\begin{align*} T ( b ) - T ( a ) = \int _ I D T ( p ( t ) ) u \ d t . \end{align*}


\begin{align*} \frac { \partial } { \partial x _ i } ( \phi ^ { - 1 } ) _ 0 ( x _ j ) & = \delta _ { i j } \\ \frac { \partial } { \partial x _ i } ( \phi ^ { - 1 } ) _ 1 ( x _ j ) & = 0 \end{align*}


\begin{align*} M : = G - \frac { g _ { 1 , 1 } + \cdots + g _ { d , d } } { d } \ , I _ d , \end{align*}


\begin{align*} \boldsymbol { \theta } ^ { i + 1 } = \boldsymbol { \theta } ^ { i } + \tau ^ { i } \nabla _ { \boldsymbol { \theta } } Q ( \boldsymbol { \theta } ^ { i } ; \boldsymbol { \theta } ^ { i } ) , \end{align*}


\begin{align*} R ^ { n , 4 } : = n ^ { 2 \alpha + 1 } \int _ { \tau _ 1 } ^ { \tau _ 2 } \gamma _ s \left [ \left ( \int ^ { s } _ 0 \psi _ { n , 1 } ( u , s ) d W _ u \right ) ^ 2 - \int ^ { s } _ 0 \psi _ { n , 1 } ^ 2 ( u , s ) d u \ , \right ] d s \rightarrow 0 , \end{align*}


\begin{align*} h | _ { D ^ \vee _ 2 } = h _ 1 : D ^ \vee _ 2 \rightarrow \mathbb C \end{align*}


\begin{align*} \begin{aligned} ( k , v ) \cdot _ 0 ( x , 0 ) \cdot _ 0 ( k , v ) ^ { - 1 } = & ( k , v ) \cdot _ 0 ( x , 0 ) \cdot _ 0 ( k ^ { - 1 } , - k \cdot v ) \\ = & ( k \cdot x , x ^ { - 1 } \cdot v ) \cdot _ 0 ( k ^ { - 1 } , - k \cdot v ) \\ = & ( k \cdot x \cdot k ^ { - 1 } , k x ^ { - 1 } \cdot v - k \cdot v ) . \end{aligned} \end{align*}


\begin{align*} E _ + : = \bigoplus _ { i = 1 } ^ k L _ { i , + } , E _ - : = \bigoplus _ { i = 1 } ^ k L _ { i , - } . \end{align*}


\begin{align*} \left \lbrace \begin{array} { c } \lambda _ { + } \phi ( \pm a / 2 ) = - \lambda _ { - } \phi ( \pm a / 2 ) \\ \lambda _ { + } \partial _ { x } \phi ( \pm a / 2 ) = - \lambda _ { - } \partial _ { x } \phi ( \pm a / 2 ) \end{array} \right . \Rightarrow \left \lbrace \begin{array} { c } \phi ( a / 2 ) = \phi ( - a / 2 ) = 0 \\ \partial _ { x } \phi ( a / 2 ) = \partial _ { x } \phi ( - a / 2 ) = 0 \end{array} \right . , \end{align*}


\begin{align*} u ( x ) = \frac { a } { x } + b , \end{align*}


\begin{align*} \sum _ { n = \ell + 1 } ^ \infty \| \widetilde { B _ 0 } S _ n ( t ) \| _ 1 \leq \psi _ 0 * \theta _ 1 ( t ) , \end{align*}


\begin{align*} \mathrm { v o l } ( N _ d ( \mathrm { C H } ( S ) ) \cap B ( x , \rho ) ) & \leq \sum _ { r = 0 } ^ { \lfloor \rho \rfloor } \mathrm { v o l } ( \mathrm { C H } ( K ) \cap \mathrm { A n } ( r ) ) \\ & \lesssim e ^ { a \rho \beta } V \left ( 1 + C '' \right ) , \end{align*}


\begin{align*} \forall t > t _ 0 , \int _ { t _ 0 } ^ t \dot \kappa ^ { \epsilon , L } _ s \ , d s \geq \log t - \log t _ 0 - \bar \chi / t _ 0 = \log ( t / t _ 0 ) - C ( \lambda , \mu , \bar \chi ) . \end{align*}


\begin{align*} M _ b = \nu _ b . \end{align*}


\begin{align*} q ( A \times Y ) = \mu ( A ) , q ( X \times B ) = \nu ( B ) , \end{align*}


\begin{align*} \gamma _ s : = \mathbf { 1 } _ { [ 0 , t _ 1 \wedge t _ 2 ] } ( s ) ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } . \end{align*}


\begin{align*} E _ c ( \Psi _ { t , \eta _ i } ^ { q ( t ) } ) ( 0 , h ) = v ' \Longleftrightarrow \begin{pmatrix} 0 \\ h \end{pmatrix} = P _ c ( \Psi _ { t , \eta _ i } ^ { q ( t ) } ) \cdot v ' , \end{align*}


\begin{align*} { \rm w t } ^ 0 _ { \rm c l } ( A ) & = { \rm c l } ( { \rm w t } ^ 0 ( A ) ) = \left ( \sum _ { j \in \Z } a _ { 1 j } \right ) { \rm c l } ( \epsilon _ 1 ) + \dots + \left ( \sum _ { j \in \Z } a _ { m j } \right ) { \rm c l } ( \epsilon _ m ) \in P _ { \rm c l } ^ 0 , \end{align*}


\begin{align*} L _ 0 ^ \lambda = - \Delta _ z + \tfrac 1 4 \lambda ^ 2 | z | ^ 2 - i \lambda \sum _ { j = 1 } ^ n ( a _ j \partial _ { b _ j } - b _ j \partial _ { a _ j } ) , \end{align*}


\begin{align*} \tilde { \lambda } _ i ^ k = \lambda _ { i } ( \sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) ) \to \lambda _ { i } ( - X ( x ^ * ) ) = 0 , \end{align*}


\begin{align*} \begin{bmatrix} P A + A ^ { T } P & P B - A ^ { T } C ^ { T } \\ B ^ { T } P - C A & - ( C B + B ^ { T } C ^ { T } ) \end{bmatrix} = \begin{bmatrix} - L ^ { T } L & - L ^ { T } W \\ - W ^ { T } L & - W ^ { T } W \end{bmatrix} \leq 0 . \end{align*}


\begin{align*} \dot x ( t ) = X _ H ( x ( t ) ) , \end{align*}


\begin{align*} \mathfrak { M } _ { i } ( a _ { i } , q _ { i } ) = \left \{ \alpha _ { i } : \left | \alpha _ { i } - \frac { a _ { i } } { q _ { i } } \right | \leq \frac { 1 } { q _ { i } Q _ { i } } \right \} \end{align*}


\begin{align*} \textrm { t a n } \Theta ( Q , \widetilde Q ) = \{ \tan ( \cos ^ { - 1 } ( \zeta _ 1 ) ) , \tan ( \cos ^ { - 1 } ( \zeta _ 2 ) ) , \cdots , \tan ( \cos ^ { - 1 } ( \zeta _ r ) ) \} . \end{align*}


\begin{align*} v ( \tau , w ) \leq v ( \tau ' , w ) & v ( \Delta ( \tau \rightarrow \tau ' ) , w ) = 1 , \\ v ( \tau , w ) > v ( \tau ' , w ) & v \left ( { \sim } \Delta ( \tau ' \rightarrow \tau ) , w \right ) = 1 . \end{align*}


\begin{align*} S _ 2 = O _ { \alpha , l } \Big ( \frac { 1 } { \sqrt { n } } \prod \limits _ { 1 \leq j \leq l } \tau _ 2 ( u _ j / d _ j ) \Big ) . \end{align*}


\begin{align*} H _ 0 ( t ) = 2 \sqrt { \pi } t ^ { s } \frac { \Gamma ( \frac { s + 1 } { 2 } ) } { \Gamma ( \frac s 2 ) } . \end{align*}


\begin{align*} \lim _ { n \to \infty } \int _ 0 ^ L \abs { u ^ n } ^ 2 d x = 0 . \end{align*}


\begin{align*} \sum \limits _ { i } \alpha _ i = 0 \end{align*}


\begin{align*} \lim _ { \gamma _ { + } \rightarrow 0 ^ { + } } \mathrm { P } _ { \Phi + \Phi ^ { \mathfrak { b } _ { + } , \gamma _ { + } } + \Phi ^ { - \mathfrak { b } _ { - } , \gamma _ { - } } } = \mathrm { P } _ { ( \Phi + \Phi ^ { - \mathfrak { b } _ { - } , \gamma _ { - } } , \mathfrak { a } _ { \mathfrak { b } _ { + } } ) } ^ { \sharp } \end{align*}


\begin{align*} \langle \nabla _ j \nabla _ i \Psi _ t - \nabla _ l \Psi _ t \Gamma _ { i j } ^ l , \nabla _ m \Psi _ t \rangle = - \Gamma ^ m _ { i j } + O ( t ) . \end{align*}


\begin{align*} \P ( \widetilde { X } _ { 1 + d } = 1 , \widetilde { T } _ Y \leq d ) \pi ( ( 0 , k ] ) \leq \P ( \widehat { X } _ { 1 + d } = 1 ) \pi ( ( 0 , k ] ) = \big ( \P _ { \pi } ( X _ 1 = 1 ) \big ) ^ 2 . \end{align*}


\begin{align*} | g _ { 2 3 } | = 1 . \end{align*}


\begin{align*} \mathbf { u } ' _ 1 + \cdots + \mathbf { u } ' _ { \ell } = \begin{cases} \boldsymbol { 0 } , & \\ a _ 1 \mathbf { u } _ 1 + \cdots + a _ { r } \mathbf { u } _ { r } , & { } \end{cases} \end{align*}


\begin{align*} \underline { g } = \frac { 1 } { p } \sum _ { g ' \in \mathbb { Z } _ p } \sum _ { k = 0 } ^ { p - 1 } \xi _ p ^ { ( g - g ' ) k } \underline { g ' } = \sum _ { k = 0 } ^ { p - 1 } \xi _ p ^ { g k } e _ k \end{align*}


\begin{align*} \frac { \sinh ( k ' a ) } { k ' a } = \mp 1 , \end{align*}


\begin{align*} [ ( x _ i , X _ \lambda ) , ( x _ j , X _ \beta ) ] _ \mathcal { A } : = \left ( [ x _ i , x _ j ] _ \mathfrak { g } , [ X _ \lambda , \varrho ( x _ j ) ] - [ X _ \beta , \varrho ( x _ i ) ] - \varphi ( x _ i , x _ j ) + [ X _ \lambda , X _ \beta ] \right ) \end{align*}


\begin{align*} \| \varphi \| = \lim \limits _ { n \rightarrow \infty } \limsup \limits _ { t \rightarrow \infty } \| P _ { \delta } ( W _ { n , m _ 0 ; \textrm { s u r } } ) e ^ { - i t H _ 0 } \varphi + P _ { \delta } ( W _ { n , m _ 0 ; \textrm { f a r } } ) e ^ { - i t H _ 0 } \varphi \| < \varepsilon \end{align*}


\begin{align*} N _ t ^ { n , 1 } = n ^ { \alpha + \frac 1 2 } \int _ 0 ^ t \psi _ { n , 1 } ( s , t ) d W _ s . \end{align*}


\begin{align*} 1 - \frac { \Psi x _ i } { x _ i } + 1 - \frac { y _ i } { \Psi x _ i } - \left ( 1 - \frac { y _ i } { x _ i } \right ) = \frac { \Psi ( 1 - \Psi ) x _ i - ( 1 - \Psi ) y _ i } { \Psi x _ i } \geq \frac { \Psi ( 1 - \Psi ) x _ i - ( 1 - \Psi ) \Psi ^ 2 x _ i } { \Psi x _ i } = ( 1 - \Psi ) ^ 2 \geq 0 \end{align*}


\begin{align*} A = \sum _ { k = 0 } ^ n A _ k \end{align*}


\begin{align*} \varphi _ { i } ( \gamma _ { i } ) : = a _ { i } \gamma ^ { b _ { i } } _ { i } + c _ { i } \gamma _ { i } + d _ { i } , \forall i \in \mathcal { N } , \end{align*}


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


\begin{align*} ( - 1 ) ^ { ( m - n ) / 2 } \frac { \mathbf { c } _ n ( \lambda ) } { \mathbf { c } _ m ( \lambda ) } \varphi ( \lambda ) = \varphi ( - \lambda ) , \ ; \ ; \lambda \in \mathfrak { a } ^ * _ \mathbb { C } , n , m \in \mathbb { Z } . \end{align*}


\begin{align*} \mathcal H = \mathcal H _ + \oplus \mathcal H _ - , \end{align*}


\begin{align*} f ( w , x , y , z ) = f ( y , z , w , x ) = f ( y , z , x , w ) = f ( x , w , y , z ) . \end{align*}


\begin{align*} \Delta f ( \lambda ) = - 2 \sum _ { k = 1 } ^ { N - 1 } \det \left ( ( \lambda I _ N - H ) _ { k k + 1 | k k + 1 } \right ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : p o s t c o l l i s i o n s i g m a } \left \{ \begin{array} { r c l } v ' & = & \displaystyle { \frac { v + v _ * } { 2 } + \frac { | v - v _ * | } { 2 } \sigma } \\ v _ * ' & = & \displaystyle { \frac { v + v _ * } { 2 } - \frac { | v - v _ * | } { 2 } \sigma } \end{array} \right . , \end{align*}


\begin{align*} 0 \le \delta \leq ( d - \alpha ) / 2 \kappa = \kappa _ \delta = \frac { 2 ^ \alpha \Gamma ( ( \delta + \alpha ) / 2 ) \Gamma ( ( d - \delta ) / 2 ) } { \Gamma ( \delta / 2 ) \Gamma ( ( d - \delta - \alpha ) / 2 ) } . \end{align*}


\begin{align*} V ( f ^ { 2 } ) g ( \operatorname { g r a d } ( \| X \| ^ { 2 } ) , W ) = W ( f ^ { 2 } ) g ( \operatorname { g r a d } ( \| X \| ^ { 2 } ) , V ) . \end{align*}


\begin{align*} C _ { ( m ) } : = \big ( - \Delta ^ { \epsilon } + m ^ 2 \big ) ^ { - 1 } , \end{align*}


\begin{align*} f = f _ 0 + f _ 1 Q + \ldots + f _ r Q ^ r \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } n ^ { \alpha + \frac 1 2 } \int _ 0 ^ { \tau } ( t - s ) ^ \alpha \sigma ' ( X _ s ) \left ( \int ^ s _ { \eta _ n ( s ) } \left ( s - \eta _ n ( u ) \right ) ^ { \alpha } \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) d s = 0 \end{align*}


\begin{align*} f ( \mathbf { n } ) = \sum _ { i = 1 } ^ e a _ i u _ i ( \mathbf { n } ) , \end{align*}


\begin{align*} \mathcal { T } _ { \left \{ C _ { i } \right \} _ { i = 1 } ^ { r } } : = 2 ^ { - 1 } \left ( I d + \mathcal { V } _ { \left \{ C _ { i } \right \} _ { i = 1 } ^ { r } } \right ) . \end{align*}


\begin{align*} t r ( A ^ n ) = t ^ n + \sum _ { r = 1 } ^ { [ n / 2 ] } ( - 1 ) ^ r \frac { n } { r } { n - r - 1 \choose r - 1 } t ^ { n - 2 r } \delta ^ r , \end{align*}


\begin{align*} t _ i & = \prod _ { \pi ( x ) = i } t _ x , \end{align*}


\begin{align*} \lim _ { t \to 0 } \frac { \sinh ( \alpha ( t v ) ) } { t } = \alpha ( v ) . \end{align*}


\begin{align*} \limsup _ { R \to 0 ^ + } \frac { e ( B _ R ( x _ 0 ) ) } { u ( B _ R ( x _ 0 ) ) } = \lim _ { \ell \to \infty } \frac { e ( B _ { \rho _ \ell } ( x _ 0 ) ) } { u ( B _ { \rho _ \ell } ( x _ 0 ) ) } , \end{align*}


\begin{align*} \begin{aligned} \mathcal { W } _ t - \mathcal { W } _ { r r } & \leq A _ 1 \mathcal { W } _ r + B ( \mathcal { W } + m ) + C _ 1 ( \mathcal { Z } + n ) + E \\ & \leq K _ 1 | \mathcal { W } _ r | + B \mathcal { W } + C _ 1 \mathcal { Z } \end{aligned} \end{align*}


\begin{align*} \frac { C } { 2 } \log ( M ) + \frac { C } { 2 } & \le \frac { C } { 2 } \log ( M ) + \frac { C } { 2 } \log ( M ) \\ & = C \log ( M ) \le n . \end{align*}


\begin{align*} Z ^ { ( n ) } _ t : = n ^ { - 1 } \sum _ { i = 1 } ^ n \delta _ { X _ t ^ i } . \end{align*}


\begin{align*} [ \textbf { v } \ , , \ , \textbf { w } ] = \sum _ { j = 1 } ^ n v _ j w _ j , \end{align*}


\begin{align*} F _ { v - i , i } = \mathrm { c o n s t . } = C _ v \end{align*}


\begin{align*} J ( u ) = \int _ { 0 } ^ { \infty } \left \{ x ^ { T } Q _ c x + 2 x ^ { T } N _ c u + u ^ { T } R _ c u \right \} d t . \end{align*}


\begin{align*} H _ n ( s ) = \xi _ 2 ( s ) + \Omega ( s ) n ^ { - 2 } + O \Bigl ( n ^ { - 4 } \Bigr ) . \end{align*}


\begin{gather*} L ( f \otimes D , 1 ) : = { 2 \pi } \int _ 0 ^ \infty ( f \otimes D ) ( i t ) { \rm d } t . \end{gather*}


\begin{align*} \mu ( \varepsilon y _ { u + t h } - \varepsilon y _ u , \varepsilon v ) + \nu ( \mathsf { m } _ \delta ( \varepsilon y _ { u + t h } ) - \mathsf { m } _ \delta ( \varepsilon y _ { u } ) , \varepsilon v ) = ( t h , v ) , \forall v \in Y . \end{align*}


\begin{align*} \widehat { \lambda } ( n P ) = n ^ 2 \widehat { \lambda } ( P ) + v ( \psi _ { n } ( P ) ) - \frac { n ^ 2 - 1 } { 1 2 } v ( \Delta ) , \end{align*}


\begin{align*} \left | S _ { \kappa , 3 } ^ A \right | & = \sum _ { i , j \in \mathbb N \colon \atop { 3 ( i + j ) \leq \kappa } } \left | \left \{ [ n _ 0 , n _ 7 ] \colon n _ 0 + n _ 7 = \kappa - 3 ( i + j ) \right \} \right | = \sum _ { i , j \in \mathbb N \colon \atop { 3 ( i + j ) \leq \kappa } } ( \kappa - 3 ( i + j ) + 1 ) = a _ \kappa . \end{align*}


