\begin{align*} \mathcal { F } = \left \{ f \in \mathcal { H } ( \mathbb { C } ) : \frac { 1 } { \pi } \iint _ { \mathbb C } | f ( z ) | ^ 2 e ^ { - | z | ^ 2 } d A ( z ) < \infty \right \} , \end{align*}


\begin{align*} \mathbf { T } _ { x } = ( \mathbf { x } ) - \mathbf { U } _ { x } , \mathbf { U } _ { x } \mathbf { e } = 0 . \end{align*}


\begin{align*} \left \| \sum \limits _ { j = 1 } ^ d A _ { j } ^ * A _ { j } \right \| = \sum \limits _ { j = 1 } ^ d \left \| X _ { j } ^ * X _ { j } \right \| . \end{align*}


\begin{align*} \delta _ \rho G ( y , z ) = - \delta _ \rho h ( y , z ) = \int _ { \partial \Omega } \frac { \partial G ( y , x ) } { \partial \mathbf { n } _ x } \frac { \partial G ( z , x ) } { \partial \mathbf { n } _ x } \rho ( x ) d \sigma _ x , \end{align*}


\begin{align*} \displaystyle \int _ { \Omega } n ( \cdot , t ) = \int _ { \Omega } n _ { 0 } , n ( x , t ) \ge 0 , 0 < c ( x , t ) < \gamma \mbox { f o r } x \in \Omega . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty \frac { q ^ { n ^ 2 } X ^ n } { \prod _ { i = 0 } ^ { n - 1 } ( q ^ n - q ^ i ) } = \left ( \sum _ { n = 0 } ^ \infty \frac { q ^ { n ^ 2 - n } X ^ n } { \prod _ { i = 0 } ^ { n - 1 } ( q ^ n - q ^ i ) } \right ) \cdot \left ( \sum _ { n = 0 } ^ \infty X ^ n \right ) . \end{align*}


\begin{align*} & f ( x , y _ { \bar u + \theta ( u - \bar u ) } ) - f ( x , \bar y ) = \frac { \partial f } { \partial y } ( x , y _ 1 ) ( y _ { \bar u + \theta ( u - \bar u ) } - \bar y ) \ , \ , y _ 1 = \bar y + \theta _ 1 ( y _ { \bar u + \theta ( u - \bar u ) } - \bar y ) , \\ & f ( x , y _ u ) - f ( x , \bar y ) = \frac { \partial f } { \partial y } ( x , y _ 2 ) ( y _ u - \bar y ) y _ 2 = \bar y + \theta _ 2 ( y _ u - \bar y ) . \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x & \leq C \int _ { B _ { 2 \rho } } r ^ { 2 - n } u _ r ^ 2 \d x + C \varepsilon ^ 2 \rho \int _ { B _ { 2 \rho } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \\ & \rho \leq 1 / 2 \varepsilon \leq \varepsilon _ 0 . \end{align*}


\begin{align*} \int _ { X _ n } \sum _ { x _ 1 , \ldots , x _ k \in \Z ^ n \atop \mathrm { i n d e p . } } f ( x _ 1 g , \ldots , x _ k g ) d \mu ( g ) = \int _ { ( \R ^ n ) ^ k } f ( x _ 1 , \ldots , x _ k ) d x _ 1 \ldots d x _ k . \end{align*}


\begin{align*} \phi ( \alpha , \tau ) = \widetilde \alpha \widetilde \tau { \widetilde \alpha } ^ { - 1 } \cdot { \widetilde \tau } ^ { - 1 } = \alpha ( x _ { i _ 1 } \cdots x _ { i _ k } ) \cdot ( x _ { i _ 1 } \cdots x _ { i _ k } ) ^ { - 1 } = x _ { \alpha ( i _ 1 ) } \cdots x _ { \alpha ( i _ k ) } \cdot ( x _ { i _ 1 } \cdots x _ { i _ k } ) ^ { - 1 } = 1 . \end{align*}


\begin{align*} \delta = \frac { 6 ( \alpha - 1 ) } { 3 \alpha - 5 } > 2 \ , \ , \mbox { a n d } \ , \ , \gamma = \frac { 6 ( \beta - 1 ) } { 3 \beta - 5 } > 2 . \end{align*}


\begin{align*} \Theta ( \cdot , \mu , \tilde { \mu } ) : = \sum _ { \substack { k = 1 , \\ \beta _ { k } > 0 } } ^ { l } \zeta _ { k } \big [ \null _ { k } \eta \big ] _ { \frac { p } { 1 - \alpha _ { k } } } \vartheta ( \mu , \tilde { \mu } ) ^ { \beta _ { k } } + \hat { \zeta } _ { k } ^ { 2 } \big [ \null _ { k } \hat { \eta } \big ] _ { \frac { p } { 1 - \alpha _ { k } } } ^ { 2 } \vartheta ( \mu , \tilde { \mu } ) ^ { 2 \beta _ { k } } . \end{align*}


\begin{align*} a \cdot b : = \bigvee \{ c \in A \ : | \ : c \wedge a \le b \} . \end{align*}


\begin{align*} Q _ { e u c l } ( f ) = T _ { e u c l } [ f ] \end{align*}


\begin{align*} ( \underline { \Delta ^ { J - 1 } } , \underline { \partial \Delta ^ { J - 1 } } ) & = \{ ( \Delta ^ { j _ i - 1 } , \partial \Delta ^ { j _ i - 1 } ) \} _ { i \in [ m ] } , \\ ( \underline { \mathrm { p t } ^ J } , \{ \emptyset \} ) & = \left \{ \left ( { \bigsqcup } _ { j _ i } \mathrm { p t } , \{ \emptyset \} \right ) \right \} _ { i \in [ m ] } . \end{align*}


\begin{align*} \lim _ { \tau \to - \infty } \| \varphi ( t , \tau , \omega ) x _ { \tau } - \varphi _ { \infty } ( t , \omega ) x _ 0 \| _ { \mathbb { U } } = 0 , t \geq 0 \omega \in \Omega , \end{align*}


\begin{align*} u _ n = r u _ { n - 1 } + s u _ { n - 2 } , \end{align*}


\begin{align*} r = z q - a n . \end{align*}


\begin{align*} \Theta ( \cdot , \mu , \tilde { \mu } ) : = \eta _ { 2 } \vartheta ( \mu , \tilde { \mu } ) + \hat { \eta } _ { 2 } ^ { 2 } \vartheta ( \mu , \tilde { \mu } ) ^ { 2 } . \end{align*}


\begin{align*} \delta ^ \nabla ( | | R ^ \nabla | | ^ { p - 2 } R ^ \nabla ) = 0 \end{align*}


\begin{align*} e ^ { 2 \varphi ( x ) } : = \int _ { T _ x N } e ^ { 2 w } \circ \exp _ x ( v ) \chi ( \Vert v \Vert ) \dd V _ x , \end{align*}


\begin{align*} D q _ k = \left [ \begin{array} { c c c c } \ 1 _ { n - 1 } ' u _ k + 2 \ 1 _ { n - 1 } ' v _ k \\ 2 J _ { n - 1 } u _ k - 2 I _ { n - 1 } u _ k + S v _ k \\ S ' u _ k + T v _ k \end{array} \right ] . \end{align*}


\begin{align*} u = \left ( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right ) , \end{align*}


\begin{align*} \frac { d } { d t } S _ X ( t , s ) v & = L _ { X _ t } S _ X ( t , s ) v \\ S _ X ( s , s ) v & = v . \end{align*}


\begin{align*} Q _ { j , \pm } ( x ; k ) = \mathcal { C } \left ( Q _ { j , - } ( x ; k ) J + D _ j \right ) ( z ) , z \in \mathbb { C } ^ { \pm } . \end{align*}


\begin{align*} \frac { 1 } { f ( x ) } \sigma _ { k } ( x ) \varphi ( h ) G ( \nabla h ) = 1 \end{align*}


\begin{align*} d X _ t ^ i = \frac { 1 } { N } \sum \limits _ { j = 1 } ^ N \hat b ( t , X _ t ^ i , X _ t ^ j , u _ t ^ i ) d t + \sigma d W _ t , \ , \ , i = 1 , \cdots , N , \end{align*}


\begin{align*} \rho _ L ( \theta ) : = \sum _ { n \in \Z } \rho ( L ( \theta + n ) ) L = L ( x ) \geq 1 , \end{align*}


\begin{align*} F ( x + y F ( x ) ) = F ( x ) F ( y ) . \end{align*}


\begin{align*} \oint s _ { 1 } ( z ) s _ 2 ( z ) d z = 0 . \end{align*}


\begin{align*} \left ( v ( a ) , v ' ( a ) , v ( b ) , v ' ( b ) \strut \right ) = \left ( u ( a ) , u ' ( a ) , u ( b ) , u ' ( b ) \strut \right ) . \end{align*}


\begin{align*} B ( v _ 1 , v _ 2 ) + B ( v _ 2 , v _ 1 ) & = \frac { k } { \ell } ( - \alpha \sin \ell x _ 1 + \beta \cos \ell x _ 1 ) \cos k x _ 2 - \frac { \ell } { k } \cos k x _ 2 ( - \alpha \sin \ell x _ 1 + \beta \cos \ell x _ 1 ) \\ & = \left ( \frac { k } { \ell } - \frac { \ell } { k } \right ) ( - \alpha \sin \ell x _ 1 + \beta \cos \ell x _ 1 ) \cos k x _ 2 . \end{align*}


\begin{align*} \begin{aligned} Z ^ { \star } _ { I } ( 1 , \{ \{ 1 \} ^ { m - 1 } , 2 \} ^ { n } ; ( \alpha , \beta ) ) = Z ( m n | n + 1 ; ( \alpha , \beta ) ) + m Z ( m n + 1 | n ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \bigvee _ { n = 0 } ^ { N - 1 } \widetilde \Gamma _ { p } ^ n \bigvee _ { ( i , j ) \in \Lambda _ N ^ { \vec { v } } ( 1 ) } T ^ { - ( i , j ) } \{ A , A ^ c \} \times [ 0 , 1 ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \Vert X _ n ( t , \omega ) - X ( t , \omega ) \Vert _ H = 0 , \forall \ , ( t , \omega ) \in \Gamma , \end{align*}


\begin{align*} \mathcal { N } _ { \epsilon } ( x , y ) : = \int _ { \mathbb { R } } \rho _ { \epsilon } ( y - z ) \mathcal { N } ( x , z ) \ , d z , \end{align*}


\begin{align*} \tilde { \Delta } _ i ^ n A : = \int _ { ( i - 1 ) \Delta _ n } ^ { i \Delta _ n } \mathcal S ( i \Delta _ n - s ) \alpha _ s d s , \\ \tilde { \Delta } _ i ^ n M : = \int _ { ( i - 1 ) \Delta _ n } ^ { i \Delta _ n } \mathcal S ( i \Delta _ n - s ) \sigma _ s d W _ s . \end{align*}


\begin{align*} & M ^ { \pm } _ 2 ( x ; z ) - e _ { 2 } = \mathcal { P } ^ { \pm } \left ( M _ { - } ( x ; \cdot ) R ( x ; \cdot ) \right ) _ { 2 1 } , z \in \mathbb { R } , \\ & 2 i k ( M ^ { \pm } _ 2 ( x ; z ) - e _ { 2 } ) = \mathcal { P } ^ { \pm } \left ( 2 i k M _ - ( x ; \cdot ) R ( x ; \cdot ) \right ) _ { 2 1 } , z \in \mathbb { R } , \end{align*}


\begin{align*} \Delta _ \Gamma G ' ( c ) + c V H & = A : D _ \Gamma ^ 2 c + B \cdot \nabla _ \Gamma c + C c \end{align*}


\begin{align*} \bar { N } ( x ) = \{ u \in G \mid u = v ( u , v ) \in E \} . \end{align*}


\begin{align*} - \Delta _ y \varphi = \mu ^ { 2 } \left ( \varphi - 1 - \frac { \gamma } { r ^ { 2 } } \right ) _ { + } \textrm { i n } \ \mathbb { R } ^ { 5 } . \end{align*}


\begin{align*} \widetilde { T } = \{ \{ k _ { i , 1 } , k _ { i , 2 } \} : 1 \le i \le m \} & \mbox { s u c h t h a t } k _ { i , 1 } < k _ { i , 2 } , \mbox { f o r a l l } 1 \le i \le m , \\ & \mbox { a n d } k _ { i , 1 } < k _ { i + 1 , 1 } , \mbox { f o r a l l } 1 \le i < m . \end{align*}


\begin{align*} \nabla _ { \frac \partial { \partial r } } X = \frac \partial { \partial r } \ , , \end{align*}


\begin{align*} H ( x ) = \sum _ { i = 1 } ^ n \sum _ { j = 1 } ^ n \frac { f ( x + \sigma u _ i + \sigma u _ j ) - f ( x + \sigma u _ i ) - f ( x + \sigma u _ j ) + f ( x ) } { \sigma ^ 2 } u _ i u _ j ^ \intercal , \end{align*}


\begin{align*} d ( X ( \bar u ) ) - L _ X g ( \nabla \bar u , \cdot ) & = \nabla ^ 2 \bar u ( X , \cdot ) - \nabla X ( \nabla \bar u , \cdot ) . \end{align*}


\begin{align*} L _ \delta \omega _ \delta = l _ { 1 , \delta } + l _ { 2 , \delta } + R _ \delta ( \omega _ \delta ) , \end{align*}


\begin{align*} { S _ { \max } } = \frac { { W I } } { { K L } } { \widetilde \varepsilon _ K } \left ( { \frac { { P { { \left | { { h _ s } } \right | } ^ 2 } } } { { W { N _ 0 } } } } \right ) , \end{align*}


\begin{align*} X _ G : = X _ G ( g _ 0 , \ldots , g _ { N - 1 } ) : = \left \{ x \in X ( K ) \colon g _ 0 ( x ) , g _ 1 ( x ) , \dots , g _ { N - 1 } ( x ) \in G \right \} \end{align*}


\begin{align*} B = \sum _ { j } c _ j \bold { B } _ j , c _ j = \int _ { \Omega } B \cdot \bold { B } _ { j } \dd x , A = \sum _ { j } c _ j \bold { A } _ { j } . \end{align*}


\begin{align*} I _ 2 ( x ) = p + q \int _ { - \alpha ( x ) } ^ { \alpha ( x ) } f ( x - y ) f ( y ) / f ( x ) d y \to 1 , \end{align*}


\begin{align*} \nabla _ H \varphi _ n ( p _ n ) = \Bigg ( \frac { x _ \xi } { n } + 2 n y _ \xi ( x _ { p _ n } y _ \xi - y _ { p _ n } x _ \xi ) - n y _ { p _ n } z _ { p _ n } , \frac { y _ \xi } { n } - 2 n x _ \xi ( x _ { p _ n } y _ \xi - y _ { p _ n } x _ \xi ) + n x _ { p _ n } z _ { p _ n } \Bigg ) . \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } J _ { 1 } ( \overline { u } _ { k } ) = \widetilde { E } _ { 1 } , \ \ \ \lim _ { k \rightarrow \infty } J _ { \omega } \overline { v } _ { k } = \widetilde { E } _ { \omega } . \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ g \textbf { \textit { U } } = 0 & \ \Omega , \\ \Lambda _ { g } ( \textbf { \textit { U } } ) = \tau \textbf { \textit { U } } & \ \partial \Omega , \end{cases} \end{align*}


\begin{align*} \prescript L { } { } \ell { } ^ R ( x , - w ^ { - 1 } \alpha ) = \prescript L { } { } \ell { } ^ R ( x , v ( - ( w v ) ^ { - 1 } \alpha ) ) \geq 0 \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - L u = f ( u ) & B _ 1 \\ u = 0 & \partial B _ 1 \end{array} \right . \end{align*}


\begin{align*} | g ( x , t ) | & = \left | \int _ { \R } \frac { \partial _ u ^ M \varphi ( x , u ) - \partial _ u ^ M \varphi ( x , t ) } { | u - t | ^ { \frac 1 { 2 s } - M + 1 } } d u \right | = \left | \int _ { I _ Q } \frac { \partial _ u ^ M \varphi ( x , u ) } { | u - t | ^ { \frac 1 { 2 s } - M + 1 } } d u \right | \approx \left | \int _ { I _ Q } \frac { \partial _ u \varphi ( x , u ) } { | u - t | ^ { \frac 1 { 2 s } } } d u \right | . \end{align*}


\begin{align*} \beta _ 1 ( u ) = \beta _ 2 ( u ) = \beta _ 3 ( u ) = 0 \Longrightarrow \beta _ 3 ( u _ { \varepsilon } ) = 0 . \end{align*}


\begin{align*} V ( r ) = \frac { L ( L + 1 ) } { r ^ 2 } - \frac { Q } { r } f + \kappa Q \frac { r } { f } + \kappa \frac { L ( L + 1 ) } { f ^ 2 } , \end{align*}


\begin{align*} & | \epsilon ( x _ a , t ) | \le \bar \epsilon ( L _ r , M _ r , \delta ) : = 2 L _ r \delta + M _ r \ , . \end{align*}


\begin{align*} F ^ { \prime } ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) \delta ^ \nabla R ^ \nabla - i _ { \big ( F ^ { \prime } ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) \big ) } R ^ \nabla = 0 \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { \frac { q - 1 } { 4 } } k _ 3 ( \langle H \rangle , x _ i ) + \sum _ { i = 1 } ^ { \frac { q - 1 } { 4 } } k _ 3 ( \langle H \rangle , y _ i ) = 3 \times k _ 3 ( \langle H \rangle ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m U _ i < \sum _ { i = 1 } ^ m V _ i , \end{align*}


\begin{align*} \phi _ x ( t _ v ) = \phi _ z ( t _ v ) \end{align*}


\begin{align*} h _ { 2 n } = f ^ n h _ { 2 n + 1 } = H \circ f ^ n . \end{align*}


\begin{align*} \frac { \sum \limits _ { i = 1 } ^ { j + 1 } \chi _ i - r ( j + 1 ) } { \chi } < \sum \limits _ { i = 1 } ^ { j + 1 } w _ i < \frac { \sum \limits _ { i = 1 } ^ { j + 1 } \chi _ i - r j } { \chi } . \end{align*}


