\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I I I \leq \lim _ { M \rightarrow \infty } \Big [ C _ M \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ { T } \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s \Big ] = 0 . \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) \| _ { \mathcal { A } } & \le \| x _ { n } ( t ) \| _ { \mathcal { A } } + \| x ( t ) - x _ { n } ( t ) \| \\ & \le \beta ( \| x ( n \bar { t } / \gamma ) \| _ { \mathcal { A } } , \gamma t - n \bar { t } ) + \tilde { d } \\ & \le \beta ( \| x ( n \bar { t } / \gamma ) \| _ { \mathcal { A } } , 0 ) + \tilde { d } & & \forall \ , t \in I _ n , \ , \forall \ , n \in \mathbb { N } \end{aligned} \end{align*}


\begin{align*} \alpha = \alpha _ { ( 0 ) } + \alpha _ { ( 1 ) } \end{align*}


\begin{align*} \big | M ^ * _ v ( t ) - M ^ * _ v ( \delta n ) \big | & \le e ^ { - ( \lambda ^ * - \lambda ) t } \big | M _ v ( t ) - M _ v ( \delta n ) \big | + \big ( e ^ { - ( \lambda ^ * - \lambda ) \delta n } - e ^ { - ( \lambda ^ * - \lambda ) t } \big ) | M _ v ( \delta n ) | \\ & \le \delta e ^ { - ( \lambda ^ * - \lambda ) ( t - \delta n ) } + \delta \big ( 1 - e ^ { - ( \lambda ^ * - \lambda ) ( t - \delta n ) } \big ) = \delta , \end{align*}


\begin{align*} \mathbb { E } \left [ \sum _ { n = 0 } ^ { \infty } z ^ n e ^ { i t S _ n } e ^ { i { s } \cdot { C } _ n } \right ] = \frac { 1 } { 1 - z \phi _ { \xi _ 1 , { \eta _ 1 } } ( t , { s } ) } = f ^ { - 1 } _ { + } ( z , t , s ) f _ { - } ( z , t , \underline { s } ) \end{align*}


\begin{align*} \boldsymbol { \alpha } _ t ^ * : = ( \alpha _ t ^ { * , i } ) _ { 1 \leq i \leq N } = \left ( - R ^ { - 1 } B ( P _ t x _ t ^ { * , i } + \varphi _ t ^ { * , i } ) - h \Big ( k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } ^ * _ t } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } ^ * _ t } , \lambda ^ { N , 1 } _ { \boldsymbol { \varphi } ^ * _ t } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \varphi } ^ * _ t } ) \Big ) \right ) _ { 1 \leq i \leq N } . \end{align*}


\begin{align*} \boldsymbol { \alpha } _ t ^ * = ( \alpha _ t ^ { * , i } ) _ { 1 \leq i \leq N } = \left ( - R ^ { - 1 } B y _ t ^ { * , i } - h \big ( \rho _ i ^ N ( \Delta ^ { N , 1 } _ { * , t } , \ldots , \Delta ^ { N , N } _ { * , t } ) \big ) \right ) _ { 1 \leq i \leq N } . \end{align*}


\begin{align*} \left [ c _ { \chi _ 1 \chi ^ { - 1 } } \right ] = \left [ \beta _ \eta \right ] = 0 \in \mathrm { H } ^ 2 ( Q , S ^ 1 ) . \end{align*}


\begin{align*} \mathbf { c } ^ { ( n - 2 k + 1 ) } = ( \diamond , \diamond , \cdots , \diamond , \underbrace { 1 , 1 , \cdots , 1 } _ { 2 k } ) . \end{align*}


\begin{align*} - n ^ p - \sum _ { K = 1 } ^ { p } \binom { p } { K } \ , B _ K \ , n ^ { p - K } . \end{align*}


\begin{align*} C u r \mathcal { G } = \mathbb { C } [ \partial ] \otimes \mathcal { G } , [ a _ \lambda b ] = [ a , b ] , \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*} \frac { \sum _ { i = 1 } ^ p \log R _ i - m _ n } { s _ n } \overset { d } { \longrightarrow } S _ \alpha , n \to \infty \ , , \end{align*}


\begin{align*} G ^ \prime = \begin{pmatrix} 1 & 1 & \cdots & 1 \\ g _ { 2 1 } ^ \prime & g _ { 2 2 } ^ \prime & \cdots & g _ { 2 n } ^ \prime \\ \vdots & \vdots & & \vdots \\ g _ { k 1 } ^ \prime & g _ { k 2 } ^ \prime & \cdots & g _ { k n } ^ \prime \end{pmatrix} = \begin{pmatrix} G _ 1 ^ \prime & G _ 2 ^ \prime & \cdots & G _ n ^ \prime \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*} v : = - d + \frac { 1 } { 8 b _ { n - 1 } } d ^ 2 . \end{align*}


\begin{align*} - \varepsilon ^ 2 ( K ( x ) \nabla u ) + V ( x ) u = u ^ { p } , \ \ \ \ x \in \mathbb { R } ^ n , \end{align*}


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


\begin{align*} F f ( x ; z ) = ( h _ { 1 } , h _ { 2 } ) ^ T , \end{align*}


\begin{align*} h _ 2 ( t ) = \frac { | A \oplus _ 2 ( \sqrt { t } B ) | ^ 2 } { | P _ { u ^ \bot } ( A \oplus _ 2 ( \sqrt { t } B ) ) | ^ 2 _ { n - 1 } } \end{align*}


\begin{align*} \ddot { x } + \frac { 3 } { t } \dot { x } + \nabla f ( x ) = 0 \end{align*}


\begin{align*} \int _ { X _ j } | u _ { \epsilon , j } | ^ 2 _ { h } e ^ { v _ \epsilon ( \Psi ) - \delta \max \{ \psi + T , 2 \log | F | \} } c ( - v _ \epsilon ( \Psi ) ) \le \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } < + \infty . \end{align*}


\begin{align*} \sum _ { k = 2 } ^ n \frac { \mu ( k ) } { k } ( I - S ) h _ k \to 1 - z \ H ^ p \end{align*}


\begin{align*} \begin{array} { l } R _ { \max } = \mathop { \max } \limits _ { \bf p } { R _ T } \\ { \rm { s } } { \rm { . t } } { \rm { . } } \\ C 2 , \ , C 3 , \ , C 8 , \ , { \rm a n d } \ , C 9 . \end{array} \end{align*}


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


\begin{align*} { \bf { \bar U } } ^ { \star } = { \left [ { { \bf { \tilde \Sigma } } { { \bf { \tilde B } } ^ H } } \right ] _ { 1 : { M _ t } , : } } , \end{align*}


\begin{align*} g \big | _ { U _ \alpha } = D ^ 2 u _ \alpha . \end{align*}


\begin{align*} \Lambda ( s , \rho ) = \epsilon ( \rho ) c ( \rho ) ^ { \frac { 1 } { 2 } - s } \Lambda ( 1 - s , \overline { \rho } ) , \end{align*}


\begin{align*} \sum _ { \tau \in D ( n - 1 ) } \operatorname { s i g n } ( \tau ) \prod _ { j = 1 } ^ { n - 1 } \dfrac { 1 } { 1 - \zeta ^ { j - \tau ( j ) } } = \frac { ( - 1 ) ^ { \frac { n - 1 } { 2 } } } { n } \left ( \frac { n - 1 } { 2 } ! \right ) ^ { 2 } , \end{align*}


\begin{align*} S _ k ^ { i j } ( T _ \alpha u ) _ { i j } = T _ \alpha f . \end{align*}


\begin{align*} D = \max \left \{ D _ { x , 1 } \sqrt { \delta ' } + D _ { x , 2 } , D _ { y , 1 } \sqrt { \delta ' } + D _ { y , 2 } \right \} , \end{align*}


\begin{align*} C ( z ) = \sum _ { i } c _ { i } g _ { i } ( z ) \end{align*}


\begin{align*} \partial \omega = \partial _ 1 \omega \cup \partial _ 2 \omega \cup \partial _ 3 \omega , \end{align*}


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


\begin{align*} \Bigg ( \sum _ { v \in \mathbb { N } ^ n } \frac { c ( v , q ) } { \mathit { g l } ( v , q ) } X ^ v \Bigg ) \ ! \circ \mathrm { E x p } \Bigg ( \sum _ { v \in \mathbb { N } ^ n \backslash \{ 0 \} } \ ! \frac { s ( v , q ) } { 1 - q } X ^ v \Bigg ) = 1 , \\ \end{align*}


\begin{align*} \div ( A ( x ) \nabla v ) + h ( x ) = 0 B _ 1 . \end{align*}


\begin{align*} g ( \overline { t } ) = \mathop { \max } \limits _ { t > 0 } g ( t ) . \end{align*}


\begin{align*} K _ { \lambda , \varrho ( I ) } = \sum _ { J \subset I } \kappa _ { \lambda , J } , \end{align*}


\begin{align*} v _ 1 ( x , t ) = v _ 0 ( x - \lambda t ) + \int _ { t _ { 1 , e x } ( x , t ) } ^ t v ( s , x - \lambda t + \lambda s ) \dd s + \int _ { 0 } ^ { t _ { 1 , e n } ( x , t ) } v ( s , x - \lambda t + \lambda s ) \dd s . \end{align*}


\begin{align*} W ( y | x ) = b _ k - a + \sum _ { n = 2 } ^ \infty ( a _ n - a ) \ , \end{align*}


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


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


\begin{align*} \theta ' _ i = \pi _ i ( X _ 2 , Y _ 2 , \dots , X _ { i - 1 } , Y _ { i - 1 } ) , i \ge 2 . \end{align*}


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


\begin{align*} ( \nabla _ X \beta ) ( Y , Z ) = ( \nabla _ Y \beta ) ( X , Z ) . \end{align*}


\begin{align*} g _ 1 ( x ) = \frac { x ^ 2 } { 1 + \cos x } , g _ 2 ( x ) = \frac { x \sinh x } { 1 + \cosh x } . \end{align*}


\begin{align*} \begin{aligned} v ' = \xi + v \ , , \ v _ 1 = v + \xi + y \ , . \end{aligned} \end{align*}


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


\begin{align*} \mathcal { F } _ { \geq p } C F ^ { k } ( H _ { t } ) : = \lbrace x \in C F ^ { k } ( H _ { t } ) \mid \mathcal { A } ( x ) \geq p \rbrace \end{align*}


\begin{align*} S H ^ { * } _ { M } ( K ; \Lambda _ { 0 } ) : = H ^ { * } ( \widehat { t e l } ( \mathcal { C } ; \Lambda _ { 0 } ) ) . \end{align*}


\begin{align*} \varphi = ( \tilde \varphi , \varphi ^ * ) : ( U _ x , \mathcal { O } _ M | _ { U _ x } ) \rightarrow \mathcal { U } . \end{align*}


\begin{align*} X _ t ^ i - X _ { t _ n } ^ i = \int _ { t _ n } ^ t b ^ i ( X _ s ) \d s + \sigma ( W _ t ^ i - W _ { t _ n } ^ i ) . \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{align*} \mu ^ i + h ( \mu ^ i ) = - R ^ { - 1 } B \Big [ P _ t \nu ^ { N , i } _ { \boldsymbol { x } } + \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) \Big ] . \end{align*}


\begin{align*} \big ( h ( t _ j ) , v ( t _ j ) \big ) & = \big ( \hat h ( t _ j ) , \hat v ( t _ j ) \big ) - \big ( L _ V g _ { t _ j } , V ( u _ { t _ j } ) \big ) \\ ( p , z ) & = ( \hat p , \hat z ) - \Big ( L _ V L _ X g _ a , V \big ( X ( u _ a ) \big ) \Big ) \end{align*}


\begin{align*} \int _ K | | \tau _ 0 | | _ { \omega _ { \epsilon _ 1 } } \le & \liminf _ { \epsilon \to 0 } \int _ K | | \tau _ { \epsilon } | | _ { \omega _ { \epsilon _ 1 } } \\ \le & \liminf _ { \epsilon \to 0 } \int _ K | | \tau _ { \epsilon } | | _ { \omega _ { \epsilon } } \\ \le & \liminf _ { \epsilon \to 0 } \int _ M | | \tau _ { \epsilon } | | _ { \omega _ { \epsilon } } . \end{align*}


\begin{align*} \alpha _ m = \max \{ t \ , | \ , Q - t J - Y \succeq 0 , Y \in \R ^ { Z _ m \times Z _ m } _ { \geq 0 } \} . \end{align*}


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


\begin{align*} L u : = \big ( a _ { j k } ^ { \alpha \beta } \partial _ j \partial _ k u _ \beta \big ) _ { 1 \leq \alpha \leq N } , \end{align*}


\begin{align*} \underline { h } _ { \mu } ( f _ { 1 , \infty } , x ) : = \lim _ { \epsilon \to 0 } \liminf _ { n \to \infty } \dfrac { - \log \mu ( B _ { n } ( x , \epsilon ) ) } { n } . \end{align*}


\begin{align*} \mathbf { Q } _ { 0 , n } f & = \frac { q t } { q t - 1 } h _ n [ ( 1 - q t ) X / ( q t ) ] \cdot f \\ \mathbf { Q } _ { 1 , n } f & = \mathbf { D } _ n f . \end{align*}


\begin{align*} ( \nu ( u ) ) ' = \nu ' ( h ) ( \bar u ) + \nu ( v ) \end{align*}


\begin{align*} W ( y | x ) = \sum _ { n \in \N } \left [ \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ' ) \right ] \end{align*}


\begin{align*} \varphi ( f \circ g ) = ( ( f \circ g ) _ { 1 1 } , \ldots , ( f \circ g ) _ { n n } ) = ( f _ { 1 1 } \circ g _ { 1 1 } , \ldots , f _ { n n } \circ g _ { n n } ) \\ = ( f _ { 1 1 } , \ldots , f _ { n n } ) \circ ( g _ { 1 1 } , \ldots , g _ { n n } ) = \varphi ( f ) \circ \varphi ( g ) . \end{align*}


\begin{align*} \frac { d } { d t } z _ t = B ( z _ t , z _ t ) \end{align*}


\begin{align*} \begin{aligned} \tfrac { \chi ( \sqrt { | V _ \shortparallel | ^ 2 + | V _ \perp ' - \zeta _ \perp | ^ 2 } ) } { ( | V _ \shortparallel | ^ 2 + | V _ \perp ' - \zeta _ \perp | ^ 2 ) ^ { \frac { 1 - \gamma } { 2 } } } \leq \tfrac { 2 ^ { 1 - \gamma } } { ( 1 + | V _ \shortparallel | ^ 2 + | V _ \perp ' - \zeta _ \perp | ^ 2 ) ^ { \frac { 1 - \gamma } { 2 } } } \ , . \end{aligned} \end{align*}


\begin{align*} n = \frac { \beta ^ { 3 } - 3 } { 6 } \end{align*}


\begin{align*} \tilde X _ { n + 1 } ^ i = \tilde X _ n ^ i + \bigg ( b ( \tilde X _ n ^ i ) + \frac 1 { N - 1 } \sum _ { j \neq i } K ( \tilde X _ n ^ i - \tilde X _ n ^ j ) \bigg ) \tau + \sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , \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*} \operatorname { R } = \frac { 1 } { 2 } \mathcal { H } ^ 2 \end{align*}