\begin{align*} \Omega ( \alpha , A , z + a f _ k ) = \{ z + a f _ k \} \cup \bigcup _ { n = 1 } ^ \infty \Omega ( \alpha _ n , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \end{align*}


\begin{align*} u ( x , t ) & = \pm { \sf U } _ \infty ( x ) + \Theta _ J ( x , t ) \\ & = \pm { \sf U } _ \infty ( x ) + K ( T - t ) ^ { \frac { \gamma } { 2 } + J } e _ J ( z ) | x | \sim \sqrt { T - t } . \end{align*}


\begin{align*} \Vert f \Vert _ { H _ w ^ 2 } ^ 2 : = \Vert f '' \Vert _ { L ^ 2 _ z } ^ 2 + \Vert ( 1 + z ) f ' \Vert _ { L ^ 2 _ z } ^ 2 + \Vert ( 1 + z ) f \Vert _ { L ^ 2 _ z } ^ 2 . \end{align*}


\begin{align*} \delta ( a ) = \left \{ \begin{aligned} & \left [ \sum _ { j = 0 } ^ \infty \beta _ j P _ { j , j + n } , a \right ] & & \textrm { i f } n \ge 0 \\ & \left [ \sum _ { j = 0 } ^ \infty \beta _ j P _ { j - n , j } , a \right ] & & \textrm { i f } n < 0 \end{aligned} \right . \end{align*}


\begin{align*} \delta _ R ( x , u ) : = ( R x , R ^ 2 u ) . \end{align*}


\begin{align*} E ( G / H ) = E ( G / ( G ^ 0 H ) ) E ( ( G ^ 0 H ) / H ) = 0 . \end{align*}


\begin{align*} \langle f , \eta ^ { \odot m } \rangle = 0 \mbox { f o r a l l } \eta \in A , \end{align*}


\begin{align*} d \Theta _ n ( A d _ { \phi ^ { - 1 } } ( e _ { \phi ( \lambda ) } ) ) = \frac { t ( \lambda , \lambda ) } { 2 } e _ { \theta ( \lambda ) } ^ \vee . \end{align*}


\begin{align*} u ( x , t ) \ge M _ n \left \{ \frac { t - x - R } { ( 1 + t - x ) ^ { 1 + b } } \right \} ^ { a _ n } \left \{ \frac { ( 2 x ) ^ { 1 - a } } { 1 + t + x } \right \} ^ { b _ n } \quad \mbox { i n } \ D \end{align*}


\begin{align*} \phi _ { \pm } = e ^ { k _ { \pm } x } , \end{align*}


\begin{align*} F ( c ) : = a _ 0 + a _ 1 N ( \beta ( c ) , c ) - c , c \in [ a _ 0 , \infty ) \end{align*}


\begin{align*} \bar A ( x ) & : = A ( x _ 0 + \Theta x ) \\ \bar f ( x ) & : = \Phi \Theta ^ { 2 } f ( x _ 0 + \Theta x ) . \end{align*}


\begin{align*} X _ { \Sigma , \Lambda } = \bigsqcup _ { i = 1 } ^ s \widetilde U _ { \tau _ i } / H , \end{align*}


\begin{align*} \mu : = \max ( \{ \abs { y } \mid y \in Y \} \cup \{ 2 \abs { g _ j } + 1 \mid 1 \leq j \leq m \} ) . \end{align*}


\begin{align*} ( L x ) ( v ) = \sum \limits _ { e \in E _ v } \frac { \delta _ E ( e ) } { \delta _ V ( v ) } \frac { 1 } { | e | ^ 2 } \sum \limits _ { u \in e } ( x ( u ) - x ( v ) ) , \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \infty } \frac { g ( n ) } { n } = \sum _ { n _ 1 , \ldots , n _ k = 1 } ^ { \infty } \frac { ( \mu * F ) ( n _ 1 , \ldots , n _ k ) } { n _ 1 \cdots n _ k } , \end{align*}


\begin{align*} \dim \mathcal { N } _ { } ( 2 m , \alpha , L ) = 2 \dim \mathcal { M } _ { } ( 2 m , \alpha , L ) = 2 m ( 2 m + 1 ) ( g - 1 ) + 2 m ^ 2 n . \end{align*}


\begin{align*} N ^ * = \bar { \rho } V _ F N ( g _ 0 + g _ 1 N ^ * , a _ 0 + a _ 1 N ^ * ) \geq 0 . \end{align*}


\begin{align*} \kappa _ t ( \alpha _ 1 , \dots , \alpha _ l ) : = \kappa ( w _ { \alpha _ 1 } ( t ) , \dots , w _ { \alpha _ l } ( t ) ) \end{align*}


\begin{align*} \tilde { \mathbf S } \psi _ k = \sum _ { k = 0 } ^ 1 [ \tilde { \mathbf S } ] _ { j , k } \psi _ j \ , \end{align*}


\begin{align*} \lim _ { | x | \rightarrow \infty } | x | ^ { s } \bar { u } _ { j } = 1 , \quad \ ; \ , \min _ { \overline { B } _ { 1 } } \bar { u } _ { j } = \varepsilon _ { 0 } ^ { s } j \rightarrow \infty , \quad \end{align*}


\begin{align*} \Delta _ { r a d } : = \frac { 1 } { 2 } \frac { \partial ^ 2 } { \partial r ^ 2 } + \frac { \alpha } { 2 } \frac { 1 } { \tanh ( \alpha r ) } \frac { \partial } { \partial r } . \end{align*}


\begin{align*} \inf \left \{ t \in [ 0 , 1 ] : \left ( \sqrt { x } \right ) ^ t \left ( \frac { x + 1 } { 2 } \right ) ^ { 1 - t } \le \frac { x - 1 } { \log x } , ( x > 0 ) \right \} = \frac { 2 } { 3 } \end{align*}


\begin{align*} \lim _ { n \to + \infty } \abs { \lambda ^ n } ^ { - 1 } \abs { \Lambda ^ n _ x ( 0 ) } \abs { u ^ n _ x ( 0 ) } = 0 . \end{align*}


\begin{align*} a _ i = \frac { ( \sigma _ Z ^ 2 \lambda ^ 2 / 2 + \rho / 2 ) ( \omega _ i ^ 2 - \omega _ { i - 1 } ^ 2 ) } { \lambda ( \omega _ i - \omega _ { i - 1 } ) } = \left ( \sigma _ Z ^ 2 \lambda + \frac { \rho } { \lambda } \right ) \cdot \frac { \omega _ i + \omega _ { i - 1 } } { 2 } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s a _ i ^ j A _ { a _ i } = f _ j | C | - 1 , \ \ j = 0 , 1 , \ldots , s , \end{align*}


\begin{align*} V _ { \widehat { a } } \ = \ V _ { \widehat { b m ( a ) } } , \end{align*}


\begin{align*} \Psi ( y ) = \int _ 0 ^ y \psi ( v ) \ d v \ , y \geq 0 \ , \end{align*}


\begin{align*} \int _ { \R ^ 3 } \omega \cdot A ( x ) e ^ { i x \cdot \xi } e ^ { i N ^ { - 1 } _ \omega ( - \omega \cdot A ) ( x ) } d x = \int _ { \R ^ 3 } \omega \cdot A ( x ) e ^ { i x \cdot \xi } d x , \end{align*}


\begin{align*} c _ { k + 1 } = c _ k \cdot \dfrac { A ( k ) } { B ( k + 1 ) } \ \ \ ( k \ge 0 ) \enspace . \end{align*}


\begin{align*} L ( t \circ \psi ( \lambda _ 1 , \cdot ) ) = l ^ \tau ( t ) \circ \psi ( \lambda _ 2 , \cdot ) . \end{align*}


\begin{align*} \lim _ { \zeta _ i \to z _ i } \left ( \frac { e _ n ( z , \zeta ) } { ( z _ i - \zeta _ i ) } \bigg | _ { z _ l = \zeta _ l , l \neq i } \right ) = e _ { n , i } ( z , z ) , \ ; z \in \Omega _ 0 . \end{align*}


\begin{align*} W ^ { \{ \alpha \} } : = W + \alpha p p ^ * , \alpha \in \R . \end{align*}


\begin{align*} \dot V ( t ) = - A _ n ^ 2 V ( t ) + A _ n F _ n ( U ( t ) ) , V ( 0 ) = U ( 0 ) , \end{align*}


\begin{align*} \label [ I e q ] { e q : n o n l i n e a r j u m p o p } L _ \mu \varphi ( x ) : = \int _ E \{ \varphi ( y ) - \varphi ( x ) \} P _ \mu ( x , \dd y ) . \end{align*}


\begin{align*} F _ { v - i , i } = 1 . \end{align*}


\begin{align*} F _ { n , \lambda } ( x ) = \sum _ { k = 0 } ^ { n } k ! S _ { 2 , \lambda } ( n , k ) x ^ { k } , ( n \ge 0 ) , ( \mathrm { s e e } \ [ 1 5 ] ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } n ^ { 2 \alpha + 1 } \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' ( X _ s ) ) ^ 2 \Psi ^ { n , 6 } _ s d s = 0 , \end{align*}


\begin{align*} \Phi ( \xi x ) ( \Omega _ \psi y ) & = L ( \eta ) ^ \ast ( \xi x \otimes \Omega _ \psi y ) \\ & = L ( \eta ) ^ \ast ( \xi \otimes x \Omega _ \psi y ) \\ & = L ( \eta ) ^ \ast ( \xi \otimes \Omega _ \psi \sigma ^ \psi _ { i / 2 } ( x ) y ) \\ & = L ( \eta ) ^ \ast L ( \xi \otimes \Omega _ \psi ) \Omega _ \psi \sigma ^ \psi _ { i / 2 } ( x ) y \\ & = \Phi ( \xi ) ( x \Omega _ \psi y ) \end{align*}


\begin{align*} \widehat { A f } ( \zeta ) = \hat A \hat g ( \zeta ) , \end{align*}


\begin{align*} \tau _ { x y } \eta \ : \ z \in V \ \longmapsto \ \begin{cases} \eta ( x ) - 1 & z = x \ , , \\ \eta ( y ) + 1 & z = y \ , , \\ \eta ( z ) & \end{cases} \end{align*}


\begin{align*} p ( n - k ( 3 k + 1 ) / 2 ) = M _ k ( n ) + \widetilde { \mathcal { P } } _ k ( n ) . \end{align*}


\begin{align*} B Z ^ { \nu } _ i + Z ^ { \nu t } _ i B = C _ i ^ { \nu } . \end{align*}


\begin{align*} \mathcal { A } _ { s } ( u , v ) = \lambda \int _ \Omega f ( x , u ) v d x \end{align*}


\begin{align*} \sum \limits _ { i } ^ { N } \alpha _ i = 0 \end{align*}


\begin{align*} X _ { t , s } ^ \theta ( x ) = x + \int _ { t } ^ s \mu ( X _ { t , r } ^ \theta ( x ) ) \ , d r + \int _ t ^ s \sigma ( X _ { t , r } ^ \theta ( x ) ) \ , d W _ r ^ \theta , \end{align*}


\begin{align*} \sigma u ( e _ 1 , \cdots , e _ m ) = \frac { 1 } { m ! } \sum \limits _ { \pi \in \Pi _ m } u ( e _ { \pi ( 1 ) } , \cdots , e _ { \pi ( m ) } ) . \end{align*}


\begin{align*} \partial _ { x _ { k } x _ { l } } ^ { 2 } \left ( \frac { 1 } { | x - y | ^ { n + \sigma p } } \right ) ( 0 ) = \frac { ( n + \sigma p ) [ ( n + \sigma p + 2 ) y _ { k } y _ { l } - \delta _ { k l } | y | ^ { 2 } ] } { | y | ^ { n + \sigma p + 4 } } . \end{align*}


\begin{align*} & b = z \sqrt { L / 2 } \\ \iff & z / b = \sqrt { 2 / L } \\ \iff & ( b - z ) / b = 1 - \sqrt { 2 / L } \end{align*}


\begin{align*} \Phi _ \rho ( u ) = A ( u ) - \rho B ( u ) , \rho \in I , \end{align*}


\begin{align*} L ^ { n , 2 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) ( s - \eta _ n ( s ) ) ^ \alpha ( W _ s - W _ { \eta _ n ( s ) } ) \left ( \int _ 0 ^ { \eta _ n ( s ) } \psi _ { n , 1 } ( u , s ) d W _ u \right ) d s , \end{align*}


\begin{align*} \mbox { f i n d \ } u _ { m } ( \alpha ) \in V _ { m } ( \alpha ) \mbox { s . t . } a ( u _ { m } ( \alpha ) , v _ { m } ; \alpha ) = f ( v _ { m } ; \alpha ) \forall v _ { m } \in V _ { m } ( \alpha ) , \end{align*}


\begin{align*} \mathbf { H } ^ { r } & = \mathbf { A } ( \boldsymbol { \theta } ) ( \boldsymbol { x } ^ { r } ) \mathbf { A } ( \boldsymbol { \theta } ) ^ { H } , \\ \mathbf { h } ^ { c } & = \mathbf { A } ( \boldsymbol { \theta } ) \boldsymbol { x } ^ { c } , \end{align*}


\begin{align*} \kappa _ 0 ( A ^ { \tt s t ' } ) = ( P _ 0 ^ { \tt s t ' } , Q ^ { \tt s t } _ 0 , \rho ) , \end{align*}


\begin{align*} F & = \bigcap _ { N = 1 } ^ { \infty } \bigcup _ { n = N } ^ { \infty } \left \{ ( x , y ) \in [ 0 , 1 ] ^ 2 : \begin{aligned} | T _ { \beta _ 1 } ^ n x - f _ 1 ( x ) | < \beta _ 1 ^ { - n \tau _ 1 ( x ) } \\ | T _ { \beta _ 2 } ^ n y - f _ 2 ( y ) | < \beta _ 2 ^ { - n \tau _ 2 ( y ) } \end{aligned} \right \} \\ & = \bigcap _ { N = 1 } ^ { \infty } \bigcup _ { n = N } ^ { \infty } \bigcup _ { w \in \Sigma _ { \beta _ 1 } ^ n , v \in \Sigma _ { \beta _ 2 } ^ n } J _ { n , \beta _ 1 } ( w ) \times J _ { n , \beta _ 2 } ( v ) . \end{align*}


\begin{align*} ( \alpha _ m , \beta _ m ) _ { \rho , E _ { \rho } ^ m , m } = \Phi _ { E _ { \rho } ^ m } ( \beta _ m ) ( \xi ) - \xi = \Phi _ { E _ { \rho } ^ n } ( \operatorname { N } _ { m , n } ( \beta _ m ) ) ( \xi ) - \xi = ( \alpha , \beta ) _ { \rho , E _ { \rho } ^ n , n } , \end{align*}


\begin{align*} d _ { 1 , \mathrm { r e l } } ^ G = \sum _ { n > 0 } x _ { i , - n } \frac { \partial } { \partial \varphi _ { i , - n } } - \frac { 1 } { 2 } \sum _ { p , q > 0 } ( - 1 ) ^ { \bar { x } _ i \bar { x } _ k + \bar { x } _ i + \bar { x } _ j } c _ { i , j } ^ k \varphi _ { k , - p - q } \frac { \partial } { \partial \varphi _ { i , - p } } \frac { \partial } { \partial \varphi _ { j , - q } } . \end{align*}


\begin{align*} \iota _ { \Lambda } ^ * \mathcal { W } = F _ \Lambda ^ * ( \mathcal { W } _ { v _ 1 ' } \boxplus \mathcal { W } _ { v _ 2 } ) . \end{align*}


\begin{align*} \sum _ { i = 0 , 1 } \big | T _ { k , j ; n } ^ { \mu , i } ( \mathfrak { m } , E ) ( t , x , \zeta ) + \hat { \zeta } \times T _ { k , j ; n } ^ { \mu , i } ( \mathfrak { m } , B ) ( t , x , \zeta ) \big | + \big | \widetilde { T } _ { k , j ; n } ^ { b i l ; \mu , 0 } ( \mathfrak { m } , E ) ( t , x , \zeta ) + \hat { \zeta } \times \widetilde { T } _ { k , j ; n } ^ { b i l ; \mu , 0 } ( \mathfrak { m } , B ) ( t , x , \zeta ) \big | \end{align*}


\begin{align*} T ( X , Y ) = \rho ( Y ) X - \rho ( X ) Y \end{align*}


\begin{align*} u ^ s _ 1 : = \mathcal { M } _ { - A _ 1 , q _ 1 } \mathcal { S } f _ 1 , u ^ s _ 2 : = \mathcal { M } _ { A _ 2 , q _ 2 } \mathcal { S } f _ 2 \mbox { i n } \R ^ 3 , \end{align*}


\begin{align*} \mathcal { I } = \left \{ [ a _ { k } , b _ { k } ] : k \in K \right \} \end{align*}


\begin{align*} - \lim _ { \epsilon \rightarrow 0 ^ { + } } \epsilon ^ { - 1 } \Im \varphi _ \mu ( x + i \epsilon ) = \int _ \mathbb { R } \frac { 1 + s ^ 2 } { ( x - s ) ^ 2 } \ , d \sigma _ \mu ( s ) , x \in \mathbb { R } , \end{align*}


\begin{align*} \mathsf { F } ' ( E ; H ) = \dfrac { H } { \abs { E } } - \dfrac { E } { \abs { E } ^ 3 } ( E : H ) . \end{align*}


\begin{align*} \rho ( \varepsilon ) : = \frac { 2 b d k \log ( \frac { 4 b d } { \varepsilon } ) } { \varepsilon ^ 2 } + \frac { 2 k ^ d \log ( \frac { 4 k ^ d } { \varepsilon } ) } { \varepsilon ^ 2 } + \frac { \log ( \frac { 3 } { \delta } ) } { 2 \varepsilon ^ 2 } . \end{align*}


\begin{align*} \max ( x , d ) + \max ( - x , d ) = d + \max ( | x | , | d | ) = \begin{cases} d + | x | & \ | x | \ge | d | , \\ d + | d | & \ | x | \le | d | . \\ \end{cases} \end{align*}


\begin{align*} D ( s ) \varphi ( x ) : = \int _ \Gamma \left [ \partial _ { \nu _ y } \mathcal { K } ( | x - y | , s ) \right ] \varphi ( y ) d \Gamma _ y x \in \R ^ d \setminus \Gamma , \end{align*}


\begin{align*} e ^ { - a | x | } = \frac { a } { \pi } \int _ { - \infty } ^ { + \infty } \frac { e ^ { - j q x } \mbox { d } q } { q ^ 2 + a ^ 2 } , ~ ~ ~ ~ a > 0 \end{align*}


