\begin{align*} I _ 2 = \int _ 0 ^ r \left [ - \frac { z } { 2 } ( \bar \psi \bar \psi ' \cdot \phi + \bar \psi \bar \phi ' \cdot \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 } \Vert z \bar \psi \Vert _ { L ^ \infty } , \end{align*}


\begin{align*} \psi _ { L , n } ( \beta ) = \eta ^ n y \mod \eta ^ n \mathfrak { X } _ L . \end{align*}


\begin{align*} \mathrm { P } _ { \Phi + \Phi ^ { \mathfrak { b } , \gamma } } = - \inf _ { \rho \in E _ { 1 } } \left \{ f _ { \Phi } \left ( \rho \right ) + e _ { \Phi ^ { \mathfrak { b } , \gamma } } \left ( \rho \right ) \right \} \ , \end{align*}


\begin{align*} \varphi _ { \beta , a } ( P ) = \frac { 1 } { P ^ { a + \beta } } \int _ { 0 } ^ { P } u ^ \beta \kappa ( u ) d u , \end{align*}


\begin{align*} \nu = \mathcal { D } \Big ( \Big ( \frac { 1 } { 1 + \alpha } \Big ) ^ { 1 / p } \sum _ { j = 1 } ^ \infty A _ { \bullet , j } Z _ j + \sigma _ { p , \alpha } ( I _ k - A A ^ T ) ^ { 1 / 2 } N _ k \Big ) \end{align*}


\begin{align*} \underline d ( S ) & = \liminf _ { n \to \infty } \frac { \# ( S \cap \{ 1 , \dots , n \} ) } { n } , \\ \overline d ( S ) & = \limsup _ { n \to \infty } \frac { \# ( S \cap \{ 1 , \dots , n \} ) } { n } . \end{align*}


\begin{align*} \bar F ( x ) = P ( B > x ) = L ( x ) x ^ { - \alpha } . \end{align*}


\begin{align*} \mathcal { T } ( \Lambda ( G ) ) : = \ , \Lambda \cup \{ P _ i [ 1 ] : i \in Q _ 0 \} \ , . \end{align*}


\begin{align*} A _ Y X = - A _ X Y - \lambda ^ 2 g ( X , Y ) \left ( \nabla _ \nu \frac { 1 } { \lambda ^ 2 } \right ) . \end{align*}


\begin{align*} s _ { \theta } = ( 1 - \theta ) s _ 0 + \theta s _ 1 = \frac { t - r } { t - z } z + \frac { r - z } { t - z } t = r . \end{align*}


\begin{align*} z ( x y ) = ( - 1 ) ^ { \vert y \vert ( \vert x \vert + \vert z \vert ) } ( y z ) x . \end{align*}


\begin{align*} \bigg ( \sum _ { k = - \infty } ^ { M - 1 } \bigg ( \sum _ { i = - \infty } ^ { k } 2 ^ { - i \frac { r } { p } } V _ r ( x _ { i - 1 } , x _ i ) ^ r a _ i ^ r \bigg ) ^ { \frac { q } { r } } \int _ { x _ k } ^ { x _ { k + 1 } } u \bigg ) ^ { \frac { 1 } { q } } \leq C ' \bigg ( \sum _ { k = - \infty } ^ { M - 1 } a _ k ^ p \bigg ) ^ { \frac { 1 } { p } } \end{align*}


\begin{align*} \partial _ t g _ t ( \partial _ i , \partial _ j ) = & 2 \bar { \nabla } ^ 2 _ { i j } t , \\ \bar { g } | _ { t = 0 } = & g , \\ \partial _ t ( \bar { \nabla } ^ 2 _ { i j } t ) - ( \bar { \nabla } ^ 2 t ) ^ 2 _ { i j } = & 0 , \\ \bar { \nabla } ^ 2 _ { i j } t | _ { t = 0 } = & k _ { i j } \end{align*}


\begin{align*} d K ^ { \ell + 1 } - K ^ { \ell + 1 } d & = d K ^ { \ell } K - K ^ { \ell + 1 } d = K ^ \ell d K + \ell S K ^ \ell - K ^ { \ell + 1 } d \\ & = K ^ { \ell } S + \ell S K ^ { \ell } = ( \ell + 1 ) S K ^ \ell , \end{align*}


\begin{align*} \pi ( x ) C _ \pi \pi ( x ) ^ * = \Delta _ G ( x ) ^ { 1 / 2 } C _ \pi , x \in G , \end{align*}


\begin{align*} \widetilde { f } ^ { | u \rangle } _ { \mu } ( p ) = \frac { u _ \mu } { u \cdot p } \end{align*}


\begin{align*} \delta ( f ^ t Q ^ { ( t ) } ) = \delta _ { t G } ( f ^ t ) + \delta _ { - t G } ( Q ^ { ( t ) } ) \le \delta _ { - t G } ( Q ^ { ( t ) } ) \leq \delta _ G ( Q ) \ , \end{align*}


\begin{align*} \mathcal { A } _ { 2 m } = H ^ 0 ( X , K ^ { 2 m } ( D ^ { 2 m - 1 } ) ) \subset \mathcal { A } . \end{align*}


\begin{align*} t _ e = t ' _ e - \omega ( e ) z = t ' _ e - \omega ( e ) \lfloor \tfrac { n } { 4 m } \rfloor \ , , \end{align*}


\begin{align*} \Vert J _ { 2 2 } \Vert _ { p } = \mathcal { O } \left ( \left ( v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) \right ) \left ( \frac { \omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \frac { \omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \right ) \right ) . \end{align*}


\begin{align*} D _ + = \underbrace { h _ + + \cdots + h _ + } _ { r - r ^ \prime } , D _ - = \underbrace { ( \xi _ - - h _ - ) + \cdots + ( \xi _ - - h _ - ) } _ { r - r ^ \prime } . \end{align*}


\begin{gather*} ( f \otimes D ) ( \tau ) = \sum _ n c _ f ( n ) \left ( \frac { D } { n } \right ) q ^ n \end{gather*}


\begin{align*} \Theta = \dfrac { \prod _ { i = 1 } ^ m \alpha ^ r _ i \prod _ { s = 0 } ^ { r - 1 } a _ { 0 , s } ^ m } { ( n - 1 ) ! ^ { r ^ 2 m } } \Psi ( { \hat { P } } _ { n , n } ) \enspace . \end{align*}


\begin{align*} \mu ( \tilde { T } _ { \vec s , k } ) = \left ( \frac 1 2 \deg ( \tilde { T } _ { \vec s , k } ) + \# \{ i : s _ i < 0 \} - \frac 1 2 \dim _ { \mathbb C } X \right ) \tilde { T } _ { \vec s , k } . \end{align*}


\begin{align*} u ( x ) = \frac { - 1 } { f ( x ) } \int _ { 0 } ^ { q x } f ( t ) r ( t ) d _ q t \end{align*}


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


\begin{align*} \mathbb { E } [ \ell , k ] = \left . \left [ \dfrac { \ell + 1 } { 2 } \dbinom { \ell } { k } + \sum \limits _ { i = 0 } ^ { k / 2 - 1 } \dbinom { \ell } { k - 2 i - 2 } \right ] \middle / \dbinom { \ell } { k } \right . \end{align*}


\begin{align*} ( 1 - r ^ 2 ) m ( r ) - 2 r = 0 . \end{align*}


\begin{align*} W ( \mathbf { t } ) = \bigcap _ { N = 1 } ^ { \infty } \bigcup _ { n = N } ^ { \infty } \bigcup _ { w \in \Sigma _ { \beta _ 1 } ^ n , v \in \Sigma _ { \beta _ 2 } ^ n } B \left ( \overline { x } _ { n , w } , \beta _ 2 ^ { - n ( 1 + \theta _ { 1 } + \epsilon ) \log _ { \beta _ 2 } \beta _ 1 } \right ) \times B \left ( \overline { y } _ { n , v } , \beta _ 2 ^ { - n ( 1 + \theta _ { 2 } + \epsilon ) } \right ) , \end{align*}


\begin{align*} J _ v ( v , g ) = - g _ L v + g ( V _ E - v ) , \end{align*}


\begin{align*} \Lambda = \big ( \Lambda ( x , y , z , u , w ) , d \big ) , \end{align*}


\begin{align*} A _ m = 1 + m + ( 1 - ( - 1 ) ^ m ) \kappa _ 1 + 2 \kappa _ 2 . \end{align*}


\begin{align*} \int _ { \Xi } e ^ { W ' / \hbar } d x _ 1 \wedge \cdots \wedge d x _ n = \int _ { \Xi } \psi ^ * \left ( e ^ { W ' / \hbar } d x _ 1 \wedge \cdots \wedge d x _ n \right ) = \int _ { \Xi } e ^ { \psi ( W ' ) / \hbar } d x _ 1 \wedge \cdots \wedge d x _ n . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { \min \{ N _ h , m p \} } ( \tilde \lambda _ i - \lambda _ i ) ^ 2 \le \lVert E \rVert _ F . \end{align*}


\begin{align*} A _ n ( x ) = \frac 1 n \sum _ { i = 0 } ^ { n - 1 } \delta _ { T ^ i ( x ) } \end{align*}


\begin{align*} & \frac { \partial } { \partial t _ { \vec s , k } } ( L ( t , - z ) \alpha , L ( t , z ) \beta ) \\ = & \frac { 1 } { z } ( \tilde { T } _ { \vec s , k } \star _ t L ( t , - z ) \alpha , L ( t , z ) \beta ) - \frac { 1 } { z } ( L ( t , - z ) \alpha , \tilde { T } _ { \vec s , k } \star _ t L ( t , z ) \beta ) \\ = & 0 \end{align*}


\begin{align*} k _ { \pm } = \sqrt { \frac { 1 + ( | a | \pm i | b | ) } { 4 \beta \hslash ^ { 2 } / 3 } } = \sqrt { \frac { ( 1 + | a | ) \pm i | b | } { 4 \beta \hslash ^ { 2 } / 3 } } \end{align*}


\begin{align*} 0 = \tilde { \nabla } ^ 2 _ { i j } u | _ { T M ^ 3 } = \nabla ^ 2 _ { i j } u + _ { i j } \hat { N } ( u ) = ( _ { i j } - k _ { i j } ) | \nabla u | \end{align*}


\begin{align*} \Gamma ^ { k , \tau } _ p & : = \{ \tau = \tau _ 1 - \tau _ 2 + \cdots + \tau _ p , \ , \ , k = k _ 1 - k _ 2 + \cdots k _ p \} , \end{align*}


\begin{align*} I _ { X } ( y , z ) = z e ^ { t / z } \sum _ { d \in \mathbb K } y ^ { d } \left ( \prod _ { i = 0 } ^ { m } \frac { \prod _ { a \leq 0 , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } { \prod _ { a \leq D _ i \cdot d , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } \right ) \textbf { 1 } _ { [ - d ] } , \end{align*}


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


\begin{align*} T _ { N , t } \boldsymbol { \mu } _ N ^ \mathrm { T } - \boldsymbol { \mu } _ N ^ \mathrm { T } T _ { \infty , t } & = - \int _ 0 ^ t \frac { \dd } { \dd s } { \left [ T _ { N , t - s } \boldsymbol { \mu } _ N ^ \mathrm { T } T _ { \infty , s } \right ] } \dd s \\ & = \int _ 0 ^ t T _ { N , t - s } { \left [ \mathcal { L } _ N \boldsymbol { \mu } _ N ^ \mathrm { T } - \boldsymbol { \mu } _ N ^ \mathrm { T } \mathcal { L } _ { \infty } \right ] } T _ { \infty , s } \dd s , \end{align*}


\begin{align*} \frac { 1 } { \pi } \sum _ { n = 1 } ^ { \infty } \frac { A ( n ) } { n } \sin ( 2 \pi n x ) = - \sum _ { n = 1 } ^ { \infty } \frac { a ( n ) } { n } \left ( \{ n x \} - \frac { 1 } { 2 } \right ) . \end{align*}


\begin{align*} \textup { L H S } = & ( 1 + \lambda _ h ) \Vert u _ { \mathrm { n c } } \Vert _ { \delta } ^ 2 + \lambda \Vert u - u _ { \mathrm { n c } } \Vert ^ 2 _ { L ^ 2 ( \Omega ) } + 2 \lambda b ( u , u _ { \mathrm { n c } } - J u _ { \mathrm { n c } } ) + 2 a _ { \mathrm { p w } } ( u , J u _ { \mathrm { n c } } - u _ { \mathrm { n c } } ) . \end{align*}


\begin{align*} \frac { \partial E } { \partial z ^ i _ { \epsilon } } = 2 b ^ 2 \Big [ z ^ 3 _ { \tilde { \epsilon } } \sum \limits _ { \tau } ^ { } ( - 1 ) ^ { { \tilde { \epsilon } } + \tau } z ^ i _ { \tilde { \tau } } z ^ 3 _ { \tau } + \delta _ { i 3 } \sum \limits _ { \tau , k } ^ { } ( - 1 ) ^ { \epsilon + \tau } z ^ 3 _ { \tau } z ^ k _ { \tilde { \tau } } z ^ k _ { \tilde { \epsilon } } \Big ] , \end{align*}


\begin{align*} \int F ( x ) r ( x ) y ( x ) d _ q x = - F ( x ) D _ { q ^ { - 1 } } y ( x ) . \end{align*}


\begin{align*} \mathcal { F } _ \perp : = \mathbb { C } \oplus \bigoplus _ { s = 1 } ^ \infty \Big ( \mathrm { s p a n } \{ u _ 1 , u _ 2 \} ^ \perp \Big ) ^ { \otimes _ { \mathrm { s y m } } s } . \end{align*}


\begin{align*} ( A ) = \sum \limits _ { j = 1 } ^ n ( \lambda _ j ) \prod \limits _ { i \neq j } \dfrac { ( A - \lambda _ i ) } { ( \lambda _ j - \lambda _ i ) } . \end{align*}


\begin{align*} u _ t = \Delta u + a ( x ) u ^ p \ \ \R ^ { N } , t > 0 \end{align*}


\begin{align*} g ( z ) = \frac { \alpha z + \beta } { \bar \beta z + \bar \alpha } \end{align*}


\begin{align*} [ B , M ] ^ * = \{ N \in \mathcal { D } : \max ( B ) \sqsubseteq \max ( r _ m ( N ) ) \mathrm { \ a n d \ } N \le M \} . \end{align*}


\begin{align*} f = \pi _ { i _ 1 } \sum _ { J \subseteq [ n - 1 ] } c _ { J } \pi _ J = \sum _ { J \subseteq [ n - 1 ] } c _ { J } \pi _ { i _ 1 } \pi _ J . \end{align*}


\begin{align*} \forall t > 0 , \| E \| _ { L ^ 1 \cap L ^ \infty } \leq c ( \mu ) \lambda , \lim _ { \mu \rightarrow \infty } c ( \mu ) = 0 . \end{align*}


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


\begin{align*} \lim _ { n \rightarrow \infty } \int \limits _ 0 ^ \infty \| \chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \perp } P _ { \delta } ^ \perp ( W _ { n ; \textrm { o u t } } ^ \perp ) \| _ { \textrm { o p } } \ , d t = 0 \end{align*}


\begin{align*} c _ { n } ^ { \ast \ast } = \frac { \Gamma ( n / 2 ) } { ( 2 \pi ) ^ { n / 2 } } \left [ \int _ { 0 } ^ { \infty } s ^ { n / 2 - 1 } \overline { \mathcal { G } } _ { n } ( s ) \mathrm { d } s \right ] ^ { - 1 } . \end{align*}


\begin{align*} & \Omega _ X ^ p : = \iota _ * \Omega _ { U } ^ p 1 \leq p \leq n , \\ & T _ X : = ( \Omega _ X ^ 1 ) ^ * , \end{align*}


\begin{align*} x \in \widetilde \Sigma _ i \ \Rightarrow \ | x - \rho \tilde \eta _ { \ell } | \ge \frac \rho 2 | \tilde \eta _ { \ell } - \tilde \eta _ { i } | \ \hbox { i f } \ \ell \not = i , \end{align*}


\begin{align*} P ( x ) = \sum \limits _ { i = 1 } ^ k \left ( \sum \limits _ { p = 0 } ^ { m _ i - 1 } \dfrac { c _ i ^ p } { p ! } ( x - \lambda _ i ) ^ p \right ) L _ i ( x ) . \end{align*}


\begin{align*} S = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \quad C = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} . \end{align*}


\begin{align*} \begin{cases} w = ( 1 + u _ { q , r } ) ^ { 1 + \delta } \\ \nu = \xi _ { \tilde K } ^ 2 ( 1 + u _ { q , r } ) ^ { 1 + 2 \delta } \end{cases} \end{align*}


\begin{align*} C _ { m , r } \left ( n \right ) \left ( j \right ) : = C _ { f \left ( \left ( r - 1 \right ) n + j - 1 \right ) } \end{align*}


\begin{align*} T _ { } \triangleq \sum _ { m = 1 } ^ { M } w _ m T _ m \overset { \mathcal { H } _ 1 } { \underset { \mathcal { H } _ 0 } { \gtrless } } \lambda \end{align*}


\begin{align*} f ( x , y ) = g ^ { h _ s ( x , y ) } , \end{align*}


\begin{align*} \begin{cases} i u _ t + \bigtriangleup u - | u | ^ { p - 1 } u + i \gamma u = f \\ u ( x , 0 ) = u _ 0 \in H ^ 1 ( \mathbb { T } ) , \end{cases} \end{align*}


\begin{align*} \delta ( p , ( \pi _ j ) _ j ) S ^ { d } ( p ) \lesssim _ { n } \frac h \lambda + \prod _ { j = 1 } ^ d s _ { k _ j , \lambda } ( p , \pi _ j ) \end{align*}


