\begin{align*} \mu _ 1 = \mathcal { N } \left ( 0 , \left [ \begin{array} { c c } 2 & - 1 \\ - 1 & 4 \end{array} \right ] \right ) , \mu _ 2 = \mathcal { N } \left ( 0 , \left [ \begin{array} { c c } 4 & - 2 \\ - 2 & 4 \end{array} \right ] \right ) , \mu _ 3 = \mathcal { N } \left ( 0 , \left [ \begin{array} { c c } 2 & - 2 \\ - 2 & 4 \end{array} \right ] \right ) . \end{align*}


\begin{align*} G _ { \omega , E , \Lambda _ L ( x ) } - G _ { \omega , E _ 0 , \Lambda _ L ( x ) } = ( E _ 0 - E ) G _ { \omega , E , \Lambda _ L ( x ) } G _ { \omega , E _ 0 , \Lambda _ L ( x ) } . \end{align*}


\begin{align*} [ \alpha ^ k ] \Gamma _ F ( \alpha ) = ( - 1 ) ^ { | F | } [ \alpha ^ k ] \chi _ F ( \alpha ) . \end{align*}


\begin{align*} \widehat { \mathcal { I } } ( \delta ) & : = \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \left [ \langle \lambda , \delta \rangle + \sup _ { \varpi \in \Pi ( \widehat { \mu } _ { 1 3 } , \widehat { \mu } _ { 2 3 } ) } \int _ { \mathcal { V } } f _ { \theta , \lambda } \ , d \varpi \right ] . \end{align*}


\begin{align*} E [ S _ j ( \tau _ N ) ] = \frac { \nu N } { \lambda } \cdot \frac 1 { j ( j + 1 ) } , j = 2 , \ldots , N - 1 . \end{align*}


\begin{align*} f ^ \mathcal { K } ( \cdot , t ) = f _ 0 \circ \big ( \tilde { y } _ t ^ { \mathcal { K } } \big ) ^ { - 1 } , \end{align*}


\begin{align*} \varphi ( q _ { n * } ( \xi , \ldots , \xi , a , b ) ) = q _ { n * } ( \varphi ( \xi ) , \ldots , \varphi ( \xi ) , \varphi ( a ) , \varphi ( b ) ) \end{align*}


\begin{align*} b \ast x = s ( b ) i ( x ) , x \ast b = i ( x ) s ( b ) , \forall b \in B , \ ; \forall x \in X \end{align*}


\begin{align*} \langle \frac { \partial } { \partial \theta _ i } , \frac { \partial } { \partial r } \rangle = 0 , \ \ \langle \frac { \partial } { \partial r } , \frac { \partial } { \partial r } \rangle = 1 , \ \ \langle \frac { \partial } { \partial \theta _ i } , \frac { \partial } { \partial \theta _ j } \rangle = \phi ^ 2 ( r ) s _ { i j } . \end{align*}


\begin{align*} P U = U \Lambda , \end{align*}


\begin{align*} I I _ { \delta } = 0 . \end{align*}


\begin{align*} ( f _ i ) ^ n _ * A = A + n R _ i + \frac { n ( n - 1 ) t _ { i i } } { 2 } D _ 0 , \end{align*}


\begin{align*} Q _ r ( \theta ) = \frac { 2 r \sin ( \theta ) } { 1 - 2 r \cos ( \theta ) + r ^ 2 } \end{align*}


\begin{align*} \begin{aligned} P _ b = & \Pr ( - 2 \Re \{ y \sqrt { P _ s } \zeta \eta \} + | \sqrt { P _ s } \zeta \eta | ^ 2 \\ & > - 2 \Re \{ y \sqrt { P _ s } \zeta \hat \eta \} + | \sqrt { P _ s } \zeta \hat \eta | ^ 2 ) \\ = & \Pr \left ( 2 \Re \{ y \sqrt { P _ s } \zeta ( \hat \eta - \eta ) \} \right . \\ & + \left . | \sqrt { P _ s } \zeta \eta | ^ 2 - | \sqrt { P _ s } \zeta \hat \eta | ^ 2 > 0 \right ) . \end{aligned} \end{align*}


\begin{align*} \tau ( X ) : = \int _ { 0 } ^ { \infty } \lambda d \tau ( e _ { \lambda } ) , \end{align*}


\begin{align*} \begin{aligned} & \frac { 1 } { \binom { N } { 2 } } \log \P \left ( N ^ { - 1 } \max _ { i \in [ N ] } d _ i \leq \beta \right ) \geq \frac { 1 } { \binom { N } { 2 } } \sum _ { i = 1 } ^ N \log \P ( N ^ { - 1 } d _ i \leq \beta ) \\ & = \frac { 2 N } { N - 1 } \frac { 1 } { N } \sum _ { i \in [ N ] } \frac { 1 } { N } \log \P ( N ^ { - 1 } d _ i \leq \beta ) = \frac { 2 N } { N - 1 } \int _ { [ 0 , 1 ] } \d x \ , \frac { 1 } { N } \log \P ( N ^ { - 1 } d _ { i _ x } \leq \beta ) . \end{aligned} \end{align*}


\begin{align*} \begin{dcases} \dfrac { \partial ^ 2 u } { \partial t ^ 2 } ( t , x ) = \dfrac { \partial ^ 2 u } { \partial x ^ 2 } ( t , x ) + u ( t , x ) \dot { W } ( t , x ) , & ( t , x ) \in \R _ + \times \R , \\ u ( 0 , x ) = 1 , \ \dfrac { \partial u } { \partial t } ( 0 , x ) = 0 , & \forall x \in \R . \end{dcases} \end{align*}


\begin{align*} \delta _ i ( \xi ) = \begin{cases} 1 & \mbox { i f } \xi = i \\ 0 & \quad \mbox { o t h e r w i s e , } \end{cases} \qquad \mbox { f o r } i = 0 , 1 . \end{align*}


\begin{align*} J _ { 2 k - 1 } \left ( \frac { 4 \pi \sqrt { I L J M } } { c } \right ) = \sum _ { \ell = 0 } ^ { \infty } \frac { ( - 1 ) ^ \ell } { \ell ! \ , \ \Gamma ( \ell + 2 k ) } \left ( \frac { 2 \pi \sqrt { I J L M } } { c } \right ) ^ { 2 k + 2 \ell - 1 } . \end{align*}


\begin{align*} \lambda ^ k \in \partial I _ K ( \zeta ^ k ) \forall \ k = 2 , 3 , \dots . \end{align*}


\begin{align*} \mathfrak { c } ( \tau ) _ { \sigma } = \begin{cases} p & \tau \subseteq \sigma \\ 1 & \mbox { e l s e } . \end{cases} \end{align*}


\begin{align*} \frac { 1 } { L ^ d } \sum _ { k \in \frac { 2 \pi } { L } \Z ^ d } \frac { | k | ^ p \hat \gamma ( k ) ^ n } { ( 1 + \hat \gamma ( k ) ) ^ m } = \frac { 1 } { L ^ d } \sum _ { k \in \frac { 2 \pi } { L } \Z ^ d } \frac { | k | ^ p e ^ { - n \beta | k | ^ 2 } } { \left ( 1 + e ^ { - \beta | k | ^ 2 } \right ) ^ m } \leq C \rho _ 0 ^ { 1 + p / d } \end{align*}


\begin{align*} \mathbb { E } \left [ \rho _ 2 ( \widetilde { Y } _ 2 , y _ 2 ) \right ] - 1 \leq \mathbb { E } \left [ \boldsymbol { d } _ { \mathcal { S } _ 2 } ( \widetilde { S } _ 2 , s _ 2 ) ^ { p _ 2 } \right ] = \boldsymbol { W } _ { p _ 2 } \left ( \mu _ 2 , \delta _ { s _ 2 } \right ) ^ { p _ 2 } . \end{align*}


\begin{align*} ( y , r ) \times { \rm T r e e } _ { F , H _ { 0 } } & \to \mathbb { R } \\ x = ( y , r , H ) & \mapsto H _ { \alpha _ { x , g } \parallel \alpha _ { x , e } } \left ( \mathcal { P } _ { x , n } \right ) \end{align*}


\begin{align*} r _ n ^ { [ 1 ] } ( h ) : = \frac { 2 ^ p \cdot y _ { 2 n } \left ( \frac { h } { 2 } \right ) - y _ n ( h ) } { 2 ^ p - 1 } \end{align*}


\begin{align*} \frac { 1 } { n } \sum _ { j = 0 } ^ { n - 1 } \psi \left ( F ^ j ( \omega , x ) \right ) = \frac { 1 } { n } \sum _ { j = 0 } ^ { n - 1 } \varphi \left ( f _ { \overline { \omega | _ { [ 1 , j ] } } } ( x ) \right ) . \end{align*}


\begin{align*} ( \ast ) = w _ j C ^ \varphi _ { k j i } R ^ \varphi _ { i k } + \frac { 1 } { 2 } \langle \nabla w , \nabla | T ^ \varphi | ^ 2 \rangle + \frac { 1 } { 2 } ( 1 + w ) | C ^ \varphi | ^ 2 + w _ { t i } R ^ \varphi _ { i k , t k } . \end{align*}


\begin{align*} \check { H } ^ 1 ( \{ U \} , \mathcal F ) = H ^ 1 ( X , \mathcal F ) \rlap { . } \end{align*}


\begin{align*} I : = \int _ M J \theta \wedge d \Omega ^ { n - 1 } , \end{align*}


\begin{align*} J E _ I ( z , 0 ) = I d . \end{align*}


\begin{align*} d _ { i j } = \begin{cases} - \frac 1 2 & j = 2 i - 1 \mbox { o r } 2 i \\ 0 & . \end{cases} \end{align*}


\begin{align*} h _ { \mu } ( Z , \lambda ) = h _ { \mu } ( Y , \nu ) + \mathrm { I } \left ( \xi _ { 1 } , \mathcal { T } | X , \eta \right ) . \end{align*}


\begin{align*} X = \bigcup _ { S \in \mathcal { P } _ { m } } X _ { S } , \end{align*}


\begin{align*} { \rm I } ( \xi _ { 1 } , \xi _ { n } ) = H \left ( \mu ^ { ( n ) } \right ) - H \left ( \mu ^ { ( n - 1 ) } \right ) \end{align*}


\begin{align*} \mathcal { I } ( \delta ) = \mathcal { I } _ { \mathrm { D } } ( \delta ) & = \mathbb { E } [ Y _ 2 ] - \mathbb { E } [ Y _ 1 ] + \sqrt { \delta _ 1 } + \sqrt { \delta _ 2 } . \end{align*}


\begin{align*} \theta _ f : \mathcal { X } \ni x \mapsto ( f _ j ( x ) ) _ { j = 1 } ^ n \in \ell ^ p ( [ n ] ) , \theta _ \tau : \ell ^ p ( [ n ] ) \ni ( a _ j ) _ { j = 1 } ^ n \mapsto \sum _ { j = 1 } ^ { n } a _ j \tau _ j \in \mathcal { X } . \end{align*}


\begin{align*} \lim _ { t \uparrow \tau _ { \max } } \left \| \left ( \sigma ( t ) , \partial _ t \sigma ( t ) , \gamma ( t ) , \partial _ t \gamma ( t ) \right ) \right \| _ { \R \times \R \times H ^ 4 _ N \times H ^ 2 _ N } = \infty . \end{align*}


\begin{align*} \left \{ \begin{aligned} & D _ u \theta ( u ^ { k + 1 } ) + S ^ * [ \lambda ^ k + \beta ( S u ^ { k + 1 } - \zeta ^ { k + 1 } ) ] = 0 ; \\ & I _ K ( \zeta ) - I _ K ( \zeta ^ { k + 1 } ) - \langle \lambda ^ k + \beta ( S u ^ { k + 1 } - \zeta ^ { k + 1 } ) , \zeta - \zeta ^ { k + 1 } \rangle \geq 0 \forall \ \zeta \in \mathcal { X } . \end{aligned} \right . \end{align*}


\begin{align*} \texttt { L H S } \eqref { R 4 . 1 } \le \int _ { B - l _ 2 } ^ B ( B - z ) ^ { \alpha + \beta - 1 / 2 } \ , d z \simeq l _ 2 ^ { \alpha + \beta + 1 / 2 } = \texttt { R H S } \eqref { R 4 . 1 } . \end{align*}


\begin{align*} L i _ 3 ( - 1 ) = - \frac { 3 } { 4 } \zeta ( 3 ) , \end{align*}


\begin{align*} \overline { C } _ { X ^ { \ast } } ( h , p ) & = \nu _ 2 ( \det p , g _ h ) \overline { c } _ { X ^ { \ast } } ( ( g _ h ) ^ { \det p } , g _ p ) = \nu _ 2 ( \det p , g _ h ) = \left \{ \begin{array} { c l } ( \det p , a ) _ { \R } & \textrm { i f } c = 0 , \\ 1 & \textrm { i f } c \neq 0 . \end{array} \right . \end{align*}


\begin{align*} \sup _ { \gamma \in \bar { \mathcal { P } } } I _ { \lambda } [ \gamma ] = \sup _ { \gamma \in \bar { \mathcal { P } } } \sup _ { \pi \in \Pi \left ( \mu _ { 1 3 } , \mu _ { 2 3 } , \gamma \right ) } \int _ { \mathcal { V } \times \mathcal { S } } \phi _ \lambda ( v , s ^ \prime ) \ , d \pi ( v , s ^ \prime ) . \end{align*}


\begin{align*} \kappa = \frac { \tau } { 2 ^ { \frac { 2 0 n } { s } } \times \left ( 2 0 \sqrt { n } \right ) ^ { 1 0 n } ( c _ { n } + 1 ) } , \end{align*}


\begin{align*} { { \boldsymbol { y } } _ k } = { { \boldsymbol { \hat y } } _ { k | k - 1 } } + { { \boldsymbol { H } } _ k } { { \boldsymbol { \varepsilon } } _ { k | k - 1 } } + { { \boldsymbol { r } } _ k } + { { \boldsymbol { v } } _ k } , \end{align*}


\begin{align*} x ^ { w _ i / \tilde w _ i } - t ^ { w _ i } \varphi ( t ) = 0 . \end{align*}


\begin{align*} J ( \phi , \omega , r ; t ) = \left | \frac { \partial z } { \partial \omega } ( \phi , \omega , r ; t ) \det ( M _ 1 ) - \frac { \partial z } { \partial \phi } ( \phi , \omega , r ; t ) \det ( M _ 2 ) + \frac { \partial z } { \partial r } ( \phi , \omega , r ; t ) \det ( M _ 3 ) \right | \end{align*}


\begin{align*} \tilde { S } _ { \gamma _ i \gamma _ j } & = J _ { 2 | \alpha _ i | } \tilde { S } _ { \gamma _ j \gamma _ j } J ^ T _ { 2 | \alpha _ j | } + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} q _ \bullet ( \xi _ u ) + q _ \bullet ( \xi _ v ) = q _ \bullet ( \xi _ { u v } ) \end{align*}


\begin{align*} J _ r ^ { ( k ) } ( x , \beta ) = \theta ^ { ( k - 1 ) } ( x , \beta ) . \end{align*}


\begin{align*} \left \| \sum _ { j = 1 } ^ n a _ j \tau _ j \right \| = \left ( \sum _ { j = 1 } ^ n | a _ j | ^ p \right ) ^ \frac { 1 } { p } . \end{align*}


\begin{align*} N ( P ) = M ( P ) + E _ 1 ( P ) + E _ 2 ( P ) , \end{align*}


\begin{align*} \partial _ k ( c _ { i _ 1 , \dots , i _ k } ) = \sum _ { j = 1 } ^ k ( - 1 ) ^ { j - 1 } ( s _ j - 1 ) \ , c _ { i _ 1 , \dots , \check { i } _ j , \dots , i _ k } \end{align*}


\begin{align*} M : = \left \{ m \in \N : m \textrm { h a s a t l e a s t } \frac { a \log { N } } { | \log ( 2 \sigma - 1 ) | } \textrm { p r i m e f a c t o r s i n } P \right \} \end{align*}


\begin{align*} \Gamma \left ( s - \frac { 1 } { 2 } \right ) = \frac { 1 } { s - \frac { 1 } { 2 } } - \gamma + O \left ( s - \frac { 1 } { 2 } \right ) \end{align*}


\begin{align*} G ( x , y , t ) = g ( x , t ) . \end{align*}


\begin{align*} \langle S _ N ( \ell ) ^ 2 \rangle = \ell \sum _ { i , j = 1 } ^ N \sum _ { m \in \mathbb { Z } } \Delta \left ( \frac { x _ i - x _ j + m } { \ell } \right ) . \end{align*}


\begin{align*} \underset { t \rightarrow \infty } { \lim } \ , \frac { 1 } { r ( t ) } \log P \left ( n _ t < \gamma _ t p _ t e ^ { \beta t } \right ) = - \kappa . \end{align*}


\begin{align*} \Big \Vert \sup _ { k \leq 2 ^ { L - m ( L ) } } \Big | \sum _ { \ell = 1 } ^ k ( \widetilde U _ { \ell , L } - V _ { \ell , L } ) \Big | \Big \Vert ^ 2 _ 2 \leq 1 6 \sum _ { \ell = 1 } ^ { 2 ^ { L - m ( L ) } } \Vert U _ { \ell , L } - V _ { \ell , L } \Vert ^ 2 _ 2 \ , . \end{align*}


\begin{align*} P ^ \omega \left ( | \widehat { Z } _ s | < e ^ { - \sqrt { \beta / 2 } \ , \widehat { R } ( s ) } p _ s e ^ { \beta s } \ : \big | \ : C _ t \right ) & \leq P _ { y _ 0 } \left ( \big | Z _ s ^ { \widehat { B } _ s } \big | < e ^ { - \sqrt { \beta / 2 } \ , \widehat { R } ( s ) } p _ s e ^ { \beta s } \right ) \\ & = \exp \left [ - \sqrt { \beta / 2 } \ , \widehat { R } ( s ) ( 1 + o ( 1 ) ) \right ] . \end{align*}


\begin{align*} \lambda _ { 1 } = \lambda _ { 1 } \left ( N , \varepsilon , \Omega , T , c \right ) \geq \lambda _ { 0 } \geq 1 \end{align*}


\begin{align*} \int \otimes _ { i = 1 } ^ { n + 1 } f _ i \otimes g \ , d \tilde \rho _ { n + 1 } = \int V f _ { n + 1 } \otimes \bigl ( \otimes _ { i = 1 } ^ n f _ i \bigr ) \otimes g \ , d \rho _ n . \end{align*}