\begin{align*} [ X , X ^ { - 1 } ] = \sum _ { g \in T _ 0 ( \alpha ) } \epsilon _ g \{ [ \alpha ] g [ \alpha ] ^ { - 1 } g ^ { - 1 } \} \end{align*}


\begin{align*} A _ 1 = \frac { 1 } { 3 } ( \frac { 1 } { 2 } S _ g \cdot g - \mathrm { R i c } _ g ) , \end{align*}


\begin{align*} \begin{pmatrix} 0 & \cdots & 0 & n \end{pmatrix} , n > 0 . \end{align*}


\begin{align*} \tau _ i = 1 0 ^ { - 3 - \frac { 2 i } { I } } , ~ i = 1 , \dots , I . \end{align*}


\begin{align*} \dim ( M _ i \cap \ker A _ i ^ s ) & = \sum _ { r \leq s } \Phi ( x _ i ^ r ) . \end{align*}


\begin{align*} p ( h _ i , \kappa ) = \left \{ \def \arraystretch { 2 } \begin{array} { l l } 2 h _ i - h _ i ^ 2 \frac { - \kappa } { 1 - \kappa } & h _ i \in \big ( 0 , 1 \big ] \\ \frac { h _ i ( 2 - h _ i ) ( 2 - \kappa h _ i ) } { 2 - ( 1 + \kappa ) h _ i } & h _ i \in \big [ 1 , \bar { h } ( \kappa ) \big ] \end{array} \right . \end{align*}


\begin{align*} \norm { f } _ { C ^ { 0 , \beta } [ a , b ] } : = \sup _ { x \neq y \in [ a , b ] } \frac { | f ( x ) - f ( y ) | } { | x - y | ^ { \beta } } . \end{align*}


\begin{align*} \psi ^ { P , G } [ p ] \cap M = p , \psi ^ { Q , G } [ q ] \cap N = q . \end{align*}


\begin{align*} \mathrm { e } ^ { - H _ { 0 } t } f ( \mathbf { x } ) = \int _ { \mathbb { R } ^ { d } } K _ { t } ( \mathbf { x } , \mathbf { y } ) f ( \mathbf { y } ) \ , \textnormal { d } \mathbf { y } . \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } \limsup _ { n \rightarrow \infty } E [ \Lambda ^ { ( 3 ) } _ { n , \delta } ] = 0 . \end{align*}


\begin{align*} c _ { n - 2 , \xi _ { \boldsymbol { s } k } , \xi _ { \boldsymbol { t } l } } ^ { \ast \ast } = \frac { \Gamma \left ( ( n - 2 ) / 2 \right ) } { ( 2 \pi ) ^ { ( n - 2 ) / 2 } } \left [ \int _ { 0 } ^ { \infty } x ^ { ( n - 4 ) / 2 } \overline { \mathcal { G } } _ { n } \left ( \frac { 1 } { 2 } \xi _ { \boldsymbol { s } k } ^ { 2 } + \frac { 1 } { 2 } \xi _ { \boldsymbol { t } l } ^ { 2 } + x \right ) \mathrm { d } x \right ] ^ { - 1 } . \end{align*}


\begin{align*} \sum _ { j \in \Z } \chi _ j ( \lambda ) = 1 \lambda \neq 0 , \end{align*}


\begin{align*} \epsilon _ 1 = \left ( \delta ( \theta _ 1 ) \cdot \sqrt { c _ 2 } \right ) ^ 2 = \epsilon _ 1 ( \theta _ 1 , c _ 2 ) = \epsilon _ 1 ( c _ 1 , c _ 2 ) = \epsilon _ 1 ( r , g , d , l ) , \end{align*}


\begin{align*} \Gamma _ c : = \begin{bmatrix} \Big ( \sum _ l \Gamma _ { i j } ^ l \Gamma _ { m b } ^ l \Big ) & \Big ( \sum _ l \Gamma _ { i j } ^ l \Gamma _ { m m } ^ l \Big ) \\ \Big ( \sum _ l \Gamma _ { m b } ^ l \Gamma _ { i i } ^ l \Big ) & \Big ( \sum _ l \Gamma _ { i i } ^ l \Gamma _ { m m } ^ l \Big ) \end{bmatrix} , i \neq j , \ , m \neq b . \end{align*}


\begin{align*} \eta ( e _ 1 , e _ 2 ) = - 4 \ , \mu \otimes _ \mathcal O \dfrac { \partial } { \partial y } . \end{align*}


\begin{align*} L ( x ) = \sum _ { j = 1 } ^ n c _ j \prod _ { i \neq j } \frac { x - \lambda _ i } { \lambda _ j - \lambda _ i } \end{align*}


\begin{align*} - \frac { 2 } { \sigma } = k ( \sigma ) + l ( \sigma ) \end{align*}


\begin{align*} \ell _ 2 \left ( \mu \otimes _ \mathcal O \dfrac { \partial } { \partial x } , \mu \otimes _ \mathcal O \dfrac { \partial } { \partial y } \right ) : = \frac { \partial \varphi } { \partial x } \ , \mu \otimes _ \mathcal O \dfrac { \partial } { \partial y } - \frac { \partial \varphi } { \partial y } \ , \mu \otimes _ \mathcal O \dfrac { \partial } { \partial x } \end{align*}


\begin{align*} E _ { n s } ( \dot { x } , x , t ) = - \left [ 1 + g _ 1 \dot x L _ { n s } ( \dot { x } , x , t ) \right ] L _ { n s } ( \dot { x } , x , t ) \ , \end{align*}


\begin{align*} \lambda _ { \pm } = \frac { \sqrt { 1 + \left ( \frac { 4 \beta \hslash ^ { 2 } } { 3 a ^ { 2 } } \right ) ^ { 2 } } \pm 1 } { 4 \beta \hslash ^ { 2 } / 3 a ^ { 2 } } . \end{align*}


\begin{align*} \Phi _ { i } ( x , R \setminus \varepsilon ) : = \Phi _ { i } ( x , R ) - \Phi _ { i } ( x , \varepsilon ) , \quad \mathcal { M } : = \| u \| _ { C ^ { \sigma p + \alpha } ( B _ { 2 R } ( 0 ) ) } + 1 , \end{align*}


\begin{align*} p ( 0 , g ) = p _ { } ( g ) , g \in \mathbb { R } , \end{align*}


\begin{align*} A ( \rho ) & = \tilde { \tau } ^ { - 1 } \rho ( \tilde { \tau } ) \zeta _ \phi ^ { p , \infty } ( \rho ) \\ & = \tilde { \tau } _ G ^ { - 1 } \zeta _ \phi ^ { p , \infty } ( \rho ) \rho ( \tilde { \tau } _ G ) \zeta _ \phi ^ { p , \infty } ( \rho ) ^ { - 1 } \zeta _ \phi ^ { p , \infty } ( \rho ) \\ & = \tilde { \tau } _ G ^ { - 1 } \zeta _ \phi ^ { p , \infty } ( \rho ) \rho ( \tilde { \tau } _ G ) . \end{align*}


\begin{align*} Q _ i ^ { - 1 } : = \{ k a ( t ) n g _ i : \ , k \in K , t \ge 0 , n \in N _ 0 \} . \end{align*}


\begin{align*} \nu _ Q ( a _ { i j k } Q '^ k ) \geq \nu _ Q ( a _ i h ^ j ) = \nu ( a _ i Q ^ i ) + ( j - i ) \nu ( Q ) > \nu ( a _ l ) + ( l + j - i ) \nu ( Q ) . \end{align*}


\begin{align*} d \sigma _ { \mu } ^ { - } ( s ) = ( q / p ) \sum \nolimits _ { k = 1 } ^ { \infty } \sigma _ { \mu } ^ { + } \big ( [ k ^ m , ( k + 1 ) ^ m ) \big ) \ , \delta _ { - k ^ m } . \end{align*}


\begin{align*} H _ n ( s ) & : = \pi ^ { - s } \Gamma ( s ) \Bigg ( \zeta ( \widetilde { \Delta } _ n , s ) - V _ \alpha ( s ) \widetilde a ( s ) n ^ { 2 - 2 s } \Bigg ) \\ & = \pi ^ { - s } \Gamma ( s ) \Bigg ( \zeta ( \Delta , s ) + \frac { s \pi ^ 2 } { 3 } \zeta ( \Delta , s - 1 ) + O \Bigl ( n ^ { - 4 } \Bigr ) \Bigg ) . \end{align*}


\begin{align*} \bigl [ V _ { u } ( f ) h ^ { e _ { n } } _ { u } \bigr ] ( \gamma ) = f ( \gamma ) e _ { n } . \end{align*}


\begin{align*} \| E \| _ { L ^ 1 \cap L ^ \infty } & \leq c \big ( \lambda \eta _ t + \lambda ^ 2 \gamma _ t + | \mu - m ^ 2 | \big ) \Big ( \frac { 1 } { m ^ 2 _ t } { \bf 1 } _ { d = 2 } + \frac { 1 } { m _ t } { \bf 1 } _ { d = 3 } + \frac { 1 } { m ^ 4 _ t } \Big ) \\ & + \frac { c \| E \| _ { L ^ 1 \cap L ^ \infty } } { m ^ 2 _ t } \big ( \lambda \eta _ t + \lambda ^ 2 \gamma _ t + | \mu - m ^ 2 | \big ) + c \sum _ { i = 1 } ^ 3 \lambda ^ i R ^ { ( i ) } _ { t } \big ( \| E \| _ { L ^ 1 \cap L ^ \infty } \big ) , \end{align*}


\begin{align*} \mathcal { G } ( n + 1 ) \oplus \mathcal { G } ( n + 1 ) \oplus \mathcal { G } ( 2 a c + 2 ) = \mathcal { G } ( n ) \oplus \mathcal { G } ( n ) \oplus \mathcal { G } ( 2 a c + 4 ) , \ ; a = 2 \bmod 6 \end{align*}


\begin{align*} [ w ] _ { Q B _ { \beta , p } ( I ) } = [ \widetilde { w } ] _ { Q B _ { \beta , p } } . \end{align*}


\begin{align*} \overline { G } _ { ( 2 ) } ( u ) = \ln \left [ 1 + \exp ( - u ) \right ] . \end{align*}


\begin{align*} \big ( ( \hat { \mu } _ n - \mu ) * \gamma _ { \sigma } \big ) ( f ) = ( \hat { \mu } _ n - \mu ) ( f * \phi _ { \sigma } ) . \end{align*}


\begin{align*} Q ( u , v ) = \langle \gamma ( - \Delta ) ^ { s / 2 } u , ( - \Delta ) ^ { s / 2 } v \rangle + \langle q u , v \rangle . \end{align*}


\begin{align*} - Y \circ u + \nabla _ t w - \nabla ( F + X ) ( u ) [ w ] = v . \end{align*}


\begin{align*} b ( n ) : = f ( n ) / \log n , \end{align*}


\begin{align*} e ^ { x } \phi _ { p , \lambda } ( x ) = \sum _ { n = 1 } ^ { \infty } \frac { ( n ) _ { p , \lambda } } { n ! } x ^ { n } , ( p \in \mathbb { Z } ) . \end{align*}


\begin{align*} q _ { \vec { \alpha } } \Vdash ` ` \dot { z } _ { \alpha _ 0 } < \dots < \dot { z } _ { \alpha _ n } \dot { f } ( \{ \dot { z } _ { \alpha _ 0 } , \dots , \dot { z } _ { \alpha _ n } \} ) = i ( \vec { \alpha } ) \end{align*}


\begin{align*} X _ + = \mathbb P _ { \mathbb P ^ r } ( \mathcal O ( - 1 ) ^ { r ^ \prime + 1 } \oplus \mathcal O ) X _ - = \mathbb P _ { \mathbb P ^ { r ^ \prime } } ( \mathcal O ( - 1 ) ^ { r + 1 } \oplus \mathcal O ) . \end{align*}


\begin{align*} f ( t , x , v ) = \tilde R _ + ( x ) M _ { T _ w } ( x , v ) , \tilde R _ + ( x ) = \int _ { \{ v \cdot n _ x > 0 \} } f ( t , x , v ) | v \cdot n _ x | \d v , \end{align*}


\begin{align*} \varpi _ 1 = 2 \lambda - 2 - 2 \beta _ 0 ^ 2 - 4 \left \| \beta \right \| ^ 2 \end{align*}


\begin{align*} \mu \int _ { \Omega } \varepsilon S ' ( \overline { u } , h ) : \varepsilon p + \nu \int _ { \Omega } \mathsf { m } ' \left ( \varepsilon \overline { y } ; \varepsilon S ' ( \overline { u } , h ) \right ) : \varepsilon p = - \alpha \int _ { \Omega } h \cdot \overline { u } = - J _ u ( \overline { y } , \overline { u } ) h . \end{align*}


\begin{align*} \Lambda _ k ^ \lambda g = \sum _ { | \nu | _ 1 = k } \sum _ { \nu ' \in \N ^ n } ( g , \Phi _ { \nu , \nu ' } ^ \lambda ) \Phi _ { \nu , \nu ' } ^ \lambda , g \in L ^ 2 ( \R ^ n ) , \end{align*}


\begin{align*} \psi _ { i _ { \sigma ( 1 ) } } \cdots \psi _ { i _ { \sigma ( k ) } } = \psi _ { i _ { \sigma ' \tau ( 1 ) } } \cdots \psi _ { i _ { \sigma ' \tau ( k ) } } = \psi _ { i _ { \sigma ' ( 1 ) } } \cdots \psi _ { i _ { \sigma ' ( k ) } } . \end{align*}


\begin{align*} \left ( | f | ^ 2 \right ) ^ 2 = \left ( \sum _ { \mathcal { I } } \left ( f ^ \mathcal { I } \right ) ^ 2 \right ) ^ 2 = \sum _ { \mathcal { I } } \sum _ \mathcal { J } \left ( f ^ \mathcal { I } \right ) ^ 2 \left ( f ^ \mathcal { J } \right ) ^ 2 \end{align*}


\begin{align*} ( - \Delta ) ^ { \sigma } _ { p } u ( x ) = c _ { n , \sigma p } P . V . \int _ { \mathbb { R } ^ { n } } \frac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) } { | x - y | ^ { n + \sigma p } } d y , \end{align*}


\begin{align*} \int _ { 0 } ^ { x } f ( t ) h ( t / q ) \left ( \frac { 1 - t ( 1 + q ) } { q ( 1 - q ) t ^ 2 } u ( t / q ) + \frac { 1 } { ( 1 - q ) ^ 2 t ^ 2 } \right ) y ( t ) d _ q t = G ( x ) - \lim _ { n \rightarrow \infty } G ( q ^ n x ) . \end{align*}


\begin{align*} p _ \star & = \sup \left \{ p > 0 : \sum _ { n \geq 0 } S _ n ^ { ( p ) } < \infty \right \} . \end{align*}


\begin{align*} \hat { V } ( \xi , t ) : = \frac { 1 } { ( 2 \pi ) ^ { n / 2 } } \int d ^ n x , e ^ { - i x \cdot \xi } V ( x , t ) . \end{align*}


\begin{align*} \mathcal { G } : = K \times \mathfrak { p } \times [ 0 , 1 ] . \end{align*}


\begin{align*} \| u \| _ { X ^ { s , b } } = \| \langle k \rangle ^ { s } \langle \tau + k ^ 2 \rangle ^ { b } \widehat { u } \| _ { L ^ 2 _ \tau \ell ^ 2 _ k } = \| \langle k \rangle ^ s \langle \tau \rangle ^ b \widehat { u } ( k , \tau - k ^ 2 \rangle ) \| _ { \ell ^ 2 _ k L ^ 2 _ \tau } = \| W _ { - t } u \| _ { H ^ s _ x H ^ b _ t } , \end{align*}


\begin{align*} \widehat { \xi ^ \mathbf { G } } ( 0 , \dots , 0 ) = \int _ { \R ^ { d ( d - 1 ) } } \phi ( { y ^ \star } ) \prod _ { j = 1 } ^ { d - 1 } g _ { \tilde { a } ( y ^ \star ) } ( z ^ { j + 1 } ) d z ^ 2 \dots d z ^ d = \phi ( { y ^ \star } ) \end{align*}


\begin{align*} f ( q ^ n x ) = q ^ { \frac { n ( n - 1 ) } { 2 } } x ^ n f ( x ) ( n \in \mathbb { N } _ 0 ) . \end{align*}


\begin{gather*} W ( x , t ) = \max \{ h _ { i j } ( x , t ) \zeta ^ i \zeta ^ j : g _ { i j } ( x ) \zeta ^ i \zeta ^ j = 1 \} , \\ p ( u ) = - \log ( u - \frac { 1 } { 2 } \min u ) , \end{gather*}


\begin{align*} a ( { w } , \varphi ) = b ( u - J z _ { \mathrm { n c } } , \varphi ) \varphi \in V . \end{align*}


\begin{align*} ( - 1 ) ^ { n - k } a _ k = \sum _ { 1 \leq i _ 1 < i _ 2 < \dots < i _ { n - k } \leq n } \lambda _ { i _ 1 } \lambda _ { i _ 2 } \cdots \lambda _ { i _ { n - k } } , \ , \ , \ , k = 0 , 1 , \dots , n - 1 , \end{align*}


\begin{align*} \mu _ t = \mu _ 0 + t _ 1 \varphi _ 1 + \cdots + t _ n \varphi _ n , \end{align*}


\begin{align*} \mu _ x : = e ^ \gamma \log x \prod _ { p < x } \Big ( 1 - \frac { 1 } { p } \Big ) . \end{align*}


\begin{align*} C _ { r } : = 2 ( \pi - \phi ( r ) ) \sinh ( \beta r ) . \end{align*}


\begin{align*} f ( A _ n ) \le \frac { \pi } { 4 } \frac { M _ q } { m _ q ^ 2 } e ^ \gamma { \rm d } ( { \rm L } _ n ) = \frac { q } { n } \ , b _ q f ( q ) \end{align*}


\begin{align*} G \mbox { i s t o r s i o n - f r e e } \ \Longleftrightarrow \ | E ( G ) | = 1 \end{align*}


\begin{align*} \left | \mathcal S _ { \kappa , \gamma } \right | = \binom { \kappa + 2 ^ \gamma - 1 } { \kappa } = \binom { \kappa + 2 ^ \gamma - 1 } { 2 ^ \gamma - 1 } . \end{align*}


\begin{align*} a _ j ( \tau ) \lesssim e ^ { - ( \frac { \gamma } { 2 } + j ) \tau } e ^ { ( j - J - { \sf c } _ 1 ) \tau } = e ^ { - ( \frac { \gamma } { 2 } + J + { \sf c } _ 1 ) \tau } = ( T - t ) ^ { \frac { \gamma } { 2 } + J + { \sf c } _ 1 } . \end{align*}


