\textrm { i f } \\ell > 3 , \\\\ \\textrm { e i t h e r $ 2 $ o r $ 6 $ } & \\textrm { i f } \\ell = 3 . \\end{cases} \\end{align*}"}
+{"id": "8287.png", "formula": "\\begin{align*} V ( \\textbf { x } _ 0 ) : = \\min _ { \\textbf { u } _ E ( \\cdot ) } \\ \\max _ { \\textbf { u } _ i ( \\cdot ) } J ( \\textbf { u } _ E ( \\cdot ) , \\textbf { u } _ i ( \\cdot ) ; \\textbf { x } _ 0 ) \\end{align*}"}
+{"id": "3206.png", "formula": "\\begin{align*} \\mathbb { P } ^ { X _ n } ( A _ \\epsilon ) \\ge 1 - \\epsilon , \\qquad \\lim _ { h \\to 0 } \\sup _ { u \\in A _ \\epsilon } \\sup _ { | t - s | \\le h } \\| u ( t ) - u ( s ) \\| _ E = 0 . \\end{align*}"}
+{"id": "9116.png", "formula": "\\begin{align*} \\rho _ 1 = \\sqrt { \\frac { h _ 1 } { h _ 2 } } \\rho _ 2 = \\sqrt { \\frac { h _ 2 } { h _ 1 } } . \\end{align*}"}
+{"id": "7186.png", "formula": "\\begin{align*} A _ { \\beta \\alpha } ( \\mathcal { S } ) y _ { \\alpha } = C _ \\beta ( \\mathcal { S } ) . \\end{align*}"}
+{"id": "2056.png", "formula": "\\begin{align*} \\arg Q _ { \\alpha , \\omega } ( x ) = \\begin{cases} \\arg \\omega + \\frac \\pi 2 ( 1 - \\alpha ) & \\ x > 0 , \\\\ [ 0 . 5 e x ] \\arg \\omega + \\frac \\pi 2 ( 1 + \\alpha ) & \\ x < 0 , \\end{cases} \\end{align*}"}
+{"id": "4504.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k v _ i \\leq \\prod _ { i = 1 } ^ k u _ i \\end{align*}"}
+{"id": "7677.png", "formula": "\\begin{align*} [ \\bigoplus _ { I } \\eta _ { I } - \\bigoplus _ { I } \\eta _ { I , - \\iota _ { E } ( \\omega ) } ] = \\nabla _ { \\omega } ( [ \\phi ( ( \\iota _ { E } ( \\bigoplus _ { I } \\eta _ { I } ) ) ) ] ) \\end{align*}"}
+{"id": "2839.png", "formula": "\\begin{align*} \\lambda = \\frac { \\tau + \\sigma } { 2 \\tau } . \\end{align*}"}
+{"id": "7888.png", "formula": "\\begin{align*} f '' + A ( z ) f = 0 \\end{align*}"}
+{"id": "3075.png", "formula": "\\begin{align*} \\int _ Y ( \\partial _ { s t } ^ 2 w ) ( \\partial _ i w ) = 0 \\forall 1 \\leq i , s , t \\leq n \\end{align*}"}
+{"id": "8926.png", "formula": "\\begin{align*} S _ 1 = \\sum \\limits _ { p = 0 } ^ { n - 1 } \\tau _ { p } ^ { ( \\nu , n ) } \\left ( - 1 \\right ) ^ { p } \\left ( \\frac { d } { d t } \\right ) ^ { p } \\left [ \\vartheta _ { 3 } ( t ) \\right ] . \\end{align*}"}
+{"id": "4594.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\frac { 1 } { b _ i } < \\sum _ { i = 1 } ^ { n } \\frac { 1 } { a _ i } . \\end{align*}"}
+{"id": "3613.png", "formula": "\\begin{align*} 0 \\ = \\ R _ * ^ \\frac { 7 } { 2 } \\left ( - \\frac { 1 0 5 } { 2 } \\sqrt { \\alpha _ 0 } + 5 \\left ( \\alpha _ 0 + \\tfrac { 1 4 7 } { 4 } \\right ) \\Big ( \\tfrac { \\pi } { 2 } - \\arctan ( \\tfrac { 7 } { 2 \\sqrt { \\alpha _ 0 } } ) \\Big ) \\right ) - \\frac { 1 2 } { 7 } R _ * ^ 2 L _ * \\alpha _ 0 \\beta _ 0 ^ { 3 / 2 } + \\frac { 6 } { 7 \\pi } \\beta _ 0 ^ { 7 / 2 } . \\end{align*}"}
+{"id": "2572.png", "formula": "\\begin{align*} T _ e N & = \\sum _ { \\lambda \\in \\Lambda _ 0 } ( \\mathfrak { g } _ 0 \\cap S _ \\lambda ) + \\sum _ { \\alpha \\in \\Delta ^ + } \\sum _ { \\lambda \\in \\Lambda _ \\alpha } ( \\mathfrak { g } _ \\alpha \\cap S _ \\lambda ) , \\\\ T ^ \\perp _ e N & = \\mathfrak { g } _ 0 \\cap T ^ \\perp _ e N + \\sum _ { \\alpha \\in \\Delta ^ + } ( \\mathfrak { g } _ \\alpha \\cap T ^ \\perp _ e N ) , \\end{align*}"}
+{"id": "2622.png", "formula": "\\begin{align*} 1 8 & = 2 ( a _ { 2 , 3 } + a _ { 2 , 4 } + a _ { 3 , 4 } ) - ( ( a _ { 1 , 2 , 3 } + a _ { 1 , 2 , 4 } + a _ { 2 , 3 , 4 } + a _ { 1 , 2 , 3 } + a _ { 1 , 3 , 4 } + a _ { 2 , 3 , 4 } + a _ { 1 , 2 , 4 } + a _ { 1 , 3 , 4 } + a _ { 2 , 3 , 4 } ) - ( a _ { 1 , 2 , 3 } + a _ { 1 , 2 , 4 } + a _ { 1 , 3 , 4 } ) ) \\\\ & = 2 ( a _ { 2 , 3 } + a _ { 2 , 4 } + a _ { 3 , 4 } ) - ( a _ { 1 , 2 , 3 } + a _ { 1 , 2 , 4 } + a _ { 1 , 3 , 4 } ) - 3 a _ { 2 , 3 , 4 } . \\end{align*}"}
+{"id": "4233.png", "formula": "\\begin{align*} \\lim \\limits _ { \\varepsilon \\longrightarrow 0 } \\phi _ 0 & = \\lim \\limits _ { \\varepsilon \\longrightarrow 0 } \\frac { \\theta ( x ) ( ( 1 + \\beta v \\sqrt { c } ) \\sinh ( \\gamma ) \\sinh ( \\beta \\gamma ) + ( \\beta + v \\sqrt { c } ) \\cosh ( \\gamma ) \\cosh ( \\beta \\gamma ) } { 2 ( 1 + ( \\cosh ( \\gamma ) \\sinh ( \\beta \\gamma ) + \\beta \\sqrt { c } \\sinh ( \\gamma ) \\cosh ( \\beta \\gamma ) ) ^ 2 ) } \\\\ & = C _ 1 \\theta ( x ) . \\end{align*}"}
+{"id": "5371.png", "formula": "\\begin{align*} { \\rm S o l S e t } ( p ) = \\{ ( x _ 1 , x _ 1 ^ 2 - p _ 1 ^ 2 + p _ 2 , p _ 3 ) \\} \\cup \\{ ( x _ 1 , x _ 1 ^ 2 - p _ 1 ^ 2 + p _ 2 , - p _ 3 ) \\} . \\end{align*}"}
+{"id": "561.png", "formula": "\\begin{align*} K _ n = \\frac { \\sigma _ n } { \\gamma } \\ , ( A X _ { t _ n } ^ \\dagger ) ^ { \\rm T } + \\mathcal { O } ( \\Delta t ) \\end{align*}"}
+{"id": "1516.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { ( - 1 ) ^ { n - 1 } } { n ^ s } = ( 1 - 2 ^ { 1 - s } ) \\zeta ( s ) \\end{align*}"}
+{"id": "1878.png", "formula": "\\begin{align*} \\alpha _ { z _ i } ( \\sigma ) = \\left \\{ \\begin{array} { l l } \\frac { \\pi } { 2 } & \\mbox { i f $ z _ i - \\psi _ k + p _ { z _ i } \\sigma + q _ { z _ i } \\ge 0 $ } \\\\ \\pi - \\sin ^ { - 1 } \\left ( \\frac { - ( z _ i - \\psi _ k + p _ { z _ i } \\sigma + q _ { z _ i } ) } { \\sqrt { \\dot { z } _ i ^ 2 + ( p _ { z _ i } \\sigma + q _ { z _ i } ) ^ 2 } } \\right ) - \\beta & \\mbox { i f $ z _ i - \\psi _ k + p _ { z _ i } \\sigma + q _ { z _ i } \\le 0 $ } \\end{array} \\right . \\end{align*}"}
+{"id": "2768.png", "formula": "\\begin{align*} \\partial _ { ( 1 ) } ^ { A } \\otimes \\partial _ { ( 2 ) } ^ { A } & = \\partial ^ { A } \\otimes 1 + ( \\mathcal { L } _ { \\partial } ) { ^ { A } } _ { B } \\otimes \\partial ^ { B } , \\\\ \\hat { \\partial } _ { ( 1 ) } ^ { A } \\otimes \\hat { \\partial } _ { ( 2 ) } ^ { A } & = \\hat { \\partial } ^ { A } \\otimes 1 + ( \\mathcal { \\bar { L } } _ { \\partial } ) { ^ { A } } _ { B } \\otimes \\hat { \\partial } ^ { B } . \\end{align*}"}
+{"id": "3103.png", "formula": "\\begin{align*} c _ j ^ { k l } ( \\tilde { A } ) = \\bar { \\gamma } \\left ( c _ j ^ { k l } ( A ) + \\bar { a } _ { k l } \\int _ Y r A e _ j \\cdot \\nabla w \\right ) , \\end{align*}"}
+{"id": "8025.png", "formula": "\\begin{align*} U _ \\alpha = \\frac { 1 } { 2 } \\sum _ { i \\ne j } [ \\mu _ i \\cdot \\mu _ j - 3 ( \\mu _ i \\cdot \\mathbf { \\hat r } _ { i j } ) ( \\mu _ j \\cdot \\mathbf { \\hat r } _ { i j } ) ] / r ^ 3 _ { i j } - \\sum \\mathbf E \\cdot \\mu _ i , \\end{align*}"}
+{"id": "1945.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\int \\| z \\| \\ , d \\mu _ N ( z ) & = \\lim _ { N \\to \\infty } \\int _ { \\| z \\| \\leq r } \\| z \\| \\ , d \\mu _ N ( z ) + \\lim _ { N \\to \\infty } \\int _ { \\| z \\| > r } \\| z \\| \\ , d \\mu _ N ( z ) \\\\ & = \\int _ { \\| z \\| \\leq r } \\| z \\| \\ , d P ( z ) + \\lim _ { N \\to \\infty } \\int _ { \\| z \\| > r } \\| z \\| \\ , d \\mu _ N ( z ) . \\end{align*}"}
+{"id": "5158.png", "formula": "\\begin{align*} \\mathcal { P } ( K _ { \\rho , \\rho } ) = \\dfrac { \\binom { 2 \\rho } { \\rho } } { 2 ^ { 2 \\rho } } . \\end{align*}"}
+{"id": "3737.png", "formula": "\\begin{align*} \\mathcal { S } : = \\mathcal { S } ( X _ { P _ \\ell } ^ \\circ ( F ) ) + \\mathcal { F } \\left ( \\mathcal { S } ( X _ { P _ \\ell } ^ \\circ ( F ) ) \\right ) < L ^ 2 ( X _ { P _ { \\ell } } ( F ) ) . \\end{align*}"}
+{"id": "4975.png", "formula": "\\begin{align*} q _ 1 : = d ^ { - 1 } ( f - \\theta ^ * ( q ) ) = d ^ { - 1 } \\xi ^ * ( q ) \\in b x + \\rho ( y ) + d ^ { p - 2 } ( b F ) ^ { p - 1 } y R [ f , a F , y ] . \\end{align*}"}
+{"id": "5276.png", "formula": "\\begin{align*} \\tilde { s e c } ( U , V ) & = s e c ( U , V ) + \\frac { \\lambda ^ 2 } { 2 } \\left ( - g ( T _ U U , \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } ) - g ( T _ V V , \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } ) \\right ) - \\frac { \\lambda ^ 4 } { 4 } \\mid \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\mid ^ 2 \\end{align*}"}
+{"id": "6557.png", "formula": "\\begin{align*} R _ k = \\int _ 0 ^ \\infty { \\log _ 2 \\left ( { 1 + \\gamma } \\right ) { f _ { \\gamma _ k } } \\left ( \\gamma \\right ) { \\rm d } \\gamma } . \\end{align*}"}
+{"id": "9098.png", "formula": "\\begin{align*} \\mathfrak { d } _ 2 & = [ \\mathfrak { d } _ 1 , \\mathfrak { n } ] + J [ \\mathfrak { d } _ 1 , \\mathfrak { n } ] \\\\ & = \\mathrm { s p a n } \\left \\{ [ T , X ] + J [ T ' , X ' ] : \\forall T , T ' \\in \\mathfrak { d } _ 1 , \\forall X , X ' \\in \\mathfrak { n } \\right \\} . \\end{align*}"}
+{"id": "8870.png", "formula": "\\begin{align*} \\partial _ t \\varphi ( t , z ) = \\Delta _ { \\nu } \\varphi ( t , z ) , t > 0 , \\ z \\in \\mathbb { C } ^ n , \\end{align*}"}
+{"id": "7369.png", "formula": "\\begin{align*} \\tilde { \\psi } ^ e ( x ) \\coloneqq \\tilde { \\psi } ^ { e _ 1 } ( x _ 1 ) \\dots \\tilde { \\psi } ^ { e _ n } ( x _ n ) , x = ( x _ 1 , \\dots , x _ n ) ; \\end{align*}"}
+{"id": "1594.png", "formula": "\\begin{align*} \\Phi ( \\delta _ x ) : = \\delta _ { \\psi ( x ) } \\qquad ( x \\in X ) \\end{align*}"}
+{"id": "7366.png", "formula": "\\begin{align*} L ^ { p , r } ( \\nu ) = [ L ^ { p _ 0 } ( \\nu ) , L ^ { p _ 1 } ( \\nu ) ] _ { \\theta , r } \\end{align*}"}
+{"id": "6565.png", "formula": "\\begin{align*} A = A _ C ( q , r ) : = \\exp \\left ( \\frac { \\log ( D + 1 0 0 ) \\log \\log \\log ( D + 1 0 0 ) } { 8 C \\log \\log ( D + 1 0 0 ) } + 1 \\right ) . \\end{align*}"}
+{"id": "4958.png", "formula": "\\begin{align*} D = 1 + \\frac { w z D ^ 2 } { 1 - z D } = 1 + w z D ^ 2 + w z ^ 2 D ^ 3 + w z ^ 3 D ^ 4 + \\cdots . \\end{align*}"}
+{"id": "3806.png", "formula": "\\begin{align*} F = \\frac { 1 } { 2 } \\begin{pmatrix} ( \\partial _ { \\xi _ j } \\partial _ { x _ k } q ( x , \\xi ) ) _ { j , k = 1 } ^ d & ( \\partial _ { \\xi _ j } \\partial _ { \\xi _ k } q ( x , \\xi ) ) _ { j , k = 1 } ^ d \\\\ - ( \\partial _ { x _ j } \\partial _ { x _ k } q ( x , \\xi ) ) _ { j , k = 1 } ^ d & - ( \\partial _ { x _ j } \\partial _ { \\xi _ k } q ( x , \\xi ) ) _ { j , k = 1 } ^ d \\end{pmatrix} . \\end{align*}"}
+{"id": "7768.png", "formula": "\\begin{align*} \\mu \\widetilde \\rho ( s ) & = - \\frac { \\widetilde f \\ , ^ \\prime ( s | \\mu ) } { \\widetilde f ( s | \\mu ) } \\quad { \\rm a n d } \\mu \\ , \\widetilde r _ \\alpha ( s ) = - \\frac { \\widetilde m _ \\alpha \\ , ^ \\prime ( s | \\mu ) } { \\widetilde m _ \\alpha ( s | \\mu ) } = - \\frac { \\widetilde f \\ , ^ \\prime ( s ^ \\alpha | \\mu ) } { \\widetilde f ( s ^ \\alpha | \\mu ) } \\end{align*}"}
+{"id": "8226.png", "formula": "\\begin{align*} X = \\frac { 1 } { \\sqrt 2 } \\left [ \\begin{array} { c c c c c c c } \\sqrt 2 & 0 & 1 & 1 & 1 & 1 & 1 \\\\ 0 & \\sqrt 2 & 1 & \\omega & \\omega ^ 2 & \\omega ^ 3 & \\omega ^ 4 \\\\ \\end{array} \\right ] \\end{align*}"}
+{"id": "6733.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\leq x \\\\ q \\mid n } } \\mu ( n + a ) \\ll \\frac { 1 } { q } \\frac { x } { ( \\log x / q ) ^ D } = \\frac { 1 } { d ^ 2 t } \\frac { x } { ( \\log x / d ^ 2 t ) ^ D } , \\end{align*}"}
+{"id": "3364.png", "formula": "\\begin{align*} \\hat { \\Phi } : = g \\Phi , \\end{align*}"}
+{"id": "1389.png", "formula": "\\begin{align*} D _ { \\rho | \\cdot | ^ { x + \\epsilon j } } ^ { ( k _ { j } + 1 ) } \\circ \\left ( D _ { \\rho | \\cdot | ^ { x + \\epsilon ( j - 1 ) } } ^ { ( k _ { j - 1 } ) } \\circ \\dots \\circ D _ { \\rho | \\cdot | ^ x } ^ { ( k _ 0 ) } \\right ) ( \\pi ) = 0 \\end{align*}"}
+{"id": "730.png", "formula": "\\begin{align*} d s = \\begin{cases} | d z | & z \\in \\Omega \\cup \\partial \\Omega , \\\\ | S ^ \\prime ( { z } ) | | d { z } | \\quad & { z } \\in \\C \\setminus \\overline { \\Omega } , \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "5329.png", "formula": "\\begin{align*} \\nu \\times E = 0 \\mbox { o n } \\partial \\Omega . \\end{align*}"}
+{"id": "8204.png", "formula": "\\begin{align*} \\gamma \\cdot \\frac { \\Phi ( t , - \\lambda ) - 1 } { \\gamma + \\lambda } = - \\lambda \\cdot \\frac { \\gamma } { \\gamma + \\lambda } \\cdot \\int _ 0 ^ t k ( t , s ) \\Phi ( s , - \\lambda ) d s , \\forall t > 0 , \\ \\forall \\lambda \\in \\mathbb { C } : \\Re \\lambda \\geq - c . \\end{align*}"}
+{"id": "2112.png", "formula": "\\begin{align*} \\overline c _ { 1 2 } ( x ) \\overline g ( x ) \\cdot g ( x ) = \\overline c _ { 1 2 } ( x ) e _ { C _ { 1 2 } } ( x ) = \\overline c _ { 1 2 } ( x ) . \\end{align*}"}
+{"id": "5319.png", "formula": "\\begin{align*} \\Phi = \\sum _ { i = 1 } ^ 2 t _ i \\mathrm { A d } _ { U _ i } \\circ \\Phi = \\sum _ { i = 1 } ^ 2 \\mathrm { A d } _ { T _ i } \\circ \\Phi , \\end{align*}"}
+{"id": "3149.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( A ) & = \\frac { \\int _ Y a } { c - \\int _ Y a } c _ 1 ^ { 2 2 } ( A ) = \\frac { \\int _ Y a } { c } ( Q _ 1 + Q _ 2 ) , \\\\ c _ 2 ^ { 1 1 } ( A ) & = \\frac { \\int _ Y a } { c - \\int _ Y a } c _ 2 ^ { 2 2 } ( A ) = \\frac { \\int _ Y a } { c } ( Q _ 2 - Q _ 1 ) , \\end{align*}"}
+{"id": "2458.png", "formula": "\\begin{align*} F ( z ) = L ( z , \\pi ) L ( z + 1 - \\beta _ { \\chi } , \\pi \\otimes \\chi ) , G _ k ( z ) = \\frac { ( - 1 ) ^ k } { k ! } \\Big ( \\frac { F ' } { F } \\Big ) ^ { ( k ) } ( z ) . \\end{align*}"}
+{"id": "9166.png", "formula": "\\begin{align*} C ( s , m ^ 2 ) = \\Gamma _ { \\leq j _ { f } } ( s , m ^ 2 ) + \\sum _ { j = j _ f + 1 } ^ { N - 1 } \\Gamma _ { j } ( s , m ^ 2 ) + \\Gamma _ N ^ { \\Lambda _ N } ( s , m ^ 2 ) + t _ N ( s , m ^ 2 ) Q _ N , \\end{align*}"}
+{"id": "2526.png", "formula": "\\begin{align*} \\partial _ t g + v \\cdot \\nabla _ x g = \\varrho _ g ^ \\varphi M _ g ^ \\varphi - g \\ , , \\end{align*}"}
+{"id": "1778.png", "formula": "\\begin{align*} \\rho ( x ) : = \\frac { \\sup \\mathsf { L } ( x ) } { \\min \\mathsf { L } ( x ) } . \\end{align*}"}
+{"id": "1.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { n \\rightarrow \\infty } \\bigg [ & \\sup _ { t \\in \\mathbb { R _ + } } \\mathbb { E } e ^ { - \\beta t } | y _ { ( n , t ) } - \\hat { y } ^ n _ t | ^ 2 + \\mathbb { E } \\int _ 0 ^ \\infty \\mathbb { I } _ { [ 0 , n ] } e ^ { - \\beta t } \\Big ( | y _ { ( n , t ) } - \\hat { y } ^ n _ t | ^ 2 + | z _ { ( n , t ) } - \\hat { z } ^ n _ t | ^ 2 \\\\ & \\quad + | \\tilde { z } _ { ( n , t ) } - \\hat { \\tilde { z } } ^ n _ t | ^ 2 + | | \\gamma _ { ( n , t ) } - \\hat { \\gamma } ^ n _ t | | ^ 2 \\Big ) d t \\bigg ] = 0 . \\end{aligned} \\end{align*}"}
+{"id": "2002.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 1 & 1 \\\\ 1 & - 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "8272.png", "formula": "\\begin{align*} I ( z , s ) : = \\int _ 1 ^ \\infty \\frac { x ^ { z - s } - x ^ { z - 1 - s } } { \\log x } d x = \\log \\left ( \\frac { s - z } { s - z - 1 } \\right ) ( \\Re s > 1 ) . \\end{align*}"}
+{"id": "5816.png", "formula": "\\begin{align*} \\beta _ { r s } = \\beta _ { s r } , 1 \\leqslant r , s \\leqslant p . \\end{align*}"}
+{"id": "2875.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 \\pi i } \\left ( \\int _ { 2 - i t _ j } ^ { 2 + i t _ j } + \\int _ { 2 + i t _ j } ^ { \\frac { 1 } { 4 } - \\sigma + i t _ j } + \\int _ { \\frac { 1 } { 4 } - \\sigma + i t _ j } ^ { \\frac { 1 } { 4 } - \\sigma - i t _ j } + \\int _ { \\frac { 1 } { 4 } - \\sigma - i t _ j } ^ { 2 - i t _ j } \\right ) \\frac { L _ f ' } { L _ f } ( s + w ) \\Gamma ( w ) Y ^ w d w \\\\ & = \\frac { L _ f ' } { L _ f } ( s ) + \\sum _ { - t _ j < \\gamma < t _ j } \\Gamma ( \\rho - s ) Y ^ { \\rho - s } . \\\\ \\end{align*}"}
+{"id": "3587.png", "formula": "\\begin{align*} v _ { A ( 1 ) } & = - \\frac { 1 } { 2 } E ^ { \\gamma _ { A ( 1 ) } } \\Omega _ \\lambda - \\frac { 1 } { 2 } E ^ { \\gamma _ { A ( 2 ) } } \\Omega _ \\lambda - \\frac { 1 } { 1 2 } E ^ { \\gamma _ { A ( 3 ) } } \\Omega _ \\lambda \\\\ v _ { A ( 2 ) } & = - \\frac { 1 } { 3 } E ^ { \\gamma _ { A ( 2 ) } } \\Omega _ \\lambda - \\frac { 1 } { 1 2 } E ^ { \\gamma _ { A ( 3 ) } } \\Omega _ \\lambda \\\\ v _ { A ( 3 ) } & = - \\frac { 1 } { 1 2 } E ^ { \\gamma _ { A ( 3 ) } } \\Omega _ \\lambda . \\end{align*}"}
+{"id": "1683.png", "formula": "\\begin{align*} \\imath ( t ) = \\left ( \\begin{array} { c c } t & \\\\ & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "8730.png", "formula": "\\begin{align*} G ( u ) = \\sum _ { m = 0 } ^ { \\infty } G _ m u ^ m = i _ { u , \\beta } \\left ( \\frac { 1 } { 1 + \\beta / u } \\right ) E ( \\beta ) H ( u ) . \\end{align*}"}
+{"id": "1863.png", "formula": "\\begin{align*} \\beta = \\displaystyle \\inf _ { k } \\{ \\beta _ k \\} \\ge 0 . \\end{align*}"}
+{"id": "5781.png", "formula": "\\begin{align*} u = ( \\ ! ( E , U ) \\ ! ) = ( \\ ! ( x , \\widetilde C _ q U ) \\ ! ) . \\end{align*}"}
+{"id": "7354.png", "formula": "\\begin{align*} d : = \\left \\lfloor k / \\epsilon \\right \\rfloor + 1 , \\end{align*}"}
+{"id": "5857.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } \\psi '' - \\Delta \\psi + b ( \\psi ) = - e ^ { \\lambda h } ( e , A U ) & \\hbox { i n } ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu \\psi = 0 & \\hbox { o n } ( 0 , T ) \\times \\Gamma , \\\\ t = 0 : \\psi = 0 , \\psi ' = 0 & \\hbox { i n } ~ \\Omega \\end{array} \\right . \\end{align*}"}
+{"id": "1226.png", "formula": "\\begin{align*} W _ \\nu ( \\mu _ 1 , \\mu _ 0 ) : = \\sqrt { \\inf _ { \\gamma \\in \\Gamma } \\int _ { X \\times X \\times X } | x _ 0 - x _ 1 | ^ 2 d \\gamma ( y , x _ 0 , x _ 1 ) } , \\end{align*}"}
+{"id": "4704.png", "formula": "\\begin{align*} \\mathbb { R } ^ { G _ { I I } } = \\mathbb { C } [ \\varphi _ 8 , \\varphi _ { 2 4 } ] \\end{align*}"}
+{"id": "7450.png", "formula": "\\begin{align*} \\widehat M ^ * _ { \\chi _ k } = \\big ( \\widehat Z ^ { ( 0 ) } _ k \\big ) ^ * . \\end{align*}"}
+{"id": "1749.png", "formula": "\\begin{align*} F ( \\infty ) - F ( 0 ) = ( - 1 ) ^ { m + M } 2 ( 2 M + 1 ) ( 2 M + 2 ) \\binom { 2 M } { k _ { \\rm i d } - 2 } \\binom { k _ { \\rm i d } - 2 } { \\frac { k _ { \\rm i d } - 2 } { 2 } - m _ { \\rm i d } } ^ { - 1 } \\binom { k _ { c } - 2 } { \\frac { k _ { c } - 2 } { 2 } - m _ { c } } ^ { - 1 } . \\end{align*}"}
+{"id": "1168.png", "formula": "\\begin{align*} & C = b A \\longleftrightarrow M X _ { m \\times n , p } ( C ) \\wedge \\forall i < m , j < n \\ , \\ , C _ { i j } = b \\cdot _ { ( m o d \\ , p ) } A _ { i j } . \\end{align*}"}
+{"id": "7058.png", "formula": "\\begin{align*} ( S ^ 0 \\subset S ^ { n \\sigma } \\to M U ( \\mathbf { C } ^ n ) ) ^ { C _ 2 } = ( S ^ { n \\sigma } \\to M U ( \\mathbf { C } ^ n ) ) ^ { C _ 2 } , \\end{align*}"}
+{"id": "1229.png", "formula": "\\begin{align*} d ^ 2 _ { L W } ( \\mu _ 0 , \\mu _ 1 ) : = W _ 2 ^ 2 ( \\mu _ 0 ^ V , \\mu _ 1 ^ V ) + \\int _ 0 ^ 1 W _ 2 ^ 2 ( \\tilde \\mu _ 0 ^ l , \\tilde \\mu _ 1 ^ l ) d l \\end{align*}"}
+{"id": "3252.png", "formula": "\\begin{align*} u ( x , t ) = o \\left ( | x | ^ { \\frac { 2 } { \\gamma + 1 } } \\right ) , \\ , \\ , \\ , \\ , \\ , \\ , | x | \\rightarrow \\infty , \\end{align*}"}
+{"id": "2559.png", "formula": "\\begin{align*} \\begin{aligned} & Z \\left ( z _ 1 ( k _ 1 ) \\left \\{ \\prod _ { i = 2 } ^ { p - 1 } z _ { 0 } ( k _ i ) \\right \\} z _ { 0 } ( k _ p + r ) ; \\alpha \\right ) \\\\ = & \\sum _ { \\begin{subarray} { c } \\sum _ { i = 1 } ^ { q - 1 } c ^ { ' } _ { i } r _ i + r _ { q } = r \\\\ c ^ { ' } _ { i } r _ i , r _ q \\ge 0 \\end{subarray} } Z \\left ( \\left \\{ \\prod _ { i = 1 } ^ { q - 1 } z _ { c ^ { ' } _ { i - 1 } } ( 1 + c ^ { ' } _ i r _ i ) \\right \\} z _ { c ^ { ' } _ { q - 1 } } ( 2 + r _ q ) ; \\alpha \\right ) \\end{aligned} \\end{align*}"}
+{"id": "469.png", "formula": "\\begin{align*} I ( G ; x ) = \\sum _ { k = 0 } ^ { \\alpha ( G ) } { s _ k } x ^ { k } = { s _ 0 } + { s _ 1 } x + { s _ 2 } x ^ { 2 } + . . . + { s _ { \\alpha ( G ) } } x ^ { \\alpha ( G ) } \\end{align*}"}
+{"id": "1028.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty | \\widehat { p } _ t ^ s ( \\xi ) | ^ 2 t ^ { a } d t = C ( n , s , a ) | \\xi | ^ { - ( a + 1 ) } . \\end{align*}"}
+{"id": "1960.png", "formula": "\\begin{align*} G = \\langle a , b | a ^ p , b ^ q , b a = a b ^ k \\rangle . \\end{align*}"}
+{"id": "2045.png", "formula": "\\begin{align*} k \\displaystyle \\sum _ { n = 0 } ^ { + \\infty } ( n + 1 ) ^ { k - 1 } \\mathbb { P } [ X \\geq n + 1 ] \\leq \\mathbb { E } [ X ^ k ] \\end{align*}"}
+{"id": "2399.png", "formula": "\\begin{align*} E ( t ) \\dot x = A ( t ) x \\end{align*}"}
+{"id": "4861.png", "formula": "\\begin{align*} X _ { n j } = \\{ \\xi \\in S _ n \\ , | \\ , \\xi _ 1 > \\xi _ 2 > \\ . > \\xi _ j = 1 \\textrm { a n d } \\ , \\xi _ j < \\xi _ { j + 1 } < \\ . < \\xi _ n \\} , \\end{align*}"}
+{"id": "7424.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ N \\langle K ( Z _ i , Z _ j ) ( P _ i P _ j ^ * ) y _ j , y _ i \\rangle \\ge 0 \\end{align*}"}
+{"id": "4333.png", "formula": "\\begin{align*} M ( 1 1 1 0 0 0 ) & = 0 , \\ ; M ( 1 0 1 1 0 0 ) = 1 , \\\\ M ( 1 0 1 0 1 0 ) & = 1 , \\ ; M ( 1 0 1 0 0 1 ) = 1 . \\end{align*}"}
+{"id": "6972.png", "formula": "\\begin{gather*} ( x + 1 ) ^ { \\beta } P _ { m , n } ^ { ( \\alpha , \\beta ) } ( x ) \\\\ { } = ( - 1 ) ^ { m } \\frac { ( \\beta + n ) ( \\alpha + n - 2 m + 1 ) } { ( \\alpha + n - m + 1 ) } \\int _ { c } ^ x ( t + 1 ) ^ { \\beta - 1 } P _ m ^ { ( - \\alpha - 1 , \\beta - 1 ) } ( t ) P _ { n - m } ^ { ( \\alpha + 1 , \\beta - 1 ) } ( t ) \\ , { \\rm d } t \\end{gather*}"}
+{"id": "7798.png", "formula": "\\begin{align*} ( \\Psi _ { c , b _ 2 } [ n _ 2 ] \\Psi _ { c , b _ 1 } [ n _ 1 ] ) ( y ^ { n _ 1 } ) & = \\Psi _ { c , b _ 2 } [ n _ 2 ] ( y ^ { n _ 1 } ) \\\\ & = y ^ { n _ 1 } ( 1 + q ^ { c + b _ 2 } y ^ { n _ 2 } ) , \\end{align*}"}
+{"id": "1942.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { f } _ j ^ Q ( u , \\eta _ j ) & = \\int _ { \\mathcal { X } } f _ j ( u , \\eta _ j , x ) \\ , { Q } ( d x ) , \\ \\ \\ \\ j = 1 , \\cdots , k \\\\ \\bar { f } _ { k + 1 } ^ Q ( u ) & = \\int _ { \\mathcal { X } } f _ { k + 1 } ( u , x ) \\ , { Q } ( d x ) \\end{aligned} \\end{align*}"}
+{"id": "8608.png", "formula": "\\begin{align*} f _ 1 & : = \\frac { 1 } { 2 } [ R _ { 1 0 } + R _ { 1 1 } ] , \\ \\ & f _ 3 : = \\frac { 1 } { 2 } [ 2 R _ 1 + R _ 2 + R _ 4 + 2 R _ 5 + 2 R _ 6 + 2 R _ 7 + 2 R _ 8 + R _ 9 + R _ { 1 2 } ] , \\\\ f _ 2 & : = [ R _ { 9 } + R _ { 1 0 } ] , \\ \\ & f _ 4 : = \\frac { 1 } { 2 } [ 2 R _ 1 + 2 R _ 2 + 2 R _ 3 + 2 R _ 4 + 2 R _ 5 + R _ 6 + R _ 8 + R _ 9 + R _ { 1 2 } ] . \\end{align*}"}
+{"id": "3261.png", "formula": "\\begin{align*} Y _ { \\mathbf { d } , \\mathcal { A } } ( a b c ) = P _ { \\mathbf { d } , \\mathcal { A } } ( a b ) \\frac { P _ { \\mathbf { d } - \\mathbf { e } _ a - \\mathbf { e } _ b , \\mathcal { A } } ( b c ) - Y _ { \\mathbf { d } - \\mathbf { e } _ a - \\mathbf { e } _ b , \\mathcal { A } } ( a b c ) } { 1 - P _ { \\mathbf { d } - \\mathbf { e } _ a - \\mathbf { e } _ b , \\mathcal { A } } ( a b ) } . \\end{align*}"}
+{"id": "3647.png", "formula": "\\begin{align*} \\partial _ { t } w - \\Delta _ { \\mathbb { H } ^ { n } } w = e ^ { \\mu t } w ^ { p + 1 } \\geq e ^ { \\mu t } w \\big ( e ^ { \\beta _ 0 w ^ { p } } - 1 \\big ) , \\end{align*}"}
+{"id": "2989.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\pi & = & \\frac { 1 } { 2 } \\pi ^ { a b } ( x , y ) e _ { a b } ^ { ( 2 ) } \\wedge \\theta - \\pi ^ a ( x , y ) e _ a ^ { ( 3 ) } \\end{array} \\end{align*}"}
+{"id": "8957.png", "formula": "\\begin{align*} R _ { p , q } ^ { \\gamma } ( \\xi ) : = | \\xi | ^ { | p - q | } e ^ { i ( p - q ) \\arg \\xi } R _ { \\min ( p , q ) } ^ { ( \\gamma , | p - q | ) } ( 2 | \\xi | ^ { 2 } - 1 ) . \\end{align*}"}
+{"id": "6335.png", "formula": "\\begin{align*} \\varphi ' ( \\nu _ 0 ) + \\int _ { 1 } ^ { t } f _ { \\nu } ( x , \\nu _ 0 ) d x - \\int _ { h ( t ) } ^ { 1 } f _ \\nu ( x , \\nu _ 0 ) d x = 0 , \\end{align*}"}
+{"id": "2213.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { \\textrm { s p t } } _ \\omega { \\left ( 2 \\times 5 ^ { 2 m } n + 5 ^ { 2 m } \\right ) } q ^ { n - 2 } & = q ^ { - 2 } \\delta \\sum _ { i = 1 } ^ \\infty x _ { 2 m , i } \\xi ^ { i - 1 } . \\end{align*}"}
+{"id": "9169.png", "formula": "\\begin{align*} u _ j & = \\begin{cases} 0 & ( j < j _ f ) \\\\ \\gamma f + \\Gamma _ { \\leq j _ f } ( f + s \\gamma \\Delta f ) & ( j = j _ f ) \\\\ \\Gamma _ { j } ( f + s \\gamma \\Delta f ) & ( N > j > j _ f ) \\\\ \\Gamma _ N ^ { \\Lambda _ N } ( f + s \\gamma \\Delta f ) & ( j = N ) . \\end{cases} \\end{align*}"}
+{"id": "3694.png", "formula": "\\begin{align*} \\frac { \\partial F _ { s ' , s } } { \\partial s } = \\frac { \\partial G _ { s ' , s } } { \\partial s ' } - \\{ F _ { s ' , s } , G _ { s ' , s } \\} , \\end{align*}"}
+{"id": "9070.png", "formula": "\\begin{align*} & R ^ { \\vee } ( m _ { 0 } ) \\mu _ { z , X } ( \\phi ) \\\\ & = \\sum _ { \\substack { \\beta \\\\ \\kappa _ { \\beta } = \\kappa } } \\int _ { M } \\int _ { A } \\int _ { N _ { P } } \\int _ { w \\overline { N } _ { P } w ^ { - 1 } \\cap \\overline { N } _ { P } } a ^ { - \\lambda + \\rho _ { P } } \\Big ( \\sigma ^ { \\vee } ( m ) c _ { \\beta } , \\partial ^ { \\beta } \\big ( R ( m _ { 0 } ^ { - 1 } ) \\phi \\big ) ( m a n \\overline { n } ) \\Big ) \\ , d \\overline { n } \\ , d n \\ , d a \\ , d m . \\end{align*}"}
+{"id": "734.png", "formula": "\\begin{align*} \\{ \\tilde { w } , w \\} _ 0 & = \\lim _ { z \\to w } \\log \\frac { \\tilde { z } - \\tilde { w } } { z - w } , \\\\ \\{ \\tilde { w } , w \\} _ 1 & = 2 \\lim _ { z \\to w } \\frac { \\partial } { \\partial z } \\log \\frac { \\tilde { z } - \\tilde { w } } { z - w } , \\\\ \\{ \\tilde { w } , w \\} _ 2 & = 6 \\lim _ { z \\to w } \\frac { \\partial ^ 2 } { \\partial z \\partial w } \\log \\frac { \\tilde { z } - \\tilde { w } } { z - w } . \\end{align*}"}
+{"id": "5126.png", "formula": "\\begin{align*} \\Omega _ { n } ^ { \\pm } ( \\lambda , b ) & : = \\frac { 1 - b ^ { 2 } } { 2 b } \\Lambda _ { 1 } ( \\lambda , b ) + \\frac { 1 } { 2 } \\Big ( \\Omega _ { n } ( \\lambda ) - \\Omega _ { n } ( \\lambda b ) \\Big ) \\\\ & \\quad \\pm \\frac { 1 } { 2 b } \\sqrt { \\Big ( b \\big [ \\Omega _ { n } ( \\lambda ) + \\Omega _ { n } ( \\lambda b ) \\big ] - ( 1 + b ^ { 2 } ) \\Lambda _ { 1 } ( \\lambda , b ) \\Big ) ^ { 2 } - 4 b ^ { 2 } \\Lambda _ { n } ^ { 2 } ( \\lambda , b ) } \\end{align*}"}
+{"id": "2650.png", "formula": "\\begin{align*} ( B _ t ) _ { 0 \\leq t \\leq T } : = \\big \\{ ( W _ t ) _ { 0 \\leq t \\leq T } | W _ T = 0 \\big \\} . \\end{align*}"}
+{"id": "1422.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( | \\mathcal C _ { \\max } | < n ^ { 2 / 3 } / A \\right ) = O ( A ^ { - 3 / 5 } ) \\mathbb { P } \\left ( | \\mathcal C _ { \\max } | > A n ^ { 2 / 3 } \\right ) = O ( A ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "9090.png", "formula": "\\begin{align*} \\mathfrak { p } _ 0 = \\mathfrak { n } \\mathfrak { p } _ j = [ \\mathfrak { p } _ { j - 1 } , \\mathfrak { n } ] + [ J \\mathfrak { p } _ { j - 1 } , \\mathfrak { n } ] j \\geq 1 . \\end{align*}"}
+{"id": "6897.png", "formula": "\\begin{align*} \\chi ( V , W ) = \\langle \\underline { d } ( V ) , \\underline { d } ( W ) \\rangle . \\end{align*}"}
+{"id": "5868.png", "formula": "\\begin{align*} t = 0 : U = 0 , U ' = e \\theta \\end{align*}"}
+{"id": "9207.png", "formula": "\\begin{align*} h \\partial _ { t } u + \\Delta _ { f } u = 0 , u _ { \\vert t = 0 } = u _ { 0 } , \\end{align*}"}
+{"id": "2696.png", "formula": "\\begin{align*} \\frac { d M } { d t } & = 1 \\\\ \\frac { d P } { d t } & = f ' ( x ( t ) ) / f ( x ( t ) ) = \\frac { d } { d x } \\left [ \\log ( f ( x ( t ) ) ) \\right ] . \\end{align*}"}
+{"id": "1006.png", "formula": "\\begin{align*} u \\mapsto \\lVert u \\rVert _ { \\infty , \\alpha } = \\sup _ { n \\in \\Z } | n | ^ \\alpha | u [ n ] | < \\infty . \\end{align*}"}
+{"id": "2479.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( T ^ n x _ 1 , T ^ n x _ 2 ) = ( X x _ 1 , X x _ 2 ) x _ 1 , x _ 2 \\in \\mathcal H \\end{align*}"}
+{"id": "2894.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow 1 } \\int _ { \\mathbb { T } ^ \\infty } \\mathbf { P } _ \\zeta \\log ^ { + } | F _ { [ r ] } | d m _ \\infty = \\int _ { \\mathbb { T } ^ \\infty } \\mathbf { P } _ \\zeta \\log ^ { + } | F ^ { * } | d m _ \\infty . \\end{align*}"}
+{"id": "1298.png", "formula": "\\begin{align*} \\vec { 1 } _ k : = \\underbrace { ( 1 , 1 , \\ldots , 1 ) } _ { k } \\in \\C ^ k . \\end{align*}"}
+{"id": "9220.png", "formula": "\\begin{align*} \\phi _ { + } = f - f ( \\Gamma ) + \\frac { \\ell _ { 0 , x } ^ { 2 } } { 2 } \\ell _ { 0 , x } ( y , z ) = \\Big ( 2 \\int _ { 0 } ^ { 1 } ( 1 - t ) \\partial ^ { 2 } _ { z , z } g ( y , t z ) \\ , d t \\Big ) ^ { 1 / 2 } z . \\end{align*}"}
+{"id": "522.png", "formula": "\\begin{align*} \\min _ { x = ( \\widetilde { x } , x _ 0 ) \\in \\mathbb { R } ^ { p + 1 } } \\Big \\{ F _ { \\lambda , \\mu } ( x ) : = \\sum _ { i = 1 } ^ n \\log \\big ( 1 + \\exp ( - b _ i ( a _ i ^ { \\mathbb { T } } \\widetilde { x } + x _ 0 ) ) \\big ) + \\frac { \\mu } { 2 } \\| x \\| ^ 2 + \\lambda \\| \\widetilde { x } \\| _ 0 \\Big \\} , \\end{align*}"}
+{"id": "4936.png", "formula": "\\begin{align*} \\begin{cases} u _ t + v _ { x x } + ( u ^ 2 + v ^ 2 ) v = 0 , \\\\ - v _ t + u _ { x x } + ( u ^ 2 + v ^ 2 ) u = { 0 } . \\end{cases} \\end{align*}"}
+{"id": "6347.png", "formula": "\\begin{align*} h ' ( t ) = - { f _ \\nu ( t , \\nu _ 0 ) } / { f _ \\nu ( h ( t ) , \\nu _ 0 ) } . \\end{align*}"}
+{"id": "1919.png", "formula": "\\begin{align*} - a \\Delta _ g u + V u = 0 , \\end{align*}"}
+{"id": "6641.png", "formula": "\\begin{align*} \\delta _ y A ( \\nabla u ) = a _ y \\nabla \\delta _ y u \\end{align*}"}
+{"id": "2886.png", "formula": "\\begin{align*} \\lvert L _ U \\rvert - 1 = q ^ { \\ell t } ( q ^ { t - 1 } + \\ldots + q ) , \\end{align*}"}
+{"id": "2515.png", "formula": "\\begin{align*} \\frac { \\theta _ n ^ d } { ( R _ n ^ d ) } \\frac { 1 } { d ! } = \\frac { \\theta _ n ^ d } { ( R _ n ^ d ) } \\frac { ( R _ n ^ d ) } { n ^ d } = \\frac { \\theta _ n ^ d } { n ^ d } \\le \\frac { \\beta _ n ^ d } { n ^ d } . \\end{align*}"}
+{"id": "1663.png", "formula": "\\begin{align*} \\ell ( f _ 1 , \\chi ) \\cdot \\ell ( f _ 2 , \\chi ^ { - 1 } ) = \\Lambda ( 1 / 2 , \\Pi , \\chi ) \\cdot \\frac { \\langle f _ 1 , f _ 2 \\rangle } { \\langle \\mathfrak { f } _ 1 , \\mathfrak { f } _ 2 \\rangle } \\cdot \\prod _ v \\beta _ v ( f _ { 1 , v } , f _ { 2 , v } ) \\alpha _ v ( W _ { \\mathfrak { f } _ 1 , v } , W ^ - _ { \\mathfrak { f } _ 2 , v } ) . \\end{align*}"}
+{"id": "2417.png", "formula": "\\begin{align*} E _ { 1 1 } ^ T = E _ { 1 1 } , E _ { 2 2 } ^ T = E _ { 2 2 } , 0 = \\dot E _ { 1 1 } , 0 = - A _ { 1 2 } - \\dot E _ { 1 2 } , A _ { 2 2 } ^ T = - A _ { 2 2 } - \\dot E _ { 2 2 } . \\end{align*}"}
+{"id": "5578.png", "formula": "\\begin{align*} \\int _ { A _ j ( 0 ) \\cap \\mathcal { O } } u _ { ( x , 0 ) , r } ^ 2 d V & \\ge \\int _ { r _ j } ^ { R _ j } \\int _ { \\partial B _ \\rho ^ n ( 0 , 0 ) \\cap \\mathcal { O } } u _ { ( x , 0 ) , r } ^ 2 d \\sigma d \\rho \\\\ & = \\int _ { r _ j } ^ { R _ j } H ( \\rho , ( 0 , 0 ) , u _ { ( x , 0 ) , r } ) d \\rho \\\\ & \\ge \\int _ { r _ j } ^ { R _ j } \\frac { \\rho ^ { n - 1 } } { r _ 1 ^ { n - 1 } } H ( r _ 1 , ( 0 , 0 ) , u _ { ( x , 0 ) , r } ) d \\rho \\\\ & \\ge C ( n ) \\frac { j ^ { n - 1 } } { r _ 1 ^ { n - 1 } } H ( r _ 1 , ( 0 , 0 ) , u _ { ( x , 0 ) , r } ) . \\end{align*}"}
+{"id": "3140.png", "formula": "\\begin{align*} A ( y ) = \\frac { c } { 2 } \\frac { 1 } { r ( y ) } B ( y ) , B ( y ) : = \\mathrm { d i a g } ( 1 + b ( y ) , 1 - b ( y ) ) \\end{align*}"}
+{"id": "7659.png", "formula": "\\begin{align*} \\eta _ { I } = \\sum _ { \\substack { J \\subseteq [ n ] \\\\ | J | = j } } \\frac { g _ { I , J } d x _ { J } } { f } \\end{align*}"}
+{"id": "3244.png", "formula": "\\begin{align*} \\lambda _ 1 L _ { 1 } ( s , z ) + \\lambda _ 2 L _ { 1 } ( t , z ) & = \\lambda _ 1 L _ { 1 } ( s , t ) + ( \\lambda _ 1 + \\lambda _ 2 ) L _ { 1 } ( t , z ) \\\\ \\lambda _ 1 L _ { - 1 } ( s , z ) + \\lambda _ 2 L _ { - 1 } ( t , z ) & = \\lambda _ 1 L _ { - 1 } ( s , t ) + ( \\lambda _ 1 + \\lambda _ 2 ) L _ { - 1 } ( t , z ) \\\\ \\mu _ 1 L _ { 0 } ( s , z ) + \\mu _ 2 L _ { 0 } ( t , z ) & = \\mu _ 1 L _ { 0 } ( s , t ) + ( \\mu _ 1 + \\mu _ 2 ) L _ { 0 } ( t , z ) . \\end{align*}"}
+{"id": "7201.png", "formula": "\\begin{align*} q \\ , = \\ , | q | \\Big ( \\cos \\vartheta e ^ { i \\phi } + \\sin \\vartheta e ^ { i \\psi } j \\Big ) \\qquad \\mbox { w h e r e } \\vartheta \\in \\left [ 0 , \\frac { \\pi } { 2 } \\right ] \\qquad \\mbox { a n d } \\phi , \\ , \\psi \\in [ 0 , \\ , 2 \\pi ] . \\end{align*}"}
+{"id": "5074.png", "formula": "\\begin{align*} H = \\sum _ { \\emptyset \\ne I \\subseteq [ 1 , d ] } \\sum _ { \\substack { ( j _ i ) _ { i \\in I } \\\\ j _ i \\in [ 0 , m ] } } u _ { I , ( j _ i ) _ { i \\in I } } H _ { I , ( j _ i ) _ { i \\in I } } \\big ( ( x _ \\mu ) _ { \\mu \\in [ 1 , d ] \\setminus I } \\big ) , \\end{align*}"}
+{"id": "2684.png", "formula": "\\begin{align*} g _ m & = { \\sf P } \\big ( { \\sf b r } _ { \\S ( \\varphi , T ) } ( v ) = m \\ , \\big | \\ , \\S ( \\varphi , T ) \\not = \\phi , \\ , { \\sf b r } _ { \\S ( \\varphi , T ) } ( v ) \\not = 1 \\big ) \\\\ & = ( 1 - p _ t ) ^ { - m } p _ t ^ { m - 1 } { \\sum \\limits _ { k = m } ^ \\infty \\binom { k } { m } p ^ k q _ k \\over 1 - ( 1 - p _ t ) ^ { - 1 } \\sum \\limits _ { k = 2 } ^ \\infty k p ^ k q _ k } ~ = { p _ t ^ { m - 1 } \\over m ! } Q ^ { ( m ) } ( 1 - p _ t ) \\ , ( 1 - Q ' ( 1 - p _ t ) ) ^ { - 1 } . \\end{align*}"}
+{"id": "2566.png", "formula": "\\begin{align*} b \\cdot ( a K ) = ( b a ) K . \\end{align*}"}
+{"id": "69.png", "formula": "\\begin{align*} o b j _ f ( \\mathbf { z } ) = \\sum _ { i = 1 } ^ { n } { z _ i } \\end{align*}"}
+{"id": "5780.png", "formula": "\\begin{align*} \\begin{cases} u '' + L u = - ( \\ ! ( E , A U ) \\ ! ) , \\\\ t = 0 : u = \\theta , \\ u ' = 0 , \\end{cases} \\end{align*}"}
+{"id": "8177.png", "formula": "\\begin{align*} X ^ { \\alpha , \\beta } _ t : = \\sqrt { A _ \\beta } B ^ { \\alpha / 2 } _ t , \\end{align*}"}
+{"id": "1245.png", "formula": "\\begin{align*} \\mathcal H ^ { m - 1 } ( X _ = ( y , k ) ) \\geq C \\mu ( X _ \\geq ( y , k ) ) . \\end{align*}"}
+{"id": "4906.png", "formula": "\\begin{align*} \\partial ^ 2 _ s \\rho = 1 - e ^ { - \\rho } - r b . \\end{align*}"}
+{"id": "1877.png", "formula": "\\begin{align*} \\beta = \\sin ^ { - 1 } \\left ( \\frac { - ( p _ { z _ i } \\sigma + q _ { z _ i } ) } { \\sqrt { \\dot { z } _ i ^ 2 + ( p _ { z _ i } \\sigma + q _ { z _ i } ) ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "1728.png", "formula": "\\begin{align*} I _ { n _ 1 , n _ 2 , m _ 1 , m _ 2 } : = \\int _ { S ^ 3 } a ^ { n _ 1 } b ^ { m _ 1 } \\bar a ^ { n _ 2 } \\bar b ^ { m _ 2 } d ( a , b ) = \\left \\{ \\begin{array} { l c } \\frac { 1 } { n _ 1 + m _ 1 + 1 } \\binom { n _ 1 + m _ 1 } { n _ 1 } ^ { - 1 } , & n _ 1 = n _ 2 , \\ ; m _ 1 = m _ 2 , \\\\ 0 , & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
+{"id": "8691.png", "formula": "\\begin{align*} \\tilde { \\tilde { F } } _ { ( 2 ) } ( x _ 1 , \\dots , x _ n ; t ) & = \\frac { ( 1 - t ) ^ { n } } { \\prod _ { i = 1 } ^ { n - 1 } ( 1 - t ^ i ) } \\sum _ { \\sigma \\in S _ n } \\sigma \\left ( ( x _ 1 ^ 2 + a ) \\prod _ { i = 1 } ^ { n } \\prod _ { i < j } \\frac { x _ { i } - t x _ { j } } { x _ { i } - x _ { j } } \\right ) = F _ { ( 2 ) } + a ( 1 - t ^ n ) . \\end{align*}"}
+{"id": "6629.png", "formula": "\\begin{align*} \\Bigl \\lVert \\frac { x _ j ^ r } { 1 + \\epsilon x ^ { 2 m } } ( t ) D _ j ^ k ( t ) \\psi \\Bigl \\lVert & = \\Bigl \\langle \\psi , D _ j ^ k ( t ) \\frac { x _ j ^ { 2 r } } { ( 1 + \\epsilon x ^ { 2 m } ) ^ 2 } ( t ) D _ j ^ k ( t ) \\psi \\Bigr \\rangle ^ { 1 / 2 } \\\\ & \\leq \\sum _ { p = 0 } ^ k \\binom { k } { p } \\Bigl \\lvert \\Bigl \\langle \\psi _ t , D _ j ^ p \\frac { x _ j ^ { 2 r } } { ( 1 + \\epsilon x ^ { 2 m } ) ^ 2 } D _ j ^ { 2 k - p } \\psi _ t \\Bigr \\rangle \\Bigr \\rvert ^ { 1 / 2 } \\end{align*}"}
+{"id": "6244.png", "formula": "\\begin{align*} A \\leq c B , \\quad c : = c ( A _ 1 , \\ldots A _ k ) . \\end{align*}"}
+{"id": "1077.png", "formula": "\\begin{align*} \\omega ' ( \\eta ) = \\omega \\left ( \\frac { \\eta } { 1 + \\eta } \\right ) = \\sup \\{ | \\theta ( R _ 0 ' ) - \\theta ( R _ 1 ' ) | : \\ , \\mathrm { T V } ( R _ 0 ' , R _ 1 ' ) \\leq \\eta , \\ , R _ 0 ' , \\ , R _ 1 ' \\in \\mathcal { P } _ k \\} , \\end{align*}"}
+{"id": "6790.png", "formula": "\\begin{align*} d ( \\mathcal { U } ( ( \\overline { w } _ { \\i , \\widehat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } ) , \\mathcal { U } ( ( \\overline { v } _ { \\i , \\widehat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } ) ) \\leq ~ c K _ { \\eta } ( ( \\overline { w } _ { \\i , \\widehat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } , ( \\overline { v } _ { \\i , \\widehat { \\eta } } ) _ { \\i = 1 } ^ { \\infty } ) ) , \\end{align*}"}
+{"id": "6404.png", "formula": "\\begin{align*} w _ j ( X , Y , t , \\tilde X , \\tilde Y , \\tilde t ) = u ( X , Y , t ) - \\phi ( \\tilde X , \\tilde Y , \\tilde t ) + \\Psi ( X , Y , t , \\tilde X , \\tilde Y , \\tilde t ) \\end{align*}"}
+{"id": "6070.png", "formula": "\\begin{align*} u ( x ) = u _ 0 \\left ( x ^ g - \\sum _ { i = 1 } ^ g \\left ( \\int _ { b _ i } ^ { a _ { i + 1 } } \\frac { y ^ g \\dd y } { ( w \\tilde w ) ( y ) } \\right ) \\ell _ i ( x ) \\right ) = c \\prod _ { i = 1 } ^ g ( x - x _ i ) ( 1 - x _ i x ) , \\end{align*}"}
+{"id": "7605.png", "formula": "\\begin{align*} \\lambda _ k = k ^ 2 \\pi ^ 2 \\ \\ w _ k ( \\xi ) = \\sqrt { 2 } \\sin ( k \\pi x ) , \\ k = 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "1555.png", "formula": "\\begin{align*} \\tilde { \\zeta } _ { \\mathrm { u } } ( t _ { 1 } ) = \\lambda \\int _ { M } | \\mathrm { u } ( z ) | ^ { r ( z ) } \\ , \\ , d v _ { g } ( z ) = \\tilde { \\zeta } _ { \\mathrm { u } } ( t _ { 2 } ) , \\end{align*}"}
+{"id": "2342.png", "formula": "\\begin{align*} \\Omega _ 0 : = [ 0 , \\tfrac 1 2 ] \\times [ 0 , 1 ] \\ni ( k , s ) , \\end{align*}"}
+{"id": "1196.png", "formula": "\\begin{align*} \\tilde { \\eta } ( X ) & : = \\eta ( \\tilde { \\xi } ) ^ { - 1 } \\eta ( X ) , \\\\ \\tilde { \\Phi } ( X ) & : = \\Phi ( X - \\tilde { \\eta } ( X ) \\tilde { \\xi } ) , \\\\ \\tilde { g } _ L ( X , Y ) & : = \\eta ( \\tilde { \\xi } ) ^ { - 1 } g _ L ( X - \\tilde { \\eta } ( X ) \\tilde { \\xi } , Y - \\tilde { \\eta } ( Y ) \\tilde { \\xi } ) + \\tilde { \\eta } ( X ) \\tilde { \\eta } ( Y ) . \\end{align*}"}
+{"id": "110.png", "formula": "\\begin{align*} k ( p , n ) = \\int _ { S ^ { n - 1 } } | e \\cdot \\omega | ^ p d \\omega = \\frac { 2 \\Gamma ( ( p + 1 ) / 2 ) \\pi ^ { ( n - 1 ) / 2 } } { \\Gamma ( ( n + p ) / 2 ) } . \\end{align*}"}
+{"id": "4109.png", "formula": "\\begin{align*} \\delta ( i _ { \\nabla f } H ) _ { i j } = \\nabla _ l \\phi H _ { l i j } + \\nabla _ l f ( d K ) _ { l i j } - h _ { l k } H _ { k i j } \\nabla _ l f . \\end{align*}"}
+{"id": "8520.png", "formula": "\\begin{align*} \\phi : \\mathbb { C } \\setminus \\{ 0 \\} \\to L \\phi ( t ) = ( z ( t ) , w ( t ) ) , \\end{align*}"}
+{"id": "4140.png", "formula": "\\begin{align*} D h ( X , Y , Z ) = \\frac { 1 } { \\sqrt { 3 } } ( \\nabla _ X h ( Y , Z ) + \\nabla _ Y h ( Z , X ) + \\nabla _ Z h ( X , Y ) ) . \\end{align*}"}
+{"id": "8195.png", "formula": "\\begin{align*} u ( t , x ) = u _ 0 ( x ) - \\int _ 0 ^ t k ( t , s ) \\left ( - L _ 0 - V \\right ) ^ \\gamma u ( s , x ) d s . \\end{align*}"}
+{"id": "7031.png", "formula": "\\begin{align*} H \\underline { R } ^ { C _ 2 } _ \\diamond = R [ \\mu , \\tau ] / ( 2 \\mu ) \\end{align*}"}
+{"id": "7975.png", "formula": "\\begin{align*} x + ( h _ 1 + h _ 2 ) = y + ( h _ 1 + h _ 2 ) . \\end{align*}"}
+{"id": "4624.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { k + 1 } , \\frac { 1 } { k } \\right ] & = \\left ( \\frac { 1 } { k + 1 } , \\frac { 1 } { k + 1 } + \\frac { 1 } { k ( k + 1 ) } \\right ] \\\\ & = \\bigcup _ { \\ell = k ( k + 1 ) } ^ { \\infty } \\left ( \\frac { 1 } { k + 1 } + \\frac { 1 } { \\ell + 1 } , \\frac { 1 } { k + 1 } + \\frac { 1 } { \\ell } \\right ] , \\end{align*}"}
+{"id": "420.png", "formula": "\\begin{align*} U : \\ell ^ p ( \\mathbb { N } ) \\ni \\sum _ { n = 1 } ^ { \\infty } \\zeta _ n ( x ) e _ n \\mapsto \\sum _ { n = 1 } ^ { \\infty } \\zeta _ n ( x ) \\tau _ n \\in \\mathcal { X } . \\end{align*}"}
+{"id": "7154.png", "formula": "\\begin{align*} \\displaystyle { w ^ { \\{ k \\} } _ m = 0 \\ , . } \\end{align*}"}
+{"id": "3317.png", "formula": "\\begin{align*} p _ { k } \\left ( \\mathbf { x } \\right ) = p _ { k } ( x _ { 0 } , x _ { 1 } , \\ldots , x _ { n - 1 } ) = \\sum _ { i \\in \\mathbb { Z } _ { n } } \\left ( x _ { i } \\right ) ^ { k } \\equiv \\begin{cases} 0 & 0 \\leq k \\leq n - 1 \\\\ n & k = n \\end{cases} \\end{align*}"}
+{"id": "7018.png", "formula": "\\begin{align*} R _ c ( R _ p ) \\leq I ( X , Y ; W ) & = h ( X , Y ) - h ( N _ X , N _ Y ) \\\\ & = \\frac { 1 } { 2 } \\log ^ + \\frac { N ^ 2 ( X , Y ) } { ( 1 - \\rho ) \\left ( 2 \\Delta e ^ { R _ p } + \\rho - 1 \\right ) } , \\end{align*}"}
+{"id": "3755.png", "formula": "\\begin{align*} & Z _ { r _ 1 } ( d _ 2 ( f ) , s + 1 ) \\\\ & = \\int _ { F ^ \\times } \\int _ { F } f ( 0 , 0 , 0 , y , 0 , 0 ) \\psi _ v ( a y ) d ( y , \\psi _ v ) | a | ^ { s + 1 } d ^ \\times ( a , \\psi _ v ) \\\\ & = q _ v ^ { e ( \\psi _ v ) ( s + \\frac { 1 } { 2 } ) } \\int _ { F ^ \\times } \\int _ { F } f ( 0 , 0 , 0 , y , 0 , 0 ) \\psi ' ( a y ) d ( y , \\psi _ v ) | a | ^ { s + 1 } d ^ \\times ( a , \\psi ' ) . \\end{align*}"}
+{"id": "1934.png", "formula": "\\begin{align*} \\eta _ { 1 , g } = R _ \\delta \\circ \\eta _ { 1 , { f } } \\eta _ { 2 , { g } } = \\eta _ { 2 , { f } } . \\end{align*}"}
+{"id": "3134.png", "formula": "\\begin{align*} \\partial _ 1 [ r ( 1 + a ) ] = \\partial _ 1 [ 1 + \\partial _ { 1 1 } ^ 2 w - \\partial _ { 2 2 } ^ 2 w - \\Delta w + \\bar { a } ] = - 2 \\ , \\partial _ { 1 2 2 } ^ 3 w , \\end{align*}"}
+{"id": "8321.png", "formula": "\\begin{align*} v ( s , y ) = ( | y | ^ 2 - s ^ 2 ) ^ { - \\frac { d - 1 } { 2 } } w \\left ( \\frac { s } { | y | ^ 2 - s ^ 2 } , \\frac { y } { | y | ^ 2 - s ^ 2 } \\right ) \\end{align*}"}
+{"id": "1152.png", "formula": "\\begin{align*} x | y \\iff \\exists H \\leq y ( x z = y ) . \\end{align*}"}
+{"id": "5667.png", "formula": "\\begin{align*} & \\frac { \\partial ^ 2 B _ j ^ { ( 1 ) } } { \\partial r ^ 2 } + \\frac { 2 } { r } \\frac { \\partial B _ j ^ { ( 1 ) } } { \\partial r } + \\frac { 1 } { r ^ 2 \\sin \\theta } \\frac { \\partial } { \\partial \\theta } \\left ( \\sin \\theta \\frac { \\partial B _ j ^ { ( 1 ) } } { \\partial \\theta } \\right ) = 0 , \\ r > 1 , \\\\ & B _ j ^ { ( 1 ) } \\sim b _ j r \\cos \\theta \\mbox { a s } r \\to \\infty ; B _ j ^ { ( 1 ) } = 0 \\mbox { o n } r = \\ell _ j . \\end{align*}"}
+{"id": "7707.png", "formula": "\\begin{align*} \\lambda ^ { - \\frac { 1 } { 2 } } = \\frac { 4 \\sqrt { \\tau } } { \\pi } \\left ( I ^ { ( 1 ) } ( \\lambda ) + I ^ { ( 2 ) } ( \\lambda ) \\right ) , \\end{align*}"}
+{"id": "3053.png", "formula": "\\begin{align*} c _ j ^ { k l } ( A ) : = \\int _ { Y } r A e _ j \\cdot \\nabla v ^ { k l } , \\end{align*}"}
+{"id": "1674.png", "formula": "\\begin{align*} C ( \\underline k , \\underline m ) = ( - 1 ) ^ { \\left ( \\sum _ { \\sigma \\not \\in \\Sigma _ B } \\frac { k _ { \\sigma } - 2 } { 2 } \\right ) } 4 ^ { r _ \\R } \\cdot ( 3 2 \\pi ) ^ { r _ \\C } \\left ( \\frac { 1 } { \\pi } \\right ) ^ { d - r } \\prod _ { \\sigma \\mid \\infty } \\prod _ { \\tilde \\sigma \\mid \\sigma } \\frac { \\Gamma ( \\frac { k _ { \\tilde \\sigma } } { 2 } - m _ { \\tilde \\sigma } ) \\Gamma ( \\frac { k _ { \\tilde \\sigma } } { 2 } + m _ { \\tilde \\sigma } ) } { ( - 1 ) ^ { m _ { \\tilde \\sigma } } . ( 2 \\pi ) ^ { k _ { \\tilde \\sigma } } } . \\end{align*}"}
+{"id": "7423.png", "formula": "\\begin{align*} K ( Z , W ) ( [ P _ { i j } ] ) = H ( Z ) [ \\pi ( P _ { i j } ) ] H ( W ) ^ * . \\end{align*}"}
+{"id": "4527.png", "formula": "\\begin{align*} \\frac { 1 } { x _ 1 } + \\frac { 1 } { x _ 2 } = \\frac { 1 } { y _ 1 } + \\frac { 1 } { y _ 2 } \\end{align*}"}
+{"id": "4729.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { m } ( a _ i ( e _ i , f _ i ) + b _ i ( g _ i , h _ i ) ) = - \\sum \\limits _ { j = 1 } ^ { e + d } c _ j ( v _ { j } , w _ { j } ) - \\sum \\limits _ { k = 1 } ^ { e } d _ k ( x _ k , y _ k ) \\in \\mathcal { C } . \\end{align*}"}
+{"id": "5448.png", "formula": "\\begin{align*} ( L V ) ( t , x , \\mu , i ) & \\leq \\lambda _ 1 V ( x , i ) + \\lambda _ 2 \\int _ { \\mathbb R ^ d } \\varphi ( x ) \\mu ( d x ) , \\\\ V _ R & : = \\inf _ { | x | \\geq R , i \\in \\mathcal M } V ( x , i ) \\to \\infty ~ a s ~ R \\to \\infty , \\end{align*}"}
+{"id": "4727.png", "formula": "\\begin{align*} h ^ { ( \\ell ) } _ { i 1 } = ( v _ i , 0 ^ { \\ell } , w _ i , 0 ^ k , - \\beta _ j , 0 ^ { \\ell - 1 - k } ) \\ \\ \\ \\ h ^ { ( \\ell ) } _ { i 2 } = ( x _ i , 0 ^ { k } , z _ i \\gamma _ j , 0 ^ { \\ell - 1 - k } , y _ i , 0 ^ { \\ell } ) \\end{align*}"}
+{"id": "4574.png", "formula": "\\begin{align*} H ( x ) = \\{ g \\in G : g ^ { - 1 } x g = x \\} . \\end{align*}"}
+{"id": "2142.png", "formula": "\\begin{align*} e ^ { - t } \\sum _ { i = N ( t ) } ^ { + \\infty } \\frac { t ^ i } { i ! } \\le e ^ { - t } \\sum _ { i = N ( t ) } ^ { + \\infty } \\left ( \\frac { e t } { i } \\right ) ^ i \\le e ^ { - t } \\sum _ { i = N ( t ) } ^ { + \\infty } \\left ( \\frac { e t } { N ( t ) } \\right ) ^ i = e ^ { - t } \\left ( \\frac { e t } { N ( t ) } \\right ) ^ { N ( t ) } \\frac { N ( t ) } { N ( t ) - e t } . \\end{align*}"}
+{"id": "9051.png", "formula": "\\begin{align*} D _ { \\mathbb { H } ^ 2 } ( P , Q ) = \\inf \\left \\{ \\left ( \\int _ { \\mathbb { H } ^ 2 \\times \\mathbb { H } ^ 2 } \\| y - y ^ { \\prime } \\| _ { \\mathbb { H } ^ 2 } ^ 2 R ( d y , d y ^ { \\prime } ) \\right ) ^ { 1 / 2 } \\right \\} , \\end{align*}"}
+{"id": "6906.png", "formula": "\\begin{align*} \\limsup _ { t \\rightarrow + \\infty } \\dfrac { < M _ t , M _ t > } { t } = \\limsup _ { t \\rightarrow + \\infty } \\dfrac { \\sigma ^ 2 } { t } \\int _ 0 ^ t u ^ 2 ( s ) d s \\leq \\sigma ^ 2 \\left ( \\dfrac { \\Lambda } { \\mu } \\right ) ^ 2 < \\infty . \\end{align*}"}
+{"id": "6820.png", "formula": "\\begin{gather*} \\begin{bmatrix} q _ { m + 1 } ^ { [ 1 ] } \\\\ q _ { m + 1 } ^ { [ 2 ] } \\end{bmatrix} = \\begin{bmatrix} q _ { m - 1 } ^ { [ 1 ] } \\\\ q _ { m - 1 } ^ { [ 2 ] } \\end{bmatrix} + 2 h \\begin{bmatrix} q _ { m } ^ { [ 2 ] } \\\\ - s i n ( q _ { m } ^ { [ 1 ] } ) \\end{bmatrix} , \\end{gather*}"}
+{"id": "4065.png", "formula": "\\begin{align*} \\int _ { \\Omega } f \\ d x & < \\frac { 1 } { t } \\int _ { 0 } ^ t \\int _ { \\Omega } f \\exp \\Big ( \\frac { \\partial u } { \\partial t } \\Big ) \\ d x \\ d \\tau \\\\ & = \\frac { 1 } { t } \\int _ { 0 } ^ t \\int _ { \\Omega } f ^ * ( Y u ) \\det D Y u \\ d x \\ d \\tau \\\\ & = \\frac { 1 } { t } \\int _ { 0 } ^ t \\int _ { \\Omega ^ * } f ^ * \\ d y \\ d \\tau \\\\ & = \\int _ { \\Omega } f ( x ) \\ d x . \\end{align*}"}
+{"id": "3857.png", "formula": "\\begin{align*} T _ { 3 n + 2 + j } ^ 2 = 1 + 3 \\delta _ { j , 2 } + \\sum _ { k = 1 } ^ n \\left \\{ T _ { 3 k + j + 1 } ^ 2 + 3 T _ { 2 k + j } ^ 2 + 4 \\ ! \\ ! \\ ! \\sum _ { i = 0 } ^ { 3 k + j - 3 } ( T _ { 3 k + j - i } + T _ { 3 k + j - i - 1 } ) T _ { i + 2 } ^ 2 \\right \\} . \\end{align*}"}
+{"id": "7091.png", "formula": "\\begin{align*} \\{ \\nu \\mid _ { \\mathbf { C P } ( k ) } ^ { \\mathbf { C P } ( k + \\ell ) } \\} = - k \\{ \\gamma ^ 1 \\} + ( k + \\ell ) \\{ \\gamma ^ 1 \\} = \\ell \\{ \\gamma ^ 1 \\} . \\end{align*}"}
+{"id": "3247.png", "formula": "\\begin{align*} I _ { D _ { i _ 1 } } ^ { + , i _ 1 } ( z ) = \\frac { 1 } { i _ 1 ! j _ 1 ! k _ 1 ! } \\int _ { 0 < t _ 1 < z } L _ 1 ( t _ 1 , z ) ^ { i _ 1 } L _ { - 1 } ( t _ 1 , z ) ^ { j _ 1 } L _ 0 ( t _ 1 , z ) ^ { k _ 1 } \\frac { d t _ 1 } { 1 - t _ 1 } , \\\\ I _ { D _ { i _ 1 } } ^ { - , i _ 1 } ( z ) = \\frac { 1 } { i _ 1 ! j _ 1 ! k _ 1 ! } \\int _ { 0 < t _ 1 < z } L _ 1 ( t _ 1 , z ) ^ { i _ 1 } L _ { - 1 } ( t _ 1 , z ) ^ { j _ 1 } L _ 0 ( t _ 1 , z ) ^ { k _ 1 } \\frac { - d t _ 1 } { 1 + t _ 1 } . \\end{align*}"}
+{"id": "8602.png", "formula": "\\begin{align*} \\Phi ( D ) = \\min \\{ D \\cdot F \\mid F ~ \\textrm { i s a h a l f - f i b e r o n } ~ S \\} \\end{align*}"}
+{"id": "4409.png", "formula": "\\begin{align*} \\prod _ { i = m + 1 } ^ { m + k } x _ i < \\prod _ { i = m + 1 } ^ { m + k } a _ i . \\end{align*}"}
+{"id": "4344.png", "formula": "\\begin{align*} \\sigma ( x , y ) = ( x y = 0 ) \\wedge ( x + y = 1 ) \\end{align*}"}
+{"id": "3210.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ E \\phi ( x ) \\mu _ n ( d x ) = \\int _ E \\phi ( x ) \\mu ( d x ) . \\end{align*}"}
