\begin{align*} \partial ^ 2 _ { x _ i x _ j } \phi = \partial ^ 2 _ { x _ i x _ j } \psi + \partial ^ 2 _ { x _ i y } \psi \Gamma _ j + \partial ^ 2 _ { x _ j y } \psi \Gamma _ i + \partial ^ 2 _ y \psi \Gamma _ j \Gamma _ i + \partial _ y \psi \partial ^ 2 _ { x _ j x _ i } T - \partial _ y T \partial ^ 3 _ { x _ j x _ i y } T + \partial _ { y } \Gamma _ i \partial _ { y } \Gamma _ j \end{align*}


\begin{align*} \begin{aligned} \gamma _ 2 ^ \star : = \dfrac { \bar c } { 3 e ^ { L _ r \bar { t } } } \gamma _ 3 ^ \star : = \dfrac { \bar c L _ r } { 3 \bar k ( L _ r , M _ r , \delta ) ( e ^ { L _ r \bar { t } } - 1 ) } \end{aligned} \end{align*}


\begin{align*} \tau ( [ L ] ) : = \log _ q [ \mathcal { O } _ F ^ d : \pi ^ i L ] , \end{align*}


\begin{align*} \tilde { u } ^ { [ n ] } ( x ) = \sum _ i w ^ { [ n ] } _ i \sigma _ i ( x ) , \end{align*}


\begin{align*} \Rightarrow \quad \frac 1 r \big ( g - d r \otimes d r \big ) = H e s s ( r ) ; \end{align*}


\begin{align*} \tilde { h } _ { i j } = h _ { i j } - B _ { i j } . \end{align*}


\begin{align*} \tilde { \rho } ( X _ { \alpha } ) = \sum _ { k = 1 } ^ L \frac { \partial } { \partial z _ { s , k } } r _ { t , k } ( 1 \pm \alpha _ { t , k } ) . \end{align*}


\begin{align*} V & = \Big ( Z ( T ' , M _ n ) + e _ 1 M _ n ( F ) e _ 3 + e _ 3 M _ n ( F ) e _ 2 \Big ) \cap \Lambda ^ \perp \\ & = \left ( Z ( T ' , M _ n ) \cap \Lambda ^ \perp \right ) + e _ 1 M _ n ( F ) e _ 3 + e _ 3 M _ n ( F ) e _ 2 \end{align*}


\begin{align*} F ^ { q , \dagger } _ v ( 0 , p ) [ u _ k ] = c _ 1 ( q , k , p ) u _ k . \end{align*}


\begin{align*} K _ { v , m } ( t ) = \sum \limits _ { \eta _ 1 ( v ) \leq u \leq t \atop { ( u , v ) = 1 } } e \left ( \frac { m \overline { u } _ { | v | } } { v } \right ) \ , . \end{align*}


\begin{align*} & \int _ { X _ j } | u _ { m , m ' , \epsilon , j } | ^ 2 _ { h _ { m ' } } e ^ { v _ \epsilon ( \Psi _ m ) - \delta M _ { \eta _ m } } c ( - v _ \epsilon ( \Psi _ m ) ) + \int _ { X _ j } | h _ { m , m ' , \epsilon , j } | ^ 2 _ { h _ { m ' } } e ^ { - \phi - \delta M _ { \eta _ m } } \\ \le & \int _ { X _ j } v '' _ { \epsilon } ( \Psi _ m ) | f F ^ { 1 + \delta } | ^ 2 _ { h _ { m ' } } e ^ { - \phi - \delta M _ { \eta _ m } } < + \infty . \end{align*}


\begin{align*} ( \phi x ) _ \epsilon = x _ { \phi ( \epsilon ) } \end{align*}


\begin{align*} \frac { u ^ k - u ^ { k - 1 } } { \tau _ k } + \mathcal { L } u ^ k = F ^ k , \end{align*}


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


\begin{align*} \hat { q } _ i = q ( z _ i ) + \frac { \hat { q } _ i } { \ln \frac { R } { \varepsilon } } g _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ i ) - \Sigma _ { j \neq i } \frac { \hat { q } _ j } { \ln \frac { R } { \varepsilon } } \bar { G } _ { z _ j } ( T _ { z _ j } z _ i , T _ { z _ j } z _ j ) , \end{align*}


\begin{align*} & \psi _ x + i k ^ 2 [ \sigma _ 3 , \psi ] = k P _ x \psi , \\ & \psi _ t + i \eta ^ 2 [ \sigma _ 3 , \psi ] = H \psi . \end{align*}


\begin{align*} \langle z , w \rangle _ { \mathbb C } : = \frac { 1 } { 2 } \left ( \overline z w + \overline w z \right ) = \frac { 1 } { 2 } \left ( z \overline w + w \overline z \right ) . \end{align*}


\begin{align*} J ( x ; k ) = \begin{cases} \begin{pmatrix} | r ( k ) | ^ 2 & \overline { r ( k ) } \mathrm { e } ^ { - 2 i k ^ 2 x } \\ r ( k ) e ^ { 2 i k ^ 2 x } & 0 \end{pmatrix} , k \in \mathbb { R } , \\ \\ \begin{pmatrix} - | r ( k ) | ^ 2 & - \overline { r ( k ) } e ^ { - 2 i k ^ 2 x } \\ r ( k ) e ^ { 2 i k ^ 2 x } & 0 \end{pmatrix} , k \in i \mathbb { R } . \end{cases} \end{align*}


\begin{align*} \frac { 2 \pi } { N } = \int _ { \alpha _ k } ^ { \beta _ k } \frac { s b ' ( s ) } { b ( s ) } d m ( s ) \geq | b ' ( s _ k ) | | \beta _ k - \alpha _ k | \ , k = 1 , \ldots , N . \end{align*}


\begin{align*} f _ { j , \theta } ( x ' , x _ 3 ) = \int _ { I _ { K ^ { - 1 } } } \int _ { \pi _ 3 ( \theta ) } \widehat { f } _ { j , \theta } ( \nu ' , \nu _ 3 ) e ^ { 2 \pi i x ' \cdot \xi ' } e ^ { 2 \pi i x _ 3 \xi _ 3 } d \nu ' d \nu _ 3 . \end{align*}


\begin{align*} \alpha _ 2 : = \underset { x \in K _ 2 } { \inf } d ^ \mathcal Z ( x , \partial \mathcal Z ) \beta _ 2 : = \underset { x \in K _ 2 } { \sup } \ ; d ^ \mathcal Z ( x , \partial \mathcal Z ) . \end{align*}


\begin{align*} u ^ k = \arg \min _ { w \in H ^ 1 ( \Omega ) } \Big ( \frac { 1 } { 2 } \| w - u ^ { k - 1 } \| _ { L _ 2 ( \Omega ) } ^ 2 + \tau _ k N ( w ) \Big ) , \end{align*}


\begin{align*} \frac { d ( \mu \circ G _ { \xi _ { \Delta } } ^ { - 1 } ) } { d \mu } ( \eta ) = \mathbf { 1 } _ { [ \xi _ { \Delta } ] } ( \eta ) \sum _ { \zeta _ { \Delta } } \frac { \gamma _ { \Delta } ( \zeta _ { \Delta } | \eta _ { \Delta ^ c } ) } { \gamma _ { \Delta } ( \xi _ { \Delta } | \eta _ { \Delta ^ c } ) } , \eta \in \Omega . \end{align*}


\begin{align*} \d ( x , y ) = \left \{ \begin{array} { l l } \mbox { $ 0 $ ( i . e . $ x = y $ ) } & \mbox { i f $ | D | = 1 $ } \\ \frac { n } { 2 } & \mbox { i f $ | D | = 2 $ , } \end{array} \right . \end{align*}


\begin{align*} \sum _ { \substack { n \\ r _ { Q } ( n ) = 0 } } n \ll \max \left \{ \frac { p ^ { 3 + \epsilon } } { ( \min Q ^ { * } ) ^ { 2 } } , p ^ { 5 / 2 + \epsilon } \right \} . \end{align*}


\begin{align*} ( A ^ q \eta ) ( x ) = \int _ \S W _ r ( x - y ) ( \eta ( y ) - \eta ( x ) ) \cos ( 2 \pi q ( y - x ) ) \ \d y - \int _ \S W _ r ( y ) \eta ( y ) \cos ( 2 \pi q y ) ) \ \d y . \end{align*}


\begin{align*} \widehat { a } ( Z ) = \widehat { a } \big ( ( Z ^ * Z ) ^ \frac { 1 } { 2 } \big ) \end{align*}


\begin{align*} z : = x , ~ k = 2 ^ h , ~ { \rm a n d } ~ f _ k ( z ) : = p _ h ( x ) , \end{align*}


\begin{align*} \| u \| _ { \mathcal { C } ( \R ^ n _ + ) } : = \sup _ { Q \subset \R ^ { n - 1 } } \bigg ( \frac { 1 } { | Q | } \iint _ { T _ Q } | \nabla u ( x ) | ^ 2 \ , t \ , d x ' d t \bigg ) ^ { \frac 1 2 } , \end{align*}


\begin{align*} \abs { c _ 1 ( S ^ 1 \setminus S _ i ) } \le \abs { c ( S \setminus S _ i ) } + \sum _ { h \in [ \chi ] \setminus \{ i \} } \abs { c _ 1 ( P _ i ' ) } \le \abs { S \setminus S _ i } + \sum _ { h \in [ \chi ] \setminus \{ i \} } q _ h = \abs { S } - \abs { S \cap S _ i } + q - q _ i . \end{align*}


\begin{align*} \hat { \mu } _ { A ^ * } = \hat { \mu } _ A \ . \end{align*}


\begin{align*} \xi ' = - L - 2 , \eta ' = \frac { Q } { 2 ( L + 2 ) } , \zeta ' = \frac { \kappa ( L + 2 ) B _ 1 } { Q } , \end{align*}


\begin{align*} g ( \left ( D ^ W _ Z J \right ) X , J Y ) = - 2 g ( N ( X , Y ) , Z ) , \end{align*}


\begin{align*} \left \langle \left ( x , \xi \right ) , \left ( y , \eta \right ) \right \rangle = \tfrac { 1 } { 2 } \left ( \left \langle x , \eta \right \rangle + \left \langle y , \xi \right \rangle \right ) \end{align*}


\begin{align*} R S _ { H ' } ( F ) = S _ { H ' } ( F ) = \varinjlim _ n S _ { H ' , n } ( F ) , \end{align*}


\begin{align*} \langle - \Delta z + \zeta z , p \rangle = \langle \nabla z , \nabla p \rangle + \langle \zeta z , p \rangle = \langle - \Delta p , z \rangle + \langle \zeta p , z \rangle = \langle \overline { y } - g , z \rangle , \ ; \ ; z \in Y . \end{align*}


\begin{align*} \Pi _ V J _ { z / | z | } ^ \perp x = \lambda \Pi _ V x \forall x \in \C ^ n . \end{align*}


\begin{align*} \mathbb E \left [ B _ i ( t ) B _ j ( s ) \right ] = \frac { \rho _ { i , j } } 2 ( | s | ^ { \mathfrak H _ { i , j } } + | t | ^ { \mathfrak H _ { i , j } } - | t - s | ^ { \mathfrak H _ { i , j } } ) , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \langle K _ t ( R ) u _ n , u _ n - v \rangle = \langle K _ t ( R ) u , u - v \rangle , \end{align*}


\begin{align*} \theta _ 0 ( x ) = \frac { 1 } { 2 \pi i } \int _ { 2 - i \infty } ^ { 2 + i \infty } \frac { x ^ s } { s } \bigg \{ \sum _ p \frac { \log p } { p ^ s } \bigg \} d s . \end{align*}


\begin{align*} \det ( e ^ { x _ i z _ j / \sqrt { n } } ) _ { i , j = 1 } ^ d & \sim C n ^ { - d ( d - 1 ) / 2 - d / 2 } \Delta ( x ) \Delta ( z ) . \end{align*}


\begin{align*} \chi ( a ) = \prod _ \tau \tau ( a ) ^ w \end{align*}


\begin{align*} \gamma _ \Theta ^ { 1 / 2 } ( E ) = \sup | \langle \nu , 1 \rangle | , \end{align*}


\begin{align*} h _ 1 ( x ) = \prod _ { i = 1 } ^ n ( \eta _ i ( x + r ) - \eta _ i ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( x + r ) - \tilde \Psi ( x ) ) \\ h _ 2 ( x ) = - \prod _ { i = 1 } ^ n ( \eta _ i ( x - r ) - \eta _ i ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( x - r ) - \tilde \Psi ( x ) ) \end{align*}


\begin{align*} k t ' = k ( t - p ) > \sum _ { i \in [ \chi ] } x _ i . \end{align*}


\begin{align*} \begin{aligned} T _ 1 & : = T ^ e _ A ( x ) \\ T _ 2 & : = T ^ r _ { \bar A } ( x ) . \end{aligned} \end{align*}


\begin{align*} b ( n , d , k ) = \max \{ \vartheta ( G ) : \ : \} \end{align*}


\begin{align*} \alpha _ i + \bar { \alpha } _ j = 1 . \end{align*}


\begin{align*} ( r + \nu ) V ( s , i ) = H \left ( s , i , p , q \right ) . \end{align*}


\begin{align*} \| \nabla u \| _ 6 \le C \| \nabla u \| ^ { 1 - \kappa } \| \Lambda ^ { s + 1 } u \| ^ \kappa = C \| \nabla u \| ^ { \frac { s - 1 } { s } } \| \Lambda ^ { s + 1 } u \| ^ { \frac { 1 } { s } } . \end{align*}


\begin{align*} \pi = \pi ^ \infty \otimes \pi _ \infty \end{align*}


\begin{align*} f ( 2 x + y ) + f \left ( \frac { x + y } { 2 } \right ) = \frac { 2 f ( x ) f ( y ) } { f ( x ) + f ( y ) } + \frac { 2 f ( x + y ) f ( y - x ) } { 3 f ( y - x ) - f ( x + y ) } \end{align*}


\begin{align*} \left \{ \begin{array} { r l l l } - L \varphi & \geq & f ' ( u ) \varphi & \Omega , \\ \varphi & > & 0 & \Omega , \\ \varphi & = & 0 & \partial \Omega . \end{array} \right . \end{align*}


\begin{align*} C _ p ( \mathcal { H } _ { \ast } ) = B _ p ( \mathcal { H } _ { \ast } ) \oplus s _ { _ { p } } ( B _ { p - 1 } ( \mathcal { H } _ { \ast } ) ) . \end{align*}


\begin{align*} M _ n ( \theta ) = n ^ { - 1 } \sum _ { i = 1 } ^ n \log f ( X _ i ; \theta ) . \end{align*}


\begin{align*} \varphi ( y ) = \frac { 1 } { 8 \pi ^ { 2 } } \int _ { \{ \varphi > W / 2 \} } \frac { \mu ^ { 2 } } { | y - w | ^ { 3 } } \left ( \varphi - \frac { W } { 2 } \right ) _ { + } \dd w . \end{align*}


\begin{align*} x = \sum _ { m = 0 } ^ \infty \sum _ { i = 0 } ^ { p - 1 } \frac { a _ { m p + i } } { ( \beta _ 0 \cdots \beta _ { p - 1 } ) ^ m \beta _ 0 \cdots \beta _ i } . \end{align*}


\begin{align*} \lim _ { \Lambda \uparrow \Z ^ d } \inf _ { \xi : \xi _ { \Lambda } = \eta _ { \Lambda } } f ( \xi ) = f ( \eta ) , \lim _ { \Lambda \uparrow \Z ^ d } \sup _ { \xi : \xi _ { \Lambda } = \eta _ { \Lambda } } f ( \xi ) = f ( \eta ) . \end{align*}


\begin{align*} g _ { k , \ell } : = \begin{cases} \max ( e ^ { \delta _ n q _ n } , | \ell | , 1 ) e ^ { - ( \beta _ n - 6 \varepsilon ) q _ n } \ & \beta _ n \geq \delta _ n + 2 0 0 \varepsilon \\ e ^ { 2 \varepsilon k } & \beta _ n < \delta _ n + 2 0 0 \varepsilon \end{cases} \end{align*}


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


\begin{align*} ( n - ( j + 2 ) ) r < \sum \limits _ { i = j + 2 } ^ n \chi _ i , \end{align*}


\begin{align*} X _ i \left ( t , x _ { 0 } , t _ { 0 } \right ) : = \lambda _ i ( t - t _ { 0 } ) + x _ { 0 } , t \in [ 0 , T ] . \end{align*}


\begin{align*} \mathbf { h } = \lambda _ { 1 } \mathbf { e } _ 1 + \lambda _ { 2 } \mathbf { e } _ 2 + \lambda _ { 3 } \mathbf { e } _ 3 + \lambda _ { 1 2 } \mathbf { e } _ { 1 2 } + \lambda _ { 1 2 3 ' } \mathbf { e } _ { 1 2 3 ' } , \end{align*}


\begin{align*} b - b _ 1 = \sum _ { i = 2 } ^ r b _ i = l ( p - q ) + ( p - q r ) + 1 . \end{align*}


\begin{align*} K _ { t } ( x , y ) = \int _ { 0 } ^ { 1 } \lambda ^ { - 1 / 2 } K _ { t } ^ { * } ( x , y , \lambda ) d \lambda . \end{align*}


\begin{align*} \deg ( F ) = \max \{ i \ : \ A _ i \ne 0 \} \end{align*}


\begin{align*} d _ H ( D , E ) = \max \left \{ \sup _ { p \in D } d _ H ( p , E ) , \ \sup _ { p \in E } d _ H ( p , D ) \right \} . \end{align*}


\begin{align*} J ( x , y , [ x , z ] ) = [ J ( x , y , z ) , x ] , \end{align*}


\begin{align*} P _ A ( \lambda ) : = \pi _ A ( P ( \lambda ) ) S _ A ( \lambda ) = \pi _ A ( S ( \lambda ) ) . \end{align*}


\begin{align*} ( \theta _ 1 , \theta _ 2 ) = ( a ^ x \alpha ^ y , b ^ x \beta ^ y ) \end{align*}


\begin{align*} t _ { 2 K - 1 } = \min \bigl \{ t \colon d ( \gamma _ { a _ 1 , b _ 1 } ( t ) , b _ 1 ) = R \bigr \} , \end{align*}


\begin{align*} F ^ { q , \dagger } ( 0 , p ) = 0 \end{align*}


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


\begin{align*} \forall \theta > 0 \ \Rightarrow \sup _ { p \ge 1 } \left [ \ \frac { L ( p ^ { \theta } ) } { L ( p ) } \ \right ] = C ( \theta ) < \infty . \end{align*}


