\begin{align*} \int _ 1 ^ \infty g ( t ) d t \sim \chi ^ { - ( \gamma + 1 ) } \ln ( \chi ^ { - 1 } ) . \end{align*}


\begin{align*} N _ m = ( T _ m ( K \cdot m ) ) ^ { \perp } \cong T _ m M / T _ m ( K \cdot m ) \end{align*}


\begin{align*} K ( T ) : = \sup _ { t \in [ 0 , T ] } \sup _ { x \in \R ^ d } \| u ( t \ , , x ) \| _ k < \infty . \end{align*}


\begin{align*} \frac { \zeta \dim P + \sum _ { k = 1 } ^ { n } \eta _ { k } \dim P \cap F _ { k } } { \dim P } \le \frac { \sum _ { k = 1 } ^ { n } \eta _ { k } \dim F _ { k } } { n + 1 } . \end{align*}


\begin{align*} \mathrm { T r } \left ( \gamma \left ( y _ { 0 } \alpha ^ { j _ { 0 } } + \cdots + y _ { t - 1 } \alpha ^ { j _ { t - 1 } } \right ) \right ) = 0 , \end{align*}


\begin{align*} \forall \omega > 0 , - \mathcal { C } ( f ) ( \omega ) = \phi ( \omega ) , \end{align*}


\begin{align*} \theta \sum _ { i = 1 } ^ n \Bigl \| I _ i \sum _ { j = 1 } ^ m x _ j \Bigr \| _ { T _ \theta } \leqslant \theta \sum _ { i = 1 } ^ n \sum _ { j \in A _ i } \| x _ j \| _ { T _ \theta } + \sum _ { i = 1 } ^ n \sum _ { j \in B _ i \setminus A _ i } \| x _ j \| _ { T _ \theta } \leqslant \theta a m + 2 a n . \end{align*}


\begin{align*} \Psi ^ * \mathcal { L } _ 1 ^ { \operatorname { f r a c } } f = \mathcal { L } _ 2 ^ { \operatorname { f r a c } } \Psi ^ * f . \end{align*}


\begin{align*} \Lambda _ p = \lambda _ 0 + \lambda _ 1 F _ p ( \alpha _ 1 ) + \lambda _ 2 F _ p ( \alpha _ 2 ) + \ldots + \lambda _ k F _ p ( \alpha _ k ) , \end{align*}


\begin{align*} S _ \iota = \bigsqcup _ { \jmath \in J ( S _ \iota ) } S _ { \iota , \jmath } , \end{align*}


\begin{align*} \beta _ 4 ( \hat V ) & = \beta _ 4 ( V ) \qquad \\ \beta _ 2 ( \hat V ) & = \beta _ 2 ( V ) + d . \end{align*}


\begin{align*} w _ n ( y , s ) = \frac { 1 } { M _ n } \ , u _ n \left ( \bar x _ n + r _ n y , \bar t _ n + r _ n ^ 2 s \right ) , ( y , s ) \in \frac { \Omega - \bar x _ n } { r _ n } \times \left ( - \frac { \bar t _ n } { r _ n ^ 2 } , \frac { T - \bar t _ n } { r _ n ^ 2 } \right ) = Q _ n . \end{align*}


\begin{align*} \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 3 , v _ j \rangle = 0 = \sum \limits _ { j = 1 } ^ m e ^ { i \theta _ j } \langle T _ { \theta _ j } e _ 4 , v _ j \rangle . \end{align*}


\begin{align*} \partial _ t n - \alpha \Delta n = - n \sum _ { i = 1 } ^ { \ell } \beta _ i \rho _ i , \end{align*}


\begin{align*} D _ { \mu , n } ( f ) = \frac { 1 } { n ! ( n - 1 ) ! } \displaystyle \int _ { \mathbb D } \big | f ^ { ( n ) } ( z ) \big | ^ 2 P _ { \ ! \mu } ( z ) ( 1 - | z | ^ 2 ) ^ { n - 1 } d A ( z ) = \int _ { \mathbb \mathbb T } D _ { \lambda , n } ( f ) d \mu ( \lambda ) . \end{align*}


\begin{align*} 0 = c _ 0 ( Y , \lambda ) < c _ 1 ( Y , \lambda ) \le c _ 2 ( Y , \lambda ) \le \cdots \le + \infty . \end{align*}


\begin{align*} \left \langle \partial \mathbb { A } \left ( I + \mathbb { A } \right ) ^ { - 1 } a , \left ( I + \mathbb { A } \right ) ^ { - 1 } a \right \rangle & = - \left \langle a , \left ( I + \mathbb { A } \right ) ^ { - 1 } a \right \rangle \left \langle a , \left ( I + \mathbb { A } \right ) ^ { - 1 } a \right \rangle \\ & = - \left \langle a , \left ( I + \mathbb { A } \right ) ^ { - 1 } a \right \rangle ^ { 2 } . \end{align*}


\begin{align*} l ( Z _ \jmath ) = \sum _ j \nu _ j ' . \end{align*}


\begin{align*} X = \frac { u + C v + B } { \gamma } . \end{align*}


\begin{align*} S ( \overline \nabla u ) ^ 2 : = \sum _ { i = 1 } ^ d S ( \partial _ { x _ i } u ) ^ 2 + \sum _ { d < i , j \leq n } S ( \partial _ { \varphi _ { i j } } u ) ^ 2 + S ( \partial _ r u ) ^ 2 , \end{align*}


\begin{align*} ( x | x ) \phi ( x ) = x . \end{align*}


\begin{align*} \frac { \tilde q } { q - \tilde q } = \left ( \frac { q } { p } + \frac { n } { p - n } \right ) ^ { - 1 } , \quad \tilde q = \frac { p q ( p - n ) } { q ( p - n ) + p ^ 2 } \ , . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ 4 \alpha _ i \ , \Pi ^ { - 1 } S _ i . \end{align*}


\begin{align*} u _ + ( x ) : = \frac { 1 } { \sqrt 2 } \left ( \left ( \Delta _ { g } ^ 2 + m \right ) ^ { 1 / 4 } \varphi + i \left ( \Delta _ { g } ^ 2 + m \right ) ^ { - 1 / 4 } \psi \right ) \ , \\ u _ - ( x ) : = \frac { 1 } { \sqrt 2 } \left ( \left ( \Delta _ { g } ^ 2 + m \right ) ^ { 1 / 4 } \varphi - i \left ( \Delta _ { g } ^ 2 + m \right ) ^ { - 1 / 4 } \psi \right ) \ , \end{align*}


\begin{align*} \chi _ { s , N } ( \boldsymbol { x } ) = \begin{cases} 1 \quad & \| \boldsymbol { x } \| _ \infty \leq s / N ^ { 1 / d } , \\ 0 \quad & \end{cases} \end{align*}


\begin{align*} \left \{ \begin{array} { l } \dot { x } = a ( t ) x , \textnormal { f o r $ t \neq \tau _ { k } $ } \\ \\ \Delta ^ { + } x ( \tau _ { k } ) = b _ { k } x ( \tau _ { k } ) , \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} \lim _ { n \to \infty } \int _ { 0 } ^ { \pi } | p ( R _ { n } , \theta ) - \bar { p } ( R _ { n } ) | ^ { 2 } d \theta = 0 . \end{aligned} \end{align*}


\begin{align*} c _ k ( Y , \lambda ) = \inf _ { f : Y \to \R ^ { > 0 } } c _ k \left ( Y , e ^ f \lambda \right ) = \sup _ { f : Y \to \R ^ { < 0 } } c _ k \left ( Y , e ^ f \lambda \right ) , \end{align*}


\begin{align*} \| M \| : = \sup \left \{ | M v | : v \in \R ^ n , \ | v | = 1 \right \} , \end{align*}


\begin{align*} \| ( H _ { q } [ \omega , 0 ] \upharpoonright \Lambda ' - E _ { i , \Lambda } ( \omega ) ) \widetilde \psi ' \| & \leq \| ( H _ { q } [ \omega , 0 ] \upharpoonright \Lambda ' - E ' ) \widetilde \psi ' \| + | E ' - E _ { i , \Lambda } ( \omega ) | \| \widetilde { \psi } ' \| \\ & < 4 \delta _ { 0 } + | E ' - E | + | E - E _ { i , \Lambda } ( \omega ) | < 9 \delta _ { 0 } . \end{align*}


\begin{align*} \langle g _ 1 ( x ) , g _ 2 ( x ) \rangle _ { \mathrm { M S } } = \mathrm { M S } ( g _ 1 ) \star \mathrm { M S } ( g _ 2 ) , g _ 1 ( x ) , g _ 2 ( x ) \in R _ F , \end{align*}


\begin{align*} \log \mathbb P \left \{ \sup _ { | z | = R _ k } \log | W _ 0 ( z ) | \ge \frac { R _ k ^ 2 } 2 - \frac { 6 a } { 1 6 } R _ k \log ^ { b } R _ k \right \} \lesssim - \log ^ { 2 ( b - 1 ) } R _ k . \end{align*}


\begin{align*} \tau _ y F ( x ) = F ( x - y ) . \end{align*}


\begin{align*} x ^ 2 + y ^ 2 + z ^ 2 = \alpha ^ 2 + \beta ^ 2 + \gamma ^ 2 , \end{align*}


\begin{align*} \beta _ { c } ( I ) & = \sum _ { w \in \mathcal { L } _ t ^ f ( u ) } \binom { n - \min ( w ) - d + 1 } { n - d - 1 } - \sum _ { \substack { w \in \mathcal { L } _ t ^ f ( v ) \\ w \ne v } } \binom { \max ( w ) - d } { n - d - 1 } \\ & = ( d - \ell ) \binom { a } { a - 1 } + m \binom { a - 1 } { a - 1 } - ( d - \ell ) \binom { a } { a - 1 } = m > 0 . \end{align*}


\begin{align*} f \circ \rho _ M ( R ) ( g ) = \rho _ { P } ( R ) ( g ) \circ f : M \rightarrow P . \end{align*}


\begin{align*} \left | \frac { d E _ { \varkappa } } { d \varkappa } \right | = \left | \left \langle \frac { d H _ { q } [ \varkappa ] } { d \varkappa } \psi _ { \varkappa } , \psi _ { \varkappa } \right \rangle \right | \leq \sum _ { x \in \Z _ { q } } ( | \psi _ { \varkappa } ( x - \overline { 1 } _ { q } ) | + | \psi _ { \varkappa } ( x + \overline { 1 } _ { q } ) | ) | \psi _ { \varkappa } ( x ) | \leq 2 \end{align*}


\begin{align*} \mathbb { X } ^ { N } _ t = 1 + \int _ 0 ^ t \mathbb { X } ^ { N } _ { s ^ - } \otimes d \mathbf { X } _ s + \sum _ { 0 < s \leq t } \mathbb { X } ^ { N } _ { s ^ - } \otimes ( \exp ^ { ( N ) } ( \Delta X _ s ) - 1 - \Delta X _ s - \frac 1 2 ( \Delta X _ s ) ^ { \otimes 2 } ) , \end{align*}


\begin{align*} \lambda _ 0 = \frac { 1 } { 2 } \sum _ { i = 1 } ^ 2 \left ( x _ i \ , d y _ i - y _ i \ , d x _ i \right ) \end{align*}


\begin{align*} I _ h z _ R ( t ) = & I _ h z _ 0 + \int _ 0 ^ t i \Delta I _ h z _ R ( s ) - i \nu I _ h z _ R ( s ) - \epsilon I _ h ( \gamma z _ R ( s ) - \mu \overline { z _ R } ( s ) ) \d s \\ & - \tfrac { 1 } { 2 } \int _ 0 ^ t I _ h ( z _ R ( s ) F _ \Phi ) \d s + i \kappa \int _ 0 ^ t \theta _ R ( | z _ R | _ { X _ s } ) I _ h ( | z _ R ( s ) | ^ 2 z _ R ( s ) ) \d s \\ & - i \int _ 0 ^ t I _ h ( z _ R ( s ) \d W ( s ) ) . \end{align*}


\begin{align*} X _ \Omega = \mu ^ { - 1 } ( \Omega ) . \end{align*}


\begin{align*} h ^ { { \rm e m p } _ { N } ^ { \epsilon } - \rho ^ { \epsilon } } = \int _ { K _ { j } } g ^ { \epsilon } ( x - y ) d ( { \rm e m p } _ { N } - \rho ) ( y ) + \sum _ { i \neq j } \int _ { K _ { i } } g ^ { \epsilon } ( x - y ) d ( { \rm e m p } _ { N } - \rho ) ( y ) , \end{align*}


\begin{align*} F _ { x + \sigma } ( t ) & = - t \arctan \left ( \tfrac { t } { x + \sigma } \right ) + \tfrac 1 2 ( x + \sigma ) \log \left ( 1 + \tfrac { t ^ 2 } { ( x + \sigma ) ^ 2 } \right ) \\ & = - \frac { t ^ 2 } { 2 x } \left ( 1 + O ( x ^ { - 1 / 2 } + t ^ 2 x ^ { - 2 } ) \right ) . \end{align*}


\begin{align*} a _ 1 ( k ) = \frac { k - i \frac { A } { 2 } } { k } , a _ 2 ( k ) = \frac { k } { k - \frac { i A } { 2 } } , \end{align*}


\begin{align*} \Theta ^ \mathrm { p a i r } [ n + 1 ] ( x , y ) : = \frac { 1 } { N } \upsilon _ N ( x - y ) \Big ( \frac { k _ { n + 1 } } { \delta - k _ { n + 1 } \circ \overline { k _ { n + 1 } } } \Big ) ( x , y ) . \end{align*}


\begin{align*} C = T ^ { - 1 } ( B [ \underline { y } ] ) = T ^ { - 1 } ( ( S ^ { - 1 } ( A [ \underline { x } ] ) ) [ \underline y ] ) . \end{align*}


\begin{align*} \mu ' ( n , n - 1 ) = \begin{cases} \mu ( n - 1 , n ) + \mu ( q , n ) & W _ { n - 1 } \geq W _ q \ ; , \\ 0 & \ ; , \end{cases} \end{align*}


\begin{align*} \tilde { \nabla } _ E T _ { \theta } e _ 6 = - \omega _ { 5 6 } ( E ) T _ { \theta } e _ 5 + \frac { i \mu _ 2 } { \kappa _ 1 } \left ( T _ { \theta } e _ 3 + i T _ { \theta } e _ 4 \right ) , \end{align*}


\begin{align*} \Omega _ \varepsilon ( t ) = \{ y + r \nu ( y , t ) \mid y \in \Gamma ( t ) , \ , \varepsilon g _ 0 ( y , t ) < r < \varepsilon g _ 1 ( y , t ) \} , t \in [ 0 , T ] \end{align*}


\begin{align*} & S ( k ) = \begin{pmatrix} a _ 1 ( k ) & - \sigma b ( k ) \\ b ( k ) & a _ { 2 } ( k ) \end{pmatrix} , k \in \mathbb { R } \backslash \{ 0 \} . \end{align*}


\begin{align*} D _ i & = \nabla _ { \psi _ { i + 1 } } \psi _ { i + 1 } - \nabla _ { \psi _ i } \psi _ i & \tilde { D } _ i & = \nabla _ { f _ { i + 1 } } f _ { i + 1 } - \nabla _ { f _ i } f _ i . \end{align*}


\begin{align*} \partial _ t f = \alpha \partial _ x ^ 2 f + \beta ( \partial _ x f ) ^ 2 + \gamma \xi , \end{align*}


\begin{align*} H _ { k , \beta } ' = \frac { 1 } { s P _ { k - 1 } L _ k ^ \beta } \bigg ( - \Big ( \sum _ { j = 1 } ^ { k - 1 } \frac 1 { P _ j } + \frac { \beta } { P _ k } \Big ) \Big ( 1 - \sum _ { j = 1 } ^ { k - 1 } \frac 1 { P _ j } - \frac { \beta } { P _ k } \Big ) + \sum _ { j = 1 } ^ { k - 1 } \frac 1 { P _ j } \Big ( \sum _ { i = 1 } ^ j \frac 1 { P _ i } + \frac { \beta } { P _ k } \Big ) \bigg ) . \end{align*}


\begin{align*} \lambda _ U ^ \eta ( \sigma _ v = \cdot , v \in \overline U ) : = \lambda ( \sigma = \cdot \mid \sigma = \eta ) , \end{align*}


\begin{align*} \Theta = \{ \theta _ 1 , \theta _ 2 , \ldots , \theta _ n \} \end{align*}


\begin{align*} f ( A _ 2 , \theta _ 2 , \omega _ 2 , S ) = d ^ { - \deg ( w ' + w '' ) / 2 } \cdot f ( A _ 1 , \theta _ 1 , \omega _ 1 , S ) . \end{align*}


\begin{align*} \lambda _ j = - 2 m + p + j ( j + m - 1 ) , \end{align*}


\begin{align*} \Lambda ( \omega , 1 ) = \Lambda ( T \omega , 0 ) \end{align*}


\begin{align*} \lim _ { \tau \to 0 } \sup _ { C \in { \rm C o n f i g } ( K _ { i } ) } d _ { \rm C o n f i g } ( C , \mathcal { R } C ) = 0 . \end{align*}


\begin{align*} ( h \circ g ) \circ f = h \circ ( g \circ f ) \ ; . \end{align*}


\begin{align*} A = \bigoplus _ { i = 1 } ^ t A _ i \end{align*}


\begin{align*} \mathbf { \Pi } _ { N } \left ( \overline { \mathbf { F } } _ { N } ( C ) \in B ( \overline { \mathbf { P } } , \delta ) \right ) \geq \mathbf { \Pi } _ { N } \left ( \frac { | C | } { N } \in ( 1 - \epsilon , 1 + \epsilon ) \right ) \min _ { \frac { j } { N } \in ( 1 - \epsilon , 1 + \epsilon ) } \mathbf { \Pi } _ { N } \left ( \overline { \mathbf { F } } _ { N } ( C ) \in B ( \overline { \mathbf { P } } , \delta ) \Big | | C | = j \right ) . \end{align*}


\begin{align*} & \sup _ { v _ 1 , v _ 2 \in \R ^ d } \prod _ { j = 1 } ^ d \left \{ \langle v _ { 1 , j } \rangle ^ { - 1 + \epsilon } \langle v _ { 2 , j } \rangle ^ { - 1 + \epsilon } \right \} \left | \nu ( q + v _ 1 ) - E \pm i \eta \right | ^ { - 1 } \left | \nu ( q + \sigma v _ 1 + v _ 2 ) - E \pm i \eta \right | ^ { - 1 } \\ & \leq K _ { d , \epsilon , E } ( 1 + \eta ^ { - 2 } ) \prod _ { j = 1 } ^ d \langle q _ j \rangle ^ { - 1 + \epsilon } , \end{align*}