+{"id": "8918.png", "formula": "\\begin{align*} c _ { d } ^ { \\left ( \\nu , n \\right ) } = \\left \\{ \\begin{array} { c } \\frac { ( n - d - 1 ) ! } { ( n - 1 ) ! } \\gamma _ { n - d - 1 } ^ { ( \\nu , n ) } , 0 \\leq d \\leq n - 1 \\\\ \\sum \\limits _ { p = 0 } ^ { n - 1 } \\frac { \\gamma _ { p } ^ { ( \\nu , n ) } } { ( n - 1 ) ! } \\Omega _ { p } ^ { \\left ( \\nu \\right ) } \\left ( d - n \\right ) , d \\geq n \\end{array} \\right . \\end{align*}"}
+{"id": "3821.png", "formula": "\\begin{align*} \\frac { \\sum _ { \\lambda \\in \\mathcal { N } _ i } \\delta ( \\lambda ) q ^ { \\mid \\lambda \\mid } } { \\prod _ { j \\geq 3 - i } ( 1 - q ^ j ) ^ 2 } = \\frac { \\mathcal { H P } _ { \\textbf { K } [ x _ j , j \\geq 3 - i ] / < x _ j ^ 2 , x _ j x _ { j + 1 } , j \\geq 3 - i > } ( q ) } { \\prod _ { j \\geq 3 - i } ( 1 - q ^ j ) } . \\end{align*}"}
+{"id": "1320.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | & \\geq \\frac { \\frac { 1 } { d } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ 2 } { n ^ 2 - n } \\\\ & = \\frac { \\frac { 1 } { ( \\mathcal { X } ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ 2 } { n ^ 2 - n } , \\end{align*}"}
+{"id": "6192.png", "formula": "\\begin{align*} x _ l ^ { ( k ) } = \\widehat x _ l ^ { ( k ) } + \\widetilde x _ l ^ { ( k ) } \\hbox { w i t h } \\widehat x _ l ^ { ( k ) } \\in \\hbox { I m } ( C _ p ^ T ) , \\widetilde x _ l ^ { ( k ) } \\in \\hbox { K e r } ( C _ p ) , \\end{align*}"}
+{"id": "4277.png", "formula": "\\begin{align*} \\omega > \\frac { - E _ \\gamma ( \\phi ) } { M ( \\phi ) } = - \\frac { I ^ m _ \\gamma ( c ) } { c } > - \\omega ^ 0 _ \\gamma . \\end{align*}"}
+{"id": "719.png", "formula": "\\begin{align*} \\begin{cases} w _ 1 = w + u , \\\\ w _ 2 = w - u . \\end{cases} \\end{align*}"}
+{"id": "1100.png", "formula": "\\begin{align*} D ^ { \\beta - 1 / 2 , \\infty } _ \\infty = \\left \\{ f \\in B ^ { \\beta - 1 / 2 , \\infty } _ \\infty , \\ , f \\geq 0 , \\ , \\mathrm { s u p p } ( f ) \\subseteq [ 0 , 1 ] , \\ , \\int _ { [ 0 , 1 ] } f ( x ) \\ , \\mathrm { d } x = 1 \\right \\} , \\end{align*}"}
+{"id": "6001.png", "formula": "\\begin{align*} Y _ { r , 0 } ^ h : & = \\{ ( L , H ) \\in X | L \\subset < e _ 1 , \\dots e _ h > , \\ : < e _ 1 , \\dots e _ r > \\subset H \\} \\\\ & = X ( h , r + 1 ) \\end{align*}"}
+{"id": "1349.png", "formula": "\\begin{align*} G _ { f , \\tau } ^ { \\circ ^ m } = [ ( f _ j ( \\tau _ k ) ) ^ m ] _ { 1 \\leq j , k \\leq n } = [ ( f _ j ^ { \\otimes m } ( \\tau ^ { \\otimes m } _ k ) ) ] _ { 1 \\leq j , k \\leq n } . \\end{align*}"}
+{"id": "9181.png", "formula": "\\begin{align*} \\Gamma _ { j + 1 } = \\Gamma _ { j , j + 1 / M } + \\cdots + \\Gamma _ { j + ( M - 1 ) / M , j + 1 } . \\end{align*}"}
+{"id": "3540.png", "formula": "\\begin{align*} [ E _ { k l } , B _ j ^ + ] = \\delta _ { j l } B _ k ^ + , \\end{align*}"}
+{"id": "2084.png", "formula": "\\begin{align*} w ^ { \\prime } = ( 1 - \\sqrt { w } ) ^ 2 = w + 1 - 2 \\sqrt { w } , \\end{align*}"}
+{"id": "4563.png", "formula": "\\begin{align*} a _ n ^ 2 - a _ n + 1 & = a _ n ( a _ n - 1 ) + 1 = a _ n \\left ( \\prod _ { i = 1 } ^ { n - 1 } a _ i \\right ) + 1 \\\\ & = \\left ( \\prod _ { i = 1 } ^ n a _ i \\right ) + 1 = a _ { n + 1 } . \\end{align*}"}
+{"id": "5014.png", "formula": "\\begin{align*} \\xi ( [ X , Y ] , Z ) = c _ 1 B _ 1 ( [ X , Y ] _ 1 , Z _ 1 ) + c _ 2 B _ 2 ( [ X , Y ] _ 2 , Z _ 2 ) , \\end{align*}"}
+{"id": "7328.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\infty } \\int _ { 0 } ^ { 1 } f ( x ) G ^ 2 _ { m _ \\gamma } ( x ) j _ \\gamma ( d x ) = f ( 0 ) \\end{align*}"}
+{"id": "189.png", "formula": "\\begin{align*} \\sum _ { n } \\frac { c _ { n + 1 } } { h _ { n + 1 } } \\leq \\sum _ { n } \\frac { h _ { n } + 2 c _ { n } + 2 r _ { n } - 2 } { r _ { n } ( h _ { n } + c _ { n } ) } \\leq \\sum _ { n } \\frac { 2 ( h _ { n } + c _ { n } ) } { r _ { n } ( h _ { n } + c _ { n } ) } = 2 \\sum _ { n } \\frac { 1 } { r _ { n } } \\end{align*}"}
+{"id": "4777.png", "formula": "\\begin{align*} \\Pr _ { \\rho \\sim \\lambda _ { S } } \\big [ | \\Omega _ \\rho ^ S | > k \\big ] & = \\Pr _ { \\rho \\sim \\lambda _ { S } } \\big [ | \\Omega _ \\rho ^ S | - 1 \\geq k \\big ] \\\\ & \\leq \\frac { \\mathsf { C o l l _ H } ( S ) - 1 } { k } . \\end{align*}"}
+{"id": "1075.png", "formula": "\\begin{align*} D _ \\gamma ( P | | Q ) = \\frac { 1 } { \\gamma - 1 } \\log \\int \\Bigl ( \\frac { d P } { d Q } \\Bigr ) ^ \\gamma \\ , \\mathrm { d } Q . \\end{align*}"}
+{"id": "8563.png", "formula": "\\begin{align*} \\lambda t ! = ( t - i ) d _ { w } ( i ) + ( i + 1 ) d _ { w } ( i + 1 ) \\end{align*}"}
+{"id": "718.png", "formula": "\\begin{align*} d s _ { \\rm r o b i n } ^ 2 \\ = e ^ { - 2 h _ 0 ( w ) } | d w | ^ 2 \\qquad \\qquad \\qquad \\end{align*}"}
+{"id": "3236.png", "formula": "\\begin{align*} L ' ( v ) = \\begin{cases} C ' & , \\cr L ( v ) & . \\end{cases} \\end{align*}"}
+{"id": "6037.png", "formula": "\\begin{align*} \\| f - g _ n \\| _ 2 ^ 2 = \\| f ^ + - g _ n ^ + \\| _ 2 ^ 2 + \\| f ^ - - g _ n ^ - \\| _ 2 ^ 2 , \\end{align*}"}
+{"id": "2504.png", "formula": "\\begin{align*} \\sigma _ { \\mathbf u } = { \\mathbf h } \\sigma _ { \\mathbf v } . \\end{align*}"}
+{"id": "8065.png", "formula": "\\begin{align*} \\abs [ \\Big ] { L \\Bigl ( \\frac { 1 } { 2 } - i t , f \\Bigr ) } ^ 2 & = \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { \\eta \\Bigl ( n , \\dfrac { 1 } { 2 } + i t \\Bigr ) A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } V _ - ( m ^ 2 n , t ) + { } \\\\ & \\qquad { } + \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { \\eta \\Bigl ( n , \\dfrac { 1 } { 2 } + i t \\Bigr ) A ( m , n ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } V _ + ( m ^ 2 n , t ) . \\end{align*}"}
+{"id": "7122.png", "formula": "\\begin{align*} \\left ( \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) - \\gamma ( \\rho _ 1 , \\dots , \\rho _ n ) \\right ) \\cap x ^ { \\rho _ 1 } & = \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) \\cap x ^ { \\rho _ 1 } - \\gamma ( \\rho _ 1 , \\dots , \\rho _ n ) \\cap x ^ { \\rho _ 1 } \\\\ & = \\beta ( \\rho _ 2 , \\dots , \\rho _ n ) - \\gamma ( \\rho _ 2 , \\dots , \\rho _ n ) = 0 \\end{align*}"}
+{"id": "8408.png", "formula": "\\begin{align*} \\| P _ { \\alpha } ^ { k } ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\| _ { C ^ i } \\leq C ( 1 - \\theta ) ^ k \\| P _ { \\alpha } ( \\hat \\Psi _ { \\alpha } h _ { \\alpha } ) \\| _ { C ^ i } i = 1 , 2 \\end{align*}"}
+{"id": "5115.png", "formula": "\\begin{align*} \\mbox { I m } \\left \\lbrace \\mathbf { v } ( t , \\gamma ( t , s ) ) \\overline { \\partial _ { s } \\gamma ( t , s ) } \\right \\rbrace = \\mbox { I m } \\left \\lbrace \\mathbf { v } ( 0 , \\gamma ( 0 , s ) ) \\overline { \\partial _ { s } \\gamma ( 0 , s ) } \\right \\rbrace . \\end{align*}"}
+{"id": "1542.png", "formula": "\\begin{align*} p _ i ^ { b c } \\Omega ^ i _ { a b } + \\delta ^ c _ a \\frac 1 2 p _ i ^ { d e } \\Omega ^ i _ { d e } = 2 \\rho ^ { b } _ { i , f } \\eta ^ { f c } \\Omega ^ i _ { a b } + \\delta ^ c _ a \\rho ^ { d } _ { i , f } \\eta ^ { f e } \\Omega ^ i _ { d e } \\end{align*}"}
+{"id": "7048.png", "formula": "\\begin{align*} d _ { i , j } - \\sum _ { \\ell = 0 } ^ { m - 1 } c _ { i , j + \\ell } u ^ \\ell & = d _ { i , j + m } u ^ m \\\\ q _ j - \\sum _ { \\ell = 0 } ^ { m - 1 } p _ { j + \\ell } u ^ \\ell & = q _ { j + m } u ^ m \\end{align*}"}
+{"id": "614.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } \\sigma _ t = - \\frac { \\sigma _ t ^ 2 } { \\gamma } ( A ^ { \\rm T } A ) : \\tilde C \\end{align*}"}
+{"id": "8055.png", "formula": "\\begin{align*} H ^ - ( x ) = \\frac { 4 } { \\pi } \\int _ { - \\infty } ^ \\infty K _ { 2 i t } ( x ) \\sinh ( \\pi t ) h ( t ) t \\ , d t , \\end{align*}"}
+{"id": "4323.png", "formula": "\\begin{align*} d ( a _ 0 , a _ 1 , \\dots , a _ { p + 1 } ) = \\sum _ { i = 0 } ^ { p + 1 } ( - 1 ) ^ i ( a _ 0 , \\dots , a _ i + a _ { i + 1 } , \\dots , a _ { p + 1 } ) . \\end{align*}"}
+{"id": "998.png", "formula": "\\begin{align*} \\S = \\left \\{ f \\in L ^ 2 ( \\Sigma ) : \\int _ 0 ^ 1 f \\big ( \\tfrac 1 3 , y \\big ) \\sin ( \\pi y ) \\ , d y = \\int _ 0 ^ 1 f \\big ( \\tfrac 2 3 , y \\big ) \\sin ( \\pi y ) \\ , d y = 0 \\right \\} , \\end{align*}"}
+{"id": "3492.png", "formula": "\\begin{align*} \\begin{aligned} y \\left [ n \\right ] & = \\Big ( { \\sum \\nolimits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ l } } } \\Big ) s \\left [ { n - { n _ { \\max } } } \\right ] + \\\\ & \\sum \\nolimits _ { l = 1 } ^ L { \\sum \\nolimits _ { l ' \\ne l } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ { l ' } } s \\left [ { n - { n _ { \\max } } + { \\Delta _ { l ' , l } } } \\right ] } } + z \\left [ n \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "2957.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla _ { j - 1 , j } \\overrightarrow { W } ^ \\ell _ j = \\ell ^ { - 1 } ( W ^ + _ j - W _ j ) \\end{aligned} \\end{align*}"}
+{"id": "4996.png", "formula": "\\begin{align*} \\lim _ { \\| Q - q \\| \\rightarrow + \\infty } \\frac { S ( q , Q ) } { \\| Q - q \\| } = + \\infty . \\end{align*}"}
+{"id": "7320.png", "formula": "\\begin{align*} g _ { m } ( a \\lambda ; b x ) & = g _ { a m ^ b } ( b \\lambda ; x ) , \\\\ \\psi _ { m } ( a \\lambda ; b x ) & = b \\psi _ { a m ^ b } ( b \\lambda ; x ) \\end{align*}"}
+{"id": "1398.png", "formula": "\\begin{align*} \\pi _ 2 ^ { ( + , + , + ) } = L ( \\Delta [ 0 , - 2 ] , \\Delta [ 0 , - 1 ] ; \\pi ( 0 ^ + , 0 ^ + , 1 ^ + ) ) . \\end{align*}"}
+{"id": "873.png", "formula": "\\begin{align*} \\mathcal { X } ^ { t + 1 } = \\mathcal { X } ^ { t } - \\mathcal { Q } ^ { - 1 } * \\mathcal { A } ^ { T } * \\mathcal { S } * ( \\mathcal { S } ^ { T } * \\mathcal { A } * \\mathcal { Q } ^ { - 1 } * \\mathcal { A } ^ { T } * \\mathcal { S } ) ^ { \\dag } * \\mathcal { S } ^ { T } * ( \\mathcal { A } * \\mathcal { X } ^ { t } - \\mathcal { B } ) , \\end{align*}"}
+{"id": "4320.png", "formula": "\\begin{align*} \\phi | _ { S ^ { 0 } \\times ( - \\epsilon , \\epsilon ) } = \\gamma | _ { S ^ { 0 } \\times ( - \\epsilon , \\epsilon ) } , \\end{align*}"}
+{"id": "2394.png", "formula": "\\begin{align*} \\Delta ^ T = - \\Delta , \\quad \\Sigma _ { 1 1 } ^ T = \\Sigma _ { 1 1 } \\ . + \\dot \\Delta , \\Sigma _ { 2 1 } ^ T = \\Sigma _ { 1 2 } \\ . , \\quad \\Sigma _ { 2 2 } ^ T = \\Sigma _ { 2 2 } \\ . , A _ { 4 1 } ^ T = A _ { 1 4 } \\ . + \\dot E _ { 1 4 } . \\end{align*}"}
+{"id": "1170.png", "formula": "\\begin{align*} - \\Delta u + \\mathbf { c } \\cdot \\nabla u = f \\end{align*}"}
+{"id": "3039.png", "formula": "\\begin{align*} \\Delta { w } ^ { s , j } _ { k } = 0 \\ \\ \\mathrm { i n } \\ \\ \\Omega _ { 2 ^ { - s _ 0 + j + 3 } } ( y _ k ) . \\end{align*}"}
+{"id": "4834.png", "formula": "\\begin{align*} | E ( t ) | = \\Big | \\frac { \\phi ^ { ( n + 1 ) } ( \\bar s ) } { ( n + 1 ) ! } \\Big | \\leq C | t | ^ { n + 1 } , \\end{align*}"}
+{"id": "1037.png", "formula": "\\begin{align*} \\mathcal { R } _ { n , \\alpha } ( \\varepsilon ) = \\inf _ { Q \\in \\mathcal { Q } _ \\alpha } \\inf _ { \\phi \\in \\Phi _ Q } \\bigg \\{ \\sup _ { P \\in \\mathcal { P } _ \\varepsilon ( P _ 0 ) } \\mathbb { E } _ { P , Q } ( \\phi ) + \\sup _ { P ' \\in \\mathcal { P } _ \\varepsilon ( P _ 1 ) } \\mathbb { E } _ { P ' , Q } ( 1 - \\phi ) \\bigg \\} , \\end{align*}"}
+{"id": "2926.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Phi _ n ( \\rho ) = g ^ \\prime ( 0 ) \\rho . \\end{align*}"}
+{"id": "1810.png", "formula": "\\begin{align*} \\dot { W } - C + W A + A ^ T W + W B W = 0 , \\end{align*}"}
+{"id": "3848.png", "formula": "\\begin{align*} \\mu _ l ^ { [ 1 2 ] } & = \\mu _ { l - 1 } ^ { [ 1 2 ] } + \\mu _ { l - 1 } ^ { [ 1 3 ] } + \\delta _ { l , 2 } , \\\\ \\mu _ l ^ { [ 1 3 ] } & = \\mu _ { l - 1 } ^ { [ 1 2 ] } + \\mu _ { l - 1 } ^ { [ 2 3 ] } + \\delta _ { l , 3 } , \\\\ \\mu _ l ^ { [ 2 3 ] } & = \\mu _ { l - 1 } ^ { [ 1 2 ] } , \\end{align*}"}
+{"id": "5507.png", "formula": "\\begin{align*} \\log | \\Gamma ( s ) | = \\left ( \\sigma - \\frac 1 2 \\right ) \\log t - { \\pi t \\over 2 } + O ( 1 ) \\end{align*}"}
+{"id": "1375.png", "formula": "\\begin{align*} M ( t ) : = - \\displaystyle \\int _ { t _ 1 } ^ t b ' ( s ) \\int _ { \\Omega } | \\Delta u ( t ) - \\Delta u ( t - s ) | ^ 2 d x d s \\leq - c E ' ( t ) . \\end{align*}"}
+{"id": "4539.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 9 } = \\frac { 4 } { 9 } < \\frac { 1 } { 4 } + \\frac { 1 } { 5 } = \\frac { 9 } { 2 0 } < \\theta \\end{align*}"}
+{"id": "7711.png", "formula": "\\begin{align*} t _ { 0 , 1 } ^ { ( 1 ) } = \\pm 2 \\left ( \\frac { \\tau } { \\lambda } \\right ) ^ { \\frac { 1 } { 2 } } i - 1 , \\end{align*}"}
+{"id": "806.png", "formula": "\\begin{align*} A _ i = \\left ( \\sigma ^ { - i } \\left ( \\bigcup _ { j = 1 } ^ m \\Sigma _ { B _ j } \\right ) \\backslash \\bigcup _ { k = 0 } ^ { i - 1 } \\sigma ^ { - k } \\left ( \\bigcup _ { j = 1 } ^ m \\Sigma _ { B _ j } \\right ) \\right ) \\cap Y \\end{align*}"}
+{"id": "6265.png", "formula": "\\begin{align*} Y _ { 0 } ( x , y , z ) = \\begin{cases} e ^ { - \\frac { 1 } { y } } \\frac { \\partial } { \\partial y } \\big | _ { ( x , y , z ) } & y > 0 \\\\ 0 & y \\leq 0 \\end{cases} \\end{align*}"}
+{"id": "4458.png", "formula": "\\begin{align*} u _ { n , A } ( \\theta ) = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ 1 , \\ldots , x _ n ) \\in A ^ n , x _ 1 \\geq \\cdots \\geq x _ n , \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < \\theta \\right \\} . \\end{align*}"}
+{"id": "1085.png", "formula": "\\begin{align*} \\bigl | \\mathbb { E } ( \\hat { \\mu } | J = j _ 0 - 1 ) - \\mu \\bigr | \\leq \\frac { 5 } { 2 } \\varepsilon M + ( 1 - \\varepsilon ) \\frac { 6 ^ k } { M ^ { k - 1 } } . \\end{align*}"}
+{"id": "5393.png", "formula": "\\begin{align*} { \\sf I d } + \\sum _ { j = 2 } ^ k P _ j , \\end{align*}"}
+{"id": "6694.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\mu ( n ) \\mu ( n + a ) = O \\left ( \\frac { x } { ( \\log x ) ^ { c } } \\right ) . \\end{align*}"}
+{"id": "5992.png", "formula": "\\begin{align*} X ( i , j ) = \\left \\{ \\left ( [ x _ 1 : \\dots : x _ i : 0 : \\dots : 0 ] , [ 0 : \\dots 0 : y _ j : \\dots : y _ n ] \\right ) \\in \\mathbb { P } ^ { n - 1 } \\times \\mathbb { P } ^ { n - 1 } \\biggr \\rvert \\sum x _ k y _ k = 0 \\right \\} . \\end{align*}"}
+{"id": "8770.png", "formula": "\\begin{align*} x & = [ u ^ * _ 0 , \\dots , u ^ * _ { N - 2 } , u ^ * _ { N - 1 } ] ^ * \\in \\mathbb { R } ^ { r N } \\\\ \\varepsilon & = [ v _ { 2 T } ^ * , \\dots , v ^ * _ N ] ^ * \\in \\mathbb { R } ^ { p \\bar { N } } \\\\ y & = [ x ^ * , \\varepsilon ^ * ] ^ * \\in \\mathbb { R } ^ { r N + p \\bar { N } } \\\\ U _ l & = [ u _ { 2 T - l } , u _ { 2 T + 1 - l } , \\dots , u _ { N - l } ] \\in \\mathcal { M } _ { p \\times \\bar { N } } ( \\mathbb { R } ) . \\end{align*}"}
+{"id": "4872.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ { n - 1 } } w _ { z _ n ; k _ n } ( t _ n ) \\ . w _ { z _ 2 ; k _ 2 } ( t _ 2 ) d \\L ^ { n - 1 } = 0 . \\end{align*}"}
+{"id": "1656.png", "formula": "\\begin{align*} e ( \\omega ) \\cap \\xi = \\int _ { T ( F _ \\infty ) _ 0 / \\Lambda } \\omega , \\omega \\in H ^ 0 ( \\Lambda , \\Omega _ M ^ u ) . \\end{align*}"}
+{"id": "3263.png", "formula": "\\begin{align*} \\tilde { P } & \\colon \\mathbb { Z } ^ n _ { \\geq 0 } \\times 2 ^ { \\binom { [ n ] } { 2 } } \\times [ n ] ^ 2 \\rightarrow \\mathbb { R } , \\\\ \\tilde { Y } & \\colon \\mathbb { Z } ^ n _ { \\geq 0 } \\times 2 ^ { \\binom { [ n ] } { 2 } } \\times [ n ] ^ 3 \\rightarrow \\mathbb { R } . \\end{align*}"}
+{"id": "3979.png", "formula": "\\begin{align*} D Y \\overline { u } = [ D X v _ \\rho ] ^ { - 1 } \\\\ D Y \\overline { u } = Y _ p [ D ^ 2 \\overline { u } - A ( \\cdot , \\overline { u } , D \\overline { u } ) ] \\\\ D X v _ \\rho = X _ q [ D ^ 2 v _ \\rho - A ^ * ( \\cdot , v _ \\rho , D v _ \\rho ) ] . \\end{align*}"}
+{"id": "276.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 6 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\tau } - \\beta \\tau ^ 2 \\leqslant 0 , \\end{align*}"}
+{"id": "8377.png", "formula": "\\begin{align*} A _ 0 = ( 2 \\pi ) ^ { \\frac { 1 } { 2 } ( p - q ) } \\kappa ^ { - \\frac { 1 } { 2 } - \\vartheta } \\prod _ { r = 1 } ^ p \\alpha _ r ^ { a _ r - \\frac { 1 } { 2 } } \\prod _ { r = 1 } ^ q \\beta _ r ^ { \\frac { 1 } { 2 } - b _ r } . \\end{align*}"}
+{"id": "6002.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( [ 1 : 0 ] ) = \\{ [ 1 : 0 ] \\} \\times g \\cdot X _ { i , j } ^ { k , l } & & \\mathrm { a n d } & & \\pi ^ { - 1 } ( [ 0 : 1 ] ) = \\{ [ 0 : 1 ] \\} \\times X ( i + k - n , j + p - 1 ) ; \\end{align*}"}
+{"id": "3578.png", "formula": "\\begin{align*} E ^ { \\gamma _ A } \\Omega _ { \\lambda _ A } = E _ { 3 1 } ^ 2 \\Omega _ { ( 4 , 2 , 0 ) } = 1 2 \\{ B _ 3 ^ + , B _ 1 ^ - \\} ^ 2 [ B _ 1 ^ + , B _ 2 ^ + ] ^ 2 ( B _ 1 ^ + ) ^ 2 v _ 0 \\end{align*}"}
+{"id": "4015.png", "formula": "\\begin{align*} - \\delta / 2 & \\geq W [ u ] ( \\overline { x } ) - W [ \\underline { u } ] ( \\overline { x } ) \\\\ & = [ A _ { \\alpha \\beta } ( x , \\phi , D ' \\phi , D _ n \\underline { u } ) - A _ { \\alpha \\beta } ( x , \\phi , D ' \\phi , D _ n u ) ] \\xi _ \\alpha \\xi _ \\beta . \\end{align*}"}
+{"id": "6997.png", "formula": "\\begin{align*} \\mathcal { R } ( \\Delta _ x , \\Delta _ y ) = \\{ & ( R _ { \\sc c } , R _ x , R _ y ) : R _ c \\ge I ( X , Y ; W ) , \\\\ & R _ x \\ge R _ { X | W } ( \\Delta _ x ) , R _ y \\ge R _ { Y | W } ( \\Delta _ y ) \\} , \\end{align*}"}
+{"id": "3399.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } \\Omega + v \\cdot \\nabla \\Omega = 0 , & \\\\ \\Omega _ { | t = 0 } = \\Omega _ 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "5681.png", "formula": "\\begin{align*} \\Omega _ { b } ^ { a } \\theta ^ { b } = 0 ; D \\Omega _ { b } ^ { a } = 0 . \\end{align*}"}
+{"id": "3293.png", "formula": "\\begin{align*} \\begin{aligned} & v ( t , x ) = \\alpha \\Theta ( t , x ) + w ( t , x ) , & & \\tilde { p } ( t , x ) = \\alpha ^ 2 \\Pi ( t , x ) + q ( t , x ) , \\\\ & \\ell _ v ( t , x ) = \\ell _ w ( t , x ) & & \\omega _ v ( t , x ) = \\alpha g ( t , 1 ) + \\omega _ w ( t , x ) . \\end{aligned} \\end{align*}"}
+{"id": "5973.png", "formula": "\\begin{align*} \\chi : K _ \\circ ( X ) \\rightarrow \\Z , & & [ \\mathcal { F } ] \\rightarrow \\chi ( \\mathcal { F } ) = \\sum _ { i } ( - 1 ) ^ i h ^ i ( \\mathcal { F } ) , \\end{align*}"}
+{"id": "6172.png", "formula": "\\begin{align*} C _ 1 A e _ 1 u = 0 \\hbox { i n } [ T , + \\infty ) \\times \\Omega \\end{align*}"}
+{"id": "7616.png", "formula": "\\begin{align*} I _ 1 : & = \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { 1 } \\chi _ { \\{ | u _ n | > M \\} } | h _ n ( u _ n ) - h ( u _ n ) | \\d x \\d t \\\\ & \\leq \\int _ { 0 } ^ { T } \\big ( | u _ n | + | u _ n | ^ { \\delta + 1 } + | u _ n | ^ { 2 \\delta + 1 } \\big ) \\d x \\d t < + \\infty , \\end{align*}"}
+{"id": "417.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty | g _ n ( x ) | ^ p = \\sum _ { n = 1 } ^ \\infty | f _ n ( V ^ { - 1 } x ) | ^ p = \\| V ^ { - 1 } x \\| ^ p = \\| x \\| ^ p , \\forall x \\in \\mathcal { X } , \\end{align*}"}
+{"id": "8437.png", "formula": "\\begin{align*} \\sup _ { \\nu \\in \\widehat { \\mathcal E } ^ + _ A } \\ , \\nu ( X ) \\leqslant \\sup _ { \\nu \\in \\widehat { \\mathcal E } ^ + _ A } \\ , G ( \\nu ) \\leqslant \\sup _ { \\nu \\in \\mathcal E ^ + _ A } \\ , G ( \\nu ) = c _ * ( A ) . \\end{align*}"}
+{"id": "9059.png", "formula": "\\begin{align*} w _ 2 ' ( x ) : = \\sum _ { n = 1 } ^ { n ' _ { \\max } } c _ n ' G ( x ' _ n , x ) \\end{align*}"}
+{"id": "1581.png", "formula": "\\begin{align*} \\Big ( \\big ( \\psi \\circ \\chi \\big ) _ \\# ( \\mu ) \\Big ) ( x ) & = \\mu \\Big ( \\big ( \\psi \\circ \\chi \\big ) ^ { - 1 } ( x ) \\Big ) = \\mu \\Big ( \\chi ^ { - 1 } \\big ( \\psi ^ { - 1 } ( x ) \\big ) \\Big ) \\\\ & = \\Big ( \\chi _ \\# ( \\mu ) \\Big ) \\big ( \\psi ^ { - 1 } ( x ) \\big ) = \\Big ( \\psi _ \\# \\big ( \\chi _ \\# ( \\mu ) \\big ) \\Big ) ( x ) . \\end{align*}"}
+{"id": "4363.png", "formula": "\\begin{align*} G ( \\theta ) = \\left \\lfloor \\frac { 1 } { \\theta } \\right \\rfloor + 1 . \\end{align*}"}
+{"id": "2899.png", "formula": "\\begin{align*} S ( K ) = \\inf _ { q \\in \\mathcal { P } _ 0 } \\int _ { \\mathbb { T } ^ \\infty } | h - q h | ^ p d m _ { \\infty } \\leq \\int _ { \\mathbb { T } ^ \\infty } | h - ( h - \\tilde { h } ( 0 ) ) | ^ p d m _ { \\infty } = | \\tilde { h } ( 0 ) | ^ p . \\end{align*}"}
+{"id": "5134.png", "formula": "\\begin{align*} 4 b ^ { 2 } \\Big ( \\Lambda _ { n } ( \\lambda , b ) - \\Lambda _ { n + 1 } ( \\lambda , b ) \\Big ) \\Big ( \\Lambda _ { n } ( \\lambda , b ) + \\Lambda _ { n + 1 } ( \\lambda , b ) \\Big ) \\underset { n \\rightarrow \\infty } { = } o _ { \\lambda , b } \\left ( \\frac { 1 } { n ^ { 2 } } \\right ) . \\end{align*}"}
+{"id": "4643.png", "formula": "\\begin{align*} g : = ( s h _ { \\rho _ T } , s h _ { \\rho _ G } ) : \\ , X \\dashrightarrow \\mathrm { S h } _ { \\rho _ T } ( X ) \\times \\mathrm { S h } _ { \\rho _ G } ( X ) \\end{align*}"}
+{"id": "3120.png", "formula": "\\begin{align*} A ( y ) : = \\mathrm { d i a g } ( 1 + a ( y ) , 1 - a ( y ) ) \\quad y \\in \\R ^ 2 . \\end{align*}"}
+{"id": "3955.png", "formula": "\\begin{align*} \\lambda & \\leq \\det D Y u \\leq \\Lambda D , \\\\ u & = g ( \\cdot , y _ 0 , z _ 0 ) \\partial D . \\end{align*}"}
+{"id": "3678.png", "formula": "\\begin{align*} \\sum _ { m = 3 } ^ { + \\infty } \\vert U _ m ( E , f ) \\vert \\leq \\frac { 2 } { \\sqrt { q } ( 1 - q ^ { - 1 / 2 } ) ^ 2 } + \\frac { ( 4 g + 2 ) ( 1 - q ^ { - 1 } ) ^ { - 1 } } { ( q - 1 ) ( 1 - q ^ { - 1 / 2 } ) } , \\end{align*}"}
+{"id": "7508.png", "formula": "\\begin{align*} S _ { k , r } ( j ) = { \\left \\{ \\begin{array} { r l } 0 , \\ \\ & 1 \\le j \\le k , \\\\ \\binom { k + r } { j } \\sum \\limits _ { i = 0 } ^ k ( - 1 ) ^ i \\binom { j } { i } , \\ \\ & k + 1 \\le j \\le k + r . \\end{array} \\right . } \\end{align*}"}
+{"id": "1460.png", "formula": "\\begin{align*} 2 \\sum _ { i = 1 } ^ m \\binom { x _ i } { 3 } = 2 \\sum _ { j = 1 } ^ m \\binom { y _ j } { 3 } = \\frac { k ( k - 1 ) ( k - 2 ) ( m - 2 ) } { 3 ( m + 1 ) ( m ^ 2 - 2 ) } ; \\end{align*}"}
+{"id": "2250.png", "formula": "\\begin{align*} \\lambda \\cdot ( 1 + \\varpi _ D ^ i a ) U ^ { i + 1 } _ D : = ( 1 + \\varpi _ D ^ i [ \\l ] a ) U ^ { i + 1 } _ D , \\end{align*}"}
+{"id": "2169.png", "formula": "\\begin{align*} \\dfrac { z F ' ( z ) } { F ( z ) } & = \\dfrac { ( 1 - 2 \\alpha - \\beta ) z ^ 2 + ( 2 - 2 \\alpha + \\beta ) z + 1 } { 1 - z ^ 2 } . \\end{align*}"}
+{"id": "761.png", "formula": "\\begin{align*} S _ 2 ( U _ 0 , \\psi , \\mathcal { M } ( \\mathsf { X } , A ) ) = \\varprojlim S _ 2 ( U _ n , \\psi , A ) , \\end{align*}"}
+{"id": "2542.png", "formula": "\\begin{align*} \\Gamma _ { \\varphi } = \\frac { \\varphi _ 0 ^ 3 + \\| \\nabla \\varphi \\| _ \\infty ^ 2 } { r ^ { 5 d } \\varphi _ 0 ^ 2 } \\ , , N _ \\varphi = \\frac { \\varphi _ 0 ^ 4 } { r ^ { 6 d } } + \\frac { \\| \\nabla \\varphi \\| _ \\infty ^ 8 } { ( r ^ { 5 d } \\varphi _ 0 ^ 2 ) ^ 4 } \\ , , \\end{align*}"}
+{"id": "953.png", "formula": "\\begin{align*} \\textrm { G } _ C \\textrm { - d i m } _ R ( M ^ C ) \\ , = \\ , \\textrm { G } _ C \\textrm { - d i m } _ R ( \\textrm { T r } _ C ( M ) ) - 2 \\ , = \\ , t - 2 . \\end{align*}"}
+{"id": "6417.png", "formula": "\\begin{align*} \\mathcal { K } _ 2 u ( X , Y , t ) = 0 \\mbox { i n } D . \\end{align*}"}
+{"id": "7493.png", "formula": "\\begin{align*} y ^ r z ^ l \\mathbb { H } _ r ^ k ( y , t z - y ) = & z ^ l \\sum _ { i = 0 } ^ { k } { \\dbinom { k + r } { i + r } y ^ { i + r } ( t z - y ) ^ { k - i } } \\\\ = & z ^ l \\sum _ { i = r } ^ { k + r } { \\dbinom { k + r } { i } y ^ i ( t z - y ) ^ { k + r - i } } . \\end{align*}"}
+{"id": "5348.png", "formula": "\\begin{align*} \\langle d \\rvert _ { \\varepsilon = \\tilde { \\varepsilon } } S _ \\varepsilon [ \\eta ] [ \\tilde { u } ] , \\tilde { v } \\rangle _ { \\tilde { \\varepsilon } } = \\tilde { \\lambda } ( \\tilde { \\lambda } + 1 ) ^ { - 2 } \\int _ \\Omega \\eta \\tilde { u } \\cdot \\tilde { v } \\ , d x \\end{align*}"}
+{"id": "6835.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) = \\mu _ c ^ { \\pm } ( R ^ { - 1 } Y _ 1 , R ^ { - 1 } Y _ 2 , \\dots , R ^ { - 1 } Y _ m ) , \\nu _ c ^ { \\pm } ( Y _ 1 , Y _ 2 , \\dots , Y _ m ) = \\nu _ c ^ { \\pm } ( R ^ { - 1 } Y _ 1 , R ^ { - 1 } Y _ 2 , \\dots , R ^ { - 1 } Y _ m ) . \\end{aligned} \\end{align*}"}
+{"id": "5684.png", "formula": "\\begin{align*} \\theta ^ { a } = ( L ^ { - 1 } ) _ { b } ^ { a } d x ^ { b } , \\end{align*}"}
+{"id": "5718.png", "formula": "\\begin{align*} ( \\mathbf { L } _ { 1 } ^ { - 1 } ) _ { b } ^ { a } & = \\delta _ { b } ^ { a } + \\omega _ { 1 b } ^ { a } \\mathbf { m } , \\mathbf { L } _ { 1 b } ^ { a } = \\delta _ { b } ^ { a } - \\omega _ { 1 b } ^ { a } \\mathbf { m } , \\\\ \\mathbf { x } _ { 1 } ^ { a } & = ( _ { c } \\mathbf { L } ^ { - 1 } ) _ { b } ^ { a } \\mathbf { x } ^ { b } . \\end{align*}"}
+{"id": "8429.png", "formula": "\\begin{align*} \\mu ^ A ( X ) = \\min _ { \\nu \\in \\Gamma _ { A , \\mu } ^ + } \\ , \\nu ( X ) . \\end{align*}"}
+{"id": "7323.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\infty } \\int _ { 0 } ^ { 1 } G ^ { d } _ { m _ \\gamma } ( x ) d m _ \\gamma ( x ) = 0 \\end{align*}"}
+{"id": "1208.png", "formula": "\\begin{align*} \\mathfrak { L } ^ { k , t } = \\sum _ { i = 1 } ^ { k + 1 } \\sum _ { \\alpha \\in I _ { i , k } } \\biggl \\{ \\nabla ^ i X | _ { \\Phi ^ t } \\circ \\nabla ^ { \\alpha } \\Phi ^ t _ * \\circ \\mathfrak { S } _ { \\alpha } + \\sum _ { j = 0 } ^ { i - 2 } ( ( \\nabla ^ j _ \\bullet R ) ( \\nabla ^ { i - 2 - j } _ \\bullet X , \\bullet ) \\bullet ) | _ { \\Phi ^ t } \\circ \\nabla ^ { \\alpha } \\Phi ^ t _ * \\circ \\mathfrak { S } _ { \\alpha , j } \\biggr \\} . \\end{align*}"}
+{"id": "4151.png", "formula": "\\begin{align*} \\int _ M \\langle \\triangle _ G h , h \\rangle d V _ g & = \\int _ M \\langle \\triangle h , h \\rangle + 3 R _ { i j k l } h _ { i l } h _ { j k } + R _ { j l } h _ { l k } h _ { j k } d V _ g \\leq - \\| h \\| ^ 2 _ { L ^ 2 } \\leq 0 . \\end{align*}"}
+{"id": "2576.png", "formula": "\\begin{align*} T _ { a } N & = d l _ { a } ( \\ \\ \\mathfrak { g } _ 0 + \\sum _ { \\substack { \\alpha \\in \\Delta ^ + \\\\ \\langle \\alpha , w \\rangle \\notin 2 \\pi \\mathbb { Z } } } \\mathfrak { g } _ { \\alpha } \\ \\ ) , \\\\ T ^ \\perp _ { a } N & = d l _ { a } ( \\ \\mathfrak { t } + \\sum _ { \\substack { \\alpha \\in \\Delta ^ + \\\\ \\langle \\alpha , w \\rangle \\in 2 \\pi \\mathbb { Z } } } \\mathfrak { g } _ { \\alpha } \\ \\ ) . \\end{align*}"}
+{"id": "6703.png", "formula": "\\begin{align*} \\mu ( n ) ^ 2 = \\sum _ { d ^ 2 \\mid n } \\mu ( d ) . \\end{align*}"}
+{"id": "8301.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\frac { \\partial V ( \\textbf { x } ) } { \\partial x _ E } \\ ! \\ ! \\ ! \\ ! & = \\frac { \\nu _ E x _ E - \\frac { 1 } { 2 } [ ( y _ 2 - y _ 3 ) R _ 1 ^ 2 + ( y _ 3 - y _ 1 ) R _ 2 ^ 2 + ( y _ 1 - y _ 2 ) R _ 3 ^ 2 ] } { \\Lambda } \\\\ & ~ ~ - \\mu _ E \\frac { \\nu _ E R _ E ^ 2 - \\nu _ 1 R _ 1 ^ 2 - \\nu _ 2 R _ 2 ^ 2 - \\nu _ 3 R _ 3 ^ 2 } { 2 \\Lambda ^ 2 } . \\end{array} \\right . \\end{align*}"}
+{"id": "2177.png", "formula": "\\begin{align*} ( 1 - 2 \\alpha - \\beta ) r ^ 2 - ( 2 - 2 \\alpha + \\beta ) r + 1 & = 2 ( \\sqrt { 2 } - 1 ) ( 1 - r ^ 2 ) , \\intertext { a n d } ( 1 - 2 \\alpha - \\beta ) r ^ 2 + ( 2 - 2 \\alpha + \\beta ) r + 1 & = 2 ( 1 - r ^ 2 ) . \\end{align*}"}
+{"id": "4865.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 f ( s _ j ) g ( s _ j ) d s _ j & = h ( 1 ) g ( 1 ) - \\int _ 0 ^ 1 h ( s _ j ) g ' ( s _ j ) d s _ j , \\end{align*}"}
+{"id": "6505.png", "formula": "\\begin{align*} ( \\check { V } \\check { V } ^ \\top ) & = \\underset { i = 1 } { \\overset { d } { \\sum } } \\underset { k = \\lfloor d / 2 \\rfloor + 1 } { \\overset { d } { \\sum } } ( v _ k ) _ i ^ 2 = \\lceil d / 2 \\rceil . \\end{align*}"}
+{"id": "4379.png", "formula": "\\begin{align*} a _ { k + 1 } = G \\left ( \\frac { p } { q } - \\sum _ { i = 1 } ^ k \\frac { 1 } { a _ i } \\right ) = q \\prod _ { i = 1 } ^ k a _ i + 1 \\end{align*}"}
+{"id": "2812.png", "formula": "\\begin{align*} \\min ( \\lambda u _ n , \\lambda v _ n ) = \\lambda \\min ( v _ n , u _ n ) . \\end{align*}"}
+{"id": "4980.png", "formula": "\\begin{align*} V _ M ( x ) : = \\max \\{ V ( x ) , - M \\} \\ , . \\end{align*}"}
+{"id": "4067.png", "formula": "\\begin{align*} 1 & < \\frac { 1 } { t } \\int _ 0 ^ t \\exp \\Big ( - \\frac { \\partial u } { \\partial t } ( x , \\tau ) \\Big ) \\ d \\tau \\\\ & = \\frac { 1 } { t } \\int _ 0 ^ t \\exp \\Big ( - g _ z \\frac { \\partial v } { \\partial t } ( y , \\tau ) \\Big ) \\ d \\tau , \\end{align*}"}
+{"id": "2982.png", "formula": "\\begin{align*} E _ { \\nu ^ n _ \\rho } [ W _ { j + i - 1 } \\big ( \\sigma _ { j + i - 1 , j + i } ( \\overrightarrow { W } _ { j + 1 } ^ \\ell ) ^ 2 \\big ) ^ 2 ] = O ( \\ell ^ { - 2 } ) , \\end{align*}"}
+{"id": "1176.png", "formula": "\\begin{align*} S _ { i i } = \\frac { 1 } { n } \\sum _ { j = 1 } ^ p A _ { i j } ^ 2 \\sum _ { t = 1 } ^ n Z _ { j t } ^ 2 + \\frac { 1 } { n } \\sum _ { t = 1 } ^ n \\sum _ { j _ 1 \\neq j _ 2 = 1 } ^ p A _ { i j _ 1 } A _ { i j _ 2 } Z _ { j _ 1 t } Z _ { j _ 2 t } = : S _ i ( 1 ) + S _ i ( 2 ) \\ , . \\end{align*}"}
+{"id": "4012.png", "formula": "\\begin{align*} L x _ n & = - w ^ { i j } A _ { i j , p _ n } - B _ { p _ n } \\\\ L x _ n ^ 2 & = 2 w ^ { n n } - 2 w ^ { i j } A _ { i j , p _ n } x _ n - 2 B _ { p _ n } x _ n \\end{align*}"}
+{"id": "1322.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) f _ k ( \\tau _ j ) | \\geq \\frac { n - d } { d ( n - 1 ) } = \\frac { n - ( \\mathcal { X } ) } { ( \\mathcal { X } ) ( n - 1 ) } \\end{align*}"}
+{"id": "6979.png", "formula": "\\begin{align*} 2 H _ { \\lambda } ' ( x ) \\big ( x H _ { | \\lambda | , n } ^ { ( \\lambda ) } ( x ) - H _ { | \\lambda | , n } ^ { ( \\lambda ) } ( x ) ' \\big ) + H _ { \\lambda } '' ( x ) H _ { | \\lambda | , n } ^ { ( \\lambda ) } ( x ) = Q _ { n , \\lambda } ( x ) H _ { \\lambda } ( x ) . \\end{align*}"}
+{"id": "7948.png", "formula": "\\begin{align*} \\pi ( m _ 1 ) + \\pi ( c _ 1 ) = \\pi ( m _ 2 ) + \\pi ( c _ 2 ) . \\end{align*}"}
+{"id": "6191.png", "formula": "\\begin{align*} C _ p ^ T ( C _ p C _ p ^ T ) ^ { - 1 } C _ p x = x , \\forall x \\in \\hbox { I m } ( C _ p ^ T ) . \\end{align*}"}
+{"id": "6707.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\mu ( n ) = O \\left ( \\frac { x } { \\log ^ { C } x } \\right ) . \\end{align*}"}
+{"id": "4875.png", "formula": "\\begin{align*} \\frac { 1 } { k _ n } = q ( k _ n ) > q ( k _ { n - 1 } ) = \\frac { 1 } { k _ { n - 1 } } , \\end{align*}"}
+{"id": "2442.png", "formula": "\\begin{align*} f _ 1 ( x ) & = 1 + x ^ 6 , & f _ 2 ( x ) & = 1 + 2 x ^ 6 , & f _ 3 ( x ) & = 2 , \\\\ f _ 4 ( x ) & = 2 + x ^ 6 , & f _ 5 ( x ) & = 2 + 2 x ^ 6 . \\end{align*}"}
+{"id": "4249.png", "formula": "\\begin{align*} v ( t , x ) : = u ( t , x _ 1 \\cos ( \\Omega t ) + x _ 2 \\sin ( \\Omega t ) , - x _ 1 \\sin ( \\Omega t ) + x _ 2 \\cos ( \\Omega t ) , x _ 3 , \\cdots , x _ N ) , \\end{align*}"}
+{"id": "333.png", "formula": "\\begin{align*} \\phi ( \\xi , w ) = \\int _ 0 ^ \\infty \\widetilde \\Phi _ { t } ( \\frac { \\xi } { t } ) t ^ { w - 1 } \\dd t \\end{align*}"}
+{"id": "1720.png", "formula": "\\begin{align*} \\langle \\Upsilon \\rangle ' = \\sum _ { j = 0 } ^ { k - 2 } \\binom { k - 2 } { j } ( - 1 ) ^ j \\langle \\mu _ { \\frac { k - 2 } { 2 } - j } , \\mu _ { j - \\frac { k - 2 } { 2 } } \\rangle ` = \\sum _ { j = 0 } ^ { k - 2 } 1 = ( k - 1 ) \\end{align*}"}
+{"id": "2725.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } d _ { J } ( \\vec { u } ( t _ n ) - \\vec { v } _ L ( t _ n ) ) = 0 . \\end{align*}"}
+{"id": "8066.png", "formula": "\\begin{align*} H _ { m , n } = \\frac { 2 } { \\pi } \\int _ 0 ^ \\infty k ( t ) V ( m ^ 2 n , t ) \\tanh ( \\pi t ) t \\ , d t , \\end{align*}"}
+{"id": "743.png", "formula": "\\begin{align*} \\{ \\frac { \\partial ^ 2 H ( z , w ) } { \\partial z \\partial w } \\} _ { z = w } = \\frac { 1 } { 2 } \\frac { \\partial h _ 1 ( w ) } { \\partial w } - h _ { 2 } ( w ) , \\end{align*}"}
+{"id": "7924.png", "formula": "\\begin{align*} C : = - K a ^ { - 1 } \\mathrm { o s c } _ { \\Omega } u + a ^ { - 1 } L \\ , > \\ , 0 \\ , . \\end{align*}"}
+{"id": "3019.png", "formula": "\\begin{align*} \\pi _ I = - \\pi { _ I } ^ { a k } \\theta ^ { ( 3 ) } _ a \\wedge \\bar { \\theta } ^ { ( N ) } _ k + \\frac { 1 } { 2 } \\pi { _ I } ^ { j k } \\theta ^ { ( 4 ) } \\wedge \\bar { \\theta } ^ { ( N - 1 ) } _ { j k } \\end{align*}"}
+{"id": "2419.png", "formula": "\\begin{align*} E _ { 3 3 } ^ T = E _ { 3 3 } , E _ { 4 4 } ^ T = E _ { 4 4 } , 0 = - A _ { 1 3 } - \\dot E _ { 1 3 } , 0 = - A _ { 1 4 } - \\dot E _ { 1 4 } , 0 = A _ { 2 3 } , 0 = A _ { 2 4 } , \\end{align*}"}
+{"id": "5063.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n Q _ j ( 0 ) f _ j ( \\omega ) = 0 \\end{align*}"}
+{"id": "2657.png", "formula": "\\begin{align*} u ( y _ 1 , \\dots , y _ N , 0 , 0 , \\dots ) : = u _ N ( y _ 1 , \\dots , y _ N ) . \\end{align*}"}
+{"id": "2880.png", "formula": "\\begin{align*} \\mu ^ { j - 1 } a _ { j - 1 } = \\ldots = \\mu a _ 1 = a _ 0 , \\end{align*}"}
+{"id": "1921.png", "formula": "\\begin{align*} m _ \\mathrm { A D M } ( \\mathcal E , \\tilde g ) = m _ \\mathrm { A D M } ( \\mathcal E , g ) - \\frac { 1 } { 2 ( n - 1 ) \\omega _ { n - 1 } } \\int _ M V u \\ , d \\mu _ g . \\end{align*}"}
+{"id": "1354.png", "formula": "\\begin{align*} n ^ 2 \\max _ { 1 \\leq j , k \\leq n } | f _ j ( \\tau _ k ) | ^ 2 \\geq P F P ( \\{ f _ j \\} _ { j = 1 } ^ n , \\{ \\tau _ j \\} _ { j = 1 } ^ n ) \\geq \\frac { n ^ 2 } { { d } } . \\end{align*}"}
+{"id": "3480.png", "formula": "\\begin{align*} { \\bf { x } } \\left [ n \\right ] = \\sum \\nolimits _ { l = 1 } ^ L { { { \\bf { f } } _ l } s \\left [ { n - { \\kappa _ l } } \\right ] } , \\end{align*}"}
+{"id": "3889.png", "formula": "\\begin{align*} \\det D Y u = \\frac { f ( \\cdot ) } { f ^ * ( Y u ( \\cdot ) ) } , \\\\ Y u ( \\Omega ) = \\Omega ^ * , \\end{align*}"}
+{"id": "2679.png", "formula": "\\begin{align*} \\lim \\limits _ { y \\to \\infty } { y ^ { \\alpha + 1 } g ( 1 - 1 / y ) \\over \\int \\limits _ a ^ y s ^ \\alpha g ( 1 - 1 / s ) \\ , d s } & = \\lim \\limits _ { y \\to \\infty } { ( \\alpha + 1 ) y ^ \\alpha g ( 1 - 1 / y ) + y ^ { \\alpha - 1 } g ' ( 1 - 1 / y ) \\over y ^ \\alpha g ( 1 - 1 / y ) } \\\\ & = \\alpha + 1 + \\lim \\limits _ { y \\to \\infty } { y ^ { \\alpha - 1 } g ' ( 1 - 1 / y ) \\over y ^ \\alpha g ( 1 - 1 / y ) } \\\\ & = \\alpha + 1 + \\lim \\limits _ { x \\rightarrow 1 - } { ( 1 - x ) g ' ( x ) \\over g ( x ) } ~ = \\alpha + 1 + L . \\end{align*}"}
+{"id": "8959.png", "formula": "\\begin{align*} d \\pi _ N ( u ) \\big ( u _ t + ( - \\Delta ) ^ { 1 / 2 } u \\big ) = 0 \\ \\hbox { o n } S ^ 1 \\times [ 0 , \\infty [ , \\end{align*}"}
+{"id": "7867.png", "formula": "\\begin{align*} f '' + a _ 0 ( z ) f = P _ 1 ( z ) e ^ { P _ 0 ( z ) } \\end{align*}"}
+{"id": "1380.png", "formula": "\\begin{align*} \\beta _ m ^ R ( R x _ 1 ) = ( n - 1 ) ^ m \\mbox { f o r a l l } m \\ge 1 . \\end{align*}"}
+{"id": "7842.png", "formula": "\\begin{align*} m ( r , 1 / f ) = O ( \\log r ) . \\end{align*}"}
+{"id": "9017.png", "formula": "\\begin{align*} \\operatorname { p } ^ { ( O W ) } _ { f _ K } ( \\phi ) = \\sup _ { \\eta \\in \\mathbb { U } _ X } \\operatorname { p } ^ { ( O W ) } _ { f _ K } [ \\phi , \\eta _ k ] \\leq \\sup _ { \\eta \\in \\mathbb { U } _ X } \\operatorname { p } ^ { ( O W ) } _ { f _ K } [ \\phi , \\eta _ K ] \\leq \\sup _ { \\epsilon \\in \\mathbb { U } _ X } \\operatorname { p } ^ { ( O W ) } _ { f _ K } [ \\phi , \\epsilon ] = \\operatorname { p } ^ { ( O W ) } _ { f _ K } ( \\phi ) \\end{align*}"}
+{"id": "647.png", "formula": "\\begin{align*} | B _ { c } | \\geq \\delta ^ { N ^ { 2 } \\tau _ { n + 1 } } | \\bar { B } _ { c } | & \\stackrel { \\eqref { d e f T a u } } { \\geq } \\delta ^ { N ^ { 2 } N ^ { 4 N - 3 ( n + 1 ) } \\epsilon + \\epsilon } | B _ { n } | \\\\ & \\ , \\ , \\ , \\geq \\delta ^ { ( N ^ { 4 N - 3 n - 1 } + 1 ) \\epsilon } | B _ { n } | \\\\ & \\ , \\ , \\ , \\geq \\delta ^ { N ^ { 4 N - 3 n } \\epsilon } | B _ { n } | = \\delta ^ { \\tau _ { n } } | B _ { n } | . \\end{align*}"}
+{"id": "4512.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k \\lambda v _ i = \\lambda ^ k \\prod _ { i = 1 } ^ k v _ i \\leq \\lambda ^ k \\prod _ { i = 1 } ^ k \\lambda u _ i = \\prod _ { i = 1 } ^ k \\lambda u _ i \\end{align*}"}
+{"id": "6952.png", "formula": "\\begin{gather*} L _ { n - m } ^ { ( \\alpha - 1 ) } ( x ) \\big ( x L _ { m - 1 } ^ { ( \\alpha + 1 ) } ( - x ) + \\alpha L _ m ^ { ( \\alpha ) } ( - x ) \\big ) + ( n - m ) L _ m ^ { ( \\alpha - 1 ) } ( - x ) L _ { n - m } ^ { ( \\alpha - 1 ) } ( x ) \\\\ { } = L _ { n - m } ^ { ( \\alpha - 1 ) } ( x ) \\big ( x L _ { m - 1 } ^ { ( \\alpha + 1 ) } ( - x ) + \\alpha L _ m ^ { ( \\alpha ) } ( - x ) + ( n - m ) L _ m ^ { ( \\alpha - 1 ) } ( - x ) \\big ) . \\end{gather*}"}
+{"id": "2068.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } \\rho _ 1 & = \\sigma , & \\rho _ 2 & = \\frac { 1 + \\alpha } { 1 - \\alpha } \\sigma & & \\ \\alpha \\ge 0 , \\\\ \\rho _ 1 & = \\frac { 1 - \\alpha } { 1 + \\alpha } \\sigma , & \\rho _ 2 & = \\sigma & & \\ \\alpha < 0 , \\end{alignedat} \\end{align*}"}
+{"id": "8096.png", "formula": "\\begin{align*} H _ { m , n } ^ + ( x ) = \\frac { 4 } { \\pi } \\int _ { t = 0 } ^ \\infty \\tanh ( \\pi t ) \\biggl ( \\int _ { \\zeta = - \\infty } ^ \\infty \\cos ( x \\cosh \\zeta ) e \\Bigl ( \\frac { t \\zeta } { \\pi } \\Bigr ) \\ , d \\zeta \\biggr ) k ( t ) V ( m ^ 2 n , t ) t \\ , d t . \\end{align*}"}
+{"id": "617.png", "formula": "\\begin{align*} \\tilde C = C + \\mathcal { O } ( \\delta ) . \\end{align*}"}
+{"id": "3085.png", "formula": "\\begin{align*} - A : D ^ 2 v ^ { k l } & = a \\left ( - B : D ^ 2 v ^ { k l } _ B \\right ) + \\bar { b } _ { k l } \\left ( - A : D ^ 2 w \\right ) \\\\ & = a ( b _ { k l } - \\bar { b } _ { k l } ) + \\bar { b } _ { k l } ( a - \\bar { a } ) \\\\ & = a _ { k l } - \\bar { a } _ { k l } \\end{align*}"}
+{"id": "3041.png", "formula": "\\begin{align*} F ( D ^ 2 u , x ) = f ( x ) \\ \\ \\mathrm { i n } \\ \\ \\Omega _ 1 \\ \\ \\mathrm { w i t h } \\ \\ f \\in L ^ { p } ( \\Omega _ 1 ) . \\end{align*}"}
+{"id": "3139.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\int _ 0 ^ 1 a ( y _ 1 , y _ 2 ) R _ 1 ' ( y _ 1 + y _ 2 ) \\ , \\mathrm { d } y _ 1 \\ , \\mathrm { d } y _ 2 = \\int _ 0 ^ 1 \\int _ 0 ^ 1 a ( y _ 1 , y _ 2 ) R _ 2 ' ( y _ 1 - y _ 2 ) \\ , \\mathrm { d } y _ 1 \\ , \\mathrm { d } y _ 2 = 0 , \\end{align*}"}
+{"id": "6374.png", "formula": "\\begin{align*} x _ \\lambda ' ( u _ 1 ) ^ 2 - x _ \\lambda ' ( u _ 2 ) ^ 2 = 4 / 3 \\cdot \\lambda ( 1 + \\lambda ) \\cdot \\sin ^ 2 u _ 2 \\cdot \\cos ^ 3 u _ 2 . \\end{align*}"}
+{"id": "7522.png", "formula": "\\begin{align*} \\dbinom { k + r } { k + 1 } = \\dbinom { k + r } { r - 1 } = \\dfrac { r } { k + 1 } \\dbinom { k + r } { r } . \\end{align*}"}
+{"id": "4518.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n v _ i = \\prod _ { i = 1 } ^ n u _ i \\end{align*}"}
+{"id": "2439.png", "formula": "\\begin{align*} R ( \\alpha ) = \\psi ( \\alpha ) \\sum _ { \\substack { ( i _ 1 , i _ 2 , \\ldots , i _ { M } ) \\in [ 0 , h - 1 ] ^ { M } \\\\ ( i _ 1 , i _ 2 , \\dots , i _ M ) \\neq ( 0 , 0 , \\ldots , 0 ) } } \\chi _ h ^ { i _ 1 } ( \\alpha ^ { - \\ell _ 1 } f _ 1 ( \\alpha ) ) \\cdots \\chi _ h ^ { i _ M } ( \\alpha ^ { - \\ell _ M } f _ M ( \\alpha ) ) . \\end{align*}"}
+{"id": "1758.png", "formula": "\\begin{align*} \\langle \\delta s ( \\Upsilon ) \\rangle = \\frac { 2 } { 3 } \\binom { 2 M } { k _ { \\rm i d } - 2 } ^ 2 \\binom { 2 M + 2 } { k _ { \\rm i d } - 1 } ^ { - 1 } ( 2 M + 2 ) ^ { 2 } ( 2 M + 1 ) ^ { 2 } . \\end{align*}"}
+{"id": "2714.png", "formula": "\\begin{align*} \\beta _ j ( t ) = - \\int ( \\Lambda W ) _ { [ \\lambda _ j ( t ) ] } \\partial _ t U ( t ) \\ , d x . \\end{align*}"}
+{"id": "5403.png", "formula": "\\begin{align*} ( \\beta _ { 2 ^ n } , \\beta _ { 2 ^ n + 1 } , \\ldots , \\beta _ { 2 ^ { n + 1 } - 1 } ) = ( \\overline { \\beta _ 0 } , \\overline { \\beta _ 1 } , \\ldots , \\overline { \\beta _ { 2 ^ n - 1 } } ) . \\end{align*}"}
+{"id": "8043.png", "formula": "\\begin{align*} W _ s ( z ) = 2 \\abs { y } ^ { \\frac { 1 } { 2 } } K _ { s - \\frac { 1 } { 2 } } ( 2 \\pi \\abs { y } ) e ( x ) , \\end{align*}"}
+{"id": "748.png", "formula": "\\begin{align*} ( \\alpha ( a ) ) _ p = \\alpha _ p ( a ) = \\alpha ( a _ p ) . \\end{align*}"}
+{"id": "2500.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\varepsilon _ k A _ k ^ 2 < \\infty . \\end{align*}"}
+{"id": "3953.png", "formula": "\\begin{align*} | \\hat { y } | = | g _ x ( 0 , \\hat { y } , 0 ) | \\leq | g _ x ( 0 , \\hat { y } , h ) | + | g _ { x z } ( 0 , \\hat { y } , \\tau h ) | h . \\end{align*}"}
+{"id": "4629.png", "formula": "\\begin{align*} h ( g \\cdot x ) = h ( x ) + C \\end{align*}"}
+{"id": "3735.png", "formula": "\\begin{align*} I ( f _ { v _ 1 } ) ( 0 ) = I ( f _ { v _ 1 ' } ) ( 0 ) = I ( \\mathcal { F } _ { X _ \\ell } ( f _ { v _ 2 } ) ) ( 0 ) = I ( \\mathcal { F } _ { X _ \\ell } ( f _ { v _ 2 ' } ) ) ( 0 ) = 0 \\end{align*}"}
+{"id": "8559.png", "formula": "\\begin{align*} v \\alpha _ { s } ( w ) = \\sum _ { w \\in [ v ] } \\alpha _ { s } ( w ) = \\sum _ { \\pi \\in X } \\sum _ { i = 0 } ^ { v - 1 } i ^ { s } = t ! \\lambda \\sum _ { i = 0 } ^ { v - 1 } i ^ { s } \\end{align*}"}
+{"id": "4577.png", "formula": "\\begin{align*} 1 = \\frac { | G | } { | H ( x _ 1 ) | } + \\cdots + \\frac { | G | } { | H ( x _ k ) | } . \\end{align*}"}
+{"id": "1621.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\lim _ { n \\to \\infty } ( \\hat S ^ { ( t ) } ) \\leq \\frac { 1 } { C } \\sum _ { c = 1 } ^ C \\pi ^ * ( \\tau ^ { ( \\infty ) } _ c , M ) \\end{align*}"}
+{"id": "2778.png", "formula": "\\begin{align*} & ( u _ 1 , u _ 2 , \\cdots \\ ! , u _ n ) \\\\ & = ( 0 , 0 , 1 , 1 , 1 , 2 , 2 , 2 , 1 , 1 , 0 , 0 , 1 , 1 , 0 , 0 ) . \\end{align*}"}
+{"id": "1483.png", "formula": "\\begin{align*} \\mathcal S _ 2 ( x ) = e ^ x \\prod _ { n = 1 } ^ \\infty \\left \\{ \\left ( \\frac { 1 - \\frac x n } { 1 + \\frac x n } \\right ) ^ n e ^ { 2 x } \\right \\} . \\end{align*}"}
+{"id": "7025.png", "formula": "\\begin{align*} d \\cap x ^ { m + n \\sigma } = 0 . \\end{align*}"}
+{"id": "6761.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\lim _ { t \\rightarrow \\infty } \\frac { \\left | z _ N ( - \\sigma + j \\omega , t ) \\right | } { \\left | z _ N ( \\sigma - j \\omega , t ) \\right | } = \\frac { \\left | \\left ( \\frac { 1 } { 2 } - \\sigma \\right ) + j \\omega \\right | } { \\left | \\left ( \\frac { 1 } { 2 } + \\sigma \\right ) - j \\omega \\right | } \\end{align*}"}
+{"id": "4193.png", "formula": "\\begin{align*} \\begin{cases*} \\partial _ { \\alpha } ( m ^ { \\alpha \\beta } \\partial _ { \\beta } \\Psi ) = F ( \\Psi , \\partial \\Psi ) \\\\ ( \\Psi , \\partial _ t \\Psi ) | _ { \\{ t = 0 \\} } = ( \\Psi _ 0 , \\Psi _ 1 ) \\in \\mathcal { H } . \\end{cases*} \\end{align*}"}
+{"id": "2927.png", "formula": "\\begin{align*} \\mathcal { X } ^ { n } _ t ( \\varphi ) = \\frac { 1 } { \\sqrt { n } } \\sum _ { j \\in \\mathbb { Z } } ( \\eta ^ { n } _ j ( t ) - \\rho ) \\varphi \\bigg ( \\frac { j - f _ n t } { n } \\bigg ) \\end{align*}"}
+{"id": "71.png", "formula": "\\begin{align*} f _ 1 ( \\mathbf { z } ) = | \\{ f ( z _ i ) = - 1 \\land ( ( \\nexists j ) \\ ; j \\in N ( i ) \\land f ( z _ j ) = 2 ) \\ ; | \\ ; i \\in \\{ 1 , . . . , n \\} \\} | \\end{align*}"}
+{"id": "1527.png", "formula": "\\begin{align*} \\operatorname { d i v } ^ { L C } T _ { \\mu \\nu } = - ( \\star \\d A ) _ { \\mu \\nu } \\end{align*}"}
+{"id": "7350.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\infty } \\frac { K ( \\gamma ) ^ { d - N } \\int _ { 1 } ^ { \\gamma } \\frac { K ( x ) ^ { \\alpha } } { x } d x } { u ( \\gamma ) ^ { d + 1 } } = 0 . \\end{align*}"}
+{"id": "6925.png", "formula": "\\begin{align*} \\tilde { u } ^ p ( t ) = \\left \\{ \\begin{array} { l l } \\tilde { u } ( t ) & \\ \\ 0 < \\tilde { u } ( t ) < 1 , \\\\ 0 & \\ \\ \\tilde { u } ( t ) \\leq 0 , \\\\ 1 & \\ \\ \\tilde { u } ( t ) \\geq 1 , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "414.png", "formula": "\\begin{align*} \\| V x \\| = \\left \\| \\sum _ { n = 1 } ^ { \\infty } f _ n ( x ) \\omega _ n \\right \\| = \\left ( \\sum _ { n = 1 } ^ \\infty | f _ n ( x ) | ^ p \\right ) ^ \\frac { 1 } { p } = \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "7367.png", "formula": "\\begin{align*} \\| f \\| _ { [ B _ 0 , B _ 1 ] _ { \\theta , \\infty } } \\coloneqq \\sup _ { 0 < t < \\infty } t ^ { - \\theta } \\inf _ { f = f _ 0 + f _ 1 } ( \\| f _ 0 \\| _ { B _ 0 } + t \\| f _ 1 \\| _ { B _ 1 } ) \\end{align*}"}
+{"id": "745.png", "formula": "\\begin{align*} \\tilde { h } _ 1 ( \\tilde { w } ) d \\tilde { w } = h _ 1 ( w ) d w + \\frac { 1 } { 2 } \\{ \\tilde { w } , w \\} _ 1 d w , \\end{align*}"}
+{"id": "9126.png", "formula": "\\begin{align*} f ( x ) = x ^ { e _ { 0 } } \\Phi _ { d _ { 1 } } ( x ) ^ { e _ { 1 } } \\Phi _ { d _ { 2 } } ( x ) ^ { e _ { 2 } } \\cdots \\Phi _ { d _ { m } } ( x ) ^ { e _ { m } } \\end{align*}"}
+{"id": "1119.png", "formula": "\\begin{align*} \\mathbb { E } \\{ Z _ j ^ 2 \\} = \\left ( \\frac { e ^ { \\alpha } + 1 } { e ^ { \\alpha } - 1 } \\right ) ^ 2 , j \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
+{"id": "4532.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { x _ 2 } \\leq \\frac { 1 } { 3 } + \\frac { 1 } { 3 } = \\frac { 2 } { 3 } < \\frac { 1 } { 2 } + \\frac { 1 } { x _ 2 - 1 } \\end{align*}"}
+{"id": "8026.png", "formula": "\\begin{align*} \\mathcal { T } : = \\{ \\alpha _ 1 , \\ldots , \\alpha _ s , \\beta _ 1 , \\ldots , \\beta _ { t } , \\gamma _ 1 , \\ldots , \\gamma _ u , \\delta _ 1 , \\ldots , \\delta _ v \\} \\subseteq \\overline { K } , \\end{align*}"}
+{"id": "6200.png", "formula": "\\begin{align*} \\hbox { r a n k } ( C _ p \\mathcal R ) = N - p . \\end{align*}"}
+{"id": "4591.png", "formula": "\\begin{align*} \\prod _ { i = m + 1 } ^ { m + k } a _ i < \\prod _ { i = m + 1 } ^ { m + k } b _ i \\end{align*}"}
+{"id": "2706.png", "formula": "\\begin{align*} \\| f \\| _ { Z _ { \\alpha } } = \\sup _ { R > 0 } \\frac { R ^ { - 3 - \\alpha } } { \\langle \\log R \\rangle } \\| f \\| _ { L ^ 2 ( R \\leq r \\leq 2 R ) } , \\end{align*}"}
+{"id": "3379.png", "formula": "\\begin{align*} t ^ { - u _ { i + 1 } } \\Big ( \\sum _ { j \\leq i } x _ j \\Big ) + t ^ { - u _ { i + 2 } + 1 } x _ { i + 1 } = \\sum _ { j = i + 2 } ^ { n - 1 } t ^ { - u _ j } x _ j + x _ n + t ^ { - u _ n } \\ , . \\end{align*}"}
+{"id": "8291.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { E } = \\cap _ i \\ \\mathcal { E } _ i . \\end{array} \\right . \\end{align*}"}
+{"id": "1508.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\C _ r \\left ( \\frac x { 2 \\pi } \\right ) & = \\left ( \\frac x { 2 \\pi } \\right ) ^ { r - 1 } \\left ( \\log \\left ( \\cos \\frac x 2 \\right ) + ( r - 1 ) \\sum _ { m = 1 } ^ \\infty \\frac { ( 2 ^ { 2 m } - 1 ) \\zeta ( 2 m ) } { m ( 2 m + r - 1 ) } \\left ( \\frac x { 2 \\pi } \\right ) ^ { 2 m } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "8434.png", "formula": "\\begin{align*} \\sup _ { \\nu \\in \\mathcal E ^ + _ A } \\ , G ( \\nu ) = \\Bigl [ \\inf _ { \\mu \\in \\breve { \\mathcal E } ^ + _ A } \\ , \\kappa ( \\mu , \\mu ) \\Bigr ] ^ { - 1 } = c _ * ( A ) . \\end{align*}"}
+{"id": "760.png", "formula": "\\begin{align*} \\mu | _ { \\gamma } ( f ( x , y ) ) = \\mu ( \\gamma \\cdot f ( x , y ) ) , \\end{align*}"}
+{"id": "2010.png", "formula": "\\begin{align*} \\frac { \\int _ { 0 } ^ { \\pi / 4 } \\cos ( x ) d x } { \\pi / 4 } = \\frac { 2 \\sqrt { 2 } } { \\pi } . \\end{align*}"}
+{"id": "8743.png", "formula": "\\begin{align*} \\lambda : = c \\phi \\sigma _ u ^ 2 \\left ( \\sqrt { \\frac { N _ 0 } { N } } \\vee \\frac { \\log ( T ) N _ 0 } { N } \\vee \\frac { \\sqrt { \\log { \\frac { 1 } { \\delta } } } } { \\sqrt { N } } \\vee \\frac { \\log ( T ) \\log { \\frac { 1 } { \\delta } } } { N } \\right ) , \\end{align*}"}
+{"id": "5842.png", "formula": "\\begin{align*} t = 0 : U = \\widehat { U } _ 0 , U ' = \\widehat { U } _ 1 \\hbox { i n } \\Omega , \\end{align*}"}
+{"id": "1702.png", "formula": "\\begin{align*} F ( \\infty ) - F ( 0 ^ + ) = ( - 1 ) ^ { m } 2 ^ { k - 1 } ( k - 1 ) \\binom { k - 2 } { \\frac { k - 2 } { 2 } - m } ^ { - 1 } . \\end{align*}"}
+{"id": "6433.png", "formula": "\\begin{align*} \\Gamma ^ 1 & : = \\partial U _ X \\times \\mathbb R ^ m \\times [ 0 , T ) , \\ \\Gamma ^ 2 : = ( U _ X \\times \\mathbb R ^ m ) \\times \\{ 0 \\} . \\end{align*}"}
+{"id": "8340.png", "formula": "\\begin{align*} \\int _ { | x | > A + t , t = M _ 0 } \\left ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 \\right ) \\mathrm { d } x \\le \\delta _ 0 , \\end{align*}"}
