\begin{align*} \mathcal { V } f ( x ) = x ^ { \lambda } \mathcal { U } \big ( ( \cdot ) ^ { - \lambda } f \big ) ( x ) , x > 0 . \end{align*}


\begin{align*} y ^ { \left ( n \right ) } \left ( x \right ) = f \left ( x , y \right ) + \int \limits _ { x _ { 0 } } ^ { x } K d t , x > x _ { 0 } \end{align*}


\begin{align*} x : = \tilde r ( y ) , \end{align*}


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


\begin{align*} w = a + \zeta \xi \end{align*}


\begin{align*} \intop _ { 1 } ^ { \infty } \frac { d t } { ( t ^ { 2 } - 1 ) ^ { s } } = \frac { \Gamma \left ( 1 - s \right ) \Gamma \left ( s - \frac { 1 } { 2 } \right ) } { 2 \sqrt { \pi } } , \ , \ , \ , \frac { 1 } { 2 } < ( s ) < 1 , \end{align*}


\begin{align*} \varphi _ { \lambda } ( x \otimes a , y \otimes b ) = \sum _ { t \in T } \varphi _ { \lambda t } ( x \otimes a , y \otimes b ) \otimes b _ t \end{align*}


\begin{align*} h _ { \mu } ( X , \eta ) = - \int _ { G } \int _ { X } \log \frac { d g ^ { - 1 } \eta } { d \eta } ( x ) d \eta ( x ) d \mu ( g ) . \end{align*}


\begin{align*} & \left \| V ( \gamma _ 1 , \sigma _ 1 , t ) - V ( \gamma _ 2 , \sigma _ 2 , t ) \right \| _ { L ^ 2 _ N } \leq \| \psi ( t ) \| _ { W ^ { 1 , \infty } } \left ( \| \gamma _ 1 - \gamma _ 2 \| _ { L ^ 2 _ N } + \frac { 1 } { \sqrt { N } } | \sigma _ 1 - \sigma _ 2 | \right ) \end{align*}


\begin{align*} \nu ( g ' ( a _ i ) ) - \nu _ j ( g ) = \nu ( g ' ) - \nu _ j ( g ) = \beta _ j < \alpha _ i = - \nu ( x - a _ i ) . \end{align*}


\begin{align*} \widetilde V ( x ) = \begin{cases} V ( x ) , \ & | x | < M _ 0 ; \\ \max \{ V ( x ) , | x | ^ 2 \} , \ & | x | \geq M _ 0 . \end{cases} \end{align*}


\begin{align*} \Sigma _ { \mathrm { R D J } } ( \delta ) : = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \ \boldsymbol { K } _ 1 ( \widehat { \mu } _ { 1 3 } , \gamma _ { 1 3 } ) \leq \delta _ 1 , \ \boldsymbol { K } _ 2 ( \widehat { \mu } _ { 2 3 } , \gamma _ { 2 3 } ) \leq \delta _ 2 \right \} , \end{align*}


\begin{align*} Q _ { 3 , \vec { v } _ { s , t , f , 0 } ^ { ( 2 ) } } = \left ( \left ( \begin{array} { r r r } t ^ { 2 } + s & t & 1 \\ s t + 1 & s & 0 \\ t & 1 & 0 \end{array} \right ) , \left ( \begin{array} { r r r } t & 1 & 0 \\ s & 0 & 1 \\ 1 & 0 & 0 \end{array} \right ) , \left ( \begin{array} { r r r } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right ) \right ) . \end{align*}


\begin{align*} C _ { n } = \left \{ \omega : \log \left | \omega _ { n } ^ { - 1 } \omega _ { n + 1 } \right | \le n \epsilon , H _ { 0 } \omega _ { n } \in A _ { n } , x _ { 0 } \in \left [ H _ { 0 } \omega _ { n } ; \omega _ { n } ^ { - 1 } \omega _ { n + 1 } \right ] \right \} . \end{align*}


\begin{align*} d u + A u \ , d t = f ( u ) \ , d t + \sigma ( u ) B \ , d W ( t ) , u ( 0 ) = u _ 0 , \end{align*}


\begin{align*} \rho _ { \theta } ( \theta , f ( \theta ) ) + 2 { \rm R e } ( e ^ { c ( \theta , f ( \theta ) ) + i d ( \theta , f ( \theta ) ) } \frac { \partial g } { \partial \theta } ( \theta ) ) = 0 . \end{align*}


\begin{align*} \mathcal { A } [ g ] = \{ u \in W ^ { 2 , 2 } ( B _ 1 ) : u = g \partial B _ 1 D u = D g \partial B _ 1 \} \end{align*}


\begin{align*} \nu _ n = \inf _ { M \in \mathcal { F } _ n } \sup _ { u \in M } G ( u ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } \mathrm { d } X _ { s } & = & u _ { s } \mathrm { d } W _ { s } , \\ - \mathrm { d } Y _ { s } & = & \Big [ \left \langle Q _ { s } X _ { s } , X _ { s } \right \rangle + \left \langle R _ { s } u _ { s } , u _ { s } \right \rangle + \kappa Z _ { s } \Big ] \mathrm { d } s - Z _ { s } \mathrm { d } W ( s ) , \\ X _ { 0 } & = & x , Y _ { T } = \left \langle G X _ { T } , X _ { T } \right \rangle , \end{array} \right . \end{align*}


\begin{align*} k : = L ^ D \ , ( = K ^ D ) . \end{align*}


\begin{align*} M _ b ( \theta ) = \mathbb { E } _ { \Phi } \{ P _ s ^ b ( \theta ) \} \end{align*}


\begin{align*} \mathcal A _ N ( \omega _ 0 , \omega _ 1 , \rho _ 1 ) = \begin{pmatrix} - 2 N + 2 - 2 N \frac { \omega _ 1 } { \omega _ 0 } & - \frac { 2 \rho _ 1 } { \omega _ 0 } - 2 \\ - 2 & - 2 N - 4 + \frac 2 { \omega _ 0 } - ( 2 N + 2 ) \frac { \omega _ 1 } { \omega _ 0 } \end{pmatrix} . \end{align*}


\begin{align*} \lim _ { n \to \infty } \norm { S _ n ( \nu _ k ) - \overline { \nu } } = 0 . \end{align*}


\begin{align*} \| f \| _ { p , q , \mu } = \Big ( p \int _ 0 ^ \infty t ^ { q - 1 } f _ { * \mu } ( t ) ^ \frac { q } { p } \dd t \Big ) ^ { \frac 1 { q } } \end{align*}


\begin{align*} \left | | \beta _ j | - \frac { 1 } { 2 } \right | < \frac { \beta _ 0 } { 2 } , \ \ \ j = \pm 1 . \end{align*}


\begin{align*} ( E = N \oplus N ^ { - 1 } K _ X \oplus N ^ { - 1 } \oplus N K _ X ^ { - 1 } , \quad \theta = \begin{pmatrix} 0 & 0 & \nu & q _ 2 \\ 0 & 0 & q _ 2 & \mu \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} ) . \end{align*}


\begin{align*} a _ n = \sum _ { i = 0 } ^ n a ^ { \frac { 1 } { p ^ i } } \end{align*}


\begin{align*} \delta _ { 1 / r } ^ { A } \delta _ { r } ^ { B } \{ x \in G \colon \| x \| = 1 \} \subseteq \mathcal { B } ^ { A } ( e , D ) \subseteq \mathcal { B } ( e , R ) , \end{align*}


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


\begin{align*} \lim _ { a \to \infty } \sup _ { x \in e M _ + e , x \neq 0 } \frac { \abs { ( M _ a ( \mu ) - \bar { \mu } ) ( x ) } } { \tau ( x ) } = 0 . \end{align*}


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


\begin{align*} \langle a v , v \rangle = \phi ( a ) , a \in M . \end{align*}


\begin{align*} X ( K ) \cap \Gamma = \left \{ p ^ n Q \colon n \ge 0 \right \} . \end{align*}


\begin{align*} U ( t , \xi ) : = \left ( \begin{array} { c } i \langle \xi \rangle \widehat { u } ( t , \xi ) \\ \partial _ { t } \widehat { u } ( t , \xi ) \end{array} \right ) , U _ { 0 } ( \xi ) : = \left ( \begin{array} { c } i \langle \xi \rangle \widehat { u } _ { 0 } ( \xi ) \\ \widehat { u } _ { 1 } ( \xi ) \end{array} \right ) , \end{align*}


\begin{align*} \Pi _ \chi ^ { \rm s t i f f } \widehat { P } _ \chi \vect e _ j = \Pi _ 0 ^ { \rm s t i f f } \widehat { P } _ 0 \vect e _ j + \mathcal { O } ( | \chi | ) = \Pi _ 0 ^ { \rm s t i f f } \vect e _ j + \mathcal { O } ( | \chi | ) = \vect e _ j + \mathcal { O } ( | \chi | ) , j = 1 , 2 , 3 , \end{align*}


\begin{align*} \left | \frac { a _ i } { r } - \frac { a _ i ' } { r ' } \right | \geq \frac { | c _ i ^ \bot | } { | r r ' | } i = 1 , 2 \quad \max _ { i = 1 , 2 } \left \{ | c _ i | \left | \frac { a _ i } { r } - \frac { a _ i ' } { r ' } \right | \right \} \geq 1 . \end{align*}


\begin{align*} g ( P \otimes \lbrace \alpha , \beta \rbrace ) = ( g P ) \otimes \lbrace g ( \alpha ) , g ( \beta ) \rbrace . \end{align*}


\begin{align*} \phi _ 0 = e ^ { \eta _ 0 } b . \end{align*}


\begin{align*} [ w , x ] : = \{ \sigma ^ m y \in \{ 0 , 1 \} ^ \Z : w \leq y \leq x , \ m \in \Z \} . \end{align*}


\begin{align*} F _ X ( x ) = \frac { 1 } { 2 } + \frac { \alpha ( x - \mu ) } { 2 } \bigg [ K _ { \nu } ( \alpha | x - \mu | ) \mathbf { L } _ { \nu - 1 } ( \alpha | x - \mu | ) + \mathbf { L } _ { \nu } ( \alpha | x - \mu | ) K _ { \nu - 1 } ( \alpha | x - \mu | ) \bigg ] , \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle \dfrac { a _ 2 } { m _ 1 } \int _ { \mathbb { R } ^ N } | \nabla \xi _ n | ^ 2 d x + \dfrac { a _ 1 } { 2 m _ 2 } \int _ { \mathbb { R } ^ N } | \nabla \eta _ n | ^ 2 d x - \dfrac { N } { 2 } a _ 1 a _ 2 \int _ { \mathbb { R } ^ N } \eta _ n \xi _ n ^ 2 d x = 0 . \end{array} \right . \end{align*}


\begin{align*} \left ( \frac { 1 } { T _ i ( u ) } , \int _ 0 ^ { + \infty } e ^ { 2 \rho _ i ( u + v ) - 2 \rho _ i ( u ) } d v \right ) _ { i \in V } = \left ( \frac { 1 } { T _ i ^ { \infty } } + \frac { 1 } { \theta _ i ^ 2 T _ i ^ * ( u ) } , \frac { Z _ i ^ * ( u ) } { e _ i ^ * ( u ) } + \frac { T _ i ^ { \infty } } { \theta _ i ^ 2 e _ i ^ * ( u ) ^ 2 } \right ) _ { i \in V } . \end{align*}


\begin{align*} \mathsf { B } \mathbb { P } & = \{ B \in \mathsf { F } \mathbb { P } : \mathbb { P } = B ^ \lessgtr \} . \\ \mathsf { C } \mathbb { P } & = \{ C \in \mathsf { P } \mathbb { P } : \exists B \in \mathsf { B } \mathbb { P } \ ( B \leq C ) \} . \end{align*}


\begin{align*} \aligned \left \{ \begin{array} { l l l } - i \partial _ t \Psi _ 1 = \Delta \Psi _ 1 + ( K ( x ) \ast | \Psi _ 1 | ^ { r _ 1 } ) | \Psi _ 1 | ^ { r _ 1 - 2 } \Psi _ 1 + \nu p ( K ( x ) \ast | \Psi _ 2 | ^ { q } ) | \Psi _ 1 | ^ { p - 2 } \Psi _ 1 , \\ - i \partial _ t \Psi _ 2 = \Delta \Psi _ 2 + ( K ( x ) \ast | \Psi _ 2 | ^ { r _ 2 } ) | \Psi _ 2 | ^ { r _ 2 - 2 } \Psi _ 2 + \nu q ( K ( x ) \ast | \Psi _ 1 | ^ { p } ) | \Psi _ 2 | ^ { q - 2 } \Psi _ 2 , \end{array} \right . \endaligned \end{align*}


\begin{align*} f = \sum _ { n = 0 } ^ \infty \sum _ { ( i _ 1 , \ldots , i _ n ) \in \{ 1 , 2 \} ^ n } f _ { i _ 1 , \ldots , i _ n } \end{align*}


\begin{align*} Z ( { \mathcal { G } ( \mathbb { Q } _ p ) } ) \mathcal { G } ( \mathbb { Q } ) \cap \mathbf { T } _ { K ^ \circ } ( \mathbb { Q } _ p ) = Z ( { \mathcal { G } ( \mathbb { Q } _ p ) } ) \mathbf { T } _ { K ^ \circ } ( \mathbb { Q } ) , \end{align*}


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


\begin{align*} \int _ 0 ^ \infty \frac { 1 } { ( 1 + t ) ^ a t ^ { \beta } } d t = \int _ 0 ^ 1 ( 1 - t ) ^ { - \beta } t ^ { a + \beta - 2 } d t = B ( 1 - \beta , a + \beta - 1 ) . \\ \end{align*}


\begin{align*} f _ { \tilde { \textbf { Q } } } = \frac { g } { b } \cdot ( b h ) . \end{align*}


\begin{align*} \left | \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } \leq x \right ) - \Phi ( x ) \right | & = \left | \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } > x \right ) - ( 1 - \Phi ( x ) ) \right | \\ & \leq \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } > x \right ) + 1 - \Phi ( x ) . \end{align*}


\begin{align*} \sum _ { \ell = 1 } ^ { L } \phi _ \ell ( x _ \ell ) \geq f ( x ) , \forall x = ( x _ 1 , \ldots , x _ L ) \in \mathcal { X } . \end{align*}


\begin{align*} \mathcal { E } ^ { C H } ( c ) : = \int _ { \Omega } \frac { 1 } { 2 } | \nabla c | ^ 2 + f ( c ) \ : x . \end{align*}


\begin{align*} \mathcal { O } _ i ^ + : = \left \{ \left ( ( A ^ + , \underline { A } ^ + ) , ( B ^ + , \underline { B } ^ + ) \right ) , \left ( ( A ^ + , \underline { A } ^ + ) , ( B ^ - , \underline { B } ^ - ) \right ) : A , B \in \binom { [ n ] } { k } \mbox { a n d } | A \cap B | = k - i \right \} , \end{align*}


\begin{align*} t = \tfrac { 1 } { \sqrt { N } } N \ge L \ , . \end{align*}


\begin{align*} \ker \bigl ( \Pi _ \chi ^ { \rm s t i f f ( s o f t ) } \bigr ) = \left \{ 0 \right \} , \mathcal { D } \bigl ( \mathcal { A } _ { 0 , \chi } ^ { \rm s t i f f ( s o f t ) } \bigr ) \cap \mathcal { R } \bigl ( \Pi _ \chi ^ { \rm s t i f f ( s o f t ) } \bigr ) = \{ 0 \} . \end{align*}


\begin{align*} \mathcal { I } ( 0 ) = \sup _ { \gamma \in \mathcal { F } ( \mu _ { 1 3 } , \mu _ { 2 3 } ) } \int _ \mathcal { S } f \ , d \gamma . \end{align*}


\begin{align*} \begin{vmatrix} \wp ( z ) & \wp ' ( z ) & 1 \\ \wp ( w ) & \wp ' ( w ) & 1 \\ \wp ( z + w ) & - \wp ' ( z + w ) & 1 \end{vmatrix} = 0 , \end{align*}


\begin{align*} \chi _ g \cdot ( \chi _ { f _ i } \circ \bar g ^ { \flat } ) = \chi _ { f _ i ' } \cdot ( \chi _ { g _ i } \circ \theta _ i ' ) \end{align*}


\begin{align*} P _ { * , j } & = \frac { ( n + m ) ( \beta _ j - \gamma _ j - \pi ) + m ( \alpha _ j - \gamma _ j - \pi ) } { 2 \pi } \textrm { a n d } \\ \omega _ { * , j } & = \frac { ( n + m ) w _ 1 - m w _ 2 } { 2 \pi } , \end{align*}


\begin{align*} f ( x + i y ) & = \lim _ { k \to \infty } \frac { 1 } { \pi } \langle F _ k ( \cdot , 0 ) , P ( x - \cdot , y ) \rangle = \lim _ { k \to \infty } \frac { 1 } { \pi } \int _ { - \infty } ^ \infty F _ k ( t , 0 ) P ( x - t , y ) \ , d t \\ & = \lim _ { k \to \infty } \frac { 1 } { \pi } \int _ { - \infty } ^ \infty f ( t + i y _ k ) P ( x - t , y ) \ , d t = \frac { 1 } { \pi } \langle f _ b , P ( x - \cdot , y ) \rangle . \end{align*}


\begin{align*} \beta _ i = \nu ( g ' ) - \nu _ i ( g ) \mbox { f o r e v e r y } i \in I ^ * \mbox { w i t h } i \geq i _ 0 . \end{align*}


