\begin{align*} V ( A , - C ) = V ( A , C ) + \sqrt { | A | \ , | C | } . \end{align*}


\begin{align*} H ^ { t } f ( x ) = \sum _ { n = 0 } ^ { \infty } e ^ { - N t } P _ { n } f ( x ) \end{align*}


\begin{align*} \rho _ x & = \rho _ { \omega \ast r _ { a _ 1 } \ast \cdots \ast r _ { a _ n } } = \rho _ { r _ { a _ n } } \circ \cdots \circ \rho _ { r _ { a _ 1 } } \circ \rho _ \omega . \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*} ( X _ 4 \cup \{ c _ 4 \} ) \cap { \cal N } = ( X _ 1 \cup \{ c _ 1 \} ) \cap { \cal N } = \{ c _ 2 , c _ 3 \} \cap { \cal N } = \emptyset . \end{align*}


\begin{align*} \int e ^ { W ( y | x ) } d \nu _ { A ^ { * } } ( y ) = \varphi _ A ( x ) \end{align*}


\begin{align*} \begin{aligned} \mathbf { \Phi } _ { \mathrm { r } } = \mathsf { b l k d i a g } ( \mathbf { \Phi } _ { \mathrm { r } , 1 } , \ldots , \mathbf { \Phi } _ { \mathrm { r } , G } ) , \\ \mathbf { \Phi } _ { \mathrm { t } } = \mathsf { b l k d i a g } ( \mathbf { \Phi } _ { \mathrm { t } , 1 } , \ldots , \mathbf { \Phi } _ { \mathrm { t } , G } ) , \end{aligned} \end{align*}


\begin{align*} \sigma _ { k } ( \lambda ) = \sum _ { i _ { 1 } < \ldots < i _ { k } } \lambda _ { i _ { 1 } } \ldots \lambda _ { i _ { k } } , \ \ k = 1 , \ldots , n , \end{align*}


\begin{align*} V ^ { - 1 } ( \rho \circ \varphi ) V = V ^ { - 1 } \circ \rho \circ V ( V ^ { - 1 } \circ \varphi \circ V ) = 0 _ s \oplus \begin{bmatrix} ( V ^ { - 1 } \varphi V ) ^ { s + 1 } ( z ) \\ \vdots \\ ( V ^ { - 1 } \varphi V ) ^ n ( z ) \\ \end{bmatrix} . \end{align*}


\begin{align*} \mathcal { D } + \mathcal { E } = 0 . \end{align*}


\begin{align*} 1 - n + m - t = 0 , \end{align*}


\begin{align*} & \omega ( \alpha ( x ) , \alpha ( y ) ) = \omega ( x , y ) , \displaystyle \circlearrowleft _ { x , y , z } \omega ( [ x , y ] , \alpha ( z ) ) = 0 . \end{align*}


\begin{align*} \lim _ { s \to \infty } \phi _ s ( z ) = \lim _ { s \to \infty } \frac { s } { 2 \Gamma ( 1 / s ) } \exp ( - | z | ^ s ) = \begin{cases} \frac { 1 } { 2 } , & | z | < 1 , \\ 0 , & \end{cases} \end{align*}


\begin{align*} \phi _ 3 = - c \phi _ 1 - \frac { \epsilon } { \kappa } \phi _ 4 - \frac { c } { \kappa } \phi _ 2 + \frac { c } { \kappa } \rightarrow 0 , \ \phi _ 1 \rightarrow \phi _ 1 ^ * , \ \phi _ 2 \rightarrow 0 \ \ \phi _ 4 \rightarrow 0 \ \ z \rightarrow - \infty \end{align*}


\begin{align*} \widetilde \mu _ { ( x , 0 ) } = \widehat \mu _ { x } \times \delta _ 0 \end{align*}


\begin{align*} ( i _ X S _ { e , \mathcal { Y M } ^ 0 } ) = \langle S _ { e , \mathcal { Y M } ^ 0 } , \nabla X ^ { \flat } \rangle + ( S _ { e , \mathcal { Y M } ^ 0 } ) ( X ) \ , , \forall \ , X \in \Gamma ( M ) \ , . \end{align*}


\begin{align*} \sigma _ { n } ^ M : = & T \wedge \inf \Big \{ t \geq 0 : \Vert X ( t , x _ n ) \Vert _ H > M \Big \} \\ & \wedge \inf \Big \{ t \geq 0 : \int _ 0 ^ t \Vert X ( s , x _ n ) \Vert _ { V } ^ { \alpha } d s > M \Big \} \\ & \wedge \inf \Big \{ t \geq 0 : \Vert X ( t , x ) \Vert _ H > M \Big \} \\ & \wedge \inf \Big \{ t \geq 0 : \int _ 0 ^ t \Vert X ( s , x ) \Vert _ { V } ^ { \alpha } d s > M \Big \} . \end{align*}


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


\begin{align*} \mathcal S ( t ) h ( x ) : = \begin{cases} h ( x + t ) , & x + t \leq 1 , \\ 0 , & x + t > 1 . \end{cases} \end{align*}


\begin{align*} \alpha _ t ^ { \xi } = - R ^ { - 1 } B y _ t ^ { \xi } - h ( \mu _ t ) . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \mathbb { P } ( \tau _ u ^ M < T ) = 0 . \end{align*}


\begin{align*} & \sum _ { A = 0 } ^ { 2 n - 1 } \left [ d _ 1 \left ( u - v \right ) \wedge ( \triangle v ) ^ { p - 1 } \wedge ( \triangle u ) ^ { m - p } \wedge \beta _ n ^ { n - m } \right ] _ A \cdot \nabla _ { A 0 ' } \varrho ( q ) \ , \Omega _ { 2 n } \\ = & d _ 0 \varrho ( q ) \wedge d _ 1 \left ( u - v \right ) \wedge ( \triangle v ) ^ { p - 1 } \wedge ( \triangle u ) ^ { m - p } \wedge \beta _ n ^ { n - m } , \end{align*}


\begin{align*} \mathcal { L } w = - \mathcal { L } H = \partial ^ \square H - \big ( \Delta _ \Gamma V + V \big | \nabla _ \Gamma \nu \big | ^ 2 \big ) = 0 \ , [ 0 , T ] \times M \end{align*}


\begin{align*} A \cdot B = \sum _ { k = 1 } ^ 3 \frac { 1 } { | V _ k | ^ 2 } ( A \cdot V _ k ) ( B \cdot V _ k ) + \sum _ { k = 1 } ^ 6 \frac { 1 } { | W _ k | ^ 2 } ( A \cdot W _ k ) ( B \cdot W _ k ) . \end{align*}


\begin{align*} ( \bigoplus _ { i \in I _ j } V _ i \to W _ j , \bigoplus _ { j \in J } W _ j \to U ) \mapsto ( \bigoplus _ { i \in I } V _ i = \bigoplus _ { j \in J } \bigoplus _ { i \in I _ j } V _ i \to \bigoplus _ { j \in J } W _ j \to U ) \end{align*}


\begin{align*} \mu ( d x ) = e ^ { - \lambda } \delta _ 0 ( d x ) + ( 1 - e ^ { - \lambda } ) f ( x ) d x , \end{align*}


\begin{align*} \mathbf { h } = \lambda _ { 1 } \mathbf { e } _ 1 + \lambda _ { 2 } \mathbf { e } _ 2 + \lambda _ { 3 } \mathbf { e } _ 3 + \lambda _ { 1 2 3 ' } \mathbf { e } _ { 1 2 3 ' } \end{align*}


\begin{align*} \left | \bigcup _ { { F \in C _ 2 } } F \cap M \right | & \le \left ( c _ 2 - q ^ { d - 1 } \right ) ( \Delta + z q ^ { d ^ 2 - d } \theta _ { d - 1 } ) + q ^ { d - 1 } ( \Delta + z ' q ^ { d ^ 2 - d } \theta _ { d - 1 } ) \\ & = c _ 2 ( \Delta + z q ^ { d ^ 2 - d } \theta _ { d - 1 } ) + \underbrace { ( z ' - z ) } _ { \le \theta _ d } q ^ { d ^ 2 - 1 } \theta _ { d - 1 } . \end{align*}


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


\begin{align*} \beta = \alpha + \alpha _ j j \geq s . \end{align*}


\begin{align*} \delta \leq \min \{ \delta _ * , \delta _ \circ \} , \delta < \epsilon _ \circ / ( \lambda _ \varphi ( 1 + \lambda _ L ) ) , \lambda _ L = \max \{ \| l \| \ , | \ , l \in L \} . \end{align*}


\begin{align*} g ' _ 2 = - \frac { 1 } { \nu } I \widehat { g _ 1 ' } I + \frac { 2 i } { \mu \nu } h _ z J g _ 1 ' \end{align*}


\begin{align*} h \cdot ( g \cdot \gamma ) & = \theta _ n \circ ( h \times g \cdot \gamma ) \circ i , \\ ( h g ) \cdot \gamma & = \theta _ n \circ ( h g \times \gamma ) \circ i . \end{align*}


\begin{align*} F ( m ; z ) = \sum _ { n = 0 } ^ \infty b _ 5 \left ( \frac { m n - 1 } { 6 } \right ) q ^ n \in S _ { 2 m - 2 } ( \Gamma _ 0 ( 1 8 0 ) , \chi _ 5 ) _ m . \end{align*}


\begin{align*} \frac { f ( x + y ) } { f ( x ) } & = \frac { \sup _ { t \ge x + y } f ( t ) } { \inf _ { s \in [ x _ 0 , x ] } f ( s ) } \frac { f ( x + y ) } { \sup _ { t \ge x + y } f ( t ) } \frac { \inf _ { s \in [ x _ 0 , x ] } f ( s ) } { f ( x ) } \\ & \ge \frac { F ( x + y + \Delta ) } { F ( x - c + \Delta ) } \frac { f ( x + y ) } { \sup _ { t \ge x + y } f ( t ) } \frac { \inf _ { s \in [ x _ 0 , x ] } f ( s ) } { f ( x ) } \end{align*}


\begin{align*} \omega = \sum _ { i = 1 } ^ n d x ^ i \wedge d \theta _ i . \end{align*}


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


\begin{gather*} \lim _ k \sup _ { P \in M } P \bigl \{ \abs { f } \ , 1 ( \abs { f } > k ) \bigr \} = 0 \quad \quad f \in F . \end{gather*}


\begin{align*} X ( q , r ) = \frac 1 q \iota ( q r ) \end{align*}


\begin{align*} \begin{aligned} \theta ^ { [ 1 ] } _ 1 = & \pi ^ { [ 1 ] } _ 1 ( \xi _ 0 , x ) \\ : = & \begin{cases} 1 , \\ 0 , \end{cases} \end{aligned} \end{align*}


\begin{align*} \lambda _ { 1 } = R ( \mathbf R , \mathbf { v } _ { \lambda _ 1 } ) = \frac { { \mathbf { v } _ { \lambda _ 1 } } ^ T \mathbf R \mathbf { v } _ { \lambda _ 1 } } { { \mathbf { v } _ { \lambda _ 1 } } ^ T \mathbf { v } _ { \lambda _ 1 } } \end{align*}


\begin{align*} d ( v _ 1 \Rightarrow w _ 2 v _ 2 ) = & d ( v _ 1 \Rightarrow w _ 2 v _ 2 ' ) + d ( w _ 2 v _ 2 ' \Rightarrow w _ 2 v _ 2 ) . \end{align*}


\begin{align*} \varepsilon ( h , y ) \varepsilon ( x , y + z ) [ [ \alpha ( y ) , h \cdot z ] , \alpha ^ 2 ( x ) ] = & 2 \varepsilon ( x , y + z ) [ \alpha ( h ) \cdot [ y , z ] , \alpha ^ 2 ( x ) ] \\ & - \varepsilon ( x , y + z ) [ [ h \cdot y , \alpha ( z ) ] , \alpha ^ 2 ( x ) ] = 0 . \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 + } \sup _ { n \in \mathbb { N } } \mathbb { E } \int _ 0 ^ { T - \delta } \Vert Y _ n ^ M ( t + \delta ) - Y _ n ^ M ( t ) \Vert _ { H } ^ { \alpha } d t = 0 , \end{align*}


\begin{align*} \lim _ { n \to \infty } \sqrt [ n ] { | B _ { F , \{ x , y \} } ( n ) | } = \lim _ { n \to \infty } \sqrt [ n ] { | B _ { G , S } ( G ) | } = 3 . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 2 } } | b | ^ { 2 } \dd x ' = \int _ { \mathbb { R } ^ { 2 } } \left ( | \nabla \phi | ^ { 2 } + | G | ^ { 2 } \right ) \dd x ' . \end{align*}


\begin{align*} M ^ \sharp ( T ) = \left ( \frac { 1 } { D + 1 } + \lambda \right ) D \theta M ^ \otimes ( T ) . \end{align*}


\begin{align*} \left ( \frac { N - 2 } { 2 } - \frac { N + \alpha } { 2 p } \right ) \int | \nabla u | ^ 2 d x = \lambda \left ( \frac { N } { 2 } - \frac { N + \alpha } { 2 p } \right ) \int | u | ^ 2 d x . \end{align*}


\begin{align*} \psi ^ { \pm } ( x , t ; k ) = I + k \int _ { \pm \infty } ^ { x } e ^ { - 2 i k ^ 2 ( x - y ) \widehat { \sigma } _ 3 } P _ y ( y ) \psi ^ { \pm } ( y , t ; k ) d y . \end{align*}


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


\begin{align*} H ^ n ( G , M ) = H ^ n \mathcal C ^ \bullet , . \end{align*}


\begin{align*} \prod _ { \ell = \ell _ 1 } ^ { \ell _ 2 } | \cos ( \pi ( \theta + \ell \alpha ) ) | \leq C ( \varepsilon ) e ^ { ( \ell _ 2 - \ell _ 1 ) ( - \ln { 2 } + \varepsilon ) } \inf _ { j = \ell _ 1 } ^ { \ell _ 2 } | \cos ( \pi ( \theta + j \alpha ) ) | , \end{align*}


\begin{align*} R i c ^ g + \nabla \tau = 0 , \end{align*}


\begin{align*} \nabla _ X ^ \perp \eta _ i = \sum _ { j } \phi _ { i j } ( X ) \varphi _ j \eta _ j . \end{align*}


\begin{align*} \xi _ { k , i } ( x ) = \left \{ \begin{array} { l l } 2 \sqrt { C _ { d } \kappa } \ , \theta _ { k } { \rm R e } ( \sigma _ { k , i } ( x ) ) , & k \in \Z _ { + } ^ d , \\ 2 \sqrt { C _ { d } \kappa } \ , \theta _ { k } { \rm I m } ( \sigma _ { k , i } ( x ) ) , & k \in \Z _ { - } ^ d , \end{array} \right . \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*} X _ { i ; j k } = - R _ { \ell k j i } X ^ \ell + \tfrac { 1 } { 2 } ( h _ { i j ; k } + h _ { i k ; j } - h _ { j k ; i } ) . \end{align*}


\begin{align*} \begin{aligned} \| f '' \| _ 1 & \leq \limsup _ { n \rightarrow \infty } | f ( n + 1 ) - f ( n ) | + \limsup _ { n \rightarrow \infty } | f ( - n - 1 ) - f ( - n ) | \\ & + 2 \sum _ { n \in \partial _ l S _ - f } f ( n ) - f ( n - 1 ) + 2 \sum _ { n \in \partial _ r S _ - f } f ( n ) - f ( n + 1 ) . \end{aligned} \end{align*}


\begin{align*} N _ { k , l } = \frac 1 { \sqrt 2 } \sum _ { c , b = 1 } ^ d \int _ 0 ^ t \hat { \sigma } _ { k l , b c } ( s ) + \hat { \sigma } _ { l k , b c } ( s ) d B ^ { c b } _ s . \end{align*}


\begin{align*} Q ( r , t ) : = \int _ { B _ { r } ( 0 ) } n ( x , t ) \ , d x = \sigma _ { d } \int _ { 0 } ^ { r } \rho ^ { d - 1 } n ( \rho , t ) \ , d \rho \end{align*}


\begin{align*} \phi _ j \cap \psi _ j = ( \alpha _ j \setminus \{ x \} ) \cap \alpha _ j = \alpha _ j \setminus \{ x \} = \psi _ j \setminus \{ x \} . \end{align*}


\begin{align*} \begin{aligned} \varphi _ t & = { \mathcal { L } } \varphi + \mathbb { P } \nabla \cdot F \textrm { o n } \ L ^ { 2 } _ { \sigma } ( \mathbb { R } ^ { 3 } ) . \\ \end{aligned} \end{align*}


\begin{align*} s _ \lambda = \det ( h _ { \lambda _ j - j + i } ) _ { i , j = 1 } ^ \ell = \sum _ { \pi \in S _ \ell } \operatorname { s g n } ( \pi ) h _ { \pi ( \lambda ) } \end{align*}


\begin{align*} \sigma _ { S , W } \colon [ 0 , l ] _ \mathbb { Z } \to \mathbb { Z } , k \mapsto - \sum _ { j = 1 } ^ k \rho \big ( s _ { \iota ( j ) } \big ) . \end{align*}