\begin{align*} V _ { m } ( \alpha ) = \Big \{ v _ { m } ( x , { \bf y } ; \alpha ) = \displaystyle \sum _ { k = 1 } ^ { m } \sum _ { j = 1 } ^ { N _ h } { \tilde v } _ { k , j } ^ \alpha \vartheta _ j ( x ) \varphi _ k ( \psi _ x ( { \bf y } ) ) , \ ( x , { \bf y } ; \alpha ) \in \Omega _ { 1 D } \times \Sigma _ x \times { \mathcal P } \Big \} , \end{align*}


\begin{align*} \psi _ s = \sum _ { j = 1 } ^ { w - 1 } \sum _ { k = 1 } ^ { r _ j } { { p } } _ { j , k } \varphi _ { \zeta _ j , k } + \sum _ { k = 1 } ^ { s _ w } { { p } } _ { w , k } \varphi _ { \zeta _ w , k } \enspace . \end{align*}


\begin{align*} \mathrm { E } [ \mathbf { Y } | \boldsymbol { a } < \mathbf { Y } \leq \boldsymbol { b } ] & = \mathrm { E } \left [ \left ( \mathbf { \Sigma } ^ { \frac { 1 } { 2 } } \mathbf { Z } + \boldsymbol { \mu } \right ) | \boldsymbol { \xi _ { a } } < \mathbf { Z } \leq \boldsymbol { \xi _ { b } } \right ] \\ & = \boldsymbol { \mu } + \mathbf { \Sigma } ^ { \frac { 1 } { 2 } } \mathrm { E } \left [ \mathbf { Z } | \boldsymbol { \xi _ { a } } < \mathbf { Z } \leq \boldsymbol { \xi _ { b } } \right ] , \end{align*}


\begin{align*} f _ * = \min _ { x \in \mathbb { R } ^ d } f ( x ) = \min _ { i \in \mathcal { I } } \big \{ f _ i - \tfrac { 1 } { 2 L } \| g _ i \| ^ 2 \big \} \end{align*}


\begin{align*} \Delta \bar { \phi } - \bar { \phi } + p U _ 0 ^ { p - 1 } \bar { \phi } = 0 \quad \textrm { i n } \ \ \mathbb { R } ^ N . \end{align*}


\begin{align*} n _ k \coloneqq | \overline { J _ k } | = | J _ k | + 1 . \end{align*}


\begin{align*} \Big | \Big ( \frac { d } { d \lambda } \Big ) ^ \alpha F ^ { ( \iota ) } ( \lambda ) \Big | & = \Big | \Big ( \frac { d } { d \lambda } \Big ) ^ \alpha \int _ { - 2 } ^ 2 2 ^ \iota F ( \tau ) \check \chi ( 2 ^ \iota ( \lambda - \tau ) ) \ , d \tau \Big | \\ & \lesssim _ N 2 ^ { \iota ( \alpha + 1 ) } \int _ { - 2 } ^ 2 \frac { | F ( \tau ) | } { ( 1 + 2 ^ \iota | \lambda - \tau | ) ^ N } \ , d \tau . \end{align*}


\begin{align*} \log F ( z \ , | \ , \overline \omega _ 1 , \omega _ 2 ) = & \log F ( c z \ , | \ , c \overline \omega _ 1 , c \omega _ 2 ) \sim \int _ C \frac { e ^ { c z s } } { e ^ { c \overline \omega _ 1 s } - 1 } \sum _ { k \geq 0 } \frac { B _ k \cdot ( c \omega _ 2 ) ^ { k - 1 } ( s ) ^ { k - 2 } } { k ! } \ , d s \\ & = \sum _ { k \geq 0 } \frac { B _ k \cdot \omega _ 2 ^ { k - 1 } } { k ! } \cdot \int _ C \frac { e ^ { c z s } \ , ( c s ) ^ { k - 2 } } { e ^ { c \overline \omega _ 1 s } - 1 } \ d ( c s ) . \end{align*}


\begin{align*} u _ { A , q } ( \cdot , y ) = \Phi ( \cdot , y ) + u _ { A , q } ^ s ( \cdot , y ) \mathrm { i n } \ ; \R ^ 3 , \end{align*}


\begin{align*} O _ i ( Q ) : = \left \langle \prod _ { \ell \in I _ i } \tau ^ { ( a _ \ell , b _ \ell ) } _ { d _ \ell } \sigma _ 1 ^ { k _ 1 ( i ) } \sigma _ 2 ^ { k _ 2 ( i ) } \sigma _ { 1 2 } \right \rangle ^ { \mathbf { s } ^ { \Gamma _ { 0 , k _ 1 ( i ) , k _ 2 ( i ) , 1 , I _ i } } , o } \end{align*}


\begin{align*} \tilde { \nabla } u = \tilde { g } ^ { u i } \partial _ i = \partial _ \tau . \end{align*}


\begin{align*} S ^ { n , 1 } _ \tau : = n ^ { \alpha + \frac 1 2 } \int _ 0 ^ { \tau } ( t - s ) ^ \alpha \sigma ' ( X _ s ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) d s . \end{align*}


\begin{align*} \begin{bmatrix} 0 & - 1 & 1 & 1 \\ 1 & 1 & 0 & 0 \end{bmatrix} . \end{align*}


\begin{align*} \tau _ x ( f ) \colon = \int _ { ( k , v ) \in \mathcal { O } _ x } f ( k , v ) \ ; d k d v . \end{align*}


\begin{align*} \gamma _ x ^ \alpha : \mathbb { R } \to N : t \mapsto \varphi ( \exp ( t v _ \alpha ) , x ) , \alpha = 1 , \dots , r , \end{align*}


\begin{align*} \kappa _ \beta = \frac { 2 ^ \alpha \Gamma ( ( \beta + \alpha ) / 2 ) \Gamma ( ( d - \beta ) / 2 ) } { \Gamma ( \beta / 2 ) \Gamma ( ( d - \beta - \alpha ) / 2 ) } , \end{align*}


\begin{align*} r R \Sigma ^ n Y = \textbf { R } r \Sigma ^ n Y \cong \Sigma ^ n \textbf { R } r Y = \Sigma ^ n r R Y , \end{align*}


\begin{align*} \bigl [ U ( \sigma ) \bigr ] \ , ( \widehat { \tau } ) = \widehat { \sigma \tau } , \ \ \forall \ , \tau \in S _ { \infty } . \end{align*}


\begin{align*} \sideset { } { ' } \sum _ { a } 2 f _ { i a } T _ { i a } ^ i + \sideset { } { ' } \sum _ { a < b } f _ { a b } \left ( T _ { i b } ^ a + T _ { a b } ^ i + T _ { a i } ^ b \right ) = - 2 \ , \partial _ i F , \end{align*}


\begin{align*} M _ 5 = f _ k ( i ) f _ k ( i + 1 ) + O _ { \alpha } \Big ( \frac { \tau _ 3 ( i ) \tau _ 3 ( i + 1 ) } { \sqrt { i } } \Big ) . \end{align*}


\begin{align*} \chi _ { \sigma _ i } ( h _ k ) = \sum _ { l = 0 } ^ { k } \binom { k } { l } \cdot \binom { n - k } { i - l } ( - 1 ) ^ l . \end{align*}


\begin{align*} \Lambda ' _ i = \Lambda _ i + \pi ^ { \nu } Z ^ { \nu } _ i . \end{align*}


\begin{align*} a _ { 2 } ^ j : = 2 ^ j ( \kappa _ 2 + 1 / 2 ) _ { \lfloor ( j + 1 ) / 2 \rfloor } ( \gamma _ 2 + 1 ) _ { \lfloor j / 2 \rfloor } . \end{align*}


\begin{align*} \lim _ { R \to + \infty } \left ( \limsup _ { n \to \infty } \ \lambda _ n ^ { - \frac { 1 } { p - 2 } } \max _ { d _ n ( x ) \ge R \lambda _ n ^ { - 1 / 2 } } u _ n ( x ) \right ) = 0 . \end{align*}


\begin{align*} \left ( \begin{array} { c } \frac 1 2 \\ l \end{array} \right ) = \frac { ( - 1 ) ^ { l - 1 } ( 2 l - 3 ) ! ! } { 2 ^ l l ! } \in \left [ - \frac 1 8 , \frac 1 2 \right ] . \end{align*}


\begin{align*} f _ \lambda ( \lambda _ i ) = \prod _ { j ( \neq i ) } ( \lambda _ i - \lambda _ j ) , \ f _ { \lambda \lambda } ( \lambda _ i ) = 2 \sum _ { j ( \neq i ) } \prod _ { k ( \neq i , j ) } ( \lambda _ i - \lambda _ k ) . \end{align*}


\begin{align*} \sin \theta ( \mathbf { x } , { \mathcal C } _ 1 ) = \min _ { \mathbf { y } \in { \mathcal C } _ 1 } \lVert \mathbf { x } - \mathbf { y } \rVert _ 2 , \end{align*}


\begin{align*} a _ n ^ N = ( \lambda _ n ^ 2 - \mu _ n ^ 2 ) \prod _ { \substack { k = 1 \\ k \ne n } } ^ N \left ( \frac { \lambda _ n ^ 2 - \mu _ k ^ 2 } { \lambda _ n ^ 2 - \lambda _ k ^ 2 } \right ) n \le N , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ \infty \frac { ( - 1 ) ^ k } { ( k + z ) ^ n } = \frac { ( - 1 ) ^ { n } } { 2 ^ { n } \Gamma ( n ) } \left ( \psi _ { n - 1 } \left ( \frac { z } { 2 } \right ) - \psi _ { n - 1 } \left ( \frac { z + 1 } { 2 } \right ) \right ) . \end{align*}


\begin{align*} \begin{aligned} & \mbox { u p s t e p s } \ , \ , \ , ( n , m ) \rightarrow ( n + 1 , m + 1 ) , \\ & \mbox { d o w n s t e p s } \ , \ , \ , ( n , m ) \rightarrow ( n + 1 , m - p ) \ , \ , \ , \mbox { b y $ p $ u n i t s } . \end{aligned} \end{align*}


\begin{align*} g _ i = ( x _ i - i t ) \left ( \prod _ { k < i , j \leq i } ( x _ k - x _ j - t ) \right ) \left ( \prod _ { j > i } ( x _ i - x _ j - t ) \right ) . \end{align*}


\begin{align*} \lim _ { k \to \infty } \sum _ { 1 \le i \le k } \frac { \sqrt { v _ i } - \sqrt { v _ { i - 1 } } } { 1 + v _ { i - 1 } } = \lim _ { k \to \infty } \sum _ { 1 \le i \le k } \int _ { v _ { i - 1 } } ^ { v _ i } \frac { \dd } { \dd v } \Big [ \sqrt { v } \Big ] \frac { \dd { v } } { 1 + v _ { i - 1 } } = \int _ 0 ^ 1 \frac { \dd { v } } { 2 \sqrt { v } ( 1 + v ) } = \frac { \pi } { 4 } . \end{align*}


\begin{align*} \label [ I e q ] { e q : b r o w n i a n g i r s a n o v } B ^ i _ t : = \overline { B } ^ i _ t - \int _ 0 ^ t \mathsf { X } ^ i _ s \dd s \end{align*}


\begin{align*} \int _ 0 ^ \infty \Phi ( K ^ { \sigma ^ 2 } _ t f ( x ) ) \nu _ { \sigma ^ 2 } ( x ) d x & = \int _ 0 ^ \infty \frac { 1 } { \sigma ^ 2 } \Phi \left ( K _ t d _ { \sigma ^ 2 } f \left ( \frac { x } { \sigma ^ 2 } \right ) \right ) \nu \left ( \frac { x } { \sigma ^ 2 } \right ) d x \\ & = \int _ 0 ^ \infty \Phi ( K _ t d _ { \sigma ^ 2 } f ( x ) ) \nu ( x ) d x . \end{align*}


\begin{align*} \alpha = \alpha _ 0 \beta _ 1 \alpha _ 1 \cdots \beta _ { n } \alpha _ n \end{align*}


\begin{align*} P _ 1 \unlhd { P _ 1 P _ 2 } \unlhd { P _ 1 P _ 2 P _ 3 } \unlhd \cdots \unlhd { P _ 1 P _ 2 \cdots P _ s } = G \end{align*}


\begin{align*} \langle \nabla ^ s u , \nabla ^ s \phi \rangle _ { L ^ 2 ( \R ^ { 2 n } ) } = \langle ( - \Delta ) ^ { s / 2 } u , ( - \Delta ) ^ { s / 2 } \phi \rangle _ { L ^ 2 ( \R ^ n ) } \end{align*}


\begin{align*} F _ { m , Q } ( \gamma _ 2 ^ { - 1 } z , s ) = \sum _ { \gamma \neq I , \gamma _ 2 ^ { - 1 } } f _ m ( \gamma z , s ) \end{align*}


\begin{align*} v ( \Box \phi , w ) & = \inf \limits _ { w ' \in W } \{ w R w ' \rightarrow _ \mathsf { G } v ( \phi , w ' ) \} , & v ( \lozenge \phi , w ) & = \sup \limits _ { w ' \in W } \{ w R w ' \wedge _ \mathsf { G } v ( \phi , w ' ) \} . \end{align*}


\begin{align*} C ( \tau , t , x ) = \begin{cases} \frac { \partial \tau ( t , x ) } { \partial x } & \boldsymbol { R } ( \tau ; t , x ) = \tau \\ \frac { \partial \boldsymbol { R } ( \tau ; t , x ) } { \partial x } & \boldsymbol { R } ( \tau ; t , x ) \neq \tau \end{cases} \end{align*}


\begin{align*} v = W _ t ^ \gamma u _ 0 - T [ W _ t ^ \gamma u _ 0 , v , \cdots , v ] + G + z , \end{align*}


\begin{align*} Q _ d ( \alpha , A , z + a f _ k ) = K ( \alpha , A , z + a f _ k ) \cap \left ( d \widetilde { z } + \operatorname { s p a n } \{ \widetilde { x } ^ t _ s : \ ; s \in \mathbb { N } , \ : t \in A \cup \{ k \} \} \right ) . \end{align*}


\begin{align*} D _ r : = Q _ r ( h _ \mathrm { M F } - \mu _ + ) Q _ r , D _ \ell : = Q _ \ell ( h _ \mathrm { M F } - \mu _ + ) Q _ \ell \end{align*}


\begin{align*} \sigma ^ { * } ( \mathcal { B } _ { 2 , k } ) & = \left \{ 2 \cos \left ( \frac { h } { k + 1 } \pi \right ) \colon 1 \le h \le k \right \} . \end{align*}


\begin{align*} \widehat E ^ \dag = \widehat E - \{ e _ 0 \} = \{ e _ 1 , \cdots , e _ r , \widehat e _ { r + 1 } , \cdots , \widehat e _ { r + s } \} . \end{align*}


\begin{align*} \Lambda _ i ^ t A _ i \Lambda _ i = ( \Lambda _ i ' ) ^ t A _ i \Lambda ' _ i = B , \end{align*}


\begin{align*} \frac { 1 } { \lambda } < \hat { \Lambda } \cdot F = \frac { \mu } { n } . \end{align*}


\begin{align*} n = x ^ 2 + x y + y ^ 2 + 1 6 z ^ 2 , \end{align*}


\begin{align*} c _ 3 = \frac 1 4 d ^ 2 - \left ( 1 - \frac { c _ 1 } { 2 d } \right ) \left ( \frac 1 2 d + c _ 2 d \right ) ^ 2 - \frac { c _ 1 } { 2 d } \left ( \frac 1 2 d + c _ 2 d - \frac 1 2 \right ) ^ 2 > 0 . \end{align*}


\begin{align*} \begin{aligned} & \int _ { { \C } _ 1 } \left ( \sum _ { i = 1 } ^ 4 f ( | y - h _ n t _ { i } | ) ( y - h _ n t _ { i } ) \cdot t _ { i } \right ) ^ 2 d y \\ & \ge \int _ { B _ 1 ( 0 ) } \left ( f ( | z | ) ( z \cdot t _ { 1 } ) \right ) ^ 2 d z + o _ n ( 1 ) \ge c _ 0 > 0 \ \textrm { f o r s o m e c o n s t a n t } \ c _ 0 > 0 , \end{aligned} \end{align*}


\begin{align*} f ( z ) = \int _ { 0 } ^ { z } \frac { G ( t ) \psi ( \omega ( t ) ) } { t } d t . \end{align*}


\begin{align*} E [ ( N ^ { n , 1 } _ { t } ) ^ 2 ] = \sum ^ { \lfloor n t \rfloor } _ { k = 0 } \int ^ { 1 \wedge ( n t - k ) } _ { 0 } \left [ ( n t - k ) ^ { \alpha } - ( n t - k - y ) ^ { \alpha } \right ] ^ 2 \ , d y , \end{align*}


\begin{align*} T x - T y = ( c ( x ) E x , s ( x ) F x ) - ( c ( y ) E y , s ( y ) F y ) \end{align*}


\begin{align*} { { \tilde { \psi } } } _ { \alpha , i , s } = { \rm { E v a l } } _ { t _ { i , s } \rightarrow \alpha } \bigcirc _ { { { w } } = 0 } ^ { s } ( \theta _ { t _ { i , s } } + \gamma _ { r - s + { { w } } } ) ^ { - 1 } : K [ t _ { i , s } ] _ { \substack { 1 \leq i \leq m \\ 0 \leq s \leq r - 1 } } \longrightarrow K [ t _ { i ' , s ' } ] _ { ( i ' , s ' ) \neq ( i , s ) } ; \ \ t ^ k _ { i , s } \mapsto \dfrac { \alpha ^ k } { \prod _ { { { w } } = 0 } ^ s ( k + \gamma _ { r - s + { { w } } } ) } \enspace . \end{align*}


\begin{align*} L = L ( B ) = \left \{ B x \mid x \in \mathbb { Z } ^ { n } \right \} , \end{align*}


\begin{align*} \rho ' _ a = r \circ \rho _ a \circ r ^ { - 1 } \end{align*}


\begin{align*} F ( z + \omega _ 2 \ , | \ , \overline \omega _ 1 , \omega _ 2 ) F ( z \ , | \ , \overline \omega _ 1 , - \omega _ 2 ) = \exp \Bigg ( \int _ { C } \frac { e ^ { ( z + \omega _ 2 ) s } } { ( e ^ { \overline \omega _ 1 s } - 1 ) ( e ^ { \omega _ 2 s } - 1 ) } \frac { d s } { s } + \int _ { c \cdot C } \frac { e ^ { z s } } { ( e ^ { \overline \omega _ 1 s } - 1 ) ( e ^ { - \omega _ 2 s } - 1 ) } \frac { d s } { s } \Bigg ) . \end{align*}


\begin{align*} H ( t ) : = ( F * G ) ( t ) = \int _ { 0 } ^ { t } F ( t - s ) G ( s ) \ , \textnormal { d } s . \end{align*}


\begin{align*} \mathbb { E } [ \ell , k ] = \left . \left [ k \dbinom { \ell } { k } + \sum \limits _ { i = 0 } ^ { ( \ell - k - 1 ) / 2 } \dbinom { \ell } { k + 2 i + 1 } \right ] \middle / \dbinom { \ell } { k } \right . \end{align*}


\begin{align*} P ^ f _ { H , d } \colon = R _ d ^ { T ^ * } ( T P ^ f _ H ) \ , . \end{align*}


\begin{align*} S ^ { i , 1 } ( \psi _ 1 , \psi _ 2 , \psi _ 3 ) & = \textstyle \sum _ \ell ( d ( x ^ \ell \psi _ \ell ) - x ^ \ell d \psi _ \ell ) = \sum _ \ell d x ^ \ell \wedge \psi _ \ell , \\ S ^ { i , 2 } \omega & = ( d x ^ 1 \wedge \omega , d x ^ 2 \wedge \omega , d x ^ 3 \wedge \omega ) . \end{align*}


\begin{align*} \bar c _ i ( \alpha , g ) = \bar c _ i ( \alpha , h ) . \end{align*}