\begin{align*} T ( M , \bar { u } ) = \{ v \in \mathcal { U } : \ \exists \{ u _ n \} \subset M , \ \{ t _ n \} \subset \mathbb { R } ^ + , \ u _ n \rightarrow \bar { u } , \ t _ n \rightarrow 0 ^ + , \frac { 1 } { t _ n } ( u _ n - \bar { u } ) \rightarrow v \} , \end{align*}


\begin{align*} \hat { u } _ k ( \mu ) = \int _ { \tilde { \gamma } ^ 3 _ k } \frac { 1 } { i ( \lambda + \mu ) } \frac { d } { d \lambda } \log \zeta _ 1 ( \lambda ) d \lambda \end{align*}


\begin{align*} 1 \ = \ L ( f ) \ \le \ m + 1 - \beta m . \end{align*}


\begin{align*} \partial _ v ( ( v ^ * ) ^ { \otimes d } ) = d \langle v , v ^ * \rangle ( v ^ * ) ^ { \otimes ( d - 1 ) } , \end{align*}


\begin{align*} \begin{aligned} I + I I : = & \int _ { \R ^ N } \left ( \nabla u _ n \nabla v _ n + \widetilde V _ \varepsilon u _ n v _ n - \bar f ( u _ n ) v _ n - \lambda _ n u _ n v _ n \right ) \\ & + 4 \Phi _ { \varepsilon _ n } ( u _ n ) ^ { \frac 1 2 } \xi _ 1 ( \Upsilon ( u _ n ) ) \int _ { \R ^ { N } } \chi _ { u _ n } u _ n v _ n \mathrm d x = o _ n ( 1 ) . \end{aligned} \end{align*}


\begin{align*} g ^ { i j } = \dfrac { s ^ { i j } } { \phi ^ 2 } - \dfrac { 1 } { \phi ^ 2 } \cdot \dfrac { \rho ^ 2 u _ k u _ l s ^ { i k } s ^ { j l } } { D ^ 2 } , \end{align*}


\begin{align*} \left \{ \begin{array} { l } \partial _ { t } ^ { 2 } \tilde { u } ( t , k ) + a ( t ) \mathcal { H } _ { \hbar , V } \tilde { u } ( t , k ) + q ( t ) \tilde { u } ( t , k ) = f ( t , k ) , \quad ( t , k ) \in ( 0 , T ] \times \hbar \mathbb { Z } ^ { n } , \\ \tilde { u } ( 0 , k ) = u _ { 0 } ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \\ \partial _ { t } \tilde { u } ( 0 , k ) = u _ { 1 } ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \end{array} \right . \end{align*}


\begin{align*} \widetilde { \rho } ( t , w ) = A t ^ { \beta _ 1 } + { \rm I m } ( e ^ { i \pi ( 1 - \beta _ 1 ) } w ) + B t ^ { 1 + \beta _ 1 } + C t ^ { 1 - \beta _ 1 } | w | ^ 2 + \end{align*}


\begin{align*} \varphi : = \log \sigma _ 1 = \left . \frac { d } { d \alpha } \right | _ { \alpha = 0 } \exp { \alpha \varphi } = \left . \frac { d } { d \alpha } \right | _ { \alpha = 0 } \sigma _ 1 ( \alpha A ) , \end{align*}


\begin{align*} g = \begin{bmatrix} t e ^ { i \gamma } & 1 \\ 0 & 0 \end{bmatrix} \end{align*}


\begin{align*} \Gamma ( \mathcal { G } _ 1 , \mathcal { G } _ 2 ) : = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \gamma _ { 1 3 } \in \mathcal { G } _ 1 , \gamma _ { 2 3 } \in \mathcal { G } _ 2 \right \} , \end{align*}


\begin{align*} \lim _ { R \to \infty } \lambda _ { } ( B _ R ) = \frac { ( n - 1 ) ^ 4 } { 1 6 } , \end{align*}


\begin{align*} \mathcal { H } ^ 1 ( J _ { w _ 1 ^ h } \setminus J _ u ) = 0 \ , . \end{align*}


\begin{align*} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot z = \begin{cases} \frac { a z + b } { c z + d } & a d - b c = 1 , \\ \frac { a \overline { z } + b } { c \overline { z } + d } & a d - b c = - 1 . \end{cases} \end{align*}


\begin{align*} H ( T ( n , k ) ) = ( 1 - o ( 1 ) ) c _ k n ^ { 1 / 2 } E ( n , { \textstyle \frac { k - 1 } { k } } ) \end{align*}


\begin{align*} s ^ 2 = \frac { 1 } { 2 } \pm \frac { 1 } { 2 } \sqrt { \frac { C - 4 } { C } } . \end{align*}


\begin{align*} \mathrm { S p a n } _ { \mathbb C } \{ g _ n \ , | \ , n \geq 0 \} + \partial ( O [ g _ 1 ] ) = O [ g _ 1 ] ; \end{align*}


\begin{align*} S = \{ \lceil 1 2 \log _ q n \rceil \le d \le n : \gcd ( k , q ^ d - 1 ) < q ^ { d / 3 } \} \end{align*}


\begin{align*} | V _ e | = \sum _ { v \in V _ i } | \Phi ^ { - 1 } ( v ) | . \end{align*}


\begin{align*} \mu _ { X _ 2 } ( x _ 2 , y _ 2 ) > 0 \wedge \mu _ { X _ 2 } ( y _ 2 , x _ 2 ) > 0 \Rightarrow x _ 2 = y _ 2 \end{align*}


\begin{align*} P _ s ( \theta ) & \overset { \Delta } { = } \left ( > \theta , \Phi ( \mathcal { A } ) > 0 | \Phi \right ) \\ & = \left ( > \theta | \Phi , \Phi ( \mathcal { A } ) > 0 \right ) \mathbf { 1 } ( \Phi ( \mathcal { A } ) > 0 | \Phi ) , \end{align*}


\begin{align*} ( C _ V ) _ { u v } = c ( u ) \delta _ { u v } , ( C _ E ) _ { e f } = c ( e ) \delta _ { e f } \end{align*}


\begin{align*} P = \frac { 4 n q } { ( p + \gamma - 1 ) ( 2 q + n - 2 - \theta ( q - 1 ) ) } , R = 2 , Q = \frac { 2 } { p + \gamma - 1 } = S , \end{align*}


\begin{align*} \mu ( \partial Q ) = 0 \mbox { f o r e v e r y d y a d i c c u b e $ Q $ . } \end{align*}


\begin{align*} H ^ * \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} H = \begin{bmatrix} X ^ * A X + Y ^ * B Y & \star \\ \star & Y ^ * A Y + X ^ * B X \end{bmatrix} \end{align*}


\begin{align*} \mathcal { Q } ( v , \gamma ) = ( 1 - \gamma _ x ) \mathcal { J } \left [ \left ( \begin{array} { c c } 3 v _ r ^ 2 + v _ i ^ 2 & 2 v _ r v _ i \\ 2 v _ r v _ i & v _ r ^ 2 + 3 v _ i ^ 2 \end{array} \right ) \phi + | v | ^ 2 v \right ] , \end{align*}


\begin{align*} \rho _ { s t } ( t ^ { 1 / \alpha } x , t ^ { 1 / \alpha } y ) = t ^ { - ( d + 2 \beta ) / \alpha } \rho _ s ( x , y ) , x , y \in \Gamma , \ ; s , t > 0 . \end{align*}


\begin{align*} y = \sqrt { P _ s } \mathbf { h } \boldsymbol { \Phi } \mathbf { g } _ { n _ t } x + n _ 0 , \end{align*}


\begin{align*} \mathcal L = \{ ( x _ 1 , x _ 2 ) \in V ^ 2 \colon x _ 1 , x _ 2 \in N ( e _ { \ell - 2 } ) \setminus R x _ 1 x _ 2 \in E ( L _ v ) \} \ , . \end{align*}


\begin{align*} | V | = \left | \bigcup _ { y H \in \mathcal { S } } \left \{ ( x H , y H ) : \ x H \in \Omega \setminus \mathcal { S } \right \} \right | = | \mathcal { S } | A ( X _ i / \pi ) _ { 1 2 } = | \mathcal { S } | ( k _ i - a _ i ) . \end{align*}


\begin{align*} E & = \big \{ ( t , x , z ) \in \mathbb { R } ^ 3 _ + : | t - x | < z < t + x \big \} , \\ F & = \big \{ ( t , x , z ) \in \mathbb { R } ^ 3 _ + : z < t - x \big \} . \end{align*}


\begin{align*} \mu _ 0 = \min _ { x \in O ^ { 3 \delta _ 0 } } V ( x ) . \end{align*}


\begin{align*} \check { H } ^ 2 ( \{ U \} , \mathcal F ) = H ^ 2 ( X , \mathcal F ) \rlap { . } \end{align*}


\begin{align*} A _ { s } ( \mathrm { i , j } ) \perp B _ s ( \mathrm { i , j } ) \quad \hbox { a n d } s _ k ( A _ s ( \mathrm { i , j } ) ) = 0 \end{align*}


\begin{align*} \frac { n m } { n + m } _ { n , m } ^ { 2 } \rightarrow \sum _ { l = 1 } ^ { \infty } \lambda _ { l } ( z _ { l } ^ { 2 } - 1 ) n , m \rightarrow \infty , \end{align*}


\begin{align*} n l ' - k l ' - l n ' = ( n - k ) \frac { l } { ( l , n - k ) } - l \frac { n - k } { ( l , n - k ) } = 0 , \end{align*}


\begin{align*} f ( R ) + M ( R ) \leq g ( R ) : = c _ * \ln ^ \alpha \ln ^ \frac 1 2 ( 1 / R ) \end{align*}


\begin{align*} U _ { k + i + 1 } = U _ { k + i } \oplus \begin{cases} z ^ { i + 1 } , & n \notin J _ { i + 1 } , \\ z ^ { n + i + 2 - x } , & n \in J _ { i + 1 } , J _ { i + 2 } = s _ 1 ( J _ { i + 1 } \setminus \{ n \} ) \cup \{ x \} . \end{cases} \end{align*}


\begin{align*} a _ { i , j } : = a _ { i , j } ( m , x ) : = \binom { m + x } { m - i + j } - \binom { m + x } { m - i - j + 1 } . \end{align*}


\begin{align*} \mathcal { B } ^ { A } ( x _ 0 , r ) = x _ 0 \mathcal { B } ^ A ( e , r ) , \mathcal { B } ^ { A } ( e , r ) = \delta _ { r } ^ { A } \mathcal { B } ^ { A } ( e , 1 ) . \end{align*}


\begin{align*} \begin{gathered} \sum _ { M = 0 } ^ \infty \sum _ { N = \lfloor w \rfloor + M } ^ \infty N ^ k e ^ { w - N } \leq \sum _ { M = 0 } ^ \infty ( M + 1 ) ( w + M ) ^ k e ^ { - M } \\ \lesssim \sum _ { M = 0 } ^ \infty M ^ { k + 1 } e ^ { - M } \end{gathered} \end{align*}


\begin{align*} \left \| \sum _ { j = 1 } ^ n a _ j \tau _ j \right \| = \sum _ { j = 1 } ^ n | a _ j | ^ p . \end{align*}


\begin{align*} \sum _ { A = 1 } ^ { \infty } \frac { G _ k ( A ^ 2 \slash X ) } { A } = \frac { \left ( ( k - 1 ) ! \right ) ^ 2 } { 2 } \log X + c _ 0 + O \left ( \frac { 1 } { X } \right ) , \end{align*}


\begin{align*} - \gamma = \inf _ { n \in \N } \{ - \nu ( x - a _ n ) \} = \inf _ { n \in \N } \alpha _ n = \inf _ { n \in \N } \beta _ n = v ( p ) - p \gamma . \end{align*}


\begin{align*} S _ { \omega , \mathbf { c } } ( U ^ { \lambda _ 0 } ) \ge \widetilde { \mu } _ { \omega , \mathbf { c } } = S _ { \omega , \mathbf { c } } ( \Phi ) \end{align*}


\begin{align*} g t g ^ { - 1 } ( g h _ i g ^ { - 1 } ) \cdot K = ( g h _ j g ^ { - 1 } ) \cdot K . \end{align*}


\begin{align*} \left \{ \begin{aligned} - d y _ { ( n , t ) } & = \bigg [ G \big ( y _ { ( n , t ) } , z _ { ( n , t ) } , \tilde { z } _ { ( n , t ) } , \gamma _ { ( n , t , e ) } \big ) + \varphi ( t ) \bigg ] d t - z _ { ( n , t ) } d W _ t - \tilde { z } _ { ( n , t ) } d \xi _ t \\ & \quad - \int _ { \mathcal { E } } \gamma _ { ( n , t , e ) } \tilde { N } ( d e , d t ) , \ t \in [ 0 , n ] , \\ y _ { ( n , n ) } & = 0 , \end{aligned} \right . \end{align*}


\begin{align*} \texttt { w p t e r m [ k ] } = \max \Bigl \{ \texttt { m a g } ( 4 0 ( L + 2 ) ) , \texttt { m a g } \Bigl ( 6 8 \frac { ( L + 2 ) ^ 2 \ , A } { \varepsilon _ 3 } \frac { \Gamma ( k + \frac 1 2 ) ^ { 1 / 2 } } { ( B _ 1 a \sqrt { \pi } ) ^ k } \Bigr ) \Bigr \} . \end{align*}


\begin{align*} J _ r ( \widetilde h ) = \inf _ { \phi \in \mathcal { M } } I _ r ( h ^ \phi ) , \end{align*}


\begin{align*} \beta ( \xi , 0 ) + \beta ( 0 , 0 ) - \beta ( 0 , 2 ) - 2 \beta ( 0 , 3 ) + \sum _ { \mathbf { n } \in \N _ 0 ^ 4 } \beta ( \mathbf { n } ) = 1 , \end{align*}


\begin{align*} \rho _ \ell ( y _ \ell , y _ \ell ^ \prime ) = \min _ { ( x _ \ell , x ^ \prime _ \ell ) \in \mathcal { X } _ \ell \times \mathcal { X } _ \ell } \left ( s _ { \ell } - s _ { \ell } ^ { \prime } \right ) ^ { \top } V _ { \ell } ^ { - 1 } \left ( s _ { \ell } - s _ { \ell } ^ { \prime } \right ) \leq c _ \ell ( s _ \ell , s _ \ell ^ \prime ) , \forall s _ \ell , s _ \ell ^ \prime \in \mathcal { S } _ \ell . \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle ( u _ { n } ) _ { \lambda } = \lambda ^ { \frac { N } { 2 } } u _ { n } ( \lambda x ) , ( v _ { n } ) _ { \lambda } = \lambda ^ { \frac { N } { 2 } } v _ { n } ( \lambda x ) . \end{array} \right . \end{align*}


\begin{align*} X _ z \subseteq \overline { [ \eta ^ { * } , \eta ' ] } = X _ { \varphi ' } , \end{align*}


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


\begin{align*} n ^ { - 2 } g u ^ { p \bar p } u ^ { q \bar q } u _ { p \bar p \bar j } u _ { q \bar q j } - n ^ { - 1 } g u ^ { p \bar p } u ^ { q \bar q } | u _ { p \bar q j } | ^ 2 + n ^ { - 1 } g u ^ { p \bar p } u _ { p \bar p j \bar j } = 2 R e ( g _ { v j } v _ { \bar j } ) + g _ v v _ { j \bar j } + g _ { j \bar j } , \end{align*}


\begin{align*} I ( v ) & = \{ j \in I ( u ) : a _ { i j } > f ( u ) \} \cup \{ i \} \quad \\ I ( w ) & = \{ j \in I ( u ) \setminus \{ i \} : a _ { i j } = f ( u ) \} . \end{align*}


\begin{align*} \widehat \psi _ r ( \beta - \varepsilon ) = & \inf _ { x \in [ 0 , 1 ] } J _ r ( x , \beta - \varepsilon ) \\ \geq \ , & \inf _ { x \in [ 0 , 1 ] } [ J _ r ( x , \beta ) - \theta ( x , \beta ) \varepsilon ] \\ \geq \ , & \widehat \psi _ r ( \beta ) - \sup _ { x \in [ 0 , 1 ] } \theta ( x , \beta ) \varepsilon \rightarrow \widehat \psi _ r ( \beta ) , \varepsilon \downarrow 0 . \end{align*}


\begin{align*} | | \sum _ { n } c _ n a _ n | | _ { a t } : = \left ( \inf \sum _ { n } | c _ n | ^ p \right ) ^ { 1 / p } , \end{align*}


\begin{align*} \tilde { G } ( a ) = \left \{ \begin{array} { l l } a + \bar { k } , & \hbox { $ 1 \leq a \leq \bar { k } $ ; } \\ a - \bar { l } + \bar { k } , & \hbox { $ r + 1 \leq a \leq r + \bar { l } - \bar { k } $ ; } \\ a + \underline { k } + \bar { k } - \bar { l } \equiv a + \underline { l } , & \hbox { $ r + \bar { l } - \bar { k } + 1 \leq a \leq r + \underline { k } $ . } \end{array} \right . \end{align*}


\begin{align*} E [ W ^ k _ { \ell \Delta , t } ( j ) | { \cal F } _ { ( \ell + 1 ) \Delta } ] & = E \left [ E \left [ W ^ k _ { \ell \Delta , t } ( j ) | X _ { \ell , \Delta } , { \cal F } _ { ( \ell + 1 ) \Delta } \right ] | { \cal F } _ { ( \ell + 1 ) \Delta } \right ] \\ & = p ^ k _ { j , + } ( t - \ell \Delta ) E \left [ X _ { \ell , \Delta } | { \cal F } _ { ( \ell + 1 ) \Delta } \right ] \\ & = \Delta \nu p ^ k _ { j , + } ( t - \ell \Delta ) Z _ 0 ( \ell \Delta ) , \end{align*}


\begin{align*} \overline { L ( \mathcal { K } , \bar { u } ) } = T ( M , \bar { u } ) = T _ w ( M , \bar { u } ) . \end{align*}


