\begin{align*} \partial _ t \rho ^ \varepsilon ( x , t ) - k _ d \Delta \rho ^ \varepsilon ( x , t ) = f ^ \varepsilon ( x , t ) = f + O ( \varepsilon ) , ( x , t ) \in Q _ { \varepsilon , T } . \end{align*}


\begin{align*} U ( t , s ) = I + \int _ { s } ^ { t } { \rm d } [ A ( r ) ] U ( r , s ) , \end{align*}


\begin{align*} H _ N ( \alpha , \beta ; q ) = \sum _ { r = 0 } ^ \infty \left ( - \frac { 1 } { N } \right ) ^ r \sum _ { s = 0 } ^ r q ^ s \langle \omega _ \alpha f _ s f _ { r - s } \omega _ \beta \rangle . \end{align*}


\begin{align*} \nabla \rho ^ \varepsilon ( x , t ) & = \varepsilon ^ { - 1 } \partial _ r \eta _ 0 ( r ) \nu + \nabla _ \Gamma \eta _ 0 ( r ) + \partial _ r \eta _ 1 ( r ) \nu \\ & + \varepsilon \{ r W \nabla _ \Gamma \eta _ 0 ( r ) + \nabla _ \Gamma \eta _ 1 ( r ) + \partial _ r \eta _ 2 ( r ) \nu \} + O ( \varepsilon ^ 2 ) . \end{align*}


\begin{align*} ( \Lambda ^ { ( N ) } ) ^ * J _ { p , q , \omega } ^ { ( N ) } \Lambda ^ { ( N ) } = J _ { | p | , q , \omega } ^ { ( N ) } . \end{align*}


\begin{align*} a \otimes ( b \otimes c ) = ( a \otimes b ) \otimes c \ ; , \end{align*}


\begin{align*} \left | \int _ \Lambda f d x - \sum _ { i = 1 } ^ N f ( \xi _ j ) \Delta x _ j \right | \leq \sqrt { d } \| f ' \| _ \infty | \Lambda | \delta . \end{align*}


\begin{align*} [ J _ \omega \psi ] ( n ) = \overline { p ( T ^ { n - 1 } \omega ) } \psi ( n - 1 ) + q ( T ^ n \omega ) \psi ( n ) + p ( T ^ n \omega ) \psi ( n + 1 ) . \end{align*}


\begin{align*} \varphi _ { l , k } = \psi _ { l , k } , \ \forall l > k \in \mathcal { N } _ T . \end{align*}


\begin{align*} ( L _ z + t L _ { z , j } ) ( f ) = \int _ \Omega { f } \ , \overline { g _ z + t g _ { z , j } } , \ \ \ \forall \ , { f \in { A ^ p ( \Omega ) } } , \end{align*}


\begin{align*} X _ M X _ N = X _ E + X _ { E ' } . \end{align*}


\begin{align*} A _ { \infty } ( x ) = \sum _ { \gamma = 1 , 2 } \int _ { \mathbb { R } ^ 3 } \mathrm { d } k \ ; \frac { \chi _ { \Lambda } ( k ) } { 2 \pi | k | ^ { 1 / 2 } } \mathbf { e } _ { \gamma } ( k ) ( a _ { \gamma } ( k ) e ^ { i k x } + a ^ { \dagger } _ { \gamma } ( k ) e ^ { - i k x } ) . \end{align*}


\begin{align*} \textbf { $ \kappa \in [ 0 , \kappa ^ * ] $ i s f i x e d a n d $ \delta \in [ 0 , ( d - \alpha ) / 2 ] $ i s s u c h t h a t $ \kappa _ { \delta } = \kappa $ } \ , . \end{align*}


\begin{align*} \mathcal S ' _ A ( D ; \ell ) = D \sum _ { ( n , 2 ) = 1 } W \left ( \frac n N \right ) \frac { \tau _ A ( n ) } { \sqrt { n } } \frac { 1 } { 2 n \ell } \sum _ { ( c , 2 n \ell ) = 1 \atop c \le Y } \frac { \mu ( c ) } { c ^ 2 } \sum _ { k = - \infty } ^ \infty ( - 1 ) ^ k G _ k ( n \ell ) \tilde \Psi \left ( \frac { k D } { 2 c ^ 2 n \ell } \right ) . \end{align*}


\begin{align*} c h L ^ W ( \nu , \ell _ 0 ) = \sum _ { w \in \widehat W ^ \natural } d e t ( w ) c h M ^ W ( w . \widehat \nu _ h ) . \end{align*}


\begin{align*} & - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \xi w + \partial _ \tau w \leq \frac { ( \delta b ) ^ { \frac { 1 } { \alpha _ 0 } } } { K } ( 1 - \eta ) ^ { \alpha _ 0 } , \\ & - C _ 1 ( 1 - \eta ) ^ { \alpha _ 0 } \leq \partial _ \tau w \leq b \delta ( 1 - \eta ) ^ { \alpha _ 0 } i n [ 0 , T ^ * ] \times [ 0 , X ] \times [ 0 , 1 ] . \end{align*}


\begin{align*} \left | I \right \rangle = B ( x _ 1 ) B ( x _ 2 ) \dots B ( x _ m ) \left | \Omega \right \rangle \ , , \end{align*}


\begin{align*} d ( x , y ) = \rho ( \delta _ x , \delta _ y ) . \end{align*}


\begin{align*} \phi ( t ) = \begin{cases} t , & t \in [ 0 , 1 ] , \\ 1 , & t \in \left [ 1 , r - 1 \right ] , \\ r - t , & t \in \left [ r - 1 , r \right ] . \end{cases} \end{align*}


\begin{align*} [ M _ A ( v ) ] _ j : = \begin{cases} v _ j & : j \in I _ A \\ - \sum \limits _ { l \in a ( j ) \setminus \{ j \} } v _ l & : j \in J _ A , \end{cases} \end{align*}


\begin{align*} X _ { N + 1 } = a _ { N + 1 } X _ N + X _ { N - 1 } \end{align*}


\begin{align*} \mathfrak { S } ( f _ k ) = n - \sum _ { e \in E ( T ) } ( | e | - 1 ) , \end{align*}


\begin{align*} c _ \star : = - \liminf _ { r \to \infty } \bigg \{ \frac { \overline { R } _ r } { \bar h _ 4 ( r ) } \bigg \} \ , , \end{align*}


\begin{align*} \dfrac { 2 } { q } = \dfrac { 1 } { 2 } - \dfrac { 1 } { p } ~ ~ ~ ~ ~ ~ \eta q = 3 . \end{align*}


\begin{align*} \phi ( s , t , x _ 0 ) = x _ 0 - ( \omega _ t - \omega _ s ) + \int _ s ^ t f ( r , \phi ( s , r , x _ 0 ) ) \ , d r , \end{align*}


\begin{align*} \begin{aligned} & \int _ I \int _ { A _ { ( t , s ) } } \| D _ x X ( t , s , x ) \| ^ p \dd x \dd s \\ & \leq \left ( \int _ { I } \int _ { A } \ell _ A ^ { 1 - q ' } ( v , y ) \exp \left ( \ell p q ' \| D _ x b ( v , y ) \| \right ) \dd y \dd v \right ) ^ { 1 / q ' } \times \\ & \phantom { A A A A A A A A A A A A A A A A A A A A A } \times \left ( \int _ I \int _ { I } \int _ { A } \| D _ y X ( s , v , y ) \| ^ { n q } \dd y \dd v \dd s \right ) ^ { 1 / q } . \end{aligned} \end{align*}


\begin{align*} \underline { \Lambda } | \xi | ^ { 2 } \leq \varrho ( x , y ) \varrho ^ { \top } ( x , y ) \xi \cdot \xi = | \varrho ( x , y ) ^ { \top } \xi | ^ { 2 } \leq \overline { \Lambda } | \xi | ^ { 2 } . \end{align*}


\begin{align*} J _ 1 = \langle B \partial _ t \partial _ y u , \partial _ y u \rangle _ { H ^ { 2 , 0 } } + \langle \partial _ y B \partial _ t u , \partial _ y u \rangle _ { H ^ { 2 , 0 } } = : J _ 1 ^ 1 + J _ 1 ^ 2 . \end{align*}


\begin{align*} \widehat \nu _ h + \widehat \rho - ( s _ 0 s _ 1 ) ^ m ( \widehat \nu _ h + \widehat \rho ) ( x + d ) & = \widehat \nu _ h + \widehat \rho - s _ 1 ( s _ 0 s _ 1 ) ^ m ( \widehat \nu _ h + \widehat \rho ) ( x + d ) \\ & = ( m ( - k m + 2 r + 1 ) ) = - k m ^ 2 + ( 2 r + 1 ) m , \end{align*}


\begin{align*} \mathcal { F } u = \frac { \mathcal { F } f } { \lambda _ { \xi } - \lambda } , \end{align*}


\begin{align*} \lambda \big ( \mathrm { t r } ( \epsilon ) \big ) ^ 2 + 2 \mu | \epsilon | ^ 2 = \frac { 1 + \nu } { E } \big ( | \sigma | ^ 2 - \nu ( \mathrm { t r } ( \sigma ) ) ^ 2 \big ) \ , . \end{align*}


\begin{align*} e ( \hat V ) & = e ( V ) + s \\ & = e ( V _ { t } ) + 2 s . \end{align*}


\begin{align*} I _ A = \bigcup _ { l \in J _ A } a ( l ) \setminus \{ l \} . \end{align*}


\begin{align*} A _ j \Phi _ { \mu } = \sqrt { 2 ( \mu _ j + 1 ) } \Phi _ { \mu + e _ j } , \end{align*}


\begin{align*} T _ { j , z _ j } ( \Gamma ( x ) ) e _ M = \Gamma \bigg ( \tilde { W } _ j ( z _ j ) x \tilde { W } _ j ( z _ j ) ^ * \bigg ) e _ { M } = \gamma _ 0 \bigg ( \tilde { W } _ j ( z _ j ) x \tilde { W } _ j ( z _ j ) ^ * \bigg ) e _ { M } , \end{align*}


\begin{align*} \pi ^ * ( N ) = \overline { \pi ^ { - 1 } ( N - C ) } \subset B l _ C ( M ) . \end{align*}


\begin{align*} \| f \| _ { L ^ p ( \mathbb R ^ { d + 1 } ) } \simeq \bigg \| \bigg ( \sum _ { j = 0 } ^ \infty | \Delta _ j f | ^ 2 \bigg ) ^ { 1 / 2 } \bigg \| _ { L ^ p ( \mathbb R ^ { d + 1 } ) } . \end{align*}


\begin{align*} \tilde H _ 1 '' ( 0 ) = h '' \frac { d } { d s } \Big \lvert _ { s = 0 } g _ H ( T _ * N , N ^ \perp ) g _ H ( T _ * N , N ) + h ' \frac { d ^ 2 } { d s ^ 2 } \Big \lvert _ { s = 0 } g _ H ( T _ * N , N ) . \end{align*}


\begin{align*} u ( a , t ) = u ( b , t ) = 0 , \end{align*}


\begin{align*} \bar { \mu } [ \dd v ] = \frac { \exp ( - \frac { \lambda _ 2 } { h _ 2 } \bar { \mathcal { E } } ( v ) ) \dd v } { \int _ { \mathbb { S } ^ 2 } \exp ( - \frac { \lambda _ 2 } { h _ 2 } \bar { \mathcal { E } } ( z ) ) \dd z } \ , , \end{align*}


\begin{align*} R _ { \alpha } ( x , y ) = \frac { 1 } { | \Gamma ( \frac { \alpha } { 2 } ) | } \int _ { 0 } ^ { + \infty } k _ t ( x , y ) t ^ { - 1 + \frac { \alpha } { 2 } } \ , d t , x , y \in G . \end{align*}


\begin{align*} - \cos ( \beta ) \kappa _ \eta = g _ H \Bigl ( \nabla _ { \dot \eta } \dot \eta , X \Bigr ) = \partial _ t \bigl ( \sin ( \beta ) \bigr ) - g _ H \Bigl ( \dot \eta , \nabla _ { \dot \beta } X \Bigr ) . \end{align*}


\begin{align*} \Delta ( \gamma _ \ell ) = \sum _ { i + j = \ell } \gamma _ i \otimes \gamma _ j S ( \gamma _ j ) = ( - 1 ) ^ j \gamma _ j . \end{align*}


\begin{align*} m _ J : = { } _ { \prec } \{ a \in \mathbb Z ^ N : { \bf f } _ S ^ a ( e _ 1 \wedge \dots \wedge e _ \ell ) = e _ { j _ 1 } \wedge \dots \wedge e _ { j _ \ell } \} . \end{align*}


\begin{align*} \int _ \R ( \lambda ^ 2 - z ^ 2 ) ^ { - 1 } f _ 0 ( \lambda ) \lambda \tanh ( \frac { \pi \lambda } { 2 } ) \ , d \lambda = \frac { 1 } { 2 } \int _ \R \left ( \frac { 1 } { \lambda - z } + \frac { 1 } { \lambda + z } \right ) f _ 0 ( \lambda ) \tanh ( \frac { \pi \lambda } { 2 } ) \ , d \lambda \ , . \end{align*}


\begin{align*} 0 = \varphi ( a / 1 ) = \varphi ( \theta ( a ) ) = \theta ' ( a ) . \end{align*}


\begin{align*} \underset { \tau \in ( 0 , \infty ) } \sup ~ \bigl ( \frac { 1 } { ( 1 + \tau ) ^ n } - e ^ { - n \tau } \bigr ) = \frac { 1 } { ( 1 + \tau _ n ) ^ n } - e ^ { - n \tau _ n } = \frac { \tau _ n } { ( 1 + \tau _ n ) ^ { n + 1 } } \le \frac { \tau _ n } { ( n + 1 ) \tau _ n } \le \frac { 1 } { n + 1 } . \end{align*}


\begin{align*} \nu \cdot \nabla _ \Gamma \eta = 0 , P \nabla _ \Gamma \eta = \nabla _ \Gamma \eta \quad \Gamma \end{align*}


\begin{align*} \Delta \log ( 1 - K ) = 4 K + 2 K _ 1 ^ { \ast } . \end{align*}


\begin{align*} K _ { \mathbb { E } ^ p _ q } ( x , y ) = \sum _ { k \in \N } \mathrm { e } ^ { - q k ^ p } h _ k ( x ) h _ k ( y ) . \end{align*}


\begin{align*} p _ a ( C ) \leq \binom { m } { 2 } s + m \epsilon . \end{align*}


\begin{gather*} \gamma _ { \mathfrak n } ( v , w , u ) = B \left ( d ' ( v , w ) , u \right ) = B \left ( B _ { \mathfrak a } \left ( \rho ( \cdot ) v , w \right ) , u \right ) = 0 . \end{gather*}


\begin{align*} & [ \eta \otimes ( r l , r v ) , g ] = \eta \otimes [ r v , g ] = \eta \otimes r [ v , g ] = r \eta \otimes [ v , g ] = [ r \eta \otimes ( l , v ) , g ] . \end{align*}


\begin{align*} A = \{ [ n \alpha ] \} _ { n = 0 } ^ \infty B = \{ [ n \beta ] \} _ { n = 0 } ^ \infty . \end{align*}


\begin{align*} \begin{aligned} H \equiv \Delta ( p - \bar { p } ) \in L _ { 1 } , i n ~ ~ \Omega _ { 0 } . \end{aligned} \end{align*}


\begin{align*} - u \times ( u \times \partial ^ 2 _ x u ) = \partial _ x ^ 2 u + u | \partial _ x u | ^ 2 \ , . \end{align*}


\begin{align*} a _ i x _ i - b _ { i + 1 } x _ { i + 1 } = h _ i . \end{align*}


\begin{align*} \mathcal { K } _ { n , m } [ 1 , z _ i ^ a \overline { z } _ i ^ b ] = \delta _ { a b } \frac { a ! } { \pi ^ a } . \end{align*}


\begin{align*} \begin{aligned} \mathcal { G } ( T ) = & \{ ( u , T ( u ) ) : u \in I \} \\ \subseteq & \{ ( u , T ( u ) + F ( u ) - F ( u ) ) : u \in I \} \\ \subseteq & \{ ( u , T ( u ) + F ( u ) ) : u \in I \} + \{ ( 0 , - F ( u ) ) : u \in I \} \\ = & \mathcal { G } ( F + T ) ) + \{ ( 0 , - F ( u ) ) : u \in I \} . \end{aligned} \end{align*}


\begin{align*} \Upsilon _ { \mu , s } Y ^ { \mu , t } ( b , z ) = Y ^ { \mu + 2 s , t + s } ( b , z ) \Upsilon _ { \mu , s } \end{align*}


\begin{align*} c _ 1 ( L _ { j , j } ) = - \omega , \end{align*}


\begin{align*} G \partial _ t p = \sum _ { i = 1 } ^ { \ell } c _ i \big ( \frac { d } { d t } \bar { G } _ i ( p , n ) - \partial _ n \bar { G } _ i ( p , n ) \partial _ t n \big ) , \end{align*}


\begin{align*} | \det M | \le \Pi _ { i = 1 } ^ n | M e _ i | \le \| M \| ^ n . \end{align*}


\begin{align*} X _ k ( t ) - X _ k ( \xi ) = \int _ { \xi } ^ { t } { \rm d } [ A ( s ) ] X _ k ( s ) , \end{align*}


\begin{align*} \underset { \epsilon \rightarrow 0 } { \lim } \left \vert \int _ { m } ^ { M } d \breve { F } _ { \epsilon } ( y ) - \int _ { m } ^ { M } y d F _ { \epsilon } ( y ) \right \vert = 0 . \end{align*}


\begin{align*} B _ m ( \theta _ 1 , x + i y ) = \sum _ { 0 \leq j \leq m - 1 } \left \{ \log \left | 1 + e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 } { m } } + \frac { x + i y } { m } \right | - \int _ { - 1 / 2 } ^ { 1 / 2 } \log \left | 1 + e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 + \phi } { m } } + \frac { x + i y } { m } \right | \mathrm { d } \phi \right \} \end{align*}


\begin{align*} \int _ { \Omega } f ( x ) | \nabla u | ^ { p ( x ) - 2 } \nabla u \nabla \phi d x = \gamma \int _ { \Omega } h ( x ) \phi d x . \end{align*}


\begin{align*} S ^ 1 _ { \gamma _ t } < \frac { \psi ( t ) \eta _ t } { 1 + \delta } : = \Delta _ t \log _ 2 t \ , . \end{align*}