\begin{align*} K _ 1 = & \Big | f _ \epsilon ( V + \phi _ 2 ) - f _ \epsilon ( V + \phi _ 1 ) - f ^ { ' } _ \epsilon ( V ) ( \phi _ 2 - \phi _ 1 ) \Big | _ { \frac { 2 n } { n + 2 } } \\ = & \Big | \Big ( f _ \epsilon ^ { ' } ( V + \phi _ \varrho ) - f ^ { ' } _ \epsilon ( V ) \Big ) ( \phi _ 2 - \phi _ 1 ) \Big | _ { \frac { 2 n } { n + 2 } } . \end{align*}


\begin{align*} \begin{aligned} & - M z _ { i j } \leq \lambda _ { i j } \leq M z _ { i j } \\ & - M ( 1 - z _ { i j } ) \leq v _ { i j } \leq M ( 1 - z _ { i j } ) \end{aligned} \end{align*}


\begin{align*} d H _ r ( \xi , & \sigma _ 1 , \ldots , \sigma _ r , \alpha _ 1 , \ldots , \alpha _ r , \beta _ 1 , \ldots , \beta _ r ) \ , = \\ & \textbf { e } ( ( \xi - \sigma _ r ) \beta _ r ) \hat { F } ( \sigma _ r - \xi , \alpha _ r ) H _ { r - 1 } ( \xi , \sigma _ 1 , \ldots , \sigma _ { r - 1 } , \alpha _ 1 , \ldots , \alpha _ { r - 1 } , \beta _ 1 - \beta _ 2 , \ldots , \beta _ { r - 1 } - \beta _ { r } ) \ , . \end{align*}


\begin{align*} \Gamma _ { s } ^ { N , \phi } ( Y ^ N ) : = L _ N \big ( Y _ { s } ^ N ( \phi ) ^ 2 \big ) - 2 Y _ { s } ^ N ( \phi ) L _ N Y _ { s } ^ N ( \phi ) . \end{align*}


\begin{align*} \left ( s , \nu k \right ) _ { k } : = \frac { 1 } { \Gamma ( s ) } \ , \intop _ { 0 } ^ { \infty } x ^ { s - 1 } e ^ { - x } \left ( \frac { k \nu - x } { k \nu + x } \right ) ^ { k } \ , d x . \end{align*}


\begin{align*} \Delta u + f ( u ) = 0 \ \ \ \ \ D . \end{align*}


\begin{align*} C ( X , t ) = o \left ( \frac { b } { \sqrt { g } } \right ) . \end{align*}


\begin{align*} \partial _ { \mathbf { n } } \mu = 0 \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \partial \Omega \times ( 0 , T ) . \end{align*}


\begin{align*} \left [ \Sigma ^ { - 1 } \right ] _ { i , j } ^ { } = \begin{cases} \phantom { - } 1 & j = i ; \\ - 1 & j = i + 1 ; \\ \phantom { - } 0 & . \end{cases} \end{align*}


\begin{align*} u ^ { \ast } \left ( t , x , \mu \right ) = - \frac { 1 } { 2 } \Pi _ { 1 } ^ { - 1 } \left ( t \right ) \Pi _ { 3 } ^ { \top } \left ( t \right ) \left ( x - \bar { \mu } \right ) - \Pi _ { 2 } ^ { - 1 } \left ( t \right ) \Pi _ { 4 } \left ( t \right ) \bar { \mu } - \frac { 1 } { 2 } \Pi _ { 2 } ^ { - 1 } \left ( t \right ) \Pi _ { 5 } \left ( t \right ) . \end{align*}


\begin{align*} \limsup _ { k \rightarrow \infty } \frac { \log \ ! \norm { S ( k ) \varphi } _ { \sup } } { k } \ ! \leq \ ! - l \lambda _ 1 + \lambda ( D ^ l \varphi ) \lambda ( D ^ l \varphi ) \ ! = \ ! \lim _ { k \rightarrow \infty } \ ! \frac { \log \ ! \norm { Q ( k ) D ^ l \varphi } } { k } . \end{align*}


\begin{align*} A ' = \widetilde { X ^ { - 1 } } A X . \end{align*}


\begin{align*} \mathrm { I } _ { \mathrm { p } } ( 2 t ) = \sum _ { \mathrm { q } \geq 0 , p + q \geq 0 } \frac { 1 } { \mathrm { q } ! ( \mathrm { p } + \mathrm { q } ) ! } t ^ { 2 \mathrm { q } + \mathrm { p } } \end{align*}


\begin{align*} v _ h : B _ 1 \to \R ^ k , v _ h : = t _ h \widetilde { u } _ h , \end{align*}


\begin{align*} \Delta _ h v : = v ( h e _ n ) + v ( - h e _ n ) - 2 v ( 0 ) \geq 2 h _ 0 . \end{align*}


\begin{align*} { \rm I m } ( f ( \xi ) ) = 0 { \rm \ f o r \ } \xi \in L \end{align*}


\begin{align*} \texttt { P } = \sum _ { k = 0 } ^ n ( V _ k + \alpha _ k ) ( W _ { n - k } + \beta _ { n - k } ) ( 1 + \eta _ 3 ) ( 1 + \eta _ { n - k + 1 } ) \end{align*}


\begin{align*} X _ \eta \subseteq \overline { [ \eta ^ * , \eta ] } \subseteq \overline { [ \underline { \varphi } , \varphi ] } = X _ \varphi . \end{align*}


\begin{align*} ( x _ 1 , x _ 2 , x _ 3 ) = ( \cos ( \theta ) , \sin ( \theta ) , \dot \theta ) . \end{align*}


\begin{align*} h = \sum _ { n } c _ n a _ n . \end{align*}


\begin{align*} \chi _ j \Pi _ \lambda \chi _ j u = \chi _ j \Pi _ { j , \lambda } \chi _ j u = \chi _ j u _ j = 0 . \end{align*}


\begin{align*} \dfrac { 1 } { 2 } \dfrac { d } { d t } \| \vec U \| _ { P \otimes P _ { \Omega } } ^ 2 + \sum _ { j = 1 , N } \lbrack \vec W ^ T \Lambda \vec W + 2 ( \vec W ^ - ) ^ T \Sigma ( \sqrt { | \Lambda ^ - | } \vec W ^ - - R \sqrt { \Lambda ^ + } \vec W ^ + - S \vec G ) \rbrack _ j d s _ j = 0 . \end{align*}


\begin{align*} \mathbf { v } = 0 \ \ \ \ \ , \partial D . \end{align*}


\begin{align*} \Delta \check \S ^ { 3 1 2 4 } = 1 \otimes \check \S ^ { 3 1 2 4 } + \check \S ^ { 1 } \otimes \check \S ^ { 1 2 3 } + \check \S ^ { 1 } \otimes \check \S ^ { 2 1 3 } + \check \S ^ { 1 2 } \otimes \check \S ^ { 1 2 } + \check \S ^ { 2 1 } \otimes \check \S ^ { 1 2 } + \check \S ^ { 3 1 2 } \otimes \check \S ^ { 1 } + \check \S ^ { 3 1 2 4 } \otimes 1 , \end{align*}


\begin{align*} v _ 3 = 0 \Gamma _ 0 , \end{align*}


\begin{align*} M _ q ( \Gamma ) = \inf \limits _ { \rho \in \ , { \rm a d m } \ , \Gamma } \int \limits _ D \rho ^ { \ , q } ( x ) \ , d m ( x ) \ , . \end{align*}


\begin{align*} \mathcal { B } _ { W } ( X _ { 0 } ) : = \{ A \in \mathcal { B } ( X _ { 0 } ) : g . A = A \mbox { f o r a l l } g \in W \} . \end{align*}


\begin{align*} \alpha _ r : = \frac 1 C \ , \inf _ { B _ { r / 2 } ( x _ r ) } u \qquad { \mbox { a n d } } \psi ( x ) : = \alpha _ r \ , \varphi \left ( \frac { x - x _ r } { r } \right ) . \end{align*}


\begin{align*} \Omega _ 0 = \Omega \cap \{ - 1 \leq \Re \lambda \leq 1 \} , \Omega _ { \pm k } = \Omega \cap \{ k \leq \pm \Re \lambda \leq k + 1 \} , k > 0 . \end{align*}


\begin{align*} \phi _ { \sigma } : \sigma _ U ^ * M ^ { \otimes r } \stackrel { \sigma ^ * _ U \phi } { \longrightarrow } f _ \sigma ^ * \sigma ^ * L \stackrel { f _ { \sigma } ^ * ( \widehat { \sigma _ L } ) } { \longrightarrow } f _ { \sigma } ^ * L \end{align*}


\begin{align*} \mathcal S _ { \ell , A } = \sum _ { p } \sum _ \nu ( p , \nu ) ^ { \eta ( p , \nu ) } , \end{align*}


\begin{align*} \bar y _ \ast & : = \inf _ { y _ \ast \in Y } y _ \ast , \end{align*}


\begin{align*} f \left ( \frac { a } { c } - \frac { t } { i c } \right ) = \sum _ { n \ge 1 } f _ { n } n ^ { \frac { \l - 1 } { 2 } } e ^ { - 2 \pi n \frac { t } { c } } e \left ( n \frac { a } { c } \right ) . \end{align*}


\begin{align*} \left ( { \nabla } _ { X } \omega \right ) Y + h ( X , \phi Y ) = C h ( X , Y ) \end{align*}


\begin{align*} \vec { e _ i } ^ T ( T _ \lambda ^ { \vec { v } } ) ^ T \vec { v } = \lambda v _ i . \end{align*}


\begin{align*} \Lambda _ { j + 1 } ^ + = \{ \lambda \in L _ { j + 1 } ' \mid q ( \lambda ) > 0 \} \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta _ 1 , \delta _ 2 ) - \mathcal { I } _ { \mathrm { D } } ( 0 , 0 ) = \sup _ { \widetilde { \gamma } \in \Sigma _ { \mathrm { D } } ( \delta ) } \int _ { \mathcal { V } } g \ , d \widetilde { \gamma } - \mathcal { I } _ { \mathrm { D } } ( 0 , 0 ) \leq \Psi ( \delta _ 1 + \epsilon , \delta _ 2 + \epsilon ) . \end{align*}


\begin{align*} \mathring { v } ( x , t ) = \psi \left ( x , t \right ) - \phi \left ( x + \gamma _ { \mathrm { n l } } ( x , t ) \right ) . \end{align*}


\begin{align*} J _ { T _ { * } } = - 2 \pi \int _ { 0 } ^ { \infty } \left [ \exp \left ( - \frac { 1 } { T _ { * } } \left [ \left ( \frac { \sigma } { r } \right ) ^ { n } - \left ( \frac { \sigma } { r } \right ) ^ { m } \right ] \right ) \right ] \ , r ^ { 2 } d r , \end{align*}


\begin{align*} \mu = \frac { \sqrt { \pi } L E ( \hat \beta _ l ) } { 2 } , \ \ \sigma ^ 2 = \frac { L [ 8 - \pi E ^ 2 ( \hat \beta _ l ) ] } { 4 } . \end{align*}


\begin{align*} \begin{cases} \Phi ' ( t ) = A ( t ) \Phi ( t ) , \\ \Phi ( t _ 0 ) = I _ { 2 n \times 2 n } ; \end{cases} \end{align*}


\begin{align*} \mu _ m ^ i \int _ { \mathcal { A } _ m ^ i } f _ 0 ^ { ' } \Big ( U _ { \mu ^ i _ m , \xi ^ i _ m } ( x ) \Big ) \psi _ { \mu ^ i _ m , \xi ^ i _ m } ^ h ( x ) u _ m ( x ) d x = o ( 1 ) . \end{align*}


\begin{align*} R _ { I } & : = \left \{ \exp ( s ) : s \in \mathfrak { a } , \alpha _ { 1 } ( s ) \le 0 , \alpha _ { 2 } ( s ) < 0 \mbox { f o r a l l } \alpha _ { 1 } \in \Delta \mbox { a n d } \alpha _ { 2 } \in \Delta \setminus I \right \} , \\ D _ { I } & : = R _ { I } \cap A _ { I } = \left \{ \exp ( s ) : s \in \mathfrak { a } _ { I } , \alpha ( s ) < 0 \mbox { f o r a l l } \alpha \in \Delta \setminus I \right \} . \end{align*}


\begin{align*} L ( \alpha ) - L ( \alpha ^ 2 ) = \frac { 1 } { 7 } , \end{align*}


\begin{align*} \lambda _ j = \frac { j } { n } , 0 \le j \le n . \end{align*}


\begin{align*} r _ x : = \max \left \{ \frac { \lambda _ x } { 2 ^ k } : k \in \N , \ , \ , \mathcal { H } ^ 1 \left ( J \cap B _ { \lambda _ x / 2 ^ k } ( x ) \right ) \geq \eta \frac { \lambda _ x } { 2 ^ k } \right \} . \end{align*}


\begin{align*} 2 \lambda _ j ^ { \downarrow } ( | X | ) \le \lambda _ j ^ { \downarrow } \left ( \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} \right ) \end{align*}


\begin{align*} ( Y _ 1 , Y _ 2 , X ) = \gamma ^ { \eta _ 0 , \delta } , ( \widetilde { Y } _ 1 , \widetilde { X } ) = \mu _ 1 , ( \tilde { Y } _ 2 , \widetilde { X } ) = \mu _ 2 , \end{align*}


\begin{align*} H _ 1 = \Big | f _ \epsilon ( V + \phi ) - f _ \epsilon ( V ) - f ^ { ' } _ \epsilon ( V ) \phi ) \Big | _ { \frac { 2 n } { n + 2 } } = \Big | f _ \epsilon ( V + t \phi ) - f ^ { ' } _ \epsilon ( V ) \phi ) \Big | _ { \frac { 2 n } { n + 2 } } . \end{align*}


\begin{align*} B _ i - A _ i ^ T = 0 , C + C ^ T = 0 , i = 1 , 2 , . . , k \end{align*}


\begin{align*} \imath ( a _ { - l } \ldots a _ { r } ) : = a _ 0 . \end{align*}


\begin{align*} \phi ( e ^ { i \theta } ) = \lambda I - \frac { 1 } { \Lambda ( \theta ) } M ( \theta ) , \end{align*}


\begin{align*} \bar { M } _ { + , \Delta } ( t ) : = \nu \Delta \sum _ { \ell = 0 } ^ { \lfloor t / \Delta \rfloor } Z _ 0 ( \ell \Delta ) . \end{align*}


\begin{gather*} c a p \times c u p = ( a _ { 1 } b _ { 1 } + \dots + a _ { n } b _ { n } ) , c u p \times c a p ( a _ { 1 } b _ { 1 } , \dots , a _ { n } b _ { n } ) . \end{gather*}


\begin{align*} X ( t ) = x ^ i ( t ) E _ i ( t ) \longmapsto ( x ^ 1 ( t ) , \ldots , x ^ n ( t ) ) , \end{align*}


\begin{align*} w \left ( x , t \right ) = v _ { t } \left ( x , t \right ) , r \left ( x , t \right ) = q _ { t } \left ( x , t \right ) . \end{align*}


\begin{align*} X _ i ( T _ i ( u ) ) = e ^ { \rho _ i ( u ) } \end{align*}


\begin{align*} F _ { n , m } : = \{ s \in \R \ , : \ , \phi + \omega s \in E _ { n , m } \} . \end{align*}


\begin{align*} \int _ { \mathbb { S } ^ n } u \mathrm { d } A = - \int _ { \mathbb { S } ^ n } \dfrac { n } { 2 } u ^ 2 \mathrm { d } A + O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 , \end{align*}


\begin{align*} ( \gamma \ast f ) ( \mathbf { g } ) = \gamma \ast ( f ( \gamma ^ { - 1 } \mathbf { g } ) ) . \end{align*}


\begin{align*} ( a _ 1 ' ) ^ * \iota _ s ^ * \mathcal { I C } _ M = \iota _ v ^ * a _ 1 ^ * \mathcal { I C } _ M = \iota _ v ^ * \mathcal { I C } _ U = \mathcal { I C } _ { U _ v } . \end{align*}


\begin{align*} & \int _ { \Omega \setminus B ( \xi _ m , \rho ) } \bigg | f _ 0 ^ { ' } \Big ( U _ { \mu ^ i _ m , \xi ^ i _ m } ( x ) \Big ) \psi _ { \mu ^ i _ m , \xi ^ i _ m } ^ h ( x ) u _ m ( x ) \bigg | d x \\ \leq & C | \psi _ { \mu ^ i _ m , \xi ^ i _ m } ^ h | _ { \frac { 2 n } { n - 2 } } | u _ m | _ { \frac { 2 n } { n - 2 } } \Big ( \int _ { \Omega \setminus B ( \xi _ m , \rho ) } U _ { \mu ^ i _ m , \xi ^ i _ m } ^ { \frac { 2 n } { n - 2 } } d x \Big ) ^ { \frac { 2 } { n } } = O ( \mu ^ i _ m ) . \end{align*}


\begin{align*} \| f \| _ { \Lambda ^ \beta } : = & \sup _ { x \ne y } \frac { | f ( x ) - f ( y ) | } { \| x - y \| ^ \beta } < \infty . \end{align*}


\begin{align*} \rho ( t ) = ( \log t ) ^ { 2 / 3 } , t > 1 . \end{align*}


\begin{align*} F ( k , t + 1 , n ) = F ( k , t , n ) - \frac { n - k } { n } f ( k , t , n ) , \end{align*}


\begin{align*} \sigma _ { \beta _ N } ^ 2 = \frac { \hat { \beta } ^ 2 } { R _ N } \sim \frac { \hat { \beta } ^ 2 \ , \pi } { \log N } \ , , \hat \beta \in ( 0 , \infty ) \ , . \end{align*}


\begin{align*} | \mathcal { Q } _ k | = ( 2 k + 1 ) ! ! . \end{align*}


\begin{align*} \partial _ t G ( t , s ) & = \sum _ { k = 1 } ^ \infty k s ^ { k - 1 } \left ( q R _ { k + 1 } ( t ) + \varphi ( t ) R _ k ( t ) \right ) \\ & = \partial _ s \left ( \sum _ { k = 1 } ^ \infty s ^ k \left ( q R _ { k + 1 } ( t ) + \varphi ( t ) R _ k ( t ) \right ) \right ) \\ & = \partial _ s \left ( q G ( t , s ) + \varphi ( t ) s G ( t , s ) \right ) . \end{align*}


\begin{align*} \left \{ \begin{array} { c } \left [ A x - \overline { b } \right ] _ { + } = h , \\ A ^ { \prime } h = 0 _ { n } . \end{array} \right . \end{align*}