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


\begin{align*} \varphi _ I ( \sigma ) = \left ( \xi _ I - \sigma \cdot \xi _ I , \ \sigma \right ) \ . \end{align*}


\begin{align*} \mathcal { J } _ { p } ( x , s ) = \intop _ { 0 } ^ { \infty } \ , \frac { y ^ { s - \frac { 1 } { 2 } } J _ { s - \frac { 1 } { 2 } } ( 2 \pi x y ) } { \sigma \left ( y \right ) e ^ { 2 \pi y } - 1 } \ , d y = \frac { 2 \ , \pi ^ { - s } x ^ { \frac { 1 } { 2 } - s } } { \Gamma \left ( 1 - s \right ) } \ , \intop _ { 1 } ^ { \infty } \frac { 1 } { \left ( t ^ { 2 } - 1 \right ) ^ { s } } \ , \intop _ { 0 } ^ { \infty } \frac { \sin ( 2 \pi x y t ) } { \sigma \left ( y \right ) e ^ { 2 \pi y } - 1 } \ , d y \ , d t , \end{align*}


\begin{align*} 0 \geq ( b - b ^ 2 r ^ 2 ) \sum _ { p = 1 } ^ n u ^ { p \bar p } - \frac { A _ 1 ( 1 + \Vert \nabla v \Vert ^ 2 + \Delta v ) } { g \Delta u } , \end{align*}


\begin{align*} \mathbb { E } ( X ) \leq 0 + \frac { t ^ 2 } { \ell } + \sum _ { \substack { h < i < j \leq k \\ j - i \leq t } } \mathbb { P } ( s _ i = s _ j ) . \end{align*}


\begin{align*} f _ a ( q x ) = \left ( 1 - \dfrac { x } { a } \right ) f _ a ( x ) , \end{align*}


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


\begin{align*} \begin{aligned} \left [ P ^ t \Sigma \right ] _ { r , k } ^ { } & = \left [ U \Lambda ^ t U ^ { - 1 } \Sigma \right ] _ { r , k } ^ { } = \left [ U \Lambda ^ t ( U ^ { - 1 } \Sigma ) \right ] _ { r , k } ^ { } \\ & = [ U ] _ { r , \mathbf { : } } ^ { } \ , \Lambda ^ t \ , [ V ^ { - 1 } ] _ { \mathbf { : } , k } ^ { } \\ & = \frac { 1 } { n ^ t } \sum _ { j = r } ^ k ( - 1 ) ^ { k - j } \binom { n - j - 1 } { n - k - 1 } \binom { n - r } { n - j } j ^ t . \end{aligned} \end{align*}


\begin{align*} \langle T _ i ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) , f \rangle = \langle ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) , \ , T _ i f \rangle \ , \ i = 1 , \ldots , D . \end{align*}


\begin{align*} \frac { 1 } { \sum _ { j \in \N } f ( j ) ^ 2 } \sum _ { v \in L } f ( v ) ^ 2 = \frac { 1 } { \prod _ { p \in P } ( 1 + f ( p ) ^ 2 ) } \sum _ { v \in L ' } f ( v ) ^ 2 \end{align*}


\begin{align*} & \zeta = 1 \mbox { o n } B _ { \sigma '' } , \norm { \nabla \zeta } _ { L ^ \infty ( B _ { \sigma ' } , \R ^ { n \times K } ) } \leq 2 ( \sigma ' - \sigma '' ) ^ { - 1 } , \\ & \varphi = 1 \mbox { o n } B _ { \sigma } , \norm { \nabla \varphi } _ { L ^ \infty ( B _ { \overline { \sigma } } , \R ^ { n \times K } ) } \leq 2 ( \overline { \sigma } - \sigma ' ) ^ { - 1 } . \end{align*}


\begin{align*} \overline { F } ( k , t + 1 , \mathbf { p } _ { n + 1 } ) = p _ { k } ^ { } \overline { F } ( k - 1 , t , \mathbf { p } _ { n + 1 } ) + ( 1 - p _ { k } ^ { } ) \overline { F } ( k , t , \mathbf { p } _ { n + 1 } ) , \end{align*}


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


\begin{align*} \overline { B ( \overline { x _ 1 } , \delta _ 0 ) } \cap | \gamma _ 2 | = \varnothing = \overline { U _ 2 } \cap | \gamma _ 1 | \ , , \overline { B ( \overline { x _ 1 } , \delta _ 0 ) } \cap \overline { U _ 2 } = \varnothing \ , . \end{align*}


\begin{align*} M ( t ) = M _ + ( t ) - M _ - ( t ) , \end{align*}


\begin{align*} M _ q ( \Gamma ^ { \ , * } _ m ) \leqslant \sum \limits _ { i = 0 } ^ { N _ 0 } M _ q ( \Gamma _ { i m } ) \leqslant \sum \limits _ { i = 1 } ^ { N _ 0 } \frac { \omega _ { n - 1 } } { I _ i ^ { q - 1 } } + \frac { \omega _ { n - 1 } } { I _ 0 ^ { p - 1 } } : = C _ 0 \ , , m = 1 , 2 , \ldots \ , . \end{align*}


\begin{align*} & \alpha _ k : = \frac { 2 } { k + 1 } , \ \beta _ k : = \frac { 1 } { 2 L _ { E f } } , \ \lambda _ k : = \frac { k \beta _ k } { 2 } , m _ k : = \left \lceil \frac { L _ { E f } k } { \tilde { D } ^ 2 } \right \rceil , \ a n d \\ & \Pr ( R = k ) : = \frac { \Gamma _ k ^ { - 1 } \beta _ k ( 1 - L _ { E f } \beta _ k ) } { \sum _ { { \tau } = 1 } ^ N \Gamma _ { \tau } ^ { - 1 } \beta _ { \tau } ( 1 - L _ { E f } \beta _ { \tau } ) } , \mathrm { f o r } \ k = 1 , 2 , \dots , N , \end{align*}


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


\begin{align*} { } ^ c \theta _ m : = { } ^ c \Phi ( \theta _ m ) . \end{align*}


\begin{align*} f ( \omega ) & = \omega ( Q ( \Phi ) - E ( \Phi ) ) - ( E ( U _ 0 ) + \omega Q ( U _ 0 ) + \sqrt { \omega } \ \mathbf { c } _ 0 \cdot \mathbf { P } ( U _ 0 ) ) \\ & = \omega ( Q ( \Phi ) - E ( \Phi ) - Q ( U _ 0 ) ) - \sqrt { \omega } \ \mathbf { c } _ 0 \cdot \mathbf { P } ( U _ 0 ) - E ( U _ 0 ) . \end{align*}


\begin{align*} f ^ * ( t . [ \varrho ] ) = t . [ f ^ * \varrho ] . \end{align*}


\begin{align*} \left | \frac { \partial z } { \partial \omega } \det ( M _ 1 ) \right | ( \phi , \omega , r ; t ) = C ( 1 + O ( \varepsilon ) ) | t | ^ 5 , \forall \ , t \in [ - \rho , \rho ] \end{align*}


\begin{align*} \partial _ { t } \mu ( t , x ) - \mathrm { d i v } ( \mu \nabla V \star \mu ) ( t , x ) = h [ \mu ] ( t , x ) , \mu ( 0 , \cdot ) = \mu ^ { i n } . \end{align*}


\begin{align*} = h ^ { - 2 } \sum _ { a , a ' , h , h ' ~ : ~ ( a + h ) ( a ' + h ' ) = 1 } A ( a ) A ( a ' ) H ( h ) H ( h ' ) \chi ( a + h ) + \mathcal { E } = \sigma + \mathcal { E } \ , , \end{align*}


\begin{align*} f = \sum _ { j = 1 } ^ r \tilde b _ j \tilde { \textbf { Q } } ^ { \lambda _ j } \end{align*}


\begin{align*} \mathrm { d } R ^ x _ s = \frac { 1 } { R ^ x _ s } \ , \mathrm { d } s + \mathrm { d } B _ s , R ^ x _ 0 = x . \end{align*}


\begin{align*} \abs { \hat { w _ 1 } ( \xi ) } \leq \frac { 1 } { \abs { \xi } } \int _ 0 ^ \infty \abs { B ( \tau ) - A ( \tau ) } \ , d \tau \leq \frac { 1 } { \abs { \xi } } \int _ 0 ^ \infty A ( \tau ) - \frac { 1 } { 3 } B ( \tau ) \ , d \tau = \frac { 1 } { 3 \abs { \xi } } \ , . \end{align*}


\begin{align*} \kappa _ \ell = \ell \chi + K + \Theta ( e ^ { - c \ell } ) \ , , \end{align*}


\begin{align*} \left ( \mathord { \operatorname { p r o j } } _ { K _ N \cap Q } \circ { \mathrm { p r o j } _ { Q } } ^ { - 1 } \right ) \# \gamma = \left ( \mathord { \operatorname { p r o j } } _ { K _ N \cap Q } \circ { \mathrm { p r o j } _ { K _ \ell } } ^ { - 1 } \right ) \# \mu _ \ell . \end{align*}


\begin{align*} w _ { 1 : k } : = \sum _ { i = 1 } ^ k w _ i \end{align*}


\begin{align*} y _ { B 2 } = h _ { C B } z + n _ { B 2 } , \end{align*}


\begin{align*} \lim _ { n \to \infty } \norm { e ( X _ n - X ) e } = 0 . \end{align*}


\begin{align*} | \sigma ( r , x _ 3 , t ) | = r | v _ \theta ( r , x _ 3 , t ) | \leq \frac { c C _ 1 } { l n ^ 3 ( e / r ) } \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*} \min \limits _ { u \in \mathcal { U } , \zeta \in \mathcal { X } } \theta ( u ) + I _ K ( \zeta ) \ \ \mbox { s . t . } \ \ S u = \zeta , \end{align*}


\begin{align*} c _ { 1 } = c _ 1 ( \omega ) : = \min \left \{ \frac { \mu _ { \omega , 0 } } { 2 \| \Phi _ { \omega } \| _ { \mathcal { H } ^ 1 } ^ 2 } , \sqrt { \frac { 2 \omega } { \sigma } } \right \} . \end{align*}


\begin{align*} \mathbb { F } _ n : = \{ \mathcal { A } \subset \mathcal { F } : \enspace \left . h : S ^ { n - 1 } \rightarrow \mathcal { A } \right \} , \end{align*}


\begin{align*} \exp \left ( - 2 \lambda \left ( \delta \right ) \beta b ^ { 2 } \right ) = \delta ^ { \left ( 2 \rho \beta b ^ { 2 } \right ) / d } . \end{align*}


\begin{align*} \begin{aligned} \bar P _ b & = \int _ 0 ^ \infty Q \left ( \sqrt { \frac { \rho \zeta ^ 2 x } { 2 ( 1 + \rho ( 1 - \zeta ^ 2 ) \sigma _ e ^ 2 L ) } } \right ) f ( x ) d x , \end{aligned} \end{align*}


\begin{align*} y \bigl ( \omega , y ^ { * } ( \omega , t ) \bigr ) = y ^ { * } \bigl ( \omega , y ( \omega , t ) \bigr ) = t \tau _ { y ^ { * } } ( X ) ( t ) = X \Bigl ( y ^ { * } ( t ) \Bigr ) , t \ge 0 , \ \mu \end{align*}


\begin{align*} P e r ( A _ G ) = \sum _ { k = 0 } ^ { \infty } 2 ^ k F _ k ( G ) \ ; . \end{align*}


\begin{align*} N ( U ) = - 2 S _ { \omega , \mathbf { c } } ( U ) + K _ { \omega , \mathbf { c } } ( U ) \end{align*}


\begin{align*} H _ 0 : P _ X = P _ Y H _ a : P _ X \neq P _ Y . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ D \sum _ { j = 1 } ^ D \alpha _ i \bar { \alpha } _ j \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( i ) \lambda _ f ( j ) \big | L ( f , n + 1 ) \big | ^ 2 = \large { S _ { \mathrm { o f f } } + S _ { \mathrm { m a i n } } } . \end{align*}


\begin{align*} \textsf { E } f ( \delta _ { 1 } h _ { 1 } , \cdots , \delta _ { n } h _ { n } ) = \textsf { E } \textsf { E } _ { \delta } f ( \delta _ { 1 } h _ { 1 } , \cdots , \delta _ { n } h _ { n } ) \ge \textsf { E } f ( h _ { 1 } / c , \cdots , h _ { n } / c ) . \end{align*}


\begin{align*} \mathcal { M } : = \textrm { s u p p } ( f ) \setminus M \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 _ \ell ) \right ] = \left ( \sum _ { 1 \leq \ell \leq 2 } \lambda _ { \ell } V _ { \ell , X X } ^ { - 1 } \right ) ^ { - 1 } \left ( \sum _ { 1 \leq \ell \leq 2 } a _ { \ell } V _ { \ell , X X } ^ { - 1 } V _ { \ell , X Y } \right ) , \end{align*}


\begin{align*} | x | \bar { \otimes } | y | = \int _ { [ 0 , \infty ) ^ 2 } \lambda _ 1 \lambda _ 2 \ , e _ { | x | , | y | } ( d \lambda _ 1 , d \lambda _ 2 ) \end{align*}


\begin{align*} \frac { c _ v } { c ( v _ 1 ) } u ' _ 1 + \cdots + \frac { c _ v } { c ( v _ n ) } u ' _ n = 0 . \end{align*}


\begin{align*} \displaystyle \Sigma : = \left \{ u : ~ ~ u \in H ^ 1 ( \mathbb { R } ^ N ) , ~ ~ x u \in L ^ 2 ( \mathbb { R } ^ N ) \right \} . \end{align*}


\begin{align*} \mathrm { R } _ { \mathrm { D } } ( \lambda , \delta ) = \langle \lambda , \delta \rangle + \sum _ { 1 \leq \ell \leq 2 } \frac { V _ \ell / V _ { \ell , X X } } { 4 \lambda _ \ell } + \frac { 1 } { 4 } V _ o ^ { \top } { \underbrace { \left ( \lambda _ 1 V _ { 1 , X X } ^ { - 1 } + \lambda _ 2 V _ { 2 , X X } ^ { - 1 } \right ) } _ { = \Lambda _ \lambda } } ^ { - 1 } V _ o \end{align*}


\begin{align*} \| f ^ { = d } \| _ q \leq \gamma ' \rho ^ { - d } = \gamma \left ( 9 9 \cdot 2 \ , r \ , r ' q \ , ( r ' ) ^ { - 2 / q } \sqrt { \frac { \log ( 1 / \gamma ) } { d } } \right ) ^ d = \gamma \left ( 9 9 \cdot 2 \cdot \sqrt { 2 } \ , r \ , q \ , ( r ' ) ^ { - 2 / q } \right ) ^ d . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ^ \star ( \lambda ) = \sup _ { \pi \in \mathcal { G } _ { \mathrm { D } , \lambda } } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda d \pi . \end{align*}


\begin{align*} \mathcal { L } \varphi = \lim _ { t \to 0 ^ + } \frac { \mathcal { K } _ t \varphi - \varphi } { t } . \end{align*}


\begin{align*} K _ { \omega , \mathbf { c } } ( \lambda U ) = L _ { \omega , \mathbf { c } } ( \lambda U ) + 3 N ( \lambda U ) = \lambda ^ 2 L _ { \omega , \mathbf { c } } ( U ) + 3 \lambda ^ 3 N ( U ) . \end{align*}


\begin{align*} \beta = ( 3 \rho ' ) \cdot \frac { 1 } { 3 } \left ( 1 + \frac { 2 ( q - 2 ) } { \log ( 1 / \rho ' ) } \right ) \le \frac { 3 \rho ' q } { \min \{ q , \log ( 1 / \rho ' ) \} } \le 3 \rho ' q . \end{align*}


\begin{align*} \Sigma _ { \mathrm { D M R } } ( \delta ) : = \left \{ \gamma \in \mathcal { P } ( \mathcal { X } ) : \boldsymbol { K } _ c ( \mu , \gamma ) \le \delta \right \} . \end{align*}


\begin{align*} \mathcal { E } = \{ ( \mathbf { z } , t ) \in K ' : | \alpha ( \mathbf { z } ) - H ( t ) \beta ( \mathbf { z } + \mathbf { p } ( t ) ) | \leq \varepsilon \} , \end{align*}


\begin{align*} \dot { x } = f ( x ) + \sum _ { i = 1 } ^ m g _ i ( x ) u _ i , x \in \mathbb { R } ^ d , \ u _ i \in \mathbb { R } , \end{align*}


\begin{align*} \forall x , y \in \mathcal { A } \ \exists i \in [ n ] : \ x _ i = y _ i . \end{align*}


\begin{align*} p _ { n } = A _ { n } ^ { \ast } , \end{align*}


\begin{align*} \zeta ( s ) & = i \beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \overline { \beta _ h ( s ) } K _ 2 - i \overline { \beta _ \sigma ( s ) } K \\ & = 2 s h ^ 2 ( K _ 1 + K _ 3 + K _ 2 ) + 2 s \sigma ^ 2 K + \mathcal { O } ( s ^ 2 \sigma ^ 4 ) . \end{align*}


\begin{align*} N \coloneqq ( d + 1 ) \cdot \min \left \{ \prod _ { i = 1 } ^ n ( h _ i + 1 ) - \prod _ { i = 1 } ^ n h _ i , \binom { n + h } { n } - \binom { h } { n } \right \} \ , . \end{align*}


