\begin{align*} \widetilde { E } _ { \omega } \leq J _ { \omega } ( u ) \leq \liminf _ { n \rightarrow \infty } J _ { \omega } ( u _ { n } ) = \widetilde { E } _ { \omega } . \end{align*}


\begin{align*} A \cdot ( v _ { 1 } , \dots , v _ { k } ) : = ( A v _ { 1 } , \dots , A v _ { k } ) . \end{align*}


\begin{align*} R _ { k , j } ^ m = B _ { \rm R F } \log _ 2 \left ( 1 + \frac { p _ { k , j } ^ m \left | G _ { k , j } ^ m \right | ^ 2 } { { \sum \limits _ { j ' \ne j } \sum \limits _ { k ' \ne { { k } } } { p _ { k ' , j ' } ^ { { m } } { { { \left | G _ { k ' , j } ^ m \right | ^ 2 } } } } } + N _ { \rm R F } B _ { \rm R F } } \right ) , k \ne 0 , \end{align*}


\begin{align*} n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } - \left ( \beta n + \beta ^ { 2 } + \beta + 1 \right ) ^ { 3 } + \left ( \beta n + \beta ^ { 2 } + 1 \right ) ^ { 3 } = 0 \end{align*}


\begin{align*} \begin{aligned} \Bigg | & \dfrac { \partial \epsilon _ 1 ( \eta , t ) } { \partial \eta } \Bigg | _ { \eta = \eta _ a } \dot { \eta } _ a \Bigg | \le \ , \gamma | a _ { 0 , \delta } ( x _ a ) / 2 - \bar { y } _ a | \\ & + \gamma \left | \left . \dfrac { \partial \varepsilon _ \delta ( x , \bar { y } _ a , t ) - b _ { 1 , \delta } ( x ) / 2 } { \partial x } \right | _ { x = x _ a } \right | \dfrac { | b _ { 1 , \delta } ( x _ a ) | } { 2 } . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \Theta _ \gamma ( v ) = 1 \textrm { i f } 0 \leq \gamma \leq 1 \ , , \textrm { a n d } \Theta _ \gamma ( v ) = \nu ^ { - 1 } ( v ) \textrm { i f } - \tfrac { 3 } { 2 } < \gamma < 0 \ , . \end{aligned} \end{align*}


\begin{align*} \frac { 1 } { N } \underset { i = 1 } { \overset { N } { \sum } } \mathbb { E } \Big [ \sup _ { 0 \leq s \leq t } | x _ s ^ i - x _ s ^ { * , i } | ^ 2 \Big ] \leq \frac { C } { N ^ 2 } \Big ( 1 + \frac { 1 } { N ^ 2 } \sum _ { i , j = 1 } ^ N \mathbb E \big [ | \xi ^ i - \xi ^ j | ^ 2 \big ] \Big ) . \end{align*}


\begin{align*} D _ { e _ i } e _ j = 0 , h _ { i j } = \delta _ { i j } \kappa _ i , \kappa _ 1 \geq \cdots \geq \kappa _ n . \end{align*}


\begin{align*} \varsigma _ { \rm u p p e r } = \log _ 2 ( 1 + \gamma ) . \end{align*}


\begin{align*} \begin{aligned} \| f _ { j } \| _ { C ^ { \alpha } ( S _ h ) } & \leq k ^ 2 \left ( \| q \| _ { C ^ { \alpha } ( \overline S _ h ) } \sup _ { S _ h } \vert v _ { j } \vert + \| v _ { j } \| _ { C ^ { \alpha } ( \overline S _ h ) } \sup _ { S _ h } \vert q - 1 \vert \right ) . \end{aligned} \end{align*}


\begin{align*} M _ { \alpha _ { 2 1 } } ( \varepsilon _ 1 e _ 5 ) = \langle \alpha _ { 2 1 } , \rho ( \varepsilon _ 1 \alpha _ { 1 2 } ) \rangle \otimes \varepsilon _ 2 \alpha _ { 2 1 } \varepsilon _ 1 ^ 2 - \langle \alpha _ { 2 1 } , \rho ( \varepsilon _ 1 ^ 2 \alpha _ { 1 2 } ) \rangle \otimes \varepsilon _ 2 \alpha _ { 2 1 } \varepsilon _ 1 = - \varepsilon _ 2 \alpha _ { 2 1 } \varepsilon _ 1 . \end{align*}


\begin{align*} F _ n ( \eta ) : = T _ n - \frac { 1 } { \eta } I _ n , \eta \in \Omega . \end{align*}


\begin{align*} A ( p , s ) = 2 ^ { - p / 2 } \left ( \frac { 3 s } { 2 } - p \right ) \frac { 1 2 s - \left ( \frac { p } { 2 } + 2 \right ) \left ( \frac { p } { 2 } + 3 \right ) } { 1 4 4 } \end{align*}


\begin{align*} \mu ( u ) = \frac { \pi ( u ) } { \sum _ { v \in V } \pi ( v ) } , u \in V . \end{align*}


\begin{align*} \int _ \sigma ^ r d t \int _ { \varrho \leq t } ( \Delta \varrho ) ^ { n - k } \wedge \Delta u _ 1 \wedge \dots \wedge \Delta u _ k = - \int _ { \varrho \leq r } u _ k ( \Delta \varrho ) ^ { n - k + 1 } \wedge \Delta u _ 1 \wedge \dots \wedge \Delta u _ { k - 1 } . \end{align*}


\begin{align*} C _ 2 = \sum _ { \beta \in F ^ { \times } } \sum _ { X \in S ( \beta ) } \Theta _ { \theta } \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = q ^ 8 ( 1 - q ) ^ 2 ( 1 - q ^ 2 ) . \end{align*}


\begin{align*} \begin{cases} \partial _ t \mathbf { w } + ( \mathbf { v } \cdot \nabla ) \mathbf { w } = ( \mathbf { w } \cdot \nabla ) \mathbf { v } , \ \ & D \times ( 0 , T ) , \\ \mathbf { w } ( \cdot , 0 ) = \nabla \times \mathbf { v } _ 0 ( \cdot ) , \ \ & D . \end{cases} \end{align*}


\begin{align*} R = \frac { n - 1 } { n } \mathcal { H } ^ 2 . \end{align*}


\begin{align*} \sum _ { r } T _ { j k r } T _ { i \ell r } = \sum _ { r } T _ { i j r } T _ { k \ell r } \end{align*}


\begin{align*} \left \{ \begin{aligned} \triangle u & = | x | ^ a v ^ p & & \quad \mbox { i n } \ B _ { R } , \\ \triangle v & = | x | ^ { b } v ^ q | \nabla u | ^ { s } & & \quad \mbox { i n } \ B _ { R } . \end{aligned} \right . \end{align*}


\begin{align*} \omega ( X _ v , X _ w ) = 0 \end{align*}


\begin{align*} \rho { u _ x } = \sum \limits _ { i = 0 } ^ { 1 8 } { { c _ { i x } } { f _ i } + \frac { { \Delta t } } { 2 } { F _ x } } , \end{align*}


\begin{align*} 2 \bar \pi = \sum _ { v \in V _ 1 } \pi _ v . \end{align*}


\begin{align*} v = 0 , ( \nu ' ) ( h ) ( \bar u ) + \nu ( v ) = 0 , A ' ( h ) = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \frac { \partial \tilde y } { \partial y } = g ^ { \frac { 1 } { 4 } } + y \cdot \biggl ( \frac { 1 } { 4 } g ^ { - \frac { 3 } { 4 } } \cdot \frac { 1 } { 6 } \int \limits _ { 0 } ^ 1 s f _ { y ^ { 5 } } ^ { ( 5 ) } ( x , s y ) ( 1 - s ) ^ 3 \ , d s \biggr ) . \end{align*}


\begin{align*} A ^ * ( y ) : = A ^ * ( y | x ) = A ( \tau _ y ( x ) ) + W \circ \hat { \sigma } ^ { - 1 } ( y | x ) - W ( y | x ) \ , \end{align*}


\begin{align*} ( x , & y , z , \zeta ) ( x ' , y ' , z ' , \zeta ' ) \\ & = ( x + x ' , y + y ' , z + z ' + x y ' - x ' y , \zeta + \zeta ' + \xi ( x y ' - x ' y ) ) . \end{align*}


\begin{align*} 0 = H _ i = ( \Delta u ) _ i + B x _ i \end{align*}


\begin{align*} & M = \begin{pmatrix} D & * \\ 0 & * \end{pmatrix} , D = d i a g ( d _ 1 , d _ 2 , \ldots , d _ { l _ 0 - 1 } ) \\ & \deg ( \det ( D ) ) \geq \deg ( \det ( D _ 0 ) ) , \deg ( \det ( M ) ) = \deg ( \det ( M _ 0 ) ) \end{align*}


\begin{align*} \begin{aligned} & \| u - \tilde { u } \| _ { H ^ 3 \left ( \mathbb { R } ^ { + } \right ) \cap H ^ { 2 , 1 } \left ( \mathbb { R } ^ + \right ) } \\ \leq & c \left ( \left \| z ^ { - 1 } ( r _ 1 - \tilde { r } _ 1 ) \right \| _ { H ^ 1 \cap L ^ { 2 , 1 } } + \left \| r _ 1 - \tilde { r } _ 1 \right \| _ { H ^ 1 \cap L ^ { 2 , 1 } } + \left \| r _ 2 - \tilde { r } _ 2 \right \| _ { H ^ 1 \cap L ^ { 2 , 1 } } \right ) . \end{aligned} \end{align*}


\begin{align*} k t ' = k ( t - p ) > \sum _ { i \in [ \chi ] } x _ i . \end{align*}


\begin{align*} \displaystyle \int _ { B _ \rho } \dfrac { | \tau _ h ( \tau _ h v ( x ) ) | ^ { p } } { | h | ^ { ( 1 + \gamma ) p } } d x \leq & c | h | ^ p \displaystyle \int _ { B _ R } \dfrac { | \tau _ h D v ( x ) | ^ { p } } { | h | ^ { ( 1 + \gamma ) p } } d x \\ = & c \displaystyle \int _ { B _ R } \dfrac { | \tau _ h D v ( x ) | ^ { p } } { | h | ^ { \gamma p } } d x \le c [ D v ] _ { B ^ { \gamma } _ { p , \infty } ( B _ R ) } ^ p , \end{align*}


\begin{align*} \int _ { t _ 1 } ^ { t _ 2 } \sum _ { i = 1 } ^ { \infty } \sum _ { j = 1 } ^ { i } g _ j V _ { i , j } \psi _ i ( s ) \psi _ j ( s ) d s < \infty \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I I _ 1 \leq C \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ { T } ( 1 + \Vert X _ n ( s ) \Vert _ H ^ { \lambda } ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s = 0 . \end{align*}


\begin{align*} \omega _ { i j } V _ i V _ j = \tfrac { 1 } { 2 } h _ { i j } V _ i V _ j . \end{align*}


\begin{align*} { \cal V } ( r ) = - \frac { Q } { r } \sqrt { 1 + \lambda r ^ 2 } , Q > 0 . \end{align*}


\begin{align*} \begin{cases} \Psi - \frac { 1 } { \lambda } \omega = 0 , & \mathrm { o n } \ \{ \omega > \delta \} , \\ \Psi - \frac { 1 } { \lambda } \omega \le 0 , & \mathrm { o n } \ \{ 0 \le \omega \le \delta \} . \\ \end{cases} \end{align*}


\begin{align*} u = u _ { \infty } , B ( x , t ) = U ( x - u _ { \infty } t ) . \end{align*}


\begin{align*} [ i , j ] : = ( | \cdot | ^ i , | \cdot | ^ { i + 1 } , \ldots , | \cdot | ^ j ) \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta w _ { n } & = f ( \cdot , y ) - \mathcal { N } _ { n } ( \cdot , y _ { n } ) , \ ; \ ; \Omega , \\ w _ n & = 0 , \ , \ ; \ ; \qquad \qquad \qquad \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} - \varepsilon ^ 2 ( K ( x ) \nabla u ) + V ( x ) u = u ^ { p } , \ \ \ \ x \in \mathbb { R } ^ n , \end{align*}


\begin{align*} z ^ \top \nabla ^ 2 f ( x ) z & \le \sum _ { i = 1 } ^ n z _ i ^ 2 V '' ( x _ i ) \le - \kappa | z | ^ 2 , \end{align*}


\begin{align*} m ( s , t ) = \frac { s } { s - t } , s \in \mathbb { R } _ { > 0 } , t \in \mathbb { R } _ { < 0 } . \end{align*}


\begin{align*} g ( \omega ( Y ) , Z ) + g ( \omega ( Z ) , Y ) = h ( Y , Z ) \end{align*}


\begin{align*} \gamma ^ i ( x ) : = \frac 1 { N - 1 } \sum _ { j \neq i } K ( x ^ i - x ^ j ) , ~ ~ ~ ~ \forall x \in \mathbb R ^ { N d } , \end{align*}


\begin{align*} N ( M ) = \left ( \begin{array} { c c c } a ^ 2 & b ^ 2 t ^ { - 1 } & 2 a b t ^ n \\ c ^ 2 t & d ^ 2 & 2 c d t ^ { n + 1 } \\ a c t ^ { - n } & b d t ^ { - ( n + 1 ) } & a d + b c \end{array} \right ) \in \mathrm { G L } _ 3 ( \C [ t ^ { \pm 1 } ] ) . \end{align*}


\begin{align*} \mathbf { y } _ k = & a _ c s _ c \mathbf { H } _ k \mathbf { p } _ c + \mathbf { H } _ k \sum _ { i \in \mathcal { M } _ k } a _ i s _ i \mathbf { p } _ i + \mathbf { H } _ k \sum \limits _ { \substack { l = 1 \\ l \neq k } } ^ { K } \sum \limits _ { j \in \mathcal { M } _ l } a _ j s _ j \mathbf { p } _ j + \mathbf { n } _ k . \end{align*}


\begin{align*} L K ^ L _ { \cdot \beta } = 0 \ , \ , \ , \ , \R ^ n _ { + } \beta \in \{ 1 , \dots , N \} , \end{align*}


\begin{align*} \left \{ \begin{aligned} S ^ \prime _ t & = - \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t , & S _ 0 & = s _ 0 \\ I ^ \prime _ t & = \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t - \gamma I _ t - I _ t \phi \left ( I _ t \right ) , & I _ 0 & = i _ 0 , \end{aligned} \right . \end{align*}


\begin{align*} f ( x ) = ( 2 \pi ) ^ { - n / 2 } \int _ { - \infty } ^ \infty \int _ { S ^ { n - 1 } } \widehat { f } ( \lambda , \omega ) e _ { \lambda , \omega } ( x ) | \lambda | ^ { n - 1 } d \omega \ , d \lambda \end{align*}


\begin{align*} \tilde { h } _ d '' ( x ) = \sum _ { n = 0 } ^ \infty \left ( \frac { 1 } { ( x + n ) ^ 2 } - \frac { 1 } { \left ( x + n + \frac { 1 } { 2 } \right ) ^ 2 } - \frac { 1 } { \left ( x + n + \frac { d - 1 } { 2 } \right ) ^ 2 } \right ) . \end{align*}


\begin{align*} F _ m ( n ) = 0 , n \leq m . \end{align*}


\begin{align*} \psi \left ( d _ - ^ { \ell + 1 } L ( 1 v , 0 w ) \right ) & = \psi \left ( t ^ { - \ell } d _ - ^ { \ell + 1 } d _ = L ( v 1 , w \bullet ) \right ) \end{align*}


\begin{align*} \nabla _ X \psi _ 2 = - \frac { 1 } { 2 } X _ 1 \cdot \psi _ 2 - \frac { 1 } { 2 } S ( X ) \cdot N \cdot \psi _ 2 \end{align*}


\begin{align*} f _ \ell ( x ) = \sqrt { \frac { \mu _ d } { n _ { \ell ; d } } } \sum _ { m = 1 } ^ { n _ { \ell ; d } } c _ { \ell , m } Y _ { \ell , m ; d } ( x ) , x \in \mathbb { S } ^ d , \end{align*}


\begin{align*} \begin{cases} \Delta _ \Psi u = f ( u ) & A ( r _ { 1 } , r _ { 2 } ) \\ u \equiv c _ { 1 } & \{ r _ { 1 } \} \times N \\ u \equiv c _ { 2 } & \{ r _ { 2 } \} \times N . \end{cases} \end{align*}


\begin{align*} { { \bf { \bar X } } ^ { \star } } = { \left [ { { \bf { \tilde A } } } \right ] _ { : , 1 : { M _ t } } } , \end{align*}


\begin{align*} \tau = \tau ' _ 1 v _ 1 \cdots \tau ' _ z v _ z \end{align*}


\begin{align*} r i \ , C : = \{ x \in \textit { a f f } \ , C \ , | \ , ( \exists \ , \epsilon > 0 ) \ , ( x + \epsilon B ) \cap ( \textit { a f f } \ , C ) \subset C \} \end{align*}


\begin{align*} \gamma _ 1 ^ \star : = \dfrac { \bar c } { \bar \epsilon ( L _ r , M _ r , \delta ) } \end{align*}


\begin{align*} { \rm V o l } ( { \tilde \Sigma } ^ { 3 r } ) \leq C ( n ) \sum _ { i = 1 } ^ s { \rm V o l } ( A \cap B _ { p _ i } ( r ) ) \leq C ( n ) \ , r \sum _ { i = 1 } ^ s { \rm V o l } ( \Sigma \cap B _ { p _ i } ( 3 r ) ) . \end{align*}


\begin{align*} \begin{aligned} d _ \alpha F = \sum _ { I } \sum _ { A = 0 } ^ { 2 n - 1 } \nabla _ { A \alpha } f _ I \omega ^ A \wedge \omega ^ I , \end{aligned} \end{align*}


\begin{align*} \sum _ { 1 \leq i , j \leq n } \Psi ( v _ i v _ i ^ * v ' _ j ) { v ' _ j } ^ * = \sum _ { j = 1 } ^ n \Psi \left ( \sum _ { i = 1 } ^ n v _ i v _ i ^ * v ' _ j \right ) { v ' _ j } ^ * = \sum _ { j = 1 } ^ n \Psi \left ( p v ' _ j \right ) { v ' _ j } ^ * = \sum _ { j = 1 } ^ n \Psi ( v ' _ j ) { v ' _ j } ^ * . \end{align*}