\begin{align*} I = \mathfrak { s } \int _ 0 ^ t \sum _ { k \in \mathbb { N } } | z ( \partial _ x \Phi e _ k ) | ^ 2 _ { L _ x ^ 2 } \d s , \end{align*}


\begin{align*} \Theta ^ \mathrm { p a i r } [ n + 1 ] ( x , y ) : = \frac { 1 } { N } \upsilon _ N ( x - y ) \Big ( \frac { k _ { n + 1 } } { \delta - k _ { n + 1 } \circ \overline { k _ { n + 1 } } } \Big ) ( x , y ) . \end{align*}


\begin{align*} M _ A : \left ( \R ^ d \right ) ^ { | I _ A | } \to \left ( \R ^ d \right ) ^ { 2 n } [ M _ A ( u ) ] _ j = \begin{cases} u _ j & : j \in I _ A , \\ - \sum \limits _ { l \in a ( j ) \setminus \{ j \} } u _ l & : j \in J _ A . \end{cases} \end{align*}


\begin{align*} \begin{aligned} T _ { \overline { \Omega } } ( [ z , \lambda ] ) = T _ { \overline { \Omega } } ( [ \lambda , z ] ) ^ { - 1 } & = 1 + \int _ \lambda ^ z \overline { \Omega } + \int _ \lambda ^ z \overline { \Omega } \ , \overline { \Omega } + \cdots \cr & = 1 + \sum _ { m \ge 1 } \frac { ( \log z / \lambda ) ^ m } { m ! } ( - L ) ^ m \cr & = ( z / \lambda ) ^ { - L } . \end{aligned} \end{align*}


\begin{align*} F ^ { \ , \prime } ( x ) = \psi ^ { \ , \prime } ( f ( \psi ( x ) ) \circ f ^ { \ , \prime } ( \psi ( x ) ) \circ \psi ^ { \ , \prime } ( x ) \ , . \end{align*}


\begin{align*} \lim _ { { \rm R e } z \to 0 ^ + } ( 2 { \rm R e } z ) ^ 2 | S f ( z ) | = 0 . \end{align*}


\begin{gather*} \theta ' = \theta + d ^ 1 \tau \end{gather*}


\begin{align*} \pi ( p ) = k ( p ) \left [ \frac { \partial \psi } { \partial y } \frac { \partial } { \partial x } \wedge \frac { \partial } { \partial z } + \frac { \partial \psi } { \partial z } \frac { \partial } { \partial x } \wedge \frac { \partial } { \partial y } - \frac { \partial \psi } { \partial x } \frac { \partial } { \partial y } \wedge \frac { \partial } { \partial z } \right ] . \end{align*}


\begin{align*} \sum _ { k \ge 1 } c _ k r ^ k \cos ( \xi k ) \sim A _ 0 ( r ) - R _ 0 + R _ 1 , R _ i = \sum _ { k > \chi ^ { - 1 - \delta / 2 } } c _ k r ^ k \cos ( \xi k ) ^ i , \ , i = 0 , 1 . \end{align*}


\begin{align*} \mathring { R } _ { o s c } ^ B = \mathring { R } _ { o s c . 1 } ^ B + \mathring { R } _ { o s c . 2 } ^ B + \mathring { R } _ { o s c . 3 } ^ B + \mathring { R } _ { o s c . 4 } ^ B , \end{align*}


\begin{align*} 2 \nabla G \cdot \nabla p = \sum _ { i = 1 } ^ { \ell } 2 \nabla c _ i \cdot \nabla p G _ i ( p , n ) + 2 c _ i \partial _ p G _ i ( p , n ) | \nabla p | ^ 2 + 2 c _ i \partial _ n G _ i ( p , n ) \nabla p \cdot \nabla n \end{align*}


\begin{align*} \partial _ t f _ \epsilon = \partial _ x ^ 2 f _ \epsilon + \sqrt { \epsilon } F ( \partial _ x f _ \epsilon ) + \xi , \end{align*}


\begin{align*} \left ( \mathcal { R } _ { S } ( \mathbf { x } ) \right ) _ k = x _ k \left ( 1 + f _ k ( \mathbf { x } ) - \sum \limits _ { i = 1 } ^ m x _ i f _ i ( \mathbf { x } ) \right ) , \forall \ k \in \mathbf { I } _ m . \end{align*}


\begin{align*} \mathring { R } _ { o s c } ^ u = \mathring { R } _ { o s c . 1 } ^ u + \mathring { R } _ { o s c . 2 } ^ u + \mathring { R } _ { o s c . 3 } ^ u , \end{align*}


\begin{align*} \nu _ \varepsilon ( x , t ) = \frac { ( - 1 ) ^ { i + 1 } } { \sqrt { 1 + \varepsilon ^ 2 | \bar { \tau } _ \varepsilon ^ i ( x , t ) | ^ 2 } } \{ \bar { \nu } ( x , t ) - \varepsilon \bar { \tau } _ \varepsilon ^ i ( x , t ) \} . \end{align*}


\begin{align*} - \widetilde { B _ s } ^ { - T } = \left ( \begin{smallmatrix} - 1 & - 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & - 1 \end{smallmatrix} \right ) - \widetilde { B _ { s ' } } ^ { - T } = \left ( \begin{smallmatrix} - 1 & 1 & - 1 \\ - 1 & 0 & 0 \\ - 1 & 0 & - 1 \end{smallmatrix} \right ) . \end{align*}


\begin{align*} \limsup _ { x \rightarrow 1 ^ { - } } \frac { H _ X ( x ) } { H _ Y ( x ) } = \infty \end{align*}


\begin{align*} \tilde { p } ^ { \epsilon , \Delta t } ( t ) = e ^ { - \frac { t - t _ n } { \epsilon ^ 2 } } \tilde { p } ^ { \epsilon , \Delta t } ( t _ n ) + \epsilon ( 1 - e ^ { - \frac { t - t _ n } { \epsilon ^ 2 } } ) f ( \tilde { q } ^ { \epsilon , \Delta t } ( t _ n ) ) + \frac { 1 } { \epsilon } \sigma ( \tilde { q } ^ { \epsilon , \Delta t } ( t _ n ) ) \int _ { t _ n } ^ { t } e ^ { - \frac { t - s } { \epsilon ^ 2 } } d \beta ( s ) . \end{align*}


\begin{align*} | H | = \prod _ { i = 1 } ^ r \ell ^ { m _ i } . \end{align*}


\begin{align*} \varepsilon \norm { \xi _ \varepsilon } { H ^ 2 } < \varepsilon C \rightarrow 0 ~ ~ ~ ~ \varepsilon \rightarrow 0 ^ + . \end{align*}


\begin{align*} | M _ k | _ * - \epsilon & < \left | M _ k \cap \cup _ { i = 1 } ^ N B _ i \right | _ * , \\ \sum _ { i = 1 } ^ N | B _ i | & < ( 1 + \epsilon ) | M _ k | _ * . \end{align*}


\begin{align*} \Phi = \varphi _ 1 ( x _ 1 , \dots , x _ k ) \land y \leq z \ , \& \dots \& \ , \varphi _ n ( x _ 1 , \dots , x _ k ) \land y \leq z \Longrightarrow y \leq z \end{align*}


\begin{align*} \breve { a } _ 1 ( k ) \breve { a } _ 2 ( k ) = 1 - b ^ 2 ( k ) , k \in \mathbb { R } . \end{align*}


\begin{align*} ( z - q _ i ^ { c _ { i j } } w ) \psi ^ \epsilon _ i ( z ) e _ j ( w ) = ( q _ i ^ { c _ { i j } } z - w ) e _ j ( w ) \psi ^ \epsilon _ i ( z ) \ , , \end{align*}


\begin{align*} G = a _ { 1 } g _ { \theta _ { 1 } } \oplus \cdots \oplus a _ { m } g _ { \theta _ { m } } , \end{align*}


\begin{align*} ( - 1 ) ^ { \mathrm { 1 } _ { i \in \mathcal { O } } } \frac { T ^ { \mathrm { 1 } _ { i \in \mathcal { B } } } } { n } \sum _ { w ^ n = ( - 1 ) ^ { \theta _ 2 } } \frac { w ^ { \mathrm { 1 } _ { i \in \mathcal { B } } + h _ i - h _ j } e ^ { - ( T w + \theta _ 1 \pi i + \lambda ) [ t _ j - t _ i ] _ i } } { 1 - ( - 1 ) ^ { \theta _ 1 } e ^ { - ( T w + \lambda ) } } = \mathrm { 1 } _ { x _ i = x _ j } - K _ n ^ { \lambda , \theta , T } ( x _ i , x _ j ) , \end{align*}


\begin{align*} \frac { \omega ( \eta _ j ) } { \omega ( k _ j ) } \ = \ \frac { \omega ( s _ j k _ j ) } { \omega ( k _ j ) } \ \ge \ C _ \alpha ( s _ j ) ^ \alpha \ \rightarrow \ \infty , \end{align*}


\begin{align*} ( \Delta - \lambda _ w ) \wedge ^ a E _ w ( z ) & = ( \Delta - \lambda _ w ) \left ( H ( a - y ) \cdot ( y ^ w + c _ w y ^ { 1 - w } ) \right ) \\ & = ( y ^ 2 \frac { \partial ^ 2 } { \partial y ^ 2 } - w ( w - 1 ) ) \left ( H ( a - y ) \cdot ( y ^ w + c _ w y ^ { 1 - w } ) \right ) \\ & = y ^ 2 \left ( \delta ' _ a \cdot ( y ^ w + c _ w y ^ { 1 - w } ) - 2 \delta _ a \cdot ( w y ^ { w - 1 } + ( 1 - w ) c _ w y ^ { - w } ) \right ) . \end{align*}


\begin{align*} 2 m q \cdot d _ { r , s } ( m , i - 2 q ) = m q \cdot d _ { r , s } ( m , i ) , \end{align*}


\begin{align*} \| X \| _ { \psi _ 2 } = \inf \left \{ t > 0 : \ ; \mathbb { E } \exp \left ( \frac { X ^ 2 } { t ^ 2 } \right ) \leq 2 \right \} . \end{align*}


\begin{align*} B ( { \bf { v } } ) : = S ( { \bf { v } } ) B _ 0 ( { \bf { v } } ) = \left ( \begin{array} { c c c } 2 \theta ^ 2 q & 0 & 0 \\ 0 & 2 \theta q & 0 \\ 0 & 0 & \theta ^ 2 \end{array} \right ) \end{align*}


\begin{align*} M ( s , g ) = \sum _ { n \leq y } \frac { \mu ( n ) \ , g ( \frac { \log y / n } { \log y } ) } { n ^ s } , \end{align*}


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


\begin{align*} \eta ( p \cdot g ) & = | \det p g | \cdot h ( e _ r \cdot p g ) ^ { - r } = | \det p g | \cdot h ( d e _ r \cdot g ) ^ { - r } \\ & = | \det p | \cdot | \det g | \cdot | d | ^ { - r } \cdot h ( e _ r \cdot g ) ^ { - r } = 1 \cdot | \det g | \cdot h ( e _ r \cdot g ) ^ { - r } = \eta ( g ) . \end{align*}


\begin{align*} N + M = \binom { u + v - 1 } { u } + N ' + M = \binom { u + v - 1 } { u } + \binom { u + v - 1 } { u - 1 } = \binom { u + v } { u } . \end{align*}


\begin{align*} \Gamma : = b ^ { - 1 } ( x ) = \{ x \} \times _ { \kappa } ( \P ^ n _ { \kappa } ) ^ * \subset X \times _ { \kappa } ( \P ^ n _ { \kappa } ) ^ * . \end{align*}


\begin{align*} v _ \lambda ( x ) : = u ( \lambda x ) , x \in \R ^ N . \end{align*}


\begin{align*} \widehat { \psi } _ { 0 , q , \# } ( u _ 0 ) = \int _ { \Lambda _ L } d x \psi _ { 0 , q } ( x ) e ^ { - 2 \pi i x \cdot u _ 0 } = \int _ { \Lambda _ L } d x \psi _ { 0 , 0 } ( x ) e ^ { - 2 \pi i ( x \cdot u _ 0 - x \cdot q ) } = : \widehat { \psi } _ { 0 , 0 , \# } ( u _ 0 - q ) , \end{align*}


\begin{align*} H _ T & : = \{ ( \textbf { \textit { x } } , \textbf { \textit { y } } ) \in \R ^ { p + n } \mid y _ i = 0 \ \forall i \in T \} \mbox { a n d } \\ \bar H _ T & : = \{ ( \textbf { \textit { x } } , \textbf { \textit { y } } ) \in \R ^ { p + n } \mid y _ j = 0 \ \forall j \in \delta ( T ) \} . \end{align*}


\begin{align*} Y _ { m _ 1 , m _ 2 } ( A ) : = \sum _ { \sigma \in \mathrm { S y m } ( \mathbb { T } _ { m _ 1 , m _ 2 } ) } \prod _ { x \in \mathbb { T } _ { m _ 1 , m _ 2 } } \left ( \alpha _ x \mathbf { 1 } _ { \sigma ( x ) = x } + \beta _ x \mathbf { 1 } _ { \sigma ( x ) = x + \mathbf { e } ^ 1 } + \gamma _ x \mathbf { 1 } _ { \sigma ( x ) = x + \mathbf { e } ^ 2 } \right ) . \end{align*}


\begin{align*} y ( t ) = \varphi ( 0 ) + \int _ s ^ t L ( \tau ) y _ \tau + G ( \tau , y _ \tau ) d \tau , t \geq s , \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \Delta ^ { 2 } u - \Delta u + u = ( R _ { \alpha } \ast | u | ^ { p } ) | u | ^ { p - 2 } u , & ~ \Omega , \\ u = 0 , & ~ \partial \Omega . \end{array} \right . \end{align*}


\begin{align*} W ( z ) : = \int _ 0 ^ \infty e ^ { - k z } w ( k ) \ , d k , \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { N \to \infty } \frac { 1 } { N } \log \left ( \int _ { \overline { \mathbf { P } } _ { N } \in B ( \overline { \mathbf { P } } , \delta ) } \Pi _ { i = 1 } ^ { N } \mu _ { \theta } ( x _ { i } ) \ , d x _ { i } \right ) \leq - \overline { { \rm E n t } } [ \overline { \mathbf { P } } | \overline { \mathbf { \Pi } } ^ { \mu _ { \theta } } ] . \end{align*}


\begin{align*} \partial _ t d ( x , t ) = \partial _ t d ( \pi ( x , t ) , t ) = - V _ \Gamma ( \pi ( x , t ) , t ) = - \overline { V } _ \Gamma ( x , t ) . \end{align*}


\begin{align*} L _ { \widetilde j } = \prod _ { k = 1 } ^ j D _ { \widetilde k } . \end{align*}


\begin{align*} \phi \left ( x ; z \right ) \partial _ { x } P _ { n + 1 } = \left ( n + 1 \right ) P _ { n + 2 } + \lambda _ { n } P _ { n } + \tau _ { n } P _ { n - 2 } , \end{align*}


\begin{align*} L _ 0 ( \partial _ \xi w + \partial _ \tau w ) = & [ - ( \partial _ \xi w + \partial _ \tau w ) ] ( 2 w \partial _ { \eta } ^ 2 w ) \\ \geq & 4 \delta [ - ( \partial _ \xi w + \partial _ \tau w ) ] \\ \geq & - 2 4 \delta ^ 2 w a t z _ { m a x } , \end{align*}


\begin{align*} \overline { L } ^ { n - 1 } \cdot \pi _ i ^ { * } \overline { L } _ i = ( \overline { L } ^ { n - 1 } | \mathrm { d i v } ( \pi _ i ^ * 1 _ a ) ) - ( n - 1 ) ! \sum _ { k = 1 } ^ n \sum _ { v \in M ( K ) } \int _ { H ^ { \mathrm { a n } } _ { \overline { K } _ v } } \log \| \pi _ i ^ * 1 _ a \| _ { \pi _ i ^ { * } \overline { L } _ i , v } \prod _ { j \neq k } c _ 1 ( \pi _ j ^ * \overline { L } _ { j , v } ) . \end{align*}


\begin{align*} t \mapsto \Phi _ f ( t ) = \int _ { 0 } ^ t S ( t - s ) f ( s ) \d s t \in [ 0 , T ) , \end{align*}


\begin{align*} \overline { Z } _ { q } ( x ) : = \int _ 0 ^ x Z _ { q } ( y ) d y , x \in \mathbb { R } . \end{align*}


\begin{align*} E _ M ( e _ N ) = [ M : N ] ^ { - 1 } 1 . \end{align*}


\begin{align*} 2 \mathrm { R e } \langle u _ { \alpha } \otimes 2 \alpha ^ { 3 / 2 } \tilde { \Phi } _ * ^ 1 \ , | \ , H _ y \Phi ^ y _ { \# } \rangle = O ( \alpha ^ 4 ) , \end{align*}


\begin{align*} \underset { l } { \lim \sup } | x ^ * _ l ( x _ { l _ 1 } ) | & = \underset { l } { \lim \sup } | x ^ * _ l ( Q z _ 1 ) | \leqslant \underset { l } { \lim \sup } \| x ^ * _ l - P ^ { \textsf { E } ^ * } _ { I _ l } x ^ * _ l \| + \underset { l } { \lim \sup } | P ^ { \textsf { E } ^ * } _ { I _ l } x ^ * _ l ( z _ 1 ) | \leqslant \delta . \end{align*}


\begin{align*} B _ { q } ( v ' , u ' ) : = \textbf { M } _ { x } ( v , u ) . \end{align*}


\begin{align*} \mathcal { E } _ { V } ( \mu ) = \mathcal { E } ( \mu ) + \int _ { M } V d \mu . \end{align*}


\begin{align*} Q _ n f = \sum _ { k = 0 } ^ n \langle f , u _ k \rangle _ n \ , S _ n ^ { - 1 } u _ k \ , . \end{align*}


\begin{align*} \Omega ^ { } = \{ m \in \Omega : S ( m ) = m \} \end{align*}


\begin{align*} \mathcal { H } _ { N } ( X _ { N } ) & = N ^ { 2 } \left ( \mathcal { E } ^ { \neq } ( { \rm e m p } _ { N } ) + \int _ { M } V \ , d { \rm e m p } _ { N } \right ) \\ & = N ^ { 2 } \left ( \mathcal { E } ( \mu _ { \theta } ) + 2 \mathcal { G } ( \mu _ { \theta } , { \rm e m p } _ { N } - \mu _ { \theta } ) + \mathcal { E } ^ { \neq } ( { \rm e m p } _ { N } - \mu _ { \theta } ) + \int _ { M } V \ , d { \rm e m p } _ { N } \right ) \end{align*}


