\begin{align*} g ( p ) = r = \log \left ( \frac { | \log p | ^ 2 } { 4 } \right ) + o ( 1 ) \sim 2 \log | \log p | , \qquad \textrm { a s } p \to 0 . \end{align*}


\begin{align*} K _ { 1 / 3 } ^ { ( 0 ) } = \pi \left ( \left \{ x \in \Lambda ^ \mathbb { N } : \liminf _ { k \to + \infty } \frac { S _ k \phi ( x ) } { k } = 0 \right \} \right ) , \end{align*}


\begin{align*} \boxed { \exp \Bigl ( \sum \alpha _ k \Bigr ) = \prod \exp ( \alpha _ k ) . } \end{align*}


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


\begin{align*} F _ { H } [ \partial _ { \Sigma , \epsilon } \chi ^ { * } \phi ] = F _ { H } [ \rho ^ { * } _ { ( 1 ) } \partial _ { \widetilde { \Sigma } , \widetilde { \epsilon } } \phi ] , \end{align*}


\begin{align*} \partial _ t u + u \cdot \nabla u + \nabla \big ( \beta ( L - 1 ) \eta + \bar { \mathfrak { z } } ( \eta ) ^ { 2 } \big ) - \mu _ 1 \Delta u - \mu _ 2 \nabla \mathrm { d i v } u - \frac { \beta } { k } \mathrm { d i v } \tau = - \nabla \pi _ 1 , \end{align*}


\begin{align*} \bigl ( ~ _ r \mathcal { F } _ s ( \alpha ) \bigr ) ^ { q ^ d } = \sum _ { n = 0 } ^ { \infty } \biggl ( \prod _ { m = 1 } ^ { n + d - 1 } ( \theta ^ { q ^ m } - t ) ^ { c ( m - n ) q ^ { n + d - m } } \biggr ) \alpha ^ { q ^ { n + d } } \end{align*}


\begin{align*} J _ { t } = J - U \otimes \zeta _ { t } - V \otimes ( \zeta _ { t } \circ J ) . \end{align*}


\begin{align*} \mathcal { U } ( h , k ) = \frac { 1 } { 2 } \sum _ { \substack { 1 \leq c \leq C \\ ( c , h k ) = 1 } } \mu ( c ) \sum _ { \substack { 1 \leq m , n < \infty \\ ( m n , c ) = 1 \\ m h \neq n k } } \frac { \tau _ A ( m ) \tau _ B ( n ) } { \sqrt { m n } } V \left ( \frac { m } { X } \right ) V \left ( \frac { n } { X } \right ) \sum _ { \substack { 1 \leq d < \infty \\ ( d , m h n k ) = 1 \\ d | m h \pm n k } } \phi ( d ) W \left ( \frac { c d } { Q } \right ) . \end{align*}


\begin{gather*} f ( x ) = \sum _ { n \in \mathbb { Z } } f _ n e ^ { i n x } \ ; \textrm { a n d } \ ; \tilde { \phi } ( x ) = \sum _ { n \in \mathbb { Z } } a _ n e ^ { i n x } , \end{gather*}


\begin{align*} \boldsymbol { \mu } _ n ^ { \sigma } \coloneqq \big ( \mu _ n ^ { \sigma } ( i ) \big ) _ { i \in \mathcal { I } _ n ^ { \sigma } } , \ ; \mu _ n ^ { \sigma } ( i ) \coloneqq \sum _ { j \in \mathcal { I } : \ , \Pi _ n ^ { \sigma } j = i } \mu ( j ) . \end{align*}


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


\begin{align*} F _ \delta ( 0 ) = \int _ { - \infty } ^ 0 \left ( \int _ x ^ 0 f _ \delta ( y ) d y \right ) p _ \delta ( x ) d x - \int _ 0 ^ { + \infty } \left ( \int _ 0 ^ { x } f _ \delta ( y ) d y \right ) p _ \delta ( x ) d x . \end{align*}


\begin{align*} \sup _ { S \in H ^ 1 } \mathcal L ( \rho , m , S ) = \mathcal A ( \rho , m ) + \mathbb I _ { C _ F ( \rho ^ a , \rho ^ b ) } ( \rho , m ) , \end{align*}


\begin{align*} & b _ 1 = \frac { \left ( L - \mu \right ) \left ( \alpha + \mu _ p \left ( 1 - L t _ 1 \right ) \right ) } { \alpha \left ( L - \mu + \mu _ p \right ) } \\ & b _ 2 = b _ 1 - \left ( \frac { \alpha } { L - \mu } b _ 1 \right ) ^ 2 , \end{align*}


\begin{align*} f _ 2 ( s , y ) : = s ^ { 2 H - 1 } \Big ( \alpha _ 2 \frac { 1 } { \tilde { \theta } ^ { \rho } } ( - \tilde { \theta } \rho + \tilde { \theta } + 1 ) y ^ { - \tilde { \theta } \rho + \tilde { \theta } } + H \tilde { \sigma } ( \tilde { \theta } + 1 ) y ^ { - 2 } \Big ) . \end{align*}


\begin{align*} K _ { \alpha \alpha } ^ { ( 1 ) } = C _ b + C _ W K _ { \alpha \beta } ^ { ( 0 ) } , K _ { \alpha \beta } ^ { ( 0 ) } : = \frac { 1 } { n _ 0 } \sum _ { j = 1 } ^ { n _ 0 } x _ { j ; \alpha } x _ { j ; \beta } , x _ \alpha , x _ \beta \in \R ^ { n _ 0 } . \end{align*}


\begin{align*} \nu ( ( - \infty , z ] ) = \frac { R } { L ^ { 1 / \rho } } ( z \vee 0 ) ^ { 1 / \rho } , z \in \R , \end{align*}


\begin{align*} B _ { \xi } \big ( 1 + K _ l , 1 + \widehat { K ' } \big ) ( 1 + K _ l + \widehat { K ' } ) ! = \xi K _ l ! \widehat { K ' } ! . \end{align*}


\begin{align*} e _ \alpha ( z ) = \left ( \frac { \pi ^ { | \alpha | } } { \alpha ! } \right ) ^ { 1 / 2 } z ^ \alpha = \prod _ { k = 1 } ^ d \left ( \frac { \pi ^ { \alpha _ k } } { \alpha _ k ! } \right ) ^ { 1 / 2 } z _ k ^ { \alpha _ k } , \end{align*}


\begin{align*} ( a ^ { \dagger } a ) _ { k , \lambda } = \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) ( a ^ { \dagger } ) ^ { l } a ^ { l } , \end{align*}


\begin{align*} W _ s : = \{ \psi < s \} . \end{align*}


\begin{align*} \Gamma ^ { ( 3 ) } \cap U _ { \delta } ( \eta ) = \Gamma ^ { \eta , \delta } _ { j } , j = 1 , 2 , 3 , 4 . \end{align*}


\begin{align*} \omega _ 1 \triangleq \mathbb { E } [ T ^ 2 ] = K ' \mathbb { E } [ Z ^ 2 ] , \end{align*}


\begin{align*} \partial _ k \bigl ( \alpha ( \beta _ 1 , \ldots , \beta _ n ) \bigr ) = \sum _ { i = 1 } ^ n ( \partial _ k \beta _ i ) a _ i \beta _ 1 ^ { a _ 1 } \ldots \beta _ i ^ { a _ i - 1 } \ldots \beta _ n ^ { a _ n } = \sum _ { i = 1 } ^ n ( \partial _ k \beta _ i ) ( \partial _ i \alpha ) ( \beta _ 1 , \ldots , \beta _ n ) . \end{align*}


\begin{align*} \begin{aligned} 0 \leq p _ i ^ L = u _ { i 0 } ( x ) \leq \cdots \leq u _ { i n } ( x ) = p _ i ( x ) , \\ u _ { i j } ( x ) \leq a _ { i j } j = 0 , \ldots , n , \\ u _ { i 0 } ( x ) = a _ { i 0 } \leq \cdots \leq a _ { i n } , \\ \deg ( u _ { i j } ) \leq \tau . \end{aligned} \end{align*}


\begin{align*} \varepsilon _ j = \| \phi _ { j , b , \beta } \| ^ { - 2 } _ { L ^ 2 _ { \rho _ { \beta } } } \langle \varepsilon , \phi _ { j , b , \beta } \rangle _ { L ^ 2 _ { \rho _ \beta } } . \end{align*}


\begin{align*} \begin{array} { l l } w _ 1 2 + w _ 2 ( 1 + i ) = \theta \\ w _ 1 ( 3 + i ) + w _ 2 3 = \theta \\ w _ 1 + w _ 2 = 1 \end{array} \end{align*}


\begin{align*} \vartheta _ 2 ( z , \tau ) = \sum _ { k \in \Z } e ^ { \pi i ( k + 1 / 2 ) ^ 2 \tau } e ^ { ( 2 k + 1 ) \pi i z } \vartheta _ 4 ( z , \tau ) = \sum _ { k \in \Z } ( - 1 ) ^ k e ^ { \pi i k ^ 2 \tau } e ^ { 2 k \pi i z } . \end{align*}


\begin{align*} H X _ b ^ 1 ( X , A ) = \mathcal C _ f ( X , A ) / A = \begin{cases} A ^ { e ( X ) - 1 } & e ( X ) < \infty \\ \bigoplus _ \N A & e ( X ) = \infty \end{cases} \end{align*}


\begin{align*} { \mathcal L } ( \varphi ) = \sum _ { ( d , e ) \in D _ { 1 } ( P G ) } \varepsilon ( d , e ) \ell _ { D } ( d , e ) . \end{align*}


\begin{align*} \omega _ n ( T _ { \mathfrak u } ( x _ { i , k } ) & , T _ { \mathfrak u } ( x _ { j , l } ) \\ & = \omega _ n \Big ( \big ( \frac { 2 ^ { k - 1 } } { ( k - 1 ) ! } u _ i \log ^ { k - 1 } | u _ i | , \ldots , u _ i , \ldots , 0 \big ) , \big ( \frac { 2 ^ { l - 1 } } { ( l - 1 ) ! } u _ j \log ^ { l - 1 } | u _ j | , \ldots , u _ j , \ldots , 0 \big ) \Big ) \\ & = ( - 1 ) ^ k \langle u _ i , u _ j \rangle = \omega _ n ( x _ { i , k } , x _ { j , l } ) . \end{align*}


\begin{align*} & ( - \mathcal { L } B _ { i j , 0 } ) ' = - a _ { i j } \sigma _ i \Lambda Q + 2 p \kappa _ 0 \sigma _ j ( y Q ^ { p - 1 } ) ' , \\ & \int B _ { i j , 0 } Q ' = 0 , \quad \lim _ { y \rightarrow + \infty } B _ { i j , 0 } ( y ) = 0 , \\ & \lim _ { y \rightarrow - \infty } B _ { i j , 0 } ( y ) = - \frac { a _ { i j } \sigma _ i ( p - 2 ) } { p - 1 } \int Q , \end{align*}


\begin{align*} P _ 3 ( t ) = \vartheta _ 3 ( 0 , i t ) \vartheta _ 3 ( 0 , i / t ) . \end{align*}


\begin{align*} \sum _ { k \in \Z ^ d } f ( t + k ) = \sum _ { l \in \Z ^ d } \widehat { f } ( l ) e ^ { 2 \pi i l \cdot t } . \end{align*}


\begin{align*} \frac { \Gamma ' ( z ) } { \Gamma ( z ) } = \log z - \frac { 1 } { 2 z } - \int _ 0 ^ \infty \left ( \frac { 1 } { 2 } - \frac { 1 } { t } + \frac { 1 } { e ^ t - 1 } \right ) e ^ { - t z } \ , d t , \qquad \Re z > 0 , \end{align*}


\begin{align*} - ( a ( \vert \nabla u \vert ^ { p } ) \vert \nabla u \vert ^ { p - 2 } \nabla u ) + V ( x ) b ( \vert u \vert ^ { p } ) \vert u \vert ^ { p - 2 } = K ( x ) f ( u ) , \ \ \ \mathbb { R } ^ { d } , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \frac { 1 } { t _ { n } } \int _ { 0 } ^ { t _ { n } } \left \{ \sum _ { k = 0 } ^ { n - 1 } \varphi \left ( X _ { t _ { k } } \right ) \mathbf { 1 } _ { \left [ t _ { k } , t _ { k + 1 } \right ) } ( s ) \right \} \mathrm { d } s = \mathbb { E } ( \varphi ( \bar { X } ) ) , \end{align*}


\begin{align*} S _ { \widehat { S } g , S \L } = \widehat { S } S _ { g , \L } \widehat { S } ^ { - 1 } \end{align*}


\begin{align*} \hat { \sigma } ( \xi ) = \int _ 0 ^ \xi ( \xi - \zeta ) \partial _ \xi ^ 2 \hat { \sigma } ( \zeta ) d \zeta . \end{align*}


\begin{align*} \Theta _ { \alpha } ( s ) : = \int _ 0 ^ s \tilde { E } _ { k , N } ( \lceil s \rceil - \sigma ) ( s - \sigma ) ^ { - \alpha } \ , \dd W ( \sigma ) . \end{align*}


\begin{align*} { \scriptstyle \begin{pmatrix} 1 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & - c & c \cdot \xi & c \cdot ( \xi - 1 ) \\ 0 & 0 & 1 & 1 & - c \cdot ( \xi + 1 ) & - c \cdot ( 1 + \xi ^ { - 1 } ) & - c \end{pmatrix} } ^ { \mathtt { T } } , c \in \mathbb { C } . \end{align*}


\begin{align*} - \Delta u _ { \varepsilon } = P _ { \epsilon } ( \mu _ T - T ) . \end{align*}


\begin{align*} f \big ( \Re \{ \mathbf { y } \} ; \mathcal { H } _ k \big ) = \ ! \int \ ! f \big ( \Re \{ \mathbf { y } \} | h ' ; \mathcal { H } _ k \big ) f _ { H ' } ( h ' ) \mathrm { d } h ' , \ \ f \big ( \Im \{ \mathbf { y } \} ; \mathcal { H } _ k \big ) = \ ! \int \ ! f \big ( \Im \{ \mathbf { y } \} | h ' ; \mathcal { H } _ k \big ) f _ { H ' } ( h ' ) \mathrm { d } h ' \end{align*}


\begin{align*} \operatorname { t r } _ \gamma ( f ) = \sum _ { \alpha \in O _ \gamma } f ( \alpha ) = \sum _ { \beta \in G / C _ \gamma } f ( \beta \gamma \beta ^ { - 1 } ) , \end{align*}


\begin{align*} \psi ( x ) = 1 + x - x \coth ( x ) . \end{align*}


\begin{align*} \gamma _ { k } ^ { \langle n \rangle } = \frac { \left ( \sum _ { m \in \mathbb { M } _ k } \sqrt { \eta _ { m k } } \alpha _ { m k } \right ) ^ 2 } { \sum _ { m \in \mathbb { M } _ k } \eta _ { m k } \beta _ { m k } \alpha _ { m k } + \frac { 1 } { \gamma _ t } } . \end{align*}


\begin{align*} t _ 0 = 0 , \dotsc , \ t _ { n - 1 } = 0 , \ t _ i - t _ j = 0 \textrm { f o r } 0 \leq i < j \leq n - 1 . \end{align*}


\begin{align*} \begin{cases} - \Delta u + \tilde { \lambda } _ 0 \underline { m } \ , u = 0 \R ^ n \\ u ( \mathbf { 0 } ) = 1 \ ; u > 0 \ ; \R ^ n \end{cases} \end{align*}


\begin{align*} \| ( a , b ) - ( p , 0 ) \| _ { H ^ 1 _ x \times L ^ 2 _ x } = \varepsilon \leq \tilde \varepsilon , \end{align*}


\begin{align*} \chi _ i = \begin{cases} \frac { \bar \alpha _ { i - 1 } } { \lambda _ i } , & i \in \{ 1 , \dots , n \} \setminus J , \\ \frac { \bar \alpha _ i } { \mu _ i } , & i \in J . \end{cases} \end{align*}


\begin{align*} \norm { X g } _ 2 \norm { P g } _ 2 = \norm { X g } _ 2 \norm { X \widehat { g } } _ 2 \geq \frac { 1 } { 4 \pi } \norm { g } _ 2 ^ 2 , \end{align*}


\begin{align*} H _ { - \alpha } ( x ) = \mathfrak { D } ^ { - \alpha } _ { { \rm r i g h t } , 2 x } 1 ( x ) = \frac 1 { \Gamma ( \alpha ) } \int _ 0 ^ \infty e ^ { - 2 x t - t ^ 2 } \ , \frac { d t } { t ^ { 1 - \alpha } } \end{align*}


\begin{align*} m ^ { ( 1 ) } ( x , t , k ) = m ( x , t , k ) e ^ { i t [ g ( k , \xi ) - \theta ( k , \xi ) ] \sigma _ 3 } . \end{align*}


\begin{align*} \Phi ( d , e , t ) = e ^ { \gamma t } d + \varphi _ { \gamma } ( t ) e \end{align*}


\begin{align*} \delta ( \mathfrak { r i c } ) = - \frac { 1 } { 2 } d ( \mathfrak { s } ) , \end{align*}


\begin{align*} \mu _ { s s } = \frac { \Gamma ( m _ { s s } + 1 / 2 ) } { \Gamma ( m _ { s s } ) } \sqrt { \frac { \Omega _ { s s } } { m _ { s s } } } , \mu _ { s 1 } = \frac { \Gamma ( m _ { s 1 } + 1 / 2 ) } { \Gamma ( m _ { s 1 } ) } \sqrt { \frac { \Omega _ { s 1 } } { m _ { s 1 } } } . \end{align*}


\begin{align*} \Delta ( K ; \alpha , \beta ) = \Big [ \frac { ( K ^ { \alpha + \beta } - 1 ) ^ 3 } { \alpha + \beta } \Big ] ^ 2 - \frac { ( K ^ { 2 \alpha } - 1 ) ^ 3 } { 2 \alpha } \cdot \frac { ( K ^ { 2 \beta } - 1 ) ^ 3 } { 2 \beta } \leq \frac { K ^ { 6 \alpha + 6 \beta } } { ( \alpha + \beta ) ^ 2 } - \frac { K ^ { 6 \alpha } } { 2 5 6 | \alpha \beta | } . \end{align*}


\begin{align*} \tilde N _ { n } ^ 2 ( k ) = \sum \limits _ { \substack { \mathbf { p } _ { k , 1 } , \mathbf { p } _ { k , 2 } \in \mathcal { P } ( d , k ) } } \tilde N _ { n , \mathbf { p } _ { k , 1 } } ( k ) \tilde N _ { n , \mathbf { p } _ { k , 2 } } ( k ) = \sum _ { \mu = 0 } ^ k \tilde N _ { n , \mu } ( k ) , \end{align*}


\begin{align*} \partial _ t B _ q + a \Delta _ q ( B J _ x ) + b \Delta _ q ( J B _ x ) + \mu \Lambda ^ \alpha B _ q = 0 . \end{align*}


\begin{align*} ( - \Delta ) ^ s \bar u + k \bar u & = \bar f B _ 1 . \end{align*}


\begin{align*} \Gamma ( \pi ) & = \lim _ { t \to \infty } \frac { 1 } { N } \sum _ { \ell = 1 } ^ { N } \frac { \sum _ { i = 1 } ^ { R _ \ell ^ \pi ( t ) } ( \rho _ \ell ( \frac { ( T _ { \ell i } ^ \pi ) ^ 2 } { 2 } + T _ { \ell i } ^ \pi Z _ { \ell i } ^ \pi ) + c _ \ell ) } { t } , \end{align*}


\begin{align*} w _ R = - \chi _ R ( v + \nabla \Psi _ 0 ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ t a _ i ^ \ell = \sum _ { a \in S } a ^ \ell = \frac { 1 } { \ell + 1 } t n ^ { \ell } + O _ \ell \left ( t n ^ { \ell - 1 } \right ) - \sum _ { k = 1 } ^ { n } k ^ { \ell - 1 } E _ k . \end{align*}


\begin{align*} \widehat { K ' } _ \mathrm { a - c p t - d } ^ \mathbb { R } & = K ' + \sum _ { i \in \mathcal { K } ' } \frac { X _ i / Y _ i } { \sqrt { 2 N _ 1 \varrho \vartheta _ i } } + V = K ' + \sum _ { i \in \mathcal { K } ' } \frac { Z _ i } { \sqrt { 2 N _ 1 \varrho \vartheta _ i } } + V , \end{align*}


