\begin{align*} \mathrm { v a r } \big [ \widehat { K } ^ \mathbb { R } _ { \mathrm { u - c p t } } \big ] & = ( K \bar { p } + \sigma ^ 2 / N ) ^ 2 / \bar { p } ^ 2 = K ^ 2 + \frac { 2 K } { N \bar { \gamma } } + \frac { 1 } { N ^ 2 \bar { \gamma } ^ 2 } \end{align*}


\begin{align*} \nabla _ { \gamma _ 1 , \ldots , \gamma _ 2 } f = \partial _ { \gamma _ 1 } \nabla _ { \gamma _ 2 , \ldots , \gamma _ n } f - \sum _ { k = 1 } ^ n \nabla _ { \gamma _ 2 , \ldots , \gamma _ { k - 1 } , \alpha , \gamma _ { k + 1 } , \ldots , \gamma _ n } f \Gamma ^ \alpha _ { \gamma _ 1 \gamma _ k } \end{align*}


\begin{align*} \Gamma ( t ) = F ( \Sigma , t ) . \end{align*}


\begin{align*} \varphi ( u ) & = \langle J ^ { ' } ( u ) , u \rangle = \int _ { \mathbb { R } ^ { d } } \int _ { \mathbb { R } ^ { d } } g \left ( \frac { u ( x ) - u ( y ) } { \vert x - y \vert ^ { \alpha } } \right ) \frac { u ( x ) - u ( y ) } { \vert x - y \vert ^ { \alpha } } \frac { d x d y } { \vert x - y \vert ^ { d } } \\ & + \int _ { \mathbb { R } ^ { d } } g ( u ) u d x - \int _ { \mathbb { R } ^ { d } } K ( x ) f ( x , u ) u d x . \end{align*}


\begin{align*} \prod _ { \ell = 1 } ^ { k } \tilde I ^ { ( p _ { \ell , 1 } ) } _ { i , i } \tilde I ^ { ( p _ { \ell , 2 } ) } _ { j , j } = \Big ( \prod _ { p \in \mathbf { p } _ { k , 1 } \cap \mathbf { p } _ { k , 2 } } \tilde I ^ { ( p ) } _ { i , i } \tilde I ^ { ( p ) } _ { j , j } \Big ) \cdot \Big ( \prod _ { p \in \mathbf { p } _ { k , 1 } \setminus \mathbf { p } _ { k , 2 } } \tilde I ^ { ( p ) } _ { i , i } \Big ) \cdot \Big ( \prod _ { p \in \mathbf { p } _ { k , 2 } \setminus \mathbf { p } _ { k , 1 } } \tilde I ^ { ( p ) } _ { j , j } \Big ) , \end{align*}


\begin{align*} C _ { \mathbf { F } _ N } = \begin{pmatrix} \cos ( 2 \pi 0 / N ) & \sin ( 2 \pi 0 / N ) \\ \vdots & \vdots \\ \cos ( 2 \pi ( N - 1 ) / N ) & \sin ( 2 \pi ( N - 1 ) / N ) \end{pmatrix} . \end{align*}


\begin{align*} X _ { s , t } ( x , \Psi _ { s , t } ( x , v ) ) = x - ( t - s ) v . \end{align*}


\begin{align*} e ^ { - i \theta } u = ( 1 + \alpha ) e ^ { i t } Q + e ^ { i t } h = e ^ { i t } Q + e ^ { i t } \left ( \alpha Q + h \right ) . \end{align*}


\begin{align*} \hat { \gamma } _ { \Sigma } ( \xi ) = \exp \left ( - \dfrac { \langle \xi ; \Sigma ( \xi ) \rangle } { 2 } \right ) . \end{align*}


\begin{align*} \lim _ { n \longrightarrow \infty } & \frac { 1 } { n } { \bf E } \Big \{ \sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \mathbb R } ^ { n _ x } } ^ 2 \Big \} = \lim _ { n \longrightarrow \infty } \frac { 1 } { n } \sum _ { t = 1 } ^ { n } t r \Big ( \Gamma P _ t \Gamma ^ T + D K _ { Z } D ^ T \Big ) \\ = & t r \Big ( \Gamma P \Gamma ^ T + D K _ { Z } D ^ T \Big ) , \ ; \forall P _ 1 \succeq 0 . \end{align*}


\begin{align*} R e ( e ^ { i ( \theta - \frac { \pi } { 2 } ) } A ) & = \frac { e ^ { i ( \theta - \frac { \pi } { 2 } ) } A + e ^ { - i ( \theta - \frac { \pi } { 2 } ) } A ^ { \ast } } { 2 } \\ & = \frac { - i e ^ { i \theta } A + i e ^ { - i \theta } A ^ { \ast } } { 2 } \\ & = \frac { e ^ { i \theta } A - e ^ { - i \theta } A ^ { \ast } } { 2 i } \\ & = I m ( e ^ { i \theta } A ) . \end{align*}


\begin{align*} J ' ( \bar \alpha , \alpha ) = \sum _ { t \neq 0 , 1 } \alpha ( ( 1 - t ) / t ) = \sum _ { y \neq 0 , - 1 } \alpha ( y ) = - \alpha ( - 1 ) . \end{align*}


\begin{align*} \log \Gamma ( s ) = \left ( s - \frac { 1 } { 2 } \right ) \log s - s - \frac { 1 } { 2 } \log ( 2 \pi ) + o ( 1 ) \end{align*}


\begin{align*} K ^ { \varphi _ 0 } _ { \xi , \Psi , \lambda } ( t ) = \frac { 1 } { \| f _ t \| _ { \lambda , t } ^ 2 } . \end{align*}


\begin{align*} \varphi ( r , t ) = \inf \big \{ \varphi ( \gamma ( t ) , 0 ) \mid \gamma ( 0 ) = r , | \dot { \gamma } ( s ) | \le \pi \min \{ R ^ 2 , \gamma ( s ) ^ 2 \} \ \ s \in [ 0 , t ] \big \} . \end{align*}


\begin{align*} { { E } _ { \mu } ^ { [ \alpha , \gamma ] , \rho , \varepsilon } = \left \{ x \in \mathbb { R } ^ d : \ \underline \dim ( \mu , x ) \in [ \alpha , \gamma ] \forall r \leq \rho , \ \mu ( B ( x , r ) ) \leq r ^ { \underline { \dim } ( \mu , x ) - \varepsilon } \right \} } , \end{align*}


\begin{align*} A ( p _ 1 , q _ 1 , p _ 2 , q _ 2 , q _ 3 , q _ 4 ) = \sum _ { \mathbf i \in \mathcal J } \Psi _ { \mathbf i } \cdot \varphi _ 2 ( \mathbf i _ { 1 : 4 } ) \varphi _ 2 ( \mathbf i _ { \{ 1 , 2 , 5 , 6 \} } ) \varphi _ 2 ( \mathbf i _ { \{ 1 , 2 , 7 , 8 \} } ) , \end{align*}


\begin{align*} u ( \cdot + x ( t _ n ) , t _ n ' ) = u ( \cdot + x ( t _ n ) , t _ n + t _ n ' - t _ n ) \to e ^ { i t _ 0 } e ^ { i \theta _ 0 } Q ( \cdot - x _ 0 ) . \end{align*}


\begin{align*} ( - 1 ) ^ { n + 1 } \sum _ { m = 1 } ^ M f ( m ) B _ { 2 n + \nu + 1 } \left ( \dfrac { m } { M } \right ) \end{align*}


\begin{align*} c ( f , p ) = \lim _ { p \to \infty } \frac { \mathcal { S } _ p } { p } . \end{align*}


\begin{align*} \left | \xi \left ( \frac { 1 } { 2 } - \epsilon + i t \right ) \right | = \left | \xi \left ( \frac { 1 } { 2 } + \epsilon - i t \right ) \right | = \left | \overline { \xi \left ( \frac { 1 } { 2 } + \epsilon + i t \right ) } \right | = \left | \xi \left ( \frac { 1 } { 2 } + \epsilon + i t \right ) \right | \end{align*}


\begin{align*} | | F ( x ) | | \leq \tau + \tau \cdot | | F | | _ { C ^ { 1 + \beta } ( C _ { R + 1 / 2 } ) } = W \tau , \end{align*}


\begin{align*} D ^ { \alpha - 1 } \left ( ( P _ t ^ { \nu _ \alpha } ) ^ * ( f ) \right ) ( x ) + \frac { 1 } { p _ \alpha ( x ) } R ^ \alpha \left ( p _ \alpha , ( P _ t ^ { \nu _ \alpha } ) ^ * ( f ) \right ) ( x ) & = \dfrac { - x e ^ { - \alpha t } } { ( 1 - e ^ { - \alpha t } ) } ( P ^ { \nu _ \alpha } _ t ) ^ { * } ( f ) ( x ) \\ & \quad + \dfrac { e ^ { - t } } { \left ( 1 - e ^ { - \alpha t } \right ) } ( P ^ { \nu _ \alpha } _ t ) ^ { * } ( x f ) ( x ) , \end{align*}


\begin{align*} \overline { u } _ n ( x ) = \frac { ( - 1 ) ^ n } { n ! } \frac { d ^ n } { d x ^ n } \left [ \alpha \delta ( x ) + \mathrm { P V } \left ( \frac { 1 } { x } \right ) \right ] \end{align*}


\begin{align*} = \int _ { U _ t ( \mathbb { A } ) P ^ t _ n ( \Q ) \backslash G ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \int _ { U _ t ( \Q ) \backslash U _ t ( \mathbb { A } ) } \sum _ { \xi , \gamma } \phi ( \tilde { \tau } _ t ( \xi n g \times \gamma h ) , s ) \mathbf { f } ( n h ) \mathbf { d } n \mathbf { d } h . \end{align*}


\begin{align*} N _ t ^ { R _ { \alpha } \nu } = \alpha \int _ 0 ^ t R _ { \alpha } \nu ( X _ s ) { \rm d } s - A _ t ^ { \nu } , t \in [ 0 , \zeta [ \end{align*}


\begin{align*} A ' ( \bar { x } , \bar { y } , t ) = \bar { x } \bar { y } e ^ { ( \bar { x } + \bar { y } ) t } { ( \bar { x } - \bar { y } ) ^ 2 \over ( \bar { x } e ^ { \bar { y } t } - \bar { y } e ^ { \bar { x } t } ) ^ 2 } . \end{align*}


\begin{align*} | e ^ { e _ 0 t } \alpha _ + ( t ) | \lesssim \sum _ { n = [ t ] } ^ \infty \Bigg | \int _ n ^ { n + 1 } e ^ { e _ 0 s } B ( R , \mathcal { Y } _ - ) d s \Bigg | \le \sum _ { n = [ t ] } ^ \infty e ^ { ( e _ 0 - c _ 1 ) n } \lesssim e ^ { ( e _ 0 - c _ 1 ) t } , \end{align*}


\begin{align*} & \tilde { \phi } ^ 1 = \phi ^ 0 + \tau \dot { \phi } \big | _ { t = 0 } + \frac { \tau ^ 2 } { 2 } \ddot { \phi } \big | _ { t = 0 } = \phi ^ 0 + \frac { \tau ^ 2 } { 2 } \left ( - G ( \phi ^ 0 ) + \lambda ^ 0 \phi ^ 0 \right ) , \\ & \phi ^ 1 = \tilde { \phi } ^ 1 / \| \tilde { \phi } ^ 1 \| . \end{align*}


\begin{align*} \Xi _ { k + 1 } & = \mathcal { A } \Xi _ k + \mathcal { B } \mathcal { G } ( \hat { \Psi } _ k ( \mathbf { u } , { \tilde { \Xi } } ) + \hat { E } _ k ( \mathbf { u } , { \tilde { \Xi } } ) + \hat { D } _ k ( \omega ) ) , \\ { \mathbf { \tilde { y } } _ k } & { = \mathcal { C } \Xi _ k + \mathbf { w } _ k } , \end{align*}


\begin{align*} \left \vert 1 - \frac { \gamma } { \gamma ' } \right \vert = \left \vert \frac { \gamma ' - \gamma } { \gamma ' } \right \vert \leq \frac { \vert \gamma ' - \gamma \vert } { \gamma ' } \leq \frac { 2 ^ { - j } } { 2 ^ { - l ' } } . \end{align*}


\begin{gather*} \varphi _ 0 [ T ] = - \phi _ 0 [ T ] + ( p , 0 ) + \left ( \phi _ 0 [ T ] - ( p , 0 ) \right ) ^ p . \end{gather*}


\begin{align*} & \Gamma _ k : = \bigg \{ x ^ * \in X _ 0 \mid \exists y ^ * \bigg ( \vert \norm { y ^ * } - \norm { T ^ \circ x ^ * } \vert \leq \frac { 1 } { k + 1 } \land H ^ * \left [ y ^ * , T x ^ * , \frac { 1 } { k + 1 } \right ] \\ & \qquad \qquad \qquad \qquad \qquad \qquad \land \forall i \leq k \left ( \norm { x ^ * - J ^ S _ { \mu _ i } ( x ^ * + \mu _ i y ^ * ) } \leq \frac { 1 } { k + 1 } \right ) \bigg ) \bigg \} \end{align*}


\begin{align*} \Phi ^ { Q ( J ) } : = \bigl \{ ( s , \phi ) \bigm | \phi = \phi ( s _ { 1 n _ 1 } , \ldots , s _ { d n _ d } ) , \ s \in Q ^ J \bigr \} . \end{align*}


\begin{align*} c ^ w _ i ( e ) \coloneqq \begin{cases} w ( e ) & o ( e ) = \eta _ i \\ 0 & \end{cases} \end{align*}


\begin{align*} \varphi = \varphi \circ S . \end{align*}


\begin{align*} \mathcal { N } ( \mathcal { A } ; \mathcal { H } ) : = \left \{ ( m , \omega ) : \ , \omega \in \mathcal { N } _ { \mathcal { I } ; \mathcal { A } } m \in \mathcal { N } ^ { \mathcal { I } ; \mathcal { H } } \right \} . \end{align*}


\begin{align*} \frac { \mathbf { P } \Big ( a _ n S _ n \geq x \sqrt { v _ n + a _ n ^ 2 \sum _ { i = 1 } ^ n ( Z _ i - 1 ) ^ 2 } \ , \Big ) } { 1 - \Phi \left ( x \right ) } = 1 + o ( 1 ) \end{align*}


\begin{align*} k ( s , 0 ) = s \bigg ( \frac { 1 } { h ( s ) } - s \bigg ) = \frac { n ( s ) } { d ( s ) } \end{align*}


\begin{align*} \langle \operatorname { t r } _ e , \operatorname { I n d } ( D _ \mu ) \rangle = \langle \widehat { A } ( \mathfrak { g } , K ) \wedge \operatorname { c h } ( V _ \mu ) _ { \mathfrak { p } ^ * } , [ V ] \rangle , \end{align*}


\begin{align*} \mathrm { R e } \langle \phi , \ddot { \phi } \rangle + \eta ( t ) \mathrm { R e } \langle \phi , \dot { \phi } \rangle & = - \int _ { \mathbb { R } ^ d } \left ( \frac 1 2 | \nabla \phi | ^ 2 + V | \phi | ^ 2 + \beta | \phi | ^ 4 - \Omega \overline { \phi } L _ z \phi \right ) \mathrm { d } \mathbf { x } + \lambda _ { \phi } ( t ) \| \phi \| ^ 2 . \end{align*}


\begin{align*} \mathfrak { I } ( z , \eta , \delta ) = \left \lbrace \rho _ c ( \gamma _ M ( g ) ) \middle | g \in \mathfrak { G } ( z , \eta , \delta ) \right \rbrace . \end{align*}


\begin{align*} H X _ b ^ { q + 1 } ( X , A ) = H Y _ b ^ q ( X , A ) \end{align*}


\begin{align*} C = C ' \big ( 1 + ( \theta d ) ^ { 2 s } \| c \| _ { L ^ \infty ( B _ { \rho } ( e _ 1 ) ) } \big ) , \end{align*}


\begin{align*} \dot { B } _ m ^ j ( t ) = \frac { B ^ j ( [ t ] _ m ^ + ) - B ^ j ( [ t ] _ m ^ - ) } { \delta _ m } , t \in [ 0 , T ] . \end{align*}


\begin{align*} \begin{aligned} A ^ { 0 } _ { 1 } ( u , v ) \partial _ { t } \hat { u } & + A ^ { i } _ { 1 1 } ( u , v ) \partial _ { i } \hat { u } = f _ { 1 } ( U , D _ { x } v ) , \\ A ^ { 0 } _ { 2 } ( u , v ) \partial _ { t } \hat { v } & - B ^ { i j } _ { 2 2 } ( u , v ) \partial _ { i } \partial _ { j } \hat { v } = f _ { 2 } ( U , D _ { x } U ) , \end{aligned} \end{align*}


\begin{align*} \phi ( \mathfrak { g } , s ) = \chi _ v ( \det ( d _ p ) ) ^ { - 1 } | \det ( d _ p ) | _ { v } ^ { - s } j ( \mathfrak { g } , z _ 0 ) ^ { - k } | j ( \mathfrak { g } , z _ 0 ) | ^ { k - s } . \end{align*}


\begin{align*} \sum _ { | \Im ( \rho ) | \leq T } \frac { x ^ { \Re ( \rho ) - 1 } } { | \Im ( \rho ) | } = \sum _ { \substack { | \Im ( \rho ) | \leq T \\ \Re ( \rho ) \leq \sigma } } \frac { x ^ { \Re ( \rho ) - 1 } } { | \Im ( \rho ) | } + \sum _ { \substack { | \Im ( \rho ) | \leq T \\ \Re ( \rho ) > \sigma } } \frac { x ^ { \Re ( \rho ) - 1 } } { | \Im ( \rho ) | } . \end{align*}


\begin{align*} | \varphi ( z ) | \leq \sum _ { n = 0 } ^ { \infty } | \varphi _ { n } C _ { n } ^ { \lambda , \beta } ( z ) | & \leq \sum _ { n = 0 } ^ { \infty } | \varphi _ { n } | | C _ { n } ^ { \lambda , \beta } ( z ) | \leq \mathrm { e } ^ { \varepsilon | z | } C _ { \varepsilon } \sum _ { n = 0 } ^ { \infty } n ! | \varphi _ { n } | \sigma _ { \varepsilon } ^ { - n } . \end{align*}


\begin{align*} \ell C = ( \pi ^ { - 1 } ( U _ 1 ) \cap \ell C ) \cup ( \pi ^ { - 1 } ( U _ 2 ) \cap \ell C ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ l \frac { | S _ i | ^ 2 } { 1 6 h ^ 2 } \leq 2 5 h ^ 2 \varepsilon n ^ 2 . \end{align*}


\begin{align*} \langle x \rangle _ { 0 , \lambda } = 1 , \langle x \rangle _ { n , \lambda } = x ( x + \lambda ) \cdots ( x + ( n - 1 ) \lambda ) , ( n \ge 1 ) . \end{align*}