\begin{align*} g = g _ M + \textbf { H } \left ( d \varphi \otimes d \theta + d \theta \otimes d \varphi \right ) + \frac { 1 } { 2 } ( 1 + \textbf { H } ^ 2 ) \ , d \theta ^ 2 + 2 \ , d \varphi ^ 2 . \end{align*}


\begin{align*} \begin{tabular} [ c ] { l } $ x ^ { n } = P _ { n } ( x ; z ) + c _ { n } \left ( z \right ) P _ { n - 2 } ( x ; z ) $ \\ $ - \left [ d _ { n } \left ( z \right ) - c _ { n } \left ( z \right ) c _ { n - 2 } \left ( z \right ) \right ] P _ { n - 4 } ( x ; z ) + O \left ( x ^ { n - 6 } \right ) . $ \end{tabular} \end{align*}


\begin{align*} \begin{cases} \partial _ t u + b \cdot D _ x u = 0 & \\ u ( 0 , \cdot ) = \bar u \ , . \end{cases} \end{align*}


\begin{align*} \int _ { \Lambda _ L ^ * } f ( p ) d p : = \frac { 1 } { | \Lambda _ L | } \sum _ { p \in \Lambda _ L ^ * } f ( p ) . \end{align*}


\begin{align*} \partial _ y ^ 2 u | _ { y = 0 } = - \frac { 1 } { h } \partial _ x \tilde { h } | _ { y = 0 } . \end{align*}


\begin{align*} \sigma _ { \widetilde { i + 1 } } & = L _ { \widetilde { i + 1 } } \pi \stackrel { \eqref { p i L x } } { = } L _ { \widetilde 1 } \pi L _ { \widetilde i } = \sigma _ { \widetilde 1 } \sigma _ { \widetilde i } \pi ^ { - 1 } = \sigma _ { \widetilde 1 } \sigma ^ i _ { \widetilde 1 } \sigma _ { \widetilde 0 } ^ { 1 - i } \sigma _ { \widetilde 0 } ^ { - 1 } = \sigma ^ { i + 1 } _ { \widetilde 1 } \sigma _ { \widetilde 0 } ^ { - i } , \end{align*}


\begin{align*} \langle \Lambda _ 1 , \Lambda _ 2 \rangle = ( | ( \Lambda _ 1 ) ^ * | + | Z ^ * | ) ( | ( \Lambda _ 2 ) ^ * | + | Z ^ * | ) + | ( \Lambda _ 1 ) ^ * \cap ( \Lambda _ 2 ) ^ * \cap Z _ * | + | ( \Lambda _ 1 ) _ * \cap ( \Lambda _ 2 ) _ * \cap Z ^ * | \pmod 2 . \end{align*}


\begin{align*} \frac { \d L _ h } { \d t } & = - 2 \sum \limits _ { j = 1 } ^ N \Big ( \frac { 4 \sin ^ 2 \big ( \frac { \alpha _ j } { 2 } \big ) } { q _ j + q _ { j + 1 } } - \frac { \pi } { L _ h } \sin \alpha _ j \Big ) \le \frac { - 2 } { L _ h } \Big ( 2 \big ( \sum _ { j = 1 } ^ N \sin \big ( \frac { \alpha _ j } { 2 } \big ) \big ) ^ { 2 } - \pi \sum _ { j = 1 } ^ N \sin \alpha _ j \Big ) \le 0 , \end{align*}


\begin{align*} \begin{cases} ( \partial _ t ^ 2 + P ) u = f \ , ( 0 , \infty ) \times M , \\ f | _ { C ( T , p ) } = 0 , \\ u | _ { B ( p , T ) \times \{ t = 0 \} } = \partial _ t u | _ { B ( p , T ) \times \{ t = 0 \} } = 0 . \end{cases} \end{align*}


\begin{align*} \tau _ { n } \left ( z \right ) = 2 \gamma _ { n + 1 } \gamma _ { n } \gamma _ { n - 1 } . \end{align*}


\begin{align*} \partial _ t \rho _ i - \nabla \cdot ( \rho _ i \nabla p ) = \rho _ i G _ i ( p , n ) \end{align*}


\begin{align*} \gamma ( S e f ) & = \rho ( e f , e f , e ) \gamma ( S e ) & S e f \subseteq S e \\ & = \rho ( e f , e f , e ) \rho ( e , u , e ) \\ & = \rho ( e f , e f u , e ) \\ & = \rho ( e f , f u , e ) & u \in e S e e f = f e . \end{align*}


\begin{align*} d ^ \star ( j \circ u ) ( t ) & = A _ 0 ^ { \odot \star } T _ 0 ^ { \odot \star } ( t - s ) j \varphi + A _ 0 ^ { \odot \star } \int _ s ^ t T _ 0 ^ { \odot \star } ( t - \tau ) [ B ( \tau ) u ( \tau ) + f ( \tau ) ] d \tau + B ( t ) u ( t ) + f ( t ) \\ & = A _ 0 ^ { \odot \star } j u ( t ) + B ( t ) u ( t ) + f ( t ) . \end{align*}


\begin{align*} \C \setminus [ 0 , + \infty ) \ ; \ni \zeta \longrightarrow R _ H ( \zeta ) = ( H - \zeta ) ^ { - 1 } \in { \rm H o m } ( C ^ \infty _ c ( \R ^ n ) , { C ^ \infty _ c } ( \R ^ n ) ^ * ) \ , , \end{align*}


\begin{align*} \log ( ( k n + k ) ! ) & = \left ( k n + k \right ) \log ( k n + k ) - ( k n + k ) + O ( \log n ) = k n \log n + k n \log \frac { k } { e } + O ( \log n ) . \end{align*}


\begin{align*} u ( y ) = \int _ 1 ^ { \infty } a ^ 2 \cdot u _ a ( y ) f ( a ) d a , \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { N } \log \left ( \mathbf { \Pi } _ { N } ( | C | = N ) \right ) = 0 , \end{align*}


\begin{align*} \gamma _ { \mathfrak n } ( x , y , v ) & = B \left ( [ x , y ] _ { \theta , \gamma } , v \right ) = B _ { \mathfrak a } ( \theta ( x , y ) , v ) ; \\ \gamma _ { \mathfrak n } ( x , v , y ) & = B \left ( [ x , v ] _ { \theta , \gamma } , y \right ) = B _ { \mathfrak a } ( \theta ( x , y ) , v ) . \end{align*}


\begin{align*} \sum _ { \gamma _ 1 , . . . , \gamma _ n = 1 } ^ M \tilde { \chi } _ A ( \gamma _ 1 , . . . , \gamma _ n ) & = 1 _ { | A | \leq M } \frac { M ! } { ( M - | A | ) ! } , \end{align*}


\begin{align*} \mathcal { F } ( \Delta ) = \mathcal { G } \cup \mathcal { H } . \end{align*}


\begin{align*} e ( \overline V ) & = e ( \hat V ) + 2 s _ 1 \\ & = e ( V _ { t } ) + 2 s + 2 s _ 1 . \end{align*}


\begin{align*} P \ , ( A - \lambda B ) \ , Q = \left [ \begin{array} { c c } R ( \lambda ) & 0 \\ 0 & S ( \lambda ) \end{array} \right ] , P U = \left [ \begin{array} { c } 0 \\ U _ 2 \end{array} \right ] , Q ^ * V = \left [ \begin{array} { c } V _ 1 \\ V _ 2 \end{array} \right ] , \end{align*}


\begin{align*} { \downarrow } ^ { \mathbb { X } } ( p ^ { - 1 } [ { \downarrow } ^ { \mathbb { Y } } U ] ) = { \downarrow } ^ { \mathbb { X } } ( X \smallsetminus { \downarrow } ^ { \mathbb { X } } ( p ^ { - 1 } [ Y \smallsetminus U ] ) ) . \end{align*}


\begin{align*} \left \{ D X _ t = 0 \right \} \Rightarrow \left \{ \langle D X _ t , h \rangle _ { \mathcal { H } } = 0 \right \} = \left \{ n I _ { n - 1 } ( \langle f _ t , h \rangle _ { \mathcal { H } } ) = 0 \right \} . \end{align*}


\begin{align*} m _ n ^ { \epsilon , \Delta t } = \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ n } m _ 0 ^ \epsilon + \frac { } { \epsilon } \sum _ { \ell = 0 } ^ { n - 1 } \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ { n - \ell } } \Delta \beta _ \ell , \end{align*}


\begin{gather*} \lim _ { n \to \infty } \sqrt { n } \ , a _ n = \lim _ { n \to \infty } \sqrt { n } \int _ { - \sqrt { 1 2 / n } } ^ 0 F _ n ( x ) \ , d x \\ = \lim _ { n \to \infty } \sqrt { n } \ , \sqrt { 1 2 / n } \ \phi ( 0 ) = \sqrt { 1 2 } / \sqrt { 2 \pi } = \sqrt { 6 / \pi } . \end{gather*}


\begin{align*} \left \lbrace \begin{aligned} d q ^ \epsilon ( t ) & = \frac { p ^ \epsilon ( t ) } { \epsilon } d t \\ d p ^ \epsilon ( t ) & = - \frac { p ^ \epsilon ( t ) } { \epsilon ^ 2 } d t + \frac { f ( q ^ \epsilon ( t ) ) } { \epsilon } d t + \frac { \sigma ( q ^ \epsilon ( t ) ) } { \epsilon } d \beta ( t ) , \end{aligned} \right . \end{align*}


\begin{align*} \sum _ { \substack { c _ { \nu \mu } \\ ( \nu , \mu ) \not = ( i , j ) } } \Big | \sum _ { c _ { i j } } ( . . . ) \Big | \ll \big ( \max _ { 1 \leq j \leq n } | \psi _ j | ^ { - 1 / 2 } _ p \big ) p ^ { r _ 1 + \ldots + r _ n - m _ { i j } / 2 + O ( \mu _ { i j } ) } . \end{align*}


\begin{align*} R _ { i j } ( x , t ) = \delta _ { i j } + \varepsilon r W _ { i j } ( y , t ) + O ( \varepsilon ^ 2 ) , \partial _ i R _ { i j } ( x , t ) = ( \nu _ i W _ { i j } ) ( y , t ) + O ( \varepsilon ) \end{align*}


\begin{align*} L ^ { * } = S _ { n m } L ^ { \perp } , \mbox { w h e r e } S _ { n m } = \left ( \begin{array} { l c } 0 & - I _ { m } \\ I _ { n } & 0 \end{array} \right ) , \end{align*}


\begin{align*} O _ U = \{ t \in G \ , : \ , ( \alpha ( t , x ) , x ) \in U \mbox { f o r a l l } x \in X \} = \bigcap _ { x \in X } O _ { x , U } \end{align*}


\begin{align*} Z = \overline { \mathcal { O } _ M } : & \ , \dim Z = \dim ( \textbf { d } ) - \dim ( M ) < \dim ( \textbf { d } ) \\ Z = \displaystyle { \overline { \bigcup _ { M \in C } \mathcal { O } _ M } } : & \ , \dim Z = \dim ( \textbf { d } ) - \dim ( M ) + 1 \leq \dim ( \textbf { d } ) \end{align*}


\begin{align*} ( \pi \circ \alpha ) ^ * \big ( D \big ) + \sum _ { i = 1 } ^ k a _ i E _ i \sim _ { \mathbb { Q } } ( \lambda n - 1 ) \beta ^ * \big ( - K _ X \big ) + \sum _ { i = 1 } ^ m b _ i F _ i . \end{align*}


\begin{align*} S ^ { v - 1 } ( m ) = 2 \cdot 3 ^ { v - 1 } w - 1 \end{align*}


\begin{align*} \tilde { x } _ i = ( [ [ t _ i m ] ] , h _ i ) \tilde { y } _ i = ( [ [ t _ i m ] ] + \mathrm { 1 } _ { i \in \mathcal { O } } , [ h _ i + \mathbf { 1 } _ { i \in \mathcal { B } } ] ) . \end{align*}


\begin{align*} \psi ( \xi ) = \psi ( - \xi ) = \Re \psi ( \xi ) \ge 0 . \end{align*}


\begin{align*} \hat k ( t ) : = \{ \Vert \mu ( z ) \Vert _ { \infty } : \ , 1 < | z | \leq 1 + t \} \leq \hat \sigma ( t ) / 2 \end{align*}


\begin{align*} m _ { \kappa , \delta } ( x ) : = \begin{cases} \phi _ \kappa ( x ) & x \in B _ \delta ( a _ 0 ) , \\ p \left ( \phi ' _ \kappa + r _ 0 \kappa R ' _ 0 u _ \delta + v _ \kappa \right ) & x \in \R ^ 2 \setminus B _ \delta ( a _ 0 ) , \end{cases} \end{align*}


\begin{align*} & a _ n \sin \omega _ n + b _ n \cos \omega _ n \\ = & \sqrt { a _ n ^ 2 + b _ n ^ 2 } \left ( \cos \xi _ n \sin \omega _ n + \sin \xi _ n \cos \omega _ n \right ) \\ = & \sqrt { a _ n ^ 2 + b _ n ^ 2 } \sin ( \omega _ n + \xi _ n ) , \end{align*}


\begin{align*} \nabla _ \Gamma \eta ( y ) = P ( y ) \nabla \tilde { \eta } ( y ) , \underline { D } _ i \eta ( y ) = \sum _ { i = 1 } ^ n P _ { i j } ( y ) \partial _ j \tilde { \eta } ( y ) , y \in \Gamma , \ , i = 1 , \dots , n \end{align*}


\begin{align*} \tau _ N \left ( Q _ 1 ( a _ { i _ 1 } ^ N ) \dots Q _ p ( a _ { i _ p } ^ N ) \right ) = \tau _ { N } \left ( u _ 1 ^ N u _ 2 ^ N \dots u _ { 2 p - 1 } ^ N u _ { 2 p } ^ N \right ) + \mathcal { O } \left ( \frac { 1 } { N } \max \limits _ { i \neq j } | y _ i ^ N - y _ j ^ N | ^ 2 \right ) . \end{align*}


\begin{align*} N ( T ) : = \sum _ { 0 < \gamma \le T } 1 = \sum _ { 0 < \gamma _ d \le T } m ( \gamma _ d ) = \frac { T } { 2 \pi } \log \frac { T } { 2 \pi } - \frac { T } { 2 \pi } + \frac { 7 } { 8 } + S ( T ) + O \Big ( \frac { 1 } { T } \Big ) \end{align*}


\begin{align*} \| \varphi \circ \psi - \varphi \circ \tilde \psi \| _ { \C ^ k } \leq & C \| \varphi \| _ { \C ^ { k + 1 } } \| D \psi \| _ { \C ^ { k } } ^ k \\ & \times \left ( \| \psi - \tilde \psi \| _ { \C ^ 0 } + \| D \tilde \psi \| _ { \C ^ { k } } ^ k \| D \psi - D \tilde \psi \| _ { \C ^ { k - 1 } } \right ) . \end{align*}


\begin{align*} X ^ \epsilon ( t ) = \varphi ( \zeta ^ \epsilon ( t ) , x _ 0 ^ \epsilon ) . \end{align*}


\begin{align*} K _ { \mu } ( r ) = \frac { 1 } { 2 } \left ( \frac { r } { 2 } \right ) ^ { \mu } \int _ 0 ^ { \infty } u ^ { - \mu - 1 } \exp \left ( - u - \frac { r ^ 2 } { 4 u } \right ) \d u , \mu > 0 , \ r > 0 , \end{align*}


\begin{align*} \overline { m } _ { \alpha , X _ 2 } = \max \{ ( 1 - f ( c _ { \alpha , X _ 2 } ) , F ( c _ { \alpha , X _ 2 } ) - 1 ) \} . \end{align*}


\begin{align*} \int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } e ^ { i \lambda x } \ , d x = \frac { \sin ( \lambda / 2 ) } { \lambda / 2 } . \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } M _ t ( \omega ) = M _ { \infty } ( \omega ) \mbox { f o r e v e r y } \omega \in \Omega _ M . \end{align*}


\begin{align*} & { \bf E } _ L \left ( \prod _ { j = 1 } ^ n \hat { V } _ { L , \omega } ( p _ j - p _ { j + 1 } ) \right ) = \sum _ { A \in \mathcal { A } _ n } \prod _ { a \in A } \left \{ m _ { | a | } \delta _ { * , L } \left ( \sum _ { l \in a } ( p _ l - p _ { l + 1 } ) \right ) \prod _ { l \in a } \hat { B } _ \# ( p _ l - p _ { l + 1 } ) \right \} . \end{align*}


\begin{align*} \left | K _ { \theta , \tilde { \theta } _ { ( z , \eta ) , ( z _ 0 , \eta _ 0 ) } , \Phi } ( ( y , \omega ) , ( z , \eta ) ) \right | = \sqrt { \frac { w ( \Phi ( \eta ) ) } { w ( \Phi ( \omega ) ) } } \cdot L _ { \Phi ( \eta ) } [ \theta , \tilde { \theta } _ { ( z , \eta ) , ( z _ 0 , \eta _ 0 ) } ] ( A ^ { T } ( \Phi ( \eta ) ) \langle z - y \rangle , \Phi ( \omega ) - \Phi ( \eta ) ) , \end{align*}


\begin{align*} \delta ( k , \xi ) = \exp \left \{ \frac { 1 } { 2 \pi i } \int _ { ( - \infty , - k _ 0 ) \cup ( k _ 0 , + \infty ) } \frac { { \log \left ( 1 + r _ 1 ( s ) r _ 2 ( s ) \right ) } } { s - k } d s \right \} . \end{align*}


\begin{align*} \sum _ { k \ge 1 } g ( k ) = \int _ 1 ^ \infty g ( t ) d t + \frac { g ( 1 ) } { 2 } + \int _ 1 ^ \infty g ' ( x ) P _ 1 ( x ) d x . \end{align*}


\begin{align*} \mathcal { B } _ { r , s , \alpha } ( x , t ) : = t ^ { r ( x ) } + \mu ( x ) ^ { \alpha ( x ) } t ^ { s ( x ) } ( x , t ) \in \overline { \Omega } \times [ 0 , \infty ) , \end{align*}


\begin{align*} U ( x , y , z ) = x ^ 2 + y ^ 2 . \end{align*}


\begin{align*} \Lambda _ i \Lambda _ { i + 1 } ^ * = 1 _ g \ ; \ ; \ ; \forall i \in \{ 0 , \ldots , n - 1 \} . \end{align*}