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


\begin{align*} M \vec { v } = \lambda \vec { v } , \end{align*}


\begin{align*} C _ { f _ * \mathcal F } = \frac { \lambda _ \Delta ^ + ( f , x , \delta ) C _ { \mathcal F } + t ( E _ x , \Delta , \delta ) t ( f _ * E _ x , \Delta ' , \delta ' ) \| D f \| _ { C ^ { \theta } } } { \lambda _ \Delta ^ - ( f , x , \delta ) \lambda _ { \mathcal F } ( f , x , \delta ) ^ { \theta } } , \end{align*}


\begin{align*} ( 1 + w ) C ^ \varphi _ { k j i } = w _ t R _ { i j k t } + \frac { S ^ \varphi } { m - 1 } ( w _ i \delta _ { j k } - w _ j \delta _ { i k } ) - ( w _ i R ^ \varphi _ { j k } - w _ j R ^ \varphi _ { i k } ) \ , . \end{align*}


\begin{align*} M = N \boldsymbol { W } _ { p _ 2 } \left ( \mu _ 2 , \delta _ { s _ 2 } \right ) ^ { p _ 2 } + N \left [ \delta ^ { 1 / p _ 2 } + \boldsymbol { W } _ { p _ 2 } \left ( \mu _ 2 , \delta _ { s _ 2 } \right ) \right ] ^ { p _ 2 } < \infty . \end{align*}


\begin{align*} \int _ { | \theta | < \widehat { Z } ^ { - 1 } } \psi ( x \theta ) \dd \theta = \begin{cases} \widehat { Z } ^ { - 1 } & | x | < \widehat { Z } \\ 0 & \end{cases} \end{align*}


\begin{align*} v ( a ^ { - 1 } ) = \mu ( q ) . \end{align*}


\begin{align*} & I _ 1 ( \ell _ 1 , \ell _ 2 ) - I _ 2 ( \ell _ 1 , \ell _ 2 ) \\ & \leq \Delta \nu p ^ k _ { j , + } ( t - \Delta \ell _ 2 ) \\ & \quad \ ; \cdot \sum _ { n = 1 } ^ \infty \sum _ { m = 1 } ^ \infty \sum _ { i = 1 } ^ m i P ( Z _ 0 ( \Delta \ell _ 2 ) = m , Z _ 0 ( \Delta \ell _ 1 ) = n , X _ { \ell _ 1 , \Delta } = 1 , B _ { \ell _ 1 \Delta } ( \Delta \ell _ 2 ) = i ) . \end{align*}


\begin{align*} [ f , g ] ( l ) = ( f \otimes g ) \Delta ( l ) = f ( l _ 1 ) g ( l _ 2 ) ; l , l _ 1 , l _ 2 \in L . \end{align*}


\begin{align*} \sum \limits _ { i = 1 } ^ k \sum \limits _ { l = 0 } ^ n c ^ { i l } _ m \langle P \psi ^ l _ { \mu _ m ^ i , \xi _ m ^ i } , P \psi ^ h _ { \mu _ m ^ j , \xi _ m ^ j } \rangle = \int _ \Omega f _ \epsilon ^ { ' } ( V _ m ) \phi _ m P \psi ^ h _ { \mu _ m ^ j , \xi _ m ^ j } d x . \end{align*}


\begin{align*} \intop _ { 0 } ^ { \infty } \frac { \sin ( 2 \pi x y t ) } { \sigma \left ( y \right ) e ^ { 2 \pi y } - 1 } \ , d y = \frac { \sigma _ { p } ( 2 \pi x t ) } { 2 } + \frac { 1 } { 4 } \ , \frac { 1 } { 1 + \frac { 1 } { \pi p } } - \frac { 1 } { 4 \pi x t } , \end{align*}


\begin{align*} R ( \lambda ^ i ) = \{ \lambda ^ j : \lambda ^ j = ( \lambda ^ i _ s , r ) \lambda ^ i ( \lambda ^ i _ s , r ) \in \rm { r e v } ( \lambda ^ i ) \} . \end{align*}


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


\begin{align*} \left . \begin{array} { l l } c _ 1 | \xi | \leq \left | g ^ * _ 1 ( \xi , \eta ) \right | = \left | g _ 1 ( \xi , - \eta ) \right | \leq d _ 1 | \xi | , c _ 2 | \eta | \leq \left | g ^ * _ 2 ( \xi , \eta ) \right | = \left | - g _ 2 ( \xi , - \eta ) \right | \leq d _ 2 | \eta | . \end{array} \right . \end{align*}


\begin{align*} \widehat { \mathcal { I } } ( \delta _ 1 , \delta _ 2 ) = \sup _ { \gamma \in \widehat { \Sigma } ( \delta _ 1 , \delta _ 2 ) } \int _ { \mathcal { V } } f ( v ) \ , d \gamma ( v ) . \end{align*}


\begin{align*} a ( s , s ' ) & : = \frac { z ( s ) } { 1 + 4 i s z ( s ) } + \frac { \widetilde z ( s ' ) } { 1 - 4 i s ' \widetilde z ( s ' ) } , \\ b ( s , s ' ) & : = \frac { \zeta ( s ) } { 1 + 4 i s z ( s ) } + \frac { \widetilde \zeta ( s ' ) } { 1 - 4 i s ' \widetilde z ( s ' ) } , \\ c ( s , s ' ) & : = \frac { i s } { 1 + 4 i s z ( s ) } \zeta ( s ) \cdot \zeta ( s ) + \frac { i s ' } { 1 - 4 i s ' \widetilde z ( s ' ) } \widetilde \zeta ( s ' ) \cdot \widetilde \zeta ( s ' ) + \gamma ( s ) + \widetilde \gamma ( s ' ) . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } \Delta w \leq \Delta v _ { \lambda _ j } & \textnormal { i n } & \Omega \\ w = v _ { \lambda _ j } = 0 & \textnormal { o n } & \partial \Omega . \end{array} \right . \end{align*}


\begin{align*} \Lambda ( x ) = \frac { S ^ \varphi } { m - 1 } = \frac { \mu } { 1 - e ^ { \mu f } } ( m \lambda ( x ) - S ^ \varphi ) \ , \Omega . \end{align*}


\begin{align*} 2 ^ { - 2 } \left | \sum _ { a , b } c _ A ( a ) c _ B ( b ) \sum _ { g \in G ~ : ~ g a = b } \chi ( \gamma a + \d ) \right | \end{align*}


\begin{align*} V ( t _ 1 , \dots , t _ n ) = \sum _ { i = 1 } ^ d \mu _ i \sum _ { k = 1 } ^ n \frac { t _ k ^ { i } } { i } = \sum _ { i = 0 } ^ d \mu _ i V _ { i } \ . \end{align*}


\begin{align*} | I _ { 1 , j } | \le \sum _ { k = 1 } ^ d \int _ { \R ^ d } | \nabla \varphi _ j ^ { ( k ) } ( x ) | p \theta _ { \epsilon } ( x ) | \varphi _ j ^ { ( k ) } ( x ) | d x \le p \left ( \frac { \delta _ j } { 2 } \| \theta _ { \epsilon } ^ { \frac { 1 } { 2 } } | \nabla \varphi _ j | \| _ { L ^ 2 } ^ 2 + \frac { 1 } { 2 \delta _ j } \| \theta _ { \epsilon } ^ { \frac { 1 } { 2 } } | \varphi _ j | \| _ { L ^ 2 } ^ 2 \right ) \end{align*}


\begin{align*} H = P = \left ( \begin{array} { c c c } \ast & \ast & \ast \\ 0 & \ast & \ast \\ 0 & 0 & \ast \\ \end{array} \right ) , U = \left ( \begin{array} { c c c } 1 & 0 & 0 \\ \ast & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right ) , V = \left ( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 1 & 0 \\ \ast & \ast & 1 \\ \end{array} \right ) , W = \left ( \begin{array} { c c c } 1 & 0 & \ast \\ 0 & 1 & \ast \\ 0 & 0 & 1 \\ \end{array} \right ) , \end{align*}


\begin{align*} c \beta _ i - c \beta _ j + \alpha _ { i j } + \gamma _ i = 0 \implies \gamma _ i = c \beta _ j - c \beta _ i - \alpha _ { i j } \leq 2 p _ { \max } , \end{align*}


\begin{align*} R \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} R = \begin{bmatrix} ( 1 - x ) A + x B & \star \\ \star & x A + ( 1 - x ) B \end{bmatrix} \end{align*}


\begin{align*} \sum _ { \delta = 0 } ^ \infty \delta \varepsilon ^ \delta & = \frac { \varepsilon } { ( 1 - \varepsilon ) ^ 2 } . \end{align*}


\begin{align*} \sum _ { 1 \leq j \leq n } u _ { j p } u _ { \bar j } + u _ p u _ { p \bar p } = - \beta ( u u _ p + B \rho _ p ) , \end{align*}


\begin{align*} W _ 0 ^ { \sharp } & : = ( W _ 0 ^ { \flat } \times \{ \theta = 0 \} ) \cup N ( \mathring { \Lambda } _ 0 ) , \\ W _ 1 ^ { \sharp } & : = ( W _ 1 ^ { \flat } \times \{ \theta = 0 \} ) \cup N ( \mathring { \Lambda } _ 0 ) . \end{align*}


\begin{align*} \mathcal { E C S } _ { { \rm { ( r a n k \ , 2 ) } } } = \mathcal { E C S } _ { { \rm { I } } } \sqcup \mathcal { E C S } _ { { \rm { I I } } } \sqcup \mathcal { E C S } _ { { \rm { I I I } } } . \end{align*}


\begin{align*} \begin{aligned} - d y _ t & = \big [ g ( x _ t , y _ t , z _ t , \tilde { z } _ t , \gamma _ { ( t , e ) } ) - g ( x _ t , 0 , 0 , 0 , 0 ) + g ( x _ t , 0 , 0 , 0 , 0 ) \big ] d t \\ & - z _ t d W _ t - \tilde { z } _ t d \xi _ t - \int _ { \mathcal { E } } \gamma _ { ( t , e ) } \tilde { N } ( d e , d t ) \\ & : = \big [ G ( y _ t , z _ t , \tilde { z } _ t , \gamma _ { ( t , e ) } ) + \varphi ( t ) \big ] d t - z _ t d W _ t - \tilde { z } _ t d \xi _ t - \int _ { \mathcal { E } } \gamma _ { ( t , e ) } \tilde { N } ( d e , d t ) . \\ \end{aligned} \end{align*}


\begin{align*} \begin{cases} - \Delta u = f ( u ) , \ \ u > 0 , & \ \ \Omega _ 2 , \\ u = 0 , \ \ | \nabla u | = c _ 2 , & \ \ , \partial \Omega _ 2 . \end{cases} \end{align*}


\begin{align*} \bar { \rm P } _ { j , j + i } = & \sum ^ { M - j } _ { m = i + 1 } { M - j \choose m } \mathbb { P } _ { \rm T X } ^ m \left ( 1 - \mathbb { P } _ { \rm T X } \right ) ^ { M - j - m } \bar { \rm P } _ { m , i } , \end{align*}


\begin{align*} \int _ { \mathcal { V } \times \mathcal { V } } g \ , d \pi = \lim _ { n \rightarrow \infty } \int _ { \mathcal { V } \times \mathcal { V } } g \ , d \pi _ n \leq \sup _ { \gamma \in \mathcal { P } _ { \mathrm { D } } } \int _ { \mathcal { V } } g \ , d \gamma . \end{align*}


\begin{align*} \mathcal { M } ( P _ X , P _ Y ) = \sup _ { f \in \mathcal { F } } \left | E \{ f ( X ) \} - E \{ f ( Y ) \} \right | , \end{align*}


\begin{align*} T _ M ( a ) : = \int _ \mathbb { R } \theta ^ M _ t ( a ) \ , d t , a \in \widetilde { M } _ + . \end{align*}


\begin{align*} \sum _ { \substack { n \le x \\ n \equiv v ( \bmod d ) } } \beta _ n = \frac 1 d \sum _ { n \le x } \beta _ n + O ( 1 ) . \end{align*}


\begin{align*} ( P \otimes I _ M ) \vec U _ t + { \bf D _ { x _ i } } { \bf A _ i } \vec U + { \bf A _ i ^ T } { \bf D _ { x _ i } } \vec U + { \bf C } \vec U + { \vec L _ D } = 0 , \vec U ( 0 ) = \vec F \end{align*}


\begin{align*} \int _ { \Omega } w u \ : x = \int _ { \Omega } \mathcal { L } ^ { \Omega } _ \varepsilon c _ \varepsilon u \ : x + \int _ { \Omega } \Delta c u \ : x + \int _ { \Omega } \big ( f ^ \prime ( c _ \varepsilon ) - f ^ \prime ( c ) \big ) u \ : x . \end{align*}


\begin{align*} ( - ) ^ \ast \colon \mathcal { B } \rightarrow \mathcal { B } , b = ( b _ { i , j } ) _ { i , j = 1 } ^ n \mapsto b ^ \ast = ( b _ { i , j } ^ \ast ) ^ { \mathrm { t } } . \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( \frac { 1 } { 1 - y ^ m z ^ n } \right ) ^ { \frac { m ^ 2 } { n ^ 3 } } = \left ( \frac { 1 } { 1 - y z } \right ) ^ { \frac { y } { 1 - y } } \end{align*}


\begin{align*} c _ { \beta } ( \sigma ) = ( \sqrt { 2 } + o ( 1 ) ) ( 1 - \beta ) ^ { 1 - \sigma } \sqrt { | \log { ( \sigma - 1 / 2 ) } | } , \textrm { a s } \sigma \to 1 / 2 ^ { + } . \end{align*}


\begin{align*} e ^ { i s \Delta } ( \overline { e ^ { i s \Delta } g _ { K , \sigma } } e ^ { i s \Delta } g _ { K _ 1 , h } \overline { e ^ { i s \Delta } g _ { K _ 2 , h } } ) & = \frac { | \beta _ h ( s ) | ^ 2 \overline { \beta _ \sigma ( s ) } } { ( 2 \pi ) ^ 6 \big ( 1 + 4 i s z ( s ) \big ) } \\ & \quad \times e ^ { - \frac { z ( s ) } { 1 + 4 i s z ( s ) } | x | ^ 2 + \frac { 1 } { 1 + 4 i s z ( s ) } x \cdot \zeta ( s ) + \frac { i s } { 1 + 4 i s z ( s ) } \zeta ( s ) \cdot \zeta ( s ) } e ^ { \gamma ( s ) } \end{align*}


\begin{align*} \frac { \zeta ' ( s ) \zeta ( s ) } { 2 z ( s ) } = \big ( i \partial _ s \beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \overline { \partial _ s \beta _ h ( s ) } K _ 2 - i \overline { \partial _ s \beta _ \sigma ( s ) } K \big ) \frac { \zeta ( s ) } { 2 z ( s ) } \end{align*}


\begin{align*} C _ n ( x ) = \left \{ \begin{array} { l r r } K - v x + G _ n ( x + B ) & x \leq s _ m - B & \mbox { ( s a t u r a t e d ) } \\ K + \min _ { x \leq y \leq x + B } \{ G _ n ( y ) - v x \} & s _ m - B < x \leq s _ m & \mbox { ( u n s a t u r a t e d o r s a t u r a t e d ) } \\ - v x + G _ n ( x ) & x > s _ m & \mbox { ( n o o r d e r ) } , \end{array} \right . \end{align*}


\begin{align*} \varphi ^ + ( M _ I ) = \widetilde { M _ I ( - X ) } , \varphi ^ - ( M _ I ) = V \widetilde { M _ I ( - X ) } , \end{align*}


\begin{align*} c _ { Y _ \ell } ( y _ \ell , y _ \ell ' ) & = \min _ { x _ \ell , x _ \ell ' \in \mathcal { X } _ \ell ' } c _ { \ell } ( s _ \ell , s _ \ell ' ) = ( y _ \ell - y _ \ell ' ) ^ { \top } V _ { \ell , Y Y } ^ { - 1 } ( y _ \ell - y _ \ell ' ) , \end{align*}


\begin{align*} \det ( I _ n - A + ( Q - I _ n ) ) = \det L = 0 , \end{align*}


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


\begin{align*} { \rm R e } \left ( \xi ^ { - N } e ^ { i ( v + i H v ) } \frac { \partial \kappa } { \partial \theta } \right ) = - e ^ { - u } e ^ { - ( H v ) } \widetilde { \rho } _ { \theta } ( \theta , \kappa ) . \end{align*}


\begin{align*} \mu _ i \tilde { S } _ { \gamma _ i \gamma _ j } & = \mu _ j J _ { 2 | \alpha _ i | } \tilde { S } _ { \gamma _ i \gamma _ j } J ^ T _ { 2 | \alpha _ j | } + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} \mathcal { F } \left ( b \right ) & : = \left \{ x \in \mathbb { R } ^ { n } : a _ { t } ^ { \prime } x \leq b _ { t } t \in T \right \} , \\ \mathcal { F } ^ { o p } \left ( c , b \right ) & : = \arg \min \left \{ c ^ { \prime } x : x \in \mathcal { F } \left ( b \right ) \right \} . \end{align*}


\begin{align*} \lbrace \alpha , \beta \rbrace + \lbrace \beta , \gamma \rbrace + \lbrace \gamma , \alpha \rbrace = 0 \end{align*}