\begin{align*} V : = \frac { 1 } { 2 \Lambda } \left \{ \left ( \varphi _ s - \frac { 2 } { r } \phi _ r \right ) \phi _ { s z } - \left ( \varphi _ z - 2 \phi _ { x ^ 0 } \right ) \phi _ { s s } \right \} . \end{align*}


\begin{align*} d X ( t ) = \frac 1 2 \sigma ' ( X ( t ) ) \sigma ( X ( t ) ) d t + \sigma ( X ( t ) ) d \beta ( t ) . \end{align*}


\begin{align*} & - r _ 1 ( k ) \delta ^ { - 2 } ( k , \xi ) e ^ { 2 i t \theta } = \frac { 2 i } { A \delta ^ 2 ( 0 , \xi ) } k + O ( k ^ 2 ) , & U _ 2 \ni k \rightarrow 0 , \\ & r _ 2 ( k ) \delta ^ { 2 } ( k , \xi ) e ^ { - 2 i t \theta } = \frac { A \delta ^ 2 ( 0 , \xi ) } { 2 i k } + O ( 1 ) , & U ^ * _ 2 \ni k \rightarrow 0 . \end{align*}


\begin{align*} H ( x , \xi _ 0 , \xi ) = \frac { 1 } { 2 } \sum _ { j , k = 0 } ^ n g ^ { j k } ( x ) \xi _ j \xi _ k \end{align*}


\begin{align*} \psi ( \varpi ) = \varpi . \end{align*}


\begin{align*} x _ n = f ^ n ( x ) \ : \ : \ : \ : \textrm { f o r } \ : \ : \ : n \in \Z . \end{align*}


\begin{align*} 0 = \widehat { Q } _ 0 ^ 1 + Q '^ 1 _ 1 \circ F ^ 1 - F ^ 1 \circ Q + \sum _ { n = 2 } ^ \infty \frac { 1 } { n ! } Q '^ 1 _ n \circ ( F ^ 1 ) ^ { \star n } . \end{align*}


\begin{align*} f = \sum _ { j = 1 } ^ \infty \lambda _ j a _ j , \end{align*}


\begin{align*} ( u , v ) = \langle u , v \rangle , \ \ \ \forall \ , u \in H , \ \ \forall \ , v \in V . \end{align*}


\begin{align*} [ w ( \tau _ 0 ) ] ^ { - 1 } \cdot \left | \frac { \partial ^ { n - m } } { \partial \upsilon _ j ^ { n - m } } w ( \upsilon + \tau _ 0 ) \right | \leq D _ { n - m } \cdot [ v _ 0 ( \upsilon ) ] ^ { d } \leq \left ( \max _ { 0 \leq m \leq n } D _ { n - m } \right ) \cdot [ v _ 0 ( \upsilon ) ] ^ d = : \widetilde { D } _ { n } \cdot [ v _ 0 ( \upsilon ) ] ^ d , \end{align*}


\begin{align*} \begin{array} { l l } \theta \left ( \alpha ( x ) , [ y , z ] \right ) + \theta \left ( \alpha ( y ) , [ x , z ] \right ) + \theta \left ( \alpha ( z ) , [ x , y ] \right ) \\ + \rho ( \alpha ( x ) ) \theta ( y , z ) + \rho ( \alpha ( y ) ) \theta ( x , z ) + \rho ( \alpha ( z ) ) \theta ( x , y ) = 0 \end{array} \end{align*}


\begin{align*} \div ( f A ) = f \div A + A \nabla f . \end{align*}


\begin{align*} \tilde { \nabla } _ E T _ { \theta } e _ 4 = - \omega _ { 3 4 } ( E ) T _ { \theta } e _ 3 + \frac { i \kappa _ 2 } { \kappa _ 1 } T _ { \theta } e _ 5 + \frac { \mu _ 2 } { \kappa _ 1 } T _ { \theta } e _ 6 + i \kappa _ 1 e ^ { i \theta } d g _ { \theta } ( \overline { E } ) , \end{align*}


\begin{align*} \frac { 2 } { \gamma } + \frac { d } { p } = 1 \end{align*}


\begin{align*} \frac { 1 } { \alpha } + \frac { 1 } { \beta } = 1 \end{align*}


\begin{align*} \Delta _ k p _ k ( r ) = 0 . \end{align*}


\begin{align*} ( A \otimes B ) ( \vec x ) = A ( \vec x \circ \Phi _ { + 0 } ^ { a , b } ) B ( \vec x \circ \Phi _ { + 1 } ^ { a , b } ) \ ; , \end{align*}


\begin{align*} c ' + m ' d ' \geq c ' = d \geq b = a ' . \end{align*}


\begin{align*} | \Gamma _ 1 ^ h ( u ) | + | \Gamma _ 2 ^ h ( u ) | = & | \langle ( I _ h - I ) z , I _ h ( z u ) \rangle _ { H _ x ^ \mathfrak { s } } | + | \langle z , ( I _ h - I ) ( z u ) \rangle _ { H _ x ^ \mathfrak { s } } | \\ = & | \langle I _ h ( I _ h - I ) z , z u \rangle _ { H _ x ^ \mathfrak { s } } | + | \langle ( I _ h - I ) z , z u \rangle _ { H _ x ^ \mathfrak { s } } | , \end{align*}


\begin{align*} \kappa _ 2 = \frac { \kappa _ 1 } { \lambda + 1 } \ \mu _ 2 = \frac { \lambda \kappa _ 1 } { \lambda + 1 } . \end{align*}


\begin{align*} L ^ 2 _ a ( \Gamma \backslash G / K ) = \{ f \in L ^ 2 ( \Gamma \backslash G / K ) : c _ P f ( g ) = 0 \eta ( g ) \geq a \} , \end{align*}


\begin{align*} & d ^ { 3 } _ { r } B _ { \mathfrak a } ( f \wedge g ) ( x , y , z , t ) = B \left ( d ^ 2 f ( x , y , z ) , g ( t ) \right ) \\ + & B \left ( d ^ 2 _ { c } f ( x , y , t ) , g ( z ) \right ) + B \left ( d ^ 2 _ { c } f ( x , z , t ) , g ( y ) \right ) + B \left ( d ^ 2 _ { c } f ( y , z , t ) , g ( x ) \right ) \\ + & B \left ( ( f \circ \alpha ) \wedge d ^ 1 g \right ) ( x , y , z , a ) \end{align*}


\begin{align*} x \to y = \max \{ u \colon L ( [ x ) \cap [ y ) ) \cap ( u ] = ( y ] \} . \end{align*}


\begin{align*} \mathcal { M } _ { \ell } : = \{ \pi \in S _ n : \sum _ { j = 1 } ^ n | x _ j - y _ { \pi ( j ) } | _ { \infty } = \ell \} . \end{align*}


\begin{align*} ( \mathcal H _ { \mu , n } ) = \{ \varphi \in \mathcal O ( \mathbb D ) : \varphi f \in \mathcal H _ { \mu , n } \ , \ , \ , f \in \mathcal H _ { \mu , n } \} . \end{align*}


\begin{align*} \Lambda _ { n } : = \frac { 1 } { \sum _ { k = 0 } ^ n \left | h _ { k } \right | ^ 2 } . \end{align*}


\begin{align*} \langle \Omega | \prod _ { j = 1 } ^ N \left ( S _ + ^ { [ j ] } \right ) ^ { k _ j } \left ( S _ - ^ { [ j ] } \right ) ^ { k _ j } | \Omega \rangle = \prod _ { j = 1 } ^ N \frac { ( 2 s ) ! k _ j ! } { ( 2 s - k _ j ) ! } \end{align*}


\begin{align*} x = a _ 1 ^ { \varepsilon _ 1 } a _ 2 ^ { \varepsilon _ 2 } \cdots a _ s ^ { \varepsilon _ s } b ^ m \ \ \ \ y = a _ 1 ^ { \theta _ 1 } a _ 2 ^ { \theta _ 2 } \cdots a _ s ^ { \theta _ s } b ^ n \end{align*}


\begin{align*} \eta _ a ( v _ w ) = \frac { a ^ w + c _ w a ^ { 1 - w } } { 1 - 2 w } \cdot a ^ { 1 - w } . \end{align*}


\begin{align*} \Sigma _ { L , M } = \underbrace { \Sigma _ { 0 , M } \times _ { Z _ M } \dots \times _ { Z _ M } \Sigma _ { 0 , M } } _ { ( L + 1 ) \times } . \end{align*}


\begin{align*} \partial _ t u + A ( x , t ) \partial _ x u + B ( x , t ) u = 0 , 0 < x < 1 , t > 0 , \end{align*}


\begin{align*} f _ * ( f ^ * y \cdot \mu ) = y \cdot f _ * \mu . \end{align*}


\begin{align*} \mathcal { L } ( y ) \mathcal { R } ( x ) L ( x - y ) = L ( x - y ) \mathcal { R } ( x ) \mathcal { L } ( y ) \ , , \end{align*}


\begin{align*} \frac { \partial \hat { z } _ i } { \partial \hat { t } } = \left ( \frac { \partial \hat { t } } { \partial \hat { z } _ i } \right ) ^ { - 1 } = \frac { \partial z } { \partial t } \in \left [ \frac { 3 } { 4 } , \frac { 4 } { 3 } \right ] . \end{align*}


\begin{align*} M _ 0 : = \dfrac { 8 ( \gamma - 1 ) } { c ^ 2 } M , ~ M _ 1 : = 4 M . \end{align*}


\begin{align*} I _ A : = \{ 1 , . . . , 2 n \} \setminus J _ A . \end{align*}


\begin{align*} D _ { \lambda , n } ( g ) = D _ { \sigma , n - 1 } \Big ( \frac { g ( z ) - g ( \lambda ) } { z - \lambda } \Big ) . \end{align*}


\begin{align*} \pm k _ { 0 } = \pm i \sqrt { \frac { x } { 1 2 t } } . \end{align*}


\begin{align*} \begin{cases} \dot { x } ( t ) = 0 , & t \geq 0 , \\ x _ 0 = \varphi , & \varphi \in X , \end{cases} \end{align*}


\begin{align*} \left \| \sigma ^ { k * } f \right \| _ 2 ^ 2 & \leq \int _ { \R ^ d } \left ( \int _ { \R ^ d } \sigma ^ { k * } ( x - y ) d y \right ) \left ( \int _ { \R ^ d } \sigma ^ { k * } ( x - y ) | f ( y ) | ^ 2 d y \right ) d x \\ & = | \sigma | ^ k \int _ { \R ^ d } \left ( \int _ { \R ^ d } \sigma ^ { k * } ( x - y ) d x \right ) | f ( y ) | ^ 2 d y = | \sigma | ^ { 2 k } \left \| f \right \| _ 2 ^ 2 . \end{align*}


\begin{align*} e ^ { i t \Delta } u _ 0 ^ \sigma ( x ) = [ e ^ { i \sigma ^ { - 2 } t \Delta } u _ 0 ^ 1 ] ( \sigma ^ { - 1 } x ) . \end{align*}


\begin{align*} \mathcal B = ( \Omega , { \mathcal F } , ( { \mathcal F } _ t ) _ { t \in { T } } , P ) \end{align*}


\begin{align*} J ( v ) : = \iint _ \Omega | \nabla \Delta ^ h _ i v | ^ 2 \frac { d t } { | t | ^ { n - d - 1 } } d x . \end{align*}


\begin{align*} u \geq v \geq C x _ n ~ ~ ~ ~ \mbox { o n } ~ ~ ~ ~ \{ x ' = 0 , 0 < x _ n < r / 2 , t = 0 \} , \end{align*}


\begin{align*} \dot x & = A x + B u , \\ y & = C x + D u , \end{align*}


\begin{align*} \mu \{ U _ 1 ^ { p _ 1 } \circ \ldots \circ U _ n ^ { p _ n } ( x ) = x \} = 0 . \end{align*}


\begin{align*} f ( \alpha , t ) = \frac { 1 } { \alpha - 1 } \left [ C ( t ) - \frac { 1 } { 2 t } \left ( \frac { 1 } { ( 1 + t ) ^ { 2 \alpha - 2 } } - \frac { 1 } { ( 1 - t ) ^ { 2 \alpha - 2 } } \right ) \right ] \ , \end{align*}


\begin{align*} \int g ( \upsilon ) f ( \upsilon ) d \upsilon = \int \tilde { g } ( \upsilon ) \frac { \partial } { \partial \upsilon _ j } f ( \upsilon ) d \upsilon = - \int \frac { \partial } { \partial \upsilon _ j } \tilde { g } ( \upsilon ) f ( \upsilon ) d \upsilon , \end{align*}


\begin{align*} R ( X _ 1 , \ldots , X _ s ) = \frac { P ( X _ 1 , \ldots , X _ s ) } { Q _ 1 ( X _ 1 ) \cdots Q _ s ( X _ s ) } , \end{align*}


\begin{align*} E _ { \infty } : = \inf \sigma ( H _ { \infty } ) , \end{align*}


\begin{align*} d ^ 2 \theta ( x , y , z ) & = 0 \\ d _ { r } ^ { 3 } \gamma ( x , y , z , \alpha ( a ) ) + \frac { 1 } { 2 } B _ { \mathfrak a } \left ( \theta \wedge ( \theta \circ \alpha ) \right ) ( x , y , z , a ) & = 0 \end{align*}


\begin{align*} U _ { \delta } : = \{ x \in U \colon d ( x , X \setminus U ) > \delta \} \quad \textrm { a n d } U ( \delta ) : = \{ x \in X \colon d ( x , U ) < \delta \} . \end{align*}


\begin{align*} 0 \in H ( x , y ) : = f ( x , y ) + N _ D \big ( g ( x , y ) \big ) , \end{align*}


\begin{align*} \langle f , g \rangle : = \sum _ { x \in \mathbb { T } _ { m _ 1 , m _ 2 } } f ( x ) \overline { g ( x ) } \end{align*}


\begin{align*} \hat { Z } _ i & = r _ i ^ 2 Z & \hat { \partial } _ i & = A r _ i \nabla _ { f _ i } . \end{align*}


\begin{align*} D ( \lambda , T ) : = \frac { 1 } { N ( T ) } \sum _ { \substack { 0 < \gamma \le T \\ \gamma ^ + - \gamma \le \frac { 2 \pi \lambda } { \log T } } } \ ! \ ! \ ! \ ! \ ! 1 D _ d ( \lambda , T ) : = \frac { 1 } { N ( T ) } \sum _ { \substack { 0 < \gamma _ d \le T \\ \gamma _ d ^ + - \gamma _ d \le \frac { 2 \pi \lambda } { \log T } } } \ ! \ ! \ ! \ ! \ ! 1 . \end{align*}


\begin{align*} \partial _ x ( u \times ( u \times g ( u ) ) ) \cdot \partial _ x u & = u \times ( \partial _ x u \times g ( u ) ) \cdot \partial _ x u + u \times ( u \times A \partial _ x u ) \cdot \partial _ x u \\ & = [ \partial _ x u ( u \cdot g ( u ) ) - g ( u ) ( \partial _ x u \cdot u ) ] \cdot \partial _ x u + u \times ( u \times A \partial _ x u ) \cdot \partial _ x u \\ & = | \partial _ x u | ^ 2 ( u \cdot g ( u ) ) - A \partial _ x u \cdot \partial _ x u \ , , \end{align*}


\begin{align*} \widetilde { Q } _ i = X _ i + \ell \widetilde { R } _ i ( X _ 1 , \ldots , X _ n ) , \widetilde { R } _ i \in \Z _ \ell [ X _ 1 , \ldots , X _ n ] . \end{align*}


\begin{align*} M _ s : = \sum _ { r = 1 } ^ s \sum _ { z \in \Z } \Delta ( x - z , t - r ) \xi ( z , r ) \Gamma ( z , r ) , \end{align*}


\begin{align*} \mu _ { \theta } : = { \rm a r g m i n } _ { \mu \in \mathcal { P } ( M ) } \mathcal { E } _ { V } ^ { \theta } ( \mu ) . \end{align*}


\begin{align*} A \otimes B \# C : = A C B , A \otimes B \widetilde { \# } C : = B C A , m ( A \otimes B ) : = B A . \end{align*}


\begin{align*} m = p \cdot q ^ v w + r \in \Omega _ p \end{align*}


\begin{align*} : a a : ^ { \mu } _ 0 v _ \mu = \mu ^ 2 v _ \mu . \end{align*}


\begin{align*} \Phi _ k ( x , x ' ) = \sum _ { | \mu | = k } \Phi _ \mu ( x ) \Phi _ \mu ( x ' ) . \end{align*}


\begin{align*} \int _ 0 ^ { \pi } \ , \mathrm { d } \theta \ ; \sin \theta \cos ^ 2 \theta \cos ( 2 \rho y \cos \theta ) & = \frac { \sin ( 2 \rho y ) } { \rho y } + \frac { \cos ( 2 \rho y ) } { \rho ^ 2 y ^ 2 } - \frac { \sin ( 2 \rho y ) } { 2 \rho ^ 3 y ^ 3 } \\ & = \left ( \frac { - i } { 2 \rho y } + \frac { 1 } { 2 \rho ^ 2 y ^ 2 } + \frac { i } { 4 \rho ^ 3 y ^ 3 } \right ) e ^ { 2 i \rho y } + , \end{align*}


\begin{align*} f ( \Phi _ x ( t , s ) ) = \Phi ^ u _ { \Phi ^ c _ { x _ 1 } ( \lambda ^ c _ x s ) } \left ( \lambda ^ u _ x \beta _ { x _ 1 } ( \lambda ^ c _ x s ) t \right ) . \end{align*}


\begin{align*} m ( B ) = \int _ { B } \mu ( x ) \ , d x . \end{align*}


\begin{align*} \| x ( t , s _ 0 , x ( s _ 0 ) ) \| = \| U ( t , s _ 0 ) x ( s _ 0 ) \| < \dfrac { \varepsilon \delta _ 0 } { 2 } , t \geq s _ 0 + T ( \varepsilon ) . \end{align*}


\begin{align*} c ' _ { \ge \ell } ( z , \zeta ) = e _ \ell ( \zeta ) + e _ { \ell - 1 } ( \zeta ) c _ { \ge 2 } ^ z + e _ { \ell - 2 } ( \zeta ) c _ { \ge 3 } ^ z + \ldots + e _ { \ell - k + 1 } ( \zeta ) c _ { \ge k } ^ z \ / . \end{align*}


\begin{align*} \overline { N ( t ) } = \{ x \in \mathbb { R } ^ n \mid - \delta \leq d ( x , t ) \leq \delta \} \end{align*}


\begin{align*} D _ { \lambda } ( f ) : = \int _ { \mathbb T } \frac { | f ^ { * } ( \zeta ) - f ^ { * } ( \lambda ) | ^ 2 } { | \zeta - \lambda | ^ 2 } d \sigma ( \zeta ) . \end{align*}