\begin{gather*} \Gamma _ 0 = \bigl \{ P \in \Gamma : ( \pi _ 1 , \ldots , \pi _ n ) P \bigr \} . \end{gather*}


\begin{align*} u _ t = - \tfrac { 1 } { h _ 3 } ( f _ 3 p _ t + g _ 3 q _ t ) , \end{align*}


\begin{align*} ( \mu ) _ \infty = \prod _ { n = 1 } ^ \infty ( 1 - z ^ { n - 1 } \mu ) . \end{align*}


\begin{align*} \sigma = \sum \limits _ { d = 1 } ^ \infty \frac { \mu ( d ) \lambda ( d ^ 2 ) } { d ^ 2 } \ , . \end{align*}


\begin{align*} \begin{aligned} 0 & \to \left ( T _ { \mu _ g ( \kappa ) ^ { - 1 } ( X , D , \alpha , m ) } \right ) _ { ( X , D , \alpha , m ) } \to \left ( T _ { \mathcal B _ g ( \kappa ) } \right ) _ { ( X , D , \alpha , p , m ) } \to \left ( T _ { \mathcal A _ g ( \kappa ) } \right ) _ { ( X , D , \alpha , m ) } \\ & \xrightarrow { \left ( d \mathfrak c _ g ( \kappa ) \right ) _ { ( X , D , \alpha , m ) } } H ^ 1 \left ( X , K _ X ^ { \otimes 2 } ( - D ) \right ) \to 0 \end{aligned} \end{align*}


\begin{align*} \omega _ { 2 2 } = 1 \omega _ { 1 2 } = 0 . \end{align*}


\begin{align*} g _ n ( t , \omega ) : = & \langle A ( t , X _ n ( t , \omega ) ) , X _ n ( t , \omega ) - X ( t , \omega ) \rangle , \\ F _ n ( t , \omega ) : = & f ( t ) ( 2 + \Vert X _ n ( t , \omega ) \Vert _ H ^ 2 ) + C \Vert X ( t , \omega ) \Vert _ V ^ { \alpha } \\ & + C \Vert X _ n ( t , \omega ) \Vert _ H ^ { \beta ( \alpha - 1 ) } \Vert X ( t , \omega ) \Vert _ V ^ { \alpha } . \end{align*}


\begin{gather*} H = \{ \pi _ 1 \} \cup \bigl \{ \bigl ( \pi _ { i + 1 } - \pi _ i \bigr ) \ , g ( \pi _ 1 , \ldots , \pi _ i ) : 1 \le i < n , \ , \ , g \in B ( \mathbb { R } ^ i , \mathcal { B } ( \mathbb { R } ^ i ) ) \bigr \} ; \end{gather*}


\begin{align*} \operatorname { g r a d } f = \nabla ^ i f \frac { \partial } { \partial x _ i } = g ^ { i j } \frac { \partial f } { \partial x _ i } \frac { \partial } { \partial x _ j } , f \in C ^ { \infty } ( \Omega ) , \end{align*}


\begin{align*} \begin{aligned} \| x ( 2 \bar { t } / \gamma ) \| _ { \mathcal { A } } & \le \beta ( \| x ( \bar { t } / \gamma ) \| _ { \mathcal { A } } , \bar { t } ) + \tilde { d } \\ & \le \beta ( \underline { \rho } , \bar { t } ) + \tilde { d } \\ & \le \beta ( \rho , \bar { t } ) + \tilde { d } \\ & \le \underline { \rho } \end{aligned} \end{align*}


\begin{align*} f ' ( x , y ) & = \alpha ( f \circ L ( x , y ) ) , \\ & = \alpha \left ( a ( t x + u y ) ^ { q + 1 } + b ( t x + u y ) ^ q ( v x + w y ) + c ( t x + u y ) ( v x + w y ) ^ q + d ( v x + w y ) ^ { q + 1 } \right ) , \end{align*}


\begin{align*} x \cdot ( x \wedge y ) = x \cdot y \in I . \end{align*}


\begin{align*} \mathrm { d i v } ( \widetilde { \mathcal { R } } q ) : = \pi _ { \widehat { Q } _ h ^ \perp \cap \widetilde { P } _ { k - 1 } ^ { \mathrm { d i s c } } ( \mathcal { T } ) } q = \begin{cases} \pi _ { \widehat { Q } _ h ^ \perp } q & k \geq d , \\ \pi _ { \widetilde { P } _ { k - 1 } ^ { \mathrm { d i s c } } ( \mathcal { T } ) } q & k < d , \end{cases} \quad q \in Q , \end{align*}


\begin{align*} \delta _ \square ( W _ 1 , W _ 2 ) : = \inf _ { \varphi } d _ \square ( W _ 1 , W _ 2 ^ \varphi ) , \end{align*}


\begin{align*} \mathfrak { X } = \frac { \prod _ { j = 1 } ^ d \Gamma ( j / 2 ) } { \pi ^ { d / 2 } \prod _ { j = 1 } ^ { d - 1 } j ! } . \end{align*}


\begin{align*} U \cdot \nabla f = 0 , \end{align*}


\begin{align*} \Im ( \varrho _ z ^ 2 + \varrho _ w ^ 2 ) = \Im ( \varrho _ { Z Z } ( L , L ) ) / a = 0 . \end{align*}


\begin{align*} \int _ { \mathbb { R } } \frac { r ' ( r r ' ) ^ { 2 \tilde { \tau } + 1 } } { | r - r ' | ^ { 4 \tilde { \tau } - 2 / q } r ' } \dd r ' = C r ^ { 2 / q } . \end{align*}


\begin{align*} \lim _ { x \to \infty } f _ { + a } ( x ) / f _ s ( x ) = 1 - e ^ { - \lambda _ s } . \end{align*}


\begin{align*} & N ( x ; k ) = I + \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } \cup i \mathbb { R } } \frac { \varrho ( x ; s ) J } { s - k } d s = I + \mathcal { C } ( \varrho J ) ( z ) , \end{align*}


\begin{align*} - n ^ p - \sum _ { K = 1 } ^ { p } \binom { p } { K } \ , B _ K \ , n ^ { p - K } . \end{align*}


\begin{align*} t _ { k + 1 } = \max \bigl \{ t \colon d ( \gamma _ { a _ 1 , b _ 1 } ( t ) , u _ { k } ) = s \bigr \} \end{align*}


\begin{align*} B ( L X ) + D ( L X ) + ( n - 1 ) [ \beta ( L X ) + \gamma ( L X ) ] = 0 . \end{align*}


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


\begin{align*} \underline { h } _ { \mu } ( f _ { 1 , \infty } ) : = \int \underline { h } _ { \mu } ( f _ { 1 , \infty } , x ) \ d \mu ( x ) . \end{align*}


\begin{align*} \Theta ^ q ( x ) = 2 \pi q x + \alpha . \end{align*}


\begin{align*} \ \begin{cases} \partial _ t v _ n + \partial _ x v _ n \cdot B + \frac { 1 } { 2 } \mathrm { T r } \big [ \partial _ { x x } v _ n \Sigma \Sigma ^ \top \big ] + f _ n ( t , \partial _ { x } v _ n \Sigma , x ) [ 0 , 1 ) \times \R ^ \ell = 0 , \\ v _ n ( 1 , x ) = F ( x ) x \in \R ^ \ell \end{cases} \end{align*}


\begin{align*} \partial _ t \Phi _ { j } + u _ j \cdot \nabla \Phi _ { j } = \mu _ j \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi _ j . \end{align*}


\begin{align*} | x | _ { A } : = | A x | \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { d } [ a _ i ( t , x , u , z ) - a _ i ( t , x , u , \tilde { z } ) ] ( z _ i - \tilde { z } _ i ) > 0 . \end{align*}


\begin{align*} \prescript { } { l } { v } _ j ( \lambda ) : = \prescript { } { l } { \tilde { h } } _ j ( \lambda ) q _ { l , n _ j } ( \lambda ) = \prescript { } { l } { \beta } ( \lambda ) \prescript { } { l } { R } _ j ( \lambda ) q _ { l , n _ j } ( \lambda ) . \end{align*}


\begin{align*} H ( \rho ^ \varepsilon ) ( 0 , z _ l ) = \frac { - 1 } { R + \varepsilon \rho _ 0 } \phantom { x x x } \phantom { x x x } H ( \rho ^ \varepsilon ) ( 0 , z _ r ) = \frac { 1 } { R - \varepsilon \rho _ 0 } \end{align*}


\begin{align*} & \sup _ { x \in \left ( x _ { 0 } , \infty \right ) } \left \| \langle x \rangle \mathcal { P } ^ { + } \left ( z ^ { - i } \bar { r } _ 1 ( z ) e ^ { - 2 i z x } \right ) \right \| _ { L _ z ^ 2 } \leq \left \| z ^ { - i } \bar r _ 1 ( z ) \right \| _ { H ^ 1 _ z } , i = 0 , 1 , \\ & \sup _ { x \in \left ( x _ 0 , \infty \right ) } \left \| \langle x \rangle \mathcal { P } ^ - \left ( r _ 2 ( z ) e ^ { 2 i z x } \right ) \right \| _ { L _ { z } ^ 2 } \leq \left \| r _ 2 ( z ) \right \| _ { H ^ 1 _ z } . \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*} L ( E , \epsilon ) = \max \{ \log | \lambda | + | \epsilon | , L ( E , 0 ) \} . \end{align*}


\begin{align*} ( \prod _ { I } \widehat { \bigoplus } _ { I } A ^ + ) [ \frac { 1 } { p } ] = ( \prod _ { I } \widehat { \bigoplus } _ { I } A ^ { H ' , + } ) [ \frac { 1 } { p } ] \oplus ( \prod _ { I } \widehat { \bigoplus } _ { I } X ^ + _ { H ' , n } ) [ \frac { 1 } { p } ] . \end{align*}


\begin{align*} \begin{aligned} \int _ { \mathbb { R } ^ { 2 } _ { + } } \nabla \phi \cdot \nabla \tilde { \phi } \frac { 2 \pi ^ { 2 } } { r } \dd z \dd r = \int _ { \mathbb { R } ^ { 5 } } \nabla _ y \varphi \cdot \nabla _ y \tilde { \varphi } \dd y , \varphi = \frac { \phi } { r ^ { 2 } } , \tilde { \varphi } = \frac { \tilde { \phi } } { r ^ { 2 } } . \end{aligned} \end{align*}


\begin{align*} i _ { k + 1 } ( B _ { I , [ m ] } ) & = { \rm a r g m a x } _ { i \in I \backslash I _ k ( B _ { I , [ m ] } ) } | \langle v _ { k + 1 } ( B _ { I , [ m ] } ) , B _ { i , [ m ] } ^ \top \rangle | \\ & = { \rm a r g m a x } _ { i \in [ n ] \backslash I _ k ( B ) } | \langle v _ { k + 1 } ( B ) , B _ { i , [ m ] } ^ \top \rangle | = i _ { k + 1 } ( B ) . \end{align*}


\begin{align*} \left . \begin{aligned} & \boldsymbol { y } _ t ^ { * } : = ( y _ t ^ { * , i } ) _ { 1 \leq i \leq N } : = ( P _ t x _ t ^ { * , i } + \varphi _ t ^ { * , i } ) _ { 1 \leq i \leq N } , \\ & \boldsymbol { z } _ t ^ { * } : = ( z _ t ^ { * , i , j } ) _ { 1 \leq i , j \leq N } : = ( P _ t \sigma + \Lambda _ t ^ { * , i , j } ) _ { 1 \leq i , j \leq N } , \boldsymbol { z } _ t ^ { 0 , * } : = ( z _ t ^ { 0 , * , i } ) _ { 1 \leq i \leq N } : = ( P _ t \sigma _ 0 + \Lambda _ t ^ { 0 , * , i } ) _ { 1 \leq i \leq N } . \end{aligned} \right . \end{align*}


\begin{align*} \psi _ 1 ^ - ( x , t ; k ) = a ( k ) \psi _ 1 ^ + ( x , t ; k ) + b ( k ) e ^ { 2 i k ^ 2 x + 2 i \eta ^ 2 t } \psi _ 2 ^ + ( x , t ; k ) , k \in \mathbb { R } \cup i \mathbb { R } , \end{align*}


\begin{align*} \max _ { v \in \Lambda } e ( \alpha ^ { ( v ) } , G ^ { ( v ) } ) = e ( 0 , G ) . \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*} \phi = \frac { 1 } { 2 } \left ( u _ { , p } \xi ^ p + v _ q x ^ q \right ) . \end{align*}


\begin{align*} \| f | \sigma _ 1 \| _ { B V ( \sigma _ 1 ) } = \lim _ { n \to \infty } \| p _ n | \sigma _ 1 \| _ { B V ( \sigma _ 1 ) } \leq \lim _ { n \to \infty } \| p _ n \| _ { B V ( \sigma ) } = \| f \| _ { B V ( \sigma ) } . \end{align*}


\begin{align*} T - \mathcal { P } ^ + \left ( T R _ + \right ) - \mathcal { P } ^ - \left ( T R _ - \right ) = F , \end{align*}


\begin{align*} V _ n ( \varphi ) = \begin{bsmallmatrix} x \\ \Phi _ n \end{bsmallmatrix} ^ { \top } \mathbf { P } _ n \begin{bsmallmatrix} x \\ \Phi _ n \end{bsmallmatrix} \leq 0 , \ ; \forall n \geq \mathcal { N } ( \mathcal { E } ( \eta _ 0 ) ) , \end{align*}


\begin{align*} ( S _ { \omega } ) ^ { \delta } : = \{ u \in H _ { r } ^ { 1 } ( \R ^ { 2 } ) | u = \widetilde { u } + \overline { u } , \widetilde { u } \in S _ { \omega } , \| \overline { u } \| _ { H _ { r } ^ { 1 } ( \R ^ { 2 } ) } \leq \delta \} , \end{align*}


\begin{align*} f ( x ) = \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \ , \phi ( x ) \ , \Phi ( x ) ^ { \alpha - 1 } \ , [ 1 - \Phi ( x ) ] ^ { \beta - 1 } , \end{align*}


\begin{align*} x ^ { ( n + 1 ) } = T \left ( x ^ { ( n ) } \right ) ( \forall n \in \mathbb { N } ) \end{align*}


\begin{align*} ( H _ { \lambda , \alpha , \theta } u ) _ n = u _ { n + 1 } + u _ { n - 1 } + \lambda \tan ( \pi ( \theta + n \alpha ) u _ n ) , \end{align*}


\begin{align*} e ( z ) = 1 + \left ( z + 1 \right ) \sum _ { n = 0 } ^ { \infty } \left ( - 1 \right ) ^ { n } z ^ { n } a _ { n } ( 0 ) , \end{align*}


\begin{align*} \kappa ( s ) : = \sup _ { \stackrel { k \in \N } { k \neq \ell } } c _ 1 ( q , k , p ( s ) ) \end{align*}


\begin{align*} \tau ^ * = \bar { \tau } \Leftrightarrow \dfrac { | \lambda _ i | } { | \lambda _ { i + 1 } | } = \dfrac { | \lambda _ { i + 1 } | } { | \lambda _ { i + 2 } | } \quad \forall i \in [ 1 , p - 2 ] \quad \forall i \in [ p + 1 , n - 2 ] ) , \end{align*}


\begin{align*} \mu _ w ( \bigoplus _ { j = l } ^ { n - 1 } K _ j ' ) = \frac { ( \bigoplus _ { j = l } ^ { n - 1 } K _ j ' ) } { ( \sum \limits _ { j = l } ^ { n - 1 } w _ j ) r } \geq \frac { \chi ( E ) } { r } \end{align*}


\begin{align*} \rho _ { \Sigma } ( m ) = \mathbb E [ U ^ { \otimes m } ] . \end{align*}


\begin{align*} [ a ] \ast [ b ] & = \{ [ d ] \in G / H : \mbox { t h e r e e x i s t } a ' , b ' , d ' \in G \\ & \mbox { w i t h } [ d ' ] = [ d ] , \ , [ a ' ] = [ a ] , \ , [ b ' ] = [ b ] \mbox { a n d } d ' \in a ' \ast b ' \} . \end{align*}


\begin{align*} \langle \alpha ^ \vee , 2 \rho \rangle = 2 + \sum _ { \substack { \alpha \neq \beta \in \Phi ^ + \\ s _ \alpha ( \beta ) \in \Phi ^ - } } \langle \alpha ^ \vee , \beta \rangle \geq 2 + \# \{ \beta \in \Phi ^ + \setminus \{ \alpha \} \mid s _ \alpha ( \beta ) \in \Phi ^ - \} = 1 + \ell ( s _ \alpha ) . \end{align*}


\begin{align*} \langle x | A _ j x \rangle & = \langle U x | U A _ j U ^ * ( U x ) \rangle \\ & = \left \langle \left ( \sqrt { s _ 1 } y _ 1 , \dots , \sqrt { s _ m } y _ m , 0 , \dots , 0 \right ) | \left ( \sqrt { s _ 1 } A _ { j , 1 } y _ 1 , \dots , \sqrt { s _ m } A _ { j , m } y _ m , 0 , \dots , 0 \right ) \right \rangle \\ & = \sum \limits _ { k = 1 } ^ m s _ k \langle y _ k | A _ { j , k } y _ k \rangle . \end{align*}