\begin{align*} \nu _ z = - \frac { 1 } { 2 } h _ z - ( \alpha - I \beta ) h _ { \overline { z } } . \end{align*}


\begin{align*} u = ( Q _ { \delta } ( - \delta ^ 2 ( K ( x ) \nabla ) ) ) ^ { - 1 } p Q _ \delta ( V _ { \delta , Z } - q ) ^ { p - 1 } _ + u + ( Q _ { \delta } ( - \delta ^ 2 ( K ( x ) \nabla ) ) ) ^ { - 1 } h , \ \ u \in E _ { \delta , Z } . \end{align*}


\begin{align*} \gamma < \frac { 1 0 } { 1 1 } \alpha \beta , \alpha = \min \{ \alpha _ 1 , 1 / 2 \} , \end{align*}


\begin{align*} | | \mathcal { A } _ q ^ p | | ^ 2 _ { L ^ 2 ( \mathbb R , \mathbb { H } ) } = E _ q \left ( | p | ^ 2 \right ) . \end{align*}


\begin{align*} \beta _ n : = ( n + 1 ) h _ A + n ( \widetilde b + a ) + a \end{align*}


\begin{align*} E _ 1 = i \sum _ \alpha \frac { \partial ( q _ 1 - b _ 1 ) } { \partial \xi _ \alpha } \frac { \partial q _ 1 } { \partial x _ \alpha } + b _ 0 q _ 1 + \frac { \partial q _ 1 } { \partial x _ n } - c _ 1 . \end{align*}


\begin{align*} \int _ { \theta _ { m } } ^ { \theta _ M } \frac { 1 } { \left ( \mathbf d \cdot \mathbf { \hat x } + \mathrm i \mathbf { d } ^ \perp \cdot \mathbf { \hat x } \right ) ^ 2 } \mathrm d \theta \geq \frac { \theta _ M - \theta _ m } { 2 } \end{align*}


\begin{align*} v = 0 \Omega _ { 1 } \setminus D \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \langle K _ t ( R ) u _ n , u _ n - v \rangle = \langle K _ t ( R ) u , u - v \rangle , \end{align*}


\begin{align*} v _ 1 \cdot v _ 2 = - \int _ X v _ 1 \wedge v _ 2 ^ \vee . \end{align*}


\begin{align*} X _ { E _ k } = \prod _ { i = 1 } ^ n v _ i ^ { - \langle \alpha _ i , \alpha _ k \rangle _ H } + \prod _ { i = 1 } ^ n v _ i ^ { - \langle \alpha _ k , \alpha _ i \rangle _ H } = x _ k ^ { - 1 } \left ( \prod _ { i = 1 } ^ n x _ i ^ { [ b _ { i k } ] _ + } + \prod _ { i = 1 } ^ n x _ i ^ { [ - b _ { i k } ] _ + } \right ) . \end{align*}


\begin{align*} I _ \varepsilon ( u _ \varepsilon ) = c _ \varepsilon , \ \ I ' _ \varepsilon ( u _ \varepsilon ) = 0 . \end{align*}


\begin{align*} \int _ { B _ R } \dfrac { | \tau _ h D v | ^ p } { | h | ^ { p \frac { 2 \gamma } { p } } } = \int _ { B _ R } \dfrac { | \tau _ h D v | ^ p } { | h | ^ { 2 \gamma } } d x \le c \int _ { B _ R } \dfrac { | \tau _ h V _ p ( D v ) | ^ 2 } { | h | ^ { 2 \gamma } } d x h . \end{align*}


\begin{align*} { \rm e } ^ * _ { j i } = u _ { i , j } + \epsilon _ { i j m } \ , \vartheta _ m , \qquad \mathfrak { K } _ { j i } = \vartheta _ { j , i } . \end{align*}


\begin{align*} L ( s , \Delta ) \coloneqq \sum _ { n = 1 } ^ { \infty } \frac { \tau ( n ) } { n ^ { s } } = \prod _ { p } \left ( 1 - \frac { \tau ( p ) } { p ^ { s } } - \frac { 1 } { p ^ { 2 s - 1 1 } } \right ) ^ { - 1 } , \Re ( s ) > \frac { 1 3 } { 2 } . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \inf _ { k \in { \mathbb Z } } \left ( \frac { \mu ( f ^ { k - n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \tag * { $ \mathcal { U E } 2 $ } \end{align*}


\begin{align*} \eta _ i : = \frac { \alpha ^ g ( X _ i , X _ i ) } { \langle A X _ i , X _ i \rangle } \in \Gamma ( T ^ \perp _ g M \otimes \mathbb { C } ) . \end{align*}


\begin{align*} F ( x ) + F ( x + a ) = b \end{align*}


\begin{align*} M _ { - , 1 } ( x ; z ) - e _ 1 = \mathcal { P } ^ - \left ( r _ 2 ( z ) e ^ { 2 i z x } M _ { + , 2 } ( x ; z ) \right ) ( z ) , z \in \mathbb { R } . \end{align*}


\begin{align*} D ( f \circ l _ g ) = ( D f ) \circ l _ g , \ ; \ ; \ ; f \in C ^ \infty ( G ) , \forall g \in G , \end{align*}


\begin{align*} \lim \limits _ { i \to \infty } \left | \ , ^ { k _ i } \ ! \lambda / \left ( \ , ^ { k _ i } \ ! \lambda _ 1 \right ) \right | = \infty . \end{align*}


\begin{align*} \lambda _ 1 = n - 1 , \lambda _ 2 = n - 3 , \cdots , \lambda _ { i } = n + 1 - 2 i , \cdots , \lambda _ n = 1 - n . \end{align*}


\begin{align*} & \mathbb { E } \int _ 0 ^ T \psi ( t ) \Big [ \Vert X ( t \wedge \tau _ u ^ M ) \Vert _ H ^ 2 - \Vert x \Vert _ H ^ 2 \Big ] d t \\ = & \mathbb { E } \int _ 0 ^ T \psi ( t ) \int _ 0 ^ { t \wedge \tau _ { u } ^ M } \Big [ 2 \langle \mathcal { A } ( s ) , X ( s ) \rangle + \Vert \mathcal { B } ( s ) \Vert _ { L _ 2 } ^ 2 \Big ] d s d t . \end{align*}


\begin{align*} & \inf \bigg \{ k > 0 : ~ ( L ) \int _ { \mathcal { J } } \theta \bigg ( t , ( \xi _ n - \xi ) ( t ) \bigg ) d \overline { \mu } \leq 1 \bigg \} \to 0 \\ & \implies ~ ( L ) \int _ { \mathcal { J } } \theta \big ( t , ( \xi _ n - \xi ) ( t ) \big ) \to 0 \\ & ~ i . e . , ~ \theta ( t , ( \xi _ n - \xi ) ( t ) = 0 ~ \overline { \mu } - a . e . ~ a s ~ n \to \infty \\ & \xi _ n - \xi = 0 ~ a . e . ~ a s ~ n \to \infty \end{align*}


\begin{align*} \begin{cases} \partial _ t \mathbf { v } + ( \mathbf { v } \cdot \nabla ) \mathbf { v } = - \nabla P , \ \ & D \times ( 0 , T ) , \\ \nabla \cdot \mathbf { v } = 0 , \ \ & D \times ( 0 , T ) , \\ \mathbf { v } \cdot \mathbf { n } = v _ n , \ \ & \partial D \times ( 0 , T ) , \\ \mathbf { v } ( x , 0 ) = \mathbf { v } _ 0 ( x ) , \ \ & D , \end{cases} \end{align*}


\begin{align*} Q ^ 2 - B Q - \Bigl [ I _ { n + 1 } \frac { \partial } { \partial x _ n } , Q \Bigr ] + C = 0 , \end{align*}


\begin{align*} \varphi ^ L _ { \sigma , s ' } ( D ) = \{ d \} \varphi ^ L _ { \sigma , s ' } ( E ) = \varphi ^ L _ { \sigma , s ' } ( C _ i ) . \end{align*}


\begin{align*} \Gamma _ k = \{ \lambda \in \mathbb { R } ^ n : \sigma _ j ( \lambda ) > 0 , 1 \leq j \leq k \} . \end{align*}


\begin{align*} ( g _ t , u _ t ) = \psi _ t ^ * \big ( \left . \big ( \bar g , \bar u \big ) \right | _ { M \setminus \Omega _ t } \big ) . \end{align*}


\begin{align*} \partial _ y T ( x _ 0 , y ) = 1 . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s | c _ { i } | \leq \sqrt { s } \sqrt { \sum _ { i = 1 } ^ s | c _ { i } | ^ { 2 } } . \end{align*}


\begin{align*} \begin{cases} \partial _ t \mathbf { w } + ( \mathbf { v } \cdot \nabla ) \mathbf { w } = ( \mathbf { w } \cdot \nabla ) \mathbf { v } , \ \ & D \times ( 0 , T ) , \\ \nabla \cdot \mathbf { v } = 0 , \ \ & D \times ( 0 , T ) , \\ \mathbf { v } \cdot \mathbf { n } = v _ n , \ \ & \partial D \times ( 0 , T ) , \\ \mathbf { w } ( x , 0 ) = \nabla \times \mathbf { v } _ 0 ( x ) , \ \ & D . \end{cases} \end{align*}


\begin{align*} L = a _ 0 x + a _ 1 x ^ q + a _ 2 x ^ { q ^ 2 } + \cdots + a _ n x ^ { q ^ n } \in F [ x ] . \end{align*}


\begin{align*} \psi _ \epsilon ( x , y ) = A _ { \epsilon } \| y \| ^ 2 + \tilde { v } ( x ) \| y \| \ge R + 1 \end{align*}


\begin{align*} f ( x ) & = \sum _ { i = 1 } ^ n V ( x _ i ) + \sum _ { 1 \le i < j \le n } J _ { i j } K ( x _ i - x _ j ) , \end{align*}


\begin{align*} - \Gamma ^ k + \bar u ^ { - 1 } \bar g ^ { i k } \frac { \partial \bar u } { \partial x _ i } = 0 . \end{align*}


\begin{align*} \zeta : = \zeta _ q : = \exp \Big ( \frac { 2 \pi \sqrt { - 1 } } { q } \Big ) . \end{align*}


\begin{align*} \widetilde { f } ( \lambda , b ) = \pi _ \lambda ( f ) Y _ 0 ( b ) , \ , \ , \ , \pi _ \lambda ( f ) = \int _ G f ( g ) \pi _ \lambda ( g ) d g . \end{align*}


\begin{align*} \liminf _ { i \to \infty } \frac { w _ { n _ i } } { n _ i ^ { r - \frac { 1 } { 2 } } } = 0 \limsup _ { i \to \infty } \frac { w _ { n _ i } } { n _ i ^ { r - \frac { 1 } { 2 } } } = \infty \end{align*}


\begin{align*} ( a ^ \alpha _ \beta ) : = \begin{pmatrix} a ^ 1 _ 1 & \dots & a ^ 1 _ { n - 1 } \\ \vdots & \ddots & \vdots \\ a ^ { n - 1 } _ 1 & \dots & a ^ { n - 1 } _ { n - 1 } \\ \end{pmatrix} , \end{align*}


\begin{align*} \mathcal { W } ( K ) = \sum _ { m = 0 } ^ n V _ m ( K ) . \end{align*}


\begin{align*} h _ { 0 j ; k _ 1 k _ 2 } = 0 , h _ { i j ; k _ 1 a } = 0 , h _ { i j ; a k _ 1 } = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \left \| \sum _ { k = 1 } ^ n a _ k \xi _ k \right \| _ q \geq c _ d ( q ) \left \| \sum _ { k = 1 } ^ n a _ k \xi _ k \right \| _ 2 . \end{align*}


\begin{align*} \frac { d } { d t } \mathbb { E } [ u _ t ] \bigg | _ { t = 0 } & = \int _ U \left ( \langle W _ u ( u ^ { - 1 } ( y ) ) , v ( y ) \rangle + W _ \xi ( u ^ { - 1 } ( y ) ) : [ \nabla _ y v ] ( y ) [ \nabla _ x u ] ( u ^ { - 1 } ( y ) ) \right ) d y \\ & = \int _ U \left ( \langle W _ u ( u ^ { - 1 } ( y ) ) , v ( y ) \rangle + W _ \xi ( u ^ { - 1 } ( y ) ) [ \nabla _ x u ] ^ t ( u ^ { - 1 } ( y ) ) : [ \nabla _ y v ] ( y ) \right ) d y . \end{align*}


\begin{align*} \frac { 1 } { n } \sum _ { i = 1 } ^ n \delta _ { X _ i } \to \int _ 0 ^ 1 \mu ^ * _ u \ , d u , . \end{align*}


\begin{align*} \nabla _ X \Psi _ 1 = - \frac { 1 } { 2 } \left ( X - \langle X , T _ 1 \rangle ( T _ 1 - f _ 1 ) - \langle X , T _ 2 \rangle ( T _ 2 - f _ 2 ) \right ) \cdot \Psi _ 2 - \frac { 1 } { 2 } S ( X ) \cdot \Psi _ 1 \end{align*}


\begin{align*} X _ M ( x _ 1 , \dots , x _ n ) = X _ { M ' } ( x _ 1 ' , \dots , x _ n ' ) , \end{align*}


\begin{align*} \begin{aligned} A _ { t } + B \times u + \nabla Q & = - \mu \nabla \times B . \end{aligned} \end{align*}


\begin{align*} \sigma _ 0 = \min \{ \frac { p _ 0 } { \nu _ 0 } , q _ 0 \} . \end{align*}


\begin{align*} ( \nabla _ V ( \nabla X ) ) ^ i _ j = V _ k \ , g ( \nabla _ { e _ k } \nabla _ { e _ j } X - \nabla _ { \nabla _ { e _ k } e _ j } X , e _ i ) : = V _ k X _ { i ; j k } . \end{align*}


\begin{align*} v ( t ) = \sum _ { \mathbf { k } = 1 } ^ { \infty } \widehat { v } _ { \mathbf { k } } { \rm e } ^ { { \rm i } \mathbf { k } t } \sum _ { \mathbf { k } = 1 } ^ { \infty } | \widehat { v } _ { \mathbf { k } } | < \infty , \end{align*}


\begin{align*} m \left | \sum _ { k = 1 } ^ m \frac { \mu ( k ) } { k } \right | \leq \frac { 1 } { 2 } . \end{align*}


\begin{align*} R _ 1 = \sum _ { X \in S ( 0 ) } \Theta \begin{bmatrix} m _ 1 ' & X \\ 0 & m _ 2 ' \end{bmatrix} \overline { \psi _ A ( { m _ 1 ' } ^ { - 1 } X } ) = - \theta ( a ) q ^ 8 ( q - 1 ) ^ 2 . \end{align*}


\begin{align*} \mathbb { E } _ { \xi _ x ^ k } \langle \nabla f _ { \delta } ( x _ g ^ k , \xi _ x ^ k ) - \nabla f ( x ^ * ) , x ^ { k + 1 } - x ^ * \rangle & = \mathbb { E } _ { \xi _ x ^ k } \langle \nabla f _ { \delta } ( x _ g ^ k , \xi _ x ^ k ) - \nabla f _ { \delta } ( x _ g ^ k ) , x ^ { k + 1 } - x ^ * \rangle \\ & + \langle \nabla f _ { \delta } ( x _ g ^ k ) - \nabla f ( x ^ * ) , x ^ { k + 1 } - x ^ * \rangle . \end{align*}


\begin{align*} l ( y ) - l ( x ) & \leq h ( y ) \\ & \leq \sup _ { g \in A } g ( y ) - g ( x ) \\ & = \sup _ { g \in A } g ( y ) - f ( x ) \\ f ( x ) + l ( y ) - l ( x ) & = \sup _ { g \in A } g ( y ) = f ( y ) \end{align*}


\begin{align*} A _ t : = t A _ 1 + ( 1 - t ) A _ 0 t \in [ 0 , 1 ] \end{align*}


\begin{align*} \bigg ( \frac { 1 } { | Q | } \iint _ { T _ Q } | \Theta f ( x ' , t ) | ^ 2 \frac { d x ' d t } { t } \bigg ) ^ { \frac 1 2 } = \bigg ( \frac { 1 } { | Q | } \iint _ { T _ Q } | \Theta ( f - f _ Q ) ( x ' , t ) | ^ 2 \frac { d x ' d t } { t } \bigg ) ^ { \frac 1 2 } \le { \rm I } + { \rm I I } , \end{align*}


\begin{align*} a ( u _ n , v _ n ) = \lambda _ n ( u _ n , v _ n ) v _ n \in X _ n . \end{align*}


\begin{align*} \sum _ { r _ { Q } ( n ) = 0 } n \ll \max \left \{ \frac { p ^ { 3 + \epsilon } } { ( \min Q ^ { * } ) ^ { 2 } } , p ^ { 5 / 2 + \epsilon } \right \} . \end{align*}


\begin{align*} n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } - \left ( \beta n + \beta ^ { 2 } + \beta + 1 \right ) ^ { 3 } + \left ( \beta n + \beta ^ { 2 } + 1 \right ) ^ { 3 } = 0 \end{align*}


\begin{align*} d v v ^ { - 1 } = \frac { 1 } { 2 } a I + \frac { 1 } { \sqrt { \tau _ 0 } } \left ( \frac { \tau _ 0 } { 4 } d \overline { z } + Q _ 0 d z \right ) i J . \end{align*}


\begin{align*} \mathsf { M } ^ n = & \{ \mathsf { m } ^ n _ 1 , \cdots , \mathsf { m } ^ n _ n \} \\ = & \left \{ g \bigl ( \xi _ 0 \bigr ) , g \bigl ( \xi _ 1 ^ { \mathsf { M } ^ n } \bigr ) , \cdots , g \bigl ( \xi _ { n - 1 } ^ { \mathsf { M } ^ n } \bigr ) \right \} \in \Theta _ n , \end{align*}