\begin{align*} W _ { \mathrm { t r } , p } ^ - ( \mu , \nu ) = \inf _ { \beta \in \Phi _ { \mathrm { t r } } ^ - ( \nu , m ) } W _ { \mathrm { t r } , p } ( \mu , \beta ) , \\ W _ { \mathrm { t r } , p } ^ + ( \mu , \nu ) = \inf _ { \alpha \in \Phi _ { \mathrm { t r } } ^ + ( \mu , n ) } W _ { \mathrm { t r } , p } ( \alpha , \nu ) . \end{align*}


\begin{align*} ( [ U ] \mid \gamma ) ( v ) = [ U ] ( v \gamma ^ { - 1 } ) = [ U \gamma ] ( v ) . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta _ { A } ) = \inf _ { \lambda \in \mathbb { R } _ { + } ^ L } \left [ \langle \lambda , \delta _ { A } \rangle + \sup _ { \pi \in \Pi ( \mu _ 1 , \dotsc , \mu _ L ) } \int _ { \mathcal { V } } g _ { \lambda , A } \ , d \pi \right ] . \end{align*}


\begin{align*} x _ p = \frac { l } { ( l , n - k ) } u _ p , y _ p = \frac { n - k } { ( l , n - k ) } u _ p \end{align*}


\begin{align*} | K _ { \alpha } ( x , \vec { y } ) - K _ { \alpha } ( x ' , \vec { y } ) | & \le \dfrac { A } { \Big ( \sum \limits _ { j = 1 } ^ { m } | x - y _ { j } | \Big ) ^ { m n - \alpha } } \omega \Big ( \frac { | x - x ' | } { | x - y _ { 1 } | + \cdots + | x - y _ { m } | } \Big ) \end{align*}


\begin{align*} | K _ \alpha ( x , y ) - K _ \alpha ( x , y ' ) | & \leqslant c \int _ 0 ^ \infty | h _ t ( x , y ) - h _ t ( x , y ' ) | \frac { d t } { t ^ { 1 - \alpha / 2 } } \\ & \leqslant c \bigg ( \int _ 0 ^ { d ( x , y ) ^ 2 } + \int _ { d ( x , y ) ^ 2 } ^ \infty \bigg ) | h _ t ( x , y ) - h _ t ( x , y ' ) | \frac { d t } { t ^ { 1 - \alpha / 2 } } \\ & = : I \ ! I _ 1 + I \ ! I _ 2 . \end{align*}


\begin{align*} B ( t ) = \int _ 0 ^ t b ( s ) \ , d s \end{align*}


\begin{align*} ( k , 2 n ) = _ { 2 n } / P . \end{align*}


\begin{align*} \left \{ \alpha _ 0 + \sum _ { i = 1 } ^ r F ^ { k n _ i } ( \alpha _ i ) \colon n _ i \in \mathbb { N } \right \} , \end{align*}


\begin{align*} \langle x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 : x _ i x _ j x _ i ^ { - 1 } = x _ { 3 ( i + j ) \bmod 5 } \rangle . \end{align*}


\begin{align*} B _ { n , k } ( q ) = q ^ { 2 n k - n ^ 2 } B _ { n , n - k } ( q ) \end{align*}


\begin{align*} ( g P ) ( X , Y ) = P ( d X - b Y , - c X + a Y ) . \end{align*}


\begin{align*} \widetilde { S } _ { \alpha , \beta } f ( x ) & = x ^ { - \beta } S _ { \alpha , \beta } \big ( ( \cdot ) ^ { \beta } f \big ) ( x ) \\ & \simeq x ^ { - \beta } \sup _ { t > 2 x } \int _ { t / 2 } ^ { t } ( t + x - z ) ^ { - \alpha - 1 / 2 } ( t - z ) ^ { \alpha + \beta - 1 / 2 } \abs { f ( z ) } \ , d z , x > 0 . \end{align*}


\begin{align*} \tilde { \mu } _ \delta \{ r \to s \} ( p U ) = \tilde { \mu } _ \delta \{ r \to s \} ( U ) . \end{align*}


\begin{align*} \| P _ { T \to 0 } ^ c \nabla \hat { f } ( p ) - \nabla \hat { f } ( p ) \| = \| \int _ { 0 } ^ T P _ { t \to 0 } ^ c \nabla ^ 2 \hat { f } ( c ( t ) ) c ' ( t ) d t \| \leq L T . \end{align*}


\begin{align*} \mathbb { P } _ { \mu } ^ { y } ( g , A ) = \int _ { G } { \bf 1 } _ { A } \left ( g s \right ) \frac { \varphi _ { g s } ( y ) } { \varphi _ { g } ( y ) } d \mu ( s ) . \end{align*}


\begin{align*} \chi ( x ) : = C ^ { - 1 } _ 1 \int _ { r _ 0 } ^ { \Lambda _ { 0 } } \phi ( x ; r ) \ , f ( r ) \ , r ^ 2 \ , \mathrm { d } r = C ^ { - 1 } _ 1 \int _ { ( x - \varepsilon _ 2 ) \vee r _ 0 } ^ { ( x + \varepsilon _ { 2 } ) \wedge \Lambda _ 0 } \phi ( x ; r ) \ , f ( r ) \ , r ^ 2 \ , \mathrm { d } r , x \in [ r _ 0 - \varepsilon _ 2 , \Lambda _ 0 + \varepsilon _ 2 ] . \end{align*}


\begin{align*} 0 = \sum _ { d _ 1 , d _ 2 < D } \frac { 1 } { \sqrt { d _ 1 d _ 2 } } \sum _ { I d _ 1 , J d _ 2 < D } \alpha _ { I d _ 1 } \bar { \alpha } _ { J d _ 2 } \sum _ { L , M \geq 1 } \frac { G _ k ( L M d _ 1 d _ 2 \slash p ) } { \sqrt { L M } } \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( I L ) \lambda _ f ( J M ) . \end{align*}


\begin{align*} E _ t \cap S _ t = \bigcap _ { j = 1 } ^ { t / h ( t ) } ( E _ { j , t } \cap S _ { j , t } ) . \end{align*}


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


\begin{align*} \alpha : = c _ 1 / ( 8 c _ 2 ) \end{align*}


\begin{align*} \mathcal { I } _ { + } ^ { ( m , \mathbf { k } ) } & = \int _ { \frac { \mu ^ 2 _ { } \Gamma _ 0 } { \rho } } ^ { \infty } \exp \Big ( - \frac { \Big ( { \sqrt { \frac { \rho x } { \Gamma _ 0 } } - \mu _ { \textrm { D } } } \Big ) ^ 2 } { 2 ( \sigma ^ { ( m , \mathbf { k } ) } _ { } ) ^ 2 } \Big ) f _ { \Gamma _ { } ^ { ( m ) } } ( x ) d x . \end{align*}


\begin{align*} \pi ( a ) & = 0 \Rightarrow \pi ( a ^ * a ) = 0 \Rightarrow g \pi ( a ^ * a ) = 0 \Rightarrow f \phi ( a ^ * a ) = 0 \\ & \Rightarrow f ( \phi ( a ) ^ * \phi ( a ) ) = 0 \Rightarrow \phi ( a ) = 0 \end{align*}


\begin{align*} w ^ T M _ { h } = 0 , \end{align*}


\begin{align*} P ' = ( P ' - [ a ] Q ) + [ a ] Q \in \ker \phi _ 1 + \langle Q \rangle = \langle P \rangle + \langle Q \rangle , \end{align*}


\begin{align*} \{ \underline { a } \colon \underline { a } / \varpi ^ k \in L ( \varpi ^ m \underline { c } ) \} = & \{ a \underline { c } ^ \bot + \varpi ^ { k - m } \underline { d } \colon | a | < | \varpi | ^ { k - m } , ( a , \varpi ) = 1 , | \underline { d } | < | \varpi | ^ m \} \setminus \\ & \{ a \underline { c } ^ \bot + \varpi ^ { k - m + 1 } \underline { d } \colon | a | < | \varpi | ^ { k - m + 1 } , | \underline { d } | < | \varpi | ^ { m - 1 } , ( a , \varpi ) = 1 \} \end{align*}


\begin{align*} ( \rho * f _ 0 ) ( x ) = ( \rho * f _ 0 ) ( x + 2 \tau _ k e _ { i _ k } ) . \end{align*}


\begin{align*} N ( U _ n ( t _ n ) ) \ \rightarrow \ - 2 h ( 0 ) = - 2 \mu _ { \omega , \mathbf { c } } . \end{align*}


\begin{align*} \psi ( \ldots , \zeta ( k - 2 ) , \zeta ( k - 1 ) , x ( k ) , v ( k ) , v ( k + 1 ) , \ldots ) = 0 \ , . \end{align*}


\begin{align*} [ \underline { \varphi } , \varphi ] : = \{ x \in \{ 0 , 1 \} ^ \Z : \underline { \varphi } ( h ) \leq x \leq \varphi ( h ) h \in H \} . \end{align*}


\begin{align*} ( \Delta _ s ) = o ( N ^ { - 2 } p ^ { - 1 } ) s = 2 , \ldots , l - 1 . \end{align*}


\begin{align*} z ^ { i } = g ^ { i } \ , , i = 1 , \ldots , l \end{align*}


\begin{align*} 1 & = \sum _ { k = 1 } ^ n | \omega ( b _ k ) | ^ 2 \leq \sum _ { k = 1 } ^ n \psi ( b _ k b _ k ^ * ) = \psi \left ( \sum _ { k = 1 } ^ n b _ k b _ k ^ * \right ) \leq 1 \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , \gcd ( m , n ) = 1 } } \left ( \frac { 1 } { 1 - y ^ m z ^ n } \right ) ^ { \frac { 1 } { m ^ a n ^ b } } = \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( \sum _ { m = 1 } ^ { n } \frac { y ^ m } { m ^ a } \right ) \frac { z ^ n } { n ^ b } \right \} . \end{align*}


\begin{align*} A = \begin{pmatrix} - a _ i & a _ { i + 1 } \\ - b _ i & b _ { i + 1 } \end{pmatrix} \end{align*}


\begin{align*} H _ \infty : = \{ h \in H : R ^ n h \in W \setminus \underline { W } \} \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \det T _ n ( \phi ) } { G [ \phi _ 0 ] ^ n \prod _ { k = 1 } ^ R \det T _ n ( \phi _ k ) } = E \end{align*}


\begin{align*} \forall s , t \in [ 1 / 2 , 1 ) , \widetilde { z } _ h ^ { ( s ) } = \widetilde { z } _ h ^ { ( t ) } = \widetilde { u } _ h \mbox { a . e . o n } B _ 1 \setminus \left ( \cup _ { \mathcal { F } ^ { ( s ) } _ h } B \bigcup \cup _ { \mathcal { F } ^ { ( t ) } _ h } B \right ) . \end{align*}


\begin{align*} \Phi = \begin{bmatrix} \phi ( y _ 1 ) & \phi ( y _ 2 ) & \cdots & \phi ( y _ n ) \end{bmatrix} , \end{align*}


\begin{align*} ( - \Delta ) ^ s u + \rho \ , c u = 0 \quad \ , \ , \ , \R ^ N \ , . \end{align*}


\begin{align*} \mu _ { \omega , c } = - \frac { 1 } { 2 } N ( \Phi ) \le C \| \Phi \| _ { \mathcal { H } ^ 1 } ^ 3 , \end{align*}


\begin{align*} \mathfrak { h } _ { 0 } & \Big ( \varphi - \sum _ { 1 \leq i \leq g } d ^ + _ { i , l - 2 } ( D ^ 2 \varphi ) h _ { i , 2 } - \sum _ { 1 \leq i \leq g } d ^ + _ { i , l - 1 } ( D \varphi ) h _ { i , 1 } - \sum _ { 1 \leq s < \gamma } d ^ 0 _ { s , l - 1 } ( D \varphi ) c _ { s , 1 } \Big ) \\ & = \sum _ { 1 \leq i \leq g } d ^ + _ { i , l } ( \varphi ) h _ i + \sum _ { 1 \leq s < \gamma } d ^ 0 _ { s , l } ( \varphi ) c _ s + \sum _ { 1 \leq j \leq g } d ^ - _ { - j , l } ( \varphi ) h _ { - j } . \end{align*}


\begin{align*} T _ l ' ( f _ 1 , f _ 2 ) ( x , y ) = \int _ \mathbb { R } \ ! f _ 1 \left ( x + \widetilde { P } _ 1 ( t ) , y \right ) f _ 2 \left ( x , y + \widetilde { P } _ 2 ( t ) \right ) \psi ( t ) t ^ { - 1 } \ , \mathrm { d } t . \end{align*}


\begin{align*} \eta _ { t } ( x ) = \begin{cases*} 0 & i f $ r ( x ) \leq t ^ 2 $ \\ ( \log t ^ 2 - \log r ) / \log ( t ) & $ t \leq r ( x ) \leq t ^ 2 $ \\ 1 & i f $ r ( x ) \geq t $ . \end{cases*} \end{align*}


\begin{align*} \mathbf { m } _ \lambda = \max \left ( \lambda ^ 2 , \mathbf { b } _ \lambda \right ) \lambda ^ 5 \mathbf { e } _ \lambda ^ 2 . \end{align*}


\begin{align*} \left ( \frac { M ^ { ( s ) } _ { i , j } } { M ^ { ( s ) } _ { k , j } } \right ) _ { s = P } ^ { + \infty } \end{align*}


\begin{align*} \phi ( t ) = D C _ D ^ \frac 1 { D } t ^ { \frac { D - 1 } { D } } ( - u _ \mu ^ * ) ' ( t ) , \ t \in ( 0 , \infty ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l } \partial _ { t } ^ { 2 } u ( t , k ) + a ( t ) \mathcal { H } _ { \hbar , V } u ( t , k ) + q ( t ) u ( t , k ) = f ( t , k ) , \quad ( t , k ) \in ( 0 , T ] \times \hbar \mathbb { Z } ^ { n } , \\ u ( 0 , k ) = u _ { 0 } ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \\ \partial _ { t } u ( 0 , k ) = u _ { 1 } ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \end{array} \right . \end{align*}


\begin{align*} ( b , x ) \ast _ \varphi ( b ' , x ' ) = ( b b ' , x x ' + b \ast x ' + b ' \ast x ) , \forall ( b , x ) , ( b ' , x ' ) \in B \oplus X . \end{align*}


\begin{align*} T ^ \ast T \psi _ j = \mu _ j \psi _ j . \end{align*}


\begin{align*} \Pi _ { \mu , \xi } \Big ( V + \phi - i ^ * [ f _ \epsilon ( V + \phi ) ] \Big ) = 0 . \end{align*}


\begin{align*} \langle u _ 1 \otimes \cdots \otimes u _ i , v _ 1 \otimes \cdots \otimes v _ i \rangle _ { \otimes i } = \prod _ { j = 1 } ^ i \langle u _ j , v _ j \rangle _ 1 . \end{align*}


\begin{align*} \mathcal { S } _ { q , \lambda } ^ 1 = \{ u \in H ^ 2 ( \mathrm { M } ) ; \ ; ( \Delta + \lambda - q ) u = 0 , \ ; ( \partial _ \nu - i a ) u \in H ^ { 1 / 2 } _ \Gamma ( \partial \mathrm { M } ) \} . \end{align*}


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


\begin{align*} \lim _ { | x | \rightarrow \infty } \varphi _ 3 ( x ) = 0 . \end{align*}


\begin{align*} 0 = \sum _ { i = 1 } ^ D \sum _ { j = 1 } ^ D \alpha _ i \bar { \alpha } _ j \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( i ) \lambda _ f ( j ) \big | L ( f , k ) \big | ^ 2 . \end{align*}


\begin{gather*} \omega ( ( q , S q ) , ( r , S r ) ) = - \langle q , S r \rangle + \langle S q , r \rangle = \langle q , ( S ^ { \mathsf { T } } - S ) r \rangle = 0 , \end{gather*}


\begin{align*} \chi _ { \tau , i } ( z ) = z ^ { a _ { \tau , i } } \overline { z } ^ { b _ { \tau , i } } \end{align*}


\begin{align*} \sigma = \varphi _ { S } ( \pi ) = ( \varphi _ { x _ { k } } \circ \cdots \circ \varphi _ { x _ { 2 } } \circ \varphi _ { x _ { 1 } } ) ( \pi ) . \end{align*}


\begin{align*} S _ { \omega , \mathbf { c } } ( V ) = \frac { 1 } { 2 } L _ { \omega , \mathbf { c } } ( V ) + N ( V ) > \mu _ { \omega , \mathbf { c } } , \end{align*}


\begin{align*} J \left ( t , \mu ; u ^ { t , \omega ^ { 0 } } \right ) = \left . J \left ( t , \mu ; u \right ) \right \vert _ { t , \mu , u ^ { t , \omega ^ { 0 } } } , \mathbb { P } ^ { 0 } \end{align*}


\begin{align*} \lambda _ { 1 } = \inf _ { v \in S _ { 0 } ^ { 1 , p } ( \Omega ) \setminus \{ 0 \} } \left \{ \frac { \int _ { \Omega } | X v | ^ { p } d x } { \int _ { \Omega } | v | ^ { p } d x } \right \} = \frac { \int _ { \Omega } | X u _ { 1 } | ^ { p } d x } { \int _ { \Omega } | u _ { 1 } | ^ { p } d x } , \end{align*}


\begin{align*} E _ 2 & \ll \widehat { P } ^ { n - 5 } \widehat { Y } ^ { 4 - n / 2 - 2 / n + \varepsilon } \widehat { Z } ^ { 3 - n / 2 } \widehat { V } ^ { n - 2 } \\ & = \widehat { P } ^ { - 3 } \widehat { Z } ^ { 1 + n / 2 } \widehat { Y } ^ { 2 + n / 2 - 2 / n + \varepsilon } \\ & \ll \widehat { P } ^ { 5 n / 6 - 3 + 5 / 3 } \widehat { Y } ^ { 1 - 2 / n + \varepsilon } \\ & \ll \widehat { P } ^ { 5 n / 6 - 3 + 1 0 / 3 + \varepsilon } , \end{align*}


\begin{align*} \langle T _ i ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) , f \rangle = \lambda _ f ( i ) \ , \langle ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) , \ , f \rangle \ , \ i = 1 , \ldots , D . \end{align*}


