formulas_finally.jsonl

{"id": "224.png", "formula": "\\begin{align} \\begin{split} K ' _ 1 \\ & = \\ ( \\bigcup _ { k > i + 1 } \\ \\{ k - , k + \\} \\times \\Z [ 2 ] ) \\cup \\{ \\infty \\} , \\\\ K ' _ 2 \\ & = \\ \\{ 1 - , 2 - , \\dots , ( i - 1 ) - , ( i - , \\bar d ) , ( i + 1 ) - , \\\\ & ( ( i + 2 ) - , a ) , ( ( i + 3 ) - , a ) , \\dots , \\infty \\} . \\\\ K ' _ 3 \\ & = \\ \\{ ( i - , \\bar d ) \\} \\cup \\{ ( i + , k ) : j - 1 \\le k \\le \\infty \\} \\cup ( \\bigcup _ { k < j - 1 } \\ \\{ i + \\} \\times \\{ z + A _ k \\} ) . \\end{split} \\end{align}"}
{"id": "154.png", "formula": "\\begin{equation} \\int _ { 3 U _ 1 \\times 3 V _ 1 } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y = 3 ^ 4 \\cdot \\int _ { U _ 1 \\times V _ 1 } J \\bigg ( \\frac { x - y } { ( t / 3 ) } \\bigg ) \\ d x d y . \\end{equation}"}
{"id": "386.png", "formula": "\\begin{equation} \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } p ] { p a b K _ p } = \\begin{cases} 2 ^ d p ^ e r ' s \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } p ] { K _ p } , & \\textnormal { f o r F L T 2 C a s e I } , \\\\ 2 ^ d p ^ e r _ 0 s ' \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } p ] { K _ p } , & \\textnormal { f o r F L T 2 C a s e I I } . \\end{cases} \\end{equation}"}
{"id": "230.png", "formula": "\\begin{equation} \\frac { d x ^ i } { d t } = X ^ i ( x ^ 1 , \\ldots , x ^ n ) , i = 1 , \\ldots , n , \\end{equation}"}
{"id": "15.png", "formula": "\\begin{equation} d m | _ 0 ( g ) = R _ 0 ( \\kappa ) C _ + g . \\end{equation}"}
{"id": "162.png", "formula": "\\begin{equation} \\rho _ k ( x _ 1 , \\ldots , x _ k ) = \\det { \\big ( K ( x _ i , x _ j ) \\big ) } _ { 1 \\leq i , j \\leq k } . \\end{equation}"}
{"id": "304.png", "formula": "\\begin{equation} \\dot { z } _ l = F _ l ( z ) = \\sum _ { k = 1 } ^ r a _ { l , k } \\ , z ^ { \\alpha ( k ) } , l = 1 , \\dots , n . \\end{equation}"}
{"id": "372.png", "formula": "\\begin{equation} ( x - a ) ^ n = f _ n ( x , b , a ) + g _ n ( b , a ) , \\end{equation}"}
{"id": "218.png", "formula": "\\begin{equation} ( x , x ' ) \\in \\widehat { A } ^ { \\circ } \\ \\ \\text { i f } \\ \\ x _ { i + 2 } = x ' _ { i + 2 } \\ \\ \\text { a n d } \\ \\ ( x ' , x ) \\in \\widehat { A } ^ { \\circ } \\ \\ \\text { o t h e r w i s e } . \\end{equation}"}
{"id": "182.png", "formula": "\\begin{align*} i _ { c , d } ( x ) \\triangleright j _ { a , b } ( b _ { i } ) & = j _ { a - k , b + k } ( 1 _ { X ^ { \\otimes c - a + k } } \\otimes x \\triangleright 1 _ { X ^ k } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } ) \\\\ & = j _ { a - k , b + k } ( 1 _ { X ^ { \\otimes c - a + k } } \\otimes x \\triangleright 1 _ { X ^ { d - b } } \\otimes b _ { i } \\otimes 1 _ { X ^ { d - b - c } } ) \\\\ & = ( 1 _ { X ^ { k + b - a } } \\otimes c ^ { * } _ { Z , X ^ { k } } ) \\circ ( 1 _ { X ^ { c - a + k } } \\otimes x ) \\circ ( 1 _ { X ^ { k + b - a } } \\otimes c _ { Z , X ^ { k } } ) \\circ ( 1 _ { X ^ { k } } \\otimes b _ i \\otimes 1 _ { X ^ { k } } ) \\\\ & = j _ { a - k , b + k } ( ( 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } ) \\circ ( x \\otimes 1 _ { X ^ { c - a + k } } ) ) \\\\ & = j _ { a , b } ( b _ { i } ) \\triangleleft i _ { c , d } ( x ) \\end{align*}"}
{"id": "41.png", "formula": "\\begin{equation} \\tilde d _ j ( \\sum _ { I \\in P ^ { j , n } _ + } \\omega _ I d x ^ I ) = \\sum _ { I \\in P ^ { j , n } _ + } ( \\tilde d _ 0 \\omega _ I ) \\wedge d x ^ I . \\end{equation}"}
{"id": "89.png", "formula": "\\begin{align*} \\begin{cases} c _ 1 = w _ 1 , & c _ 2 = 0 , \\\\ c _ 3 = w _ 2 , & c _ 4 = - w _ 2 \\gamma \\theta . \\end{cases} \\end{align*}"}
{"id": "235.png", "formula": "\\begin{equation} \\left \\{ \\begin{array} { r c l } \\dfrac { d q ^ i } { d t } & = & v ^ i \\\\ \\dfrac { d v ^ i } { d t } & = & f ^ i ( q , v ) \\end{array} \\right . i = 1 , \\ldots , n , \\end{equation}"}
{"id": "70.png", "formula": "\\[ \\begin{split} \\Sigma = \\{ \\sigma \\in C ^ { 1 } ( \\R ) \\ , : \\ , \\sigma \\text { i s e v e n } , \\ , \\sigma ( 0 ) = \\sigma ' ( 0 ) = 0 , \\text { a n d } \\sigma ( r ) > 0 \\text { f o r a l l } r \\neq 0 \\} . \\end{split} \\]"}
{"id": "352.png", "formula": "\\begin{equation} \\begin{aligned} I ^ { [ k ] } \\ & = \\ I _ 0 ^ { [ k ] } + \\sum _ { \\substack { p \\in [ s ] \\\\ \\nu ( I _ p ) \\ge k - 1 } } u _ p I _ p ^ { [ k - 1 ] } , & & & I ^ { [ \\ell ] } \\ & = \\ I _ 0 ^ { [ \\ell ] } + \\sum _ { \\substack { q \\in [ s ] \\\\ \\nu ( I _ q ) \\ge \\ell - 1 } } u _ q I _ q ^ { [ \\ell - 1 ] } . \\end{aligned} \\end{equation}"}
{"id": "138.png", "formula": "\\begin{equation} \\widehat { \\vec { u } } _ { \\infty } = 2 | \\xi | ^ { - 2 } ( I - \\Pi _ \\xi ) \\widehat { \\vec { f } } , \\widehat { p } = | \\xi | ^ { - 2 } \\xi ^ * \\widehat { \\vec { f } } . \\end{equation}"}
{"id": "405.png", "formula": "\\[ M _ { 2 } ( \\xi ; a ) : = \\bigcap \\{ M _ { 2 } \\left ( M _ { 2 } ( \\xi ; b ) \\cap M _ { 3 } ( \\nu ) \\right ) : \\nu < \\xi , b < a \\} . \\]"}
{"id": "190.png", "formula": "\\begin{equation} R \\cap R ^ { - 1 } \\ = \\ 1 _ X , \\text { a n d } R \\cup R ^ { - 1 } \\ = \\ X \\times X . \\end{equation}"}
{"id": "153.png", "formula": "\\begin{equation} \\begin{split} \\int _ { \\Omega \\times \\Omega ^ c } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y = \\sum _ { i = 1 } ^ { 1 2 } \\int _ { U _ i \\times V _ i } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y + \\sum _ { i \\neq j } \\int _ { U _ i \\times V _ j } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y . \\end{split} \\end{equation}"}
{"id": "381.png", "formula": "\\[ g _ p ( b , a ) = - p a b \\sum _ { i = 1 } ^ { p - 1 } \\dfrac { ( - 1 ) ^ i } { i } \\binom { p - 1 } { i - 1 } b ^ { p - i - 1 } a ^ { i - 1 } , \\]"}
{"id": "139.png", "formula": "\\begin{equation} \\langle A _ \\lambda ^ h \\vec { u } _ h , \\vec { v } _ h \\rangle = a _ \\lambda ( \\vec { u } _ h , \\vec { v } _ h ) = \\langle \\vec { f } , \\vec { v } _ h \\rangle \\quad \\forall \\vec { v } \\in V _ h . \\end{equation}"}
{"id": "101.png", "formula": "\\begin{align} u _ t - | D u | ^ { \\gamma } \\big ( \\Delta u + ( p - 2 ) \\Delta _ \\infty ^ N u \\big ) = 0 . \\end{align}"}
{"id": "99.png", "formula": "\\begin{align} c _ 3 = 2 + 2 \\kappa \\geq 2 > 0 . \\end{align}"}
{"id": "229.png", "formula": "\\begin{equation} d t = r \\ , d \\tau \\Longleftrightarrow \\frac { d \\tau } { d t } = \\frac 1 r , \\end{equation}"}
{"id": "174.png", "formula": "\\begin{align*} u ^ { F , G } _ { X , Y } ( b ^ { \\prime } _ { i } \\boxtimes c ^ { \\prime } _ { j } ) & = u ^ { F , G } _ { X , Y } \\left ( \\sum _ { l , k } b _ { l } \\langle b _ { l } \\ | \\ b ^ { \\prime } _ { i } \\rangle \\boxtimes c _ { k } \\langle c _ { k } \\ | \\ c ^ { \\prime } _ { j } \\rangle \\right ) \\\\ & = u ^ { F , G } _ { X , Y } \\left ( \\sum _ { l , k } b _ { l } \\boxtimes c _ { k } \\ \\langle b _ { l } \\ | \\ b ^ { \\prime } _ { i } \\rangle \\langle c _ { k } \\ | \\ c ^ { \\prime } _ { j } \\rangle \\right ) \\\\ & = \\sum _ { l , k } c _ { k } \\boxtimes b _ { l } \\ \\langle b _ { l } \\ | \\ b ^ { \\prime } _ { i } \\rangle \\langle c _ { k } \\ | \\ c ^ { \\prime } _ { j } \\rangle \\\\ & = c ^ { \\prime } _ { j } \\boxtimes b ^ { \\prime } _ { i } . \\\\ \\end{align*}"}
{"id": "420.png", "formula": "\\begin{equation} \\bold { \\Sigma } = \\bold { d } \\widetilde { \\bold { d } } ^ { T } , \\end{equation}"}
{"id": "140.png", "formula": "\\begin{equation} \\langle A _ \\lambda ^ h \\vec { u } _ h , \\vec { v } _ h \\rangle = a ( \\vec { u } _ h , \\vec { v } _ h ) + \\lambda b ( \\vec { v } _ h , \\Pi _ h { \\rm d i v } \\vec { u } _ h ) = \\langle \\vec { f } , \\vec { v } _ h \\rangle \\quad \\forall \\vec { v } \\in V _ h , \\end{equation}"}
{"id": "65.png", "formula": "\\begin{align} u _ t - | D u | ^ { \\gamma } \\big ( \\Delta u + ( p - 2 ) \\Delta _ \\infty ^ N u \\big ) = 0 \\end{align}"}
{"id": "53.png", "formula": "\\begin{align} ( F H _ h F ^ * - z ) ^ { - 1 } ( \\xi ) & = \\frac { 1 } { r _ z ( \\xi ) } F H _ h F ^ * + \\frac { z } { r _ z ( \\xi ) } \\end{align}"}
{"id": "136.png", "formula": "\\begin{equation} \\widehat { \\vec { u } } _ { \\lambda } = 2 | \\xi | ^ { - 2 } \\left ( I - \\frac { 2 \\lambda + 1 } { 2 ( \\lambda + 1 ) } \\Pi _ \\xi \\right ) \\widehat { \\vec { f } } , \\widehat { \\vec { u } } _ { 0 } = 2 | \\xi | ^ { - 2 } \\left ( I - \\frac { 1 } { 2 } \\Pi _ \\xi \\right ) \\widehat { \\vec { f } } . \\end{equation}"}
{"id": "318.png", "formula": "\\begin{equation} u _ t = \\Delta u ^ m + u ^ p , \\end{equation}"}
{"id": "317.png", "formula": "\\begin{equation} u ( x , 0 ) = u _ 0 ( x ) , x \\in \\real ^ N . \\end{equation}"}
{"id": "110.png", "formula": "\\begin{align} a = - \\frac { 1 } { 4 } ( G - P ) ^ 2 , b = P \\cdot E + \\frac { 1 } { 2 } ( G - P ) \\Big ( \\frac { G } { n - 1 } + K - \\frac { ( n - 2 ) P } { n - 1 } \\Big ) \\nonumber \\end{align}"}
{"id": "314.png", "formula": "\\begin{equation} Q _ { j k } = \\dfrac { K ^ 2 \\left | \\left \\langle L _ { F ' } e _ k ' , e _ j ' \\right \\rangle \\right | ^ 2 } { \\xi ^ 2 \\ , \\left | \\Re \\left ( \\left \\langle L _ { F ' } e _ j ' , e _ j ' \\right \\rangle \\right ) \\right | \\left | \\Re \\left ( \\left \\langle L _ { F ' } e _ k ' , e _ k ' \\right \\rangle \\right ) \\right | } j < k . \\end{equation}"}
{"id": "214.png", "formula": "\\begin{equation} E _ 1 \\ \\not = \\ E _ 2 \\Leftrightarrow F _ 1 \\ \\not = \\ F _ 2 \\Leftrightarrow [ ( E _ 1 \\cap F _ 2 ) \\cup ( E _ 2 \\cap F _ 1 ) ] \\ \\not = \\ \\emptyset . \\end{equation}"}
{"id": "211.png", "formula": "\\begin{equation} \\tilde \\pi ( 0 , x ) \\ = \\ x , \\text { a n d } \\tilde \\pi ( 1 , x ) \\ = \\ \\infty . \\end{equation}"}
{"id": "205.png", "formula": "\\begin{align} \\begin{split} Q \\ = \\ \\{ ( x , y , z ) : & ( x , z ) , ( z , y ) \\in R \\} \\ \\cup \\\\ & \\{ ( x , y , z ) : ( y , z ) , ( z , x ) \\in R \\} , \\\\ Q ^ { \\circ } \\ = \\ \\{ ( x , y , z ) : & ( x , z ) , ( z , y ) \\in R ^ { \\circ } \\} \\ \\cup \\\\ & \\{ ( x , y , z ) : ( y , z ) , ( z , x ) \\in R ^ { \\circ } \\} \\end{split} \\end{align}"}
{"id": "117.png", "formula": "\\begin{align*} a ( \\vec { u } , \\vec { v } ) & = ( \\varepsilon ( \\vec { u } ) , \\varepsilon ( \\vec { v } ) ) , \\\\ b ( \\vec { v } , q ) & = ( \\operatorname { d i v } \\vec { v } , q ) . \\end{align*}"}
{"id": "276.png", "formula": "\\begin{equation} \\frac { d ^ 2 y } { d \\tau ^ 2 } + \\alpha \\frac { d y } { d \\tau } + B y + C = 0 , \\end{equation}"}
{"id": "157.png", "formula": "\\begin{equation} 2 \\bigg [ F \\bigg ( \\frac { t } { ( 1 - ( 1 / \\eta ) ) / 2 } \\bigg ) \\cdot { \\bigg ( \\frac { 1 - ( 1 / \\eta ) } { 2 } \\bigg ) } ^ { 2 n } + F \\bigg ( \\frac { t } { ( 1 / \\eta ) } \\bigg ) \\cdot { \\bigg ( \\frac { 1 } { \\eta } \\bigg ) } ^ { 2 n } \\bigg ] = F ( t ) + R ( t ) , \\end{equation}"}
{"id": "321.png", "formula": "\\begin{equation} L : = \\sigma ( m - 1 ) + 2 ( p - 1 ) > 0 . \\end{equation}"}
{"id": "294.png", "formula": "\\begin{equation} e _ { 2 k + 1 } e _ { 2 l } = C _ { k + l } ^ { k } e _ { 2 k + 2 l + 1 } . \\end{equation}"}
{"id": "326.png", "formula": "\\begin{equation} u ( x , t ) = ( T - t ) ^ { - \\alpha } f ( | x | ( T - t ) ^ { \\beta } ) , \\ \\alpha = \\frac { \\sigma + 2 } { L } , \\ \\beta = \\frac { m - p } { L } \\end{equation}"}
{"id": "305.png", "formula": "\\begin{equation} \\dot { z } _ l = F _ l ( z ) = \\sum _ { k = 1 } ^ \\infty a _ { l , k } \\ , z ^ { \\alpha ( k ) } , l = 1 , \\dots , n , \\end{equation}"}
{"id": "263.png", "formula": "\\[ \\bar { A } ( \\bar { x } ) \\bar { v } + \\bar { b } ( \\bar { x } ) = \\frac { c } { h ( x ) } \\left ( A ( x ) v + b ( x ) \\right ) . \\]"}
{"id": "313.png", "formula": "\\begin{equation} \\begin{cases} b _ { j j } = ( 1 - \\xi ) \\\\ b _ { j k } = \\dfrac { \\xi } { 2 K } & \\textrm { i f } j \\neq k \\textrm { w i t h } \\left \\langle L _ { F ' } e _ k ' , e _ j ' \\right \\rangle \\neq 0 \\ , \\textrm { o r } \\left \\langle L _ { F ' } e _ j ' , e _ k ' \\right \\rangle \\neq 0 \\\\ b _ { j k } = 0 , & \\textrm { i f } j \\neq k \\textrm { w i t h } \\left \\langle L _ { F ' } e _ k ' , e _ j ' \\right \\rangle = 0 \\ , \\textrm { o r } \\left \\langle L _ { F ' } e _ j ' , e _ k ' \\right \\rangle = 0 , \\end{cases} \\end{equation}"}
{"id": "400.png", "formula": "\\begin{equation} \\begin{split} f _ 5 ( x , b , a ) & = 5 a b ( 4 x ^ 3 - 6 a x ^ 2 + 4 a ^ 2 x + 6 b x ^ 2 + 4 b ^ 2 x - 6 a b x ) , \\\\ g _ 5 ( b , a ) & = 5 a b ( b ^ 3 - 2 a b ^ 2 + 2 a ^ 2 b - a ^ 3 ) , \\end{split} \\end{equation}"}
{"id": "26.png", "formula": "\\begin{align*} q _ + ( t , z ) = \\lim _ { y \\to \\infty } \\tfrac { 1 } { 2 \\pi i } \\bigl \\langle U ( t ) ^ * \\chi _ y , \\big ( X - t \\kappa R ( \\kappa ; q ^ 0 ) ^ 2 - z \\big ) ^ { - 1 } q ^ 0 _ + \\bigr \\rangle . \\end{align*}"}
{"id": "390.png", "formula": "\\begin{equation} ( x - a ) ^ 2 = - ( b - a ) ^ 2 + b ^ 2 + a ^ 2 = 2 a b , \\end{equation}"}
{"id": "382.png", "formula": "\\[ ( x - a ) ^ p = f _ p ( x , b , a ) + g _ p ( b , a ) \\textnormal { a n d } K _ p = \\dfrac { f _ p ( x , b , a ) } { p a b } + \\dfrac { g _ p ( b , a ) } { p a b } . \\]"}
{"id": "241.png", "formula": "\\begin{equation} X ^ i ( x , v ) = \\sum _ { j = 1 } ^ n ( A ^ i \\ , _ j \\ , x ^ j + B ^ i \\ , _ j \\ , v ^ j ) , i = 1 , \\ldots , n . \\end{equation}"}
{"id": "356.png", "formula": "\\begin{equation} x ^ n + y ^ n = z ^ n \\end{equation}"}
{"id": "417.png", "formula": "\\begin{equation} \\begin{aligned} \\mathcal { E } _ { S } & = \\{ ( m _ x , m _ y ) \\in \\mathbb { Z } ^ { 2 } | ( \\frac { m _ x } { L _ { S , x } } ) ^ 2 + ( \\frac { m _ y } { L _ { S , y } } ) ^ 2 \\le 1 \\} , \\\\ \\mathcal { E } _ { R } & = \\{ ( l _ x , l _ y ) \\in \\mathbb { Z } ^ { 2 } | ( \\frac { l _ x } { L _ { r R x } } ) ^ 2 + ( \\frac { l _ y } { L _ { R , y } } ) ^ 2 \\le 1 \\} . \\end{aligned} \\end{equation}"}
{"id": "277.png", "formula": "\\[ \\ddot { x } + \\frac { 2 } { x } \\dot { x } ^ 2 + \\frac { \\omega ^ 2 } { x ^ 3 } = 0 , \\omega \\in \\mathbb { R } , \\]"}
{"id": "38.png", "formula": "\\begin{equation} \\tilde d _ { 0 } \\omega = \\sum _ { l = 1 } ^ n ( \\mathcal { D } _ { l } \\omega ) d x _ l . \\end{equation}"}
{"id": "397.png", "formula": "\\begin{equation} \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } 3 ] { 3 a b K _ 3 } = \\begin{cases} 2 ^ d 3 ^ e r ' s \\sqrt [ \\leftroot { - 2 } \\uproot { 3 } 3 ] { K _ 3 } , & \\textnormal { f o r F L T 2 C a s e I } , \\\\ 2 ^ d 3 ^ e r _ 0 s ' \\sqrt [ \\leftroot { - 2 } \\uproot { 3 } 3 ] { K _ 3 } , & \\textnormal { f o r F L T 2 C a s e I I } . \\end{cases} \\end{equation}"}
{"id": "373.png", "formula": "\\begin{equation} \\begin{aligned} f _ n ( x , b , a ) & = - \\sum _ { i = 1 } ^ { n - 1 } \\binom { n } { i } b ^ i \\bigg ( ( x - a ) ^ { n - i } - ( x ^ { n - i } + ( - a ) ^ { n - i } ) \\bigg ) , \\\\ g _ n ( b , a ) & = - ( b - a ) ^ n + b ^ n + ( - a ) ^ n = - \\sum _ { i = 1 } ^ { n - 1 } \\binom { n } { i } b ^ { n - i } a ^ i , \\end{aligned} \\end{equation}"}
{"id": "220.png", "formula": "\\begin{align} \\begin{split} P [ j , k ] \\ = \\ \\widehat { A } + \\ & \\cup \\ \\widehat { A } - \\ \\cup \\\\ [ ( D _ j - \\times D _ j + ) \\cup ( \\bar D _ j - \\times \\bar D _ j + ) ] \\ & \\cup \\ [ ( D _ j + \\times \\bar D _ j - ) \\cup ( \\bar D _ j + \\times D _ j - ) ] . \\end{split} \\end{align}"}
