\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}


\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}


\begin{equation} \frac { d x ^ i } { d t } = X ^ i ( x ^ 1 , \ldots , x ^ n ) , i = 1 , \ldots , n , \end{equation}


\begin{equation} d m | _ 0 ( g ) = R _ 0 ( \kappa ) C _ + g . \end{equation}


\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}


\begin{equation} \dot { z } _ l = F _ l ( z ) = \sum _ { k = 1 } ^ r a _ { l , k } \ , z ^ { \alpha ( k ) } , l = 1 , \dots , n . \end{equation}


\begin{equation} ( x - a ) ^ n = f _ n ( x , b , a ) + g _ n ( b , a ) , \end{equation}


\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}


\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*}


\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}


\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*}


\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}


\[ \begin{split} \Sigma = \{ \sigma \in C ^ { 1 } ( \R ) \ , : \ , \sigma \text { i s e v e n } , \ , \sigma ( 0 ) = \sigma ' ( 0 ) = 0 , \text { a n d } \sigma ( r ) > 0 \text { f o r a l l } r \neq 0 \} . \end{split} \]


\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}


\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}


\begin{equation} R \cap R ^ { - 1 } \ = \ 1 _ X , \text { a n d } R \cup R ^ { - 1 } \ = \ X \times X . \end{equation}


\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}


\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}


\begin{align} u _ t - | D u | ^ { \gamma } \big ( \Delta u + ( p - 2 ) \Delta _ \infty ^ N u \big ) = 0 . \end{align}


\begin{align} c _ 3 = 2 + 2 \kappa \geq 2 > 0 . \end{align}


\begin{equation} d t = r \ , d \tau \Longleftrightarrow \frac { d \tau } { d t } = \frac 1 r , \end{equation}


\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*}


\begin{equation} \bold { \Sigma } = \bold { d } \widetilde { \bold { d } } ^ { T } , \end{equation}


\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}


\begin{align} u _ t - | D u | ^ { \gamma } \big ( \Delta u + ( p - 2 ) \Delta _ \infty ^ N u \big ) = 0 \end{align}


\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}


\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}


\begin{equation} u _ t = \Delta u ^ m + u ^ p , \end{equation}


\begin{equation} u ( x , 0 ) = u _ 0 ( x ) , x \in \real ^ N . \end{equation}


\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}


\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}


\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}


\begin{equation} \tilde \pi ( 0 , x ) \ = \ x , \text { a n d } \tilde \pi ( 1 , x ) \ = \ \infty . \end{equation}


\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*}


\begin{equation} \frac { d ^ 2 y } { d \tau ^ 2 } + \alpha \frac { d y } { d \tau } + B y + C = 0 , \end{equation}


\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}


\begin{equation} L : = \sigma ( m - 1 ) + 2 ( p - 1 ) > 0 . \end{equation}


\begin{equation} e _ { 2 k + 1 } e _ { 2 l } = C _ { k + l } ^ { k } e _ { 2 k + 2 l + 1 } . \end{equation}


\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}


\begin{equation} \dot { z } _ l = F _ l ( z ) = \sum _ { k = 1 } ^ \infty a _ { l , k } \ , z ^ { \alpha ( k ) } , l = 1 , \dots , n , \end{equation}


\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}


\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}


\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*}


\begin{equation} ( x - a ) ^ 2 = - ( b - a ) ^ 2 + b ^ 2 + a ^ 2 = 2 a b , \end{equation}


\[ ( x - a ) ^ p = f _ p ( x , b , a ) + g _ p ( b , a ) \textnormal { a n d } K _ p = \dfrac { f _ p ( x , b , a ) } { p a b } + \dfrac { g _ p ( b , a ) } { p a b } . \]


\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}


\begin{equation} x ^ n + y ^ n = z ^ n \end{equation}


\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}


\begin{equation} \tilde d _ { 0 } \omega = \sum _ { l = 1 } ^ n ( \mathcal { D } _ { l } \omega ) d x _ l . \end{equation}


\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}


\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}


\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}


\begin{equation} X ^ i ( x , v ) = \sum _ { j = 1 } ^ n B ^ i \ , _ j ( x ) \ , v ^ j , i = 1 , \ldots , n . \end{equation}