\begin{align*} r i \ , C : = \{ x \in \textit { a f f } \ , C \ , | \ , ( \exists \ , \epsilon > 0 ) \ , ( x + \epsilon B ) \cap ( \textit { a f f } \ , C ) \subset C \} \end{align*}


\begin{align*} \Sigma ^ 2 = \left ( \frac { 1 } { 2 L _ { x } } + \frac { \omega } { L _ { x y } } \right ) \frac { \sigma _ { F , r } ^ 2 } { n } + \left ( \frac { 1 } { 2 L _ { y } } + \frac { 1 } { L _ { x y } \omega } \right ) \frac { \sigma _ { G , r } ^ 2 } { n } . \end{align*}


\begin{align*} g ( z ) = ( 2 \pi ) ^ { - n } \ , | \lambda | ^ n \ , \sum _ { k = 0 } ^ \infty g \ast _ \lambda \varphi _ { k , \lambda } ^ { n - 1 } ( z ) . \end{align*}


\begin{align*} \tilde Y _ { n + 1 } ^ i = \tilde Y _ n ^ i + \bigg ( b ( \tilde Y _ n ^ i ) + \frac 1 { p - 1 } \sum _ { j \neq i , j \in \C } K ( \tilde Y _ n ^ i - \tilde Y _ n ^ j ) \bigg ) \tau + \sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , ~ ~ i \in \C . \end{align*}


\begin{align*} \nabla _ { z , r } \eta ( z , r ) = \int _ { 0 } ^ { \infty } \nabla _ { z , r } { \mathcal { G } } * _ 1 G ( \cdot , r ' ) \frac { 1 } { r ' } \dd r ' . \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta y + \mathcal { N } ( \cdot , y ) & = u , \ ; \ ; \Omega , \\ y & = 0 , \ ; \ ; \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} G _ { \ell ; d } ( t ) ^ r = \sum _ { j = 0 } ^ { r \ell } \gamma _ { j , \ell ; r } G _ { j ; d } ( t ) \end{align*}


\begin{align*} M = \left ( \begin{array} { c c } \varepsilon _ 1 \mu & - \varepsilon _ 1 \mu \\ \varepsilon _ 3 \mu & - \varepsilon _ { 3 } \mu \end{array} \right ) \end{align*}


\begin{align*} \dot { { x } } _ a & = - \gamma \int _ { 0 } ^ { 1 } \left ( y _ \delta ( { x _ a } , \tau ) - \bar { y } _ a \right ) \ , u ( \tau ) \ , d \tau \\ \dot { \bar { y } } _ a & = - \gamma \ , \bar { y } _ a + \gamma \int _ { 0 } ^ { 1 } y _ \delta ( { x } _ a , \tau ) \ , d \tau . \end{align*}


\begin{align*} g ( \omega ( Y ) , Z ) + g ( \omega ( Z ) , Y ) = h ( Y , Z ) \end{align*}


\begin{align*} \begin{aligned} | h ( x + & \delta u ( \tau ) ) - h ( x _ a + \delta u ( \tau ) ) | \le L _ r | x - x _ a | \\ & \le L _ r ( | x - z _ 1 | + \gamma | \epsilon _ 1 ( \eta _ a , \tau ) | ) \\ & \le L _ r ( | x - z _ 1 | + \gamma \delta L _ r ( 3 + L _ r ) ) . \end{aligned} \end{align*}


\begin{align*} W ( y | x ) = \ , \sum _ { n = 1 } ^ \infty \left [ A \circ \hat { \sigma } ^ { - n } ( y | x ) - A \circ \hat { \sigma } ^ { - n } ( y | x ' ) \right ] , \end{align*}


\begin{align*} \nabla _ X \varphi = \frac { 1 } { 2 } ( \sqrt { c _ 1 } \ X _ 1 \cdot \nu _ 1 + \sqrt { c _ 2 } \ X _ 2 \cdot \nu _ 2 ) \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi \end{align*}


\begin{align*} I ( A ^ { 1 , \dots , \ell } ) = \langle p _ { ( u , s ) } - p _ { ( u , r ) } \mid u \in [ \beta ] , s , r \in \{ 0 , \dots , x _ u Y _ u - 1 \} \rangle + J . \end{align*}


\begin{align*} \mathrm { Q u a d } ^ i : = \{ z ^ i _ { j k } z ^ i _ { j ' k ' } - z ^ i _ { j k ' } z ^ i _ { j ' k } \mid j , j ' \in [ s _ i ] , k , k ' \in [ t _ i ] \} . \end{align*}


\begin{align*} F ( x ) = \sum _ { { \bf k } \in \Gamma _ { r } \backslash \Lambda _ { 1 } } \begin{pmatrix} a _ { \bf k } & 0 \\ 0 & - a _ { \bf k } \end{pmatrix} { \rm e } ^ { { \rm i } \langle { \bf k } , x \rangle } + \sum _ { { \bf k } \in \Gamma _ { r } \backslash \Lambda _ { 2 } } \begin{pmatrix} 0 & b _ { \bf k } \\ c _ { \bf k } & 0 \end{pmatrix} { \rm e } ^ { { \rm i } \langle { \bf k } , x \rangle } . \end{align*}


\begin{align*} g _ d ( \ell ) = \begin{cases} \ell , & d = 1 , \\ \log \ell , & d = 2 , \\ 1 , & d \geq 3 , \end{cases} \end{align*}


\begin{align*} 4 \lambda + 2 \mu = \frac { \hat W _ r ( 0 ) + \hat W _ r ( 2 q ) - 2 \hat W _ r ( q ) } { \hat W _ r ( q ) } = : H ( q , r ) . \end{align*}


\begin{align*} \frac { \left | \sum _ { i = 1 } ^ n [ 0 , z _ i ] \right | } { \left | \sum _ { i = 1 } ^ { n - 1 } [ 0 , z _ i ] \right | } = d ( z _ n , H _ n ) . \end{align*}


\begin{align*} \zeta ( x ) : = \begin{cases} \frac { \xi ( x ) } { p ( x ) } , & \{ p ( x ) \ne 0 \} , \\ f _ { - } ' ( \overline { y } ( x ) ) , & \{ p ( x ) = 0 \} , \end{cases} \end{align*}


\begin{align*} ( \mathcal { R } _ \lambda ( \sigma s _ i ) v _ T , v _ T ) = \pm ( \mathcal { R } _ \lambda ( \sigma ) v _ T , v _ T ) . \end{align*}


\begin{align*} G = \begin{pmatrix} 1 & 0 & \cdots & 0 & a _ 1 & 0 & \cdots & 0 & 1 \\ 0 & 1 & \cdots & 0 & a _ 2 & 0 & \cdots & 1 & 0 \\ \vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & a _ k & 1 & \cdots & 0 & 0 \\ \end{pmatrix} _ { k \times ( 2 k + 1 ) } , \end{align*}


\begin{align*} \ell ( x ) = a _ n x ^ n + a _ { n - 1 } x ^ { n - 1 } + \cdots + a _ 0 \end{align*}


\begin{align*} ( \sum \limits _ { i = 1 } ^ j w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + r ( j - 1 ) \leq \chi _ j \leq ( \sum \limits _ { i = 1 } ^ j w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + r j , \end{align*}


\begin{align*} Z G \left ( X , Y \right ) = G \left ( \nabla _ { Z } X , Y \right ) + G \left ( X , \nabla _ { Z } ^ { \dag } Y \right ) , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n \binom n k \frac { 2 ^ k F _ { j k } } { L _ j ^ { k } } B _ { n - k } = \frac { n } { \sqrt 5 } \Big ( \frac { \sqrt 5 F _ j } { L _ j } \Big ) ^ { n - 1 } , \qquad \mbox { $ n $ e v e n } , \end{align*}


\begin{align*} | | A _ q ^ z | | ^ 2 _ { L ^ 2 ( \mathbb R ) } = E _ q \left ( | z | ^ 2 \right ) . \end{align*}


\begin{align*} N = n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } = \left ( n + 3 \right ) ^ { 3 } + \left ( n + \alpha \right ) ^ { 3 } \end{align*}


\begin{align*} \begin{cases} - \Delta \psi + \psi = \lambda ( \psi - W x _ 2 ) _ + , & \mathrm { i n } \ \ \Pi , \\ \psi = 0 , & \mathrm { o n } \ \partial \Pi , \\ \psi ( x ) \rightarrow 0 , & \mathrm { a s } \ \ | x | \rightarrow \infty . \\ \end{cases} \end{align*}


\begin{align*} \Phi ( B _ \phi v , p ) & = Q F ^ q ( B _ \phi v + \psi ( B _ \phi v , p ) , p ) \\ & = Q F ^ q ( B _ \phi v + B _ \phi \psi ( v , p ) , p ) \\ & = Q F ^ q ( B _ \phi [ v + \psi ( v , p ) ] , p ) \\ & = Q B _ \phi F ^ q ( v + \psi ( v , p ) , p ) \\ & = B _ \phi Q F ^ q ( v + \psi ( v , p ) , p ) \\ & = B _ \phi \Phi ( v , p ) , \end{align*}


\begin{align*} \mathcal S ( f _ 2 \circ f _ 1 ) = \left ( \mathcal S ( f _ 2 ) \circ f _ 1 \right ) ( f _ 1 ' ) ^ 2 + \mathcal S ( f _ 1 ) \end{align*}


\begin{align*} w = u _ 1 + u _ 2 + \cdots + u _ l . \end{align*}


\begin{align*} \int _ { \mathfrak X } | \omega | = { \mathfrak h _ 0 } _ ! \int _ { \mathfrak Y } | \omega | , \end{align*}


\begin{align*} \Psi ( t , \cdot ) : = - \Delta ^ { - 1 } \omega ( t , \cdot ) \end{align*}


\begin{align*} ( I - Q ^ \dagger ) F ^ { q , \dagger } _ v ( v + \psi ^ \dagger ( v , p ) , p ) \psi _ p ( v , p ) + ( I - Q ^ \dagger ) F ^ { q , \dagger } _ p ( v + \psi ^ \dagger ( v , p ) , p ) = 0 . \end{align*}


\begin{align*} \int _ 0 ^ \infty \left | \frac { G ( e ^ { \tau } , \delta ) } { e ^ { \sigma \tau } } \right | ^ 2 d \tau = \int _ 1 ^ \infty \left | \frac { G ( y , \delta ) } { y ^ { \sigma + \frac { 1 } { 2 } } } \right | ^ 2 d y & = \int _ 1 ^ \infty \left | \frac { G ( y , \delta ) } { y ^ { 1 + \frac { \alpha } { \log ( 1 / \delta ) } } } \right | ^ 2 d y > \frac { 1 } { e ^ { \alpha b } } \int _ 1 ^ { 1 / \delta ^ b } \left | \frac { G ( y , \delta ) } { y } \right | ^ 2 d y \end{align*}


\begin{align*} f _ \xi ( z ) = \begin{cases} \frac { f ( z ) } { \xi - z } & , \\ 0 & \end{cases} \end{align*}


\begin{align*} T : L ^ 2 ( D ) \rightarrow L ^ 2 ( D ) T f = u . \end{align*}


\begin{align*} i _ { \chi \theta } ( f ) ( F _ \bullet ) = \sum _ { F ' _ \bullet \in \partial _ { \chi \theta } ^ { - 1 } ( F _ \bullet ) } f ( F ' _ \bullet ) . \end{align*}


\begin{align*} & d _ { \mathrm { s t } } = \int d z \sum _ { \alpha \in \Delta _ + } e _ { \alpha } ( z ) \otimes \psi _ { \alpha _ i } ^ * ( z ) - \frac { 1 } { 2 } \sum _ { \alpha , \beta , \gamma \in \Delta _ + } c _ { \alpha , \beta } ^ \gamma : \psi _ \gamma ( z ) \psi _ \alpha ^ * ( z ) \psi _ \beta ^ * ( z ) : , \\ & d _ { \chi } = \int d z \sum _ { \alpha \in \Pi } \psi ^ * _ \alpha ( z ) = \sum _ { \alpha \in \Pi } \psi ^ * _ { \alpha , 1 } . \end{align*}


\begin{align*} t ( v ' ) t ( w '' ) = t ( w ' ) t ( v '' ) \end{align*}


\begin{align*} \psi ( x , t ; k ) = \phi ( x , t ; k ) e ^ { i ( k ^ 2 x + \eta ^ 2 t ) \sigma _ 3 } , \end{align*}


\begin{align*} \begin{aligned} | \bar { y } ( t ) & - z _ 2 ( t ) | \le | \bar { y } ( 0 ) - z _ 2 ( 0 ) | + t \gamma ^ 2 \bar { k } _ 2 ( L _ r , \delta ) \\ & + \gamma \int _ 0 ^ t L _ r | x - z _ 1 | + | \bar { y } ( \tau ) - z _ 2 ( \tau ) | \ , d \tau \end{aligned} \end{align*}


\begin{align*} \gamma ( x ) : = p ( x , e ) \equiv p ( x , 1 ) . \end{align*}


\begin{align*} X : p \mapsto \left . \frac { d } { d t } \right \vert _ { t = 0 } \Big ( \exp ( t \frak { X } ) ( p ) \Big ) . \end{align*}


\begin{gather*} \delta _ 1 ( X ) = 2 \mu F ( \mu ) - 2 \mu + 2 \int _ { \mu } ^ { \infty } x f ( x ) \mathrm { d } x \delta _ 2 ( X ) = 2 \int _ { m } ^ { \infty } x f ( x ) \mathrm { d } x - \mu , \end{gather*}


\begin{align*} \mathbf { y } ' = \begin{bmatrix} y _ c \\ y _ 1 \\ \vdots \\ y _ M \end{bmatrix} = \begin{bmatrix} \sum _ { i = 1 } ^ { M } y _ i \\ y _ 1 \\ \vdots \\ y _ M \end{bmatrix} , \end{align*}


\begin{align*} \frac { \tilde { u } ^ { [ n ] } ( x ) - \tilde { u } ^ { [ n - 1 ] } ( x ) } { \Delta t } = \theta \left [ \mathcal { L } ( \tilde { u } ^ { [ n ] } ) + f ( t _ n , x ) \right ] + ( 1 - \theta ) \left [ \mathcal { L } ( \tilde { u } ^ { [ n - 1 ] } ) + f ( t _ { n - 1 } , x ) \right ] , \end{align*}


\begin{align*} \begin{aligned} | \langle \boldsymbol { \nu } , G ( \lambda | \bar U , x , \tau ) \rangle | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; , \end{aligned} \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ H q = 0 , \\ \nabla ^ { \perp } q \cdot \nu | _ { \partial \Omega } = - v _ n | _ { \partial \Omega } . \\ \end{cases} \end{align*}


\begin{align*} \Delta _ { \Psi } u = e ^ { \Psi } \div ( e ^ { - \Psi } \nabla u ) = \Delta u - g ( \nabla \Psi , \nabla u ) . \end{align*}


\begin{align*} \varphi = ( \tilde \varphi , \varphi ^ * ) : ( U _ x , \mathcal { O } _ M | _ { U _ x } ) \rightarrow \mathcal { U } . \end{align*}


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


\begin{align*} \varepsilon ( z ) \Lambda ( z ^ { - 1 } ) = - \Lambda ( - z ^ { - 1 } ) , \dot { s } \Lambda ( z ) = - \Lambda ( - z ^ { - 1 } ) , \quad \\ \dot { s } \varepsilon ( x ) \Lambda ( y ) = \varepsilon ( - x ^ { - 1 } ) \Lambda ( x ( x y - 1 ) ) + \Lambda ( x ) - \Lambda ( y ^ { - 1 } ( x y - 1 ) ) , \end{align*}


\begin{align*} H = ( p \wedge q ^ { \perp } ) H \oplus ( p ^ { \perp } \wedge q ) H \oplus ( p \wedge q ) H \oplus ( p ^ { \perp } \wedge q ^ { \perp } ) H \oplus ( e _ 1 + e _ 2 ) H . \end{align*}


\begin{align*} B _ q ( \psi _ m ) ( z ) & = \int \limits _ { \mathbb R } \overline { A _ q ( z , x ) } \psi _ m ( x ) d x \\ & = \int \limits _ { \mathbb R } \left ( \sum _ { j = 0 } ^ { \infty } \frac { z ^ j } { \sqrt { \Gamma ( q j + 1 ) } } \psi _ j ( x ) \right ) \psi _ m ( x ) d x , \\ & = \sum _ { j = 0 } ^ { \infty } \frac { z ^ j } { \sqrt { \Gamma ( q j + 1 ) } } \left ( \int \limits _ { \mathbb R } \psi _ j ( x ) \psi _ m ( x ) d x \right ) . \\ \end{align*}


\begin{align*} Q _ { r } \ge 0 , c _ { r } = r ^ { 1 - d } \int _ { 0 } ^ { r } \rho ^ { d - 1 } n c \ , d \rho \ge 0 , r < R , t < T _ { \rm m a x } . \end{align*}


\begin{align*} L : = \max \bigg ( \frac { 1 } { \tilde \varepsilon } , 8 0 p \ , \frac { C ( \tilde { \varepsilon } ) } { c ( \tilde { \varepsilon } ) } \bigg ) . \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { 1 } { \ln \frac { 1 } { \varepsilon } } \int _ { \Omega } ( K _ H ( x ) \nabla v ^ \tau _ \varepsilon | \nabla v ^ \tau _ \varepsilon ) d x = 2 \pi q ^ 2 \sqrt { d e t ( K _ { H } ) } ( \bar { x } ) \end{align*}


\begin{align*} \sum _ { r _ { Q } ( n ) = 0 } n \ll p ^ { 1 + \epsilon } \frac { m } { \min Q ^ { * } } \ll \frac { p ^ { 3 + \epsilon } } { \left ( \min Q ^ { * } \right ) ^ { 2 } } . \end{align*}


\begin{align*} k ^ 2 f _ j ( 0 ) \int _ { S _ h } e ^ { \rho \cdot \mathbf x } \mathrm d \mathbf x = I _ 1 + I _ 2 + I _ 3 + J _ 1 + J _ 2 , \end{align*}


\begin{align*} V ( J ) = \{ x \in \R ^ n \ , : \ , p ( x ) = 0 \ , , \ , \forall p \in J \} \ , . \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta z _ { h } + F ' ( y ; z _ { h } ) & = h , \ ; \ ; \Omega , \\ z _ { h } & = 0 , \ ; \ ; \partial \Omega , \end{aligned} \right . \end{align*}