\begin{align*} E ^ L _ Y ( u ) \subset E ^ L _ Y ( s \sigma ( v ) w ) = \varphi ^ L _ { \sigma , s } ( E ^ L _ X ( v ) ) \end{align*}


\begin{align*} & k _ \lambda ( M ) = k _ \lambda ( M / K ) + k _ \lambda ( K ' ) \mbox { i f } \nu \neq \lambda \\ & k _ \lambda ( M ) = k _ \lambda ( M / K ) + k _ \lambda ( K ' ) + 1 \mbox { i f } \nu = \lambda \end{align*}


\begin{align*} \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { y } ' \right ] = a _ c \sum _ { j = 1 } ^ { M } \mathbf { h } _ { j , * } \mathbf { p } _ c + \sum _ { l = 1 } ^ M a _ l \mathbf { h } _ { l , * } \mathbf { p } _ l . \end{align*}


\begin{align*} D _ { n + 1 } ( x ) & = \sum _ { \lambda _ { n } \in \Lambda _ { n } } \sum _ { l \in L } e ^ { 2 \pi i \tau ( R ^ t \lambda _ { n } + l ) \cdot x } \\ & = \sum _ { l \in L } e ^ { 2 \pi i ( \tau l ) \cdot x } \sum _ { \lambda _ { n } \in \Lambda _ { n } } e ^ { 2 \pi i \tau ( R ^ t \lambda _ { n } ) \cdot x } \\ & = m _ \tau ( x ) D _ n ( R ^ t x ) . \end{align*}


\begin{align*} H = \displaystyle \sum _ { i = 1 } ^ { D N } \left ( \frac { p _ { i } ^ { 2 } } { 2 m } + \frac { 1 } { 2 } m \omega ^ { 2 } x _ { i } ^ { 2 } \right ) , \end{align*}


\begin{align*} \alpha \wedge \beta : = \{ \gamma \in R ^ + \mid \gamma = \alpha + \beta - \delta \delta \in \alpha \vee \beta \} \end{align*}


\begin{align*} t _ { k } = \max \bigl \{ t \in \bigcup _ { j \in J } T _ j \colon t > t _ { k - 1 } \bigr \} . \end{align*}


\begin{align*} P ^ g ( x _ 1 , \ldots , x _ n ) = P ( g x _ 1 , \ldots , g x _ n ) . \end{align*}


\begin{align*} \| x \| = \sup _ { p \in \Gamma _ { \tau } } p ( x ) , x \in X ; \end{align*}


\begin{align*} { { P } } _ k ^ { \mathrm { d i s c } } ( \mathcal { T } ) : = \left \{ q _ h \in L ^ 2 ( \Omega ) : q _ h | _ T \in P _ k ( T ) T \in \mathcal { T } \right \} . \end{align*}


\begin{align*} ( x _ p , y _ p , z _ p ) \cdot ( x _ q , y _ q , z _ q ) = \left ( x _ p + x _ q , y _ p + y _ q , z _ p + z _ q + \frac { 1 } { 2 } ( x _ p y _ q - x _ q y _ p ) \right ) , \end{align*}


\begin{align*} \Gamma ( G \times H ) = \Gamma ( G ) \boxtimes \Gamma ( H ) . \end{align*}


\begin{align*} \sigma _ W ^ 2 = 3 2 / \alpha ^ 2 \sigma _ Z ^ 2 = 3 2 T ^ 2 / \alpha ^ 2 \end{align*}


\begin{align*} ( I _ { n } + p ^ { i + \varepsilon } a ) ^ { - 1 } = I _ { n } + \sum _ { k = 1 } ^ \infty ( - 1 ) ^ { k } p ^ { ( i + \varepsilon ) k } a ^ { k } \equiv I _ { n } - p ^ { i + \varepsilon } a \bmod p ^ { i + 1 + \varepsilon } . \end{align*}


\begin{align*} [ g ' k g h , \psi k \varphi h ] = [ g ' g \underbrace { c _ g ( k ) h } _ { \in H } , \psi k \varphi h ] = [ g ' g , \psi k \varphi h ( c _ { g } ( k ) h ) ^ { - 1 } ] = [ g ' g , \psi k \varphi c _ { g ^ { } } ( k ^ { - 1 } ) ] = [ g ' g , \psi \varphi ] \end{align*}


\begin{align*} \begin{aligned} { F _ { \Phi _ { ( 3 ) } } } _ x ( v ) = \sum _ { i = 1 } ^ { m } \big ( 6 \langle B ( v , e _ { i } ) , B ( v , e _ { i } ) \rangle _ { \mathbb R ^ q } - \langle B ( v , v ) , B ( e _ { i } , e _ { i } ) \rangle _ { \mathbb R ^ q } \big ) < 0 \end{aligned} \end{align*}


\begin{align*} J ( \varphi ) = \inf \left \{ \int ^ { 1 } _ { 0 } \Lambda ^ { * } ( \phi ( r ) ) d r ; ~ \varphi ( t ) = \int ^ { t } _ { 0 } \phi ( r ) d r \in \mathcal { A C } , ~ t \in [ 0 , 1 ] \right \} , \end{align*}


\begin{align*} \forall \ N \leq k , M ^ N ( g h ) = \sum ^ { 2 N } _ { j = 0 } A _ { j , 2 N - j } ( g , h ) , \end{align*}


\begin{align*} A ( 1 + B ) ^ k - B ( 1 + A ) ^ k = 0 \end{align*}


\begin{align*} \lim _ { k \to \infty } \sup _ { z _ 0 \in \mathbb { R } } \int _ { D ( z _ 0 , R ) } \rho _ { n _ { k } } \dd z \dd r = 0 . \end{align*}


\begin{align*} \theta ^ j _ { \alpha \beta } ( x ' , t ; y ' ) : = t \ , \partial _ j K ^ L _ { \alpha \beta } ( x ' - y ' , t ) , x ' , y ' \in \R ^ { n - 1 } , \ , t > 0 . \end{align*}


\begin{align*} \begin{cases} \partial _ t \psi = - i H _ N \psi , t \in { \mathbb { R } } , \\ \psi ( 0 ) = \psi \in \ell ^ 2 ( \mathbb { N } ) . \end{cases} \end{align*}


\begin{align*} \rho _ v = \begin{pmatrix} \chi _ 1 & 0 \\ 0 & \chi _ 2 \end{pmatrix} \end{align*}


\begin{align*} \prod _ { k = 1 } ^ { m } \langle \tilde { \Delta } _ { i _ k } ^ n Y , e _ { j _ k } \rangle = & \prod _ { k = 1 } ^ { m } \left ( \langle \Delta _ { i _ k } ^ n S , e _ { j _ k } \rangle + \Delta _ n ^ { ( i _ k - 1 ) \Delta _ n } a ^ { j _ k } \right ) \\ = & \prod _ { k = 1 } ^ { m } \langle \Delta _ { i _ k } ^ n S , e _ { j _ k } \rangle + \sum _ { y = 1 , x = 2 } ^ m b _ { x y } ( i _ 1 , . . . , i _ m , j _ 1 , . . . , j _ m ) , \end{align*}


\begin{align*} R S _ { H ' } ( F ) = S _ { H ' } ( F ) = \varinjlim _ n \varprojlim _ k A _ { H ' , n } \widehat { \otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V _ k ) . \end{align*}


\begin{align*} \Theta = \left ( { \theta } ^ { 1 } \mathbf { I } _ { n _ 1 } , \cdots , { \theta } ^ { k } \mathbf { I } _ { n _ k } \right ) . \end{align*}


\begin{align*} f ( x ) = c \ , \int _ { - \infty } ^ \infty \int _ { K / M } \widetilde { f } ( \lambda , b ) e ^ { ( i \lambda + \rho ) A ( x , b ) } | c ( \lambda ) | ^ { - 2 } d b \ , d \lambda \end{align*}


\begin{align*} \tilde { h } _ { x } ^ { V } : = \sum _ { n = 1 } ^ { | V | } \varphi _ { n } ( x ) Z _ { n } , x \in \mathbb { Z } ^ { d } \end{align*}


\begin{align*} \beta ( x , y ) = \beta ( x ' , y ' ) \ , . \end{align*}


\begin{align*} \widehat { f } ( a , z ) = \rho _ k ^ \lambda ( f ) e _ a ( z , 0 ) , \ , \ , \ , \rho _ k ^ \lambda ( f ) = \int _ { G _ n } f ( g ) \rho _ k ^ \lambda ( g ) d g . \end{align*}


\begin{align*} \Theta ( \cdot , \mu , \tilde { \mu } ) = [ \null _ { 2 } \eta ] _ { p } \vartheta ( \mu , \tilde { \mu } ) + [ \null _ { 2 } \hat { \eta } ] _ { p } ^ { 2 } \vartheta ( \mu , \tilde { \mu } ) ^ { 2 } \end{align*}


\begin{align*} 0 \in ( F _ k + B ) ( x ) = ( F + B ) ( x ) + \frac { 1 } { \rho _ k } ( x - z ^ k ) . \end{align*}


\begin{align*} \xi _ 2 = \kappa _ 0 \xi _ 1 ^ 2 + O ( | \xi _ 1 | ^ 3 ) \mbox { a s } \xi _ 1 \rightarrow 0 \end{align*}


\begin{align*} - \Delta _ { y } \varphi = \mu ^ { 2 } \left ( \varphi - \frac { W } { 2 } \right ) _ { + } \textrm { i n } \ \mathbb { R } ^ { 5 } . \end{align*}


\begin{align*} I ( m , a ) = \frac { 1 } { m ! } \int _ 0 ^ { \infty } \ldots \int _ 0 ^ { \infty } e ^ { - t _ 1 - \ldots - t _ { m } } \prod _ { j = 1 } ^ { m } t _ j ^ { a } \Delta ( t _ 1 , \ldots , t _ { m } ) ^ 2 d t _ 1 \ldots d t _ { m } . \end{align*}


\begin{align*} Y _ \infty = \left ( \frac { m } { \textrm { G e V } } \right ) ^ { - 3 } \left [ 1 + \frac { 3 \ln ( m / \textrm { G e V } ) } { 1 5 } + \frac { \ln ( c _ 2 / 5 ) } { 1 5 } \right ] \end{align*}


\begin{align*} \prod _ { j = 1 } ^ { m - 1 } | h _ { j } ( x ) | \leq | h _ { m } ( x ) | ^ \delta \end{align*}


\begin{align*} \left \langle l _ 1 , m _ 1 \right | X ( r , \theta , e ^ { i \phi } ; a , M ) \left | l _ 2 , m _ 2 \right \rangle = \end{align*}


\begin{align*} \partial _ \tau \gamma = \frac { c \bar { K } } { 4 \pi } \mathbf { b } _ { \gamma ( \tau ) } . \end{align*}


\begin{align*} R \cdot R ( u , J u , x , J x ; x , J x ) = \frac { 1 } { 2 } \ , L ( p , \bar { \pi } ^ h ) \ , Q ^ c ( g , R ) ( u , J u , x , J x ; x , J x ) . \end{align*}


\begin{align*} [ x , y ] = x \cdot y - \varepsilon ( x , y ) y \cdot x . \end{align*}


\begin{align*} g _ \mu : = \arg \max _ { \psi \in \mathcal { H } } \left \{ - \frac { \mathrm { d } } { \mathrm { d } \gamma } \mathrm { K L } \left ( T _ { \gamma } { \# } \mu \| \pi \right ) \Big | _ { \gamma = 0 } \quad \| \psi \| _ { \mathcal { H } } \leq 1 \right \} . \end{align*}


\begin{align*} W _ { 1 } \left ( \mu , \nu \right ) = \sup _ { \left \Vert f \right \Vert _ { L i p } = 1 } \int f \left ( d \mu - d \nu \right ) , \end{align*}


\begin{align*} \int _ 0 ^ \pi d \theta \sin \theta \sin ( n \theta ) P _ l ( \cos \theta ) = \end{align*}


\begin{align*} | \mu | = m _ 1 + \dots + m _ n , \end{align*}


\begin{align*} ( \alpha + \beta ) ^ { q ^ i } = \alpha ^ { q ^ i } + \beta ^ { q ^ i } \end{align*}


\begin{align*} \begin{aligned} \left ( \sup _ { 0 \le s \le \tau } \int _ 0 ^ \tau \mathbf { 1 } _ { s \le t } \| S _ { Z } ( t , s ) \| ^ 2 d t \right ) ^ { 1 / 2 } \le e ^ { C _ 0 } \left ( \int _ 0 ^ { \tau } e ^ { ( t - \tau _ 1 ) \epsilon | x _ 0 | } d t \right ) ^ { 1 / 2 } \le \frac { e ^ { C _ 0 } } { \sqrt { \epsilon } } | x _ 0 | ^ { \epsilon - 1 / 2 } \end{aligned} \end{align*}


\begin{align*} \mu _ t ^ { * , \xi } = \mathbb E [ \alpha ^ { * , \xi } _ t | \mathcal { F } _ t ^ { W ^ 0 } ] \quad \nu _ t ^ { * , \xi } = \mathbb E [ x _ t ^ { \xi , \alpha ^ { * , \xi } } | \mathcal { F } _ t ^ { W ^ 0 } ] . \end{align*}


\begin{align*} R _ { i j } = g ^ { k l } R _ { i k l j } , \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } \Psi ^ - _ { 1 1 } ( x ; z ) = 1 + \frac { 1 } { 2 i } \int _ { - \infty } ^ { x } | u _ y ( y ) | ^ 2 \lim _ { | z | \rightarrow \infty } \Psi ^ - _ { 1 1 } ( y ; z ) \mathbf { d } y , \end{align*}


\begin{align*} \int _ 0 ^ \infty \left | \frac { G ( e ^ { \tau } , \delta ) } { e ^ { \sigma \tau } } \right | ^ 2 d \tau = \int _ 1 ^ \infty \left | \frac { G ( y , \delta ) } { y ^ { \sigma + \frac { 1 } { 2 } } } \right | ^ 2 d y & = \int _ 1 ^ \infty \left | \frac { G ( y , \delta ) } { y ^ { 1 + \frac { \alpha } { \log ( 1 / \delta ) } } } \right | ^ 2 d y > \frac { 1 } { e ^ { \alpha b } } \int _ 1 ^ { 1 / \delta ^ b } \left | \frac { G ( y , \delta ) } { y } \right | ^ 2 d y \end{align*}


\begin{align*} \left ( \frac { \left | \sum _ { i = 1 } ^ n [ 0 , z _ i ] \right | } { \left | \sum _ { i = 1 } ^ { n - 1 } [ 0 , z _ i ] \right | } \right ) ^ 2 \ge \left ( \frac { \left | \sum _ { i = 1 } ^ n [ 0 , P _ E z _ i ] \right | } { \left | \sum _ { i = 1 } ^ { n - 1 } [ 0 , P _ E z _ i ] \right | } \right ) ^ 2 + \left ( \frac { \left | \sum _ { i = 1 } ^ n [ 0 , P _ { E ^ \bot } z _ i ] \right | } { \left | \sum _ { i = 1 } ^ { n - 1 } [ 0 , P _ { E ^ \bot } z _ i ] \right | } \right ) ^ 2 . \end{align*}


\begin{align*} 0 & \leq P _ s * ( \chi _ { B _ 0 } \mu ) ( \bar x ) \lesssim \int _ { \bar y \in B _ 0 } \frac 1 { | \bar x - \bar y | _ p ^ N } \ , d \mu ( \bar y ) \\ & \leq \int _ { | \bar x - \bar y | _ p \leq 4 d } \frac 1 { | \bar x - \bar y | _ p ^ N } \ , d \mu ( \bar y ) \lesssim d ^ { 2 s - 1 } = | t - u | ^ { 1 - \frac 1 { 2 s } } . \end{align*}


\begin{align*} & \sum _ { x , y = 0 } ^ n ( \delta _ { x , y } - M _ { x , y } ) f _ y f _ x \ge 2 C _ * \sum _ { x = 0 } ^ { n - 1 } ( \nabla f _ x ) ^ 2 \end{align*}


\begin{align*} \forall \eta _ \Lambda \in \Omega _ \Lambda : \sum _ { \Delta \subset \Lambda } \sum _ { \xi _ \Delta } \left ( c _ \Delta ( \eta _ \Lambda , \xi _ \Delta ) - \frac { \mu ( \xi _ \Delta \eta _ { \Lambda \setminus \Delta } ) } { \mu ( \eta _ \Lambda ) } c _ \Delta ( \xi _ \Delta \eta _ { \Lambda \setminus \Delta } , \eta _ \Delta ) \right ) = 0 . \end{align*}


\begin{align*} \omega _ { S ^ 1 } = ( x - i y ) ( d x + i d y ) = ( x - i y ) d x + ( y + i x ) d y . \end{align*}


\begin{align*} | \theta ^ j _ { \alpha \beta } ( x ' , t ; y ' ) | & = t | \partial _ j K ^ L _ { \alpha \beta } ( x ' - y ' , t ) | \lesssim \frac { t } { | ( x ' - y ' , t ) | ^ n } , \\ | \nabla _ { y ' } \theta ^ j _ { \alpha \beta } ( x ' , t ; y ' ) | & \le t | \nabla ^ 2 K ^ L _ { \alpha \beta } ( x ' - y ' , t ) | \lesssim \frac { t } { | ( x ' - y ' , t ) | ^ { n + 1 } } . \end{align*}


\begin{align*} \varphi ^ L _ { \sigma , s } ( x ) = a = \varphi ^ L _ { \sigma , s } ( y ) \end{align*}


\begin{align*} d \varphi _ { r i } \left ( i \right ) = \left . \frac { d } { d t } \right \vert _ { r } \left ( l \left ( t \right ) i , e ^ { \theta \left ( t \right ) i } \right ) = \left ( l ^ { \prime } \left ( r \right ) i , \theta ^ { \prime } \left ( r \right ) i e ^ { \theta \left ( r \right ) i } \right ) \end{align*}


\begin{align*} \begin{aligned} F ( \eta , t ) & : = \left ( \begin{array} { c } - \varepsilon _ \delta ( { x } , \bar { y } , t ) u ( t ) \\ \varepsilon _ \delta ( { x } , \bar { y } , t ) \end{array} \right ) \end{aligned} \end{align*}