\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}


\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}


\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*}


\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}


\begin{align*} \| N \| _ { L ^ \infty ( \Omega _ T ) } : = \sup \{ | N ( x , t ) | : ( x , t ) \in \Omega _ T \} \end{align*}


\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*}


\begin{align} \begin{cases} w _ 1 = p - \gamma , & w _ 2 = 2 , \\ w _ 3 = 4 - p + \gamma , & w _ 4 = 2 , \end{cases} \end{align}


\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}


\begin{equation} P [ j , k ] ^ { \circ } ( z + ) \ = \ \bar D _ k - \ \cup \ [ \bigcup _ { i \in \N } \ ( z + A _ i ) + ] . \end{equation}


\begin{equation} Z _ X ( S ) : = \bigcup _ { v \in S } Z _ X ( v ) . \end{equation}


\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}


\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}


\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}


\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*}


\begin{equation} \Delta v _ k ( x , t ) \geq - \frac { K } { t } , v _ k = \frac { m } { m - 1 } u _ k ^ { m - 1 } . \end{equation}


\begin{equation} \bold { p } _ { j } = \bold { B } _ { j } \bold { p } _ { j } + \bold { q } _ { j } + \bold { v } _ { j } , \end{equation}


\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}


\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}


\begin{equation} \frac { d \varphi } { d x } = c \ , h . \end{equation}


\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}


\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}


\begin{align*} w _ 1 = \big ( \sqrt { p - 1 } - \sqrt { 1 - \gamma } \big ) ^ 2 + \eta . \end{align*}


\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}


\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}


\begin{equation} G : = p - 1 + s - \gamma \quad \text { a n d } E : = s + 1 + \frac { p - 1 } { n - 1 } . \end{equation}


\begin{equation} \tilde d ^ * = ( - 1 ) ^ { n i + 1 } * \tilde d * . \end{equation}


\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}


\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}


\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}


\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}


\begin{equation} x ^ n = z ^ n - y ^ n = ( z - y ) \phi _ n ( z , y ) . \end{equation}


\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}


\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}


\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}


\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}


\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}


\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}


\begin{equation} ( I , x _ n ) ^ { [ k ] } = I ^ { [ k ] } + x _ n I ^ { [ k - 1 ] } \subset S \end{equation}


\begin{equation} A _ { \lambda } ^ { - 1 } \eqsim \frac { \lambda } { \lambda + 1 } P A ^ { - 1 } + \frac { 1 } { \lambda + 1 } A ^ { - 1 } = : M _ \lambda , \end{equation}


\begin{equation} v _ t = ( m - 1 ) v \Delta v + | \nabla v | ^ 2 + K ( m , p ) v ^ { ( m + p - 2 ) / ( m - 1 ) } . \end{equation}


\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}


\begin{equation} X _ N ( \varphi ) \equiv \sum _ { i = 1 } ^ N \varphi ( z _ i ) \end{equation}


\begin{equation} x - a = x + y - z , 2 x + b - a = x + y , y = x + b - a , \end{equation}


\[ \left ( A \frac { d } { d x } - 3 A ' \right ) \left ( A \frac { d } { d x } + \gamma \ , A - A ' \right ) b = ( \Delta _ \R - 3 ) \Delta _ \R b = 0 , \]


\begin{equation} D _ j \ = \ \{ x \in \Z [ 2 ] : x _ j = 0 \} , \bar D _ j \ = \ \{ x \in \Z [ 2 ] : x _ j = 1 \} . \end{equation}


\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}


\begin{equation} z ( x , t ) = ( T - t ) ^ { - \alpha } f ( ( 1 + | x | ) ( T - t ) ^ { \beta } ) , \end{equation}


\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}


\begin{equation} \ddot x + f ( x ) \ , \dot x + g ( x ) = 0 , \end{equation}


\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}


\begin{equation} \begin{aligned} 2 c _ 1 + c _ 2 \geq \min \{ 2 ( p - \gamma ) , 8 \} > 0 . \end{aligned} \end{equation}