\begin{align*} \begin{aligned} \nabla ^ 2 _ U \eta ( U , x , t ) - ( \nabla _ V \tilde { \eta } ) & ( A ( U , x , t ) , x , t ) \cdot \nabla ^ 2 _ U A ( U , x , t ) = \\ & = ( \nabla _ V ^ 2 \tilde { \eta } ) ( A ( U , x , t ) , x , t ) : ( \nabla _ U A ( U , x , t ) , \nabla _ U A ( U , x , t ) ) \ ; . \end{aligned} \end{align*}


\begin{align*} \kappa ( t ) = h _ { \alpha , \rho } ( t ) , \ 0 < \alpha < 1 , \ \rho \ge 0 , \end{align*}


\begin{align*} \sum _ { k = 1 } ^ \infty \frac { \mu ( k ) } { k } = 0 \ \ \mathrm { a n d } \ \ \sum _ { k = 1 } ^ \infty \frac { \mu ( k ) \log k } { k } = - 1 \end{align*}


\begin{align*} P _ { m ; l } \otimes P _ { m ' ; l ' } \cong \bigoplus \limits _ { \nu = \vert l - l ' \vert \mathrm { s t e p } 2 } ^ { \mathrm { m i n \left ( l + l ' , 2 d - 4 - l - l ' \right ) } } P _ { m + m ' + \frac { 1 } { 2 } \left ( l + l ' - \nu \right ) ; \nu } \end{align*}


\begin{gather*} \begin{aligned} & 2 \langle A ( t , u ) - A ( t , v ) , u - v \rangle + \Vert B ( t , u ) - B ( t , v ) \Vert _ { L _ 2 } ^ 2 \\ \leq & [ f ( t ) + \rho ( u ) + \eta ( v ) ] \Vert u - v \Vert _ { H } ^ 2 , \\ \end{aligned} \\ | \rho ( u ) | + | \eta ( u ) | \leq C ( 1 + \Vert u \Vert _ V ^ { \alpha } ) ( 1 + \Vert u \Vert _ H ^ { \gamma } ) , \end{gather*}


\begin{align*} Z _ { I } ^ { \star } ( 1 , \{ 2 \} ^ { s - 1 } ; ( \alpha , \beta , \gamma ) ) = Z ( s - 1 | s - 1 | 1 ; ( \alpha , \beta , \gamma ) ) + Z ( s - 1 | s - 1 | 1 ; ( \alpha , \beta , \alpha + \beta - \gamma ) ) \end{align*}


\begin{align*} \frac { 1 } { \ln \frac { R } { s _ { \delta , j } } } = \frac { 1 } { \ln \frac { R } { \varepsilon } } + O \left ( \frac { \ln | \ln \varepsilon | } { | \ln \varepsilon | ^ 2 } \right ) . \end{align*}


\begin{align*} y ^ { \left ( \prod _ { i = 1 } ^ d r _ i \right ) n } = \sum _ { a _ 1 , \ldots , a _ d \geq 1 } S _ { d , ( a _ 1 r _ 1 , \ldots , a _ d r _ d n ) } ( x ) . \end{align*}


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


\begin{align*} \begin{aligned} \frac { 1 } { 2 } \sum _ { \beta = 1 } ^ { \alpha } b _ \beta \bar { x } _ \beta ^ 2 \leq C b _ \alpha \bar { x } _ n + b _ \alpha O ( | \bar { x } ' | ^ 3 ) \leq C b _ \alpha \bar { x } _ n + C \delta ^ 3 b _ \alpha ^ 2 . \end{aligned} \end{align*}


\begin{align*} \epsilon _ 1 ( \eta _ a , t ) = \int _ { N } ^ { t } \varepsilon _ \delta ( { x } _ a , \bar { y } _ a , \tau ) u ( \tau ) - b _ { 1 , \delta } ( x _ a ) / 2 \ , d \tau \end{align*}


\begin{align*} z ^ k & \mapsto f _ k ( s ) = - \frac { 1 } { s } \left ( ( k + 1 ) ^ { 1 - s } - k ^ { 1 - s } \right ) & & ( k \in \mathbb { N ^ * } ) \\ 1 & \mapsto - \frac { 1 } { s } \end{align*}


\begin{align*} h ( x ) = \begin{cases} - P ^ { - 1 } ( x ) & ( x \geq 0 ) \\ x & ( x < 0 ) , \end{cases} \end{align*}


\begin{align*} \alpha _ { k } ^ { \pm } = \frac { \left ( d _ { k } \right ) ^ { \pm 1 } } { i a ^ { \prime } ( \rho _ { k } ) } \end{align*}


\begin{align*} f : = \varinjlim _ i f _ i \colon \varinjlim _ i \mathcal A _ i \to \varinjlim _ i \mathcal B _ i \end{align*}


\begin{align*} \mathbf { \Xi } _ g ^ v = - \triangledown _ { \mathcal { M } _ g } \tilde { f } _ g ( \mathbf { \Phi } _ { g } ^ v ) + \mu _ g ^ v \mathsf { P r } _ { \mathbf { \Phi } _ { g } ^ v } ( \mathbf { \Xi } _ g ^ { v - 1 } ) , \forall g \in \mathcal { G } , \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} \widetilde { m } ( x ; z ) & = z \int _ { - \infty } ^ { x } e ^ { 2 i z ( x - y ) } w ( y ) d y + \frac { 1 } { 2 i } w ( x ) \\ & - \frac { 1 } { 2 i } \int _ { - \infty } ^ x e ^ { 2 i z ( x - y ) } \left [ \left ( 2 i \bar { u } _ { y y } ( y ) + \bar { u } _ y ( y ) | u _ y ( y ) | ^ { 2 } \right ) \widehat { \Psi } ^ - _ { 1 1 } ( y ) + | u _ y ( y ) | ^ 2 \widehat { \Psi } ^ - _ { 2 1 } ( y ) \right ] d y , \end{aligned} \end{align*}


\begin{align*} \Theta ^ M ( t , x ) = \phi _ i ( t ) x \in \left [ \frac { i - 1 } M , \frac i M \right ) . \end{align*}


\begin{align*} e ^ { \prime } ( \rho , x ) = e ^ { i \rho x } \left ( \frac { z - 1 } { 2 ( z + 1 ) } - \frac { 1 } { 2 } \int _ { x } ^ { \infty } q ( t ) d t + ( z + 1 ) \sum _ { n = 0 } ^ { \infty } \left ( - 1 \right ) ^ { n } z ^ { n } d _ { n } ( x ) \right ) \end{align*}


\begin{align*} G ^ { D _ { N } } ( [ x N ] , [ y N ] ) = \tilde { G } ^ { D } ( x , y ) + o ( 1 ) \end{align*}


\begin{align*} \Delta x _ k + \bar u ^ { - 1 } \nabla \bar u \cdot \nabla x _ k = - \Gamma ^ k + \bar u ^ { - 1 } \bar g ^ { i k } \frac { \partial \bar u } { \partial x _ i } . \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*} \phi _ q ( n ) : = ( \pi _ H ( q ^ { - 1 } n ) ) ^ { - 1 } \phi ( \pi _ N ( q ^ { - 1 } n ) ) , \mbox { f o r a l l } n \in E _ q , \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*} \sup _ { z \in \mathbb { D } } \bigg | \frac { 1 - \overline { b ( a ) } b ( z ) } { 1 - \overline { a } z } \bigg | = o \left ( \frac { 1 } { 1 - | a | ^ 2 } \right ) \ , | b ( a ) | \to 1 , \end{align*}


\begin{align*} \left \{ \begin{aligned} \sigma _ k \big ( \lambda ( D ^ { 2 } u ) \big ) & = f \ ; \ ; \mbox { i n } ~ \Omega , \\ u & = \varphi \ ; \ ; \mbox { o n } ~ \partial \Omega , \end{aligned} \right . \end{align*}


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


\begin{align*} \frac { g ( t ) - g ( r ) } { t - r } = \lambda \frac { g ( t ) - g ( s ) } { t - s } + ( 1 - \lambda ) \frac { g ( s ) - g ( r ) } { s - r } . \end{align*}


\begin{align*} \bigl ( | u _ t | ^ { \ell - 2 } u _ t \bigr ) _ t - a \nabla \cdot \bigl ( | \nabla u | ^ { q - 2 } \nabla u \bigr ) + b | u _ t | ^ { m - 2 } u _ t = 0 , \end{align*}


\begin{align*} { _ { n + 1 } } F _ n \left ( \begin{array} { c c c c } A _ 0 , & A _ 1 , & \ldots , & A _ n \\ & B _ 1 , & \ldots , & B _ n \end{array} \mid x \right ) : = \frac { q } { q - 1 } \sum _ { \chi \in \widehat { \mathbb { F } _ q ^ \times } } { A _ 0 \chi \choose \chi } { A _ 1 \chi \choose B _ 1 \chi } \cdots { A _ n \chi \choose B _ n \chi } \chi ( x ) . \end{align*}


\begin{align*} \lim _ { \ell \to + \infty } \lambda ^ \ell ( p ) = \lambda ^ \infty ( p ) \end{align*}


\begin{align*} \frac { 1 } { 1 - y } = \sum _ { n = 0 } ^ \infty y ^ n . \end{align*}


\begin{align*} \chi _ i = \left \{ \begin{aligned} & \psi _ i & & ( i \neq j ) \\ & \sigma & & ( i = j ) . \end{aligned} \right . \end{align*}


\begin{align*} - \Delta z _ { n } + \frac { F ( y _ { n } ) - F ( y ) } { t _ { n } } = h _ { n } . \end{align*}


\begin{align*} v ( x ) = \left | \sum _ { i = 1 } ^ k x _ i B _ i \right | \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) \| _ { \mathcal { A } } & \le \| x _ { n } ( t ) \| _ { \mathcal { A } } + \| x ( t ) - x _ { n } ( t ) \| \\ & \le \beta ( \| x ( n \bar { t } / \gamma ) \| _ { \mathcal { A } } , \gamma t - n \bar { t } ) + \tilde { d } \\ & \le \beta ( \| x ( n \bar { t } / \gamma ) \| _ { \mathcal { A } } , 0 ) + \tilde { d } & & \forall \ , t \in I _ n , \ , \forall \ , n \in \mathbb { N } \end{aligned} \end{align*}


\begin{align*} \hat W _ r ( k ) = \begin{cases} \frac { 2 \sin ( 2 \pi k r ) } { \pi k } & k \neq 0 \\ 4 r & \quad k = 0 \end{cases} . \end{align*}


\begin{align*} \ell ( x _ \ast ) = \ell ( x _ 1 ' ) + \ell ( x _ 2 ) = \ell ( x _ 1 ) + \ell ( x _ 2 ) - d ( v _ 1 \Rightarrow w _ 2 v _ 2 ) . \end{align*}


\begin{align*} \Psi ( t , x , y ) = ( x e ^ { - t } , y e ^ { - t } ) . \end{align*}


\begin{align*} p ( x ) \Longleftrightarrow t _ { c , \rho } ( y ) = p \Big ( \frac { x - c } { \rho } \Big ) , \end{align*}


\begin{align*} \left ( - ( 1 - \kappa r ^ 2 ) \frac { d ^ 2 } { d r ^ 2 } - ( d - 1 - d \kappa r ^ 2 ) \frac { 1 } { r } \frac { d } { d r } + \frac { l ( l + d - 2 ) } { r ^ 2 } - \frac { Q } { r } \sqrt { 1 - \kappa r ^ 2 } - { \cal E } \right ) R ( r ) = 0 , \end{align*}


\begin{align*} \nu ( b _ x ) = \max _ J \pi _ J ( \lambda ( b _ x ) ) , \end{align*}


\begin{align*} \langle \theta , \vartheta \rangle _ { \mathcal { L } ^ 2 , { G } } = \langle \theta , G \vartheta \rangle _ { \mathcal { L } ^ 2 } = \beta r \langle x , u \rangle _ { \mathcal { L } ^ 2 } + \beta \langle y , v \rangle _ { \mathcal { L } ^ 2 } + \frac { 1 } { \beta } \langle \lambda , w \rangle _ { \mathcal { L } ^ 2 } , \end{align*}


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


\begin{align*} \begin{aligned} \int _ B u _ { p + N } \Delta u _ p \wedge \alpha & = B _ { p , N } + \int _ B u _ p \Delta u _ { p + N } \wedge \alpha \geq B _ { p , N } + \int _ B u _ { p + N } \Delta u _ { p + N } \wedge \alpha , \end{aligned} \end{align*}


\begin{align*} & \lim _ { m \to \infty } \lim _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \sup _ { t \in [ 0 , 1 ] } | \mathcal { R } _ { 4 , \epsilon } ( t ) | \geq \sup _ { u \in [ 0 , 1 ] } | \varphi ( u ) - \varphi _ m ( u ) | ^ { 1 / 2 } \Big ) \\ & \leq - \lim _ { m \to \infty } \frac { c \eta } { L ^ { 2 } \sup _ { t \in [ 0 , 1 ] } | \varphi ( t ) - \varphi _ m ( t ) | } = - \infty . \end{align*}


\begin{align*} e _ { a _ j } \left [ t \right ] = a _ j \left [ t \right ] - a _ j ^ { \left ( o \right ) } , \end{align*}


\begin{align*} 0 = \lim _ { n \to \infty } \ \sum _ { i = 1 } ^ { n } \mathbb E \left [ \| ( \mathcal S ( \Delta _ n ) - I ) Y _ { ( i - 1 ) \Delta _ n } ) \| ^ 4 \right ] \geq & \lim _ { n \to \infty } \ \sum _ { i = 1 } ^ { n } \mathbb E \left [ \| ( \mathcal S ( \Delta _ n ) - I ) Y _ { ( i - 1 ) \Delta _ n } ) \| ^ 2 \right ] ^ 2 \\ = & \lim _ { n \to \infty } \Delta _ n ^ { 2 + 4 \mathfrak H } \sum _ { i = 1 } ^ n ( i - 1 ) ^ 2 > 0 , \end{align*}


\begin{align*} \lambda | ^ * _ { [ w _ 1 , w _ 2 ] } ( a ) = \lambda ( w _ 1 a ) \end{align*}


\begin{align*} \varphi _ { | N } = \frac { 1 } { n } ( \psi _ 1 + \cdots + \psi _ n ) \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow 0 } I I _ 3 = 0 . \end{align*}


\begin{align*} e ^ { - t \Lambda _ g } = \frac { i } { 2 \pi } \int _ { \mathcal { C } } e ^ { - t \tau } ( \Lambda _ g - \tau I ) ^ { - 1 } \ , d \tau , \end{align*}


\begin{align*} a ( Z ) = a \big ( ( Z ^ * Z ) ^ \frac { 1 } { 2 } \big ) \end{align*}


\begin{align*} u '' + q ( t ) u = 0 , t \in ( a , b ) \end{align*}


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


\begin{align*} \lim _ { x \to \infty } \frac { h ( x ) } { 1 / x } = & \lim _ { x \to \infty } \frac { 2 \theta } { \operatorname { B } ( a , b ) } \frac { 1 } { x ^ 3 } \frac { \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } \exp \left ( - \frac { a \theta } { x ^ 2 } \right ) } { \operatorname { I } _ { 1 - \exp ( - \theta / x ^ 2 ) } ( b , a ) } x . \end{align*}


\begin{align*} \chi _ 1 = \psi _ 1 , \ldots , \chi _ { j - 1 } = \psi _ { j - 1 } , \chi _ j = \sigma < _ K \psi _ j . \end{align*}


\begin{align*} \tau = \frac { \varrho ^ 2 _ z } { | w | ^ 2 } < 0 . \end{align*}


\begin{align*} \Sigma ^ { \pm } ( s - w , \phi , F _ { \pm } ) = & \lim _ { z \downarrow 0 } \sum _ { \varepsilon = \pm } \int _ 0 ^ \infty w _ z ( \lambda ) \lambda ^ { 4 ( 1 - s + w ) - 1 } \int _ 0 ^ \infty \\ & \times \sum _ { \ell , m = 1 } ^ \infty \rho _ \phi ( 3 \ell m ) K _ \nu ( 6 \pi \ell m t ^ 2 ) \hat { F } _ 4 \left ( 0 , 0 , \frac { 3 m \lambda } { t } , \frac { \varepsilon 3 \ell m t ^ 3 } { \lambda } \right ) \frac { d \lambda } { \lambda } t d t . \end{align*}


\begin{align*} \mathbf { H } = | \mathbf { H } | \odot e ^ { j \angle \mathbf { H } } , \end{align*}


\begin{align*} \| x _ { k + 1 } - x ^ \star \| ^ 2 \leq \tfrac { 1 } { 2 } \| x _ k - x ^ \star \| ^ 2 , \textnormal { f o r a l l } k = 0 , 1 , 2 , \ldots , T . \end{align*}


\begin{align*} B _ { x , \gamma x } ( \gamma ^ + ) = B _ { y , \gamma y } ( \gamma ^ + ) + B _ { x , y } ( \gamma ^ + ) - B _ { \gamma x , \gamma y } ( \gamma ^ + ) . \end{align*}


\begin{align*} \begin{aligned} \epsilon ( \eta _ a , t ) = & \sum _ { n = 1 } ^ N \int _ { n - 1 } ^ { n } F ( \eta _ a , \tau ) - F _ a ( \eta _ a ) \ , d \tau \\ & + \int _ { N } ^ { t } F ( \eta _ a , \tau ) - F _ a ( \eta _ a ) \ , d \tau \\ = & \int _ { N } ^ { t } F ( \eta _ a , \tau ) - F _ a ( \eta _ a ) \ , d \tau . \end{aligned} \end{align*}


\begin{align*} [ F ( \gamma , \beta ) : F ] = [ F ( \gamma , \beta ) : F ( \beta ) ] [ F ( \beta ) : F ] = [ F ( \gamma , \beta ) : F ( \beta ) ] ( q ^ n - 1 ) . \ \end{align*}