{"id": "242.png", "formula": "\\begin{equation} X ^ i ( x , v ) = \\sum _ { j = 1 } ^ n B ^ i \\ , _ j ( x ) \\ , v ^ j , i = 1 , \\ldots , n . \\end{equation}"}
{"id": "125.png", "formula": "\\begin{equation} \\begin{aligned} \\inf _ { q \\in Q } \\sup _ { \\vec { v } \\in V } \\frac { ( \\operatorname { d i v } \\vec { v } , q ) } { \\| \\varepsilon ( \\vec { v } ) \\| \\| q \\| } = \\inf _ { \\vec { v } \\in W ^ { \\perp } } \\sup _ { q \\in Q } \\frac { ( \\operatorname { d i v } \\vec { v } , q ) } { \\| \\varepsilon ( \\vec { v } ) \\| \\| q \\| } \\ge \\beta > 0 . \\end{aligned} \\end{equation}"}
{"id": "184.png", "formula": "\\begin{align*} & u _ { F _ { a , b } ( Z , c ) , F _ { a , b } ( W , d ) } ( j _ { a , b } ( e _ { i } ) \\boxtimes j _ { a , b } ( f _ { j } ) ) \\\\ & = u _ { F _ { a , b } ( Z , c ) , F _ { a , b } ( W , d ) } \\left ( \\sum _ { l } j _ { a , b } ( e _ { i } ) \\boxtimes \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\langle k _ { 0 } ( f ^ { \\prime } _ { l } ) \\ | j _ { a , b } ( f _ { j } ) ) \\rangle \\right ) \\\\ & = \\sum _ { l } \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\boxtimes j _ { a , b } ( e _ { i } ) \\langle \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\ | j _ { a , b } ( f _ { j } ) ) \\rangle \\\\ & = \\sum _ { l , s } j _ { a , b } ( f _ { s } ) \\langle j _ { a , b } ( f _ s ) \\ | \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\rangle \\ \\boxtimes \\ j _ { a , b } ( e _ { i } ) \\langle \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\ | j _ { a , b } ( f _ { j } ) ) \\rangle \\\\ & = \\sum _ { s , l } j _ { a , b } ( f _ { s } ) \\ \\boxtimes \\ \\langle j _ { a , b } ( f _ s ) \\ | \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\rangle j _ { a , b } ( e _ { i } ) \\langle \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\ | j _ { a , b } ( f _ { j } ) ) \\rangle \\end{align*}"}
{"id": "50.png", "formula": "\\begin{equation} ( \\tilde { U } _ { j , h } ^ * f ) ( \\mu ) : = h ^ { j } \\sum _ { l = 1 } ^ { \\binom { n } { j } } f _ l ( \\mu ) d x ^ { I ^ j _ l } \\end{equation}"}
{"id": "269.png", "formula": "\\[ \\frac { d B } { d x } = \\frac { 1 } { A _ 0 ^ 4 } \\left ( A _ 0 \\frac { d } { d x } - 3 A ' _ 0 \\right ) \\left ( A _ 0 \\frac { d } { d x } + \\gamma _ 0 A _ 0 - A ' _ 0 \\right ) b _ 0 = 0 . \\]"}
{"id": "58.png", "formula": "\\begin{align*} \\prescript { } { i _ 0 } { ( \\hat { s } ^ * } ) ( i ) = & \\begin{cases} \\hat { s } ^ * ( i ) & i < i _ 0 \\\\ \\hat { s } ^ * ( i + 1 ) & i _ 0 \\leq i \\end{cases} \\\\ = & \\begin{cases} \\hat { s } ( j - i + 1 ) & i < i _ 0 \\\\ \\hat { s } ( j - i ) & i _ 0 \\leq i \\end{cases} \\end{align*}"}
{"id": "249.png", "formula": "\\begin{equation} \\frac { d \\bar v } { d \\tau } = f f ' ( q ) \\ , 2 ( E - \\mathcal { V } ) - f ^ 2 \\ , \\mathcal { V } ' ( q ) = \\frac d { d q } \\left ( f ^ 2 ( E - \\mathcal { V } ) \\right ) , \\end{equation}"}
{"id": "74.png", "formula": "\\begin{align*} \\| N \\| _ { L ^ \\infty ( \\Omega _ T ) } : = \\sup \\{ | N ( x , t ) | : ( x , t ) \\in \\Omega _ T \\} \\end{align*}"}
{"id": "412.png", "formula": "\\[ M h ^ { a } _ { c } ( f ) : = \\bigcap \\{ M h ^ { a } _ { d } ( f ( d ) ) : d \\in { \\rm s u p p } ( f ^ { c } ) \\} = \\bigcap \\{ M h ^ { a } _ { d } ( f ( d ) ) : c \\leq d \\in { \\rm s u p p } ( f ) \\} . \\]"}
{"id": "173.png", "formula": "\\begin{align*} \\langle u ^ { F , G } _ { X , Y } ( \\sum b _ i \\boxtimes c _ { j } a _ { i j } ) \\ | \\ u ^ { F , G } _ { X , Y } ( \\sum b _ { i } \\boxtimes c _ { j } a _ { i j } ) \\rangle & = \\langle \\sum c _ { j } \\boxtimes _ { A } b _ { i } \\ a _ { i j } | \\sum c _ { j } \\boxtimes _ { A } b _ { i } a _ { i j } \\rangle \\\\ & = \\sum a ^ { * } _ { i j } \\langle b _ i \\ | \\langle c _ { j } | c _ { k } \\rangle b _ l \\rangle a _ { l k } \\\\ & = \\sum a ^ { * } _ { i j } \\ \\langle b _ i \\ | \\ b _ { l } \\rangle \\ \\langle c _ { j } \\ | c _ { k } \\rangle \\ a _ { l k } \\\\ & = \\sum \\langle c _ { j } a _ { i j } \\ | \\langle b _ i \\ | \\ b _ { l } \\rangle c _ { k } a _ { l k } \\rangle \\\\ & = \\langle \\sum b _ i \\boxtimes c _ { j } \\ a _ { i j } \\ | \\ \\sum b _ { i } \\boxtimes c _ { j } \\ a _ { i j } \\rangle \\end{align*}"}
{"id": "95.png", "formula": "\\begin{align} \\begin{cases} w _ 1 = p - \\gamma , & w _ 2 = 2 , \\\\ w _ 3 = 4 - p + \\gamma , & w _ 4 = 2 , \\end{cases} \\end{align}"}
{"id": "244.png", "formula": "\\begin{equation} \\bar X ^ i \\left ( \\bar x , \\frac { d \\bar x } { d \\tau } \\right ) = { f ^ 2 } \\ , X ^ i \\left ( \\bar x , \\frac 1 f \\frac { d \\bar x } { d \\tau } \\right ) + \\frac d { d \\tau } ( \\log f ) \\ , \\frac { d \\bar x ^ i } { d \\tau } , \\end{equation}"}
{"id": "221.png", "formula": "\\begin{equation} P [ j , k ] ^ { \\circ } ( z + ) \\ = \\ \\bar D _ k - \\ \\cup \\ [ \\bigcup _ { i \\in \\N } \\ ( z + A _ i ) + ] . \\end{equation}"}
{"id": "64.png", "formula": "\\begin{equation} Z _ X ( S ) : = \\bigcup _ { v \\in S } Z _ X ( v ) . \\end{equation}"}
{"id": "349.png", "formula": "\\begin{align*} g _ { I ( G ) } ( k + 1 ) - g _ { I ( G ) } ( k ) \\ & \\le \\ g _ { I ( G _ 1 ) } ( k + 1 ) + 1 - ( g _ { I ( G _ 1 ) } ( k ) + 1 ) \\\\ & = \\ g _ { I ( G _ 1 ) } ( k + 1 ) - g _ { I ( G _ 1 ) } ( k ) \\le 0 , \\end{align*}"}
{"id": "7.png", "formula": "\\begin{equation} R ( \\kappa ) - R _ 0 ( \\kappa ) = R _ 0 ( \\kappa ) C _ + q \\sqrt { R _ 0 ( \\kappa ) } \\cdot \\sqrt { R _ 0 ( \\kappa ) } \\bigl [ 1 + C _ + q R ( \\kappa ) \\bigr ] . \\end{equation}"}
{"id": "183.png", "formula": "\\begin{align} \\mu ^ { [ a , b ] } _ { ( Z , c ) , ( W , d ) } \\circ u _ { F _ { a , b } ( Z , c ) , F _ { a , b } ( W , d ) } ( j _ { a , b } ( e _ { i } ) \\boxtimes j _ { a , b } ( f _ { j } ) ) = j _ { a , b } \\left ( 1 _ { X ^ { b - a } } \\otimes c _ { Z , W } ) \\circ ( e _ { j } \\otimes 1 _ { W } ) \\circ f _ { j } \\right ) \\end{align}"}
{"id": "252.png", "formula": "\\begin{equation} \\frac { d ^ 2 x } { d t ^ 2 } + \\gamma _ 0 ( x ) \\left ( \\frac { d x } { d t } \\right ) ^ 2 + A _ 0 ( x ) \\frac { d x } { d t } + b _ 0 ( x ) = 0 , \\end{equation}"}
{"id": "172.png", "formula": "\\begin{align*} \\langle b _ { i } \\ | \\ a b _ { j } \\rangle b & = \\langle b _ { i } \\ | \\ a b _ { j } b \\rangle \\\\ & = \\langle b _ { i } \\ | \\ b a b _ { j } \\rangle \\\\ & = \\langle b ^ { * } b _ { i } \\ | \\ a b _ { j } \\rangle \\\\ & = \\langle b _ { i } b ^ { * } \\ | \\ a b _ { j } \\rangle \\\\ & = b \\langle b _ { i } \\ | \\ a b _ { j } \\rangle \\end{align*}"}
{"id": "335.png", "formula": "\\begin{equation} \\Delta v _ k ( x , t ) \\geq - \\frac { K } { t } , v _ k = \\frac { m } { m - 1 } u _ k ^ { m - 1 } . \\end{equation}"}
{"id": "427.png", "formula": "\\begin{equation} \\bold { p } _ { j } = \\bold { B } _ { j } \\bold { p } _ { j } + \\bold { q } _ { j } + \\bold { v } _ { j } , \\end{equation}"}
{"id": "410.png", "formula": "\\begin{equation} \\xi = \\sum _ { i \\leq m } \\tilde { \\theta } _ { b _ { i } } ( \\xi _ { i } ) \\cdot a _ { i } = _ { N F } \\tilde { \\theta } _ { b _ { m } } ( \\xi _ { m } ) \\cdot a _ { m } + \\cdots + \\tilde { \\theta } _ { b _ { 0 } } ( \\xi _ { 0 } ) \\cdot a _ { 0 } \\end{equation}"}
{"id": "44.png", "formula": "\\begin{align} \\langle \\sum _ I ( \\Delta \\omega _ I ) d x ^ I ; \\sum _ I \\omega _ I d x ^ I \\rangle = & \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I ( \\Delta \\omega _ I ( \\mu ) ) \\overline { \\omega _ I ( \\mu ) } \\nonumber \\\\ = & \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I \\sum _ { \\alpha = 1 } ^ n ( ( \\mathcal { D } _ { \\alpha } ^ 2 \\omega _ I ) ( \\mu - e _ \\alpha ) ) \\overline { \\omega _ I ( \\mu ) } \\nonumber \\\\ = & \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I \\sum _ { \\alpha = 1 } ^ n | \\mathcal { D } _ { \\alpha } \\omega _ I ( \\mu ) | ^ 2 \\ . \\end{align}"}
{"id": "258.png", "formula": "\\begin{equation} \\frac { d \\varphi } { d x } = c \\ , h . \\end{equation}"}
{"id": "345.png", "formula": "\\begin{align*} J _ 1 & = x _ { n } x _ { n - 1 } [ I ( G _ 3 ) ^ { [ k ] } + x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } ] , \\\\ J _ 2 & = x _ { n } x _ { n - 1 } \\sum _ { j = 1 } ^ t x _ { i _ j } I ( G _ 2 ) ^ { [ k - 1 ] } = x _ { n } x _ { n - 1 } ( x _ { i _ 1 } , \\dots , x _ { i _ t } ) I ( G _ 2 ) ^ { [ k - 1 ] } . \\end{align*}"}
{"id": "146.png", "formula": "\\begin{equation} J ( z ) = J ( | z | ) = \\frac { 1 } { 2 } \\frac { d ^ 2 } { d r ^ 2 } \\bigg | _ { r = | z | } r ^ 2 \\big [ \\coth { ( r ) } - 1 \\big ] . \\end{equation}"}
{"id": "207.png", "formula": "\\begin{equation} f ' _ i \\circ q _ { i + 1 } \\ = \\ q _ i \\circ f _ i , \\text { a n d } q _ i \\circ h _ i \\ = \\ h _ i ' . \\end{equation}"}
{"id": "93.png", "formula": "\\begin{align*} w _ 1 = \\big ( \\sqrt { p - 1 } - \\sqrt { 1 - \\gamma } \\big ) ^ 2 + \\eta . \\end{align*}"}
{"id": "0.png", "formula": "\\begin{align} q ( t , x ) \\mapsto q _ \\lambda ( t , x ) = \\lambda q ( \\lambda ^ 2 t , \\lambda x ) \\qquad \\text { f o r $ \\lambda > 0 $ } \\end{align}"}
{"id": "23.png", "formula": "\\begin{equation} q _ + ( t , z ) = \\tfrac { 1 } { 2 \\pi i } I _ + \\Big ( \\big ( X - t \\kappa R ( \\kappa ; q ^ 0 ) ^ 2 - z \\big ) ^ { - 1 } q ^ 0 _ + \\Big ) \\end{equation}"}
{"id": "108.png", "formula": "\\begin{equation} G : = p - 1 + s - \\gamma \\quad \\text { a n d } E : = s + 1 + \\frac { p - 1 } { n - 1 } . \\end{equation}"}
{"id": "155.png", "formula": "\\begin{equation} { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ i F ( t / 3 ^ i ) = { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ { i + 1 } F ( t / 3 ^ { i + 1 } ) + { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ i \\cdot R ( t / 3 ^ i ) . \\end{equation}"}
{"id": "43.png", "formula": "\\begin{equation} \\tilde d ^ * = ( - 1 ) ^ { n i + 1 } * \\tilde d * . \\end{equation}"}
{"id": "34.png", "formula": "\\begin{equation} ( d _ h f ) ( s ) = \\sum _ { r \\subset s } f ( r ) ; \\quad ( d _ h ^ * f ) ( s ) = \\frac 1 { h ^ 2 } \\sum _ { s \\subset r } f ( r ) \\ . \\end{equation}"}
{"id": "192.png", "formula": "\\begin{align} \\begin{split} X _ 2 \\ = \\ X _ 1 \\times \\{ Y _ x \\} \\ = \\ & \\ \\bigcup _ { x \\in X } \\ \\{ x \\} \\times Y _ x , \\\\ R _ 2 \\ = \\ R _ 1 \\ltimes \\{ S _ x \\} \\text { w h e r e } \\ & \\text { f o r } \\ ( x _ 1 , y _ 1 ) , ( x _ 2 , y _ 2 ) \\in X _ 2 , \\\\ ( ( x _ 1 , y _ 1 ) , ( x _ 2 , y _ 2 ) ) \\in R _ 2 \\Longleftrightarrow & \\begin{cases} ( x _ 1 , x _ 2 ) \\in R _ 1 ^ { \\circ } \\text { o r } \\\\ x _ 1 = x _ 2 \\ \\text { a n d } \\ ( y _ 1 , y _ 2 ) \\in S _ { x _ 1 } . \\end{cases} \\end{split} \\end{align}"}
{"id": "288.png", "formula": "\\begin{equation} \\frac { d ^ 2 x } { d t ^ 2 } + \\gamma ( x ) \\Bigl ( \\frac { d x } { d t } \\Bigr ) ^ 2 + A ( x ) \\frac { d x } { d t } + b ( x ) = 0 , \\end{equation}"}
{"id": "198.png", "formula": "\\begin{align} \\begin{split} \\widehat { A } \\ = \\ \\{ ( x , y ) : x ^ { - 1 } y & \\in A \\} \\text { s o t h a t } \\ \\ \\widehat { A } ^ { - 1 } \\ = \\ \\widehat { A ^ { - 1 } } , \\\\ \\text { a n d s o } \\ \\ \\widehat { A } ^ { \\circ } \\ & = \\ \\{ ( x , y ) : x ^ { - 1 } y \\in A ^ { \\circ } \\} . \\end{split} \\end{align}"}
{"id": "180.png", "formula": "\\begin{align*} & \\sum _ { i } | 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } \\rangle _ { A _ { [ a - k , b + k ] } } \\ \\langle 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } | \\\\ & = \\sum _ { i } ( 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } ) \\circ ( 1 _ { X ^ { k } } \\otimes b ^ { * } _ { i } \\otimes 1 _ { X ^ { k } } ) \\\\ & = 1 _ { X ^ { 2 k + b - a + 1 } \\otimes Z } = i d _ { F ^ { k } _ { [ a , b ] } ( Z , c ) } \\end{align*}"}
{"id": "212.png", "formula": "\\begin{align} \\begin{split} ( Y _ { i + } , S _ { i + } ) \\ = \\ & ( Z + , P + ) , \\\\ ( Y _ { i - } , S _ { i - } ) \\ = \\ & ( Z - , P - ) \\end{split} \\end{align}"}
{"id": "362.png", "formula": "\\begin{equation} x ^ n = z ^ n - y ^ n = ( z - y ) \\phi _ n ( z , y ) . \\end{equation}"}
{"id": "12.png", "formula": "\\begin{equation} m ( x ; \\kappa , q ) : = R ( \\kappa , q ) q _ + = R _ 0 ( \\kappa ) \\sum _ { \\ell \\geq 1 } [ C _ + q R _ 0 ( \\kappa ) ] ^ { \\ell - 1 } q _ + \\end{equation}"}
{"id": "247.png", "formula": "\\begin{equation} \\frac { d ^ 2 \\bar x ^ i } { d \\tau ^ 2 } = \\left ( { \\displaystyle \\sum _ { j = 1 } ^ n } \\frac { 1 } { f } \\frac { \\partial f } { \\partial \\bar x ^ { j } } \\frac { d \\bar x ^ j } { d \\tau } \\right ) \\frac { d \\bar x ^ i } { d \\tau } + f ^ { 2 } X ^ { i } \\left ( \\bar x , \\frac 1 f \\frac { d \\bar x } { d \\tau } \\right ) . \\end{equation}"}
{"id": "246.png", "formula": "\\begin{equation} \\bar { \\Gamma } ( \\bar x , \\bar v ) = { \\displaystyle \\sum _ { j = 1 } ^ n \\bar { v } ^ { j } \\frac { \\partial } { \\partial \\bar { x } ^ { j } } } + { \\displaystyle \\sum _ { i = 1 } ^ n } \\left ( { \\displaystyle \\sum _ { j = 1 } ^ n } \\frac { 1 } { f } \\bar { v } ^ { j } \\frac { \\partial f } { \\partial \\bar x ^ { j } } \\bar { v } ^ { i } + f ^ { 2 } X ^ { i } \\left ( \\bar x , \\dfrac 1 f \\ , \\bar v \\right ) \\right ) \\frac { \\partial } { \\partial \\bar { v } ^ { i } } , \\end{equation}"}
{"id": "426.png", "formula": "\\begin{equation} \\begin{aligned} \\bold { p } _ { j } & = [ \\psi _ { j , 1 } , \\psi _ { j , 2 } , . . . , \\psi _ { j , j } , \\rho ^ 2 \\widetilde { t } _ { 1 , 1 } ^ 2 \\psi _ { j , 1 } + \\Theta _ { j , 1 } , \\\\ & \\rho ^ 2 \\widetilde { t } _ { 2 , 2 } ^ 2 \\psi _ { j , 2 } + \\Theta _ { j , 2 } , . . . , \\rho ^ 2 \\widetilde { t } _ { j , j } ^ 2 \\psi _ { j , j } + \\Theta _ { j , j } ] ^ { T } , \\\\ \\bold { q } _ { j } & = M [ \\Gamma _ { j , 1 } , \\Gamma _ { j , 2 } , . . . , \\Gamma _ { j , j } , \\Pi _ { j , 1 } , \\Pi _ { j , 2 } , . . . , \\Pi _ { j , j } ] ^ { T } . \\end{aligned} \\end{equation}"}
{"id": "368.png", "formula": "\\begin{equation} \\begin{aligned} \\phi _ p ( x , - y ) & = \\dfrac { x ^ p + y ^ p } { x + y } = \\sum _ { i = 0 } ^ { p - 1 } x ^ { p - i - 1 } ( - y ) ^ i \\\\ & = A _ p ( x , - y ) + ( - 1 ) ^ k ( x y ) ^ k = D _ p ( x , - y ) + ( x y ) ^ k , \\\\ \\end{aligned} \\end{equation}"}
{"id": "325.png", "formula": "\\begin{equation} ( P _ k ) \\ \\left \\{ \\begin{array} { l l } w _ t = \\Delta w ^ m + \\min \\{ ( 1 + | x | ) ^ { \\sigma } , k \\} f _ k ( w ) , & ( x , t ) \\in \\real ^ N \\times ( 0 , \\infty ) , \\\\ w ( x , 0 ) = u _ 0 ( x ) , & x \\in \\real ^ N \\end{array} \\right . \\end{equation}"}
{"id": "337.png", "formula": "\\begin{equation} \\begin{split} \\frac { m - 1 } { 1 - p } & \\int _ { \\tau _ 0 } ^ { \\tau _ 1 } \\int _ { \\real ^ N } ( v ^ { ( 1 - p ) / ( m - 1 ) } ) _ t \\varphi \\ , d x \\ , d t \\geq ( m - 1 ) \\int _ { \\tau _ 0 } ^ { \\tau _ 1 } \\int _ { \\real ^ N } v ^ { ( 1 - p ) / ( m - 1 ) } \\Delta v \\varphi \\ , d x \\ , d t \\\\ & + K ( m , p ) \\int _ { \\tau _ 0 } ^ { \\tau _ 1 } \\int _ { \\real ^ N } \\varphi \\ , d x \\ , d t \\\\ & \\geq - \\frac { N ( m - 1 ) } { N ( m - 1 ) + 2 } \\int _ { \\tau _ 0 } ^ { \\tau _ 1 } \\int _ { \\real ^ N } \\frac { 1 } { t } v ^ { ( 1 - p ) / ( m - 1 ) } \\varphi \\ , d x \\ , d t + K ( m , p ) \\int _ { \\tau _ 0 } ^ { \\tau _ 1 } \\int _ { \\real ^ N } \\varphi \\ , d x \\ , d t . \\end{split} \\end{equation}"}