+{"id": "1901.png", "formula": "\\begin{align*} d { v } ( \\cdot , t ) = [ \\mathcal { A } { v } ( \\cdot , t ) - p _ t ( \\cdot , t ) ] d t + \\sigma [ { v } ( \\cdot , t ) + p ( \\cdot , t ) ] d B ( t ) , \\end{align*}"}
+{"id": "6556.png", "formula": "\\begin{align*} { F _ { Y _ w } } \\left ( y \\right ) = \\int _ 0 ^ \\infty { { F _ { T _ w } } \\left ( { y - t } \\right ) \\frac { 1 } { { { C _ { 1 w } } } } { f _ { X _ w } } \\left ( { \\frac { t } { { { C _ { 1 w } } } } } \\right ) { \\rm d } t } . \\end{align*}"}
+{"id": "5755.png", "formula": "\\begin{align*} c _ j ^ { T ^ * X } \\left ( C C ( \\varphi ) \\right ) = ( - 1 ) ^ j \\cdot c _ j ( \\varphi ) \\in A _ j ( X ) , \\ \\ j = 0 , \\ldots , \\dim ( X ) , \\end{align*}"}
+{"id": "2059.png", "formula": "\\begin{align*} y _ 2 = - \\frac { 1 } { h _ 2 } \\Bigl ( \\frac { 1 } { z } y _ 1 ' + h _ 3 y _ 1 \\Bigr ) . \\end{align*}"}
+{"id": "3331.png", "formula": "\\begin{align*} a _ 1 = b _ 1 = 0 , a _ 2 = - \\frac { \\sqrt { \\delta } } { 2 \\nu } p , b _ 2 = \\sqrt { \\delta } q , c = \\sqrt { \\delta } , n = 2 , \\end{align*}"}
+{"id": "8711.png", "formula": "\\begin{align*} \\tilde F _ k = \\sum _ { s = 1 } ^ k F _ k = ( 1 - t ) \\sum _ i ( x _ i + \\dots + x _ i ^ k ) \\prod _ { j \\ne i } \\frac { x _ i - t x _ j } { x _ i - x _ j } = ( 1 - t ) \\sum _ i \\frac { x _ i ( 1 - x _ i ^ { k } ) } { 1 - x _ i } \\prod _ { j \\ne i } \\frac { x _ i - t x _ j } { x _ i - x _ j } . \\end{align*}"}
+{"id": "8343.png", "formula": "\\begin{align*} v ^ { ( M _ 0 ) } _ 0 ( y ) = | y | ^ { - d + 1 } { w ^ { ( M _ 0 ) } } ( 0 , \\frac { y } { | y | ^ 2 } ) , v ^ { ( M _ 0 ) } _ 1 = | y | ^ { - d - 1 } \\partial _ t { w ^ { ( M _ 0 ) } } ( 0 , \\frac { y } { | y | ^ 2 } ) . \\end{align*}"}
+{"id": "2387.png", "formula": "\\begin{align*} \\dot x _ 1 + E _ { 1 2 } ( t ) \\dot x _ 2 ( t ) = A _ { 2 2 } ( t ) x _ 2 ( t ) . \\end{align*}"}
+{"id": "2972.png", "formula": "\\begin{align*} \\frac { a } { \\sqrt { n } } W _ j \\eta _ j + \\frac { b } { n } W _ j \\eta _ j ^ 2 = a \\frac { \\eta _ j ^ 2 } { \\sqrt { n } } + \\bigg ( a \\frac { g ^ { \\prime \\prime } ( 0 ) } { 2 g ^ \\prime ( 0 ) } + b \\bigg ) \\frac { \\eta _ j ^ 3 } { n } + O ( n ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "1306.png", "formula": "\\begin{align*} \\int _ { \\Omega \\times \\Omega } | \\langle \\tau _ \\alpha , \\tau _ \\beta \\rangle | ^ { 2 m } \\ , d ( \\mu \\times \\mu ) ( \\alpha , \\beta ) = \\int _ { \\Omega } \\int _ { \\Omega } | \\langle \\tau _ \\alpha , \\tau _ \\beta \\rangle | ^ { 2 m } \\ , d \\mu ( \\alpha ) \\ , d \\mu ( \\beta ) \\geq \\frac { \\mu ( \\Omega ) ^ 2 } { { d + m - 1 \\choose m } } , \\forall m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "7311.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } G ^ k _ { m _ n } ( x ) = 0 \\end{align*}"}
+{"id": "623.png", "formula": "\\begin{align*} m _ t = 1 - \\frac { \\sigma _ t } { \\sigma _ 0 } \\end{align*}"}
+{"id": "9125.png", "formula": "\\begin{align*} \\{ r ^ { p _ { i } } \\mod q , r ^ { p _ { i } ^ { 2 } } \\mod q , \\dots , r ^ { p _ { i } ^ { ( g ( x ) ) + 1 } } \\mod q \\} \\subseteq ( r ^ { l } \\mod q ) _ { ( l , q - 1 ) = 1 } \\end{align*}"}
+{"id": "8847.png", "formula": "\\begin{align*} \\tau _ { m } : = \\left ( 2 m + 1 \\right ) \\nu + m \\left ( m + 1 \\right ) \\end{align*}"}
+{"id": "1232.png", "formula": "\\begin{align*} W ^ * _ { \\mu , \\sigma , c , p } ( \\nu _ 0 , \\nu _ 1 ) : = | | D v _ 0 - D v _ 1 | | _ { L ^ p ( \\sigma ) } \\end{align*}"}
+{"id": "8977.png", "formula": "\\begin{align*} \\| \\nabla ^ { k _ 0 + 1 } u \\| ^ 2 _ { L ^ 2 ( B ) } & + \\| \\nabla ^ { k _ 0 } u \\| ^ 2 _ { L ^ 4 ( B ) } + \\sum _ { 1 \\le j \\le k _ 0 - 1 } \\| \\nabla ^ { j } u \\| ^ 2 _ { L ^ { \\infty } ( B ) } \\\\ & \\le C _ { k _ 0 } \\| \\nabla ^ { k _ 0 + 1 } u _ 0 \\| ^ 2 _ { L ^ 2 ( B ) } + C _ { k _ 0 } \\le C _ k < \\infty \\end{align*}"}
+{"id": "2988.png", "formula": "\\begin{align*} e ^ { ( 4 ) } = e ^ 0 \\wedge e ^ 1 \\wedge e ^ 2 \\wedge e ^ 3 , e _ a ^ { ( 3 ) } = \\frac { 1 } { 3 ! } \\epsilon _ { a b c d } e ^ { b } \\wedge e ^ { c } \\wedge e ^ { d } \\hbox { a n d } e _ { a b } ^ { ( 2 ) } = \\frac { 1 } { 2 ! } \\epsilon _ { a b c d } e ^ { c } \\wedge e ^ { d } \\end{align*}"}
+{"id": "5615.png", "formula": "\\begin{align*} \\begin{gathered} \\hat { R } _ { 0 1 } = \\hat { R } _ { 1 0 0 1 } + \\hat { R } _ { 3 0 3 1 } = R _ { 1 0 0 1 } - g ( A ( l , k ) , A ( k , l ) ) + g ( A ( k , k ) , A ( l , l ) ) \\\\ + R _ { 3 0 3 1 } - g ( A ( m _ 3 , m _ 3 ) , A ( k , l ) ) + g ( A ( k , m _ 3 ) , A ( m _ 3 , l ) ) = { R } _ { 1 0 0 1 } + { R } _ { 3 0 3 1 } - 2 \\theta ^ 2 , \\end{gathered} \\end{align*}"}
+{"id": "6902.png", "formula": "\\begin{align*} m a x \\{ b _ 1 , b _ 2 , b _ 1 + b _ 2 - 1 \\} & \\leq - ( a + 1 ) \\\\ 0 & \\leq b _ 1 , b _ 2 \\end{align*}"}
+{"id": "6941.png", "formula": "\\begin{align*} L _ { m , n } ^ { I , ( \\alpha ) } ( x ) = L _ m ^ { ( \\alpha ) } ( - x ) L _ { n - m } ^ { ( \\alpha - 1 ) } ( x ) + L _ { m } ^ { ( \\alpha - 1 ) } ( - x ) L _ { n - m - 1 } ^ { ( \\alpha ) } ( x ) \\end{align*}"}
+{"id": "919.png", "formula": "\\begin{align*} 2 ^ { - 6 ^ { \\dots { - ( ( n - 1 ) n ) ^ { - T _ n } } } } = 2 ^ { - 6 ^ { \\dots { - ( ( n - 1 ) n ) ^ { - ( n ( n + 1 ) ) ^ { - T _ n } } } } } \\end{align*}"}
+{"id": "5738.png", "formula": "\\begin{align*} \\mathbf { x } _ { 1 } ^ { a } = ( _ { c } \\mathbf { L } ^ { - 1 } ) _ { b } ^ { a } \\mathbf { x } ^ { b } \\longleftrightarrow ( _ { c } \\mathbf { L } ^ { - 1 } ) _ { B ^ { \\prime } } ^ { A ^ { \\prime } } ( _ { c } \\mathbf { L } ^ { - 1 } ) _ { B } ^ { A } \\mathbf { x } ^ { B B ^ { \\prime } } . \\end{align*}"}
+{"id": "7524.png", "formula": "\\begin{align*} T _ { t _ 1 , \\cdots , t _ n } ( x _ 1 , \\cdots , x _ n ) : = ( \\pi ^ { t _ 1 } y _ 1 , \\cdots , \\pi ^ { t _ n } y _ n ) . \\end{align*}"}
+{"id": "7487.png", "formula": "\\begin{align*} \\kappa ( G ) = \\kappa ( F ) + \\dim P . \\end{align*}"}
+{"id": "4310.png", "formula": "\\begin{align*} \\mathcal V _ t = \\sup \\{ v | ~ v \\in P S H ( X , \\hat \\omega _ t ) , \\ , v \\le 0 \\} . \\end{align*}"}
+{"id": "644.png", "formula": "\\begin{align*} | B \\cap B ( x , r ) | \\leq r ^ { \\gamma } | B | , x \\in \\R , \\ , \\delta \\leq r \\leq \\delta ^ { \\epsilon _ { 0 } } = \\delta ^ { \\bar { \\epsilon } _ { 0 } } . \\end{align*}"}
+{"id": "4807.png", "formula": "\\begin{align*} \\Pr _ { c \\sim \\mathcal { D } ( n , d ) } [ | c | \\leq \\frac { 1 - 2 ^ { - j } } { 2 } N ] & \\leq 2 ^ { - ( 1 - h ( \\frac { 1 - 2 ^ { - j } } { 2 } ) ) \\cdot \\binom { n } { \\leq d } } \\\\ & \\leq 2 ^ { - \\frac { 2 ^ { - 2 j } } { 2 \\ln 2 } \\cdot \\binom { n } { \\leq d } } . \\end{align*}"}
+{"id": "5505.png", "formula": "\\begin{align*} \\pi ( x ; q , a ) = { 1 \\over \\varphi ( q ) } \\int _ 2 ^ x { \\mathrm d t \\over \\log t } + O _ A \\left \\{ x \\exp \\left ( - C _ A \\sqrt { \\log x } \\right ) \\right \\} \\end{align*}"}
+{"id": "7815.png", "formula": "\\begin{align*} \\Psi _ { a , b } [ n ] ^ c ( x ^ { \\tilde m } ) & = \\exp \\Biggl ( \\ , c \\sum _ { j = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { j + 1 } } { j [ j a ] _ q } q ^ { j b } X _ { j n } \\Biggr ) ( x ^ { \\tilde m } ) \\\\ & = x ^ { \\tilde m } \\exp \\Biggl ( c \\sum _ { j = 1 } ^ { \\infty } \\frac { q ^ { 2 j \\langle n , \\tilde m \\rangle _ 1 } - 1 } { q ^ { 2 j a } - 1 } \\frac { ( - 1 ) ^ { j + 1 } } { j } q ^ { j a } q ^ { j b } x ^ { j p _ 1 ^ * ( n ) } \\Biggr ) . \\end{align*}"}
+{"id": "4614.png", "formula": "\\begin{align*} \\frac { 1 } { q } \\left ( 1 - \\frac { 1 } { \\prod _ { i = 1 } ^ n a _ i } \\right ) & = \\sum _ { i = 1 } ^ n \\frac { 1 } { a _ i } \\leq \\sum _ { i = 1 } ^ n \\frac { 1 } { b _ i } = \\frac { r } { \\prod _ { i = 1 } ^ n b _ i } \\\\ & \\leq \\frac { 1 } { q } \\left ( 1 - \\frac { 1 } { \\prod _ { i = 1 } ^ n b _ i } \\right ) . \\end{align*}"}
+{"id": "181.png", "formula": "\\begin{align*} & ( t + r + 1 ) \\cdots ( t + n ) \\left [ \\underset { i = 1 } { \\overset { s } { \\sum } } \\binom { m _ i + r - 1 } { r } + \\dim _ K ( I _ { \\alpha } ) _ t \\right ] = ( t + r + 1 ) \\cdots ( t + n ) \\binom { t + r } { r } \\\\ & = ( r + 1 ) \\cdots ( n - 1 ) n \\binom { t + n } { n } = ( r + 1 ) \\cdots ( n - 1 ) n \\left [ \\underset { i = 1 } { \\overset { s } { \\sum } } \\binom { m _ i + n - 1 } { n } + \\dim _ K I _ t \\right ] . \\end{align*}"}
+{"id": "6230.png", "formula": "\\begin{align*} N _ p = N - p \\end{align*}"}
+{"id": "7818.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} _ { \\frac { 1 } { 4 } , 0 } \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} _ { \\frac { 1 } { 2 } , 0 } \\equiv \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} _ { \\frac { 1 } { 2 } , 0 } \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} _ { \\frac { 1 } { 4 } , - \\frac { 1 } { 2 } } \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} _ { \\frac { 1 } { 4 } , \\frac { 1 } { 2 } } \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} _ { \\frac { 1 } { 4 } , 0 } . \\end{align*}"}
+{"id": "7228.png", "formula": "\\begin{align*} W _ { i j } & = \\begin{cases} C & i = j \\\\ H _ { i j } R & i \\neq j , \\end{cases} \\end{align*}"}
+{"id": "9219.png", "formula": "\\begin{align*} \\forall x \\in \\pi _ { x } ( K _ { \\pm } ) , \\phi _ { \\pm } ( x ) = f ( x ) - f ( \\Gamma ) . \\end{align*}"}
+{"id": "5418.png", "formula": "\\begin{align*} { _ i r } ^ { m } ( x y ) = \\sum _ { t = 0 } ^ { m } v ^ { ( \\nu - t i , ( m - t ) i ) + t ( m - t ) } \\frac { [ m ] _ { v } ! } { [ t ] _ { v } ! [ m - t ] _ { v } ! } { _ i r } ^ { t } ( x ) { _ i r } ^ { m - t } ( y ) \\end{align*}"}
+{"id": "3270.png", "formula": "\\begin{align*} f _ { \\lambda } \\left ( x \\right ) \\equiv F \\left ( x \\right ) - \\lambda G \\left ( x \\right ) = 0 , \\ F \\left ( x \\right ) = \\lambda G \\left ( x \\right ) , \\ x \\in X \\end{align*}"}
+{"id": "3145.png", "formula": "\\begin{align*} w _ A ( y _ 1 , y _ 2 ) = \\frac { 1 } { 2 } \\left ( R _ 1 ( y _ 1 + y _ 2 ) + R _ 2 ( y _ 1 - y _ 2 ) \\right ) \\end{align*}"}
+{"id": "8673.png", "formula": "\\begin{align*} \\Gamma ( u _ 1 ) \\dots \\Gamma ( u _ l ) \\ , ( f ) = \\mathcal F ( u _ 1 , \\dots , u _ l ) . \\end{align*}"}
+{"id": "7289.png", "formula": "\\begin{align*} ( - 1 ) ^ { k } G ^ { k } _ m ( x ) & = \\int _ { 0 } ^ { x } z ( - 1 ) ^ { k - 1 } G ^ { k - 1 } _ m ( z ) d m ( z ) + x \\int _ { x } ^ { 1 } ( - 1 ) ^ { k - 1 } G ^ { k - 1 } _ m ( z ) d m ( z ) \\\\ & \\geq x \\int _ { x } ^ { 1 } ( - 1 ) ^ { k - 1 } G ^ { k - 1 } _ m ( z ) d m ( z ) . \\end{align*}"}
+{"id": "1360.png", "formula": "\\begin{align*} \\Omega = \\Omega _ 1 \\cup \\cdots \\cup \\Omega _ n , \\Omega _ j \\cap \\Omega _ k = \\emptyset , \\forall 1 \\leq j , k \\leq n , j \\neq n . \\end{align*}"}
+{"id": "5582.png", "formula": "\\begin{align*} [ \\nabla _ { a } k _ { b } ] X ^ { a } z ^ { b } = 0 , \\end{align*}"}
+{"id": "7851.png", "formula": "\\begin{align*} n _ { \\{ z _ n \\} } ( r ) = C _ 0 r ^ q + o ( r ^ q ) \\end{align*}"}
+{"id": "6614.png", "formula": "\\begin{align*} H = \\C { A } + Q \\end{align*}"}
+{"id": "1194.png", "formula": "\\begin{align*} \\Upsilon ^ { * } r _ { i } = r _ { i } + O ( r _ { i } ^ { 1 - ( 1 / c _ i ) } ) \\ ; \\ , \\textrm { w i t h f o u r $ g _ { 0 , i } $ - d e r i v a t i v e s . } \\end{align*}"}
+{"id": "8537.png", "formula": "\\begin{align*} \\bigg ( \\frac { n ^ 2 } { 2 } \\bigg ) ^ p 2 ^ { Q ( n , n - 2 p ) - Q ( n , n ) } & = \\bigg ( \\frac { n ^ 2 } { 2 } \\bigg ) ^ p 2 ^ { ( 1 - n ) p } \\\\ & = \\bigg ( \\frac { n ^ 2 } { 2 ^ n } \\bigg ) ^ p . \\end{align*}"}
+{"id": "4804.png", "formula": "\\begin{align*} \\Pr _ { c \\sim \\mathcal { D } ( n , d ) } [ | c | \\leq 2 ^ { - j } \\cdot 2 ^ n ] & \\leq 2 ^ { - \\big ( 1 - 1 7 ( \\frac { j } { 1 - \\frac { d } { n } } + \\frac { 2 - \\frac { d } { n } } { ( 1 - \\frac { d } { n } ) ^ 2 } ) ( \\frac { d } { n } ) ^ { j - 1 } \\big ) \\binom { n } { \\leq d } + O ( n ^ 4 ) } . \\end{align*}"}
+{"id": "3607.png", "formula": "\\begin{align*} \\widehat u ( \\xi _ 1 , 0 ) \\ = \\ \\frac 1 { ( 2 \\pi ) ^ { \\frac { n - 1 } 2 } } \\widehat A ( \\xi _ 1 ) . \\end{align*}"}
+{"id": "6812.png", "formula": "\\begin{align*} Y _ 1 & = h a _ { 1 1 } f ( Y _ 1 ) + h a _ { 1 2 } f ( Y _ 2 ) + u _ { 1 1 } y _ 1 ^ { [ m - 1 ] } + u _ { 1 2 } y _ 2 ^ { [ m - 1 ] } \\\\ Y _ 2 & = h a _ { 2 1 } f ( Y _ 1 ) + h a _ { 2 2 } f ( Y _ 2 ) + u _ { 2 1 } y _ 1 ^ { [ m - 1 ] } + u _ { 2 2 } y _ 2 ^ { [ m - 1 ] } \\\\ y _ 1 ^ { [ m ] } & = h b _ { 1 1 } f ( Y _ 1 ) + h b _ { 1 2 } f ( Y _ 2 ) + y _ 1 ^ { [ m - 1 ] } \\\\ y _ 2 ^ { [ m ] } & = h b _ { 2 1 } f ( Y _ 1 ) + h b _ { 2 2 } f ( Y _ 2 ) - y _ 2 ^ { [ m - 1 ] } \\end{align*}"}
+{"id": "3449.png", "formula": "\\begin{align*} 4 n = ( 2 \\lambda - \\delta ) ^ 2 + ( 2 \\mu - \\delta ) ^ 2 + 1 0 \\delta ^ 2 \\ , . \\end{align*}"}
+{"id": "8253.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { l } \\sum _ { l = 1 } ^ { m } \\left \\{ e _ { k l } \\log ( \\eta { k l } ) + ( 1 - e _ { k l } ) \\log ( 1 - \\eta _ { k l } ) \\right \\} \\end{align*}"}
+{"id": "3046.png", "formula": "\\begin{align*} E _ { \\mu _ \\tau } \\left ( f \\left ( \\left ( \\frac { 1 } { t ^ { 3 / 2 } } \\nu _ { n _ { t s } } \\circ \\pi \\right ) _ { s \\geq 0 } \\right ) \\right ) & = \\int _ M \\int _ 0 ^ { \\tau ( x ) } f \\left ( \\left ( \\frac { 1 } { t ^ { 3 / 2 } } \\nu _ { n _ { t s } } \\circ \\pi ( x , u ) \\right ) _ { s \\geq 0 } \\right ) g ( x , u ) d \\lambda d \\mu \\\\ & = \\int _ M f \\left ( \\left ( \\frac { 1 } { t ^ { 3 / 2 } } \\nu _ { n _ { t s } } ( x , 0 ) \\right ) _ { s \\geq 0 } \\right ) \\int _ 0 ^ { \\tau ( x ) } g ( x , u ) d \\lambda d \\mu . \\end{align*}"}
+{"id": "6300.png", "formula": "\\begin{align*} \\pi _ i \\pi _ { i + 1 } \\pi _ i ( T ) = \\pi _ { i + 1 } \\pi _ i \\pi _ { i + 1 } ( T ) . \\end{align*}"}
+{"id": "1557.png", "formula": "\\begin{align*} \\psi '' _ { \\mathrm { u } } ( t _ { 1 } ) = t _ { 1 } ^ { 1 - r ^ { - } } \\zeta ' _ { \\mathrm { u } } ( t _ { 1 } ) > 0 . \\end{align*}"}
+{"id": "7912.png", "formula": "\\begin{align*} & \\P ( B _ s \\geq - y , s \\leq t ) = \\P ( | B _ 1 | \\leq y / \\sqrt { t } ) \\leq C \\frac { y \\wedge \\sqrt { t } } { \\sqrt { t } } . \\end{align*}"}
+{"id": "6238.png", "formula": "\\begin{align*} A e _ r = \\sum _ { s = 1 } ^ p \\widehat \\alpha _ { r s } e _ s , B e _ r = \\sum _ { s = 1 } ^ p \\widehat \\beta _ { r s } e _ s , 1 \\leqslant r \\leqslant p . \\end{align*}"}
+{"id": "1559.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow + \\infty } \\inf \\psi ' _ { \\mathrm { u } _ { m } } ( t _ { 1 } ) > \\psi ' _ { \\tilde { \\mathrm { u } } ^ { * } } ( t _ { 1 } ) = 0 . \\end{align*}"}
+{"id": "2023.png", "formula": "\\begin{align*} P ( x , C _ j ) = P ( y , C _ j ) x , y \\in C _ i . \\end{align*}"}
+{"id": "3516.png", "formula": "\\begin{align*} E _ { i j } \\omega _ { A ^ \\tau } = \\delta _ { i j } \\frac { p } { 2 } \\omega _ A + \\sum _ { \\substack { ( k , l ) \\in \\lambda , \\\\ A ( k , l ) = j } } \\omega _ { ( A _ { ( k , l ) \\to i } ) ^ \\tau } . \\end{align*}"}
+{"id": "6656.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\eta ^ k ( \\phi ( z ^ k ) - \\phi ^ { \\ast } ) < \\infty a . s . \\end{align*}"}
+{"id": "859.png", "formula": "\\begin{align*} \\frac { \\partial f ( w ) } { \\partial w _ i } = [ \\frac { \\partial F ( w ) } { \\partial w _ i } ] ^ \\vee = [ e ^ { \\widehat { w } } \\frac { \\partial \\widehat { w } } { \\partial w _ i } J _ d + J _ d \\frac { \\partial \\widehat { w } } { \\partial w _ i } ( e ^ { \\widehat { w } } ) ^ T ] ^ \\vee \\end{align*}"}
+{"id": "5436.png", "formula": "\\begin{align*} \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} . Z = ( A Z + B ) ( C Z + D ) ^ { - 1 } . \\end{align*}"}
+{"id": "9160.png", "formula": "\\begin{align*} \\avg { \\cdot } _ { J , \\beta } ^ { \\Z ^ 2 } : = \\lim _ { N \\to \\infty } \\avg { \\cdot } _ { J , \\beta } ^ { \\Lambda _ N } . \\end{align*}"}
+{"id": "8297.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } x ^ * ( \\textbf { x } ) \\ ! \\ ! \\ ! \\ ! & = \\frac { \\mu _ E R _ E ^ 2 - \\mu _ 1 R _ 1 ^ 2 - \\mu _ 2 R _ 2 ^ 2 - \\mu _ 3 R _ 3 ^ 2 } { 2 \\Lambda } \\end{array} \\right . \\end{align*}"}
+{"id": "7722.png", "formula": "\\begin{align*} \\hat { \\rho } = 4 C _ { \\beta } ^ 4 \\tau ( n - 1 ) ^ 4 \\cong \\rho ^ { \\star } . \\end{align*}"}
+{"id": "744.png", "formula": "\\begin{align*} \\{ \\frac { \\partial ^ 2 H ( z , w ) } { \\partial z \\partial \\bar { w } } \\} _ { z = w } = \\frac { 1 } { 2 } \\frac { \\partial h _ 1 ( w ) } { \\partial \\bar { w } } - h _ { 1 1 } ( w ) . \\end{align*}"}
+{"id": "6222.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p \\beta _ r E _ r = 0 , \\end{align*}"}
+{"id": "6346.png", "formula": "\\begin{align*} \\psi ' ( \\nu _ 0 ) - \\int ^ { t } _ 1 f _ \\nu ( x , \\nu _ 0 ) d x + \\int _ { h ( t ) } ^ 1 f _ \\nu ( x , \\nu _ 0 ) d x = 0 , \\end{align*}"}
+{"id": "7673.png", "formula": "\\begin{align*} [ \\bigoplus _ { I } \\sum _ { p \\in \\mathbb { Z } } \\eta _ { I , p } - \\bigoplus _ { I } \\eta _ { I , - \\iota _ { E } ( \\omega ) } ] = [ \\nabla _ { \\omega } ( \\bigoplus _ { I } \\phi ( \\iota _ { E } ( \\eta _ { I } ) ) ) ] . \\end{align*}"}
+{"id": "5018.png", "formula": "\\begin{align*} H = h _ 1 e ^ { 1 2 3 } + h _ 2 e ^ { 1 4 5 } , \\end{align*}"}
+{"id": "7312.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\varphi ^ d _ { m _ n } ( \\lambda ; x ) = 1 . \\end{align*}"}
+{"id": "5692.png", "formula": "\\begin{align*} \\mathbf { \\mathbf { e } } ^ { a } & \\mathbf { = } \\theta ^ { a } - \\tau ^ { a } \\mathbf { m } - \\overline { \\tau } ^ { a } \\overline { \\mathbf { m } } + \\zeta ^ { a } \\mathbf { m } \\overline { \\mathbf { m } } ; \\\\ \\mathbf { A } _ { b } ^ { a } & = \\alpha _ { b } ^ { a } - \\beta _ { b } ^ { a } \\mathbf { m } - \\overline { \\beta } _ { b } ^ { a } \\overline { \\mathbf { m } } + \\gamma _ { b } ^ { a } \\mathbf { m } \\overline { \\mathbf { m } } . \\end{align*}"}
+{"id": "2710.png", "formula": "\\begin{align*} \\| \\vec { u } ( t _ 0 ) \\| _ { \\dot { H } ^ 1 _ { R _ { 0 } } \\times L ^ 2 _ { R _ { 0 } } } = \\varepsilon \\leq \\varepsilon _ 0 , \\end{align*}"}
+{"id": "6301.png", "formula": "\\begin{align*} \\pi _ i \\pi _ { i + 1 } ( s _ i ( T ) ) = \\pi _ { i + 1 } ( s _ i ( T ) ) , \\end{align*}"}
+{"id": "8599.png", "formula": "\\begin{align*} F _ i \\cdot F _ j = 1 - \\delta _ { i j } \\end{align*}"}
+{"id": "402.png", "formula": "\\begin{align*} S _ { f , \\tau } : \\mathcal { X } \\ni x \\mapsto S _ { f , \\tau } x \\coloneqq \\sum _ { n = 1 } ^ \\infty f _ n ( x ) \\tau _ n \\in \\mathcal { X } \\end{align*}"}
+{"id": "3774.png", "formula": "\\begin{align*} { \\varepsilon _ 2 ' } _ \\# { \\varepsilon _ 1 } _ \\# ( v ) & = { \\varepsilon _ 1 ' } _ \\# { \\varepsilon _ 2 } _ \\# ( v ) = \\begin{cases} v & v \\in V _ 1 ( \\varepsilon _ 1 ) \\cap V _ 1 ( \\varepsilon _ 2 ) ; \\\\ x _ { e _ 1 } ^ { - 1 } v x _ { e _ 1 } & v \\in V _ 1 ( \\varepsilon _ 2 ) \\setminus V _ 1 ( \\varepsilon _ 1 ) ; \\\\ x _ { e _ 2 } ^ { - 1 } v x _ { e _ 2 } & v \\in V _ 1 ( \\varepsilon _ 1 ) \\setminus V _ 1 ( \\varepsilon _ 2 ) . \\end{cases} \\end{align*}"}
+{"id": "3933.png", "formula": "\\begin{align*} \\overline { u } ( q ) : = \\frac { g _ z ( 0 , 0 , 0 ) } { g _ z ( x ( q ) , 0 , 0 ) } [ u ( x ( q ) ) - g ( x ( q ) , 0 , 0 ) ] , \\end{align*}"}
+{"id": "5097.png", "formula": "\\begin{align*} a c ^ n = \\sum _ { b \\in B } b g _ b ( n ) \\qquad . \\end{align*}"}
+{"id": "5772.png", "formula": "\\begin{align*} 0 = n _ 0 < n _ 1 < n _ 2 < \\cdots < n _ p = N \\end{align*}"}
+{"id": "8712.png", "formula": "\\begin{align*} \\prod _ { i } \\frac { 1 - x _ i t u } { 1 - x _ i u } = 1 + ( 1 - t ) \\sum _ { r = 1 } ^ \\infty \\ , \\sum _ { i } \\frac { x _ i ( 1 - x _ i ^ { r } ) } { ( 1 - x _ i ) } \\prod _ { i \\ne j } \\frac { x - t x _ j } { x _ i - x _ j } \\ , ( u ^ r - u ^ { r + 1 } ) , \\end{align*}"}
+{"id": "9190.png", "formula": "\\begin{align*} D _ { j } ( Y , \\varphi ) : = K _ j ( Y , \\varphi ; ( \\Psi _ k ) _ { k < j } ) - K _ j ( Y , \\varphi ; 0 ) . \\end{align*}"}
+{"id": "7207.png", "formula": "\\begin{align*} - q = ( - 1 ) ( \\cos \\theta + \\omega \\sin \\theta ) . \\end{align*}"}
+{"id": "5355.png", "formula": "\\begin{align*} \\Lambda _ { F , s } [ \\varepsilon ] = \\sum _ { k = 0 } ^ s ( - 1 ) ^ { s - k } \\binom { \\ , \\abs { F } - k } { s - k } \\hat { \\Lambda } _ { F , k } [ \\varepsilon ] , \\end{align*}"}
+{"id": "5749.png", "formula": "\\begin{align*} \\mu ^ { A A ^ { \\prime } } = \\frac { 1 } { 2 } \\theta ^ { A A ^ { \\prime } } ; \\nu ^ { A A ^ { \\prime } } = \\frac { i } { 2 } ( \\omega _ { B } ^ { A } \\theta ^ { B A ^ { \\prime } } - \\omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } \\theta ^ { A B ^ { \\prime } } ) , \\end{align*}"}
+{"id": "4457.png", "formula": "\\begin{align*} \\frac { 1 } { A } = \\left \\{ \\frac { 1 } { x } : x \\in A \\right \\} \\end{align*}"}
+{"id": "1488.png", "formula": "\\begin{align*} \\C _ 1 ( x ) = 2 \\cos ( \\pi x ) \\\\ = 2 \\prod _ { n = 1 , n } ^ \\infty \\left ( 1 - \\frac { x ^ 2 } { ( \\frac n 2 ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "1095.png", "formula": "\\begin{align*} \\| \\hat { f } - f \\| _ 2 ^ 2 = \\sum _ { j = 1 } ^ k ( \\theta _ j - \\overline { Z } _ j ) ^ 2 + \\sum _ { j > k } \\theta _ j ^ 2 = ( I ) + ( I I ) . \\end{align*}"}
+{"id": "6762.png", "formula": "\\begin{align*} I _ n ( t ) = e ^ { t / 2 } \\int _ { - \\infty } ^ { \\infty } | \\Delta x | \\exp \\left [ - \\frac { 1 } { 4 } e ^ { t } \\left ( \\Delta x \\pm \\alpha _ n ( t ) \\right ) ^ 2 \\right ] d \\Delta x \\end{align*}"}
+{"id": "4371.png", "formula": "\\begin{align*} \\frac { p } { q } = \\sum _ { i = 1 } ^ k \\frac { 1 } { a _ i } + \\frac { 1 } { q \\prod _ { i = 1 } ^ k a _ i } . \\end{align*}"}
+{"id": "7210.png", "formula": "\\begin{align*} \\omega ' = \\frac { | x ' | } { | x | } w - \\frac { | x | ' } { | x | } \\omega , \\qquad \\mbox { w h e r e } w = \\frac { x ' } { | x ' | } . \\end{align*}"}
+{"id": "4587.png", "formula": "\\begin{align*} \\prod _ { i = m + 1 } ^ n a _ i \\leq \\prod _ { i = m + 1 } ^ n b _ i . \\end{align*}"}
+{"id": "8875.png", "formula": "\\begin{align*} \\varphi ( t , z ) = \\sum \\limits _ { m = 0 } ^ { + \\infty } e ^ { \\beta _ m t } \\int _ { \\mathbb { C } ^ n } K _ { \\nu , m } ( z , w ) g ( w ) d \\mu _ n ( w ) . \\end{align*}"}
+{"id": "4755.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ b x _ i ( t ) \\geq \\sum _ { i = 1 } ^ b \\hat { z } _ i = \\sum _ { i = 1 } ^ b \\tilde { z } _ i \\forall t \\geq 0 . \\end{align*}"}
+{"id": "2578.png", "formula": "\\begin{align*} \\nu _ \\xi ( p ) = p , \\nu _ \\xi ( N ) = N , d \\nu _ \\xi ( \\xi ) = - \\xi . \\end{align*}"}
+{"id": "4260.png", "formula": "\\begin{align*} I _ \\gamma ( c ) = \\lim _ { n \\rightarrow \\infty } E _ \\gamma ( f _ n ) & = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { 2 } \\| ( \\nabla - i A ) f _ n \\| ^ 2 _ { L ^ 2 } + \\int _ { \\R ^ N } V _ \\gamma ( x ) | f _ n ( x ) | ^ 2 d x \\\\ & \\geq \\lim _ { n \\rightarrow \\infty } \\omega ^ 0 _ \\gamma \\| f _ n \\| ^ 2 _ { L ^ 2 } = \\omega ^ 0 _ \\gamma c . \\end{align*}"}
+{"id": "6776.png", "formula": "\\begin{align*} \\frac { \\left | z \\left ( \\left ( \\frac { 1 } { 2 } - \\sigma \\right ) + j \\omega \\right ) \\right | } { \\left | z \\left ( \\left ( \\frac { 1 } { 2 } + \\sigma \\right ) - j \\omega \\right ) \\right | } = \\frac { \\left | \\left ( \\frac { 1 } { 2 } - \\sigma \\right ) + j \\omega \\right | } { \\left | \\left ( \\frac { 1 } { 2 } + \\sigma \\right ) - j \\omega \\right | } \\end{align*}"}
+{"id": "3174.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( A ) = \\int _ { [ 0 , 1 ] ^ 3 } r _ A \\ , a _ 1 \\ , \\partial _ 1 v ^ { 1 1 } _ A = \\int _ { [ 0 , 1 ] ^ 2 } r _ B \\ , b _ 1 \\ , \\partial _ 1 v ^ { 1 1 } _ B = c _ 1 ^ { 1 1 } ( B ) \\neq 0 , \\end{align*}"}
+{"id": "3279.png", "formula": "\\begin{align*} f _ { \\lambda } \\left ( u \\right ) \\equiv - \\nabla \\left ( \\left \\vert \\nabla u \\right \\vert ^ { p - 2 } \\nabla u \\right ) - \\lambda \\left \\vert u \\right \\vert ^ { p _ { 0 } - 2 } u \\left \\vert \\nabla u \\right \\vert ^ { p _ { 1 } } = h \\left ( x \\right ) \\end{align*}"}
+{"id": "973.png", "formula": "\\begin{align*} ( x , y , t ) * ( \\tilde x , \\tilde y , \\tilde t ) = ( x + \\tilde x , y + \\tilde y , t + \\tilde t + 2 \\sum _ { i = 1 } ^ { n } \\tilde x _ i y _ i - x _ i \\tilde y _ i ) . \\end{align*}"}
+{"id": "3749.png", "formula": "\\begin{align*} & \\zeta ( s + 1 ) ^ { - 1 } Z _ { r _ 1 } ( d _ 2 ( f ) , s + 1 ) \\\\ & = f ( 0 ) q ^ { A s } + \\zeta ( 1 ) ^ { - 1 } \\sum _ { j = - A } ^ { A - 1 } f ( 0 , 0 , 0 , \\varpi ^ j , 0 , 0 ) q ^ { j s } ( 1 - \\zeta ( - 1 ) \\zeta ( - s - 1 ) ^ { - 1 } ) \\\\ & = f ( 0 ) + s ( \\log q ) A f ( 0 ) - s ( \\log q ) \\sum _ { j = - A } ^ { A - 1 } f ( 0 , 0 , 0 , \\varpi ^ j , 0 , 0 ) + O _ f ( s ^ 2 ) . \\end{align*}"}
+{"id": "5036.png", "formula": "\\begin{align*} H _ t = \\lambda _ t H _ o = \\lambda _ t \\left ( q \\ , e ^ { 1 2 3 } + p \\ , e ^ { 1 4 5 } \\right ) , \\end{align*}"}
+{"id": "8091.png", "formula": "\\begin{align*} J _ { \\nu } ( z ) = 2 \\frac { \\Bigl ( \\dfrac { z } { 2 } \\Bigr ) ^ \\nu } { \\Gamma \\Bigl ( \\nu + \\dfrac { 1 } { 2 } \\Bigr ) \\Gamma \\Bigl ( \\dfrac { 1 } { 2 } \\Bigr ) } \\int _ 0 ^ { \\pi / 2 } \\sin ( \\theta ) ^ { 2 \\nu } \\cos ( z \\cos \\theta ) \\ , d \\theta , \\end{align*}"}
+{"id": "1560.png", "formula": "\\begin{align*} | | D \\mathrm { u } | | _ { q ( z ) , \\mu ( z ) } ^ { q ^ { + } } = - | | D \\mathrm { u } | | _ { p ( z ) } ^ { p ^ { + } } + \\int _ { M } g ( z ) | \\mathrm { u } ( z ) | ^ { 1 - \\gamma ( z ) } \\ , \\ , d v _ { g } ( z ) + \\lambda | | \\mathrm { u } | | _ { r ( z ) } ^ { r ^ { - } } , \\end{align*}"}
+{"id": "1709.png", "formula": "\\begin{align*} F ( 0 ^ + ) = ( - 1 ) ^ { m - 1 } ( k - 1 ) \\binom { k - 2 } { \\frac { k - 2 } { 2 } - m } ^ { - 1 } \\sum _ { s \\geq 0 } \\binom { k } { \\frac { k - 2 } { 2 } - m - 2 s } . \\end{align*}"}
+{"id": "7074.png", "formula": "\\begin{align*} \\langle \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) , x ^ { \\rho _ 1 + \\cdots + \\rho _ { j - 1 } } \\rangle = \\begin{cases} 1 & i = j \\\\ 0 & i \\neq j . \\end{cases} \\end{align*}"}
+{"id": "904.png", "formula": "\\begin{align*} \\begin{bmatrix} F _ { n + 1 } & F _ { n } \\\\ F _ { n } & F _ { n - 1 } \\end{bmatrix} = \\begin{bmatrix} 1 & 0 \\\\ 1 & 1 \\end{bmatrix} ^ n , \\end{align*}"}
+{"id": "5060.png", "formula": "\\begin{align*} F = \\frac { c _ 0 u _ 0 } { ( 1 - c _ 1 u _ 1 ) \\cdots ( 1 - c _ l u _ l ) } . \\end{align*}"}
+{"id": "5500.png", "formula": "\\begin{align*} s \\mapsto \\mathcal T _ s : = s \\cdot \\jmath ^ * ( ^ { - 1 } \\circ F _ { \\lambda _ 0 } \\circ ) + ( 1 - s ) \\cdot ^ { - 1 } \\circ ( \\jmath ^ * F _ { \\lambda _ 0 } ) \\circ . \\end{align*}"}
+{"id": "3462.png", "formula": "\\begin{align*} & f _ j ^ { ( n ) } ( x _ 1 , \\ldots , x _ { n - 1 } , z , x ; t ) = f _ n ( x _ 1 , \\ldots , x _ { j - 1 } , z , x _ { j } , \\ldots , x _ { n - 1 } , x ; t ) \\\\ & = \\int _ { \\{ 0 < t _ 1 < \\ldots < t _ { j - 1 } < r < t _ j < \\ldots < t _ { n - 1 } < t \\} } G _ { t - t _ { n - 1 } } ( x - x _ { n - 1 } ) \\ldots G _ { t _ j - r } ( x _ j - z ) G _ { r - t _ { j - 1 } } ( z - x _ { j - 1 } ) \\ldots \\\\ & G _ { t _ 2 - t _ 1 } ( x _ 2 - x _ 1 ) d t _ 1 \\ldots d t _ { n - 1 } d r . \\end{align*}"}
+{"id": "4654.png", "formula": "\\begin{align*} \\mathcal { T } _ { i i } = \\sum _ { m = 0 } ^ { \\infty } \\frac { \\sigma ^ { 2 m } _ { i } } { 2 ^ m m ! } \\partial _ i ^ { 2 m } , \\ \\ i = 1 , \\ldots , N , \\end{align*}"}
+{"id": "4648.png", "formula": "\\begin{align*} \\tilde { \\ell } _ { i i } = \\ell _ { i i } , \\ \\ \\ \\ \\tilde { \\ell } _ { i j } = \\ell _ { i j } + \\ell _ { j i } \\ \\ \\ \\ i \\neq j , \\end{align*}"}
+{"id": "4346.png", "formula": "\\begin{align*} \\sigma ( x , y ) = ( x + y = 0 ) \\wedge ( x \\cdot y = 1 ) \\end{align*}"}
+{"id": "6205.png", "formula": "\\begin{align*} t = 0 : u _ r = ( E _ r , \\widehat U _ 0 ) , \\ u _ r ' = ( E _ r , \\widehat U _ 1 ) \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "447.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n \\in \\mathbb { M } } f _ n ( \\cdot ) \\tau _ n - \\sum _ { n = 1 } ^ { \\infty } r _ n f _ n ( \\cdot ) \\tau _ n \\right \\| \\leq c \\varepsilon ^ \\frac { 1 } { d } . \\end{align*}"}
+{"id": "4606.png", "formula": "\\begin{align*} \\frac { 1 } { q + 1 } = \\frac { 1 } { a _ 1 } \\leq \\frac { 1 } { b _ 1 } < \\frac { 1 } { q } \\end{align*}"}
+{"id": "1186.png", "formula": "\\begin{align*} | \\nabla ^ { k } _ { g _ { 0 , i } } ( \\Phi ^ { * } J - J _ { 0 } ) | _ { g _ { 0 , i } } = O _ { K , \\epsilon } ( r _ { i } ^ { \\lambda + \\epsilon - k } ) \\ ; \\ , \\end{align*}"}
+{"id": "7121.png", "formula": "\\begin{align*} \\beta ( \\rho _ 1 , \\dots , \\rho _ n ) = \\beta - \\epsilon ( \\beta ) \\beta ( \\rho _ 1 ) . \\end{align*}"}
+{"id": "3194.png", "formula": "\\begin{align*} \\tilde { A } ( y ) : = A ( y ) + I _ 2 \\quad y \\in \\R ^ 2 , \\end{align*}"}
+{"id": "1973.png", "formula": "\\begin{align*} \\left \\vert A _ { j } ( z ) \\right \\vert \\leq \\exp _ { p } \\left [ b \\left \\{ \\log _ { q - 1 } \\left ( r \\right ) \\right \\} ^ { \\rho - \\delta } \\right ] , j = 0 , 1 , . . . , k , j \\neq l \\end{align*}"}
+{"id": "732.png", "formula": "\\begin{align*} r _ { \\rm M V } ( z ) = \\frac { 1 } { 2 } \\{ S ( z ) , z \\} _ 1 = - \\frac { T ' ( z ) } { T ( z ) } ( z \\in \\partial \\Omega ) . \\end{align*}"}
+{"id": "5394.png", "formula": "\\begin{align*} | h ( r ) | = 1 , r \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "2455.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { | \\Lambda _ { \\pi \\otimes \\chi } ( n ) | } { n ^ { 1 + \\eta } } \\leq \\frac { 1 } { \\eta } + \\frac { m } { 2 } \\log C ( \\pi ) + O ( m ^ 2 ) . \\end{align*}"}
+{"id": "5256.png", "formula": "\\begin{align*} \\frac { d } { d t } g ( U , U ) = 2 a \\sin \\omega ( t ) \\cos \\omega ( t ) \\frac { d \\omega } { d t } . \\end{align*}"}
+{"id": "8500.png", "formula": "\\begin{align*} \\frac { \\partial C ( p , ~ t ) } { \\partial t } = C _ { \\xi ^ 2 } ( p , ~ t ) , C ( \\cdot , ~ 0 ) = C _ 0 ( \\cdot ) . \\end{align*}"}
+{"id": "3098.png", "formula": "\\begin{align*} r _ B b _ { 1 1 } & = r _ B + ( r _ B b - \\bar { b } ) + \\bar { b } = 1 + \\partial _ { 1 1 } ^ 2 w _ B - \\partial _ { 2 2 } ^ 2 w _ B - \\Delta w _ B + \\bar { b } = ( 1 + \\bar { b } ) - 2 \\ , \\partial _ { 2 2 } ^ 2 w _ B , \\\\ r _ B b _ { 2 2 } & = r _ B - ( r _ B b - \\bar { b } ) - \\bar { b } = 1 + \\partial _ { 1 1 } ^ 2 w _ B - \\partial _ { 2 2 } ^ 2 w _ B + \\Delta w _ B - \\bar { b } = ( 1 - \\bar { b } ) + 2 \\ , \\partial _ { 1 1 } ^ 2 w _ B , \\end{align*}"}
+{"id": "1790.png", "formula": "\\begin{align*} a ^ \\sharp \\left ( \\frac { [ g ] - 1 } { \\xi } \\right ) ^ \\sharp = b ^ \\sharp - g ( b ^ \\sharp ) . \\end{align*}"}
+{"id": "8565.png", "formula": "\\begin{align*} d _ { I , w } = \\frac { d _ { w } ( i ) } { \\binom { t } { i } } . \\end{align*}"}
+{"id": "6064.png", "formula": "\\begin{align*} \\det ( \\mathsf W ) = \\int _ { b _ 1 } ^ { a _ 2 } \\cdots \\int _ { b _ g } ^ { a _ { g + 1 } } V \\big ( y _ 1 , \\ldots , y _ g \\big ) \\frac { ( y _ g \\cdots y _ 1 ) ( x _ g ^ g \\cdots x _ 1 ^ g ) } { ( w \\tilde w ) ( x _ g ) \\cdots ( w \\tilde w ) ( x _ 1 ) } \\dd x _ g \\cdots \\dd x _ 1 , \\end{align*}"}
+{"id": "7544.png", "formula": "\\begin{align*} a _ { k , r } ( i ) = & \\dfrac { ( k + r ) ! } { ( r + i ) ! ( k - i ) ! } \\cdot \\dfrac { ( i + r - 1 ) ! } { ( r - 1 ) ! i ! } \\\\ = & \\dfrac { r } { r + i } \\cdot \\dfrac { k ! } { ( k - i ) ! i ! } \\cdot \\dfrac { ( k + r ) ! } { r ! k ! } \\\\ = & \\dfrac { r } { r + i } \\dbinom { k } { i } \\dbinom { k + r } { r } . \\end{align*}"}
+{"id": "2393.png", "formula": "\\begin{align*} \\widetilde E ^ T = Q ^ T E ^ T Q = - Q ^ T E Q = - \\widetilde E \\end{align*}"}
+{"id": "2897.png", "formula": "\\begin{align*} S ( K ) = \\exp \\left ( \\int _ { \\mathbb { T } ^ \\infty } \\log K d m _ { \\infty } \\right ) \\end{align*}"}
+{"id": "2374.png", "formula": "\\begin{align*} U ^ T E V = \\begin{bmatrix} \\Sigma & 0 \\\\ 0 & 0 \\end{bmatrix} \\end{align*}"}
+{"id": "3385.png", "formula": "\\begin{align*} \\gamma _ n ( [ 0 , \\lambda ] ) = 2 ^ { n - 1 } + \\gamma _ { n - 1 } ( [ 0 , \\lambda - ( u _ n + 1 ) ] ) \\ , . \\end{align*}"}
+{"id": "9093.png", "formula": "\\begin{align*} \\mathfrak { d } _ { j _ 0 - s + 1 } & = [ \\mathfrak { d } _ { j _ 0 - s } , \\mathfrak { n } ] + J [ \\mathfrak { d } _ { j _ 0 - s } , \\mathfrak { n } ] \\\\ & \\subseteq [ \\mathfrak { d } ^ { s - 1 } , \\mathfrak { n } ] + J [ \\mathfrak { d } ^ { s - 1 } , \\mathfrak { n } ] \\subseteq \\mathfrak { d } ^ { s - 2 } . \\end{align*}"}
+{"id": "4453.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 1 3 2 } = \\frac { 1 } { 4 } + \\frac { 1 } { 1 1 } = \\frac { 1 5 } { 4 4 } . \\end{align*}"}
+{"id": "6818.png", "formula": "\\begin{align*} \\textbf { M } = \\{ y ; g ( y ) = 0 \\} . \\end{align*}"}
+{"id": "7171.png", "formula": "\\begin{align*} \\mathcal { C } = \\bigcup _ { t \\in [ 0 , \\ , \\infty ] } \\{ t \\} \\times C _ t , \\end{align*}"}
+{"id": "127.png", "formula": "\\begin{align*} E _ { \\lambda , K } = \\left \\{ ( x , y ) \\in \\mathbb R ^ n \\times \\mathbb R ^ n : \\ x \\neq y , \\frac { | f ( x ) - f ( y ) | } { | | x - y | | _ K ^ { \\frac { n } { p } + 1 } } \\geq \\lambda \\right \\} , \\end{align*}"}
+{"id": "2906.png", "formula": "\\begin{align*} \\sup _ { \\sigma > 0 } \\int _ { \\mathbb { T } ^ k } \\log ( 1 + | F _ { \\{ \\sigma \\} } | ) d m _ k = \\sup _ { 0 < r < 1 } \\int _ { \\mathbb { T } ^ k } \\log ( 1 + | F _ { [ r ] } | ) d m _ k , \\end{align*}"}
+{"id": "3891.png", "formula": "\\begin{align*} v ( y _ 0 ) = g ^ * ( x _ 0 , y _ 0 , u ( x _ 0 ) ) . \\end{align*}"}
+{"id": "3600.png", "formula": "\\begin{align*} B _ j ^ + \\Omega _ \\lambda = \\sum _ { i = 1 } ^ j \\sum _ { s = 1 } ^ { j - i + 1 } \\sum _ { I \\in \\mathcal { I } _ { i j } ( s ) } ( - 1 ) ^ { \\sum _ { \\alpha = j } ^ n ( \\alpha + 1 ) ( \\lambda _ \\alpha - \\lambda _ { \\alpha + 1 } ) } \\frac { \\lambda _ i - \\lambda _ j + 1 } { \\lambda _ i + 1 } d _ I ^ + ( \\mu ) \\Omega _ { \\lambda + \\epsilon _ i } , \\end{align*}"}
+{"id": "6499.png", "formula": "\\begin{align*} \\int q _ u ( x _ 1 , x _ 2 ) \\ ; d P _ 0 ( x _ 1 ) = \\underset { y \\in \\mathcal { Y } } { \\overset { } { \\sum } } \\frac { K ( y | x _ 2 , u ) } { \\P _ 0 ^ { Y | U = u } ( y ) } \\int K ( y | x _ 1 , u ) \\ ; d P _ 0 ( x _ 1 ) = \\underset { y \\in \\mathcal { Y } } { \\overset { } { \\sum } } K ( y | x _ 2 , u ) = 1 , \\end{align*}"}
+{"id": "4126.png", "formula": "\\begin{align*} \\Gamma ( S ^ 2 M ) \\times \\Omega ^ 2 = ( \\mathcal { V } + \\mathcal { V } _ 1 ) \\oplus ( \\mathcal { V } ^ \\perp \\cap \\mathcal { V } _ 1 ^ \\perp ) \\end{align*}"}
+{"id": "4608.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n - 1 } \\frac { 1 } { b _ i } < \\sum _ { i = 1 } ^ { n - 1 } \\frac { 1 } { a _ i } . \\end{align*}"}
+{"id": "3143.png", "formula": "\\begin{align*} - \\Delta w _ B = b - \\int _ Y b = \\frac { 2 } { c } \\left ( a - \\int _ Y a \\right ) \\quad Y , w _ B Y , \\int _ Y w _ B = 0 . \\end{align*}"}
+{"id": "4200.png", "formula": "\\begin{align*} E _ [ \\Lambda , \\phi ] ( t ) : = \\int \\left ( \\dfrac { 1 } { 2 } \\left ( ( \\partial _ t \\Lambda ) ^ 2 - 1 + ( \\partial _ x \\Lambda ) ^ 2 \\right ) + 2 \\sinh ^ 2 ( \\Lambda ) ( ( \\partial _ t \\phi ) ^ 2 + ( \\partial _ x \\phi ) ^ 2 ) \\right ) , \\end{align*}"}
+{"id": "5125.png", "formula": "\\begin{align*} D _ { f _ { 1 } } \\mathcal { I } _ { 2 } ( \\lambda , b , 0 , 0 ) ( h _ { 1 } ) ( w ) = - \\sum _ { n = 0 } ^ { \\infty } ( n + 1 ) I _ { n + 1 } ( \\lambda b ) K _ { n + 1 } ( \\lambda ) a _ { n } e _ { n + 1 } ( w ) . \\end{align*}"}
+{"id": "7396.png", "formula": "\\begin{align*} \\omega ( \\varpi _ F ) + \\chi ^ 2 \\chi '^ { - 1 } ( \\varpi _ L ) = 0 , \\end{align*}"}
+{"id": "2371.png", "formula": "\\begin{align*} { \\mathcal J } ( x , u ) = \\frac 1 2 x ( t _ f ) ^ T M _ f x ( t _ f ) + \\frac 1 2 \\int _ { t _ 0 } ^ { t _ f } \\left ( x ^ T W x + x ^ T S u + u ^ T S ^ T x + u ^ T R u \\right ) \\ , d t , \\end{align*}"}
+{"id": "5533.png", "formula": "\\begin{align*} I _ { k } ( x ) : = \\frac { 1 } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } \\frac { \\Gamma ( s ) } { \\zeta ( k - 2 s ) } x ^ { - s } { \\rm d } s . \\end{align*}"}
+{"id": "70.png", "formula": "\\begin{align*} p e n _ f ( \\mathbf { z } ) = ( 1 + f _ 1 ( \\mathbf { z } ) ) ( 1 + f _ 2 ( \\mathbf { z } ) ) - 1 + f _ 3 ( \\mathbf { z } ) \\end{align*}"}
+{"id": "7769.png", "formula": "\\begin{align*} \\mu \\widetilde r _ \\alpha ( s ) & = - \\frac { 1 } { \\widetilde f ( s ^ \\alpha | \\mu ) } \\frac { d } { d s } { \\widetilde f ( s ^ \\alpha | \\mu ) } = - \\frac { \\alpha s ^ { \\alpha - 1 } } { \\widetilde f ( s ^ \\alpha | \\mu ) } \\frac { d } { d s ^ \\alpha } { \\widetilde f ( s ^ \\alpha | \\mu ) } = \\mu \\alpha s ^ { \\alpha - 1 } \\widetilde \\rho ( s ^ \\alpha ) \\end{align*}"}
+{"id": "5797.png", "formula": "\\begin{align*} \\sum _ { j = n _ { s - 1 } + 1 } ^ { n _ { s } } d _ { i j } = 0 , n _ { r - 1 } + 1 \\leqslant i \\leqslant n _ { r } , 1 \\leqslant r , s \\leqslant p . \\end{align*}"}
+{"id": "457.png", "formula": "\\begin{align*} | \\langle x , \\tau _ n \\rangle | = | \\langle y , \\tau _ n \\rangle | , \\forall n \\in \\mathbb { N } , \\end{align*}"}
+{"id": "8705.png", "formula": "\\begin{align*} ( A ^ - ) _ { i j } = \\begin{cases} 1 , & i \\le j < 0 \\\\ 0 , & . \\end{cases} ( A ^ + ) _ { i , j } = \\begin{cases} 1 , & i = j > 0 , \\\\ - 1 , & j = i + 1 > 1 \\\\ 0 , & . \\end{cases} \\end{align*}"}
+{"id": "6843.png", "formula": "\\begin{align*} \\tilde Q _ k = \\tilde Q _ k ^ T = \\tilde U _ k \\tilde X _ k ^ { - 1 } , \\ , \\tilde Y _ k = R _ { \\alpha } ^ { - 1 } Y _ k = \\binom { \\tilde X _ k } { \\tilde U _ k } . \\end{align*}"}
+{"id": "55.png", "formula": "\\begin{align*} f ( i ) = - 1 \\implies ( ( \\exists j ) \\ ; j \\in N ( i ) \\land f ( j ) = 2 ) , i \\in V \\end{align*}"}
+{"id": "3231.png", "formula": "\\begin{align*} \\begin{aligned} m _ { { i _ 0 } + 1 } & = \\sum _ { c \\in C - F } m _ { i _ 0 + 1 } ( c ) \\\\ & = \\sum _ { c \\in C } m _ { i _ 0 + 1 } ( c ) - \\sum _ { c \\in F _ 1 \\cup F _ 2 } m _ { i _ 0 + 1 } ( c ) \\\\ & \\ge k | R _ { { i _ 0 } + 1 } | - ( | R _ { { i _ 0 } + 1 } | - 1 ) ( k - { i _ 0 } ) - | R _ { { i _ 0 } + 1 } | { i _ 0 } = k - { i _ 0 } , \\end{aligned} \\end{align*}"}
+{"id": "431.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } | \\langle h , \\tau _ j \\rangle | ^ 2 \\leq b \\| h \\| ^ 2 , \\forall h \\in \\mathcal { H } . \\end{align*}"}
+{"id": "7649.png", "formula": "\\begin{align*} \\Gamma _ { Q } ^ { 0 } ( M ) = \\{ m \\in M \\mid Q ^ { \\ell } m = 0 \\ell \\in \\mathbb { N } \\} . \\end{align*}"}
+{"id": "1595.png", "formula": "\\begin{align*} d _ p ( \\delta _ { x _ 1 } , \\Phi ( \\mu ) ) = d _ p ( \\Phi ( \\delta _ { x _ 1 } ) , \\Phi ( \\mu ) ) = d _ p ( \\delta _ { x _ 1 } , \\mu ) \\leq \\sqrt [ p ] { s } d _ p ( \\delta _ { x _ 1 } , \\delta _ { x _ 2 } ) , \\end{align*}"}
+{"id": "983.png", "formula": "\\begin{align*} \\Vert u - ( g \\otimes a ) u \\Vert _ { \\epsilon } & \\leq \\Vert u - ( g \\otimes a ) u \\Vert _ { \\pi } \\\\ & \\leq \\sum _ { j = 1 } ^ { n } ( \\Vert g _ j - g g _ j \\Vert _ { \\infty } \\ , \\Vert a _ j \\Vert _ { \\mathcal { A } } + \\Vert g _ j \\Vert _ { \\infty } \\ , \\Vert a _ j - a a _ j \\Vert _ { \\mathcal { A } } \\\\ & + \\Vert g _ j - g g _ j \\Vert _ { \\infty } \\ , \\Vert a _ j - a a _ j \\Vert _ { \\mathcal { A } } \\\\ & < \\epsilon \\end{align*}"}
+{"id": "2919.png", "formula": "\\begin{align*} \\partial _ t h = \\nu \\partial _ x ^ 2 h + \\lambda ( \\partial _ x h ) ^ 2 + \\sqrt { D } \\dot { W } ( t , x ) \\mathbb { R } _ + \\times \\mathbb { R } . \\end{align*}"}
+{"id": "1080.png", "formula": "\\begin{align*} \\sigma _ 1 ^ k & = \\mathbb { E } _ { R _ 1 } [ | X - \\mu _ 1 | ^ k ] = ( 1 - \\eta ) \\Big ( 2 ^ { - 1 / k } \\eta ^ { 1 - 1 / k } \\Big ) ^ k + \\eta \\left \\{ ( 2 \\eta ) ^ { - 1 / k } - ( 2 ^ { - 1 / k } \\eta ^ { 1 - 1 / k } \\right \\} ^ k \\\\ & = 2 ^ { - 1 } ( 1 - \\eta ) \\eta ^ { k - 1 } + 2 ^ { - 1 } ( 1 - \\eta ) ^ k \\leq 1 , \\end{align*}"}
+{"id": "3539.png", "formula": "\\begin{align*} \\mathbf { p } B _ j ^ + = \\sum _ { s _ 1 , \\dots , s _ m \\in \\N _ 0 } E _ { - \\alpha _ 1 } ^ { s _ 1 } \\cdots E _ { - \\alpha _ m } ^ { s _ m } E _ { \\alpha _ m } ^ { - s _ m } \\cdots E _ { \\alpha _ 1 } ^ { - s _ 1 } B _ j ^ + H _ { s _ 1 , \\dots , s _ m } , \\end{align*}"}
+{"id": "1495.png", "formula": "\\begin{align*} \\int _ 0 ^ x \\theta ^ { r - 2 } \\log \\left ( \\cos \\frac \\theta 2 \\right ) d \\theta = \\frac { x ^ { r - 1 } } { r - 1 } \\log \\left ( \\cos \\frac x 2 \\right ) - \\frac { ( 2 \\pi ) ^ { r - 1 } } { r - 1 } \\log \\C _ r \\left ( \\frac x { 2 \\pi } \\right ) . \\end{align*}"}
+{"id": "509.png", "formula": "\\begin{align*} \\| x ^ k - \\widetilde { x } \\| \\le \\Delta _ k \\le \\left \\{ \\begin{array} { c l } \\gamma \\varrho ^ { k } & { \\rm i f } \\ \\theta \\in ( 0 , 1 / 2 ] , \\\\ \\gamma k ^ { \\frac { 1 - \\theta } { 1 - 2 \\theta } } & { \\rm i f } \\ \\theta \\in ( 1 / 2 , 1 ) \\end{array} \\right . { \\rm w i t h } \\ \\Delta _ k : = \\sum _ { j = k } ^ { \\infty } \\| x ^ { j + 1 } \\ ! - \\ ! x ^ j \\| . \\end{align*}"}
+{"id": "6155.png", "formula": "\\begin{align*} a = \\sum _ { j = 1 } ^ N a _ { i j } , i = 1 , \\cdots , N . \\end{align*}"}
+{"id": "6494.png", "formula": "\\begin{align*} f ^ L = \\underset { l = 0 } { \\overset { L } { \\sum } } \\underset { i = 0 } { \\overset { 2 ^ l - 1 } { \\sum } } \\tilde { f } _ { l i } \\psi _ { l i } \\end{align*}"}
+{"id": "260.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = & \\alpha ( w / u ) + \\beta u , \\psi _ 2 ( u , v , w ) = \\alpha ( w / v ) + \\beta u , \\\\ \\psi _ 3 ( u , v , w ) = & \\alpha ( w / u ) + \\beta v \\psi _ 4 ( v , w ) = \\alpha ( w / v ) + \\beta v . \\end{align*}"}
+{"id": "6294.png", "formula": "\\begin{align*} \\pi _ i \\pi _ { i + 1 } ( s _ i ( T ) ) = \\pi _ { i + 1 } ( s _ i ( T ) ) , \\end{align*}"}
+{"id": "6009.png", "formula": "\\begin{align*} \\langle w _ { 1 , i _ 1 + 1 } , w _ { 1 , i _ 2 + 1 } , w _ { 1 , i _ 3 + 1 } \\rangle _ d = \\chi ( \\O _ { e v _ 1 ^ { - 1 } ( g _ 1 \\cdot L _ { i _ 1 } ) \\cap e v _ 2 ^ { - 1 } ( g _ 2 \\cdot L _ { i _ 2 } ) \\cap e v _ 3 ^ { - 1 } ( g _ 3 \\cdot L _ { i _ 3 } ) } ) . \\end{align*}"}
+{"id": "6463.png", "formula": "\\begin{align*} \\frac { d } { d t } E ( m ) & = \\int \\partial _ t m \\cdot \\delta E ( m ) d x \\\\ & = \\int ( m \\wedge H ( m ) ) \\cdot \\delta E ( m ) d x - \\alpha \\int ( m \\wedge ( m \\wedge H ( m ) ) ) \\cdot \\delta E ( m ) d x = I + I I . \\end{align*}"}
+{"id": "8243.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { N } } V ( | x | ) f ( u ) d x = \\lim _ n \\int _ { \\mathbb { R } ^ { N } } V ( | x | ) f ( u _ n ) ^ 2 d x . \\end{align*}"}
+{"id": "7668.png", "formula": "\\begin{align*} d f / f \\wedge \\phi ( \\eta _ { I } ) = d f / f \\wedge \\phi ( \\eta _ { I } - \\eta _ { I , - \\iota _ { E } ( \\omega ) } ) = \\phi ( d f / f \\wedge ( \\eta _ { I } - \\eta _ { I , - \\iota _ { E } ( \\omega ) } ) ) . \\end{align*}"}
+{"id": "4833.png", "formula": "\\begin{align*} \\phi ^ { ( n ) } ( 0 ) = D ^ n _ 0 F ( w _ 1 ( t ) , \\dots , w _ { n - 1 } ( t ) , 0 ) = D ^ n _ 0 F ( w ( t ) , 0 ) . \\end{align*}"}
+{"id": "2331.png", "formula": "\\begin{align*} \\det A = 1 + t . \\end{align*}"}
+{"id": "7941.png", "formula": "\\begin{align*} 0 + 0 = 0 , 1 + 1 = 1 + 0 = 1 , ( - 1 ) + ( - 1 ) = ( - 1 ) + 0 = - 1 , 1 + ( - 1 ) = \\mathbb { S } . \\end{align*}"}
+{"id": "532.png", "formula": "\\begin{align*} 1 \\ge S _ { { \\bf A } , d } ( u ) = \\frac { 1 } { u } \\sum _ { a \\in { \\bf A } } f ( a ) \\ge \\frac { \\alpha } { u } \\left ( \\sum _ { \\substack { 1 \\le \\ell \\le u / 2 \\\\ \\ell \\wedge u = 1 } } \\ell \\right ) . \\end{align*}"}
+{"id": "2406.png", "formula": "\\begin{align*} \\widetilde E = Q ^ T E Q , \\quad \\widetilde A = Q ^ T A Q - Q ^ T E \\dot Q \\end{align*}"}
+{"id": "8051.png", "formula": "\\begin{align*} \\omega _ j = \\frac { 4 \\pi \\abs { \\rho _ j ( 1 ) } ^ 2 } { \\cosh ( \\pi t _ j ) } , \\end{align*}"}
+{"id": "8543.png", "formula": "\\begin{align*} n ^ { 3 n } 3 ^ { n } 2 ^ { - n ^ 2 / 2 + n / 2 } = 2 ^ { 3 n \\log _ 2 ( n ) + \\log _ 2 ( 3 ) n - n ^ 2 / 2 + n / 2 } , \\end{align*}"}
+{"id": "2726.png", "formula": "\\begin{align*} \\iota _ j \\iota _ { j - 1 } ( \\theta _ j - \\theta _ { j - 1 } ) = c _ j \\theta _ { j - 1 } , c _ j \\in \\left \\{ \\frac { 1 } { 2 } , 1 \\right \\} . \\end{align*}"}
+{"id": "8558.png", "formula": "\\begin{align*} \\frac { 1 } { t ! \\lambda } \\sum _ { j = 0 } ^ { v - 1 } j ^ { s } d _ { w } ( j ) = \\frac { 1 } { v } \\sum _ { i = 0 } ^ { v - 1 } i ^ { s } . \\end{align*}"}
+{"id": "8494.png", "formula": "\\begin{align*} P _ 0 & = \\frac { 1 } { 2 } \\big ( ( f _ 6 ) _ { \\xi ^ 6 } - ( f _ 5 ) _ { \\xi ^ 5 } + ( f _ 4 ) _ { \\xi ^ 4 } - ( f _ 3 ) _ { \\xi ^ 3 } + ( f _ 2 ) _ { \\xi ^ 2 } - ( f _ 1 ) _ { \\xi } \\big ) + f _ 0 , \\\\ P _ 1 & = - 3 ( f _ 6 ) _ { \\xi ^ 4 } + \\frac { 5 } { 2 } ( f _ 5 ) _ { \\xi ^ 3 } - 2 ( f _ 4 ) _ { \\xi ^ 2 } + \\frac { 3 } { 2 } ( f _ 3 ) _ { \\xi } - f _ 2 , \\\\ P _ 2 & = \\frac { 9 } { 2 } ( f _ 6 ) _ { \\xi ^ 2 } - \\frac { 5 } { 2 } ( f _ 5 ) _ { \\xi } + f _ 4 , \\\\ P _ 3 & = - f _ 6 . \\end{align*}"}
+{"id": "2230.png", "formula": "\\begin{align*} \\nu { \\left ( \\sum _ { k = 2 } ^ \\infty x _ { 2 m , k } \\beta _ { k , 1 } \\right ) } \\geq \\min _ { k \\geq 2 } \\big ( \\nu ( x _ { 2 m , k } ) + \\nu ( \\beta _ { k , 1 } ) \\big ) \\geq \\min _ { k \\geq 2 } \\nu ( x _ { 2 m , k } ) \\geq 2 m . \\end{align*}"}
+{"id": "6175.png", "formula": "\\begin{align*} t = 0 : W = C _ 1 \\widehat U _ 0 , \\ W ' = C _ 1 \\widehat U _ 1 \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "3632.png", "formula": "\\begin{align*} h _ n ( x , y , t ) : = h _ n ( r , t ) = ( 4 \\pi t ) ^ { - \\frac { n } { 2 } } e ^ { - \\frac { ( n - 1 ) ^ 2 t } { 4 } - \\frac { ( n - 1 ) r } { 2 } - \\frac { r ^ 2 } { 4 t } } ( 1 + r + t ) ^ { \\frac { ( n - 3 ) } { 2 } } ( 1 + r ) , \\end{align*}"}
+{"id": "2654.png", "formula": "\\begin{align*} W ^ s : = [ V , W ] _ { s , \\infty } \\ ; , 0 < s < 1 \\ ; . \\end{align*}"}
+{"id": "259.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\rho } { \\tau } + \\frac { \\beta \\eta } { \\rho } \\leqslant \\frac { 2 \\alpha \\rho ^ 2 } { ( 1 + 3 \\rho ^ 2 ) } \\leqslant \\frac { 2 \\alpha } { 3 } , \\end{align*}"}
+{"id": "5456.png", "formula": "\\begin{align*} P ( \\alpha ^ { n } _ { t + \\Delta t } = j | \\alpha ^ { n } _ t = i , ( X ^ n _ s , \\alpha ^ { n } _ s ) , s \\leq t ) = q _ { i j } ( X ^ { n } _ t ) \\Delta t + o ( \\Delta t ) . \\end{align*}"}
+{"id": "8577.png", "formula": "\\begin{align*} a ^ * ( x ) : = x ^ n a ( x ^ { - 1 } ) = 1 + a _ { n - 1 } \\ , x + \\ldots + a _ 1 \\ , x ^ { n - 1 } + x ^ n . \\end{align*}"}
+{"id": "5495.png", "formula": "\\begin{align*} T = L + K , \\end{align*}"}
+{"id": "4253.png", "formula": "\\begin{align*} C _ 2 : = \\min \\left \\{ \\frac { \\gamma ^ 2 - \\Omega ^ 2 } { \\gamma ^ 2 + \\Omega ^ 2 } , \\frac { \\gamma ^ 2 - \\Omega ^ 2 } { 2 } , \\gamma _ 3 ^ 2 , \\cdots , \\gamma _ N ^ 2 \\right \\} . \\end{align*}"}
+{"id": "4564.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { a _ i } < 1 . \\end{align*}"}
+{"id": "9094.png", "formula": "\\begin{align*} [ \\mathfrak { n } _ 1 , \\mathfrak { n } _ { j - 1 } ] = \\mathfrak { n } _ j j \\in \\{ 2 , \\ldots , k \\} . \\end{align*}"}
+{"id": "4793.png", "formula": "\\begin{align*} \\Pr _ { \\rho \\sim P _ \\epsilon } [ \\rho \\notin D _ k ( H \\rho ^ \\intercal ) ] & \\leq 2 e ^ { - \\frac { \\sqrt { N } } { 3 \\epsilon } } + \\frac { 2 N } { k } \\max _ { j \\in B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } [ | v H | = j ] \\cdot \\frac { 2 ^ N } { \\binom { N } { j } } \\right \\} \\\\ & + \\frac { 2 ^ { h ( \\epsilon ) N + N ^ { \\frac { 4 } { 5 } } + 1 } } { k } \\max _ { j \\notin B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } \\big [ | v H | = j \\big ] \\cdot 2 ^ { 2 \\epsilon N \\log ( 1 - \\frac { 2 j } { N } ) } \\right \\} , \\end{align*}"}
+{"id": "8891.png", "formula": "\\begin{align*} c _ { i } ^ { \\left ( \\nu , n \\right ) } = \\left \\{ \\begin{array} { c } \\frac { ( n - i - 1 ) ! } { ( n - 1 ) ! } \\gamma _ { n - i - 1 } ^ { ( \\nu , n ) } , \\quad \\qquad \\qquad \\qquad 0 \\leq i \\leq n - 1 \\\\ \\sum \\limits _ { p = 0 } ^ { n - 1 } \\frac { ( - 1 ) ^ { i - n + 1 } c _ { p } ^ { \\left ( \\nu , n \\right ) } } { ( i - n ) ! ( i - p ) p ! } B _ { 2 ( i - p ) } \\left ( \\nu + \\frac { 1 } { 2 } \\right ) , i \\geq n \\end{array} \\right . \\end{align*}"}