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


\begin{align*} { \rm a r g m a x } _ { j \in [ n ] \backslash I } | \langle v _ { k + 1 } ( B ) , ( B _ { j , [ m ] } ) ^ \top \rangle | & \le | \langle v _ { k + 1 } ( B _ { I , [ m ] } ) , ( B _ { i _ { k + 1 } ( B _ { I , [ m ] } ) , [ m ] } ) ^ \top \rangle | \\ & = { \rm a r g m a x } _ { i \in I \backslash I _ { k } ( B ) } | \langle e _ 1 , ( B _ { i , [ m ] } ) ^ \top \rangle | , \end{align*}


\begin{align*} h ( s \xi , s \eta , s \omega ) - h ( \xi , \eta , \omega ) = 2 \hat \beta ( s , \xi ) . \end{align*}


\begin{align*} T ^ { - 1 } ( B ) = \{ \theta \in \Theta : T ( \theta ) \in B \} = \textstyle \bigcup _ { \sigma \in S _ k } B [ \sigma ] . \end{align*}


\begin{align*} u = S _ T ^ { \infty } ( u _ 0 ) \in C ( [ - T , T ] ; H ^ { \infty , 0 } ( \R ^ 2 ) ) \end{align*}


\begin{align*} \xi _ { k , i } ( x , t ) = \left \{ \begin{array} { l l } 2 \sqrt { C _ { d } \kappa } \ , \theta _ { k } ( t ) { \rm R e } ( \sigma _ { k , i } ( x ) ) , & k \in \Z _ { + } ^ d , \\ 2 \sqrt { C _ { d } \kappa } \ , \theta _ { k } ( t ) { \rm I m } ( \sigma _ { k , i } ( x ) ) , & k \in \Z _ { - } ^ d , \end{array} \right . \end{align*}


\begin{align*} ( U ^ \top U ) _ { i , i } - \frac { m } { n } = \frac { \sqrt { m - m ^ 2 / n } } { \sqrt { 1 - 1 / n } } \bigg ( t _ i ^ 2 - \frac { 1 } { n } \bigg ) \mbox { f o r a l l } i \in \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} I & = - \Vert u - \widetilde { u } \Vert _ { V } ^ 2 , \\ I I & \leq \varepsilon \Vert u - \widetilde { u } \Vert _ { V } ^ 2 + C _ { \varepsilon } \Vert u \Vert _ { V } ^ 2 \Vert u - \widetilde { u } \Vert _ { H } ^ 2 . \end{align*}


\begin{align*} H \left ( \nu \mid \mu \right ) : = \int \frac { d \nu } { d \mu } \log \frac { d \nu } { d \mu } d \mu , \end{align*}


\begin{align*} { { \bf { F } } _ i } \left ( { { \bf { x } } , t } \right ) = f _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { u } } ^ { e q } } + \Delta { \bf { u } } } \right ) - f _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { u } } ^ { e q } } } \right ) , \end{align*}


\begin{align*} C _ { \lambda } & = \chi \int _ { W ^ d } e ^ { - \sum _ { j = 1 } ^ d ( \lambda _ j - \bar { \lambda } ) z _ j } \Delta ( z ) d z _ 1 \ldots d z _ d . \end{align*}


\begin{align*} \norm { \tfrac { 1 } { p _ n } - \tfrac { 1 } { f } } _ \infty = \norm { \tfrac { p _ n - f } { p _ n f } } _ \infty \leq \tfrac { 2 } { \delta ^ 2 } \| p _ n - f \| _ \infty \to 0 . \end{align*}


\begin{align*} \mathcal { M } ^ { r } ( t ) \geq \mathcal { M } ^ { r } ( \delta ) + ( r - 1 ) \theta _ { 1 } \varrho _ { 0 } ^ { 1 - \sigma } \int _ { \delta } ^ { t } ( \mathcal { M } ^ { r } ( s ) ) ^ { \sigma + 1 } d s , 0 < \delta < t < t _ { 0 } < T _ { \mathrm { g e l } } . \end{align*}


\begin{align*} \theta _ 1 & = \frac { 1 } { 2 } ( \arg \alpha _ 1 - \arg \alpha _ 2 + \arg \beta _ 1 - \arg \beta _ 2 ) , & \delta _ 1 & = u ( \theta _ 1 ) \\ \theta _ 2 & = \frac { 1 } { 2 } ( \arg \alpha _ 1 - \arg \alpha _ 2 - \arg \beta _ 1 + \arg \beta _ 2 ) , & \delta _ 2 & = u ( \theta _ 2 ) \end{align*}


\begin{align*} \mathcal { H } ( t ) = \lambda ( t ) \mathcal { E } ( t ) + \mu \alpha ( t ) \mathcal { E } ^ r ( t ) \langle P ( u _ t ) , u \rangle \end{align*}


\begin{align*} \| \tilde { x } _ { T _ { \textnormal { i n n e r } } } - x ^ \star \| ^ 2 \leq \tfrac { 1 } { \mu ( \sum _ { k = 1 } ^ { T _ { \textnormal { i n n e r } } } \lambda _ k ) } \left ( \sum _ { k = 1 } ^ { T _ { \textnormal { i n n e r } } } \lambda _ k \langle F ( x _ k ) , x _ k - x ^ \star \rangle \right ) . \end{align*}


\begin{align*} B ( G ) _ x : = \{ [ b ] \in B ( G ) \mid I x I \cap [ b ] \neq \emptyset \} . \end{align*}


\begin{align*} \alpha \circ \mathcal { T } ( a ) & = \alpha ( T \varrho ( x ) ( a ) + [ T ( a ) , x ] ) = T ( \beta ( \varrho ( x ) ( a ) ) + \alpha ( [ T ( a ) , x ] ) ) \\ & = T ( \varrho ( \alpha ( x ) ) ( \beta ( a ) ) ) + [ \alpha ( T ( a ) ) , \alpha ( x ) ] ) ) = T ( \varrho ( x ) ( \beta ( a ) ) ) + [ T ( \beta ( a ) ) , x ] ) ) \\ & = \mathcal { T } ( \beta ( a ) ) . \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { h ( x ) } { 1 / x } = & 2 \theta ( b - 1 ) \lim _ { x \to \infty } \left [ \frac { 1 / x ^ 2 } { 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) } \right ] - 2 . \end{align*}


\begin{align*} f _ \mathcal { B } ( x ) = \frac { 1 } { | \mathcal { B } | } \sum _ { d \in \mathcal { B } } l ( x , d ) \ \ g _ \mathcal { B } ( x ) = \frac { 1 } { | \mathcal { B } | } \sum _ { d \in \mathcal { B } } \nabla _ x l ( x , d ) . \end{align*}


\begin{align*} \frac { \partial \hat { q } _ { \delta , i } } { \partial z _ { i , h } } = O ( 1 ) , \end{align*}


\begin{align*} \prod _ { \ell = z _ { j + 1 } } ^ { z _ j ' } | \cos ( \pi ( \theta + \ell \alpha ) ) | \leq e ^ { | z _ j ' - z _ { j + 1 } | ( - \ln 2 + \varepsilon ) } . \end{align*}


\begin{align*} & \gamma \colon \mathbb { R } \rightarrow \mathbb { C } , \\ & \gamma ( t ) = p - i e ^ { i \alpha } | t | , \quad t \in ( - \infty , 0 ] , \\ & \gamma ( t ) = p + i e ^ { - i \alpha } | t | , \quad t \in ( 0 , + \infty ] , \end{align*}


\begin{align*} \mathcal { Z } \left ( \hat { \mu } \right ) = \bigcup _ { i = 1 } ^ { r } \mathcal { Z } \left ( \hat { \nu } _ { i } \right ) , \ \mathcal { Z } \left ( \hat { \nu } _ { i } \right ) = \rho ^ { - i } \bigcup _ { j = 0 } ^ { \infty } \left ( \frac { q } { p } \right ) ^ { j } \frac { a _ { j } } { N _ { j } } , \ a _ { j } \in \mathbb { Z } \setminus N _ { j } \mathbb { Z } . \end{align*}


\begin{align*} \frac { \mu ( \eta _ { \Lambda _ n } ) } { \mu ( \xi _ \Delta \eta _ { \Lambda _ n \setminus \Delta } ) } = \frac { \mu ( \eta _ \Delta | \eta _ { \Lambda _ n \setminus \Delta } ) } { \mu ( \xi _ \Delta | \eta _ { \Lambda _ n \setminus \Delta } ) } . \end{align*}


\begin{align*} \nabla \times ( a _ { t } ^ { \varepsilon } + ( B \times u ) ^ { \varepsilon } + \mu \nabla \times b ^ { \varepsilon } ) = 0 \textrm { o n } \ ( L ^ { 2 } _ { \sigma } \cap H ^ { 1 } ) ( \mathbb { R } ^ { 3 } ) ^ { * } , \end{align*}


\begin{align*} & \varepsilon ( i , j ) = ( - 1 ) ^ { i j } , \\ & e _ 1 \cdot e _ 2 = e _ 2 \cdot e _ 1 = - 2 e _ 3 , \\ & \begin{array} [ t ] { l } \alpha ( e _ 1 ) = \sqrt { 2 } e _ 1 , \alpha ( e _ 2 ) = e _ 3 - e _ 2 , \\ \alpha ( e _ 3 ) = e _ 3 . \end{array} \end{align*}


\begin{align*} \begin{aligned} \epsilon \Theta _ { \epsilon } ( t ) = \epsilon ^ { 2 } \int ^ { \epsilon ^ { - 2 } t } _ { 0 } \xi ( r ) d r , \end{aligned} \end{align*}


\begin{align*} D \Lambda ( x ) = D Z ( \Lambda ( x ) ) ^ { - 1 } \end{align*}


\begin{align*} \mathcal { G } \triangleq \left \{ \Theta \mathbf { G } : \mathbf { G } = \left ( { { \mathbf { G } } ^ { 1 } } , \cdots , { { \mathbf { G } } ^ { k } } \right ) , \mathbf { G } ^ i \in \mathbf { R } ^ { n _ i \times n _ i } , ( \mathbf { G } ^ i ) ^ T \mathbf { Q } ^ i \mathbf { G } ^ i = \mathbf { Q } ^ i \right \} \end{align*}


\begin{align*} \tilde { u } ^ 2 - \tilde { v } ^ 2 & = \tilde { y } ^ 2 - \tilde { x } ^ 2 , \\ \tilde { u } \tilde { v } & = - \tilde { x } \tilde { y } . \end{align*}


\begin{align*} M _ { \delta , \pm } ( x ; z ) : = M _ { \pm } ( x ; z ) \delta _ { \pm } ( z ) ^ { - \delta _ 3 } , \end{align*}


\begin{align*} j ( \tau , s ) = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } ( U + \tau \tilde { U } + s U _ 0 ) \cdot ( U + \tau \tilde { U } + s U _ 0 ) \dd x . \end{align*}


\begin{align*} a _ c \left [ t + 1 \right ] = & a _ c \left [ t \right ] - 4 \mu a _ c \left [ t \right ] \sum _ { j = 1 } ^ { M } \lvert \phi ^ { \left ( j , c \right ) } \rvert ^ 2 - 2 \mu \sum _ { l = 1 } ^ { M } \Re \left \{ \phi ^ { \left ( l , c \right ) } \right \} + 2 \mu a _ c \left [ t \right ] f ^ { \left ( c \right ) } , \end{align*}


\begin{align*} f ( x ) _ { i j } = p _ { 1 } \pi ( t _ { i } ^ { - 1 } ) x \pi ( t _ { j } ) p _ { 1 } \forall x \in M , \ ; \forall i , j \in \{ 1 , \ldots , n \} \end{align*}


\begin{align*} \rho ( \dot x ( J ) ) = \rho ( x ) = - \sigma ^ + , \rho ( \dot y ( J ) ) = \rho ( y ) = - \sigma ^ - + \sigma ^ + , \rho ( \dot z ( J ) ) = \rho ( z ) = \sigma ^ { - } \end{align*}


\begin{align*} \sum _ { x = 0 } ^ n | \tilde V _ x ( m ) | ^ 2 = \sum _ { x = 0 } ^ n \sum _ { x ' = 1 } ^ n M _ { x , x ' } ( m ) \tilde V _ { x ' } ( m ) \tilde V ^ { \star } _ { x } ( m ) + \sum _ { x = 0 } ^ n \tilde v _ x ( m ) \tilde V ^ { \star } _ { x } ( m ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T g _ n ( t ) d t = 0 . \end{align*}


\begin{align*} G = F Q + R , \end{align*}


\begin{align*} { \widetilde p _ { b , m } ^ * } = { \widetilde \lambda ^ * } { W _ m } - \frac { 1 } { { { { \widetilde h } _ m } } } , { \widetilde p _ { b , o } ^ * } = { \widetilde \lambda ^ * } \left ( { W - { W _ m } } \right ) - \frac { 1 } { { { { \widetilde h } _ b } } } , \end{align*}


\begin{align*} \mathfrak X _ \pi : = V ( \pi ) \subset \mathfrak X . \end{align*}


\begin{align*} \bar K _ 1 ^ { ( n ) } ( x , y ) : = \frac { 1 } { 4 ( n + 1 ) ^ 2 } \sum _ { j , j ' = - n - 1 } ^ n \Phi \left ( \frac { j } { n + 1 } , \frac { j ' } { n + 1 } \right ) \cos \left ( \frac { \pi j ( y - x ) } { n + 1 } \right ) \cos \left ( \frac { \pi j ' ( y - x ) } { n + 1 } \right ) , \end{align*}


\begin{align*} p ' _ { 1 } \circ ( p _ { 1 2 } \circ f \circ \iota _ { 1 2 } ) = p _ { 1 } \circ f \circ \iota _ { 1 2 } = p _ { 1 } \circ f \circ l _ { 1 } \circ \iota _ { 1 2 } = p _ { 1 } \circ f \circ \iota _ { 1 2 } \circ l ' _ { 1 } \\ = p ' _ { 1 } \circ ( p _ { 1 2 } \circ f \circ \iota _ { 1 2 } ) \circ l ' _ { 1 } , \end{align*}


\begin{align*} \sum _ { i = 0 } ^ { q - 1 } \binom { n + i - 1 } { i } = \frac { q ( q + 1 ) \cdots ( q + n - 1 ) } { n ! } . \end{align*}


\begin{align*} \begin{cases} \mathcal { Q } ( u ) = 0 & \ \Omega _ { T } \\ u = u _ 0 & \ \overline { \Omega } \times \{ 0 \} \\ u = \psi & \ \{ a , b \} \times [ 0 , T ] \end{cases} \end{align*}


\begin{align*} M ( \xi ) = - ( \xi _ 1 ^ 2 + \xi _ 2 ^ 2 + \cdots + \xi _ d ^ 2 ) \begin{pmatrix} 1 & 0 \\ 0 & \epsilon \end{pmatrix} + i \xi _ 1 c \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 0 & e ^ { - \kappa } \\ 0 & - \kappa e ^ { - \kappa } \end{pmatrix} , \end{align*}


\begin{align*} \forall z \in \Z ^ d : \sum _ { \Delta \Subset \Z ^ d } \sum _ { i = 1 } ^ q \norm { \nabla ^ i _ z \gamma _ \Delta ( \cdot | \cdot ) } _ \infty < \infty , \end{align*}


\begin{align*} \mathbb E \left [ \left \| \sum _ { i = 1 } ^ { n } \int _ { 0 } ^ { ( i - 1 ) \Delta _ n } ( \mathcal S ( \Delta _ n ) - I ) \Sigma _ s ^ { \mathcal S _ n } ( \mathcal S ( \Delta _ n ) - I ) ^ * d s \right \| _ { \mathcal H } ^ 2 \right ] \geq C n ^ { 1 - 4 H } . \end{align*}


\begin{align*} \overline { M } = \overline { C } \cap ( N K \cap H ) / K . \end{align*}


\begin{align*} N _ j ( t ) - N _ j ( 0 ) = \sum _ { k = 1 } ^ d \int _ 0 ^ t \Psi _ { j k } ( s ) d \beta ^ c _ k ( s ) \forall \ , 0 \le t \le T , \ ; j = 1 , \dots , d . \end{align*}


\begin{align*} b - a = \frac { 1 } { 2 } , a \le 0 \quad \mbox { a n d } b \ge 0 . \end{align*}


\begin{align*} f \ast g ( x ) = \int _ { - \infty } ^ \infty f ( x - y ) g ( y ) d y f ^ { \ast n } ( x ) . \end{align*}


\begin{align*} \partial _ x ^ \alpha \partial _ y ^ \beta \Phi _ s ( x , y ) = O ( \mu ^ { \frac 3 2 - | \alpha | - | \beta | } ) , \ | \alpha | , | \beta | \ge 1 . \end{align*}


\begin{align*} \ell ( x , v \alpha ) = \langle v ^ { - 1 } \mu , \alpha \rangle - \Phi ^ + ( - v \alpha ) + \Phi ^ + ( - w v \alpha ) \geq 0 . \end{align*}