\begin{align*} D _ { r , \epsilon } ( t , s ) : = \{ x \in D _ { r } ( t , s ) : p ( s - t , Y ( t , s , x ) ) > \epsilon \} \end{align*}


\begin{align*} \frac { \Phi _ 1 } { \Phi _ 4 } = \frac { \Phi _ 3 } { \Phi _ 2 } \end{align*}


\begin{align*} \tilde { D } ( x ) B ( y ) = \mathbf { h } ( x , y ) B ( y ) \tilde { D } ( x ) + \mathbf { k } _ { A } ( x , y ) B ( x ) A ( y ) + \mathbf { k } _ { \tilde { D } } ( x , y ) B ( x ) \tilde { D } ( y ) \ , . \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { \sum _ { r = 0 } ^ { N - 1 } \# [ \sigma ( J _ r ) \cap ( E , \infty ) ] } { \sum _ { r = 0 } ^ { N - 1 } \ell _ r } = 1 - k ( E ) . \end{align*}


\begin{align*} J _ A : = \{ \max a : a \in A \} \end{align*}


\begin{align*} E ( F _ N ; G _ N ) \ll N ^ 2 + N ^ \epsilon \displaystyle \sum _ { 1 \leq s \leq [ N ^ \epsilon ] + 1 } \sum _ { n , n _ 1 \in I _ s } \sum _ { \substack { 1 \leq n < m \leq N \\ 1 \leq n _ 1 < m _ 1 \leq N \\ f ( n ) + f ( m ) = f ( n _ 1 ) + f ( m _ 1 ) \\ g ( n ) + g ( m ) = g ( n _ 1 ) + g ( m _ 1 ) } } 1 . \end{align*}


\begin{align*} H = \bigcup _ { i , j } H _ { i , j } \subset V . \end{align*}


\begin{align*} \lim _ { t \nearrow \tau ^ * } | z | _ { X ^ \mathfrak { s } _ t } = \infty . \end{align*}


\begin{align*} X _ \infty = \Gamma _ \infty \backslash \{ x + i y \in \mathfrak { H } : y \geq a \} \approx ( \Z \backslash \R ) \times [ a , \infty ) . \end{align*}


\begin{align*} \exp \left ( \frac { \partial W } { \partial \ln X _ a } \right ) \ : = \ : 1 \end{align*}


\begin{align*} \| T _ m f ( \cdot , x ) \| _ { L ^ 2 _ \rho L ^ 2 _ x } ^ 2 & = \sum _ { k = 0 } ^ \infty \big \| m ( \tau , k ) P _ k ( \mathcal F _ \rho f ) ( \cdot , x ) \big \| _ { L ^ 2 _ \tau L ^ 2 _ x } ^ 2 \\ & \leq \| m ( \tau , k ) \| _ { L ^ \infty } \sum _ { k = 0 } ^ \infty \big \| P _ k ( \mathcal F _ \rho f ) ( \cdot , x ) \big \| _ { L ^ 2 _ \tau L ^ 2 _ x } ^ 2 \\ & \lesssim \| f \| _ { L ^ 2 ( \mathbb R ^ { d + 1 } ) } ^ 2 . \end{align*}


\begin{gather*} \theta ( x , y ) - \theta ' \left ( x , y ) \right ) = - f \left ( [ x , y ] \right ) + \rho \left ( x \right ) f ( y ) + \rho \left ( y \right ) f ( x ) . \end{gather*}


\begin{align*} ^ { \rho } I ^ { 1 - \gamma } _ { 0 _ + } u ( 0 ) = \displaystyle \frac { \Gamma ( \gamma ) } { \Gamma ( \alpha ) } \Omega \ \sum _ { i = 1 } ^ { m } \omega _ { i } \int ^ { \xi _ { i } } _ { 0 } \left ( \frac { \xi _ { i } ^ { \rho } - s ^ { \rho } } { \rho } \right ) ^ { \alpha - 1 } s ^ { \rho - 1 } \left ( f ( s , u ( s ) , ^ \rho D ^ { \alpha , \beta } u ( s ) ) - p ( s ) u ( s ) \right ) d s \end{align*}


\begin{align*} \mu ( E ) = \int _ { T ^ { - 1 } ( E ) } f ( x ) \ , d x . \end{align*}


\begin{align*} \vert \cos \theta _ { \hat x } - \cos \theta _ c \vert = \left \vert 2 \sin \left ( ( { \theta _ c - \theta _ { \hat x } } ) / 2 \right ) \sin \left ( ( { \theta _ c + \theta _ { \hat x } } ) / 2 \right ) \right \vert \le k ^ { - 1 / 2 } _ + \vert x \vert ^ { - 1 / 2 } , \end{align*}


\begin{align*} D _ { \mathcal { G } } ( T u , T w ) & = D _ { \mathcal { G } } ( ( u , F u ) , ( w , F w ) ) \\ & = \lvert u - w \rvert + H _ d ( F u , F w ) \\ D _ { \mathcal { G } } ( T u , T w ) & \geq \frac { 1 } { 2 } \lvert u - w \rvert . \end{align*}


\begin{align*} R = \sqrt { A } \end{align*}


\begin{align*} S _ { \neq \mathrm { k } } ( x ) = S ( x ) \cdot T _ 1 ( x ) , T _ 1 ( x ) = \exp \bigg \{ - \sum _ { 1 \le i \le \ell } c _ { k _ i } x ^ { k _ i } \bigg \} . \end{align*}


\begin{align*} \mathcal X ^ n = ( X ^ n _ t , p ^ { u t } _ n , T ) , \ \ \ \mbox { w h e r e } \ \ \ X ^ n = \pi _ n ( X ) , \ X ^ n _ t = p ^ { t b } ( X ^ n ) , \ p ^ { u t } _ n = p ^ { u t } | _ { X ^ n _ t } . \end{align*}


\begin{align*} T _ m \nu _ N ( \varphi _ j ) = \nu _ N ( \varphi _ j ) 1 \leq j \leq n \ , . \end{align*}


\begin{align*} R ^ + ( h ( U ) ) & = \big \{ y \in h ( S ) , R ( y ) \subset h ( U ) \big \} = \big \{ h ( x ) , \ \ x \in S \ \ R ( h ( x ) ) \subset h ( U ) \big \} \\ & = h \left ( \big \{ x \in S , \ \ R ( h ( x ) ) \subset h ( U ) \big \} \right ) . \end{align*}


\begin{align*} w ( p ) = | \lambda | ^ { n ( 1 - \alpha ) } c _ w \prod _ { j = 1 } ^ d \langle p _ j \rangle ^ { 1 + \delta } \end{align*}


\begin{align*} & W _ { s , t } a \cdot a + a \cdot W _ { s , t } a = 0 \ , , \\ & \mathbb { W } _ { s , t } a \cdot a + a \cdot \mathbb { W } _ { s , t } a + W _ { s , t } a \cdot W _ { s , t } a = - W _ { s , t } a \cdot W _ { s , t } a + W _ { s , t } a \cdot W _ { s , t } a = 0 \ , . \end{align*}


\begin{align*} \Delta ( x , t ) : = p ( x + 1 , t ) - p ( x - 1 , t ) . \end{align*}


\begin{align*} \left \lbrace \begin{aligned} d \tilde { q } ^ { \epsilon , \Delta t } ( t ) & = \frac { \tilde { p } ^ { \epsilon , \Delta t } ( t ) } { \epsilon } d t \\ d \tilde { p } ^ { \epsilon , \Delta t } ( t ) & = - \frac { \tilde { p } ^ { \epsilon , \Delta t } ( t ) } { \epsilon ^ 2 } d t + \frac { f ( \tilde { q } ^ { \epsilon , \Delta t } ( t _ n ) ) } { \epsilon } d t + \frac { \sigma ( \tilde { q } ^ { \epsilon , \Delta t } ( t _ n ) ) } { \epsilon } d \beta ( t ) , \end{aligned} \right . \end{align*}


\begin{align*} \mathcal { U } ( x ) = \left ( \begin{array} { c c } A ( x ) & B ( x ) \\ C ( x ) & D ( x ) \end{array} \right ) \ , , \end{align*}


\begin{align*} ( \tau \sigma ) ( \alpha ) ^ { a p } = \alpha ^ { q b } , \end{align*}


\begin{align*} \mbox { $ \lim _ { k \rightarrow \infty } \frac { 1 } { k } \sum _ { i = 1 } ^ k X _ i = x \mbox { a l m o s t s u r e l y } $ } \end{align*}


\begin{align*} x \sim y \Leftrightarrow \tau ( \_ \ , , x ) = \tau ( \_ \ , , y ) . \end{align*}


\begin{align*} m ^ { p c } _ { - k _ 0 } = m _ 0 ( \zeta ) \mathcal { P } \zeta ^ { - i \nu \sigma _ 3 } e ^ { \frac { i } { 4 } \zeta ^ 2 \sigma _ 3 } , \end{align*}


\begin{align*} \textrm { i f } \ : m > n , \ : \textrm { t h e n } \ : \mu ^ { c u } _ { n , x } ( A ) = \frac { \mu ^ { c u } _ { m , x } ( A ) } { \mu ^ { c u } _ { m , x } ( \xi ^ { c u } _ n ( x ) ) } , \end{align*}


\begin{align*} \min _ { \{ p _ k \} , \tau } & & \tau ^ { - 1 } & & \\ & & \frac { \tau \left ( \sum \limits _ { i = 1 , i \neq k } ^ K p _ i f _ { k , i } + n _ k \right ) } { p _ k f _ { k , k } } \leq 1 \ \ & \forall k \in \{ 1 , \cdots , K \} , \\ & & 0 \leq p _ k \leq p _ { { \max } , k } \ \ & \forall k \in \{ 1 , \cdots , K \} , \end{align*}


\begin{align*} A _ X = A _ X ^ T \ ; , \ A _ I = A _ O ^ T \ ; . \end{align*}


\begin{align*} H [ n ] ( e , \overline { e } ) - | \Theta [ n ] ( \overline { e } , \overline { e } ) | \ge c \| e \| , e \in ( \phi _ n ) _ \perp , n = 0 , 1 , 2 , \dots . \end{align*}


\begin{align*} \mathbb { E } [ \xi f ( \xi ) ] = \sum _ { k = 2 } ^ { \infty } \frac { \kappa _ k } { N ^ { k / 4 } } \mathbb { E } [ \partial ^ { k - 1 } f ( \xi ) ] , \end{align*}


\begin{align*} v ( y , s ) = \frac { u ( x , t ) - H _ { m _ 0 } ( x , t ) } { r ^ { k + 1 + \alpha } } . \end{align*}


\begin{align*} \frac { d ^ \ell } { d y ^ \ell } G ( x , y ) = G ( x , y ) \bigg ( B _ 1 ( x , y ) ^ \ell + \sum _ { 0 \le k _ 1 \le \cdots \le k _ { \ell - 1 } \atop k _ 1 + \cdots + k _ { \ell - 1 } = \ell } d _ { k _ 1 , \dots , k _ { \ell - 1 } } \prod _ { 1 \le i \le \ell - 1 } B _ { k _ i } ( x , y ) \bigg ) . \end{align*}


\begin{align*} \omega _ 1 ( t ) : = \mathrm { R e } \ , \int _ \Omega \left \{ | g _ z + t g _ { z , j } | ^ { q - 2 } \overline { ( g _ z + t g _ { z , j } ) } - | g _ z | ^ { q - 2 } \overline { g } _ z \right \} g _ { z , j } . \end{align*}


\begin{align*} \| x _ j \| _ { \mathcal { C G } _ q ^ \omega } & \ \stackrel { \eqref { n 6 } } { \gtrsim } \ \left ( \sum _ { n = 1 } ^ { \eta _ j } \frac { 1 } { n } \left ( \omega ( n ) h _ r ( k _ j ) \right ) ^ q \right ) ^ { 1 / q } \\ & \ = \ h _ r ( k _ j ) ( \widetilde { \zeta } ( \eta _ j ) ) ^ { 1 / q } , \end{align*}


\begin{align*} \frac { \prod _ { i = 1 } ^ { j } C _ i } { \omega + F _ n ( x _ 1 ) } = \frac { \omega - F _ n ( x _ 1 ) } { \prod _ { i = j + 1 } ^ { n / 2 } C _ i } , \end{align*}


\begin{align*} r ( E _ i , E _ j ) - \rho ^ \nabla _ M ( E _ i , E _ j ) & = \sum _ { l , k = 1 } ^ { 6 } g \left ( J E _ i , N ( e _ l , e _ k ) \right ) g \left ( E _ j , N ( e _ l , e _ k ) \right ) \\ & + \sum _ { l , k = 1 } ^ { 6 } g \left ( N ( E _ i , e _ k ) , e _ l \right ) g \left ( N ( E _ j , e _ k ) , J e _ l \right ) , \\ & = \sum _ { l , k = 1 } ^ { 6 } g \left ( N ( E _ i , e _ k ) , e _ l \right ) g \left ( N ( E _ j , e _ k ) , J e _ l \right ) . \end{align*}


\begin{align*} P ^ * _ { u _ i , \nu } ( A _ i ^ { c } \triangle \hat \Omega _ i ) \le \sum \limits _ { j = 1 } ^ s P ^ * _ { u _ i , \nu } ( ( A _ { i j } \times C _ { i j } ) \triangle \hat \Omega _ { i j } ^ c ) < \frac { \varepsilon } { m } , \ \ \mbox { w h e r e } \ \ A _ i ^ { c } = \bigcup \limits _ { j = 1 } ^ { r } \Omega _ { i j } ^ c . \end{align*}


\begin{align*} \binom { i + 1 } { 2 } s + ( i + 1 ) = \binom { r + i } { i } \end{align*}


\begin{align*} ( a , b ) \otimes ( c , d ) = ( a \otimes b , c \otimes d ) \ ; . \end{align*}


\begin{align*} { \bf E } _ L = { \bf E } _ M { \bf E } _ { y _ L } ^ { \otimes M } { \bf E } _ v ^ { \otimes M } \end{align*}


\begin{align*} \tilde { a } _ 1 ( k ) = a _ 1 ( k ) \frac { k ^ 2 } { ( k - i \kappa ) ( k + i ) } , \tilde { a } _ 2 ( k ) = a _ 2 ( k ) \frac { k - i \kappa } { k - i } . \end{align*}


\begin{align*} d _ 1 ( \underline { x } , \underline { x } ' ) = \sum _ { i = 1 } ^ { n } | x _ i - x ' _ i | . \end{align*}


\begin{align*} \int _ \beta \omega _ 1 \cdots \omega _ m \int _ \beta \omega _ { m + 1 } \cdots \omega _ { m + n } = \sum _ \sigma \int _ \beta \omega _ { \sigma ( 1 ) } \cdots \omega _ { \sigma ( m + n ) } , \end{align*}


\begin{align*} \frac { H ( t , x ) } { \phi _ t ( x ) } = \frac { 1 + t ^ { \delta / \alpha } | x | ^ { - \delta } } { 1 + t ^ { \delta / \alpha } \varphi ( x ) } \leq c _ 3 , \end{align*}


\begin{align*} | R ( x ) | ^ 2 = \sum _ { \alpha = 1 } ^ { n - 1 } \{ 1 - r \kappa _ \alpha ( y ) \} ^ { - 2 } + 1 , | I _ n - R ( x ) | ^ 2 = \sum _ { \alpha = 1 } ^ { n - 1 } \left ( \frac { r \kappa _ \alpha ( y ) } { 1 - r \kappa _ \alpha ( y ) } \right ) ^ 2 . \end{align*}


\begin{align*} \begin{cases} \begin{aligned} \dot { x } & = - \nabla f ( x ) , t > 0 \\ x ( 0 ) & = X _ 0 . \end{aligned} \end{cases} \end{align*}


\begin{align*} 2 z \cdot [ x , y ] = [ z \cdot x , y ] + ( - 1 ) ^ { | x | | z | } [ x , z \cdot y ] , \ x , y , z \in { \mathfrak L } _ 0 \cup { \mathfrak L } _ 1 . \end{align*}


\begin{align*} \widehat { h } : = \frac { K _ p ( \cdot , z ) - K _ p ( \cdot , z ' ) } { | z - z ' | ^ { 1 / 2 } } \end{align*}


\begin{align*} \widehat L ( s ) = L ( s ) + \widehat L . \end{align*}


\begin{align*} L ^ 2 _ a = \{ f \in L ^ 2 ( \Gamma \backslash G / K ) : g \in Y _ a , c _ P f ( g ) = 0 , P \} , \end{align*}


\begin{align*} 0 = \textstyle \left [ v ^ \sharp , \sum _ i a _ i X _ i \right ] = \sum _ i a _ i \left [ v ^ \sharp , X _ i \right ] + \sum _ i v ^ \sharp \left ( a _ i \right ) X _ i = \sum _ i v ^ \sharp \left ( a _ i \right ) X _ i , \end{align*}


\begin{align*} [ K ( P ) : K ] \gg _ A [ K ' ( P ) : K ' ] = \prod _ { \ell | n } [ K ' ( P _ \ell ) : K ' ] . \end{align*}


\begin{align*} & \lvert u - w \rvert + H _ d \big ( A + F ^ { \alpha } ( w ) , B + F ^ { \alpha } ( u ) \big ) = 0 \\ ~ & \lvert u - w \rvert = 0 H _ d \big ( A + F ^ { \alpha } ( w ) , B + F ^ { \alpha } ( u ) \big ) = 0 \\ ~ & u = w ~ H _ d \big ( A + F ^ { \alpha } ( w ) , B + F ^ { \alpha } ( u ) \big ) = H _ d ( A , B ) = 0 \\ ~ & u = w A = B \\ ~ & ( u , A ) = ( w , B ) . \end{align*}


\begin{align*} \sum _ { | j | \leq m ^ { 1 / 4 } } c _ j = \sum _ { j \in \mathbb { Z } } c _ j + O ( m ^ { - 1 / 4 } ) , \end{align*}


\begin{align*} \begin{aligned} & \| u \| _ { L ^ { \infty } ( \Omega _ 1 ) } \leq 1 ; \\ & \| f \| _ { C ^ { - 1 , \alpha } ( 0 ) } \leq \delta ; \\ & | g ( x , t ) | \leq \frac { \delta } { 2 } | ( x , t ) | ^ { 1 + \alpha } , ~ ~ ~ ~ \forall ~ ~ ( x , t ) \in ( \partial \Omega ) _ 1 ; \\ & \| ( \partial \Omega ) _ 1 \| _ { C ^ { 1 , \alpha } ( 0 ) } \leq \frac { \delta } { 2 C _ 0 } , \end{aligned} \end{align*}


\begin{gather*} 0 = D _ i Q = \sum _ { l } \omega _ { l i } \omega _ { l } , \\ 0 \geq D _ { i j } ^ 2 Q = \sum _ { l } \omega _ { l i } \omega _ { l j } + \sum _ { l } \omega _ { l } \omega _ { l i j } . \end{gather*}