\begin{align*} \ddot { \varrho } _ t ( x ) = \frac { \kappa } { 2 } \partial _ { x x } \dot { \varrho } _ t ( x ) - \lambda \partial _ x \dot { \Delta } _ t ( x ) = \frac { \kappa } { 2 } \partial _ { x x } \dot { \varrho } _ t ( x ) - \lambda \partial _ x \left ( \frac { \kappa } { 2 } \partial _ { x x } \Delta _ t ( x ) - \lambda \partial _ x \varrho _ t ( x ) - 2 \gamma \Delta _ t ( x ) \right ) . \end{align*}


\begin{align*} \begin{cases} \mathrm { d } \nu _ t ( x ) = \alpha \left ( \Psi ( \nu _ t , \mu _ t ) ( x ) - \nu _ t ( x ) \right ) \mathrm { d } t , \\ \mathrm { d } \mu _ t ( y ) = \alpha \left ( \Phi ( \nu _ t , \mu _ t ) ( y ) - \mu _ t ( y ) \right ) \mathrm { d } t , t \geq 0 , \end{cases} \end{align*}


\begin{align*} \psi ( x ) = F ( x ) + \lambda x ^ { \frac { 1 } { \beta } - \alpha - 1 } . \end{align*}


\begin{align*} \Phi _ \xi = \phi ' + \mathcal { O } ( | \xi | ) . \end{align*}


\begin{align*} \psi ( \ldots , \zeta _ { [ - 2 ] } , \zeta _ { [ - 1 ] } , x , \delta ^ { A } ( \varphi ) , \delta ^ { A + 1 } ( \varphi ) , \ldots ) = 0 \ , . \end{align*}


\begin{align*} \| f \| _ { \dot { \mathcal F } ^ { \alpha , q } _ { p , { \rm D } } } & = \bigg \{ \sum _ { k \in \Bbb Z } \Big ( { \mathfrak R } ^ { k \alpha } \Big \| q _ { k } \Big ( \sum _ { j \in \Bbb Z } \sum _ { Q \in Q ^ j } \omega ( Q ) { \widetilde D } _ j ( x , x _ { Q } ) D _ j ( f ) ( x _ { Q } ) \Big ) \Big \| _ p \Big ) ^ q \bigg \} ^ { 1 / q } . \end{align*}


\begin{align*} \| \tilde { S } - S \| = \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} w ^ h _ l : = \begin{cases} \phi ^ i ( w _ { l - 1 } ) & \mbox { o n } B ^ i _ l , \ , i \leq h , \\ w _ { l - 1 } & \mbox { o t h e r w i s e } , \end{cases} \end{align*}


\begin{align*} F _ { j , t } = \{ M _ { j h ( t ) } \leq k j h ( t ) \} , \ : \ : F _ t = F _ { 1 , t } \cap \ldots \cap F _ { t / h ( t ) , t } . \end{align*}


\begin{align*} L ( \Phi ) + \omega Q ( \Phi ) \ge C ' ( 1 + \omega ) \| \Phi \| _ { \mathcal { H } ^ 1 } ^ 2 \ge \frac { 4 C ' C _ { \omega } ^ 2 \omega ^ 2 ( 1 + \omega ) } { C ^ 2 } = : \widetilde { B } _ { \omega } , \end{align*}


\begin{align*} X _ { \gamma , 1 } ( x ) & = 2 ^ { \alpha } g _ { \alpha , 1 } ( x ) ^ { 1 + \alpha } ( \log 2 + \log g _ { \alpha , 1 } ( x ) ) \forall x \in ( 0 , 1 ] \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( 0 ) = \sup _ { \pi \in \Pi ( \mu _ 1 , \mu _ 2 ) } \inf _ { \lambda \in \mathbb { R } ^ 2 _ + } \int g _ \lambda \ , d \pi = \inf _ { \lambda \in \mathbb { R } ^ 2 _ + } \sup _ { \pi \in \Pi ( \mu _ 1 , \mu _ 2 ) } \int g _ \lambda \ , d \pi = \sup _ { \pi \in \Pi ( \mu _ 1 , \mu _ 2 ) } \int g \ , d \pi . \end{align*}


\begin{align*} \begin{aligned} - \Delta w _ t + \left [ \frac { f _ 1 ( u _ t ) - f _ 1 ( u ) } { u _ t - u } - \lambda \right ] w _ t = \frac { f _ 2 ( u _ t ) } { u _ t } u _ t - \frac { f _ 2 ( u ) } { u } u \geq \frac { f _ 2 ( u ) } { u } w _ t \geq 0 , \end{aligned} \end{align*}


\begin{align*} \frac { \partial V _ m } { \partial \bar { c } _ j } = ( - 1 ) ^ { j - 1 } c _ { m - j } \ . \end{align*}


\begin{align*} ( R \otimes 1 _ L ) ( \alpha \otimes \varphi _ L ) \Delta ( l ) & = ( R \otimes 1 _ L ) ( \alpha \otimes \varphi _ L ) ( l _ 1 \otimes l _ 2 ) \\ & = ( R \otimes 1 _ L ) ( \alpha ( l _ 1 ) \otimes \varphi _ L ( l _ 2 ) ) \\ & = R ( \alpha ( l _ 1 ) ) \otimes \varphi _ L ( l _ 2 ) \\ & \overset { \eqref { e q 3 . 3 } } { = } \alpha ( R ( l _ 1 ) ) \otimes \varphi _ L ( l _ 2 ) \\ & = ( \alpha \otimes \varphi _ L ) ( R \otimes 1 _ L ) \Delta ( l ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} ( u ^ { k + 1 } , \zeta ^ { k + 1 } ) & = \arg \min \limits _ { ( u , \zeta ) \in \mathcal { U } \times \mathcal { X } } L _ { \beta } ( u , \zeta ; \lambda ^ k ) , \\ \lambda ^ { k + 1 } & = \lambda ^ k + \beta ( S u ^ { k + 1 } - \zeta ^ { k + 1 } ) , \end{aligned} \right . \end{align*}


\begin{align*} \nu ( g ' ) - \nu _ { j _ 0 } ( g ' ) = \beta _ { j _ 0 } - \tilde { \beta } _ { j _ 0 } < \epsilon . \end{align*}


\begin{align*} T = p _ 6 ( x ) \partial ^ 6 + \cdots + p _ 1 ( x ) \partial + p _ 0 , \end{align*}


\begin{align*} \begin{aligned} \overline { F } ( k , t + 1 , \mathbf { p } _ { n + 1 } ) = \overline { F } ( k , t , \mathbf { p } _ { n + 1 } ) + p _ { k } ^ { } f ( k , t , \mathbf { p } _ { n + 1 } ) , \end{aligned} \end{align*}


\begin{align*} \left ( T _ \lambda ^ { \vec { v } } \right ) _ { 1 , j } = x _ j , \left ( T _ \lambda ^ { \vec { v } } \right ) _ { 2 , j } = x _ { j - 1 } + x _ j \alpha _ { n - 1 } \end{align*}


\begin{align*} S _ { j , + } ^ k ( t ) = \sum _ { \ell = 0 } ^ { \lfloor t / \Delta \rfloor } W ^ k _ { \ell \Delta , t } ( j ) . \end{align*}


\begin{align*} c _ { i , j } = \begin{cases} 2 & i = j , \\ - | b _ { i , j } | & . \end{cases} \end{align*}


\begin{align*} \max _ { A \in \mathcal { P } _ { x , n } } \left | 1 - \frac { \alpha _ { x , g } ( A ) } { q _ { x , g } ^ { n } ( A ) } \right | \quad \textrm { a n d } \quad \lim _ { n \to \infty } \varepsilon _ { x } ( g , n ) = 0 . \end{align*}


\begin{align*} \alpha ( \Omega ) = \inf \left \lbrace \mathrm { V o l } \left ( \Omega \Delta \overline { B } _ { \rho } ( x ) \right ) : x \in \mathbb { N } ^ { n + 1 } ( K ) , \mathrm { V o l } ( \Omega ) = \mathrm { V o l } ( \overline { B } _ { \rho } ( x ) ) \right \rbrace , \end{align*}


\begin{align*} E = \{ \{ i + \gamma , j + ( \gamma + z ( e ) ) \} : ( i , j ; z ( e ) ) \in \hat { E } ( \hat { G } ) , \gamma \in \mathbb { Z } \} . \end{align*}


\begin{align*} \widehat { T } _ k = G _ { \frac { 1 } { n } } + G _ { \frac { 2 } { n } } + \cdots + G _ { \frac { k - 1 } { n } } , \end{align*}


\begin{align*} \phi = \phi _ { e } + \sum _ { m = q _ x } ^ { \infty } C _ { m , 0 } \Delta x ^ { m } + \sum _ { m = q _ t } ^ { \infty } C _ { 0 , m } \Delta t ^ { m } + \sum _ { m = q _ x + q _ t } ^ { \infty } \sum _ { n = q _ t } ^ { m - q _ x } C _ { m - n , n } \Delta x ^ { m - n } \Delta t ^ { n } . \end{align*}


\begin{align*} c _ { Y _ { \ell } } \left ( y _ { \ell } , y _ { \ell } ^ { \prime } \right ) = \inf _ { x _ \ell , x _ \ell ^ \prime \in \mathcal { X } } c _ \ell \left ( ( y _ \ell , x _ \ell ) , ( y ^ \prime _ \ell , x ^ \prime _ \ell ) \right ) \leq c _ \ell \left ( ( y _ \ell , x _ \ell ) , ( y ^ \prime _ \ell , x ^ \prime _ \ell ) \right ) . \end{align*}


\begin{align*} \mathrm { c } ^ { D } _ n ( x ) = \mathrm { c } ^ { D , \mathrm { S A W } } _ n ( x ) & : = \sum _ { w \in \mathcal { W } _ n ^ { S A W } ( 0 , x ) } \prod _ { j = 0 } ^ { n - 1 } D ( w _ { j } , w _ { j + 1 } ) , \end{align*}


\begin{align*} \lim _ { N \to \infty } \binom { N } { 2 } ^ { - 1 } \log \mathbb { P } ^ * _ N ( \lambda _ N / N \leq \beta ) = - \psi _ r ( \beta ) , \beta \in [ 0 , C _ r ] , \end{align*}


\begin{align*} \Delta ^ { n } _ { y } A ^ { \ast } ( x ) = n ! A ^ { \ast } ( y ) . \end{align*}


\begin{align*} \phi ( t ) = \phi _ 0 ( t ) \prod _ { k = 1 } ^ R u _ { \beta _ k , \tau _ k } ( t ) \end{align*}


\begin{align*} \sigma _ n : = \frac { 1 } { \abs { W _ n } } \sum _ { a \in W _ n } \phi ( a ) , ~ n \in \N , \end{align*}


\begin{align*} a y d ^ \ast - b y ^ \ast c ^ \ast = ( a y b ^ \ast - b y ^ \ast a ^ \ast s ^ \ast ( t ^ \ast ) ^ { - 1 } ) t ^ \ast , a y b ^ \ast - b y ^ \ast a ^ \ast \in B . \end{align*}


\begin{align*} \left ( \bigcup _ { \ell < 2 } K _ { \ell } \right ) \cap K _ { 2 } = K _ 1 \cap K _ 2 = \{ 3 \} \in \bigcup _ { \ell < 2 } 2 ^ { K _ { \ell } } = 2 ^ { K _ 1 } . \end{align*}


\begin{align*} \mathcal { M } _ { \omega , c } ^ * ( \eta ) = \begin{cases} \{ \Psi \in \mathcal { M } _ { \omega , c } \ | \ \omega Q ( \Psi ) + c P ( \Psi ) \ge \eta \} , & \ d = 1 , \\ \{ \Psi \in \mathcal { M } _ { \omega , \mathbf { c } } \ | \ \mathbf { c } \cdot \mathbf { P } ( \Psi ) \ge \eta \} , & \ d = 2 , \\ \emptyset , & \ d = 3 . \end{cases} \end{align*}


\begin{align*} ( g ^ k ) ^ * ( \xi _ { g ^ l } ) = \exp \left ( - 2 \pi i \cdot \frac { k } { r } \right ) \xi _ { g ^ l } . \end{align*}


\begin{align*} W _ p ^ p ( \lambda \mu + ( 1 - \lambda ) \nu , \delta _ b ) = \lambda W _ p ^ p ( \mu , \delta _ b ) + ( 1 - \lambda ) W _ p ^ p ( \nu , \delta _ b ) \geq \lambda W _ p ^ p ( \mu , \delta _ a ) + ( 1 - \lambda ) W _ p ^ p ( \nu , \delta _ a ) , \end{align*}


\begin{align*} d _ { H } \left ( x \right ) = \dfrac { [ \left \langle a , x \right \rangle - b ] _ { + } } { \left \Vert a \right \Vert _ { \ast } } . \end{align*}


\begin{align*} E x _ { k + 1 } & = A x _ k + B u _ k , y _ k = C x _ k + D u _ k , k \geq 0 , x _ 0 = x ^ 0 , \end{align*}


\begin{align*} F ( \lambda ) = \sum _ { i = 1 } ^ { k } a _ i \lambda ^ i = 0 . \end{align*}


\begin{align*} [ a _ r ^ * a _ r , a _ p ^ * a _ q ] = \big ( \delta ( r - q ) - \delta ( r - p ) \big ) a _ p ^ * a _ q \end{align*}


\begin{align*} H ( \widetilde { d } ) = H ( { \rm R e } ( q ) ) + H ( \widetilde { e } ) \end{align*}


\begin{align*} ( f _ 1 ) ^ { n _ 1 } _ * \cdots ( f _ k ) ^ { n _ k } _ * A = A + \sum _ i n _ i R _ i + \left ( \sum _ i \frac { n _ i ( n _ i - 1 ) } { 2 } t _ { i i } + \sum _ { i < j } n _ i n _ j t _ { i j } \right ) D _ 0 . \end{align*}


\begin{align*} h ( t ) - \alpha ( g ( t ) ) & = \sum _ { \theta = 1 } ^ d ( h _ { p \theta + p - 1 } - g _ { p \theta + p - 1 } a ^ { p \theta } ) t ^ { p \theta + p - 1 } - \sum _ { \theta = 1 } ^ d r _ { \theta } ( t ) + f _ 0 ( t ) \\ & = \sum _ { \theta = 1 } ^ d ( h _ { p \theta + p - 1 } - g _ { p \theta + p - 1 } a ^ { p \theta } ) t ^ { p \theta + p - 1 } + f _ { h , g } ( t ) , \end{align*}


\begin{align*} w _ b = \sum _ { n } c _ n a _ n . \end{align*}


\begin{align*} v \cdot u = & - \cosh ( \phi - \theta ) = \left ( \begin{array} { c } \cosh ( \phi - \theta ) \\ \sinh ( \phi - \theta ) \\ \end{array} \right ) \cdot \left ( \begin{array} { c } 1 \\ 0 \\ \end{array} \right ) , \end{align*}


\begin{align*} y ( q x ) = \sin \left ( \dfrac { 2 } { x } \right ) y ( x ) . \end{align*}


\begin{align*} \displaystyle \lim _ { m \rightarrow \infty } \dfrac { C _ { F } ( p ^ { 2 m + 1 } ) } { \beta ^ { 2 m } } \not = 0 \end{align*}


\begin{align*} F _ { Y _ 2 | X } ( y | x ) = \mathbb { P } ( Y _ 2 \leq y | X = x ) = \mathbb { P } ( Y \leq y | X = x , D = 1 ) . \end{align*}


\begin{align*} \log \left ( 1 + \frac { P _ k } { 1 + P _ { k + 1 } } \right ) = R , 1 \leq k \leq K - 1 , \end{align*}


\begin{align*} b ^ { i j , k l } ( D ^ 2 u ( x ) ) = \int _ { 0 } ^ 1 \frac { \partial } { \partial { u _ { i j } } } \bigg [ a ^ { p q , k l } ( D ^ 2 u ( x ) + t [ D ^ 2 u ( x + h _ m ) - D ^ 2 u ( x ) ] ) u _ { p q } ( x ) \bigg ] d t \end{align*}


\begin{align*} l ( \alpha ( x ) , \alpha ( y ) , \beta ( u ) ) & = \beta ( l ( x , y , u ) ) , \\ m ( \alpha ( x ) , \beta ( u ) , \alpha ( z ) ) & = \beta ( m ( x , u , z ) ) , \\ r ( \beta ( u ) , \alpha ( y ) , \alpha ( z ) ) & = \beta ( r ( u , y , z ) ) \end{align*}


\begin{align*} L ( t ) = \Phi ( t ) L ( t _ 0 ) G ( t ) . \end{align*}


\begin{align*} P ^ \omega \left ( A _ t ^ c \mid S \right ) & = \frac { P ^ \omega ( A _ t ^ c \cap S ) } { P ^ \omega ( S ) } \leq c ( \omega ) P ^ \omega ( A _ t ^ c \cap S ) \\ & \leq c ( \omega ) P ^ \omega ( A _ t ^ c \cap S _ { \alpha t } ) \to 0 , \ : \ : \ : t \to \infty . \end{align*}


\begin{align*} A _ { \varepsilon } ( t ) : = \left ( \begin{array} { c c } 0 & 1 \\ a _ { \varepsilon } ( t ) & 0 \end{array} \right ) , Q _ { \varepsilon } ( t ) : = \left ( \begin{array} { c c } 0 & 0 \\ q _ { \varepsilon } ( t ) - a _ { \varepsilon } ( t ) & 0 \end{array} \right ) , F _ { \varepsilon } ( t , \xi ) : = \left ( \begin{array} { c } 0 \\ \widehat { f } _ { \varepsilon } ( t , \xi ) \end{array} \right ) . \end{align*}


\begin{align*} F _ a = \left \{ \mu \in K ( B ) \mid \mu ( a ^ * a ) = 1 \right \} . \end{align*}


\begin{align*} { \bf M } _ { - \kappa ^ 2 } ^ n = \begin{cases} \mathbb { R } ^ n \mbox { - - t h e E u c l i d e a n s p a c e } , & \mbox { i f } \kappa = 0 , \\ \mathbb H ^ n _ { - \kappa ^ 2 } \mbox { - - t h e H y p e r b o l i c s p a c e } , & \mbox { i f } \kappa > 0 . \end{cases} \end{align*}


\begin{align*} T ( z ) : = A z ^ { n + m } + B z ^ m + C \textrm { f o r a l l } z \in \mathbb { C } , \end{align*}


\begin{align*} \tilde { \varphi } ( m , e ) = ( \varphi ( m ) , g ( m ) e ) : E \to E . \end{align*}