\begin{align*} Q ^ { n , 6 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( s - \eta _ n ( s ) ) ^ { 2 \alpha + 1 } ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) d s . \end{align*}


\begin{align*} - \Delta _ p G = \delta _ { x _ 0 } \Omega \end{align*}


\begin{align*} \sum \limits _ { i = m } ^ \infty \alpha _ i \leq C _ { n , r } \alpha _ m . \end{align*}


\begin{align*} \lambda \frac { \partial ^ 2 u } { \partial t ^ 2 } + \frac { \partial ^ 2 } { \partial x ^ 2 } \Biggl ( E I \frac { \partial ^ 2 u } { \partial x ^ 2 } \Biggr ) = f ( u ) . \end{align*}


\begin{align*} \int _ { \Xi _ { \mu } } x ^ { \mu ' } e ^ { x ^ r / \hbar } \Omega = \delta _ { \mu \mu ' } ; , \end{align*}


\begin{align*} k \tanh ( k a / 2 ) = k ' \tanh ( k ' a / 2 ) , \end{align*}


\begin{align*} \Psi ^ { n , 1 } _ s = \sigma ^ 2 ( X _ s ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) ^ 2 , \end{align*}


\begin{align*} \sum _ { u = i } ^ s \alpha _ u \alpha ^ * _ u - \sum _ { v = i } ^ t \beta _ v \beta _ v ^ * . \end{align*}


\begin{align*} \varphi _ n ^ { h , m } : = \left \{ \begin{array} { c c } \varphi _ n ^ h & n \leq m \\ 0 & n > m \end{array} \right . \psi _ n ^ { h , m } : = \left \{ \begin{array} { c c } \psi _ n ^ h & n \leq m \\ 0 & n > m \end{array} \right . . \end{align*}


\begin{align*} \left \{ \begin{array} { l } H = h + E \\ h = D + E \end{array} \right . . \end{align*}


\begin{align*} M _ n ( m ) & = 2 \dbinom { m + n } { n } m > n , \\ M _ n ( n ) & = \dbinom { 2 n } { n } . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ { \infty } \frac { ( a ) _ n t ^ n } { ( q ) _ n } = \frac { ( a t ) _ { \infty } } { ( t ) _ { \infty } } . \end{align*}


\begin{align*} \lambda ( ( z _ { k + \ell + 1 } , z _ { k + 1 } ) ) + \lambda ( \{ z _ { k + 1 } \} ) \leq \overline \lambda ( z _ { k + \ell + 1 } ) - \overline \lambda ( z _ { k + 1 } ) \leq k + \ell + 1 - k = \ell + 1 , \end{align*}


\begin{align*} a ^ 4 & = a ^ 2 \gamma ^ 2 + 1 = ( a \gamma + 1 ) \gamma ^ 2 + 1 = a + \gamma , \\ a ^ 8 & = a ^ 2 + \gamma ^ 2 = a \gamma + 1 + \gamma ^ 2 = a \gamma + \gamma . \end{align*}


\begin{align*} \partial _ z \left [ \sqrt y K _ { i r } ( 2 \pi | n | y ) e ( n x ) \right ] = \pi i n \sqrt y K _ { i r } ( 2 \pi | n | y ) e ( n x ) + g ( n , y ) e ( n x ) \end{align*}


\begin{align*} F _ \ell ( \lambda , \rho ) = F ( \sqrt \lambda ) \chi _ \ell ( \lambda / \rho ) \quad \rho \neq 0 , \end{align*}


\begin{align*} \vec { v } \left ( { \bf s ' } _ { \alpha , \delta } \right ) = \vec { v } \left ( { \bf s } _ { \alpha , \delta } \right ) . \end{align*}


\begin{align*} f ^ { ( j ) } ( x ) = ( n ) _ j \prod _ { i = 1 } ^ { m } \left ( x - \lambda _ i \right ) ^ { \mu _ { i , j } } \end{align*}


\begin{align*} \sum _ { i = - \infty } ^ k 2 ^ { - i \frac { r } { p - r } } V _ r ( 0 , x _ i ) ^ { \frac { p r } { p - r } } \approx \sum _ { i = - \infty } ^ k 2 ^ { - i \frac { r } { p - r } } V _ r ( x _ { i - 1 } , x _ i ) ^ { \frac { p r } { p - r } } \end{align*}


\begin{align*} \left \{ \lambda _ { m i n } + \eta : \lambda _ { m i n } = \displaystyle \sum _ { w s _ i < w } \omega _ i \eta \in \left ( \displaystyle \bigcap _ { j \in I ' } ( \alpha _ j ^ \vee ) ^ \perp \right ) \cap P _ + \right \} \end{align*}


\begin{align*} k \cot ( k a / 2 ) - k ' \coth ( k ' a / 2 ) = 0 \end{align*}


\begin{align*} X _ k ( t ) = \begin{cases} X _ k ( t - ) & k \leq j , \\ X _ j ( t - ) & k > j Y _ { i k } = 1 , \\ X _ { k - J _ { i k } } ( t - ) & k > j Y _ { i k } = 0 , \end{cases} \end{align*}


\begin{align*} \partial _ i L ( h _ \theta ) = b _ { i 0 } + b _ { i 1 } h _ \theta + \ldots + b _ { i s } h _ \theta ^ s , \end{align*}


\begin{align*} \delta \left [ f \right ] & = \sum _ { k = 0 } ^ \infty \left ( \sum _ { \vec { k } } \delta [ p _ { \vec { k } } ] a _ { \vec { k } } \right ) \end{align*}


\begin{align*} 0 = \mathcal { E } _ 0 \to \mathcal { E } _ 1 \to \ldots \to \mathcal { E } _ N = L , \end{align*}


\begin{align*} \sigma _ N ^ 2 = \begin{cases} \sqrt { \mu _ - - \mu _ + } N \mbox { i f } \delta < 1 \mbox { i n t h e a s s u m p t i o n } T \sim N ^ { - \delta } \\ N ^ { 1 / 2 } \mbox { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} \langle \xi _ { \tilde K } \nabla w , \xi _ { \tilde K } \nabla w \rangle = \frac { ( 1 + \delta ) ^ 2 } { 1 + 2 \delta } \left ( \langle \nabla u _ { q , r } , \nabla \nu \rangle - ( 1 + u _ { q , r } ) ^ { 2 \delta + 1 } \langle \nabla u _ { q , r } , \nabla ( \xi _ { \tilde K } ) ^ 2 \rangle \right ) . \end{align*}


\begin{align*} u ^ { \dagger } \left ( \begin{array} { l r } I & O \\ O & - I \end{array} \right ) v = 0 , \end{align*}


\begin{align*} f _ { r , s } ( \kappa _ i ) = f \left ( r \kappa _ i + s \right ) = f ( \lambda _ i ) \end{align*}


\begin{align*} 0 \le \frac { d } { d x } H ( x + i f ( x ) ) = H ' ( x + i f ( x ) ) ( 1 + i f ' ( x ) ) . \end{align*}


\begin{align*} G ^ { i } | _ { \ker ( T ^ { i } ) } = 0 , \end{align*}


\begin{align*} g ' ( z ) = w \frac { h ' ( w ) } { h ( w ) } . \end{align*}


\begin{align*} d ( o , x ) \int _ { \varepsilon D ^ 2 } ^ { b D ^ 2 } \sqrt { \frac { d } { e t } } h _ { t / 2 } ( o , o ) d t & \leq d ( o , x ) ( b - \varepsilon ) D ^ 2 \sqrt { \frac { d } { e \varepsilon D ^ 2 } } h _ { \varepsilon D ^ 2 / 2 } ( o , o ) \\ & \leq C ' \frac { d ( o , x ) } { D } \frac { D ^ 2 } { n } \leq \varepsilon \frac { D ^ 2 } { n } . \end{align*}


\begin{align*} K _ k ( y ) : = \int _ { \R ^ 3 } e ^ { i y \cdot \xi } \varphi _ k ( \xi ) d \xi , \Longrightarrow | K _ k ( y ) | \lesssim 2 ^ { 3 k } ( 1 + 2 ^ k | y | ) ^ { - 1 / \epsilon ^ 3 } . \end{align*}


\begin{align*} \gamma _ { i , l } = \dfrac { P | h [ r ( i , l - 1 ) , r ( i , l ) , l ] | ^ 2 } { \sigma ^ 2 + \sum _ { j \neq i } ^ N P | h [ r ( j , l - 1 ) , r ( i , l ) , l ] | ^ 2 } , \end{align*}


\begin{align*} d f _ i ( ( x , u _ i ) , ( z , v _ i ) ) = d _ 1 f _ i ( x , u _ i , z ) + d _ 2 f _ i ( x , u _ i , v _ i ) \end{align*}


\begin{align*} \big ( ( \hat { v } + \omega ) \times B \big ) _ a = ( - 1 ) ^ { a - 1 } \big [ ( \hat { v } + \omega ) _ { 3 - a } ( \omega \cdot \mathbf { e } _ 3 ) ( B \cdot \omega ) - \big ( \sum _ { i = 1 , 2 , 3 } ( \hat { v } + \omega ) _ { 3 - a } ( \omega \times \mathbf { e } _ 3 ) \cdot \mathbf { e } _ i ( B \cdot ( \omega \times \mathbf { e } _ i ) ) + ( \hat { v } + \omega ) _ 3 B _ 2 \big ) \big ] . \end{align*}


\begin{align*} d ( A ) = { s r ( A ) + r s ( A ) - \sum _ { i = 1 } ^ l ( s \alpha _ i + r \beta _ i + r s ( d _ i - 1 ) ) \over r s } - 2 . \end{align*}


\begin{align*} T _ { i , j } = \begin{cases} - 2 / h ^ 2 , j = i , \\ 1 / h ^ 2 , j = i \pm 1 , \\ 0 , \quad ; \end{cases} \end{align*}


\begin{align*} w ( \alpha ) = ( w _ { p _ { \sigma ( i ) } } ( \alpha ) ) _ { 1 \leq i \leq \abs { J } } \end{align*}


\begin{align*} T _ { i } = \left ( \begin{array} { l l } A _ { i } & 0 _ { 3 , N - 3 } \\ 0 _ { N - 3 , 3 } & I _ { N - 3 } \end{array} \right ) , \end{align*}


\begin{align*} \rho \in \mathcal D _ k : = \left [ r _ 1 k \ln k , r _ 2 k \ln k \right ] \ \hbox { f o r s o m e } \ r _ 2 > r _ 1 > 0 . \end{align*}


\begin{align*} \begin{aligned} h _ * = \arg \max _ { 1 { } \leq { } h { } \leq { } \bar { h } ( \kappa ) } 2 h + \frac { \kappa h ^ 3 } { 2 - h \left ( 1 + \kappa \right ) } \end{aligned} . \end{align*}


\begin{align*} b _ h ( u ^ { \Delta t } _ { h } , u ^ { \Delta t } _ { h } , u ^ { \Delta t } _ { h } \phi ) & = - \left ( u ^ { \Delta t } _ { h } \frac 1 2 | u ^ { \Delta t } _ { h } | ^ 2 , \nabla \phi \right ) . \end{align*}


\begin{align*} a ( v ) = \sum _ { i = 1 } ^ { v - 1 } F _ { v - i , i } . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \overline { \varphi } _ t - A _ 1 \overline { \varphi } _ { i i } - ( B _ 1 + h ' C _ 1 + D _ 1 ) \overline { \varphi } _ i = a _ { 0 1 } \chi ^ { p } - \lambda _ { 0 1 } \sqrt { A _ 1 } ^ \alpha | \varphi _ i | ^ { \alpha } , & t > 0 , \ 0 < i < h _ 0 , \\ \overline { \varphi } _ i ( t , 0 ) = \overline { \varphi } ( t , h _ 0 ) = 0 , & t > 0 , \\ \overline { \varphi } ( 0 , i ) = \varphi _ 0 ( i ) , & 0 \leq i \leq h _ 0 , \\ \end{array} \right . \end{align*}


\begin{align*} \int _ { \Xi _ d } x ^ { n r + k } e ^ { x ^ r / \hbar } d x = { } & ( - 1 ) ^ n \hbar ^ n \left ( \prod _ { i = 1 } ^ n ( i - 1 + \frac { k + 1 } { r } ) \right ) \int _ { \Xi _ d } x ^ { k } e ^ { x ^ r / \hbar } d x \\ = { } & ( - 1 ) ^ n \hbar ^ n \frac { \Gamma \left ( n + \frac { k + 1 } { r } \right ) } { \Gamma \left ( \frac { k + 1 } { r } \right ) } \int _ { \Xi _ d } x ^ { k } e ^ { x ^ r / \hbar } d x . \end{align*}


\begin{align*} p ( t , g ) = \frac { 1 } { \sqrt { 2 \pi c ( t ) } } \exp \left ( - \frac { ( g - b ( t ) ) ^ 2 } { 2 c ( t ) } \right ) , \end{align*}


\begin{align*} \mathrm { D T E } _ { ( p , q ) } ( X ) = \mathrm { E } ( X | x _ { p } < X < x _ { q } ) \end{align*}


\begin{align*} \partial _ { v _ { b _ { n } } } & \partial _ { \mu } \bigl [ \partial _ { c } f ( \mu , v _ { 1 } , . . . , v _ { p } ) \cdot e _ { 1 } \otimes . . . \otimes e _ { i } \bigr ] ( v _ { p + 1 } ) \cdot e _ { i + 1 } \otimes e _ { { n } } \\ & = \partial _ { \mu } \partial _ { v _ { b _ { n } } } \bigl [ \partial _ { c } f ( \mu , v _ { 1 } , \cdots , v _ { p } ) , e _ { 1 } \otimes . . . \otimes e _ { i } \bigr ] ( v _ { p + 1 } ) \cdot e _ { { n } } \otimes e _ { i + 1 } , \end{align*}


\begin{align*} { \Gamma } ^ T H { \Gamma } = I , \end{align*}


\begin{align*} u ( x , t ) & = \pm { \sf U } _ \infty ( x ) + K { \sf D } _ J ( T - t ) ^ { \frac { \gamma } { 2 } + J } | z | ^ \gamma ( 1 + o ( 1 ) ) \\ & = \pm { \sf U } _ \infty ( x ) + K { \sf D } _ J ( T - t ) ^ J \eta ^ \gamma | \xi | ^ \gamma ( 1 + o ( 1 ) ) | z | \to 0 . \end{align*}


\begin{align*} \Phi _ \nu ^ \lambda ( \xi ) = \lambda ^ { n / 4 } \prod _ { j = 1 } ^ n h _ { \nu _ j } ( \lambda ^ { 1 / 2 } \xi _ j ) , \xi \in \R ^ n , \end{align*}


\begin{align*} d s \left ( \overleftarrow { [ a , b ] _ { A } } - [ \overleftarrow { a } , \overleftarrow { b } ] \right ) & = \rho ( [ a , b ] _ { A } ) - [ \rho ( a ) , \rho ( b ) ] \\ & = 0 , \end{align*}


\begin{align*} \overline F _ p : = \sum _ { i \in I \cup \{ 0 \} } a _ i Q _ \theta ^ i . \end{align*}


\begin{align*} \Psi ( E _ j ( x , a ) ) & = \begin{cases} 2 \sqrt { a } \left ( \cot \left ( \dfrac { \pi } { 2 j + 2 } \right ) - 1 \right ) , & 2 \nmid j , \\ 2 \sqrt { a } \left ( \csc \left ( \dfrac { \pi } { 2 j + 2 } \right ) - 1 \right ) , & 2 \mid j . \end{cases} \end{align*}


\begin{align*} \Pi _ 1 Q ^ { \beta } \hat U ( t ) + A \hat U ( t ) = \Pi _ 1 f ( t ) + \hat G ( t ) , t \in [ 0 , T ] , \end{align*}


\begin{align*} z \sim z ' \iff z = a \ , z ' \ , \ \ a \in \mathbb C ^ \ast \ . \end{align*}


\begin{align*} \theta ( s ) = 0 , \ | s | \le 1 , \ \theta ( s ) = 1 , \ \ | s | \ge 2 . \end{align*}


\begin{align*} \boldsymbol { V } _ { A \rightarrow B } ^ { e x t } & = \left ( ( \overline { \boldsymbol { V } } _ { A } ^ { p o s t } ) ^ { - 1 } - ( \boldsymbol { V } _ { A } ^ { p r i } ) ^ { - 1 } \right ) ^ { - 1 } , \\ \boldsymbol { x } _ { A \rightarrow B } ^ { e x t } & = \boldsymbol { V } _ { A \rightarrow B } ^ { e x t } \left ( ( \overline { \boldsymbol { V } } _ { A } ^ { p o s t } ) ^ { - 1 } \boldsymbol { x } _ { A } ^ { p o s t } - ( \boldsymbol { V } _ { A } ^ { p r i } ) ^ { - 1 } \boldsymbol { x } _ { A } ^ { p r i } \right ) , \end{align*}


\begin{align*} U _ t = \mathcal { A } _ d U , U ( 0 ) = U _ 0 , \end{align*}


\begin{align*} | Z _ 1 | = 0 ~ ~ u _ n ( x ) \longrightarrow u ( x ) ~ ~ x \in \Omega \setminus Z _ 1 . \end{align*}


\begin{align*} k = k ' = \frac { 1 } { 2 \hslash \sqrt { \beta / 3 } } . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } L ^ { n , 4 } _ \tau = \frac { \alpha } { 2 ( \alpha + 1 ) ^ 2 } \int _ 0 ^ \tau \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ s ) d s , \end{align*}


\begin{align*} \langle S \rangle : = \{ s \in S : s \notin { \rm L } _ t \ ; \forall \ ; t < s , t \in S \} . \end{align*}


\begin{align*} L _ { M } \cdot V = \int _ { V } R _ { h } ^ { T } \wedge \eta \end{align*}


\begin{align*} f _ { \mathbf { M ^ { \ast \ast } } _ { - \xi _ { \boldsymbol { s } k } , \xi _ { \boldsymbol { t } l } } } ( \boldsymbol { u } ) = c _ { n - 2 , \xi _ { \boldsymbol { s } k } , \xi _ { \boldsymbol { t } l } } ^ { \ast \ast } \overline { \mathcal { G } } _ { n } \left \{ \frac { 1 } { 2 } \boldsymbol { u } ^ { T } \boldsymbol { u } + \frac { 1 } { 2 } \xi _ { \boldsymbol { s } k } ^ { 2 } + \frac { 1 } { 2 } \xi _ { \boldsymbol { t } l } ^ { 2 } \right \} , ~ \boldsymbol { u } \in \mathbb { R } ^ { n - 2 } , \end{align*}