\begin{align*} 0 = \frac { ( \tilde { \kappa } _ 1 ) _ i } { \tilde { \kappa } _ 1 } - N \frac { ( \nu ^ { n + 1 } ) _ i } { \nu ^ { n + 1 } } , \end{align*}


\begin{gather*} P ( f ) = P ( \tilde { f } ) = \int Q ( \tilde { f } ) \ , P _ \mathcal { U } ( d Q ) = \int Q ( f ) \ , P _ \mathcal { U } ( d Q ) = 0 . \end{gather*}


\begin{align*} \mathbf { y } = \mathbf { H } \mathbf { P } ^ { \left ( \right ) } \left ( \mathbf { a } ^ { \left ( \right ) } \right ) \mathbf { s } ^ { \left ( \right ) } + \mathbf { n } , \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{gather*} P ( \abs { f } ) \le j + P \bigl \{ \abs { f } \ , 1 ( \abs { f } > j ) \bigr \} = j + \sup _ k P \bigl \{ \abs { f } \wedge k \ , 1 ( \abs { f } > j ) \bigr \} \\ = j + \sup _ k \int Q \bigl \{ \abs { f } \wedge k \ , 1 ( \abs { f } > j ) \bigr \} \ , P _ \mathcal { U } ( d Q ) \le j + \sup _ { Q \in \mathcal { U } } Q \bigl \{ \abs { f } \ , 1 ( \abs { f } > j ) \bigr \} . \end{gather*}


\begin{align*} g _ \lambda ( z ) = \begin{cases} 0 & z = \lambda , \\ ( z - \lambda ) ^ { - 1 } & z \neq \lambda , \end{cases} \end{align*}


\begin{align*} { \mathcal { E } } = \frac { 1 } { 2 } \int _ { \Omega } \left ( | u | ^ { 2 } + | B | ^ { 2 } \right ) \dd x , { \mathcal { H } } = \int _ { \Omega } A \cdot B \dd x , \end{align*}


\begin{align*} \pi ( x , \lambda ) = x \ , , \ , \forall x \in \R ^ n \ , , \ , \forall \lambda \in \R ^ { m } \ , . \end{align*}


\begin{align*} \left \{ \begin{aligned} \mathcal H _ X ^ D ( \alpha ) _ x & = \mathcal H _ { X , x } & \\ \mathcal H _ X ^ D ( \alpha ) _ x & = \alpha _ i \cap R ^ 0 _ { y _ i , n _ i } & \end{aligned} \right . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \partial _ t f + \mathcal { L } ( f ) = \Gamma ( f , f ) , \\ & f | _ { t = 0 } = f _ 0 , \end{aligned} \right . \end{align*}


\begin{align*} c _ s ^ 2 \partial _ { y 1 } T = - \frac { 1 } { \Delta t } { \varsigma } _ 2 { m } _ 2 ^ { ( 1 ) } , \end{align*}


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


\begin{align*} N _ { r s } \coloneqq \begin{cases} M _ { r s } & \\ N _ { r k } & . \end{cases} \end{align*}


\begin{align*} T _ { \delta } - \mathcal { P } ^ - \left ( T _ { \delta } R _ { \delta } \right ) = F _ { \delta } , \end{align*}


\begin{align*} \frac { \partial ^ 2 \varphi } { \partial x _ \beta ^ 2 } ( x ' , \rho ( x ' ) ) \geq b _ \beta - C \delta \tilde { b } _ \beta , \ \beta = 1 , \ldots , n - 1 \end{align*}


\begin{align*} W : = \max _ { \overline { \omega } _ \delta } ( A \Psi \pm T _ \alpha ( u - \varphi ) ) \end{align*}


\begin{align*} 2 \frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \kappa _ 1 ( \kappa _ 1 - \tilde { \kappa } _ i ) } - \frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \kappa _ 1 ^ 2 } = \frac { F ^ { i i } ( \kappa _ 1 + \tilde { \kappa } _ i ) } { \kappa _ 1 ^ 2 ( \kappa _ 1 - \tilde { \kappa } _ i ) } h _ { 1 1 i } ^ 2 \geq c ( n ) \frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \kappa _ 1 ^ 2 } , \end{align*}


\begin{align*} \mathbb E | X _ t ^ i | ^ 2 & = \mathbb E | X _ 0 ^ i | ^ 2 + 2 \int _ 0 ^ t X _ s ^ i \cdot b ^ i ( X _ s ) \d s + d \sigma ^ 2 t \\ & = \mathbb E | X _ 0 ^ i | ^ 2 + 2 \int _ 0 ^ t X _ s ^ i \cdot \big ( b ( X _ s ^ i ) + \gamma ^ i ( X _ s ) \big ) \d s + d \sigma ^ 2 t . \end{align*}


\begin{align*} { [ k + 1 + \ell ] \choose [ q ] } = \sum _ { i = - 1 } ^ { q } { [ k ] \choose [ i ] } \cdot { [ \ell ] \choose [ q - 1 - i ] } = { [ k ] \choose [ q ] } + { [ \ell ] \choose [ q ] } + \sum _ { i = 0 } ^ { q - 1 } { [ k ] \choose [ i ] } \cdot { [ \ell ] \choose [ q - 1 - i ] } , \end{align*}


\begin{align*} 0 = V _ 0 \subset V _ 1 \subset V _ 2 \subset \cdots \subset V _ n \subseteq V , \end{align*}


\begin{align*} \begin{aligned} \mathsf { M } ^ n = & \{ \mathsf { m } _ 1 , \cdots , \mathsf { m } _ n \} \\ = & \left \{ g \bigl ( \xi _ 0 \bigr ) , g \bigl ( \xi _ 1 ^ { \mathsf { M } } \bigr ) , \cdots , g \bigl ( \xi _ { n - 1 } ^ { \mathsf { M } } \bigr ) \right \} . \end{aligned} \end{align*}


\begin{align*} \mathcal { M } _ { \Gamma } = \left \lbrace p R \ , \vert \ , p _ { i } = \dfrac { 1 } { Z ( \theta ) } \prod \limits _ { F \in \mathrm { f a c e t } ( \Gamma ) } \theta ^ { F } _ { i _ { F } } \right \rbrace \end{align*}


\begin{align*} s _ i ( x ) = \left \{ \begin{array} { l l } s _ i x \in X _ { k + 1 } \subseteq ( \Sigma X ) _ { m + 1 } & \mbox { i f } 0 \leq i \leq k , \\ x \in X _ { k } \subseteq ( \Sigma X ) _ { m } & \mbox { i f } k + 1 \leq i \leq m . \end{array} \right . \end{align*}


\begin{align*} H _ { m i n } \left ( ( E _ 1 [ t ] \times E _ 1 [ t ] ^ { \vee } ) \otimes F \right ) & = H _ { m i n } \left ( E _ 1 [ t ] \otimes F \times E _ 1 [ t ] ^ { \vee } \otimes F \right ) \\ & = \min \left ( H _ { m i n } ( E _ 1 [ t ] \otimes F ) , H _ { m i n } ( E _ 1 [ t ] ^ { \vee } \otimes F ) \right ) \\ & \leq H _ { m i n } ( E _ 1 [ t ] \otimes F ) = t H _ { m i n } ( E _ 1 \otimes F ) \leq t H _ { m i n } ( E _ 1 ) H _ { m i n } ( F ) \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*} K _ { \lambda , \beta } = \sum _ { J \subset \varrho ^ { - 1 } ( \beta ) } \tilde \kappa _ { \lambda , J } , \end{align*}


\begin{align*} S _ k ^ { i j } ( T _ \alpha u ) _ { i j } = T _ \alpha f . \end{align*}


\begin{align*} \omega _ { i j } ^ k = \left \{ \begin{array} { l r } - 2 \Gamma _ { i j } ^ j & k = j \neq i , \\ 2 \Gamma _ { i j } ^ j & k = i \neq j , \\ 0 & \ , . \end{array} \right . \end{align*}


\begin{align*} \mu _ { ( \boldsymbol { x } , \boldsymbol { \Psi } ( t , \boldsymbol { x } ) ) } ^ { N , i } : = k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } } , \lambda ^ { N , 1 } _ { \boldsymbol { \Psi } ( t , \boldsymbol { x } ) } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \Psi } ( t , \boldsymbol { x } ) } ) , \end{align*}


\begin{align*} \lim _ { n \to \infty } ( T _ i ( x ^ \pm _ n ) - x ^ \pm _ n ) Q _ i ( x ^ \pm _ n ) = 0 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { t \leq T } \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H = 0 , \ \mathbb { P } . \end{align*}


\begin{align*} M ( x ; z ) : = \begin{cases} e ^ { i c _ + ( x ) \sigma _ 3 } \left ( \frac { \Psi _ 1 ^ - ( x ; z ) } { a ( z ) } , \Psi ^ + _ 2 ( x ; z ) \right ) , \ ; z \in \mathbb { C } ^ + , \\ [ 5 p t ] e ^ { i c _ + ( x ) \sigma _ 3 } \left ( \Psi _ 1 ^ + ( x ; z ) , \frac { \Psi ^ - _ 2 ( x ; z ) } { \bar { a } ( z ) } \right ) , \ ; z \in \mathbb { C } ^ - , \end{cases} \end{align*}


\begin{align*} h _ { 1 1 i i } = h _ { i i 1 1 } + \kappa _ 1 ^ 2 \kappa _ i - \kappa _ 1 \kappa _ i ^ 2 - \kappa _ 1 + \kappa _ i . \end{align*}


\begin{align*} f _ { a } ( x ) = \dfrac { a } { a ^ 2 + x ^ 2 } . \end{align*}


\begin{align*} K _ { \lambda , \varrho ( I ) } = \sum _ { J \subset I } \kappa _ { \lambda , J } , \end{align*}


\begin{align*} \left ( \sqrt { q ' _ j } - \sqrt { p ' _ j } \right ) ^ 2 = q ' _ j \left ( 1 - \sqrt { 1 - \frac { q ' _ j - p ' _ i } { q ' _ j } } \right ) ^ 2 \leq \frac { ( q ' _ j - p ' _ i ) ^ 2 } { q ' _ j } . \end{align*}


\begin{align*} G _ { \hat { x } } ( x , y ) = \frac { 1 } { 2 \pi } \ln \frac { 1 } { | x - y | } - h _ { \hat { x } } ( x , y ) , \ \ \forall x , y \in T _ { \hat { x } } ( \Omega ) . \end{align*}


\begin{align*} y _ n = \rho ( x ) - \rho ( x _ 0 ) \geq \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( x ' _ 0 ) y _ \beta + \kappa | y ' | ^ 2 , \end{align*}


\begin{align*} q , r , \tilde q , \tilde r \geq 2 \quad \tfrac { 2 } { q } + \tfrac { 1 } { r } = \tfrac { 1 } { 2 } = \tfrac { 2 } { \tilde q } + \tfrac { 1 } { \tilde r } . \end{align*}


\begin{align*} \tilde M ^ { \tau } _ { \mu , s } ( t ) = M ^ { \tau } _ { \mu , s } ( T _ { \mu , s } ^ { \tau } ( t ) ) . \end{align*}


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


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


\begin{align*} a ( u , v ) : = ( \triangle u , \Delta v ) . \end{align*}


\begin{align*} \lambda | ^ * _ { [ w _ 1 , w _ 2 ] } ( a ) = \lambda ( w _ 1 a ) \end{align*}


\begin{align*} n ! F _ { n + 1 } \ & = \ \sum _ { k = 1 } ^ { n } \begin{bmatrix} n \\ k \end{bmatrix} A _ k , \\ n ! F _ { n + 2 } \ & = \ \sum _ { k = 1 } ^ { n } \begin{bmatrix} n \\ k \end{bmatrix} B _ k . \\ \end{align*}


\begin{align*} G r _ w ( E ) = \bigoplus _ { j = 1 } ^ { k } E ^ j / E ^ { j - 1 } , \end{align*}


\begin{align*} \mathrm { L i f t } ( f ) = \left \lbrace \prod _ { \alpha = 1 } ^ d z ^ { i _ { \alpha } } _ { j _ { \alpha } k _ { \alpha } } - \prod _ { \alpha = 1 } ^ d z ^ { i _ { \alpha } } _ { j ' _ { \alpha } k _ { \alpha } } \mid k _ { \alpha } \in [ t _ { i _ { \alpha } } ] \alpha \in [ d ] \right \rbrace . \end{align*}


\begin{align*} g ( \upsilon ) = \frac 1 4 \left ( f ( \upsilon ) + \frac 1 3 f ( 3 \upsilon ) - 2 f ( \upsilon ) - u ( \upsilon ) f ( \upsilon ) \right ) . \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*} \begin{aligned} \nabla \times U = f U , \nabla \cdot U & = 0 \textrm { i n } \ \Omega , \\ U \cdot n & = 0 \textrm { o n } \ \partial \Omega . \end{aligned} \end{align*}


\begin{align*} d X ^ i _ t = \alpha _ i ( t , X _ t ) d t + d B ^ i _ t , X ^ i _ 0 = 0 , \ \ i = 1 , \ldots , n , \end{align*}


\begin{align*} & \limsup _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \frac { 1 } { \inf _ { I _ { i , n } \backslash U _ k } c ( t ) } \int _ { I _ { i , n } \backslash U _ k } c ( s ) e ^ { - s } d s \\ \le & \limsup _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \frac { \sup _ { I _ { i , n } \backslash U _ k } c ( t ) } { \inf _ { I _ { i , n } \backslash U _ k } c ( t ) } \int _ { I _ { i , n } \backslash U _ k } e ^ { - s } d s \\ = & \int _ { ( t ' _ 2 , t ' _ 1 ] \backslash U _ k } e ^ { - s } d s . \end{align*}


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


\begin{align*} \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*} \mathcal { M } _ 1 = \sqrt { \frac { \pi } { 2 } } \ , e ^ { \frac { \psi _ { 1 } ^ 2 \tau _ { \rm d } ^ 2 } { 2 N _ { \rm a } ^ 2 } } \left [ \mathrm { e r f } \left ( \frac { \kappa _ { \rm t } + \frac { \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } } { \sqrt { 2 } } \right ) - \mathrm { e r f } \left ( \frac { \kappa _ { \rm b } + \frac { \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } } { \sqrt { 2 } } \right ) \right ] . \end{align*}


\begin{align*} \begin{array} { r l } h _ = & ( \lambda _ 1 \begin{bmatrix} 2 x _ 1 \\ - 1 \end{bmatrix} + \lambda _ 2 \begin{bmatrix} - 2 x _ 1 \\ 4 \end{bmatrix} + \lambda _ 3 \begin{bmatrix} 0 \\ - 1 \end{bmatrix} , \\ \\ & \lambda _ 1 ( x _ 1 ^ 2 - x _ 2 ) , \lambda _ 2 ( - x _ 1 ^ 2 + 4 x _ 2 ) , \lambda _ 3 ( - x _ 2 + 1 ) , 1 - \lambda _ 0 ^ 2 - \lambda _ 1 ^ 2 - \lambda _ 2 ^ 2 - \lambda _ 3 ^ 2 ) \ , . \end{array} \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } P ^ L ( x ' ) \ , d x ' = I _ { N \times N } , \end{align*}


\begin{align*} \epsilon ' = \sum _ { i \in I } \bar { \nu } _ { j ! } & \ & ( \delta _ k ^ n ) _ ! = \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*} & W _ 0 = \{ ( p , q ) \in \hat A \times \hat B \mid d ( q , w ) \geq d ( p , w ) \} ; \\ & W _ 1 = \{ ( p , q ) \in \hat A \times \hat B \mid d ( p , \tilde { w } ) \geq d ( q , \tilde { w } ) \} . \end{align*}