\begin{align*} \lambda _ \ell = q ^ { \ell j } \frac { { k + j - \ell \choose j } _ q } { { k + j \choose j } _ q } \frac { { n - k - \ell \choose j } _ q } { { n - k \choose j } _ q } \approx q ^ { - \ell j } . \end{align*}


\begin{align*} \pi _ { * } ( \nabla y ^ { j } ) = \bar { \nabla } z ^ { j } , j = 1 , \dots , m , \end{align*}


\begin{align*} \dot { x } _ i \sim \mu _ i , \quad \dot { \mu } _ i + \sum ^ n _ { \substack { j = 1 , \\ j \not = i } } \frac { a _ { i j } } { ( x _ i - x _ j ) ^ 3 } \sim 0 , \end{align*}


\begin{align*} \sum _ { \substack { d \leq Q ^ { \vartheta } \\ ( d , g _ 3 g _ 4 c a e ) = 1 } } \frac { 1 } { d ^ { 1 - \varepsilon } } \cdot d \ll Q ^ { \varepsilon } . \end{align*}


\begin{align*} \begin{aligned} z _ { i 0 } = 1 z _ { i j } = \frac { s _ { i j } - s _ { i j - 1 } } { a _ { i j } - a _ { i j - 1 } } j = 1 , \ldots , n . \end{aligned} \end{align*}


\begin{align*} L _ { A _ { S } } : = - i \partial _ { t } - A _ { S } : C ^ { \infty } ( S ^ { 1 } ; \mathbb { C } ) \to C ^ { \infty } ( S ^ { 1 } ; \mathbb { C } ) . \end{align*}


\begin{align*} A _ m = \{ ( x , \xi ) \in S ^ * B ^ n _ 1 ; \ ; ( m \sqrt { \lambda } ) ^ { s } \leq n ( x , \xi ) < ( ( m + 1 ) \sqrt { \lambda } ) ^ { s } \} . \end{align*}


\begin{align*} \forall x \in \mathbb X \colon \Phi ( x ) : = \bigl ( \Omega - x ^ 1 , S ( x ^ 1 ) - x ^ 2 \bigr ) , \end{align*}


\begin{align*} A = \partial _ k R _ k ^ \alpha ( p _ \alpha , f ) ( x ) , B = \partial _ k \left ( p _ \alpha ( x ) D _ k ^ { \alpha - 1 } ( f ) \right ) ( x ) , C = \partial _ k \left ( f ( x ) D _ k ^ { \alpha - 1 } ( p _ \alpha ) \right ) ( x ) . \end{align*}


\begin{align*} H = \left \{ 1 , a / b , b / a \right \} , \end{align*}


\begin{align*} \begin{aligned} & \begin{pmatrix} \lambda Y _ q & 0 \\ 0 & ( 1 - \lambda ) Z _ q \end{pmatrix} \\ & \geq \begin{pmatrix} \lambda ^ 2 X _ q & \lambda ( 1 - \lambda ) X _ q \\ \lambda ( 1 - \lambda ) X _ q & ( 1 - \lambda ) ^ 2 X _ q \end{pmatrix} - \varepsilon \begin{pmatrix} \lambda ^ 2 X _ q & \lambda ( 1 - \lambda ) X _ q \\ \lambda ( 1 - \lambda ) X _ q & ( 1 - \lambda ) ^ 2 X _ q \end{pmatrix} ^ 2 , \end{aligned} \end{align*}


\begin{align*} \left \{ \begin{aligned} & \mu \phi ( \mathbf { x } ) = - \frac 1 2 \Delta \phi ( \mathbf { x } ) + V ( \mathbf { x } ) \phi ( \mathbf { x } ) + \beta | \phi ( \mathbf { x } ) | ^ 2 \phi ( \mathbf { x } ) - \Omega L _ z \phi ( \mathbf { x } ) , \mathbf { x } \in \mathbb { R } ^ d , \\ & \| \phi \| ^ 2 = 1 , \end{aligned} \right . \end{align*}


\begin{align*} a \circ ( b + c ) = a \circ b - a + a \circ c \end{align*}


\begin{align*} \sigma ^ \prime _ { \mathcal { X } } = \left ( \mathcal { E } ^ \prime _ p , \mathcal { E } _ { p _ 1 } , \mathcal { E } _ { p _ 2 } , \mathcal { E } _ { p _ 3 } , \mathcal { E } _ { p _ 4 } , \Phi ( \sigma _ { \widetilde { X } } ) \right ) \end{align*}


\begin{align*} \Phi _ i = g _ i ^ { - 1 } \cdot b \cdot \phi ( g _ i ) . \end{align*}


\begin{align*} \begin{aligned} y ^ * & \in \mathcal N _ { \mathcal T _ D ( g ( \bar x ) ) } ( \nabla g ( \bar x ) u ) \cap \ker \nabla g ( \bar x ) ^ * , \\ z ^ * & \in \mathcal T _ { \mathcal N _ { \mathcal T _ D ( g ( \bar x ) ) } ( \nabla g ( \bar x ) u ) } ( y ^ * ) , \end{aligned} \end{align*}


\begin{align*} K ( z ) = | 1 + z | ^ { p - 2 } ( 1 + z ) - \left ( 1 + \frac { p } { 2 } z + \frac { p - 2 } { 2 } \bar { z } \right ) , z \in \mathbb { C } \end{align*}


\begin{align*} \langle U _ { L ' _ 1 , L ' _ 2 } g _ 1 , T U _ { L _ 1 , L _ 2 } f _ 1 \rangle & \quad = \sum \limits _ { k = L _ 1 } ^ { L _ 2 } \sum \limits _ { k ' = L ' _ 1 } ^ { L ' _ 2 } \Big \langle D ^ { M } _ { k ' } g _ 1 , D ^ * _ { k ' } T D _ { k } D ^ { M } _ { k } f _ 1 \Big \rangle . \end{align*}


\begin{align*} M _ \omega T _ x = e ^ { 2 \pi i \omega x } T _ x M _ \omega . \end{align*}


\begin{align*} \textup { ( v ) } \ \Big | \int _ { \R ^ N } f _ 2 ( y ) d \omega ( y ) \Big | = \Big | - \int _ { \R ^ N } f _ 1 ( y ) d \omega ( y ) - \int _ { \R ^ N } f _ 3 ( y ) d \omega ( y ) \Big | \lesssim \Big ( \frac { r } { R } \Big ) ^ \gamma . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{align*}


\begin{align*} ( \star ) \qquad \lim _ { m \to \infty } \frac { B _ { k _ m } } { k _ m } = \lim _ { m \to \infty } \frac { B _ { 2 k _ m - 2 } } { 2 k _ m - 2 } = \frac { ( 1 - p ) B _ 2 } { 2 } = \frac { 1 - p } { 1 2 } . \end{align*}


\begin{align*} ( - 1 ) \cdot * _ \Sigma i ^ * _ \Sigma \Pi _ \ell d \phi ( s ) = ( \partial _ \Sigma \phi ) ( s ) = \frac { 1 } { \sqrt { \gamma ' ( s ) } } ( \partial _ u \phi ) ( - s , \gamma ( s ) ) . \end{align*}


\begin{align*} - \triangle _ { g } ( u ) + K _ { g _ { \textnormal { p o i n } } } \ , e ^ { - 2 u } = K _ g \textnormal { o n } \ , \ , \ , \Sigma \end{align*}


\begin{align*} A ^ { 0 } ( U ) U _ { t } + A ^ { i } ( U ) \partial _ { i } U - B ^ { i j } ( U ) \partial _ { i } \partial _ { j } U + D ( U ) U = F ( U ; D _ { x } U ) , \end{align*}


\begin{align*} \frac { \delta E ( \phi ) } { \delta \overline { \phi } } = - \frac 1 2 \Delta \phi + V \phi + \beta | \phi | ^ 2 \phi - \Omega L _ z \phi \end{align*}


\begin{align*} ( P X - X P ) g = \frac { 1 } { 2 \pi i } g , \end{align*}


\begin{align*} C ( \vect { H } , { \sf S N R } ) = \sum _ { n = 1 } ^ { N } \log _ 2 \ ! \left ( 1 + \left ( \nu - \frac { 1 } { \lambda _ n ( \vect { H } ) } \right ) ^ { \ ! \ ! + } \ ! \ ! \ ! \lambda _ n ( \vect { H } ) \right ) \end{align*}


\begin{align*} \begin{aligned} \max _ { \mathbf { w } _ 1 } & \Bigl | \textbf { a } _ 1 ^ T ( \theta _ I ) \mathbf { w } _ 1 \Bigl | ^ 2 \\ \textrm { s . t . } & \| \mathbf { w } _ 1 \| ^ 2 \leqslant 1 . \end{aligned} \end{align*}


\begin{align*} \frac { d } { d t } \int B ( t , x ) \ , d x = & \int \left ( - a B J _ x - b J B _ x - \mu \Lambda ^ \alpha B \right ) \ , d x \\ = & \int ( a - b ) B _ x J \ , d x \\ = & \ ( a - b ) \int J \mathcal H J \ , d x \\ = & \ 0 \end{align*}


\begin{align*} \mathfrak { B } _ J = \begin{cases} \mathfrak { S } _ { p _ 2 } \times \cdots \times \mathfrak { S } _ { n - p _ k } , & p _ 1 = 0 , \\ \mathfrak { B } _ { p _ 1 } \times \mathfrak { S } _ { p _ 2 - p _ 1 } \cdots \times \mathfrak { S } _ { n - p _ k } , & p _ 1 > 0 . \end{cases} \end{align*}


\begin{align*} h ^ { ( \varepsilon ) } ( x ) = - \varepsilon \zeta _ 2 ( x ) \geqslant - \varepsilon \R ^ n \setminus B _ 1 ( 2 e _ 1 ) . \end{align*}


\begin{align*} u _ { t } = \left ( u _ { 0 } ^ { * } + t ( \phi - R ) \right ) ^ { * } . \end{align*}


\begin{align*} \varphi \Bigl ( u ^ { \frac { l } { s + n h + 1 } } \Bigr ) \cdot u = \varphi \Bigl ( u ^ { \frac { l + 1 } { s + n h + 1 } } u ^ { \frac { s + n h } { s + n h + 1 } } \Bigr ) \in \varphi \Bigl ( u ^ { \frac { l + 1 } { s + n h + 1 } } \Bigr ) \cdot \prod _ { i = 1 } ^ n I ^ { ( s _ i + 1 ) } . \end{align*}


\begin{align*} \mathrm { d e g } ( 0 ) : = 0 , \quad \mathrm { d e g } ( \tau ( \rho ) ) : = \max \{ \mathrm { d e g } ( \tau ) , \mathrm { d e g } ( \rho ) + 1 \} . \end{align*}


\begin{align*} v _ p \left ( \sum \limits _ { k _ 1 = 0 \atop p \neq 2 k _ 1 + 1 } ^ { n - r } \frac { c _ { k _ 1 } } { ( 2 k _ 1 + 1 ) ^ { s _ 1 } } \right ) \geq \min \limits _ { 0 \leq k _ 1 \leq n - r \atop p \neq 2 k _ 1 + 1 } v _ p \left ( \frac { c _ { k _ 1 } } { ( 2 k _ 1 + 1 ) ^ { s _ 1 } } \right ) = \min \limits _ { 0 \leq k _ 1 \leq n - r \atop p \neq 2 k _ 1 + 1 } v _ p ( c _ { k _ 1 } ) > v _ p \left ( \frac { c _ { \frac { p - 1 } { 2 } } } { p ^ { s _ 1 } } \right ) . \end{align*}


\begin{align*} \psi ( x ) = x - \sum _ { | \gamma | \le T ^ { \star } } \frac { x ^ { \varrho } } { \varrho } + O ^ { \star } \left ( M \frac { x } { T } ( \log x ) ^ { 1 - \omega } \right ) \ \ x \ge x _ M . \end{align*}


\begin{align*} D ( f g ) = f D ( g ) + D ( f ) g . \end{align*}


\begin{align*} \| T _ j T ^ * _ k f \| _ { L ^ 2 ( \R ^ N , \omega ) } ^ 2 & = \int _ { \mathbb R ^ N } \bigg | \int _ { \mathbb R ^ N } T _ j T ^ * _ k ( x , y ) f ( y ) d \omega ( y ) \bigg | ^ 2 d \omega ( x ) \end{align*}


\begin{align*} x _ t = x _ 0 + \int _ 0 ^ t \big ( \alpha _ { - 1 } x _ s ^ { - 1 } - \alpha _ 0 + \alpha _ 1 x _ s - \alpha _ 2 s ^ { 2 H - 1 } x _ s ^ { \rho } \big ) d s + \int _ 0 ^ t \sigma x ^ { \theta } _ s d B _ s ^ H , \end{align*}


\begin{align*} ( \mathcal { S } ( 0 ) ) \ll & \sum _ { 2 < p \leq X } \sum ^ { \mathcal { J } } _ { l = 1 } \Big ( \alpha ^ { 3 / 4 } _ { 1 } { | \mathcal M } _ { 1 , l } ( p ) | \Big ) ^ { 2 \lceil 1 / ( 1 0 \alpha _ { 1 } ) \rceil } \Phi \left ( \frac { p } X \right ) . \end{align*}


\begin{align*} \tilde { x } ^ 0 & = u \ , , \\ \tilde { x } ^ 1 & = v - \frac { 1 } { 2 } h _ { i j } \dot { M } ^ i _ k M ^ j _ l x ^ k x ^ l \ , , \\ \tilde { x } ^ i & = M ^ i _ j x ^ j \ , , \end{align*}


\begin{align*} p _ { k , m } = f _ { k , m + 1 } - \tilde f _ { k , m + 1 } , k = \overline { m + 1 , 2 m } . \end{align*}


\begin{align*} & \ \int _ { B ( x , r ) } ( | \nabla u | ^ { p - 2 } \nabla u - | \nabla v | ^ { p - 2 } \nabla v ) \cdot ( \nabla u - \nabla v ) \ , d \mu \\ & = \int _ { B ( x , r ) } | \nabla u | ^ { p - 2 } \nabla u \cdot ( \nabla u - \nabla v ) - | \nabla v | ^ { p - 2 } \nabla v \cdot ( \nabla u - \nabla v ) \ , d \mu \\ & = \int _ { B ( x , r ) } | \nabla u | ^ { p - 2 } \nabla u \cdot ( \nabla u - \nabla v ) \ , d \mu . \end{align*}


\begin{align*} \varepsilon ^ { - 2 } \int _ 0 ^ \tau \int _ { \Omega _ R } p ( \rho ) \left ( \rho - \frac { 1 } { | \Omega _ R | } \int _ { \Omega _ R } \rho \right ) = \sum _ { i = 1 } ^ 4 I _ i , \end{align*}


\begin{align*} u \in \prod _ { i = 1 } ^ n I ^ { ( s _ i + 1 ) } B . \end{align*}


\begin{align*} z _ i ^ { \widetilde k } \in { } \ , \{ 1 , z _ i , \dots , z _ i ^ { { \widetilde k } - 1 } \} + { R a n } \ , M _ { z _ { n + 1 } } \ , , \ ; i = 1 , \dots , n . \end{align*}


\begin{align*} \frac { f ( x ) - g ( x ) } { h ( x ) + 1 } \ = \ \frac { r } { s } . ] \end{align*}


\begin{align*} h ( p ) = \sum _ { i = 1 } ^ d c _ i h ( p _ i ) + \beta ^ { - 1 } \lambda > \sum _ { i = 1 } ^ d c _ i h ( p _ i ) , \end{align*}


\begin{align*} M _ { n , \mathbf p _ k } = \frac 1 n \sum _ { i \ne j } ^ n \prod _ { \ell = 1 } ^ k I _ { i , j } ^ { ( p _ \ell ) } , N _ { n , \mathbf p _ k } = \frac 1 n \sum _ { i = 1 } ^ n \prod _ { \ell = 1 } ^ k I _ { i , i } ^ { ( p _ \ell ) } . \end{align*}


\begin{align*} \overline { \zeta } \left ( A , T ^ d _ N , u \right ) = \det \Big ( I _ { 2 d N ^ d } - u M _ A \Big ) ^ { - 1 / N ^ d } . \end{align*}


\begin{align*} \int _ 0 ^ t g ( x _ s ) d B ^ H _ s = \int _ 0 ^ t - H s ^ { 2 H - 1 } g ^ { \prime } ( x _ s ) d s + \int _ 0 ^ t g ( x _ s ) d ^ { \circ } B _ s ^ H \end{align*}


\begin{align*} \theta ( 1 - \delta ) \chi / ( \chi - \delta ) = \min \{ 1 / 2 e , \theta / 2 \} . \end{align*}


\begin{align*} g _ 1 ( t ) \Omega ^ n \mathcal { L i } _ { K , n } ( \alpha ) + g _ 2 ( t ) \Omega ^ n \mathcal { L i } _ { C , n } ( \beta ) = 0 . \end{align*}


\begin{align*} \int _ { U ( \Q ) \backslash U ( \mathbb { A } ) } \mathbf { f } ( u g ) \mathbf { d } u = 0 , \end{align*}


\begin{align*} & P _ { i , 1 } = ( \dot { x } _ i - \mu _ i ) \sigma _ i \partial _ y \widetilde { R } _ i , \\ & P _ { i , 2 } = - \bigg ( \dot { \mu } _ i + \sum ^ n _ { \substack { j = 1 , \\ j \not = i } } \frac { a _ { i j } } { x ^ 3 _ { i j } } + \sum ^ n _ { \substack { k , j = 1 , \\ j \not = i } } \frac { b _ { i j k } \mu _ k } { x ^ 3 _ { i j } } \bigg ) \frac { \sigma _ i \Lambda \widetilde { R } _ i } { 1 + \mu _ i } . \end{align*}


\begin{align*} z ^ 3 y ^ 3 + z y ^ 5 + x ^ 6 = 0 \ , . \end{align*}


\begin{align*} \lim \limits _ { n \to \infty } \frac { 1 } { n } \log \prod \limits _ { i = 1 } ^ { n } p _ { w _ i } = \sum \limits _ { k \in T } p _ k \log p _ k . \end{align*}


\begin{align*} u ( r , t ) = \frac { 1 } { T - t } w \left ( \frac { r } { \sqrt { T - t } } , \tau \right ) , \tau = - \log ( T - t ) . \end{align*}


\begin{align*} \sum _ { \ell = 1 } ^ { N } \bigg ( \frac { 2 \rho _ { \ell } \mu _ { \ell } } { p _ \ell } + \frac { c _ \ell p _ \ell } { \mu _ \ell } \bigg ) \le \sum _ { \ell = 1 } ^ { N } \bigg ( \frac { 2 \rho _ { \ell } \mu _ { \ell } } { f _ \ell ^ \star } + \frac { c _ \ell f _ \ell ^ \star } { \mu _ \ell } \bigg ) . \end{align*}


\begin{align*} Z ( x , y ) = x + i Q ( x , y ) = x + i \int _ 0 ^ y P ( x , \tau ) d \tau \ , . \end{align*}