\begin{align*} \nabla _ { [ x , y ] _ \mathfrak g } - [ \nabla _ x , \nabla _ y ] = \gamma ( x , y ) \circ \ell _ 1 - \ell _ 1 \circ \gamma ( x , y ) + \ell _ 2 ( \eta ( x , y ) , \cdot \ , ) , \end{align*}


\begin{align*} \sigma \left ( \sum _ { i = 1 } ^ { n } c _ { i } x _ { i } \right ) & \geq \rho \left ( \varphi \left ( \sum _ { i = 1 } ^ { n } c _ { i } x _ { i } \right ) \right ) = \rho \left ( \sum _ { i = 1 } ^ { n } c _ { i } \varphi ( x _ { i } ) \right ) \\ & = \rho \left ( \sum _ { i = 1 } ^ { n } c _ { i } y _ { i } \right ) \geq \delta \sum _ { i = 1 } ^ { n } \lvert c _ { i } \rvert . \end{align*}


\begin{align*} y ^ * = \sum _ { w \in \mathcal { W } } y ^ * ( w ) \widetilde { w } , \end{align*}


\begin{align*} < L _ { n } ( \phi _ { n } ) , \ \psi > = o _ n ( 1 ) \| \phi _ n \| \| \psi \| \ \textrm { f o r a n y } \ \psi \in E _ n . \end{align*}


\begin{align*} \left ( \frac q p - 1 \right ) + \frac p q \left ( \frac q r - 1 \right ) = \frac r p - 1 \ . \end{align*}


\begin{align*} n _ 1 = \displaystyle \prod _ { p | \gcd ( n , \binom { n + s } { s } ) } p ^ { _ p ( n ) } . \end{align*}


\begin{align*} \psi ( t ) = \exp \left ( - { \rm i } \beta ( t ) \widehat { H } _ 3 \right ) \exp \left ( - { \rm i } \alpha ( t ) \widehat { H } _ 2 \right ) \exp \left ( - { \rm i } \tau ( t ) ( \widehat { H } _ 1 + \widehat { H } _ 4 ) \right ) \psi ' ( 0 ) . \end{align*}


\begin{align*} \overline B : = \sup _ { s > 0 } \left ( \int _ 0 ^ s V ( t ) ^ { - \frac 1 { p - 1 } } \ , d t \right ) ^ { p - 1 } \left ( \int _ s ^ \infty W ( t ) \ , d t \right ) \end{align*}


\begin{align*} \zeta _ Q ( s ) : = \sum _ { k \in \Z ^ \alpha \setminus \{ 0 \} } ( k ^ T Q k ) ^ { - s } , \textup { R e } ( s ) > \frac { \alpha } { 2 } . \end{align*}


\begin{align*} \vartheta ( G ) \vartheta ( \overline { G } ) = | G | . \end{align*}


\begin{align*} H ^ \mu \varphi _ k ^ { | \mu | } = [ k ] | \mu | \varphi _ k ^ { | \mu | } \end{align*}


\begin{align*} w _ t = \nabla \lambda _ t \cdot \nabla p + \nabla \lambda \cdot \nabla p _ t + \Delta p _ t . \end{align*}


\begin{align*} \frac { s ^ r e ^ { z s } } { \prod _ { i = 1 } ^ { r } ( e ^ { \omega _ i s } - 1 ) } = \sum _ { n = 0 } ^ { \infty } B _ { n , r } ( z \ , | \ , \omega _ 1 , \cdots , \omega _ r ) \frac { s ^ n } { n ! } . \end{align*}


\begin{align*} \begin{aligned} T _ 3 ( h _ { N - 1 } { } \leq { } 1 ) = 0 \end{aligned} \end{align*}


\begin{align*} \begin{aligned} & \int _ { { \C } _ 1 } \left ( | \nabla \phi _ n | ^ 2 + \phi _ n ^ 2 \right ) d y = \frac { 1 } { 4 } . \end{aligned} \end{align*}


\begin{align*} E = b ^ 2 \sum \limits _ { k = 1 } ^ { 3 } ( - 1 ) ^ { \gamma + \tau } z ^ { k } _ { \tilde { \gamma } } z ^ { k } _ { \tilde { \tau } } z ^ { 3 } _ { \gamma } z ^ { 3 } _ { \tau } . \end{align*}


\begin{align*} \psi _ { n , 2 } ( u , s ) = ( s - \eta _ n ( u ) ) ^ { \alpha } - ( \eta _ n ( s ) - \eta _ n ( u ) ) ^ { \alpha } . \end{align*}


\begin{align*} d f ( x , y ) : = ( D _ y f ) ( x ) : = \lim _ { t \to 0 } \frac { f ( x + t y ) - f ( x ) } { t } \end{align*}


\begin{align*} a _ j ( t ) = \frac 1 { \Gamma ( 2 - \beta ) } ( t - t _ j ) ^ { 1 - \beta } , t \geq t _ j , \end{align*}


\begin{align*} F ( A x + e ) = G ( x ) + d \end{align*}


\begin{align*} p _ n ( i ) - p _ n ( i + 3 ) = a _ i n + b _ i . \end{align*}


\begin{align*} \varphi _ k ( \lambda ) : = \sum _ { l = 0 } ^ m h _ l ( \lambda ^ 2 + k ^ 2 ) \cdot ( k \lambda ) ^ l , \ ; \ ; \ ; k = - m , \dots , m \end{align*}


\begin{align*} \varrho _ 1 ( t ) = 2 ^ { \lfloor \log _ 2 \varrho ( t ) \rfloor } . \end{align*}


\begin{align*} c \Big ( \lfloor T \rfloor _ i , \Upsilon , \lfloor Y \rfloor _ i \Big ) = c \Big ( T , \Upsilon , Y \Big ) \end{align*}


\begin{align*} 1 - \sum \limits _ { n \geq 1 } \frac { a _ n ^ N } { \lambda _ { n } ^ 2 } = \prod \limits _ { k = 1 } ^ { N } \left ( \frac { \mu _ k ^ 2 } { \lambda _ k ^ 2 } \right ) > 0 , \end{align*}


\begin{align*} \lim _ { t \to T } u ( x , t ) = 0 k ^ { - 1 } \eta ( t ) < | x | < k \sqrt { T - t } \end{align*}


\begin{align*} g ( x ) = \min \{ 1 , G ( x , x _ 0 ) \} , x \in \Omega , \end{align*}


\begin{align*} \begin{aligned} 0 & = \int _ { \mathbb { R } ^ N } \left ( \nabla \bar { \phi } ( y ) \cdot \nabla { \bar \psi } ( y ) + \bar { \phi } ( y ) { \bar \psi } ( y ) - p \left ( U _ 0 ( y ) \right ) ^ { p - 1 } \bar { \phi } ( y ) { \bar \psi } ( y ) \right ) d y . \end{aligned} \end{align*}


\begin{align*} k \tan ( k a / 2 ) + k ' \tanh ( k ' a / 2 ) = 0 , \end{align*}


\begin{align*} \lvert \tilde \lambda _ i - \lambda _ i \rvert \le \lVert E \rVert _ S i = 1 , \ldots , \min \{ N _ h , m p \} . \end{align*}


\begin{align*} D _ { q ^ { - 1 } } k ( x ) = u ( x ) k ( x ) x \in I . \end{align*}


\begin{align*} \beta \ , \| y \| _ H \le \| b ( \bullet , y ) \| _ { H ^ * } = \| x _ h \| _ H = 1 , \quad \| y \| _ H \le 1 / \beta \end{align*}


\begin{align*} - \Delta u _ i + V ( x ) u _ i = | u _ i | ^ { p - 1 } u _ i + \beta | u _ i | ^ { q - 1 } u _ i \sum \limits _ { j = 1 \atop j \not = i } ^ d | u _ j | ^ { r - 1 } u _ j \ \hbox { i n } \ \mathbb R ^ n , \ i = 1 , \dots , d \end{align*}


\begin{align*} F ( v , x ) : = \mathcal { F } ^ { - 1 } _ x ( L _ t [ v ] \widehat { f } ) . \end{align*}


\begin{align*} B _ q \colon H ^ s ( \R ^ n ) \times H ^ s ( \R ^ n ) \to \R , B _ q ( u , v ) \vcentcolon = \int _ { \R ^ n } ( - \Delta ) ^ { s / 2 } u \ , ( - \Delta ) ^ { s / 2 } v \ , d x + \int _ { \R ^ n } q u v \ , d x \end{align*}


\begin{align*} A _ m ( \alpha ) { \bf u } _ m ( \alpha ) = { \bf f } _ m ( \alpha ) , \end{align*}


\begin{align*} & \mathrm { D T V } _ { ( p , q ) } ( X ) = - \mathrm { D T E } _ { ( p , q ) } ^ { 2 } ( X ) + \mu ^ { 2 } + 2 \mu \sigma \mathrm { D T E } _ { ( p , q ) } ( Y ) + \sigma ^ { 2 } \left ( L _ { 1 } + 1 \right ) , \end{align*}


\begin{align*} U _ \beta ^ 0 : = \frac { 1 } { h ^ N } \int _ { x _ \beta + R _ h } u _ 0 ( x ) \dd x \quad G _ \beta ^ j : = \frac { 1 } { h ^ N \Delta t _ j } \int _ { t _ { j - 1 } } ^ { t _ j } \int _ { x _ \beta + R _ h } g ( x , \tau ) \dd x \dd \tau . \end{align*}


\begin{align*} \big ( \sqrt { n } T _ n ^ B , \sqrt { n } T _ n \big ) & = \big ( \sqrt { n } ( \hat { \mu } _ n ^ B - \hat { \mu } _ n ) * \gamma _ \sigma + \sqrt { n } ( \hat { \mu } _ n - \mu ) * \gamma _ \sigma , \sqrt { n } ( \hat { \mu } _ n - \mu ) * \gamma _ \sigma \big ) \\ & \stackrel { d } { \to } \big ( G _ { \mu } ' + G _ { \mu } , G _ \mu \big ) \dot { H } ^ { - 1 , p } ( \mu * \gamma _ \sigma ) \times \dot { H } ^ { - 1 , p } ( \mu * \gamma _ \sigma ) \end{align*}


\begin{align*} \begin{array} { l l l } \mbox { C o n d a t - V \ ~ u } & \sigma = L _ 1 / 2 & \tau = 1 / ( 2 L _ 1 ) \\ \mbox { P D 3 O } & \sigma = L _ 2 / 4 & \tau = 1 / L _ 2 . \end{array} \end{align*}


\begin{align*} \left | \sum _ { k = 1 } ^ \infty \frac { \beta ( \sigma , t ) ^ { k } } { k ! } \right | \leq \left | \frac { \beta ( \sigma , t ) } { 1 - \beta ( \sigma , t ) } \right | : = \beta _ 1 ( \sigma , t ) . \end{align*}


\begin{align*} \varphi ( \alpha , \beta ) = ( p ^ { 2 } - 1 ) ( q ^ { 2 } - 1 ) . \end{align*}


\begin{align*} \begin{array} { l l } \partial _ t ^ 2 u = \partial _ t \partial _ x ^ k u + \partial _ x ^ { 2 k } u \end{array} \end{align*}


\begin{align*} \begin{dcases} & d X ( t ) = d B ( t ) + \dfrac { \alpha - 1 } { 2 } \frac { 1 } { X ( t ) } d t , \ t < T _ 0 ^ { x , \alpha } \\ & X ( 0 ) = x \end{dcases} , \end{align*}


\begin{align*} H _ \sigma = \mathcal { Z } _ K \cap \left ( \bigcap _ { \rho _ i \in \sigma } \{ z _ i = 0 \} \right ) \cap \left ( \bigcap _ { \rho _ j \notin \sigma } \{ z _ j \ne 0 \} \right ) \subset \C ^ m . \end{align*}


\begin{align*} { \rm k } _ { m , n } ( z , w ) = \sum _ { k = 0 } ^ { n - 1 } P _ { m , k } ( z ) \overline { P _ { m , k } ( w ) } \end{align*}


\begin{align*} G / B = N ' \rtimes P , \end{align*}


\begin{align*} M f ( x ) \vcentcolon = \sup _ { r > 0 } \frac { 1 } { | B _ r ( x ) | } \int _ { B _ r ( x ) } | f ( y ) | \ , d y . \end{align*}


\begin{align*} \hat { \pi } _ t ^ i = \hat { F } ^ i ( \hat { \pi } ^ i _ { t - 1 } , e ^ i _ t , x ^ i _ t ) , \end{align*}


\begin{align*} w ' _ { c } = 0 . \end{align*}


\begin{align*} g _ b ( x ) = \sum _ { j = 1 } ^ n b _ j ( g \circ S ^ { 1 - j } ) ( x ) + d ' . \end{align*}


\begin{align*} \max _ { 0 \leq l \leq n } { n \choose l } \alpha ^ l ( 1 - \alpha ) ^ { n - l } = O _ { \alpha } \Big ( \frac { 1 } { \sqrt n } \Big ) , \end{align*}


\begin{align*} \frac { A _ 1 + \sqrt { n P } } { \sqrt { \frac { 1 } { n - 1 } \sum _ { i = 2 } ^ n A _ i ^ 2 } } , \end{align*}


\begin{align*} \textstyle C = C _ 0 \bigcup \Big ( \bigcup _ { \ell = \lceil \frac { 1 } { 2 } \mu _ q ( n ) \rceil } ^ { \frac { n - 1 } { 2 } } \bigcup _ { L \in \Omega _ { 2 \ell } } \tilde L ^ * \Big ) ; ~ ~ ~ ~ ~ | \Omega _ { 2 \ell } | \le n ^ { 2 \ell / \mu _ q ( n ) } . \end{align*}


\begin{align*} I _ { ( Y _ + , D _ { Y , + } ) } ( y , z ) = z e ^ { t _ + / z } \sum _ { d \in \mathbb K _ { + } } y ^ { d } \left ( \prod _ { i \in M _ 0 } \frac { \prod _ { a \leq 0 , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } { \prod _ { a \leq D _ i \cdot d , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } \right ) \\ \left ( \prod _ { i = 1 } ^ k \prod _ { 0 < a \leq v _ i \cdot d } ( v _ i + a z ) \right ) \textbf { 1 } _ { [ - d ] } I _ { D _ { Y , + } , d } , \end{align*}


\begin{align*} \vartheta ( x , y , z ) = \Phi _ 1 \left ( [ x , y ] _ \mathfrak { g } , z \right ) - [ \Phi _ 0 ( x ) , \Phi _ 1 ( y , z ) ] + \circlearrowleft ( x , y , z ) . \end{align*}


\begin{align*} U & \le m \left ( L + \frac { L ( m - 1 ) } { 2 } \right ) \\ & = \frac { L m ( m + 1 ) } { 2 } . \end{align*}


\begin{align*} F ' ( U , V ) ( H , K ) & = \langle R ( U , V ) , H \Sigma V ^ { T } + U \Sigma K ^ { T } - P ( H \Sigma V ^ { T } + U \Sigma K ^ { T } ) \rangle \\ & = \langle R ( U , V ) , H \Sigma V ^ { T } + U \Sigma K ^ { T } \rangle \\ & = \langle R ( U , V ) V \Sigma ^ { T } , H \rangle + \langle \Sigma ^ { T } U ^ { T } R ( U , V ) , K ^ { T } \rangle . \end{align*}


\begin{align*} { \mathcal C } _ X ( A , z ) & = \sum _ { n = 0 } ^ \infty z ^ n = \frac 1 { 1 - z } , \\ { \mathcal C } _ X ( L , z ) & = \sum _ { n = 1 } ^ \infty \frac { z ^ n } n = - \ln ( 1 - z ) , \\ { \mathcal C } _ X ( { \bf { N } } _ s , z ) & = \sum _ { n = 1 } ^ s \frac { z ^ n } n . \end{align*}


\begin{align*} t ^ { - 1 } \left ( x ^ { k } g ( x ) \right ) = H ^ { k } t ^ { - 1 } ( g ( x ) ) = H ^ { k } \overline { g } , 0 \leq k \leq n - 1 . \end{align*}


\begin{align*} \bigg | \Big \langle \mathbb { H } - \mathbb { H } _ \mathrm { r i g h t } ^ { ( M _ \Lambda ) } & - \mathbb { H } _ \mathrm { l e f t } ^ { ( M _ \Lambda ) } - \mathrm { d } \Gamma _ \perp \left ( Q _ { > M _ \Lambda } \left ( h _ { \mathrm { M F } } - \mu _ + \right ) Q _ { > M _ \Lambda } \right ) \Big \rangle _ \Phi \bigg | \\ \le \ ; & C _ \Lambda o _ N ( 1 ) + \frac { C } { \left ( \mu _ { 2 M _ \Lambda + 2 } - \mu _ + \right ) ^ { 1 / 2 } } \end{align*}


\begin{align*} P ( G ) : = \prod _ { e \in \mathcal { E } ( G ) } p ( e ) , p ( e ) : = p _ { i _ 1 i _ 2 } , \end{align*}


\begin{align*} S : = \{ a _ 1 , \ldots , a _ m \} . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ N \frac { 1 } { p _ k } > N - 1 , \end{align*}


\begin{align*} P _ { \kappa } ( x ) = 1 - \left [ \ , R _ { \kappa } ( x ) + Q _ { \kappa } ( x ) \sqrt { 1 + \kappa ^ 2 x ^ 2 } \ , \ , \right ] \exp _ { \kappa } ( - x ) \ \ , \end{align*}


\begin{align*} S \simeq \sum \limits _ { n = 1 } ^ \infty 2 ^ n \frac { 1 } { A ( 2 ^ { n } ) } \simeq \sum \limits _ { i = 1 } ^ \infty \frac { 1 } { A ( i ) } = \infty . \end{align*}


\begin{align*} \| e _ Q a J b J x J c J d e _ Q \| _ { \infty , 1 } = \sup _ { y , z \in ( Q ) _ 1 } | \tau ( a x d y c ^ * b ^ * z ) | \leq \| b c \| _ 2 \| a x d \| _ 2 . \end{align*}


\begin{align*} P _ { f X } Y = f P _ X Y , P _ X f Y = f P _ X Y . \end{align*}


\begin{align*} D _ + = \sum _ { i \in ( M _ + \cup \{ j \} ) } \bar { D } _ i \subset X _ + D _ - = \sum _ { i \in M _ + } \bar { D } _ i \subset X _ - . \end{align*}