{"id": "371.png", "formula": "\\[ \\begin{aligned} z - y & = r ^ p , & & \\phi _ p ( z , y ) = r _ 1 ^ p , & & x = r r _ 1 , \\\\ z - x & = s ^ p , & & \\phi _ p ( z , x ) = s _ 1 ^ p , & & y = s s _ 1 , \\\\ x + y & = t ^ p , & & \\phi _ p ( x , - y ) = t _ 1 ^ p , & & z = t t _ 1 . \\end{aligned} \\]"}
{"id": "340.png", "formula": "\\begin{equation} ( I , x _ n ) ^ { [ k ] } = I ^ { [ k ] } + x _ n I ^ { [ k - 1 ] } \\subset S \\end{equation}"}
{"id": "121.png", "formula": "\\begin{equation} A _ { \\lambda } ^ { - 1 } \\eqsim \\frac { \\lambda } { \\lambda + 1 } P A ^ { - 1 } + \\frac { 1 } { \\lambda + 1 } A ^ { - 1 } = : M _ \\lambda , \\end{equation}"}
{"id": "118.png", "formula": "\\[ \\lambda = \\frac { \\nu } { 1 - 2 \\nu } , 0 \\le \\nu < \\frac 1 2 . \\]"}
{"id": "336.png", "formula": "\\begin{equation} v _ t = ( m - 1 ) v \\Delta v + | \\nabla v | ^ 2 + K ( m , p ) v ^ { ( m + p - 2 ) / ( m - 1 ) } . \\end{equation}"}
{"id": "422.png", "formula": "\\begin{equation} \\begin{aligned} \\frac { 1 } { N } C _ { M } ( \\zeta ) \\xrightarrow { N \\xlongrightarrow { c } \\infty } \\frac { 1 } { N } \\overline { C } _ { M } ( \\zeta ) , \\end{aligned} \\end{equation}"}
{"id": "165.png", "formula": "\\begin{equation} X _ N ( \\varphi ) \\equiv \\sum _ { i = 1 } ^ N \\varphi ( z _ i ) \\end{equation}"}
{"id": "375.png", "formula": "\\begin{equation} x - a = x + y - z , 2 x + b - a = x + y , y = x + b - a , \\end{equation}"}
{"id": "68.png", "formula": "\\[ \\begin{split} f ( 0 ) = f ' ( 0 ) = f '' ( 0 ) = 0 , \\ f '' ( r ) > 0 \\text { f o r a l l } r > 0 , \\end{split} \\]"}
{"id": "286.png", "formula": "\\[ \\left ( A \\frac { d } { d x } - 3 A ' \\right ) \\left ( A \\frac { d } { d x } + \\gamma \\ , A - A ' \\right ) b = ( \\Delta _ \\R - 3 ) \\Delta _ \\R b = 0 , \\]"}
{"id": "219.png", "formula": "\\begin{equation} D _ j \\ = \\ \\{ x \\in \\Z [ 2 ] : x _ j = 0 \\} , \\bar D _ j \\ = \\ \\{ x \\in \\Z [ 2 ] : x _ j = 1 \\} . \\end{equation}"}
{"id": "395.png", "formula": "\\[ x - a = r s \\sqrt [ \\leftroot { - 4 } \\uproot { 3 } 3 ] { 3 K _ 3 } . \\]"}
{"id": "137.png", "formula": "\\begin{equation} \\begin{pmatrix} \\frac 1 2 | \\xi | ^ { 2 } ( I + \\Pi _ \\xi ) & \\xi \\\\ \\xi ^ * & 0 \\end{pmatrix} \\begin{pmatrix} \\widehat { \\vec { u } } _ \\infty \\\\ \\widehat { p } \\end{pmatrix} = \\begin{pmatrix} \\widehat { \\vec { f } } \\\\ 0 \\end{pmatrix} . \\end{equation}"}
{"id": "408.png", "formula": "\\[ M _ { 2 } ( \\bar { \\nu } ) \\cap M _ { 2 } ( \\bar { \\mu } ) = M _ { 2 } ( \\max \\{ \\bar { \\nu } , \\bar { \\mu } \\} ) . \\]"}
{"id": "285.png", "formula": "\\[ \\ddot { x } + \\frac { 1 } { x } \\dot { x } ^ 2 + x \\ , \\dot { x } + b ( x ) = 0 . \\]"}
{"id": "328.png", "formula": "\\begin{equation} z ( x , t ) = ( T - t ) ^ { - \\alpha } f ( ( 1 + | x | ) ( T - t ) ^ { \\beta } ) , \\end{equation}"}
{"id": "374.png", "formula": "\\[ ( u \\pm v ) ^ p = u ^ p \\pm \\binom { p } { 1 } u ^ { p - 1 } v + \\binom { p } { 2 } u ^ { p - 2 } v ^ 2 \\pm \\cdots + \\binom { p } { p - 1 } u v ^ { p - 1 } \\pm v ^ p , \\]"}
{"id": "299.png", "formula": "\\begin{align*} \\psi ( u _ i ) ( a _ 1 , \\ldots , a _ r ) & = \\psi ( \\pi ( t _ i ) ) ( p ( x _ 1 ) , \\ldots , p ( x _ r ) ) \\\\ & = \\psi ( \\pi ( t _ i ) ) ( ( p \\circ \\pi ) ( x _ 1 ) , \\ldots , ( p \\circ \\pi ) ( x _ r ) ) \\\\ & = ( p \\circ \\pi ) ( t _ i ) \\\\ & = p ( u _ i ) , \\end{align*}"}
{"id": "228.png", "formula": "\\begin{align} \\begin{split} \\max ( \\bar u ( H x , H y ) , \\bar u ( H y , H z ) ) \\ = \\ & \\max ( u ( x _ 1 , y _ 1 ) , u ( y _ 1 , z _ 1 ) ) \\\\ \\ge \\ u ( x _ 1 , z _ 1 ) \\ \\ge \\ & \\bar u ( H x , H z ) . \\end{split} \\end{align}"}
{"id": "365.png", "formula": "\\begin{equation} \\begin{aligned} \\phi _ p ( z , y ) & = ( z y ) ^ k + \\sum _ { i = 0 } ^ { k - 1 } ( z y ) ^ { i } ( z ^ { 2 ( k - i ) } + y ^ { 2 ( k - i ) } ) \\\\ & = p ( z y ) ^ k + \\sum _ { i = 0 } ^ { k - 1 } ( z y ) ^ { i } ( z ^ { k - i } - y ^ { k - i } ) ^ 2 . \\end{aligned} \\end{equation}"}
{"id": "287.png", "formula": "\\begin{equation} \\ddot x + f ( x ) \\ , \\dot x + g ( x ) = 0 , \\end{equation}"}
{"id": "407.png", "formula": "\\[ M _ { k } ( ( \\nu _ { k } , \\nu _ { k + 1 } , \\ldots , \\nu _ { N - 1 } ) ) : = \\bigcap _ { i \\geq k } M _ { i } ( \\nu _ { i } ) . \\]"}
{"id": "419.png", "formula": "\\begin{equation} \\begin{aligned} \\mathcal { E } _ { l _ x , l _ y } & = [ \\frac { 2 \\pi l _ x } { L _ { R , x } } , \\frac { 2 \\pi ( l _ x + 1 ) } { L _ { R , x } } ] \\times [ \\frac { 2 \\pi l _ y } { L _ { R , y } } , \\frac { 2 \\pi ( l _ y + 1 ) } { L _ { R , y } } ] \\\\ \\mathcal { E } _ { m _ x , m _ y } & = [ \\frac { 2 \\pi m _ x } { L _ { S , x } } , \\frac { 2 \\pi ( m _ x + 1 ) } { L _ { S , x } } ] \\times [ \\frac { 2 \\pi m _ y } { L _ { S , y } } , \\frac { 2 \\pi ( m _ y + 1 ) } { L _ { S , y } } ] . \\end{aligned} \\end{equation}"}
{"id": "98.png", "formula": "\\begin{equation} \\begin{aligned} 2 c _ 1 + c _ 2 \\geq \\min \\{ 2 ( p - \\gamma ) , 8 \\} > 0 . \\end{aligned} \\end{equation}"}
{"id": "312.png", "formula": "\\begin{equation} \\left \\langle L _ { F ' } e _ { k } ' , e _ { j } ' \\right \\rangle = \\begin{cases} \\mu ^ { | \\alpha ( j ) | - | \\alpha ( k ) | } \\sum _ { l = 1 } ^ n \\alpha _ l ( k ) \\ , a _ { l , ( \\alpha ( j ) - \\alpha ( k ) ) _ l } & \\text { i f } | \\alpha ( j ) | > | \\alpha ( k ) | \\\\ \\sum _ { l = 1 } ^ n \\alpha _ l ( k ) \\ , a _ { l , l } & \\text { i f } j = k \\\\ 0 & \\text { o t h e r w i s e } . \\end{cases} \\end{equation}"}
{"id": "295.png", "formula": "\\begin{equation} e _ { 2 k + 1 } e _ { 2 l + 1 } = C _ { k + l } ^ { l } e _ { 2 k + 2 l + 2 } . \\end{equation}"}
{"id": "240.png", "formula": "\\begin{equation} \\left \\{ \\begin{array} { r c l } { \\displaystyle \\frac { d x ^ i } { d t } } & = & v ^ i \\\\ { \\displaystyle \\frac { d v ^ i } { d t } } & = & X ^ i ( x ^ 1 , \\ldots , x ^ n , v ^ 1 , \\ldots , v ^ n ) \\end{array} \\right . i = 1 , \\ldots , n , \\end{equation}"}
{"id": "343.png", "formula": "\\begin{equation} I ( G _ 1 ) ^ { [ k ] } = x _ { n - 1 } x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } + \\sum _ { j = 1 } ^ t x _ { n - 1 } x _ { i _ j } I ( G _ 2 ) ^ { [ k - 1 ] } + I ( G _ 2 ) ^ { [ k ] } . \\end{equation}"}
{"id": "37.png", "formula": "\\begin{equation} \\langle \\omega , \\eta \\rangle _ { \\Omega _ c ^ j ( \\Z ^ n ) } = \\sum _ { \\mu \\in \\Z ^ n } \\sum _ { I \\in P ^ { j , n } _ + } \\omega _ { I } ( \\mu ) \\overline { \\eta _ { I } ( \\mu ) } \\ . \\end{equation}"}
{"id": "142.png", "formula": "\\begin{equation} \\begin{aligned} a ( P _ h \\vec { v } _ h , \\vec { w } _ h ) + b ( \\vec { w } _ h , p _ h ) & = a ( \\vec { v } _ h , \\vec { w } _ h ) , & & \\forall \\vec { w } _ h \\in V _ h , \\\\ b ( P _ h \\vec { v } _ h , q _ h ) & = 0 , & & \\forall q _ h \\in Q _ h . \\end{aligned} \\end{equation}"}
{"id": "145.png", "formula": "\\begin{equation} { | | \\varphi | | } ^ p _ { W ^ { t , p } } = \\int _ A \\int _ A \\frac { { | \\varphi ( x ) - \\varphi ( y ) | } ^ p } { { | x - y | } ^ { n + t p } } \\ d x d y \\end{equation}"}
{"id": "9.png", "formula": "\\begin{align} I _ + ( f ) : = \\lim _ { y \\to \\infty } 2 \\pi y f ( i y ) = \\lim _ { y \\to \\infty } \\bigl \\langle \\chi _ y , f \\bigr \\rangle = \\lim _ { \\xi \\downarrow 0 } \\sqrt { { 2 \\pi } } \\widehat f ( \\xi ) \\ \\text { w i t h } \\ \\chi _ y ( x ) = \\tfrac { i y } { x + i y } . \\end{align}"}
{"id": "210.png", "formula": "\\begin{align} \\begin{split} \\N + \\ = \\ ( 2 L ) ^ { \\circ - 1 } ( \\infty ) , & \\N - \\ = \\ ( 2 L ) ^ { \\circ } ( \\infty ) , \\\\ ( i - , i + ) \\ \\in \\ 2 L & \\text { f o r a l l } \\ \\ i \\in \\N . \\end{split} \\end{align}"}
{"id": "402.png", "formula": "\\[ x - a = y - b = z - ( a + b ) = \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } 5 ] { 5 a b K _ 5 } = r s \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } 5 ] { 5 K _ 5 } . \\]"}
{"id": "169.png", "formula": "\\begin{align*} & \\eta _ { X } ( a \\triangleright _ { \\alpha } x \\triangleleft _ { \\alpha } b ) \\\\ & = \\left ( \\prod _ { i \\in J _ { N } } u ^ { * } _ { i } \\right ) \\left ( \\left ( \\prod _ { i \\in J _ { I } } u _ { i } \\right ) a \\left ( \\prod _ { i \\in J _ { I } } u ^ { * } _ { i } \\right ) x \\left ( \\prod _ { i \\in J _ { K } } u _ { i } \\right ) b \\left ( \\prod _ { i \\in J _ { K } } u ^ { * } _ { i } \\right ) \\right ) \\left ( \\prod _ { i \\in J _ { N } } u _ { i } \\right ) \\\\ & = \\left ( \\prod _ { i \\in J _ { N } } u ^ { * } _ { i } \\right ) \\left ( \\left ( \\prod _ { i \\in J _ { N } } u _ { i } \\right ) a \\ \\text { A d } \\left ( \\prod _ { i \\in J _ { N } } u ^ { * } _ { i } \\right ) ( x ) \\ b \\left ( \\prod _ { i \\in J _ { N } } u ^ { * } _ { i } \\right ) \\right ) \\left ( \\prod _ { i \\in J _ { N } } u _ { i } \\right ) \\\\ & = a \\left ( \\prod _ { i \\in J _ { N } } u ^ { * } _ { i } \\right ) x \\left ( \\prod _ { i \\in J _ { N } } u _ { i } \\right ) b \\\\ & = a \\ \\eta _ { X } ( x ) \\ b \\end{align*}"}
{"id": "83.png", "formula": "\\begin{equation} \\begin{aligned} \\begin{cases} w _ 1 = p - \\gamma - 2 \\sqrt { ( p - 1 ) ( 1 - \\gamma ) } + \\eta , & w _ 2 = 2 , \\\\ w _ 3 = 0 , & w _ 4 = 0 , \\end{cases} \\end{aligned} \\end{equation}"}
{"id": "200.png", "formula": "\\begin{equation} A \\ = \\ \\bigcap _ { i \\in \\N } \\ \\pi _ { i } ^ { - 1 } ( A _ i ) \\ = \\ \\overleftarrow { L i m } \\{ A _ i \\} \\end{equation}"}
{"id": "148.png", "formula": "\\begin{equation} K ( n ) = \\frac { \\Gamma ( \\frac { n } { 2 } ) } { \\sqrt { \\pi } \\Gamma ( \\frac { n + 1 } { 2 } ) } . \\end{equation}"}
{"id": "204.png", "formula": "\\begin{equation} Q \\ = \\ \\bigcup _ { n = 2 } ^ { \\infty } \\ ( \\{ t \\} \\times A _ n ) \\times ( \\{ t _ n \\} \\times A _ 1 ) . \\end{equation}"}
{"id": "266.png", "formula": "\\[ b _ 2 ( x _ 2 ) = \\frac { b _ 1 ( x _ 1 ) } { | A _ 1 ( x _ 1 ) | } , \\]"}
{"id": "380.png", "formula": "\\begin{equation} x - a = y - b = z - ( a + b ) = \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } p ] { p a b K _ p } = r s \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } p ] { p K _ p } , \\end{equation}"}
{"id": "334.png", "formula": "\\begin{equation} \\begin{split} & v _ t = ( m - 1 ) v \\Delta v + | \\nabla v | ^ 2 + K ( m , p ) v ^ { ( m + p - 2 ) / ( m - 1 ) } , \\\\ & K ( m , p ) = m \\left ( \\frac { m - 1 } { m } \\right ) ^ { ( m + p - 2 ) / ( m - 1 ) } . \\end{split} \\end{equation}"}
{"id": "396.png", "formula": "\\[ x - a = y - b = z - ( a + b ) = \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } 3 ] { 3 a b K _ 3 } , \\]"}
{"id": "185.png", "formula": "\\begin{align*} & \\sum _ { l } \\langle j _ { a , b } ( f _ s ) \\ | \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\rangle j _ { a , b } ( e _ { i } ) \\langle \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\ | j _ { a , b } ( f _ { j } ) ) \\rangle \\\\ & = j _ { a , b } \\left ( ( f ^ { * } _ { s } \\otimes 1 _ { Z } ) \\circ 1 _ { X ^ { b - a } } \\otimes c _ { Z , W } ) \\circ ( e _ { i } \\otimes 1 _ { W } ) \\circ f _ { j } \\right ) \\end{align*}"}
{"id": "71.png", "formula": "\\begin{equation} \\begin{aligned} S : = & w _ 1 G D _ 1 ( p - 2 + s ) + w _ 2 G D _ 2 ( p - 2 + s ) \\\\ & + \\epsilon w _ 3 G D _ 1 ( p - 4 + s ) + \\epsilon w _ 4 G D _ 2 ( p - 4 + s ) \\end{aligned} \\end{equation}"}
{"id": "47.png", "formula": "\\begin{align*} \\langle \\omega , \\eta \\rangle _ { \\ell ^ 2 ( h \\Z ^ n ; \\bigwedge ^ { j } ( h \\Z ^ n ) ) } & = \\frac 1 { h ^ { 2 j } } \\sum _ { \\mu \\in h \\Z ^ n ; I \\in P ^ { j , n } _ + } \\omega _ I ( \\mu ) \\overline { \\eta } _ I ( \\mu ) , \\\\ \\langle f , g \\rangle _ { \\ell ^ 2 \\Big ( h \\Z ^ n ; \\C ^ { \\binom { n } { j } } \\Big ) } & = \\sum _ { \\mu \\in h \\Z ^ n ; 1 \\leq l \\leq \\binom { n } { j } } f _ l ( \\mu ) \\overline { g } _ l ( \\mu ) . \\end{align*}"}
{"id": "22.png", "formula": "\\begin{equation} \\lim _ { \\kappa \\to \\infty } \\ , \\sup _ { q \\in Q _ * } \\ , \\sup _ { | t | \\leq T } \\norm { \\tfrac { d n } { d t } } _ { H ^ { - 2 } } = 0 , \\end{equation}"}
{"id": "415.png", "formula": "\\[ [ { \\tt Q } ] _ { A ^ { ( \\rho ) } } J = [ { \\tt Q } ] _ { \\lnot A ^ { ( \\rho ) } } J = [ { \\tt Q } ] _ { A } J \\cap [ \\partial { \\tt Q } ] J \\cap [ \\rho ] J \\]"}
{"id": "107.png", "formula": "\\begin{equation} P : = p - 1 \\quad \\text { a n d } K : = \\gamma + 1 , \\end{equation}"}
{"id": "36.png", "formula": "\\begin{equation} \\omega ( \\mu ) = \\sum _ { I \\in P ^ { j , n } _ + } \\omega _ { I } ( \\mu ) d x ^ { I } , \\end{equation}"}
{"id": "403.png", "formula": "\\begin{equation} x ^ p + y ^ p - z ^ p = R ^ p + \\sum _ { i = 1 } ^ { p - 1 } \\binom { p } { i } \\bigg ( ( z - y ) ^ { p - i } R ^ i + ( - z ) ^ { p - i } y ^ i \\bigg ) , \\end{equation}"}
{"id": "300.png", "formula": "\\begin{equation} \\dot { z } = F ( z ) , z \\in \\mathbb { D } ^ n ( \\rho ) , \\end{equation}"}
{"id": "63.png", "formula": "\\begin{align*} B _ l ( A _ i ) ( s ) = B _ l ( ( - 1 ) ^ { j - i } ( \\lfloor s \\rfloor ; \\prescript { } { i } { \\hat { s } } ) = & B _ l ( ( \\lceil s \\rceil - \\delta _ i ; ( \\prescript { } { i } { \\hat { s } } ) ^ * ) \\\\ = & ( - 1 ) ^ { l } ( \\lfloor s \\rfloor ; ( \\prescript { } { l } { ( ( \\prescript { } { i } { \\hat { s } } ) ^ * } ) ) ^ * ) \\\\ = & ( \\lfloor s \\rfloor ; \\prescript { } { j - l } { ( \\prescript { } { i } { \\hat { s } } ) } ) = \\begin{cases} ( \\lfloor s \\rfloor ; \\prescript { } { j - l } { ( \\prescript { } { i } { \\hat { s } } ) } ) & \\text { i f } j - l < i \\\\ ( \\lfloor s \\rfloor ; \\prescript { } { i } { ( \\prescript { } { j - l + 1 } { \\hat { s } } ) } ) & \\text { i f } i \\leq j - l \\end{cases} \\ . \\end{align*}"}
{"id": "290.png", "formula": "\\begin{equation} \\frac { d ^ 2 x } { d \\tau ^ 2 } + \\tilde g ( x ) \\frac { d x } { d \\tau } + \\tilde h ( x ) = 0 , \\end{equation}"}
{"id": "303.png", "formula": "\\begin{equation} \\sum _ { k = 1 } ^ { + \\infty } | \\alpha ( k ) | \\ , \\epsilon _ k \\ , \\rho ^ { 2 | \\alpha ( k ) | } \\end{equation}"}
{"id": "251.png", "formula": "\\begin{equation} \\frac { d ^ 2 x } { d t ^ 2 } = X \\left ( x , \\frac { d x } { d t } \\right ) , \\end{equation}"}
{"id": "421.png", "formula": "\\begin{equation} \\begin{aligned} \\bold { d } & = [ \\sigma ^ 2 _ { R } ( l _ { x , 1 } , l _ { y , 1 } ) , . . . , \\sigma ^ 2 _ { R } ( l _ { x , n _ R } , l _ { y , n _ R } ) ] ^ { T } , \\\\ \\widetilde { \\bold { d } } & = [ \\sigma ^ 2 _ { S } ( m _ { x , 1 } , m _ { y , 1 } ) , . . . , \\sigma ^ 2 _ { S } ( m _ { x , n _ S } , m _ { y , n _ S } ) ] ^ { T } . \\end{aligned} \\end{equation}"}
{"id": "429.png", "formula": "\\begin{equation} \\small \\begin{aligned} & C _ 2 = \\det ( \\\\ & \\begin{bmatrix} \\bold { I } _ { j } ^ { [ 1 : j - 1 ] } \\ ! - \\ ! \\bold { \\Pi } _ { j } ^ { [ 1 : 1 - j ] } & - \\bold { \\Pi } _ { j } ^ { [ j ] } & - \\bold { \\Gamma } _ { j } ^ { [ 1 : j - 1 ] } & - \\bold { \\Gamma } _ { j } ^ { [ j ] } \\\\ - \\bold { \\Xi } _ { j - 1 } - \\widetilde { \\bold { \\Lambda } } _ { j - 1 } & - \\bold { \\Xi } _ { j } ^ { [ j ] } & \\bold { I } _ { j - 1 } \\ ! - \\ ! \\bold { \\Pi } _ { j - 1 } ^ { T } & - \\bold { \\Pi } _ { j } ^ { ( j ) , T } \\\\ - \\bold { \\Xi } _ { j } ^ { ( j ) } & 0 & - \\bold { \\Pi } _ { j } ^ { [ j ] , T } & - \\Pi _ { j , j } \\end{bmatrix} ) . \\end{aligned} \\end{equation}"}