\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}


\begin{equation} e _ { 2 k + 1 } e _ { 2 l + 1 } = C _ { k + l } ^ { l } e _ { 2 k + 2 l + 2 } . \end{equation}


\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}


\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}


\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}


\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}


\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}


\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}


\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*}


\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}


\begin{equation} A \ = \ \bigcap _ { i \in \N } \ \pi _ { i } ^ { - 1 } ( A _ i ) \ = \ \overleftarrow { L i m } \{ A _ i \} \end{equation}


\begin{equation} K ( n ) = \frac { \Gamma ( \frac { n } { 2 } ) } { \sqrt { \pi } \Gamma ( \frac { n + 1 } { 2 } ) } . \end{equation}


\begin{equation} Q \ = \ \bigcup _ { n = 2 } ^ { \infty } \ ( \{ t \} \times A _ n ) \times ( \{ t _ n \} \times A _ 1 ) . \end{equation}


\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}


\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}


\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*}


\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}


\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*}


\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}


\begin{equation} P : = p - 1 \quad \text { a n d } K : = \gamma + 1 , \end{equation}


\begin{equation} \omega ( \mu ) = \sum _ { I \in P ^ { j , n } _ + } \omega _ { I } ( \mu ) d x ^ { I } , \end{equation}


\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}


\begin{equation} \dot { z } = F ( z ) , z \in \mathbb { D } ^ n ( \rho ) , \end{equation}


\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*}


\begin{equation} \frac { d ^ 2 x } { d \tau ^ 2 } + \tilde g ( x ) \frac { d x } { d \tau } + \tilde h ( x ) = 0 , \end{equation}


\begin{equation} \sum _ { k = 1 } ^ { + \infty } | \alpha ( k ) | \ , \epsilon _ k \ , \rho ^ { 2 | \alpha ( k ) | } \end{equation}


\begin{equation} \frac { d ^ 2 x } { d t ^ 2 } = X \left ( x , \frac { d x } { d t } \right ) , \end{equation}


\begin{align*} I ( i ) = & I ' ( i ) , i < l , \\ I ( l ) < & I ' ( l ) . \end{align*}


\begin{equation} U _ N = \max \{ U _ R , U _ S \} , ~ ~ U _ M = \min \{ U _ R , U _ S \} . \end{equation}


\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}


\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}


\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*}


\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}


\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*}


\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}


\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*}


\begin{equation} \Delta _ j ( X ) = d _ j ^ * d _ j + d _ { j - 1 } d _ { j - 1 } ^ * \ . \end{equation}


\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}


\begin{align*} \Delta u : = \sum _ { i = 1 } ^ n u _ { x _ i x _ i } \end{align*}


\begin{equation} \psi ( E ) = \phi ( E ) + E \phi ' ( E ) . \end{equation}


\begin{equation} \frac { d ^ 2 y } { d \tau ^ 2 } = 0 \end{equation}


\begin{equation} u _ t = \Delta u ^ m + | x | ^ { \sigma } u ^ p , \end{equation}


\[ \ddot { x } = 2 \dot { x } ^ 2 \cot x + \sin x \cos x , \]


\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*}


\begin{equation} e _ 1 e _ i = e _ { i + 1 } . \end{equation}


\begin{equation} T \ = \ S \cup R \cup ( \bigcup _ i [ ( E _ i \times C _ i ) \cup ( C _ i \times F _ i ) ] ) . \end{equation}


\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*}


\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}


\begin{align} c _ 3 = w _ 2 + w _ 4 \kappa \geq c , \end{align}


\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}


\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}


\begin{equation} a ( P \vec { v } , \vec { w } ) = a ( \vec { v } , \vec { w } ) , \forall \vec { w } \in W . \end{equation}


\begin{equation} X = \sum _ { i \in I } \delta _ { x _ i } , \end{equation}


\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*}


\[ \begin{split} \lim _ { x \to 0 , x \neq 0 } F ( D g ( x ) , D ^ 2 g ( x ) ) = 0 . \end{split} \]


\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}


\begin{equation} F ' _ l ( z ' ) = \sum _ { k = 1 } ^ { r } \mu ^ { | \alpha ( k ) | - 1 } a _ { l , k } z '^ { \alpha ( k ) } . \end{equation}


