\begin{align*} \begin{aligned} \mathbf { \hat h } ^ { L o S } ( \varphi ) = & [ 1 , e ^ { j \frac { 2 \pi d } { \lambda } \sin \varphi } , \cdots , e ^ { j \frac { 2 \pi d } { \lambda } l \sin \varphi } , \\ & \cdots , e ^ { j \frac { 2 \pi d } { \lambda } ( L - 1 ) \sin \varphi } ] ^ T , \end{aligned} \end{align*}


\begin{align*} d v _ \lambda ^ h + A v _ \lambda ^ h = \bigl ( \sigma ( u _ \lambda ) Q h + f ' _ \lambda ( u _ \lambda ) v _ \lambda ^ h \bigr ) \ , d t + \sigma ' ( u _ \lambda ) v _ \lambda ^ h B \ , d W , v _ \lambda ^ h ( 0 ) = 0 . \end{align*}


\begin{align*} \log ( 1 / \mu _ p ( g ) ) = q > c / p , \end{align*}


\begin{align*} \operatorname { s u p p } \chi _ \lambda = [ - n - 1 , n + 1 ] , \bigl . \chi _ n \bigr \vert _ { [ - n , n ] } = 1 . \end{align*}


\begin{align*} E _ \alpha = \inf \Big \{ J ( u ) = \frac 1 2 \int _ { \R ^ N } | \nabla u | ^ 2 - \int _ { \R ^ N } F ( u ) \ \Big | \ { u \in \mathcal M _ \alpha } \Big \} , \end{align*}


\begin{align*} \begin{array} { c } V _ { H , w } ^ { y , \Gamma } = \displaystyle \frac { 1 } { H } \ ; \left ( \pi _ 1 ( X ^ { y , \Gamma } _ { \tau _ { H , w } } ) - \pi _ 1 ( y ) \right ) . \end{array} \end{align*}


\begin{align*} \delta = \delta _ { j } , \delta = \delta _ { j } \pm 1 , \dotsc , \delta = \delta _ { j } \pm \lfloor k / 2 \rfloor \mod m , \delta \in \Delta _ { j } . \end{align*}


\begin{align*} & \quad \Pr \left ( > \theta | \Phi , \Phi ( \mathcal { A } ) > 0 \right ) \\ & \approx \sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } \prod \limits _ { x _ { i } \in \Phi \backslash \{ x _ 1 \} \cap { \mathcal { A } } } \frac { 1 } { \left ( 1 + \frac { m \eta \theta r _ 1 ^ { \alpha } } { M r _ i ^ { \alpha } } \right ) ^ M } . \end{align*}


\begin{align*} \psi ( b _ { k } d ) & = \langle \pi ( d ) \xi , \pi ( b _ k ) ^ * \xi \rangle = \psi ( b _ { k } ) \psi ( d ) = \omega ( b _ k ) \psi ( d ) \end{align*}


\begin{align*} \left \langle \Lambda _ \chi ^ { \rm s t i f f } \vect g , \vect v \right \rangle _ { H ^ { - 1 / 2 } ( \Gamma ; \C ^ 3 ) , H ^ { 1 / 2 } ( \Gamma ; \C ^ 3 ) } = \lambda _ \chi ^ { \rm s t i f f } ( \vect g , \vect v ) , \forall \vect g , \vect h \in H ^ { 1 / 2 } ( \Gamma ; \C ^ 3 ) . \end{align*}


\begin{align*} \widehat { \gamma } = \left ( \eta / \eta _ 0 \right ) \gamma ^ { \eta _ 0 , \delta } + \left ( 1 - \eta / \eta _ 0 \right ) \widetilde { \gamma } , \ \widetilde { \gamma } = ( \widetilde { Y } _ 1 , \widetilde { Y } _ 2 , \widetilde { X } ) . \end{align*}


\begin{align*} K ^ \circ = \{ x \in K ^ \times \mid c ( x ) x \in \mathbb { Q } ^ \times \} . \end{align*}


\begin{align*} ( \lambda C _ \lambda ) ^ { p - 1 } \cdot ( \lambda ( p - 1 ) + n - p ) & = c _ 1 C _ \mu ^ { k _ 1 } \cdot ( \lambda C _ \lambda ) ^ \alpha , \\ ( \mu C _ \mu ) ^ { p - 1 } \cdot ( \mu ( p - 1 ) + n - p ) & = c _ 2 C _ \mu ^ { k _ 2 } \cdot ( \lambda C _ \lambda ) ^ { k _ 3 } . \end{align*}


\begin{align*} \mathcal { P } _ { x , n } = \left \{ \mho ( v ) \right \} _ { v \in S _ { x } ( n ) } , \end{align*}


\begin{align*} \mathbb { P } \left ( \left | D _ { m , n } \right | > \alpha _ { n } \right ) \leq \displaystyle \frac { \mathbb { E } \left | D _ { m , n } \right | } { \alpha _ { n } } = & \frac { \sqrt { n } \ , \mathbb { E } \left | \log W _ { n } - \log W _ { m } \right | } { \sigma \sqrt { n } } \leq \displaystyle C \gamma ^ { m } \\ \leq & \ , C \frac { 1 } { n ^ { \delta / 2 } } \frac { 1 } { 1 + | x | ^ { 2 + \delta } } . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{align*}


\begin{align*} b _ j : = ( x - y ) ^ 2 \cdot z ^ j \cdot w ^ { n - 1 - j } . \end{align*}


\begin{align*} \mathcal { E } _ \varepsilon ( c ) : = \frac { 1 } { 4 } \int _ { \Omega } \int _ { \Omega } J _ \varepsilon ( x - y ) \big | c ( x ) - c ( y ) \big | ^ 2 \ : y x \end{align*}


\begin{align*} \mu ^ * ( y ) = \frac { 1 } { Z ' ( \nu ^ * , \mu ^ * ) } \exp { \left ( \frac { 2 } { \sigma ^ 2 } \frac { \delta F } { \delta \mu } ( \nu ^ * , \mu ^ * , y ) - U ^ { \rho } ( y ) \right ) } , \end{align*}


\begin{align*} \mathcal { C } _ { K , r } : = \{ x _ n \geq K | x ' | \} \cap B _ r ( 0 ) \subset U . \end{align*}


\begin{align*} \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( r ) \lambda _ f ( s ) = \delta _ { r s } + 2 \pi i ^ { 2 k } \sum _ { p | c } \frac { \mathcal { S } ( r , s , c ) } { c } \ , J _ { 2 k - 1 } \left ( \frac { 4 \pi \sqrt { r s } } { c } \right ) , \end{align*}


\begin{align*} I _ { \omega , \mathbf { c } } ( V ) = 0 , \ \ \mathbf { c } \cdot \mathbf { P } ( V ) \ge 0 . \end{align*}


\begin{align*} I _ { 2 } = \mathbb { P } \left ( W _ { n } \geq \exp \left \{ x ^ { 2 } \right \} \right ) \leq \exp \left \{ - x ^ { 2 } \right \} \mathbb { E } W _ { n } = \exp \left \{ - x ^ { 2 } \right \} . \end{align*}


\begin{align*} & \{ 0 \} = U _ 1 \subset U _ 2 \subset \ldots \subset U _ { g } \subset U _ { g + 1 } \subset U _ { - g } \subset \ldots \subset U _ { - 1 } \subset U _ 0 = \Gamma \\ & U _ { g + 1 } \subset H ( \pi ) E _ j ( \pi , \lambda ) \oplus U _ j = \Gamma - g \leq j \leq g + 1 . \end{align*}


\begin{align*} W ^ { 1 , p ( \cdot ) } \left ( \Omega , \R ^ k \right ) : = \left \{ u \in \left ( W ^ { 1 , 1 } \cap L ^ { p ( \cdot ) } \right ) \left ( \Omega , \R ^ k \right ) : \nabla u \in L ^ { p ( \cdot ) } \left ( \Omega , \R ^ { k \times n } \right ) \right \} . \end{align*}


\begin{align*} \delta y ( x ) = 0 , \delta = m _ 0 ( x ) + m _ 1 ( x ) \sigma _ q + . . . + m _ n ( x ) \sigma _ q ^ n , \end{align*}


\begin{align*} \pi = \alpha _ 1 \cdots \alpha _ k \beta \qquad \qquad \phi ( \pi , x ) = \alpha _ 1 ^ { \prime } \cdots \alpha _ m ^ { \prime } \beta ^ { \prime } \end{align*}


\begin{align*} u ( t , x ) : = u _ 0 ( x - c t ) \ , , c \in \R \ , . \end{align*}


\begin{align*} S ( n , \alpha \cdot \chi _ { 0 } ) = S ( n , \iota _ \alpha ) \end{align*}


\begin{align*} T ( f ^ { 2 } ) = 2 f T ( f ) + 2 B ( A ( f ) , A ( f ) ) \end{align*}


\begin{align*} M _ a ( \mu ) ( x ) \leq \frac { \chi _ d } { ( [ a ] + 1 ) _ d } \sum _ { j = 0 } ^ { ( [ a ] + 1 ) _ d - 1 } U ^ j ( M _ 1 ( \mu ) ) ( x ) \leq \epsilon \rho ( x ) x \in ( e M e ) _ + . \end{align*}


\begin{align*} S _ { i j } : = & \ , \pounds _ X g _ { i j } = \nabla _ i X _ j + \nabla _ j X _ i , \\ A _ { i j } : = & \ , ( d X ) _ { i j } = \nabla _ i X _ j - \nabla _ j X _ i . \end{align*}


\begin{align*} & M ^ { \rm O P T } _ { n m } : = b _ { n m } \times \underset { \eqref { e q : O T S _ N P _ F l o w _ S } - \eqref { e q : O T S _ N P _ s l a c k } \ , \cap \ , \mathcal { X } _ { n m } } { \max } ( \theta _ n - \theta _ m ) \\ & M ^ { \rm O P T } _ { m n } : = b _ { n m } \times \underset { \eqref { e q : O T S _ N P _ F l o w _ S } - \eqref { e q : O T S _ N P _ s l a c k } \ , \cap \ , \mathcal { X } _ { n m } } { \max } ( \theta _ m - \theta _ n ) \end{align*}


\begin{align*} \int _ { S _ { y } } D \left ( \alpha _ { x , g } \parallel \alpha _ { x , e } \right ) d \eta ^ { y } ( x ) = \lim _ { n } \int _ { S _ { y } } H _ { q _ { x , g } ^ { n } \parallel q _ { x , e } ^ { n } } \left ( \mathcal { P } _ { x , n } \right ) d \eta ^ { y } ( x ) . \end{align*}


\begin{align*} \| \delta '' \| _ { L ^ { \infty } [ a , b ] } = O \left ( ( \alpha \overline { h } ) ^ 2 \right ) \max \left \{ \| \sigma ''' \| _ { L ^ { \infty } ( [ a , b ] \setminus X ) } , \| \sigma '' \| _ { L ^ { \infty } ( X ) } \right \} ; \alpha \to 0 . \end{align*}


\begin{align*} N D ^ { d } _ { \lambda } ( \mathcal { H } \times J ) = \left \{ \mathcal { K } \times I \in N D _ { \lambda } ^ { d } \ , : P ^ { d } \mathcal { K } \times I \subset \mathcal { H } \times J \right \} \quad ( d = 1 , 2 ) \end{align*}


\begin{align*} d f _ { i j } + \sum _ { k } f _ { k j } \circ f _ { i k } = 0 \end{align*}


\begin{align*} U = \sum _ { u \in Z _ + ^ d } a ( u ) S _ 1 ^ { u _ 1 } \cdots S _ d ^ { u _ d } \end{align*}


\begin{align*} \widetilde { f } _ h ( \xi ) : = \gamma _ h \sigma _ h f \left ( x _ j , ( \gamma _ h \sigma _ h ) ^ { - 1 / p _ h ^ 0 } \xi \right ) = \abs { \xi } ^ { p ^ 0 _ h } ( \xi \in \R ^ k ) , \end{align*}


\begin{align*} \lim _ { x \to \infty } W ( x ) = 1 , \ \ \lim _ { x \to 0 ^ + } W ( x ) = \frac 1 3 . \end{align*}


\begin{align*} \omega ( t ; x , y ) = \sum _ { i = 1 } ^ N e _ { k _ i } ( x - y ) \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle \dfrac { d } { d t } G ( t ) = 4 I m \int _ { \mathbb { R } ^ N } \left ( \dfrac { a _ 2 } { 2 m _ 1 } x \overline { \phi } \nabla \phi + \dfrac { a _ 1 } { 2 m _ 2 } x \overline { \psi } \nabla \psi \right ) d x , \end{array} \right . \end{align*}


\begin{align*} \int _ \Omega \Delta ( V + \phi ) P \psi ^ h _ { \mu _ j , \xi _ j } d x + \int _ \Omega f _ \epsilon ( V + \phi ) P \psi ^ h _ { \mu _ j , \xi _ j } d x . = 0 , \end{align*}


\begin{align*} f ' _ m ( p ) g ( p ) - f ( p ) g ' _ m ( p ) = 0 , \end{align*}


\begin{align*} x ^ 3 - y ^ 2 z - z = 0 \end{align*}


\begin{align*} \partial _ 3 v _ 3 = ( \delta _ { j i } - a _ { j i } ) \partial _ j v _ i - \partial _ 1 v _ 1 - \partial _ 2 v _ 2 , \end{align*}


\begin{align*} \int _ { B _ { 2 s } ( x _ 0 ) } b ^ { i j , k l } v _ { i j } \eta _ { k l } d x = \int _ { B _ { 2 s } ( x _ 0 ) } b ^ { i j , k l } w _ { i j } \eta _ { k l } d x . \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ s u = V ( x ) u & , \\ u = 0 & . \end{cases} \end{align*}


\begin{align*} f ( z ) = \sum _ { n \geq 1 } f _ n n ^ { \frac { \lambda - 1 } { 2 } } e ( n z ) , \end{align*}


\begin{align*} | S ( k ) ( \widetilde { v } _ 0 \circ T - \widetilde v _ 0 ) ( x ) | = | \widetilde { v } _ 0 ( T ^ { Q _ \alpha ( k ) } x ) - \widetilde v _ 0 ( x ) | \leq \| v _ 0 \| _ { \sup } | x - T ^ { Q _ \alpha ( k ) } x | . \end{align*}


\begin{align*} \tilde { \lambda } ( \zeta ) = \lim \limits _ { k \rightarrow + \infty } \langle \lambda ^ { k } , \zeta \rangle \forall \ \zeta \in \mathrm { R } ( S ) . \end{align*}


\begin{align*} a = \varliminf _ { m \to \infty } \big ( d ( m S , \N ^ n \setminus m S ) \big ) ^ { 1 / { m } } \in [ 0 , 1 ] . \end{align*}


\begin{align*} \lim _ { r \to \infty } \frac { u ( r ) } { u _ 0 ( r ) } = \lim _ { r \to \infty } \frac { v ( r ) } { v _ 0 ( r ) } = 1 . \end{align*}


\begin{align*} \sigma = \sigma _ { H H } + \sigma _ { P P } + \sigma _ { H P } + \sigma _ { P H } \end{align*}


\begin{align*} u _ { t } + \mathcal { L } _ { A } ^ { \Phi } u = ( - \Delta ) ^ { \frac { s } { 2 } } f + g \quad Q _ { 2 } . \end{align*}


\begin{align*} \mathcal { J } _ { \mathrm { D } } ( \delta ) & = \inf _ { \substack { \lambda \in \mathbb { R } _ { + } ^ L , \ \psi _ \ell > - \infty \\ ( \psi _ \ell ) _ { \ell \in [ L ] } \in \prod _ { \ell \in [ L ] } L ^ 1 ( \mu _ \ell ) } } \Bigg \{ \langle \lambda , \delta \rangle + \sum _ { \ell = 1 } ^ L \int \psi _ \ell d \mu _ \ell : \sum _ { \ell = 1 } ^ L \psi _ \ell ( s _ \ell ) \ge g _ { \lambda , [ L ] } ( s ) , \forall s \Bigg \} . \end{align*}


\begin{align*} E _ m = \omega _ Q ^ { \frac { 1 } { \bar { p } _ 2 } - \frac { 1 } { \bar { p } _ 1 } } \int _ { \mathbb H ^ n } K ( e , y ) | y | _ h ^ { - Q / p } d y \| f _ j \| _ { L _ { | x | _ h } ^ p L _ \theta ^ { \bar { p } _ 2 } ( \mathbb H ^ n ) } . \end{align*}


\begin{align*} \langle \cdot , \cdot \rangle : W ^ { \otimes d } \times ( W ^ * ) ^ { \otimes d } \to \C \ , \ \langle v _ 1 \otimes \cdots \otimes v _ d , v _ 1 ^ * \otimes \cdots \otimes v _ d ^ * \rangle : = \prod _ { i = 1 } ^ d \langle v _ { d - i + 1 } , v _ i ^ * \rangle , \end{align*}


\begin{align*} C _ T = \sum _ { T ' \le T } X _ { T ' } \end{align*}


\begin{align*} d = \bigcap \{ e \in C \cup D : x _ d \in e \} \qquad d \neq \{ x _ d \} \ \Rightarrow \ d \notin C . \end{align*}


\begin{align*} \sum _ { n _ { j } \le x } n _ { j } ^ { - \sigma } = A \frac { x ^ { 1 - \sigma } - 1 } { 1 - \sigma } + O \Bigl ( \frac { \sigma } { \sigma - 1 / 2 } \Bigr ) , \end{align*}


\begin{align*} J : = \langle j _ f \rangle \subseteq G _ f ^ d . \end{align*}


\begin{align*} v ( x , t ) = \psi \left ( x - \gamma _ { \mathrm { n l } } ( x , t ) , t \right ) - \phi ( x ) , \end{align*}


\begin{align*} \begin{bmatrix} \partial _ { x } f & \partial _ { u } f \\ \partial _ { x } g & \partial _ { u } g \end{bmatrix} \end{align*}


\begin{align*} B \overline { x } = ( B ' [ x ] ) \cap _ { 2 n } / P \end{align*}


\begin{align*} | f ( \alpha , x ) - f _ \xi ( \alpha , x ) | = & | h ( x ) - f _ \xi ( \alpha , x ) | \\ = & | h ( x ) - f _ \xi ( \gamma , x ) + f _ \xi ( \gamma , x ) - f _ \xi ( \alpha , x ) | \\ \leq & | h ( x ) - h _ \xi ( x ) | + | f _ \xi ( \gamma , x ) - f _ \xi ( \alpha , x ) | \\ < & \frac { \epsilon } { 2 } + \frac { \epsilon } { 2 } = \epsilon . \end{align*}