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


\begin{align*} A ( Y , Z ) = \sum _ { i = 1 } ^ m g _ X ( Y , e _ i ) A ( e _ i , Z ) = \sum _ { i = 1 } ^ m \sum _ { j = 1 } ^ m g _ X ( Y , e _ i ) g _ X ( Z , e _ j ) A ( e _ i , e _ j ) . \end{align*}


\begin{align*} L _ n ^ { \alpha } ( x ; q ) : = \frac { 1 } { ( q ; q ) _ n } \ , _ 2 \phi _ 1 \left ( \begin{array} { c c c c } q ^ { - n } , - x \\ 0 \end{array} \mid q ; q ^ { n + \alpha + 1 } \right ) , \alpha > - 1 , n \in \mathbb { N } \end{align*}


\begin{align*} \begin{aligned} \left | \int _ { \Omega } f ( x , v _ n ) ( v _ n - v ) d x \right | & \leqslant c _ 1 \int _ { \Omega } \left | v _ n \right | ^ { q ( x ) - 1 } \left | v _ n - v \right | d x \\ & \leqslant c _ 1 \left | \left | | v _ n | ^ { q ( x ) - 1 } \right | \right | _ { \frac { q ( x ) } { q ( x ) - 1 } } \left | \left | v _ n - v \right | \right | _ { q ( x ) } \longrightarrow 0 . \end{aligned} \end{align*}


\begin{align*} T ^ n _ \tau & : = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s \left ( \sigma ' ( X _ s ) \right ) ^ 2 \sigma ( X _ s ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) \\ & \times \left ( \int _ 0 ^ { \eta _ n ( s ) } \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) d s . \end{align*}


\begin{align*} \widehat { \mathfrak { R } } _ { k , n } ( \xi ) : = \int _ { \R ^ 3 } e ^ { i y \cdot \xi } 2 ^ { 3 k + 2 n } ( 1 + 2 ^ k | y \cdot \tilde { \zeta } | ) ^ { - N _ 0 ^ 3 } ( 1 + 2 ^ { k + n } ( | y \cdot \tilde { \zeta } _ 1 | + | y \cdot \tilde { \zeta } _ 2 | ) ) ^ { - N _ 0 ^ 3 } d y = \widehat { \mathfrak { R } } ( 2 ^ { - k } ( \xi \cdot \tilde { \zeta } ) , 2 ^ { - k - n } ( \xi \cdot \tilde { \zeta } _ 1 ) , 2 ^ { - k - n } ( \xi \cdot \tilde { \zeta } _ 2 ) ) , \end{align*}


\begin{align*} \tilde { \Omega } _ { L _ i } \geq 0 , \int _ { \mathcal { M } } \tilde { \Omega } _ { L _ i } = \int _ { L _ i } \Omega , i = 0 , 1 , \ldots N . \end{align*}


\begin{align*} { \rm r o w } ( A ) = \left ( \ \sum _ { j \in \Z } a _ { 1 j } , \ \dots , \ \sum _ { j \in \Z } a _ { m j } \right ) , { \rm c o l } ( A ) = \left ( \ \sum _ { i \in \Z } a _ { i 1 } , \ \dots , \ \sum _ { i \in \Z } a _ { i n } \right ) \end{align*}


\begin{align*} \left \{ \begin{array} { c } ( w _ 1 , \ldots , w _ d , 0 , \ldots , 0 , \ldots ) , \mbox { w i t h $ d \geq 2 $ a n d w i t h } \\ \mbox { $ w _ 1 \geq w _ 2 \geq \cdots \geq w _ d > 0 $ s u c h t h a t $ \sum _ { i = 1 } ^ d w _ d = 1 $ . } \end{array} \right . \end{align*}


\begin{align*} r _ k ( M ) = \bigcup _ { n < k } M ( n ) . \end{align*}


\begin{align*} ( \partial _ s + \Delta ) w = I _ 1 + I _ 2 . \end{align*}


\begin{align*} \mathrm { F i x } ( Q ) = \cap _ { i = 1 } ^ { m } C _ { i } \not = \emptyset . \end{align*}


\begin{align*} Q B _ { \beta , \infty } : = \bigcup _ { p > 0 } Q B _ { \beta , p } \end{align*}


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


\begin{align*} { } ^ L H _ S : = ( H ( E ) \rtimes { } ^ L ( Z _ H ) _ S ) / Z _ H ( E ) , \end{align*}


\begin{align*} a ( \xi ) = \frac { \abs { \xi _ n } ^ s } { \abs { \xi } ^ s } = \left ( \frac { \xi _ n ^ 2 } { \xi _ 1 ^ 2 + \cdots + \xi _ n ^ 2 } \right ) ^ { s / 2 } , \xi \neq 0 , \end{align*}


\begin{align*} f ( q ) = \frac { 1 } { q \log q } = \frac { 1 } { q } \frac { e ^ \gamma } { \mu _ q } \prod _ { p < q } \Big ( 1 - \frac { 1 } { p } \Big ) = \frac { e ^ \gamma } { \mu _ q } { \rm d } ( { \rm L } _ q ) . \end{align*}


\begin{align*} \phi ^ { ( 0 ) } \Big ( + , \frac { k } { 2 } \Big ) \Big \vert _ { D _ { - k } \oplus D _ k } = 0 , \ ; \ ; \ ; k \in 2 \mathbb { Z } + 1 . \end{align*}


\begin{align*} u _ { p + \frac { [ j _ 1 + j _ 2 ] } { T } } w _ 1 ( n ) w _ 2 = \sum _ { i \in \Lambda } x _ i ( n _ i ) w _ 2 , \end{align*}


\begin{align*} A ^ k = \sum _ { i = 0 } ^ { n - 1 } \binom { k } { i } N ^ i \end{align*}


\begin{align*} L _ { { \bf k } } = L _ 1 ^ { k _ 1 } \otimes \ldots \otimes L _ m ^ { k _ m } \end{align*}


\begin{align*} \psi _ { \mu } ( z ) = \int _ { \mathbb { R } _ { + } } \frac { z t } { 1 - z t } \ , d \mu ( t ) , \quad \eta _ { \mu } ( z ) = \frac { \psi _ { \mu } ( z ) } { 1 + \psi _ { \mu } ( z ) } , z \in \mathbb { C } \backslash \mathbb { R } _ { + } , \end{align*}


\begin{align*} I _ { g , k , \ell } ( c ) = h ^ 4 \int _ { \mathbb { R } ^ 4 } w ( x ) H ( h ^ { - 1 } k , F ( x ) ) e _ { k \ell } ( - h c \cdot g x ) \ , d x = I _ { k , \ell } ( { } ^ t g c ) . \end{align*}


\begin{align*} & \| P _ { \delta } ( W _ { n ; \textrm { i n } } ) e ^ { - i H _ 0 t _ n } \psi \| = \| P _ { \delta } ( W _ { n ; \textrm { i n } } ) e ^ { - i H _ 0 t _ n } \chi _ { S _ R } \psi \| \\ & \leq \| \chi _ { S _ R } e ^ { i H _ 0 t _ n } P _ { \delta } ( W _ { n ; \textrm { i n } } ) \| _ \textrm { o p } \| \psi \| \xrightarrow { n \rightarrow \infty } 0 \end{align*}


\begin{align*} v _ 1 ( \triangle ( p \rightarrow { \sim } \neg p ) ) & = \begin{cases} 1 & $ p $ \\ 0 & $ p $ \end{cases} \end{align*}


\begin{align*} \begin{cases} & \frac { \partial X } { \partial t } = \psi u ^ { \alpha } \rho ^ \delta \sigma _ { k } ^ { \frac { \beta } { k } } ( \l ) \nu ( x , t ) , \\ & X ( \cdot , 0 ) = X _ 0 , \end{cases} \end{align*}


\begin{align*} \mathbb { \hat { E } } [ \varphi ( X ) ] = \mathbb { \hat { E } } [ \varphi ( Y ) ] . \end{align*}


\begin{align*} a _ { i j } ( x ) D _ { i j } ( D _ { r } u ) ( x ) = D _ { r } g ( x ) \quad \ | x | > r _ 0 , \end{align*}


\begin{align*} A ( \cdot , u , p ) = g _ { x x } ( \cdot , Y ( \cdot , u , p ) , Z ( \cdot , u , p ) ) , \ B ( \cdotp , u , p ) = \det E ( \cdot , Y ( \cdot , u , p ) , Z ( \cdot , u , p ) ) \psi ( \cdot , u , p ) . \end{align*}


\begin{align*} \overline { f } ^ { \otimes m } = \overbrace { \overline { f } \otimes \overline { f } \otimes \cdots \otimes \overline { f } } ^ { m } , \ \ m \in \mathbb { Z } , \ \ m \geq 1 . \end{align*}


\begin{align*} \mathcal E ^ s : = \{ \mathcal E _ \varphi ^ s \} . \end{align*}


\begin{align*} u _ { i } ( t ) = \beta _ { i } u _ { i } ( t _ { k } ) + \alpha _ { i } ( t - t _ { k } ) , \forall t \in ( t _ { k } , t _ { k + 1 } ] , \end{align*}


\begin{align*} \left [ \frac { 1 } { 2 } \mathfrak { a } ^ { T } K \mathfrak { a } , \prod \limits _ { l = 1 } ^ m f _ { l } ^ { T } \mathfrak { a } \right ] = \sum \limits _ { i = 1 } ^ m \prod \limits _ { l = 1 } ^ m f _ { l } ^ { T } ( J K ) ^ { \delta _ { i l } } \mathfrak { a } . \end{align*}


\begin{align*} { M } _ i ( e _ i ) = \frac { 1 } { \abs { G L _ { e _ i } ( \kappa _ i ) } } \left ( \frac { \abs { W _ i ^ \tau } \abs { H ^ 1 ( G , V _ i ^ \vee ) } } { \abs { H ^ 1 ( G , V _ i ) } } \right ) ^ { e _ i } . \end{align*}


\begin{align*} \norm { u } _ { H ^ { s , p } ( \Omega ) } \vcentcolon = \inf \{ \ , \norm { w } _ { H ^ { s , p } ( \R ^ n ) } \ , ; \ , w \in H ^ { s , p } ( \R ^ n ) , w | _ \Omega = v \ , \} . \end{align*}


\begin{align*} u - T _ { A , q } u = v \mbox { i n } H ^ 1 ( D ) . \end{align*}


\begin{align*} \| u \| ^ p _ { W ^ { k , p } _ \delta ( M ) } : = \sum _ { j = 0 } ^ { k } \left \vert \left \vert \rho ^ { - \delta - \frac { n } { p } + j } \left \vert \hat \nabla ^ j u \right \vert _ { \hat g } \right \vert \right \vert ^ p _ { L ^ p ( M , \hat g ) } \end{align*}


\begin{align*} \mathfrak { T } ( t ) e ^ { z t } = \sum _ { n = 0 } ^ { \infty } \frac { w _ n ( z ) } { n ! } t ^ n , \end{align*}


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


\begin{align*} r = \frac { \dim M - \dim \left ( \ker \omega _ x \right ) } { 2 } + \dim ( T _ x { \mathcal L } \cap \ker \omega _ x ) , \hbox { f o r a l l } \ ; x \in M \ , . \end{align*}


\begin{align*} \psi _ n ( y ) = \bar \psi \left ( y - h _ n t _ 1 \right ) + \bar \psi \left ( T _ 5 \left ( y - h _ n t _ 1 \right ) \right ) + \bar \psi \left ( T _ 9 \left ( y - h _ n t _ 1 \right ) \right ) . \end{align*}


\begin{align*} \langle \Theta \nabla ^ s u , \nabla ^ s \phi \rangle _ { L ^ 2 ( \R ^ { 2 n } ) } = & \frac { 1 } { 2 } \int _ { \R ^ n } \gamma ^ { 1 / 2 } ( x ) \left [ \phi ( x ) ( - \Delta ) ^ s ( \gamma ^ { 1 / 2 } u ) ( x ) + u ( x ) ( - \Delta ) ^ s ( \gamma ^ { 1 / 2 } \phi ) ( x ) \right . \\ & \quad \left . - ( - \Delta ) ^ { s } ( \gamma ^ { 1 / 2 } u \phi ) ( x ) - u ( x ) \phi ( x ) ( - \Delta ) ^ s m ( x ) \right ] \ , d x . \end{align*}


\begin{align*} z ^ { - \mu } z ^ { \rho } : = \exp ( - \mu \log z ) \exp ( \rho \log z ) \end{align*}


\begin{align*} D _ { m , n } ( x , y ) : = \sigma _ { m , 2 m ; n , 2 n } ( f ; x , y ) - f ( x , y ) . \end{align*}


\begin{align*} Q ( g ) : = D L _ { ( g , P ( g ) ) } ( 0 , \cdot ) . \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 , 4 } _ s d s \\ & = \frac { \alpha } { 2 ( \alpha + 1 ) ^ 2 } \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*} g \left ( x \right ) = \sum _ { z \in \mathbb { Z } ^ { d } } \gamma ^ { d } f \left ( x + \gamma z \right ) \ , x \in \mathbb { R } ^ { d } \ . \end{align*}


\begin{align*} J = J ( n ) : = \left \lfloor 4 \frac { \log \log n } { \log ( b ( n ) ) } - 1 \right \rfloor , \end{align*}


\begin{align*} E ( X _ 2 ^ 2 ) = \varepsilon ^ 2 \big [ P ( \boldsymbol { X } = \mathbf { s } ( 2 ) ) + P ( \boldsymbol { X } = \mathbf { s } ( 5 ) ) \big ] , \end{align*}


\begin{align*} \int _ C \frac { 1 } { e ^ { 2 \pi i ( s + \bar { f } ) } - 1 } q ^ { s + \bar { f } } \frac { \Gamma ( 1 + \frac { d H } { 2 \pi i } + s d + \bar { f } d ) } { \Gamma ( 1 - \frac { d ^ \prime H } { 2 \pi i } - s d ^ \prime + \bar { f } d ^ \prime ) \prod _ { i = 1 } ^ N \Gamma ( 1 + \frac { w _ i H } { 2 \pi i } + w _ i s + \bar { f } w _ i ) } , \end{align*}


\begin{align*} \langle v , K _ { 1 1 } u \rangle = \ ; & \frac { 1 } { 2 } \langle v \otimes u _ 1 \ , , \ , w \ , u _ 1 \otimes u \rangle \\ \langle v , K _ { 2 2 } u \rangle = \ ; & \frac { 1 } { 2 } \langle v \otimes u _ 2 \ , , \ , w \ , u _ 2 \otimes u \rangle \\ [ 2 m m ] \langle v , K _ { 1 2 } u \rangle = \ ; & \langle v \otimes u _ 1 \ , , \ , w \ , u _ 2 \otimes u \rangle . \end{align*}


\begin{align*} \Bar \Phi \circ Q _ \mathfrak { g } = \Bar { Q } \circ \Bar \Phi . \end{align*}


\begin{align*} & \varphi _ { n ; \textrm { o u t } } = P _ { \delta } ( W _ { n , m ; \textrm { o u t } } ) \varphi _ n & & \varphi _ { n ; \textrm { i n } } = P _ { \delta } ( W _ { n , m ; \textrm { i n } } ) \varphi _ n \\ & \varphi _ { n ; \textrm { s u r } } = P _ { \delta } ( W _ { n , m ; \textrm { s u r } } ) \varphi _ n \end{align*}


\begin{align*} a ( u , J I { w } ) - a _ { \mathrm { p w } } ( { w } , z _ { \mathrm { n c } } ) & = b ( \lambda u , J I { w } ) - a _ { \mathrm { p w } } ( I { w } , z _ { \mathrm { n c } } ) = \lambda b ( u , J I { w } - I { w } - \delta I { w } ) . \end{align*}