\begin{align*} B _ { \bf { A } _ \tau } = \left ( \begin{array} { c c } 0 _ { d \times d } & ( \tau - \frac 1 2 ) I _ { d \times d } \\ ( \tau - \frac 1 2 ) I _ { d \times d } & 0 _ { d \times d } \end{array} \right ) , \end{align*}


\begin{align*} a = D ^ 2 h ( x - T x ) - \frac 1 2 \ , \ ( \dfrac { \partial D h ( x - T x ) } { \partial x } + \ ( \dfrac { \partial D h ( x - T x ) } { \partial x } \ ) ^ t \ ) \end{align*}


\begin{align*} m ^ { \mathcal { M } \times \mathcal { N } } ( a ) = ( m ^ { \mathcal { M } } ( a ) , m ^ { \mathcal { N } } ( a ) ) \end{align*}


\begin{align*} M = ( d ^ \top \otimes I _ p ) \bar { \Gamma } = \Gamma \left ( \bar { D } \otimes I _ m \right ) , \end{align*}


\begin{align*} x \left ( \phi - 1 \right ) P _ { k } & = \left [ \allowbreak x ^ { 3 } - \left ( z ^ { 2 } + 1 \right ) x \right ] P _ { k } = P _ { k + 3 } + \gamma _ { k } \gamma _ { k - 1 } \gamma _ { k - 2 } P _ { k - 3 } \\ & + \left [ \gamma _ { k } + \gamma _ { k + 1 } + \gamma _ { k + 2 } - \left ( z ^ { 2 } + 1 \right ) \right ] P _ { k + 1 } \\ & + \gamma _ { k } \left [ \gamma _ { k } + \gamma _ { k + 1 } + \gamma _ { k - 1 } - \left ( z ^ { 2 } + 1 \right ) \right ] P _ { k - 1 } . \end{align*}


\begin{align*} H ( t , x ) = 1 + ( t ^ { - 1 / \alpha } | x | ) ^ { - \delta } . \end{align*}


\begin{align*} q > q _ 0 = ( N + 2 ) \frac { \gamma - 1 } { \gamma } , \end{align*}


\begin{align*} \lim _ { n \to \infty } n \mathbb { P } ( | X | > b _ n ) = 0 . \end{align*}


\begin{align*} T _ { \theta } = \left ( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - \frac { i e ^ { - i \theta } } { \sqrt { 2 } } & \frac { e ^ { i \theta } } { \sqrt { 2 } } \\ 0 & - \frac { e ^ { - i \theta } } { \sqrt { 2 } } & \frac { i e ^ { i \theta } } { \sqrt { 2 } } \\ \end{array} \right ) \end{align*}


\begin{align*} \beta _ { 0 } = \frac { \mu _ { 1 } } { \mu _ { 0 } } . \end{align*}


\begin{align*} \varphi ^ { o u t } = h ( 0 , \varphi ^ { i n } ) . \end{align*}


\begin{align*} F ( z , s ) : = \lim _ { n \to \infty } F _ n ( z , s ) = \sum _ { j \in \mathbb { Z } } \frac { e ^ { 2 \pi i j s } } { 2 \pi i j + z } \end{align*}


\begin{align*} a \otimes b = a \times b \end{align*}


\begin{align*} j ^ * \theta = ( j ^ * \circ ( \Delta - \lambda _ w ) \circ j ) u _ w = ( \tilde T ^ { \# } _ { \geq a } - \lambda _ w ) u _ w . \end{align*}


\begin{align*} + \epsilon ^ { 2 } \underset { \mathbb { R } ^ { 2 } } { \int } ( V \star ( \partial _ { t } \mathfrak { U } ) ) \mathfrak { U } ( t , x ) d x + \frac { \epsilon } { 2 N } \stackrel [ i = 1 ] { N } { \sum } \mathrm { t r a c e } ( u ^ { k } ( t , x _ { i } ) \lor x _ { i } ^ { k } R _ { N } ( t ) ) . \end{align*}


\begin{align*} \partial _ j \bar { \eta } ( x ) = \sum _ { k = 1 } ^ n R _ { j k } ( x ) \overline { \underline { D } _ k \eta } ( x ) , x \in \overline { N } . \end{align*}


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


\begin{align*} \tilde { g } _ { \theta } \circ \sigma = \Psi _ { \theta } ( \sigma ) \circ \tilde { g } _ { \theta } . \end{align*}


\begin{align*} \alpha ( t , x ) \oplus \alpha ( 0 , x ) = \alpha ( t , x ) \ , . \end{align*}


\begin{align*} \left \{ \int _ { 0 } ^ { 1 } g _ t d D X _ t = 0 , \ g _ t \neq 0 \right \} \Rightarrow \left \{ c ( \omega ) \cdot \norm { D X _ { a + \epsilon } - D X _ a } _ \mathcal { H } \leq C \norm { g _ t ( \omega ) } _ \tau \norm { D X _ r ( \omega ) } _ { \rho } \epsilon ^ { \tau + \rho } \right \} . \end{align*}


\begin{align*} \mathfrak X _ { T _ \varepsilon } : = \bigl \lbrace ( \xi , v ) | ( \xi , v ) \in \mathfrak Y _ { T _ \varepsilon } ~ ~ \eqref { l e - p r i o r i - e s t i m a t e } ~ \bigr \rbrace . \end{align*}


\begin{align*} \gamma _ m ( x ) \ : = \ \sup _ { \pi } \| x - G _ m ^ \pi ( x ) \| . \end{align*}


\begin{align*} \Delta ( C ^ i ) \subset \bigoplus _ { j + k = i } C ^ j \otimes C ^ k . \end{align*}


\begin{align*} \lambda ( P \oplus _ { [ m ] } P _ { \mathcal { G } } ) & = \max \left \{ \lambda _ { \mathcal { E } } ( x ) + \lambda ( \mathcal { G } ) \left ( \sum _ { i = 1 } ^ m x _ i ^ r \right ) : x = ( x _ 1 , \dots , x _ { m } ) \in \Delta _ { m - 1 } \right \} \\ & = 1 - ( 1 - \lambda ( \mathcal { G } ) ) \max _ { x \in \Delta _ { m - 1 } } \sum _ { i = 1 } ^ m x _ i ^ r \\ & = 1 - \frac { 1 - \lambda ( \mathcal { G } ) } { m ^ { r - 1 } } . \end{align*}


\begin{align*} \begin{pmatrix} h [ n + 1 ] & \Theta ^ \mathrm { p a i r } [ n + 1 ] \\ \overline { \Theta ^ \mathrm { p a i r } [ n + 1 ] } & h [ n + 1 ] \end{pmatrix} \circ \begin{pmatrix} \phi _ { n + 1 } \\ \overline { \phi _ { n + 1 } } \end{pmatrix} = \mu _ { n + 1 } \begin{pmatrix} \phi _ { n + 1 } \\ \overline { \phi _ { n + 1 } } \end{pmatrix} , \end{align*}


\begin{align*} F & = \vert \overline { y } \vert \phi \left ( x ^ 0 , \frac { y ^ 0 } { \vert \overline { y } \vert } \right ) , \\ F & = \vert \overline { y } \vert \phi \left ( \frac { y ^ 0 } { \vert \overline { y } \vert } , \vert \overline { x } \vert \right ) , \end{align*}


\begin{align*} x \to y = \max \{ u \ge y \colon u \perp _ y x \} . \end{align*}


\begin{align*} \lim _ { i \to \infty } \int _ { \R } \int _ { \R } \frac { | f ( x ) - f ( y ) | } { | x - y | } \rho _ i ( x , y ) \ , d \mathcal L ^ 1 ( x ) \ , d \mathcal L ^ 1 ( y ) = \Vert D f \Vert ( \R ) . \end{align*}


\begin{align*} \frac { 1 } { s _ 1 \cdots s _ { n } } \sum _ { j _ 1 = 0 } ^ { s _ 1 - 1 } \cdots \sum _ { j _ n = 0 } ^ { s _ n - 1 } f \left ( U _ 1 ^ { j _ 1 } \cdots U _ n ^ { j _ n } x \right ) \end{align*}


\begin{align*} G _ s ' : = \prod _ { [ a ] \neq [ 1 ] } G _ { s , [ a ] } , \end{align*}


\begin{align*} d ( ( 0 , 0 ) , \partial ^ + Q _ n ) = d \left ( 0 , \frac { \partial \Omega - \bar x _ n } { r _ n } \right ) + \frac { ( T - \bar t _ n ) ^ { 1 / 2 } } { r _ n } = \frac { d _ n } { r _ n } \to + \infty \end{align*}


\begin{align*} \Delta e _ i = - \lambda _ i e _ i , e _ i | _ { \partial D } = 0 , i \in \mathbb { N } . \end{align*}


\begin{align*} \begin{gathered} X \begin{pmatrix} G _ 0 & G _ 1 \end{pmatrix} \begin{pmatrix} G _ 0 ^ - \\ G _ 1 ^ - \end{pmatrix} = X \ ; , \\ \begin{pmatrix} H _ 0 ^ - & H _ 1 ^ - \end{pmatrix} \begin{pmatrix} H _ 0 \\ H _ 1 \end{pmatrix} Y = Y \ ; , \\ \end{gathered} \end{align*}


\begin{align*} Z = \overline { Z _ 1 \oplus \ldots \oplus Z _ m } \end{align*}


\begin{align*} \Delta _ g ^ s u = f \end{align*}


\begin{align*} y ^ 2 = - \nu \mathfrak P + \nu \alpha \beta \gamma - \nu \gamma ^ 2 + \beta ^ 2 , \end{align*}


\begin{align*} k = \psi \circ ( \delta + \overline \psi \circ \psi ) ^ { - 1 / 2 } = ( \delta + \psi \circ \overline { \psi } ) ^ { - 1 / 2 } \circ \psi , \end{align*}


\begin{align*} H ^ i _ { P } ( R ) & \cong H ^ i _ P ( W ) \otimes _ W R \\ & \cong H ^ i _ P ( W ) \otimes _ W \left ( W \otimes _ A A [ X _ { t + 1 } , \ldots , X _ n ] \right ) \\ & \cong H ^ i _ P ( W ) \otimes _ A A [ X _ { t + 1 } , \ldots , X _ n ] \\ & = \begin{cases} A [ X _ 1 ^ { - 1 } , \ldots , X _ t ^ { - 1 } , X _ { t + 1 } , \ldots , X _ n ] ( - 1 , - 1 , \cdots , - 1 , 0 , \cdots , 0 ) & \mbox { i f } i = t \\ 0 & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} P \bigg ( \liminf _ { n \to \infty } ( r ( n ) ^ { - 1 } \sup _ { t < n } | B _ t | ) \le 1 \bigg ) = 1 . \end{align*}


\begin{align*} \varphi _ g ( \tau , z ) = \eta _ g ( \tau ) \phi _ { - 2 , 1 } ( \tau , z ) . \end{align*}


\begin{align*} \displaystyle \Phi ( t , s ) = e ^ { \int _ { s } ^ { t } a ( \tau ) \ , d \tau } \leq K _ { 0 } e ^ { - \alpha ( t - s ) } \textnormal { f o r a l l $ t \geq s \geq 0 $ } . \end{align*}


\begin{align*} d X _ t = f ( t , X _ t ) d t + d W _ t . \end{align*}


\begin{align*} q \left ( x , t \right ) = - 2 \partial _ { x } ^ { 2 } \log \det \left ( \delta _ { m n } + c _ { n } ^ { 2 } \dfrac { \mathrm { e } ^ { - \left ( \kappa _ { m } + \kappa _ { n } \right ) x + 8 \left ( \kappa _ { m } ^ { 3 } + \kappa _ { n } ^ { 3 } \right ) t } } { \kappa _ { m } + \kappa _ { n } } \right ) . \end{align*}


\begin{align*} Q _ { i j } ( u , v ) & = \partial _ i u \partial _ j v - \partial _ j u \partial _ i v , \ , Q _ { 0 } ( u , v ) = \partial _ t u \partial _ t v - \nabla u \cdot \nabla v , \end{align*}


\begin{align*} N = N ^ { ( 1 ) } \supseteq N ^ { ( 2 ) } \supseteq N ^ { ( 3 ) } \supseteq \dots , \end{align*}


\begin{align*} \begin{cases} \dot { y } ( t ) & = B ( t ) y ( t ) , \\ y ( s ) & = \mathrm { I d } , \end{cases} \qquad \begin{cases} \dot { J } ( t ) & = \beta ( t ) J ( t ) , \\ J ( s ) & = 1 . \end{cases} \end{align*}


\begin{align*} u ( x , t ) = 2 i \lim _ { k \rightarrow \infty } k M _ { 1 2 } ( x , t , k ) , u ( - x , - t ) = 2 i \lim _ { k \rightarrow \infty } k M _ { 2 1 } ( x , t , k ) , \end{align*}


\begin{align*} \begin{aligned} | \left | k \right | _ g ^ 2 - \left | \ell \right | _ g ^ 2 | & = \beta \Big | \sum _ { i j } \overline g _ { i j } \left ( k _ i k _ j - \ell _ i \ell _ j \right ) \Big | \\ & \geq \beta _ 1 \dfrac { \gamma } { \Big ( \sum _ { i , j } | k _ i k _ j - \ell _ i \ell _ j | \Big ) ^ \tau } \ , . \end{aligned} \end{align*}


\begin{align*} \omega _ { s } = D _ { x } ^ { - 1 } \big ( \sum _ { i = 1 } ^ { 3 N } m _ { \lfloor \frac { i - 1 } { 3 } \rfloor + 1 } x _ { i } d x _ { i } \big ) \end{align*}


\begin{align*} K = R \left [ \frac { r ' _ 1 } { r _ 1 } , . . . \frac { r ' _ m } { r _ m } \right ] = R \left [ \frac { 1 } { r _ 1 } , . . . \frac { 1 } { r _ m } \right ] = R \left [ \frac { 1 } { f } \right ] \end{align*}


\begin{align*} \left \{ \begin{array} { r @ { \ , = \ , } l } \dot { \gamma } ( t ) & b ( t , \gamma ( t ) ) \\ \gamma ( s ) & x \ , , \end{array} \right . \end{align*}


\begin{align*} t = \mathcal { H } ^ { - 1 } _ * ( x , \mathcal { H } _ * ( x , t ) ) \leq p ^ * ( x ) \left [ \mathcal { H } _ * ( x , t ) \right ] ^ { \frac { 1 } { p ^ * ( x ) } } ( x , t ) \in \overline { \Omega } \times [ 0 , \infty ) , \end{align*}


\begin{align*} \pi ( x ; m , a ) : = \sum _ { \substack { p \leq x \\ p \equiv a \pmod m } } 1 , \theta ( x ; m , a ) : = \sum _ { \substack { p \leq x \\ p \equiv a \pmod m } } \log { p } \end{align*}


\begin{align*} f ( A _ 2 , \theta _ 2 , \omega _ 2 , S ) = d ^ { \frac { g \cdot \deg ( w ) } { 2 } } \cdot f ( A _ 1 , \theta _ 1 , \omega _ 1 , S ) . \end{align*}


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


\begin{align*} \mathcal { E } _ { m , n , 2 , 1 } ^ { \epsilon , \Delta t } = \mathcal { E } _ { m , n , 2 , 1 , 1 } ^ { \epsilon , \Delta t } + \mathcal { E } _ { m , n , 2 , 1 , 2 } ^ { \epsilon , \Delta t } , \end{align*}


\begin{align*} \log ^ { ( N ) } ( \Delta \mathbb { X } _ t ^ N ) = \log ^ { ( [ p ] ) } ( \Delta \mathbf { X } _ t ) \ t \in [ 0 , 1 ] , \end{align*}


\begin{align*} W _ L ^ { } \simeq \begin{dcases} - \frac { D _ 1 ( \alpha ) } { L ^ 7 } , & \\ - \frac { D _ 2 ( \alpha ) } { L ^ 4 } , & , \end{dcases} \end{align*}


\begin{align*} U _ { n } P _ { n } = P _ { n - 1 } , n \in \mathbb { N } . \end{align*}


\begin{align*} f ( \mu ( h ) m ) = \mu ( h ) f ( m ) = ( h \cdot f ) ( m ) = f ( \tau ( h ) m ) = f ( h m ) = f ( \omega ( h ) m ) , \ \forall m \in M _ \omega . \end{align*}


\begin{align*} \frac { \Delta t } { \epsilon ^ 2 } \sum _ { \ell = 0 } ^ { n - 1 } \Bigl ( \frac { 1 } { ( 1 + \frac { \Delta t } { \epsilon ^ 2 } ) ^ { n - \ell } } - e ^ { - \frac { ( n - \ell ) \Delta t } { \epsilon ^ 2 } } \Bigr ) ^ 2 & \le C \frac { \Delta t } { \epsilon ^ 2 } \sum _ { \ell = 0 } ^ { n - 1 } \frac { 1 } { ( n - \ell ) ^ 2 } \le C \frac { \Delta t } { \epsilon ^ 2 } . \end{align*}


\begin{align*} g _ 0 ( { \bf { v } } ) = \left ( 0 , r _ 2 , r _ 3 \right ) ^ T . \end{align*}


\begin{align*} \operatorname { T r } ( M ^ 2 ) = \sum _ { i , j } M _ { i j } M _ { j i } = 3 \ ; . \end{align*}


\begin{align*} E _ n = \{ x \in \Gamma : d ( x v _ 0 , v _ 0 ) = n \} , \chi _ n = \chi _ { E _ n } , \varphi _ { n } = \sum _ { i = 0 } ^ { \lfloor n / 2 \rfloor } \chi _ { n - 2 i } . \end{align*}


\begin{align*} \det \limits _ { x , y \in \mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x , y ) = \sum _ { \sigma } \mathrm { s g n } ( \sigma ) \prod _ { x \in \mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x , \sigma ( x ) ) , \end{align*}


\begin{align*} \Lambda : = \{ \lambda \in ( \mathbb P ^ n _ { k ' } ) ^ * ( k ' ) \ , | \ , ( I ) , ( I I ) \} . \end{align*}


\begin{align*} \partial ^ \alpha _ j \big ( b _ i ( a ) \big ) = b _ i \big ( \widetilde { a } \big ) , \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta u = - u + \frac { u } { v } \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; u > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; u \lfloor _ { \partial \Omega } = 0 , \\ - \Delta v = - \alpha v + \frac { u } { v } \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; v > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; v \lfloor _ { \partial \Omega } = 0 , \end{alignedat} \right . \end{align*}