\begin{align*} \tilde { R } _ h ( z ) f = \frac { i } { h } \int _ 0 ^ \infty e ^ { - i t h ^ { - 1 } \tilde { P } _ h ( z ) } f d t = \frac { i } { h } \int _ 0 ^ \infty e ^ { - i t h ^ { - 1 } ( - i h X - z - i ( q _ 1 + W ) ) } f d t \end{align*}


\begin{align*} \varphi _ { [ 0 , R ] } = \Psi ( x , \varphi _ { c } , \varphi _ { [ A , R ] } ) \end{align*}


\begin{align*} \mathbb { E } \left \{ \sup \limits _ { t } \left \Vert \langle X - x \rangle ^ { b d } e ^ { - i t H _ \omega } P _ \omega ( I ) \delta _ x \right \Vert ^ s \right \} & \leq C \sum \limits _ k 2 ^ { \frac { s b d } { 2 } + ( k + 1 ) s b d - k ( p d - s \nu ) } \\ & \leq C \sum \limits _ k \left ( 2 ^ { s b d - p d + s \nu } \right ) ^ k < \infty . \end{align*}


\begin{align*} M = G / Q \times _ { \beta } \left ( \Omega _ { 0 } \times _ { \sigma } M _ { F } \right ) . \end{align*}


\begin{align*} \underset { x ^ \prime \in \mathbb { R } ^ d } { \arg \max } \left [ \sum _ { 1 \leq \ell \leq 2 } \varphi _ \ell ( x ^ { \prime } , \lambda _ { \mathrm { D } , \ell } ^ { \star } ) \right ] = \underset { x ^ \prime \in \mathbb { R } ^ d } { \arg \max } \ \varphi _ \ell ( x ^ { \prime } , \lambda _ { \mathrm { D } , \ell } ^ { \star } ) . \end{align*}


\begin{align*} \frac { z f ^ { \prime } ( z ) } { f ( z ) } = w ( z ) + \sqrt { 1 + w ^ 2 ( z ) } , \end{align*}


\begin{align*} \partial _ z q ^ i ( z ) + \langle Z , \alpha _ i \rangle q ^ i ( z ) = \Big [ L _ i ( z ) { p ^ i ( z ) } \Big ] ^ { 2 } \prod _ { j \neq i , j \in I _ w \cup I _ g } p ^ j ( z ) ^ { 2 a _ { j i } } \prod _ { j \neq i , j \in I _ b } p ^ j ( z ) ^ { a _ { j i } } . \end{align*}


\begin{align*} \left \{ \begin{aligned} & - \epsilon ^ 2 \Delta u + V ( x ) u = \lambda u + u \log u ^ 2 , \hbox { i n } \mathbb { R } ^ N , \\ & \int _ { \mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } \epsilon ^ N , \end{aligned} \right . \end{align*}


\begin{align*} \Big \Vert \sup _ { k \leq n } | S _ k - T _ k | \Big \Vert _ 2 = O ( n ^ { 1 / p } ( \log n ) ^ { ( 1 / 2 - 1 / p ) } ) \ , . \end{align*}


\begin{align*} \lambda _ { j _ 2 } = \lambda _ { j _ 2 ^ * } = \lambda _ { j _ 3 } = \lambda _ { j _ 3 ^ * } = \lambda _ { j _ 1 } = \lambda _ { j _ 1 ^ * } , \end{align*}


\begin{align*} E _ { j - 1 } = \dfrac { \alpha h _ { j } \cosh ( \alpha h _ { j } ) - \sinh ( \alpha h _ { j } ) } { 2 \alpha ^ 2 h _ { j } ( \tanh ( \alpha x _ j ) - \tanh ( \alpha x _ { j - 1 } ) - \alpha h _ j \tanh ( \alpha x _ { j - 1 } ) \tanh ( \alpha x _ j ) ) } , \end{align*}


\begin{align*} \ , c > 0 \ , , & \liminf _ { r \to \infty } \frac { 1 } { r } \log \left ( \int _ { B _ r \cap \Omega } w ^ 2 \psi _ + ^ 2 \right ) > 0 \ , ; \\ \ , c = 0 \ , , & \liminf _ { r \to \infty } \frac { 1 } { r ^ 2 } \int _ { B _ r \cap \Omega } w ^ 2 \psi _ + ^ 2 > 0 \end{align*}


\begin{align*} \hat { R } _ { i j } = & \ , 2 \hat \omega _ { i k } \hat \omega _ { j } ^ { \ ; k } + \lambda \hat { g } _ { i j } , \\ \hat { \nabla } ^ i \hat \omega _ { i j } = & \ , 0 , \\ \hat \omega _ { i j } \hat \omega ^ { i j } = & \ , \frac { \alpha ^ 2 } { m } + \lambda . \end{align*}


\begin{align*} \left \| \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} \right \| _ { \infty } & = \left \| \begin{bmatrix} F & 0 \\ 0 & F \end{bmatrix} \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} \begin{bmatrix} F & 0 \\ 0 & F \end{bmatrix} \right \| _ { \infty } \\ & = \left \| \begin{bmatrix} F A F & F X F \\ F X ^ * F & F B F \end{bmatrix} \right \| _ { \infty } . \end{align*}


\begin{align*} \texttt { a = A + T u r i n g ( e p s ) } \end{align*}


\begin{align*} \avg { z _ h ^ { ( s ) } } _ { B _ s } = t _ h \avg { \widetilde { z } _ h ^ { ( s ) } } _ { B _ s } , \mbox { s o t h a t } \lambda _ h ^ { ( s ) } : = \abs { \avg { { z } _ h ^ { ( s ) } } _ { B _ s } } = t _ h \widetilde \lambda _ h ^ { ( s ) } . \end{align*}


\begin{align*} \gamma = \begin{bmatrix} a / \alpha & 0 \\ 0 & d \end{bmatrix} \begin{bmatrix} 1 & \alpha b / a \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \alpha & 0 \\ 0 & 1 \end{bmatrix} , \end{align*}


\begin{align*} \tilde \beta _ 0 ( a , b , c ) = ( \sigma _ 2 \sigma _ { 1 , 3 } \sigma _ 2 ) \sigma _ 2 ^ { a - 1 } \sigma _ 1 ^ { b - 1 } \sigma _ 3 ^ { c - 1 } \quad \tilde \beta ( a , b , c ) = ( \sigma _ 2 \sigma _ { 1 , 3 } \sigma _ 2 ) ^ 3 \sigma _ 2 ^ { a - 1 } \sigma _ 1 ^ { b + 1 } \sigma _ 3 ^ { c + 1 } , \end{align*}


\begin{align*} \underline { d } ( x , Y ) : = \inf _ { y \in Y } \underline { d } ( x , y ) \underline { d } ^ H ( X , Y ) : = \max \{ \sup _ { x \in X } \underline { d } ( x , Y ) , \sup _ { y \in Y } \underline { d } ( y , X ) \} \end{align*}


\begin{align*} \mathbb { E } ( \overline { S } _ n ) ^ 2 = c _ n ^ 2 + O ( \log ^ { - A } n ) . \end{align*}


\begin{align*} A _ 1 = A + h , \end{align*}


\begin{align*} D \lambda ^ { p } - E \mu ^ { 2 p } = F \nu , \end{align*}


\begin{align*} \sigma ( t ) : = \frac { p + t } { p - t } , \ , \ , \ , \sigma ^ { \prime } ( t ) : = \frac { p ^ { \prime } + t } { p ^ { \prime } - t } . \end{align*}


\begin{align*} \left ( \tilde { W } _ 1 ^ { \sharp } ; \ , \tilde { \mathcal { L } } \cup \mathcal { L } ' , \ , \mathcal { L } '' \cup L \right ) = \left ( \tilde { W } _ 1 ^ { \sharp } ; \ , \tilde { \mathcal { L } } \cup \mathcal { L } \cup L \right ) \end{align*}


\begin{align*} \sigma ( \hat { A } ) = \sigma ( ( 1 , 2 , \ldots , k ) ) = \sigma _ { | \{ 1 , 2 \ldots , k \} } ( ( 1 , 2 , \ldots , k ) ) = ( \sigma ( 1 ) , \sigma ( 2 ) , \ldots , \sigma ( k ) ) . \end{align*}


\begin{align*} X - a Y & = h _ m \\ - a X + Y & = h _ { \lambda _ m } \end{align*}


\begin{align*} \theta _ { 3 / 2 } ( \tfrac { a z + b } { c z + d } , \epsilon ) = \lambda ^ { \pm } ( \gamma , \epsilon ) ( \sqrt { \det \gamma ( c z + d ) } ) ^ 3 \sum _ { n \in \Z } n e ^ { i ( \det \gamma ) \epsilon \pi n ^ 2 z } , \end{align*}


\begin{align*} { \mathrm { T r \ , } } f \left ( \sum _ { i = 1 } ^ m Z ^ * _ i A _ i Z _ i \right ) \le { \mathrm { T r \ , } } \sum _ { i = 1 } ^ m Z ^ * _ i f ( A ) _ i Z _ i . \end{align*}


\begin{gather*} \sum _ { ( v _ \nu ) \in \mathcal { U } _ \gamma } \prod _ { \nu \in S } \min ( 1 , e ^ { v _ \nu } ) \\ \lesssim \sum _ { M = 0 } ^ \infty \sum _ { k = 1 } ^ { \# S - 1 } \begin{pmatrix} \# S - 1 \\ k \end{pmatrix} e ^ { - M } ( M ^ { k + 1 } + M ^ { \# S } ) \\ \lesssim \sum _ { M = 0 } ^ \infty e ^ { - M } M ^ { \# S } < \infty \end{gather*}


\begin{align*} \bar { \mu } = \norm { \cdot } _ 1 - \lim _ { n \to \infty } S _ n ( \mu ) , \end{align*}


\begin{align*} S _ { \omega , \mathbf { c } } ( \Psi _ { \omega } ) = \omega ^ { 2 - \frac { d } { 2 } } S _ { 1 , \frac { \mathbf { c } } { \sqrt { \omega } } } ( \Psi ) . \end{align*}


\begin{align*} \left \{ \begin{array} { @ { \ , } l l l } \mathcal { E C S } _ { 0 0 1 } = \{ S ( p , q , a , b , c , d ) \in \mathcal { E C } : p = a = 0 , c \ne 0 \} \\ \mathcal { E C S } _ { 0 1 0 } = \{ S ( p , q , a , b , c , d ) \in \mathcal { E C } : p = c = 0 , a \ne 0 \} \\ \mathcal { E C S } _ { 1 0 0 } = \{ S ( p , q , a , b , c , d ) \in \mathcal { E C } : a = c = 0 , p \ne 0 \} . \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} \mathbf { h } _ t ( i ) = - \sum _ { \zeta \in \mathbb { F } _ q } & \left ( s _ i = \zeta \mid \mathbf { y } _ t , \dots , \mathbf { y } _ 0 \right ) \log _ 2 \left ( s _ i = \zeta \mid \mathbf { y } _ t , \dots , \mathbf { y } _ 0 \right ) . \end{aligned} \end{align*}


\begin{align*} \varphi _ { k , \mathrm { e s t } } = \frac { 1 } { \mu } \mathrm { u n w r a p } \left ( \arg \left ( \sum _ { k ' = k - K } ^ { k + K } z _ { k ' } ^ \mu \right ) \right ) , \end{align*}


\begin{align*} L _ t [ f ] ( h ) & = 2 \pi \int _ { \Z ^ d } | \hat V ( k ) | ^ 2 \Big ( \rho ^ { H } _ t ( h - k , k ) f ( h - k ) \widetilde f ( h ) - \rho ^ { H } _ t ( h , k ) f ( h ) \widetilde f ( h + k ) \Big ) \d k . \end{align*}


\begin{align*} \sum _ { m , n \in \mathcal { M } ' } r ( m ) r ( n ) \Phi \left ( T \log { \frac { m } { n } } \right ) \le \sum _ { m \in \mathcal { M } ' } r ( m ) ^ 2 + \sum _ { \substack { m , n \in \mathcal { M } ' \\ m \not = n } } r ( m ) r ( n ) \Phi \left ( T \log { \frac { m } { n } } \right ) . \end{align*}


\begin{align*} X _ 0 \rho ( x ) = 0 \Rightarrow X _ 0 ^ 2 \rho ( x ) < 0 \end{align*}


\begin{align*} & \lim _ { N \to \infty } \frac { 1 } { N } \log \P \left ( N ^ { - 1 } \max _ { i \in [ N ] } d _ i \geq \beta \right ) = \lim _ { N \to \infty } \sup _ { x \in [ 0 , 1 ] } \frac { 1 } { N } \log \P \left ( N ^ { - 1 } d _ { i _ x } \geq \beta \right ) \\ & = \sup _ { x \in [ 0 , 1 ] } \lim _ { N \to \infty } \frac { 1 } { N } \log \P \left ( N ^ { - 1 } d _ { i _ x } \geq \beta \right ) = - \inf _ { x \in [ 0 , 1 ] } J _ r ( x , \beta ) = - \widehat { \psi } _ r ( \beta ) . \end{align*}


\begin{align*} \Pr ( T _ { r : k { + } 1 } > t ) = \Pr ( X _ t \le k \ , \vert \ , X _ 0 = r ) . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c \underline u ) ^ n = ( 1 + \mu _ 1 M ) ^ n f ^ n \omega ^ n & \textnormal { i n } & \Omega , \\ \underline u = 0 & \textnormal { o n } & \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} \| x \| = \max \{ | x _ 1 | , \ldots , | x _ k | \} . \end{align*}


\begin{align*} \mathcal { I } _ { A J P A } = \{ I _ { 1 , j , k } , I _ { 2 , j , k } , I _ { 3 , j , k } \colon j , k \in \N _ 3 \} \end{align*}


\begin{align*} \pi \cot ( \pi x ) = \lim _ { N \to \infty } \sum _ { k = - N } ^ N \frac 1 { x + n } = \frac 1 x + \lim _ { N \to \infty } \sum _ { k = 1 } ^ N \frac { 2 x } { x ^ 2 - n ^ 2 } \end{align*}


\begin{align*} v ( p ^ \alpha ) \ : = \ p + \iota ( \alpha ) \end{align*}


\begin{align*} \mathbb { M } _ 2 = ( F \slash R ) \slash ( F \slash R ) _ { \mathrm { t o r } } , \end{align*}


\begin{align*} g _ 0 = [ u _ 1 , \ldots , u _ { r + 1 } , a _ 0 , \ldots , a _ r , g _ { r + 1 } ] \in P _ { \le r } . \end{align*}


\begin{align*} L ^ p ( \Omega ) ( w ) ' = L ^ { p ' } ( \Omega ) ( w ^ { - 1 } ) , \end{align*}


\begin{align*} \int _ \Omega \ ! | f g | \ , \mathrm { d } x = | \iota ( g ) ( \widetilde { f } ) | \leq \| \iota ( g ) \| _ { X ^ \ast } \| \widetilde { f } \| _ X \leq \| \iota ( g ) \| _ { X ^ \ast } \| f \| _ X . \end{align*}


\begin{align*} H _ \Gamma ^ s ( \partial \mathrm { M } ) = \{ \varphi \in H ^ s ( \partial \mathrm { M } ) ; \ ; \mathrm { s u p p } ( \varphi ) \subset \Gamma \} . \end{align*}


\begin{align*} F _ 0 = [ & - \det \Delta \cdot ( \Delta ^ { - 1 } ( \partial F _ i / \partial x _ 0 ) ) \\ & - \det \Delta \cdot ( \Delta ^ { - 1 } ( \partial F _ i / \partial x _ 1 ) ) ] \end{align*}


\begin{align*} ( \underline { \varphi } \otimes \varphi ) _ \ast ( m _ H ) ( \underline { A } \times A ) = m _ H ( \underline { \varphi } ^ { - 1 } ( \underline { A } ) \cap \varphi ^ { - 1 } ( A ) ) = m _ H ( \underline { C } \cap C ) . \end{align*}


\begin{align*} \lim _ { N \to \infty } N ^ { - 1 } S _ j ( \tau _ N ) & = \frac { \nu q } { \lambda } \int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } ( 1 - y ) y ^ { j - 1 } d y \\ & = \frac { \nu q } { \lambda } \sum _ { k = 0 } ^ \infty \frac { p ^ k } { ( j + k ) ( j + k + 1 ) } , j \geq 1 . \end{align*}


\begin{align*} \theta _ { \epsilon } ( x ) : = \exp \left ( \frac { p \sqrt { 1 + | x | ^ 2 } } { 1 + \epsilon \sqrt { 1 + | x | ^ 2 } } \right ) . \end{align*}


\begin{align*} ( \textsf { E } \Vert B A ^ { \prime } \Vert ^ { p } ) ^ { 1 / p } = ( \textsf { E } _ { \tilde { g } } \textsf { E } _ { g } \Vert B A ^ { \prime } \Vert ^ { p } ) ^ { 1 / p } \lesssim \sqrt { n } p . \end{align*}


\begin{align*} \eta ( p , \nu ) = \eta ( p , j + 1 - \nu - p ) . \end{align*}


\begin{align*} ( a t + b ) ^ { p \theta + \ell } & = ( a t + b ) ^ { p \theta } ( a t + b ) ^ \ell \\ & = \begin{cases} a ^ { p \theta + \ell } t ^ { p \theta + \ell } & b = 0 , \\ ( a ^ { p \theta } t ^ { p \theta } + b ^ { p \theta } ) \left ( \sum _ { m = 0 } ^ \ell \binom { \ell } { m } a ^ { \ell - m } b ^ m t ^ { \ell - m } \right ) & . \end{cases} \end{align*}


\begin{align*} { \rm R e } ( i e ^ { - i \pi \beta _ 1 } w ) = - { \rm I m } ( e ^ { - i \pi \beta _ 1 } w ) = { \rm I m } ( e ^ { i \pi ( 1 - \beta _ 1 ) } w ) \end{align*}


\begin{align*} b _ p + k x _ p + l y _ p > a _ p + n x _ p = c _ p + m y _ p . \end{align*}


\begin{align*} \beta ' _ n = \frac { ( b q ) _ n } { ( b ) _ n } \beta _ n . \end{align*}


\begin{align*} p ^ 0 _ h : = p _ h ( 0 ) = p ( x _ h ) , \end{align*}


\begin{align*} r x ^ { r - 1 } \cdot ( x ^ q y - x y ^ q ) + x ^ r \cdot ( - y ^ q ) + k r x ^ { r - 1 } \cdot ( z ^ q x - z x ^ q ) + ( z ^ r + k x ^ r ) \cdot z ^ q = 0 \end{align*}


\begin{align*} P _ { N / 8 \leq \cdot \leq 8 N } P _ N u = P _ N u . \end{align*}