\begin{align*} \frac { \tilde { u } ^ { [ n ] } ( x ) - \tilde { u } ^ { [ n - 1 ] } ( x ) } { \Delta t } = \theta \left [ \mathcal { L } ( \tilde { u } ^ { [ n ] } ) + f ( t _ n , x ) \right ] + ( 1 - \theta ) \left [ \mathcal { L } ( \tilde { u } ^ { [ n - 1 ] } ) + f ( t _ { n - 1 } , x ) \right ] , \end{align*}


\begin{align*} \prescript { } { l } { u } - \prescript { } { l } { v } _ 1 = Q _ 1 \prescript { } { l } { f } , \end{align*}


\begin{align*} J _ { l , 0 } \left ( \rho \right ) : = \int _ { \mathbb { R } } \frac { \varphi _ { 0 } \left ( \tau \right ) \tau ^ { l - 1 } } { \left ( \tau - \rho \right ) ^ { l + 1 } } d \tau . \end{align*}


\begin{align*} w / v = \mu ^ { - 1 } ( \infty ) \textrm { i s a z e r o o f $ f ' $ } \iff \infty \textrm { i s a z e r o o f } f \end{align*}


\begin{align*} & M i n i m i z e _ { x , y \in \mathbb { R } ^ { n } } \ \ \alpha _ J ^ T y + 2 b _ J ^ T x \\ & s . t . \ \ \alpha _ k ^ T y + 2 b _ k ^ T x + c _ j \le 0 \ \ ( \forall \ , k \in \{ 1 , . . . , m \} ) \\ & s . t . \ \ x _ i ^ 2 - y _ i = 0 , \ \ \forall \ , i \end{align*}


\begin{align*} \phi _ \alpha ( e _ \alpha ) & = f _ \alpha \\ \phi _ \alpha ( f _ \alpha ) & = e _ \alpha . \end{align*}


\begin{align*} x x ' + x y ' + y y ' = n \ , . \end{align*}


\begin{align*} \lambda = \sum _ { s = 1 } ^ q \langle \sum _ { r = 1 } ^ p \alpha _ { r s } e _ r , \sum _ { \mu = 1 } ^ p \alpha _ { \mu s } X e _ { \mu } \rangle , \end{align*}


\begin{align*} \left \lbrace \begin{matrix} A _ k ( \theta , E ) = A ( \theta + ( k - 1 ) \alpha , E ) \cdots A ( \theta + \alpha , E ) A ( \theta , E ) \ \mathrm { f o r } \ k \geq 1 , \\ A _ 0 ( \theta , E ) = \mathrm { I d } , \\ A _ k ( \theta , E ) = ( A _ { - k } ( \theta + k \alpha , E ) ) ^ { - 1 } \ \mathrm { f o r } \ k \leq - 1 . \end{matrix} \right . \end{align*}


\begin{align*} \varprojlim _ { n \in \Delta } \mathcal F ( Y _ n ) = \varprojlim _ { k \in \Delta } \mathcal C ^ k . \end{align*}


\begin{align*} F _ { q } ^ { ( k ) } ( s ) = \frac { 1 } { Q _ { 1 } ( 2 - q ) } \left [ \displaystyle \sum _ { n = 0 } ^ { \infty } \frac { 1 } { \left ( \frac { 3 - 2 q } { 1 - q } \right ) _ { n } } \frac { \left ( \pm \frac { \sigma _ { \epsilon } } { 1 - q } \right ) ^ { n } } { n ! } ( - 1 ) ^ { k } \frac { \Gamma ( n + k + 1 ) } { s ^ { n + k + 1 } } \right ] . \end{align*}


\begin{align*} \lim _ { n \to \infty } \| T _ n p _ n f - p _ n T f \| = 0 f \in X . \end{align*}


\begin{align*} \Phi _ k : = \sup \{ \zeta \in U S C ( \overline { \Omega } \times \mathbb R ^ m ) : \zeta | _ { \Omega \times \mathbb R ^ m } F \star \mathcal P \zeta | _ { \partial \Omega \times \mathbb R ^ m } \le \phi _ { \tau , k } \} \end{align*}


\begin{align*} L _ X g ( \nu , \cdot ) = h ( \nu , \cdot ) \mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \Sigma . \end{align*}


\begin{align*} \Delta u \wedge \Delta v _ 1 \wedge \dots \wedge \Delta v _ { m - 1 } \wedge \beta _ n ^ { n - m } ( \omega ) = \int u \Delta v _ 1 \wedge \dots \wedge \Delta v _ { m - 1 } \wedge \beta _ n ^ { n - m } \wedge \Delta \omega , \omega \in C _ 0 ^ { \infty } ( \Omega ) , \end{align*}


\begin{align*} x _ k = \begin{cases} \sqrt { P _ { k , 1 } } s _ { k , 1 } + \sqrt { P _ { k , 2 } } s _ { k , 2 } , & \textrm { i f } \ ; k \in \mathcal { K } \setminus \{ K \} , \\ \sqrt { P _ { K } } s _ { K } , & \textrm { i f } \ ; k = K . \end{cases} \end{align*}


\begin{align*} h g ( h g e _ n h g ) ( h g ^ 2 e _ n h g ^ 2 ) \cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) = ( h g e _ n h g ) ( h g ^ 2 e _ n h g ^ 2 ) \cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) . \end{align*}


\begin{align*} | g ( a ) - g ( p _ i ) | & = | f ( a ) - f ( q _ i ) + d ( p _ i , q _ i ) | \leq d ( a , q _ i ) + d ( p _ i , q _ i ) \\ & \leq d ( a , p _ i ) + 2 d ( p _ i , q _ i ) \leq d ( a , p _ i ) + 2 \theta r \leq ( 1 + 2 \theta ) d ( a , p _ i ) . \end{align*}


\begin{align*} & l \sum h _ { k } \nabla _ { k } \chi - w ^ { 1 1 } \nabla _ { 1 1 } \chi \\ & \leq \widetilde { C } _ { 1 } l + \widetilde { C } _ { 2 } d + \widetilde { C } _ { 3 } w ^ { 1 1 } + \widetilde { C } _ { 4 } w _ { 1 1 } + \widetilde { C } _ { 5 } . \end{align*}


\begin{align*} \mbox { \tt h e t = P h i 1 . z e t } . \end{align*}


\begin{align*} D _ j ( x ; k ) : = V _ j ( k ) J = R ( x ; z ) V _ j ( k ) , \ j = 1 , 2 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \Vert X _ n ( t , \omega ) - X ( t , \omega ) \Vert _ H = 0 , \ a . e . \ ( t , \omega ) . \end{align*}


\begin{align*} \begin{cases} \chi _ 1 \leq w _ 1 \chi + r \\ \chi _ 1 \geq w _ 1 \chi . \end{cases} \end{align*}


\begin{align*} \widetilde W = N _ G ( T ) ( L ) / ( T ( L ) \cap I ) \end{align*}


\begin{align*} b _ k : = \begin{cases} 1 & \min ( \abs { k } , M - \abs { k } ) \le k _ 0 \\ r M - k _ 0 & \min ( \abs { k } , M - \abs { k } ) = k _ 0 + 1 \\ 0 & \end{cases} \end{align*}


\begin{align*} T _ x T _ y = \sum _ { z \in \widetilde W } f _ { x , y , z } ( v - v ^ { - 1 } ) T _ z \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n p _ { i } = 1 _ { Q } = 1 _ { M } p _ i p _ j = 0 \forall i \neq j \end{align*}


\begin{align*} f ( z ) = & \sum _ { n = 0 } ^ { \infty } \left ( x - \frac { \psi _ 0 } { \psi _ 1 } y \right ) ^ n a _ n = \sum _ { n = 0 } ^ { \infty } \left ( \frac { z + \bar z } { 2 } - \frac { \psi _ 0 } { \psi _ 1 } \frac { z - \bar z } { 2 } \right ) ^ n a _ n \\ = & \sum _ { n = 0 } ^ { \infty } \left ( ( 1 - \frac { \psi _ 0 } { \psi _ 1 } ) z + ( 1 + \frac { \psi _ 0 } { \psi _ 1 } ) \bar z \right ) ^ n \frac { a _ n } { 2 ^ n } , \end{align*}


\begin{align*} \alpha _ 1 : = \underset { x \in K _ 1 } { \inf } d ^ \mathcal Z ( x , \partial \mathcal Z ) \beta _ 1 : = \underset { x \in K _ 1 } { \sup } \ ; d ^ \mathcal Z ( x , \partial \mathcal Z ) \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \inf _ { k \in { \mathbb Z } } \left ( \frac { \mu ( f ^ { k + n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \tag * { $ \mathcal { U E } 1 $ } \end{align*}


\begin{align*} f _ s ( Z ^ n _ s ) = C ^ { ( n ) } _ s \end{align*}


\begin{align*} \Phi ( x ) : = \eta ( r ) s ( x ) \end{align*}


\begin{align*} \widetilde { P } _ { k } ^ { \mathrm { d i s c } } ( \mathcal { T } ) : = \left \{ q _ h \in { P } _ { k } ^ { \mathrm { d i s c } } ( \mathcal { T } ) : ( q _ h , 1 ) _ T = 0 T \in \mathcal { T } \right \} . \end{align*}


\begin{align*} \mathcal { H } _ m ( u ) ( q _ 0 ) = \lambda _ { n } ( q _ 0 ) \sum \limits _ { 1 \leq j _ 2 < \dots < j _ m \leq n - 1 } \lambda _ { j _ 1 } ( q _ 0 ) \cdots \lambda _ { j _ { m - 1 } } ( q _ 0 ) + \sum \limits _ { 1 \leq j _ 1 < \dots < j _ m \leq n - 1 } \lambda _ { j _ 1 } ( q _ 0 ) \cdots \lambda _ { j _ m } ( q _ 0 ) . \end{align*}


\begin{align*} y _ { k } ' ( s ) = F _ { k } ( s , y _ { 1 } ( s ) , \ldots , y _ { a } ( s ) ) \mbox { w i t h $ y _ { k } ( 0 ) = \hat { y } _ k $ f o r $ 1 \le k \le a $ , } \end{align*}


\begin{align*} { \tt w = e v a l u ( P s i 1 , \tilde z , { ' v ' } ) } \end{align*}


\begin{align*} a ( U D ( x ) V ) = a \circ \widehat { \varphi } ( [ U , V ] , x ) = f ( [ V ] , x ) = a \circ \widehat { \varphi } ( [ I _ n , V ] , x ) = a ( D ( x ) V ) \end{align*}


\begin{align*} d _ r = \sum _ { j = 0 } ^ r \binom { r } { j } g _ j \ \geq \ \sum _ { j = 0 } ^ r \binom { r } { j } = 2 ^ r \end{align*}


\begin{align*} \gamma _ { u , v } ( t ) : = ( t ^ { \alpha } u ( t \cdot ) , t ^ { \alpha } v ( t \cdot ) ) , \ \ t \geq 0 , \end{align*}


\begin{align*} f _ { G } ( x , y | \rho ) = f _ { g } ( x | \beta ) f _ { g } ( y | \beta ) \sum _ { j \geq 0 } \rho ^ { n } l _ { n } ( x | \beta ) l _ { n } ( y | \beta ) , \end{align*}


\begin{align*} d ( v _ 1 \Rightarrow w _ 2 v _ 2 ' ) = & d ( v _ 1 \Rightarrow w _ 2 v _ 2 ) + d ( w _ 2 v _ 2 \Rightarrow w _ 2 v _ 2 ' ) . \end{align*}


\begin{align*} \hat \beta _ { A C } & = A C + \pi C D + \pi ^ 2 D A \\ \hat \beta _ { C B } & = C B + \pi ^ 4 B D + \pi ^ 2 D C \\ \hat \beta _ { B A } & = B A + \pi ^ 4 A D + \pi ^ 5 D B . \end{align*}


\begin{align*} \begin{aligned} \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \geq \bar b & \forall \ , x \geq x ^ \star + \delta \\ \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \leq - \bar b & \forall \ , x \leq x ^ \star - \delta \ , . \end{aligned} \end{align*}


\begin{align*} \mathcal { C } _ n ^ { ( p ) } \ = \ A _ p F _ n + B _ p F _ { n + 1 } - \sum _ { k = 0 } ^ p \binom { p } { k } B _ k n ^ { p - k } . \end{align*}


\begin{align*} \varphi _ { - } ( F ( z ) ) = T _ 1 ( \varphi _ { - } ( z ) ) \end{align*}


\begin{align*} \beta = \alpha + \alpha _ s a _ s = 1 \end{align*}


\begin{align*} C \le M _ n ^ { \psi } ( Q ^ n ) / n = T _ { W _ { n J } , \psi } ( \mu _ n ( Q ^ n ) ) - I ( \mu _ n ( Q ^ n ) ) \le - I ( \mu _ n ( Q ^ n ) ) . \end{align*}


\begin{align*} \begin{aligned} v _ { j } ( t ) & = i \int _ { t } ^ { \infty } U ( t - \tau ) \left \{ \tilde { { \mathcal N } } _ { j } ( v _ { 1 } + u _ { \mathrm { a p } , 1 } , v _ { 2 } + u _ { \mathrm { a p } , 2 } ) - \tilde { { \mathcal N } } _ { j } ( u _ { \mathrm { a p } , 1 } , u _ { \mathrm { a p } , 2 } ) \right \} ( \tau ) d \tau \\ & + i \int _ { t } ^ { \infty } U ( t - \tau ) \mathcal { E } _ { j } ( \tau ) d \tau + \mathcal { R } _ j . \end{aligned} \end{align*}


\begin{align*} \nabla c \cdot \nu = ( \gamma - c ) g \quad \mbox { o n } \partial \Omega . \end{align*}


\begin{align*} \P _ x ( \tau > n ) \sim K _ { \lambda } n ^ { - d ( d - 1 ) / 2 - d / 2 } e ^ { - \gamma n } e ^ { \sum _ { i = 1 } ^ d ( \lambda _ i - \bar { \lambda } ) x _ i } h ^ { ( \bar { \lambda } , \ldots , \bar { \lambda } ) } ( x ) , n \rightarrow \infty . \end{align*}


\begin{align*} \Pi _ \lambda [ \psi _ j ^ { - } ] ( x , y ) & = C _ d \ , \overline { \Pi _ { \lambda } [ \psi _ j ^ { + } ] } ( x , y ) , \\ [ 4 p t ] \Pi _ \lambda [ \psi _ j ^ { \pm \pi } ] ( x , y ) & = C _ d ' \ , \Pi _ { \lambda } [ \psi _ j ^ { \mp } ] ( x , - y ) , \end{align*}


\begin{align*} \chi ( g ) = \lim _ { l \to \infty } \chi _ { _ { ^ { k ( l ) } \ ! { \scriptstyle \lambda } } } ( g ) ~ ~ g \in \mathfrak { S } _ { \widehat { \mathbf { n } } } . \end{align*}


\begin{align*} \mathcal { S } _ n ^ { ( p ) } \ & = \ \sum _ { i = 0 } ^ { n } \ , i ^ p \cdot F _ { i } , \\ \intertext { a n d r e p l a c i n g $ i $ w i t h $ n - i $ w e h a v e } \mathcal { S } _ n ^ { ( p ) } \ & = \ \sum _ { i = 0 } ^ { n } \ , ( n - i ) ^ p \cdot F _ { n - i } \ = \ \sum _ { i = 0 } ^ { n } \left ( \sum _ { k = 0 } ^ p \binom { p } { k } n ^ { p - k } ( - 1 ) ^ k i ^ k \right ) \cdot F _ { n - i } . \end{align*}


\begin{align*} \begin{array} { l l l } f _ L ^ M ( ( L ' , R ' ) \star ( L , R ) ) \circ u \\ & = & f _ L ^ M ( L ' \circ L , R \circ R ' ) \circ f _ L ( s ) \circ t \\ & = & f _ L ( L ' \circ L \circ s ) \circ t \\ & = & f _ L ^ M ( L ' , R ' ) \circ f _ L ( L \circ s ) \circ t \\ & = & f _ L ^ M ( L ' , R ' ) \circ f _ L ^ M ( L , R ) \circ f _ L ( s ) \circ t \\ & = & f _ L ^ M ( L ' , R ' ) \circ f _ L ^ M ( L , R ) \circ u . \end{array} \end{align*}


\begin{align*} G ( t , s ) = \frac { 1 } { \Gamma ( \alpha ) } \left \{ \begin{array} { l r } \frac { ( t - a ) ^ { \alpha - 1 } ( b - s ) ^ { \alpha - 1 - \beta } } { ( b - a ) ^ { \alpha - 1 - \beta } } - ( t - s ) ^ { \alpha - 1 } , & a \le s \le t \le b , \\ \\ \frac { ( t - a ) ^ { \alpha - 1 } ( b - s ) ^ { \alpha - 1 - \beta } } { ( b - a ) ^ { \alpha - 1 - \beta } } , & a \le t \le s \le b . \end{array} \right . \end{align*}


\begin{align*} \begin{cases} \partial _ t F + v \cdot \partial _ x F = Q _ L ( F , \ F ) , \\ F | _ { t = 0 } = F _ 0 , \end{cases} \end{align*}


\begin{align*} \begin{aligned} F ^ { i j } \left ( \sqrt { b _ { n - 1 } } ( A _ 1 v - A _ 2 w ) \pm \nabla _ \alpha ( u - \varphi ) \right ) _ { i j } \geq \ , & 0 \mbox { i n } \omega _ 1 \\ A \sqrt { b _ { n - 1 } } ( A _ 1 v - A _ 2 w ) \pm \nabla _ \alpha ( u - \varphi ) \leq \ , & 0 \mbox { o n } \partial \omega _ 1 \end{aligned} \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta v _ { l } + u _ { l } v _ { l } = f , & x \in \Omega , \\ \nabla v _ { l } \cdot \nu + v _ { l } = \eta , & x \in \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} \partial _ x \left ( e ^ { i c _ - } \Psi ^ - _ { 1 1 } ( x ; z ) \right ) = \frac { 1 } { 2 i } u _ x ( x ) e ^ { i c _ - ( x ) } \Psi ^ - _ { 2 1 } ( x ; z ) . \end{align*}


