\begin{align*} \sum \limits _ { i = 1 } ^ { m - 1 } \sum \limits _ { j = 1 } ^ { m - 1 } \sum \limits _ { k = 1 } ^ { m - 1 } l _ { j k } \tau _ { i j } \tau _ { i k } = t r ( \ { \Gamma } ^ T L { \Gamma } ) = t r ( L \ { \Gamma } { \Gamma } ^ T ) \\ = \sum \limits _ { i = 1 } ^ { m - 1 } \sum \limits _ { j = 1 } ^ { m - 1 } l _ { i j } h ^ { j i } = \sum \limits _ { i = 1 } ^ { m - 1 } \sum \limits _ { j = 1 } ^ { m - 1 } l _ { i j } h ^ { i j } . \end{align*}


\begin{align*} \Re H ' _ { \mu } ( z ) > 0 z = s + i t , \ , t \ge f _ { \mu } ( s ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : m a r t i n g a l e P o i s s o n m e a s u r e } M _ t = \int _ 0 ^ t \int _ \Theta a ( s , \theta ) \mathcal { N } ( \dd s , \dd \theta ) - \int _ 0 ^ t \int _ \Theta a ( s , \theta ) \nu ( \dd \theta ) \dd s , \end{align*}


\begin{align*} \sigma = \left ( \sigma ^ 2 - | \det ( \sigma ^ 2 I _ n - A ^ H A ) | \left ( \frac { n - 1 } { ( n + 1 ) \sigma ^ 2 - \| A \| _ F ^ 2 } \right ) ^ { n - 1 } \right ) ^ { 1 / 2 } \end{align*}


\begin{align*} \mathcal { R } ( \psi ) = \left ( \begin{array} { l l l } m _ { 0 0 } & m _ { 0 1 } & 0 \\ m _ { 1 0 } & m _ { 1 1 } & 0 \\ E & F & 1 \end{array} \right ) , \ \left ( \begin{array} { l l } m _ { 0 0 } & m _ { 0 1 } \\ m _ { 1 0 } & m _ { 1 1 } \end{array} \right ) = M _ \psi \ \ \ \ E , F \in \mathbb { N } . \end{align*}


\begin{align*} \gamma = e _ { 1 } e _ { 2 } \cdots e _ { k } , \end{align*}


\begin{align*} \mathcal { F } _ \tau f ( \lambda , k ) = \int _ G e ^ \tau _ { \lambda , k } ( g ) f ( g ) \ ; d g = \int _ { G / K } e ^ \tau _ { \lambda , k } ( g ) f ( g ) \ ; d g , \end{align*}


\begin{align*} \mathcal { A D } _ k = \{ r _ k ( M ) : M \in \mathcal { D } \} , \end{align*}


\begin{align*} \lim \limits _ { t \rightarrow \infty } \| \chi _ { S _ { v t } ^ c } e ^ { - i t H } \psi \| \leq \lim \limits _ { t \rightarrow \infty } \sup _ { \tau \geq 0 } \| \chi _ { S _ { v t } ^ c } e ^ { - i \tau H } \psi \| = \lim \limits _ { R \rightarrow \infty } \sup _ { \tau \geq 0 } \| \chi _ { S _ { R } ^ c } e ^ { - i \tau H } \psi \| \end{align*}


\begin{align*} \mathrm { e } ^ { - t \operatorname { S p e c } ( A _ j ) } = \operatorname { S p e c } ( T ( t , A _ j ) ) \end{align*}


\begin{align*} \log \left ( 1 + \frac { \log ( 1 + x ) } { u } \right ) = \sum _ { n = 1 } ^ \infty \ell _ n ( u ) x ^ n , \end{align*}


\begin{align*} \emph { \textsf { K e r } } ( l ) = B \emph { \textsf { K e r } } ( r ) = A . \end{align*}


\begin{align*} S = \{ \beta \in B ( 3 ) \slash \exists n \in \mathbb { N } , \exists i , j \in \{ 1 , 2 \} \ \ \mbox { s u c h t h a t } \ \ \ s i g n \big ( ( \beta \sigma _ i \beta ^ { - 1 } \sigma _ j ) ^ n \big ) \neq 0 \} \end{align*}


\begin{align*} \sigma _ k ( \lambda ( D ^ 2 \underline { u } ) ) & = ( \underline { u } ' ) ^ k + \underline { u } '' ( \underline { u } ' ) ^ { k - 1 } \Sigma _ { i = 1 } ^ n \sigma _ { k - 1 ; i } ( a ) ( a _ i x _ i ^ 2 ) \\ & \geq ( \underline { u } ' ) ^ k + 2 h _ k ( a ) \underline { u } '' ( \underline { u } ' ) ^ { k - 1 } \tau \\ & = \overline { f } \quad \ \mathbb { R } ^ n \setminus \overline { D } _ { s _ 0 } . \end{align*}


\begin{align*} | P - Q | \leq 2 \sqrt { \epsilon _ 0 } + 2 \sqrt { \epsilon _ 0 } = 4 \sqrt { \epsilon _ 0 } . \end{align*}


\begin{align*} { B } \overline { u } ^ \prime ( t ) + A \overline { u } ( t ) + B Q \overline { u } ( t ) = - H ( { u } ^ \prime ( t ) + Q u ( t ) ) , \ t \in \ , [ 0 , T ] , \quad \overline { u } ( 0 ) = 0 . \end{align*}


\begin{align*} \Phi ^ * = \Phi \setminus \mathcal { H } \subseteq \Phi \setminus \bigcup _ { v \in \overline { \Phi } \cap \mathbb { S } ^ 1 } H ^ v \end{align*}


\begin{align*} \int _ { \Psi _ { j + 1 } - \Psi _ j } x ^ k e ^ { x ^ r / \hbar } d x = \frac { 1 } { r } e ^ { \pi i \frac { k + 1 } { r } + 2 \pi i j \frac { k + 1 } { r } } \hbar ^ { ( k + 1 ) / r } \Gamma \left ( \frac { k + 1 } { r } \right ) \left ( e ^ { 2 \pi i ( k + 1 ) / r } - 1 \right ) = \frac { 1 } { r } C _ k \zeta ^ { j ( k + 1 ) } \end{align*}


\begin{align*} \frac { x _ { k + 1 } - x _ k } { h } = f _ h ( t _ k , x _ k , x _ { k + 1 } ) , \end{align*}


\begin{align*} \phi ( q _ \rho ) = g _ \rho \rtimes \rho , \qquad \phi _ 0 ( q _ \rho ) = e \phi ( q _ \rho ) = e _ \rho g _ \rho \rtimes \rho \end{align*}


\begin{align*} \begin{array} [ c ] { c } R ^ { T } ( t ) = - \Delta _ { B } u - t r _ { \omega } \widehat { \omega } _ { \infty } . \end{array} \end{align*}


\begin{align*} | E l l ^ { a } _ { k , j , n } ( s , x , \zeta ) | \lesssim \sum _ { l \in [ - M _ t , n + \epsilon M _ t ] \cap \Z } | E l l ^ { a ; l } _ { k , j , n } ( s , x , \zeta ) | , E l l ^ { a ; l } _ { k , j , n } ( s , x , \zeta ) : = 2 ^ { 3 k + 4 n + 3 \epsilon M _ t } \int _ { \R ^ 3 } \int _ { \R ^ 3 } f ( s , x - y , v ) \end{align*}


\begin{align*} \epsilon _ 0 = \alpha _ 1 \psi _ 1 \alpha _ 1 = - \int _ 0 ^ \infty e ^ { \mu _ 1 s _ 1 } ( \mathcal { R } _ 1 + \mathcal { R } _ 2 , \psi _ 1 ) _ { L _ y ^ 2 ( B _ { 4 { \sf R } _ { \sf i n } } ) } d s _ 1 . \end{align*}


\begin{align*} L ^ \mu ( g \circ T _ { \bar \mu } ^ { - 1 } ) ( T _ { \bar \mu } z ) = L _ 0 ^ { | \mu | } g ( z ) , \end{align*}


\begin{align*} & \widetilde \xi ( s ) = ( s - 1 ) \Gamma \left ( \frac s 2 + 1 \right ) \pi ^ { s / 2 } \zeta _ R ( s ) , \\ & \xi ( s , \chi ) = \left ( \frac \pi 4 \right ) ^ { - s / 2 } \Gamma \left ( \frac s 2 \right ) \beta ( s ) , \end{align*}


\begin{align*} \left ( \frac { t - 1 } t , 1 \right ) = \frac { t - 1 } t ( 1 , 0 ) + \frac 1 t ( 0 , t ) \end{align*}


\begin{align*} \int _ { \R ^ d } \left ( \prod _ { i = 1 } ^ d \abs { K _ { h _ { \sigma ( i ) } } ( x _ { \sigma ( i ) } - y _ { \sigma ( i ) } ^ i ) } \right ) \abs { d _ { ( w _ i ) _ i , \phi } ^ { ( l ) } ( y ^ 1 _ { \sigma ( 1 ) } , \dots , y ^ d _ { \sigma ( d ) } ) } d y _ { \sigma ( 1 ) } ^ 1 \dots d y _ { \sigma ( d ) } ^ d \le C \sqrt { \Delta _ n } . \end{align*}


\begin{gather*} ( W , W ' ) : = ( \phi ^ W , \phi ^ { W ' } ) \end{gather*}


\begin{align*} d _ \sigma A ( \sigma ) = A ( \sigma ) \ne 0 \end{align*}


\begin{align*} w = e ^ { - \kappa \phi } ( u - u _ 0 ) , \end{align*}


\begin{align*} [ \operatorname { s o c } _ { G } H ^ { 1 } ( \lambda ) : L ( \sigma ) ] = [ \operatorname { s o c } _ { L _ { \alpha } } R ^ { 1 } \operatorname { i n d } _ { B } ^ { P _ { \alpha } } L _ { \mathfrak f } ( \lambda ) : L _ { P _ { \alpha } } ( \bar { \sigma } ) ] . \end{align*}


\begin{align*} \hat A : = | \zeta | ^ 2 - \tfrac 1 4 \Delta _ \zeta + i \sum _ { j = 1 } ^ n ( \partial _ { \alpha _ j } \beta _ j - \partial _ { \beta _ j } \alpha _ j ) \end{align*}


\begin{align*} \deg _ { \vec { s } } \vec { v } = \max _ k \{ \deg v _ k + s _ k \} \ . \end{align*}


\begin{align*} \ker ( \kappa _ f ) \cong & \langle \ , x _ i \frac { \partial f } { \partial x _ i } , i = 1 , . . . , n \\ & \ , \ , \ , w _ { - \alpha } ( f ) , \alpha \in R ( N , \Sigma _ { \Delta } ) \ , \rangle . \end{align*}


\begin{align*} \small { \begin{aligned} \dot { \hat { \tau } } & = \dot { \hat { \tau } } ^ * \left ( K _ 0 ( \hat { \tau } ^ * ) - K _ { \bar { \tau } } ( \hat { \tau } ^ * ) \right ) + \hat { \tau } ^ * ( \dot { K } _ 0 ( \hat { \tau } ^ * ) - \dot { K } _ { \bar { \tau } } ( \hat { \tau } ^ * ) ) + \bar \tau \dot { K } _ { \bar \tau } ( \hat \tau ^ * ) , \end{aligned} } \normalsize \end{align*}


\begin{align*} f ( r , \theta ) & : = \begin{cases} \displaystyle \frac { \alpha \sinh ( \alpha r ) } { 2 \pi ( \cosh ( \alpha R ) - 1 ) } , & , \\ [ 2 e x ] 0 , & \end{cases} \end{align*}


\begin{align*} & \tau ^ \mathrm { a n } ( M / S , \rho ) - \tau ^ \mathrm { t o p } ( M / S , \rho ) \\ & = \frac { 1 } { [ K : \Q ] } \sum _ { g \in \mathrm { G a l } ( K / \Q ) } \Big ( \tau ^ \mathrm { a n } ( M / S , g . \rho ) - \tau ^ \mathrm { t o p } ( M / S , g . \rho ) \Big ) = 0 \ ; . \end{align*}


\begin{align*} \gamma _ 2 : = \inf \{ k : k \in \R _ + , | V ( x , v , \tau _ { \ast } , 0 ) | | \leq 2 ^ { k M _ t } \} . \end{align*}


\begin{align*} \sin \theta \Big ( R ( \tilde \Xi _ 1 ) , R ( \Xi _ 1 ) \Big ) \le \sin \theta ( \tilde { \boldsymbol { \xi } } _ 1 , \boldsymbol { \xi } _ 1 ) = { \mathcal P } _ { B 2 } , \end{align*}


\begin{align*} \varepsilon = s - \frac { p - 5 } { 2 ( p - 1 ) } . \end{align*}


\begin{align*} B = \bigcup \limits _ { l = 0 } ^ m D ' _ l . \end{align*}


\begin{align*} d _ { M K } ( \mu , \nu ) = \sup \left \{ \left | \int _ { X } f \mathrm { d } \mu - \int _ { X } f \mathrm { d } \nu \right | : f \in \operatorname { L i p } _ { 1 } ( X , \mathbb { R } ) \right \} , \end{align*}


\begin{align*} \lim \limits _ { \ell \rightarrow \infty } \dfrac { \mathbb { E } [ \ell ] } { \ell } = \dfrac { 1 } { 2 } \end{align*}


\begin{align*} C ( s , \Omega ) = C ( s , A \Omega ) \leq C ( s , \Omega _ { \infty } ) \leq C ( s , \omega ) \leq \frac { ( \omega _ k \abs { \omega } ) ^ { s / k } } { ( 2 \pi ) ^ s } \sqrt { \frac { \delta ^ { 1 + \frac { 2 s } { k } } } { \delta - 1 } } = \vcentcolon c ( s , \Omega ) , \delta = 1 + \frac { k } { 2 s } , \end{align*}


\begin{align*} \mathcal { F } ( f ) ( \xi ) = \int e ^ { - i x \cdot \xi } f ( x ) d x . \end{align*}


\begin{align*} \begin{cases} & \frac { \partial X } { \partial t } ( x , t ) = \frac { \phi ( t ) \psi u ^ \alpha \rho ^ \delta } { K ^ \frac { \beta } { n } } \nu - X , \\ [ 4 p t ] & X ( x , 0 ) = X _ 0 ( x ) , \end{cases} \end{align*}


\begin{align*} \lim _ { k \rightarrow - \infty } V _ r ( 0 , x _ k ) = 0 . \end{align*}


\begin{align*} h _ n ( X ) P _ { n m _ 0 - 1 } ( X ) = P _ { n m _ 0 } ( X ) \end{align*}


\begin{align*} A _ { n } = \sum _ { \gamma \in \mathcal { M } _ { n } } w ( \gamma ) . \end{align*}


\begin{align*} & \Psi ^ * \circ L \circ ( \Psi ^ * ) ^ { - 1 } = - a ^ { i j } ( s , x ) \partial _ { x _ i } \partial _ { x _ j } + b ^ { j } ( s , x ) \partial _ { x _ j } + c ( s , x ) , \\ & \textup { w h e r e } \Lambda ^ { - 1 } \| \xi \| ^ 2 \leq a ^ { i j } ( s , x ) \xi _ i \xi _ j \leq \Lambda \| \xi \| ^ 2 , \\ & \textup { a n d } \| b ( s , x ) \| \leq \Lambda ^ { - 1 } , \ 0 \leq c ( s , x ) \leq \Lambda ^ { - 1 } , \end{align*}


\begin{align*} \int _ 0 ^ 1 \frac { 1 } { u ( 1 \vee \log \Phi ^ { - 1 } ( u ) ) } \mathrm { d } u = \infty \sum _ { n = 1 } ^ \infty \frac { 1 } { ( 1 \vee \log \phi ^ { - 1 } ( n ) ) } = \infty . \end{align*}


\begin{align*} h _ J ( j ) = \begin{cases} j + 1 & , \\ j & . \end{cases} \end{align*}


\begin{align*} R ^ { n , 3 } _ \tau = \frac 1 n \sum _ { j = 0 } ^ { \lfloor n \tau \rfloor } \int _ { j } ^ { ( j + 1 ) \wedge n \tau } \gamma _ { \frac x n } ( \sigma ' \sigma ) ^ 2 ( X _ { \frac x n } ) \sum _ { i = 0 } ^ { \lfloor x \rfloor } \int _ { i } ^ { ( i + 1 ) \wedge x } [ ( x - i ) ^ \alpha - ( x - v ) ^ \alpha ] ^ 2 d v d x . \end{align*}


\begin{align*} H _ m ( t ) = \frac { 2 \sqrt \pi \Gamma ( \frac { s + 1 } { 2 } ) t ^ { 1 / 2 } } { \Gamma ( \frac s 2 ) } J _ { s - \frac 1 2 } ( 2 \pi | m | t ) \end{align*}


\begin{align*} \int _ 0 ^ \infty \mathrm { e } ^ { - \omega s } \psi ^ { [ n * ] } _ 1 ( s ) \ , \textnormal { d } s = \left ( \int _ 0 ^ \infty \mathrm { e } ^ { - \omega s } \psi _ 1 ( s ) \ , \textnormal { d } s \right ) ^ n , \end{align*}


\begin{align*} \phi _ { j } ^ { ( q ) } ( z ) & : = \langle ( z I - \mathcal { H } _ { q } ) ^ { - 1 } e _ { j } , e _ { 0 } \rangle = \sum _ { n = 0 } ^ { \infty } \frac { \langle \mathcal { H } _ { q } ^ { n } \ , e _ { j } , e _ { 0 } \rangle } { z ^ { n + 1 } } , 0 \leq j \leq p , q \geq 0 , \\ \beta _ { j } ^ { ( q ) } ( z ) & : = \langle ( z I - \mathcal { B } _ { q } ) ^ { - 1 } e _ { j } , e _ { 0 } \rangle = \sum _ { n = 0 } ^ { \infty } \frac { \langle \mathcal { B } _ { q } ^ { n } \ , e _ { j } , e _ { 0 } \rangle } { z ^ { n + 1 } } , 0 \leq j \leq p , q \geq 0 , \end{align*}