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


\begin{align*} \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*} F _ m ( n ) = 1 - G ( n , m ) + m \left \lfloor \frac { n } { m } \right \rfloor \sum _ { k = 1 } ^ m \frac { \mu ( k ) } { k } . \end{align*}


\begin{align*} \int _ { a x } ^ { x } | \theta ( t + \delta t ) - \theta ( t ) - \delta t | ^ 2 d t & > \int _ I | \theta ( t + \delta t ) - \theta ( t ) - \delta t | ^ 2 d t = \delta ^ 2 \int _ I t ^ 2 d t \\ & > \delta ^ 2 \int _ { a x } ^ { x / 2 } t ^ 2 d t = \frac { \delta ^ 2 x ^ 3 } { 3 } \left ( \frac { 1 } { 8 } - a ^ 3 \right ) . \end{align*}


\begin{align*} \Gamma ( x , y ) & = - \frac { 1 } { e ^ { - { \psi } } } \left ( d \phi ( x ) \int _ { - \infty } ^ y e ^ { - { \psi ( x , v ) } } d v - \int _ { - \infty } ^ y e ^ { - { \psi ( x , v ) } } d _ x { \psi } ( x , v ) d v \right ) \\ & = \frac { 1 } { e ^ { - \psi } } \left ( d \phi ( x ) \int _ { y } ^ \infty e ^ { - { \psi ( x , v ) } } d v - \int _ { y } ^ \infty e ^ { - { \psi ( x , v ) } } d _ x { \psi } ( x , v ) d v \right ) . \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 \{ \begin{aligned} - \Delta z _ { h } + F ' ( y ; z _ { h } ) & = h , \ ; \ ; \Omega , \\ z _ { h } & = 0 , \ ; \ ; \partial \Omega , \end{aligned} \right . \end{align*}


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


\begin{align*} f ( x _ 1 , . . . , x _ n ) = x _ n \end{align*}


\begin{align*} \mathbf { v } \cdot \overrightarrow { \zeta } = 0 , \end{align*}


\begin{align*} N + E _ N ^ + - E _ N ^ - = f ^ * ( f _ * N ) . \end{align*}


\begin{align*} \tilde \Sigma = \{ x \in M \mid { \rm V o l } ( A \cap B _ x ( r ) ) = { \rm V o l } ( B \cap B _ x ( r ) ) \} , \end{align*}


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


\begin{align*} F ' _ m ( n ) = 1 - G ' ( n , m ) + m \left \lfloor \frac { n } { m } \right \rfloor \sum _ { \begin{smallmatrix} k = 1 \\ k | m \end{smallmatrix} } ^ m \frac { \mu ( k ) } { k } . \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*} \tau = \sum _ { j = 1 } ^ \infty \frac { c _ j } { 2 ^ j } \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*} q _ 0 ( x ) : = q ( x ) , ~ 2 \sqrt x q _ { h + 1 } ( x ) : = q _ h ( \sqrt x ~ ) p _ h ( \sqrt { - x } ~ ) - q _ h ( \sqrt { - x } ~ ) p _ h ( \sqrt x ~ ) , \end{align*}


\begin{align*} ( \alpha _ { \underline { S } } \times \mathrm { i d } _ { \underline { X } _ { \mathbb { C } } } ) \circ \Psi ^ { - 1 } \circ ( \tau ^ { * } \alpha _ { \underline { S } } \times \mathrm { i d } _ { \underline { X } _ { \mathbb { C } } } ) = ( \mathrm { i d } _ { \tau ^ { * } \underline { S } _ { \mathbb { C } } } \times \tau ^ { * } \alpha _ { \underline { X } } ) \circ \tau ^ { * } \Psi \circ ( \mathrm { i d } _ { \tau ^ { * } \underline { S } _ { \mathbb { C } } } \times \alpha _ { \underline { X } } ) , \end{align*}


\begin{align*} \aligned e _ 1 & \ , : = \ , ( 1 - y ) \ , \partial _ x - z \ , \partial _ y + x \ , \partial _ u , \\ e _ 2 & \ , : = \ , ( 1 - y ) \ , \partial _ y - 2 z \ , \partial _ z + u \ , \partial _ u , \\ e _ 3 & \ , : = \ , u \ , \partial _ x - \tfrac { 4 } { 3 } \ , x \ , \partial _ y + ( 1 - y ) \ , \partial _ z , \\ e _ 4 & \ , : = \ , x \ , \partial _ x - z \ , \partial _ z + 2 \ , u \ , \partial _ u . \endaligned \end{align*}


\begin{align*} X _ \ell = m _ { \ell ; d } + \sum _ { q \geq 2 } \frac { b _ q } { q ! } \int _ { \mathbb S ^ d } H _ q ( T _ \ell ( x ) ) d x , \end{align*}


\begin{gather*} K _ 2 = \bigl \{ P \in M : ( \pi _ 1 , \ldots , \pi _ n ) P \bigr \} \quad \quad \\ K _ 3 = \bigl \{ P \in M : P \ll P ^ * ( \pi _ 1 , \ldots , \pi _ n ) P \bigr \} . \end{gather*}


\begin{align*} M _ n = n \sup _ { Q \in \P ( \R ) } \left ( \int _ { \R } V \ , d Q + \frac 1 2 \int _ { \R } \int _ { \R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \right ) , \end{align*}


\begin{align*} \ T _ { \tau } ( t ) \stackrel { d e f } { = } { \bf P } ( | \tau | \ge t ) , \ t \ge 0 . \end{align*}


\begin{align*} s _ 3 = \frac { ( \lambda + \mu ) ( \lambda + 2 \mu ) } { \lambda + 3 \mu } , s _ 4 = \frac { 2 \mu ( \lambda + 2 \mu ) } { \lambda + 3 \mu } , t _ 1 = \frac { \mu } { 2 | \xi | } \biggl ( \frac { 1 - 2 s _ 1 } { \tau - 2 \mu s _ 1 | \xi | } - \frac { 1 } { \tau - 2 \mu | \xi | } \biggr ) . \end{align*}


\begin{align*} D = \sum _ { i = 1 } ^ r n _ i \cdot y _ i \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*} { \mathbb { E } } \left [ { { { \left \| { { \bf { x } } \left [ n \right ] } \right \| } ^ 2 } } \right ] = \sum \limits _ { l = 1 } ^ L { { \mathbb { E } } \left [ { { { \left \| { { { \bf { f } } _ l } s \left [ { n - { \kappa _ l } } \right ] } \right \| } ^ 2 } } \right ] } { \rm = } \sum \limits _ { l = 1 } ^ L { { { \left \| { { { \bf { f } } _ l } } \right \| } ^ 2 } } \le P , \end{align*}


\begin{align*} \chi ( s ) : = \beta ^ { - 1 } ( s , 0 ) . \end{align*}


\begin{align*} \nabla _ V ( \nabla X ) = - R ( X , V ) + T _ h ( V , \cdot ) \end{align*}


\begin{align*} \mathbf { Y } = \mathbf { F X F } ^ \top , \end{align*}


\begin{align*} \hat W _ r ( k ) = \begin{cases} \frac { 2 \sin ( 2 \pi k r ) } { \pi k } & k \neq 0 \\ 4 r & \quad k = 0 \end{cases} . \end{align*}


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


\begin{align*} \varphi _ n ( t ) : = \exp \left ( - \int _ 0 ^ t \big [ f ( r ) + \rho ( X ( r , x _ n ) ) + \eta ( X ( r , x ) ) \big ] d r \right ) . \end{align*}


\begin{align*} a ( u , v ) = \int _ D \nabla u \cdot \nabla \bar { v } ~ d x , ( f , v ) = \int _ D f \bar { v } ~ d x . \end{align*}


\begin{align*} o _ { k } ( 1 ) = \int _ { \R ^ { 2 } } | \nabla \widetilde { u } _ { k } | ^ { 2 } + \widetilde { u } _ { k } ^ { 2 } + 3 B ( \widetilde { u } _ { k } ) - \int _ { \R ^ { 2 } } \widetilde { u } _ { k } ^ { 2 p } , \end{align*}


\begin{align*} Y _ n ( t ) = P _ n x + \int _ 0 ^ t P _ n A ( s , Y _ n ( s ) ) d s + \int _ 0 ^ t P _ n B ( s , Y _ n ( s ) ) Q _ n d W ( s ) , \end{align*}


\begin{align*} \mathbb { P } ( S _ k = 0 ) ^ { 1 / p _ k } \leq \mathbb { P } ( S _ k = - k ) ^ { 1 / p _ k } + \mathbb { P } ( S _ k = k ) ^ { 1 / p _ k } \end{align*}


\begin{align*} \hat \Phi ^ \dagger _ { a a } ( 0 , 0 ) & = \langle z _ 1 ^ * , \Phi ^ \dagger _ { v v } ( 0 , p _ 0 ) [ u _ \ell , u _ \ell ] \rangle = \langle z _ 1 ^ * , Q ^ \dagger F ^ { q , \dagger } _ { v v } ( 0 , p _ 0 ) [ u _ \ell , u _ \ell ] \rangle = \langle z _ 1 ^ * , 0 \rangle = 0 . \end{align*}


\begin{align*} h _ { i } = e _ { i } \cdot F ( x ) . \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty \left | \frac { \theta _ 0 ( e ^ { \kappa + \tau } ) - \theta _ 0 ( e ^ \tau ) - \delta e ^ \tau } { e ^ { \sigma \tau } } \right | ^ 2 d \tau = \frac { 1 } { 2 \pi } \int _ { - \infty } ^ \infty \left | \frac { e ^ { \kappa s } - 1 } { s } \right | ^ 2 \left | \frac { \zeta ' } { \zeta } ( s ) + g ( s ) \right | ^ 2 d t . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \langle A _ 0 ( u _ n ) , u _ n - u \rangle = 0 . \end{align*}


\begin{align*} 1 - e ^ { c _ 2 } = \frac { 1 - e ^ { a _ { \alpha + n + 1 } + a _ { \alpha + n + 2 } } } { \left ( e ^ { - a _ { \alpha + 1 } } - 1 \right ) \left ( 1 - e ^ { a _ { n + 1 } } \right ) \exp \left [ \sum _ { j = 2 } ^ n ( a _ j - a _ { \alpha + j } ) \right ] } \ , \ \forall n \in \N \ . \end{align*}


\begin{align*} B _ n ( 1 + x ) - B _ n ( x ) = n x ^ { n - 1 } , \end{align*}


\begin{align*} { \sigma } _ { j i } & = \mu ^ * ( u _ { i , j } + u _ { j , i } ) + \varkappa \ , ( u _ { i , j } - \epsilon _ { j i s } \vartheta _ s ) + \lambda \ , u _ { k , k } \delta _ { j i } , \\ * m _ { j i } & = \beta \ , \vartheta _ { j , i } + \gamma \ , \vartheta _ { i , j } + \alpha \ , \vartheta _ { k , k } \delta _ { j i } \ , \end{align*}


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


\begin{align*} \Vert f \Vert _ { H ^ s _ \gamma ( G ) } : = \left ( \int _ { G ^ \wedge } \left ( 1 + \gamma ( \xi ) ^ 2 \right ) ^ s \vert \widehat { f } ( \xi ) \vert ^ 2 d \nu ( \xi ) \right ) ^ { 1 / 2 } . \end{align*}


\begin{align*} ( I - Q ^ \dagger ) F ^ { q , \dagger } _ v ( 0 , p _ 0 ) \psi ^ \dagger _ p ( 0 , p _ 0 ) + ( I - Q ^ \dagger ) F ^ { q , \dagger } _ p ( 0 , p _ 0 ) = 0 . \end{align*}


\begin{align*} \begin{aligned} I _ 4 ^ { \prime } ( x ) & = - 4 \pi ^ { - 1 } \int _ { - \infty } ^ { \infty } r _ 2 ( z ) e ^ { - 2 i z x } \left ( M _ { + , 2 2 } ( x ; z ) - 1 \right ) d z \\ & - 2 i \pi ^ { - 1 } \int _ { - \infty } ^ { \infty } s ^ { - 1 } r _ 2 ( z ) e ^ { - 2 i z x } \partial _ { x } M _ { + , 2 2 } ( x ; z ) d z . \end{aligned} \end{align*}


\begin{align*} F \subseteq \bigcup _ { n = 0 } ^ \infty [ x _ n , y _ n ] \end{align*}


\begin{align*} \int _ { a x } ^ { x } | \theta ( t + \delta t ) - \theta ( t ) - \delta t | ^ 2 d t & > \int _ I | \theta ( t + \delta t ) - \theta ( t ) - \delta t | ^ 2 d t = \delta ^ 2 \int _ I t ^ 2 d t \\ & > \delta ^ 2 \int _ { a x } ^ { x / 2 } t ^ 2 d t = \frac { \delta ^ 2 x ^ 3 } { 3 } \left ( \frac { 1 } { 8 } - a ^ 3 \right ) . \end{align*}


\begin{align*} \mathcal { L } _ 1 f = - \sum _ { i , j = 1 } ^ 3 \left \{ \partial _ i [ ( a _ { i j } * \mu ) \partial _ j f ] + ( a _ { i j } * \mu ) \frac { v _ i } { 2 } \frac { v _ j } { 2 } f - \partial _ i \left [ ( a _ { i j } * \mu ) \frac { v _ j } { 2 } \right ] f \right \} , \end{align*}


\begin{align*} \tilde { \kappa } _ i = \begin{cases} \kappa _ 1 , & i = 1 , \\ \kappa _ i - 1 , & i > 1 . \end{cases} \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n { \binom n k L _ { j k + m } B _ { n - k } ( x ) z ^ k } = \alpha ^ m B _ n ( x + \alpha ^ j z ) + \beta ^ m B _ n ( x + \beta ^ j z ) . \end{align*}


\begin{align*} J ( q ) = \int _ { 0 } ^ { I _ { \Phi \left ( \frac { q - \mu } { \sigma } \right ) } ( \alpha , \beta ) } \ , Q ( u ) \mathrm { d } u = \sum _ { i = 0 } ^ { \infty } \frac { f _ i \ , \ , \ , I _ { \Phi \left ( \frac { q - \mu } { \sigma } \right ) } ( \alpha , \beta ) ^ { ( i / \alpha + 1 ) } } { i / \alpha + 1 } . \end{align*}