\begin{align*} { { q _ \delta { q _ \delta } } ^ { \ast } } = \left [ \begin{array} { c c c c } 0 & \ 0 & \ 0 \\ \ 0 & u _ { \delta } u _ { \delta } ' & O \\ \ 0 & O & O \end{array} \right ] = \left [ \begin{array} { c c c c } 0 & \ 0 & \ 0 \\ \ 0 & T & O \\ \ 0 & O & O \end{array} \right ] . \end{align*}


\begin{align*} M ^ { \pm } = \left \{ s \in \left ( 0 , \infty \right ) \mid \left \vert \rho \mp \beta ( s ) \right \vert \leq \Lambda ^ { - \frac { 1 } { 2 } } \left ( s \right ) \beta ( s ) \right \} . \end{align*}


\begin{align*} \delta = \lim _ { N \to \infty } \dfrac { | A \cap \Phi _ N | } { | \Phi _ N | } \end{align*}


\begin{align*} \eta \left ( { { \mathbf { s } } , \widehat { \mathbf { s } } } \right ) = \frac { { { \mathbf { B } } \left ( { \mathbf { s } } \right ) { \mathbf { B } } { { \left ( { \widehat { \mathbf { s } } } \right ) } ^ T } } } { { \left \| { { \mathbf { B } } \left ( { \mathbf { s } } \right ) } \right \| \left \| { { \mathbf { B } } { { \left ( { \widehat { \mathbf { s } } } \right ) } } } \right \| } } , \end{align*}


\begin{align*} s w _ n = ( s x _ { w _ n } , s y _ { w _ n } , 0 ) \quad \ w _ n = ( x _ { w _ n } , y _ { w _ n } , 0 ) , \end{align*}


\begin{align*} x ^ { ( k ) } _ { i } = x ^ { ( n ) } _ { i } ( \forall i \in \{ 1 , \dots , D \} ) ( \forall k > n ) . \end{align*}


\begin{align*} Z = U P = Q V . \end{align*}


\begin{align*} v _ { n } ( y ) = w _ { n } ( y ) - w _ { n } ( 0 ) \forall y \in ( A _ { n } , B _ { n } ) , \end{align*}


\begin{align*} E ( g _ 1 , w ) = \int _ { - \infty } ^ \infty \int _ 0 ^ 1 \int _ 0 ^ \infty \int _ 0 ^ \infty e ^ { i \Phi \left ( \lambda , \theta , t , u + \left ( \frac { t _ 1 } { t } \right ) ^ 2 u _ 1 \right ) } W ( \lambda , t , u ) d \lambda d \theta d t d u . \end{align*}


\begin{align*} 4 \lambda + 2 \mu = \frac { \hat W _ r ( 0 ) + \hat W _ r ( 2 q ) - 2 \hat W _ r ( q ) } { \hat W _ r ( q ) } = : H ( q , r ) . \end{align*}


\begin{align*} A \oplus _ 2 \left ( \sqrt { ( 1 - \lambda ) s + \lambda t } B \right ) = \left ( \sqrt { 1 - \lambda } ( A \oplus _ 2 \sqrt { s } B ) \right ) \oplus _ 2 \left ( \sqrt { \lambda } ( A \oplus _ 2 \sqrt { t } B ) \right ) . \end{align*}


\begin{align*} L = a _ 0 x + a _ 1 x ^ q + a _ 2 x ^ { q ^ 2 } + \cdots + a _ n x ^ { q ^ n } . \end{align*}


\begin{align*} u _ E = U _ C ( x + B _ { \infty } t ) - B _ { \infty } , \end{align*}


\begin{align*} Y = f ^ { * } ( X ) + \xi \end{align*}


\begin{align*} \lambda _ S ( G ) = \lim _ { n \rightarrow \infty } \sqrt [ n ] { | B _ S ( n ) | } = \inf \big \{ \sqrt [ n ] { | B _ S ( n ) | } \mid n \in \mathbb { N } _ 0 \big \} \end{align*}


\begin{align*} L ( \gamma ) = \sup \bigl \{ \sum _ { i = 0 } ^ { n - 1 } d ( \gamma ( t _ i ) , \gamma ( t _ { i + 1 } ) ) \colon n \in \mathbb { N } , \ , \alpha = t _ 0 \leq \dotsb \leq t _ n = \beta \bigr \} . \end{align*}


\begin{align*} Y _ { l , m } ( \theta , \phi ) = \sqrt { \frac { ( 2 l + 1 ) ( l - m ) ! } { 4 \pi ( l + m ) ! } } P _ l ^ m ( \cos \theta ) e ^ { i m \phi } \ , , \end{align*}


\begin{align*} \begin{aligned} & H ( P ( x ) , 0 ) = ( Q , 0 ) \left ( H ( x , 0 ) \right ) \\ & \Psi \left ( t , H ( x , y ) \right ) = H \left ( \Phi ( t , x , y ) \right ) \\ & \Psi _ s \left ( t , H ( x , 0 ) \right ) = H \left ( \Phi _ s ( t , x , 0 ) \right ) \end{aligned} \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) = \lambda _ n ( u _ n , v _ n ) \quad \ , v _ n \in X _ n . \end{align*}


\begin{align*} \Phi ^ + ( \alpha ) : = \begin{cases} 1 , & \alpha \in \Phi ^ + , \\ 0 , & \alpha \in \Phi ^ - . \end{cases} \end{align*}


\begin{align*} \begin{aligned} | \langle \boldsymbol { \nu } , f _ \alpha ( \lambda | \bar U ; x , t ) \rangle | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; , \end{aligned} \end{align*}


\begin{align*} e ( C ) = 2 - I ( C ) - s ( C ) ; \end{align*}


\begin{align*} x ' _ 1 = \varphi _ { \sigma , s } ^ L ( x _ 1 ) , x ' _ 2 = \varphi _ { \sigma , s } ^ L ( x _ 2 ) , y ' _ 1 = \varphi _ { \sigma , p } ^ R ( y _ 1 ) y ' _ 2 = \varphi _ { \sigma , p } ^ R ( y _ 2 ) . \end{align*}


\begin{align*} d X _ t ^ i = \left ( b _ 0 ( t , X _ t ^ i ) + \frac { 1 } { n - 1 } \sum _ { j = 1 , j \neq i } ^ n b \left ( t , X _ t ^ i , X _ t ^ j \right ) \right ) d t + d B ^ { H , i } _ t , i = 1 , \cdots , n , \end{align*}


\begin{align*} | B _ S ( n + D ' ) | > \frac { q ( n ) } { 2 } \ , \sum _ { \ell > q ( n ) } \left | R _ { q , \ell } ^ \flat ( n ) \right | = \frac { q ( n ) } { 2 } \ , \left | R _ { q } ^ \flat ( n ) \right | , \end{align*}


\begin{align*} \xi ^ { 2 ^ k } = \left \{ \begin{array} { l l } \xi + 1 & \textrm { i f $ k $ i s o d d , } \\ \xi & \textrm { i f $ k $ i s e v e n } . \end{array} \right . \end{align*}


\begin{align*} S = \int d ^ 4 x \sqrt { - g } \left [ \frac { R } { 1 6 \pi } - \frac { 1 } { 2 } g _ { \mu \nu } \partial ^ \mu \Phi \partial ^ \nu \Phi - \frac { V ( \Phi ) } { 1 6 \pi } \right ] + S ( \psi _ m , \mathcal { A } ( \Phi ) ^ 2 g _ { \mu \nu } ) \ , . \end{align*}


\begin{align*} N = \begin{pmatrix} Q & 0 _ { k \times k } \\ 0 _ { k \times k } & Q \end{pmatrix} . \end{align*}


\begin{align*} \overline { \mathrm { s p a n } } _ { H ^ 2 ( \mathbb { D } ) } \{ ( I - S ) h _ k \mid k \geq 2 \} = H ^ 2 ( \mathbb { D } ) . \end{align*}


\begin{align*} P _ { \min } ^ { { \rm { O } } } \left ( { \overline S , \overline \varepsilon , \overline R } \right ) = \bigcap { \left \{ { P \in { { \mathbb { R } } ^ + } : \left ( { \overline S , \overline R } \right ) \in { \mathcal { R } } _ { { \rm { S v B } } } ^ { { \rm { O } } } \left ( { W , P , K , \overline \varepsilon } \right ) } \right \} } , \end{align*}


\begin{align*} \Psi ^ { \pm } ( x ; z ) = I + \int _ { \pm \infty } ^ x e ^ { - i z ( x - y ) \widehat { \sigma } _ 3 } \widetilde { Q } \Psi ^ { \pm } ( y ; z ) d y . \end{align*}


\begin{align*} \frac { 1 } { 3 } \frac { d } { d t } \int _ { \Omega } n ^ { 3 } + \frac { 8 } { 9 } \int _ { \Omega } | \nabla n ^ { \frac { 3 } { 2 } } | ^ { 2 } = \frac { 4 } { 3 } \int _ { \Omega } n ^ { \frac { 3 } { 2 } } \nabla n ^ { \frac { 3 } { 2 } } \cdot ( S ( x , n , c ) \cdot \nabla c ) . \end{align*}


\begin{align*} \gamma _ 1 ^ \star : = \dfrac { \bar c } { \bar \epsilon ( L _ r , M _ r , \delta ) } \end{align*}


\begin{align*} \begin{aligned} & \Gamma _ p = 2 \left ( \gamma ^ 2 / 2 + ( 1 - \nu ) ^ 2 k \right ) ^ { 1 / 2 } / ( 1 - 3 \gamma ) . \end{aligned} \end{align*}


\begin{align*} \beta = ( \beta _ 1 , \ldots , \beta _ k ) \textrm { w i t h } \sum _ i \beta _ i = m . \end{align*}


\begin{align*} f ( t ) = \frac { 1 } { 2 \pi i } \lim _ { \tau \rightarrow \infty } \int _ { \sigma - i \tau } ^ { \sigma + i \tau } \exp ( s t ) F ( s ) d s . \end{align*}


\begin{align*} \dot { V } ( x _ t ) = - \norm { x ( t - h ) } ^ 2 , \end{align*}


\begin{align*} T \circ \pi _ 1 ( h ) = \pi _ 2 ( h ) \circ T , \end{align*}


\begin{align*} n ^ 2 = p ^ { m _ 1 } + p ^ { m _ 2 } + \cdots + p ^ { m _ r } . \end{align*}


\begin{align*} S _ k [ r ] = \sigma _ k ( \lambda ( r ) ) \end{align*}


\begin{align*} E : y ^ { 2 } = x ( x - 4 ^ { m } p ) ( x + p ) \end{align*}


\begin{align*} h _ { \Lambda } ( \nu \lvert \mu ) = \sum _ { \eta _ { \Lambda } } \nu ( \eta _ { \Lambda } ) \log \left ( \nu ( \eta _ { \Lambda } ) \right ) - \sum _ { \eta _ { \Lambda } } \nu ( \eta _ { \Lambda } ) \log \left ( \mu ( \eta _ { \Lambda } ) \right ) . \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } A \cdot B \dd x & \leq \frac { 1 } { f _ 1 ^ { + } } \int _ { \Omega } | B | ^ { 2 } \dd x , \\ - \int _ { \Omega } A \cdot B \dd x & \leq \frac { 1 } { - f _ 1 ^ { - } } \int _ { \Omega } | B | ^ { 2 } \dd x , \end{aligned} \end{align*}


\begin{align*} S ( k ) = \begin{pmatrix} a ( k ) & b ( k ) \\ [ 3 p t ] - \overline { b ( \bar { k } ) } & \overline { a ( \bar { k } ) } \end{pmatrix} , \ \ \det S ( k ) = 1 . \end{align*}


\begin{align*} L ( x ) = \sum _ { i = 0 } ^ r a _ i x ^ { 2 ^ i } \end{align*}


\begin{align*} x _ 1 = \left ( \begin{matrix} 1 \\ \alpha + \alpha ^ 2 \\ \alpha \end{matrix} \right ) , x _ 2 = \left ( \begin{matrix} 1 \\ \sigma ( \alpha ) + \sigma ( \alpha ) ^ 2 \\ \sigma ( \alpha ) \end{matrix} \right ) , x _ 3 = \left ( \begin{matrix} 1 \\ \sigma ^ 2 ( \alpha ) + \sigma ^ 2 ( \alpha ) ^ 2 \\ \sigma ^ 2 ( \alpha ) \end{matrix} \right ) , \end{align*}


\begin{align*} 0 \rightarrow \mathcal { O } _ C \rightarrow \bigoplus _ { j = 1 } ^ n \mathcal { O } _ { C _ j } \rightarrow \mathcal { T } \rightarrow 0 . \end{align*}


\begin{align*} \lambda ^ { \prime } ( t ) \mathcal { E } ( t ) + \mu \alpha ^ { \prime } ( t ) \mathcal { E } ^ r ( t ) \langle P ( u _ t ) , u \rangle = \frac { \lambda ^ { \prime } ( t ) } { \lambda ( t ) } \mathcal { H } ( t ) + \mu \bigl ( \frac { \alpha ( t ) } { \lambda ( t ) } \bigr ) ^ { \prime } \lambda ( t ) \mathcal { E } ^ r ( t ) \langle P ( u _ t ) , u \rangle . \end{align*}


\begin{align*} Q ( t ) ^ { - 1 } P ^ { - 1 } \begin{pmatrix} w ( \xi _ 0 ) \\ \overline { w ( \xi _ 0 ) } \end{pmatrix} = \begin{pmatrix} \beta _ 1 ( \xi _ 0 ) \\ \beta _ 2 ( \xi _ 0 ) \end{pmatrix} + O ( \varepsilon _ 1 ^ 2 \varepsilon _ 2 t ^ { - 1 / 4 + C _ 1 \varepsilon ^ 2 _ 1 + 2 \mu C _ 2 \varepsilon ^ 2 _ 1 + C _ 4 \varepsilon _ 1 ^ 2 } ) \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) = ( f _ n , v _ n ) v _ n \in X _ n , \end{align*}


\begin{align*} \frac { x _ i } { g } ( x _ 1 - x _ k ) = \begin{cases} x _ i & ( i = k ) \\ 0 & ( i \neq k ) . \end{cases} \end{align*}


\begin{align*} & \Psi ^ - _ 1 ( x ; z ) : = e _ 1 + \int _ { - \infty } ^ x \left ( 1 , e ^ { 2 i z ( x - y ) } \right ) \widetilde { Q } \Psi ^ - _ 1 ( y ; z ) d y , \\ & \Psi ^ - _ 2 ( x ; z ) : = e _ 2 + \int _ { - \infty } ^ x \left ( e ^ { - 2 i z ( x - y ) } , 1 \right ) \widetilde { Q } \Psi ^ - _ 2 ( y ; z ) d y . \end{align*}


\begin{align*} \cos S _ c ( x , y ) = \tau ^ - ( x , y ) . \end{align*}


\begin{align*} \omega ^ { - 1 } ( d x ^ i , d x ^ j ) = \omega ^ { - 1 } ( d \theta _ i , d \theta _ j ) = 0 , \omega ^ { - 1 } ( d \theta _ i , d x ^ j ) = \delta _ i ^ j , \end{align*}


\begin{align*} \langle \ 1 , f ^ { i } \rangle = 0 ~ ~ ~ i = 1 , \dotsc , n - 1 \end{align*}


\begin{align*} { \bf { \bar q } } \left [ i \right ] = \sum \limits _ { l = 1 } ^ { L ' } { { { { \bf { F } } } _ l } { \bf { d } } \left [ { i - { \kappa _ l } } \right ] } , \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) - x _ a ( t ) \| & \ , \le \| x ( t ) - z ( t ) \| + \| z ( t ) - x _ a ( t ) \| \\ & \ , \le \| x ( t ) - z ( t ) \| + \gamma \| \epsilon ( x _ a ( t ) , t ) \| \\ & \ , \le \| x ( t ) - z ( t ) \| + \gamma 2 M _ \mathcal { D } T . \end{aligned} \end{align*}


\begin{align*} K _ { N } : = \frac { N ^ { 4 } } { \sqrt { \log N } } e ^ { - \frac { a _ { N } ^ { 2 } } { 2 \gamma \log N } } , \end{align*}


\begin{align*} \left ( \sum _ { n = 0 } ^ \infty a ( n ) q ^ n \right ) \ | \ U ( j ) : = \sum _ { n = 0 } ^ \infty a ( j n ) q ^ n . \end{align*}


\begin{align*} \mu ( X ( s _ 1 ) \in B _ 1 , \cdots , X ( s _ r ) \in B _ r ) = \frac { 1 } { ( 2 \pi ) ^ { \frac { r } { 2 } } . | C | } \int _ { B _ 1 \times \cdots \times B _ r } \exp \big ( - \frac { 1 } { 2 } . { } ^ { t } { x } . C ^ { - 1 } . x \big ) d x , \end{align*}


\begin{align*} c _ \lambda = \sum _ { T } \operatorname { s g n } ( T ) \end{align*}


\begin{align*} \ell ( x _ 1 \ast x _ 2 ) = \ell ( y ) + \ell ( x _ 2 ) \leq \ell ( x _ 1 ) + \ell ( x _ 2 ) - d ( v _ 1 \Rightarrow w _ 2 v _ 2 ) - d ( w ' w _ 2 v _ 2 \Rightarrow w _ 1 v _ 1 ) . \end{align*}


\begin{align*} R _ { \ell } ^ + : = [ \ell q _ n + m _ n + 1 , \ell q _ n + b _ n ] R _ { \ell } ^ - : = [ \ell q _ n - b _ n , \ell q _ n + m _ n - 1 ] , \end{align*}


\begin{align*} \begin{aligned} & T ( k _ 1 , \ldots , k _ n ; ( \alpha , \beta ) ) - T ( k _ n , k _ 1 , \ldots , k _ { n - 1 } ; ( \alpha , \beta ) ) \\ = & Z _ { I I } ( k _ n , k _ 1 , \ldots , k _ { n - 2 } , k _ { n - 1 } + 1 ; ( \alpha , \beta ) ) \\ & - \sum _ { j = 0 } ^ { k _ n - 2 } Z _ { I } ( j + 1 , k _ 1 , \ldots , k _ { n - 1 } , k _ n - j ; ( \alpha , \beta ) ) , \end{aligned} \end{align*}