\begin{align*} \frac { 2 } { \gamma } + \frac { 3 } { p } = 1 . \end{align*}


\begin{align*} | \sigma | = \int _ { \R ^ d } \sigma ( x ) \d x = m \sigma ( x ) \leq \frac { c } { | x | ^ { d + \alpha - 2 } } , x \in \R ^ d \setminus \left \{ 0 \right \} , \end{align*}


\begin{align*} \nu ^ { M | H } : = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } A ( e _ i ) e _ i , \end{align*}


\begin{align*} \langle \epsilon _ I , \hat { \mathbb { X } } ^ N _ t \rangle = \int _ 0 ^ t \langle \epsilon _ { I ' } , \hat { \mathbb { X } } ^ N _ { s ^ - } \rangle \ d s . \end{align*}


\begin{align*} = C _ 2 \cdot \int \limits _ { 1 / R < | x | < 1 } \varphi \left ( \frac { \Vert f ^ { \ , \prime } ( x ) \Vert } { | x | ^ { 2 n } } \ , \right ) d m ( x ) \leqslant C _ 2 R ^ { 2 n } \cdot \int \limits _ { 1 / R < | x | < 1 } \varphi ( | \nabla f ( x ) | ) \ , d m ( x ) < \infty \ , . \end{align*}


\begin{align*} - \Delta _ L : D ( - \Delta _ L ) \to L ^ 2 ( \Lambda _ L ) , f \mapsto - \Delta _ L f = ( 2 \pi ) ^ 2 ( \nu \hat { f } ) ^ \vee . \end{align*}


\begin{align*} S _ { \iota , \jmath } : = S _ \iota \cap S _ \jmath = \{ s \in S _ \iota : V ^ s = V ^ \jmath \} . \end{align*}


\begin{align*} \forall \omega > 0 , J _ E ( \mathfrak f ) ( \omega ) = 0 . \end{align*}


\begin{align*} \min _ { \tilde { \psi } _ { t , n } , \tilde { \psi } _ { r , n } } & \mathrm { R e } ( \tilde { \vartheta } _ { t , n } ^ * \tilde { \psi } _ { t , n } ) + \mathrm { R e } ( \tilde { \vartheta } _ { r , n } ^ * \tilde { \psi } _ { r , n } ) \\ [ - 0 . 5 e m ] \mathrm { s . t . } & \cos ( \tilde { \phi } _ { t , n } - \tilde { \phi } _ { r , n } ) = 0 , \\ & | \tilde { \psi } _ { t , n } | = 1 , | \tilde { \psi } _ { r , n } | = 1 . \end{align*}


\begin{align*} A \delta ^ { 0 } + ( B - \nu D ) \delta = 0 \end{align*}


\begin{align*} \varphi ( 0 ) : = 2 \pi \sum _ { j \in \mathbb { Z } } f ( 2 \pi j ) = \sum _ { j \in \mathbb { Z } } \hat { f } ( j ) . \end{align*}


\begin{align*} \begin{aligned} & \left | J \right | ^ 2 \equiv | j | ^ 2 : = \sum _ { i = 1 } ^ d | j _ i | ^ 2 \ , , | J | ^ 2 _ g \equiv | j | _ g ^ 2 : = g ( j , j ) \ , . \end{aligned} \end{align*}


\begin{align*} q & = L ^ n q _ { 2 d } \ / , X _ a = \exp \left ( L \sigma _ a \right ) = 1 + L \sigma _ a + \frac { L ^ 2 } { 2 } { \sigma _ a } ^ 2 + \cdots , \\ T _ i & = \exp \left ( L m _ i \right ) = 1 + L m _ i + \frac { L ^ 2 } { 2 } { m _ i } ^ 2 + \cdots \end{align*}


\begin{align*} \partial E : = \{ \exp _ o ( \rho ( x ) ) : x \in S ^ { n - 1 } \subset T _ o M \} , \end{align*}


\begin{align*} P : = \Delta ^ E + A \end{align*}


\begin{align*} ( \tilde { \psi } _ { t , n } ^ \star , \tilde { \psi } _ { r , n } ^ \star ) = \ ! \ ! \ ! \ ! \operatorname * { a r g m i n } _ { ( \tilde { \psi } _ { t , n } , \tilde { \psi } _ { r , n } ) \in \chi _ { \psi } ^ n } \ ! \ ! \ ! \ ! \mathrm { R e } ( \tilde { \vartheta } _ { t , n } ^ * \tilde { \psi } _ { t , n } ) + \mathrm { R e } ( \tilde { \vartheta } _ { r , n } ^ * \tilde { \psi } _ { r , n } ) , \end{align*}


\begin{align*} \theta ( s ) : = \begin{cases} \exp \displaystyle { \left ( \frac { \bar s } { \log \bar s } \right ) } & 0 \le \ , s \le \ , \bar s \ , ; \\ \exp \displaystyle { \left ( \frac { s } { \log s } \right ) } & s \ge \ , \bar s \ , . \end{cases} \end{align*}


\begin{align*} M ( A ) ( i ) = A ( \rho ^ a ( i ) ) \ ; . \end{align*}


\begin{align*} \omega _ 2 ( t ) : = \mathrm { R e } \ , \int _ \Omega \left \{ | g _ { z + t e _ j } - t g _ { z , j } | ^ { q - 2 } \overline { ( g _ { z + t e _ j } - t g _ { z , j } ) } - | g _ z | ^ { q - 2 } \overline { g } _ z \right \} g _ { z , j } . \end{align*}


\begin{align*} \Omega _ { n , \mathrm { k } } : = \{ ( N _ 1 , \dots , N _ n ) \in \Omega _ n : \forall 1 \le i \le \ell : N _ { k _ i } \ge 1 \} . \end{align*}


\begin{align*} \mathcal { C } _ { 1 , j , k } ^ { J } = \frac { 1 + 2 x _ { 1 } } { 2 x _ { 1 } } \left [ \frac { 1 } { 2 + 2 x _ { 1 } } \mathcal { A } _ { 1 , j } ^ { J \setminus k } + \tilde { \mathcal { D } } _ { 1 , j } ^ { J \setminus k } \right ] { \mathcal { A } } _ { 1 , k } ^ { J } + \frac { 1 + 2 x _ { 1 } } { 2 x _ { 1 } } \left [ \frac { 1 } { 2 x _ { 1 } } \mathcal { A } _ { 1 , k } ^ { J \setminus j } + \tilde { \mathcal { D } } _ { 1 , k } ^ { J \setminus j } \right ] { \mathcal { A } } _ { 1 , j } ^ { J } \ , . \end{align*}


\begin{align*} a \circ b = a + 7 ^ a b . \end{align*}


\begin{align*} \frac { d ^ \ell _ x \lambda ^ c _ { x ^ u } ( \tau ' ( \ell ) + k ) } { d ^ \ell _ y \lambda ^ c _ { y ^ u } ( \tau ' ( \ell ) ) } = \frac { d ^ \ell _ x } { d ^ \ell _ y } \times \frac { \lambda ^ c _ { x ^ u } ( \tau ' ( \ell ) ) } { \lambda ^ c _ { y ^ u } ( \tau ' ( \ell ) ) } \times \lambda ^ c _ { x ^ u _ { \tau ' ( \ell ) } } ( k ) . \end{align*}


\begin{align*} G _ 2 = \left ( \begin{array} { c c c c c c } 1 & 0 & 1 & 0 & \omega & \omega \\ 0 & 1 & 0 & 1 & \omega & \omega \\ \end{array} \right ) . \end{align*}


\begin{align*} \begin{aligned} & \dfrac { 1 } { 2 } \norm { v ( t ) } { H ^ 2 } ^ 2 + \dfrac { c ^ 2 } { 4 ( \gamma - 1 ) } \norm { \xi ( t ) } { H ^ 2 } ^ 2 + \dfrac { \min \lbrace \mu , \lambda , 1 \rbrace c _ 1 } { 4 } \int _ 0 ^ t \norm { \nabla v ( s ) } { H ^ 2 } ^ 2 \ , d s \\ & ~ ~ ~ ~ \leq M + C _ { \delta ' } \mathcal H _ 2 ( C _ 0 M _ 0 , C _ 1 M _ 1 ) t < 2 M , \end{aligned} \end{align*}


\begin{align*} \frac { \| e _ { m + 1 } \| _ { \mathcal G _ q ^ { \omega } } } { \| e _ { m + 1 } \| _ { \mathcal { P G } _ q ^ { \omega } } } \le \frac { 1 } { \omega ( 1 ) \left ( \sum _ { n = 1 } ^ { m } \frac { 1 } { n } \right ) ^ { \frac { 1 } { q } } } \xrightarrow [ m \to \infty ] { } 0 . \end{align*}


\begin{align*} \Delta = D _ { ( 0 , 0 ) } = \sum _ { \substack { ( j , k ) \in J \\ l > k / 2 } } s _ { j , k , l + 1 } \ , \frac { \partial } { \partial s _ { j , k , l } } \ , , \end{align*}


\begin{align*} \Theta _ 1 & = \vartheta ( s _ 1 ) - \tilde \alpha _ 1 - \frac \pi 2 = \vartheta ( s _ 2 ) + \alpha _ 2 - ( \tilde \alpha _ 1 + \alpha _ 2 ) - \frac \pi 2 \\ & = \vartheta ( s _ 1 ) + \vartheta ( s _ 2 ^ * ) - \pi - ( \tilde \alpha _ 1 + \alpha _ 2 ) \\ & = \vartheta ( s _ 1 ) - \vartheta ( - s _ 2 ^ * ) - ( \tilde \alpha _ 1 + \alpha _ 2 ) \\ & = - \int _ { s _ 1 } ^ { s ^ * _ 1 } \dot \vartheta ( s ) \ , d s - ( \tilde \alpha _ 1 + \alpha _ 2 ) . \end{align*}


\begin{align*} H _ { \overline { \mu } + 2 t } ( \Upsilon _ { \mu , t } ( m ) , \Upsilon _ { \mu , t } ( m ' ) ) = H _ \mu ( m , m ' ) . \end{align*}


\begin{align*} \zeta ( x ) = h ^ { \mu _ { V } } + \frac { V } { 2 } + c . \end{align*}


\begin{align*} \frac { w ( \upsilon + \tau ) } { w ( \upsilon ) } = \det \left ( [ A ( \upsilon ) ] ^ { - 1 } A ( \upsilon + \tau ) \right ) \leq \big \| [ A ( \upsilon ) ] ^ { - 1 } A ( \upsilon + \tau ) \big \| ^ d = \big \| [ \phi _ { \upsilon } ( \tau ) ] ^ T \big \| ^ d \leq [ v _ 0 ( \tau ) ] ^ d . \end{align*}


\begin{align*} & X ( t , s , \cdot ) \in W ^ { 1 , p } ( \Omega ; \R ^ n ) \quad \\ & \int _ { \Omega } \| D _ x X ( t , s , x ) \| ^ p \dd x \leq \frac 1 { \ell } \int _ { I } \int _ \Omega \exp \left ( \frac { \ell p ^ 2 } { p - n } \| D _ x b ( v , y ) \| \right ) \dd y \dd v . \end{align*}


\begin{align*} A = R R ^ T \ ; , \end{align*}


\begin{align*} m ^ \epsilon ( t ) - m ^ \epsilon ( 0 ) = - \frac { 1 } { \epsilon ^ 2 } \int _ { 0 } ^ { t } m ^ \epsilon ( s ) d s + \frac { 1 } { \epsilon } \beta ( t ) , \end{align*}


\begin{align*} \int _ { H ^ 1 ( \mathbb { S } ^ 2 ) } \mathcal { A } \phi ( x ) \dd \mu ( x ) = 0 \ , . \end{align*}


\begin{align*} \mathring { R } _ { o s c } ^ u & = \mathring { R } _ { o s c . 1 } ^ u + \mathring { R } _ { o s c . 2 } ^ u + \mathring { R } _ { o s c . 3 } ^ u + \mathring { R } _ { o s c . 4 } ^ u , \end{align*}


\begin{align*} s _ p ( E , \Omega ) = \frac { \int _ E | f _ E | ^ p } { \int _ \Omega | f _ E | ^ p } . \end{align*}


\begin{align*} W _ { q } ( 0 + ) & = \left \{ \begin{array} { l l } 0 & \textrm { i f $ X $ i s o f u n b o u n d e d v a r i a t i o n , } \\ 1 / c & \textrm { i f $ X $ i s o f b o u n d e d v a r i a t i o n . } \end{array} \right . \end{align*}


\begin{align*} \lim _ { \chi \to 0 } \chi ^ \beta g ( t ) = t ^ { \gamma - \beta } , t \ge 1 . \end{align*}


\begin{align*} \nu _ { G _ { E } } ^ { K V } ( \gamma ) = e ( E / F ) \cdot \nu _ { G } ^ { K V } ( \gamma ) , \end{align*}


\begin{align*} ( \sum ^ n _ { j = 1 } a _ { i j } \mu _ j \frac { \partial } { \partial \mu _ j } - c _ i ) \Phi = 0 & , i = 1 , \ldots , r \\ \left ( \frac { \partial } { \partial \mu } \right ) ^ { u } \Phi = \left ( \frac { \partial } { \partial \mu } \right ) ^ { v } \Phi & , u , v \in \N ^ n u - v \in \mathbb { L } . \end{align*}


\begin{align*} W ^ { } _ L = - \frac { \alpha ^ 2 } { L ^ 3 } + O \Big ( \frac { \alpha ^ 2 } { L ^ 5 } \Big ) , \end{align*}


\begin{align*} \frac { d w ( z , t ) } { d t } = G ( w ( z , t ) ) , w ( z , 0 ) = z . \end{align*}


\begin{align*} c _ i \mapsto \begin{cases} c _ i , & ; \\ b _ i , & , \end{cases} d _ j \mapsto \begin{cases} d _ j , & ; \\ a _ { j + 1 } , & . \end{cases} \end{align*}


\begin{align*} e _ { i } ( 1 - x _ 1 , \ldots , 1 - x _ n ) = \sum _ { s = 0 } ^ i ( - 1 ) ^ s { n - s \choose i - s } e _ s ( x _ 1 , \ldots , x _ n ) \ / . \end{align*}


\begin{align*} ( T _ 0 ( t ) \varphi ) ( \theta ) : = \begin{cases} \varphi ( t + \theta ) , & - h \leq t + \theta \leq 0 , \\ \varphi ( 0 ) , & t + \theta \geq 0 , \end{cases} \forall \varphi \in X , \ t \geq 0 , \ \theta \in [ - h , 0 ] . \end{align*}


\begin{align*} \mathfrak { S } ( f _ k ) = \mathfrak { S } ( f _ k | H ^ { \prime } ) \geq \tilde { k } + \tilde { r } - 1 - l ^ { \prime } + l _ { + } . \end{align*}


\begin{align*} d _ { r , s } ( - n , 0 ) = - d _ { r , s } ( n , 0 ) , \mbox { i f } s \ne 0 . \end{align*}


\begin{align*} \partial ^ { 2 } \log \det \left ( I + \mathbb { A } \right ) & = \partial \operatorname * { t r } \left ( I + \mathbb { A } \right ) ^ { - 1 } \partial \mathbb { A } \\ & = \operatorname * { t r } \left ( I + \mathbb { A } \right ) ^ { - 1 } \partial \mathbb { A } . \end{align*}


\begin{align*} \mathcal { E } ( \mu ) = \int _ { M \times M } g ( x - y ) \ , \mathrm d \mu \otimes \mu ( x , y ) . \end{align*}


\begin{align*} \nabla ^ { | \beta | } z _ N ^ { \beta } = | \beta | ! \cdot d z _ N ^ { \odot \beta } . \end{align*}


\begin{align*} \alpha ( x ) = ( q + x ) \left ( x + \frac { 1 } { 2 } + s \right ) ^ { 2 N } \ , , \tilde \delta ( x ) = \frac { 2 x } { 2 x + 1 } ( q - x - 1 ) \left ( x + \frac { 1 } { 2 } - s \right ) ^ { 2 N } \ , . \end{align*}


\begin{align*} \mathcal { R } _ N ( \boldsymbol { v } ) : = \# \{ 1 \leq m \neq n \leq N : a _ n ^ { ( i ) } - a _ m ^ { ( i ) } = v _ i , \ : 1 \leq i \leq h \} . \end{align*}


\begin{align*} T _ { f , p } ^ { X } ( g ) : = B _ p ^ X ( f \cdot B _ p ^ X g ) . \end{align*}


\begin{align*} d _ { r , 0 } ( 0 , i ) = 0 , \mbox { i f } r \ne 0 . \end{align*}


\begin{align*} g ( \theta ) : = a \cos 2 \theta + b | \sin 2 \theta | . \end{align*}


\begin{align*} \delta w _ { s , t } = \mathcal { D } ( w ) _ { s , t } + \int _ { s } ^ { t } w _ r \times h _ 1 \circ \dd B _ r \ , . \end{align*}


\begin{align*} c d + a b = \frac { ( c + a ) ( d + b ) + ( c - a ) ( d - b ) } { 2 } \in \S ~ ~ \textup { a n d } ~ ~ c d - a b = \frac { ( c + a ) ( d - b ) + ( c - a ) ( d + b ) } { 2 } \in \S . ~ ~ \end{align*}


\begin{align*} \xi _ { 1 } ( r ) = \left \{ \begin{aligned} & 0 , r \leq \frac { 1 } { 2 } ( r _ { 0 } + r _ { 1 } ) \\ & 1 , r \geq r _ { 1 } \end{aligned} \right . , \xi _ { 2 } ( r ) = \left \{ \begin{aligned} 1 , r \leq 1 \\ 0 , r \geq 2 \end{aligned} \right . . \end{align*}


\begin{align*} M ( A ) ( \vec { \alpha } ) = A _ P \operatorname { P f } \left ( A _ X \rvert _ { \vec \alpha , \vec \alpha } \right ) \ ; , \end{align*}


\begin{align*} F _ 0 ( D ^ 2 H ) = 0 , \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ { \frac { n } { 2 } } u ( x ) = v ^ p ( x ) , & x \in \Omega , \\ ( - \Delta ) ^ { \frac { n } { 2 } } v ( x ) = u ^ q ( x ) , & x \in \Omega , \\ u ( x ) = - \Delta u ( x ) = \cdots = ( - \Delta ) ^ { \frac { n } { 2 } } u ( x ) = 0 , & x \in \partial \Omega , \\ v ( x ) = - \Delta v ( x ) = \cdots = ( - \Delta ) ^ { \frac { n } { 2 } } v ( x ) = 0 , & x \in \partial \Omega . \end{cases} \end{align*}