+{"id": "4210.png", "formula": "\\begin{align*} & ( \\partial _ x \\phi ) ^ 2 - ( \\partial _ t \\phi ) ^ 2 = Q _ 0 ( \\phi , \\phi ) = 2 L \\phi \\underline { L } \\phi , \\\\ & \\partial _ x \\phi \\partial _ x \\tilde { \\Lambda } - \\partial _ t \\phi \\partial _ t \\tilde { \\Lambda } = Q _ 0 ( \\phi , \\tilde { \\Lambda } ) = \\dfrac { 1 } { 2 } L \\phi \\underline { L } \\tilde { \\Lambda } + \\dfrac { 1 } { 2 } L \\tilde { \\Lambda } \\underline { L } \\phi . \\end{align*}"}
+{"id": "6579.png", "formula": "\\begin{align*} \\frac { 1 } { r ^ { n - 1 } } \\mu ( B _ s \\cap B _ r ( x ) ) \\leq \\sum _ { i = 1 } ^ { m } \\frac { 1 } { r ^ { n - 1 } } \\mu ( B _ s ' ( y _ i ) ) \\leq m K ( \\mu , s , s ' ) . \\end{align*}"}
+{"id": "1018.png", "formula": "\\begin{align*} \\beta _ 0 = \\sup \\left \\{ p \\geq 0 , \\ \\underset { | \\xi | \\rightarrow 0 } { \\lim \\sup } \\frac { | \\Psi ( \\xi ) | } { | \\xi | ^ p } < \\infty \\right \\} . \\end{align*}"}
+{"id": "6426.png", "formula": "\\begin{align*} g _ k ( X , Y ) = g ( X , \\hat Y ) , \\end{align*}"}
+{"id": "2978.png", "formula": "\\begin{align*} \\begin{aligned} 1 0 W _ { j - 1 } W _ j ^ 2 = & - 2 W _ { j - 2 } W _ { j - 1 } W _ j - 6 W _ { j - 2 } W _ { j - 1 } W _ { j + 1 } - 9 W _ { j - 2 } W _ { j } W _ { j + 1 } + 2 7 W _ { j - 1 } W _ j W _ { j + 1 } \\\\ & + 1 0 W _ { j - 1 } W _ { j } + E _ j ^ { ( 8 ) } . \\end{aligned} \\end{align*}"}
+{"id": "5395.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t u & = \\mathcal { A } u + f ( x , t ) , \\\\ u | _ { t = 0 } & = 0 , \\end{aligned} \\end{align*}"}
+{"id": "7057.png", "formula": "\\begin{align*} \\{ \\pi _ 1 , \\pi _ 2 , \\pi _ 3 , \\dots \\} = \\begin{Bmatrix} c _ { i , j } - d _ { i , j } & i \\geq 1 \\\\ p _ j - q _ { j } & j \\geq 0 \\end{Bmatrix} . \\end{align*}"}
+{"id": "4493.png", "formula": "\\begin{align*} \\frac { 1 } { n } = \\frac { 1 } { n _ 1 } + \\cdots + \\frac { 1 } { n _ k } \\end{align*}"}
+{"id": "7805.png", "formula": "\\begin{align*} \\{ \\tilde n , \\tilde n ' \\} _ 1 & : = \\{ n , n ' \\} + \\langle n ' , m \\rangle - \\langle n , m ' \\rangle , \\\\ \\{ \\tilde m , \\tilde m ' \\} _ 1 & : = \\{ n , n ' \\} + \\langle n ' , m \\rangle - \\langle n , m ' \\rangle , \\end{align*}"}
+{"id": "6007.png", "formula": "\\begin{align*} [ \\mathcal { O } _ { X ( i , j ) \\cup X ( i - 1 , j - 1 ) } ] & = [ \\mathcal { O } _ { Y _ { h , p } ^ h } \\oplus \\mathcal { O } _ { Y _ { h - 1 , p } ^ { h - 1 } } ] - [ \\widetilde { S \\otimes T } _ { / < I _ 1 , I _ 2 } ] \\\\ & = [ \\mathcal { O } _ { Y _ { h , p } ^ h } ] + [ \\mathcal { O } _ { Y _ { h - 1 , p } ^ { h - 1 } } ] - [ \\mathcal { O } _ { Y _ { h + 1 , p } ^ { h - 1 } } ] . \\end{align*}"}
+{"id": "5550.png", "formula": "\\begin{align*} A ( x ) : = \\sum _ { 1 \\leq n \\leq x } \\mu ( n ) = O _ { \\epsilon } ( x ^ { \\frac { 1 } { 2 } + \\epsilon } ) . \\end{align*}"}
+{"id": "5101.png", "formula": "\\begin{align*} [ u _ 0 \\vec u ^ { \\vec n } ] F = \\sum _ { i = 1 } ^ l c _ { i , 0 } \\vec c _ i ^ { \\vec n } = g _ 1 ( \\vec n ) + \\cdots + g _ r ( \\vec n ) , \\end{align*}"}
+{"id": "8332.png", "formula": "\\begin{align*} \\Box _ { s , y } v & = 5 I ^ { - 2 } \\overline { Q _ l } ^ 4 v + 1 0 I ^ { - 1 } \\overline { Q _ l } ^ 3 v ^ 2 + 1 0 \\overline { Q _ l } ^ 2 v ^ 3 + 5 I \\overline { Q _ l } v ^ 4 + I ^ 2 v ^ 5 \\\\ & = : \\sum _ { i = 1 } ^ 5 G _ j ^ { ( 3 ) } v ^ j . \\end{align*}"}
+{"id": "6758.png", "formula": "\\begin{align*} \\rho ( t ) = 1 - e ^ { - t } . \\end{align*}"}
+{"id": "4156.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda ( u g , \\triangle \\omega ) = 0 \\Longleftrightarrow \\int _ M - ( \\mu _ i + 1 2 ) a _ i ^ 2 - 1 0 a _ i b _ i \\mu _ i - ( \\mu _ i ^ 2 + 2 \\mu _ i ) b _ i ^ 2 = 0 \\end{align*}"}
+{"id": "6692.png", "formula": "\\begin{align*} \\frac { 1 } { \\log x } \\sum _ { n \\leq x } \\frac { \\mu ( n ) \\mu ( n + a ) } { n } = O \\left ( \\frac { 1 } { ( \\log \\log \\log x ) ^ c } \\right ) , \\end{align*}"}
+{"id": "7150.png", "formula": "\\begin{align*} \\displaystyle { \\frac { d z _ m } { d t _ k } = - \\frac { 1 } { c } \\sum \\limits _ { \\substack { | I | = k \\\\ m \\in I } } \\prod \\limits _ { \\substack { i \\in I \\\\ j \\notin I } } \\phi ( z _ j - z _ i ) \\prod _ { i \\in I } e ^ { - v _ i / c } \\ , . } \\end{align*}"}
+{"id": "8777.png", "formula": "\\begin{align*} \\dot { \\phi } ^ N _ i = & \\omega _ i ^ N ( t ) + \\frac { 1 } { N } \\sum _ { j = 1 } ^ N W ^ N _ { i j } g ( \\phi ^ N _ j - \\phi ^ N _ i ) , 0 < t \\le T ^ * , \\\\ \\dot { W } ^ N _ { i j } = & - \\varepsilon ( W ^ N _ { i j } + h ( \\phi _ j ^ N - \\phi ^ N _ i ) ) , 0 < t \\le T ^ * \\\\ \\phi ^ N _ i ( 0 ) = & \\varphi ^ N _ { i } , W ^ N _ { i j } ( 0 ) = W ^ N _ { i , j , 0 } , i , j = 1 , \\ldots , N , \\end{align*}"}
+{"id": "4357.png", "formula": "\\begin{align*} L u = f , \\end{align*}"}
+{"id": "5721.png", "formula": "\\begin{align*} L _ { C } ^ { A } L _ { D } ^ { B } \\epsilon _ { A B } = \\epsilon _ { C D } \\end{align*}"}
+{"id": "818.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n ^ { 1 / 4 } } \\sum _ { k = 0 } ^ { n ^ { 1 / 4 } } \\sigma _ \\ast ^ k \\widehat { \\nu } ( C _ { n , t _ 1 , \\ldots , t _ d } ^ { \\star } ( x ) ) = F _ { t _ 1 , \\ldots , t _ d } ( x ) , \\end{align*}"}
+{"id": "7577.png", "formula": "\\begin{align*} A _ 0 ( x ) = \\left \\{ \\begin{array} { l l } k _ 1 , & x \\in \\Omega _ 1 , \\\\ k _ 2 , & x \\in \\Omega \\backslash \\overline { \\Omega } _ 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "7480.png", "formula": "\\begin{align*} \\mathcal { P } _ { A ^ c } ( u \\to v ) = \\{ P \\in \\mathcal { P } ( u \\to v ) : w P w \\notin A \\} . \\end{align*}"}
+{"id": "8738.png", "formula": "\\begin{align*} \\begin{array} { l l l } X _ l : = [ u ^ * _ { l - 1 } , u ^ * _ { l - 2 } , \\dots , u ^ * _ { l - 2 T + 1 } ] ^ * \\in \\mathbb { R } ^ { ( 2 T - 1 ) r } \\\\ \\bar { X } _ l : = [ u ^ * _ { l - 2 T } , u ^ * _ { l - 2 T - 1 } , \\dots , u ^ * _ 0 , 0 , \\dots , 0 ] ^ * \\in \\mathbb { R } ^ { \\bar { N } r } \\\\ W _ l : = [ w ^ * _ { l - 1 } , w ^ * _ { l - 2 } , \\dots , w ^ * _ 0 , 0 , \\dots , 0 ] ^ * \\in \\mathbb { R } ^ { N d _ 0 } , \\end{array} \\end{align*}"}
+{"id": "4472.png", "formula": "\\begin{align*} u = \\frac { n + 1 } { 4 } \\end{align*}"}
+{"id": "3168.png", "formula": "\\begin{align*} r ( y ) = \\left ( \\int _ 0 ^ 1 \\frac { \\mathrm { d } t } { a _ { 1 1 } ( t ) } \\right ) ^ { - 1 } \\frac { 1 } { \\tilde { a } _ { 1 1 } ( y _ 1 ) } \\quad y = ( y _ 1 , \\dots , y _ n ) \\in \\R ^ n . \\end{align*}"}
+{"id": "7814.png", "formula": "\\begin{align*} \\tilde B = \\begin{pmatrix} B \\\\ I \\end{pmatrix} . \\end{align*}"}
+{"id": "3418.png", "formula": "\\begin{align*} P _ { G } ( \\phi ) : = \\limsup \\limits _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\log Z _ n ( \\phi , a ) , \\end{align*}"}
+{"id": "8481.png", "formula": "\\begin{align*} x _ { \\xi ^ { n + 1 } } = \\varphi _ 1 x _ { \\xi ^ n } + \\lambda x _ { \\xi ^ { n - 1 } } + \\varphi _ 3 x _ { \\xi ^ { n - 2 } } + \\cdots + \\varphi _ n x _ { \\xi } . \\end{align*}"}
+{"id": "3049.png", "formula": "\\begin{align*} - D ^ 2 : ( r A ) = 0 \\quad Y , r Y , r > 0 , \\int _ Y r = 1 , \\end{align*}"}
+{"id": "1172.png", "formula": "\\begin{align*} \\mu _ { r + 1 } = \\mu _ { r + 1 } ( n , p ) : = \\binom { n } { r + 1 } ( 1 - p ) ^ { \\binom { r + 1 } { 2 } } \\end{align*}"}
+{"id": "4311.png", "formula": "\\begin{align*} j ( g g ' , z ) = j ( g , g ' . z ) j ( g ' , z ) . \\end{align*}"}
+{"id": "5762.png", "formula": "\\begin{align*} q \\left ( \\xi + \\sum _ { i = 1 } ^ r s _ i d \\log f _ i + \\sum _ { i = 1 } ^ { r } s _ i ( - d \\log t _ i ) \\right ) ( v ) = & \\left ( \\xi + \\sum _ { i = 1 } ^ r s _ i d \\log f _ i \\right ) ( v ) - \\sum _ { i = 1 } ^ { r } s _ i \\frac { d f _ i } { t _ i } ( v ) \\\\ = & \\xi ( v ) + \\sum _ { i = 1 } ^ r s _ i \\left ( \\frac { d f _ i } { f _ i } ( v ) - \\frac { d f _ i } { t _ i } ( v ) \\right ) \\\\ = & \\xi ( v ) \\\\ = & 0 \\end{align*}"}
+{"id": "8866.png", "formula": "\\begin{align*} \\tau _ { p , q , \\gamma } : = \\frac { ( p + q + \\gamma + 1 ) ( \\gamma + 1 ) _ { p } ( \\gamma + 1 ) _ { q } } { ( \\gamma + 1 ) p ! q ! } . \\end{align*}"}
+{"id": "4905.png", "formula": "\\begin{align*} \\partial ^ 2 _ s ( \\rho _ 2 - \\rho _ 1 ) = e ^ { - \\rho _ 2 } \\big ( e ^ { \\rho _ 2 - \\rho _ 1 } - 1 \\big ) . \\end{align*}"}
+{"id": "1098.png", "formula": "\\begin{align*} \\mathbb { E } [ ( \\theta _ j - \\overline { Z } _ j ) ^ 2 ] = \\left ( \\mathbb { E } [ \\overline { Z } _ j ] - \\theta _ j \\right ) ^ 2 + \\mathrm { V a r } ( Z _ { i , j } ) / n \\leq 2 \\varepsilon ^ 2 ( B _ 0 ^ 2 + \\theta _ j ^ 2 ) + B ^ 2 / n . \\end{align*}"}
+{"id": "2712.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } d _ { J } ( \\vec { u } ( t _ n ) - \\vec { v } _ L ( t _ n ) ) = 0 . \\end{align*}"}
+{"id": "707.png", "formula": "\\begin{align*} d A = d a + d \\oint _ \\alpha * d G ^ \\omega = d a + \\sum _ { k = 1 } ^ n \\Gamma _ k d U _ \\alpha ( w _ k ) , \\end{align*}"}
+{"id": "7059.png", "formula": "\\begin{align*} \\Omega ^ { C _ 2 } _ { * - n \\sigma } = \\dfrac { \\Omega ^ { C _ 2 } _ * \\{ 1 , \\dots , u ^ n \\} } { \\begin{matrix} u ^ { k - 1 } ( d _ { i , j } - c _ { i , j } ) = u ^ k d _ { i , j + 1 } \\\\ u ^ { k - 1 } ( q _ j - p _ j ) = u ^ k q _ { j + 1 } \\end{matrix} } \\end{align*}"}
+{"id": "1551.png", "formula": "\\begin{align*} \\lambda \\int _ { M } | \\mathrm { u } ( z ) | ^ { r ( z ) } \\ , \\ , d v _ { g } ( z ) = & \\int _ { M } | D \\mathrm { u } ( z ) | ^ { p ( z ) } \\ , \\ , d v _ { g } ( z ) + \\int _ { M } \\mu ( z ) \\ , | D \\mathrm { u } ( z ) | ^ { q ( z ) } \\ , \\ , d v _ { g } ( z ) \\\\ & - \\int _ { M } g ( z ) \\ , | \\mathrm { u } ( z ) | ^ { 1 - \\gamma ( z ) } \\ , \\ , d v _ { g } ( z ) . \\end{align*}"}
+{"id": "2090.png", "formula": "\\begin{align*} \\dim _ P ( \\mathcal { S } _ K ^ { \\ast } ( w ) ) \\geq \\dim _ P ( \\mathcal { S } _ K ^ { ( b ) \\ast } ( w ) ) \\geq \\left ( 1 - \\frac { 1 } { m } \\right ) \\dim ( K ) = \\left ( 1 - \\frac { 1 } { m } \\right ) \\frac { \\sum _ { i = 1 } ^ { m } \\log | W _ i | } { \\log b } . \\end{align*}"}
+{"id": "3789.png", "formula": "\\begin{align*} ( \\alpha _ { i } , \\alpha _ { i } ) & = ( \\alpha _ i , \\alpha _ 3 ) = ( \\alpha _ 3 , \\alpha _ i ) = \\varepsilon _ i , & ( \\alpha _ 1 , \\alpha _ 2 ) & = ( \\alpha _ 2 , \\alpha _ 1 ) = \\operatorname { I d } , & ( \\alpha _ 3 , \\alpha _ 3 ) & = \\varepsilon _ 1 \\varepsilon _ 2 . \\end{align*}"}
+{"id": "1917.png", "formula": "\\begin{align*} - \\Delta _ g u + V u & = f \\quad M _ \\sigma \\\\ \\nu _ g ( u ) & = 0 \\quad \\partial M _ \\sigma , \\end{align*}"}
+{"id": "1388.png", "formula": "\\begin{align*} \\pi ^ \\epsilon \\coloneqq D _ { \\rho | \\cdot | ^ { x + \\epsilon ( t - 1 ) } } ^ { ( k _ { t - 1 } ) } \\circ \\dots \\circ D _ { \\rho | \\cdot | ^ x } ^ { ( k _ 0 ) } ( \\pi ) \\not = 0 , \\end{align*}"}
+{"id": "5724.png", "formula": "\\begin{align*} \\Omega _ { B } ^ { A } & = d \\omega _ { B } ^ { A } + \\omega _ { C } ^ { A } \\omega _ { B } ^ { C } \\\\ \\Omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } & = d \\omega _ { B ^ { \\prime } } ^ { A ^ { \\prime } } + \\omega _ { C ^ { \\prime } } ^ { A ^ { \\prime } } \\omega _ { B ^ { \\prime } } ^ { C ^ { \\prime } } , \\end{align*}"}
+{"id": "7395.png", "formula": "\\begin{align*} \\omega ( \\varpi _ F ) q _ F ^ { - 2 s } + \\chi ^ 2 \\chi '^ { - 1 } ( \\varpi _ L ) q _ L ^ { - s } = 0 . \\end{align*}"}
+{"id": "835.png", "formula": "\\begin{align*} \\tau _ { n p } ( g = g _ 0 g _ 1 g _ 2 : g _ 1 \\in E ( t + C n ^ { - 1 / 2 } \\log n ) ) \\tau _ { n p } ( g = g _ 0 g _ 1 g _ 2 : g _ 1 \\in E ( t - C n ^ { - 1 / 2 } \\log n ) ) \\end{align*}"}
+{"id": "2998.png", "formula": "\\begin{align*} \\partial _ b ^ \\gamma \\textbf { F } ^ { a b } = \\frac { \\int _ 0 ^ q \\partial _ b ^ \\gamma \\textbf { F } ^ { a b } \\hbox { d } s } { \\int _ 0 ^ q \\hbox { d } s } = \\frac { \\int _ 0 ^ q \\partial _ s p ^ a \\hbox { d } s } { q } = 0 \\end{align*}"}
+{"id": "1348.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j , k \\leq n , j \\neq k } | f _ j ( \\tau _ k ) | ^ m \\geq \\sqrt { \\frac { \\frac { 1 } { ( G _ { f , \\tau } ^ { \\circ ^ m } ) } \\left ( \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) ^ m \\right ) ^ 2 - \\sum _ { j = 1 } ^ n | f _ j ( \\tau _ j ) | ^ { 2 m } } { n ^ 2 - n } } . \\end{align*}"}
+{"id": "736.png", "formula": "\\begin{align*} q ( z , w ) = \\frac { 1 } { 2 } \\big ( \\frac { \\partial r ( z ) } { \\partial z } + \\frac { \\partial r ( w ) } { \\partial w } - r ( z ) r ( w ) \\big ) , \\end{align*}"}
+{"id": "6539.png", "formula": "\\begin{align*} f = \\sum _ { j = j _ 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { 2 ^ { j } - 1 } f _ { j k } \\psi _ { j k } , \\end{align*}"}
+{"id": "2337.png", "formula": "\\begin{align*} C ( q ) = \\omega _ { N - 1 } ^ { 1 - N / q } C ( N , q ) ^ { - N } = \\omega _ { N - 1 } ^ { 1 - N / q } \\ ( \\frac { \\Gamma \\ ( \\frac { q } { q - N } \\ ) \\Gamma \\ ( \\frac { N ( q - 1 ) } { q - N } \\ ) } { \\Gamma \\ ( \\frac { N q } { q - N } \\ ) } \\ ) ^ { 1 - N / q } \\ ( \\frac { q ( N - 1 ) } { N } \\ ) ^ { N / q } . \\end{align*}"}
+{"id": "4191.png", "formula": "\\begin{align*} \\Lambda ( t , x ) = h ( x \\pm t ) = h ( s ) , \\phi ( t , x ) = k ( x \\pm t ) = k ( s ) . \\end{align*}"}
+{"id": "4449.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 3 0 } = \\frac { 1 } { 5 } + \\frac { 1 } { 6 } = \\frac { 1 1 } { 3 0 } \\end{align*}"}
+{"id": "1340.png", "formula": "\\begin{align*} \\left ( \\frac { \\sum _ { j = 1 } ^ n f _ j ( \\tau _ j ) } { d } \\right ) ^ r = \\left ( \\frac { \\operatorname { T r a } ( S _ { f , \\tau } ) } { d } \\right ) ^ r = \\left ( \\frac { \\sum _ { k = 1 } ^ { d } \\lambda _ k } { d } \\right ) ^ r \\leq \\frac { \\sum _ { k = 1 } ^ { d } \\lambda _ k ^ r } { d } = \\frac { 1 } { d } \\operatorname { T r a } ( S ^ r _ { f , \\tau } ) \\end{align*}"}
+{"id": "1867.png", "formula": "\\begin{align*} \\beta = \\sin ^ { - 1 } \\left ( \\frac { \\ddot { s } _ i } { \\sqrt { \\dot { s } _ i ^ 2 + \\ddot { s } _ i ^ 2 } } \\right ) = \\sin ^ { - 1 } \\left ( \\frac { p _ { s _ i } \\sigma + q _ { s _ i } } { \\sqrt { \\dot { s } _ i ^ 2 + ( p _ { s _ i } \\sigma + q _ { s _ i } ) ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "2624.png", "formula": "\\begin{align*} ( a _ { 1 , 2 } , a _ { 1 , 3 } , a _ { 1 , 4 } ) = ( 3 , 3 , 3 ) . \\end{align*}"}
+{"id": "1102.png", "formula": "\\begin{align*} \\mathbb { P } ( \\tilde { V } = B _ 0 ) = 1 - \\mathbb { P } ( \\tilde { V } = - B _ 0 ) = \\frac { 1 } { 2 } + \\frac { v _ j } { 2 B _ 0 } . \\end{align*}"}
+{"id": "2467.png", "formula": "\\begin{align*} W = \\left \\{ s \\in ( - R ^ 2 , - \\frac { 1 } { 4 } R ^ 2 ) : \\| h _ { \\delta } \\| _ { L ^ { 1 } ( B ( \\frac { 1 } { 2 } R ) ) } \\geq N _ { 4 } R ^ 3 \\left ( 1 + A ( R ) \\right ) ^ { - 1 } \\right \\} , \\end{align*}"}
+{"id": "3094.png", "formula": "\\begin{align*} v ^ { 1 1 } _ B = w _ B , v ^ { 2 2 } _ B = - w _ B , v ^ { 1 2 } _ B = v ^ { 2 1 } _ B \\equiv 0 ; \\end{align*}"}
+{"id": "2412.png", "formula": "\\begin{align*} E _ { 3 3 } ^ T = E _ { 3 3 } \\ . , A _ { 3 3 } ^ T = - A _ { 3 3 } - \\dot E _ { 3 3 } \\ . \\end{align*}"}
+{"id": "6277.png", "formula": "\\begin{align*} d _ { \\mathcal { M } } \\big ( \\varphi ^ { t } ( p '' ) , \\mathcal { S } \\big ) = d _ { \\mathcal { M } } \\big ( h \\circ \\varphi _ { 0 } ^ { t } \\circ h ^ { - 1 } ( p '' ) , \\mathcal { S } \\big ) \\leq d _ { \\mathcal { M } _ { 0 } } \\big ( \\varphi _ { 0 } ^ { t } \\circ h ^ { - 1 } ( p '' ) , \\mathcal { S } _ { 0 } \\big ) . \\end{align*}"}
+{"id": "5103.png", "formula": "\\begin{align*} c _ { i , 0 } = \\sum _ { j \\in J _ \\tau } g _ j ( \\vec n ) / \\vec c _ i ^ { \\vec n } \\qquad \\end{align*}"}
+{"id": "7305.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { 0 } ^ { 1 } G ^ d _ { m _ n } ( x ) d m _ n ( d x ) = 0 . \\end{align*}"}
+{"id": "3513.png", "formula": "\\begin{align*} E _ { i j } \\omega _ A = \\delta _ { i j } \\frac { p } { 2 } \\omega _ A + \\sum _ { \\substack { ( k , l ) \\in \\lambda , \\\\ A ( k , l ) = j } } \\omega _ { A _ { ( k , l ) \\to i } } , \\end{align*}"}
+{"id": "1517.png", "formula": "\\begin{align*} \\begin{aligned} \\textbf { I } _ 3 & = \\sum _ { n = 1 } ^ \\infty \\frac { ( - 1 ) ^ { n - 1 } } { n } \\left ( - r ! \\frac 1 { n ^ { r + 1 } } \\right ) \\sin \\left ( \\frac { r \\pi } { 2 } \\right ) \\\\ & = - r ! \\sin \\left ( \\frac { r \\pi } { 2 } \\right ) \\sum _ { n = 1 } ^ \\infty \\frac { ( - 1 ) ^ { n - 1 } } { n ^ { r + 2 } } \\\\ & = - r ! \\sin \\left ( \\frac { r \\pi } { 2 } \\right ) \\left ( 1 - \\frac 1 { 2 ^ { r + 1 } } \\right ) \\zeta ( r + 2 ) . \\end{aligned} \\end{align*}"}
+{"id": "4805.png", "formula": "\\begin{align*} P r [ | c | \\leq 2 ^ { - j } \\cdot 2 ^ n ] & \\leq 2 ^ { - \\big ( 1 - h ( 2 ^ { - j } ) \\big ) \\binom { n } { \\leq d } } \\\\ & \\leq 2 ^ { - ( 1 - 2 j \\cdot 2 ^ { - j } ) \\binom { n } { \\leq d } } , \\end{align*}"}
+{"id": "6027.png", "formula": "\\begin{align*} & P _ { l _ 1 + l _ 2 } ( h _ 1 , w _ { 1 , p } ) = 0 & \\mathrm { i f } \\ : p < n \\end{align*}"}
+{"id": "7214.png", "formula": "\\begin{align*} \\mathcal R ( p , \\ , q ) = 0 . \\end{align*}"}
+{"id": "6396.png", "formula": "\\begin{align*} \\partial _ 1 & : = \\partial U _ X \\times \\overline { U _ Y } \\times [ 0 , T ) , \\\\ \\partial _ 2 & : = \\{ ( X , Y ) \\in \\overline { U _ X } \\times \\partial U _ Y : \\ X \\cdot N _ Y > 0 \\} \\times [ 0 , T ) , \\\\ \\partial _ 3 & : = ( U _ X \\times U _ Y ) \\times \\{ 0 \\} , \\end{align*}"}
+{"id": "4063.png", "formula": "\\begin{align*} \\exp { [ - g _ z ( X v , \\cdot , u ( X v ) ) v _ t ( \\cdot ) ] } & = \\frac { f ( X v ) \\det D X v ( \\cdot ) } { f ^ * ( \\cdot ) } \\overline { \\Omega ^ * } \\times [ 0 , T ] \\\\ X v ( \\Omega ^ * ) & = \\Omega . \\end{align*}"}
+{"id": "5733.png", "formula": "\\begin{align*} \\mathbf { A } _ { B } ^ { A } = ( \\mathbf { L } ^ { - 1 } ) _ { C } ^ { A } d \\mathbf { L } _ { B } ^ { C } ; \\mathbf { A } _ { B ^ { \\prime } } ^ { A ^ { \\prime } } = ( \\mathbf { L } ^ { - 1 } ) _ { C ^ { \\prime } } ^ { A ^ { \\prime } } d \\mathbf { L } _ { B ^ { \\prime } } ^ { C ^ { \\prime } } \\end{align*}"}
+{"id": "1588.png", "formula": "\\begin{align*} s \\alpha \\geq d _ p ^ p ( \\mu , \\xi ) \\geq \\big | \\xi ( u ) - \\mu ( u ) | \\geq | \\eta ( u ) + ( 1 - s ) \\alpha - \\big ( \\eta ( u ) + \\alpha \\delta _ u ( u ) \\big ) \\big | = s \\alpha . \\end{align*}"}
+{"id": "2480.png", "formula": "\\begin{align*} \\mathcal H = \\mathcal H _ n \\dotplus \\vee _ { k \\neq n } \\mathcal H _ k \\ n \\geq 0 , \\end{align*}"}
+{"id": "5940.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } \\displaystyle \\phi _ r '' - \\Delta \\phi _ r + \\sum _ { s = 1 } ^ p \\alpha _ { r s } \\phi _ s = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\displaystyle \\partial _ \\nu \\phi _ r + \\sum _ { s = 1 } ^ p \\beta _ { r s } \\phi _ s = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma , \\\\ t = 0 : \\phi _ r = ( E _ r , \\widehat { U } _ 0 ) , \\phi _ r ' = ( E _ r , \\widehat { U } _ 1 ) & \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "3829.png", "formula": "\\begin{align*} \\sum \\limits _ { m \\geq 0 } c ( m ) q ^ m = 2 q ^ { - 1 } + 2 0 - 1 2 8 q ^ 3 + 2 1 6 q ^ 4 - 1 0 2 6 q ^ 7 + 1 6 1 8 q ^ 8 + \\cdots . \\end{align*}"}
+{"id": "6288.png", "formula": "\\begin{align*} K \\times \\mathcal { O } \\ni ( x , p ) \\mapsto ( t \\mapsto \\frac { d } { d \\tau } \\big | _ { \\tau = s } f \\circ \\Phi ^ { p _ 0 } _ { \\tau , t _ 0 } \\circ ( \\Phi ^ { p } _ { s , t _ 0 } ) ^ { - 1 } ( x ) ) \\in L ^ 1 _ { } ( | t _ 0 , t _ 1 | ; \\mathbb { R } ) \\end{align*}"}
+{"id": "4536.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { x _ 2 } \\leq \\frac { 1 } { 3 } + \\frac { 1 } { 5 } = \\frac { 8 } { 1 5 } < \\frac { 1 } { 2 } + \\frac { 1 } { x _ 2 - 1 } \\end{align*}"}
+{"id": "4103.png", "formula": "\\begin{align*} \\lambda ( g , H ) = R - \\frac { 1 } { 1 2 } | H | ^ 2 + 2 \\triangle f - | \\nabla f | ^ 2 \\end{align*}"}
+{"id": "5930.png", "formula": "\\begin{align*} \\widetilde C _ 1 = \\begin{pmatrix} C _ 2 \\\\ c _ 3 \\end{pmatrix} . \\end{align*}"}
+{"id": "2607.png", "formula": "\\begin{align*} k + 1 = ( r - 1 ) ( k + 1 ) - 2 \\sum _ { l , t \\in [ r ] \\setminus \\{ 1 \\} , l < t } a _ { l , t } . \\end{align*}"}
+{"id": "249.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 5 ( i \\rho , i \\tau , \\xi + i \\eta ) = - ( 1 - \\alpha ) \\tau ^ 2 - \\beta \\tau \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "2958.png", "formula": "\\begin{align*} \\begin{aligned} W _ { j - 1 } ( W ^ + _ j - W _ j ) = ( W _ { j - 1 } W _ j ^ + - W _ j ^ 2 ) + ( W _ j ^ 2 - W _ { j - 1 } W _ j ) . \\end{aligned} \\end{align*}"}
+{"id": "3825.png", "formula": "\\begin{align*} a _ 2 ( h ) & = 8 1 6 \\\\ a _ { 1 , 1 } ( h ) & = 3 3 4 8 0 . \\end{align*}"}
+{"id": "5818.png", "formula": "\\begin{align*} \\begin{cases} u _ r '' + L u _ r + \\sum _ { s = 1 } ^ p \\beta _ { r s } u _ s = 0 , \\\\ t = 0 : u _ r = ( \\ ! ( U _ 0 , e _ r ) \\ ! ) / \\| e _ r \\| , u _ r ' = ( \\ ! ( U _ 1 , e _ r ) \\ ! ) / \\| e _ r \\| \\end{cases} \\end{align*}"}
+{"id": "5154.png", "formula": "\\begin{align*} \\partial _ { \\Omega } D _ { f _ { j } } G _ { j } ( \\lambda , b , \\Omega , f _ { j } ) ( h _ { j } ) ( w ) & = \\partial _ { \\Omega } D _ { f _ { j } } \\mathcal { S } _ { j } ( \\lambda , b , \\Omega , f _ { j } ) ( h _ { j } ) ( w ) \\\\ & = \\mbox { I m } \\left \\lbrace h _ { j } ( w ) \\overline { w } \\overline { \\Phi _ { j } ' ( w ) } + \\Phi _ { j } ( w ) \\overline { w } \\overline { h _ { j } ' ( w ) } \\right \\rbrace . \\end{align*}"}
+{"id": "4296.png", "formula": "\\begin{align*} \\widehat { \\mathbf { H } } _ { k } = f _ ( f _ ( \\mathbf { H } _ { k } ) ) \\end{align*}"}
+{"id": "7766.png", "formula": "\\begin{align*} m _ \\alpha ( x | \\mu ) \\equiv \\Pr ( x | \\mu ) & = \\int _ 0 ^ \\infty \\Pr ( x , y | \\mu ) d y = \\int _ 0 ^ \\infty f _ \\alpha ( x | y ) f ( y | \\mu ) d y \\end{align*}"}
+{"id": "8351.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ { 1 , M , B } } \\left ( \\frac 1 2 \\left | ( \\partial _ t - \\partial _ r ) u \\right | ^ 2 + \\frac 1 2 \\left | \\frac { \\partial _ \\omega u } { r } \\right | ^ 2 - \\frac { d - 2 } { 2 d } | u | ^ { \\frac { 2 d } { d - 2 } } \\right ) \\mathrm { d } t \\mathrm { d } S = \\varepsilon ( M ) . \\end{align*}"}
+{"id": "1697.png", "formula": "\\begin{align*} \\delta s ( \\mu _ m ) = \\frac { 1 } { 2 } \\mu _ m ( P _ { k - 2 } ) ( k - 1 ) f _ { \\frac { k } { 2 } } + \\frac { 1 } { 2 } \\mu _ m ( P _ { 0 } ) ( k - 1 ) f _ { - \\frac { k } { 2 } } \\end{align*}"}
+{"id": "6305.png", "formula": "\\begin{align*} \\rho ( f ) \\ , ( x _ 1 , \\ldots , x _ n ) = f ( x _ n , \\ldots , x _ 1 ) . \\end{align*}"}
+{"id": "1128.png", "formula": "\\begin{align*} c _ { ( a , 1 ^ b ) ( c ^ k ) } ^ { ( a + c , c ^ { k - 1 } , 1 ^ b ) } \\ge c _ { ( a , 1 ^ b ) ( c ) } ^ { ( a + c , 1 ^ b ) } \\ge c _ { ( a ) ( c ) } ^ { ( a + c ) } = 1 \\end{align*}"}
+{"id": "3593.png", "formula": "\\begin{align*} \\Omega _ \\lambda = \\kappa ( \\lambda ) | m _ \\lambda ) , ( \\lambda \\in \\mathbb { P } , \\ell ( \\lambda ) \\leq p ) . \\end{align*}"}
+{"id": "6429.png", "formula": "\\begin{align*} v _ { k + 2 } ( X , Y , t ) - v _ { k + 1 } ( X , Y , t ) = ( \\mathcal { T } v _ { k } ) ( X , Y , t ) - ( \\mathcal { T } v _ { k } ) ( X , Y , t ) = 0 . \\end{align*}"}
+{"id": "6839.png", "formula": "\\begin{align*} \\nu ^ { \\pm } _ c ( Y _ 1 , Y _ 2 , Y _ 3 , Y _ 4 ) = \\nu ^ { \\pm } _ c ( Y _ 1 , Y _ 2 , Y _ 3 ) + \\nu ^ { \\pm } _ c ( Y _ 1 , Y _ 3 , Y _ 4 ) = \\nu ^ { \\pm } _ c ( Y _ 1 , Y _ 2 , Y _ 4 ) + \\nu ^ { \\pm } _ c ( Y _ 2 , Y _ 3 , Y _ 4 ) , \\end{align*}"}
+{"id": "2550.png", "formula": "\\begin{align*} Z ( \\sigma _ { r } ( v ) ; \\alpha ) = Z ( \\sigma _ { r } \\tau ( v ) ; \\alpha ) \\end{align*}"}
+{"id": "7335.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { m ( x ) G _ m ( x ) j ( x , \\infty ) } { m ( x ) ^ 2 \\int _ { 0 } ^ { x } j ( y , \\infty ) d y } = 0 . \\end{align*}"}
+{"id": "172.png", "formula": "\\begin{align*} \\beta _ p = \\sum _ { j \\leq \\log n / \\log p } \\# \\{ 1 \\leq k \\leq n : p ^ j \\mid k \\} = \\sum _ { j \\leq \\log n / \\log p } \\left \\lfloor \\dfrac { n } { p ^ j } \\right \\rfloor , \\end{align*}"}
+{"id": "9251.png", "formula": "\\begin{align*} \\tilde { d } = d _ { ( 1 , \\ldots , n ) , ( v _ 1 , \\ldots , v _ { w ^ { - 1 } ( k ) } , \\ldots , v _ { h ( w ^ { - 1 } ( k ) ) } , X v _ { w ^ { - 1 } ( k ) } , \\ldots \\widehat { v _ { \\ell } } , \\ldots v _ n ) } \\neq 0 , \\end{align*}"}
+{"id": "8058.png", "formula": "\\begin{align*} \\Psi ^ \\pm ( x ) = \\int _ { ( \\sigma ) } y ^ { - s } \\gamma ^ { \\pm } ( s ) \\tilde g ( - s ) \\frac { d s } { 2 \\pi i } \\end{align*}"}
+{"id": "7810.png", "formula": "\\begin{align*} - \\{ p _ 1 ^ * ( n ) , \\tilde m ' \\} _ 1 & = \\{ ( m ' , n ' ) , ( p ^ * ( n ) , n ) \\} _ 1 \\\\ & = \\{ n ' , n \\} + \\langle n , m ' \\rangle - \\langle n ' , p ^ * ( n ) \\rangle \\\\ & = \\langle n , m ' \\rangle = \\langle n , \\tilde m ' \\rangle _ 1 . \\end{align*}"}
+{"id": "1742.png", "formula": "\\begin{align*} \\frac { \\partial P _ n } { \\partial x } ( - \\bar \\beta , \\bar \\alpha ) = \\left ( ( n + \\lambda ) ( \\bar \\beta y + \\bar \\alpha x ) \\beta - ( n - \\lambda ) ( \\alpha y - \\beta x ) \\bar \\alpha \\right ) P _ { n - 1 } ( - \\bar \\beta , \\bar \\alpha ) = \\left ( ( n + \\lambda ) y - 2 n ( \\alpha y - \\beta x ) \\bar \\alpha \\right ) P _ { n - 1 } ( - \\bar \\beta , \\bar \\alpha ) , \\end{align*}"}
+{"id": "5924.png", "formula": "\\begin{align*} \\widehat { \\Lambda } ^ T E = \\widehat { \\Lambda } E = \\mu E \\hbox { a n d } \\widehat D ^ T E = 0 . \\end{align*}"}
+{"id": "9141.png", "formula": "\\begin{align*} N _ n ( m ) \\ ; = \\ ; m ( n + m ) ^ { n - 1 } . \\end{align*}"}
+{"id": "3702.png", "formula": "\\begin{align*} | D | ^ { 3 / 2 } | D | ^ { - 1 / 2 } | D | ^ { - 1 / 2 } \\cdot \\zeta ( 2 ) = | D | ^ { 1 / 2 } \\zeta ( 2 ) . \\end{align*}"}
+{"id": "2097.png", "formula": "\\begin{align*} \\dim _ H ( ( A ^ { \\ast } ( w ) + B ) \\cap K ) = \\dim ( K ) . \\end{align*}"}
+{"id": "2557.png", "formula": "\\begin{align*} \\frac { ( \\beta ) _ { m _ 1 } } { ( \\beta ) _ { m _ q + 1 } } = \\left ( \\prod _ { i = 1 } ^ { q - 1 } \\frac { ( \\beta ) _ { m _ i + c ^ { ' } _ { i } } } { ( \\beta ) _ { m _ { i + 1 } } } \\right ) \\left ( \\prod _ { i = 1 } ^ { q - 1 } \\frac { 1 } { ( m _ i + \\beta ) ^ { c ^ { ' } _ { i } } } \\right ) \\frac { 1 } { m _ q + \\beta } , \\end{align*}"}
+{"id": "7905.png", "formula": "\\begin{align*} f '' + A ( z ) f = 0 , \\end{align*}"}
+{"id": "1773.png", "formula": "\\begin{align*} j _ B ( S ) \\cap T \\ = \\ j _ T ( S \\cap T ) \\end{align*}"}
+{"id": "4612.png", "formula": "\\begin{align*} q r \\leq \\prod _ { i = 1 } ^ n b _ i - 1 . \\end{align*}"}
+{"id": "1271.png", "formula": "\\begin{align*} h & \\geq | L ( T ) | - m - 2 + | E ( H ) | = | L ( T ) | - m - 2 + | V ( H ) | - 1 \\\\ & = | L ( T ) | - m - 2 + | B ( T ) | - 1 = | L ( T ) | + | B ( T ) | - m - 3 \\geq k . \\end{align*}"}
+{"id": "8231.png", "formula": "\\begin{align*} I ( u ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { N } } | \\nabla u | ^ 2 d x + \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { N } } V ( | x | ) f ( u ) ^ 2 d x - \\int _ { \\mathbb { R } ^ { N } } K ( | x | ) G ( f ( u ) ) \\ , d x \\end{align*}"}
+{"id": "3547.png", "formula": "\\begin{align*} B _ j ^ - = \\mathbf { p } B _ j ^ - + \\sum _ { i = j + 1 } ^ { n } \\sum _ { s = 2 } ^ { i - j + 1 } \\sum _ { I \\in \\mathcal { I } _ { j i } ( s ) } E ^ { e _ I } \\mathbf { p } B _ i ^ - \\frac { 1 } { \\prod _ { \\ell \\in I , \\ell \\neq i } ( h _ i - h _ \\ell ) } . \\end{align*}"}
+{"id": "2188.png", "formula": "\\begin{align*} \\sigma _ { 0 } & = 2 ( \\mathit { e } - 1 ) \\big ( ( 2 - 2 \\alpha + \\beta ) ( \\mathit { e } + 1 ) \\\\ & + \\sqrt { ( ( \\mathit { e } + 1 ) ( 2 - 2 \\alpha + \\beta ) ) ^ 2 - 4 ( \\mathit { e } - 1 ) ( 3 + \\mathit { e } - 2 \\alpha - 2 \\alpha \\mathit { e } - \\beta - \\beta \\mathit { e } ) } \\big ) ^ { - 1 } . \\end{align*}"}
+{"id": "7413.png", "formula": "\\begin{align*} q _ { \\alpha } = q _ F , q _ { \\alpha ^ * } = 1 \\end{align*}"}
+{"id": "1060.png", "formula": "\\begin{align*} \\sum _ { j \\geq - 1 } ( 2 ^ j ) ^ { 2 \\beta } \\| \\beta _ { j \\cdot } \\| _ 2 ^ 2 = \\sum _ { j \\geq - 1 } ( 2 ^ j ) ^ { 2 \\beta } \\left ( \\sum _ { k \\in \\mathcal { N } _ j } \\beta ^ 2 _ { j k } \\right ) < \\infty . \\end{align*}"}
+{"id": "2294.png", "formula": "\\begin{gather*} \\omega _ 0 - \\omega _ 2 = O ( \\Delta x ^ 4 ) , \\\\ \\omega _ k - d _ k = O ( \\Delta x ^ 3 ) . \\end{gather*}"}
+{"id": "2073.png", "formula": "\\begin{align*} & \\left | p _ 1 \\exp ( i \\eta x _ 1 ) Z _ 1 + p _ { - 1 } \\exp ( - i \\eta x _ 1 ) Z _ { - 1 } \\right | ^ 2 \\\\ & = p _ 1 ^ 2 | Z _ 1 | ^ 2 + p _ { - 1 } ^ 2 | Z _ { - 1 } | ^ 2 + p _ 1 p _ { - 1 } ( Z _ 1 \\overline { Z _ { - 1 } } \\exp ( i 2 \\eta x _ 1 ) + \\overline { Z _ 1 } Z _ { - 1 } \\exp ( - i 2 \\eta x _ 1 ) ) \\\\ & = p _ 1 ^ 2 | Z _ 1 | ^ 2 + p _ { - 1 } ^ 2 | Z _ { - 1 } | ^ 2 + 2 p _ 1 p _ { - 1 } | Z _ 1 | | Z _ { - 1 } | \\cos ( 2 \\phi _ 1 + 2 x _ 1 \\eta ) . \\end{align*}"}
+{"id": "6152.png", "formula": "\\begin{align*} C _ 1 = \\begin{pmatrix} 1 & - 1 \\\\ & 1 & - 1 \\\\ & & \\ddots & \\ddots \\\\ & & & 1 & - 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "6932.png", "formula": "\\begin{align*} S ( x ) \\log x & = S _ 1 + S _ 2 + S _ 3 \\\\ & \\leq \\left ( 1 + A h _ 1 ( x ) + B h _ 2 ( x ) \\right ) x L ( x ) . \\\\ \\end{align*}"}
+{"id": "3081.png", "formula": "\\begin{align*} r = d \\ , \\frac { r _ B } { a } \\quad d : = \\left ( \\int _ Y \\frac { r _ B } { a } \\right ) ^ { - 1 } = \\bar { a } , \\end{align*}"}
+{"id": "4646.png", "formula": "\\begin{align*} H _ { \\ell , k } = \\frac { \\ell ! } { 2 ^ k k ! ( \\ell - 2 k ) ! } , \\ \\ k = 0 , \\ldots , \\lfloor \\ell / 2 \\rfloor , \\end{align*}"}
+{"id": "258.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\rho } { \\tau } + \\frac { \\beta \\eta } { \\tau } \\leqslant \\frac { 2 \\alpha \\rho ^ 2 } { ( 1 + 3 \\rho ^ 2 ) } \\leqslant \\frac { 2 \\alpha } { 3 } , \\end{align*}"}
+{"id": "5855.png", "formula": "\\begin{align*} \\psi = e ^ { \\lambda h } \\phi . \\end{align*}"}
+{"id": "8131.png", "formula": "\\begin{align*} L _ 2 = \\frac { T ^ 6 c ^ 3 m ^ 3 p n _ 1 ^ 3 c '^ 3 } { \\pi ^ 6 N ^ 2 l _ 1 ^ 2 } . \\end{align*}"}
+{"id": "1962.png", "formula": "\\begin{align*} \\rho ( i ) = \\begin{cases} 0 , \\ldots , k & i = 0 , \\ldots , k , \\\\ 1 - k , \\ldots , - 1 & i = k + 1 , \\ldots , 2 k - 1 , \\end{cases} \\end{align*}"}
+{"id": "6302.png", "formula": "\\begin{align*} \\pi _ i ( s _ { i + 1 } ( T ) ) = \\pi _ { i + 1 } \\pi _ i ( s _ { i + 1 } ( T ) ) , \\end{align*}"}
+{"id": "1079.png", "formula": "\\begin{align*} \\mu _ 1 = \\mathbb { E } _ { R _ 1 } ( X ) = ( 1 - \\eta ) ( D - 1 ) + \\eta \\left ( \\{ D - 1 + \\left ( 2 \\eta \\right ) ^ { - 1 / k } \\} \\right ) = D - 1 + 2 ^ { - 1 / k } \\eta ^ { 1 - 1 / k } \\leq D \\end{align*}"}
+{"id": "7336.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } \\left ( \\frac { d ^ 2 } { d x ^ 2 } + b ( x ) \\frac { d } { d x } \\right ) . \\end{align*}"}
+{"id": "2691.png", "formula": "\\begin{align*} \\frac { \\mu _ 0 ( U ) } { | U | ^ s } = \\mathcal { H } ^ s ( K _ 0 ) ^ { - 1 } \\end{align*}"}
+{"id": "6391.png", "formula": "\\begin{align*} ( ( p - 2 ) \\Delta _ { \\infty , X } ^ N + \\Delta _ X ) u ( X , t ) - ( m + p ) \\partial _ t u ( X , t ) = 0 , \\end{align*}"}
+{"id": "8146.png", "formula": "\\begin{align*} \\tilde H _ { m , n } ^ { - , 1 } ( x ) & = 4 T \\sum _ { 0 \\leq l \\leq L } \\sum _ { 0 \\leq l _ 1 \\leq l } d ' _ { l , l _ 1 } \\frac { x ^ l } { M ^ { 5 l - 2 l _ 1 } } { k ^ * } ^ { ( 5 l - 2 l _ 1 ) } \\Bigl ( \\frac { x - 2 T } { 2 M } \\Bigr ) + O \\Bigl ( \\frac { T \\abs { x } ^ { L + 1 } } { M ^ { 3 L + 3 } } + \\frac { T \\abs { x } } { M ^ 7 } \\Bigr ) \\end{align*}"}
+{"id": "6633.png", "formula": "\\begin{align*} \\Bigl \\lVert \\frac { x _ j ^ m } { 1 + \\epsilon x ^ { 2 m } } \\psi \\Bigr \\rVert ^ { 1 / m } \\leq \\norm { x ^ m \\psi } ^ { 1 / m } + C t \\sum _ { k = 0 } ^ m \\norm { H ^ k \\psi } ^ { 1 / m } , \\end{align*}"}
+{"id": "4331.png", "formula": "\\begin{align*} M = \\sum \\limits _ { u \\in V } M ( u ) \\end{align*}"}
+{"id": "1540.png", "formula": "\\begin{align*} \\Omega ^ A = \\frac 1 2 \\Omega ^ A _ { B C } \\varpi ^ B \\wedge \\varpi ^ C \\end{align*}"}
+{"id": "7115.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ k L T ( q _ t ) ( u x _ { i _ t } + p _ { i _ t } ) = \\sum _ { t = 1 } ^ k L T ( q _ t ) p _ { i _ t } & = \\sum _ { t = 1 } ^ k c _ t M ( q _ t ) p _ { i _ t } \\\\ & = \\sum _ { t = 1 } ^ { k - 1 } \\left ( c _ 1 + \\dots + c _ t \\right ) ( M ( q _ t ) p _ { i _ t } - M ( q _ { t + 1 } ) p _ { i _ { t + 1 } } ) \\\\ & = \\sum _ { t = 1 } ^ { k - 1 } \\left ( c _ 1 + \\dots + c _ t \\right ) \\frac { M ( q _ t ) } { x _ { i _ { t + 1 } } } ( x _ { i _ { t + 1 } } p _ { i _ t } - x _ { i _ t } p _ { i _ { t + 1 } } ) . \\end{align*}"}
+{"id": "929.png", "formula": "\\begin{align*} \\lim _ { \\delta \\downarrow 0 } \\frac { \\tau _ { \\mu , \\alpha } ( h + \\delta ) - \\tau _ { \\mu , \\alpha } ( h ) } { \\delta } & = \\lim _ { \\delta \\downarrow 0 } \\int _ { 0 } ^ { 1 } \\frac { f ( Y _ { \\alpha } ^ { h + \\delta } ( t ) ) - f ( Y _ { \\alpha } ^ { h } ( t ) ) } { \\delta } \\mathrm { d } t \\\\ & = \\int _ { 0 } ^ { 1 } \\lim _ { \\delta \\downarrow 0 } \\frac { f ( Y _ { \\alpha } ^ { h + \\delta } ( t ) ) - f ( Y _ { \\alpha } ^ { h } ( t ) ) } { \\delta } \\mathrm { d } t , \\end{align*}"}
+{"id": "7645.png", "formula": "\\begin{align*} d _ h ( \\lambda ) = \\lambda \\times \\Pi _ { j = 2 } ^ { n _ 0 } ( \\lambda - \\beta _ { j , h } ) + O ( h ^ { 5 / 2 } h ^ { 2 ( n _ 0 - 1 ) } ) = \\det F _ h ( \\lambda ) ( 1 + o _ h ( 1 ) ) . \\end{align*}"}
+{"id": "4842.png", "formula": "\\begin{align*} V _ n = \\bigcap _ { i = 1 } ^ n U _ i . \\end{align*}"}
+{"id": "4706.png", "formula": "\\begin{align*} \\mathbb { R } ^ { G _ { I V } } = \\mathbb { C } [ \\varphi _ 2 , \\varphi _ 6 ] \\end{align*}"}
+{"id": "1787.png", "formula": "\\begin{align*} X ^ \\diamondsuit ( S ) = \\{ ( S ^ \\sharp , \\iota , f ) \\} / \\mathrm { i s o m . } \\end{align*}"}
+{"id": "4446.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } + \\frac { 1 } { 1 2 } = \\frac { 1 } { 4 } + \\frac { 1 } { 6 } = \\frac { 5 } { 1 2 } \\end{align*}"}
+{"id": "7839.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { j = 1 } ^ m P _ j ( z ) e ^ { Q _ j ( z ) } , \\end{align*}"}
+{"id": "8178.png", "formula": "\\begin{align*} u ( t , x ) = u _ 0 ( x ) + \\frac { \\alpha } { \\beta \\Gamma ( \\beta ) } \\int _ 0 ^ t s ^ { \\frac { \\alpha } { \\beta } - 1 } \\left ( t ^ { \\frac { \\alpha } { \\beta } } - s ^ { \\frac { \\alpha } { \\beta } } \\right ) ^ { \\beta - 1 } \\frac 1 2 \\Delta u ( s , x ) d s . \\end{align*}"}
+{"id": "6638.png", "formula": "\\begin{align*} d ( ( t , x ) , ( t ' , x ' ) ) : = | t - t ' | ^ { \\frac { 1 } { 2 } } + | x - x ' | \\end{align*}"}
+{"id": "6725.png", "formula": "\\begin{align*} V _ 0 ( x ) = \\frac { 1 } { q } \\sum _ { n \\leq x } \\mu ( n + a ) \\ll \\frac { 1 } { q } \\frac { x } { ( \\log x ) ^ { D _ 0 } } , \\end{align*}"}
+{"id": "4043.png", "formula": "\\begin{align*} D _ { x _ i } Q _ j = - \\frac { E _ { i j } } { g _ z } . \\end{align*}"}
+{"id": "6168.png", "formula": "\\begin{align*} ( E _ 1 , \\widehat U _ 0 ) = ( E _ 1 , \\widehat U _ 1 ) = 0 \\end{align*}"}
+{"id": "4301.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { H } ( u ; t ) = 0 , \\end{align*}"}
+{"id": "475.png", "formula": "\\begin{align*} \\phi ^ * ( b ( u ) ) = b ( u ) u - \\phi ( u ) \\mbox { f o r a l l } u \\geq 0 . \\end{align*}"}
+{"id": "7333.png", "formula": "\\begin{align*} I _ 1 ( \\gamma ) & = \\int _ { 0 } ^ { \\gamma } m ( x ) ^ 2 d x \\int _ { 0 } ^ { x } j ( y , \\infty ) d y , I _ 2 ( \\gamma ) = \\int _ { 0 } ^ { \\gamma } m ( x ) G _ m ( x ) j ( x , \\infty ) d x , \\\\ I _ 3 ( \\gamma ) & = - m ( \\gamma ) G _ m ( \\gamma ) \\int _ { 0 } ^ { \\gamma } j ( x , \\infty ) d x , \\\\ I _ 4 ( \\gamma ) & = j ( \\gamma , \\infty ) \\left ( \\gamma m ( \\gamma ) G _ m ( \\gamma ) - ( 1 / 2 ) G _ m ( \\gamma ) ^ 2 - \\int _ { 0 } ^ { \\gamma } x m ( x ) ^ 2 d x \\right ) . \\end{align*}"}
+{"id": "5473.png", "formula": "\\begin{align*} \\tilde L \\tilde V ( x - y ) : = & ( b ( t , x , \\mu , 1 ) - b ( t , y , \\nu , 1 ) ) \\nabla \\tilde V ( x - y ) \\\\ & + \\frac { 1 } { 2 } t r ( \\nabla ^ 2 \\tilde V ( x - y ) A ( t , x , y , \\mu , \\nu , 1 ) ) , \\end{align*}"}
+{"id": "4517.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k v _ i \\leq \\prod _ { i = 1 } ^ k u _ i \\end{align*}"}
+{"id": "5404.png", "formula": "\\begin{align*} v _ { n + 1 } = A _ n v _ n \\end{align*}"}
+{"id": "9103.png", "formula": "\\begin{align*} ( \\hat { \\mathsf { R } } _ 1 , \\hat { \\mathsf { R } } _ 2 ) = \\left ( \\hat { \\mathsf { R } } _ { \\mathrm { F O } , 1 } - \\frac { \\hat { \\mathsf { R } } _ { \\mathrm { S O } , 1 } } { \\sqrt { n } } , \\hat { \\mathsf { R } } _ { \\mathrm { F O } , 2 } - \\frac { \\hat { \\mathsf { R } } _ { \\mathrm { S O } , 2 } } { \\sqrt { n } } \\right ) , \\end{align*}"}
+{"id": "4776.png", "formula": "\\begin{align*} c = c + \\rho + \\rho \\in d ( c + \\rho ) . \\end{align*}"}
+{"id": "640.png", "formula": "\\begin{align*} \\epsilon _ { 0 } : = \\bar { \\epsilon } _ { 0 } \\epsilon : = \\min \\left \\{ \\frac { \\bar { \\epsilon } } { 2 } , \\frac { \\kappa \\bar { \\epsilon } _ { 0 } } { 8 } , \\frac { \\beta - \\bar { \\beta } } { 2 } \\right \\} . \\end{align*}"}
+{"id": "7089.png", "formula": "\\begin{align*} \\Phi M U ^ { C _ 2 } _ * = M U _ * [ b _ 1 ' , b _ 2 ' , \\dots ] [ u ^ { \\pm 1 } ] . \\end{align*}"}
+{"id": "3276.png", "formula": "\\begin{align*} - \\nabla \\circ \\left ( \\left \\vert u \\right \\vert ^ { p - 2 } \\nabla u \\right ) - \\lambda \\left \\vert u \\right \\vert ^ { p _ { 0 } - 2 } u = 0 , u \\left \\vert _ { \\ \\partial \\Omega } \\right . = 0 , \\ \\lambda \\in \\mathbb { C } , \\end{align*}"}
+{"id": "5143.png", "formula": "\\begin{align*} J _ \\nu ( x ) = \\frac { 1 } { \\pi } \\int _ 0 ^ \\pi \\cos \\big ( x \\sin \\theta - \\nu \\theta \\big ) d \\theta . \\end{align*}"}
+{"id": "4559.png", "formula": "\\begin{align*} \\gcd ( a ' , b ' ) = 1 \\qquad a ' a n - b . \\end{align*}"}
+{"id": "2389.png", "formula": "\\begin{align*} E ( t ) Q ( t ) \\left [ \\begin{array} { c } \\dot x _ 1 ( t ) \\\\ \\dot x _ 2 ( t ) \\end{array} \\right ] = ( A ( t ) Q ( t ) - E ( t ) \\dot Q ( t ) ) \\left [ \\begin{array} { c } x _ 1 ( t ) \\\\ x _ 2 ( t ) \\end{array} \\right ] \\end{align*}"}
+{"id": "4335.png", "formula": "\\begin{align*} \\tilde { b } _ { s ( X ) , s ( g ) } & = \\tilde { b } _ { X \\otimes g ^ i \\boxtimes g ^ { - i } , g ^ { i k + 1 } \\boxtimes g ^ { - i k } } \\\\ & = b _ { X \\otimes g ^ i , g ^ { i k + 1 } } b _ { g ^ { - i } , g ^ { - i k } } ^ { - 1 } \\\\ & = b _ { X , g ^ { i k + 1 } } b _ { g ^ i , g ^ { i k + 1 } } b _ { g ^ { - i } , g ^ { - i k } } ^ { - 1 } \\\\ & = t ^ { i k + 1 } t ^ { i k ( i k + 1 ) } t ^ { - ( - i k ) ( - i k ) } \\\\ & = t ^ { 2 i k + 1 } \\end{align*}"}
+{"id": "6652.png", "formula": "\\begin{align*} \\partial _ t \\delta _ y w = \\nabla \\cdot a ( t ' , x ' ) \\nabla \\delta _ y w + \\nabla \\cdot g \\end{align*}"}
+{"id": "5951.png", "formula": "\\begin{align*} ( E _ r , B \\widehat { U } ) = 0 \\hbox { o n } ( 0 , T ) \\times \\Gamma , r = 1 , \\cdots , p . \\end{align*}"}
+{"id": "1646.png", "formula": "\\begin{align*} \\Sigma _ B = \\Sigma _ T ^ \\R \\cup \\Sigma _ T ^ \\C ; \\Sigma _ T ^ \\R = \\{ \\sigma \\in \\Sigma _ B ; \\ ; T ( F _ \\sigma ) = \\R ^ \\times \\} , \\qquad \\Sigma _ T ^ \\C = \\{ \\sigma \\in \\Sigma _ B ; \\ ; T ( F _ \\sigma ) = \\C ^ \\times \\} , \\end{align*}"}
+{"id": "7076.png", "formula": "\\begin{align*} \\Pi _ { \\rho _ 1 + \\cdots + \\rho _ n } = \\sum _ { i = 1 } ^ n \\pi _ { \\rho _ i + \\cdots + \\rho _ n } \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) . \\end{align*}"}
+{"id": "7428.png", "formula": "\\begin{align*} \\phi _ K \\colon P \\mapsto K ( Z ^ { ( 0 ) } , Z ^ { ( 0 ) } ) ( P ) : = [ K ( Z _ i , Z _ j ) ( P _ { i j } ) ] _ { 1 \\le i , j \\le N } \\end{align*}"}
+{"id": "987.png", "formula": "\\begin{align*} \\Vert \\overline { f } \\Vert _ { \\infty , \\mathcal { A } } & = \\sup \\lbrace \\Vert \\overline { f } ( z ) \\Vert _ { \\mathcal { A } } : \\ , z \\in \\beta X \\rbrace \\\\ & = \\Vert \\overline { f } ( z _ 0 ) \\Vert _ { \\mathcal { A } } \\end{align*}"}
+{"id": "2587.png", "formula": "\\begin{align*} & \\{ u : \\mathbb { R } \\rightarrow \\mathfrak { g } \\mid u \\in H ^ 0 , \\ u ( t + 2 \\pi ) = \\sigma ( u ( t ) ) \\ \\} \\\\ \\cong \\ & \\{ u : [ 0 , 2 \\pi ] \\rightarrow \\mathfrak { g } \\mid u \\in H ^ 0 \\} . \\end{align*}"}
+{"id": "5466.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { m \\geq n } E ( \\sup _ { 0 \\leq t \\leq T } | X ^ { n } _ { t \\wedge \\tau _ N ^ { n , m } } - X ^ { m } _ { t \\wedge \\tau _ N ^ { n , m } } | ^ 2 ) \\leq \\lim _ { n \\to \\infty } \\sup _ { m \\geq n } e ^ { C ( N , T ) T } C ( N , T , M , \\tilde M ) \\epsilon _ { n , m } = 0 . \\end{align*}"}
+{"id": "6326.png", "formula": "\\begin{align*} \\langle X Q , Y Q \\rangle = \\sum _ { e \\in E _ { \\Lambda } ^ { + } } ( X _ { e } Y _ { e } ^ { * } ) . \\end{align*}"}
+{"id": "9161.png", "formula": "\\begin{align*} \\nabla ^ n f ( x ) = ( \\nabla ^ { \\mu _ 1 } \\cdots \\nabla ^ { \\mu _ n } f ( x ) : \\mu _ k \\in \\hat { e } k ) , \\end{align*}"}
+{"id": "3447.png", "formula": "\\begin{align*} \\{ x ^ 4 + y ^ 4 + z ^ 4 + t ^ 4 - 6 \\left ( x ^ 2 y ^ 2 + x ^ 2 z ^ 2 + x ^ 2 t ^ 2 + y ^ 2 z ^ 2 + y ^ 2 t ^ 2 + z ^ 2 t ^ 2 \\right ) = 0 \\} . \\end{align*}"}
+{"id": "1821.png", "formula": "\\begin{align*} p ( 0 ) - R ^ T p \\bigg ( \\frac { T } { M } \\bigg ) = 0 , \\end{align*}"}
+{"id": "3059.png", "formula": "\\begin{align*} A ( y ) : = \\mathrm { d i a g } ( a _ 1 ( y ) , a _ 2 ( y ) ) \\quad y \\in \\R ^ 2 . \\end{align*}"}
+{"id": "3932.png", "formula": "\\begin{align*} q ( x ) & : = g _ z ( 0 , 0 , 0 ) \\left [ \\frac { g _ y } { g _ z } ( x , 0 , 0 ) - \\frac { g _ y } { g _ z } ( 0 , 0 , 0 ) \\right ] , \\\\ p ( y ) & : = g _ x ( 0 , y , g ^ * ( 0 , y , h ) ) - g _ x ( 0 , 0 , 0 ) . \\end{align*}"}
+{"id": "5734.png", "formula": "\\begin{align*} \\theta ^ { A A ^ { \\prime } } = ( \\mathbf { L } ^ { - 1 } ) _ { b } ^ { a } d \\mathbf { x } ^ { b } = ( \\mathbf { L } ^ { - 1 } ) _ { B ^ { \\prime } } ^ { A ^ { \\prime } } ( \\mathbf { L } ^ { - 1 } ) _ { B } ^ { A } d \\mathbf { x } ^ { B B ^ { \\prime } } , \\end{align*}"}
+{"id": "2512.png", "formula": "\\begin{align*} \\{ m : \\ , m \\mid n , \\ , 1 \\leq m < n \\} = \\bigcup _ { i = 1 } ^ t \\{ m : \\ ; m \\mid n _ i , \\ , m \\geq 1 \\} = A _ 1 \\cup \\ldots \\cup A _ t . \\end{align*}"}
+{"id": "5824.png", "formula": "\\begin{align*} \\langle L u , \\phi \\rangle = \\int _ \\Omega \\nabla u \\cdot \\nabla \\phi d x , \\langle g v , \\phi \\rangle = \\int _ { \\Gamma _ 1 } v \\phi d \\Gamma . \\end{align*}"}
+{"id": "1676.png", "formula": "\\begin{align*} g = u \\left ( \\begin{array} { c c } y & x \\\\ & y ^ { - 1 } \\end{array} \\right ) \\kappa ( \\theta ) , y \\in \\R ^ \\times , \\ ; u \\in \\R _ + , \\ ; x \\in \\R , \\ ; \\theta \\in S ^ 1 . \\end{align*}"}
+{"id": "5623.png", "formula": "\\begin{align*} _ { t _ { 0 } } ^ { A B C } D _ { t } ^ { \\alpha } f ( t ) = \\dfrac { B ( \\alpha ) } { 1 - \\alpha } \\int ^ { t } _ { t _ { 0 } } f ^ { ' } ( x ) E _ \\alpha \\left [ - \\dfrac { \\alpha } { 1 - \\alpha } ( t - x ) ^ { \\alpha } \\right ] d x . \\end{align*}"}
+{"id": "7430.png", "formula": "\\begin{align*} \\phi ( \\alpha \\cdot P \\cdot \\beta ^ * ) = L _ \\alpha \\ , \\phi ( P ) \\ , L _ { \\beta ^ * } . \\end{align*}"}
+{"id": "8168.png", "formula": "\\begin{align*} \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } H _ { m , p } & = \\frac { 1 } { \\pi } \\int _ { - \\infty } ^ \\infty k ( t ) \\Bigl ( \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } U ( m ^ 2 p , t ) \\Bigr ) \\tanh ( \\pi t ) t \\ , d t . \\end{align*}"}
+{"id": "1438.png", "formula": "\\begin{align*} \\tau _ 1 = \\tau _ 1 ( x ) \\coloneqq \\inf \\{ t \\in \\mathbb { N } : D _ t \\geq x \\} \\tau _ 2 = \\tau _ 2 ( x ) \\coloneqq \\inf \\{ t \\in \\mathbb { N } : D _ { \\tau _ h + t } \\geq x \\} . \\end{align*}"}
+{"id": "8850.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\frac { \\partial ^ { 2 } } { \\partial z _ { j } \\partial \\bar { z } _ { j } } \\end{align*}"}
+{"id": "5800.png", "formula": "\\begin{align*} K e r ( D ) \\cap I m ( C _ p ^ T ) = K e r ( C _ p ) \\cap I m ( D ) = \\{ 0 \\} , \\end{align*}"}
+{"id": "3089.png", "formula": "\\begin{align*} \\bar { a } : = \\int _ Y r a , \\bar { b } : = \\int _ Y r _ B b \\end{align*}"}
+{"id": "1766.png", "formula": "\\begin{align*} \\big [ X _ n ^ { ( a , b ) } \\big ] _ { ( i , j ) } = \\begin{cases} \\big [ X _ 4 \\big ] _ { ( i - a + 1 , j - b + 1 ) } , & ( i , j ) \\in A \\times B ; \\\\ 0 , & \\end{cases} \\end{align*}"}
+{"id": "318.png", "formula": "\\begin{align*} \\begin{aligned} 1 \\rightarrow \\prod _ { \\Delta } \\mathbb { G } _ a \\rightarrow \\prod _ { \\Delta } \\mathrm { R e s } _ { E / F } \\mathbb { G } _ { a , E } \\rightarrow \\prod _ { \\Delta } \\mathbb { G } _ a ^ { \\mathrm { o p } } \\rightarrow 1 . \\end{aligned} \\end{align*}"}
+{"id": "5569.png", "formula": "\\begin{align*} C _ { \\alpha , r } ( x , 0 ) : = \\{ ( x ' , y ' ) \\in B _ r ^ n ( x , 0 ) : | y ' | \\le \\alpha ( n , \\gamma ) | x ' | \\} . \\end{align*}"}
+{"id": "6091.png", "formula": "\\begin{align*} H _ n ( z ) = 1 + o ( 1 ) \\end{align*}"}
+{"id": "4486.png", "formula": "\\begin{align*} 2 = \\sum _ { \\substack { d | n \\\\ d \\geq 1 } } \\frac { d } { N } = \\sum _ { \\substack { d | n \\\\ d \\geq 1 } } \\frac { 1 } { d } . \\end{align*}"}
+{"id": "3847.png", "formula": "\\begin{align*} \\mu _ l ^ { [ 1 2 ] } & = \\mu _ { l - 1 } ^ { [ 1 2 ] } + \\mu _ { l - 2 } ^ { [ 1 2 ] } + \\mu _ { l - 3 } ^ { [ 1 2 ] } + \\delta _ { l , 2 } + \\delta _ { l , 4 } , \\mu _ { l < 2 } ^ { [ 1 2 ] } = 0 , \\\\ \\mu _ l ^ { [ 1 3 ] } & = \\mu _ { l - 1 } ^ { [ 1 3 ] } + \\mu _ { l - 2 } ^ { [ 1 3 ] } + \\mu _ { l - 3 } ^ { [ 1 3 ] } + 2 \\delta _ { l , 3 } , \\mu _ { l < 3 } ^ { [ 1 3 ] } = 0 , \\\\ \\mu _ l ^ { [ 2 3 ] } & = \\mu _ { l - 1 } ^ { [ 2 3 ] } + \\mu _ { l - 2 } ^ { [ 2 3 ] } + \\mu _ { l - 3 } ^ { [ 2 3 ] } + \\delta _ { l , 3 } + \\delta _ { l , 5 } , \\mu _ { l < 3 } ^ { [ 2 3 ] } = 0 . \\end{align*}"}
+{"id": "3286.png", "formula": "\\begin{align*} v ( t , x ) = u ( t , x - h ( t ) ) , \\tilde { p } = p ( t , x - h ( t ) ) , \\ell ( t ) = h ' ( t ) , \\omega ( t ) = \\omega ( t ) . \\end{align*}"}
+{"id": "5241.png", "formula": "\\begin{align*} ( \\nabla F _ \\ast ) ( X , Y ) = { \\nabla } _ { F _ \\ast X } ^ B F _ \\ast Y - F _ \\ast ( { \\nabla } _ X ^ M Y ) , ~ \\forall X , Y \\in \\Gamma ( T M ) , \\end{align*}"}
+{"id": "4158.png", "formula": "\\begin{align*} ( \\chi _ 2 g , \\frac { 1 } { 2 } d ^ * ( \\chi _ 2 d V _ g ) ) = ( \\chi _ 2 g + \\frac { 1 } { 4 } \\nabla ^ 2 \\chi _ 2 , \\frac { 1 } { 4 } d ^ * ( \\chi _ 2 d V _ g ) ) + ( - \\frac { 1 } { 4 } \\nabla ^ 2 \\chi _ 2 , \\frac { 1 } { 4 } d ^ * ( \\chi _ 2 d V _ g ) ) \\in \\mathcal { V } ^ \\perp + \\mathcal { V } , \\end{align*}"}
+{"id": "7145.png", "formula": "\\begin{align*} \\delta _ 0 \\varphi _ 0 ^ { s , m } = \\sum _ { k = 0 } ^ { r } \\ \\binom { r } { k } ( - 1 ) ^ { k + s } ( g _ { s - 2 k } \\alpha _ { s - 2 k + 1 } \\cdots \\alpha _ { s } - g _ { s - 2 k - 1 } \\alpha _ { s - 2 k } \\cdots \\alpha _ { s } ) , \\end{align*}"}
+{"id": "8239.png", "formula": "\\begin{align*} u '' + \\frac { N - 1 } { r } u ' + V ( r ) f ( u ) f ' ( u ) - K ( r ) g ( f ( u ) ) f ' ( u ) = 0 { \\mbox { i n } } \\mathbb { R } _ + . \\end{align*}"}
+{"id": "5130.png", "formula": "\\begin{align*} \\Delta _ { \\infty } ( \\lambda , b ) = \\delta _ { \\infty } ^ { 2 } ( \\lambda , b ) \\quad \\mbox { w i t h } \\quad \\delta _ { \\infty } ( \\lambda , b ) : = b \\big [ I _ { 1 } ( \\lambda ) K _ { 1 } ( \\lambda ) + I _ { 1 } ( \\lambda b ) K _ { 1 } ( \\lambda b ) \\big ] - ( 1 + b ^ { 2 } ) I _ { 1 } ( \\lambda b ) K _ { 1 } ( \\lambda ) . \\end{align*}"}
+{"id": "2470.png", "formula": "\\begin{align*} f ( y ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { n ! } D ^ n f ( x ) \\left [ ( y - x ) ^ n \\right ] \\end{align*}"}