{"id": "48.png", "formula": "\\begin{align*} I ( i ) = & I ' ( i ) , i < l , \\\\ I ( l ) < & I ' ( l ) . \\end{align*}"}
{"id": "423.png", "formula": "\\begin{equation} U _ N = \\max \\{ U _ R , U _ S \\} , ~ ~ U _ M = \\min \\{ U _ R , U _ S \\} . \\end{equation}"}
{"id": "72.png", "formula": "\\begin{align} \\begin{cases} c _ 1 = w _ 1 + w _ 3 \\kappa , & c _ 2 = \\big ( w _ 1 ( p - 2 + s ) + w _ 3 ( p - 4 + s ) \\kappa \\big ) \\theta , \\\\ c _ 3 = w _ 2 + w _ 4 \\kappa , & c _ 4 = \\big ( w _ 2 ( p - 2 + s - \\gamma ) + w _ 4 ( p - 4 + s - \\gamma ) \\kappa \\big ) \\theta . \\end{cases} \\end{align}"}
{"id": "292.png", "formula": "\\begin{equation} e _ { 2 k } e _ 1 = 0 \\ \\mbox { a n d } \\ e _ { 2 k + 1 } e _ 1 = e _ 1 e _ { 2 k + 1 } = e _ { 2 k + 2 } . \\end{equation}"}
{"id": "260.png", "formula": "\\begin{equation} \\bar { v } = \\frac { d \\bar { x } } { d \\bar { t } } = \\frac { 1 } { h } \\frac { d \\varphi } { d x } \\frac { d x } { d t } = \\frac { 1 } { h } \\frac { d \\varphi } { d x } v , \\end{equation}"}
{"id": "171.png", "formula": "\\begin{align*} \\mu ^ { \\alpha } _ { X , Y } ( \\eta _ { X } \\boxtimes \\eta _ { Y } ) ( x \\boxtimes y ) & = \\eta _ { X } ( x ) \\boxtimes \\eta _ { Y } ( y ) \\\\ & = \\left ( \\prod _ { i \\in J _ { H } } u ^ { * } _ { i } \\right ) x \\left ( \\prod _ { i \\in J _ { H } } u _ { i } \\right ) \\boxtimes \\left ( \\prod _ { i \\in J _ { H } } u ^ { * } _ { i } \\right ) y \\left ( \\prod _ { i \\in J _ { H } } u _ { i } \\right ) \\\\ & = \\left ( \\prod _ { i \\in J _ { H } } u ^ { * } _ { i } \\right ) \\left ( x \\boxtimes y \\right ) \\left ( \\prod _ { i \\in J _ { H } } u _ { i } \\right ) \\\\ & = \\eta _ { X \\boxtimes Y } ( \\mu ^ { \\alpha } _ { X , Y } ( x \\boxtimes y ) ) \\end{align*}"}
{"id": "201.png", "formula": "\\begin{align} \\begin{split} a _ i , b _ i \\in & U _ { i - 1 } , \\overline { U _ { i } } \\subset U _ { i - 1 } , \\\\ \\{ a _ i , z , b _ i \\} \\ \\text { i s a } \\ & 3 - \\text { c y c l e , f o r a l l } \\ z \\in \\overline { U _ i } . \\end{split} \\end{align}"}
{"id": "267.png", "formula": "\\begin{align*} B = \\frac { d x _ 1 } { d x _ 2 } \\frac { d } { d x _ 1 } \\left ( \\frac { b _ 1 } { A _ 1 } \\right ) = \\frac { 1 } { A _ 1 } \\frac { d } { d x } \\left ( \\frac { b _ 0 } { A _ 0 h _ 0 } \\right ) = \\frac { 1 } { A _ 0 ^ 3 } \\left ( A _ 0 \\frac { d } { d x } + \\gamma A _ 0 - A ' _ 0 \\right ) b _ 0 . \\end{align*}"}
{"id": "225.png", "formula": "\\begin{equation} R \\ = \\ S \\ \\cup \\ P \\ \\cup \\ \\ \\bigcup _ i [ H _ i \\times ( E _ i \\cup G _ i ) \\ \\cup \\ F _ i \\times H _ i ] . \\end{equation}"}
{"id": "170.png", "formula": "\\begin{align*} f ( \\eta _ { X } ( x ) ) & = f ( \\left ( \\prod _ { i \\in J _ { F } } u ^ { * } _ { i } \\right ) x \\left ( \\prod _ { i \\in J _ { F } } u _ { i } \\right ) ) \\\\ & = \\left ( \\prod _ { i \\in J _ { F } } u ^ { * } _ { i } \\right ) f ( x ) \\left ( \\prod _ { i \\in J _ { F } } u _ { i } \\right ) \\\\ & = \\eta _ { Y } ( f ( x ) ) \\\\ & = \\eta _ { Y } ( \\alpha _ { * } ( f ) ( x ) ) \\\\ \\end{align*}"}
{"id": "132.png", "formula": "\\begin{align*} \\| \\operatorname { d i v } \\vec { v } \\| & = \\sup _ { q \\in Q , ~ \\| q \\| = 1 } ( { \\rm d i v } \\vec { v } , q ) = \\sup _ { q \\in Q , ~ \\| q \\| = 1 } ( { \\rm d i v } ( \\vec { v } - P \\vec { v } ) , q ) \\\\ & \\leq \\left \\| { \\rm t r } [ \\varepsilon ( \\vec { v } - P \\vec { v } ) ] \\right \\| \\leq \\sqrt { d } \\left \\| \\varepsilon ( \\vec { v } - P \\vec { v } ) \\right \\| . \\end{align*}"}
{"id": "29.png", "formula": "\\begin{equation} \\Delta _ j ( X ) = d _ j ^ * d _ j + d _ { j - 1 } d _ { j - 1 } ^ * \\ . \\end{equation}"}
{"id": "387.png", "formula": "\\begin{equation} K _ p \\equiv \\begin{cases} b & \\textnormal { f o r F L T 2 C a s e I } , \\\\ a & \\textnormal { f o r F L T 2 C a s e I I } , \\end{cases} \\pmod { p } , \\end{equation}"}
{"id": "67.png", "formula": "\\begin{align*} \\Delta u : = \\sum _ { i = 1 } ^ n u _ { x _ i x _ i } \\end{align*}"}
{"id": "4.png", "formula": "\\begin{equation} \\psi ( E ) = \\phi ( E ) + E \\phi ' ( E ) . \\end{equation}"}
{"id": "274.png", "formula": "\\begin{equation} \\frac { d ^ 2 y } { d \\tau ^ 2 } = 0 \\end{equation}"}
{"id": "320.png", "formula": "\\begin{equation} u _ t = \\Delta u ^ m + | x | ^ { \\sigma } u ^ p , \\end{equation}"}
{"id": "279.png", "formula": "\\[ \\ddot { x } = 2 \\dot { x } ^ 2 \\cot x + \\sin x \\cos x , \\]"}
{"id": "114.png", "formula": "\\begin{align*} \\lim _ { G \\to P } \\frac { \\sqrt { E } - \\sqrt { \\frac { G } { n - 1 } + K } } { \\sqrt { G } - \\sqrt { P } } = \\frac { ( n - 2 ) \\sqrt { P } } { ( n - 1 ) \\sqrt { \\frac { P } { n - 1 } + K } } . \\end{align*}"}
{"id": "81.png", "formula": "\\begin{align*} \\left | D ( | D u | ^ { \\frac { p - 2 + s } 2 } D u ) \\right | & = \\frac { 1 } { 2 } \\alpha ^ { \\frac { p + s } { 2 } } ( \\alpha - 1 ) ( p + s ) | x _ 1 | ^ { \\frac { ( \\alpha - 1 ) ( p + s ) } { 2 } - 1 } \\\\ & = C ( p , s , \\gamma ) | x _ 1 | ^ { \\frac { p + s } { 2 ( \\gamma + 1 ) } - 1 } . \\end{align*}"}
{"id": "291.png", "formula": "\\begin{equation} e _ 1 e _ i = e _ { i + 1 } . \\end{equation}"}
{"id": "215.png", "formula": "\\begin{equation} T \\ = \\ S \\cup R \\cup ( \\bigcup _ i [ ( E _ i \\times C _ i ) \\cup ( C _ i \\times F _ i ) ] ) . \\end{equation}"}
{"id": "342.png", "formula": "\\begin{equation} J = x _ n x _ { n - 1 } \\big [ I ( G _ 3 ) ^ { [ k ] } + x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } + \\sum _ { j = 1 } ^ { t } x _ { i _ j } I ( G _ 2 ) ^ { [ k - 1 ] } \\big ] . \\end{equation}"}
{"id": "278.png", "formula": "\\[ y = \\omega ^ 2 x ^ 2 , d \\tau = \\frac { \\omega ^ 2 } { x } \\ , d t . \\]"}
{"id": "181.png", "formula": "\\begin{align*} i _ { c , d } ( x ) \\triangleright j _ { a , b } ( b _ { i } ) & = j _ { a - k , b + k } ( x \\otimes 1 _ { X ^ { b - d + k } } \\triangleright 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } ) \\\\ & = j _ { a - k , b + k } ( x \\otimes 1 _ { X ^ { a - d } } \\otimes 1 _ { X ^ { b - c } } \\triangleright 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } ) \\\\ & = j _ { a - k , b + k } ( ( x \\otimes 1 _ { X ^ { a - d } } \\otimes 1 _ { X ^ { b - c } } ) \\circ ( 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } ) ) \\\\ & = j _ { a - k , b + k } ( ( 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } ) \\circ ( x \\otimes 1 _ { X ^ { a - d } } \\otimes 1 _ { X ^ { b - c } } \\circ ) \\\\ & = j _ { a , b } ( b _ { i } ) \\triangleleft i _ { c , d } ( x ) \\end{align*}"}
{"id": "158.png", "formula": "\\begin{equation} 2 \\bigg [ G \\big ( s + \\log ( r _ 1 ) \\big ) \\cdot { r _ 1 } ^ { \\alpha ( \\eta ) } + G \\big ( s + \\log ( r _ 2 ) \\big ) \\cdot { r _ 2 } ^ { \\alpha ( \\eta ) } \\bigg ] = G ( s ) + e ^ { ( 2 n - \\alpha ( \\eta ) ) s } R ( e ^ { - s } ) . \\end{equation}"}
{"id": "293.png", "formula": "\\begin{equation} e _ { 2 k } e _ 2 = k e _ { 2 k + 2 } , \\ e _ { 2 k + 1 } e _ 2 = ( k + 1 ) e _ { 2 k + 3 } , \\ e _ 2 e _ { 2 k } = e _ { 2 k + 2 } , \\ e _ 2 e _ { 2 k + 1 } = 0 . \\end{equation}"}
{"id": "87.png", "formula": "\\begin{align} c _ 3 = w _ 2 + w _ 4 \\kappa \\geq c , \\end{align}"}
{"id": "54.png", "formula": "\\begin{equation} ( \\mathcal { F } H \\mathcal { F } ^ * - z ) ^ { - 1 } ( \\xi ) = \\frac { 1 } { R _ z ( \\xi ) } \\mathcal { F } H \\mathcal { F } ^ * + \\frac { z } { R _ z ( \\xi ) } \\end{equation}"}
{"id": "84.png", "formula": "\\begin{equation} \\begin{aligned} \\begin{cases} w _ 1 = p - \\gamma , & w _ 2 = 2 \\\\ w _ 3 = 4 - p + \\gamma , & w _ 4 = 2 , \\end{cases} \\end{aligned} \\end{equation}"}
{"id": "91.png", "formula": "\\begin{align*} \\sup _ \\theta X _ 1 ( \\theta ) & = \\max \\{ X _ 1 ( 0 ) , X _ 1 ( 1 ) \\} \\\\ & = \\max \\Big \\{ 0 , \\big ( \\sqrt { p - 1 } - \\sqrt { 1 - \\gamma } \\big ) ^ 2 \\Big \\} \\\\ & = \\big ( \\sqrt { p - 1 } - \\sqrt { 1 - \\gamma } \\big ) ^ 2 . \\end{align*}"}
{"id": "126.png", "formula": "\\begin{equation} a ( P \\vec { v } , \\vec { w } ) = a ( \\vec { v } , \\vec { w } ) , \\forall \\vec { w } \\in W . \\end{equation}"}
{"id": "144.png", "formula": "\\begin{equation} X = \\sum _ { i \\in I } \\delta _ { x _ i } , \\end{equation}"}
{"id": "366.png", "formula": "\\begin{equation} \\begin{aligned} A _ p ( z , y ) & = \\sum _ { i = 0 } ^ { k - 1 } ( z y ) ^ { i } ( z ^ { 2 ( k - i ) } + y ^ { 2 ( k - i ) } ) , \\\\ D _ p ( z , y ) & = \\sum _ { i = 0 } ^ { k - 1 } ( z y ) ^ { i } ( z ^ { k - i } - y ^ { k - i } ) ^ 2 \\end{aligned} \\end{equation}"}
{"id": "177.png", "formula": "\\begin{align*} u _ { X ^ { \\prime } , Y ^ { \\prime } } \\circ ( f \\boxtimes g ) ( b _ { i } \\boxtimes c _ { j } ) & = \\sum _ { l , k } u _ { X ^ { \\prime } , Y ^ { \\prime } } \\left ( b ^ { \\prime } _ { l } \\langle b ^ { \\prime } _ { l } \\ | \\ f ( b _ { i } ) \\rangle \\boxtimes c ^ { \\prime } _ { k } \\langle c ^ { \\prime } _ { k } \\ | \\ g ( c _ { j } ) \\rangle \\right ) \\\\ & = \\sum _ { l , k } u _ { X ^ { \\prime } , Y ^ { \\prime } } ( b ^ { \\prime } _ { l } \\boxtimes c ^ { \\prime } _ { k } ) \\langle b ^ { \\prime } _ { l } \\ | \\ f ( b _ { i } ) \\rangle \\langle c ^ { \\prime } _ { k } \\ | \\ g ( c _ { j } ) \\rangle \\\\ & = \\sum _ { l , k } c ^ { \\prime } _ { k } \\langle c ^ { \\prime } _ { k } \\ | \\ g ( c _ { j } ) \\rangle \\boxtimes b ^ { \\prime } _ { l } \\langle b ^ { \\prime } _ { l } \\ | \\ f ( b _ { i } ) \\rangle \\\\ & = g ( c _ j ) \\boxtimes f ( b _ i ) \\\\ & = ( g \\boxtimes f ) \\circ u _ { X , Y } ( b _ { i } \\boxtimes c _ { j } ) \\end{align*}"}
{"id": "264.png", "formula": "\\[ \\bar { A } ( \\bar { x } ) = \\frac { 1 } { h ( x ) } A ( x ) \\qquad \\text { a n d } \\bar { b } ( \\bar { x } ) = \\frac { c } { h ( x ) } b ( x ) , \\]"}
{"id": "69.png", "formula": "\\[ \\begin{split} \\lim _ { x \\to 0 , x \\neq 0 } F ( D g ( x ) , D ^ 2 g ( x ) ) = 0 . \\end{split} \\]"}
{"id": "104.png", "formula": "\\begin{equation} \\begin{aligned} & ( \\lambda - 2 \\eta ) \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p - 2 + s } { 2 } } | D ^ 2 u | ^ 2 \\phi ^ 2 d x d t \\\\ & \\leq \\frac { C } { \\eta } \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p - 2 + s } { 2 } } | D u | ^ 2 | D \\phi | ^ 2 d x d t + C \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p + s - \\gamma } { 2 } } | \\phi _ t | \\phi d x d t , \\end{aligned} \\end{equation}"}
{"id": "311.png", "formula": "\\begin{equation} F ' _ l ( z ' ) = \\sum _ { k = 1 } ^ { r } \\mu ^ { | \\alpha ( k ) | - 1 } a _ { l , k } z '^ { \\alpha ( k ) } . \\end{equation}"}
{"id": "56.png", "formula": "\\begin{align*} \\left ( 1 - Q _ h ^ * Q _ h \\right ) g ( \\xi ) & = g ( \\xi ) - \\sum _ { \\mu \\in \\{ 0 , \\pm 1 \\} ^ n } \\hat { \\varphi } ( h \\xi ) \\overline { \\hat { \\varphi } ( h \\xi + \\mu ) } g ( \\xi + h ^ { - 1 } \\mu ) \\\\ & = ( 1 - | \\hat \\varphi ( h \\xi ) | ^ 2 ) g ( \\xi ) - \\sum _ { 0 \\neq \\mu \\in \\{ 0 , \\pm 1 \\} ^ n } \\hat { \\varphi } ( h \\xi ) \\overline { \\hat { \\varphi } ( h \\xi + \\mu ) } g ( \\xi + h ^ { - 1 } \\mu ) . \\end{align*}"}
{"id": "150.png", "formula": "\\begin{equation} { [ f ] } _ { W ^ { 1 , p } ( A ) } = { \\bigg ( \\int _ A { | \\nabla f | } ^ p \\ d x \\bigg ) } ^ { \\frac { 1 } { p } } , \\end{equation}"}
{"id": "25.png", "formula": "\\begin{equation} \\big ( X - t \\kappa R ( \\kappa ; q ^ 0 ) ^ 2 - z \\big ) ^ { - 1 } q ^ 0 _ + = U ( t ) ^ * ( X - z ) ^ { - 1 } U ( t ) q ^ 0 _ + \\end{equation}"}
{"id": "135.png", "formula": "\\begin{equation} \\vec { u } _ { \\lambda } = \\frac { \\lambda } { \\lambda + 1 } \\vec { u } _ { \\infty } + \\frac { 1 } { \\lambda + 1 } \\vec { u } _ { 0 } . \\end{equation}"}
{"id": "327.png", "formula": "\\begin{equation} ( f ^ m ) '' ( \\xi ) - \\alpha f ( \\xi ) + \\beta \\xi f ' ( \\xi ) + \\xi ^ { \\sigma } f ( \\xi ) ^ p = 0 , \\xi = | x | ^ { \\sigma } ( T - t ) ^ { \\beta } . \\end{equation}"}
{"id": "151.png", "formula": "\\begin{equation} \\lim _ { t \\to 0 ^ { + } } \\frac { 1 } { t ^ { n + p } } \\int _ A \\int _ A { | f ( x ) - f ( y ) | } ^ p \\cdot J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y = K ( n , p ) \\cdot { \\big ( { [ f ] } _ { W ^ { 1 , p } ( A ) } \\big ) } ^ p , \\end{equation}"}
{"id": "307.png", "formula": "\\begin{equation} F ( z _ 1 , z _ 2 ) = \\begin{cases} a \\left ( z _ 1 - \\frac { 1 } { a c } z _ 2 \\right ) \\\\ a \\left ( z _ 2 - \\frac { 1 } { a c } z _ 1 + b z _ 1 ^ 2 \\right ) , \\end{cases} \\end{equation}"}
{"id": "79.png", "formula": "\\begin{align*} D \\varphi ( x _ 0 , t _ 0 ) = D u ( x _ 0 , t _ 0 ) \\neq 0 , \\phi _ t ( x _ 0 , t _ 0 ) = u _ t ( x _ 0 , t _ 0 ) = 0 \\end{align*}"}
{"id": "152.png", "formula": "\\begin{equation} 2 \\cdot { \\bigg ( \\frac { \\eta - 1 } { 2 \\eta } \\bigg ) } ^ { \\alpha ( \\eta ) } + 2 \\cdot { \\bigg ( \\frac { 1 } { \\eta } \\bigg ) } ^ { \\alpha ( \\eta ) } = 1 . \\end{equation}"}
{"id": "141.png", "formula": "\\begin{equation} \\left ( A ^ h _ { \\lambda } \\right ) ^ { - 1 } \\eqsim M ^ h _ { \\lambda } : = \\frac { \\lambda } { 1 + \\lambda } P _ h A _ h ^ { - 1 } + \\frac { 1 } { \\lambda + 1 } A _ h ^ { - 1 } . \\end{equation}"}
{"id": "353.png", "formula": "\\begin{equation} \\begin{aligned} I ^ { [ k ] } : I ^ { [ \\ell ] } \\ & = \\ ( I _ 0 ^ { [ k ] } : I ^ { [ \\ell ] } ) + \\ ! \\ ! \\ ! \\ ! \\ ! \\sum _ { \\substack { p \\in [ s ] \\\\ \\nu ( I _ p ) \\ge k - 1 } } \\ ! \\ ! \\ ! \\ ! ( u _ p I _ p ^ { [ k - 1 ] } : I ^ { [ \\ell ] } ) . \\end{aligned} \\end{equation}"}
{"id": "105.png", "formula": "\\begin{equation} \\begin{aligned} & \\lambda \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\gamma / 2 } | D ^ 2 u | ^ 2 \\phi ^ 2 d x d t \\\\ & \\leq C \\Big ( \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\gamma / 2 } | D ^ 2 u | | D u | | D \\phi | \\phi d x d t + \\int _ { Q _ { 2 r } } | u _ t | | D u | | D \\phi | \\phi d x d t \\\\ & + \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) | \\phi _ t | \\phi d x d t + \\epsilon \\int _ { Q _ { 2 r } } \\big | \\ln ( | D u | ^ 2 + \\epsilon ) \\big | | \\phi _ t | \\phi d x d t \\\\ & + \\epsilon \\int _ { B _ { 2 r } } \\abs { \\ln \\big ( | D u ( x , t _ 0 ) | ^ 2 + \\epsilon \\big ) } \\phi ^ 2 ( x , t _ 0 ) d x \\Big ) , \\end{aligned} \\end{equation}"}
{"id": "186.png", "formula": "\\begin{align*} & \\mu ^ { [ a , b ] } _ { ( Z , c ) , ( W , d ) } \\circ u _ { F _ { a , b } ( Z , c ) , F _ { a , b } ( W , d ) } ( j _ { a , b } ( e _ { i } ) \\boxtimes j _ { a , b } ( f _ { j } ) ) \\\\ & = j _ { a , b } \\circ \\mu ^ { 0 ; [ a , b ] } _ { ( Z , c ) , ( W , d ) } \\left ( \\sum _ { s } f _ { s } \\boxtimes ( f ^ { * } _ { s } \\otimes 1 _ { Z } ) \\circ 1 _ { X ^ { b - a } } \\otimes c _ { Z , W } ) \\circ ( e _ { i } \\otimes 1 _ { W } ) \\circ f _ { j } \\right ) \\\\ & = j _ { a , b } \\left ( ( 1 _ { X ^ { b - a } } \\otimes c _ { Z , W } ) \\circ ( e _ { j } \\otimes 1 _ { W } ) \\circ f _ { j } \\right ) \\end{align*}"}
{"id": "209.png", "formula": "\\begin{align} \\begin{split} ( a _ 1 , b _ 1 ) , \\ \\ ( b _ 2 , a _ 2 ) , ( a _ 1 , b _ 2 ) , & \\ \\ ( a _ 2 , b _ 1 ) , ( a _ 1 , a _ 2 ) , \\ \\ ( b _ 1 , b _ 2 ) , \\\\ ( c , a _ 1 ) , \\ \\ ( c , a _ 2 ) , & ( b _ 1 , c ) , \\ \\ ( b _ 2 , c ) . \\end{split} \\end{align}"}