\begin{align*} \mathcal { X } = \{ T _ { j } \in \mathbb { F } _ { q } ^ { t } : j \in \Omega [ q + 1 , t ] \} . \end{align*}


\begin{align*} \partial _ \xi w + \partial _ \tau w \leq \frac { ( \delta b ) ^ { \frac { 1 } { \alpha _ 0 } } } { 2 e ^ X } ( 1 - \eta ) \leq e ^ { - X } \frac { b \delta } { 2 } ( 1 - \eta ) \leq e ^ { - X } \frac { \delta } { 2 } w o n \tau = 0 a n d \xi = 0 . \end{align*}


\begin{align*} | \lambda | = \left | 1 + \frac { M _ 1 } { \lambda ^ { N _ 1 - 1 } } + \cdots + \frac { M _ k } { \lambda ^ { N _ k - 1 } } \right | \end{align*}


\begin{align*} \begin{gathered} A _ X ( i , j ) = - A _ X ( j , i ) \\ A _ i = A _ o \ ; , \ A _ I ( i , l ) = - A _ O ( l , i ) \ ; , \end{gathered} \end{align*}


\begin{align*} \mathrm { r a n k } \left ( \begin{bmatrix} H _ 1 ( x _ { [ 0 , N - L ] } ) \\ H _ L ( u ) \end{bmatrix} \right ) = m L + n , \end{align*}


\begin{align*} S _ 2 : = \{ b \mid \textrm { $ b $ - o r b i t e x i t s } \Gamma \textrm { v i a } h = \frac { \pi } { 2 } \textrm { w i t h } \Omega ( b ) \leq \tfrac { 3 } { 2 } \} . \end{align*}


\begin{align*} \frac { d } { d t } \det A ( t ) = \det A ( t ) \cdot \mathop { \operatorname { t r a c e } } ( [ A ( t ) ] ^ { - 1 } \cdot A ' ( t ) ) , \end{align*}


\begin{align*} \Omega ^ { } = \{ m \in \Omega : S ^ k ( m ) = m \} \end{align*}


\begin{align*} \mathcal E ( f ) ^ M = \det ( M ) ^ { \deg ( w ) } \cdot d ^ { - \frac { g \cdot \deg ( w ) } { 2 } } \cdot \mathcal E ( f ) ; \end{align*}


\begin{align*} \sqrt { \frac { v } { u } } = 2 \alpha ^ 3 \exp ( \alpha ^ 2 ) \leq \exp ( 2 \alpha ^ 2 ) \implies \frac { 1 } { 2 } \log \left ( \frac { v } { u } \right ) \leq \alpha ^ 2 . \end{align*}


\begin{align*} T _ n ( 1 , t ) = t ( t + 1 ) ^ { n - 1 } . \end{align*}


\begin{align*} H _ b ^ E : = \overline { \tilde { H } _ b } , b \geq 0 . \end{align*}


\begin{align*} f _ { s , a , b } ( x ) : = \frac { e ^ { i s x } } { a ^ 2 + ( x + b ) ^ 2 } . \end{align*}


\begin{align*} \sum _ { 0 < \gamma , \gamma ' \le T } r \Big ( ( \gamma - \gamma ' ) \frac { \log T } { 2 \pi } \Big ) \ , w ( \gamma - \gamma ' ) = N ( T ) \int _ { \mathbb R } \widehat r ( \alpha ) \ , F ( \alpha ) \ , d \alpha , \end{align*}


\begin{align*} \langle X + \xi , Y + \eta \rangle : = \frac { 1 } { 2 } \left ( \eta ( X ) + \xi ( Y ) \right ) . \end{align*}


\begin{align*} m = p \cdot q ^ v w + r \in \Omega _ p \end{align*}


\begin{align*} \hat { f } _ \theta ( k ) = e ^ { - d \theta / 2 } \left ( \tilde { Q } _ { \sigma , a } ( e ^ { - \theta } k ) e ^ { - \frac { \pi } { \sigma } \frac { \delta } { 4 } | k | ^ 2 } \right ) \left ( e ^ { - \frac { \pi } { \sigma } e ^ { - 2 \theta } | k | ^ 2 } e ^ { \frac { \pi } { \sigma } \frac { \delta } { 2 } | k | ^ 2 } \right ) \left ( e ^ { - e ^ { - \theta } k \cdot u } e ^ { - \frac { \pi } { \sigma } \frac { \delta } { 4 } | k | ^ 2 } \right ) . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ n ( - 1 ) ^ { n - k } \sum _ { \sigma \in S h ( k , n - k ) } \chi ( \sigma ) Q ^ a _ { n - k + 1 } ( Q ^ a _ k ( x _ { \sigma ( 1 ) } \wedge \dots \wedge x _ { \sigma ( k ) } ) \wedge x _ { \sigma ( k + 1 ) } \wedge \dots \wedge x _ { \sigma ( n ) } ) = 0 \end{align*}


\begin{align*} \phi \circ \alpha = \alpha ' \circ \phi \\ \phi \left ( [ x , y ] \right ) = \left [ \phi ( x ) , \phi ( y ) \right ] ' \end{align*}


\begin{align*} \mathcal { E } ( x , t ) = \widetilde { \mathcal E } ( x , t ) | _ { x _ 1 > 0 } , \mathcal B ( x , t ) = \widetilde { \mathcal B } ( x , t ) | _ { x _ 1 > 0 } , \mathcal A ( x , t ) = \widetilde { \mathcal A } ( x , t ) | _ { x _ 1 > 0 } , \end{align*}


\begin{align*} & \gcd ( 2 ^ h + 1 , 2 ^ n - 1 ) = \begin{cases} 3 & \mbox { i f } n \mbox { i s e v e n , } \\ 1 & \mbox { i f } n \mbox { i s o d d , } \end{cases} \end{align*}


\begin{align*} 2 \sum _ { u = 1 } ^ p E \underline { Z } _ u = ( 1 + o ( 1 ) ) \frac { \pi ^ 2 } { 8 } \bar h _ 4 ( n ) \end{align*}


\begin{align*} i _ \omega \ = \ \sup _ { M > 1 } \frac { \ln ( \varphi _ \omega ( M ) ) } { \ln M } \mbox { a n d } I _ \omega \ = \ \inf _ { M > 1 } \frac { \ln ( \Phi _ \omega ( M ) ) } { \ln M } . \end{align*}


\begin{align*} Y _ { m _ 1 , m _ 2 } ( \alpha , \beta , \gamma ) = \frac { 1 } { 2 } ( - 1 ) ^ { ( \theta _ 1 + m _ 1 + 1 ) ( \theta _ 2 + m _ 2 + 1 ) } \prod _ { j \in \mathbb { T } _ { m _ 1 , m _ 2 } } \left ( \alpha + \beta \exp \left \{ 2 \pi i \frac { j _ 1 + \theta _ 1 / 2 } { m _ 1 } \right \} + \gamma \exp \left \{ 2 \pi i \frac { j _ 2 + \theta _ 2 / 2 } { m _ 2 } \right \} \right ) . \end{align*}


\begin{align*} \min J ( x ) : = \ell ( x ( a ) , x ( b ) ) + \int _ a ^ b f ( t , x ( t ) , \dot { x } ( t ) ) d t , \end{align*}


\begin{align*} u _ { z _ 0 } \left ( \bar z ^ { - 1 } \right ) = \frac { 2 z } { 1 - \bar z _ 0 z } = u _ { z _ 0 } ( z ) , \end{align*}


\begin{align*} 2 ^ { - i } = \int _ I g ( s ) \ , d s = \int _ I g _ { i + 1 } ( s ) \ , d s , \end{align*}


\begin{align*} M : = & \Big ( \sup _ { 0 \leq t \leq T } \big \| u ^ h ( t ) \big \| \Big ) \vee \Big ( \sup _ { 0 \leq t \leq T } \big \| v ^ h ( t ) \big \| \Big ) \vee \Big ( \int _ { 0 } ^ { T } \big \| u ^ h ( s ) \big \| _ V ^ 2 d s \Big ) \vee \Big ( \int _ { 0 } ^ { T } \big \| v ^ h ( s ) \big \| _ V ^ 2 d s \Big ) . \end{align*}


\begin{align*} ( a \star b , c ) _ { s m } = \sum _ { d \ge 0 } q ^ d \langle a , b , c \rangle _ d \ / , \end{align*}


\begin{align*} \Sigma ( z ) = \{ P \in E _ { n } ( \mathbb { Q } ) \mid [ 2 ] P = \pm \psi ( z ) \} = \{ P \in E _ { n } ( \mathbb { Q } ) \mid x ( [ 2 ] P ) = z ^ { 2 } \} , \end{align*}


\begin{align*} I : = \frac { \alpha } { 4 \pi ^ 2 } \int _ { \mathbb { R } ^ 3 } \mathrm { d } k \ ; \frac { \chi ^ 2 _ { \Lambda } ( k ) } { | k | } ( - 1 - \hat { k } _ 1 ^ 2 + \hat { k } _ 2 ^ 2 + \hat { k } _ 3 ^ 2 ) \cos ( 2 k _ 1 y ) \left \langle x u _ { \alpha } \ , \bigg | \ , \frac { ( h _ { \alpha } - e _ { \alpha } ) | k | } { 3 ( h _ { \alpha } - e _ { \alpha } + | k | ) } \ , \bigg | \ , x u _ { \alpha } \right \rangle . \end{align*}


\begin{align*} L _ n \le \big ( d ^ { \alpha _ 0 } _ n + | T - \bar t _ n | ^ { \frac { \alpha _ 0 } \gamma } \big ) \frac { M _ n ^ { \frac 2 \gamma + \alpha _ 0 \frac { \gamma - 1 } \gamma } } { r _ n ^ { \alpha _ 0 } } = \big ( d ^ { \alpha _ 0 } _ n + | T - \bar t _ n | ^ { \frac { \alpha _ 0 } \gamma } \big ) \frac { M _ n } { r _ n ^ { \alpha _ 0 } } \le 2 L _ n . \end{align*}


\begin{align*} \mathcal { D } ( S ) : = \{ ( t , s , \varphi ) \in \Omega _ J \times X \ : \ t \in I _ { \varphi } \} , S ( t , s , \varphi ) : = u _ \varphi ( t ) , \end{align*}


\begin{align*} \tau ( \pi _ 1 , \pi _ 2 ) = X ^ { \langle \pi _ 1 , \pi _ 2 \rangle } . \end{align*}


\begin{align*} \mathcal { F } _ d ^ { \mathbf { t } } ( \mathbf { x } , \mathbf { y } ) = G _ d + \frac { q } { 2 } \mathbf { t } \cdot \mathbf { y } ; ~ \mathcal { G } _ d ^ { \mathbf { t } } ( \mathbf { x } , \mathbf { y } ) = G _ { d } + \frac { q } { 2 } x _ { m - 2 } + \frac { q } { 2 } \mathbf { t } \cdot \mathbf { y } , \end{align*}


\begin{align*} \int \limits _ { a } ^ b \alpha ^ { \ , \prime } ( t ) \ , d t = \alpha ( b ) - \alpha ( a ) \ , . \end{align*}


\begin{align*} _ Q ( I , \textbf { d } , \textbf { r } ) = \{ M \in _ { Q } ( I , \textbf { d } ) \ , | \ , M ( a ) \leq r _ a , \forall a \in Q ^ * _ 1 \} \end{align*}


\begin{align*} \left \{ \begin{aligned} & \frac { \partial U } { \partial t } + U \frac { \partial U } { \partial x } = \gamma \frac { \partial ^ 2 U } { \partial x ^ 2 } , & & ( x , t ) \in [ 0 , L ] \times [ 0 , T ] \\ & U ( 0 , t ) = U ( L , t ) = 0 , & & t \in [ 0 , T ] \\ & U ( x , 0 ) = \sin \left ( \frac { 2 \pi x } { L } \right ) , & & x \in [ 0 , T ] \end{aligned} \right . \end{align*}


\begin{align*} \partial _ { x } P _ { n } ( x ; z ) = n P _ { n - 1 } ( x ; z ) + \left [ 2 c _ { n } \left ( z \right ) - n \gamma _ { n - 1 } \right ] P _ { n - 3 } ( x ; z ) + O \left ( x ^ { n - 5 } \right ) . \end{align*}


\begin{align*} K _ 2 : = \int _ { B _ 1 ( 0 ) } G _ { n , 1 } ( 0 , y ) \mathrm { d } y , \end{align*}


\begin{align*} a _ m \partial ^ m \omega + \cdots + a _ 1 \partial \omega + a _ 0 \omega = 0 . \end{align*}


\begin{align*} \alpha \big ( \left [ [ x , y ] , \cdot \right ] \big ) = - \left [ y , [ \alpha ( x ) , \cdot ] \right ] - \left [ x , [ \alpha ( y ) , \cdot ] \right ] \end{align*}


\begin{align*} A _ { N + 1 } = A _ N \left ( \begin{smallmatrix} 0 & 1 \\ 1 & a _ { N + 1 } \end{smallmatrix} \right ) \ , . \end{align*}


\begin{align*} A d m _ { g , N } ^ { \epsilon } ( X ^ { ( n ) } , \beta ) = \coprod _ { \underline \mu } A d m _ { g , N } ^ { \epsilon } ( X ^ { ( n ) } , \beta , \underline { \mu } ) , \end{align*}


\begin{align*} ( A \otimes B ) ( ( i , j ) ) = A ( i ) \cdot B ( j ) \end{align*}


\begin{align*} P _ 2 \left ( n + 1 \right ) - P _ 2 \left ( n \right ) = F ( n ) F ( n ) \equiv F ^ { ( k ) } _ n \end{align*}


\begin{align*} H ( x , y ) & : = \{ - \Delta _ x + V _ \mathrm { t r a p } ( x ) + ( \upsilon _ N \ast \rho ^ \mathrm { p a i r } ) ( x ) - \mu \} \delta ( x - y ) + ( \upsilon _ N n ) ( x , y ) , \\ B ( x ) & : = \big ( - \Delta _ x + V _ \mathrm { t r a p } ( x ) + \frac { 1 } { N } ( \upsilon _ N \ast \rho ) ( x ) - \mu \big ) \phi ( x ) \\ & + \int { d y \{ ( \upsilon _ N n ) ( x , y ) \phi ( y ) + ( \upsilon _ N m ) ( x , y ) \overline { \phi ( y ) } \} } , \end{align*}


\begin{align*} \nabla \int _ { v _ { 0 } \in \mathbb { B } ^ d } J ( u _ { 0 } + \epsilon v _ 0 ) d V = \frac { 1 } { \epsilon } \int _ { v _ { 0 } \in \mathbb { S } ^ { d - 1 } } J ( u _ { 0 } + \epsilon v _ 0 ) v _ 0 d S . \end{align*}


\begin{align*} \mathfrak { L } \left [ p _ { k } p _ { n } \right ] = h _ { n } \delta _ { k , n } , k , n \in \mathbb { N } _ { 0 } , h _ { n } \neq 0 , \end{align*}


\begin{align*} P = \sum _ { k = 3 } ^ { N - 1 } \widetilde { \cal K } _ m ^ { ( r ) } + \tilde { R } _ N \ , , \end{align*}


\begin{align*} d X ^ { 0 , { \rm E } } ( t ) = \sigma ( X ^ { 0 , { \rm E } } ( t ) ) d \beta ( t ) \end{align*}


\begin{align*} | B _ n ( \beta ) | = \sum _ { j = 0 } ^ { k - 1 } \binom { n } { j } . \end{align*}


\begin{align*} R ( g , T , \hat y , \hat \eta ) = \int \limits _ 0 ^ T \sqrt { \sum _ { j , k = 0 } ^ n g ^ { j k } ( y ) \eta _ j \eta _ k } \ , d t = \sqrt { 2 H ( y , \eta _ 0 , \eta ) } T . \end{align*}


\begin{align*} ( \gamma _ 1 * _ N \gamma _ 2 ) ( t ) = \begin{cases} \gamma _ 1 ( 2 t ) & \hbox { i f } 0 \leq t \leq \frac { 1 } { 2 } , \\ \gamma _ 2 ( 2 t - 1 ) & \hbox { i f } \frac { 1 } { 2 } \leq t \leq 1 \end{cases} \end{align*}


\begin{align*} & [ d ^ 0 _ { r , s } ( m , i ) L _ { m + r , i + s } , G _ { n , j } ] + [ L _ { m , i } , d ^ 1 _ { r , s } ( n , j ) G _ { n + r , j + s } ] \\ & \quad = \left ( n ( i + s + q ) - ( m + r ) \left ( j + \frac q 2 \right ) \right ) d ^ 0 _ { r , s } ( m , i ) G _ { m + n + r , i + j + s } \\ & \quad \quad + \left ( ( n + r ) ( i + q ) - m \left ( j + s + \frac q 2 \right ) \right ) d ^ 1 _ { r , s } ( n , j ) G _ { m + n + r , i + j + s } . \end{align*}


\begin{align*} B ( Z , x + v ) = Z ( x ) \end{align*}


\begin{align*} \delta a _ { s , t } = \int _ { s } ^ { t } f \dd t + A _ { s , t } a _ s + \mathbb { A } _ { s , t } a _ s + a ^ { \natural } _ { s , t } \ , , \quad \quad \quad \left ( \textrm { r e s p . } \quad \delta b _ { s , t } = \int _ { s } ^ { t } g \dd t + B _ { s , t } b _ s + \mathbb { B } _ { s , t } b _ s + b ^ { \natural } _ { s , t } \right ) \ , , \end{align*}


\begin{align*} \Delta ( x , t ) = p ( x + 1 , t ) - p ( x - 1 , t ) . \end{align*}


\begin{align*} m ( \pi , \sigma ) : = \langle \pi \otimes \nu _ { l , \psi } ^ \vee , \sigma \rangle _ { R _ l ^ F } . \end{align*}


\begin{align*} T _ t f = \Psi ( i _ { \mathsf { b } } ( t ) ) \ , \mbox { f o r a l l } t \in G \ , . \end{align*}


\begin{align*} \| \Phi _ { N k } - 1 \| _ \delta = O _ k ( N ^ { 2 - 2 k } ) \end{align*}


\begin{align*} \| \eta ' _ k - \eta _ k \| _ { C ^ { m } ( \Gamma ; \mathbb { C } ) } \le c _ m t ( m = 1 , 2 , \dots ) \end{align*}