\begin{align*} \hbox { $ \sum _ { i = 1 } ^ \ell u _ i = 0 $ i n $ \Lambda _ x / \Lambda _ { \tilde \rho _ { j _ k } } $ } \end{align*}


\begin{align*} Y _ 1 Y _ 2 = 1 + \mathcal { O } ( h ^ \infty ) L , L \subset { \rm e l l } _ h ( Z ) \end{align*}


\begin{align*} T _ m = \sum _ { \substack { m ' \le m \\ ( k ' , m ' , i ' , n ' ) \in \mathcal { N } } } 3 \cdot 2 ^ { - k ' } < 4 2 . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c u ) ^ n = \psi ( \cdot , u ) \omega ^ n & \textnormal { i n } & \Omega , \\ u = 0 & \textnormal { i n } & \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} L _ { u v } = Q _ { u v } - A _ { u v } , Q _ { u v } = \delta _ { u v } \sum _ { h \in T _ v X } \frac { c ( v ) } { c ( h ) } , A _ { u v } = \sum _ { h \in T _ v X : \ , r _ X ( \iota _ X ( h ) ) = u } \frac { c ( v ) } { c ( h ) } . \end{align*}


\begin{align*} \Delta ( 2 , 2 n ) = \{ \varepsilon _ { i , j } \ : : \ : \{ i , j \} \in J _ n ^ 2 \} \subset \mathbb { R } ^ n . \end{align*}


\begin{align*} T _ i ( P _ i ) = 0 \ , , T _ i ( \nu ^ i + P _ i ) = e _ n \ , , G ( P _ i ) \coloneqq \nu ^ i \in \mathbb { S } ^ { n - 1 } \ , , \end{align*}


\begin{align*} \prod _ { \substack { l , m , n \geq 1 \\ l , m \leq n ; \ , \gcd ( l , m , n ) = 1 } } \left ( \frac { 1 } { 1 - x ^ l y ^ m z ^ n } \right ) ^ { \frac { l m ^ 2 } { n ^ 4 } } \end{align*}


\begin{align*} \int _ { V } | u _ { k } - u | ^ { p } d \mu = \int _ { \{ x : \rho ( x ) \geq k + 1 \} } | u | ^ p d \mu = o _ k ( 1 ) . \end{align*}


\begin{align*} \deg ( F ) = - \sum _ { 1 \le i < j \le 4 } \chi ( F _ { i j } ( c ) ) . \end{align*}


\begin{align*} \varphi ^ - ( x ) & = - P _ - \big ( \varphi ( x ) + \sum _ { ( x ) } \varphi ^ - ( x ' ) \varphi ( x '' ) \big ) \\ \varphi ^ + ( x ) & = P _ + \big ( \varphi ( x ) + \sum _ { ( x ) } \varphi ^ - ( x ' ) \varphi ( x '' ) \big ) . \end{align*}


\begin{align*} \widetilde { \widetilde { C } } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) = \widetilde { C } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) \widetilde { s } ( g _ 1 ) \widetilde { s } ( g _ 2 ) \widetilde { s } ( g _ 1 g _ 2 ) ^ { - 1 } , \end{align*}


\begin{align*} \sum \limits _ { k = 0 } ^ { + \infty } \varepsilon ^ { k - 1 } \ , \mathfrak { B } _ { k - 1 } \ + \ \sum \limits _ { k = 0 } ^ { + \infty } \varepsilon ^ { \alpha + k - 1 } \ , \mathfrak { B } _ { \alpha + k - 1 } . \end{align*}


\begin{align*} \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda d \widetilde { \pi } \geq \int _ { \mathcal { V } } g \ , d \gamma - \lambda _ 1 \boldsymbol { K } _ { 1 } ( \mu _ 1 , \gamma _ 1 ) - \lambda _ 2 \boldsymbol { K } _ { 2 } ( \mu _ 2 , \gamma _ 2 ) - \epsilon = I _ { \mathrm { D } , \lambda } [ \gamma ] - \epsilon . \end{align*}


\begin{align*} \phi = \begin{bmatrix} \phi _ 1 & \cdots & \phi _ p \end{bmatrix} ^ T , \psi = \begin{bmatrix} \psi _ 1 & \cdots & \psi _ q \end{bmatrix} ^ T . \end{align*}


\begin{align*} H f ( x ) = \int _ { \mathbb { H } ^ { n } } K ( x , y ) f ( y ) d y . \end{align*}


\begin{align*} \left \langle \begin{pmatrix} \hat { P } & \ell \end{pmatrix} ^ \top S \begin{pmatrix} \hat { P } & \ell \end{pmatrix} , F _ e \right \rangle = 0 \end{align*}


\begin{align*} \Delta ( x ) : = \prod _ { 1 \leq i < j \leq k } ( x _ i - x _ j ) \quad \Delta ( x ; y ) : = \prod _ { i = 1 } ^ k \prod _ { j = 1 } ^ { \ell } ( x _ i - y _ j ) \end{align*}


\begin{align*} [ D \psi : D \varphi ] _ t & = s ( \psi ) [ D \chi : D \varphi ] _ t , \\ [ D \widetilde { \psi } : D \widetilde { \varphi } ] _ t & = s ( \psi ) [ D \widetilde { \chi } : D \widetilde { \varphi } ] _ t , \\ [ D \widetilde { \psi } : D \tau _ M ] _ t & = s ( \psi ) [ D \widetilde { \chi } : D \tau _ M ] _ t \end{align*}


\begin{align*} \frac { \rho } { \theta } ( C _ v \theta + a ( \theta ) q ^ 2 ) _ t + \frac { \rho u } { \theta } ( C _ v \theta + a ( \theta ) q ^ 2 ) _ x + R \rho u _ x + \frac { q _ x } { \theta } = \frac { 1 } { \mu \theta } S ^ 2 . \end{align*}


\begin{align*} \nabla _ j \left ( | X | ^ 2 \right ) = 2 X ^ i \nabla _ j X _ i , \end{align*}


\begin{align*} f ( \alpha ) = f ( 0 ) + f ^ { ' } ( 0 ) \alpha + o ( \alpha ) = \left [ \sum _ { i = 1 } ^ { k } \lambda _ i ^ { \gamma - 1 } ( A ) b _ { i } \gamma \right ] \alpha + o ( \alpha ) , \end{align*}


\begin{align*} \partial _ t u = - \mathcal { L } ^ \Omega _ \varepsilon c _ \varepsilon - f ^ \prime ( c _ \varepsilon ) - \Delta c + f ^ \prime ( c ) . \end{align*}


\begin{align*} \div ( T ^ \varphi ( \nabla w , \ , \cdot \ , ) ^ \sharp ) = \alpha ( 1 + w ) | \tau ( \varphi ) | ^ 2 + ( 1 + w ) | T ^ \varphi | ^ 2 \ , . \end{align*}


\begin{align*} P ( x ) = x ^ n - \left ( \frac { \alpha ^ 2 } { m ( n + 1 ) } + \lambda \right ) \frac { x ^ { n + 1 } } { n } + c , \end{align*}


\begin{align*} [ [ x _ \mu [ { y _ 1 } _ \eta y _ 2 ] ] _ { \mu + \gamma } \varphi _ { \lambda } ( w , v ) ] = [ \varphi _ \lambda ( [ x _ \mu y _ 1 ] , y _ 2 ) _ { \lambda + \mu } [ w _ \gamma v ] ] + ( - 1 ) ^ { x y _ 1 } [ \varphi _ \lambda ( y _ 1 , [ x _ \mu y _ 2 ] ) _ { \lambda + \mu } [ w _ \gamma v ] ] , \end{align*}


\begin{align*} A _ x = \big \{ y : \ , 1 - \xi < | y - x | < 1 + \xi \big \} \ \end{align*}


\begin{align*} \| 1 + \rho d X \| _ { q } ^ { q } \le \left ( \rho \left ( 1 + \frac { 1 } { \omega } \right ) \right ) ^ q \| d X \| _ { q } ^ { q } + \| \left ( 1 + \rho d X \right ) 1 _ { \rho | d X | < \omega } \| _ { q } ^ { q } = \beta ^ q \| d X \| _ { q } ^ { q } + \| \left ( 1 + \rho d X \right ) 1 _ { \rho | d X | < \omega } \| _ { q } ^ { q } . \end{align*}


\begin{align*} c ^ { \sigma / \tau } = c ^ { \sigma / \tau } ( K ) : ( \underline { X } , \underline { A } ) ^ K \longrightarrow ( \underline { X } , \underline { A } ) ^ K \end{align*}


\begin{align*} S _ { \chi } ^ { \rm s t i f f } ( \varepsilon ^ 2 z ) = \Bigl ( I + \varepsilon ^ 2 z \bigl ( \mathcal { A } _ { 0 , \chi } ^ { \rm s t i f f } - \varepsilon ^ 2 z I \bigr ) ^ { - 1 } \Bigr ) \Pi ^ { \rm s t i f f } _ \chi = \Pi ^ { \rm s t i f f } _ \chi + \mathcal { O } ( \varepsilon ^ 2 ) , \end{align*}


\begin{align*} T ( z ) = \mathcal { O } ( h ^ { - 2 n - 2 } ) . \end{align*}


\begin{align*} \left ( f _ { \mathcal { S } } \right ) _ \lambda ( s _ 1 , s _ 2 ) = \sup _ { ( y _ 1 ^ \prime , y _ 2 ^ \prime , x ^ \prime ) \in \mathcal { S } } \left \{ f _ 1 ( y _ 1 ^ \prime ) + f _ 2 ( y _ 2 ^ \prime ) - \sum _ { 1 \leq \ell \leq 2 } \lambda _ \ell c _ \ell \left ( ( y _ \ell , x _ \ell ) , ( y ^ \prime _ \ell , x ^ \prime ) \right ) \right \} . \end{align*}


\begin{gather*} \lambda ( g _ { - s } E _ { T , c } \cap c ^ { - 1 } g _ { - t } E _ { T , c } ) = \lambda \left ( g _ { \tau } \left ( g _ { - s } E _ { T , c } \cap c ^ { - 1 } g _ { - t } E _ { T , c } \right ) \right ) \\ = \lambda ( g _ { \tau - s } E _ { T , c } \cap c ^ { - 1 } g _ { \tau - t } E _ { T , c } ) \end{gather*}


\begin{align*} \gamma = \frac { 2 ( n + 2 s ) } { n + 4 s } . \end{align*}


\begin{align*} \exp \left \{ - \frac { x ^ { 2 } } { 2 } + C \frac { x ^ { 3 } } { \sqrt { n } } \right \} = \exp \left \{ - \frac { x ^ { 2 } } { 2 } \left ( 1 - C \frac { x } { \sqrt { n } } \right ) \right \} \leq \exp \left \{ - \frac { x ^ { 2 } } { 2 ( 1 + C x / \sqrt { n } ) } \right \} . \end{align*}


\begin{gather*} \mathfrak { h } _ { 0 } ( \varphi ) = 0 , \mathfrak { h } _ { - g , 2 } ( D \varphi ) = 0 , \mathfrak { h } _ { 2 } ( D ^ 2 \varphi ) = 0 \\ \| S ( k ) D \varphi \| _ { \sup } = O ( e ^ { - \rho r ( 0 , k ) } ) c ( D \varphi ) \rho > 0 , \end{gather*}


\begin{align*} u = \prod _ { p \in { \cal P } _ 1 } p ^ { u _ p } \cdot \prod _ { p \in { \cal P } _ 4 } p ^ { u _ p } , v = \prod _ { p \in { \cal P } _ 2 } p ^ { v _ p } \cdot \prod _ { p \in { \cal P } _ 5 } p ^ { v _ p } . \end{align*}


\begin{align*} \mathcal { C } _ { i , j } = \mathbf { 1 } _ { G _ { j , l } \cap I _ { j , l } } \int _ { y _ { j - 1 } } ^ { x _ i } \Biggl | \sum _ { \substack { n \leq x _ i / t \\ P ( n ) \leq y _ { j - 1 } } } \frac { f ( n ) } { \sqrt { n } } \Biggr | ^ 2 \sum _ { \substack { t / ( 1 + 1 / V ) \leq m \leq t \\ p | m \Rightarrow p \in ( y _ { j - 1 } , y _ j ] } } \frac { V \tau _ q ( m ) } { m ^ 2 } d t \ , . \end{align*}


\begin{align*} \begin{gathered} \langle \alpha _ i , Z \rangle + \partial _ z \log \Big [ F _ i ( z ) ( z - w ^ i _ \ell ) ^ 2 \Big ] \Bigg | _ { z = w ^ i _ \ell } = 0 , \\ i = 1 , \dots , r ; \ell = 1 , \dots , \deg ( q ^ i _ + ( z ) ) . \end{gathered} \end{align*}


\begin{align*} \left ( \beta _ 0 \Gamma _ 0 + \beta _ 1 \Gamma _ 1 \right ) \vect u = 0 , \end{align*}


\begin{align*} \left | A _ t ^ { - 1 } ( x ) - \left ( x + \gamma ( x , t ) + \frac { 1 } { N } \sigma ( t ) \right ) \right | & = \left | \gamma ( A _ t ^ { - 1 } ( x ) , t ) - \gamma ( x , t ) \right | \\ & \leq \| \gamma _ x ( t ) \| _ { L ^ \infty } \left | A _ t ^ { - 1 } ( x ) - x \right | \\ & \leq \| \gamma _ x ( t ) \| _ { L ^ \infty } \left ( \left | \gamma \left ( A _ t ^ { - 1 } ( x ) , t \right ) \right | + \frac { 1 } { N } | \sigma ( t ) | \right ) , \end{align*}


\begin{align*} \sup _ { \varpi \in \Pi ( \mu _ { 1 3 } , \mu _ { 2 3 } ) } \int ( f _ { \mathcal { S } } ) _ { \lambda , \ell } \ , d \varpi = \mathbb { E } [ Y _ 2 ] - \mathbb { E } [ Y _ 1 ] + \frac { V _ { \ell } / V _ { \ell , X X } } { 4 \lambda _ \ell } , \ell \in [ 2 ] . \end{align*}


\begin{align*} \int _ { \mathrm { M } } | \nabla u | ^ 2 d \mu = \int _ { \mathrm { M } } ( \lambda - q ) u ^ 2 d \mu + \int _ { \mathrm { M } } f u d \mu . \end{align*}


\begin{align*} s _ h ( m , n ) = \sum _ { k = 1 } ^ { \infty } \left ( \gamma + \Psi ( k + 1 ) \right ) ^ m \frac { ( - 1 ) ^ { k + 1 } } { ( k + 1 ) ^ n } , \end{align*}


\begin{align*} \Gamma _ T ( X _ { q , t } ) = \omega ( \Gamma _ T ) \left ( \frac { 1 - q t | } { | 1 - q } \right ) . \end{align*}


\begin{align*} \lim _ { r \downarrow 0 } \frac { h ( r ) } { r ^ k } = 0 \ , , \end{align*}


\begin{align*} & ( \alpha \oplus \beta ) ( T u + u ) = \alpha ( T u ) + \beta ( u ) , \\ & [ T u + u , T v + v , T w + w ] _ { \rho } = [ T u , T v , T w ] + \rho ( T u , T v ) w , \end{align*}


\begin{align*} C = \begin{pmatrix} \underline { a } / r - \underline { b } / r _ 1 \\ \underline { a } / r - \underline { a } ' / r ' \end{pmatrix} . \end{align*}


\begin{align*} V ' ( \rho ) = - S ( \rho ) \end{align*}


\begin{align*} \varphi ( p ( \mathcal J ) ) _ { k + 1 + i } = \varphi ( p ( \mathcal J ) ) _ { k + i } \oplus z ^ { i + 1 } , \end{align*}


\begin{align*} P U _ t + ( A _ i U ) _ { x _ i } + A ^ T _ i U _ { x _ i } + C U + L _ C = 0 , t \geq 0 , \vec x = ( x _ 1 , x _ 2 , . . , x _ k ) \in \Omega . \end{align*}


\begin{align*} F _ { Y _ 1 | X } ( y | x ) = \mathbb { P } ( Y _ 1 \leq y | X = x ) = \mathbb { P } ( Y \leq y | X = x , D = 0 ) \end{align*}


\begin{align*} G ( 0 ) = \int _ { \mathbb { R } ^ N } | x | ^ 2 \left ( a _ 2 | \phi _ 0 | ^ 2 + a _ 1 | \psi _ 0 | ^ 2 \right ) d x \geq 0 , \end{align*}


\begin{align*} \tau _ M ( W _ M ^ + ( h ) ) = W _ M ^ + ( h ) \mbox { a n d } \tau _ M ( W ( \gamma ) ) = W ( \gamma ) . \end{align*}


\begin{align*} \binom { z } { n } = \frac { ( - 1 ) ^ n ( - z ) _ n } { n ! } . \end{align*}


\begin{align*} f ' + a [ x ; \R , W ] f = b [ x ; \R , W ] , \end{align*}


\begin{align*} \mathcal { P } _ { \textrm { o u t } } & = 1 - \mathcal { P } _ 1 - \mathcal { P } _ 3 = e ^ { - \frac { \mu ^ 2 _ { } \Gamma _ 0 } { \rho \lambda _ { } } } + \sum ^ M _ { m = 1 } \mathcal { V } ^ { ( M , m ) } I _ + ^ { ( m ) } . \end{align*}


\begin{align*} \Phi M & = M ^ { - 1 } \Phi . \\ \intertext { M u l t i p l y i n g b y $ \Phi ^ { - 1 } $ o n t h e l e f t a n d r i g h t o f b o t h s i d e s y i e l d s a n a n a l o g o u s s t a t e m e n t f o r $ \Phi ^ { - 1 } $ : } M \Phi ^ { - 1 } & = \Phi ^ { - 1 } M ^ { - 1 } . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c ) = 1 \\ a , b , c \geq 1 } } \left ( \frac { 1 } { 1 - x ^ a y ^ b z ^ c } \right ) ^ { \frac { c ^ 4 } { a ^ 2 b ^ 3 } } = \exp \left \{ L i _ 2 ( x ) L i _ 3 ( y ) L i _ { - 4 } ( z ) \right \} \end{align*}


\begin{align*} \overline { u } _ h : = \zeta ( v + m _ h ) + ( 1 - \zeta ) \left ( \varphi z _ h + ( 1 - \varphi ) u _ h \right ) \end{align*}


\begin{align*} \mathcal { D } _ s ^ { ( j ) } f _ { j , m _ j } ( x , y ) = \sum _ { k \in \mathbb { Z } ^ 2 } a _ { j , m _ j , k , s } e ^ { \pi i \lambda ^ \gamma k \cdot ( x , y ) } \textrm { f o r $ ( x , y ) \in Q _ { m _ j } $ } , \end{align*}