+{"id": "7875.png", "formula": "\\begin{align*} u ' = \\frac { \\sum _ { k = 0 } ^ { n } A _ k ( z ) u ^ k } { \\sum _ { k = 0 } ^ { m } B _ k ( z ) u ^ k } , \\end{align*}"}
+{"id": "675.png", "formula": "\\begin{align*} { \\rm c o n s t a n t } = \\frac { 1 } { V } \\int _ M \\omega , \\end{align*}"}
+{"id": "5794.png", "formula": "\\begin{align*} \\sum _ { j = n _ { s - 1 } + 1 } ^ { n _ { s } } a _ { i j } = \\alpha _ { r s } , n _ { r - 1 } + 1 \\leqslant i \\leqslant n _ { r } , 1 \\leqslant r , s \\leqslant p , \\end{align*}"}
+{"id": "3942.png", "formula": "\\begin{align*} c _ { q _ i q _ j } ( t q , t p ) & = c _ { q _ i q _ j } ( t q , 0 ) + c _ { q _ i q _ j , p _ k } ( t q , 0 ) p _ k + a ^ { ( 1 ) } _ { i j , k l } ( q , p ) p _ k p _ l , \\\\ c _ { p _ i p _ j } ( t q , t p ) & = c _ { p _ i p _ j } ( 0 , t p ) + c _ { q _ k , p _ i p _ j } ( 0 , t p ) q _ k + a ^ { ( 2 ) } _ { i j , k l } ( q , p ) q _ k q _ l \\end{align*}"}
+{"id": "474.png", "formula": "\\begin{align*} \\phi ^ * ( x ) : = \\sup _ { v \\geq 0 } ( x v - \\phi ( v ) ) , \\mbox { f o r a l l } x \\geq 0 . \\end{align*}"}
+{"id": "8229.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } f ' ( t ) = \\frac { 1 } { \\sqrt { 1 + 2 f ( t ) ^ 2 } } \\mathbb { R } \\\\ f ( 0 ) = 0 \\end{array} \\right . \\end{align*}"}
+{"id": "7778.png", "formula": "\\begin{align*} m _ { 1 / 2 , \\lambda } ( x | \\mu ) & = \\frac { \\lambda ^ \\mu } { 2 \\Gamma ( \\mu ) \\sqrt { \\pi x ^ 3 } } \\int _ 0 ^ \\infty y ^ \\mu \\ , e ^ { - y ^ 2 / 4 x - \\lambda y } \\ , d y \\\\ & = \\sqrt { \\frac { 2 } { \\pi } } \\ , \\mu \\lambda ^ \\mu \\ , ( 2 x ) ^ { \\mu / 2 - 1 } \\ , e ^ { \\lambda ^ 2 x / 2 } \\ , D _ { - \\mu - 1 } ( \\lambda \\sqrt { 2 x } \\ , ) \\end{align*}"}
+{"id": "2668.png", "formula": "\\begin{align*} \\beta : = \\left ( \\frac 2 { q _ 1 } - \\frac { 1 } { 2 } \\right ) \\frac { \\alpha } { \\alpha + \\delta } , \\delta : = \\frac 2 { q _ 1 } - \\frac 2 { q _ 2 } . \\end{align*}"}
+{"id": "211.png", "formula": "\\begin{align*} q _ { \\iota } ( x ) ^ { n } = \\alpha _ { \\iota } \\int _ { X _ { \\iota } } x ^ { 2 n } \\end{align*}"}
+{"id": "2633.png", "formula": "\\begin{align*} | g _ j ( n ) | & = \\begin{cases} \\sqrt { N ( n ) } & , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "8206.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty g ( a ) { \\nu } ^ c _ t ( d a ) = \\int _ 0 ^ t k ( t , s ) \\int _ 0 ^ \\infty g ( a ) e ^ { c a } \\mathcal { P } _ { A ( s ) } ( d a ) d s , \\end{align*}"}
+{"id": "3689.png", "formula": "\\begin{align*} ( \\phi ^ H _ s ) ^ { - 1 } \\circ \\phi ^ { H ' } _ s = \\phi ^ { \\overline { H } \\sharp H ' } _ s , \\end{align*}"}
+{"id": "8660.png", "formula": "\\begin{align*} i _ { u , v } \\left ( \\frac { 1 } { u - v } \\right ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { v ^ k } { u ^ { k + 1 } } . \\end{align*}"}
+{"id": "320.png", "formula": "\\begin{align*} L ( \\frac { 1 } { 2 } , f \\otimes \\chi _ d ) = L ( \\frac { 1 } { 2 } , g \\otimes \\chi _ d ) \\end{align*}"}
+{"id": "432.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\right \\| \\leq b \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "2381.png", "formula": "\\begin{align*} ( E , A ) \\sim ( [ \\ > E \\Phi \\ > \\ > E \\Phi ' \\ > ] , [ \\ > A \\Phi \\ > \\ > A \\Phi ' \\ > ] - [ \\ > E \\dot \\Phi \\ > \\ > E \\dot \\Phi ' \\ > ] ) = ( [ \\ > E _ 1 \\ > \\ > E _ 2 \\ > ] , [ \\ > 0 \\ > \\ > A _ 2 \\ > ] ) . \\end{align*}"}
+{"id": "6279.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ^ { t } ( p '' ) & & = & h \\circ \\varphi _ { 0 } ^ { t } \\circ h ^ { - 1 } ( p '' ) \\\\ & & = & h \\circ \\varphi _ { 0 } ^ { t } ( 0 , y '' , - 1 ) \\\\ & & = & h ( 0 , \\gamma '' ( t ) , - 1 ) \\\\ & & = & \\big ( 0 , \\gamma '' ( t ) , \\sqrt { r \\circ \\gamma '' ( t ) } \\big ) , \\end{aligned} \\end{align*}"}
+{"id": "6506.png", "formula": "\\begin{align*} \\big ( ( \\sqrt { \\bar { \\Gamma } } ^ \\top \\Xi _ u \\sqrt { \\bar { \\Gamma } } ) ^ 2 \\big ) = \\left ( ( \\check { V } ^ \\top \\Xi \\check { V } ) ^ 2 \\right ) = \\underset { i = \\lfloor d / 2 \\rfloor + 1 } { \\overset { d } { \\sum } } \\xi _ i ^ 2 \\leq d \\xi _ { \\lfloor d / 2 \\rfloor } ^ 2 \\leq \\frac { 4 } { d } ( \\Xi ) ^ 2 , \\end{align*}"}
+{"id": "5548.png", "formula": "\\begin{align*} G _ { 1 , 2 } ^ { 1 , 1 } \\left ( \\begin{matrix} 1 \\\\ \\frac { k } { 2 } , \\frac { k + 1 } { 2 } \\end{matrix} \\Big | \\frac { 1 } { X _ n } \\right ) = \\frac { \\Gamma \\left ( \\frac { k } { 2 } \\right ) } { X _ n ^ { \\frac { k } { 2 } } \\sqrt { \\pi } } { } _ 1 F _ { 1 } \\left ( \\frac { k } { 2 } ; \\frac { 1 } { 2 } ; - \\frac { 1 } { X _ n } \\right ) . \\end{align*}"}
+{"id": "4868.png", "formula": "\\begin{align*} w _ { z ; k } ( t ) = a \\cos ( 2 k \\pi t ) + b \\sin ( 2 k \\pi t ) , t \\in I . \\end{align*}"}
+{"id": "4566.png", "formula": "\\begin{align*} 1 - \\sum _ { i = 1 } ^ { n } \\frac { 1 } { a _ i } = \\frac { 1 } { \\prod _ { i = 1 } ^ n a _ i } . \\end{align*}"}
+{"id": "8512.png", "formula": "\\begin{align*} \\frac { \\bar { g } _ t } { \\bar { g } } = \\alpha _ { \\sigma } - \\frac { 2 } { 3 } \\beta \\mu + \\frac { 1 } { 3 } \\beta _ { \\sigma ^ 2 } . \\end{align*}"}
+{"id": "8760.png", "formula": "\\begin{align*} x & = [ u ^ * _ 0 , u ^ * _ 1 , \\dots , u ^ * _ { N - 2 } , u ^ * _ { N - 1 } ] ^ * \\in \\mathbb { R } ^ { N r } \\\\ U _ l & = [ u _ { 2 T - l } , u _ { 2 T + 1 - l } , \\dots , u _ { N - l } ] \\in \\mathcal { M } _ { r \\times \\bar { N } } ( \\mathbb { R } ) . \\end{align*}"}
+{"id": "9057.png", "formula": "\\begin{align*} \\Gamma ( n + l + d / 2 ) = \\frac { \\sqrt { \\pi } ( 2 n + 2 l + d - 1 ) ! } { 2 ^ { 2 n + 2 l + d - 2 } ( n + l + \\frac { d - 1 } { 2 } ) ! } \\ , , \\end{align*}"}
+{"id": "6976.png", "formula": "\\begin{gather*} P _ { m , n } ^ { ( \\alpha , \\beta ) } ( x ) = \\frac { ( - 1 ) ^ m 2 ^ { - n } \\binom { n } { m } ( n \\ ! + \\ ! \\delta \\ ! - \\ ! 2 m ) \\Gamma ( \\beta \\ ! - \\ ! \\delta \\ ! + \\ ! 2 m ) \\Gamma ( \\delta \\ ! + \\ ! \\beta \\ ! + \\ ! 2 ( n \\ ! - \\ ! m ) ) ( n \\ ! + \\ ! \\beta ) } { n ! ( \\delta + n - m ) \\Gamma ( \\beta \\ ! - \\ ! \\delta \\ ! + \\ ! m ) \\Gamma ( \\delta \\ ! + \\ ! \\beta \\ ! + \\ ! n \\ ! - \\ ! m ) \\prod _ { i < j } \\big ( W ^ { ( \\alpha , \\beta ) } _ { j , m , n } - W ^ { ( \\alpha , \\beta ) } _ { i , m , n } \\big ) } \\det ( M _ n ) . \\end{gather*}"}
+{"id": "7206.png", "formula": "\\begin{align*} \\langle q , \\ , \\omega q \\rangle = 0 , \\qquad \\mbox { a n d } \\qquad \\langle q , \\ , q j \\rangle = 0 , \\end{align*}"}
+{"id": "5402.png", "formula": "\\begin{align*} y _ n = \\sum _ { m = 0 } ^ { 2 ^ n - 1 } \\frac { \\big \\lceil \\ \\overline { \\beta _ m } \\ \\big \\rceil - 1 } { \\prod _ { k = 0 } ^ { m } \\overline { \\beta _ k } } . \\end{align*}"}
+{"id": "5852.png", "formula": "\\begin{align*} \\nabla h = \\nu \\hbox { o n } \\Gamma , \\end{align*}"}
+{"id": "5112.png", "formula": "\\begin{align*} \\mbox { I m } \\left \\lbrace \\big [ \\partial _ { t } \\gamma ( t , s ) - \\mathbf { v } ( t , \\gamma ( t , s ) ) \\big ] \\overline { \\partial _ { s } \\gamma ( t , s ) } \\right \\rbrace = 0 , \\end{align*}"}
+{"id": "9038.png", "formula": "\\begin{align*} \\mathcal { W } _ p ^ p ( L _ n [ \\textbf { x } ] , L _ n [ \\textbf { 0 } ] ) \\le \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } | x _ j | ^ p . \\end{align*}"}
+{"id": "913.png", "formula": "\\begin{align*} t _ n = ( n ( n + 1 ) ) ^ { - ( ( n + 1 ) ( n + 2 ) ) ^ { - ( ( n + 2 ) ( n + 3 ) ) ^ { \\dots } } } \\end{align*}"}
+{"id": "3743.png", "formula": "\\begin{align*} \\tilde { c } _ 2 ( b _ 2 ) = 1 = \\tilde { c } _ 1 ( b _ 1 ) = \\tilde { c } _ 1 ( \\tilde { d } _ 2 ( b _ 2 ) ) . \\end{align*}"}
+{"id": "3786.png", "formula": "\\begin{align*} R _ 2 & = \\{ \\alpha \\varepsilon _ 1 = \\varepsilon _ 2 \\alpha \\} , & R _ 3 & = \\{ \\alpha ^ 2 \\} \\end{align*}"}
+{"id": "3251.png", "formula": "\\begin{align*} u ( x , t ) = o \\left ( \\| ( x , t ) \\| ^ { \\frac { 2 } { \\gamma + 1 } } \\right ) , \\ , \\ , \\ , \\ , \\ , \\ , \\| ( x , t ) \\| \\rightarrow \\infty , \\end{align*}"}
+{"id": "3988.png", "formula": "\\begin{align*} D _ { i i } u _ \\epsilon ( x ) & = \\int _ { \\mathbf { R } ^ n \\cap \\{ y ; x - y \\not \\in E _ u \\} } D _ { i i } u _ 0 ( x - y ) \\rho _ \\epsilon ( y ) \\ d y \\\\ & \\quad \\quad + \\int _ { \\{ y ; x - y \\in E _ u \\} } \\rho _ \\epsilon ( y ) \\ d \\mu ^ { i i } , \\\\ & \\geq \\int _ { \\mathbf { R } ^ n \\cap \\{ y ; x - y \\not \\in E _ u \\} } D _ { i i } u _ 0 ( x - y ) \\rho _ \\epsilon ( y ) \\ d y \\end{align*}"}
+{"id": "8726.png", "formula": "\\begin{align*} \\tilde s _ \\lambda ( x _ 1 , \\dots , x _ n ) = \\frac { \\det [ ( x _ i - 1 ) ^ { \\lambda _ j - 1 } x _ i ^ { n - j + 1 } ] } { { \\prod _ { i < j } ( x _ i - x _ j ) } . } \\end{align*}"}
+{"id": "4214.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) = \\mathcal { E } _ 0 ( t ) + \\mathcal { E } _ 1 ( t ) . \\end{align*}"}
+{"id": "3333.png", "formula": "\\begin{align*} \\eta _ o = \\sqrt { \\delta } \\dd \\kappa + \\frac { k } { y } \\dd x + \\nu ( - p \\dd q + q \\dd p ) . \\end{align*}"}
+{"id": "2425.png", "formula": "\\begin{align*} q ^ \\dag \\in \\mathcal Q : = \\{ \\psi \\in C ( \\overline \\Omega ) : 0 \\le \\psi \\le M _ 1 \\} . \\end{align*}"}
+{"id": "6525.png", "formula": "\\begin{align*} \\| \\tilde { f ^ L } \\| _ 2 ^ 2 = \\| f ^ L \\| _ { L _ 2 } ^ 2 \\geq \\rho ^ 2 C _ \\alpha ^ 2 / 2 - R ^ 2 2 ^ { - 2 L s } \\geq \\rho ^ 2 ( C _ \\alpha ^ 2 / 2 - { R ^ 2 } ) . \\end{align*}"}
+{"id": "4850.png", "formula": "\\begin{align*} v _ 1 = v _ { 0 , 1 / 2 } \\textrm { a n d } v _ 2 = v _ { 1 / 2 , 1 / 2 } . \\end{align*}"}
+{"id": "4386.png", "formula": "\\begin{align*} a _ k = G \\left ( \\theta - \\sum _ { i = 1 } ^ { k - 1 } \\frac { 1 } { a _ i } \\right ) . \\end{align*}"}
+{"id": "2433.png", "formula": "\\begin{align*} \\begin{aligned} u _ h ^ n ( q ) & = F _ \\tau ^ h ( n ; q ) \\big ( \\mathcal { I } _ h v - D _ h ( q ) \\mathcal { I } _ h ^ \\partial b \\big ) + D _ h ( q ) \\mathcal { I } _ h ^ \\partial b + \\tau \\sum _ { j = 1 } ^ n E _ \\tau ^ h ( j ; q ) P _ h f \\\\ & = F _ \\tau ^ h ( n ; q ) \\big ( \\mathcal { I } _ h v - D _ h ( q ) \\mathcal { I } _ h ^ \\partial b \\big ) + D _ h ( q ) \\mathcal { I } _ h ^ \\partial b + ( I - F _ \\tau ^ h ( n ; q ) ) A _ h ( q ) ^ { - 1 } P _ h f , \\end{aligned} \\end{align*}"}
+{"id": "3200.png", "formula": "\\begin{align*} L ( E _ { r + 2 } ( \\psi , \\tau ) , g _ \\ell , 1 + j ) = L ( g _ \\ell , \\psi , 1 + j ) \\cdot L ( g _ \\ell , \\tau , j - r ) . \\end{align*}"}
+{"id": "3237.png", "formula": "\\begin{align*} f ' ( v ) = \\begin{cases} f ( v ) & , \\cr c _ i & , \\cr g ( v ) & . \\end{cases} \\end{align*}"}
+{"id": "6891.png", "formula": "\\begin{align*} & \\nabla _ { y _ { n + 1 } } \\mathcal { L } ( y ^ * , \\alpha ^ * , \\beta ^ * , \\gamma ^ * , ( \\delta ^ \\pm ) ^ * , \\pi ^ * ) = - y ^ * _ n - \\beta ^ * _ { n + 1 } + \\gamma ^ * = 0 , \\\\ & \\nabla _ { y _ { n + 2 } } \\mathcal { L } ( y ^ * , \\alpha ^ * , \\beta ^ * , \\gamma ^ * , ( \\delta ^ \\pm ) ^ * , \\pi ^ * ) = - y ^ * _ n - ( \\delta ^ - ) ^ * + ( \\delta ^ + ) ^ * = 0 \\end{align*}"}
+{"id": "2662.png", "formula": "\\begin{align*} \\Delta ^ { { \\rm I } } _ m : = \\ I _ m - I _ { m - 1 } , \\end{align*}"}
+{"id": "5574.png", "formula": "\\begin{align*} r _ 0 : = r _ 0 ( n , \\sup _ { \\partial B _ 2 ^ n ( 0 , 0 ) } u _ 0 , \\norm { \\nabla u } _ { L ^ 2 ( B ^ n _ 2 ( 0 , 0 ) ) } , \\epsilon _ 0 ) \\end{align*}"}
+{"id": "427.png", "formula": "\\begin{align*} \\theta _ g \\theta _ \\omega = I _ { \\ell ^ p ( \\mathbb { N } ) } . \\end{align*}"}
+{"id": "3.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { - \\beta t } y _ t & = \\int _ t ^ \\infty \\beta e ^ { - \\beta s } y _ s d s + \\int _ t ^ \\infty e ^ { - \\beta s } g ( x _ s , y _ s , z _ s , \\tilde { z } _ s , \\gamma _ { ( s , e ) } ) d s - \\int _ t ^ \\infty e ^ { - \\beta s } z _ s d W _ s \\\\ & \\quad - \\int _ t ^ \\infty e ^ { - \\beta s } \\tilde { z } _ s d \\xi _ s - \\int _ t ^ \\infty \\int _ { \\mathcal { E } } e ^ { - \\beta s } \\gamma _ { ( s , e ) } \\tilde { N } ( d e , d s ) , \\ t \\in [ 0 , \\infty ) . \\end{aligned} \\end{align*}"}
+{"id": "1014.png", "formula": "\\begin{align*} F ( x , y ) = \\frac { ( 1 + | y - x | ) ^ \\alpha } { ( 1 + | x | ) ^ \\alpha ( 1 + | y | ) ^ \\alpha } . \\end{align*}"}
+{"id": "2060.png", "formula": "\\begin{align*} y _ 1 ' & = z ( - h _ 3 y _ 1 - h _ 2 y _ 2 ) , \\\\ [ 0 . 5 e x ] y _ 2 ' & = z ( h _ 1 y _ 1 + h _ 3 y _ 2 ) . \\end{align*}"}
+{"id": "4240.png", "formula": "\\begin{align*} \\lim _ { t \\nearrow T ^ * } \\| \\nabla u ( t ) \\| _ { L ^ 2 } = \\infty \\left ( \\lim _ { t \\searrow - T _ * } \\| \\nabla u ( t ) \\| _ { L ^ 2 } = \\infty \\right ) . \\end{align*}"}
+{"id": "7608.png", "formula": "\\begin{align*} \\| v ( t ) \\| _ { \\L ^ 2 } ^ 2 = \\| v _ 0 \\| _ { \\L ^ 2 } ^ 2 - 2 \\int _ 0 ^ t \\langle \\mathfrak { A } _ 0 ( s ) , v ( s ) \\rangle \\d s , \\end{align*}"}
+{"id": "4218.png", "formula": "\\begin{align*} \\begin{aligned} I _ { 3 5 } : = I _ 3 + I _ 5 = & ~ { } \\iint _ { D _ t } \\left ( ( 1 + \\left | \\underline { u } \\right | ^ 2 ) ^ { 1 + \\delta } \\abs { \\underline { L } \\partial _ x \\tilde { \\Lambda } } \\right ) \\\\ & ~ { } \\left ( 2 | \\partial _ x \\tilde { \\Lambda } \\cosh ( 2 \\lambda + \\tilde { \\Lambda } ) | | Q _ 0 ( \\phi , \\phi ) | + 2 | \\sinh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) | | Q _ 0 ( \\phi , \\partial _ x \\phi ) | \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "1247.png", "formula": "\\begin{align*} \\alpha _ 0 & : = 1 - \\frac { \\sqrt { \\alpha } } { 1 - \\alpha ^ 2 } , \\alpha _ 1 : = 1 - \\frac { 4 \\alpha } { ( 1 - \\alpha ) ( 1 + 6 \\alpha + \\alpha ^ 2 ) } , \\\\ \\tilde { \\alpha } _ 0 & : = 1 + \\frac { \\sqrt { \\alpha } } { 1 - \\alpha ^ 2 } , \\tilde { \\alpha } _ 1 : = 1 + \\frac { 4 \\alpha } { ( 1 - \\alpha ) ( 1 + 6 \\alpha + \\alpha ^ 2 ) } . \\end{align*}"}
+{"id": "1250.png", "formula": "\\begin{align*} R _ { \\mathcal { B S } ( \\alpha ) } ( \\mathcal { S ^ { * } ( \\beta ) } ) & = \\begin{dcases} \\dfrac { 2 \\sqrt { \\alpha } } { ( 1 + \\alpha ) \\sqrt { 1 + 1 6 \\alpha ( 1 - \\beta ) ^ 2 } } , & \\ 0 \\leq \\beta < \\max \\left \\{ 0 ; \\dfrac { 9 \\alpha - 1 } { 8 \\alpha } \\right \\} , \\\\ \\dfrac { 1 } { 1 + 2 ( 1 - \\alpha ) ( 1 - \\beta ) } , & \\ \\max \\left \\{ 0 ; \\dfrac { 9 \\alpha - 1 } { 8 \\alpha } \\right \\} \\leq \\beta < 1 . \\end{dcases} \\end{align*}"}
+{"id": "2347.png", "formula": "\\begin{align*} F _ T ( Q ( k ) ) = Q ( - k ) . \\end{align*}"}
+{"id": "239.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = & \\beta u w + \\alpha v ( 1 + u ) , \\psi _ 2 ( u , v , w ) = \\beta v w + \\alpha u ( 1 + v ) , \\\\ \\psi _ 3 ( u , v , w ) = & \\beta v w + \\alpha v \\left ( 1 + u \\right ) \\psi _ 4 ( v , w ) = \\beta u w + \\alpha u ( 1 + v ) . \\end{align*}"}
+{"id": "3338.png", "formula": "\\begin{align*} \\Phi _ { \\kappa z } = - \\frac { 1 } { \\sqrt { \\delta } } ( \\frac { k } { y } \\Phi _ { x z } - \\nu p \\Phi _ { q z } + \\nu q \\Phi _ { p z } ) , z = x , y , q , p . \\end{align*}"}
+{"id": "2292.png", "formula": "\\begin{align*} \\begin{aligned} \\gamma ^ + _ 0 & = \\frac { 1 } { 2 1 } , & \\gamma ^ + _ 1 & = \\frac { 1 9 } { 2 1 } , & \\gamma ^ + _ 2 & = \\frac { 1 } { 2 1 } ; \\\\ \\gamma ^ - _ 0 & = \\frac { 4 } { 2 7 } , & \\gamma ^ - _ 1 & = \\frac { 1 9 } { 2 7 } , & \\gamma ^ + _ 2 & = \\frac { 4 } { 2 7 } , \\end{aligned} \\end{align*}"}
+{"id": "8647.png", "formula": "\\begin{align*} F _ i ( x _ 1 , \\ldots , x _ i ) & : = f ( t _ i ( x _ 1 , \\ldots , x _ { i - 1 } ) ) , \\\\ t _ i ( x _ 1 , \\ldots , x _ i ) & : = \\min \\{ d ( x _ i , x _ j ) : j = 1 , \\ldots , i - 1 \\} . \\end{align*}"}
+{"id": "9106.png", "formula": "\\begin{align*} \\mathsf { C } _ { \\rho , 1 } : = p \\mathsf { C } \\left ( h _ { \\rho } \\mathsf { P } \\right ) + \\bar { p } \\mathsf { C } \\left ( h _ 1 \\mathsf { P } \\right ) + \\frac { \\bar { p } \\log e } { 2 } \\left ( \\frac { h _ { \\rho } } { 1 + h _ { \\rho } \\mathsf { P } } - \\frac { h _ 1 } { 1 + h _ 1 \\mathsf { P } } \\right ) \\mathsf { P } \\end{align*}"}
+{"id": "8378.png", "formula": "\\begin{align*} [ B , \\epsilon ] = \\{ ( f , g ) : | f ( x ) - g ( x ) | < \\epsilon x \\in B \\} \\ \\ \\ ( B \\in \\mathcal { B } , \\epsilon > 0 ) . \\end{align*}"}
+{"id": "5289.png", "formula": "\\begin{align*} \\tau ^ { ( k e r F _ \\ast ) ^ \\bot } ( F ) = - \\frac { \\lambda ^ 2 } { 2 } \\sum _ { j = 1 } ^ { n } \\left \\{ 2 \\lambda ^ 2 X _ j \\left ( \\frac { 1 } { \\lambda ^ 2 } \\right ) \\tilde { X _ j } - g ( \\lambda X _ j , \\lambda X _ j ) F _ \\ast \\left ( \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\right ) \\right \\} . \\end{align*}"}
+{"id": "7443.png", "formula": "\\begin{align*} \\alpha = \\sum _ { i = 1 } ^ M \\alpha _ i P _ i \\end{align*}"}
+{"id": "141.png", "formula": "\\begin{align*} D : & = \\{ \\phi : [ - \\beta , 0 ] \\rightarrow \\mathbb { X } : \\phi \\ \\mbox { i s p i e c e w i s e c o n t i n u o u s h a v i n g j u m p d i s c o n t i n u i t y a t f i n i t e n u m b e r } \\ \\\\ & \\qquad \\mbox { o f f i x e d p o i n t s } \\ \\{ \\beta _ 0 , \\beta _ 1 , . . . . . . . . \\beta _ { l + 1 } \\} \\subset [ - \\beta , 0 ] , \\ \\mbox { s u c h t h a t } \\ - \\beta = \\beta _ 0 \\leq \\beta _ 1 < \\beta _ 2 . . . . . . < \\beta _ l \\\\ & \\qquad < \\beta _ { l + 1 } = 0 \\} , \\end{align*}"}
+{"id": "7442.png", "formula": "\\begin{align*} \\widehat \\Phi _ { \\rm e x t } \\colon \\begin{bmatrix} P _ { 1 1 } & P _ { 1 2 } \\\\ P _ { 2 1 } ^ * & P _ { 2 2 } \\end{bmatrix} \\mapsto \\begin{bmatrix} \\Phi _ { 1 1 } ( P _ { 1 1 } ) & \\phi ( P _ { 1 2 } ) \\\\ \\phi ( P _ { 2 1 } ^ * ) & \\Phi _ { 2 2 } ( P _ { 2 2 } ) \\end{bmatrix} \\end{align*}"}
+{"id": "960.png", "formula": "\\begin{align*} \\mathcal { R } ( V , K ) = \\{ 0 \\} \\Longleftrightarrow W _ q ( V , K ) = 0 \\ \\mbox { f o r } q \\geq \\mbox { d i m } ( V ) - 3 . \\end{align*}"}
+{"id": "2255.png", "formula": "\\begin{align*} 1 + \\varpi _ D [ \\lambda ] + \\varpi _ D [ \\mu ] + p [ \\lambda ^ q \\mu ] = ( 1 + \\varpi _ D [ \\lambda ] + \\varpi _ D [ \\mu ] ) ( 1 + p [ \\lambda ^ q \\mu ] + x ) \\end{align*}"}
+{"id": "4069.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } g _ t = h , g _ 0 = g , \\\\ & \\frac { \\partial } { \\partial t } \\Big | _ { t = 0 } H _ t = d K , H _ 0 = H \\end{align*}"}
+{"id": "7835.png", "formula": "\\begin{align*} g ( z ) = F _ 1 e ^ { \\lambda _ 1 z } + \\cdots + F _ m e ^ { \\lambda _ m z } , \\end{align*}"}
+{"id": "5205.png", "formula": "\\begin{align*} H = \\begin{pmatrix} \\cosh ( s ) & 0 & 0 & \\sinh ( s ) \\cos ( d ) & \\sinh ( s ) \\sin ( d ) \\\\ 0 & \\cosh ( s ) & 0 & \\sinh ( s ) \\sin ( d ) & - \\sinh ( s ) \\cos ( d ) \\\\ 0 & 0 & 1 & 0 & 0 \\\\ \\sinh ( s ) \\cos ( d ) & \\sinh ( s ) \\sin ( d ) & 0 & \\cosh ( s ) & 0 \\\\ \\sinh ( s ) \\sin ( d ) & - \\sinh ( s ) \\cos ( d ) & 0 & 0 & \\cosh ( s ) \\end{pmatrix} \\ , , \\end{align*}"}
+{"id": "4531.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 3 0 } = \\frac { 1 } { 3 } + \\frac { 1 } { 5 } = \\frac { 8 } { 1 5 } < \\theta . \\end{align*}"}
+{"id": "8306.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\frac { \\partial V ( \\textbf { x } ) } { \\partial \\textbf { x } } \\cdot \\textbf { f } ( \\textbf { x } , \\textbf { u } _ E ^ * , \\textbf { u } _ i ^ * * ) = \\frac { d _ E } { \\Lambda } [ \\nu _ 1 + \\nu _ 2 + \\nu _ 3 - \\nu _ E ] . \\end{array} \\right . \\end{align*}"}
+{"id": "3865.png", "formula": "\\begin{align*} T _ { n + 1 } T _ { n - 1 } = 2 \\sum _ { l = 3 } ^ n T _ { l - 1 } T _ { n - l + 2 } ^ 2 . \\end{align*}"}
+{"id": "7274.png", "formula": "\\begin{align*} A ( t ) : = \\int _ { 0 } ^ { t } 1 _ { ( 0 , \\infty ) } ( X _ s ) d s . \\end{align*}"}
+{"id": "7180.png", "formula": "\\begin{align*} W _ t = \\left \\{ z _ { \\alpha , t } ( x _ { j } , \\ , t ) , \\ , z _ { \\alpha , j t } ( x _ { j } , \\ , t ) , \\ , z _ { \\alpha , j k } ( x _ { j } , \\ , t ) \\right \\} , \\end{align*}"}
+{"id": "1998.png", "formula": "\\begin{align*} \\zeta _ { \\lambda , s } \\left ( \\pmb k , \\pmb l \\right ) \\cdot \\zeta _ { e ^ t } \\left ( \\pmb m \\right ) = \\sum _ { p + q = t } \\sum _ { ( \\pmb u , \\pmb v ) } c _ { \\pmb u , \\pmb v } \\zeta _ { \\nu _ p , q } \\left ( \\pmb u , \\pmb v \\right ) , \\end{align*}"}
+{"id": "1725.png", "formula": "\\begin{align*} z _ 1 = R \\cos \\theta e ^ { i \\alpha } , z _ 2 = R \\sin \\theta e ^ { i \\beta } , \\bar z _ 1 = R \\cos \\theta e ^ { - i \\alpha } , \\bar z _ 2 = R \\sin \\theta e ^ { - i \\beta } . \\end{align*}"}
+{"id": "4940.png", "formula": "\\begin{align*} F _ 2 = - 2 u _ x u _ t - 2 v _ x v _ t , G _ 2 = u ^ 2 _ x + v ^ 2 _ x - \\frac { 1 } 2 ( u ^ 2 + v ^ 2 ) ^ 2 , \\end{align*}"}
+{"id": "3807.png", "formula": "\\begin{align*} M : = \\frac { \\sqrt { \\abs { Q } } } { \\norm { f } _ { L ^ 2 ( Q ) } } \\cdot \\sup _ { z \\in Q + D _ { 4 l } } \\abs { f ( z ) } , \\end{align*}"}
+{"id": "97.png", "formula": "\\begin{align*} ( x \\ast y ) \\ast z & = u ( ( u ( x y ) ) z ) - u ( ( ( u x ) y ) z ) - u ( ( x ( u y ) ) z ) \\\\ & - ( u ( u ( x y ) ) ) z + ( u ( ( u x ) y ) ) z + ( u ( x ( u y ) ) ) z \\\\ & - ( u ( x y ) ) ( u z ) + ( ( u x ) y ) ( u z ) + ( x ( u y ) ) ( u z ) \\end{align*}"}
+{"id": "8395.png", "formula": "\\begin{align*} & \\alpha \\mapsto \\int \\hat \\Psi _ { \\alpha } ^ 2 \\ : d \\mu _ { \\alpha } \\in C ^ 1 ( [ \\alpha _ - , \\alpha _ + ] ) \\\\ & \\alpha \\mapsto \\int \\hat \\Psi _ { \\alpha } \\cdot \\hat \\Psi _ { \\alpha } \\circ F _ { \\alpha } ^ k \\ : d \\mu _ { \\alpha } \\in C ^ 1 ( [ \\alpha _ - , \\alpha _ + ] ) , \\ : k \\geq 1 . \\end{align*}"}
+{"id": "8464.png", "formula": "\\begin{align*} \\kappa \\gamma _ A = \\min _ { \\nu \\in \\Gamma _ A ^ + } \\ , \\kappa \\nu \\end{align*}"}
+{"id": "5885.png", "formula": "\\begin{align*} ( C _ p ) \\cap ( D ) = ( C _ p ) \\cap \\{ ( C _ p ) \\} ^ { \\perp } = \\{ 0 \\} . \\end{align*}"}
+{"id": "3506.png", "formula": "\\begin{align*} S _ \\lambda : = S _ { \\lambda _ 1 } \\times \\dots \\times S _ { \\lambda _ { \\ell ( \\lambda ) } } \\end{align*}"}
+{"id": "3224.png", "formula": "\\begin{align*} f ' ( x ) + f ' ( y ) \\geq f ( x ) + f ( y ) - 1 = 2 k _ 3 + 2 k _ 2 + k _ 1 - 1 = 2 k _ 3 ' + 2 k ' _ 2 + k _ 1 ' . \\end{align*}"}
+{"id": "4057.png", "formula": "\\begin{align*} f ^ * ( Y u ( \\cdot ) ) \\det D Y u & = f ( \\cdot ) \\overline { \\Omega } \\\\ Y u ( \\Omega ) & = \\Omega ^ * , \\end{align*}"}
+{"id": "6927.png", "formula": "\\begin{align*} S _ 1 & : = \\sum _ { n \\leq x \\atop P ^ + ( n ) \\leq \\mathcal { L } } n ^ { - a } \\\\ & = \\psi ( x , \\mathcal { L } ) x ^ { - a } + a \\int _ 1 ^ x \\psi ( w , \\mathcal { L } ) w ^ { - 1 - a } \\ d w \\\\ & \\ll 1 + \\int _ 1 ^ x w ^ { - a } \\exp \\bigg ( - \\bigg ( \\frac { \\log w } { \\log \\mathcal { L } } \\bigg ) \\bigg \\{ \\log \\bigg ( \\frac { \\log w } { \\log \\mathcal { L } } \\bigg ) \\\\ & \\qquad + \\log _ 2 \\bigg ( \\frac { \\log w } { \\log \\mathcal { L } } \\bigg ) - 1 \\bigg \\} \\bigg ) \\ d w . \\end{align*}"}
+{"id": "319.png", "formula": "\\begin{align*} \\begin{aligned} e _ { \\psi _ { N _ 0 ( E ) } } = - \\sigma ( e _ { \\psi _ { N _ 0 ( E ) } } ) . \\end{aligned} \\end{align*}"}
+{"id": "7973.png", "formula": "\\begin{align*} y + ( \\sum f ( a _ i ) + \\sum f ( b _ i ) + \\sum f ( x _ i ) + \\sum f ( y _ i ) ) = ( \\sum f ( a _ i ) + \\sum f ( b _ i ) + \\sum f ( x _ i ) + \\sum f ( y _ i ) ) , \\end{align*}"}
+{"id": "7035.png", "formula": "\\begin{align*} F _ n \\widehat { E } ^ { C _ 2 } _ * = E ^ { C _ 2 } _ { * + | n \\sigma | - n \\sigma } \\end{align*}"}
+{"id": "5161.png", "formula": "\\begin{align*} P _ { 1 } ( v _ { 1 } ) & = \\frac { 1 } { n _ { 1 } } = \\frac { 1 } { d e g ( v _ { 0 } ) } , \\\\ P _ { 2 } ( v _ { 2 } ) & = \\frac { 1 } { n _ { 2 } } = \\frac { 1 } { | N ( v _ { 0 } ) | \\cap | N ( v _ { 1 } ) | } \\end{align*}"}
+{"id": "2430.png", "formula": "\\begin{align*} \\begin{aligned} u ^ n ( q ) & = F _ \\tau ( n ; q ) ( v - D ( q ) b ) + D ( q ) b + \\tau \\sum _ { j = 1 } ^ { n } E _ \\tau ( j ; q ) f \\\\ & = F _ \\tau ( n ; q ) ( v - D ( q ) b ) + D ( q ) b + ( I - F _ \\tau ( n ; q ) ) A ( q ) ^ { - 1 } f . \\end{aligned} \\end{align*}"}
+{"id": "8281.png", "formula": "\\begin{align*} \\mathcal { T } : = \\mathcal { T } _ p \\ \\bigcup \\ \\mathcal { T } _ e \\end{align*}"}
+{"id": "6476.png", "formula": "\\begin{align*} ( L v , v ) & = ( L w , w ) + 2 ( v , \\phi ) ( L \\phi , w ) + ( v , \\phi ) ^ 2 ( L \\phi , \\phi ) \\\\ & = ( L w , w ) \\ge b \\| w \\| _ { H ^ 1 } ^ 2 = b \\| v - ( v , \\phi ) \\phi \\| _ { H ^ 1 } ^ 2 \\\\ & \\ge b ( \\| v \\| _ { H ^ 1 } - | ( v , \\phi ) | \\| \\phi \\| _ { H ^ 1 } ) ^ 2 \\ge b \\left ( \\frac { 1 } { 2 } \\| v \\| _ { H ^ 1 } ^ 2 - | ( v , \\phi ) | ^ 2 \\| \\phi \\| _ { H ^ 1 } ^ 2 \\right ) . \\end{align*}"}
+{"id": "1981.png", "formula": "\\begin{align*} y _ { k , i } & = r _ { k , i } + { \\textbf { w } } _ k ^ { \\mathsf { H } } \\left ( \\sqrt { \\alpha _ k } { \\textbf { h } } _ k x _ { k , i } + \\sqrt { \\alpha _ { k ' } } { \\textbf { h } } _ { k ' } x _ { { k ' } , i } \\right ) \\\\ & + \\sum \\nolimits _ { j \\neq k } { \\textbf { w } } _ k ^ { \\mathsf { H } } \\left ( \\sqrt { \\alpha _ j } { \\textbf { h } } _ j x _ { j , i } + \\sqrt { \\alpha _ { j ' } } { \\textbf { h } } _ { j ' } x _ { { j ' } , i } \\right ) + n _ { k , i } , \\end{align*}"}
+{"id": "5453.png", "formula": "\\begin{align*} P ( \\alpha ^ { n } _ { t + \\Delta t } = j | \\alpha ^ { n } _ t = i , ( X ^ n _ s , \\alpha ^ { n } _ s ) , s \\leq t ) = q _ { i j } ( X ^ { n } _ t ) \\Delta t + o ( \\Delta t ) \\end{align*}"}
+{"id": "4867.png", "formula": "\\begin{align*} g ' ( s _ j ) = u ^ { j - 1 } ( s _ { j } ) \\int _ { A _ { j - 2 } ( s _ j ) } u ^ { j - 2 } ( s _ { j - 2 } ) \\dots u ^ 1 ( s _ { 1 } ) d \\L ^ { j - 2 } , \\end{align*}"}
+{"id": "1071.png", "formula": "\\begin{align*} \\mathbb { E } _ { P , Q } \\Big [ \\frac { \\tilde { N } _ 0 } { n } \\Big ] = \\frac { e ^ \\alpha + 1 } { e ^ \\alpha - 1 } \\left ( \\frac { \\mathbb { E } _ { P , Q } [ \\hat { N } _ 0 ] } { n } - \\frac { 1 } { e ^ \\alpha + 1 } \\right ) = P ( A ) , \\end{align*}"}
+{"id": "3306.png", "formula": "\\begin{align*} F _ 1 ( t ) & = \\int _ { t _ 0 } ^ { t _ { m i d } } S ( t - s ) \\mathbb P { \\rm d i v } [ \\mathbf { 1 } _ { \\mathcal F _ 0 } ( w _ a - \\ell _ a ) \\otimes w _ b ] { \\rm d } s \\\\ F _ 2 ( t ) & = \\int _ { t _ { m i d } } ^ { t } S ( t - s ) \\mathbb P [ w _ a \\cdot \\nabla w _ b ] { \\rm d } s \\\\ F _ 3 ( t ) & = \\int _ { t _ { m i d } } ^ { t } S ( t - s ) \\mathbb P [ \\ell _ { a } \\cdot \\nabla w _ b ] { \\rm d } s . \\end{align*}"}
+{"id": "4827.png", "formula": "\\begin{align*} X = \\ker ( d _ 0 F ) \\oplus \\R ^ { m - \\ell } , \\R ^ m = \\R ^ { \\l } \\oplus \\mathrm { I m } ( d _ 0 F ) , \\end{align*}"}
+{"id": "2564.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta ( v ; \\alpha ) = & \\sum _ { \\begin{subarray} { c } 0 { \\le } m _ 1 < _ { c ^ { ' } _ 1 } \\cdots < _ { c ^ { ' } _ { q - 1 } } m _ q < \\infty \\end{subarray} } \\frac { ( m _ q + 1 ) ! } { ( \\alpha ) _ { m _ q + 1 } } \\left \\{ \\prod _ { i = 1 } ^ { q } \\frac { 1 } { ( m _ i + 1 ) ^ { k ^ { ' } _ i } } \\right \\} \\\\ = & H ^ { * } _ { ( \\{ 0 \\} _ { i = 1 } ^ q ) } ( \\tau ( v ) ; \\alpha ) \\end{aligned} \\end{align*}"}
+{"id": "1860.png", "formula": "\\begin{gather*} \\sin ^ 2 ( \\alpha ) - 2 ( 1 - \\cos ( \\alpha ) ) = - ( 1 - \\cos ( \\alpha ) ) ^ 2 , \\\\ \\sin ^ 2 ( \\alpha ) \\ge 1 - \\cos ( \\alpha ) \\ge \\frac { 1 } { 2 } \\sin ^ 2 ( \\alpha ) . \\end{gather*}"}
+{"id": "3698.png", "formula": "\\begin{align*} c _ i ( f ) : = \\begin{cases} Z _ { r _ i } ( f , 2 - \\frac { \\dim V _ i } { 2 } ) & \\textrm { i f } Z _ { r _ i } ( f , s ) \\textrm { i s h o l o m o r p h i c a t } 2 - \\frac { \\dim V _ i } { 2 } \\\\ \\lim _ { s \\to 0 } \\frac { d } { d s } ( s Z _ { r _ i } ( f , s + 2 - \\frac { \\dim V _ i } { 2 } ) ) & \\textrm { i f } Z _ { r _ i } ( f , s ) \\textrm { h a s a p o l e a t } 2 - \\frac { \\dim V _ i } { 2 } . \\end{cases} \\end{align*}"}
+{"id": "5679.png", "formula": "\\begin{align*} D \\theta ^ { a } \\equiv d \\theta ^ { a } + \\omega _ { b } ^ { a } \\theta ^ { b } = 0 \\end{align*}"}
+{"id": "1909.png", "formula": "\\begin{align*} 2 ^ { 2 T } = ( \\log ( a + d L ) ) ^ { ( 2 \\log 2 ) \\left ( 1 + ( \\log \\log ( a + d L ) ) ^ { - \\frac 1 3 } \\right ) } < ( \\log ( a + d L ) ) ^ { 2 \\log 2 + \\epsilon / 3 } . \\end{align*}"}
+{"id": "6315.png", "formula": "\\begin{align*} { \\bar { v } } = u - r _ * = - P , \\end{align*}"}
+{"id": "2913.png", "formula": "\\begin{align*} ( \\hat { \\mathcal { B } } f ) ( \\zeta ) = \\sum _ { n = 1 } ^ \\infty c _ n \\zeta ^ { \\alpha ( n ) } , \\quad \\zeta \\in \\mathbb { D } _ { 1 } ^ \\infty \\end{align*}"}
+{"id": "2264.png", "formula": "\\begin{align*} \\psi _ 1 : = [ \\chi ] = [ \\xi ] ^ { a + 2 + ( p + 1 ) b } , ~ \\psi _ { 2 } : = [ \\xi ] ^ { a + 3 + ( p + 1 ) ( b - 1 ) } , ~ \\psi _ { 3 } : = [ \\xi ] ^ { a + 1 + ( p + 1 ) b } . \\end{align*}"}
+{"id": "7093.png", "formula": "\\begin{align*} \\pi _ { 1 + \\sigma } = - q _ 2 \\end{align*}"}
+{"id": "7100.png", "formula": "\\begin{align*} \\Pi _ { \\rho _ 1 + \\cdots + \\rho _ n } & = \\sum _ { i = 1 } ^ n \\langle \\Pi _ { \\rho _ 1 + \\cdots + \\rho _ n } , x ^ { \\rho _ 1 + \\cdots + \\rho _ { i - 1 } } \\rangle \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) \\\\ & = \\sum _ { i = 1 } ^ n \\pi _ { \\rho _ i + \\cdots + \\rho _ n } \\beta ( \\rho _ 1 , \\dots , \\rho _ i ) . \\end{align*}"}
+{"id": "6132.png", "formula": "\\begin{align*} \\begin{cases} \\Phi '' - { \\Delta } \\Phi + A ^ T \\Phi = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ \\Phi = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\cr \\partial _ \\nu \\Phi + B ^ T \\Phi = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{cases} \\end{align*}"}
+{"id": "8600.png", "formula": "\\begin{align*} \\mathsf { H F } ( G , N ) = \\{ c _ N ( [ v ] ) \\mid v \\in L _ G ~ \\textrm { i s a n e l l i p t i c v e c t o r } \\} \\subseteq N . \\end{align*}"}
+{"id": "981.png", "formula": "\\begin{align*} C _ 0 ( X , \\mathcal { A } ) & = C _ 0 ( X ) \\check { \\otimes } \\mathcal { A } \\\\ & = C _ 0 ( X ) \\check { \\otimes } C _ 0 ( Y ) \\\\ & = C _ 0 ( X \\times Y ) , \\end{align*}"}
+{"id": "2268.png", "formula": "\\begin{align*} & \\gamma ^ - f ( t , x , v ) = g ( t , x , v ) , ( t , x , v ) \\in ( 0 , T ) \\times \\Sigma ^ - , \\\\ & \\Phi ( t , x ) = 0 , ( t , x ) \\in ( 0 , T ) \\times \\partial \\Omega , \\end{align*}"}
+{"id": "7397.png", "formula": "\\begin{align*} q _ { \\alpha } = q _ F , q _ { \\alpha ^ * } = q _ L = q _ F ^ { f ( L / F ) } . \\end{align*}"}
+{"id": "5735.png", "formula": "\\begin{align*} d s ^ { 2 } = \\eta _ { a b } d \\mathbf { x } ^ { a } \\otimes d \\mathbf { x } ^ { b } = \\epsilon _ { A B } \\epsilon _ { A ^ { \\prime } B ^ { \\prime } } d \\mathbf { x } ^ { A A ^ { \\prime } } \\otimes d \\mathbf { x } ^ { B B ^ { \\prime } } . \\end{align*}"}
+{"id": "4397.png", "formula": "\\begin{align*} U _ n ^ { ( k ) } ( \\theta ) = \\left \\{ \\left ( x _ i \\right ) _ { i = 1 } ^ n \\in U _ n ( \\theta ) : x _ i < x _ i ^ * i = 1 , \\ldots , k \\right \\} . \\end{align*}"}
+{"id": "8552.png", "formula": "\\begin{align*} z \\left ( n ; \\beta \\right ) = \\left \\{ \\begin{array} { l l } x \\left ( n ; \\beta \\right ) & \\left ( n ; \\beta \\right ) \\in F _ { \\downarrow } \\\\ 0 & \\end{array} \\right . \\end{align*}"}
+{"id": "7231.png", "formula": "\\begin{align*} ( W W ^ \\top ) _ { i j } & = \\sum _ { 1 \\leq \\ell \\leq m } ( N ^ { c _ { i \\ell } } C _ 2 ) ( N ^ { c _ { j \\ell } } C _ 2 ) ^ \\top \\\\ & = \\sum _ { 1 \\leq \\ell \\leq m } N ^ { c _ { i \\ell } } C _ 2 C _ 2 ^ \\top N ^ { - c _ { j \\ell } } \\\\ & = k \\sum _ { 1 \\leq \\ell \\leq m } N ^ { c _ { i \\ell } - c _ { j \\ell } } \\\\ & = \\begin{cases} m k I _ n & \\\\ O & i \\neq j , \\end{cases} \\end{align*}"}
+{"id": "422.png", "formula": "\\begin{align*} a \\left ( \\sum _ { n = 1 } ^ { m } | c _ n | ^ p \\right ) ^ \\frac { 1 } { p } \\leq \\left \\| \\sum _ { n = 1 } ^ { m } c _ n \\tau _ n \\right \\| \\leq b \\left ( \\sum _ { n = 1 } ^ { m } | c _ n | ^ p \\right ) ^ \\frac { 1 } { p } , \\forall c _ 1 , \\dots , c _ m \\in \\mathbb { K } \\end{align*}"}
+{"id": "2350.png", "formula": "\\begin{align*} \\overline { Q } ( - k ) = Q ( k ) . \\end{align*}"}
+{"id": "5268.png", "formula": "\\begin{align*} \\mathcal { H } { \\tilde \\nabla } _ { { \\tilde U } _ i } { \\tilde U } _ i & = \\mathcal { H } \\nabla _ { \\frac { 1 } { \\phi } U _ i } ( \\frac { 1 } { \\phi } U _ i ) - \\frac { 1 } { \\phi ^ 2 } g ( U _ i , U _ i ) \\mathcal { H } \\nabla \\log \\phi \\\\ & = \\frac { 1 } { \\phi ^ 2 } ( - \\nabla f - \\nabla \\log \\phi ) . \\end{align*}"}
+{"id": "7147.png", "formula": "\\begin{align*} D _ k = \\sum \\limits _ { \\substack { | I | = k } } \\prod \\limits _ { \\substack { i \\in I \\\\ j \\notin I } } \\phi ( z _ j - z _ i ) \\prod _ { i \\in I } p _ { i } , k = 1 , \\dots , N \\ , . \\end{align*}"}
+{"id": "1419.png", "formula": "\\begin{align*} F ( x , y ) = \\begin{cases} \\max { \\{ x , y \\} } & \\\\ x & , x , y \\in A _ j \\ , \\ , \\epsilon ( j ) = 1 \\\\ y & , x , y \\in A _ j \\ , \\ , \\epsilon ( j ) = 2 \\end{cases} \\end{align*}"}
+{"id": "2671.png", "formula": "\\begin{align*} \\varphi ( T ) = \\textsc { l e n g t h } ( T ) . \\end{align*}"}
+{"id": "2386.png", "formula": "\\begin{align*} E _ { 2 2 } ( t ) \\dot x _ 2 = A _ { 2 2 } ( t ) x _ 2 \\end{align*}"}
+{"id": "4702.png", "formula": "\\begin{align*} \\mathfrak { E } ^ { G _ { I V } } = \\mathbb { C } [ \\varphi _ 2 , \\varphi _ { 6 } ] \\end{align*}"}
+{"id": "4423.png", "formula": "\\begin{align*} ( 0 , 1 ] = \\bigcup _ { a _ 1 = 2 } ^ { \\infty } \\bigcup _ { a _ 2 = a _ 1 ^ 2 - a _ 1 + 1 } ^ { \\infty } \\left ( \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 } , \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 - 1 } \\right ] . \\end{align*}"}
+{"id": "7244.png", "formula": "\\begin{align*} \\int _ { \\partial B _ n } f ( x ( u ) ) ^ { 1 - b - \\frac { 2 } { n - 1 } } & \\int _ 0 ^ { \\phi _ f ( u , 1 ) } s ^ { N + b - 2 + \\frac { 2 } { n - 1 } } \\ , d s \\ , d \\mu _ { \\partial B _ n } ( u ) \\\\ & = \\int _ { \\partial B _ n } f ( x ( u ) ) ^ { 1 - b - \\frac { 2 } { n - 1 } } \\int _ { 1 - \\phi _ f ( u , 1 ) } ^ { 1 } ( 1 - w ) ^ { N + b - 2 + \\frac { 2 } { n - 1 } } \\ , d w \\ , d \\mu _ { \\partial B _ n } ( u ) . \\end{align*}"}
+{"id": "6142.png", "formula": "\\begin{align*} t = 0 : u _ r = ( E _ r , \\widehat U _ 0 ) , u _ r ' = ( E _ r , \\widehat U _ 1 ) \\hbox { i n } \\Omega . \\end{align*}"}
+{"id": "5866.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l l } \\phi '' - \\Delta \\phi = 0 & \\hbox { i n } ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu \\phi = 0 & \\hbox { o n } ( 0 , T ) \\times \\Gamma , \\\\ t = 0 : \\phi = \\theta , \\phi ' = 0 & \\hbox { i n } \\Omega \\end{array} \\right . \\end{align*}"}
+{"id": "3093.png", "formula": "\\begin{align*} r _ B = 1 + \\partial _ { 1 1 } ^ 2 w _ B - \\partial _ { 2 2 } ^ 2 w _ B ; \\end{align*}"}
+{"id": "5799.png", "formula": "\\begin{align*} \\hbox { r a n k } ( D ) = \\hbox { r a n k } ( C _ p D ) = N - p \\end{align*}"}
+{"id": "8468.png", "formula": "\\begin{align*} \\Gamma _ { A , \\gamma _ Q } ^ + = \\Gamma _ A ^ + . \\end{align*}"}
+{"id": "2816.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { n + 1 } x _ { \\sigma ( i ) } ^ { b _ i } = \\prod _ { i = 1 } ^ { n + 1 } x _ { \\sigma ( i ) } ^ { \\log v _ i } = e ^ { \\log v _ j } = v _ j . \\end{align*}"}
+{"id": "9050.png", "formula": "\\begin{align*} \\underset { n \\to \\infty } \\lim \\ , \\| Y ^ { i , n } - Y ^ i \\| _ { \\mathcal { H } ^ { p , 1 } } = 0 . \\end{align*}"}
+{"id": "4487.png", "formula": "\\begin{align*} \\frac { 2 } { n } = \\frac { 1 } { n } + \\frac { 1 } { 2 n } + \\frac { 1 } { 3 n } + \\frac { 1 } { 6 n } . \\end{align*}"}
+{"id": "5092.png", "formula": "\\begin{align*} F _ n \\coloneqq \\sum _ { i = 1 } ^ s \\beta _ i \\lambda _ i ^ n \\in L . \\end{align*}"}
+{"id": "8969.png", "formula": "\\begin{align*} | \\partial _ r u | ^ 2 = | d \\pi _ N ( u ) \\partial _ r u | ^ 2 + | d \\pi ^ { \\perp } _ N ( u ) \\partial _ r u | ^ 2 = | d \\pi _ N ( u ) \\partial _ r u | ^ 2 + | \\partial _ r ( d i s t _ N ( u ) ) | ^ 2 . \\end{align*}"}
+{"id": "2813.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ j b _ i = \\sum _ { i = 1 } ^ j \\log v _ i = \\log \\prod _ { i = 1 } ^ j v _ i \\leq \\log \\prod _ { i = 1 } ^ j u _ i = \\sum _ { i = 1 } ^ j \\log u _ i = \\sum _ { i = 1 } ^ j a _ i \\end{align*}"}
+{"id": "6723.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\mu ( n + a ) e ^ { i 2 \\pi u n / q } = O \\left ( \\frac { x } { ( \\log x / q ) ^ { D } } \\right ) , \\end{align*}"}
+{"id": "6490.png", "formula": "\\begin{align*} d X ^ { j } _ t = f ( t ) d t + \\sqrt { \\frac { m } { n } } d W _ t ^ { j } \\end{align*}"}
+{"id": "3459.png", "formula": "\\begin{align*} u ( t , x ) = 1 + \\sum _ { n \\geq 1 } I _ n ( f _ n ( \\cdot , x ; t ) ) , \\end{align*}"}
+{"id": "5599.png", "formula": "\\begin{align*} S _ { 1 ^ { \\prime } 2 ^ { \\prime } } = S _ { 1 2 } - z ( S _ { 0 1 } - S _ { 2 2 } ) . \\end{align*}"}
+{"id": "4167.png", "formula": "\\begin{align*} \\mathcal { L } ( ( g , b ) , \\mathcal { P } ( g , b ) ) = 0 . \\end{align*}"}
+{"id": "5171.png", "formula": "\\begin{align*} \\rho = - 2 \\eta \\otimes \\eta , \\end{align*}"}
+{"id": "6560.png", "formula": "\\begin{align*} \\frac { { { m _ w } ^ { { m _ w } } } } { { \\Gamma \\ ! \\left ( { { m _ w } } \\right ) } } \\ ! \\sum \\limits _ { j = 0 } ^ M \\ ! { \\frac { { { K _ w } ^ j { \\alpha _ { w j } } } } { { j ! } } } { \\left ( { { \\varepsilon _ k } \\ ! - \\ ! { \\kappa ^ 2 } } \\right ) ^ { q + \\ ! { m _ { f w } } \\ ! - 2 } } \\ ! \\geqslant \\ ! { \\left ( { { \\varepsilon _ { t h } } \\ ! - \\ ! { \\kappa ^ 2 } } \\right ) ^ { q + { m _ { f w } } - 2 } } \\ ! . \\end{align*}"}
+{"id": "3579.png", "formula": "\\begin{align*} z _ 1 ^ + & = B _ 1 ^ + \\\\ z _ 2 ^ + & = B _ 2 ^ + ( h _ 1 - h _ 2 ) - E _ { 2 1 } B _ 1 ^ + \\\\ z _ 3 ^ + & = B _ 3 ^ + ( h _ 1 - h _ 3 ) ( h _ 2 - h _ 3 ) - E _ { 3 2 } B _ 2 ^ + ( h _ 1 - h _ 3 ) - E _ { 3 1 } B _ 1 ^ + ( h _ 2 - h _ 3 - 1 ) + E _ { 2 1 } E _ { 3 2 } B _ 1 ^ + \\\\ z _ 1 ^ - & = B _ 1 ^ - ( h _ 1 - h _ 2 ) ( h _ 1 - h _ 3 ) + E _ { 2 1 } B _ 2 ^ - ( h _ 1 - h _ 3 ) + E _ { 3 1 } B _ 3 ^ - ( h _ 1 - h _ 2 ) + E _ { 2 1 } E _ { 3 2 } B _ 3 ^ - \\\\ z _ 2 ^ - & = B _ 2 ^ - ( h _ 2 - h _ 3 ) + E _ { 3 2 } B _ 3 ^ - \\\\ z _ 3 ^ - & = B _ 3 ^ - \\end{align*}"}
+{"id": "2833.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ n } t _ { \\sigma } = 1 \\end{align*}"}
+{"id": "4725.png", "formula": "\\begin{align*} \\mathcal { C } ^ { \\perp _ { \\mathrm { s } , t } } \\ = \\ \\{ ( v ' , w ' ) \\in \\mathbb { Z } _ { p ^ a } ^ { 2 n } ~ : ~ \\langle ( v ' , w ' ) \\mid ( v , w ) \\rangle _ { \\mathrm { s } } \\equiv 0 \\ ! \\ ! \\ ! \\ ! \\pmod { p ^ { a - t } } ( v , w ) \\in \\mathcal { C } \\} . \\end{align*}"}
+{"id": "3654.png", "formula": "\\begin{align*} \\bar u ( x , t ) \\leq e ^ { - \\gamma t ^ { \\frac { q } { \\alpha } } } , \\hbox { w h e r e } ~ \\gamma = \\delta ^ { - \\frac { 1 } { \\alpha } } . \\end{align*}"}
+{"id": "5678.png", "formula": "\\begin{align*} d s ^ { 2 } = \\eta _ { a b } \\theta ^ { a } \\otimes \\theta ^ { b } , \\end{align*}"}
+{"id": "3249.png", "formula": "\\begin{align*} I ^ { \\pm } ( z ) & = \\sum _ { i _ 1 + j _ 1 = q _ 1 } ( I _ { D _ { i _ 1 } } ^ + ( z ) + I _ { D _ { i _ 1 } } ^ - ( z ) ) = \\sum _ { i _ 1 + j _ 1 = q _ 1 } \\sum _ { \\sigma \\in \\mathfrak { S } _ { q _ 1 + k _ 1 } } ( I _ { D _ \\sigma } ^ + ( z ) + I _ { D _ { \\sigma } } ^ - ( z ) ) . \\end{align*}"}
+{"id": "8545.png", "formula": "\\begin{align*} n ^ { 3 n } 2 ^ { - 3 n ^ 2 / 2 + 9 n / 2 - 4 } \\frac { 2 ^ { n ( n + 1 ) } - 1 } { 2 ^ n - 1 } & = 2 ^ { 3 n \\log _ 2 ( n ) - n ^ 2 / 2 + 1 1 n / 2 - 4 } \\frac { 2 ^ { - n ^ 2 - n } ( 2 ^ { n ^ 2 + n } - 1 ) } { 2 ^ n - 1 } \\\\ & = 2 ^ { 3 n \\log _ 2 ( n ) - n ^ 2 / 2 + 1 1 n / 2 - 4 } \\frac { 1 - 2 ^ { - n ^ 2 - n } } { 2 ^ { n } - 1 } . \\end{align*}"}
+{"id": "4615.png", "formula": "\\begin{align*} \\prod _ { i = m } ^ n a _ i \\leq \\prod _ { i = m } ^ n b _ i . \\end{align*}"}
+{"id": "6746.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\rho } z _ N ( \\rho ) = \\frac { 4 } { \\sqrt { \\pi } } \\frac { 1 } { \\sqrt { 1 - \\rho } } \\left [ \\frac { 1 } { 2 } + \\sum _ { n = 1 } ^ { N } \\exp \\left [ - \\frac { { \\alpha _ n } ^ 2 } { 4 \\left ( 1 - \\rho \\right ) } \\right ] \\right ] . \\end{align*}"}
+{"id": "518.png", "formula": "\\begin{align*} \\beta _ { k , 0 } = { ( t _ { k - 1 } - 1 ) } / { t _ k } \\ \\ { \\rm w i t h } \\ \\ t _ { k + 1 } = \\big ( 1 + \\ ! \\sqrt { 1 + 4 t _ k ^ 2 } \\big ) / 2 \\ \\ { \\rm f o r } \\ \\ t _ { - 1 } = t _ 0 = 1 . \\end{align*}"}
+{"id": "3519.png", "formula": "\\begin{align*} \\gamma = \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ i \\gamma _ { i j } e _ { i j } = \\gamma _ \\lambda + \\sum _ { i = 2 } ^ n \\sum _ { j = 1 } ^ { i - 1 } \\gamma _ { i j } ( e _ { i j } - e _ { j j } ) . \\end{align*}"}
+{"id": "914.png", "formula": "\\begin{align*} t _ { n } = ( n ( n + 1 ) ) ^ { - t _ { n + 1 } } \\end{align*}"}
+{"id": "7138.png", "formula": "\\begin{align*} u _ 0 ( a e _ i ) & = u _ 2 ( a e _ i ) = a b , \\\\ u _ 1 ( a e _ i ) & = f ( a \\otimes b ) e _ j + a c \\end{align*}"}
+{"id": "1110.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\{ Y _ i , W _ i \\} _ { i = 1 } ^ n } \\big \\{ R ( \\widehat { \\theta } ) - R ( \\theta ( P ) ) \\big \\} \\lesssim \\frac { r } { \\sqrt { n \\alpha ^ 2 } } \\vee \\varepsilon r , \\end{align*}"}
+{"id": "6159.png", "formula": "\\begin{align*} ( E , U _ n ) = ( x , C _ 1 U _ n ) \\rightarrow 0 \\hbox { i n } C ^ 0 _ { l o c } ( [ T , + \\infty ) ; \\mathcal H _ { 0 } ) \\cap C ^ 1 _ { l o c } ( [ T , + \\infty ) ; \\mathcal H _ { - 1 } ) \\end{align*}"}
+{"id": "6339.png", "formula": "\\begin{align*} \\varphi ' ( \\nu _ 0 ) + \\int _ 1 ^ { r ( q ) } f _ \\nu ( x , \\nu _ 0 ) d x - \\int _ { r ( q _ c ) } ^ 1 f _ \\nu ( x , \\nu _ 0 ) d x = 0 \\end{align*}"}
+{"id": "9089.png", "formula": "\\begin{align*} \\mathfrak { d } _ s & = [ \\mathfrak { d } _ { s - 1 } , \\mathfrak { n } ] + J [ \\mathfrak { d } _ { s - 1 } , \\mathfrak { n } ] \\\\ & \\subseteq [ \\mathfrak { d } ^ { j _ 0 - s + 1 } , \\mathfrak { n } ] + J [ \\mathfrak { d } ^ { j _ 0 - s + 1 } , \\mathfrak { n } ] \\\\ & \\subseteq \\mathfrak { d } ^ { j _ 0 - s } + J \\mathfrak { d } ^ { j _ 0 - s } = \\mathfrak { d } ^ { j _ 0 - s } . \\end{align*}"}
+{"id": "5522.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq n \\leq x } a _ n f ( n ) = A ( x ) f ( x ) - \\int _ { 1 } ^ { x } A ( t ) f ' ( t ) { \\rm d } t . \\end{align*}"}
+{"id": "4736.png", "formula": "\\begin{align*} \\mathfrak { q } = \\bigl ( ( A : _ A x ) : _ A r \\bigr ) = ( A : _ A r x ) , \\end{align*}"}
+{"id": "5049.png", "formula": "\\begin{align*} m _ i = p _ i / N + \\sum _ { j = 1 } ^ d n _ j q _ { i , j } / N \\end{align*}"}
+{"id": "8282.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\mathcal { T } _ p : = \\ ! \\big \\{ \\textbf { x } \\ | \\ \\| \\textbf { x } _ 1 \\ ! - \\ ! \\textbf { x } _ E \\| _ 2 = 0 \\big \\} \\cup \\big \\{ \\textbf { x } \\ | \\ \\| \\textbf { x } _ 2 \\ ! - \\ ! \\textbf { x } _ E \\| _ 2 = 0 \\big \\} \\\\ \\cup \\big \\{ \\textbf { x } \\ | \\ \\| \\textbf { x } _ 3 \\ ! - \\ ! \\textbf { x } _ E \\| _ 2 = 0 \\big \\} \\end{array} \\right . \\end{align*}"}
+{"id": "136.png", "formula": "\\begin{align*} H _ { \\lambda , K , x , r } ^ + = H _ { \\lambda , K , x } ^ + \\setminus r K \\subseteq H _ { \\lambda , K , x } ^ + \\subseteq H _ { \\lambda , K , x , r } ^ + \\cup r K . \\end{align*}"}
+{"id": "5427.png", "formula": "\\begin{align*} \\bar { \\alpha } _ t \\alpha _ t = q _ t g _ t ^ { - 1 } \\bar { q } _ t g _ t ^ { - 1 } = g _ t ^ { \\frac { 1 } { 2 } } g _ t ^ { - \\frac { 1 } { 2 } } q _ t g _ t ^ { - 1 } \\bar { q } _ t g _ t ^ { - \\frac { 1 } { 2 } } g _ t ^ { - \\frac { 1 } { 2 } } \\end{align*}"}
+{"id": "2772.png", "formula": "\\begin{align*} u _ i \\triangleq \\sum _ { \\ell = i } ^ n x _ \\ell - \\sum _ { \\ell = i } ^ n x ' _ \\ell . \\end{align*}"}
+{"id": "7223.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\alpha _ n = \\alpha _ 0 \\qquad \\mbox { a n d } \\qquad \\lim _ { n \\to \\infty } s _ n = \\lim _ { n \\to \\infty } t _ n = t _ 0 \\in I \\end{align*}"}
+{"id": "3859.png", "formula": "\\begin{align*} A _ { n - 2 } - T _ n = \\sum _ { k = 2 } ^ { n - 2 } \\sum _ { l = 2 } ^ k \\mu _ l A _ { k - l } T _ { n - 2 - k } . \\end{align*}"}
+{"id": "3504.png", "formula": "\\begin{align*} E ^ \\gamma : = \\prod _ { k = 2 , \\dots , n } ^ \\rightarrow ( E _ { k 1 } ^ { \\gamma _ { k 1 } } \\cdots E _ { k , k - 1 } ^ { \\gamma _ { k , k - 1 } } ) , \\end{align*}"}
+{"id": "3898.png", "formula": "\\begin{align*} D _ { x _ j } ( E _ { a b } ) & = D _ { x _ j } \\left [ g _ { a , b } - \\frac { g _ { a , z } g _ { , b } } { g _ z } \\right ] \\\\ & = g _ { a j , b } - \\frac { g _ { a j , z } g _ { , b } } { g _ z } - \\frac { g _ { a , z } g _ { j , b } } { g _ z } + \\frac { g _ { j , z } g _ { a , z } g _ { , b } } { g _ z ^ 2 } \\\\ & = - \\frac { g _ { a , z } } { g _ z } E _ { j b } + E _ { l , b } D _ { p _ l } g _ { a j } . \\end{align*}"}
+{"id": "5960.png", "formula": "\\begin{align*} ( \\sum _ { r = 1 } ^ p a _ r P e _ r , \\sum _ { s = 1 } ^ p a _ s P e _ s ) = 0 , \\end{align*}"}
+{"id": "1288.png", "formula": "\\begin{align*} ~ p _ { \\ell - 1 } ^ { ( k - 1 ) } ( x ) = m _ { k - 1 , \\ell - 1 } ' ( x ) q _ { \\ell - 1 } ^ { ( k - 1 ) * } ( x ) . \\end{align*}"}
+{"id": "1776.png", "formula": "\\begin{align*} \\omega ( M ) : = \\sup \\{ \\omega ( a ) \\mid a \\in \\mathcal { A } ( M ) \\} . \\end{align*}"}
+{"id": "5368.png", "formula": "\\begin{align*} B _ 0 : = \\left \\{ \\eta \\in W ^ { 1 , \\infty } \\left ( \\Omega \\right ) \\cap \\mathrm { S y m } _ 3 ( \\Omega ) : \\| \\eta \\| _ { W ^ { 1 , \\infty } ( \\Omega ) } \\leq \\delta \\right \\} \\end{align*}"}
+{"id": "8167.png", "formula": "\\begin{align*} \\mathcal { D } & = \\frac { A ( p , 1 ) } { 2 p ^ { \\frac { 1 } { 2 } } } \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } H _ { m , p } - \\frac { 1 } { 2 p ^ { \\frac { 3 } { 2 } } } \\sum _ { m \\geq 1 } \\frac { A ( 1 , m ) } { m } H _ { m p , p } . \\end{align*}"}
+{"id": "1907.png", "formula": "\\begin{align*} | X _ 2 | & \\ll 2 ^ { 2 T } \\sum _ { \\frac { e a ^ 2 } { L ^ 2 } < x _ 1 < \\sqrt { a + d L } } 2 ^ { - \\omega ( x _ 1 ) } \\cdot \\frac { d ^ 3 L ^ 3 } { x _ 1 a \\phi ( d ) ^ 3 } ( \\log x _ 1 ) ^ { - \\frac 1 2 } ( \\log ( a + d L ) ) ^ { - 1 } \\\\ & = \\frac { d ^ 3 L ^ 3 } { a \\phi ( d ) ^ 3 } 2 ^ { 2 T } ( \\log ( a + d L ) ) ^ { - 1 } \\sum _ { \\frac { e a ^ 2 } { L ^ 2 } < x _ 1 < \\sqrt { a + d L } } 2 ^ { - \\omega ( x _ 1 ) } \\frac { ( \\log x _ 1 ) ^ { - \\frac 1 2 } } { x _ 1 } . \\end{align*}"}
+{"id": "5210.png", "formula": "\\begin{align*} & \\begin{cases} b _ { 1 1 } = b _ { 2 2 } \\\\ b _ { 1 6 } = b _ { 2 4 } \\end{cases} \\quad \\mathrm { o r } \\frac { b _ { 1 1 } } { b _ { 1 6 } } + \\frac { b _ { 1 6 } } { b _ { 1 1 } } + \\frac { b _ { 2 2 } } { b _ { 2 4 } } = 0 \\\\ & \\mathrm { o r } \\frac { b _ { 1 6 } } { b _ { 1 1 } } + \\frac { b _ { 2 4 } } { b _ { 2 2 } } + \\frac { b _ { 1 6 } b _ { 2 4 } } { b _ { 1 1 } b _ { 2 2 } } = 0 \\ , . \\end{align*}"}
+{"id": "7377.png", "formula": "\\begin{align*} \\theta = \\prod _ { i = - 1 } ^ d \\phi _ i , \\end{align*}"}
+{"id": "4292.png", "formula": "\\begin{align*} - \\Delta \\ , ( { \\psi } \\widetilde { \\textbf { \\textit { u } } } ) + \\nabla \\ , ( { \\psi } \\widetilde { \\pi } ) = \\textbf { \\textit { f } } _ { 1 } \\mathrm { a n d } \\mathrm { d i v } \\ , ( { \\psi } \\widetilde { \\textbf { \\textit { u } } } ) = \\chi _ { 1 } \\quad \\quad \\R ^ 3 , \\end{align*}"}
+{"id": "6930.png", "formula": "\\begin{align*} S ( x ) \\log x & = \\sum _ { n \\leq x } g ( n ) \\log x \\\\ & = \\sum _ { n \\leq x } g ( n ) \\log \\left ( \\frac { x } { n } \\right ) + \\sum _ { n \\leq x } g ( n ) \\sum _ { p | | n } \\log p + \\sum _ { n \\leq x } g ( n ) \\sum _ { \\alpha \\geq 2 \\atop p ^ { \\alpha } | | n } \\log p ^ { \\alpha } \\\\ & = S _ 1 + S _ 2 + S _ 3 . \\end{align*}"}
+{"id": "7904.png", "formula": "\\begin{align*} f ^ { ( n ) } + A _ { n - 1 } ( z ) f ^ { ( n - 1 ) } + \\cdots + A _ 1 ( z ) f ' + A _ 0 ( z ) f = 0 \\end{align*}"}