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


\begin{align*} \varphi ( x , t ; k ) = I + k \begin{pmatrix} 0 & u \\ v & 0 \end{pmatrix} + \mathcal { O } \left ( k ^ 2 \right ) , k \rightarrow 0 , \end{align*}


\begin{align*} \mathbf { h } _ { k l } = \beta _ { k l , 0 } \mathbf { a } ( \theta _ { k l , 0 } ) + \sum _ { p = 1 } ^ { P } \beta _ { k l , p } \mathbf { a } ( \theta _ { k l , p } ) , \end{align*}


\begin{align*} \lim _ { n \to + \infty } \int _ { a } ^ { b } | u _ { n } ' ( x ) - w _ { n } ' ( x ) | \ , d x = 0 , \end{align*}


\begin{align*} B = \begin{bmatrix} F & M \\ W & Q \end{bmatrix} , \end{align*}


\begin{align*} y \left [ n \right ] = \left ( { \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { \bf { H } } _ l ^ \bot { { \bf { b } } _ l } } } \right ) s \left [ { n - { n _ { \max } } } \right ] + z \left [ n \right ] . \end{align*}


\begin{align*} \mathbb E [ Z f ( Y _ n ) ] \to \tilde { \mathbb E } [ Z f ( Y ) ] = \int _ { \Omega } Z ( \omega ) \int _ { \mathcal D ( [ 0 , T ] , H ) } f ( x ) K ( \omega , d x ) \mathbb P [ d \omega ] , n \to \infty \end{align*}


\begin{align*} a _ c \left [ t + 1 \right ] & = a _ c \left [ t \right ] - \mu \frac { \partial \mathbb { E } \left [ \lvert \varepsilon \rvert ^ 2 | \hat { \mathbf { H } } ^ { } \right ] } { \partial a _ c } , \\ a _ i \left [ t + 1 \right ] & = a _ i \left [ t \right ] - \mu \frac { \partial \mathbb { E } \left [ \lvert \varepsilon \rvert ^ 2 | \hat { \mathbf { H } } ^ { } \right ] } { \partial a _ i } , \end{align*}


\begin{align*} g \in 1 \ast f & \mbox { i f f } g ( x ) \in 1 ( x ) \ast f ( x ) = 1 \ast f ( x ) \\ & \mbox { i f f } g ( x ) = f ( x ) \\ & \mbox { i f f } f = g \end{align*}


\begin{align*} u _ n = T _ n f _ n . \end{align*}


\begin{align*} M _ I = \bigoplus _ { J \subset I } R _ J . \end{align*}


\begin{align*} | \tau | = | \tau ' | . \end{align*}


\begin{align*} X X ^ \top = l \ , V ^ \top V \mbox { a n d } X ^ \top X = k \ , W ^ \top W . \end{align*}


\begin{align*} \lim _ { m \to + \infty } \Big ( 1 - \Theta _ m ( c , c ' ) \Big ) = 1 \end{align*}


\begin{align*} \varrho ( x , z ) = ( z + 1 ) f ( x ) - z b ( x ) \end{align*}


\begin{align*} \lim _ { n \rightarrow 0 } g _ { n } ^ { + } ( t , \omega ) = 0 , ( t , \omega ) . \end{align*}


\begin{align*} & \gamma \beta ( \gamma ) = \frac { \gamma _ { t - 1 } ( 1 - \kappa _ t ) } { ( 1 - \kappa _ { t - 1 } ) } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 3 } { 2 } \gamma _ 0 , \\ & \alpha ( \gamma ) = \frac { \kappa _ { t - 1 } \gamma \beta ( \gamma ) } { \gamma _ { t - 1 } } = \frac { \kappa _ { t - 1 } ( 1 - \kappa _ { t } ) } { 1 - \kappa _ { t - 1 } } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 1 } { 2 } . \end{align*}


\begin{align*} \hat U ( \xi , t ) = \exp ( E ( \xi ) t ) \hat w _ 0 ( \xi ) . \end{align*}


\begin{align*} V = g ( c ) H , \end{align*}


\begin{align*} [ x , y ] = x \diamond y - \varepsilon ( x , y ) y \diamond x , \quad \forall x , y \in \mathcal { H } ( A ) . \end{align*}


\begin{align*} c ( t ) = - w ( t ) = 0 M t \in ( 0 , t _ 0 ] \phantom { x } \phantom { x } c ( t ) = - w ( t ) > 0 M t \in ( t _ 0 , T ] . \end{align*}


\begin{align*} \int _ { \Omega } | \phi | ^ { 2 } \dd x + 2 \mu \int _ { 0 } ^ { t } \int _ { \Omega } | \nabla \phi | ^ { 2 } \dd x \dd s = \int _ { \Omega } | \phi _ 0 | ^ { 2 } \dd x , \end{align*}


\begin{align*} \begin{aligned} & \eta ( z , r ) = \int _ { \mathbb { R } ^ { 2 } _ { + } } { \mathcal { G } } ( z , r , z ' , r ' ) \frac { G ( z ' , r ' ) } { r ' } \dd z ' \dd r ' , \\ & { \mathcal { G } } ( z , r , z ' , r ' ) = \frac { r r ' } { 2 \pi } \int _ { 0 } ^ { \pi } \frac { \cos \theta \dd \theta } { \sqrt { | z - z ' | ^ { 2 } + r ^ { 2 } + r '^ { 2 } - 2 r r ' \cos \theta } } . \end{aligned} \end{align*}


\begin{align*} P _ { n , k } ( z ) = z ^ k A _ n ( z ) . \end{align*}


\begin{align*} & \sum _ { y \in \mathbb { F } _ q } \chi _ 4 ( y ) \overline { \chi _ 4 } ( a - y ) = \sum _ { y ' \in \mathbb { F } _ q } \chi _ 4 ( a y ' ) \overline { \chi _ 4 } ( a - a y ' ) \\ & = \sum _ { y ' \in \mathbb { F } _ q } \chi _ 4 ( y ' ) \overline { \chi _ 4 } ( 1 - y ' ) \\ & = \sum _ { y ' \in \mathbb { F } _ q } \chi _ 4 \left ( y ' ( 1 - y ' ) ^ { - 1 } \right ) \\ & = \sum _ { y '' \in \mathbb { F } _ q , y '' \neq - 1 } \chi _ 4 ( y '' ) = - 1 , \end{align*}


\begin{align*} \beta ( a + 1 ) = \begin{cases} 1 , & a + 1 \in \mathbb { P } , \\ 0 , & a + 1 \not \in \mathbb { P } . \end{cases} \end{align*}


\begin{align*} \rho _ { E _ 1 } = \sum _ { Z \in \mathcal I } m _ Z \rho _ { Z } \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T | g _ n ( t ) | d t = 0 . \end{align*}


\begin{align*} \left ( d \theta \right ) ^ { ( 2 , 0 ) + ( 0 , 2 ) } _ { J \cdot , \cdot } = g \left ( J \left ( \mathcal { L } _ { J \theta ^ \sharp } J \right ) ^ { a n t i - s y m } \cdot , \cdot \right ) . \end{align*}


\begin{align*} G : = D \rtimes \langle ( A _ 1 , ( \alpha , 1 ) ) , ( A _ 2 , ( 1 , \sigma ( \alpha ) ) ) \rangle \end{align*}


\begin{align*} \Theta _ { { \mathrm { E u c l } } } ^ { \Sigma _ \infty } ( 0 , 1 0 \lambda K ) - \Theta _ { { \mathrm { E u c l } } } ^ { \Sigma _ \infty } ( 0 , 1 / ( 3 K ) ) = 0 . \end{align*}


\begin{align*} & = \sup _ { ( x ' , y ' ) \in X \times \mathbb R } \{ \psi ( x ' , y ) - \frac { 1 } { 2 \epsilon } \left ( \| x - x ' \| ^ 2 - \| x \| ^ 2 \right ) - \frac { 1 } { 2 } \left ( | y - y ' | ^ 2 - \| y \| ^ 2 \right ) \} \\ & = \sup _ { ( x ' , y ' ) \in X \times \mathbb R } \{ \psi ( x ' , y ) + \frac { 1 } { \epsilon } x \cdot x ' + \frac { 1 } { \epsilon } y \cdot y ' - \frac { 1 } { 2 \epsilon } \| x ' \| ^ 2 - \frac { 1 } { 2 \epsilon } \| y ' \| ^ 2 \} \end{align*}


\begin{align*} \log \leq \log ^ + : = \max \left \{ 0 , \log ( \cdot ) \right \} \end{align*}


\begin{align*} a ^ i _ S ( T ) = \begin{cases} 1 , & T \cap F _ i = S \\ 0 , & \end{cases} \end{align*}


\begin{align*} \int _ { B _ \rho ( x _ 0 ) } \big ( \exp ( \frac { | | R ^ \nabla | | ^ 2 } 2 ) - 1 \big ) \ , d v = o ( \rho ^ \lambda ) \rho \rightarrow \infty \ , , \end{align*}


\begin{align*} d ( J ) \leq d ( \mathcal { G } ) = n ^ { 2 } . \end{align*}


\begin{align*} \begin{cases} - \varepsilon ^ 2 ( K _ H ( x ) \nabla u ) = \left ( u - \left ( \frac { \alpha | x | ^ 2 } { 2 } + \beta \right ) \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ & x \in B _ { R ^ * } ( 0 ) , \\ u = 0 , \ \ & x \in \partial B _ { R ^ * } ( 0 ) , \end{cases} \end{align*}


\begin{align*} & \log 4 \mathbf { e } _ 1 + \log 4 \mathbf { e } _ 2 + \log 4 \mathbf { e } _ 3 + \log \frac { 3 } { 2 } \mathbf { e } _ { 1 2 } + \log \frac { 3 } { 2 } \mathbf { e } _ { 1 2 3 ' } \\ & = [ \log 9 , \log 9 , \log 6 , \log 5 4 , \log 5 4 , \log 5 4 , \log 2 1 6 ] ^ \intercal \end{align*}


\begin{align*} \frac { \partial z } { \partial \alpha } = & \frac { \phi ( z ) \ , [ 1 - \Phi ( z ) ] } { \Phi ( z ) \ , [ 1 - \Phi ( z ) ] + z \ , \phi ( z ) \left [ \alpha - ( \alpha + \beta ) \ , \Phi ( z ) \right ] - ( 2 - \alpha - \beta ) \ , \phi ^ 2 ( z ) } . \end{align*}


\begin{align*} \mathbb { E } \left [ H \left ( \mu [ t ] \mid P _ { Y ^ 1 , \cdots , Y ^ k } ^ { ( k + 1 \mid k ) } [ t ] \right ) \right ] = H \left ( \mu ^ { \otimes k + 1 } [ t ] \mid P ^ { ( k + 1 ) } [ t ] \right ) - H \left ( \mu ^ { \otimes k } [ t ] \mid P ^ { ( k ) } [ t ] \right ) = H _ t ^ { k + 1 } - H _ t ^ k . \end{align*}


\begin{align*} \left \{ \begin{aligned} d x _ t ^ { t _ 0 , x _ 0 , \xi , \alpha } = & ~ [ A x _ t ^ { t _ 0 , x _ 0 , \xi , \alpha } + B \alpha _ t + f ( \nu _ t ^ { * , t _ 0 , \xi } ) + b ( \mu _ t ^ { * , t _ 0 , \xi } ) ] d t + \sigma d W _ t + \sigma _ 0 d W ^ 0 _ t , \\ x _ { t _ 0 } ^ { t _ 0 , x _ 0 , \xi , \alpha } = & ~ x _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} & \hphantom { { } = { } } \{ f , g ; F , G \} ( \xi , \eta ; \dot { \xi } , \dot { \eta } ) \\ * & = ( f ( \xi ) , D f ( \xi ) \eta + g ( \xi ) ; F ( \xi ) \dot { \xi } , ( D F ( \xi ) ^ i { } _ { \alpha j } \eta ^ \alpha + G ( \xi ) ) \dot { \xi } + F ( \xi ) \dot { \eta } ) , \end{align*}


\begin{align*} x = x _ 1 , \dotsc , x _ n \in \widetilde W \end{align*}


\begin{align*} \widehat { f } ( \lambda , \omega ) = ( 2 \pi ) ^ { - n / 2 } \int _ { \R ^ n } f ( x ) e ^ { - i \lambda x \cdot \omega } d x , \ , \ , \ , \lambda \in \R , \omega \in S ^ { n - 1 } . \end{align*}


\begin{align*} \xi ( \mu , 0 ) = \xi ^ { \mathrm { i n } } ( \mu ) \ge 0 , \ \mu \in \mathbb { R } _ { > 0 } . \end{align*}


\begin{align*} g ' ( \theta ) = ( \cos \theta - \sin \theta ) \left ( C _ { 0 } - 6 C _ { 1 } \cos \theta \sin \theta ( \cos \theta + \sin \theta ) \right ) . \end{align*}


\begin{align*} \C = \left \{ ( \lambda + \mu \theta ^ { i _ 1 } , \lambda + \mu \theta ^ { i _ 2 } , \cdots , \lambda + \mu \theta ^ { i _ n } ) \ , | \ , \lambda , \mu \in \mathbb { F } _ { p ^ e } \right \} \end{align*}


\begin{align*} O : = \{ f \in X : f ( x ) = - f ( - x ) \} . \end{align*}


\begin{align*} & \sum _ { a = 0 } ^ t \pi _ N ( x ) ^ a ( \log \log x ) ^ { t - a } L ^ { 2 r - a } L ^ { 2 r + x ' - 2 t } \sum _ { ( b _ 1 , b _ 2 , \dots , b _ { t } ) \atop { b _ u \in \mathcal J _ { \mathcal I ( s _ u ) } \atop { a ( \underline { b } ) = a } } } L ^ { - a _ 3 ( \underline { b } ) } \\ & \ll _ r \sum _ { a = 0 } ^ t \pi _ N ( x ) ^ a ( \log \log x ) ^ { t - a } L ^ { 2 r - a } L ^ { 2 r + x ' - 2 t } , \end{align*}


\begin{align*} \int _ { B ( 0 , R ) } \partial _ t a _ { j } \cdot \xi \dd x + \int _ { B ( 0 , R ) } B _ j \times u _ j \cdot \mathbb { P } \xi \dd x = - \mu _ j \int _ { B ( 0 , R ) } b _ j \cdot \nabla \times \xi \dd x . \end{align*}


\begin{align*} R i c ^ W & = R i c ^ g - \frac { 1 } { 2 } \left ( \| \theta \| ^ 2 g - \theta \otimes \theta \right ) , \\ & = \frac { s ^ W } { 4 } g + ( R i c ^ g ) ^ { J , - } + \frac { 1 } { 2 } ( \theta \otimes \theta ) ^ { J , - } . \end{align*}


\begin{align*} f _ { y , \mathbf { X } } ( \beta ) : = \sum _ { i = 1 } ^ p V ( \beta _ i ) - \frac { 1 } { 2 \sigma ^ 2 } \left | y - \mathbf { X } \beta \right | ^ 2 . \end{align*}


\begin{align*} H ( t , z , r , s ) = \min \limits _ { u \in \mathbb U } \big ( \langle f ( t , z , r , u ) , s \rangle + f ^ 0 ( t , z , r , u ) \big ) , \ t \in [ 0 , \vartheta ] , \ z , r , s \in \mathbb R ^ n , \end{align*}


\begin{gather*} m ( f _ 1 ) + m ( f _ 2 ) = \lim _ { \epsilon \rightarrow 0 } \ , \Bigl \{ \int _ \epsilon ^ 1 f _ 1 ( x ) \ , d x + \int _ 0 ^ { 1 - \epsilon } f _ 2 ( x ) \ , d x \Bigr \} \\ = \lim _ { \epsilon \rightarrow 0 } \int _ 0 ^ { 1 - \epsilon } \bigl \{ f _ 1 ( x + \epsilon ) + f _ 2 ( x ) \bigr \} \ , d x \le \lim _ { \epsilon \rightarrow 0 } \int _ 0 ^ { 1 - \epsilon } c ( x + \epsilon , x ) \ , d x = 0 . \end{gather*}


\begin{align*} \mathbf { R } _ y = \frac { 1 } { K } \sum \limits _ { n = 1 } ^ { K } \mathbf { y } ( n ) \mathbf { y } ^ { H } ( n ) , \end{align*}


\begin{align*} p _ { S } ( x ) : = \int _ S \frac { 1 _ { B _ y } ( x ) } { U ( B _ y ) } d U ( y ) , \end{align*}


\begin{align*} \beta _ { \bar { \mathbf { g } } } \mathbf { g } ( \partial _ t ) & = 0 \\ \beta _ { \bar { \mathbf { g } } } \mathbf { g } ( \partial _ a ) & = \beta _ { \bar g } g ( \partial _ a ) + \bar u ^ { - 2 } u \partial _ a u - \bar u ^ { - 1 } g ( \nabla _ { \bar g } \bar u , \partial _ a ) \mbox { f o r $ a = 1 , \dots , n $ } . \end{align*}


\begin{align*} \gamma - m _ { 0 } < \gamma - c ( R ) = c _ { r } ( R ) < \frac { M _ { 0 } m _ { 0 } } { \sigma _ { d } } R . \end{align*}


\begin{align*} \ell _ n ( \theta ) \ ! = \ ! \begin{bmatrix} l _ 0 ( \theta ) I _ m & l _ 1 ( \theta ) I _ m & \dots & l _ { n - 1 } ( \theta ) I _ m \end{bmatrix} ^ { \ ! \top } \ ! \ ! \ ! , \ ; \forall \theta \in [ - h , 0 ] . \end{align*}


\begin{align*} x ( t ) = \int ^ { T _ 2 } _ { T _ 1 } G ( t , \tau ) f ( x ( \tau ) ) \ ; d \tau , t \in [ T _ 1 , T _ 2 ] , \end{align*}


\begin{align*} { f _ i } \left ( { { \bf { x } } + { { \bf { c } } _ i } \Delta t , t + \Delta t } \right ) - { f _ i } \left ( { { \bf { x } } , t } \right ) = { \left ( { { { { \bf { \hat M } } } ^ { - 1 } } { \bf { \hat \Lambda \hat M } } } \right ) _ { i j } } \left [ { { f _ i } \left ( { { \bf { x } } , t } \right ) - f _ i ^ { \left ( { e q } \right ) } \left ( { { \bf { x } } , t } \right ) } \right ] + { { \bf { F } } _ i } \left ( { { \bf { x } } , t } \right ) , \end{align*}


\begin{align*} \Lambda ^ \ell g \cdot \ , \Lambda ^ \ell h : = \sum ^ { 2 } _ { j _ 1 = 1 } \cdot \cdot \cdot \sum ^ { 2 } _ { j _ \ell = 1 } ( \Lambda _ { j _ 1 } \cdots \Lambda _ { j _ \ell } g ) \ , ( \Lambda _ { j _ 1 } \cdots \Lambda _ { j _ \ell } h ) . \end{align*}


\begin{align*} \displaystyle \sum _ { i = 1 } ^ { n } d _ i ^ 2 \leq \frac { 4 m ^ 2 } { n } + \frac { n } { 4 } { ( \triangle ( G ) - \delta ( G ) ) } ^ 2 . \end{align*}