\begin{align*} s ^ \prime x ^ \prime y ^ \prime & = s ^ \prime \Delta ^ n & ( \ ; x ^ \prime y ^ \prime = \Delta ^ n ) \\ & = \Delta ^ n t ^ \prime \quad \ ; t ^ \prime \in T & ( \ ; T \Delta = \Delta T ) \\ & = x ^ \prime y ^ \prime t ^ \prime & ( \ ; \Delta ^ n = x ^ \prime y ^ \prime ) . \end{align*}


\begin{align*} \mathrm { c a r d } ( \mathcal { S } _ { [ m ( p + 1 ) + j , j ] } ) = \frac { j + 1 } { p m + j + 1 } \binom { m ( p + 1 ) + j } { m } . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { s \in [ 0 , T ] } n ^ { 2 \alpha + 1 } \int _ 0 ^ { \eta _ n ( s ) } \psi _ { n , 2 } ^ 2 ( u , s ) E [ | \sigma ( X ^ n _ { \eta _ n ( u ) } ) - \sigma ( X _ { \eta _ n ( u ) } ) | ^ 2 ] d u = 0 . \end{align*}


\begin{align*} \hat { P } \phi _ { n } ^ { \pm } ( x ) = \hat { T } \phi _ { n } ^ { \pm } ( x ) = \phi _ { n } ^ { \mp } ( x ) . \end{align*}


\begin{align*} c _ 1 ( \alpha , h ) = c _ 2 ( \alpha , g ) = r \ , \ , \mbox { a n d } \ , \ , c _ 1 ( \alpha , g ) = c _ 2 ( \alpha , h ) = t , \end{align*}


\begin{align*} u ( x , t ) \ge M _ n \left \{ \frac { ( 2 x ) ^ { 1 - a } } { 1 + t + x } \log \left ( \frac { 1 + t - x } { 1 + R } \right ) \right \} ^ { a _ n } \quad \mbox { i n } \ D \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } J ( y _ n , u _ n ) = \inf _ { ( y , u ) \in Y \times U } J ( y , u ) : = \hat { J } . \end{align*}


\begin{align*} ( r , s ) \cdot ( a \otimes b \otimes \dots \otimes y \otimes z ) \cdot ( r ' , s ' ) = ( r \cdot a \cdot r ' ) \otimes b \otimes \dots \otimes y \otimes ( \bar s \cdot z \cdot \bar s ' ) \ , , \end{align*}


\begin{align*} Z = \cap _ { i \in M _ + } \{ z _ i = 0 \} \subset X \end{align*}


\begin{align*} { \rm { H o r } } P _ U U = { \rm { H o r } } P _ V V . \end{align*}


\begin{align*} h ( 0 ) \varphi _ 0 = E \varphi _ 0 \end{align*}


\begin{align*} \ell ^ 2 ( \mathcal { F } _ \perp ) : = \bigoplus _ { d \in \mathbb { Z } } \mathcal { F } _ \perp . \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \infty } \frac { g _ k ( n ) \tau ^ * ( n ) } { n ^ 2 } = \prod _ { p } \Big ( 1 - \frac { 2 } { p ^ { 2 k } } \Big ) , \end{align*}


\begin{align*} ( - \Delta ) ^ s m - \frac { ( - \Delta ) ^ s m _ 1 } { \gamma _ 1 ^ { 1 / 2 } } m & = 0 \quad \Omega , \\ m & = m _ 0 \quad \Omega _ e . \end{align*}


\begin{align*} \mathfrak { H } ^ N = \ ; & \Big ( \mathrm { s p a n } \{ u _ 1 \} \oplus \mathrm { s p a n } \{ u _ 2 \} \oplus \bigoplus _ { m \ge 3 } ^ \infty \mathrm { s p a n } \{ u _ m \} \Big ) ^ { \otimes _ { \mathrm { s y m } } N } \end{align*}


\begin{align*} \begin{aligned} \beta _ { r r } & = \varphi '' ( r ) \big [ \log \cos r - f ( s , 0 ) \big ] - 2 \varphi ' ( r ) \tan r - \varphi ( r ) \sec ^ 2 r \\ & \leq | \varphi '' ( r ) | \big [ - \log \cos R + \bar { f } \big ] \end{aligned} \end{align*}


\begin{align*} \overline { G } _ { ( 2 ) } ( u ) = ( 3 + 2 u + 3 \sqrt { 2 u } ) \exp ( - \sqrt { 2 u } ) . \end{align*}


\begin{align*} h ( r , \dot { r } , t ) = \frac { 1 } { 2 } ( 1 - b ^ 2 ) ^ \frac { 3 } { 2 } r ^ 2 + \lambda ( \sqrt { r ^ 2 + \dot { r } ^ 2 } + b ( \dot { r } \cos ( \theta - t ) + r \sin ( \theta - t ) ) ) . \end{align*}


\begin{align*} \bigg ( \sum _ { k = - \infty } ^ { M - 1 } 2 ^ { - k \frac { q } { p } } B ( x _ k , x _ { k + 1 } ) ^ q a _ k ^ q \bigg ) ^ { \frac { 1 } { q } } \leq C '' \bigg ( \sum _ { k = - \infty } ^ { M - 1 } a _ k ^ p \bigg ) ^ { \frac { 1 } { p } } \end{align*}


\begin{align*} Q _ j ^ k ( \alpha _ i ) + \tilde { \eta } k = 1 , \ldots , m , \ j = 1 , \ldots , L , \ i = 1 , \ldots , p . \end{align*}


\begin{align*} f ( z ) = \sum _ { n = 0 } ^ \infty a _ n q ^ n , \textrm { w i t h } q = e ^ { 2 \pi i z } \end{align*}


\begin{align*} \mathcal O _ { x , y } \simeq \begin{cases} \mathbb C P ^ 2 \ , \ \ \mbox { i f } \ \ x = 0 \ \ \mbox { o r } \ \ y = 0 \\ \mathcal E \ , \ \ \mbox { i f } \ \ x , y > 0 \end{cases} \ . \end{align*}


\begin{align*} P ^ n _ \tau = P ^ { n , 1 } _ \tau + P ^ { n , 2 } _ \tau , \end{align*}


\begin{align*} \left \{ \begin{aligned} & ( \varphi _ y , \psi _ y ) ( x , 0 ) = ( \overline u _ e ^ i , \overline h _ e ^ i ) ( x ) , ( \varphi , \psi ) | _ { y \rightarrow \infty } = ( 0 , 0 ) , \\ & \varphi ( 1 , y ) = \int _ y ^ \infty u _ 0 ^ i ( \theta ) { \rm d } \theta , \psi ( 1 , y ) = \int _ y ^ \infty h _ 0 ^ i ( \theta ) { \rm d } \theta . \end{aligned} \right . \end{align*}


\begin{align*} v _ t & : = \left ( ( t + T ) \wedge \lambda _ { \ell , T } \wedge \rho _ { r , T } - T \right ) \ 1 { T < \rho _ \infty } ; \\ \zeta ^ { ( f , \theta ) } _ t & : = \left ( f ( \kappa _ { T + v _ t } ) - \theta v _ t \right ) \ 1 { T < \rho _ \infty } . \end{align*}


\begin{align*} P _ X Y = P _ Y X , \end{align*}


\begin{align*} 1 _ A ^ { - 1 } ( \mathbb { R } _ { > t } ) = \{ g \in G \mid 1 _ A ( g ) > t \} = \left \{ \begin{aligned} & A & & \ ; t < 1 \\ & \emptyset & & \ ; t \geq 1 \end{aligned} \right . . \end{align*}


\begin{align*} I _ { D _ { Y , + } , d } = \frac { 1 } { \prod _ { i \in I _ + , D _ i \cdot d > 0 } ( \bar D _ i + ( D _ i \cdot d ) z ) } [ \textbf { 1 } ] _ { ( - D _ i \cdot d ) _ { i \in I _ + } } ; \end{align*}


\begin{align*} E _ { 1 } \doteq \bigcap \limits _ { x \in \mathbb { Z } ^ { d } , A \in \mathcal { U } } \{ \rho \in \mathcal { U } ^ { \ast } : \rho ( \mathfrak { 1 } ) = 1 , \ ; \rho ( | A | ^ { 2 } ) \geq 0 , \ ; \rho = \rho \circ \alpha _ { x } \} \ , \end{align*}


\begin{align*} \Delta _ k = y ^ 2 \left ( \partial _ x ^ 2 + \partial _ y ^ 2 \right ) - i k y \partial _ x \end{align*}


\begin{align*} \log F ( z \ , | \ , \overline \omega _ 1 , \omega _ 2 ) \sim & \int _ C \frac { e ^ { z s } } { e ^ { \overline \omega _ 1 s } - 1 } \sum _ { k \geq 0 } \frac { B _ k \cdot ( \omega _ 2 ) ^ { k - 1 } ( s ) ^ { k - 2 } } { k ! } \ , d s \\ = & \sum _ { k \geq 0 } \frac { B _ k \cdot \omega _ 2 ^ { k - 1 } } { k ! } \cdot f _ { k - 2 } ( z , \overline \omega _ 1 ) . \end{align*}


\begin{align*} \eta _ n ( [ 0 , T ] ) : = \bigcup _ { t \in [ 0 , T ] } \eta _ n ( t ) . \end{align*}


\begin{align*} c _ p ^ 2 \geq \frac { 3 L + 3 \epsilon } { 3 L + \epsilon } = 1 + \frac { 2 \epsilon } { 3 L + \epsilon } . \end{align*}


\begin{align*} ( \partial _ s + \Delta ) \varphi = - \rho \lambda ^ 2 e ^ { \lambda u } ( 1 + \rho e ^ { \lambda u } ) \varphi | \nabla u | _ g ^ 2 + \rho \lambda e ^ { \lambda u } \varphi ( \partial _ s + \Delta ) u . \end{align*}


\begin{align*} C _ { 6 , 1 } & \approx \sum _ { k = - \infty } ^ { M - 1 } \bigg ( \int _ { x _ k } ^ { x _ { k + 1 } } u \bigg ) \bigg ( \int _ { x _ k } ^ { \infty } u \bigg ) ^ { \frac { q } { p - q } } \Phi ( 0 , x _ k ) . \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { \partial ^ 2 C ^ 2 } { \partial z ^ i _ { \epsilon } \partial z ^ j _ { \eta } } \frac { \partial ^ 2 \varphi ^ j } { \partial x ^ { \epsilon } \partial x ^ { \eta } } v ^ i = f ( x ^ 1 ) \left [ - f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \right ] , \end{align*}


\begin{align*} \Vert \widehat { \lambda } _ h \widehat { u } _ { \mathrm { n c } } - \lambda _ h u _ { \mathrm { n c } } \Vert _ { L ^ 2 ( \Omega ) } \le & \lambda _ h \Vert u - u _ { \mathrm { n c } } \Vert _ { L ^ 2 ( \Omega ) } + \widehat { \lambda } _ h \Vert u - \widehat { u } _ { \mathrm { n c } } \Vert _ { L ^ 2 ( \Omega ) } + \vert \widehat { \lambda } _ h - \lambda _ h \vert . \end{align*}


\begin{align*} \mathrm { F i x } \left ( \mathcal { T } _ { \left \{ C _ { i } \right \} _ { i = 1 } ^ { r } } \right ) = \cap _ { i = 1 } ^ { r } C _ { i } . \end{align*}


\begin{align*} \operatorname { s g n } ( Z ) = - \operatorname { s g n } ( g ( \operatorname { g r a d } ( \| Z \| ^ { 2 } ) + 2 \nabla _ { Z } Z , 2 \nabla _ { Z } Z ) ) \end{align*}


\begin{align*} f ( x ) & = x ( x - a ) ( x - b ) ( x - c ) . \end{align*}


\begin{align*} \lambda \left [ 1 + b \dot { \theta } \sin ( \theta - t ) \right ] = - a ( 1 - b ^ 2 ) ^ { \frac { 3 } { 2 } } . \end{align*}


\begin{align*} F _ m ( A ) = \bigoplus _ { j = 1 } ^ K \Q ^ { d ( m , j ) } d ( m , j ) = \begin{cases} 1 & : 1 \leq m \leq 2 n _ j - 1 , m , \\ 0 & : \end{cases} \end{align*}


\begin{align*} \mathcal { N } _ { A , q } h = ( \mathcal { M } _ { A , q } \mathcal { S } h ) | _ { \partial B } . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { s \in [ 0 , T ] } n ^ { 2 \alpha + 1 } E [ | \widehat \Theta ^ n _ s - \Theta ^ n _ s | ^ 2 ] = 0 . \end{align*}


\begin{align*} \begin{array} { l l } \frac { 1 } { 2 } ( L _ Z g ) ( X , Y ) + g ( X , Y ) d i v ( H ' ) + \frac { 1 } { \lambda ^ 2 } R i c ^ N ( \tilde { X } , \tilde { Y } ) \\ - \frac { 1 } { 4 } \lambda ^ 4 g ( X , Y ) | \nabla _ { \nu } \frac { 1 } { \lambda ^ 2 } | ^ 2 + \mu g ( X , Y ) + \frac { n \lambda ^ 2 } { 2 } g ( X , Y ) \left ( H ' \frac { 1 } { \lambda ^ 2 } \right ) = 0 . \end{array} \end{align*}


\begin{align*} \widetilde { \mathcal { F } } = \{ r _ k ( F ) : F \in \mathcal { F } \mathrm { \ a n d \ } k \le | F | \} . \end{align*}


\begin{align*} \epsilon ( g , d ) : = \min _ { 0 \leq r \leq g } \{ \epsilon ( r , g , d ) \} , c ( g , d ) : = \max _ { 0 \leq r \leq g } \{ c ( r , g , d ) \} . \end{align*}


\begin{align*} \omega _ 1 \equiv \dfrac { i } { 2 } \ , \lambda _ 3 + \dfrac { i } { 2 \sqrt { 3 } } \ , \lambda _ 8 = \dfrac { i } { 3 } \small { \begin{pmatrix} 2 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{pmatrix} } \ \ , \ \ \ \omega _ 2 = \dfrac { i } { \sqrt { 3 } } \ , \lambda _ 8 = \dfrac { i } { 3 } \small { \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 2 \end{pmatrix} } \ . \end{align*}


\begin{align*} N ^ t _ s ( b ) : = \# \{ i \in [ N ^ t _ s ] : | \pi _ i ^ t ( s ) | \geq b \} . \end{align*}


\begin{align*} f _ { \mathbf { M ^ { \ast } } _ { - \xi _ { \boldsymbol { s } k } } } ( \boldsymbol { w } ) = c _ { n - 1 , \xi _ { \boldsymbol { s } k } } ^ { \ast } \exp \left \{ - \frac { 1 } { 2 } \boldsymbol { w } ^ { T } \boldsymbol { w } - \frac { 1 } { 2 } \xi _ { \boldsymbol { s } k } ^ { 2 } \right \} = \phi _ { n - 1 } ( \boldsymbol { w } ) , ~ \boldsymbol { w } \in \mathbb { R } ^ { n - 1 } , \end{align*}


\begin{align*} A _ 2 & \lesssim C _ 1 + \sup _ { k \leq M - 1 } \sup _ { x \in ( x _ k , x _ { k + 1 } ) } \bigg ( \int _ { x } ^ { \infty } u \bigg ) ^ { \frac { 1 } { q } } \bigg ( \int _ { 0 } ^ { x } \bigg ( \int _ 0 ^ t w \bigg ) ^ { - \frac { p } { p - r } } w ( t ) V _ r ( 0 , t ) ^ { \frac { p r } { p - r } } d t \bigg ) ^ { \frac { p - r } { p r } } \\ & = C _ 1 + C _ 3 , \end{align*}


\begin{align*} G = N \rtimes P \end{align*}


\begin{align*} b _ { m + 1 } = \sum _ { i \in M _ + } ( D _ i \cdot e ) b _ i = \sum _ { i \in M _ - } - ( D _ i \cdot e ) b _ i . \end{align*}


\begin{align*} | \mathcal { S } _ { \kappa , 3 } ^ C | = \binom { \kappa + 7 } { 7 } - a _ k - 3 b _ k = 6 c _ k . \end{align*}


\begin{align*} d H e ^ { \frac { H } { 2 \pi i } } \sum _ { m \geq 1 } ( - 1 ) ^ m \frac { ( 2 \pi i ) e ^ { \frac { 2 \pi i m } { d } } } { e ^ { - H } - e ^ { \frac { 2 \pi i m } { d } } } \frac { q ^ { - \frac m d } } { \Gamma ( m ) \Gamma ( 1 + \frac { d ^ \prime } { d } m ) \prod _ { i = 1 } ^ N \Gamma ( 1 - q _ i m ) } . \end{align*}


\begin{align*} \sigma _ k ( \lambda ( D ^ 2 u ) ) = f \end{align*}


\begin{align*} P ( T , x ) & = \dfrac { \displaystyle \prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) } } { \displaystyle \prod _ { j = 0 } ^ { l ( T ) - 1 } W _ j ^ { n ( T , l ( T ) - j ) } } . \end{align*}


\begin{align*} \langle \nabla _ i \nabla _ j \Psi _ t - \nabla _ l \Psi _ t \Gamma _ { i j } ^ l , \nabla _ m \nabla _ b \Psi _ t - \nabla _ l \Psi _ t \Gamma _ { m b } ^ l \rangle = \frac { 1 } { 2 t } O ( t ) + \sum _ l \Gamma _ { i j } ^ l \Gamma _ { m b } ^ l . \end{align*}


\begin{align*} \omega _ j ( t ) & = \frac 1 { \Gamma ( 1 - \beta ) } \int _ { t _ j } ^ { \min ( t , t _ { j + 1 } ) } ( t - \tau ) ^ { - \beta } \ , d \tau \\ & = \frac 1 { \Gamma ( 2 - \beta ) } \left ( ( t - t _ j ) ^ { 1 - \beta } - ( t - \min ( t , t _ { j + 1 } ) ) ^ { 1 - \beta } \right ) \end{align*}


\begin{align*} A _ n ( t ) = t ^ { n - 1 } A _ n ( 1 / t ) . \end{align*}


\begin{align*} \sum _ { z \in S ^ 1 _ N } \exp \Big ( - \kappa ( z - w - y ) ^ 2 \Big ) & \leq \sum _ { z = w - 1 } ^ { w + 1 } \exp \Big ( - \kappa ( z - w - y ) ^ 2 \Big ) \\ & + \int _ { - \infty } ^ { \infty } \exp \Big ( - \kappa ( z - w - y ) ^ 2 \Big ) d z . \end{align*}


\begin{align*} \zeta ( t ) = t ^ { \sigma } , \ ; \ ; \sigma > - 1 . \end{align*}


\begin{align*} \tau ^ \mathrm { t o p } ( M ' / S , \mathbb { 1 } ) = \tau ^ \mathrm { t o p } \big ( M / S , \mathrm { I n d } ^ G _ H \mathbb { 1 } \big ) \ ; . \end{align*}