\begin{align*} u ( x , 0 ) = f ( x ) \end{align*}


\begin{align*} \mathcal { I } ( a ) = \int _ { | w | = 1 } \frac { \mathrm { d } z } { 2 \pi i z } \log | a + z | = \int _ { | w | = 1 } \frac { \mathrm { d } z } { 2 \pi i z } \log ( a + z ) \end{align*}


\begin{align*} \frac { \partial X } { \partial t } ( p , t ) \ ; = \ ; \triangle _ { \Gamma _ t } X ( p , t ) . \end{align*}


\begin{align*} \nabla _ { p } ^ 2 u ^ \epsilon ( t , q , p ) : a ( q ) = \sum _ { i , j = 1 } ^ { d } \partial _ i \partial _ j u ^ \epsilon ( t , q , p ) a _ { i j } ( q ) . \end{align*}


\begin{align*} | f | ^ 2 _ { V ^ n } = \langle S ^ n f , f \rangle _ { L ^ 2 } . \end{align*}


\begin{align*} \partial _ t \pi ( x , t ) = - \partial _ t d ( x , t ) \nabla d ( x , t ) - d ( x , t ) \partial _ t \nabla d ( x , t ) . \end{align*}


\begin{align*} \omega _ j : = \sqrt { \left | j \right | ^ 4 _ g + m } \ . \end{align*}


\begin{align*} \| b _ i - p _ i \| _ 2 \leq \frac { \varepsilon C } { k } i = \overline { 1 , k } . \end{align*}


\begin{align*} | \bar { H } _ \pi | = \sum _ { x ' \in V _ i } \left ( | V _ j | | V _ k | - d _ { j , k } ( x ' ) \right ) \geq \sum _ { x ' \in S _ x } \left ( | V _ j | | V _ k | - d _ { j , k } ( x ' ) \right ) \geq | S _ x | \cdot \frac { 1 } { 2 } c n ^ 2 \geq \frac { 1 } { 2 } c n ^ { \frac { 5 } { 2 } } , \end{align*}


\begin{align*} \rho \left ( h _ a \right ) = a \ , , D \left ( h _ a \right ) = a - a ^ { - 1 } \ , . \end{align*}


\begin{align*} M ( x , t , k ) = \left ( \begin{array} { c c } \frac { k + v _ 1 ( x , t ) } { k - \frac { i A } { 2 } } & \frac { v _ 1 ( x , t ) } { k } \\ \frac { - \overline { v _ 1 ( - x , - t ) } } { k - \frac { i A } { 2 } } & \frac { k - \overline { v _ 1 ( - x , - t ) } } { k } . \end{array} \right ) \end{align*}


\begin{align*} \zeta _ 2 = k _ d ^ { - 1 } ( \partial ^ \circ \zeta + k _ d ^ { - 1 } V _ \Gamma ^ 2 \zeta - k _ d \Delta _ \Gamma \zeta - V _ \Gamma H \zeta ) \quad \overline { S _ T } , \end{align*}


\begin{align*} \omega ^ { a } = \psi ^ { a } + \sum _ { i = 1 } ^ { 3 N - 6 } \wedge ^ { a } _ { i } d q ^ { i } \end{align*}


\begin{align*} V _ \ell ( A ) = \bigoplus _ { i = 1 } ^ n V _ \ell ( A _ i ^ { m _ i } ) \end{align*}


\begin{align*} \widehat { Q } ^ 1 _ n ( F _ 1 \vee \cdots \vee F _ n ) = ( Q ' ) ^ 1 _ n \circ ( F _ 1 \star F _ 2 \star \cdots \star F _ n ) , \end{align*}


\begin{align*} I _ { q } \cap \mathbb { F } _ { \geqslant q \boldsymbol { w } } = R \cap \mathbb { F } _ { \geqslant q \boldsymbol { w } } \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u = f ( x , u + \epsilon , v , \nabla u , \nabla v ) \ ; \ ; & \mbox { i n $ \R ^ N $ , } \ ; \ ; u > 0 \ ; \ ; & & \mbox { i n $ \R ^ N $ , } \\ - \Delta _ q v = g ( x , u , v + \epsilon , \nabla u , \nabla v ) \ ; \ ; & \mbox { i n $ \R ^ N $ , } \ ; \ ; v > 0 \ ; \ ; & & \mbox { i n $ \R ^ N $ , } \end{alignedat} \right . \end{align*}


\begin{align*} P ^ { r - 1 , 1 } _ k \cdot \begin{pmatrix} * & . . . & * \\ \vdots & \ddots & \vdots \\ c _ 1 & . . . & c _ { r } \end{pmatrix} \mapsto k ^ \times \cdot ( c _ 1 , . . . , c _ { r } ) . \end{align*}


\begin{gather*} \langle \partial _ t \bar { u } , z \rangle _ { V ^ * , V } + \int _ \Omega \nabla \bar { \mu } \cdot \nabla z \ , d x = 0 , \\ ( \bar { \mu } , z ) _ H = ( \nabla \bar { u } , \nabla z ) _ { H } - \langle \partial _ { \boldsymbol { \nu } } \bar { u } , z _ { | _ \Gamma } \rangle _ { Z _ \Gamma ^ * , Z _ \Gamma } + ( \bar { \xi } , z ) _ H + \bigl ( \pi ( u _ 1 ) - \pi ( u _ 2 ) - \bar { f } , z \bigr ) _ { \ ! H } \end{gather*}


\begin{align*} \langle x u _ { \alpha } \ , | \ , ( h _ { \alpha } - e _ { \alpha } ) \ , | \ , x u _ { \alpha } \rangle = 3 . \end{align*}


\begin{align*} \| f \| _ { V ^ q ( 0 , 1 ; E ) } ^ q \lesssim \sum _ { N } \Bigl ( N ^ \epsilon \sum _ { k = 0 } ^ { N - 1 } \| f ( t _ { k + 1 } ^ N ) - f ( t _ k ^ N ) \| _ E ^ q \Bigr ) , \end{align*}


\begin{align*} F ^ { d _ 1 , . . . , d _ m } = \left ( k ^ { d _ 1 } \subset k ^ { d _ 1 + d _ 2 } \subset . . . . \subset k ^ { d _ 1 + d _ 2 + . . . + d _ m } \right ) . \end{align*}


\begin{align*} \Phi _ 1 : \varphi \in M _ 2 ( \Gamma ) \mapsto \widetilde { \varphi } \in M _ 2 ( G ) , \widetilde { \varphi } = \chi _ { \Omega } * ( \varphi \mu _ \Gamma ) * \chi _ { \Omega } ^ * , \end{align*}


\begin{align*} \bar { \Gamma } _ r \coloneqq \begin{bmatrix} \Gamma _ { r } & 0 & \dots & 0 \\ \Gamma _ { r + 1 } & \Gamma _ { r } & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ \Gamma _ n & \dots & \Gamma _ { r + 1 } & \Gamma _ { r } \end{bmatrix} . \end{align*}


\begin{align*} G _ { \omega , \varkappa , \Lambda , E , q } = ( H _ { q } [ \omega , \varkappa ] \upharpoonright \Lambda - E ) ^ { - 1 } \end{align*}


\begin{align*} \nu ^ { c u } _ { n , x } ( A ) = \int _ { \pi ( A ) } \int _ { \xi ^ u _ n ( y ) } \mathbf { 1 } _ A ( s , t ) \varrho _ { n , x } ( s ) \ , d \mathrm { L e b } _ { n , x } ( s , t ) , \end{align*}


\begin{align*} \dot { f } _ t ( z ) = z p ( z , t ) f _ t ' ( z ) \end{align*}


\begin{align*} S _ \Lambda ( C ) = \sum _ { \substack { 1 \leq j \leq k \\ 1 \leq i _ 1 < \cdots < i _ j \leq k } } { } ( - 1 ) ^ j \ , S _ \Lambda ( C _ { i _ 1 } \cap \cdots \cap C _ { i _ j } ) . \end{align*}


\begin{align*} \bigcup _ { j \in J _ A } a ( j ) = \bigcup _ { a \in A } a = \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} L ^ { + } ( z ) \textbf { 1 } = I \textbf { 1 } , \end{align*}


\begin{align*} \mathcal F _ { \nu , \gamma } : = \{ a \in A : \nu ( a ) \ge \gamma \} \left ( \mathcal F _ { \nu , > \gamma } : = \{ a \in A : \nu ( a ) > \gamma \} \right ) . \end{align*}


\begin{align*} & a _ 1 ( k ) = \frac { k - i \kappa } { k } \exp \left \{ \frac { 1 } { 2 \pi i } \int _ { - \infty } ^ { \infty } \frac { { \rm l o g } \left ( 1 - b ^ 2 ( s ) \right ) } { s - k } d s \right \} , \\ & a _ 2 ( k ) = \frac { k } { k - i \kappa } \exp \left \{ - \frac { 1 } { 2 \pi i } \int _ { - \infty } ^ { \infty } \frac { { \rm l o g } \left ( 1 - b ^ 2 ( s ) \right ) } { s - k } d s \right \} . \end{align*}


\begin{align*} ( e _ { \alpha } + \alpha \| \lambda _ y \| ^ 2 ) \ , 2 \mathrm { R e } \langle \kappa \Phi _ { \# } ^ y \ , | \ , R ^ { \# } _ y \rangle & \geq - C \varepsilon _ 6 \alpha \| R ^ { \# } _ y \| ^ 2 - C \varepsilon _ 6 ^ { - 1 } \alpha | \kappa | ^ 2 \| \Phi _ { \# } ^ y \| ^ 2 \\ & = - C \varepsilon _ 6 \alpha \| R ^ { \# } _ y \| ^ 2 + O ( \alpha ^ 4 \log ( \alpha ^ { - 1 } ) ) . \end{align*}


\begin{align*} J _ k ( x ) = \frac { 1 } { 2 \pi } \int _ 0 ^ \infty e ^ { i x \sin \theta } e ^ { - i k \theta } d \theta \end{align*}


\begin{align*} \lambda \cdot \mu ( i , j ) = \frac { \partial f ( \mathbf { x } ) } { \partial \mathbf { x } _ { i , j } } ( \mu ) \cdot \mu ( i , j ) \leq 3 \sum _ { P \in \mathbf { P ^ * } ( \{ i , j \} ) } \mu ( P ) \ ; . \end{align*}


\begin{align*} y ^ { ' 1 } - x ^ { ' 1 } = \Re \frac { \zeta ' - \zeta ' _ 0 } { \partial _ { \gamma } \eta ' _ i ( l _ 0 ) } . \end{align*}


\begin{align*} G = \begin{bmatrix} F \\ F ' B \end{bmatrix} \end{align*}


\begin{align*} x \star y = \gamma _ x ( y ) - y . \end{align*}


\begin{align*} \deg ( \Phi _ { K _ S } ) = \deg ( \Phi _ { \hat { M } } ) = \hat { M } ^ 2 . \end{align*}


\begin{align*} ( \omega ^ j ) ^ { \varphi ( a ^ k b ^ \ell ) } = \zeta ^ { k 2 ^ j } \omega ^ { j - \ell } \ j , \ell \in \{ 0 , 1 , 2 \} k \in \{ 0 , 1 , \dots , 6 \} . \end{align*}


\begin{align*} d _ { 0 , s } ( n , 0 ) = 0 , \mbox { i f } s \ne 0 . \end{align*}


\begin{align*} t ^ m _ i = \left \{ \begin{array} { l l } x _ i & \mbox { i f } i \in I \setminus A , \\ z ^ m _ i & \mbox { i f } i \in A . \end{array} \right . \end{align*}


\begin{align*} U [ \varphi _ 1 , \ldots , \varphi _ n ; \epsilon ] : = \{ ( \omega , \varpi ) \ , : \ , | \omega ( \varphi _ j ) - \varpi ( \varphi _ j ) | < \epsilon 1 \leq j \leq n \} \subseteq U \ , . \end{align*}


\begin{align*} d _ { \mathcal { P } ( \Omega \times { \rm C o n f i g } ) } ( \overline { \mathbf { P } } _ { 1 } , \overline { \mathbf { P } } _ { 2 } ) = \sup _ { F \in { \rm L i p } _ { 1 } ( \Omega \times { \rm C o n f i g } ) } \int F d ( \overline { \mathbf { P } } _ { 1 } - \overline { \mathbf { P } } _ { 2 } ) , \end{align*}


\begin{align*} e ^ d = \left ( \begin{array} { l l l l l } e ^ \alpha & 0 & \ldots & 0 & \beta ' \\ 0 & e ^ { 2 \alpha } & \ldots & 0 & ( e ^ { 2 \alpha } - e ^ { 2 ^ { n - 1 } \alpha } ) a _ { 1 n } \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \ldots & e ^ { 2 ^ { n - 2 } \alpha } & ( e ^ { 2 ^ { n - 2 } \alpha } - e ^ { 2 ^ { n - 1 } \alpha } ) a _ { n - 2 , n } \\ 0 & 0 & \ldots & 0 & e ^ { 2 ^ { n - 1 } \alpha } \end{array} \right ) , \end{align*}


\begin{align*} h _ j ( x _ j ) ( \omega ) = p ^ { j i } h _ i ( x _ i ) ( \omega ) \ \mbox { a . s . o n } \ \Omega _ i \cap \Omega _ j \ \ \mbox { i f } \ \ \ x _ j = p ^ { j i } x _ i \ \ ( i \ge j ) . \end{align*}


\begin{align*} E [ \widetilde { V } _ { 0 , n } ^ \ell ] \le 2 ^ \ell \sum _ { \underline { x } , \underline { y } } E \Big [ \prod _ { i = 1 } ^ \ell L _ n ( x _ i ) \Big ] E \Big [ \prod _ { i = 1 } ^ \ell L _ \infty ( y _ i ) \Big ] \prod _ { i = 1 } ^ \ell G ( x _ i , y _ i ) \ , , \end{align*}


\begin{align*} a = z w , \ , b = w z . \end{align*}


\begin{align*} U _ { k , \varepsilon } : = \big \{ ( x _ 1 , \ldots , x _ k ) \in X ^ k \mid \sqrt { d ^ 2 ( x _ 1 , x _ 2 ) + d ^ 2 ( x _ 2 , x _ 3 ) + \ldots + d ^ 2 ( x _ k , x _ 1 ) } < \varepsilon \big \} \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { \log \left ( K _ { N , \beta } \right ) } { N } = 0 . \end{align*}


\begin{align*} \partial ^ \circ \nu ( y , t ) = - \nabla _ \Gamma V _ \Gamma ( y , t ) . \end{align*}


\begin{align*} \frac { d } { d t } \big ( p \circ X ) = \big ( \partial _ t p - | \nabla p | ^ 2 \big ) \circ X = \big ( \gamma p ( \Delta p + G ) \big ) \circ X . \end{align*}


\begin{align*} J e _ { 2 i - 1 } = e _ { 2 i } , \end{align*}


\begin{align*} \underset { n \rightarrow + \infty } { \lim } \| u _ n \| ^ { 2 } _ { E _ \lambda } = \frac { 2 p c } { p - 1 } . \end{align*}


\begin{align*} \mathcal { G } = \bigcup \limits _ { i = 1 } ^ \infty ( a _ i , b _ i ) \subseteq [ 0 , T ] , \end{align*}


\begin{align*} \left \{ \begin{aligned} & ( u _ 2 ) _ t - F _ 3 ( D ^ 2 u _ 2 , y , s ) = f _ 2 & & ~ ~ \mbox { i n } ~ ~ \tilde { \Omega } _ 1 ; \\ & u _ 2 = g _ 2 & & ~ ~ \mbox { o n } ~ ~ ( \partial \tilde \Omega ) _ 1 , \end{aligned} \right . \end{align*}


\begin{align*} \frac { 1 } { \tau } \int _ { t } ^ { t + \tau } \int _ 0 ^ s | E _ { \delta } ( \theta ) | \ , d \theta \ , d s = 0 . \end{align*}


\begin{align*} s _ 1 | \nabla _ { s _ 1 } u | _ { 2 } ^ { 2 } + s _ 2 | \nabla _ { s _ 2 } u | _ { 2 } ^ { 2 } - \displaystyle \int _ { \mathbb { R } ^ { d } } W ( x ) u ^ 2 d x = d \int _ { \mathbb { R } ^ { d } } \widetilde { G } ( u ) d x . \end{align*}


\begin{align*} H _ i ( \Gamma , C _ c ( \Q _ q , \Z ) ) = \varinjlim _ { k \to \infty } H _ i ( \Gamma _ k , C _ c ( \Q _ q , \Z ) ) \end{align*}


\begin{align*} \begin{cases} \mathcal { E } x _ { k + 1 } = [ A _ 1 + A _ 2 + B ( K _ 1 + K _ 2 ) \\ \ \ \ \ \ \ \ \ \ \ \ + \bar { \alpha } B ( \Delta K _ { 1 , k } + \Delta K _ { 2 , k } ) ] \mathcal { E } x _ k , \\ \mathcal { E } x _ { 0 } = x _ 0 = \xi , \ k \in { \mathcal N } _ { T - 1 } . \end{cases} \end{align*}


\begin{align*} U ( f < a , \ , g \ge b ) : = \bigcap _ { i = 1 } ^ m \left \{ | f _ i | < | a _ i | \right \} \bigcap _ { j = 1 } ^ n \left \{ | g _ j | \ge | b _ j | \right \} ~ . \end{align*}


\begin{align*} \Psi : \{ \pm 1 \} \cdot ( c _ 1 , . . . , c _ { r } ) \mapsto ( P \cap \Gamma ) \cdot \begin{pmatrix} * & . . . & * \\ \vdots & \ddots & \vdots \\ c _ 1 & . . . & c _ { r } \end{pmatrix} . \end{align*}


\begin{align*} x _ 1 + 2 x _ 2 = \binom n 2 . \end{align*}


\begin{align*} Z _ { \left \lfloor T r \right \rfloor } ( d ) = T ^ { 1 / 2 - d } \sum _ { n = 0 } ^ { \left \lfloor T r \right \rfloor - 1 } \pi _ { n } ( d ) \xi _ { \left \lfloor T r \right \rfloor - n } \Rightarrow \Gamma ( d ) ^ { - 1 } \int _ { 0 } ^ { r } ( r - s ) ^ { d - 1 } \mathsf { d } W ( s ) = W ( r ; d ) , \end{align*}


\begin{align*} A _ A ( l ) = \sum _ { l _ d \in L ^ { V _ { } } } [ A _ A ] ( ( l \sqcup l _ d ) \circ \sim ) \ ; . \end{align*}