\begin{align*} h '^ 2 ( x ) \geq \tfrac { 1 } { p - 1 } \big ( \tfrac { m - 1 } { 2 } - \mbox { m a x } \big [ ( m - 1 ) \cos ^ 2 h ( x _ a ) , ( \tfrac { m - 1 } { p - 1 } + m - 1 ) ^ { 1 - \tfrac { p } { 2 } } \epsilon \big ] \big ) = : c _ 1 \end{align*}


\begin{align*} S _ T = \bigcup _ { t \in ( 0 , T ] } \Gamma ( t ) \times \{ t \} \end{align*}


\begin{align*} f _ 1 ( P _ 1 ( Y ^ N , A ^ N ) ) \cdots f _ k ( P _ k ( Y ^ N , A ^ N ) ) = \int _ { \R ^ k } Q \left ( Y ^ N , A ^ N \right ) \ d \mu _ 1 ( y _ 1 ) \dots d \mu _ k ( y _ k ) . \end{align*}


\begin{align*} \hat { f } _ \theta ( k ) = e ^ { - d \theta / 2 } \hat { f } ( e ^ { - \theta } k ) = e ^ { - d \theta / 2 } \tilde { Q } _ { \sigma , a } ( e ^ { - \theta } k ) e ^ { - \frac { \pi } { \sigma } e ^ { - 2 \theta } | k | ^ 2 } e ^ { - e ^ { - \theta } k \cdot u } . \end{align*}


\begin{align*} H _ N ( \alpha , \beta ; q ) = \sum _ { \substack { \lambda \vdash d \\ \ell ( \lambda ) \leq N } } \frac { ( \dim \mathsf { V } ^ \lambda ) ^ 2 } { d ! } \omega _ \alpha ( \lambda ) \Omega _ { \frac { q } { N } } ^ { - 1 } ( \lambda ) \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) \omega _ \beta ( \lambda ) . \end{align*}


\begin{align*} \nu ( a b ) = \nu ( a ) + \nu ( b ) , \nu ( a + b ) \le \max \{ \nu ( a ) , \nu ( b ) \} , \nu ( c a ) = \nu ( a ) . \end{align*}


\begin{align*} \alpha ( t , x ) \oplus \alpha ( - t , x ) = \alpha ( 0 , x ) \ , . \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ s u ( x ) = v ^ p ( x ) , & x \in \Omega , \\ ( - \Delta ) ^ t v ( x ) = u ^ q ( x ) , & x \in \Omega , \\ u ( x ) = - \Delta u ( x ) = \cdots = ( - \Delta ) ^ { s _ 0 } u ( x ) = 0 , & x \in \mathbb { R } ^ n \backslash \Omega , \\ v ( x ) = - \Delta v ( x ) = \cdots = ( - \Delta ) ^ { t _ 0 } v ( x ) = 0 , & x \in \mathbb { R } ^ n \backslash \Omega , \end{cases} \end{align*}


\begin{align*} \frac { \partial } { \partial \upsilon _ j } f ( \upsilon ) = - 2 \pi i \cdot \left \langle A ^ { - T } ( \tau ) \langle x \rangle , \frac { \partial } { \partial \upsilon _ j } \Phi ^ { - 1 } ( \upsilon + \tau ) \right \rangle \cdot e _ { \tau } ( x , \upsilon ) = - 2 \pi i \cdot ( \phi _ { \tau } ( \upsilon ) \cdot x ) _ j \cdot e _ { \tau } ( x , \upsilon ) . \end{align*}


\begin{align*} \mu ( z ) = \left ( \pi | z _ 1 | ^ 2 , \pi | z _ 2 | ^ 2 \right ) . \end{align*}


\begin{align*} M \geq u ( x _ v ) & = C _ 1 \int _ \Omega \ln \frac { 1 } { | x _ v - y | } v ^ p ( y ) \mathrm { d } y - \int _ \Omega h ( x _ v , y ) v ^ p ( y ) \mathrm { d } y \\ & \geq C _ 1 \int _ \Omega \ln \frac { 1 } { | x _ v - y | } v ^ p ( y ) \mathrm { d } y - \gamma _ 2 \int _ \Omega v ^ p ( y ) \mathrm { d } y \\ & \geq C _ 1 \int _ \Omega \ln \frac { 1 } { | x _ v - y | } v ^ p ( y ) \mathrm { d } y - \gamma _ { 1 } \gamma _ 2 , \end{align*}


\begin{align*} 2 \lambda s ^ 2 + ( \beta ^ 2 - \nu ^ 2 + ( \lambda + 1 / 2 ) ^ 2 ) s + \beta ^ 2 = 0 , \end{align*}


\begin{align*} ( R _ { v _ b r _ i } E _ { i , e } ^ { - 1 } ) ^ a , \ a , b \in \mathfrak F , \ i = 1 , \ldots , l , \end{align*}


\begin{align*} 0 & = \frac { d } { d s } [ ( \Delta - \lambda _ s ) \left ( s - 1 \right ) E ^ P _ { s , \varphi } ] = ( \Delta - \lambda _ s ) \frac { d } { d s } [ \left ( s - 1 \right ) E ^ P _ { s , \varphi } ] - r ( r - 1 ) ( 2 s - 1 ) ( s - 1 ) E ^ P _ { s , \varphi } \\ & = ( \Delta - \lambda _ s ) \frac { d } { d s } [ ( s - 1 ) E ^ P _ { s , \varphi } ] - r ( r - 1 ) ( 2 s - 1 ) [ R + ( s - 1 ) E _ 1 ^ * + O ( s - 1 ) ] . \end{align*}


\begin{align*} \varepsilon _ 1 = f _ 1 ( x ^ 0 ) + f _ 2 ( r ) . \end{align*}


\begin{align*} ^ \rho D ^ { \alpha , \beta } u ( t ) + p ( t ) u ( t ) = f ( t , u ( t ) , ^ \rho D ^ { \alpha , \beta } u ( t ) ) , t \in I = [ 0 , T ] \end{align*}


\begin{align*} \psi ^ { x _ 1 } _ { - \ell } ( \lambda ^ c _ x s ) = \Phi ^ c _ { x _ { - \ell + 1 } } \left ( ( \lambda ^ c _ { x _ { - \ell + 1 } } \cdots \lambda ^ c _ { x _ { - 1 } } \lambda ^ c _ x ) ^ { - 1 } \lambda ^ c _ x s \right ) = \psi ^ x _ { - \ell + 1 } ( s ) . \end{align*}


\begin{align*} \dim ( \textbf { d } ) = \dim ( \textbf { d } ) \cdot M + \dim _ { ( \textbf { d } ) } ( M ) = \dim ( \textbf { d } ) \cdot M + m . \end{align*}


\begin{align*} U _ 0 ^ \star ( s , t ) = U _ 0 ( t , s ) ^ \star = ( U _ 0 ( t , \tau ) U _ 0 ( \tau , s ) ) ^ \star = U _ 0 ( \tau , s ) ^ \star U _ 0 ( t , \tau ) ^ \star = U _ 0 ^ \star ( s , \tau ) U _ 0 ^ \star ( \tau , s ) , \end{align*}


\begin{align*} R = \mathbb F _ { p ^ r } [ x ] / \langle f ( x ^ { p ^ k } ) \rangle = \mathbb F _ { p ^ r } [ x ] / \left \langle \left ( \prod _ { i = 1 } ^ { s } g _ i ( x ) \right ) ^ { p ^ k } \right \rangle \end{align*}


\begin{align*} V _ b ( b ) = \frac { I _ F ' ( b ) \varphi _ F ( b ) - I _ F ( b ) \varphi _ F ' ( b ) } { \psi _ b ' ( b ) \varphi _ F ( b ) - \psi _ b ( b ) \varphi _ F ' ( b ) } . \end{align*}


\begin{align*} & d ^ { 2 ^ n + 1 } = 1 . \end{align*}


\begin{align*} \cot \sigma = \frac { \Vert \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) \Vert ^ 2 - \Vert \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \Vert ^ 2 } { 2 \langle \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) , \alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \rangle } . \end{align*}


\begin{align*} \partial _ { t _ 1 } \partial _ { t _ 2 } \delta \Phi ( t _ 1 = 0 , t _ 2 = 0 , x ) = 0 . \end{align*}


\begin{align*} \overline { \mathbf { F } } _ { N } ^ { i } ( C ) = \frac { 1 } { | K _ { i } | } \int _ { K _ { i } } \delta _ { \left ( x , \theta _ { N ^ { \frac { 1 } { d } } x } C \right ) } \ , d x . \end{align*}


\begin{align*} \Phi = \varphi _ 1 ( x _ 1 , \dots , x _ n ) \land y \leq z \ , \& \dots \& \ , \varphi _ m ( x _ 1 , \dots , x _ n ) \land y \leq z \Longrightarrow y \leq z \end{align*}


\begin{align*} { \rm e n t } [ \mu ] = \begin{cases} \int _ { M } \log ( { d \mu } ) \ , d { \mu } \mu \\ \infty \end{cases} \end{align*}


\begin{align*} [ \alpha _ { j - 1 } , \alpha _ j ] \subset \left ( \tau _ j - \delta ( \tau _ j ) , \tau _ j + \delta ( \tau _ j ) \right ) , \ ; \ ; j = 1 , \ldots , \nu ( P ) . \end{align*}


\begin{align*} \begin{gathered} + : K \times K \rightarrow K \ ; , \\ 1 \in K \ ; , \\ \cdot : K \times K \rightarrow K \ ; , \\ \end{gathered} \end{align*}


\begin{align*} \int _ { \mathbb { T } ^ { d } } \rho & = \int _ { \mathbb { T } ^ { d } } { \rm i n t } [ \overline { \mathbf { P } } ^ { x } ] \ , d x \\ & = 1 . \end{align*}


\begin{align*} & I ( a , b , g _ 1 , g _ 2 , Q _ 1 , Q _ 2 ) \\ & \qquad = \sum _ { T < \gamma \leq 2 T } Q _ 1 \Big ( - \frac { d } { d a } \Big ) \zeta \Big ( \rho + \frac { a } { \log T } \Big ) \ , Q _ 2 \Big ( - \frac { d } { d b } \Big ) \zeta \Big ( 1 - \rho + \frac { b } { \log T } \Big ) \ , M ( \rho , g _ 1 ) \ , M ( 1 - \rho , g _ 2 ) . \end{align*}


\begin{align*} \left [ ( S R ) ^ q z \right ] ( x , t ) = \left [ ( S R ) ^ q w \right ] ( x , t + T ) , t \ge T . \end{align*}


\begin{gather*} d ^ 2 _ { Q } ( \theta , \gamma ) = d ^ 2 _ { Q } ( \theta ' , \gamma ' ) \iff \left \{ \begin{array} { l l } \theta ' = \theta + d ^ 1 \tau \\ d _ { r } ^ 3 \gamma ' = d _ { r } ^ 3 \gamma - \frac { 1 } { 2 } d _ { r } ^ 3 B ( \tau \wedge d ^ 1 \tau ) - d _ { r } ^ 3 B ( \tau \wedge \theta ) \\ \end{array} \right . \end{gather*}


\begin{align*} x _ 0 + 2 x _ 1 + 3 x _ 2 = n | C | \end{align*}


\begin{align*} n ^ { e _ \pm } ( b , T , \mp \mu ) & = B _ { F _ { F D } ^ { ( e _ \pm ) } } \end{align*}


\begin{align*} \mu ^ s _ x \left \{ y \in \xi ^ s ( x ) : \alpha ^ s ( x , y ) = 0 \right \} = 0 . \end{align*}


\begin{align*} e _ { T , S } \cdot \langle R ^ G _ { T , \chi } \otimes \omega ^ \vee _ \psi , R ^ G _ { S , \eta } \rangle _ { G ^ F } = \sum _ { \nu , \nu ' } ( - 1 ) ^ { \sum _ j \nu _ j ' } \prod _ j { | I _ j | \choose \nu _ j } { | I _ j ' | \choose \nu _ j ' } = \begin{cases} 2 ^ r , & \textrm { i f } I _ j ' = \varnothing \textrm { f o r a l l } j , \\ 0 , & \textrm { o t h e r w i s e , } \end{cases} \end{align*}


\begin{align*} m ^ { p c } _ { - k _ 0 , + } ( \zeta ) = m ^ { p c } _ { - k _ 0 , - } ( \zeta ) J ^ { p c } ( \zeta ) , \zeta \in \Sigma ^ { p c } , \end{align*}


\begin{align*} t = \mathcal { H } ^ { - 1 } _ * ( x , \mathcal { H } _ * ( x , t ) ) \leq q ^ * ( x ) \mu ( x ) ^ { - \frac { 1 } { q ( x ) } } \left [ \mathcal { H } _ * ( x , t ) \right ] ^ { \frac { 1 } { q ^ * ( x ) } } \end{align*}


\begin{align*} \sigma _ { \sigma _ y ( x ) } = \sigma _ { \sigma _ z ( x ) } . \end{align*}


\begin{align*} \textbf { M } _ { b x } ( S c _ { * } u , S c _ { * } v ) = \textbf { M } _ { b x } ( b u , b v ) = b ^ { 2 } \textbf { M } _ { b x } ( v , u ) \end{align*}


\begin{align*} R ( s ) = ( c - 1 ) s ^ 2 + \left [ c ( \nu - \lambda + 1 ) - \nu - \lambda \right ] s + c ( \nu + \lambda ) \end{align*}


\begin{align*} \hat { \mathcal { K } } ( x ) = \left ( \begin{array} { c c } q + x & 0 \\ 0 & q - x \end{array} \right ) \ , . \end{align*}


\begin{align*} \mathbb { E } \left ( \sup _ { t \leq \tau } | z ( t ) | ^ 2 _ { L _ x ^ 2 } \right ) & = \mathbb { E } \left ( | z _ 0 | ^ 2 _ { L _ x ^ 2 } \right ) + 2 \epsilon ( \gamma + \mu ) \mathbb { E } \left ( \sup _ { t \leq \tau } \int _ 0 ^ t | z ( s ) | ^ 2 _ { L _ x ^ { 2 } } \d s \right ) \\ & \leq \mathbb { E } \left ( | z _ 0 | ^ 2 _ { L _ x ^ 2 } \right ) + 2 \epsilon ( \gamma + \mu ) \int _ 0 ^ \tau \mathbb { E } \left ( \sup _ { s \leq t } | z ( s ) | ^ 2 _ { L _ x ^ { 2 } } \right ) \d t . \end{align*}


\begin{align*} \beta _ { a , s } = \chi _ { [ a , \infty ) } ( y ) \cdot ( A _ s y ^ s + B _ s y ^ { 1 - s } - y ^ s ) . \end{align*}


\begin{align*} \mathcal { C } _ { 1 , j , k } ^ { J } = \frac { 1 + 2 x _ { 1 } } { 2 + 2 x _ { 1 } } \left [ \mathcal { A } _ { 1 , j } ^ { J \setminus k } - \frac { 1 } { 2 + 2 x _ { 1 } } \tilde { \mathcal { D } } _ { 1 , j } ^ { J \setminus k } \right ] \tilde { \mathcal { D } } _ { 1 , k } ^ { J } + \frac { 1 + 2 x _ { 1 } } { 2 + 2 x _ { 1 } } \left [ \mathcal { A } _ { 1 , k } ^ { J \setminus j } - \frac { 1 } { 2 + 2 x _ { 1 } } \tilde { \mathcal { D } } _ { 1 , k } ^ { J \setminus j } \right ] \tilde { \mathcal { D } } _ { 1 , j } ^ { J } \ , , \end{align*}


\begin{align*} \| \xi ' \| ^ 2 = \sum _ { \alpha \in \Gamma , i \in I } | a _ { \alpha , i } | ^ 2 \| \eta ' \| = \sum _ { \beta \in \Gamma , j \in I } | b _ { \beta , j } | ^ 2 . \end{align*}


\begin{align*} \left ( \begin{smallmatrix} c & 0 \\ 0 & c ' \end{smallmatrix} \right ) A = B \ \ { \rm m o d } \left ( \begin{smallmatrix} Y \\ Y ' \end{smallmatrix} \right ) \ , . \end{align*}


\begin{align*} \Phi = \lnot x \land y \leq z \ , \& \ , \lnot \lnot x \land y \leq z \Longrightarrow y \leq z \end{align*}


\begin{align*} \mathcal { H } _ * ^ { \frac { N - 1 } { N } } ( x , t ) : = \left [ \mathcal { H } _ * ( x , t ) \right ] ^ { \frac { N - 1 } { N } } ( x , t ) \in \overline { \Omega } \times [ 0 , \infty ) . \end{align*}


\begin{align*} N ' = [ n _ { u - 1 } , n _ { u - 2 } , \dots , b ] . \end{align*}