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


\begin{align*} \| ( \mathcal { M } ( \tilde u _ 0 , a , b _ n ) u ) ( t , \cdot ) - ( \mathcal { M } ( \tilde u _ 0 , a , b _ 0 ) u ) ( t , \cdot ) \| _ { C ^ 0 ( \bar \Omega ) } & = \| \int _ s ^ t e ^ { - A ( t - \tau ) } u ^ 2 ( \tau , \cdot ) \big [ b _ n ( \tau , \cdot ) - b _ 0 ( \tau , \cdot ) \big ] d \tau \| _ { C ^ 0 ( \bar \Omega ) } \\ & \le R ^ 2 \| b _ n - b _ 0 \| _ { \mathcal { X } _ T } T . \end{align*}


\begin{align*} U = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \ , : \ , \mathbb C ^ m \oplus \mathbb C ^ n \rightarrow \mathbb C ^ m \oplus \mathbb C ^ n . \end{align*}


\begin{align*} \gamma \circ ( \mu ^ T \Gamma ) = ( \Gamma \mu ^ S ) \circ ( \gamma S ) \circ ( T \gamma ) \end{align*}


\begin{align*} D = \frac { 4 \pi ^ 2 } { A ^ 2 } . \end{align*}


\begin{align*} \sum _ { j _ = 1 } ^ N \lambda _ j \partial _ { x _ j } Q + \lambda _ { N + 1 } i Q + \lambda _ { N + 2 } \mathcal { Y } _ + + \lambda _ { N + 3 } f = 0 , \end{align*}


\begin{align*} \norm { J ^ A _ \gamma x - J ^ A _ \gamma y } & \leq \norm { ( J ^ A _ \gamma x - J ^ A _ \gamma y ) + s ( \gamma ^ { - 1 } ( x - J ^ A _ \gamma x ) - \gamma ^ { - 1 } ( y - J ^ A _ \gamma y ) ) } \\ & = \norm { ( J ^ A _ \gamma x - J ^ A _ \gamma y ) + s \gamma ^ { - 1 } ( ( x - y ) - ( J ^ A _ \gamma x - J ^ A _ \gamma y ) ) } \\ & = \norm { s \gamma ^ { - 1 } ( x - y ) + ( 1 - s \gamma ^ { - 1 } ) ( J ^ A _ \gamma x - J ^ A _ \gamma y ) } \end{align*}


\begin{align*} \big \{ ( x , x ^ \prime , y , y ^ \prime ) \ \ y ^ \prime = f ^ \prime ( x ^ \prime ) \big \} \approx \Gamma _ { f ^ \prime } \times \R ^ { n } \times \R ^ { m } . \end{align*}


\begin{align*} \sum _ { \substack { c > C , d \geq 1 \\ c d = q } } \mu ( c ) = \sum _ { c | q } \mu ( c ) - \sum _ { \substack { c \le C , d \geq 1 \\ c d = q } } \mu ( c ) = \left \lfloor \frac { 1 } { q } \right \rfloor - \sum _ { \substack { c \le C , d \geq 1 \\ c d = q } } \mu ( c ) . \end{align*}


\begin{align*} L _ t ^ h = \exp \left ( M _ t - \frac { \ , 1 \ , } { 2 } \ < M ^ c \ > _ t \right ) \prod _ { 0 < s \le t } \frac { h ( X _ s ) } { h ( X _ { s - } ) } \exp \left ( 1 - \frac { h ( X _ s ) } { h ( X _ { s - } ) } \right ) , \end{align*}


\begin{align*} u _ k : = \frac { x _ k - \bar x } { \norm { x _ k - \bar x } } , v _ k : = \frac { y _ k - \bar y } { \norm { y _ k - \bar y } } , y _ k ^ * : = \frac { \norm { y _ k - \bar y } } { \norm { x _ k - \bar x } } \lambda _ k \end{align*}


\begin{align*} B ^ { i j } = \left ( \begin{array} { c c c } 0 & 0 & 0 \\ 0 & B _ { 0 } ^ { i j } & 0 \\ 0 & 0 & 0 \\ \end{array} \right ) \in \mathbb { M } _ { N \times N } , \end{align*}


\begin{align*} \bar { \rho } \circ d _ { \Sigma } = d _ { \Sigma _ 0 } \circ \bar { \rho } , \end{align*}


\begin{align*} \dfrac { p _ \alpha ( v ) } { p _ { \alpha } ( x ) ^ 2 } \dfrac { 1 } { ( 1 - e ^ { - \alpha t } ) ^ { \frac { d } { \alpha } } } D ^ { \alpha - 1 } ( p _ \alpha ) ( x ) p _ \alpha \left ( \dfrac { x - v e ^ { - t } } { ( 1 - e ^ { - \alpha t } ) ^ { \frac { 1 } { \alpha } } } \right ) = \dfrac { p _ \alpha ( v ) } { p _ { \alpha } ( x ) } \dfrac { ( - x ) } { ( 1 - e ^ { - \alpha t } ) ^ { \frac { d } { \alpha } } } p _ \alpha \left ( \dfrac { x - v e ^ { - t } } { ( 1 - e ^ { - \alpha t } ) ^ { \frac { 1 } { \alpha } } } \right ) , \end{align*}


\begin{align*} W ( n , k + 1 ) = M ( n , k ) . \end{align*}


\begin{align*} & \partial _ t \rho _ i = \nabla \cdot \left [ D _ i ( x ) \left ( \nabla \rho _ i + z _ i \rho _ i \nabla \phi \right ) \right ] , i = 1 , \cdots , s , \\ & - \nabla \cdot ( \epsilon ( x ) \nabla \phi ) = f ( x ) + \sum _ { i = 1 } ^ { s } z _ i \rho _ i . \end{align*}


\begin{align*} R H _ \infty \subsetneq \bigcap _ { s > 1 } R H _ s \subset \bigcup _ { s > 1 } R H _ s = A _ \infty . \end{align*}


\begin{align*} g = 2 \alpha , \end{align*}


\begin{align*} \frac { y _ k } { \norm { x _ k - \bar x } } = \frac { g ( x _ k ) - z _ k } { \norm { x _ k - \bar x } } = \frac { g ( x _ k ) - g ( \bar x ) } { \norm { x _ k - \bar x } } - \frac { z _ k - g ( \bar x ) } { \norm { x _ k - \bar x } } \to v - v = 0 , \end{align*}


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


\begin{align*} \lambda _ { I J K } = \sum _ { i \in I } \sum _ { j \in J } \lambda _ { i j k } \end{align*}


\begin{align*} \mathcal { B C } _ { ( s _ 1 , s _ 2 , \dots , s _ p ) } = \mathcal { C S } _ { ( s _ 1 , s _ 2 , \dots , s _ p ) } = \mathcal { D D } _ { ( s _ 1 , s _ 2 , \dots , s _ p ) } . \end{align*}


\begin{align*} \rho _ c ( x ) = \frac { \left \langle c , z - x \right \rangle } { \| c \| } . \end{align*}


\begin{align*} \Upsilon _ { \omega } ^ { \sigma ( + ) } = \left \{ \tau ^ { 2 \cdot \varepsilon _ { 2 } } - 1 , C ^ { - 1 } \right \} , \Upsilon _ { \omega } ^ { \sigma ( - ) } = \left \{ \tau - 1 , - \tau ^ { \varepsilon _ { 1 } } - 1 \right \} . \end{align*}


\begin{align*} \partial _ t ^ 2 \phi - \partial _ x ^ 2 \phi + \phi + V ( x ) \phi \pm \phi ^ 3 = 0 , ( u , u _ t ) ( 0 ) = ( u _ 0 , u _ 1 ) , \end{align*}


\begin{align*} \left ( \sum \limits _ { j = 1 } ^ k \norm { f _ j } \right ) ^ 2 & \leq k \sum \limits _ { j = 1 } ^ k \norm { f _ j } ^ 2 \\ & \leq k \langle f , f \rangle - k \sum \limits _ { i \neq j \neq 0 } \langle f _ i , f _ j \rangle \\ & \leq k \langle f , f \rangle + c \gamma \sum \limits _ { i \neq j \neq 0 } \norm { f _ i } \norm { f _ j } \\ & \leq k \langle f , f \rangle + c \gamma \left ( \sum \limits _ { j = 1 } ^ k \norm { f _ j } \right ) ^ 2 \end{align*}


\begin{align*} & \frac { \rho ^ { n + 1 } _ i - \rho ^ n _ i } { \tau } = \nabla \cdot \left ( D _ i ( \nabla \rho ^ { n + 1 } _ i + z _ i \nabla \phi ^ { n + 1 } ) \right ) + O ( \tau ) , \\ & \nabla \cdot ( \epsilon ( x ) \nabla \phi ^ { n + 1 } ) = f + \sum _ { i = 1 } ^ s z _ i \rho _ i ^ { n + 1 } . \end{align*}


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


\begin{align*} \Pi ( S ) = C ( \Sigma ) . \end{align*}


\begin{align*} e ^ { e _ 0 t } \alpha _ + ( t ) = \alpha _ + ( 0 ) + \int _ 0 ^ t e ^ { e _ 0 s } B ( R , \mathcal { Y } _ - ) d s . \end{align*}


\begin{align*} \langle T _ x \ell , f \rangle = \langle \ell , T _ { - x } f \rangle \langle M _ \omega \ell , f \rangle = \langle \ell , M _ { - \omega } f \rangle . \end{align*}


\begin{align*} L _ + = - \Delta + 1 - p \left ( | \cdot | ^ { - ( N - \gamma ) } * \left ( Q ^ { p - 1 } \cdot \right ) \right ) Q ^ { p - 1 } - ( p - 1 ) \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ { p - 2 } , \end{align*}


\begin{align*} V ^ \nu _ { 2 , p } ( x ) = \int _ M g _ \lambda ( x , y ) \left ( \int _ M g _ \lambda ( z , y ) \nu ( d z ) \right ) ^ { q - 1 } \mu ( d y ) , x \in M , \end{align*}


\begin{align*} q ' ( x ) = 1 - \frac { e } { \frac { e } { 2 } x + e + \frac { 1 } { 2 } } > 0 \end{align*}


\begin{align*} r _ 2 = \frac { 1 } { 3 } ( - \lambda + 4 i l \pi - 2 i m \pi ) , \ \ \ r _ 3 = \frac { 1 } { 3 } ( - \lambda - 2 i l \pi + 4 i m \pi ) . \end{align*}


\begin{align*} \tilde { y } _ { k , n } [ t ] = \sum _ { m = 1 } ^ M \tilde { g } _ { m k , n } [ t ] \tilde { x } _ { m , n } [ t ] + \tilde { z } _ { k , n } [ t ] , \ : \ : k \in \{ 1 , \ldots , K \} . \end{align*}


\begin{gather*} q _ { 1 } = \theta _ { 1 } \beta + \beta K + \theta _ { 1 } ^ { 1 - \beta } | b ( 0 ) | ^ { \beta } + \frac { C _ { d , \alpha } \beta ( 3 - \beta ) \sqrt { d } \sigma _ { d - 1 } } { 2 ( 2 - \alpha ) } + \frac { C _ { d , \alpha } \beta \sigma _ { d - 1 } } { \alpha - \beta } + \left ( \frac { \theta _ { 1 } } { 4 } \right ) ^ { 1 - \beta } \left ( \frac { C _ { d , \alpha } \sigma _ { d - 1 } } { \alpha - 1 } \right ) ^ { \beta } , \end{gather*}


\begin{align*} x ^ 3 = \gamma _ 1 ( x ) x ^ 2 + \gamma _ 2 ( x ) x \end{align*}


\begin{align*} \iota ( z _ 1 , z _ 2 ) = \end{align*}


\begin{align*} \quad \psi '' ( \eta ) = \tfrac { 1 } { n ^ 2 } f '' ( \tfrac { \eta } { n } ) - f '' ( \eta ) , \eta \in ( \eta _ n / 2 , 2 \eta _ n ) . \end{align*}


\begin{align*} \mathcal { M } _ { k } ^ { J } ( \alpha , \beta ) : = \{ \ , u \in \mathcal { M } ^ { J } ( \alpha , \beta ) \ , | \ , I ( u ) = k \ , \ , \} \end{align*}


\begin{align*} I ( \omega ) = e ^ { - \frac { \pi \omega ^ 2 } { M } + C } = C _ 1 \ , e ^ { - \frac { \pi \omega ^ 2 } { M } } . \end{align*}


\begin{align*} \mu _ { \alpha , \infty } = \prod _ { k = 1 } ^ d \left ( \int _ 0 ^ \infty \left ( \frac { s } { \pi } \right ) ^ { \alpha _ k } e ^ { - s } \ , d s \right ) = \prod _ { k = 1 } ^ d \frac { \Gamma ( \alpha _ k ) } { \pi ^ { \alpha _ k } } = \prod _ { k = 1 } ^ d \frac { \alpha _ k ! } { \pi ^ { \alpha _ k } } = \frac { \alpha ! } { \pi ^ { | \alpha | } } . \end{align*}


\begin{align*} \sum _ { l , k } ( \partial ^ 2 _ { x _ l x _ k } f _ \alpha ) ( \partial _ { t _ j } \varphi ^ l ) ( \partial _ { t _ i } \varphi ^ k ) + \sum _ { l } ( \partial _ { x _ l } f _ \alpha ) ( \partial ^ 2 _ { t _ i t _ j } \varphi ^ l ) = 0 . \end{align*}


\begin{align*} - \Delta u = { } & f ( u ) , \textrm { i n } \Omega , \\ u = { } & 0 , \textrm { o n } \partial \Omega . \end{align*}


\begin{align*} S ( e , z ^ * ) = [ ( Q \widetilde { T } ) ^ * ( z ^ * ) ] ( e \otimes \cdot ) \ , \ , ( e , z ^ * ) \in E \times Q ( X ) ^ * . \end{align*}


\begin{align*} \frac { \tau ( n ) } { \tau ( n ) \log n } = \frac { 1 } { \log n } . \end{align*}


\begin{gather*} \omega ( x _ i ^ \pm ( u ) ) = x _ { i } ^ \mp ( u ) , \omega ( h _ i ( u ) ) = h _ i ( u ) \\ \varsigma ( x _ i ^ \pm ( u ) ) = x _ i ^ \pm ( - u ) , \varsigma ( h _ i ( u ) ) = h _ i ( - u ) \end{gather*}


\begin{align*} h \rhd ( a \star b ) = ( h _ { \widehat { ( 1 ) } } \rhd a ) \star ( h _ { \widehat { ( 2 ) } } \rhd b ) , \end{align*}


\begin{align*} \tfrac { 1 } { n } A ( \tfrac { \eta } { n } , \tfrac { 1 } { \eta } ) = \tanh ( \tfrac { \eta } { 2 } ) \ , \psi ( \eta ) . \end{align*}


\begin{align*} \langle \Phi ^ P _ { t } , \operatorname { I n d } ( D ) \rangle = \frac { \sum _ { w \in W _ K } ( - 1 ) ^ w e ^ { w \cdot \mu } ( t ) } { \Delta ^ { M } _ T ( t ) } . \end{align*}


\begin{align*} \int _ { \{ q \Psi < - t \} } | f | ^ 2 e ^ { - \varphi _ 0 } \geq e ^ { - t } C ( q \Psi , \varphi _ 0 , J _ q , f ) = e ^ { - t } C ( \Psi , \varphi _ 0 , J _ q , f ) . \end{align*}


\begin{align*} | \langle T \chi _ 0 , \psi \rangle | & = | - \langle T \chi _ 1 , \psi \rangle | \\ & = \Big | \int _ { \R ^ N } \int _ { \R ^ N } [ K ( x , y ) - K ( x _ 0 , y ) ] \chi _ 1 ( y ) \psi ( x ) d \omega ( y ) d \omega ( x ) \Big | . \end{align*}


\begin{align*} w _ M ( 2 e _ 1 ) & = - C M \int _ { \R ^ n \setminus B _ 1 ( 2 e _ 1 ) } \frac { \zeta _ 2 ( y ) } { ( \vert y \vert ^ 2 - 1 ) ^ s \vert y \vert ^ n } \dd y . \end{align*}


\begin{align*} \omega ( z ) = z - \frac { | \lambda | ^ 2 } { z + H _ 1 ( \omega ( z ) ) } \end{align*}