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


\begin{align*} E & = \left ( \bigcup _ { ( k , m ) \in \mathcal { D } } [ t _ { k , m } , t _ { k , m } + 2 ^ { - 2 k } ] \right ) \cup \left \{ t _ { k , m } + 2 ^ { 2 - 2 k } : ( k , m ) \in \mathcal { D } \right \} \\ & \subset [ 0 , 1 ] , \end{align*}


\begin{align*} \lambda _ { 1 , n } = - a ^ { - 2 } \langle J ' _ \nu ( u _ n , v _ n ) , ( u _ n , 0 ) \rangle + o _ n ( 1 ) , \ \lambda _ { 2 , n } = - b ^ { - 2 } \langle J ' _ \nu ( u _ n , v _ n ) , ( 0 , v _ n ) \rangle + o _ n ( 1 ) . \end{align*}


\begin{align*} g _ F ( t ) : = - \frac { m _ 0 } { 2 } \ , t ^ 2 + i B _ F t + \sum _ { \gamma \not = 0 } m _ \gamma \frac { e ^ { - i \gamma t } - 1 } { \gamma ^ 2 } , B _ F : = i \frac { \xi _ F ^ \prime } { \xi _ F } \left ( \frac { 1 } { 2 } \right ) \end{align*}


\begin{align*} g \mapsto \varphi _ { g } ( \pi ( x ) ) = \frac { d g \nu } { d \nu } ( \pi ( x ) ) \end{align*}


\begin{align*} \mu ( f ) = \min \limits _ { i \in \{ 1 , \dots , s \} } v ( a _ i ) . \end{align*}


\begin{align*} \widetilde { g } \bigl ( \widetilde { y } , F ( \widetilde { y } ) \bigr ) = \bigl ( \widetilde { f } ( \widetilde { y } ) , F ^ \prime ( \widetilde { y } ) \widetilde { f } ( \widetilde { y } ) \bigr ) , \end{align*}


\begin{align*} f ( y , \xi ( \omega ) ) : = y ^ 2 \delta _ 0 ( \xi ( \omega ) ) + ( 1 - y ) ^ 2 \delta _ 1 ( \xi ( \omega ) ) . \end{align*}


\begin{align*} \mathbb E [ \xi _ { \beta } ^ J ] : = \prod _ { i : \ , | J ^ i | \geq 2 } \mathbb E [ \xi _ { \beta } ^ { | J ^ i | } ] \ , . \end{align*}


\begin{align*} \varphi ( q _ { n * } ( \xi , \ldots , \xi , a , b ) ) = q _ { n * } ( \varphi ( \xi ) , \ldots , \varphi ( \xi ) , \varphi ( a ) , \varphi ( b ) ) \end{align*}


\begin{align*} C _ { \mathrm { U B F } } = \left ( \frac { e } { \theta } \right ) ^ { \lceil \frac { 2 \eta } { b } \rceil } = \left ( \frac { 5 e ^ 2 \Delta ^ 2 } { b ^ 2 } \right ) ^ { \lceil \frac { 2 \eta } { b } \rceil } . \end{align*}


\begin{align*} \mathcal { S } \left ( x , \left ( L _ { x } \omega _ { 1 } , L _ { x } \omega _ { 2 } , \ldots \right ) \right ) = \left ( x , \left ( L _ { x } \omega _ { 2 } , L _ { x } \omega _ { 3 } , \ldots \right ) \right ) , \end{align*}


\begin{align*} { \wedge } = { \wedge } \circ { \vartriangleleft } . \end{align*}


\begin{align*} Z _ q ( G _ { n , \ell } ) & = \left \{ \begin{aligned} & n ( \ell - 1 ) & \mbox { i f } q = 0 , \\ & n \ell - 2 & \mbox { i f } q \geq 1 . \end{aligned} \right . \end{align*}


\begin{align*} [ { \hat n _ t } ] = \arg \min \limits _ { n _ t \in \{ 1 , \cdots , N _ t \} } \left | y - \sqrt { P _ s } \zeta \sum \nolimits _ { l = 1 } ^ L \alpha _ { l , { n _ t } } \hat \beta _ l \right | ^ 2 . \end{align*}


\begin{align*} \pi ( \lambda ) = \lambda _ { L _ 1 } - \frac { ( \lambda , z ) } { N } \zeta _ { L _ 1 } , \lambda \in M ' \end{align*}


\begin{align*} | a d - b c | = d ( v , w ) - 1 . \end{align*}


\begin{align*} \partial ^ { l o g } f ( t ) = f ^ { \prime } ( t ) \cdot f ( t ) ^ { - 1 } , \end{align*}


\begin{align*} \nabla _ g u _ \delta = ( \phi ' ( \rho ) - s \phi ( \rho ) ) e ^ { - s \rho } \nabla _ g \rho , \end{align*}


\begin{align*} K = K ( \omega ) : = \bigcup _ { x _ i \in \ , ( \Pi ) } \bar { B } ( x _ i , a ) , \end{align*}


\begin{align*} \left ( g . \overline { \mathbb { P } } _ { \mu } \right ) _ { x } = \overline { \mathbb { P } } _ { \mu , x , g } , \end{align*}


\begin{align*} ( \ref { i n t e g r a l } ) = M _ N ( 2 k ) \det ( s _ { i , j } ) _ { i , j = 1 , \ldots , n } , \end{align*}


\begin{align*} \mathcal { S } ^ \vee ( A ) = \{ ( p _ i , n _ i + ( p _ 1 - 1 ) ) ^ { \eta ( p _ i , n _ i ) } \} . \end{align*}


\begin{align*} & E [ W ^ k _ { \ell _ 2 \Delta , t } ( j ) W ^ k _ { \ell _ 1 \Delta , t } ( j ) ; W ^ k _ { \ell _ 2 \Delta , t } ( j ) > 1 ] + E [ W ^ k _ { \ell _ 2 \Delta , t } ( j ) W ^ k _ { \ell _ 1 \Delta , t } ( j ) ; W ^ k _ { \ell _ 1 \Delta , t } ( j ) > 1 ] \\ & = O \left ( e ^ { \lambda \Delta \ell _ 1 } e ^ { 2 \lambda \Delta \ell _ 2 } \Delta ^ 3 \right ) , \end{align*}


\begin{align*} ( B + i ) ( \phi _ + + U \phi _ + ) = 2 i \phi _ + \qquad \qquad ( B _ n + i ) ( \phi _ + ^ n + U _ n \phi _ + ^ n ) = 2 i \phi _ + ^ n \end{align*}


\begin{align*} \Theta _ { \mathrm { D } } ( \delta ) & : = \left \{ \int _ { \mathcal { S } _ 1 \times \mathcal { S } _ 2 } g \ , d \gamma : \gamma \in \Sigma _ { \mathrm { D } } ( \delta ) \right \} \Theta ( \delta ) : = \left \{ \int _ { \mathcal { S } } f \ , d \gamma : \gamma \in \Sigma ( \delta ) \right \} . \end{align*}


\begin{align*} \widetilde { T } _ { \eta } f ( x ) = x ^ { \eta - 1 } T _ { \eta } \big ( ( \cdot ) ^ { 1 - \eta } f \big ) ( x ) = x ^ { \eta - 1 } \sup _ { t > 2 x } \int _ { t / 2 } ^ t \frac { \abs { f ( z ) } } { ( t - z + x ) ^ { \eta } } \ , d z , x > 0 . \end{align*}


\begin{align*} \Lambda = \{ ( \lambda _ 1 , \dots , \lambda _ n ) \in \Z ^ n \ | \ \lambda _ 1 + \dots + \lambda _ n = 0 \} \subset \Z ^ n \ . \end{align*}


\begin{align*} P _ \phi \preceq _ { \mu } \frac { 1 } { k } \sum _ { j = 1 } ^ k K _ j = P _ { \mathcal { B } } . \end{align*}


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


\begin{align*} R _ { [ j + 1 ] } & = \begin{pmatrix} 1 & 0 \\ 0 & w _ { j + 1 } ^ T J _ { 2 | \alpha _ i | } z _ { j + 1 } \end{pmatrix} . \end{align*}


\begin{align*} 0 \leq \mu ( f ) = \min \left \{ \mu \left ( f _ i q ^ i \right ) \right \} \leq \mu ( f _ i ) , \mbox { f o r e v e r y } i , 1 \leq i \leq r , \end{align*}


\begin{align*} \alpha = \alpha ( \alpha _ i \mid i \in I \} ) \mbox { a n d } \beta = \alpha ( \{ \beta _ i \mid i \in I \} ) \end{align*}


\begin{align*} \| z \| _ A = \mathcal { O } ( \kappa ^ { \frac { 1 } { 2 } } n ) , \end{align*}


\begin{align*} r \ge \frac { b ^ { 5 + 4 \kappa } } { k \Delta ^ { 2 + 4 \kappa } \cdot e ^ { 8 \kappa } } , \end{align*}


\begin{align*} Y _ 0 = \dfrac { 1 } { \sqrt { \mathrm { A r e a } ( \mathbb { S } ^ n ) } } , \ \ Y _ 1 = \dfrac { \sqrt { n + 1 } } { \sqrt { \mathrm { A r e a } ( \mathbb { S } ^ n ) } } x \cdot v , \end{align*}


\begin{align*} \eta - \hat \eta = \sum _ { l = 1 } ^ L \hat \beta _ l \left ( \alpha _ { l , { { n } _ t } } - \alpha _ { l , { \hat { n } _ t } } e ^ { - j \omega } \right ) . \end{align*}


\begin{align*} \prod _ { x : U _ { i j } } \frac { m _ i ( x ) } { f ( x ) ^ k } = \varphi ( x ) = \frac { m _ j ( x ) } { f ( x ) ^ k } \end{align*}


\begin{align*} B ( T _ { * } ) = J _ { T _ { * } } - \lim \limits _ { \Lambda \rightarrow \infty } J _ { \Lambda } . \end{align*}


\begin{align*} \left \{ \begin{aligned} & D _ u \theta ( u ^ { k + 1 } ) + S ^ * \lambda ^ { k + 1 } = 0 ; \\ & I _ K ( \zeta ) - I _ K ( \zeta ^ { k + 1 } ) - \langle \lambda ^ { k + 1 } , \zeta - \zeta ^ { k + 1 } \rangle \geq 0 \forall \ \zeta \in \mathcal { X } ; \\ & \beta r ^ { k + 1 } = \lambda ^ { k + 1 } - \lambda ^ k , \end{aligned} \right . \end{align*}


\begin{align*} \zeta ( 5 , 1 ) = \frac { 5 } { 2 } \zeta ( 6 ) - \zeta ( 2 ) \zeta ( 3 ) . \end{align*}


\begin{align*} \rho _ { h , g } = \prod _ { \substack { k = 1 , \dots , N \\ \lambda ' _ k \neq 1 } } \lambda _ k \end{align*}


\begin{align*} \sum _ { m = n } ^ { \infty } \overline { \mathbb { P } } _ { \mu } \left ( L _ { x } \omega _ { m } \in A _ { n } \left ( x , \zeta \right ) , ( x , L _ { x } \omega ) \in V | \left ( x , \zeta \right ) \in U \right ) \le c e ^ { - \frac { 1 } { 2 } \epsilon n } . \end{align*}


\begin{align*} K _ { \omega , c } ( U _ n ( t _ n ) ) = N ( U _ n ( t _ n ) ) + 2 S _ { \omega , \mathbf { c } } ( U _ n ( t _ n ) ) \ \rightarrow \ 0 . \end{align*}


\begin{align*} \Vert x _ 1 \bar { \otimes } x _ 2 \Vert _ { p , \varphi _ 1 \bar { \otimes } \varphi _ 2 , \eta } = \Vert x _ 1 \Vert _ { p , \varphi _ 1 , \eta } \Vert x _ 2 \Vert _ { p , \varphi _ 2 , \eta } \end{align*}


\begin{align*} a \otimes b = a b ( 1 + \eta _ 3 ) . \end{align*}


\begin{align*} \rho ( t ) = \big \| \hat \lambda _ t \big \| _ * , \dot \psi ( t ) = \frac { h _ 1 ( t ) \dot h _ 2 ( t ) - \dot h _ 1 ( t ) h _ 2 ( t ) } { \rho ^ 2 ( t ) } . \end{align*}


\begin{align*} Q = Q _ { [ 1 ] } \oplus ^ { \operatorname { s } } \cdots \oplus ^ { \operatorname { s } } Q _ { [ r ] } , \end{align*}


\begin{align*} S _ r ( \beta ) = \{ x \in [ 0 , 1 ] \colon \ , d _ r ( x ) \geq \beta \} \end{align*}


\begin{align*} B ( ( - \Delta ) ^ b ) \tilde { P } ( ( - \Delta ) ^ { \tilde { s } } ) u ( x ) = \sum _ { j = 1 } ^ { J } c _ j ( - \Delta ) ^ { n _ j } u ( x ) + c _ r ( - \Delta ) ^ r u ( x ) = 0 \quad x \in \mathbb { R } ^ n . \end{align*}


\begin{align*} R _ { c l } = \sum _ { c = 1 } ^ { C } R _ { c } \end{align*}


\begin{align*} F _ \gamma ( \hat { x } ) - F _ \gamma ( \bar { y } ) = \int _ { 0 } ^ 1 F _ \gamma ( \bar { y } + t ( \hat { x } - \bar { y } ) ) ( \hat { x } - \bar { y } ) \ : t = : A _ \gamma ( \hat { x } , \bar { y } ) ( \hat { x } - \bar { y } ) . \end{align*}


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


\begin{align*} \mathcal { I } ( \delta ) : = \sup _ { \gamma \in \Sigma ( \delta ) } \int _ \mathcal { S } f \ , d \gamma , \ \delta \in \mathbb { R } _ { + } ^ 2 , \end{align*}


\begin{align*} \mathcal { R } ( A \circ B ) = \mathcal { R } A \circ \mathcal { R } B - \mathcal { I } A \circ \mathcal { I } B . \end{align*}


\begin{align*} \left . \begin{array} { l l } Q ( u , v ) = & \displaystyle \dfrac { a _ 2 } { m _ 1 } \int _ { \mathbb { R } ^ N } | \nabla u | ^ 2 d x + \dfrac { a _ 1 } { 2 m _ 2 } \int _ { \mathbb { R } ^ N } | \nabla v | ^ 2 d x + \dfrac { N } { 2 } a _ 1 a _ 2 \int _ { \mathbb { R } ^ N } v u ^ 2 d x , \end{array} \right . \end{align*}


\begin{align*} { \rm R e } \ W _ { \vec { k } , \vec { l } } ( \vec { x } , \vec { y } ) - { \rm R e } \ W _ { \vec { k } , \vec { l } } ( \vec { y } , \vec { x } ) = 0 , \end{align*}


\begin{align*} f ( | x | \bar { \otimes } | y | ) = \int _ { [ 0 , \infty ) ^ 2 } f ( \lambda _ 1 ) f ( \lambda _ 2 ) \ , e _ { | x | , | y | } ( d \lambda _ 1 , d \lambda _ 2 ) = f ( | x | ) \bar { \otimes } f ( | y | ) . \end{align*}


\begin{align*} M _ { \chi , \varepsilon } ( z ) = \Lambda _ { \chi , \varepsilon } + \mathcal { B } _ { \chi , \varepsilon } ( z ) = \varepsilon ^ { - 2 } \Lambda _ { \chi } ^ { \rm s t i f f } + \Lambda _ { \chi } ^ { \rm s o f t } + \mathcal { B } _ { \chi , \varepsilon } ( z ) , \end{align*}


\begin{align*} \widetilde s _ { p , N } ( t ) = \chi ( t ) s _ { p , N } ( t ) , \widetilde { S } _ N ( t ) = ( 1 - \chi ( t ) ) \left ( \mathcal { P } _ { 0 , N } + \phi ' \widetilde s _ { p , N } ( t ) \right ) + { S } _ { N } ( t ) . \end{align*}


\begin{align*} ( x , u ) = \psi ( x ^ { + } , \zeta ) \ , , \end{align*}


\begin{gather*} \dim W ^ { \mathrm { u } } ( p ^ - ) = 2 n - k ^ - , \dim W ^ { \mathrm { s } } ( p ^ + ) = k ^ + . \end{gather*}


\begin{align*} a _ { I + p - 1 } = \frac { b _ I } { a _ I \cdots a _ { I + p - 2 } } ; \end{align*}


\begin{align*} \int _ { B _ 1 } \Theta \bigg ( g ^ { i j } \partial _ { i j } \eta - g ^ { j p } \Theta _ q u _ { p q } \partial _ j \eta \bigg ) d \mu _ g = 0 . \end{align*}


\begin{align*} h ( u , v ) = u ^ N \phi ( v / u ) , \end{align*}


\begin{align*} d ' ( a _ 0 \otimes \ldots \otimes a _ n ) & = \sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ i a _ 0 \otimes \ldots \otimes a _ i a _ { i + 1 } \otimes \ldots \otimes a _ n & & n \geq 1 . \end{align*}