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


\begin{align*} Z _ { R _ 2 \ , \parallel \ , L } ( f ) = \frac { \textrm { j } 2 \pi f a _ { \textrm { R } } \ , R _ 2 } { c + \textrm { j } 2 \pi f a _ { \textrm { R } } } \ , [ \Omega ] . \end{align*}


\begin{align*} \left ( ^ { \rho } D _ { a ^ + } ^ { \alpha } g \right ) ( x ) = \frac { \rho ^ { \alpha - n - 1 } } { \Gamma ( n - \alpha ) } \left ( x ^ { 1 - \rho } \frac { d } { d x } \right ) ^ { n } \int _ { a } ^ { x } \ ( x ^ { \rho } - \tau ^ { \rho } ) ^ { n - \alpha + 1 } \tau ^ { \rho - 1 } g ( \tau ) d \tau , \end{align*}


\begin{align*} ( g ( \xi _ 0 , \xi ) , ( \xi _ 0 , \xi ) ) = \Big ( g ^ { - 1 } \Big ( \frac { d x _ 0 } { d t } , \frac { d x } { d t } \Big ) , \Big ( \frac { d x _ 0 } { d t } , \frac { d x } { d t } \Big ) \Big ) , \end{align*}


\begin{align*} \tau ^ { \delta } : = & \inf \Big \{ t \in ( 0 , T ] : \big \| u ^ h ( t ) - v ^ h ( t ) \big \| > \delta \Big \} . \end{align*}


\begin{align*} x = \lambda g \sigma ( s ) = \big ( \lambda g \pmb { \sigma } _ { 1 } ( s ) , . . . , \lambda g \pmb { \sigma } _ { N - 1 } ( s ) \big ) \end{align*}


\begin{align*} e _ { n , P , 0 } ^ { \epsilon , \Delta t } = \bigl ( \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ n } - e ^ { - \frac { t _ n } { \epsilon ^ 2 } } \bigr ) P _ 0 ^ { \epsilon , \Delta t } = \epsilon \bigl ( \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ n } - e ^ { - \frac { t _ n } { \epsilon ^ 2 } } \bigr ) p _ 0 ^ { \epsilon } , \end{align*}


\begin{align*} \ell _ i : = i \vert i + 1 \vert i , \end{align*}


\begin{align*} B = \begin{pmatrix} 0 & \cos \theta & 0 & - \sin \theta \cr 0 & 0 & 1 & 0 \cr x _ 1 & x _ 2 \sin \theta & 0 & x _ 2 \cos \theta \cr y _ 1 & y _ 2 \sin \theta & 0 & y _ 2 \cos \theta \cr \end{pmatrix} , \hbox { w h e r e } | x _ 1 | ^ 2 + | x _ 2 | ^ 2 \le 1 , \ | y _ 1 | ^ 2 + | y _ 2 | ^ 2 \le 1 . \end{align*}


\begin{align*} \eta _ k ( y , t , r ) , ( y , t ) \in \overline { S _ T } , \ , r \in [ g _ 0 ( y , t ) , g _ 1 ( y , t ) ] , \ , k = 0 , 1 , \dots \end{align*}


\begin{align*} \mathcal { P } : = \{ ( p , q ) \in \mathbb { L } ^ \infty \colon \| p \| _ \infty , \| q \| _ \infty \leq R \} . \end{align*}


\begin{align*} \left \{ \begin{aligned} & v _ t - \mathcal { M } ^ - ( D ^ 2 v ) \leq 0 & & ~ ~ \mbox { i n } ~ ~ Q _ { r } ( r e _ n , 0 ) \backslash Q ; \\ & v \leq C _ 0 & & ~ ~ \mbox { o n } ~ ~ \partial Q \cap Q _ { r } ( r e _ n , 0 ) ; \\ & v \leq 0 & & ~ ~ \mbox { o n } ~ ~ \partial Q _ { r } ( r e _ n , 0 ) \backslash \bar { Q } . \end{aligned} \right . \end{align*}


\begin{align*} \theta _ t ( v ) : = \frac { \eta _ t v ^ 2 } { ( 1 + \delta ) ^ 2 } + \frac { ( 1 + \delta - v \eta _ t ) ^ 2 } { ( 1 + \delta ) ^ 2 ( 1 - \eta _ t ) } - 1 \ge \theta _ t ( 1 ) \ge \frac { \eta _ t \delta ^ 2 } { ( 1 + \delta ) ^ 2 } \ , . \end{align*}


\begin{align*} a ( t ) : = \begin{bmatrix} 1 & & \\ & e ^ { t / 2 } & \\ & & e ^ { - t / 2 } \end{bmatrix} \in G . \end{align*}


\begin{align*} \Delta ( A , \nu ) : = \left ( \bigcup _ { j > 0 } \{ \nu ( f ) / j : f \in A _ j \} \right ) \end{align*}


\begin{align*} M _ { ( a , v ) , ( b , w ) } = \begin{cases} \sigma _ { v , w } & \\ 0 & \end{cases} \end{align*}


\begin{align*} \omega _ j : = \left | j \right | _ g ^ 2 + \hat V _ j \ , \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } \| f _ k - f \| _ { L ^ q ( E ) } = 0 . \end{align*}


\begin{align*} f _ i = \sum _ { j = 1 } ^ d L _ { i j } \tau _ j ' + E _ i + p J _ i . \end{align*}


\begin{align*} & \beta \left ( f ( [ x , y ] , t ) \right ) + f ( y , [ \alpha ( x ) , t ] ) + f ( x , [ \alpha ( y ) , t ] ) \\ & = d ^ 2 _ { c } f ( y , z , t ) - \rho ( y ) f ( \alpha ( z ) , t ) - \rho ( y ) f ( \alpha ( z ) , t ) - \beta \left ( \rho ( t ) f ( y , z ) \right ) , \end{align*}


\begin{align*} Z _ { N , \beta } = \int _ { M ^ { N } } \exp \left ( - \beta \mathcal { H } _ { N } ( X _ { N } ) \right ) d X _ { N } \end{align*}


\begin{align*} \pi ^ { - s } \Gamma ( s ) \zeta ( 2 s ) E ^ P _ { 2 s / r , \varphi } ( g ) = \pi ^ { - ( \frac { r } { 2 } - s ) } \Gamma \left ( \frac { r } { 2 } - s \right ) \zeta ( r - 2 s ) { E ^ P _ { 1 - 2 s / r , \varphi } \left ( ( g ^ \top ) ^ { - 1 } \right ) } . \end{align*}


\begin{align*} \mathcal { G } ( \overline { \mathbf { P } } ) = \begin{cases} \overline { \rm E n t } [ \overline { \mathbf { P } } | \overline { \mathbf { \Pi } ^ { 1 } } ] \ { \rm i f } \ { \rm i n t } [ \overline { \mathbf { P } } ^ { x } ] = \mu _ { V } \ { \rm a . e . } \\ \infty \ { \rm o . w . } \end{cases} \end{align*}


\begin{align*} \begin{aligned} \sup _ { r \in [ 0 , t ] } \mathbb { E } \left [ \| \partial _ x u _ r \| ^ 2 _ { L ^ 2 } \right ] + 2 \lambda _ 2 \int _ { 0 } ^ { t } \mathbb { E } \left [ \| u _ r \times \partial ^ 2 _ x u _ r \| ^ 2 _ { L ^ 2 } \right ] \dd r \leq \mathbb { E } \left [ \| \partial _ x u ^ 0 \| _ { L ^ 2 } ^ 2 \right ] + t \| \partial _ x h \| ^ 2 _ { L ^ 2 } \ , . \end{aligned} \end{align*}


\begin{align*} \mathfrak { D } _ { i n t } : \ \mathcal { E } ( M ) \times \mathcal { E } ' ( M ' ) \mapsto [ 0 , + \infty ) , \mathfrak { D } _ { i n t } ( \xi , \xi ' ) : = | \xi ' - \xi | ^ 2 ; \end{align*}


\begin{align*} \eta _ a ( u _ w ) = \frac { 1 } { ( \lambda _ 1 - \lambda _ w ) \langle 1 , 1 \rangle } + \frac { 1 } { 2 \pi i } \int _ { ( \frac { 1 } { 2 } ) } \frac { a ^ { 1 - s } \cdot E _ s ( z ) } { \lambda _ s - \lambda _ w } d s . \end{align*}


\begin{align*} \mathcal { A } = V _ C ^ T \mathcal { A } _ { F T 2 } \mathcal { D } _ L , \end{align*}


\begin{align*} g ( z ) : = \begin{cases} h _ 1 ( z ) , \ \ \ z \in { E } , \\ h _ 2 ( z ) , \ \ \ z \in \Omega \setminus { E } . \end{cases} \end{align*}


\begin{align*} | a | ^ { - 1 } g _ k ( a ^ { - 1 } r e ^ { i \theta } ) = \begin{cases} r ^ { - 1 } e ^ { i k \theta } & \\ ( - 1 ) ^ k r ^ { - 1 } e ^ { i k \theta } & \ , . \end{cases} \end{align*}


\begin{align*} \left \{ \begin{aligned} & - \partial _ \tau \partial _ { \tau , \xi } w - \eta \partial _ \xi \partial _ { \tau , \xi } w + w ^ 2 \partial _ { \eta } ^ 2 \partial _ { \tau , \xi } w + ( 2 w \partial _ { \eta } ^ 2 w ) \partial _ { \tau , \xi } w = 0 ( \tau , \xi , \eta ) \in D , \\ & \partial _ \eta \partial _ { \tau , \xi } w \mid _ { \eta = 0 } = 0 , \displaystyle \lim _ { \eta \to 1 } \partial _ { \tau , \xi } w = 0 , \end{aligned} \right . \end{align*}


\begin{align*} | K l _ p ( \psi _ p , \psi _ p ' ; \tilde { c } , w _ { G _ 4 } ) | & \leq C _ 8 \cdot \min ( p ^ { r + \sigma + \varrho / 2 + 3 m } , p ^ { \varrho + 3 \sigma / 2 + r / 2 + 3 m } ) . \end{align*}


\begin{align*} z ^ { 2 } \left [ \gamma _ { n } ^ { \prime \prime } + 2 \left ( 6 \gamma _ { n } - n \right ) \left ( 2 \gamma _ { n } - n \right ) \right ] ^ { 2 } = 4 \left ( z ^ { 2 } + 2 \gamma _ { n } - n \right ) ^ { 2 } \left [ \left ( \gamma _ { n } ^ { \prime } \right ) ^ { 2 } + 4 \gamma _ { n } \left ( 2 \gamma _ { n } - n \right ) ^ { 2 } \right ] . \end{align*}


\begin{align*} x \to y : = ( x \vee y ) * y . \end{align*}


\begin{align*} \limsup _ { n \to \infty } c _ 2 ( \overline { \alpha } ; R ) ^ n ( k n + k ) ! \cdot ( k n + k ) \prod _ { p \in R } \left | ( k n ) ! n ! \right | _ p = 0 \end{align*}


\begin{align*} w : = \frac { d \Psi _ \# \mu } { d \mu } \in L _ { \rm l o c } ^ 1 ( \R ^ n , \mu ) \ , . \end{align*}


\begin{align*} \langle \alpha _ { i } , \alpha _ { j } \rangle = \langle \beta _ { i } , \beta _ { j } \rangle = 0 , \quad \langle \alpha _ { i } , \beta _ { j } \rangle = \delta _ { i j } ( i , j = 1 , 2 , \ldots , g ) . \end{align*}


\begin{align*} H _ 0 [ \phi , k ] : = H [ \phi , k ; \mu ] + \mu \delta ( x - y ) , \end{align*}


\begin{align*} \frac { 1 } { a ^ 2 } & = A _ a \cdot \left ( - c ( e ^ { 2 c a } - e ^ { 2 c } ) e ^ { - c a } - c ( e ^ { c a } + e ^ { 2 c } e ^ { - c a } \right ) = A _ a \cdot ( - 2 c e ^ { c a } ) . \end{align*}


\begin{align*} m ( \pi , \sigma ) : = \langle \pi \otimes \nu _ { l , \psi } ^ \vee , \sigma \rangle _ { R _ l ^ F } . \end{align*}


\begin{align*} g ^ { \epsilon } = g \ast \delta ^ { \epsilon } . \end{align*}


\begin{align*} m ( a ) = a \times \{ 0 , \ldots n - 1 \} \ ; . \end{align*}


\begin{align*} H = [ n - ( d - 1 ) t - 1 , n ] \setminus \{ n - \ell t - 1 \} \in \Delta . \end{align*}


\begin{align*} \limsup _ { n \to \infty } \frac { \overline { V } _ { d ' } ( n ) } { \psi _ { d ' } ( n ) } = 1 \ , . \end{align*}


\begin{align*} F = \phi _ { i j } ^ 2 + q _ i q _ j r _ { i j } s _ { i j } . \end{align*}


\begin{align*} \langle \psi ( u ) , u \rangle = 0 . \end{align*}


\begin{align*} \{ \omega = F - x _ 0 x _ 1 , x _ 0 + x _ 1 = i ( x _ 2 + x _ 3 ) \} \subset V \end{align*}


\begin{align*} m ^ { \Phi } ( - A ^ { - T } ( \tau _ 0 ) \langle x \rangle , \tau ) \cdot \sqrt { w _ 0 ( \tau ) } = m ^ { \Phi } ( 0 , \tau ) \cdot \sqrt { w _ 0 ( \tau ) } \leq M ( x , \tau ) . \end{align*}


\begin{align*} \Big ( \sum _ { p \le x } \frac { \log p } { p ^ { 1 / 2 + i t } } \Big ( 1 - \frac { \log p } { \log x } \Big ) \Big ) ^ k & = \sum _ { \substack { \mu _ 1 , \mu _ 2 , \ldots , \mu _ R \\ \sum _ r \mu _ r = k } } \binom { k } { \mu _ 1 \ \mu _ 2 \ \cdots \ \mu _ R } \prod _ { r = 1 } ^ R \Big ( \frac { \log p _ r } { p _ r ^ { 1 / 2 + i t } } \Big ( 1 - \frac { \log p _ r } { \log x } \Big ) \Big ) ^ { \ ! \mu _ r } \\ & = \sum _ n c ( n ) \ , n ^ { - i t } . \end{align*}


\begin{align*} \varphi : = e ^ { - \psi } + \kappa : = \chi + \kappa , \end{align*}


\begin{align*} \langle Q _ n f , g \rangle = \langle f , S _ n ^ { - 1 } g \rangle _ n \ , . \end{align*}


\begin{align*} \Omega _ \varepsilon ( t ) = \{ y + r \nu ( y , t ) \mid y \in \Gamma ( t ) , \ , \varepsilon g _ 0 ( y , t ) < r < \varepsilon g _ 1 ( y , t ) \} , t \in [ 0 , T ] \end{align*}


\begin{align*} \mathcal { E } _ { V } ^ { \theta } ( \mu ) = \mathcal { E } _ { V } ( \mu ) + \frac { 1 } { \theta } { \rm e n t } [ \mu ] , \end{align*}


\begin{align*} x _ t = x _ 0 + B _ t ^ H + \int _ 0 ^ t f ( s , x _ s ) \ , d s , t \leq T . \end{align*}


\begin{align*} \bar { \tau } _ \varepsilon ^ i ( x , t ) = ( I _ n - \varepsilon g _ i W ) ^ { - 1 } \nabla _ \Gamma g _ i = \nabla _ \Gamma g _ i + \varepsilon g _ i W \nabla _ \Gamma g _ i + O ( \varepsilon ^ 2 ) \end{align*}


\begin{align*} \overline { R [ I t , t ^ { - 1 } ] } = \bigoplus _ { n = - \infty } ^ { \infty } \overline { I ^ n } t ^ n , \end{align*}


\begin{align*} T \left ( k \right ) = \frac { 2 \mathrm { i } k } { W \left \{ \psi _ { - } \left ( x , k \right ) , \psi _ { + } \left ( x , k \right ) \right \} } , \end{align*}


\begin{align*} x = ( a \otimes b ) \otimes ( c \otimes d ) \ ; . \end{align*}


\begin{align*} \frac { \varphi f ( z ) - \varphi f ( \lambda ) } { z - \lambda } = \varphi ( z ) \frac { f ( z ) - f ^ * ( \lambda ) } { z - \lambda } + f ^ * ( \lambda ) \frac { \varphi ( z ) - \varphi ( \lambda ) } { z - \lambda } , \ , \ , \ , \ , z \in \mathbb D , \end{align*}


\begin{align*} \int _ { \R ^ d } \left ( \frac { 1 - \cos ( \xi \cdot h ) } { \| h \| ^ { d + a } } \right ) \d h = c \| \xi \| ^ { a } \qquad , \end{align*}


\begin{align*} \lim _ { N \to \infty } \mathcal { E } _ { V } ^ { \theta } ( \mu _ { N } ) = \inf _ { \mu \in \mathcal { P } ( M ) } \mathcal { E } _ { V } ^ { \theta } ( \mu ) . \end{align*}


\begin{align*} C ( x _ { 1 } ) \left | J \right \rangle = \sum _ { j \in J } \mathcal { C } _ { 1 , j } ^ { J } \left | J \setminus j \right \rangle + \sum _ { \substack { j , k \in J \\ j < k } } \mathcal { C } _ { 1 , j , k } ^ { J } \left | J \setminus \left ( j , k \right ) \cup 1 \right \rangle \ , , \end{align*}


\begin{align*} n : = \max \left \{ a \in \mathbb { Z } : N _ 1 \left ( a , R ' \right ) \geq 0 \right \} . \end{align*}


\begin{align*} T _ i ^ { - 1 } = T _ i + q - q ^ { - 1 } . \end{align*}


\begin{align*} \mathcal { R } _ N : = P _ N ( ( P _ N u \cdotp \nabla ) P _ N u ) - P _ N ( ( u \cdotp \nabla ) u ) . \end{align*}


\begin{align*} \int _ 0 ^ 1 g ( s ) \ , d s = \int _ 0 ^ 1 g _ i ( s ) \ , d s = 1 \quad \textrm { f o r a l l } i \in \N . \end{align*}


\begin{align*} \Pi = \left ( g \cdot \sigma + h \cdot \lambda + \tau \right ) \pi _ F \end{align*}


\begin{align*} \int _ { \Omega } | \nabla u _ { i } | ^ { p _ { i } ( x ) - 2 } \nabla u _ { i } \nabla \varphi _ { i } \ d x = \int _ { \Omega } f _ { i } ( x , u _ { 1 } , u _ { 2 } , \nabla u _ { 1 } , \nabla u _ { 2 } ) \varphi _ { i } \ d x , \end{align*}


\begin{align*} x ^ 2 = ( \nu - 1 ) \mathfrak P + \alpha ^ 2 + \alpha \beta \gamma ( 1 - \nu ) + \nu \gamma ^ 2 \end{align*}


\begin{align*} e ^ { i t H } = \mu ( \chi _ t ) O p _ w ( b _ t ) \end{align*}


\begin{align*} \widetilde { \psi } \left ( \mathrm { i } \alpha \right ) & = \psi \left ( \mathrm { i } \alpha \right ) - \int K \left ( \alpha , s \right ) \mathrm { d } \mu \left ( s \right ) \\ & = \psi \left ( \mathrm { i } \alpha \right ) - \int K \left ( \alpha , s \right ) y \left ( s \right ) \mathrm { d } \sigma \left ( s \right ) \\ & = \psi - \mathbb { K } y = y . \end{align*}


\begin{align*} f _ 0 : = \frac { d } { d t } P _ N u + \nu A P _ N u + P _ N B ( P _ N u , P _ N u ) , \end{align*}


\begin{align*} \omega _ j ( \beta ) = \beta ^ 2 \Omega _ j , \Omega _ j : = \sqrt { | j | ^ 4 _ { \overline g } + \frac { m } { \beta ^ 4 } } \ , . \end{align*}


\begin{align*} m _ - n _ + - m _ + n _ - = 1 . \end{align*}


\begin{align*} \mathcal { A } _ { 2 n } = \sqcup _ { B = \{ b _ 1 < b _ 2 < . . . < b _ n \} \subset [ 2 n ] } \mathcal { A } _ B , \end{align*}


\begin{align*} A ( x ) : = h '^ 2 ( x ) + ( m - 1 ) \cos ^ 2 h ( x ) \end{align*}


\begin{align*} \| X \| _ { p - v a r } : = \sup _ { \mathcal { D } \subset [ 0 , 1 ] } \left ( \sum _ { t _ i \in \mathcal { D } } d ( X _ { t _ i } , X _ { t _ { i + 1 } } ) ^ p \right ) ^ { \frac { 1 } { p } } . \end{align*}