\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*}


\begin{equation} { [ f ] } _ { W ^ { 1 , p } ( A ) } = { \bigg ( \int _ A { | \nabla f | } ^ p \ d x \bigg ) } ^ { \frac { 1 } { p } } , \end{equation}


\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}


\begin{equation} \vec { u } _ { \lambda } = \frac { \lambda } { \lambda + 1 } \vec { u } _ { \infty } + \frac { 1 } { \lambda + 1 } \vec { u } _ { 0 } . \end{equation}


\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}


\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}


\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}


\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}


\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}


\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}


\begin{equation} D _ t u _ 0 = S ( t ) u _ 0 . \end{equation}


\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}


\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*}


\begin{equation} d _ j ^ * d _ j f = - d _ { j - 1 } d _ { j - 1 } ^ * f \ . \end{equation}


\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}


\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}


\begin{align*} c _ 3 + c _ 4 = & 2 ( 1 - \gamma ) - 2 \kappa + 2 ( 2 + \gamma ) \kappa ^ 2 . \end{align*}


\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}


\begin{equation} x ^ p + y ^ p = z ^ p , \end{equation}


\begin{equation} L _ \mu = \sum _ { l = 1 } ^ n \sum _ { k = 1 } ^ \infty \mu ^ { \vert \alpha ( k ) \vert } \left | a _ { l , k } \right | . \end{equation}


\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*}


\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}


\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*}


\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}


\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}


\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*}


\begin{equation} M _ { \lambda } = \frac { \lambda } { 1 + \lambda } P A ^ { - 1 } + \frac { 1 } { \lambda + 1 } A ^ { - 1 } , \end{equation}


\begin{align*} 2 c _ 1 + c _ 2 = & 2 ( p - \gamma ) + 2 ( 4 - p + \gamma ) \kappa ^ 2 . \end{align*}


\begin{equation} u _ t - \Delta _ p u = 0 \end{equation}


\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}


\begin{equation} \tilde { d } \circ U = U \circ d \ . \end{equation}


\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}


\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*}


\begin{equation} ( I + t P ) ^ { - 1 } = I - \frac { t } { t + 1 } P , \quad \forall t \in \mathbb { R } \backslash \{ - 1 \} . \end{equation}


\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}


\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}


\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}


\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}


\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}


\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}


\begin{equation} q ( t , x ) = \widetilde q ( t , x - 2 c t ) + c . \end{equation}


\begin{equation} i ( \Gamma ) \omega _ L = d E _ L . \end{equation}


\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}


\begin{equation} \ddot x + f ( x ) \ , \dot x ^ 2 + g ( x ) \ , \dot x + h ( x ) = 0 \end{equation}


\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*}


\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}


\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}


\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}


\begin{equation} J _ 1 \cap J _ 2 = ( x _ { i _ 1 } , \dots , x _ { i _ t } ) J _ 1 \end{equation}


\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*}


\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}


\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}


\begin{equation} y = \varphi ( x ) , d \tau = h ( x ) \ d t , \end{equation}


\begin{align*} b ^ 2 - 4 a c = G \cdot P \cdot E \Big ( \frac { G } { n - 1 } + K \Big ) . \end{align*}


\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*}


\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}


\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}


\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}


\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}


\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*}


\begin{equation} m ' = - i \kappa m + i C _ + [ q ( m + 1 ) ] , \end{equation}


\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}


\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*}


\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}


\begin{equation} \gcd ( z - y , \phi _ p ( z , y ) ) = 1 . \end{equation}


\begin{equation} A \cap A ^ { - 1 } \ = \ \{ e \} , \text { a n d } A \cup A ^ { - 1 } \ = \ G . \end{equation}


\[ \mathfrak { R } = \left \{ \vec { c } + \mathfrak { m } \vec { x } \ ; | \ ; \vec { c } \in \mathbb { R } ^ d , \mathfrak { m } \in \mathfrak { s o } ( d ) \right \} , \]