\begin{align*} \widehat { P } ( s ) = N ^ { 1 - s } \cdot \max \big \{ N a ^ t b ^ { 1 - t } + c ^ t d ^ { 1 - t } , N b + d \big \} , \end{align*}


\begin{align*} \partial _ \Sigma ^ * \Psi ( f ) \star _ H \partial _ \Sigma ^ * \Psi ( g ) = \partial _ \Sigma ^ * \Psi ( f ) \cdot \partial _ \Sigma ^ * \Psi ( g ) + \hbar \left \langle ( \partial _ \Sigma \otimes \partial _ \Sigma ) [ \tfrac { i } { 2 } E + H ] , f \otimes g \right \rangle _ { \Sigma ^ 2 } . \end{align*}


\begin{align*} S _ 0 ^ \infty ( - \bar x ) + \bar x \cdot \bar c + \sum _ { t = 0 } ^ { T - 1 } x _ t \Delta s _ { t + 1 } > 0 \end{align*}


\begin{align*} \Lambda ( M , \Gamma ) : = \{ \ , \ , \alpha \ , \ , | \ , \ , \alpha \ , \ , \mathrm { i s \ , \ , a d m i s s i b l e \ , \ , o r b i t \ , \ , s e t \ , \ , s u c h \ , \ , t h a t \ , \ , } [ \alpha ] = \Gamma \mathrm { \ , \ , a n d } \ , \ , A ( \alpha ) < M \ , \ , \} \end{align*}


\begin{align*} \begin{dcases} \rho _ t + V _ 0 \rho _ x + Q _ 0 u _ x = 0 \ & ( 0 , + \infty ) \times ( 0 , L ) , \\ u _ t - \frac { \nu } { Q _ 0 } u _ { x x } + V _ 0 u _ x + a \gamma Q _ 0 ^ { \gamma - 2 } \rho _ x = 0 \ & ( 0 , + \infty ) \times ( 0 , L ) . \end{dcases} \end{align*}


\begin{align*} r ^ n = q ^ * r ^ o q , \end{align*}


\begin{align*} \lambda _ i ( U D ) = \frac { R ( k - 1 , i ) } { R ( k , i ) } . \end{align*}


\begin{align*} S = \left \{ R ( s , t ) = \left ( s , t , z ( s , t ) \right ) ; \ ( s , t ) \in \R ^ { 2 } \right \} , \end{align*}


\begin{align*} d ( ( a , e _ H ) , ( \gamma _ a , \gamma _ a ^ * ) ) = \max \{ d _ G ( a , \gamma _ a ) , d _ H ( \gamma _ a ^ * , e _ H ) \} \leq D , \end{align*}


\begin{align*} u = \alpha x _ d ^ + + \beta x _ d ^ - \end{align*}


\begin{align*} \kappa ( g ) = \sum _ { ( d , e ) \in D _ { 2 } ( H G ) } | g ( d ) \cap g ( e ) | = 7 \equiv 1 \pmod 2 . \end{align*}


\begin{align*} \zeta _ { h _ { \vert ( V _ f ^ I ) ^ { \langle ( \underline { \lambda } , \sigma ) \rangle } / C _ { G \rtimes S } ( \underline { \lambda } , \sigma ) } } ( t ) = ( 1 - t ^ { m _ I } ) ^ { r _ I ( \underline { \lambda } , \sigma ) } \ , , \end{align*}


\begin{align*} D _ { U _ { i } } Y _ { s } ^ { x ; l ^ { \epsilon } } = \frac { s - l _ { 0 } ^ { \epsilon } } { t - l _ { 0 } ^ { \epsilon } } \ , \nabla _ { v _ i } Y _ { s } ^ { x ; l ^ { \epsilon } } , l _ { 0 } ^ { \epsilon } \leq s \leq t . \end{align*}


\begin{align*} M = \begin{pmatrix} m _ 0 & m _ { - 1 } & m _ { - 2 } & \ldots & \ldots & m _ { - k } \\ m _ 1 & m _ 0 & m _ { - 1 } & m _ { - 2 } & \ldots & m _ { - ( k - 1 ) } \\ \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ m _ l & \ldots & \ldots & \ldots & \ldots & \cdot \end{pmatrix} \end{align*}


\begin{align*} E _ { \psi } ( x ) \le \varepsilon _ s ( x ) : = \varepsilon _ 1 ( x , T ) + 2 ( S _ 0 + B _ 1 ( T _ 0 , T ) ) x ^ { - 1 / 2 } \end{align*}


\begin{align*} \sigma _ \gamma ( a \otimes b ) = \widetilde { \phi } _ \gamma ( a ) ( b ) \ ; , \end{align*}


\begin{align*} \nu ( j ) = \min \left \lbrace k \geq 0 : y _ { j } \in \mathcal C _ k ^ j \right \rbrace . \end{align*}


\begin{align*} 1 \leq \sum _ { i } \frac { \partial f ^ { * } } { \partial \kappa _ { i } } ( \kappa ^ { * } ) = \frac { 1 } { ( f ( \kappa ) ) ^ { 2 } } \frac { \partial f } { \partial \kappa _ { i } } ( \kappa ) \kappa _ { i } ^ { 2 } . \end{align*}


\begin{align*} ( \sigma _ T , v _ T ) & = \sum _ { k \geq K _ 0 } a _ k \Phi _ { \lambda ^ p _ { k } } + \sum _ { | k | \geq K _ 0 } b _ k \Phi _ { \lambda ^ h _ k } \\ & = \sum _ { k \geq K _ 0 } a _ k ( \xi _ { \lambda ^ p _ k } , \eta _ { \lambda ^ p _ k } ) + \sum _ { | k | \geq K _ 0 } b _ k ( \xi _ { \lambda ^ h _ k } , \eta _ { \lambda ^ h _ k } ) , \end{align*}


\begin{align*} \begin{aligned} | \tilde Y _ { s , t } ( x , v ) | & \lesssim \frac { \delta ( t - s ) } { 1 + \frac { \langle x ^ \perp \rangle ^ 2 } { \langle v ^ \perp \rangle ^ 2 } } , \\ | \nabla _ v \tilde Y _ { s , t } ( x , v ) | & \lesssim \log ( 2 + t ) \frac { \delta ( t - s ) } { 1 + \frac { \langle x ^ \perp \rangle } { \langle v ^ \perp \rangle } } . \end{aligned} \end{align*}


\begin{align*} \mathfrak { b c } = \sum _ { i = 0 } ^ { s / 2 } \binom { ( s + d - 1 ) / 2 } { \lfloor i / 2 \rfloor , ( d + 1 ) / 2 + \lfloor ( i - 1 ) / 2 \rfloor , s / 2 - i } . \end{align*}


\begin{align*} \left [ \begin{array} { c } \iota ( g _ 1 z _ 2 , g _ 2 z _ 2 ) \\ 1 \end{array} \right ] = R ^ { - 1 } \left [ \begin{array} { c c } U ( g _ 1 z _ 1 ) & 0 \\ 0 & \mathfrak { J } \overline { U ( g _ 2 z _ 2 ) } \end{array} \right ] B ( g _ 1 z _ 1 , g _ 2 z _ 2 ) ^ { - 1 } S \end{align*}


\begin{align*} \begin{aligned} \bar { \nabla } z ^ { j } ( h _ { s t } ) ( \bar { q } ) & = \pi _ { * } ( \nabla y ^ { j } ) ( h ( \bar { \nabla } z ^ { s } , \bar { \nabla } z ^ { t } ) ) ( \pi ( q ) ) = \nabla y ^ { j } ( g ( \nabla y ^ { s } , \nabla y ^ { t } ) ) ( q ) \\ & = y ^ { s } ( \nabla y ^ { j } , \nabla y ^ { t } ) ( q ) + y ^ { t } ( \nabla y ^ { s } , \nabla y ^ { j } ) ( q ) \end{aligned} \end{align*}


\begin{align*} g z = ( a z + b u _ 0 ) ( c z + d u _ 0 ) ^ { - 1 } u _ 0 , \lambda ( g , z ) = u _ 0 ^ { - 1 } ( c z + d u _ 0 ) , g = \left [ \begin{array} { c c } a & b \\ c & d \end{array} \right ] \in G _ { \infty } . \end{align*}


\begin{align*} \det { \mathfrak { S } } = A _ N ^ K B _ K ^ N \prod _ { i = 1 } ^ { N } \prod _ { j = 1 } ^ { K } ( \alpha _ i - \beta _ j ) . \end{align*}


\begin{align*} ( 2 k - 1 ) \ , \left ( \sum \limits ^ { 2 k } _ { i = 1 } \ , | \lambda _ i | \right ) ^ 2 & \geq 4 k \ , \sum \limits _ { i < j } \ , | \lambda _ i | \ , | \lambda _ j | , \\ & = 2 k \ , \left [ \left ( \sum \limits ^ { 2 k } _ { i = 1 } \ , | \lambda _ i | \right ) ^ 2 - \sum \limits ^ { 2 k } _ { i = 1 } \ , \lambda ^ 2 _ i \right ] , \end{align*}


\begin{align*} G _ { 0 ^ + } ( u ) : = \int _ { 0 } ^ { + \infty } P _ t ( u ) d t , u \in H _ 0 , \end{align*}


\begin{align*} R ( \alpha ) = - \Big ( \pi \mu \frac { \Gamma ( \frac { \gamma ^ 2 } { 4 } ) } { \Gamma ( 1 - \frac { \gamma ^ 2 } { 4 } ) } \Big ) ^ { 2 \frac { ( Q - \alpha ) } { \gamma } } \frac { \Gamma ( - \frac { \gamma ( Q - \alpha ) } { 2 } ) \Gamma ( - \frac { 2 ( Q - \alpha ) } { \gamma } ) } { \Gamma ( \frac { \gamma ( Q - \alpha ) } { 2 } ) \Gamma ( \frac { 2 ( Q - \alpha ) } { \gamma } ) } . \end{align*}


\begin{align*} \Delta _ 1 ( \Gamma ) = \sum _ { b \in B ( \Gamma ) } \sum _ { c _ 1 , c _ 2 \in C ( b ) } e _ { c _ 1 c _ 2 } ( \Gamma ) . \end{align*}


\begin{align*} U \ , \pi _ 1 ( \mathbf { h } ) \ , U ^ { - 1 } = \pi _ 2 ( \mathbf { h } ) \forall \mathbf { h } \in \mathbf { H } . \end{align*}


\begin{align*} e ^ { - i t - i \theta ( t ) } u ( x + X ( t ) , t ) = ( 1 + \alpha ( t ) ) Q ( x ) + h ( t , x ) , \ ; \forall t \in D _ { \delta _ 0 } , \end{align*}


\begin{align*} \Theta ^ { Q , j } = 0 , \\ W ^ { Q , j } = 0 . \end{align*}


\begin{align*} \partial _ { \alpha , \beta } ( f ) ( x ) : = \sqrt { 1 - x ^ 2 } f ' ( x ) . \end{align*}


\begin{align*} I ( \alpha , \beta ) - J _ { 0 } ( \alpha , \beta ) = \sum _ { i } \mu _ { \mathrm { d i s c } } ( \alpha _ { i } ^ { m _ { i } } ) - \sum _ { j } \mu _ { \mathrm { d i s c } } ( \beta _ { j } ^ { n _ { j } } ) . \end{align*}


\begin{align*} \| \hat { V } ^ { T } \hat { V } - I \| _ F = \| R ^ { T } R - I \| _ F \leq \xi . \end{align*}


\begin{align*} S = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in S p ( \R , 2 d ) \det ( A ) \neq 0 . \end{align*}


\begin{align*} L ( s , \mathbf { f } , \chi ) \mathbf { f } ( g ) = c _ k ( s ) \int _ { G ( \Q ) \backslash G ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \mathfrak { E } ( g \times h , s ) \mathbf { f } ( h ) \mathbf { d } h . \end{align*}


\begin{align*} \P ( \theta \in A | x ) & = \max ( 0 , \alpha ^ A - 0 . 4 \delta ) , \\ \P ( \theta \in B | x ) & = \max ( 0 , \alpha ^ B - 0 . 4 \delta ) , \\ \P ( \theta \in A \cup B | x ) & = \min ( 1 , \alpha ^ A + \alpha ^ B - 0 . 8 \delta ) , \end{align*}


\begin{align*} \frac { 2 } { e _ { \lambda } ( t ) + 1 } e _ { \lambda } ^ { x } ( t ) = \sum _ { n = 0 } ^ { \infty } \mathcal { E } _ { n , \lambda } ( x ) \frac { t ^ { n } } { n ! } , ( \mathrm { s e e } \ [ 2 ] ) . \end{align*}


\begin{align*} \mathcal J = \mathcal J _ n = \left \{ \mathbf { i } = ( i _ 1 , i _ 2 , i _ 3 , i _ 4 ) \in \{ 1 , \dots , n \} ^ 4 : i _ 1 < i _ 2 , i _ 3 < i _ 4 \right \} . \end{align*}


\begin{align*} x + u = _ X J ^ A _ 1 ( x + u ) + ( ( x + u ) - J ^ A _ 1 ( x + u ) ) \end{align*}


\begin{align*} \left \{ ( \lambda , \delta ) \in \Lambda \times \{ 0 , 1 \} ^ { \lceil \log _ 2 l _ i \rceil } \left | ( \gamma _ i , \delta _ i ) \in M , \ \sum _ { j \in \varpi _ i ( t ) } \lambda _ { i j } \geq \gamma _ { i t } \ ; t \in \{ 1 , \ldots , l _ i \} , \ \gamma _ { i t } = 0 \{ l _ i + 1 , \ldots 2 ^ { \lceil \log _ 2 l _ i \rceil } \right . \right \} \end{align*}


\begin{align*} \Sigma _ { d , n } = \left \{ P \in \R [ X _ 1 , \ldots , X _ n ] _ { \leq d } \ ; \mid \ ; P = \sum _ { 0 \le i \le d } a _ i \sigma _ { i , n } , a _ i \in \R , 0 \leq i \leq d \right \} . \end{align*}


\begin{align*} W = ( \nabla F , \nabla ^ 2 F ) , \end{align*}


\begin{align*} A _ { u , v } = \begin{cases} 1 & u , v \\ 0 & \end{cases} \end{align*}


\begin{align*} \alpha \alpha ' \alpha ^ { - 1 } \alpha '^ { - 1 } = 1 + ( \alpha _ m \alpha ' _ { m ' } - \alpha ' _ { m ' } \alpha _ { m } ) t ^ { m + m ' } + \cdots \end{align*}


\begin{align*} \partial _ t [ U _ { \frac { t } { 2 } } F ( \varphi ) ] = \frac { 1 } { 2 } \Delta U _ { \frac { t } { 2 } } F ( \varphi ) - \frac { 1 } { 2 } \left ( A _ { \epsilon } \varphi , \nabla U _ { \frac { t } { 2 } } F ( \varphi ) \right ) , \end{align*}


\begin{align*} 0 \le \int _ s ^ \tau ( v - y ) \dd ( y - u ) = - \varepsilon ( ( y - u ) ( \tau ) - ( y - u ) ( s ) ) . \end{align*}


\begin{align*} H _ n ^ { \star } ( \mathbf { s } ) = H _ n ^ { \star } ( s _ 1 , s _ 2 , \ldots , s _ r ) : = \sum \limits _ { 1 \leq k _ 1 \leq k _ 2 \leq \cdots \leq k _ r \leq n } \prod _ { j = 1 } ^ { r } \frac { 1 } { k _ j ^ { s _ j } } . \end{align*}


\begin{align*} h = \lambda ^ { - 1 } \phi ( \lambda ) . \end{align*}


\begin{align*} H _ u ( x ' , u ( x ' , x _ d ) ) = x _ d , \quad u ( x ' , H _ u ( x ' , y _ d ) ) = y _ { d } \ , , \end{align*}


\begin{align*} \sqrt { 2 \pi } \mathcal { K } ( x , k ) = \mathcal { K } _ { S } ( x , k ) + \mathcal { K } _ { R } ( x , k ) , \end{align*}


\begin{align*} X ' { \theta ) } = \lambda \Lambda ( \theta ) X ( \theta ) . \end{align*}


\begin{align*} \hat q = q + { \epsilon \over 2 } q p ^ b , \end{align*}


\begin{align*} A b _ 0 ( x , \omega ) = \begin{cases} \frac { \sin ( \pi \omega ( 1 - | x | ) ) } { \pi \omega } , & | x | \leq 1 \\ 0 , & \end{cases} \end{align*}


\begin{align*} \min \{ | A ( \beta _ j ) | , | B ( \alpha _ i ) | , 1 \leq i \leq N , 1 \leq j \leq K \} = \delta > 0 . \end{align*}


\begin{align*} \Delta u = \varepsilon ^ { - 2 } \left ( e ^ { u } - \left ( x ^ { 2 } + y ^ { 2 } \right ) e ^ { - u } \right ) \mathbb { R } ^ 2 \end{align*}


\begin{align*} \left . \begin{aligned} & \nabla g ( \bar x ) ^ * y ^ * = 0 , \ , \nabla g ( \bar x ) ^ * \hat z ^ * = 0 , \\ & y ^ * \in \mathcal N _ D ( g ( \bar x ) ; \nabla g ( \bar x ) u ) , \ , \hat z ^ * \in D \mathcal N _ D ( g ( \bar x ) , y ^ * ) ( 0 ) \end{aligned} \right \} \quad \Longrightarrow \hat z ^ * = 0 \end{align*}


\begin{align*} \sum _ { p \leq x } \log p \cdot \chi ( p ) = \delta _ { \chi = \chi _ 0 } x + O ( \sqrt { x } \left ( \log 2 q x ) ^ 2 \right ) , \end{align*}


\begin{align*} \mathcal { L } ^ \sigma _ m ( f ) ( x ) = ( 1 + \| x \| ^ 2 ) \Delta ( f ) ( x ) + 2 \left ( 1 - m - \frac { d } { 2 } \right ) \langle x ; \nabla ( f ) ( x ) \rangle , \end{align*}


\begin{align*} s \star a = ( \overline { \mathcal { R } } _ 1 \rhd a ) \star ( \overline { \mathcal { R } } _ 2 \rhd s ) \end{align*}


\begin{align*} G ( \mathbb { A } ) = \bigcup _ { j } G ( \Q ) t _ j K _ 1 ( \mathfrak { n } ) G ( \R ) . \end{align*}


\begin{align*} \sum _ { \ell = 1 } ^ k d _ \ell \leq k ( k - 1 ) + \sum _ { \ell = k + 1 } ^ n \min ( k , d _ \ell ) . \end{align*}


\begin{align*} f '' ( \xi ) = a ( \xi ) f ' ( \xi ) + + b ( \xi ) f ( \xi ) \end{align*}


\begin{align*} \sum _ { n = p _ { i } + 1 } ^ { p _ { i + 1 } } | \Lambda _ { ( n , \infty ) } ( M , \Gamma ) | \geq N | \Lambda _ { ( p _ { i + 1 } , \infty ) } ( M , \Gamma ) | \end{align*}