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


\begin{align*} \forall s \in S : C ( s ) = b ( s ) \enspace . \end{align*}


\begin{align*} \frac { \partial ^ { k } \varphi _ { 0 } ^ { - } \left ( \rho , x \right ) } { \partial x ^ { k } } = \frac { \left ( 2 i \right ) ^ { k } } { 2 \pi i } \int _ { 0 } ^ { \infty } \left ( q ( s ) L _ { k } ^ { \left ( 1 \right ) } \left ( s , \rho \right ) + Q ^ { \prime } ( s ) L _ { k } ^ { \left ( 2 \right ) } \left ( s , \rho \right ) \right ) d s \end{align*}


\begin{align*} \Psi _ { \rho } ( v , w ) : = \Phi _ { \rho } ^ { - 1 } \big ( \Phi _ { \rho } ( v ) + w \big ) \end{align*}


\begin{align*} \omega _ \epsilon : = \{ x \in \Omega : | x _ \beta | < \epsilon \delta \frac { b _ \alpha } { \sqrt { b _ \beta } } , \beta = \alpha + 1 , \ldots , n - 1 , x _ n < \epsilon ^ 2 \delta ^ 2 b _ \alpha \} \end{align*}


\begin{align*} \begin{aligned} v ^ { ( 1 ) } = \textup { R e } \ , u ^ { ( 1 ) } & = \big ( 1 , \cos ( \pi / N ) , \cos ( 2 \pi / N ) , \ldots , \cos ( ( N - 1 ) \pi / N ) \big ) \\ v ^ { ( 2 ) } = \textup { I m } \ , u ^ { ( 1 ) } & = \big ( 0 , \sin ( \pi / N ) , \sin ( 2 \pi / N ) , \ldots , \sin ( ( N - 1 ) \pi / N ) \big ) \end{aligned} \end{align*}


\begin{align*} & \int _ { \mathbb { R } ^ { 3 } } \Phi _ { + } ^ { 2 } \chi _ R \dd x - \int _ { \mathbb { R } ^ { 3 } } \Phi _ { 0 , + } ^ { 2 } \chi _ R \dd x + 2 \mu \int _ { 0 } ^ { t } \int _ { \mathbb { R } ^ { 3 } } | \nabla \Phi _ { + } | ^ { 2 } \chi _ R \dd x \dd s \\ & = \mu \int _ { 0 } ^ { t } \int _ { \mathbb { R } ^ { 3 } } \Phi _ { + } ^ { 2 } \left ( \Delta + \frac { 2 } { r } \partial _ r \right ) \chi _ R \dd x \dd s + \int _ { 0 } ^ { t } \int _ { \mathbb { R } ^ { 3 } } u \cdot \nabla \chi _ R \Phi _ { + } ^ { 2 } \dd x \dd s . \end{align*}


\begin{align*} F ( x ) = \sum _ { n = 0 } ^ \infty c _ n ( a , b ) \ , G ( x ; ( a + n ) \theta ) . \end{align*}


\begin{align*} h _ m = F _ m \circ F _ { m - 1 } \circ \dots \circ F _ 1 , \end{align*}


\begin{align*} K _ { p , d } = 2 ^ { - p } \pi ^ { - d / 2 } \frac { \Gamma \left ( \frac { d - p } { 2 } \right ) } { \Gamma \left ( \frac { p } { 2 } \right ) } . \end{align*}


\begin{align*} \bar { J } _ 0 ( t , x ) : = \int _ { Y ^ { \ast } } J _ 0 ( t , x , y ) d y . \end{align*}


\begin{align*} \begin{cases} - \delta ^ 2 ( K ( \hat { x } ) \nabla v ) = ( V _ { \delta , \hat { x } , \hat { q } , z } - \hat { q } ) ^ { p } _ + , \ \ & \ \Omega , \\ v = 0 , \ \ & \ \partial \Omega . \end{cases} \end{align*}


\begin{align*} x = x _ a + x _ s + x _ p \end{align*}


\begin{align*} \omega : = \nabla \times u = \partial _ { x _ 1 } u _ { 2 } - \partial _ { x _ 2 } u _ { 1 } \end{align*}


\begin{align*} r _ d \le d \Big | \frac { p _ 0 } { p _ 1 } \Big | = d \Big | \frac { p ( 0 ) } { p ' ( 0 ) } \Big | . \end{align*}


\begin{align*} \beta _ { q } ( n ) = \lim _ { v \to \infty } \frac { \# \{ \vec { x } \in ( \Z / p ^ { v } \Z ) ^ { 4 } | Q ( \vec { x } ) \equiv n \pmod { p ^ { v } } \} } { p ^ { 3 v } } , \end{align*}


\begin{align*} \overline { x } = x + \sqrt { x ^ 2 - 1 } . \end{align*}


\begin{align*} \varpi _ 1 ( \mu , \nu ) = \nu \varpi ' ( \mu ) - \varpi ( \nu ) , \end{align*}


\begin{align*} M ' ( S ) = \varinjlim _ i N ( T _ i ) = \varinjlim _ i \varprojlim _ { n \in \Delta } N ( T _ { i , n } ) = \varprojlim _ { n \in \Delta } \varinjlim _ i N ( T _ { i , n } ) = \varprojlim _ { n \in \Delta } M ' ( S _ n ) , \end{align*}


\begin{align*} \dot { { x } } _ a = \ , - \gamma c _ 1 \delta \left . \dfrac { \partial h } { \partial x } \right | _ { { x } _ a } - \gamma \sum _ { k = 2 } ^ { \infty } c _ k \ , { \delta ^ { 2 k - 1 } } \ , \left . \dfrac { \partial ^ { 2 k - 1 } h } { \partial x ^ { 2 k - 1 } } \right | _ { { x } _ a } \end{align*}


\begin{align*} \begin{aligned} & ( C _ { a j } - C _ j - C _ a ) \ , [ C _ { i k } , C _ { k \ell } ] + ( C _ { a i } - C _ i - C _ a ) [ C _ { k \ell } , C _ { j k } ] + ( C _ { a \ell } - C _ \ell - C _ a ) [ C _ { i j } , C _ { j k } ] \\ & \qquad + ( C _ { a k } - C _ k - C _ a ) [ C _ { i \ell } , C _ { j \ell } ] = 0 \ , . \end{aligned} \end{align*}


\begin{align*} \langle \Pi _ m Q \Pi _ m h - Q h , h \rangle \leq \| ( I - \Pi _ m ) Q h \| + \| ( I - \Pi _ m ) Q h _ m \| = ( 1 ) _ m + ( 2 ) _ m . \end{align*}


\begin{align*} g ( \Phi ) _ { t } + u \cdot \nabla g ( \Phi ) = 0 , \end{align*}


\begin{align*} \left \langle l _ 1 , m _ 1 \right | \sin ( n \theta ) \left | l _ 2 , m _ 2 \right \rangle = 0 \ , , \end{align*}


\begin{align*} \Theta ^ { 2 ( 1 ) } & = d \theta ^ i { } _ j + \theta ^ i { } _ { \alpha j } \wedge \theta ^ \alpha + \theta ^ i { } _ \alpha \wedge \theta ^ \alpha _ j \\ * & = \Omega ^ i { } _ j + \theta ^ i { } _ { \alpha j } \wedge \theta ^ \alpha , \\ * \Theta ^ { 2 ( 2 ) } & = d \theta ^ i { } _ j + \theta ^ i { } _ { j \alpha } \wedge \theta ^ \alpha + \theta ^ i { } _ \alpha \wedge \theta ^ \alpha { } _ j \\ * & = \Omega ^ i { } _ j + \theta ^ i { } _ { j \alpha } \wedge \theta ^ \alpha . \end{align*}


\begin{align*} p _ 0 = - 2 k ( t _ k - \xi _ k ) - \lambda _ \varphi \| x _ k - z \| , p = - 2 k ( x _ k - y _ k ) . \end{align*}


\begin{align*} \phi = 2 \pi \frac { d } { \lambda } { s i n } \theta _ 0 = 2 \pi Z \end{align*}


\begin{align*} \chi _ D ( x _ 1 ) = \varphi \left ( \frac { \sqrt { K } | x _ 1 | } { 4 R } \right ) , \chi _ T ( x _ 1 ) = 1 - \varphi \left ( \frac { \sqrt { K } | x _ 1 | } { R } \right ) . \end{align*}


\begin{align*} \left \Vert \varphi \right \Vert _ { p } = \max _ { 0 \leq q \leq p } | \sup _ { x \in \mathbb { R } } \left ( e ^ { p \left \vert x \right \vert } \varphi ^ { ( q ) } \left ( x \right ) \right ) | \end{align*}


\begin{align*} x ( t ) - z ( t ) = x ( 0 ) - z ( 0 ) + \int _ 0 ^ t \dot { x } ( \tau ) - \dot { z } ( \tau ) \ , d \tau \end{align*}


\begin{align*} \begin{aligned} x _ n ^ * - x _ n ^ 0 = \ , & x _ n - \sum _ { \beta = 1 } ^ \alpha x _ \beta \rho _ \beta ( 0 , \tilde { x } ) - \rho ( 0 , \tilde { x } ) \\ > \ , & \rho ( \hat { x } , \tilde { x } ) - \sum _ { \beta = 1 } ^ \alpha x _ \beta \rho _ \beta ( 0 , \tilde { x } ) - \rho ( 0 , \tilde { x } ) \geq 0 \end{aligned} \end{align*}


\begin{align*} p : w = w _ 1 \rightarrow w _ 2 \rightarrow \cdots \rightarrow w _ { \ell + 1 } = w ' . \end{align*}


\begin{align*} - ( K _ H ( x ) \nabla u _ { \bar { x } } ) = \frac { 1 } { \varepsilon ^ 2 } \left ( u _ { \bar { x } } - q _ { \bar { x } } \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + , \ \ \ \ \ \Omega _ { \bar { x } } . \end{align*}


\begin{align*} \mathcal { Y M } _ e ( \nabla ) = \int _ M \exp ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) \ , d v \ , , \end{align*}


\begin{align*} i \partial _ t v _ 2 & = \lambda _ 6 t ^ { - 1 } \mathcal { F } M ( t ) ^ { - 1 } \mathcal { F } ^ { - 1 } \mathcal { N } _ 2 ( \mathcal { F } M ( t ) { { \mathcal F } } ^ { - 1 } v _ 1 , \mathcal { F } M ( t ) { { \mathcal F } } ^ { - 1 } v _ 2 ) \\ & = \lambda _ 6 t ^ { - 1 } \mathcal { N } _ 2 ( v _ 1 , v _ 2 ) + i R _ 2 ( s ) . \end{align*}


\begin{align*} A ( y _ 1 x ) - A ( y _ 1 x ' ) = A ( 1 ^ \infty ) - A ( 1 0 ^ \infty ) = c - d \ . \end{align*}


\begin{align*} g ( \rho , x ) = e ^ { - i \rho x } \left ( 1 + \left ( z + 1 \right ) \sum _ { n = 0 } ^ { \infty } \left ( - 1 \right ) ^ { n } z ^ { n } b _ { n } ( x ) \right ) \end{align*}


\begin{align*} h _ { x , t } : = b \vert _ { E ^ s } + \varphi ( t ) b \vert _ { E ^ u } + d t ^ 2 \end{align*}


\begin{align*} \Phi ( \mathcal { P } ( Z ) ) = \mathcal { P } ' ( \Phi ( Z ) ) \end{align*}


\begin{align*} L ( p ) = \log F ( p , 8 / 3 ) , R ( p ) = \log \left ( e ^ { - p / 6 } \tilde G ( p , 2 ) \right ) \end{align*}


\begin{align*} \lim _ { k \to \infty } \deg _ p ( N _ k ' ) = \lim _ { k \to \infty } \deg _ p ( N _ k '' ) ; \end{align*}


\begin{align*} M _ { \pm } ( x ; z ) = \left ( M _ { \pm , 1 } ( x ; z ) , M _ { \pm , 2 } ( x ; z ) \right ) . \end{align*}


\begin{align*} \overline { I x I } = \bigsqcup _ { y \leq x } I y I \subseteq G ( \breve F ) . \end{align*}


\begin{align*} \widetilde { C } = C \times \{ t \} . \end{align*}


\begin{align*} \mathbf { x } ^ { ( i ) } \circ \mathbf { s } ^ { ( i ) } = \left ( { \left ( \mathbf { x } ^ { ( i ) } \right ) } ^ T \mathbf { s } ^ { ( i ) } , \mathbf { x } _ 1 ^ { ( i ) } \mathbf { s } _ { 2 : n _ i } ^ { ( i ) } + \mathbf { s } _ 1 ^ { ( i ) } \mathbf { x } _ { 2 : n _ i } ^ { ( i ) } \right ) , \forall \mathbf { x } ^ { ( i ) } , \mathbf { s } ^ { ( i ) } \in \mathcal { R } ^ { n _ i } \end{align*}


\begin{align*} v = 0 , ( \nu ' ) ( h ) ( \bar u ) + \nu ( v ) = 0 , A ' ( h ) = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \lim _ { n \to + \infty } \| u _ { n } - z _ { n } \| _ { L ^ { p } ( ( a , b ) ) } = 0 \forall p \in [ 1 , + \infty ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta u & = | x | ^ { a } v ^ { p } & & \quad \mbox { i n } \Omega , \\ \Delta v & = | x | ^ { b } v ^ { q } f ( | \nabla u | ) & & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \langle f ( x ' ) , h ( x ' ) \rangle d x ' = \lim _ { \varepsilon \to 0 ^ + } \int _ { \R ^ { n - 1 } } \langle f ( x ' ) , \widetilde { \Psi } _ { \varepsilon , \varepsilon ^ { - 1 } } * h ( x ' ) \rangle d x ' . \end{align*}


\begin{align*} \frac { \partial u } { \partial t } = \mathcal { L } ( u ) + f ( t , x ) , \mathcal { B } ( u ) = g ( t , x ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} S ^ \prime _ t & = - \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t , & S _ 0 & = s _ 0 \\ I ^ \prime _ t & = \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t - \gamma I _ t - I _ t \phi \left ( I _ t \right ) , & I _ 0 & = i _ 0 , \end{aligned} \right . \end{align*}


\begin{align*} \frac { d } { d \tau } \mathcal { U } _ { \lambda } ( \tau ) = C _ { 1 } \Theta _ { \lambda } ( \tau ) \geq C _ { 1 } \mathcal { U } _ { \lambda } ( \tau ) . \end{align*}


\begin{align*} \tau = \frac { ( 1 / 2 + r + \epsilon ) \log ( | z _ 0 | ) } { \lambda _ R | z _ 0 | } \end{align*}


\begin{align*} \sigma _ { k } ^ { i i } = \frac { \partial \sigma _ k } { \partial \kappa _ i } , \sigma _ { k } ^ { i i , j j } = \frac { \partial ^ 2 \sigma _ { k } } { \partial \kappa _ i \partial \kappa _ j } . \end{align*}


\begin{align*} \Gamma ( g , g ' ) = \tilde { \Gamma } ( g ^ { - 1 } \circ g ' ) \end{align*}


\begin{align*} \mathcal { E } ( \eta ) = - \frac { \kappa _ 1 + \kappa _ 2 } { \kappa _ 2 + 1 } + \sqrt { \left ( \frac { \kappa _ 1 + \kappa _ 2 } { \kappa _ 2 + 1 } \right ) ^ 2 + \frac { \eta } { h ( \kappa _ 2 + 1 ) } } , \end{align*}


\begin{align*} \begin{aligned} & z '' _ { b e n } + ( M ^ S _ { b e n } - P ) z _ { b e n } = 0 \\ & z '' _ { n b } + ( M ^ S _ { n b } - P ) z _ { n b } = 0 , \end{aligned} \end{align*}