\begin{align*} I ( v ) : = \int _ { \overline { \Omega } } | \nabla v | ^ { p } \ , d \mu - p \int _ { \partial { \Omega } } v f \ , d \nu \end{align*}


\begin{align*} \Box \tilde \phi _ { k + 1 } = \left ( | \tilde \phi _ { k + 1 , t } | ^ 2 - | \tilde \phi _ { k + 1 , x } | ^ 2 \right ) \tilde \phi _ { k + 1 } + \mathbf { 1 } _ { \omega } f _ { k } ^ { \tilde \phi ^ { \perp } _ { k + 1 } } + e _ k , \ ; \tilde \phi _ { k + 1 } [ 0 ] = ( a , b ) \end{align*}


\begin{align*} \sum _ { k \in \Z ^ d } f ( k - x ) e ^ { 2 \pi i \omega ( k - x ) } = \sum _ { l \in \Z ^ d } \widehat { f } ( l - \omega ) e ^ { - 2 \pi i x \cdot l } . \end{align*}


\begin{align*} \begin{aligned} \phi ( x + h ) - 2 \phi ( x ) + \phi ( x - h ) & \leq u ( x + h ) - 2 u ( x ) + u ( x - h ) \\ & \leq C \left ( 1 + \frac { 2 } { C _ 0 } \right ) | h | ^ 2 \end{aligned} \end{align*}


\begin{align*} 1 - F _ N ( x s _ N ) = \exp \left ( - \frac { 1 } { 2 } x ^ 2 Q _ N ( x ) \right ) \left ( 1 - \Phi ( x ) + \vartheta \lambda _ N e ^ { - x ^ 2 / 2 } \right ) . \end{align*}


\begin{align*} V ( r , p ) = \beta r ^ { - 1 } | p | ^ 2 + a ( 1 - \beta ) ^ { \alpha - 1 } r ^ { \beta ( 1 - \alpha ) } | p | ^ \alpha \end{align*}


\begin{align*} x _ 1 ^ { \alpha _ 1 } ( 1 - x _ 1 ) ^ { \beta _ 1 } \cdots x _ n ^ { \alpha _ n } ( 1 - x _ n ) ^ { \beta _ n } , \sum _ { i = 1 } ^ n ( \alpha _ i + \beta _ i ) = \delta , \end{align*}


\begin{align*} \phi ( u , v ) = \int _ 0 ^ u \psi ( - u ' ) \ , \mathrm { d } u ' . \end{align*}


\begin{align*} I _ 2 ( u ) = \int _ { \R } u ^ 2 ( x , t ) d x \end{align*}


\begin{align*} ( v _ 1 * \ldots * v _ k ) . u = v _ 1 ( \ldots ( v _ k u ) \ldots ) , \mbox { f o r a l l } v _ 1 , \ldots , v _ k \in V \mbox { a n d } u \in L ( A ) . \end{align*}


\begin{align*} \forall x ^ X , y ^ X , { x ' } ^ X , { y ' } ^ { X } \left ( x = _ X x ' \land y = _ X y ' \rightarrow \chi _ A x y = _ 0 \chi _ A x ' y ' \right ) . \end{align*}


\begin{align*} L i _ { K , \mathfrak { s } } ( { \bf z } ) : = \sum _ { i _ 1 > \cdots > i _ r > 0 } \frac { z _ 1 ^ { i _ 1 } \cdots z _ r ^ { i _ r } } { ( \theta ^ { q ^ { i _ 1 } } - \theta ) ^ { s _ 1 } \cdots ( \theta ^ { q ^ { i _ r } } - \theta ) ^ { s _ r } } \in \mathbb { C } _ { \infty } . \end{align*}


\begin{align*} \inf _ { ( \rho , m ) \in C _ F ( \rho ^ 0 , \rho ^ 1 ) } \{ \mathcal A ( \rho , m ) \} = \sup _ { S \in H ^ 1 } \inf _ { ( \rho , m ) } \mathcal L ( \rho , m , S ) \end{align*}


\begin{align*} \begin{aligned} \bar x _ t & = \alpha _ t x _ t + ( 1 - \alpha _ t ) \bar x _ { t - 1 } \\ x _ { t + 1 } & = x _ t - \alpha _ t \eta _ t g _ t , \end{aligned} \end{align*}


\begin{align*} \dot { H } ^ \alpha = \big \{ v \in \dot { H } : | v | _ \alpha < \infty \big \} , H ^ \alpha = \big \{ v \in H : \| v \| _ \alpha < \infty \big \} , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { n } \frac { \partial f } { \partial \kappa _ { i } } \kappa _ { i } ^ { 2 } \geq H _ { 1 } H _ { k } ^ { \frac { 1 } { k } } \geq H _ { k } ^ { \frac { 2 } { k } } = f ^ { 2 } . \end{align*}


\begin{align*} Z \varphi _ s ( x , - \omega ) = \sum _ { k \in \Z } e ^ { - \pi s ( x + k ) ^ 2 } e ^ { 2 \pi i k \omega } . \end{align*}


\begin{align*} S _ { 2 , \lambda } ( n + 1 , k ) = S _ { 2 , \lambda } ( n , k - 1 ) + ( k - n \lambda ) S _ { 2 , \lambda } ( n , k ) , ( \mathrm { S e e } \ [ 4 ] ) , \end{align*}


\begin{align*} \mathcal N _ Q ( \bar x ; u ) : = \left \{ \eta \in \mathbb X \ , \middle | \ , \begin{aligned} & \exists \{ u _ k \} _ { k \in \N } \subset \mathbb X , \ , \exists \{ t _ k \} _ { k \in \N } \subset \R _ + , \ , \exists \{ \eta _ k \} _ { k \in \N } \subset \mathbb X \colon \\ & u _ k \to u , \ , t _ k \searrow 0 , \ , \eta _ k \to \eta , \ , \eta _ k \in \widehat { \mathcal N } _ Q ( \bar x + t _ k u _ k ) \ , \forall k \in \N \end{aligned} \right \} \end{align*}


\begin{align*} \begin{aligned} & { 1 \over q } ( \lambda h _ q + ( 1 - \lambda ) k _ q ) + w _ q ( x _ q , t _ q ) ^ { q - 1 } \mu ( w _ q ( x _ q , t _ q ) ) \\ & \geq - \lambda v ( y _ q , t _ q ) ^ { 1 - { 1 \over q } } \omega ( | X _ q | ) - ( 1 - \lambda ) v ( z _ q , t _ q ) ^ { 1 - { 1 \over q } } \omega ( | X _ q | ) , \end{aligned} \end{align*}


\begin{align*} \tilde M _ n ( k ) = \sum \limits _ { \substack { A \subset \{ 1 , \dots , d \} \\ | A | = k } } \tilde M _ { n , A } , \tilde N _ n ( k ) = \sum \limits _ { \substack { A \subset \{ 1 , \dots , d \} \\ | A | = k } } \tilde N _ { n , A } . \end{align*}


\begin{align*} \forall \gamma ^ 1 , x ^ X , p ^ X \left ( \gamma > _ \mathbb { R } 0 \land p = _ X J ^ A _ \gamma x \rightarrow \gamma ^ { - 1 } ( x - _ X p ) \in A p \right ) . \end{align*}


\begin{align*} \left | \frac { 1 } { \mathbb { E } [ f ] } ( D ^ k _ i f ) \right | = \left | \underset { P _ i } { \mathbb { E } } [ D ^ k _ i f ] - \mathbb { E } [ f ] \right | \leq \| D ^ k _ i f - \mathbb { E } [ f ] \| _ { \infty } . \end{align*}


\begin{align*} \sum _ { \rho \in R _ \epsilon } ( \rho ) = - \frac { \epsilon T } { 2 \pi } \log \pi + \frac { \epsilon T } { 2 \pi } \log \left ( \frac { T } { 2 } \right ) - \frac { \epsilon T } { 2 \pi } + \frac { 1 } { 2 \pi } \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \arg \zeta ( \sigma + i T ) \ d \sigma + \left ( \frac { \epsilon } { \pi } \right ) \mathcal { O } ( \log T ) \end{align*}


\begin{align*} \Phi _ 1 ( h _ 1 ) \ge c \| h _ 1 \| _ 2 ^ 2 , \forall \ ; h _ 1 \int \left ( \partial _ { x _ j } Q \right ) h _ 1 = \int ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p ) Q ^ { p - 1 } h _ 1 = 0 , \forall 1 \le j \le N . \end{align*}


\begin{align*} \bar { \Delta } _ { \rm R e a c t i v e } = \frac { \mathbb { E } \sum _ { t \in \mathcal { I } _ j } \Delta { \left ( t \right ) } } { \mathbb { E } T ^ { \rm R e a c } _ j } = \frac { \mathbb { E } T ^ { \rm P r o a c } _ j } { 2 \mathbb { E } \left ( T ^ { \rm R e a c } _ j \right ) ^ 2 } + \mathbb { E } \tau ^ { \rm R e a c } _ { V _ j } - \tau _ { \rm f } - \frac { 1 } { 2 } . \end{align*}


\begin{align*} F ( z _ 1 , z _ 2 ) = ( \alpha z _ 1 + t z _ 2 ^ m , \beta z _ 2 ) , 0 < | \alpha | < | \beta | < 1 , t ( \alpha - \beta ^ m ) = 0 , \ m \ge 1 . \end{align*}


\begin{align*} \eta _ m ^ { j , k } : = \frac { 1 } { \delta _ m } \big ( B ^ j ( k \delta _ m ) - B ^ j ( ( k - 1 ) \delta _ m ) \big ) , j = 1 , \ldots , r k = 1 , \ldots , m \end{align*}


\begin{align*} s - t = \frac { 1 - \sqrt { 1 - 4 | \lambda | ^ 2 h ( s ) ^ 2 } } { 2 h ( s ) } . \end{align*}


\begin{align*} 0 = \langle J ^ { ' } ( \widehat { w } ) , \widehat { w } \rangle + \lambda _ + \langle \varphi ^ { * } _ { \widehat { w } ^ + } , \widehat { w } \rangle + \lambda _ - \langle \varphi ^ { * } _ { \widehat { w } ^ - } , \widehat { w } \rangle = \lambda _ + \langle \varphi ^ { * } _ { \widehat { w } ^ + } , \widehat { w } \rangle + \lambda _ - \langle \varphi ^ { * } _ { \widehat { w } ^ - } , \widehat { w } \rangle , \end{align*}


\begin{align*} f _ { \zeta } ( x ) = \begin{cases} \eta \circ \xi ( x ) & x \in [ \eta ( 0 ) , 0 ] , \\ \eta ( x ) & x \in [ 0 , \xi \circ \eta ( 0 ) ] \end{cases} \end{align*}


\begin{align*} f ( x ) - f ( y _ 1 ) & \geq d ( x , y _ 1 ) - \epsilon r \\ & = r - \epsilon r \\ & = r ( 1 - \epsilon ) \\ & > 0 . \end{align*}


\begin{align*} S p e c ( U D ) \in \{ 1 \} \cup \bigcup \limits _ { i = 1 } ^ k [ \lambda _ i ( U D ) + O _ k ( \gamma ) , \lambda _ i ( U D ) - O _ k ( \gamma ) ] , \end{align*}


\begin{align*} q ( x , t ) = 2 i D ^ { - 2 } _ { \infty } ( \xi ) \left [ E _ { 1 } ^ { ( 1 2 ) } + \lim _ { k \rightarrow \infty } k \Delta _ { L } ^ { ( 1 2 ) } ( k ) \right ] , \end{align*}


\begin{align*} u ( x ) + | u ' ( x ) | ^ p - f ( x ) = 0 \end{align*}


\begin{align*} \mu _ { T } \{ \lambda \in \mathbb { C } : | \lambda | \leq r \} = \begin{cases} 0 , & r < \lambda _ 1 ( \mu _ { | T | } ) ; \\ 1 + S ^ { \langle - 1 \rangle } _ { \mu _ { T ^ * T } } ( r ^ { - 2 } ) , & \lambda _ 1 ( \mu _ { | T | } ) < r < \lambda _ 2 ( \mu _ { | T | } ) ; \\ 1 , & r \geq \lambda _ 2 ( \mu _ { | T | } ) . \end{cases} \end{align*}


\begin{align*} u _ { \lambda _ 0 , t _ 0 , x _ 0 } ( t , x ) = \frac { 1 } { \lambda _ 0 ^ { 1 / ( p - 1 ) } } u \bigg ( \frac { t - t _ 0 } { \lambda _ 0 ^ 2 } , \frac { x - x _ 0 } { \lambda _ 0 } \bigg ) . \end{align*}


\begin{align*} S ( a , b , 2 f ) = s ( a , b , 2 f ) \end{align*}


\begin{align*} [ a , q * w ] _ + \in [ \Gamma _ { [ c - k ] } ( B ) , \Gamma _ { k } ( B ) ] _ + = 0 . \end{align*}


\begin{align*} P _ 0 = \frac { n _ 0 } { \gamma } , e _ 0 = n _ 0 \left ( \frac { K _ 3 ( \gamma ) } { K _ 2 ( \gamma ) } - \frac { 1 } { \gamma } \right ) = n _ 0 \left ( \frac { K _ 1 ( \gamma ) } { K _ 2 ( \gamma ) } + \frac { 3 } { \gamma } \right ) , \end{align*}


\begin{align*} \Gamma ( F ( f _ 1 , . . . , f _ n ) , g ) = \sum _ { i = 1 } ^ n \frac { \partial F } { \partial x _ i } ( f _ 1 , . . . , f _ n ) \Gamma ( f _ i , g ) \end{align*}


\begin{align*} \mathcal { P } f ( t ) = \mathcal { P } _ 1 f ( t ) . \end{align*}


\begin{align*} \mathcal { P } ( d , k ) = \left \{ \mathbf p _ k = ( p _ 1 , \dots , p _ k ) \in \{ 1 , \dots , d \} ^ k : p _ 1 < \cdots < p _ k \right \} . \end{align*}


\begin{align*} | T ( a ) ( x ) | & = \bigg | \int _ Q K ( x , y ) a ( y ) d \omega ( y ) \bigg | = \bigg | \int _ Q [ K ( x , y ) - k ( x , x _ Q ) ] a ( y ) d \omega ( y ) \bigg | \\ & \leqslant C \Big ( \frac { \l ( Q ) } { \| x - x _ Q \| } \Big ) ^ \varepsilon \frac { 1 } { \omega ( x , d ( x , x _ Q ) ) } \| a \| _ 1 . \end{align*}


\begin{align*} & \inf _ { \rho , v } \ ; [ \int _ 0 ^ 1 \frac 1 2 \ < v _ t , v _ t \ > _ { \theta ( \rho _ t ) } d t ] \\ & \ ; \ ; d \rho ( t ) + d i v _ G ^ { \theta } ( \rho ( t ) v ( t ) ) + d i v _ G ^ { \theta } ( \rho ( t ) \nabla _ G \Sigma ) \circ d W _ t = 0 \\ & \ ; \ ; \rho ( 0 , \omega ) = \rho _ a , \ ; \rho ( 1 , \omega ) = \rho _ b , \end{align*}


\begin{align*} \tau _ K = \tau _ K ( L ) = \inf \left \{ t \geq 0 : \exists i \in \{ 0 , 1 , \cdots , { 1 \over \delta _ K } - 1 \} , \ , | \beta ^ K _ { i + 1 } ( t ) - \beta ^ K _ i ( t ) | > L \delta _ K \right \} , \end{align*}


\begin{align*} [ \bar { b } _ n \mapsto \bar { a } _ n ] ( G ) \cup [ \bar { b } _ n \mapsto \bar { a } _ n ] ( H ) \cup [ \bar { b } _ n \mapsto \bar { a } _ n ] ( I ) & = [ \bar { b } _ n \mapsto \bar { a } _ n ] ( G \cup H \cup I ) \\ & = [ \bar { b } _ n \mapsto \bar { a } _ n ] ( \mathbb { N } ) = \{ \bar { a } _ n \} \end{align*}


\begin{align*} \alpha = C _ k = c _ k = C _ \Phi = c _ \Phi = 1 . \end{align*}


\begin{align*} \| f \| _ { X ^ a _ { \xi _ 0 } } = \sup _ { \xi \in [ 0 , \xi _ 0 ] } \sum _ { i = 0 } ^ 2 \frac { \left | ( \xi \partial _ \xi ) ^ i f ( \xi ) \right | } { \langle \xi \rangle ^ { a } } , \end{align*}


\begin{align*} ( T _ 2 ( t ) - T _ 1 ( t ) ) f ) ( x _ n ) = \langle T _ 2 ( t ) f - T _ 1 ( t ) f , \delta _ { x _ n } \rangle \geq 0 \end{align*}


\begin{align*} N _ t ^ { h _ n } : = - \int _ 0 ^ t h _ n ( X _ s ) { \rm d } A _ s ^ { \overline { \mu } ^ n } , M _ t ^ { h _ n } : = h _ n ( X _ t ) - h _ n ( X _ 0 ) - N _ t ^ { h _ n } . \end{align*}


\begin{align*} - \triangle _ { g _ { \textnormal { p o i n } , j } } ( u _ { j } ) = e ^ { 2 u _ { j } } \ , K _ { \tilde F _ { t _ { j } } ^ * ( g _ { \textnormal { e u c } } ) } \textnormal { o n } \ , \ , \ , \Sigma . \end{align*}