\begin{align*} \label [ I e q ] { e q : m c k e a n v l a s o v - l i m i t s y n c h r o n o u s c o u p l i n g } \dd \overline { X } { } ^ i _ t = b \big ( \overline { X } { } ^ i _ t , f _ t \big ) \dd t + \sigma \big ( \overline { X } { } ^ i _ t , f _ t \big ) \dd B ^ i _ t , \end{align*}


\begin{align*} H ^ k ( \mathbb { P } , \Omega _ { \mathbb { P } } ^ { n - 1 } \otimes \mathcal { O } ( - m Y - m K _ { \mathbb { P } } ) ) \cong & H ^ { n - k } ( \mathbb { P } , \Omega _ { \mathbb { P } } ^ 1 \otimes \mathcal { O } ( m Y + m K _ { \mathbb { P } } ) ) = 0 \end{align*}


\begin{align*} Q ( p , q ) ^ \ast = Q ( q , p ) \ . \end{align*}


\begin{align*} F _ m ( \psi _ n ) ( x , y ) = ( 2 x , y ) . \end{align*}


\begin{align*} S _ Q ( f ) = \{ i , 0 \leq i \leq r \mid \nu _ Q ( f ) = \nu ( f _ i Q ^ i ) \} \mbox { a n d } \delta _ Q ( f ) = \max S _ Q ( f ) . \end{align*}


\begin{align*} h _ \mathrm { M F } : = - \Delta + V _ { \mathrm { D W } } + \lambda w * | u _ + | ^ 2 . \end{align*}


\begin{align*} A ^ 1 _ Q = \varnothing ; A ^ 2 _ Q = \bigcup _ { S _ 0 = 0 } ^ { s - 1 } A ^ 2 _ { Q , j , S _ 0 } . \end{align*}


\begin{align*} \Big ( \gamma _ j - \gamma ' _ j \big | \boldsymbol { \gamma } _ { - j } , \boldsymbol { \gamma } ' _ { - j } \Big ) \overset { d } { = } \big ( \gamma _ j - \gamma _ j ' \big ) \ , \end{align*}


\begin{align*} s = \min _ { A \in \mathcal { A } } s _ A = \min \left \{ \frac { 2 + \theta _ { 1 } } { 1 + \theta _ { 1 } + \epsilon } , \frac { 2 + \theta _ { 2 } - ( \theta _ { 1 } + 2 \epsilon ) \log _ { \beta _ 2 } \beta _ 1 } { 1 + \theta _ { 2 } + \epsilon } \right \} . \end{align*}


\begin{align*} h ^ * ( X , G ) = \inf _ { k \in \mathbb { N } } \ \sup _ { m \in \mathbb { N } } h _ G ( \P _ m | \P _ k ) . \end{align*}


\begin{align*} E _ { 2 - \mathrm { m o d e } } : = \inf \Big \{ \left \langle \psi _ N , H _ N \psi _ N \right \rangle , \ ; | \ ; \psi _ N \in P ^ { \otimes N } \mathcal { H } _ N , \ ; \int _ { \mathbb { R } ^ { d N } } | \psi _ N | ^ 2 = 1 \Big \} . \end{align*}


\begin{align*} \mathcal { P } _ { ( A _ 1 , A _ 2 , q ) } v : = i \textrm { d i v } ( A v ) + i A \cdot \nabla v + ( \abs { A _ 2 } ^ 2 - \abs { A _ 1 } ^ 2 + q ) v , v \in H ^ 1 ( \R ^ 3 ) , \end{align*}


\begin{align*} Z _ j = \begin{cases} \dot { x } = - y + d _ j x + l _ j \ , x ^ 2 + m _ j \ , x \ , y + n _ j \ , y ^ 2 , \\ \dot { y } = x ( 1 - y ) , \end{cases} \ \ ( x , y ) \in \Sigma ^ { \mathcal { H } } _ j , \end{align*}


\begin{align*} \lim \limits _ { | \zeta | \to \infty } \partial _ { \zeta _ i } f ( \zeta ) = 0 . \end{align*}


\begin{align*} \widetilde { Y } ^ n _ t = \widetilde { A } ^ n _ t + \widetilde { C } ^ n _ t + \int _ 0 ^ t ( t - s ) ^ { \alpha } \sigma ' ( X _ s ) \widetilde { Y } ^ n _ s \ , d W _ s . \end{align*}


\begin{align*} S _ n & = \sum _ { p = 1 } ^ { n - 1 } A _ p + b _ p - \epsilon _ p \\ & = \int _ 1 ^ n \log x \ , d x + \frac { 1 } { 2 } \log n - \sum _ { p = 1 } ^ { n - 1 } \epsilon _ p \\ & = \big ( n + \frac { 1 } { 2 } \big ) \log n - n + 1 - \sum _ { p = 1 } ^ { n - 1 } \epsilon _ p . \end{align*}


\begin{align*} g ^ { \lambda } _ t : = \lambda \cdot \psi _ { s _ + } ^ * \big [ g _ { t / \lambda } \big ] , t \leq 0 . \end{align*}


\begin{align*} D _ { q ^ { - 1 } } A _ q ( x ) = \frac { q } { 1 - q } A _ q ( q x ) , \end{align*}


\begin{align*} c ( 1 - m ) \int _ 0 ^ L \abs { \omega _ x } ^ 2 d x - \frac { c m } { 2 } \int _ 0 ^ L \int _ 0 ^ { \infty } \sigma ' ( s ) \abs { \eta _ x } ^ 2 d s d x = - \Re \left ( \left < { \mathcal { A } } _ m U ^ n , U ^ n \right > _ { \mathcal { H } } \right ) \leq \| F _ n \| _ { \mathcal { H } } \| U \| _ { \mathcal { H } } . \end{align*}


\begin{align*} \| \chi _ { S _ { r _ 0 } } e ^ { - i t H _ 0 } P _ { \delta } ( W _ { n ; \textrm { o u t } } ) \| _ { \textrm { o p } } = \| e ^ { - i t H _ 0 ^ \parallel } \otimes ( \chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \perp } P ^ \perp _ \delta ( W _ { n ; \textrm { o u t } } ^ \perp ) ) \| _ \textrm { o p } = \| \chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \perp } P ^ \perp _ \delta ( W _ { n ; \textrm { o u t } } ^ \perp ) \| _ \textrm { o p } \end{align*}


\begin{align*} \dfrac { ( \Tilde { \mathcal { N } } ( f ) \circ \rho _ { \eta } ( X ) ) ' } { \Tilde { \mathcal { N } } ( f ) \circ \rho _ { \eta } ( X ) } = \sum _ { w \in W _ { \rho } ^ { m _ 0 } } \dfrac { ( f ( X + w ) ) ' } { f ( X + w ) } = \sum _ { w \in W _ { \rho } ^ { m _ 0 } } \dfrac { f ' ( X + w ) } { f ( X + w ) } . \end{align*}


\begin{align*} X : = X _ - , \tilde X : = X _ + , \quad , D : = D _ - , \tilde D : = D _ + . \end{align*}


\begin{align*} \int _ { K / M } e _ { w \lambda , k } ^ \tau ( g ) \beta ( w \lambda , k ) d k = \int _ { K / M } e _ { - \lambda , k } ^ \tau ( g ) \beta ( \lambda , k ) d k , \ ; \ ; \ ; \ ; \ ; w \in W _ A . \end{align*}


\begin{align*} \mathrm { T C K } _ { q } ( X ) = \frac { \mathrm { E } \left [ ( X - \mathrm { T C E } _ { q } ( X ) ) ^ { 4 } | X > x _ { q } \right ] } { \mathrm { T V } _ { q } ^ { 2 } ( X ) } - 3 , \end{align*}


\begin{align*} \mathcal { Q } u = \mathbf { S } _ { d , 0 } ( t ) u _ 0 + \int _ { 0 } ^ t \mathbf { S } _ { d , - \alpha } ( t - s ) | u ( s ) | ^ p d s + \int _ { 0 } ^ t \mathbf { S } _ { d , 0 } ( t - s ) \zeta ( s ) \mathbf { w } d s \end{align*}


\begin{align*} K ( \nu ; z ) = - { \textstyle \frac { 4 \pi } { \nu } } z ^ 2 e ^ { z } \sum \limits _ { n = - \infty } ^ { + \infty } { \textstyle \frac { [ \nu + i ( 2 n + 1 - z ) ] ^ { n - 1 } } { [ \nu + i ( 2 n + 1 + z ) ] ^ { n + 2 } } } \end{align*}


\begin{align*} Q = \begin{pmatrix} Q _ { 0 } & 0 \\ 0 & I \end{pmatrix} , \end{align*}


\begin{align*} t _ { n + 1 } - t _ n \leq \sum \limits _ { j = x _ n + 1 } ^ { x _ { n + 1 } } \frac { 1 } { A ( j ) } ; \end{align*}


\begin{align*} y = \frac { t ( t ' - y ) + t ' ( y - t ) } { t ' - t } = \frac { t ( t ' - y ) + t ' f _ t ( y ) } { t ' - t } . \end{align*}


\begin{align*} \norm { \pi ( \beta ) } ^ 2 = \norm { \pi ( \beta ^ \ast \ast \beta ) } = \abs { \frac { 1 } { m _ S ( B ) } \int _ { B } \omega _ \pi ( g ) \ , d m _ S ( g ) } ^ 2 \le \frac { 1 } { m _ S ( B ) } \int _ { B } \abs { \omega _ \pi ( g ) } ^ 2 \ , d m _ S ( g ) , \end{align*}


\begin{align*} G D ^ k \widetilde { G } = \widetilde { G } \widetilde { D } ^ k G D G ^ k \widetilde { D } = \widetilde { D } \widetilde { G } ^ k D . \end{align*}


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


\begin{align*} \tilde { K } _ { \lambda , \tau } ( q ) = q ^ { n ( \tau ) } K _ { \lambda , \tau } ( q ^ { - 1 } ) . \end{align*}


\begin{align*} \sum _ { \nu = 1 } ^ k \beta \Big ( f _ 1 ( x + t _ n y ) , \ldots , f _ { \nu - 1 } ( x + t _ n y ) , \frac { f _ \nu ( x + t _ n y ) - f _ \nu ( x ) } { t _ n } , f _ { \nu + 1 } ( x ) , \ldots , f _ k ( x ) \Big ) , \end{align*}


\begin{align*} \dim ( ( \mathbb { C } \phi ^ 1 ) _ 2 \cap \ker A _ 2 ) + \dim ( ( \mathbb { C } \phi ^ 2 ) _ 2 \cap \ker A _ 2 ) + \dim ( ( \mathbb { C } \phi ^ 3 ) _ 2 \cap \ker A _ 2 ) & = 1 + 1 + 1 \\ & = \dim ( M _ 2 \cap \ker A _ i ) . \end{align*}


\begin{align*} \partial _ { \zeta _ i } h ( \zeta ) = s { \zeta _ i \over | \zeta | } h ( \zeta ) ( 1 + o ( 1 ) ) , \end{align*}


\begin{align*} R ^ { n , M , 1 } _ \tau = n ^ { 2 \alpha + 1 } \sum _ { i = 0 } ^ { M - 1 } ( \sigma ' \sigma ) ^ 2 ( X _ { \tau _ i } ) \int _ { \tau _ i } ^ { \tau _ { i + 1 } } \gamma _ s \Xi ^ { n , 1 } _ s d s \end{align*}


\begin{align*} \sup _ { \| Y \| _ 2 = 1 } Y ^ T \Sigma _ t ( \varphi ) Y = Y ^ T _ 0 \Sigma _ t ( \varphi ) Y _ 0 \leq Y ^ T _ 0 \Sigma _ t ( 0 ) Y _ 0 \leq \| \Sigma _ t ( 0 ) \| = \sup _ { \| Y \| _ 2 = 1 } Y ^ T \Sigma _ t ( 0 ) Y . \end{align*}


\begin{align*} f ( x ) & = f ( x ) - f ( c ) + f ( c ) \\ & = \int _ c ^ x \partial _ y f ( y ) d y + \int _ 0 ^ c \partial _ y f ( y ) d y + f ( 0 ) \\ \Rightarrow \left | f ( x ) \right | & \le \int _ 0 ^ 1 \left | \partial _ y f ( y ) \right | d y + \left | f ( 0 ) \right | . \end{align*}


\begin{align*} ( x ) _ j = x ( x - 1 ) \cdots ( x - j + 1 ) . \end{align*}


\begin{align*} \mathcal { H } _ { - \tilde { A } _ 1 , q _ 1 } \tilde { u } _ 1 & = e ^ { - i \varphi / 2 } \mathcal { H } _ { - A _ 1 , q _ 1 } u _ 1 = k ^ 2 \tilde { u } _ 1 , \cr \mathcal { H } _ { \tilde { A } _ 2 , q _ 2 } \tilde { u } _ 2 & = e ^ { - i \varphi / 2 } \mathcal { H } _ { A _ 2 , q _ 2 } u _ 1 = k ^ 2 \tilde { u } _ 2 , \textrm { i n } \ , B . \end{align*}


\begin{align*} & \min _ { \delta , P } \delta ^ 2 \\ & \ \ \\ & \begin{bmatrix} - P A - A ^ { T } P & - P B + A ^ { T } C ^ { T } \\ - B ^ { T } P + C A & ( C B + B ^ { T } C ^ { T } ) \end{bmatrix} + \delta I _ { n + m } \geq 0 , \end{align*}


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


\begin{align*} \sup _ { A \textnormal { L - p r i m i t i v e } } f ( A ) \ & = \ \sum _ p \max \{ f ( p ) , e ^ \gamma { \rm d } ( { \rm L } _ p ) \} , \lim _ { x \to \infty } \sup _ { \substack { A \subset [ x , \infty ) \\ A \textnormal { L - p r i m i t i v e } } } f ( A ) \ = \ e ^ \gamma . \end{align*}


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


\begin{align*} A + B \ , \sqrt { 1 + \kappa ^ 2 x ^ 2 } = 0 \ \ . \end{align*}


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


\begin{align*} M _ f : = v _ { f ( 1 ) } \otimes v _ { f ( 2 ) } \otimes \cdots \otimes v _ { f ( m ) } \otimes w _ { f ( m + 1 ) } \otimes \cdots \otimes w _ { f ( m + n ) } . \end{align*}


\begin{align*} \theta = \omega ^ i \ , \ , \mbox { f o r s o m e } i \in \Z , i \neq 0 \end{align*}


\begin{align*} r = \theta - \eta . \end{align*}


\begin{align*} x _ i = x _ 0 + \tfrac { 1 } { L } \sum \limits _ { k = 0 } ^ { i - 1 } h _ k g _ k \end{align*}


\begin{align*} A = \left [ \begin{matrix} 1 + \delta & \ & \ & \ \\ \ & 1 & \ & \ \\ \ & \frac { 1 } { 2 } & 1 - \delta _ 1 & \ \\ \ & \ & \ & ( 1 - 2 \delta _ 1 ) I _ { n - 3 } \end{matrix} \right ] , \Delta A = \left [ \begin{matrix} 0 & \cdots & 0 \\ \epsilon & \vdots & \vdots \\ 0 & \vdots & \vdots \\ \vdots & \vdots & \vdots \\ 0 & \cdots & 0 \end{matrix} \right ] . \end{align*}


\begin{align*} \tilde { \lambda } _ i ^ k = \lambda _ { i } ^ k ( \sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) ) \geq \sigma _ { k - 1 } \lambda _ { 1 } ( Z _ { k - 1 } ) + \lambda _ { i } ( - X ( x _ k ) ) . \end{align*}