\begin{align*} \begin{aligned} \det \left ( \frac { \partial ^ 2 u } { \partial \overline { q _ l } \partial { q _ k } } \right ) , \end{aligned} \end{align*}


\begin{align*} L _ 1 \circ F \circ L _ 2 = F ' , \end{align*}


\begin{align*} \partial _ t w + ( \mathbf { v } \cdot \nabla ) w = 0 . \end{align*}


\begin{align*} \varphi _ { n } = \varphi _ { 1 } \circ \psi _ { 2 } \cdots \circ \psi _ { n } \colon D _ { n } \to D _ { n - 1 } \to \cdots D _ { 2 } \to D _ { 1 } \to { \mathbf { A } } _ { k } ^ { 2 } \end{align*}


\begin{align*} \mathcal { F } _ { h _ { f } } ( x ) = \frac { 2 \left ( \varrho \beta \right ) ^ { \left ( \varrho + \beta \right ) / 2 } } { \Gamma ( \varrho ) \Gamma ( \beta ) } { x } ^ { \left ( \varrho + \beta \right ) / 2 - 1 } \mathcal { K } _ { \varrho - \beta } \left ( 2 \sqrt { \varrho \beta x } \right ) , \end{align*}


\begin{align*} \textbf { \textit { K } } ( t , x ^ { \prime } , y ^ { \prime } ) = e ^ { - t \Lambda _ g } \delta ( x ^ { \prime } - y ^ { \prime } ) = \sum _ { k = 1 } ^ { \infty } e ^ { - t \tau _ k } \textbf { \textit { U } } _ k ( x ^ { \prime } ) \otimes \textbf { \textit { U } } _ k ( y ^ { \prime } ) . \end{align*}


\begin{align*} ( T _ 1 + f _ 1 ) \cdot ( T _ 2 - f _ 2 ) \cdot \Psi _ 2 = i \Psi _ 2 \end{align*}


\begin{align*} \pi ( u , v ) = \pi ( v , u ) \in [ 0 , \infty ) . \end{align*}


\begin{align*} 8 ^ { n - m } ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } = n ! \det ( A , \dots , A , 8 I , \dots , 8 I ) \Omega _ { 2 n } , \end{align*}


\begin{align*} \underline { u } = \frac { 1 } { 4 } \left ( \sum _ { j = 1 } ^ p e _ j \cdot u ( e _ j ) + \sum _ { j = p + 1 } ^ n e _ j \cdot ( - u ^ * ( e _ j ) ) \right ) . \end{align*}


\begin{align*} \phi = \frac { 1 } { 2 } ( u _ { , p } \xi ^ p + v _ q x ^ q ) . \end{align*}


\begin{align*} 0 = S ' ( h , v ) - S ' \big ( L _ X \bar g , X ( \bar u ) \big ) = S ' ( 0 , v - X ( \bar u ) ) = 0 \mbox { i n } M \setminus \Omega . \end{align*}


\begin{align*} A _ { \gamma _ j , \gamma _ { j + 1 } } = \{ i : p _ i / q _ i \in [ \gamma _ j , \gamma _ { j + 1 } ) \} = \{ i : \delta _ i \in [ \nu _ j , \nu _ { j + 1 } ) \} . \end{align*}


\begin{align*} \lim _ { \lambda \to + 0 } \frac { 1 } { m ( B ( \mathbf x , \rho ) ) } \int _ { B ( \mathbf x , \lambda ) } \vert u ( \mathbf x ) \vert \mathrm d \mathbf x \not = 0 , \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*} \alpha ^ * & = 1 + \lim _ { z \rightarrow 0 } z \frac { \Phi ( z ) } { \phi ( z ) } \frac { 1 - \Phi ( z ) } { 1 - 2 \Phi ( z ) } = 1 + \frac { \sqrt { 2 \pi } } { 4 } \lim _ { z \rightarrow 0 } \frac { z } { 1 - 2 \Phi ( z ) } \\ & = 1 + \frac { \sqrt { 2 \pi } } { 4 } \lim _ { z \rightarrow 0 } \frac { 1 } { - 2 \phi ( z ) } = 1 - \frac { \pi } { 4 } \approx 0 . 2 1 4 6 . \end{align*}


\begin{align*} \begin{aligned} G ^ { \alpha \beta } _ 0 \sigma _ { \alpha \beta } \geq \ , & G ( \sigma _ { \alpha \beta } - \theta \delta _ { \alpha \beta } ) + \theta \sum _ { \alpha = 1 } ^ { n - 1 } G ^ { \alpha \alpha } _ 0 \\ \geq \ , & \gamma + \theta \sum _ { \alpha = 1 } ^ { n - 1 } G ^ { \alpha \alpha } _ 0 \end{aligned} \end{align*}


\begin{align*} \delta _ H ( k ) = 2 \phi _ H ( k + 1 ) - ( \phi _ H ( 2 k ) + \phi _ H ( 2 ) ) \end{align*}


\begin{align*} \bar \beta _ n ( G ) = O ( n ^ { e ( G ) } ) , \end{align*}


\begin{align*} & \lambda _ x | _ { E _ { d ^ * ( y ) } } = \lambda _ y \\ & \lambda _ x | _ { E _ { [ w b ^ { - n } , w b ^ { - n + 1 } a ] } } = \phi _ n \end{align*}


\begin{align*} T x = \frac { 1 } { \lambda } x . \end{align*}


\begin{align*} c _ \varepsilon \leq \inf \limits _ { w \in \mathcal { H } ( \Omega ) , w _ + \neq 0 } \max \limits _ { t \geq 0 } I _ \varepsilon ( t w ) = \inf \limits _ { w \in \mathcal { N } _ \varepsilon } I _ \varepsilon ( w ) . \end{align*}


\begin{align*} \begin{aligned} K f = K ^ { 1 - \chi } f + K ^ \chi f \ , , \ K ^ \Lambda f = K _ 1 ^ \Lambda f - K _ 2 ^ \Lambda f \textrm { f o r } \Lambda = 1 - \chi \textrm { a n d } \chi \ , , \end{aligned} \end{align*}


\begin{align*} \lfloor x \rfloor + \Big \lfloor x + \frac 1 n \Big \rfloor + \ldots + \Big \lfloor x + \frac { n - 1 } n \Big \rfloor = \lfloor n x \rfloor . \end{align*}


\begin{align*} \nabla _ { V } \hat X & = \omega ( V ) \\ \nabla _ V \omega & = - R ( \hat X , V ) + T _ h ( V ) . \end{align*}


\begin{align*} \begin{aligned} \frac { v ' ( r ) ^ { 2 } } { 2 } & \leq \int _ { 0 } ^ { r } t ^ { b } v ^ { q } ( t ) v ' ( t ) f ( w ( t ) ) d t \\ & \leq r ^ { b } f ( w ( r ) ) \int _ { 0 } ^ { r } \left ( \frac { v ^ { q + 1 } ( t ) } { q + 1 } \right ) ' d t \\ & \leq C f ( w ( r ) ) v ^ { q + 1 } ( r ) \quad \mbox { f o r a l l } \ 0 < r < R , \end{aligned} \end{align*}


\begin{align*} \frac { \partial f ( 0 ) } { \partial z } = \frac { \partial ^ 2 f ( 0 ) } { \partial z ^ 2 } = \frac { \partial ^ 3 f ( 0 ) } { \partial z ^ 3 } = 0 \frac { \partial ^ 4 f ( 0 ) } { \partial z ^ 4 } & = \frac { 4 \sqrt { 2 } } { \pi } \frac { \Gamma ( 3 / 2 - \pi / 4 ) } { \Gamma ( 1 - \pi / 4 ) } \left ( \frac { 3 } { \pi } - 1 \right ) < 0 . \end{align*}


\begin{align*} f ^ \lambda ( z ) = ( 2 \pi ) ^ { - n } \ , | \lambda | ^ n \ , \sum _ { k = 0 } ^ \infty R _ k ^ { n - 1 } ( \lambda , f ) \ , \varphi _ { k , \lambda } ^ { n - 1 } ( z ) . \end{align*}


\begin{align*} \varphi ( t ) = 1 + A ( t ) - B ( t ) \end{align*}


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


\begin{align*} b ^ \Lambda _ \lambda = \eta ( \tau ) ^ \ell c ^ \Lambda _ \lambda \end{align*}


\begin{align*} F ( \lambda ) x = 0 . \end{align*}


\begin{align*} \pi _ { 2 } \circ \lambda _ { f \times i d _ { X } } [ ( f \times i d _ { X } ) ( r , x ) , \alpha ] ( t ) = \pi _ { 2 } ( f ( r ) , \alpha ( t ) ) = \alpha ( t ) . \end{align*}


\begin{align*} 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} , \ i \mapsto \begin{pmatrix} \sqrt { a } & 0 \\ 0 & - \sqrt { a } \end{pmatrix} , \ j \mapsto \begin{pmatrix} 0 & \sqrt { b } \\ \sqrt { b } & 0 \end{pmatrix} , \ i j \mapsto \begin{pmatrix} 0 & \sqrt { a b } \\ - \sqrt { a b } & 0 \end{pmatrix} . \end{align*}


\begin{align*} \lambda _ k = \frac 1 8 \sum _ { \ell \in \Z } \Big [ & \hat W _ r ( - k + \ell - q ) \hat W _ r ( \ell + q ) + \hat W _ r ( - k + \ell + q ) \hat W _ r ( \ell - q ) \\ & + \hat W _ r ( \ell - q ) \hat W _ r ( k + q + \ell ) + \hat W _ r ( \ell + q ) \hat W _ r ( k - q + \ell ) \\ & - 4 \hat W _ r ( \ell - q ) \hat W _ r ( \ell + q ) \Big ] \end{align*}


\begin{align*} H _ r ( E [ t ] ) = t H _ r ( E ) . \end{align*}


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


\begin{align*} \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { s } ^ { \left ( \right ) } \right ] = & \mathbb { E } \left [ s _ c ^ * s _ c \right ] + \mathbb { E } \left [ s _ 1 ^ * s _ 1 \right ] \cdots + \mathbb { E } \left [ s _ M ^ * s _ M \right ] , = M + 1 . \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*} N _ r \coloneqq \begin{cases} M _ r , & , \\ N _ k , & \end{cases} \end{align*}


\begin{align*} { \rm e } ^ * _ { j i } = u _ { i , j } + \epsilon _ { i j m } \ , \vartheta _ m , \qquad \mathfrak { K } _ { j i } = \vartheta _ { j , i } . \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \theta ) = E _ P \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] , \end{align*}


\begin{align*} \displaystyle T ( f ) ( x ) = \int _ { \mathbb { C } } { A _ q ( z , x ) } f ( z ) | z | ^ { \frac { 2 } { q } - 2 } e ^ { - \frac { | z | ^ 2 } { q } } d A ( z ) , \end{align*}


\begin{align*} \max _ { | \eta - \lambda | \le \delta } \max _ { x \in G ( \lambda ) } \| F _ n ( \eta ) p _ n x - p _ n F ( \eta ) x \| = \max _ { x \in G ( \lambda ) } \left \| ( T _ n p _ n - p _ n T ) x \right \| \le \left \| T _ n p _ n - p _ n T \right \| . \end{align*}


\begin{align*} \langle e _ { i } , A f _ { j } \rangle = \langle A e _ { i } , A f _ { j } \rangle = \langle e _ { i } , f _ { j } \rangle = \delta _ { i j } . \end{align*}


\begin{align*} \Big ( f ( u _ 2 ) - f ( u _ 1 ) - f ' ( u _ 2 ) \omega _ + \Big ) \omega _ + = 0 \end{align*}


\begin{align*} n ! F _ { n + 1 } \ & = \ \sum _ { k = 1 } ^ { n } \begin{bmatrix} n \\ k \end{bmatrix} A _ k , \\ n ! F _ { n + 2 } \ & = \ \sum _ { k = 1 } ^ { n } \begin{bmatrix} n \\ k \end{bmatrix} B _ k . \\ \end{align*}


\begin{align*} | | B _ q ( \psi ) | | _ { M L _ q ( \mathbb { C } ) } = | | \psi | | _ { L ^ 2 ( \mathbb { R } ) } . \end{align*}


\begin{align*} \mathbf { v } \cdot \overrightarrow { \zeta } = 0 , \end{align*}


\begin{align*} \partial _ t \mu ^ \phi = - \partial _ x \Big \{ \Big [ b ( t , \cdot , \mu ^ \phi ) + \sigma \phi ( t , \cdot , \mu ^ \phi ) \Big ] \mu ^ \phi \Big \} . \end{align*}


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


\begin{align*} \sum _ { \substack { U \subset F _ s : \\ U \cap F _ { \ell + 1 } = T } } \alpha ^ s _ U . \end{align*}


\begin{align*} D ( A ^ { \circ } ) & = \{ x ^ { \circ } \in X ^ { \circ } \ | \ \exists y ^ { \circ } \in X ^ { \circ } \ \forall x \in D ( A ) : \langle - A x , x ^ { \circ } \rangle = \langle x , y ^ { \circ } \rangle \} , \\ - A ^ { \circ } x ^ { \circ } & = y ^ { \circ } \quad ( x ^ { \circ } \in D ( A ^ { \circ } ) ) , \end{align*}


\begin{align*} F _ m ( n ) = 0 , n \leq m . \end{align*}


\begin{align*} D ^ k \phi ( Z ) = \sum _ { q \geq 2 \vee k } \frac { | b _ { q } | } { ( q - k ) ! } H _ { q - k } ( Z ) D ^ k L \phi ( Z ) = - \sum _ { q \geq 2 \vee k } q \frac { | b _ { q } | } { ( q - k ) ! } H _ { q - k } ( Z ) \end{align*}


\begin{align*} T h _ k = \Psi ( r _ k ) . \end{align*}


\begin{align*} \begin{aligned} & \xi y \left [ n \right ] = \sqrt P s \left [ { n - { n _ { \max } } } \right ] + \\ & \sqrt P \sum \limits _ { l = 1 } ^ L { \sum \limits _ { l ' \ne l } ^ L { \frac { { { \bf { h } } _ l ^ H { { \bf { h } } _ { l ' } } } } { { \sum \nolimits _ { k = 1 } ^ L { { { \left \| { { { \bf { h } } _ k } } \right \| } ^ 2 } } } } s \left [ { n - { n _ { \max } } + { n _ { l ' } } - { n _ l } } \right ] } } + \xi z \left [ n \right ] . \end{aligned} \end{align*}