{"id": "416.png", "formula": "\\begin{equation} D _ t u _ 0 = S ( t ) u _ 0 . \\end{equation}"}
{"id": "309.png", "formula": "\\begin{equation} F ( z _ 1 , z _ 2 ) = \\begin{cases} a \\left ( z _ 1 - \\dfrac { 2 z _ 2 ^ 2 } { c - z _ 2 } \\right ) \\\\ a \\left ( z _ 2 - \\dfrac { b z _ 1 ^ 2 } { ( d - z _ 1 ) ^ 2 } \\right ) , \\end{cases} \\end{equation}"}
{"id": "109.png", "formula": "\\begin{align*} \\operatorname { d e t } ( M ) = a \\Big ( w _ 2 - \\frac { n - 2 } { n - 1 } \\Big ) ^ 2 + b \\Big ( w _ 2 - \\frac { n - 2 } { n - 1 } \\Big ) + c \\end{align*}"}
{"id": "30.png", "formula": "\\begin{equation} d _ j ^ * d _ j f = - d _ { j - 1 } d _ { j - 1 } ^ * f \\ . \\end{equation}"}
{"id": "2.png", "formula": "\\begin{align} q ( \\vec t ; q _ 0 ) = \\Bigl [ \\exp \\bigl \\{ { \\textstyle \\sum } t _ i J \\nabla H _ i \\bigr \\} q _ 0 \\Bigr ] ( x = 0 ) \\end{align}"}
{"id": "268.png", "formula": "\\[ \\frac { d } { d x } ( A _ 0 ^ { - 3 } z ) = A _ 0 ^ { - 3 } z ' - 3 A _ 0 ^ { - 4 } A ' _ 0 z = A _ 0 ^ { - 4 } ( A _ 0 z ' - 3 A ' _ 0 z ) , \\]"}
{"id": "119.png", "formula": "\\begin{equation} \\begin{aligned} \\langle A _ \\lambda \\vec { u } , \\vec { v } \\rangle & : = a _ \\lambda ( \\vec { u } , \\vec { v } ) , \\\\ \\langle B \\vec { v } , q \\rangle & : = b ( \\vec { v } , q ) = ( \\operatorname { d i v } \\vec { v } , q ) . \\end{aligned} \\end{equation}"}
{"id": "100.png", "formula": "\\begin{align*} c _ 3 + c _ 4 = & 2 ( 1 - \\gamma ) - 2 \\kappa + 2 ( 2 + \\gamma ) \\kappa ^ 2 . \\end{align*}"}
{"id": "301.png", "formula": "\\begin{equation} \\left \\langle L _ F e _ { k } , e _ { j } \\right \\rangle = \\begin{cases} \\sum _ { l = 1 } ^ n \\alpha _ l ( k ) \\ , a _ { l , ( \\alpha ( j ) - \\alpha ( k ) ) _ l } & \\textrm { i f } | \\alpha ( j ) | \\geq | \\alpha ( k ) | \\\\ 0 & \\textrm { i f } | \\alpha ( j ) | < | \\alpha ( k ) | . \\end{cases} \\end{equation}"}
{"id": "363.png", "formula": "\\begin{equation} x ^ p + y ^ p = z ^ p , \\end{equation}"}
{"id": "306.png", "formula": "\\begin{equation} L _ \\mu = \\sum _ { l = 1 } ^ n \\sum _ { k = 1 } ^ \\infty \\mu ^ { \\vert \\alpha ( k ) \\vert } \\left | a _ { l , k } \\right | . \\end{equation}"}
{"id": "310.png", "formula": "\\[ L _ \\mu = \\left ( 1 0 + 2 \\sum _ { k = 0 } ^ \\infty \\dfrac { 1 0 ^ { k + 2 } } { 3 0 ^ { k + 1 } } \\right ) + \\left ( 1 0 + 4 \\sum _ { k = 0 } ^ \\infty \\dfrac { ( k + 1 ) 1 0 ^ { k + 2 } } { 2 0 ^ { k + 2 } } \\right ) = 7 3 / 3 , \\]"}
{"id": "92.png", "formula": "\\begin{align*} \\inf _ \\theta X _ 2 ( \\theta ) & = \\min \\Big \\{ X _ 2 ( 0 ) , X _ 2 \\Big ( \\frac { p - 2 - \\gamma } { ( p - 2 ) \\gamma } \\Big ) , X _ 2 ( 1 ) \\Big \\} \\\\ & = \\min \\Big \\{ 4 , \\frac { p - 2 } { \\gamma } + \\frac { \\gamma } { p - 2 } + 2 , \\big ( \\sqrt { p - 1 } + \\sqrt { 1 - \\gamma } \\big ) ^ 2 \\Big \\} \\\\ & = \\min \\Big \\{ 4 , \\big ( \\sqrt { p - 1 } + \\sqrt { 1 - \\gamma } \\big ) ^ 2 \\Big \\} . \\end{align*}"}
{"id": "369.png", "formula": "\\begin{equation} \\begin{aligned} A _ p ( x , - y ) & = \\sum _ { i = 0 } ^ { k - 1 } ( - 1 ) ^ i ( x y ) ^ { i } ( x ^ { 2 ( k - i ) } + y ^ { 2 ( k - i ) } ) , \\\\ D _ p ( x , - y ) & = \\sum _ { i = 0 } ^ { k - 1 } ( - 1 ) ^ i ( x y ) ^ { i } ( x ^ { k - i } - y ^ { k - i } ) ^ 2 . \\end{aligned} \\end{equation}"}
{"id": "116.png", "formula": "\\begin{equation} \\begin{aligned} & a _ { \\lambda } ( \\vec { u } , \\vec { v } ) = a ( \\vec { u } , \\vec { v } ) + \\lambda b ( \\vec { v } , \\operatorname { d i v } \\vec { u } ) = \\langle \\vec { f } , \\vec { v } \\rangle , \\end{aligned} \\end{equation}"}
{"id": "168.png", "formula": "\\begin{align*} \\eta _ { X } ( x ) & = \\left ( \\prod _ { i } u ^ { * } _ i \\right ) x \\left ( \\prod _ { i } u _ { i } \\right ) \\\\ & = \\left ( \\prod _ { i \\in J _ { F } } u ^ { * } _ { i } \\right ) x \\left ( \\prod _ { i \\in J _ { F } } u ^ { * } _ { i } \\right ) \\\\ & = \\left ( \\prod _ { i \\in J _ { G } } u ^ { * } _ { i } \\right ) x \\left ( \\prod _ { i \\in J _ { G } } u ^ { * } _ { i } \\right ) \\ \\text { f o r a n y $ F \\subseteq G $ f i n i t e } \\end{align*}"}
{"id": "361.png", "formula": "\\begin{equation} \\phi _ n ( z , y ) = \\dfrac { z ^ n - y ^ n } { z - y } = \\sum _ { i = 0 } ^ { n - 1 } z ^ { n - i - 1 } y ^ i , \\end{equation}"}
{"id": "330.png", "formula": "\\begin{equation} ( f ^ m ) '' ( \\xi ) + \\frac { N - 1 } { \\xi } ( f ^ m ) ' ( \\xi ) - \\alpha f ( \\xi ) + \\beta \\xi f ' ( \\xi ) + \\xi ^ { \\sigma } f ( \\xi ) ^ p = 0 , \\xi = | x | ^ { \\sigma } ( T - t ) ^ { \\beta } . \\end{equation}"}
{"id": "399.png", "formula": "\\[ ( x + y - z ) ^ 3 = 3 ( z - y ) ( z - x ) ( x + y ) , \\]"}
{"id": "411.png", "formula": "\\[ M _ { c } ( f ) : = \\bigcap \\{ M _ { d } ( f ( d ) ) : d \\in { \\rm s u p p } ( f ^ { c } ) \\} = \\bigcap \\{ M _ { d } ( f ( d ) ) : c \\leq d \\in { \\rm s u p p } ( f ) \\} . \\]"}
{"id": "222.png", "formula": "\\begin{equation} Z _ { z + } \\ = \\ \\{ z + \\} \\ \\cup \\ \\{ \\bar d _ k - \\} \\ \\cup \\ \\{ i + : i \\ge j - 1 \\} \\ \\cup \\ [ \\bigcup _ { i = 1 } ^ { j - 2 } \\ ( z + A _ i ) + ] . \\end{equation}"}
{"id": "166.png", "formula": "\\begin{align*} \\alpha ^ { - 1 } ( Z _ { B } ( B _ { F ^ { c } } ) ) & = Z _ { A } ( \\alpha ^ { - 1 } ( B _ { F ^ { c } } ) ) \\\\ & \\subseteq Z _ { A } ( A _ { ( F ^ { \\prime } ) ^ { c } } ) \\\\ & \\subseteq A _ { F ^ { \\prime \\prime } } \\end{align*}"}
{"id": "167.png", "formula": "\\begin{align*} \\alpha \\circ \\beta \\circ \\alpha ^ { - 1 } ( x ) & = \\alpha \\left ( \\left ( \\prod u _ { i } \\right ) \\alpha ^ { - 1 } ( x ) \\left ( \\prod u ^ { * } _ { i } \\right ) \\right ) \\\\ & = \\left ( \\prod \\alpha ( u _ { i } ) \\right ) x \\left ( \\prod \\alpha ( u _ { i } ) ^ { * } \\right ) \\\\ & = \\left ( \\prod _ { i _ 1 \\in I _ { 1 } } w _ { i _ 1 } \\right ) . . . \\left ( \\prod _ { i _ { S + 1 } \\in I _ { S + 1 } } w _ { i _ { S + 1 } } \\right ) x \\left ( \\prod _ { i _ { S + 1 } \\in I _ { S + 1 } } w ^ { * } _ { i _ { S + 1 } } \\right ) . . . \\left ( \\prod _ { i _ { 1 } \\in I _ { 1 } } w ^ { * } _ { i _ { 1 } } \\right ) \\\\ & = \\beta _ { 1 } \\circ \\dots \\circ \\beta _ { S + 1 } ( x ) \\end{align*}"}
{"id": "128.png", "formula": "\\begin{equation} M _ { \\lambda } = \\frac { \\lambda } { 1 + \\lambda } P A ^ { - 1 } + \\frac { 1 } { \\lambda + 1 } A ^ { - 1 } , \\end{equation}"}
{"id": "97.png", "formula": "\\begin{align*} 2 c _ 1 + c _ 2 = & 2 ( p - \\gamma ) + 2 ( 4 - p + \\gamma ) \\kappa ^ 2 . \\end{align*}"}
{"id": "66.png", "formula": "\\begin{equation} u _ t - \\Delta _ p u = 0 \\end{equation}"}
{"id": "339.png", "formula": "\\begin{equation} \\beta _ { i , j } ( I ) = \\beta _ { i , j } ( I _ 1 ) + \\beta _ { i , j } ( I _ 2 ) + \\beta _ { i - 1 , j } ( I _ 1 \\cap I _ 2 ) , \\ \\ \\ \\text { f o r a l l } \\ i , j \\ge 0 . \\end{equation}"}
{"id": "384.png", "formula": "\\[ \\begin{aligned} z - y & = 2 ^ { p d } p ^ { p e - 1 } r '^ p , & & \\phi _ p ( z , y ) = p r _ 1 ^ p , & & x = 2 ^ { d } p ^ { e } r ' r _ 1 , \\\\ z - x & = s ^ p , & & \\phi _ p ( z , x ) = s _ 1 ^ p , & & y = s s _ 1 , \\\\ x + y & = t ^ p , & & \\phi _ p ( x , - y ) = t _ 1 ^ p , & & z = t t _ 1 , \\end{aligned} \\]"}
{"id": "42.png", "formula": "\\begin{equation} \\tilde { d } \\circ U = U \\circ d \\ . \\end{equation}"}
{"id": "401.png", "formula": "\\begin{equation} K _ 5 = 4 x ^ 3 + 6 ( b - a ) x ^ 2 + 2 ( 2 b ^ 2 - a b + 2 a ^ 2 ) x - 2 a b c + b ^ 3 - a ^ 3 , \\end{equation}"}
{"id": "413.png", "formula": "\\begin{equation} f _ { d _ { 1 } } = h _ { d _ { 1 } } \\ , \\& \\ , f ^ { d _ { 1 } } < ^ { d _ { 1 } } h ^ { \\prime } ( d _ { 1 } ) \\end{equation}"}
{"id": "280.png", "formula": "\\[ y = \\frac { 1 } { 2 \\sin ^ 2 ( x ) } , d \\tau = | \\cot ( x ) | \\ d t , \\]"}
{"id": "94.png", "formula": "\\begin{align} \\begin{cases} 2 c _ 1 + c _ 2 & = 2 ( w _ 1 + w _ 3 \\kappa ) - 2 w _ 3 \\theta \\kappa \\geq c , \\\\ c _ 3 & = w _ 2 + w _ 4 \\kappa \\geq c , \\\\ \\det ( M ) & = c _ 3 ( c _ 3 + c _ 4 ) P _ \\theta - \\frac { \\left ( c _ 3 P _ \\theta + ( c _ 3 + c _ 4 ) - ( 2 c _ 1 + c _ 2 ) \\right ) ^ 2 } { 4 } \\geq c \\end{cases} \\end{align}"}
{"id": "272.png", "formula": "\\begin{align*} \\left ( A J \\frac { d } { d x } - 3 J A ' \\right ) P & = J \\left ( A \\frac { d } { d x } - 3 A ' \\right ) P . \\end{align*}"}
{"id": "129.png", "formula": "\\begin{equation} ( I + t P ) ^ { - 1 } = I - \\frac { t } { t + 1 } P , \\quad \\forall t \\in \\mathbb { R } \\backslash \\{ - 1 \\} . \\end{equation}"}
{"id": "206.png", "formula": "\\begin{align} \\begin{split} \\{ x \\} \\ = \\ \\ & Q ( x , x ) , \\emptyset \\ = \\ Q ^ { \\circ } ( x , x ) . \\\\ \\{ x , y \\} \\ \\subset \\ \\ & Q ( x , y ) \\ = \\ Q ( \\{ x , y \\} \\times \\{ x , y \\} ) . \\\\ Q ( x , y ) \\setminus \\{ x , y \\} \\ = \\ \\ & Q ^ { \\circ } ( x , y ) \\ = \\ Q ^ { \\circ } ( \\{ x , y \\} \\times \\{ x , y \\} ) . \\end{split} \\end{align}"}
{"id": "57.png", "formula": "\\begin{equation} \\left ( Q _ h ^ * \\dfrac { a _ l } { r _ z } Q _ h - Q _ h ^ * Q _ h \\dfrac { A _ l } { R _ z } \\right ) \\psi = \\sum _ { \\mu \\in \\{ 0 , \\pm 1 \\} ^ n } \\hat { \\varphi } ( h \\xi ) \\overline { \\hat { \\varphi } ( h \\xi + \\mu ) } \\mathcal B _ h ( \\xi + h ^ { - 1 } \\mu ) \\psi ( \\xi + h ^ { - 1 } \\mu ) , \\end{equation}"}
{"id": "122.png", "formula": "\\begin{equation} \\| \\vec { u } \\| \\lesssim \\left ( \\| \\vec { u } \\| _ { H ^ { - 1 } ( \\Omega ) } ^ 2 + \\sum _ { j = 1 } ^ d \\left \\| \\frac { \\partial \\vec { u } } { \\partial x _ j } \\right \\| _ { H ^ { - 1 } ( \\Omega ) } ^ 2 \\right ) ^ { 1 / 2 } \\end{equation}"}
{"id": "130.png", "formula": "\\begin{equation} \\begin{aligned} M _ { \\lambda } ^ { - 1 } & = \\left ( \\frac { 1 } { \\lambda + 1 } \\left ( I + \\lambda P \\right ) A ^ { - 1 } \\right ) ^ { - 1 } \\\\ & = A \\left ( ( \\lambda + 1 ) I - \\lambda P \\right ) \\\\ & = A + \\lambda A ( I - P ) . \\end{aligned} \\end{equation}"}
{"id": "46.png", "formula": "\\begin{align} \\langle ( \\tilde d ^ * \\tilde d ) \\omega , \\omega \\rangle & = \\langle \\tilde d \\omega ; \\tilde d \\omega \\rangle \\nonumber \\\\ & = \\langle \\sum _ I \\sum _ { \\alpha = 1 } ^ n { \\mathcal D _ { \\alpha } \\omega _ I } d x ^ \\alpha \\wedge d x ^ J ; \\sum _ I \\sum _ { \\alpha = 1 } ^ n { \\mathcal D _ { \\alpha } \\omega _ I } d x ^ \\alpha \\wedge d x ^ J \\rangle \\nonumber \\\\ & = \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I \\sum _ { \\substack { \\alpha \\neq I ( i ) \\\\ 1 \\leq i \\leq j } } | \\mathcal D _ { \\alpha } \\omega _ I ( \\mu ) | ^ 2 \\nonumber \\\\ & = \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I \\sum _ { i = j + 1 } ^ n | \\mathcal { D } _ { J _ I ( i ) } \\omega _ I ( \\mu ) | ^ 2 . \\end{align}"}
{"id": "35.png", "formula": "\\begin{equation} \\langle d x ^ I ; d x ^ { I ' } \\rangle _ { \\bigwedge \\nolimits ^ j ( \\Z ^ n ) } : = \\begin{cases} 1 & \\text { i f } I = I ' \\\\ 0 & \\text { i f e l s e } \\end{cases} \\ . \\end{equation}"}
{"id": "394.png", "formula": "\\begin{equation} ( x - a ) ^ 3 = 6 a b x + 3 a b ^ 2 - 3 a ^ 2 b = 3 a b ( 2 x + b - a ) . \\end{equation}"}
{"id": "1.png", "formula": "\\begin{equation} q ( t , x ) = \\widetilde q ( t , x - 2 c t ) + c . \\end{equation}"}
{"id": "385.png", "formula": "\\[ \\begin{aligned} z - y & = 2 ^ { p d } r _ 0 ^ p , & & \\phi _ p ( z , y ) = r _ 1 ^ p , & & x = 2 ^ { d } r _ 0 r _ 1 , \\\\ z - x & = p ^ { p e - 1 } s '^ p , & & \\phi _ p ( z , x ) = p s _ 1 ^ p , & & y = p ^ e s ' s _ 1 , \\\\ x + y & = t ^ p , & & \\phi _ p ( x , - y ) = t _ 1 ^ p , & & z = t t _ 1 , \\end{aligned} \\]"}
{"id": "355.png", "formula": "\\begin{equation} \\begin{aligned} ( u _ p I _ p ^ { [ k - 1 ] } : I ^ { [ \\ell ] } ) \\ & = \\ ( u _ p I _ p ^ { [ k - 1 ] } : I _ p ^ { [ \\ell ] } ) \\cap ( u _ p I _ p ^ { [ \\ell - 1 ] } : u _ p I _ p ^ { [ \\ell - 1 ] } ) \\cap ( u _ p I _ p ^ { [ k - 1 ] } : J ) \\\\ & = \\ u _ p I _ p ^ { [ k - 1 ] } , \\end{aligned} \\end{equation}"}
{"id": "236.png", "formula": "\\begin{equation} i ( \\Gamma ) \\omega _ L = d E _ L . \\end{equation}"}
{"id": "254.png", "formula": "\\begin{equation} \\gamma _ 1 = \\gamma _ 0 + \\frac { h ' } { h } , A _ 1 = \\frac { A _ 0 } { h } \\qquad \\text { a n d } b _ 1 = \\frac { b _ 0 } { h ^ 2 } . \\end{equation}"}
{"id": "289.png", "formula": "\\begin{equation} \\ddot x + f ( x ) \\ , \\dot x ^ 2 + g ( x ) \\ , \\dot x + h ( x ) = 0 \\end{equation}"}
{"id": "75.png", "formula": "\\begin{align*} S = w _ 1 G D _ 1 ( p - 2 + s ) + w _ 2 G D _ 2 ( p - 2 + s ) + \\epsilon w _ 3 G D _ 1 ( p - 4 + s ) + \\epsilon w _ 4 G D _ 2 ( p - 4 + s ) \\end{align*}"}
{"id": "45.png", "formula": "\\begin{align} \\langle ( \\tilde d \\tilde d ^ * ) \\omega , \\omega \\rangle & = \\langle \\tilde d * \\omega ; \\tilde d * \\omega \\rangle \\nonumber \\\\ & = \\langle \\tilde d \\sum _ I { \\rm s i g n } ( I J _ I ) \\omega _ I d x ^ J ; \\tilde d \\sum _ I { \\rm s i g n } ( I J _ I ) \\omega _ I d x ^ { J _ I } \\rangle \\nonumber \\\\ & = \\langle \\sum _ I { \\rm s i g n } ( I J _ I ) \\sum _ { \\alpha = 1 } ^ n { \\mathcal D _ { \\alpha } \\omega _ I } d x ^ \\alpha \\wedge d x ^ { J _ I } ; \\sum _ I { \\rm s i g n } ( I J _ I ) \\sum _ { \\alpha = 1 } ^ n { \\mathcal D _ { \\alpha } \\omega _ I } d x ^ \\alpha \\wedge d x ^ { J _ I } \\rangle \\nonumber \\\\ & = \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I \\sum _ { \\substack { \\alpha \\neq J _ I ( i ) \\\\ j + 1 \\leq i \\leq n } } | \\mathcal D _ { \\alpha } \\omega _ I ( \\mu ) | ^ 2 \\nonumber \\\\ & = \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I \\sum _ { i = 1 } ^ j | \\mathcal { D } _ { I ( i ) } \\omega _ I ( \\mu ) | ^ 2 . \\end{align}"}
{"id": "406.png", "formula": "\\[ M _ { 2 } ( \\bar { \\xi } * ( \\xi ) ; \\bar { a } * ( a ) ) = M _ { 2 } ( \\bar { \\xi } ; \\bar { a } ) \\cap \\bigcap \\{ M _ { 2 } \\left ( M _ { 2 } ( \\bar { \\xi } * ( \\xi ) ; \\bar { a } * ( b ) ) \\cap M _ { 3 } ( \\nu ) \\right ) : \\nu < \\xi , b < a \\} \\]"}
{"id": "160.png", "formula": "\\begin{equation} \\lim _ { s \\to + \\infty } G ( s ) = \\frac { 1 } { 2 r _ 1 ^ { \\alpha ( \\eta ) } \\log ( r _ 1 ) + 2 r _ 2 ^ { \\alpha ( \\eta ) } \\log ( r _ 2 ) } \\cdot \\int _ { 0 } ^ { + \\infty } e ^ { ( 2 n - \\alpha ( \\eta ) ) s } R ( e ^ { - s } ) \\ d s . \\end{equation}"}
{"id": "331.png", "formula": "\\begin{equation} z ( x , t ) = ( T - t ) ^ { - \\alpha } f ( ( 1 + | x | ) ( T - t ) ^ { \\beta } ) , f ( \\xi ) = \\left \\{ \\begin{array} { l l } f _ 1 ( \\xi ; A ) , & \\xi \\in [ 0 , \\overline { \\xi } ] , \\\\ f _ 2 ( \\xi ) , & \\xi \\geq \\overline { \\xi } , \\end{array} \\right . \\end{equation}"}
{"id": "346.png", "formula": "\\begin{equation} J _ 1 \\cap J _ 2 = ( x _ { i _ 1 } , \\dots , x _ { i _ t } ) J _ 1 \\end{equation}"}
{"id": "187.png", "formula": "\\begin{align*} u ^ { F , G } _ { X , Y } ( a ( b _ { i } \\boxtimes c _ { j } ) ) & = \\sum _ { k } c _ { k } \\boxtimes b _ { i } \\langle c _ { k } \\ | \\ a c _ { j } \\rangle \\\\ & = \\sum _ { k } c _ { k } \\boxtimes \\langle c _ { k } \\ | \\ a c _ { j } \\rangle b _ { i } \\\\ & = \\sum _ { k } c _ { k } \\langle c _ { k } \\ | \\ a c _ { j } \\rangle \\boxtimes b _ { i } \\\\ & = a c _ { j } \\boxtimes b _ { i } = a ( c _ { j } \\boxtimes b _ i ) \\\\ & = a u ^ { F , G } _ { X , Y } ( b _ { i } \\boxtimes c _ { j } ) . \\end{align*}"}