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


\begin{align*} L ( s , \varphi \times \chi _ d ) = \sum _ { n = 1 } ^ \infty \frac { a ( n ) \chi _ d ( n ) } { n ^ s } . \end{align*}


\begin{align*} f \circ \varphi = \psi \circ f \ \mbox { ( e q u a l i t y o f m a p s f r o m $ B _ { \pi } $ t o $ \{ 1 , \ldots , \ell + 1 \} $ ) , } \end{align*}


\begin{align*} L k ( \overline { { } ^ J U ^ - _ { v , w , > 0 } } ) = \overline { { } ^ J U ^ - _ { v , w , > 0 } } \bigcap \{ | | x | | = 1 \vert x \in L \} \end{align*}


\begin{align*} \omega _ i ^ * = \min \left ( \max ( \omega _ i ^ 0 , - \bar { \omega } ) , \bar { \omega } \right ) , \end{align*}


\begin{align*} \kappa ( Y ) = \min ( n - 1 , \dim \ , F ( \Delta ) ) . \end{align*}


\begin{align*} \P ( T _ { d e t } \geq \ss ) = \exp \big ( { - } \Theta ( \ss ^ { \frac { 1 } { 2 \alpha } } ) \big ) . \end{align*}


\begin{align*} A B x = x , \ x \in Y . \end{align*}


\begin{align*} \langle \vartheta _ { p _ 1 } , \ldots , \vartheta _ { p _ r } \rangle ^ { n a i v e } : = \sum _ { \textbf { A } \in N E ( \tilde { X } ) } t ^ { \pi _ * ( \textbf { A } ) } N _ \textbf { A } ^ { n a i v e } ( \textbf { p } ) \in \mathbb { Z } [ N E ( X ) ] . \end{align*}


\begin{align*} U ^ m _ { S ; j , l } ( t , x ) = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } ( t - s ) \big ( E ( s , x + ( t - s ) \omega ) + \hat { v } \times B ( s , x + ( t - s ) \omega ) \big ) \cdot \nabla _ v \big ( \frac { m _ U ( v , \omega ) \varphi _ { j , l } ( v , \omega ) } { 1 + \hat { v } \cdot \omega } \big ) \end{align*}


\begin{align*} z _ n ^ * = y _ n ^ * - \sum _ { m = 1 } ^ \infty a _ m f _ { q ( m ) } ( y _ n ^ * ) \widetilde { u } _ m . \end{align*}


\begin{align*} { \rm R C } ( I ) : = \mathbb { R } _ + \{ e _ 1 , \ldots , e _ s , ( v _ 1 , 1 ) , \ldots , ( v _ q , 1 ) \} . \end{align*}


\begin{align*} \| V _ \alpha ( \hat { x } ) \| ^ 2 = \tau ( \rho _ \alpha ( x ) x ^ * ) + \frac { 1 } { \alpha } \tau ( \Delta \circ \rho _ \alpha ( x ) x ^ * ) = \| x \| _ 2 ^ 2 , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ \infty \frac { ( - 1 ) ^ k } { k + z } = - \frac { 1 } { 2 } \left ( \psi \left ( \frac { z } { 2 } \right ) - \psi \left ( \frac { z + 1 } { 2 } \right ) \right ) , z \in \mathbb { C } \setminus - \mathbb { N } _ 0 . \end{align*}


\begin{gather*} \begin{pmatrix} x _ 0 \ , g _ { 0 0 } + x _ 1 g _ { 1 0 } & x _ 0 \ , g _ { 0 1 } + x _ 1 g _ { 1 1 } \\ x _ 0 \ , h _ { 0 0 } + x _ 1 h _ { 1 0 } & x _ 0 \ , h _ { 0 1 } + x _ 1 h _ { 1 1 } \end{pmatrix} \begin{pmatrix} y _ 0 \\ y _ 1 \end{pmatrix} \xmapsto { t _ j \mapsto f _ j } \begin{pmatrix} 0 \\ 0 \end{pmatrix} \end{gather*}


\begin{align*} a _ { \lambda } ( m ' _ i ) = \sum ^ { n } _ { j = 1 } h ' _ { j } ( a _ { \lambda } ( m ' _ i ) ) h _ j , 1 \leq i \leq r . \end{align*}


\begin{align*} ( 1 + u _ p ^ 0 ) - ( \sigma + h _ p ^ 0 ) \geq 1 - \sigma - \mathcal { O } ( \delta , \sigma ) : = c _ 1 > 0 . \end{align*}


\begin{align*} X _ t = X _ 0 + \int ^ t _ 0 ( t - s ) ^ { \alpha } \sigma ( X _ s ) \ , d W _ s , t \in [ 0 , T ] , \end{align*}


\begin{align*} & x _ i \frac { \partial } { \partial x _ i } : x ^ m \mapsto m _ i \cdot x ^ m , i = 1 , . . . , n , \\ & \frac { \partial e _ { - \alpha } ^ { \lambda } } { \partial \lambda _ i } _ { \vert { \lambda = 0 } } : x ^ m \mapsto h t _ { - \alpha } ( m ) \cdot x ^ { m - \alpha } , \alpha \in R ( N , \Sigma _ { C ( \Delta ) } ) . \end{align*}


\begin{align*} N _ 0 ( ( R f ^ { i _ 1 \dots i _ k } ) _ { p _ 1 q _ 1 \dots p _ { m - k } q _ { m - k } } ) = 0 \mbox { i n } U . \end{align*}


\begin{align*} \frac { x } { y } + \frac { y } { z } + \frac { z } { x } = n \end{align*}


\begin{align*} \int _ { A _ m } \Omega = \int _ { L _ m } \Omega > 0 , \sum _ 1 ^ m \int _ { A _ i } \Omega \geq \sum _ 1 ^ m \ \int _ { L _ i } \Omega , \int _ L \Omega = \sum _ 1 ^ N \int _ { L _ i } \Omega . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { t \in [ 0 , T ] } E [ | Z ^ n _ t - \widetilde { Z } ^ n _ t | ^ 2 ] = 0 , \end{align*}


\begin{align*} & I ( L ( - 1 ) w _ 1 , z ) = \frac { d } { d z } I ( w _ 1 , z ) . \end{align*}


\begin{align*} \lbrack \omega _ { 0 } ] _ { B } - T c _ { 1 } ^ { B } ( M ) = [ \psi ^ { \ast } \omega _ { N } ] _ { B } . \end{align*}


\begin{align*} z ^ { 2 } = a ^ { 2 } - b ^ { 2 } + 2 i a b = 1 \pm 1 6 i m \beta \lambda / 3 \Rightarrow \left \lbrace \begin{array} { l } a ^ { 2 } - b ^ { 2 } = 1 \\ 2 a b = \pm 1 6 m \beta \lambda / 3 \end{array} \right . . \end{align*}


\begin{align*} { \bf u } _ m ( \alpha ) = \big [ { \tilde u } _ { 1 , 1 } ^ \alpha , \ldots , { \tilde u } _ { 1 , N _ h } ^ \alpha , { \tilde u } _ { 2 , 1 } ^ \alpha , \ldots , { \tilde u } _ { 2 , N _ h } ^ \alpha , \ldots , { \tilde u } _ { m , 1 } ^ \alpha , \ldots , { \tilde u } _ { m , N _ h } ^ \alpha \big ] ^ T \in \mathbb { R } ^ { m N _ h } \end{align*}


\begin{align*} b ( \alpha , \lambda ) = \langle \check { \alpha } , \lambda \rangle Q ( \alpha ) , ( \alpha \in \Delta , \lambda \in \Lambda ) . \end{align*}


\begin{align*} \left \lbrace \begin{array} { c } \phi ( + a / 2 ) = - \phi ( - a / 2 ) \\ \partial _ { x } \phi ( + a / 2 ) = - \partial _ { x } \phi ( - a / 2 ) \\ \partial _ { x } ^ { 2 } \phi ( + a / 2 ) = - \partial _ { x } ^ { 2 } \phi ( - a / 2 ) \\ \partial _ { x } ^ { 3 } \phi ( + a / 2 ) = - \partial _ { x } ^ { 3 } \phi ( - a / 2 ) \end{array} \right . , \end{align*}


\begin{align*} \lambda = \sqrt { \sum _ { i } \lambda _ { i } ^ 2 } . \end{align*}


\begin{align*} u _ { k + 1 } = B _ k u _ k + A _ k u _ { k - 1 } , \end{align*}


\begin{align*} \widetilde { H } ^ { S ; U ; m , i } _ { j , n , l , r } ( t , x , \zeta ) = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \big ( 1 + ( t - s ) \omega \cdot \nabla \big ) \big [ \big ( E ( s , x + ( t - s ) \omega ) + \hat { v } \times B ( s , x + ( t - s ) \omega ) \big ) \cdot \nabla _ v \big ( \frac { m _ { U } ( v , \omega ) \varphi _ { j , n } ^ { i ; r } ( v , \zeta ) \varphi _ { l ; r } ( \tilde { v } + \omega ) } { 1 + \hat { v } \cdot \omega } \big ) \end{align*}


\begin{align*} \mathbf { C } _ { \Lambda } \doteq \left \{ \begin{array} { l l } a _ { x , \uparrow } a _ { x , \downarrow } & \Lambda = \left \{ x \right \} x \in \mathfrak { L } \\ 0 & \end{array} \right . , \Lambda \in \mathcal { P } _ { \mathrm { f } } \ . \end{align*}


\begin{align*} \partial ^ \alpha _ x K _ j ( x - y ) = ( - 1 ) ^ { | \alpha | } \partial ^ \alpha _ y K _ j ( x - y ) \end{align*}


\begin{align*} T _ { k , j ; n } ^ { \mu , i } ( \mathfrak { m } , U ) ( t , x , \zeta ) : = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } e ^ { i x \cdot \xi + i \mu ( t - s ) | \xi | - i s \hat { v } \cdot \xi } | \xi | ^ { - 1 } \mathfrak { m } ( \xi , \zeta ) m ^ { 0 } _ U ( \xi , v ) \hat { g } ( s , \xi , v ) \varphi _ k ( \xi ) \varphi _ { n ; - M _ t } \big ( \tilde { \xi } + \mu \tilde { \zeta } \big ) \varphi _ { j , n } ^ i ( v , \zeta ) \end{align*}


\begin{align*} \Psi _ { A B } = \theta _ { A \ , N + 2 } \wedge \theta _ { N + 2 \ , B } \end{align*}


\begin{align*} E _ { k , t } ^ { } = \tau p _ { k , t } , ~ \forall k \in { \cal K } . \end{align*}


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


\begin{align*} H : = \left \{ \small { \begin{pmatrix} \det ( U ) ^ { - 1 } & 0 \\ 0 & U \end{pmatrix} } \ , \ \ U \in U ( 2 ) \right \} \simeq S \big ( U ( 2 ) \times U ( 1 ) \big ) \simeq U ( 2 ) \ , \end{align*}


\begin{align*} g . \rho : \pi _ 1 ( M ) & \rightarrow \mathrm { G L } _ n ( K ) \\ \gamma & \mapsto \big ( g \big ( { \rho ( \gamma ) } _ { i , j } \big ) \big ) _ { 1 \leqslant i , j \leqslant n } \ ; , \end{align*}


\begin{align*} Y = X \beta + e \end{align*}


\begin{align*} \sum _ { \substack { a \in \overline { E } \\ \mathrm { h e a d } ( a ) = i } } \epsilon ( a ) a a ^ * = 0 . \end{align*}


\begin{align*} & = \sum \limits _ { v \in e } \frac { \delta _ E ( e ) } { | e | } [ ( a v g ( x ) ) ( e ) - x ( v ) ) ] x ( v ) \\ & = \sum \limits _ { i = 1 } ^ k \frac { \delta _ E ( e ) } { | e | ^ 2 } \left [ \left \{ \sum \limits _ { j = 1 } ^ k x ( v _ j ) \right \} - | e | x ( v _ i ) ) \right ] x ( v _ i ) \\ & = - \frac { 1 } { 2 } \frac { \delta _ E ( e ) } { | e | ^ 2 } \sum _ { i , j = 1 } ^ k ( x ( v _ i ) - x ( v _ j ) ) ^ 2 \le 0 . \end{align*}


\begin{align*} \varphi ( a ^ * J b ^ * J T J b J a ) \leq \| T \| \varphi ( a ^ * J b ^ * J e _ A J b J a ) = 0 \end{align*}


\begin{align*} \left ( I + E ( g ) \right ) ^ { - 1 } = \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ k \left ( E ( g ) \right ) ^ { k } \left \| \left ( I + E ( g ) \right ) ^ { - 1 } \right \| _ { \mathcal B _ 3 } \le \frac { 1 } { 1 - \left \| E ( g ) \right \| _ { \mathcal B _ 3 } } . \end{align*}


\begin{align*} & { \bf T } _ { w _ 0 ^ W } P ( \mu ) = T ^ r ( \mu ' ) , \end{align*}


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


\begin{align*} B _ P ( v , w ) \vcentcolon = \langle ( - \Delta ) ^ { s / 2 } v , ( - \Delta ) ^ { s / 2 } w \rangle + \sum _ { | \alpha | \leq m } \langle a _ \alpha , ( D ^ \alpha v ) w \rangle \end{align*}


\begin{align*} U : = \sqrt { \frac { 2 L \ \Delta } { \sum \limits _ { i = 0 } ^ { N - 1 } h _ i \big ( 2 - h _ i \frac { - \kappa } { 1 - \kappa } \big ) } } . \end{align*}


\begin{align*} P ^ { \mathrm { M F } } \left ( \eta \right ) = \inf _ { c \in \mathbb { C } } \left \{ \eta \left \vert c \right \vert ^ { 2 } + P \left ( c , \eta \right ) \right \} \ , \eta \in \mathbb { R } _ { 0 } ^ { + } \ , \end{align*}


\begin{align*} \upsilon _ \pm ( \eta ) = \left \{ \begin{aligned} & 1 , { } \pm \eta \geq 0 , \\ & 0 , { } \pm \eta < 0 , \end{aligned} \right . \end{align*}


\begin{align*} \Psi _ N = \frac { W _ { N _ 0 } T _ \nu e ^ { A _ \nu } \Omega } { \| e ^ { A _ { \nu } } \Omega \| ^ 2 } . \end{align*}


\begin{align*} C _ { u , m } = \prod _ { i = 1 } ^ m \alpha _ i ^ { r ( u + 1 ) + r ^ 2 n + { r \choose 2 } } \left ( \Psi _ { \boldsymbol { 1 } } \left ( Q _ u \right ) \right ) \enspace . \end{align*}


\begin{align*} \lim _ { \gamma \rightarrow 0 ^ { + } } \mathfrak { F } _ { \gamma } \left ( \Phi , f \right ) = 0 \ , \Phi \in \mathcal { W } _ { 0 } \subseteq \mathcal { W } _ { 1 } \ , \end{align*}


\begin{align*} C _ { R } ( J ) R [ x ] = C _ { R [ x ] } ( J ' ) \textrm { a n d } C _ { R [ x ] } ( J ' ) \cap R = C _ R ( J ) . \end{align*}


\begin{align*} \mathsf { F r } _ x ( B ) = \lim _ { n \to \infty } \frac 1 n \# \big \{ i \in \{ 1 , 2 , \dots , n \} : ( x _ i , x _ { i + 1 } , \dots , x _ { i + k - 1 } ) = B \big \} , \end{align*}


\begin{align*} q _ A = k \frac { \displaystyle \sum _ { m = 0 } ^ \infty [ p ^ \# _ m ( o , o ) - \pi ( o ) ] } { \displaystyle \sum _ { m = 0 } ^ \infty [ \tilde p ^ \# _ m ( A , A ) - \pi ( A ) ] } \end{align*}


\begin{align*} S ( \delta ) = o ( 1 ) \end{align*}


\begin{align*} ( 1 + 4 \sqrt { c _ 1 } ) ^ \rho \cdot r N M = ( 1 + 4 \sqrt { c _ 1 } ) ^ \rho \cdot r \cdot c ( r - 1 , g , d ^ { 2 g } ) ^ { \rho + 1 } \cdot M ( c _ 1 ) \leq c _ 3 ^ { 1 + \rho } \end{align*}


\begin{align*} \{ . , . \} \ , = \ , \beta _ * \circ C ^ \infty ( U , [ . , . ] ) \circ ( D \times D ) \end{align*}


\begin{align*} \Lambda _ { \sigma } = ( \kappa _ 1 , \kappa _ 2 , \ldots , \kappa _ { m } ) \end{align*}


\begin{align*} \dim ( \ker \bar { \partial } \subset W ^ { 1 , 2 ; \mu } ) = \deg q - \sum _ 1 ^ k \deg p _ i + \frac { n \mu } { \pi } . \end{align*}


\begin{align*} \int _ { B _ { 1 / 2 } } Q ( 0 ) \lambda ( U _ 0 ) \ , d x & \le \int _ { B _ { 1 / 2 } } Q ( 0 ) \liminf _ { k \to \infty } \lambda ( U _ k ) \ , d x \le \liminf _ { k \to \infty } \int _ { B _ { 1 / 2 } } Q ( 0 ) \lambda ( U _ k ) \ , d x . \\ \end{align*}


\begin{align*} \check f ( x ) = ( 2 \pi ) ^ { - n } \int _ { \R ^ n } f ( \xi ) e ^ { i x \xi } \ , d \xi , x \in \R ^ n . \end{align*}


\begin{align*} \mathfrak { M } = \mathfrak { M } _ { 1 } \times \mathfrak { M } _ { 2 } = \{ ( \alpha _ { 1 } , \alpha _ { 2 } ) : \alpha _ { 1 } \in \mathfrak { M } _ { 1 } , \alpha _ { 2 } \in \mathfrak { M } _ { 2 } \} , \ \mathfrak { m } = \left [ \frac { 1 } { Q _ { i } } , 1 + \frac { 1 } { Q _ { i } } \right ] ^ 2 \setminus \mathfrak { M } . \end{align*}


\begin{align*} ( - \Delta ) ^ { \sigma } _ { p } u _ { j } ( x ) = K _ { j } ( x ) u _ { j } ^ { q ( p - 1 ) } ( x ) , \ ; \ , x \in \mathbb { R } ^ { n } , \quad | x | ^ { s } u _ { j } ( x ) \rightarrow 1 , \quad , \end{align*}


\begin{align*} E ( \mathbf { R } _ n ) _ 3 = \varepsilon n \left [ \frac { n - 1 } { 2 } a _ 3 + b _ 3 \right ] . \end{align*}


\begin{align*} v _ r \left ( a _ s g ^ s \right ) = v _ r ( f ) < v _ r ( f - h ) \le v _ r \left ( ( a _ s - a ' _ s ) g ^ s \right ) . \end{align*}


\begin{align*} v _ { \mathrm { n c } } : = \sum _ { { \ell } = 1 } ^ { j } \xi _ { \ell } \phi _ { \mathrm { n c } } ( { \ell } ) , v _ { \mathrm { p w } } : = \sum _ { { \ell } = 1 } ^ { j } \xi _ { \ell } \phi _ { \mathrm { p w } } ( { \ell } ) , w _ { \mathrm { p w } } : = \sum _ { { \ell } = 1 } ^ { j } \xi _ { \ell } \lambda _ h ( \ell ) \phi _ { \mathrm { p w } } ( { \ell } ) . \end{align*}


\begin{align*} c _ \rho : = \inf _ { \gamma \in \Gamma } \ \max _ { t \in [ 0 , 1 ] } \Phi _ \rho ( \gamma ( t ) ) > \max \{ \Phi _ \rho ( w _ 1 ) , \Phi _ \rho ( w _ 2 ) \} , \rho \in I . \end{align*}


\begin{align*} f ( \alpha ) & = \sum _ { \{ B \in { H } : \alpha \in B \} } a _ B \\ & = \sum _ { \{ D _ B \in F _ 1 : \alpha \in D _ B \} } a _ B + \sum _ { B \in S _ \gamma } a _ B \\ & = \sum _ { \{ D _ B \in F _ 1 : \alpha \in D _ B \} } a _ B \end{align*}


\begin{align*} \psi _ { A _ { k } } = A _ { k } \left [ \frac { \cosh ( k x ) } { \cosh ( k a / 2 ) } - \frac { \cosh ( k ' x ) } { \cosh ( k ' a / 2 ) } \right ] \end{align*}