\begin{align*} \left \langle E ^ { i j } _ 0 , { t _ { c } } _ { * } f \otimes { t _ { c } } _ { * } g \right \rangle = \left ( \mathfrak { P } _ { \ell } t _ c \left \{ \Psi ^ i ( f ) , \Psi ^ j ( g ) \right \} _ { \ell } ^ { \Sigma _ 0 } \right ) [ 0 ] . \end{align*}


\begin{align*} \sum _ { \substack { 1 \leq w \leq h \\ 1 \leq z \leq s } } ( - 1 ) ^ { w + z } \sum _ { \substack { y _ 1 + . . . + y _ w = h \\ x _ 1 + . . . + x _ z = s } } \sum _ { \substack { 0 \leq p _ w \leq . . . \leq p _ 1 \leq i - h d \\ i - h d + 1 \leq v _ 1 \leq . . . \leq v _ z \leq r + 1 - j d } } \prod _ { a = 1 } ^ w i _ a \alpha ^ { i _ a } _ { r - ( h - y _ 1 - . . . - y _ a ) d - p _ a } \prod _ { b = 1 } ^ z x _ b \alpha ^ { x _ b } _ { r - ( j - x _ b - . . . - x _ z ) d - v _ b } . \end{align*}


\begin{align*} \begin{gathered} A ^ * y + c = p , \\ A x - b \in K , y \in K ^ * , ( A x - b ) \cdot y = 0 , \end{gathered} \end{align*}


\begin{align*} \sum _ { c \in D ( \Gamma _ { x y } ) } \prod _ { \iota \in 2 ^ { V ' } } h _ { x y } ^ \iota ( c ) = \sum _ { c \in D ( \Gamma _ { x y } ) } \prod _ { \iota \in 2 ^ { V ' } } \sum _ { b \in \Lambda _ { x y } ( V ' , \iota ) ( c _ \iota ) } g ^ \iota ( b ) \end{align*}


\begin{align*} \textstyle S _ 0 ( \bigwedge ^ 2 V ) = U _ 1 \oplus U _ 2 \oplus U _ 3 , \end{align*}


\begin{align*} f ( s , r ; x ) z : = [ \Gamma ( \gamma ) ] ^ { - 1 } ( z , ( s - r ) ^ { \gamma - 1 } [ S ( s - r ) ] ^ * x ) _ H , z \in H . \end{align*}


\begin{align*} \varphi ( a ^ 3 \cdot v _ 0 ) = [ \varphi ( a ) , a ^ 2 \cdot v _ 0 ] + [ a , \varphi ( a ^ 2 \cdot v _ 0 ) ] = 3 \lambda ( a \cdot v _ 0 ) a ^ 2 \cdot v _ 0 , \end{align*}


\begin{align*} \lim \limits _ { n \to \infty } \frac { 1 } { n } \log \prod \limits _ { i = 1 } ^ { n } C _ { w _ i } = \sum \limits _ { k \in T } p _ k \log C _ k . \end{align*}


\begin{align*} \mathcal A ( \rho ^ { * , \infty } , S ^ { * , \infty } ) = \sup _ { S \in H ^ 1 } \mathcal L ( \rho ^ { * , \infty } , m ^ { * , \infty } , S ) \ge \inf _ { ( \rho , m ) } \sup _ { S \in H ^ 1 } \mathcal L ( \rho , m , S ) , \end{align*}


\begin{align*} \sum _ { s , t = 1 } ^ { Q } \mu ( L _ { B , s } \cap L _ { B , t } ) \leq \frac { C } { \mu ( B ) } \left ( \sum _ { n = 1 } ^ { Q } \mu ( L _ { B , n } ) \right ) ^ 2 . \end{align*}


\begin{align*} L ( 1 , \chi ) = \frac { \pi } { 2 f } \sum _ { a = 1 } ^ { f - 1 } \chi ( a ) \cot \left ( \frac { \pi a } { f } \right ) \ \ \ \ \ ( \chi \in X _ f ^ - ) . \end{align*}


\begin{align*} \int _ U \varphi ( x ) \bar { \nu } ( x ) d x & \geq \bigg ( \frac { R + T } { R } \bigg ) ^ n \cdot \int _ { Q _ 1 } \varphi _ { + } ( x ) \bar { \nu } ( x ) \ d x - C ( n ) \cdot ( T / R ) \cdot ( \sup _ U \varphi ) \cdot | | \bar { \nu } | | _ { L ^ \infty ( Q _ 1 ) } \cdot | Q _ 1 | \\ & - \omega _ \varphi ( \delta ) \cdot | | \bar { \nu } | | _ { L ^ \infty ( Q _ 1 ) } \cdot | Q _ 1 | . \end{align*}


\begin{align*} \partial _ t B _ q + a B J _ { x , q } + b J B _ { x , q } + \mu \Lambda ^ \alpha B _ q = a [ B , \Delta _ q ] J _ x - b [ J , \Delta _ q ] B _ x . \end{align*}


\begin{align*} \sum _ l ( \partial ^ 2 _ { t _ i t _ j } \varphi ^ l ) ( \partial _ { t _ s } \varphi ^ l ) = 0 , \forall \ i , j , s = 1 , \dots , n - m . \end{align*}


\begin{align*} \| V _ l - Z \| _ F ^ 2 = { \rm T r } [ ( V _ l - Z ) ^ { T } ( V _ l - Z ) ] = 2 l - 2 { \rm T r } ( Z ^ { T } V _ l ) \leq 2 l - 2 \left [ l - \frac { 2 0 l \epsilon \| C \| _ F ^ 2 } { \eta } \right ] = \frac { 4 0 l \epsilon \| C \| _ F ^ 2 } { \eta } . \end{align*}


\begin{align*} \tilde { \Lambda } = \begin{bmatrix} \Lambda & \Theta ( v ) \\ \Theta ^ T ( v ) & \Lambda \end{bmatrix} \end{align*}


\begin{align*} \phi ( p , - \lambda ) = \phi ( p , \lambda ) , \ , \ , \tau ( p , - \lambda ) = - \tau ( p , \lambda ) \end{align*}


\begin{align*} \mathfrak { p } = ( \mathfrak { p } \cap \mathfrak { m } ) \oplus \mathfrak { a } \oplus ( \mathfrak { k } / ( \mathfrak { k } \cap \mathfrak { m } ) ) , \end{align*}


\begin{align*} X ( t _ j ) = E ( t _ j - s ) X ( s ) - \int _ { s } ^ { t _ j } E ( t _ j - \sigma ) A F ( X ( \sigma ) ) \ , \dd \sigma + \int _ { s } ^ { t _ j } E ( t _ j - \sigma ) \ , \dd W ( \sigma ) , \end{align*}


\begin{align*} \mathrm { S p i n } ( 2 p ) \cdot \mathrm { S p i n } ( 2 q + 1 ) = & \big \{ \psi \in \mathrm { S p i n } ( 2 p + 2 q + 1 ) \ , ; \ , \psi = \varphi \ , \phi \ , , \\ & \qquad \varphi \in \mathrm { S p i n } ( 2 p ) \ , , \phi \in \mathrm { S p i n } ( 2 q + 1 ) \big \} \ , , \end{align*}


\begin{align*} \log x ( 1 + 2 \log \log t ) & = \frac { \log ^ { 5 / 3 } t ( \log \log x ) ^ { 1 / 3 } ( 1 + 2 \log \log t ) } { B _ 0 ^ { 5 / 3 } } \\ & = 3 c \log ^ { 5 / 3 } t ( \log \log t ) ^ { 4 / 3 } \cdot \left ( 1 + \frac { 1 } { 2 \log \log t } \right ) \cdot \left ( \frac { \frac { 3 } { 5 } \log \log x } { \log \log t } \right ) ^ { 1 / 3 } \\ & \geq 3 c \log ^ { 5 / 3 } t ( \log \log t ) ^ { 4 / 3 } \cdot \left ( \frac { \frac { 3 } { 5 } \log \log x } { \log \log t } \right ) ^ { 1 / 3 } , \end{align*}


\begin{align*} s ^ 2 + r ^ 2 - \frac { s } { h ( s ) } = 0 , \end{align*}


\begin{align*} \frac { d ^ { n } } { d \hbar ^ n } \omega _ { H , \psi } ( \Psi ^ i ( z _ 1 ) \star \Psi ^ j ( z _ 2 ) ) | _ { \hbar = 0 } = \int _ { \Sigma ^ { 2 n } } \big [ W _ { \Sigma } ( y _ 1 , y _ 2 ) \cdots W _ { \Sigma } ( y _ { 2 n - 1 } , y _ { 2 n } ) \\ K ^ i _ { n , \psi } ( z , y _ 1 , \ldots , y _ { 2 n - 1 } ) K ^ j _ { n , \psi } ( z , y _ 2 , \ldots , y _ { 2 n } ) \big ] \ , \mathrm { d } y _ 1 \cdots \mathrm { d } y _ { 2 n } . \end{align*}


\begin{align*} \tilde \sigma ( B ) = \textstyle { \frac 1 2 } ( \sigma ( B ) + \sigma ( - B ) ) , \end{align*}


\begin{align*} \begin{cases} - \Delta P _ { \Omega _ { \varepsilon } } w + \tilde { \lambda } _ 0 \underline { m } P _ { \Omega _ { \varepsilon } } w = f _ 0 ( w ) & \Omega _ { \varepsilon } \ ; , \\ P _ { \Omega _ { \varepsilon } } w = 0 & \partial \Omega _ { \varepsilon } \ ; , \end{cases} \end{align*}


\begin{align*} | H _ k f ( x ) | & = \bigg | \int _ { \Bbb R ^ N } H _ k ( x , y ) [ f ( y ) - f ( x ) ] d \omega ( y ) \bigg | \\ & \leqslant C \int _ { | x - y | \leqslant r ^ { 6 - ( k - M ) } } r ^ { - M } ( V _ { k - M - 1 } ( x ) ) ^ { - 1 } \| x - y \| ^ s \| f \| _ s d \omega ( y ) \\ & \leqslant C r ^ { - M } r ^ { - ( k - M ) s } \| f \| _ s . \end{align*}


\begin{align*} s h ( s ) = \frac { s ^ 2 } { s ^ 2 + r ^ 2 } . \end{align*}


\begin{align*} & \int _ 0 ^ { \frac { t } { 2 } } t ^ { \frac { \beta } { \alpha } } ( t - \tau ) ^ { - 1 + \frac { 2 \gamma - \beta } { \alpha } } \tau ^ { - \frac { 2 \gamma } { \alpha } } \ , d \tau \\ = & \int _ 0 ^ { \frac { 1 } { 2 } } ( 1 - \tau ' ) ^ { - 1 + \frac { 2 \gamma - \beta } { \alpha } } ( \tau ' ) ^ { - \frac { 2 \gamma } { \alpha } } \ , d \tau ' \\ \lesssim & \int _ 0 ^ { \frac { 1 } { 2 } } ( \tau ' ) ^ { - \frac { 2 \gamma } { \alpha } } \ , d \tau ' \lesssim 1 \end{align*}


\begin{align*} \eta _ t \norm { \nabla f ( \bar x _ t ) } ^ 2 & \leq f ( x _ t ) - f ( x _ { t + 1 } ) + { L \eta _ t ^ 2 } \norm { \nabla f ( \bar x _ t ) } ^ 2 + \frac { L } { 2 } ( 1 - \alpha _ t ) ^ 2 \Gamma _ { t - 1 } \sum _ { k = 1 } ^ { t - 1 } \frac { \alpha _ k } { \Gamma _ k } \frac { ( \eta _ k - \gamma _ k ) ^ 2 } { \alpha _ k ^ 2 } \norm { \nabla f ( \bar x _ k ) } ^ 2 . \end{align*}


\begin{align*} H : C = \Pi _ d , \end{align*}


\begin{align*} \mathbf { y } _ k [ t ] = \sum _ { m = 1 } ^ M \mathbf { g } _ { m k } ^ N [ t ] \otimes \mathbf { x } _ m [ t ] + \mathbf { z } _ k [ t ] , \end{align*}


\begin{align*} ( s - t ) ^ 2 - \frac { s - t } { h ( s ) } + | \lambda | ^ 2 = 0 , \end{align*}


\begin{align*} E [ ( S _ n ) ^ 2 ] \sim \begin{cases} \dfrac { n } { 1 - 2 \alpha } & ( - 1 < \alpha < 1 / 2 ) , \\ [ 3 m m ] n \log n & ( \alpha = 1 / 2 ) , \\ [ 1 m m ] \dfrac { n ^ { 2 \alpha } } { ( 2 \alpha - 1 ) \Gamma ( 2 \alpha ) } & ( 1 / 2 < \alpha < 1 ) . \end{cases} \end{align*}


\begin{align*} \eta ( V ) = \eta ( B ) \times \frac { \mu ( B ^ { [ V ] } ) } { \sum _ { V ^ { \prime } \in \mathcal { F } ^ { B } } \mu ( B ^ { [ V ^ { \prime } ] } ) } . \end{align*}


\begin{align*} S E ( G ) = n - 2 k + \sum \limits ^ { 2 k } _ { i = 1 } \ , | \lambda _ i | , \end{align*}


\begin{align*} g z = ( a z + b ) ( c z + d ) ^ { - 1 } , \lambda ( g , z ) = c z + d , g = \left [ \begin{array} { c c } a & b \\ c & d \end{array} \right ] . \end{align*}


\begin{align*} \begin{aligned} z _ { i 0 } = 1 z _ { i j } = \frac { s _ { i j } - s _ { i j - 1 } } { a _ { i j } - a _ { i j - 1 } } j = 1 , \ldots , n _ i . \end{aligned} \end{align*}


\begin{align*} J _ 1 = & - \sum _ { \substack { i , j = 1 , \\ i \not = j } } ^ n \Bigg [ \partial _ y \bigg ( \frac { p \sigma _ j ( \tau _ i Q ) ^ { p - 1 } } { x _ { i j } ^ 2 } - \frac { 2 p \kappa _ 0 \sigma _ j \partial _ y [ \tau _ i ( y Q ^ { p - 1 } ) ] } { x _ { i j } ^ 3 } \bigg ) + \frac { a _ { i j } \sigma _ i \tau _ i ( \Lambda Q ) } { x _ { i j } ^ 3 } \Bigg ] . \end{align*}


\begin{align*} & \omega ( x , y , z ) + \omega ( y , z , x ) + \omega ( z , x , y ) \\ = & \varphi ( [ x , y ] , z ) - \rho ( z ) \varphi ( x , y ) + \varphi ( [ y , z ] , x ) - \rho ( x ) \varphi ( y , z ) + \varphi ( [ z , x ] , y ) - \rho ( y ) \varphi ( z , x ) \\ = & \partial _ { \rho } ( \varphi ) ( x , y , z ) = 0 . \end{align*}


\begin{align*} J _ { \text r } ^ - ( \kappa _ x , \kappa _ y ) = J _ + ( \kappa _ x , \kappa _ y ) R _ - ( \kappa _ x , \kappa _ y ) . \end{align*}


\begin{align*} 2 ^ { - 2 \beta / ( \beta - 1 ) } H _ R ^ { - 2 \beta } = H _ { R / 2 } ^ { - 2 \beta } \le \omega ( x , y ) \le H _ R ^ { - 2 \beta } . \end{align*}


\begin{align*} f ( x ) = \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } \omega ( Q ) \psi _ { j } ( x , x _ { Q } ) q _ { j } h ( x _ { Q } ) , \end{align*}


\begin{align*} \mathcal W _ { [ a , a + T ] } = \big ( \{ 1 \} \cup \{ e ^ { \lambda ^ h _ k t } \} _ { | k | \geq k _ 0 } \cup \{ e ^ { \widehat \lambda _ { n _ l } t } \} _ { l = 1 } ^ { l _ 0 } \big ) \ L ^ 2 ( a , a + T ) , \ a \in \mathbb R , \end{align*}


\begin{align*} { E _ { m } ^ { [ \alpha , \beta ] , \varepsilon } = \bigcup _ { n \geq 1 } E _ { m } ^ { [ \alpha , \beta ] , \frac { 1 } { n } , \varepsilon } } . \end{align*}


\begin{align*} \Psi ^ { A i } ( \zeta ) \sim \zeta ^ { - \frac { \sigma _ 3 } { 4 } } \Psi ^ { A i } _ { 0 } \left ( I + \sum _ { j = 1 } ^ { \infty } \frac { \left ( \Psi ^ { A i } _ { 0 } \right ) ^ { - 1 } \Psi ^ { A i } _ { j } \left ( \frac { 2 } { 3 } \right ) ^ { - j } } { \zeta ^ { 3 j / 2 } } \right ) , \zeta \rightarrow \infty . \end{align*}


\begin{align*} Y _ m ^ j : = \sup _ { t \in [ 0 , T ] } | \dot { B } _ m ^ j ( t ) | , j = 1 , \ldots , r . \end{align*}


\begin{align*} \phi = \alpha + \frac { r } { 1 - n } ( 1 - \delta _ e ) \end{align*}


\begin{align*} \widetilde { P } _ { n + 1 } ^ { \alpha , \beta } ( x ) = ( A _ 1 x + A _ 0 ) \widetilde { P } _ n ^ { \alpha , \beta } ( x ) + B \widetilde { P } _ { n - 1 } ^ { \alpha , \beta } ( x ) . \end{align*}


\begin{align*} v _ x ( \tau , x ) ~ = ~ u _ x \bigl ( t , x + y _ 2 ( t ) \bigr ) , v _ { \tau } ( \tau , x ) ~ = ~ { u _ t \bigl ( t , x + y _ 2 ( t ) \bigr ) + \dot { y } _ 2 ( t ) \cdot u _ x \bigl ( t , x + y _ 2 ( t ) \bigr ) \over \dot { y } _ 1 ( t ) - \dot { y } _ 2 ( t ) } . \end{align*}


\begin{align*} \pi ( p ) = \sum _ { i = 1 } ^ d c _ i \pi ( p _ i ) h ( p ) < \sum _ { i = 1 } ^ d c _ i h ( p _ i ) \end{align*}


\begin{align*} \lim _ { t \to s + } \| u ( t , \cdot ; s , u _ 0 ) - u _ 0 ( \cdot ) \| _ { C ^ 0 ( \bar \Omega ) } = 0 , \end{align*}


\begin{align*} \langle A , B \rangle = \langle A \rangle \langle B \rangle + \langle A '' \rangle \langle B ' \rangle . \end{align*}


\begin{align*} \begin{cases} - \nu \Delta \mathbf { u } + \operatorname { d i v } ( \mathbf { u } \otimes \mathbf { u } ) + \nabla \pi = 0 , \\ - \Delta \mathbf { b } + \operatorname { d i v } ( \mathbf { u } \otimes \mathbf { b } ) - \operatorname { d i v } ( \mathbf { b } \otimes \mathbf { u } ) = 0 , \\ \operatorname { d i v } \mathbf { u } = 0 = \operatorname { d i v } \mathbf { b } , \end{cases} \end{align*}