\begin{gather*} \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \ge \inf _ { Q \in \mathcal { U } } Q \Bigl ( \ , \sum _ { i = 1 } ^ n f _ i \circ \pi _ i \Bigr ) \end{gather*}


\begin{align*} \int _ { R _ { 0 } } ^ { \infty } \nu \xi ( \nu , s ) \ d \nu = 0 , s \in ( 0 , t _ { 0 } ) . \end{align*}


\begin{align*} \mu ^ n _ g ( t ) : = \sum _ { i = 1 } ^ n g _ i \psi _ i ( t ) , \end{align*}


\begin{align*} \nabla G ( U , x , t ) ^ T \nabla A ( U , x , t ) & = \nabla A ( U , x , t ) ^ T \nabla G ( U , x , t ) \\ \nabla f _ \alpha ( U , x , t ) ^ T \nabla A ( U , x , t ) & = \nabla f _ \alpha ( U , x , t ) ^ T \nabla G ( U , x , t ) \qquad \alpha = 1 , \dots , d , \end{align*}


\begin{align*} \lim _ { h \to 0 } \mathcal R \bigl [ \eqref { a h d u h } \bigr ] = \eqref { d u h a m e l 1 } \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta y + \tilde { f } ( y ) & = u _ { 1 } - u _ { 2 } , \ ; \ ; \Omega , \\ y & = 0 , \ ; \quad \quad \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} | \nabla r | = 1 \end{align*}


\begin{align*} u _ { \varepsilon } ( x ' , t ) : = u ( x ' , t + \varepsilon ) \quad f _ { \varepsilon } ( x ' ) : = u ( x ' , \varepsilon ) , x ' \in \R ^ n , \ , t > 0 . \end{align*}


\begin{align*} \begin{array} { l l l l l l } \partial _ q [ \mathbf a ] & = \partial _ q [ a _ 0 , \dots , a _ q , a _ { q + 1 } ] \\ & = \sum \limits _ { i = 0 } ^ { q + 1 } ( - 1 ) ^ i \cdot [ a _ 0 , \dots , \widehat { a _ i } , \dots , a _ { q + 1 } ] \in O [ m ] _ { q } \\ \end{array} \end{align*}


\begin{align*} ( D _ { a ^ + } ^ { \alpha } 1 ) ( x ) = \frac { ( x - a ) ^ { - \alpha } } { \Gamma [ 1 - \alpha ] } , \ \forall x \in [ a , b ] . \end{align*}


\begin{align*} \ln \frac { \sigma } { \tau } \int _ { \Omega } q ^ 2 ( K _ H ( x ) \nabla w ^ { \sigma , \tau } _ \varepsilon | \nabla w ^ { \sigma , \tau } _ \varepsilon ) d x = \frac { 1 } { \varepsilon ^ 2 } \int _ { \Omega } \left ( u _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p } q d x . \end{align*}


\begin{align*} \Phi ( a u _ \ell + b w _ \ell , p ) = c u _ \ell + d w _ \ell \end{align*}


\begin{align*} ( I - Q ) F ^ q ( v + \psi ( v , p ) , p ) = 0 \end{align*}


\begin{align*} F _ 1 = \frac { 1 } { \sqrt { c _ 1 } } \langle \langle \nu _ 1 \cdot \varphi , \varphi \rangle \rangle . \end{align*}


\begin{align*} \tau = \frac { \partial _ v \Gamma _ { u v } ^ v - 2 \Gamma _ { u v } ^ u \Gamma _ { v u } ^ v } { \partial _ u \Gamma _ { v u } ^ u - 2 \Gamma _ { u v } ^ u \Gamma _ { v u } ^ v } \neq 1 \end{align*}


\begin{align*} \Delta _ 1 ( m , n ) = \delta ( m , n ) + O ( ( m n ) ^ { 3 / 8 + \epsilon } k ^ { - 1 3 / 1 2 } ) . \end{align*}


\begin{align*} \L ^ n _ { \overline { A } } f \to \int f \ d \mu \ , \ \ \ \L ^ n _ { \overline { A } ^ * } f \to \int f \circ \theta ^ { - 1 } \ d \mu ^ * = \int f \ d \left ( \theta ^ { - 1 } _ * \mu ^ * \right ) \ , \end{align*}


\begin{align*} { \left ( { { { { D \mathord { \left / { \vphantom { D D } } \right . \kern - \nulldelimiterspace } D } } _ 0 } } \right ) ^ 2 } = 1 - K t , \end{align*}


\begin{align*} \frac { d ^ 2 \ , } { d \ , \tau ^ 2 } W ( \mathbf { e } + \tau \ , \xi \otimes \eta , \boldsymbol { \mathfrak { K } } + \tau \ , \zeta \otimes \eta ) \bigg | _ { \tau = 0 } > 0 \forall \ \eta , \xi , \zeta \in \mathbb { R } ^ 3 , \ \ \lVert \eta \rVert = \lVert \xi \rVert = \lVert \zeta \rVert = 1 . \end{align*}


\begin{align*} \Gamma \left ( X , Y , Z \right ) = C \left ( X , J Y , Z \right ) + g \left ( X , \left ( \nabla _ { Z } J \right ) Y \right ) . \end{align*}


\begin{align*} F ( x ) = \sum _ { { \bf k } \in \Lambda _ { 1 } } \begin{pmatrix} a _ { \bf k } & 0 \\ 0 & - a _ { \bf k } \end{pmatrix} { \rm e } ^ { { \rm i } \langle { \bf k } , x \rangle } + \sum _ { { \bf k } \in \Lambda _ { 2 } } \begin{pmatrix} 0 & b _ { \bf k } \\ c _ { \bf k } & 0 \end{pmatrix} { \rm e } ^ { { \rm i } \langle { \bf k } , x \rangle } , \end{align*}


\begin{align*} [ \partial a _ \lambda b ] & = - \lambda [ a _ \lambda b ] , \ \ [ a _ \lambda \partial b ] = ( \partial + \lambda ) [ a _ \lambda b ] \ \ \mbox { ( c o n f o r m a l \ s e s q u i l i n e a r i t y ) } , \\ { [ a _ \lambda b ] } & = - [ b _ { - \lambda - \partial } a ] \ \ \mbox { ( s k e w - s y m m e t r y ) } , \\ { [ a _ \lambda [ b _ \mu c ] ] } & = [ [ a _ \lambda b ] _ { \lambda + \mu } c ] + [ b _ \mu [ a _ \lambda c ] ] \ \ \mbox { ( J a c o b i \ i d e n t i t y ) } . \end{align*}


\begin{align*} w ( x ' , t ) = W ( \bar { R } _ { - \alpha | \ln \varepsilon | t } ( x ' ) ) , \ \ \varphi ( x ' , t ) = \varPhi ( \bar { R } _ { - \alpha | \ln \varepsilon | t } ( x ' ) ) , \end{align*}


\begin{align*} \begin{cases} \partial _ t \mathbf { v } + ( \mathbf { v } \cdot \nabla ) \mathbf { v } = - \nabla P , \ \ & D \times ( 0 , T ) , \\ \nabla \cdot \mathbf { v } = 0 , \ \ & D \times ( 0 , T ) , \\ \mathbf { v } ( \cdot , 0 ) = \mathbf { v } _ 0 ( \cdot ) , \ \ & D , \end{cases} \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \mathsf { M } ^ n ) = W ( 1 - \xi _ 0 , n , x , \mathsf { M } ^ n ) , ~ \forall \xi _ 0 \in [ 0 , 1 ] , \forall x \in \mathbb { R } . \end{align*}


\begin{align*} q ( A | E _ i ) = p ( A | E _ i ) , i q ( E _ i ) > 0 A \in { \bf A } , \end{align*}


\begin{align*} A = \begin{pmatrix} \sigma & 0 \\ 0 & 1 _ { 2 ( n - p ) } \end{pmatrix} \end{align*}


\begin{align*} \underline { a } : = \begin{pmatrix} a \\ & a \\ & & \ddots \\ & & & a \\ & & & & a \end{pmatrix} , \end{align*}


\begin{align*} B _ { 2 } = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 2 } } \Theta _ { \theta } \begin{bmatrix} m _ { 1 } & X \\ 0 & m _ { 2 } \end{bmatrix} \overline { \psi _ { A } ( { m _ { 1 } } ^ { - 1 } X } ) = q ^ { 8 } ( 1 - q ) . \end{align*}


\begin{align*} \sum _ { \tau \in D ( n - 1 ) } \operatorname { s i g n } ( \tau ) \prod _ { j = 1 } ^ { n - 1 } \dfrac { 1 } { 1 - \zeta ^ { j - \tau ( j ) } } = \frac { ( - 1 ) ^ { \frac { n - 1 } { 2 } } } { n } \left ( \frac { n - 1 } { 2 } ! \right ) ^ { 2 } , \end{align*}


\begin{align*} \int _ { \Omega } | \nabla ( v - \tilde { v } ) | ^ { 2 } + \int _ { \partial \Omega } | v - \tilde { v } | ^ { 2 } + \int _ { \Omega } u | v - \tilde { v } | ^ { 2 } = 0 , \end{align*}


\begin{align*} \sup _ { z \in \mathbb { B } _ n } \beta ( \varphi _ { k j } ( z ) , \rho ( z ) ) & \leq \sup _ { z \in \mathbb { B } _ n } k _ { V ^ { - 1 } \mathbb { B } _ n } ( V ^ { - 1 } \varphi _ { k j } ( z ) , V ^ { - 1 } \rho ( z ) ) \\ & = \sup _ { z \in V ^ { - 1 } \mathbb { B } _ n } k _ { V ^ { - 1 } \mathbb { B } _ n } ( V ^ { - 1 } \varphi _ { k j } V ( z ) , V ^ { - 1 } \rho V ( z ) ) . \end{align*}


\begin{align*} D ( p ) = \frac { \Gamma ( 2 x ) } { \Gamma ( x + \frac 1 2 ) ^ 2 \Gamma ( x + \frac 3 2 ) } = \frac { 2 ^ { 2 x - 1 } \Gamma ( x ) } { \sqrt { \pi } \Gamma ( x + \frac { 1 } { 2 } ) \Gamma ( x + \frac { 3 } { 2 } ) } . \end{align*}


\begin{align*} \begin{aligned} \Phi _ 2 ( \gamma ) : = \{ b _ 0 | \ , - 2 ( 2 - b _ 0 - \gamma ) > - 3 \ , , 2 \gamma > - 3 \} . \end{aligned} \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } \sup _ { n \in \mathbb { N } } \mathbb { E } | \langle Y _ n ^ M ( \zeta _ n + \delta ) - Y _ n ^ M ( \zeta _ n ) , e \rangle | ^ { \alpha } = 0 . \end{align*}


\begin{align*} ( p - L _ X h ) ^ \intercal & = - 2 \zeta A ' ( h ) \\ H ' ( p - L _ X h ) & = \zeta A \cdot A ' ( h ) . \end{align*}


\begin{align*} u _ { n n } \sigma _ { k - 1 } ( b ) - \sum _ { \alpha = 1 } ^ { n - 1 } u _ { \alpha n } ^ 2 \sigma _ { k - 2 ; \alpha } ( b ) + \sigma _ k ( b ) = f . \end{align*}


\begin{align*} [ S ] = \mu \left ( \frac { 1 } { 2 } d \overline { z } + ( \alpha - I \beta ) d z \right ) J \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } \Big ( \sup _ { t \leq T } \Vert Y _ n ( t ) \Vert _ H > \sqrt { M } \Big ) \leq \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } ( \tau _ { n } ^ M < T ) = 0 , \end{align*}


\begin{align*} \Gamma ^ { \dag } \left ( X , Y , Z \right ) \equiv \left ( \nabla _ { Z } ^ { \dag } G \right ) \left ( X , Y \right ) = - \Gamma \left ( X , Y , Z \right ) . \end{align*}


\begin{align*} \boldsymbol { A ^ * A } x = \| \boldsymbol { A } \| ^ 2 x . \end{align*}


\begin{align*} \begin{aligned} K f ( v ) = K _ 1 f ( v ) - K _ 2 f ( v ) \ , . \end{aligned} \end{align*}


\begin{align*} \frac { \lVert y - x \rVert _ 2 } { \lVert x \rVert _ 2 } = O \big ( \delta \ , \kappa ( A ) \big ) \end{align*}


\begin{align*} \rho { u _ z } = \sum \limits _ { i = 0 } ^ { 1 8 } { { c _ { i z } } { f _ i } + \frac { { \Delta t } } { 2 } { F _ z } } . \end{align*}


\begin{align*} \left \{ \begin{aligned} S ^ \prime _ t & = - \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t , & S _ 0 & = s _ 0 , \\ I ^ \prime _ t & = \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t - \gamma I _ t - I _ t \phi \left ( I _ t \right ) , & I _ 0 & = i _ 0 , \\ R ^ \prime _ t & = \gamma I _ t , & R _ 0 & = r _ 0 , \\ D ^ \prime _ t & = I _ t \phi \left ( I _ t \right ) , & D _ 0 & = d _ 0 . \\ \end{aligned} \right . \end{align*}


\begin{align*} b - b _ 1 = \sum _ { i = 2 } ^ r b _ i = l ( p - q ) + ( p - q r ) + 1 . \end{align*}


\begin{align*} f _ * \mathcal L ^ \vee ( d ) [ 2 d ] = ( f _ * \mathcal L ) ^ \vee \end{align*}


\begin{align*} | G ( n , m ) | \leq \left ( m ^ s + \frac { n ^ s } { \sqrt { \log ( n ) } } + \left ( \frac { n } { m } \right ) ^ { \frac { s } { 1 - s } } \right ) \cdot \omega \left ( m \left | \sum _ { k = 1 } ^ m \frac { \mu ( k ) } { k } \right | \right ) . \end{align*}


\begin{align*} \begin{aligned} | \langle \boldsymbol { \nu } , f _ \alpha ( \lambda | \bar U ; x , t ) \rangle | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; , \end{aligned} \end{align*}


\begin{align*} \eta ( q _ 1 ) \eta ( q _ 2 ) \overline { \pi } ( q _ 1 ) \overline { \pi } ( q _ 2 ) = \eta ( q _ 1 q _ 2 ) c _ { \chi _ 1 } ( q _ 1 , q _ 2 ) \overline { \pi } ( q _ 1 q _ 2 ) . \end{align*}


\begin{align*} \tilde { g } = ( \mathrm { i d } _ { S _ \C } \times d _ \C ^ { - 1 } ) \circ g \circ ( \mathrm { i d } _ { S _ \C } \times d _ \C ) \colon ( S _ \C \times _ \C \tilde { Y } _ n ^ \star , \sigma _ S \times \tilde { \sigma } ) \to ( S _ \C \times _ \C \tilde { Y } _ n ^ \star , \tilde { \Sigma } ) \end{align*}


\begin{align*} \Gamma \left ( X , Y , Z \right ) \equiv \left ( \nabla _ { Z } G \right ) \left ( X , Y \right ) = Z G \left ( X , Y \right ) - G \left ( \nabla _ { Z } X , Y \right ) - G \left ( X , \nabla _ { Z } Y \right ) . \end{align*}


\begin{align*} \gamma _ { w , k , m } \le \epsilon = 1 / 2 ^ b ~ { \rm f o r } ~ k \ge ( b + 2 ) \log _ { \Delta _ { v , m } } ( 2 ) ~ { \rm i f } ~ 2 \Delta _ { v , m } ^ k \le 1 . \end{align*}


\begin{align*} \left \| \sum _ { k = 1 } ^ n a _ k \xi _ k \right \| _ q \geq c _ d ( q ) \left \| \sum _ { k = 1 } ^ n a _ k \xi _ k \right \| _ 2 . \end{align*}


\begin{align*} \log \det ( u _ { , i j } ) = - v _ j x ^ j + u _ { , i } \xi ^ i + c , \xi \in \R ^ n , v \in ( \R ^ n ) ^ * . \end{align*}


\begin{align*} W _ { 1 } = \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle , W _ { 2 } = \left \langle \left [ \vartheta _ { 1 } \right ] , \left [ \vartheta _ { 2 } \right ] , \dots , \left [ \vartheta _ { s } \right ] \right \rangle \in G _ { s } \left ( { \rm H ^ { 2 } } \left ( { \bf A } , \mathbb C \right ) \right ) , \end{align*}


\begin{align*} \widetilde { \tau _ 2 } { \widetilde { \sigma } } ^ { - 1 } { \widetilde { \tau _ 2 } } ^ { - 1 } { \widetilde { \tau _ 1 } } ^ { - 1 } = { \widetilde \sigma } ^ { - 1 } { \widetilde { \tau _ 1 } } ^ { - 1 } [ \widetilde \sigma , \widetilde { \tau _ 2 } ] . \end{align*}


\begin{align*} Z _ { V , [ \alpha ] } ( t ) = - \int _ { V } \alpha + \sqrt { - 1 } t \int _ { V } \omega . \end{align*}


\begin{align*} \begin{aligned} \tfrac { \partial \nu ( v ) } { \partial | v | } \geq 0 \ \textrm { f o r } 0 \leq \gamma \leq 1 \ , , \textrm { a n d } \tfrac { \partial \nu ( v ) } { \partial | v | } < 0 \ \textrm { f o r } - 3 < \gamma < 0 \ , . \end{aligned} \end{align*}


\begin{align*} \tilde { B } = \{ z \in \R ^ n : | \Pi _ { \mathrm { k e r } A ^ \perp } z | ^ 2 \le \delta | \Pi _ { \mathrm { k e r } A } z | ^ { 2 r } \} , \end{align*}


\begin{align*} \begin{aligned} \chi ( s ) \equiv 0 \textrm { f o r } 0 < s \leq r \ , , \ \chi ( s ) \equiv 1 \textrm { f o r } s \geq 2 r \ , , \ 0 \leq \chi ( s ) \leq 1 \textrm { f o r a l l } s > 0 \ , . \end{aligned} \end{align*}