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


\begin{align*} \mbox { $ b _ { m a x p r i m } ( G ) : = \max \{ b ( G ^ \Omega ) \mid G \Omega \} $ . } \end{align*}


\begin{align*} & \mathbf { v } ^ { 0 } = [ \cdots , 1 , 0 , \cdots , 1 , \cdots , 1 , \cdots , 1 , \cdots ] ^ { T } , \\ & m , n , k , l = 1 , \cdots , N . \end{align*}


\begin{align*} \beta \left ( R ( r + s ) \right ) & = \int ^ { T _ 2 } _ { T _ 1 } G ( T _ 2 , \tau ) f _ \uparrow \left ( r ( \tau ) + s ( \tau ) \right ) d \tau \\ & \le f _ \uparrow \left ( b + 4 ^ k c \right ) \int ^ { T _ 2 } _ { T _ 1 } G ( T _ 2 , \tau ) \ ; d \tau \\ & < b . \end{align*}


\begin{align*} G ( x ; \theta ) = \exp \left ( - \frac { \theta } { x ^ 2 } \right ) , x > 0 , \theta > 0 . \end{align*}


\begin{align*} \begin{cases} d \tilde { Y } _ t = | z | J ^ \perp \tilde { Y } _ t d t + P ^ { - 1 } \Pi _ { \mathrm { k e r } A ^ \perp } \sigma d W _ s \\ \tilde { Y } _ 0 = P ^ { - 1 } Y _ 0 \in \C ^ n . \end{cases} \end{align*}


\begin{align*} | F ^ { i j } ( \nabla _ \alpha \varphi ) _ { i j } | \leq C \sqrt { b _ { n - 1 } } \sum _ { i = 1 } ^ n F ^ { i i } \mbox { i n } \omega _ 1 . \end{align*}


\begin{align*} \alpha ^ 2 + \gamma ^ 2 = \frac { 1 - a ^ 2 } { a \tau ^ 2 - a + \tau ^ 2 + 1 } . \end{align*}


\begin{align*} G \cdot R - \bar G \cdot \bar R - \bar R \cdot \nabla \bar G ( \nabla & \bar A ) ^ { - 1 } ( A - \bar A ) - \bar G \cdot ( P - \bar P ) = ( G - \bar G ) \cdot ( R - \bar R ) \\ & + \bar R \cdot G ( U | \bar U ; x , t ) + \bar G \cdot \left [ A _ t - \bar A _ t + f _ { \alpha , x _ \alpha - } \bar f _ { \alpha , x _ \alpha } \right ] \ , . \end{align*}


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


\begin{align*} \nabla _ { 1 1 } \Theta - \frac { k } { k + 1 } \frac { ( \nabla _ { 1 } \Theta ) ^ { 2 } } { \Theta } + ( k + 1 ) \Theta = ( k + 1 ) \Theta ^ { \frac { k } { k + 1 } } \left ( \Theta ^ { \frac { 1 } { k + 1 } } + ( \Theta ^ { \frac { 1 } { k + 1 } } ) _ { \iota \iota } \right ) = ( k + 1 ) \Theta ( 1 + \Theta ^ { - \frac { 1 } { k + 1 } } \Theta ^ { \frac { 1 } { k + 1 } } _ { \iota \iota } ) . \end{align*}


\begin{align*} \frac { \partial } { \partial t } u ^ h ( t , x ) = \frac { \partial } { \partial t } \left ( \sum _ i w _ i ( t ) \sigma _ i ( x ) \right ) = \sum _ i \sigma _ i ( x ) \frac { d } { d t } w _ i ( t ) , \end{align*}


\begin{align*} & g _ { j k } ( x _ 0 ) = g ^ { j k } ( x _ 0 ) = \delta _ { j k } , 1 \leqslant j , k \leqslant n , \\ & \frac { \partial g _ { j k } } { \partial x _ \alpha } ( x _ 0 ) = \frac { \partial g ^ { j k } } { \partial x _ \alpha } ( x _ 0 ) = 0 , 1 \leqslant \alpha \leqslant n - 1 , \end{align*}


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


\begin{align*} \lim _ { \lambda _ 1 , \ldots , \lambda _ d \rightarrow 1 } \frac { h ^ { ( \lambda _ 1 , \ldots , \lambda _ d ) } ( x ) } { \Delta ( \lambda ) } = \frac { h ( x ) } { \prod _ { j = 1 } ^ { d - 1 } j ! } . \end{align*}


\begin{align*} y = U ( t , 0 ) x - \lambda i \int _ 0 ^ t U ( t , s ) e ^ { ( \alpha - 1 ) W ( s ) } g ( y ( s ) ) d s . \end{align*}


\begin{align*} E [ M ^ * _ v ( t ) ^ 2 ] \le e ^ { - 2 ( \lambda ^ * - \lambda ) t } E [ M _ v ( 0 ) ^ 2 ] + \left \{ \begin{aligned} & 4 | v | ^ 2 \| x \| t e ^ { - 2 ( \lambda ^ * - \lambda ) t } & & \lambda \ge 1 , \\ & 4 | v | ^ 2 \| x \| \frac { 1 } { 1 - \lambda } e ^ { - 2 ( \lambda ^ * - 1 ) t } & & \lambda < 1 . \end{aligned} \right . \end{align*}


\begin{align*} r _ { \ell } \leq e ^ { 3 9 \varepsilon q _ n } \frac { e ^ { \beta _ n q _ n } } { \max ( | \ell | , 1 ) } e ^ { - q _ n L } \max \begin{cases} c _ { n , \ell - 1 } r _ { \ell - 1 } \\ \max ( | \ell | , e ^ { \delta _ n q _ n } , 1 ) e ^ { - \beta _ n q _ n } r _ { \ell } \\ c _ { n , \ell } r _ { \ell + 1 } \end{cases} . \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { f ( x ) } { g _ 1 ( x ) } = \lim _ { x \to \infty } \frac { f ( x ) } { f _ u ( x ) } \frac { f _ u ( x ) } { g _ u ( x ) } \frac { g _ u ( x ) } { g _ 1 ( x ) } = \nu ( ( 1 , \infty ) ) . \end{align*}


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


\begin{align*} M _ { x , y } : = \sum _ { j , j ' = 0 } ^ n \Theta ( \mu _ j , \mu _ { j ' } ) \psi _ j ( x ) \psi _ { j ' } ( x ) \psi _ j ( y ) \psi _ { j ' } ( y ) . \end{align*}


\begin{align*} \bigcap _ { i = 1 } ^ { n } \left ( B \left ( X _ { i } , r \right ) ^ { c } \cap C _ { j } \right ) \neq \emptyset . \end{align*}


\begin{align*} r _ a = s _ \alpha \varepsilon ^ { k \alpha ^ \vee } \in \widetilde W . \end{align*}


\begin{align*} ( \mathbf { v } ^ { 0 } ) ^ { H } = \frac { \sum \limits _ { j = 1 } ^ { N } \mathbf { R } _ { . j } } { \mathbf { S ( R ) } } \end{align*}


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


\begin{align*} \varphi _ { n } = \varphi _ { 1 } \circ \psi _ { 2 } \cdots \circ \psi _ { n } \colon D _ { n } \to D _ { n - 1 } \to \cdots D _ { 2 } \to D _ { 1 } \to { \mathbf { A } } _ { k } ^ { 2 } \end{align*}


\begin{align*} B S ( m , n ) = \langle a , b \mid a b ^ m = b ^ n a \rangle \end{align*}


\begin{align*} r _ { \ell } ^ + \leq & e ^ { 5 0 \varepsilon q _ n } \frac { e ^ { - q _ n L } } { \max ( | \ell | , 1 ) } \max ( c _ { n , \ell - 1 } r _ { \ell - 1 } ^ - , r _ { \ell - 1 } ^ + , r _ { \ell } ^ + , r _ { \ell + 1 } ^ - , r _ { \ell + 1 } ^ + ) \times \begin{cases} \max ( | \ell | , e ^ { \delta _ n q _ n } , 1 ) , & \beta _ n \geq \delta _ n + 2 0 0 \varepsilon \\ e ^ { \beta _ n q _ n } , & \beta _ n < \delta _ n + 2 0 0 \varepsilon \end{cases} . \end{align*}


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


\begin{align*} ( x , y ) = ( x ' , y ' ) = 1 \end{align*}


\begin{align*} u ( x ) \geq l ( y ' ) + \frac { 1 } { 4 } \sum _ { \beta = 1 } ^ { n - 1 } b _ \beta y _ \beta ^ 2 - C b _ \alpha | \hat { y } | ^ 2 \end{align*}


\begin{align*} \ell ( x ' , s _ \alpha u \beta ) = \ell ( x , u \beta ) + \ell ( s _ \alpha , - u \beta ) = \begin{cases} \ell ( x , u \beta ) , & u \beta \neq \pm \alpha , \\ - \ell ( x , \alpha ) + 1 > 0 , & u \beta = - \alpha , \\ \ell ( x , \alpha ) - 1 < 0 , & u \beta = \alpha . \end{cases} \end{align*}


\begin{align*} p = 1 \oplus 0 \oplus 0 \oplus \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} , q = 0 \oplus 1 \oplus 0 \oplus \begin{pmatrix} a ^ 2 & a b \\ a b & b ^ 2 \end{pmatrix} , \end{align*}


\begin{align*} & \lim _ { j \rightarrow \infty } \sup _ { z \in V ^ { - 1 } \mathbb { B } _ n } \Big { | } ( ( V ^ { - 1 } \varphi _ { k j } V ) ^ 1 , . . . , ( V ^ { - 1 } \varphi _ { k j } V ) ^ s ) ( z ) \Big { | } \\ & \qquad = \lim _ { j \rightarrow \infty } \| V ^ { - 1 } ( \varphi _ { k j } - \rho ) \| _ \infty \\ & \leq C \lim _ { j \rightarrow \infty } \| \varphi _ { k j } - \rho \| _ \infty = 0 . \end{align*}


\begin{align*} \Phi ( x ) ^ { \delta } = \sum _ { r = 0 } ^ \infty \ , s _ r ( \delta ) \ , \Phi ( x ) ^ r , \end{align*}


\begin{align*} v = D '' u _ \epsilon + P _ { \omega _ { \epsilon } , h } ( \sqrt { \lambda } \tau _ \epsilon ) \end{align*}


\begin{align*} I _ h \leq E [ b ] \leq \liminf _ { n \to \infty } E [ b _ n ] = I _ h . \end{align*}


\begin{align*} E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 2 } ^ { n } Z _ i ^ { \theta [ u ] } \Bigr ) \right ] = E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \Bigr ) \right ] . \end{align*}


\begin{align*} f _ u ( x ) = ( e ^ { \Lambda _ u } - 1 ) ^ { - 1 } \sum _ { n = 1 } ^ \infty ( \Lambda _ u ^ n / n ! ) g _ u ^ { \ast n } ( x ) . \end{align*}


\begin{align*} T ^ 2 [ f ] ( p ) = \begin{cases} z ^ 2 - 1 & \ | z | \ge 1 , \\ 0 & \ | z | < 1 . \end{cases} \end{align*}


\begin{align*} \lim \limits _ { \varepsilon \to 0 } \frac { | A \oplus _ p ( \varepsilon ^ { \frac { 1 } { p } } \ , B ) | - | A | } { \varepsilon } = \frac { 1 } { p } \int \limits _ { S ^ { n - 1 } } h ^ p _ B ( u ) h _ A ^ { 1 - p } d S _ A ( u ) , \end{align*}


\begin{align*} \sum _ { \Delta \Subset \Z ^ d } \sum _ { \xi _ \Delta } \sum _ { i = 1 } ^ q \norm { \nabla ^ i _ z \hat { c } _ \Delta ( \cdot , \xi _ \Delta ) } _ \infty < \infty . \end{align*}


\begin{align*} { \bf { c } } _ j = [ x _ j , y _ j ] ^ { } \in \mathbb { R } ^ 2 , j \in [ 1 , C ] , \end{align*}


\begin{align*} { F _ { \Phi _ { ( 3 ) } } } _ x ( v ) = \sum _ { i = 1 } ^ { m } \big ( 6 \langle B ( v , e _ { i } ) , B ( v , e _ { i } ) \rangle _ { \mathbb R ^ q } - \langle B ( v , v ) , B ( e _ { i } , e _ { i } ) \rangle _ { \mathbb R ^ q } \big ) , \end{align*}


\begin{align*} \mathcal { U } = \{ u \in \hat { \mathcal { N } } _ p ( q ) : u \} \subseteq \hat { \mathcal { N } } _ p ( q ) \cap \{ u = ( u _ i ) _ i : 1 + \sum _ i u _ i = 0 \} \cong \mathbb { R } ^ { p + 1 - k } . \end{align*}


\begin{align*} G r _ w ( E ) = \bigoplus _ { j = 1 } ^ { k } E ^ j / E ^ { j - 1 } , \end{align*}


\begin{align*} \tilde z = \Psi _ 1 ( w ) \end{align*}


\begin{align*} \mathrm { E } ( Z ^ s ) = \int _ { 0 } ^ { 1 } \ , \left ( \sum _ { i = 0 } ^ { \infty } f _ i \ , \ , u ^ { i / \alpha } \right ) ^ s \mathrm { d } u = \sum _ { i = 0 } ^ { \infty } \ , \frac { g _ { s , i } } { i / \alpha + 1 } , \end{align*}


\begin{align*} K ( x ) = \frac { 1 } { 2 \pi } \frac { x ^ \perp } { | x | ^ 2 } + \frac { 1 } { 2 \pi } \sum _ { k \in \mathbb { Z } ^ 2 , k \neq 0 } \frac { ( x - k ) ^ \perp } { | x - k | ^ 2 } , \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*} Q _ \lambda ( \zeta _ \lambda ) = O ( b - \lambda ) ~ ~ ~ ~ \lambda \to b ^ - \end{align*}


\begin{align*} \mathbf { R } = \frac { 1 } { K } \sum \limits _ { n = 1 } ^ { K } \overline { \mathbf { X } } ( n ) \overline { \mathbf { X } } ^ { H } ( n ) , \end{align*}


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


\begin{align*} S _ - = T _ - \partial _ { p _ 0 } ^ 2 - p _ 0 \partial _ { p _ 0 } . \end{align*}


\begin{align*} F ( x ) = \frac { 1 } { \operatorname { B } ( a , b ) } \sum ^ { \infty } _ { n = 0 } \frac { ( - 1 ) ^ n \ , \Gamma ( b ) } { ( a + n ) \ , \Gamma ( b - n ) \ , n ! } \exp \left \{ - \frac { ( a + n ) \theta } { x ^ 2 } \right \} . \end{align*}