\begin{align*} F ^ { ( \iota ) } : = ( \hat F \chi _ \iota ) ^ \vee \iota \in \N , \end{align*}


\begin{align*} \frac { d f } { d t } = [ f , h ] _ { \star } \ , \end{align*}


\begin{align*} \lim _ { n \to \infty } \sup _ { V \in \mathbb { V } _ { n , k } } \rho _ \mathrm { L P } ( \mu _ { V X ^ { ( n ) } } , \tilde { \mu } _ { V X ^ { ( n ) } } ) = 0 . \end{align*}


\begin{align*} \Phi _ { n } ( \eta _ { \nu _ { n } } ( z ) ) = z \end{align*}


\begin{align*} \d u ( t ) = \left [ ( 1 + i \alpha ) \Delta u ( t ) - ( \gamma ( t ) + i \beta ) | u ( t ) | ^ { 2 } u ( t ) + f ( t , u ( t ) ) \right ] \d t + g ( t , u ( t ) ) \d W ( t ) , ~ t \in \R , \end{align*}


\begin{align*} - \Delta _ { g _ o } ^ { f _ o } \varphi _ i = \lambda _ i \varphi _ i , \int _ M \varphi _ i \varphi _ j d \nu _ * = \delta _ { i j } . \end{align*}


\begin{align*} { \Delta W _ h - W _ h + \sum _ { i = 1 } ^ 4 U _ { h , i } ^ p = 0 . } \end{align*}


\begin{align*} D _ { n , v } = \begin{cases} \displaystyle { \sum _ { w \in \Z [ \zeta ] ^ 3 + v \ ; : \ ; \langle w , w \rangle = n } w ^ \perp } & \\ [ 8 m m ] \displaystyle { \frac { 1 } { 2 } \sum _ { w \in \Z [ \zeta ] ^ 3 + v \ ; : \ ; \langle w , w \rangle = n } w ^ \perp } & \end{cases} \end{align*}


\begin{align*} \log n ! = n \log n - n \log e + o ( n ) \ ; . \ , \end{align*}


\begin{align*} f _ { n , u } ( \alpha , \beta , \underline { X } , \underline { Y } ) = f ( \alpha , \beta , \underline { X } , \underline { Y } ) & = \displaystyle c ( t _ { i , s } ) \prod _ { k \geq 3 } \prod _ { s { { = 0 } } } ^ { { r - 1 } } \prod _ { 1 \leq i , j \leq r } ( t _ { k , s } - X _ i ) ( t _ { k , s } - Y _ j ) \\ & \cdot \displaystyle \prod _ { s = { { 0 } } } ^ { { r - 1 } } ( X _ s Y _ s ) ^ u [ ( X _ s - \alpha ) ( X _ s - \beta ) ( Y _ s - \alpha ) ( Y _ s - \beta ) ] ^ { r n } \enspace , \end{align*}


\begin{align*} \L I _ m ^ * g , f \R = \L g , I _ m f \R \end{align*}


\begin{align*} 1 \geq \int \cosh _ 2 f \ d \mu \geq \frac 1 { ( 2 k ) ! } \int f ^ { 2 k } \ d \mu \end{align*}


\begin{align*} \begin{aligned} A _ 1 & : = \int _ { - \infty } ^ { \lambda } \frac { 1 } { \sqrt { 2 \pi } } \exp ( - g ^ 2 / 2 ) d g , \\ A _ 2 & : = \frac { \frac { 1 } { 2 \sqrt { 2 \pi } } \exp ( - \lambda ^ 2 / 2 ) } { \frac { 1 } { \sqrt { 2 \pi } } \exp ( - \lambda ^ 2 / 2 ) + \lambda \int _ { - \lambda } ^ { + \infty } \frac { 1 } { \sqrt { 2 \pi } } \exp ( - g ^ 2 / 2 ) d g } . \end{aligned} \end{align*}


\begin{align*} Z = f ( r ) \partial _ { \theta } . \end{align*}


\begin{align*} \widetilde { w } ( t ) = \chi _ { ( 0 , s ) } ( t ) t ^ \alpha . \end{align*}


\begin{align*} \frac { y _ { n } } { x _ { n } } = \prod _ { k \leq n } \left ( 1 - \frac { \chi ( p _ k ) } { p _ k ^ 2 } \right ) ^ { - 1 } = \prod _ { k \leq n } \frac { p _ k ^ 2 } { p _ k ^ 2 - \chi ( p _ k ) } . \end{align*}


\begin{align*} \beta = \frac { \chi } { 1 + \chi } , \gamma \leq \min \left \{ \frac { \tilde { \lambda } \chi ( 1 - \chi ) \xi } { 8 ( 1 + \chi ) } , \frac { 1 + \chi } { 2 ( 2 + \chi ) R _ 0 \| \nabla _ x M \| _ \infty } \right \} , \mbox { w i t h } \xi : = \begin{cases} \tilde C ^ { k - 2 } , & \mbox { i f } k < 2 , \\ 1 , & \mbox { i f } k = 2 , \\ \| \nabla _ x M \| _ \infty ^ { k - 2 } , & \mbox { i f } k > 2 . \end{cases} \end{align*}


\begin{align*} \mathrm { d } _ { \mathrm { T V } } ( X _ n , X _ \infty ) = \| p _ { X _ n } - p _ { X _ \infty } \| _ { L ^ 1 ( \R ) } . \end{align*}


\begin{align*} \Phi _ { \mathrm { t r i a l } , s } : = \frac { \big ( \mathbb { U } _ \mathrm { l e f t } \mathbb { U } _ \mathrm { r i g h t } \Omega \big ) _ { s } } { \sqrt { \sum _ { s = 0 } ^ N \left \| \big ( \mathbb { U } _ \mathrm { l e f t } \mathbb { U } _ \mathrm { r i g h t } \Omega \big ) _ { s } \right \| ^ 2 } } \in \mathcal { F } _ \perp . \end{align*}


\begin{align*} \Phi = ( \phi , \psi ) \in H _ 0 ^ 1 ( 0 , r ) \times H _ 0 ^ 1 ( 0 , r ) \triangleq X , ~ { \rm { a n d } } ~ F = ( f _ 1 , f _ 2 ) \in H ^ { - 1 } \times H ^ { - 1 } \triangleq X ^ { - 1 } . \end{align*}


\begin{align*} u = 0 y = 0 \omega _ c . \end{align*}


\begin{align*} \Omega _ { 2 \ell } = \{ L \ , | \ , \mbox { $ L $ i s a d i r e c t s u m o f s o m e o f $ C _ 1 , \cdots , C _ r $ , \ , $ \dim L = 2 \ell $ } \} ; \end{align*}


\begin{align*} S = L \otimes \O \left ( - \sum _ { i \in I _ 0 } [ z _ i ] \right ) , \end{align*}


\begin{align*} g ( M ) = \sum _ { 0 \leq 2 n \leq M - 1 } ( M - 2 n ) \binom { M - n - 1 } n . \end{align*}


\begin{align*} - \Delta _ p ( \Gamma + H _ \lambda ) + \Delta _ p \Gamma = \lambda G _ \lambda ^ { p - 1 } \Omega \setminus \{ x _ 0 \} \end{align*}


\begin{align*} \| \lambda _ j ^ { - 1 } \partial _ \nu u _ { \lambda _ j } \| _ { L ^ 2 ( \Gamma ) } = o ( 1 ) . \end{align*}


\begin{align*} \widetilde { G } ( x , y ) = \frac { G ( x , y ) } { g ( x ) g ( y ) } \end{align*}


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


\begin{align*} n ( \tau , v ) = \rho _ { } ( v - \tau ) , \end{align*}


\begin{align*} & \dot { x } ( t , s ) = \lambda x ( t , s ) + \partial _ s ^ 2 x ( t , s ) , s \in [ 0 , 1 ] , \\ & \dot { \hat x } ( t , s ) = \lambda \hat x ( t , s ) + \partial _ s ^ 2 \hat x ( t , s ) \\ & \qquad + \int _ 0 ^ 1 l ( s ) \left ( \partial _ s ^ 2 x ( t , \theta ) - \partial _ s ^ 2 \hat x ( t , \theta ) \right ) d \theta , \\ & x ( t , 0 ) = 0 , x ( t , 1 ) = 0 , \\ & \hat x ( t , 0 ) = 0 , \hat x ( t , 1 ) = 0 . \end{align*}


\begin{align*} & x _ i \frac { \partial } { \partial x _ i } : x ^ m \mapsto m _ i \cdot x ^ m , i = 1 , . . . , n , \\ & \frac { \partial e _ { - \alpha } ^ { \lambda } } { \partial \lambda } _ { \vert { \lambda = 0 } } : x ^ m \mapsto h t _ { - \alpha } ( m ) \cdot x ^ { m - \alpha } , \alpha \in R ( N , \Sigma _ { C ( \Delta ) } ) . \end{align*}


\begin{align*} t _ { i } ( n ) = \sum \limits _ { \omega N _ { i } < p _ { 1 } , p _ { 2 } \leq N _ { i } \atop p _ { 1 } - p _ { 2 } = n } \log p _ { 1 } \log p _ { 2 } . \end{align*}


\begin{align*} \mathrm { T r } _ \perp ( A ) : = \sum _ { m \ge 1 } \langle u _ m , A u _ m \rangle . \end{align*}


\begin{align*} A = \begin{pmatrix} 1 + ( f ' ( x ^ 1 ) ) ^ 2 & 0 \\ 0 & ( f ( x ^ 1 ) ) ^ 2 \\ \end{pmatrix} . \end{align*}


\begin{align*} - \Delta \int _ M \chi ( y ) G ( x , y ) d \mathrm { v o l } ( y ) = \chi ( x ) \geq \left \lbrace \begin{matrix} 1 & N _ { C + 1 } ( K ) , \\ 0 & M . \end{matrix} \right . \end{align*}


\begin{align*} \iota ( \mathbb { H } ^ 2 ) = \bigcup _ { z \in S ^ 1 } \iota ( [ - z , z ] ) . \end{align*}


\begin{align*} I _ 1 & = \int _ 0 ^ r \left [ \frac { z } { 2 } ( \bar \phi + e _ \delta - \delta ) ( \bar \phi ' , \bar \psi ' ) \cdot ( \phi , \psi ) \right ] { \rm { d } } z \\ & \leq \Vert ( \phi , \psi ) \Vert _ { L ^ 2 } \cdot \Vert ( \bar \phi ' , \bar \psi ' ) \Vert _ { L ^ 2 } \cdot \frac { 1 } { 2 } \left [ \Vert z \bar \phi \Vert _ { L ^ \infty } + \Vert z ( e _ \delta - \delta ) \Vert _ { L ^ \infty } \right ] , \end{align*}


\begin{align*} \frac { \sum _ { k = 1 } ^ N f ( S _ k \alpha ) - E ( f ) N } { \sqrt { N } } \overset { d } { \to } \mathcal { N } ( 0 , \sigma ^ 2 ) \end{align*}


\begin{align*} G ( x , u , p ) = \phi ^ * \circ Y ( x , u , p ) , \end{align*}


\begin{align*} \mathcal { E } = \{ R \in S l ( \mathbb { Z } , 3 ) : R C _ 1 \subset C _ 1 , \ R ^ { - 1 } C _ 2 \subset C _ 2 , \ R C _ 3 \subset C _ 3 \} , \end{align*}


\begin{align*} J ^ { e s s ; 1 } _ { k , m ' } ( t , x ) = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } e ^ { i x \cdot \xi + i ( t - s ) \omega \cdot \xi } \mathcal { F } [ ( E + \hat { v } \times B ) f ] \cdot \nabla _ v \big [ \frac { \varphi ^ { e s s } ( \omega , \xi ) } { i ( \omega + \hat { v } ) \cdot \xi } \varphi _ k ( \xi ) \varphi _ j ( v ) \big ] \end{align*}


\begin{align*} y _ { p , \lambda } ( x ) = e ^ { x } \int _ { 0 } ^ { x } \phi _ { p , \lambda } ( t ) d t . \end{align*}


\begin{align*} \Phi ( x ) = \sum _ { n = \lceil C \rceil } ^ \infty e ^ { - a n } \phi _ n - \int _ { M } \chi ( y ) G ( x , y ) d \mathrm { v o l } ( y ) . \end{align*}


\begin{align*} \lim _ { n \to + \infty } c _ n = 0 , \ \ \int _ { \mathbb { R } ^ N } \sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \frac { \partial U _ { h _ n , i } } { \partial h _ n } \left ( \psi _ n + c _ n \sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \frac { \partial U _ { h _ n , i } } { \partial h _ n } \right ) ~ d y = 0 . \end{align*}


\begin{align*} b _ { \pm } = \frac { 1 \pm | a | } { 4 } = \frac { 1 \pm \frac { \sqrt { 1 + \sqrt { 1 + \left ( \frac { 1 6 \beta m \lambda } { 3 } \right ) ^ { 2 } } } } { \sqrt { 2 } } } { 4 } , \end{align*}


\begin{align*} A \ , \ , = \ , \ , \begin{small} \begin{pmatrix} \lambda _ 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & \lambda _ 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & \lambda _ 3 & 1 \end{pmatrix} . \end{small} \end{align*}


\begin{align*} \cap _ { i = 1 } ^ { m } C _ { i } \subset \cap _ { n = 0 } ^ { \infty } \mathrm { F i x } ( S _ { n } ) \subset \cap _ { n = 0 } ^ { j _ { f } } \mathrm { F i x } ( S _ { n } ) . \end{align*}


\begin{align*} ( - \Delta ) ^ s u _ 1 + \sum _ { | \alpha | \leq m } a _ { 1 , \alpha } D ^ \alpha u _ 1 & = 0 \mbox { i n } \ ; \ ; \Omega , u _ 1 - f _ 1 \in \widetilde H ^ s ( \Omega ) \end{align*}


\begin{align*} \mathcal { X } \cap [ \emptyset , N ] = \bigcup _ { j \in J } [ B _ j , N ] ^ * , \end{align*}


\begin{align*} f _ { \lambda } ( \lambda _ i ) = \prod _ { j ( \neq i ) } ( \lambda _ i - \lambda _ j ) \neq 0 , \end{align*}


\begin{align*} \pi : E : = \mathbb P ( N _ { Z / X } ) \rightarrow Z . \end{align*}


\begin{align*} g ' ( z ) = \Phi ( z ) f ' ( z ) + \Phi ' ( z ) f ( z ) . \end{align*}


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


\begin{align*} \sum \limits _ { i = n } ^ \infty \alpha _ i \leq \sum \limits _ { i = n } ^ \infty r ^ n \alpha _ n = \frac { \alpha _ n } { 1 - r } . \end{align*}


\begin{align*} f _ { k } ( \lambda ) = ( \lambda - a _ k ) f _ { k - 1 } ( \lambda ) - b _ { k - 1 } ^ 2 f _ { k - 2 } ( \lambda ) , \ \ 2 \le k \le n , \end{align*}


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


\begin{align*} \det ( x I - B _ 1 ( T ) ) = \det ( x I - B _ 2 ( T ) ) & = \prod _ { v \in V ( T ) } \mathcal { F } ( v , x ) . \end{align*}


\begin{align*} & ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \ldots , y _ n ) = 0 . \end{align*}


\begin{align*} \abs { h } _ { Y \cup \mathcal { O } } \leq \abs { \iota ( h ) } \leq \lambda \abs { h } _ { X \cup \mathcal { P } } + \lambda c = \lambda \abs { g _ j ^ { - 1 } p g _ k } _ { X \cup \mathcal { P } } + \lambda c \leq \lambda ( \mu + c ) \end{align*}


\begin{align*} ( \partial _ { \eta _ 1 } - \partial _ { \eta _ 2 } ) e ^ { i t \Phi ( \eta _ 1 , \eta _ 2 , \xi ) } = ( p ' ( \eta _ 1 ) - p ' ( \eta _ 2 ) ) i t e ^ { i t \Phi ( \eta _ 1 , \eta _ 2 , \xi ) } , \end{align*}


\begin{align*} b _ q = \frac { \pi } { 4 } \frac { M _ q } { m _ q ^ 2 } \mu _ q \ > \ \frac { \log q } { \log ( 2 q ) } \quad \qquad q \le 2 3 . \end{align*}


\begin{align*} \frac { d x } { d t } = X ( t , x ) , x \in N , X \in \mathfrak { X } _ t ( N ) , \end{align*}


\begin{align*} X ^ 0 _ { ( 0 , 0 , 0 ) , 0 } \equiv 1 \end{align*}


\begin{align*} A _ { [ n , j ] } ^ { ( q ) } & : = \sum _ { \gamma \in \mathcal { D } _ { [ n , j ] } } w ( \gamma + q ) , \\ B _ { [ n , j ] } ^ { ( q ) } & : = \sum _ { \gamma \in \widehat { \mathcal { D } } _ { [ n , j ] } } w ( \gamma - q ) . \end{align*}


\begin{gather*} V = \bigoplus _ { i \in I } V _ i \end{gather*}


\begin{align*} 1 \leq a _ { i } \leq q _ { i } \leq P _ { i } , \ \ ( a _ { i } , q _ { i } ) = 1 , \end{align*}


\begin{align*} D ( A ^ { 3 } ) & = \left \{ x : = \left ( x _ k \right ) _ { k \in \N } \in D ( A ^ 2 ) \ , \middle | \ , A ^ 2 x \in D ( A ) \right \} \\ & = \left \{ x : = \left ( x _ k \right ) _ { k \in \N } \in c _ 0 \ , \middle | \ , \left ( w ^ { k } w ^ { k + 1 } w ^ { k + 2 } x _ { k + 3 } \right ) _ { k \in \N } \in c _ 0 \right \} \end{align*}


\begin{align*} ( \alpha , \beta ) : = \int _ X \alpha ^ \vee \cup \beta . \end{align*}