\begin{align*} \left \{ \begin{aligned} \mathcal ( \delta f ) ^ { - 1 } \left ( j ^ 2 _ { f ( x ) } \phi \right ) & = j ^ 2 _ x ( \phi \circ f ) & \\ \mathcal ( \delta f ) ^ { - 1 } \left ( j ^ 2 _ { f ( x ) } \phi \right ) & = j ^ { 2 ( n _ i + 1 ) } _ x ( \phi \circ f ) & \end{aligned} \right . \end{align*}


\begin{align*} \begin{aligned} f _ { 1 } ^ { + } & = \inf \left \{ \int _ { \Omega } | B | ^ { 2 } \dd x \ \middle | \ B \in L ^ { 2 } _ { \sigma } ( \Omega ) , \ \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x = 1 \right \} , \\ - f _ { 1 } ^ { - } & = \inf \left \{ \int _ { \Omega } | B | ^ { 2 } \dd x \ \middle | \ B \in L ^ { 2 } _ { \sigma } ( \Omega ) , \ \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x = - 1 \right \} . \end{aligned} \end{align*}


\begin{align*} E ' _ 0 = \kappa \left ( \frac { B _ 3 } { 2 \sqrt { \kappa B _ 4 } } + L + 5 \right ) ^ 2 - \left ( \frac { Q } { 2 ( L + 2 ) } - \sqrt { \kappa B _ 4 } \right ) ^ 2 , \end{align*}


\begin{align*} \displaystyle \int _ { 0 } ^ { 1 } y _ \delta ( x _ a , \tau ) \ , d \tau & = \dfrac { a _ { 0 , \delta } ( x _ a ) } { 2 } \\ \displaystyle \int _ { 0 } ^ { 1 } \left ( y _ \delta ( { x _ a } , \tau ) - \bar { y } _ a \right ) u ( \tau ) \ , d \tau & = \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \end{align*}


\begin{align*} g ( f ) ( x ) = \Big ( \int _ { 0 } ^ { \infty } \Big | \frac { d } { d t } e ^ { - t L } ( f ) ( x ) \Big | ^ 2 t d t \Big ) ^ { 1 / 2 } , \end{align*}


\begin{align*} & - \alpha \leq \theta _ 1 \leq \theta \leq \theta _ 2 \leq \alpha , \\ & \omega ( s ) = - s e ^ { - i \left ( \theta _ 1 + \theta _ 2 \right ) / 2 } , \\ & s \geq 0 . \end{align*}


\begin{align*} E _ q : = \{ n \in N \ , : \ , \pi _ N ( q ^ { - 1 } n ) \in E \} \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta w _ { n } & = f ( \cdot , y ) - \mathcal { N } _ { n } ( \cdot , y _ { n } ) , \ ; \ ; \Omega , \\ w _ n & = 0 , \ , \ ; \ ; \qquad \qquad \qquad \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} \phi _ p ( x ) = ( 1 + x ) ^ { - p / 2 } \end{align*}


\begin{align*} N _ + ( x ; k ) = I + \mathcal { P } ^ + \left ( N _ - J \right ) ( z ) , z \in \mathbb { R } . \end{align*}


\begin{align*} F _ k ( x ) = \max _ { i = 1 , \ldots , 2 ^ k } x _ i . \end{align*}


\begin{align*} \nu = \int _ Y \nu _ y \ , \mathrm { d } \mu . \end{align*}


\begin{align*} F ( x ) = & \frac { 1 } { \operatorname { B } ( a , b ) } \sum ^ { \infty } _ { n = 0 } \frac { ( - 1 ) ^ n \Gamma ( b ) } { \Gamma ( b - n ) n ! } \int ^ { \infty } _ { \frac { \theta } { x ^ 2 } } \exp \{ - ( a + n ) u \} \mathrm { d } u \end{align*}


\begin{align*} w ( t , \xi _ 0 ) = \beta ( \xi _ 0 ) + O ( t ^ { - 1 / 4 + C _ 4 \varepsilon _ 1 ^ 2 } ) . \end{align*}


\begin{align*} d _ { \widetilde K } = \# \{ 1 \leq r \leq j : C _ r \in O _ { \widetilde K } , \ \lambda _ r = 1 \} . \end{align*}


\begin{align*} [ F ( \gamma ) : F ] = m , \end{align*}


\begin{align*} T ^ { \ast } : = \sup \left \{ T > 0 : \sup _ { 0 \le t < T } \langle t \rangle ^ { - \delta } ( \| u _ 2 ( t ) \| _ { H ^ 1 } + \| J u _ 2 ( t ) \| _ { L ^ 2 } ) < 4 \varepsilon _ 2 \right \} . \end{align*}


\begin{align*} ( c _ { i j k } ) _ { i k } = \theta ^ { j - 1 } \end{align*}


\begin{align*} \mathrm { E } ( X ^ n ) = \int _ { - \infty } ^ \infty x ^ n g ( x ) \mathrm { d } x = \frac { 1 } { \sigma \mathrm { B } ( \alpha , \beta ) } \sum _ { j = 0 } ^ { \infty } ( - 1 ) ^ j \binom { \beta - 1 } { j } \ , \int _ { - \infty } ^ \infty x ^ n \ , h _ j ^ { ( s ) } \mathrm { d } x . \end{align*}


\begin{align*} | \bar K _ 1 ^ { ( n ) } ( x , y ) | \le \frac { C } { 1 + ( x - y ) ^ 2 } , y = 0 , \ldots , n , \ , x \in ( \delta n , ( 1 - \delta ) n ) \end{align*}


\begin{align*} H _ 0 ^ 2 ( D ) : = \left \{ v \in H ^ 2 ( D ) : v = \frac { \partial v } { \partial \nu } = 0 \partial D \right \} , \end{align*}


\begin{align*} 2 H = \alpha \frac { \nu _ 3 } { z } + \varpi , \end{align*}


\begin{align*} x ^ { t + 1 } = ( I + \gamma _ t B ) ^ { - 1 } \left ( x ^ t + \alpha _ t ( x ^ t - x ^ { t - 1 } ) - \gamma _ t ( F ( x ^ t ) + \beta _ t ( F ( x ^ t ) - F ( x ^ { t - 1 } ) ) ) \right ) , \end{align*}


\begin{align*} \phi \rightarrow \phi e ^ { i \chi } , \ \ A _ { \mu } = A _ { \mu } - \partial _ { \mu } \chi , \end{align*}


\begin{align*} \sup _ { y \in \Z ^ d } \sum _ { \Delta \ni y : \ c _ \Delta > 0 } \sum _ { z \neq y } \sum _ { i = 1 } ^ q \norm { \nabla ^ i _ z \gamma _ \Delta ( \cdot | \cdot ) } _ \infty < \infty , \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } \int _ 0 ^ T \int _ { \Omega } u _ { \epsilon } ( t , x ) \phi \left ( t , x , \frac { x } { \epsilon } \right ) d x d t = \int _ 0 ^ T \int _ { \Omega } \int _ Y u _ 0 ( t , x , y ) \phi ( t , x , y ) d y d x d t . \end{align*}


\begin{align*} \epsilon _ { f \otimes \chi _ D } = \begin{cases} - \lambda _ f ( p ) p ^ { 1 / 2 } & ( p , D ) = 1 D \equiv \square \mod 4 p ; \\ 1 & p | D . \end{cases} \end{align*}


\begin{align*} & ( i ) \ \ \ \nabla ( J + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ) ( x ^ * ) = 0 \\ & ( i i ) \ \ \gamma _ k f _ k ( x ^ * ) = 0 \ \ k \in \{ 1 , . . . , m \} \\ & ( i i i ) \ A _ J + \sum \limits _ { k = 1 } ^ m \gamma _ k A _ k \succeq 0 \end{align*}


\begin{align*} \mathcal { I } ( x , t ) = \bigcup _ { i = p + 1 } ^ { n } [ t _ { i , e n } ( x , t ) , t _ { i , e x } ( x , t ) ] . \end{align*}


\begin{align*} \gamma ( s , \tau ) = \left ( r _ * \cos \left ( \frac { - s - a _ 1 \tau } { \sqrt { k ^ 2 + r _ * ^ 2 } } \right ) , r _ * \sin \left ( \frac { - s - a _ 1 \tau } { \sqrt { k ^ 2 + r _ * ^ 2 } } \right ) , \frac { k s - b _ 1 \tau } { \sqrt { k ^ 2 + r _ * ^ 2 } } \right ) ^ t , \end{align*}


\begin{align*} A \circ \widetilde \Phi \circ \widetilde { \mathcal { P } } \circ \widetilde \Phi ^ { - 1 } \circ A ^ { - 1 } = \mathcal { P } ' \end{align*}


\begin{gather*} \mathcal { K } _ { \psi } ( \Omega ) : = \{ w \in W ^ { 1 , p } ( \Omega ) : w \geq \psi \ \ \Omega \} \end{gather*}


\begin{align*} \mathrm { D E } ( \Sigma , c ) ( V , w ) & = \frac { \mathrm { d } } { \mathrm { d } t } \mathrm { E } \big ( \Gamma _ t , c _ t \big ) _ { \ , | t = 0 } = \frac { \mathrm { d } } { \mathrm { d } t } \int _ { \Gamma _ t } G ( c _ t ) \ , \mathrm { d } \mathcal { H } ^ d _ { \ , | t = 0 } \\ & = \int _ { \Sigma } \partial ^ \square \big ( G ( c _ t ) \big ) _ { \ , | t = 0 } - G ( c ) H V \ , \mathrm { d } \mathcal { H } ^ d = \int _ \Sigma G ' ( c ) w - G ( c ) H V \ , \mathrm { d } \mathcal { H } ^ d \end{align*}


\begin{align*} \sigma ( \alpha + \beta ) = \sigma ( \alpha ) + \sigma ( \beta ) \end{align*}


\begin{align*} g ( \xi , \hat s ) = ( g \xi , \hat s + \beta ( g ^ { - 1 } o , o , \xi ) . \end{align*}


\begin{align*} y _ { k } \left [ l \right ] & = \mathbf { h } _ { k } ^ { H } \mathbf { x } \left [ l \right ] + z _ { k } \left [ l \right ] \\ & = \mathbf { h } _ { k } ^ { H } \mathbf { p } _ { \mathrm { c } } s _ { \mathrm { c } } \left [ l \right ] + \mathbf { h } _ { k } ^ { H } \sum _ { i \in \mathcal { K } } \mathbf { p } _ { \mathrm { p } , i } s _ { \mathrm { p } , i } \left [ l \right ] + z _ { k } \left [ l \right ] , \end{align*}


\begin{align*} \begin{cases} d u = \nabla \cdot [ a ( u ) \nabla u + b ( u ) ] d t , ( 0 , T ] \times \mathbb { T } ^ d , \\ u ( 0 ) = u _ 0 , \end{cases} \end{align*}


\begin{align*} ( Z _ d ( n ) ) _ { n \geq 0 } \stackrel { d } { = } ( L ( d , n ) ) _ { n \geq 0 } . \end{align*}


\begin{align*} C ( G , x ) : = 1 + \sum _ { k = 1 } ^ { \omega ( G ) } c _ { k } ( G ) , \end{align*}


\begin{align*} N _ { i t } = - \log \left ( \max \left ( \left \| \mathbf { r } _ p ( \mathbf { q } _ w ) \right \| , \left \| \mathbf { r } _ d ( \mathbf { q } _ w ) \right \| , \mu ( \mathbf { q } _ w ) \right ) / \epsilon _ u \right ) / \log \left ( 1 - \delta / \sqrt { 2 k + 1 } \right ) . \end{align*}


\begin{align*} \langle T f , g \rangle _ \lambda = \langle T ^ { ( \lambda ) } _ a f , g \rangle _ \lambda \end{align*}


\begin{align*} K \oplus _ 2 L \supset K \widetilde { + } _ 2 L : = \cup _ { u \in S ^ { n - 1 } } \left [ 0 , \sqrt { \rho _ K ^ 2 ( u ) + \rho _ L ^ 2 ( u ) } u \right ] . \end{align*}


\begin{align*} \left \| \div \big ( | \nabla u | \nabla u \big ) \right \| _ { L ^ 1 ( B _ { 3 / 4 } ) } \leq C \| \nabla u \| _ { L ^ 2 ( B _ { 1 } ) } ^ 2 . \end{align*}


\begin{align*} R S _ { H ' } ( C ^ \bullet ) : = R \Gamma ( H ' , C ^ { \bullet } ) ^ { R \Gamma _ { H ' } - l a } . \end{align*}


\begin{align*} H [ b _ n ] = 4 \pi \int _ { D } ( \phi _ n - \phi _ \infty ) _ + G _ n \frac { 1 } { r } \dd z \dd r + 4 \pi \int _ { \mathbb { R } ^ { 2 } _ { + } \backslash D } ( \phi _ n - \phi _ \infty ) _ + G _ n \frac { 1 } { r } \dd z \dd r . \end{align*}


\begin{gather*} P ( c ) = \inf _ { Q \in K } Q ( c ) \quad \quad P \in K \end{gather*}


\begin{align*} { \rm { t r } ( \pi _ { \lambda , c _ 1 } ' ( w ) ) } = \pm { \rm { t r } } ( \pi ( p ( w ) ) ) . \end{align*}


\begin{align*} y = \sum _ { m = 0 } ^ \infty \frac { \psi ( \eta _ m ( y ) ) } { \gamma ^ { m + 1 } } . \end{align*}


\begin{align*} \phi ( U | \bar U ; x , t ) & : = ( \nabla \bar A ) ^ { - 1 } ( A - \bar A ) - ( U - \bar U ) \ ; , \\ G _ 1 ( U | \bar U ; x , t ) & : = \nabla G - \nabla \bar G - \nabla ^ 2 \bar G \cdot ( \nabla \bar A ) ^ { - 1 } ( A - \bar A ) \ ; , \\ G _ 2 ( U | \bar U ; x , t ) & : = G _ { x _ \alpha } - \bar G _ { x _ \alpha } - \nabla \bar G _ { x _ \alpha } \cdot ( \nabla \bar A ) ^ { - 1 } ( A - \bar A ) \ ; , \end{align*}


\begin{align*} a = \frac { u _ 1 - u _ 0 \beta } { \alpha - \beta } , b = \frac { u _ 0 \alpha - u _ 1 } { \alpha - \beta } \end{align*}


\begin{align*} l = \max \{ \textnormal { d i s t } ( \rho , v ) : v \in T \} . \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } \alpha } \log \mathrm { B } ( \alpha , \beta ) & = \psi ( \alpha ) - \psi ( \alpha + \beta ) . \end{align*}


\begin{align*} \lambda _ { 1 2 } + \lambda _ { 1 2 3 ' } & \geq \log \lceil 2 ^ { \lambda _ { 1 2 3 ' } } \rceil , \\ \lambda _ { 1 2 3 ' } & = \log m , m \in \mathbb { N } . \end{align*}


\begin{align*} | g ( x , t ) | & \leq \int _ { 5 I _ Q } \frac { | \partial _ u ^ M \varphi ( x , u ) - \partial _ u ^ M \varphi ( x , t ) | } { | u - t | ^ { \frac 1 { 2 s } - M + 1 } } d u + \| \partial _ t ^ M \varphi \| _ \infty \int _ { ( 5 I _ Q ) ^ c } \frac { d u } { | u - s | ^ { \frac 1 { 2 s } - M + 1 } } = T _ 1 + T _ 2 . \end{align*}


\begin{align*} \deg _ H ( S ) \leq \binom { p - \ell } { k - \ell } \leq d ^ { \frac { k - \ell } { k - 1 } } . \end{align*}


\begin{align*} \partial ^ 2 _ { x x } p ^ { \varepsilon } ( x , t ) = \exp \left [ - \frac { Y _ t h ( x ) } { \varepsilon } \right ] \partial ^ 2 _ { x x } q ^ { \varepsilon } ( x , t ) & - \frac { 2 Y _ t h ' ( x ) } { \varepsilon } p ^ { \varepsilon } ( x , t ) \\ & - p ^ { \varepsilon } ( x , t ) \left [ \frac { \left ( Y _ t h ' ( x ) \right ) ^ 2 } { \varepsilon } + \frac { Y _ t h '' ( x ) } { \varepsilon } . \right ] \end{align*}


\begin{align*} \log \int _ { \R ^ n } e ^ f \ , d \gamma _ t \le \sup _ { y \in \R ^ n } \left ( \int _ { \R ^ n } f \ , d \gamma _ { y , t } - H ( \gamma _ { y , t } \ , | \ , \gamma _ t ) \right ) + \frac { t ^ 2 } { 2 } \sum _ { i , j = 1 } ^ n \int _ { \R ^ n } | \partial _ { i j } f | ^ 2 \ , d \gamma _ { y ^ * , t } . \end{align*}


\begin{align*} \frac { \partial f ( z ) } { \partial z } = & \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \phi ( z ) [ \Phi ( z ) ] ^ { \alpha - 2 } [ 1 - \Phi ( z ) ] ^ { \beta - 2 } \times \\ & \Big \{ ( \alpha - 1 ) \phi ( z ) [ 1 - \Phi ( z ) ] - ( \beta - 1 ) \phi ( z ) \Phi ( z ) - z \Phi ( z ) [ 1 - \Phi ( z ) ] \Big \} . \end{align*}


\begin{align*} \frak f _ x & : = \frac 1 { 4 \gamma } \left ( q _ { x + 1 } - q _ x \right ) \left ( p _ x + p _ { x + 1 } \right ) + \frac 1 4 \left ( q _ { x + 1 } - q _ x \right ) ^ 2 , x = 0 , \dots , n - 1 , \\ \mathfrak F _ x & = p _ x ^ 2 + \left ( q _ { x + 1 } - q _ x \right ) \left ( q _ { x } - q _ { x - 1 } \right ) - \omega _ 0 ^ 2 q _ x ^ 2 , x = 0 , \dots , n , \end{align*}


\begin{align*} S ( t ) = \left [ \begin{array} { c c } { S ^ { ( q ) } ( t ) } & S ^ { ( q , p ) } ( t ) \\ S ^ { ( p , q ) } ( t ) & S ^ { ( p ) } ( t ) \end{array} \right ] , \end{align*}