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


\begin{align*} D = \sum _ { i = 1 } ^ r n _ i \cdot y _ i \end{align*}


\begin{align*} \mathfrak { W } _ 2 ^ { [ k , k ] } ( 1 + \varepsilon , 1 - \varepsilon ) & = 1 - \frac { k - 1 } { 2 ( 2 k - 1 ) } \varepsilon ^ 2 + O ( \varepsilon ^ 3 ) , \\ \mathfrak { M } _ 2 ^ { [ r ] } ( 1 + \varepsilon , 1 - \varepsilon ) & = 1 - \frac { 1 - r } { 2 } \varepsilon ^ 2 + O ( \varepsilon ^ 3 ) \end{align*}


\begin{align*} \hat \beta ( s , \xi ) = \varphi _ s ( \xi ) + \frac { f ( s \xi , s \eta ) - f ( \xi , \eta ) - \varphi _ s ( \xi ) - \varphi _ s ( \eta ) } { 2 } . \end{align*}


\begin{align*} ( n - 2 ) r < \sum \limits _ { i = 2 } ^ n \chi _ i , \end{align*}


\begin{align*} \left | \Pi _ { \mathrm { k e r } A ^ \perp } \frac { d ^ j } { d t ^ j } X _ t | _ { t = 0 } \right | \geq \frac { 1 } { C _ J } K ^ { j + 1 } . \end{align*}


\begin{align*} V : = U ( \mathfrak g ) \otimes _ { U ( \mathfrak g _ { \geq 0 } ) } \mathbb C _ x . \end{align*}


\begin{align*} A _ i f ( x ) : = \sum _ { \substack { y \sim x \\ \tau ( y ) = \tau ( x ) + i } } f ( y ) , \end{align*}


\begin{align*} \Delta t b _ k \mathcal { L } ( \tilde { u } ^ { [ n + k ] } ( x ) ) + a _ k \tilde { u } ^ { [ n + k ] } ( x ) = - \sum _ { j = 0 } ^ { k - 1 } a _ j \tilde { u } ^ { [ n + j ] } ( x ) + \Delta t b _ k f ( t _ { n + k } , x ) , \end{align*}


\begin{align*} \tau _ y = \tau ' _ y v _ y , v _ y = \sum _ { \substack { 1 \leq r \leq j \\ C _ r \in O _ y } } \lambda _ r ( e _ { i _ { r , 1 } } + \dots + e _ { i _ { r , d _ r } } ) \end{align*}


\begin{align*} \left \{ \begin{array} { c l } \partial _ t u - D ^ { 2 } _ x \partial _ { x } u - \partial _ { x } ^ { - 1 } \partial _ y ^ 2 u & = u \partial _ x u , ( t , x , y ) \in \R \times \R \times \R , \\ u ( 0 ) & = u _ 0 \in H ^ { s _ 1 , s _ 2 } ( \R ^ 2 ) , \end{array} \right . \end{align*}


\begin{align*} A , { \tilde A } \subset \mathbb A _ 0 : = \big \{ x : | x | ^ { - 1 } x \in \mathbb S \big \} . \end{align*}


\begin{align*} \begin{cases} \gamma & \leq { ( \alpha - 1 ) } \min \{ 1 , { 1 } / { L _ 1 } \} \left [ { \alpha B ^ 2 \left ( C _ 0 + 1 \right ) } \right ] ^ { - 1 } ; \\ \gamma & \leq B ^ { - 2 } \Big [ \alpha ^ 2 + ( e - 1 ) \Big ( L _ { 0 } + L _ 1 C _ 0 \Big ) \Big ] ^ { - 1 } . \end{cases} \end{align*}


\begin{gather*} E ( X - Y ) ^ { 2 k } \allowbreak = \allowbreak E ( X + \rho Y - ( 1 + \rho ) Y ) ^ { 2 k } \allowbreak = \\ \allowbreak \sum _ { j = 0 } ^ { 2 k } \binom { 2 k } { j } ( - 1 ) ^ { j } ( 1 + \rho ) ^ { 2 k - j } E ( X + \rho Y ) ^ { j } Y ^ { 2 k - j } \allowbreak = \\ \allowbreak \sum _ { j = 0 } ^ { 2 k } \binom { 2 k } { j } ( - 1 ) ^ { j } ( 1 + \rho ) ^ { 2 k - j } E ( E ( X + \rho Y ) ^ { j } | Y ) Y ^ { 2 k - j } \allowbreak . \end{gather*}


\begin{align*} \psi _ * \mu \big ( B ( \psi ( y ) , r ) \cap \psi ( Y ) \big ) = \mu ( \psi ^ { - 1 } \big ( B ( \psi ( y ) , r ) \cap \psi ( Y ) \big ) ) = \mu ( \pi \big ( B ( \psi ( y ) , r ) \cap \psi ( Y ) \big ) ) , \end{align*}


\begin{align*} \Pi _ { \mathrm { s p a n } \{ P _ x ^ { - 1 } v \} } J _ x ^ \perp y = \lambda J _ x \Pi _ { \mathrm { s p a n } \{ P _ x ^ { - 1 } v \} } y \forall y \in \C ^ n . \end{align*}


\begin{align*} ( { } _ a D _ g ^ { \alpha } f ) ( x ) = & \left ( \frac { 1 } { g ' ( y ) } \frac { d } { d y } \right ) ( { } _ { a } I _ g ^ { 1 - \alpha } f ) ( y ) \\ = & \frac { \left ( \displaystyle \frac { 1 } { g ' ( y ) } \displaystyle \frac { d } { d y } \right ) } { \Gamma ( 1 - \alpha ) } \int _ a ^ x \frac { f ( y ) g ' ( y ) } { ( g ( x ) - g ( y ) ) ^ { \alpha } } d y . \end{align*}


\begin{align*} \tau ' _ y ( i ) = \begin{cases} \tau ' ( i ) & i \in I _ y \\ i & i \notin I _ y . \end{cases} \end{align*}


\begin{align*} \dot { x } = - \gamma \ , y _ \delta ( x , t ) \ , u ( t ) x ( 0 ) = x _ 0 \end{align*}


\begin{align*} C _ k ( t ) ^ { 2 \beta - m + 1 } \sum _ { i \geq 0 } ( - 1 ) ^ i \binom { \alpha - m - ( k - 1 ) i } { i } t ^ i - C _ k ( t ) ^ { \beta - m } \sum _ { i \geq 0 } ( - 1 ) ^ i \binom { \alpha - m - 1 - ( k - 1 ) i } { i } t ^ i . \end{align*}


\begin{align*} [ T ( a ) , T ( b ) ] _ { \alpha } & = \alpha [ T ( a ) , T ( b ) ] = \alpha \big ( T ( \varrho ( T ( a ) ) b - \varrho ( T ( b ) ) a \big ) , \\ T \big ( \tilde { \varrho } ( T ( a ) ) b - \tilde { \varrho } ( T ( b ) ) a \big ) & = T \big ( ( \varrho ( \alpha ( T ( a ) ) ) \beta ( b ) - \varrho ( \alpha ( T ( b ) ) ) \beta ( a ) \big ) . \end{align*}


\begin{align*} { R ^ { \rm { N } } } = \min \left \{ { R _ { b \to s } ^ { \rm { N } } , R _ { b \to b } ^ { \rm { N } } } \right \} \triangleq W { \log _ 2 } \left ( { 1 + \frac { { { p _ b } { { \left | { { { \widetilde h } _ b } } \right | } ^ 2 } } } { { { p _ s } { { \left | { { { \widetilde h } _ b } } \right | } ^ 2 } + W { N _ 0 } } } } \right ) , \end{align*}


\begin{align*} \left \{ \begin{array} { c c l } \frac { d } { d t _ n } L & = & \left [ ( L ^ n ) _ D ( t ) , L ( t ) \right ] = - \left [ ( L ^ n ) _ S ( t ) , L ( t ) \right ] \\ L ( 0 ) & = & L _ 0 \\ \end{array} \right . \ ; , \end{align*}


\begin{align*} B _ 1 + \cdots + B _ k \subset \{ n \in \N : T ^ n x _ { \vec 0 } \in E \} = A \end{align*}


\begin{align*} \hat { c } _ { \Delta } ( \eta , \xi _ { \Delta } ) : = c _ { \Delta } ( \xi _ { \Delta } \eta _ { \Delta ^ c } , \eta _ { \Delta } ) \frac { \gamma _ { \Delta } ( \xi _ { \Delta } | \eta _ { \Delta ^ c } ) } { \gamma _ { \Delta } ( \eta _ { \Delta } | \eta _ { \Delta ^ c } ) } , \end{align*}


\begin{align*} \left \langle \left ( x , y \right ) , \left ( x ^ { \prime } , y ^ { \prime } \right ) \right \rangle = \tfrac { 1 } { 2 } \left ( \left \langle x , x ^ { \prime } \right \rangle - \left \langle y , y ^ { \prime } \right \rangle \right ) \end{align*}


\begin{align*} \Pi _ { \mathcal { S } } ( x ) : = \Big \{ \bar x \in \mathcal { S } \ \Big | \ \| x - \bar x \| _ { \mathcal { H } } = \inf _ { y \in \mathcal { S } } \| x - y \| _ { \mathcal { H } } \Big \} . \end{align*}


\begin{align*} r _ { b } w \varepsilon ^ \lambda = s _ \beta w \varepsilon ^ { \lambda + \Phi ^ + ( - \beta ) w ^ { - 1 } \beta ^ \vee } \in \widetilde W _ J \cdot w \varepsilon ^ \lambda \end{align*}


\begin{align*} \begin{aligned} & y _ \delta ( x _ a , t ) = \dfrac { a _ { 0 , \delta } ( x _ a ) } { 2 } + \\ & \sum _ { k = 1 } ^ { \infty } a _ { k , \delta } ( x _ a ) \cos \left ( k 2 \pi t \right ) + b _ { k , \delta } ( x _ a ) \sin \left ( k 2 \pi t \right ) \end{aligned} \end{align*}


\begin{align*} v + \dfrac { 1 } { 2 } \varphi _ \alpha \circ ( u - v ) & { } = \dfrac { u + h ( u , v ) } { 2 } , & u - \dfrac { 1 } { 2 } \varphi _ \alpha \circ ( u - v ) & { } = \dfrac { v + k ( u , v ) } { 2 } . \end{align*}


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


\begin{align*} F ^ { q , \dagger } _ v ( 0 , p _ 0 ) \psi ^ \dagger _ { v v } ( 0 , p _ 0 ) [ u _ \ell , u _ \ell ] = - c _ 2 ( q , \ell , p _ 0 ) u _ { 2 \ell } . \end{align*}


\begin{align*} \langle \alpha ^ \vee , 2 \rho \rangle = 2 + \sum _ { \substack { \alpha \neq \beta \in \Phi ^ + \\ s _ \alpha ( \beta ) \in \Phi ^ - } } \langle \alpha ^ \vee , \beta \rangle . \end{align*}


\begin{align*} U = \left \{ \begin{pmatrix} 1 & u \\ 0 & 1 \end{pmatrix} \colon u \in \Z _ { p } \right \} \ , \end{align*}


\begin{align*} f ( x ) = \begin{cases} G ( H ( x ) ) & \ x \in T , \\ H ( x ) & \ x \notin T . \end{cases} \end{align*}


\begin{align*} A ( y ) : = A i ( \gamma ( y - y _ c ) ) , \end{align*}


\begin{align*} \mathcal { E } ( t ) = \mathcal { E } ( 0 ) - \int ^ t _ 0 \langle B ( s , x , u _ t ( s ) ) , u _ t ( s ) \rangle d s t \in J . \end{align*}


\begin{align*} H = \sum _ \alpha \kappa _ \alpha = - \Gamma ^ { \alpha } _ { n \alpha } = - \frac { 1 } { 2 } \frac { \partial g _ { \alpha \alpha } } { \partial x _ n } , \end{align*}


\begin{align*} Y _ { i \Delta _ n } ( j \Delta _ n ) : = Y _ { t _ i } ( x _ j ) , i , j = 1 , . . . , n , \end{align*}


\begin{align*} \begin{aligned} J \mathcal { N } _ 2 ( u _ 1 , u _ 2 ) & = 2 | u _ 1 | ^ 2 J u _ 2 - u ^ 2 _ 1 \overline { J u _ 2 } \\ & + 2 ( \overline { u } _ 1 u _ 2 J u _ 1 - u _ 1 u _ 2 \overline { J u _ 1 } ) + 2 u _ 1 \overline { u } _ 2 J u _ 1 . \end{aligned} \end{align*}


\begin{align*} \ell ( \mathbf { \theta } ) = & - \log \mathrm { B } ( \alpha , \beta ) - \log \sigma + ( \alpha - 1 ) \log \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \\ & + ( \beta - 1 ) \log \left [ 1 - \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \right ] + \log \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) . \end{align*}


\begin{align*} [ x , y ] = x \cdot y - \varepsilon ( x , y ) y \cdot x . \end{align*}


\begin{align*} \widehat { \mathcal D } _ { \hat { A } } = \left \{ P = \sum _ { \alpha \in \mathbb { Z } } a _ { \alpha } \ , D ^ { \alpha } \ ; | \ ; P \in \widehat { \Psi } ( \hat { A } ) \mbox { a n d } a _ \alpha = 0 \mbox { f o r } \alpha < 0 \right \} \ ; . \end{align*}


\begin{align*} w _ n = \sum _ { i = 0 } ^ { r + \tilde { r } - 1 } c _ i w _ { n + i } \end{align*}


\begin{align*} ( I - F ) \left ( \Psi _ - - e ^ { - i c _ - ( x ) \sigma _ 3 } \right ) = \begin{pmatrix} 0 & n \\ m & 0 \end{pmatrix} , \end{align*}


\begin{align*} & \sup _ { x \in \mathbb { R } } \left \| \mathcal { P } ^ + \left ( z ^ { - i } \bar { r } _ 1 ( z ) e ^ { 2 i z x } \right ) \right \| _ { L _ z ^ \infty } \leq \frac { 1 } { \sqrt { 2 } } \left \| z ^ { - i } r _ 1 ( z ) \right \| _ { H ^ 1 } , i = 0 , 1 , \\ & \sup _ { x \in \mathbb { R } } \left \| \mathcal { P } ^ - \left ( r _ 2 ( z ) \mathrm { e } ^ { - 2 i z x } \right ) \right \| _ { L _ { z } ^ { \infty } } \leq \frac { 1 } { \sqrt { 2 } } \left \| r _ 2 ( z ) \right \| _ { H ^ { 1 } } . \end{align*}


\begin{align*} \Gamma ( X ) = \sigma X + \mathcal { O } \left ( X ^ { \frac { 4 } { 5 } + \varepsilon } \right ) \end{align*}


\begin{align*} \nabla _ V X & = \omega ( V ) \\ \nabla _ V \omega & = - R ( X , V ) + T _ h ( V , \cdot ) . \end{align*}


\begin{align*} Z _ 1 = L : = \varrho _ { w } \partial _ z - \varrho _ z \partial _ w , \end{align*}


\begin{align*} \rho ^ * ( d x \wedge d z ) & = ( ( \cos \theta + 1 ) d t - t \sin \theta \ d \theta ) \wedge ( \sin \theta \ d t + t \cos \theta \ d \theta ) \\ & = t \cos \theta ( \cos \theta + 1 ) d t \wedge d \theta - t \sin ^ 2 \theta d \theta \wedge d t \\ & = ( t \cos ^ 2 \theta + t \cos \theta + t \sin ^ 2 \theta ) d t \wedge d \theta \\ & = t ( 1 + \cos \theta ) d t \wedge d \theta \end{align*}


\begin{align*} \bigg | \frac { 1 } { k } \int _ { r } ^ { r + k } g ( t ) d t - g ( r ) \bigg | & = \bigg | \frac { 1 } { k } \int _ { r } ^ { r + k } ( g ( t ) - g ( r ) ) d t \bigg | \\ & \leq \frac { 1 } { k } \int _ { r } ^ { r + k } | g ( t ) - g ( r ) | d t \\ & = \frac { 1 } { k } \int _ { [ r , r + k ] \cap B _ r } | g ( t ) - g ( r ) | d t + \frac { 1 } { k } \int _ { [ r , r + k ] \setminus B _ r } | g ( t ) - g ( r ) | d t \end{align*}


\begin{align*} U & = b + B _ { \infty } = \nabla \times ( ( \phi - \phi _ { \infty } ) \nabla \theta ) + G \nabla \theta , \\ \nabla \times U & = \nabla \times ( G \nabla \theta ) - L \phi \nabla \theta . \end{align*}


\begin{align*} \Re \mathcal { P } [ \varrho ] = \frac { a } { 4 } | w | ^ 6 \left ( 4 \tau ^ 6 + ( 8 a + 8 ) \tau ^ 4 + ( 4 + 6 a + 5 a ^ 2 ) \tau ^ 2 - 2 a \right ) , \end{align*}


\begin{align*} S ( \zeta ) = \mathrm { s t a b } _ N ( \psi _ \zeta ) \end{align*}


\begin{align*} \mathbb { H } : = H \times [ H ^ 1 ( \mathcal { O } ) ^ 3 ] , \mathbb { V } : = V \times \Big \{ n \in H ^ 2 ( \mathcal { O } ) ^ 3 : \frac { \partial n } { \partial \nu } = 0 \Big \} , \end{align*}


\begin{align*} { \frac { \partial { R _ n ( s ) } } { \partial { s } } } = - { \frac { \mathcal { A } ^ { 2 } _ { \gamma } ( s ) } { ( q - s ) ^ { 2 } } } \ , \beta ^ n \cdot \bigl ( { 1 + o ( 1 ) } \bigr ) n \to \infty , \end{align*}


\begin{align*} \langle F _ 1 , \ldots , F _ t \rangle = \{ \sigma \subset V \ | \ \sigma \subset F _ i i \in [ t ] \} . \end{align*}


\begin{align*} \sum _ { l = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M \mathbb { E } \left [ y _ l ^ * y _ j \right ] = & 2 \sum _ { i = 1 } ^ { M - 1 } \sum _ { q = i + 1 } ^ { M } \sum _ { r = 1 } ^ M a _ r ^ 2 \Re \left [ \phi ^ { \left ( i , r \right ) ^ * } \phi ^ { \left ( q , r \right ) } \right ] + \sum _ { l = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M a _ c ^ 2 \phi ^ { \left ( l , c \right ) ^ * } \phi ^ { \left ( j , c \right ) } . \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*} S t _ { D N M } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( 1 , z ) - 1 | | _ { \infty } = 0 , \end{align*}