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


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


\begin{align*} S & \lesssim \sum \limits _ { j = 1 } ^ \infty \int _ { t 2 ^ { j - 1 } \leqslant d ( x , y ) < t 2 ^ j } \frac 1 { \omega ( x , t + d ( x , y ) ) } \bigg ( \frac { t } { t + d ( x , y ) } \bigg ) ^ { \varepsilon } d \omega ( x ) \\ & + \int _ { d ( x , y ) < t } \frac 1 { \omega ( x , t + d ( x , y ) ) } d \omega ( x ) . \end{align*}


\begin{align*} \left \{ \begin{array} [ c ] { l } u \in C \left ( \left [ 0 , T \right ] , \mathcal { H } _ { s } \right ) \cap C ^ { 1 } \left ( \left [ 0 , T \right ] , \mathcal { H } _ { s } \right ) ; \\ \\ u _ { t } = - \gamma \boldsymbol { D } ^ { \alpha } u - \beta u , t \in \left [ 0 , T \right ] ; \\ \\ u ( x , 0 ) = f _ { 0 } ( x ) \in \mathcal { H } _ { s } , \end{array} \right . \end{align*}


\begin{align*} \langle f , S ^ { - 1 } f \rangle = \sum _ { \gamma \in \Gamma } a _ \gamma \langle e _ \gamma , S ^ { - 1 } f \rangle = \sum _ { \gamma \in \Gamma } | a _ \gamma | ^ 2 . \end{align*}


\begin{align*} X ( v \otimes a b ) = ( X ( v \otimes a ) ) ( 1 \otimes b ) , v \in V , a , b \in A . \end{align*}


\begin{align*} \lim _ { N \to \infty } \mu ^ { v _ N } _ { B _ N + \theta A _ N } = \left ( \lim _ { N \to \infty } \mu ^ { v _ N } _ { \theta A _ N } \right ) \rhd \left ( \lim _ { N \to \infty } \mu ^ { v _ N } _ { B _ N } \right ) = \delta _ \theta \rhd \mu _ { s c } . \end{align*}


\begin{align*} \det M _ { t } = \frac { \left [ ( 1 - t ) + ( 2 + 4 t ) \left \vert w \right \vert ^ { 2 } + ( 1 - t ) \left \vert w \right \vert ^ { 4 } \right ] ^ 2 } { \left ( 1 + \left \vert w \right \vert ^ { 2 } \right ) ^ { 8 } } . \end{align*}


\begin{align*} ( n - 1 ) r ^ 2 & = \theta r \\ \sum _ k S _ { i , k } & = 0 \\ S _ { i , j } ^ 2 + \sum _ k S _ { i , k } S _ { k , j } & = \theta S _ { i , j } i < j . \end{align*}


\begin{align*} \mu _ { R , 2 } ( k , \ell , m , n ) : = \int \overline { \mathcal { K } _ S ( x , k ) } \mathcal { K } _ S ( x , \ell ) \overline { \mathcal { K } _ S ( x , m ) } \mathcal { K } _ S ( x , n ) \ , d x - \mu _ { S } ( k , \ell , m , n ) . \end{align*}


\begin{align*} x \circ y = \alpha _ 3 ^ { - 1 } ( \alpha _ 1 x \ast \alpha _ 2 y ) \end{align*}


\begin{align*} \theta _ { \rho } ( x , y ) = \rho ( y ) \rho ( x ) . \end{align*}


\begin{align*} \max \{ n _ { \max } , m _ { \max } \} = O \left ( \sqrt { R _ n ^ { - 1 } ( \mathcal { N } _ n ^ 2 + \mathcal { M } _ n ^ 2 ) } \right ) . \end{align*}


\begin{align*} T = ( 1 - \alpha ) I d + \alpha N \end{align*}


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


\begin{align*} M ( f , H ) : = { 1 \over \# X _ f ^ - ( H ) } \sum _ { \chi \in X _ f ^ - ( H ) } \vert L ( 1 , \chi ) \vert ^ 2 . \end{align*}


\begin{align*} \sum _ i c _ i d _ i = a , \sum _ j e _ j f _ j = b . \end{align*}


\begin{align*} \lim _ { i \rightarrow \infty } \mathrm { A r e a } ( \Gamma _ i ) = + \infty \ , , \lim _ { i \rightarrow \infty } \mathrm { i n d e x } ( \Gamma _ i ) = + \infty \ , . \end{align*}


\begin{align*} \tilde { \phi } ^ { n + 1 } = \frac { ( \tau \eta ^ n - 2 ) \phi ^ { n - 1 } + 4 \phi ^ { n } + 2 \tau ^ 2 ( - G ( \phi ^ n ) + \lambda ^ n \phi ^ n ) } { 2 + \tau \eta ^ n } \end{align*}


\begin{align*} \varphi ( k , l ) = \varphi ( k , 0 ) + \kappa ( 0 , l ) = \left ( \varphi ( 0 , 0 ) + \kappa ( k , 0 ) \right ) + \kappa ( 0 , l ) . \end{align*}


\begin{align*} \Lambda = \left ( \begin{array} { c c c } \mathbb { I } _ { n \times n } & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \mathbb { I } _ { p \times p } \end{array} \right ) \end{align*}


\begin{align*} D ^ { \alpha - 1 } ( p _ \alpha ) ( x ) = - x p _ \alpha ( x ) , ( D ^ { \alpha - 1 } ) ^ * ( p _ \alpha ) ( x ) = x p _ \alpha ( x ) . \end{align*}


\begin{align*} \nabla _ { i } h _ { j l } = \nabla _ { l } h _ { i j } + \bar { R } _ { \nu j l i } , \end{align*}


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


\begin{align*} { \begin{bmatrix} H + D & - A ^ T \\ A & 0 \end{bmatrix} } \begin{bmatrix} \Delta \mathbf { x } \\ \Delta \mathbf { y } \end{bmatrix} = \begin{bmatrix} f _ { \mathbf { x } } \\ f _ { \mathbf { y } } \end{bmatrix} , \end{align*}


\begin{align*} s ( b , c , d ) + s ( d , b , c ) + s ( c , d , b ) = { b ^ 2 + c ^ 2 + d ^ 2 - 3 \vert b c d \vert \over 1 2 b c d } . \end{align*}


\begin{align*} \left \{ \begin{pmatrix} B & A \\ - A ^ * & C \end{pmatrix} \mid B ^ * = - B , C ^ * = - C \right \} . \end{align*}


\begin{align*} \Sigma _ 1 \ll p ^ { ( \frac { 3 } { 2 } - \epsilon ) h _ p } \sum _ { m = 0 } ^ { k _ p - h _ p - 1 } \frac { p ^ { \frac { 1 } { 2 } - \epsilon } } { p ^ { m ( - \frac { 1 } { 2 } - \varepsilon + \epsilon ) } } \end{align*}


\begin{align*} \lambda \varphi ( b b ^ * ) = \varphi ( b b ^ * \delta _ { \varphi } ) = \varphi ( \nu \delta _ { \varphi } b b ^ * ) = \nu _ r \lambda \varphi ( b b ^ * ) . \end{align*}


\begin{align*} \sum _ { y < m \leq M } g ( m ) e ^ { 2 \pi i f ( m ) } = \int _ y ^ M \frac { 1 } { u ^ { \overline { s _ 1 } } ( u + n ) ^ { \overline { s _ 3 } } } d u + O \left ( \frac { 1 } { y ^ { \sigma _ 1 } ( y + n ) ^ { \sigma _ 3 } } \right ) . \end{align*}


\begin{align*} & \ ; [ T u + u , T v + v , T w + w ] _ { L \oplus V } = [ T u , T v , T w ] + \theta ( T v , T w ) u - \theta ( T u , T w ) v + D ( T u , T v ) w . \end{align*}


\begin{align*} F _ 2 ( x ) : = D ( T ^ \circ x , S x ) . \end{align*}


\begin{align*} S _ n = n ^ { 1 / 2 } + O ( n ^ { 1 1 / 4 0 + \varepsilon } ) . \end{align*}


\begin{align*} L _ { n + 1 } ( x , y ) = \sum _ { k = 0 } ^ n { n \choose k } L _ { k } ( x , y ) M _ { n - k } ( x , y ) , \end{align*}


\begin{align*} q _ \alpha ( t ) = \left ( e ^ { - \alpha t } \dfrac { ( 1 - e ^ { - \alpha t } ) ^ { \alpha - 1 } + e ^ { - \alpha t } } { ( 1 - e ^ { - \alpha t } ) ^ { \alpha - 1 } } \right ) ^ { \frac { 1 } { \alpha } } & = \dfrac { e ^ { - t } } { ( 1 - e ^ { - \alpha t } ) ^ { 1 - \frac { 1 } { \alpha } } } \left ( ( 1 - e ^ { - \alpha t } ) ^ { \alpha - 1 } + e ^ { - \alpha t } \right ) ^ { \frac { 1 } { \alpha } } . \end{align*}


\begin{align*} \Psi ^ { ( - 1 ) } = \Phi \Psi \end{align*}


\begin{align*} L ( s , \mathbf { f } , \chi ) = \Lambda _ { \mathfrak { n } } ( s , \chi ) D ( s , \mathbf { f } , \chi ) , \Lambda _ { \mathfrak { n } } ( s , \chi ) = \prod _ { i = 0 } ^ { n - 1 } L _ { \mathfrak { n } } ( 2 s - 2 i , \chi ^ 2 ) . \end{align*}


\begin{align*} \bar \pi = \{ \mathcal F \in \hat S : \{ \mathcal F \} \curlywedge \pi \} . \end{align*}


\begin{align*} & \forall x ^ X , { x ' } ^ X , \gamma ^ 1 , { \gamma ' } ^ 1 \big ( \gamma > _ \mathbb { R } 0 \land \tilde { \rho } > _ \mathbb { R } - \gamma \land x \in \mathrm { d o m } ( J ^ A _ \gamma ) \\ & \qquad \qquad \qquad \qquad \land \gamma = _ \mathbb { R } \gamma ' \land x = _ X x ' \rightarrow J ^ A _ \gamma x = _ X J ^ A _ { \gamma ' } x ' \big ) . \end{align*}


\begin{align*} H _ c = \left \{ x \in \mathbb { R } ^ n \middle | \left \langle c , x - z \right \rangle = 0 \right \} , \end{align*}


\begin{align*} \forall x ^ X , p ^ X \left ( x \in \mathrm { d o m } ( J ^ A _ 1 ) \land p = _ X J ^ A _ { 1 } x \rightarrow ( x - _ X p ) \in A p \right ) . \end{align*}


\begin{align*} \int _ { W ' } \langle a \nabla y ^ k , \nabla \phi \rangle \ ; d x = 0 \end{align*}


\begin{align*} a _ i & = ^ \# \{ ( \alpha , \beta ) \in S \cap { \Bbb Z } ^ 2 \mid \alpha = i \} \\ b _ i & = ^ \# \{ ( \alpha , \beta ) \in T \cap { \Bbb Z } ^ 2 \mid \alpha = i \} . \end{align*}


\begin{align*} \rho ^ 0 = \rho ^ { i n } , \rho ^ { n + 1 } = \arg \min _ { \rho \in [ \mathcal { P } ( \Omega ) ] ^ 2 } \left \{ \frac { 1 } { 2 \tau } d ^ 2 ( \rho ^ n , \rho ) + \mathcal { E } ( \rho ) \right \} . \end{align*}


\begin{align*} x _ 1 = 0 , \dotsc , \ x _ n = 0 , \ x _ i - x _ j = 0 \textrm { f o r } 1 \leq i < j \leq n . \end{align*}


\begin{align*} M ^ { ( 2 ) } _ n = 0 . \end{align*}


\begin{align*} \mathbb { E } \left [ \left \| \overline { \mathcal { M } } \left ( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } G _ { i } ^ { \prime } \mathcal { X } _ { i } \right ) \right \| _ { } \right ] \leq C _ { 9 } K \left ( \sqrt { \frac { p _ { 2 } \lceil \sqrt { p _ 3 } \rceil } { n } } + \sqrt { \frac { p _ { 1 } \lceil \sqrt { p _ 3 } \rceil } { n } } + \sqrt [ 4 ] { \frac { p _ { 1 } \lceil \sqrt { p _ 3 } \rceil ^ 2 p _ { 3 } } { n ^ 2 } } \right ) . \end{align*}


\begin{align*} \sum _ { z \leq p } \left ( \frac { 1 } { p } \sum _ { \gcd ( k , p - 1 ) = 1 } 1 \right ) \left ( \frac { 1 } { p } \sum _ { \gcd ( n , p - 1 ) = 1 } 1 \right ) = \left ( \frac { \varphi ( p - 1 ) } { p - 1 } \right ) ^ 2 p + O \left ( 1 \right ) . \end{align*}


\begin{align*} | \nabla ^ j _ \xi \frac { d ^ \ell } { d ^ \ell p } \hat { \psi } _ \xi ( p ) | \lesssim _ { j , l , M } \begin{cases} \frac { 1 } { 1 + | \xi | ^ { 2 + j } } \frac { 1 } { 1 + | p | ^ M } , & , \\ \frac { 1 } { 1 + | p | ^ M } & . \end{cases} \end{align*}


\begin{align*} d _ K ( \eta _ r ( t ; x ) ) = d _ K ( \xi ( r + t ; r , x ) ) \leq \Phi _ { \beta + L , 0 } ( d _ K ( x ) , \epsilon , t ) , ( t , x ) \in [ 0 , u - r ] \times X . \end{align*}


\begin{align*} \partial _ { t _ j } ( \Phi \circ \varphi ) = \sum _ { l } ( \partial _ { x _ l } \Phi ) ( \partial _ { t _ j } \varphi ^ { l } ) , \end{align*}


\begin{align*} \begin{aligned} \xi ^ { \prime } ( x ) + \eta ^ { \prime } ( x ) & = \lambda \xi ( x ) , x \in ( 0 , 1 ) , \\ \eta ^ { \prime \prime } ( x ) + \eta ^ { \prime } ( x ) + \xi ^ { \prime } ( x ) & = \lambda \eta ( x ) , x \in ( 0 , 1 ) , \\ \xi ( 0 ) & = \xi ( 1 ) , \\ \eta ( 0 ) = 0 , \ \ \eta ( 1 ) & = 0 . \end{aligned} \end{align*}


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


\begin{align*} \phi ( s ) = \prod _ { i \in 1 } ^ d \Bigl ( \sum _ { j \in 1 } ^ m \alpha _ { i j } s _ { i j } + \beta _ i \Bigr ) , \end{align*}


\begin{align*} \Big \{ \alpha = Q + i p \in \Sigma \ , | \ , \forall j \leq \ell , { \rm I m } \sqrt { p ^ 2 - 2 j } \in ( \beta / 2 - \gamma , \beta / 2 ) \Big \} \end{align*}


\begin{align*} T & = t _ 0 x ^ { \nu _ 2 ( t _ 0 ) } \\ & = \exp ( \theta \sqrt { \log x } ) \cdot \exp \left ( \frac { \log x } { 3 . 3 5 9 \theta \sqrt { \log x } } \left ( 1 - \frac { 8 . 0 2 \log ( \theta \sqrt { \log x } ) } { \theta \sqrt { \log x } } \right ) \right ) \\ & = \exp \left ( \sqrt { \log x } \left ( \theta + \frac { 1 } { 3 . 3 5 9 \theta } \left ( 1 - \frac { 8 . 0 2 \log ( \theta \sqrt { \log x } ) } { \theta \sqrt { \log x } } \right ) \right ) \right ) . \end{align*}


\begin{align*} \liminf _ { R \rightarrow \infty } M ^ 3 _ { \ , p , \ , q } ( R ) \leq \delta D ( \mathbf { u } ) \end{align*}


\begin{align*} M ( X ; \chi , u ) = R e s _ { z = 1 } L ( z , \chi ) L ( 2 z , \chi ^ { 2 } ) G ( z , \chi ) F _ { u } ( z , \chi ) \frac { X ^ { z } } { z } + \sum _ { j = 1 } ^ 4 \int _ j - \int _ 5 + O \left ( \frac { X ^ { 1 + \varepsilon } \log X } { T } \right ) , \end{align*}


\begin{align*} \| X ( t _ m ) - X _ m ^ { M , N } \| _ { L ^ p ( \Omega ; H ) } = O ( \lambda _ N ^ { - \frac \gamma 2 } + k ^ { \frac \gamma 4 } ) , \ ; \gamma \in \big ( \tfrac d 2 , 4 \big ] , \end{align*}


\begin{align*} \frac { \partial } { \partial x } \frac { \sinh ( x ) } { x ^ 2 } = \frac { \cosh ( x ) x ^ 2 - 2 x \sinh ( x ) } { x ^ 4 } = \frac { \sinh ( x ) } { x ^ 3 } ( x \coth ( x ) - 2 ) . \end{align*}


\begin{align*} E _ n = \{ \{ u , v \} : u , v \in [ n ] , u \ne v \} , \end{align*}


\begin{align*} \widetilde { H } _ n ' ( x ) = - 2 n \widetilde { H } _ { n - 1 } ( x ) \end{align*}


\begin{align*} s ( A , B ) : = \sup _ { x \in A } d _ M ( x , B ) : = \sup _ { x \in A } \inf _ { y \in B } d _ M ( x , y ) , \end{align*}


\begin{align*} \mathbb { H } = \R \oplus \R \mathbf { i } \oplus \R \mathbf { j } \oplus \R \mathbf { i j } , \ , \ , \ , \ , \ , \mathbf { i } ^ 2 = \mathbf { j } ^ 2 = - 1 , \mathbf { i j } = - \mathbf { j i } , \end{align*}


\begin{align*} f ( x ) = \tau _ x ( f ) = \int f d m = \int f d \lambda D _ y = \lambda f ( y ) = \lambda \tau _ y ( f ) \end{align*}


\begin{align*} \frac { 2 N ( \sigma , T ) } { T } & \le 2 \left [ C _ 1 ( \sigma ) T ^ { ( 5 - 8 \sigma ) / 3 } \log ^ { 5 - 2 \sigma } T + \frac { C _ 2 ( \sigma ) \log ^ 2 T } { T } \right ] , \\ & \le 2 \bigg [ C _ 1 ( \sigma ) \exp \left ( \frac { B _ 2 ( 5 - 8 \sigma ) } { 3 } \sqrt { \log x } \right ) \left ( B _ 2 \sqrt { \log x } \right ) ^ { 5 - 2 \sigma } \\ & \qquad \qquad \qquad + C _ 2 ( \sigma ) \exp \left ( - B _ 2 \sqrt { \log x } \right ) ( B _ 2 ) ^ 2 \log x \bigg ] \\ & = s _ 2 ' ( x , \sigma ) , \ . \end{align*}


\begin{align*} \Psi ^ { A i } ( \zeta ) \sim \frac { e ^ { \frac { i \pi } { 1 2 } } } { 2 \sqrt { \pi } } \zeta ^ { - \frac { \sigma _ 3 } { 4 } } \sum _ { j = 0 } ^ { \infty } \begin{pmatrix} ( - 1 ) ^ { j } s _ { j } & s _ { j } \\ - ( - 1 ) ^ { j } \nu _ j & \nu _ j \end{pmatrix} e ^ { - \frac { i \pi } { 4 } \sigma _ 3 } \left ( \frac { 2 } { 3 } \zeta ^ { \frac { 3 } { 2 } } \right ) ^ { - j } , \zeta \rightarrow \infty . \end{align*}


\begin{align*} \operatorname { T o r } ^ S _ \ast ( k [ K ] , k ) _ { \subseteq J } = \bigoplus _ { I \subseteq J } \operatorname { T o r } ^ S _ \ast ( k [ K ] , k ) _ I \end{align*}


\begin{align*} \check H ^ q _ { c t } ( X ; A _ X ) = \check H ( \nu ( X ) ; A _ { \nu ( X ) } ) \end{align*}


\begin{align*} | G _ k ( f ) ( x ) | & = | \int _ { \R ^ N } D _ k ( x , y ) D ^ M _ k ( f ) ( y ) d \omega ( y ) | \leqslant C \frac { 1 } { V _ k ( x ) } \| f \| _ \infty \\ & \lesssim r ^ k \frac { 1 } { V _ 0 ( x ) } \| f \| _ \infty \lesssim r ^ k \| f \| _ \infty \end{align*}


\begin{align*} \sum _ { \substack { 1 \leq e < \infty \\ ( e , g ) = 1 } } \frac { \mu ( e ) } { e } \sum _ { \substack { a | g \\ ( e a , \frac { m h } { g } \cdot \frac { n k } { g } ) = 1 } } \frac { \mu ( a ) } { \phi ( e a ) } R ( 0 ; a e , m n h k / g ^ 2 ) = \frac { \phi ( m n h k ) } { m n h k } . \end{align*}


\begin{align*} \widehat { g } ( \omega ) = C \ , e ^ { - \gamma \omega ^ 2 } \prod _ { k = 1 } ^ \infty \frac { e ^ { 2 \pi i \delta _ k \omega } } { 1 + 2 \pi i \delta _ k \omega } . \end{align*}


\begin{align*} S _ { \mu _ { T ^ * T } } ^ { \langle - 1 \rangle } \left ( \frac { 1 } { | \lambda | ^ { 2 } } \right ) = \psi _ { \mu _ { T ^ * T } } \left ( - \frac { 1 } { s ( | \lambda | , 0 ) ^ 2 } \right ) . \end{align*}