\begin{align*} s _ i ( z ) = \sum _ { 1 \leq k _ 1 \leq k _ 2 \dots \leq k _ i \leq n } z _ { k _ 1 } \dots z _ { k _ i } \ ; , \ ; i = 1 , \dots , n - 1 p ( z ) = \prod _ { i = 1 } ^ { n } z _ i \ , . \end{align*}


\begin{align*} U ^ { - 1 } = \sum _ { k = 0 } ^ \infty ( I - U ) ^ k . \end{align*}


\begin{align*} g ( k + 1 ) \ = \ \begin{cases} \ 3 g ( k ) 6 g ( k ) + 3 \leq h ( k , 2 \cdot 3 ^ { k + 1 } - 1 ) \\ [ 8 p t ] \ 3 g ( k ) + 2 \end{cases} , \end{align*}


\begin{align*} F ( x , \sigma ) = \left [ \left ( \frac { \log x } { R } \right ) ^ { B } \exp \left ( - C ' \sqrt { \frac { \log x } { R } } \right ) \right ] ^ { - 1 } \end{align*}


\begin{align*} f ( t ) : = \exp \left ( - \sigma ( t ) \right ) \geqq A _ { 0 } ( t - \alpha ) ( \alpha \leqq { t } < \beta ) , \end{align*}


\begin{align*} d _ { \alpha } d ^ { c } _ { \alpha } F = 0 , \end{align*}


\begin{align*} x ^ \prime = a _ 2 b _ 1 x _ 1 \ , , y ^ \prime = a _ 1 b _ 1 y \ , , z ^ \prime = a _ 1 b _ 2 z \ , , w ^ \prime = a _ 2 b _ 2 w \ , . \end{align*}


\begin{align*} \begin{aligned} \rho ^ 0 = \rho ^ { \rm i n } , ( \rho ^ { n + 1 } , \phi ^ { n + 1 } ) = & \arg \min _ { ( \rho , \phi ) \in \mathcal { A } } \left \{ \frac { 1 } { 2 \tau } d ^ 2 ( \rho ^ n , \rho ) + E ( \rho , \phi ) \right \} . \end{aligned} \end{align*}


\begin{align*} x _ { i } ( t ) = \alpha _ i \sqrt { t } + \beta _ i \log t + \gamma _ i \end{align*}


\begin{align*} R _ F ( X ( s ) , X ( t _ j ) ) = \int _ 0 ^ 1 F '' \big ( X ( s ) + \lambda ( X ( t _ j ) - X ( s ) ) \big ) ( X ( t _ j ) - X ( s ) , X ( t _ j ) - X ( s ) ) ( 1 - \lambda ) \ , \dd \lambda . \end{align*}


\begin{align*} m ( f ) \leq \prod _ { i ' = 1 } ^ d f _ { i ' } ^ L + \sum _ { i = 1 } ^ d \Biggl ( \prod _ { i ' \in \{ \omega _ 1 , \ldots , \omega _ { i - 1 } \} } f ^ U _ { i ' } \Biggr ) \cdot \Biggl ( \prod _ { i ' \in \{ \omega _ { i + 1 } , \ldots , \omega _ { d } \} } f ^ L _ { i ' } \Biggr ) \cdot ( f _ { \omega _ i } - f _ { \omega _ i } ^ L ) , \end{align*}


\begin{align*} ( \Psi _ { \Sigma } ( s ) \star _ { H , \ell } \Psi _ { \Sigma } ( s ' ) ) [ \psi ] = \psi ( s ) \psi ( s ' ) - \frac { \hbar } { 4 \pi } \frac { 1 } { ( s - s ' ) ^ 2 } . \end{align*}


\begin{align*} \langle \partial ^ \alpha \ell , f \rangle = ( - 1 ) ^ \alpha \langle \ell , \partial ^ \alpha f \rangle . \end{align*}


\begin{align*} J ( x \cdot y ) \cdot y = J x \end{align*}


\begin{align*} { } ( T ^ n ) = \# \{ { \bf x } \in X \mid T ^ n ( { \bf x } ) = { \bf x } \} = { } \left ( \begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix} \right ) ^ n = L _ n \end{align*}


\begin{align*} a ^ { + } ( \hat { n } ) _ { k , \lambda } a = ( \hat { n } - 1 ) _ { k , \lambda } \hat { n } = \hat { n } ( \hat { n } - 1 ) _ { k , \lambda } , \quad ( k \in \mathbb { N } ) . \end{align*}


\begin{align*} E _ { \lambda } = p _ { 1 } d \rho . \end{align*}


\begin{align*} 0 < 1 < \det ( S ^ T S + I ) & = \det ( S ^ T ( S + J S J ^ { - 1 } ) ) \\ & = \det ( S ) \det ( E + i F ) \det ( E - i F ) \\ & = \det ( S ) \det ( E + i F ) \ , \overline { \det ( E + i F ) } \\ & = \det ( S ) | \det ( E + i F ) | ^ 2 \end{align*}


\begin{align*} \Lambda ( s , \operatorname { s y m } ^ 2 f ) = & \pi ^ { - 3 s / 2 } \Gamma \left ( \frac { s + 1 } { 2 } \right ) \Gamma \left ( \frac { s + \kappa - 1 } { 2 } \right ) \Gamma \left ( \frac { s + \kappa } { 2 } \right ) L ( s , \operatorname { s y m } ^ 2 f ) . \end{align*}


\begin{align*} I ( \alpha , \beta , Z ) + I ( \beta , \gamma , Z ' ) = I ( \alpha , \gamma , Z + Z ' ) . \end{align*}


\begin{align*} \mathcal { N } _ q ( u ' ) - \mathcal { N } _ q ( u '' ) = \mathcal { B } _ q ( u '' , u ' - u '' ) + ( 1 / 2 ) \ , \mathcal { B } _ q ( u ' - u '' , u ' - u '' ) . \end{align*}


\begin{align*} \lim _ { z \downarrow 0 } z \tilde f ( - i z ) & = \lim _ { z \downarrow 0 } \int _ 0 ^ \infty f ( t ) z e ^ { - z t } \dd t = \lim _ { z \downarrow 0 } \int _ 0 ^ \infty f ' ( t ) e ^ { - z t } \dd t - \left [ f e ^ { - z t } \right ] _ 0 ^ \infty \\ & = \lim _ { z \downarrow 0 } \int _ 0 ^ \infty f ' ( t ) e ^ { - z t } \dd t - \left [ f e ^ { - z t } \right ] _ 0 ^ \infty = \int _ 0 ^ \infty f ' ( t ) \dd t - f ( 0 ) = \lim _ { t \to \infty } f ( t ) . \end{align*}


\begin{align*} \beta _ { k + 1 } : = \beta _ k + X ^ k \nu , & & \gamma _ { k + 1 } : = \gamma _ k + X ^ k \mu \end{align*}


\begin{align*} | \lambda _ { k n } | = ( \theta _ 0 \theta _ 2 \cdots \theta _ { n - 1 } ) ^ k , \end{align*}


\begin{align*} V _ i : = \bigcup _ { ( \varepsilon _ 1 , \ldots , \varepsilon _ n ) \in \{ 1 , 2 \} ^ n , \varepsilon _ i = 2 } A _ 1 ^ { \varepsilon _ 1 } \cap \cdots \cap A _ n ^ { \varepsilon _ n } . \end{align*}


\begin{align*} U ( z _ 1 ) ^ { \ast } H U ( z _ 2 ) = \lambda ( \alpha , z _ 1 ) ^ { \ast } U ( \alpha z _ 1 ) ^ { \ast } H U ( \alpha z _ 2 ) \lambda ( \alpha , z _ 2 ) , \end{align*}


\begin{align*} \| h _ \perp ( t ) \| _ { H _ x ^ 1 } ^ 2 \simeq B ( h _ \perp , h _ \perp ) = \Phi ( h ) - 2 \alpha _ + \alpha _ - \lesssim e ^ { - ( c _ 0 + c _ 1 ) t } + e ^ { - 2 c _ 1 ^ - t } \lesssim e ^ { - ( c _ 0 + c _ 1 ) t } \end{align*}


\begin{align*} L ( g , g \cdot \xi _ 1 , \dots , g \cdot \xi _ k ) = \frac { 1 } { 2 } \mathcal { G } ( \xi _ 1 , \xi _ 1 ) + \dots + \frac { 1 } { 2 } \mathcal { G } ( \xi _ k , \xi _ k ) = \ell ( \xi _ 1 , \dots , \xi _ k ) . \end{align*}


\begin{align*} F ( q ) : = \sum _ { n = 0 } ^ { \infty } ( q ) _ n \end{align*}


\begin{align*} I ( \delta _ { 1 3 } , \delta _ { 1 2 } ) - J ( \delta _ { 1 3 } , \delta _ { 1 2 } ) = \mu _ { \mathrm { d i s c } } ( \gamma _ { 1 } \cup { \gamma _ { 4 } } ) - \mu _ { \mathrm { d i s c } } ( \gamma _ { 1 } \cup { \gamma _ { 3 } } ) = 2 \end{align*}


\begin{align*} \begin{pmatrix} 1 & \omega _ 1 & \ldots & \omega _ d & \tau \\ 0 & 1 & \ldots & 0 & x _ 1 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \ldots & 1 & x _ d \\ 0 & 0 & \ldots & 0 & 1 \end{pmatrix} \end{align*}


\begin{align*} \check H _ { c t } ^ q ( M , A ) = H _ { s i n g } ^ q ( S ^ { n - 1 } ; A ) \end{align*}


\begin{align*} v ( x ) : = \tau d ^ { - n - 2 } \bigg ( 1 - \frac \theta 2 \bigg ) ^ { - n - 2 } \zeta ( x ) - u ( x ) x \in \R ^ n . \end{align*}


\begin{align*} \lambda _ w & = 0 , ~ w \in 1 : K - 1 , \\ \nu & = y _ { K - 1 } / N , \end{align*}


\begin{align*} \widehat \beta ( x ) = 1 - \widehat { \Phi ^ { - 1 } } \big ( x { \widehat { \Phi } } ( x ) \big ) . \end{align*}


\begin{align*} \cosh ( \eta ) - \tfrac { n } { \eta ^ 2 } > n - \tfrac { n } { 1 } = 0 . \end{align*}


\begin{align*} \int _ { a } ^ { b } K ( s , t ) f ( t ) d t = g ( s ) , \ \ c \leq s \leq d , \end{align*}


\begin{align*} \langle h , h ' \rangle = \langle \theta h , h ' \rangle + \langle h , \theta h ' \rangle \ ; . \end{align*}


\begin{align*} \forall x \in \mathbb { Z } , \overleftarrow { \eta } ^ { \ell } ( x ) - \overleftarrow { \eta } ^ { \varepsilon n } ( x ) = \frac { 1 } { m \ell } \sum _ { j = 1 } ^ { m - 1 } \sum _ { z = x - \ell } ^ { x - 1 } [ \eta ( z ) - \eta ( z - j \ell ) ] . \end{align*}


\begin{align*} \begin{cases} \partial _ z g _ { i j } ^ z = - 2 A ^ z _ { i j } ; \\ \partial _ z A _ { i j } ^ z = - A ^ z _ { i k } A ^ z _ { j l } g _ z ^ { k l } + R ^ z _ { i j } , \end{cases} \end{align*}


\begin{align*} \mathcal { U } _ k = \left \lbrace j : \frac { | e _ j ^ \intercal ( A x _ k - b ) | ^ 2 } { \norm { A ^ \intercal e _ j } _ 2 ^ 2 \norm { A x _ k - b } _ 2 ^ 2 } \geq \epsilon _ k \right \rbrace ; \end{align*}


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


\begin{align*} \mathcal { L } u = \bar { g } ( \lambda \partial _ { r } , \nabla F ) + F ^ { i j } \bar { R } _ { \nu j l i } \bar { g } ( \lambda \partial _ { r } , e _ { l } ) + \lambda ' F - u F ^ { i j } h _ { i l } h _ { j l } . \end{align*}


\begin{align*} G = \{ u \rightarrow 2 u v , \ ; \ ; v \rightarrow u \} , \end{align*}


\begin{align*} h ( t ) = t ^ \theta , 0 \le \theta < d . \end{align*}


\begin{align*} \sum _ { r , q } s _ r s _ q = & \sum _ { r , q \notin \{ i , j \} } s _ r s _ q + s _ i \sum _ { r \notin \{ i , j \} } s _ r + s _ j \sum _ { r \notin \{ i , j \} } s _ r + s _ i s _ j \\ < & \underbrace { \sum _ { r , q \notin \{ i , j \} } s _ r s _ q + ( s _ i + 1 ) \sum _ { r \notin \{ i , j \} } s _ r + ( s _ j - 1 ) \sum _ { r \notin \{ i , j \} } s _ r + ( s _ i + 1 ) ( s _ j - 1 ) } _ { = A } - 1 . \end{align*}


\begin{align*} P ( E , m ) = \sum _ { j = 0 } ^ d \alpha _ { d - j } ( E ) \frac { m ^ { d - j } } { ( d - j ) ! } , \end{align*}


\begin{align*} \begin{aligned} F ( x ) \otimes G ( x ) & \preceq A ^ 0 ( x ) \otimes B ^ 0 ( x ) + \sum _ { t : \omega _ t = 1 } \Bigl ( U ^ { p ^ t _ 1 } ( x ) - U ^ { p ^ { t - 1 } _ 1 } ( x ) \Bigr ) \otimes B ^ { p ^ t _ 2 } ( x ) \\ & + \sum _ { t : \omega _ t = 2 } A ^ { p ^ t _ 1 } ( x ) \otimes \Bigl ( V ^ { p ^ t _ 2 } ( x ) - V ^ { p ^ { t - 1 } _ 2 } ( x ) \Bigr ) x \in X . \end{aligned} \end{align*}


\begin{align*} c _ G = 4 n ^ 2 + 5 n - 9 , \end{align*}


\begin{align*} \norm { M _ \omega T _ x f } _ p = \norm { f } _ p \ . \end{align*}


\begin{align*} \sum _ { \substack { 0 \leq b < p \\ \gcd ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { e m } - v ) } { p } } = 0 \end{align*}


\begin{align*} \partial _ k \partial _ \ell \left ( f _ h \right ) ( x ) = - \int _ 0 ^ { + \infty } e ^ { - 2 t } P ^ { \Sigma } _ t ( \partial _ k \partial _ \ell ( h ) ) ( x ) d t . \end{align*}


\begin{align*} y _ R ( t ) = \int _ { \mathbb { R } ^ N } \varphi _ R ( x ) | u ( x , t ) | ^ 2 d x = \int _ { \mathbb { R } ^ N } R ^ 2 \varphi \left ( \frac { x } { R } \right ) | u ( x , t ) | ^ 2 d x , \end{align*}


\begin{align*} g _ 1 ( t ) \Omega ^ w \mathcal { L i } _ { K , \mathfrak { s } } ( { \boldsymbol \alpha } ) + g _ 2 ( t ) \cdot 1 = 0 \end{align*}


\begin{align*} K ( I ) \subseteq \{ 0 , 1 , \ldots , n - 2 \} , K ( I ) = \varnothing & , \\ \left . \begin{array} { r c l } \nu = 2 k \in K ( I ) & \Leftrightarrow & i _ { 2 k } = m - k , \\ \nu = 2 k + 1 \in K ( I ) & \Leftrightarrow & i _ { 2 k + 1 } = m - k - 1 \end{array} \right \} & . \end{align*}


\begin{align*} \mathbb { E } _ { U , \xi } \left [ \phi _ { \mu } ^ { 1 / \bar { \rho } } ( x _ { t + 1 } ) \right ] \leq & \ \mathbb { E } _ { U , \xi } \left [ \phi _ { \mu } ^ { 1 / \bar { \rho } } ( x _ { t } ) \right ] - \frac { \alpha _ t ( \bar { \rho } - \rho ) } { \bar { \rho } } \mathbb { E } _ { U , \xi } \left [ \left \| \nabla \phi _ { \mu } ^ { 1 / \bar { \rho } } ( x _ t ) \right \| _ 2 ^ 2 \right ] \\ & + 2 ( n ^ 2 + 2 n ) \bar { \rho } \alpha _ t ^ 2 L _ { f , 0 } ^ 2 , \end{align*}


\begin{align*} A ^ { 0 } U _ { t } + A ^ { i } \partial _ { i } U - B ^ { i j } \partial _ { i } \partial _ { j } U + D u = F \end{align*}


\begin{align*} ( F _ 1 * F _ 2 ) ( x _ 0 , \omega _ 0 , \tau _ 0 ) = \int _ { \R ^ { 2 d } } \int _ { \R } F _ 1 ( x , \omega , \tau ) F _ 2 ( x _ 0 - x , \omega _ 0 - \omega , \tau _ 0 - \tau + \tfrac { 1 } { 2 } ( x \cdot \omega _ 0 - x _ 0 \cdot \omega ) ) \ , d \tau \ , d ( x , \omega ) . \end{align*}


\begin{align*} \sum _ { r \in \mathbb { Z } } c ^ F _ r ( \widetilde { W _ v ^ l } , \widetilde { U _ v ^ l } ) q _ F ^ { - r / 2 } X ^ { - r } = \omega _ { \sigma ^ { ( l ) } } ( - 1 ) ^ { n - 2 } \gamma ( X , \pi ^ { ( l ) } , \sigma ^ { ( l ) } , \psi ^ l ) \sum _ { r \in \mathbb { Z } } c ^ F _ r ( W _ v ^ l , U _ v ^ l ) q _ F ^ { r / 2 } X ^ r . \end{align*}


\begin{align*} H ( P , Q ) : = \max \left \{ \sup _ { p \in P } \inf _ { q \in Q } \norm { p - q } , \sup _ { q \in Q } \inf _ { p \in P } \norm { p - q } \right \} \end{align*}


\begin{align*} 2 ^ { \mu _ * k _ * t } \cdot 2 ^ { ‐ m _ t ( s _ * + 1 / q ‐ 1 / p _ 1 ) } = 2 ^ { ‐ \alpha _ * k _ * t } \cdot 2 ^ { ‐ m _ t ( 1 / q ‐ 1 / p _ 0 ) } \stackrel { \eqref { h a t _ t h e t a _ d e f } } { = } n ^ { ‐ \hat \theta } ; \end{align*}


\begin{align*} x _ i = x _ 2 - \frac { 2 s _ 2 } { 1 + \lambda } x _ 3 - \frac { 2 s _ 3 } { 1 + \lambda } x _ 5 - \cdots - \frac { 2 s _ { \frac { i } { 2 } } } { 1 + \lambda } x _ { i - 1 } . \end{align*}


\begin{align*} A = \frac { \kappa } { 2 } \frac { d ^ 2 } { d x ^ 2 } + \frac { d } { d x } \end{align*}


\begin{align*} G _ \lambda \nu ( x ) : = \int _ M g _ \lambda ( x , y ) \nu ( d y ) , x \in M , \end{align*}