\begin{align*} \lim \limits _ { t \rightarrow \infty } \| e ^ { - i t H _ 0 } \varphi - e ^ { - i t H } \psi \| = 0 \end{align*}


\begin{align*} \Lambda _ { \mathrm { m o n } } ^ 2 : = \frac { 1 + 1 / \alpha + ( 1 + \alpha ) ( \Lambda _ 1 ^ 2 + \Lambda _ 2 ^ 2 ) ( \Lambda _ 3 + \epsilon _ 3 ) } { 1 - ( 1 + \alpha ) ( \Lambda _ 1 ^ 2 + \Lambda _ 2 ^ 2 ) \widehat \Lambda _ 3 } . \end{align*}


\begin{align*} X _ j ^ \mu = \partial _ { x _ j } + \tfrac i 2 \mu ( [ x , X _ j ] ) . \end{align*}


\begin{align*} \beta _ i ( z ) & : = \beta _ { i - 1 } ( z ) + \sum _ { m = 1 } ^ { s _ i } \frac 1 m \sum _ { a | m } \mu \left ( \frac m a \right ) \left ( \alpha _ { i - 1 } ( z ^ { m / a } ) \right ) ^ a \ ! \ ! , \\ \alpha _ i ( z ) & : = 1 + ( k z - 1 ) \cdot { \mathcal E } ( \beta _ i ( z ) ) , 1 \le i \le q . \end{align*}


\begin{align*} f = a _ 0 + a _ 1 Q + \ldots + a _ r Q ^ r = l ( Q ) \end{align*}


\begin{align*} \Phi ( S ) = \left \{ \begin{array} { l l } S \setminus \{ p \} , & \mbox { i f $ p \in S $ , } \\ S \cup \{ p \} , & \mbox { i f $ p \not \in S $ } \end{array} \right \} , \ \ \mbox { f o r } S \subseteq \{ 1 , \ldots , k \} . \end{align*}


\begin{align*} \nabla ^ \Sigma _ { \alpha \beta } F = | \nabla u | ^ { - 2 } A _ { \alpha \beta } . \end{align*}


\begin{align*} \phi _ \mathrm { d c v } ( x , z ) = \phi _ + ( x , z ) - h ( x ) , \phi _ \mathrm { p c v } ( x , z ) = \phi _ - ( x , z ) - h ( x ) . \end{align*}


\begin{align*} \gamma _ { p , i } ^ k & = 2 ^ { \gamma _ { p , i } ^ \star } - 1 , i \in \mathcal K , & s ^ k = 2 ^ { s ^ \star } - 1 \end{align*}


\begin{align*} x \tilde \delta _ \alpha ( a ) = \alpha ^ { - 1 / 2 } \delta ( x \zeta _ \alpha ( a ) ) - \alpha ^ { - 1 / 2 } \delta ( x ) \zeta _ \alpha ( a ) = : A _ 1 - A _ 2 . \end{align*}


\begin{align*} X \otimes ^ G Y ( \underbrace { \omega _ g , . . . } _ { g \in G } ) = X ( \underbrace { \omega _ { \psi ^ { P , G } [ p ] } , . . . } _ { p \in P } ) \otimes Y ( \underbrace { \omega _ { \psi ^ { Q , G } [ q ] } , . . . } _ { q \in Q } ) . \end{align*}


\begin{align*} \mu \int _ \Omega \varepsilon ( w _ \delta - z _ \delta ) : \varepsilon v + \nu \int _ \Omega ( \mathsf { m } _ \delta ( \varepsilon w _ \delta ) - \mathsf { m } _ \delta ( \varepsilon z _ \delta ) ) : \varepsilon v = \langle u _ 1 - u _ 2 , v \rangle , \forall v \in Y . \end{align*}


\begin{align*} \partial _ \mu ^ 2 E ( \lambda , \mu ) & = - \lim _ { T \to \infty } \frac 1 T \partial _ \mu ^ 2 ( \ln Z ^ { ( W ) } _ T ( \lambda , \mu ) + 1 ) \\ & = - \lim _ { T \to \infty } \frac 1 T \frac { \partial _ \mu ^ 2 Z ^ { ( W ) } _ T ( \lambda , \mu ) } { Z ^ { ( W ) } _ T ( \lambda , \mu ) } . \end{align*}


\begin{align*} & a _ \ell ( c _ i ^ 2 - c _ i ) - a _ k ( b _ i ^ 2 - b _ i ) = 0 i > \ell \\ & a _ \ell c _ i c _ j - a _ k b _ i b _ j = 0 i > \ell , j > i . \end{align*}


\begin{align*} \{ f , g \} ( x ) \ ; : = \ ; \langle [ f ' ( x ) , g ' ( x ) ] , x \rangle \mbox { f o r $ x \in U $ , } \end{align*}


\begin{align*} \varphi ( \exp ( \lambda _ \alpha v _ \alpha ) , x ) = \varphi _ \alpha ( \lambda _ \alpha , x ) \forall \ , \alpha = 1 , \dots , r , \forall x \in N , \end{align*}


\begin{align*} v = ( v _ 1 , \ldots , v _ n ) ^ t : = \delta ^ { ( s ) } c : = ( \delta ^ { ( s ) } c _ 1 , \ldots , \delta ^ { ( s ) } c _ n ) ^ t \in R ^ n \end{align*}


\begin{align*} P ( u ) \cdot w = \begin{bmatrix} 0 \\ g \end{bmatrix} , w ( x ) \perp \mathrm { K e r } P ( u ) ( x ) , \end{align*}


\begin{align*} \int s W _ + ( \omega , \omega ) d \mu = \int \Big [ 8 | W _ + | ^ 2 - 4 | W _ + ( \omega ) | ^ 2 + 2 [ W _ + ( \omega , \omega ) ] ^ 2 \Big ] d \mu . \end{align*}


\begin{align*} \delta _ n ( \beta ) : = \dfrac { f ' ( v _ n ) } { \beta } \mod \mathcal { D } _ n , \end{align*}


\begin{align*} \underset { n \rightarrow \infty } { } \int _ { \Omega } \varepsilon ( z + t _ n w ) : \varepsilon v = \int _ { \Omega } \varepsilon ( z + t w ) : \varepsilon v . \end{align*}


\begin{align*} \Lambda = ( \lambda _ 1 , \lambda _ 2 ) \subseteq K \end{align*}


\begin{align*} e _ { \lambda } ^ { x } ( t ) = \sum _ { n = 0 } ^ { \infty } ( x ) _ { n , \lambda } \frac { t ^ { n } } { n ! } , ( \mathrm { s e e } \ [ 1 1 , 1 4 ] ) . \end{align*}


\begin{align*} \sum _ { u = i } ^ s q _ { \alpha _ u } \alpha _ u \alpha ^ * _ u - \sum _ { v = i } ^ t \beta _ v \beta _ v ^ * \end{align*}


\begin{align*} \mu _ { \beta , N } ( d x ) = \frac { 1 } { Z _ { \beta , N } } e ^ { - \frac { \beta } { 4 } \sum _ { i = 1 } ^ N x _ i ^ 2 } \prod _ { 1 \le i < j \le N } | x _ i - x _ j | ^ { \beta } \prod _ { i = 1 } ^ N d x _ i , \end{align*}


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


\begin{align*} \Vert g _ { \le \iota } ^ { ( 1 ) } \Vert _ p = \bigg \Vert \sum _ { \ell = - 1 } ^ \iota \sum _ { m = 1 } ^ { M _ { \ell } } \tilde g _ { m } ^ { ( \ell ) } \bigg \Vert _ p \lesssim 2 ^ { - \varepsilon \iota } \norm { F ^ { ( \iota ) } } _ { L ^ 2 _ s } \norm { f } _ p , \end{align*}


\begin{align*} \label [ I e q ] { e q : k a c c h a o s } \lim _ { N \to + \infty } \langle f ^ N , \varphi _ k \otimes 1 ^ { \otimes N - k } \rangle = \langle f ^ { \otimes k } , \varphi _ k \rangle . \end{align*}


\begin{align*} \left ( g _ u \circ f _ u \right ) ( b _ n ) = k _ n . \end{align*}


\begin{align*} [ f ] \in \mathbb K / \mathbb L \mapsto v _ f = \sum _ { i = 1 } ^ m \lceil - ( D _ i \cdot f ) \rceil b _ i \in N , \end{align*}


\begin{align*} \mathcal { J } = \mathcal { J } _ \delta : = \{ i \in [ N ] : \ , \gamma _ i ( 0 ) \in \mathcal { I } _ \delta \} , \mathcal { I } _ \delta : = ( - 2 + \delta , 2 - \delta ) , \end{align*}


\begin{align*} s _ i ( h _ i , \kappa ) : = 2 h _ i - h _ i ^ 2 \frac { - \kappa } { 2 \min \left ( 1 , \frac { 1 } { h _ i } \right ) - ( 1 + \kappa ) } \end{align*}


\begin{align*} \alpha _ { a } ( e ) & = \theta ( a ) e \theta ( a ) ^ { * } & & \\ \theta ( \sigma a \sigma ^ { - 1 } ) & = \overline { \alpha } _ { \sigma } ( \theta ( a ) ) & & \end{align*}


\begin{align*} \left | \iint _ { Q _ T } ( \vec { b } \cdot \nabla m ) \Delta p \right | = \left | \iint _ { Q _ T } - \nabla ( \vec { b } \cdot \nabla m ) \cdot \nabla p \right | \leq C \iint _ { Q _ T } | \nabla p | \leq C . \end{align*}


\begin{align*} \left | \mathcal T _ { \kappa , 2 } \right | = \sum _ { i = 0 } ^ { \lfloor \kappa / 2 \rfloor } ( \kappa - 2 i + 1 ) . \end{align*}


\begin{align*} \frac { d \beta } { d t } ( t ) = \beta ^ 2 ( t ) + \omega ^ 2 ( t ) , \frac { d \alpha } { d t } ( t ) + 2 \beta ( t ) = 0 , \end{align*}


\begin{align*} \hat { P } \phi _ { 0 } ( x ) = \hat { T } \phi _ { 0 } ( x ) = \phi _ { 0 } ( x ) \end{align*}


\begin{align*} \begin{vmatrix} \det \left ( ( \lambda I _ N - H ) _ { k | k } \right ) & \det \left ( ( \lambda I _ N - H ) _ { k | k + 1 } \right ) \\ \det \left ( ( \lambda I _ N - H ) _ { k + 1 | k } \right ) & \det \left ( ( \lambda I _ N - H ) _ { k + 1 | k + 1 } \right ) \end{vmatrix} = f ( \lambda ) \det \left ( ( \lambda I _ N - H ) _ { k k + 1 | k k + 1 } \right ) , \end{align*}


\begin{align*} \abs { h ( z ) - z } \leq 6 \sum _ { n = 2 } ^ \infty \left ( \frac { 3 } { 2 } \abs { z } \right ) ^ n < 6 \cdot \frac { 9 } { 4 } \abs { z } ^ 2 \sum _ { n = 0 } ^ \infty \left ( \frac { 3 } { 4 } \right ) ^ n < 1 0 0 \abs { z } ^ 2 . \end{align*}


\begin{align*} \mathcal { S } ( L ) = \int _ L f _ L ( e ^ { - i \hat { \theta } } \Omega ) - \int _ { L _ 0 } f _ { L _ 0 } ( e ^ { - i \hat { \theta } } \Omega ) - \int _ { \mathcal { C } } \lambda \wedge e ^ { - i \hat { \theta } } \Omega . \end{align*}


\begin{align*} \mathbb { E } ( X _ i X _ j ) = \mathbb { P } ( P _ i , P _ j \ { \rm { a r e \ b o t h } } \ k - { \rm { f r e e } } ) . \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*} e ( t _ { k + 1 } ) = \left \{ \begin{aligned} & A _ { 1 } e ( t _ { k } ) & & \beta ^ { \top } e ( t _ { k } ) \geq 0 \\ & A _ { 2 } e ( t _ { k } ) & & \beta ^ { \top } e ( t _ { k } ) < 0 . \end{aligned} \right . \end{align*}


\begin{align*} V = \left [ - \frac { m \nabla p _ { \infty } } { ( m - \varrho ^ E ) } - \vec { b } \right ] \cdot \vec { \nu } , \end{align*}


\begin{align*} \rho ( \varepsilon ) = \frac { 2 b d k \log ( \frac { 4 b d } { \varepsilon } ) } { \varepsilon ^ 2 } + \frac { 2 k ^ d \log ( \frac { 4 k ^ d } { \varepsilon } ) } { \varepsilon ^ 2 } + \frac { \log ( \frac { 3 } { \delta } ) } { 2 \varepsilon ^ 2 } = n . \end{align*}


\begin{align*} e _ \delta ( z ) = \frac { \delta } { \sqrt \pi } \int _ 0 ^ z e ^ { - \frac { t ^ 2 } { 4 } } { \rm { d } } t , e _ \sigma ( z ) = \frac { \sigma } { \sqrt \pi } \int _ 0 ^ z e ^ { - \frac { t ^ 2 } { 4 } } { \rm { d } } t , \end{align*}


\begin{align*} \chi _ o ( Q _ d ) = \begin{cases} 2 & d \ , ; \\ 4 & d \ , . \end{cases} \end{align*}


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


\begin{align*} \psi ( u _ t T ) & = \varphi ( e _ M u _ t T e _ M ) = \varphi ( e _ M ( u _ t J u _ t J ) T ( J u _ t J ) ^ * e _ M ) = \varphi ( \alpha _ t ^ 0 e _ M T ( J u _ t J ) ^ * e _ M ) \\ & = \varphi ( e _ M T ( J u _ t J ) ^ * e _ M \alpha _ t ^ 0 ) = \varphi ( e _ M T ( J u _ t J ) ^ * u _ t J u _ t J e _ M ) = \psi ( T u _ t ) . \end{align*}


\begin{align*} d F = \Omega ( \cdot , v _ 1 , \ldots , v _ { n - 1 } ) . \end{align*}


\begin{align*} \eta ( e _ 1 , e _ 2 ) : = \dfrac { \partial \{ F , G \} } { \partial y } \ , \mu \otimes _ \mathcal { O } \dfrac { \partial } { \partial x } - \dfrac { \partial \{ F , G \} } { \partial x } \ , \mu \otimes _ \mathcal { O } \dfrac { \partial } { \partial y } . \end{align*}


\begin{align*} \ker \left ( \Gamma ( \varphi ^ * E _ { - 2 } ) \stackrel { \varphi ^ * \dd ^ { ( 2 ) } } { \longrightarrow } \Gamma ( \varphi ^ * E _ { - 1 } ) \right ) & \longrightarrow \ker \left ( d t _ 1 | _ { \Gamma ( \ker d s _ 1 ) } \right ) = \Gamma ( \ker d s _ 1 ) \cap \Gamma ( \ker d t _ 1 ) \subset \mathfrak { X } ( B _ 2 ) \end{align*}


\begin{align*} \mu \int _ { \Omega } \varepsilon z : \varepsilon v + \nu \int _ { \Omega } \mathsf { m } ' \left ( \varepsilon y ; \varepsilon z \right ) : \varepsilon v = \int _ { \Omega } h \cdot v , \forall v \in Y , \end{align*}


\begin{align*} \phi ( t ) = \| T ( t , A ) \| _ { \infty } \psi ( t ) = \| \tilde { B } T ( t , A ) \| _ q , \end{align*}


\begin{align*} ( v ( E ) , v ( F ) ) : = \chi ( E ^ \vee \otimes F ) \end{align*}


\begin{align*} \delta \in [ 0 , ( d - \alpha ) / 2 ] , \kappa = \kappa _ \delta , h ( x ) = h _ \delta ( x ) = | x | ^ { - \delta } , q ( x ) = q _ \delta ( x ) = \kappa | x | ^ { - \alpha } . \end{align*}


\begin{align*} \mathit { M } _ { \Phi } \doteq \left \{ \omega \in E _ { 1 } : f _ { \Phi } \left ( \omega \right ) = \inf \ , f _ { \Phi } ( E _ { 1 } ) = - \mathrm { P } _ { \Phi } \right \} \end{align*}


\begin{align*} \rho = \sum _ { k = 1 } ^ n a _ k \delta _ { \xi _ k } , n = \deg \theta . \end{align*}


\begin{align*} \mathcal { N } \bigg ( \begin{bmatrix} \mathbf { m } _ { j , } ^ k \\ [ 2 m m ] \mathbf { m } ^ k _ { j , } \end{bmatrix} , \begin{bmatrix} Z ^ k _ { j , } & Z ^ k _ { j , } \\ [ 2 m m ] Z ^ { k } _ { j , } & Z ^ k _ { j , } \end{bmatrix} \bigg ) , \end{align*}


\begin{align*} \sum _ { 1 \leq i < j \leq n } \mathbb { E } ( X _ i X _ j ) = \frac { 1 } { 2 } \Big ( \sum _ { 1 \leq i \leq n } f _ k ( i ) \Big ) ^ 2 + O _ { \alpha , \varepsilon } ( n ^ { 3 / 2 + \varepsilon } ) . \end{align*}


\begin{align*} \int _ { \Omega _ e } ( - \Delta ) ^ s m _ 0 f g \ , d x = 0 \end{align*}


\begin{align*} ( b \rightharpoonup f ) ( m ) = \sum \limits _ { ( b ) } b _ 0 f ( S ^ { - 1 } b _ { - 1 } \rightharpoonup m ) . \end{align*}


\begin{align*} A = \left \{ k \geq k _ 0 : \frac { 1 } { k } | B \cap \{ k _ 0 , \ldots , k \} | \geq \frac { 1 } { 2 } \right \} . \end{align*}