\begin{align*} & e ^ { - \lambda t } \prod _ { i = 1 } ^ { N } \textsf { E } \exp ( C _ { 0 } \lambda \Vert B _ { i } \Vert _ { 2 } ^ { 2 } ( \max _ { j \le n } \eta _ { j } - \textsf { E } \max _ { j \le n } \eta _ { j } ) ) \le \exp ( - \lambda t + C _ { 1 } ^ { 2 } \lambda ^ { 2 } \sum _ { i \le N } \Vert B _ { i } \Vert _ { 2 } ^ { 4 } ) . \end{align*}


\begin{align*} X \# Y = X ^ { 1 / 2 } W Y ^ { 1 / 2 } \end{align*}


\begin{align*} p \mapsto H ( \xi _ { n } | X , \eta _ { p } ) = \int _ { X } H ( \xi _ { n } ^ { x } ) d \eta _ { p } ( x ) = \int _ { Y } \int _ { \pi ^ { - 1 } ( y ) } H ( \theta _ { L _ { x } \ast } \mathbb { P } _ { \mu , n } ^ { y } ) d \eta _ { p } ^ { y } ( x ) d \nu ( y ) \end{align*}


\begin{align*} \lim _ { t \to \infty } \frac { S _ j ( t ) } { M _ j ( t ) } = \lim _ { N \to \infty } \frac { S _ j ( \tau _ N ) } { M _ j ( \tau _ N ) } = \frac { \int _ 0 ^ \infty e ^ { - \lambda s } p _ j ( s ) d s } { \int _ 0 ^ \infty e ^ { - \lambda s } \big ( \sum _ { k = j } ^ \infty p _ k ( s ) \big ) d s } , j \geq 1 , \end{align*}


\begin{align*} \tilde \beta _ 0 ( a , b , b ) = ( \sigma _ 2 \sigma _ { 1 , 3 } \sigma _ 2 ) \sigma _ 2 ^ { a - 1 } \sigma _ { 1 , 3 } ^ { b - 1 } \quad \tilde \beta ( a , b , b ) = ( \sigma _ 2 \sigma _ { 1 , 3 } \sigma _ 2 ) ^ 3 \sigma _ 2 ^ { a - 1 } \sigma _ { 1 , 3 } ^ { b - 1 } , \end{align*}


\begin{align*} \tilde { \delta } = ( d _ 1 , d _ p , \dots , d _ 2 ) \quad \quad \tilde { \delta } ^ { \prime } = ( x , \pi _ { l - 1 } , \dots , \pi _ { j + 1 } , \pi _ j , d _ p , \dots , d _ 2 , d _ 1 , a _ q , \dots , a _ 2 , a _ 1 ) . \end{align*}


\begin{align*} \mathcal { G } _ { \overline { \psi } _ 0 } ^ { 1 , \mathbf { K } } : = \mathcal { G } ^ 1 ( \mathbb { Q } ) \bigcap \mathbf { K } _ { { \overline { \psi } _ 0 } } ^ p \end{align*}


\begin{align*} \Theta ( \delta ) : = \left \{ \int _ { \mathcal { S } } f ( y _ 1 , y _ 2 ) \ , d \gamma ( y _ 1 , y _ 2 , x ) : \gamma \in \Sigma ( \delta ) \right \} , \end{align*}


\begin{align*} \dim _ { \mathbb { C } } ( { \rm S p } \Sigma _ { \{ i , j \} } ) = \begin{cases} p ( i ) + p ( j ) - 3 , & p ( i ) + p ( j ) < 2 n + 1 \\ p ( i ) + p ( j ) - 4 , & p ( i ) + p ( j ) > 2 n + 1 , \end{cases} \end{align*}


\begin{align*} | \Phi ( \Pi ) | = | \Pi | = | \Pi ^ { \prime } | \qquad \qquad | \Phi ( \hat { \Pi } ) | = | \hat { \Pi } | = | \hat { \Pi } ^ { \prime } | , \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle a _ 2 \int _ { \mathbb { R } ^ N } | \phi | ^ 2 d x + a _ 1 \int _ { \mathbb { R } ^ N } | \psi | ^ 2 d x = a _ 2 \int _ { \mathbb { R } ^ N } | \phi _ 0 | ^ 2 d x + a _ 1 \int _ { \mathbb { R } ^ N } | \psi _ 0 | ^ 2 d x , \end{array} \right . \end{align*}


\begin{align*} E _ { 1 , b } ( Y , \Theta _ 1 , \Theta _ 2 ) & \ll \widehat { P } ^ n \widehat { Y } ^ { 5 / 2 - n / 2 + \varepsilon } \widehat { \Theta } _ 1 \widehat { \Theta } _ 2 \widehat { Z } ^ { 1 - n / 2 - \mu } \left ( \widehat { V } ^ n + \widehat { Y } ^ { n / 3 } \right ) \\ & = \widehat { Y } ^ { 5 / 2 + n / 2 + \varepsilon } \widehat { \Theta } _ 1 \widehat { \Theta } _ 2 \widehat { Z } ^ { 1 + n / 2 - \mu } + \widehat { P } ^ { n - 5 } \widehat { Y } ^ { 5 / 2 - n / 6 + \varepsilon } \widehat { Z } ^ { 3 - n / 2 - \mu } . \end{align*}


\begin{align*} f _ t ( \lambda ) : = \begin{cases} \lambda ^ { i t } = e ^ { i t \log \lambda } & ( \lambda > 0 ) , \\ 0 & ( \lambda = 0 ) . \end{cases} \end{align*}


\begin{align*} I _ { \omega , \mathbf { c } } ( U ( t ) ) \le 2 ( S _ { \omega , \mathbf { c } } ( \Phi _ 0 ^ { \lambda _ 0 } ) - S _ { \omega , \mathbf { c } } ( \Phi _ 0 ) ) = - 2 ( \mu _ { \omega , \mathbf { c } } - S _ { \omega , \mathbf { c } } ( \Phi _ 0 ^ { \lambda _ 0 } ) ) < 0 . \end{align*}


\begin{align*} \mathfrak { M } _ J = \left ( \mathfrak { M } _ { J _ 1 } , \mathfrak { M } _ { J _ 2 } , \ldots , \mathfrak { M } _ { J _ n } \right ) , \end{align*}


\begin{align*} M _ q ( G ) = Z _ q ( G ) = T + \max _ { 1 \le i _ 1 < \ldots < i _ q \le s } \sum _ { j = 1 } ^ q a _ { i _ j } . \end{align*}


\begin{align*} Z _ L ( S ) = \{ x \in L : [ x _ \lambda y ] = 0 , ~ ~ ~ ~ \forall y \in S \} . \end{align*}


\begin{align*} \tilde { T } ( x , \omega ) = \left ( \omega _ { 1 } ^ { - 1 } . x , \left ( \omega _ { 1 } ^ { - 1 } \omega _ { n + 1 } \right ) _ { n \in \mathbb { Z } } \right ) . \end{align*}


\begin{align*} \left ( \frac { 1 } { T _ i ^ { \infty } } + \frac { 1 } { \theta _ i ^ 2 T _ i ^ * ( u ) } , \frac { Z _ i ^ * ( u ) ^ 2 } { T _ i ^ * ( u ) } + \frac { T _ i ^ { \infty } Z _ i ^ * ( u ) ^ 2 } { \theta _ i ^ 2 T _ i ^ * ( u ) ^ 2 } \right ) _ { i \in V } & \overset { l a w } = \left ( 2 \delta _ i + \frac { 1 } { \theta _ i ^ 2 T _ i ^ * ( u ) } , \frac { z _ i ^ 2 } { \theta _ i ^ 2 T _ i ^ * ( u ) } + \frac { z _ i ^ 2 } { 2 \delta _ i ( \theta _ i ^ 2 T _ i ^ * ( u ) ) ^ 2 } \right ) _ { i \in V } \end{align*}


\begin{align*} 0 = \sum _ { i = 1 } ^ D \sum _ { j = 1 } ^ D \alpha _ i \bar { \alpha } _ j \sum _ { l , \ , m \geq 1 } G _ k \left ( \frac { l m } { p } \right ) \frac { 1 } { \sqrt { l m } } \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( i ) \lambda _ f ( j ) \lambda _ f ( l ) \lambda _ f ( m ) . \end{align*}


\begin{align*} \mathbf { e } _ \lambda = \max \left ( \frac { 1 } { \mathrm { d i s t } ( \lambda , \sigma _ \lambda ) } , 1 \right ) , \end{align*}


\begin{align*} \sqrt { 2 H ( \Phi ( p ) ) } = \ell ( \gamma ) = \delta ( p ) \in ( 0 , \epsilon ) , \end{align*}


\begin{align*} \mathcal { R } ( t ) = \bigcup _ { 0 \leq s \leq t } ( Z _ s ) . \end{align*}


\begin{align*} \frac { a } { a + b } \ : x + \frac { b } { a + b } \ : y = z \end{align*}


\begin{align*} y _ { i + 1 } & = y _ { i } + h f \left ( x _ { i } , y _ { i } \right ) + h \int \limits _ { x _ { 0 } } ^ { x _ { i } } K d t \\ & = y _ { i } + h f \left ( x _ { i } , y _ { i } \right ) + \frac { h ^ { 2 } } { 2 } \left ( \sum \limits _ { j = 0 } ^ { j = i } 2 K _ { j } - \left ( K _ { 0 } + K _ { i } \right ) \right ) . \end{align*}


\begin{align*} W _ { [ j + 1 ] } = M _ { [ j + 1 ] } - \left ( S _ { [ 1 ] } \diamond \cdots \diamond S _ { [ j ] } \right ) J ^ T _ { 2 j } \left ( S _ { [ 1 ] } \diamond \cdots \diamond S _ { [ j ] } \right ) ^ T J _ { 2 | \alpha _ i | } M _ { [ j + 1 ] } . \end{align*}


\begin{align*} \mathcal { Q } _ { x _ 0 , L _ { \hat { n } } } = \bigcap _ { k = 0 } ^ { \hat { n } } Y _ { k } ( \omega _ { k - 1 } ) \end{align*}


\begin{align*} \sum _ { u \in V ^ * , u \not = v } \left ( d ( v , u ) - 1 \right ) + \sum _ { 1 \le i \le r ' } \left ( d ( u _ i , v ) - 1 \right ) \ge ( r - 1 ) ^ 2 . \end{align*}


\begin{align*} T \oplus ^ { \operatorname { s } } T ' & = \begin{pmatrix} W \oplus W ' & X \oplus X ' \\ Y \oplus Y ' & Z \oplus Z ' \end{pmatrix} . \end{align*}


\begin{align*} L ( x ) = \frac { 6 } { \pi ^ 2 } \left ( L i _ 2 ( x ) + \frac { 1 } { 2 } \log ( x ) \log ( 1 - x ) \right ) = \frac { 6 } { \pi ^ 2 } \left ( \sum _ { k = 1 } ^ { \infty } \frac { x ^ k } { k ^ 2 } + \frac { 1 } { 2 } \log ( x ) \log ( 1 - x ) \right ) . \end{align*}


\begin{align*} d _ k ( y ) = x _ F \wedge y \ . \end{align*}


\begin{align*} t ^ 3 + 2 t ^ 2 - t - 1 = 0 , \end{align*}


\begin{align*} \begin{aligned} \abs { \xi _ { = 1 } } & \leq C I _ g \rho _ 0 ^ { 3 + 2 / d } | x _ 1 - x _ 2 | ^ 2 . \end{aligned} \end{align*}


\begin{align*} s a R _ 2 : \left ( \begin{array} { c c c } x = 0 : & 0 & 0 \\ x = 1 : & 0 & 0 \\ x = \infty : & 1 / 2 & 1 / 2 \end{array} \right ) . \end{align*}


\begin{align*} \Sigma _ 0 ( \delta _ 0 ) & = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \boldsymbol { K } ( \mu , \gamma ) \le \delta _ 0 \right \} \\ \Sigma ( \delta ) & = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \boldsymbol { K } _ \ell ( \mu _ { \ell , 3 } , \gamma _ { \ell , 3 } ) \le \delta _ \ell , \ \forall \ell = 1 , 2 \right \} . \end{align*}


\begin{align*} \alpha ^ 2 \log ( 1 / \alpha ) = p ^ 2 \log ( 1 / p ) < W ^ { = 1 } [ f ] . \end{align*}


\begin{align*} \mathcal { H } ^ 1 \left ( \Omega \cap \left ( \overline { J _ u } \setminus J _ u \right ) \right ) = 0 . \end{align*}


\begin{align*} \| U ( t ) \| _ { \mathcal { H } ^ 1 } ^ 2 \le \frac { 6 } { C } S _ { \omega , \mathbf { c } } ( U ( t ) ) = \frac { 6 } { C } S _ { \omega , \mathbf { c } } ( U _ 0 ) \end{align*}


\begin{align*} A _ i & = \left \{ ( t _ 1 , \dots , t _ \ell ) \in \Delta ^ \ell \ | \ t _ i \geq 1 - \delta _ \ell \right \} 1 \leq i \leq \ell \ , \\ A _ 0 & = \left \{ ( t _ 1 , \dots , t _ \ell ) \in \Delta ^ \ell \ | \ \sum _ { j = 1 } ^ n t _ j \leq \delta _ \ell \right \} \ . \end{align*}


\begin{align*} \psi ( b _ k b _ k ^ * ) = | \omega ( b _ k ) | ^ 2 = | \psi ( b _ k ) | ^ 2 , 1 \leq k \leq n . \end{align*}


\begin{align*} [ \beta _ k , \mathcal D _ \ell ] = \int _ { \Lambda ^ { * } } \ 1 _ { \S ( 1 ) } ( p - \ell ) a _ { p + k - \ell } a _ p \d p + \int _ { \Lambda ^ { * } } \ 1 _ { \S ( 1 ) } ( p ) a _ { p + k - \ell } a _ p \d p \ . \end{align*}


\begin{align*} \rho _ j = \begin{cases} \mathfrak { s } _ j ^ 2 - \varepsilon _ j & j \leq \log | h - h ' | ^ { - 1 } , \\ \varepsilon _ j & j > \log | h - h ' | ^ { - 1 } . \end{cases} \end{align*}


\begin{align*} H _ { p , p ^ { \prime } } ( 1 , c ) = \frac { 4 \pi } { \sqrt { c } } \ , \sum _ { n = 1 } ^ { \infty } \frac { p ^ { \prime 2 } + \lambda _ { n } ^ { \prime 2 } } { p ^ { \prime } \left ( p ^ { \prime } + \frac { 1 } { \pi } \right ) + \lambda _ { n } ^ { \prime 2 } } \cdot \frac { \lambda _ { n } ^ { \prime - 1 } } { \sigma \left ( \sqrt { c } \lambda _ { n } ^ { \prime } \right ) e ^ { 2 \pi \sqrt { c } \lambda _ { n } ^ { \prime } } - 1 } \end{align*}


\begin{align*} _ { s = 1 } \tilde { \zeta } _ { p , p ^ { \prime } } ( s , c ) = \frac { \pi } { \sqrt { c } \left ( 1 + \frac { 1 } { \pi p } \right ) } . \end{align*}


\begin{align*} s _ + = \sqrt { \frac { 1 } { 2 } + \frac { 1 } { 2 } \sqrt { \frac { C - 4 } { C } } } \end{align*}


\begin{align*} K = \Q ( \alpha _ 1 , \dots , \alpha _ n ) : = \bigcap \{ T | T \mathbb { C } , \alpha _ 1 , \dots , \alpha _ n \in T \} . \end{align*}


\begin{align*} H ^ p _ A = \{ f \in \mathcal { S } ' \colon M _ { \phi , A } ^ 0 f \in L ^ p \} \end{align*}


\begin{align*} \left ( \Pi _ { i = 1 } ^ t \sigma _ { ( 1 , 1 ) } \right ) \star \sigma _ \mu = \left ( \Pi _ { i = 1 } ^ t \tau _ { ( 1 , 1 ) } \right ) \star \tau _ \mu = \tau _ { ( \mu _ 1 + t , \mu _ 2 + t ) } = \sigma _ { ( \mu _ 1 + t , \mu _ 2 + t ) } . \end{align*}


\begin{align*} Y _ i = \mu ( X _ i ) + \sigma ( X _ i ) \epsilon _ i , i = 1 , 2 , \cdots , n , \end{align*}


\begin{align*} \Lambda ^ 3 _ t - \widetilde { \Lambda } ^ 3 _ t = \max _ { t _ 0 \le s \le t } \ , | \Lambda ^ 3 _ s - \widetilde { \Lambda } ^ 3 _ s | > 0 . \end{align*}


\begin{align*} V _ e = \{ r ^ i ( v ) : 0 \le i < | V _ e | \} . \end{align*}


\begin{align*} ( a _ 0 \cdots a _ m ) _ { 1 0 } \ = \ f ( b _ m \cdots b _ 0 ) _ { 1 0 } . \end{align*}