+{"id": "6969.png", "formula": "\\begin{gather*} \\vec { h } = \\begin{pmatrix} h _ { 1 , n } \\\\ h _ { 2 , n } \\\\ \\vdots \\\\ h _ { n , n } \\end{pmatrix} \\ ! , \\vec { b } _ { I I I } = \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ P _ n ( x ) \\end{pmatrix} \\ ! , \\end{gather*}"}
+{"id": "4134.png", "formula": "\\begin{align*} u = \\sum a _ i \\chi _ i , \\omega = \\sum b _ j \\chi _ j \\end{align*}"}
+{"id": "8254.png", "formula": "\\begin{align*} \\int \\frac { \\partial ^ 2 \\log f ( u , \\theta ) } { \\partial \\theta _ i \\partial \\theta _ j } . f ( u , \\theta ) d \\nu = - \\int \\frac { \\partial \\log f ( u , \\theta ) } { \\partial \\theta _ i } . \\frac { \\partial \\log f ( u , \\theta ) } { \\partial \\theta _ j } . f ( u , \\theta ) d \\nu \\end{align*}"}
+{"id": "2698.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u - \\Delta u = | u | u , \\end{align*}"}
+{"id": "3273.png", "formula": "\\begin{align*} x - \\lambda F ^ { - 1 } \\circ G \\left ( x \\right ) = 0 \\end{align*}"}
+{"id": "5396.png", "formula": "\\begin{align*} \\mathsf { E } ( G ) = D ( G ) + | G | - 1 . \\end{align*}"}
+{"id": "5286.png", "formula": "\\begin{align*} ( m - n ) \\nabla f = ( n - 2 ) \\nabla \\log { \\lambda } . \\end{align*}"}
+{"id": "6913.png", "formula": "\\begin{align*} \\alpha = \\frac { k } { \\rho c _ { \\rho } } , \\end{align*}"}
+{"id": "990.png", "formula": "\\begin{align*} \\hat { a } ( \\varphi _ j ) = \\varphi _ j ( a ) & = \\sum _ { x \\in G } \\varphi _ j ( f ( x ) ) \\\\ & = \\int _ G \\varphi _ j \\circ f ( x ) d x \\\\ & = \\int _ G \\overline { I ( x ) } \\varphi _ j \\circ f ( x ) d x \\\\ & = I \\otimes \\varphi _ j ( f ) \\end{align*}"}
+{"id": "1618.png", "formula": "\\begin{align*} \\hat S ^ { ( t ) } _ i ( \\theta _ { c ( i ) } ) = 1 [ | \\hat U _ i ^ { ( t ) } | ^ 2 > \\theta _ { c ( i ) } ] \\end{align*}"}
+{"id": "5867.png", "formula": "\\begin{align*} \\| \\theta \\| ^ 2 _ { L ^ 2 ( \\Omega ) } = \\int _ 0 ^ T \\int _ \\Omega { \\cal L } \\theta \\phi d x + \\int _ 0 ^ T \\int _ { \\Gamma } { \\cal R } \\theta \\phi d \\Gamma . \\end{align*}"}
+{"id": "2086.png", "formula": "\\begin{align*} \\dim _ H ( C _ { b , W } ) = \\dim _ P ( C _ { b , W } ) = \\frac { \\log | W | } { \\log b } , \\end{align*}"}
+{"id": "2903.png", "formula": "\\begin{align*} S ( K ) = \\inf _ { f \\in H _ 0 ^ p ( K d m _ \\infty ) } \\int _ { \\mathbb { T } ^ \\infty } | 1 - f | ^ p K d m _ { \\infty } = \\| \\pi ( 1 ) \\| _ { \\mathcal { Q } } ^ p = \\sup _ { g \\in \\mathbf { B } \\cap \\mathcal { Q } ^ { * } } \\left | \\int _ { \\mathbb { T } ^ \\infty } g K d m _ \\infty \\right | ^ p , \\end{align*}"}
+{"id": "6114.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\dfrac { \\left [ n ^ { - h } \\right ] ( 2 n L ( j , n ) ) } { \\left [ n ^ { - h } \\right ] ( 2 n R ( j , n ) ) } = 1 . \\end{align*}"}
+{"id": "3733.png", "formula": "\\begin{align*} Z _ { r _ \\ell } ( f , 2 - \\ell ) & = \\int _ { F ^ \\times } f ( 0 , 0 , a ) | a | ^ s d ^ \\times a \\bigg | _ { s = 2 - \\ell } \\\\ & = \\int _ { | a | \\le q ^ { - A } } f ( 0 ) | a | ^ s d ^ \\times a \\bigg | _ { s = 2 - \\ell } + \\sum _ { j = - B } ^ { A - 1 } q ^ { j ( \\ell - 2 ) } f ( 0 , 0 , \\varpi ^ j ) \\\\ & = \\frac { q ^ { A ( \\ell - 2 ) } } { 1 - q ^ { \\ell - 2 } } f ( 0 ) + \\sum _ { j = - B } ^ { A - 1 } q ^ { j ( \\ell - 2 ) } f ( 0 , 0 , \\varpi ^ j ) , \\end{align*}"}
+{"id": "7087.png", "formula": "\\begin{align*} \\Omega ^ { C _ 2 } _ * = M U _ * [ d _ { i , j } , q _ j ] / I \\end{align*}"}
+{"id": "7843.png", "formula": "\\begin{align*} f ( z ) = F _ 1 ( z ) e ^ { w _ 1 z ^ q } + \\cdots + F _ n ( z ) e ^ { w _ n z ^ q } , \\end{align*}"}
+{"id": "677.png", "formula": "\\begin{align*} G ^ { { \\rm v o l } } = 0 . \\end{align*}"}
+{"id": "8189.png", "formula": "\\begin{align*} \\kappa _ g ( \\tau , \\sigma ) : = k ( g ( \\tau ) , g ( \\sigma ) ) \\dot { g } ( \\sigma ) . \\end{align*}"}
+{"id": "6563.png", "formula": "\\begin{align*} D ( q , r ) : = \\frac { \\max \\left ( r \\psi ( q ) , q \\psi ( r ) \\right ) } { \\gcd ( q , r ) } , q , r \\in \\mathbb { N } . \\end{align*}"}
+{"id": "5481.png", "formula": "\\begin{align*} & E \\hat V ( X _ t - Y _ t ) \\\\ = & E \\hat V ( X _ 0 - Y _ 0 ) + E \\int _ 0 ^ t \\tilde L ^ { ( \\alpha _ s ) } \\hat V ( X _ s - Y _ s ) d s \\\\ \\leq & E \\hat V ( X _ 0 - Y _ 0 ) - \\theta E \\int _ 0 ^ t \\hat V ( X _ s - Y _ s ) d s . \\end{align*}"}
+{"id": "9118.png", "formula": "\\begin{align*} \\frac { d P _ { Y ^ { n _ 1 } } ( y ^ { n _ 1 } ) } { d Q _ { Y ^ { n _ 1 } } ( y ^ { n _ 1 } ) } \\leq \\tilde { \\mathsf { K } } : = 7 2 9 \\cdot \\frac { \\pi } { 8 } \\cdot \\frac { ( 1 + \\mathsf { P } ) ^ 2 } { 1 + 2 \\mathsf { P } } . \\end{align*}"}
+{"id": "238.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\tau ^ 2 - \\beta \\tau \\eta + \\alpha \\leqslant \\alpha , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho ^ 2 - \\beta \\tau \\eta + \\alpha \\leqslant \\alpha , \\end{align*}"}
+{"id": "2124.png", "formula": "\\begin{align*} \\frac { 1 } { \\left ( \\frac { j } { \\alpha _ k } \\right ) ^ \\frac { l } { 2 \\mu _ k } } { H _ { 2 \\mu _ k } ^ { \\beta _ k } } ^ { ( m ) } \\left ( Y _ { n , j , k } \\right ) l \\in \\N ^ * , m \\in \\N , Y _ { n , j , k } : = \\frac { n \\alpha _ k - j } { \\left ( \\frac { j } { \\alpha _ k } \\right ) ^ \\frac { 1 } { 2 \\mu _ k } } . \\end{align*}"}
+{"id": "7671.png", "formula": "\\begin{align*} \\sum _ { m \\in M } ( - 1 ) ^ { \\sigma _ { M \\setminus \\{ m \\} } ( m ) } \\eta _ { M \\setminus \\{ m \\} , p } = 0 . \\end{align*}"}
+{"id": "5437.png", "formula": "\\begin{align*} A ( Z , S ) = \\Im ( Z ) ^ { - \\frac { 1 } { 2 } } S \\Im ( Z ) ^ { - 1 } \\bar { S } \\Im ( Z ) ^ { - \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "5002.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u - \\nabla \\cdot ( S ( u , v ) \\nabla v ) + g ( u , v ) , \\\\ v _ t = \\Delta v + h ( u , v ) , \\end{cases} \\end{align*}"}
+{"id": "2788.png", "formula": "\\begin{align*} x _ 1 ^ { a + b } + x _ 2 ^ { a + b } - x _ 1 ^ a x _ 2 ^ b - x _ 1 ^ b x _ 2 ^ a = ( x _ 1 ^ a - x _ 2 ^ a ) ( x _ 1 ^ b - x _ 2 ^ b ) > 0 . \\end{align*}"}
+{"id": "3265.png", "formula": "\\begin{align*} | \\chi _ S \\tilde { P } ' _ { \\mathbf { d } , \\mathcal { A } } ( a b ) | \\leq & \\frac { d } { n } ( 2 4 d ) ^ { - | S | } ( 4 | S | ) ! \\cdot \\left ( C _ 8 + 2 C _ 8 | S | \\binom { 4 | S | } { 4 } ^ { - 1 } + \\frac { 4 | S | ^ 2 } { 7 0 } \\binom { 4 | S | } { 8 } ^ { - 1 } + o ( 1 ) \\right ) \\\\ \\leq & \\frac { d } { n } ( 2 4 d ) ^ { - | S | } ( 4 | S | ) ! \\cdot ( 2 C _ 8 + 1 / 2 ) , \\end{align*}"}
+{"id": "8230.png", "formula": "\\begin{align*} - \\Delta u + V \\left ( \\left | x \\right | \\right ) f ( u ) f ' ( u ) = K ( | x | ) g ( f ( u ) ) f ' ( u ) \\mathbb { R } ^ { N } , \\end{align*}"}
+{"id": "7505.png", "formula": "\\begin{align*} Z _ { \\tilde { g } } ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) = & \\int _ { ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 } \\chi ( a c \\tilde { g } ( u , v ) ) | \\tilde { g } ( u , v ) | ^ s | d u d v | \\\\ = & q ^ { - e _ 0 s } \\int _ { ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 } \\chi ( a c \\tilde { g } ( u , v ) ) | d u d v | \\\\ : = & q ^ { - e _ 0 s } c ( \\chi ) . \\end{align*}"}
+{"id": "8583.png", "formula": "\\begin{align*} \\frac { g ( x ) } { F ( x ) } = \\left ( 1 + x ^ 2 + x ^ 5 + x ^ 6 + x ^ 8 + x ^ 9 + x ^ { 1 0 } + x ^ { 1 1 } \\right ) + x ^ { 1 5 } \\left ( \\frac { g ( x ) } { F ( x ) } \\right ) , \\end{align*}"}
+{"id": "224.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - \\beta \\rho \\eta \\leqslant - \\alpha \\frac { ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\alpha } { 2 } \\end{align*}"}
+{"id": "8105.png", "formula": "\\begin{align*} \\phi ( \\xi ) = - \\frac { T \\xi } { M } \\pm \\frac { x } { 2 \\pi } \\cosh \\frac { \\xi \\pi } { M } , \\end{align*}"}
+{"id": "4938.png", "formula": "\\begin{align*} D _ x F _ \\ell + D _ t G _ \\ell = 0 , \\ell = 1 , 2 , \\end{align*}"}
+{"id": "934.png", "formula": "\\begin{align*} \\theta _ { \\alpha } = \\frac { e _ { 1 } + \\alpha e _ { 2 } } { \\sqrt { 1 + | \\alpha | ^ { 2 } } } \\end{align*}"}
+{"id": "1132.png", "formula": "\\begin{align*} T ' ( m ) & = \\frac { 1 } { \\mu } ( - e T + T e ) ( m ) \\\\ & = \\frac { 1 } { \\mu } ( - E T ( m ) + T ( E m ) ) . \\end{align*}"}
+{"id": "8013.png", "formula": "\\begin{align*} e & = ( F _ 3 , F _ 2 ) , \\\\ T & = \\left ( \\{ F _ 3 , F _ 1 , F _ 2 \\} , \\{ ( F _ 3 , F _ 1 ) , ( F _ 3 , F _ 2 ) \\} \\right ) , \\\\ F & = [ \\emptyset ] . \\end{align*}"}
+{"id": "1058.png", "formula": "\\begin{align*} \\{ \\phi _ k = \\phi ( \\cdot - k ) : k \\in \\mathbb { Z } \\} \\cup \\{ \\psi _ { j k } = 2 ^ { j / 2 } \\psi ( 2 ^ j ( \\cdot ) - k ) : j \\in \\mathbb { N } _ + \\cup \\{ 0 \\} , k \\in \\mathbb { Z } \\} . \\end{align*}"}
+{"id": "2228.png", "formula": "\\begin{align*} \\nu { \\left ( \\sum _ { k = 2 } ^ \\infty x _ { 2 m - 1 , k } \\alpha _ { k , 1 } \\right ) } \\geq \\min _ { k \\geq 2 } \\big ( \\nu ( x _ { 2 m - 1 , k } ) + \\nu ( \\alpha _ { k , 1 } ) \\big ) \\geq \\min _ { k \\geq 2 } \\nu ( x _ { 2 m - 1 , k } ) \\geq 2 m - 1 . \\end{align*}"}
+{"id": "2409.png", "formula": "\\begin{align*} \\dot x _ 2 = S ^ { - 1 } J ( t ) x _ 2 + S ^ { - 1 } \\widetilde f _ 2 ( t ) . \\end{align*}"}
+{"id": "2677.png", "formula": "\\begin{align*} r ( y + \\varepsilon _ t ) = r ( y ) + r ( \\varepsilon _ t ) \\forall t \\geq 0 , \\end{align*}"}
+{"id": "6813.png", "formula": "\\begin{align*} \\delta Y _ { i } & = ( - 1 ) ^ { n - 1 } u _ { i 2 } z _ { m - 1 } , \\\\ \\delta F _ { i } & = ( - 1 ) ^ { n - 1 } \\frac { \\partial f } { \\partial y } u _ { i 2 } z _ { m - 1 } . \\end{align*}"}
+{"id": "67.png", "formula": "\\begin{align*} \\min _ { \\mathbf { z } } \\sum _ { i = 1 } ^ { n } { z _ i } \\end{align*}"}
+{"id": "4206.png", "formula": "\\begin{align*} \\begin{cases*} \\partial _ { \\alpha } ( a ^ { \\alpha \\beta } \\partial _ { \\beta } \\Psi ^ { ( i ) } ) = F ( \\Psi ^ { ( i - 1 ) } , \\partial \\Psi ^ { ( i - 1 ) } ) \\\\ ( \\Psi ^ { ( i ) } , \\partial _ t \\Psi ^ { ( i ) } ) | _ { \\{ t = 0 \\} } = ( \\Psi _ 0 , \\Psi _ 1 ) \\in \\mathcal { H } . \\end{cases*} \\end{align*}"}
+{"id": "2158.png", "formula": "\\begin{align*} r _ a & = \\begin{dcases} a - \\frac { 1 } { \\mathit { e } } & \\ \\frac { 1 } { \\mathit { e } } < a \\leqslant \\frac { \\mathit { e } + \\mathit { e } ^ { - 1 } } { 2 } \\\\ \\mathit { e } - a & \\ \\frac { \\mathit { e } + \\mathit { e } ^ { - 1 } } { 2 } \\leqslant a < \\mathit { e } . \\end{dcases} \\end{align*}"}
+{"id": "3963.png", "formula": "\\begin{align*} \\overline { g } ( \\overline { x } , \\overline { y } , z ) = \\overline { x } \\cdot \\overline { y } - z + O ( | \\overline { y } | ^ 2 ) + O ( | \\overline { y } | | z | ) + O ( | z | ^ 2 ) \\end{align*}"}
+{"id": "223.png", "formula": "\\begin{align*} \\psi _ 1 ( u , v , w ) = & u ( \\alpha v + \\beta w ) , \\\\ \\psi _ 2 ( u , v , w ) = & v ( \\alpha u + \\beta w ) , \\\\ \\psi _ 3 ( u , v , w ) = & u ( \\alpha u + \\beta w ) \\shortintertext { a n d } \\psi _ 4 ( v , w ) = & v ( \\alpha v + \\beta w ) . \\end{align*}"}
+{"id": "7538.png", "formula": "\\begin{align*} Z _ g \\big ( s , \\chi , S ( \\Delta _ { \\gamma _ 2 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) ) \\big ) = & \\sum _ { m = 0 } ^ { j _ 0 - 1 } q ^ { - m - w ( m ) } \\dfrac { \\tilde { G } _ 2 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } \\sum _ { b = 1 } ^ { \\infty } q ^ { - b ( i _ 0 + j _ 0 + i _ 0 j _ 0 s ) } \\\\ = & \\dfrac { G _ 2 ( q ^ { - s } ) } { ( 1 - q ^ { - 1 - s } ) ( 1 - q ^ { - i _ 0 - j _ 0 - i _ 0 j _ 0 s } ) } , \\end{align*}"}
+{"id": "8857.png", "formula": "\\begin{align*} F ( r \\omega ) = \\frac { 1 } { \\left ( 1 + r ^ { 2 } \\right ) ^ { ( m + \\nu ) } } \\sum _ { \\substack { 0 \\leq p \\leq m \\\\ 0 \\leq q \\leq m + 2 \\nu } } r ^ { p + q } { } _ { 2 } F _ { 1 } \\left ( \\begin{array} { c } p - m , q - m - 2 \\nu \\\\ n + p + q \\end{array} \\mid - r ^ { 2 } \\right ) \\sum _ { j = 1 } ^ { d ( n , p , q ) } a _ { j } ^ { \\nu , p , q } h _ { p , q } ^ { j } ( \\omega , \\bar { \\omega } ) , \\end{align*}"}
+{"id": "8221.png", "formula": "\\begin{align*} P \\lbrace N ( s ) \\ \\ | \\ \\mathcal { T } ( s ) = x \\rbrace = \\frac { ( a _ 1 - a _ 2 ) s \\ , I _ 1 \\bigl ( \\frac { 2 \\lambda } { a _ 1 - a _ 2 } A ( s , x ) \\bigr ) } { 2 A ( s , x ) \\ , I _ 0 \\bigl ( \\frac { 2 \\lambda } { a _ 1 - a _ 2 } A ( s , x ) \\bigr ) + ( a _ 1 - a _ 2 ) s \\ , I _ 1 \\bigl ( \\frac { 2 \\lambda } { a _ 1 - a _ 2 } A ( s , x ) \\bigr ) } \\end{align*}"}
+{"id": "6886.png", "formula": "\\begin{align*} J _ \\mu ( u ) \\geq \\ell _ \\mu > 0 u \\in X \\ \\| u \\| = \\delta _ \\mu . \\end{align*}"}
+{"id": "1667.png", "formula": "\\begin{align*} C _ \\sigma = \\left \\{ \\begin{array} { l l } 4 ( - 1 ) ^ { m _ \\sigma } ( 2 \\pi ) ^ { - k _ \\sigma } \\Gamma ( \\frac { k _ \\sigma } { 2 } - m _ \\sigma ) \\Gamma ( \\frac { k _ \\sigma } { 2 } + m _ \\sigma ) , & \\chi _ { 0 , \\sigma } = \\lambda _ \\sigma ; \\\\ 0 , & \\chi _ { 0 , \\sigma } \\neq \\lambda _ \\sigma . \\end{array} \\right . \\end{align*}"}
+{"id": "7896.png", "formula": "\\begin{align*} f ( z ) = H _ 0 ( z ) + H _ 1 ( z ) e ^ { w _ 1 z ^ q } + \\cdots + H _ m ( z ) e ^ { w _ m z ^ q } . \\end{align*}"}
+{"id": "6376.png", "formula": "\\begin{align*} P _ \\lambda ( u ) : = Q ' _ \\lambda ( u ) ( x _ \\lambda ' ( u ) ^ 2 + y ' ( u ) ^ 2 ) - 2 Q _ \\lambda ( u ) ( x _ \\lambda ' ( u ) ^ 2 + y ' ( u ) ^ 2 ) ' . \\end{align*}"}
+{"id": "3365.png", "formula": "\\begin{align*} \\dd \\eta ( X , Y ) = g ( X , \\Phi ( Y ) ) , \\end{align*}"}
+{"id": "2782.png", "formula": "\\begin{align*} u _ i \\ ! = \\ ! \\left \\{ \\ ! \\begin{aligned} & 0 , ~ ~ ~ ~ ~ ~ ~ ~ \\ ! ~ ~ ~ ~ ~ i \\ ! \\in \\ ! [ 1 , d _ 1 ] ; \\\\ & x _ { i } - x ' _ { d _ 2 } , ~ ~ i \\in [ d _ 1 + 1 , \\lambda _ 1 - 1 ] ; \\\\ & x _ { i } + x _ { \\lambda _ 2 } - x ' _ { d _ 2 } - x ' _ { \\lambda _ 2 } , ~ ~ i \\in [ \\lambda _ 1 , d _ 2 ] ; \\\\ & x _ { \\lambda _ 2 } - x ' _ { \\lambda _ 2 } , ~ ~ ~ ~ i \\in [ d _ 2 + 1 , \\lambda _ 2 ] ; \\\\ & 0 , ~ ~ ~ ~ ~ ~ ~ ~ \\ ! ~ ~ ~ ~ ~ i \\in [ \\lambda _ 2 + 1 , n ] . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4894.png", "formula": "\\begin{align*} C _ { s _ 1 s _ 2 u _ 0 } C _ { u _ l s _ 2 s _ 1 } = \\xi ^ 3 ( C _ { s _ 1 } C _ { s _ 2 } - 1 ) C _ { u _ l s _ 2 s _ 1 } . \\end{align*}"}
+{"id": "328.png", "formula": "\\begin{align*} A ( p ^ { k + 1 } , 1 ) = A ( p , 1 ) A ( p ^ { k } , 1 ) - A ( p ^ { k - 1 } , p ) , \\\\ A ( p ^ { k - 1 } , p ) = A ( p ^ { k - 1 } , 1 ) A ( 1 , p ) - A ( p ^ { k - 2 } , 1 ) , \\end{align*}"}
+{"id": "4391.png", "formula": "\\begin{align*} U _ 1 ( \\theta ) = \\left \\{ ( x _ 1 ) : x _ 1 \\geq a _ 1 = G ( \\theta ) \\right \\} . \\end{align*}"}
+{"id": "2276.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ 0 \\in L ^ 2 ( \\Omega ) , \\rho _ 0 \\in W ^ { 1 , 2 } ( \\Omega ) , \\\\ & 0 < \\underline { \\rho } \\leq \\rho _ 0 ( x ) \\leq \\bar { \\rho } < \\infty , x \\in \\Omega , \\\\ & 0 < \\underline { \\rho } \\leq \\rho _ B ( x ) \\leq \\bar { \\rho } < \\infty , x \\in \\Gamma _ { \\rm { i n } } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3416.png", "formula": "\\begin{align*} P _ \\nu : = h ( \\nu ) + \\nu ( \\phi ) , \\end{align*}"}
+{"id": "5081.png", "formula": "\\begin{align*} \\det ( A _ n ) = \\alpha _ 1 ^ n \\cdots \\alpha _ k ^ n c _ { k _ { 1 } } ^ n \\cdots c _ { k _ p } ^ n \\prod _ { 1 \\le i < j \\le k } ( \\alpha _ j - \\alpha _ i ) \\prod _ { 1 \\le i < j \\le p } ( c _ { k _ j } - c _ { k _ i } ) \\prod _ { \\substack { 1 \\le i \\le k \\\\ 1 \\le j \\le p } } ( c _ { k _ j } - \\alpha _ i ) . \\end{align*}"}
+{"id": "8101.png", "formula": "\\begin{align*} \\widehat { k ^ * } ( \\zeta ) = \\int _ { - \\infty } ^ \\infty k ^ * ( u ) e ( - u \\zeta ) \\ , d u , \\end{align*}"}
+{"id": "6809.png", "formula": "\\begin{align*} L _ d ( q _ { m } , q _ { m + 1 } ) & = \\frac { h } { 2 } [ \\alpha ( q _ { m } ) \\cdot \\frac { q _ { m + 1 } - q _ { m } } { h } - H ( q _ { m } ) + \\alpha ( q _ { m + 1 } ) \\cdot \\frac { q _ { m + 1 } - q _ { m } } { h } - H ( q _ { m + 1 } ) ] , \\\\ L _ d ( q _ { m - 1 } , q _ { m } ) & = \\frac { h } { 2 } [ \\alpha ( q _ { m - 1 } ) \\cdot \\frac { q _ { m } - q _ { m - 1 } } { h } - H ( q _ { m - 1 } ) + \\alpha ( q _ { m } ) \\cdot \\frac { q _ { m } - q _ { m - 1 } } { h } - H ( q _ { m } ) ] . \\end{align*}"}
+{"id": "962.png", "formula": "\\begin{align*} W ( X , E ) : = W ( V , K ) , \\mbox { w h e r e } V : = H ^ 0 ( X , E ) ^ \\vee \\mbox { a n d } K : = \\ker ( d ) ^ \\perp \\subseteq \\bigwedge ^ 2 V . \\end{align*}"}
+{"id": "5661.png", "formula": "\\begin{align*} 2 \\pi \\nu \\sum _ { i = 1 } ^ N B _ { i } ( \\nu , \\lambda ) { \\mathcal G } _ { i j } ( - \\lambda ) + D ^ { - 1 } B _ { j } ( \\nu , \\lambda ) = 0 . \\end{align*}"}
+{"id": "8588.png", "formula": "\\begin{align*} \\Psi = \\mathcal { C } _ 1 \\circ \\mathcal { C } _ 2 \\circ \\cdots \\circ \\mathcal { C } _ j . \\end{align*}"}
+{"id": "2719.png", "formula": "\\begin{gather*} \\left | \\frac { 1 } { 2 } \\delta ^ 2 - \\kappa _ 1 \\sum _ { j = 1 } ^ { J - 1 } \\left ( \\frac { \\lambda _ { j + 1 } } { \\lambda _ j } \\right ) ^ 2 \\right | \\lesssim o _ n ( 1 ) + \\gamma ^ { 3 } , \\\\ \\gamma \\approx \\delta + o _ n ( 1 ) . \\end{gather*}"}
+{"id": "1124.png", "formula": "\\begin{align*} \\Theta = \\left [ \\begin{smallmatrix} \\alpha _ 0 & \\alpha _ 1 \\\\ \\beta _ 0 & \\beta _ 1 \\\\ \\gamma _ 0 & \\gamma _ 1 \\end{smallmatrix} \\right ] \\end{align*}"}
+{"id": "8740.png", "formula": "\\begin{align*} \\mathcal { C } = \\begin{bmatrix} B _ 0 & A _ 0 B _ 0 & \\cdots & A _ 0 ^ { T - 1 } B _ 0 \\end{bmatrix} \\mathcal { O } = \\begin{bmatrix} C _ 0 \\\\ C _ 0 A _ 0 \\\\ \\vdots \\\\ C _ 0 A _ 0 ^ { T - 1 } \\end{bmatrix} \\end{align*}"}
+{"id": "2953.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\bigg \\langle \\sum _ { j \\in \\mathbb { Z } } \\overline { W } _ { j - 1 } ( \\overline { W } _ j - \\overrightarrow { W } ^ \\ell _ j ) \\varphi _ j , f \\bigg \\rangle _ { L ^ 2 ( \\nu ^ n _ \\rho ) } & = 2 \\sum _ { j \\in \\mathbb { Z } } E _ { \\nu ^ n _ \\rho } [ F _ j ( W _ j - W _ { j + 1 } ) f ] \\\\ & = - 2 \\sum _ { j \\in \\mathbb { Z } } E _ { \\nu ^ n _ \\rho } [ W _ j ( \\nabla _ { j , j + 1 } f ) F _ j ] . \\end{aligned} \\end{align*}"}
+{"id": "485.png", "formula": "\\begin{align*} u ^ { ( k ) } ( \\cdot , t ) : = u _ i \\mbox { f o r } t \\in ( ( i - 1 ) h _ k , i h _ k ) \\mbox { w i t h } i \\in \\{ 0 , 1 , \\ldots , k \\} \\end{align*}"}
+{"id": "3945.png", "formula": "\\begin{align*} D : = \\{ x \\in \\Omega ; u ( x ) < g ( x , y _ 0 , z _ h ) \\} \\subset \\subset \\Omega . \\end{align*}"}
+{"id": "9203.png", "formula": "\\begin{align*} \\Delta _ { f } = d _ { f } ^ { * } \\circ d _ { f } d _ { f } = e ^ { - f / h } \\circ h \\nabla \\circ e ^ { f / h } , \\end{align*}"}
+{"id": "4858.png", "formula": "\\begin{align*} t ^ n \\Big [ \\frac { b _ 1 ^ n + \\dots + b _ { k - 1 } ^ n } { | b | ^ { n } } - | b | ^ { n } \\Big ] = 1 . \\end{align*}"}
+{"id": "7724.png", "formula": "\\begin{align*} s _ 0 = \\frac { \\sqrt { 2 } } { \\sqrt { \\sqrt { 1 + \\tau } } - 1 } + \\frac { 1 } { \\sqrt { 1 - \\frac { 2 } { \\tau } ( \\sqrt { 1 + \\tau } - 1 ) } } . \\end{align*}"}
+{"id": "8255.png", "formula": "\\begin{align*} \\mathcal { I } _ n = n D _ n ( \\hat { \\theta _ { n } } ) ' ( V _ n ( \\hat { \\theta _ { n } } ) ) ^ { - 1 } D _ n ( \\hat { \\theta _ { n } } ) \\end{align*}"}
+{"id": "2280.png", "formula": "\\begin{align*} & \\int ^ { T } _ 0 \\int _ { \\Omega } ( \\rho _ { N } \\partial _ t \\psi + \\rho _ { N } u _ { N } \\cdot \\nabla \\psi - \\varepsilon \\nabla \\rho _ { N } \\cdot \\nabla \\psi ) \\ , d x d t + \\int _ { \\Omega } \\rho _ 0 \\psi ( 0 , x ) \\ , d x \\\\ & \\qquad \\qquad \\qquad = \\int _ 0 ^ T \\int _ { \\Gamma _ { \\rm { i n } } } \\rho _ B u _ B \\cdot \\nu ( x ) \\varphi \\ , d \\sigma ( x ) d t ; \\end{align*}"}
+{"id": "8097.png", "formula": "\\begin{align*} H _ { m , n } ^ + ( x ) = \\frac { 4 } { \\pi } \\int _ { t = 0 } ^ \\infty \\int _ { \\zeta = - \\infty } ^ \\infty \\cos ( x \\cosh \\zeta ) e \\Bigl ( \\frac { t \\zeta } { \\pi } \\Bigr ) e ^ { - \\frac { ( t - T ) ^ 2 } { M ^ 2 } } V ( m ^ 2 n , t ) t \\ , d \\zeta \\ , d t + O ( T ^ { - A } ) \\end{align*}"}
+{"id": "2634.png", "formula": "\\begin{align*} \\sum _ { c \\bmod { \\varpi ^ { l - 1 } } } \\widetilde { e } \\left ( \\frac { c } { \\varpi ^ { l - k - 1 } } \\right ) = N ( \\varpi ^ k ) \\sum _ { c _ 2 \\bmod { \\varpi ^ { l - k - 1 } } } \\widetilde { e } \\left ( \\frac { c _ 2 } { \\varpi ^ { l - k - 1 } } \\right ) = 0 , \\end{align*}"}
+{"id": "7872.png", "formula": "\\begin{align*} A ( z ) = a e ^ { - w z } , B ( z ) = - w ^ 2 \\quad f ( z ) = c \\left ( 1 + \\frac { w } { a } e ^ { w z } \\right ) , \\end{align*}"}
+{"id": "4803.png", "formula": "\\begin{align*} \\max _ { j \\notin B } \\left \\{ \\Pr _ { v \\sim \\mu _ t } [ | v H | = j ] \\cdot 2 ^ { 2 \\epsilon N \\log | 1 - \\frac { 2 j } { N } | } \\right \\} & \\leq 2 ^ { - ( 1 - \\eta ) N + o ( N ) } . \\end{align*}"}
+{"id": "267.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 3 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\tau } - \\beta \\tau ^ 2 \\leqslant 0 , \\shortintertext { a n d } \\operatorname { R e } \\psi _ 4 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\eta } { \\rho } - \\beta \\tau ^ 2 \\leqslant 0 , \\end{align*}"}
+{"id": "2311.png", "formula": "\\begin{align*} u _ t + f ( u ) _ x = \\epsilon \\left ( \\nu ( u ) u _ x \\right ) _ x , ~ ~ \\epsilon \\nu ( u ) \\geqslant 0 . \\end{align*}"}
+{"id": "6990.png", "formula": "\\begin{align*} \\phi \\left ( x ^ g \\right ) = \\C _ g \\left ( \\phi ( x ) \\right ) \\ \\forall x \\in S . \\end{align*}"}
+{"id": "3169.png", "formula": "\\begin{align*} A ( y ) = \\mathrm { d i a g } ( a _ 1 ( y ) , a _ 2 ( y ) , a _ 3 ( y ) ) \\quad y \\in \\R ^ 3 \\end{align*}"}
+{"id": "3850.png", "formula": "\\begin{align*} \\mu _ l = 4 ( T _ { l } + T _ { l - 1 } ) - 2 \\delta _ { l , 2 } . \\end{align*}"}
+{"id": "7179.png", "formula": "\\begin{align*} \\mathcal { \\hat E } = \\bigcup _ { t \\in [ 0 , \\ , \\infty ] } \\{ t \\} \\times \\hat E _ t . \\end{align*}"}
+{"id": "966.png", "formula": "\\begin{align*} \\mathrm { S u p p } ( \\mathcal { D } _ \\mathfrak { R e s } ) = \\mathrm { S u p p } ( \\mathcal { D } _ \\mathfrak { K o s z } ) . \\end{align*}"}
+{"id": "6864.png", "formula": "\\begin{align*} J ( u ; F ) = \\norm { Q u - F } ^ 2 _ Y , u _ h = \\underset { v \\in X _ h } { \\arg \\min } \\ ; J ( v ; F ) . \\end{align*}"}
+{"id": "909.png", "formula": "\\begin{align*} s _ j : = [ u ^ j ] S ( z , u ) & = { \\frac { { z } ^ { 4 } + { z } ^ { 4 } g _ 0 + { z } ^ { 2 } g _ 0 - { z } ^ { 2 } + 1 } { z r _ 1 ( 1 - u / r _ 1 ) } } \\\\ & = { \\frac { { z } ^ { 4 } + { z } ^ { 4 } g _ 0 + { z } ^ { 2 } g _ 0 - { z } ^ { 2 } + 1 } { z r _ 1 ^ { j + 1 } } } . \\end{align*}"}
+{"id": "1801.png", "formula": "\\begin{align*} V = \\{ u \\in C ^ 1 ( [ 0 , T ] , \\R ^ n ) : u ( 0 ) = u ( T ) \\} , \\end{align*}"}
+{"id": "5968.png", "formula": "\\begin{align*} & ( A ^ T E _ r - \\sum _ { s = 1 } ^ p \\alpha _ { r s } E _ s , e _ k ) = ( E _ r , A e _ k ) - \\sum _ { s = 1 } ^ p \\alpha _ { r s } ( E _ s , e _ k ) \\\\ = & \\sum _ { s = 1 } ^ p \\alpha _ { s k } ( E _ r , e _ s ) - \\alpha _ { r k } = \\alpha _ { r k } - \\alpha _ { r k } = 0 , \\end{align*}"}
+{"id": "2321.png", "formula": "\\begin{align*} u ( y ) = v ( z ) , \\ , \\ , \\ , G _ { \\Omega , a } ( y ) = G _ { B _ 1 ^ N , O } ( z ) . \\end{align*}"}
+{"id": "3594.png", "formula": "\\begin{align*} d _ { j } ^ \\pm ( \\lambda ) = \\frac { \\kappa ( \\lambda ) } { \\kappa ( \\lambda \\pm \\epsilon _ j ) } c _ { j } ^ \\pm ( \\lambda ) , \\end{align*}"}
+{"id": "1151.png", "formula": "\\begin{align*} \\langle x _ 1 , x _ 2 , . . . , x _ k \\rangle = \\langle . . . \\langle \\langle x _ 1 , x _ 2 \\rangle , x _ 3 \\rangle , . . . , x _ k \\rangle , \\end{align*}"}
+{"id": "589.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\mathbb { E } ^ \\dagger \\left [ A ^ { \\rm T } : ( X _ { t _ n , t _ { n + 1 } } ^ \\dagger \\otimes X _ { t _ n , t _ { n + 1 } } ^ \\dagger ) \\right ] = \\frac { \\Delta t \\ , \\gamma } { 2 } \\mbox { t r } \\ , ( A ) + \\mathcal { O } ( \\Delta t ^ 2 ) , \\end{align*}"}
+{"id": "6165.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } u '' - \\Delta u + \\alpha u = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\cr u = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\cr \\partial _ \\nu u + \\beta u = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "751.png", "formula": "\\begin{align*} \\varphi ( b \\tilde { b } r ) = \\widetilde { \\chi } ^ { - 1 } ( r ) \\varphi ( \\tilde { b } ) , \\end{align*}"}
+{"id": "4733.png", "formula": "\\begin{align*} p \\alpha & = ( a _ 0 a _ 1 ^ { n _ 1 } a _ 2 ^ { n _ 2 } \\cdots a _ m ^ { n _ m } ) ( p a _ { m + 1 } ^ { n _ { m + 1 } } ) \\in A \\\\ p \\beta & = ( p a _ 1 ^ { n _ 1 } a _ 2 ^ { n _ 2 } \\cdots a _ m ^ { n _ m } ) ( p a _ { m + 1 } ^ { n _ { m + 1 } } ) \\in A \\end{align*}"}
+{"id": "5636.png", "formula": "\\begin{align*} { \\| { \\psi ( x ) \\ , \\psi ( x ) ^ * } \\| \\leq \\| { \\psi } \\| ^ 2 = \\| { \\psi ( e ) } \\| = 1 } \\end{align*}"}
+{"id": "3099.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( A ) & = \\bar { a } ( 1 + \\bar { b } ) \\int _ Y r _ B b _ { 1 1 } \\partial _ 1 w _ A = - 2 \\bar { a } ( 1 + \\bar { b } ) \\int _ Y ( \\partial _ 1 w _ A ) ( \\partial _ { 2 2 } ^ 2 w _ B ) , \\\\ c _ 2 ^ { 1 1 } ( A ) & = \\bar { a } ( 1 + \\bar { b } ) \\int _ Y r _ B b _ { 2 2 } \\partial _ 2 w _ A = 2 \\bar { a } ( 1 + \\bar { b } ) \\int _ Y ( \\partial _ 2 w _ A ) ( \\partial _ { 1 1 } ^ 2 w _ B ) , \\end{align*}"}
+{"id": "5570.png", "formula": "\\begin{align*} u _ { ( x , 0 ) , r } ( x ' , y ' ) : = \\frac { u ( r ( x ' , y ' ) + ( x , 0 ) ) } { r ^ { \\gamma + 1 } } . \\end{align*}"}
+{"id": "2977.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { L } _ n ( \\eta _ { j - 1 } \\eta _ j ) & = W _ { j - 2 } W _ { j } + W _ { j - 1 } ^ 2 - 2 W _ { j - 1 } W _ j - W _ { j - 1 } + E ^ { ( 5 ) } _ { j } , \\\\ \\tilde { L } _ n ( \\eta _ { j - 1 } \\eta _ { j + 1 } ) & = W _ { j - 2 } W _ { j + 1 } - 2 W _ { j - 1 } W _ { j + 1 } + W _ { j - 1 } W _ j + E ^ { ( 6 ) } _ { j } , \\\\ \\tilde { L } _ n \\eta _ j ^ 2 & = 2 W _ { j - 1 } W _ j - 2 W _ j ^ 2 + W _ { j - 1 } + W _ j + E ^ { ( 7 ) } _ { j } . \\end{aligned} \\end{align*}"}
+{"id": "142.png", "formula": "\\begin{align*} \\begin{aligned} \\Gamma ^ { \\theta _ { j + 1 } } _ { \\lambda _ { j } } & : = \\int _ { \\lambda _ { j } } ^ { \\theta _ { j + 1 } } \\mathrm { T } ( \\theta _ { j + 1 } - \\tau ) B B ^ { * } \\mathrm { T } ( \\theta _ { j + 1 } - \\tau ) ^ { * } d \\tau , \\ j = 0 , \\dots , n , \\end{aligned} \\end{align*}"}
+{"id": "1574.png", "formula": "\\begin{align*} \\Delta _ f \\beta _ + + \\mu \\beta _ + = - f ' ( 0 ) + \\liminf \\limits _ { t \\rightarrow \\infty } \\int _ { 0 } ^ { t } R i c \\big ( \\gamma ' ( s ) , \\gamma ' ( s ) \\big ) d s . \\end{align*}"}
+{"id": "3965.png", "formula": "\\begin{align*} G & : = \\{ u \\equiv g ( \\cdot , y _ 0 , z _ 0 ) \\} \\\\ G ^ h & : = \\{ u < g ( \\cdot , y _ 0 , z _ 0 - h ) \\} , \\end{align*}"}
+{"id": "854.png", "formula": "\\begin{align*} g ( F _ k ) = F _ k J _ d - J _ d F _ k ^ T - \\widehat { h \\Pi _ k } \\end{align*}"}
+{"id": "4506.png", "formula": "\\begin{align*} x _ { \\sigma ( i ) } ^ { a _ i } = x _ j ^ { a _ { \\sigma ^ { - 1 } ( j ) } } . \\end{align*}"}
+{"id": "7518.png", "formula": "\\begin{align*} Z _ h ( s , \\chi , D ) = \\dfrac { L ( q ^ { - s } ) } { ( 1 - q ^ { - 1 - s } ) ( 1 - q ^ { - p - ( k + 1 ) - p ( k + 1 ) s } ) } , \\end{align*}"}
+{"id": "3113.png", "formula": "\\begin{align*} c _ 1 ^ { 1 1 } ( A ^ 1 ) = \\int _ Y r ^ 1 A ^ 1 e _ 1 \\cdot \\nabla v ^ { 1 1 } _ { A ^ 1 } = 0 , \\end{align*}"}
+{"id": "1510.png", "formula": "\\begin{align*} \\textrm { L i } _ k ( x ) = \\sum _ { n = 1 } ^ \\infty \\frac { x ^ n } { n ^ k } , \\end{align*}"}
+{"id": "209.png", "formula": "\\begin{align*} T ^ { k _ { n } ( h _ { n } + c _ { n } ) } ( I _ { n , b } ^ { [ i ] } ) & = T ^ { k _ { n } ( h _ { n } + c _ { n } ) } ( I _ { n + 1 , b + i ( h _ { n } + c _ { n } ) + \\frac { 1 } { 2 } i ( i - 1 ) } ) \\\\ & = T ^ { k _ { n } ( h _ { n } + c _ { n } ) + i ( h _ { n } + c _ { n } ) + \\frac { 1 } { 2 } i ( i - 1 ) } ( I _ { n + 1 , b } ) = T ^ { h _ { n + 1 } } ( I _ { n + 1 , b + h _ { n } + 2 c _ { n } - \\frac { 1 } { 2 } r _ { n } ( r _ { n } - 1 ) } ) . \\end{align*}"}
+{"id": "8114.png", "formula": "\\begin{align*} W _ { m , n } ^ - ( x ) = 4 T e \\Bigl ( - \\frac { x } { 2 \\pi } \\Bigr ) \\int _ { - \\infty } ^ \\infty \\xi ^ 6 k _ 0 ^ * ( \\xi ) e \\Bigl ( - \\frac { T \\xi } { M } - \\frac { \\pi x \\xi ^ 2 } { 4 M ^ 2 } \\Bigr ) \\ , d \\xi . \\end{align*}"}
+{"id": "2907.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | \\mathcal { B } _ k f _ \\sigma | ) d m _ \\infty = \\int _ { \\mathbb { T } ^ \\infty } \\log ( 1 + | \\mathcal { B } f _ \\sigma | ) d m _ \\infty = \\| \\mathcal { B } f _ \\sigma \\| _ 0 . \\end{align*}"}
+{"id": "7601.png", "formula": "\\begin{align*} \\| \\pi _ n u - \\pi _ n v \\| _ { \\L ^ p } & = \\bigg \\| \\frac { n } { \\| u \\| _ { \\L ^ p } } u - \\frac { n } { \\| v \\| _ { \\L ^ p } } v \\bigg \\| _ { \\L ^ p } \\\\ & = \\frac { n } { \\| u \\| _ { \\L ^ p } \\| v \\| _ { \\L ^ p } } \\| ( u - v ) \\| v \\| _ { \\L ^ p } + v ( \\| v \\| _ { \\L ^ p } - \\| u \\| _ { \\L ^ p } ) \\| _ { \\L ^ p } \\\\ & \\leq \\| u - v \\| _ { \\L ^ p } + ( \\| v \\| _ { \\L ^ p } - \\| u \\| _ { \\L ^ p } ) \\leq 2 \\| u - v \\| _ { \\L ^ p } , \\end{align*}"}
+{"id": "7731.png", "formula": "\\begin{align*} g ^ { ( 1 ) } ( \\tau , \\hat { \\rho } ) = g ^ { ( 2 ) } ( \\tau , \\rho _ 0 ) . \\end{align*}"}
+{"id": "9245.png", "formula": "\\begin{align*} B _ { j , k } = \\left \\lbrace v _ 1 , \\ldots v _ { h ( j ) } , X v _ j , v _ { h ( j ) + 1 } , \\ldots \\widehat { v _ { k } } , \\ldots v _ n \\right \\rbrace \\end{align*}"}
+{"id": "1369.png", "formula": "\\begin{align*} \\begin{array} { l l } E ' ( t ) & = \\frac { 1 } { 2 } ( b ' \\circ \\Delta u ) ( t ) - \\frac { 1 } { 2 } b ( t ) \\left \\| \\Delta u \\right \\| _ { 2 } ^ { 2 } - \\int _ { \\Omega } u _ { t } h ( u _ { t } ) \\medskip \\\\ & \\leq \\frac { 1 } { 2 } ( b ' \\circ \\Delta u ) ( t ) - \\int _ { \\Omega } u _ { t } h ( u _ { t } ) \\medskip \\\\ & \\leq 0 . \\end{array} \\end{align*}"}
+{"id": "6381.png", "formula": "\\begin{align*} K ^ { s + 1 } = ( K ^ s _ { 1 } , \\cdots , K ^ s _ { r - 1 } , K ^ s _ { r + 2 } , \\cdots , K ^ s _ { n _ s } ) \\end{align*}"}
+{"id": "3012.png", "formula": "\\begin{align*} \\begin{array} { r l l } \\partial ^ { \\gamma , \\mathbf { A } } _ b \\mathbf { F } { _ i } ^ { a b } & = & \\partial _ k p { _ i } ^ { a k } \\\\ \\partial ^ { \\gamma , \\mathbf { A } } _ b p { _ i } ^ { j b } + \\partial _ k p { _ i } ^ { j k } + \\frac { 1 } { 2 } c ^ j _ { k \\ell } p { _ i } ^ { k \\ell } & = & \\frac { 1 } { 2 } \\| \\mathbf { F } \\| ^ 2 \\delta { _ i } ^ j + \\frac { 1 } { 2 } \\mathbf { F } { _ i } ^ { a b } \\mathbf { F } { ^ j } _ { a b } \\end{array} \\end{align*}"}
+{"id": "1687.png", "formula": "\\begin{align*} s ( \\mu _ m ) = \\sum _ { \\frac { 2 - k } { 2 } \\leq n \\leq \\frac { k - 2 } { 2 } } \\mu _ m \\left ( P _ { n + \\frac { k - 2 } { 2 } } \\right ) f _ n . \\end{align*}"}
+{"id": "57.png", "formula": "\\begin{align*} s ( i ) = \\sum _ { j \\in N [ i ] } { f ( j ) } \\geq 1 , i \\in V . \\end{align*}"}
+{"id": "8796.png", "formula": "\\begin{align*} s _ 6 ( x _ 4 y _ 5 - y _ 4 x _ 5 ) = { s _ 4 ( y _ 5 x _ 6 - x _ 5 y _ 6 ) + s _ 5 ( x _ 4 y _ 6 - y _ 4 x _ 6 ) } . \\end{align*}"}
+{"id": "1371.png", "formula": "\\begin{align*} I ( t ) : = I ( u ( t ) ) = \\left \\| \\Delta u _ { t } \\right \\| _ { 2 } ^ { 2 } + ( 1 - \\int _ { 0 } ^ { t } b ( s ) d s ) \\left \\| \\Delta u \\right \\| _ { 2 } ^ { 2 } + \\left \\| u \\right \\| _ { 2 } ^ { 2 } + ( b \\circ \\Delta u ) - \\int _ { \\Omega } u ^ { 2 } \\ln \\left | u \\right | ^ { k } , \\end{align*}"}
+{"id": "6731.png", "formula": "\\begin{align*} x _ 1 x _ 0 = x ^ { \\varepsilon _ 0 + \\varepsilon _ 1 } < x ^ { 1 / 2 } / ( \\log x ) ^ B , \\end{align*}"}
+{"id": "1699.png", "formula": "\\begin{align*} \\langle \\mu _ m , \\mu _ { m ' } \\rangle ' = \\left \\{ \\begin{array} { l l } ( - 1 ) ^ { m + \\frac { k - 2 } { 2 } } \\binom { k - 2 } { \\frac { k - 2 } { 2 } - m } ^ { - 1 } , & m = - m ' , \\\\ 0 , & m \\neq - m ' . \\end{array} \\right . \\end{align*}"}
+{"id": "7365.png", "formula": "\\begin{align*} Z _ { \\vec { \\gamma } , \\ell } ^ f = \\bigcup _ { \\vec { p } \\in \\{ 0 , \\ldots , 5 \\} ^ d } \\left ( T _ { \\vec { p } } \\cap Z ^ { f _ { J \\Tilde { K } ^ { \\vec { p } } \\Tilde { L } ^ { \\vec { p } } } } _ { \\vec { \\gamma } , \\ell } \\right ) . \\end{align*}"}
+{"id": "4455.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { 1 } { a _ i } = \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } < \\theta ? \\end{align*}"}
+{"id": "1734.png", "formula": "\\begin{align*} \\rho ( \\varphi _ n ( ( x ^ { 2 n } ) ^ \\vee ) ) ( x ^ { \\underline m } y ^ { \\underline k - 2 - \\underline m } ) = \\left \\{ \\begin{array} { l l } \\frac { ( - 1 ) ^ { m _ { \\rm i d } + m _ c } } { n + \\frac { k _ { \\rm i d } - 1 + k _ c - 1 } { 2 } } \\binom { n + \\frac { k _ { \\rm i d } - 2 + k _ c - 2 } { 2 } } { m _ { \\rm i d } } ^ { - 1 } , & n = m _ { \\rm i d } - m _ c - \\frac { k _ { \\rm i d } - k _ { c } } { 2 } , \\\\ 0 , & n \\neq m _ { \\rm i d } - m _ c - \\frac { k _ { \\rm i d } - k _ { c } } { 2 } , \\end{array} \\right . \\end{align*}"}
+{"id": "6032.png", "formula": "\\begin{align*} \\O _ u \\star \\O _ v & = Q _ j \\mathcal { O } _ { u ' } \\star \\mathcal { O } _ v = \\sum _ { w \\in W ^ P , \\beta \\in E ( X ) } N _ { u ' , v } ^ { w , \\beta + l _ j } Q ^ { \\beta } \\mathcal { O } _ w \\\\ & = \\sum _ { w \\in W ^ P , \\ : \\beta \\in E ( X ) } N _ { u ' , v } ^ { w , \\beta + l _ j } Q ^ { \\beta } \\end{align*}"}
+{"id": "3205.png", "formula": "\\begin{align*} F ^ { - 1 } ( A ) = \\bigcup _ { n = 1 } ^ \\infty F ^ { - 1 } ( A ) \\cap B _ { X ^ \\ast } ( n ) = \\bigcup _ { n = 1 } ^ \\infty ( F _ { | B _ { X ^ \\ast } ( n ) } ) ^ { - 1 } ( A ) , \\end{align*}"}
+{"id": "1144.png", "formula": "\\begin{align*} \\tilde { T } _ { p ^ { r + 2 } } & = \\tilde { T } _ p \\tilde { T } _ { p ^ { r + 1 } } - p ^ { k - 1 } \\tilde { T } _ { p ^ r } \\end{align*}"}
+{"id": "2856.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n b _ i = \\sum _ { i = 1 } ^ n a _ i = a _ k + a _ { \\ell } + \\sum _ { \\substack { i = 1 \\\\ i \\neq k , \\ell } } ^ n a _ i = c _ k + c _ { \\ell } + \\sum _ { \\substack { i = 1 \\\\ i \\neq k , \\ell } } ^ n c _ i = \\sum _ { i = 1 } ^ n c _ i . \\end{align*}"}
+{"id": "2746.png", "formula": "\\begin{align*} \\varphi \\circ D ( a b ) = & \\varphi \\circ \\psi ( a ) \\varphi \\circ D ( b ) + \\varphi \\circ \\psi ( b ) \\varphi \\circ D ( a ) \\\\ = & \\varphi ( \\psi ( a ) D b ) + \\varphi ( \\psi ( b ) D a ) \\\\ = & \\varphi ( \\psi ( a ) D b + \\psi ( b ) D a ) . \\end{align*}"}
+{"id": "4353.png", "formula": "\\begin{align*} { \\downarrow } z = \\{ x \\in X \\mid x \\leq z \\} = z X z . \\end{align*}"}
+{"id": "6771.png", "formula": "\\begin{align*} \\Psi _ { \\chi } ( y ) = \\sum _ { n = 1 } ^ { \\infty } \\chi ( n ) \\exp \\left [ - \\pi n ^ 2 y \\right ] . \\end{align*}"}
+{"id": "5528.png", "formula": "\\begin{align*} G _ { p , q } ^ { \\ , m , n } \\ ! \\left ( \\ , \\begin{matrix} a _ 1 , \\cdots , a _ p \\\\ b _ 1 , \\cdots , b _ q \\end{matrix} \\ ; \\Big | z \\right ) : = \\frac { 1 } { 2 \\pi i } \\int _ L \\frac { \\prod _ { j = 1 } ^ m \\Gamma ( b _ j - s ) \\prod _ { j = 1 } ^ n \\Gamma ( 1 - a _ j + s ) z ^ s } { \\prod _ { j = m + 1 } ^ q \\Gamma ( 1 - b _ j + s ) \\prod _ { j = n + 1 } ^ p \\Gamma ( a _ j - s ) } \\mathrm { d } s , \\end{align*}"}
+{"id": "5377.png", "formula": "\\begin{align*} F ( x _ 1 , \\ldots , x _ N ) = ( x _ 1 , \\ldots , x _ \\rho , 0 , \\dots , 0 ) \\in \\mathbb { R } ^ M \\end{align*}"}
+{"id": "1474.png", "formula": "\\begin{align*} x _ i = \\begin{cases} 1 & i = 1 , a + 1 \\\\ 2 & 2 \\leq i \\leq a \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "229.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = - \\alpha \\rho \\tau - \\beta \\rho \\eta , \\leqslant - \\alpha \\frac { ( 1 + 3 \\rho ^ 2 ) } { 2 } \\leqslant - \\frac { \\alpha } { 2 } , \\end{align*}"}
+{"id": "3746.png", "formula": "\\begin{align*} & \\frac { d } { d s } \\left ( - \\zeta ( s ) ^ { - 1 } Z _ { r _ 2 } ( f , s ) + \\zeta ( s + 1 ) ^ { - 1 } Z _ { r _ 1 } ( d _ 2 ( f ) , s + 1 ) \\right ) \\Big | _ { s = 0 } \\\\ & = ( \\log q ) \\left ( \\tilde { c } _ 2 ( I ( f ) ) - \\tilde { a } _ 2 ( I ( f ) ) \\right ) . \\end{align*}"}
+{"id": "6599.png", "formula": "\\begin{align*} | V _ \\phi f ( x , \\xi ) | & \\leq C ' _ { N _ 0 } ( 1 + | \\xi | ^ 2 ) ^ { N _ 0 } \\sum _ { | \\alpha | \\leq N _ 0 } \\sum _ { \\gamma \\leq \\alpha } \\binom { \\alpha } { \\gamma } | | e ^ { - r _ 0 | \\cdot - x | ^ { 1 / s } } e ^ { r | \\cdot | ^ { 1 / s } } | | _ 2 . \\end{align*}"}
+{"id": "7797.png", "formula": "\\begin{align*} & \\Psi _ { c , b _ 2 } [ n _ 2 ] \\Psi _ { c , b _ 1 } [ n _ 1 ] = \\Psi _ { c , b _ 1 } [ n _ 1 ] \\Psi _ { c , b _ 1 + b _ 2 } [ n _ 1 + n _ 2 ] \\Psi _ { c , b _ 2 } [ n _ 2 ] . \\end{align*}"}
+{"id": "3657.png", "formula": "\\begin{align*} \\Phi ( x , t ) : = \\int \\limits _ { M } \\mathcal { K } _ M ( x , y , T - t ) u ( y , t ) ~ { \\rm d } v _ { M } ( y ) \\end{align*}"}
+{"id": "1890.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } k _ { x x } ( x , \\zeta ) - k _ { \\zeta \\zeta } ( x , \\zeta ) = ( c + a ( \\zeta ) ) k ( x , \\zeta ) , \\cr k _ \\zeta ( x , 0 ) = 0 , \\cr \\frac { d k ( x , x ) } { d x } + k _ { x } ( x , x ) + k _ { \\zeta } ( x , x ) = - a ( x ) - c . \\end{array} \\right . \\end{align*}"}
+{"id": "4806.png", "formula": "\\begin{align*} \\Pr _ { c \\sim \\mathcal { D } ( n , d ) } [ | c | \\leq \\frac { 1 - \\delta } { 2 } N ] & \\leq 2 ^ { - ( 1 - h ( \\frac { 1 - \\delta } { 2 } ) ) \\cdot \\binom { n } { \\leq d } } \\\\ & \\leq e ^ { - \\frac { \\delta ^ 2 } { 2 } \\cdot \\binom { n } { \\leq d } } . \\end{align*}"}
+{"id": "4756.png", "formula": "\\begin{align*} A ( t + 1 ) = F ( A ( t ) ) , \\end{align*}"}
+{"id": "2948.png", "formula": "\\begin{align*} \\begin{aligned} \\| F \\| ^ 2 _ { - 1 , n } = \\sup _ { f \\in L ^ 2 ( \\nu ^ n _ \\rho ) , \\ , } \\big \\{ 2 \\langle F , f \\rangle _ { L ^ 2 ( \\nu ^ n _ \\rho ) } - \\| f \\| ^ 2 _ { 1 , n } \\big \\} . \\end{aligned} \\end{align*}"}
+{"id": "2852.png", "formula": "\\begin{align*} c _ { k - 1 } = a _ { k - 1 } \\geq a _ k = \\rho + \\tau > \\rho + \\sigma = c _ k \\geq b _ k \\geq b _ { k + 1 } = a _ { k + 1 } = c _ { k + 1 } . \\end{align*}"}
+{"id": "7334.png", "formula": "\\begin{align*} I _ 4 ( \\gamma ) = \\left \\{ \\begin{aligned} & O ( I _ 3 ( \\gamma ) ) & ( \\alpha > 2 ) , \\\\ & o ( I _ 3 ( \\gamma ) ) & ( \\alpha = 2 ) , \\end{aligned} \\right . ( \\gamma \\to \\infty ) . \\end{align*}"}
+{"id": "4199.png", "formula": "\\begin{align*} g ^ { ( 0 ) } = \\left [ \\begin{array} { c c } e ^ { \\Lambda ^ { ( 0 ) } } & 0 \\\\ 0 & e ^ { - \\Lambda ^ { ( 0 ) } } \\end{array} \\right ] . \\end{align*}"}
+{"id": "3106.png", "formula": "\\begin{align*} w = - \\bar { \\gamma } \\sum _ { i , j = 1 } ^ n c _ { i j } v ^ { i j } . \\end{align*}"}
+{"id": "4893.png", "formula": "\\begin{align*} t _ { s _ 1 s _ 2 u _ 0 } t _ { u _ l s _ 2 s _ 1 } = t _ { x _ { l + 1 } } + t _ { x _ l } . \\end{align*}"}
+{"id": "4025.png", "formula": "\\begin{align*} D _ i ( G _ p \\cdot \\gamma ) - K \\phi _ { k } ^ * D _ i Y ^ k = 0 & i = 1 , \\dots , n - 1 , \\\\ D _ i ( G _ p \\cdot \\gamma ) - K \\phi _ { k } ^ * D _ i Y ^ k \\leq C & i = n . \\end{align*}"}
+{"id": "7282.png", "formula": "\\begin{align*} \\varphi ^ d _ m ( \\lambda ; x ) = 1 + \\sum _ { k = 1 } ^ { d } \\lambda ^ { k } G ^ { k } _ m ( x ) + \\sum _ { k = 1 } ^ { \\infty } \\lambda ^ { d + k } ( s \\bullet m \\bullet ) ^ k G ^ { d } _ m ( x ) . \\end{align*}"}
+{"id": "7618.png", "formula": "\\begin{align*} | w _ n ( t , x ) | _ + = 0 , x \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "462.png", "formula": "\\begin{align*} \\theta _ f S _ { f , \\tau } ^ { - 1 } \\theta _ \\tau = I _ { \\mathcal { L } ^ p ( \\Omega , \\mu ) } . \\end{align*}"}
+{"id": "3118.png", "formula": "\\begin{align*} - A : D ^ 2 \\phi = - \\frac { A ^ 1 : D ^ 2 \\phi } { 1 + A ^ 1 : D ^ 2 \\phi } = \\gamma - 1 = \\gamma - \\bar { \\gamma } . \\end{align*}"}
+{"id": "2150.png", "formula": "\\begin{align*} \\vartheta \\eta ' ( t ) = C \\left ( 1 - U ( c t + s _ 0 ) \\right ) + ( C + \\mu ) q ( t ) > 0 , \\end{align*}"}
+{"id": "6127.png", "formula": "\\begin{align*} N - d \\leqslant \\hbox { r a n k } ( \\mathcal R ) = N - \\hbox { d i m K e r } ( \\mathcal R ^ T ) \\leqslant N - \\hbox { d i m } ( V ) , \\end{align*}"}
+{"id": "5043.png", "formula": "\\begin{align*} X _ 1 + \\cdots + X _ m = 0 \\end{align*}"}
+{"id": "7727.png", "formula": "\\begin{align*} D _ { \\beta } = \\left ( 1 + 2 \\rho _ 0 \\cos ( \\beta \\pi ) + \\rho _ 0 ^ 2 \\right ) ^ { \\frac { 1 } { 4 } } , \\end{align*}"}
+{"id": "7537.png", "formula": "\\begin{align*} Z _ { \\tilde { g } _ { 2 , a } } ( s , \\chi , ( \\mathcal { O } _ K ^ { \\times } ) ^ 2 ) = c _ 2 ( \\chi , a , m ) q ^ { - ( e _ 1 + m i _ 0 ) s } , \\end{align*}"}
+{"id": "3429.png", "formula": "\\begin{align*} \\frac { \\mu _ { t } ( \\underline { \\gamma } ) } { \\mu _ { t } ( [ x _ 0 ] ) } = \\exp \\left ( ( t \\phi - P _ G ( t \\phi ) ) ( \\underline { \\gamma } ) \\right ) . \\end{align*}"}
+{"id": "3485.png", "formula": "\\begin{align*} \\gamma = \\frac { { { { \\Big | { \\sum \\nolimits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ l } } } \\Big | } ^ 2 } } } { { { \\sigma ^ 2 } } } . \\end{align*}"}
+{"id": "6146.png", "formula": "\\begin{align*} \\mu = \\hbox { d i m } ( V ) \\leqslant N - \\hbox { d i m K e r } ( D ^ T ) = \\hbox { r a n k } ( D ) . \\end{align*}"}
+{"id": "3009.png", "formula": "\\begin{align*} \\hbox { d } ^ \\mathbf { A } p _ i = \\frac { 1 } { 2 } \\| p \\| ^ 2 e ^ { ( 4 ) } \\wedge \\bar { e } _ i ^ { ( r - 1 ) } \\end{align*}"}
+{"id": "9048.png", "formula": "\\begin{align*} \\P ^ { k , n } : = \\ , ( \\Theta ^ { 1 , n } , \\Theta ^ { 2 , n } , \\ldots , \\Theta ^ { k , n } ) , \\P _ { \\Theta } ^ { \\otimes k } : = \\ , ( \\Theta ^ 1 , \\Theta ^ 2 , \\ldots , \\Theta ^ k ) . \\end{align*}"}
+{"id": "3759.png", "formula": "\\begin{align*} c _ 2 ( f ) & = \\lim _ { s \\to 0 } \\left ( \\frac { d } { d s } s \\zeta ( s ) \\zeta ( s ) ^ { - 1 } Z _ { r _ 2 } ( f , s ) \\right ) \\\\ & = \\big ( \\mathrm { R e s } _ { s = 0 } \\zeta ( s ) \\big ) \\lim _ { s \\to 0 } \\left ( \\frac { d } { d s } \\zeta ( s ) ^ { - 1 } Z _ { r _ 2 } ( f , s ) \\right ) + \\kappa _ 1 \\left ( \\zeta ( s ) ^ { - 1 } Z _ { r _ 2 } ( f , s ) \\right ) \\big | _ { s = 0 } \\\\ & = - \\epsilon ( 0 ) \\kappa _ 0 \\lim _ { s \\to 0 } \\left ( \\frac { d } { d s } \\zeta ( s ) ^ { - 1 } Z _ { r _ 2 } ( f , s ) \\right ) + \\kappa _ 1 \\tilde { c } _ 2 ( I ( f ) ) . \\end{align*}"}
+{"id": "3620.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } u = \\Delta _ { \\mathbb { H } ^ { n } } u + f ( u , t ) & \\hbox { i n } ~ \\mathbb { H } ^ { n } \\times ( 0 , T ) , \\\\ \\\\ u = u _ { 0 } & \\hbox { i n } ~ \\mathbb { H } ^ { n } \\times \\{ 0 \\} . \\end{array} \\right . \\end{align*}"}
+{"id": "8872.png", "formula": "\\begin{align*} H _ { \\nu } ( t , z , w ) = N _ t ^ { ( n , \\nu ) } ( z , w ) \\sum \\limits _ { m = 0 } ^ { + \\infty } ( 2 m + 2 \\nu + n ) \\frac { \\Gamma ( m + 2 \\nu + n ) } { \\Gamma ( m + 2 \\nu + 1 ) } e ^ { - 4 t ( m + \\nu + \\frac { n } { 2 } ) ^ 2 } P _ { m } ^ { ( n - 1 , 2 \\nu ) } \\left ( \\cos 2 d _ { F S } ( z , w ) \\right ) \\end{align*}"}
+{"id": "6509.png", "formula": "\\begin{align*} \\left ( \\bigg | \\frac { 1 } { \\sqrt { k } l } \\underset { i = 1 } { \\overset { k } { \\sum } } \\left ( \\underset { j = 1 } { \\overset { l } { \\sum } } ( B _ i ^ j - \\frac { 1 } { 2 } ) \\right ) ^ 2 - \\sqrt { k } / 4 \\bigg | \\leq c _ { \\alpha , n } \\right ) \\leq \\frac { 1 / 2 + 1 6 \\eta _ { p , l , k } / \\sqrt { k } } { \\eta _ { p , l , k } ^ 2 } . \\end{align*}"}
+{"id": "6679.png", "formula": "\\begin{align*} \\begin{aligned} & x ^ { k + 1 } - { \\bf 1 } \\otimes \\bar { x } ^ { k + 1 } = ( I + \\gamma ^ k W \\otimes I _ d ) x ^ k - { \\bf 1 } \\otimes \\bar { x } ^ { k } \\\\ & \\quad + \\gamma ^ k \\left ( \\zeta _ w ^ k - \\frac { 1 } { m } \\sum _ { i = 1 } ^ m { \\bf 1 } \\otimes \\zeta ^ k _ { w , i } \\right ) - \\lambda ^ k \\left ( g ^ k - \\frac { 1 } { m } \\sum _ { i = 1 } ^ m { \\bf 1 } \\otimes g _ i ^ k \\right ) \\end{aligned} \\end{align*}"}
+{"id": "3259.png", "formula": "\\begin{align*} C _ 4 a ^ t ( 2 4 a ) ^ { k _ 0 } b ^ { k _ 0 + k _ 1 } \\prod \\limits _ { i = 1 } ^ { k _ 1 } ( 4 s _ i ) ! \\end{align*}"}
+{"id": "3717.png", "formula": "\\begin{align*} \\int _ { [ N ] } & \\sum _ { ( \\xi _ 1 ' , \\xi _ 2 ' ) / \\sim } r _ i ( n g ) \\mathcal { F } _ 2 ( f ) \\left ( \\xi , \\xi ' _ 1 , \\xi ' _ 2 \\right ) d n \\\\ & = \\int _ { [ N ] } r _ i ( n g ) \\mathcal { F } _ 2 ( f ) \\left ( \\xi , 0 , 1 \\right ) d n + \\int _ { [ N ] } \\sum _ { \\alpha \\in F } r _ i ( n g ) \\mathcal { F } _ 2 ( f ) \\left ( \\xi , 1 , \\alpha \\right ) d n . \\end{align*}"}
+{"id": "4562.png", "formula": "\\begin{align*} 1 - \\sum _ { i = 1 } ^ { n + 1 } \\frac { 1 } { a _ i } = \\frac { 1 } { \\prod _ { i = 1 } ^ n a _ i } - \\frac { 1 } { a _ { n + 1 } } = \\frac { a _ { n + 1 } - \\prod _ { i = 1 } ^ n a _ i } { a _ { n + 1 } \\prod _ { i = 1 } ^ n a _ i } = \\frac { 1 } { \\prod _ { i = 1 } ^ { n + 1 } a _ i } . \\end{align*}"}
+{"id": "2234.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": "100.png", "formula": "\\begin{align*} x \\ast y & = u ( x y ) - ( u x ) y - x ( u y ) = u ( x y ) + ( x y ) u = u ( x y + y x ) = y \\ast x . \\end{align*}"}
+{"id": "4943.png", "formula": "\\begin{align*} \\mathrm { i } \\delta _ n ^ + z _ { m , n } = \\delta _ m ^ { ( 2 ) } \\mu _ n z _ { m , n } + \\mu _ n ( | z _ { n , m } | ^ 2 ) \\mu _ n z _ { n , m } . \\end{align*}"}
+{"id": "1726.png", "formula": "\\begin{align*} \\det \\left ( \\begin{array} { c c c c } \\cos \\theta e ^ { i \\alpha } & - R \\sin \\theta e ^ { i \\alpha } & i R \\cos \\theta e ^ { i \\alpha } & \\\\ \\sin \\theta e ^ { i \\beta } & R \\cos \\theta e ^ { i \\beta } & & i R \\sin \\theta e ^ { i \\beta } \\\\ \\cos \\theta e ^ { - i \\alpha } & - R \\sin \\theta e ^ { - i \\alpha } & - i R \\cos \\theta e ^ { - i \\alpha } & \\\\ \\sin \\theta e ^ { - i \\beta } & R \\cos \\theta e ^ { - i \\beta } & & - i R \\sin \\theta e ^ { - i \\beta } \\end{array} \\right ) = - R ^ 3 \\sin 2 \\theta . \\end{align*}"}
+{"id": "278.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\tau } { \\rho } + \\beta \\tau \\eta \\geqslant \\frac { \\alpha ( 1 + 3 \\rho ^ 2 ) } { 2 \\rho ^ 2 } \\geqslant \\frac { 3 \\alpha } { 2 } \\end{align*}"}
+{"id": "3994.png", "formula": "\\begin{align*} G ( x , u , D u ) & < 0 \\Omega \\\\ G ( x , u , D u ) & = 0 \\partial \\Omega \\\\ D _ { p _ k p _ l } G ( x , u , D u ) & \\geq c _ 0 I \\Omega . \\end{align*}"}
+{"id": "8202.png", "formula": "\\begin{align*} \\left ( \\mathcal { L } \\left [ \\mathcal { D } ^ h _ t v \\right ] \\right ) ( \\sigma ) = h ( \\sigma ) ( \\mathcal { L } v ) ( \\sigma ) - \\frac { h ( \\sigma ) } { \\sigma } v ( + 0 ) . \\end{align*}"}
+{"id": "8983.png", "formula": "\\begin{align*} \\partial _ t a ^ { ( m ) } _ l = ( \\varphi _ l , & u ^ { ( m ) } _ t ) _ { L ^ 2 ( S ^ 1 ) } = - \\big ( \\varphi _ l , ( \\varepsilon + d \\pi _ N ( v ) ) u ^ { ( m ) } _ r \\big ) _ { L ^ 2 ( S ^ 1 ) } \\\\ & = - \\sum _ { j = 0 } ^ m a ^ { ( m ) } _ j \\lambda _ j \\big ( \\varphi _ l , ( \\varepsilon + d \\pi _ N ( v ) ) \\varphi _ j \\big ) _ { L ^ 2 ( S ^ 1 ) } , \\ 0 \\le l \\le m . \\end{align*}"}
+{"id": "378.png", "formula": "\\begin{align*} Y ( t _ { 2 k } ( \\mathfrak { w } ^ { ( n ) } ) ; n , \\mathfrak { w } ^ { ( n ) } ) = \\widetilde { Z } ( t _ { 2 k } ( \\mathfrak { w } ^ { ( n ) } ) ; n , \\mathfrak { w } ^ { ( n ) } ) \\ , \\end{align*}"}
+{"id": "366.png", "formula": "\\begin{align*} u ^ { ( n ) } _ k : = u _ k \\textrm { f o r } k \\in \\{ 1 , \\ldots , n \\} \\textrm { a n d } u ^ { ( n ) } _ k : = 2 \\beta _ n \\textrm { f o r } k \\in \\{ n + 1 , n + 2 , \\ldots , \\} \\ , . \\end{align*}"}
+{"id": "7117.png", "formula": "\\begin{align*} r ^ V = r ^ { \\rho _ 1 } \\cdots r ^ { \\rho _ k } . \\end{align*}"}