\begin{align*} \langle \Delta _ G v , v \rangle & \leq \langle \Delta _ { F _ { \leq k } } v , v \rangle \\ & = \langle P _ { F _ { \leq k - 1 } } \Delta _ X v , v \rangle + \langle P _ { F _ k } \Delta _ { F _ { \leq k } } P _ { F _ { k - 1 } \cup F _ k } v , v \rangle \\ & = \eta \langle P _ { F _ { \leq k - 1 } } v , v \rangle + \norm { P _ { F _ k } \Delta _ { F _ { \leq k } } } \cdot \norm { P _ { F _ { k - 1 } \cup F _ k } v } \cdot \norm v \\ & \leq 2 ^ \frac 3 2 d h ^ { - 1 } \eta ^ \frac 3 2 + 2 d \delta . \end{align*}


\begin{align*} \frac { \langle v _ 1 , v _ 2 \rangle } { | v _ 1 | | v _ 2 | } = \frac { | v _ 1 | ^ 2 } { | v _ 1 | | v _ 2 | } + \frac { \langle v _ 1 , v _ 2 - v _ 1 \rangle } { | v _ 1 | | v _ 2 | } \geq 1 - \frac { | v _ 1 | | v _ 2 - v _ 1 | } { | v _ 1 | | v _ 2 | } \geq 1 - \frac { \epsilon \cdot c } { c - \epsilon \cdot c } = \cos \theta . \end{align*}


\begin{align*} c ( 1 - m ) \int _ 0 ^ L \abs { \omega _ x } ^ 2 d x + \frac { c m d _ { \sigma } } { 2 } \int _ 0 ^ L \int _ 0 ^ { \infty } \sigma ( s ) \abs { \eta _ x } ^ 2 d s d x = - \Re \left ( \left < { \mathcal { A } } _ m U , U \right > _ { \mathcal { H } } \right ) \leq \| F \| _ { \mathcal { H } } \| U \| _ { \mathcal { H } } , \end{align*}


\begin{align*} \pi \mid _ { C _ 2 ^ n } = \left ( m _ 0 + \sum _ { i = 1 } ^ n m _ i \right ) \mathbb { 1 } \bigoplus _ { i = 1 } ^ n r _ { i } \rho _ i , \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { C _ N } { D _ N } = \lambda ( [ 1 , 1 ] ) . \end{align*}


\begin{align*} \sum _ { m = 1 } ^ \infty \prod _ { i = 1 } ^ m \mu ( [ 0 , A ( m ) ^ { { \rho } i } ] ) < \infty . \end{align*}


\begin{align*} & \int \frac { x ^ { n - 2 } } { ( - x ^ 2 ; q ^ 2 ) _ \infty } \widetilde { h } _ n ( x ; q ) d _ q x = \dfrac { x ^ n } { [ n - 1 ] _ q ( - x ^ 2 ; q ^ 2 ) _ \infty } \left ( \frac { \widetilde { h } _ n ( x ; q ) } { x } - \widetilde { h } _ { n - 1 } ( x ; q ) \right ) . \end{align*}


\begin{align*} G _ { ( t , \rho ) } \left ( \frac { \partial } { \partial t } , \frac { \partial } { \partial t } \right ) = 1 . \end{align*}


\begin{align*} \frac { ( k + n - j ) ! } { ( n - j ) ! } = \mathrm O ( n ^ k ) \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d t } \left ( \mu \int _ 0 ^ L \abs { y _ t } ^ 2 d x + \beta \int _ 0 ^ L \abs { y _ x } ^ 2 d x \right ) - \gamma \beta \Re \left ( \int _ 0 ^ L u _ x \overline { y _ { x t } } \right ) = 0 . \end{align*}


\begin{align*} \partial _ { \zeta _ i } f = g \partial _ { \zeta _ i } h + h \partial _ { \zeta _ i } g \end{align*}


\begin{align*} \left ( \bigcup _ { i = 1 } ^ { d } S _ { d , i } \right ) \cup \left ( \bigcup _ { i = 1 } ^ { e } T _ { e , i } \right ) = [ 1 , 4 k N ] , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , t ] } E [ | P ^ { n , 2 } _ \tau | ^ 2 ] = 0 . \end{align*}


\begin{align*} \omega ( \frac { R _ 0 } { S ^ j } ) \leq \theta ^ j [ \omega ( R _ 0 ) + \tau R _ 0 ^ \sigma \sum _ { k = 0 } ^ { j - 1 } ( \theta S ^ \sigma ) ^ { - k } ] . \end{align*}


\begin{gather*} ( f | _ { k } ( \gamma , \upsilon ) ) ( \tau ) : = f \left ( \frac { a \tau + b } { c \tau + d } \right ) \frac 1 { \upsilon ( \tau ) ^ { 2 k } } \end{gather*}


\begin{align*} - i \frac { 3 a ^ { 2 } } { 2 \beta \hslash ^ { 2 } } \lambda a _ { 2 } + a _ { 1 } = \lambda ^ { 2 } a _ { 1 } \ , \Rightarrow \ , a _ { 2 } = i \frac { 2 \beta \hslash ^ { 2 } } { 3 a ^ { 2 } } . \frac { \lambda ^ { 2 } - 1 } { \lambda } a _ { 1 } \end{align*}


\begin{align*} \Delta _ F ( X _ { n } ) = n , \Delta _ F ( \varphi _ { i , n } ) = | n | , \Delta _ F ( \varphi ^ * _ { i , n } ) = - | n | , \Delta _ F | _ { M _ \Delta } = - \Delta \end{align*}


\begin{align*} \alpha = \sum _ { i = 0 } ^ { n - 1 } a _ { i } \theta ^ { i } \in E \stackrel { \tau } { \longrightarrow } \overline { \alpha } = \begin{pmatrix} a _ 0 \\ a _ 1 \\ \vdots \\ a _ { n - 1 } \end{pmatrix} \in Q ^ { n } \end{align*}


\begin{align*} \limsup _ { t \downarrow 0 } \frac { c \eta _ t } { m _ t } \leq 0 { \bf 1 } _ { d = 2 } + c _ 0 { \bf 1 } _ { d = 3 } , \lim _ { t \downarrow 0 } \frac { \gamma _ t } { m _ t } = 0 . \end{align*}


\begin{align*} \omega ^ { - 1 } h _ { i j } = - \rho _ { ; i j } + \frac { 1 } { \rho } g _ { i j } - \frac { 1 } { \rho } \rho _ { i } \rho _ { j } . \end{align*}


\begin{align*} l ( h _ \rho ) = a _ 0 + a _ 1 h _ \rho + \ldots + a _ r h _ \rho ^ r . \end{align*}


\begin{align*} \Big | \bigotimes _ { i = 1 } ^ n x _ i - \bigotimes _ { i = 1 } ^ n \hat { x } _ i \Big | \leq \sum _ { k = 1 } ^ n \bigg ( \prod _ { i = 1 } ^ { k - 1 } | x _ i | \bigg ) \cdot \big | x _ k - \hat { x } _ k \big | \cdot \bigg ( \prod _ { i = k + 1 } ^ n \big | \hat { x } _ i \big | \bigg ) \end{align*}


\begin{align*} t \wedge s = \tau ( A \cap B ) , t \vee s = \tau ( A \cup B ) , \end{align*}


\begin{align*} \left \{ \begin{array} { c l } i \partial _ t u + \Delta u & = \pm | u | ^ 2 u , ( t , x ) \in \R \times \R ^ 2 , \\ u ( 0 ) & = u _ 0 \in M ^ s _ { 4 , 2 } ( \R ^ 2 ) + L ^ 2 ( \R ^ 2 ) \end{array} \right . \end{align*}


\begin{align*} \sum _ { \gamma \in \mathcal { S } _ { 1 } ( n + 1 , r ) } w ( \gamma ) & = \sum _ { \gamma \in \mathcal { S } _ { 1 } ( n , r - 1 ) } w ( \gamma ) + \sum _ { k = 0 } ^ { p } a _ { r + q } ^ { ( k ) } \Big ( \sum _ { \gamma \in \mathcal { S } _ { 1 } ( n , r + k ) } w ( \gamma ) \Big ) \\ & = \langle \mathcal { H } _ { q } ^ { n } e _ { r - 1 } , e _ { 0 } \rangle + \sum _ { k = 0 } ^ { p } a _ { r + q } ^ { ( k ) } \langle \mathcal { H } _ { q } ^ { n } e _ { r + k } , e _ { 0 } \rangle \\ & = \langle \mathcal { H } _ { q } ^ { n + 1 } e _ { r } , e _ { 0 } \rangle \end{align*}


\begin{align*} \Delta ( \Gamma ) : = \# Z ( H ^ { \Gamma } ) . \end{align*}


\begin{align*} f ^ \wedge \colon U \times E \to F \ , , f ^ \wedge ( x , v ) \ , : = \ , f ( x ) ( v ) \ , ? \end{align*}


\begin{align*} | \hat { h } ( y ) | \leq \begin{cases} C | y | ^ { 2 - \delta } , & \ \delta \neq n , \\ C | y | ^ { 2 - n } \ln | y | , & \ \delta = n . \end{cases} \end{align*}


\begin{align*} \bar { R } _ { \alpha \beta \gamma \delta } = 0 . \end{align*}


\begin{align*} \begin{aligned} \frac { d B ( t ) } { d t } & = g _ 0 + g _ 1 N ( t ) - B ( t ) , \\ \frac { d C ( t ) } { d t } & = 2 ( a _ 0 + a _ 1 N ( t ) - C ( t ) ) . \end{aligned} \end{align*}


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


\begin{align*} \frac { \partial h } { \partial r } - \frac { d } { d t } \frac { \partial h } { \partial \dot { r } } = 0 . \end{align*}


\begin{align*} s _ 1 = { \mathcal B } _ { - 1 } ( \sigma _ 1 ) = \begin{pmatrix} 1 & 0 \\ - 1 & 1 \end{pmatrix} \ \ \ \ s _ 2 = { \mathcal B } _ { - 1 } ( \sigma _ 2 ) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \end{align*}


\begin{align*} \tilde { \tau } _ { \mathcal { C } } ( x , y , t ) \ , = \ , \frac { 1 } { E _ 2 E _ 4 \cdots E _ { 2 g } } \biggl ( \displaystyle \sum _ { I } \displaystyle \prod _ { i < j \in I } ( \kappa _ i - \kappa _ j ) \displaystyle \prod _ { \{ i \ , : \ , c _ i = 1 \} } \lambda _ i \prod _ { i \in I } E _ i \biggr ) , \end{align*}


\begin{align*} \lim \limits _ { n \rightarrow \infty } \liminf _ { t \rightarrow \infty } \| \chi _ { B _ n } \psi _ t \| = \| \psi \| \end{align*}


\begin{align*} & \lim _ { n \rightarrow \infty } n ^ { 2 \alpha + 1 } \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' ( X _ s ) ) ^ 2 \Psi ^ { n , 3 } _ s d s \\ & = \kappa _ 4 \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' \sigma ) ^ 2 ( X _ s ) d s , \end{align*}


\begin{align*} \overline { \mathcal { G } } _ { n } ( u ) = \int _ { u } ^ { \infty } { G } _ { n } ( v ) \mathrm { d } v , \end{align*}


\begin{align*} \int _ { \Theta _ { d } } f ( \theta ) g ( \theta ) \mathrm { d } ( \mathrm { F } _ { \ast } \mathrm { P } ) \left ( \theta \right ) = \int _ { \Theta _ { d } } f ( \theta ) \mathrm { d } ( \mathrm { F } _ { \ast } \mathrm { P } ) \left ( \theta \right ) \int _ { \Theta _ { d } } g ( \theta ) \mathrm { d } ( \mathrm { F } _ { \ast } \mathrm { P } ) \left ( \theta \right ) \ . \end{align*}


\begin{align*} a _ i ( t , s ) = { \rm L o g } ( U _ i ( t , s ) + \kappa I ) \end{align*}


\begin{align*} \psi _ { B } = B \left [ \frac { \sinh ( k ' x ) } { \sinh ( k ' a / 2 ) } - \frac { x } { a / 2 } \frac { \cosh ( k ' x ) } { \cosh ( k ' a / 2 ) } \right ] , \end{align*}


\begin{align*} w _ 4 ( \pi ) = \begin{cases} \sum \limits _ { i = 1 } ^ 4 t _ i ^ 2 , & , \\ 0 , & . \end{cases} \end{align*}


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


\begin{align*} { \bf U } _ { { \rm H i P O D } } ^ k ( \alpha ^ * ) = [ u _ { { \rm P O D } , k , 1 } ^ { \alpha ^ * } , \ldots , u _ { { \rm P O D } , k , N _ h } ^ { \alpha ^ * } ] ^ T = \displaystyle \sum _ { j = 1 } ^ { L } T _ j ^ k ( \alpha ^ * ) { \boldsymbol \xi } _ j k = 1 , \ldots , m \end{align*}


\begin{align*} \dot { x } = A x + B u . \end{align*}


\begin{align*} \widetilde { T } _ { k , j ; n , l , r } ^ { T ; \mu , m , i } ( \mathfrak { m } , U ) ( t , x , \zeta ) = \sum _ { \tilde { j } \in [ j - 2 , j + 2 ] } \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \big [ i \mu \omega ^ { m ; U } _ { j , l , r } ( t - s , v , \omega ) K _ { k ; n } ^ { } ( \mathfrak { m } ) ( y , \zeta ) + c ^ { m , U } _ { j , l , r ; q } ( t - s , v , \omega ) K _ { k ; n } ^ { q } ( \mathfrak { m } ) ( y , \zeta ) \end{align*}


\begin{align*} S _ { k } ( N , \alpha ) = \sum \limits _ { p \sim N } ( \log p ) e ( p ^ { k } \alpha ) \end{align*}


\begin{align*} \frac { 1 } { e ^ { \omega _ 2 s } - 1 } = \sum _ { k \geq 0 } \frac { B _ k \cdot ( \omega _ 2 s ) ^ { k - 1 } } { k ! } , \end{align*}


\begin{align*} \pi _ { v , w } = \langle M _ v ^ t \cdot \alpha _ w , M _ w ^ t \cdot \alpha _ v \rangle \subset V _ { n + m } . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d } } | V ( \mathbf { x } ) \mathrm { e } ^ { - H _ { 0 } t } f ( \mathbf { x } ) | ^ { 2 } \ , \textnormal { d } \mathbf { x } = \int _ { \mathbb { R } ^ { d } } | V ( \mathbf { x } ) | ^ { 2 } \bigg | \int _ { \mathbb { R } ^ { d } } K _ { t } ( \mathbf { x } , \mathbf { y } ) f ( \mathbf { y } ) \ , \textnormal { d } \mathbf { y } \bigg | ^ { 2 } \ , \textnormal { d } \mathbf { x } . \end{align*}


\begin{align*} \Delta _ 1 ( k , m ) = z \Delta _ 1 ( k , m - 1 ) + a _ { m } \Delta _ 2 ( k , m - 1 ) + a _ { m } ^ 2 \Delta _ 1 ( k , m - 2 ) . \end{align*}


\begin{align*} { } ^ 1 _ 3 L a s t E l l ^ i ( t _ 1 , t _ 2 ) = \int _ { t _ 1 } ^ { t _ 2 } \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \R ^ 3 } e ^ { i X ( s ) \cdot ( \xi + \eta ) + i s \mu _ 1 | \eta | - i s \hat { v } \cdot \xi } K ( s , X ( s ) , V ( s ) ) \cdot \nabla _ \zeta \big [ i ( \hat { \zeta } \cdot ( \xi + \eta ) + \mu _ 1 | \eta | - \hat { v } \cdot \xi ) ^ { - 1 } \end{align*}


\begin{align*} y ^ * _ i = \left ( ( i - n + m ) L + U / m - L ( m + 1 ) / 2 \right ) _ + , \end{align*}


\begin{align*} \sigma _ N ^ 2 : = \begin{cases} \sqrt { \mu _ - - \mu _ + } \ , N \qquad & \delta > 2 \\ C \qquad & . \end{cases} \end{align*}


\begin{align*} \times \varphi _ k ( \xi ) d { \omega } d v d s \big | \lesssim \sup _ { s \in [ 0 , t ] } \sup _ { g \in L ^ 2 , \| g \| _ { L ^ 2 } = 1 } 2 ^ { 2 m + k + l } 2 ^ { 2 \tilde { c } ( m , k , l ) } \int _ { \R ^ 3 } \int _ { \R ^ 3 } \big | \hat { f } ( s , \xi , v ) \big | \varphi _ k ( \xi ) \psi _ { [ l - 2 , l + 2 ] } ( \tilde { v } \times \tilde { \xi } ) \big | \overline { \hat { g } ( \xi ) } \big | \varphi _ j ( v ) d v d \xi \end{align*}


\begin{align*} ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \ldots , y _ i ) = 0 . \end{align*}


\begin{align*} H ( \rho ) = \rho \int _ 1 ^ { \rho } \frac { p ( s ) } { s ^ 2 } d s , \end{align*}


\begin{align*} \log L & \geq n \log \frac { A } { \sqrt { a } } - \frac { 1 } { 4 } ( 1 + b ) \ , n \log n + \frac { 1 } { 2 } n \log n - n \log e + o ( n ) \\ & = \left ( \frac { 1 - b } { 4 } \right ) \ , n \log n + n ( \log \frac { A } { e \sqrt { a } } ) + o ( n ) \ ; , \ , \end{align*}


\begin{align*} h = f + \lambda g , \end{align*}


\begin{align*} G ( o , o ) = \int _ 0 ^ { \infty } h _ t ( o , o ) d t \lesssim _ d 1 + \frac { 1 } { R } + \frac { \log ( D / R ) } { R } + \frac { D ^ 2 } { n } \lesssim _ { d , C } \frac { \log n } { R } = \frac { 1 } { n } D ^ 2 \log n , \end{align*}


\begin{align*} \xi _ { \mathcal { I } } \left ( z \right ) \doteq \left ( y _ { 1 } , \dots , y _ { d } \right ) , z = \left ( z _ { 1 } , \dots , z _ { n } \right ) \in \mathbb { Z } ^ { n } \ , \end{align*}