\begin{align*} h : = \left ( \prod \limits _ { j = 1 } ^ s \frac { 1 } { y _ j } \right ) \left ( \sum \limits _ { k , k ' = 1 } ^ s \frac { 1 } { y _ k y _ { k ' } } d x _ k \otimes d x _ { k ' } + d y _ k \otimes d y _ { k ' } \right ) + d x _ { s + 1 } ^ 2 + d y _ { s + 1 } ^ 2 , \end{align*}


\begin{align*} u _ 2 ( t ) = M ( t ) D ( t ) \mathcal { F } M ( t ) \mathcal { F } ^ { - 1 } v _ 2 ( t ) . \end{align*}


\begin{align*} \min _ { \substack { \omega _ { n , k } , \ , { { \bf { u } } } _ k \\ { \alpha _ z , \ss _ { z } , \gamma ( . ) } } } & \mathcal { L } \\ & \omega _ { n , k } \in \{ 0 , 1 \} , \forall n , \forall k . \end{align*}


\begin{align*} Y ^ T _ t : = & \mathcal S ( T ) Y _ 0 + \int _ 0 ^ t \mathcal S ( T - s ) \alpha _ s d s + \int _ 0 ^ t \mathcal S ( T - s ) \sigma _ s d W _ s \\ = & \tilde Y _ 0 + \int _ 0 ^ t \tilde { \alpha } _ s d s + \int _ 0 ^ t \tilde { \sigma } _ s d W _ s , t \in [ 0 , T ] , \end{align*}


\begin{align*} E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] = E _ { \xi _ 0 } \left [ h \bigl ( x , Z _ 1 ^ \theta \bigr ) \right ] , \forall x \in \mathbb R , \end{align*}


\begin{align*} \cos ( n \theta ) = \sum \limits _ { l = 0 } ^ n a ^ n _ l P _ l ^ 0 ( \cos \theta ) \ , , \qquad \sin ( n \theta ) = \sum \limits _ { l = 0 } ^ n b ^ n _ l P _ l ^ 1 ( \cos \theta ) \ , . \end{align*}


\begin{align*} \omega ( E , \epsilon ) = - 1 , L ( E , \epsilon ) = - \epsilon + \log | \lambda | . \end{align*}


\begin{align*} K _ { \alpha } \left ( t \right ) = L _ { \alpha } ^ { M } \left ( t \right ) + L _ { \alpha } ^ { P } \left ( t \right ) \end{align*}


\begin{align*} \sum \limits _ { x , y \in \mathbb { F } _ q , x \neq 1 } \overline { \chi _ 4 } ( x ) \chi _ 4 ( y ) \chi _ 4 ( 1 - y ) \chi _ 4 ( x - y ) = - 2 \rho \end{align*}


\begin{align*} M _ p = \max \{ d ( x _ 0 , p ) , d ( u , p ) \} . \end{align*}


\begin{align*} \d X _ t ^ i = b ^ i ( X _ t ) \d t + \sigma \d W _ t ^ i , ~ ~ \tilde X _ { n + 1 } ^ i = \tilde X _ n ^ i + b ^ i ( \tilde X _ n ) \tau + \sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , ~ ~ i = 1 , \cdots , N . \end{align*}


\begin{align*} \| \mathbf { c } _ X \| _ 2 \| \mathbf { c } _ Y \| _ 2 \leq N L \| \mathbf { x } \| _ 2 \| \mathbf { y } \| _ 2 = N L , \end{align*}


\begin{align*} \theta _ s ( f ^ \zeta ) ( \gamma _ 1 , \dots , \gamma _ s ) = \theta _ r ( f ) ( \gamma _ { \zeta ( 1 ) } , \dots , \gamma _ { \zeta ( r ) } ) , \end{align*}


\begin{align*} h _ \Lambda ( \nu | \mu ) : = \begin{cases} \sum _ { \omega _ \Lambda \in \Omega _ \Lambda } \nu ( \omega _ { \Lambda } ) \log \frac { \nu ( \omega _ { \Lambda } ) } { \mu ( \omega _ \Lambda ) } , & \nu \ll \mu , \\ \ \infty , & \end{cases} \end{align*}


\begin{align*} A ^ { 1 , \dots , k } : = A ^ { 1 } A ^ { 2 } \dots A ^ { k } . \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } t } \int _ { \Gamma ( t ) } f \ , \mathrm { d } \mathcal { H } ^ d = \int _ { \Gamma ( t ) } \partial ^ \square f - f H V \ , \mathrm { d } \mathcal { H } ^ d . \end{align*}


\begin{align*} \Sigma ^ { \pm } ( s - w , \phi , F _ { \pm } ) = & \lim _ { z \downarrow 0 } \int _ 0 ^ \infty w _ z ( \lambda ) \lambda ^ { 4 ( 1 - s + w ) } \int _ { - \infty } ^ \infty \int _ { 0 } ^ \infty \phi ( n _ { u } ^ t a _ { \frac { 1 } { t } } ) \\ & \times \sum _ { m = 1 } ^ \infty \sum _ { n = 0 } ^ { 3 m - 1 } \hat { F } ( \lambda a _ { \frac { 1 } { t } } n _ u \cdot ( 0 , 0 , 3 m , n ) ) \frac { d \lambda } { \lambda } \frac { d t } { t ^ 3 } d u . \end{align*}


\begin{align*} \log \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) & = \log \left [ \frac { s } { 2 \Gamma ( 1 / s ) } \exp \left ( - \left | \frac { x - \mu } { \sigma } \right | ^ s \right ) \right ] \\ & = \log s - \log ( 2 \Gamma ( 1 / s ) ) - \left | \frac { x - \mu } { \sigma } \right | ^ s . \end{align*}


\begin{align*} k _ 3 ( \langle H \rangle ) = \frac { q - 1 } { 1 2 } [ k _ 3 ( \langle H \rangle , 1 ) + k _ 3 ( \langle H \rangle , g ) ] . \end{align*}


\begin{align*} c _ i : = 2 a _ 0 - ( a _ { - i } + a _ i ) \end{align*}


\begin{align*} r ^ 2 + r q + q ^ 2 = n \ , . \end{align*}


\begin{align*} \begin{aligned} & d _ 2 ( \mathbf { x } , \mathbf { s } , \kappa , \tau ) = \sqrt { 2 } \left \| \mathbf { w } _ { x s \kappa \tau } - \mu \mathbf { e } \right \| , \\ & d _ \infty ( \mathbf { x } , \mathbf { s } , \kappa , \tau ) = \max \left ( d _ \infty ( \mathbf { x } , \mathbf { s } ) , \left | \kappa \tau - \mu \right | \right ) , \end{aligned} \end{align*}


\begin{align*} N _ f ( X ) = C _ f X + \Delta \end{align*}


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


\begin{align*} \forall \varepsilon > 0 , \lim _ { n \to \infty } \sum _ { m = 1 } ^ n \mathbb { E } _ { n , m } ' [ \| \chi _ { n , m } ' \| _ { L ^ 2 ( \mu ) } ^ 2 : \| \chi _ { n , m } ' \| _ { L ^ 2 ( \mu ) } > \varepsilon ] = 0 \end{align*}


\begin{align*} \Phi _ { k , r } ( \omega ) = \{ x \in C ( \omega ) \cap \bar { B } ( 0 , r ) \mid \langle \mathcal { F } ( x ^ * ) ( \omega ) , x - x ^ * ( \omega ) \rangle \le - \frac { 1 } { k } \} . \end{align*}


\begin{align*} i \leq N _ v / 2 , \ \beta _ i ^ v = \beta _ i ^ { v _ l } \oplus \beta _ i ^ { v _ r } , i > N _ v / 2 , \ \beta _ i ^ v = \beta ^ { v _ r } _ { i - N _ v / 2 } . \end{align*}


\begin{align*} D _ { a , k } : = \begin{pmatrix} D _ { a } \\ & \ddots \\ & & D _ { a } \\ & & & 1 _ { 2 n - 2 k } \end{pmatrix} = \left ( \bigoplus _ { 1 } ^ { k } D _ { a } \right ) \bigoplus 1 _ { 2 ( n - k ) } , D _ { a } : = \begin{pmatrix} a & 0 \\ 0 & a ^ { - 1 } \end{pmatrix} . \end{align*}


\begin{align*} \begin{aligned} & d _ 2 ( \mathbf { x } , \mathbf { s } ) = \sqrt { 2 } \left \| \mathbf { w } _ { x s } - \mu \mathbf { e } \right \| , \\ & d _ \infty ( \mathbf { x } , \mathbf { s } ) = \max _ { i = 1 , \cdots , k , j = 0 , 1 } \left ( \left | ( \mathbf { w } _ { x s } ) _ 0 ^ { ( i ) } + \| ( \mathbf { w } _ { x s } ) _ 1 ^ { ( i ) } \| - \mu \right | , \left | ( \mathbf { w } _ { x s } ) _ 0 ^ { ( i ) } - \| ( \mathbf { w } _ { x s } ) _ 1 ^ { ( i ) } \| - \mu \right | \right ) . \end{aligned} \end{align*}


\begin{align*} A _ { t } \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) + \nabla Q \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) = - \mu \nabla \times B \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \textrm { o n } \ L ^ { 1 } ( \mathbb { R } ^ { 3 } ) , \end{align*}


\begin{align*} L ( \gamma _ { u _ { k - 1 } , u _ k } ) & < d ( u _ { k - 1 } , u _ k ) + \varepsilon \delta = d ( u _ { k - 1 } , u _ k ) + \varepsilon \min \bigl \{ 1 , \frac { d ( u _ { k - 1 } , u _ k ) } { C _ { n + 1 } \beta _ k } \bigr \} . \end{align*}


\begin{align*} \mathcal { F } _ i : = \mathcal { H } o m ( \pi ^ * _ i ( \iota ^ * _ i ( \mathcal { P } _ i ) ) , \pi ^ * _ { i + 1 } ( { \iota ' } ^ * _ i ( \mathcal { P } _ { i + 1 } ) ) ) . \end{align*}


\begin{align*} f _ { i : n } ( x ) = \frac { f ( x ) } { \operatorname { B } ( i , n - i + 1 ) } \sum _ { l = 0 } ^ { n - i } \binom { n - i } { l } ( - 1 ) ^ l F ( x ) ^ { i + l - 1 } . \end{align*}


\begin{align*} 0 = \phi ^ \theta ( e ) = \tau ( u _ { l - 1 } , e ) \theta ( u _ { l - 1 } ) + \phi ( e ) + \tau ( u _ l , e ) \theta ( u _ l ) , \end{align*}


\begin{align*} \phi _ m ( x ) = \begin{cases} 1 & x \in A _ m ^ { ( 1 ) } \\ 0 & x \notin A _ m ^ { ( 2 ) } . \end{cases} \end{align*}


\begin{align*} \mathfrak { t } : = \sum _ { j = 0 } ^ 2 \sum _ { p \in C \cap H _ j } ( I _ p ( C , H _ { j } ) - 1 ) . \end{align*}


\begin{align*} \frac { \dd } { \dd t } p ^ { \varepsilon } ( x , t ) = Y _ t h ' ( x ) \partial _ { x } p ^ { \varepsilon } ( x , t ) + \frac { p ^ { \varepsilon } ( x , t ) } { \varepsilon } \left ( - \frac { h ^ 2 ( x ) } { 2 } + \frac { \left ( Y _ t h ' ( x ) \right ) ^ 2 } { 2 } + \frac { \varepsilon } { 2 } Y _ t h '' ( x ) \right ) + \frac { \varepsilon } { 2 } \partial ^ 2 _ { x x } p ^ { \varepsilon } ( x , t ) , \end{align*}


\begin{align*} \log Z = \log \int _ { \R ^ n } e ^ f \ , d \mu ^ { \otimes n } = \sup _ { Q \in \P ( \R ^ n ) } \left ( \int _ { \R ^ n } f \ , d Q - H ( Q \ , | \ , \mu ^ { \otimes n } ) \right ) , \end{align*}


\begin{align*} \forall t > 0 , \ H _ { m i n } ( E [ t ] ) = t H _ { m i n } ( E ) . \end{align*}


\begin{align*} \begin{aligned} Z _ { I } ( 1 , \{ 2 \} ^ { n } ; ( \alpha , \beta ) ) = Z _ { I I } ( \{ 2 \} ^ { n - 1 } , 3 ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \int _ { \hat { h } _ { P } ^ { - 1 } ( b ) ( F ) } f _ { \hat { \alpha } } \abs { \omega _ { \hat { h } _ { P } } } = \int _ { \check { h } _ { P } ^ { - 1 } ( b ) ( F ) } f _ { \check { \alpha } } \abs { \omega _ { \check { h } _ { P } } } . \end{align*}


\begin{align*} C ^ { n + 1 } _ c ( G , V ) : = C ^ 0 _ c ( G , C ^ n _ c ( G , V ) ) , n \geq 0 . \end{align*}


\begin{align*} [ M ] & = \frac { \prod \limits _ { i = 0 } ^ N v _ i ^ { x _ i } } { \prod \limits _ { n = 1 } ^ \infty ( 1 - z ^ n ) ^ { N } \prod \limits _ { n = 1 } ^ \infty \prod \limits _ { 1 \leq i < j \leq N } ( 1 - v _ j v _ i ^ { - 1 } z ^ { n - 1 } ) ( 1 - v _ i v _ j ^ { - 1 } z ^ n ) } . \end{align*}


\begin{align*} \begin{pmatrix} A _ { \mathrm { c } \mathrm { c } } - A _ { \mathrm { R } \mathrm { c } } ^ { \top } R + R ^ { \top } A _ { \mathrm { R } \mathrm { c } } & B _ 0 ^ { \top } \\ B _ 0 & 0 \\ \end{pmatrix} \begin{pmatrix} U _ \mathrm { c } \\ P _ 0 \end{pmatrix} = \begin{pmatrix} F _ \mathrm { c } - R ^ \top F _ \mathrm { R } \\ 0 \end{pmatrix} , \end{align*}


\begin{align*} \langle \mathcal { F } ( x ^ * ) , \tilde { \eta } - x ^ * \rangle _ { \mathcal { L } ^ 2 } = & \int _ { \Omega } \langle \mathcal { F } ( x ^ * ) ( \omega ) , \tilde { \eta } ( \omega ) - x ^ * ( \omega ) \rangle P ( d ( \omega ) ) \\ = & \int _ { A _ { k , r } } \langle \mathcal { F } ( x ^ * ) ( \omega ) , \eta ( \omega ) - x ^ * ( \omega ) \rangle P ( d ( \omega ) ) \\ < & - \frac { 1 } { k } P ( A _ { k , r } ) , \end{align*}


\begin{align*} \{ A ( \xi ) \} \cap \mathrm { K e r } ( B ) = \{ 0 \} , \forall \xi \in \R ^ * . \end{align*}


\begin{align*} c _ { k + 1 } = \sum _ { i = 1 } ^ { t - 1 } c _ i a _ i + c _ { k } \sum _ { i \in \Omega _ t ^ e } a _ i + c _ { k + 1 } \sum _ { i \in \Omega _ t ^ o } a _ i , \end{align*}


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


\begin{align*} ( \varphi \psi ) ( h ) = \varphi ( h _ { ( 1 ) } ) \psi ( h _ { ( 2 ) } ) = \left < h _ { ( 1 ) } , \varphi \right > \left < h _ { ( 2 ) } , \psi \right > \end{align*}


\begin{align*} \xi ' = - L - 2 , \eta ' = \frac { Q } { 2 ( L + 2 ) } - \sqrt { \kappa B _ 4 } , \zeta ' = \kappa \left ( \frac { B _ 3 } { 2 \sqrt { \kappa B _ 4 } } + 3 \right ) , \sigma ' = \sqrt { \kappa B _ 4 } , \end{align*}


\begin{align*} p _ k = \log _ 2 \binom { 2 k } { k } . \end{align*}


\begin{align*} \Sigma \big ( X , q \big ) = X \frac { \lambda ( q ) } { q } + \mathcal { O } \big ( X ^ \varepsilon \big ) \ , . \end{align*}


\begin{align*} g _ 1 ^ { \prime + } ( 0 ) & = \sum _ { i = 1 } ^ n \int _ { \R ^ n } \partial _ i f ( x ) \big ( T _ i ( x _ i ) - x _ i \big ) Q ( x ) d x , \\ g _ 2 ^ { \prime + } ( 0 ) & = - \sum _ { i = 1 } ^ n \int _ { \R } \left ( T _ i ' ( x _ i ) - 1 \right ) Q _ i ( x _ i ) d x _ i . \end{align*}


\begin{align*} H _ \alpha ( X ) = \frac { 1 } { 1 - \alpha } \log \left ( \int ^ \infty _ 0 f ( x ) ^ { \alpha } \mathrm { d } x \right ) , \alpha > 0 , \ ; \alpha \neq 1 , \end{align*}


\begin{align*} \mathcal { O } ( x ) : = \{ \Phi ( t , x ) \mid ( t , x ) \in \mathrm { d o m } \ , \Phi \} . \end{align*}