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


\begin{align*} H _ 1 = \sum _ { X \in S ( 0 ) } \Theta _ { \theta } \begin{pmatrix} m _ 1 & X \\ 0 & m _ 2 \end{pmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = - q ^ 8 ( 1 - q ) ^ 2 ( 1 - q ^ 2 ) . \end{align*}


\begin{align*} & \frac { 1 } { 2 } ( V + V ^ H ) = \left ( \begin{array} { c c } 1 + | r ( k ) | ^ 2 & \overline { r ( k ) } e ^ { - 2 i k ^ 2 x } \\ r ( k ) e ^ { 2 i k ^ 2 x } & 1 \end{array} \right ) , \ \ \ k \in \mathbb { R } , \\ & \frac { 1 } { 2 } \left ( V + V ^ H \right ) = \left ( \begin{array} { c c } 1 - | r ( k ) | ^ 2 & 0 \\ 0 & 1 \end{array} \right ) , \ \ \ k \in i \mathbb { R } , \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*} 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*} j ( \tau , s ( \tau ) ) = h . \end{align*}


\begin{align*} \mu ( A _ { h , j } ) \geq c h ^ d , j = 1 , \ldots , N _ n , \end{align*}


\begin{align*} \omega = \eta _ 1 ^ { * } \widetilde \omega ^ 0 \wedge \eta _ 1 ^ { * } \widetilde \omega ^ 1 \wedge \cdots \wedge \eta _ k ^ { * } \widetilde \omega ^ 0 \wedge \eta _ k ^ { * } \widetilde \omega ^ 1 , \end{align*}


\begin{align*} \Gamma = \bigoplus _ { n = 1 } ^ \infty \Gamma _ n , F = \prod _ { n = 1 } ^ \infty F _ n \quad G = \Gamma \rtimes F \end{align*}


\begin{align*} x = \sum _ { n = 0 } ^ \infty \frac { \varepsilon _ n } { \gamma ^ { n + 1 } } . \end{align*}


\begin{align*} I _ 2 = - \frac { 1 } { 2 i z } ( 1 - e ^ { 2 i z \delta } ) \nu ( x ; z ) . \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*} \begin{aligned} \partial _ t u - \nabla \cdot ( A \nabla u ) = & \ F , & ( 0 , T ] \times \Omega , \\ u ( 0 , \cdot ) = & \ u _ 0 , \ , & \Omega , \\ u = & \ g _ D ^ { } , & ( 0 , T ] \times \Gamma _ D ^ { } , \\ \mathbf { n } \cdot A \nabla u = & \ g _ N ^ { } , & ( 0 , T ] \times \Gamma _ N ^ { } , \end{aligned} \end{align*}


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


\begin{align*} { \bf { h } } \left [ n \right ] = \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l \delta \left [ { n - { n _ l } } \right ] } , \end{align*}


\begin{align*} \int _ { 0 } ^ { t _ { 0 } } \bigg ( \int _ { R _ { 0 } } ^ { \infty } \nu \xi ( \nu , s ) d \nu \bigg ) ^ { 2 } \ d s = 0 \end{align*} % \end{align*}


\begin{align*} \sum _ { k = 0 } ^ 3 x _ k \psi _ k \rightarrow ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) . \end{align*}


\begin{align*} A \cdot B = 2 \Phi \frac { G } { r ^ { 2 } } + \nabla \cdot ( \Phi \nabla \theta \times A ) . \end{align*}


\begin{align*} \mathcal { H } ^ 2 ( t ; c , f , H ) : = \Bigg \{ \tilde { f } : \int _ { \{ \Psi < - t \} } | \tilde { f } | ^ 2 _ h c ( - \Psi ) < + \infty , \ \tilde { f } \in H ^ 0 ( \{ \Psi < - t \} , \mathcal { O } ( K _ M \otimes E ) ) \\ \& ( \tilde { f } - f ) _ { z _ 0 } \in \mathcal { O } ( K _ M ) _ { z _ 0 } \otimes ( J _ { z _ 0 } \cap H _ { z _ 0 } ) , \ z _ 0 \in Z _ 0 \Bigg \} , \end{align*}


\begin{align*} \phi _ { i j } ( X ) = \langle \nabla ^ \perp _ X \eta _ i , \eta _ j \rangle . \end{align*}


\begin{align*} \begin{cases} - \Delta h _ { \hat { x } } ( x , y ) = 0 , \ \ & x \in T _ { \hat { x } } ( \Omega ) , \\ h _ { \hat { x } } ( x , y ) = \frac { 1 } { 2 \pi } \ln \frac { 1 } { | x - y | } , \ \ & x \in \partial T _ { \hat { x } } ( \Omega ) . \end{cases} \end{align*}


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


\begin{align*} \frac { g ' ( x + \gamma ) } { g ' ( x ) } = e ^ { a \gamma } > 1 , \end{align*}


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


\begin{align*} v ( \mathbf 0 ) = 0 . \end{align*}


\begin{align*} \tilde { \sigma } ( u _ 1 , \dots , u _ n ) : = ( \tilde { \sigma } _ 1 ( u _ 1 , u _ 2 ) , \dots , \tilde { \sigma } _ { n - 1 } ( u _ { n - 1 } , u _ n ) ) . \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*} \nu ^ * A = \mu ^ * h ^ * D + E . \end{align*}


\begin{align*} \mathcal D _ { \psi } f ( t z ) = \psi _ 0 \frac { \partial f } { \partial x } ( t z ) + \psi _ 1 \frac { \partial f } { \partial y } ( t z ) = 0 \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*} \partial _ t w + ( \mathbf { v } \cdot \nabla ) w = 0 . \end{align*}


\begin{align*} h ' ( x ) = \beta ^ { q + 1 } ( x + v ) ^ { q + 1 } h ( \mu _ { \alpha , \beta , u , v } ( x ) ) , \end{align*}


\begin{align*} a \cdot b a & = b , \\ a b \cdot c & = a \cdot ( b \cdot c ) , \\ ( a \cdot b ) a & = ( b \cdot a ) b , \end{align*}


\begin{align*} \mu _ t ^ { * , \xi } = k ( t , \nu _ t ^ { * , \xi } , \varphi _ t ^ { * , \xi } ) . \end{align*}


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


\begin{align*} d = 4 + 4 n = 8 + 8 l \equiv 0 \mod 8 \end{align*}


\begin{align*} C _ m ( K ) = \int _ { K } ( \Delta \omega ^ * ( q , K , \Omega ) ) ^ m \wedge \beta _ n ^ { n - m } . \end{align*}


\begin{align*} \mathcal { R } = \left ( N \left ( \frac { b } { c t s } + \frac { b } { a s d } + \frac { 1 } { t d } \right ) \right ) ^ { - 1 } . \end{align*}


\begin{align*} 1 = N _ 0 \rhd N _ 1 \rhd \cdots \rhd N _ r = G , \end{align*}


\begin{align*} \sum _ { \deg ( v ) \mid \ell } \deg ( v ) \mathrm { t r } \left ( \rho ( \mathrm { F r o b } _ { v } ) ^ { \frac { \ell } { \deg ( v ) } } \right ) = - \sum _ { \lambda } \lambda ^ { \ell } . \end{align*}


\begin{align*} \frac { 1 } { \rho _ K } \| x ^ { K + 1 } - x ^ K \| \leq \frac { r _ 0 + \theta } { \rho _ 0 \zeta ^ K } \overset { \eqref { K - s p } } { \leq } \frac { \varepsilon } { 2 } , \eta _ K = \eta _ 0 \sigma ^ K \overset { \eqref { K - s p } } { \leq } \frac { \varepsilon } { 2 } , \end{align*}


\begin{align*} f _ n \colon \mathcal { X } _ n = ( S _ \C \times _ \C Y _ n ) / \Sigma \to S \end{align*}


\begin{align*} V _ 1 ( r ) = V ( r ) - E _ 0 , \end{align*}


\begin{align*} T _ 2 = \sum _ { \beta \in F ^ { \times } } \sum _ { X \in S ( \beta ) } \Theta \begin{bmatrix} m _ 1 ' & X \\ 0 & m _ 2 ' \end{bmatrix} \overline { \psi _ A ( { m _ 1 ' } ^ { - 1 } X } ) = \theta ( a ) q ^ 8 ( q - 1 ) ^ 2 . \end{align*}


\begin{align*} \nabla _ X \varphi = - \frac { 1 } { 2 } \sqrt { | c _ 1 | } \ X _ 1 \cdot \nu _ 1 \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi \end{align*}


\begin{align*} \Sigma _ t = \{ ( x ' , x _ n ) \in \mathbb { R } ^ n \ | \ x ' \in \mathbb { S } _ t ^ { n - 1 } , \ x _ n \in \mathbb { R } \} \end{align*}


\begin{align*} \phi ( \sigma , \tau ) = [ \widetilde \sigma , \widetilde \tau ] = \widetilde \sigma \widetilde \tau { \widetilde \sigma } ^ { - 1 } { \widetilde \tau } ^ { - 1 } \in \langle \epsilon \rangle \cong \mathbb Z / 2 \mathbb Z . \end{align*}


\begin{align*} \Delta _ { g } f = g ^ { i j } \Bigl ( \frac { \partial ^ 2 f } { \partial x _ i \partial x _ j } - \Gamma ^ k _ { i j } \frac { \partial f } { \partial x _ k } \Bigr ) , f \in C ^ { \infty } ( \Omega ) . \end{align*}


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


\begin{align*} E _ { \xi ^ { \bar { \theta } } _ 1 ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 2 } ^ { n } Z ^ { { \bar { \theta } } [ u ] } _ i \biggr ) \biggr ] = & E _ { \xi ^ { \bar { \theta } } _ 1 ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z ^ { \theta ^ { [ n - 1 ] } } _ i \biggr ) \biggr ] \\ = & V \bigg ( \frac { \xi _ 0 f _ 1 ( u ) } { \xi _ 0 f _ 1 ( u ) + ( 1 - \xi _ 0 ) f _ 2 ( u ) } , n - 1 , x + u \bigg ) . \end{align*}


\begin{align*} c = \sum _ { [ \mathbf a ] \in B [ m ] _ q } c _ { [ \mathbf a ] } \cdot [ \mathbf a ] \end{align*}


\begin{align*} \| u \| _ { 2 , h _ n } = \sum _ { K \in \mathcal { T } _ n } ( u , u ) _ { H ^ 2 ( K ) } , \end{align*}


\begin{align*} \sum _ { i = 0 } ^ { q - 1 } \binom { n + i - 1 } { i } = \frac { q ( q + 1 ) \cdots ( q + n - 1 ) } { 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*} d ^ D _ t = \sum _ { u , v : t ( u , v ) = t } \sum _ { s = 0 } ^ { \omega _ { u v } - 1 } d _ { ( u , v , s ) } . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \int _ { \R ^ { 2 } } ( | \nabla u _ { n } | ^ { 2 } + u _ { n } ^ { 2 } ) d x = 0 , \end{align*}


\begin{align*} h \cdot ( g \cdot \gamma ) & = \theta _ n \circ ( h \times g \cdot \gamma ) \circ i , \\ ( h g ) \cdot \gamma & = \theta _ n \circ ( h g \times \gamma ) \circ i . \end{align*}


\begin{align*} [ \sqrt { - 1 } \theta \wedge \bar { \theta } , \Lambda _ \omega ] \alpha = \bar { \theta } \wedge ( \alpha \llcorner ( \bar { \theta } ) ^ \sharp ) , \end{align*}


\begin{align*} S ^ { i j } _ k = \frac { \partial S _ k [ D ^ 2 u ] } { \partial r _ { i j } } . \end{align*}


\begin{align*} \max _ { | \eta - \lambda | \le \delta } \max _ { x \in G ( \lambda ) } \| F _ n ( \eta ) p _ n x - p _ n F ( \eta ) x \| = \max _ { x \in G ( \lambda ) } \left \| ( T _ n p _ n - p _ n T ) x \right \| \le \left \| T _ n p _ n - p _ n T \right \| . \end{align*}


\begin{align*} \operatorname { T r } \textbf { \textit { K } } ( t , x ^ { \prime } , x ^ { \prime } ) = \frac { 1 } { ( 2 \pi ) ^ { n - 1 } } \int _ { \mathbb { R } ^ { n - 1 } } \bigg ( \frac { i } { 2 \pi } \int _ { \mathcal { C } } e ^ { - t \tau } \sum _ { j \leqslant - 1 } \operatorname { T r } \phi _ { j } \ , d \tau \bigg ) \ , d \xi . \end{align*}


\begin{align*} f ( p , s ) = s \log a + \log A _ k - \left ( \frac { p } { 2 } + 2 \right ) \log s + \log \Gamma \left ( \frac { p } { 2 } + 2 \right ) . \end{align*}


\begin{align*} \lim _ { j \rightarrow \infty } M _ j ( C _ \varphi ) = \frac { 1 } { k } \sum _ { i = 0 } ^ { k - 1 } C _ { \rho \circ \varphi _ i } . \end{align*}


\begin{align*} \chi _ P ( P _ a , P _ b ) & = \left | \sum _ { i = 1 } ^ k k P _ a ( i ) P _ b ( i ) - 1 \right | \\ & = \left | \sum _ { i = 1 } ^ k k \frac { 1 } { k } \left ( 1 + \epsilon v _ { a , i } \right ) \frac { 1 } { k } \left ( 1 + \epsilon v _ { b , i } \right ) - 1 \right | \\ & = \left | \sum _ { i = 1 } ^ k \frac { 1 } { k } \left ( 1 + \epsilon v _ { a , i } + \epsilon v _ { b , i } + \epsilon ^ 2 v _ { a , i } v _ { b , i } \right ) - 1 \right | \\ & = 0 , \end{align*}


\begin{align*} \begin{aligned} P ( s I + x ) = \det ( s I + \operatorname { d i a g } ( \lambda _ 1 , \dots , \lambda _ n ) ) = \prod \limits _ 1 ^ n ( s + \lambda _ k ) . \end{aligned} \end{align*}


\begin{align*} Q ( u ) = \sum _ { i = 0 } ^ { \infty } f _ i \ , \ , u ^ { i / \alpha } , \end{align*}


\begin{align*} [ - T , T ] = \bigcup _ { j = - [ T / \tau ] - 1 } ^ { [ T / \tau ] } I _ j , I _ j = [ j \tau , ( j + 1 ) \tau ] \cap [ - T , T ] , \end{align*}


\begin{align*} \hat \Phi \left ( \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) = \hat h ( ( a , b ) ^ \top , s ) \begin{pmatrix} a \\ b \end{pmatrix} , \end{align*}


\begin{align*} N _ { c o m p } ^ i & = N ( r _ { i , f } + r _ { i , g } ) \\ & = 2 N + \mathcal { O } \left ( \frac { \max \{ \omega , \omega ^ { - 1 } \} } { n L _ { x y } ( 1 - \theta ) ^ 2 \epsilon } \left ( \left ( \frac { 1 } { L _ { x } } + \frac { \omega } { L _ { x y } } \right ) \sigma _ { f , i } ^ 2 + \left ( \frac { 1 } { L _ { y } } + \frac { 1 } { L _ { x y } \omega } \right ) \sigma _ { g , i } ^ 2 \right ) \log \left ( \frac { \Psi ^ 0 \nu } { \epsilon } \right ) \right ) . \end{align*}


\begin{align*} - \Delta w = w ^ p _ + , \ \ \ \mathbb { R } ^ 2 . \end{align*}


\begin{align*} & s _ { B S } ( d ( \lambda ) ) = \star = d ^ 0 ( s _ { \Lambda } ) \\ & r _ { B S } ( d ( \lambda ) ) = \star = d ^ 0 ( r _ { \Lambda } ) \end{align*}


\begin{align*} \dim H & \geq x - \lfloor \tilde \varepsilon x \rfloor + \ell _ { i + 1 } '' - \lfloor \tilde \varepsilon x \rfloor + \sum _ { j = 1 } ^ i \big ( \ell _ { j } - \lfloor 2 ^ { j - i - 1 } \tilde \varepsilon x \rfloor \big ) \\ & \geq r + x - \ell _ { i + 1 } ' - 3 \tilde \varepsilon x \\ & \geq r + x - ( 1 + \tilde \varepsilon ) ^ { - i - 1 } ( r + x ) . \end{align*}


\begin{align*} & ( r + \nu ) v ( s , i ) - H \left ( s , i , \partial _ s v ( s , i ) , \partial _ i v ( s , i ) \right ) = 0 , \\ - & ( r + \nu ) v ( s , i ) + H \left ( s , i , \partial _ s v ( s , i ) , \partial _ i v ( s , i ) \right ) = 0 , \end{align*}


\begin{align*} d _ n ^ j = \mu _ n ^ { 1 / 2 } \bigg ( p _ j ( \bar t _ n ) + \sum _ { l = 1 } ^ L \sum _ { k = 1 } ^ { m _ l } \frac { c _ { l k } } { ( \bar t _ n - \bar \gamma _ l ) ^ { k } } \bigg ) + u _ n ^ j , \end{align*}