\begin{align*} ( ^ { \rho } I ^ { \alpha } _ { b ^ - } g ) ( x ) = \dfrac { \rho ^ { 1 - \alpha } } { \Gamma ( \alpha ) } \int _ { x } ^ { b } \ ( x ^ { \rho } - \tau ^ { \rho } ) ^ { \alpha - 1 } \tau ^ { \rho - 1 } g ( \tau ) d \tau , x < b . \end{align*}


\begin{align*} A ( x ) | \Omega \rangle = \alpha ( x ) | \Omega \rangle \ , , \tilde D ( x ) | \Omega \rangle = \tilde \delta ( x ) | \Omega \rangle \ , , C ( x ) | \Omega \rangle = 0 \ , . \end{align*}


\begin{align*} L = - ( - \Delta + m ^ { 2 / \alpha } ) ^ { \alpha / 2 } + m . \end{align*}


\begin{align*} \tilde { u } = u , \tilde { \theta } = \theta - \chi ( y ) \Theta - ( 1 - \chi ( y ) ) \theta ^ * , \tilde { q } = q - H ^ 2 / 2 . \end{align*}


\begin{align*} Z _ i \cap Z _ j = \alpha ^ { - 1 } ( X _ i ) \cap \alpha ^ { - 1 } ( X _ j ) = \alpha ^ { - 1 } ( X _ i \cap X _ j ) . \end{align*}


\begin{align*} H [ n ] \circ k _ { n + 1 } + k _ { n + 1 } \circ H [ n ] ^ T + \Theta [ n ] + k _ { n + 1 } \circ \overline { \Theta [ n ] } \circ k _ { n + 1 } = 0 , \end{align*}


\begin{align*} \left \{ \begin{array} { c } \partial _ { t } f ^ { \epsilon } + \xi \cdot \nabla _ { x } f ^ { \epsilon } + \frac { 1 } { \epsilon } ( - \nabla \Phi ^ { \epsilon } + \xi ^ { \bot } ) \cdot \nabla _ { \xi } f ^ { \epsilon } = 0 , \\ \rho ^ { \epsilon } = 1 - \Delta \Phi ^ { \epsilon } . \end{array} \right . \end{align*}


\begin{align*} \beta _ A = \max _ \ell \gamma _ { A , \ell } , \end{align*}


\begin{align*} X _ { n + 1 } ^ { 0 , \Delta t } = \Phi ( \Delta \beta _ n , X _ n ^ { 0 , \Delta t } ) , \end{align*}


\begin{align*} & \frac { \sqrt { 2 } } { \sqrt { k _ + } e ^ { i \frac { \pi } { 4 } } } \sqrt { - s _ b } \sqrt { - s _ b ^ { * } } H _ { \theta _ c } ( 0 ) H _ { \pi - \theta _ c } ( 0 ) \\ & = \mathcal S ( \cos ( \zeta ( 0 ) ) , n ) = - i \sqrt { n ^ 2 - \cos ^ 2 \theta _ { \hat x } } \ne 0 \textrm { f o r } ~ \theta _ { \hat x } \in ( \theta _ c , \pi / 2 ) . \end{align*}


\begin{align*} u ( t ) = U ( t , s ) \varphi + j ^ { - 1 } \int _ s ^ t U ^ { \odot \star } ( t , \tau ) R ( \tau , u ( \tau ) ) d \tau , \varphi \in X , \end{align*}


\begin{align*} s _ i x _ i \ne 0 \quad & 1 \leq i \leq h - 1 , \\ x _ i - x _ j \ne 0 \quad & 1 \leq i < j \leq h - 1 . \end{align*}


\begin{align*} \tilde { P } ^ { \epsilon , \Delta t } ( t ) - \tilde { P } ^ { \epsilon , \Delta t } ( t _ n ) = \bigl ( e ^ { - \frac { t - t _ n } { \epsilon ^ 2 } } - 1 \bigr ) \tilde { P } ^ { \epsilon , \Delta t } ( t _ n ) + \int _ { t _ n } ^ { t } e ^ { - \frac { t - s } { \epsilon ^ 2 } } f ( q _ n ^ { \epsilon , \Delta t } ) d s + \int _ { t _ n } ^ { t } e ^ { - \frac { t - s } { \epsilon ^ 2 } } \sigma ( q _ n ^ { \epsilon , \Delta t } ) d \beta ( s ) . \end{align*}


\begin{align*} \sum _ { k \ge 1 } \frac { k ^ { - ( \beta - \gamma ) } } { \gamma - \beta } = \int _ 1 ^ \infty \frac { t ^ { - ( \beta - \gamma ) } } { \gamma - \beta } d t + \frac { 1 } { 2 ( \gamma - \beta ) } + \int _ 1 ^ \infty t ^ { - ( \beta - \gamma + 1 ) } P _ 1 ( t ) d t . \end{align*}


\begin{align*} \frac { \bar { t } ( r ) } { r } = \frac { 1 } { Q } - \frac { A } { Q \sqrt r } , \end{align*}


\begin{align*} \chi ( y ) = \begin{cases} 0 , y \in [ 0 , 1 ] , \\ 1 , y \ge 2 , \end{cases} \end{align*}


\begin{align*} R ^ s ( X , \phi ) = R ^ u ( X , \phi ^ { - 1 } ) . \end{align*}


\begin{align*} \mathcal { U } _ d : = \{ & U \subseteq K \colon U = \{ x \in K \colon \exists y \in K , f ( x , y ) = 0 , g ( x , y ) \neq 0 , \frac { \partial f } { \partial Y } ( x , y ) \neq 0 \} \\ & \} . \end{align*}


\begin{align*} j _ M ^ 0 = j _ M ^ \infty + \int _ 0 ^ \infty \omega ^ { 1 / 2 } \phi \ , \dd \omega . \end{align*}


\begin{align*} & = \sum _ { j = 0 } ^ { k + 1 } \omega ^ j \sup _ { | \alpha | = k + 1 - j } \| ( \partial ^ { \alpha } \varphi ) \circ \psi - ( \partial ^ { \alpha } \varphi ) \circ \tilde \psi \| _ { \C ^ 0 } \prod _ { i = j } ^ { k } \| ( D \psi ) ^ t \| _ { \C ^ i } \\ & + \sum _ { j = 1 } ^ { k + 1 } \sup _ { | \alpha | = k + 1 - j } \| \partial ^ { \alpha } \varphi \circ \tilde \psi \| _ { \C ^ { j - 1 } } \| ( D \psi ) ^ t - ( D \tilde \psi ) ^ t \| _ { \C ^ { j - 1 } } \prod _ { i = j } ^ { k } \| ( D \psi ) ^ t \| _ { \C ^ { i } } . \end{align*}


\begin{align*} & \left ( \left ( I _ r \otimes \left [ \begin{array} { c c c c } C _ 1 \\ \vdots \\ C _ n \end{array} \right ] \right ) A \left ( I _ r \otimes \left [ \begin{array} { c c c c } D _ 1 \\ \vdots \\ D _ n \end{array} \right ] \right ) ' \right ) \\ = & \sum ^ n _ { i = 1 } C _ i ( A ) D _ i ' . \end{align*}


\begin{align*} \lim _ { r \to \infty } r ^ { \frac { 1 } { 2 } } | \omega ( r , \theta ) | = 0 , \end{align*}


\begin{align*} P \{ y \circ \alpha _ n \notin K _ \gamma \} = P \alpha _ n ^ { - 1 } \{ x _ m \notin K _ \gamma \} < \gamma \ \ \ \mbox { f o r a l l } \ \ \ n \ge n _ 1 . \end{align*}


\begin{align*} \left \{ \begin{array} { r l } U & = 0 , \\ ( r ^ 2 - s ^ 2 ) \frac { \phi _ z } { \phi } V & = \frac { \phi _ z } { 2 \phi \Lambda } \Omega ( \varphi _ z - 2 \phi _ { x ^ 0 } ) , \\ ( r ^ 2 - s ^ 2 ) \left ( \frac { \phi - z \phi _ z } { \phi } \right ) V & = \frac { \Omega } { 2 \phi \Lambda } ( \phi - z \phi _ z ) ( \varphi _ z - 2 \phi _ { x ^ 0 } ) . \end{array} \right . \end{align*}


\begin{align*} \begin{cases} P _ x - H H _ x = 0 , \\ \Theta _ t - \frac { a P _ t \Theta } { P + \frac 1 2 ( 1 - 2 a ) H ^ 2 } = 0 , \\ H _ t - \frac { P _ t H } { ( P + \frac 1 2 ( 1 - 2 a ) H ^ 2 ) ( R + 1 ) } = 0 . \end{cases} \end{align*}


\begin{align*} \widehat { \gamma } \left ( a , z \right ) = a ^ { - 1 } z ^ { a } e ^ { - z } \ _ { 1 } F _ { 1 } \left ( \begin{array} [ c ] { c } 1 \\ a + 1 \end{array} ; z \right ) , \end{align*}


\begin{align*} d _ n = d ( ( \bar x _ n , \bar t _ n ) , \partial ^ + Q ) & \le d ( x , \partial \Omega ) + ( T - t ) ^ { \frac 1 2 } + | x - \bar x _ n | + | t - \bar t _ n | ^ { \frac 1 2 } \\ & \le d ( x , \partial \Omega ) + ( T - t ) ^ { \frac 1 2 } + 2 r _ n R \\ & = d ( ( x , t ) , \partial ^ + Q ) + 2 r _ n R . \end{align*}


\begin{align*} P _ 0 = \mathsf D ^ 2 + a \mathsf D + \Delta _ { g _ F } , \end{align*}


\begin{align*} ( G \circ F ) ^ \pi = G ^ S \circ F ^ \pi \end{align*}


\begin{align*} \acute { \mu } ^ s ( \alpha , e ) = \begin{cases} ( d + s e ) / r , & r > 0 , \\ + \infty , & r = 0 . \end{cases} \end{align*}


\begin{align*} F ( [ a _ { j } ^ i ] ) = f ( \mu _ 1 , \cdots , \mu _ n ) , \end{align*}


\begin{align*} L \left [ \partial _ { x } \left ( x P _ { n } ^ { 2 } \right ) \right ] = 2 z P _ { n } ^ { 2 } \left ( z ; z \right ) e ^ { - z ^ { 2 } } + 2 \left [ h _ { n + 1 } \left ( z \right ) + \gamma _ { n } ^ { 2 } \left ( z \right ) h _ { n - 1 } \left ( z \right ) \right ] . \end{align*}


\begin{align*} S ( n ) = \frac { 3 n + 1 } { 4 } \equiv - 1 \pmod { 4 } . \end{align*}


\begin{align*} A ( \varphi ) = \omega _ { n - 1 } \int \limits _ 0 ^ \varphi \frac { r ^ { n - 2 } } { x _ n } d r \ , . \end{align*}


\begin{align*} \int _ { s } ^ { t } \gamma ( u _ r ) \circ \dd W _ r = \frac { 1 } { 2 } \int _ { s } ^ { t } c ( u _ r ) d r + \int _ { s } ^ { t } \gamma ( u _ r ) d W _ r \ , . \end{align*}


\begin{align*} \bar { \ell } _ { k } ( u ) = t r _ { 1 , \cdots , k } ~ A _ k L _ 1 ^ { - } ( u _ 1 ) \cdots L _ k ^ { - } ( u _ k ) . \end{align*}


\begin{align*} R [ t ] \ni g ' ( t ) = g ( t ) + \sum _ { m = 1 } ^ { r } \pi _ m t ^ { N + m } . \end{align*}


\begin{align*} F ( x , t ) = A ( t ) x + g ( t ) . \end{align*}


\begin{align*} S ^ v ( m ) = S \left ( 2 \cdot 3 ^ { v - 1 } w - 1 \right ) = \frac { 3 ^ v w - 1 } { 2 ^ e } \end{align*}


\begin{align*} y \sqrt { y } \sqrt { a } - x ^ 2 ~ = ~ & x \sqrt { y } \sqrt { a } - x ^ 2 - x \sqrt { y } \sqrt { a } + y \sqrt { y } \sqrt { a } \\ [ . 5 e m ] = ~ & x ( \sqrt { y } \sqrt { a } - x ) + \sqrt { y } \sqrt { a } ( y - x ) \\ [ . 5 e m ] \leq ~ & x ( y - x ) + \sqrt { y } \sqrt { a } ( y - x ) \\ [ . 5 e m ] = ~ & ( x + \sqrt { y } \sqrt { a } ) ( y - x ) , \end{align*}


\begin{align*} T _ { n , \infty } [ z ; \psi _ 1 , \psi _ 2 ] : = \lim _ { L \to \infty } T _ { n , L } [ z ; \psi _ 1 , \psi _ 2 ] , \end{align*}


\begin{align*} \Omega ( b ) = - \frac { 1 } { \pi } \big ( \theta ( x _ e ( b ) , b ) - \theta ( 0 , b ) \big ) . \end{align*}


\begin{align*} ( a ) _ m = \frac { \Gamma ( a + m ) } { \Gamma ( a ) } \end{align*}


\begin{align*} \frac { d ^ \ell } { d y ^ \ell } G ( x , y ) \bigg | _ { y = 1 } = G ( x , y ) \bigg ( A _ { 0 , 1 } ( x ) ^ \ell + \sum _ { 0 \le k _ 1 \le \cdots \le k _ { \ell - 1 } \atop k _ 1 + \cdots + k _ { \ell - 1 } \le \ell } d ' _ { k _ 1 , \dots , k _ { \ell - 1 } } \prod _ { 1 \le i \le \ell - 1 } A _ { 0 , k _ i } ( x ) \bigg ) . \end{align*}


\begin{align*} \tilde { B } ( v ) : = \hat { B } _ \# \left ( - v + u _ 0 + \sum _ { l \in \{ 1 , . . . , s - 1 \} \cap I _ A } \sigma _ { A , s } ( l ) u _ l \right ) . \end{align*}


\begin{align*} \sum _ { | j | > m _ 0 } q _ { \delta , j , m } = O ( 1 / m _ 0 ) , \end{align*}


\begin{align*} \tilde { A } _ i ( s ) & = \int _ s ^ { s _ 0 } U \mathbf { D } ^ { \ell } F _ { i \ell } U ^ { - 1 } \dd s ' + U A _ i ( s _ 0 ) U ^ { - 1 } - \partial _ i U U ^ { - 1 } \\ & = \int _ s ^ { s _ 0 } \tilde { \mathbf { D } } ^ { \ell } \tilde { F } _ { i \ell } ( s ' ) \dd s ' + \tilde { A } _ i ( s _ 0 ) \end{align*}


\begin{align*} f ( X _ i ) \subseteq X _ { i + r } , i = \overline { 1 , m } , \end{align*}


\begin{align*} \left ( \sum _ { i } \alpha _ i \ , \Pi ^ { - 1 } S _ i \right ) . \ , L _ p = 0 \end{align*}


\begin{align*} \frac { d x ( t ) } { d t } = \sigma ( x ( t ) ) , \end{align*}


\begin{align*} \lim _ { w \to - i \infty } \lambda _ { k + 1 , D } ( \tau , w ) = - 2 \pi i \varphi _ { k + 1 , D } ( \tau ) , \lim _ { \tau \to i \infty } \lambda _ { k + 1 , D } ( \tau , w ) = 0 . \end{align*}


\begin{align*} P \in X F , F ' \in A F \cap F ' = P \end{align*}


\begin{align*} \varphi ( v ) = \varphi \circ \alpha ( v ' ) = \beta \circ ( \varphi \times _ k k ' ) ( v ' ) = \beta \circ ( \psi \times _ k k ' ) ( v ' ) = \psi \circ \alpha ( v ' ) = \psi ( v ) . \end{align*}


\begin{align*} a _ \ell ( x ) : = x f _ \ell ' ( x ) , b _ \ell ( x ) : = x ^ 2 f _ \ell '' ( x ) + x f _ \ell ' ( x ) \quad c _ \ell ( x ) : = x ^ 3 f _ \ell ''' ( x ) + 3 x ^ 2 f _ \ell '' ( x ) + x f _ \ell ' ( x ) \enspace . \end{align*}


\begin{align*} \beta _ { p - v a r } ( \mathbf { X } , \mathbf { Y } ) : = \lim _ { \delta \to 0 } \inf _ { \lambda \in \Lambda } \{ | \lambda | \vee d _ p ( \mathbf { X } ^ { \phi , \delta } _ \lambda , \mathbf { Y } ^ { \phi , \delta } ) \} . \end{align*}


\begin{align*} F _ { a , b } ( s ) & = \frac { 1 } { 2 \pi } \sum _ { k \in \mathbb { Z } } \frac { \pi e ^ { - i b ( s + k ) } } { a } e ^ { - a | s + k | } \end{align*}


\begin{align*} L _ 0 g = - ( g \frac { L v } { v } + 2 \frac { w ^ 2 } { v } \partial _ { \eta } v \partial _ { \eta } g ) = - g \frac { L v } { v } > 0 a t z _ { m a x } . \end{align*}


\begin{align*} \kappa ( C ^ 1 ) = \coth ( \ell _ 1 ^ 1 ) = \frac { \cos ( \alpha _ 1 ) } { \tanh ( \ell _ 3 ^ 1 ) } \leq \frac { \cos ( \alpha _ 2 ) } { \tanh ( \ell _ 3 ^ 2 ) } = \coth ( \ell _ 1 ^ 2 ) = \kappa ( C ^ 2 ) . \end{align*}


\begin{align*} \Delta ^ { + } \varphi ( t ) : = \varphi ( t ^ { + } ) - \varphi ( t ) , \Delta ^ { - } \varphi ( t ) : = \varphi ( t ) - \varphi ( t ^ { - } ) , \end{align*}


\begin{align*} \frac { \Delta t m _ { n + 1 } ^ { \epsilon , \Delta t } } { \epsilon } = \Delta \beta _ n + \epsilon ( m _ n ^ { \epsilon , \Delta t } - m _ { n + 1 } ^ { \epsilon , \Delta t } ) . \end{align*}


\begin{align*} \mathrm { d } U _ t = ( \mu - F ( U _ t ) ) \mathrm { d } t + \sigma \mathrm { d } W _ t . \end{align*}


\begin{align*} X ( s , t , \cdot ) = X ( s , u , X ( u , t , \cdot ) ) \end{align*}


\begin{align*} & \theta _ 1 ( v , u ; q ; z ) : = \sum _ { k \geq 0 } \theta _ { 1 , k } ( v , u ; q ) z ^ k = - 2 e ^ { z v } \sum _ { m \geq 0 } \Bigl ( \gamma - \frac 1 2 u + \psi ( m + 1 ) \Bigr ) q ^ m e ^ { m u } \frac { z ^ { 2 m } } { m ! ^ 2 } , \\ & \theta _ 2 ( v , u ; q ; z ) : = \sum _ { k \geq 0 } \theta _ { 2 , k } ( v , u ; q ) z ^ k = z ^ { - 1 } \biggl ( \sum _ { m \geq 0 } q ^ m e ^ { m u + z v } \frac { z ^ { 2 m } } { m ! ^ 2 } - 1 \biggr ) , \end{align*}