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


\begin{align*} \frac { 1 } { T _ { * } } = \frac { 2 7 \varepsilon } { 4 k T } . \end{align*}


\begin{align*} L _ { \Sigma } = \Delta + | A | ^ 2 + \mathrm { R i c } ( \nu , \nu ) . \end{align*}


\begin{align*} c _ { \ell } ( s _ \ell , s _ \ell ' ) = ( s _ \ell - s _ \ell ' ) ^ { \top } V _ \ell ^ { - 1 } ( s _ \ell - s _ \ell ' ) . \end{align*}


\begin{align*} \gamma = \frac { 1 } { 8 } \sigma = \sigma ( d , p ) \in \left ( 0 , \frac { 1 } { 2 } \right ) , \end{align*}


\begin{align*} f ^ K ( \cdot , t ) = f _ 0 \circ \big ( \tilde { y } _ t ^ K \big ) ^ { - 1 } , \end{align*}


\begin{align*} \chi _ j R ( \lambda ) \chi _ j = \chi _ j R _ j ( \lambda ) \chi _ j \end{align*}


\begin{align*} \left \| P _ 1 u \right \| ^ 2 + \left \| P _ 2 u \right \| ^ 2 + 2 \left ( P _ 1 u , P _ 2 u \right ) = \left \| e ^ { - s \sigma } f \right \| ^ 2 . \end{align*}


\begin{align*} m _ j : = m ^ { ( s _ { h _ j } ) } _ { \rho _ { h _ j } } \to m \mbox { a s } j \to + \infty . \end{align*}


\begin{align*} \overline { \overline { C } } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) = \overline { C } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) \overline { s } ( g _ 1 ) \overline { s } ( g _ 2 ) \overline { s } ( g _ 1 g _ 2 ) ^ { - 1 } . \end{align*}


\begin{align*} \pi _ { \psi } ( g ) f = \theta _ { L , X ^ { \ast } } [ \pi _ { \psi } ( g ) ( f ' ) ] = \theta _ { L , X ^ { \ast } } [ \pi _ { \psi } ( g ) \theta _ { X ^ { \ast } , L } ( f ) ] . \end{align*}


\begin{align*} \begin{bmatrix} I \\ S ( t ) \end{bmatrix} = \begin{bmatrix} \cos t I & - \sin t I \\ \sin t I & \cos t I \end{bmatrix} \begin{bmatrix} I \\ S ( 0 ) \end{bmatrix} G ( t ) . \end{align*}


\begin{align*} U _ N ( t ) = \bigoplus _ { k = 0 } ^ N q ( t ) ^ { \otimes k } \left ( \frac { a ( u _ N ( t ) ) ^ { N - k } } { \sqrt { ( N - k ) ! } } \right ) \end{align*}


\begin{align*} \mathcal { L } : = \Big \{ ( \underbar { x } , \underbar { y } ) : = ( x _ 1 , \ldots , x _ N , y _ 1 , \ldots , y _ N ) \in R : x _ i , y _ i \geq 0 \ \sum _ { i = 1 } ^ { N } ( x _ i + y _ i ) \leq 1 \Big \} . \end{align*}


\begin{align*} P _ V a | _ V = \omega ( a ) I , a \in M \end{align*}


\begin{align*} \textbf { D } _ { n } = \begin{bmatrix} \sqrt { p _ { 1 } } & 0 & \cdots & 0 \\ 0 & \sqrt { p _ { 2 } } & \cdots & 0 \\ \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & \cdots & \sqrt { p _ { n } } \end{bmatrix} \end{align*}


\begin{align*} & \frac { 2 } { \pi } \int _ { - ( \log { t } ) ^ { 2 } } ^ { ( \log { t } ) ^ { 2 } } \log \zeta ( \sigma + i ( t + u ) ) \left ( \frac { \sin { \psi u } } { u } \right ) ^ 2 e ^ { i H u } d u \\ & = \sum _ { n = 1 } ^ { \infty } \frac { \Lambda ( n ) \max ( 0 , \psi - | H - \log { n } | ) } { n ^ { \sigma + i t } \log { n } } + O \left ( \frac { e ^ { | H | + 2 \psi } } { ( \log { t } ) ^ { 2 } } \right ) . \end{align*}


\begin{align*} f = - \log ( 1 + w ) \end{align*}


\begin{align*} \bar \Psi _ n = \sum _ { | T | = n } X _ T . \end{align*}


\begin{align*} [ X _ 1 , \dots , X _ i ] : = \phi _ i ( X _ 1 \otimes \cdots \otimes X _ i ) . \end{align*}


\begin{align*} & F \ ! \left ( \sum _ { j = 1 } ^ n q _ j ( \xi _ I - \rho \cdot \xi _ I ) t _ j \right ) \cdot ( \rho \ast \kappa _ I ( x ) ) \\ = \ & \psi ( c _ I ( \rho ) ) \cdot ( \rho \ast \kappa _ I ( x ) ) = \varphi _ I ( \rho ) \cdot \kappa _ I ( x ) \ , \end{align*}


\begin{align*} \Delta \left ( \frac { m } { \ell } \right ) = \max \left \{ 1 - \left | \frac { m } { \ell } \right | , 0 \right \} = \begin{cases} 1 , & m = 0 \\ 0 , & m \neq 0 . \end{cases} \end{align*}


\begin{align*} f ^ 2 = Q - \sum \nolimits _ { i } \eta ^ i \otimes \xi _ i , \eta ^ i ( \xi _ j ) = \delta ^ i _ j \ , . \end{align*}


\begin{align*} \begin{aligned} X \cdot \left ( \left | X u \right | ^ { p - 2 } X u \right ) & = - \lambda | u | ^ { p - 2 } u & & \Omega , \\ u & = 0 & & \partial \Omega , \end{aligned} \end{align*}


\begin{align*} \langle z ^ n \otimes \lbrace 0 , \infty \rbrace , f \rangle = \langle T ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) , T f \rangle = 0 . \end{align*}


\begin{align*} \mathrm { I } ( \xi _ { 1 } ^ { x } , \mathcal { T } _ { x } ) & = \int _ { G ^ { \mathbb { N } } } \log \frac { d \left ( \omega _ { 1 } \lambda \right ) _ { x } } { d \lambda _ { x } } ( \psi _ { x } ( \omega ) ) d \mathbb { P } _ { \mu } ^ { \pi ( x ) } ( \omega ) \\ & = \int _ { G } \int _ { Z _ { x } } \log \frac { d ( g \lambda ) _ { x } } { d \lambda _ { x } } ( z ) d ( g \lambda ) _ { x } ( z ) \varphi _ { g } ( \pi ( x ) ) d \mu ( g ) . \end{align*}


\begin{align*} \begin{bmatrix} u _ { 1 , k } ^ 1 \\ u _ { 2 , k } ^ 1 \end{bmatrix} = - \begin{bmatrix} D _ 1 ^ { 1 1 } & - I \\ - I & D _ 2 ^ { 1 1 } \end{bmatrix} ^ { - 1 } \begin{bmatrix} C _ 1 ^ 1 & 0 \\ 0 & C _ 2 ^ 1 \end{bmatrix} \begin{bmatrix} x _ { 1 , k } \\ x _ { 2 , k } \end{bmatrix} . \end{align*}


\begin{align*} D = \left \{ x \in X \ | \ A x = b \right \} , \end{align*}


\begin{align*} \phi : M \to \R ; \phi ( z ) : = \frac 1 2 \delta ^ 2 ( z ) . \end{align*}


\begin{align*} A _ { n } = \left \{ f \in C _ { s } \left ( X \right ) : \underset { x \in X } { \sup } \left \vert f \left ( x \right ) \right \vert \leq n \right \} \end{align*}


\begin{align*} ( ( W ^ { \flat } , \lambda ^ { \flat } , \phi ^ { \flat } ) ; \ , \mathcal { L } = ( L _ 1 , \dots , L _ m ) ) , \end{align*}


\begin{align*} X _ 1 \cdots X _ N = \sum _ { n = 1 } ^ N X _ n + \sum _ { p \geq 2 } \sum _ { ( n _ 1 , \dots , n _ p ) \in \mathbb { I } _ N ^ p } \beta _ { ( n _ 1 , \dots , n _ p ) } [ X _ { n _ 1 } , \dots , X _ { n _ p } ] . \end{align*}


\begin{align*} x _ \tau = | { \rm A u t } ( \tau ) | \sum _ { \bar T = \tau } X _ T \end{align*}


\begin{align*} w _ b = \sum _ { n } c _ n a _ n + T ( f ) _ b \end{align*}


\begin{align*} \mathcal Z ( f ) \coloneqq \{ n \in \mathbb N : a ( n ) = 0 \} \end{align*}


\begin{align*} \phi = \phi _ { e } + \sum _ { m = 1 } ^ { \infty } C _ { q _ m } h ^ { q _ m } , \end{align*}


\begin{align*} R _ \ell + b \leq R _ { n + 1 } + b \leq R _ n + 1 + b = R _ n + k . \end{align*}


\begin{align*} y _ \lambda ( t ) = v _ { 0 , \lambda } ( t ) + \int _ 0 ^ t S ( t - s ) \sigma ' ( u _ \lambda ( s ) ) y _ \lambda ( s ) B \ , d W ( s ) . \end{align*}


\begin{align*} D _ { L , 2 } = O ( \lambda _ L ) = O ( 2 ^ { L / p } L ^ { 1 / 2 + \varepsilon / p } ) \end{align*}


\begin{align*} \sigma _ j ( a ) = \begin{cases} a - 1 , & a = j \mod n , \\ a + n - 1 , & \end{cases} . \end{align*}


\begin{align*} R ( t ) : = \sum _ { m \in \mathcal { M } ' } \frac { r ( m ) } { m ^ { i t } } . \end{align*}


\begin{align*} \| T _ { \eta } f \| _ { L ^ p ( x ^ \gamma d x ) } ^ p & \lesssim \int _ 0 ^ \infty x ^ { - \tau - 1 + \gamma } \int _ x ^ \infty \abs { f ( z ) } ^ p z ^ { \tau } \ , d z \ , d x = \int _ 0 ^ \infty \abs { f ( z ) } ^ p z ^ { \tau } \int _ 0 ^ z x ^ { - \tau - 1 + \gamma } \ , d x \ , d z \\ & \simeq \| f \| _ { L ^ p ( x ^ \gamma d x ) } ^ p , \end{align*}


\begin{align*} \pi \big ( ( \ell _ E ) _ { E \in \mathbf { E } } , ( n _ E ) _ { E \in \mathbf { E } } \big ) = \big ( \ell _ E \mathbf { u } ( E , v _ E ) - n _ E ) _ { E \in \mathbf { E } } , \end{align*}


\begin{align*} \rho ( \xi , w ) = \rho ( \overline { \xi } , \overline { w } ) { \rm \ f o r \ } \xi \ne \pm 1 . \end{align*}


\begin{align*} | \eta ( x , y ) | \le e ^ { \gamma } \left ( \eta _ 1 ( y _ 0 ) + \sum _ { k = 2 } ^ { \infty } \xi _ { k } ( y _ 0 ) \right ) R ( y ) \end{align*}


\begin{align*} P _ r ( \theta ) = \frac { 1 - r ^ 2 } { 1 - 2 r \cos ( \theta ) + r ^ 2 } , \end{align*}


\begin{align*} M [ \phi ] = 2 \left ( \begin{array} { c c } - 2 \phi _ r \phi _ i & \phi _ r ^ 2 - \phi _ i ^ 2 \\ \phi _ r ^ 2 - \phi _ i ^ 2 & 2 \phi _ r \phi _ i \end{array} \right ) , \end{align*}


\begin{align*} f ( x , y ) = \sum _ { m , n \geq 0 } f _ { m , n } \frac { x ^ m y ^ n } { m ! n ! } \end{align*}


\begin{align*} 0 \le \sqrt { t } \ , \frac { w _ n ( t , \Lambda _ t ^ n \pm y ) } { y } \le \frac { \varepsilon } { 2 } \Big ( 1 + \sup _ { 0 < t \le \upsilon _ 0 } \ ; | \sqrt { t } \ , ( \Lambda _ t ^ n ) ' | ^ 2 \Big ) \le \frac \varepsilon 2 ( 1 + C _ \varepsilon ^ 2 ) \le C _ \varepsilon , \ ; y \in ( 0 , y _ 0 ] , \ ; t \in ( 0 , \upsilon \wedge \upsilon _ 0 = \upsilon _ 0 ) . \end{align*}


\begin{align*} A ^ h : = \begin{pmatrix} y ^ { \alpha _ y } & 0 \\ 0 & ( x + h ) ^ { \alpha _ x } \end{pmatrix} , \end{align*}


\begin{align*} \int \chi ( x ) e ^ { - i x \cdot \xi _ 0 / h \Tilde { h } } \tilde { R } _ h ( z ) ( \psi ( x ) e ^ { i x \cdot r \eta _ 0 / h \Tilde { h } } ) d x = \mathcal { O } ( h ^ \infty \Tilde { h } ^ \infty ) . \end{align*}


\begin{align*} \overline { q } ^ { f } _ L ( x ) : = \max _ { 1 \le n \le L } q _ n ^ { f } ( x ) \ , . \end{align*}


\begin{align*} E \big | M _ + ( t ) - \hat { M } _ + ( t ) \big | = O ( t e ^ { \lambda t / 2 } ) , \end{align*}


\begin{align*} K ( y , z ) = \sum _ { k , \ell \in \mathbb N _ 0 } c _ { k \ell } \ , y ^ k z ^ \ell , \end{align*}


\begin{align*} E _ 1 ( h ; y , u ) = e ^ { \gamma } \log y \int _ { 1 } ^ { h } t ^ { - 1 } y ^ { t - u } \ , d t < e ^ { \gamma } y ^ { h - u } . \end{align*}


\begin{align*} u ( z ) = \frac { P ( z ) } { Q ( z ) } \ , , P \in \C _ { \leq N } [ z ] \ , , \end{align*}


\begin{align*} Z _ L ( L ) = Z ( L ) = \{ x \in L : [ x _ \lambda y ] = 0 , ~ ~ \forall y \in L \} . \end{align*}


\begin{align*} \mu _ { \omega , c } = \frac { 1 } { 6 } L _ { \omega , c } ( \Phi ) \ge C _ { \omega } ( 4 \omega - \sigma c ^ 2 ) \| \Phi \| _ { \mathcal { H } ^ 1 } ^ 2 \end{align*}


\begin{align*} \mathfrak { M } _ { J \otimes I } = \left ( \mathfrak { M } _ { J _ 1 \otimes I } , \ldots , \mathfrak { M } _ { J _ n \otimes I } \right ) . \end{align*}


\begin{align*} \int _ { X _ { 0 } } \mathbb { E } _ { \lambda } \left [ \tilde { f } ( g , \cdot ) | \mathcal { F } ^ { s } \right ] d \lambda = \int _ { X _ { 0 } } \tilde { f } \left ( g , x _ { 0 } \right ) d \lambda ( x _ { 0 } ) = \int _ { X } f \left ( g . x _ { 0 } \right ) d \lambda ( x _ { 0 } ) = \int _ { X } f \left ( x \right ) d g . \lambda ( x ) . \end{align*}


\begin{align*} \tfrac { 1 } { 2 } \nabla ^ 2 _ \xi H ^ 2 ( \xi ) = H ( \xi ) \ , \nabla ^ 2 _ \xi H ( \xi ) + \nabla _ \xi H ( \xi ) \otimes \nabla _ \xi H ( \xi ) \end{align*}


\begin{align*} \frac { \partial m } { \partial \mathbf { n } } \bigg | _ { \partial \Omega } = \frac { \partial c } { \partial \mathbf { n } } \bigg | _ { \partial \Omega } = \frac { \partial d } { \partial \mathbf { n } } \bigg | _ { \partial \Omega } = 0 , \end{align*}


\begin{align*} t \ge t _ { 0 } = e ^ { - \frac { \tilde { c } \sqrt { N } } { d } } \end{align*}


\begin{align*} D ^ \lambda w _ j ^ * = \sum _ { i = 1 } ^ { d _ \lambda } w _ i ^ * \partial _ { w _ i } w _ j ^ * = d ! \sum _ { i = 1 } ^ { d _ \lambda } w _ i ^ * \langle w _ i , w _ j ^ * \rangle = d ! w _ j ^ * . \end{align*}


\begin{align*} \log \Pr [ d ( \hat { M } ' , M ) \leq n ^ { 1 - \beta } ] ^ { - 1 } & \geq \log \Pr [ \hat { M } ' = M ] ^ { - 1 } - n ^ { 1 - \beta } ( 2 + \max ( 1 , \alpha + \beta ) \log n ) \\ & \geq \omega ( n ^ { 1 - \beta } \log n ) - O ( n ^ { 1 - \beta } \log n ) \\ & \geq \omega ( n ^ { 1 - \beta } \log n ) \geq \omega ( \log n ) . \end{align*}


\begin{align*} L _ { \mu , \xi } ( \phi ) = \phi - \Pi ^ { \bot } _ { \mu , \xi } \Big ( i ^ * [ f _ \epsilon ^ { ' } ( V ) \phi ] \Big ) , \end{align*}


\begin{align*} - c _ 2 F _ 1 + x _ 0 c _ 1 F _ 2 = g h \end{align*}


\begin{align*} \begin{aligned} | f ( x ) - f ( x ' ) | & \le C \Big ( \frac { \| x - x ' \| } { r } \Big ) ^ \beta \Big \{ \frac { 1 } { V ( x , x _ 0 , r + d ( x , x _ 0 ) ) } \Big ( \frac { r } { r + { \| x - x _ 0 \| } } \Big ) ^ \gamma \\ & \qquad + \frac { 1 } { V ( x ' , x _ 0 , r + d ( x ' , x _ 0 ) ) } \Big ( \frac { r } { r + { \| x ' - x _ 0 \| } } \Big ) ^ \gamma \Big \} ; \end{aligned} \end{align*}


\begin{align*} & \forall \ , x : n \forall \ , y : m \forall \ , z : p ( ( x \ast _ { n , m } y ) \ast _ { ( m + n ) , p } z = x \ast _ { n , ( m + p ) } ( y \ast _ { m , p } z ) ) \\ & \forall \ , x : n \forall \ , y : m ( x \ast _ { n , m } y = y \ast _ { m , n } x ) \end{align*}


\begin{align*} ( \underline { X } , \underline { A } ) ^ { K } = \bigcup _ { \sigma \subseteq K } ( \underline { X } , \underline { A } ) ^ { \sigma } \subseteq \prod ^ m _ { i = 1 } X _ i . \end{align*}


\begin{align*} \begin{cases} \partial _ { t } z - \sum _ { i , j = 1 } ^ n \partial _ { x _ i } ( A _ { i j } \partial _ { x _ j } z ) + b z = \chi _ { \omega } g , & \mbox { i n } \ Q , \\ z = 0 \ \mbox { o r } \ A \nabla z \cdot \nu = 0 , & \mbox { o n } \ \Sigma , \\ z ( 0 ) = z _ { 0 } , & \mbox { i n } \ \Omega . \end{cases} \end{align*}


\begin{align*} f & = \int _ { \Theta } T _ { w } ( f ) d \mu ( w ) \\ & = \int _ { \Theta } v ( w ) s ( w ) K \pi _ { G ( w ) } \xi ^ { \ast } \chi _ { w } \pi _ { F ( w ) } f d \mu ( w ) \\ & = K \int _ { \Theta } v ( w ) s ( w ) \pi _ { G ( w ) } \xi _ { w } ^ { \ast } \chi _ { w } \pi _ { F ( w ) } f d \mu ( w ) \\ & = K S _ { \xi \chi } f . \end{align*}


\begin{align*} \langle f _ b , \varphi \rangle : = \lim _ { r \nearrow 1 } \int _ 0 ^ { 2 \pi } f ( r e ^ { i \theta } ) \ , \varphi ( \theta ) \ , d \theta \end{align*}


\begin{align*} - Y '' + R ( \dot \gamma , X ) \dot \gamma = \lambda X , Y ( a ) = Y ( b ) = 0 , \end{align*}


\begin{align*} \varphi ^ + ( Y _ T ) & = P _ + ( \varphi ^ + ( Y _ F ) a ) , \\ \varphi ^ - ( Y _ T ) & = - P _ - ( \varphi ^ + ( Y _ F ) a ) . \end{align*}