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


\begin{align*} \langle R ( x \wedge y ) , z \wedge w \rangle + \langle R ( z \wedge x ) , y \wedge w \rangle + \langle R ( y \wedge z ) , x \wedge w \rangle = 0 . \end{align*}


\begin{align*} \forall x ^ X , y ^ X , { x ' } ^ X , { y ' } ^ { X } \left ( x = _ X x ' \land y = _ X y ' \rightarrow \chi _ A x y = _ 0 \chi _ A x ' y ' \right ) \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - A ^ \intercal e _ { i _ k } \frac { e _ { i _ k } ^ \intercal ( A x _ k - b ) } { \norm { A ^ \intercal e _ { i _ k } } _ 2 ^ 2 } . \end{align*}


\begin{align*} \mathcal I ( \rho ^ 0 , \rho ^ T ) = \inf \{ \mathcal S ( \rho _ t , \Phi _ t ) | ( - \Delta _ { \rho _ t } ) ^ { \dagger } \Phi _ t \in \mathcal T _ { \rho _ t } \mathcal P _ { o } ( G ) , \rho ( 0 ) = \rho ^ 0 , \rho ( T ) = \rho ^ T \} \end{align*}


\begin{align*} \mathfrak { A } _ { \omega } ( \Gamma ) = \{ h \in \mathfrak { A } _ { \omega } : h | _ { \omega } \gamma = h \gamma \in \Gamma \} , \end{align*}


\begin{align*} b c _ s ( n ) = d d _ s ( 2 n ) + d d _ s ( 2 n - s ) + d d _ s ( 2 n - 4 s ) + \cdots = \sum _ { i \geq 0 } d d _ s ( 2 n - i ^ 2 s ) , \end{align*}


\begin{align*} \delta = \underset { m \geq 2 , n } { \sup } ~ \sum _ { k = 0 } ^ { \infty } \bigg ( \frac { w _ { k + m } \cdots w _ { k + n } } { w _ { m } \cdots w _ { n } } \bigg ) ^ { 2 } < \infty . \end{align*}


\begin{align*} \mbox { m i n i m i z e } \ ; \ ; f ( x ) : = \varphi ( x ) + g ( x ) \mbox { s u b j e c t t o } \ ; \ ; x \in \R ^ n . \end{align*}


\begin{align*} \int _ { B _ \varepsilon ( x ) } y _ 1 \eta _ \varepsilon ( x - y ) \dd y & = \int _ { B _ \varepsilon ( x ) } ( y _ 1 - x _ 1 ) \eta _ \varepsilon ( x - y ) \dd y + x _ 1 \int _ { B _ \varepsilon ( x ) } \eta _ \varepsilon ( x - y ) \dd y = x _ 1 \end{align*}


\begin{align*} d X _ t = - \beta _ t \nabla U ( X _ t ) \ , d t + \sqrt { 2 } d W _ t , \end{align*}


\begin{align*} X _ { n , r } : = X _ { n , r } ( k ) : = \begin{cases} \frac 1 { \delta _ n ( k ) } \sum \limits _ { \substack { \mathbf p _ k \in \mathcal P ( d , k ) \\ p _ k = r } } \tilde M _ { n , \mathbf { p } _ k } & , r \ge k , \\ 0 & , r < k . \end{cases} \end{align*}


\begin{align*} \Psi ^ n _ { ( \Sigma , U ) } ( f ) [ \rho ^ { * } _ { ( 1 ) } \psi ] = \Psi ^ n _ { ( \widetilde { \Sigma } , \widetilde { U } ) } ( \rho _ { * } ( \omega _ { \ell } | _ { \Sigma } ^ { \mu - 1 } f ) ) [ \psi ] . \end{align*}


\begin{align*} B _ + ( y , u ) : = \{ t \in [ 0 , T ] \colon y ( t ) = r \exists \varepsilon > 0 y - u = \mathrm { c o n s t } [ t , t + \varepsilon ) \} , \end{align*}


\begin{align*} H \rhd u ^ i = \lambda u ^ i , ~ ~ ~ ~ ~ ~ ~ ~ ~ E ^ \pm \rhd u ^ i = \delta _ 0 ^ i \frac { 1 } { \sqrt { a } } u ^ \pm - \delta ^ i _ \mp \sqrt { a } u ^ 0 . \end{align*}


\begin{align*} \alpha = P ^ { - 1 } \Big ( \frac n \beta \Big ) \in ( c , n ] , c : = \max \{ q _ 2 , P ^ { - 1 } ( 1 / \Gamma ) \} . \end{align*}


\begin{align*} D ( k , \xi ) : = \exp \left \{ \frac { X _ { \eta } ( k ) } { 2 \pi i } \left ( \int _ { - \eta } ^ { - C _ R } + \int _ { C _ R } ^ { \eta } \right ) \frac { \log r _ { + } ( s ) } { X _ { \eta + } ( s ) ( s - k ) } d s \right \} , \end{align*}


\begin{align*} \Xi _ k = \begin{bmatrix} x _ { 1 , k } & x _ { 1 , k } + T _ s x _ { 2 , k } & x _ { 3 , k } & x _ { 3 , k } + T _ s x _ { 4 , k } \end{bmatrix} ^ \top . \end{align*}


\begin{align*} \tau _ n : = \inf \{ t \in [ 0 , T ] : \| S ^ n \| _ { \mathcal C ( [ 0 , t ] ; \mathbb R ^ N \times \mathbb R ^ N ) } \ge n \} \wedge \inf \{ t \in [ 0 , T ] : \min _ { i = 1 } ^ N \min _ { s \in [ 0 , t ] } \rho _ i ^ n ( s ) \le \frac 1 { n } \} , \end{align*}


\begin{align*} \partial _ t V = \mathcal L V + g + \rho \max \{ V ^ * - V , 0 \} , \end{align*}


\begin{align*} z ^ 3 y ^ 3 = f _ 6 ( x , y ) \ , . \end{align*}


\begin{align*} \begin{cases} u _ 0 | x | u _ 0 \in L ^ 2 , \\ u _ 0 \in L _ { r a d } ^ 2 ( \mathbb { R } ^ N ) ( N , p , \gamma ) ( \ref { p a r a m e t e r : r a n g e f o r r a d i a l c a s e } ) , \end{cases} \end{align*}


\begin{align*} M _ { d _ 0 } ( p , H ) = \frac { 2 \pi ^ 2 \mu ( d _ 0 ) \phi ( d _ 0 ) } { d _ 0 ^ 2 p } \sum _ { \delta \mid d _ 0 } \frac { \delta \mu ( \delta ) } { \phi ( \delta ) } S ( H _ \delta , \delta p ) , \hbox { w h e r e } S ( H _ \delta , \delta f ) = \sum _ { h \in H _ \delta } s ( h , \delta f ) , \end{align*}


\begin{align*} \left ( \prescript { ( 1 \ , 2 ) } { } N _ r ^ A \right ) ^ { - 1 } = \prescript { ( 1 \ , 2 ) } { } N _ l ^ A . \end{align*}


\begin{align*} t = \sum _ { i = 1 } ^ { R _ \ell ^ { \pi } ( t ) } T _ { \ell i } ^ \pi + \eta _ \ell ^ \pi ( t ) , \end{align*}


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


\begin{align*} \frac { d } { d t } X ( x _ 0 , t ) = J ( X ( x _ 0 , t ) , t ) = - H B _ x ( X ( x _ 0 , t ) , t ) = - \Lambda B ( X ( x _ 0 , t ) , t ) . \end{align*}


\begin{align*} \mathcal { M } _ { k } ^ { J } ( \alpha , \beta ) : = \{ \ , u \in \mathcal { M } ^ { J } ( \alpha , \beta ) \ , | \ , I ( u ) = k \ , \ , \} \end{align*}


\begin{align*} T _ n ( t ) f ( x ) = \begin{cases} \Big ( \frac { x - t } { x } \Big ) ^ { \frac { 1 } { n } } f ( x - t ) & \mbox { f o r } x > t \\ 0 & \mbox { f o r } x \leq t . \end{cases} \end{align*}


\begin{align*} \Delta v - \varepsilon ^ { - 2 } \left ( e ^ { v } - \left ( ( { x ^ { \lambda } } ) ^ { 2 } + y ^ { 2 } \right ) e ^ { - v } \right ) = \Delta v - \varepsilon ^ { - 2 } \left ( e ^ { v } - \left ( { x } ^ { 2 } + y ^ { 2 } \right ) e ^ { - v } \right ) = 0 . \end{align*}


\begin{align*} \langle \phi _ 0 , \partial _ \tau \phi _ 0 \rangle _ { L ^ 2 _ \rho } = - \frac { \gamma b _ \tau } { b } \| \phi _ 0 \| ^ 2 _ { L ^ 2 _ \rho } ( 1 + O ( b ^ { 1 - \frac { \epsilon } { 2 } } ) ) . \end{align*}


\begin{align*} \varphi ( a _ 1 \cdots a _ n ) & = \frac { 1 } { \tau ' ( p ) } \tau ' ( p a _ 1 \cdots a _ n ) = \frac { 1 } { \tau ' ( p ) } \tau ' ( p ) \tau ( a _ 1 \cdots a _ n ) = 0 . \end{align*}


\begin{align*} \theta ( x ) = \sum _ { p \le x } \log p , \ E _ { \theta } ( x ) = \Big | \frac { \theta ( x ) - x } { x } \Big | , \ \pi ( x ) = \sum _ { p \le x } 1 , \ \ \ E _ { \pi } ( x ) = \Big | \frac { \pi ( x ) - \textrm { L i } ( x ) } { \frac { x } { \log x } } \Big | , \end{align*}


\begin{align*} \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } b ( \cos \theta ) | v - v _ * | ^ \gamma f ( v ' ) d \sigma d v = \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } b ( \cos \theta ) \frac 1 { \cos ^ { 3 + \gamma } ( \theta / 2 ) } | v - v _ * | ^ \gamma f ( v ) d \sigma d v . \end{align*}


\begin{align*} H ( Q ( \cdot - x _ 1 ) + \sigma Q ( \cdot - x _ 2 ) ) = 2 H ( Q ) - \frac { \sigma \kappa _ 0 } { ( x _ 1 - x _ 2 ) ^ 2 } + O \bigg ( \frac { 1 } { | x _ 1 - x _ 2 | ^ 3 } \bigg ) , \end{align*}


\begin{align*} \nu : = 2 + \theta < 5 / 2 + a . \end{align*}


\begin{align*} X ^ { y _ { i } } _ H - \pi _ 1 ( y _ { i } ) = X ^ { y _ { i } } _ { r \frac { H } { r } } - \pi _ 1 ( y _ { i } ) \leq \Big ( v _ + - \frac { \theta } { 2 r } \Big ) r \frac { H } { r } = \Big ( v _ + - \frac { \theta } { 2 r } \Big ) H . \end{align*}


\begin{align*} \Psi _ { \alpha , \nu , \tilde { \nu } } = \lim _ { t \to + \infty } e ^ { t ( \Delta _ { \alpha } + | \nu | + | \tilde { \nu } | ) } e ^ { - t { \bf H } } \Psi _ { \alpha , \nu , \tilde { \nu } } ^ 0 \end{align*}


\begin{align*} & . \ \liminf _ { \alpha \to 0 } | \tilde { x } - \tilde { y } | > 0 , \\ & . \ \liminf _ { \alpha \to 0 } | \tilde { x } - \tilde { y } | = 0 , \ \ \exists \alpha _ i \to 0 \ \ \lim _ { \alpha _ i \to 0 } | \tilde { x } - \tilde { y } | = 0 . \end{align*}


\begin{align*} A ( z ) = \abs { \{ n \in [ 1 , N ] : a _ { n } = z \} } \end{align*}


\begin{align*} \left ( b _ 2 \mathbb { N } + b _ 1 \mathbb { Z } \right ) \ , \cap \ , \left ( m b _ 1 \mathbb { N } + \pi \mathbb { Z } \right ) \ , = \ , \emptyset . \end{align*}


\begin{align*} A : x \mapsto \begin{cases} \partial \varphi ( x ) & x \in ( 0 , \pi / 2 ) , \\ \emptyset & , \end{cases} \end{align*}


\begin{align*} w = \delta ^ { \beta } \langle \check \tau _ { t , x } \rangle ( v ^ \perp - v ^ \perp _ \ast ( t , x , v _ 1 ) ) , \end{align*}


\begin{align*} ( m + 1 ) P ( x ) + x P ' ( x ) = \mu _ { n , m } ( 1 - x ) ^ n . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ d } d x \ , \mathbb { E } \bigg [ \Big | \mathcal { F } ^ { - 1 } \Big ( \big ( 1 + | . | ^ 2 \big ) ^ { - s + \frac { \kappa } { 2 } } \mathcal { F } \Big ( \rho \Big [ \ < I P s i 2 > _ { n } ( t , . ) - \ < I P s i 2 > _ { n } ( s , . ) \Big ] \Big ) \Big ) ( x ) \Big | ^ { 2 p } \bigg ] \lesssim \Bigg ( \sum _ { k = 1 } ^ { 8 } | \tilde { I _ k } | \Bigg ) ^ p , \end{align*}


\begin{align*} - \frac { \partial ^ 2 } { \partial t ^ 2 } \left [ \log ( \cosh ( t ) - 1 ) \right ] & = - \frac { \partial } { \partial t } \frac { \sinh ( t ) } { \cosh ( t ) - 1 } \\ & = - \frac { \cosh ( t ) \ , ( \cosh ( t ) - 1 ) - \sinh ( t ) \sinh ( t ) } { ( \cosh ( t ) - 1 ) ^ 2 } \\ & = \frac { \cosh ( t ) } { ( \cosh ( t ) + 1 ) ^ 2 } > 0 . \end{align*}


\begin{align*} \Delta _ { \mathbf { R } _ { 0 } ( \tau ) } ( \mathcal { I } ) = \left ( 1 + \tau \cdot \mathbf { 1 } ^ { \mathtt { T } } \cdot \left ( \mathbf { r } _ { 0 } ( 0 ) _ { \kappa ( \mathcal { I } ) } ^ { \kappa ( \mathcal { I } ) } \right ) ^ { - 1 } \cdot \mathbf { 1 } \right ) \cdot \Delta _ { \mathbf { R } _ { 0 } ( 0 ) } ( \mathcal { I } ) = \Delta _ { \mathbf { R } _ { 0 } ( 0 ) } ( \mathcal { I } ) \in \mathbb { C } \end{align*}


\begin{align*} T B _ { 9 } ( f ) = 3 T B _ { 5 } ( f ) . \end{align*}


\begin{align*} x _ j = \lambda y _ j + ( 1 - \lambda ) z _ j , w _ \star ( x _ j , t _ j ) = \min \{ u ( y _ j , t _ j ) , u ( z _ j , t _ j ) \} . \end{align*}


\begin{align*} \mathcal { T } _ \star ^ { p , r } : = \underbrace { \Omega _ \star \otimes _ { \mathcal { X } _ \star } \ldots \otimes _ { \mathcal { X } _ \star } \Omega _ \star } _ { p } \otimes _ { \mathcal { X } _ \star } \underbrace { \Xi _ \star \otimes _ { \mathcal { X } _ \star } \ldots \otimes _ { \mathcal { X } _ \star } \Xi _ \star } _ { r } \end{align*}


\begin{align*} \sum _ { j = 1 } ^ \infty \Big \| \Big ( q - \sum _ { k = 1 } ^ { L } q _ k \widetilde { \phi _ k } \Big ) e _ j \Big \| _ { H ^ 2 } ^ 2 & \le \frac { C } { \delta ^ { 2 \max m _ j } } \Big ( \frac { 1 } { \delta ^ 2 } \log \frac { 1 } { \delta } \Big ) ^ 2 . \end{align*}


\begin{align*} \alpha _ d = \left \{ \begin{array} { l } 1 / 4 \ d = 1 \\ 5 / 6 \ d = 2 \\ 1 7 / 1 2 \ d = 3 \end{array} \right . \ , . \end{align*}


\begin{align*} \lim _ { s \rightarrow 0 ^ + } \frac { h ( s ) } { s } = \lim _ { s \rightarrow 0 ^ + } \int _ \mathbb { R } \frac { 1 } { s ^ 2 + u ^ 2 } d \mu ( u ) = \frac { 1 } { \lambda _ 1 ( \mu ) ^ 2 } , \end{align*}