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


\begin{align*} L _ { q } [ f ( t ) ] ( s ) = F _ { q } ( s ) = \int _ { 0 } ^ { \infty } f ( t ) \exp _ { q } ( - s t ) d t . \end{align*}


\begin{align*} \begin{aligned} \pi _ 1 ( M ) & = \pi _ 2 ( M ) = 0 \\ \big ( \operatorname { r e s p . } \ , \pi _ 1 ( M ) & = \cdots = \pi _ { [ p ] } = 0 \ , \big ) . \end{aligned} \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*} a ( x _ 1 , x _ 2 ) = \nabla \times ( \eta ( x _ 1 , x _ 2 ) \nabla z ) + \phi ( x _ 1 , x _ 2 ) \nabla z , \end{align*}


\begin{align*} \lfloor x \rfloor + \Big \lfloor x + \frac 1 n \Big \rfloor + \ldots + \Big \lfloor x + \frac { n - 1 } n \Big \rfloor = \lfloor n x \rfloor . \end{align*}


\begin{align*} X _ { t _ { n + 1 } } ^ i = X _ { t _ n } ^ i + \int _ { t _ n } ^ { t _ { n + 1 } } b ^ i ( X _ t ) \d t + \sigma W _ \tau ^ i , ~ ~ ~ ~ \tilde X _ { n + 1 } ^ i = \tilde X _ n + \int _ { t _ n } ^ { t _ { n + 1 } } b ^ i ( \tilde X _ n ) \d t + \sigma W _ \tau ^ i , \end{align*}


\begin{align*} A _ k ^ { \epsilon } : = \left \lbrace h \in H : \sum _ { l \geq N _ k ^ { \epsilon } } \langle h , e _ l \rangle ^ 2 \leq \frac 1 { l _ k } \right \rbrace . \end{align*}


\begin{align*} O _ { \gamma } ( f _ 0 ) = 2 \frac { q ^ { d _ \gamma + 1 } - 1 } { q - 1 } . \end{align*}


\begin{align*} \overline { \pi } _ 1 ( q _ 1 ) \overline { \pi } _ 1 ( q _ 2 ) = c _ { \chi _ 1 } ( q _ 1 , q _ 2 ) \overline { \pi } _ 1 ( q _ 1 q _ 2 ) \forall q _ 1 , q _ 2 \in Q \end{align*}


\begin{align*} \int _ 0 ^ { \theta } { \frak G } _ { x , y } ( s ) \dd s = M _ { x , y } \end{align*}


\begin{align*} \prod _ { p \leq X } \left ( 1 - \frac { 1 } { p } \right ) \left ( 1 + \frac { \Delta _ K ( p ) \iota _ { K , f } ( p ) } { p } \right ) \asymp \exp \left ( \sum _ { p \leq X } \frac { \Delta _ K ( p ) \iota _ { K , f } ( p ) - 1 } { p } \right ) & = \exp \left ( \sum _ { p \leq X } \frac { \Delta _ K ( p ) - 1 } { p } + O _ { K , f } ( 1 ) \right ) \\ & \asymp _ { K , f } \prod _ { p \leq X } \left ( 1 + \frac { \Delta _ K ( p ) - 1 } { p } \right ) , \end{align*}


\begin{align*} \int _ { \mathfrak U } | \omega | = { \phi _ s } _ ! \int _ { \mathfrak Y } | \omega | , \end{align*}


\begin{align*} & B _ i = \{ \omega : \| u _ i - u _ i ' \| \leq q \| u _ { i - 1 } - u _ { i - 1 } ' \| \} \\ & \boldsymbol { A } _ { 0 } = B _ { m } ^ c \\ & \boldsymbol { A } _ k = \big ( \cap _ { i = m } ^ { k + m - 1 } B _ { i } \big ) \cap B _ { k + m } ^ c , \forall 1 \leq k \leq n - 1 , \\ & \boldsymbol { A } _ { n } = \cap _ { i = m } ^ { n + m - 1 } B _ i . \end{align*}


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


\begin{align*} \lambda _ j = 2 \theta ^ { \alpha _ X } _ { \gamma } ( 0 ) \lambda _ H ( 1 + 2 \lambda _ g ) + \lambda _ { j + 1 } ( 2 + \lambda _ g ) . \end{align*}


\begin{align*} l _ { x y } ( i ) = { \lambda _ i \over \lambda _ 0 } \prod _ { z \in \partial x \setminus \{ y \} } { a _ { i 0 } + \sum _ { j \in \Z _ 0 } a _ { i j } l _ { z x } ( j ) \over a _ { 0 0 } + \sum _ { j \in \Z _ 0 } a _ { 0 j } l _ { z x } ( j ) } . \end{align*}


\begin{align*} \begin{cases} t _ { i , e n } ( x _ 0 , t _ 0 ) = \max \left ( 0 , t _ 0 - \dfrac { x _ 0 + R } { \lambda _ i } \right ) \forall \ : i \in [ 1 , p ] , \\ t _ { i , e n } ( x _ 0 , t _ 0 ) = \max \left ( 0 , t _ 0 - \dfrac { x _ 0 - R } { \lambda _ i } \right ) \forall \ : i \in [ p + 1 , n ] . \end{cases} \end{align*}


\begin{align*} \left \{ \begin{aligned} d x ^ { * , i } _ t = & ~ [ A x ^ { * , i } _ t + B \alpha ^ { * , i } _ t ] d t + \sigma d W ^ i _ t + \sigma _ 0 d W ^ 0 _ t , \\ x ^ { * , i } _ 0 = & ~ \xi ^ i . \end{aligned} \right . \end{align*}


\begin{align*} \mathbf { h } ( H _ { \rho } ( x ) ) = R _ \rho \mathbf { h } ( x ) \end{align*}


\begin{align*} \varrho _ N ^ { \rm i n i } ( u ) : = \varrho _ * + N ^ { - \alpha } \rho ^ { \rm i n i } ( u ) , \end{align*}


\begin{align*} | f _ \alpha ( U | \bar U ; x , t ) | & \le | f _ \alpha - \bar f _ \alpha - \nabla \bar f _ \alpha ( U - \bar U ) ) | \\ & + \left | \nabla \bar f _ \alpha \nabla \bar A ^ { - 1 } \big ( A ( U , x , t ) - \bar A - \nabla \bar A ( U - \bar U ) \big ) \right | \\ & \le k _ 4 | A ( U , x , t ) - \bar A | ^ 2 \ , , \end{align*}


\begin{align*} x ^ { \ast , \sigma , n } = x ^ { \ast , \sigma , n - 1 } \ast \left ( \prescript { \sigma ^ { n - 1 } } { } x \right ) = x ^ { \ast , \sigma , n - 1 } \cdot \left ( \prescript { \sigma ^ { n - 1 } } { } x _ n \right ) . \end{align*}


\begin{align*} \lim _ { n , m \rightarrow \infty } \Omega ( m , n ) = 0 \end{align*}


\begin{align*} F ^ { q , \dagger } _ v ( v ( \tau ) , p ( s ( \tau ) ) ) ( u _ \ell + \omega ( \tau ) ) = \nu ( \tau ) ( u _ \ell + \omega ( \tau ) ) , \end{align*}


\begin{align*} q ^ { k s } \pmod { N _ { n _ { 1 } } } = 1 \end{align*}


\begin{align*} M ( X ) = \max _ z p ( z ) = p ( 0 ) \leq q ( 0 ) = \max _ z q ( z ) = M ( Y ) . \end{align*}


\begin{align*} H _ N ( t ) = H ( f _ { N , t } ; \nu _ { N , t } ) : = \int _ { \Omega _ N } f _ { N , t } \log f _ { N , t } d \nu _ { N , t } . \end{align*}


\begin{align*} x '' : = w '' \varepsilon ^ { \mu '' } : = x ' r _ { a } = w ' s _ \alpha \varepsilon ^ { \mu ' - \langle \mu ' , \alpha \rangle \alpha ^ \vee } < x ' . \end{align*}


\begin{align*} \underline { u } = \frac { 1 } { 2 } \sum _ { 1 \leq j < k \leq N } \langle u ( e _ j ) , e _ k \rangle \ e _ j \cdot e _ k . \end{align*}


\begin{align*} D = \max \left \{ D _ { x , 1 } \sqrt { \delta ' } + D _ { x , 2 } , D _ { y , 1 } \sqrt { \delta ' } + D _ { y , 2 } \right \} , \end{align*}


\begin{align*} \psi ( S ( r + s ) ) & = \int ^ { T _ 2 } _ { T _ 1 } G ( T _ 2 , \tau ) f _ \downarrow \left ( r ( \tau ) + s ( \tau ) \right ) d \tau \\ & \ge f _ \downarrow \left ( b + d \right ) \int ^ { T _ 2 } _ { T _ 1 } G ( T _ 2 , \tau ) \ ; d \tau \\ & > d . \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*} A _ k ( a _ j ) = A _ k ( b _ j ) = 0 . \end{align*}


\begin{align*} \mathcal { F } ^ { ( k ) } \left ( \mathbf { x } ^ { ( k ) } \right ) = s o f t \left ( \mathcal { F } ^ { ( k ) } \left ( \mathbf { r } ^ { ( k ) } \right ) , \theta ^ { ( k ) } \right ) , \end{align*}


\begin{align*} & H _ { \gamma } [ b _ n ] = H _ { \gamma } [ b _ { 0 , n } ] = h _ n , \\ & I _ { h _ n , \gamma } \leq { \mathcal { E } } [ v _ n , b _ n ] \leq { \mathcal { E } } [ v _ { 0 , n } , b _ { 0 , n } ] . \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } ( 1 - ( 1 - q ) k y ) ^ { \frac { 1 } { 1 - q } - k } y ^ { k } d y = \frac { \Gamma ( k + 1 ) \Gamma \left ( \frac { 2 - q } { 1 - q } - k \right ) } { \Gamma \left ( \frac { 3 - 2 q } { 1 - q } \right ) } . \end{align*}


\begin{align*} k _ 3 ( P ^ \ast ( q ) ) & = \frac { 1 } { 3 } \times q \times k _ 3 ( P ^ \ast ( q ) , 0 ) \\ & = \frac { q } { 3 } \times \langle H \rangle . \end{align*}


\begin{align*} l = \left \{ \begin{array} { l l l } 1 , & \hbox { i f $ i $ i s o d d , } \\ - 1 , & \hbox { ; } \end{array} \right . \end{align*}


\begin{align*} Q ( u ) = F ^ { - 1 } ( u ) = \sqrt { - \frac { \theta } { \log ( \operatorname { I } ^ { - 1 } _ { u } ( a , b ) ) } } , 0 < u < 1 , \end{align*}


\begin{align*} \phi ( x , y ) = \sum _ { i = 1 } ^ L c _ i a _ i ( x ) b _ i ( y ) , \end{align*}


\begin{align*} J ( P _ 0 \Psi _ { R ( \lambda ) } ) = O ( | b - \lambda | ^ { 2 p } ) ~ ~ \lambda \to b . \end{align*}


\begin{align*} H _ { m i n } ( E \otimes F _ { \ell - 1 } ) = H _ { m i n } ( E \otimes F _ 1 ) . \end{align*}


\begin{align*} \alpha _ y = \mu _ y , \end{align*}


\begin{align*} \Phi ^ + _ J ( \alpha ) : = \begin{cases} 1 , & \alpha \in \Phi ^ + _ J , \\ 0 , & \alpha \in \Phi \setminus \Phi ^ + _ J . \end{cases} . \end{align*}


\begin{align*} R ( x ; z ) = \begin{pmatrix} \bar { r } _ 1 ( z ) r _ 2 ( z ) & \bar { r } _ 1 ( z ) \mathrm { e } ^ { - 2 i z x } \\ r _ 2 ( z ) \mathrm { e } ^ { 2 i z x } & 0 \end{pmatrix} , \end{align*}


\begin{align*} ( \langle \gamma _ 1 \rangle , \langle \gamma _ 2 \rangle ) \in R _ \alpha \qquad \Longleftrightarrow \ell _ \alpha ( \langle \gamma _ 1 \rangle ) = \ell _ \alpha ( \langle \gamma _ 2 \rangle ) . \end{align*}


\begin{align*} \mathcal { C } _ { i j } & \simeq \Omega _ { X } | _ { D _ { i j } } & \mathcal { C } _ { i j } ^ l & \simeq \omega _ l | _ { D _ { i j } } , l = i , j \\ \omega _ X | _ { D _ i } & \simeq \mathcal { C } _ i \ \otimes \omega _ i , & \det ( \mathcal { C } _ { i j } ) & \simeq \mathcal { C } _ { i j } ^ l \otimes \mathcal { C } _ l | _ { D _ { i j } } l = i , j \end{align*}


\begin{align*} S : = \{ 1 \} \cup \{ g \in G : \phi _ { g , 1 } \neq 0 \} \cup \{ g \in G : \psi _ { g , 1 } \neq 0 \} . \end{align*}


\begin{align*} \begin{aligned} \left . \dfrac { \partial h ( s ) } { \partial s } \right | _ { s = x + \delta \sin ( 2 \pi t ) } = & \ , \left . \dfrac { \partial h ( s ) } { \partial s } \right | _ x \\ & + \left . \dfrac { \partial ^ 2 h ( s ) } { \partial s ^ 2 } \right | _ { x } \delta \sin ( 2 \pi t ) + O ( \delta ^ 2 ) \\ \end{aligned} \end{align*}


\begin{align*} \sum _ { \xi _ { \Delta } } \int _ { \Omega } c _ { \Delta } ( \omega , \xi _ { \Delta } ) f ( \omega ) g ( \xi _ { \Delta } \omega _ { \Delta ^ c } ) \mu ( d \omega ) = \sum _ { \zeta _ { \Delta } } \int _ { \Omega } \hat { c } _ { \Delta } ( \omega , \zeta _ { \Delta } ) f ( \zeta _ { \Delta } \omega _ { \Delta ^ c } ) g ( \omega ) \mu ( d \omega ) , \end{align*}


\begin{align*} g ( z ) = \sum _ { m = 0 } ^ { \infty } a _ m z ^ { q ( m + 1 ) } = \sum _ { n = 1 } ^ { \infty } a _ { n - 1 } z ^ { q n } = \sum _ { n = 0 } ^ { \infty } b _ n z ^ { q n } , b _ n = \begin{cases} a _ { n - 1 } & , n \geq 1 , \\ 0 & , . \end{cases} \end{align*}


\begin{align*} K _ r ( B ) = K _ r ( B _ { [ n ] , [ r ] } ) \times \R ^ { m - r } \mbox { a n d } \sigma _ m ( K _ r ( B ) ) = \sigma _ r ( K _ r ( B _ { [ n ] , [ r ] } ) ) , r \in [ m - 1 ] , \end{align*}


\begin{align*} j _ 1 \alpha - \frac { p } { q _ n } = j _ 1 \frac { p _ n } { q _ n } - \frac { p } { q _ n } + j _ 1 ( \alpha - \frac { p _ n } { q _ n } ) = k + j _ 1 ( \alpha - \frac { p _ n } { q _ n } ) , \end{align*}


\begin{align*} & u _ { x t } + \alpha \beta ^ 2 u - 2 i \alpha \beta u _ x - \alpha u _ { x x } - i \alpha \beta ^ 2 | u | ^ 2 u _ x = 0 , \\ & u ( x , t = 0 ) = u _ 0 ( x ) , \end{align*}


\begin{align*} V ( x ) = & v o l ( B _ 1 ( x ) \cap \tau _ 1 ) + v o l ( B _ 2 ( x ) \cap \tau _ 2 ) \\ = & V ( 0 ) \frac { e ^ { 2 x } + e ^ { - 2 x } } { 2 } = V ( 0 ) \cosh \left ( 2 x \right ) \end{align*}


\begin{align*} A ^ { \tilde \Psi + \eta _ n } [ \eta _ 1 , \dots , \eta _ { n - 1 } ] - A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ { n - 1 } ] - A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ n ] = g _ 1 ( x ) + g _ 2 ( x ) . \end{align*}


\begin{align*} \frac { k l } { m } , k \ , \sqrt { \frac { l - m } { m ( l - 1 ) } } , l \ , \sqrt { \frac { k - m } { m ( k - 1 ) } } . \end{align*}


\begin{align*} \begin{aligned} & \mathbf { A } \mathbf { d } _ { h x } = 0 , \\ & { { \mathbf { A } } ^ { T } } \mathbf { d } _ { h y } + \mathbf { C } d _ { h x } + \mathbf { d } _ { h s } = 0 , \\ & \bar { \mathbf { X } } \mathbf { D } \mathbf { d } _ { h s } + \bar { \mathbf { S } } \mathbf { D } ^ { - T } \mathbf { d } _ { h x } = 0 , \end{aligned} \end{align*}


\begin{align*} \textstyle 1 , \binom { n } { 1 } , \binom { n } { 2 } , \dotsc , \binom { n } { n - 1 } , 1 . \end{align*}


\begin{align*} & \frac { d } { d r } \log _ { m + 1 } \frac { e } { r } = \frac { 1 } { \prod _ { k = 1 } ^ m \log _ k \frac { e } { r } } \left ( - \frac { 1 } { r } \right ) , \\ & \frac { d } { d r } \log _ { m + 1 } \frac { 1 } { C r ^ \alpha } = \frac { 1 } { \prod _ { k = 1 } ^ m \log _ k \frac { 1 } { C r ^ \alpha } } \left ( - \frac { \alpha } { r } \right ) . \end{align*}


\begin{align*} \delta _ { p , c } : = \sum _ { k = 1 } ^ { l } ( 2 - \alpha _ { k } - \beta _ { k } ) c _ { k } ^ { \beta _ { k } } \zeta _ { k } ^ { \frac { p } { 2 - \alpha _ { k } - \beta _ { k } } } \big [ \null _ { k } \eta \big ] _ { \frac { p } { 2 - \alpha _ { k } } } , \end{align*}