\begin{align*} & ( i ) \ \ \ \nabla ( \gamma _ 0 J + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ) ( x ^ * ) = 0 \\ & ( i i ) \ \ \gamma _ k f _ k ( x ^ * ) = 0 \ \ k \in \{ 1 , . . . , m \} \\ & ( i i i ) \ \gamma _ { 0 } A _ J + \sum \limits _ { k = 1 } ^ m \gamma _ k A _ k \succeq 0 \end{align*}


\begin{align*} v _ \epsilon { ' } ( t ) = \int _ { - \infty } ^ { t } ( \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ) ( s ) d s . \end{align*}


\begin{align*} g ( \nabla \Psi , \vec { \nu } ) = c o n s t \ \ \textnormal { o n e a c h l e a f } \Sigma _ { t } \end{align*}


\begin{align*} [ V ^ W ] = [ V \otimes F ' ] \end{align*}


\begin{align*} q _ X ( \alpha ) : = \frac { n } { 2 } \int _ X ( \sigma \overline \sigma ) ^ { n - 1 } \alpha ^ 2 + ( 1 - n ) \int _ X \sigma ^ n \overline \sigma ^ { n - 1 } \alpha \int _ X \sigma ^ { n - 1 } \overline \sigma ^ n \alpha \end{align*}


\begin{align*} \underset { 0 \leq t \leq T } { \sup } \Bigg [ \frac { 1 } { N } \underset { i = 1 } { \overset { N } { \sum } } \mathbb { E } ^ { \boldsymbol { \xi } } [ | \phi ^ { N , i } ( t , \boldsymbol { x } ^ * _ t ) - \Psi ^ { N , i } ( t , \boldsymbol { x } ^ * _ t ) | ^ 2 ] \Bigg ] \leq \frac { C } { N ^ 2 } \Big ( 1 + \frac { 1 } { N ^ 2 } \underset { i , j = 1 } { \overset { N } { \sum } } | \xi ^ { i } - \xi ^ { j } | ^ 2 \Big ) \end{align*}


\begin{align*} \left \{ \begin{array} { l } - \Delta \Psi = \lambda \left ( \Psi - W x _ { 2 } \right ) _ + , \ , \Pi , \\ \Psi \rightarrow 0 r \rightarrow \infty , \Psi = 0 , \ , \ , \ , \ , \partial \Pi , \\ \Psi \left ( x _ { 1 } , x _ { 2 } \right ) = - \Psi \left ( x _ { 1 } , - x _ { 2 } \right ) , \forall \ , x \in \mathbb { R } ^ { 2 } . \end{array} \right . \end{align*}


\begin{align*} F ( \lambda ) x _ j + \frac { 1 } { 1 ! } F ^ { ( 1 ) } ( \lambda ) x _ { j - 1 } + \ldots + \frac { 1 } { j ! } F ^ { ( j ) } ( \lambda ) x _ 0 = 0 , j = 0 , 1 , \ldots , k , \end{align*}


\begin{align*} ( d _ i y ) ( [ \mathbf { a ' } , \mathbf { a } ] \otimes [ 0 , \mathbf { b } ] ) = y ( d ^ i [ \mathbf { a ' } , \mathbf { a } ] \otimes [ 0 , \mathbf { b } ] ) = 0 , \end{align*}


\begin{align*} f _ { a } ( x ) = \dfrac { a } { a ^ 2 + x ^ 2 } . \end{align*}


\begin{align*} g ( \Gamma ) = h ^ 1 ( \Gamma ) + \sum _ { v \in V } g ( v ) , h ^ 1 ( \Gamma ) = | E | - | V | + 1 . \end{align*}


\begin{align*} \theta ^ 0 { } _ u ( X ) & = v ^ i { } _ \alpha d u ^ \alpha , \\ * \theta ^ 1 { } _ u ( X ) & = v ^ i { } _ \alpha d u ^ \alpha { } _ j - v ^ i { } _ \alpha u ^ \alpha { } _ { j \beta } v ^ \beta { } _ \gamma d u ^ \gamma . \end{align*}


\begin{align*} h = \nu & = 0 \\ \lambda \varepsilon _ 1 \delta _ 1 = \lambda \varepsilon _ 1 \delta _ 4 & = 0 \\ \varepsilon _ 2 \delta _ 2 & = \varepsilon _ 3 \delta _ 3 \\ \varepsilon _ 2 \delta _ 5 & = \varepsilon _ 3 \delta _ 6 \\ \lambda ^ 2 + \rho ^ 2 - \varepsilon _ 1 \mu \delta _ 1 + \varepsilon _ 2 \rho \delta _ 2 & = 0 \\ \lambda ^ 2 + \rho ^ 2 - \varepsilon _ 1 \mu \delta _ 4 + \varepsilon _ 2 \rho \delta _ 5 & = 0 . \end{align*}


\begin{align*} \tilde h _ d ( \alpha _ d ) & \leq O ( 1 ) + \log \frac { d \cdot \Gamma ( \frac { d } { 2 } ) } { \alpha _ d d ^ { \frac { d - 1 } { 2 } } } + d \log 2 \\ & = O ( 1 ) + \log \frac { \Gamma ( \frac { d } { 2 } ) } { C e ^ { - c d } d ^ { \frac { d - 1 } { 2 } } } + d \log 2 . \end{align*}


\begin{align*} V _ 2 ( r ) = \frac { L ' ( L ' + 1 ) } { r ^ 2 } - \frac { Q ' } { r } f + \kappa B ' _ 1 \frac { r } { f } + \kappa B ' _ 2 \frac { 1 } { f ^ 2 } + R , \end{align*}


\begin{align*} h ^ \intercal = 0 , A ' ( h ) = 0 , ( \nabla ^ k _ \nu A ) ' ( h ) = 0 \mbox { o n } \hat \Sigma \end{align*}


\begin{align*} u ^ s ( \mathbf x ) = \frac { e ^ { i k \vert \mathbf x \vert } } { \vert \mathbf x \vert ^ { ( n - 1 ) / 2 } } \left ( u ^ \infty ( \hat { \mathbf x } ) + \mathcal O \left ( \frac { 1 } { \vert \mathbf x \vert ^ { ( n - 1 ) / 2 } } \right ) \right ) , \vert \mathbf x \vert \to \infty , \end{align*}


\begin{align*} H _ 2 ^ { [ 2 ] } \bigl ( \alpha ( t ) \bigr ) = \pm \int _ { - \pi } ^ \pi 4 \sin ^ 2 ( \theta ) | \widehat { \alpha } ( t , \theta ) | ^ 2 \ , \tfrac { d \theta } { 2 \pi } , \end{align*}


\begin{align*} v ( \mathbf 0 ) = 0 . \end{align*}


\begin{align*} & \mathcal { P } ^ - ( f ( z ) e ^ { 2 i z x } ) = - \int _ { 2 x } ^ \infty \widehat { f } ( \xi ) e ^ { - i z ( \xi - 2 x ) } \mathrm { d } \xi , \end{align*}


\begin{align*} \left \{ \begin{aligned} S ^ \prime _ t & = - \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t , & S _ 0 & = s _ 0 \\ I ^ \prime _ t & = \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t - \gamma I _ t - I _ t \phi \left ( I _ t \right ) , & I _ 0 & = i _ 0 , \end{aligned} \right . \end{align*}


\begin{align*} \sigma _ { k , i } ( x ) = a _ { k , i } e _ { k } ( x ) , \end{align*}


\begin{align*} K _ q ( z , w ) = E _ q ( z \overline { w } ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { ( z \overline { w } ) ^ n } { \Gamma ( q n + 1 ) } , \end{align*}


\begin{align*} Z ^ i ( v ( s ) , s ) - Z ( v ( s ) , s ) = \mathcal O ( \norm { ( v ( s ) , s ) } ^ { i + 1 } ) = \mathcal O ( s ^ \frac { i + 1 } { 2 } ) . \end{align*}


\begin{align*} [ X \times _ k \mathbb A _ k ^ n \to S , \sigma ] = [ X \times _ k \mathbb A _ k ^ n \to S , \sigma ' ] \end{align*}


\begin{align*} \begin{aligned} \int _ B \varphi _ 0 \Delta u _ 1 \wedge \dots \wedge \Delta u _ m \wedge \beta _ n ^ { n - m } \leq \gamma \left ( \| \varphi _ 0 \| _ S + \int _ B \varphi _ 0 ( \Delta \varphi _ 0 ^ { + } ) ^ m \wedge \beta _ n ^ { n - m } \right ) ^ \kappa , \end{aligned} \end{align*}


\begin{align*} h ( \nu , \cdot ) = 0 , ( \nabla _ { \nu } h ) ( \nu , \cdot ) = 0 . \end{align*}


\begin{align*} x ( t ) = \frac { 1 } { \sqrt { 2 t + 1 } } \end{align*}


\begin{align*} \lambda _ n = ( \lambda { m } ) ^ n . \end{align*}


\begin{align*} X ^ \sigma : = \lim X \in \mathcal C \end{align*}


\begin{align*} & \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi _ { j , + } ^ { 2 } ( \partial _ t \varphi + ( v _ j + u _ { \infty } ) \cdot \nabla \varphi ) \dd x \dd s + \int _ { \mathbb { R } ^ { 3 } } \Phi _ { 0 , + } ^ { 2 } \varphi _ 0 \dd x \\ & = - \mu _ j \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \left ( \Phi _ { j , + } ^ { 2 } \Delta \varphi + \frac { 2 } { r } \partial _ r \Phi _ { j , + } ^ { 2 } \varphi - 2 | \nabla \Phi _ { j , + } | ^ { 2 } \varphi \right ) \dd x \dd s . \end{align*}


\begin{align*} & \sup \{ | \varphi _ { 2 k j _ 0 } ( z ) - \rho ( z ) | ; \ z \in \mathbb { B } _ n , \ | \varphi _ { k j _ 0 } ( z ) | \leq r \} \\ & = \sup \{ | \varphi _ { k j _ 0 } \circ \varphi _ { k j _ 0 } ( z ) - \rho \circ \varphi _ { k j _ 0 } ( z ) | ; \ z \in \mathbb { B } _ n , \ | \varphi _ { k j } ( z ) | \leq r \} \\ & \leq \sup \{ | \varphi _ { k j _ 0 } ( z ) - \rho ( z ) | ; \ z \in \mathbb { B } _ n , \ | z | \leq r \} \leq \varepsilon \end{align*}


\begin{align*} E _ { \omega } : = \inf _ { u \in \mathcal { N } _ { \omega } } J _ { \omega } ( u ) , \end{align*}


\begin{align*} A '' _ n ( j ) = A _ n ( j ) A ' _ n ( j ) \end{align*}


\begin{align*} H [ \nabla \times ( \phi \nabla \theta ) + G \nabla \theta ] = - H [ \nabla \times ( \phi \nabla \theta ) - G \nabla \theta ] \end{align*}


\begin{align*} \div X = 0 . \end{align*}


\begin{align*} \int _ X | | \xi ( x ) | | G \ \mu ( d x ) = \int _ X \beta ( x ) \ \mu ( d x ) , \end{align*}


\begin{align*} \begin{cases} \frac { d } { d t } \tilde { X } _ t = B ( \tilde { X } _ t , \tilde { X } _ t ) \\ \tilde { X } _ t | _ { t = 0 } = \Pi _ { \mathrm { k e r } A } x _ 0 . \end{cases} \end{align*}


\begin{align*} & \sum _ { a = 0 } ^ t \pi _ N ( x ) ^ a ( \log \log x ) ^ { t - a } L ^ { 2 r - a } L ^ { 2 r + x ' - 2 t } \sum _ { ( b _ 1 , b _ 2 , \dots , b _ { t } ) \atop { b _ u \in \mathcal J _ { \mathcal I ( s _ u ) } \atop { a ( \underline { b } ) = a } } } L ^ { - a _ 3 ( \underline { b } ) } \\ & \ll _ r \sum _ { a = 0 } ^ t \pi _ N ( x ) ^ a ( \log \log x ) ^ { t - a } L ^ { 2 r - a } L ^ { 2 r + x ' - 2 t } , \end{align*}


\begin{align*} x ( \varpi _ n ) = \varphi _ M ^ { - n } ( \frac \varpi \pi x ' ) \end{align*}


\begin{align*} n H = \eta x _ { n + 1 } ^ m + \lambda . \end{align*}


\begin{align*} \int _ { \{ \Psi < - t \} } | F _ t + \alpha q | ^ 2 _ h c ( - \Psi ) - \int _ { \{ \psi < - t \} } | F _ t | ^ 2 _ h c ( - \Psi ) \geq 0 \end{align*}


\begin{align*} H ( E ) = \left ( \N \left ( \prod _ { i = 1 } ^ { \ell } \mathfrak a _ i \right ) \prod _ { v \in V _ \infty } \det ( h _ { v } ( b _ i , b _ j ) ) ^ { e _ { v } / 2 } \right ) ^ { 1 / [ K : \mathbb { Q } ] } \end{align*}


\begin{align*} R = \left ( \prod _ { m = i + 1 } ^ { j - 1 } M _ m ^ T \bar { Q } _ m \right ) \times M _ j ^ T , \end{align*}


\begin{align*} \lim _ { j \rightarrow \infty } M _ j ( C _ \varphi ) = \frac { 1 } { k } \sum _ { i = 0 } ^ { k - 1 } C _ { \rho \circ \varphi _ i } . \end{align*}


\begin{align*} ( I - Q ) F ^ q ( v + \psi ( v , p ) , p ) = 0 \end{align*}


\begin{align*} \widehat { f } ( \lambda , \omega ) = c _ { n , m } \ , P _ m ( \lambda \omega ) \ , \int _ 0 ^ \infty g ( r ) \ , \frac { J _ { n / 2 + m - 1 } ( \lambda r ) } { ( \lambda r ) ^ { n / 2 + m - 1 } } r ^ { n + 2 m - 1 } d r . \end{align*}


\begin{align*} Z _ { V , [ \alpha ] } ( t ) = \frac { 1 } { 2 } \int _ { V } ( t ^ { 2 } \omega ^ { 2 } - \alpha ^ { 2 } ) + \sqrt { - 1 } t \int _ { V } \alpha \wedge \omega . \end{align*}


\begin{align*} \beta _ 1 ( u ) = 0 & \Longrightarrow \beta _ 1 ( u _ { \varepsilon } ) = 0 , \\ \beta _ 1 ( u ) = \beta ' _ 3 ( u ; Q ) = 0 & \Longrightarrow \beta ' _ 3 ( u _ { \varepsilon } ; Q ) = 0 . \end{align*}


\begin{align*} \begin{aligned} b ( \omega , v _ 1 - v ) = \beta ( \theta ) | v _ 1 - v | ^ \gamma \ , , \ \cos \theta = \frac { ( v _ 1 - v ) \cdot \omega } { | v _ 1 - v | } \ , , \ - 3 < \gamma \leq 1 \ , , \end{aligned} \end{align*}


\begin{align*} \lim _ { z \downarrow 0 } \hat { F } _ { \pm } ( 0 ) \int _ 0 ^ \infty w _ z ( \lambda ) \lambda ^ { 4 ( 1 - s + w ) } \frac { d \lambda } { \lambda } \int _ { G ^ 1 / \Gamma } \phi ( g ^ { - 1 } ) d g = 0 \end{align*}


\begin{align*} A u t \left ( \mathcal { R P } ( G ) \right ) = \mathbb Z _ 2 \times { \prod _ { i = 1 } ^ { t } S _ { \widehat { u _ i } } } . \end{align*}


\begin{align*} C _ 3 ( t ) = \sum _ { u : \ , i _ u \geq 3 \atop { 2 \leq d _ u \leq i _ u } } d _ u = \sum _ { i \geq 3 } \sum _ { d = 2 } ^ { i } d x _ { i , d } , \\ \end{align*}


\begin{align*} v ( r , \xi ) = \Delta ^ { N } _ \Gamma u ( r , \xi ) . \end{align*}


\begin{align*} \Q ^ t \partial _ \ell \Q \nabla y _ k = y _ k ^ { - 1 } \partial _ \ell f _ k [ w ^ k ] ^ \perp , \Q ^ t \partial _ \ell \Q \ , x = \sum _ { i = 1 } ^ d \partial _ \ell f _ i [ w ^ i ] ^ \perp , \end{align*}


\begin{align*} & f _ k ( \Tilde { x } ) \le \lambda f _ k ( x _ v ) + ( 1 - \lambda ) f _ k ( x _ w ) < \lambda v _ k + ( 1 - \lambda ) w _ k . \\ & a _ k \| \Tilde { x } \| ^ 2 + 2 b _ k ^ T \Tilde { x } + c _ k \le a _ k ( \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 ) + 2 b _ k ^ T ( \lambda x _ v + ( 1 - \lambda ) x _ w ) + c _ k . \end{align*}


\begin{align*} g _ \lambda ( z ) = \begin{cases} 0 & z = \lambda , \\ ( z - \lambda ) ^ { - 1 } & z \neq \lambda , \end{cases} \end{align*}


\begin{align*} I ( \varphi , \overline { \mathbf { R } } ) = \dd \int _ { \Omega } \left [ W _ { \rm { m p } } ( \overline { \mathbf { U } } ) + W _ { \rm { c u r v } } ( \boldsymbol { \alpha } ) \right ] d V { \to } \textrm { \ \ m i n . } { \rm w . r . t . } ( \varphi , \overline { \mathbf { R } } ) \ , , \end{align*}


\begin{align*} [ T ( f ) ] ( \xi ) : = \frac { f ( \xi ) } { \sqrt { ( 1 + \gamma ( \xi ) ^ 2 ) } } . \end{align*}


\begin{align*} 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} , \ i \mapsto \begin{pmatrix} \sqrt { a } & 0 \\ 0 & - \sqrt { a } \end{pmatrix} , \ j \mapsto \begin{pmatrix} 0 & \sqrt { b } \\ \sqrt { b } & 0 \end{pmatrix} , \ i j \mapsto \begin{pmatrix} 0 & \sqrt { a b } \\ - \sqrt { a b } & 0 \end{pmatrix} . \end{align*}


\begin{align*} H = \left \{ S \in V ^ { ( k - 1 ) } : \ : \deg _ { G } ( S ) \geq \sqrt { d } \right \} , \end{align*}