\begin{align*} \begin{bmatrix} \cosh ( \alpha x _ { j - 1 } ) & \sinh ( \alpha x _ { j - 1 } ) \\ \cosh ( \alpha x _ j ) & \sinh ( \alpha x _ j ) \end{bmatrix} \begin{bmatrix} a _ j \\ b _ j \end{bmatrix} = \begin{bmatrix} y _ { j - 1 } \\ y _ { j } \end{bmatrix} . \end{align*}


\begin{align*} F ( k , t , \mathbf { p } _ { n + 1 } ) \overset { } { = } \Pr ( X _ t \le k \ , \vert \ , X _ 0 = 0 , \mathbf { p } _ { n + 1 } ) , k \in S , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ n x _ i ^ { e _ i } = r , \end{align*}


\begin{align*} \varphi ( q _ { n * } ( \xi , \ldots , \xi , a , b ) ) = q _ { n * } ( \varphi ( \xi ) , \ldots , \varphi ( \xi ) , \varphi ( a ) , \varphi ( b ) ) \end{align*}


\begin{align*} \begin{dcases} L u = f ( x , u ) & , \\ u = 0 & , \end{dcases} \end{align*}


\begin{align*} \begin{pmatrix} A & X ^ * \\ X & B \end{pmatrix} \ge 0 \Longrightarrow | X | \le A \# ( V ^ * B V ) \le \frac { A + V ^ * B V } { 2 } , \end{align*}


\begin{align*} \check { A } ( t ) = \begin{cases} A ( t ) , & t \in [ 0 , \tau _ { \max } - \frac { \delta } { 2 } ] , \\ \widetilde A ( t ) , & t \in [ \tau _ { \max } - \frac { \delta } { 2 } , \min \{ \tau _ { \max } + \frac { \delta } { 2 } , T _ { \max } \} ] , \end{cases} \end{align*}


\begin{align*} \sigma _ { K , N } ^ { ( t ) } ( \theta ) : = \begin{cases} \displaystyle \frac { \sin ( t \theta \sqrt { K / N } ) } { \sin ( \theta \sqrt { K / N } ) } & \textrm { i f } \ N \pi ^ { 2 } > K \theta ^ { 2 } > 0 , \crcr t & \textrm { i f } \ K = 0 , \crcr \displaystyle \frac { \sinh ( t \theta \sqrt { - K / N } ) } { \sinh ( \theta \sqrt { - K / N } ) } & \textrm { i f } \ K < 0 . \end{cases} \end{align*}


\begin{align*} x ' _ { h , j } : = R _ h \left ( \cos \frac { 2 \pi j } { 2 ^ { h } } , \sin \frac { 2 \pi j } { 2 ^ { h } } \right ) , \mbox { f o r } j \in \left \{ 1 , \dots , 2 ^ { h } \right \} . \end{align*}


\begin{align*} \nu ( \N ) \equiv \int _ { \Lambda ^ { * } } \nu ( a _ p ^ * a _ p ) \d p = \int _ { \Lambda ^ { * } } f _ 0 ( p ) \d p < \infty \ , \end{align*}


\begin{align*} w _ t \varphi ^ a _ { t i } = - w _ { t i } \varphi ^ a _ t , \end{align*}


\begin{align*} \widetilde { T } _ \sharp & = \widetilde { T } _ l ( f _ { 1 , \sharp } , f _ { 2 , \sharp } ) , \\ \widetilde { T } _ \flat & = \widetilde { T } _ l ( f _ { 1 , \flat } , f _ 2 ) + \widetilde { T } _ l ( f _ { 1 , \sharp } , f _ { 2 , \flat } ) , \\ \widetilde { T } _ { } & = \widetilde { T } _ l ( f _ { 1 , \sharp } , f _ { 2 , } ) + \widetilde { T } _ l ( f _ { 1 , } , f _ 2 ) . \end{align*}


\begin{align*} f = \sum _ { j = 1 } ^ r b _ j \underline { a } ^ { \lambda _ j } \tilde { \textbf { Q } } ^ { \lambda _ j } = \sum _ { j = 1 } ^ r \tilde b _ j \tilde { \textbf { Q } } ^ { \lambda _ j } \end{align*}


\begin{align*} \boldsymbol { d } _ { \mathcal { V } } ( ( s _ 1 , s _ 2 ) , ( s _ 1 ^ \prime , s _ 2 ^ \prime ) ) = \boldsymbol { d } _ { \mathcal { S } _ 1 } ( s _ 1 , s _ 1 ^ \prime ) \vee \boldsymbol { d } _ { \mathcal { S } _ 2 } ( s _ 2 , s _ 2 ^ \prime ) . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ N P _ i ( x _ 1 , \ldots , x _ m ) G _ i ( x _ 1 , \ldots , x _ m ) = 0 \end{align*}


\begin{align*} \lambda \in ( U _ r ) _ \R ^ \perp \cap L _ { j + 1 } ' = ( ( U _ r ) _ \R \oplus ( L _ { r + 1 } ) _ \R ) \cap L _ { j + 1 } ' . \end{align*}


\begin{align*} D ^ * _ k = D _ { - k } \forall k \in \Lambda ^ * \ . \end{align*}


\begin{align*} \gamma = \begin{bmatrix} \alpha & x \\ \beta & y \end{bmatrix} . \end{align*}


\begin{align*} \nabla _ \xi ^ 2 H ( \xi ) \ , \xi = 0 \end{align*}


\begin{align*} \| \varphi _ s ( a ) \varphi _ t ( b ) - \varphi _ { s t } ( a b ) \| & = \| \varphi ( \lambda _ s ( a ) ) \varphi ( \lambda _ t ( b ) ) - \varphi ( \lambda _ { s t } ( a b ) ) \| \\ & = \| \varphi ( \lambda _ s ( a ) ) \varphi ( \lambda _ t ( b ) ) - \varphi ( \lambda _ { s } ( a ) \lambda _ t ( b ) ) \| < \varepsilon . \end{align*}


\begin{align*} { { \boldsymbol { x } } _ k } { } { \left ( { { \boldsymbol { W } } _ k ^ T { { \boldsymbol { \Omega } } _ k } { { \boldsymbol { W } } _ k } } \right ) ^ { - 1 } } { \boldsymbol { W } } _ k ^ { } { { \boldsymbol { \Omega } } _ k } { { \boldsymbol { d } } _ k } , \end{align*}


\begin{align*} \widehat { \mu } ( t ) = \exp \left [ - \frac { 1 } { 2 } a t ^ 2 + i b _ 0 t + \int _ { - \infty } ^ { \infty } ( e ^ { i t \lambda } - 1 ) \ , \nu ( d \lambda ) \right ] \end{align*}


\begin{align*} f ( \xi ) = ( \xi - 1 ) ^ { \delta _ 1 } ( \xi + 1 ) ^ { \delta _ { - 1 } } \kappa ( \xi ) , \end{align*}


\begin{align*} I I = \sum _ { A \in \mathcal { P } } \alpha ' ( A ) \log \frac { \beta ' ( A ) } { \beta ( A ) } \le \sum _ { A \in \mathcal { P } } \alpha ' ( A ) \log \left ( \max _ { A \in \mathcal { P } } \frac { \beta ' ( A ) } { \beta ( A ) } \right ) = \log \left ( \max _ { A \in \mathcal { P } } \frac { \beta ' ( A ) } { \beta ( A ) } \right ) . \end{align*}


\begin{align*} \langle x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 : x _ i x _ j x _ i ^ { - 1 } = x _ { - i + 2 j \bmod 5 } \rangle . \end{align*}


\begin{align*} \left ( \int \sup _ { k \leq n } \left | \sum _ { i = 1 } ^ k ( f \circ T _ { \gamma } ^ { i } ( x ) - \nu _ { \gamma } ( f ) ) - \sum _ { i = 1 } ^ k Z _ i ( x , y ) \right | ^ 2 \ ! \ ! \ ! \nu _ { \gamma } ( d x ) d y \ ! \right ) ^ { \frac 1 2 } \ ! \ ! \ ! = o ( n ^ { \max ( \gamma , 1 / 4 ) } ( \log n ) ^ { \eta } ) \ , , \end{align*}


\begin{align*} U \Lambda = \frac { 1 } { n } \left [ \mathbf { u } _ 0 , \mathbf { u } _ 1 , 2 \mathbf { u } _ 2 , \ , . . . , \ , k \mathbf { u } _ k , \ , . . . . , \ , ( n { - } 1 ) \mathbf { u } _ { n - 1 } , n \mathbf { u } _ n \right ] . \end{align*}


\begin{align*} T _ { 1 } G _ { k , Y } = \sqrt { x } \frac { d G _ { k , Y } } { d x } - \frac { k ^ 2 + m } { 2 \sqrt { x } } G _ { k , Y } - \frac { t } { \sqrt { x } } T _ 2 G _ { k , Y } . \end{align*}


\begin{align*} f _ l ( x ) = \frac { \partial ^ l } { \partial t ^ l } G _ k ( x , t ) \Big | _ { t = 0 } , \end{align*}


\begin{align*} R _ K ( X , \varepsilon , t , a , b ) = R ^ - _ K ( X , \varepsilon , t , a , b , L ) + R ^ + _ K ( X , \varepsilon , t , a , b , L ) . \end{align*}


\begin{align*} ( \Delta + \lambda - q _ 1 ) w = ( q _ 2 - q _ 1 ) v _ 2 + [ \Delta , \psi ] v \ ; \mathrm { i n } \ ; \mathrm { M } . \end{align*}


\begin{align*} u _ { t t } ( 0 ) = - H _ { \lambda } ( u _ { 0 } ) \geq \xi > 0 . \end{align*}


\begin{align*} m _ \nu ( a , b ) = \inf _ { \mathcal { T } ( a , b ) } \max _ { t \in \mathbb { R } } J _ \nu ( t \star ( u , v ) ) , m _ { r , \nu } ( a , b ) = \inf _ { \mathcal { T } _ r ( a , b ) } \max _ { t \in \mathbb { R } } J _ \nu ( t \star ( u , v ) ) . \end{align*}


\begin{align*} \sum \limits _ { k = 0 } ^ { + \infty } \sum \limits _ { p = 0 } ^ { 1 } \varepsilon ^ { p \alpha + k - 1 } \ , N _ { p \alpha + k - 1 } \left ( \frac { x } { \varepsilon } , t \right ) \end{align*}


\begin{align*} h \circ F = h ( \partial / \partial X _ 1 , \ldots , \partial / \partial X _ r ) \circ F , \end{align*}


\begin{align*} z ^ \ast = \frac { z ^ + - z ^ - } { 2 \Vert z ^ + \Vert ^ 2 } \in L _ \R , \end{align*}


\begin{align*} | \nabla X | ^ 2 = & \ , \frac { 1 } { 4 } \left ( | S | ^ 2 + | A | ^ 2 \right ) , \\ \nabla ^ i X ^ j \nabla _ j X _ i = & \ , \frac { 1 } { 4 } \left ( | S | ^ 2 - | A | ^ 2 \right ) , \\ \left \langle S , A \right \rangle = & \ 0 . \end{align*}


\begin{align*} \xi _ { F ^ \ast } ( s ) = \overline { \xi _ F ( \bar { s } ) } . \end{align*}


\begin{align*} \psi _ { \mathrm { S l a t e r } } ( x _ 1 , \ldots , x _ N ) \equiv ( 1 / \sqrt { N ! } ) \det [ e _ { k _ i } ( x _ j ) ] _ { i , j = 1 } ^ N \end{align*}


\begin{align*} A _ t ( N T ) - A _ t ( 0 ) = N T , \end{align*}


\begin{align*} I _ 0 ( f , i ) = \{ \ell \in I \ | \ \lambda _ j ( Q _ \ell ) \neq 0 \mbox { f o r s o m e } j , \ 0 \leq j \leq r \} . \end{align*}


\begin{align*} \lambda \preceq \mu \lambda _ i \leq \mu _ i i = 1 , 2 , \dots , k . \end{align*}


\begin{align*} | \nabla _ g d _ { x _ 0 } | = 1 . \end{align*}


\begin{align*} \underline { C } _ K : = \bigcap _ { s \in \underline { S } } \Gamma _ { H , H ^ * } ^ { - 1 } ( R ^ * ) ^ { - s } ( W _ { K } ^ * ) ^ c . \end{align*}


\begin{align*} \| e ^ { - t H ^ { \beta } } g \| _ { \exp L ^ p } & = \inf \left \{ \lambda > 0 : \int _ { \R ^ d } \left ( \exp \left | \frac { e ^ { - t H ^ { \beta } } g } { \lambda } \right | ^ p - 1 \right ) \ , d x \leq 1 \right \} \\ & \leq \inf \left \{ \lambda > 0 : \int _ { \R ^ d } \left ( \exp \left | \frac { { C } g } { \lambda } \right | ^ p - 1 \right ) \ , d x \leq 1 \right \} = { C } \| g \| _ { \exp L ^ { p } } . \end{align*}


\begin{align*} & ( m , \gamma ) \mapsto m \cdot \gamma \\ & ( m , \gamma ) \mapsto m \\ & ( m , \gamma ) ( m \cdot \gamma , \gamma ' ) = ( m , \gamma \gamma ' ) \\ & ( m , \gamma ) \mapsto ( m \cdot \gamma , \gamma ^ { - 1 } ) \end{align*}


\begin{align*} F ^ { 2 } = I ( F \not = \overset { - } { + } I ) \end{align*}


\begin{align*} \psi ( X ) = \sum _ { j = 1 } ^ d \pi ( S _ j ) X \overline { T _ j } = ( I \otimes S ^ * ) \left ( \sum _ { j = 1 } ^ d \pi ( S _ j ) X Z _ j ^ * \right ) ( I \otimes S ^ { - 1 * } ) . \end{align*}


\begin{align*} { { \boldsymbol { P } } _ { k | k } } = \left ( { { \boldsymbol { I } } - { { \boldsymbol { K } } _ k } { { \boldsymbol { H } } _ k } } \right ) { { \boldsymbol { P } } _ { k | k - 1 } } { \left ( { { \boldsymbol { I } } - { { \boldsymbol { K } } _ k } { { \boldsymbol { H } } _ k } } \right ) ^ { } } + { { \boldsymbol { K } } _ k } { { \boldsymbol { R } } _ k } { \boldsymbol { K } } _ k ^ T . \end{align*}


\begin{align*} w ( z ) = \Phi ( z ) - \frac { 1 } { \pi } \iint _ D \frac { f ( \zeta ) } { \zeta - z } \ , d \xi \ , d \eta , \end{align*}


\begin{align*} \varPsi _ { \lambda } ( u \otimes a ) = \sum _ { t \in T } \varPsi _ { \lambda t } ( u \otimes a ) \otimes b _ t \end{align*}


\begin{align*} g ( t ) : = \frac { d ( s _ 1 ) } { d ( s _ 2 ) } \left [ f \left ( \alpha ( s _ 2 ) + \frac { d ( s _ 2 ) } { d ( s _ 1 ) } ( t - \alpha ( s _ 1 ) ) \right ) - s _ 2 \right ] , \qquad \forall \ , t \in \R . \end{align*}


\begin{align*} g _ n ( \pmb { t } _ n , \pmb { x } _ n , r , z , t , x ) = G _ { t - t _ { n } } ( x - x _ { n } ) \times \dots \times G _ { t _ 1 - r } ( x _ 1 - z ) . \end{align*}


\begin{align*} a + b & = \begin{cases} H _ p \mbox { i f } a = b , \ , a , b \ne 0 \\ \{ a , b \} \mbox { i f } a \ne b , \ , a , b \ne 0 \\ \{ a \} \mbox { i f } b = 0 \\ \{ b \} \mbox { i f } a = 0 \end{cases} \\ a \cdot b & = k \mbox { w h e r e } 0 \le k < p \mbox { a n d } k \equiv a b \mbox { m o d p } . \end{align*}


\begin{align*} P _ { \mu , x } ^ { n } \left ( L _ { x } g ^ { - 1 } , A \right ) = P _ { \mu , g . x } ^ { n } \left ( L _ { g . x } , g . A \right ) . \end{align*}


\begin{align*} z ' _ { k , l } = \sigma \left ( s _ l ( | z _ k | - \theta _ { 0 , l } ) \right ) \sigma \left ( s _ l ( \theta _ { 1 , l } - | z _ k | ) \right ) \frac { z _ k } { | z _ k | } , \end{align*}


\begin{align*} T _ l ( f _ 1 , f _ 2 ) ( x , y ) = \int _ { \mathbb { R } } \ ! f _ 1 \left ( x + P _ 1 ( t ) , y \right ) f _ 2 \left ( x , y + P _ 2 ( t ) \right ) \psi \left ( 2 ^ l t \right ) t ^ { - 1 } \ , \mathrm { d } t . \end{align*}


\begin{align*} \tilde W ^ T \tilde \Lambda \tilde W = \begin{bmatrix} W - \epsilon \Lambda ^ { - 1 } T ^ T \tilde F \\ \ , \ - \epsilon \tilde F _ { \tau } \end{bmatrix} ^ T \begin{bmatrix} \Lambda & 0 \\ 0 & - 1 / u _ n \end{bmatrix} \begin{bmatrix} W - \epsilon \Lambda ^ { - 1 } T ^ T \tilde F \\ \ , \ \ - \epsilon \tilde F _ { \tau } \end{bmatrix} . \end{align*}


\begin{align*} \theta _ + ( x , y , \xi ( \omega ) ) & = \frac { 1 } { 2 } \big ( x ^ 2 \delta _ 0 ( \xi ( \omega ) ) + ( 1 - x ) ^ 2 \delta _ 1 ( \xi ( \omega ) ) \big ) + f ( x , y , \xi ( \omega ) ) \\ \theta _ { - } ( x , y , \xi ( \omega ) ) & = \frac { 1 } { 2 } \big ( x ^ 2 \delta _ 0 ( \xi ( \omega ) ) - ( 1 - x ) ^ 2 \delta _ 1 ( \xi ( \omega ) ) \big ) - f ( x , y , \xi ( \omega ) ) \end{align*}


\begin{align*} \max _ { \abs { s - z } = r } \abs { \log \Bigl ( \zeta ( s ) \frac { s - 1 } { s } \Bigr ) } \le \frac { 2 r } { R - r } \max _ { \abs { s - z } = R } \log \abs { \zeta ( s ) \frac { s - 1 } { s } } + \frac { R + r } { R - r } \abs { \log \Bigl ( \zeta ( z ) \frac { z - 1 } { z } \Bigr ) } . \end{align*}