\begin{align*} I ( \alpha \cup { \gamma _ { 0 } } , \alpha ) = I ( \gamma _ { 0 } ) + 2 \# ( \mathbb { R } \times \alpha \cap u ) = 2 + 2 w ( \alpha ) . \end{align*}


\begin{align*} Z _ { r , m } : = L \sum _ { j = 1 } ^ r Y _ m ^ j . \end{align*}


\begin{align*} X ( \theta ) \log X ( \theta ) - \theta X ^ 0 \log X ^ 0 - ( 1 - \theta ) X ^ 1 \log X ^ 1 = - \theta ( 1 - \theta ) ( X ^ 1 - X ^ 0 ) ^ 2 g ( X ^ 0 , X ^ 1 ; \theta ) , \end{align*}


\begin{align*} \int _ { \beta } c _ 1 ( T _ X ) = r ( n + g - 1 ) . \end{align*}


\begin{align*} & \frac 1 2 \frac { d } { d t } \| D ^ m B \| _ { L ^ 2 } ^ 2 + \int _ { \mathbb S ^ 1 } [ D ^ m , B ] J _ x D ^ m B \ , d x + \int _ { \mathbb S ^ 1 } B D ^ m J _ x D ^ m B \ , d x \\ & - \int _ { \mathbb S ^ 1 } [ D ^ m , J ] B _ x D ^ m B \ , d x - \int _ { \mathbb S ^ 1 } J D ^ m B _ x D ^ m B \ , d x + \mu \| \Lambda ^ { \frac { \alpha } { 2 } } D ^ m B \| _ { L ^ 2 } ^ 2 = 0 . \end{align*}


\begin{align*} \omega _ { p } ( T ) & \geq \norm { \cos ( 2 \pi ) R e ( T ) - \sin ( 2 \pi ) I m ( T ) } _ { p } \\ & = \frac { 1 } { 2 } \norm { T + T ^ { \ast } } _ { p } \end{align*}


\begin{align*} \tilde { R } _ h ^ 2 ( w ) = \sum _ { i \in I _ h } \tilde { r } _ { h , i } ^ 2 ( w ) \ , , \end{align*}


\begin{align*} L _ { \eta } \psi _ { \eta } = 0 , \end{align*}


\begin{align*} \sum _ { \gamma < \alpha } b _ { \gamma } ^ { \alpha } f ^ { ( \gamma ) } ( o ) + f ^ { ( \alpha ) } ( o ) = \int _ D f \overline { { g } _ { \alpha } } e ^ { - \varphi } . \end{align*}


\begin{align*} \mathrm { L a w } ( \widehat { X } ( t ) ) = \widehat { \mu } _ t . \end{align*}


\begin{align*} u _ { q , \lambda } ( x , t ) = \inf \left \{ \left ( \lambda u ( y , t ) ^ q + ( 1 - \lambda ) u ( z , t ) ^ q \right ) ^ { { 1 \over q } } : \lambda y + ( 1 - \lambda ) z = x \right \} \end{align*}


\begin{align*} \Delta _ j f = \mathcal { F } ^ { - 1 } \Big ( \phi _ j \mathcal { F } f \Big ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ j \in \mathbb { N } _ 0 , ~ ~ f \in S ^ { \prime } ( K ) . \end{align*}


\begin{align*} E = \int _ { \Omega } \bigg ( \sum _ { i = 1 } ^ s \rho _ i \log \rho _ i + \frac { 1 } { 2 } ( f + \sum _ { i = 1 } ^ s z _ i \rho _ i ) \phi \bigg ) d x + B , \end{align*}


\begin{align*} q = \max \mathsf L ( A ) / \min \mathsf L ( A ) \ , . \end{align*}


\begin{align*} Q = \begin{pmatrix} 0 & 1 & 1 & \dots & 1 \\ 1 & 0 & 1 & \dots & 1 \\ \vdots & & \ddots & & \vdots \\ 1 & 1 & 1 & \dots & 0 \end{pmatrix} , \end{align*}


\begin{align*} f _ { R C B } = f _ { R | C B } f _ { B | C } f _ B . \end{align*}


\begin{align*} \varphi ( \delta _ { \varphi } ^ * a ) = \overline { \varphi ( a ^ * \delta _ { \varphi } ) } = \overline { \varphi ( S ( a ) ) } = \varphi ( S ^ { - 1 } ( a ^ * ) ) = \varphi ( \delta _ { \varphi } a ^ * ) . \end{align*}


\begin{align*} & \tau _ { A _ { s _ 1 } \smallsetminus \{ \alpha + s _ 1 \} \cup \{ - \beta - s _ 2 \} } ( p ^ m ) \tau _ { B _ { s _ 2 } \smallsetminus \{ \beta + s _ 2 \} \cup \{ - \alpha - s _ 1 \} } ( p ^ m ) \\ & = \tau _ { A \smallsetminus \{ \alpha \} \cup \{ - \beta - s _ 1 - s _ 2 \} } ( p ^ m ) \tau _ { B _ { s _ 1 + s _ 2 } \smallsetminus \{ \beta + s _ 1 + s _ 2 \} \cup \{ - \alpha \} } ( p ^ m ) . \end{align*}


\begin{align*} q _ 1 = \sum _ { i = 1 } ^ k \alpha _ i p _ i , q _ 2 = \beta p + \sum _ { i = k + 1 } ^ d \beta _ i p _ i . \end{align*}


\begin{align*} \int _ s ^ \tau z \dd ( y - u ) = 0 \forall \ ; 0 \le s < \tau \le T . \end{align*}


\begin{align*} U \left ( t ^ S _ { j - 1 } \right ) = t ^ G _ { j } = t ^ S _ { j - 1 } + \tau _ { \rm f } - \tau ^ { \rm R e a c } _ { V _ { j - 1 } } . \end{align*}


\begin{align*} u _ { t } = - \gamma \boldsymbol { D } _ { x } ^ { \alpha } u - u ^ { 3 } + \left ( \beta + 1 \right ) u ^ { 2 } - \beta u , \end{align*}


\begin{align*} \omega _ { p } \left ( \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix} \right ) \leq 2 ^ { \frac { 1 } { p } } \omega _ { p } ( B ) + \min ( \omega _ { p } ( A + B ) , \omega _ { p } ( A - B ) ) . \end{align*}


\begin{align*} \nabla _ { X } e _ i = A _ { i } ( X ) + \sum _ { j = 1 } ^ { r } B _ { i } ^ j ( X ) e _ j , \end{align*}


\begin{align*} \tilde { V } _ t = \{ \beta \times \gamma \in P _ { n } ^ t \times P _ { n } ^ t : \pi _ t ( \beta ) = \pi _ t ( \gamma ) \} . \end{align*}


\begin{align*} S : \mathcal { H } \to \mathcal { H } , S f = \sum _ { \gamma \in \Gamma } \langle f , e _ \gamma \rangle e _ \gamma . \end{align*}


\begin{align*} \Phi ^ { \xi , \eta } ( x , \omega ) = \langle \rho ( \xi , \eta ) \varphi , \rho ( x , \omega ) \varphi \rangle \end{align*}


\begin{align*} h _ { p , q } = \frac { 1 } { 2 } ( \delta _ { q , p - n } + \delta _ { q , p + n } ) \end{align*}


\begin{align*} \int _ { \Omega _ r } D ( u - f ) \cdot A ( x ) D ( u - f ) \ , d x & = \int _ { \Omega _ r } D u \cdot A ( x ) D u \ , d x - \int _ { \Omega _ r } D f \cdot A ( x ) D f \ , d x \\ & \le ( 1 + C _ g r ^ \alpha ) \left ( \int _ { \Omega _ r } | D u | ^ 2 \ , d x - \frac { 1 - C _ g r ^ \alpha } { 1 + C _ g r ^ \alpha } \int _ { \Omega _ r } | D f | ^ 2 \ , d x \right ) \\ & \le ( 1 + C _ g r ^ \alpha ) \left ( | \Omega _ r | + C _ g r ^ \alpha \int _ { \Omega _ r } | D f | ^ 2 \ , d x \right ) . \end{align*}


\begin{align*} 0 & = - \int _ M | \nabla \dot u | ^ 2 \ , d V _ g + \int _ M V \dot u \ , r \ , d V _ g + \int _ M V \dot u ^ 2 \ , d V _ g . \end{align*}


\begin{align*} \left ( - \mathcal { A } \right ) ^ { - \frac { r } { 2 } } f = \frac { 1 } { \Gamma ( \frac { r } { 2 } ) } \int _ 0 ^ { + \infty } t ^ { \frac { r } { 2 } - 1 } P _ t ( f ) d t . \end{align*}


\begin{align*} \varepsilon ( a b ) = \varepsilon ( a ) \varepsilon ( b ) \textrm { w h e n e v e r } a \in A _ t ^ s , b \in { } ^ s _ t A , \end{align*}


\begin{align*} P _ { n , k } ( W _ { \mathcal { S } ( n , k ; M ) } | A ^ { ( n ) } _ { M , n , l } ) = \begin{cases} 1 , \ \ l \in [ k - 1 ] ; \\ 0 , \ \ l = k . \end{cases} \end{align*}


\begin{align*} \begin{cases} \displaystyle \limsup _ { t - s \to \infty } \int _ \Omega u ( t , x ; s , u _ 0 ) \le M _ 0 ^ * \cr \displaystyle \limsup _ { t - s \to \infty } \int _ \Omega u ^ { - p } ( t , x ; s , u _ 0 ) d x \le M _ 1 ^ * \cr \displaystyle \limsup _ { t - s \to \infty } \int _ \Omega u ^ { q } ( t , x ; s , u _ 0 ) d x \le M _ 2 ^ * . \end{cases} \end{align*}


\begin{align*} \begin{aligned} & G _ \beta ( r , p , X , A ) \\ & = { 1 \over 1 - \beta } r ^ \beta F \left ( r ^ { 1 - \beta } , ( 1 - \beta ) r ^ { - \beta } p , ( 1 - \beta ) r ^ { - \beta } X + ( \beta ^ 2 - \beta ) r ^ { - \beta - 1 } p \otimes p , A \right ) \end{aligned} \end{align*}


\begin{align*} S ( \Theta ) : = \begin{bmatrix} H + \rho I + ( \delta I + ( \Theta ^ { - 1 } + \rho I ) ^ { - 1 } ) ^ { - 1 } & A ^ T \\ A & - \delta I \end{bmatrix} . \end{align*}


\begin{align*} | p - q | \leq 1 , p + q = 1 + s \geq 2 , 2 p + r = 2 + s = p + q + 1 , r = q - p + 1 , 0 \leq r \leq 2 . \end{align*}


\begin{align*} \lambda _ 0 T _ 0 ( u _ 0 ) = u _ 0 \end{align*}


\begin{align*} ( x _ t - y _ t ) ^ 2 & = 2 \int _ 0 ^ t \big ( b ( s , x _ s ) - b ( s , y _ s ) \big ) ( x _ s - y _ s ) d s \\ & \le 2 \int _ 0 ^ t K _ s ( x _ s - y _ s ) ^ 2 d s . \end{align*}


\begin{align*} \int _ M \langle \nabla v _ m , \nabla v _ m \rangle \ , d V _ g + \int _ M V \ , v _ m \ , d V _ g = 0 , \end{align*}


\begin{align*} \partial _ { \xi _ j } \left ( \varphi _ \mu \right ) ( \xi ) = - \langle \Sigma ( e _ j ) ; \xi \rangle \varphi _ \mu ( \xi ) , \end{align*}


\begin{align*} t _ i = i \Delta t , i = 0 , 1 , . . . N . \end{align*}


\begin{align*} ( - \Delta _ p ) ^ \theta v = f Z , \end{align*}


\begin{align*} \tau _ 1 : = \frac { \beta } { } \bigg ( \frac { 1 } { k } \mathbb { T } ( x , t ) - L \eta ( x , t ) \mathbb { I } \bigg ) , \end{align*}


\begin{align*} \mathcal D _ F = \{ y \colon y ^ { [ k ] } \in A C _ { l o c } ( \mathbb R _ + ) , \ , k = \overline { 0 , n - 1 } \} . \end{align*}


\begin{align*} \eta _ { \pm } ( s , t ) = e ^ { \kappa _ { \mp } s } \eta _ { \kappa _ { \mp } } ( t ) + O ( e ^ { ( \kappa _ { \mp } + \epsilon _ { \mp } ) s } ) \end{align*}


\begin{align*} \mathfrak { G } ( \mathbf { L } ) : = \left \{ \mathcal { I } \in \wp _ { k } [ n ] : \ , \Delta _ { \mathbf { L } } ( \mathcal { I } ) \neq 0 \right \} \end{align*}


\begin{align*} T ( z ) = { \rm t r } ^ \flat ( e ^ { - i t _ 0 h ^ { - 1 } \widetilde { P } _ h ( z ) } \widetilde { R } _ h ( z ) ) \end{align*}


\begin{align*} \phantom { , \quad w _ 1 , w _ 2 \in W } L ( w _ 1 \wedge w _ 2 ) = l ( w _ 1 ) \wedge l ( w _ 2 ) , \quad w _ 1 , w _ 2 \in W , \end{align*}


\begin{align*} \int _ \R g ( x + z ) K '' ( z ) \ , d z = \int _ \R g ( x - y ) \tilde K '' ( y ) \ , d z = ( \tilde K '' * g ) ( x ) = ( \tilde K * g ) '' ( x ) . \end{align*}


\begin{align*} \omega _ 2 ( i t ) = i \frac { | \lambda | ^ 2 } { s ( | \lambda | , t ) - t } = \frac { | \lambda | ^ 2 } { i t - i \omega _ 1 ( i t ) } . \end{align*}


\begin{align*} S _ d ( f ) ( x ) = \left \{ \sum \limits _ { k = - \infty } ^ { \infty } \sum \limits _ { Q } | \psi _ { Q } \ast f ( x _ Q ) | ^ 2 \chi _ Q ( x ) \right \} ^ { 1 / 2 } . \end{align*}


\begin{align*} J [ \lambda ] ( x , t , \tau ) : = \int _ { \R } G ( x - y , t - \tau ) \eta ( y , \tau ) ^ { - 1 } \lambda ( y , \tau ) d y = \left ( G ( t - \tau ) * ( \eta ^ { - 1 } \lambda \right ) ( \tau ) ) ( x ) . \end{align*}


\begin{align*} s \left ( a , \frac { a ^ d - 1 } { a - 1 } \right ) = \frac { ( f - 1 ) ( f - a ^ 2 - 1 ) } { 1 2 a f } = O \left ( f ^ { 1 - \frac { 1 } { d - 1 } } \right ) \hbox { f o r $ d \geq 3 $ o d d a n d $ a \neq - 1 , 0 , 1 $ , } \end{align*}


\begin{align*} \widetilde \Gamma ^ \lambda _ { \alpha \beta \gamma } \stackrel { ( \alpha \beta \gamma ) } { = } \partial _ \gamma \Gamma ^ \lambda _ { \alpha \beta } + \Gamma ^ \lambda _ { ( \gamma \sigma ) } \Gamma ^ \sigma _ { \alpha \beta } \end{align*}


\begin{align*} 2 Q _ { \tau } ( Z , Z _ { \gamma } ) + 2 ( \lfloor M \theta \rfloor - \lfloor ( M - p _ { i } ) \theta \rfloor ) = 0 \end{align*}


\begin{align*} \int _ s ^ \tau z \dd ( y - u ) = \int _ s ^ \tau \min \{ 0 , z \} \dd ( y - u ) = \int _ s ^ \tau \max \{ 0 , - z \} \dd ( u - y ) \geq 0 . \end{align*}


\begin{align*} \begin{aligned} \frac { 1 } { \tau } d _ W ( \rho _ i ^ 0 , \ \rho _ i ^ 1 ) ^ 2 = & \inf \left \{ \int _ 0 ^ { \tau } \int _ { \Omega } \rho _ i D ^ { - 1 } _ i | u _ i | ^ 2 d x d t \right \} , \\ & \partial _ t \rho _ i + \nabla \cdot ( \rho _ i u _ i ) = 0 , \\ & ( \rho _ i u _ i ) \cdot \mathbf { n } = 0 , x \in \partial \Omega , \\ & \rho _ i ( x , 0 ) = \rho ^ 0 _ i , \ \rho _ i ( x , \tau ) = \rho ^ 1 _ i . \end{aligned} \end{align*}


\begin{align*} M _ \mu ( \alpha , \varepsilon , \delta , T , f ) & = \lim _ { n \to \infty } M _ \mu ( n , \alpha , \varepsilon , \delta , T , f ) , \\ \underline { M } _ \mu ( \alpha , \varepsilon , \delta , T , f ) & = \liminf _ { n \to \infty } R _ \mu ( n , \alpha , \varepsilon , \delta , T , f ) , \\ \overline { M } _ \mu ( \alpha , \varepsilon , \delta , T , f ) & = \limsup _ { n \to \infty } R _ \mu ( n , \alpha , \varepsilon , \delta , T , f ) . \end{align*}


\begin{align*} \Phi ( k , n ) = \phi \left ( \left \lceil 2 C ( k + 1 ) \right \rceil - 1 , \max \{ \theta ( M \varpi ( k ) + M - 1 ) , n \} \right ) \end{align*}


\begin{align*} f = \iint _ { \R ^ { 2 d } } F ( x , \omega ) ( M _ \omega T _ x g ) \ , d ( x , \omega ) . \end{align*}


\begin{align*} \epsilon '' + ( d - 4 ) \epsilon ' - ( d - 2 ) \epsilon ( \epsilon + 1 ) ( \epsilon - 1 ) = 0 . \end{align*}


\begin{align*} Y ( \mathcal { B } ) _ { \alpha \beta } ^ { i j } = - Y ( \mathcal { B } ) _ { \alpha v } ^ { i \omega } Y ( \mathcal { B } ) _ { v \beta } ^ { i \omega } Y ( \mathcal { B } ) _ { \alpha v } ^ { \omega j } Y ( \mathcal { B } ) _ { v \beta } ^ { \omega j } \end{align*}


\begin{align*} R _ 3 ^ * = \left \{ u \in V ( F ) \colon \phi _ { x y } ( u ) \in T \right \} \mbox { a n d } R _ 4 ^ * = \left ( \cup _ { i = 1 } ^ s \left \{ c _ 2 ^ i , \ldots , c _ { 2 k - 2 } ^ i \right \} \right ) \setminus R _ 3 ^ * . \end{align*}


\begin{align*} \nabla ( | | G ( x ) | | ^ 2 ) = 2 ( g _ 1 ( x ) \nabla g _ 1 ( x ) + \dots g _ m ( x ) \nabla g _ m ( x ) ) . \end{align*}


\begin{align*} \log ( I ( \omega ) ) = - \frac { \pi \omega ^ 2 } { M } + C , \end{align*}


\begin{align*} \nabla S _ t ( \varphi ) w = | \alpha _ w ( t ) | \ge \exp \left ( \int _ 0 ^ t \dot { \lambda } _ { 2 s } d s \right ) = e ^ { \frac { \lambda _ { 2 t } } { 2 } } , \end{align*}