\begin{align*} \beta _ t = \sup _ { s \in [ 0 , t ] } \beta _ t ( s ) \leq 1 - 2 \epsilon , \alpha _ t = \sup _ { s \in [ 0 , t ] } \alpha _ t ( s ) \leq ( 2 / 3 + \iota ) - \epsilon . \end{align*}


\begin{align*} \phi _ n ( x ) = e ^ { - \gamma \ , \lambda _ n ^ { \frac { 1 } { 2 } } | x - P ^ i _ n | } + e ^ { - \gamma \ , \lambda _ n ^ { \frac { 1 } { 2 } } | x - P ^ j _ n | } , \end{align*}


\begin{align*} \Omega _ { \bullet } ( z , w ) : = & \sum _ { \lambda \in \mathcal { P } } \frac { N _ { \lambda } ( z w , z ^ 2 , w ^ 2 ) ^ { g - 1 } \tilde { N } _ { \lambda } ( z w , z ^ 2 , w ^ 2 ) } { \tilde { N } _ { \lambda } ( z ^ 2 , w ^ 2 ) } \prod ^ k _ { j = 1 } H _ { \lambda } ( \Z _ j ; z ^ 2 , w ^ 2 ) \\ \Omega _ { \ast } ( z , w ) : = & \sum _ { \lambda \in \mathcal { P } } \frac { N _ { \lambda } ( z w , z ^ 2 , w ^ 2 ) ^ { 2 g - 1 } } { N _ { \lambda } ( z ^ 2 , w ^ 2 ) } \prod ^ k _ { j = 1 } H _ { \lambda } ( \Z _ j ; z ^ 2 , w ^ 2 ) ^ 2 , \end{align*}


\begin{align*} \iint _ { Q _ T } \varphi _ t ( \varrho _ 1 - \varrho _ 2 ) + \nabla \cdot ( m \nabla \varphi ) ( p _ 1 - p _ 2 ) - \nabla \varphi \cdot \vec { b } ( \varrho _ 1 - \varrho _ 2 ) + \varphi f ( \varrho _ 1 - \varrho _ 2 ) d x d t = 0 . \end{align*}


\begin{align*} \sum \limits _ { i } ^ { N } \alpha _ i = 0 , \end{align*}


\begin{align*} \nu ( z ) : = \langle \phi ( z ) , \nabla g ( z ) \rangle , \end{align*}


\begin{align*} H ^ i ( F , A ( 1 ) ) \cong \begin{cases} H _ 0 ( F , A ) & i = 2 \\ 0 & i \ge 3 \end{cases} \end{align*}


\begin{align*} \nabla _ { \vec s , k } = \nabla _ { \frac { \partial } { \partial t _ { \vec s , k } } } = \frac { \partial } { \partial t _ { \vec s , k } } + \frac { 1 } { z } \tilde { T } _ { \vec s , k } \star _ t \end{align*}


\begin{align*} v ( \Delta \tau , w ) & = \begin{cases} 1 & v ( \tau , w ) = 1 \\ 0 & , \end{cases} & & v ( \Delta ^ \neg \phi , w ) & = \begin{cases} ( 1 , 0 ) & v ( \phi , w ) = ( 1 , 0 ) \\ ( 0 , 1 ) & . \end{cases} \end{align*}


\begin{align*} q _ A = k + o ( 1 ) . \end{align*}


\begin{align*} w _ t ^ * = g ^ { - 1 } ( s _ t | \rho , \lambda ) , ~ ~ ~ ~ ~ t = 1 , \ldots , n , \end{align*}


\begin{align*} \sup _ { \delta < | t | \leq \frac { \pi } { h } } \left \lvert \frac { \phi ( s + \mathrm { i } t ) } { \phi ( s ) } \right \rvert & = \sup _ { \delta < | t | \leq \frac { \pi } { h } } \left \lvert \prod _ { i = 1 } ^ n \frac { \phi _ i ( s + \mathrm { i } t ) } { \phi _ i ( s ) } \right \rvert \\ & \leq c _ 1 ^ n = o ( n ^ { - 1 / 2 } ) . \end{align*}


\begin{align*} K ^ { \phi } _ t p _ k ( x ) = \sum _ { l = 0 } ^ k \dbinom { k } { l } \frac { W _ \phi ( k + 1 ) } { W _ { \phi } ( l + 1 ) } e ^ { - t l } \left ( 1 - e ^ { - t } \right ) ^ { k - l } p _ l ( x ) . \end{align*}


\begin{align*} f _ - ^ * ( D _ - ) = \sum _ { i = 1 } ^ m a _ { i , - } \tilde { D } _ i - \left ( \sum _ { i = 1 } ^ m a _ { i , - } D _ i \cdot e \right ) E . \end{align*}


\begin{align*} \gamma \star \delta = \rho _ { g ^ { - 1 } \mu _ { t ^ { - 1 } } } ( \rho _ { \mu _ t g } ( \gamma ) \cup \rho _ { \mu _ t g } ( \delta ) ) . \end{align*}


\begin{align*} p ( t , v , g ) = \int _ 0 ^ { V _ F } \int _ { - \infty } ^ { \infty } e ^ t p _ { } ( t , \tilde { v } , e ^ t \tilde { g } ) \left [ \sum _ { k = - \infty } ^ { + \infty } \frac { 1 } { V _ F } e ^ { - k ^ 2 ( \frac { 2 \pi } { V _ F } ) ^ 2 D ( t ) + i k \frac { 2 \pi } { V _ F } ( v - \tilde { v } ) + i k \frac { 2 \pi } { V _ F } ( g - e ^ t \tilde { g } ) } \bar { G } _ { t , k } ( g - \tilde { g } ) \right ] d \tilde { v } d \tilde { g } . \end{align*}


\begin{align*} P _ 0 ( D ' _ k ) = \sum \limits _ { j = 0 } ^ n P _ 0 ( D _ j \cap D ' _ k ) . \end{align*}


\begin{align*} \phi _ { j } ( z ) = \int \frac { d \mu _ { j } ( x ) } { z - x } , 0 \leq j \leq p - 1 , \end{align*}


\begin{align*} H ( t , x ) \triangleq \begin{bmatrix} \vdots \\ h _ p ( \tau _ i , t , x ) \\ \vdots \end{bmatrix} , \forall \tau _ i \in \boldsymbol { M } ( t , x ) , \end{align*}


\begin{align*} \ker ( t _ 0 ^ * \rho ) & \longrightarrow \ker \left ( d t _ 0 | _ { \Gamma ( \ker d s _ 0 ) } \right ) = \Gamma ( \ker d s _ 0 ) \cap \Gamma ( \ker d t _ 0 ) \subset \mathfrak { X } ( B _ 1 ) \end{align*}


\begin{align*} \lambda ( v , v _ * ) & = \Phi ( | v - v _ * | ) \int _ 0 ^ \pi \Sigma ( \theta ) \dd \theta , \\ \Gamma ( v , v _ * , \dd z ' , \dd v _ * , \dd v _ * ' ) & = \psi ( v , v _ * , \cdot ) _ { \# } \left ( \frac { \Sigma } { \int _ 0 ^ \pi \Sigma ( \theta ) \dd \theta } \right ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : t r u e m a x w e l l m o l e c u l e s } \Phi ( | u | ) = 1 , \int _ 0 ^ \pi \Sigma ( \theta ) \dd \theta = + \infty . \end{align*}


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


\begin{align*} \mathbf t ( z ) = \sum \limits _ { l = 0 } ^ \infty t _ l z ^ l . \end{align*}


\begin{align*} \begin{array} [ c ] { c } \mathrm { R i c } _ { \omega ( t ) } ^ { T } + \sqrt { - 1 } \partial _ { B } \overline { \partial } _ { B } u ( t ) = - \widehat { \omega } _ { \infty } . \end{array} \end{align*}


\begin{align*} z = x + i y . \end{align*}


\begin{align*} \dot x & = v , \\ \dot v & = - \nabla \Phi ( x ) . \end{align*}


\begin{align*} \lim _ { n \to + \infty } \| a _ n - a \| _ \infty = 0 , \sup _ \Omega a < \lambda _ 1 , \sup _ { n \in \mathbb N } \left [ \| f _ n \| _ 1 + \| g _ n \| _ \infty \right ] < + \infty . \end{align*}


\begin{align*} \left [ \frac { \partial } { \partial z } , z \frac { \partial } { \partial w } \right ] = \frac { \partial } { \partial w } . \end{align*}


\begin{align*} H _ { ( \mathbb P ^ 3 , K 3 ) } ( y ) = e ^ { \frac { H \log y } { 2 \pi i } } \sum _ { d \geq 0 } y ^ d \frac { \Gamma ( 1 + \frac { 4 H } { 2 \pi i } + 4 d ) } { \Gamma ( 1 + \frac { H } { 2 \pi i } + d ) ^ 4 } [ \textbf { 1 } ] _ { 4 d } , \end{align*}


\begin{align*} \rho = \sum _ { k \ge 1 } a _ k \delta _ { \lambda _ k ^ 2 } \end{align*}


\begin{align*} T u ( \Omega ) : = Y ( \cdot , u , D u ) ( \Omega ) = \Omega ^ * , \end{align*}


\begin{align*} C _ { \tiny \rm F i n s l e r } ( T ) = C _ { \psi , \varphi } ( \mathbf { T } ) : = \inf _ { | \mathbf { y } ^ * | _ { f ( x ) , \psi } = 1 } | \mathbf { T } ^ * { \mathbf { y } ^ * } | _ { x , \varphi } \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , T ] } E \left [ | Q ^ { n , 5 } _ \tau | ^ 2 \right ] = 0 . \end{align*}


\begin{align*} \begin{aligned} \| \nabla f ( x _ N ) \| ^ 2 { } \leq { } 2 L \ , \big ( f ( x _ 0 ) - f _ * \big ) \max \left \{ ( 1 - h ) ^ { 2 N } \ , , \ , \frac { 1 } { 1 + 2 N h } \right \} \end{aligned} \end{align*}


\begin{align*} | \widetilde c _ p | = \frac { 1 } { | c _ p | } > 1 \ , \end{align*}


\begin{align*} A = A _ 1 \sqcup . . . \sqcup A _ \ell , \end{align*}


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


\begin{align*} S ^ n = n ^ { \alpha + \frac 1 2 } \int _ { [ \tau _ 1 , \tau _ 2 ] } ( t - s ) ^ \alpha \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) d s \end{align*}


\begin{align*} g _ { r , s , j } ( x ) = r ^ { j + \sum _ { i = 1 } ^ { m } \mu _ { i , j } } g _ j \left ( r x + s \right ) \end{align*}


\begin{align*} I I \le q \ , \mathfrak { b } \int _ \beta ^ { \infty } \lambda ^ { q - 2 } d \lambda = \frac { q } { 1 - q } \mathfrak { b } \beta ^ { q - 1 } . \end{align*}


\begin{align*} T _ { \lambda ^ 3 } : = 5 4 \lambda ^ 3 \ , C \star Q \star S , Q : = S \big ( S ^ 2 \star S ^ 2 \big ) . \end{align*}


\begin{align*} X _ j ^ \mu ( g \circ T _ { \bar \mu } ^ { - 1 } ) ( T _ { \bar \mu } z ) & = \nabla g ( z ) v ^ \mu _ j + \tfrac i 2 \omega _ \mu ( T _ { \bar \mu } z , X _ j ) g ( z ) \\ & = \nabla g ( z ) v ^ \mu _ j + \tfrac i 2 | \mu | \omega ( z , v ^ \mu _ j ) g ( z ) . \end{align*}


\begin{align*} L ^ { n , 3 } _ \tau & = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) ( s - \eta _ n ( s ) ) ^ \alpha \\ & \times \left [ ( W _ s - W _ { \eta _ n ( s ) } ) \left ( \int _ { \eta _ n ( s ) } ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) - \int _ { \eta _ n ( s ) } ^ s \psi _ { n , 1 } ( u , s ) d u \right ] d s . \end{align*}


\begin{align*} \sum _ { k = - \infty } ^ { M - 1 } 2 ^ { - k \frac { q } { p - q } } \bigg ( \int _ { x _ k } ^ { \infty } u \bigg ) ^ { \frac { p } { p - q } } V _ r ( 0 , x _ k ) ^ { \frac { p q } { p - q } } \approx \sum _ { k = - \infty } ^ { M - 1 } 2 ^ { - k \frac { q } { p - q } } \bigg ( \int _ { x _ k } ^ { \infty } u \bigg ) ^ { \frac { p } { p - q } } V _ r ( x _ { k - 1 } , x _ k ) ^ { \frac { p q } { p - q } } \end{align*}


\begin{align*} \varrho ( [ x , y ] _ \mathfrak { g } ) - [ \varrho ( x ) , \varrho ( y ) ] = \rho ( \eta ( x , y ) ) . \end{align*}


\begin{align*} \lim _ { | y | \to \infty } \| R ( x + i y , A ) \| _ { \infty } = 0 , \end{align*}


\begin{align*} \eta _ { \mu } ( z \Sigma _ { \mu } ( z ) ) = z , z \in \rho \mathbb { D } . \end{align*}


\begin{align*} I _ { X _ - } ( y , z ) = z e ^ { t _ - / z } \sum _ { d \in \mathbb K _ { - } } \tilde y ^ { d } \left ( \prod _ { i = 0 } ^ { m } \frac { \prod _ { a \leq 0 , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } { \prod _ { a \leq D _ i \cdot d , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } \right ) \textbf { 1 } _ { [ - d ] } , \end{align*}


\begin{align*} \widetilde { N _ H } ( n ) = C a r d ( \{ v : | v | = n , u \not \sqsubset v w _ 1 v u \in \mathcal { L } _ H \} ) . \end{align*}


\begin{align*} u ( 0 ) = u _ 0 , \ ; \partial _ t u ( 0 ) = v _ 0 \Omega ^ - . \end{align*}


\begin{align*} \frac { f ( x + t y ) - f ( x ) } { t } \ ; = \ ; g ^ \vee ( t ) \ ; \to \ ; g ^ \vee ( 0 ) \end{align*}


\begin{align*} ( ( \exp , 0 ) \cdot X _ 0 - x _ 0 ) ( ( f , a ) ) = \exp ( a ) . \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { \lambda _ N ^ 2 } \prod \limits _ { k = 1 } ^ { N - 1 } \Bigg ( \frac { \lambda _ N ^ 2 + \mu _ { k } ^ { 2 } } { \lambda _ N ^ 2 + \lambda _ { k } ^ { 2 } } \Bigg ) = \infty . \end{align*}


\begin{align*} \begin{aligned} ( \rho _ \alpha - \alpha ( n - 1 ) ) x _ { u _ 1 } & = ( 1 - \alpha ) ( k - 2 ) x _ { u _ 1 } + ( 1 - \alpha ) \sum _ { u u _ 1 \in E ( H ) } x _ u \\ & \leq ( 1 - \alpha ) ( k - 2 ) x _ { u _ 1 } + ( 1 - \alpha ) ( n - k + 1 ) x _ v \end{aligned} \end{align*}


\begin{align*} \forall x , y \in \Lambda , S _ t ( x , y ) : = \big < \varphi _ x \varphi _ y \big > _ { A , g , \nu + 1 / t } . \end{align*}


\begin{align*} J ( t , z ) = L ( t , z ) ^ { - 1 } [ \textbf { 1 } ] _ { \vec 0 } . \end{align*}


\begin{align*} A _ { j } ( z ) = S _ { j } ( z ) , A _ { j } ^ { ( q ) } ( z ) = S _ { j } ^ { ( q ) } ( z ) , \mbox { p r o v i d e d \eqref { e q : r e d a n k } h o l d s } . \end{align*}


\begin{align*} p _ i = \frac { 1 } { 2 } \frac { \widetilde { \Phi } ( n - 1 ) + \widetilde { \Phi } ( n + 1 ) - 2 \widetilde { \Phi } ( n ) } { \widetilde { \Phi } ( n - 1 ) - \widetilde { \Phi } ( n + 1 ) } , \end{align*}


\begin{align*} { \lambda _ k } = \mathop { \lim } \limits _ { T \to \infty } \frac 1 T \ln { D _ k } \ , \ \ \lambda _ 1 \ge \dots \ge \lambda _ k \end{align*}


\begin{align*} \mathbf { c } _ n ( \lambda ) = \mathbf { c } _ { - n } ( \lambda ) = \frac { 1 } { \sqrt { \pi } } \frac { \Gamma ( \lambda ) \Gamma ( \lambda + \frac { 1 } { 2 } ) } { \Gamma ( \lambda + \frac { 1 + n } { 2 } ) \Gamma ( \lambda + \frac { 1 - n } { 2 } ) } , \ ; \ ; \ ; \lambda \in \mathfrak { a } _ \mathbb { C } ^ * . \end{align*}


\begin{align*} ( - \Delta ) ^ { s } f _ { \epsilon } ( x ) = - \frac { C _ { n , s } } { 2 } \int _ { \R ^ n } \frac { \delta f _ { \epsilon } ( x , y ) } { | y | ^ { n + 2 s } } \ , d y \end{align*}


\begin{align*} \frac { d Y } { d t } = A ( t ) Y Y ( 0 ) = I , \end{align*}


\begin{align*} \mathrm { E } \left [ \mathbf { Z } | \boldsymbol { \xi _ { a } } < \mathbf { Z } \leq \boldsymbol { \xi _ { b } } \right ] = \frac { \boldsymbol { \delta } } { F _ { \mathbf { Z } } ( \boldsymbol { \xi _ { a } } , \boldsymbol { \xi _ { b } } ) } . \end{align*}


\begin{align*} \| T _ \phi ( x _ n ) \| _ 2 ^ 2 = \langle T _ \phi ^ * T _ \phi \widehat { x _ n } , \widehat { x _ n } \rangle \leq \| T _ \phi ^ * T _ \phi ( \widehat { x _ n } ) \| _ 1 \to 0 . \end{align*}


\begin{align*} C ( P ) : = \frac { 1 } { B \circ q ( P ) } \prod _ { k = 1 } ^ N B ( p _ k ) . \end{align*}


\begin{align*} \langle h , k \rangle = \int _ L h k \Omega . \end{align*}


\begin{align*} H _ m ( t ) = i t \int _ 0 ^ \pi e ( - m t \cos \theta ) \phi _ { 2 , m } ( t \sin \theta , s ) e ^ { - i \theta } \ , d \theta . \end{align*}