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


\begin{align*} d X _ t = \left ( b _ 0 ( t , X ) + \langle b ( t , X , \cdot ) , \mu \rangle \right ) d t + d B ^ H _ t , \quad \operatorname { L a w } ( X ) = \mu , \quad \operatorname { L a w } ( X _ 0 ) = \mu _ 0 . \end{align*}


\begin{align*} \mathbb { E } \left [ \mathbf { y ' } ^ H \mathbf { y } ' \right ] = \sum _ { l = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M \mathbb { E } \left [ y _ l ^ * y _ j \right ] + 2 \sum _ { i = 1 } ^ { M } \mathbb { E } \left [ y _ i ^ * y _ i \right ] , \end{align*}


\begin{align*} \gamma ^ i ( x ) : = \frac 1 { N - 1 } \sum _ { j \neq i } K ( x ^ i - x ^ j ) , ~ ~ ~ ~ \forall x \in \mathbb R ^ { N d } , \end{align*}


\begin{align*} \partial _ t c _ { \min \ , | t } = \partial _ t c _ { | ( t , p ) } \end{align*}


\begin{align*} \gamma _ { t - 1 } \alpha ( \gamma ) ^ 2 = ( 1 - \alpha ( \gamma ) ) \alpha _ { t - 1 } ^ 2 \gamma + \mu \gamma \gamma _ { t - 1 } \alpha ( \gamma ) . \end{align*}


\begin{align*} f ( z , t ) = \int _ \Omega \int _ { \C ^ n } \widehat { f } ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \ , d \nu ( a ) . \end{align*}


\begin{align*} & \lim _ { M \rightarrow \infty } \mathbb { E } \int _ 0 ^ T \psi ( t ) \Big [ \Vert X ( t \wedge \tau _ u ^ M ) \Vert _ H ^ 2 - \Vert x \Vert _ H ^ 2 \Big ] d t \\ = & \lim _ { M \rightarrow \infty } \liminf _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T \psi ( t ) \Big [ \Vert X _ n ( t \wedge \tau _ u ^ M ) \Vert _ H ^ 2 - \Vert P _ n x \Vert _ H ^ 2 \Big ] d t . \end{align*}


\begin{align*} \psi _ { 2 n } = \begin{pmatrix} \psi _ { 2 } \\ & \psi _ { 2 } \\ & & \ddots \\ & & & \psi _ { 2 } \end{pmatrix} = \bigoplus _ { 1 } ^ { n } \psi _ { 2 } , \psi _ { 2 } = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \end{align*}


\begin{align*} \zeta _ { x } ^ k : = F ( x ^ k ) - F ( \tilde { x } ^ k ) + \beta ( x ^ k - \tilde { x } ^ k ) , \end{align*}


\begin{align*} M _ 1 = & \bigcup _ { i = 1 } ^ { \infty } \Big ( [ \epsilon _ i \mathcal { H } _ { n _ i } , \mathcal { H } _ { n _ i } - L _ i ] \cup [ \mathcal { H } _ { n _ i } + L _ i , \mathcal { H } _ { n _ i + 1 } ] \cap \N \Big ) , \\ M _ 2 = & M \setminus M _ 1 . \end{align*}


\begin{align*} h ( q , r _ 0 ) = - \hat W _ { r _ 0 } ( q ) = - \frac { 2 } { \pi q } \sin ( 2 \pi q r _ 0 ) . \end{align*}


\begin{align*} g \cdot \overline { h } : = \overline { g h } . \end{align*}


\begin{align*} & \prod _ { j = k } ^ { x _ 2 } | \cos ( \pi \theta _ j ) ) | \leq C ( \varepsilon ) c _ { n , \ell } e ^ { ( - \ln { 2 } + \varepsilon ) | x _ 2 - k | } , \ \ \\ & \prod _ { j = x _ 1 } ^ k | \cos ( \pi \theta _ j ) ) | \leq C ( \varepsilon ) c _ { n , \ell - 1 } e ^ { ( - \ln 2 + \varepsilon ) | k - x _ 1 | } . \end{align*}


\begin{align*} L ( x ) = a _ 0 x + a _ 1 x ^ { q ^ s } + a _ 2 x ^ { q ^ { 2 s } } + \cdots + a _ m x ^ { q ^ { m s } } \in F [ x ] , \end{align*}


\begin{align*} \sum _ { k = 1 } ^ L z _ { s , k } \frac { \partial } { \partial z _ { s , k } } \Lambda _ a = \left \{ \begin{array} { l l } 0 & \mbox { f o r $ 1 \leq s \leq m - a $ , } \\ 1 & \mbox { f o r $ m - a < s \leq m $ . } \end{array} \right . \end{align*}


\begin{align*} \hat { A } ( y | x ) = A ( x ) \ , \ \forall \ ( y | x ) \in \hat { \Omega } \ . \end{align*}


\begin{align*} \tilde { Q } _ { i j } ( H ) = \partial ^ 2 _ { i j } H - \tilde { \Gamma } ^ i _ { j i } \partial _ i H - \tilde { \Gamma } ^ j _ { i j } \partial _ j H = 0 , \quad \forall i < j , \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta u & = | x | ^ { a } v ^ { p } & & \quad \mbox { i n } B _ R , \\ \Delta v & = | x | ^ { b } v ^ { q } f ( | \nabla u | ) & & \quad \mbox { i n } B _ R , \end{aligned} \right . \end{align*}


\begin{align*} & [ C _ I , C _ J ] = 0 I \cap J = \emptyset \ \ \ \ I \subseteq J \ , , \\ & C _ I = \frac { 1 } { 2 } \sum _ { i , j \in I \atop i \neq j } C _ { i j } - ( | I | - 2 ) \sum _ { i \in I } C _ i \ , , \end{align*}


\begin{align*} [ \phi _ { \lambda } \psi ] _ { \mu } a = \phi _ { \lambda } ( \psi _ { \mu - \lambda } a ) - \psi _ { \mu - \lambda } ( \phi _ { \lambda } a ) , \forall \ a \in \mathcal { A } , \end{align*}


\begin{align*} \int _ { \hat { B } _ { \delta } ( 0 ) \setminus \hat { B } _ { \varepsilon } ( 0 ) } q ^ 2 ( K _ { H } ) _ { 2 2 } ( x + \bar { x } ) \partial _ 2 \hat { U } \left ( \frac { x } { \varepsilon } \right ) \partial _ 2 \hat { U } \left ( \frac { x } { \varepsilon } \right ) d x = \frac { l _ 1 } { l _ 2 } \pi q ^ 2 ( K _ { H } ) _ { 2 2 } ( \bar { x } ) \ln \frac { \delta } { \varepsilon } + O ( 1 ) . \end{align*}


\begin{align*} \partial _ s \sigma _ k + \mathbf l ' _ k H ( \mathbf r _ k , \mathbf r _ k ) \big ( m \cdot \nabla _ u \big ) \big ( 2 ^ { - 1 } \sigma _ k ^ 2 \big ) = 0 , \forall \ , k = 1 , \ldots , n . \end{align*}


\begin{align*} \begin{aligned} & w ( j ) < w ( i _ 2 ) = w ( l ) w ( j ) > w ( i _ 1 ) i _ 1 < j < i _ 2 , \\ & w ( i _ 1 ) > w ( l + 1 ) > w ( i _ 2 ) = w ( l ) . \end{aligned} \end{align*}


\begin{align*} f ( x - T ( x ) ) & = f ( z ) . \end{align*}


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


\begin{align*} \int _ 0 ^ T \int _ { \Omega } \int _ { Y ^ { \ast } } D _ 0 ( t , x , y ) \big [ \nabla _ x u _ 0 ( t , x ) + \nabla _ y u _ 1 ( x , y ) \big ] \cdot \nabla _ y \phi ( t , x , y ) d y d x d t = 0 . \end{align*}


\begin{align*} \widetilde { \lambda } \ , ' ( m ) \ r _ n ( m ) = E ' _ { n } \ \widetilde { r } \ , ' _ { n + 1 } ( m ) + F ' _ n \ \widetilde { r } \ , ' _ { n } ( m ) + G \ , ' _ n \ \widetilde { r } \ , ' _ { n - 1 } ( m ) \ , , \end{align*}


\begin{align*} \| ( I - P _ N ) A \| _ { \mathcal H } & = \| \sum _ { k , l \leq N - 1 } \langle A , e _ k \otimes e _ l \rangle _ { \mathcal H } e _ k \otimes e _ l \| _ { \mathcal H } \\ & \leq \sum _ { k , l \leq N - 1 } | \langle A , e _ k \otimes e _ l \rangle _ { \mathcal H } | \| e _ k \otimes e _ l \| _ { \mathcal H } \\ & = \sum _ { k , l \leq N - 1 } | \langle A , e _ k \otimes e _ l \rangle _ { \mathcal H } | \\ & = \sum _ { k , l \leq N - 1 } | \langle A e _ k , e _ l \rangle | . \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } z ( \Psi ^ \pm _ 1 ( x ; z ) - e ^ { - i c _ \pm ( x ) } e _ 1 ) = \widehat { \Psi } ^ \pm _ { 1 1 } ( x ) e _ 1 + \widehat { \Psi } ^ \pm _ { 2 1 } ( x ) e _ 2 . \end{align*}


\begin{align*} \Gamma _ { ( g , u ) } ( X ) = - \Delta _ g X - u ^ { - 1 } \nabla X ( \nabla _ g u , \cdot ) + u ^ { - 2 } X ( u ) d u . \end{align*}


\begin{align*} \bigl \| x \pm \frac { y } { \| y \| } \bigr \| & \leq \bigl \| x - \sum _ { j = 1 } ^ m \alpha _ j \ , m _ { a _ j , b _ j } \bigr \| + \bigl \| \sum _ { j = 1 } ^ m \alpha _ j \ , m _ { a _ j , b _ j } \pm y \bigr \| + \bigl | \| y \| - 1 \bigr | \\ & < \frac { \varepsilon } { 5 } + \sum _ { j = 1 } ^ m \alpha _ j \| m _ { a _ j , b _ j } \pm y _ j \| + \frac { 2 \varepsilon } { 5 } \leq 1 + \varepsilon . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \sup _ { t \leq T \wedge \sigma _ n ^ M } \Big [ \varphi _ n ( t ) \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H ^ 2 \Big ] = 0 . \end{align*}


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


\begin{align*} P _ { \mu , f _ { 1 , \infty } } ( X , \psi ) : = \int P _ { \mu , f _ { 1 , \infty } } ( x , \psi ) \ d \mu ( x ) . \end{align*}


\begin{align*} \frac { \partial ^ { j } } { \partial x ^ { j } } \varphi \left ( \rho , x \right ) = \left ( 2 i \rho \right ) ^ { j } R ( \rho ) e ^ { 8 i t \rho ^ { 3 } + 2 i x \rho } . \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } l _ { n } ( x | \beta ) l _ { m } ( x | \beta ) f _ { g } ( x | \beta ) d x = \delta _ { n , m } . \end{align*}


\begin{align*} X ( q , r ) & = - \frac { 1 } { 4 q } f ( q r ) + \frac { h ( q r ) } { 4 g ( q r ) } \left ( - 4 r + \frac { 1 } { 2 q } f ( 2 q r ) \right ) \\ & = \frac { 1 } { q } \iota ( q r ) , \end{align*}


\begin{align*} \psi _ \epsilon ( x , y ) = A _ { \epsilon } \| y \| ^ 2 + \tilde { v } ( x ) \| y \| \ge R + 1 \end{align*}


\begin{align*} 0 = \frac { \dd } { \dd \tau } E [ b + \tau \tilde { b } + s ( \tau ) b _ 0 ] \Bigg | _ { \tau = 0 } & = \int _ { \mathbb { R } ^ { 3 } } b \cdot ( \tilde { b } + \dot { s } ( 0 ) b _ 0 ) \dd x \\ & = \int _ { \mathbb { R } ^ { 3 } } b \cdot \tilde { b } \dd x - \frac { 1 } { \partial _ { s } j ( 0 , 0 ) } \left ( \int _ { \mathbb { R } ^ { 3 } } b \cdot b _ 0 \dd x \right ) \partial _ { \tau } j ( 0 , 0 ) , \\ & = \int _ { \mathbb { R } ^ { 3 } } b \cdot \tilde { b } \dd x - \frac { \mu } { 2 } \partial _ { \tau } j ( 0 , 0 ) . \end{align*}


\begin{align*} \pi _ 1 : G \to M , \pi _ 1 ( s ( q ) z ) = \overline { \pi } ( q ) \chi _ 1 ( z ) \forall q \in Q , z \in Z . \end{align*}


\begin{align*} \partial _ t R _ 0 = g ( u _ 0 , R _ 0 ) \mbox { a . e . i n } ( 0 , T ) \times \Omega . \end{align*}


\begin{align*} \rho ^ \star ( \theta ^ \sharp , X ) = D ^ g \theta ( J \theta ^ \sharp , J X ) + \sum _ { i = 1 } ^ 4 D ^ g \theta ( J e _ i , \left ( D ^ g _ { e _ i } J \right ) X ) . \end{align*}


\begin{align*} E [ t ] ^ { \vee } = E ^ { \vee } [ t ^ { - 1 } ] t > 0 . \end{align*}


\begin{align*} \lambda _ 0 \alpha ^ s + \lambda _ 1 \alpha ^ { s - 1 } \beta + \cdots + \lambda _ s \beta ^ s = 0 , \end{align*}


\begin{align*} \rho ( \eta _ { i _ j } | \eta _ { [ i _ { j + 1 } , i _ k ] } \eta _ { \Lambda \setminus \Delta } \omega _ { \Lambda ^ c \cup [ i _ 1 , i _ { j - 1 } ] } ) = \gamma _ { i _ j } ( \eta _ { i _ j } | \eta _ { [ i _ { j + 1 } , i _ k ] } \eta _ { \Lambda \setminus \Delta } \omega _ { \Lambda ^ c \cup [ i _ 1 , i _ { j - 1 } ] } ) . \end{align*}


\begin{align*} \lim _ { x \to \infty } e ^ { \Lambda _ u } \frac { 2 } { \Lambda _ 2 ^ 2 } \cdot \frac { f ( x ) } { g _ u ( x ) } = \lim _ { x \to \infty } e ^ { \Lambda _ u } \frac { 2 } { \Lambda _ 2 ^ 2 } \cdot \frac { f ( x ) } { g _ 1 ( x ) } \frac { g _ 1 ( x ) } { g ( x ) } \frac { g ( x ) } { g _ u ( x ) } = e ^ { \Lambda _ u } \frac { 2 } { \Lambda _ u } . \end{align*}


\begin{align*} \int _ { \partial \Omega } { \texttt f } \sigma ^ \psi _ x \mathfrak { f } = & \int _ { \Omega } \left ( { \texttt f } { } ^ \psi \mathcal D [ \mathfrak { f } ] + { } ^ { { \psi } } \mathcal D _ r [ { \texttt f } ] \mathfrak { f } \right ) d x , \end{align*}


\begin{align*} \Psi _ { a , b } ( x ' ) : = 4 \int _ a ^ b \Phi _ t * \Phi _ t ( x ' ) \ , \frac { d t } { t } , x \in \R ^ { n - 1 } . \end{align*}


\begin{align*} \phi _ u & = \{ v \in V ( H ) \ | \ ( u , v ) \in \phi \} , \\ \overline { \phi } & = \{ u \in V ( G ) \ | \ \phi _ u \cap U \neq \emptyset \} . \end{align*}


\begin{align*} L L \left ( | \varrho _ { Z Z } ( L , L ) | ^ 2 \right ) = - 4 \varrho _ { Z Z Z } ( L , L , L ) \overline { \varrho _ { Z Z } ( N , L ) } - 2 \varrho _ { Z Z } ( L , L ) ( L \overline { \varrho _ { Z Z } ( N , L ) } ) \\ + \overline { \varrho _ { Z Z } ( L , L ) } \left ( \varrho _ { Z Z Z Z } ( L , L , L , L ) - 3 \varrho _ { Z Z Z } ( L , L , \varrho _ { Z Z } \cdot L ) \right ) . \end{align*}


\begin{align*} \frac { 1 } { n } \sum _ { i = 1 } ^ n \delta _ { X _ i } \to Q , . \end{align*}


\begin{align*} \theta _ { j , ( 2 ) } ( f ) = \sum _ { k } r _ k g \cdot \theta _ { j , ( 2 ) } ( v _ k ) . \end{align*}


\begin{align*} | \phi ^ { N , i } ( t , \boldsymbol { x } _ t ) - \Phi ( t , \bar { \nu } _ t ^ i ) | = | \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } _ t } ) - \Phi ( t , \bar { \nu } _ t ^ i ) | \leq C | \nu ^ { N , i } _ { \boldsymbol { x } _ t } - \bar { \nu } _ t ^ i | . \end{align*}


\begin{align*} \begin{aligned} \Delta K _ { m , \epsilon } = d _ 0 \left ( \frac { \kappa _ m d _ 1 | q | ^ 2 } { ( | q | ^ 2 + \epsilon ) ^ { \kappa _ m + 1 } } \right ) = - \frac { \kappa _ m ( \kappa _ m + 1 ) } { ( | q | ^ 2 + \epsilon ) ^ { \kappa _ m + 2 } } d _ 0 | q | ^ 2 \wedge d _ 1 | q | ^ 2 + \frac { 8 \kappa _ m \beta _ n } { ( | q | ^ 2 + \epsilon ) ^ { \kappa _ m + 1 } } = : A + B . \end{aligned} \end{align*}


\begin{align*} \left [ Z _ { \kappa } \left ( x , \xi \right ) , Z _ { \kappa } \left ( y , \eta \right ) \right ] = Z _ { \kappa } \left ( C _ { \xi } y - C _ { \eta } x , \kappa x \times y + \xi \times \eta \right ) \end{align*}


\begin{align*} \mathbb E \left [ \| \Delta _ i \mathcal S B ^ { \mathfrak H } \| ^ 4 \right ] \geq \left ( \int _ 0 ^ { 1 - i \Delta _ n } \mathbb E \left [ \left ( B ^ { \mathfrak H } _ { i \Delta _ n + y } - B ^ { \mathfrak H } _ { ( i - 1 ) \Delta _ n + y } \right ) ^ 2 \right ] d y \right ) ^ 2 = & ( n - i ) ^ 2 \Delta _ n ^ { 2 + 4 \mathfrak H } . \end{align*}


\begin{align*} B _ t ^ { k , i } = \left \{ \begin{array} { l l } { \rm R e } ( W ^ { k , i } _ { t } ) , & k \in \Z _ { + } ^ d , \\ - { \rm I m } ( W ^ { k , i } _ { t } ) , & k \in \Z _ { - } ^ d . \end{array} \right . \end{align*}


\begin{align*} \Pr [ \exists ( S , b ) : Z _ { S , b } < 2 ^ { - ( k + 1 ) } ( n - k ) ] \le \binom { n } { k } 2 ^ k \cdot \exp \left ( - 2 ^ { - ( k + 3 ) } ( n - k ) \right ) \enspace . \end{align*}