\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}


\begin{align} 2 c _ 1 + c _ 2 = 2 ( w _ 1 + w _ 3 \kappa ) - 2 w _ 3 \theta \kappa \geq c \end{align}


\begin{equation} x = a + \sqrt { 2 a b } , y = b + \sqrt { 2 a b } , z = a + b + \sqrt { 2 a b } . \end{equation}


\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}


\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}


\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*}


\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}


\[ \left ( A \frac { d } { d x } + \gamma A - A ' \right ) b = \left ( x \frac { d } { d x } \right ) ( 1 / 2 ) = 0 , \]


\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}


\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}


\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*}


\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}


\begin{equation} x ^ p + y ^ p = z ^ p \Longrightarrow ( x - a ) ^ p - p a b K _ p = 0 . \end{equation}


\begin{equation} q ( \phi ; q _ 0 ) = \bigl ( e ^ { J \nabla H _ \phi } q _ 0 \bigr ) ( x = 0 ) . \end{equation}


\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}


\begin{equation} E ( x , t ) = t ^ { 1 / ( 1 - p ) } \varphi ( | x | t ^ { - \gamma } ) , \gamma = \frac { m - p } { 2 ( 1 - p ) } \end{equation}


\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}


\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}


\begin{equation} J _ i \left ( \frac { j } { n } \right ) = \delta _ { i j } . \end{equation}


\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*}


\begin{equation} \ddot { x } + \frac { 1 } { x } \dot { x } ^ 2 + x \ , \dot { x } + \frac { 1 } { 2 } = 0 , \end{equation}


\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}


\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}


\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}


\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*}


\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}


\begin{equation} \lim _ { \kappa \to \infty } \ , \sup _ { q \in Q _ * } \ , \sup _ { | t | \leq T } \norm { n ( t ) - n ( 0 ) } _ { H ^ { - 2 } } = 0 . \end{equation}


\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}


\begin{align*} a _ { h , l } ( \xi ) : = \frac { ( - 1 + e ^ { - 2 \pi i h \xi _ l } ) } h , 1 \leq l \leq n . \end{align*}


\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}


\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}


\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}


\begin{equation} \beta ( \lambda \kappa ; q _ \lambda ) = \beta ( \kappa ; q ) \quad \text { f o r a n y $ \lambda > 0 $ . } \end{equation}


\begin{equation} \Delta v \geq - \frac { K } { t } , K = \frac { N } { N ( m - 1 ) + 2 } , \end{equation}


\begin{equation} G ( s ) = - e ^ { ( 2 n - \alpha ( \eta ) ) s } R ( e ^ { - s } ) + \int _ { 0 } ^ s G ( s - s ' ) \ d \mu ( s ' ) . \end{equation}


\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}


\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*}


\begin{equation} d _ j ^ * g ( r ) = \sum _ { r \subset s } \frac { m ( s ) } { m ( r ) } g ( s ) \end{equation}


\begin{equation} \mathcal { L } _ \Delta S = - S . \end{equation}


\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}


\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}


\begin{equation} \frac { d \tau } { d t } = \frac 1 { f ( \gamma ( t ) ) } , \end{equation}


\begin{equation} u _ t = \Delta u + | x | ^ { \sigma } u ^ p , \end{equation}


\begin{equation} z ( x , 0 ) = T _ 0 ^ { - \alpha } f ( ( 1 + | x | ) T _ 0 ^ { \beta } ) \geq \| u _ 0 \| _ { \infty } \geq u _ 0 ( x ) \end{equation}


\begin{equation} \mathcal { L } _ \Gamma \theta _ L = d L , \end{equation}


\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}


\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*}


\[ \mathfrak { R } = \left \{ \vec { c } + \mathfrak { m } \vec { x } \ ; | \ ; \vec { c } \in \mathbb { R } ^ d , \mathfrak { m } \in \mathfrak { s o } ( d ) \right \} , \]


\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}


\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}


\begin{equation} \frac { d ^ 2 y } { d \tau ^ 2 } + 1 = 0 \end{equation}


\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}


\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}


\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*}


\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}


\begin{equation} R ' | X \ = \ R , \text { a n d } R '^ { \circ } ( u ) \ = \ F , \ \ R '^ { \circ - 1 } ( u ) \ = \ E . \end{equation}