{"id": "424.png", "formula": "\\begin{equation} \\begin{aligned} & \\sigma ^ 2 ( l _ x , l _ y , m _ x , m _ y ) = \\sigma ^ 2 _ { R } ( l _ x , l _ y ) \\sigma ^ 2 _ { S } ( m _ x , m _ y ) \\\\ & \\times \\exp ( - \\frac { ( l _ x - m _ x ) ^ 2 + ( l _ y - m _ y ) ^ 2 } { a } ) , \\end{aligned} \\end{equation}"}
{"id": "193.png", "formula": "\\begin{equation} X \\ = \\ \\{ x \\in \\prod _ { i \\in \\N } \\ X _ i : f _ i ( x _ { i + 1 } ) \\ = \\ x _ i \\ \\ \\text { f o r a l l } \\ \\ i \\in \\N \\} , \\end{equation}"}
{"id": "256.png", "formula": "\\begin{equation} \\frac { d ^ 2 x _ 1 } { d t _ 1 ^ 2 } + A _ 1 ( x _ 1 ) \\frac { d x _ 1 } { d t _ 1 } + b _ 1 ( x _ 1 ) = 0 , \\end{equation}"}
{"id": "250.png", "formula": "\\begin{equation} y = \\varphi ( x ) , d \\tau = h ( x ) \\ d t , \\end{equation}"}
{"id": "111.png", "formula": "\\begin{align*} b ^ 2 - 4 a c = G \\cdot P \\cdot E \\Big ( \\frac { G } { n - 1 } + K \\Big ) . \\end{align*}"}
{"id": "59.png", "formula": "\\begin{align*} ( \\prescript { } { j - i _ 0 + 1 } { \\hat { s } } ) ^ * ( i ) = & \\prescript { } { j - i _ 0 + 1 } { \\hat { s } } ( j - i ) \\\\ = & \\begin{cases} \\hat { s } ( j - i ) & j - i < j - i _ 0 + 1 \\\\ \\hat { s } ( j - i + 1 ) & j - i _ 0 + 1 \\leq j - i \\end{cases} \\\\ = & \\begin{cases} \\hat { s } ( j - i ) & i _ 0 \\leq i \\\\ \\hat { s } ( j - i + 1 ) & i < i _ 0 \\end{cases} \\end{align*}"}
{"id": "323.png", "formula": "\\begin{equation} \\lim \\limits _ { t \\to 0 } u ( t ) = u _ 0 , { \\rm w i t h \\ c o n v e r g e n c e \\ i n } \\ L ^ 1 _ { \\rm l o c } ( \\real ^ N ) \\end{equation}"}
{"id": "103.png", "formula": "\\begin{equation} \\begin{aligned} & \\lambda \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p - 2 + s } { 2 } } | D ^ 2 u | ^ 2 \\phi ^ 2 d x d t \\\\ & \\leq C \\Big ( \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p - 2 + s } { 2 } } | D ^ 2 u | | D u | | D \\phi | \\phi d x d t \\\\ & + \\int _ { Q _ { 2 r } } | u _ t | ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p - 2 + s - \\gamma } { 2 } } | D u | | D \\phi | \\phi d x d t \\\\ & + \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p + s - \\gamma } { 2 } } | \\phi _ t | \\phi d x d t \\Big ) , \\end{aligned} \\end{equation}"}
{"id": "338.png", "formula": "\\begin{equation} \\underline { U } ( x , t ) = \\left ( \\frac { 1 } { 1 - p } \\right ) ^ { 1 / ( p - 1 ) } t ^ { 1 / ( 1 - p ) } ( 1 + | x | ) ^ { \\sigma / ( 1 - p ) } , \\end{equation}"}
{"id": "302.png", "formula": "\\begin{equation} \\left \\langle L _ F e _ { k } , e _ { j } \\right \\rangle = \\begin{cases} \\sum _ { l = 1 } ^ { n } \\ , \\alpha _ l ( j ) \\ , a _ { l , \\alpha ( l ) } & \\textrm { i f } j = k \\\\ \\alpha _ { l } ( k ) \\ , a _ { l , \\alpha ( r ) } & \\textrm { i f } \\alpha ( j ) = ( \\alpha _ 1 ( k ) , \\cdots , \\alpha _ { l } ( k ) - 1 , \\cdots , \\\\ & \\alpha _ { r } ( k ) + 1 , \\cdots , \\alpha _ n ( k ) ) , \\\\ 0 & \\textrm { o t h e r w i s e } . \\end{cases} \\end{equation}"}
{"id": "60.png", "formula": "\\begin{align*} \\prescript { } { i _ 1 } { ( \\prescript { } { i _ 0 } { \\hat { s } } ) } ( i ) = & \\begin{cases} \\prescript { } { i _ 0 } { \\hat { s } } ( i ) & i < i _ 1 \\\\ \\prescript { } { i _ 0 } { \\hat { s } } ( i + 1 ) & i _ 1 \\leq i \\end{cases} \\\\ = & \\begin{cases} \\hat { s } ( i ) & i < i _ 1 \\\\ \\hat { s } ( i + 1 ) & i _ 1 \\leq i \\leq i _ 0 - 2 \\\\ \\hat { s } ( i + 2 ) & i _ 0 - 1 \\leq i \\end{cases} \\end{align*}"}
{"id": "347.png", "formula": "\\begin{equation} g _ { I ( G ) } ( k ) = \\min \\{ g _ { I ( G _ 1 ) } ( k ) + 1 , g _ { I ( G _ 2 ) } ( k - 1 ) , g _ { I ( G _ 3 ) } ( k - 1 ) , g _ { I ( G _ 3 ) } ( k ) + 1 \\} . \\end{equation}"}
{"id": "11.png", "formula": "\\begin{equation} m ' = - i \\kappa m + i C _ + [ q ( m + 1 ) ] , \\end{equation}"}
{"id": "216.png", "formula": "\\begin{equation} C _ 1 \\ = \\ E _ 1 \\ = \\ F _ 2 \\ = \\ E , \\text { a n d } C _ 2 \\ = \\ F _ 1 \\ = \\ E _ 2 \\ = \\ F . \\end{equation}"}
{"id": "344.png", "formula": "\\begin{equation} I ( G _ 2 ) ^ { [ k ] } + x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } = I ( G _ 3 ) ^ { [ k ] } + x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } \\end{equation}"}
{"id": "360.png", "formula": "\\[ \\begin{aligned} x ^ 2 = ( z - y ) ( z + y ) & = a ( 2 x + 2 b - a ) \\\\ x ^ 2 - 2 a x + a ^ 2 & = 2 a b \\\\ ( x - a ) ^ 2 & = 2 a b , \\end{aligned} \\]"}
{"id": "261.png", "formula": "\\begin{equation} \\frac { d ^ 2 \\bar { x } } { d \\bar { t } ^ 2 } = \\frac { 1 } { h } \\left [ \\frac { d } { d x } \\left ( \\frac { 1 } { h } \\frac { d \\varphi } { d x } \\right ) v ^ 2 + \\left ( \\frac { 1 } { h } \\frac { d \\varphi } { d x } \\right ) \\frac { d ^ 2 x } { d t ^ 2 } \\right ] . \\end{equation}"}
{"id": "16.png", "formula": "\\begin{align*} Q _ { * * } = \\Bigl \\{ q ( b ) \\Big | \\ , q : [ a , b ] \\to H ^ s \\text { i s c o n t i n u o u s , } q ( a ) \\in Q , \\text { a n d } \\beta ( z ; q ( t ) ) \\equiv \\beta ( z ; q ( a ) ) \\Bigr \\} , \\end{align*}"}
{"id": "409.png", "formula": "\\[ M h _ { k } ^ { a } ( \\vec { \\nu } ) = \\bigcap _ { i \\geq k } M h _ { i } ^ { a } ( \\nu _ { i } ) . \\]"}
{"id": "322.png", "formula": "\\begin{equation} \\begin{split} \\int _ { \\real ^ N } u ( x , t ) \\eta ( x , t ) \\ , d x & + \\int _ 0 ^ t \\int _ { \\real ^ N } \\left ( - u ( x , \\tau ) \\eta _ t ( x , \\tau ) - u ^ m ( x , \\tau ) \\Delta \\eta ( x , \\tau ) \\right ) d x \\ , d \\tau \\\\ & = \\int _ 0 ^ t \\int _ { \\real ^ N } ( 1 + | x | ) ^ { \\sigma } u ^ p ( x , \\tau ) \\eta ( x , \\tau ) d x \\ , d \\tau . \\end{split} \\end{equation}"}
{"id": "370.png", "formula": "\\begin{equation} \\gcd ( z - y , \\phi _ p ( z , y ) ) = 1 . \\end{equation}"}
{"id": "197.png", "formula": "\\begin{equation} A \\cap A ^ { - 1 } \\ = \\ \\{ e \\} , \\text { a n d } A \\cup A ^ { - 1 } \\ = \\ G . \\end{equation}"}
{"id": "123.png", "formula": "\\[ \\mathfrak { R } = \\left \\{ \\vec { c } + \\mathfrak { m } \\vec { x } \\ ; | \\ ; \\vec { c } \\in \\mathbb { R } ^ d , \\mathfrak { m } \\in \\mathfrak { s o } ( d ) \\right \\} , \\]"}
{"id": "202.png", "formula": "\\begin{equation} ( x _ n ) _ A \\ = \\ \\begin{cases} 1 \\ \\ \\text { i f } \\ \\ n \\in A , \\\\ 0 \\ \\ \\text { i f } \\ \\ n \\not \\in A . \\end{cases} \\end{equation}"}
{"id": "86.png", "formula": "\\begin{align} 2 c _ 1 + c _ 2 = 2 ( w _ 1 + w _ 3 \\kappa ) - 2 w _ 3 \\theta \\kappa \\geq c \\end{align}"}
{"id": "359.png", "formula": "\\begin{equation} x = a + \\sqrt { 2 a b } , y = b + \\sqrt { 2 a b } , z = a + b + \\sqrt { 2 a b } . \\end{equation}"}
{"id": "27.png", "formula": "\\begin{align} \\langle d _ j f , g \\rangle _ { X ^ { j + 1 } } = & \\frac 1 2 \\sum _ { s \\in X ^ { j + 1 } } m ( s ) d f ( s ) \\overline { g ( s ) } \\nonumber \\\\ = & \\frac 1 2 \\sum _ { s \\in X ^ { j + 1 } } m ( s ) \\overline { g ( s ) } \\sum _ { r \\subset s } f ( r ) \\\\ = & \\frac 1 2 \\sum _ { r \\in X ^ { j } } m ( r ) f ( r ) \\overline { \\sum _ { r \\subset s } \\frac { m ( s ) } { m ( r ) } g ( s ) } \\ . \\end{align}"}
{"id": "147.png", "formula": "\\begin{equation} \\begin{split} & \\int _ { \\Omega _ 1 } \\int _ { \\Omega _ 2 } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y \\geq \\int _ { N _ { ( a _ J / 2 ) \\cdot t } ( E ) } \\int _ { B ( z , ( a _ J / 2 ) \\cdot t ) \\cap \\Omega _ 2 } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d y d x \\\\ & \\geq \\int _ { N _ { ( a _ J / 2 ) \\cdot t } ( E ) } \\int _ { B ( z , ( a _ J / 2 ) \\cdot t ) \\cap \\Omega _ 2 } c _ J \\ d y d x \\geq c _ J D _ E { \\bigg ( \\frac { a _ J t } { 2 } \\bigg ) } ^ n \\cdot \\big | N _ { ( a _ J / 2 ) \\cdot t } ( E ) \\big | \\\\ & \\geq C _ 2 c _ J D _ E { \\bigg ( \\frac { a _ J t } { 2 } \\bigg ) } ^ { 2 n - \\alpha } . \\end{split} \\end{equation}"}
{"id": "133.png", "formula": "\\begin{align*} \\beta \\left \\| \\varepsilon \\left ( \\vec { v } - P \\vec { v } \\right ) \\right \\| & \\leq \\sup _ { q \\in Q , ~ \\| q \\| = 1 } \\big ( { \\rm d i v } ( \\vec { v } - P \\vec { v } ) , q \\big ) \\\\ & = \\sup _ { q \\in Q , ~ \\| q \\| = 1 } \\big ( { \\rm d i v } \\vec { v } , q \\big ) = \\left \\| \\operatorname { d i v } \\vec { v } \\right \\| , \\end{align*}"}
{"id": "179.png", "formula": "\\begin{align*} ( \\mu ^ { \\alpha } _ { X , Y } ) ^ { * } \\circ \\alpha _ { * } ( u _ { X , Y } ) \\circ \\mu ^ { \\alpha } _ { X , Y } ( b _ { i } \\boxtimes _ { B } c _ { j } ) & = ( \\mu ^ { \\alpha } _ { X , Y } ) ^ { * } ( u ^ { F , G } _ { X , Y } ( b _ i \\boxtimes _ { A } c _ { j } ) ) \\\\ & = ( \\mu ^ { \\alpha } _ { X , Y } ) ^ { * } ( c _ { j } \\boxtimes _ { A } b _ { i } ) \\\\ & = c _ { j } \\boxtimes _ { B } b _ { i } \\\\ & = u ^ { F ^ { \\prime } , G ^ { \\prime } } _ { \\alpha _ { * } ( X ) , \\alpha _ { * } ( Y ) } ( b _ { i } \\boxtimes _ { B } c _ { j } ) \\\\ & = u _ { \\alpha _ { * } ( X ) , \\alpha _ { * } ( Y ) } ( b _ { i } \\boxtimes _ { B } c _ { j } ) \\end{align*}"}
{"id": "20.png", "formula": "\\begin{equation} \\lim _ { \\kappa \\to \\infty } \\ , \\sup _ { q \\in Q _ * } \\ , \\sup _ { | t | \\leq T } \\norm { n ( t ) - n ( 0 ) } _ { H ^ { s + 1 } } = 0 . \\end{equation}"}
{"id": "282.png", "formula": "\\[ \\left ( A \\frac { d } { d x } + \\gamma A - A ' \\right ) b = \\left ( x \\frac { d } { d x } \\right ) ( 1 / 2 ) = 0 , \\]"}
{"id": "164.png", "formula": "\\begin{equation} \\bigg | e ^ { N s } - \\sum _ { m = 0 } ^ N \\frac { N ^ m } { m ! } { s } ^ m \\bigg | \\leq \\frac { e ^ { N s } } { C \\sqrt { N } } \\cdot \\delta ^ N . \\end{equation}"}
{"id": "85.png", "formula": "\\begin{align} \\begin{cases} c _ 1 = w _ 1 + w _ 3 \\kappa , & c _ 2 = - 2 w _ 3 \\theta \\kappa , \\\\ c _ 3 = w _ 2 + w _ 4 \\kappa , & c _ 4 = - ( w _ 2 \\gamma + w _ 4 \\kappa ( 2 + \\gamma ) ) \\theta , \\end{cases} \\end{align}"}
{"id": "178.png", "formula": "\\begin{align*} ( 1 _ { Y } \\boxtimes u _ { X , Z } ) \\circ ( u _ { X , Y } \\boxtimes 1 _ { Z } ) ( b _ { i } \\boxtimes c _ { j } \\boxtimes d _ { k } ) & = c _ { j } \\boxtimes d _ { k } \\boxtimes b _ { i } \\\\ & = u ^ { F , K } _ { X , Y \\boxtimes Z } ( b _ { i } \\boxtimes c _ { j } \\boxtimes d _ { k } ) \\\\ & = u _ { X , Y \\boxtimes Z } ( b _ { i } \\boxtimes c _ { j } \\boxtimes d _ { k } ) , \\end{align*}"}
{"id": "33.png", "formula": "\\begin{equation} \\partial ( s ) : = \\cup _ { i = 1 } ^ j \\{ ( - 1 ) ^ { j - i } ( \\lfloor s \\rfloor ; \\prescript { } { i } { \\hat { s } } ) \\} \\bigcup \\cup _ { i = 1 } ^ j \\{ ( - 1 ) ^ { i } ( \\lceil s \\rceil ; ( \\prescript { } { i } { \\hat { s } } ) ^ * ) \\} \\ . \\end{equation}"}
{"id": "392.png", "formula": "\\begin{equation} x ^ p + y ^ p = z ^ p \\Longrightarrow ( x - a ) ^ p - p a b K _ p = 0 . \\end{equation}"}
{"id": "3.png", "formula": "\\begin{equation} q ( \\phi ; q _ 0 ) = \\bigl ( e ^ { J \\nabla H _ \\phi } q _ 0 \\bigr ) ( x = 0 ) . \\end{equation}"}
{"id": "414.png", "formula": "\\[ [ { \\tt Q } ] _ { A ^ { ( \\rho ) } } J = [ { \\tt Q } ] _ { \\lnot A ^ { ( \\rho ) } } J = [ { \\tt Q } ] _ { A } J \\cap [ \\partial { \\tt Q } ] J \\cap [ \\rho ] J \\]"}
{"id": "199.png", "formula": "\\begin{align} \\begin{split} h _ x ( y ) \\ = \\ & x y ^ { - 1 } x \\text { s o t h a t } \\ \\ h _ x ( x ) \\ = \\ x , \\\\ \\text { a n d } & x ^ { - 1 } h _ x ( y ) \\ = \\ y ^ { - 1 } x , \\\\ \\text { a n d } & h _ x \\circ h _ x = 1 _ G . \\end{split} \\end{align}"}
{"id": "332.png", "formula": "\\begin{equation} E ( x , t ) = t ^ { 1 / ( 1 - p ) } \\varphi ( | x | t ^ { - \\gamma } ) , \\gamma = \\frac { m - p } { 2 ( 1 - p ) } \\end{equation}"}
{"id": "308.png", "formula": "\\begin{equation} \\widehat { F } ( \\widehat { z } _ 1 , \\widehat { z } _ 2 ) = \\begin{cases} ( a - \\frac { 1 } { c } ) \\widehat { z } _ 1 + \\frac { a ^ 2 b } { 2 } \\left ( \\widehat { z } _ 1 ^ 2 - 2 \\widehat { z } _ 1 \\widehat { z } _ 2 + \\widehat { z } _ 2 ^ 2 \\right ) \\\\ ( a + \\frac { 1 } { c } ) \\widehat { z } _ 2 + \\frac { a ^ 2 b } { 2 } \\left ( \\widehat { z } _ 1 ^ 2 - 2 \\widehat { z } _ 1 \\widehat { z } _ 2 + \\widehat { z } _ 2 ^ 2 \\right ) . \\end{cases} \\end{equation}"}
{"id": "243.png", "formula": "\\begin{equation} \\frac { d } { d t } = \\frac 1 f \\frac { d } { d \\tau } , \\frac { d ^ 2 } { d t ^ 2 } = \\frac 1 f \\frac { d } { d \\tau } \\left ( \\left ( \\frac 1 f \\right ) \\frac { d } { d \\tau } \\right ) = \\frac 1 { f ^ 2 } \\frac { d ^ 2 } { d \\tau ^ 2 } - \\frac 1 { f ^ 3 } \\frac { d f } { d \\tau } \\frac { d } { d \\tau } , \\end{equation}"}
{"id": "298.png", "formula": "\\begin{equation} J _ i \\left ( \\frac { j } { n } \\right ) = \\delta _ { i j } . \\end{equation}"}
{"id": "188.png", "formula": "\\begin{align*} u ^ { F , G } _ { X , Y } ( a ( b _ { i } \\boxtimes c _ { j } ) ) & = \\sum _ { l } c _ { j } \\boxtimes b _ { l } \\langle b _ { l } \\ | \\ a b _ { i } \\rangle \\\\ & = c _ { j } \\boxtimes a b _ { i } = c _ { j } a \\boxtimes b _ { i } \\\\ & = a c _ { j } \\boxtimes b _ { i } = a ( c _ { j } \\boxtimes b _ { i } ) \\\\ & = a u ^ { F , G } _ { X , Y } ( b _ { i } \\boxtimes c _ { j } ) . \\end{align*}"}
{"id": "281.png", "formula": "\\begin{equation} \\ddot { x } + \\frac { 1 } { x } \\dot { x } ^ 2 + x \\ , \\dot { x } + \\frac { 1 } { 2 } = 0 , \\end{equation}"}
{"id": "425.png", "formula": "\\begin{equation} T \\xlongrightarrow [ M \\rightarrow \\infty ] { \\mathcal { P } } \\sum _ { j = 1 } ^ { M } ( \\frac { \\rho ^ 2 \\widetilde { t } _ { j , j } ^ 2 \\psi _ { j , j } } { M } + \\frac { 2 \\Theta _ { j , j } } { M } ) : = K _ { M } , \\end{equation}"}
{"id": "238.png", "formula": "\\begin{equation} N _ T ( X _ 1 , X _ 2 ) = [ T ( X _ 1 ) , T ( X _ 2 ) ] + T ^ 2 ( [ X _ 1 , X _ 2 ] ) - T ( [ T ( X _ 1 ) , X _ 2 ] ) - T ( [ X _ 1 , T ( X _ 2 ) ] ) , \\ \\forall X _ 1 , X _ 2 \\in \\mathfrak { X } ( M ) , \\end{equation}"}
{"id": "191.png", "formula": "\\begin{align} \\begin{split} x \\ \\ \\text { i s r i g h t b a l a n c e d } & \\Longleftrightarrow \\overline { R ^ { \\circ } ( x ) } = R ( x ) \\\\ x \\ \\ \\text { i s l e f t b a l a n c e d } & \\Longleftrightarrow \\overline { R ^ { \\circ - 1 } ( x ) } = R ^ { - 1 } ( x ) \\\\ x \\ \\ \\text { i s b a l a n c e d } & \\Longleftrightarrow x \\ \\ \\text { i s b o t h l e f t a n d r i g h t b a l a n c e d } . \\end{split} \\end{align}"}
{"id": "163.png", "formula": "\\begin{equation} \\begin{split} & e ^ { N s } - \\sum _ { m = 0 } ^ { N - 1 } \\frac { N ^ m } { m ! } { s } ^ m = \\sum _ { m \\geq N } \\frac { N ^ m } { m ! } { s } ^ m \\\\ & = \\frac { { ( N s ) } ^ { N } } { N ! } \\cdot \\bigg ( 1 + \\sum _ { m \\geq 1 } \\frac { { ( N s ) } ^ { m } } { ( N + 1 ) \\cdots ( N + m ) } \\bigg ) \\leq C ( \\lambda ) \\cdot \\frac { { ( N s ) } ^ { N } } { N ! } , \\end{split} \\end{equation}"}
{"id": "76.png", "formula": "\\begin{align*} \\big ( M + \\lambda ( N - M ) \\big ) _ { 1 1 } = M _ { 1 1 } + \\lambda ( N _ { 1 1 } - M _ { 1 1 } ) \\geq c - \\lambda ( \\| N \\| _ { L ^ { \\infty } ( \\Omega _ T ) } + \\| M \\| _ { L ^ \\infty ( \\Omega _ T ) } ) , \\end{align*}"}
{"id": "194.png", "formula": "\\begin{align} \\begin{split} R \\cap R ^ { - 1 } \\ = \\ & \\bigcap _ { i } \\ ( \\pi _ i \\times \\pi _ i ) ^ { - 1 } ( R _ i \\cap R _ i ^ { - 1 } ) \\\\ = \\ \\bigcap _ { i } \\ & ( \\pi _ i \\times \\pi _ i ) ^ { - 1 } ( 1 _ { X _ i } ) \\ = \\ 1 _ X . \\end{split} \\end{align}"}