\begin{align*} F : \R ^ n \to \R ; F ( x ) : = f \circ \varphi ^ { - 1 } ( x ) - C \frac { | x | ^ 2 } { 2 } , \end{align*}


\begin{align*} { \rm W F } _ h ( A ) \subset { \rm e l l } _ h ( B _ 1 ) , L \subset { \rm e l l } _ h ( A ) , L \cap { \rm W F } _ h ( B ) = \varnothing \\ q \geq 0 { \rm W F } _ h ( B _ 1 ) , { \rm I m } \ , P \lesssim - h H ^ s _ h L . \end{align*}


\begin{align*} \underset { t \to \infty } { \lim } \ : P ^ \omega ( E _ t \cap S _ t ) = 0 . \end{align*}


\begin{align*} \Theta ^ { \omega } ( x ) : = \omega ^ { - \frac { 1 } { 2 } } \Theta ( \omega ^ { - \frac { 1 } { 2 } } x ) . \end{align*}


\begin{align*} \frac { 3 5 } { 2 } \zeta ( 3 ) - \pi ^ 2 \log 2 = 3 6 L i _ 3 \left ( \frac { 1 } { 2 } \right ) - 1 8 L i _ 3 \left ( \frac { 1 } { 4 } \right ) - 4 L i _ 3 \left ( \frac { 1 } { 8 } \right ) + L i _ 3 \left ( \frac { 1 } { 6 4 } \right ) , \end{align*}


\begin{align*} \lim _ { t \to 0 } \sup \left \{ d ( x ) : x \in M _ 1 ( t ) \cup M _ 2 ( t ) \right \} = 0 . \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle G ( t ) = \int _ { \mathbb { R } ^ N } | x | ^ 2 \left ( a _ 2 | \phi | ^ 2 + a _ 1 | \psi | ^ 2 \right ) d x , \end{array} \right . \end{align*}


\begin{align*} \begin{array} { r c c } \mathrm { d } x ^ { 1 } \\ \mathrm { d } x ^ { 2 } \\ \mathrm { d } x ^ { 3 } \\ \mathrm { d } \delta ( \varphi ^ { 1 } ) & = & \mathrm { d } x ^ { 1 } + \mathrm { d } u ^ { 1 } \end{array} \end{align*}


\begin{align*} \Phi ( x , y ) = \sum _ { d \mid P ( y ) } \mu ( d ) \left \lfloor \frac { x } { d } \right \rfloor , \end{align*}


\begin{align*} \begin{cases} \partial _ t m = \nabla \cdot \left ( D ( m ) \nabla m \right ) - \chi \nabla \cdot \left ( f ( m ) \nabla c \right ) + M ( m ) , \\ \tau \partial _ t c = \alpha \Delta c + \lambda d - c + \beta m , \\ \partial _ t d = r h ( m ) \left ( 1 - d \right ) , \end{cases} \end{align*}


\begin{align*} \sum _ { \{ j , j ^ * \} \in M } \lambda _ j \ , \lambda _ { j ^ * } a _ { j , j ^ * } = 0 . \end{align*}


\begin{align*} \mathfrak S ( h ) = \{ \mathcal G _ k ( h ) \in [ L _ k - 1 , U _ k + 1 ] , \forall n _ 0 < k \leq n _ \mathcal L \} , h \in [ - 1 , 1 ] . \end{align*}


\begin{align*} & \frac { 2 ^ \beta ( y - \tfrac 1 2 ) ^ { \beta } } { b _ 1 } \biggl [ \frac { \beta - 1 } { \beta 2 ^ \beta ( y - \tfrac 1 2 ) ^ \beta } + \frac { b _ 1 } { y \beta 2 ^ { \beta } ( y - \tfrac 1 2 ) ^ { \beta - 1 } } \biggr ] \\ & = \frac { 1 } { b _ 1 } ( 1 - \frac { 1 } { \beta } ) + \frac { 1 } { \beta } - \frac { 1 } { 2 y \beta } \le 1 , \end{align*}


\begin{align*} S = \left \{ ( X _ i , \underline { X _ i } ) ^ + : \ 0 \leq i \leq k \right \} \cup \left \{ ( X _ i , \underline { X _ i } ) ^ - : \ 0 \leq i \leq k \right \} \end{align*}


\begin{align*} W = B A = B _ { I } A _ { I } + B _ { I ^ { c } } A _ { I ^ { c } } . \end{align*}


\begin{align*} B : = 4 n \sup _ { \bar \Omega \times I } \vert \nabla \psi ^ { 1 / n } \vert + 1 , \end{align*}


\begin{align*} | \Gamma ( x , y ) | \le c \frac { 1 } { d ( x , y ) ^ { q - 2 } } , | X _ j \Gamma ( x , y ) | \le c \frac { 1 } { d ( x , y ) ^ { q - 2 + d e g ( X _ j ) } } , \\ | X _ i X _ j \Gamma ( x , y ) | \le c \frac { 1 } { d ( x , y ) ^ { q - 2 + d e g ( X _ j ) + d e g ( X _ i ) } } \end{align*}


\begin{align*} J _ { r } ( g _ { r } ) = \int \limits _ { { \cal { X } } } \ldots \int \limits _ { { \cal { X } } } g _ { r } ( x _ 1 , \ldots , x _ r ) W ( d x _ 1 ) \ldots W ( d x _ r ) , \end{align*}


\begin{align*} \frac { x ' } { y ' } = \frac { \alpha ( \alpha x - y ) - x } { \alpha x - y } = \alpha - \frac { x } { \alpha x - y } = \alpha - \frac { x / y } { \alpha ( x / y ) - 1 } . \end{align*}


\begin{align*} \mathcal { F } ( \mathcal { S } ; \mu _ 1 , \mu _ 2 ) = \left \{ \pi \in \mathcal { P } ( \mathcal { X } _ 1 \times \mathcal { X } _ 2 ) : \pi \circ \operatorname { p r o j } _ { \{ j \} } ^ { - 1 } = \mu _ j , \ \forall j = 1 , 2 \right \} . \end{align*}


\begin{align*} \beta S ^ * r ^ { k + 1 } = S ^ * ( \lambda ^ { k + 1 } - \lambda ^ k ) = S ^ * \lambda ^ { k + 1 } - S ^ * \lambda ^ k = D _ u \theta ( u ^ { k } ) - D _ u \theta ( u ^ { k + 1 } ) \end{align*}


\begin{align*} \tilde { \delta } = ( d _ 1 , d _ l , d _ { l - 1 } , \dots , d _ 3 , d _ 2 ) . \end{align*}


\begin{align*} X = Y _ \mathbb { Q } Y _ f ^ { - 1 } , \overline { \lambda } _ g = Y _ \mathbb { Q } ^ \vee \overline { \lambda } _ 0 Y _ \mathbb { Q } , \overline { \lambda } _ g = Y _ q ^ \vee \overline { \lambda } _ 0 Y _ q , \ , \forall q . \end{align*}


\begin{align*} \bigcup _ { \substack { \omega \ge 1 \\ 1 \le k \le n - 1 } } { \rm G r } _ { k \omega } ( \overline T _ { n , \omega } ) = \bigsqcup _ { i = 0 } ^ { n - 1 } \widehat { S L } _ n / P _ i . \end{align*}


\begin{align*} \| \partial _ k \varphi _ 1 \| _ { H ^ m } & = \| \partial _ k ( - \alpha \Delta + 2 \omega + i \mathbf { c } \cdot \nabla ) ^ { - 1 } \{ ( \nabla \cdot \varphi _ 3 ) \varphi _ 2 \} \| _ { H ^ m } \lesssim \| ( \nabla \cdot \varphi _ 3 ) \varphi _ 2 \| _ { H ^ m } \lesssim \| \varphi _ 3 \| _ { H ^ { m + 1 } } \| \varphi _ 2 \| _ { H ^ m } < \infty \end{align*}


\begin{align*} \langle V ^ 1 \rangle _ { K , \sigma } & = \frac { \sigma ^ 2 } { ( 2 \pi ) ^ 5 } \sum _ { K _ 1 , K _ 2 , K _ 3 } \eta _ { K _ 1 } \overline { \eta _ { K _ 2 } } \eta _ { K _ 3 } \int _ 0 ^ t \frac { \pi } { z ( s ) } | \beta _ h ( s ) | ^ 2 \beta _ h ( s ) \overline { \beta _ { \sigma } ( s ) } e ^ { \gamma ( s ) + \frac { \zeta ( s ) \cdot \zeta ( s ) } { 4 z ( s ) } } \dd s . \end{align*}


\begin{align*} \deg ( q ) \leq \deg ( f ) \mbox { a n d } \nu ( f ) = \nu _ q ( f ) . \end{align*}


\begin{align*} \nu _ F ( d \lambda ) = \sum _ { \gamma \not = 0 } \frac { m _ \gamma } { \gamma ^ 2 } \ , \delta _ { - \gamma } ( d \lambda ) , \end{align*}


\begin{align*} h ( t ) - \alpha ( g ( t ) ) = \sum _ { \theta = 1 } ^ { d } ( h _ { p \theta + p - 1 } - g _ { p \theta + p - 1 } a ^ { p \theta } ) t ^ { p \theta + p - 1 } + f _ { h , g } ( t ) , \end{align*}


\begin{align*} \int _ { \partial B _ 1 } \frac { ( \rho ^ 2 - \rho _ 0 ^ 2 ) ^ s } { | \rho e - \rho _ 0 \omega | ^ { n } } \ , d H ^ { n - 1 } _ \omega \ , G ( \rho _ 0 e , 0 ) = \rho ^ { 2 s - n } \int _ { \partial B _ 1 } G ( \rho _ 0 e , \rho _ 0 \omega ) \ , d H ^ { n - 1 } _ \omega . \end{align*}


\begin{align*} \frac { N _ m } { N } = 1 + O ( 1 / m ) , \end{align*}


\begin{align*} u _ { \alpha \bar \beta } ( 0 ) = - u _ { x _ n } ( 0 ) \rho _ { \alpha \bar \beta } ( 0 ) , ~ ~ \alpha , \beta \leq n - 1 . \end{align*}


\begin{align*} - \Delta \psi = f _ 0 ^ { ' } ( U _ { \mu , \xi } ) \psi \quad \mbox { i n } \ \mathbb { R } ^ n , \end{align*}


\begin{align*} \hat { D } _ \epsilon ( z ) = \begin{cases} D _ \epsilon ( 0 ) & z < 0 , \\ D _ \epsilon ( z ) & 0 \leq z \leq K + 1 , \\ D _ \epsilon ( K + 1 ) & z > K + 1 , \end{cases} \end{align*}


\begin{align*} \phi \left ( \bigotimes \limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \otimes X _ m E _ { j _ { m } j _ { m } } X _ m ^ * \right ) = U _ { X _ m } ( E _ { j _ 1 j _ 1 } \otimes \cdots \otimes E _ { j _ m j _ m } ) V _ { X _ m } ^ * \end{align*}


\begin{align*} Y _ k = x ^ * ( X _ k ) \textrm { a n d } T _ k = \sum \limits _ { i = 1 } ^ k Y _ i . \end{align*}


\begin{align*} \mu _ n = \sup _ { M \in \mathcal { G } _ n } \inf _ { u \in M } F ( u ) . \end{align*}


\begin{align*} P ( x _ 1 ^ k x _ 2 , x _ 3 , \dots , x _ n ) = 0 , \end{align*}


\begin{align*} p ( \theta ) : = \mbox { l c m } \{ p ( a , b ) : a , b \mbox { a r e a p e r i o d i c p a i r } \} , \end{align*}


\begin{align*} \sigma _ { \mathrm { e s s } } ( S ) = \sigma ( S ) \setminus \sigma _ d ( S ) . \end{align*}


\begin{align*} k _ j = \frac { \epsilon M } { 2 ^ j } j \geq 1 \end{align*}


\begin{align*} D ^ { k - 1 } ( f ) = D ^ { k - 1 } ( h ) + d g . \end{align*}


\begin{align*} \mathcal { L } ( v ) = \sum _ { i = 1 } ^ n c p _ i \max \{ - x _ i - y _ i , - d _ i \} + \sum _ { i \neq j } \alpha _ { i j } v _ { i j } - \langle \lambda , v \rangle + \sum _ { i \in I } \gamma _ i ( \sum _ { j \neq i } v _ { i j } - x _ { 0 i } ) . \end{align*}


\begin{align*} \sum _ { k = m } ^ { J - 1 } 2 ^ k \delta _ k < \frac { 1 } { 3 } \frac { \varepsilon _ m } { m ! } , 0 \le m < M . \end{align*}


\begin{align*} \tilde { S } = \begin{pmatrix} \tilde { W } & \tilde { X } \\ \tilde { Y } & \tilde { Z } \end{pmatrix} , \end{align*}


\begin{align*} V _ k ^ 1 ( t ) = & \int _ 0 ^ t R _ k ( s , \varphi , \varphi , \varphi ) \dd s , \\ V _ k ^ 2 ( t ) = & 2 \int _ 0 ^ t \int _ { 0 } ^ { s } R _ k ( s , \varphi , \varphi , R ( s ' , \varphi , \varphi , \varphi ) ) \dd s ' \dd s + \int _ 0 ^ t \int _ { 0 } ^ { s } R _ k ( s , \varphi , R ( s ' , \varphi , \varphi , \varphi ) , \varphi ) \dd s ' \dd s . \end{align*}


\begin{align*} \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda \ , d \pi = \int _ { \mathcal { V } } g \ , d \gamma - \int _ { \mathcal { V } \times \mathcal { V } } \left ( \lambda _ 1 c _ 1 + \lambda _ 2 c _ 2 \right ) \ , d \pi < \infty , \end{align*}


\begin{align*} x _ p = x ^ 0 _ p + \frac { m } { \gcd ( n , m ) } u _ p , y _ p = y ^ 0 _ p + \frac { n } { \gcd ( n , m ) } u _ p u _ p \geq 0 . \end{align*}


\begin{align*} \mathcal { O } _ i = \left \{ \left ( \mathcal { B } , \mathcal { B } ^ \prime \right ) \in U _ { k } \times U _ k : \ \mbox { t h e r e e x i s t s $ B \in \mathcal { B } , \ B ^ \prime \in \mathcal { B } ^ \prime $ s u c h t h a t $ | B \cap B ^ \prime | = k - i $ } \right \} , \end{align*}


\begin{align*} P U _ t + ( A _ i ( V ) U ) _ { x _ i } + A ^ T _ i ( V ) U _ { x _ i } + C ( V ) U + L _ C ( U ) = 0 , t \geq 0 , \vec x = ( x _ 1 , x _ 2 , . . , x _ k ) \in \Omega . \end{align*}


\begin{align*} \Pi _ { \psi } ( g ) f = \theta _ { L , X ^ { \ast } } [ \Pi _ { \psi } ( g ) ( f ' ) ] = \theta _ { L , X ^ { \ast } } [ \Pi _ { \psi } ( g ) \theta _ { X ^ { \ast } , L } ( f ) ] . \end{align*}


\begin{align*} | K _ \alpha ( x , y ) | & \lesssim \int _ 0 ^ \infty \frac 1 { V ( x , y , \sqrt t ) } \frac t { \| x - y \| ^ 2 } e ^ { - c d ( x , y ) ^ 2 / t } \frac { d t } { t ^ { 1 - \alpha / 2 } } \\ & \leqslant \frac 1 { \| x - y \| ^ 2 } \bigg ( \int _ 0 ^ { d ( x , y ) ^ 2 } + \int _ { d ( x , y ) ^ 2 } ^ \infty \bigg ) \frac 1 { V ( x , y , \sqrt t ) } e ^ { - c d ( x , y ) ^ 2 / t } t ^ \alpha d t \\ & = : I _ 1 + I _ 2 . \end{align*}


\begin{align*} \int _ { \Omega } \partial _ t u u \ ; x = \frac { } { t } \frac { 1 } { 2 } \| u \| _ { L ^ 2 ( \Omega ) } ^ 2 . \end{align*}


\begin{align*} \left ( \mathcal { H } _ { \hbar , V } + I \right ) ^ { - 1 } = \left ( \mathcal { H } _ { \hbar , V } + I \right ) ^ { - 1 } \left ( - \hbar ^ { - 2 } \mathcal { L } _ { \hbar } \right ) \left ( V + I \right ) ^ { - 1 } + \left ( V + I \right ) ^ { - 1 } . \end{align*}


\begin{align*} F ^ * ( \sigma , V ) : = ( \sigma \circ F , V ) . \end{align*}


\begin{align*} R = \partial ^ \alpha ( a \nabla \eta ) - \partial ^ \alpha a \nabla \eta - a \partial ^ \alpha \nabla \eta . \end{align*}


\begin{align*} v _ \lambda ( t ) = v _ { 0 , \lambda } ( t ) + \int _ 0 ^ t S ( t - s ) f ' _ \lambda ( u _ \lambda ( s ) ) v _ \lambda ( s ) \ , d s + \int _ 0 ^ t S ( t - s ) \sigma ' ( u _ \lambda ( s ) ) v _ \lambda ( s ) B \ , d W ( s ) . \end{align*}


\begin{align*} a ^ { \sigma } ( z ) = \sum _ { r \in \Z } a ^ { ( r ) } \left ( \frac { r } { T } \right ) z ^ { - \frac { r } { T } - 1 } , \end{align*}


\begin{align*} B _ \ell = \{ ( ( s _ 1 , s _ 2 ) , ( y _ 1 , y _ 2 , x ) ) \in \mathcal { V } \times \mathcal { S } : c _ \ell ( s _ \ell , ( y _ \ell , x ) ) < \infty \} , \end{align*}


\begin{align*} \sigma \left ( \overline { b } \right ) = \left \{ \left \langle a _ { i } , x \right \rangle \leq \overline { b } _ { i } , i = 1 , . . . , m \right \} \end{align*}


\begin{align*} d \widetilde { v } _ \lambda + A \widetilde { v } _ \lambda \ , d t = f ' _ \lambda ( u _ \lambda ) \bigl ( \widetilde { v } _ \lambda + v _ { 0 , \lambda } \bigr ) \ , d t + \sigma ' ( u _ \lambda ) \bigl ( \widetilde { v } _ \lambda + v _ { 0 , \lambda } \bigr ) B \ , d W \end{align*}