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


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


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


\begin{align*} \widetilde \theta _ 1 & = 0 , \\ \widetilde \theta _ i & = \pi ^ { [ n - 1 ] } _ { i - 1 } ( \xi ^ { \widetilde \theta } _ 1 ( Y _ 1 ) , x + Y _ 1 , X _ 2 , Y _ 2 , \dots , X _ { i - 1 } , Y _ { i - 1 } ) , \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*} \sup \limits _ i \int _ K | \tilde { F } _ i - ( 1 - b _ { t _ 0 , B } ( \Psi ) ) f F ^ { 1 + \delta } | ^ 2 _ h < + \infty . \end{align*}


\begin{align*} \epsilon ( \eta _ a , t ) : = \int _ 0 ^ t F ( \eta _ a , \tau ) - F _ a ( \eta _ a ) \ , d \tau \end{align*}


\begin{align*} \lim _ { n \rightarrow + \infty } \int _ { \{ \Psi < - l \} } | f | _ h ^ 2 c _ t ^ n ( - \Psi ) = \int _ { \{ - t \leq \Psi < - l \} } | f | _ h ^ 2 . \end{align*}


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


\begin{align*} m = \left \lceil \frac { \log C _ 0 + 1 } { \beta \tau } \right \rceil . \end{align*}


\begin{align*} \Delta ^ 2 u = f \qquad & D , \\ [ 1 m m ] u = \frac { \partial u } { \partial \nu } = 0 & \partial D . \end{align*}


\begin{align*} \left | \frac { \partial ^ 2 \varphi } { \partial x _ \beta \partial x _ \mu } ( x ' , \rho ( x ' ) ) \right | \leq C \delta \sqrt { \tilde { b } _ \beta \tilde { b } _ \mu } , \ \beta , \mu = 1 , \ldots , n - 1 ; \beta \neq \mu \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*} \mu _ { \rm a } ( x [ n ] ) = N _ { \rm a } \mu _ { \rm s } ( x [ n ] ) = { \lambda _ \mathrm { a } [ n ] T _ { \rm s } } \exp \left ( - \frac { \lambda _ \mathrm { a } [ n ] \tau _ { \rm d } } { N _ { \rm a } } \right ) . \end{align*}


\begin{align*} { \bf { \hat \Lambda } } = d i a g \left [ { 1 , 1 , 1 , 1 , { { \hat s } _ e } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ \pi } , { { \hat s } _ \pi } , { { \hat s } _ \pi } } \right ] , \end{align*}


\begin{align*} \begin{aligned} \beta ( \arccos \tfrac { | v _ 1 - v | } { \sqrt { | v _ 1 - v | ^ 2 + | y | ^ 2 } } ) \leq \beta _ 0 | v _ 1 - v | \big ( | v _ 1 - v | ^ 2 + | y | ^ 2 \big ) ^ { - \frac { 1 } { 2 } } \ , . \end{aligned} \end{align*}


\begin{align*} \left | \Pi _ { \mathrm { k e r } A ^ \perp } \frac { d ^ j } { d t ^ j } X _ t | _ { t = 0 } \right | \geq \frac { 1 } { C _ J } K ^ { j + 1 } . \end{align*}


\begin{align*} v = \frac { 1 } { \sqrt { b _ \alpha } } \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta y _ \beta ^ 2 + \sqrt { b _ \alpha } \bar { y } _ n . \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*} \tau = \frac { ( 1 / 2 + r + \epsilon ) \log ( | z _ 0 | ) } { \lambda _ R | z _ 0 | } \end{align*}


\begin{align*} \mathbb { E } \left [ y _ l ^ * y _ j | \hat { \mathbf { H } } \right ] = & a _ c ^ 2 \hat { \phi } ^ { \left ( l , c \right ) ^ * } \hat { \phi } ^ { \left ( j , c \right ) } + \sum _ { q = 1 } ^ M a _ q ^ 2 \hat { \phi } ^ { \left ( l , q \right ) ^ * } \hat { \phi } ^ { \left ( j , q \right ) } . \end{align*}


\begin{align*} B ( L X ) + D ( L X ) + ( n - 1 ) [ \beta ( L X ) + \gamma ( L X ) ] = 0 . \end{align*}


\begin{align*} \begin{aligned} | \L ^ { - 1 } g ( v ) | \lesssim ( 1 + | v | ) ^ { - m } \| ( 1 + | v | ) ^ m g \| _ { L ^ \infty } \ ( \forall v \in \R ^ 3 ) \end{aligned} \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { \ln \frac { ( A _ \varepsilon , \partial \Omega ) } { ( A _ \varepsilon ) } } { \ln \frac { 1 } { \varepsilon } } = 1 . \end{align*}


\begin{align*} p _ { \gamma _ Z } ( F v + G w ) = p _ { \gamma _ Z } ( - \norm { v } _ { V } z ) = \norm { v } _ { V } p _ { \gamma _ Z } ( z ) \leq \norm { v } _ { V } \frac { \varepsilon } { \norm { v } _ { V } } = \varepsilon , \end{align*}


\begin{align*} d ( x , y ) = d ( y , x ) = d ( x , x ) = d ( y , y ) \implies x = y . \end{align*}


\begin{align*} \operatorname { d e t } \left ( M _ { n , 1 } \right ) = ( - 1 ) ^ { \frac { n - 1 } { 2 } } \left ( \frac { n - 1 } { 2 } ! \right ) ^ { 2 } . \end{align*}


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


\begin{align*} \begin{aligned} \widetilde { m } ( x ; z ) & = z \int _ { - \infty } ^ { x } e ^ { 2 i z ( x - y ) } w ( y ) d y + \frac { 1 } { 2 i } w ( x ) \\ & - \frac { 1 } { 2 i } \int _ { - \infty } ^ x e ^ { 2 i z ( x - y ) } \left [ \left ( 2 i \bar { u } _ { y y } ( y ) + \bar { u } _ y ( y ) | u _ y ( y ) | ^ { 2 } \right ) \widehat { \Psi } ^ - _ { 1 1 } ( y ) + | u _ y ( y ) | ^ 2 \widehat { \Psi } ^ - _ { 2 1 } ( y ) \right ] d y , \end{aligned} \end{align*}


\begin{align*} ( W \times T \times L ) / D = ( V \times U \times T \times L ) / D \cong V \times K \end{align*}


\begin{align*} L ( v ) & = \int _ U \left ( \langle W _ u ( u ^ { - 1 } ( y ) ) , \phi ( u ^ { - 1 } ( y ) ) \rangle + W _ \xi ( u ^ { - 1 } ( y ) ) : [ \nabla _ x \phi ] ( u ^ { - 1 } ( y ) ) \right ) d y \\ & = \int _ \Omega \left ( \langle W _ u ( x ) , \phi ( x ) \rangle + W _ \xi ( x ) : [ \nabla _ x \phi ] ( x ) \right ) d x . \end{align*}


\begin{align*} d | \tilde { x } _ t | ^ 2 = 2 \lambda _ A | \tilde { x _ t } | ^ 2 d t - 2 A \tilde { x } _ t \cdot \tilde { x } _ t d t + 2 e ^ { \lambda _ A t } \tilde { x } _ t \cdot \sigma d W _ t + C _ \sigma e ^ { 2 \lambda _ A t } d t . \end{align*}


\begin{align*} \langle e _ { i } , A f _ { j } \rangle = \langle A e _ { i } , A f _ { j } \rangle = \langle e _ { i } , f _ { j } \rangle = \delta _ { i j } . \end{align*}


\begin{align*} \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { \infty } h _ { n } ( x ) h _ { m } ( x ) \exp ( - x ^ { 2 } / 2 ) d x = \delta _ { n m } , \end{align*}


\begin{align*} \tau = \sum _ { j = 1 } ^ \infty \frac { c _ j } { 2 ^ j } \end{align*}


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


\begin{align*} \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ 0 \cdot B _ 0 \dd x , \end{align*}


\begin{align*} ( p - L _ X h ) ^ \intercal & = - 2 \zeta A ' ( h ) \\ H ' ( p - L _ X h ) & = \zeta A \cdot A ' ( h ) . \end{align*}


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


\begin{align*} M _ 1 = \ker ( { _ 1 H _ 2 \otimes N _ 2 } \xrightarrow { N _ { 1 2 } } N _ 1 ) , \end{align*}


\begin{align*} J ( z ) = \int _ 0 ^ z x f ( x ) \mathrm { d } x = \frac { \sqrt \theta } { B ( a , b ) } \int _ { \theta / z ^ 2 } ^ \infty y ^ { - 1 / 2 } \exp ( - a y ) \left [ 1 - \exp ( - y ) \right ] ^ { b - 1 } \mathrm { d } y . \end{align*}


\begin{align*} F _ n - n ^ 3 \ = \ \sum _ { i = 1 } ^ { n } \left ( i ^ 3 - 2 \left ( \binom { 3 } { 2 } i ^ 2 + \binom { 3 } { 0 } i ^ 0 \right ) \right ) \cdot F _ { n - i } . \end{align*}


\begin{align*} \nabla _ X ^ \perp \eta _ i = \sum _ { j } \phi _ { i j } ( X ) \varphi _ j \eta _ j . \end{align*}


\begin{align*} \widehat { \iota } ( D ) = \frac { \beta - e } { d } \end{align*}


\begin{align*} a ( u , v ) = ( f , v ) v \in H _ 0 ^ 2 ( D ) . \end{align*}


\begin{align*} K ^ { ( 1 ) } ( S _ t , I _ t ) & = \frac { w } { 2 \theta ( \partial _ i V ( S _ t , I _ t ) - \partial _ s V ( S _ t , I _ t ) ) } , \\ K ^ { ( 2 ) } ( S _ t , I _ t ) & = \frac { w } { 2 \theta ( 1 - \theta \bar L ) ( \partial _ i V ( S _ t , I _ t ) - \partial _ s V ( S _ t , I _ t ) ) } . \end{align*}


\begin{align*} \lim _ { m _ i \in M _ 2 , m _ i \to \infty } \mu ( T ^ { m _ i } A \cap B ) = \mu ( A ) \mu ( B ) . \end{align*}


\begin{align*} e _ { n + 1 } ^ i = e _ n ^ i + \int _ { t _ n } ^ { t _ { n + 1 } } ( b ^ i ( X _ t ) - b ^ i ( \tilde X _ n ) ) \d t . \end{align*}


\begin{align*} - \sum _ { K = 0 } ^ { p } \binom { p } { K } \ , B _ K \ , n ^ { p - K } \end{align*}


\begin{align*} \sigma ( \sqrt { a } ) = - \sqrt { a } , \ & \sigma ( \sqrt { d } ) = \sqrt { d } \\ \tau ( \sqrt { a } ) = \sqrt { a } , \ & \tau ( \sqrt { d } ) = - \sqrt { d } . \end{align*}


\begin{align*} \aligned e _ 1 & \ , : = \ , ( 1 - y ) \ , \partial _ x + x \ , \partial _ u , \\ e _ 2 & \ , : = \ , ( 1 - y ) \ , \partial _ y + u \ , \partial _ u , \\ e _ 3 & \ , : = \ , x \ , \partial _ x + 2 \ , u \ , \partial _ u , \\ e _ 4 & \ , : = \ , - \ , u \ , \partial _ x + x \ , \partial _ y , \endaligned \end{align*}


\begin{align*} \left \{ e _ { m , q } ( z ) = \frac { z ^ m } { \sqrt { \Gamma ( m q + 1 ) } } \ ; ; \ ; m \geq 0 \right \} \end{align*}


\begin{align*} \begin{cases} a \mapsto a b , \\ c \mapsto c , & c \neq a , \end{cases} \begin{cases} a \mapsto b a , \\ c \mapsto c , & c \neq a . \end{cases} \end{align*}


\begin{align*} \frac { \partial \textbf { \textit { U } } } { \partial x _ n } + Q \textbf { \textit { U } } = \textbf { \textit { W } } \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*} T _ 1 = \sum _ { X \in S ( 0 ) } \Theta \begin{bmatrix} m _ 1 ' & X \\ 0 & m _ 2 ' \end{bmatrix} \overline { \psi _ A ( { m _ 1 ' } ^ { - 1 } X } ) = \theta ( a ) q ^ 8 ( q - 1 ) ^ 3 . \end{align*}


\begin{align*} \mathcal { F } _ { \geq p } C F ^ { k } ( H _ { t } ) : = \lbrace x \in C F ^ { k } ( H _ { t } ) \mid \mathcal { A } ( x ) \geq p \rbrace \end{align*}


\begin{align*} \Phi _ { n , e } ^ { ( i ) } ( x , y ) = \prod ^ { n e } _ { \substack { j = 1 \\ ( j , n e ) = 1 \\ j \equiv i \bmod e } } ( x - \zeta ^ j _ { n e } y ) . \end{align*}


\begin{align*} e _ 2 ( C ) = e ( C ) , \end{align*}


\begin{align*} \widehat { C } _ { \psi } = C _ { \widehat { \psi } } , \end{align*}


\begin{align*} S _ { h _ C , W , 0 } = \left \{ U _ { C } ( \cdot + z e _ z ) - B _ { \infty } \ \big | \ z \in \mathbb { R } \ \right \} . \end{align*}


\begin{align*} \epsilon ( x _ a , t ) = \int _ N ^ t h ( x _ a + \delta u ( \tau ) ) u ( \tau ) - \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \ , d \tau . \end{align*}