\begin{align*} A _ a - \lambda B _ a : = \left [ \begin{array} { c c } A & U T _ A \\ S _ A V ^ * & 0 \end{array} \right ] - \lambda \ , \left [ \begin{array} { c c } B & U T _ B \\ S _ B V ^ * & 0 \end{array} \right ] , \end{align*}


\begin{align*} \dfrac { \max _ { 1 \le j \le n } \left | \sum _ { i = 1 } ^ { j } X _ i \right | } { b _ n } \overset { \mathbb { P } } { \to } 0 \ n \to \infty , \end{align*}


\begin{align*} \left [ z ^ { 2 k } \right ] \left ( \gamma _ { n - 1 } - \gamma _ { n + 1 } + 1 \right ) \gamma _ { n } = \eta _ { n , k } + { \displaystyle \sum \limits _ { j = 1 } ^ { k - 1 } } \left ( \eta _ { n - 1 , j } - \eta _ { n + 1 , j } \right ) \eta _ { n , k - j } , \end{align*}


\begin{align*} \mathbb { P } \left \{ \left \vert \left ( \sum _ { i = 1 } ^ n a _ i X _ i - \mathbb { M } \sum _ { i = 1 } ^ n a _ i X _ i \right ) - \left ( \sum _ { i = 1 } ^ n a _ i X _ i ' - \mathbb { M } \sum _ { i = 1 } ^ n a _ i X _ i ' \right ) \right \vert > C _ q \left ( t \left \vert a \right \vert + e ^ { t ^ 2 / ( 2 q ) } \left \vert a \right \vert _ q \right ) \right \} \end{align*}


\begin{align*} \rho = { \rm i n t } [ \overline { \mathbf { P } } ^ { x } ] . \end{align*}


\begin{align*} & \Gamma _ { k _ 0 , \epsilon } : = \Gamma \cap U _ { \epsilon } ( k _ 0 ) = \Gamma _ { j , \epsilon } \cup \Gamma _ { j , \epsilon } ^ { * } , & j = 1 , 2 , \\ & \Gamma _ { - k _ 0 , \epsilon } : = \Gamma \cap U _ { \epsilon } ( - k _ 0 ) = \Gamma _ { j , \epsilon } \cup \Gamma _ { j , \epsilon } ^ { * } , & j = 3 , 4 , \\ & \Gamma _ \epsilon : = \Gamma _ { k _ 0 , \epsilon } \cup \Gamma _ { - k _ 0 , \epsilon } . \end{align*}


\begin{align*} \begin{tabular} [ c ] { l } $ \phi C _ { n } \partial _ { x } ^ { 2 } P _ { n } - 2 x \left [ \left ( \phi - 1 \right ) C _ { n } + \phi \right ] \partial _ { x } P _ { n } + \left ( n - 2 \gamma _ { n } \right ) \left [ \left ( 2 x ^ { 2 } - 1 \right ) C _ { n } + 2 x ^ { 2 } \right ] P _ { n } $ \\ $ + \left [ 4 \gamma _ { n } \left ( C _ { n } + l _ { n - 1 } \right ) - \left ( n - 2 \gamma _ { n } \right ) ^ { 2 } \right ] C _ { n } P _ { n } = 0 . $ \end{tabular} \end{align*}


\begin{align*} \gamma ^ 0 = \begin{bmatrix} I _ { 2 \times 2 } & \mathbf 0 \\ \mathbf 0 & - I _ { 2 \times 2 } \end{bmatrix} , \ \gamma ^ j = \begin{bmatrix} \mathbf 0 & \sigma ^ j \\ - \sigma ^ j & \mathbf 0 \end{bmatrix} , \end{align*}


\begin{align*} \int _ { G } | \Delta u _ n | ^ 2 \ , d \mu - \int _ { G } | \Delta ( u _ n - u ) | ^ 2 \ , d \mu = \int _ { G } | \Delta u | ^ 2 \ , d \mu + o ( 1 ) . \end{align*}


\begin{align*} x = ( x ^ 0 , \overline { x } ) , \ ; \overline { x } = ( x ^ 1 , \ldots , x ^ n ) , \ ; y = ( y ^ 0 , \overline { y } ) \ ; \mbox { a n d } \overline { y } = ( y ^ 1 , \ldots , y ^ n ) . \end{align*}


\begin{gather*} \theta ' \left ( x ' , y ' \right ) = f \left ( \left [ x ' , y ' \right ] \right ) - \rho ( x ' ) f ( y ' ) ) - \rho ( y ' ) f ( x ' ) \end{gather*}


\begin{align*} \sum _ { \sigma } s g n ( \sigma ) l _ { \sigma ( j ) , i } ^ { + } ( u ) \cdots l _ { \sigma ( 1 ) , 1 } ^ { + } ( u + ( i - 1 ) h ) \mid 0 \rangle = 0 , \end{align*}


\begin{align*} \lim \limits _ { \varepsilon \rightarrow 0 } P \Big ( \rho _ T \big ( Y ^ { h _ \varepsilon } , X ^ { h _ \varepsilon } \big ) > \delta \Big ) = 0 \end{align*}


\begin{align*} A ( { \bf { v } } ) : = S ( { \bf { v } } ) A _ 0 ( { \bf { v } } ) = \left ( \begin{array} { c c c } \frac { \theta ( P - q ) } { R } u & 0 & - \theta ^ 2 \\ 0 & \frac { P - q } { a } u & \theta u \\ - \theta ^ 2 & \theta u & \frac { \theta ^ 2 } { 2 q } \frac { P + q } { P - q } u \end{array} \right ) \end{align*}


\begin{gather*} d ^ 2 \theta = d ^ 2 \theta ' \end{gather*}


\begin{align*} S _ { \mathbf { t } } ' = \left \{ \Psi _ { 2 ^ { m - 2 } - 1 } \left ( \mathcal { G } _ d ^ { \mathbf { t } } ( \mathbf { x } , \mathbf { y } ) \right ) : d \in \{ 1 , 2 \} , \mathbf { y } \in \mathbb { Z } _ { 2 } ^ { n } \right \} , \end{align*}


\begin{align*} \mathbb { E } ( R _ { 2 , 2 } ^ { ( d ) } ( s , . , N ) ) = \omega _ d s ^ d \frac { N - 1 } { N } . \end{align*}


\begin{align*} | \varphi _ { t , t ' } ( z ) - z | & = | \varphi _ { t , t ' } ( z ) - \varphi _ { t , t } ( z ) | \\ & = \left | \int _ { t } ^ { t ' } \dot { \varphi } _ { t , s } ( z ) d s \right | = \left | \int _ { t } ^ { t ' } p ( \varphi _ { t , s } ( z ) , s ) d s \right | \\ & \leq \int _ { t } ^ { t ' } | p ( \varphi _ { t , s } ( z ) , s ) | d s \leq K ( t ' - t ) \end{align*}


\begin{align*} \mathrm { D } \Phi _ i ^ { - 1 } ( \tau ) = \mathrm { d i a g } ( e ^ { \tau _ 1 } , \dots , e ^ { \tau _ d } ) w ( \tau ) = \exp ( \tau _ 1 + \cdots + \tau _ d ) , \end{align*}


\begin{align*} \biggl ( \frac { \mu ^ 2 } { 2 \mu - 1 } \biggr ) ^ { t } = \biggl ( 1 + O \biggl ( \frac { 1 } { N } \biggr ) \biggr ) ^ t = O ( 1 ) , \end{align*}


\begin{align*} f ^ q - g ^ q = c . \end{align*}


\begin{align*} u ( t ) = e ^ { i t \Delta } u _ - - i \int _ { - \infty } ^ t e ^ { i ( t - s ) \Delta } F ( u ( s ) ) \ , d s . \end{align*}


\begin{align*} \lim _ { k \to \infty } h _ d ( n _ k ) ^ { - 1 } \big [ \overline { R } _ { n _ k } - \sigma _ d B _ { \rho _ { n _ k } ^ 2 } \big ] = 0 \ , , \end{align*}


\begin{align*} \mathcal { F } ( \overline { \mathbf { P } } ) = \mathcal { G } ( \overline { \mathbf { P } } ) - \inf _ { \overline { \mathbf { P } } ^ * \in \mathcal { P } _ { s , 1 } ( \Sigma \times { \rm C o n f i g } ) } \mathcal { G } ( \overline { \mathbf { P } } ^ * ) , \end{align*}


\begin{align*} [ \varphi ( a \otimes x ) , b \otimes f ] & = \varphi ( a ) b \cdot f ( x ) = \varphi ( a \varphi ^ { - 1 } ( b ) \cdot f ( x ) ) = a p ^ { - 1 } \varphi ^ { - 1 } ( b ) f ( x ) \\ & = [ a \otimes x , p ^ { - 1 } \varphi ^ { - 1 } ( b \otimes f ) ] . \end{align*}


\begin{align*} u ^ { \odot \beta } ( v ^ { \odot \alpha } ) = \begin{cases} 0 , \alpha \neq \beta , \\ \frac { \alpha ! } { k ! } , \end{cases} \end{align*}


\begin{align*} \Delta _ { + } ^ { d _ { p } } \tilde { X } _ { t } = - \alpha \beta ^ { \prime } ( \Delta _ { + } ^ { - b } - 1 ) \Delta _ { + } ^ { d _ { p } } \tilde { X } _ { t } + \varepsilon _ { t } , t = 1 , \dots , T . \end{align*}


\begin{align*} t ' _ { i x } = t _ { i \Phi _ 0 ( x ) \Phi _ 1 ( x ) } \ ; . \end{align*}


\begin{align*} G ^ { ( 4 ) } _ { \mathcal R } ( x , y ) = G _ { \mathcal R , \zeta _ o , 1 } ( x , y ) + G _ { \mathcal R , \zeta _ o , 2 } ( x , y ) . \end{align*}


\begin{align*} r ^ { \rho - \rho ( r ) } \left | \sum _ { R _ n < | z _ k | \le r } \frac 1 { z _ k ^ \rho } \right | = o ( 1 ) , \ ; \ ; \ ; { r \to \infty } . \end{align*}


\begin{align*} \mu = \Lambda f ^ { 2 } + f \Delta _ { B } f + ( m - 1 ) | \nabla _ { B } f | ^ { 2 } - f \nabla _ { B } \varphi ( f ) . \end{align*}


\begin{align*} \sigma { } ( u + \ell ) = \sigma { } ( u ) \exp ( L ( u + \frac { 1 } { 2 } \ell , \ell ) ) \chi ( \ell ) , \end{align*}


\begin{align*} ( s + 1 ) ( s - 1 ) ^ { - 1 } \begin{pmatrix} x \\ - x \end{pmatrix} = ( s + 1 ) \begin{pmatrix} 0 \\ x \end{pmatrix} = \begin{pmatrix} x \\ x \end{pmatrix} \ , . \end{align*}


\begin{align*} \int _ { s } ^ { t } h \partial _ x u _ r \times \circ \dd W _ r + \int _ { s } ^ { t } \partial _ x h u _ r \times \circ \dd W _ r & = \int _ { s } ^ { t } h \partial _ x u _ r \times \dd W _ r - \int _ { s } ^ { t } h ^ 2 \partial _ x u _ r \dd r \\ & + \int _ { s } ^ { t } \partial _ x h u _ r \times \dd W _ r - \int _ { s } ^ { t } \partial _ x h ^ 2 u _ r \dd r \ , . \end{align*}


\begin{align*} \lim _ { m \to \infty } \frac { 1 } { m } \sum _ { j = - m / 2 + 1 } ^ { m / 2 } \frac { e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \lfloor s m \rfloor } } { 1 - \exp \left \{ - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \right \} + \frac { z } { m } } = e ^ { \pi i \theta _ 1 s } F ( z + \pi i \theta _ 1 , s ) \end{align*}


\begin{align*} \sum _ 1 : = \sum _ { \ell = \tilde { D } } ^ { B _ { d , \mu } n } C _ d ^ n \left ( \frac { 2 \ell + n } { n } \right ) ^ { d n } e ^ { - \mu \ell } , \end{align*}


\begin{align*} \mathbf { y } ^ { ( w ) } _ { i , j } = \mathbf { L } ^ { - 1 } _ { b } \mathbf { y } _ { i , j } , \mathbf { \hat { H } } ^ { ( w ) } _ { i , j } = \mathbf { L } ^ { - 1 } _ { b } \mathbf { \hat { H } } _ { i , j } , ( i , j ) \in \mathbb { S } _ b \end{align*}


\begin{align*} \begin{aligned} & P _ t - F _ 0 ( D ^ 2 P ) - P _ f = 0 , \\ & \mathbf { \Pi } _ { 2 } \left ( P ( x ' , P _ { \Omega } ( x ' , t ) , t ) \right ) \equiv \mathbf { \Pi } _ { 2 } \left ( P _ g ( x ' , P _ { \Omega } ( x ' , t ) , t ) \right ) , \end{aligned} \end{align*}


\begin{align*} A _ S ( \cdot , i ) A _ S ( \cdot , j ) = A _ S ( \cdot , j ) A _ S ( \cdot , i ) \ ; . \end{align*}


\begin{align*} \partial ^ { m } _ { z ' } [ w ' _ j \circ w _ i ^ { ' - 1 } ] ( z ' ) = \partial ^ { m } _ { z ' } ( \pi _ j \xi ' ) = \frac { m ! } { 2 \pi i } \int \limits _ { \Gamma } \frac { \eta ' _ j d \eta ' _ i } { ( \eta ' _ i - z ' ) ^ { m + 1 } } . \end{align*}


\begin{align*} R _ { \xi } ( x ) = R _ { \phi } ( x ) = - \phi \times x \end{align*}


\begin{align*} K = \{ x _ 4 Q + R = 0 \} , \end{align*}


\begin{align*} \bar { \mu } ( x ) \cdot ( m - 1 ) \cdot \beta ( \mu ; P _ m ) = \sum _ { \substack { P \in \mathbf { C } ( P _ m , n ) \\ V ( P ) \ni x } } \deg _ { P } ( x ) \mu ( P ) \ ; . \end{align*}


\begin{align*} h _ n ( i ) & : = 1 _ { \{ S _ i \notin \R ( i , n ] \} } \hat { P } ^ { S _ i } ( \hat { \tau } _ { \R _ n } = \infty ) \le g _ { n , \alpha } ( i ) \end{align*}


\begin{align*} \langle 0 \mid l _ { j i } ^ { - } ( u ) = 0 f o r \ a l l \ j > i a n d \quad \langle 0 \mid l _ { i i } ^ { - } ( u ) = \langle 0 \mid \varkappa _ { i } ^ { - } ( u ) f o r \end{align*}


\begin{align*} G _ { \mathcal R } ( x , y ) = & \frac { i } { 4 \pi } \bigg \{ \int ^ { \pi } _ { 0 } F ( \eta , x , y ) d \eta + \int _ { + \infty } ^ { 0 } F ( i \eta , x , y ) i d \eta + \int ^ { - \infty } _ { 0 } F ( i \eta + \pi , x , y ) i d \eta \bigg \} \\ = & \frac { i } { 4 \pi } \int _ { \mathcal L } F ( \zeta , x , y ) d \zeta , \end{align*}


\begin{align*} \Pi _ { | \ell | \le K } M = \begin{cases} M _ j ^ { j - \pi ( \ell ) } ( \ell ) & \mbox { i f } \ ; | \ell | \le K \\ 0 & \mbox { o t h e r w i s e } \end{cases} , \Pi _ { | \ell | > K } M : = M - \Pi _ { | \ell | \le K } M . \end{align*}


\begin{align*} \lim _ { m \to \infty } I _ m = \mathbf { 1 } _ { \mathrm { R e } ( z ) > 0 , \delta = 0 } - \mathbf { 1 } _ { \mathrm { R e } ( z ) < 0 , \delta = 1 } + F ( z + \pi i \theta _ 1 , 0 ) - \int _ { - 1 / 2 } ^ { 1 / 2 } F ( z + \pi i \theta _ 1 + 2 \pi i u , 0 ) \mathrm { d } u . \end{align*}


\begin{align*} \alpha _ 1 = \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 } = - \alpha _ 7 \end{align*}


\begin{align*} \frac { \max _ { 1 \le i \le n } | X _ i | } { b _ n } \overset { \mathbb { P } } { \to } 0 \sum _ { i = 1 } ^ n \mathbb { P } ( | X _ i | > b _ n \varepsilon ) \to 0 \varepsilon > 0 . \end{align*}


\begin{align*} R ^ + ( h ( U ) ) = h \left ( \left \{ x \in S , \ \ h ( Q ( x ) ) \subset h ( U ) \right \} \right ) = h \left ( \left \{ x \in S , \ \ Q ( x ) \subset U \right \} \right ) = h \left ( Q ^ + ( U ) \right ) \end{align*}


\begin{align*} \displaystyle \lim _ { \lambda \rightarrow 1 / 2 } s _ * ( \lambda ) = + \infty , \ , \displaystyle \lim _ { \lambda \rightarrow 2 } s _ * ( \lambda ) = \nu + \lambda . \end{align*}


\begin{align*} \mathcal { F } ( \overline { \mathbf { P } } ) = \mathcal { G } ( \overline { \mathbf { P } } ) - \left ( { \rm e n t } [ \mu _ { V } ] - 1 + | \Sigma | \right ) , \end{align*}


\begin{align*} f ( y ) = g ( - 2 \tilde \alpha ( I - \tilde \alpha ) ^ { - 1 } y ) , \ \ y \in Y . \end{align*}


\begin{align*} H _ { q } [ \omega , \varkappa ] & : \C ^ { \Z _ { q } } \rightarrow \C ^ { \Z _ { q } } \\ ( H _ { q } [ \omega , \varkappa ] \psi ) ( x ) = e ^ { i \varkappa } \psi ( x - \overline { 1 } _ { q } & ) + e ^ { - i \varkappa } \psi ( x + \overline { 1 } _ { q } ) + V _ { \omega , q } ( x ) \psi ( x ) , \end{align*}


\begin{align*} \ln N \leq \frac { 1 } { n ( n - 2 ) \ln 2 } \left [ \frac { ( n + 2 ) ( 1 + \gamma _ 1 \gamma _ 2 ) } { C _ { 4 } } \right ] ^ { 2 } = : \ln \gamma _ { 9 } ( n , \Omega , \kappa ) , \end{align*}


\begin{align*} j ( x ; q ) = j ( q / x ; q ) , \\ j ( q x ; q ) = - x ^ { - 1 } j ( x ; q ) , \end{align*}