\begin{align*} & \xi ( \xi + 1 ) = L ( L + 1 ) , \\ & - \kappa \xi ^ 2 + 2 \xi \zeta + \eta ^ 2 - \frac { \zeta ^ 2 } { \kappa } = - E _ 0 , \\ & 2 \xi \eta = - Q , \\ & 2 \eta \zeta = \kappa B _ 1 , \\ & \frac { \zeta } { \kappa } \left ( \frac { \zeta } { \kappa } - 1 \right ) = B _ 2 . \end{align*}


\begin{align*} & ( \nexists \ , x \in \mathbb { R } ^ n ) \ \ s . t . \\ & f ( x ) < 0 \\ & g ( x ) \le 0 \end{align*}


\begin{align*} \int _ { \R ^ { n - 1 } } \langle f ( x ' ) , h ( x ' ) \rangle d x ' = \lim _ { \varepsilon \to 0 ^ + } \int _ { \R ^ { n - 1 } } \langle f ( x ' ) , \widetilde { \Psi } _ { \varepsilon , \varepsilon ^ { - 1 } } * h ( x ' ) \rangle d x ' . \end{align*}


\begin{align*} \xi [ X ] \stackrel { d e f } { = } \int _ X \xi ( x ) \ \mu ( d x ) \ - \end{align*}


\begin{align*} \frac { 1 } { 2 } \int _ { \Omega } | U | ^ { 2 } \dd x = \frac { 1 } { 2 } \int _ { \Omega } U \cdot \nabla \times ( \textrm { c u r l } ^ { - 1 } U ) \dd x & = \frac { 1 } { 2 } \int _ { \Omega } \nabla \times U \cdot \textrm { c u r l } ^ { - 1 } U \dd x \\ & = \frac { f _ { 1 } ^ { + } } { 2 } \int _ { \Omega } U \cdot \textrm { c u r l } ^ { - 1 } U \dd x = { \mathcal { I } } _ 1 . \end{align*}


\begin{align*} \Delta \rho _ 1 ( x ) \leq \psi _ { K , - H } ( \rho _ 1 ( x ) ) = ( n - 1 ) \sqrt K \cdot \frac { ( n - 1 ) \sqrt K \sinh \sqrt K \rho _ 1 - H \cosh \sqrt K \rho _ 1 } { ( n - 1 ) \sqrt K \cosh \sqrt K \rho _ 1 - H \sinh \sqrt K \rho _ 1 } . \end{align*}


\begin{align*} h ( G ) & = h ( G ' ) + f ( y ) + \sum _ { j = 1 } ^ y f ( x _ j ) - \sum _ { j = 1 } ^ y f ( x _ j - 1 ) \\ & \le h ( K _ { 1 , m - y } ) + f ( y ) + y \left ( f \left ( \frac { m } { y } \right ) - f \left ( \frac { m } { y } - 1 \right ) \right ) . \end{align*}


\begin{align*} \P \left ( \frac { 1 } { m } \sum _ { i = 1 } ^ m V _ i \leq \left ( \frac { \pi } { 4 } + \frac { 1 9 9 \pi } { 3 2 0 0 k } \right ) \| \mathbf { x } \| \right ) \leq 2 \exp \left ( - \frac { c _ 5 m } { k } \right ) . \end{align*}


\begin{align*} L ( p \mid u ) = \prod _ { i = 1 } ^ m p _ j ^ { u _ j } . \end{align*}


\begin{align*} d _ { i j } : = \langle \eta _ i , \eta _ j \rangle = 1 + \frac { \delta _ { i j } } { \varphi _ i } , \end{align*}


\begin{align*} X _ { \mathbf e ' ; s } ( M ' ) = Z _ { \mathbf e ' ; r } ( M ) r + s = \sum _ { j \in \Omega ( k , - ) } b _ { k j } e _ j . \end{align*}


\begin{align*} \lim _ { n \to \infty } \| T _ n p _ n f - p _ n T f \| = 0 f \in X . \end{align*}


\begin{align*} \mathbb { E } _ { n , m } ' [ \beta \gamma ] \\ = \mathbb { E } _ { \mathcal { M } } [ \mathbb { 1 } _ { X _ 0 ( M ) } ( x _ 1 ) \mathbb { 1 } _ { X _ 1 ( M ) } ( x _ 2 ) \mathbb { E } _ { n , m } [ \chi _ { n , m } ( w , x _ 1 ) \chi _ { n , m } ( w , x _ 2 ) | M ] \end{align*}


\begin{align*} | | \tau | | G \psi \stackrel { d e f } { = } \sup _ { p \in ( 1 , b ) } \left \{ \ \frac { | | \tau | | L _ p ( \Omega ) } { \psi ( p ) } \ \right \} . \end{align*}


\begin{align*} d I _ 3 \cap ( I _ 1 \cup I _ 2 \cup I _ 3 ) = \emptyset , \end{align*}


\begin{align*} \sigma ( \lambda \alpha ) = \sigma ( \lambda ) \sigma ( \alpha ) = \lambda \sigma ( \alpha ) . \end{align*}


\begin{align*} a \in \sigma _ l ( A ) \mbox { w i t h e i g e n v e c t o r s } \left \{ \begin{aligned} v = ( x , 1 ) ^ T , \ , \forall x \in S _ L ( c , a - d ) \mbox { p r o v i d e d } S _ L ( c , a - d ) \neq \emptyset ; \\ v = ( 1 , x ) ^ T , \ , \forall x \in S _ L ( a - d , c ) \mbox { p r o v i d e d } S _ L ( a - d , c ) \neq \emptyset . \end{aligned} \right . \end{align*}


\begin{align*} f _ \ell ( x ) = \sqrt { \frac { \mu _ d } { n _ { \ell ; d } } } \sum _ { m = 1 } ^ { n _ { \ell ; d } } c _ { \ell , m } Y _ { \ell , m ; d } ( x ) , x \in \mathbb { S } ^ d , \end{align*}


\begin{align*} N = \begin{pmatrix} Q & 0 _ { k \times k } \\ 0 _ { k \times k } & Q \end{pmatrix} . \end{align*}


\begin{align*} \left ( t _ i ^ 2 + c - c \ , ( U ^ \top U ) _ { i , i } \right ) U ^ \top _ { i , h } = \mu _ h U ^ \top _ { i , h } . \end{align*}


\begin{align*} 0 = R _ { 4 1 } ^ \delta - R _ { 6 3 } ^ \delta = h ^ 2 + \varepsilon _ 2 \delta _ 2 \left ( \lambda - \rho \right ) . \end{align*}


\begin{align*} & \lim _ { n \rightarrow \infty } \int _ { \mathbb { T } ^ d } \langle a ( u _ n ( x ) ) \nabla u _ n ( x ) + b ( u _ n ( x ) ) , \nabla u _ n ( x ) - \nabla v ( x ) \rangle d x \\ = & \int _ { \mathbb { T } ^ d } \langle a ( u ( x ) ) \nabla u ( x ) + b ( u ( x ) ) , \nabla u ( x ) - \nabla v ( x ) \rangle d x . \end{align*}


\begin{align*} \psi _ { \tau } ( p ) = | | \tau | | _ p \le C _ 1 ( \beta , \gamma , L ) \ ( \beta - p ) ^ { - ( \gamma + 1 ) / \beta } \ L ^ { 1 / \beta } ( 1 / ( \beta - p ) ) , \end{align*}


\begin{align*} \int _ W | \Delta \rho | \leq \int _ W \left ( | C - \Delta \rho | + C \right ) = \int _ W ( 2 C - \Delta \rho ) = 2 C \ , _ n ( W ) + _ { n - 1 } ( \partial W ) . \end{align*}


\begin{align*} w _ \varepsilon ( x _ 1 , x _ 2 , 0 , t ) = \frac { 1 } { \varepsilon ^ 2 } \left ( u _ \varepsilon ( \bar { R } _ { - \alpha | \ln \varepsilon | t } ( x _ 1 , x _ 2 ) ) - \left ( \frac { \alpha | ( x _ 1 , x _ 2 ) | ^ 2 } { 2 } + \beta \right ) \ln \frac { 1 } { \varepsilon } \right ) ^ p _ + . \end{align*}


\begin{align*} \mathcal { M _ { \mathbb { N } } } & = \{ \omega = \frac { \sum _ { h = 1 } ^ { H } c _ h \delta _ { a _ h } } { \sum _ { h = 1 } ^ { H } c _ h } : \ ; c _ h \in \mathbb { N } ^ + \} , \\ \mathcal { M _ { \mathbb { Q } } } & = \{ \omega = \frac { \sum _ { h = 1 } ^ { H } c _ h \delta _ { a _ h } } { \sum _ { h = 1 } ^ { H } c _ h } : \ ; c _ h \in \mathbb { Q } ^ + \} , \\ \mathcal { M _ { \mathbb { R } } } & = \{ \omega = \frac { \sum _ { h = 1 } ^ { H } c _ h \delta _ { a _ h } } { \sum _ { h = 1 } ^ { H } c _ h } : \ ; c _ h \in \mathbb { R } ^ + \} , \end{align*}


\begin{align*} C ^ - ( k ) = e ^ { - i c \sigma _ 3 } , C ^ + ( k ) = { I } . \end{align*}


\begin{align*} \mathbf { c } ^ { ( 2 ) } = ( \star , \underbrace { 1 , 1 , \cdots , 1 } _ { 2 k } , \star , \star , \cdots , \star ) . \end{align*}


\begin{align*} \int | f | ^ 4 \lesssim \sum _ { \substack { R ^ { \frac { 1 } { 2 } } \le W \le R \\ W \ , } } \sum _ { \ell ( \tau ) = \frac { W } { R } } \sum _ { U \| U _ { \tau , R } } | U | ^ { - 1 } \Big ( \int \sum _ { \theta \subset \tau } | f _ \theta | ^ 2 W _ U \Big ) ^ 2 . \end{align*}


\begin{align*} M _ { f _ { 1 , \infty } } ( n , \alpha , \epsilon , Z , \psi ) : = \inf \bigg \{ \sum _ { i } ^ { - \alpha n _ { i } + S _ { 1 , n _ { i } } \psi ( x _ { i } , \epsilon ) } : Z \subseteq \bigcup _ { i } B _ { n _ { i } } ( x _ { i } , \epsilon ) \bigg \} , \end{align*}


\begin{align*} \varepsilon _ { f , n } = \sum _ { j = 1 } ^ { \infty } ( - 1 ) ^ { j - 1 } \zeta _ { ( f ) } ^ { \varphi ^ { - ( n + 1 + 2 j ) } } p ^ j \end{align*}


\begin{align*} A _ n & : = \big \{ M ^ * _ v ( \delta n ) \in ( - \delta , \delta ) \big \} , \\ B _ n & : = \big \{ | M _ v ( t ) - M _ v ( \delta n ) | < \delta e ^ { ( \lambda ^ * - \lambda ) \delta n } t \in [ \delta n , \delta ( n + 1 ) ] \big \} . \end{align*}


\begin{align*} g _ \mu : = \arg \max _ { \psi \in \mathcal { H } } \left \{ - \frac { \mathrm { d } } { \mathrm { d } \gamma } \mathrm { K L } \left ( T _ { \gamma } { \# } \mu \| \pi \right ) \Big | _ { \gamma = 0 } \quad \| \psi \| _ { \mathcal { H } } \leq 1 \right \} . \end{align*}


\begin{align*} G _ n / \langle \epsilon \rangle = G _ n / \{ 1 , \epsilon \} \cong ( \mathbb Z / 2 \mathbb Z ) ^ n \rtimes S _ n = B _ n . \end{align*}


\begin{align*} H ^ { ( k ) } ( E ) = \min \limits _ { \substack { 0 \neq F \subset E \\ \dim F = k } } H _ r ( F ) \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { c _ \varepsilon } { \ln \frac { 1 } { \varepsilon } } = \pi \min \limits _ { \overline { \Omega } } q ^ 2 \sqrt { d e t ( K _ H ) } . \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } \sup _ { n \in \mathbb { N } } \mathbb { P } \Big ( | \langle Y _ n ( \zeta _ n + \delta ) - Y _ n ( \zeta _ n ) , e \rangle | > \varepsilon \Big ) = 0 , \end{align*}


\begin{align*} C ^ { \ast } \left ( X , Y , Z \right ) \equiv \left ( \nabla _ { Z } ^ { \ast } g \right ) \left ( X , Y \right ) = - C \left ( X , Y , Z \right ) \end{align*}


\begin{align*} V \cdot \psi _ 1 = \psi _ 2 . \end{align*}


\begin{align*} S ( X ; \pi , \hat { \phi } , W ) : = \sum _ { ( d , 2 ) = 1 } \mu ^ 2 ( d ) \sum _ { p } \frac { a _ { \pi } ( p ) \log p } { \sqrt { p } } \left ( \frac { 8 d } { p } \right ) \hat { \phi } \left ( \frac { \log p } { M \log X } \right ) W \left ( \frac { d } { X } \right ) . \end{align*}


\begin{align*} h ( T ' ) & = y q \log ( y q ) + r \Delta ( q + 1 ) + f ( r ) \\ & = q y \log ( q ) + q y \log ( y ) + r \log ( q ) + r + f ( r ) + o ( 1 ) \\ & = m \log ( m - r ) - r \log ( y ) + r + r \log ( r ) + o ( 1 ) \\ & = m \log ( m ) - r \log \left ( \frac y r \right ) + o ( 1 ) , \end{align*}


\begin{align*} Q = \ln \kappa _ 1 - N \ln \nu ^ { n + 1 } , \end{align*}


\begin{gather*} g ( X , \nabla \Psi ) \equiv c o n s t \\ \Updownarrow \\ \div _ \Psi ( X ) = \div ( X ) - g ( X , \nabla \Psi ) \equiv c o n s t . \end{gather*}


\begin{align*} \begin{array} { r l } \min & \sum _ { j = 1 } ^ n z _ j ^ { p } \\ & z \ge 0 \ , , \ , M z + q \ge 0 \ , , \ , z ^ \top ( M z + q ) = 0 \ , , \end{array} \end{align*}


\begin{align*} \dot x = \varphi _ { \epsilon , r _ 0 } ( x , h ( x ) , t ) x ( 0 ) = x _ 0 , \end{align*}


\begin{align*} y = \sum _ { m = 0 } ^ \infty \frac { \psi ( \eta _ m ( y ) ) } { \gamma ^ { m + 1 } } . \end{align*}


\begin{align*} \Phi \ , : \ , \left ( \C ^ { Z _ m \times Z _ m } \right ) ^ { G _ m } \to \bigoplus _ { i = 1 } ^ k \C ^ { m _ i \times m _ i } , \end{align*}


\begin{align*} [ \mathbf a ] = [ a _ 0 , a _ 1 , \dots , a _ q ] \end{align*}


\begin{align*} H ( p , s ) = G ( p , s ) - F ( p , s ) . \end{align*}


\begin{align*} \dot { x } = - \gamma h ( x + \delta u ) u = - \gamma ( h ( x ) + R ( x , \delta u ) ) u \end{align*}


\begin{align*} s _ i ( A ) : = \sqrt { \lambda _ i ( A ^ * A ) } , i = 1 , 2 , \dots , n , \end{align*}


\begin{align*} ( \mathbf { x } , - \mathbf { y } ) M _ { I J } = 0 . \end{align*}


\begin{align*} \xi _ y ( x ) = e ^ { 2 \pi i \ , y \cdot x } , \end{align*}


\begin{align*} & | \epsilon ( x _ a , t ) | \le \bar \epsilon ( L _ r , M _ r , \delta ) : = 2 L _ r \delta + M _ r \ , . \end{align*}


\begin{align*} \chi ^ g = \chi \Rightarrow \chi ( n ^ g ) = \chi ( n ) \forall n \in N \Rightarrow n ^ g = n \forall n \in N . \end{align*}


\begin{align*} \begin{cases} \partial _ t \mathbf { w } + ( \mathbf { v } \cdot \nabla ) \mathbf { w } = ( \mathbf { w } \cdot \nabla ) \mathbf { v } , \ \ & D \times ( 0 , T ) , \\ \nabla \cdot \mathbf { v } = 0 , \ \ & D \times ( 0 , T ) , \\ \mathbf { v } \cdot \mathbf { n } = v _ n , \ \ & \partial D \times ( 0 , T ) , \\ \mathbf { w } ( x , 0 ) = \nabla \times \mathbf { v } _ 0 ( x ) , \ \ & D . \end{cases} \end{align*}


\begin{align*} p ( x ) : = \sum _ { i = 0 } ^ d p _ i x ^ i = p _ d \prod _ { j = 1 } ^ d ( x - x _ j ) , ~ p _ d \neq 0 , \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) = \lambda _ n ( u _ n , v _ n ) \quad \ , v _ n \in X _ n . \end{align*}


\begin{align*} W _ { U _ { \tau _ k , R } } ( x ) = W \circ R _ { \tau _ k } ( x ) \end{align*}


\begin{align*} H ^ { ( k ) } ( E ) = \min \limits _ { \substack { 0 \neq F \subset E \\ \dim F = k } } H _ r ( F ) \end{align*}


\begin{align*} { y _ { \rm f } } \left [ { m , k } \right ] = { { { \bf { \tilde h } } } ^ H } \left [ k \right ] { { \bf { u } } _ k } s \left [ { m , k } \right ] + z \left [ m , k \right ] , \ \forall k \in \left [ 0 , { K } - 1 \right ] . \end{align*}


\begin{align*} V ( \xi _ 0 , n , x ) : = \sup _ { \theta \in \Theta _ n } W ( \xi _ 0 , n , x , \theta ) = \sup _ { \theta \in \Theta _ n } E _ P \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] . \end{align*}


\begin{align*} & M i n i m i z e _ { \mathbf { x } \in \mathbb { R } ^ { 3 n } } \ \ \sum \limits _ { i = 1 } ^ n ( a _ J , a _ J , b _ { J i } ) ^ T \mathbf { x } _ i \\ & \sum \limits _ { i = 1 } ^ n ( a _ k , a _ k , b _ { k i } ) ^ T \mathbf { x } _ i + c _ k \le 0 \ \ \forall \ , k \in { 1 , . . . , m } \\ & \mathbf { x } _ i \in \mathcal { L } _ i \end{align*}