\begin{align*} g ( y ) = f ( y ) - ( f - g ) ( y ) \geq \| y \| - \frac { 4 ( 1 + \theta ) } { n } , \end{align*}


\begin{align*} & \nabla _ V \left ( \nabla _ Y \hat X - \omega ( Y ) \right ) \\ & = \nabla _ Y \nabla _ V \hat X + R ( V , Y ) \hat X - ( \nabla _ V \omega ) ( Y ) - \omega ( \nabla _ V Y ) \\ & = \nabla _ Y ( \omega ( V ) ) + R ( V , Y ) \hat X + R ( \hat X , V ) Y - T _ h ( V , Y ) - \omega ( \nabla _ V Y ) \mbox { ( b y \eqref { e q : O D E } ) } \\ & = ( \nabla _ Y \omega ) ( V ) - R ( Y , \hat X ) V - T _ h ( V , Y ) \mbox { ( b y B i a n c h i i d e n t i t y ) } \end{align*}


\begin{align*} \frac { ( p ^ 3 - p ^ 2 ) } { ( p ^ 2 - p ) } = p \end{align*}


\begin{align*} \begin{aligned} I _ 3 ' ( x ) = 4 \pi ^ { - 1 } \int _ { \mathbb { R } } z r _ 2 ( z ) e ^ { 2 i z x } d z = 4 \pi ^ { - 2 } \widehat { z r _ 2 ( z ) } ( - 2 x ) , \end{aligned} \end{align*}


\begin{align*} f _ { k } ( x , y ) : = \begin{cases} f ( x , - k ) , & y < - k , \\ f ( x , y ) , & | y | \le k , \\ f ( x , k ) , & y > k . \end{cases} \end{align*}


\begin{align*} \chi _ 0 : Y \rightarrow \R , \chi _ 0 ( y ) = \chi ( d _ { \Gamma } ( y ) ) , \end{align*}


\begin{align*} \chi ( E ) = \sum \limits _ { j = 1 } ^ n \chi ( E _ j ) - r ( n - 1 ) . \end{align*}


\begin{align*} \hat { k } _ d ( \Delta ) & = \prod _ { q , i } x _ { q , i } ^ { e ( q , i ) } \prod _ { q = 1 } ^ { d + 1 } \prod _ { \rho \in V ' _ 1 \times \cdots \times \widehat { V ' _ q } \times \cdots \times V ' _ { d + 1 } } D _ { q , n _ q } \\ & = \prod _ { q = 1 } ^ { d + 1 } \left [ \left ( \prod _ { i = 1 } ^ { n _ i } x _ { q , i } \right ) ^ { e ( q , i ) } ( x _ { q , 1 } + \cdots + x _ { q , n _ q } ) ^ { \prod _ { r \neq q } ( n _ { r } - 1 ) } \right ] \end{align*}


\begin{align*} \Theta ( f ) = \Delta f + \partial _ t f = 0 , \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \mathsf { L } ^ n ) = W ( 1 - \xi _ 0 , n , x , \mathsf { R } ^ n ) , ~ \forall \xi _ 0 \in [ 0 , 1 ] , \forall x \in \mathbb { R } . \end{align*}


\begin{align*} \gamma _ Z ( y , \varphi ( y ) ) = \left ( - y _ 1 \frac { \partial \varphi ( y ) } { \partial y _ 1 } : \cdots : - y _ { n } \frac { \partial \varphi ( y ) } { \partial y _ { n } } : \varphi ( y ) \right ) \end{align*}


\begin{align*} w = 0 , \ z = \sqrt { \frac { 1 } { 1 + a } } \cos ( t ) + i \sqrt { \frac { 1 } { 1 - a } } \sin ( t ) , t \in [ 0 , 2 \pi ) . \end{align*}


\begin{align*} X _ 1 & \mapsto \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} , & X _ 2 & \mapsto \begin{pmatrix} 0 & 0 & 0 \\ \lambda & 0 & 0 \\ \mu & \lambda & 0 \end{pmatrix} . \end{align*}


\begin{align*} L = \frac { 1 } { 2 } \frac { m } { 1 + \lambda r ^ 2 } \left ( \sum _ i v _ i ^ 2 + \lambda \sum _ { i < j } ( x _ i v _ j - x _ j v _ i ) ^ 2 \right ) - { \cal V } ( r ) , { \cal V } ( r ) = \frac { 1 } { 2 } m \alpha ^ 2 \frac { r ^ 2 } { 1 + \lambda r ^ 2 } . \end{align*}


\begin{align*} u ( t ) = U ( t , 0 ) u _ 0 + \int _ 0 ^ t U ( t , s ) f ( s ) d s , 0 \le t \le T , \end{align*}


\begin{align*} \frac { \left | \sum _ { i = 1 } ^ n [ 0 , z _ i ] \right | } { \left | \sum _ { i = 1 } ^ { n - 1 } [ 0 , z _ i ] \right | } = d ( z _ n , H _ n ) . \end{align*}


\begin{align*} h ( \lambda ) = \frac { ( \lambda , \lambda + 2 \rho ) } { 2 ( \kappa + h ^ \vee ) } + \frac { ( \lambda , \lambda + 2 \rho ) } { 2 ( \kappa ^ * + h ^ \vee ) } - ( \lambda , \rho ^ \vee ) = \frac { ( \lambda , \lambda ) r ^ \vee n } { 2 } + ( \lambda , n r ^ \vee \rho - \rho ^ \vee ) . \end{align*}


\begin{align*} \rho & : = \inf \{ n \geq 1 : S ( n - 1 ) \nprec S ( n ) \} \\ \tau & : = \inf \{ n \geq 1 : S ( n ) \notin W ^ d \} . \end{align*}


\begin{align*} d ( J ^ * d \theta _ i ) = 0 , \mbox { l o c a l l y } J ^ * d \theta _ i = d y _ i . \end{align*}


\begin{align*} - [ M ^ k , \ , \mathcal { L } ] f & = \Big ( M ^ k \Gamma ( \sqrt \mu , f ) - \Gamma ( \sqrt \mu , M ^ k f ) \Big ) + \Big ( M ^ k \Gamma ( f , \sqrt \mu ) - \Gamma ( M ^ k f , \sqrt \mu ) \Big ) \\ & = \sum _ { 1 \leq j \leq 6 } \Big ( M ^ k L _ j ( \sqrt \mu , f ) - L _ j ( \sqrt \mu , M ^ k f ) \Big ) + \sum _ { 1 \leq j \leq 6 } \Big ( M ^ k L _ j ( f , \sqrt \mu ) - L _ j ( M ^ k f , \sqrt \mu ) \Big ) \end{align*}


\begin{align*} \Pi _ S ( X ) \rightarrow \bigoplus _ { i \in I } \Pi _ S ( Z _ i , N _ i ) \rightrightarrows \cdots \rightrightarrows \bigoplus _ { J \subset I , \sharp J = n } \Pi _ S ( Z _ J , N _ J ) \rightrightarrows \cdots \end{align*}


\begin{align*} \sum _ { J } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } q \ , \partial _ { x ^ \prime } ^ { J } q - \sum _ { J } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } b \ , \partial _ { x ^ \prime } ^ { J } q - \frac { \partial q } { \partial x _ n } + c = 0 . \end{align*}


\begin{align*} & [ \alpha ( y ) , \alpha ( z ) ] \cdot \alpha ( x \cdot t ) + [ [ x , y ] , \alpha ( z ) ] \cdot \alpha ^ { 2 } ( t ) \\ & + \alpha ^ { 2 } ( y ) \cdot ( [ x , z ] \cdot \alpha ( t ) ) - \alpha ^ { 2 } ( x ) \cdot ( \alpha ( y ) \cdot ( z \cdot t ) ) + \alpha ^ { 2 } ( z ) \cdot ( \alpha ( x ) \cdot ( y \cdot t ) ) = 0 . \end{align*}


\begin{align*} | \rho ^ \prime | = \left | \frac { 1 } { 1 + h ^ \prime } \right | \leq \frac { 1 } { \varepsilon _ 0 } . \end{align*}


\begin{align*} I _ n : = \left [ n \dfrac { \bar { t } } { \gamma } , \ , ( n + 1 ) \dfrac { \bar { t } } { \gamma } \right ) n \in { \mathbb { N } } \end{align*}


\begin{align*} k ( t ) = h _ { 1 - \alpha } ( t ) \cdot k _ 1 ( t ) , \ k _ 1 ( t ) = \sum _ { k = 0 } ^ { + \infty } \ , b _ k t ^ k , \end{align*}


\begin{align*} h _ c ( G ) & = h _ c ( G \backslash v ) + f _ c ( b ) - f _ c ( 0 ) + \sum _ { i = 1 } ^ b \Delta _ c ( d _ i ) \\ & \le h _ c ( G \backslash v ) + f _ c ( b ) - f _ c ( 0 ) + b \cdot \Delta _ c \left ( \frac { m + \binom { b } { 2 } } { b } \right ) \\ & < h _ c ( K _ { 1 , m - b } ) + f _ c ( m ) - f _ c ( m - b ) + b \Delta _ c \left ( 1 \right ) \\ & = h _ c ( K _ { 1 , m } ) . \end{align*}


\begin{align*} \Lambda _ { g } ( \textbf { \textit { V } } ) : = \begin{pmatrix} \lambda \nu \operatorname { d i v } + \mu \nu S & - \beta \nu \\ 0 & \alpha \partial _ \nu \end{pmatrix} \textbf { \textit { U } } \ \partial \Omega , \end{align*}


\begin{align*} V _ { | ( t , x , \varphi ) } = \frac { 1 } { \sqrt { 1 + | \partial _ x w ( t , x ) | ^ 2 } } \partial _ t w ( t , x ) . \end{align*}


\begin{align*} ( \delta _ n ^ k ) _ ! = \sum _ { J = \{ i _ 0 < \hdots < \widehat { i _ k } < \hdots < i _ n \} \subset K = \{ i _ 0 < \hdots < i _ n \} } ( \nu _ { K } ^ J ) _ ! \end{align*}


\begin{align*} \mathcal { E } _ { m } = \displaystyle \sum _ { n = x _ { m } + 1 } ^ { y _ { m } } e _ { n } \ ; \ ; \ ; \mbox { a n d } \ ; \ ; \ ; \mathcal { F } _ { m } = \displaystyle \sum _ { n = x _ { m } + 1 } ^ { y _ { m } } g _ { n } . \\ \end{align*}


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


\begin{align*} \partial ^ \square c & = \Delta _ \Gamma \big ( G ' ( c ) \big ) + c H V . \end{align*}


\begin{align*} \left \{ \begin{aligned} \partial _ x z & = 0 , \\ 0 & \leq \partial _ t z - \frac { \varepsilon } { 2 } \partial _ { x } ^ 2 z . \end{aligned} \right . \end{align*}


\begin{align*} g ( z ) = 1 + \left ( z + 1 \right ) \sum _ { n = 0 } ^ { \infty } \left ( - 1 \right ) ^ { n } z ^ { n } b _ { n } ( 0 ) , \end{align*}


\begin{align*} H [ \nabla \times ( \phi \nabla \theta ) + G \nabla \theta ] = - H [ \nabla \times ( \phi \nabla \theta ) - G \nabla \theta ] \end{align*}


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


\begin{align*} \alpha = 1 + z \frac { \Phi ( z ) } { \phi ( z ) } \frac { 1 - \Phi ( z ) } { 1 - 2 \Phi ( z ) } , . \end{align*}


\begin{align*} x _ { n + 1 } & = ( 1 - \beta _ n ) U y _ n + \beta _ n y _ n , \\ y _ { n } & = ( 1 - \alpha _ n ) T x _ n + \alpha _ n u , \end{align*}


\begin{align*} \kappa ( t ) = h _ { 1 - \beta + \alpha } ( t ) \ , + \ , h _ { 1 - \beta } ( t ) , \ 0 < \alpha < \beta < 1 , \end{align*}


\begin{align*} & \hphantom { { } = { } } ( x ^ i , ( a ^ i { } _ j , a ^ i { } _ { j \alpha } a ^ \alpha { } _ k ) , ( b ^ i { } _ j , b ^ i { } _ { j k } ) ) . ( g , X ) \\ * & = ( x ^ i , ( a ^ i { } _ \alpha g ^ \alpha { } _ j , a ^ i { } _ { \beta \alpha } g ^ \beta { } _ j a ^ \alpha { } _ \beta g ^ \beta { } _ k ) , ( h ^ i { } _ \alpha b ^ \alpha { } _ \beta g ^ \beta { } _ j + X ^ i { } _ j , h ^ i { } _ \alpha b ^ \alpha { } _ { \beta k } g ^ \beta { } _ j ) ) . \end{align*}


\begin{align*} k = \sum _ { p \in C } I _ p ( C , H _ { j } ) = \# ( C \cap H _ { j } ) + \sum _ { p \in C \cap H _ { j } } ( I _ p ( C , H _ { j } ) - 1 ) \end{align*}


\begin{align*} \frac 1 N \sum _ { k = 1 } ^ N ( \sigma \circ w ) ^ k \mu . \end{align*}


\begin{align*} \phi _ { \lambda } = \lambda ^ { - \alpha } \phi ( \lambda ^ { - 1 } x , \lambda ^ { - \frac { \alpha + 2 } { 2 } } y ) . \end{align*}


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \xi _ 2 = \dot { s } ( \varepsilon ( z - 1 ) - \varepsilon ( z ) ) \xi _ 2 + \dot { s } \varepsilon ( z ) \varpi . \end{align*}


\begin{align*} \P _ x ( S ( n ) \in d z , \tau > n ) & = \P _ x ( \widehat S _ i ( n + d - i ) \in d z _ i , i = 1 , \ldots , d , \widehat \tau = \infty ) \\ & = \sum _ { \pi \in \mathcal S _ d } ( \pi ) \P _ x ( \widehat S _ i ( n + d - \pi ( i ) ) \in d z _ { \pi ( i ) } , i = 1 , \ldots , d , \widehat \tau = \infty ) . \end{align*}


\begin{align*} a d ^ \star ( x ) \xi = a d ^ * ( \alpha ( x ) ) ( \alpha ^ { - 2 } ) ^ * ( \xi ) , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 3 ( I _ N ^ i ( p ) - J _ N ^ i ( p ) ) = O ( p ^ { - \nu } ) \quad \mbox { f o r l a r g e } p > 0 , \end{align*}


\begin{align*} [ \mathbf { h } ' _ 3 , \mathbf { h } _ 3 ] = 1 . \end{align*}


\begin{align*} e m + \gamma = \widehat { \iota } ( D ) c \geq \widehat { \iota } ( D ) ( b _ 1 d + 1 ) . \end{align*}


\begin{align*} \mathfrak M ( x , y ) = - \sin S _ c ^ l ( x , y ) \partial _ x \partial _ y ^ { \intercal } \Phi _ s ( x , y ) . \end{align*}


\begin{align*} \mathfrak g = \mathfrak g _ + \oplus \mathfrak h \oplus \mathfrak g _ - \end{align*}


\begin{align*} & \psi ^ { \pm } ( x , t ; k ) = \mathrm { e } ^ { - i c _ \pm ( x ) \sigma _ 3 } + \mathcal { O } ( k ^ { - 1 } ) , k \rightarrow \infty , \end{align*}


\begin{align*} \psi ( v , p ) = \psi ^ \dagger ( v , p ) \in O \end{align*}


\begin{align*} A \cdot ( v _ { 1 } , \dots , v _ { k } ) : = ( A v _ { 1 } , \dots , A v _ { k } ) . \end{align*}


\begin{align*} \| \psi _ { n + m } - \psi _ n \| _ 1 \leq \sum _ { i = 1 } ^ m \frac { 1 } { 2 \pi } \int _ 0 ^ { 2 \pi } \left | \arg \frac { \hat { b } _ { n + i } \left ( e ^ { i \theta } \right ) } { e ^ { i \theta } } \right | \ , d \theta . \end{align*}


\begin{align*} f ( v _ { \ell , m , n } ) = \sqrt { q } ^ { 3 \ell + m - n } & \sum _ { \substack { ( i , j ) \in \{ 1 , 3 \} ^ 2 \\ i \neq j } } \{ F _ { i i j } z _ i ^ { \ell + m } z _ j ^ n + F _ { i j i } z _ i ^ { \ell + n } z _ j ^ m + F _ { i j j } z _ i ^ \ell z _ j ^ { m + n } \} , \\ \end{align*}


\begin{align*} \lim _ { k \to + \infty } \int _ { \tau } ^ { \tau + T } Q _ 1 ( t ) \d t = 0 . \end{align*}


\begin{align*} | | \tau | | G \psi \stackrel { d e f } { = } \sup _ { p \in ( 1 , b ) } \left \{ \ \frac { | | \tau | | L _ p ( \Omega ) } { \psi ( p ) } \ \right \} . \end{align*}


\begin{align*} \dot y = a y '' + M y , y ( 0 , x ) = y _ 0 ( x ) , \end{align*}


\begin{align*} \aligned \frac d { d t } \mathcal { Y M } _ e ^ 0 ( \nabla ^ t ) _ { \big { | } _ { s = 0 } } & = \frac d { d t } \mathcal { Y M } _ e ( \nabla ^ t ) _ { \big { | } _ { s = 0 } } \\ & = \int _ M \ e x p \ , ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) \langle R ^ \nabla , d ^ \nabla A \rangle \ , d v \\ & = \int _ M \langle \delta ^ \nabla \big ( \exp \ , ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) R ^ \nabla \big ) , A \rangle \ , d v \ , . \endaligned \end{align*}