\begin{align*} \overline { A s y _ { n } ^ { \vec { v } } ( X , T ) \cap \left ( \operatorname { s u p p } \mu _ { x } \right ) ^ { ( n ) } } = \left ( \operatorname { s u p p } \mu _ { x } \right ) ^ { ( n ) } \end{align*}


\begin{align*} y _ n ^ 2 = \left ( \bar { y } _ n + \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( x ' _ 0 ) y _ \beta \right ) ^ 2 \leq 2 \bar { y } _ n ^ 2 + C \epsilon ^ 2 \delta ^ 2 b _ \alpha | y ' | ^ 2 . \end{align*}


\begin{align*} \kappa ( A ) : = \lVert A \rVert \ , \lVert A ^ { - 1 } \rVert = \frac { s _ { \max } ( A ) } { s _ { \min } ( A ) } . \end{align*}


\begin{align*} e _ 2 ( C ) = 2 - I ( C ) - s ( C ) ; \end{align*}


\begin{align*} 1 = - \sum _ { \emptyset \neq I \subset \{ 1 , \dots , m \} } ( - 1 ) ^ { | I | } , \end{align*}


\begin{align*} w _ \varepsilon ( x _ 1 , x _ 2 , 0 , t ) = \frac { 1 } { \varepsilon ^ 2 } \left ( u _ \varepsilon ( \bar { R } _ { - \alpha | \ln \varepsilon | t } ( x _ 1 , x _ 2 ) ) - \left ( \frac { \alpha | ( x _ 1 , x _ 2 ) | ^ 2 } { 2 } + \beta \right ) \ln \frac { 1 } { \varepsilon } \right ) ^ p _ + . \end{align*}


\begin{align*} | \left \{ z \in \{ 1 , 2 , \ldots , n \} \mid \d ( x , z ) = d \right \} | = \left \{ \begin{array} { l l } 1 & \mbox { i f $ d = \frac { n } { 2 } $ } \\ 2 & \mbox { i f $ d < \frac { n } { 2 } $ , } \end{array} \right . \end{align*}


\begin{align*} a _ k ( x ) f _ k '' ( x ) + b _ k ( x ) f _ k ' ( x ) + p _ k c _ k ( x ) f _ k ( x ) = 0 , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ \infty Q ^ { ( m ) } _ \nu ( k ) q ^ { - k n } = \zeta _ \nu ( n ) \zeta _ \nu ( n - 1 ) \cdots \zeta _ \nu ( n - m + 1 ) . \end{align*}


\begin{align*} \begin{aligned} ( \Delta u ) ^ { m } \wedge \beta _ n ^ { n - m } = m ! ( n - m ) ! \sum _ { 1 \leq i _ 1 \leq \dots \leq i _ m \leq n } \lambda _ { i _ 1 } \dots \lambda _ { i _ m } \Omega _ { 2 n } . \end{aligned} \end{align*}


\begin{align*} ( \mathcal { R } _ \lambda ( \sigma s _ i ) v _ T , v _ T ) = \pm ( \mathcal { R } _ \lambda ( \sigma ) v _ T , v _ T ) . \end{align*}


\begin{align*} \beta _ i ' = s _ { \beta _ \ell } ( \beta _ i ) , i = 1 , \dotsc , \ell ' . \end{align*}


\begin{align*} & \lim _ { \delta \rightarrow 0 + } \sup _ { n \in \mathbb { N } } \mathbb { E } \int _ 0 ^ { T - \delta } \Vert Y _ n ^ M ( t + \delta ) - Y _ n ^ M ( t ) \Vert _ { H } ^ { \alpha } d s \\ \leq & C \bigg ( \lim _ { \delta \rightarrow 0 + } \sup _ { n \in \mathbb { N } } \mathbb { E } \int _ 0 ^ { T - \delta } \Vert Y _ n ^ M ( t + \delta ) - Y _ n ^ M ( t ) \Vert _ { H } ^ { 2 } d s \bigg ) ^ { \frac { \alpha } { 2 } } = 0 . \end{align*}


\begin{align*} ( H _ { \lambda v , \alpha , x } \psi ) _ { n } = \psi _ { n + 1 } + \psi _ { n - 1 } + \lambda v ( x + n \alpha ) \psi _ { n } , \end{align*}


\begin{align*} \sum _ { X _ k } \sum _ { X _ k ' } \int \widehat { h _ { X _ k } } \overline { \widehat { h _ { X _ k ' } } * \widehat { w _ U } } & = \sum _ { X _ k } \sum _ { X _ k ' \sim X _ k } \int h _ { X _ k } \overline { h _ { X _ k ' } } w _ U \\ & \le \sum _ { X _ k } \sum _ { X _ k ' \sim X _ k } \int ( | h _ { X _ k } | ^ 2 + | h _ { X _ k ' } | ^ 2 ) w _ U \\ & \le \sum _ { X _ k } \sum _ { X _ k ' \sim X _ k } \int ( | h _ { X _ k } | ^ 2 + | h _ { X _ k ' } | ^ 2 ) w _ U \le 2 L \sum _ { X _ k } \int | h _ { X _ k } | ^ 2 w _ U . \end{align*}


\begin{align*} & \xi _ { k } = \inf \Big \{ t \geq s _ { k } ; | X ^ { \epsilon } ( t ) | \geq K \Big \} . \end{align*}


\begin{align*} \sigma = t _ { \mu _ \sigma } \circ \sigma _ 1 \circ \sigma _ 2 , \end{align*}


\begin{align*} & g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor \big ) \geq 1 6 \ , g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j - 1 } ) r \rfloor \big ) , j = 1 , 2 , \dots , i - 1 ; \\ & g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - i } ) r \rfloor \big ) \leq \sqrt { x } . \end{align*}


\begin{align*} G = \begin{pmatrix} 1 & 0 & \cdots & 0 & a _ 1 & 0 & \cdots & 0 & 1 \\ 0 & 1 & \cdots & 0 & a _ 2 & 0 & \cdots & 1 & 0 \\ \vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & a _ k & 1 & \cdots & 0 & 0 \\ \end{pmatrix} _ { k \times ( 2 k + 1 ) } , \end{align*}


\begin{align*} { \left ( { { { { D \mathord { \left / { \vphantom { D D } } \right . \kern - \nulldelimiterspace } D } } _ 0 } } \right ) ^ 2 } = 1 - K t , \end{align*}


\begin{align*} { { \bf { \bar x } } _ { \rm t } } \left [ m , n \right ] { \rm = } \frac { 1 } { \sqrt K } \sum \limits _ { k = 0 } ^ { K - 1 } { { { \bf { u } } _ k } s \left [ m , k \right ] { e ^ { j \frac { { 2 \pi } } { K } k n } } } , \ n { \rm = } - { { N } _ { \rm C P } } . \cdots , K - 1 . \end{align*}


\begin{align*} \phi _ { i s } ( \partial _ s ) = \frac { \Gamma _ { s i } ^ i } { \varphi _ s } , \quad \forall s \neq i , \end{align*}


\begin{align*} \begin{cases} d Y _ t = L _ { z } ^ \perp Y _ t d t + \Pi _ { \mathrm { k e r } A ^ \perp } \sigma d W _ t \\ Y _ t | _ { t = 0 } = Y _ 0 \in \mathrm { k e r } A ^ \perp . \end{cases} \end{align*}


\begin{align*} L _ { q } ^ { - 1 } [ L _ { q } [ f ( t ) ] ] = \int _ { 0 } ^ { \infty } d t ^ { \prime } f ( t ^ { \prime } ) \delta ( t - t ^ { \prime } ) \equiv f ( t ) . \end{align*}


\begin{align*} \langle H \rangle = \frac { 1 } { 2 } \times \mathop { \sum \sum } _ { \chi _ 4 ( x - y ) \in \{ 1 , \chi _ 4 ( g ) \} } 1 , \end{align*}


\begin{align*} \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi ( \partial _ t \varphi + ( v + u _ { \infty } ) \cdot \nabla \varphi ) \dd x \dd s = - \int _ { \mathbb { R } ^ { 3 } } \Phi _ 0 \varphi _ 0 \dd x , \end{align*}


\begin{align*} & \max \Big \{ \ ! \left \lVert \mu ( s , x ) - \mu ( t , x ) \right \rVert ^ 2 , \left \lVert \sigma ( s , x ) - \sigma ( t , x ) \right \rVert ^ 2 \Big \} \leq c \Big [ \lvert t - s \rvert + \ ! \ ! \sup _ { \substack { u , v \in [ 0 , t ] \colon \\ \lvert u - v \rvert \leq \lvert t - s \rvert } } \left \lVert x ( u ) - x ( v ) \right \rVert ^ 2 \Big ] , \end{align*}


\begin{align*} F ^ * = \min _ { x } \ \{ F ( x ) \coloneqq f ( x ) + P ( x ) \} , \end{align*}


\begin{align*} R ( t ) = F ^ { - 1 } ( t ) \rightarrow F ^ { - 1 } ( T ) = F ^ { - 1 } \big ( F ( 0 ) \big ) = 0 \end{align*}


\begin{align*} \begin{aligned} | \epsilon _ 2 ( \eta _ a , t ) | \le \int _ N ^ t \left | h ( x _ a + \delta u ( \tau ) ) - h ( \bar { x } _ 1 ) \right | \ , d \tau \le L _ r 2 \delta . \end{aligned} \end{align*}


\begin{align*} \lambda ( Y ) : = \sup \lbrace \lambda ( Y , X ) : Y \subset X \rbrace . \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } z ( \Psi ^ - _ { 1 1 } ( x ; z ) - e ^ { - i c _ - ( x ) } ) = \widehat { \Psi } ^ - _ { 1 1 } ( x ) . \end{align*}


\begin{align*} \lim _ { s \rightarrow \infty } s L _ { q } [ f ( t ) ] = \lim _ { t \rightarrow 0 } \frac { f ( t ) } { 1 + ( 1 - q ) } . \end{align*}


\begin{align*} \begin{aligned} \| f \| _ { B M O } & \leq C \| f \| _ { \dot B ^ 0 _ { \infty , 2 } } \\ & \leq C \big ( 1 + \| f \| _ { \dot B ^ 0 _ { \infty , \infty } } \ln ^ { \frac { 1 } { 2 } } ( 1 + \| f \| _ { H ^ s } ) \big ) , \end{aligned} \end{align*}


\begin{align*} \lim _ { s \to \infty } b _ \eta ( \xi ( s ) ) = - \infty , \lim _ { s \to - \infty } b _ \eta ( \xi ( s ) ) = \infty , \end{align*}


\begin{align*} \liminf _ { r \to \infty } \frac { \log \log M ( r , f ) } { \log \log r } = \infty \end{align*}


\begin{align*} \mathcal { I } & = \{ ( k , l ) \in \Z ^ 2 : k \leq i \mbox { o r } l \leq j \} \subseteq \Z ^ 2 \\ m _ { k , l } & = n _ { k , l } \mbox { i f } k \leq i \\ m _ { k , l } & = 0 \mbox { i f } l \leq j \end{align*}


\begin{align*} \| \tilde { F } \| _ { h } \leq \| F \| _ { h } \| P \| ^ { 2 } \leq { \rm e } ^ { 2 | \log \varepsilon | ^ { \frac { 2 } { 2 + \eta } } } \varepsilon = : \tilde { \varepsilon } . \end{align*}


\begin{align*} \tilde { \kappa } ( p ) \cdot \tilde { k _ 1 } ( p ) \cdot \tilde { k _ 2 } ( p ) \ , = \ , \frac { 1 } { p } , \ \Re ( p ) > c \end{align*}


\begin{align*} & \sum \limits _ { n = - \infty } ^ { \infty } ( - 1 ) ^ { n } q ^ { n ( 3 n + 1 ) / 2 } \cdot \sum \limits _ { n = 0 } ^ { \infty } d _ { e } ( n ) q ^ { n } \\ & = \sum \limits _ { n = - \infty } ^ { \infty } ( - 1 ) ^ { n } q ^ { 2 n ( 3 n + 1 ) } \sum \limits _ { n = 1 } ^ { \infty } \sigma _ { 0 } ( n ) q ^ { 4 n } . \end{align*}


\begin{align*} \sum _ { y \neq x } \mu ( y ) L _ { y x } = \sum _ { y \neq x } \mu ( x ) L _ { x y } = \mu ( x ) L _ { x x } . \end{align*}


\begin{align*} P = P _ w = _ \gamma + \gamma I - B _ w : T L \rightarrow T L , \end{align*}


\begin{align*} \varphi ( \vartheta , z , w ( \cdot ) ) = \sigma ( z , w ( \cdot ) ) , ( z , w ( \cdot ) ) \in \mathbb R ^ n \times \mathrm { P L i p } . \end{align*}


\begin{align*} \langle B ^ { \circ } x ^ { \circ } , u \rangle = \langle x ^ { \circ } , B u \rangle = \lim _ { n \to \infty } \langle x ^ { \circ } , B u _ { n } \rangle , \end{align*}


\begin{align*} D _ n ( x ) = \prod _ { k = 0 } ^ { n } m _ \tau ( ( R ^ { t } ) ^ { k } x ) . \end{align*}


\begin{align*} v ( x , y ) : = ( x , - y ) . \end{align*}


\begin{align*} \Phi _ t ( x ' ) = t ^ { 1 - n } ( \partial _ n K ^ L ) ( x ' / t , 1 ) = t \ , ( \partial _ n K ^ L ) ( x ' , t ) = t \partial _ t K ^ L ( x ' , t ) = t \partial _ t P _ t ^ L ( x ' ) , \end{align*}


\begin{align*} f _ { i : n } ( x ) = \frac { f ( x ) } { \operatorname { B } ( i , n - i + 1 ) } F ( x ) ^ { i - 1 } \{ 1 - F ( x ) \} ^ { n - i } , \end{align*}


\begin{align*} Y _ { j , ( 1 ) } = \begin{cases} \log ( \frac { [ T _ { j , ( 1 ) } ^ { \flat } ] } { T _ { j , ( 1 ) } } ) & \mbox { i f } 1 \leq j \leq e \\ \log ( \frac { [ S _ { j , ( 1 ) } ^ { \flat } ] } { S _ { j , ( 1 ) } } ) & \mbox { i f } e + 1 \leq j \leq d \end{cases} \end{align*}


\begin{align*} \nabla \cdot ( \nabla n \otimes \nabla n ) = \frac { 1 } { 2 } \nabla ( | \nabla n | ^ 2 ) + \nabla n \cdot \Delta n . \end{align*}


\begin{align*} X + \xi \in E = E ( \mathfrak { g } ) \coloneqq \mathfrak { g } \oplus \mathfrak { g } ^ * \end{align*}


\begin{align*} \beta _ 1 \oplus \beta _ 2 : = \{ ( \gamma _ 1 , \gamma _ 2 ) \mid ( \gamma _ 1 , \gamma _ 2 ) \in \mathcal { P } ( \beta _ 1 + \beta _ 2 ) \mid ( \beta _ 1 , \beta _ 2 ) < ( \gamma _ 1 , \gamma _ 2 ) \} . \end{align*}


\begin{align*} M _ { f _ { 1 , \infty } } ( n , \alpha , \epsilon , Z , \psi ) : = \inf \bigg \{ \sum _ { i } ^ { - \alpha n _ { i } + S _ { 1 , n _ { i } } \psi ( x _ { i } , \epsilon ) } : Z \subseteq \bigcup _ { i } B _ { n _ { i } } ( x _ { i } , \epsilon ) \bigg \} , \end{align*}


\begin{align*} \mathcal { O } _ { s } ( k ) = { { 2 s - k - 1 } \choose { k } } ( C _ { s - k } ) ^ 2 . \end{align*}


\begin{align*} \frac { d } { d t } \bar { y } _ t = - Z _ { t , 3 } \bar { y } _ t - \bar { z } _ { t , 3 } y _ t , \end{align*}


\begin{align*} \left ( \lambda + \rho \right ) ^ 2 = \left ( \mu + \nu \right ) ^ 2 . \end{align*}


\begin{align*} S t _ { D N D C } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( z ^ { 2 } , z ) - z ^ { 2 } | | _ { \infty } = 0 . \\ \end{align*}


\begin{align*} \mathcal P ( m ) = & \left \{ p \subset \lbrace 1 , . . . , m \rbrace ^ 2 : \# p = \frac m 2 ( x , y ) , ( x ' , y ' ) \in p , \right . \\ & \qquad \left . x < y , x ' < y ' \right \} . \end{align*}


\begin{align*} | \xi | _ p \le \psi ( p ) = \psi _ { p _ 0 } [ \eta ] ( p ) , \ p \in U . \end{align*}


\begin{align*} d _ \textrm { H S } ( A , B ) = \| A - B \| _ \textrm { H S } \forall A , B \in \mathrm { U } ( n ) . \end{align*}


\begin{align*} F ( x ) = I _ { G ( x ) } ( \alpha , \beta ) = \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \int _ 0 ^ { G ( x ) } t ^ { \alpha - 1 } ( 1 - t ) ^ { \beta - 1 } \mathrm { d } t . \end{align*}


\begin{align*} ( \C ^ n , \omega _ { \mathrm { s t } } ) , \omega _ { \mathrm { s t } } = \i \sum d z ^ i \wedge d \bar { z } ^ i \end{align*}


\begin{align*} \sigma ( B ) \leq \sum _ { j = 1 } ^ N \sigma ( B ^ j ) \mbox { w h e n e v e r } N \in \mathbb { Z } ^ + , \{ B ^ j \} _ { j = 1 } ^ N \subset \mathcal { B } , B \subset \bigcup _ { j = 1 } ^ N B ^ j . \end{align*}


\begin{align*} \mathcal { T } \circ \beta & = \alpha \circ \mathcal { T } , \\ [ T ( a ) , \mathcal { T } ( b ) ] + [ \mathcal { T } , ( a ) , T ( b ) ] & = T ( \varrho ( \mathcal { T } ( a ) ) ( b ) - \varrho ( \mathcal { T } ( b ) ) ( a ) ) + \mathcal { T } ( \varrho ( T ( a ) ) ( b ) - \varrho ( T ( b ) ) ( a ) ) , \\ [ \mathcal { T } ( a ) , \mathcal { T } ( b ) ] & = \mathcal { T } ( \varrho ( \mathcal { T } ( a ) ) ( b ) - \varrho ( \mathcal { T } ( b ) ) ( a ) ) . \end{align*}