\begin{align*} \overline { \partial } f = \mu \ , \partial f a . e . \end{align*}


\begin{align*} R _ { \xi } ( x ) = \sum _ { i } ( \sum _ { j } \xi _ { i j } x _ { j } ) \pmb { e } _ { i } \end{align*}


\begin{align*} \tau ( \theta ^ 0 , \ , h , \ , \theta ^ S ) = ( \theta ^ 0 , \ , h , \ , \theta ^ S + \theta ^ 0 ) . \end{align*}


\begin{align*} \| T _ m f ( \cdot , x ) \| _ { L ^ 2 _ \rho } = \big \| \sum _ { k = 0 } ^ \infty m ( \tau , k ) P _ k ( \mathcal F _ \rho f ) ( \cdot , x ) \big \| _ { L ^ 2 _ \tau } . \end{align*}


\begin{align*} H ^ { \lambda , \theta , T } ( y , y ' ) = \frac { T } { n } \sum _ { z ^ n = ( - 1 ) ^ { \theta _ 2 } } z ^ { 1 + h - h ' } \frac { e ^ { - ( \lambda + \theta _ 1 \pi i + T z ) [ t ' - t ] } } { 1 - e ^ { - ( \lambda + \theta _ 1 \pi i + T z ) } } , \end{align*}


\begin{align*} T ( x ) = 2 \frac { 1 + x } { 1 + 2 x } ( p + x ) A ( x ) + ( p - x - 1 ) \tilde D ( x ) \ , . \end{align*}


\begin{align*} g ( t , x ) : = \sup _ { r > 0 } \frac { 1 } { | B _ r | } \int _ { B _ r ( x ) } | \nabla p ( t , y ) | \ , d y . \end{align*}


\begin{align*} \{ x _ 4 Q + R = 0 \} , \end{align*}


\begin{align*} \Delta ^ { 2 } u - \Delta u + ( 1 + \lambda a ( x ) ) u = ( R _ { \alpha } \ast | u | ^ { p } ) | u | ^ { p - 2 } u \end{align*}


\begin{align*} \frac { \int _ { \mathbb { D } } f e ^ { - \varphi } } { \int _ { \mathbb { D } } e ^ { - \varphi } } = \frac { \int _ { \mathbb { D } } f e ^ { - \psi } - \varepsilon \int _ { \mathbb { D } } f \widetilde { \eta } } { \int _ { \mathbb { D } } e ^ { - \psi } - \varepsilon \int _ { \mathbb { D } } \widetilde { \eta } } = \frac { \int _ { \mathbb { D } } f e ^ { - \psi } } { \int _ { \mathbb { D } } e ^ { - \psi } } = f ( 0 ) , \ \ \ \forall \ , f \in { A ^ 2 ( \mathbb { D } ) } , \end{align*}


\begin{align*} \psi ( u , v ) = \frac { u + v } { 2 } + \sqrt { 1 + ( u - v ) ^ 2 } . \end{align*}


\begin{align*} ( x + y + x \star y ) \star z = x \star z + y \star z + x \star ( y \star z ) , x \star ( y + z ) = x \star y + x \star z \end{align*}


\begin{align*} H ( x ( t ) , \xi _ 0 ( t ) , \xi ( t ) ) = H ( y , \eta _ 0 , \eta ) , \ \ \forall t . \end{align*}


\begin{align*} \Phi _ g ( \tau , z , \omega ) = q \zeta s \prod _ { ( n , r , m ) > 0 } \exp \left ( - \sum _ { a = 1 } ^ \infty \frac { c _ { g ^ a } ( 4 n m - r ^ 2 ) } { a } \Big ( q ^ n \zeta ^ r s ^ m \Big ) ^ a \right ) , \end{align*}


\begin{align*} 1 _ i = \begin{cases} 1 & \ i = 1 \\ 0 & \end{cases} \ ; . \end{align*}


\begin{align*} \alpha - 1 & = \frac { \alpha } { \alpha - 1 } - ( t + 1 ) \end{align*}


\begin{align*} d \left ( e ^ s \left ( d t + \frac { 1 } { 2 } r ^ 2 \ , d \theta \right ) \right ) = e ^ s \left ( d s \ , d t + \frac { 1 } { 2 } r ^ 2 \ , d s \ , d \theta + r \ , d r \ , d \theta \right ) . \end{align*}


\begin{align*} \psi _ N ( r ) = \sum _ { h = 0 } ^ { + \infty } c _ { h , k } \Delta _ k ^ \ast \psi _ N ( 2 ^ h r ) . \end{align*}


\begin{align*} \psi _ { \sigma } \left ( x , t , k \right ) & = \psi \left ( x , t , k \right ) \\ & - { \displaystyle \sum \limits _ { n = 1 } ^ { N } } c _ { n } ^ { 2 } e ^ { 8 \kappa _ { n } ^ { 3 } t } y _ { n } \left ( x , t \right ) \int _ { x } ^ { \infty } \psi \left ( s , t , k \right ) \psi \left ( s , t , \mathrm { i } \kappa _ { n } \right ) \mathrm { d } s \end{align*}


\begin{align*} \phi _ { r z } - r \phi _ { x ^ 0 s } = 0 , \end{align*}


\begin{align*} \left \{ \begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = f & & ~ ~ \mbox { i n } ~ ~ \Omega \cap Q _ 1 ; \\ & u = g & & ~ ~ \mbox { o n } ~ ~ \partial \Omega \cap Q _ 1 , \end{aligned} \right . \end{align*}


\begin{align*} A = \left \{ \begin{pmatrix} a _ 1 & \\ & \ddots \\ & & a _ r \end{pmatrix} : a _ i \in \R \right \} , \end{align*}


\begin{align*} \partial _ \eta w ( z _ { m i n } ) = 0 , \end{align*}


\begin{align*} u = s ^ \beta , \ v = t ^ \beta , \ g _ 1 = a ^ 4 c ^ 5 , \ g _ 2 = a ^ 2 c ^ 3 d ^ 2 , \end{align*}


\begin{align*} F _ { \theta } ( x ^ { \star } ) - F _ { \theta } ( x _ { \theta } ^ { \star } ) \leq F ( x ^ { \star } ) - F _ { \theta } ( x _ { \theta } ^ { \star } ) \leq F ( x _ { \theta } ^ { \star } ) - F _ { \theta } ( x _ { \theta } ^ { \star } ) = g ( x _ { \theta } ^ { \star } ) - g _ { \theta } ( x _ { \theta } ^ { \star } ) \leq \frac { L _ 0 } { 2 } \theta . \end{align*}


\begin{align*} F _ { 0 m } ( D ^ 2 H ) = 0 , \end{align*}


\begin{align*} P _ { i } + P ^ { \perp } _ { i } = I , i = 1 , 2 , \end{align*}


\begin{align*} g = \beta \bar g , \beta \in { \cal B } : = ( \beta _ 1 , \beta _ 2 ) , 0 < \beta _ 1 < \beta _ 2 < + \infty \ , . \end{align*}


\begin{align*} a _ { n } ^ { i } \mid 0 \rangle & = b _ { n } ^ { i j } \mid 0 \rangle = c _ { n } ^ { i j } \mid 0 \rangle = 0 \quad ( n > 0 ) , \\ p _ { a ^ i } \mid 0 \rangle & = p _ { b ^ { i j } } \mid 0 \rangle = p _ { c ^ { i j } } \mid 0 \rangle = 0 . \end{align*}


\begin{align*} X ( x , t ) = 1 + \sum _ { z \in \Z } \sum _ { s = 1 } ^ { t } p ( x - z , t - s ) \xi ( z , s ) \Gamma ( z , s ) . \end{align*}


\begin{align*} \Phi _ N : = \max \left \{ \frac { 1 + m + \frac { Q _ { N - 1 } } { Q _ N } } { k + \ell m + \ell \frac { Q _ { N - 1 } } { Q _ N } } \ : \ \left ( \begin{smallmatrix} \ell \\ k \end{smallmatrix} \right ) \in \mathcal { S } ( N , m ) \ \ 0 \leq m < a _ { N + 1 } \right \} \ , . \end{align*}


\begin{align*} | A ( L ) [ \ell ^ \infty ] | = | A ( L _ \ell ) [ \ell ^ \infty ] | \leq C \cdot [ L _ \ell : K ] ^ { \gamma _ { A , \ell } } \leq C \cdot [ L _ \ell : K ] ^ { \xi _ { A } } \end{align*}


\begin{align*} y ' _ i = \left \{ \begin{array} { l l } x _ i & \mbox { i f } i \in I \setminus A , \\ y _ i & \mbox { i f } i \in A . \end{array} \right . \end{align*}


\begin{align*} d x ( t ) = ( V ^ 0 x ) ( t ) d t + \sum \limits _ { j = 1 } ^ m ( V ^ j ) ( t x ) d W _ j ( t ) \ ( t \in [ a , b ] ) \ \ \ \mbox { a n d } \ \ \ x ( a ) = x _ 0 , \end{align*}


\begin{align*} L \left [ \partial _ { x } \left ( \phi P _ { n } P _ { n + 1 } \right ) \right ] = L \left [ \psi P _ { n } P _ { n + 1 } \right ] . \end{align*}


\begin{align*} \mu : = \sum _ { n \in \N } n \ , \delta _ { 5 ^ { n + 1 } \Z + 2 \cdot 5 ^ n } \end{align*}


\begin{align*} f ( x _ 1 , y _ 1 , [ n \alpha ] ) = ( [ n \beta ] - y _ 1 ) - ( [ n \alpha ] - x _ 1 ) \end{align*}


\begin{align*} g ( x ) : = f ( x ) - \sum _ { h \le k } \frac { f ^ { ( h ) } ( 0 ) } { h ! } x ^ { h } . \end{align*}


\begin{align*} \alpha ( t + s , x ) = \alpha ( t , \alpha ( s , x ) ) = \alpha ( t , \alpha ( s ' , x ) ) = \alpha ( s ' , \alpha ( t , x ) ) = \alpha ( s ' , \alpha ( t ' , x ) ) = \alpha ( t ' + s ' , x ) \ , . \end{align*}


\begin{align*} = \Re \left \langle \Pi ^ { l } \overline { \Pi ^ { k } } \varphi , \Pi ^ { l } \Pi ^ { k } \varphi \right \rangle \geq - \frac { 1 } { 2 } ( | | \Pi ^ { l } \overline { \Pi ^ { k } } \varphi | | _ { 2 } ^ { 2 } + | | \Pi ^ { l } \Pi ^ { k } \varphi | | _ { 2 } ^ { 2 } ) . \end{align*}


\begin{align*} \small \mathbf { y } _ { i , j , t } = \begin{cases} \mathbf { H } _ { i , j , t } \mathbf { x } _ { i , j , t } + \mathbf { n } _ { i , j , t } , ( i , j ) \in \mathbb { S } _ b , b \notin \mathbb { I } , \\ \mathbf { H } _ { i , j , t } \mathbf { x } _ { i , j , t } + \mathbf { H _ I } _ { i , j , t } \mathbf { x _ I } _ { i , j , t } + \mathbf { n } _ { i , j , t } , ( i , j ) \in \mathbb { S } _ b , b \in \mathbb { I } , \end{cases} \end{align*}


\begin{align*} q _ { n + 1 } ^ { 0 , \Delta t } = q _ n ^ { 0 , \Delta t } + \Delta t f ( q _ n ^ { 0 , \Delta t } ) + \sigma ( q _ n ^ { 0 , \Delta t } ) \Delta \beta _ n , \end{align*}


\begin{align*} \begin{aligned} & | \bar u ( x , t ) - \bar { H } ( x , t ) | \leq C _ k | ( x , t ) | ^ { k + 1 + \alpha } , ~ ~ \forall ~ ( x , t ) \in Q _ { 1 } ^ + , \\ & \| \bar { H } \| \leq C _ k , \\ & | \bar { H } _ t - \bar F ( D ^ 2 \bar { H } , x , t ) | \leq C _ k | ( x , t ) | ^ { k } , ~ ~ \forall ~ ( x , t ) \in Q _ { 1 } ^ + , \\ \end{aligned} \end{align*}


\begin{align*} \nabla _ y \left ( \frac { i } { 4 } H ^ { ( 1 ) } _ 0 ( k _ { + } \vert x - y \vert ) \right ) = \frac { e ^ { i k _ { + } \vert x \vert } } { \sqrt { \vert x \vert } } e ^ { - i \frac { \pi } 4 } { \sqrt { \frac { k _ + } { 8 \pi } } } \left ( \hat x e ^ { - i k _ { + } \hat x \cdot y } + O \left ( { \vert x \vert ^ { - 1 } } \right ) \right ) \end{align*}


\begin{align*} v = ( 1 - \eta ) ^ \alpha \end{align*}


\begin{align*} f _ i ^ + : = \max _ { x \in \overline { \Omega } _ i } f ( x ) f ^ { - } _ { i } : = \min _ { x \in \overline { \Omega } _ i } f ( x ) . \end{align*}


\begin{align*} K _ N ( ( a _ 1 , \dots , a _ N ) , ( b _ 1 , \dots , b _ N ) ) = e ^ { i \sum _ { x = 1 } ^ N a _ x b _ x } \end{align*}


\begin{align*} \mathcal { H } ( a + \mathbf { i } b + \mathbf { j } c + \mathbf { k } d ) = \begin{pmatrix} a & - b & - c & - d \\ b & a & - d & c \\ c & d & a & - b \\ d & - c & b & a \end{pmatrix} \ ; . \end{align*}


\begin{align*} \lim \limits _ { t \rightarrow + \infty } P _ \infty ( t * u ) = - \infty . \end{align*}


\begin{align*} K [ z , z ' , z '' , \ldots ] = K [ z , z ^ { \phi } , z ^ { \phi ^ 2 } , \ldots ] : = K [ z _ i , z ^ { \phi } _ i , z ^ { \phi ^ 2 } _ i , \ldots | i \in \{ 0 , \ldots , n \} ] . \end{align*}


\begin{align*} M ( x ) : = \max \{ a ( x ) , b ( x ) \} , x \in \R ^ N . \end{align*}


\begin{align*} p ( x ) \leq ^ { \mathbb { Y } } y , z \in \mathsf { d o m } ( p ) x \leq ^ { \mathbb { X } } z y = p ( z ) ; \end{align*}


\begin{align*} W = W ' \cap ( W ' ) ^ { - 1 } \ , . \end{align*}


\begin{align*} \begin{cases} \begin{aligned} d X ( t ) & = F ( t , X ( t ) ) d t + G ( t , X ( t ) ) d W ( t ) , t \geq 0 , \\ X ( 0 ) & = X _ 0 , \end{aligned} \end{cases} \end{align*}


\begin{align*} a \leq b \ , \ , \Longleftrightarrow \ , \ , a \land b = a . \end{align*}


\begin{align*} T _ t f = \Psi ( i _ { \mathsf { b } } ( t ) ) \mbox { f o r a l l } t \in G \ , . \end{align*}


\begin{align*} \| \phi \| _ { L ^ { \frac { 2 n p } { 2 n - p } } ( \mathbb C ^ n ) } \le \frac { C _ n } { p - 1 } \sum _ { j = 1 } ^ n \| \partial \phi / \partial \bar { z } _ j \| _ { L ^ p ( \mathbb C ^ n ) } , \ \ \ \forall \ , \phi \in C ^ 1 _ 0 ( \mathbb C ^ n ) , \end{align*}


\begin{align*} \Phi ( z ) = \frac { - \lambda _ { 2 } e ^ { \lambda _ { 1 } ( z - R ) } + \lambda _ { 1 } e ^ { \lambda _ { 2 } ( z - R ) } } { - \lambda _ { 2 } e ^ { \lambda _ { 1 } ( n - R ) } + \lambda _ { 1 } e ^ { \lambda _ { 2 } ( n - R ) } } , \end{align*}


\begin{align*} \begin{array} { l } d _ R ^ \nu ( \hat \alpha _ i , \alpha _ i ^ c ) = E ^ \nu \{ \min | \alpha _ i - \alpha _ i ^ c | ; 1 \} \le P ^ * _ { u _ i , \nu } \{ \alpha ^ c _ i \ne \alpha _ i \} < \frac { \varepsilon } { m } , \\ d _ E ^ \nu ( \hat x , \hat x ^ c ) = E ^ \nu \{ \min \| \alpha _ i - \alpha _ i ^ c \| _ E ; 1 \} < { \varepsilon } , \end{array} \end{align*}


\begin{align*} & \langle O p _ \mathcal { A } ( a ) f , g \rangle = \langle a , \mu ( \mathcal { A } ) ( f \otimes \bar { g } ) \rangle = \langle \mu ( \mathcal { A } ^ { - 1 } ) a , f \otimes \bar { g } \rangle , \\ & \langle O p _ \mathcal { B } ( b ) f , g \rangle = \langle b , \mu ( \mathcal { B } ) ( f \otimes \bar { g } ) \rangle = \langle \mu ( \mathcal { B } ^ { - 1 } ) b , f \otimes \bar { g } \rangle . \end{align*}


\begin{align*} h _ { \alpha } = - \Delta _ x - \frac { \alpha } { | x | } , u _ { \alpha } ( x ) = \frac { \alpha ^ { 3 / 2 } } { \sqrt { 8 \pi } } e ^ { - \alpha \frac { | x | } { 2 } } , e _ { \alpha } = - \frac { \alpha ^ 2 } { 4 } , \end{align*}


\begin{align*} \beta ^ r ( \xi ) = \frac { k _ 0 } { k _ 0 + i \kappa } \beta ( \xi ) , \gamma ^ r ( \xi ) = \frac { k _ 0 + i \kappa } { k _ 0 } \gamma ( \xi ) , \end{align*}


\begin{align*} E _ 1 ( s ) : = \exp ( s ) , E _ { k + 1 } ( s ) : = \exp \left ( E _ k ( s ) \right ) s \in \R , \ , k \ge \ , 1 . \end{align*}


\begin{align*} X _ { 0 , 0 } ( x , x _ { 1 } ) | I _ 1 \rangle \simeq \sum _ { n = 0 } ^ { | I _ 1 | } \sum _ { \substack { J ' \subseteq I _ 1 \\ | { J ' } | = n } } \mathcal { G } ^ { n } ( 0 , I _ 1 , { J ' } ) W _ { n + 1 , n + 1 } ( x ) \left | I _ 1 \setminus { J ' } \right \rangle \ , . \end{align*}


\begin{align*} \sum _ { \pi \in S _ n } \prod _ { j = 1 } ^ n | m _ { j \ , \pi ( j ) } | \leq C _ { \mu , d } ^ n \ , e ^ { - \frac { \mu } { 2 \sqrt { d } } D _ s ( X , Y ) } . \end{align*}