\begin{align*} \Delta _ R : = \{ ( x , y ) \in X \times X | d ( x , y ) \le R \} \end{align*}


\begin{align*} \tilde { Y } _ { k + 1 } = ( 1 - \eta ) ^ { k + 1 } x + \frac { \eta ^ { 1 / \alpha } } { \sigma } \sum _ { i = 0 } ^ { k } ( 1 - \eta ) ^ { i } \widetilde { Z } _ { k + 1 - i } . \end{align*}


\begin{align*} 3 r _ 1 + 2 i l \pi + 2 i m \pi = - \lambda , \ \ \ r _ 1 = \frac { 1 } { 3 } ( - \lambda - 2 i l \pi - 2 i m \pi ) , \end{align*}


\begin{align*} \langle W \rangle = w _ 1 \langle W ' \rangle + \langle ( W ' ) ' \rangle \mbox { f o r } | W | \geq 2 . \end{align*}


\begin{align*} \mathcal { T } ^ \mathcal { F } _ \star ( X , Y ) = \langle X \otimes _ { \mathcal { X } _ \star } Y , \mathrm { T } ^ \mathcal { F } \rangle _ \star , ~ \mathrm { R } ^ \mathcal { F } _ \star ( X , Y , Z ) = \langle X \otimes _ { \mathcal { X } _ \star } Y \otimes _ { \mathcal { X } _ \star } Z , \mathrm { R } ^ \mathcal { F } \rangle _ \star \end{align*}


\begin{align*} | M ^ { n } | = \int _ { M ^ { n } } H ^ { 1 - \alpha } . \end{align*}


\begin{align*} ( a b ) ^ { - 1 } \sum _ { k , l \in \Z } V _ g g \left ( \frac { k } { a } , \frac { l } { b } \right ) e ^ { 2 \pi i ( k x + l \omega ) } = \sum _ { k = 0 } ^ { n - 1 } \left | Z _ b g \left ( \frac { x + k } { n } , \omega \right ) \right | ^ 2 . \end{align*}


\begin{align*} e ( \vect { r } ) = \begin{cases} e _ { \text i } ( \vect { r } ) + e _ { \text r } ( \vect { r } ) & r _ z < d _ 1 \\ e _ { \text t } ( \vect { r } ) & r _ z > d _ 1 . \end{cases} \end{align*}


\begin{align*} \int _ X g _ { u , \rho } ^ 2 \ , d \mu _ \omega = \int _ X g _ u ^ 2 \ , d \mu _ X . \end{align*}


\begin{align*} N _ { \ell , n } = X _ 1 ^ \ell + \cdots + X _ n ^ \ell . \end{align*}


\begin{align*} x = \left [ \begin{array} { c c c } a _ 1 & \frac { b _ 1 } { 2 } + b _ 2 \zeta ^ { - 1 } & a _ 2 \end{array} \right ] , y = \left [ \begin{array} { c c c } - c _ 1 & - \frac { b _ 1 } { 2 } + b _ 2 \zeta ^ { - 1 } & c _ 2 \end{array} \right ] , x \phi _ 1 x ^ { \ast } = y \phi _ 2 y ^ { \ast } . \end{align*}


\begin{align*} \frac { m } { ( \log n ) ^ m } \displaystyle \sum ^ n _ { j = 1 } \frac { ( \log j ) ^ { m - 1 } } { j } & = 1 + O ( ( \log n ) ^ { - m } ) ( n \to \infty ) . \end{align*}


\begin{align*} b _ { \beta } ^ { \alpha } g _ { \beta } ^ { ( \beta ) } ( o ) + \sum _ { \beta < \gamma < \alpha } b _ { \gamma } ^ { \alpha } g _ { \beta } ^ { ( \gamma ) } ( o ) = - g _ { \beta } ^ { ( \alpha ) } ( o ) , \forall \beta < \alpha , \end{align*}


\begin{align*} Q _ { b ( \tau ) } ( y ) = \frac { 1 } { b ( \tau ) } Q \left ( \frac { | y | } { \sqrt { b ( \tau ) } } \right ) , \end{align*}


\begin{align*} \varphi | _ { A _ { i _ 0 } \cap B _ { j _ 0 } \times \cdots \times A _ { i _ q } \cap B _ { j _ q } } = \psi | _ { A _ { i _ 0 } \cap B _ { j _ 0 } \times \cdots \times A _ { i _ q } \cap B _ { j _ q } } \end{align*}


\begin{align*} v = \rho _ 2 + \rho _ 3 + 4 v _ 0 \end{align*}


\begin{align*} \lambda ( \alpha , z _ 1 ) ^ { \ast } \eta ( \alpha z _ 1 , \alpha z _ 2 ) \lambda ( \alpha , z _ 2 ) = \eta ( z _ 1 , z _ 2 ) , \end{align*}


\begin{align*} \begin{aligned} \lambda \rho + \rho _ x + u _ x - \epsilon \rho _ { x x } = f , \\ \lambda u + \rho _ x + u _ x - u _ { x x } = g , \end{aligned} \end{align*}


\begin{align*} d _ { q - 1 } ( h \varphi ) + h ( d _ q \varphi ) = \beta ^ * \varphi - \alpha ^ * \varphi . \end{align*}


\begin{align*} \frac { B } { A } = e ^ { 2 \gamma a } \coth \left ( \frac { \gamma n a } { 2 } \right ) ^ 2 . \end{align*}


\begin{align*} \left \{ \aligned & \partial _ { t } u + ( u \cdot \nabla ) u + ( - \Delta ) ^ { \alpha } u + \nabla p = ( b \cdot \nabla ) b , x \in \mathbb { R } ^ { 2 } , \ , t > 0 , \\ & \partial _ { t } b + ( u \cdot \nabla ) b + ( - \Delta ) ^ { \beta } b = ( b \cdot \nabla ) u , \\ & \nabla \cdot u = 0 , \ \ \ \nabla \cdot b = 0 , \\ & u ( x , 0 ) = u _ { 0 } ( x ) , \ \ b ( x , 0 ) = b _ { 0 } ( x ) , \endaligned \right . \end{align*}


\begin{align*} & [ T u , T v , T w ] = T \Big ( \theta ( T v , T w ) u - \theta ( T u , T w ) v + D ( T u , T v ) w \Big ) . \end{align*}


\begin{align*} \frac { \partial C ( x , t ) } { \partial t } = D \frac { \partial ^ { 2 } C ( x , t ) } { \partial x ^ { 2 } } \end{align*}


\begin{align*} O ( \widetilde { \delta } ^ s ) = \Phi ( \alpha Q + h ) = \alpha ^ 2 \Phi ( Q ) + \Phi ( h ) + 2 \alpha B ( Q , h ) = \alpha ^ 2 \Phi ( Q ) + \Phi ( h ) , \end{align*}


\begin{align*} & n ^ { \gamma - 2 } \sum _ { | x | \leq 2 b _ G n } \sum _ { | y | \leq 2 b _ G n } \sup _ { s \in [ 0 , T ] } C _ { G , \delta } | x - y | ^ { \delta } n ^ { - \delta } p ( y - x ) \\ \lesssim & n ^ { \gamma - 2 - \delta } \sum _ { | x | \leq 2 b _ G n } \sum _ { z \neq 0 } | z | ^ { \delta - \gamma - 1 } = 2 ( 2 b _ G + 1 ) n ^ { \gamma - 1 - \delta } \sum _ { z = 1 } ^ { \infty } z ^ { \delta - \gamma - 1 } . \end{align*}


\begin{align*} \begin{cases} \mathcal { C } ^ \omega \vdash F ( \underline { a } ) \Rightarrow \\ \mathcal { C } ^ { \omega - } + ( \mathrm { B R } ) \vdash \forall \underline { a } , \underline { y } ( F ' ) _ D ( \underline { t } \underline { a } , \underline { y } , \underline { a } ) \end{cases} \end{align*}


\begin{align*} \epsilon N _ 0 ( T ) = - \frac { \epsilon T } { 2 \pi } \log \pi + \frac { \epsilon T } { 2 \pi } \log \left ( \frac { T } { 2 } \right ) - \frac { \epsilon T } { 2 \pi } + \frac { \epsilon } { \pi } \ \mathcal { O } ( \log T ) \end{align*}


\begin{align*} ( s - t ) ^ 2 - \frac { s - t } { h ( s ) } + | \lambda | ^ 2 = 0 . \end{align*}


\begin{align*} \phi = c \alpha + \bar c \bar \alpha . \end{align*}


\begin{align*} \epsilon \big ( X , \rho , \psi _ K \big ) \epsilon \big ( q _ K ^ { - 1 } X ^ { - 1 } , \rho ^ \vee , \psi _ K \big ) = \rm d e t \big ( \rho ( - 1 ) \big ) , \end{align*}


\begin{align*} ( f | _ k \gamma ) ( z ) = \sum _ { \tau \in S _ + } c ( \tau , \gamma , f ; v , w ) e ( \lambda ( \mathfrak { i } ( \tau ) u ) ) , z = ( u , v , w ) \in \mathfrak { Z } , \end{align*}


\begin{align*} \epsilon \int _ { 0 } ^ { 1 } \rho ^ { \epsilon } _ x \bar { \sigma } _ x + \int _ { 0 } ^ { 1 } u ^ { \epsilon } _ x \bar { \sigma } - \int _ { 0 } ^ { 1 } \rho ^ { \epsilon } \bar { \sigma } _ x + \lambda \int _ { 0 } ^ { 1 } \rho ^ { \epsilon } \bar { \sigma } = \int _ { 0 } ^ { 1 } f \bar { \sigma } . \end{align*}


\begin{align*} \mathbb { E } [ \zeta _ \mathcal { K } ] & \stackrel { ( a ) } { = } \mathbb { E } \big [ \Re \{ \zeta _ \mathcal { K } \} \big ] = \sum _ { i \in \mathcal { K } } \sqrt { \bar { p } / \beta _ i } \mathbb { E } \big [ | \hat { h } _ i | ] \stackrel { ( b ) } { = } \frac { \sqrt { \pi \bar { p } } } { 2 } \sum _ { i \in \mathcal { K } } \sqrt { 1 + \frac { 1 } { N _ 1 \bar { \gamma } _ i } } \stackrel { ( c ) } { \approx } \frac { K \sqrt { \pi \bar { p } } } { 2 } \sqrt { 1 + \frac { 1 } { N _ 1 \bar { \gamma } ' } } , \end{align*}


\begin{align*} \left \{ \Phi ( \mathbf { x } ) , \Pi ( \mathbf { y } ) \right \} ^ \Sigma _ { \mathrm { c a n } } = \delta _ \Sigma ( \mathbf { x } , \mathbf { y } ) , \end{align*}


\begin{align*} L _ + f _ * = \lambda _ 0 \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ { p } \right ) Q ^ { p - 1 } + \sum _ { j = 1 } ^ { N } \lambda _ j \partial _ { x _ j } Q + \lambda _ { N + 1 } f _ * . \end{align*}


\begin{align*} P _ { \gamma } ^ { \mathrm { i n } } ( M ) : = P _ { \theta } ^ { \mathrm { i n } } ( M ) , \ , \ , \ , \ , P _ { \gamma } ^ { \mathrm { o u t } } ( M ) : = P _ { \theta } ^ { \mathrm { o u t } } ( M ) . \end{align*}


\begin{align*} & \mbox { i f $ k > 0 $ } , \widetilde { f } ( k ) = - \frac { \alpha k } { \sqrt { 2 \pi } } \int f ( x ) \psi _ + ( x , 0 ) \ , d x + O ( k ^ 2 ) \\ & \mbox { i f $ k < 0 $ } , \widetilde { f } ( k ) = \frac { \alpha k } { \sqrt { 2 \pi } } \int f ( x ) \psi _ - ( x , 0 ) \ , d x + O ( k ^ 2 ) , \end{align*}


\begin{align*} \Gamma ( s , x ) = \int _ x ^ { \infty } t ^ { s - 1 } e ^ { - t } d t , \ \ \Re s > 0 . \end{align*}


\begin{align*} \begin{aligned} & \lim _ { N \to \infty } \lim _ { n \to \infty } \\ & \sum _ { j = 1 } ^ { n - N } \sum _ { l = 1 } ^ k \frac { \binom { c _ n k n } l ( k ) _ l ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \frac k { k ( n - j + 1 ) - l } \frac { E _ { n , k } ( \tau ( A ^ { ( n ) } _ { c _ n k n , j , l } ) | \mathcal { W A } ^ { ( n , k ) } _ { M _ n , j , l } ) } { n k } = 0 . \end{aligned} \end{align*}


\begin{align*} g ( g \varphi ^ { j } _ { s } + \lambda \psi ^ { j } _ { s } ) + \lambda ( g \varphi ^ { j } _ { t } + \lambda \psi ^ { j } _ { t } ) = - \eta ^ { t } \Lambda \begin{pmatrix} \varphi ^ { j } \\ \psi ^ { j } \end{pmatrix} + g ^ { 2 } F ^ { j } + g \lambda G ^ { j } + \lambda ^ { 2 } H ^ { j } , \end{align*}


\begin{align*} R _ - ( \kappa _ x ) & = \frac { 1 - \alpha } { 1 + \alpha } = \frac { \mu _ 2 \kappa _ { 1 z } - \mu _ 1 \kappa _ { 2 z } } { \mu _ 2 \kappa _ { 1 z } + \mu _ 1 \kappa _ { 2 z } } \\ T _ + ( \kappa _ x ) & = \frac { 2 } { 1 + \alpha } = \frac { 2 \mu _ 2 \kappa _ { 1 z } } { \mu _ 2 \kappa _ { 1 z } + \mu _ 1 \kappa _ { 2 z } } . \end{align*}


\begin{align*} Y _ { l _ { t } ^ { \epsilon } } ^ { x ; l ^ { \epsilon } } = X _ { t } ^ { x ; l ^ { \epsilon } } \quad \nabla _ { v } Y _ { l _ { t } ^ { \epsilon } } ^ { x ; l ^ { \epsilon } } = \nabla _ { v } X _ { t } ^ { x ; l ^ { \epsilon } } . \end{align*}


\begin{align*} \widehat { 0 } : = 0 , \ ; \widehat { X } : = 0 , \ ; \widehat { \tau ( \xi ) } : = \widehat { \tau } ( \widehat { \xi } ) . \end{align*}


\begin{align*} \Delta ( A ) ( 1 \otimes A ) = \Delta ( A ) ( A \otimes 1 ) = \Delta ( 1 ) ( A \otimes A ) , ( A \otimes 1 ) \Delta ( A ) = ( 1 \otimes A ) \Delta ( A ) = ( A \otimes A ) \Delta ( 1 ) . \end{align*}


\begin{align*} ( F \setminus \{ \bar { b } _ n \} ) \cup D _ 1 \cup D _ 2 & = ( F \setminus \{ \bar { b } _ n \} ) \cup ( D \setminus \{ \bar { b } _ n \} ) \\ & = ( F \cup D ) \setminus \{ \bar { a } _ n \} = \mathbb { N } \setminus \{ \bar { a } _ n \} \end{align*}


\begin{align*} \omega _ 0 = \sup \big \{ \Re ( \lambda ) : \lambda \in \sigma ( A ) \big \} < 0 . \end{align*}


\begin{align*} & v _ j \coloneqq u _ { j + 1 } - u _ j , \ , \\ & v _ r \coloneqq u _ 1 - u _ r , \end{align*}


\begin{align*} & A _ { p , q } = \partial _ p \partial _ q - \partial _ { p + 1 } \partial _ { q - 1 } { \rm w h e r e } ( p , q ) \in [ 1 , k - 1 ] \times [ 2 , k ] \\ & \mathcal { T } ^ m : = \partial _ 1 \partial _ { m - 1 } + \partial _ m E { \rm w h e r e } E : = \sum _ { h = 1 } ^ k \sigma _ h \partial _ h { \rm a n d } m \in [ 2 , k ] \end{align*}


\begin{align*} y & \in \partial V ( u - \sum _ { t = 0 } ^ { T - 1 } x _ t \Delta s _ { t + 1 } - \bar c \cdot \bar x + S _ 0 ( \bar x ) ) , \\ p _ t + y \Delta s _ { t + 1 } & \in N _ { D _ t } ( x _ t ) t = 0 , \dots , T , \\ p _ { - 1 } + y \bar c & \in \partial ( y S _ 0 ) ( \bar x ) \end{align*}


\begin{align*} \mathbf { f } ( \alpha g _ { \mathbf { h } } k , \alpha z ) = j ( \alpha , z ) ^ k \mathbf { f } ( g _ { \mathbf { h } } , z ) . \end{align*}


\begin{align*} & I _ { 1 } = - \frac { 1 } { 2 \pi i } \oint _ { \partial U _ \xi } \left ( J ^ E ( s ) - I \right ) d s , \\ & I _ { 2 } = - \frac { 1 } { 2 \pi i } \int _ { \Gamma ^ { ( 3 ) } \backslash U _ \xi } \left ( J ^ E ( s ) - I \right ) d s , \\ & I _ { 3 } = - \frac { 1 } { 2 \pi i } \int _ { \Gamma _ E } \mu ( s ) ( J ^ E ( s ) - I ) d s . \end{align*}


\begin{align*} T _ { t , j , 0 } = \{ \Omega _ { t , j } \} , { \rm c a r d } \ , T _ { t , j , m } \le c \cdot 2 ^ m , \end{align*}


\begin{align*} { E _ { m } ^ { [ \alpha , \beta ] , \rho , \varepsilon } = \left \{ x \in \widetilde { E } _ { m } ^ { [ \alpha , \beta ] , \rho , \varepsilon } : \forall r \leq \rho , \ \frac { 3 } { 4 } m ( B ( x , r ) ) \leq m ( B ( x , r ) \cap \widetilde { E } _ { m } ^ { [ \alpha , \beta ] , \rho , \varepsilon } ) \right \} } \end{align*}


\begin{align*} \frac { \Gamma ( \pi _ { s r } ) } { \Gamma ( \pi _ { t p } ) } = \frac { 1 } { 2 } \left ( \frac { \sigma ^ 2 } { \mu ^ 2 } + 1 \right ) . \end{align*}


\begin{align*} \begin{array} { c } d _ { C _ 1 C n } ( M , \ , Y _ q ( \Omega ) ) \underset { \mathfrak { Z } _ 0 } { \lesssim } \sum \limits _ { t = 0 } ^ { \hat t ( n ) } \sum \limits _ { m = 0 } ^ \infty d _ { k _ { t , m } } ( W _ { t , m } , \ , l _ q ^ { \nu _ { t , m } } ) + \\ + \sum \limits _ { t = 0 } ^ { \hat t ( n ) } d _ { s _ t } ( 2 ^ { ‐ \alpha _ * k _ * t } B _ { p _ 0 } ^ { \nu _ { t , 0 } } , \ , l _ q ^ { \nu _ { t , 0 } } ) + \sup _ { f \in M } \| f \| _ { Y _ q ( \tilde \Omega _ { \hat t ( n ) + 1 } ) } . \end{array} \end{align*}