\begin{align*} \rho ^ { \rm ( s t ) } = \exp \biggl ( \frac 1 2 \mathfrak { c } ^ { T } M \mathfrak { c } + s \biggr ) , \end{align*}


\begin{align*} N ( t ) = \Lambda _ \Omega ( t ) + O _ \Omega \big ( t ^ { d - \delta } \big ) . \end{align*}


\begin{align*} | \mathcal K _ { \kappa , 2 } | & = \left | \mathcal T _ { \kappa , 2 } \right | + \frac 1 2 \left | \mathcal S _ { \kappa , 2 } \right | - \frac 1 2 \left | \mathcal T _ { \kappa , 2 } \right | = \frac 1 2 \left ( \left | \mathcal S _ { \kappa , 2 } \right | + \left | \mathcal T _ { \kappa , 2 } \right | \right ) = \frac 1 2 \left ( \sum _ { i = 0 } ^ { \lfloor \kappa / 2 \rfloor } ( \kappa - 2 i + 1 ) + \binom { \kappa + 3 } { 3 } \right ) . \end{align*}


\begin{align*} N ( \pi ) = \# \{ ( x _ i , y _ j ) \ , | \ , \hbox { t h e $ s $ - p o i n t $ y _ j $ p r e c e d e s t h e $ r $ - p o i n t $ x _ i $ i n t h e $ \pi $ - o r d e r } \} . \end{align*}


\begin{align*} \Tilde { \mathcal { N } } ( f ^ { \tilde { \phi } ^ { - ( n + 1 ) } } ) \circ \rho _ { \eta } ( X ) = \Pi _ { w \in W _ { \rho } ^ { m _ 0 } } f ^ { \tilde { \phi } ^ { - ( n + 1 ) } } ( X + w ) , \end{align*}


\begin{align*} { \rm R C } ( I ^ \vee ) = \mathbb { R } _ + \{ e _ 1 , \ldots , e _ s , ( u _ 1 , 1 ) , \ldots , ( u _ m , 1 ) \} . \end{align*}


\begin{align*} E ( G / H ^ 0 ) = E ( G / H ) E ( H / H ^ 0 ) \neq 0 \end{align*}


\begin{align*} A _ i u : = \begin{cases} \frac { \partial ^ 2 } { \partial x ^ 2 } \Bigl ( a \frac { \partial ^ 2 u } { \partial x ^ 2 } \Bigr ) & \ , \ , i = 1 , \\ a \frac { \partial ^ 4 u } { \partial x ^ 4 } & \ , \ , i = 2 , \end{cases} \end{align*}


\begin{align*} A = ( J - R ) Q , , B = - ( J - R ) C ^ T \end{align*}


\begin{align*} \begin{aligned} \log \hat Z _ { N , 2 , 1 } & \geq - ( I _ { 1 , 2 , 1 } + I _ { 2 , 2 , 1 } ) \\ & \geq - C \left [ \gamma N ^ { 2 } \big ( ( \beta ^ { - 1 / 2 } a ^ { - 1 } N \log N ) \vee 1 \big ) + \beta N ^ 2 a ^ 2 \right ] . \end{aligned} \end{align*}


\begin{align*} \int _ { \Xi _ { r ( I ' ) , s ( I ' ) } } e ^ { W ^ { \nu } / \hbar } \Omega = \int _ { \Xi _ { r ( I ' ) , s ( I ' ) } } e ^ { W ^ { g ( \nu ) } / \hbar } \Omega , \end{align*}


\begin{align*} f _ { \boldsymbol { X } } ( \boldsymbol { x } ) : = \frac { c _ { n } } { \sqrt { | \boldsymbol { \Sigma } | } } g _ { n } \left \{ \frac { 1 } { 2 } ( \boldsymbol { x } - \boldsymbol { \mu } ) ^ { T } \mathbf { \Sigma } ^ { - 1 } ( \boldsymbol { x } - \boldsymbol { \mu } ) \right \} , ~ \boldsymbol { x } \in \mathbb { R } ^ { n } , \end{align*}


\begin{align*} I ( q ^ { \frac { k } { 2 } } \cdot y _ { \gamma _ e + \gamma _ m } ) = { \rm e x p } ( \pi i \tau k + 2 \pi i \theta ( \gamma _ e ) ) \cdot y _ { \gamma _ m } \end{align*}


\begin{align*} x _ { k + 1 } = \varphi ( Y _ { k + 1 } , x _ k ) \end{align*}


\begin{align*} \epsilon ^ { \bullet } ( k ) = \epsilon ^ { \le n } ( k ) \epsilon ^ { \le n } ( s ( \alpha _ u ) ) ^ { - 1 } q _ { \alpha _ u } \end{align*}


\begin{align*} \begin{aligned} \mathcal L ( Y _ \rho ) = W _ \rho \sum _ { i = 2 } ^ d W ^ 2 _ \rho ( \hat \Theta _ i \cdot ) - \gamma _ \rho \partial _ \rho W _ \rho \ \hbox { i n } \ \mathbb R ^ n \end{aligned} \end{align*}


\begin{align*} f _ k ( n ) = \sum _ { w \mid n } \frac { 1 } { w } \sum _ { \substack { r d = w \\ d \ { \rm { i s } } \ k - { \rm { f r e e } } } } \mu ( r ) = \sum _ { w \mid n } \frac { g _ k ( w ) } { w } . \end{align*}


\begin{align*} E [ | Q ^ { n , 4 } _ \tau | ^ 2 ] = n ^ { 4 \alpha + 2 } \sum _ { j = 0 } ^ { \lfloor n \tau \rfloor } E [ \xi _ { j , n } ^ 2 ] . \end{align*}


\begin{align*} f _ { x _ k } ( \lambda ) = - \sqrt { 2 } \det \left ( ( \lambda I _ N - H ) _ { k | k } \right ) , \ \ f _ { y _ { k k + 1 } } ( \lambda ) = 2 \det \left ( ( \lambda I _ N - H ) _ { k | k + 1 } \right ) , \end{align*}


\begin{align*} \mathbb { H } ^ n ( X , ( \Omega _ X ^ \bullet , d + \hbar ^ { - 1 } d W \wedge - ) ) = \Omega _ { \C ^ n } ^ n / \delta ( \Omega _ { \C ^ n } ^ { n - 1 } ) . \end{align*}


\begin{align*} E _ { g n } ( \dot { x } , x , t ) = - \left [ \frac { { \dot h _ 1 } ( t ) } { h _ 1 ( t ) } - \frac { \dot h _ 2 ( t ) } { h _ 2 ( t ) } \right ] \Phi _ { g n } ( x , t ) \end{align*}


\begin{align*} \beta ( \ell ) = \min _ { g \in \mathcal { \mathcal { C } _ { \ell } } } \left \{ \frac { \langle ( I - P _ { B ( o , \ell ) } ) g , g \rangle _ { \pi _ { \ell } } } { \mathbb { E } _ { \pi _ { \ell } } [ g ^ 2 ] } \right \} . \end{align*}


\begin{align*} p \rightarrow { \textstyle \frac { 1 } { \sqrt { 4 \pi } } } ( p \cdot v ) ^ { i \nu - 1 } , \ , \ , \ , \ , v = ( 1 , 0 , 0 , 0 ) , \end{align*}


\begin{align*} \Gamma = \left ( \gamma _ { j + k } \right ) _ { j , k \ge 0 } \ , . \end{align*}


\begin{align*} L _ { i j } ( v , v ) \ , d x ^ i \otimes d x ^ j : = \nabla ^ l ( \nabla _ i v \ , d x ^ i ) \cdot \nabla _ l ( \nabla _ j v \ , d x ^ j ) + \{ \frac { 1 } { 2 } e \nabla _ i v \cdot \nabla _ j v + \frac { 1 } { 2 } ( R _ i { } ^ k \nabla _ j v + R _ j { } ^ k \nabla _ i v ) \cdot \nabla _ k v \} d x ^ i \otimes d x ^ j - \Delta v \cdot \nabla _ i ( \nabla _ j v \ , d x ^ j ) d x ^ i . \end{align*}


\begin{align*} t _ - = \sum _ { a = 1 } ^ \mathrm { r } \bar p _ a ^ - \log \tilde y _ a , \tilde y ^ d = \tilde y _ 1 ^ { p _ 1 ^ - \cdot d } \cdots \tilde y _ \mathrm { r } ^ { p _ \mathrm { r } ^ - \cdot d } . \end{align*}


\begin{align*} \mathcal { P } _ { [ n , j , k ] } = \{ \gamma \in \mathcal { P } _ { [ n , j ] } : \kappa _ { i } ( \gamma ) = k \} i \leq k \leq n - j + i . \end{align*}


\begin{align*} \frac { \partial E } { \partial z ^ i _ { \epsilon } } = 2 b ^ 2 \Big [ z ^ 1 _ { \tilde { \epsilon } } ( - 1 ) ^ { { \tilde { \epsilon } } + \tau } z ^ i _ { \tilde { \tau } } z ^ 1 _ { \tau } + \delta _ { i 1 } \sum \limits _ { k } ^ { } ( - 1 ) ^ { \epsilon + \tau } z ^ 1 _ { \tau } z ^ k _ { \tilde { \tau } } z ^ k _ { \tilde { \epsilon } } \Big ] , \end{align*}


\begin{align*} F ^ { \{ \alpha \} } ( z ) : = ( ( W ^ { \{ \alpha \} } - z I ) ^ { - 1 } p , p ) = \int _ \R \frac { d \rho ^ { \{ \alpha \} } ( s ) } { s - z } \forall z \in \C \setminus \R ; \end{align*}


\begin{align*} H _ { k , j ; n , l , r ; p ; \star } ^ { \mu , m , i ; n o n , a } ( t , x , \zeta ) = \int _ { 0 } ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } E B ^ i ( t , s , x - y + ( t - s ) \omega , \omega , v ) \cdot \nabla _ v \big ( ( t - s ) \mathcal { H } ^ { \star ; \mu , E , i } _ { k , j ; n , l , r } ( y , \omega , v , \zeta ) + \mathcal { H } ^ { \star ; e r r ; \mu , E , i } _ { k , j ; n , l , r } ( y , v , \omega , \zeta ) \big ) \end{align*}


\begin{align*} \delta \psi ( x , y ) \vcentcolon = \psi ( x + y ) + \psi ( x - y ) - 2 \psi ( x ) \end{align*}


\begin{align*} r _ { k , t } = B _ k \log _ 2 \left ( 1 + \frac { h _ { k , t } p _ { k , t } } { \Gamma } \right ) , \end{align*}


\begin{align*} { \bf T } _ j ( \alpha _ i ) \cong \displaystyle \sum _ { k = 1 } ^ { \mu _ j } Q _ j ^ k ( \alpha _ i ) { \bf r } _ j ^ k j = 1 , \ldots , L , \ i = 1 , \ldots , p . \end{align*}


\begin{align*} \rho _ t ( x , y ) = \frac { t ^ { - d / \alpha } \tilde p \big ( 1 , t ^ { - 1 / \alpha } x , t ^ { - 1 / \alpha } y \big ) } { t ^ { - \delta / \alpha } h ( t ^ { - 1 / \alpha } x ) t ^ { - \delta / \alpha } h ( t ^ { - 1 / \alpha } y ) } = t ^ { \frac { 2 \delta - d } { \alpha } } \rho _ 1 \big ( t ^ { - 1 / \alpha } x , t ^ { - 1 / \alpha } y \big ) , \end{align*}


\begin{align*} \left \langle \xi \left ( { { t _ 1 } } \right ) . . . \xi \left ( { { t _ n } } \right ) \right \rangle _ c = { W ^ { \left ( n \right ) } } \left ( { { t _ 1 } - { t _ 2 } , \ldots , { t _ 1 } - { t _ n } } \right ) \end{align*}


\begin{align*} \langle f , I ^ { \circ } ( w _ 1 , z _ 2 ) u _ { p + \frac { j _ 2 } { T } } w _ 2 \rangle = 0 \end{align*}


\begin{align*} \psi _ N = \ ; & \sum _ { s = 0 } ^ N \ ; \sum _ { d = - N + s , \ , - N + s + 2 , \ , \dots } ^ { \dots , \ , N - s - 2 , \ , N - s } u _ 1 ^ { \otimes ( N - s + d ) / 2 } \otimes _ \mathrm { s y m } u _ 2 ^ { \otimes ( N - s - d ) / 2 } \otimes _ \mathrm { s y m } \Phi _ { s , d } \end{align*}


\begin{align*} \int _ { \Omega } c \abs { u } ^ 2 d x = \int _ { \Omega } c \abs { y } ^ 2 d x . \end{align*}


\begin{align*} \frac { d ^ { 2 } \omega } { d t ^ { 2 } } + \omega - \frac { \mu a ^ { 2 } } { \lambda } = 0 . \end{align*}


\begin{align*} C _ { t , \mu , \varphi } = \min _ { r _ 0 \geq 0 } \left ( e ^ { r _ 0 t } ( r _ 0 t ) ^ { 1 - \mu } + \frac 1 { \pi } ( \cos \varphi ) ^ { \mu - 1 } \Gamma ( 1 - \mu , r _ 0 \cos \varphi ) \right ) . \end{align*}


\begin{align*} \begin{aligned} \kappa h _ i ^ 2 - 2 ( 1 + \kappa ) h _ i + 3 { } \geq { } 0 , \\ \end{aligned} \end{align*}


\begin{align*} \displaystyle \lim _ { n \to + \infty } c _ n = 0 . \end{align*}


\begin{align*} \psi _ I ( g ( W ) ) = \widetilde \psi _ I ( g ) ( \psi _ I ( W ) ) . \end{align*}


\begin{align*} \| P _ j ( a \otimes 1 ) P _ k ^ \perp \| = \max _ { 0 \leq i \leq j } \| Q _ i a Q _ { k - j + i } ^ \perp \| \leq \| Q _ j a Q _ { k - j } ^ \perp \| < \varepsilon . \end{align*}


\begin{align*} W _ i = | \hat \theta _ i | - | \hat \theta ' _ i | \ , 1 \leq i \leq p \ . \end{align*}


\begin{align*} \mathrm { F i x } ( S _ { n } ) = \mathrm { F i x } \left ( \mathcal { T } _ { \left \{ C _ { m , r } \left ( n \right ) \left ( j \right ) \right \} _ { j = 1 } ^ { r } } \right ) = \mathrm { F i x } \left ( \mathcal { V } _ { \left \{ C _ { m , r } \left ( n \right ) \left ( j \right ) \right \} _ { j = 1 } ^ { r } } \right ) \supset \cap _ { i = 1 } ^ { m } C _ { i } . \end{align*}


\begin{align*} f = l ( Q ) = l ( Q _ \rho + h _ \rho ) = Q _ \rho p ( x ) + l ( h _ \rho ) \mbox { f o r s o m e } p ( x ) \in K [ x ] . \end{align*}


\begin{align*} \rho = \frac { 1 } { [ K : \Q ] } \sum _ { k = 1 } ^ m \mathrm { I n d } _ { H _ k } ^ G \Big ( \sum _ { g \in \mathrm { G a l } ( K / \Q ) } g . \varphi _ k \Big ) \ ; . \end{align*}


\begin{align*} \theta _ 4 ( x ) : = \left \{ \begin{array} { c l c } 1 & & x \in ( c _ 1 + \delta , c _ 2 - \delta ) , \\ 0 & & x \in ( 0 , c _ 1 ) \cup ( c _ 2 , L ) , \\ 0 \leq \theta _ 4 \leq 1 & & . \end{array} \right . \end{align*}


\begin{align*} X = p \partial _ { 1 } + q \partial _ { 2 } . \end{align*}


\begin{align*} k \cdot ( g ; g _ 1 , \ldots , g _ n ) = ( g k ^ { - 1 } ; k g _ 1 k ^ { - 1 } , \ldots , k g _ n k ^ { - 1 } ) \end{align*}


\begin{align*} u & : = \ell ( \gamma _ 2 \gamma _ 1 \gamma _ 2 \gamma _ 1 \gamma _ 1 \gamma _ 1 \gamma _ 2 { \gamma _ 1 } ^ k \gamma _ 2 ) \\ w _ 1 & : = \ell ( \gamma _ 2 \gamma _ 1 \gamma _ 1 \gamma _ 2 \gamma _ 1 \gamma _ 1 \gamma _ 2 { \gamma _ 1 } ^ k \gamma _ 2 ) \\ w _ 2 & : = \ell ( \gamma _ 2 \gamma _ 1 \gamma _ 1 \gamma _ 1 \gamma _ 2 \gamma _ 1 \gamma _ 2 { \gamma _ 1 } ^ k \gamma _ 2 ) \\ \end{align*}


\begin{align*} d \Theta _ n ( e _ \lambda ^ \vee ) = d ( \rho \circ \tilde { \iota } ) ( e _ \lambda ^ \vee ) . \end{align*}


\begin{align*} F _ { \mu } ( H _ { \mu } ( z ) ) = z , z \in \Gamma _ { \alpha , \beta } ; \end{align*}


\begin{align*} b _ { \xi } ( p , p _ 0 ) = \lim _ n d ( p _ n , p ) - d ( p _ n , p _ 0 ) . \end{align*}


\begin{align*} P _ N \begin{pmatrix} 1 & 0 & 0 \\ w & K & u \\ q & v ^ { { } } & p \end{pmatrix} : = \begin{cases} \begin{pmatrix} 1 & 0 \\ w + \frac { q } { 1 - p } u & K + \frac { 1 } { 1 - p } u v ^ { } \\ \end{pmatrix} , & p < 1 , \\ & \\ \begin{pmatrix} 1 & 0 \\ w + u & K \\ \end{pmatrix} , & p = 1 , \end{cases} \end{align*}


\begin{align*} \left ( \mathrm { c u r l } \ \boldsymbol { u } , \mathrm { c u r l } _ w \ \boldsymbol { v } _ h \right ) _ { \mathcal { T } _ h } + \left ( \boldsymbol { u } , \boldsymbol { v } ^ 0 _ h \right ) _ { \mathcal { T } _ h } - \left ( \boldsymbol { f } , \boldsymbol { v } ^ 0 _ h \right ) _ { \mathcal { T } _ h } - E ( \boldsymbol { u } , \boldsymbol { v } _ h ) = 0 , \end{align*}


\begin{align*} B A ^ { - 1 } = A ^ { - 1 } B \end{align*}


\begin{align*} \pi _ { i _ 1 } \pi _ J = \pi _ { i _ 1 } \cdot \pi _ { J _ 1 } \cdots \pi _ { J _ m } = \pi _ { J ' } \end{align*}


\begin{align*} g ( \pi ) : = \sum _ { j = 1 } ^ m \max \{ F ( x ) : x \in P ( \chi _ { H ^ j ( \pi ) } ) \} , \end{align*}