\begin{align*} d '^ { 2 } \circ d '^ { 1 } = 0 . \end{align*}


\begin{align*} C ( x ) : = \sum _ { k \ge 1 } c _ k x ^ k = \sum _ { k \ge 1 } \frac { C _ k } { k ! } x ^ k S ( x ) : = e ^ { C ( x ) } . \end{align*}


\begin{align*} \mathfrak { Z } _ { h } ( \mathfrak { g l } _ n ) = \{ v \in V _ { h } ( \mathfrak { g l } _ n ) \mid L ^ { + } ( z ) v = I v \} . \end{align*}


\begin{align*} \lim _ { t \searrow 0 } \Psi _ { k , t } c = 0 \end{align*}


\begin{align*} Y _ s ^ { t , x } = \phi ( B _ T ^ { t , x } ) + \int _ s ^ T Y _ r ^ { t , x } W ( d r , B _ { r - t } ^ { t , x } ) - \int _ s ^ t Z ^ { t , x } _ r d B _ r \ , , \end{align*}


\begin{align*} \breve { M } ^ r _ { k _ 0 } ( x , t , k ) = \overline { \breve { M } ^ r _ { - k _ 0 } ( x , t , - \bar { k } ) } , \vert k - k _ 0 \vert < \epsilon . \end{align*}


\begin{align*} V = \bigoplus _ { \alpha \in J _ { 1 } } \bigoplus _ { i = 1 } ^ { k _ { \alpha } } \C _ { e _ { \alpha } q ^ { i } } , W _ { 0 } = \bigoplus _ { \alpha \in J _ { 0 } } \C _ { e _ { \alpha } } , W _ { 1 } = \bigoplus _ { \alpha \in J _ { 1 } } \C _ { e _ { \alpha } } , \end{align*}


\begin{align*} \hat U c ^ \dagger _ i \hat U ^ \dagger = \sum _ j U _ { i j } c ^ \dagger _ j \ ; , \end{align*}


\begin{align*} \mu ( 1 / \bar z ) = - \frac { 1 } { 2 } \left ( z / \bar z \right ) ^ 2 ( 1 - | z | ^ 2 ) ^ 2 S f ( z ) ; \end{align*}


\begin{align*} A ( z , t ) = e ^ { z L ( t ) _ 1 } z ^ { - 2 H ( 0 ) } g , \end{align*}


\begin{align*} | \psi _ \zeta ^ \varepsilon | \leq c \varepsilon \sum _ { \xi = \zeta , \zeta _ 2 } \left ( | \bar { \xi } | + \Bigl | \overline { \nabla _ \Gamma \xi } \Bigr | \right ) \quad \partial _ \ell Q _ { \varepsilon , T } . \end{align*}


\begin{align*} \mathcal { A } ( x , t ) = \left ( { \begin{array} { c c } \mathcal { A } _ 1 ( x , t ) & \mathcal { A } _ 2 ( x , t ) \frac { t } { | t | } \\ 0 & I d _ { ( n - d ) \times ( n - d ) } \\ \end{array} } \right ) , \end{align*}


\begin{align*} g _ H \Bigl ( A X + B X ^ { \perp } , \nabla _ { A X + B X ^ \perp } X \Bigr ) = B ^ 2 g _ H \Bigl ( X ^ { \perp } , \nabla _ { X ^ \perp } X \Bigr ) = \cos ( \beta ) ^ 2 K _ 1 ( \eta ( t ) ) . \end{align*}


\begin{align*} c _ i ( \mathcal { E } _ \eta ) = e _ i ( \eta _ 0 c _ 1 ( \mathcal { L } _ 0 ) , \ldots , \eta _ N c _ 1 ( \mathcal { L } _ N ) ) \ , , \end{align*}


\begin{align*} \frac { d } { d s } \psi ^ x _ { - \ell } ( s ) = ( \lambda ^ c _ { x _ { - \ell } } \cdots \lambda ^ c _ { x _ { - 1 } } ) ^ { - 1 } D \Phi ^ c _ { x _ { - \ell } } \left ( ( \lambda ^ c _ { x _ { - \ell } } \cdots \lambda ^ c _ { x _ { - 1 } } ) ^ { - 1 } s \right ) e _ 1 . \end{align*}


\begin{align*} \left ( \frac { n } { 2 } - g _ { n } \right ) \allowbreak \left ( g _ { n } + g _ { n + 1 } \right ) \allowbreak \left ( g _ { n } + g _ { n - 1 } \right ) = z ^ { 2 } g _ { n } ^ { 2 } . \end{align*}


\begin{align*} \bar { \rho } ( x ) _ { \mid T } : = \bar { \rho } _ { \mid T } : = \min _ { x \in T } \rho ( x ) \forall T \in \mathcal { T } _ { h } . \end{align*}


\begin{align*} \widehat \mu ( \alpha ) \propto \prod _ { e \in V ( \widehat G ) } e ^ { w _ e \alpha _ e } \prod _ { x \in V ( G ) } \left ( \mathbf { 1 } _ { \{ \sum _ { e : e \ni x } \alpha _ e = 1 \} } + e ^ { \nu _ x } \mathbf { 1 } _ { \{ \sum _ { e : e \ni x } \alpha _ e = 0 \} } \right ) , \alpha \in \{ 0 , 1 \} ^ { V ( \widehat G ) } . \end{align*}


\begin{align*} Z _ n ( \lambda , T ) & : = \lim _ { m \to \infty , } Z _ { m , n } \left ( 1 - \frac { \lambda } { m } , 1 , \frac { T } { m } \right ) = \frac { e ^ { - \lambda } } { 2 } \sum _ { \theta \in \{ 0 , 1 \} ^ 2 } \prod _ { w ^ n = ( - 1 ) ^ { \theta _ 2 } } \left | e ^ { T w + \lambda } - ( - 1 ) ^ { \theta _ 1 } \right | . \end{align*}


\begin{align*} \lambda _ { \mathcal { E } } ( x _ 1 , \dots , x _ m ) : = r ! \ , \sum _ { E \in \mathcal { E } } \ ; \prod _ { i = 1 } ^ m \ ; \frac { x _ i ^ { E ( i ) } } { E ( i ) ! } . \end{align*}


\begin{align*} & \left [ \Psi ( x + v + Z ) , \Psi ( y + w + Z ' ) \right ] _ { \mathfrak J } \\ & = \left [ s ' s ( x ) + s ' i ( v ) + ( s ' s ) ^ { * } ( Z ) , s ' s ( y ) + s ' i ( w ) + ( s ' s ) ^ { * } ( Z ' ) \right ] _ { \mathfrak J } \\ & = \Psi \left ( [ x + v + Z , y + w + Z ' ] _ { \mathfrak d } \right ) . \end{align*}


\begin{align*} \mathfrak { m } _ 1 = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} , \mathfrak { m } _ 2 = \begin{pmatrix} i f / { \sqrt { 2 } } & - 1 \\ 1 & - i f / { \sqrt { 2 } } \end{pmatrix} \mathfrak { m } _ 3 = \begin{pmatrix} - j & - j \frac { i } { \sqrt { 2 } } \\ - j \frac { i } { \sqrt { 2 } } & j f \end{pmatrix} , \mathfrak { m } _ 4 = \begin{pmatrix} - j i & j \frac { 1 } { \sqrt { 2 } } \\ j \frac { 1 } { \sqrt { 2 } } & j i f \end{pmatrix} . \end{align*}


\begin{align*} \varphi _ p ( x ) = { | \Lambda _ L | } ^ { - 1 / 2 } e ^ { i 2 \pi p \cdot x } . \end{align*}


\begin{align*} \dim Z = \dim ( \textbf { d } ) \cdot M + m . \end{align*}


\begin{align*} L _ { z , j } ( f ) = \widetilde { L } _ { z , j } ( f ) = \int _ \Omega { f } \overline { g } _ { z , j } , \ \ \ \forall \ , f \in { A ^ p ( \Omega ) } . \end{align*}


\begin{align*} y ( \Psi _ f ( p ) ) = f ( p ) . \end{align*}


\begin{align*} g \geq 0 o n ( \{ \tau = 0 \} \cup \{ \xi = 0 \} \cup \{ \eta = 1 \} ) \cap \overline { D } . \end{align*}


\begin{align*} ( x , y ) = \eta _ a ( \bullet ) \ ; . \end{align*}


\begin{align*} Z _ r - Z _ { t _ i } & = ( D _ r ^ B \xi - D _ { t _ i } ^ B \xi ) + \int _ r ^ { t _ i } D _ r ^ B Y _ s { { W } } ( d s , B _ s ) \\ & + \int _ r ^ { t _ i } Y _ s \nabla _ x { { W } } ( d s , B _ s ) - \int _ r ^ { t _ i } D _ r ^ B Z _ s d B _ s , \ 0 \leq t _ i \leq r \leq T \\ & = \tilde Z _ 1 + \tilde Z _ 2 + \tilde Z _ 3 + \tilde Z _ 4 \ , . \end{align*}


\begin{align*} & W \notin C \implies \mbox { ( $ W = R $ a n d $ X = f ( S , R ) $ ) } \\ & W \in C \implies W = b ^ { - 1 } ( X ) \end{align*}


\begin{align*} \int _ { M } ( 2 h ^ { \mu _ { \theta } } + V + \frac { 1 } { \theta } \log \mu _ { \theta } ) g \ , d x = 0 . \end{align*}


\begin{align*} | \nu | = 1 , \ , \tau _ \varepsilon ^ i \cdot \nu = 0 \quad \overline { S _ T } , \bar { \nu } \cdot \nabla \overline { V _ \Gamma } = 0 \quad \overline { N _ T } , d = \varepsilon \bar { g } _ i \quad \Gamma _ \varepsilon ^ i ( t ) , \end{align*}


\begin{align*} \tilde { M } ( x , t , k ) = M ( x , t , k ) \delta ^ { - \sigma _ 3 } ( \xi , k ) . \end{align*}


\begin{align*} \| f \| _ { S _ K ^ p } : = \sup _ { y \in G } \left ( \frac { 1 } { | K | } \int _ { y + K } | f ( t ) | ^ p \dd t \right ) ^ { \frac { 1 } { p } } < \infty \ , . \end{align*}


\begin{align*} ( K _ { \theta _ 1 , \Phi } \cdot K _ { \theta _ 3 , \Phi } ) \cdot ( K _ { \theta _ 3 , \Phi } \cdot K _ { \theta _ 2 , \Phi } ) = \langle \theta _ 1 , \theta _ 3 \rangle \overline { \langle \theta _ 2 , \theta _ 3 \rangle } \cdot \ K _ { \theta _ 1 , \theta _ 3 , \Phi } \cdot K _ { \theta _ 3 , \theta _ 2 , \Phi } . \end{align*}


\begin{align*} \widetilde { F } ^ 1 \circ Q ( X ) = Q '^ 1 \circ \left ( \exp ( \alpha ) \vee \widetilde { F } ( X ) \right ) , \end{align*}


\begin{align*} \begin{aligned} & | P _ t - F _ 0 ( D ^ 2 P , x , t ) - P _ f | \leq C | ( x , t ) | ^ { k + l - 1 } , \\ & \mathbf { \Pi } _ { k + l } \left ( P ( x ' , P _ { \Omega } ( x ' , t ) , t ) \right ) \equiv \mathbf { \Pi } _ { k + l } \left ( P _ g ( x ' , P _ { \Omega } ( x ' , t ) , t ) \right ) , \end{aligned} \end{align*}


\begin{align*} \begin{array} { l } I _ { \epsilon } = \int _ { m } ^ { M } \left [ \int _ { \Omega } a _ { \epsilon , A ( y ) } ( x ) \ , | \nabla u | ^ { p ( x ) - 2 } \nabla u \nabla \phi _ { \epsilon } d x \right ] d y \\ = \int _ { m } ^ { M } \left [ \int _ { \Omega } | \nabla u | ^ { p ( x ) - 2 } \nabla u \nabla \Phi _ { \epsilon , y } d x \right ] d y \\ = \int _ { m } ^ { M } \left [ \int _ { \Omega } h ( x ) \Phi _ { \epsilon , y } d x \right ] d y \geq 0 . \end{array} \end{align*}


\begin{align*} \Omega _ { \frac { q } { N } } ^ { - 1 } ( \lambda ) \Omega _ { \frac { 1 } { N } } ^ { - 1 } ( \lambda ) = \sum _ { r = 0 } ^ \infty \left ( - \frac { 1 } { N } \right ) ^ r \sum _ { s = 0 } ^ r q ^ s f _ s ( \lambda ) f _ { r - s } ( \lambda ) . \end{align*}


\begin{align*} \left | \int _ { D } [ ( u _ t - B _ t ) \cdot ( u _ t - B _ t ) - \alpha ] \dd x \right | = & \bigg | - \int _ { t } ^ { + \infty } \int _ { D } b ( u _ r ) \cdot ( u _ r - B _ r ) \dd x \dd r \bigg | \ , . \end{align*}


\begin{align*} V ( s ) = \sup _ { a \in A ( s ) } [ r ( s , a ) + \gamma \varsigma ( V ( T ( s , a , W ) ) ] , s \in S , \end{align*}


\begin{gather*} L \left [ \partial _ { x } \left ( \phi p \right ) \right ] = { \displaystyle \int \limits _ { - z } ^ { z } } \partial _ { x } \left ( \phi p \right ) e ^ { - x ^ { 2 } } d x \\ = \left [ \phi \left ( x \right ) p \left ( x \right ) e ^ { - x ^ { 2 } } \right ] _ { - z } ^ { z } - { \displaystyle \int \limits _ { - z } ^ { z } } - 2 x \phi \left ( x \right ) p \left ( x \right ) e ^ { - x ^ { 2 } } d x = L \left [ 2 x \phi p \right ] , \end{gather*}


\begin{align*} I = ( \alpha _ 1 , . . . , \alpha _ r ) . \end{align*}


\begin{align*} E _ { k , n } ( x _ 1 , \ldots , x _ k ) : = \prod _ { 1 \leq r \leq k } x _ r ^ n \prod _ { 1 \leq r \ne s \leq k } ( x _ r - q ^ { - 2 } x _ s ) \ , . \end{align*}


\begin{align*} b _ { i , j } \ , \coloneqq \ , \begin{cases} d _ { i , 0 , j } + d _ { i , 1 , j } + \dots + d _ { i , N , j } , & j \ne i , \\ d _ { i , 0 , i } + d _ { i , 1 , i } + \dots + d _ { i , N , i } - m _ i - 1 , & j = i . \end{cases} \end{align*}


\begin{align*} \Tilde { X } : = \left ( X \setminus D ^ 2 \times \S ^ 2 \right ) \cup _ \tau D ^ 2 \times \S ^ 2 , \end{align*}


\begin{align*} \mathcal { A } = \begin{pmatrix} A _ { 1 1 } & I _ { d \times d } - A _ { 1 1 } & A _ { 1 3 } & A _ { 1 3 } \\ A _ { 2 1 } & - A _ { 2 1 } & I _ { d \times d } - A _ { 1 1 } ^ T & - A _ { 1 1 } ^ T \\ 0 _ { d \times d } & 0 _ { d \times d } & I _ { d \times d } & I _ { d \times d } \\ - I _ { d \times d } & I _ { d \times d } & 0 _ { d \times d } & 0 _ { d \times d } \end{pmatrix} , \end{align*}


\begin{align*} w ( x ) : = \max \left \{ \ell ^ { n / ( n - p ) } , \frac { ( { \rm d i s t } ( x , \partial \Omega ) ) ^ { n / ( n - p ) } } { ( \sup | b | ) ^ { n / ( n - p ) } } \right \} , \end{align*}


\begin{align*} d \pi ( y ) = \frac { 1 } { { z } ^ { * } } \exp \left ( - V ( y ) \right ) \ , d x , \end{align*}


\begin{align*} \prod _ { i = 1 } ^ l v _ { \gamma _ { j _ i } } = v _ { \gamma _ { j _ 1 } } ^ l , \end{align*}


\begin{align*} \mathcal { R } _ N = f _ 0 - P _ N f . \end{align*}


\begin{align*} g _ 1 ( x ) = P ( x ) \exp \big ( - 2 \pi i \vartheta \eta ( 1 - x ) \big ) \quad g _ 2 ( x ) = P ( x ) \exp \big ( 2 \pi i \vartheta \eta ( 1 - x ) \big ) . \end{align*}


\begin{align*} & v ^ * _ { \tilde T , j } = q ^ * _ { \tilde T } \cdot y ^ * _ j + ( 1 - q ^ * _ { \tilde T } ) \cdot 0 = q ^ * _ { \tilde T } \cdot v ^ 3 _ { \tilde T , j } + ( 1 - q ^ * _ { \tilde T } ) \cdot v ^ 2 _ { \tilde T , j } , \\ & \bar v ^ * _ { \tilde T , j } = q ^ * _ { \tilde T } \cdot 0 + \bar q ^ * _ { \tilde T } \cdot y ^ * _ j = q ^ * _ { \tilde T } \cdot \bar v ^ 3 _ { \tilde T , j } + ( 1 - q ^ * _ { \tilde T } ) \cdot \bar v ^ 2 _ { \tilde T , j } , \end{align*}


\begin{align*} Y ^ { ( r ) } = \biggl ( \prod _ { v \in P ( r ) } K _ v \biggr ) / R _ r \end{align*}


\begin{align*} \overline \nabla = ( \nabla _ x , \nabla _ \varphi , \partial _ r ) . \end{align*}


\begin{align*} \dim Z = \dim ( \textbf { d } ) \cdot M + m . \end{align*}


\begin{align*} z _ 1 z _ 2 = \prod _ { j = 0 } ^ 2 f _ j . \end{align*}


\begin{align*} \rho _ \zeta ^ \varepsilon ( x , t ) = \bar { \zeta } ( x , t ) - k _ d ^ { - 1 } d ( x , t ) \Bigl ( \overline { V _ \Gamma \zeta } \Bigr ) ( x , t ) + \frac { 1 } { 2 } d ( x , t ) ^ 2 \bar { \zeta } _ 2 ( x , t ) , ( x , t ) \in \overline { Q _ { \varepsilon , T } } , \end{align*}


\begin{align*} \begin{gathered} | A _ i | = \operatorname { r a n k } ( A _ I ) \leq | i | \ ; , \\ | A _ o | = \operatorname { r a n k } ( A _ O ) \leq | o | \ ; . \end{gathered} \end{align*}


\begin{align*} \sum _ { k = 1 } ^ n \alpha ^ k L \left ( \beta ^ k \right ) \le C \alpha ^ n L ( \beta ^ n ) \alpha > 1 , \beta \ge 1 . \end{align*}


\begin{align*} \dim ( M ) = m = \dim _ { ( \textbf { d } ) } ( M ) . \end{align*}