\begin{align*} E _ 0 ( x ) = \sum _ { x \leq p \leq 2 x } \frac { 1 } { p } \sum _ { \substack { 0 \leq a < p \\ \gcd ( n , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi a ( \tau ^ { n } - u ) } { p } } \cdot \frac { 1 } { p } \sum _ { \substack { 0 < \leq b < p \\ g c d ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { m } - v ) } { p } } = 0 . \end{align*}


\begin{align*} \tau _ n = \tau _ { n - 1 } + k _ { n - 1 } ( \tau _ 0 - 1 ) . \end{align*}


\begin{align*} \begin{aligned} A _ { 1 } ^ { 0 } ( u , v ) \partial _ { t } u & + A ^ { i } _ { 1 1 } ( u , v ) \partial _ { i } u = f _ { 1 } ( U , D _ { x } v ) , \\ A _ { 2 } ^ { 0 } ( u , v ) \partial _ { t } v & - B _ { 2 2 } ^ { i j } ( u , v ) \partial _ { i } \partial _ { j } v = f _ { 2 } ( U , D _ { x } U ) . \end{aligned} \end{align*}


\begin{align*} R _ H ( s _ 1 , s _ 2 ) = \mathbb { E } [ B _ { s _ 1 } ^ H \cdot B _ { s _ 2 } ^ H ] \end{align*}


\begin{align*} S _ { t } ( l ) : = l _ { t } \end{align*}


\begin{align*} \chi = \sum _ { i = 0 } ^ { l - 1 } \zeta _ l ^ i \varphi ^ i ( \xi ) \neq 0 , \end{align*}


\begin{align*} f ( t ) = \int _ \R \widehat { f } ( \omega ) e ^ { 2 \pi i \omega t } \ , d t , \end{align*}


\begin{align*} \flat _ { \omega } ( \pmb { X } ) = d H . \end{align*}


\begin{align*} \begin{aligned} x _ k & \to \bar x , & y _ k & \to \bar y , & \lambda _ k & \to \lambda , & \\ \eta _ k & \to 0 , & \frac { x _ k - \bar x } { \norm { x _ k - \bar x } } & \to u , & \frac { y _ k - \bar y } { \norm { x _ k - \bar x } } & \to 0 , & \end{aligned} \end{align*}


\begin{align*} \operatorname { d i v } \mathbf { v } _ A = & \ ; \operatorname { d i v } ( \zeta \circ d _ \Gamma ) ( \mathbf { v } _ A ^ { i n } - \mathbf { v } _ A ^ + \chi _ + - \mathbf { v } _ A ^ - \chi _ - ) + ( \zeta \circ d _ \Gamma ) \operatorname { d i v } \ , \mathbf { v } _ A ^ { i n } \\ & + ( 1 - \zeta \circ d _ \Gamma ) ( \operatorname { d i v } \ , \mathbf { v } _ A ^ + \chi _ + + \operatorname { d i v } \ , \mathbf { v } _ A ^ - \chi _ - ) \triangleq I _ 1 + I _ 2 + I _ 3 . \end{align*}


\begin{align*} \gamma ( X , \pi _ E , \widehat { \sigma _ E } , \psi _ E ) = \prod _ { j = 1 } ^ t \gamma ( X , \pi _ E , Q ( \widehat { D _ i } ) , \psi _ E ) \end{align*}


\begin{align*} \mathbb { E } \Big [ \ < P s i 2 > _ n ( t _ 1 , y ) \overline { \ < P s i 2 > _ n ( t _ 2 , \tilde { y } ) } \Big ] = \Big | \mathbb { E } \Big [ \ < P s i > _ n ( t _ 1 , y ) \overline { \ < P s i > _ n ( t _ 2 , \tilde { y } ) } \Big ] \Big | ^ 2 + \Big | \mathbb { E } \Big [ \ < P s i > _ n ( t _ 1 , y ) \ < P s i > _ n ( t _ 2 , \tilde { y } ) \Big ] \Big | ^ 2 . \end{align*}


\begin{align*} & \sum _ { i , j \geq 1 } \frac { \omega _ { i , j } } { \lambda ^ { i + 1 } \mu ^ { j + 1 } } = \frac { { \rm t r } \ , R ( \lambda ) R ( \mu ) - 1 } { ( \lambda - \mu ) ^ 2 } , \end{align*}


\begin{align*} E [ u ] = E [ u _ 0 ] = E [ Q ] \end{align*}


\begin{align*} ( \tilde K * g ) ( \xi ) = \int _ { B ( 0 , \delta ) ^ c } g ( \xi - y ) \tilde K ( y ) \ , d z = \int _ { B ( 0 , \delta ) ^ c } g ( \xi - y ) K ( y ) \ , d z = ( K * g ) ( \xi ) . \end{align*}


\begin{align*} P _ { n + 2 } ( x ) = 2 \sum _ { k = 0 } ^ n { n \choose k } P _ k ( x ) P _ { n + 1 - k } ( x ) . \end{align*}


\begin{align*} { \rm t r } ^ \flat ( P ( h ) ) = \mathcal { O } ( h ^ { - 2 n - m } ) . \end{align*}


\begin{align*} x ^ n _ 1 = \mathbb { E } [ x ^ n _ 1 ] + \int _ 0 ^ 1 \mathbb { E } [ D _ s x ^ n _ 1 \vert \mathcal { F } _ s ] d W _ s , \end{align*}


\begin{align*} \beta p = \sum _ { i = 1 } ^ k \alpha _ i p _ i - \sum _ { i = k + 1 } ^ d \beta _ i p _ i + \lambda e _ d . \end{align*}


\begin{align*} \phi ( - d ( L \phi ) ) = \phi ( - d ( \eta _ 0 ) ) = \phi ( 0 ) = L \phi = \eta _ 0 . \end{align*}


\begin{align*} t _ i + x _ j \ = \ v _ { i j } \end{align*}


\begin{align*} \int _ \R \partial _ k \overline { \mathcal { K } ^ \# _ 0 ( x , k ) } f ( x ) \ , d x = T ( 0 ) \int _ \R m _ + ( x , 0 ) ( - i x ) e ^ { i x k } f ( x ) \ , d x \end{align*}


\begin{align*} \Sigma _ 1 \mathrm { E R } : \quad \frac { \exists y ^ \sigma A _ 0 ( y ) \rightarrow s = _ \rho t } { \exists y ^ \sigma A _ 0 ( y ) \rightarrow r [ s / x ^ \rho ] = _ \tau r [ t / x ^ \rho ] } \end{align*}


\begin{align*} f = \sum _ { \gamma \in \Gamma } \langle f , e _ \gamma \rangle S ^ { - 1 } e _ \gamma , \end{align*}


\begin{align*} \underline { \lambda } = x - c _ { 2 } \theta ( x ) \big ( x ^ 2 ( \epsilon _ n + \delta _ n ) + x \delta _ n \sqrt { | \ln \delta _ n | } \big ) \in [ 0 , \ , o ( \min \{ \epsilon _ n ^ { - 1 } , \delta _ n ^ { - 1 } \} \ , ] , \end{align*}


\begin{align*} \psi _ { Q _ S } ( x ) = \det ( Q _ S - x I ) = \det \Big ( Q _ { J - 2 A } - ( x + 1 ) I \Big ) = \psi _ { Q _ { J - 2 A } } ( x + 1 ) \end{align*}


\begin{align*} ( R \# T ) ( e _ i \wedge e _ j ) = \frac { 1 } { 2 } \sum _ k ( r _ { i , k } t _ { j , k } + t _ { i , k } r _ { j , k } ) e _ i \wedge e _ j . \end{align*}


\begin{align*} \sum _ { n \geq 0 } \mu ( L _ { B , n } ) = + \infty \end{align*}


\begin{align*} M = \sum \limits _ { Y \in \mathcal Y } \alpha _ Y Y \end{align*}


\begin{align*} \partial _ t \left ( e ^ { e _ 0 t } \alpha _ + \right ) = e ^ { e _ 0 t } B ( R , \mathcal { Y } _ - ) , \end{align*}


\begin{align*} \begin{array} { l l } A & = ( L ^ { 2 } R \times \overline { L } R ) \cdot ( L R \times \overline { L } R ) = O ( r ^ { 5 m _ { j } - 1 1 } ) , \\ B & = ( L ^ { 2 } R \times L R ) \cdot ( L R \times \overline { L } R ) = O ( r ^ { 5 m _ { j } - 1 1 } ) , \\ C & = ( L R \times \overline { L } R ) \cdot ( L R \times \overline { L } R ) = O ( r ^ { 4 m _ { j } - 8 } ) . \end{array} \end{align*}


\begin{align*} T B _ { 1 0 } ( f ) = 5 T B _ { 6 } ( f ) , \end{align*}


\begin{align*} \widehat { \beta } = ( X ' X ) ^ { - 1 } X ' y . \end{align*}


\begin{align*} V _ { g _ 0 } f ( x , - \omega ) = e ^ { \pi i x \cdot \omega } e ^ { - \frac { \pi } { 2 } | z | ^ 2 } B f ( z ) . \end{align*}


\begin{align*} \nabla ^ \mathcal { F } _ X Y : = \nabla _ { \overline { \mathcal { F } } _ 1 \rhd X } ( \overline { \mathcal { F } } _ 2 \rhd Y ) , ~ X , Y \in \Xi _ \star \end{align*}


\begin{align*} \frac { 1 } { \int _ { \R ^ N } m ' u ^ 2 } = \tilde { \lambda } _ 0 \ ; . \end{align*}


\begin{align*} D _ u x _ t ^ n & = \int ^ t _ u \exp \Big ( \int _ s ^ t \tilde { f } ' ( r , x _ r ^ n ) d r \Big ) \tilde { f } ' _ n ( s , x _ s ^ n ) K _ H ( s , u ) d s + K _ H ( t , u ) \\ & = J _ 1 ^ n ( t , u ) + J _ 2 ^ n ( t , u ) + K _ H ( t , u ) , u < t , \lambda \times , \end{align*}


\begin{align*} { L ^ { \phantom { s } } \ ! \ ! } ^ { \scriptscriptstyle { | | } } : = A + i | V | = A + i \frac { \alpha ^ 2 } { 2 } . \end{align*}


\begin{align*} \| u \| _ { S ' ( \dot { H } ^ { - s } ) } = \inf \left \lbrace \| u \| _ { L _ t ^ { q ' } L _ x ^ { r ' } } : ( q , r ) ( \ref { S t r i c h a r t z p a i r } ) ( \ref { r e s t r i c t i o n o n d u a l S t r i c h a r t z p a i r } ) \right \rbrace . \end{align*}


\begin{align*} ( - \Delta ) ^ s u ( x ) = c _ { n , s } \lim _ { \varepsilon \to 0 ^ + } \int _ { \R ^ n _ + \setminus B _ \varepsilon ( x ) } \left ( \frac 1 { \vert x - y \vert ^ { n + 2 s } } - \frac 1 { \vert x _ \ast - y \vert ^ { n + 2 s } } \right ) \big ( u ( x ) - u ( y ) \big ) \dd y + \frac { c _ { 1 , s } } s u ( x ) x _ 1 ^ { - 2 s } \end{align*}


\begin{align*} \left ( \sqrt { j ( 2 j - 1 ) } s ^ { j - 1 } \right ) L \psi ^ { 1 } - i \sqrt { k ( 2 k - 1 ) } t ^ { k - 1 } L \varphi ^ { 1 } = 0 . \end{align*}


\begin{align*} D ( \mathbf { u } , \mathbf { b } ) : = \int _ { \mathbb { R } ^ 3 } | \nabla \mathbf { u } | ^ 2 + | \nabla \mathbf { b } | ^ 2 \ , d x \leq \mathcal { C } \liminf _ { R \rightarrow \infty } M ^ 3 _ { \ , p , \ , q } ( R ) . \end{align*}


\begin{align*} x ^ r + \gamma _ 1 \omega ( x ) x ^ { r - 1 } + \cdots + \gamma _ { r - 1 } \omega ( x ) ^ { r - 1 } x = 0 , \end{align*}


\begin{align*} \hat { h } _ i = \mathbf { v } ^ H \mathbf { z } _ i / ( N _ 1 \sqrt { p } ) \sim \mathcal { C N } \Big ( 0 , \beta _ i + \frac { 1 } { N _ 1 \varrho } \Big ) , \end{align*}


\begin{align*} \tau _ t ( a _ 1 \cdots a _ k ) = \tau ( a _ 1 \ldots a _ k ) + o ( 1 ) . \end{align*}


\begin{align*} U ^ { - 1 } \rho ( x , \omega , \tau ) U = \pi ( x , \omega , \tau ) . \end{align*}


\begin{align*} \lim _ { t \to 0 ^ + } \frac { e ^ { - t \mathfrak { D } _ { { \rm l e f t } , a } } u ( x ) - u ( x ) } { t } & = - \mathfrak { D } _ { { \rm l e f t } , a } u ( x ) \\ \lim _ { t \to 0 ^ + } \frac { e ^ { - t \mathfrak { D } _ { { \rm r i g h t } , a } } v ( x ) - v ( x ) } { t } & = - \mathfrak { D } _ { { \rm r i g h t } , a } v ( x ) . \end{align*}


\begin{align*} \| \operatorname { H e s s } y ^ j \| _ { C ^ { 0 , \alpha } ( B _ { r } ( p ) ) } \leq \tau ( n , r , \alpha , Q ) , \ \ j = 1 , \dots , m , \end{align*}


\begin{align*} x ^ m \bigl ( ( m + 1 ) P ( x ) + x P ' ( x ) \bigr ) = ( 1 - x ) ^ n \bigl ( ( n + 1 ) Q ( x ) - ( 1 - x ) Q ' ( x ) \bigr ) . \end{align*}


\begin{align*} \Phi ( z , t ) = ( x , y ) : = \big ( z + t , Q _ z ( t ) \big ) , \ \ ( z , t ) \in E . \end{align*}


\begin{align*} \begin{dcases} \lambda \sigma _ n - ( \sigma _ n ) _ x - ( u _ n ) _ x = f _ n & ( 0 , 1 ) , \\ \lambda u _ n - ( \sigma _ n ) _ x - ( u _ n ) _ x - ( u _ n ) _ { x x } = g _ n & ( 0 , 1 ) , \\ \sigma _ n ( 0 ) = \sigma _ n ( 1 ) , \ \ u _ n ( 0 ) = u _ n ( 1 ) = 0 . \end{dcases} \end{align*}


\begin{align*} \langle \ ! \langle \varphi , \rho _ { \pi _ { \lambda , \beta } } ( z , \cdot ) \rangle \ ! \rangle _ { \pi _ { \lambda , \beta } } = \int _ { \mathbb { R } } \varphi ( x - z ) \ , \mathrm { d } \pi _ { \lambda , \beta } ( x ) , \quad \varphi \in ( N ) _ { \pi _ { \lambda , \beta } } ^ { 1 } . \end{align*}


\begin{align*} \mathrm { W F } \left ( \left \langle E _ { \Sigma } , F ^ { ( k + 1 ) } [ \psi ] \otimes G ^ { ( m + 1 ) } [ \psi ] \right \rangle \right ) \cap ( \Xi ^ { ( k + m ) } _ + \cup \Xi ^ { ( k + m ) } _ - ) = \emptyset , \end{align*}


\begin{align*} m ^ { ( 1 ) } \rightarrow \Delta _ { R } ( k ) , \vert J ^ { ( 1 ) } ( x , t , k ) - I \vert = \mathcal { O } \left ( e ^ { - 8 t k _ 2 ( 3 k _ 1 ^ 2 - k _ 2 ^ 2 + 3 \xi ) } \right ) . \end{align*}


\begin{align*} V \left ( \frac { g _ 2 g _ 4 v } { K X } \right ) = \frac { 1 } { 2 \pi i } \Psi \Big ( \frac { v } { X Q ^ { \vartheta } } \Big ) \int _ { ( \varepsilon ) } \left ( \frac { X K } { g _ 2 g _ 4 v } \right ) ^ { s _ 4 } \widetilde { V } ( s _ 4 ) \ , d s _ 4 . \end{align*}


\begin{align*} ( \varphi , \phi ) _ M : = \sum _ { i j } M _ { i j } \varphi _ i \phi _ j , ( \varphi ) _ M ^ 2 : = ( \varphi , \varphi ) _ M , \Delta _ M F : = ( \nabla , \nabla ) _ M F ; \end{align*}


\begin{align*} \mathcal { H } ^ { s } _ { t } ( E ) = \inf \left \{ \sum _ { n \in \mathbb { N } } \vert B _ n \vert ^ s : \ ( B _ n ) _ { n \in \N } \mbox { c l o s e d b a l l s , } \vert B _ n \vert \leq t E \subset \bigcup _ { n \in \mathbb { N } } B _ n \right \} . \end{align*}


\begin{align*} \Re { \left ( \int _ y ^ M g ( u ) e ^ { 2 \pi i \left ( f ( u ) - u \right ) } d u \right ) } & = \Re { \left ( \int _ y ^ M \frac { e ^ { - 2 \pi i w ( u ) } } { u ^ { \sigma _ 1 } ( u + n ) ^ { \sigma _ 3 } } d u \right ) } \\ & = \int _ { w ( y ) } ^ { w ( M ) } \frac { 1 } { u ^ { \sigma _ 1 } ( u + n ) ^ { \sigma _ 3 } } \frac { \cos ( 2 \pi w ) d w } { 1 - \frac { 1 } { 2 \pi } \left ( \frac { t _ 1 } { u } + \frac { t _ 3 } { u + n } \right ) } . \end{align*}


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


\begin{align*} \Big | \frac { 1 } { 2 \varepsilon n ^ 2 } \sum _ { x } \Phi _ n ( s , \tfrac { x } { n } ) \sum _ { r = 1 } ^ { \varepsilon n } \int _ { \Omega } \overleftarrow { \eta } ^ { \ell } ( x ) [ \eta ( x + 1 ) - \eta ( x + 1 + r ) ] [ f ( \eta ) - f ( \eta ^ { x + 1 , x + 1 + r } ) ] d \nu _ { b } \Big | . \end{align*}


\begin{align*} x _ t - y _ t = \int _ 0 ^ t \big ( b ( s , x _ s ) - b ( s , y _ s ) \big ) d s . \end{align*}


\begin{align*} H ^ * [ P , Q , \varepsilon ] : = \forall p \in P \exists q \in Q \left ( \norm { p - q } \leq \varepsilon \right ) . \end{align*}


\begin{align*} D i f f ^ { i + j - 1 } & \ni [ m _ a , D ^ i ] \circ D ^ j + D ^ i \circ [ m _ a , D ^ j ] = m _ a \circ D ^ i \circ D ^ j - D ^ i \circ m _ a \circ D ^ j \\ & + D ^ i \circ m _ a \circ D ^ j - D ^ i \circ D ^ j \circ m _ a = m _ a \circ D ^ i \circ D ^ j - D ^ i \circ D ^ j \circ m _ a = [ m _ a , D ^ i \circ D ^ j ] \ \end{align*}