\begin{align*} \Pi _ { \psi } [ \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} ] f ( [ \epsilon , y ] ) = \psi ( \tfrac { 1 } { 2 } \langle y , \epsilon y b \rangle ) f ( [ \epsilon , y ] ) . \end{align*}


\begin{align*} \mathcal { E C S } _ { { \rm { I I } } } = \mathcal { E C S } _ { 0 1 1 } \cup \mathcal { E C S } _ { 1 0 1 } \cup \mathcal { E C S } _ { 1 1 0 } , \end{align*}


\begin{align*} \rho _ t ( x , y ) = \frac { p _ t ^ { \Gamma } ( x , y ) } { M _ { \Gamma } ( x ) M _ { \Gamma } ( y ) } , x , y \in \Gamma , \ ; t > 0 . \end{align*}


\begin{align*} \lim _ { \delta \to \infty } L ^ { ( k ) } ( \delta ) = \begin{cases} p s , & \mbox { i f } k = 0 , \\ 0 , & \mbox { i f } k \ge 1 . \end{cases} \end{align*}


\begin{align*} \textsf { E } f ( x _ { 1 } \varepsilon _ { 1 } , x _ { 2 } \varepsilon _ { 2 } , \cdots , x _ { n } \varepsilon _ { n } ) = \textsf { E } f ( - x _ { 1 } \varepsilon _ { 1 } , x _ { 2 } \varepsilon _ { 2 } , \cdots , x _ { n } \varepsilon _ { n } ) \end{align*}


\begin{align*} H ' = \frac { A '' \circ A ^ { - 1 } } { H } = H F - 1 . \end{align*}


\begin{align*} y ^ { * } ( N ^ { a } ) ( t ) & = N ^ { a } \left ( y ^ { * } ( N , t ) \right ) \\ & = N \bigl ( y ^ { * } ( N , t ) \wedge a \bigr ) \end{align*}


\begin{align*} \big | A _ { k _ \epsilon , \varrho } ( t ) \big | & = \big | A _ { k _ \epsilon , ( 1 - \sigma ) \varrho } ( t ) \cup ( A _ { k _ \epsilon , \varrho } ( t ) \setminus A _ { k _ \epsilon , ( 1 - \sigma ) \varrho } ( t ) ) \big | \\ & \leq \big | A _ { k _ \epsilon , ( 1 - \sigma ) \varrho } ( t ) \big | + | K _ \varrho \setminus K _ { ( 1 - \sigma ) \varrho } | \\ & \leq \big | A _ { k _ \epsilon , ( 1 - \sigma ) \varrho } ( t ) \big | + N \sigma | K _ \varrho | . \end{align*}


\begin{align*} f ^ { ( 2 ) } ( \boldsymbol \beta ) \triangleq & - \log p ^ { ( 2 ) } ( \mathbf R ; \boldsymbol \beta ) - L \log \pi \\ = & \log | \boldsymbol \Sigma ^ { ( 2 ) } _ { \boldsymbol \beta } | + \left ( \boldsymbol \Sigma _ { \boldsymbol \beta } ^ { ( 2 ) - 1 } \widehat { \mathbf \Sigma } _ { \mathbf R } \right ) . \end{align*}


\begin{align*} \eta _ { 0 } = \eta _ 0 ( \omega ) : = \frac { B _ { \omega } } { 2 } , \ \ c _ 0 = c _ 0 ( \omega ) : = \min \left \{ c _ 1 ( \omega ) , c _ 2 ( \omega ) , \frac { B _ { \omega } } { 6 A _ { \omega } ^ 2 } \right \} > 0 . \end{align*}


\begin{align*} \tilde { y } _ t ^ { K + 2 } = \tilde { y } _ t ^ K . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ p \norm { u } _ { H ^ { \alpha _ i } ( \mathbb { R } ^ n ) } \leq C \left ( \norm { F } _ { ( H ^ \alpha ( \Omega ) ) ^ * } + \norm { f } _ { ( H ^ \alpha ( \Omega ^ c ) ) ^ * } \right ) , \end{align*}


\begin{align*} \sum _ i a _ i d ( k ) _ i = c _ k ( M ) ; \end{align*}


\begin{align*} \lim _ { n \to \infty } - \frac { 1 } { n } \log P _ { \mu , { x } } ^ { n } \left ( L _ { x } , L _ { x } \omega _ { n } \right ) = \lim _ { n \to \infty } \frac { 1 } { n } H ( \xi _ { n } | X ) = h _ { \mu } ( Z , \lambda ) - h _ { \mu } ( X , \eta ) , \end{align*}


\begin{align*} ( d d ^ c u ) ^ n = e ^ { B u + \gamma ( v ) } f \omega ^ n . \end{align*}


\begin{align*} \frac { d } { d \lambda } S _ { \omega , \mathbf { c } } ( \Phi ^ { \lambda } ) | _ { \lambda = 1 } & = \frac { d } { d \lambda } \left . \left ( L ( \Phi ^ { \lambda } ) + N ( \Phi ^ { \lambda } ) + \omega Q ( \Phi ^ { \lambda } ) + \mathbf { c } \cdot \mathbf { P } ( \Phi ^ { \lambda } ) \right ) \right | _ { \lambda = 1 } \\ & = 2 L ( \Phi ) + \left ( \frac { d } { 2 } + 1 \right ) N ( \Phi ) + \mathbf { c } \cdot \mathbf { P } ( \Phi ) \end{align*}


\begin{align*} ( ( D F ) ( g , \mu , u ) ) ( f , \nu , w ) = F ( f , \nu , u ) + F ( g , \mu , w ) . \end{align*}


\begin{align*} \mathcal { E } _ { \varrho , s } : C ( Y ) & \to \tilde { L } ^ { \infty } \left ( Y \right ) , \\ f & \mapsto \mathcal { E } _ { \varrho , s } f , \mbox { w h e r e } \left ( \mathcal { E } _ { \varrho , s } f \right ) ( \bar { u } , \bar { v } , \cdot ) = \mathbb { E } _ { \lambda } \left [ \tilde { f } \left ( \bar { u } , \cdot \right ) | \mathcal { F } ^ { s } \right ] , \end{align*}


\begin{align*} F ( t ) = \sum _ { i = 0 } ^ d \mu _ i t ^ i \ . \end{align*}


\begin{align*} { \rm I m } ( f _ { \ast } ( \xi ) ) = 0 { \rm \ \ f o r \ \ } \xi \in L \end{align*}


\begin{align*} \| 1 + d Z \| _ { q } ^ { q } \ge 1 + \binom { \lfloor q \rfloor } { 2 } d ^ 2 \ge 1 + \frac { q ^ 2 } { 9 } d ^ 2 , \end{align*}


\begin{align*} N : = \left \{ ( x , y ) \in \Omega \times \R ^ k : w ( x ) - y \in \mathcal { X } \right \} \end{align*}


\begin{align*} \mathcal L _ \xi g = 2 g . \end{align*}


\begin{align*} \limsup _ { n \rightarrow \infty } ( n \log n ) ^ { - 1 / p } \big | S _ n - \sum _ { k = 1 } ^ n g _ k \big | > 0 \mbox { a . s . } \end{align*}


\begin{align*} c _ k ( X _ \Omega ) = c _ k ( \sqcup _ i B ( a _ i ) ) . \end{align*}


\begin{align*} \max _ { b , C } f ( b , C ) = f \left ( \frac { ( n - 1 ) \kappa } { 2 } , \frac { ( n - 1 ) ^ 2 \kappa ^ 2 } { 4 } \right ) = \frac { ( n - 1 ) ^ 2 \kappa ^ 2 } { 4 } , \end{align*}


\begin{align*} \mathcal { M } _ { F _ 1 , \alpha } f = \mathcal { M } _ { F _ 2 , \alpha } f , f \in H ^ { s _ M } ( \Omega ^ c ) \end{align*}


\begin{align*} \{ ( \bar { x } , \bar { y } ) \in \R ^ { m + n } : 1 - \epsilon \leq \| N \bar { y } \| \leq e + \epsilon , | N x _ i | < \vartheta _ { i } ( \epsilon ) \| N \bar { y } \| ^ { - w _ i } i = 1 , \ldots , m \} . \end{align*}


\begin{align*} \begin{gathered} \left \| f ^ { n + 1 , \nu } - f ^ { n , \nu } \right \| _ { L ^ \infty ( [ T _ { \mathcal { K } _ { n + 1 } } , T _ { \mathcal { K } _ { n + 1 } } + 3 \cdot 2 ^ { - k } ] ; L ^ 1 ( \mathbb { T } ^ 2 ) ) } \\ \le 2 \left \| f _ 0 \right \| _ { L ^ \infty } 2 ^ { - \lfloor k / 2 \rfloor } + C \left \| f _ 0 \right \| _ { L ^ \infty } \sqrt { \nu 2 ^ { - k } } , \end{gathered} \end{align*}


\begin{align*} p _ 1 \ge \cdots \ge p _ s \qquad { \nu _ k \le \nu _ { k + 1 } } { p _ k = p _ { k + 1 } } . \end{align*}


\begin{align*} \Omega \cap \overline { S _ u } = \Omega \cap \overline { J _ u } = \Omega \setminus \Omega _ u . \end{align*}


\begin{align*} a _ 0 \ = \ \begin{cases} 4 , & , \\ 9 , & . \end{cases} \end{align*}


\begin{align*} _ { s = 1 } \zeta _ { p , p ^ \prime } ( s , c ) = \frac { \pi } { \sqrt { c } } . \end{align*}


\begin{align*} S _ { \mathrm { m a i n } } : = \sum _ { i = 1 } ^ D \sum _ { j = 1 } ^ D \alpha _ i \bar { \alpha } _ j \ , \ \delta _ { i j } \ , \big | L ( f , n + 1 ) \big | ^ 2 \geq \sum _ { i = 1 } ^ D | \alpha _ i | ^ 2 \ , c , \end{align*}


\begin{align*} x + x ^ 2 y - y z ^ 2 = 0 . \end{align*}


\begin{align*} K _ { \omega , \mathbf { c } } ( V _ n ) = K _ { \omega , \mathbf { c } } ( U _ n ) \rightarrow 0 , \end{align*}


\begin{align*} \langle ( f \ast - , - \ast f ) , ( g \ast - , - \ast g ) \rangle = ( f \ast ( g \ast - ) , ( - \ast f ) \ast g ) \end{align*}


\begin{align*} s _ 1 = \prod _ { \substack { \varpi ^ k \parallel s \\ \varpi ^ { 1 + v _ \varpi } \nmid a _ 1 } } \varpi ^ k \quad s _ 2 = \prod _ { \substack { \varpi ^ k \parallel s \\ \varpi ^ { 1 + v _ \varpi } \mid a _ 1 } } \varpi ^ k , \end{align*}


\begin{align*} F _ { c o n s e r v a t i v e } ^ - = F _ { d i s s i p a t i v e } ^ - , \end{align*}


\begin{align*} \frac { c } { 2 a ( 1 - a ) } & | ( y - l _ \alpha ) ^ { 1 - a } - ( x - l _ \alpha ) ^ { 1 - a } | = \int _ x ^ { y } \frac { c } { 2 a ( s - l _ \alpha ) ^ a } d s \leq \Big | \int _ x ^ { y } D ^ { n } \varphi ( s ) d s \Big | \\ & \leq | D ^ { n - 1 } \varphi ( x ) - D ^ { n - 1 } \varphi ( y ) | \leq \| D ^ { n - 1 } \varphi \| _ { C ^ { 1 - a + \tau } } | ( y - l _ \alpha ) - ( x - l _ \alpha ) | ^ { 1 - a + \tau } . \end{align*}


\begin{align*} \mathrm { d i v } _ T \Big ( & \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \big ) \Big ) \ , \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \cdot \nu \\ & - \nabla _ T \big [ \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \big ] \ , \big [ \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla w ) \big ] _ T \cdot \nu = H ^ 2 ( \nabla w ) \ , H ( \nu ) \ , \mathrm { t r } \ , \mathcal { B } ^ H \ , . \end{align*}


\begin{align*} \pi _ \sigma ( a _ 1 \otimes a _ 2 ) \lambda ^ \sigma ( t , t ) = \pi _ { \varphi _ 1 \bar { \otimes } \varphi _ 2 } ( a _ 1 \otimes a _ 2 ) \lambda ^ { \varphi _ 1 \bar { \otimes } \varphi _ 2 } ( t ) , a \in M _ 1 , b \in M _ 2 , ( t , t ) \in H \end{align*}


\begin{align*} \Xi _ 3 ( x ) = & \sum _ { x ' \le n \le x } \beta _ n \sum _ { r \mid n } \mu ( r ) \Omega ( r , n ) \end{align*}


\begin{align*} p _ { } ( \mathcal { D } _ a , C ) = p _ { } ( \mathcal { D } _ a , C ) + p _ { } ( \mathcal { D } _ a , C ) \forall C \in \mathcal { C } . \end{align*}


\begin{align*} \sum _ { t \in ( a ' , b ' ) } \dim ( \sigma _ 0 ( t ) \cap \sigma ) - \sum _ { \lambda \in ( \lambda _ 0 , \lambda ' ) } \dim ( \sigma _ { \lambda } ( b ) \cap \sigma ) = 0 . \end{align*}


\begin{align*} p ( x ) = \sum _ { i = 1 } ^ k q _ i ( x ) ^ 2 . \end{align*}


\begin{align*} \lim _ { \lambda \rightarrow \infty } K _ { \omega , \mathbf { c } } ( V ^ { \lambda } ) = - \infty \end{align*}


\begin{align*} p \vartriangleleft q \qquad \Leftrightarrow \qquad \exists C \in \mathsf { C } \mathbb { P } \ ( C p = \{ q \} ) . \end{align*}


\begin{align*} \mathcal { A } ( f , g ) = T ( f \cdot g ) - f T ( g ) - T ( f ) g - 2 B ( A ( f ) , A ( g ) ) \left ( f , g \in P \right ) . \end{align*}


\begin{align*} \phi ( E _ { j _ 1 j _ 1 } \otimes \cdots \otimes E _ { j _ { m } j _ { m } } ) = E _ { j _ 1 j _ 1 } \otimes \cdots \otimes E _ { j _ m j _ m } . \end{align*}


\begin{align*} M ^ { 0 } _ { \phi , A } f ( x ) = \sup _ { t > 0 } | f \ast \phi ^ A _ { t } ( x ) | , x \in G . \end{align*}


\begin{align*} \frac { h '' ( 0 ) } { 2 } - Q ( \Phi ) = \frac { 1 } { 2 \omega } \left \{ ( 4 - 2 d ) \omega Q ( \Phi ) + ( 3 - d ) \mathbf { c } \cdot \mathbf { P } ( \Phi ) ) \right \} \ge \frac { \eta } { 2 \omega } \ ( > 0 ) \end{align*}


\begin{align*} \theta _ i ^ 2 \frac { Z _ i ^ * ( u ) } { e _ i ^ * ( u ) } + \frac { T _ i ^ { \infty } } { e _ i ^ * ( u ) ^ 2 } & = \theta _ i ^ 2 \frac { Z _ i ( u ) } { \theta _ i } \left ( \frac { \theta _ i } { e ^ { \rho _ i ( u ) } } - \frac { \theta _ i Z _ i ( u ) } { T _ i ^ { \infty } } \right ) + T _ i ^ { \infty } \left ( \frac { \theta _ i } { e ^ { \rho _ i ( u ) } } - \frac { \theta _ i Z _ i ( u ) } { T _ i ^ { \infty } } \right ) ^ 2 \\ & = \theta _ i ^ 2 \left ( \frac { T _ i ^ { \infty } } { e ^ { 2 \rho _ i ( u ) } } - \frac { Z _ i ( u ) } { e ^ { \rho _ i ( u ) } } \right ) . \end{align*}


\begin{align*} \mathcal { I } ( \delta _ 1 , 0 ) = \widehat { \mathcal { I } } ( \delta _ 1 , \delta _ 2 ) = \inf _ { \lambda _ 1 \in \mathbb { R } _ { + } } \left [ \langle \lambda , \delta \rangle + \sup _ { \varpi \in \Pi \left ( \mu _ { 1 3 } , \mu _ { 2 3 } \right ) } \int _ { \mathcal { V } } f _ { \lambda } ( s _ 1 , s _ 2 ) \ , d \varpi ( s _ 1 , s _ 2 ) \right ] , \end{align*}


\begin{align*} H ' = \left \lceil H ^ { 1 / 2 } \right \rceil . \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle R e \mathcal { S } \left ( \bar { \phi } ( t ) , \psi ( t ) \right ) = R e \mathcal { S } ( \bar { \phi } _ 0 , \psi _ 0 ) . \end{array} \right . \end{align*}


\begin{align*} W _ { \mathrm { t r } , p } ^ + ( \mu , \nu ) & = \inf _ { \phi \in \mathcal { M } _ { \mathrm { t r } } } \{ W _ { \mathrm { t r } , p } ( \alpha , \nu ) : \mu = \phi ( \alpha ) \} \geq \inf _ { \phi \in \mathcal { M } _ { \mathrm { t r } } } \{ W _ { \mathrm { t r } , p } ( \mu , \phi ( \nu ) ) \} = W _ { \mathrm { t r } , p } ^ - ( \mu , \nu ) . \end{align*}


\begin{align*} | f ( x ) - f ( y ) | \leqslant C _ n \cdot \frac { ( \Vert Q \Vert _ 1 ) ^ { \frac { 1 } { q } } } { \log ^ { \frac { 1 } { n } } \left ( 1 + \frac { r _ 0 } { 2 | x - y | } \right ) } , \ , \ , r _ 0 = d ( K , \partial { \mathbb B } ^ n ) , \end{align*}


\begin{align*} h ( \mp \tau _ 0 ) = h ( 0 ) \mp \tau _ 0 h ' ( 0 ) + \frac { \tau _ 0 ^ 2 } { 2 } h '' ( \theta ) . \end{align*}


\begin{align*} D ^ k ( y ) = y _ k + \sum _ { n \geq 1 } y _ { k + n } \frac { x ^ n } { n ! } . \end{align*}


\begin{align*} \left . \begin{array} { l l } E ( \phi ( t , x ) , \psi ( t , x ) ) = E ( \phi _ 0 ( x ) , \psi _ 0 ( x ) ) , \end{array} \right . \end{align*}


\begin{align*} E ( t , \xi ) : = ( S ( t ) U ( t , \xi ) , U ( t , \xi ) ) , \end{align*}


\begin{align*} p ^ n \to p \ \Rightarrow \ p ^ n = p \quad n \geq \bar n . \end{align*}


\begin{align*} \mbox { $ \sigma ( w ) = \tau ( w ) \Leftrightarrow $ $ v _ { \sigma ( \mathcal { S } ) } = v _ { \tau ( S ) } \Leftrightarrow \tau ^ { - 1 } \sigma ( \mathcal { S } ) = \mathcal { S } $ . } \end{align*}


\begin{align*} G \cdot H ^ \star / 2 = \frac { \nu } { 2 } \sum _ { i = 1 } ^ n | \lambda _ i | . \end{align*}


\begin{align*} T _ t \circ \beta = & \alpha \circ T _ t , \\ [ T _ t u , T _ t v , T _ t w ] = & T _ t \rho ( T _ t u , T _ t v ) w , \end{align*}


\begin{align*} R _ N ^ 2 ( L , \alpha , \Delta ) \leq ( 1 + C / m ) ( L + o ( 1 ) ) = L + o ( 1 ) . \end{align*}


\begin{align*} \rho _ w ( \xi , w ) w = \rho _ { \overline { w } } ( \overline { \xi } , \overline { w } ) w = \overline { \rho _ w ( \overline { \xi } , \overline { w } ) \overline { w } } . \end{align*}


\begin{align*} \operatorname { s u p p } \chi _ \lambda = [ - 1 / \lambda - 1 , 1 / \lambda + 1 ] , \bigl . \chi _ \lambda \bigr \vert _ { [ - 1 / \lambda , 1 / \lambda ] } = 1 , \end{align*}


\begin{align*} P _ { \le r } = \bigcap _ { 0 \leq j \leq r } P _ { U _ j } \end{align*}


\begin{align*} _ { J } ( K _ { 1 } , K _ { 2 } ) : = \{ y \in \mathbb { S } ^ { n - 1 } \cap \mathbb { R } ^ { J } : \frac { K _ { 1 } } { \sqrt { d } } \le \vert y _ { k } \vert \le \frac { K _ { 2 } } { \sqrt { d } } , \forall k \in J \} , \end{align*}


\begin{align*} = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \prod _ { j = 1 } ^ { M } L i _ { s _ j } \left ( e ^ { i \theta } \right ) \prod _ { k = 1 } ^ { N } L i _ { t _ k } \left ( e ^ { - i \theta } \right ) d \theta . \end{align*}


\begin{align*} \big ( \mathcal { T } _ q \big ) _ { k , l } : = \delta _ { k , l + q } = \begin{cases} 1 & { \rm i f } \ k = l + q , \\ 0 & { \rm o t h e r w i s e } \end{cases} \mathrm { f o r } \ k , l \in [ m ] . \end{align*}