{"id": "21.png", "formula": "\\begin{equation} \\lim _ { \\kappa \\to \\infty } \\ , \\sup _ { q \\in Q _ * } \\ , \\sup _ { | t | \\leq T } \\norm { n ( t ) - n ( 0 ) } _ { H ^ { - 2 } } = 0 . \\end{equation}"}
{"id": "431.png", "formula": "\\begin{equation} \\begin{aligned} & \\| ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } - \\bold { \\Xi } _ { j - 1 } ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } ^ { T } ) ^ { - 1 } \\bold { \\Gamma } _ { j - 1 } ) ^ { - 1 } _ { ( m ) } \\| _ { 1 } \\\\ & \\| [ \\bold { S } _ { j } ^ { - 1 } ] ^ { ( 1 , 1 ) } _ { ( m ) } \\| _ { 1 } \\le \\| [ \\bold { S } _ { j } ^ { - 1 } ] ^ { ( m ) } \\| _ { 1 } < \\infty . \\end{aligned} \\end{equation}"}
{"id": "13.png", "formula": "\\begin{equation} m ( x + h ; \\kappa , q ) = m ( x ; \\kappa , q ( \\cdot + h ) ) \\quad \\text { f o r a l l } h \\in \\R . \\end{equation}"}
{"id": "51.png", "formula": "\\begin{align*} a _ { h , l } ( \\xi ) : = \\frac { ( - 1 + e ^ { - 2 \\pi i h \\xi _ l } ) } h , 1 \\leq l \\leq n . \\end{align*}"}
{"id": "19.png", "formula": "\\begin{equation} \\beta ( \\kappa ; q + c ) + { \\textstyle \\int } ( q + c ) \\ , d x = \\tfrac { \\kappa ^ 2 } { ( \\kappa - c ) ^ 2 } \\Bigl [ \\beta ( \\kappa - c ; q ) + { \\textstyle \\int } q \\ , d x \\Bigr ] + \\tfrac { c \\kappa } { \\kappa - c } . \\end{equation}"}
{"id": "96.png", "formula": "\\begin{align} \\begin{cases} 2 c _ 1 + c _ 2 & = 2 ( w _ 1 + w _ 3 \\kappa ) - 2 w _ 3 \\theta \\kappa \\geq c , \\\\ c _ 3 & = w _ 2 + w _ 4 \\kappa \\geq c , \\\\ c _ 3 + c _ 4 & = w _ 2 + w _ 4 \\kappa - \\big ( w _ 2 \\gamma + w _ 4 \\kappa ( 2 + \\gamma ) \\big ) \\theta \\geq c \\end{cases} \\end{align}"}
{"id": "428.png", "formula": "\\begin{equation} \\bold { F } _ { j } = \\begin{bmatrix} \\bold { \\Pi } _ { j } + \\bold { \\Gamma } _ { j } \\widetilde { \\bold { \\Lambda } } _ { j } & \\bold { \\Gamma } _ { j } \\\\ \\bold { \\Pi } _ { j } ^ { T } \\widetilde { \\bold { \\Lambda } } _ { j } + \\bold { \\Xi } _ { j } & \\bold { \\Pi } _ { j } ^ { T } \\end{bmatrix} , \\end{equation}"}
{"id": "18.png", "formula": "\\begin{equation} \\beta ( \\lambda \\kappa ; q _ \\lambda ) = \\beta ( \\kappa ; q ) \\quad \\text { f o r a n y $ \\lambda > 0 $ . } \\end{equation}"}
{"id": "324.png", "formula": "\\begin{equation} \\Delta v \\geq - \\frac { K } { t } , K = \\frac { N } { N ( m - 1 ) + 2 } , \\end{equation}"}
{"id": "159.png", "formula": "\\begin{equation} G ( s ) = - e ^ { ( 2 n - \\alpha ( \\eta ) ) s } R ( e ^ { - s } ) + \\int _ { 0 } ^ s G ( s - s ' ) \\ d \\mu ( s ' ) . \\end{equation}"}
{"id": "208.png", "formula": "\\begin{equation} h _ { n + 1 } ( x ) \\ = \\ ( h _ n ( x ) , \\pi _ { n h _ n ( x ) } ( x ) ) \\text { f o r a l l } \\ \\ x \\in X . \\end{equation}"}
{"id": "24.png", "formula": "\\begin{align*} [ \\kappa R ( \\kappa ) - 1 ] \\chi _ y & = R ( \\kappa ) C _ + q \\kappa R _ 0 ( \\kappa ) \\chi _ y + [ \\kappa R _ 0 ( \\kappa ) - 1 ] \\chi _ y \\\\ & = R ( \\kappa ) q _ + - R ( \\kappa ) C _ + ( q - q \\chi _ y ) + [ R ( \\kappa ) C _ + q + 1 ] [ \\kappa R _ 0 ( \\kappa ) - 1 ] \\chi _ y . \\end{align*}"}
{"id": "354.png", "formula": "\\begin{equation} I _ 0 ^ { [ k ] } : I ^ { [ \\ell ] } = ( I _ 0 ^ { [ k ] } : I _ 0 ^ { [ \\ell ] } ) \\cap \\big ( I _ 0 ^ { [ k ] } : \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\sum _ { \\substack { q \\in [ s ] \\\\ \\nu ( I _ q ) \\ge \\ell - 1 } } \\ ! \\ ! \\ ! \\ ! u _ q I _ q ^ { [ \\ell - 1 ] } \\big ) = I _ 0 ^ { [ k ] } , \\end{equation}"}
{"id": "189.png", "formula": "\\begin{equation} R _ 1 \\times R _ 2 \\ = \\ \\{ ( x _ 1 , x _ 2 ) , ( y _ 1 , y _ 2 ) ) : ( x _ 1 , y _ 1 ) \\in R _ 1 \\ \\ \\text { a n d } \\ \\ ( x _ 2 , y _ 2 ) \\in R _ 2 \\} . \\end{equation}"}
{"id": "28.png", "formula": "\\begin{equation} d _ j ^ * g ( r ) = \\sum _ { r \\subset s } \\frac { m ( s ) } { m ( r ) } g ( s ) \\end{equation}"}
{"id": "234.png", "formula": "\\begin{equation} \\mathcal { L } _ \\Delta S = - S . \\end{equation}"}
{"id": "383.png", "formula": "\\begin{equation} \\begin{split} K _ p & \\equiv 2 \\sum _ { i = 1 } ^ { k } \\dfrac { 1 } { 2 i - 1 } \\binom { 2 k } { 2 i - 2 } b ^ { 2 ( i - 1 ) } a ^ { 2 ( k - i ) + 1 } \\\\ & - \\sum _ { j = 1 } ^ { 2 k } \\dfrac { ( - 1 ) ^ j } { j } \\binom { 2 k } { j - 1 } b ^ { 2 k - j } a ^ { j - 1 } \\pmod { p } , \\end{split} \\end{equation}"}
{"id": "17.png", "formula": "\\begin{equation} \\int w ( x ; \\kappa , q ) \\ , d x = 2 \\beta ( \\kappa ) - \\kappa \\frac { \\partial \\beta } { \\partial \\kappa } + 2 \\int q \\ , d x . \\end{equation}"}
{"id": "239.png", "formula": "\\begin{equation} \\frac { d ^ 2 x ^ i } { d t ^ 2 } = X ^ i ( x ^ 1 , \\ldots , x ^ n , \\dot x ^ 1 , \\ldots , \\dot x ^ n ) , i = 1 , \\ldots , n , \\end{equation}"}
{"id": "233.png", "formula": "\\begin{equation} \\frac { d \\tau } { d t } = \\frac 1 { f ( \\gamma ( t ) ) } , \\end{equation}"}
{"id": "319.png", "formula": "\\begin{equation} u _ t = \\Delta u + | x | ^ { \\sigma } u ^ p , \\end{equation}"}
{"id": "329.png", "formula": "\\begin{equation} z ( x , 0 ) = T _ 0 ^ { - \\alpha } f ( ( 1 + | x | ) T _ 0 ^ { \\beta } ) \\geq \\| u _ 0 \\| _ { \\infty } \\geq u _ 0 ( x ) \\end{equation}"}
{"id": "237.png", "formula": "\\begin{equation} \\mathcal { L } _ \\Gamma \\theta _ L = d L , \\end{equation}"}
{"id": "120.png", "formula": "\\begin{equation} \\langle A \\vec { u } , \\vec { v } \\rangle = a ( \\vec { u } , \\vec { v } ) : = \\left ( \\varepsilon \\left ( \\vec { u } \\right ) , \\varepsilon \\left ( \\vec { v } \\right ) \\right ) . \\end{equation}"}
{"id": "257.png", "formula": "\\[ \\frac { d ^ 2 x _ 1 } { d t _ 1 ^ 2 } = 0 . \\]"}
{"id": "62.png", "formula": "\\begin{align*} \\partial ( \\overline { s } ) = & \\cup _ { i = 1 } ^ j \\{ ( - 1 ) ^ { j - i } ( \\lfloor \\overline { s } \\rfloor ; \\prescript { } { i } { ( \\hat { s } ^ * ) } ) \\} \\bigcup \\cup _ { i = 1 } ^ j \\{ ( - 1 ) ^ { i } ( \\lceil \\overline { s } \\rceil ; ( \\prescript { } { i } { ( \\hat { s } ^ * ) } ) ^ * ) \\} \\\\ = & \\cup _ { i = 1 } ^ j \\{ ( - 1 ) ^ { j - i } ( \\lceil s \\rceil \\lfloor ; ( \\prescript { } { j - i + 1 } { \\hat { s } } ) ^ * ) \\} \\bigcup \\cup _ { i = 1 } ^ j \\{ ( - 1 ) ^ { i } ( \\lfloor s \\rfloor ; \\prescript { } { j - i + 1 } { \\hat { s } } ) \\} \\\\ = & \\cup _ { m = 1 } ^ j \\{ ( - 1 ) ^ { m - 1 } ( \\lceil s \\rceil \\lfloor ; ( \\prescript { } { m } { \\hat { s } } ) ^ * ) \\} \\bigcup \\cup _ { m = 1 } ^ j \\{ ( - 1 ) ^ { j - m + 1 } ( \\lfloor s \\rfloor ; \\prescript { } { m } { \\hat { s } } ) \\} \\\\ = & \\cup _ { m = 1 } ^ j \\overline { \\{ ( - 1 ) ^ { m } ( \\lceil s \\rceil \\lfloor ; ( \\prescript { } { m } { \\hat { s } } ) ^ * ) \\} } \\bigcup \\cup _ { m = 1 } ^ j \\overline { \\{ ( - 1 ) ^ { j - m } ( \\lfloor s \\rfloor ; \\prescript { } { m } { \\hat { s } } ) \\} } = \\overline { \\partial ( s ) } \\ . \\end{align*}"}
{"id": "143.png", "formula": "\\[ \\mathfrak { R } = \\left \\{ \\vec { c } + \\mathfrak { m } \\vec { x } \\ ; | \\ ; \\vec { c } \\in \\mathbb { R } ^ d , \\mathfrak { m } \\in \\mathfrak { s o } ( d ) \\right \\} , \\]"}
{"id": "217.png", "formula": "\\begin{align} \\begin{split} x \\in A _ i \\ \\ \\ \\text { a n d } \\ \\ x ' \\in A _ j \\cup - A _ j \\ \\ & \\Rightarrow \\ \\ ( x ' , x ) \\in \\widehat { A } ^ { \\circ } , \\\\ x \\in - A _ i \\ \\ \\text { a n d } \\ \\ x ' \\in A _ j \\cup - A _ j \\ & \\Rightarrow \\ \\ ( x , x ' ) \\in \\widehat { A } ^ { \\circ } , \\\\ x \\in A _ i \\ \\ \\text { a n d } \\ \\ x ' \\in A _ { i + 1 } \\cup - A _ { i + 1 } \\ \\ & \\Rightarrow \\ \\ ( x , x ' ) \\in \\widehat { A } ^ { \\circ } , \\\\ x \\in - A _ i \\ \\ \\text { a n d } \\ \\ x ' \\in A _ { i + 1 } \\cup - A _ { i + 1 } \\ \\ & \\Rightarrow \\ \\ ( x , x ' ) \\in \\widehat { A } ^ { \\circ } . \\end{split} \\end{align}"}
{"id": "49.png", "formula": "\\begin{equation} ( \\tilde U _ { j , h } \\omega ) _ { l } ( \\mu ) : = \\frac 1 { h ^ { { j } } } \\omega _ { I ^ j _ l } ( \\mu ) , 1 \\leq l \\leq \\binom { n } { j } . \\end{equation}"}
{"id": "275.png", "formula": "\\begin{equation} \\frac { d ^ 2 y } { d \\tau ^ 2 } + 1 = 0 \\end{equation}"}
{"id": "32.png", "formula": "\\begin{equation} P ^ { j , n } : = \\{ I : \\{ 1 , \\dots , j \\} \\to \\{ 1 , \\dots , n \\} ; I \\ , \\ , { \\rm s t r i c t l y \\ , \\ , m o n o t o n e } \\} \\end{equation}"}
{"id": "106.png", "formula": "\\begin{equation} \\begin{aligned} & ( \\lambda - 2 \\eta ) \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\gamma / 2 } | D ^ 2 u | ^ 2 \\phi ^ 2 d x d t \\\\ & \\leq \\frac { C } { \\eta } \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\gamma / 2 } | D u | ^ 2 | D \\phi | ^ 2 d x d t + C \\Big ( \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) | \\phi _ t | \\phi d x d t \\\\ & + \\epsilon \\int _ { Q _ { 2 r } } \\big | \\ln ( | D u | ^ 2 + \\epsilon ) \\big | | \\phi _ t | \\phi d x d t + \\epsilon \\int _ { B _ { 2 r } } \\big | \\ln \\big ( | D u ( x , t _ 0 ) | ^ 2 + \\epsilon \\big ) \\big | \\phi ^ 2 ( x , t _ 0 ) d x \\Big ) \\end{aligned} \\end{equation}"}
{"id": "73.png", "formula": "\\begin{align*} \\begin{cases} N _ { 1 1 } & = c _ 2 - c _ 1 \\\\ N _ { 1 2 } = N _ { 2 1 } & = \\tfrac { 1 } { 2 } \\big ( c _ 3 P _ \\theta + ( c _ 3 + c _ 4 ) - ( 2 c _ 1 + c _ 2 ) \\big ) \\\\ N _ { 2 2 } & = ( c _ 3 + c _ 4 ) P _ \\theta - c _ 1 . \\end{cases} \\end{align*}"}
{"id": "227.png", "formula": "\\begin{align} \\begin{split} A + \\ = \\ \\{ i + : i & \\leq n \\} , A - \\ = \\ \\{ i - : i \\leq n \\} , \\\\ B + \\ = \\ \\{ i + : n < i & \\leq n + m \\} , B - \\ = \\ \\{ i - : n < i \\leq n + m \\} , \\\\ C \\ = \\ K \\setminus ( A + \\cup \\ A - \\cup \\ & B + \\cup B - ) \\ = \\ \\{ i + , i - : n + m < i \\} \\cup \\{ \\infty \\} . \\end{split} \\end{align}"}
{"id": "113.png", "formula": "\\begin{align*} R o o t _ { - } - R o o t _ + & = - \\frac { \\sqrt { b ^ 2 - 4 a c } } { a } \\\\ & = 4 \\frac { \\sqrt { G \\cdot P \\cdot E ( \\frac { G } { n - 1 } + K ) } } { ( G - P ) ^ 2 } > 0 . \\end{align*}"}
{"id": "284.png", "formula": "\\[ \\frac { d ^ 2 y } { d \\tau ^ 2 } + \\frac { d y } { d \\tau } + \\frac { 1 } { 2 } = 0 . \\]"}
{"id": "115.png", "formula": "\\begin{equation} { \\rm d i v } \\big ( 2 \\mu \\varepsilon ( \\vec { u } ) + \\tilde { \\lambda } \\mathrm { t r } ( \\varepsilon ( \\vec { u } ) ) I \\big ) = \\tilde { \\vec { f } } . \\end{equation}"}
{"id": "213.png", "formula": "\\begin{equation} R ' | X \\ = \\ R , \\text { a n d } R '^ { \\circ } ( u ) \\ = \\ F , \\ \\ R '^ { \\circ - 1 } ( u ) \\ = \\ E . \\end{equation}"}
{"id": "176.png", "formula": "\\begin{align*} u _ { X , Y } ( a b _ { i } \\boxtimes c _ { k } ) & = u _ { X , Y } ( \\sum _ { j } b _ { j } \\langle b _ { j } \\ | a b _ { i } \\rangle \\boxtimes c _ { k } ) \\\\ & = u _ { X , Y } ( \\sum _ { j } b _ { j } \\boxtimes c _ { k } \\langle b _ { j } \\ | a b _ { i } \\rangle ) \\\\ & = u _ { F , G } ( \\sum _ { j } b _ { j } \\boxtimes c _ { k } ) \\langle b _ { j } \\ | a b _ { i } \\rangle \\\\ & = \\sum _ { j } c _ { k } \\boxtimes b _ { j } \\langle b _ { j } \\ | a b _ { i } \\rangle \\\\ & = c _ { k } \\boxtimes a b _ { i } \\\\ & = a c _ { k } \\boxtimes b _ { i } \\\\ & = a u _ { X , Y } ( b _ { i } \\boxtimes c _ { k } ) \\end{align*}"}
{"id": "88.png", "formula": "\\begin{align} \\det ( M ) = c _ 3 ( c _ 3 + c _ 4 ) P _ \\theta - \\frac { \\big ( c _ 3 P _ \\theta + ( c _ 3 + c _ 4 ) - ( 2 c _ 1 + c _ 2 ) \\big ) ^ 2 } { 4 } \\geq c \\end{align}"}
{"id": "315.png", "formula": "\\begin{equation} Q _ { j k } = \\begin{cases} \\dfrac { \\sum _ { l = 1 } ^ { \\infty } \\left | \\left \\langle L _ { F ' } e _ l ' , e _ j ' \\right \\rangle \\right | \\sum _ { l = 1 } ^ { \\infty } \\left | \\left \\langle L _ { F ' } e _ k ' , e _ l ' \\right \\rangle \\right | } { \\kappa ^ 2 \\left | \\Re \\left ( \\left \\langle L _ { F ' } e _ j ' , e _ j ' \\right \\rangle \\right ) \\right | \\left | \\Re \\left ( \\left \\langle L _ { F ' } e _ k ' , e _ k ' \\right \\rangle \\right ) \\right | } & \\textrm { i f } | \\alpha ( k ) | \\neq | \\alpha ( j ) | \\\\ & \\textrm { a n d } \\left \\langle L _ { F ' } e _ k ' , e _ j ' \\right \\rangle \\neq 0 \\\\ 0 & \\textrm { o t h e r w i s e . } \\end{cases} \\end{equation}"}
{"id": "348.png", "formula": "\\begin{align} g _ { I ( G ) } ( k + 1 ) & = \\min \\{ g _ { I ( G _ 1 ) } ( k + 1 ) + 1 , g _ { I ( G _ 2 ) } ( k ) , g _ { I ( G _ 3 ) } ( k ) , g _ { I ( G _ 3 ) } ( k + 1 ) + 1 \\} , \\\\ g _ { I ( G ) } ( k ) & = \\min \\{ g _ { I ( G _ 1 ) } ( k ) + 1 , g _ { I ( G _ 2 ) } ( k - 1 ) , g _ { I ( G _ 3 ) } ( k - 1 ) , g _ { I ( G _ 3 ) } ( k ) + 1 \\} . \\end{align}"}
{"id": "297.png", "formula": "\\begin{align*} B _ 0 & = \\{ s \\in B \\ , | \\ , \\text { t h e r e i s $ t \\in C $ w i t h $ t \\sqsubseteq s \\subset \\textstyle \\bigcup C $ } \\} , \\\\ C ' & = \\{ t \\in C \\ , | \\ , \\text { t h e r e i s $ s \\in B $ w i t h $ s \\sqsubset t $ } \\} \\cup \\{ s ^ \\frown n \\ , | \\ , s \\in B _ 0 \\text { a n d } n \\in { \\textstyle \\bigcup C } / s \\} . \\end{align*}"}
{"id": "391.png", "formula": "\\[ x - a = \\sqrt { 2 a b K _ 2 } = 2 ^ { e } r ' s , \\]"}
{"id": "203.png", "formula": "\\begin{equation} ( ( x , t ) , ( y , s ) ) \\in R _ 2 ^ { \\circ } \\Longleftrightarrow \\begin{cases} ( x , y ) \\in L _ 1 ^ { \\circ } , \\text { o r } \\\\ x = y \\ \\ \\text { a n d } \\ \\ ( t , s ) \\in S _ x ^ { \\circ } . \\end{cases} \\end{equation}"}
{"id": "223.png", "formula": "\\begin{align} \\begin{split} P [ j ] \\ = \\ \\widehat { A } + \\ & \\cup \\ \\widehat { A } - \\ \\cup \\\\ ( D _ j - \\times D _ j + ) \\ \\cup \\ ( \\bar D _ j - \\times \\bar D _ j + ) \\ & \\cup \\ ( D _ j + \\times \\bar D _ j - ) \\ \\cup \\ ( \\bar D _ j + \\times D _ j - ) . \\end{split} \\end{align}"}
{"id": "149.png", "formula": "\\begin{equation} \\begin{split} & 2 \\int _ { \\Omega _ 1 \\cap A } \\int _ { \\Omega _ 2 \\cap A } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y \\\\ & = \\int _ { \\Omega _ 1 \\cap A } \\int _ { \\Omega _ 1 ^ c \\cap A } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y + \\int _ { \\Omega _ 2 \\cap A } \\int _ { \\Omega _ 2 ^ c \\cap A } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y \\\\ & - \\int _ { ( \\Omega _ 1 \\cup \\Omega _ 2 ) \\cap A } \\int _ { { ( \\Omega _ 1 \\cup \\Omega _ 2 ) } ^ c \\cap A } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y . \\end{split} \\end{equation}"}
{"id": "39.png", "formula": "\\begin{equation} ( { U } _ j f ) ( \\mu ) : = \\sum _ { I \\in P ^ { j , i } _ + } f ( \\mu ; \\delta _ { I ( 1 ) } , \\hdots , \\delta _ { I ( j ) } ) \\ , d x ^ I \\ . \\end{equation}"}
{"id": "351.png", "formula": "\\[ \\big ( \\sum _ i I _ i \\big ) : ( \\sum _ j J _ j ) \\ = \\ \\sum _ i \\big ( I _ i : \\big ( \\sum _ j J _ j \\big ) \\big ) \\ = \\ \\sum _ i \\bigcap _ j ( I _ i : J _ j ) . \\]"}
{"id": "78.png", "formula": "\\begin{equation} S = w _ 1 G D _ 1 ( p - 2 + s ) + w _ 2 G D _ 2 ( p - 2 + s ) \\end{equation}"}
{"id": "55.png", "formula": "\\begin{align} & \\sum _ { \\mu \\in \\Z ^ n } | \\hat { \\varphi } ( \\xi + \\mu ) | ^ 2 = 1 , \\xi \\in \\R ^ n , \\\\ & { \\rm s u p p } ( \\hat { \\varphi } ) \\subset ( - 1 , 1 ) ^ n . \\end{align}"}
{"id": "393.png", "formula": "\\begin{equation} ( x + y - z ) ^ p - p a b K _ p = x ^ p + y ^ p - z ^ p = 0 . \\end{equation}"}