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


\begin{align*} Q ^ * Q R x = Q ^ * Q R Q ^ * Q x = R x - R ^ { - 1 / 2 } p ( R ^ { - 1 / 2 } p ) ^ * x = R x ; \end{align*}


\begin{align*} \mathcal { F } _ = \{ f \in \mathcal { F } : f \circ \gamma = \gamma \circ f \gamma \in ( \Omega ) \} . \end{align*}


\begin{align*} u ( 0 , t ) = u ( L , t ) = y ( 0 , t ) = y ( L , t ) = 0 , \ t \in ( 0 , \infty ) , \end{align*}


\begin{align*} & \sum _ { n = 0 } ^ { \infty } q ^ { \frac { n ( n - 3 ) } { 2 } } x ^ n \left ( 1 - q + q ^ { n + 2 } x \right ) A _ q ( q ^ n x ) = ( 1 - q ) A _ q ( \frac { x } { q } ) - x A _ q ( q x ) , \end{align*}


\begin{align*} \sum _ { k = \ell } ^ { \ell + m } \delta ^ 2 ( \mathcal { T } _ k , \mathcal { T } _ { k + 1 } ) \leq \Lambda _ { \mathrm { q o } } \eta ^ 2 _ \ell \quad \ell , m \in \mathbb { N } _ 0 . \end{align*}


\begin{align*} \max _ { 1 \le i \le n } \bigg | \sum _ { k = 1 } ^ i \widehat \eta _ { n , k } \bigg | = o _ P ( n ^ { - 1 / 2 + p / 4 } ) \quad \mbox { a s \ } n \to \infty \end{align*}


\begin{align*} \Phi _ K ( u ) ( \omega ) = \rho _ { u ^ { - 1 } } ( \omega ) \end{align*}


\begin{align*} f _ x | _ { U \cap V _ x \cap V _ y } \ ; = \ ; f | _ { U \cap V _ x \cap V _ y } \ ; = \ ; f _ y | _ { U \cap V _ x \cap V _ y } \ , , \end{align*}


\begin{align*} q = q _ s + q _ 0 , q _ s \in M _ \Omega ( H ^ s \to H ^ { - s } ) , q _ 0 \in M _ { \geq 0 } ( H ^ s \to H ^ { - s } ) . \end{align*}


\begin{align*} 0 = \sum _ { x \in \Gamma } G ( o , x ) & = \sum _ { \underset { G ( o , x ) \geq 0 } { x \in \Gamma } } G ( o , x ) + \sum _ { \underset { 0 > G ( o , x ) > - \tfrac { C _ 1 C _ 3 } { 2 } \tfrac { D ^ 2 } { n } } { x \in \Gamma } } G ( o , x ) + \sum _ { \underset { - \tfrac { C _ 1 C _ 3 } { 2 } \tfrac { D ^ 2 } { n } \geq G ( o , x ) } { x \in \Gamma } } G ( o , x ) \\ & = : A _ 1 + A _ 2 + A _ 3 . \end{align*}


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


\begin{align*} P ( E _ { x \rightarrow y } ) = \sum _ { i = 1 } ^ { m i n ( m _ x , k _ x \gamma _ x ) } P ( E _ { x \rightarrow y } | S _ i ) P ( S _ i ) , \end{align*}


\begin{align*} f _ { \sigma } ( x ) = \sum _ { k = 0 } ^ n \sigma ( c _ k ) x ^ k \in K [ x ] . \end{align*}


\begin{align*} \varphi _ { \mu } ( z ) = H _ { \mu } ( z ) - z \end{align*}


\begin{align*} x = \frac { \epsilon _ 1 ( x , \beta ) } { \beta } + \frac { \epsilon _ 2 ( x , \beta ) } { \beta ^ { 2 } } + \cdots + \frac { \epsilon _ n ( x , \beta ) } { \beta ^ n } + \cdots , \end{align*}


\begin{align*} T _ m \triangleq \frac { 1 } { \bar { N } \sigma ^ 2 } \sum _ { i = 0 } ^ { \bar { N } - 1 } | y _ m [ i ] | ^ 2 , m = 1 , 2 , \ldots , M . \end{align*}


\begin{align*} f ( A ) = \sum _ { a \in A } \frac { 1 } { a \log a } \ \le \ \frac { e ^ \gamma } { m _ q } \frac { 1 } { 1 + v } \sum _ { a \in A } { \rm d } ( { \rm L } _ a ) = \frac { e ^ \gamma } { m _ q } \frac { { \rm d } ( { \rm L } _ A ) } { 1 + v } . \end{align*}


\begin{align*} \ 1 _ { \{ \sigma ^ r _ \infty = \infty \} } = \ 1 _ { \{ \theta _ \infty = \infty \} } . \end{align*}


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


\begin{align*} \psi ( x ) = { } & x \left ( 1 + \sum _ { n = 1 } ^ { \infty } f _ { x , n } ( x ^ a y ^ b ) ^ n \right ) = x f _ x \\ \psi ( y ) = { } & y \left ( 1 + \sum _ { n = 1 } ^ { \infty } f _ { y , n } ( x ^ a y ^ b ) ^ n \right ) = y f _ y \end{align*}


\begin{align*} \frac { d P } { d t } = P B R ^ { - 1 } B ^ \textsf { T } P - P A - A ^ \textsf { T } P - Q = P ^ 2 + 4 P - 1 , \end{align*}


\begin{align*} I _ 1 \leq \ & - \rho \lambda e ^ { \lambda u } \Bigl ( \lambda - 2 c \Bigr ) | \nabla u | _ g ^ 2 \varphi v - 3 \left ( \frac { f ' ( u ) } { f ( u ) } \right ) ^ 2 | \nabla u | _ g ^ 2 \varphi v \\ & + 2 c \left | \frac { f ' ( u ) } { f ( u ) } \right | \varepsilon ^ { - 1 } \varphi v ^ 3 + \frac { \varepsilon ' } { ( 1 - \varepsilon ' ) } \rho ^ 2 \lambda ^ 2 e ^ { 2 \lambda u } | \nabla u | ^ 2 _ g \varphi v . \end{align*}


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


\begin{align*} H _ { \psi , r } = \frac { k - 1 } { r } + \delta ' ( r ) , \end{align*}


\begin{align*} \frac { d g } { d t } = \sum _ { \alpha = 1 } ^ { r } b _ \alpha ( t ) X _ \alpha ^ \mathrm { R } ( g ) = \widehat { X } ^ G _ R ( t , g ) , \ , \end{align*}


\begin{align*} \tilde { h } '' - \dfrac { \tilde { h } ^ { 2 } - 1 } { r ^ { 2 } } \tilde { h } = \left ( g '^ { 2 } \sigma ^ { 2 } - B ^ { 2 } \right ) \tilde { h } \end{align*}


\begin{align*} f \circ S ^ { ( 1 - i ) k } ( A x + e ) = f ( A S ^ { ( 1 - i ) k } x + e ) \quad g \circ S ^ { ( 1 - i ) k } ( x ) + d ' , \end{align*}


\begin{align*} \begin{cases} \frac { d } { d t } X ( t , x _ 0 ) = - \vec { b } ( X ( t , x _ 0 ) , t ) ( 0 , \infty ) \\ X ( 0 , x _ 0 ) = x _ 0 \end{cases} . \end{align*}


\begin{align*} D _ q u ( x ) + u ( x ) u ( q x ) = 0 . \end{align*}


\begin{align*} 2 K _ X & \equiv f ^ * \left ( 2 E + \sum \limits _ { i = 1 } ^ { n } { \left ( F _ { i } + F _ { i } ^ { ' } \right ) } \right ) . \end{align*}


\begin{align*} \int \frac { \widetilde { h } _ n ( x ; q ) } { ( - x ^ 2 ; q ^ 2 ) _ \infty } d _ q x = - \dfrac { q ^ { 1 - n } ( 1 - q ) } { ( - x ^ 2 ; q ^ 2 ) _ \infty } \widetilde { h } _ { n - 1 } ( x ; q ) . \end{align*}


\begin{align*} y t Z ^ { \prime \prime } - ( y + 1 ) Z ^ \prime = 0 \end{align*}


\begin{align*} \{ x \in M \mid \| X _ { 2 } \| = 0 \} \end{align*}


\begin{align*} \P ( Z \ge 1 2 \rho L _ { } \tau ) & \le e ^ { - 1 2 h \rho L _ { } \tau } 4 ^ { \tau } = \big ( 4 / e ^ { 3 } \big ) ^ { \tau } \leq e ^ { - \tau } . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \frac { \partial } { \partial t } \omega = - \mathrm { R i c } ^ { T } ( \omega ) \mathrm { w i t h } \omega ( T _ { 0 } ) = \omega ^ { \prime } , \end{array} \end{align*}


\begin{align*} \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } \ln ( \vert \varphi ' _ { 0 , 1 } ( x _ k ) \vert ) \to \ln ( 2 ) \end{align*}


\begin{align*} f \mapsto \sum _ { n = 0 } ^ { \infty } { v ^ n ( f ) \over n ! } , \end{align*}


\begin{align*} \cap _ { i = 1 } ^ { m } C _ { i } \subset \cap _ { n = 0 } ^ { \infty } \mathrm { F i x } ( S _ { n } ) \subset \cap _ { n = 0 } ^ { j _ { f } } \mathrm { F i x } ( S _ { n } ) , \end{align*}


\begin{align*} \mathbb { E } ( \overline { S } _ n ) = \frac { 1 } { n } \sum _ { 1 \leq i \leq n } f _ k ( i ) + O _ \alpha ( n ^ { - 1 / 2 + \varepsilon } ) \end{align*}


\begin{align*} k = \frac { 1 } { 2 \hslash \sqrt { \beta / 3 } } \sqrt { \frac { \sqrt { \frac { 1 6 } { 3 } m \beta | E | } - 1 } { 2 } } \end{align*}


\begin{align*} Q _ { 3 , \tilde { q } , \tilde { p } } = \sigma _ 2 ^ { \lambda ( 1 ) } \sigma _ 1 ^ { \lambda ( 2 ) } \sigma _ 2 ^ { \lambda ( 3 ) } \sigma _ 1 ^ { \lambda ( 4 ) } . . . \sigma _ 2 ^ { \lambda ( \tilde { q } - 2 ) } \sigma _ 1 ^ { \lambda ( \tilde { q } - 1 ) } \end{align*}


\begin{align*} d x \wedge d y = \psi ^ * ( d x \wedge d y ) = ( f ^ { b - a } + ( a - b ) x ^ a y ^ b f ' f ^ { b - a - 1 } ) d x \wedge d y . \end{align*}


\begin{align*} \mathrm { ( \widetilde { I } ) } ~ \mathrm { M T E } _ { \boldsymbol { q } } ( \mathbf { Y } ) & = \boldsymbol { \mu } + \frac { \mathbf { \Sigma } ^ { \frac { 1 } { 2 } } \boldsymbol { \delta } } { \overline { F } _ { \mathbf { Z } } ( \boldsymbol { \xi _ { q } } ) } , \end{align*}


\begin{align*} \sum \limits _ { r = 0 } ^ { m - 2 } \sum \limits _ { k = 0 } ^ { m - 1 } \prod \limits _ { i = 0 } ^ { r } \rho _ { i + k } = \sum \limits _ { r = 0 } ^ { m - 2 } \sum \limits _ { k = 0 } ^ { m - 1 } \prod \limits _ { i = 0 } ^ { r } \rho _ { \sigma ( i + k ) } \end{align*}


\begin{align*} A = I _ { r + 1 } - \left [ \begin{matrix} 0 & & & & & \\ - 1 & \delta & & & & \\ & - 1 & 2 \delta & & & \\ & & \ddots & \ddots & & \\ & & & - 1 & ( r - 1 ) \delta & \\ & & & & 0 & r \delta \\ \end{matrix} \right ] , \Delta A = \left [ \begin{matrix} 0 & \cdots & \cdots & \cdots & 0 \\ \vdots & \cdots & \cdots & \cdots & \vdots \\ 0 & \cdots & 0 & \epsilon & 0 \end{matrix} \right ] . \end{align*}


\begin{align*} B ( N , \tilde { q } , \tilde { p } ) = Q _ { N , \tilde { q } , \tilde { p } } \sigma _ 2 ^ { \epsilon ( 2 ) } Q _ { N , \tilde { q } , \tilde { p } } ^ { - 1 } \sigma _ 1 ^ { \epsilon ( 1 ) } \end{align*}


\begin{align*} \max _ { \sum _ { k = 1 } ^ { r - 1 } N _ { k } + 1 \le j \le \sum _ { k = 1 } ^ { r } N _ { j } } M _ j \lesssim \sigma ^ { - \lfloor d / 2 \rfloor } \exp \left ( \frac { ( p - 1 ) r ^ 2 \delta ^ 2 } { 2 \sigma ^ 2 ( 1 - \eta ) } \right ) \end{align*}


\begin{align*} D ( \partial _ t ) \psi ( x , t ) : = \int _ 0 ^ t \int _ \Gamma \left [ \partial _ { \nu _ y } k ( | x - y | , t - \tau ) \right ] \psi ( y , \tau ) d \Gamma _ y d \tau x \in \R ^ d \setminus \Gamma , \end{align*}


\begin{align*} \kappa ( c _ 1 ( \chi ^ j \circ \det \mid _ M ) ) & = \kappa ( ( q + 1 ) j u ) \\ & = ( q + 1 ) j \kappa ( u ) \\ & = 0 . \end{align*}


\begin{align*} - u '' + \lambda u = | u | ^ { p - 2 } u \end{align*}


\begin{align*} & \partial _ t \langle \tilde { B } : \phi ( t ) \rangle _ t = \pm \langle \phi ( t ) , C ^ * C \phi ( t ) \rangle + g ( t ) \\ & g ( t ) \in L ^ 1 ( d t ) , C ^ * C \geq 0 . \end{align*}


\begin{align*} J _ v ( 0 , g ) p ( t , 0 , g ) = J _ v ( V _ F , g ) p ( t , V _ F , g ) , g > g _ F , t > 0 . \end{align*}


\begin{align*} \frac { d \tau ( t , x ) } { d t } = \frac { \partial \tau ( t , x ) } { \partial x } g ( t , x ) ( u - \mu ( t , x ) ) \ , . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \Re ( \lambda _ { n } ) = \infty . \end{align*}


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


\begin{align*} & \limsup \limits _ { t \rightarrow \infty } \| P _ \delta ( W _ { n , m ; \textrm { f a r } } ) e ^ { - i t H } \psi - P _ \delta ( W _ { n , m ; \textrm { f a r } } ) e ^ { - i t H _ 0 } \varphi \| \\ & \leq \limsup \limits _ { t \rightarrow \infty } \| P _ \delta ( W _ { n , m ; \textrm { f a r } } ) \| _ { \textrm { o p } } \| e ^ { - i t H } \psi - e ^ { - i t H _ 0 } \varphi \| = 0 \end{align*}


\begin{align*} \left ( \frac { p + 1 } { 2 } \right ) ^ 2 \sum _ { \substack { k = k _ 1 - k _ 2 + \cdots - k _ { p - 1 } + k _ { 2 p - 1 } \\ | k _ 1 | \gg k _ 2 ^ * \\ | \Phi ( k _ 1 - k _ 2 + \cdots + k _ p , k _ 1 \cdots , k _ p ) | \gtrsim | k _ 1 | } } \frac { \widehat { ( W _ t u _ 0 ) } _ { k _ 1 } } { \Phi ( k _ 1 - k _ 2 + \cdots + k _ p , k _ 1 \cdots , k _ p ) } \prod _ { \substack { j = 3 \\ o d d } } ^ { 2 p - 1 } \widehat { v } _ { k _ i } \prod _ { \substack { l = 2 \\ e v e n } } ^ { 2 p - 1 } \overline { \widehat { v } _ { k _ l } } , \end{align*}


\begin{align*} \begin{cases} f ^ k \to f F ^ s _ { p _ 1 , q } , \ , & \| f ^ k \| _ { L ^ { \infty } ( \R ^ n ) } \leq C \\ g ^ k \to g F ^ s _ { p _ 1 , q } , \ , & \| g ^ k \| _ { L ^ { \infty } ( \R ^ n ) } \leq C \end{cases} \end{align*}


\begin{align*} K ( x _ 1 ; x _ 2 ) : = v ( x _ 1 - x _ 2 ) \varphi ( x _ 1 ) \varphi ( x _ 2 ) \ , , \end{align*}


\begin{align*} \alpha = \alpha _ 1 + \gamma ^ 2 \beta . \end{align*}


\begin{gather*} A \otimes D : = A _ F , \end{gather*}


\begin{align*} \begin{cases} \varphi ( t ) = e ^ { \eta t } & \alpha = 1 - \delta - \beta , \\ \varphi ( t ) = ( 1 + ( 1 - \beta - \delta - \alpha ) \eta t ) ^ { \frac { 1 } { 1 - \beta - \delta - \alpha } } & \alpha \not = 1 - \delta - \beta , \end{cases} \end{align*}


\begin{align*} \langle M ^ { ( n ) } \rangle _ \tau = n ^ { 2 \alpha + 1 } \int _ 0 ^ { \tau } \left | \sum _ { i = 1 } ^ Q \rho _ i \mathbf { 1 } _ { [ 0 , t _ i ] } ( \tau ) ( t _ i - s ) ^ { \alpha } \sigma ' ( X _ s ) \Theta ^ n _ s \right | ^ 2 d s , \end{align*}


\begin{align*} \lim \limits _ { t \rightarrow \infty } \| \chi _ { S _ { 2 v t } } e ^ { - i t H } \psi \| = \| \psi \| \end{align*}


\begin{align*} \nu _ \sigma ( f ) = \nu ( l ( h _ \sigma ) ) \mbox { f o r e v e r y } \sigma , \rho < \sigma < \lambda . \end{align*}