\begin{align*} & \sum _ { x , y = 0 } ^ n ( \delta _ { x , y } - M _ { x , y } ) f _ y f _ x = \sum _ { x , y = 0 } ^ n \sum _ { j , j ' = 0 } ^ n \left ( 1 - \Theta ( \mu _ j , \mu _ { j ' } ) \right ) \psi _ j ( x ) \psi _ { j ' } ( x ) \psi _ j ( y ) \psi _ { j ' } ( y ) f _ y f _ x \\ & = \sum _ { j , j ' = 0 } ^ n \left ( 1 - \Theta ( \mu _ j , \mu _ { j ' } ) \right ) \left ( \sum _ { x = 0 } ^ n \psi _ j ( x ) f _ x \psi _ { j ' } ( x ) \right ) ^ 2 . \end{align*}


\begin{align*} \begin{aligned} E _ { D ^ 2 } ( e ^ { 2 \pi i t } ) & = E _ { D ^ 2 } \left ( M ^ { - 1 } \circ M ( e ^ { 2 \pi i t } ) \right ) \\ & = M ^ { - 1 } \circ Q \left ( M ( e ^ { 2 \pi i t } ) \right ) \\ & = \hat { P } ( e ^ { 2 \pi i t } ) = e ^ { 2 \pi i P ( t ) } , \end{aligned} \end{align*}


\begin{align*} N _ f ( z ) & = \dfrac { 1 - z } { 1 - m z } . \end{align*}


\begin{align*} p ( \eta , j ) & = p ( \eta , \ell _ { m } ) + r - m - 1 \quad \\ | \pi ( \eta \cup j ) | & = | \pi ( \eta \cup \ell _ { m } ) | + r - m , \end{align*}


\begin{align*} f \left ( \left ( a + 1 + \frac { b } { y } \right ) ( y - 1 ) \right ) - f \left ( \left ( a + \frac { b } { y } \right ) ( y - 1 ) \right ) & = y - 1 + ( y - 1 ) \int _ { u = a + \frac b y } ^ { a + 1 + \frac b y } \left ( \log ( u ) + \log ( y - 1 ) \right ) \d u . \end{align*}


\begin{align*} { \cal L } { \bf T } : = ( { \frak G } _ 0 { \bf T } , \ldots , { \frak G } _ n { \bf T } ) , \end{align*}


\begin{align*} \left ( I - \mathcal { P } ^ - \right ) s = 0 \end{align*}


\begin{align*} \begin{cases} \lambda + 2 < p , & \alpha \leq 2 , \\ \lambda + 2 < \alpha + p - 2 , & \alpha > 2 . \end{cases} \end{align*}


\begin{align*} [ l , r ] \otimes [ l ' , r ' ] = \bigoplus \limits _ { \nu = \vert l - l ' \vert \mbox { s t e p } 2 } ^ { \mathrm { m i n \left ( l + l ' , 2 d - 4 - l - l ' \right ) } } [ \nu , r + r ' ] . \end{align*}


\begin{align*} { S ( t ) = \int _ { 0 } ^ { + \infty } e ^ { - A s } \Sigma _ 2 \big ( \overline { \frak p ^ 2 } ( t - s ) \big ) e ^ { - A ^ T s } \dd s } , t \ge 0 \end{align*}


\begin{align*} \begin{aligned} \biggr { | } \int _ { \partial \Omega \cap B _ { \eta } } \nabla c \cdot \nu c \psi ^ { 2 } \biggr { | } & = \biggr { | } \int _ { \partial \Omega \cap B _ { \eta } } ( \gamma - c ) c \psi ^ { 2 } \biggr { | } \\ & \le \gamma ^ { 2 } \int _ { \partial \Omega \cap B _ { \eta } } \psi ^ { 2 } \\ & \le \gamma ^ { 2 } \int _ { \partial \Omega } \psi ^ { 2 } . \end{aligned} \end{align*}


\begin{align*} ( \alpha - \alpha ^ 2 ) ( 1 - K _ { 2 } ) ^ 2 \sum \limits _ { k = 0 } ^ { m } \norm { \theta ^ k - \tilde { \theta } ^ k } _ { \mathcal { L } ^ 2 , G } ^ 2 & \leq \norm { \theta ^ 0 - \theta ^ * } _ { \mathcal { L } ^ 2 , G } ^ 2 - \norm { \theta ^ { m + 1 } - \theta ^ * } _ { \mathcal { L } ^ 2 , G } ^ 2 \\ & \leq \norm { \theta ^ 0 - \theta ^ * } _ { \mathcal { L } ^ 2 , G } ^ 2 , \forall \ m \in \mathbb { N } . \end{align*}


\begin{align*} \rho ( g ) & = g & g \in G \\ \rho ^ { \vee } ( g ) & = ( g ^ { - 1 } ) ^ { \vee } ( \varphi ) & g \in G . \end{align*}


\begin{align*} \hat \Phi ^ \dagger ( a , s ) = h ( a , s ) . \end{align*}


\begin{align*} \bigg | \frac { \partial G _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) } { \partial z _ { i , \hbar } } \bigg | \leq C , \ \ 1 \leq i \neq j \leq m , \hbar = 1 , 2 . \end{align*}


\begin{align*} \aligned \mathcal { Y M } _ p ( \nabla ) & = \int _ M \frac 1 p | | R ^ \nabla | | ^ p \ , d v \ , \\ \big ( r e s p . \mathcal { Y M } _ F ( \nabla ) & = \int _ M F ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) \ , d v \ , \big ) , \endaligned \end{align*}


\begin{align*} w \varepsilon ^ \mu = x > w \varepsilon ^ \mu s _ \alpha \varepsilon ^ { ( \langle \mu , \alpha \rangle - 1 ) \alpha ^ \vee } = w s _ \alpha \varepsilon ^ { \mu - \alpha ^ \vee } . \end{align*}


\begin{align*} \widetilde { W } _ { B _ { r _ 1 } } ( x ) = \sum _ { k = 0 } ^ \infty \frac { 1 } { ( 2 ^ k ) ^ { 1 0 } } \int H ( x - y ) \chi _ { A _ k } ( y ) d y . \end{align*}


\begin{align*} T ( T _ { + } ) = \left ( \begin{array} [ c ] { c c c c c c } t _ { 0 } & 0 & 0 & \ldots & 0 & \ldots \\ t _ { 1 } & t _ { 0 } & 0 & \ldots & 0 & \ldots \\ t _ { 2 } & t _ { 1 } & t _ { 0 } & \ldots & 0 & \ldots \\ & \ldots & & \ddots & & \\ t _ { n } & t _ { n - 1 } & t _ { n - 2 } & \ldots & t _ { 0 } & \ldots \\ & \ldots & & \ldots & & \ddots \end{array} \right ) \end{align*}


\begin{align*} \gamma _ t \beta _ t = \frac { \gamma _ { t - 1 } ( 1 - \kappa _ t ) } { ( 1 - \kappa _ { t - 1 } ) } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 3 } { 2 } \gamma _ 0 , \quad \alpha _ t = \frac { \kappa _ { t - 1 } \gamma _ t \beta _ t } { \gamma _ { t - 1 } } = \frac { \kappa _ { t - 1 } ( 1 - \kappa _ { t } ) } { 1 - \kappa _ { t - 1 } } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } \leq \frac { 1 } { 2 } . \end{align*}


\begin{align*} h ( x ) : = \begin{cases} x \sin ( 1 / x ) & x \neq 0 \\ 0 & x = 0 . \end{cases} \end{align*}


\begin{align*} \| \widetilde { Q } \| _ { L ^ 1 } : = \sum _ { i = 1 } ^ 2 \sum _ { j = 1 } ^ 2 \left \| \widetilde { Q } _ { i j } \right \| _ { L ^ 1 } . \end{align*}


\begin{align*} e _ { a _ j } \left [ t + 1 \right ] = & e _ { a _ j } \left [ t \right ] + 2 \mu \Re { \left \{ \hat { \phi } ^ { \left ( j , j \right ) } \right \} } - 4 \mu \left ( \sum _ { l = 1 } ^ M \lvert \hat { \phi } ^ { \left ( l , j \right ) } \rvert ^ 2 + M \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 \right . \left . + \sum _ { q = 1 } ^ { M - 1 } \sum _ { r = q + 1 } ^ M f _ { q , r } ^ { \left ( j \right ) } \right ) a _ j \left [ t \right ] . \end{align*}


\begin{align*} \begin{aligned} \iota ( C ' _ w ) & = C ' _ w , \\ q ^ { \frac { \ell ( w ) } { 2 } } C ' _ w & = \sum _ { z \leq w } P _ { z , w } ( q ) T _ z , \end{aligned} \end{align*}


\begin{align*} A _ { k , r } = \{ \o \in \O \mid \exists ~ x \in C ( \omega ) \cap \bar { B } ( 0 , r ) ~ s . t . ~ \langle \mathcal { F } ( x ^ * ) ( \omega ) , x - x ^ * ( \omega ) \rangle \le - \frac { 1 } { k } \} \end{align*}


\begin{align*} \lim _ { n \to \infty } n J _ n \varphi \left ( 1 - \frac 1 n \right ) = J \varphi ( 1 ) . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty \frac { a _ { i , n } } { \prod _ { k = 0 } ^ n \beta _ { i + k } } = \frac { 1 } { q _ i } , \end{align*}


\begin{align*} F ^ - ( t , x ) : = \left ( \begin{array} { c } - R ^ - ( t , x ) \cos \pi t , \\ - R ^ - ( t , x ) \sin \pi t \end{array} \right ) , \end{align*}


\begin{align*} L ( \gamma _ { u _ { k - 1 } , u _ k } ) & < d ( u _ { k - 1 } , u _ k ) + \varepsilon \delta = d ( u _ { k - 1 } , u _ k ) + \varepsilon \min \bigl \{ 1 , \frac { d ( u _ { k - 1 } , u _ k ) } { C _ { n + 1 } \beta _ k } \bigr \} . \end{align*}


\begin{align*} \nu ( H ' | _ g ( h ) ) & = - \tfrac { 1 } { 2 } R ' ( h ) + \tfrac { 1 } { 2 } R '^ \Sigma | _ { g ^ \intercal } ( h ^ \intercal ) + A _ g \cdot ( A _ g \circ h ) \\ & - A _ g \cdot A ' ( h ) - H _ g H ' ( h ) \\ & + \tfrac { 1 } { 4 } \big ( - R _ g + R ^ \Sigma _ g - | A _ g | ^ 2 - H _ { g } ^ 2 \big ) h ( \nu , \nu ) \\ & - \tfrac { 1 } { 2 } \Delta _ \Sigma h ( \nu , \nu ) + g ^ { a b } \omega ( e _ a ) e _ b ( H _ g ) . \end{align*}


\begin{align*} \widetilde { \nu } _ t ^ { * , \xi } = \eta _ t \nu _ t ^ { * , \xi } , \widetilde { \varphi } _ t ^ { * , \xi } = \eta ^ { - 1 } _ t \varphi _ t ^ { * , \xi } , \quad \widetilde { \Lambda } _ t ^ { 0 , * , \xi } = \eta ^ { - 1 } _ t \Lambda _ t ^ { 0 , * , \xi } , t \in [ 0 , T ] . \end{align*}


\begin{align*} | | z | | : = \sum _ { k = 1 } ^ { \infty } k | z _ k | , \end{align*}


\begin{align*} \rho ( g ) = X \rho ( g ) \psi ( g ) X ^ { - 1 } \end{align*}


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


\begin{align*} 0 = \oint q _ 1 ( z ) q _ 2 ( z ) d z = \int _ { \mathbb { R } } \left [ Q _ { j , - } \mathcal { P } ^ - ( D _ j ) + Q _ { j , - } J \right ] Q _ { j , - } ^ H \mathrm { d } z , \end{align*}


\begin{align*} \forall \varepsilon > 0 , \lim _ { n \to \infty } \sum _ { m = 1 } ^ n \mathbb { E } _ { n , m } [ \| \chi _ { n , m } \| _ { L ^ 2 ( \mu ) } ^ 2 : \| \chi _ { n , m } \| _ { L ^ 2 ( \mu ) } > \varepsilon ] = 0 \end{align*}


\begin{align*} A ^ U _ \xi : = I ^ * ( { \ge } \xi , { < } \nu ) \end{align*}


\begin{align*} L L ( b , c ) & = b \left ( f _ c ( b ) - f _ c ( b - 1 ) \right ) + f _ c ( b ) - f _ c ( 0 ) \\ & = b \Delta _ c ( b ) + f _ c ( b ) - f _ c ( 0 ) \\ & = b \Delta _ c ( b ) + \sum _ { i = 1 } ^ b \left [ f _ c ( i ) - f _ c ( i - 1 ) \right ] \\ & = b \Delta _ c ( b ) + \sum _ { i = 1 } ^ b \Delta _ c ( i ) . \end{align*}


\begin{align*} \psi ( x , w ) = ( \gamma h + \nabla \gamma ) ( x ) + w . \end{align*}


\begin{align*} Q = \ln \kappa _ 1 - N \ln \nu ^ { n + 1 } , \end{align*}


\begin{align*} \lim _ { x \to \infty } e ^ { \gamma x } f _ 0 ( x ) = 0 . \end{align*}


\begin{align*} & \Upsilon _ { 0 , j ' } ( \ell ) = 0 , j ' = 1 , \ldots , n , \\ & \Upsilon _ { j , j ' } ( \ell ) = \frac { 1 - ( - 1 ) ^ { j + j ' } } { n + 1 } \cos \left ( \frac { \pi j ' \ell } { 2 ( n + 1 ) } \right ) \cos \left ( \frac { \pi j \ell } { 2 ( n + 1 ) } \right ) , j , j ' = 1 , \ldots , n , \ , j \not = j ' \\ & \Upsilon _ { j , j } ( \ell ) = \cos \left ( \frac { \pi \ell j } { n + 1 } \right ) , j = 0 , \ldots , n . \end{align*}


\begin{align*} A = \begin{pmatrix} 1 & - 1 \\ 0 & 1 \\ 0 & - 1 \end{pmatrix} B = \begin{pmatrix} 1 & 1 & - 1 \end{pmatrix} . \end{align*}


\begin{align*} \frac { 1 } { m } \sum _ { i = 1 } ^ m V _ i > \left ( \frac { \pi } { 4 } + \frac { 1 9 9 \pi } { 3 2 0 0 k } \right ) \| \mathbf { x } \| . \end{align*}


\begin{align*} \begin{array} { l l l } f _ L ^ M ( \mathcal { M } _ { \mathbb { C } , \mathsf { S } } ( a ) ) \circ u & = & G ( L _ a , R _ a ) ( f _ L ( s ) \circ t ) \\ & = & f _ L ( L _ a \circ s ) \circ t \\ & = & f _ L ( a \bullet _ { \mu _ S } s ) \circ t \\ & = & f _ L ( a ) \circ f _ L ( s ) \circ t \\ & = & f _ L ( a ) \circ u . \end{array} \end{align*}


\begin{align*} \mu \ast \mu _ a = \mu _ s \ast \mu _ r , \end{align*}


\begin{align*} P ( \lambda ) + P ( \mu ) = P ( \lambda + \mu ) S ( \lambda ) + S ( \mu ) = S ( \lambda + \mu ) , \end{align*}


\begin{align*} \lim _ { n \to \infty } \int _ { D } ( \phi _ n - \phi _ \infty ) _ + G _ n \frac { 1 } { r } \dd z \dd r = \int _ { D } ( \phi - \phi _ \infty ) _ + G \frac { 1 } { r } \dd z \dd r . \end{align*}


\begin{align*} [ S ] = \mu \left ( \frac { 1 } { 2 } d \overline { z } + ( \alpha - I \beta ) d z \right ) J \end{align*}


\begin{align*} \theta ^ 0 { } _ u ( X ) & = v ^ i { } _ \alpha d u ^ \alpha , \\ * \theta ^ 1 { } _ u ( X ) & = v ^ i { } _ \alpha d u ^ \alpha { } _ j - v ^ i { } _ \alpha u ^ \alpha { } _ { j \beta } v ^ \beta { } _ \gamma d u ^ \gamma . \end{align*}


\begin{align*} V _ 1 ( k ) : = \begin{pmatrix} 1 & 0 \\ 0 & 2 i k \end{pmatrix} , V _ 2 ( k ) : = \begin{pmatrix} ( 2 i k ) ^ { - 1 } & 0 \\ 0 & 1 \end{pmatrix} , \end{align*}


\begin{align*} V _ 1 \oplus V _ 2 = \mathrm { s p a n } ( \cos \ell x _ 1 , \sin \ell x _ 1 ) \oplus \mathrm { s p a n } ( \cos k x _ 2 , \sin k x _ 2 ) , \end{align*}


\begin{align*} \Delta _ n ^ t a ^ j & : = \int _ t ^ { t + \Delta _ n } \int _ t ^ u \frac d { d u } \langle \alpha _ s , \mathcal S ( u - s ) ^ * e _ j \rangle d s d u \\ & \qquad \qquad + \int _ t ^ { t + \Delta _ n } \int _ t ^ u \langle \sigma _ s , \mathcal A ^ * \mathcal S ( u - s ) ^ * e _ j \rangle d W _ s d u . \end{align*}


\begin{align*} \langle \tilde { \Delta } _ { i } ^ n Y , e _ { j } \rangle = \langle \Delta _ { i } ^ n S , e _ { j } \rangle + \Delta _ n ^ { ( i - 1 ) \Delta _ n } a ^ { j } . \end{align*}


\begin{align*} v = D '' u _ 0 + P _ { \omega , h } ( \sqrt { \lambda } \tau _ 0 ) . \end{align*}


\begin{align*} - \Delta u + V ( x ) u + \frac { \gamma } { 2 \pi } \left ( \log ( | \cdot | ) \ast u ^ 2 \right ) u = b | u | ^ { p - 2 } u \qquad \ \mathbb { R } ^ 2 . \end{align*}


\begin{align*} ( K * \mu ) ( x ) = \int _ { \mathbb R ^ d } K ( x - z ) \mu ( \d z ) . \end{align*}


\begin{align*} \tilde { h } _ d ' ( x ) = \log d + \left ( \psi ( x ) - \psi \left ( x + \frac 1 2 \right ) \right ) - \psi \left ( x + \frac { d - 1 } { 2 } \right ) \end{align*}