\begin{align*} \Phi ( x , y ) = \Phi \left ( x , x ^ { 1 / 3 } \right ) + \sum _ { y < p \le x ^ { 1 / 3 } } \Phi ( x / p , p ^ - ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r l l l } v _ t + \mathcal { L } ^ { \Phi } _ { A _ { 2 } ( t ) } v & = 0 \quad \mbox { i n } Q _ 2 , \\ v & = w \quad \mbox { i n } ( \mathbb { R } ^ n \setminus B _ 2 ) \times \Lambda _ 2 \cup B _ 2 \times \{ - 2 ^ { 2 s } \} . \end{array} \right . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { M } } ( \mu _ { 1 3 } , \mu _ { 2 3 } ) : = \sup _ { \gamma \in \mathcal { F } ( \mu _ { 1 3 } , \mu _ { 2 3 } ) } \int _ { \mathcal { S } } f \ , d \gamma . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c u ) ^ n = ( - \lambda _ 1 u ) ^ n f ^ n \omega ^ n & \textnormal { o n } & \Omega \\ u = 0 & \textnormal { i n } & \partial \Omega . \\ \end{array} \right . \end{align*}


\begin{align*} \left \{ \begin{aligned} - d \hat { y } ^ n _ t & = g \big ( x _ t , \hat { y } ^ n _ t , \hat { z } ^ n _ t , \hat { \tilde { z } } ^ n _ t , \hat { \gamma } ^ n _ { ( t , e ) } \big ) d t - \hat { z } ^ n _ t d W _ t - \hat { \tilde { z } } ^ n _ t d \xi _ t - \int _ { \mathcal { E } } \hat { \gamma } ^ n _ { ( t , e ) } \tilde { N } ( d e , d t ) , \ t \in [ 0 , n ] , \\ \hat { y } _ n ^ n & = \zeta _ n : = \mathbb { E } [ \zeta | \mathcal { F } _ n ] , \end{aligned} \right . \end{align*}


\begin{align*} \begin{cases} \mathcal { K } \to \R ^ V \times \R ^ V \\ ( \rho , t ) \mapsto \bigg ( - \frac { 1 } { 2 } - e ^ { \rho _ i } \Big ( W K _ t ^ { - 1 } ( e ^ \rho + t \eta ) + \eta \Big ) _ i \ , , \ , e ^ { 2 \rho _ i } \bigg ) _ { i \in V } \end{cases} \end{align*}


\begin{align*} \Psi ( X ) = J ^ * ( \oplus ^ 6 X ) J \end{align*}


\begin{align*} \widetilde { X } = \begin{pmatrix} a ^ 2 & b ^ 2 & a b & a b \\ c ^ 2 & d ^ 2 & c d & c d \\ a c & b d & a d & b c \\ a c & b d & b c & a d \end{pmatrix} , \end{align*}


\begin{align*} \eta _ { p } ( s ) : = \sum _ { k = 1 } ^ { \infty } \frac { ( s , 2 \pi p k ) _ { k } } { k ^ { s } } = \left ( 2 ^ { 1 - s } - 1 \right ) \zeta ( s ) + \sum _ { k = 1 } ^ { \infty } e ^ { 2 \pi p k } \sum _ { \ell = 1 } ^ { k } \left ( \begin{array} { c } k \\ \ell \end{array} \right ) \left ( 4 \pi p \right ) ^ { \ell } ( - 1 ) ^ { k - \ell } \ , \intop _ { k } ^ { \infty } t ^ { - s } e ^ { - 2 \pi p t } \ , \frac { ( t - k ) ^ { \ell - 1 } } { ( \ell - 1 ) ! } \ , d t , \end{align*}


\begin{align*} N _ k & = 1 1 \leq k \leq k _ 2 \\ N _ k & = 2 k _ 2 < k \leq k _ 3 \\ N _ k & = 3 k _ 3 < k \leq k _ 4 \\ \ldots & \ldots . \end{align*}


\begin{align*} \mathcal { J } _ { N } ( x , s ) : = \intop _ { 0 } ^ { \infty } \ , \frac { y ^ { s - \frac { 1 } { 2 } } \left \{ J _ { s - \frac { 1 } { 2 } } ( 2 \pi x y ) - \sum _ { k = 0 } ^ { N } \frac { \left ( - 1 \right ) ^ { k } \left ( \pi x y \right ) ^ { s + 2 k - \frac { 1 } { 2 } } } { k ! \Gamma \left ( s + k + \frac { 1 } { 2 } \right ) } \right \} } { \sigma \left ( y \right ) e ^ { 2 \pi y } - 1 } \ , d y \end{align*}


\begin{align*} Z = Z ^ { ( u , v ) } : t \mapsto \Big ( X _ i \big ( ( T _ i ( u ) + t ) \wedge T _ i ( v ) \big ) \Big ) _ { i \in V } . \end{align*}


\begin{align*} \mathfrak { G } = \{ \mathcal { P } \subset \mathcal { O } : \mathcal { P } \mathcal { P } \mid r \mathcal { P } \mid 2 n \sqrt { { - m } } \} . \end{align*}


\begin{align*} \lambda _ { 1 + 2 j } \left ( \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} \right ) & \le \inf _ { \dim { \mathcal { S } } = n - j } \lambda _ { 1 } \left ( A _ { \mathcal { S } } + B _ { \mathcal { S } } \right ) + \delta _ 2 ( X ) \\ & = \lambda _ { 1 + j } ( A + B ) + \delta _ 2 ( X ) \end{align*}


\begin{align*} 2 L ( \Phi ) + \left ( \frac { d } { 2 } + 1 \right ) N ( \Phi ) + \mathbf { c } \cdot \mathbf { P } ( \Phi ) = 0 . \end{align*}


\begin{align*} P _ R ( x _ 1 , \ldots , x _ m , n ) g _ { n + R } + \cdots + P _ 0 ( x _ 1 , \ldots , x _ m , n ) g _ n = 0 \end{align*}


\begin{align*} 2 { \rm R e } ( e ^ { i ( d ( \theta , f ( \theta ) ) + i H d ( \theta , f ( \theta ) ) ) } \frac { \partial g } { \partial \theta } ( \theta ) ) = - \rho _ { \theta } ( \theta , f ( \theta ) ) e ^ { - c ( \theta , f ( \theta ) ) - H d ( \theta , f ( \theta ) ) } . \end{align*}


\begin{align*} \left ( Z _ i ( u ) , \frac { 1 } { T _ i ( v ) } \right ) _ { u \geq 0 , v \geq 0 , i \in V } = \left ( \theta _ i Z ^ * _ i ( u ) , \frac { 1 } { T _ i ^ { \infty } } + \frac { 1 } { \theta _ i ^ 2 T _ i ^ * ( v ) } \right ) _ { u \geq 0 , v \geq 0 , i \in V } . \end{align*}


\begin{align*} T ^ 0 = \{ ( p , q , r ) \in X \times X \times X \setminus \Delta _ T \ , | \ , X \cdot H ^ p _ { q , r } = 4 p + q + r \in Z _ 0 ( X ) \} \end{align*}


\begin{align*} ( u _ { \kappa } ) _ \mu ^ * ( t ) = u _ \mu ^ * ( \kappa ^ D t ) \end{align*}


\begin{align*} U _ k & = \frac { y } { 1 0 } + \alpha ( k - n _ 0 ) - 1 0 \mathcal { S } _ { \mathcal { L } } ( x \mapsto \log ( x ) ) ( k ) , \\ L _ k & = - 1 0 y + \alpha ( k - n _ 0 ) - \mathcal { S } _ { \mathcal { L } } ( x \mapsto x ^ { 3 / 4 } ) ( k ) . \end{align*}


\begin{align*} | g ( z ) | & \leq C e ^ { c | z | } , z \in Q _ 1 , \\ | g ( z ) | & \leq 1 , z \in \partial Q _ 1 = \R _ + \cup i \R _ + . \end{align*}


\begin{align*} ( \partial _ t ^ 2 - \Delta ) u = 0 \R \times \R ^ d , \end{align*}


\begin{align*} v _ { s } v _ { t } & = \lim _ { i } \lim _ { j } \psi ( \mu _ j \delta _ { s } \mu _ i \delta _ { t } ) = \lim _ { i } \lim _ { j } \psi ( \alpha _ { s } ( \mu _ i ) u ( s , t ) \mu _ j \delta _ { s t } ) \\ & = \lim _ { i } \pi ( \alpha _ { s } ( \mu _ i ) u ( s , t ) ) v _ { s t } = \overline { \pi } ( u ( s , t ) ) v _ { s t } . \end{align*}


\begin{align*} w = H + \widetilde { T } ( f ) . \end{align*}


\begin{align*} f ( k , t , \mathbf { p } _ { n + 1 } ) = F ( k , t , \mathbf { p } _ { n + 1 } ) - F ( k - 1 , t , \mathbf { p } _ { n + 1 } ) . \end{align*}


\begin{align*} J _ r ' ( x , \beta ) = \theta ( x , \beta ) . \end{align*}


\begin{align*} h ^ { - 1 } _ { \bar u ( t ) } ( \lambda _ t ) = \max _ { u \in \R ^ k } h ^ { - 1 } _ u ( \lambda _ t ) = H ( \lambda _ t ) , \qquad \forall \ , t \in [ 0 , 1 ] , \end{align*}


\begin{align*} \mu ( \sigma ) = \frac { e ^ { - H ( \sigma ) } } { Z } , \end{align*}


\begin{align*} p _ a ( Y ) = - \sum _ { j = 1 } ^ { k } { ( \alpha + 1 + d _ i ) } . \end{align*}


\begin{align*} t _ { \mathrm { s u c } _ K ( k , m ) } = \begin{cases} t _ { k , m } + 2 ^ { - 2 k } & k < K , \\ t _ { k , m } + 2 ^ { 1 - 2 k } & k = K . \\ \end{cases} \end{align*}


\begin{align*} D _ k ( t ) = \int _ { ( \Lambda ^ * ) ^ 2 } d _ t ( k , p , q ) a _ { p } ^ * a _ { q } \d p \d q \end{align*}


\begin{align*} \mathcal E _ 2 \coloneqq \left \{ ( a _ 1 , \ldots , a _ n , b ) \in \mathbb N ^ { n + 1 } : b \leq d , \sum _ { i = 1 } ^ n a _ i \leq h , \exists i , a _ i = 0 \right \} \ , . \end{align*}


\begin{align*} \lim _ { i \to \infty } \frac { 1 } { \ell _ i } \sum _ { \ell \leq \ell _ i } \delta _ { \sigma ^ \ell \eta } = \nu _ \eta , \end{align*}


\begin{align*} \int _ \Omega \big ( \mathrm { d i v } \ , V \big ) ^ 2 \ , d x = \int _ \Omega \mathrm { t r } \big ( ( \nabla \ , V ) ^ 2 \big ) d x + \int _ { \partial \Omega } \Big \{ ( \mathrm { d i v } _ T V ) \ , V \cdot \nu - \nabla _ T V \ , V _ T \cdot \nu \Big \} d \mathcal { H } ^ { n - 1 } . \end{align*}


\begin{align*} h ^ \ast \left ( \xi _ { h g h ^ { - 1 } } \right ) = \rho _ { h , g } \xi _ { g } . \end{align*}


\begin{align*} { \rm T r e e } _ { F } ^ { H _ { 0 } } = \left \{ H \in { \rm T r e e } _ { F } : H < H _ { 0 } ^ { \gamma } \mbox { f o r s o m e } \gamma \in F \right \} , \end{align*}


\begin{align*} D _ { S , x } [ f ^ { = T } ] = ( L _ S E _ { T ^ c } L _ T [ f ] ) _ { S \rightarrow x } = ( E _ { T ^ c } L _ T [ f ] ) _ { S \rightarrow x } \end{align*}


\begin{align*} \sup _ { \gamma \in \mathcal { P } ( \mathcal { V } ) } \left \{ \int _ { \mathcal { V } } g d \gamma - \lambda _ 1 \boldsymbol { K } _ 1 ( \mu _ 1 , \gamma _ 1 ) - \lambda _ 2 \boldsymbol { K } _ 2 ( \mu _ 2 , \gamma _ 2 ) : \boldsymbol { K } _ \ell ( \mu _ \ell , \gamma _ \ell ) < \infty \ell = 1 , 2 \right \} \end{align*}


\begin{align*} \rho _ { i } = 2 ^ { j _ { 0 } } \rho _ { z _ { i } } \quad i . \end{align*}


\begin{align*} \frac { \partial f } { \partial t } + \nabla \cdot ( u f ) = 0 , \end{align*}


\begin{align*} \epsilon : = \beta _ j - \tilde { \beta } _ j = \nu ( g ' ) - \nu _ j ( g ' ) \in \Delta \mbox { f o r e v e r y } j \in I ' , j \geq j _ 0 . \end{align*}


\begin{align*} i \partial _ t u + \partial _ x ^ 2 u \ , \pm \ , \frac { 2 } { i } \ , \partial _ x \Pi ( | u | ^ 2 ) u = 0 \ , , x \in \mathbb { T } \ , , \end{align*}


\begin{align*} \partial _ { x _ n } c ( x ^ \prime , x _ n ) = \partial _ { x _ n } c ( x ^ \prime , x _ n ) - \partial _ { x _ n } c ( x ^ \prime , 0 ) = \int _ 0 ^ 1 \partial _ { x _ n } ^ 2 c ( x ^ \prime , t x _ n ) x _ n \ : t . \end{align*}


\begin{align*} \lambda _ { j _ 1 } = \lambda _ { j _ 1 ^ { - 1 } } = \dots = \lambda _ { j _ r } = \lambda _ { j _ r ^ { - 1 } } = \pm 1 \end{align*}


\begin{align*} t ^ { \ell + y } f ( t ) - a t ^ { 2 k + \ell + x } f ( t ^ { - 1 } ) & = \sum _ { m \in \Z } \left ( f _ m t ^ { m + \ell + y } - a f _ m t ^ { 2 k + \ell + x - m } \right ) \\ & = \sum _ { m \in \Z } ( f _ { m - \ell - y } - a f _ { 2 k + \ell + x - m } ) t ^ m \\ & = \sum _ { m \in \Z } h _ m t ^ m = h ( t ) , \end{align*}


\begin{align*} ( t _ 1 , t _ 2 , \dots , t _ n ) \sum _ { i = 0 } ^ m [ a _ { i } e _ { \lambda ^ i } ] = \sum _ { i = 0 } ^ m [ t _ { \lambda ^ i } a _ { i } e _ { \lambda ^ i } ] \end{align*}


\begin{align*} L = K ( \eta ) = K [ x ] / ( g ) \end{align*}


\begin{align*} | \mathbf { n } | _ { \mathfrak { s } } : = \sum _ { i = 1 } ^ d \mathfrak { s } _ i \mathbf { n } ( i ) ; \end{align*}


\begin{align*} t _ n = \frac { \Gamma ( n ) \ , \Gamma ( n + x ) \ , \Gamma ( 2 n + x - m ) \ , \Gamma ( 2 n + x + m - 1 ) } { \Gamma ( n + m ) \ , \Gamma ( n + x - m ) \ , \Gamma ( 2 n + x ) \ , \Gamma ( 2 n + x - 1 ) } . \end{align*}


\begin{align*} x _ 0 = ( A _ 0 , \lambda _ 0 , \psi _ 0 ) \leftrightarrow ( \Lambda _ 0 , \xi _ 0 , \psi _ { 0 } ) \end{align*}


\begin{align*} \mathcal { D } \bigl ( \mathcal { A } _ { \chi , \varepsilon } ^ { \rm a p p } \bigr ) : = \bigl \{ ( \vect u , \widehat { \vect u } ) \in \mathcal { H } ^ { \rm s o f t } \oplus \widehat { \mathcal { H } } _ \chi ^ { \rm s t i f f } , \vect u \in \mathcal { D } ( \widehat { \mathcal { A } } _ \chi ^ { \rm s o f t } ) , \widehat { \vect u } = \widehat { \Pi } _ \chi ^ { \rm s t i f f } \widehat { \Gamma } _ { 0 , \chi } ^ { \rm s o f t } \vect u \bigr \} , \end{align*}


\begin{align*} \overline { \mathcal G } _ { j } = \frac { { \mathcal G } _ { j } + \mathcal G ' _ { j } } { 2 } , \mathcal G ^ \perp _ { j } = \frac { { \mathcal G } _ { j } - \mathcal G ' _ { j } } { 2 } , n _ 0 < j \leq n _ \mathcal L . \end{align*}


\begin{align*} ( R ^ i \overline { f } _ * ( \mathcal { I C } _ { M } ) ) [ d i m ( S ) ] = { } ^ p \mathcal { H } ^ { d i m ( S ) + i } ( R \overline { f } _ * ( \mathcal { I C } _ { M } ) ) . \end{align*}


\begin{align*} & \Big \| \sum _ { | j | > \ell } \sum _ { Q \in Q ^ j } w ( Q ) \psi _ { j } ( x , x _ { Q } ) q _ { j } h ( x _ { Q } ) \Big \| _ { \dot { \mathcal F } ^ { \alpha , q } _ { p , { \rm D } } } \\ & = C \Big \| \Big \{ \sum _ { k \in \mathbb Z } \sum _ { Q \in Q ^ k } \Big ( { \mathfrak R } ^ { k \alpha } \Big | q _ k \Big ( \sum _ { | j | > \ell } \sum _ { Q \in Q ^ j } w ( Q ) \psi _ { j } ( \cdot , x _ { Q } ) q _ { j } h ( x _ { Q } ) \Big ) ( x _ Q ) \Big | \Big ) ^ q \chi _ Q \Big \} ^ { 1 / q } \Big \| _ { L ^ p _ \omega } , \end{align*}


\begin{align*} & \left ( \frac { \sqrt { p } } { 2 \pi } \right ) ^ { k + t } \Gamma ( k + t ) \ , L ( f , k + t ) = i ^ { - 2 k } \epsilon _ f \left ( \frac { \sqrt { p } } { 2 \pi } \right ) ^ { k - t } \Gamma ( k - t ) \ , L ( f , k - t ) \\ \implies & \left ( \frac { p } { 4 \pi ^ 2 } \right ) ^ t \Gamma ( k + t ) ^ 2 L ( f , k + t ) ^ 2 = \left ( \frac { p } { 4 \pi ^ 2 } \right ) ^ { - t } \Gamma ( k - t ) ^ 2 L ( f , k - t ) ^ . \end{align*}


\begin{align*} \Lambda ^ S _ { h , t } = \{ \exp _ x ( h \phi _ t ( x ) \nu ( x ) ) \in M \ ; | \ ; x \in \Sigma _ 0 \setminus B _ t \} . \end{align*}


\begin{align*} H '' = \bigg \{ x \notin H ' : \ , \frac { x \cdot z ^ + } { | z ^ + | } \leq \frac 1 2 \ , R ^ + \bigg \} \ , , & & H ''' = \bigg \{ x \notin H ' : \ , \frac { x \cdot z ^ + } { | z ^ + | } > \frac 1 2 \ , R ^ + \bigg \} \ , , \end{align*}