\begin{align*} C : = C _ t , S : = S _ t , Q : = Q _ t , E : = S - C . \end{align*}


\begin{align*} ( X ^ \vee , h = ( h _ 1 , h _ 2 ) : X ^ \vee \rightarrow \mathbb C ^ 2 ) , \end{align*}


\begin{align*} \psi ( x ) = x + v ( x ) + \cdots = x + ( b + 1 ) g x ^ { a + 1 } y ^ b + \cdots , \end{align*}


\begin{align*} 2 ^ { - 1 } \int _ 0 ^ T g _ { \beta , T } ( t ) & \| u ( t ) \| ^ 2 \ , d t + \int _ 0 ^ T | u ( t ) | ^ 2 _ 1 d t \\ & \leq \int _ 0 ^ T \langle f ( t ) , u ( t ) \rangle \ , d t + \frac { 1 } { \Gamma ( 1 - \beta ) } \int _ 0 ^ T t ^ { - \beta } \langle u ^ 0 , u ( t ) \rangle \ , d t , \end{align*}


\begin{align*} H _ \mu ( \P ) = - \sum _ { A \in \P } \mu ( A ) \log ( \mu ( A ) ) . \end{align*}


\begin{align*} \phi _ { n } ^ { \pm } ( x ) = \frac { e ^ { i \left [ \left ( 1 \pm 2 n \right ) \pi x / a \mp \theta \right ] } } { \sqrt { a } } ; E _ { n } ^ { \pm } = \frac { \hslash ^ { 2 } } { 2 m } \left ( \frac { 2 n \pi \pm \pi } { a } \right ) ^ { 2 } + \frac { \beta \hslash ^ { 4 } } { 3 m } \left ( \frac { 2 n \pi \pm \pi } { a } \right ) ^ { 4 } , \end{align*}


\begin{align*} \partial _ { \theta } f = 0 . \end{align*}


\begin{align*} \phi _ D ( n ) = \langle m _ \sigma , n \rangle , \end{align*}


\begin{align*} Z _ { j , , r s } ^ k = z ^ k _ { j , } ( \alpha _ r , \alpha _ s ) = \exp \Big [ - \frac { 1 } { 2 } ( \alpha _ r - \alpha _ s ) ^ 2 \Big ] r , s = 1 , \ldots , p , \end{align*}


\begin{align*} R _ i = ( \underbrace { m _ i , \dots , m _ i } _ { i } ) ( 1 \le i \le l ) , \end{align*}


\begin{align*} \overline { X } _ { k } = \bigcup _ { i = 0 } ^ { k } X _ { i } . \end{align*}


\begin{align*} \hat { y } ^ { \hat { d } } = y ^ { d _ - } \cdot ( y ^ \prime ) ^ { d ^ \prime } . \end{align*}


\begin{align*} \Vert ( w , \Omega ) \Vert _ { \mathcal { Q } ( \sigma _ 0 ) } : = \Vert ( w , \Omega ) \Vert _ { \mathcal { Q } ( \sigma _ 0 , 0 ) } + \Vert ( w , \Omega ) \Vert _ { \mathcal { Q } ( \sigma _ 0 , 1 ) } , \end{align*}


\begin{align*} & \tilde \varphi \big ( \tilde x \circ a ( x ) \big ) = ( a _ { n - 1 } + a _ 0 \tilde x + \cdots + a _ { n - 2 } \tilde x ^ { n - 1 } ) \varphi \\ & = \big ( \tilde x a ( \tilde x ) - a _ { n - 1 } ( \tilde x ^ n - 1 ) \big ) \varphi = \tilde x a ( \tilde x ) \varphi = \tilde x \tilde \varphi \big ( a ( x ) \big ) . \end{align*}


\begin{align*} v _ r \left ( b _ { 0 , 0 } g ^ { s _ 0 + \ell _ 0 } \right ) = v _ r ( f ) + v _ r ( h ) . \end{align*}


\begin{align*} C _ { \alpha } ( n , s ) : = { n \choose s } \alpha ^ s ( 1 - \alpha ) ^ { n - s } . \end{align*}


\begin{align*} n \otimes y \leftharpoonup a \# b = \sum _ { ( y ) } n ( y _ { - 1 } \rightharpoonup a ) \otimes y _ 0 b \end{align*}


\begin{align*} | \varphi ( 2 \pi x ) - \omega ^ n | & \le K \left | x - \frac { n } { D } \right | , \\ | \varphi ( 2 \pi x ) | & \le 1 - L \left ( x - \frac { n } { D } \right ) ^ 2 , \end{align*}


\begin{align*} u = 0 ~ ~ \Omega \end{align*}


\begin{align*} \gamma ( t ) = ( x ^ { 1 } ( t ) , x ^ { 2 } ( t ) ) = ( r ( t ) \cos t , r ( t ) \sin t ) . \end{align*}


\begin{align*} b _ l = \sum _ { i - j + k = l } a _ { i j k } . \end{align*}


\begin{align*} \lim _ { R \rightarrow \infty } \mathfrak { G } ( x , R ) = 0 . \end{align*}


\begin{align*} \mathcal { L } _ { [ n , j ] } : = \{ \gamma \in \mathcal { D } _ { [ n , 0 ] } : \mbox { t h e l a s t s t e p o f } \ , \ , \gamma \ , \ , \mbox { i s } \ , \ , ( n - 1 , j ) \rightarrow ( n , 0 ) \} , 0 \leq j \leq p . \end{align*}


\begin{align*} ( N S ) ^ g = N ^ g S ^ g = N S ^ x \end{align*}


\begin{align*} \mu ( T _ 0 , \lambda ) \ge \sum _ { j = 1 } ^ { k } \mu ( T _ j , \lambda ) ( \forall \lambda \in \mathbb { R } ) . \end{align*}


\begin{align*} I = \int _ { B ( x , t ) } ( h _ 1 ( y ) ) ^ q d \sigma ( y ) \le C _ 1 ( q , \kappa ) \ , [ \varkappa ( B ( x , 2 \kappa t ) ) ] ^ { \frac { q } { 1 - q } } , \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { N } \left \langle \left ( \mathcal { N } _ { u _ 1 } - \frac { N } { 2 } \right ) ^ 2 \right \rangle _ { \psi _ \mathrm { g s } } = 0 . \end{align*}


\begin{align*} [ K ] & = \bigsqcup _ { i \in [ m ] } I _ i = \bigsqcup _ { j \in [ n ] } J _ j , \\ \Z & = \bigsqcup _ { i \in Z } I _ i = \bigsqcup _ { j \in Z } J _ j . \end{align*}


\begin{align*} a _ { + } = \left ( \begin{array} { c } a _ { 1 } \\ a _ { 2 } \\ a _ { 3 } \\ a _ { 4 } \end{array} \right ) . \end{align*}


\begin{align*} a _ \infty ( s ) = \int _ 1 ^ \infty z ^ { 2 \alpha - 2 s - 1 } \int _ 0 ^ 1 \int _ 0 ^ 1 \Big ( \frac { \sin ^ 2 ( \pi x ) } { \pi ^ 2 } + \frac { \sin ^ 2 ( \pi y ) } { \pi ^ 2 } + z ^ 2 \Big ) ^ { - \alpha } d x d y d z . \end{align*}


\begin{align*} \widetilde { W } _ { n } ( t ) : = \frac { 1 } { \sqrt { n } } \bigg [ \sum _ { j = 0 } ^ { [ n t ] - 1 } \tilde { v } \circ T ^ j + ( n t - [ n t ] ) \tilde { v } \circ T ^ { [ n t ] } \bigg ] , t \in [ 0 , 1 ] . \end{align*}


\begin{align*} x y '' + y ' - x y = 0 , \end{align*}


\begin{align*} \mathrm { P } _ { \Phi + \mathcal { K } _ { \gamma } \left ( \Psi , f \right ) } = - \inf _ { \rho \in E _ { 1 } } \left \{ f _ { \Phi } \left ( \rho \right ) + e _ { \mathcal { K } _ { \gamma } \left ( \Psi , f \right ) } \left ( \rho \right ) \right \} \ , \gamma \in \left ( 0 , 1 \right ) \ , \end{align*}


\begin{align*} \L J _ m ^ k f , g \R = \L f , ( J _ m ^ k ) ^ * g \R \end{align*}


\begin{align*} \mathbf { P } ( A _ k \setminus \cup _ { i = k _ 1 } ^ { k - 1 } A _ i ) & \geq \frac { c } { 1 + \log \psi ^ { - 1 } ( a ( k ) ) } \left ( \mathbf { P } ( C _ k \cap G _ k \cap B _ k \setminus \cup _ { i = k _ 1 } ^ { k - 1 } A _ i ) - \sqrt { e ^ { - a ( k ) + a ( k - 1 ) + 1 } } \right ) \vee 0 . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \frac { \partial } { \partial t } \varphi = \log \frac { ( \widehat { \omega } _ { t } + \sqrt { - 1 } \partial _ { B } \overline { \partial } _ { B } \varphi ) ^ { 2 } \wedge \eta _ { 0 } } { \Omega \wedge \eta _ { 0 } } , \varphi ( 0 ) = 0 . \end{array} \end{align*}


\begin{align*} \psi _ n = \psi _ { n , \textrm { b o u n d e d } } + \psi _ { n , \textrm { i n } } + \psi _ { n , \textrm { o u t } } + o ( 1 ) \end{align*}


\begin{align*} p ^ { 1 / 2 } s _ { \sqrt p } ( \sqrt q ) = \frac 2 { 1 + \int \sqrt { p q } \ d \mu } \left ( \sqrt { \frac q p } - \int \sqrt { \frac q p } \ p \ d \mu \ \right ) \end{align*}


\begin{align*} x _ 0 \ , a _ 1 - x _ 1 \ , a _ 0 = \begin{vmatrix} x _ 0 & x _ 1 \\ a _ 0 & a _ 1 \end{vmatrix} \ , \ y _ 0 \ , b _ 1 - y _ 1 \ , b _ 0 = \begin{vmatrix} y _ 0 & y _ 1 \\ b _ 0 & b _ 1 \end{vmatrix} \ , \end{align*}


\begin{align*} H ^ { \sigma , \lambda } _ { \infty , ( m ) } : = H ^ { \sigma } _ { [ \lambda ] } / m _ \lambda ^ { m + 1 } H ^ { \sigma } _ { [ \lambda ] } , \end{align*}


\begin{align*} \label [ I e q ] { e q : m c k e a n v l a s o v - l i m i t } \dd \overline { X } _ t = b { \left ( \overline { X } _ t , f _ t \right ) } \dd t + \sigma { \left ( \overline { X } _ t , f _ t \right ) } \dd B _ t . \end{align*}


\begin{align*} u = 0 \widetilde { \Omega } . \end{align*}


\begin{align*} \Omega _ { \mu } = \{ x + i y : x \in \mathbb { R } , y > f _ { \mu } ( x ) \} , \end{align*}


\begin{align*} H _ f ( m _ \ast ) + H _ f ( m ^ \ast ) = H _ f ( m _ 1 ) + H _ f ( m _ 2 ) . \end{align*}


\begin{align*} \partial _ { t } \mu _ { t } + \nabla \cdot ( v _ { t } \mu _ { t } ) = 0 \ \ \R ^ { d } \times I \end{align*}


\begin{align*} \lambda = \lambda b ( u , u _ { \mathrm { n c } } ) + \lambda b ( u , u - u _ { \mathrm { n c } } ) = \lambda b ( u , u _ { \mathrm { n c } } - J u _ { \mathrm { n c } } ) + a _ { \mathrm { p w } } ( u , J u _ { \mathrm { n c } } ) + \lambda / 2 \ \Vert u - u _ { \mathrm { n c } } \Vert ^ 2 _ { L ^ 2 ( \Omega ) } . \end{align*}


\begin{align*} E ( k _ u u ) = k _ u ^ 2 E ( u ) . \end{align*}


\begin{align*} F ( x _ 1 , x _ 2 , \ldots , x _ n ) = \left ( f ( x _ 1 , \ldots , x _ k ) , f ( x _ 2 , \ldots , x _ { k + 1 } ) , \ldots , f ( x _ { k } , x _ 1 , \ldots , x _ { k - 1 } ) \right ) . \end{align*}


\begin{align*} \mathbb { O } _ { k , l } \left ( A _ { 1 } , A _ { 2 } \right ) \doteq \left \{ \begin{array} { c c c } A _ { 1 } A _ { 2 } & & k < l , \\ - A _ { 2 } A _ { 1 } & & k > l , \\ 0 & & k = l . \end{array} \right . \end{align*}


\begin{align*} \textstyle \lim \limits _ { t \to \infty } \Pr ( \mbox { \rm $ A $ h a s f u l l r a n k } ) \ge \lim \limits _ { t \to \infty } \prod _ { i = 0 } ^ m ( 1 - \frac { q ^ { d _ i k } } { q ^ { d _ i t } } ) = 1 . \end{align*}


\begin{align*} P _ c ( \Psi _ t ) = P ( \Psi _ t ) - \frac { 1 } { n } \begin{bmatrix} 0 & 0 \\ 0 & J _ n \end{bmatrix} P ( \Psi _ t ) , \end{align*}


\begin{align*} \Psi - \Phi = \Bar Q \circ H + H \circ Q _ { \mathfrak { g } } \end{align*}


\begin{align*} E ( G / H ) = E ( N / N _ N ( S ) ) = \pm 1 \end{align*}


\begin{align*} c _ { i , i } = ( d - 1 ) / d ^ 2 \mbox { f o r $ 1 \leq i \leq d $ a n d } c _ { i , j } = c _ { j , i } = - 1 / d ^ 2 \mbox { f o r $ 1 \leq i < j \leq d $ . } \end{align*}


\begin{align*} \frac { \partial E } { \partial z ^ i _ { \epsilon } } v ^ i = 2 b ^ 2 f ( x ^ 1 ) \cos \left ( x ^ 2 \right ) \left [ - \delta _ { \epsilon 1 } f ( x ^ 1 ) f ' ( x ^ 1 ) \cos \left ( x ^ 2 \right ) + \delta _ { \epsilon 2 } \left \lbrace 1 + f '^ 2 \left ( x ^ 1 \right ) \right \rbrace \sin \left ( x ^ 2 \right ) \right ] . \end{align*}


\begin{align*} \omega ( s ) = \sum \nolimits _ { n = 0 } ^ { \infty } a _ { n } ( s - s _ { 0 } ) ^ { n } , \end{align*}


\begin{align*} \widehat { W } _ T \stackrel { \mathrm { d } } { = } \sqrt { 1 - c T } \widetilde { W } + \sqrt { c T } U , \end{align*}


\begin{align*} P ( \phi , f ) = \limsup _ { t \to \infty } \frac 1 t \log \sum _ { \tau : p ( \tau ) \leq t } e ^ { \ell _ \tau ( f ) } . \end{align*}


\begin{align*} \mathrm { ( \widetilde { I I } ) } ~ & \mathrm { M D T C o v } _ { ( \boldsymbol { a } , \boldsymbol { b } ) } ( \mathbf { Y } ) = \mathbf { \Sigma } ^ { \frac { 1 } { 2 } } \left [ \frac { \boldsymbol { \Omega } } { \overline { F } _ { \mathbf { Z } } ( \boldsymbol { \xi _ { q } } ) } - \frac { \boldsymbol { \delta } \boldsymbol { \delta } ^ { T } } { \overline { F } _ { \mathbf { Z } } ^ { 2 } ( \boldsymbol { \xi _ { q } } ) } \right ] \mathbf { \Sigma } ^ { \frac { 1 } { 2 } } , \end{align*}


\begin{align*} \eta \in C _ c ^ { \infty } ( \mathbb { R } ) , 0 \leq \eta \leq 1 , \eta ( s ) = \begin{cases} 1 \mbox { i f } | s | \leq 1 , \\ 0 \mbox { i f } | s | \geq 2 , \end{cases} \zeta \in C _ c ^ { \infty } ( \mathbb { R } ^ 2 ) , 0 \leq \zeta \leq 1 , \zeta ( s ) = \begin{cases} 1 \mbox { i f } | x ' | \leq 1 , \\ 0 \mbox { i f } | x ' | \geq 2 . \end{cases} \end{align*}


\begin{align*} R ( W , U , Y , Z ) = \bar { R } ( W , U , Y , Z ) + h ( W , Z ) h ( U , Y ) - h ( W , Y ) h ( U , Z ) \end{align*}


\begin{align*} \rho ( \varepsilon ) = \frac { 2 b d k \log ( \frac { 4 b d k } { \varepsilon } ) } { \varepsilon ^ 2 } + \frac { \log ( \frac { 3 } { \delta } ) } { 2 \varepsilon ^ 2 } = n . \end{align*}


\begin{align*} \widetilde { C } ^ n _ t = n ^ { \alpha + \frac 1 2 } \int ^ t _ 0 ( t - s ) ^ { \alpha } \sigma ' ( X _ s ) \left ( X ^ n _ { \eta _ n ( s ) } - X ^ n _ s \right ) \ , d W _ s \end{align*}


\begin{align*} \phi = ( - 1 ) ^ k \gamma _ k \rho _ { u _ k } \gamma _ { k - 1 } \rho _ { u _ { k - 1 } } \cdots \gamma _ 1 \rho _ { u _ 1 } \gamma _ 0 , \end{align*}


\begin{align*} \Sigma _ { \mu \boxtimes \nu } ( z ) = \Sigma _ { \mu } ( z ) \Sigma _ { \nu } ( z ) \end{align*}


\begin{align*} G ( x ) : = G ( x , n , s ) = \displaystyle \sum _ { j = 0 } ^ { n } a _ j b _ j \frac { x ^ j } { j ! } \end{align*}


\begin{align*} \int _ 0 ^ { c _ 1 } \left ( \abs { v } ^ 2 + a \abs { u _ x } ^ 2 \right ) d x = o ( 1 ) \int _ { c _ 2 } ^ L \left ( \abs { z } ^ 2 + \abs { y _ x } ^ 2 \right ) d x = o ( 1 ) . \end{align*}


\begin{align*} ( f + g ) ^ \dagger = f ^ \dagger + g ^ \dagger , ( \lambda f ) ^ \dagger = \lambda ^ \dagger f ^ \dagger = \lambda f ^ \dagger , f ^ { \dagger \dagger } = f , ( f g ) ^ \dagger = g ^ \dagger f ^ \dagger , \end{align*}


\begin{align*} D _ { c _ 1 , \dots , c _ r } ( x _ i , y _ { i j } ) = ( c _ i x _ i , c _ i c _ j y _ { i j } ) , i = 1 , \dots , r , i < j . \end{align*}