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


\begin{align*} \Phi ( A \circ B ) = \Phi ( A ) \circ \Phi ( B ) . \end{align*}


\begin{align*} \int _ { B _ 1 } \Theta \Delta _ g \eta d \mu _ g = 0 . \end{align*}


\begin{align*} ( c ^ { \wedge \sigma / \tau } ) ^ * ( x ) = \left ( \prod _ { i \in \tau } \frac { c _ i ^ { \sigma } } { c _ i ^ { \tau } } \right ) x \end{align*}


\begin{align*} P ^ t \Sigma = U \Lambda ^ t U ^ { - 1 } \Sigma = U \Lambda ^ t V ^ { - 1 } , \end{align*}


\begin{align*} \Delta _ k = a _ { k - 1 } a _ { k - 2 } ^ 2 \cdots a _ 0 ^ { k } , k \ge 1 . \end{align*}


\begin{align*} \varphi ^ \star ( \lambda ) = \sup _ { x \in \mathbb { R } ^ n _ + } \left \{ \varphi ( x ) - \langle \lambda , x \rangle \right \} . \end{align*}


\begin{align*} \begin{dcases} L ( u - w ) \ge 0 & B _ r ( x _ r ) \setminus \overline { B _ { r / 2 } ( x _ r ) } , \\ u - w \ge 0 & \R ^ n \setminus \big ( B _ r ( x _ r ) \setminus \overline { B _ { r / 2 } ( x _ r ) } \big ) , \end{dcases} \end{align*}


\begin{align*} ( a \delta _ s ) ( b \delta _ t ) & = a b \delta _ { s t } \\ ( a \delta _ s ) ^ * & = a ^ * \delta _ { s ^ { - 1 } } . \end{align*}


\begin{align*} \frac { \partial } { \partial t } G _ k ( x , t ) = T _ 2 G _ k ( x , t ) = G _ { k , Y _ { 2 , 1 } } - G _ { k , Y _ { 2 , 2 } } . \end{align*}


\begin{align*} \overline { \overline { C } } _ { X ^ { \ast } } ( g , k ) = \overline { \overline { C } } _ { X ^ { \ast } } ( p _ g k _ g , k ) & = \overline { \overline { C } } _ { X ^ { \ast } } ( p _ g , k _ g ) ^ { - 1 } \overline { \overline { C } } _ { X ^ { \ast } } ( p _ g , k _ g k ) \overline { \overline { C } } _ { X ^ { \ast } } ( k _ g , k ) = 1 . \end{align*}


\begin{align*} & T \circ \beta = \alpha \circ T , \\ & [ T u , T v , T w ] = T ( \rho ( T u , T v ) w ) , \end{align*}


\begin{align*} \| \xi _ i \| ^ p & = \sum _ { t \in G } \| v _ { t ^ { - 1 } } \pi ( g _ i ( t ) ) \xi \| ^ p = \sum _ { t \in G } \int | \pi _ 0 ( g _ i ( t ) ) \xi | ^ p \ , d \mu = \int \pi _ 0 \left ( \sum _ { t \in G } g _ i ( t ) ^ p \right ) | \xi | ^ p \ , d \mu \\ & \longrightarrow \int | \xi | ^ p \ , d \mu = \| \xi \| ^ p . \end{align*}


\begin{align*} \phi _ { \omega , c } ( x ) : = \left [ \frac { \sqrt { \omega } } { 4 \omega - c ^ 2 } \left \{ \cosh ( \sqrt { ( 4 \omega - c ^ 2 ) x } ) - \frac { c } { \sqrt { 4 \omega } } \right \} \right ] ^ { - 1 / 2 } \end{align*}


\begin{align*} m : = \min \{ i \leq N \ , : \ , \exists \ , z \in V a _ i ( z ) \neq 0 \} . \end{align*}


\begin{align*} \sup _ { \bar \Omega } \vert u \vert \leq C _ 0 : = \sup _ { \bar \Omega } \vert \underline u \vert , \end{align*}


\begin{align*} \mathbf H _ { n , m } \triangleq \begin{bmatrix} h _ { n , m , 1 } & h _ { n , m , L } & \cdots & h _ { n , m , 2 } \\ h _ { n , m , 2 } & h _ { n , m , 1 } & \cdots & h _ { n , m , 3 } \\ \vdots & \vdots & \ddots & \vdots \\ h _ { n , m , L } & h _ { n , m , L - 1 } & \cdots & h _ { n , m , 1 } \\ \end{bmatrix} \in \mathbb { C } ^ { L \times L } , \end{align*}


\begin{align*} \frac { \partial M ( c , x , y ) } { \partial y } = 0 \ ; \mbox { i m p l i e s t h a t } \ ; 8 ( 4 - c ^ 2 ) ^ 2 ( 1 - x ^ 2 ) [ c x ( 1 + x ) + 2 ( 1 - x ) ( 8 - x ) y ] = 0 \end{align*}


\begin{align*} n ( 1 ) \leq \frac { 1 } { \log 2 } \int _ 1 ^ 2 \frac { n ( r ) } { r } d r = \mathcal { O } ( h ^ { - n } ) . \end{align*}


\begin{align*} R _ N ^ 2 ( L , \alpha , \Delta ) \geq \frac { N _ { m - 1 } } { N } R _ { N _ { m - 1 } } ^ 2 ( L N _ { m - 1 } / N , \alpha , \Delta ) = L - o ( 1 ) . \end{align*}


\begin{align*} \Pr ( \Phi ( \mathcal { A } ) > 0 ) = 1 - \exp \left ( - 2 \pi \lambda R _ { m i n } R _ S \right ) . \end{align*}


\begin{align*} \sigma ( m _ { i j } ) = m _ { \sigma i \sigma j } . \end{align*}


\begin{align*} l _ R ( a ) l _ R ( - d ) - l _ R ( b ) l _ R ( - c ) & = l _ R ( c ) l _ R ( a d ) - l _ R ( a d ) l _ R ( d ) = \\ & = l _ R ( c ) l _ R ( a d ) - l _ R ( d ) l _ R ( a d ) = l _ R ( - c d ) l _ R ( a d ) . \end{align*}


\begin{align*} 0 = \sum _ { d = 0 } ^ m \left ( - \frac { p } { 1 - p } \right ) ^ d \langle h ^ { = d } , g \rangle \geq \mu _ { p } ( h ) \mu _ { p } ( g ) - \sum _ { d = 1 } ^ m \left ( \frac { p } { 1 - p } \right ) ^ d | \langle h ^ { = d } , g \rangle | . \end{align*}


\begin{align*} D _ j S _ { \omega , \mathbf { c } } ( \Psi ) = \eta D _ j K _ { \omega , \mathbf { c } } ( \Psi ) , \ \ j = 1 , 2 , 3 . \end{align*}


\begin{align*} \| \varphi \| _ { H ^ s ( \Sigma ) } = \inf \{ \| h \| _ { H ^ s ( \partial \mathrm { M } ) } ; \ ; h _ { | \Sigma } = \varphi \} . \end{align*}


\begin{align*} & \limsup _ { r \to \infty } \frac { \log \log M ( r , h ) } { \log r } = \rho ( h ) , \\ \implies & | h ( z ) | \leq e ^ { r ^ { \rho ( h ) + o ( 1 ) } } , \end{align*}


\begin{align*} \begin{aligned} & \mathbb { E } \left [ c _ 1 \left ( ( \widehat { Y } _ 1 , \widehat { X } ) , ( Y _ 1 , X ) \right ) \right ] \leq \left ( 1 - \frac { \eta } { \eta _ 0 } \right ) \mathbb { E } \left [ c _ 1 \left ( ( \widetilde { Y } _ 1 , \widetilde { X } ) , ( Y _ 1 , X ) \right ) \right ] \\ \end{aligned} \leq ( \eta - \eta _ 0 ) \end{align*}


\begin{align*} \int _ { B _ { 1 } } \left \vert D ^ { 2 } \bar { w } \right \vert ^ { p _ 0 } = \int _ { B _ { 1 } } \left \vert D ^ { 2 } w - \left ( D ^ { 2 } w \right ) _ { 1 } \right \vert ^ { p _ 0 } . \end{align*}


\begin{align*} g ( N ^ { \ , ( 1 ) } ( X , Y ) , \xi _ i ) = 2 \ , g ( \widetilde { Q } X , { f } Y ) . \end{align*}


\begin{align*} H = \langle h , P _ r ( \theta - \cdot ) + i Q _ r ( \theta - \cdot ) \rangle \in H ^ p ( D ) . \end{align*}


\begin{align*} x _ p = x ^ 0 _ p + m ' v _ p , y _ p = y ^ 0 _ p + k ' v _ p v _ p \geq 0 , \end{align*}


\begin{align*} \sum _ { l \equiv v ( \bmod a ) } 1 _ { I } ( l ) = 0 ~ ~ 1 . \end{align*}


\begin{align*} d ( J _ { \mathrm { s e c t } ( \delta / 2 ) } ( [ \ , t \ , | \left ( \begin{smallmatrix} n \\ a \end{smallmatrix} \right ) ] ) = J _ { \mathrm { s e c t } ( \delta / 2 ) } ( t ) \cdot \omega _ { n , a } \end{align*}


\begin{align*} \pm \prod _ { p \in { \cal P } _ 3 } p ^ { a _ p - b _ p + \sum _ { i = 1 } ^ n ( \alpha _ i - \beta _ i ) z _ { i p } } \pm \prod _ { p \in { \cal P } _ 2 } p ^ { b _ p - c _ p + \sum _ { i = 1 } ^ n ( \beta _ i - \gamma _ i ) z _ { i p } } \pm \prod _ { p \in { \cal P } _ 1 } p ^ { c _ p - a _ p + \sum _ { i = 1 } ^ n ( \gamma _ i - \alpha _ i ) z _ { i p } } = 0 . \end{align*}


\begin{align*} \begin{aligned} p ^ { ( t ) } _ { r , k } = \frac { 1 } { n ^ t } \ , \binom { n - r } { n - k } \sum _ { j = 0 } ^ { k - r } ( - 1 ) ^ { k - r - j } \binom { k - r } { j } ( j + r ) ^ t . \\ \end{aligned} \end{align*}


\begin{align*} { { \boldsymbol { K } } _ { k ; t } } = { \left ( { { \boldsymbol { W } } _ k ^ T { { \boldsymbol { \Omega } } _ { k ; t } } { { \boldsymbol { W } } _ k } } \right ) ^ { - 1 } } \left ( { { \boldsymbol { \bar P } } _ { k | k - 1 ; t } ^ { y x } + { \boldsymbol { H } } _ k ^ T { \boldsymbol { \bar P } } _ { k | k - 1 ; t } ^ y } \right ) . \end{align*}


\begin{align*} \iota _ \mathbf { n } ( U _ 1 U _ 2 ) = \rho ( U _ 1 ) \iota _ \mathbf { n } ( U _ 2 ) + \sum _ \mathbf { m } \tbinom { \mathbf { n } + \mathbf { m } } { \mathbf { m } } \iota _ { \mathbf { n } + \mathbf { m } } ( U _ 1 ) \epsilon _ \mathbf { m } ( U _ 2 ) , \end{align*}


\begin{align*} \begin{cases} \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) \tilde { u } = 0 & \Omega , \\ \tilde { u } = 0 & \Omega ^ c , \end{cases} \end{align*}


\begin{align*} \mathcal { T } = B _ 1 ^ { ( 0 ) } \otimes L + B _ 2 ^ { ( 0 ) } \otimes ( - 2 I _ { m } ) , \end{align*}


\begin{align*} \int _ { \Omega ' } b ^ { i j , k l } f _ { i j } \eta _ { k l } d x = 0 \forall \eta \in C _ 0 ^ \infty ( \Omega ' ) , \end{align*}


\begin{align*} \omega _ { \pm } = \omega _ { \pm } ( \tau _ 0 ) : = ( \sqrt { \omega } \pm \tau _ 0 ) ^ 2 , \ \ \mathbf { c } _ { \pm } = \mathbf { c } _ { \pm } ( \tau _ 0 ) : = \frac { \mathbf { c } } { \sqrt { \omega } } ( \sqrt { \omega } \pm \tau _ 0 ) , \end{align*}


\begin{align*} \sum _ { p = 1 } ^ n u ^ { p \bar p } u _ { p \bar p j } = g _ j , \end{align*}


\begin{align*} \frac { d } { d \lambda } S _ { \omega , \mathbf { c } } ( \Phi _ 0 ^ { \lambda } ) = \frac { 5 } { 2 } N ( \Phi _ 0 ) \lambda ^ { \frac { 3 } { 2 } } + 2 \lambda L ( \Phi _ 0 ) + \mathbf { c } \cdot \mathbf { P } ( \Phi _ 0 ) = \frac { I _ { \omega , \mathbf { c } } ( \Phi _ 0 ^ { \lambda } ) } { \lambda } . \end{align*}


\begin{align*} \tilde { S } = M Q + \mathcal { O } ( \| H \| ) , \end{align*}


\begin{align*} \left \langle \mathcal { A } x , x \right \rangle = \left \langle \Re \mathcal { A } x , x \right \rangle + i \left \langle \Im \mathcal { A } x , x \right \rangle \end{align*}


\begin{align*} v ^ { ( r , z ) } ( t , x ) = G _ { t - r } ( x - z ) + \sum _ { n \geq 1 } I _ n ( g _ n ( \cdot , r , z , t , x ) ) , \end{align*}


\begin{align*} \Tilde { \phi } _ { e } & = \phi _ 1 - C _ { \Tilde { q } } h ^ { \Tilde { q } } , \\ \Tilde { q } & = \ln \Big ( \frac { \phi _ 3 - \phi _ 2 } { \phi _ 2 - \phi _ 1 } \Big ) / \ln r , \\ C _ { \Tilde { q } } h ^ { \Tilde { q } } & = \frac { \phi _ 2 - \phi _ 1 } { r ^ { \Tilde { q } } - 1 } . \end{align*}


\begin{align*} \Delta _ p ( C _ \lambda r ^ \lambda ) = ( \lambda C _ \lambda ) ^ { p - 1 } \ , ( \lambda ( p - 1 ) + n - p ) \cdot r ^ { ( p - 1 ) \lambda - p } . \end{align*}


\begin{align*} \langle e ^ { \rho ( u ) } , \partial _ u ( \widetilde { W } ^ { ( u ) } ) e ^ { \rho ( u ) } \rangle & = \langle e ^ { \rho ( u ) } , W ( H ^ { ( u ) } ) ^ { - 1 } T ( u ) ^ { - 1 } e ^ { 2 \rho ( u ) } T ( u ) ^ { - 1 } ( H ^ { ( u ) } ) ^ { - 1 } W e ^ { \rho ( u ) } \rangle \\ & = \langle e ^ { \rho ( u ) } , \widetilde { W } ^ { ( u ) } e ^ { 2 \rho ( u ) } \widetilde { W } ^ { ( u ) } e ^ { \rho ( u ) } \rangle = \sum _ { i \in V } ( \widetilde { W } ^ { ( u ) } e ^ { \rho ( u ) } ) _ i ^ 2 e ^ { 2 \rho _ i ( u ) } , \end{align*}


\begin{align*} | \varphi \circ d _ q ( x ) - ( d _ q ( x ) - r ) | & = \frac \epsilon { b } ( d _ q ( x ) - r ) ^ 2 \le \frac \epsilon { b } d ^ 2 _ p ( x ) \le 2 \epsilon \cdot d _ p ( x ) . \end{align*}


\begin{align*} u _ * ( \theta , \dot \theta ) = 2 1 2 . 5 7 5 5 \cos ( \theta ) \sin ( \theta ) + 5 4 . 1 2 9 6 \cos ( \theta ) \dot \theta . \end{align*}


\begin{align*} Z ( t ) = N \Bigl ( y ( Z , t ) \Bigr ) \end{align*}


\begin{align*} f ( b , C ) : = 2 \sqrt { C b ( n - 1 ) \kappa - C b ^ 2 } - C . \end{align*}


\begin{align*} \Big \| \sum _ { j = 1 } ^ \infty \alpha _ j u _ j \Big \| _ { p ^ * , q , \mu } \geq \Bigg ( \frac { \lambda } { 1 + \varepsilon _ 1 } - \varepsilon _ 2 \Bigg ) \Big ( \sum _ { j = 1 } ^ \infty | \alpha _ j | ^ q \Big ) ^ { \frac 1 { q } } . \end{align*}


\begin{align*} 1 0 ^ m - 6 \ \le \ v ( f ) \ = \ v ( 2 ^ \eta ) + v ( 3 ^ \delta ) + v ( 5 ^ \gamma ) \ \le \ 1 0 + \eta + \delta + \gamma . \end{align*}