\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*}


\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}


\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}


\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*}


\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}


\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}


\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}


\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}


\[ \big ( \sum _ i I _ i \big ) : ( \sum _ j J _ j ) \ = \ \sum _ i \big ( I _ i : \big ( \sum _ j J _ j \big ) \big ) \ = \ \sum _ i \bigcap _ j ( I _ i : J _ j ) . \]


\begin{equation} S = w _ 1 G D _ 1 ( p - 2 + s ) + w _ 2 G D _ 2 ( p - 2 + s ) \end{equation}


\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}


\begin{equation} ( x + y - z ) ^ p - p a b K _ p = x ^ p + y ^ p - z ^ p = 0 . \end{equation}


\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}


\begin{equation} \frac { d \bar v } { d \tau } = \frac 1 f f ' ( q ) \ , \bar v \ , \bar v + f ^ 2 \ , F ( q ) , \end{equation}


\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}


\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*}


\[ \frac { d ^ 2 y } { d \tau ^ 2 } + \alpha \frac { d y } { d \tau } + \beta ( y ) = 0 , \]


\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}


\begin{align} \lim _ { y \to 0 , y \neq 0 } F \big ( D g ( y ) , D ^ 2 g ( y ) \big ) = 0 . \end{align}


\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}


\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}


\begin{equation} z ^ n - y ^ n = ( z - y ) ( z ^ { n - 1 } + y ^ { n - 1 } ) + z y ( z ^ { n - 2 } - y ^ { n - 2 } ) \end{equation}


\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*}


\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}


\begin{equation} \rho _ k ( x _ 1 , \ldots , x _ k ) = \rho _ k ( x _ { \sigma ( 1 ) } , \ldots , x _ { \sigma ( k ) } ) , \end{equation}


\[ \frac { d ^ 2 x } { d t ^ 2 } + \gamma ( x ) \left ( \frac { d x } { d t } \right ) ^ 2 + A ( x ) \frac { d x } { d t } + b ( x ) = 0 , \]


\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}


\begin{equation} q ( x ) _ { i } \ = \ g _ { i + 1 } ( \pi _ { i + 1 } ( x ) ) , \ \ \text { f o r } \ \ i \in \Z _ + \end{equation}


\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}


\[ \limsup _ { \delta \to 0 } \ \sup _ { q \in Q } \ \sup _ { | y | < \delta } \| q ( \cdot + y ) - q ( \cdot ) \| _ { H ^ \sigma } = 0 . \]


\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}


\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}


\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*}


\begin{equation} x - a = y - b = z - ( a + b ) = x + y - z . \end{equation}


\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}


\[ W : = \operatorname { K e r } ( B ) = \left \{ \vec { v } \in V \ ; \big | \ ; B \vec { v } = \operatorname { d i v } \vec { v } = 0 \right \} . \]


\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}


\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*}


\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*}


\begin{equation} ( x - a ) ^ p = ( y - b ) ^ p = \big ( z - ( a + b ) \big ) ^ p \equiv 0 \pmod { p } \end{equation}


\begin{equation} 2 x = c - b + a , 2 y = c + b - a , 2 z = c + b + a , \end{equation}


\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}


\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}


\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}


\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}


\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*}


\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}


\begin{equation} \frac { d ^ 2 x } { d t ^ 2 } + A ( x ) \frac { d x } { d t } + b ( x ) = 0 \end{equation}


\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}


\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}


\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*}


\begin{equation} d t = f ( x ^ 1 , \ldots , x ^ n ) \ d \tau , f ( x ^ 1 , \ldots , x ^ n ) > 0 , \end{equation}


\[ A _ \lambda \vec { u } _ \lambda = \vec { f } , A _ \infty \vec { u } _ \infty = \vec { f } , \text { a n d } A \vec { u } _ 0 = \vec { f } , \]


\begin{equation} ( x - a ) ^ p = ( y - b ) ^ p = \big ( z - ( a + b ) \big ) ^ p = p a b K _ p , \end{equation}


\[ x - a = \sqrt [ \leftroot { - 4 } \uproot { 5 } p ] { p a b K _ p } \]


\begin{equation} z - y = a , z - x = b . \end{equation}