\begin{align*} \langle { \boldsymbol { \nu } } _ { x , t } , g ( A ( \lambda , x , t ) , x , t ) \rangle = \langle { \boldsymbol { \nu } } _ { x , t } , f ( \lambda , x , t ) \rangle = \langle { \boldsymbol { N } } _ { x , t } , g ( \rho , x , t ) \rangle \end{align*}


\begin{align*} \xi \left ( ( a ) \right ) = \left ( \frac { a } { | a | } \right ) ^ { k - 1 } \end{align*}


\begin{align*} \exists C _ 3 = C _ 3 ( m ) \in ( 0 , \infty ) \ \Rightarrow T [ M ^ * n ] ( t ) \le \exp ( - C _ 3 ( m ) \ t ^ m ) , \ t \ge 0 , \end{align*}


\begin{align*} V _ { \delta , Z } ( x ) - q ( x ) = V _ { \delta , z _ i , \hat { q } _ { \delta , i } , z _ i } ( x ) - \hat { q } _ { \delta , i } + O \left ( \frac { \ln | \ln \varepsilon | } { | \ln \varepsilon | ^ 2 } \right ) . \end{align*}


\begin{align*} w _ \Gamma ( x ) = j p ( x ) / q ( x ) \in [ \gamma _ { j } , \gamma _ { j + 1 } ) . \end{align*}


\begin{align*} \Pi _ S ( X ) \rightarrow \oplus _ { i \in I } \Pi _ S ( D _ i , N _ i ) \rightarrow \cdots \rightarrow \oplus _ { J \subset I , \sharp J = n } \Pi _ S ( D _ J , N _ J ) \rightarrow \cdots \end{align*}


\begin{align*} 0 \le c _ { r } = r ^ { 1 - d } \int _ { 0 } ^ { r } \rho ^ { d - 1 } n c \ , d \rho \le \frac { \gamma M _ { 0 } } { \sigma _ { d } } r \quad \mbox { f o r } r \in ( 0 , R ] . \end{align*}


\begin{align*} \begin{aligned} | \mathfrak { a } ( | v | ) | + | \mathfrak { b } ( | v | ) | \leq C ( 1 + | v | ) \ \forall v \in \R ^ 3 \ , . \end{aligned} \end{align*}


\begin{align*} \exists j \leq J , \ ; \Pi _ { \mathrm { k e r } A ^ { \perp } } \frac { d ^ j } { d t ^ j } X _ t | _ { t = 0 } \neq 0 . \end{align*}


\begin{align*} \lambda _ x | _ { E _ { 2 , m } } = \lambda _ y , \lambda _ x ( m + v _ i ) = f \lambda _ x ( m + e _ i ) = s ( f ) \end{align*}


\begin{align*} \langle A ( t , u ) , v \rangle = - & \int _ { \mathcal { O } } \Big \{ \sum _ { i = 1 } ^ { d } a _ i \big ( t , x , u ( x ) , \nabla u ( x ) \big ) \partial _ i u ( x ) \\ & ~ ~ ~ ~ ~ ~ + a _ 0 \big ( t , x , , u ( x ) , \nabla u ( x ) \big ) v ( x ) \Big \} d x . \end{align*}


\begin{align*} | \xi | _ p \le \psi ( p ) = \psi _ { p _ 0 } [ \eta ] ( p ) , \ p \in U . \end{align*}


\begin{align*} d = 2 n + 1 = 4 l + 3 \equiv 3 \mod 4 \end{align*}


\begin{align*} g ^ { q p } u _ { , p q i } = - v _ { i } + u _ { , i q } \xi ^ q \end{align*}


\begin{align*} 0 = \int _ { B _ 1 ( 0 ) } \phi ^ { p - 1 } _ + \frac { \partial \phi } { \partial x _ h } \hat { u } _ i . \end{align*}


\begin{align*} \forall x \in E _ v \ , , \ \Vert x \Vert _ v = \begin{cases} \sqrt { h _ v ( x , x ) } v \in V _ { \infty } \\ \inf \left \lbrace \vert \alpha \vert _ v \ , , \ \alpha \in K _ v \ , , \ x \in \alpha L \right \rbrace v \in V _ f . \end{cases} \end{align*}


\begin{align*} & \lim _ { | z | \rightarrow 0 } \Psi ^ { \pm } ( x ; z ) = \begin{pmatrix} 1 & - u _ x ( x ) \\ - \bar { u } _ x ( x ) & 1 \end{pmatrix} , \\ [ 4 p t ] & \lim _ { | z | \rightarrow \infty } \Psi ^ { \pm } ( x ; z ) = e ^ { - i c _ \pm \sigma _ 3 } . \end{align*}


\begin{align*} u _ { n n } \sigma _ { k - 1 } ( b ) - \sum _ { \alpha = 1 } ^ { n - 1 } u _ { \alpha n } ^ 2 \sigma _ { k - 2 ; \alpha } ( b ) + \sigma _ k ( b ) = f . \end{align*}


\begin{align*} K _ q ( p , s ) = E _ q ^ * ( p \overline { s } ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { p ^ n \overline { s } ^ n } { \Gamma ( q n + 1 ) } , \end{align*}


\begin{align*} \theta _ 0 ( x ) = \frac { 1 } { 2 \pi i } \int _ { 2 - i \infty } ^ { 2 + i \infty } \frac { x ^ s } { s } \bigg \{ \sum _ p \frac { \log p } { p ^ s } \bigg \} d s . \end{align*}


\begin{align*} \partial _ i \Gamma ^ j _ { k j } + \Gamma ^ j _ { k j } \Gamma ^ j _ { i j } - \Gamma ^ j _ { k j } \Gamma ^ k _ { i k } - \Gamma ^ j _ { i j } \Gamma ^ i _ { k i } + g _ { i k } = 0 . \end{align*}


\begin{align*} \begin{aligned} \Delta u _ 1 ^ { t } \wedge \dots \wedge \Delta u _ k ^ { t } \wedge \beta _ n ^ { n - m } \rightarrow \Delta u _ 1 \wedge \dots \wedge \Delta u _ k \wedge \beta _ n ^ { n - m } . \end{aligned} \end{align*}


\begin{align*} \sum _ { i , j = 1 } ^ 3 \int _ { \mathbb R ^ 3 } \partial _ i [ ( a _ { i j } * \mu ) v _ j ] f ^ 2 d v & = \sum _ { i , j = 1 } ^ 3 \int _ { \mathbb R ^ 3 } \partial _ i \bigg ( \int _ { \mathbb R ^ 3 } a _ { i j } ( v - v ' ) v _ j ' \mu ( v ' ) d v ' \bigg ) f ^ 2 d v \\ & = \sum _ { i , j = 1 } ^ 3 \int _ { \mathbb R ^ 3 } \partial _ i [ a _ { i j } * ( v _ j \mu ) ] f ^ 2 d v . \end{align*}


\begin{align*} I _ { h , \gamma } & = \inf \left \{ E [ b ] \ \Bigg | \ b \in L ^ { 2 } _ { \sigma , \textrm { a x i } } ( \mathbb { R } ^ { 3 } ) , \ H [ b ] = h \right \} , \\ S _ { h , \gamma } & = \left \{ b \in L ^ { 2 } _ { \sigma , \textrm { a x i } } ( \mathbb { R } ^ { 3 } ) \ \middle | \ E [ b ] = I _ { h , \gamma } , \ H [ b ] = h \right \} . \end{align*}


\begin{align*} \partial _ x \left ( e ^ { i c _ - } \Psi ^ - _ { 1 1 } ( x ; z ) \right ) = \frac { 1 } { 2 i } u _ x ( x ) e ^ { i c _ - ( x ) } \Psi ^ - _ { 2 1 } ( x ; z ) . \end{align*}


\begin{align*} \sigma _ r ( C _ t ) = \overline { \cup _ { p _ 1 , \ldots , p _ r \in C _ t } \langle p _ 1 , \ldots , p _ r \rangle } . \end{align*}


\begin{align*} \beta _ 1 ( u ) = \beta _ 2 ( u ) = \beta _ 3 ( u ) = 0 \Longrightarrow \beta _ 3 ( u _ { \varepsilon } ) = 0 . \end{align*}


\begin{align*} T ^ { ( N ) } : M _ N ( E _ 1 ) \times \ldots \times M _ N ( E _ n ) \to M _ N ( E ) \end{align*}


\begin{align*} \left ( K _ H ( x ) \nabla q | \nabla \left ( \frac { u ^ 2 } { q } \right ) \right ) = \left ( K _ H ( x ) \nabla u | \nabla u \right ) - q ^ 2 \left ( K _ H ( x ) \nabla \left ( \frac { u } { q } \right ) | \nabla \left ( \frac { u } { q } \right ) \right ) . \end{align*}


\begin{align*} L ( s , \pi ) = \prod _ p L ( s , \pi _ p ) = \prod _ p \prod ^ M _ { j = 1 } ( 1 - \alpha _ { \pi } ( p , j ) p ^ { - s } ) ^ { - 1 } . \end{align*}


\begin{align*} \begin{aligned} | F ^ { i j } ( \nabla _ \beta u ) _ { i j } | \leq \ , & | \nabla _ \beta f ^ { 1 / k } | + C \left ( f ^ { 1 / k } + b _ \alpha \sum _ { i = 1 } ^ n F ^ { i i } \right ) \\ \leq \ , & C m ^ { 1 / k } m ^ { - 1 / 2 ( k - 1 ) } + C b _ \alpha \sum _ { i = 1 } ^ n F ^ { i i } \\ \leq \ , & C b _ \alpha ^ { - 1 / 2 } m ^ { 1 / k } + C b _ \alpha \sum _ { i = 1 } ^ n F ^ { i i } \end{aligned} \end{align*}


\begin{align*} c _ k = \sum _ { i = 1 } ^ { t - 1 } c _ i a _ i + c _ { k } \sum _ { i \in \Omega _ t ^ o } a _ i + c _ { k + 1 } \sum _ { i \in \Omega _ t ^ e } a _ i . \end{align*}


\begin{align*} \xi ( \mu , 0 ) = \xi ^ { \mathrm { i n } } ( \mu ) \ge 0 , \ \mu \in \mathbb { R } _ { > 0 } . \end{align*}


\begin{align*} \omega : = - p _ 1 ^ * ( \log \sqrt { \tau _ 0 } ) _ z d z + p _ 2 ^ * \omega _ 0 . \end{align*}


\begin{align*} 0 \rightarrow \mathcal { O } _ C \rightarrow \bigoplus _ { j = 1 } ^ n \mathcal { O } _ { C _ j } \rightarrow \mathcal { T } \rightarrow 0 . \end{align*}


\begin{align*} u _ { g , n } = \sum _ { i = 0 } ^ n ( - 2 ) ^ i w _ { g , n - i } , \end{align*}


\begin{align*} \int _ { \Omega } | \phi | ^ { 2 } \dd x + 2 \mu \int _ { 0 } ^ { t } \int _ { \Omega } | \nabla \phi | ^ { 2 } \dd x \dd s = \int _ { \Omega } | \phi _ 0 | ^ { 2 } \dd x , \end{align*}


\begin{align*} \sigma ^ L ( G ^ L ( X ) ) = G ( \{ \varphi ^ L _ { \sigma , s } ( E ^ L _ X ( v ) ) \mid ( s , v ) \in D \} ) . \end{align*}


\begin{align*} \begin{aligned} \Big | \langle \boldsymbol { \nu } , A _ t \rangle - \bar A _ t + \langle \boldsymbol { \nu } , f _ { \alpha , x _ \alpha } \rangle - \bar f _ { \alpha , x _ \alpha } ) \Big | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; . \end{aligned} \end{align*}


\begin{align*} \mu ^ { N , i } _ { \boldsymbol { \alpha } ^ * _ t } + \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } h ( \mu ^ { N , j } _ { \boldsymbol { \alpha } ^ * _ t } ) = \Delta _ { * , t } ^ { N , i } : = - R ^ { - 1 } B \lambda ^ { N , i } _ { \boldsymbol { y } _ t ^ { * } } , \end{align*}


\begin{align*} B _ a = \{ a , 2 a , 3 a , 4 a , 5 a , 6 a \} . \end{align*}


\begin{align*} X _ { M _ \mathbf i } ( x _ 1 , \dots , x _ n ) = x _ \mathbf i . \end{align*}


\begin{align*} \underline { a } : = \begin{pmatrix} a \\ & a \\ & & \ddots \\ & & & a \\ & & & & a \end{pmatrix} , \end{align*}


\begin{align*} u _ + = \max \{ u , 0 \} \end{align*}


\begin{align*} \begin{aligned} \dfrac { | b _ { 1 , \delta } ( x _ a ) | } { 2 } = & \ , \left | \int _ 0 ^ 1 h ( x _ a + \delta u ( t ) ) u ( t ) \ , d t \right | \\ = & \ , \left | \int _ 0 ^ 1 ( h ( x _ a + \delta u ( t ) ) - h ( x _ a ) ) u ( t ) \ , d t \right | \\ \le & \ , \int _ 0 ^ 1 \left | h ( x _ a + \delta u ( t ) ) - h ( x _ a ) \right | \ , d t \le L _ r \delta . \end{aligned} \end{align*}


\begin{align*} - \sum \limits _ { j = 3 } ^ { n - 1 } k _ j \geq - ( n - 3 ) r \end{align*}


\begin{align*} y = \exp { \left ( \frac { \lambda } { \sigma - \frac { 1 } { 2 } } \right ) } , x = y ^ { - 1 } \log ^ { 2 } { q } , \end{align*}


\begin{align*} ( \tilde { \kappa } _ 1 ) _ i = \tilde { h } _ { 1 1 i } = h _ { 1 1 i } , \end{align*}


\begin{align*} \Delta u _ 0 + k ^ 2 \widetilde q u _ 0 = 0 \mbox { i n } \mathbb R ^ n , \widetilde q ~ \big | _ { S _ h } = q . \end{align*}


\begin{align*} \mathcal { I } ( x , t ) = \bigcup _ { i = p + 1 } ^ { n } [ t _ { i , e n } ( x , t ) , t _ { i , e x } ( x , t ) ] . \end{align*}


\begin{align*} \vec { h } = \mathcal { H } ( \vec { r } , h _ 1 ) . \end{align*}


\begin{align*} \Omega _ 0 \cap \ ( - \mathbb { R } _ + ^ { m + 1 } ) = \emptyset . \end{align*}


\begin{align*} ( F f ) ( x ; z ) : = \int _ { - \infty } ^ x \left ( 1 , e ^ { 2 i z ( x - y ) } \right ) \widetilde { Q } f ( y ) \mathrm { d } y , \end{align*}


\begin{align*} \sum \limits _ { i = 1 } ^ { j + 1 } \chi _ i - r ( j + 1 ) < \chi = \sum \limits _ { i = 1 } ^ n \chi _ i - ( n - 1 ) r \end{align*}


\begin{align*} F ( z ) ( 0 ) = \lambda _ { \Theta } [ ( \Theta \circ h ) ( z ) , \alpha _ z ] ( 0 ) = ( \Theta \circ h ) ( z ) \end{align*}


\begin{align*} & x ^ { t + 1 } ( \gamma ) = ( I + \gamma B ) ^ { - 1 } \left ( x ^ t + \alpha ( \gamma ) ( x ^ t - x ^ { t - 1 } ) - \gamma \left ( F ( x ^ t ) + \beta ( \gamma ) ( F ( x ^ t ) - F ( x ^ { t - 1 } ) ) \right ) \right ) \forall \gamma > 0 , \\ & L _ { t + 1 } ( \gamma ) = \| F ( x ^ { t + 1 } ( \gamma ) ) - F ( x ^ { t } ) - \kappa _ t \gamma ^ { - 1 } ( x ^ { t + 1 } ( \gamma ) - x ^ t ) \| / \| x ^ { t + 1 } ( \gamma ) - x ^ t \| \forall \gamma > 0 , \end{align*}


\begin{align*} x \ , h _ s ( i ) ^ { \tilde \varepsilon x / 6 4 } \le x \exp \bigg ( - \frac { 1 2 8 \ , \log n } { \tilde \varepsilon ^ 2 ( 1 + \tilde \varepsilon ) ^ { - i } m _ s } \frac { \tilde \varepsilon x } { 6 4 } \bigg ) \le x \exp ( - 2 \log n ) = n ^ { - 1 } , \end{align*}


\begin{align*} \norm { \Theta _ { \norm { \cdot } } ( \nu ) } _ { ( C ( [ 0 , T ] ; ( U , \norm { \cdot } _ { U } ) ) , \norm { \cdot } _ \infty ) ' } = \norm { \nu } _ { v a r } ( \nu \in M ( [ 0 , T ] ; U ' ) ) \end{align*}


\begin{align*} d + c x \in \sigma _ l ( A ) , \ , v = ( x , 1 ) ^ T \in V _ l ( \lambda ) , \forall x \in S _ Q ( c , d - a , - b ) . \end{align*}


\begin{align*} n = \frac { - 3 \left ( a ^ { 2 } + b ^ { 2 } - 1 \right ) \pm \sqrt { 3 \left ( \left ( a - b \right ) ^ { 4 } - \left ( a ^ { 4 } + b ^ { 4 } + \left ( a - 1 \right ) ^ { 4 } + \left ( b - 1 \right ) ^ { 4 } \right ) + 1 \right ) } } { 6 \left ( a + b - 1 \right ) } \end{align*}


\begin{align*} & f _ k ( \Tilde { x } ) \le \lambda f _ k ( x _ v ) + ( 1 - \lambda ) f _ k ( x _ w ) < \lambda v _ k + ( 1 - \lambda ) w _ k . \\ & a _ k \| \Tilde { x } \| ^ 2 + 2 b _ k ^ T \Tilde { x } + c _ k \le a _ k ( \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 ) + 2 b _ k ^ T ( \lambda x _ v + ( 1 - \lambda ) x _ w ) + c _ k . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { r _ n ^ 2 } { h _ n } = 0 . \end{align*}


\begin{align*} C _ { B _ n } ( \sigma ) = \{ \tau \in B _ n : \sigma \tau = \tau \sigma \} \end{align*}