\begin{align*} \mathbf { \Phi } _ { \mathrm { t } , g } ^ \star = [ \mathbf { \Phi } _ { g } ^ \star ] _ { 1 : \bar { M } , : } , ~ ~ ~ \mathbf { \Phi } _ { \mathrm { r } , g } ^ \star = [ \mathbf { \Phi } _ { g } ^ \star ] _ { \bar { M } + 1 : 2 \bar { M } , : } , \forall g \in \mathcal { G } . \end{align*}


\begin{align*} \lim _ { x \to \infty } e ^ { \gamma x } f _ r ( x ) = 0 . \end{align*}


\begin{align*} w ( t , x ) = 1 + \rho \big ( t , \gamma ( x , \varphi ) \big ) \end{align*}


\begin{align*} \begin{cases} \frac { d } { d t } X _ t = B ( X _ t , X _ t ) \\ X _ 0 = x _ 0 \end{cases} \end{align*}


\begin{align*} J _ b ( F ) = \{ g \in G ( L ) \mid g ^ { - 1 } b \sigma ( g ) = b \} . \end{align*}


\begin{align*} 0 \leq \varphi '' ( s ) = 2 \beta M [ ( M s - A ) + ( B - M s ) ] . \end{align*}


\begin{align*} \lim _ { \tau \to - \infty } \mathrm { d i s t } _ { \mathbb { U } } ( \mathcal { A } ( \tau , \omega ) , \mathcal { A } _ { \infty } ( \omega ) ) = 0 \ \emph { f o r a l l } \ \omega \in \Omega , \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } & n ^ { p } ( \cdot , t ) \\ & \le \int _ { \Omega } n _ { 0 } ^ { p } + \frac { p ( p - 1 ) } { 4 } \| S _ { 0 } \| _ { \mathcal { C } ( [ 0 , \gamma ] ) } ^ { 2 } \| \nabla c \| _ { L ^ { \infty } ( 0 , T ; L ^ { \infty } ( \Omega ) ) } ^ { 2 } \int _ { 0 } ^ { t } \int _ { \Omega } n ^ { p } \quad \mbox { f o r } t \le T . \end{aligned} \end{align*}


\begin{align*} \nu ^ G ( [ b ] ) = & \nu ^ { \tilde G } ( [ b \gamma ] ) - \frac 1 { \# W } \sum _ { u \in W } u \mu _ \sigma , \\ \implies \nu _ x = & \nu ^ { \tilde G } ( [ b _ { x \gamma } ] ) - \frac 1 { \# W } \sum _ { u \in W } u \mu _ \sigma . \end{align*}


\begin{align*} \ell ( x , \alpha ) = \langle \mu , v \beta \rangle + \Phi ^ + ( \alpha ) - \Phi ^ + ( w v \beta ) \geq \Phi ^ + ( \alpha ) . \end{align*}


\begin{align*} \mathbb { E } \left [ Z _ { i , j } \right ] = \frac { \pi } { 4 } \| \mathbf { x } \| . \end{align*}


\begin{align*} \begin{pmatrix} \begin{array} { c | c c c c | c } J L ( 2 ) & J M ( 2 ) & O & \dots & O & \\ \vdots & O & \ddots & \ddots & \vdots & J R \\ \vdots & \vdots & \ddots & \ddots & O & \\ J L ( n ) & O & \dots & O & J M ( n ) & \\ \end{array} \end{pmatrix} , \end{align*}


\begin{align*} C \left ( X , Y , Z \right ) \equiv \left ( \nabla _ { Z } g \right ) \left ( X , Y \right ) = Z g \left ( X , Y \right ) - g \left ( \nabla _ { Z } X , Y \right ) - g \left ( X , \nabla _ { Z } Y \right ) \end{align*}


\begin{align*} \int _ { \mathbb { R } } \frac { e ^ { - \frac { \pi | \xi | ^ 2 } { t ^ 2 } } } { | \xi | ^ { 2 ( 1 - \beta ) } } \ , d \xi = \pi ^ { 1 / 2 - \beta } \Gamma \biggr ( \beta - \frac { 1 } { 2 } \biggr ) t ^ { 2 \beta - 1 } \leq \Gamma \biggr ( \beta - \frac { 1 } { 2 } \biggr ) t ^ { 2 \beta - 1 } , \end{align*}


\begin{align*} \gamma _ { t - 1 } \alpha _ t ^ 2 = ( 1 - \alpha _ t ) \alpha _ { t - 1 } ^ 2 \gamma _ t + \mu \alpha _ t \gamma _ t \gamma _ { t - 1 } , \end{align*}


\begin{align*} \begin{aligned} \det M = \sum \limits _ { \sigma \in S _ n } \sigma { M } _ { n _ { 1 1 } n _ { 1 2 } } \cdots { M } _ { n _ { 1 l _ 1 } n _ { 1 1 } } M _ { n _ { 2 1 } M _ { 2 2 } } \cdots { M } _ { n _ { r l _ r } n _ { r 1 } } . \end{aligned} \end{align*}


\begin{align*} ( Q ( \xi ) ) _ { i j } = Q _ { i j } ( \xi ) = \partial ^ 2 _ { i j } \xi + a _ { i j } ^ j \partial _ j \xi + a _ { j i } ^ i \partial _ i \xi + b _ { i j } \xi = 0 \quad \forall \ , 0 \leq i < j \leq p , \end{align*}


\begin{align*} \rho ( [ x , y ] ) \beta ( v ) = \rho ( \alpha ( x ) ) \rho ( y ) v - \varepsilon ( x , y ) \rho ( \alpha ( y ) ) \rho ( x ) v . \end{align*}


\begin{align*} R = \frac { 1 } { 2 J ( X _ { J } ) } \left [ 2 \delta ( X _ { J } ) g \wedge S + 2 \omega ( X _ { J } ) g \wedge g + E \wedge g + F \wedge S \right ] , \end{align*}


\begin{align*} \mathfrak h = \frac { _ { n - 1 } ( \Sigma ) } { \min \{ _ n ( A ) , _ n ( B ) \} } . \end{align*}


\begin{align*} \left ( \frac { d } { d \lambda } \right ) ^ r \left [ I ( \lambda ) - \lambda ^ { - \frac { n } { 2 } } \sum _ { j = 0 } ^ N a _ j \lambda ^ { - j } \right ] = O \left ( \lambda ^ { - \frac { n } { 2 } - r - N - 1 } \right ) . \end{align*}


\begin{align*} \ < g , \gamma \ > = \ < M _ { \omega \circ f } ( g ) , \gamma \ > = \lim \limits _ { \alpha } \ < M _ { \omega \circ f } ( h _ \alpha \circ f ) , \gamma \ > = \lim \limits _ { \alpha } \ < M _ { \omega } ( h _ \alpha ) \circ f , \gamma \ > . \end{align*}


\begin{align*} h _ { j _ p + 1 } = \rho _ p h _ { j _ p } + \frac { \rho _ p ( \rho _ p - 1 ) } { 2 } \end{align*}


\begin{align*} i _ { \hat { q } } = N ( \hat { k } - 1 ) + \hat { \imath } , j _ { \hat { q } } = N ( \hat { \ell } - 1 ) + \hat { \jmath } \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x & \leq C \int _ { B _ { 8 \rho } \setminus B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x \rho \leq 1 / 8 \varepsilon \leq \varepsilon _ 0 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \Big [ \sup _ { t \leq T } \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H ^ p \Big ] = 0 , \end{align*}


\begin{align*} K = \{ k _ n : n \in \N \} \subset M ^ c . \end{align*}


\begin{align*} M _ { f _ { 1 , \infty } } ( \alpha , \epsilon , Z , \psi ) : = \lim _ { n \to \infty } M _ { f _ { 1 , \infty } } ( n , \alpha , \epsilon , Z , \psi ) . \end{align*}


\begin{align*} Y ' _ { k , ( 2 ) } = Y _ { k , ( 2 ) } - z _ { k } t \mbox { w i t h } z _ k \in B _ { \infty , \infty } . \end{align*}


\begin{align*} \prescript J { } \pi ( w _ 1 ) = ( w _ 1 ' , \mu _ 1 ) , \prescript J { } \pi ( w _ 2 ) = ( w _ 2 ' , \mu _ 2 ) . \end{align*}


\begin{align*} S _ { R } ^ { \alpha } f ( x ) = \sum _ { n = 0 } ^ { \infty } \Big ( 1 - \frac { 2 n + d } { R } \Big ) _ { + } ^ { \alpha } P _ { n } f ( x ) , \end{align*}


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


\begin{align*} N _ n ( v ) : = \{ v ' \in V : d ( v , v ' ) \leq n \} \end{align*}


\begin{align*} \rho { u _ x } = \sum \limits _ { i = 0 } ^ { 1 8 } { { c _ { i x } } { f _ i } + \frac { { \Delta t } } { 2 } { F _ x } } , \end{align*}


\begin{align*} \lambda : = \frac { \beta } { \log C _ 0 + \beta / ( 2 \alpha ) + 1 } , \end{align*}


\begin{align*} \sigma ( x , t ) & = f ( x - t ) + f ( - x - t ) , \\ \mu ( x , t ) & = f ( x - t ) - f ( - x - t ) , \end{align*}


\begin{align*} S _ m ( \lambda ) = \sum \limits _ { 1 \leq j _ 1 < \cdots < j _ m \leq n } \lambda _ { j _ 1 } \dots \lambda _ { j _ m } , \end{align*}


\begin{align*} \rho _ { r _ { ( \alpha , k ) } } ( v ) = \begin{cases} v , & v ^ { - 1 } \alpha \in \Phi ^ + , \\ s _ \alpha v , & v ^ { - 1 } \alpha \in \Phi ^ - . \end{cases} \end{align*}


\begin{align*} f ( x ) & \in h ( x ) \ast ( g ( x ) ) ^ { - 1 } = h ( x ) \ast r ( g ) ( x ) \mbox { i m p l i e s } f \in h \ast r ( g ) \\ g ( x ) & \in ( f ( x ) ) ^ { - 1 } \ast h ( x ) = r ( f ) ( x ) \ast h ( x ) \mbox { i m p l i e s } g \in r ( f ) \ast h \end{align*}


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


\begin{align*} \lambda _ 1 = q \sqrt { q } \sigma _ 1 ( \textbf { z } ) , \lambda _ 2 = q ^ 2 \sigma _ 2 ( \textbf { z } ) , \lambda _ 3 = q \sqrt { q } \sigma _ 3 ( \textbf { z } ) . \end{align*}


\begin{align*} \int ^ 1 _ 0 | ( 1 - t ) ^ { \beta - 1 } t ^ \delta | d t = \int ^ 1 _ 0 ( 1 - t ) ^ { R e \beta - 1 } t ^ \delta d t < \infty . \end{align*}


\begin{align*} \Gamma ( X ) = \sum \limits _ { 1 \leq d \leq \sqrt { X ^ 2 + X + 1 } } \mu ( d ) \sum \limits _ { 1 \leq n \leq X \atop { n ^ 2 + n + 1 \equiv 0 \ , ( d ^ 2 ) } } 1 = \Gamma _ 1 ( X ) + \Gamma _ 2 ( X ) \ , , \end{align*}


\begin{align*} \left \langle X , \Gamma _ { ( g , u ) } ( X ) \right \rangle = - \tfrac { 1 } { 2 } \Delta | X | ^ 2 - \tfrac { 1 } { 2 } u ^ { - 1 } { \nabla u } \cdot \nabla | X | _ g ^ 2 + | \nabla X | ^ 2 + u ^ { - 2 } ( X ( u ) ) ^ 2 . \end{align*}


\begin{align*} \begin{aligned} & \| \mathbf { w } _ { x s } - \nu \mu \mathbf { e } \| ^ 2 = \| \mathbf { w } _ { x s } - \mu \mathbf { e } \| ^ 2 + 2 \| \mu \mathbf { e } - \nu \mu \mathbf { e } \| ^ 2 + 2 ( 1 - \nu ) \mu ( \mathbf { w } _ { x s } - \mu \mathbf { e } ) ^ T \mathbf { e } \\ & \le \left ( \gamma ^ 2 / 2 + ( 1 - \nu ) ^ 2 k \right ) \mu ^ 2 . \end{aligned} \end{align*}


\begin{align*} \ \alpha \ \lim _ { x \to \infty } f ( x ) / \alpha ( x ) = 1 . \end{align*}


\begin{align*} D ^ q _ * ( f ) ( z ) = \sum _ { n = 1 } ^ { \infty } a _ n \frac { \Gamma ( q n + 1 ) } { \Gamma ( q ( n - 1 ) + 1 ) } z ^ { q ( n - 1 ) } . \end{align*}


\begin{align*} O _ { n , n } ( R ) \subseteq O _ { n + 1 , n + 1 } ( R ) : A \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & A \end{pmatrix} . \end{align*}


\begin{align*} I ( \tilde { A } ^ { 1 , \dots , \ell } ) = \langle q _ { ( u , s ) } - q _ { ( u , r ) } \mid u \in [ \beta ] , s , r \in \{ 0 , \dots , x _ u - 1 \} \rangle + \tilde { J } . \end{align*}


\begin{align*} P _ N v \cdot \partial _ x P _ N ( P _ { \gtrsim N } v \cdot P _ { \gtrsim N } u _ 1 ) = \sum _ { N _ 2 \sim N _ 1 \gtrsim N } P _ N v \cdot \partial _ x P _ N ( P _ { N _ 1 } v \cdot P _ { N _ 2 } u _ 1 ) . \end{align*}


\begin{align*} \frac { d } { d \varepsilon } J ( u + \varepsilon v ) \Big | _ { \varepsilon = 0 } \geq 0 . \end{align*}


\begin{align*} x = q _ i ' - p _ i \end{align*}


\begin{align*} W ( t , \xi ) & = \sum _ { j = 1 } ^ N \mu _ j e _ j ( \xi ) \beta _ j ( t ) , t \ge 0 , \ \xi \in \mathbb { R } ^ d , \\ \mu ( \xi ) & = \frac { 1 } { 2 } \sum _ { j = 1 } ^ N | \mu _ j | ^ 2 e _ j ^ 2 ( \xi ) , \xi \in \mathbb { R } ^ d . \end{align*}


\begin{align*} p ^ { \kappa , \varepsilon } ( x , t ) = \mathrm { e x p } \Big ( - \frac { 1 } { \varepsilon } y ( t ) h ( x ) \Big ) q ^ { \kappa , \varepsilon } ( x , t ) , \end{align*}


\begin{align*} \tau \lim _ { n \to \infty } T ( t ) ( x _ n - x ) = 0 \end{align*}


\begin{align*} \varphi ^ L _ { \sigma , s ' } ( D ) = \{ a , b \} \varphi ^ L _ { \sigma , s ' } ( E ) = \varphi ^ L _ { \sigma , s ' } ( C _ i ) \setminus \{ a \} = C \setminus \{ a \} . \end{align*}


\begin{align*} & \frac { \varpi _ n } { \varpi _ { n + 1 } } x ( \varpi _ { n + 1 } ) = \frac { \varpi _ n } { \varpi _ { n + 1 } } \varphi _ M ^ { - n - 1 } ( \frac \varpi \pi x ' ) = \varphi _ M ^ { - n - 1 } ( \frac { \varpi ^ p } { \varpi } \frac \varpi \pi x ' ) = \varphi _ M ^ { - n } ( \varphi _ M ^ { - 1 } ( \frac { \varpi ^ p } \pi x ' ) ) = \\ & \qquad = \varphi _ M ^ { - n } ( \frac \varpi \pi x ' ) = x ( \varpi _ n ) . \end{align*}


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


\begin{align*} X _ 1 = \frac { \partial } { \partial x } - \frac { y } { 2 } \frac { \partial } { \partial z } , X _ 2 = \frac { \partial } { \partial y } + \frac { x } { 2 } \frac { \partial } { \partial z } , X _ 3 = \frac { \partial } { \partial z } . \end{align*}


\begin{align*} \sum _ { \Delta \Subset \Z ^ d } \sum _ { \xi _ { \Delta } } \nabla _ { \Lambda } \left ( c _ { \Delta } ( \cdot , \xi _ { \Delta } ) - \hat { c } ( \cdot , \xi _ { \Delta } ) \right ) ( \eta ) = 0 , \end{align*}


\begin{align*} d ( \lambda _ v ) = e = \iota _ { \star } = \lambda _ { d ^ 0 ( v ) } \end{align*}


\begin{align*} I ^ \alpha : = X ^ \alpha + \sum _ { \beta , \gamma \in S _ + } { f ^ { \alpha \beta } } _ \gamma : \varphi _ \gamma \varphi ^ \beta : + \frac { 1 } { 2 } \sum _ { \beta \in S _ { \frac { 1 } { 2 } } } { f ^ { \beta \alpha } } _ \gamma \Phi ^ \beta \Phi _ \gamma \end{align*}


\begin{gather*} \mathcal { A } _ j ( x , \xi ) = D _ { \xi } F _ j ( x , \xi ) , \end{gather*}


\begin{align*} P \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \beta _ 1 ( \xi _ 0 ) \\ \beta _ 2 ( \xi _ 0 ) \end{pmatrix} & { } = P \lim _ { t \to \infty } t ^ { - \mu | W _ 1 ( \xi _ 0 ) | ^ 2 } P ^ { - 1 } \begin{pmatrix} w ( \xi _ 0 ) \\ \overline { w ( \xi _ 0 ) } \end{pmatrix} \\ & { } = \lim _ { t \to \infty } t ^ { - \mu | W _ 1 ( \xi _ 0 ) | ^ 2 } \begin{pmatrix} w ( \xi _ 0 ) \\ \overline { w ( \xi _ 0 ) } \end{pmatrix} \in Z , \end{align*}


\begin{align*} \nabla \hat X = \omega \quad \mbox { i n } \Omega . \end{align*}