\begin{align*} A _ { i j } = \begin{cases} O _ { m _ i \times m _ j } , & a _ { i , j } = 0 \\ E _ { m _ i \times m _ j } , & a _ { i , j } = 1 \end{cases} \end{align*}


\begin{align*} F _ \ell & = F / x ^ \ell F , \\ G _ \ell & = G / y ^ \ell G , \\ A _ \ell & = A / ( x ^ \ell F \cap A ) , \\ B _ \ell & = B / ( y ^ \ell G \cap B ) . \end{align*}


\begin{align*} \begin{aligned} & \frac 1 { \alpha _ k } \frac k { k + 1 } - \frac { 1 - \alpha _ k } { \alpha _ k } \le \liminf _ { n \to \infty } E _ { n , k } ( X _ { n , k } ^ { \mathcal { T } } | W _ { \mathcal { T } ( n , k ; R _ n ) } ) \le \\ & \limsup _ { n \to \infty } E _ { n , k } ( X _ { n , k } ^ { \mathcal { T } } | W _ { \mathcal { T } ( n , k ; R _ n ) } ) \le \frac 1 { \alpha _ k } \frac k { k + 1 } . \end{aligned} \end{align*}


\begin{align*} \tilde { \lambda } _ 0 = \lim _ { n \to + \infty } \min _ { B ^ 1 ( \mathbf { x } ) \subset B ^ { A _ n } ( \mathbf { 0 } ) } \lambda ^ 1 ( B ^ 1 ( \mathbf { x } ) , B ^ { A _ n } ( \mathbf { 0 } ) ) = \lim _ { n \to + \infty } \lambda ^ 1 ( B ^ 1 ( \mathbf { 0 } ) , B ^ { A _ n } ( \mathbf { 0 } ) ) , \end{align*}


\begin{align*} \partial _ { J } \langle \alpha \rangle = \sum _ { \beta : \mathrm { a d m i s s i b l e \ , \ , o r b i t \ , \ , s e t \ , \ , w i t h \ , \ , } [ \beta ] = \Gamma } \# ( \mathcal { M } _ { 1 } ^ { J } ( \alpha , \beta ) / \mathbb { R } ) \cdot \langle \beta \rangle . \end{align*}


\begin{align*} - | D | Q - Q + Q ^ p = 0 . \end{align*}


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


\begin{align*} \nabla _ { \beta _ 1 } \cdots \nabla _ { \beta _ { n - 1 } } \partial _ { \beta _ n } = \Gamma ^ \alpha _ { \beta _ 1 , \ldots , \beta _ n } \partial _ \alpha \end{align*}


\begin{align*} \left \{ \begin{aligned} & x ( s ) = \frac { 1 } { 2 } + A e ^ { 2 t } + B e ^ { - t } , \\ & y ( s ) = 2 A e ^ { 2 t } - B e ^ { - t } . \end{aligned} \right . \end{align*}


\begin{align*} & \lambda { } ^ 3 + ( 2 \beta - \beta \mu + 5 \delta ) \lambda { } ^ 2 + ( 2 \delta ( \beta + 4 \delta + 2 ) + d \beta { } ^ 2 - \beta ( \beta + 4 \delta ) \mu ) \lambda \\ & + 2 \delta ( 2 \delta ( \delta + 1 ) + d \beta ^ 2 - \beta ( \beta + 2 \delta ) \mu ) = 0 , \ \ \mu \in { \rm S p e c } ( { \bf A } ( G ) ) , \end{align*}


\begin{align*} \left ( \frac { \sin ( t \sqrt { \Delta _ { E _ 1 } } ) } { \sqrt { \Delta _ { E _ 1 } } } f \right ) ( x ) = \left ( \frac { \sin ( t \sqrt { \Delta _ { E _ 2 } } ) } { \sqrt { \Delta _ { E _ 2 } } } f \right ) ( x ) , \end{align*}


\begin{align*} \phi ( p ) ( T ( a ) ) = \tau ( p , 1 ) p ( a ) \end{align*}


\begin{align*} \rho \Bigg ( \begin{bmatrix} \theta \\ p ( \theta ) \\ \end{bmatrix} \Bigg ) = \begin{bmatrix} \theta - \alpha \Delta \theta \\ p ( \theta ) - \alpha \Delta p ( \theta ) \\ \end{bmatrix} , \end{align*}


\begin{align*} X _ n + 1 = \varepsilon _ { n - 1 } u _ n ^ 2 ( a + b ) . \end{align*}


\begin{align*} E = I + k ^ { - 1 } E _ 1 + \mathcal { O } ( k ^ { - 2 } ) , k \rightarrow \infty . \end{align*}


\begin{align*} \lim _ { | x _ n | + | \omega _ n | + | \tau _ n | \to 0 } \norm { e ^ { 2 \pi i \tau _ n } M _ { \omega _ n / 2 } T _ { x _ n } M _ { \omega _ n / 2 } f - f } _ 2 = 0 . \end{align*}


\begin{align*} \mathfrak { D } _ { \omega } ^ d f = ( \omega \otimes \tau ^ d ) ( \eta ( z ) ) ^ { - 1 } \mathfrak { C } ^ d [ \omega ( \eta ( z ) ) f ] . \end{align*}


\begin{align*} \overset { \bullet } { v } _ 1 ( t ) & = - \mu v _ 1 ( t ) \\ \overset { \bullet } { v } _ 2 ( t ) & = C v _ 2 ( t - \Delta ) - \mu v _ 2 ( t ) \\ & \vdots \\ \overset { \bullet } { v } _ N ( t ) & = C v _ N ( t - \Delta ) - \mu v _ N ( t ) \end{align*}


\begin{align*} \abs { A ( x ) - \alpha } \ = \ \abs { A ( x _ 0 ) - \alpha } \ \leq \ \frac { 1 } { x _ 0 + 1 } \ \leq \ \frac { 1 } { x + 1 } . ] \end{align*}


\begin{align*} \lim _ { k \to \infty } \sup _ { t \in [ \tau _ 0 , 0 ] } \left \| w ^ { ( k ) } ( t , \cdot ) - w ( t , \cdot ) \right \| _ { H ^ 2 ( \R \backslash \{ t , 0 \} ) } ~ = ~ 0 . \end{align*}


\begin{align*} \operatorname { A S } ( D ) : = \widehat { A } \Big ( \frac { R _ X } { 2 \pi i } \Big ) \operatorname { t r } \Big ( \exp \big ( \frac { R _ F } { 2 \pi i } \big ) \Big ) . \end{align*}


\begin{align*} \lambda \mathbf { \mathrm { v } } _ u = \sum _ { w \sim u } \mathbf { \mathrm { v } } _ w . \end{align*}


\begin{align*} \nu ^ { \epsilon , L } ( d \varphi ) = e ^ { - V _ 0 ^ { \epsilon , L } ( \varphi ) } \gamma ^ { \epsilon , L } ( d \varphi ) . \end{align*}


\begin{align*} \beta = \frac 1 2 \ \ | \gamma | \le H _ 0 \end{align*}


\begin{align*} a _ n : = \prod _ { j = 1 } ^ { n - 1 } \left ( 1 + \dfrac { \alpha } { j } \right ) = \prod _ { j = 1 } ^ { n - 1 } \dfrac { j + \alpha } { j } = \dfrac { \Gamma ( n + \alpha ) } { \Gamma ( n ) \Gamma ( 1 + \alpha ) } . \end{align*}


\begin{align*} \hat z ( 0 , \tau ) = 0 , \tau \in [ - T , T ] . \end{align*}


\begin{align*} \sum _ { \xi , \eta } \langle X _ \alpha , [ X _ \xi , X _ \eta ] \rangle \langle X _ \beta , [ \tilde R X _ \xi , \tilde T X \eta ] \rangle = \sum _ { \xi , \eta } r _ \xi t _ \eta \langle X _ \alpha , [ X _ \xi , X _ \eta ] \rangle \langle X _ \beta , [ X _ \xi , X \eta ] \rangle = 0 , \end{align*}


\begin{align*} D T \left [ f ''' ( \eta ) \right ] = ( k + 1 ) ( k + 2 ) ( k + 3 ) F ( k ) \end{align*}


\begin{align*} T B _ { k } ( f ) = q \cdot T B _ { j } ( f ) . \end{align*}


\begin{align*} h ( s ) = \int _ \mathbb { R } \frac { s } { s ^ 2 + u ^ 2 } d \mu ( u ) , s > 0 . \end{align*}


\begin{align*} \| 1 _ A \| \ = \ \| z - 1 _ { B \backslash E } - P _ E ( z ) \| & \ \leqslant \ \mathbf C _ \ell \| z - P _ E ( z ) \| \\ & \ \leqslant \ \mathbf C _ \ell \mathbf C _ { \lambda , r p } \sigma ^ { \mathcal { U } _ X , R } _ m ( z ) \ \leqslant \ \mathbf C _ \ell \mathbf C _ { \lambda , r p } \| 1 _ B \| . \end{align*}


\begin{align*} \rho ^ * _ + \Psi _ \Sigma ( f ) = \frac { 1 } { 2 } \left ( \Pi ( f ) + \Phi ( * d f ) \right ) . \end{align*}


\begin{align*} \zeta ^ \wedge _ { \mathcal { L } _ \Delta , p } ( s ) = \int _ { G _ \Delta ^ + ( \Q _ p ) } | \det g | _ p ^ s d \mu , \end{align*}


\begin{align*} \chi _ Q ( x ) = 1 \operatorname { d i s t } ( x , \partial Q ) > \alpha \varepsilon \chi _ Q ( x ) = 0 \operatorname { d i s t } ( x , \partial Q ) < \frac { \alpha } { 2 } \varepsilon , \\ \psi _ Q ( x ) = 1 \operatorname { d i s t } ( x , \partial Q ) > \frac { \alpha } { 2 } \varepsilon \psi _ Q ( x ) = 0 \operatorname { d i s t } ( x , \partial Q ) < \frac { \alpha } { 4 } \varepsilon , \end{align*}


\begin{align*} \phi ( p , \lambda ) ( T ( a ) ) = \tau ( p , \lambda ) , a \in F _ { p , \lambda } . \end{align*}


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


\begin{align*} \omega ( z ) = z + H _ 2 ( z + H _ 1 ( \omega ( z ) ) ) , z \in \mathbb { C } ^ + \end{align*}


\begin{align*} \lambda ( q , t ) = t + q \frac { t ^ 3 } { 3 ! } + ( q + q ^ 2 ) \frac { t ^ 4 } { 4 ! } + \dotsb \end{align*}


\begin{align*} J ^ A _ \gamma : = ( I d + \gamma A ) ^ { - 1 } . \end{align*}


\begin{align*} G _ m ( r ) : = \begin{cases} \log r & m = k \\ - r ^ { - 2 \left ( \frac { k } { m } - 1 \right ) } & m < k , \end{cases} \end{align*}


\begin{align*} A _ { n } & = \frac { z } { \sqrt { n } } A _ { n - 1 } = \frac { z } { \sqrt { n } } \frac { z } { \sqrt { n - 1 } } A _ { n - 2 } = \cdots = \frac { z ^ { n } } { \sqrt { n ! } } A _ { 0 } . \end{align*}


\begin{align*} p : = \iota \circ \pi : I _ 0 & \to I _ 0 \\ ( x , y ) & \mapsto ( x + y , 0 ) \end{align*}


\begin{align*} L ' \otimes \Z _ v & = \begin{cases} \displaystyle { \bigoplus _ { j = 0 } ^ 1 } L ' _ { 2 , j } ( \pi ^ j ) & ( v = 2 ) , \\ L ' _ { v , 0 } & ( \mathrm { o t h e r w i s e } ) , \\ \end{cases} \\ K _ { \ell } \otimes \Z _ v & = \begin{cases} \displaystyle { \bigoplus _ { j = 0 } ^ 1 } K _ { \ell , 2 , j } ( \pi ^ j ) & ( v = 2 ) , \\ K _ { \ell , v , 0 } & ( \mathrm { o t h e r w i s e } ) , \end{cases} \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d } } ( h , U ^ { t } ) _ { \mathbb { R } ^ { N } } d x & = \lambda _ { h } ( U ^ { t } ) = \langle \psi ( h ) , U ^ { t } \rangle _ { s } = \lim _ { j \rightarrow \infty } \langle \psi ( h ) , U ^ { j } ( t ) \rangle _ { s } \\ & = \lim _ { j \rightarrow \infty } \lambda _ { h } ( U ^ { j } ( t ) ) = \lim _ { j \rightarrow \infty } \int _ { \mathbb { R } ^ { d } } ( h , U ^ { j } ( t ) ) _ { \mathbb { R } ^ { N } } d x = \int _ { \mathbb { R } ^ { d } } ( h , U ( t ) ) _ { \mathbb { R } ^ { N } } d x , \end{align*}


\begin{align*} \frac { n ^ 2 } { \gamma } \int _ \R \int _ \R K ( \alpha [ x - y ] ) \ , d \rho ( y ) d \rho ( x ) = \frac { n ^ 2 } { \gamma \alpha } \int _ 0 ^ 1 \int _ 0 ^ 1 \alpha K ( \alpha [ x - y ] ) \ , d y d x . \end{align*}


\begin{align*} & T \left ( - k \right ) = T ( k ) , \ \ \ R _ { \pm } \left ( - k \right ) = \overline { R _ { \pm } ( k ) , } \\ & \left | R _ { \pm } ( k ) \right | ^ { 2 } + \left | T ( k ) \right | ^ { 2 } = 1 , \ \ T ( k ) \overline { R _ { - } ( k ) } + \overline { T ( k ) } R _ { + } ( k ) = 0 . \end{align*}


\begin{align*} & e ^ { \gamma ( t - s ) } \varphi _ { \gamma } ( s - r ) + \varphi _ { \gamma } ( t - s ) = e ^ { \gamma ( t - s ) } \int _ 0 ^ { s - r } e ^ { \gamma ( s - r - u ) } d u + \int _ 0 ^ { t - s } e ^ { \gamma ( t - s - u ) } d u \\ & = \int _ 0 ^ { s - r } e ^ { \gamma ( t - r - u ) } d u + \int _ { s - r } ^ { t - r } e ^ { \gamma ( t - r - u ) } d u = \int _ 0 ^ { t - r } e ^ { \gamma ( t - r - u ) } d u = \varphi _ { \gamma } ( t - r ) \end{align*}


\begin{align*} | z \rangle = \sum _ { n = 0 } ^ { \infty } A _ { n } | n \rangle . \end{align*}


\begin{align*} J _ 2 & \leq C ( \lambda _ N ^ { - \frac { 2 + \ell ( \gamma ) } 2 } + k ^ { \frac { 2 + \ell ( \gamma ) } 4 } ) \| F ( X ( t _ m ) ) \| _ { L ^ p ( \Omega ; H ^ { \ell ( \gamma ) } ) } \\ & \leq C ( \lambda _ N ^ { - \frac { 2 + \ell ( \gamma ) } 2 } + k ^ { \frac { 2 + \ell ( \gamma ) } 4 } ) \sup _ { s \in [ 0 , T ] } \| F ( X ( s ) ) \| _ { L ^ p ( \Omega ; H ^ { \ell ( \gamma ) } ) } \leq C ( \lambda _ N ^ { - \frac \gamma 2 } + k ^ { \frac \gamma 4 } ) , \end{align*}


\begin{align*} [ \bar { a } _ n \mapsto \bar { b } _ n ] ( J ) \cup [ \bar { a } _ n \mapsto \bar { b } _ n ] ( K ) \cup [ \bar { a } _ n \mapsto \bar { b } _ n ] ( L ) & = [ \bar { a } _ n \mapsto \bar { b } _ n ] ( J \cup K \cup L ) \\ & = [ \bar { a } _ n \mapsto \bar { b } _ n ] ( \mathbb { N } ) = \{ \bar { b } _ n \} \end{align*}


\begin{align*} c _ { M a p } = 4 n ^ 2 m + 2 m ^ 2 n + 1 6 n m - 1 2 m . \end{align*}


\begin{align*} | \sigma _ n ( x ) - \sigma _ n ( y ) | = \Bigl | \frac { \sigma ( x ) - \sigma ( y ) } { 1 + n ^ { - 1 / 2 } | x | ^ { \ell _ \mu } } + f _ n ( x , y ) \Bigr | \leq c \cdot ( 1 + | x | ^ { \ell _ { \sigma } } + | y | ^ { \ell _ { \sigma } } ) \cdot | x - y | + | f _ n ( x , y ) | . \end{align*}


\begin{align*} \psi : E _ d & \to K \\ g = \sum _ { F \subset [ n ] , | F | = d } \beta _ F x ^ F & \mapsto \sum _ { F \subset [ n ] , | F | = d } \beta _ F \alpha _ F = g ( \partial _ { 1 } , \cdots , \partial _ n ) f \end{align*}


\begin{align*} \bar x = \arg \min \left \{ \rho _ c ( x ) \middle | x \in M \right \} . \end{align*}


\begin{align*} \min \left \{ I _ b [ \gamma ] ; \gamma \in \mathrm { A C } \left ( [ 0 , b ] ; [ - 1 , 1 ] \right ) , \gamma ( 0 ) = x _ 0 , \gamma ( b ) = \gamma ( b ) \right \} . \end{align*}


\begin{align*} E : = \{ v \in V ( H _ k ) \setminus L ' : | N _ 1 ( v ) \cap L ' | \leq k - 1 \} . \end{align*}


\begin{align*} K _ 1 ( \mathfrak { n } ) \xi K _ 1 ( \mathfrak { n } ) = \bigcup _ { y \in Y } K _ 1 ( \mathfrak { n } ) y . \end{align*}


\begin{align*} E _ t \Delta y _ { t + 1 } & \in \partial _ { x } H _ t ( x _ { t } , y _ t ) , \\ \Delta x _ t & \in \partial _ { y } [ - H _ t ] ( x _ { t } , y _ t ) , \end{align*}


\begin{align*} \sum _ { h \in H _ { d _ 0 } } s ( h , \delta f ) = \frac { \phi ( d _ 0 ) } { \phi ( \delta ) } \sum _ { h \in H _ { \delta } } s ( h , \delta f ) \end{align*}


\begin{align*} | \Psi _ { s , t } ( x , v ) - v | = | \Psi _ { s , t } ( x , v ) - \zeta ( \Psi _ { s , t } ( x , v ) ) | \leq \frac { | \tilde Y _ { s , t } ( x , \Psi _ { s , t } ( x , v ) ) | } { t - s } \leq \frac { | \tilde Y _ { s , t } ( x , v ) | } { t - s } + \frac 1 2 | \Psi _ { s , t } ( x , v ) - v | , \end{align*}