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


\begin{align*} a \big ( X ^ { 6 ^ k } \big ) b \big ( X ^ { 6 ^ k } \big ) = \big ( X ^ { 2 \cdot 3 ^ k } + X ^ { 3 ^ k } + 1 \big ) ^ { 2 ^ k } \end{align*}


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


\begin{align*} \Psi ( \xi ) = i \langle a , \xi \rangle + \frac 1 2 \langle \xi , Q \xi \rangle + \int _ { \R ^ d } \left ( 1 - e ^ { i \langle x , \xi \rangle } + i \langle x , \xi \rangle 1 _ { \{ | x | < 1 \} } \right ) \nu ( d x ) . \end{align*}


\begin{align*} g ( \psi _ G ) = \begin{cases} 4 , \ i f \ a t \ l e a s t \ o n e \ o f \ t h e \ ( P _ 1 ) \ c o n d i t i o n s \ h o l d s \\ 8 , \ i f \ n o n e \ o f \ t h e \ ( P _ 1 ) \ c o n d i t i o n s \ h o l d s \end{cases} . \end{align*}


\begin{align*} \Pr ( \Delta ( C _ M ) \le \delta ) = \Pr ( X \ge 1 ) . \end{align*}


\begin{align*} \mu ( A ) = \Theta ( n e ^ { - \alpha R } \kappa \sqrt { \ss } ) \int _ { \widehat { r } } ^ R e ^ { - \beta r _ 0 } \sinh ( \alpha r _ 0 ) d r _ 0 \end{align*}


\begin{align*} \sum _ { n = 1 } ^ \infty \frac { 1 } { 1 \vee \log \left ( \phi ^ { - 1 } ( n ) \right ) } = \infty . \end{align*}


\begin{align*} \begin{aligned} T _ 1 \left ( h _ i { } \leq { } 1 \right ) = - 1 + h _ i { } \leq { } 0 \end{aligned} \end{align*}


\begin{align*} \psi ( \Phi v , w ) = - \psi ( v , \Phi w ) . \end{align*}


\begin{align*} N ( t ) & = \int _ { \mathbb { R } } g _ + p ( t , g ) d g = \int _ { \mathbb { R } } g _ + d g \int _ { \mathbb { R } } G _ t ( g - y ) p _ { t , 0 } ( y ) d y \\ & = \int _ { \mathbb { R } } p _ { t , 0 } ( y ) d y \int _ { \mathbb { R } } g _ + G _ t ( g - y ) d g \\ & = \int _ { \mathbb { R } } ( N ( B ( t ) + y , C ( t ) ) ) p _ { t , 0 } ( y ) d y . \end{align*}


\begin{align*} \left ( - \frac { \hslash ^ { 2 } } { 2 m } \partial _ { x } ^ { 2 } + \frac { \beta \hslash ^ { 4 } } { 3 m } \partial _ { x } ^ { 4 } \right ) \phi _ { \pm } = \pm i \lambda \phi _ { \pm } . \end{align*}


\begin{align*} & a _ \kappa = \sum _ { i , j \in \mathbb N \colon \atop { 3 ( i + j ) \leq \kappa } } ( \kappa - 3 ( i + j ) + 1 ) , b _ \kappa = \sum _ { i , j \in \mathbb N \colon \atop { 2 ( i + j ) \leq \kappa } } \binom { \kappa - 2 ( i + j ) + 3 } { 3 } - a _ \kappa , \\ & c _ \kappa = \frac 1 6 \left ( \binom { \kappa + 7 } { 7 } - 3 b _ \kappa - a _ \kappa \right ) . \end{align*}


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


\begin{align*} ( z + a f _ k ) ( y ^ * ) = \lim _ n ( z + a f _ k ) ( y _ n ^ * ) = 0 \end{align*}


\begin{align*} E _ { \ell ^ \infty \Gamma } ( T _ \xi ^ * T _ \eta ) ( t ) = E _ { \ell ^ \infty \Gamma } ( T _ { \eta \otimes \overline { \xi } } ) ( t ) = \langle T _ { \eta \otimes \overline { \xi } } \delta _ t , \delta _ t \rangle = \langle u _ t ^ * ( \eta \otimes \overline { \xi } ) u _ t , \eta \otimes \overline { \xi } \rangle . \end{align*}


\begin{align*} \big < F ; G \big > ^ h _ { A , g , \nu } = \big < F G \big > ^ h _ { A , g , \nu } - \big < F \big > ^ h _ { A , g , \nu } \big < G \big > ^ h _ { A , g , \nu } , F , G : \R ^ \Lambda \rightarrow \R , h \in \R ^ \Lambda . \end{align*}


\begin{align*} \binom { k } { d } d \mid \gcd ( n , k ) \end{align*}


\begin{align*} w _ 2 ( \pi ) = \dbinom { \sum \limits _ { i = 1 } ^ n c _ i N _ i } { 2 } \left ( \sum _ { l = 1 } ^ n v _ l ^ 2 \right ) + \left ( \sum _ { i = 1 } ^ n c _ i N _ i '' \right ) \left ( \sum \limits _ { 1 \leq l < k \leq n } v _ l v _ k \right ) . \end{align*}


\begin{align*} \mathfrak { F } ( v ) & = \mathcal { E } [ v , \cdots , v ] + \mathcal { H L } _ B [ T [ W _ t u _ 0 , v , \cdots , v ] , v , \cdots , v ] \\ & \qquad - \frac { p - 1 } { 2 } T [ W _ t u _ 0 , v , \left ( \mathcal { R } ^ 2 + \mathcal { N R } \right ) [ v , \cdots , v ] , v , \cdots , v ] \\ & \qquad + \frac { p - 1 } { 2 } T [ W _ t u _ 0 , \left ( \mathcal { R } ^ 2 + \mathcal { N R } \right ) [ v , \cdots , v ] , v , \cdots , v ] , \end{align*}


\begin{align*} T ( t , A + B ) x - T ( t , A ) x = \sum _ { n = 1 } ^ { \infty } S _ { n } ( t ) x , \end{align*}


\begin{align*} \mathsf { F r } _ x ( a ) = \mathsf { F r } _ { x | _ S } ( a ) \end{align*}


\begin{align*} & \zeta ( \Delta _ n , s ) = V _ \alpha ( s ) \Bigg ( a ( s ) \ , n ^ { \alpha - 2 s } + \sum _ { m = 0 } ^ { M - 1 } b _ m ( s ) \ , n ^ { - 2 m } \Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) , \\ & \zeta ( \widetilde { \Delta } _ n , s ) = V _ \alpha ( s ) \Bigg ( \widetilde { a } ( s ) \ , n ^ { \alpha - 2 s } + \sum _ { m = 0 } ^ { M - 1 } \widetilde { b } _ m ( s ) \ , n ^ { - 2 m } \Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) . \end{align*}


\begin{align*} e ^ { x \sqrt { n } ( \sqrt { \kappa _ n } - 1 ) } = \sum _ { \ell = 0 } ^ { L - 1 } \beta _ \ell ( x ) \cdot n ^ { - \ell / 2 } + O ( n ^ { - L / 2 } ) . \end{align*}


\begin{align*} \lim \limits _ { n \rightarrow \infty } \lim \limits _ { t \rightarrow \infty } \| P _ \delta ( W _ { n , m _ 0 ; \textrm { s u r } } ) e ^ { - i t H _ 0 } \varphi \| = 0 \end{align*}


\begin{align*} & \int _ { D _ j } | F _ { m , \epsilon , j } - ( 1 - v ' _ { \epsilon } ( \Psi _ m ) ) { f F ^ { 1 + \delta } } | ^ { 2 } e ^ { v _ { \epsilon } ( \Psi _ m ) - \varphi _ { m } - \Psi _ m } \\ & \leq \int _ { D _ j } ( v '' _ { \epsilon } ( \Psi _ m ) ) | f F ^ { 1 + \delta } | ^ 2 e ^ { - \phi - \varphi _ { m } - \Psi _ m } . \end{align*}


\begin{align*} \label [ I e q ] { e q : h a r d s p h e r e B } \Phi ( | u | ) = | u | , \Sigma ( \theta ) = 1 . \end{align*}


\begin{align*} \int _ 0 ^ L \abs { w ^ n } ^ 2 d x = o ( 1 ) \int _ 0 ^ { \infty } \int _ 0 ^ L \sigma ( s ) \abs { \eta ^ n _ x } ^ 2 d s d x = o ( 1 ) . \end{align*}


\begin{align*} \inf h _ { \gamma _ { - } } \left ( E _ { 1 } \right ) = \inf \Gamma ( h _ { \gamma _ { - } } ) \left ( E _ { 1 } \right ) \ , \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d } } \mathrm { e } ^ { - 2 b | \mathbf { x } - \mathbf { y } | ^ 2 / t } \ , \textnormal { d } \mathbf { y } = \int _ { \mathbb { R } ^ { d } } \mathrm { e } ^ { - 2 b | \mathbf { z } | ^ 2 / t } \ , \textnormal { d } \mathbf { z } = \left ( \frac { t \pi } { 2 b } \right ) ^ { d / 2 } . \end{align*}


\begin{align*} v _ 1 ( \alpha ) & = f _ \alpha ( v _ 1 ( l _ 1 ) , \ldots , v _ 1 ( l _ n ) ) \\ v _ 2 ( \alpha ) & = f _ \alpha ( v _ 2 ( l _ 1 ) , \ldots , v _ 2 ( l _ n ) ) \end{align*}


\begin{align*} H _ f ( m ) = m ^ { 1 - \frac { p } { N } } H _ f ( 1 ) , 0 < H _ f ( 1 ) < + \infty . \end{align*}


\begin{align*} \left ( U ^ * H _ u \big | _ { E ( s ) } U f \right ) ( \xi ) = s \bar \xi \overline { f ( \xi ) } . \end{align*}


\begin{align*} \| B _ { k ; j , l } ^ { e s s ; m } ( t , x ) \| _ { L ^ 2 } \lesssim \sup _ { g \in L ^ 2 , \| g \| _ { L ^ 2 } = 1 } \big | \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } ( t - s ) \big | { ( \hat { v } \times \xi ) } \big | \big | \overline { \hat { g } ( \xi ) } \big | \big | \hat { f } ( s , \xi , v ) \big | \varphi _ { m , k , l } ^ { e s s } ( \omega , \xi ) \varphi _ { m ; - 1 0 M _ t } ( t - s ) \varphi _ k ( \xi ) \end{align*}


\begin{align*} P _ { k } ( x ) & = x ^ { \mathrm { p a r } _ k } \sum _ { i = 0 } ^ n c _ i ( x ^ 2 - 1 ) ^ i , \end{align*}


\begin{align*} U ^ * = \{ t ^ * _ i : i \le \mathbf { i } \} \cup \{ u ^ * : u \in \max ( D ) ^ + \setminus B \} , \end{align*}


\begin{align*} \begin{cases} - \Delta _ p G - \lambda G ^ { p - 1 } = \delta _ { x _ 0 } & \Omega \\ G \geq 0 & \Omega \\ G = 0 & \partial \Omega . \end{cases} \end{align*}


\begin{align*} \sum \limits _ { 1 \leq n \leq N } f _ k ( n ) f _ k ( n + 1 ) = N \sum _ { \substack { w _ 1 \leq N , w _ 2 \leq N + 1 \\ ( w _ 1 , w _ 2 ) = 1 } } \frac { g _ k ( w _ 1 ) g _ k ( w _ 2 ) } { ( w _ 1 w _ 2 ) ^ 2 } + O _ { \varepsilon } ( N ^ { \varepsilon } ) . \end{align*}


\begin{align*} T _ \xi ^ f ( \phi , \varepsilon , \tau ( \phi ) ) = T _ \xi ^ f ( \gamma _ 0 , a , [ b _ 0 ] ) . \end{align*}


\begin{align*} \eta _ \epsilon = \eta \Omega \setminus B _ { \epsilon } ( 0 ) , \eta _ \epsilon = 0 B _ \frac { \epsilon } { 2 } ( 0 ) , | \eta _ \epsilon | + \epsilon | \nabla \eta _ \epsilon | \leq C B _ { \epsilon } ( 0 ) \setminus B _ \frac { \epsilon } { 2 } ( 0 ) \end{align*}


\begin{align*} \sum _ { \varepsilon _ j = 0 , 1 } \prod _ { j = 1 } ^ l ( - 1 ) ^ { \varepsilon _ j } r _ { k _ j } \left ( \alpha \oplus \frac { \varepsilon _ j } { 2 ^ { k _ j } } \right ) = \prod _ { j = 1 } ^ l \left ( r _ { k _ j } \left ( \alpha \right ) - r _ { k _ j } \left ( \alpha \oplus 2 ^ { - k _ j } \right ) \right ) = \pm 2 ^ l . \end{align*}


\begin{align*} \Delta | \nabla u | = \frac 1 { | \nabla u | } ( - K | \nabla u | + | k | ^ 2 | \nabla u | ^ 2 - \langle \div k , \nabla u \rangle | \nabla u | ) \end{align*}


\begin{align*} \kappa _ j ( x _ 1 , \cdots , x _ { j - 1 } , B ) = & \frac { \overline { V } ( X _ 1 ) - \bar { f } ( \frac { j - 1 } { n } , \hat { W ^ { ( n ) } } ) ^ 2 } { \overline { V } ( X _ 1 ) - \underline { V } ( X _ 1 ) } \underline { P } ( B ) \\ & + \frac { \bar { f } ( \frac { j - 1 } { n } , \hat { W ^ { ( n ) } } ) ^ 2 - \underline { V } ( X _ 1 ) } { \overline { V } ( X _ 1 ) - \underline { V } ( X _ 1 ) } \overline { P } ( B ) , \end{align*}


\begin{align*} \label [ I e q ] { e q : c o m p a t i b i l i t y m e a s u r e s } \forall \varphi _ j \in C _ b ( E ^ j ) , \langle f ^ k , \varphi _ j \otimes 1 ^ { \otimes ( k - j ) } \rangle = \langle f ^ j , \varphi _ j \rangle . \end{align*}


\begin{align*} X _ { T , \Lambda } = \bigg \{ u \in C ( [ 0 , T ] ; C _ { \Lambda } ( \mathbb { R } ^ { N } ) ) : \| \Lambda u ( t ) \| _ { L ^ { \infty } ( \mathbb { R } ^ { N } ) } \leq \delta , t \in ( 0 , T ) \bigg \} \end{align*}


\begin{align*} v = - f _ 1 z = - f _ 3 . \end{align*}


\begin{align*} c ( \mathcal E ) = 1 + n H + \binom { n } { 2 } H ^ 2 + \binom { n } { 3 } H ^ 3 + \binom { n } { 4 } H ^ 4 \end{align*}


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


\begin{align*} \rho ( t , v ) : = \frac { 1 } { V _ F } \sum _ { k = - \infty } ^ { + \infty } c _ k \exp \left ( i k \frac { 2 \pi } { V _ F } ( v - \int _ 0 ^ t g _ { } ( s ) d s ) \right ) , \end{align*}


\begin{align*} N _ k ( t ) : = \frac { 1 } { V _ F } \int _ { 0 } ^ { \infty } g p _ k ( t , g ) d g , k \in \mathbb { Z } , \end{align*}


\begin{align*} T = \sum _ { d _ 1 , \ldots , d _ k \le x } g _ k ( d _ 1 , \ldots , d _ k ) \Big ( \frac { x } { d _ 1 \cdots d _ k } P _ { 2 k - 1 } \Big ( \log \frac { x } { d _ 1 \cdots d _ k } \Big ) + O \Big ( \Big ( \frac { x } { d _ 1 \cdots d _ k } \Big ) ^ { \theta _ { 2 k } + \varepsilon } \Big ) , \end{align*}


\begin{align*} \textstyle \sum _ { j = 1 } ^ t p _ { i , j } = p _ i , ~ ~ i = 1 , \cdots , s ; \sum _ { i = 1 } ^ s p _ { i , j } = p ' _ j , ~ ~ j = 1 , \cdots , t . \end{align*}


\begin{align*} H ' _ { \mu } ( s + i t ) = 1 - \int _ { \mathbb { R } } \frac { 1 + x ^ { 2 } } { ( s + i t - x ) ^ { 2 } } d \sigma ( x ) , \end{align*}


\begin{align*} \begin{aligned} y '' ( \delta ) & = y '' ( 1 ) - \int _ \delta ^ 1 \Biggl ( y ''' ( 1 ) - \int _ x ^ 1 y '''' ( s ) d s \Biggr ) d x \\ & = y '' ( 1 ) - y ''' ( 1 ) + \delta y ''' ( 1 ) + \int _ \delta ^ 1 y '''' ( s ) ( s - \delta ) d s . \end{aligned} \end{align*}


\begin{align*} \Theta _ n ( \phi ) = B _ { \delta / 2 } \circ \widetilde { \Theta } _ n ( \phi ) \circ B _ { - \delta / 2 } , \end{align*}


\begin{align*} \mathcal { N } _ 1 = a ^ * _ 1 a _ 1 , \qquad \mathcal { N } _ 2 = a ^ * _ 2 a _ 2 , \qquad \mathcal { N } _ \perp = \sum _ { m \ge 3 } a ^ * _ m a _ m = N - \mathcal { N } _ 1 - \mathcal { N } _ 2 . \end{align*}


\begin{align*} \Lambda = \left ( \lambda _ 1 , \lambda _ 2 , \ldots , \lambda _ m \right ) \end{align*}