\begin{align*} \norm { h _ 1 } _ { L ^ 2 } ^ 2 & = \int _ \S \left [ \prod _ { i = 1 } ^ n ( \eta _ i ( x + r ) - \eta _ i ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( x + r ) - \tilde \Psi ( x ) ) \right ] ^ 2 \ \d x \\ & \le \int _ \S \prod _ { i = 1 } ^ n ( \eta _ i ( x + r ) - \eta _ i ( x ) ) ^ 2 \ \d x \\ & \le \prod _ { i = 1 } ^ n \norm { \partial \eta _ i } _ { L ^ 2 } ^ 2 . \end{align*}


\begin{align*} \inf \Bigg \{ \int _ { \{ \Psi < - t \} } | \tilde { f } | ^ 2 _ h c ( - \Psi ) : \tilde { f } \in H ^ 0 ( \{ \Psi < - t \} , \mathcal { O } ( K _ M \otimes E ) ) \\ \& \ , ( \tilde { f } - f ) _ { z _ 0 } \in \mathcal { O } ( K _ M ) _ { z _ 0 } \otimes J _ { z _ 0 } , z _ 0 \in Z _ 0 \Bigg \} \end{align*}


\begin{align*} d u = \Delta _ p u \ , d t + S ( u ) \ , d t + \sum _ { n = 0 } ^ { + \infty } \xi _ { n } \cdot \nabla u \ , d B _ t ^ { n } , \end{align*}


\begin{align*} f _ { j , \theta } ( x ' , x _ 3 ) = \int _ { I _ { K ^ { - 1 } } } \int _ { \pi _ 3 ( \theta ) } \widehat { f } _ { j , \theta } ( \nu ' , \nu _ 3 ) e ^ { 2 \pi i x ' \cdot \xi ' } e ^ { 2 \pi i x _ 3 \xi _ 3 } d \nu ' d \nu _ 3 . \end{align*}


\begin{align*} { \bf E } \left ( { \bf A } , { \mathbb V } \right ) = \left \{ { \bf A } _ { \theta } : \theta \left ( x , y \right ) = \sum _ { i = 1 } ^ { s } \theta _ { i } \left ( x , y \right ) e _ { i } \ \ \ \ \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle \in { \bf T } _ { s } ( { \bf A } ) \right \} . \end{align*}


\begin{align*} e ( C ) = 2 - I ( C ) - s ( C ) ; \end{align*}


\begin{align*} \begin{aligned} \int _ G \partial _ j \zeta ( 0 , \omega ) \ ; \overline { f ( 0 , \omega ) } d \mu ( \omega ) & = \int _ { G ^ \wedge } 2 \pi i \ ; y _ j ( \xi ) \widehat { \zeta ( 0 , \xi ) } \ ; \overline { \widehat { f ( 0 , \xi ) } } d \nu ( \xi ) \\ [ 5 p t ] & = - \int _ { G ^ \wedge } \widehat { \zeta ( 0 , \xi ) } \ ; \overline { 2 \pi i \ ; y _ j ( \xi ) \widehat { f ( 0 , \xi ) } } d \nu ( \xi ) . \end{aligned} \end{align*}


\begin{align*} X ( t ) = X ( 0 ) + \int _ 0 ^ t F ( X ( s ) ) d s + \int _ 0 ^ t G ( X ( s ) ) d W ( s ) , \end{align*}


\begin{align*} \partial _ s \sigma _ k + \mathbf l ' _ k H ( \mathbf r _ k , \mathbf r _ k ) \big ( m \cdot \nabla _ u \big ) \big ( 2 ^ { - 1 } \sigma _ k ^ 2 \big ) = 0 , \forall \ , k = 1 , \ldots , n . \end{align*}


\begin{align*} \psi ( p ) = \psi _ { p _ 0 } [ \eta ] ( p ) \stackrel { d e f } { = } \inf _ { r \in R ( p ) } g ( p , r , | \eta | _ r ) , \ p > p _ 0 ; \end{align*}


\begin{align*} ( \tilde { \kappa } _ 1 ) _ { i i } = \tilde { h } _ { 1 1 i i } + 2 \sum _ { p \neq 1 } \frac { \tilde { h } _ { 1 p i } ^ 2 } { \tilde { \kappa } _ 1 - \tilde { \kappa } _ p } = h _ { 1 1 i i } + 2 \sum _ { p \neq 1 } \frac { h _ { 1 p i } ^ 2 } { \kappa _ 1 - \tilde { \kappa } _ p } . \end{align*}


\begin{align*} \Psi ^ - _ 1 = e _ 1 + F \Psi ^ - _ 1 . \end{align*}


\begin{align*} M _ { I } = \bigoplus _ { J \subset I } R _ { J } , \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I I = 0 . \end{align*}


\begin{align*} \gamma _ { t - 1 } \alpha _ t ^ 2 = ( 1 - \alpha _ t ) \alpha _ { t - 1 } ^ 2 \gamma _ t + \mu \alpha _ t \gamma _ t \gamma _ { t - 1 } \leq ( 1 - \alpha _ t ) \alpha _ { t - 1 } ^ 2 \gamma _ t + \alpha _ t \gamma _ t \alpha _ { t - 1 } ^ 2 = \gamma _ t \alpha _ { t - 1 } ^ 2 , \end{align*}


\begin{align*} a _ { n , k } = \frac 1 n \sum _ { j \in [ n ] ^ k } \rho _ H ( j _ 1 - j _ 2 ) \rho _ H ( j _ 2 - j _ 3 ) \cdots \rho _ H ( j _ { k - 1 } - j _ k ) \rho _ H ( j _ k - j _ 1 ) . \end{align*}


\begin{align*} a ( u _ n , v _ n ) = ( f _ n , v _ n ) v _ n \in X _ n , \end{align*}


\begin{align*} D '' u _ { \epsilon , j } = D '' \left ( ( 1 - v ' _ { \epsilon } ( \Psi ) ) f F ^ { 1 + \delta } \right ) . \end{align*}


\begin{align*} \sigma ( P Q ) ( x , \xi ) = \sum \limits _ { \beta } \frac { ( - i ) ^ { | \beta | } } { | \beta | ! } \partial ^ { \xi } _ { \beta } \sigma ( P ) ( x , \xi ) \partial _ { \beta } \sigma ( Q ) ( x , \xi ) , \end{align*}


\begin{align*} ( I - Q ^ \dagger ) F ^ { q , \dagger } ( v + \psi ^ \dagger ( v , p ) , p ) = 0 \end{align*}


\begin{align*} \beta ' = \beta \pi _ { 2 3 } , \delta ' = \delta \pi _ { 2 3 } , \gamma ' = \gamma \pi _ { 1 2 } , \gamma '' = \pi _ { 2 3 } \gamma \pi _ { 2 3 } . \end{align*}


\begin{align*} \varphi ( x , t , k ) = \phi ( x , t ; k ) e ^ { i ( k ^ 2 x + \eta ^ 2 t ) \sigma _ 3 } , \end{align*}


\begin{align*} \beta _ { p , d } = \frac { \sqrt { \pi } \Gamma \left ( \frac { d - p } { 2 } \right ) } { \Gamma \left ( \frac { 1 - p } { 2 } \right ) \Gamma \left ( \frac { d } { 2 } \right ) } . \end{align*}


\begin{align*} \mu | _ { E _ { 2 , m + e _ i - e _ j } } = \lambda ^ j = \lambda _ x | _ { E _ { 2 , m + e _ i - e _ j } } \end{align*}


\begin{align*} \nabla _ V \big ( \nabla _ Y \hat X - \omega ( Y ) \big ) & = \big ( \nabla _ Y \omega - R ( Y , \hat X ) \big ) V - T _ h ( V , Y ) \\ \nabla _ V \left ( \nabla _ Y \omega - R ( Y , \hat X ) \right ) & = - R \big ( \nabla _ Y \hat X - \omega ( Y ) , V \big ) \\ & - R ( \omega ( Y ) , V ) + \nabla _ Y ( T _ h ( V ) ) + R ( V , Y ) \omega \\ & + ( \nabla _ { \hat X } R ) ( V , Y ) - R ( Y , \omega ( V ) ) . \end{align*}


\begin{align*} \min _ { \xi , \eta } p ( \xi , \eta ) = \min \left \{ \min _ { \xi } p \left ( \xi , \frac 1 2 \right ) , \min _ { \xi } p \left ( \xi , \frac { 9 9 } { 1 0 0 } \right ) \right \} . \end{align*}


\begin{align*} \frac { x _ i } { g } ( x _ 1 - x _ k ) = \begin{cases} x _ i & ( i = k ) \\ 0 & ( i \neq k ) . \end{cases} \end{align*}


\begin{align*} ( \Delta K _ { m , \epsilon } ) ^ p \wedge \beta _ n ^ { n - p } = ( p A \wedge B ^ { p - 1 } + B ^ p ) \wedge \beta _ n ^ { n - p } , p = 1 , \ldots , m , \end{align*}


\begin{align*} h ( a , s ) : = \hat \Phi ( ( a , 0 ) ^ \top , s ) = \hat \Phi ^ \dagger ( a , s ) : = \langle z _ 1 ^ * , \Phi ^ \dagger ( a u _ \ell , p ( s ) ) \rangle . \end{align*}


\begin{align*} a _ { \ell } = R _ 1 ( \ell ) \beta _ 1 ^ { \ell } + \cdots + R _ s ( \ell ) \beta _ s ^ { \ell } \end{align*}


\begin{align*} h _ { s , t } ^ { m , n } = \frac { ( m t - n s ) ^ 2 - ( s - t ) ^ 2 } { 4 s t } . \end{align*}


\begin{align*} j ' ( t ) = & - \frac { 1 } { t ^ 2 \varepsilon ^ 2 } \int _ { \Omega } \left ( p ( t u - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p - 1 } t u ^ 2 - \left ( t u - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p } u ) d x \\ = & \frac { 1 } { t ^ 2 \varepsilon ^ 2 } \int _ { \Omega } \left ( t u - q \ln \frac { 1 } { \varepsilon } \right ) _ + ^ { p - 1 } u [ ( 1 - p ) ( t u - q \ln \frac { 1 } { \varepsilon } ) - p q \ln \frac { 1 } { \varepsilon } ] d x \\ \leq & 0 . \end{align*}


\begin{align*} \Q [ \mathsf { f } ] ( y ) = \left ( \begin{array} { c c c c c } { \bf R } [ f _ 1 ( y ) ] & 0 & \dots & 0 & 0 \\ 0 & { \bf R } [ f _ 2 ( y ) ] & \dots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & { \bf R } [ f _ { d - 1 } ( y ) ] & 0 \\ 0 & 0 & \dots & 0 & { \bf R } [ f _ d ( y ) ] \end{array} \right ) , \end{align*}


\begin{align*} \sum _ { \lambda , \mu } \exp \Big ( \frac { - \pi ( u ^ 2 + v ^ 2 ) } { v } S [ \lambda ] - \frac { \pi } { v } S [ \mu ] - \frac { 2 \pi } { v } u \lambda ^ t S \mu \Big ) = \sum _ { \lambda , \mu } \exp \Big ( - \pi v S [ \lambda ] - \frac { \pi } { v } S [ u \lambda + \mu ] \Big ) . \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) = ( f _ n , v _ n ) v _ n \in X _ n \end{align*}


\begin{align*} \begin{aligned} & z '' _ { b e n } + ( M _ { b e n } - E ) A _ { b e n } ^ { - 1 } z _ { b e n } = 0 \\ & z '' _ { n b } + ( M _ { n b } - E ) A _ { n b } ^ { - 1 } z _ { n b } = 0 , \end{aligned} \end{align*}


\begin{align*} \eta \{ F \} \stackrel { d e f } { = } { \bf E } \eta / F . \end{align*}


\begin{align*} L = \frac { 1 } { \delta } \det A . \end{align*}


\begin{align*} \Pi _ { \tilde { V } } e ^ { J _ { z / | z | } ^ \perp t } x = e ^ { \lambda t } ( \Pi _ { \tilde { V } } x + t \Pi _ { V } x ) . \end{align*}


\begin{align*} \mu ^ { \rm N } & = \mu ^ * + \frac { \varkappa } { 2 } , \varkappa ^ { \rm N } = \frac { \varkappa } { 2 } , \qquad \ , \lambda ^ { \rm N } = \lambda , \\ \gamma ^ { \rm N } & = \frac { \beta + \gamma } { 2 } , \qquad \ \ \beta ^ { \rm N } = \frac { \gamma - \beta } { 2 } , \ \ \ \ \ \ \ \alpha ^ { \rm N } = \alpha \ , . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 5 } } \nabla _ y \varphi \cdot \nabla _ y \tilde { \varphi } \dd y = \mu ^ { 2 } \int _ { \mathbb { R } ^ { 5 } } \left ( \varphi - 1 - \frac { \gamma } { r ^ { 2 } } \right ) _ { + } \tilde { \varphi } \dd y , \end{align*}


\begin{align*} a _ 0 ( x ) = \frac { \Gamma ( n - 1 ) \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) } { ( 2 \pi ) ^ { n - 1 } } \biggl [ \biggl ( \frac { \lambda + 3 \mu } { 2 \mu ( \lambda + \mu ) } \biggr ) ^ { n - 1 } + \frac { 1 } { ( 2 \mu ) ^ { n - 1 } } + \frac { n - 2 } { \mu ^ { n - 1 } } + \frac { 1 } { \alpha ^ { n - 1 } } \biggr ] , \end{align*}


\begin{align*} { \bf T } _ { s } ( { \bf A } ) = \left \{ W = \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle \in G _ { s } \left ( { \rm H ^ { 2 } } \left ( { \bf A } , \mathbb C \right ) \right ) : \bigcap \limits _ { i = 1 } ^ { s } \operatorname { A n n } ( \theta _ { i } ) \cap \operatorname { A n n } ( { \bf A } ) = 0 \right \} , \end{align*}


\begin{align*} d ( \phi \wedge \beta ) = d \phi \wedge \beta - \phi \wedge d \beta = d \phi \wedge \beta = \alpha \wedge \beta . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \int _ { \mathbb { T } ^ d } \langle a ( u _ n ( x ) ) \nabla u ( x ) , \nabla u _ n ( x ) - \nabla u ( x ) \rangle d x = 0 . \end{align*}


\begin{align*} \sigma \cdot X : = \{ S ^ k \sigma ( x ) \mid x \in X , 0 \leq k < | \sigma ( x _ 0 ) | \} . \end{align*}


\begin{align*} c _ \varepsilon \leq \inf \limits _ { w \in \mathcal { H } ( \Omega ) , w _ + \neq 0 } \max \limits _ { t \geq 0 } I _ \varepsilon ( t w ) = \inf \limits _ { w \in \mathcal { N } _ \varepsilon } I _ \varepsilon ( w ) . \end{align*}


\begin{align*} & M i n i m i z e _ { x \in \mathbb { R } ^ n } \ , T r ( A _ J , X ) + 2 b _ J ^ T x \\ & s . t . \ \ T r ( A _ k , X ) + 2 b _ k ^ T x + c _ k \le 0 \ , k = 1 , . . . , m \\ & \begin{pmatrix} 1 & x ^ T \\ x & X \end{pmatrix} \succeq 0 \end{align*}


\begin{align*} \mathcal { N } _ { \epsilon } ( x , y ) : = \int _ { \mathbb { R } } \rho _ { \epsilon } ( y - z ) \mathcal { N } ( x , z ) \ , d z , \end{align*}


\begin{align*} \mathcal { W } ( K ) = \sum _ { m = 0 } ^ n V _ m ( K ) . \end{align*}


\begin{align*} \begin{cases} \Delta _ { \Psi } w = A w & \O \\ w = 0 & \partial \O . \end{cases} \end{align*}


\begin{align*} X ^ \Delta ( t _ { n + 1 } ) = X ^ \Delta ( t _ { n } ) + F ( X ^ \Delta ( t _ { n } ) ) \Delta + \Delta ^ { 1 / 2 } G ( X ^ \Delta ( t _ { n } ) ) \xi _ { n + 1 } . \end{align*}


\begin{align*} \tau ( [ L ] ) : = \log _ q [ \mathcal { O } _ F ^ d : \pi ^ i L ] , \end{align*}


\begin{align*} \frac { \lVert y - x \rVert _ 2 } { \lVert x \rVert _ 2 } = O \big ( \delta \ , \kappa ( A ) \big ) \end{align*}


\begin{align*} \delta ^ 2 \int _ { \Omega } \left ( K ( x ) \nabla u | \nabla \varphi \right ) = \int _ { \Omega } \hat { h } \varphi , \ \ \ \ \forall \varphi \in H ^ 1 _ 0 ( \Omega ) . \end{align*}


\begin{align*} | | z | | : = \sum _ { k = 1 } ^ { \infty } k | z _ k | , \end{align*}


\begin{align*} 0 = D f | _ { \Gamma _ 0 } ( \Delta ) = \phi ( C \Delta + \Delta ^ T C ^ T + \Delta ^ T D \Gamma _ 0 + \Gamma _ 0 ^ T D \Delta ) . \end{align*}


\begin{align*} B _ q ( \psi _ m ) ( z ) = \sum _ { j = 0 } ^ { \infty } \frac { z ^ j } { \sqrt { \Gamma ( q j + 1 ) } } ( \delta _ { j , m } ) = \frac { z ^ m } { \sqrt { \Gamma ( q j + 1 ) } } : = e _ { m , q } ( z ) . \end{align*}


\begin{align*} \begin{aligned} \| f ( z ( \tau ) , \tau ) - f ( x _ a ( \tau ) , \tau ) \| \le & \ , L _ \mathcal { D } \| z ( \tau ) - x _ a ( \tau ) \| \\ \le & \ , \gamma 2 L _ \mathcal { D } M _ \mathcal { D } T \\ \| f ( x ( \tau ) , \tau ) - f ( z ( \tau ) , \tau ) \| \le & \ , L _ \mathcal { D } \| x ( \tau ) - z ( \tau ) \| . \end{aligned} \end{align*}


\begin{align*} | C _ G ( x ) | = 1 + \deg ( x ) = { \frac { | G | } { | x ^ G | } } , \textrm { f o r a l l } x \in G . \end{align*}