+{"id": "4207.png", "formula": "\\begin{align*} \\sum _ { \\gamma = 0 , 1 } \\sup _ { | x | , | p | \\leq \\frac { \\lambda } { 2 } } | \\partial ^ { \\gamma } _ { x , p } F _ j | ( x , p ) \\leq C _ { j , \\frac 1 2 \\lambda } . \\end{align*}"}
+{"id": "99.png", "formula": "\\begin{align*} J ( x , y , z ) _ \\ast = 4 [ ( ( ( x y ) u ) z ) + ( ( ( z x ) u ) y ) + ( ( ( y z ) u ) x ) ] u \\end{align*}"}
+{"id": "3466.png", "formula": "\\begin{align*} D _ { w , z } ^ 2 u ( t , x ) = \\sum _ { n \\geq 2 } n ( n - 1 ) I _ { n - 2 } ( \\widetilde { f } _ n ( \\cdot , w , z , x ; t ) ) = : \\sum _ { n \\geq 2 } B _ n ( w , z , x ; t ) \\end{align*}"}
+{"id": "3726.png", "formula": "\\begin{align*} \\mathcal { S } ( X _ { P _ \\ell } ( F ) ) = \\mathcal { S } _ { \\textrm { B K } } ( X _ { \\tilde { P } _ \\ell } ( F ) ) . \\end{align*}"}
+{"id": "5111.png", "formula": "\\begin{align*} \\Omega _ { \\mathbf { m } } ^ { \\pm } ( \\lambda , b ) & = \\frac { 1 - b ^ { 2 } } { 2 b } \\Lambda _ { 1 } ( \\lambda , b ) + \\frac { 1 } { 2 } \\Big ( \\Omega _ { \\mathbf { m } } ( \\lambda ) - \\Omega _ { \\mathbf { m } } ( \\lambda b ) \\Big ) \\\\ & \\quad \\pm \\frac { 1 } { 2 b } \\sqrt { \\Big ( b \\big [ \\Omega _ { \\mathbf { m } } ( \\lambda ) + \\Omega _ { \\mathbf { m } } ( \\lambda b ) \\big ] - ( 1 + b ^ { 2 } ) \\Lambda _ { 1 } ( \\lambda , b ) \\Big ) ^ { 2 } - 4 b ^ { 2 } \\Lambda _ { \\mathbf { m } } ^ { 2 } ( \\lambda , b ) } , \\end{align*}"}
+{"id": "7784.png", "formula": "\\begin{align*} [ X _ n , X _ { n ' } ] : = \\{ n , n ' \\} X _ { n + n ' } . \\end{align*}"}
+{"id": "3771.png", "formula": "\\begin{align*} \\varepsilon ' _ \\# \\alpha _ \\# & = \\alpha _ \\# \\varepsilon _ \\# , & & & { \\varepsilon _ 2 ' } _ \\# { \\varepsilon _ 1 } _ \\# & = { \\varepsilon _ 1 ' } _ \\# { \\varepsilon _ 2 } _ \\# \\end{align*}"}
+{"id": "2563.png", "formula": "\\begin{align*} \\begin{aligned} & \\zeta ( x _ { 1 } x _ { e _ 1 } { \\cdots } x _ { e _ { n - 1 } } x _ { - 1 } ; \\alpha ) \\\\ = & \\idotsint \\displaylimits _ { \\begin{subarray} { c } 0 < t _ 0 < \\cdots < t _ n < 1 \\end{subarray} } t _ 0 ^ { \\alpha - 1 } \\omega _ 1 ( t _ 0 ) \\left \\{ \\prod _ { i = 1 } ^ { n - 1 } \\omega _ { e _ i } ( t _ i ) \\right \\} \\omega _ { - 1 } ( t _ n ) \\mathrm { d } t _ { 0 } \\cdots \\mathrm { d } t _ { n } \\end{aligned} \\end{align*}"}
+{"id": "2081.png", "formula": "\\begin{align*} \\ln ( \\tfrac { 2 ^ { 2 \\kappa } } { ( 1 + \\kappa ) ^ { 1 + \\kappa } } ) \\geq \\ln ( \\tfrac { 4 ^ { \\kappa } } { \\exp ( \\kappa ( 1 + \\kappa ) ) } ) > 0 , \\end{align*}"}
+{"id": "4925.png", "formula": "\\begin{align*} u ( t + \\Delta t ) - u ( t ) = \\Delta t u ' ( t + \\tfrac { \\Delta t } 2 ) + \\mathcal { O } ( \\Delta t ^ 2 ) . \\end{align*}"}
+{"id": "4056.png", "formula": "\\begin{align*} \\exp { ( u _ t ) } & = \\frac { f ^ * ( Y ( \\cdot , u , D u ) ) \\det D Y ( \\cdot , u , D u ) } { f ( \\cdot ) } \\Omega \\times ( 0 , T ) , \\\\ Y u ( \\Omega ) & = \\Omega ^ * t \\in ( 0 , T ) , \\\\ u ( \\cdot , 0 ) & = u _ 0 ( \\cdot ) \\Omega \\times \\{ 0 \\} . \\end{align*}"}
+{"id": "6752.png", "formula": "\\begin{align*} z _ N ( 1 - s ) = \\sqrt { \\frac { 1 6 } { \\pi } } ( 1 - s ) \\left ( \\int _ { 0 } ^ { 1 } y ^ { \\frac { s } { 2 } - 1 } \\left [ \\sum _ { n = 1 } ^ { N } \\exp \\left [ - n ^ 2 \\pi y \\right ] \\right ] d y + \\frac { 1 } { 1 - s } \\right ) \\end{align*}"}
+{"id": "26.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { \\epsilon \\rightarrow 0 } \\big ( \\mathbf { D } _ { 2 6 } + \\mathbf { D } _ { 2 7 } \\big ) = \\mathbb { E } \\int _ 0 ^ \\infty e ^ { - \\beta t } \\bar { f } \\mathcal { Z } _ 1 d t + \\mathbb { E } \\int _ 0 ^ \\infty e ^ { - \\beta t } \\bar { F } _ u v d t . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "6958.png", "formula": "\\begin{gather*} L _ { m , n } ^ { I , ( \\alpha ) } ( x ) = \\frac { ( - 1 ) ^ { n - m } ( \\alpha + n ) } { m ! ( n - m ) ! \\prod _ { 1 \\leq i < j \\leq n } \\big ( X ^ { ( \\alpha ) } _ { j , m , n } - X ^ { ( \\alpha ) } _ { i , m , n } \\big ) } \\det \\big ( M _ n ^ I \\big ) . \\end{gather*}"}
+{"id": "2241.png", "formula": "\\begin{align*} & \\alpha _ 1 + \\tau = \\tilde { a } _ 1 , \\ \\alpha _ 2 + \\alpha _ 1 \\tau + \\frac { 1 } { 2 } \\tau ^ 2 = \\tilde { a } _ 2 , \\ \\alpha _ 3 + \\alpha _ 2 \\tau + \\frac { 1 } { 2 } \\alpha _ 1 \\tau ^ 2 + \\frac { 1 } { 6 } \\tau ^ 3 = \\tilde { a } _ 3 , \\\\ & \\alpha _ 4 + \\alpha _ 3 \\tau + \\frac { 1 } { 2 } \\alpha _ 2 \\tau ^ 2 + \\frac { 1 } { 6 } \\alpha _ 1 \\tau ^ 3 + \\frac { 1 } { 2 4 } \\tau ^ 4 = \\tilde { a } _ 4 . \\end{align*}"}
+{"id": "3235.png", "formula": "\\begin{align*} | F ' | \\ge \\frac { k | V | - ( k + 1 ) | C | } { p _ 1 + 1 } = \\frac { k \\lambda - | C | } { p _ 1 + 1 } . \\end{align*}"}
+{"id": "3304.png", "formula": "\\begin{align*} \\liminf _ { t \\to \\infty } \\| V ( t ) \\| _ { \\mathcal L ^ 2 } = 0 . \\end{align*}"}
+{"id": "2636.png", "formula": "\\begin{align*} h ( r , s ; \\chi ) = \\sum _ { \\substack { ( n , r ) = 1 \\\\ n } } \\frac { \\chi ( n ) g _ 6 ( r , n ) } { N ( n ) ^ s } . \\end{align*}"}
+{"id": "5001.png", "formula": "\\begin{align*} \\begin{cases} v _ t = \\Delta v - \\kappa v + f ( x , t ) & , \\\\ \\partial _ \\nu v = 0 & , \\\\ v ( \\cdot , 0 ) = v _ 0 & . \\end{cases} \\end{align*}"}
+{"id": "6385.png", "formula": "\\begin{align*} \\xi _ 0 ^ 3 - \\xi _ 0 ^ 2 + g ^ { d - 1 } \\xi _ 0 - g ^ d = 0 \\implies \\xi _ 0 ^ { - 3 } - g ^ { - 1 } \\xi _ 0 ^ { - 2 } + g ^ { - d } \\xi _ 0 ^ { - 1 } - g ^ { - d } = 0 , \\end{align*}"}
+{"id": "7685.png", "formula": "\\begin{align*} \\omega ^ { \\prime } = \\omega - \\sum _ { k } \\lambda _ { k } \\frac { d u _ { k } } { u _ { k } } = \\omega - \\sum \\lambda _ { k } ( d \\log u _ { k } ) \\end{align*}"}
+{"id": "5324.png", "formula": "\\begin{align*} ( Z T _ i ^ { - 1 } ) ^ * Z T _ i ^ { - 1 } = ( T _ i ^ * ) ^ { - 1 } \\Psi ( I ) T _ i ^ { - 1 } = ( T _ i ^ * ) ^ { - 1 } T _ i ^ * T _ i T _ i ^ { - 1 } = I , \\end{align*}"}
+{"id": "8278.png", "formula": "\\begin{align*} b = 3 , \\varepsilon = 0 . 0 1 , f = 0 . 5 , g = 0 . 6 , s = 0 . 3 , c = 4 . 8 , \\alpha = 0 . 7 8 2 5 , \\end{align*}"}
+{"id": "6439.png", "formula": "\\begin{align*} f ( X , Y , t ) & : = u ( X , Y , t ) - \\phi ( X , Y , t ) , \\\\ f _ j ( X , Y , t ) & : = u _ { \\epsilon _ j } ( X , Y , t ) - \\phi ( X , Y , t ) . \\end{align*}"}
+{"id": "4890.png", "formula": "\\begin{align*} C _ { s _ 2 } C _ { u _ l } = C _ { s _ 2 u _ l } + \\Box \\in C _ { s _ 2 u _ l } + H ^ { < 0 1 3 } . \\end{align*}"}
+{"id": "5607.png", "formula": "\\begin{align*} \\nabla _ { 1 } S _ { 2 2 } = \\nabla _ { 2 } S _ { 1 2 } = \\nabla _ { 3 } S _ { 1 3 } = \\nabla _ { 2 } S _ { 1 3 } = \\nabla _ { 0 } S _ { 1 3 } \\equiv 0 , \\end{align*}"}
+{"id": "8972.png", "formula": "\\begin{align*} \\| \\partial _ { \\phi } u \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } = \\| \\partial _ r u \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } = \\frac 1 2 \\| \\nabla u \\| ^ 2 _ { L ^ 2 ( S ^ 1 ) } \\end{align*}"}
+{"id": "2693.png", "formula": "\\begin{align*} M & = \\displaystyle \\sum _ i \\int _ { a _ i } ^ { b _ i } f \\\\ P & = \\displaystyle \\sum _ i f ( a _ i ) + f ( b _ i ) \\end{align*}"}
+{"id": "6109.png", "formula": "\\begin{align*} j _ i = p _ i n _ i \\end{align*}"}
+{"id": "6073.png", "formula": "\\begin{align*} | D _ \\lambda ( z ) | ^ 2 = | z - e | ^ \\alpha | D _ * ( z ) | ^ 2 , \\end{align*}"}
+{"id": "899.png", "formula": "\\begin{align*} \\begin{pmatrix} 3 & 1 & 2 \\\\ 1 & 2 & 3 \\\\ 1 & 2 & 3 \\end{pmatrix} \\begin{pmatrix} 1 & 2 & 3 \\\\ 2 & 3 & 1 \\\\ 3 & 2 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "8846.png", "formula": "\\begin{align*} \\Phi _ { k } ^ { \\nu , m } ( z ) : = \\sqrt { \\frac { \\left ( 2 \\nu + 2 m + 1 \\right ) \\left ( 2 \\nu + m \\right ) ! m ! } { ( m + k ) ! ( 2 \\nu + m - k ) ! } } \\left ( 1 + z \\overline { z } \\right ) ^ { - \\nu } z ^ { k } P _ { m } ^ { \\left ( k , 2 \\nu - k \\right ) } \\left ( \\frac { 1 - z \\overline { z } } { 1 + z \\overline { z } } \\right ) , \\end{align*}"}
+{"id": "1140.png", "formula": "\\begin{align*} d ^ 0 ( m ) & = \\sum _ { s \\in S } ( s m - m s ) e _ s \\\\ d ^ 1 ( \\sum _ { s \\in S } m _ s e _ s ) & = \\sum _ { s _ 0 , s _ 1 \\in S } ( s _ 0 m _ { s _ 1 } - m _ { s _ 1 } s _ 0 - s _ 1 m _ { s _ 0 } + m _ { s _ 0 } s _ 1 ) e _ { s _ 0 , s _ 1 } \\end{align*}"}
+{"id": "3517.png", "formula": "\\begin{align*} E _ { i j } \\Omega _ { A } = \\delta _ { i j } \\frac { p } { 2 } \\Omega _ A + \\sum _ { \\substack { ( k , l ) \\in \\lambda , \\\\ A ( k , l ) = j } } \\Omega _ { A _ { ( k , l ) \\to i } } . \\end{align*}"}
+{"id": "5624.png", "formula": "\\begin{align*} _ { t } ^ { A B } I _ { t _ { 0 } } ^ { \\alpha } f ( t ) = \\dfrac { 1 - \\alpha } { B ( \\alpha ) } f ( t ) + \\dfrac { \\alpha } { B ( \\alpha ) } \\ , _ { t } ^ { R L } I _ { t _ { 0 } } ^ { \\alpha } f ( t ) . \\end{align*}"}
+{"id": "3096.png", "formula": "\\begin{align*} c _ j ^ { k l } ( A ) = \\bar { a } \\left ( c _ j ^ { k l } ( B ) + \\bar { b } _ { k l } \\int _ Y r _ B B e _ j \\cdot \\nabla w _ A \\right ) = \\bar { a } \\ , \\bar { b } _ { k l } \\int _ Y r _ B B e _ j \\cdot \\nabla w _ A , \\end{align*}"}
+{"id": "632.png", "formula": "\\begin{gather*} \\beta _ { 0 } ( ( I + ( w ) ) ^ h ) = \\sum _ { \\ell = 1 } ^ h \\beta _ { 0 } ( I ^ { \\ell } ) + 1 , \\\\ \\beta _ { i } ( ( I + ( w ) ) ^ h ) = \\sum _ { \\ell = 1 } ^ h \\left [ \\beta _ { i } ( I ^ { \\ell } ) + \\beta _ { i - 1 } ( I ^ { \\ell } ) , \\right ] i \\in \\mathbb { N } _ { > 0 } . \\end{gather*}"}
+{"id": "4913.png", "formula": "\\begin{align*} \\frac { \\partial z } { \\partial t } = - \\mathrm { i } \\frac { \\delta \\mathcal H } { \\delta z ^ * } , ( x , t ) \\in ( a , b ) \\times ( 0 , T ) . \\end{align*}"}
+{"id": "4020.png", "formula": "\\begin{align*} \\phi ^ * _ i D _ { p _ k } Y ^ i w _ { k j } = \\chi \\gamma _ j , \\end{align*}"}
+{"id": "5287.png", "formula": "\\begin{align*} \\tau ( F ) = \\tau ^ { k e r F _ \\ast } ( F ) + \\tau ^ { ( k e r F _ \\ast ) ^ \\bot } ( F ) . \\end{align*}"}
+{"id": "4840.png", "formula": "\\begin{align*} X = L ^ 2 ( I ; \\R ^ d ) . \\end{align*}"}
+{"id": "5008.png", "formula": "\\begin{align*} \\begin{cases} u _ t \\ge \\Delta u - \\nabla \\cdot ( u \\nabla v ) + g & , \\\\ \\nu \\cdot ( \\nabla u - u \\nabla v ) \\ge 0 & , \\\\ u ( \\cdot , 0 ) \\ge u _ 0 & \\end{cases} \\end{align*}"}
+{"id": "8286.png", "formula": "\\begin{align*} J ( \\textbf { u } _ E ( t ) , \\textbf { u } _ i ( t ) ; \\textbf { x } _ 0 ) = \\Phi _ p ( \\textbf { x } ( t _ f ) ) \\end{align*}"}
+{"id": "4999.png", "formula": "\\begin{align*} \\begin{cases} v _ { \\varepsilon t } = \\Delta v _ \\varepsilon + f _ \\varepsilon ( x , t ) & , \\\\ \\partial _ \\nu v _ \\varepsilon = 0 & , \\\\ v _ \\varepsilon ( \\cdot , 0 ) = v _ { 0 \\varepsilon } & , \\end{cases} \\end{align*}"}
+{"id": "6795.png", "formula": "\\begin{align*} d ( \\mathcal { U } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) , \\mathcal { U } ( ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) \\leq ~ \\beta ( \\mathcal { M } '^ { * } _ { \\eta } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } , ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) ) \\mathcal { M } '^ { * } _ { \\eta } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } , ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) \\end{align*}"}
+{"id": "1431.png", "formula": "\\begin{align*} c _ r \\frac { m + 1 } { k + 1 } \\mathbb { P } ( \\sum _ { i = 1 } ^ { k + 1 } ( X _ i - 1 ) \\geq m + 1 ) + c _ r ( k + 1 ) ^ { - 1 } \\sum _ { j \\geq m + 2 } ^ { } \\mathbb { P } ( \\sum _ { i = 1 } ^ { k + 1 } ( X _ i - 1 ) \\geq j ) . \\end{align*}"}
+{"id": "2052.png", "formula": "\\begin{align*} & \\min \\{ \\rho _ 1 , \\rho _ 2 \\} = \\sigma , \\\\ [ 2 m m ] & \\alpha = \\begin{cases} \\dfrac { \\rho _ 2 - \\rho _ 1 } { \\rho _ 2 + \\rho _ 1 } & \\ \\max \\{ \\rho _ 1 , \\rho _ 2 \\} < \\infty , \\\\ [ 2 e x ] + 1 & \\ \\rho _ 2 = \\infty , \\\\ - 1 & \\ \\rho _ 1 = \\infty , \\end{cases} \\end{align*}"}
+{"id": "1475.png", "formula": "\\begin{align*} y _ j = \\begin{cases} 2 & 1 \\leq j \\leq a \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "3375.png", "formula": "\\begin{align*} \\bigvee _ { j = 1 } ^ { i - 1 } ( - u _ i + z _ j ) \\vee ( - u _ { i + 1 } + 1 + z _ i ) & \\leq \\bigvee _ { j = i + 1 } ^ { n - 1 } ( - u _ j + z _ j ) \\vee ( - u _ n ) & & \\enspace i = 1 , \\dots , n - 2 \\ , . \\end{align*}"}
+{"id": "5230.png", "formula": "\\begin{align*} & f ^ 2 _ { 0 1 0 } = 0 \\\\ & f ^ 1 _ { 1 0 0 } f ^ 3 _ { 0 0 1 } - f ^ 3 _ { 1 0 0 } f ^ 1 _ { 0 0 1 } = 0 \\\\ & f ^ 2 _ { 1 0 0 } - f ^ 3 _ { 0 1 0 } + f ^ 1 _ { 0 1 0 } f ^ 2 _ { 0 0 1 } = 0 \\ , . \\end{align*}"}
+{"id": "7579.png", "formula": "\\begin{align*} \\widetilde { \\sigma } = \\begin{cases} k _ 1 , & - d \\leq r \\leq 0 , \\\\ \\sigma , & 0 < r < \\delta , \\\\ k _ 2 , & \\delta \\leq r < d , \\end{cases} \\widetilde { \\mu } = \\begin{cases} k _ 1 , & - d \\leq r \\leq 0 , \\\\ \\mu , & 0 < r < \\delta , \\\\ k _ 2 , & \\delta \\leq r < d . \\end{cases} \\end{align*}"}
+{"id": "1445.png", "formula": "\\begin{align*} \\frac { 1 } { 2 T } \\sum _ { v \\in [ n ] } ^ { } \\mathbb { E } [ k _ v \\mathbb { 1 } _ { \\{ k _ v > T \\} } ] & = \\frac { 1 } { 2 T } \\sum _ { v \\in [ n ] } ^ { } [ T \\mathbb { P } ( k _ v > T ) + \\sum _ { \\ell > T } ^ { } \\mathbb { P } ( k _ v \\geq \\ell ) ] . \\end{align*}"}
+{"id": "3972.png", "formula": "\\begin{align*} L w : = a ^ { i j } D _ { i j } w + b ^ k D _ k w + c w \\leq 0 , \\end{align*}"}
+{"id": "2196.png", "formula": "\\begin{align*} \\dfrac { ( \\phi \\ast ( H _ { k } G _ { n } ) ) ( z ) } { ( \\phi \\ast G _ { n } ) ( z ) } = \\dfrac { 2 z ( \\phi \\ast f _ { k } ) ' ( z ) } { ( \\phi \\ast F ) _ { n } ( z ) - ( \\phi \\ast F ) _ { n } ( - z ) } \\end{align*}"}
+{"id": "3358.png", "formula": "\\begin{align*} & \\flat : \\alpha _ i = - B _ i + p _ i ( p _ j A _ j - C ) , \\beta _ i = A _ i , \\gamma = C - p _ j A _ j , \\\\ & \\sharp = \\flat ^ { - 1 } : A _ i = \\beta _ i , B _ i = - \\alpha _ i - \\gamma p _ i , C = p _ i \\beta _ i + \\gamma . \\end{align*}"}
+{"id": "5714.png", "formula": "\\begin{align*} \\mathbf { L } _ { b } ^ { a } = ( \\delta _ { b } ^ { a } - \\omega _ { b } ^ { a } \\mathbf { m } ) . \\end{align*}"}
+{"id": "7352.png", "formula": "\\begin{align*} D ( F , X ) : = \\big | | F \\cap X | - | F | \\cdot \\lambda ( X ) \\big | . \\end{align*}"}
+{"id": "9071.png", "formula": "\\begin{align*} w \\cdot \\Im \\lambda = \\Im \\lambda . \\end{align*}"}
+{"id": "149.png", "formula": "\\begin{align*} L _ { F } & = \\max \\bigg \\{ \\max _ { 1 \\leq j \\leq n } \\bigg ( \\bigg ( 1 + \\frac { M ^ { 2 } K ^ { 2 } b } { \\delta ^ { j } } \\bigg ) \\bigg ( K K _ { 1 } \\gamma b + K L _ { \\nu _ { j } } \\bigg ) \\bigg ) , \\bigg ( \\bigg ( 1 + \\frac { M ^ { 2 } K ^ { 2 } b } { \\delta ^ { 0 } } \\bigg ) \\\\ & \\qquad \\qquad \\qquad \\bigg ( K _ { 1 } K \\gamma b + K C _ { \\nu } \\bigg ) \\bigg ) , \\max _ { 1 \\leq j \\leq n } L _ { \\nu _ { j } } \\bigg \\} . \\end{align*}"}
+{"id": "4290.png", "formula": "\\begin{align*} \\textbf { \\textit { F } } = - \\Delta \\ , \\widetilde { \\textit { \\textbf { u } } } + \\nabla \\widetilde { \\pi } \\quad e = \\mathrm { d i v } \\ , \\widetilde { \\textit { \\textbf { u } } } . \\end{align*}"}
+{"id": "7472.png", "formula": "\\begin{align*} \\Delta _ v ( t ) = \\Delta _ v ( B _ { n - 1 } ) + t - B _ { n - 1 } \\ , . \\end{align*}"}
+{"id": "972.png", "formula": "\\begin{align*} h ^ 0 ( C , E ) = h ^ 0 ( C , N ) + h ^ 0 ( C , M ) \\le \\frac { \\mathrm { d e g } ( N ) + \\mathrm { d e g } ( M ) } { 2 } + 2 . \\end{align*}"}
+{"id": "2472.png", "formula": "\\begin{align*} D ^ 3 \\ ! F ( 0 ) \\left [ x ^ 3 \\right ] \\ ! = \\ ! 3 D ^ 2 \\ ! f ( 0 ) \\left [ x ^ 2 \\right ] x \\ \\ T _ x ( D ^ 3 \\ ! F ( 0 ) [ x ^ 3 ] ) \\ ! = \\ ! 3 D ^ 2 \\ ! f ( 0 ) [ x ^ 2 ] \\| x \\| . \\end{align*}"}
+{"id": "3783.png", "formula": "\\begin{align*} \\iota ( \\varepsilon _ 1 ) _ \\# ( \\varepsilon _ 2 ) _ \\# = ( \\varepsilon _ 1 ) ^ { - 1 } _ \\# ( \\varepsilon _ 2 ) ^ { - 1 } _ \\# \\iota . \\end{align*}"}
+{"id": "6765.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\mathbb { P } \\left ( \\left | \\Delta x - \\mathfrak { R e } ( \\alpha _ n ( t ) ) \\right | < \\epsilon \\right ) = 1 . \\end{align*}"}
+{"id": "4076.png", "formula": "\\begin{align*} & \\langle F u , F v \\rangle ' = \\langle u , v \\rangle . \\\\ & [ F u , F v ] ' = F [ u , v ] . \\\\ & \\pi ' \\circ F = \\pi . \\end{align*}"}
+{"id": "4070.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\Big | _ { t = 0 } \\lambda = \\int _ { M } \\langle \\mathcal { N } ( h , K , v _ { h , K } ) , ( h , K , v _ { h , K } ) \\rangle e ^ { - f } d V _ g = \\Big ( \\mathcal { N } ( h , K , v _ { h , K } ) , ( h , K , v _ { h , K } ) \\Big ) _ { f } \\end{align*}"}
+{"id": "2278.png", "formula": "\\begin{align*} - 2 \\int _ { \\Omega \\times \\mathbb { R } ^ 3 } \\nabla \\Phi ^ * \\cdot v f _ N \\ , d x d v & = - 2 \\sigma \\int _ \\Omega \\nabla \\Phi _ N \\cdot \\int _ { \\mathbb { R } ^ 3 } v f _ N \\ , d v d x \\\\ & = - 2 \\sigma \\int _ \\Omega \\Phi _ N \\ , \\partial _ t \\int _ { \\mathbb { R } ^ 3 } f _ N \\ , d v d x \\\\ & = - \\sigma \\frac { d } { d t } \\int _ \\Omega ( \\varepsilon | \\nabla ^ { 2 m + 1 } \\Phi _ N | ^ 2 + | \\nabla \\Phi _ N | ^ 2 ) \\ , d x . \\end{align*}"}
+{"id": "7705.png", "formula": "\\begin{align*} \\lambda ^ { - \\frac { 1 } { 2 } } = \\frac { 4 \\sqrt { \\tau } } { \\pi } \\left ( \\int _ { - 1 } ^ 1 \\frac { 1 } { 4 \\tau + \\lambda ( t + 1 ) ^ 2 } d t + \\int _ { - 1 } ^ 1 \\frac { 1 } { \\tau ( t + 1 ) ^ 2 + 4 \\lambda } d t \\right ) , \\end{align*}"}
+{"id": "6276.png", "formula": "\\begin{align*} \\varphi ^ { t } ( p '' ) = h \\circ \\varphi _ { 0 } ^ { t } \\circ h ^ { - 1 } ( p '' ) \\end{align*}"}
+{"id": "2534.png", "formula": "\\begin{align*} d \\Big ( \\sqrt { \\bar T _ N ^ \\varphi } - \\sqrt { T _ g ^ \\varphi } \\Big ) ^ 2 = d \\bigg ( \\frac { \\bar T _ N ^ \\varphi - T _ g ^ \\varphi } { \\sqrt { T _ N ^ \\varphi } + \\sqrt { T _ g ^ \\varphi } } \\bigg ) ^ 2 \\le \\frac { 2 d \\big ( \\bar T _ N ^ \\varphi - \\tilde T _ N ^ \\varphi \\big ) ^ 2 } { A _ t } + \\frac { 2 d \\big ( \\tilde T _ N ^ \\varphi - T _ g ^ \\varphi \\big ) ^ 2 } { A _ t } \\ , , \\end{align*}"}
+{"id": "995.png", "formula": "\\begin{align*} H ^ s _ \\rho ( \\Sigma ) : = \\left \\{ \\phi : \\rho \\phi \\in H ^ s ( \\Sigma ) \\right \\} , \\left < \\phi _ 1 , \\phi _ 2 \\right > _ { H ^ s _ \\rho ( \\Sigma ) } : = \\left < \\rho \\phi _ 1 , \\rho \\phi _ 2 \\right > _ { H ^ s ( \\Sigma ) } . \\end{align*}"}
+{"id": "6628.png", "formula": "\\begin{align*} F _ \\epsilon ( x ) \\coloneqq \\frac { x ^ { 2 m } } { 1 + \\epsilon x ^ { 2 m } } = \\frac { x _ 1 ^ { 2 m } + \\dots + x _ d ^ { 2 m } } { 1 + \\epsilon ( x _ 1 ^ { 2 m } + \\dots + x _ d ^ { 2 m } ) } , \\quad \\epsilon > 0 . \\end{align*}"}
+{"id": "2521.png", "formula": "\\begin{align*} M _ { u , T } ( v ) = \\frac 1 { ( 2 \\pi T ) ^ { d / 2 } } \\exp \\left ( - \\frac { | v - u | ^ 2 } { 2 T } \\right ) . \\end{align*}"}
+{"id": "8430.png", "formula": "\\begin{align*} \\mu ^ A \\ne \\mu \\mu ^ A ( \\mathbb R ^ n ) = \\mu ( \\mathbb R ^ n ) , \\end{align*}"}
+{"id": "4398.png", "formula": "\\begin{align*} u _ n ( \\theta ) = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ^ { ( k ) } ( \\theta ) \\right \\} . \\end{align*}"}
+{"id": "6905.png", "formula": "\\begin{align*} d V ( t , X ( t ) ) & = L V ( t , X ( t ) ) d t + \\partial _ x V ( t , X ( t ) ) \\cdot \\sigma I ( t ) S ( t ) d W _ t \\\\ & + \\int _ U \\left ( V ( X ( t - ) + J ( u ) ) - V ( X ( t - ) ) \\right ) \\check { N } ( d t , d u ) \\\\ & = L V ( t , X ( t ) ) d t + \\sigma S ( t ) d W _ t + \\int _ U \\left ( \\log ( 1 + J S ) \\right ) \\check { N } ( d t , d u ) , \\end{align*}"}
+{"id": "9123.png", "formula": "\\begin{align*} p \\mid ( b ^ { p - 1 } - 1 ) = \\prod _ { k \\mid ( p - 1 ) } \\Phi _ { k } ( b ) , \\end{align*}"}
+{"id": "8134.png", "formula": "\\begin{align*} H _ { m , n } ^ - ( x ) = \\frac { 4 } { \\pi } \\int _ { - \\infty } ^ \\infty K _ { 2 i t } ( x ) \\sinh ( \\pi t ) k ( t ) V ( m ^ 2 n , t ) t \\ , d t . \\end{align*}"}
+{"id": "1149.png", "formula": "\\begin{align*} & S S ( B , A ) \\iff | A | = | B | \\wedge \\forall i < | B | \\ , ( B ( i ) \\rightarrow A ( i ) ) . \\end{align*}"}
+{"id": "8779.png", "formula": "\\begin{align*} \\eta ^ { x } _ { N , 0 } = \\frac { 1 } { N } \\sum _ { j = 1 } ^ N W _ { i , j , 0 } ^ N \\delta _ { \\frac { 2 j - 1 } { 2 N } } , \\nu _ { N , t } ^ { x } = \\delta _ { \\phi _ i ^ N ( t ) } . \\end{align*}"}
+{"id": "7157.png", "formula": "\\begin{align*} \\displaystyle { D _ k = D _ k ^ { [ 0 ] } + \\eta D _ k ^ { [ 1 ] } + \\eta ^ 2 D _ k ^ { [ 2 ] } + O ( \\eta ^ 3 ) \\ , , k = 1 , . . . , N \\ , . } \\end{align*}"}
+{"id": "6655.png", "formula": "\\begin{align*} \\min \\limits _ { \\theta \\in \\mathbb { R } ^ d } F ( \\theta ) \\triangleq \\frac { 1 } { m } \\sum _ { i = 1 } ^ m f _ i ( \\theta ) \\end{align*}"}
+{"id": "7317.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { 0 } ^ { 1 } \\Phi ^ d _ { m _ n } d j _ n = 0 . \\end{align*}"}
+{"id": "2384.png", "formula": "\\begin{align*} [ \\ > E _ 1 ( t ) \\ > \\ > E _ 2 ( t ) \\ > ] \\dot x = [ \\ > 0 \\ > \\ > A _ 2 ( t ) \\ > ] x + f ( t ) \\end{align*}"}
+{"id": "556.png", "formula": "\\begin{align*} \\sigma _ t = \\pi _ t \\left [ ( \\theta - \\pi _ t [ \\theta ] ) ^ 2 \\right ] . \\end{align*}"}
+{"id": "603.png", "formula": "\\begin{align*} \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ \\dagger ) ^ { \\rm T } \\diamond { \\rm d } X _ t ^ \\dagger = \\lim _ { \\epsilon \\to 0 } \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ { ( \\epsilon ) } ) ^ { \\rm T } { \\rm d } X _ t ^ { ( \\epsilon ) } , \\end{align*}"}
+{"id": "5246.png", "formula": "\\begin{align*} A _ X X + \\nu \\nabla _ X U + T _ U X + { \\nu \\nabla _ U U } = 0 , \\end{align*}"}
+{"id": "8228.png", "formula": "\\begin{align*} - \\Delta w + V \\left ( \\left | x \\right | \\right ) w - w \\left ( \\Delta w ^ 2 \\right ) = K ( | x | ) g ( w ) \\mathbb { R } ^ { N } \\end{align*}"}
+{"id": "6149.png", "formula": "\\begin{align*} \\begin{cases} u '' - \\Delta u = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ v '' - \\Delta v + u = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ u = v = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu u = \\partial _ \\nu v = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 \\end{cases} \\end{align*}"}
+{"id": "2313.png", "formula": "\\begin{align*} V _ N ( x , y ) : = \\frac { 1 } { \\ ( | x | ^ 2 + ( 1 - y ) ^ 2 \\ ) \\ ( \\frac { | x | ^ 2 + ( 1 + y ) ^ 2 } { 4 } \\ ) \\ ( \\log \\sqrt { \\frac { | x | ^ 2 + ( 1 + y ) ^ 2 } { | x | ^ 2 + ( 1 - y ) ^ 2 } } \\ ) ^ 2 } . \\end{align*}"}
+{"id": "4640.png", "formula": "\\begin{align*} N ( r , R _ 2 ) \\leq N ( r , \\pi ^ * _ Y R _ C ) : = \\frac { 1 } { \\mathrm { d e g } ( p _ Y ) } \\int ^ r _ 0 \\# \\left ( \\pi ^ * _ Y R _ C \\cap Y ( t ) \\right ) \\frac { d t } { t } = \\frac { \\mathrm { d e g } ( \\pi _ Y ) } { \\mathrm { d e g } ( p _ Y ) } \\int ^ r _ 0 \\# \\left ( R _ C \\cap C ( t ) \\right ) \\frac { d t } { t } = O ( \\log r ) . \\end{align*}"}
+{"id": "3347.png", "formula": "\\begin{align*} \\Phi _ { x x } ^ 2 + \\zeta \\Phi _ { x q } ( \\Phi _ { y q } - \\Phi _ { y p } ) + 1 & = - \\Phi _ { x y } \\Phi _ { y x } , \\\\ \\Phi _ { q q } ^ 2 + \\zeta \\Phi _ { x q } ( \\Phi _ { y q } - \\Phi _ { y p } ) + 1 & = - \\Phi _ { q p } \\Phi _ { p q } . \\end{align*}"}
+{"id": "866.png", "formula": "\\begin{align*} | I _ { R _ i } | = | V ( R _ i ) \\setminus ( Q _ { R _ i } \\cup S _ { R _ i } ) | & = | V ( R _ i ) | + | S _ { R _ i } | - ( | Q _ { R _ i } | + 2 | S _ { R _ i } | ) \\\\ & > | V ( R _ i ) | + | S _ { R _ i } | - \\left ( \\frac { 1 } { 3 } - 3 \\sqrt \\delta \\right ) v ( F ) . \\end{align*}"}
+{"id": "1406.png", "formula": "\\begin{align*} \\pi _ 1 ^ { ( + , - , - ) } = L ( \\Delta [ 0 , - 2 ] , \\Delta [ 1 , - 2 ] ; \\pi ( 0 ^ + , 1 ^ - , 1 ^ - ) ) . \\end{align*}"}
+{"id": "2709.png", "formula": "\\begin{align*} Z '' + \\frac { 1 } { 1 6 s ^ { 3 / 2 } } Z ^ 2 = 0 , \\lim _ { s \\to 0 } \\frac { 1 } { s } Z ( s ) = \\lim _ { s \\to 0 } Z ' ( s ) = - c _ W . \\end{align*}"}
+{"id": "379.png", "formula": "\\begin{align*} \\frac { 4 } { 3 T } L ^ { - 2 } v ^ 2 _ 0 T ^ 3 \\big | 1 - \\frac { 2 \\widetilde { n } } { n } \\big | ^ 3 = \\frac { 4 } { 3 } \\frac { T _ * } { L ^ 2 _ * } \\frac { T _ * } { n ^ { 3 } } \\ , . \\end{align*}"}
+{"id": "8921.png", "formula": "\\begin{align*} \\mathcal { \\wp } ' = \\prod \\limits _ { l = \\nu } ^ { \\frac { n } { 2 } + \\nu - 1 } \\left ( r ^ { 2 } - l ^ { 2 } \\right ) \\prod \\limits _ { s = 1 - \\nu } ^ { \\frac { n } { 2 } - \\nu - 1 } \\left ( r ^ { 2 } - s ^ { 2 } \\right ) \\end{align*}"}
+{"id": "9266.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\prod _ { \\substack { N v _ j = v _ { \\ell } \\\\ i \\leq \\ell \\leq h ( i - 1 ) } } \\left ( \\prod _ { j = j _ { \\ell } } ^ { i - 1 } { d _ { ( 1 , \\ldots , n ) , ( v _ 1 , v _ 2 , \\ldots , v _ { h ( j ) } , N v _ j , \\ldots v _ n ) } } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "8805.png", "formula": "\\begin{align*} \\mathbf { K } ^ { \\operatorname { e p i } ( - f ) } _ { - t } = \\rho \\mathbf { K } ^ { \\operatorname { h y p o } ( f ) } _ t \\rho , \\end{align*}"}
+{"id": "5515.png", "formula": "\\begin{align*} \\sqrt { \\alpha } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n } \\exp \\left ( { - \\left ( \\frac { \\alpha } { n } \\right ) ^ 2 } \\right ) - \\sqrt { \\beta } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ( n ) } { n } \\exp \\left ( { - \\left ( \\frac { \\beta } { n } \\right ) ^ 2 } \\right ) = - \\frac { 1 } { 2 \\sqrt { \\beta } } \\sum _ { \\rho } \\frac { \\Gamma \\left ( \\frac { 1 - \\rho } { 2 } \\right ) \\beta ^ \\rho } { \\zeta ' ( \\rho ) } . \\end{align*}"}
+{"id": "5694.png", "formula": "\\begin{align*} \\digamma _ { b } ^ { a } = d \\alpha _ { b } ^ { a } + \\alpha _ { c } ^ { a } \\alpha _ { b } ^ { c } , \\end{align*}"}
+{"id": "2012.png", "formula": "\\begin{align*} | E ' | \\leq \\frac { k ^ { n / 2 } } { 1 - \\frac { k } { k - 1 } \\frac { s - 1 } { s } } = \\frac { s ( k - 1 ) k ^ { n / 2 } } { k - s } . \\end{align*}"}
+{"id": "4008.png", "formula": "\\begin{align*} L ( \\log \\eta ) \\geq - \\frac { C } { \\eta } - C w ^ { i i } - C \\sum _ { i = 1 } ^ n w ^ { i i } \\left ( \\frac { D _ i \\eta } { \\eta } \\right ) ^ 2 . \\end{align*}"}
+{"id": "6208.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ d \\beta _ r ( E _ r , \\widehat U _ 0 ) = \\sum _ { r = 1 } ^ d \\beta _ r ( E _ r , \\widehat U _ 1 ) = 0 \\end{align*}"}
+{"id": "1179.png", "formula": "\\begin{align*} | \\nabla _ { g _ 0 } ^ j ( \\Phi ^ * g - g _ 0 ) | _ { g _ 0 } = O ( r ^ { \\lambda - j } ) , \\end{align*}"}
+{"id": "4314.png", "formula": "\\begin{align*} E _ { x _ j } = E _ j ^ 1 \\supsetneq E _ j ^ 2 \\supsetneq \\cdots \\supsetneq E _ j ^ { n _ j + 1 } = 0 \\end{align*}"}
+{"id": "410.png", "formula": "\\begin{align*} V : \\mathcal { X } \\ni \\sum _ { n = 1 } ^ { \\infty } f _ n ( x ) \\tau _ n \\mapsto \\sum _ { n = 1 } ^ { \\infty } f _ n ( x ) \\omega _ n \\in \\mathcal { X } \\end{align*}"}
+{"id": "7654.png", "formula": "\\begin{align*} \\epsilon ( \\iota _ { E } ( Q _ { I } \\eta _ { I } ) ) = \\epsilon ( Q _ { I } ( \\iota _ { E } ( \\eta _ { I } ) ) ) & = \\sum _ { \\ell \\notin I } ( - 1 ) ^ { \\sigma _ { I } ( \\ell ) } Q _ { I \\cup \\{ \\ell \\} } \\iota _ { E } ( \\eta _ { I } ) \\\\ & = \\iota _ { E } ( \\sum _ { \\ell \\notin I } ( - 1 ) ^ { \\sigma _ { I } ( \\ell ) } Q _ { I \\cup \\{ \\ell \\} } \\eta _ { I } ) = \\iota _ { E } ( \\epsilon ( Q _ { I } \\eta _ { I } ) ) . \\end{align*}"}
+{"id": "3596.png", "formula": "\\begin{align*} c _ { j } ^ - ( \\lambda ) = \\bigg ( \\prod _ { \\ell = 1 } ^ { j - 1 } \\frac { \\lambda _ j - \\lambda _ \\ell - j + \\ell } { \\lambda _ j - \\lambda _ \\ell - j + \\ell - 1 } \\bigg ) ^ { \\frac { 1 } { 2 } } F _ j ( \\lambda _ 1 , \\dots , \\lambda _ { j - 1 } , \\lambda _ j - 1 , \\lambda _ { j + 1 } \\dots , \\lambda _ n ) , \\end{align*}"}
+{"id": "8392.png", "formula": "\\begin{align*} F _ { \\alpha } ( x ) = f _ { \\alpha } ^ { \\tau _ { \\alpha } ( x ) } ( x ) , \\end{align*}"}
+{"id": "5378.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } A ( n ) z ^ n & = \\sum _ { m \\ge 0 } \\frac { 1 } { m ! } \\Big ( \\sum _ { k \\ge 1 } \\frac { X ( k ) } { \\sqrt { k } } z ^ k \\Big ) ^ m \\\\ & = \\sum _ { m \\ge 0 } \\frac { 1 } { m ! } \\sum _ { n \\ge 0 } z ^ n \\sum _ { k _ 1 + \\dots + k _ m = n } \\frac { X ( k _ 1 ) \\cdots X ( k _ m ) } { \\sqrt { k _ 1 \\cdots k _ m } } \\end{align*}"}
+{"id": "6769.png", "formula": "\\begin{align*} \\chi ( - n ) = \\chi ( n ) \\end{align*}"}
+{"id": "5232.png", "formula": "\\begin{align*} A _ { X _ 1 } X _ 2 = \\mathcal { H } \\nabla _ { \\mathcal { H } X _ 1 } \\nu X _ 2 + \\nu \\nabla _ { \\mathcal { H } X _ 1 } \\mathcal { H } X _ 2 , \\end{align*}"}
+{"id": "2847.png", "formula": "\\begin{align*} b _ k = \\rho + \\sigma , b _ { \\ell } = \\rho - \\sigma . \\end{align*}"}
+{"id": "1793.png", "formula": "\\begin{align*} \\lvert \\mu _ \\bullet \\rvert = \\mu _ 1 + \\dots + \\mu _ n \\end{align*}"}
+{"id": "2380.png", "formula": "\\begin{align*} \\Phi ' = U \\left [ \\begin{array} { c } 0 \\\\ I _ { a } \\end{array} \\right ] \\end{align*}"}
+{"id": "5309.png", "formula": "\\begin{align*} f ^ { 2 } \\ , g ( A _ { \\xi } ( X ) , A _ { \\xi } ( Y ) ) = f ^ { 2 } \\ , g ( A ^ { 2 } _ { \\xi } ( X ) , Y ) , \\end{align*}"}
+{"id": "2043.png", "formula": "\\begin{align*} \\displaystyle \\int _ { \\epsilon } ^ { \\epsilon } R e \\left ( \\dfrac { 1 } { 1 - \\hat { \\mu } ( t ) } \\right ) d t = + \\infty , \\end{align*}"}
+{"id": "3024.png", "formula": "\\begin{align*} \\int _ \\textsf { f } \\partial _ k p { _ I } ^ { b k } \\hat { e } ^ { ( N + 1 ) } = \\int _ \\textsf { f } \\hbox { d } ( p { _ I } ^ { b k } \\hat { e } ^ { ( N ) } _ k ) = 0 \\end{align*}"}
+{"id": "1273.png", "formula": "\\begin{align*} \\mathcal { C } _ 1 = \\left \\{ \\big ( c ^ { ( x ^ 0 ) } | c ^ { ( x ^ 1 ) } | \\cdots | c ^ { ( x ^ { s - 1 } ) } \\big ) : c ( x , y , z ) \\in \\mathcal { C } \\right \\} , \\end{align*}"}
+{"id": "428.png", "formula": "\\begin{align*} ( \\{ f _ n \\} _ { n } , \\{ \\tau _ n \\} _ { n } ) = ( \\{ f ^ { ( 1 ) } _ n \\} _ { n } , \\{ \\tau ^ { ( 1 ) } _ n \\} _ { n } ) \\cup \\cdots \\cup ( \\{ f ^ { ( m ) } _ n \\} _ { n } , \\{ \\tau ^ { ( m ) } _ n \\} _ { n } ) , \\end{align*}"}
+{"id": "8487.png", "formula": "\\begin{align*} \\epsilon \\frac { \\partial g ^ 2 } { \\partial t } = 3 E - 1 0 F + 1 2 G , \\end{align*}"}
+{"id": "1220.png", "formula": "\\begin{align*} ( \\mathfrak { g } _ { 1 } ) _ { \\beta \\alpha } ( p ) = \\sum _ { i = 1 } ^ { n - 1 } \\frac { \\partial F ^ { i } _ { \\beta \\alpha } } { \\partial z _ { \\beta } ^ { n } } \\biggr | _ { z _ \\beta ( p ) } \\frac { \\partial } { \\partial z ^ { i } _ { \\alpha } } \\biggr | _ { z _ \\alpha ( p ) } \\otimes d z _ { \\beta } ^ { n } | _ { z _ \\beta ( p ) } \\end{align*}"}
+{"id": "7013.png", "formula": "\\begin{align*} R _ c ( R _ p ) \\geq \\ell \\left ( 1 \\right ) = \\frac { 1 } { 2 } \\log ^ + \\frac { N ^ 2 ( X , Y ) } { \\Delta ^ 2 e ^ { 2 R _ p } } , \\end{align*}"}
+{"id": "2640.png", "formula": "\\begin{align*} M _ 0 = A y \\widehat { W } ( 1 ) + O \\left ( \\left ( \\frac { y } { z } \\right ) ^ { - 1 / 3 + \\varepsilon } y \\right ) . \\end{align*}"}
+{"id": "5266.png", "formula": "\\begin{align*} g ( T _ W W , X ) + g ( T _ V V , X ) - 2 g ( T _ W V , X ) + g ( W , W ) g ( X , \\nabla f ) + g ( V , V ) g ( X , \\nabla f ) = 0 , \\end{align*}"}
+{"id": "7684.png", "formula": "\\begin{align*} \\omega = \\sum _ { k } \\lambda _ { k } \\frac { d u _ { k } } { u _ { k } } + \\lambda _ { k } \\frac { d f _ { k ^ { \\prime } } } { f _ { k } ^ { \\prime } } . \\end{align*}"}
+{"id": "3691.png", "formula": "\\begin{align*} d _ H ( \\varphi , \\varphi ' ) \\coloneqq \\inf \\left \\{ \\| H \\| _ { \\mathrm { o s c } } \\mid \\phi ^ H _ 1 = \\varphi ^ { - 1 } \\varphi ' \\right \\} \\end{align*}"}
+{"id": "1113.png", "formula": "\\begin{align*} \\theta ( R _ 0 ) = r , \\theta ( R _ 1 ) = - r \\mbox { a n d } \\mathrm { T V } ( R _ 0 , R _ 1 ) = \\delta . \\end{align*}"}
+{"id": "906.png", "formula": "\\begin{align*} \\mu _ n ( \\alpha ) = - \\frac { \\log \\left | \\alpha - q _ n / p _ n \\right | } { \\log q _ n } , \\end{align*}"}
+{"id": "2867.png", "formula": "\\begin{align*} f ( z ) & = \\sum _ { \\gamma \\in U _ { n - 1 } ( \\mathbb { Z } ) \\backslash S L ( n - 1 , \\mathbb { Z } ) } \\sum _ { m _ 1 = 1 } ^ { \\infty } \\dots \\sum _ { m _ { n - 2 } = 1 } ^ { \\infty } \\sum _ { m _ { n - 1 } \\neq 0 } \\frac { A ( m _ 1 , \\dots , m _ { n - 1 } ) } { \\prod _ { j = 1 } ^ { n - 1 } \\abs { m _ j } ^ \\frac { j ( n - j ) } { 2 } } \\\\ & \\times W _ J \\left ( M \\cdot \\begin{pmatrix} \\gamma & \\\\ & 1 \\end{pmatrix} z , v , \\psi _ { 1 , \\dots , 1 , \\frac { m _ { n - 1 } } { \\abs { m _ { n - 1 } } } } \\right ) , \\end{align*}"}
+{"id": "4459.png", "formula": "\\begin{align*} u _ n ( \\theta ) = \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } . \\end{align*}"}
+{"id": "2237.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u \\leq 0 \\ \\ & i n \\ B _ 1 \\\\ u \\geq 0 \\ \\ & i n \\ B _ 1 \\cap \\{ x _ { n + 1 } = 0 \\} \\\\ \\Delta u = 0 \\ & i n \\ B _ 1 \\cap ( \\{ u > 0 \\} \\cup \\{ x _ { n + 1 } \\neq 0 \\} ) . \\end{cases} \\end{align*}"}
+{"id": "3893.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } = ( \\dot { x _ \\theta } ) _ i D _ { x _ i } \\end{align*}"}
+{"id": "5767.png", "formula": "\\begin{align*} \\mathfrak { X } ( Z ) \\cap E = \\overline { \\Lambda } ^ { \\log } . \\end{align*}"}
+{"id": "9073.png", "formula": "\\begin{align*} & \\mu _ { z } ( P : \\sigma : \\lambda : \\eta ) ( \\phi ) \\\\ & \\qquad = \\int _ { N _ { Q } } \\int _ { M / M \\cap H _ { z } } \\int _ { A / A \\cap H _ { z } } a ^ { - \\lambda + \\rho _ { P } - 2 \\rho _ { Q } } \\Big ( \\sigma ^ { \\vee } ( m ) \\eta , \\phi ( n m a \\cdot z ) \\Big ) \\ , d m \\ , d n \\ , d a . \\end{align*}"}
+{"id": "8439.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\ , c ( K _ j ) = c _ * ( A ) = \\infty . \\end{align*}"}
+{"id": "7611.png", "formula": "\\begin{align*} ( v _ 1 ( t ) - v _ 2 ( t ) , \\zeta ) = - \\int _ 0 ^ t \\langle \\mathfrak { A } ( v _ 1 ( s ) ) - \\mathfrak { A } ( v _ 2 ( s ) ) , \\zeta \\rangle \\d s , \\end{align*}"}
+{"id": "2885.png", "formula": "\\begin{align*} s _ 1 - s _ 2 = b \\left ( \\frac { u ' _ 1 } { u ' _ 2 } - \\frac { u _ 1 } { u _ 2 } \\right ) . \\end{align*}"}
+{"id": "5853.png", "formula": "\\begin{align*} \\phi = ( e , U ) , \\end{align*}"}
+{"id": "1701.png", "formula": "\\begin{align*} \\delta \\mu _ m = \\frac { d } { d x } ( \\imath ( e ^ x ) \\mu _ m ) \\mid _ { x = 0 } = \\frac { d } { d x } ( e ^ { - m x } \\mu _ m ) \\mid _ { x = 0 } = - m \\cdot \\mu _ m . \\end{align*}"}
+{"id": "2345.png", "formula": "\\begin{align*} \\langle \\psi , T \\psi \\rangle = \\langle T ^ 2 \\psi , T \\psi \\rangle = - \\langle \\psi , T \\psi \\rangle , \\langle \\psi , T \\psi \\rangle = 0 . \\end{align*}"}
+{"id": "4516.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\lambda v _ i \\leq \\sum _ { i = 1 } ^ n \\lambda u _ i . \\end{align*}"}
+{"id": "4520.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k v _ i \\leq \\prod _ { i = 1 } ^ k u _ i \\end{align*}"}
+{"id": "2965.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { S } ^ n _ t ( \\varphi ) = \\frac { g ^ \\prime ( 0 ) } { 2 \\sqrt { n } } \\int _ 0 ^ t \\sum _ { j \\in \\mathbb { Z } } \\eta ^ n _ j ( s ) \\Delta ^ n \\varphi ^ n _ j ( s ) d s + \\frac { 1 } { 2 \\sqrt { n } } \\int ^ t _ 0 \\sum _ { j \\in \\mathbb { Z } } [ g _ n ( \\eta ^ n _ j ( s ) ) - g ^ \\prime ( 0 ) \\eta ^ n _ j ( s ) ] \\Delta ^ n \\varphi ^ n _ j ( s ) d s . \\end{aligned} \\end{align*}"}
+{"id": "7278.png", "formula": "\\begin{align*} 0 = d ( g _ m \\psi ^ + _ { m } - g ^ + _ m \\psi _ m ) = g _ m d \\psi ^ + _ { m } - \\psi _ { m } d g ^ + _ m = \\lambda g _ m \\psi _ { m } d m - \\psi _ { m } d g ^ + _ m . \\end{align*}"}
+{"id": "739.png", "formula": "\\begin{align*} H ( z , w ) = h _ 0 ( w ) + \\frac { 1 } { 2 } \\left ( h _ 1 ( w ) ( z - w ) + \\overline { h _ 1 ( w ) } ( \\bar { z } - \\bar { w } ) \\right ) + \\end{align*}"}
+{"id": "4147.png", "formula": "\\begin{align*} | \\Lambda ^ 2 | & = h _ { a j } H _ { j i b } \\Lambda _ { a i b } = \\frac { 1 } { 3 } h _ { a j } h _ { a k } H _ { j i b } H _ { k i b } + \\frac { 2 } { 3 } h _ { a j } h _ { b k } H _ { b j i } H _ { k a i } \\\\ & = \\frac { 4 } { 3 } \\mu | h | ^ 2 + \\frac { 8 } { 3 } \\langle \\mathring { R } ( h ) , h \\rangle , \\end{align*}"}
+{"id": "6427.png", "formula": "\\begin{align*} | \\mathcal { T } v ( X , Y _ 1 , t ) - \\mathcal { T } v ( X , Y _ 2 , t ) | = | F ( X , Y _ 1 , t ) - F ( X , Y _ 2 , t ) | < \\eta , \\end{align*}"}
+{"id": "918.png", "formula": "\\begin{align*} T _ n = e ^ { - W ( \\log ( n ^ 2 + n ) ) } \\end{align*}"}
+{"id": "48.png", "formula": "\\begin{align*} T ( n _ i ) + T ( n _ { i + 1 } ) + T ( n _ { i + 2 } ) & < \\left ( \\frac { 8 5 } { 2 7 } + \\frac { 3 2 0 } { 2 4 3 } \\right ) \\cdot \\frac { 1 } { X _ 0 } = \\frac { 1 0 8 5 } { 2 4 3 } \\cdot \\frac { 1 } { X _ 0 } \\\\ & < 6 \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } = ( k _ i + k _ { i + 1 } + k _ { i + 2 } ) \\cdot \\frac { 3 } { 4 } \\cdot \\frac { 1 } { X _ 0 } . \\end{align*}"}
+{"id": "7026.png", "formula": "\\begin{align*} F _ \\bullet A = ( F _ 0 A \\subseteq F _ 1 A \\subseteq F _ 2 A \\subseteq \\cdots \\subseteq A ) . \\end{align*}"}
+{"id": "2218.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty \\sum _ { j = 1 } ^ \\infty \\alpha _ { i , j } x ^ i y ^ j & = \\dfrac { N _ \\alpha } { D } , \\end{align*}"}
+{"id": "805.png", "formula": "\\begin{align*} E _ n ( \\epsilon ) = \\left \\{ ( x _ m ) _ { m = 0 } ^ \\infty \\in Y _ \\infty : \\left | \\frac { \\log \\| \\rho ( \\lambda ( \\ast , x _ 1 ) \\ldots \\lambda ( x _ { n - 1 } , x _ n ) ) \\| } { n } - \\Lambda \\right | > \\epsilon \\right \\} . \\end{align*}"}
+{"id": "740.png", "formula": "\\begin{align*} h _ 1 ( w ) = \\frac { \\partial h _ 0 ( w ) } { \\partial w } , \\end{align*}"}
+{"id": "256.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 2 ( i \\rho , i \\tau , \\xi + i \\eta ) = \\frac { \\alpha \\tau } { \\rho } + \\frac { \\beta \\eta } { \\tau } \\geqslant \\frac { \\alpha } { 2 } \\frac { ( 1 + 3 \\rho ^ 2 ) } { \\rho ^ 2 } \\geqslant \\frac { 3 \\alpha } { 2 } , \\end{align*}"}
+{"id": "5779.png", "formula": "\\begin{align*} t = 0 : U = \\theta E , U ' = 0 \\end{align*}"}
+{"id": "7602.png", "formula": "\\begin{align*} \\phi _ n ( r ) = \\chi _ { [ 0 , n ^ p ] } ( r ) + n r ^ { - \\frac { 1 } { p } } \\chi _ { ( n ^ p , \\infty ) } ( r ) . \\end{align*}"}
+{"id": "7384.png", "formula": "\\begin{align*} X _ \\alpha ( \\chi _ s ) = | \\det ( h _ \\alpha ) | _ F ^ s . \\end{align*}"}
+{"id": "4830.png", "formula": "\\begin{align*} F \\left ( \\Phi ( r , t ) \\right ) = d _ 0 F ( r ) + D _ 0 ^ n F ( w ( t ) , 0 ) + R ( r , t ) , \\end{align*}"}
+{"id": "6651.png", "formula": "\\begin{align*} \\nabla \\tilde { f } ( t , x ) = \\nabla f ( t , x ) - ( B ( x - x ' ) + b ) \\end{align*}"}
+{"id": "5151.png", "formula": "\\begin{align*} \\forall \\lambda > 0 , \\quad \\forall b \\in ( 0 , 1 ] , I _ { n } ( \\lambda b ) K _ { n } ( \\lambda ) \\underset { n \\rightarrow \\infty } { \\sim } \\frac { b ^ { n } } { 2 n } \\left ( \\sum _ { m = 0 } ^ { \\infty } \\frac { b _ { m } ( \\lambda b ) } { n ^ { m } } \\right ) \\left ( \\sum _ { m = 0 } ^ { \\infty } ( - 1 ) ^ { m } \\frac { b _ { m } ( \\lambda ) } { n ^ { m } } \\right ) , \\end{align*}"}
+{"id": "7500.png", "formula": "\\begin{align*} \\Delta _ { \\gamma _ 4 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) = & \\bigcup _ { m = 0 } ^ { i _ 0 - 1 } \\Big \\{ \\big ( \\frac { m j _ 0 + n _ m ^ { ' } } { i _ 0 } , m \\big ) + \\big ( a j _ 0 + b , a i _ 0 \\big ) | a , b \\in \\mathbb { Z } ^ { + } \\Big \\} \\\\ = & \\bigcup _ { m = 0 } ^ { i _ 0 - 1 } \\Big \\{ \\big ( \\frac { m j _ 0 + n _ m ^ { ' } } { i _ 0 } + a j _ 0 + b , m + a i _ 0 \\big ) | a , b \\in \\mathbb { Z } ^ { + } \\Big \\} . \\end{align*}"}
+{"id": "3435.png", "formula": "\\begin{align*} \\frac { \\mu _ { t } ( [ b ] ) } { \\mu _ { t } ( [ a ] ) } = \\frac { 1 - p _ { a a } ^ t } { 1 - p _ { b b } ^ t } . \\end{align*}"}
+{"id": "6045.png", "formula": "\\begin{align*} \\varphi ( z ) : = \\exp \\left \\{ \\pi \\frac { 1 - a b } { 2 K } \\int _ 1 ^ z \\frac { \\dd s } { ( w \\tilde w ) ( s ) } \\right \\} , \\end{align*}"}
+{"id": "801.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\lim _ { m \\to \\infty } \\int 1 _ { F ( t , n ) } \\ d \\mu _ { ( z , m ) } = \\lim _ { n \\to \\infty } \\mu _ j ( F ( t , n ) ) = \\frac { 1 } { \\sigma _ j \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ t e ^ { - \\frac { s ^ 2 } { 2 \\sigma _ j ^ 2 } } \\ d s \\end{align*}"}
+{"id": "8717.png", "formula": "\\begin{align*} \\sum _ { k > 0 } \\frac { ( x | a ) _ { k - 1 } } { ( y | a ) _ k } = \\frac { 1 } { y - x } . \\end{align*}"}
+{"id": "6253.png", "formula": "\\begin{align*} \\left \\{ k = ( k _ 1 , k _ 2 , k _ 3 ) \\in \\Z ^ { 3 } : \\frac 3 5 M \\le k _ 1 \\le M , \\ , \\frac 4 5 M \\le k _ 2 \\le M , \\ , 0 \\le k _ 3 \\le c _ { \\delta ^ { * } } | k _ H | \\right \\} . \\end{align*}"}
+{"id": "534.png", "formula": "\\begin{align*} 1 = \\sum _ { j = 1 } ^ n \\frac { a _ j } { d } = \\sum _ { m = 1 } ^ k \\sum _ { j \\in I _ m } \\frac { a _ j } { d } = \\sum _ { m = 1 } ^ k \\sum _ { j \\in I _ m } \\frac { a _ j } { u _ m ( d \\wedge a _ j ) } = \\sum _ { m = 1 } ^ k S _ { { \\bf A _ m } , d } ( u _ m ) . \\end{align*}"}
+{"id": "2672.png", "formula": "\\begin{align*} \\varphi ( T ) = \\textsc { l e a v e s } ( T ) , \\end{align*}"}
+{"id": "4038.png", "formula": "\\begin{align*} a ^ { i j } D _ { i j } v + b ^ i D _ i v - c v & = 0 \\Omega \\\\ \\alpha v + \\beta \\cdot D v & = 0 \\partial \\Omega , \\end{align*}"}
+{"id": "4952.png", "formula": "\\begin{align*} \\widetilde { A } ^ { A V F } : = \\left ( \\begin{array} { c } \\delta _ n ^ + u _ { m , n } + \\delta _ m ^ { ( 2 ) } \\mu _ n v _ { m , n } + \\mu _ n ( v _ { m , n } ^ 2 + \\frac { 2 } 3 u _ { m , n } ^ 2 ) \\mu _ n v _ { m , n } + \\frac { 1 } 3 \\mu _ n ( u _ { m , n } ^ 2 v _ { m , n } ) \\\\ - \\delta _ n ^ + v _ { m , n } + \\delta _ m ^ { ( 2 ) } \\mu _ n u _ { m , n } + \\mu _ n ( u _ { m , n } ^ 2 + \\frac { 2 } 3 v _ { m , n } ^ 2 ) \\mu _ n u _ { m , n } + \\frac { 1 } 3 \\mu _ n ( v _ { m , n } ^ 2 u _ { m , n } ) \\end{array} \\right ) = \\mathbf { 0 } . \\end{align*}"}
+{"id": "3832.png", "formula": "\\begin{align*} E \\exp \\biggl \\{ i \\sum _ { j = 1 } ^ n \\theta _ j X _ j \\biggr \\} = \\exp \\biggl \\{ - \\int _ { S ^ n } \\biggl | \\sum _ { j = 1 } ^ d \\theta _ j s _ j \\biggr | \\ , \\Gamma ( d s _ 1 , \\ldots , d s _ n ) \\biggr \\} . \\end{align*}"}
+{"id": "395.png", "formula": "\\begin{align*} \\chi _ { \\delta , \\varepsilon } ( x ) : = \\eta _ \\delta ( | x | / \\varepsilon ) \\ , , x \\in \\R ^ { n - 1 } \\ , , \\end{align*}"}
+{"id": "5263.png", "formula": "\\begin{align*} g ( T _ U U , X ) + g ( U , U ) g ( X , \\nabla f ) = 0 \\end{align*}"}
+{"id": "5248.png", "formula": "\\begin{align*} \\nabla _ { \\dot { \\alpha } } \\dot { \\alpha } = A _ X X + \\nu \\nabla _ X U + T _ U X + \\nu \\nabla _ U U + \\mathcal { H } \\nabla _ X X + 2 A _ X U + T _ U U . \\end{align*}"}
+{"id": "7561.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , A _ b ) = \\sum _ { a = 0 } ^ { \\omega - 1 } { Z _ f ( s , \\chi , A _ b ^ a ) } . \\end{align*}"}
+{"id": "6551.png", "formula": "\\begin{align*} \\mathop { \\max } \\limits _ { \\{ { P _ { j k } } \\} } & \\sum \\limits _ { k = 1 } ^ K { \\ln \\left ( { { R _ k } - R _ k ^ { { \\rm { t h } } } } \\right ) } \\\\ { \\rm s . t . } & \\eqref { s t 1 } , \\eqref { s t 2 } , \\eqref { s t 3 } . \\end{align*}"}
+{"id": "4786.png", "formula": "\\begin{align*} & \\mathop { \\max } _ { \\substack { S \\subseteq \\{ \\epsilon N \\pm N ^ { 3 / 4 } \\} \\\\ 1 \\leq | S | \\leq 2 } } \\Big \\{ \\frac { 1 } { \\binom { N } { S } } \\sum _ { j \\notin B } \\Pr _ { v \\sim \\mu _ { t } } \\big [ | v H | = j \\big ] K _ S ( j ) ^ 2 \\Big \\} . \\end{align*}"}
+{"id": "2305.png", "formula": "\\begin{align*} u ( x , 0 ) = \\begin{cases} 1 , & x \\in ( - 3 . 7 , - 0 . 7 ) \\cup ( 0 . 7 , 3 . 7 ) , \\\\ 0 , & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
+{"id": "682.png", "formula": "\\begin{align*} - d * d G ( \\cdot , w ) = \\delta _ w - \\frac { 1 } { V } { \\rm { v o l } } . \\end{align*}"}
+{"id": "7761.png", "formula": "\\begin{align*} \\beta _ { \\alpha , 1 - \\alpha } ( t ) & = \\frac { 1 } { \\pi } \\mathrm { I m } \\left \\{ \\frac { ( e ^ { - i \\pi } t ) ^ { \\alpha - 1 } } { ( 1 + e ^ { - i \\pi } t ) ^ { \\alpha } } \\right \\} = \\begin{cases} \\frac { \\sin \\pi \\alpha } { \\pi } \\ , t ^ { \\alpha - 1 } ( 1 - t ) ^ { - \\alpha } & 0 \\le t \\le 1 \\\\ 0 & t > 1 \\end{cases} \\end{align*}"}
+{"id": "2773.png", "formula": "\\begin{align*} x _ i = \\ ! \\left \\{ \\ ! \\begin{aligned} & x ' _ i , ~ ~ ~ ~ ~ i \\in [ 1 , d _ 1 - 1 ] \\backslash \\{ \\lambda _ 1 , \\lambda _ 2 \\} ; \\\\ & x ' _ { i - 1 } , ~ ~ i \\in [ d _ 1 + 1 , d _ 2 ] ; \\\\ & x ' _ i , ~ ~ ~ ~ ~ i \\in [ d _ 2 + 1 , n ] . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7902.png", "formula": "\\begin{align*} k ( q \\theta ) = h _ f ( \\theta ) . \\end{align*}"}
+{"id": "8457.png", "formula": "\\begin{align*} c _ * ( A ) = \\lim _ { j \\to \\infty } \\ , c _ * ( A _ j ) . \\end{align*}"}
+{"id": "1313.png", "formula": "\\begin{align*} \\theta _ \\tau : \\mathbb { K } ^ n \\ni ( a _ j ) _ { j = 1 } ^ n \\mapsto \\theta _ \\tau ( a _ j ) _ { j = 1 } ^ n \\coloneqq \\sum _ { j = 1 } ^ n a _ j \\tau _ j \\in \\mathcal { X } . \\end{align*}"}
+{"id": "3545.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ { j } \\sum _ { s = 1 } ^ { j - i + 1 } \\sum _ { I \\in \\mathcal { I } _ { i j } ( s ) } E _ { k l } E ^ { e _ I } B _ i ^ + H _ I ^ + . \\end{align*}"}
+{"id": "7900.png", "formula": "\\begin{align*} f ( z ) ^ n + q ( z ) e ^ { Q ( z ) } f ^ { ( k ) } ( z + c ) = P ( z ) , \\end{align*}"}
+{"id": "1835.png", "formula": "\\begin{align*} p \\le s \\le t \\le q , \\ p + q = s + t \\Rightarrow f ( s ) + f ( t ) \\le f ( p ) + f ( q ) . \\end{align*}"}
+{"id": "3658.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ { t } u = \\Delta _ { \\mathbb { H } ^ { 3 } } u + f ( u , t ) & \\hbox { i n } ~ \\mathbb { H } ^ { 3 } \\times ( 0 , T ) , \\\\ \\\\ u = u _ { 0 } \\in C ( \\mathbb { H } ^ { 3 } ) \\cap L ^ { \\infty } ( \\mathbb { H } ^ { 3 } ) & \\hbox { i n } ~ \\mathbb { H } ^ { 3 } \\times \\{ 0 \\} , \\end{array} \\right . \\end{align*}"}
+{"id": "5354.png", "formula": "\\begin{align*} \\hat { \\Lambda } _ { F , s } [ \\varepsilon ] = \\frac { M _ { F , \\ , | F | - s } [ \\varepsilon ] } { M _ { F , \\ , | F | } [ \\varepsilon ] } \\end{align*}"}
+{"id": "3666.png", "formula": "\\begin{align*} \\mathcal { A } ^ k : = \\{ x \\in \\Omega : \\lambda ^ k ( x ) + c ( u ^ k ( x ) - g ( x ) ) > 0 \\} . \\end{align*}"}
+{"id": "6442.png", "formula": "\\begin{align*} f _ j > c _ \\rho / 2 = c _ \\rho / 4 + c _ \\rho / 4 > f _ j ( \\hat X , \\hat Y , \\hat t ) + \\epsilon _ j ^ 3 \\geq \\inf _ { W _ { 2 \\rho } } f _ j + \\epsilon _ j ^ 3 , \\textrm { o n } K _ \\rho . \\end{align*}"}
+{"id": "138.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow 0 ^ + } \\lambda ^ p \\mathcal { L } ^ { 2 n } ( H _ { \\lambda , K } ^ + ) = | K | \\| f \\| ^ p _ { L ^ p ( \\mathbb R ^ n ) } . \\end{align*}"}
+{"id": "4229.png", "formula": "\\begin{align*} E _ [ \\Lambda , \\phi ] ( t ) = 0 . \\end{align*}"}
+{"id": "7517.png", "formula": "\\begin{align*} Z _ { g _ 3 } ( s , \\chi , D ) = M _ { 2 , 5 } ( q ^ { - s } ) , \\end{align*}"}
+{"id": "7889.png", "formula": "\\begin{align*} - 4 A ( z ) = e ^ { 2 \\varphi ( z ) } + { \\varphi } ' ( z ) ^ 2 - 2 \\varphi ^ { \\prime \\prime } ( z ) , \\end{align*}"}
+{"id": "7189.png", "formula": "\\begin{align*} z _ { \\alpha , j k } ( x _ { j } , \\ , t _ 0 ) = z _ { \\alpha \\ , 0 , j k } ( { x _ j } ) , \\end{align*}"}
+{"id": "2790.png", "formula": "\\begin{align*} \\rho = \\frac { a _ 1 + a _ 2 } { 2 } = \\frac { b _ 1 + b _ 2 } { 2 } \\end{align*}"}
+{"id": "195.png", "formula": "\\begin{align*} \\liminf \\frac { c _ { n } ( \\log ( c _ { n } ) ) ^ { \\epsilon } } { h _ { n } } & \\geq \\liminf \\frac { ( d ( n - 1 ) ^ { 1 + \\alpha } ) ^ { \\epsilon } } { n ^ { 1 + \\epsilon } } \\geq \\liminf \\frac { d ^ { \\epsilon } ( n - 1 ) ^ { \\epsilon + \\alpha \\epsilon } } { n ^ { 1 + \\epsilon } } \\\\ & \\geq \\liminf \\frac { d ^ { \\epsilon } ( n - 1 ) ^ { 1 + 2 \\epsilon } } { n ^ { 1 + \\epsilon } } = \\liminf d ^ { \\epsilon } \\Big { ( } 1 - \\frac { 1 } { n } \\Big { ) } ^ { 1 + \\epsilon } ( n - 1 ) ^ { \\epsilon } = \\infty . \\end{align*}"}