\begin{align*} s = \frac { \tau } { a } , \ \ y = \frac { R } { a } , \ \ a > 0 . \end{align*}


\begin{align*} a _ 0 ^ 2 = O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 + O ( \varepsilon ) \| \nabla u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 . \end{align*}


\begin{align*} \mathcal { A } _ i = B ( \xi , \sqrt { \mu _ i \mu _ { i - 1 } } ) \setminus B ( \xi , \sqrt { \mu _ i \mu _ { i + 1 } } ) , i = 1 , \cdots , k , \end{align*}


\begin{align*} \nu _ n ( g ) = p \nu ( x - a _ n ) \end{align*}


\begin{align*} D ^ n _ { x _ 1 } G = \left ( \sum _ { j = 0 } ^ n { n \choose j } \alpha _ 1 ^ { n - j } D _ { x _ 1 } ^ j D _ { x _ m } ^ { n - j } F \right ) \Biggr \rvert _ { \substack ( x _ 1 , x _ 2 , \ldots , \alpha _ 1 x _ 1 + \cdots + \alpha _ { m - 1 } x _ { m - 1 } ) } . \end{align*}


\begin{align*} x _ p = x ^ 0 _ p + l ' u _ p , y _ p = y ^ 0 _ p + n ' u _ p u _ p \geq 0 , \end{align*}


\begin{align*} \mu _ { D D , \delta } \{ \infty \to \tfrac { \alpha } { \beta } \} ( f ) = \sum _ { D \mid N } n _ D \cdot C ( \alpha , \beta / D , a , b D ) . \end{align*}


\begin{align*} \mathrm { V a r } \big ( \big \langle D F _ R ( t _ i ) , - D L ^ { - 1 } F _ R ( t _ j ) \big \rangle _ { \mathcal { H } } \big ) \leq C R , \mbox { f o r a n y $ i , j = 1 , \ldots , m $ } . \end{align*}


\begin{align*} h _ i \sum \limits _ { j = 1 } ^ { s _ { { i _ { } i } } } b _ { i _ { } i j } { \textbf { X } } ^ { \lambda _ j } \in \mathcal I _ 2 . \end{align*}


\begin{align*} 2 ^ n \sigma = 2 \end{align*}


\begin{align*} \int _ { \Omega } F ^ { i j } ( D ^ { 2 } u ) \eta _ { i j } d x = 0 , \end{align*}


\begin{align*} \partial _ t { w } _ { p \alpha + k - 1 } ^ { ( i ) } ( x _ i , t ) + \Big ( v ^ { ( i ) } _ i ( x _ i ) \ , w ^ { ( i ) } _ { p \alpha + k - 1 } ( x _ i , t ) \Big ) ^ \prime = \Big ( w ^ { ( i ) } _ { p \alpha + k - 2 } ( x _ i , t ) \Big ) ^ { \prime \prime } , ( x _ i , t ) \in I _ \varepsilon ^ { ( i ) } \times ( 0 , T ) . \end{align*}


\begin{align*} \mathbf { K } \overline { \psi } _ { g } ^ { \square } Y _ f Z _ f ^ { - 1 } = \mathbf { K } \psi ( X ) ^ { \square } Y _ f = \mathbf { K } \overline { \psi } _ { g } ^ { \square } Y _ f \end{align*}


\begin{align*} 1 = \| \varphi ( a ) \| ^ 2 \leq \| \varphi ( a ^ * a ) \| \leq 1 \end{align*}


\begin{align*} S = \{ m \le d \le n : \gcd ( k , q ^ d - 1 ) < q ^ { d / 3 } \} . \end{align*}


\begin{align*} F _ \beta ( v . u ) = \lim _ n \beta \theta ^ n ( v _ { - 1 } . u _ 0 ) & = \lim _ n \eta ^ n ( \beta ( v _ { - 1 } ) . \beta ( u _ 0 ) ) \\ & = \eta ( \lim _ n \eta ^ n ( \beta ( v _ { - 1 } ) . \beta ( u _ 0 ) ) ) = \eta ( F _ \beta ( v . u ) ) . \end{align*}


\begin{align*} ( \det u _ { j \bar k } ) ^ \frac { 1 } { n } = \inf \Big \{ L _ a u \ , ; \ , ~ a \in \mathcal { A } ( \Omega ) \Big \} . \end{align*}


\begin{align*} \Pi _ x = \Gamma _ { x y } ^ * \Pi _ y , \ , \ , \ , \Gamma _ { x y } ^ * = \Gamma _ { x z } ^ * \Gamma _ { z y } ^ * \end{align*}


\begin{align*} N ( s , t , \mu ) : = \bigcup _ { r \in [ s , t ] } \partial R ( \Sigma _ r ' , \mu ) . \end{align*}


\begin{align*} g _ \beta ( x , t _ 1 , t _ 2 , \ldots , t _ { d } ) = \sum _ { n _ 1 = 0 } ^ \infty \cdots \sum _ { n _ d = 0 } ^ \infty \frac { t _ 1 ^ { n _ 1 } \cdots t _ d ^ { n _ d } } { n _ 1 ! \cdots n _ d ! } I _ { \beta + \sum _ { i = 1 } ^ { d } ( i + 1 ) n _ i } ( 2 \sqrt { x } ) , \end{align*}


\begin{align*} X ^ k C _ { k i j } = - \frac { 1 } { m ( n - 2 ) } | X | ^ 2 \left ( \nabla _ i X _ j - \nabla _ j X _ i \right ) = - \frac { 1 } { m ( n - 2 ) } | X | ^ 2 ( d X ) _ { i j } . \end{align*}


\begin{align*} \mathbf { X } \simeq \left \{ z = z _ 1 + i z _ 2 \in \mathrm { S y m } _ { n } ( \mathbb { C } ) \mid z _ 1 , z _ 2 \in \mathrm { S y m } _ { n } ( \mathbb { R } ) , \ , z _ 2 \right \} . \end{align*}


\begin{align*} ( \vec U , { \vec L _ D } ) = \sum _ { j = 1 , N } \lbrack 2 ( \vec W ^ - ) ^ T \Sigma ( \sqrt { | \Lambda ^ - | } \vec W ^ - - R \sqrt { \Lambda ^ + } \vec W ^ + - S \vec G ) \rbrack _ j d s _ j . \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } S _ { j , + } ( t ) = \hat { S } \end{align*}


\begin{align*} g \sum _ { p = 1 } ^ n u ^ { p \bar p } u _ { p \bar p j \bar j } \geq - A _ 1 ( 1 + \Vert \nabla v \Vert ^ 2 + \Delta v ) , \end{align*}


\begin{align*} \int _ { B _ { \delta } ( x ^ \prime ) } J _ \varepsilon ( | x - \hat { y } | ) \nabla _ { x ^ \prime } c ( x ) \cdot ( x ^ \prime - y ^ \prime ) y ^ \prime = 0 , \end{align*}


\begin{align*} & P ( Z _ 0 ( \Delta \ell _ 2 ) = m | W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 , Z _ 0 ( \Delta \ell _ 1 ) = n ) \\ & \cdot P ( B _ { \ell _ 1 \Delta } ( \ell _ 2 \Delta ) = i | Z _ 0 ( \Delta \ell _ 2 ) = m , Z _ 0 ( \Delta \ell _ 1 ) = n , W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 ) \\ & = \frac { P ( B _ { \ell _ 1 \Delta } ( \ell _ 2 \Delta ) = i , Z _ 0 ( \Delta \ell _ 2 ) = m , Z _ 0 ( \Delta \ell _ 1 ) = n , W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 ) } { P ( W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 , Z _ 0 ( \Delta \ell _ 1 ) = n ) } \end{align*}


\begin{align*} \lim _ { u \to + \infty } T _ i ( u ) = \tau _ i , \end{align*}


\begin{align*} \langle \phi ^ { + } ( t ) v , v \rangle = \int _ { t } ^ { \infty } | \langle H ( s ) , v \rangle | ^ 2 d s , \end{align*}


\begin{align*} f = \sum _ { i = 1 } ^ s a _ i \textbf { Q } ^ { \lambda _ i } \end{align*}


\begin{align*} \widetilde { Y } _ { t } = U _ { t } ^ { - 1 } \left ( U _ 0 \widetilde { Y } _ 0 + \int _ { 0 } ^ { t } U _ { s } d \widetilde { \mu } _ { s } \right ) \end{align*}


\begin{align*} \begin{cases} \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) u = 0 & \Omega , \\ u = 0 & \Omega ^ c , \end{cases} \end{align*}


\begin{align*} L i _ 2 \left ( \frac { x } { 1 - x } . \frac { y } { 1 - y } \right ) = L i _ 2 \left ( \frac { x } { 1 - y } \right ) + L i _ 2 \left ( \frac { y } { 1 - x } \right ) - L i _ 2 ( x ) - L i _ 2 ( y ) - \log ( 1 - x ) \log ( 1 - y ) . \end{align*}


\begin{align*} ( \rho ( g ) ( \varphi ) ) ( x ) = \varphi ( x g ) . \end{align*}


\begin{align*} l = \frac { m } { 2 ^ { j } } j \geq 1 Q = K _ { \rho } \times J \end{align*}


\begin{align*} \nu _ i \left ( f ' _ j Q _ i ^ j \right ) = \nu _ i \left ( f _ j Q _ i ^ j \frac { f ' _ j } { f _ j } \right ) > \nu _ i ( f ) + \alpha _ i . \end{align*}


\begin{align*} \ell _ t ( x , y ) : = \rho _ { 1 - e ^ { - t } } ( e ^ { - t / \alpha } x , y ) , x , y \in \Gamma , \ ; t > 0 , \end{align*}


\begin{align*} \frac { 1 } { n ^ d } \sum _ { 0 \leq i _ 1 < n } \cdots \sum _ { 0 \leq i _ d < n } T _ 1 ^ { i _ 1 } \cdots T _ d ^ { i _ d } f \leq \frac { \chi _ d } { n _ d } \sum _ { j = 0 } ^ { n _ d - 1 } U ^ j f \end{align*}


\begin{align*} { \rm c a p } _ { q , Q } \ , ( E , F , D ) \quad = \quad \inf \limits _ { u \ , \in \ , W _ 0 ( E , F ) } \quad \int \limits _ D \ , Q ( x ) \cdot | \nabla u | ^ q \ , \ , d m ( x ) \ , . \end{align*}


\begin{align*} - \varepsilon \ , \partial _ { \boldsymbol { \nu } _ \varepsilon } u _ \varepsilon + u _ \varepsilon \ , \overrightarrow { V _ \varepsilon } \boldsymbol { \cdot } \boldsymbol { \nu } _ \varepsilon = 0 \end{align*}


\begin{align*} f _ \lambda ( x ) : = f ( 0 ) + \int _ 0 ^ x f ' ( y ) \chi _ \lambda ( y ) \ , d y , \end{align*}


\begin{align*} E _ 2 = - E _ 1 = \begin{bmatrix} - I & 0 \\ 0 & 0 \end{bmatrix} \end{align*}


\begin{align*} f ( k ) = \sum _ { \xi \in \mathcal { I } _ { \hbar } } \widehat { f } ( \xi ) u _ { \xi } ( k ) , f \in \mathrm { H } ^ { \infty } _ { \mathcal { H } _ { \hbar , V } } . \end{align*}


\begin{align*} \beta \ \le \ \frac { m } { m - 1 } \ = \ 1 + \frac { 1 } { m - 1 } , \end{align*}


\begin{align*} \int _ V \hat f - f _ j = \int _ { U ^ c } f _ j - \hat f = \eta \ , . \end{align*}


\begin{align*} \nabla | \nabla f | ^ { 2 q - 2 } = ( q - 1 ) | \nabla f | ^ { 2 q - 4 } \nabla | \nabla f | ^ 2 , \end{align*}


\begin{align*} \left ( \sum _ { n \ge 0 } ( - 1 ) ^ n X _ { \bullet \bullet \cdots \bullet } \right ) ^ { - 1 } = \sum _ { n \ge 0 } \sum _ { | F | = n } X _ F . \end{align*}


\begin{align*} \int _ { B _ r } c _ { 0 } ^ { i j , k l } w _ { i j } \eta _ { k l } d x = 0 \forall \eta \in C _ { 0 } ^ { \infty } ( B _ { r } ( 0 ) ) , \end{align*}


\begin{align*} X = Y _ \mathbb { Q } Y _ f ^ { - 1 } Y _ \mathbb { Q } \in \mathcal { R } ^ 1 ( \mathbb { Q } ) Y _ f = ( Y _ q ) _ q \in \mathcal { R } ^ 1 ( \widehat { \mathbb { Z } } ) . \end{align*}


\begin{align*} \bigl ( \widehat { S } _ \chi ^ { \rm s t i f f ( s o f t ) } ( z ) \bigr ) ^ * = \widehat { P } _ \chi \bigl ( S _ \chi ^ { \rm s t i f f ( s o f t ) } \bigr ) ^ * , \bigl ( \widehat { \Pi } _ \chi ^ { \rm s t i f f ( s o f t ) } \bigr ) ^ * = \widehat { P } _ \chi \bigl ( \Pi _ \chi ^ { \rm s t i f f ( s o f t ) } \bigr ) ^ * . \end{align*}


\begin{align*} R _ { n m } ^ { 1 2 } ( z + u - v + a h ) = \prod _ { i = 1 , \dots , n } ^ { \longrightarrow } \prod _ { j = n + 1 , \ldots , n + m } ^ { \longleftarrow } R _ { i j } ( z + u _ i - v _ { j - n } + a h ) . \end{align*}


\begin{align*} \alpha | \xi | ^ 2 + 2 \omega - \mathbf { c } \cdot \xi = \alpha \left | \xi - \frac { \mathbf { c } } { 2 \alpha } \right | ^ 2 + 2 \left ( \omega - \frac { | \mathbf { c } | ^ 2 } { 8 \alpha } \right ) \sim 1 + | \xi | ^ 2 \end{align*}


\begin{align*} U & ( \vec { a } , \Lambda ) \left ( S , f _ \alpha \otimes \rho ( E ^ \alpha ) \right ) \\ & : = \left ( \Lambda S + \vec { a } , e ^ { - i [ \vec { a } \cdot ( \hat { H } ( \rho _ n ) \hat { f } _ 0 + \hat { P } ( \rho _ n ) \hat { f } _ 1 ) ] } f _ { \alpha } ( \Lambda ^ { - 1 } ( \cdot - \vec { a } ) ) \otimes \rho ( E ^ \alpha ) \right ) . \end{align*}


\begin{align*} | \det ( I - d _ x \varphi ^ { - t } ( x _ 0 ) ) | = ( - 1 ) ^ s \det ( I - d _ x \varphi ^ { - t } ( x _ 0 ) ) . \end{align*}


\begin{align*} \partial _ { n t } \Omega : = \{ x \in \partial \Omega \cap \overline { \Omega \cap \partial U } : \langle D u ( x ) - x , z \rangle \leq 0 \ \ \forall z \in \Omega \} . \end{align*}


\begin{align*} c _ x = \prod _ { p \le \exp ( \log ^ \beta x ) } ( 1 - p ^ { - 1 } ) . \end{align*}


\begin{align*} \overline { [ \underline { \varphi ^ { \prime } } , \varphi ^ { \prime } ] } = X _ { \varphi ^ { \prime } } = X _ { \eta ' } \subseteq X _ \eta \subseteq X _ \varphi = \overline { [ \underline { \varphi } , \varphi ] } . \end{align*}


\begin{align*} \psi ^ { \alpha _ \theta , n } ( h ) & = \phi ^ { \alpha _ \theta , n } ( h ) ^ \ast , \ 1 \leq \theta \leq r , \\ \psi ^ { \beta _ { \theta - r } , n } ( h ) & = \phi ^ { \beta _ { \theta - r } , n } ( h ) ^ \ast , \ r + 1 \leq \theta \leq r + s , \end{align*}


\begin{align*} M ( U ) _ f = \left ( \prod _ { x : U } ( M _ x ) _ { f ( x ) } \right ) \to \left ( \prod _ { x : U } ( M _ x ) ^ { D ( f ( x ) ) } \right ) = \left ( \prod _ { x : D ( f ) } M _ x \right ) = M ( D ( f ) ) \end{align*}


\begin{align*} T _ 6 = - \frac { s } { 2 } \iint _ Q \sigma _ { t t } | u | ^ 2 d x d t \geq - C s \iint _ Q \xi ^ { 3 / 2 } | u | ^ 2 d x d t . \end{align*}


\begin{align*} | K | _ { q } = \int ^ { [ n ] \in \mathcal { A } } K _ n . ( \{ 0 < 1 \} ^ n ) ^ { c o f } \end{align*}


\begin{align*} \Lambda _ { , 1 } ( k , p ) = \prod _ { s = 1 } ^ k \left ( \frac { n } { p } - 2 s \right ) ^ p \left ( \frac { n ( p - 1 ) } { p } + 2 s - 2 \right ) ^ p . \end{align*}


\begin{align*} p _ 0 \left ( \xi _ 0 \right ) = 0 \end{align*}


\begin{align*} \Delta \circ T = ( T \otimes T ) \Delta . \end{align*}


\begin{align*} T ( p ) : = \sum _ { e \in p } t _ e , \end{align*}


\begin{align*} T ( f ) = & \int _ { I ^ 2 } f ^ { \hat { S } ^ \flat } ( \sigma ( \hat { t } ) ) h ( \hat { t } ) \ d \hat { t } . \end{align*}


\begin{align*} \| \tilde { A } \tilde { R } _ h ( z ) \tilde { B } \| _ { L ^ 2 \to L ^ 2 } = \mathcal { O } ( h ^ { - 1 } ) , \end{align*}


\begin{align*} n ^ { - 2 } g u ^ { k \bar l } u _ { k \bar l \bar j } u ^ { p \bar q } u _ { p \bar q j } + n ^ { - 1 } g ( u ^ { p \bar q } ) _ { \bar j } u _ { p \bar q j } + n ^ { - 1 } g u ^ { p \bar q } u _ { p \bar q j \bar j } = 2 R e ( g _ { v j } v _ { \bar j } ) + g _ v v _ { j \bar j } + g _ { v v } | v _ j | ^ 2 + g _ { j \bar j } . \end{align*}


\begin{align*} \begin{aligned} u '''' & = f , \\ u = u ' & = 0 , \end{aligned} \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \mathrm { d } \Psi _ { s } & = & \kappa \Psi _ { s } \mathrm { d } W _ { s } , \\ \Psi _ { 0 } & = & 1 . \end{array} \right . \end{align*}