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


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


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


\begin{align*} \int _ 1 ^ \infty \frac { d s } { \sqrt { F ( s ) } } < \infty \quad \mbox { w h e r e } \ ; F ( s ) = \int _ 0 ^ s f ( t ) d t . \end{align*}


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


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


\begin{align*} \frac { \partial } { \partial t } u ^ h ( t , x ) = \frac { \partial } { \partial t } \left ( \sum _ i w _ i ( t ) \sigma _ i ( x ) \right ) = \sum _ i \sigma _ i ( x ) \frac { d } { d t } w _ i ( t ) , \end{align*}


\begin{align*} m : = \inf _ { \partial \Omega } \frac { G ( ( D ^ 2 u ) ' ) } { \widetilde { f } } , \end{align*}


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


\begin{align*} \langle \delta _ \chi , \delta _ \pi \rangle _ { L ^ 2 ( G ^ \wedge ) } = \int _ { G ^ \wedge } \delta _ \chi ( \xi ) \ ; \delta _ \pi ( \xi ) \ , d \nu ( \xi ) = \left \{ \begin{array} { c c l } 1 , & & \chi = \pi , \\ 0 , & & \chi \not = \pi . \end{array} \right . \end{align*}


\begin{align*} \lim _ { t \to 0 } \frac { \pi ^ { - } ( a _ t \cdot \psi _ t - a \cdot \psi ) } { t } & = \pi ^ { - } \left ( \lim _ { t \to 0 } \frac { a _ t \cdot \psi _ t - a _ t \cdot \psi } { t } + \lim _ { t \to 0 } \frac { a _ t - a } { t } \cdot \psi \right ) \\ & = \pi ^ { - } ( b \cdot \phi + a \cdot \psi ) . \end{align*}


\begin{align*} \boldsymbol { \lambda } = \sum \limits _ { k = 1 } ^ m s _ k \boldsymbol { \lambda } _ k . \end{align*}


\begin{align*} \left | \frac { e ^ { \kappa s } - 1 } { s } \right | & = \left | \sum _ { n = 0 } ^ \infty \frac { \kappa ^ { n + 1 } s ^ n } { ( n + 1 ) ! } \right | \leq e ^ { \frac { 3 } { 4 } \kappa } \kappa , \end{align*}


\begin{align*} \nabla p = B \cdot \nabla B - \nu ( - \Delta ) ^ { \kappa } u . \end{align*}


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


\begin{align*} a \cdot b a & = b , \\ a b \cdot c & = a \cdot ( b \cdot c ) , \\ ( a \cdot b ) a & = ( b \cdot a ) b , \end{align*}


\begin{align*} X _ { i ; j k } - X _ { i ; k j } & = R _ { k j \ell i } X ^ \ell \\ X _ { j ; k i } - X _ { j ; i k } & = R _ { i k \ell j } X ^ \ell \\ X _ { k ; i j } - X _ { k ; j i } & = R _ { j i \ell k } X ^ \ell . \end{align*}


\begin{align*} G ^ { ( 1 ) } \vee . . . \vee G ^ { ( k ) } = \vee _ { \gamma \in \Gamma } G ^ { ( \gamma ) } . \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*} 0 = S ' ( h , v ) - S ' \big ( L _ X \bar g , X ( \bar u ) \big ) = S ' ( 0 , v - X ( \bar u ) ) = 0 \mbox { i n } M \setminus \Omega . \end{align*}


\begin{align*} N _ 1 = ( E _ 1 ) _ 1 = H _ 1 N _ 2 = { _ 2 H _ 1 } \otimes H _ 1 = ( H _ 2 \cdot \alpha _ { 2 1 } ) \oplus ( H _ 2 \cdot \alpha _ { 2 1 } \varepsilon _ 1 ) \oplus ( H _ 2 \cdot \alpha _ { 2 1 } \varepsilon _ 1 ^ 2 ) . \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*} H _ { m i n } ( E ) = \min \limits _ { 0 \neq F \subset E } H _ r ( F ) . \end{align*}


\begin{align*} & \frac { 1 } { \varepsilon ^ 2 } \int _ \Omega \left ( v ^ \tau _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p + 1 } _ + d x = \int _ { \hat { B } _ \tau ( 0 ) } q ^ { p + 1 } ( \bar { x } + \varepsilon y ) ( \hat { U } ( y ) + \ln \tau ) ^ { p + 1 } _ + d y \\ \rightrightarrows & q ^ { p + 1 } ( \bar { x } ) \int _ { \hat { B } _ \tau ( 0 ) } ( \hat { U } ( y ) + \ln \tau ) ^ { p + 1 } _ + d y , \end{align*}


\begin{align*} F ^ 1 ( \phi _ i x ) = F ^ i x F _ 1 ( \phi _ i x ) = F _ i x \end{align*}


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


\begin{align*} \frac { d } { d t } \theta _ u = \omega _ u + \frac { K } { | V | } \sum _ { v \colon \{ u , v \} \in E } \sin ( \theta _ v - \theta _ u ) , u \in V \end{align*}


\begin{align*} a = \left ( \beta - 1 \right ) n + \beta ^ { 2 } + \beta + 1 \end{align*}


\begin{align*} 0 = \partial ^ 2 _ { x _ i x _ j } \phi - \partial ^ 2 _ { x _ i x _ j } \psi - \partial ^ 2 _ { x _ i y } \psi \partial _ { x _ j } T - \partial ^ 2 _ { x _ j y } \psi \partial _ { x _ i } T - \partial ^ 2 _ y \psi \partial _ { x _ j } T \partial _ { x _ i } T - \partial _ y \psi \partial ^ 2 _ { x _ j x _ i } T \\ + { ( \partial _ y T ) ^ { - 2 } } ( \partial _ y T \partial ^ 3 _ { x _ j x _ i y } T - \partial ^ 2 _ { x _ i y } T \partial ^ 2 _ { x _ j y } T ) \end{align*}


\begin{align*} \Theta _ { \iota } = ( f ^ { - 1 } ) _ { \iota } \varphi h G + f ^ { - 1 } G \varphi h _ { \iota } \left ( 1 + \frac { \varphi ^ { ' } } { \varphi } h \right ) + f ^ { - 1 } \varphi h G _ { \iota } . \end{align*}


\begin{align*} \begin{aligned} & M _ { \delta , + , 1 } ( x ; z ) = e _ 1 + \mathcal { P } ^ + \left ( r _ { \delta , - , 2 } e ^ { 2 i z x } M _ { \delta , - , 2 } ( x ; \cdot ) \right ) , \\ & M _ { \delta , - , 2 } ( x ; z ) = e _ 2 + \mathcal { P } ^ { - } \left ( \bar { r } _ { \delta , - , 1 } e ^ { - 2 i z x } M _ { \delta , + , 1 } ( x ; \cdot ) \right ) . \end{aligned} \end{align*}


\begin{align*} \pi _ i ( \theta _ A ) = \begin{cases} \theta _ { A } & \hbox { i f } i , i + 1 \in A \hbox { o r } i , i + 1 \notin A \\ \theta _ { A \cup \{ i + 1 \} \backslash \{ i \} } & \hbox { i f } i \in A \hbox { a n d } i + 1 \notin A \\ \theta _ { A \cup \{ i \} \backslash \{ i + 1 \} } & \hbox { i f } i + 1 \in A \hbox { a n d } i \notin A \end{cases} ~ . \end{align*}


\begin{align*} \mathcal { V } _ p = T _ p \Sigma _ t \ \ \ \ \forall p \in \Sigma _ t \end{align*}


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


\begin{align*} \begin{aligned} \left \vert \frac { \tau } { \eta \cdot \omega _ j } \right \vert & = \frac { \sqrt { 1 + s ^ 2 } } { \vert \mathbf z \cdot \omega _ j - s \omega _ 1 \cdot \omega _ j + \mathrm i \sqrt { 1 + s ^ 2 } \mathbf d ^ \perp \cdot \omega _ j \vert } \\ & = \frac { \sqrt { 1 + s ^ 2 } } { \sqrt { \vert \mathbf z \cdot \omega _ j - s \omega _ 1 \omega _ j \vert ^ 2 + ( 1 + s ^ 2 ) } } \geq \frac { 1 } { 2 } , j = 2 , 3 . \end{aligned} \end{align*}


\begin{align*} \int _ { \mathfrak X ( n ) } | \omega ( n ) | = ( \mathfrak h ( n ) _ 0 ) _ ! \int _ { \mathfrak Y ( n ) } | \mathfrak h ( n ) _ { \eta } ^ * \omega ( n ) | . \end{align*}


\begin{align*} \varphi ( x ' , \rho ( x ' ) ) = \frac { 1 } { 2 } \sum _ { \alpha \leq n - 1 } b _ \alpha x _ \alpha ^ 2 + \frac { 1 } { 6 } \sum _ { \alpha , \beta , \gamma \leq n - 1 } \varphi _ { \alpha \beta \gamma } ( 0 ) x _ \alpha x _ \beta x _ \gamma + O ( | x ' | ^ 4 ) , \end{align*}


\begin{align*} ( \nu S \textbf { \textit { u } } ) ^ i = \nu ^ j ( S \textbf { \textit { u } } ) ^ i _ j & = \nu ^ j ( \nabla ^ i u _ j + \nabla _ j u ^ i ) \end{align*}


\begin{align*} \log \det ( u _ { , i j } ) = - v _ j x ^ j + u _ { , i } \xi ^ i + c , \end{align*}


\begin{align*} \ell _ t = 2 \ , \ell _ b \qquad ( \beta = \gamma ) \Psi = \frac { 3 } { 2 } . \end{align*}


\begin{align*} \Delta x _ k + \bar u ^ { - 1 } \nabla \bar u \cdot \nabla x _ k = - \Gamma ^ k + \bar u ^ { - 1 } \bar g ^ { i k } \frac { \partial \bar u } { \partial x _ i } . \end{align*}


\begin{align*} \mathcal { U } _ \mu = \{ u \in \mathcal { U } : u \mu \} \subseteq \mathcal { U } , \end{align*}


\begin{align*} u = u _ { + } \sqcup u _ { - } \colon U _ { X } = U _ { X , + } \sqcup U _ { X , - } = R _ { \mathbb { C } / \mathbb { R } } \mathbb { G } _ { m , \mathbb { C } } \sqcup R _ { \mathbb { C } / \mathbb { R } } \mathbb { G } _ { m , \mathbb { C } } \to \mathbb { S } ^ { 1 } \sqcup \mathbb { G } _ { m , \mathbb { R } } = Z _ { + } \sqcup Z _ { - } = Z \end{align*}


\begin{align*} F ' _ m ( n ) = F _ { p _ t } ( n ) = 0 , n \leq p _ t . \end{align*}


\begin{align*} q _ d ^ * = - ( d - 1 ) + O ( d ) \left ( - \frac { 1 - \log 2 } { 2 } d \right ) , d \to \infty . \end{align*}


\begin{align*} c _ k : = \frac { N _ k ( A ) } { r ( A ) ^ k } \ \end{align*}


\begin{align*} T _ \sigma ( f , g ) ( x ) = \langle K _ \sigma ( x , y , z ) , ( f \otimes g ) ( y , z ) \rangle , \end{align*}


\begin{align*} \mathcal { A } = & \int _ { 0 } ^ { T } \int _ \Omega \bigg ( \frac { 1 } { 2 } \ , \rho \ , \| \dot { u } \| ^ 2 + \ , \rho \ , j \ , \| \dot { \vartheta } \| ^ 2 - W ) \ , d v \ , d t , \end{align*}


\begin{align*} M _ { + , 2 } ( x ; z ) - e _ { 2 } = \mathcal { P } ^ + \left ( \bar { r } _ 1 ( z ) \mathrm { e } ^ { - 2 i z x } M _ { - , 1 } ( x ; z ) \right ) ( z ) , z \in \mathbb { R } . \end{align*}


\begin{align*} - \epsilon _ 1 b _ { n - 1 } u _ n ( x ) \leq u ( 0 ) - u ( x ) + \sum _ { \alpha = 1 } ^ { n - 1 } x _ \alpha u _ \alpha ( x ) \leq C b _ { n - 1 } ^ 2 . \end{align*}


\begin{align*} | a _ { C } ( n ) | \leq \left ( \sum _ { i = 1 } ^ s | c _ { i } | \right ) \tau ( n ) \sqrt { n } \end{align*}


\begin{align*} H ( s , t ) = \frac { \kappa ( s ) } { n } . \end{align*}


\begin{align*} \prescript { } { l } { v } _ j ( \lambda ) : = \prescript { } { l } { \tilde { h } } _ j ( \lambda ) q _ { l , n _ j } ( \lambda ) = \prescript { } { l } { \beta } ( \lambda ) \prescript { } { l } { R } _ j ( \lambda ) q _ { l , n _ j } ( \lambda ) . \end{align*}


\begin{align*} \Lambda : = \{ n \in \mathbb { N } : g _ n ( t , \omega ) > 0 \} . \end{align*}


\begin{align*} \mathcal { Q } _ { | _ { x \times C } } \cong \bigoplus _ { i = 1 } ^ { n } j _ { i * } ( E _ i ) \mathcal { R } _ { | _ { x \times C } } \cong \bigoplus _ { i = 1 } ^ { n - 1 } j _ { p _ i * } ( j _ { p _ i } ^ * ( j _ { i + 1 * } ( E _ { i + 1 } ) ) ) . \end{align*}


\begin{align*} & \int _ M ( \eta + g ^ { - 1 } ) | D ^ { '' * } v | ^ 2 _ Q d V _ M + \int _ M \eta | D ^ { '' } v | ^ 2 _ Q d V _ M \\ \ge & \int _ M \langle [ \eta \sqrt { - 1 } \Theta _ Q - \sqrt { - 1 } \partial \bar { \partial } \eta - \sqrt { - 1 } g \partial \eta \wedge \bar { \partial } \eta , \Lambda _ { \omega } ] v , v \rangle _ Q d V _ M . \end{align*}


\begin{align*} \mathbf { y } _ k = \mathbf { H } _ k \sum _ { i \in \mathcal { M } _ k } a _ i s _ i \mathbf { p } _ i + \mathbf { H } _ k \sum \limits _ { \substack { l = 1 \\ l \neq k } } ^ { K } \sum \limits _ { j \in \mathcal { M } _ l } a _ j s _ j \mathbf { p } _ j + \mathbf { n } _ k \end{align*}


\begin{align*} x _ { i j } : = x _ { i i - 1 } x _ { i - 1 i - 2 } \cdots x _ { j + 1 j } \end{align*}


\begin{align*} \sum \limits _ { j = 1 } ^ d A _ { j } ^ * A _ { j } \ ( x _ 1 \otimes \cdots \otimes x _ d ) & = \sum \limits _ { j = 1 } ^ d \| X _ { j } \| ^ 2 ( x _ 1 \otimes \cdots \otimes x _ d ) \\ & = \left \| \sum \limits _ { j = 1 } ^ d A _ { j } ^ * A _ { j } \right \| ( x _ 1 \otimes \cdots \otimes x _ d ) . \end{align*}


\begin{align*} \begin{aligned} v ' = \xi + v \ , , \ v _ 1 = v + \xi + y \ , . \end{aligned} \end{align*}


\begin{align*} \mathfrak X _ \pi : = V ( \pi ) \subset \mathfrak X . \end{align*}


\begin{align*} \int _ { \mathfrak X } | \omega | = { \mathfrak h _ 0 } _ ! \int _ { \mathfrak Y } | \omega | , \end{align*}