\begin{align*} \lambda = \tfrac 1 4 + r ^ 2 r \geq 0 . \end{align*}


\begin{align*} \bar \nabla _ Z Y = \nabla _ Z Y - h ( Z , Y ) \nu , \end{align*}


\begin{align*} \kappa ^ { ( X + Y ) } ( z ) = \kappa ^ { ( X ) } ( z ) + \kappa ^ { ( Y ) } ( z ) . \end{align*}


\begin{align*} \theta ( \lambda ) = ( c ^ * ( \theta ) ) ( P _ 0 ) \ \ \ \textup { f o r } \ \ c \in R ^ n , \ \ \lambda = c ( P _ 0 ) , \ \theta \in R _ { \pi } . \end{align*}


\begin{align*} T _ { \lambda } : = 3 \lambda \big [ S ( 0 ) - C ( 0 ) + C ( 0 ) - C _ \infty ( 0 ) \big ] S \star C = 3 \lambda \big [ S ( 0 ) - C ( 0 ) - \eta _ t \big ] S \star C , \end{align*}


\begin{gather*} \phi ^ W = \sum _ { i , \chi } m _ { i , \chi } \varphi _ i \otimes \chi \end{gather*}


\begin{align*} \begin{array} { l l } \nabla _ { e _ 1 } e _ 1 = \nabla _ { X } X = e ^ { - 2 x _ 2 } e _ 2 , \\ \nabla _ { e _ 2 } e _ 2 = \nabla _ { U } U = 0 , \\ \nabla _ { e _ 1 } e _ 2 = \nabla _ { X } U = - e _ 1 , \\ \nabla _ { e _ 2 } e _ 1 = \nabla _ { U } X = - e _ 1 . \end{array} \end{align*}


\begin{align*} I _ Q ( f ) : = \sum W ( U , f ) \int _ U Q \omega . \end{align*}


\begin{align*} \psi ( r ) = C r ^ { 2 } + O \left ( r ^ { 2 + 2 \alpha } \right ) \ , ( r \rightarrow 0 ^ { + } ) . \end{align*}


\begin{align*} d \doteq d _ { - } + \mathrm { r } _ { + } ( d _ { - } ) = e _ { ( \cdot ) } ( \hat { \omega } ) \ . \end{align*}


\begin{align*} B _ x = Z _ x Y _ x , \end{align*}


\begin{align*} \dot x ( t ) = A ( t ) x ( t ) t \geq 0 , \end{align*}


\begin{align*} q _ { n } ( z ) \ , \phi _ { k } ( z ) - q _ { n , k + 1 } ( z ) = O ( z ^ { - n _ { k } - 1 } ) , z \rightarrow \infty , 0 \leq k \leq p - 1 , \end{align*}


\begin{align*} g _ 1 ( x ) : = n ! g ( x ) = n ! \displaystyle \sum _ { j = 0 } ^ { n } \binom { n + s - j } { n - j } \frac { x ^ j } { j ! } \end{align*}


\begin{align*} k a = a _ 1 k _ 1 \end{align*}


\begin{align*} ^ j ( f ) : = \sum _ { j \le i } ( - 1 ) ^ { b _ i } \varepsilon _ { f ( i ) } \in \texttt { P } , ( f ) : = ^ { 1 } ( f ) \in \texttt { P } . \end{align*}


\begin{align*} v = \frac { f ( u ) } { \sqrt { f ( u ) ^ 2 - | \widetilde { \nabla } u | ^ 2 } } . \end{align*}


\begin{align*} \left ( \pi _ z ( x , v ) s \right ) ( u ) = e ^ { i \langle \mathrm { A d } _ u z , v \rangle } \cdot s ( x ^ { - 1 } u ) , u \in K , \end{align*}


\begin{align*} Q = ( \alpha _ 0 : \alpha _ 1 ) \times ( \beta _ 0 : \beta _ 1 ) \times ( \gamma _ 0 : \gamma _ 1 ) \in X \end{align*}


\begin{align*} a ( x , y , t ) = a ( y , x , t ) ~ ~ \forall ( x , y , t ) \in \overline { \Omega } \times \overline { \Omega } \times \R , \end{align*}


\begin{align*} \sum _ { \substack { p > 2 \\ p \notin A } } \sum _ { k = 1 } ^ { K - 1 } f ( ( A ^ k ) _ { 2 ^ k p } ) & \le ( 1 - 2 ^ { 1 - K } ) \sum _ { \substack { 2 < p \le 2 3 \\ p \notin A } } b _ p f ( p ) + \sum _ { \substack { p > 2 3 \\ p \notin A } } f ( p ) \Big ( ( 1 - 2 ^ { 1 - J } ) b _ p + 2 ^ { - J } \frac { \log p } { \log ( 2 p ) } \Big ) \\ & \le ( 1 - 2 ^ { 1 - K } ) \sum _ { \substack { 2 < p \le 2 3 \\ p \notin A } } b _ p f ( p ) + \sum _ { \substack { p > 2 3 \\ p \notin A } } b _ p f ( p ) , \end{align*}


\begin{align*} Z ^ i : = Z ^ { i , 0 } \oplus Z ^ { i , 1 } \oplus \dots \oplus Z ^ { i , N } . \end{align*}


\begin{align*} \frac { d ( \mu * \gamma _ { \sigma } + t _ n \tilde { h } _ i ) } { d ( \mu * \gamma _ { \sigma } ) } \ge ( 1 - c t _ n ) \ge \frac { 1 } { 2 } , \ i = 1 , 2 . \end{align*}


\begin{align*} \prod _ { k = 1 } ^ s \cos ( 2 \pi n _ { k _ j } x ) = 2 ^ { - s } \sum _ { \varepsilon _ i = \pm 1 } \cos 2 \pi ( \varepsilon _ 1 n _ { k _ 1 } + \varepsilon _ { 2 } n _ { k _ 2 } + \ldots + \varepsilon _ s n _ { k _ s } ) x , \end{align*}


\begin{align*} \P ( T _ { d e t } \geq t ) \ , = \ , e ^ { - \Theta ( t ^ { \frac { 1 } { 2 \alpha } } ) } . \end{align*}


\begin{align*} ( - \Delta ) ^ { s / 2 } ( u \circ A ) ( x ) = ( ( - \Delta ) ^ { s / 2 } u \circ A ) ( x ) \end{align*}


\begin{align*} u ( x , t ) = \begin{cases} \lambda ( t ) ^ { - \frac { 2 } { p - 1 } } { \sf Q } ( y ) & , \\ & | y | \to \infty , \ | x | \ll 1 . \end{cases} \end{align*}


\begin{align*} \mu _ i ( x ) = r _ i x _ i \left ( 1 - \sum _ { j = 1 } ^ { d } a _ { i j } x _ j ^ + \right ) , \sigma ( x ) = \mathrm { d i a g } \left ( \frac { x _ 1 } { 1 + \lVert x \rVert ^ 2 } , \frac { x _ 2 } { 1 + \lVert x \rVert ^ 2 } , \ldots , \frac { x _ d } { 1 + \lVert x \rVert ^ 2 } \right ) , \end{align*}


\begin{align*} \mu _ 1 : = \ell \ , \Big ( \frac { \lambda _ 2 ( \mathcal { G } ) } { p - 2 } \Big ) ^ { \frac { 2 } { p - 2 } } , \end{align*}


\begin{align*} { \sim } \left ( \sum \limits _ { i = 1 } ^ { n } a _ i \cdot \mathtt { w } ^ + ( \phi _ i ) \geqslant c \right ) \dashv \vdash _ { \mathtt { w } \mathsf { B D } } \sum \limits _ { i = 1 } ^ { n } - a _ i \cdot \mathtt { w } ^ + ( \phi _ i ) > - c \end{align*}


\begin{align*} \sum _ { k = 0 } ^ m ( - 1 ) ^ k P _ k \left ( m , x \right ) \left ( j + x \right ) ^ { m - k } = \sum _ { k = 0 } ^ m ( - 1 ) ^ { m - k } P _ { m - k } ( m , x ) ( j + x ) ^ k . \end{align*}


\begin{align*} L ( n , R ) = 2 ^ { ( n \log n ) R + n ( \log \frac { A } { e \sqrt { a } } ) + o ( n ) } \ ; . \ , \end{align*}


\begin{align*} 2 g ( \nabla _ X Y , Z ) = X g ( Y , Z ) + Y g ( X , Z ) - Z g ( X , Y ) + g ( [ X , Y ] , Z ) - g ( Y , \nabla ^ * _ X Z - \nabla _ Z X ) - g ( X , [ Y , Z ] ) . \end{align*}


\begin{align*} \int _ { S U ( 3 ) } \tilde f ( g ) \ , d g = \int _ { \mathcal O _ { x , y } } f ( x ) \ , d x \end{align*}


\begin{align*} | | u | | _ { s , \varPhi _ { x , y } } = | | u | | _ { \widehat { \varPhi } _ x } + [ u ] _ { s , \varPhi _ { x , y } } , \end{align*}


\begin{align*} n = n _ 0 \cdot n _ 1 \ { \rm w i t h } \ \gcd ( n _ 0 , n _ 1 ) = 1 \end{align*}


\begin{align*} Y _ { n } = \frac { X _ { 1 } + \cdots + X _ { n } } { \sqrt { n } } \end{align*}


\begin{align*} d \left ( y _ { k } , p \right ) \leq \frac { \alpha ^ { k } } { 1 - \alpha } \ ; d \left ( x _ { 0 } , T x _ { 0 } \right ) + \sum _ { i = 1 } ^ { k } \alpha ^ { k - i } \ ; \varepsilon _ { k } , \ ; k \in \mathbb { N } , \end{align*}


\begin{align*} u ( t ) = \int _ 0 ^ t k ( t - \tau ) g ( \tau ) d \tau , \end{align*}


\begin{align*} \sup _ { t \in [ 0 , 1 ] } | W _ t - W _ { \lfloor n t \rfloor / n } | = O _ P \big ( n ^ { - 1 / 2 } ( \ln n ) ^ { 1 / 2 } \big ) , \end{align*}


\begin{align*} - \frac { 1 } { z } \prod \limits _ { k = 1 } ^ { \infty } \Bigg ( \frac { z - \mu _ { k } ^ { 2 } } { z - \lambda _ { k } ^ { 2 } } \Bigg ) = \sum \limits _ { k = 1 } ^ { \infty } \frac { a _ k } { \lambda _ k ^ 2 ( \lambda _ { k } ^ { 2 } - z ) } . \end{align*}


\begin{align*} \mathcal { A D } = \bigcup _ { k = 1 } ^ { \infty } \mathcal { A D } _ k , \end{align*}


\begin{align*} f _ { \mathfrak { m } } ^ { \sharp } = \underset { ^ { \ast } } { \underbrace { \Delta _ { \mathfrak { a } _ { + } } } } + \underset { ^ { \ast } } { \underbrace { \left ( - \Delta _ { \mathfrak { a } _ { - } } + e _ { \Phi } - \beta ^ { - 1 } s \right ) } } \ . \end{align*}


\begin{align*} \sigma ^ { * } ( \mathcal { A } _ 1 ) & = \{ 0 \} . \end{align*}


\begin{align*} \left \vert \int _ \Omega 2 R \xi _ E \xi _ { \tilde K } u ^ 2 d V _ g \right \vert = \left \vert \int _ \Omega \tilde R u ^ 2 d V _ g \right \vert \leq \epsilon \| \nabla u \| ^ 2 _ { L ^ 2 ( \Omega \cap \tilde K ) } + C ' _ { \epsilon } \| u \| ^ 2 _ { L ^ 2 ( \Omega \cap \tilde K ) } . \end{align*}


\begin{align*} \phi ( x , v ) = H ( x , v ) + \alpha \frac { x \cdot v } { \langle x \rangle } + \beta \langle x \rangle , \mbox { a n d } V ( s ) = 1 + s ^ { \frac { \xi } { 1 + \xi } } . \end{align*}


\begin{align*} A ^ 1 _ Q = \bigcup _ { R _ 0 = 0 } ^ { r - 1 } A ^ 1 _ { Q , j , R _ 0 } ; A ^ 2 _ Q = \varnothing . \end{align*}


\begin{align*} E ( \boldsymbol { X } , \boldsymbol { M } ) = ( \boldsymbol { M } , \boldsymbol { M } ) . \end{align*}


\begin{align*} G ( t , x ) \left ( e ^ { t } - 1 \right ) = \sum _ { n = 0 } ^ { \infty } \frac { \bar { P } _ n ( x ) } { n ! } t ^ n , \end{align*}


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


\begin{align*} L _ { \chi } + L _ { \chi ' } \equiv L _ { \chi \chi ' } + \sum \limits _ { \chi \left ( \sigma \right ) = \chi ' \left ( \sigma \right ) = - 1 } { D _ { \sigma } } \end{align*}


\begin{align*} C _ 6 ^ { \frac { p q } { p - q } } & \lesssim \sum _ { k = - \infty } ^ { M - 1 } 2 ^ { - k } \sup _ { i \leq k } 2 ^ { i } \bigg ( \int _ { x _ i } ^ { x _ { i + 1 } } u ( t ) \bigg ( \int _ t ^ { \infty } u \bigg ) ^ { \frac { q } { p - q } } d t \bigg ) \Phi ( 0 , x _ i ) \\ & + \sum _ { k = - \infty } ^ { M - 1 } 2 ^ { - k } \sup _ { i \leq k } 2 ^ { i } \sup _ { y \in ( x _ i , x _ { i + 1 } ) } \bigg ( \int _ y ^ { x _ { i + 1 } } u ( t ) \bigg ( \int _ t ^ { \infty } u \bigg ) ^ { \frac { q } { p - q } } d t \bigg ) \Phi ( x _ i , y ) . \end{align*}


\begin{align*} l e n g t h ( X ) \geqslant n X = D [ i _ 0 , i _ 1 , \ldots , i _ n , i _ { n + 1 } , \ldots , i _ m ] i _ { n + 1 } , \ldots , i _ m . \end{align*}


\begin{align*} \| \phi _ n \| = 1 . \end{align*}


\begin{align*} \textstyle \omega = \{ i \ , | \ , 1 \le i \le r , \ , c _ i \ne 0 \} , ~ ~ ~ \ell _ c = \sum _ { i \in \omega } k _ i , ~ ~ ~ \omega ' = \{ 1 , \cdots , r \} - \omega . \end{align*}


\begin{align*} A _ { B H } = \iint \limits _ { R } ( 1 - b ^ 2 ) ^ { \frac { 3 } { 2 } } d x ^ 1 d x ^ 2 , \end{align*}


\begin{align*} [ \widetilde { X } , s ^ { - 1 } ( \mathcal { F } ) ] & = [ \widetilde { X } , \Gamma ( \ker \dd s ) + \Gamma ( \ker \dd t ) ] \\ & = [ \widetilde { X } , \Gamma ( \ker ( \dd s ) ] + [ \widetilde { X } , \Gamma ( \ker \dd t ) ] \\ & \subset \Gamma ( \ker \dd s ) + \Gamma ( \ker \dd t ) = s ^ { - 1 } ( \mathcal { F } ) . \end{align*}


\begin{align*} \lim _ { \kappa \to 0 } \kappa ^ p \nu _ p ( \{ m _ f > \kappa \} ) = c _ { d , p } \ , \| \nabla f \| _ { L ^ p ( \R ^ d ) } ^ p \end{align*}


\begin{align*} \mathcal { V } _ { [ n , \ell ] } = \bigcup _ { k = \ell } ^ { n - 1 } \mathcal { V } _ { [ n , \ell , k ] } \end{align*}


\begin{align*} \omega [ u ] ( \Omega ^ \prime , \Omega ) = \inf _ { x _ 0 \in \Omega ^ \prime , R < R _ 0 } \omega [ u ] ( x _ 0 , R ) \end{align*}


\begin{align*} g ( x , y , g ^ * ( x , y , u ) ) = u , \end{align*}


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


\begin{align*} \Big ( x \frac { d } { d x } \Big ) _ { p , \lambda } \Big ( \frac { 1 } { 1 - x } \Big ) ^ { r + 1 } & = \sum _ { n = 0 } ^ { \infty } \binom { n + r } { n } \Big ( x \frac { d } { d x } \Big ) _ { p , \lambda } x ^ { n } = \sum _ { n = 0 } ^ { \infty } \binom { n + r } { n } ( n ) _ { p , \lambda } x ^ { n } . \\ \end{align*}


\begin{align*} \phi ( \gamma _ 1 \cup \gamma _ 2 ) = \phi ( \gamma _ 1 ) \star \phi ( \gamma _ 2 ) , \end{align*}


\begin{align*} H = \sum _ { i = 1 } ^ 3 \left ( \dfrac { 1 } { 2 m } P _ i ^ 2 + \dfrac { 1 } { 2 } m \omega ^ 2 X _ i ^ 2 \right ) \ , \end{align*}


\begin{align*} \left . \int u _ { 1 , t } d \nu ^ 2 _ { t } \ , \right | _ { t = \bar s } ^ { t = s ' } & = \int _ { \bar s } ^ { s ' } \int _ M \left ( \Box _ { g _ { 2 , t } } u _ { 1 , t } \right ) d \nu ^ 2 _ { t } d t . \end{align*}


\begin{gather*} W = \bigoplus _ D W _ D \end{gather*}


\begin{align*} a _ { \mathrm { p w } } ( \phi _ { \mathrm { n c } } ( { j } ) , v _ { \mathrm { n c } } ) = \lambda _ h ( j ) b ( \phi _ { \mathrm { p w } } ( { j } ) , v _ { \mathrm { n c } } ) \phi _ { \mathrm { p w } } ( { j } ) - \phi _ { \mathrm { n c } } ( { j } ) = \lambda _ h ( { j } ) \kappa _ { m } ^ 2 h _ { \mathcal { T } } ^ { 2 m } \phi _ { \mathrm { p w } } ( { j } ) . \end{align*}


\begin{align*} k ! e _ k ( x _ 1 , \ldots , x _ i ) = \varpi _ { i - k + 1 } \varpi _ { i - k + 2 } \cdots \varpi _ { i } . \end{align*}