\begin{align*} 1 = \psi \left ( c ^ 2 [ a _ 1 , b _ 1 ] \right ) = x \sigma _ 1 x \sigma _ 1 y g z y ^ { - 1 } g ^ { - 1 } z ^ { - 1 } = x \ldotp \sigma _ 1 x \sigma _ 1 ^ { - 1 } \ldotp A _ { 1 , 2 } \ldotp y \ldotp g z g ^ { - 1 } \ldotp g y ^ { - 1 } g ^ { - 1 } \ldotp z ^ { - 1 } . \end{align*}


\begin{align*} ( F ( f ) ) ( t ) : = f _ 0 ( t ) F _ Z ( f ( t ) ) f \in X t \in \partial A . \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty \int _ { - \infty } ^ \infty W \psi ( x , p ) \ , d p \ , d x = 1 , \end{align*}


\begin{align*} \tau ' ( b _ 0 a _ 1 c _ 1 a _ 2 \cdots c _ { n - 1 } a _ n b _ n ) = \tau ( b _ 0 \tau ' ( a _ 1 ) c _ 1 a _ 2 \cdots c _ { n - 1 } a _ n b _ n ) = 0 . \end{align*}


\begin{align*} \sum _ { \substack { 1 \leq \tilde { \jmath } < \infty \\ p | \tilde { \jmath } \Rightarrow p | j } } \frac { ( j , \tilde { \jmath } ) } { \tilde { \jmath } } & = \prod _ { p | j } \sum _ { \nu = 0 } ^ { \infty } \frac { p ^ { \min \{ j _ p , \nu \} } } { p ^ { \nu } } = \prod _ { p | j } \bigg ( j _ p + \frac { 1 } { 1 - \frac { 1 } { p } } \bigg ) \ll j ^ { \varepsilon } \end{align*}


\begin{align*} \sum _ { r s } v _ { ( 0 ; r s ) } \otimes b S ( v _ { ( 1 ; r s ) } ) a = ( 1 \otimes b ) ( X ^ - ( v \otimes a ) ) , v \in V , a , b \in A . \end{align*}


\begin{align*} M = \begin{pmatrix} \alpha & 0 & 0 & 0 \\ 0 & \frac { 1 } { \alpha } & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \frac { 1 } { \alpha } \end{pmatrix} . \end{align*}


\begin{align*} \begin{aligned} & \rho _ 1 ^ { i n } ( x ) = 2 + 1 2 ( x - 0 . 5 ) ^ 2 , \rho _ 2 ^ { i n } ( x ) = 1 + 2 x , \\ & \partial _ x \phi ( 0 , t ) = 0 , \ \partial _ x \phi ( 1 , t ) = 0 . \end{aligned} \end{align*}


\begin{align*} \begin{array} { l l } \displaystyle \sum _ { i = 1 } ^ { m } z _ i a _ { i j } = \eta , \ ; \ ; \ ; j = 1 , 2 , . . . , n \\ \displaystyle \sum _ { i = 1 } ^ { m } z _ i = 1 \\ \end{array} \end{align*}


\begin{align*} \widetilde { A } ^ { 0 } ( U ) = \left ( \begin{array} { c c c c } 1 & & & \\ & \rho \mathbb { I } _ { 3 } & & \\ & & \rho e _ { \theta } & \\ & & & \tau \mathbb { I } _ { 3 } \end{array} \right ) , \widetilde { D } ( U ) = \left ( \begin{array} { c c } \mathbb { O } _ { 5 \times 5 } & \\ & \mathbb { I } _ { 3 } \end{array} \right ) , \end{align*}


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


\begin{align*} | b ^ { ( j ) } ( S _ \varepsilon ( \rho ) ) | \leq h ^ j = \begin{cases} \max _ { [ 0 , \bar \rho - \alpha _ 0 ] } | b ^ { ( j ) } | & \rho \leq \bar \rho - \alpha _ 0 \\ | b ^ { ( j ) } ( \rho ) | & \rho > \bar \rho - \alpha _ 0 . \end{cases} \end{align*}


\begin{align*} \begin{aligned} M ( x , y ) & : = \{ ( ( \tilde x , g ( \tilde x ) ) , ( 0 , \tilde y ) ) \ , | \ , \tilde x = x , \ , \tilde y \in - D , y = g ( \tilde x ) + \tilde y \} \\ & = \begin{cases} \{ ( ( x , g ( x ) ) , ( 0 , y - g ( x ) ) ) \} & g ( x ) - y \in D , \\ \varnothing & \end{cases} \end{aligned} \end{align*}


\begin{align*} J ( u ) = \frac 1 2 \| \nabla u \| ^ 2 + \int _ D \Phi ( u ) \ , \dd x , \end{align*}


\begin{align*} S _ 1 + T ^ 2 S _ 2 \leq \sum _ { \gamma \in \mathcal { I } } \| g _ { \gamma } \| _ { L ^ 2 } ^ 2 ( 2 \N ) ^ { - p _ 3 \gamma } + T ^ 2 \sum _ { \gamma \in \mathcal { I } } \| f _ { \gamma } \| ^ 2 ( 2 \N ) ^ { - p _ 2 \gamma } : = A < \infty . \end{align*}


\begin{align*} & \theta _ g ( x ^ i , 1 ; \| \Sigma ^ i \| _ g ) - \theta ( x ^ i ; \| \Sigma ^ i \| _ g ) \leq \delta ; \\ & \theta ( x ^ 0 ; \| \Sigma ^ 0 \| _ g ) = \theta ( x ^ 1 ; \| \Sigma ^ 1 \| _ g ) = \theta ( x ^ 2 ; \| \Sigma ^ 2 \| _ g ) . \end{align*}


\begin{align*} J _ { 0 } ( \hat { \alpha } \cup { ( \gamma , N ) } , \hat { \alpha } \cup { ( \gamma , N - p _ { i } ) } \cup { ( \delta _ { 1 } , 1 ) } \cup { ( \delta _ { 2 } , 1 ) } ) = 1 . \end{align*}


\begin{align*} \rho = \sum _ i p _ i P _ i , \end{align*}


\begin{align*} \mathrm { v a r } [ \widehat { K } _ { \mathrm { a - c p t - d } } ^ \mathbb { R } ] \ ! = \ ! \frac { \psi } { 2 } \Big ( \frac { 1 } { N _ 2 \bar { \gamma } _ c ' } \ ! - \ ! \frac { \mathrm { l i } ( 1 \ ! - \ ! \xi ) } { 1 - \xi } \sum _ { i \in \mathcal { K } ' } \frac { 1 } { 1 \ ! + \ ! N _ 1 \bar { \gamma } _ i } \Big ) \end{align*}


\begin{align*} \mathcal { A } ( \phi ) = \mathcal { A } ( \phi ^ { - 1 } ) . \end{align*}


\begin{align*} A D ^ T - B C ^ T = \begin{pmatrix} \alpha ^ 2 & 0 \\ 0 & \frac { 1 } { \alpha ^ 2 } \end{pmatrix} \neq I , \alpha \in \R , \ , \alpha \neq \pm 1 . \end{align*}


\begin{align*} \Im \left \{ \log \zeta ( s ) \right \} = \sum _ { | \gamma - t | \leq 1 } \Im \left \{ \log ( s - \rho ) \right \} + \mathcal { O } ( \log t ) \end{align*}


\begin{align*} \lambda _ \mu : = v ^ \mu ( p ) - u ^ \mu ( p ) \qquad w ^ \mu : = \lambda _ \mu ^ { - 1 } ( v ^ \mu - u ^ \mu ) \in W ^ { 1 , 2 } _ { l o c } ( B _ 1 ) \ , . \end{align*}


\begin{align*} \begin{bmatrix} 2 & 1 & 0 \\ - 1 & 2 & 3 \end{bmatrix} \quad \mathrm { a n d } \begin{bmatrix} 1 & - 3 & 6 \\ 0 & 1 & - 5 \end{bmatrix} , \end{align*}


\begin{align*} g _ t ( v , v ) = d \theta ^ c ( J v , v ) + d \zeta ( J v , v ) = 1 + d \zeta ( J v , v ) > 0 \end{align*}


\begin{align*} \lim _ { N \to \infty } \left | \mathcal { F } _ N ( \mathbf { h } _ N , \mathbf { g } _ N ) - \mathcal { F } ( \mathbf { h } _ N , \mathbf { g } _ N ) \right | = 0 . \end{align*}


\begin{align*} a _ 1 ( z ) : = \prod _ { j = 1 } ^ n ( z - \xi _ j ) ^ { m _ j } . \end{align*}


\begin{align*} \log _ { \lambda } ( 1 + t ) = \sum _ { k = 1 } ^ { \infty } \lambda ^ { k - 1 } ( 1 ) _ { k , \frac { 1 } { \lambda } } \frac { t ^ { k } } { k ! } , ( \mathrm { s e e } \ [ 8 , 1 4 ] ) . \end{align*}


\begin{align*} \mathrm { W F } ( W _ { \Sigma } ) = \left \{ ( r , r ; \xi , - \xi ) \in \dot { T } ^ { * } \Sigma ^ 2 \ , | \ , \xi > 0 \right \} , \end{align*}


\begin{align*} \lambda _ h = \lambda _ 0 - i \frac { { \lambda _ 0 } ^ \frac { 5 } { 2 } } { 4 \pi } \left [ { ( \eta _ 0 ^ 1 U _ 1 + \eta _ 0 ^ 2 U _ 2 ) U _ 0 } \right ] h + \mathcal { O } ( h ^ 2 ) . \end{align*}


\begin{align*} \frac { \widetilde { f } ' ( \gamma ) } { \widetilde { f } ( \gamma ) } = a + \frac { 1 } { \gamma } - a \coth ( \gamma a ) = a + a \ , \underbrace { \left ( \frac { 1 } { \gamma a } - \coth ( \gamma a ) \right ) } _ { 0 > , \ , \searrow , \ , > - 1 } , \end{align*}


\begin{align*} ( 1 ‐ \lambda ) \left ( s _ * + \frac 1 q ‐ \frac { 1 } { p _ 1 } \right ) + \lambda \left ( \frac 1 q ‐ \frac { 1 } { p _ 0 } \right ) = ( 1 ‐ \lambda ) s _ * . \end{align*}


\begin{align*} \widetilde { \mathfrak { L } } _ \alpha f & = \widetilde { Q } _ { \alpha } ^ { - 1 } \delta _ x ^ { \alpha - 1 } ( \delta _ x ^ { \alpha - 1 } ) ^ * \widetilde { Q } _ { \alpha } f \\ & = \widetilde { Q } _ { \alpha } ^ { - 1 } ( \delta _ x ^ { \alpha } ) ^ * \delta _ x ^ { \alpha } \widetilde { Q } _ { \alpha } f + \frac 1 2 f = \widetilde { Q } _ { \alpha } ^ { - 1 } Q { \mathfrak { L } } _ \alpha Q ^ { - 1 } \widetilde { Q } _ { \alpha } f + \frac 1 2 f \end{align*}


\begin{align*} \tilde m _ { l , j + 1 / 2 } = \frac { 1 } { h } \sum _ { p = 1 } ^ j \tilde \rho _ { l , p } = \left \{ \begin{array} { l l } \frac { 1 } { h } b _ j \gamma , & l = i , \ ; j _ 1 \leq j \leq j _ { k + 1 } - 1 , \\ 0 , & , \end{array} \right . \end{align*}


\begin{align*} - V ^ \prime ( t ) = A ^ * V ( t ) + F ( t ) , \ \ \forall t \in ( 0 , T ) , V ( T ) = V _ T , \end{align*}


\begin{align*} f _ { a , b } ( x ) = e ^ { - 2 \pi i b \cdot x } f ( x + a ) = M _ { - b } T _ { - a } f ( x ) , \end{align*}


\begin{align*} \langle \tfrac { 1 } { x } , f \rangle = \lim _ { \varepsilon \to 0 } \int _ { | x | > \varepsilon } \frac { \overline { f ( x ) } } { x } \ , d x . \end{align*}


\begin{align*} \widetilde { P } _ n ^ { \alpha , \beta } ( x ) = \frac { ( - 1 ) ^ { n } } { 2 ^ n n ! } ( 1 - x ) ^ { \alpha } ( 1 + x ) ^ { \beta } \frac { d ^ n } { d x ^ n } \big ( ( 1 - x ) ^ { - \alpha + n } ( 1 + x ) ^ { - \beta + n } \big ) \end{align*}


\begin{align*} \Gamma \backslash \mathcal { B } \stackrel { \sim } \longrightarrow \mathcal { F } ( \Omega ) , \Gamma = \{ \gamma \in U ( \mathcal { T } ) : L \gamma = L , u _ i \gamma - u _ i \in L \} . \end{align*}


\begin{align*} \partial ^ \alpha t ^ \beta ( M _ \omega T _ x g ( t ) ) = \sum _ { \gamma _ 1 \leq \alpha } \sum _ { \gamma _ 2 \leq \beta } \begin{pmatrix} \alpha \\ \gamma _ 1 \end{pmatrix} \begin{pmatrix} \beta \\ \gamma _ 2 \end{pmatrix} x ^ { \gamma _ 2 } ( 2 \pi i \omega ) ^ { \gamma _ 1 } M _ \omega T _ x ( \partial ^ { \alpha - \gamma _ 1 } t ^ { \beta - \gamma _ 2 } g ( t ) ) , \end{align*}


\begin{align*} \mathcal A ( \widetilde \rho , \widetilde m ) = \frac 1 2 \int _ 0 ^ 1 \frac { ( ( \Sigma _ 1 - \Sigma _ 2 ) \theta _ { 1 2 } ( \widetilde \rho ) \dot W ^ { \delta } ) ^ 2 } { \theta _ { 1 2 } ( \widetilde \rho ) } + ( m _ { 2 3 } + \frac 1 2 \rho _ 1 ( \Sigma _ 2 - \Sigma _ 3 ) \dot W ^ { \delta } ) ^ 2 d t . \end{align*}


\begin{align*} - \norm { \xi } ^ 2 = z _ { i ; \alpha } ^ { ( \ell + 1 ) } . \end{align*}


\begin{align*} \mu _ { \mathrm { d i s c } } ( \gamma _ { 1 } ) = 5 , \ , \ , \ , \ , \mu _ { \mathrm { d i s c } } ( \gamma _ { 2 } ) = 7 \end{align*}


\begin{align*} \tau _ { \delta , k } ( x ) = \frac { 1 } { 2 \varphi _ { \delta } ( \| x \| ^ 2 ) } \int _ { \| x \| ^ 2 } ^ { + \infty } \varphi _ { \delta } ( t ) d t . \end{align*}


\begin{align*} \nabla u _ { \frac { t } { 2 } } ( \varphi ) = - Q _ t \nabla V _ t ( Q _ t \varphi ) \quad \nabla ^ 2 u _ { \frac { t } { 2 } } ( \varphi ) = - Q _ t \nabla ^ 2 V _ t ( Q _ t \varphi ) Q _ t . \end{align*}


\begin{align*} \begin{aligned} \mathbb { E } R _ j & = \frac { \epsilon _ m } { 1 - \epsilon _ m } , \\ \mathbb { E } R _ j ^ 2 & = \frac { \epsilon _ m ^ 2 + \epsilon _ m } { \left ( 1 - \epsilon _ m \right ) ^ 2 } . \end{aligned} \end{align*}


\begin{align*} \int _ { \Omega } | \nabla u | ^ { p - 2 } \nabla u \cdot \nabla \phi \ , d \mu = \int _ { \partial \Omega } \phi f \ , d \nu . \end{align*}


\begin{align*} x ^ i x ^ j = \frac 1 2 \left [ y ^ i x ^ j + y ^ j x ^ i \right ] , \end{align*}


\begin{align*} \mathcal { L } ^ \gamma ( f ) ( x ) = - \langle x ; \nabla ( f ) ( x ) \rangle + \Delta ( f ) ( x ) , \end{align*}


\begin{align*} K _ { \alpha \alpha } ^ { ( \ell ) } = : \frac { 1 } { a \ell } + \epsilon ^ { ( \ell ) } , a : = - 6 \frac { \sigma _ 3 } { \sigma _ 1 } > 0 , \end{align*}


\begin{align*} \max _ { \pi \in \Omega ' } \hat { B } \bigl ( \psi ( T ) \bigr ) \Bigl ( Z \bigl ( s ( T ) \bigr ) \Bigr ) \leq \phi \leq \min _ { \pi \in \Omega ' } \hat { B } ( \psi ) \bigl ( Z ( s ) \bigr ) , \ V _ 1 ^ { - 1 } ( s _ 1 ) \in ( \ref { e q : S O S 1 } ) , \ s _ { 2 1 } \in [ a _ { 2 0 } , a _ { 2 1 } ] , \ s _ { i 0 } = \end{align*}


\begin{align*} \zeta _ 1 ( x ) = \begin{cases} 0 & \R ^ n _ + \setminus B _ 1 ( 2 e _ 1 ) , \\ 1 & \R ^ n _ - \setminus \{ x _ 1 > - 1 \} \end{cases} \qquad { \mbox { a n d } } \zeta _ 2 ( x ) = \begin{cases} 0 & \R ^ n _ + \setminus B _ 1 ( 2 e _ 1 ) , \\ 1 & B _ { 1 / 2 } ( - 2 e _ 1 ) . \end{cases} \end{align*}


\begin{align*} \phi _ { i , \Delta } ( \Delta ) = \Phi _ i ( \Delta ) . \end{align*}


\begin{align*} \partial ^ \alpha t ^ \beta f ( t ) = \sum _ { \gamma _ 1 \leq \alpha } \sum _ { \gamma _ 2 \leq \beta } \begin{pmatrix} \alpha \\ \gamma _ 1 \end{pmatrix} \begin{pmatrix} \beta \\ \gamma _ 2 \end{pmatrix} \iint _ { \R ^ { 2 d } } F ( x , \omega ) \ , x ^ { \gamma _ 2 } ( 2 \pi i \omega ) ^ { \gamma _ 1 } M _ \omega T _ x ( \partial ^ { \alpha - \gamma _ 1 } t ^ { \beta - \gamma _ 2 } g ( t ) ) \ , d ( x , \omega ) . \end{align*}