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


\begin{align*} \alpha ( g , ( z _ 1 + 1 , z _ 2 , \ldots ) ) = ( \xi + ( z _ 1 + 1 ) \psi _ 1 ) \alpha ( g , Z ) - \sum _ { \overline { \Gamma } \in { \rm B i c } ( { g } , { Z } ) _ 1 } m ( \overline { \Gamma } ) \alpha ( \overline { \Gamma } ) . \end{align*}


\begin{align*} F ^ { \perp } : = \left \lbrace \varphi \in E ^ { \vee } \mid \varphi ( F ) = 0 \right \rbrace . \end{align*}


\begin{align*} \begin{aligned} \| x ( ( n + 1 ) \bar { t } / \gamma ) \| _ { \mathcal { A } } & \le \beta ( \| x ( n \bar { t } / \gamma ) \| _ { \mathcal { A } } , \bar { t } ) + \tilde { d } & & \forall \ , n \in \mathbb { N } . \end{aligned} \end{align*}


\begin{align*} V i r = \mathbb { C } [ \partial ] L , [ L _ \lambda L ] = ( \partial + 2 \lambda ) L . \end{align*}


\begin{align*} s _ i ( A ) : = \sqrt { \lambda _ i ( A ^ * A ) } , i = 1 , 2 , \dots , n , \end{align*}


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


\begin{align*} x _ 2 \partial _ { x _ 1 } v _ 3 - x _ 1 \partial _ { x _ 2 } v _ 3 + k \partial _ { x _ 3 } v _ 3 = 0 . \end{align*}


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


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


\begin{align*} \Theta & = d \theta ^ 0 + \theta ^ 1 \wedge \theta ^ 0 , \\ * \Omega & = d \theta ^ 1 + \theta ^ 1 \wedge \theta ^ 1 , \end{align*}


\begin{align*} & R _ + ( x ; z ) = \begin{pmatrix} 0 & \bar { r } _ 1 ( z ) \mathrm { e } ^ { - 2 i z x } \\ 0 & 0 \end{pmatrix} , R _ - ( x ; z ) = \begin{pmatrix} 0 & 0 \\ r _ 2 ( z ) \mathrm { e } ^ { 2 i z x } & 0 \end{pmatrix} , \\ & F = \left ( \mathcal { P } ^ { - } ( r _ 2 ( z ) \mathrm { e } ^ { 2 i z x } ) e _ 2 , \mathcal { P } ^ + ( \bar { r } _ 1 ( z ) e ^ { - 2 i z x } ) e _ 1 \right ) , \end{align*}


\begin{align*} \tilde M _ { \mu , s } ( t ) = M _ { \mu } ( T _ { \mu , s } ( t ) ) , \end{align*}


\begin{align*} \mu \big ( T ^ { ( r + 1 ) r _ n } A \cap A ^ c \big ) \leq & \sum _ { i = 1 } ^ { r + 1 } \mu \big ( T ^ { i r _ n } A \cap T ^ { ( i - 1 ) r _ n } A ^ c \big ) \\ < & \big ( r + 1 \big ) \big ( 1 - \beta \big ) \mu ( A ) \\ = & \Big ( \big ( r + 1 \big ) - \big ( r + 1 \big ) \beta \Big ) \mu ( A ) \\ = & \Big ( 1 - \big ( ( r + 1 ) \beta - r \big ) \Big ) \mu ( A ) \\ < & \Big ( 1 - \mu ( A ) \Big ) \mu ( A ) . \end{align*}


\begin{align*} Y = m ( X ) + \eta \end{align*}


\begin{align*} \left \{ \begin{array} { l } \lambda ^ { N _ { i - 1 } } + \sum _ { \ell = 1 } ^ { r } c _ \ell u _ { i - 1 } ^ { \gamma _ \ell } \lambda ^ { ( i - 1 ) \gamma _ \ell + N _ { i - 1 } - \ell } = 0 \\ u _ { i - 1 } = \lambda u _ { i } \end{array} \right . \end{align*}


\begin{align*} ( \mathbf { x } , - \mathbf { y } ) \underbrace { \begin{pmatrix} G _ { 4 } & G _ { 5 } & \cdots & G _ { 2 k + 2 } \\ G _ { 5 } & G _ { 6 } & \cdots & G _ { 2 k + 3 } \end{pmatrix} } _ M = 0 . \end{align*}


\begin{align*} \sum _ { A = 0 } ^ { 2 n - 1 } \nabla _ { A \alpha } \varrho \left ( d _ 1 u _ k \wedge \Theta \wedge ( \Delta \varrho ) ^ { n - k } \right ) _ A \Omega _ { 2 n } = d _ \alpha \varrho \wedge d _ 1 u _ k \wedge \Theta \wedge ( \Delta \varrho ) ^ { n - k } , \end{align*}


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


\begin{align*} E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 2 } ^ { n } Z _ i ^ { \theta [ u ] } \Bigr ) \right ] = E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \Bigr ) \right ] . \end{align*}


\begin{align*} \delta _ { \pm } ( z ) = \exp [ \mathcal { P } ^ { \pm } \log \left ( 1 + \bar { r } _ 1 r _ 2 \right ) ] , z \in \mathbb { R } . \end{align*}


\begin{align*} \sum _ { k = c _ 1 } ^ { c _ 2 } { L _ { i w _ k + m } v _ k z ^ { w _ k } } = \alpha ^ m h ( \alpha ^ i z ) - \beta ^ m h ( \beta ^ i z ) \ , . \end{align*}


\begin{align*} \int _ { X } ( \alpha + \sqrt { - 1 } \omega ) ^ { 2 } = \int _ { X } ( \alpha ^ { 2 } - \omega ^ { 2 } ) + 2 \sqrt { - 1 } \int _ { X } \alpha \wedge \omega = 3 2 + 2 6 \sqrt { - 1 } . \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } \sup _ { n \in \mathbb { N } } I I _ n = 0 . \end{align*}


\begin{align*} \hat \Phi \left ( A _ \phi \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) = A _ \phi \hat \Phi \left ( \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) , \end{align*}


\begin{align*} \begin{aligned} N _ 1 [ K [ \hat { f } ] ] < k N _ 2 [ \hat { f } ] \ , , \ N _ 3 ^ { ( r + 1 / 2 ) } [ K [ \hat { f } ] ] < k N _ 3 ^ { ( r ) } [ \hat { f } ] \end{aligned} \end{align*}


\begin{align*} - \varepsilon ^ 2 ( K ( x ) \nabla u ) + V ( x ) u = u ^ { p } , \ \ \ \ x \in \mathbb { R } ^ n , \end{align*}


\begin{align*} X _ n = ( \mathfrak X , \mathcal O _ { \mathfrak X } \otimes _ R R _ n ) , \end{align*}


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


\begin{align*} h ( T ) & = x y \log ( x y ) - \sum _ { k = 1 } ^ { r } \left ( \Delta ( i _ k ) + \Delta ( j _ k ) \right ) \\ & \le x y \log ( x y ) - \sum _ { k = 1 } ^ { r } \left ( \Delta ( x ) + \Delta ( y + 1 - k ) \right ) \\ & = h ( T ' ) , \end{align*}


\begin{align*} P _ { m ; l } \otimes P _ { m ' ; l ' } \cong \bigoplus \limits _ { \nu = \vert l - l ' \vert \mathrm { s t e p } 2 } ^ { \mathrm { m i n \left ( l + l ' , 2 d - 4 - l - l ' \right ) } } P _ { m + m ' + \frac { 1 } { 2 } \left ( l + l ' - \nu \right ) ; \nu } \end{align*}


\begin{align*} \begin{aligned} \mathfrak r _ { l k + 2 } ( 2 ) & = \mathfrak p _ { k } \mathfrak r _ { ( l - 1 ) k + 2 } ( 2 ) - i \partial _ { a } \mathfrak p _ { k } \delta _ { a } \mathfrak r _ { ( l - 1 ) k + 1 } ( 1 ) - \frac { 1 } { 2 } \partial _ { a } \partial _ { b } \mathfrak p _ { k } \delta _ { a } \delta _ { b } \mathfrak r _ { ( l - 1 ) k } = \cdots \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \left | ( h ( x _ a + \delta u ( \tau ) ) \pm h ( x _ a ) ) u ( \tau ) - \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \right | \le \\ \left | ( h ( x _ a + \delta u ( \tau ) ) - h ( x _ a ) ) u ( \tau ) \right | + \left | \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \right | \\ + | h ( x _ a ) | | u ( \tau ) | . \end{aligned} \end{align*}


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


\begin{align*} d g _ 1 ' = ( \log \sqrt { \tau _ 0 } ) _ z d z \ g ' _ 1 + \frac { 1 } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) \ J \widehat { g _ 1 ' } I , \end{align*}


\begin{align*} ( \hat h , \hat v ) : = \big ( h + L _ V \bar g , v + V ( \bar u ) \big ) \end{align*}


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


\begin{align*} \frac { d } { d t } ( f , \mu _ t ) = \frac { a ( . ) } { ( a , \mu ) } ( L ^ { \rm d i f } _ { \mu _ t } f , \mu _ t ) ; \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { n - 1 } } e ^ { - C | \xi | } \frac { \xi _ { \alpha } ^ 2 } { | \xi | ^ 2 } \ , d \xi = \frac { \Gamma ( n - 1 ) \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) } { ( n - 1 ) C ^ { n - 1 } } , C > 0 , \ n \geqslant 2 , \end{align*}


\begin{align*} S _ R ( X ; \pi , \hat { \phi } , W ) & = \sum _ { \substack { l > Z \\ ( l , 2 ) = 1 } } \mu ( l ) \sum _ { ( m , 2 ) = 1 } W \left ( \frac { l ^ 2 m } { X } \right ) E ( Y ; \chi _ { 8 l ^ 2 m } , \hat { \phi } ) \ll \sum _ { l > Z } \sum _ { X / l ^ 2 \leq m \leq 2 X / l ^ 2 } \log ^ { 3 } ( X ) \ll \frac { X \log ^ { 3 } X } { Z } . \end{align*}


\begin{align*} \begin{aligned} u _ t + u \cdot \nabla u + \nabla p & = B \cdot \nabla B + \nu \Delta u , \\ B _ t + u \cdot \nabla B & = B \cdot \nabla u + \mu \Delta B , \\ \nabla \cdot u = \nabla \cdot B & = 0 , \end{aligned} \end{align*}


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


\begin{align*} V ( \check h ) = V ( h , x _ { n + 1 } - f ) = V ( h ^ G ) \ , . \end{align*}


\begin{align*} \widetilde { G } _ { 2 } ( s ) : = & \zeta ( s ) L ( s - 2 , \chi ) \prod \limits _ { n = 1 } ^ { j } L ( s , s y m ^ { 2 n } f ) L ( s - 2 , s y m ^ { 2 n } f \otimes \chi ) , \end{align*}


\begin{align*} \beta _ { \bar { \mathbf { g } } } \mathbf { g } = \beta _ { \bar g } g + \bar u ^ { - 2 } u d u - \bar u ^ { - 1 } g ( \nabla _ { \bar g } \bar u , \cdot ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 3 } } A _ { t } \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x = - \mu \int _ { \mathbb { R } ^ { 3 } } \nabla \times B \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x . \end{align*}


\begin{align*} \begin{aligned} \nabla ^ 2 _ U \eta ( U , x , t ) - ( \nabla _ V \tilde { \eta } ) & ( A ( U , x , t ) , x , t ) \cdot \nabla ^ 2 _ U A ( U , x , t ) = \\ & = ( \nabla _ V ^ 2 \tilde { \eta } ) ( A ( U , x , t ) , x , t ) : ( \nabla _ U A ( U , x , t ) , \nabla _ U A ( U , x , t ) ) \ ; . \end{aligned} \end{align*}


\begin{align*} P _ e = \frac { 4 \big ( \sqrt { M } - 1 \big ) } { \sqrt { M } \mathrm { l o g } _ 2 ( M ) } { Q } \Bigg ( \sqrt { \frac { 3 \gamma } { M - 1 } } \Bigg ) + \frac { 4 \big ( \sqrt { M } - 2 \big ) } { \sqrt { M } \mathrm { l o g } _ 2 ( M ) } { Q } \Bigg ( 3 \sqrt { \frac { 3 \gamma } { M - 1 } } \Bigg ) . \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*} \varphi ^ L _ { \sigma , s } ( x ) = a = \varphi ^ L _ { \sigma , s } ( y ) \end{align*}


\begin{align*} M _ n ( Q ) / n & \le \int _ \R V \ , d \overline { Q } + \frac 1 2 \int _ \R \int _ \R K ( x - y ) \ , \overline { Q } ( d x ) \overline { Q } ( d y ) - H ( \overline { Q } ) = M _ n ( \overline { Q } ^ { \otimes n } ) / n . \end{align*}


\begin{align*} \left \{ \begin{array} { l r } \frac { \partial X ( x , t ) } { \partial t } = \frac { 1 } { f ( v ) } \sigma _ { k } ( x , t ) \varphi ( X \cdot v ) ( X \cdot v ) G ( X ) v - X ( x , t ) ; \\ X ( x , 0 ) = X _ { 0 } ( x ) , \end{array} \right . \end{align*}


\begin{align*} M _ \delta ( x ; z ) = I + \mathcal { C } \left ( M _ { \delta , - } ( x ; \cdot ) \hat { R } _ \delta ( x ; \cdot ) \right ) ( z ) , z \in \mathbb { C } ^ { \pm } . \end{align*}


\begin{align*} A = \frac { N ^ { \frac 1 d } } { ( \log N ) ^ { \frac 1 { 1 6 d } } } , R = 1 , T = \frac 1 { N ( \log N ) ^ { \frac 1 { 1 6 } } } . \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*} x y - ( - 1 ) ^ { p ( x ) p ( y ) } y x = 0 \end{align*}


\begin{align*} \left ( \int _ { B _ 1 } | v | ^ p \d x \right ) ^ { \frac { 1 } { p } } \leq C \left ( \int _ { B _ 1 } | \nabla v | ^ 2 \d x \right ) ^ { \frac { 1 } { 2 } } = C . \end{align*}


\begin{align*} \Re \bigg \{ \left ( \frac { L ' } { L } \right ) ' ( s , \chi ) \bigg \} = \sum _ { \gamma _ \chi } g _ { a } ( t - \gamma _ \chi ) + \sum _ { \gamma } g _ { a } ( \gamma ) + O ( 1 ) , \end{align*}


\begin{align*} h ( \nu , \cdot ) = 0 , ( \nabla _ { \nu } h ) ( \nu , \cdot ) = 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*}