\begin{align*} P = \sum _ { \abs { \alpha } \leq m } ( a _ \alpha + b _ \alpha ) D ^ \alpha , a _ { \alpha } \in M _ 0 ( H ^ { s - | \alpha | } \rightarrow H ^ { - s } ) , P _ s \vcentcolon = \sum _ { \abs { \alpha } \leq m } b _ \alpha D ^ \alpha \in \mathcal { P } _ { m , s } ( \Omega ) . \end{align*}


\begin{align*} V = V _ { \phi } = \left [ \theta _ { j } ^ { i } \right ] _ { 0 \leq i , j \leq n - 1 } , \ \ \ \ \Delta = ( V _ { \phi } ) \neq 0 . \end{align*}


\begin{align*} { { \mathcal { L } } } _ m ( { { \mathfrak { A } } } ( \boldsymbol { t } ) ) = E \cdot { \rm { d e t } } \left ( \varphi _ { \zeta _ j , s _ j } ( t ^ { u + \ell } ( t - 1 ) ^ { r n } ) \right ) _ { \substack { 0 \le \ell \le r - 1 \\ 1 \le j \le d , 1 \le s _ j \le r _ j } } \enspace . \end{align*}


\begin{gather*} u ( \{ x _ 0 \} ) = \lim _ { \ell \to \infty } \int _ { B _ { r _ \ell } ( x _ 0 ) } u _ { n _ \ell } , e ( \{ x _ 0 \} ) = \lim _ { \ell \to \infty } e _ { n _ \ell } ( B _ { R _ \ell } ( x _ 0 ) ) , \\ \shortintertext { a n d } \limsup _ { \ell \to \infty } \frac { e ( B _ { R _ \ell } ( x _ 0 ) ) } { u ( B _ { r _ \ell } ( x _ 0 ) ) } = \lim _ { \ell \to \infty } \frac { e _ { n _ \ell } ( B _ { R _ \ell } ( x _ 0 ) ) } { \int _ { B _ { r _ \ell } ( x _ 0 ) } u _ { n _ \ell } } . \end{gather*}


\begin{align*} \left \langle \tau _ 0 ^ { ( a , b ) } \prod _ { i \in I \setminus B } \tau _ { d _ i } ^ { ( a _ i , b _ i ) } \right \rangle ^ { } \not = 0 . \end{align*}


\begin{align*} \varphi ( x ) - \varphi ( y ) = \varphi ( x ) \le ( \| \varphi \| _ { \infty } \vee C _ 1 C _ 3 ^ { 2 - p } ) | x - y | ^ p , \ \forall x \in S , y \in S ^ c \end{align*}


\begin{align*} \varrho ( r ) = \int _ 0 ^ r \frac { 1 } { \omega ( \rho ) } d \rho . \end{align*}


\begin{align*} \Lambda _ { d , b , A } \triangleq \prod _ { i = 1 } ^ d \left [ \frac { A _ i - 1 } { b } , \frac { A _ i } { b } \right ) . \end{align*}


\begin{align*} \partial _ x g ( x , y ) = 1 - \gamma \frac { b ' ( x ) } { b ( x ) ^ 2 } \| y \| _ d ^ 2 , \nabla _ y g ( x , y ) = 2 \gamma \frac { y } { b ( x ) } , x > 1 . \end{align*}


\begin{align*} \tilde { B } _ 0 T ( t , A + B ) = \widetilde { B _ 0 } S _ 0 ( t ) + \cdots + \widetilde { B _ 0 } S _ { \ell } ( t ) + W ( t ) , \end{align*}


\begin{align*} \frac { \partial E } { \partial z ^ j _ { \eta } } \frac { \partial ^ 2 \varphi ^ j } { \partial x ^ { \epsilon } \partial x ^ { \eta } } = 2 b ^ 2 \delta _ { \epsilon 1 } f ( x ^ 1 ) f ' ( x ^ 1 ) . \end{align*}


\begin{align*} m = n _ { j _ 1 } + \ldots + n _ { j _ s } - n _ { k _ 1 } - \ldots - n _ { k _ t } , \end{align*}


\begin{align*} S _ { \alpha , \beta } = \left \{ \left ( x _ { 1 } , x _ { 2 } , \cdots , x _ { n } \right ) \in \mathbb { Z } ^ { n } \mid x _ { i } \in \mathbb { Z } \ \ \ \ 0 \leq x _ { i } < b _ { i } \right \} . \end{align*}


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


\begin{align*} \dim ( h ^ { - 1 } ( \mathcal { D } _ i ) \setminus ( T ^ * \mathcal { U } \cap h ^ { - 1 } ( \mathcal { D } _ i ) ) ) & = \dim ( ( \mathcal { Z } \cap h ^ { - 1 } ( \mathcal { D } _ i ) ) \cup ( T ^ * \mathcal { W } \cap h ^ { - 1 } ( \mathcal { D } _ i ) ) ) \\ & \leq 2 N - 3 \\ & = \dim h ^ { - 1 } ( \mathcal { D } _ i ) - 2 \end{align*}


\begin{align*} \mathrm { q a d j } ( \lambda _ i ( A ) E _ n - A ) = \prod _ { k = 1 ; k \ne i } ^ n ( \lambda _ i ( A ) - \lambda _ k ( A ) ) v _ i v _ i ^ * . \end{align*}


\begin{align*} \widehat { \lambda } ( R + Q ) + \widehat { \lambda } ( R - Q ) = 2 \widehat { \lambda } ( R ) + 2 \widehat { \lambda } ( Q ) + v ( x ( R ) - x ( Q ) ) - v ( \Delta ) / 6 \end{align*}


\begin{align*} x _ i = ( \phi _ { 0 } ) ^ { d - 1 } \sum _ { j \in S _ N ^ 1 \setminus \{ 0 \} } \phi _ { j } ( x _ i ) \alpha _ { j } ^ { ( i ) } , \end{align*}


\begin{align*} k \cdot \gcd \left ( C , C _ 1 , C _ 2 , \dots , C _ m \right ) & = \gcd \left ( k C , k C _ 1 , k C _ 2 , \dots , k C _ m \right ) \\ & = \gcd \left ( k C , j _ 1 C , j _ 2 C , \dots , j _ m C \right ) \\ & = C \gcd ( k , j _ 1 , j _ 2 , \dots , j _ m ) . \end{align*}


\begin{align*} \overline { p } ( n ) = \frac { 1 } { 2 \pi } \underset { 2 \nmid k } { \sum _ { k = 1 } ^ { N } } \sqrt { k } \underset { ( h , k ) = 1 } { \sum _ { h = 0 } ^ { k - 1 } } \dfrac { \omega ( h , k ) ^ 2 } { \omega ( 2 h , k ) } e ^ { - \frac { 2 \pi i n h } { k } } \dfrac { d } { d n } \biggl ( \dfrac { \sinh \frac { \pi \sqrt { n } } { k } } { \sqrt { n } } \biggr ) + R _ { 2 } ( n , N ) , \end{align*}


\begin{align*} D _ q h ( x ) = u ( x ) h ( x ) ( x \in I ) . \end{align*}


\begin{align*} S _ \pm = \left \{ i \in \{ 1 , \ldots , m \} | \overline { \{ i \} } \not \in \mathcal A _ { \pm } \} \right \} . \end{align*}


\begin{align*} \Phi ( x , y , Z ) : = r _ { \mathrm { V } } ( x ) + \kappa r _ { \mathrm { O } } ( x , y , Z ) , \Psi ( x , y , Z ) : = \kappa r _ { \mathrm { V } } ( x ) + r _ { \mathrm { O } } ( x , y , Z ) , \end{align*}


\begin{align*} \sum _ { k \ge 1 } \frac { a _ k } { \lambda _ k ^ 4 } = \infty . \end{align*}


\begin{align*} \widetilde { H } ( \varphi ^ { [ n ] } ) = \det ( \varphi ) ^ { n + 1 } \tilde { \iota } _ \varphi . \end{align*}


\begin{align*} \begin{aligned} \hat { W } ( x _ i ) = f _ i \nabla \hat { W } ( x _ i ) = g _ i . \end{aligned} \end{align*}


\begin{align*} D ^ k _ { X ^ \pm } ( n , m ) = \begin{cases} C ^ k _ { X ^ \pm } ( n , m ) , & \ X = A , C , \\ C ^ k _ { X ^ \pm } ( n , m ) ^ { \Z _ 2 } , & \ X = B , D , O . \end{cases} \end{align*}


\begin{align*} i _ { \frac { \partial } { \partial r } } \overline { \omega } ^ { n + 1 } = ( \Pi ^ { \ast } \omega _ { h } ) ^ { n } \wedge \eta . \end{align*}


\begin{align*} B R ( w , A ) = B R ( z , A ) ( I + ( z - w ) R ( w , A ) ) , \end{align*}


\begin{align*} N ( b , c ) = \frac { 1 } { V _ F } \int _ { 0 } ^ { + \infty } g \frac { 1 } { \sqrt { 2 \pi c } } \exp \left ( - \frac { ( g - b ) ^ 2 } { 2 c } \right ) d g \geq 0 , b \in \mathbb { R } , c > 0 . \end{align*}


\begin{align*} \nabla _ { z \partial z } L ( t , z ) \alpha = L ( t , z ) ( \mu \alpha - \frac { \rho } { z } \alpha ) , \end{align*}


\begin{align*} z _ s = e ^ { - \pi t + \pi \sigma i } , \cos \frac { \pi s } { 2 } = \frac { e ^ { \frac { \pi } { 2 } \left ( t - \sigma i \right ) } } { 2 } \left ( 1 + z _ s \right ) , \quad \sin \frac { \pi s } { 2 } = \frac { e ^ { \frac { \pi } { 2 } \left ( t - \sigma i \right ) } i } { 2 } \left ( 1 - z _ s \right ) . \end{align*}


\begin{align*} \max \{ | S | : S \in \bigcap _ { i = 1 } ^ m \mathcal { F } _ i \} & = \min \{ r _ 1 ( X ) + r _ 2 ( X _ 1 \backslash X ) + \dots + r _ { m - 1 } ( X _ { m - 2 } \backslash X _ { m - 3 } ) + r _ m ( E \backslash X _ { m - 2 } ) : \\ & X \subseteq X _ 1 \subseteq X _ 2 \subseteq \dots \subseteq X _ { m - 2 } \subseteq E \} \end{align*}


\begin{align*} { \sigma _ { i } ^ 2 = ( u - u ^ { - 1 } ) \sigma _ { i } + 1 } . \end{align*}


\begin{align*} \phi \cdot \psi = \phi \circ F _ * ^ e \psi . \end{align*}


\begin{align*} M _ f ( \lambda ) = ( \mu _ 0 , \mu _ 1 , \ldots , \mu _ n ) \end{align*}


\begin{align*} \frac { 1 } { 2 \pi i } \int _ \gamma \frac { h ' ( z ) } { h ( z ) } d z & = \int _ 0 ^ 1 g ' ( t + i R ) d t = 1 . \end{align*}


\begin{align*} \Sigma = I _ { n } . \end{align*}


\begin{align*} \Phi ( z ) : = \prod _ { k \ge 0 } \left ( \frac { z - \mu _ k ^ 2 } { z - \lambda _ k ^ 2 } \right ) \ , . \end{align*}


\begin{align*} \lim _ { x \to \infty } \sup _ { y : \| y \| _ d \leq b ( x ) } \bigl | b ( x ) G ( x , y ) - \gamma \sigma ^ 2 \bigr | = 0 . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \frac { \partial } { \partial t } \omega ( t ) = - \mathrm { R i c } _ { \omega ( t ) } ^ { T } , \omega ( 0 ) = \omega _ { 0 } . \end{array} \end{align*}


\begin{align*} a _ i Q ^ i = \sum _ { j = 0 } ^ i { i \choose j } a _ i h ^ j Q '^ { i - j } . \end{align*}


\begin{align*} | H _ i \cap H _ j | & \le r _ i + r _ j - r \ \tag * { ( H 1 ) } \\ | S - H _ i | + r _ i & \ge r \ \tag * { ( H 2 ) } \end{align*}


\begin{align*} \mathcal A _ 0 : = \mathcal A _ { \omega _ 0 } = \{ I \subset \{ 1 , \ldots , m \} : \omega _ 0 \in \angle _ I \} . \end{align*}


\begin{align*} M _ { u _ 1 } ^ t \cdot a _ 2 = M _ { u _ 2 } ^ t \cdot a _ 1 = 0 \ ; . \end{align*}


\begin{align*} \partial _ { t } \mathcal { P } m - \bar { \mu } \Delta \mathcal { P } m = \mathcal { P } g . \end{align*}


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


\begin{align*} g = \frac { d r ^ 2 } { \alpha ( r ) } + r ^ 2 \cdot h _ { \mathbb { S } ^ { n - 2 } } + e ^ { 2 \beta ( s , r ) } \cdot d s ^ 2 . \end{align*}


\begin{align*} E _ { i } \cdot v _ r & = \delta _ { r , i + 1 } v _ { r - 1 } , \\ F _ { i } \cdot v _ r & = \delta _ { r , i } v _ { r + 1 } , \\ K _ i \cdot v _ r & = \begin{cases} q v _ { i } & \hbox { i f } r = i , \\ q ^ { - 1 } v _ { i + 1 } & \hbox { i f } r = i + 1 , \\ v _ { r } & \hbox { e l s e } . \end{cases} \end{align*}


\begin{align*} \Phi ( x ) = \int _ 0 ^ x \phi ( u ) \ d u \ , x \geq 0 \ , \end{align*}


\begin{align*} S = \{ A > 0 : T ( A w _ 0 ) = \infty , \ h _ { \infty } ( A w _ 0 ) + g _ { \infty } ( A w _ 0 ) < \infty \ \ u ( A w _ 0 , \cdot ) , v ( ( A w _ 0 , \cdot ) \ \} . \end{align*}


\begin{align*} d \mu ( \nu ) = { \textstyle \frac { 1 } { 2 \pi ^ 2 } } \nu ^ 2 \ , K ( \nu ; z ) d \nu \end{align*}


\begin{align*} 0 = \operatorname { d i v } ( X _ { 1 } ) = \operatorname { d i v } ( h X _ { 2 } ) , \end{align*}


\begin{align*} \kappa _ 2 ^ 2 \sum _ { i = 1 } ^ Q \rho _ i \int ^ { \tau \wedge t _ i } _ 0 ( t _ i - s ) ^ { \alpha } ( \sigma ' \sigma ) ( X _ s ) d B _ s , \end{align*}


\begin{align*} Y _ O ( P , G ) \leq C ( P ) , Y _ R ( P , G ) \geq C ( P ) , C ( P ) : = \frac { B ( p _ 1 ) B ( p _ 2 ) } { B \circ q ( p _ 1 , p _ 2 ) } . \end{align*}


\begin{align*} X _ t = X _ 0 + \int _ 0 ^ t b ( X _ s ) d s + \int _ 0 ^ t a ( X _ s ) d W _ s , t \in [ 0 , T ] , \end{align*}


\begin{align*} f _ b ( x ) = a x ^ q + b x ^ r , \end{align*}


\begin{align*} f ' _ - ( x _ 0 , 0 ^ \pm , 0 ) \coloneqq \displaystyle \liminf _ { ( x , u , \xi ) \to ( x _ 0 , 0 ^ \pm , 0 ) } \frac { f ( x , u , \xi ) } { \abs { u } } \geq \limsup _ { u \to 0 ^ \pm } \sup _ { \abs { \xi } = 1 } \frac { f ( x _ 0 , u , \rho ( \abs { u } ) \xi ) } { \abs { u } } , \end{align*}


\begin{align*} \begin{aligned} Q ( p , q ) \otimes Q ( q , p ) = & \ , \bigoplus _ { n = 0 } ^ { p } \bigoplus _ { m = 0 } ^ { q } \Bigg \{ Q ( p + q - n - m , p + q - n - m ) \\ & \ \ \oplus \Bigg [ \bigoplus _ { k = 1 } ^ { \min ( p - n , q - m ) } \Bigg ( Q ( p + q - n - m - 2 k , p + q - n - m + k ) \\ & \ \ \oplus Q ( p + q - n - m + k , p + q - n - m - 2 k ) \Bigg ) \Bigg ] \Bigg \} \ . \end{aligned} \end{align*}


\begin{align*} F [ u ] : = \log \det [ D ^ 2 u - A ( \cdot , u , D u ) ] = [ \tau ( 1 - t ) + \epsilon ] ( u - u _ 0 ) + \log B _ t ( \cdot , u , D u ) , \end{align*}


\begin{align*} H ( z ) = z + ( \varphi _ { \mu } \circ F _ { \nu } ) ( z ) = z + \gamma _ { \mu } + \int _ { \mathbb { R } } \frac { 1 + F _ { \nu } ( z ) s } { F _ { \nu } ( z ) - s } \ , d \sigma _ { \mu } ( s ) \end{align*}


\begin{align*} \boldsymbol { R } ( \tau ; t , x ) \triangleq \begin{cases} \boldsymbol { R } _ 1 ( \tau ; t , x ) & h ( \tau , p ( \tau ; t , x ) ) > 0 \\ \tau & h ( \tau , p ( \tau ; t , x ) ) \leq 0 \end{cases} , \end{align*}


\begin{align*} { \rm { \Delta } } _ { i j . . . k p } ^ { \left ( n \right ) } = w _ { i j . . . k p } ^ { \left ( n \right ) } \delta \left ( { { t _ 1 } - { t _ 2 } } \right ) . . . \delta \left ( { { t _ 1 } - { t _ n } } \right ) \ , \end{align*}


\begin{align*} Q ^ { n . 4 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( s - \eta _ n ( s ) ) ^ { 2 \alpha } ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) \ , \left [ \left ( W _ s - W _ { \eta _ n ( s ) } \right ) ^ 2 - ( s - \eta _ n ( s ) ) \right ] \ , d s . \end{align*}