\begin{align*} ( \nabla F ( u ) ) ^ \# = ( v _ 1 ( u ) , v _ 2 ( u ) , \dots , v _ m ( u ) ) , \end{align*}


\begin{align*} T _ k f ( x ) = \int _ { | y | > q ^ { k } } f ( x - y ) \dfrac { \Omega ( y ) } { | y | } d y , ~ ~ ~ ~ ~ k \in \mathbb { Z } , \end{align*}


\begin{align*} E _ { \text i } ^ \pm ( \kappa _ x ) = \frac { \kappa _ 1 \eta _ 1 } { 2 } \frac { J _ \pm ( \kappa _ x ) } { \kappa _ { 1 z } } \end{align*}


\begin{align*} U : = \bigg \{ y \in B _ \rho ( e _ 1 ) u ( y ) > \frac { u ( a ) } 2 \bigg \} . \end{align*}


\begin{align*} \frac { 1 } { \inf _ { t > \tau , x \in \Omega } v ^ { q + 1 - \varepsilon _ 0 / 2 } ( t , x ) } = \sup _ { t > \tau , x \in \Omega } \frac { 1 } { v ^ { q + 1 - \varepsilon _ 0 / 2 } ( t , x ) } \le \sup _ { t > \tau } \frac { \big ( \int _ \Omega u ^ { - p } \big ) ^ { \frac { q + 1 - \varepsilon _ 0 / 2 } { p } } } { \big ( \delta _ 0 | \Omega | \big ) ^ { q + 1 - \varepsilon _ 0 / 2 } } . \end{align*}


\begin{align*} \lVert m ^ \natural f \rVert _ { B U O } & = \lVert - M ^ \natural ( - f ) \rVert _ { B U O } = \lVert M ^ \natural ( - f ) \rVert _ { B L O } \\ & \leq C ( C _ d ) \norm { - f } _ { B M O } = C \norm { f } _ { B M O } . \end{align*}


\begin{align*} ( x , y ) \ = \ \bigg [ \frac { x } { y + 1 } \bigg ] \end{align*}


\begin{align*} \chi ( x , t ) : = \frac { 1 } { \sqrt { 1 + t } } \chi _ { * } \left ( \frac { x } { \sqrt { 1 + t } } \right ) , x \in \R , \ \ t \ge 0 , \end{align*}


\begin{align*} ( f g ) _ { i _ 1 \dotsm i _ m } : = f _ { i _ 1 \dotsm i _ m } g _ 0 + f _ 0 g _ { i _ 1 \dotsm i _ m } + \sum _ { j = 1 } ^ { m - 1 } f _ { i _ 1 \dotsm i _ j } g _ { i _ { j + 1 } \dotsm i _ m } . \end{align*}


\begin{align*} \Omega _ G = \pi _ 1 ( G ) _ I \to \Omega _ { G ' } = \pi _ 1 ( G ' ) _ I \to X _ * ( T ) _ I \to 0 \end{align*}


\begin{align*} \partial M ( t ) = \Gamma ( t ) \end{align*}


\begin{align*} \mathfrak { W } _ \alpha & : \ell ^ \infty ( \N ) \to C ^ \alpha ( S ^ 1 ) , \\ [ \mathfrak { W } _ \alpha & ( c ) ] ( z ) : = \sum _ { k = 0 } ^ \infty c _ k 2 ^ { - \alpha k } z ^ { 2 ^ k } . \end{align*}


\begin{align*} \theta ^ { - 1 } \prescript { \tau \sigma \tau ^ { - 1 } } { } N _ v ^ \ast \theta \subseteq \prescript { \sigma } { } N _ l ^ \circ \iff \sigma = \varepsilon \ ; \ ; ( 1 \ , 3 ) . \end{align*}


\begin{align*} \omega _ { p } ^ { p } ( A ) \leq \underset { i , j = 1 } { \overset { 2 } { \sum } } \omega _ { p } ^ { p } ( a _ { i j } ) \end{align*}


\begin{align*} F _ { \widehat { K } _ \mathrm { a - c p t - d } ^ \mathbb { R } | K } ( \hat { k } ) & = \sum _ { k ' = 0 } ^ { K } F _ { \widehat { K } _ \mathrm { a - c p t - d } ^ \mathbb { R } | K ' } ( \hat { k } ) \ \ ! p _ { K ' | K } ( k ' ) , \end{align*}


\begin{align*} \tilde { x } ^ L : = \frac { L x } { \max _ \mathbb { R } \{ \norm { x } _ X , L \} } . \end{align*}


\begin{align*} \varphi _ { i j } ( t , y ) : = \psi \bigg ( \frac { y - x _ i ( t ) } { x _ { i j } ^ 3 ( t ) } \bigg ) , \end{align*}


\begin{align*} \frac { d \rho _ i } { d t } + d i v _ G ^ { \theta } ( \rho v ) = 0 , \end{align*}


\begin{align*} \mathbf 1 _ { ( 0 , T ) } \otimes x _ n \to \mathbf 1 _ { ( 0 , T ) } \otimes x , \\ \mathcal A ( \mathbf 1 _ { ( 0 , T ) } \otimes x _ n ) = \mathbf 1 _ { ( 0 , T ) } \otimes A x _ n \to \mathbf 1 _ { ( 0 , T ) } \otimes y . \end{align*}


\begin{align*} \int _ { 0 } ^ { T } \langle A ^ { 0 } U ^ { \delta } _ { t } + A ^ { i } \partial _ { i } U ^ { \delta } + D U ^ { \delta } - B ^ { i j } \partial _ { i } \partial _ { j } U ^ { \delta } - \delta \Lambda \Delta U ^ { \delta } , \phi \rangle d t = \int _ { 0 } ^ { T } \langle f , \phi \rangle \end{align*}


\begin{align*} f ( \gamma z ) = j ( \gamma , z ) ^ k f ( z ) . \end{align*}


\begin{align*} u \in \mathbb S _ { \mathbb X } , \ , \nabla g ( \bar x ) u \in \mathcal T _ D ( g ( \bar x ) ) , \ , \nabla g ( \bar x ) ^ * y ^ * = 0 , \ , y ^ * \in \mathcal N _ D ( g ( \bar x ) ; \nabla g ( \bar x ) u ) \Longrightarrow y ^ * = 0 \end{align*}


\begin{align*} a ( \widetilde E _ 2 ^ { ( 4 ) } , T ) = a ( \mathrm { g e n u s } \ \Theta ^ { ( 4 ) } ( S ^ { ( p ) } ) , T ) . \end{align*}


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


\begin{align*} M _ { d _ 0 } ( f , H ) = M ( d _ 0 f , H _ { d _ 0 } ) = { 2 \pi ^ 2 \over d _ 0 ^ 2 f ^ 2 } \sum _ { \delta \mid d _ 0 f } \delta \mu ( d _ 0 f / \delta ) \sum _ { h \in H _ { d _ 0 } } s ( h , \delta ) . \end{align*}


\begin{align*} \tilde \tau = 2 \beta ( \tau ) \tau , \end{align*}


\begin{align*} u _ n ( x ) = U ^ 1 ( x - x _ n ^ 1 ) + r _ n ^ 1 ( x ) \end{align*}


\begin{align*} \varPhi _ q \ , d _ q \ , \varPsi _ { \mu , q } W _ q u = \varPhi _ q \ , d _ q \ , \varPsi _ \mu B _ q ( w , u ) , \end{align*}


\begin{align*} & \lim _ { m \to \infty } F _ q ( T , q ^ { k _ m - n - 1 } ) = F _ q ( T , q ^ { 2 - n - 1 } ) . \\ \end{align*}


\begin{align*} m _ 0 = \inf \limits _ { \mathcal { N } } J \leq \inf \limits _ { \mathcal { M } } J = m _ 1 . \end{align*}


\begin{align*} \rho _ c ( x ) = \| \pi _ c ( x ) - x \| . \end{align*}


\begin{align*} r = \frac { y \pm \sqrt { y ^ 2 - 4 a v ^ 2 } } { 2 a v } . \end{align*}


\begin{align*} \| \phi \| ^ 2 : = \int _ { \mathbb { R } ^ d } | \phi ( \mathbf { x } ) | ^ 2 \mathrm { d } \mathbf { x } = 1 . \end{align*}


\begin{align*} ~ \begin{aligned} & u _ { i ' j ' } ( x ) \leq u ^ t _ { i ' j ' } ( x ) \leq \min \bigl \{ m ^ t _ { i ' j ' } , f _ { i ' } ( x ) \bigr \} , \\ & m ^ { t } _ { i ' \tau ( i ' , t ) - 1 } \leq m ^ t _ { i ' j ' } \leq m ^ { t } _ { i ' \tau ( i ' , t ) } t < t ' a _ { i ' j ' - 1 } \leq m ^ { t ' } _ { i ' j ' } \leq a _ { i ' j ' } . \end{aligned} \end{align*}


\begin{align*} & C _ 1 = 1 / \left [ ( S _ { \max } - S _ { \min } ) - \frac { \sqrt { \pi } } { 2 } \frac { 1 - \beta } { \beta } \alpha \left ( \left ( \frac { S _ { \max } - K } { \alpha } \right ) - \left ( \frac { S _ { \min } - K } { \alpha } \right ) \right ) \right ] , \\ & C _ 2 = \left [ \frac { \sqrt { \pi } } { 2 } \frac { 1 - \beta } { \beta } \alpha \left ( \frac { S _ { \min } - K } { \alpha } \right ) - S _ { \min } \right ] C _ 1 . \end{align*}


\begin{align*} K : = \Bigl \{ \lambda \begin{pmatrix} 1 \\ y \end{pmatrix} \Bigm | y \in F , \ \lambda \geq 0 \Bigr \} , \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*} A ( x , \sigma ) & : = \frac { s _ 1 ( x , \sigma ) + s _ 2 ( x , \sigma ) + s _ 3 ( x , \sigma ) } { \log ^ { \frac { 5 - 2 \sigma } { 2 } } ( x ) \exp \left ( \frac { B _ 2 ( x _ 0 ) ( 5 - 8 \sigma ) } { 3 } \sqrt { \log x } \right ) } \end{align*}


\begin{align*} & \mathcal B ^ * _ \rho \Phi _ { \lambda ^ p _ k } = \xi _ { \lambda ^ p _ k } ( 1 ) , \forall k \geq k _ 0 , \\ & \mathcal B ^ * _ \rho \Phi _ { \lambda ^ h _ k } = \xi _ { \lambda ^ h _ k } ( 1 ) , \forall | k | \geq k _ 0 . \end{align*}


\begin{align*} \dim _ { H } ( \limsup _ { \underline { i } \in \Lambda ^ * } B ( f _ { \underline { i } } ( x ) , c _ { \underline { i } } ^ { \delta } ) ) = \frac { \dim _ H ( K ) } { \delta } . \end{align*}


\begin{align*} \varphi _ R ( A , b , \lbrace x _ j : j \leq k \rbrace , \lbrace W _ j : j < k \rbrace , \lbrace \zeta _ j : j < k \rbrace ) = ( e _ { \mathrm { r e m } ( \zeta _ { k - 1 } , n ) + 1 } , \zeta _ { k - 1 } + 1 ) . \end{align*}


\begin{align*} A _ { r _ j } ( x - a _ j ) \circ A _ { r _ j } ( x - a _ j ) = - A _ { r _ j } ' ( x - a _ j ) + F _ { r _ j } ( x - a _ j ) \end{align*}


\begin{align*} \alpha _ p : = \begin{cases} ( p - 1 ) ^ { 1 / 2 } \quad p \in [ 2 , \infty ) \\ \frac { \pi } { 2 } \quad p \in [ 1 , 2 ) . \end{cases} \end{align*}


\begin{align*} h ( x , t ) = e ^ { \sqrt { \lambda } t } \psi _ \lambda ( x ) , \end{align*}


\begin{align*} Q ( z _ 1 , \dots , \widehat { z _ i } , \dots , z _ n ) = 0 : \end{align*}


\begin{align*} x \phi _ 1 x ^ { \ast } = y \phi _ 2 y ^ { \ast } = \left [ \begin{array} { c c c } 0 & 0 & e _ t \\ 0 & \zeta I _ r & 0 \\ e _ t ^ { \ast } & 0 & 0 \end{array} \right ] \left [ \begin{array} { c c c } 0 & 0 & e _ t \\ 0 & 0 _ r & 0 \\ e _ t ^ { \ast } & 0 & 0 \end{array} \right ] , 0 \leq t \leq m . \end{align*}


\begin{align*} ( \phi ^ 2 + \phi * \phi ) ( a ) & = c ^ 2 ( 1 + J ' ( \alpha , \alpha ) ) \alpha ^ 2 ( a ) + \bar c ^ 2 ( 1 + J ' ( \bar \alpha , \bar \alpha ) ) \bar \alpha ^ 2 ( a ) + 2 ( 1 + J ( \alpha , \bar \alpha ) ) \\ & = c ^ 2 ( 1 + J ' ( \alpha , \alpha ) ) \bar \alpha ( a ) + \bar c ^ 2 ( 1 + J ' ( \bar \alpha , \bar \alpha ) ) \alpha ( a ) . \end{align*}


\begin{gather*} \Box y = 2 \left ( | \phi _ t | ^ 2 - | \phi _ x | ^ 2 \right ) ( y - 1 ) + a ( x ) y _ t , \\ y ( 0 , x ) = 1 \textrm { a n d } y _ t ( 0 , x ) = 0 , \ ; \forall x \in \S , \end{gather*}


\begin{align*} A _ \alpha ( p _ \alpha f ) ( x ) = d p _ \alpha ( x ) f ( x ) + \langle x ; p _ \alpha ( x ) \nabla ( f ) ( x ) \rangle + \langle x ; f ( x ) \nabla ( p _ \alpha ) ( x ) \rangle + \sum _ { k = 1 } ^ d \partial _ k D _ k ^ { \alpha - 1 } ( p _ \alpha f ) ( x ) . \end{align*}


\begin{align*} \begin{aligned} \mathcal { T } : X _ { T _ { 0 } } ^ { s } ( g _ { 2 } , M ) & \rightarrow X _ { T _ { 0 } } ^ { s } ( g _ { 2 } , M ) \\ U & \mapsto V \end{aligned} \end{align*}


\begin{align*} \mathbf { a } _ M ( ( z , \tau ) ( z ' , \tau ' ) ) = \mathbf { a } _ M ( z + z ' , \tau + \tau ' + \tfrac { 1 } { 2 } \sigma ( z , z ' ) ) = ( M ( z + z ' ) , \tau + \tau ' + \tfrac { 1 } { 2 } \sigma ( z , z ' ) ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \dot { \gamma } ^ \ast ( s ) = - \pi R ^ 2 & \ 0 < s < \tilde { t } \\ \gamma ^ \ast ( 0 ) = r , & \end{array} \right . \quad \left \{ \begin{array} { l l } \dot { \gamma } ^ \ast ( s ) = - \pi \gamma ^ \ast ( s ) ^ 2 & \ \tilde { t } < s < t , \\ \gamma ^ \ast ( \tilde { t } ) = R . & \end{array} \right . \end{align*}


\begin{align*} \mu _ T ( \{ \lambda : | \lambda | \leq r \} ) = \frac { s ( r , 0 ) ^ 2 } { s ( r , 0 ) ^ 2 + r ^ 2 } , \end{align*}


\begin{align*} \Sigma = \bigsqcup _ { 1 \leq i \leq r } \tilde { p } _ i , \end{align*}


\begin{align*} \varphi ( P , G ) = \frac { m ( B \cap P ) } { m ( B ) } , \end{align*}


\begin{gather*} \gamma _ { X ^ * } = ( \gamma _ X ^ { - 1 } ) ^ * . \end{gather*}


\begin{align*} T _ { \max } ( s , u _ 0 ) = \infty \end{align*}


\begin{align*} 2 \geq \mu _ { \mathrm { d i s c } } ( \gamma _ { 2 } ) - \mu _ { \mathrm { d i s c } } ( \gamma _ { 1 } ) = \mu _ { \mathrm { d i s c } } ( \gamma _ { 2 } ) - n . \end{align*}


\begin{align*} P ^ { \nu _ \alpha } _ t ( f ) ( x ) = \int _ { \mathbb { R } ^ d } f ( x e ^ { - t } + ( 1 - e ^ { - \alpha t } ) ^ { \frac { 1 } { \alpha } } y ) \mu _ \alpha ( d y ) . \end{align*}


\begin{align*} v _ t + a _ n ( t , x ) \cdot v _ x ~ ~ = 0 , v ( 0 , \cdot ) ~ = ~ \bar { v } ~ \in ~ H ^ 2 \bigl ( \R \backslash [ - \delta _ 0 \tau , \delta _ 0 \tau ] \bigr ) , \end{align*}


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


\begin{align*} \lim _ { t \to 0 } t ^ { \frac { \beta } { \alpha } } \| B ( t ) \| _ { H ^ { \frac 5 2 - \alpha + \beta } ( \mathbb S ^ 1 ) } = 0 \ \ \forall \ \ \beta > 0 . \end{align*}


\begin{align*} - \partial _ r ^ { k + 2 } Q - \partial _ r ^ k \left ( \frac { N - 1 } { r } \partial _ r Q \right ) + \partial _ r ^ k Q - \partial _ r ^ k \left [ \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ { p - 1 } \right ] = 0 , \end{align*}


\begin{align*} v _ { i j } = ( a _ { i 0 } , \ldots , a _ { i j - 1 } , a _ { i j } , \ldots , a _ { i j } ) j = 0 , \ldots , n , \end{align*}


\begin{align*} \widetilde E _ 2 ^ { ( n ) } = \mathrm { g e n u s } \ \Theta ^ { ( n ) } ( S ^ { ( p ) } ) . \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { | \Lambda _ { ( n , \infty ) } ( M , \Gamma ) | } { | \Lambda ( M , \Gamma ) | } = 0 \end{align*}


\begin{align*} \partial _ t u + ( \mathbf { v } _ \varepsilon \nabla c _ { \varepsilon } - \mathbf { \tilde { v } } _ A \nabla c _ A ) + \frac { \mu _ { \varepsilon } - \mu _ A } { \varepsilon } = \mathbf { w } | _ \Gamma \ , \nabla c _ { A , 0 } - \mathbf { w } | _ \Gamma \ , \mathbf { Q } - \mathcal { C } . \end{align*}