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


\begin{align*} \sum _ { n = 1 } ^ { \infty } \frac { g _ k ( n ) } { n ^ 2 } = \prod _ { p } \Big ( 1 - \frac { 1 } { p ^ { 2 k } } \Big ) = \frac { 1 } { \zeta ( 2 k ) } . \end{align*}


\begin{align*} | W _ c ( \mu , \nu ) - W _ c ( \tilde \mu , \tilde \nu ) | & = \left ( W _ c ( \mu , \nu _ t ) ^ { \frac { 1 } { 2 } } + W _ c ( \tilde \mu , \tilde \nu ) ^ { \frac { 1 } { 2 } } \right ) \left | W _ c ( \mu , \nu ) ^ { \frac { 1 } { 2 } } - W _ c ( \tilde \mu , \tilde \nu ) ^ { \frac { 1 } { 2 } } \right | . \end{align*}


\begin{align*} u = - { \sf U } _ \infty ( x ) - \theta = - { \sf U } _ \infty ( x ) - ( \theta _ 0 + \theta _ 1 + \cdots + \theta _ L ) ( L \gg 1 ) \end{align*}


\begin{align*} \alpha = P _ 1 \widehat { d x _ 1 } \wedge \dots \wedge d x _ n + \dots + P _ n d x _ 1 \wedge \dots \wedge \ \widehat { d x _ n } . \end{align*}


\begin{align*} ( \mathsf { m } ' ( \varepsilon y ; \varepsilon z ) - \mathsf { m } ' ( \varepsilon y ; \varepsilon w ) ) : \varepsilon ( z - w ) \geq 0 | \varepsilon y | = g . \end{align*}


\begin{align*} \liminf _ { k \to \infty } a _ k + \liminf _ { k \to \infty } b _ k & \stackrel { * } { \le } \liminf _ { k \to \infty } ( a _ k + b _ k ) \\ & \le \limsup _ { k \to \infty } a _ k + \liminf _ { k \to \infty } b _ k \\ & \le \limsup _ { k \to \infty } ( a _ k + b _ k ) \\ & \stackrel { * * } { \le } \limsup _ { k \to \infty } a _ k + \limsup _ { k \to \infty } b _ k . \end{align*}


\begin{align*} a _ { n } = a _ { n } ^ { ( p ) } , n \in \mathbb { Z } . \end{align*}


\begin{align*} \Phi ( z ) = z \Sigma _ { \nu } ( z ) \Phi _ { n } ( z ) = z \Sigma _ { \nu _ { n } } ( z ) , n \in \mathbb { N } . \end{align*}


\begin{align*} \xi _ { x , y } = x \ , \omega _ 1 + y \ , \omega _ 2 \in \overline C \ , \backslash \{ 0 \} \ . \end{align*}


\begin{align*} \mathcal S _ X = \left \{ \left ( \begin{smallmatrix} a & x \\ y ^ * & b \end{smallmatrix} \right ) \mid a \in C ^ * ( M , \tilde M ) , b \in C ^ * ( N , \tilde N ) , x , y \in X \right \} . \end{align*}


\begin{align*} l _ { i j } ( \omega ) = \sum \limits _ { k = 1 } ^ d \frac { \partial ^ 2 \psi ^ k } { \partial \omega _ i \partial \omega _ j } ( \omega ) n ^ k ( \omega ) , i , j = 1 , . . . , m - 1 ; \end{align*}


\begin{align*} \begin{aligned} T : = & \left \{ \small { \begin{pmatrix} e ^ { i \theta _ 1 } & 0 & 0 \\ 0 & e ^ { i \theta _ 2 } & 0 \\ 0 & 0 & e ^ { i \theta _ 3 } \end{pmatrix} } : \theta _ 1 + \theta _ 2 + \theta _ 3 = 0 \right \} \\ \simeq \ & S \big ( U ( 1 ) \times U ( 1 ) \times U ( 1 ) \big ) \simeq U ( 1 ) \times U ( 1 ) \ . \end{aligned} \end{align*}


\begin{align*} \int _ M \rho ^ { - \delta - \frac { n } { p } + j } | \hat \nabla ^ j u | d V _ { \hat g } = \sum _ { i = 1 } ^ p \int _ { f _ i ^ { - 1 } ( E _ i ) } f _ i ^ \ast \rho ^ { - \delta - \frac { n } { p } + j } | \partial _ j f _ i ^ \ast u | \sqrt { \det \hat g _ { i j } } d \mathbf { x } \end{align*}


\begin{align*} 0 < \alpha : = \inf _ { v \in H ^ 1 _ 0 ( \Omega ) \setminus \{ 0 \} } \sup _ { v \in H ^ 1 _ 0 ( \Omega ) \setminus \{ 0 \} } \frac { a ( v , w ) } { \| \nabla v \| \ , \| \nabla w \| } . \end{align*}


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


\begin{align*} \alpha _ { j } = \phi _ 0 ^ { ( 1 - d ) } \sum _ { n = - N } ^ { N } n \phi _ j ( n ) , j \not = 0 , \textrm { a n d } \alpha _ { 0 } = 0 . \end{align*}


\begin{align*} \label [ I e q ] { e q : b a c k k o l m o g o r o v a p p e n d i x } \frac { \dd } { \dd t } T _ t \varphi = T _ t L \varphi , \frac { \dd } { \dd t } T _ t \varphi = L T _ t \varphi , \end{align*}


\begin{align*} V _ { U } = \left \lbrace v \in \mathcal { H } \left ( \begin{array} { c c } U & - I \\ 0 & 0 \end{array} \right ) v = 0 \right \rbrace . \end{align*}


\begin{align*} 0 = D _ i Q = \sum \theta _ k \theta _ { k i } , \\ 0 \ge D ^ 2 _ { i j } Q = \theta _ { k i j } \theta _ k + \theta _ { k i } \theta _ { k j } . \end{align*}


\begin{align*} g ( A _ { X _ j } U , A _ { X _ j } V ) = n ^ 2 g ( H ' , U ) g ( H ' , V ) . \end{align*}


\begin{align*} K ^ { n , 1 } _ \tau & = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ { ( \delta + \frac 1 n ) \wedge \tau } \gamma _ s \left ( \sigma ' ( X _ s ) \right ) ^ 2 \\ & \times \left [ \left ( \int _ { 0 } ^ { \eta _ n ( s ) } \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) ^ 2 - \left ( \int _ { \eta _ n ( s ) - \delta } ^ { \eta _ n ( s ) } \psi _ { n , 2 } ( u , s ) \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) ^ 2 \right ] \ , d s . \end{align*}


\begin{align*} M _ C ( H ^ s \to H ^ { - s } ) \vcentcolon = \{ \ , a \in M ( H ^ s \to H ^ { - s } ) \ , ; \ , \norm { a } _ { s , - s } < C \ , \} \end{align*}


\begin{align*} \nabla _ { z \partial z } = z \frac { \partial } { \partial z } - \frac { 1 } { z } E \star _ t + \mu , \end{align*}


\begin{align*} ( f ' \circ A ) ( z ) = ( B ' \circ g ) ( z ) g ' ( z ) \frac { ( z + i \lambda ) ^ 2 } { 2 \lambda i } . \end{align*}


\begin{align*} \mathbf { u } _ { i _ 1 i _ 2 } + \cdots + \mathbf { u } _ { i _ { \ell - 1 } i _ \ell } = \begin{cases} \boldsymbol { 0 } & i _ 1 = 0 i _ \ell = n + 1 , \\ \mathbf { u } _ { i _ 1 i _ \ell } & \end{cases} \end{align*}


\begin{align*} ( \nabla F _ \ast ) ( X , Y ) = { \nabla } _ { F _ \ast X } ^ N F _ \ast Y - F _ \ast ( { \nabla } _ X ^ M Y ) , ~ \forall X , Y \in \Gamma ( T M ) . \end{align*}


\begin{align*} \left \| e ^ { f _ o } v _ s - ( 4 \pi ) ^ { - \frac { n } { 2 } } \right \| ^ 2 _ { L ^ 2 ( d \nu _ * ) } = \sum _ { j = 1 } ^ { \infty } c _ j ^ 2 e ^ { - 2 \lambda _ j s } & \leq e ^ { - 2 \lambda _ 1 s } \sum _ { j = 1 } ^ { \infty } c _ j ^ 2 \leq \overline { C } e ^ { - 2 \lambda _ 1 s } \quad s \geq 0 . \end{align*}


\begin{align*} V ( \mathcal { T } ) : = \textit { C R } ^ 1 _ 0 ( \mathcal { T } ) : = \{ v \in P _ 1 ( \mathcal { T } ) : \ & v \textup { m i d } ( F ) F \in \mathcal { F } ( \Omega ) \\ & v ( \textup { m i d } ( F ) ) = 0 F \in \mathcal { F } ( \partial \Omega ) \} . \end{align*}


\begin{align*} \Gamma _ i ^ { ( s ) t } : = ( x ^ { ( p ^ s ) } ) ^ { - 1 } ( \delta _ i ^ { ( s ) } ) ^ G x \in \textup { M a t } _ N ( \mathcal A ) . \end{align*}


\begin{align*} B { u } _ i ^ \prime + ( A + B Q ) u _ i & \ , = - H _ i ( u ^ \prime + Q u ) , \ i = 1 , 2 , \\ ( B + H _ 1 ) \widetilde { u } \ , ^ \prime + \bigl ( A + ( B + H _ 1 ) Q \bigr ) \widetilde { u } & \ , = - H _ 2 ( u ^ \prime _ + + Q u _ + ) , \\ B \overline { \overline { u } } \ , ^ \prime + ( A + B Q ) \overline { \overline { u } } & \ , = - H _ 1 ( { u } _ 2 ' + Q { u } _ 2 ) - H _ 2 ( { u } _ 1 ' + Q { u } _ 1 ) . \end{align*}


\begin{align*} \frac { \abs { f ' ( z ) } } { \abs { f ' ( z _ 0 ) } } = \abs { \frac { g ( \textbf { z } _ 0 ) } { g ( \textbf { z } ) } } ^ 2 \abs { \frac { g ' ( \textbf { z } ) } { g ' ( \textbf { z } _ 0 ) } } \abs { \frac { \textbf { z } + \lambda i } { \textbf { z } _ 0 + \lambda i } } ^ 2 . \end{align*}


\begin{align*} \mathit { \Omega } _ { \mathfrak { m } } ^ { \flat } \doteq \left \{ \omega \in E _ { 1 } : f _ { \mathfrak { m } } ^ { \flat } \left ( \omega \right ) = \inf \ , f _ { \mathfrak { m } } ^ { \flat } ( E _ { 1 } ) = - \mathrm { P } _ { \mathfrak { m } } ^ { \flat } \right \} \end{align*}


\begin{align*} \lambda _ { a _ \delta } = \begin{cases} 1 , & { \rm i f } ~ { 0 } \in I _ { \bar { b } } ^ t + b _ k a _ \delta ; \\ 0 , & \mbox { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} r \psi ( F _ * 1 ) = \psi ( F _ * ^ e r ^ { p ^ e } ) = ( \psi \cdot r ^ { p ^ e - 1 } ) ( F _ * ^ e r ) \notin Q , \end{align*}


\begin{align*} \abs { a _ n } = \abs { \frac { 1 } { 2 \pi i } \int _ { \gamma } \frac { h ( z ) } { z ^ { n + 1 } } d z } \leq \frac { 1 } { 2 \pi } \left ( \frac { 3 } { 2 } \right ) ^ n \int _ 0 ^ { 2 \pi } \abs { h \left ( \frac { 2 } { 3 } e ^ { i \theta } \right ) } d \theta . \end{align*}


\begin{align*} \overline { M _ f } = M _ f + \sum _ { g \prec f } r _ { g f } ( q ) M _ g , \end{align*}


\begin{align*} f _ { \alpha } ( r ) = \begin{cases} r , & r \geq 0 \\ - \alpha r & r < 0 . \end{cases} \end{align*}


\begin{align*} t _ { k _ i } : = T _ k ^ { - 1 } \ , t _ i . \end{align*}


\begin{align*} | \int _ L ( e ^ { - i \hat { \theta } } \Omega ) | \leq | \int _ L \Omega | \leq \int _ L d v o l ( L ) = ( L ) . \end{align*}


\begin{align*} \| \mu ( E ) - \nu ( E ) \| _ { T V } = \frac { 1 } { 2 } \| f - g \| _ { L ^ 1 } . \footnotemark \end{align*}


\begin{align*} [ ( V \setminus L ) \times ( V \setminus L ) ] \cap [ ( V \setminus R ) \times ( V \setminus R ) ] = C \times C , \end{align*}


\begin{align*} W _ 0 \subseteq W _ 1 \subseteq \ldots \subseteq W _ n = S H _ * ( X ; \Z ) , \end{align*}


\begin{align*} \begin{aligned} Z ( x _ i ) = f _ i - \tfrac { \mu } { 2 } \| x _ i \| ^ 2 \nabla Z ( x _ i ) = g _ i - \mu x _ i . \end{aligned} \end{align*}


\begin{align*} T _ { k , j ; n , l , r } ^ { \mu , m , i } ( \mathfrak { m } , E ) ( t , x , \zeta ) = \int _ { 0 } ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \big ( ( t - s ) \mathfrak { K } ^ { \mu , E } _ { k ; n } ( y , \omega , v , \zeta ) + \mathfrak { K } ^ { e r r ; \mu , E } _ { k ; n } ( y , v , \zeta ) \big ) f ( s , x - y + ( t - s ) \omega , v ) \varphi _ { j , n } ^ { i ; r } ( v , \zeta ) \end{align*}


\begin{align*} C _ o = D _ { x y } \cap D _ z \ , \ D _ { z } \backslash C _ o \ , \ D _ { x y } \backslash C _ o \ , \ X \backslash ( D _ { z } \cup D _ { x y } ) \ . \end{align*}


\begin{align*} \tilde { \Lambda } = \left ( \tilde { \lambda } _ 1 , \tilde { \lambda } _ 2 , \ldots , \tilde { \lambda } _ { m } \right ) \end{align*}


\begin{align*} \prod _ { i = 1 } ^ m \left ( p ^ { k m } - p ^ { k ( i - 1 ) } \right ) , \end{align*}


\begin{align*} \begin{aligned} \tfrac { 2 L \sigma _ N } { B } = 1 + \tfrac { h _ { N - 1 } } { 1 - \kappa } \ , \ , , \tfrac { \alpha _ { N - 1 } } { B } = 1 \end{aligned} \end{align*}


\begin{align*} \mathfrak H _ { \vec s } : = H ^ * ( D _ { I _ { \vec s } } ) . \end{align*}


\begin{align*} \tilde { T } _ { \vec s } ^ k = [ T _ { I _ { \vec s } } ^ k ] _ { \vec s } . \end{align*}


\begin{align*} I \le q \ , \Vert \mu \Vert \int _ 0 ^ \beta \lambda ^ { q - 1 } \ , d \lambda = \beta ^ q \ , \Vert \mu \Vert . \end{align*}


\begin{align*} \zeta ( \widetilde { \Delta } _ n , s ) \sim A ( s ) n ^ { \alpha - 2 s } + B ( s ) + \sum _ { j = 1 } ^ \infty C _ j ( s ) n ^ { - 2 j } , \end{align*}


\begin{align*} z ^ i _ { \epsilon } : = \frac { \partial \varphi ^ i } { \partial x ^ { \epsilon } } = \delta _ { \epsilon 1 } [ \delta _ { i 1 } f ' ( x ^ 1 ) \cos x ^ 2 + \delta _ { i 2 } f ' ( x ^ 1 ) \sin x ^ 2 + \delta _ { i 3 } ] \\ + \delta _ { \epsilon 2 } [ - \delta _ { i 1 } f ( x ^ 1 ) \sin x ^ 2 + \delta _ { i 2 } f ( x ^ 1 ) \cos x ^ 2 ] , \end{align*}


\begin{align*} \lim _ { z _ { k } \rightarrow + \infty } z _ { k } H ( \boldsymbol { z } ) \overline { G } _ { n } \left ( \frac { 1 } { 2 } \boldsymbol { z } ^ { T } \boldsymbol { z } \right ) = 0 , ~ k \in \{ 1 , 2 , \cdots , n \} \end{align*}


\begin{align*} A = \bigcap _ \tau U _ \tau \subset \bigcap _ \tau \left ( B _ \varepsilon + B ( \delta _ \tau , 0 ) \right ) = B _ \varepsilon , \end{align*}


\begin{align*} \psi = \psi ^ \parallel \otimes \psi ^ \perp \end{align*}


\begin{align*} \Bigl \| \sum _ { \eta \in G _ { u } } j ( b ) ( \eta \gamma ) ^ { * } \ , j ( b ) ( \eta \gamma ) \Bigr \| = \Bigl \| \sum _ { \eta \in G _ { v } } j ( b ) ( \eta ) ^ { * } \ , j ( b ) ( \eta ) \Bigr \| \le \| b \| _ { r } ^ { 2 } . \end{align*}


\begin{align*} \begin{cases} \mathbf { U } ^ 0 \cdot \nabla _ { X , Y } \mathbf { U } ^ 0 - \mathbf { B } ^ 0 \cdot \nabla _ { X , Y } \mathbf { B } ^ 0 + \nabla _ { X , Y } P ^ 0 = 0 , \\ \mathbf { U } ^ 0 \cdot \nabla _ { X , Y } \mathbf { B } ^ 0 - \mathbf { B } ^ 0 \cdot \nabla _ { X , Y } \mathbf { U } ^ 0 = 0 , \\ \nabla _ { X , Y } \cdot \mathbf { U } ^ 0 = 0 , \quad \nabla _ { X , Y } \cdot \mathbf { B } ^ 0 = 0 . \end{cases} \end{align*}