\begin{align*} \frac { d L } { d t _ { k } } = \left [ ( L ^ { k / \alpha } ) _ { D } , L \right ] _ \epsilon = - \left [ ( L ^ { k } ) _ { S } , L \right ] \ ; , k \geq 1 \ ; , \end{align*}


\begin{align*} P ( d x ) = Z ^ { - 1 } e ^ { f ( x ) } \mu ^ { \otimes n } ( d x ) , \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \sum _ { 1 \le \beta \le N } ( P ^ L _ { \alpha \beta } ) _ t ( x ' - y ' ) C _ { \beta } \ , d y ' = C _ { \alpha } , \quad \forall ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} ( \sum _ { j = 1 } ^ s a _ j ( f _ j - 1 ) ) ( h \otimes 1 ) & = \sum _ { j = 1 } ^ s a _ j ( f _ j ( h \otimes 1 ) - 1 ( h \otimes 1 ) ) \\ & = \sum _ { j = 1 } ^ s a _ j ( h \otimes ( f _ j ) 1 - h \otimes 1 ) \\ & = \sum _ { j = 1 } ^ s a _ j ( h \otimes 1 - h \otimes 1 ) \\ & = 0 , \end{align*}


\begin{align*} \P _ x ( \tau > n ) \sim K _ { \lambda } n ^ { - d ( d - 1 ) / 2 - d / 2 } e ^ { - \gamma n } e ^ { \sum _ { i = 1 } ^ d ( \lambda _ i - \bar { \lambda } ) x _ i } h ^ { ( \bar { \lambda } , \ldots , \bar { \lambda } ) } ( x ) , n \rightarrow \infty \end{align*}


\begin{align*} V ( r ) = \frac { L ( L + 1 ) } { r ^ 2 } - \frac { Q } { r } f + \kappa B _ 1 \frac { r } { f } + \kappa B _ 2 \frac { 1 } { f ^ 2 } + \kappa B _ 3 \frac { r } { f ^ 3 } + \kappa B _ 4 \frac { 1 } { f ^ 4 } , B _ 4 > 0 , \end{align*}


\begin{align*} f _ a ( x ) : = \dfrac { 1 } { T } \int _ { 0 } ^ { T } f ( x , t ) \ , d t . \end{align*}


\begin{align*} \mathbf { X } - \mathbf { T } _ x = \mathbf { U } _ { x } = \left ( \mathbf { U } _ { x ^ { ( i ) } } : i = 1 , \cdots , k \right ) , \end{align*}


\begin{align*} s ^ \star - s ^ g = - 2 \delta ^ g \theta - \| \theta \| ^ 2 + 2 \| N \| ^ 2 . \end{align*}


\begin{align*} \nabla \times ( a _ { t } + ( B \times u ) + \mu \nabla \times b ) = 0 \textrm { o n } \ ( L ^ { 2 } _ { \sigma } \cap H ^ { 1 } ) ( \mathbb { R } ^ { 3 } ) ^ { * } . \end{align*}


\begin{align*} B ^ { i } _ { G ' , n } : = ( B _ { G ' , n } \widehat { \otimes } _ { A _ { n } } A _ { n } ^ i ) ^ { \wedge - u } \end{align*}


\begin{align*} & \lim _ { m ' \to + \infty } \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ { h _ { m ' } } e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } \\ = & \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } < + \infty . \end{align*}


\begin{align*} s ( n , k , \ell ) = \max \{ \vartheta ( S ) : \ : S \} \end{align*}


\begin{align*} H P M ( x , y , z , t ) & = H P L ( [ x , y ] , \alpha ( z ) , \alpha ( t ) ) - H P L ( [ y , x ] , \alpha ( z ) , \alpha ( t ) ) \\ & \quad + H P L ( \alpha ( x ) , \alpha ( y ) , [ z , t ] ) + H P L ( \alpha ( y ) , \alpha ( z ) , [ x , t ] ) \\ & \quad - [ \alpha ^ { 2 } ( z ) , H P L ( x , y , t ) ] + [ \alpha ^ { 2 } ( y ) , H P L ( x , z , t ) ] . \end{align*}


\begin{align*} \kappa \stackrel { d e f } { = } \lim _ { n \to \infty } \kappa _ n . \end{align*}


\begin{align*} \overline { W ^ { s } ( ( x , t ) , T _ { \vec { v } } ) \cap \operatorname { s u p p } \widetilde { \mu } _ { x } } = \operatorname { s u p p } \widetilde { \mu } _ { x } . \end{align*}


\begin{align*} | h ( x + \delta u ) - h ( x ) | = | R ( x , \delta u ) | \le L _ r \delta . \end{align*}


\begin{align*} & \widehat \zeta = ( \widehat y , \widehat p , \widehat z , \widehat q ) = ( y - y ' , p - p ' , z - z ' , q - q ' ) , \\ & \widehat { \overline \zeta } = ( \widehat { \overline y } , \widehat { \overline p } , \widehat { \overline z } , \widehat { \overline q } ) = ( \overline y - \overline y ' , \overline p - \overline p ' , \overline z - \overline z ' , \overline q - \overline q ' ) . \end{align*}


\begin{align*} \Big ( \frac p q \Big ) \Big ( \frac q p \Big ) = ( - 1 ) ^ { \mu + \nu } . \end{align*}


\begin{align*} \nabla _ { i j } u : = \nabla _ { e _ i } \nabla _ { e _ j } u = e _ i ^ k e _ j ^ l \partial _ { k } \partial _ l u = e _ i ^ k e _ j ^ l u _ { k l } . \end{align*}


\begin{align*} \left \Vert \varphi \right \Vert _ { p , r } = \sup _ { x \in \mathbb { R } } \left \vert x ^ { p } \varphi ^ { ( r ) } \left ( x \right ) \right \vert , \ ; p , r \in \mathbb { N } _ { 0 } \end{align*}


\begin{align*} W _ + ( r ) & = ( 2 L + 3 ) \left ( - \frac { f } { r } + \Q + \kappa \frac { r } { f } \right ) , \\ W _ - ( r ) & = - \frac { f } { r } - \Q + \kappa \frac { r } { f } , \end{align*}


\begin{align*} P _ b ( x ) = x ^ { q + 1 } + x + b , \end{align*}


\begin{align*} u = \arg \min _ { w \in L _ 2 ( 0 , T ; \mathcal { V } ) } \bigg ( | | \partial _ t w + \mathcal { L } w - F | | ^ 2 _ { L _ 2 ( 0 , T ; \mathcal { H } ) } + | | \alpha w + \beta \mathbf { n } \cdot \nabla w - g | | ^ 2 _ { L _ 2 ( 0 , T ; \partial \mathcal { H } ) } + | | w ( 0 , \cdot ) - u _ 0 | | ^ 2 _ { \mathcal { H } } \bigg ) , \end{align*}


\begin{align*} Z = U D ( x ) V = U D ( \overline { t } ) D ( x ) D ( t ) V , \end{align*}


\begin{align*} y \left [ n \right ] = \left ( { \sqrt { P \sum \nolimits _ { l = 1 } ^ L { { { \left \| { { { \bf { h } } _ l } } \right \| } ^ 2 } } } } \right ) s \left [ { n - { n _ { \max } } } \right ] + z \left [ n \right ] . \end{align*}


\begin{align*} \begin{aligned} \bar \Delta & = C _ 1 \min _ { 0 \le k \le T - 1 } \| \nabla \phi ( X _ k ) \| - C _ 2 \epsilon _ g , \\ \bar \Delta ' & = \min _ l \{ \gamma ^ l \delta _ 0 : ~ \gamma ^ l \delta _ 0 > \gamma \bar \Delta l \in \mathbb { Z } \} , \end{aligned} \end{align*}


\begin{align*} - \sum _ { i \geq 3 } \sum _ { d \geq 2 } a _ { i , d } ( \underline { b } ) = - a ( \underline { b } ) + \sum _ { i \geq 1 } \left ( a _ { i , 0 } ( \underline { b } ) + a _ { i , 1 } ( \underline { b } ) \right ) + a _ { 2 , 2 } ( \underline { b } ) . \end{align*}


\begin{align*} \forall x \in \Omega _ 4 : \zeta _ x : = \max \{ | \tau \cap W | : \tau \in \mathcal { U } _ x , \dim ( \tau \cap U ) = 1 \} . \end{align*}


\begin{align*} C u r \mathcal { G } = \mathbb { C } [ \partial ] \otimes \mathcal { G } , [ a _ \lambda b ] = [ a , b ] , \end{align*}


\begin{align*} \begin{aligned} \int _ { 0 } ^ { 1 / 2 } \left [ h ( x + \delta \sin ( 2 \pi t ) ) - h ( x ) \right ] \sin ( 2 \pi t ) \ , d t \\ \int _ { 0 } ^ { 1 / 2 } \left [ h ( x ) - h ( x - \delta \sin ( 2 \pi t ) ) \right ] \sin ( 2 \pi t ) \ , d t . \end{aligned} \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \Vert X _ n ( t , \omega ) - X ( t , \omega ) \Vert _ H = 0 , \ ( t , \omega ) . \end{align*}


\begin{align*} \Omega _ w = \{ ( F _ \bullet , V _ \bullet ) ; \dim V _ i \cap F _ j \geq r _ { i , j } ( w ) \} . \end{align*}


\begin{align*} V _ { a , b } = \mathbb { C } [ \partial ] v , L _ \lambda v = ( \partial + a \lambda + b ) v , { H _ i } _ \lambda v = 0 , \end{align*}


\begin{align*} s ( G ^ { \sigma , \tau } ) & = \sum _ { \gamma \in \Gamma } s ( G ^ { ( \gamma ) } ) + \sum _ { v \in V } \sigma ( v ) - 2 ( \abs { \Gamma } - 1 ) . \end{align*}


\begin{align*} h = 0 , \nabla h = 0 , v = 0 , \nu ( v ) = 0 \mbox { o n } \hat \Sigma \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \theta ^ j _ { \alpha \beta } ( x ' , t ; y ' ) d y ' = \int _ { \R ^ { n - 1 } } t \ , \partial _ j K ^ L _ { \alpha \beta } ( x ' - y ' , t ) d y ' = t \ , \partial _ j \int _ { \R ^ { n - 1 } } K ^ L _ { \alpha \beta } ( y ' , t ) d y ' = 0 . \end{align*}


\begin{align*} I _ 2 & = | c | | \langle f , g \rangle | = \left | c \int ( f - m _ Q f ) \ ; g \ ; d m \right | \\ & \lesssim \int _ { 2 Q } | f - m _ Q f | \ ; | g | \ ; d m + \int _ { \R ^ { N + 1 } \setminus 2 Q } ( f - m _ Q f ) \ ; g \ ; d m \\ & = I _ { 2 1 } + I _ { 2 2 } . \end{align*}


\begin{align*} - x - T ( x ) \in [ - a - T ( a ) , a - T ( - a ) ] = [ - a - T ( a ) , a - T ( a ) ] \end{align*}


\begin{align*} Y _ t ( x ) = & Y _ 0 ( x + t ) + \int _ 0 ^ t \alpha _ s ( x + t - s ) d s \int _ 0 ^ t \langle \sigma _ s , \delta _ { x + t - s } \rangle d W _ s . \end{align*}


\begin{align*} h ( a , s ) : = \hat \Phi ( ( a , 0 ) ^ \top , s ) = \hat \Phi ^ \dagger ( a , s ) : = \langle z _ 1 ^ * , \Phi ^ \dagger ( a u _ \ell , p ( s ) ) \rangle . \end{align*}


\begin{align*} \frac { \sum _ { i = 1 } ^ p \log R _ i - m _ n } { s _ n } \overset { d } { \longrightarrow } S _ \alpha , n \to \infty \ , , \end{align*}


\begin{align*} \sum _ { Y < p \leq X } \frac { 1 - ( p ^ { 2 i D t } ) } { p } & = \frac { 1 } { 2 \pi } \left ( \int _ 0 ^ { 2 \pi } ( 1 - \cos \alpha ) d \alpha \right ) \cdot \log \left ( \frac { \log ( 2 D | t | \log X ) } { \max \{ 1 , D | t | \log N _ { X , \eta } \} } \right ) - O _ K ( 1 ) \\ & \geq \left ( \frac { 1 } { 3 } - \eta \right ) \min \{ \log \log X , \log ( 1 + | t | \log X ) \} - O _ K ( 1 ) . \end{align*}


\begin{align*} P _ T = P _ { \rm P B S } \left ( \left | { \mathcal { K } } \right | - 1 \right ) + P _ { \rm M B S } + P _ { \rm V L C } \left ( \left | { \mathcal { V } } \right | \right ) + \sum \limits _ { \forall j } P _ j , \end{align*}


\begin{align*} V _ { \alpha , \beta , \gamma } = \mathbb { C } [ \partial ] v , L _ \lambda v = ( \partial + \alpha \lambda + \beta ) v , H _ \lambda v = \gamma v . \end{align*}


\begin{align*} S ( u ) = \frac { 1 } { 2 } \sum _ { n = 0 } ^ { + \infty } \xi _ { n } \cdot \nabla \left ( \xi _ { n } \cdot \nabla u \right ) . \end{align*}


\begin{align*} D ( A ' ) \coloneqq \{ x ' \in ( X , \upsilon ) ' \ ; | \ ; \exists \ ; y ' \in ( X , \upsilon ) ' \ ; \forall \ ; x \in D ( A ) : \ ; \langle A x , x ' \rangle = \langle x , y ' \rangle \} \end{align*}


\begin{align*} f ( t , q ^ { \prime } ) = \displaystyle \sum _ { n = 0 } ^ { \infty } \frac { \left ( - \frac { 1 } { 1 - q ^ { \prime } } \right ) _ { 2 n } \left ( ( 1 - q ^ { \prime } ) \ ; \alpha t \right ) ^ { 2 n } } { ( 2 n ) ! } = \cosh _ { q ^ { \prime } } \left ( \alpha t \right ) . \end{align*}


\begin{align*} \begin{aligned} z _ m ( & \phi _ { \mathrm { t } , m } , \phi _ { \mathrm { r } , m } ) = \sum _ { i \in \{ \mathrm { t , r } \} } \Bigg ( [ \mathbf { V } _ i ] _ { m , m } | \phi _ { i , m } | ^ 2 \\ & + 2 \Re \Big \{ \underbrace { \Big ( \sum _ { n \ne m } [ \mathbf { V } _ i ] _ { m , n } \phi _ { i , n } - [ \tilde { \mathbf { v } } _ i ] _ m \Big ) } _ { = \chi _ { i , m } } \phi _ { i , m } ^ * \Big \} \Bigg ) . \end{aligned} \end{align*}


\begin{align*} W ( r ) = \frac { \xi } { r } f + \eta + \zeta \frac { r } { f } , \xi \le 0 , \zeta > 0 , \end{align*}


\begin{align*} \omega _ T = \frac { d \chi _ 1 } { \chi _ 1 } \wedge \ldots \wedge \frac { d { \chi _ r } } { \chi _ r } . \end{align*}


\begin{align*} \underset { j = 1 } { \overset { k } { \sum } } \lambda _ { j } \leq \underset { j = 1 } { \overset { k } { \sum } } a _ { j } , \end{align*}


\begin{align*} a _ { 2 c - i } & = \sum _ { j = 0 } ^ { \lfloor i / 2 \rfloor } \binom { c } { j } \binom { c - j } { i - 2 j } 2 ^ j , \end{align*}


\begin{align*} \Gamma ( t , x ) : = \partial B _ { R _ 0 ( t , x ) } ( x ) . \end{align*}


\begin{align*} d _ { \Omega } ( \omega , \omega ' ) : = \sum _ { m = 1 } ^ { \infty } \frac { 1 } { 2 ^ m } \frac { \| \omega - \omega ' \| _ { m } } { 1 + \| \omega - \omega ' \| _ { m } } , \| \omega - \omega ' \| _ { m } : = \sup _ { - m \leq t \leq m } | \omega ( t ) - \omega ' ( t ) | , \end{align*}


\begin{align*} \phi ' _ { x _ 1 , \dots , x _ k } ( x ) = ( - 1 ) ^ { i } \end{align*}


\begin{align*} a _ { k } ( x ) = \frac { i } { ( 2 \pi ) ^ { n } } \int _ { \mathbb { R } ^ { n - 1 } } \int _ { \mathcal { C } } e ^ { - t \tau } \operatorname { T r } \phi _ { - 1 - k } \ , d \tau \ , d \xi , 0 \leqslant k \leqslant n - 1 . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 2 } _ { + } } \nabla \phi \cdot \nabla \tilde { \phi } \frac { 1 } { r } \dd z \dd r = \mu ^ { 2 } \int _ { \mathbb { R } ^ { 2 } _ { + } } ( \phi - \phi _ { \infty } ) _ { + } \tilde { \phi } \frac { 1 } { r } \dd z \dd r , \end{align*}


\begin{align*} \varphi ( - t ) ^ 2 H ( - t ) + \varphi ( t ) ^ 2 H ( t ) = 2 \varphi ( t ^ 4 ) ^ 2 G ( t ^ 4 ) . \end{align*}


\begin{align*} \begin{cases} \dim H ^ 1 ( S , T _ S ) = 4 0 ; \\ \dim H ^ 2 ( S , T _ S ) = 0 , \end{cases} \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta _ { p } u & = v ^ { m } | \nabla u | ^ { \alpha } & & \quad \mbox { i n } \Omega , \\ \Delta _ { p } v & = v ^ { \beta } | \nabla u | ^ q & & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


\begin{align*} a \approx a ^ \mathrm { a p p } ( s ) : = \sqrt { \frac { - \gamma _ 2 s } { \gamma _ 1 } } . \end{align*}


\begin{align*} \left ( P \circ Q \right ) = - \left ( Q \circ P \right ) \ , , \end{align*}


\begin{align*} \begin{aligned} W ( \xi _ 0 , k + 1 , x , \mathsf { L } ^ { k + 1 } ) = & E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { k + 1 } Z _ i ^ { \mathsf { L } ^ { k + 1 } } \biggr ) \biggr ] \\ = & E _ { 1 - \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { k + 1 } Z _ i ^ { \mathsf { R } ^ { k + 1 } } \biggr ) \biggr ] \\ = & W ( 1 - \xi _ 0 , k + 1 , x , \mathsf { R } ^ { k + 1 } ) . \end{aligned} \end{align*}