\begin{align*} \mathbf { H } = \left [ \mathbf { J } _ { 1 } , \ \mathbf { J } _ { 2 } \right ] \begin{bmatrix} \mathbf { \Sigma } _ { 1 } & \ \\ \ & \mathbf { \Sigma } _ { 2 } \end{bmatrix} \left [ \mathbf { J } _ { 1 } , \ \mathbf { J } _ { 2 } \right ] ^ { \mathbf { H } } , \end{align*}


\begin{align*} \big ( h ( t _ j ) , v ( t _ j ) \big ) & = \big ( \hat h ( t _ j ) , \hat v ( t _ j ) \big ) - \big ( L _ V g _ { t _ j } , V ( u _ { t _ j } ) \big ) \\ ( p , z ) & = ( \hat p , \hat z ) - \Big ( L _ V L _ X g _ a , V \big ( X ( u _ a ) \big ) \Big ) \end{align*}


\begin{align*} x \cdot y = \varepsilon ( x , y ) y \cdot x , \end{align*}


\begin{align*} F _ { p , x } ( v ) = ( p - 2 ) \langle \mathsf B ( v , v ) , B ( v , v ) \rangle _ { \mathbb R ^ q } + \langle Q ^ M _ x ( v ) , v \rangle _ M , \end{align*}


\begin{align*} \Pi _ { \mathrm { k e r } A } B ( v _ 1 , v _ 2 ) + \Pi _ { \mathrm { k e r } A } B ( v _ 2 , v _ 1 ) = 0 \end{align*}


\begin{align*} \exists \ \theta > 1 \ \Rightarrow \ K _ p ( \theta ) : = | | \kappa _ p ( \xi ) | | _ { \theta } < \infty , \end{align*}


\begin{align*} g _ 1 ( t , s ) & = \frac { ( t - a ) ^ { \alpha - 1 } ( b - s ) ^ { \alpha - 1 - \beta } } { ( b - a ) ^ { \alpha - 1 - \beta } } - ( t - s ) ^ { \alpha - 1 } \\ & = ( t - a ) ^ { \alpha - 1 } \left ( \frac { b - s } { b - a } \right ) ^ { \alpha - 1 - \beta } - ( t - a ) ^ { \alpha - 1 } \left ( \frac { t - s } { t - a } \right ) ^ { \alpha - 1 } \\ & = ( t - a ) ^ { \alpha - 1 } \left [ \left ( \frac { b - s } { b - a } \right ) ^ { \alpha - 1 - \beta } - \left ( \frac { t - s } { t - a } \right ) ^ { \alpha - 1 } \right ] . \end{align*}


\begin{align*} \mathcal { D } _ { G , \kappa } ^ { \mathrm { c h } } \simeq \bigoplus _ { \lambda \in P ^ + } \mathbb { V } ^ { \kappa } _ \lambda \otimes \mathbb { V } ^ { \kappa ^ * } _ { \lambda ^ * } , \frac { 1 } { \kappa + h ^ \vee } + \frac { 1 } { \kappa ^ * + h ^ \vee } = 0 . \end{align*}


\begin{align*} \gamma _ i ( s , \tau ) = \bar { Q } _ { \frac { 2 \pi ( i - 1 ) } { m } } \gamma ( s , \tau ) . \end{align*}


\begin{align*} \left | \mathrm { d e t } { \left ( e ^ { - i \theta _ i x _ j } \right ) } _ { i , j = 1 } ^ { d } \right | \le C _ d \Delta ( \theta ) \Delta ( x ) . \end{align*}


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


\begin{align*} f ( z ) & = \frac 1 2 h ( g ^ { - 1 } ( z ) ) . \end{align*}


\begin{align*} \left \langle l _ 1 , m _ 1 \right | e ^ { i k \phi } \cos ( n \theta ) \left | l _ 2 , m _ 2 \right \rangle = \end{align*}


\begin{align*} ( S _ j ( n ) : 1 \leq j \leq d , n \geq 0 ) = ( T _ j ( n + d - j ) : 1 \leq j \leq d , n \geq 0 ) . \end{align*}


\begin{align*} \mathcal { L } ( B _ v ) : = \begin{cases} \{ u \in B _ v : \textnormal { d i s t } ( u , v ) = r \} & \textnormal { i f } B _ v \textnormal { i s a t r e e } , \\ \{ u \in B _ v : \textnormal { d i s t } ( u , v ) \geq r \textnormal { a n d } \textnormal { d i s t } ( u , C _ v ) \geq r - h \} & \textnormal { i f } C _ v \textnormal { e x i s t s } , \end{cases} \end{align*}


\begin{align*} \tilde F _ E ( t ) = \begin{cases} F _ E ( t + \alpha ) - \alpha & ( t \in [ 0 , 1 - \alpha ) ) \\ F _ E ( t + \alpha - 1 ) - \alpha & ( t \in [ 1 - \alpha , 1 ) ) . \end{cases} \end{align*}


\begin{align*} T _ \alpha = \partial _ { \alpha } + \sum _ { \beta < n } B _ { \alpha \beta } ( x _ { \beta } \partial _ { n } - x _ { n } \partial _ { \beta } ) , \ ; \ ; \mbox { f o r } \alpha < n , \end{align*}


\begin{align*} [ \Box - \mu _ { \rm e f f } ^ 2 ( r , \theta ) ] \varphi = 0 \ , . \end{align*}


\begin{align*} M i n i m i z e _ { x \in \mathbb { R } ^ n } \ \ J ( x ) : = 1 / 2 \| x - z \| ^ 2 \ \ s . t . \ x \in A \end{align*}


\begin{align*} \mathcal { Z } _ K ( D ^ 2 , S ^ 1 ) = \mathcal { Z } _ { \mathcal { Z } ^ * _ K ( \Delta ^ 1 , \partial \Delta ^ 1 ) } ( D ^ 1 , S ^ 0 ) . \end{align*}


\begin{align*} \begin{aligned} - \nabla _ y \cdot \left ( D _ 0 \big ( \nabla _ y w _ i + e _ i \big ) \right ) & = 0 \mbox { i n } ( 0 , T ) \times \Omega \times Y ^ { \ast } , \\ - D _ 0 \big ( \nabla _ y w _ i + e _ i \big ) \cdot \nu & = 0 \mbox { o n } ( 0 , T ) \times \Omega \times \Gamma , \\ w _ i \mbox { i s } Y \mbox { - p e r i o d i c , } \int _ { Y ^ { \ast } } & w _ i ( t , x , y ) d y = 0 \mbox { f . a . e . } ( t , x ) \in ( 0 , T ) \times \Omega . \end{aligned} \end{align*}


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


\begin{align*} ( h _ \alpha \ , * \ , h _ \beta ) ( t ) = h _ { \alpha + \beta } ( t ) , \ t > 0 , \ \alpha > 0 , \ \beta > 0 \end{align*}


\begin{align*} c _ { n , \ell } : = | \cos ( \pi \theta _ { m _ n + \ell q _ n } ) | . \end{align*}


\begin{align*} g ( x ) = \lim _ { n \rightarrow { \infty } } g _ n ( x ) \end{align*}


\begin{align*} \delta ^ \nabla \big ( F ^ { \prime } ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) R ^ \nabla \big ) = 0 \ , . \end{align*}


\begin{align*} \begin{cases} \partial _ t w + \nabla ^ \perp \varphi \cdot \nabla w = 0 , \ \ & \ B _ { R ^ * } ( 0 ) , \\ w = \mathcal { L } _ H \varphi , \ \ & \ B _ { R ^ * } ( 0 ) , \\ \varphi = 0 , \ \ & \ \partial B _ { R ^ * } ( 0 ) . \end{cases} \end{align*}


\begin{align*} F _ { \delta } ( x ; z ) : = \left ( \mathcal { P } ^ + \left ( r _ { \delta , 2 } ( z ) e ^ { 2 i z x } \right ) e _ 2 , \mathcal { P } ^ - \left ( \bar { r } _ { \delta , 1 } ( z ) e ^ { - 2 i z x } \right ) e _ 1 \right ) . \end{align*}


\begin{align*} G ( \breve F ) = \bigsqcup _ { x \in \widetilde W } I x I . \end{align*}


\begin{align*} d X _ t = \left ( b _ 0 ( t , X ) + \langle b ( t , X , \cdot ) , \mu \rangle \right ) d t + d W _ t , \quad \operatorname { L a w } ( X ) = \mu , \quad \operatorname { L a w } ( X _ 0 ) = \mu _ 0 . \end{align*}


\begin{align*} \begin{aligned} u ( \bar { x } ) & = \frac { 1 } { 2 } \sum _ { \beta = 1 } ^ { n - 1 } b _ \beta \bar { x } _ \beta ^ 2 + + \frac { 1 } { 6 } \sum _ { \xi , \beta , \gamma } \varphi _ { \xi \beta \gamma } ( 0 ) \bar { x } _ \xi \bar { x } _ \beta \bar { x } _ \gamma + O ( | \bar { x } ' | ^ 4 ) \\ & \leq u ( x _ 0 ) + \frac { 1 } { 2 } \sum _ { \beta = 1 } ^ { \alpha } b _ \beta \bar { x } _ \beta ^ 2 + C \delta ^ 3 b _ \alpha ^ 2 . \end{aligned} \end{align*}


\begin{align*} \lambda _ { f \times i d _ { X } } [ ( f \times i d _ { X } ) ( r , x ) , \alpha ] ( 0 ) = ( f ( r ) , x ) = ( f \times i d _ { X } ) ( r , x ) \end{align*}


\begin{align*} \begin{aligned} & \int _ { B } ( u _ p - u _ { p + N } ) ( \Delta u _ p ) ^ k \wedge ( \Delta u _ { p + N } ) ^ { m - k } \wedge \beta _ n ^ m \\ \leq & \sigma ( v , p , N ) + \int _ B u _ p ( \Delta u _ p ) ^ m \wedge \beta _ n ^ { n - m } - \int _ B u _ { p + N } ( \Delta u _ { p + N } ) ^ m \wedge \beta _ n ^ { n - m } \end{aligned} \end{align*}


\begin{align*} \phi _ 1 = \psi _ 1 , \ldots , \phi _ { j - 1 } = \psi _ { j - 1 } , \phi _ j < _ K \psi _ j \end{align*}


\begin{align*} \frac { \partial R } { \partial t } ( t ) = - \frac { \partial \phi } { \partial r } ( R ( t ) ) . \end{align*}


\begin{align*} { \bf P } ( \lim _ { n \to \infty } \kappa _ n = \kappa ) = 1 . \end{align*}


\begin{align*} \begin{aligned} & \hat { \mathbf { r } } _ p ( \alpha ) = ( 1 - ( 1 - \nu ) \alpha ) \hat { \mathbf { r } } _ p \\ & \hat { \mathbf { r } } _ d ( \alpha ) = ( 1 - ( 1 - \nu ) \alpha ) \hat { \mathbf { r } } _ d \\ & \mu ( \alpha ) = ( 1 - ( 1 - \nu ) \alpha ) \mu , \end{aligned} \end{align*}


\begin{gather*} \sum _ { m = 0 } ^ { n } ( - 1 ) ^ { n - m } \binom { n } { m } 2 ^ { - m } \sum _ { j = 0 } ^ { m } \binom { m } { j } \left ( \beta \right ) ^ { \left ( n - j \right ) } \left ( \beta + m - j \right ) ^ { \left ( j \right ) } \\ = \left \{ \begin{array} { c c c } 0 & & n \\ \frac { n ! } { 2 ^ { n / 2 } ( n / 2 ) ! } \left ( \beta \right ) ^ { \left ( n / 2 \right ) } & & n \end{array} \right . . \end{gather*}


\begin{align*} | m _ n + \ell _ n q _ n - \tilde { m } _ n | = q _ { n + 1 } , \end{align*}


\begin{align*} & \Delta : = ( \langle u _ * , ( \lambda x _ v + ( 1 - \lambda ) x _ w ) \rangle ) ^ 2 + \lambda ( 1 - \lambda ) \| u _ * \| ^ 2 \| x _ v - x _ w \| ^ 2 \ge 0 , \\ \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { M } \mathbb { E } \left [ y _ i ^ * y _ i \right ] = \sum _ { l = 1 } ^ M a _ c ^ 2 \lvert \mathbf { h } _ { l , * } \mathbf { p } _ c \rvert ^ 2 + \sum _ { i = 1 } ^ M \sum _ { j = 1 } ^ M a _ j ^ 2 \lvert \mathbf { h } _ { i , * } \mathbf { p } _ j \rvert ^ 2 + M \sigma _ n ^ 2 , \end{align*}


\begin{align*} \begin{aligned} & ~ & & y _ 2 ^ 2 - 2 L _ 1 y _ 1 y _ 2 + ( L _ 1 \| x _ k \| ) ^ 2 \le ( \kappa _ { \rm e g } \delta _ k + \epsilon _ g ) ^ 2 \ ; \ ; & & \\ \end{aligned} \end{align*}


\begin{align*} \Gamma ( X ) = \sum \limits _ { 1 \leq n \leq X } \mu ^ 2 ( n ^ 2 + n + 1 ) \ , . \end{align*}


\begin{align*} d u = \Delta _ p u \ , d t + \kappa \Delta u \ , d t + \sum _ { k \in \Z ^ d } \sum _ { i = 1 } ^ { d - 1 } \xi _ { k , i } ( x ) \cdot \nabla u \ , d B _ t ^ { k , i } , \end{align*}


\begin{align*} b ^ { \varepsilon } = g + \frac { 1 } { 2 } \left ( \partial _ x v + \partial _ x \tilde w ^ \varepsilon \right ) . \end{align*}


\begin{align*} Z ( \theta ) = \sum \limits _ { i \in R } \prod \limits _ { F \in \mathrm { f a c e t } ( \Gamma ) } \theta ^ { F } _ { i _ { F } } . \end{align*}


\begin{align*} A _ 1 \circ F \circ A _ 2 ( x ) + A _ 3 ( x ) = F ' ( x ) , \end{align*}


\begin{align*} \Tilde { x } \in \mathcal { S } ^ n : = \{ x \in \mathbb { R } ^ n \ | \ \| x \| ^ 2 = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 \} . \end{align*}


\begin{align*} \begin{aligned} [ D _ * ^ q , M _ { z ^ q } ] & = D ^ q _ * M _ { z ^ q } ( f ) ( z ) - M _ { z ^ q } D _ * ^ q ( f ) ( z ) \\ & = D ^ q _ * ( z ^ q f ) ( z ) - z ^ q D _ * ^ q ( f ) ( z ) . \end{aligned} \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*} \begin{aligned} \det ( s I + A ) & = \prod \limits _ { k = 1 } ^ n ( s + \lambda _ k ( A ) ) = \sum \limits _ { m = 0 } ^ n \sum \limits _ { 1 \leq j _ 1 < \dots < j _ m \leq n } \lambda _ { j _ 1 } ( A ) \cdots \lambda _ { j _ m } ( A ) s ^ { n - m } = \sum \limits _ { m = 0 } ^ n \mathcal { H } _ m ( A ) s ^ { n - m } \end{aligned} \end{align*}


\begin{align*} \lim _ { j \rightarrow \infty } \lim _ { n \rightarrow \infty } \max \{ | 2 | ^ { k } \mu ( 0 , \ 2 ^ { k + 1 } x ) ; j \leq k < n + j \} = 0 \end{align*}


\begin{align*} M ( \Gamma ) = \inf _ { \rho \in \mathcal { F } ( \Gamma ) } \int _ { \mathbb { R } ^ d } \rho ^ d d m , \end{align*}


\begin{align*} G _ B ( R ) = ( B \otimes _ F R ) ^ \times \end{align*}


\begin{align*} \underset { n \rightarrow \infty } { \lim } \mathbf { v } ^ { n } = \underset { n \rightarrow \infty } { \lim } { \lambda } ^ { n } _ { 1 } \alpha _ { 1 } \mathbf { u _ { 1 } } , \end{align*}


\begin{align*} \widehat { H } : = \tau H ( x , \xi / \tau ) . \end{align*}


\begin{align*} H ( f ; \mu ) : = \int _ { \Omega _ N ^ d } f \log f d \mu = \sum _ { \eta \in \Omega _ N ^ d } f ( \eta ) \log f ( \eta ) \mu ( \eta ) . \end{align*}


\begin{align*} D / D = \{ 1 \} ^ { [ k ] } \cup ( G \setminus \{ 1 \} ) ^ { [ \lambda ] } . \end{align*}


\begin{align*} \begin{aligned} \| \eta ( t ) & - \eta _ a ( t ) \| = \| \eta ( t ) - \eta _ a ( t ) \pm z ( t ) \| \\ \le & \| \eta ( t ) - z ( t ) \| + \| z ( t ) - \eta _ a ( t ) \| \\ \le & \| \eta ( t ) - z ( t ) \| + \gamma \| \epsilon ( \eta _ a ( t ) , t ) \| \\ \le & \| \eta ( t ) - z ( t ) \| + \gamma \bar { k } _ 3 ( L _ r , \delta ) . \end{aligned} \end{align*}


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