+{"id": "681.png", "formula": "\\begin{align*} \\mathcal { E } ( \\delta _ a , \\delta _ b ) & = \\int _ M d G ^ { \\delta _ a } \\wedge * d G ^ { \\delta _ b } = - \\int _ M G ^ { \\delta _ a } \\wedge d * d G ^ { \\delta _ b } \\\\ & = \\int _ M G ^ { \\delta _ a } \\wedge \\left ( \\delta _ b - \\frac { 1 } { V } \\ , { \\rm v o l } \\right ) = G ^ { \\delta _ a } ( b ) = G ( a , b ) . \\end{align*}"}
+{"id": "4908.png", "formula": "\\begin{align*} \\big ( ( \\chi ) _ * u , \\tau - \\partial _ s \\chi \\big ) = \\Phi \\circ \\Psi \\big ( u , \\tau \\big ) . \\end{align*}"}
+{"id": "2193.png", "formula": "\\begin{align*} F ( z ) = \\sum _ { k = 1 } ^ { m } \\alpha _ k f _ k ( z ) . \\end{align*}"}
+{"id": "2454.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\Lambda _ { \\pi } ( n ) } { n ^ s } = - \\frac { L ' } { L } ( s , \\pi ) , \\sum _ { n = 1 } ^ { \\infty } \\frac { \\Lambda _ { \\pi \\otimes \\chi } ( n ) } { n ^ s } = - \\frac { L ' } { L } ( s , \\pi \\otimes \\chi ) , \\mathrm { R e } ( s ) > 1 . \\end{align*}"}
+{"id": "833.png", "formula": "\\begin{align*} \\sup _ { \\gamma } \\left | \\frac { \\pi _ { n ' } ( \\gamma ) } { \\widetilde { \\tau } _ { n p } ^ c ( \\gamma ) } - 1 \\right | = O ( n ^ { - 1 / 2 } ) \\end{align*}"}
+{"id": "3472.png", "formula": "\\begin{align*} & \\| D _ { z } F _ R ( t ) \\| _ 4 = \\left \\| \\int _ { B _ R } D _ { z } u ( t , x ) d x \\right \\| _ 4 \\leq \\int _ { B _ R } \\| D _ { z } u ( t , x ) \\| _ 4 d x \\leq C \\int _ { B _ R } \\int _ 0 ^ t G _ { t - s } ( x - z ) d s d x . \\end{align*}"}
+{"id": "7458.png", "formula": "\\begin{align*} K ( Z , W ) ( P ) = \\Gamma _ \\mu ( Z , W ) ( P ) \\end{align*}"}
+{"id": "8858.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } F ( r \\omega ) = \\sum _ { 0 \\leq p \\leq m } \\frac { \\Gamma ( m - p + 1 ) \\Gamma ( n + 2 p + 2 \\nu ) } { ( - 1 ) ^ { p - m } \\Gamma ( m + n + p + 2 \\nu ) } \\sum _ { j = 1 } ^ { d ( n , p , p + 2 \\nu ) } a _ { j } ^ { \\nu , p } h _ { p , p + 2 \\nu } ^ { j } ( \\omega , \\bar { \\omega } ) \\end{align*}"}
+{"id": "3387.png", "formula": "\\begin{align*} \\mathcal { F } = \\mathcal { T } ^ { \\perp _ 0 } \\textrm { a n d } \\mathcal { T } = { } ^ { \\perp _ 0 } \\mathcal { F } . \\end{align*}"}
+{"id": "426.png", "formula": "\\begin{align*} \\theta _ g \\theta _ \\omega \\{ a _ n \\} _ n & = \\theta _ g \\left ( \\sum _ { n = 1 } ^ { \\infty } a _ n \\omega _ n \\right ) = \\sum _ { n = 1 } ^ { \\infty } a _ n \\theta _ g \\omega _ n \\\\ & = \\sum _ { n = 1 } ^ { \\infty } a _ n \\sum _ { m = 1 } ^ { \\infty } g _ m ( \\omega _ n ) e _ m = \\sum _ { n = 1 } ^ { \\infty } a _ n e _ n \\\\ & = \\{ a _ n \\} _ n , \\forall \\{ a _ n \\} _ n \\in \\ell ^ p ( \\mathbb { N } ) . \\end{align*}"}
+{"id": "3305.png", "formula": "\\begin{align*} V ( t ) = S ( t ) V _ 0 + \\int _ 0 ^ t S ( t - s ) \\mathbb P [ \\mathbf { 1 } _ { \\mathcal F _ 0 } ( V - \\ell _ V ) \\cdot \\nabla V ] { \\rm d } s . \\end{align*}"}
+{"id": "3104.png", "formula": "\\begin{align*} - \\tilde { A } : D ^ 2 w = \\gamma - \\bar { \\gamma } \\quad Y , w Y , \\int _ Y w = 0 , \\end{align*}"}
+{"id": "5411.png", "formula": "\\begin{align*} \\delta ' ( q _ { i , j , k } , \\ell _ k ^ { ( i ) } ) = \\begin{cases} q _ { i , ( j + 1 ) \\bmod p , k + 1 } & k \\ne m _ i + n _ i - 1 \\\\ q _ { i , ( j + 1 ) \\bmod p , m _ i } & \\end{cases} \\end{align*}"}
+{"id": "7163.png", "formula": "\\begin{align*} h _ k ' = \\sum \\limits _ { \\substack { | I | = k } } \\prod \\limits _ { \\substack { i \\in I \\\\ j \\notin I } } \\sqrt { \\phi ( z _ j - z _ i ) } \\prod _ { i \\in I } e ^ { - v _ i / c } \\prod \\limits _ { \\substack { i \\in I \\\\ j \\notin I } } \\sqrt { \\phi ( z _ i - z _ j ) } k = 1 , \\dots , N \\ , , \\end{align*}"}
+{"id": "6604.png", "formula": "\\begin{align*} 4 ^ m ( \\sin \\theta ) ^ { 2 m } = \\binom { 2 m } { m } + \\sum _ { q = 0 } ^ { m - 1 } \\binom { 2 m } { q } ( - 1 ) ^ { q + m } ( 2 \\cos ( 2 m - 2 q ) \\theta ) \\ , . \\end{align*}"}
+{"id": "5915.png", "formula": "\\begin{align*} w = ( E , \\widetilde C _ { p - 1 } U ) , { \\cal L } \\theta = - ( E , \\widetilde C _ { p - 1 } A U ) , { \\cal R } \\theta = - ( E , \\widetilde C _ { p - 1 } B U ) , \\end{align*}"}
+{"id": "993.png", "formula": "\\begin{align*} \\lambda _ 1 ( \\Omega _ 1 ) = \\cdots = \\lambda _ 1 ( \\Omega _ k ) , \\end{align*}"}
+{"id": "4663.png", "formula": "\\begin{align*} m ^ { \\underline { \\ell _ i } } m ^ { \\underline { \\ell _ j } } = \\sum _ { k = 0 } ^ { \\min \\{ \\ell _ i , \\ell _ j \\} } G _ { \\ell _ i , \\ell _ j , k } m ^ { \\underline { \\ell _ i + \\ell _ j - k } } , \\end{align*}"}
+{"id": "629.png", "formula": "\\begin{gather*} \\widetilde H _ 3 ( \\Gamma _ h , \\mathbb Z _ 2 ) = \\mathbb Z _ 2 , \\widetilde H _ 4 ( \\Gamma _ h , \\mathbb Z _ 2 ) = ( \\mathbb Z _ 2 ) ^ 2 , \\\\ \\widetilde H _ 3 ( \\Gamma _ h , \\Q ) = 0 , \\widetilde H _ 4 ( \\Gamma _ h , \\Q ) = \\Q . \\end{gather*}"}
+{"id": "1967.png", "formula": "\\begin{align*} ( r _ A , r _ B ) = & ( ( 5 , 3 , 3 , 3 , 3 , 1 ) , ( 5 , 5 , 5 , 6 , 7 , 7 ) ) \\\\ & ( ( 0 , 0 , 0 , 1 , 1 , 3 ) , ( 0 , 1 , 0 , 3 , 3 , 1 ) ) , \\end{align*}"}
+{"id": "1684.png", "formula": "\\begin{align*} \\mu _ m \\left ( \\left | \\begin{array} { c c } X & Y \\\\ x & y \\end{array} \\right | ^ { k - 2 } \\right ) = x ^ { \\frac { k - 2 } { 2 } - m } y ^ { \\frac { k - 2 } { 2 } + m } , \\frac { 2 - k } { 2 } \\leq m \\leq \\frac { k - 2 } { 2 } . \\end{align*}"}
+{"id": "1846.png", "formula": "\\begin{align*} \\ddot { \\varrho } + \\frac { 4 } { 3 t } \\dot { \\varrho } - \\tilde { \\kappa } t ^ { - 2 \\gamma + \\frac { 2 } { 3 } } \\Delta \\varrho - \\frac { 2 } { 3 t ^ 2 } \\varrho = 0 , \\\\ \\end{align*}"}
+{"id": "5006.png", "formula": "\\begin{align*} \\begin{cases} \\Phi ' ( b ) \\le \\Phi ' ( s ) & a \\ge b , \\\\ \\Phi ' ( b ) \\ge \\Phi ' ( s ) & a \\le b , \\end{cases} \\end{align*}"}
+{"id": "2779.png", "formula": "\\begin{align*} x _ i = \\ ! \\left \\{ \\ ! \\begin{aligned} & x ' _ i , ~ ~ ~ ~ ~ i \\in [ 1 , d _ 1 - 1 ] \\backslash \\{ \\lambda _ 1 \\} , \\\\ & x ' _ { i - 1 } , ~ ~ i \\in [ d _ 1 + 1 , d _ 2 ] , \\\\ & x ' _ i , ~ ~ ~ ~ ~ i \\in [ d _ 2 + 1 , n ] \\backslash \\{ \\lambda _ 2 \\} . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5829.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( \\ ! ( U '' , \\Phi ) \\ ! ) d x + \\int _ { \\Omega } ( \\ ! ( \\Delta U , \\Delta \\Phi ) \\ ! ) d x + \\int _ { \\Omega } ( \\ ! ( A U , \\Phi ) \\ ! ) d x + \\int _ { \\Omega } a ( \\ ! ( D U ' , \\Phi ) \\ ! ) d x = 0 . \\end{align*}"}
+{"id": "1988.png", "formula": "\\begin{align*} { \\mathcal { R } } _ { } \\approx \\frac { N M } { L } \\left ( \\log _ 2 { p _ { } } + \\frac { 1 } { M } \\sum \\nolimits _ { m = 1 } ^ { M } \\log _ 2 \\left ( \\frac { \\lambda _ m } { M } \\right ) \\right ) . \\end{align*}"}
+{"id": "8042.png", "formula": "\\begin{align*} E ( z , s ) = y ^ s + \\phi ( s ) y ^ { 1 - s } + \\sum _ { n \\neq 0 } \\phi ( n , s ) W _ s ( n z ) . \\end{align*}"}
+{"id": "3993.png", "formula": "\\begin{align*} [ D _ { i j } \\phi - D _ { p _ k } A _ { i j } & ( x , u , D u ) D _ k \\phi ] \\xi _ i \\xi _ j \\\\ & = [ - d _ { i j } + D _ { p _ k } A _ { i j } ( x , u , D u ) D _ k d ] \\xi _ i \\xi _ j + 2 K d _ i d _ j \\xi _ i \\xi _ j . \\end{align*}"}
+{"id": "6666.png", "formula": "\\begin{align*} \\begin{aligned} x _ i ^ { k + 1 } - { x ' _ i } ^ { k + 1 } = & ( 1 - \\gamma _ 1 ^ k | R _ { i i } | ) ( x _ i ^ k - { x ' _ i } ^ k ) - \\lambda ^ k ( y _ i ^ k - { y ' _ i } ^ k ) , \\\\ y _ i ^ { k + 1 } - { y ' _ i } ^ { k + 1 } = & ( 1 - \\alpha ^ k - \\gamma ^ k _ 2 | C _ { i i } | ) ( y _ i ^ k - { y ' _ i } ^ k ) \\\\ & + ( g _ i ^ { k + 1 } - { g ' _ i } ^ { k + 1 } ) - ( 1 - \\alpha ^ k ) ( g _ i ^ k - { g ' _ i } ^ k ) , \\end{aligned} \\end{align*}"}
+{"id": "1723.png", "formula": "\\begin{align*} g = u \\left ( \\begin{array} { c c } r & x \\\\ & r ^ { - 1 } \\end{array} \\right ) \\kappa ( a , b ) , u \\in \\C ^ \\times \\ ; , x \\in \\C \\ ; , r \\in \\R ^ \\times \\ ; , ( a , b ) \\in S ^ 3 . \\end{align*}"}
+{"id": "8517.png", "formula": "\\begin{align*} \\begin{cases} z ' = P ( z , w ) \\\\ w ' = Q ( z , w ) \\end{cases} \\end{align*}"}
+{"id": "6006.png", "formula": "\\begin{align*} I _ 1 = < x _ { h + 1 } , \\dots , x _ n ; y _ 1 , \\dots , y _ { h } , y _ { n - p + 1 } , \\dots , y _ n > & & \\mathrm { a n d } & & I _ 2 = < x _ { h } , \\dots , x _ n ; y _ 1 , \\dots , y _ { h - 1 } , y _ { n - p + 1 } , \\dots , y _ n > . \\end{align*}"}
+{"id": "4399.png", "formula": "\\begin{align*} \\sup & \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ^ { ( k ) } ( \\theta ) \\right \\} \\\\ & = \\sup \\left \\{ \\sum _ { i = 1 } ^ n \\frac { 1 } { x _ i } : ( x _ i ) _ { i = 1 } ^ n \\in U _ n ^ { ( k ) } ( \\theta ) x _ i < x _ i ^ * i = 1 , \\ldots , k + 1 \\right \\} . \\end{align*}"}
+{"id": "8315.png", "formula": "\\begin{align*} Q _ l ( t , x ) = Q \\left ( \\mathbf { P } _ 2 x + \\frac { 1 } { \\sqrt { 1 - | l | ^ 2 } } \\ , \\mathbf { P } _ 1 ( x - l t ) \\right ) , \\end{align*}"}
+{"id": "2568.png", "formula": "\\begin{align*} \\Upsilon ( g * u ) & = \\Upsilon ( g u g ^ { - 1 } - g ' g ^ { - 1 } ) = \\Upsilon ( g u g ^ { - 1 } ) - \\Upsilon ( g ' g ^ { - 1 } ) \\\\ & = \\Omega ( g ) \\Upsilon ( u ) \\Omega ( g ) ^ { - 1 } - \\Omega ( g ) ' \\Omega ( g ) ^ { - 1 } = \\Omega ( g ) * \\Upsilon ( u ) . \\end{align*}"}
+{"id": "1850.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ { t } w _ 0 + \\frac { 4 } { 3 t } w _ { 0 } - \\tilde { \\kappa } t ^ { - 2 \\gamma + \\frac { 2 } { 3 } } \\delta ^ { i j } \\partial _ { j } w _ { i } - \\frac { 2 } { 3 t ^ 2 } w = 0 , \\\\ & \\tilde { \\kappa } t ^ { - 2 \\gamma + \\frac { 2 } { 3 } } \\delta ^ { i k } \\partial _ { t } w _ { i } - \\tilde { \\kappa } t ^ { - 2 \\gamma + \\frac { 2 } { 3 } } \\delta ^ { i k } \\partial _ { i } w _ { 0 } = 0 , \\\\ & \\partial _ { t } w = w _ 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7725.png", "formula": "\\begin{align*} \\tilde { \\rho } \\cong \\rho _ 0 : = \\frac { 1 - \\cos ( \\beta \\pi ) } { 1 + \\cos ( \\beta \\pi ) } = \\tan ^ 2 \\left ( \\frac { \\beta \\pi } { 2 } \\right ) . \\end{align*}"}
+{"id": "7771.png", "formula": "\\begin{align*} m _ { \\alpha , \\beta } ( x | \\mu ) \\equiv f _ { \\alpha \\beta } ( x | \\mu ) & = \\int _ 0 ^ \\infty f _ \\alpha ( x | y ) f _ \\beta ( y | \\mu ) d y \\\\ r _ { \\alpha , \\beta } ( x ) \\equiv \\rho _ { \\alpha \\beta } ( x ) & = x \\int _ 0 ^ \\infty f _ \\alpha ( x | y ) y ^ { - 1 } \\rho _ \\beta ( y ) d y \\end{align*}"}
+{"id": "8881.png", "formula": "\\begin{align*} H _ { 0 } \\left ( t ; z , w \\right ) = \\pi ^ { - n } \\sum _ { m \\geq 0 } \\left ( 2 m + n \\right ) \\frac { ( m + n - 1 ) ! } { m ! } e ^ { - 4 m ( m + n ) t } P _ { m } ^ { ( n - 1 , 0 ) } ( \\cos 2 d _ { F S } ( z , w ) ) , \\end{align*}"}
+{"id": "2185.png", "formula": "\\begin{align*} 3 ( ( 1 - 2 \\alpha - \\beta ) r ^ 2 + ( 2 - 2 \\alpha + \\beta ) r + 1 ) = 5 ( 1 - r ^ 2 ) . \\end{align*}"}
+{"id": "8163.png", "formula": "\\begin{align*} H _ { m , n } ^ - ( x ) = \\frac { 4 } { \\pi } \\int _ { - \\infty } ^ \\infty K _ { 2 i t } ( x ) \\sinh ( \\pi t ) k ( t ) U ( m ^ 2 n , t ) t \\ , d t . \\end{align*}"}
+{"id": "2022.png", "formula": "\\begin{align*} \\nu ( m ) : = \\left \\{ \\begin{array} { l l l } \\mu ' ] - \\infty , m ] & { \\rm i f } & m \\leq - 1 \\\\ \\mu [ m + 1 , + \\infty [ & { \\rm i f } & m \\geq 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "4440.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } + \\frac { 1 } { 1 2 } = \\frac { 1 } { 3 } + \\frac { 1 } { 4 } = \\frac { 7 } { 1 2 } . \\end{align*}"}
+{"id": "2747.png", "formula": "\\begin{align*} D _ { \\varphi } ( f g ) = D ( \\varphi \\circ f g ) = & D ( ( \\varphi \\circ f ) ( \\varphi \\circ g ) ) \\\\ = & \\varphi \\circ f ( x ) D ( \\varphi \\circ g ) + \\varphi \\circ g ( x ) D ( \\varphi \\circ f ) \\\\ = & \\varphi ( f ( x ) ) D _ { \\varphi } g + \\varphi ( g ( x ) ) D _ { \\varphi } f . \\end{align*}"}
+{"id": "7436.png", "formula": "\\begin{align*} \\alpha K ^ * ( Z , W ) ( P ) & = \\alpha K ( W , Z ) ( P ^ * ) ^ * = \\big ( K ( W , Z ) ( P ^ * ) \\alpha ^ * \\big ) ^ * \\\\ & = K ^ * ( \\widetilde Z , W ) ( \\alpha P ) \\end{align*}"}
+{"id": "4734.png", "formula": "\\begin{align*} J = \\bigl \\langle \\{ b _ \\beta \\} , \\{ c _ \\gamma \\} , \\{ d _ \\delta \\} \\bigm \\vert \\beta \\neq 2 n , \\gamma \\ne 3 n \\bigr \\rangle . \\end{align*}"}
+{"id": "3248.png", "formula": "\\begin{align*} I ^ { \\pm } ( z ) = \\sum _ { i _ 1 + j _ 1 = q _ 1 } \\frac { 1 } { i _ 1 ! j _ 1 ! k _ 1 ! } \\int _ { 0 < t _ 1 < z } L _ 1 ( t _ 1 , z ) ^ { i _ 1 } L _ { - 1 } ( t _ 1 , z ) ^ { j _ 1 } L _ 0 ( t _ 1 , z ) ^ { k _ 1 } \\left ( \\frac { d t _ 1 } { 1 - t _ 1 } + \\frac { - d t _ 1 } { 1 + t _ 1 } \\right ) . \\end{align*}"}
+{"id": "7496.png", "formula": "\\begin{align*} Z _ f ( s , \\chi , D ) = v ( \\bar { f } , D , \\chi ) & + \\sigma ( \\bar { f } , D , \\chi ) \\dfrac { ( 1 - q ^ { - 1 } ) q ^ { - s } } { 1 - q ^ { - 1 - s } } + Z _ f ( s , \\chi , D _ { S ( f , D ) } ) , \\end{align*}"}
+{"id": "8030.png", "formula": "\\begin{align*} f ( x ) & = \\pi ^ { - 1 } \\left ( h ( c + \\pi x ) - c \\right ) \\\\ & = \\pi ^ { - 1 } \\left ( \\sum _ { r \\ge 0 } h ^ { ( r ) } ( c ) \\pi ^ { r } x ^ r / r ! - c \\right ) \\\\ & = ( h ( c ) - c ) \\pi ^ { - 1 } + h ' ( c ) x + \\sum _ { r \\ge 2 } h ^ { ( r ) } ( c ) \\frac { \\pi ^ { r - 1 } } { r ! } x ^ r . \\end{align*}"}
+{"id": "6245.png", "formula": "\\begin{align*} r _ k ^ { 0 0 } = ( 0 , 0 , 1 , 0 ) , r _ k ^ 0 = e _ k ^ 0 , r ^ + _ k = \\frac 1 { \\sqrt 2 } \\alpha _ k , r ^ - _ k = \\overline { r ^ + _ k } . \\end{align*}"}
+{"id": "1265.png", "formula": "\\begin{align*} \\tilde { \\rho _ 0 } & = \\dfrac { 2 \\sqrt { \\alpha } } { ( 1 + \\alpha ) \\sqrt { 4 \\alpha ( A - B ) ^ 2 + B ^ 2 } } \\rho _ 0 = \\dfrac { 1 } { ( 1 - \\alpha ) ( A - B ) - B } \\end{align*}"}
+{"id": "905.png", "formula": "\\begin{align*} \\frac { p _ { n + 1 } } { q _ { n + 1 } } = \\begin{bmatrix} a _ 0 & 0 \\\\ 1 & 1 \\end{bmatrix} \\begin{bmatrix} a _ 1 & 0 \\\\ 1 & 1 \\end{bmatrix} \\begin{bmatrix} a _ 2 & 0 \\\\ 1 & 1 \\end{bmatrix} \\cdots \\begin{bmatrix} a _ { n + 1 } & 0 \\\\ 1 & 1 \\end{bmatrix} \\begin{bmatrix} p _ n & p _ { n - 1 } \\\\ q _ n & q _ { n - 1 } \\end{bmatrix} \\end{align*}"}
+{"id": "7540.png", "formula": "\\begin{align*} Z _ g \\big ( s , \\chi , S ( \\Delta _ { \\gamma _ 3 } \\bigcap ( \\mathbb { N } ^ 2 \\setminus \\{ 0 \\} ) ) \\big ) = & \\dfrac { \\tilde { G } _ 3 ( q ^ { - s } ) } { 1 - q ^ { - 1 - s } } \\sum _ { a = 1 } ^ { \\infty } q ^ { - a ( i _ 0 + j _ 0 + i _ 0 j _ 0 s ) } \\\\ = & \\dfrac { G _ 3 ( q ^ { - s } ) } { ( 1 - q ^ { - 1 - s } ) ( 1 - q ^ { - i _ 0 - j _ 0 - i _ 0 j _ 0 s } ) } , \\end{align*}"}
+{"id": "7037.png", "formula": "\\begin{align*} F _ n A & = F _ n M U ^ { C _ 2 } _ * \\cdot F _ 0 A \\\\ F _ n D & = F _ n M U ^ { C _ 2 } _ * \\cdot F _ 0 D . \\end{align*}"}
+{"id": "6940.png", "formula": "\\begin{gather*} H _ { \\lambda } = \\operatorname { W r } [ H _ { \\lambda _ m } , H _ { \\lambda _ { m - 1 } + 1 } , \\dots , H _ { \\lambda _ 2 + m - 2 } , H _ { \\lambda _ 1 + m - 1 } ] , \\end{gather*}"}
+{"id": "3740.png", "formula": "\\begin{align*} I ( f ) ( \\varpi ^ n \\xi ) = & I ( f ) ( 0 ) - q ^ { - n } \\Big ( \\frac { q ^ { A - 1 } } { 1 - q ^ { - 1 } } f ( 0 ) - q ^ { A - 1 } \\sum _ { j = 0 } ^ { B + A - 1 } q ^ { - j } f ( \\varpi ^ { A - j - 1 } \\xi , 0 , 0 ) \\Big ) . \\end{align*}"}
+{"id": "768.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\varphi ( \\xi _ n ) } { n } = \\Lambda \\end{align*}"}
+{"id": "3315.png", "formula": "\\begin{align*} \\left \\{ P _ { f } \\left ( \\mathbf { x } \\right ) \\right \\} = \\left ( G _ { f } \\right ) . \\end{align*}"}
+{"id": "2446.png", "formula": "\\begin{align*} \\begin{aligned} | \\langle u _ k , | \\varphi | ^ 2 \\rangle | \\ll _ { u _ { k } } \\lambda _ { \\varphi } ^ { - 1 / 4 } ( \\log \\lambda _ { \\varphi } ) L ( \\tfrac { 1 } { 2 } , u _ k \\otimes \\mathrm { A d } ^ 2 \\varphi ) ^ { 1 / 2 } . \\end{aligned} \\end{align*}"}
+{"id": "4372.png", "formula": "\\begin{align*} \\frac { 1 } { t } = \\frac { p } { q + 1 } < \\frac { p } { q } = \\frac { p } { p t - 1 } \\leq \\frac { 1 } { t - 1 } \\end{align*}"}
+{"id": "8497.png", "formula": "\\begin{align*} P _ 0 & = \\ ; \\frac { 8 1 \\Big ( - 4 \\sqrt { 2 } \\xi ( { \\xi } ^ 2 - 8 1 ) - 6 { \\xi } ^ 2 + 2 4 7 5 \\Big ) } { 4 { \\xi } ^ 8 } , \\\\ P _ 1 & = \\ ; \\frac { 2 7 \\Big ( - \\sqrt { 2 } \\xi ( 9 3 - 2 { \\xi } ^ 2 ) + 3 { \\xi } ^ 2 - 6 0 3 \\Big ) } { 2 { \\xi } ^ 6 } , \\\\ P _ 2 & = \\ ; \\frac { 9 \\Big ( - 1 2 \\sqrt { 2 } \\xi ( 2 { \\xi } ^ 2 - 4 3 ) + 4 { \\xi } ^ 4 - 3 6 { \\xi } ^ 2 + 2 7 8 1 \\Big ) } { 3 2 { \\xi } ^ 4 } , \\\\ P _ 3 & = \\ ; \\frac { 9 ( - 6 \\sqrt { 2 } \\xi + 2 { \\xi } ^ 2 - 2 7 ) } { 4 { \\xi } ^ 2 } . \\end{align*}"}
+{"id": "4470.png", "formula": "\\begin{align*} 8 = v _ 1 + v _ 2 > u _ 1 + u _ 2 = 6 . \\end{align*}"}
+{"id": "900.png", "formula": "\\begin{align*} \\mu = \\mu ( \\pi ^ 2 ) > 2 , \\end{align*}"}
+{"id": "370.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n _ i } u _ k \\leq \\frac { t } { T r _ i } < \\sum \\limits _ { k = 1 } ^ { n _ i + 1 } u _ k \\textrm { f o r e a c h } i \\in \\{ 1 , 2 \\} \\ , . \\end{align*}"}
+{"id": "7631.png", "formula": "\\begin{align*} \\psi ( \\cdots u _ { - 2 } u _ { - 1 } | u _ 0 u _ 1 u _ 2 \\cdots ) = \\cdots \\psi ( u _ { - 2 } ) \\psi ( u _ { - 1 } ) | \\psi ( u _ 0 ) \\psi ( u _ 1 ) \\psi ( u _ 2 ) \\cdots . \\end{align*}"}
+{"id": "4874.png", "formula": "\\begin{align*} D _ h ( t ) = \\int _ { \\Sigma _ h ( t , 1 - t ) } w _ { z _ h ; k _ h } ( t _ h ) \\ . w _ { z _ 1 ; k _ 1 } ( t _ 1 ) d \\L ^ { h } . \\end{align*}"}
+{"id": "5322.png", "formula": "\\begin{align*} V = U V _ 1 = \\sqrt { \\lambda } U T ^ \\frac { 1 } { 2 } V T _ 1 ^ { - 1 } W = U T ^ \\frac { 1 } { 2 } V Z ^ { - 1 } , \\end{align*}"}
+{"id": "8326.png", "formula": "\\begin{align*} \\begin{cases} \\square _ { s , y } \\tilde { v } = ( | y | ^ 2 - s ^ 2 ) _ + ^ \\frac { 2 } { d - 2 } | \\tilde { v } | ^ { \\frac { 4 } { d - 2 } } \\tilde { v } , \\\\ \\tilde { v } | _ { s = 0 } = v _ 0 , \\partial _ s \\tilde { v } | _ { s = 0 } = v _ 1 , \\end{cases} \\end{align*}"}
+{"id": "8706.png", "formula": "\\begin{align*} f _ k ( x ) = g _ { - k } ( x ^ { - 1 } ) = x + \\dots + x ^ k = \\frac { x ( 1 - x ^ k ) } { 1 - x } , f _ { - k } ( x ) = g _ k ( x ^ { - 1 } ) = \\frac { x - 1 } { x ^ { k + 1 } } , f _ 0 ( x ) = g _ 0 ( x ) = 1 . \\end{align*}"}
+{"id": "9136.png", "formula": "\\begin{align*} M _ 1 \\ ; : = & \\ ; m \\sum _ { k = 0 } ^ n { n \\choose k } ( m + n - 1 ) ^ { n - k - 1 } = \\\\ = & m ( m + n - 1 ) ^ { - 1 } \\sum _ { k = 0 } ^ n { n \\choose k } ( m + n - 1 ) ^ { n - k } = m ( m + n - 1 ) ^ { - 1 } ( m + n ) ^ { n } , \\\\ \\end{align*}"}
+{"id": "6263.png", "formula": "\\begin{align*} \\mathcal { M } _ { 0 } = \\{ ( x , y , z ) \\in \\mathbb { R } ^ { 3 } : x ^ { 2 } + z ^ { 2 } = 1 \\} \\end{align*}"}
+{"id": "9112.png", "formula": "\\begin{align*} \\rho _ 1 = \\sqrt { \\frac { h _ 1 } { h _ 2 } } \\rho _ 2 = \\sqrt { \\frac { h _ 2 } { h _ 1 } } . \\end{align*}"}
+{"id": "8084.png", "formula": "\\begin{align*} \\mathcal { R ^ + } = O ( M ^ { - 3 } T ^ { \\frac { 5 } { 2 } + \\varepsilon } p ^ { 1 + \\varepsilon } ) . \\end{align*}"}
+{"id": "6148.png", "formula": "\\begin{align*} \\begin{cases} u '' - \\Delta u + a u + b v = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ v '' - \\Delta v + c u + d v = 0 & \\hbox { i n } ( 0 , + \\infty ) \\times \\Omega , \\\\ u = v = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 0 , \\\\ \\partial _ \\nu u + \\alpha u = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 , \\\\ \\partial _ \\nu v + \\beta v = 0 & \\hbox { o n } ( 0 , + \\infty ) \\times \\Gamma _ 1 . \\end{cases} \\end{align*}"}
+{"id": "1522.png", "formula": "\\begin{align*} p ^ { b c } _ A = 2 \\delta ^ i _ A \\rho _ { i , d } ^ b \\eta ^ { d c } \\end{align*}"}
+{"id": "7205.png", "formula": "\\begin{align*} \\big \\langle q , \\ , e _ \\ell q \\big \\rangle = 0 , \\qquad \\mbox { w h e r e } e _ \\ell = \\big \\{ i , \\ , j , \\ , k \\big \\} . \\end{align*}"}
+{"id": "5726.png", "formula": "\\begin{align*} \\Omega _ { B } ^ { A } \\theta ^ { B A ^ { \\prime } } = 0 . \\end{align*}"}
+{"id": "5220.png", "formula": "\\begin{align*} Q _ { 1 3 } & = - r _ { 1 1 } \\ell _ 1 ' - r _ { 1 2 } \\ell _ 2 ' + c _ { 1 2 } \\ell _ 2 = c _ { 1 3 } \\\\ Q _ { 1 4 } & = r _ { 1 1 } \\ell _ 2 ' - r _ { 1 2 } \\ell _ 1 ' + c _ { 1 2 } \\ell _ 1 = c _ { 1 4 } \\\\ Q _ { 2 3 } & = - r _ { 1 2 } \\ell _ 1 ' - r _ { 2 2 } \\ell _ 2 ' - c _ { 1 2 } \\ell _ 1 = c _ { 2 3 } \\\\ Q _ { 2 4 } & = r _ { 1 2 } \\ell _ 2 ' - r _ { 2 2 } \\ell _ 1 ' + c _ { 1 2 } \\ell _ 2 = c _ { 2 4 } \\ . \\end{align*}"}
+{"id": "8135.png", "formula": "\\begin{align*} \\mathcal { R } _ 1 ^ - = \\frac { 1 } { 2 } \\sum _ { m \\geq 1 } \\sum _ { n \\geq 1 } \\frac { A ( n , m ) } { ( m ^ 2 n ) ^ { \\frac { 1 } { 2 } } } g \\Bigl ( \\frac { m ^ 2 n } { N } \\Bigr ) \\sum _ { c \\geq \\frac { C } { m } } \\frac { S ( - n , p ; c ) } { c } H _ { m , n } ^ - \\Bigl ( \\frac { 4 \\pi \\sqrt { n p } } { c } \\Bigr ) \\end{align*}"}
+{"id": "1731.png", "formula": "\\begin{align*} \\varphi _ n ( \\mu ) ( \\alpha , \\beta ) : = \\mu \\left ( \\left | \\begin{array} { c c } \\alpha & \\beta \\\\ x & y \\end{array} \\right | ^ { n + \\lambda } \\left | \\begin{array} { c c } - \\bar \\beta & \\bar \\alpha \\\\ x & y \\end{array} \\right | ^ { n - \\lambda } \\right ) . \\end{align*}"}
+{"id": "6983.png", "formula": "\\begin{align*} H _ { 2 , n } ^ { ( \\{ 1 , 1 \\} ) } ( x ) = 1 6 ( n - 1 ) \\big ( { - } 2 x H _ { n - 1 } ( x ) + \\big ( n \\big ( 2 x ^ 2 + 1 \\big ) - 2 \\big ) H _ { n - 2 } ( x ) \\big ) \\end{align*}"}
+{"id": "7908.png", "formula": "\\begin{align*} \\mathcal { E } _ \\infty = \\sum _ { j \\in \\N } \\sum _ { d \\in \\mathcal { D } _ j } \\delta _ { \\xi _ j + d + \\frac { 1 } { \\sqrt { 2 } } \\log Z _ \\infty } , \\end{align*}"}
+{"id": "1729.png", "formula": "\\begin{align*} I _ { n _ 1 , n _ 2 , m _ 1 , m _ 2 } = 2 \\int _ { S ^ 1 } \\int _ { S ^ 1 } \\int _ 0 ^ { \\pi / 2 } \\sin \\theta ^ { 1 + m _ 1 + m _ 2 } \\cos \\theta ^ { 1 + n _ 1 + n _ 2 } e ^ { ( n _ 1 - n _ 2 ) i \\alpha } e ^ { ( m _ 1 - m _ 2 ) i \\beta } d \\theta d \\alpha d \\beta , \\end{align*}"}
+{"id": "4769.png", "formula": "\\begin{align*} \\Pr _ { y \\sim \\mathcal { D } ( C ^ \\perp ) } \\big [ | y | = j \\big ] \\leq \\big ( 1 + o ( N ^ { - 1 } ) \\big ) \\frac { \\binom { N } { j } } { 2 ^ N } , \\end{align*}"}
+{"id": "3816.png", "formula": "\\begin{align*} f = \\sum _ k \\phi _ k \\otimes \\psi _ k \\end{align*}"}
+{"id": "1137.png", "formula": "\\begin{align*} ( a \\cdot f ) ( m ) & = a f ( m ) \\mbox { , \\ a n d } \\\\ ( f \\cdot a ) ( m ) & = f ( a m ) . \\end{align*}"}
+{"id": "5178.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 0 } ^ { n } p _ { v _ i ^ 0 } \\right ) C ^ * ( \\overline { L } _ { 2 n + 1 } ^ { r ; \\underline { m } } ) \\left ( \\sum _ { i = 0 } ^ { n } p _ { v _ i ^ 0 } \\right ) \\cong C ^ * ( \\overline { L } _ { 2 n + 1 } ^ { r ; \\underline { m } } ) . \\end{align*}"}
+{"id": "541.png", "formula": "\\begin{align*} | { \\mathcal N } _ j ( d , k - 1 ) | \\leq \\sum _ { r = 0 } ^ { \\ell _ j } \\frac { | ( k - 1 ) { \\mathcal P } _ j ( d ) ) | ^ r } { r ! } \\leq | k { \\mathcal P } _ j ( d ) ) | ^ { \\ell _ j } \\sum _ { r = 0 } ^ { \\ell _ j } ( \\frac { 1 0 k } { \\ell _ j } ) ^ { \\ell _ j - r } \\frac { 1 } { r ! } \\leq ( \\frac { 1 2 k ^ 2 | { \\mathcal P } _ j ( d ) ) | } { \\ell _ j } ) ^ { \\ell _ j } . \\end{align*}"}
+{"id": "1996.png", "formula": "\\begin{align*} \\underbar { { \\bf s } } = ( s { ( 1 , 2 ) } , s { ( 2 , 3 ) } , \\ldots , s { ( r , r + 1 ) } , s { ( 1 , 3 ) } , s { ( 2 , 4 ) } & , \\ldots , s { ( r - 1 , r + 1 ) } , \\\\ & s { ( 1 , r ) } , s { ( 2 , r + 1 ) } , s { ( 1 , r + 1 ) } ) . \\end{align*}"}
+{"id": "5938.png", "formula": "\\begin{align*} A ^ T E _ r = \\sum _ { s = 1 } ^ p \\alpha _ { r s } E _ s , B ^ T E _ r = \\sum _ { s = 1 } ^ p \\beta _ { r s } E _ s . \\end{align*}"}
+{"id": "5254.png", "formula": "\\begin{align*} \\frac { d } { d t } g ( U , U ) = 2 g ( \\nabla _ { \\dot { \\alpha } } U , U ) = 2 g ( \\nu \\nabla _ { U } U + \\nu \\nabla _ { X } U , U ) . \\end{align*}"}
+{"id": "5144.png", "formula": "\\begin{align*} I _ { \\nu } ( z ) = \\sum _ { m = 0 } ^ { \\infty } \\frac { \\left ( \\frac { z } { 2 } \\right ) ^ { \\nu + 2 m } } { m ! \\Gamma ( \\nu + m + 1 ) } , \\mbox { } \\quad | \\mbox { a r g } ( z ) | < \\pi \\end{align*}"}
+{"id": "3073.png", "formula": "\\begin{align*} - C : D ^ 2 w = - A : D ^ 2 w + a M : D ^ 2 w = a - \\bar { a } + ( r - 1 ) a = r a - \\bar { a } . \\end{align*}"}
+{"id": "2669.png", "formula": "\\begin{align*} \\varphi ( T ) = { \\sf o r d } ( T ) - 1 . \\end{align*}"}
+{"id": "3690.png", "formula": "\\begin{align*} \\begin{aligned} E _ + ( H ) & \\coloneqq \\int _ 0 ^ 1 \\max _ { p \\in T ^ * M } H _ s ( p ) d s , E _ - ( H ) \\coloneqq - \\int _ 0 ^ 1 \\min _ { p \\in T ^ * M } H _ s ( p ) d s , \\\\ \\| H \\| _ { \\mathrm { o s c } } & \\coloneqq E _ + ( H ) + E _ - ( H ) = \\int _ 0 ^ 1 \\left ( \\max _ { p \\in T ^ * M } H _ s ( p ) - \\min _ { p \\in T ^ * M } H _ s ( p ) \\right ) d s . \\end{aligned} \\end{align*}"}
+{"id": "8708.png", "formula": "\\begin{align*} \\tilde s _ \\lambda ( x _ 1 , \\dots , x _ n ) = \\frac { \\det [ x _ i ^ { n - j + 1 } ( 1 - x _ i ^ { \\lambda _ j } ) ( 1 - x _ i ) ^ { - 1 } ] } { \\det [ x ^ { n - j } _ i ] } . \\end{align*}"}
+{"id": "1236.png", "formula": "\\begin{align*} X ^ c _ 1 ( y , p ) : = \\{ x \\in X : D _ y c ( x , y ) = p \\} . \\end{align*}"}
+{"id": "8722.png", "formula": "\\begin{align*} s _ { \\mu ; a } ^ { [ O R V ] } = \\det \\left [ h ^ { [ O R V ] } _ { \\mu _ i - i + j ; \\tau ^ { 1 - j } a } \\right ] , \\end{align*}"}
+{"id": "1630.png", "formula": "\\begin{align*} C _ { u _ { 0 } } : = 2 A ( L ) + 3 C ( \\| u _ { 0 } \\| _ { L _ { x } ^ { 1 } ( \\R ^ d ) } + M M _ { 1 } ^ { 3 } ) , \\end{align*}"}
+{"id": "1937.png", "formula": "\\begin{align*} \\hat m _ { \\hat \\sigma ( x ) } = \\left ( \\hat g _ { \\hat x } \\right ) _ * \\hat m _ { \\hat x } . \\end{align*}"}
+{"id": "4092.png", "formula": "\\begin{align*} R m ^ + ( X , Y , Z , W ) = & R m ( X , Y , Z , W ) + \\frac { 1 } { 2 } \\nabla _ X H ( Y , Z , W ) - \\frac { 1 } { 2 } \\nabla _ Y H ( X , Z , W ) \\\\ & - \\frac { 1 } { 4 } \\langle H ( X , W ) , H ( Y , Z ) \\rangle + \\frac { 1 } { 4 } \\langle H ( Y , W ) , H ( X , Z ) \\rangle , \\end{align*}"}
+{"id": "3503.png", "formula": "\\begin{align*} ( \\gamma _ A ) _ { i j } & : = \\# \\{ k \\in \\{ 1 , \\dots , \\lambda _ j \\} : A ( k , j ) = i \\} \\\\ & = \\# \\{ \\} , \\end{align*}"}
+{"id": "5065.png", "formula": "\\begin{align*} 0 = \\sum _ { j = 1 } ^ n Q _ j ( \\pi ( \\omega ) ) f _ j ( \\omega ) = \\sum _ { j = 1 } ^ n \\sum _ { s = 0 } ^ m q _ { j , s } l ^ { \\varepsilon ( \\omega ) s } f _ j ( \\omega ) \\end{align*}"}
+{"id": "6730.png", "formula": "\\begin{align*} A ( q ) = d ( q ) \\left | \\sum _ { \\substack { m \\leq x + a \\\\ m \\equiv a \\bmod q } } \\mu ( m ) \\right | ^ { 1 / 2 } B ( q ) = \\left | \\sum _ { \\substack { m \\leq x + a \\\\ m \\equiv a \\bmod q } } \\mu ( m ) \\right | ^ { 1 / 2 } . \\end{align*}"}
+{"id": "3982.png", "formula": "\\begin{align*} u _ 1 ( x _ \\theta ) = g ( x _ \\theta , y _ b , z _ b ) , \\\\ Y u _ 1 ( x _ \\theta ) = y _ b \\theta \\in ( 0 , 1 ] . \\end{align*}"}
+{"id": "5630.png", "formula": "\\begin{align*} V _ { 1 } ( t ) = \\int ^ { S ( t ) } _ { S ^ { * } } \\dfrac { F ( x , I ^ { * } ) - F ( S ^ { * } , I ^ { * } ) } { F ( x , I ^ { * } ) } d x + \\int ^ { E ( t ) } _ { E ^ { * } } \\dfrac { x - E ^ { * } } { x } d x + \\dfrac { m _ { 1 } } { \\sigma ^ { \\alpha } } \\left ( \\int ^ { I ( t ) } _ { I ^ { * } } \\dfrac { x - I ^ { * } } { x } d x \\right ) . \\end{align*}"}
+{"id": "9091.png", "formula": "\\begin{align*} \\mathfrak { p } _ { s + 1 } = [ \\mathfrak { p } _ s , \\mathfrak { n } ] + [ J \\mathfrak { p } _ s , \\mathfrak { n } ] \\subseteq [ \\mathfrak { d } _ s , \\mathfrak { n } ] \\subseteq \\mathfrak { d } _ { s + 1 } J \\mathfrak { p } _ { s + 1 } \\subseteq J [ \\mathfrak { d } _ s , \\mathfrak { n } ] \\subseteq \\mathfrak { d } _ { s + 1 } \\end{align*}"}
+{"id": "6821.png", "formula": "\\begin{align*} \\implies y _ { m + 1 } = y _ { m - 1 } + 2 h f . \\end{align*}"}
+{"id": "5764.png", "formula": "\\begin{align*} x \\mapsto ( x , f _ 1 ( x ) , \\ldots , f _ r ( x ) ) = ( x , x _ 1 , \\ldots , x _ { r _ 0 } , f ( x _ { r _ 0 + 1 } ) , \\ldots , f ( x _ r ) ) . \\end{align*}"}
+{"id": "5854.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } \\phi '' - \\Delta \\phi = - ( e , A U ) & \\hbox { i n } ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu \\phi + \\lambda \\phi = ( e , D H ) & \\hbox { o n } ( 0 , T ) \\times \\Gamma , \\\\ t = 0 : ~ \\phi = 0 , ~ \\phi ' = 0 & \\hbox { i n } \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "7637.png", "formula": "\\begin{align*} \\mathsf P _ h = - h \\ , e ^ { - \\frac 1 h U } \\ , \\mathsf L _ h e ^ { \\frac 1 h U } . \\end{align*}"}
+{"id": "657.png", "formula": "\\begin{align*} \\frac { \\partial \\rho } { \\partial t } + \\nabla \\cdot ( \\rho { \\bf v } ) = 0 , \\end{align*}"}
+{"id": "8032.png", "formula": "\\begin{align*} f _ 1 ( v ) = ( a _ 1 - x _ 1 ) ( y _ 2 - y _ 1 ) - ( x _ 2 - x _ 1 ) ( b _ 1 - y _ 1 ) = 0 . \\end{align*}"}
+{"id": "7286.png", "formula": "\\begin{align*} c ^ { d + 1 } _ m ( \\lambda ) = c ^ d _ m ( \\lambda ) - \\lambda ^ { d + 1 } \\int _ { 0 } ^ { 1 } G ^ d _ m ( x ) d m ( x ) . \\end{align*}"}
+{"id": "3534.png", "formula": "\\begin{align*} E ^ { e _ I } = E _ { i _ 2 i _ 1 } \\cdots E _ { i _ s i _ { s - 1 } } . \\end{align*}"}
+{"id": "6789.png", "formula": "\\begin{align*} d ( \\mathcal { U } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) , \\mathcal { U } ( ( \\overline { v } _ { i } ) _ { i = 1 } ^ { \\eta } ) ) \\leq ~ c K ^ { * } _ { \\eta } ( ( \\overline { w } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } , ( \\overline { v } _ { \\i } ) _ { \\i = 1 } ^ { \\eta } ) ) , \\end{align*}"}
+{"id": "6115.png", "formula": "\\begin{align*} \\left [ j ^ k \\frac { 1 } { n ^ h } \\right ] \\left ( \\frac { n ^ j } { j ! } \\right ) \\ln \\left ( \\sum _ { h \\geq 0 } \\dfrac { \\hat { a } _ h ( j , \\left \\{ \\epsilon _ t \\right \\} ) } { n ^ h } \\right ) = 0 k \\geq h + 2 . \\end{align*}"}
+{"id": "7969.png", "formula": "\\begin{align*} y + x + f ( q ) = f ( z ) + f ( q ) = f ( z + q ) = \\sum f ( x _ i ) , f ( q ) = \\sum f ( y _ i ) . \\end{align*}"}
+{"id": "590.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c } 1 & \\beta \\\\ - \\beta & 1 \\end{array} \\right ) , \\end{align*}"}
+{"id": "2049.png", "formula": "\\begin{align*} \\mathbb E [ ( \\xi _ 1 + \\ldots + \\xi _ { t _ 1 } ) ^ p \\mid X _ 0 ^ { ( 0 ) } = x < 0 ] & \\leq \\mathbb E [ ( { \\xi _ 1 ^ + } + \\ldots + { \\xi _ { t _ 1 } ^ + } ) ^ p \\mid X _ 0 ^ { ( 0 ) } = x < 0 ] \\\\ & \\leq \\mathbb E [ ( \\xi _ 1 ^ + ) ^ p + \\ldots + ( \\xi _ { t _ 1 } ^ + ) ^ p \\mid X _ 0 ^ { ( 0 ) } = x < 0 ] \\\\ & = \\mathbb E [ ( \\xi _ 1 ^ + ) ^ p ] \\ , \\mathbb E [ t _ 1 \\mid X _ 0 ^ { ( 0 ) } = x < 0 ] < + \\infty \\end{align*}"}
+{"id": "178.png", "formula": "\\begin{align*} 1 0 \\log 2 + \\dfrac { 1 } { 4 } \\sum _ { i = 1 } ^ r \\left ( \\dfrac { 3 p _ i - 2 - p _ i ^ { - e _ i } } { ( p _ i - 1 ) ^ 2 } - \\dfrac { e _ i + 3 } { p _ i ^ { e _ i } ( p _ i - 1 ) } + \\dfrac { 4 } { p _ i ^ { e _ i / 2 } ( p _ i - 1 ) } \\right ) \\log p _ i , \\end{align*}"}
+{"id": "325.png", "formula": "\\begin{align*} S _ M ( \\chi _ { 8 d } ( l ) B ( d ) ; \\Phi ) = P ( l ) + R ( l ) , \\end{align*}"}
+{"id": "201.png", "formula": "\\begin{align*} \\mu ( S _ { n } ) = ( c _ { n } r _ n + \\frac { 1 } { 2 } r _ { n } ( r _ { n } - 1 ) ) \\mu ( I _ { n + 1 } ) = \\left ( c _ n \\frac { r _ n } { r _ { n } + 1 } + \\frac { 1 } { 2 } \\frac { r _ { n } ( r _ { n } - 1 ) } { r _ { n } + 1 } \\right ) \\mu ( I _ { n } ) \\leq \\frac { c _ { n } + r _ { n } } { h _ { n } } ~ \\mu ( C _ { n } ) , \\end{align*}"}
+{"id": "6520.png", "formula": "\\begin{align*} \\left ( \\underset { i = 1 } { \\overset { b } { \\sum } } Z _ i ^ 2 \\leq \\frac { 2 5 k N _ \\alpha } { C _ \\alpha } b \\right ) \\leq \\exp \\left ( - b \\frac { \\frac { 2 5 k N _ \\alpha } { C _ \\alpha } - 1 - \\log \\left ( \\frac { 2 5 k N _ \\alpha } { C _ \\alpha } \\right ) } { 2 } \\right ) \\leq \\alpha / 8 , \\end{align*}"}
+{"id": "6366.png", "formula": "\\begin{align*} \\frac { d u } { d r } = \\frac { 1 } { l _ \\lambda \\sqrt { x _ \\lambda ' ( u ( r ) ) ^ 2 + y ' ( u ( r ) ) ^ 2 } } \\end{align*}"}
+{"id": "1664.png", "formula": "\\begin{align*} \\delta s : V ( k _ \\sigma - 2 ) ( \\lambda _ \\sigma ) \\longrightarrow D ( k _ \\sigma ) , \\delta s ( \\mu ) = \\imath _ { \\lambda _ \\sigma } ^ { - 1 } ( \\delta ^ T _ \\sigma ( s ( \\mu ) ) - s ( \\delta ^ T _ \\sigma \\mu ) ) . \\end{align*}"}
+{"id": "4835.png", "formula": "\\begin{align*} ( r , t ) \\mapsto \\Psi ( r , t ) = F ( \\Phi ( r , \\varrho ( t ) ) ) = d _ 0 F ( r ) + D _ 0 ^ n F ( w ( \\varrho ( t ) ) , 0 ) + R ( r , \\varrho ( t ) ) \\end{align*}"}
+{"id": "5706.png", "formula": "\\begin{align*} \\mathbf { A } _ { b } ^ { a } & = \\alpha _ { b } ^ { a } - ( \\digamma _ { b } ^ { a } - \\mathfrak { f } _ { b } ^ { a } ) \\mathbf { m , } \\\\ \\mathbf { F } _ { b } ^ { a } & = \\mathfrak { f } _ { b } ^ { a } + D \\mathfrak { f } _ { b } ^ { a } \\mathbf { m , } \\end{align*}"}
+{"id": "5013.png", "formula": "\\begin{align*} c _ 1 \\ , \\omega _ 1 ( Z , X , Y ) + c _ 2 \\ , \\omega _ 2 ( Z , X , Y ) = - d \\xi ( Z , X , Y ) = \\xi ( [ X , Y ] , Z ) . \\end{align*}"}
+{"id": "4921.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { H } } ( \\mathbf { Z } ) = \\Delta x \\sum _ m \\widetilde H ( \\mathbf { Z } ) _ m . \\end{align*}"}
+{"id": "9034.png", "formula": "\\begin{align*} \\overline { V } ^ { ( n ) } = \\overline { \\Psi } ( \\overline { V } ^ { n - 1 } ) = \\underline { \\Psi } ( \\overline { V } ^ { n - 1 } ) = \\underline { \\Psi } ( \\underline { V } ^ { n - 1 } ) = \\underline { V } ^ { ( n ) } . \\end{align*}"}
+{"id": "824.png", "formula": "\\begin{align*} \\left \\{ \\xi \\in \\partial \\Gamma : \\frac { \\log \\| \\rho ( \\xi _ n ) \\| - \\Lambda n } { \\sqrt { n } } \\le x \\right \\} \\end{align*}"}
+{"id": "4759.png", "formula": "\\begin{align*} a \\left ( m ( u ) \\right ) \\int _ { \\Omega } h ( x , | \\nabla u | ) \\ \\nabla u \\ \\nabla v = \\int _ { \\Omega } f ( x , u ) v \\end{align*}"}
+{"id": "5251.png", "formula": "\\begin{align*} \\begin{array} { l l } g ( U ( t ) , U ( t ) ) g ( \\dot { \\alpha } ( t ) , ( \\nabla f ) _ { \\alpha ( t ) } ) \\\\ + \\frac { \\lambda ^ 2 } { 2 } g \\left ( \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } , U ( t ) \\right ) g ( X ( t ) , X ( t ) ) + g ( ( T _ U U ) ( t ) , X ( t ) ) = 0 , \\end{array} \\end{align*}"}
+{"id": "4946.png", "formula": "\\begin{align*} \\delta _ n ^ + \\widetilde { G } _ \\ell + \\delta _ m ^ + \\widetilde { F } _ \\ell = \\widetilde { C } _ \\ell \\widetilde { A } ^ { D V D } , \\ell = 1 , 2 , \\end{align*}"}
+{"id": "246.png", "formula": "\\begin{align*} \\operatorname { R e } \\psi _ 1 ( i \\rho , i \\tau , \\xi + i \\eta ) = - ( 1 - \\alpha ) \\rho ^ 2 - \\beta \\tau \\eta \\leqslant 0 , \\end{align*}"}
+{"id": "7615.png", "formula": "\\begin{align*} I : = \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { 1 } | h _ n ( u _ n ( t , x ) ) - h ( u ( t , x ) ) | \\d x \\d t \\to 0 , \\ \\P , \\ \\ n \\to \\infty . \\end{align*}"}
+{"id": "5479.png", "formula": "\\begin{align*} & d ( ( x , i ) , ( y , j ) ) : = \\sqrt { \\mathcal \\mathbb 1 _ { i \\neq j } + \\hat V ( x - y ) } , ~ x , y \\in \\mathbb R ^ d , ~ i , j \\in \\mathcal M , \\\\ & \\mathcal P _ d : = \\left \\{ \\mu \\in \\mathcal P ( \\mathbb R ^ d \\times \\mathcal M ) : \\int _ { \\mathbb R ^ d \\times \\mathcal M } d ( ( x , i ) , ( 0 , 1 ) ) \\mu ( d x \\times \\{ i \\} ) < \\infty \\right \\} . \\end{align*}"}
+{"id": "4528.png", "formula": "\\begin{align*} I _ 3 & = ( 2 / 3 , 7 / 1 0 ] \\\\ I _ 4 & = ( 7 / 1 2 , 1 3 / 2 2 ] \\\\ I _ 5 & = ( 8 / 1 5 , 3 1 / 5 8 ] . \\end{align*}"}
+{"id": "4926.png", "formula": "\\begin{align*} \\Delta t u ' ( t + \\tfrac { \\Delta t } 2 ) = & \\ , { \\Delta t } u ' ( t ) + \\mathcal { O } ( \\Delta t ^ 2 ) , \\\\ \\Delta t u ' ( t + \\tfrac { \\Delta t } 2 ) = & \\ , { \\Delta t } u ' ( t + \\Delta t ) + \\mathcal { O } ( \\Delta t ^ 2 ) , \\end{align*}"}
+{"id": "3483.png", "formula": "\\begin{align*} { \\bf { h } } _ l ^ H { { \\bf { f } } _ { l ' } } = 0 , \\ \\forall l \\ne l ' , \\end{align*}"}
+{"id": "4767.png", "formula": "\\begin{align*} \\Delta ( L ( m n - 1 , n ) , \\omega ) & = - \\frac { d ( u ) d ( v ) } { 4 ( - 1 ) ^ { \\sigma _ + ( L ) } } ( u ^ 2 v ^ { - 2 n } + u ^ { - 2 } v ^ { 2 n } ) ( u ^ { 2 m } v ^ { - 2 } + u ^ { - 2 m } v ^ 2 ) \\\\ & = - \\frac { d ( u ) d ( v ) } { 4 ( - 1 ) ^ { \\sigma _ + ( L ) } } ( ( u v ^ { - n } ) ^ 2 + ( u ^ { - 1 } v ^ { n } ) ^ 2 ) ( ( u ^ { m } v ^ { - 1 } ) ^ { 2 } + ( u ^ { - m } v ) ^ 2 ) \\\\ & = ( - 1 ) ^ { \\sigma _ + ( L ) + 1 } d ( u ) d ( v ) . \\end{align*}"}
+{"id": "6909.png", "formula": "\\begin{align*} \\dfrac { I ( t ) - I ( 0 ) } { t } & = \\frac { 1 } { t } \\int _ 0 ^ t ( \\left ( \\beta I ( s ) S ( s ) - ( \\rho + \\phi + \\mu ) I ( s ) + \\alpha A ( s ) + \\omega C ( s ) \\right ) d s \\\\ & + \\frac { 1 } { t } \\int _ 0 ^ t \\sigma I S d W _ s + \\frac { 1 } { t } \\int _ 0 ^ t \\int _ U J ( u ) I S \\check { N } ( d s , d u ) \\\\ & \\geq - ( \\rho + \\phi + \\mu ) < I ( t ) > + \\frac { 1 } { t } \\int _ 0 ^ t \\sigma I S d W _ s + \\frac { 1 } { t } \\int _ 0 ^ t \\int _ U J ( u ) I S \\check { N } ( d s , d u ) . \\end{align*}"}
+{"id": "566.png", "formula": "\\begin{align*} \\int _ { t _ n } ^ { t _ { n + 1 } } ( A X _ t ^ \\dagger ) ^ { \\rm T } { \\rm d } X _ t ^ { \\dagger } \\approx \\sum _ { l = 0 } ^ { L - 1 } ( A X _ { \\tau _ l } ^ \\dagger ) ^ { \\rm T } ( X _ { \\tau _ { l + 1 } } ^ { \\dagger } - X _ { \\tau _ l } ^ \\dagger ) \\end{align*}"}
+{"id": "8263.png", "formula": "\\begin{align*} I : = \\left [ Y , Y ^ { A _ { 1 0 } \\frac { \\log ( \\gamma _ 0 + 5 ) } { ( \\beta _ 0 - \\theta ) ^ 2 } } \\right ] , \\end{align*}"}
+{"id": "6184.png", "formula": "\\begin{align*} C _ p A = \\overline A _ p C _ p . \\end{align*}"}
+{"id": "7034.png", "formula": "\\begin{align*} F _ n D = \\left \\{ \\sum a _ i \\Pi _ { \\rho _ 1 + \\cdots + \\rho _ i } \\in D : a _ i \\in F _ { n + \\ell _ i } A \\right \\} , \\end{align*}"}
+{"id": "6190.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\overline d } ( r _ k - \\overline r _ k ) = p . \\end{align*}"}
+{"id": "4576.png", "formula": "\\begin{align*} | G | = | C ( x _ 1 ) | + \\cdots + | C ( x _ k ) | . \\end{align*}"}
+{"id": "4799.png", "formula": "\\begin{align*} \\max _ { j < \\beta N } \\left \\{ \\Pr _ { v \\sim \\mu _ t } [ | v H | = j ] \\cdot 2 ^ { 2 \\epsilon N \\log ( 1 - \\frac { 2 j } { N } ) } \\right \\} & \\leq 2 ^ { - \\frac { \\epsilon N } { \\ln 2 } } \\cdot 2 ^ { \\epsilon N \\log ( \\frac { 2 \\epsilon } { 1 - \\eta } ) } \\\\ & \\leq 2 ^ { - h ( \\epsilon ) N } \\cdot 2 ^ { \\epsilon N \\log ( \\frac { 2 } { 1 - \\eta } ) } , \\end{align*}"}
+{"id": "5740.png", "formula": "\\begin{align*} ( _ { c } \\mathbf { L } ^ { - 1 } ) _ { B } ^ { A } = [ \\delta _ { c } ^ { a } - ( L ^ { - 1 } ) _ { D } ^ { A } d L _ { C } ^ { D } \\mathbf { m } ] ( L ^ { - 1 } ) _ { B } ^ { C } , \\end{align*}"}
+{"id": "3382.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\log _ t ( t ^ a + t ^ b ) = \\max ( a , b ) , \\log _ t ( t ^ a \\times t ^ b ) \\equiv a + b \\enspace . \\end{align*}"}
+{"id": "7712.png", "formula": "\\begin{align*} t _ { 0 , 1 } ^ { ( 2 ) } = \\pm 2 \\left ( \\frac { \\lambda } { \\tau } \\right ) ^ { \\frac { 1 } { 2 } } i - 1 , \\end{align*}"}
+{"id": "5741.png", "formula": "\\begin{align*} \\mathbf { x } ^ { A A ^ { \\prime } } = \\xi ^ { A A ^ { \\prime } } - \\mu ^ { A A ^ { \\prime } } \\mathbf { m } - \\overline { \\mu } ^ { A A ^ { \\prime } } \\overline { \\mathbf { m } } + i \\nu ^ { A A ^ { \\prime } } \\mathbf { m } \\overline { \\mathbf { m } } , \\end{align*}"}
+{"id": "1497.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ { \\frac \\pi 2 } \\log \\left ( \\cos \\frac \\theta 2 \\right ) d \\theta & = \\frac \\pi 2 \\log \\frac 1 { \\sqrt 2 } - 2 \\pi \\log \\C _ 2 \\left ( \\frac 1 4 \\right ) \\\\ & = \\frac \\pi 2 \\log \\frac 1 { \\sqrt 2 } - \\log \\left ( \\prod _ { n = 1 , n } ^ \\infty \\left ( \\frac { 2 n - 1 } { 2 n + 1 } \\right ) ^ { \\frac n 2 } e ^ { \\frac 1 2 } \\right ) ^ { 2 \\pi } , \\end{aligned} \\end{align*}"}
+{"id": "9173.png", "formula": "\\begin{align*} \\langle e ^ { ( f _ { \\epsilon } , \\sigma ) } \\rangle ^ { \\Lambda _ N } _ { \\beta , m ^ 2 } = e ^ { \\frac { 1 } { 2 } ( f _ { \\epsilon } , \\tilde { C } ( s , m ^ 2 ) f _ { \\epsilon } ) } \\frac { \\tilde { Z } _ N ( u [ \\epsilon ] , 0 ) } { \\tilde { Z } _ N ( 0 , 0 ) } . \\end{align*}"}
+{"id": "8594.png", "formula": "\\begin{align*} [ n - r ] : = \\begin{cases} n - r & \\mbox { i f } n \\geq r , \\\\ n - r + 2 ^ n - 1 & \\mbox { i f } n < r . \\end{cases} \\end{align*}"}
+{"id": "1212.png", "formula": "\\begin{align*} ( \\tilde { U } \\times S ^ 1 \\times \\C ) / \\Gamma & \\to ( ( \\tilde { U } \\times S ^ 1 ) / \\Gamma ) \\times \\C , \\\\ [ ( \\tilde { u } , h , z ) ] & \\mapsto ( [ ( \\tilde { u } , h ) ] , h ^ { \\alpha } z ) . \\end{align*}"}
+{"id": "2415.png", "formula": "\\begin{align*} \\Phi ' = U \\left [ \\begin{array} { c } 0 \\\\ I _ { a } \\end{array} \\right ] \\end{align*}"}
+{"id": "1093.png", "formula": "\\begin{align*} \\omega ( \\varepsilon / 2 ) \\gtrsim \\begin{cases} \\varepsilon ^ { \\frac { 4 \\beta } { 2 \\beta + 1 } } , & \\mbox { f o r t h e s q u a r e d - $ L _ 2 $ - l o s s } , \\\\ \\varepsilon ^ { \\frac { 2 \\beta - 1 } { 2 \\beta + 1 } } , & \\mbox { f o r t h e $ L _ { \\infty } $ - l o s s } . \\end{cases} \\end{align*}"}
+{"id": "8067.png", "formula": "\\begin{align*} H _ { m , n } ^ + ( x ) = 2 i \\int _ { - \\infty } ^ \\infty J _ { 2 i t } ( x ) \\frac { k ( t ) V ( m ^ 2 n , t ) t } { \\cosh ( \\pi t ) } \\ , d t , \\end{align*}"}
+{"id": "7394.png", "formula": "\\begin{align*} \\begin{cases} X _ { \\alpha } ( s ) & = \\omega ( \\varpi _ F ) q _ F ^ { - 2 s } \\\\ X _ { \\alpha } ( s ) & = - \\chi ^ 2 \\chi '^ { - 1 } ( \\varpi _ L ) q _ L ^ { - s } , \\end{cases} \\end{align*}"}
+{"id": "1447.png", "formula": "\\begin{align*} \\tau : ( R _ i , C _ j ) \\to ( R _ j , C _ i ) \\mbox { f o r } i , j = 1 , \\dots , m . \\end{align*}"}
+{"id": "7965.png", "formula": "\\begin{align*} \\beta ( x ) \\vee 1 _ L = i ( m ) \\vee 1 _ L . \\end{align*}"}
+{"id": "8714.png", "formula": "\\begin{align*} ( x | a ) _ { n } & = \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { n - k } e _ { n - k } ( a _ 1 , \\dots , a _ { n } ) \\ , x ^ k , \\\\ \\frac { 1 } { ( x | a ) _ { n } } & = \\sum _ { k = n } ^ { \\infty } h _ { k - n } ( a _ 1 , \\dots , a _ { n } ) \\ , x ^ { - k } , \\\\ x ^ n & = \\sum _ { k = 0 } ^ { n } h _ { n - k } ( a _ 1 , \\dots , a _ { k + 1 } ) \\ , ( x | a ) _ k , \\\\ x ^ { - n } & = \\sum _ { k = n } ^ { \\infty } ( - 1 ) ^ { n - k } e _ { k - n } ( a _ 1 , \\dots , a _ { k - 1 } ) \\frac { 1 } { ( x | a ) _ { k } } . \\end{align*}"}
+{"id": "8082.png", "formula": "\\begin{align*} & \\frac { A ( 1 , p ) } { 2 p ^ { \\frac { 1 } { 2 } } } \\sum _ { m \\geq 1 } \\frac { A ( m , 1 ) } { m } H _ { m , p } - \\frac { 1 } { 2 p ^ { \\frac { 3 } { 2 } } } \\sum _ { m \\geq 1 } \\frac { A ( m , 1 ) } { m } H _ { m p , p } \\\\ & \\qquad = \\frac { L ( 1 , f ) \\bigl ( A ( 1 , p ) p - 1 \\bigr ) } { p ^ { \\frac { 3 } { 2 } } \\pi } \\int _ 0 ^ { \\infty } k ( t ) \\tanh ( \\pi t ) t \\ , d t + O ( M T ^ { \\frac { 1 } { 7 } + \\varepsilon } p ^ { \\varepsilon } ) \\end{align*}"}
+{"id": "5508.png", "formula": "\\begin{align*} e ^ { - y } = { 1 \\over 2 \\pi i } \\int _ { 2 - i \\infty } ^ { 2 + i \\infty } y ^ { - s } \\Gamma ( s ) \\mathrm d s \\end{align*}"}
+{"id": "4439.png", "formula": "\\begin{align*} a _ 2 = \\left ( \\frac { 1 } { 3 } + \\frac { 1 } { 4 } - \\frac { 1 } { 2 } \\right ) ^ { - 1 } = 1 2 \\end{align*}"}
+{"id": "6421.png", "formula": "\\begin{align*} \\phi ( X _ 1 , \\tilde Y - ( \\tilde t - t ) X _ 1 , \\tilde t ) & = \\min _ { \\tilde X \\in \\overline { { B _ \\epsilon ( X ) } } } \\phi ( \\tilde X , \\tilde Y - ( \\tilde t - t ) \\tilde X , \\tilde t ) = \\inf _ { \\tilde X \\in { { B _ \\epsilon ( X ) } } } \\phi ( \\tilde X , \\tilde Y - ( \\tilde t - t ) \\tilde X , \\tilde t ) . \\end{align*}"}
+{"id": "4710.png", "formula": "\\begin{align*} \\vert \\lbrace i , j , k \\vert p _ { i j } ^ k \\neq 0 \\rbrace \\vert \\leq \\dim T \\leq \\sum _ { i = 0 } ^ d \\frac { | G | } { | C _ i | } . \\end{align*}"}
+{"id": "3958.png", "formula": "\\begin{align*} a ^ + _ t : = \\sup \\{ x _ 1 = x \\cdot e _ 1 ; x \\in D _ t \\} \\rightarrow 0 . \\end{align*}"}
+{"id": "3080.png", "formula": "\\begin{align*} - A : D ^ 2 w = a - \\bar { a } \\quad Y , w Y , \\int _ Y w = 0 . \\end{align*}"}
+{"id": "4794.png", "formula": "\\begin{align*} \\Pr _ { v \\sim \\mu _ t } \\big [ | v H | = j \\big ] \\leq 2 ^ { - ( 1 - h ( \\frac { j } { N } ) ) ( 1 - \\eta ) N } . \\end{align*}"}
+{"id": "2574.png", "formula": "\\begin{align*} \\mathfrak { g } = \\sum _ { \\epsilon \\in U ( 1 ) _ { \\geq 0 } } \\mathfrak { g } _ { 0 , \\epsilon } + \\sum _ { \\alpha \\in \\Delta ^ + } \\sum _ { \\epsilon \\in U ( 1 ) } \\mathfrak { g } _ { \\alpha , \\epsilon } , \\end{align*}"}
+{"id": "1423.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( | \\mathcal C _ { \\max } | > A n ^ { 2 / 3 } \\right ) = O ( A ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "8149.png", "formula": "\\begin{align*} & \\sum _ { n \\geq 1 } A ( n , m ) e \\Bigl ( \\frac { n d } { c } \\Bigr ) \\psi ( n ) \\\\ & = c \\sum _ { \\pm } \\sum _ { n _ 1 \\mid c m } \\sum _ { n _ 2 \\geq 1 } \\frac { A ( n _ 1 , n _ 2 ) } { n _ 1 n _ 2 } S ( m \\bar d , \\pm n _ 2 ; m c n _ 1 ^ { - 1 } ) \\Psi ^ { \\pm } \\Bigl ( \\frac { n _ 2 n _ 1 ^ 2 } { c ^ 3 m } \\Bigr ) \\end{align*}"}
+{"id": "3048.png", "formula": "\\begin{align*} \\nu _ n ( x ) : = \\sum _ { k \\geq 1 } k \\sum _ { 0 \\leq i < j \\leq n - 1 } 1 _ { V _ k ^ { ( T ^ j ( x ) ) } } \\circ T ^ i ( x ) + \\sum _ { i = 0 } ^ { n - 1 } \\nu _ 1 ( T ^ i x ) , \\end{align*}"}
+{"id": "8356.png", "formula": "\\begin{align*} ( \\Box + V _ 1 ) v = 0 , V _ 1 = ( | y | ^ 2 - s ^ 2 ) _ + ^ \\frac { 2 } { d - 2 } | v | ^ { \\frac { 4 } { d - 2 } } . \\end{align*}"}
+{"id": "3617.png", "formula": "\\begin{align*} 0 \\ = \\ \\pi P - \\frac { 1 2 \\pi } { 6 } C _ 0 L _ * - 6 \\pi P + \\frac { 6 \\pi } { 7 } \\lambda _ 0 . \\end{align*}"}
+{"id": "5238.png", "formula": "\\begin{align*} ( m - n ) H = \\sum \\limits _ { i = n + 1 } ^ { m } T _ { U _ i } U _ i , \\end{align*}"}
+{"id": "1279.png", "formula": "\\begin{align*} \\eta _ { j } ^ \\ast ( y ) = \\left \\{ \\begin{array} { l l l } c _ { j } ~ \\eta _ { \\ell - 2 - j } ( y ) & { \\rm i f } & \\beta = 1 \\\\ c _ { j } ~ \\eta _ { \\ell - 1 - j } ( y ) & { \\rm i f } & \\beta = - 1 \\end{array} \\right . \\end{align*}"}
+{"id": "8748.png", "formula": "\\begin{align*} \\bar { \\mathcal { O } } _ { 1 : r ( T - 1 ) , 1 : d _ 0 } = \\bar { \\mathcal { O } } ^ + . \\end{align*}"}
+{"id": "711.png", "formula": "\\begin{align*} i ( \\xi ) \\Omega = d \\mathcal { H } . \\end{align*}"}
+{"id": "4317.png", "formula": "\\begin{align*} \\omega _ i = d z _ i - ( a _ i . d x + b _ i . d y ) & = 0 i , \\end{align*}"}
+{"id": "6823.png", "formula": "\\begin{align*} \\mu ( Y , \\hat Y ) = \\mu _ 1 ( Y , \\hat Y ) + \\mu _ 2 ( Y , \\hat Y ) , \\ , \\mu _ 1 ( Y , \\hat Y ) = { \\rm { r a n k } } \\mathcal M , \\mu _ 2 ( Y , \\hat Y ) = { \\rm { i n d } } \\mathcal P , \\end{align*}"}
+{"id": "4684.png", "formula": "\\begin{align*} \\mathfrak { R } ^ { G _ { I V } } = \\mathbb { C } [ \\varphi _ 2 , \\varphi _ 6 ] \\end{align*}"}
+{"id": "6700.png", "formula": "\\begin{align*} \\mu ( n ) = \\left \\{ \\begin{array} { l l } ( - 1 ) ^ { \\omega ( n ) } & n = p _ 1 p _ 2 \\cdots p _ v \\\\ 0 & n \\ne p _ 1 p _ 2 \\cdots p _ v , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "8246.png", "formula": "\\begin{align*} y _ 1 + \\dots + y _ k = \\sum _ { t = 1 } ^ m a _ t z _ t \\end{align*}"}