\begin{align*} p ' _ { j } \circ p _ { 1 \ldots ( n - 1 ) } \circ f \circ \iota _ { 1 \ldots ( n - 1 ) } \circ u ' _ { j } = p _ { j } \circ f \circ u _ { j } \circ \iota _ { 1 \ldots ( n - 1 ) } = p _ { j } \circ f \circ \iota _ { 1 \ldots ( n - 1 ) } \\ = p ' _ { j } \circ p _ { 1 \ldots ( n - 1 ) } \circ f \circ \iota _ { 1 \ldots ( n - 1 ) } ~ ~ ~ ~ \forall j = 1 , \ldots , n - 1 . \end{align*}


\begin{align*} \mathop { \inf } \limits _ { \| ( u , v ) \| _ { E } = r } I ( u , v ) > 0 . \end{align*}


\begin{align*} g = d r ^ 2 + e ^ { 2 \phi ( r ) } d \theta ^ 2 \end{align*}


\begin{align*} \widehat { f } ( a , z ) = R _ k ^ { n - 1 } ( - \lambda , f ) e _ { k , \lambda } ^ { n - 1 } ( z , 0 ) \end{align*}


\begin{align*} h ^ \theta ( x , u ) \le \sup _ { \rho \in \Theta _ { n - 1 } } E _ { \xi _ 1 ^ \theta ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \biggr ) \biggr ] = V \left ( \xi _ 1 ^ \theta ( u ) , n - 1 , x + u \right ) . \end{align*}


\begin{align*} U A _ j U ^ * = \begin{bmatrix} A _ { j , 1 } & 0 & \ldots & 0 \\ 0 & A _ { j , 2 } & \ddots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \ldots & 0 & A _ { j , \ell } \end{bmatrix} . \end{align*}


\begin{align*} \sum _ { r } T _ { j k r } T _ { i \ell r } = \sum _ { r } T _ { i j r } T _ { k \ell r } \end{align*}


\begin{align*} \mathbf { P } _ { x } = \left ( \begin{pmatrix} & 0 & 0 \\ & 0 & \mathbf { P } _ { x ^ { ( i ) } } \end{pmatrix} : i = 1 , \cdots , k \right ) . \end{align*}


\begin{align*} E ^ { 1 } _ { p , q } = H _ { q } ( G , C _ { p } ) \Rightarrow H _ { p + q } ( G , C _ { * } ) \end{align*}


\begin{align*} \mathbf { \Psi } = ( \mathbf { u } _ { \lambda _ { 1 1 } } ) ^ { \dag } \mathbf { u } _ { \lambda _ { 1 2 } } = ( { \mathbf { u } ^ { H } _ { \lambda _ { 1 1 } } } \mathbf { u } _ { \lambda _ { 1 1 } } ) ^ { - 1 } { \mathbf { u } ^ { H } _ { \lambda _ { 1 1 } } } \mathbf { u } _ { \lambda _ { 1 2 } } , \end{align*}


\begin{align*} \begin{cases} a _ { \alpha + n + 1 } - a _ { n + 1 } = d _ { \alpha + n } - d _ { \alpha + n + 1 } \\ b _ n = a _ { \alpha + 1 } + ( d _ { \alpha + 1 } - d _ \alpha ) = b \\ d _ { \alpha + n } = d _ \alpha + ( b _ 1 - a _ { \alpha + 1 } ) + \sum _ { j = 2 } ^ n ( a _ j - a _ { \alpha + j } ) \\ d = d _ \alpha + b - a _ { \alpha + 1 } + \sum _ { j = 2 } ^ \infty ( a _ j - a _ { \alpha + j } ) \end{cases} \ \end{align*}


\begin{align*} u _ n = a \alpha ^ n + b \beta ^ n , \end{align*}


\begin{align*} \int _ 0 ^ \infty y ^ { - r / 2 } \exp \left \{ - ( a + n ) y \right \} \mathrm { d } y = \frac { \Gamma \left ( 1 - \frac { r } { 2 } \right ) } { ( a + n ) ^ { 1 - r / 2 } } , r < 2 . \end{align*}


\begin{align*} h ( G ) & = h ( G ' ) + f ( y ) + \sum _ { j = 1 } ^ y f ( x _ j ) - \sum _ { j = 1 } ^ y f ( x _ j - 1 ) \\ & \le h ( K _ { 1 , m - y } ) + f ( y ) + y \left ( f \left ( \frac { m } { y } \right ) - f \left ( \frac { m } { y } - 1 \right ) \right ) . \end{align*}


\begin{align*} \phi _ { \gamma ^ { - 1 } \gamma _ 1 , \gamma _ 2 } = \sum _ { h \in \Gamma } \beta ^ { \gamma } _ { \gamma _ 1 , h } \phi _ { h , \gamma _ 2 } = \sum _ { h \in \Gamma } \phi _ { \gamma _ 1 , h } \beta ^ { \gamma } _ { h , \gamma _ 2 } = \phi _ { \gamma _ 1 , \gamma \gamma _ 2 } \ , , \ \ \ \end{align*}


\begin{align*} a _ N ( z ) : = \sup _ { n \in \mathbb N } \mathbb E \left [ \int _ 0 ^ T \| p _ N \alpha _ s ^ { \mathcal S _ n } \| _ { \mathcal H } ^ z d s \right ] , b _ N ( z ) : = \sup _ { n \in \mathbb N } \mathbb E \left [ \int _ 0 ^ T \| p _ N \sigma _ s ^ { \mathcal S _ n } \| _ { L _ { } ( U , H ) } ^ z d s \right ] , \end{align*}


\begin{align*} \begin{aligned} E ^ \Phi _ A ( x ) & = \Phi ( T _ 1 , x ) , \\ E ^ \Psi _ B \left ( h ( x ) \right ) & = \Psi \left ( T _ 2 , h ( x ) \right ) , \end{aligned} \end{align*}


\begin{align*} a _ 0 ( x ) = \frac { \Gamma ( n - 1 ) \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) } { ( 2 \pi ) ^ { n - 1 } } \biggl [ \biggl ( \frac { \lambda + 3 \mu } { 2 \mu ( \lambda + \mu ) } \biggr ) ^ { n - 1 } + \frac { 1 } { ( 2 \mu ) ^ { n - 1 } } + \frac { n - 2 } { \mu ^ { n - 1 } } + \frac { 1 } { \alpha ^ { n - 1 } } \biggr ] , \end{align*}


\begin{align*} \sup \limits _ i \int _ K | \tilde { F } _ i - ( 1 - b _ { t _ 0 , B } ( \Psi ) ) f F ^ { 1 + \delta } | ^ 2 _ { h _ { j _ K , 1 } } < + \infty . \end{align*}


\begin{align*} [ v _ a , v _ b ] = \varepsilon _ c L v _ c , \quad \forall \quad \mbox { c y c l i c } ( a , b , c ) \in \mathfrak { S } _ 3 . \end{align*}


\begin{align*} w = 1 + 2 \gamma _ 1 a _ 1 + 2 \gamma _ 2 a _ 2 > 0 \end{align*}


\begin{align*} D _ { \uparrow } V ( u ) ( h ) = \liminf _ { \substack { t \downarrow 0 \\ h ' \to h } } \frac { V ( u + t h ' ) - V ( u ) } { t } . \end{align*}


\begin{align*} p _ { k , 0 } = { k + 3 \choose 3 } - { k + 2 \choose 2 } - { k + 2 \choose 3 } = 0 . \end{align*}


\begin{align*} \partial _ t U + \sum _ { j = 1 } ^ d A ^ j ( U ) \frac { \partial U } { \partial x _ j } + B ( x ) V = 0 . \end{align*}


\begin{align*} d F ( X ) = \langle \langle X \cdot \varphi , \varphi \rangle \rangle , \end{align*}


\begin{align*} \nabla _ Y \mathcal { P } ( X ) = - \mathcal { P } ( B ^ * ( Y , X ) ) - B ( Y , s ( X ) ) + B ^ * ( Y , t ( X ) ) . \end{align*}


\begin{align*} c ( t , \xi ) = \sum _ { j = 1 } ^ d ( \partial _ j W ( t , \xi ) ) ^ 2 + \Delta W ( t , \xi ) - i ( \mu ( \xi ) + \tilde { \mu } ( \xi ) ) . \end{align*}


\begin{align*} v _ k ( \alpha ) = \sum _ { m = k } ^ \infty ( - 1 ) ^ { k + m } \ , \binom { \alpha } { m } \ , \binom { m } { k } \end{align*}


\begin{align*} f ( t ) = ( \alpha t ) ^ { \delta } { } _ { 0 } F _ { 1 } \left ( \ ; ; \frac { 1 } { 2 } + \delta ; \frac { 1 } { 4 } \alpha ^ { 2 } t ^ { 2 } \right ) = \left \{ \begin{array} { l l } \cosh \alpha t & \mbox { f o r } \ ; \delta = 0 \\ \sinh \alpha t & \mbox { f o r } \ ; \delta = 1 \\ \end{array} \right . \end{align*}


\begin{align*} X _ t ^ i = X _ 0 ^ i + \int _ 0 ^ t \bigg ( b ( X _ s ^ i ) + \frac 1 { N - 1 } \sum _ { j \neq i } K ( X _ s ^ i - X _ s ^ j ) \bigg ) \d s + \sigma W _ t ^ i , ~ ~ ~ ~ i = 1 , \cdots , N , \end{align*}


\begin{align*} Y \cdot ( f v ) = ( Y \cdot f ) v + f ( Y \cdot f ) . \end{align*}


\begin{align*} \sup \limits _ { X } e ^ { - u ( - v ( \Psi ) ) } \le \sup \limits _ { t \in [ T , t _ 0 + B ] } e ^ { - u ( t ) } = \frac { 1 } { \delta } c ( T ) e ^ { - T } + \int ^ { t _ 0 + B } _ T c ( t _ 1 ) e ^ { - t _ 1 } d t _ 1 . \end{align*}


\begin{align*} \frac { 1 } { \sqrt { \lambda _ t } } - 1 \geq \frac { 1 } { 2 } \alpha _ 0 \sum _ { i = 1 } ^ t \sqrt { \gamma _ i / \gamma _ 0 } \quad \Rightarrow \lambda _ t \leq 4 \left ( 2 + \alpha _ 0 \sum _ { i = 1 } ^ t \sqrt { \gamma _ i / \gamma _ 0 } \right ) ^ { - 2 } . \end{align*}


\begin{align*} I ( G ) = \{ \sigma \subset V ( G ) \ | \ \{ u , v \} \notin E ( G ) \} . \end{align*}


\begin{align*} ( x , . . . , x ) \in X ^ N = p _ X ^ { \otimes N } [ \nu _ 1 \cdot \gamma _ 1 , . . . , \nu _ N \cdot \gamma _ N ] \end{align*}


\begin{align*} \begin{aligned} 0 < 2 \sigma _ 2 ( \lambda ( D ^ 2 u ) ) = \ , & 2 \sum _ { 1 \leq i < j \leq n } ( u _ { i i } u _ { j j } - u _ { i j } ^ 2 ) \\ = \ , & ( \Delta u ) ^ 2 - \sum _ i u _ { i i } ^ 2 - \sum _ { i \neq j } u _ { i j } ^ 2 . \end{aligned} \end{align*}


\begin{align*} P ( \alpha , \alpha ^ q ) = \alpha ^ { q ( q + 1 ) } + a \alpha ^ { 2 q } + b \alpha ^ { q + 1 } = \alpha ^ q ( \alpha ^ { q ^ 2 } + a \alpha ^ { q } + b \alpha ) = 0 . \end{align*}


\begin{align*} Q _ { 1 1 } = \sum _ { k = 2 } ^ { 4 } \mathcal { Q } _ k [ \varrho ] \biggl | _ M , \end{align*}


\begin{align*} \kappa ( t ) = h _ \alpha ( t ) , \ k _ 1 ( t ) = h _ \gamma ( t ) , \ k _ 2 ( t ) = h _ { 1 - \alpha - \gamma } ( t ) , \ t > 0 \end{align*}


\begin{align*} F _ n ( \eta ) : = T _ n - \frac { 1 } { \eta } I _ n , \eta \in \Omega . \end{align*}


\begin{align*} \begin{aligned} \frac { d ( r _ n ( x ) , r _ n ( y ) ) } { d ( x , y ) } & \leq \frac { d ( x , y ) + 2 r } { d ( x , y ) } \leq 1 + \frac { 2 r } { \rho r } = 1 + \frac { 2 } { \rho } . \end{aligned} \end{align*}


\begin{align*} u _ 1 ( t , x , y ) = \sum _ { i = 1 } ^ n \partial _ { x _ i } u _ 0 ( t , x ) w _ i ( t , x , y ) , \end{align*}


\begin{align*} \widehat { \mathbf { H } } ( \varphi ) = \left \{ h _ { k + m } \right \} _ { k , m = 0 } ^ { \infty } , h _ { p } = \frac { \left ( - 1 \right ) ^ { p + 1 } } { 2 \pi } \int _ { - \infty } ^ { \infty } \frac { \varphi \left ( \tau \right ) z ^ { p + 1 } \left ( \tau \right ) d \tau } { \frac { 1 } { 4 } + \tau ^ { 2 } } . \end{align*}


\begin{align*} f _ 1 & = \mathcal { F } ( \gamma ( N _ 1 ^ { ( 5 - 2 \alpha ) + \varepsilon } t - n ) \mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ { N _ 1 } v ) , \\ f _ 2 & = \mathcal { F } ( \gamma ( N _ 1 ^ { ( 5 - 2 \alpha ) + \varepsilon } t - n ) \mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ { N _ 2 } u _ 1 ) , \\ f _ 3 & = \mathcal { F } ( \gamma ( N _ 1 ^ { ( 5 - 2 \alpha ) + \varepsilon } t - n ) \mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ N v ) . \end{align*}


\begin{align*} B ( x , x ) = \begin{pmatrix} x _ 2 x _ 3 \\ x _ 1 x _ 3 \\ - 2 x _ 1 x _ 2 \end{pmatrix} . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ d q _ i = \left ( \prod _ { i = 1 } ^ d r _ i \right ) n . \end{align*}


\begin{align*} | | \mathcal { B } _ q ( \psi _ m ) | | _ { Q M L _ q ( \mathbb { H } ) } = 1 = | | \psi _ m | | _ { L ^ 2 ( \mathbb { R , , \mathbb { H } } ) } . \end{align*}


\begin{align*} C ( G _ { ( h ) } , F ) : = C ( G _ { ( h ) } , K ) \widehat { \otimes } _ K F . \end{align*}


\begin{align*} f ( x ) = \int _ \S W _ r ( x - y ) \prod _ { i = 1 } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( y ) - \tilde \Psi ( x ) ) \ \d y \end{align*}


\begin{align*} \boldsymbol { \alpha } _ t ^ * : = ( \alpha _ t ^ { * , i } ) _ { 1 \leq i \leq N } = \left ( - R ^ { - 1 } B y _ t ^ { * , i } - h ( \mu ^ { N , i } _ { \boldsymbol { \alpha } ^ * _ t } ) \right ) _ { 1 \leq i \leq N } \end{align*}


\begin{align*} \prod _ { { \substack { p < T \\ p \nmid 2 \alpha \beta } } } | t _ l | ^ { - s } _ p = A B \end{align*}


\begin{align*} \langle G \rangle \supset \{ \mathrm { i d } , - \mathrm { i d } \} \circ \langle F _ 2 \rangle \circ ( F _ 0 \cup F _ 1 ) \supset \{ \mathrm { i d } , - \mathrm { i d } \} \circ \bigcup _ { k = 0 } ^ \infty F _ k \supset \langle G \rangle , \end{align*}


\begin{align*} \chi _ l \leqslant w _ l \chi + k _ l , ( l = 2 , \dots , n ) \end{align*}


\begin{align*} \left < \partial _ t \gamma , \nu \right > _ g = \kappa \textnormal { o n } \mathbb { R } \times ( 0 , T ) , \end{align*}


\begin{align*} B _ q ( \phi ) ( z ) & = \int _ { \mathbb { C } } f ( w ) | w | ^ { \frac { 2 } { q } - 2 } e ^ { - \frac { | w | ^ 2 } { q } } \overline { K _ q ( w , z ) } d A ( w ) \end{align*}


\begin{align*} q ( x ) = \left \{ \begin{array} [ c ] { c c } e ^ { x } \cos 4 x , & x < 0 , \\ e ^ { - x } J _ { 0 } ( 2 x ) , & x > 0 , \end{array} \right . \end{align*}


\begin{align*} \begin{cases} \mathcal { L } ( u ) = f & \ \Omega _ { T } \\ u = u _ 0 & \ \overline { \Omega } \times \{ 0 \} \\ u = \psi & \ \{ a , b \} \times [ 0 , T ] \end{cases} \end{align*}


\begin{align*} = \frac { \sqrt { \pi } \left ( \frac { 1 } { 2 } | z | \right ) ^ { \nu + \frac { 1 } { 2 } } } { \Gamma ( \nu + 1 ) } \frac { \Gamma ( 2 \nu + 1 , x ) } { x ^ { 2 \nu + 1 } } , \end{align*}


\begin{align*} \left \{ \begin{aligned} & Y ( t _ { j } ) \leq Y _ 2 , \\ & Z ( t _ j ) \leq Z _ 2 , \\ & W ( t _ j ) \leq W _ 2 . \end{aligned} \right . \end{align*}


\begin{align*} T : = T ^ e _ A ( x ) . \end{align*}


\begin{align*} \mathcal { N } _ \varepsilon = & \{ u \in \mathcal { H } ( \Omega ) \setminus \{ 0 \} \mid \langle I ' _ \varepsilon ( u ) , u \rangle = 0 \} \\ = & \left \{ u \in \mathcal { H } ( \Omega ) \setminus \{ 0 \} \mid \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x = \frac { 1 } { \varepsilon ^ 2 } \int _ { \Omega } \left ( u - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + u d x \right \} . \end{align*}