{"id": "8.png", "formula": "\\begin{align} \\lim _ { y \\to \\infty } \\pi y f ( i y ) = \\lim _ { y \\to \\infty } \\int \\frac { y ^ 2 } { x ^ 2 + y ^ 2 } f ( x ) \\ , d x = \\lim _ { \\xi \\downarrow 0 } \\tfrac { \\sqrt { 2 \\pi } } { 2 } \\widehat f ( \\xi ) \\end{align}"}
{"id": "248.png", "formula": "\\begin{equation} \\frac { d \\bar v } { d \\tau } = \\frac 1 f f ' ( q ) \\ , \\bar v \\ , \\bar v + f ^ 2 \\ , F ( q ) , \\end{equation}"}
{"id": "14.png", "formula": "\\begin{equation} d m | _ q ( g ) = \\frac { d } { d \\theta } m ( x ; \\kappa , q + \\theta g ) \\bigg | _ { \\theta = 0 } = R ( \\kappa , q ) \\bigl [ ( m + 1 ) C _ + g \\bigr ] , \\end{equation}"}
{"id": "90.png", "formula": "\\begin{align*} X _ 1 ^ \\prime ( \\theta ) = & p - 2 - \\gamma - \\frac { ( p - 2 ) R _ \\theta - \\gamma P _ \\theta } { \\sqrt { P _ \\theta R _ \\theta } } \\\\ = & \\frac { ( \\sqrt { P _ \\theta } - \\sqrt { R _ \\theta } ) \\big ( ( p - 2 ) \\sqrt { R _ \\theta } + \\gamma \\sqrt { P _ \\theta } \\big ) } { \\sqrt { P _ \\theta R _ \\theta } } . \\end{align*}"}
{"id": "273.png", "formula": "\\[ \\frac { d ^ 2 y } { d \\tau ^ 2 } + \\alpha \\frac { d y } { d \\tau } + \\beta ( y ) = 0 , \\]"}
{"id": "270.png", "formula": "\\[ \\left ( A _ 0 \\frac { d } { d x } - 3 A ' _ 0 \\right ) \\left ( A _ 0 \\frac { d } { d x } + \\gamma _ 0 A _ 0 - A ' _ 0 \\right ) b _ 0 = 0 . \\]"}
{"id": "333.png", "formula": "\\begin{equation} u _ { \\epsilon , 0 } ( x ) = \\left \\{ \\begin{array} { l l } u _ 0 ( x ) , & { \\rm f o r } \\ x \\in ( - \\infty , R _ { \\epsilon } ) , \\\\ \\frac { \\epsilon ( R _ 0 + r - x ) } { R _ 0 + r - R _ { \\epsilon } } , & \\rm { f o r } \\ x \\in [ R _ { \\epsilon } , R _ 0 + r ] , \\\\ 0 , & { \\rm f o r } \\ x \\in ( R _ 0 + r , \\infty ) , \\end{array} \\right . \\end{equation}"}
{"id": "80.png", "formula": "\\begin{align} \\lim _ { y \\to 0 , y \\neq 0 } F \\big ( D g ( y ) , D ^ 2 g ( y ) \\big ) = 0 . \\end{align}"}
{"id": "131.png", "formula": "\\begin{equation} \\begin{aligned} & \\| \\operatorname { d i v } \\vec { v } \\| ^ 2 \\eqsim \\langle A ( I - P ) \\vec { v } , \\vec { v } \\rangle \\\\ & \\quad = a \\left ( \\vec { v } - P \\vec { v } , \\vec { v } - P \\vec { v } \\right ) = \\left \\| \\varepsilon \\left ( \\vec { v } - P \\vec { v } \\right ) \\right \\| ^ 2 . \\end{aligned} \\end{equation}"}
{"id": "398.png", "formula": "\\[ z = K _ 3 - ( x - a ) , \\]"}
{"id": "226.png", "formula": "\\begin{align} \\begin{split} R \\ = \\ & 2 L \\cup \\ P \\ \\cup \\\\ \\bigcup _ i [ \\{ i + \\} \\times & ( E _ i \\cup G _ { i } ) ] \\cup [ F _ i \\times \\{ i + \\} ] \\cup \\\\ \\bigcup _ i [ \\{ i - \\} \\times & E _ i ] \\cup [ ( F _ i \\cup G _ i ) \\times \\{ i - \\} ] . \\end{split} \\end{align}"}
{"id": "357.png", "formula": "\\begin{equation} z ^ n - y ^ n = ( z - y ) ( z ^ { n - 1 } + y ^ { n - 1 } ) + z y ( z ^ { n - 2 } - y ^ { n - 2 } ) \\end{equation}"}
{"id": "350.png", "formula": "\\begin{align*} g _ { I ( P _ n ) } ( k ) & = \\min \\Big \\{ \\Big \\lceil \\frac { n - 1 } { 3 } \\Big \\rceil - k + 1 , \\Big \\lceil \\frac { n - 2 } { 3 } \\Big \\rceil - ( k - 1 ) , \\Big \\lceil \\frac { n - 3 } { 3 } \\Big \\rceil - ( k - 1 ) , \\\\ & \\phantom { = \\min \\Big \\{ . } \\Big \\lceil \\frac { n - 3 } { 3 } \\Big \\rceil - k + 1 \\Big \\} \\\\ & = \\min \\Big \\{ \\Big \\lceil \\frac { n - 1 } { 3 } \\Big \\rceil - k + 1 , \\Big \\lceil \\frac { n - 2 } { 3 } \\Big \\rceil - k + 1 , \\Big \\lceil \\frac { n } { 3 } \\Big \\rceil - k \\Big \\} \\\\ & = \\Big \\lceil \\frac { n } { 3 } \\Big \\rceil - k , \\end{align*}"}
{"id": "262.png", "formula": "\\[ \\frac { d ^ 2 \\bar { x } } { d \\bar { t } ^ 2 } = \\frac { c } { h ( x ) } \\frac { d ^ 2 x } { d t ^ 2 } , \\]"}
{"id": "255.png", "formula": "\\[ \\frac { d h } { d x } + \\gamma _ 0 ( x ) h = 0 , \\]"}
{"id": "31.png", "formula": "\\begin{equation} \\tau f = \\begin{cases} f & \\text { i f } f \\in \\ell ^ 2 ( X ) _ { \\text { e v e n } } \\\\ - f & \\text { i f } f \\in \\ell ^ 2 ( X ) _ { \\text { o d d } } \\\\ \\end{cases} \\end{equation}"}
{"id": "161.png", "formula": "\\begin{equation} \\rho _ k ( x _ 1 , \\ldots , x _ k ) = \\rho _ k ( x _ { \\sigma ( 1 ) } , \\ldots , x _ { \\sigma ( k ) } ) , \\end{equation}"}
{"id": "283.png", "formula": "\\[ y = \\frac { 1 } { 3 } x ^ 3 , d \\tau = x \\ , d t \\]"}
{"id": "271.png", "formula": "\\[ \\frac { d ^ 2 x } { d t ^ 2 } + \\gamma ( x ) \\left ( \\frac { d x } { d t } \\right ) ^ 2 + A ( x ) \\frac { d x } { d t } + b ( x ) = 0 , \\]"}
{"id": "5.png", "formula": "\\begin{align} C _ + + C _ - = 1 \\text { o n l y o n t h e l i n e ; o n t h e c i r c l e , } C _ + f + C _ - f = f + \\textstyle \\int \\ ! f . \\end{align}"}
{"id": "196.png", "formula": "\\begin{equation} q ( x ) _ { i } \\ = \\ g _ { i + 1 } ( \\pi _ { i + 1 } ( x ) ) , \\ \\ \\text { f o r } \\ \\ i \\in \\Z _ + \\end{equation}"}
{"id": "296.png", "formula": "\\begin{equation} e _ { 2 k } e _ { 2 l } = C _ { k + l - 1 } ^ { l } e _ { 2 k + 2 l } , k \\geq 1 , \\ l \\geq 1 . \\end{equation}"}
{"id": "6.png", "formula": "\\[ \\limsup _ { \\delta \\to 0 } \\ \\sup _ { q \\in Q } \\ \\sup _ { | y | < \\delta } \\| q ( \\cdot + y ) - q ( \\cdot ) \\| _ { H ^ \\sigma } = 0 . \\]"}
{"id": "430.png", "formula": "\\begin{equation} \\begin{aligned} & [ \\bold { S } _ { j - 1 } ^ { - 1 } ] ^ { ( 1 , 2 ) } = [ \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } - \\bold { \\Gamma } _ { j - 1 } \\times \\\\ & ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } ^ { T } ) ^ { - 1 } \\bold { \\Xi } _ { j - 1 } ] ^ { - 1 } \\bold { \\Gamma } _ { j - 1 } ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } ^ { T } ) ^ { - 1 } , \\end{aligned} \\end{equation}"}
{"id": "40.png", "formula": "\\begin{align} [ ( U _ 1 \\circ d _ 0 ) f ] ( \\mu ) = \\sum _ { i = 1 } ^ j ( d _ 0 f ) ( \\mu ; \\delta _ j ) d x ^ { i } = \\sum _ { i = 1 } ^ j ( d _ 0 f ) ( \\mu , \\mu + \\delta _ i ) d x ^ { i } = & \\sum _ { i = 1 } ^ j ( f ( \\mu + \\delta _ i ) - f ( \\mu ) ) d x ^ { i } \\nonumber \\\\ = & \\sum _ { i = 1 } ^ j \\mathcal { D } _ i f ( \\mu ) d x ^ { i } \\end{align}"}
{"id": "367.png", "formula": "\\begin{equation} \\phi _ p ( z , y ) = \\dfrac { z ^ p - y ^ p } { z - y } = A _ p ( z , y ) + ( z y ) ^ k = D _ p ( z , y ) + p ( z y ) ^ k . \\end{equation}"}
{"id": "77.png", "formula": "\\begin{align*} \\operatorname { d e t } \\big ( M + \\lambda ( N - M ) \\big ) & = \\operatorname { d e t } ( M ) + \\lambda \\big ( M _ { 1 1 } N _ { 2 2 } + M _ { 2 2 } N _ { 1 1 } - 2 M _ { 1 2 } N _ { 1 2 } - 2 \\det ( M ) \\big ) \\\\ & + \\lambda ^ 2 \\operatorname { d e t } ( N - M ) \\\\ & \\geq c - 2 \\lambda \\left ( 2 \\| M \\| _ { L ^ \\infty ( \\Omega _ T ) } \\| N \\| _ { L ^ \\infty ( \\Omega _ T ) } + \\| M \\| _ { L ^ \\infty ( \\Omega _ T ) } ^ 2 \\right ) - \\lambda ^ 2 \\| N - M \\| _ { L ^ \\infty ( \\Omega _ T ) } ^ 2 \\\\ & \\geq c - 4 \\lambda \\left ( \\| M \\| _ { L ^ \\infty ( \\Omega _ T ) } + \\| N \\| _ { L ^ \\infty ( \\Omega _ T ) } \\right ) ^ 2 . \\end{align*}"}
{"id": "376.png", "formula": "\\begin{equation} x - a = y - b = z - ( a + b ) = x + y - z . \\end{equation}"}
{"id": "102.png", "formula": "\\begin{equation} \\begin{aligned} \\int _ { Q _ r } & \\abs { D \\big ( ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p - 2 + s } { 4 } } D u \\big ) } ^ 2 d x d t \\\\ \\leq & \\frac { C } { r ^ 2 } \\Big ( \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p - 2 + s } { 2 } } | D u | ^ 2 d x d t + \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p + s - \\gamma } { 2 } } d x d t \\Big ) \\\\ & + \\epsilon \\Big ( \\frac { C } { r ^ 2 } \\int _ { Q _ { 2 r } } \\abs { \\ln ( | D u | ^ 2 + \\epsilon ) } d x d t + C \\int _ { B _ { 2 r } } \\abs { \\ln \\big ( | D u ( x , t _ 0 ) | ^ 2 + \\epsilon \\big ) } d x \\Big ) \\end{aligned} \\end{equation}"}
{"id": "124.png", "formula": "\\[ W : = \\operatorname { K e r } ( B ) = \\left \\{ \\vec { v } \\in V \\ ; \\big | \\ ; B \\vec { v } = \\operatorname { d i v } \\vec { v } = 0 \\right \\} . \\]"}
{"id": "316.png", "formula": "\\begin{equation} \\partial _ t u = \\Delta u ^ m + ( 1 + | x | ) ^ { \\sigma } u ^ p , ( x , t ) \\in \\real ^ N \\times ( 0 , \\infty ) , \\ N \\geq 1 , \\end{equation}"}
{"id": "112.png", "formula": "\\begin{align*} R o o t _ { \\pm } & = \\frac { - \\Big ( P \\cdot E + \\frac { 1 } { 2 } ( G - P ) \\big ( \\frac { G } { n - 1 } + K - \\frac { ( n - 2 ) P } { n - 1 } \\big ) \\Big ) \\pm \\sqrt { b ^ 2 - 4 a c } } { - \\frac { 1 } { 2 } ( G - P ) ^ 2 } \\\\ & = \\frac { \\Big ( \\sqrt { P \\cdot E } \\mp \\sqrt { G ( \\frac { G } { n - 1 } + K ) } \\Big ) ^ 2 } { ( G - P ) ^ 2 } \\\\ & = \\Bigg ( \\frac { \\sqrt { E } } { \\sqrt { G } \\pm \\sqrt { P } } - \\frac { \\sqrt { G } \\Big ( \\sqrt { E } - \\sqrt { \\frac { G } { n - 1 } + K } \\Big ) } { ( \\sqrt { G } + \\sqrt { P } ) ( \\sqrt { G } - \\sqrt { P } ) } \\Bigg ) ^ 2 \\geq 0 . \\end{align*}"}
{"id": "52.png", "formula": "\\begin{align*} ( \\tilde d _ { j , h } \\tilde U _ { j , h } ^ * f ) ( \\mu ) & = h ^ { j } \\sum _ l ^ { \\binom { n } { j } } ( \\tilde d _ { 0 , h } f _ { j , l } ) ( \\mu ) \\wedge d x ^ { I ^ j _ l } \\\\ & = h ^ { j } \\sum _ l ^ { \\binom { n } { j } } \\sum _ { \\alpha = 1 } ^ n { ( f _ { j , l } ( \\mu + h \\delta _ \\alpha ) - f _ { j , l } ( \\mu ) ) } d x ^ \\alpha \\wedge d x ^ { I ^ j _ l } \\\\ & = h ^ { j } \\sum _ { 1 \\leq \\tilde l \\leq { \\binom { n } { j } } } \\left ( \\sum _ { \\substack { \\alpha , l \\\\ d x ^ \\alpha \\wedge d x ^ { I ^ j _ l } = ( \\pm ) d x ^ { I ^ { j + 1 } _ { \\tilde l } } } } { ( f _ { j , l } ( \\mu + h \\delta _ \\alpha ) - f _ { j , l } ( \\mu ) ) } \\right ) ( \\pm ) d x ^ { I ^ { j + 1 } _ { \\tilde l } } . \\end{align*}"}
{"id": "379.png", "formula": "\\begin{equation} ( x - a ) ^ p = ( y - b ) ^ p = \\big ( z - ( a + b ) \\big ) ^ p \\equiv 0 \\pmod { p } \\end{equation}"}
{"id": "388.png", "formula": "\\begin{equation} 2 x = c - b + a , 2 y = c + b - a , 2 z = c + b + a , \\end{equation}"}
{"id": "156.png", "formula": "\\begin{equation} \\sum _ { i = 0 } ^ \\infty { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ i \\cdot | R ( t / 3 ^ i ) | \\leq \\sum _ { i = 0 } ^ \\infty { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ i \\cdot C _ J \\cdot D ( \\Omega ) \\cdot t ^ { 2 n } \\cdot { ( 3 ^ { 2 n } ) } ^ { - i } < \\infty . \\end{equation}"}
{"id": "195.png", "formula": "\\begin{equation} ( y , z ) \\in S ^ { \\circ } \\Longleftrightarrow ( y _ i , z _ i ) \\in S _ i \\ \\ \\text { f o r } \\ \\ i = \\min \\{ j : y _ j \\not = z _ j \\} , \\end{equation}"}
{"id": "10.png", "formula": "\\begin{align} f ( z ) = \\tfrac { 1 } { 2 \\pi i } I _ + \\bigl ( ( X - z ) ^ { - 1 } f \\bigr ) = \\lim _ { y \\to \\infty } \\tfrac { 1 } { 2 \\pi i } \\bigl \\langle \\chi _ y , ( X - z ) ^ { - 1 } f \\bigr \\rangle \\end{align}"}
{"id": "418.png", "formula": "\\begin{equation} \\begin{aligned} n _ S & = \\lceil \\frac { \\pi L _ { S , x } L _ { S , y } } { \\lambda ^ 2 } \\rceil + o ( \\frac { L _ { S , x } L _ { S , y } } { \\lambda ^ 2 } ) , \\\\ n _ R & = \\lceil \\frac { \\pi L _ { R , x } L _ { R , y } } { \\lambda ^ 2 } \\rceil + o ( \\frac { L _ { R , x } L _ { R , y } } { \\lambda ^ 2 } ) , \\end{aligned} \\end{equation}"}
{"id": "61.png", "formula": "\\begin{align*} \\prescript { } { i _ 0 - 1 } { ( \\prescript { } { i _ 1 } { \\hat { s } } ) } ( i ) = & \\begin{cases} \\prescript { } { i _ 1 } { \\hat { s } } ( i ) & i < i _ 0 - 1 \\\\ \\prescript { } { i _ 1 } { \\hat { s } } ( i + 1 ) & i _ 0 - 1 \\leq i \\end{cases} \\end{align*}"}
{"id": "232.png", "formula": "\\begin{equation} \\frac { d x ^ i } { d \\tau } = f ( x ^ 1 , \\ldots , x ^ n ) \\ X ^ i ( x ^ 1 , \\ldots , x ^ n ) , i = 1 , \\ldots , n . \\end{equation}"}
{"id": "259.png", "formula": "\\begin{equation} \\frac { d ^ 2 x } { d t ^ 2 } + A ( x ) \\frac { d x } { d t } + b ( x ) = 0 \\end{equation}"}
{"id": "253.png", "formula": "\\[ \\frac { d ^ 2 x _ 1 } { d t _ 1 ^ 2 } + \\left ( \\gamma _ 0 + \\frac { h ' } { h } \\right ) \\left ( \\frac { d x _ 1 } { d t _ 1 } \\right ) ^ 2 + \\frac { 1 } { h } A _ 0 \\frac { d x _ 1 } { d t _ 1 } + \\frac { 1 } { h ^ 2 } b _ 0 = 0 , \\]"}
{"id": "341.png", "formula": "\\begin{equation} I ( G ) ^ { [ k ] } = I ( G _ 1 ) ^ { [ k ] } + x _ n x _ { n - 1 } I ( G _ 2 ) ^ { [ k - 1 ] } \\end{equation}"}
{"id": "245.png", "formula": "\\begin{equation} \\left \\{ \\begin{array} { r c l } \\dfrac { d \\bar x ^ i } { d \\tau } & = & f \\ , v ^ i = \\bar v ^ i \\\\ \\dfrac { d \\bar v ^ i } { d \\tau } & = & { f ^ 2 } \\ , X ^ i \\left ( \\bar x , \\dfrac 1 f \\ , \\bar v \\right ) + \\dfrac d { d \\tau } ( \\log f ) \\ , \\bar v ^ i \\end{array} \\right . . \\end{equation}"}
{"id": "127.png", "formula": "\\begin{equation} \\begin{aligned} a ( \\vec { v } _ 0 , \\vec { w } ) + b ( \\vec { w } , p ) & = a ( \\vec { v } , \\vec { w } ) , & & \\forall \\vec { w } \\in V , \\\\ b ( \\vec { v } _ 0 , q ) & = 0 , & & \\forall q \\in Q , \\end{aligned} \\end{equation}"}
{"id": "175.png", "formula": "\\begin{align*} u ^ { F ^ { \\prime } , G ^ { \\prime } } _ { X , Y } ( b _ { i } \\boxtimes c _ { j } ) & = u _ { F ^ { \\prime } , G ^ { \\prime } } \\left ( \\sum _ { l , k } b ^ { \\prime } _ { l } \\langle b ^ { \\prime } _ { l } \\ | \\ b _ { i } \\rangle \\boxtimes c ^ { \\prime } _ { k } \\langle c ^ { \\prime } _ { k } \\ | \\ c _ { j } \\rangle \\right ) \\\\ & = \\sum _ { l , k } c ^ { \\prime } _ { k } \\boxtimes b ^ { \\prime } _ { l } \\ \\langle b ^ { \\prime } _ { l } \\ | \\ b _ { i } \\rangle \\langle c ^ { \\prime } _ { k } \\ | \\ c _ { j } \\rangle \\\\ & = c _ { j } \\boxtimes b _ { i } = u ^ { F , G } _ { X , Y } ( b _ { i } \\boxtimes c _ { j } ) , \\\\ \\end{align*}"}
{"id": "265.png", "formula": "\\[ \\frac { d ^ 2 x _ 2 } { d t _ 2 ^ 2 } + 1 = 0 . \\]"}
{"id": "378.png", "formula": "\\begin{equation} \\begin{split} K _ p & = - \\sum _ { i = 1 } ^ { p - 1 } \\dfrac { 1 } { i } \\binom { p - 1 } { i - 1 } b ^ { i - 1 } \\bigg ( \\sum _ { j = 1 } ^ { p - i - 1 } ( - 1 ) ^ j \\binom { p - i } { j } x ^ { p - i - j } a ^ { j - 1 } \\bigg ) \\\\ & - \\sum _ { i = 1 } ^ { p - 1 } \\dfrac { ( - 1 ) ^ i } { i } \\binom { p - 1 } { i - 1 } b ^ { p - i - 1 } a ^ { i - 1 } . \\end{split} \\end{equation}"}
{"id": "82.png", "formula": "\\begin{align} S & = w _ 1 G D _ 1 ( p - 2 + s ) + w _ 2 G D _ 2 ( p - 2 + s ) + \\epsilon w _ 3 G D _ 1 ( p - 4 + s ) + \\epsilon w _ 4 G D _ 2 ( p - 4 + s ) \\nonumber \\\\ & = w _ 1 G D _ 1 ( 0 ) + w _ 2 G D _ 2 ( 0 ) + w _ 3 \\epsilon G D _ 1 ( - 2 ) + w _ 4 \\epsilon G D _ 2 ( - 2 ) , \\end{align}"}
{"id": "231.png", "formula": "\\begin{equation} d t = f ( x ^ 1 , \\ldots , x ^ n ) \\ d \\tau , f ( x ^ 1 , \\ldots , x ^ n ) > 0 , \\end{equation}"}
{"id": "134.png", "formula": "\\[ A _ \\lambda \\vec { u } _ \\lambda = \\vec { f } , A _ \\infty \\vec { u } _ \\infty = \\vec { f } , \\text { a n d } A \\vec { u } _ 0 = \\vec { f } , \\]"}
{"id": "377.png", "formula": "\\begin{equation} ( x - a ) ^ p = ( y - b ) ^ p = \\big ( z - ( a + b ) \\big ) ^ p = p a b K _ p , \\end{equation}"}
{"id": "389.png", "formula": "\\[ x - a = \\sqrt [ \\leftroot { - 4 } \\uproot { 5 } p ] { p a b K _ p } \\]"}
{"id": "358.png", "formula": "\\begin{equation} z - y = a , z - x = b . \\end{equation}"}
{"id": "364.png", "formula": "\\[ ( x ^ m ) ^ p + ( y ^ m ) ^ p = ( z ^ m ) ^ p , \\forall n = m p . \\]"}