\begin{align*} E ( Y ; u , v ) = \sum _ { p , q \geq 0 } h ^ { p , q } ( Y ) ( - u ) ^ p ( - v ) ^ q \ , \in \ , \mathbb Z [ u , v ] . \end{align*}


\begin{align*} \begin{aligned} \tilde I _ { 1 , 2 , 1 } & \leq ( 2 N + 1 ) ^ 2 + C \beta ^ { - 1 / 2 } a ^ { - 1 } N ^ 3 \log N \\ & \leq \tilde C \big ( ( \beta ^ { - 1 / 2 } a ^ { - 1 } N \log N ) \vee 1 \big ) N ^ { 2 } . \end{aligned} \end{align*}


\begin{align*} \mathcal { I } _ 3 = \underbrace { \iint _ { Q _ T } ( \nabla p \cdot \vec { b } ) ( \nabla \cdot ( m \nabla p ) ) } _ { \mathcal { I } _ { 3 , 1 } } + \underbrace { \iint _ { Q _ T } m F ( \nabla p \cdot \nabla \vec { b } ) } _ { \mathcal { I } _ { 3 , 2 } } . \end{align*}


\begin{align*} \hat { H } ^ { \dagger } \phi _ { \pm } = \pm i \lambda \phi _ { \pm } \end{align*}


\begin{align*} D _ v ( n ) = \Big \{ p \mid n \ ; : \ , \prod _ { q ^ e \| n , q < p } q ^ e \ \le \ p ^ v \Big \} . \end{align*}


\begin{align*} \sigma _ j = \frac { 1 } { [ \log _ 2 \log _ 2 ( j + 3 ) ] ^ \alpha } , j = 1 , 2 , \dots , \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } s ^ { - 1 / 2 } \overline { G } _ { ( k ) } ( s ) \mathrm { d } s < \infty , ~ k = 1 , 2 \cdots , n . \end{align*}


\begin{align*} \langle \hat h , v \rangle _ { Y ' , Y } : = \mu \int _ { \Omega } \varepsilon \xi : \varepsilon v + \nu \int _ { \Omega } \mathsf { m } ' \left ( \varepsilon \overline { y } ; \varepsilon \xi \right ) : \varepsilon v , \forall v \in Y . \end{align*}


\begin{align*} \big \langle V ' ( \vec p ) \widehat { \vec p } \big ( u ' + Q u \big ) , w \rangle _ X = \int _ D \big ( \widehat { \rho } \ , \partial _ t \vec v \cdot { \vec w } + S _ 0 + S _ 1 + \cdots + S _ L \big ) \d x \end{align*}


\begin{align*} \frac { d x } { d t } = f ( t , x ) , t \in [ a , b ] , x ( t ) \in N , f \in \mathfrak { X } _ t ( N ) . \end{align*}


\begin{align*} P ( \Pi ) = \left \{ y \in [ 1 , m ] ^ m : \begin{array} { r l c } \sum _ { i \in E } y _ i & = \binom { m + 1 } { 2 } & \\ \sum _ { i \in S } y _ i & \leq \binom { m + 1 } { 2 } - \binom { m + 1 - | S | } { 2 } , & \forall S \subseteq E : S \neq \emptyset \end{array} \right \} . \end{align*}


\begin{align*} \textstyle ( C \beta ) ^ \bot = \bigoplus _ { i = 0 } ^ { r } ( C _ i \beta _ i ) ^ { \bot _ { A _ i } } , \end{align*}


\begin{align*} [ D ^ 2 u - A ( \cdot , u , D u ) ] = B ( \cdot , u , D u ) , \end{align*}


\begin{align*} \int _ 0 ^ \infty \ 1 _ { \{ \underline f > \tau \} } ( r ) g ( r ) \ , d r & = \int _ { a _ \tau } ^ \infty g ( r ) \ , d r \leq a _ \tau ^ { - \alpha } \sup _ { s > 0 } s ^ { \alpha } \int _ s ^ \infty g ( r ) \ , d r \\ & = \alpha \int _ 0 ^ \infty \ 1 _ { \{ \underline f > \tau \} } ( r ) \ , \frac { d r } { r ^ { 1 + \alpha } } \ \overline \nu _ \alpha ( g ) \ , . \end{align*}


\begin{align*} \langle \tau ^ { ( m _ 1 , m _ 2 ) } _ 0 \sigma _ 1 ^ { m _ 1 } \sigma _ 2 ^ { m _ 2 } \sigma _ { 1 2 } \rangle ^ { \mathbf { s } , o } = 1 . \end{align*}


\begin{align*} \int _ { B _ h } \phi _ w \big ( b ^ { - 1 } ( x ^ { - 1 } , u ) \big ) \ , d m _ p ( b ) = \hbox { N } _ h \left ( N , w _ x , h I \ , \hbox { m o d } \ , \ell \right ) . \end{align*}


\begin{align*} \bar { p } = a \Pi _ { \alpha \in \left ( \Delta _ 1 ^ { + } \cap { \text B } ^ { \perp } \right ) \backslash \pm { \text B } } \alpha \bar { q } = b \Pi _ { \alpha \in \left ( \Delta _ 1 ^ { + } \cap { \text B } ^ { \perp } \right ) \backslash \pm { \text B } } \alpha \end{align*}


\begin{align*} B _ { \gamma } \left ( a \right ) = \int _ { \Theta _ { d } } \hat { f } \left ( - \gamma ^ { - 1 } \theta \right ) \mathrm { e } ^ { - i \theta \cdot a } \mathrm { d } \left ( \mathrm { F } _ { \ast } \mathrm { P } \right ) \left ( \theta \right ) + R _ { f , a , \gamma } \ , \end{align*}


\begin{align*} \min _ { 0 { } \leq { } i { } \leq { } N } \big \{ \| \nabla f ( x _ i ) \| ^ 2 \big \} { } \leq { } \frac { 2 L \big [ f ( x _ 0 ) - f ( x _ N ) \big ] } { \sum \limits _ { i = 0 } ^ { N - 1 } p ( h _ i , \kappa ) } \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*} U _ 0 = G _ 0 \supset \dots \supset U _ n \supset G _ n \supset V _ n \supset D _ n \supset U _ { n + 1 } \supset \dots \ \ \ \ M _ n \subset U _ n . \end{align*}


\begin{align*} W _ { ( \rho _ 1 , \rho _ 2 ) } = \langle M _ { u _ i } ^ t \cdot a _ j \ , | \ , 1 \leq i , j \leq 2 \rangle \end{align*}


\begin{align*} R ( N , \Sigma ) : = \{ & \alpha \in M \ , | \ , \langle \alpha , n ( \alpha ) \rangle = 1 \textrm { f o r s o m e } n ( \alpha ) \in \Sigma [ 1 ] \\ & \textrm { a n d } \langle \alpha , n _ { j } \rangle \leq 0 \textrm { f o r } n _ j \in \Sigma [ 1 ] \setminus \{ n ( \alpha ) \} \} \end{align*}


\begin{align*} u _ n \coloneqq \sum _ { i = 1 } ^ { n \wedge k } m _ i \delta _ { x _ i } + \sum _ { i = 1 } ^ { ( n 2 ^ n ) ^ N } u ^ d _ + ( Q ^ n _ i ) \delta _ { x ^ n _ i } - \sum _ { i = 1 } ^ { ( n 2 ^ n ) ^ N } u ^ d _ - ( Q ^ n _ i ) \delta _ { y ^ n _ i } , \end{align*}


\begin{align*} \frac { d x _ i } { d t } = \eta _ i ( t , x ) , i = 1 , \dots , n . \end{align*}


\begin{align*} A ( \rho ) = \tau ( \phi ) ^ { - 1 } \rho ( \tau ( \phi ) ) \zeta _ \phi ^ { p , \infty } ( \rho ) \end{align*}


\begin{align*} & \int _ 0 ^ r \left [ ( - \phi '' - \frac { z } { 2 } \phi ' + \frac { z } { 2 } e _ \sigma \psi ' ) \cdot ( z ^ { 2 } \phi ) + ( - \psi '' - \frac { z } { 2 } \psi ' + \frac { z } { 2 } e _ \sigma \phi ' ) \cdot ( z ^ { 2 } \psi ) \right ] { \rm { d } } z \\ = & \int _ 0 ^ r \left ( | z ( \phi ' , \psi ' ) | ^ 2 + \frac { 3 } { 4 } | z ( \phi , \psi ) | ^ 2 + 2 z ( \phi ' \phi + \psi ' \psi ) - \frac { 3 e _ \sigma + z e _ \sigma ' } { 2 } z ^ { 2 } \phi \psi \right ) . \end{align*}


\begin{align*} \Lambda ^ { ( 1 ) } _ { n , \delta } & = \exp \left ( i \mu Y ^ { n , 1 } _ { t } + i \lambda n ^ { \alpha + \frac 1 2 } \int _ { t - \delta } ^ { \eta _ n ( t ) } \psi _ { n , 1 } ( s , \eta _ n ( t ) ) d W _ s \right ) \\ & \times \left [ \exp \left ( i \lambda n ^ { \alpha + \frac 1 2 } \int ^ { t - \delta } _ 0 \psi _ { n , 1 } ( s , \eta _ n ( t ) ) d W _ s \right ) - 1 \right ] \end{align*}


\begin{align*} C \star \big ( S ^ 3 - C ^ 3 \big ) \star S = C \star E \big ( 3 C ^ 2 + 3 E C + E ^ 2 \big ) \star ( E + C ) . \end{align*}


\begin{align*} \sum _ { i = 0 , 1 , 2 , 3 , 4 } \big | { } _ 0 ^ { 1 } E r r ^ i _ { k , j ; n } ( t _ 1 , t _ 2 ) \big | \lesssim \mathcal { M } ( C ) 2 ^ { ( \alpha ^ \star + 3 \iota + 1 4 0 \epsilon ) M _ t - ( k + 2 n ) / 2 } \lesssim 2 ^ { M _ t / 2 } \mathcal { M } ( C ) . \end{align*}


\begin{align*} \mathbb { E } ( X _ i X _ { i + 1 } ) = f _ k ( i ) f _ k ( i + 1 ) + O _ { \alpha , \varepsilon } ( i ^ { - 1 / 2 + \varepsilon } ) \end{align*}


\begin{align*} - \Delta _ x S + \alpha S = \rho ( t , x ) : = \int _ { \mathcal { V } } f ( t , x , v ) \d v . \end{align*}


\begin{align*} \phi ( G / H \times \{ 0 \} ) = G \cdot x . \end{align*}


\begin{align*} \begin{cases} i v _ t + \bigtriangleup v \pm \mathcal { R } ^ 2 [ v ] \pm \mathcal { N R } [ v ] = 0 \\ v ( x , 0 ) = v _ 0 = u _ 0 \in H ^ s _ x ( \mathbb { T } ) . \end{cases} \end{align*}


\begin{align*} \Delta = \Delta _ 1 \xrightarrow { \gamma _ 1 } \Delta _ 2 \xrightarrow { \gamma _ 2 } \cdots \xrightarrow { \gamma _ { k - 1 } } \Delta _ k = \emptyset \end{align*}


\begin{align*} \widetilde { F } ( n ) = \sum _ { d \delta = n } g ( d ) \tau _ k ( \delta ) , \end{align*}


\begin{align*} \begin{pmatrix} \cos ( \frac { N \pi } { d } ) & - \sin ( \frac { N \pi } { d } ) & 0 & 0 \\ \sin ( \frac { N \pi } { d } ) & \cos ( \frac { N \pi } { d } ) & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \end{align*}


\begin{align*} [ F r e e ~ C h a n n e l ] ~ ~ \quad & \Omega _ F ( \psi _ 0 ) \equiv s \lim _ { t \to \infty } U _ 0 ( 0 , t ) J _ c \psi ( t ) \\ & \psi ( t ) = U ( t , 0 ) \psi _ 0 , \\ & J _ c = U _ 0 ( t , 0 ) J ( | x | / t ^ { \alpha } \leq 1 ) U _ 0 ( 0 , t ) , \\ & 0 \leq \alpha \leq a ( n , p ) < 1 , \end{align*}


\begin{align*} B _ { \alpha , \beta } ^ { * } = \operatorname { d i a g } \left \{ b _ { 1 } , b _ { 2 } , \cdots , b _ { n } \right \} , \end{align*}


\begin{align*} \vec s ^ j = ( s _ 1 ^ j , \ldots , s _ n ^ j ) \in \mathbb Z ^ n , j = 1 , 2 , \ldots , m , \end{align*}


\begin{align*} \det D Y ( \cdot , u , D u ) = \psi ( \cdot , u , D u ) , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m ( D _ i \cdot e ) b _ i = 0 \end{align*}


\begin{align*} \mu = \frac { f ( u ) } { \sqrt { f ( u ) ^ 2 - | \widetilde { \nabla } u | ^ 2 } } \Bigl ( \partial _ t + \frac { 1 } { f ( u ) ^ 2 } \widetilde { g } ^ { i j } u _ j \partial _ { i } \Bigr ) . \end{align*}


\begin{align*} \mathbf { s } = \mathbf { s } ^ \Gamma = s ^ \Gamma \oplus \bigoplus _ { i \in \left [ l \right ] , ~ j \in \left [ d _ i + \delta _ { i = 1 } \right ] } s _ { i j } ^ \Gamma \end{align*}


\begin{align*} \gcd \left ( \binom { \alpha \beta } { d _ 1 } , \binom { \alpha \beta } { d _ 2 } , \dots , \binom { \alpha \beta } { d _ r } \right ) \alpha \end{align*}


\begin{align*} \int _ 0 ^ \infty y ^ { s - \frac 1 2 } K _ { i r } ( 2 \pi | n | y ) \ , d y = \tfrac 1 4 ( \pi | n | ) ^ { - s - \frac 1 2 } \Gamma ( \tfrac s 2 + \tfrac { i r } { 2 } + \tfrac 1 4 ) \Gamma ( \tfrac s 2 - \tfrac { i r } { 2 } + \tfrac 1 4 ) . \end{align*}


\begin{align*} \sup _ { \mathbb { R } ^ n } \left | \underline { u } ( x ) - \frac { 1 } { 2 } \sum _ { i = 1 } ^ n a _ i x _ i ^ 2 \right | \leq C , \end{align*}


\begin{align*} F ( \mathbf { H } _ ) = l F ( \mathbf { R } _ 1 ) + ( l - 1 ) ( F ( \mathbf { R } _ 2 ) - 2 F ( \mathbf { R } _ 1 ) ) , \\ F ( \mathbf { B } _ ) = l F ( \mathbf { Q } _ 1 ) + ( l - 1 ) ( F ( \mathbf { Q } _ 2 ) - 2 F ( \mathbf { Q } _ 1 ) ) . \end{align*}


\begin{align*} C _ { f , k } = \prod _ p \left ( 1 - \frac 1 { p } \right ) ^ k \sum _ { \nu _ 1 , \ldots , \nu _ k = 0 } ^ { \infty } \frac { f ( p ^ { \max ( \nu _ 1 , \ldots , \nu _ k ) } ) } { p ^ { ( r + 1 ) ( \nu _ 1 + \cdots + \nu _ k ) } } . \end{align*}


\begin{align*} \tilde { u } _ { n } - T _ { A _ n , q _ n } \tilde { u } _ { n } = \frac { v _ { n } } { \Vert u _ { n } \Vert _ { H ^ 1 ( D ) } } = : \tilde { v } _ { n } \textrm { i n } H ^ 1 ( D ) , \end{align*}


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


\begin{align*} L ^ { n , 4 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) ( s - \eta _ n ( s ) ) ^ \alpha \int _ { \eta _ n ( s ) } ^ s \psi _ { n , 1 } ( u , s ) d u d s . \end{align*}


\begin{align*} f ( x _ 0 , t ) = \left ( t ^ { - 1 } G ( x _ 0 , t ) \right ) ^ { 2 - \sigma } \leq C _ { 1 0 } t ^ { \sigma - 2 } . \end{align*}


\begin{align*} e _ { k \ell } ( c \cdot z ) & = e _ { k _ 1 \ell _ 1 k _ 2 \ell _ 2 } \big ( c \cdot k _ 2 \ell _ 2 \bar k _ 2 \bar \ell _ 2 z _ 1 + c \cdot k _ 1 \ell _ 1 \bar k _ 1 \bar \ell _ 1 z _ 2 \big ) \\ & = e _ { k _ 1 \ell _ 1 } \big ( ( \bar k _ 2 \bar \ell _ 2 c ) \cdot z _ 1 \big ) e _ { k _ 2 \ell _ 2 } \big ( ( \bar k _ 1 \bar \ell _ 1 c ) \cdot z _ 2 \big ) . \end{align*}


\begin{align*} \Phi _ { k - 1 } ( \xi _ { i _ 1 } , \ldots , \xi _ { i _ k } ) = \emph { p r } \circ [ \cdots [ [ Q ' , \iota _ { \xi _ { i _ 1 } } ] , \iota _ { \xi _ { i _ 2 } } ] , \ldots , \iota _ { \xi _ { i _ k } } ] \subset \mathfrak { X } ( E ) [ 1 ] , \ ; k \in \mathbb { N } , \end{align*}


\begin{align*} I ( \nu ) = - \frac { 1 } { 2 } \log \mathrm { d e t } ( I _ k - A A ^ T ) \end{align*}


\begin{align*} \partial _ t U - \partial _ { r r } U - A _ r ( \partial _ r U ) + A _ 0 U = \theta \end{align*}


\begin{align*} Q _ { n } ( z ) = \int \frac { P _ { n } ( z ) - P _ { n } ( x ) } { z - x } \ , d \mu ( x ) . \end{align*}


\begin{align*} c _ n = b _ n / | b _ n | = b _ { n - } / | b _ { n - } | \end{align*}