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


\begin{align*} \mathrm { H } _ { \mathcal { H } _ { V } } ^ { s } : = \left \{ f \in \mathcal { D } _ { \mathcal { H } _ { V } } ^ { \prime } \left ( \mathbb { R } ^ { n } \right ) : \left ( I + \mathcal { H } _ { V } \right ) ^ { s / 2 } f \in L ^ { 2 } \left ( \mathbb { R } ^ { n } \right ) \right \} , s \in \mathbb { R } , \end{align*}


\begin{align*} \mathbb { P } ( \bar { E } _ { i , 1 } ) = \mathbb { P } _ K ^ { k _ 1 } \left ( 1 - \mathbb { P } _ { e , 1 } \right ) ^ { k _ 1 } \cdots \mathbb { P } _ K ^ { k _ K } \left ( 1 - \mathbb { P } _ { e , K } \right ) ^ { k _ K } , \end{align*}


\begin{align*} c _ { i j k } c _ { i k \ell } = c _ { i j \ell } c _ { j k \ell } \end{align*}


\begin{align*} F ( \tau ) = \frac { h '' ( \tau ) } { 2 } - \sup _ { \Phi \in \mathcal { M } _ { \omega , \mathbf { c } } ^ * ( \eta ) } Q ( \Phi ) . \end{align*}


\begin{align*} J _ n ( u ) = J ( u ) , \end{align*}


\begin{align*} \Gamma \left ( s \right ) = \sqrt { \pi } - \sqrt { \pi } \left ( 2 \log ( 2 ) + \gamma \right ) \left ( s - \frac { 1 } { 2 } \right ) + O \left ( s - \frac { 1 } { 2 } \right ) ^ { 2 } . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ { \infty } k \omega ( 2 ^ { - k } ) & \approx \int _ { 0 } ^ { 1 } \omega ( t ) ( 1 + \log t ^ { - 1 } ) \frac { \mathrm { d } t } { t } < \infty . \end{align*}


\begin{align*} \overline { \psi } _ { y _ 1 , y _ 2 } = \norm { \cdot } _ 1 - \lim _ { l \to \infty } B _ l ( \psi _ { y _ 1 , y _ 2 } ) . \end{align*}


\begin{align*} B _ { L , k , 1 } & \ll T \sum _ { p \mid k } \nu _ p ( k ) \log p = T \log k . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b ) = 1 \\ a , b \geq 1 } } \left ( \frac { 1 } { 1 - y ^ a z ^ b } \right ) ^ { \frac { 1 } { a ^ s b ^ t } } = \exp \left \{ \left ( \sum _ { j = 1 } ^ { \infty } \frac { y ^ j } { j ^ s } \right ) \left ( \sum _ { k = 1 } ^ { \infty } \frac { z ^ k } { k ^ t } \right ) \right \} . \end{align*}


\begin{align*} \int _ { \Omega } \mathcal { L } ^ { \Omega } _ \varepsilon u u \ : x = \frac { 1 } { 4 } \int _ { \Omega } \int _ { \Omega } J _ \varepsilon ( x - y ) \big | u ( x ) - u ( y ) \big | ^ 2 \ : x \ : y = \mathcal { E } _ \varepsilon ( u ) \end{align*}


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


\begin{align*} [ T _ k ] _ i ^ j ( A ) = [ T _ k ] _ i ^ j ( \underbrace { A , A , \cdots , A } _ k ) . \end{align*}


\begin{align*} u _ m = i ^ * \Big [ f _ \epsilon ^ { ' } ( V _ m ) u _ m + f _ \epsilon ^ { ' } ( V _ m ) ( h _ m + \omega _ m ) \Big ] . \end{align*}


\begin{align*} \beta ^ { i j , k l } ( D ^ 2 u ( x ) ) = \int _ { 0 } ^ { 1 } \frac { \partial F ^ { i j } } { \partial u _ { k l } } ( D ^ 2 u ( x ) + t [ D ^ 2 u ( x + h _ m ) - D ^ 2 u ( x ) ] ) d t . \end{align*}


\begin{align*} \Phi ( x , y ) = \frac { x } { \log y } \left ( \omega ( u ) + O \left ( \frac { 1 } { \log y } \right ) \right ) \end{align*}


\begin{align*} \Gamma _ v = v K \mbox { a n d } \Gamma _ \mu = \mu ( K [ x ] ) . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { \infty } \frac { \binom { 2 k } { k } ^ 2 } { 3 2 ^ k } = \frac { \Gamma ( 1 / 4 ) ^ 2 } { 2 \pi \sqrt { \pi } } , \end{align*}


\begin{align*} u _ { \omega , c } ( t , x ) : = \phi _ { \omega , c } ( x - c t ) \exp \left \{ i \omega t + i \frac { c } { 2 } ( x - c t ) - \frac { 3 i } { 4 } \int _ { - \infty } ^ { x - c t } | \phi _ { \omega , c } ( \eta ) | ^ 2 d \eta \right \} . \end{align*}


\begin{align*} \beta : = - \textstyle \sum _ { i = 1 } ^ { n - 1 } y _ i \ , d x _ i - y ' \ , d x ' + \phi ( x ' ) \ , d y ' . \end{align*}


\begin{align*} \Gamma _ { H , H ^ { * } } ( \underline { W } ) = W ^ { * } \Gamma _ { H , H ^ { * } } ( H \setminus \underline { W } ) = H ^ { * } \setminus W ^ { * } . \end{align*}


\begin{align*} \hat f ( k ) & = \int _ { \R ^ 2 } e ^ { - z | x | ^ 2 + \xi \cdot x } e ^ { - i k \cdot x } \dd x = \frac { \pi } { z } e ^ { \frac { 1 } { 4 z } ( \xi - i k ) ( \xi - i k ) } = \frac { \pi } { z } e ^ { \frac { 1 } { 4 z } ( - | k | ^ 2 - 2 i \xi \cdot k + \xi \cdot \xi ) } . \end{align*}


\begin{align*} \mathbf s _ n = \mathbf F ^ H \tilde { \mathbf s } _ n \in \mathbb { C } ^ { L } , n \in \mathcal { N } . \end{align*}


\begin{align*} \frac { \rho \beta b ^ { 2 } } { d } = \frac { \rho \beta } { 3 / 2 + \left ( 1 + \beta \right ) \left ( T / 2 - \varepsilon \right ) ^ { 2 } / \left ( \varepsilon \left ( T - \varepsilon \right ) \right ) } \geq 1 - \rho . \end{align*}


\begin{align*} \| \phi _ { \omega , c } \| _ { L ^ 2 ( \R ) } ^ 2 = 8 \tan ^ { - 1 } \sqrt { \frac { \sqrt { 4 \omega } + c } { \sqrt { 4 \omega } - c } } < 4 \pi . \end{align*}


\begin{align*} \dot z ^ { * } \bigl ( Y , y ^ { * } ( N , t ) \bigr ) \times \ , \dot y ^ { * } ( N , t ) = 1 , \ , \pi - \end{align*}


\begin{align*} \inf _ { \gamma \in \Pi ( \mu _ 1 , \mu _ 2 ) } \mathbb { E } _ { \gamma } [ g ( S _ 1 , S _ 2 ) ] & = \sup _ { x \in \mathbb { R } } \max \left \{ \mu _ 1 ( x ) + \mu _ 2 ( z - x ) - 1 , 0 \right \} \ \\ \sup _ { \gamma \in \Pi ( \mu _ 1 , \mu _ 2 ) } \mathbb { E } _ { \gamma } [ g ( S _ 1 , S _ 2 ) ] & = 1 + \inf _ { x \in \mathbb { R } } \min \left \{ \mu _ 1 ( x ) + \mu _ 2 ( z - x ) - 1 , 0 \right \} . \end{align*}


\begin{align*} Q = S S ^ t + T T ^ t , A = S T ^ t + T S ^ t , L = Q - A = ( S - T ) ( S - T ) ^ t . \end{align*}


\begin{align*} c _ v & = \sup _ { x \in Q } ( v ( x ) , 1 ) , \ Q = [ 0 , 1 ] ^ d \end{align*}


\begin{align*} \overline { \mathrm { o r b } ( x ) } \supset \{ u \in \C ^ { m | n } \mid u _ i - u _ { i _ 0 } = 2 u _ i + u _ { m + j } = \pm \frac 1 2 , u _ s = x _ s \forall s \notin \{ i , i _ 0 , m + j \} \} , \end{align*}


\begin{align*} \rho ( E _ 3 [ \deg \tau ] ) = \tau \pi \psi \widehat { \tau } ( E _ 3 [ \deg \tau ] ) = \tau \pi \psi ( \ker \tau ) = \tau \pi \psi ( G ) = \tau ( G ) = 0 . \end{align*}


\begin{align*} \sum _ { j \in \mathbb { Z } } ( - 1 ) ^ j z ^ j q ^ { j ( j - 1 ) / 2 } = ( q , z , q / z ; q ) _ \infty , \end{align*}


\begin{align*} \big | I _ \varepsilon ^ 1 \big | = \left | \int _ { \mathbb { R } ^ n } J _ \varepsilon ( | x | ) \big ( f _ \xi ( x ) - f _ \xi ( 0 ) - \nabla f _ \xi ( 0 ) \cdot x - \frac { 1 } { 2 } x ^ T D ^ 2 f _ \xi ( 0 ) x \big ) \ : x \right | . \end{align*}


\begin{align*} \begin{aligned} 0 \leq \mathcal { I } ( \eta _ 0 , \delta ) - \mathcal { I } ( \eta , 0 ) & = \mathcal { I } ( \eta _ 0 , \delta ) - \mathcal { I } ( \eta , \delta ) + \mathcal { I } ( \eta , \delta ) - \mathcal { I } ( \eta , 0 ) \\ & \leq \Psi _ 1 \left ( \eta _ 0 - \eta , M ( 1 - \eta / \eta _ 0 ) \right ) + \Psi _ 2 \left ( M \delta , \delta \right ) . \end{aligned} \end{align*}


\begin{align*} \mu : = \frac 1 2 \delta _ x + \frac 1 2 \delta _ y . \end{align*}


\begin{align*} \sum \limits _ { \substack { k = 1 } } ^ { n ^ s } \psi ( k ; n ^ s ) ( k , n ^ s ) _ s & = \sum \limits _ { \substack { k = 1 } } ^ { n ^ s } \psi ( k ; n ^ s ) \sum \limits _ { d ^ s | ( k , n ^ s ) _ s } \Phi _ s ( d ^ s ) \\ & = \sum \limits _ { \substack { k = 1 } } ^ { n ^ s } \psi ( k ; n ^ s ) \sum \limits _ { \substack { d ^ s \mid k \\ d ^ s \mid n ^ s } } \Phi _ s ( d ^ s ) . \end{align*}


\begin{align*} H * w = \tilde { H } * w , \end{align*}


\begin{align*} \mathcal { S } = \left \{ A ^ \pm : \ A \in \binom { [ n ] } { k } \mbox { a n d } n \in A \right \} \mbox { a n d } \Omega \setminus \mathcal { S } . \end{align*}


\begin{align*} 1 0 ^ m - 6 \ \le \ v ( f ) \ = \ v ( 2 ^ \gamma ) \ \le \ 2 + \gamma . \end{align*}


\begin{align*} 0 < \varepsilon < \varepsilon _ L : = ( \frac \delta { 4 L } ) ^ 4 , \end{align*}


\begin{align*} \| v \| _ A = \langle A v , v \rangle ^ \frac { 1 } { 2 } = \gamma \langle e _ p , A ^ { - 1 } e _ p \rangle ^ \frac { 1 } { 2 } = \gamma ^ \frac { 1 } { 2 } = \langle e _ p , A ^ { - 1 } e _ p \rangle ^ { - \frac { 1 } { 2 } } . \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } \left | \left | u _ n \right | ^ { p - 2 } u _ n - | u | ^ { p - 2 } u \right | ^ { p ' } d x & \leq C \int _ { \Omega } \left | u _ n - u \right | ^ { p ' ( p - 1 ) } d x \\ & = C \left \| u _ n - u \right \| _ { p } ^ { p } . \end{aligned} \end{align*}


\begin{align*} \boldsymbol { K } _ 1 \left ( \gamma _ { 1 , 3 } ^ { \eta _ 1 , \delta } , \mu _ 1 \right ) = \mathbb { E } \left [ c _ 1 ( S _ 1 , \widetilde { S } _ 1 ) \right ] \leq \eta _ 0 , S _ 1 = ( Y _ 1 , X ) \widetilde { S } _ 1 = ( \widetilde { Y } _ 1 , \widetilde { X } ) . \end{align*}


\begin{align*} \frac { \delta F } { \delta \mu } ( \nu _ t , \mu _ t , y ) - \frac { \sigma ^ 2 } { 2 } \log \frac { \Phi ( \nu _ t , \mu _ t ) ( y ) } { \rho ( y ) } = \Tilde { C _ t } , \end{align*}


\begin{align*} u _ j + \iota v _ j = x _ j + \iota y _ j + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} P _ { \leq N } u = u _ { \leq N } , P _ { \geq N } u = u _ { \geq N } . \end{align*}


\begin{align*} ( f _ { \mathcal { S } } ) _ \lambda ( s _ 1 , s _ 2 ) = \sup _ { x ^ \prime \in \mathbb { R } ^ d } \left [ \varphi _ 1 ( x ^ \prime - x _ 1 , \lambda _ 1 ) + \varphi _ 2 ( x ^ \prime - x _ 2 , \lambda _ 2 ) \right ] . \end{align*}


\begin{align*} H ^ \star = \nu P ( \Lambda ) P ^ T . \end{align*}


\begin{align*} \begin{aligned} Q _ 1 ^ { ( n + 1 ) } \circ \partial & = \partial \circ Q _ 1 ^ { ( n ) } \circ f ^ { ( n ) } , \\ Q _ 2 ^ { ( n + 1 ) } \circ \partial & = \partial \circ \frac 1 { f ^ { ( n ) } } \circ Q _ 2 ^ { ( n ) } , \\ Q _ 1 ^ { ( n + 1 ) } \circ Q _ 2 ^ { ( n + 1 ) } \circ \partial & = \partial \circ Q _ 1 ^ { ( n ) } \circ Q _ 2 ^ { ( n ) } . \end{aligned} \end{align*}


\begin{align*} \langle D _ { j } S _ { \omega , \mathbf { c } } ( \Phi ) , \psi \rangle : = \lim _ { \epsilon \rightarrow 0 } \frac { S _ { \omega , \mathbf { c } } ( \Phi + \epsilon E _ j ( \psi ) ) - S _ { \omega , \mathbf { c } } ( \Phi ) } { \epsilon } , \ \ \Phi \in \mathcal { H } ^ 1 , \ \psi \in ( H ^ 1 ( \R ^ d ) ) ^ { d } , \ j = 1 , 2 , 3 , \end{align*}


\begin{align*} ( z ^ 2 - 1 ) ^ \alpha : = ( z + 1 ) ^ \alpha ( z - 1 ) ^ \alpha , \end{align*}


\begin{align*} \| v _ t \| _ { L ^ 2 ( \mathrm { M } _ 0 ) } ^ 2 = ( u | ( \Delta + \lambda - q ) w _ t ) _ { L ^ 2 ( \mathrm { M } _ 0 ) } . \end{align*}


\begin{align*} Q _ j = \left \{ \otimes ^ { j - 1 } \begin{pmatrix} 1 & 0 \\ 0 & - 1 \end{pmatrix} \right \} \otimes \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes \left \{ \otimes ^ { \beta - j } \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right \} . \end{align*}


\begin{align*} S = \{ s \in \R ^ 2 \ , ; \ , 0 \leq s _ 2 \leq f ( s _ 1 ) , 0 \leq s _ 1 \leq 1 \} . \end{align*}


\begin{align*} \left ( \sum _ { j = 1 } ^ d X _ j ^ t \otimes Y _ j \right ) ( \varphi \otimes \xi ) = \sum _ { j = 1 } ^ d ( \varphi \circ X _ j ) \otimes Y _ j \xi . \end{align*}


\begin{align*} ( P - \lambda ) ^ { - 1 } = Q ( \lambda ) + Q ( \lambda ) \sum _ { j \geq 1 } R ( \lambda ) ^ j . \end{align*}


\begin{align*} \lambda _ { j _ 1 } = \lambda _ { j _ 1 ^ { - 1 } } = \dots = \lambda _ { j _ r } = \lambda _ { j _ r ^ { - 1 } } = \pm 1 . \end{align*}


\begin{align*} & \widetilde { u } _ h : B _ 1 \to \S ^ { k - 1 } , \widetilde { u } _ h ( y ) : = u _ h ( x _ h + \sigma _ h y ) , \\ & p _ h : B _ 1 \to ( 1 , + \infty ) , p _ h ( y ) : = p ( x _ h + \sigma _ h y ) , \end{align*}


\begin{align*} \tilde \phi _ 0 ( u _ n ^ N ) = u _ n \tilde \phi ( u ^ G _ g ) = u _ g \otimes \lambda ^ { G / N } _ { q ( g ) } , n \in N , \ g \in G , \end{align*}


\begin{align*} \begin{cases} u = g & \\ D u = f & , \end{cases} \end{align*}


\begin{align*} f _ { \lvert \tau \rvert } ^ { ( m ) } ( 0 ) = \sum _ { \pi \in S _ m } \mathrm { d } y _ { \pi ( 1 ) } \otimes \cdots \otimes \mathrm { d } y _ { \pi ( m ) } , \end{align*}


\begin{align*} \Delta w _ \delta = \Delta w = \mu ( S ^ \varphi - m \lambda ( x ) ) ( 1 + w ) \ , . \end{align*}


\begin{align*} \phi ( \xi + \mathbf { h } ) = \sum _ { | \alpha | \leq 3 } \frac { \partial ^ { \alpha } \phi ( \xi ) } { \alpha ! } \mathbf { h } ^ { \alpha } + \sum _ { | \alpha | = 4 } \frac { \partial ^ { \alpha } \phi ( \xi + \theta _ { \xi } \mathbf { h } ) } { \alpha ! } \mathbf { h } ^ { \alpha } , \end{align*}