\begin{align*} M = O ( \log q / \log \log q ) \end{align*}


\begin{align*} \frac { \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } \geq x \right ) } { 1 - \Phi ( x ) } = 1 + o ( 1 ) \quad \frac { \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } \leq - x \right ) } { \Phi ( - x ) } = 1 + o ( 1 ) \end{align*}


\begin{align*} g ( \nabla _ X \ , \xi _ i + { f } X ^ \top , \ , { f } \ , Z ) = 0 . \end{align*}


\begin{align*} N ' \coloneqq ( d + 1 ) \cdot \min \left \{ \prod _ { i = 1 } ^ n ( h _ i + 1 ) , \binom { n + h } { n } \right \} \ , . \end{align*}


\begin{align*} A ^ \dagger - B ^ \dagger = - A ^ \dagger ( A - B ) B ^ \dagger + A ^ \dagger ( A ^ * ) ^ \dagger ( A ^ * - B ^ * ) ( I - B B ^ \dagger ) + ( I - A ^ \dagger A ) ( A ^ * - B ^ * ) ( B ^ * ) ^ \dagger B . \end{align*}


\begin{align*} \partial _ t \sigma + \Big ( v + 2 \frac { x ' } { | x ' | ^ 2 } \Big ) \cdot \nabla \sigma - \Delta \sigma = 0 , \end{align*}


\begin{align*} D _ { S , x } [ f ^ { = T } ] = ( L _ S E _ { T ^ c } L _ T [ f ] ) _ { S \rightarrow x } , \end{align*}


\begin{align*} a \prod _ { i = 1 } ^ n x _ i ^ { \alpha _ i } + b \prod _ { i = 1 } ^ n x _ i ^ { \beta _ i } + c \prod _ { i = 1 } ^ n x _ i ^ { \gamma _ i } = 0 , \end{align*}


\begin{align*} \left . \begin{array} { c } \max _ { \overline { Q } _ { T } } \varphi _ { \lambda } \left ( x _ { 1 } , t \right ) = \varphi _ { \lambda } \left ( b , T / 2 \right ) = e ^ { 2 \lambda b ^ { 2 } } , \\ \min _ { \overline { Q } _ { T , \varepsilon } } \varphi _ { \lambda } \left ( x _ { 1 } , t \right ) = \exp \left [ 2 \lambda \left ( a ^ { 2 } - \alpha \left ( T / 2 - \varepsilon \right ) ^ { 2 } \right ) \right ] . \end{array} \right . \end{align*}


\begin{align*} M ( P ) = c \widehat { P } ^ { n - 5 } + O \left ( \widehat { P } ^ { n - 5 - \kappa ' } \right ) \quad E _ 1 ( P ) \ll \widehat { P } ^ { n - 5 - \kappa '' } , \end{align*}


\begin{align*} 1 = \mu ( T ) = \langle \sigma ( T ) \eta , \eta \rangle = \frac { 1 } { 2 } + \frac { 1 } { 2 } \langle \Re ( \sigma ( B ) ) \eta , \eta \rangle . \end{align*}


\begin{align*} \tau ( u ) ^ 2 = \tau ( u ^ + , u ^ - ) ^ 2 = & ~ \left ( 1 - \frac { \Vert u ^ + \Vert ^ 2 - \Vert u ^ - \Vert ^ 2 } { 4 } \right ) ^ 2 + \Vert u ^ + \Vert ^ 2 = \left ( 1 - \frac { Q ( u ) } { 2 } \right ) ^ 2 + \Vert u ^ + \Vert ^ 2 \\ = & ~ \left ( 1 + \frac { \Vert u ^ + \Vert ^ 2 - \Vert u ^ - \Vert ^ 2 } { 4 } \right ) ^ 2 + \Vert u ^ - \Vert ^ 2 = \left ( 1 + \frac { Q ( u ) } { 2 } \right ) ^ 2 + \Vert u ^ - \Vert ^ 2 . \end{align*}


\begin{align*} \textrm { b o u n d e d b y } n ^ { 1 - \beta } \textrm { i f } s = 0 \textrm { a n d } x & \geq 1 + \beta \\ \textrm { b o u n d e d b y } n ^ { 1 - \beta } ( 1 + o ( 1 ) ) \textrm { i f } s \geq 1 \textrm { a n d } x & \geq 2 ( \alpha + \beta ) \end{align*}


\begin{align*} \dot { \widetilde { y } } = \widetilde { f } ( \widetilde { y } ) , \dot { \widetilde { \xi } } = \widetilde { f } ^ \prime ( \widetilde { y } ) \widetilde { \xi } , \dot { \widetilde { \eta } } = \widetilde { f } ^ \prime ( \widetilde { y } ) \widetilde { \eta } . \end{align*}


\begin{align*} \tilde { y } _ t ^ K \circ \big ( \tilde { y } _ { t _ { k , m } } ^ K \big ) ^ { - 1 } = \begin{cases} y _ { t - t _ { k , m } } ^ { ( i _ k ; L _ k ) } & t \in [ t _ { k , m } , t _ { k , m } + \tau _ k ] , \\ y _ { \tau _ k } ^ { ( i _ k ; L _ k ) } & t \in [ t _ { k , m } + \tau _ k , t _ { \mathrm { s u c } _ K ( k , m ) } ] . \end{cases} \end{align*}


\begin{align*} { \mathbb T } _ \lambda : = \nabla _ \lambda \otimes _ { \mathcal { A } _ \lambda } ( \Delta _ \lambda ^ \vee ) ^ o . \end{align*}


\begin{align*} A _ { k , \rho } ( t ) = [ u ( \cdot , t ) < k ] \cap K _ { \rho } . \end{align*}


\begin{align*} \left [ P \right ] _ { i , j } ^ { } = \begin{dcases} q _ i = \frac { n - ( n - j ) p } { n } & \mbox { i f } \ , j = i ; \\ p _ i = \frac { ( n - j ) p } { n } & \mbox { i f } \ , j = i + 1 ; \\ 0 & \mbox { o t h e r w i s e } ; \end{dcases} \end{align*}


\begin{align*} \dot { y } = f ( y ) , \dot { \eta } = f ^ \prime ( y ) \eta , \end{align*}


\begin{align*} \varphi ( q _ { n * } ( \xi , \ldots , \xi , a , b ) ) = q _ { n * } ( \varphi ( \xi ) , \ldots , \varphi ( \xi ) , \varphi ( a ) , \varphi ( b ) ) \end{align*}


\begin{align*} 1 - z ^ r x - k x ^ { r + 1 } = 0 . \end{align*}


\begin{align*} \Phi ( \rho ( 1 + u ) ) = \Phi ( \rho ) + \phi ( \rho ) \rho u + \dfrac { 1 } { 2 } \phi ' ( \rho ) \rho ^ 2 u ^ 2 + o ( u ^ 2 ) , \end{align*}


\begin{align*} ( \prod _ { i = 1 } ^ m a ( i ) \ast f ( t ( i ) ) ) \ast a ( m + 1 ) \in A \cap \sigma ( L ) . \end{align*}


\begin{align*} \mathcal { H } ^ { n - 1 } \left ( J \cap \partial B _ R ^ n \right ) = 0 \end{align*}


\begin{align*} \tilde { \sigma } _ v ( w ) - \sigma _ v ( w ) = \gamma ^ { ( m - 3 ) } ( w ) \end{align*}


\begin{align*} \frac { 1 } { | r _ 1 r _ 2 | } \leq \left | \frac { \underline { a } _ 1 } { r _ 1 } - \frac { \underline { a } _ 2 } { r _ 2 } \right | = \left | \left ( \frac { \underline { a } _ 1 } { r _ 1 } - \underline { \alpha } \right ) + \left ( \underline { \alpha } - \frac { \underline { a } _ 2 } { r _ 2 } \right ) \right | < \max \left \{ \frac { 1 } { | r _ 1 | \widehat { R } _ 2 } , \frac { 1 } { | r _ 2 | \widehat { R } _ 2 } \right \} , \end{align*}


\begin{align*} \left \{ \tilde { T } ^ { n } ( x , \omega ) \in A \right \} & = \left \{ L _ { \omega _ { n } ^ { - 1 } . x } \in S \left ( \omega _ { n } ^ { - 1 } . x , \left ( \omega _ { n } ^ { - 1 } \omega _ { m + n } \right ) _ { m \in \mathbb { Z } } \right ) \right \} \\ & = \left \{ L _ { x } \omega _ { n } \in S \left ( x , \omega \right ) \right \} \ \ \mbox { b y c o m p a t i b i l i t y ( i ) } . \end{align*}


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


\begin{align*} D ( g _ { i 1 } , \dots , g _ { i b _ i } ) = U \cap D ( f _ i ) \rlap { . } \end{align*}


\begin{align*} M : = \omega _ { d } \left ( \int _ { B _ { 4 r } } \abs { \partial \phi } ^ d \right ) ^ { \frac { 1 } { d } } . \end{align*}


\begin{align*} \Pr ( \widetilde { X } _ t = k + 1 ) + \sum _ { j = 0 } ^ k \Pr ( \widetilde { X } _ t = j ) = 1 . \end{align*}


\begin{align*} G _ { k , Y _ { 1 , 1 } } = \sqrt { x } \frac { \partial G _ { k } } { \partial x } - \frac { 1 } { 2 \sqrt { x } } k ^ 2 G _ { k } - \frac { t } { \sqrt { x } } F _ { 1 } , \end{align*}


\begin{align*} a _ h = \mathcal { L } ^ 2 \left ( \left \{ f _ h \neq g _ h \right \} \cap B _ 1 \right ) \lesssim \left ( \mathcal { H } ^ 1 \left ( \left \{ f _ h \neq g _ h \right \} \cap B _ 1 \right ) \right ) ^ 2 = { \rm o } ( \gamma _ h ) \mbox { a s } h \to + \infty \end{align*}


\begin{align*} \delta ^ { ' } = 3 + \delta - \frac { 3 + 2 \delta } { \tau } . \end{align*}


\begin{align*} \phi \left ( \bigotimes \limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \otimes B \right ) = \bigotimes \limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \otimes \varphi _ { j _ 1 , \ldots , j _ { m - 1 } , \mathrm { I } } ( B ) \hbox { f o r a l l \quad } B \in M _ { n _ m } . \end{align*}


\begin{align*} \begin{aligned} \widehat { \gamma } ( A ) & = \mathbb { P } ( \widehat { S } \in A ) = \mathbb { E } \left [ \mathbb { P } ( \widehat { S } \in A | \varepsilon ) \right ] \\ & = \left ( \eta / \eta _ 0 \right ) \mathbb { P } ( S \in A ) + \left ( 1 - \eta / \eta _ 0 \right ) \mathbb { P } ( \widetilde { S } \in A ) . \end{aligned} \end{align*}


\begin{align*} & \| x - z \| \ge \max ( \| x - \sigma ( x ) \| / 2 , { \mathfrak R } ^ k ) \quad \mbox { f o r } z \in \sigma ( \Gamma ) , \\ & 5 r \le \| \sigma ( x ) - z \| = d ( x , z ) \le \| x - z \| \quad \mbox { f o r } z \in \sigma ( \Gamma ) , z \notin { \mathcal O } ( 5 B ) . \end{align*}


\begin{align*} \psi _ u ( n ) = \psi ( \kappa ( u , \log ( n ) ) ) . \end{align*}


\begin{align*} \mathcal { I } ( \delta _ { A } ) = \inf _ { \lambda \in \mathbb { R } _ { + } ^ L } \left [ \langle \lambda , \delta _ { A } \rangle + \sup _ { \pi \in \Pi ( \mu _ { 1 , L + 1 } , \dotsc , \mu _ { L , L + 1 } ) } \int _ { \mathcal { V } } f _ { \lambda , A } \ , d \pi \right ] . \end{align*}


\begin{align*} \begin{array} { l l } & x = ( 1 - y ) / ( 1 - p y ) , \\ & d x = - \big ( q / ( 1 - p y ) ^ 2 \big ) d y , \\ & 1 - p x = q / ( 1 - p y ) . \end{array} \end{align*}


\begin{align*} \mathrm { I } ( \xi _ { 1 } ^ { x } , \mathcal { T } _ { x } ) = \int _ { ( L _ { x } \backslash G ) ^ { \mathbb { N } } } \log \frac { d \overline { \mathbb { P } } _ { \mu , x , \mathrm { \xi _ { 1 } ^ { x } } } | _ { \mathcal { T } _ { x } } } { d \overline { \mathbb { P } } _ { \mu , x , e } | _ { \mathcal { T } _ { x } } } d \overline { \mathbb { P } } _ { \mu , x , e } , \end{align*}


\begin{align*} C _ { q _ m } ( x _ o ) = \Gamma _ a C _ { q _ m } ( x _ a ) + \Gamma _ b C _ { q _ m } ( x _ b ) + \mathcal { O } ( h ^ { 2 } ) \end{align*}


\begin{align*} \Gamma ^ { \ , * } _ m = \bigcup \limits _ { i = 0 } ^ { N _ 0 } \Gamma _ { m i } \ , , \end{align*}


\begin{align*} E _ 1 ( \psi ) = ( \psi , \mathbf { 0 } , \mathbf { 0 } ) , \ E _ 2 ( \psi ) = ( \mathbf { 0 } , \psi , \mathbf { 0 } ) , \ E _ 3 ( \psi ) = ( \mathbf { 0 } , \mathbf { 0 } , \psi ) \in \mathcal { H } ^ 1 , \ \ \mathbf { 0 } = ( 0 , \cdots , 0 ) \in ( H ^ 1 ( \R ^ d ) ) ^ d . \end{align*}


\begin{align*} \alpha = \frac { n q } { 2 q + n - 2 - \theta ( q - 1 ) } \quad \mbox { a n d } \quad \beta = \frac { n q } { ( q - 1 ) ( n - 2 + \theta ) } . \end{align*}


\begin{align*} \begin{aligned} & E ( \alpha _ { n _ t , l } ) = { \sqrt { \pi } } / { 2 } , \ \ \ V a r ( \alpha _ { n _ t , l } ) = ( 4 - \pi ) / { 4 } . \end{aligned} \end{align*}


\begin{align*} h _ n = \sum \limits _ { i = 0 } ^ { 2 n } \bigg { ( } 1 - \frac { 1 } { \sqrt { n } } \bigg { ) } ^ { | i - n | } ( T ^ { \prime \prime } ) ^ i g _ n \end{align*}


\begin{align*} \sum _ { p \in \Z } \int _ { [ 0 , N ] ^ n } h ( N p + v _ { m + 1 } + \langle \bar { u } , N \bar { q } + v '' \rangle ) \ , d \bar { u } & = \int _ { \R ^ n / N \Z ^ n } \Tilde { \psi } ( S x ) \ , d \bar { u } \\ & = \int _ { \R ^ n / N \Z ^ n } \Tilde { \psi } ( x ) \ , d \bar { u } = N ^ { n - 1 } \int _ { \R } h ( u _ 1 ) \ , d u _ 1 , \end{align*}


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


\begin{align*} \overline { F } ( k , t + 1 , \mathbf { p } _ { n + 1 } ) = p _ { k } ^ { } \overline { F } ( k - 1 , t , \mathbf { p } _ { n + 1 } ) + ( 1 - p _ { k } ^ { } ) \overline { F } ( k , t , \mathbf { p } _ { n + 1 } ) , \end{align*}


\begin{align*} \pi _ { \psi } ( g ) f ( w ) = \epsilon _ g f ( w g ) . \end{align*}


\begin{align*} f ( z + 1 ) = f ( z ) \ \ f ( z + \eta ) = - e ^ { - 2 \pi i n z } f ( z ) . \end{align*}


\begin{align*} - \Delta u - g ( b , \ , \nabla u ) + c \ , u = 0 \ , \ , \ , M , \end{align*}


\begin{align*} A _ r ^ n = & \psi ^ { \alpha _ 1 , n } ( f _ 1 ) \cdots \psi ^ { \alpha _ r , n } ( f _ r ) , B _ s ^ n = \psi ^ { \beta _ 1 , n } ( g _ 1 ) \cdots \psi ^ { \beta _ s , n } ( g _ s ) . \end{align*}


\begin{align*} \widehat { \xi } _ { [ ( \sigma , k , l ) ] } ( s ) & = \frac { ( r _ \alpha - s ) ^ { \frac { k - ( m _ \sigma - 2 ) } { m _ \sigma } } } { m _ \sigma ^ 2 k ! } k \neq m _ \sigma - 2 \ \operatorname { m o d } m _ \sigma \\ \widehat { \xi } _ { [ ( \sigma , k , l ) ] } ( s ) & = - \frac { ( r _ \alpha - s ) ^ { \frac { k - ( m _ \sigma - 2 ) } { m _ \sigma } } \log ( r _ \alpha - s ) } { m _ \sigma ^ 2 k ! } k = m _ \sigma - 2 \ \operatorname { m o d } m _ \sigma . \end{align*}


\begin{align*} \mathrm { p r o j } _ { Q } \# \pi = \left ( \mathord { \operatorname { p r o j } } _ { Q } \circ { \operatorname { p r o j } _ { K _ j } } ^ { - 1 } \right ) \# \mu _ j , \forall j \in [ N ] . \end{align*}


\begin{align*} \phi ^ { - 1 } ( h _ l ) = ( \sum ^ m _ { k = 1 } ( a ^ l _ k + i b ^ l _ k ) f _ k , t _ l ) ; \end{align*}


\begin{align*} k _ 2 = \min \big \{ | \theta _ { a , z } \cdot \nu | ^ 2 , \ , a \in S , \ , z \in B _ 2 \big \} \ , , & & k _ 3 = \min \big \{ | \theta _ { a , w } \cdot \nu | ^ 2 , \ , a \in S , \ , w \in B _ 3 \big \} \ , , \end{align*}


\begin{align*} \bigcap \limits ^ { \infty } _ { k = 1 } W ^ { * } _ { ( k ) } \neq \varnothing , \end{align*}


\begin{align*} \rho _ t ( x , y ) = \frac { t ^ { - d / \alpha } p _ 1 ^ { \Gamma } ( t ^ { - 1 / \alpha } x , t ^ { - 1 / \alpha } y ) } { t ^ { 2 \beta / \alpha } M _ { \Gamma } ( t ^ { - 1 / \alpha } x ) M _ { \Gamma } ( t ^ { - 1 / \alpha } y ) } = t ^ { - ( d + 2 \beta ) / \alpha } \rho _ 1 ( t ^ { - 1 / \alpha } x , t ^ { - 1 / \alpha } y ) . \end{align*}


\begin{align*} \| a \| _ { H ^ { 1 } _ { A } } \approx \int \mathcal { M } _ { ( N ) , A } ^ { 0 } a \ , \dd \mu & = \int \lim _ { i \to \infty } \mathcal { M } _ { ( N ) , A } ^ { 0 } a _ { j _ { i } } \ , \dd \mu \\ & \leq \liminf _ { i \to \infty } \int \mathcal { M } _ { ( N ) , A } ^ { 0 } a _ { j _ { i } } \ , \dd \mu \leq C \liminf _ { i \to \infty } \| a _ { j _ { i } } \| _ { H ^ { 1 } _ { A } } \leq C ' , \end{align*}


\begin{align*} \langle \lambda _ s ( b ) \xi , \eta \rangle & = \sum _ { t \in G } \lambda _ s ( b ) \xi ( t ) ^ * \eta ( t ) = \sum _ { t \in G } ( b \xi ( s ^ { - 1 } t ) ) ^ * \eta ( t ) = \sum _ { t \in G } \xi ( s ^ { - 1 } t ) ^ * b ^ * \eta ( t ) \\ & \stackrel { r = s ^ { - 1 } t } { = } \sum _ { r \in G } \xi ( r ) ^ * b ^ * \eta ( s r ) = \sum _ { r \in G } \xi ( r ) ^ * \lambda _ { s ^ { - 1 } } ( b ^ * ) \eta ( r ) = \langle \xi , \lambda _ { s ^ { - 1 } } ( b ^ * ) \eta \rangle . \end{align*}


\begin{align*} d _ 2 \cdots d _ l x \pi _ j \cdots \pi _ n = \alpha _ { k + 1 } ^ { \prime } \cdots \alpha _ m ^ { \prime } \underset { \beta ^ { \prime } } { \underbrace { d _ p d _ { p + 1 } \cdots d _ l x \pi _ j \cdots \pi _ n } } \end{align*}


\begin{align*} h _ { \mu } \left ( G / Q , \nu _ { Q } \right ) = \sum _ { \ell = 1 } ^ { k } \sum _ { r _ { \ell - 1 } < i \le r _ { \ell } , j > r _ { \ell } } \lambda _ { i } - \lambda _ { j } = h _ { \mu } \left ( G / P , \nu _ { P } \right ) - \sum _ { \ell = 1 } ^ { k } \sum _ { r _ { \ell - 1 } \le i < j \le r _ { \ell } } \lambda _ { i } - \lambda _ { j } . \end{align*}


\begin{align*} \mp _ s = \begin{bmatrix} \mp _ { s , 1 } & \mp _ { s , 2 } & \cdots & \mp _ { s , 2 ^ { t - s } } \end{bmatrix} . \end{align*}


\begin{align*} \begin{pmatrix} f & 0 \\ 0 & p e \end{pmatrix} \ , \ , ( p \in \mathcal H ) \end{align*}


\begin{align*} \chi _ F = \frac { 1 } { t - t ^ { - 1 } } \ , \det \begin{pmatrix} F ( t ) & F ( t ^ { - 1 } ) \\ 1 & 1 \end{pmatrix} \ . \end{align*}


\begin{align*} e ^ { \tau \Phi } D _ j ^ { \alpha _ j } e ^ { - \tau \Phi } & = ( e ^ { \tau \Phi } D _ j e ^ { - \tau \Phi } ) ( e ^ { \tau \Phi } D _ j e ^ { - \tau \Phi } ) \cdots ( e ^ { \tau \Phi } D _ j e ^ { - \tau \Phi } ) ( \alpha _ j ) . \end{align*}


\begin{align*} U = \overline { \xi } ( T _ 1 ) \circ \overline { \xi } ( T _ 2 ) . \end{align*}


\begin{align*} \it { L } u - c u = f \quad \quad \ , \ , \ , \Omega \ , ; \end{align*}


\begin{align*} a _ { \varepsilon } ( t ) = \left ( a * \psi _ { \omega ( \varepsilon ) } \right ) ( t ) = \langle a , \tau _ { t } \tilde { \psi } _ { \omega ( \varepsilon ) } \rangle \geq \tilde { a } _ { 0 } > 0 , \end{align*}


\begin{align*} g _ { \mu \nu } u ^ \mu u ^ \nu = - 1 . \end{align*}


\begin{align*} \bar \kappa ^ { \pm } _ { a - i } = 2 \kappa ^ { \pm } _ { a } - \kappa ^ { \pm } _ { a + i } , i \geq 2 , \mathcal { K } ^ { \pm } = \big \{ \kappa ^ { \pm } _ { i } \big \} _ { i \geq a } \bigcup \big \{ \bar \kappa ^ { \pm } _ { a - i } \big \} _ { i \geq 1 } . \end{align*}


\begin{align*} 2 n K _ 1 \beta ^ { - 1 / 2 } | \nabla v | \psi ^ { - 1 \slash n } ( \cdot , v ) \leq 2 n K _ 1 e ^ { C _ 0 ^ 2 + 1 } \psi ^ { - 1 \slash n } ( \cdot , v ) \leq ( B - 2 K _ 1 - 1 ) \sum _ { p = 1 } ^ n u ^ { p \bar p } . \end{align*}


\begin{align*} \partial _ t d = r m \frac { m } { 1 + \delta m } \left ( 1 - d \right ) . \end{align*}


\begin{align*} d ( \tau , j ) ^ { \delta } _ { i } = \prod _ { \tau \subseteq \sigma } ( c ( \sigma , j ) ^ { \delta } _ { i } ) ^ { ( - 1 ) ^ { | \sigma \setminus \tau | } } \end{align*}


\begin{align*} p ( X _ { ( m , n ) } | s _ { n } = 0 ) = \mathcal { C N } ( X _ { ( m , n ) } ; 0 , \epsilon ) , \end{align*}


\begin{align*} { \sqsupset ^ { * * } } = { \sqsupset ^ * } = { \sqsupset * \vartriangleright } . \end{align*}


\begin{align*} H _ \Delta ( \mathrm { M } ) = \{ u \in H ^ 1 ( \mathrm { M } ) ; \ ; \Delta u \in L ^ 2 ( \mathrm { M } ) \} , \end{align*}


\begin{align*} v ( h _ i ) = - \nu _ i ( g ) . \end{align*}


\begin{align*} \eta ( t , x ) = \chi ( t , r ) x , \end{align*}


\begin{align*} \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} = U _ 1 \begin{bmatrix} ( 1 - x ) A + x B & 0 \\ 0 & 0 \end{bmatrix} U _ 1 ^ * + V _ 1 \begin{bmatrix} 0 & 0 \\ 0 & x A + ( 1 - x ) B \end{bmatrix} V _ 1 ^ * \end{align*}


\begin{align*} \sum _ { \substack { | r | = \widehat { Y } \\ d \mid r } } | b _ 2 r _ 3 | ^ { 1 / 2 } \ll \widehat { Y } ^ { 1 + \varepsilon } . \end{align*}


\begin{align*} F _ { \Gamma _ { } ^ { ( m ) } } ( x ) & = 1 - \xi Q \Bigg ( \frac { \sqrt { \frac { x } { \Gamma _ 0 } } - \mu _ { \textrm { D } } } { \sigma _ { } } \Bigg ) ~ ~ \textrm { f o r ~ } x \ge 0 , \end{align*}


\begin{align*} H ( z ) : = A z ^ { n + m } + B \overline { z } ^ m + C \textrm { f o r a l l } z \in \mathbb { C } , \end{align*}


\begin{align*} \mathcal { I } ^ \star ( \lambda ) = \sup _ { \gamma \in \bar { \mathcal { P } } } I _ { \lambda } [ \gamma ] = \sup _ { \pi \in \Gamma ( \Pi , \phi _ \lambda ) } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda d \pi = \sup _ { \pi \in \Pi } \int _ { \mathcal { V } } f _ \lambda d \pi . \end{align*}


\begin{align*} \begin{dcases} L \varphi \le - 1 & B _ 1 \setminus \overline { B _ { 1 / 2 } } , \\ \varphi \ge ( 1 - | x | ^ 2 ) _ + ^ s & \\ \varphi \le C & B _ { 1 / 2 } , \\ \varphi = 0 & \R ^ n \setminus B _ 1 . \end{dcases} \end{align*}


\begin{align*} \partial _ t G ( t , s ) = ( q + s \varphi ( t ) ) \partial _ s G ( t , s ) + \varphi ( t ) G ( t , s ) , \end{align*}


\begin{align*} \mathring { v } ( x , t ) : = \psi \left ( x , t \right ) - \phi \left ( x + \gamma ( x , t ) + \frac { 1 } { N } \sigma ( t ) \right ) , \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c u ) ^ n = \psi ( \cdot , v ) \omega ^ n & \textnormal { i n } & \Omega , \\ u = 0 & \textnormal { i n } & \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} \sqrt { \alpha } \left ( \sum _ { m , n = 1 } ^ { \infty } ( - 1 ) ^ { m } \ , K _ { 0 } \left ( \left ( 2 n - 1 \right ) m \ , \alpha \right ) - \frac { \log ( 2 ) } { 4 } \right ) = \sqrt { \beta } \left ( \sum _ { m , n = 1 } ^ { \infty } ( - 1 ) ^ { m } K _ { 0 } \left ( \left ( 2 n - 1 \right ) m \ , \beta \right ) - \frac { \log ( 2 ) } { 4 } \right ) . \end{align*}


\begin{align*} \binom { n - 1 } { 2 } - 1 \leq Z _ 1 ( G _ n ) \leq \binom { n - 1 } { 2 } . \end{align*}


\begin{align*} \theta _ { 1 / 2 } ( p _ g , \epsilon ) & = \theta _ { L , X ^ { \ast } } \Big ( J ( p _ g , i ) \overline { \Pi } _ { \psi } ( p _ g ) A \Big ) [ \epsilon , 0 ] \\ & = \sum _ { n \in \Z } J ( p _ g , i ) \overline { \Pi } _ { \psi } ( p _ g ) A ( \epsilon , n ) \\ & = \sum _ { n \in \Z } e ^ { i \epsilon \pi n ^ 2 z _ g } . \end{align*}


\begin{align*} \int _ { \Omega } \big ( f ^ \prime ( c _ \varepsilon ) - f ^ \prime ( c ) \big ) u \ : x \geq - \alpha ( c _ \varepsilon - c ) u = - \alpha | u | ^ 2 . \end{align*}


\begin{align*} \lambda \cdot ( [ x : y : z ] , [ u : v : w ] ) = ( [ \lambda x : y : z ] , [ \lambda ^ { - 1 } u : v : w ] ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \langle D _ u f ( u ^ * ) , v \rangle _ { \mathcal { U } } + \langle \bar { \lambda } , G ' ( u ^ * ) v \rangle = 0 \forall \ v \in \mathcal { U } ; \\ & - \langle \bar { \lambda } , \zeta - G ( u ^ * ) \rangle \geq 0 \forall \ \zeta \in \mathcal { K } . \\ \end{aligned} \right . \end{align*}


\begin{align*} \zeta ( s ) = \sum _ { n _ j \leq x } n _ j ^ { - s } - A \frac { x ^ { 1 - s } } { 1 - s } + O \Big ( \Big ( 1 + \frac { \abs { s } } { \sigma - \theta } \Big ) x ^ { \theta - \sigma } \Big ) , x \geq 1 , \sigma = \Re ( s ) > \theta . \end{align*}


\begin{align*} A _ { m ( t ) } = \left \{ \exists \ , z _ 0 = z _ 0 ( \omega ) \in [ - k m ( t ) , k m ( t ) ] ^ d \ : \ : \ : \ : Z _ { m ( t ) } \left ( B ( z _ 0 , c [ \log \log m ( t ) ] ^ { 1 / d } ) \right ) \geq I ( t ) \right \} \end{align*}


\begin{align*} F _ k ( x , y ) & = \frac { 1 } { \pi } \int _ { - \infty } ^ \infty F _ k ( t , 0 ) \frac { y } { ( x - t ) ^ 2 + y ^ 2 } \ , d t \\ f ( x + i ( y + y _ k ) ) & = \frac { 1 } { \pi } \langle F _ k ( \cdot , 0 ) , P ( x - \cdot , y ) \rangle . \end{align*}


\begin{align*} \log N \simeq \int _ { 1 / 2 } ^ N x ^ { - 1 } \ , d x \lesssim \int _ { N } ^ { N + 1 } \ , d x = 1 , N > 1 , \end{align*}


\begin{align*} Y _ { t + \delta } ^ { t , \mu ; u } = Y _ { t + \delta } ^ { t + \delta , \rho _ { t + \delta } ^ { t , \mu ; u } ; u } = J \left ( t + \delta , \rho _ { t + \delta } ^ { t , \mu ; u } ; u \right ) . \end{align*}


\begin{align*} \mathcal { L } : = \mathcal { M } \setminus L . \end{align*}


\begin{align*} \mathcal I _ 1 K [ \textbf { X } ] = \ker \ , ( \bf e ) \end{align*}


\begin{align*} = \exp \left \{ \frac { 1 } { 6 } L i _ 2 ( z ) + \frac { 1 } { 2 } \log \left ( \frac { 1 } { 1 - z } \right ) + \frac { 1 } { 3 } \frac { z } { 1 - z } \right \} , \end{align*}


\begin{align*} \mathcal { K } _ c ( x ; y ; t ; p , q ) = \mathcal { K } _ c ( y ; x ; t ; p , q ) = \mathcal { K } _ c ( x ; y ; t ; q , p ) = \mathcal { K } _ { - c } ( - x ; y ; t ; p , q ) . \end{align*}


\begin{align*} \zeta _ 1 ( \lambda ) = \exp \left ( - \sum \limits _ \gamma \frac { T _ \gamma ^ \# e ^ { i \lambda T _ \gamma } } { T _ \gamma | \det ( I - \mathcal { P } _ \gamma ) | } \right ) , \quad \Im \lambda \gg 1 \end{align*}


\begin{align*} D _ { 1 } S _ { \omega , \mathbf { c } } ( \Phi ) = D _ { 2 } S _ { \omega , \mathbf { c } } ( \Phi ) = D _ { 3 } S _ { \omega , \mathbf { c } } ( \Phi ) = 0 . \end{align*}


\begin{align*} W _ \vartheta ( \delta _ a , \mu ) = \inf S . \end{align*}


\begin{align*} \left ( \mathcal { F } _ { \mathcal { H } _ { \hbar , V } } \mathcal { H } _ { \hbar , V } f \right ) ( \xi ) = ( \mathcal { H } _ { \hbar , V } f , u _ { \xi } ) = ( f , \mathcal { H } _ { \hbar , V } u _ { \xi } ) = ( f , \lambda _ { \xi } u _ { \xi } ) = \lambda _ { \xi } \widehat { f } ( \xi ) , \xi \in \mathcal { I } _ { \hbar } , \end{align*}


\begin{align*} ( \prod _ { i = 1 } ^ u d ( i ) \ast h _ l ( q ( i ) ) ) \ast d ( u + 1 ) = ( \prod _ { i = 1 } ^ m a ( i ) \ast g _ { w ( l ) } ( t ( m ) ) ) \ast a ( m + 1 ) \in A _ 1 \cap \sigma ( L ) \subseteq A _ 1 \cap \sigma ( L _ 1 ) . \end{align*}


\begin{align*} \alpha _ n = ( - 1 ) ^ n q ^ { \left ( { n \atop 2 } \right ) } \mbox { a n d } \quad \beta _ n = ( q ) _ m ( - 1 ) ^ n q ^ { \left ( { n \atop 2 } \right ) } \left [ { m + n \atop m + 2 n } \right ] . \end{align*}


\begin{align*} \delta ^ \ell _ { i _ 1 , \ldots , i _ n } ( f _ { i _ 1 , \ldots , i _ n } ) = f _ { i _ 1 , \ldots , i _ { \ell - 1 } , i _ { \ell + 1 } , \ldots , i _ n } \end{align*}


\begin{align*} ( \partial _ t - \partial _ x ^ 2 ) u = ( \lambda _ 0 + \lambda _ { e _ 0 } u ) \xi . \end{align*}


\begin{align*} p ( z ) = \frac { 1 + ( \overline { \tau _ 1 } \tau _ 2 + \tau _ 1 ) z + \tau _ 2 z ^ 2 } { 1 + ( \overline { \tau _ 1 } \tau _ 2 - \tau _ 1 ) z - \tau _ 2 z ^ 2 } , \ ; \ ; z \in \mathbb { D } . \end{align*}