\begin{align*} \mathrm { V o l } _ { \omega _ 0 } ( F _ b ) = \int _ { F _ b } \Omega _ 0 ^ m = \int _ { { F _ { \tilde { \pi } ^ { - 1 } ( \gamma ( b ) ) } } } \gamma ^ * ( \Omega _ 0 ) ^ m = \int _ { { F _ { \tilde { \pi } ^ { - 1 } ( \gamma ( b ) ) } } } { { c _ \gamma ^ m } } \cdot \Omega _ 0 ^ m = { { c _ \gamma ^ m } } \int _ { F _ b } \Omega _ 0 ^ m , \end{align*}


\begin{align*} \int _ { M _ i } K _ { g _ i } \operatorname d v o l _ { g _ i } = 4 \pi = \int _ Z K _ x \operatorname d s . \end{align*}


\begin{align*} & d \ , G _ r ( \xi , \sigma _ 1 , \ldots , \sigma _ r , \alpha _ 1 , \ldots , \alpha _ r , \beta _ 1 , \ldots , \beta _ r ) \ , = \ , \\ & \textbf { e } ( ( \xi - \sigma _ r ) \beta _ r ) \ , F ( \sigma _ r - \xi , \alpha _ r - \tau \beta _ r ) \ , G _ { r - 1 } ( \xi , \sigma _ 1 , \ldots , \sigma _ { r - 1 } , \alpha _ 1 , \ldots , \alpha _ { r - 1 } , \beta _ 1 , \ldots , \beta _ { r - 1 } ) \ , d \xi , \end{align*}


\begin{align*} S ( \beta ) = \beta _ 1 \sigma _ j \beta _ 2 \end{align*}


\begin{align*} \mathcal G : = \{ ( t , r ) \ , \big | \ , 1 - g ( r ) t > 0 \} . \end{align*}


\begin{align*} a x _ i ^ n v ^ { n m ' - k m ' - l k ' } + b x _ i ^ k y _ i ^ l + c y _ i ^ m u ^ { m n ' - k l ' - l n ' } = 0 , \end{align*}


\begin{align*} \begin{cases} P ( ( - \Delta ) ^ s ) u + F ( x , u ) = 0 & \Omega , \\ u = f & \Omega ^ c . \end{cases} \end{align*}


\begin{align*} \begin{aligned} ( [ 1 , g ] , 1 ) ( [ \epsilon , 1 ] , 1 ) & = ( [ \epsilon , g ^ { s ( \epsilon ) } ] , \widetilde { C } _ { X ^ { \ast } } ( [ 1 , g ] , [ \epsilon , 1 ] ) ) \\ & = ( [ \epsilon , g ^ { s ( \epsilon ) } ] , \nu ( \epsilon , g ) \widetilde { c } _ { X ^ { \ast } } ( g ^ { s ( \epsilon ) } , 1 ) ) \\ & = ( [ \epsilon , g ^ { s ( \epsilon ) } ] , \nu ( \epsilon , g ) ) \\ & = ( [ \epsilon , 1 ] , 1 ) ( [ 1 , g ^ { s ( \epsilon ) } ] , \nu ( \epsilon , g ) ) . \end{aligned} \end{align*}


\begin{align*} \| H ( f ) \| _ { L _ { | x | _ h } ^ p L _ { \theta } ^ { \bar { p } _ 1 } ( \mathbb H ^ n ) \rightarrow L _ { | x | _ h } ^ p L _ { \theta } ^ { \bar { p } _ 2 } ( \mathbb H ^ n ) } = G . \end{align*}


\begin{align*} \int _ { \R ^ n } \nabla \psi _ { 1 , \sigma _ i } ^ h \nabla \tilde { u } ^ i d x = 0 , h = 0 , 1 , \cdots , n . \end{align*}


\begin{align*} \beta _ y ^ { - 1 } \chi _ y ( \tfrac { 1 } { s } + \tfrac { 1 } { t } ) = ( \tfrac { 1 } { s } + \tfrac { 1 } { t } , y ) _ F \gamma ( y , \psi ^ { \tfrac { 1 } { 2 } } ) . \end{align*}


\begin{align*} p \widetilde { \mathbb { S } } _ { \lambda ^ i } : = \pi ^ { * } ( \widetilde { \mathbb { S } } _ { \lambda ^ i } ) . \end{align*}


\begin{align*} \| f \| _ X : = \int _ 0 ^ 1 \ ! | f | \ , \mathrm { d } x , \end{align*}


\begin{align*} F ( k , t , n ) = \frac { ( n ) _ { k + 1 } } { n ^ t } \frac { 1 } { k ! } \Delta ^ k \left [ \frac { x ^ t } { n - x } \right ] _ { x = 0 } ^ { } , \end{align*}


\begin{align*} y _ { D 1 } ^ { } = \left \{ \begin{array} { c c c c c c c c c c } \sqrt { 1 - \alpha } h _ { A D } + \sqrt { \alpha } h _ { C D } z _ { d } + n _ { D 1 } , & \mbox { i f } x = 1 ; \\ \sqrt { \alpha } h _ { C D } z _ d + n _ { D 1 } , & \mbox { i f } x = 0 ; \\ \end{array} \right . \end{align*}


\begin{align*} \varphi ^ + ( Y _ T ) = \sum _ { F \ge T } a _ F z ^ { - r ( F ) } \end{align*}


\begin{align*} t _ 0 = \frac { ( 1 - \lambda ) ^ { p - 1 } } { \lambda ^ { p - 1 } + ( 1 - \lambda ) ^ { p - 1 } } . \end{align*}


\begin{align*} D _ \Psi = \begin{pmatrix} 3 & 2 \\ - 2 & - 1 \end{pmatrix} D _ \Theta = \begin{pmatrix} 1 & 0 \\ - 1 & 1 \end{pmatrix} \end{align*}


\begin{align*} \lambda _ t : = ( P _ { t , 1 } ^ u ) ^ * \lambda _ 1 \in T ^ * _ { \gamma ( t ) } M , \qquad \forall \ , t \in [ 0 , 1 ] , \end{align*}


\begin{align*} Q _ j = Q _ i - ( Q _ i - Q _ j ) . \end{align*}


\begin{align*} & ~ ~ ~ ~ ( \beta M _ 1 ^ H - \alpha M _ 2 ^ H ) ^ H ( \beta M _ 1 ^ H - \alpha M _ 2 ^ H ) - ( \beta M _ 1 ^ H + \alpha M _ 2 ^ H ) ^ H ( \beta M _ 1 ^ H + \alpha M _ 2 ^ H ) \\ & = - \overline { \beta } \alpha M _ 1 M _ 2 ^ H - \overline { \alpha } \beta M _ 2 M _ 1 ^ H - \overline { \beta } \alpha M _ 1 M _ 2 ^ H - \overline { \alpha } \beta M _ 2 M _ 1 ^ H \\ & = - 2 \alpha \overline { \beta } ( M _ 1 M _ 2 ^ H + M _ 2 M _ 1 ^ H ) \\ & \leq 0 . \end{align*}


\begin{align*} \Phi ( \mathcal { R } A ) = \mathcal { R } ( \Phi ( A ) ) , \end{align*}


\begin{align*} { \mathcal { C } } _ { \alpha } = ( I _ n \otimes U ) \Pi \big ( \Lambda \otimes B _ 1 ^ { ( \alpha ) } + ( - 2 I _ { m } ) \otimes B _ 2 ^ { ( \alpha ) } \big ) \Pi ^ T ( I _ n \otimes U ) ^ T . \end{align*}


\begin{align*} \int _ a ^ { b } ( A + x ) ^ { \xi } x ^ { \lambda } \ , d x \simeq \begin{cases} ( b - a ) b ^ { \lambda } ( b + A ) ^ { \xi } & \xi + \lambda > - 1 , \\ b ^ { \lambda } ( b + A ) ^ { - \lambda } \log \Big ( 1 + \frac { b - a } { a + A } \Big ) & \xi + \lambda = - 1 , \\ ( b - a ) b ^ { \lambda } ( a + A ) ^ { \xi + \lambda + 1 } ( b + A ) ^ { - \lambda - 1 } & \xi + \lambda < - 1 , \end{cases} \end{align*}


\begin{align*} \tau = \frac { \sigma q } { q - 2 } < \left ( 1 - \frac { 2 } { q } \right ) \min \{ s , 1 - s \} . \end{align*}


\begin{align*} \Sigma _ N ^ 2 ( L ) : = \langle ( S _ N ( \ell ) - L ) ^ 2 \rangle = \langle S _ N ( \ell ) ^ 2 \rangle - L ^ 2 . \end{align*}


\begin{align*} d ( x , y ) : = \max _ { 0 \le j \le n - 1 } ( x - y ) ^ 2 \cdot y ^ j \cdot x ^ { n - 1 - j } . \end{align*}


\begin{align*} v _ t = \sum _ { j \in N _ t ^ 0 } a _ j \varphi _ j . \end{align*}


\begin{align*} \| \hat { \mathbf { A } } \hat { \mathbf { x } } _ k + \hat { \mathbf { B } } \hat { \mathbf { u } } _ k \| _ { \hat { \mathcal { X } } } ^ 2 - \| \hat { \mathbf { x } } _ k \| _ { \hat { \mathcal { X } } } ^ 2 = \| \hat { \mathbf { x } } _ { k + 1 } \| _ { \hat { \mathcal { X } } } ^ 2 - \| \hat { \mathbf { x } } _ k \| _ { \hat { \mathcal { X } } } ^ 2 \leq \| \hat { \mathbf { u } } _ k \| ^ 2 - \| \hat { \mathbf { y } } _ k \| ^ 2 = \| \hat { \mathbf { u } } _ k \| ^ 2 - \| \hat { \mathbf { C } } \hat { \mathbf { x } } _ k + \hat { \mathbf { D } } \hat { \mathbf { u } } _ k \| ^ 2 , \end{align*}


\begin{align*} t ^ n = \sum _ { k = 0 } ^ { n } S _ B ( n , k ) ( t ) _ k ^ B , \end{align*}


\begin{align*} \frac { \partial u } { \partial \nu } = 0 \ , \mbox { o n } \ , \partial \Omega \times ( 0 , + \infty ) . \end{align*}


\begin{align*} t ^ { \sigma } \| \Phi ( u ) ( t ) - \Phi ( v ) ( t ) \| _ { L ^ a } \leq C \rho ( u , v ) \sum _ { k = 0 } ^ { \infty } ( C \lambda ) ^ k M ^ { p k + m - 1 } . \end{align*}


\begin{align*} X _ T ^ \ast : = \sup _ { s \in [ 0 , T ] } X _ s . \end{align*}


\begin{align*} t _ i \ , t _ j = \lambda \quad \quad \{ i , j \} \in M . \end{align*}


\begin{align*} & ( \gamma , m ) \mapsto m \\ & ( \gamma , m ) \mapsto \gamma \cdot m \\ & ( \gamma ' , \gamma \cdot m ) ( \gamma , m ) = ( \gamma ' \gamma , m ) \\ & ( \gamma , m ) \mapsto ( \gamma ^ { - 1 } , \gamma \cdot m ) \end{align*}


\begin{align*} P ^ \omega \left ( E _ { j , t } \mid A _ { j - 1 , t } \right ) \leq 1 - e ^ { - 2 h ( t ) \log h ( t ) } , j = 2 , \ldots , t / h ( t ) . \end{align*}


\begin{align*} \varphi ^ + ( Y _ T ) + \varphi ^ + ( Y _ F ) \alpha ( Y _ \bullet ) = \varphi ^ + ( Y _ T ) - \varphi ^ + ( Y _ F ) a = \varphi ^ - ( Y _ T ) \end{align*}


\begin{align*} & P ( W ^ k _ { \ell _ 2 \Delta , t } ( j ) = 1 , A | Z _ 0 ( \Delta \ell _ 2 ) = m , Z _ 0 ( \Delta \ell _ 1 ) = n , X _ { \ell _ 1 , \Delta } = 1 , B _ { \ell _ 1 \Delta } ( \ell _ 2 \Delta ) = i ) \\ & = i \nu \Delta p ^ k _ { j , + } ( t - \Delta \ell _ 2 ) . \end{align*}


\begin{align*} \| f ^ { = d } \| _ 2 ^ 2 \leq \rho ^ { - d } \gamma \cdot \gamma _ 1 \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) ^ \theta = ( e / \rho ) ^ { d } \gamma _ 1 \cdot \gamma . \end{align*}


\begin{align*} 0 = [ ( - \varphi _ { \lambda } ( x , [ { y _ 1 } _ \eta y _ 2 ] ) + \varphi _ \lambda ( [ x _ \mu y _ 1 ] , y _ 2 ) + ( - 1 ) ^ { x y _ 1 } \varphi _ \lambda ( y _ 1 , [ x _ \mu y _ 2 ] ) ) _ { \lambda + \mu } [ w _ \gamma v ] ] . \end{align*}


\begin{align*} \zeta _ { Q } \left ( s \right ) : = \zeta _ { 2 } ( s , c ) = \frac { \pi } { \sqrt { c } } \ , \frac { 1 } { s - 1 } + \frac { \pi } { \sqrt { c } } \ , \left ( 2 \gamma - \log \left ( 4 c \right ) - 4 \log \left ( \left | \eta \left ( i \sqrt { c } \right ) \right | \right ) \right ) + O ( s - 1 ) . \end{align*}


\begin{align*} \begin{aligned} \int _ { \R ^ N } | v ( x ) | \zeta ( x ) | \Delta \gamma _ R | \ , d x & \le \frac { \bar C } { R ^ 2 } \int _ { B _ R \setminus B _ { R / 2 } } | v ( x ) | \zeta ( x ) \ , d x \\ & \le \frac { \bar C C } { R ^ 2 } \int _ { B _ R \setminus B _ { R / 2 } } | v ( x ) | \psi ( x ) \ , d x , \end{aligned} \end{align*}


\begin{align*} \dfrac { 1 } { r ( x ) } = \dfrac { 1 } { p ( x ) } + \dfrac { 1 } { q ( x ) } \mbox { f o r a . e . } \ x \in \mathbb { R } ^ { n } . \end{align*}


\begin{align*} { \mathcal P } _ { k , l } ^ { ( \alpha , \beta ) } ( w , \phi ) : = w ^ { k - l } P _ l ^ { ( \alpha - \beta - 1 , \beta + k - l ) } ( 2 w ^ 2 \ ! - \ ! 1 ) C _ { k - l } ^ \beta ( \cos \phi ) , \end{align*}


\begin{align*} \overline { \overline { C } } _ { X ^ { \ast } } ( p _ g , k _ g ) & = \widetilde { C } _ { X ^ { \ast } } ( p _ g , k _ g ) \widetilde { s } ( p _ g ) \widetilde { s } ( k _ g ) \widetilde { s } ( g ) ^ { - 1 } \\ & = \widetilde { s } ( p _ g ) \widetilde { s } ( k _ g ) \widetilde { s } ( g ) ^ { - 1 } = 1 ; \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ N } ( I _ \mu * | t _ \nu \star { u _ 0 } | ^ { p } ) | t _ \nu \star { v _ 0 } | ^ { q } \leq \mathbb { A } , \ \ \mathbb { A } = \Bigl ( \frac { 2 2 ^ * _ \mu - \gamma _ p - \gamma _ q } { 2 2 ^ * _ \mu - 2 } \nu ( \gamma _ p + \gamma _ q ) D ^ { \frac { 2 } { \gamma _ p + \gamma _ q } } _ 0 \Bigr ) ^ { \frac { \gamma _ p + \gamma _ q } { 2 - \gamma _ p - \gamma _ q } } . \end{align*}


\begin{align*} \mathbb { E } \{ Z \} = \frac { \kappa } { \kappa + \beta } , \mathbb { E } \{ Z ^ 2 \} = \frac { \kappa ( \kappa + 1 ) } { \kappa + \beta ( \kappa + \beta + 1 ) } . \end{align*}


\begin{align*} 0 = \int _ M \chi ( q _ 2 - q _ 1 ) v _ 2 v _ 1 d x + \int _ M [ \Delta , \psi ] v v _ 1 d x . \end{align*}


\begin{align*} \rho ( a b ) \rho ( - a b ) & = \rho ( a ) \rho ( ( - a ) b ) + \rho ( b ) \rho ( ( - b ) a ) \\ & = \rho ( a ) \rho ( - a ) + \rho ( a ) \rho ( b ) + \rho ( b ) \rho ( - b ) + \rho ( b ) \rho ( a ) \\ & = \rho ( a ) \rho ( b ) + \rho ( b ) \rho ( a ) . \end{align*}


\begin{align*} \phi _ 0 = e ^ { \eta _ 0 } e ^ { \eta _ 1 } \cdots e ^ { \eta _ S } , \end{align*}


\begin{align*} J _ r ( x , \beta ) = \frac { 1 } { 2 v _ r ( x ) } ( \beta - d _ r ( x ) ) ^ 2 + \frac { 1 } { 6 } \theta '' ( x , \beta _ * ) ( \beta - d _ r ( x ) ) ^ 3 \end{align*}


\begin{align*} \langle x , \pi ( k , \ell ) g \rangle \overline { \langle x , \pi ( k + q , \ell + p ) g \rangle } = \\ \frac { e ^ { 2 \pi i k p / M } } { 3 } \sum _ { t = 0 } ^ { 2 } e ^ { 2 \pi i t / 3 } \vert \langle x , \pi ( k , \ell ) g _ { q p t } \rangle \vert ^ 2 \end{align*}


\begin{align*} & d \overline { \Pi } _ { \psi } ( e ^ + ) A ( [ \epsilon , x ] ) - d \overline { \Pi } _ { \psi } ( e ^ - ) A ( [ \epsilon , x ] ) \\ & = \pi i \epsilon x ^ 2 A ( [ \epsilon , x ] ) - \frac { \epsilon i } { 4 \pi } \frac { d ^ 2 } { d x ^ 2 } A ( [ \epsilon , x ] ) \\ & = \pi i \epsilon x ^ 2 A ( [ \epsilon , x ] ) - \pi i \epsilon x ^ 2 A ( [ \epsilon , x ] ) - \frac { \epsilon i } { 4 \pi } [ - 2 \epsilon \pi e ^ { - \epsilon \pi x ^ 2 } ] \\ & = \tfrac { i } { 2 } A ( [ \epsilon , x ] ) ; \end{align*}


\begin{align*} t _ \alpha ( x ) = p ( x ) + q ( x ) \tanh ( \alpha x ) , \end{align*}


\begin{align*} \nabla h ( x ) = - \frac { 1 } { m \omega _ m } \begin{cases} x & \ | x | < 1 \\ \frac { x } { | x | ^ m } & \ | x | \ge 1 \end{cases} \end{align*}